LMFDB - Embedded newform 169.4.b.f.168.3

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.f.168.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{2} +8.68466 q^{3} +5.56155 q^{4} +3.56155i q^{5} +13.5616i q^{6} -27.1771i q^{7} +21.1771i q^{8} +48.4233 q^{9} +O(q^{10})$	\(q+1.56155i q^{2} +8.68466 q^{3} +5.56155 q^{4} +3.56155i q^{5} +13.5616i q^{6} -27.1771i q^{7} +21.1771i q^{8} +48.4233 q^{9} -5.56155 q^{10} +15.2614i q^{11} +48.3002 q^{12} +42.4384 q^{14} +30.9309i q^{15} +11.4233 q^{16} -44.5464 q^{17} +75.6155i q^{18} -23.9697i q^{19} +19.8078i q^{20} -236.024i q^{21} -23.8314 q^{22} -122.739 q^{23} +183.916i q^{24} +112.315 q^{25} +186.054 q^{27} -151.147i q^{28} -219.909 q^{29} -48.3002 q^{30} -27.0928i q^{31} +187.255i q^{32} +132.540i q^{33} -69.5616i q^{34} +96.7926 q^{35} +269.309 q^{36} +94.1922i q^{37} +37.4299 q^{38} -75.4233 q^{40} +160.354i q^{41} +368.563 q^{42} +151.302 q^{43} +84.8769i q^{44} +172.462i q^{45} -191.663i q^{46} +466.948i q^{47} +99.2074 q^{48} -395.594 q^{49} +175.386i q^{50} -386.870 q^{51} -120.847 q^{53} +290.533i q^{54} -54.3542 q^{55} +575.531 q^{56} -208.169i q^{57} -343.400i q^{58} -439.633i q^{59} +172.024i q^{60} -137.305 q^{61} +42.3068 q^{62} -1316.00i q^{63} -201.022 q^{64} -206.968 q^{66} -512.280i q^{67} -247.747 q^{68} -1065.94 q^{69} +151.147i q^{70} -410.719i q^{71} +1025.46i q^{72} -308.004i q^{73} -147.086 q^{74} +975.420 q^{75} -133.309i q^{76} +414.759 q^{77} -586.462 q^{79} +40.6847i q^{80} +308.386 q^{81} -250.401 q^{82} -1354.20i q^{83} -1312.66i q^{84} -158.654i q^{85} +236.266i q^{86} -1909.84 q^{87} -323.191 q^{88} +439.882i q^{89} -269.309 q^{90} -682.617 q^{92} -235.292i q^{93} -729.164 q^{94} +85.3693 q^{95} +1626.24i q^{96} +1511.27i q^{97} -617.740i q^{98} +739.006i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9}+O(q^{10}) $ 4 * q + 10 * q^3 + 14 * q^4 + 70 * q^9	$ 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} - 14 q^{10} + 86 q^{12} + 178 q^{14} - 78 q^{16} - 38 q^{17} + 284 q^{22} - 392 q^{23} + 474 q^{25} + 670 q^{27} - 88 q^{29} - 86 q^{30} + 214 q^{35} + 500 q^{36} + 628 q^{38} - 178 q^{40} + 394 q^{42} - 574 q^{43} + 570 q^{48} - 766 q^{49} - 962 q^{51} - 236 q^{53} - 36 q^{55} + 2030 q^{56} - 2116 q^{61} + 664 q^{62} - 1538 q^{64} - 1636 q^{66} - 422 q^{68} - 1592 q^{69} + 294 q^{74} + 1032 q^{75} - 1524 q^{77} - 2016 q^{79} + 244 q^{81} - 144 q^{82} - 5116 q^{87} + 2484 q^{88} - 500 q^{90} - 1576 q^{92} - 1622 q^{94} + 292 q^{95}+O(q^{100}) $ 4 * q + 10 * q^3 + 14 * q^4 + 70 * q^9 - 14 * q^10 + 86 * q^12 + 178 * q^14 - 78 * q^16 - 38 * q^17 + 284 * q^22 - 392 * q^23 + 474 * q^25 + 670 * q^27 - 88 * q^29 - 86 * q^30 + 214 * q^35 + 500 * q^36 + 628 * q^38 - 178 * q^40 + 394 * q^42 - 574 * q^43 + 570 * q^48 - 766 * q^49 - 962 * q^51 - 236 * q^53 - 36 * q^55 + 2030 * q^56 - 2116 * q^61 + 664 * q^62 - 1538 * q^64 - 1636 * q^66 - 422 * q^68 - 1592 * q^69 + 294 * q^74 + 1032 * q^75 - 1524 * q^77 - 2016 * q^79 + 244 * q^81 - 144 * q^82 - 5116 * q^87 + 2484 * q^88 - 500 * q^90 - 1576 * q^92 - 1622 * q^94 + 292 * 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 $

$2$	1.56155i	0.552092i	0.961144	+	0.276046i	$0.0890243\pi$
$2$	1.56155i	0.552092i	−0.961144	+	0.276046i	$0.910976\pi$
$3$	8.68466	1.67136	0.835682	−	0.549214i	$-0.185073\pi$
$3$	8.68466	1.67136	0.835682	+	0.549214i	$0.185073\pi$
$4$	5.56155	0.695194
$5$	3.56155i	0.318555i	0.987234	+	0.159277i	$0.0509165\pi$
$5$	3.56155i	0.318555i	−0.987234	+	0.159277i	$0.949084\pi$
$6$	13.5616i	0.922747i
$7$	− 27.1771i	− 1.46742i	−0.679460	−	0.733712i	$-0.737786\pi$
$7$	− 27.1771i	− 1.46742i	0.679460	−	0.733712i	$-0.262214\pi$
$8$	21.1771i	0.935904i
$9$	48.4233	1.79346
$10$	−5.56155	−0.175872
$11$	15.2614i	0.418316i	0.977882	+	0.209158i	$0.0670723\pi$
$11$	15.2614i	0.418316i	−0.977882	+	0.209158i	$0.932928\pi$
$12$	48.3002	1.16192
$13$	0	0
$13$	0	0
$14$	42.4384	0.810154
$15$	30.9309i	0.532421i
$16$	11.4233	0.178489
$17$	−44.5464	−0.635535	−0.317767	−	0.948169i	$-0.602933\pi$
$17$	−44.5464	−0.635535	−0.317767	+	0.948169i	$0.602933\pi$
$18$	75.6155i	0.990153i
$19$	− 23.9697i	− 0.289422i	−0.989474	−	0.144711i	$-0.953775\pi$
$19$	− 23.9697i	− 0.289422i	0.989474	−	0.144711i	$-0.0462253\pi$
$20$	19.8078i	0.221458i
$21$	− 236.024i	− 2.45260i
$22$	−23.8314	−0.230949
$23$	−122.739	−1.11273	−0.556365	−	0.830938i	$-0.687804\pi$
$23$	−122.739	−1.11273	−0.556365	+	0.830938i	$0.687804\pi$
$24$	183.916i	1.56423i
$25$	112.315	0.898523
$26$	0	0
$27$	186.054	1.32615
$28$	− 151.147i	− 1.02014i
$29$	−219.909	−1.40814	−0.704071	−	0.710130i	$-0.748636\pi$
$29$	−219.909	−1.40814	−0.704071	+	0.710130i	$0.748636\pi$
$30$	−48.3002	−0.293946
$31$	− 27.0928i	− 0.156968i	−0.996915	−	0.0784840i	$-0.974992\pi$
$31$	− 27.0928i	− 0.156968i	0.996915	−	0.0784840i	$-0.0250080\pi$
$32$	187.255i	1.03445i
$33$	132.540i	0.699158i
$34$	− 69.5616i	− 0.350874i
$35$	96.7926	0.467455
$36$	269.309	1.24680
$37$	94.1922i	0.418516i	0.977860	+	0.209258i	$0.0671049\pi$
$37$	94.1922i	0.418516i	−0.977860	+	0.209258i	$0.932895\pi$
$38$	37.4299	0.159788
$39$	0	0
$40$	−75.4233	−0.298137
$41$	160.354i	0.610808i	0.952223	+	0.305404i	$0.0987914\pi$
$41$	160.354i	0.610808i	−0.952223	+	0.305404i	$0.901209\pi$
$42$	368.563	1.35406
$43$	151.302	0.536589	0.268295	−	0.963337i	$-0.413540\pi$
$43$	151.302	0.536589	0.268295	+	0.963337i	$0.413540\pi$
$44$	84.8769i	0.290811i
$45$	172.462i	0.571314i
$46$	− 191.663i	− 0.614329i
$47$	466.948i	1.44918i	0.689181	+	0.724589i	$0.257970\pi$
$47$	466.948i	1.44918i	−0.689181	+	0.724589i	$0.742030\pi$
$48$	99.2074	0.298320
$49$	−395.594	−1.15333
$50$	175.386i	0.496067i
$51$	−386.870	−1.06221
$52$	0	0
$53$	−120.847	−0.313199	−0.156600	−	0.987662i	$-0.550053\pi$
$53$	−120.847	−0.313199	−0.156600	+	0.987662i	$0.550053\pi$
$54$	290.533i	0.732158i
$55$	−54.3542	−0.133257
$56$	575.531	1.37337
$57$	− 208.169i	− 0.483730i
$58$	− 343.400i	− 0.777424i
$59$	− 439.633i	− 0.970090i	−0.874489	−	0.485045i	$-0.838803\pi$
$59$	− 439.633i	− 0.970090i	0.874489	−	0.485045i	$-0.161197\pi$
$60$	172.024i	0.370136i
$61$	−137.305	−0.288198	−0.144099	−	0.989563i	$-0.546028\pi$
$61$	−137.305	−0.288198	−0.144099	+	0.989563i	$0.546028\pi$
$62$	42.3068	0.0866609
$63$	− 1316.00i	− 2.63176i
$64$	−201.022	−0.392621
$65$	0	0
$66$	−206.968	−0.386000
$67$	− 512.280i	− 0.934104i	−0.884230	−	0.467052i	$-0.845316\pi$
$67$	− 512.280i	− 0.934104i	0.884230	−	0.467052i	$-0.154684\pi$
$68$	−247.747	−0.441820
$69$	−1065.94	−1.85977
$70$	151.147i	0.258078i
$71$	− 410.719i	− 0.686526i	−0.939239	−	0.343263i	$-0.888468\pi$
$71$	− 410.719i	− 0.686526i	0.939239	−	0.343263i	$-0.111532\pi$
$72$	1025.46i	1.67850i
$73$	− 308.004i	− 0.493823i	−0.969038	−	0.246912i	$-0.920584\pi$
$73$	− 308.004i	− 0.493823i	0.969038	−	0.246912i	$-0.0794158\pi$
$74$	−147.086	−0.231060
$75$	975.420	1.50176
$76$	− 133.309i	− 0.201205i
$77$	414.759	0.613847
$78$	0	0
$79$	−586.462	−0.835217	−0.417608	−	0.908627i	$-0.637132\pi$
$79$	−586.462	−0.835217	−0.417608	+	0.908627i	$0.637132\pi$
$80$	40.6847i	0.0568585i
$81$	308.386	0.423027
$82$	−250.401	−0.337222
$83$	− 1354.20i	− 1.79088i	−0.445182	−	0.895440i	$-0.646861\pi$
$83$	− 1354.20i	− 1.79088i	0.445182	−	0.895440i	$-0.353139\pi$
$84$	− 1312.66i	− 1.70503i
$85$	− 158.654i	− 0.202453i
$86$	236.266i	0.296247i
$87$	−1909.84	−2.35352
$88$	−323.191	−0.391503
$89$	439.882i	0.523904i	0.965081	+	0.261952i	$0.0843662\pi$
$89$	439.882i	0.523904i	−0.965081	+	0.261952i	$0.915634\pi$
$90$	−269.309	−0.315418
$91$	0	0
$92$	−682.617	−0.773563
$93$	− 235.292i	− 0.262351i
$94$	−729.164	−0.800080
$95$	85.3693	0.0921969
$96$	1626.24i	1.72894i
$97$	1511.27i	1.58192i	0.611869	+	0.790959i	$0.290418\pi$
$97$	1511.27i	1.58192i	−0.611869	+	0.790959i	$0.709582\pi$
$98$	− 617.740i	− 0.636747i
$99$	739.006i	0.750231i
$100$	624.648	0.624648
$101$	−336.260	−0.331278	−0.165639	−	0.986186i	$-0.552969\pi$
$101$	−336.260	−0.331278	−0.165639	+	0.986186i	$0.552969\pi$
$102$	− 604.118i	− 0.586438i
$103$	−322.712	−0.308716	−0.154358	−	0.988015i	$-0.549331\pi$
$103$	−322.712	−0.308716	−0.154358	+	0.988015i	$0.549331\pi$
$104$	0	0
$105$	840.611	0.781288
$106$	− 188.708i	− 0.172915i
$107$	1434.62	1.29617	0.648083	−	0.761570i	$-0.275571\pi$
$107$	1434.62	1.29617	0.648083	+	0.761570i	$0.275571\pi$
$108$	1034.75	0.921933
$109$	− 849.147i	− 0.746179i	−0.927795	−	0.373089i	$-0.878298\pi$
$109$	− 849.147i	− 0.746179i	0.927795	−	0.373089i	$-0.121702\pi$
$110$	− 84.8769i	− 0.0735699i
$111$	818.027i	0.699493i
$112$	− 310.452i	− 0.261919i
$113$	1614.53	1.34409	0.672044	−	0.740511i	$-0.265417\pi$
$113$	1614.53	1.34409	0.672044	+	0.740511i	$0.265417\pi$
$114$	325.066	0.267064
$115$	− 437.140i	− 0.354465i
$116$	−1223.04	−0.978931
$117$	0	0
$118$	686.509	0.535579
$119$	1210.64i	0.932599i
$120$	−655.026	−0.498295
$121$	1098.09	0.825012
$122$	− 214.409i	− 0.159112i
$123$	1392.62i	1.02088i
$124$	− 150.678i	− 0.109123i
$125$	845.211i	0.604784i
$126$	2055.01	1.45297
$127$	−865.174	−0.604502	−0.302251	−	0.953228i	$-0.597738\pi$
$127$	−865.174	−0.604502	−0.302251	+	0.953228i	$0.597738\pi$
$128$	1184.13i	0.817683i
$129$	1314.01	0.896836
$130$	0	0
$131$	−281.400	−0.187680	−0.0938400	−	0.995587i	$-0.529914\pi$
$131$	−281.400	−0.187680	−0.0938400	+	0.995587i	$0.529914\pi$
$132$	737.127i	0.486050i
$133$	−651.426	−0.424705
$134$	799.953	0.515712
$135$	662.641i	0.422452i
$136$	− 943.363i	− 0.594799i
$137$	− 2641.43i	− 1.64725i	−0.567137	−	0.823624i	$-0.691949\pi$
$137$	− 2641.43i	− 1.64725i	0.567137	−	0.823624i	$-0.308051\pi$
$138$	− 1664.53i	− 1.02677i
$139$	−1998.64	−1.21958	−0.609791	−	0.792562i	$-0.708747\pi$
$139$	−1998.64	−1.21958	−0.609791	+	0.792562i	$0.708747\pi$
$140$	538.317	0.324972
$141$	4055.28i	2.42210i
$142$	641.359	0.379026
$143$	0	0
$144$	553.153	0.320112
$145$	− 783.218i	− 0.448570i
$146$	480.964	0.272636
$147$	−3435.60	−1.92764
$148$	523.855i	0.290950i
$149$	1752.98i	0.963824i	0.876220	+	0.481912i	$0.160058\pi$
$149$	1752.98i	0.963824i	−0.876220	+	0.481912i	$0.839942\pi$
$150$	1523.17i	0.829109i
$151$	− 2794.64i	− 1.50613i	−0.657949	−	0.753063i	$-0.728576\pi$
$151$	− 2794.64i	− 1.50613i	0.657949	−	0.753063i	$-0.271424\pi$
$152$	507.608	0.270871
$153$	−2157.08	−1.13980
$154$	647.669i	0.338900i
$155$	96.4924	0.0500030
$156$	0	0
$157$	3244.87	1.64949	0.824743	−	0.565508i	$-0.191320\pi$
$157$	3244.87	1.64949	0.824743	+	0.565508i	$0.191320\pi$
$158$	− 915.792i	− 0.461117i
$159$	−1049.51	−0.523470
$160$	−666.918	−0.329528
$161$	3335.68i	1.63285i
$162$	481.562i	0.233550i
$163$	3281.47i	1.57684i	0.615139	+	0.788418i	$0.289100\pi$
$163$	3281.47i	1.57684i	−0.615139	+	0.788418i	$0.710900\pi$
$164$	891.818i	0.424630i
$165$	−472.047	−0.222720
$166$	2114.66	0.988731
$167$	− 3126.52i	− 1.44873i	−0.689418	−	0.724364i	$-0.742134\pi$
$167$	− 3126.52i	− 1.44873i	0.689418	−	0.724364i	$-0.257866\pi$
$168$	4998.29	2.29540
$169$	0	0
$170$	247.747	0.111773
$171$	− 1160.69i	− 0.519066i
$172$	841.474	0.373034
$173$	−97.5698	−0.0428792	−0.0214396	−	0.999770i	$-0.506825\pi$
$173$	−97.5698	−0.0428792	−0.0214396	+	0.999770i	$0.506825\pi$
$174$	− 2982.31i	− 1.29936i
$175$	− 3052.40i	− 1.31851i
$176$	174.335i	0.0746648i
$177$	− 3818.06i	− 1.62137i
$178$	−686.900	−0.289243
$179$	34.7150	0.0144956	0.00724782	−	0.999974i	$-0.497693\pi$
$179$	34.7150	0.0144956	0.00724782	+	0.999974i	$0.497693\pi$
$180$	959.157i	0.397174i
$181$	1229.35	0.504843	0.252422	−	0.967617i	$-0.418773\pi$
$181$	1229.35	0.504843	0.252422	+	0.967617i	$0.418773\pi$
$182$	0	0
$183$	−1192.45	−0.481684
$184$	− 2599.25i	− 1.04141i
$185$	−335.471	−0.133320
$186$	367.420	0.144842
$187$	− 679.839i	− 0.265854i
$188$	2596.96i	1.00746i
$189$	− 5056.40i	− 1.94603i
$190$	133.309i	0.0509012i
$191$	4280.80	1.62172	0.810858	−	0.585243i	$-0.199001\pi$
$191$	4280.80	1.62172	0.810858	+	0.585243i	$0.199001\pi$
$192$	−1745.81	−0.656212
$193$	472.320i	0.176157i	0.996114	+	0.0880786i	$0.0280727\pi$
$193$	472.320i	0.176157i	−0.996114	+	0.0880786i	$0.971927\pi$
$194$	−2359.93	−0.873365
$195$	0	0
$196$	−2200.12	−0.801791
$197$	4484.37i	1.62182i	0.585173	+	0.810908i	$0.301026\pi$
$197$	4484.37i	1.62182i	−0.585173	+	0.810908i	$0.698974\pi$
$198$	−1154.00	−0.414197
$199$	366.240	0.130463	0.0652314	−	0.997870i	$-0.479221\pi$
$199$	366.240	0.130463	0.0652314	+	0.997870i	$0.479221\pi$
$200$	2378.51i	0.840931i
$201$	− 4448.98i	− 1.56123i
$202$	− 525.087i	− 0.182896i
$203$	5976.49i	2.06634i
$204$	−2151.60	−0.738442
$205$	−571.110	−0.194576
$206$	− 503.932i	− 0.170440i
$207$	−5943.41	−1.99563
$208$	0	0
$209$	365.810	0.121070
$210$	1312.66i	0.431343i
$211$	2122.55	0.692524	0.346262	−	0.938138i	$-0.387451\pi$
$211$	2122.55	0.692524	0.346262	+	0.938138i	$0.387451\pi$
$212$	−672.095	−0.217734
$213$	− 3566.95i	− 1.14743i
$214$	2240.23i	0.715603i
$215$	538.870i	0.170933i
$216$	3940.08i	1.24115i
$217$	−736.303	−0.230339
$218$	1325.99	0.411960
$219$	− 2674.91i	− 0.825358i
$220$	−302.294	−0.0926392
$221$	0	0
$222$	−1277.39	−0.386185
$223$	5926.42i	1.77965i	0.456301	+	0.889826i	$0.349174\pi$
$223$	5926.42i	1.77965i	−0.456301	+	0.889826i	$0.650826\pi$
$224$	5089.04	1.51797
$225$	5438.68	1.61146
$226$	2521.17i	0.742060i
$227$	895.661i	0.261881i	0.991390	+	0.130941i	$0.0417998\pi$
$227$	895.661i	0.261881i	−0.991390	+	0.130941i	$0.958200\pi$
$228$	− 1157.74i	− 0.336286i
$229$	627.717i	0.181138i	0.995890	+	0.0905692i	$0.0288686\pi$
$229$	627.717i	0.181138i	−0.995890	+	0.0905692i	$0.971131\pi$
$230$	682.617	0.195698
$231$	3602.04	1.02596
$232$	− 4657.03i	− 1.31788i
$233$	−2303.72	−0.647734	−0.323867	−	0.946103i	$-0.604983\pi$
$233$	−2303.72	−0.647734	−0.323867	+	0.946103i	$0.604983\pi$
$234$	0	0
$235$	−1663.06	−0.461643
$236$	− 2445.04i	− 0.674401i
$237$	−5093.22	−1.39595
$238$	−1890.48	−0.514881
$239$	− 544.622i	− 0.147400i	−0.997280	−	0.0737001i	$-0.976519\pi$
$239$	− 544.622i	− 0.147400i	0.997280	−	0.0737001i	$-0.0234808\pi$
$240$	353.332i	0.0950313i
$241$	5426.10i	1.45031i	0.688584	+	0.725157i	$0.258233\pi$
$241$	5426.10i	1.45031i	−0.688584	+	0.725157i	$0.741767\pi$
$242$	1714.73i	0.455483i
$243$	−2345.23	−0.619121
$244$	−763.629	−0.200354
$245$	− 1408.93i	− 0.367400i
$246$	−2174.65	−0.563621
$247$	0	0
$248$	573.746	0.146907
$249$	− 11760.8i	− 2.99321i
$250$	−1319.84	−0.333897
$251$	5221.22	1.31299	0.656494	−	0.754331i	$-0.272039\pi$
$251$	5221.22	1.31299	0.656494	+	0.754331i	$0.272039\pi$
$252$	− 7319.02i	− 1.82958i
$253$	− 1873.16i	− 0.465472i
$254$	− 1351.02i	− 0.333741i
$255$	− 1377.86i	− 0.338372i
$256$	−3457.26	−0.844057
$257$	−658.206	−0.159758	−0.0798789	−	0.996805i	$-0.525453\pi$
$257$	−658.206	−0.159758	−0.0798789	+	0.996805i	$0.525453\pi$
$258$	2051.89i	0.495136i
$259$	2559.87	0.614141
$260$	0	0
$261$	−10648.7	−2.52544
$262$	− 439.422i	− 0.103617i
$263$	3246.45	0.761160	0.380580	−	0.924748i	$-0.375724\pi$
$263$	3246.45	0.761160	0.380580	+	0.924748i	$0.375724\pi$
$264$	−2806.81	−0.654344
$265$	− 430.401i	− 0.0997711i
$266$	− 1017.24i	− 0.234477i
$267$	3820.23i	0.875634i
$268$	− 2849.07i	− 0.649384i
$269$	−2585.80	−0.586093	−0.293047	−	0.956098i	$-0.594669\pi$
$269$	−2585.80	−0.586093	−0.293047	+	0.956098i	$0.594669\pi$
$270$	−1034.75	−0.233233
$271$	988.933i	0.221673i	0.993839	+	0.110836i	$0.0353530\pi$
$271$	988.933i	0.221673i	−0.993839	+	0.110836i	$0.964647\pi$
$272$	−508.867	−0.113436
$273$	0	0
$274$	4124.74	0.909433
$275$	1714.09i	0.375866i
$276$	−5928.30	−1.29290
$277$	−8142.40	−1.76617	−0.883086	−	0.469211i	$-0.844538\pi$
$277$	−8142.40	−1.76617	−0.883086	+	0.469211i	$0.844538\pi$
$278$	− 3120.97i	− 0.673322i
$279$	− 1311.92i	− 0.281515i
$280$	2049.78i	0.437493i
$281$	1534.21i	0.325705i	0.986650	+	0.162853i	$0.0520695\pi$
$281$	1534.21i	0.325705i	−0.986650	+	0.162853i	$0.947930\pi$
$282$	−6332.54	−1.33722
$283$	6965.00	1.46299	0.731495	−	0.681847i	$-0.238823\pi$
$283$	6965.00	1.46299	0.731495	+	0.681847i	$0.238823\pi$
$284$	− 2284.23i	− 0.477269i
$285$	741.403	0.154095
$286$	0	0
$287$	4357.96	0.896314
$288$	9067.49i	1.85523i
$289$	−2928.62	−0.596096
$290$	1223.04	0.247652
$291$	13124.9i	2.64396i
$292$	− 1712.98i	− 0.343303i
$293$	640.029i	0.127614i	0.997962	+	0.0638070i	$0.0203242\pi$
$293$	640.029i	0.127614i	−0.997962	+	0.0638070i	$0.979676\pi$
$294$	− 5364.87i	− 1.06424i
$295$	1565.77	0.309027
$296$	−1994.72	−0.391691
$297$	2839.44i	0.554750i
$298$	−2737.37	−0.532120
$299$	0	0
$300$	5424.85	1.04401
$301$	− 4111.95i	− 0.787404i
$302$	4363.99	0.831520
$303$	−2920.30	−0.553686
$304$	− 273.813i	− 0.0516587i
$305$	− 489.019i	− 0.0918070i
$306$	− 3368.40i	− 0.629276i
$307$	− 100.406i	− 0.0186660i	−0.999956	−	0.00933299i	$-0.997029\pi$
$307$	− 100.406i	− 0.0186660i	0.999956	−	0.00933299i	$-0.00297083\pi$
$308$	2306.71	0.426743
$309$	−2802.64	−0.515977
$310$	150.678i	0.0276062i
$311$	3878.92	0.707245	0.353623	−	0.935388i	$-0.384950\pi$
$311$	3878.92	0.707245	0.353623	+	0.935388i	$0.384950\pi$
$312$	0	0
$313$	−3789.39	−0.684311	−0.342155	−	0.939643i	$-0.611157\pi$
$313$	−3789.39	−0.684311	−0.342155	+	0.939643i	$0.611157\pi$
$314$	5067.04i	0.910668i
$315$	4687.02	0.838360
$316$	−3261.64	−0.580638
$317$	− 4406.81i	− 0.780791i	−0.920647	−	0.390396i	$-0.872338\pi$
$317$	− 4406.81i	− 0.780791i	0.920647	−	0.390396i	$-0.127662\pi$
$318$	− 1638.87i	− 0.289004i
$319$	− 3356.11i	− 0.589048i
$320$	− 715.950i	− 0.125071i
$321$	12459.2	2.16636
$322$	−5208.84	−0.901482
$323$	1067.76i	0.183938i
$324$	1715.11	0.294086
$325$	0	0
$326$	−5124.19	−0.870559
$327$	− 7374.55i	− 1.24714i
$328$	−3395.83	−0.571657
$329$	12690.3	2.12656
$330$	− 737.127i	− 0.122962i
$331$	4131.49i	0.686064i	0.939324	+	0.343032i	$0.111454\pi$
$331$	4131.49i	0.686064i	−0.939324	+	0.343032i	$0.888546\pi$
$332$	− 7531.47i	− 1.24501i
$333$	4561.10i	0.750591i
$334$	4882.23	0.799831
$335$	1824.51	0.297564
$336$	− 2696.17i	− 0.437762i
$337$	4560.82	0.737221	0.368611	−	0.929584i	$-0.379834\pi$
$337$	4560.82	0.737221	0.368611	+	0.929584i	$0.379834\pi$
$338$	0	0
$339$	14021.6	2.24646
$340$	− 882.365i	− 0.140744i
$341$	413.473	0.0656622
$342$	1812.48	0.286572
$343$	1429.34i	0.225007i
$344$	3204.14i	0.502196i
$345$	− 3796.41i	− 0.592441i
$346$	− 152.360i	− 0.0236733i
$347$	10069.4	1.55779	0.778896	−	0.627153i	$-0.215780\pi$
$347$	10069.4	1.55779	0.778896	+	0.627153i	$0.215780\pi$
$348$	−10621.6	−1.63615
$349$	5879.32i	0.901757i	0.892585	+	0.450878i	$0.148889\pi$
$349$	5879.32i	0.901757i	−0.892585	+	0.450878i	$0.851111\pi$
$350$	4766.49	0.727942
$351$	0	0
$352$	−2857.76	−0.432725
$353$	9142.56i	1.37850i	0.724525	+	0.689249i	$0.242059\pi$
$353$	9142.56i	1.37850i	−0.724525	+	0.689249i	$0.757941\pi$
$354$	5962.10	0.895147
$355$	1462.80	0.218696
$356$	2446.43i	0.364215i
$357$	10514.0i	1.55871i
$358$	54.2093i	0.00800293i
$359$	− 2754.32i	− 0.404924i	−0.979290	−	0.202462i	$-0.935106\pi$
$359$	− 2754.32i	− 0.404924i	0.979290	−	0.202462i	$-0.0648942\pi$
$360$	−3652.24	−0.534695
$361$	6284.45	0.916235
$362$	1919.69i	0.278720i
$363$	9536.54	1.37889
$364$	0	0
$365$	1096.97	0.157310
$366$	− 1862.07i	− 0.265934i
$367$	3040.19	0.432416	0.216208	−	0.976347i	$-0.430631\pi$
$367$	3040.19	0.432416	0.216208	+	0.976347i	$0.430631\pi$
$368$	−1402.08	−0.198610
$369$	7764.88i	1.09546i
$370$	− 523.855i	− 0.0736052i
$371$	3284.26i	0.459596i
$372$	− 1308.59i	− 0.182385i
$373$	−5384.72	−0.747481	−0.373740	−	0.927533i	$-0.621925\pi$
$373$	−5384.72	−0.747481	−0.373740	+	0.927533i	$0.621925\pi$
$374$	1061.60	0.146776
$375$	7340.37i	1.01081i
$376$	−9888.59	−1.35629
$377$	0	0
$378$	7895.84	1.07439
$379$	3424.27i	0.464097i	0.972704	+	0.232049i	$0.0745428\pi$
$379$	3424.27i	0.464097i	−0.972704	+	0.232049i	$0.925457\pi$
$380$	474.786	0.0640948
$381$	−7513.74	−1.01034
$382$	6684.69i	0.895336i
$383$	382.985i	0.0510956i	0.999674	+	0.0255478i	$0.00813301\pi$
$383$	382.985i	0.0510956i	−0.999674	+	0.0255478i	$0.991867\pi$
$384$	10283.8i	1.36665i
$385$	1477.19i	0.195544i
$386$	−737.553	−0.0972551
$387$	7326.54	0.962349
$388$	8405.00i	1.09974i
$389$	−8588.34	−1.11940	−0.559699	−	0.828696i	$-0.689083\pi$
$389$	−8588.34	−1.11940	−0.559699	+	0.828696i	$0.689083\pi$
$390$	0	0
$391$	5467.56	0.707178
$392$	− 8377.52i	− 1.07941i
$393$	−2443.87	−0.313681
$394$	−7002.57	−0.895392
$395$	− 2088.72i	− 0.266063i
$396$	4110.02i	0.521556i
$397$	− 7239.16i	− 0.915171i	−0.889166	−	0.457586i	$-0.848714\pi$
$397$	− 7239.16i	− 0.915171i	0.889166	−	0.457586i	$-0.151286\pi$
$398$	571.904i	0.0720275i
$399$	−5657.41	−0.709837
$400$	1283.01	0.160376
$401$	4269.62i	0.531708i	0.964013	+	0.265854i	$0.0856539\pi$
$401$	4269.62i	0.531708i	−0.964013	+	0.265854i	$0.914346\pi$
$402$	6947.32	0.861942
$403$	0	0
$404$	−1870.12	−0.230302
$405$	1098.33i	0.134757i
$406$	−9332.60	−1.14081
$407$	−1437.50	−0.175072
$408$	− 8192.78i	− 0.994125i
$409$	− 13562.5i	− 1.63967i	−0.572602	−	0.819834i	$-0.694066\pi$
$409$	− 13562.5i	− 1.63967i	0.572602	−	0.819834i	$-0.305934\pi$
$410$	− 891.818i	− 0.107424i
$411$	− 22939.9i	− 2.75315i
$412$	−1794.78	−0.214618
$413$	−11947.9	−1.42353
$414$	− 9280.95i	− 1.10177i
$415$	4823.06	0.570494
$416$	0	0
$417$	−17357.5	−2.03837
$418$	571.232i	0.0668418i
$419$	−14576.9	−1.69959	−0.849794	−	0.527114i	$-0.823274\pi$
$419$	−14576.9	−1.69959	−0.849794	+	0.527114i	$0.823274\pi$
$420$	4675.10	0.543147
$421$	− 15848.4i	− 1.83469i	−0.398099	−	0.917343i	$-0.630330\pi$
$421$	− 15848.4i	− 1.83469i	0.398099	−	0.917343i	$-0.369670\pi$
$422$	3314.48i	0.382337i
$423$	22611.2i	2.59904i
$424$	− 2559.18i	− 0.293124i
$425$	−5003.24	−0.571042
$426$	5569.98	0.633490
$427$	3731.55i	0.422909i
$428$	7978.70	0.901087
$429$	0	0
$430$	−841.474	−0.0943709
$431$	− 10694.7i	− 1.19524i	−0.801781	−	0.597618i	$-0.796114\pi$
$431$	− 10694.7i	− 1.19524i	0.801781	−	0.597618i	$-0.203886\pi$
$432$	2125.35	0.236703
$433$	16079.0	1.78454	0.892272	−	0.451498i	$-0.149110\pi$
$433$	16079.0	1.78454	0.892272	+	0.451498i	$0.149110\pi$
$434$	− 1149.78i	− 0.127168i
$435$	− 6801.98i	− 0.749724i
$436$	− 4722.57i	− 0.518739i
$437$	2942.01i	0.322049i
$438$	4177.01	0.455674
$439$	−6035.80	−0.656203	−0.328101	−	0.944643i	$-0.606409\pi$
$439$	−6035.80	−0.656203	−0.328101	+	0.944643i	$0.606409\pi$
$440$	− 1151.06i	− 0.124715i
$441$	−19156.0	−2.06845
$442$	0	0
$443$	10201.3	1.09409	0.547043	−	0.837105i	$-0.315753\pi$
$443$	10201.3	1.09409	0.547043	+	0.837105i	$0.315753\pi$
$444$	4549.50i	0.486283i
$445$	−1566.66	−0.166892
$446$	−9254.41	−0.982532
$447$	15224.0i	1.61090i
$448$	5463.19i	0.576141i
$449$	− 5822.54i	− 0.611988i	−0.952033	−	0.305994i	$-0.901011\pi$
$449$	− 5822.54i	− 0.611988i	0.952033	−	0.305994i	$-0.0989888\pi$
$450$	8492.78i	0.889675i
$451$	−2447.22	−0.255511
$452$	8979.27	0.934402
$453$	− 24270.5i	− 2.51728i
$454$	−1398.62	−0.144583
$455$	0	0
$456$	4408.40	0.452724
$457$	− 4621.60i	− 0.473062i	−0.971624	−	0.236531i	$-0.923990\pi$
$457$	− 4621.60i	− 0.473062i	0.971624	−	0.236531i	$-0.0760105\pi$
$458$	−980.213	−0.100005
$459$	−8288.03	−0.842816
$460$	− 2431.18i	− 0.246422i
$461$	− 5127.77i	− 0.518056i	−0.965870	−	0.259028i	$-0.916598\pi$
$461$	− 5127.77i	− 0.518056i	0.965870	−	0.259028i	$-0.0834023\pi$
$462$	5624.78i	0.566425i
$463$	6486.27i	0.651064i	0.945531	+	0.325532i	$0.105543\pi$
$463$	6486.27i	0.651064i	−0.945531	+	0.325532i	$0.894457\pi$
$464$	−2512.09	−0.251338
$465$	838.004	0.0835731
$466$	− 3597.39i	− 0.357609i
$467$	−12978.0	−1.28598	−0.642990	−	0.765875i	$-0.722306\pi$
$467$	−12978.0	−1.28598	−0.642990	+	0.765875i	$0.722306\pi$
$468$	0	0
$469$	−13922.3	−1.37073
$470$	− 2596.96i	− 0.254869i
$471$	28180.6	2.75689
$472$	9310.13	0.907910
$473$	2309.08i	0.224464i
$474$	− 7953.34i	− 0.770694i
$475$	− 2692.16i	− 0.260053i
$476$	6733.04i	0.648337i
$477$	−5851.79	−0.561709
$478$	850.456	0.0813786
$479$	− 5808.96i	− 0.554109i	−0.960854	−	0.277055i	$-0.910642\pi$
$479$	− 5808.96i	− 0.554109i	0.960854	−	0.277055i	$-0.0893583\pi$
$480$	−5791.95	−0.550761
$481$	0	0
$482$	−8473.14	−0.800707
$483$	28969.2i	2.72908i
$484$	6107.09	0.573543
$485$	−5382.46	−0.503928
$486$	− 3662.20i	− 0.341812i
$487$	5387.14i	0.501262i	0.968083	+	0.250631i	$0.0806381\pi$
$487$	5387.14i	0.501262i	−0.968083	+	0.250631i	$0.919362\pi$
$488$	− 2907.72i	− 0.269726i
$489$	28498.4i	2.63547i
$490$	2200.12	0.202839
$491$	−15259.1	−1.40251	−0.701255	−	0.712911i	$-0.747376\pi$
$491$	−15259.1	−1.40251	−0.701255	+	0.712911i	$0.747376\pi$
$492$	7745.14i	0.709711i
$493$	9796.16	0.894922
$494$	0	0
$495$	−2632.01	−0.238990
$496$	− 309.489i	− 0.0280171i
$497$	−11162.1	−1.00742
$498$	18365.1	1.65253
$499$	− 1856.04i	− 0.166509i	−0.996528	−	0.0832544i	$-0.973469\pi$
$499$	− 1856.04i	− 0.166509i	0.996528	−	0.0832544i	$-0.0265314\pi$
$500$	4700.69i	0.420442i
$501$	− 27152.8i	− 2.42135i
$502$	8153.20i	0.724891i
$503$	1049.46	0.0930283	0.0465142	−	0.998918i	$-0.485189\pi$
$503$	1049.46	0.0930283	0.0465142	+	0.998918i	$0.485189\pi$
$504$	27869.1	2.46307
$505$	− 1197.61i	− 0.105530i
$506$	2925.04	0.256984
$507$	0	0
$508$	−4811.71	−0.420246
$509$	551.106i	0.0479909i	0.999712	+	0.0239954i	$0.00763872\pi$
$509$	551.106i	0.0479909i	−0.999712	+	0.0239954i	$0.992361\pi$
$510$	2151.60	0.186813
$511$	−8370.64	−0.724649
$512$	4074.36i	0.351686i
$513$	− 4459.66i	− 0.383818i
$514$	− 1027.82i	− 0.0882010i
$515$	− 1149.36i	− 0.0983431i
$516$	7307.92	0.623475
$517$	−7126.26	−0.606214
$518$	3997.37i	0.339063i
$519$	−847.361	−0.0716667
$520$	0	0
$521$	−8995.30	−0.756413	−0.378206	−	0.925721i	$-0.623459\pi$
$521$	−8995.30	−0.756413	−0.378206	+	0.925721i	$0.623459\pi$
$522$	− 16628.5i	− 1.39427i
$523$	2663.91	0.222724	0.111362	−	0.993780i	$-0.464479\pi$
$523$	2663.91	0.222724	0.111362	+	0.993780i	$0.464479\pi$
$524$	−1565.02	−0.130474
$525$	− 26509.1i	− 2.20372i
$526$	5069.51i	0.420230i
$527$	1206.89i	0.0997586i
$528$	1514.04i	0.124792i
$529$	2897.77	0.238167
$530$	672.095	0.0550829
$531$	− 21288.5i	− 1.73981i
$532$	−3622.94	−0.295253
$533$	0	0
$534$	−5965.49	−0.483431
$535$	5109.47i	0.412900i
$536$	10848.6	0.874232
$537$	301.488	0.0242275
$538$	− 4037.86i	− 0.323577i
$539$	− 6037.30i	− 0.482458i
$540$	3685.31i	0.293686i
$541$	− 6169.23i	− 0.490270i	−0.969489	−	0.245135i	$-0.921168\pi$
$541$	− 6169.23i	− 0.490270i	0.969489	−	0.245135i	$-0.0788322\pi$
$542$	−1544.27	−0.122384
$543$	10676.5	0.843776
$544$	− 8341.52i	− 0.657426i
$545$	3024.28	0.237699
$546$	0	0
$547$	5140.42	0.401807	0.200904	−	0.979611i	$-0.435612\pi$
$547$	5140.42	0.401807	0.200904	+	0.979611i	$0.435612\pi$
$548$	− 14690.5i	− 1.14516i
$549$	−6648.76	−0.516871
$550$	−2676.64	−0.207513
$551$	5271.15i	0.407547i
$552$	− 22573.6i	− 1.74057i
$553$	15938.3i	1.22562i
$554$	− 12714.8i	− 0.975090i
$555$	−2913.45	−0.222827
$556$	−11115.5	−0.847847
$557$	2778.56i	0.211367i	0.994400	+	0.105683i	$0.0337030\pi$
$557$	2778.56i	0.211367i	−0.994400	+	0.105683i	$0.966297\pi$
$558$	2048.64	0.155422
$559$	0	0
$560$	1105.69	0.0834356
$561$	− 5904.17i	− 0.444339i
$562$	−2395.75	−0.179819
$563$	−4906.14	−0.367263	−0.183632	−	0.982995i	$-0.558785\pi$
$563$	−4906.14	−0.367263	−0.183632	+	0.982995i	$0.558785\pi$
$564$	22553.7i	1.68383i
$565$	5750.22i	0.428166i
$566$	10876.2i	0.807706i
$567$	− 8381.04i	− 0.620759i
$568$	8697.82	0.642522
$569$	9363.15	0.689849	0.344924	−	0.938631i	$-0.387905\pi$
$569$	9363.15	0.689849	0.344924	+	0.938631i	$0.387905\pi$
$570$	1157.74i	0.0850744i
$571$	−7199.32	−0.527640	−0.263820	−	0.964572i	$-0.584982\pi$
$571$	−7199.32	−0.527640	−0.263820	+	0.964572i	$0.584982\pi$
$572$	0	0
$573$	37177.3	2.71048
$574$	6805.18i	0.494848i
$575$	−13785.4	−0.999813
$576$	−9734.14	−0.704148
$577$	11449.6i	0.826086i	0.910711	+	0.413043i	$0.135534\pi$
$577$	11449.6i	0.826086i	−0.910711	+	0.413043i	$0.864466\pi$
$578$	− 4573.19i	− 0.329100i
$579$	4101.94i	0.294423i
$580$	− 4355.91i	− 0.311843i
$581$	−36803.3	−2.62798
$582$	−20495.2	−1.45971
$583$	− 1844.28i	− 0.131016i
$584$	6522.62	0.462171
$585$	0	0
$586$	−999.439	−0.0704547
$587$	5439.39i	0.382466i	0.981545	+	0.191233i	$0.0612487\pi$
$587$	5439.39i	0.382466i	−0.981545	+	0.191233i	$0.938751\pi$
$588$	−19107.3	−1.34008
$589$	−649.406	−0.0454301
$590$	2445.04i	0.170611i
$591$	38945.2i	2.71064i
$592$	1075.99i	0.0747006i
$593$	− 28405.8i	− 1.96709i	−0.180651	−	0.983547i	$-0.557820\pi$
$593$	− 28405.8i	− 1.96709i	0.180651	−	0.983547i	$-0.442180\pi$
$594$	−4433.93	−0.306273
$595$	−4311.76	−0.297084
$596$	9749.30i	0.670045i
$597$	3180.67	0.218051
$598$	0	0
$599$	−10482.3	−0.715020	−0.357510	−	0.933909i	$-0.616374\pi$
$599$	−10482.3	−0.715020	−0.357510	+	0.933909i	$0.616374\pi$
$600$	20656.6i	1.40550i
$601$	3199.54	0.217158	0.108579	−	0.994088i	$-0.465370\pi$
$601$	3199.54	0.217158	0.108579	+	0.994088i	$0.465370\pi$
$602$	6421.02	0.434720
$603$	− 24806.3i	− 1.67527i
$604$	− 15542.6i	− 1.04705i
$605$	3910.91i	0.262812i
$606$	− 4560.20i	− 0.305686i
$607$	11342.8	0.758468	0.379234	−	0.925301i	$-0.376188\pi$
$607$	11342.8	0.758468	0.379234	+	0.925301i	$0.376188\pi$
$608$	4488.44	0.299392
$609$	51903.7i	3.45361i
$610$	763.629	0.0506859
$611$	0	0
$612$	−11996.7	−0.792384
$613$	− 14385.4i	− 0.947831i	−0.880570	−	0.473916i	$-0.842840\pi$
$613$	− 14385.4i	− 0.947831i	0.880570	−	0.473916i	$-0.157160\pi$
$614$	156.789	0.0103053
$615$	−4959.89	−0.325207
$616$	8783.39i	0.574502i
$617$	− 22056.8i	− 1.43918i	−0.694401	−	0.719588i	$-0.744331\pi$
$617$	− 22056.8i	− 1.43918i	0.694401	−	0.719588i	$-0.255669\pi$
$618$	− 4376.48i	− 0.284867i
$619$	13621.4i	0.884477i	0.896898	+	0.442238i	$0.145815\pi$
$619$	13621.4i	0.884477i	−0.896898	+	0.442238i	$0.854185\pi$
$620$	536.648	0.0347618
$621$	−22836.0	−1.47565
$622$	6057.14i	0.390465i
$623$	11954.7	0.768789
$624$	0	0
$625$	11029.2	0.705866
$626$	− 5917.34i	− 0.377803i
$627$	3176.94	0.202352
$628$	18046.5	1.14671
$629$	− 4195.92i	− 0.265982i
$630$	7319.02i	0.462852i
$631$	− 18737.5i	− 1.18214i	−0.806622	−	0.591068i	$-0.798707\pi$
$631$	− 18737.5i	− 1.18214i	0.806622	−	0.591068i	$-0.201293\pi$
$632$	− 12419.6i	− 0.781683i
$633$	18433.7	1.15746
$634$	6881.46	0.431069
$635$	− 3081.36i	− 0.192567i
$636$	−5836.91	−0.363913
$637$	0	0
$638$	5240.75	0.325209
$639$	− 19888.4i	− 1.23125i
$640$	−4217.35	−0.260477
$641$	−29798.7	−1.83616	−0.918081	−	0.396394i	$-0.870261\pi$
$641$	−29798.7	−1.83616	−0.918081	+	0.396394i	$0.870261\pi$
$642$	19455.6i	1.19603i
$643$	22983.5i	1.40961i	0.709399	+	0.704807i	$0.248966\pi$
$643$	22983.5i	1.40961i	−0.709399	+	0.704807i	$0.751034\pi$
$644$	18551.5i	1.13515i
$645$	4679.90i	0.285692i
$646$	−1667.37	−0.101551
$647$	24905.4	1.51334	0.756672	−	0.653794i	$-0.226824\pi$
$647$	24905.4	1.51334	0.756672	+	0.653794i	$0.226824\pi$
$648$	6530.72i	0.395912i
$649$	6709.39	0.405804
$650$	0	0
$651$	−6394.54	−0.384980
$652$	18250.1i	1.09621i
$653$	10077.8	0.603946	0.301973	−	0.953316i	$-0.402355\pi$
$653$	10077.8	0.603946	0.301973	+	0.953316i	$0.402355\pi$
$654$	11515.7	0.688534
$655$	− 1002.22i	− 0.0597864i
$656$	1831.77i	0.109022i
$657$	− 14914.6i	− 0.885650i
$658$	19816.5i	1.17406i
$659$	12334.6	0.729116	0.364558	−	0.931181i	$-0.381220\pi$
$659$	12334.6	0.729116	0.364558	+	0.931181i	$0.381220\pi$
$660$	−2625.32	−0.154834
$661$	− 12749.1i	− 0.750202i	−0.926984	−	0.375101i	$-0.877608\pi$
$661$	− 12749.1i	− 0.750202i	0.926984	−	0.375101i	$-0.122392\pi$
$662$	−6451.54	−0.378771
$663$	0	0
$664$	28678.1	1.67609
$665$	− 2320.09i	− 0.135292i
$666$	−7122.40	−0.414395
$667$	26991.3	1.56688
$668$	− 17388.3i	− 1.00715i
$669$	51468.9i	2.97444i
$670$	2849.07i	0.164283i
$671$	− 2095.46i	− 0.120558i
$672$	44196.5	2.53708
$673$	13618.2	0.780007	0.390004	−	0.920813i	$-0.372474\pi$
$673$	13618.2	0.780007	0.390004	+	0.920813i	$0.372474\pi$
$674$	7121.96i	0.407014i
$675$	20896.7	1.19158
$676$	0	0
$677$	9655.67	0.548150	0.274075	−	0.961708i	$-0.411628\pi$
$677$	9655.67	0.548150	0.274075	+	0.961708i	$0.411628\pi$
$678$	21895.5i	1.24025i
$679$	41071.9	2.32135
$680$	3359.84	0.189476
$681$	7778.51i	0.437699i
$682$	645.660i	0.0362516i
$683$	16316.8i	0.914119i	0.889436	+	0.457060i	$0.151097\pi$
$683$	16316.8i	0.914119i	−0.889436	+	0.457060i	$0.848903\pi$
$684$	− 6455.25i	− 0.360852i
$685$	9407.61	0.524739
$686$	−2232.00	−0.124225
$687$	5451.51i	0.302748i
$688$	1728.37	0.0957753
$689$	0	0
$690$	5928.30	0.327082
$691$	− 2350.84i	− 0.129421i	−0.997904	−	0.0647106i	$-0.979388\pi$
$691$	− 2350.84i	− 0.129421i	0.997904	−	0.0647106i	$-0.0206124\pi$
$692$	−542.640	−0.0298093
$693$	20084.0	1.10091
$694$	15723.9i	0.860045i
$695$	− 7118.24i	− 0.388504i
$696$	− 40444.7i	− 2.20266i
$697$	− 7143.20i	− 0.388189i
$698$	−9180.88	−0.497853
$699$	−20007.1	−1.08260
$700$	− 16976.1i	− 0.916623i
$701$	8076.90	0.435179	0.217589	−	0.976040i	$-0.430181\pi$
$701$	8076.90	0.435179	0.217589	+	0.976040i	$0.430181\pi$
$702$	0	0
$703$	2257.76	0.121128
$704$	− 3067.87i	− 0.164239i
$705$	−14443.1	−0.771573
$706$	−14276.6	−0.761058
$707$	9138.55i	0.486125i
$708$	− 21234.3i	− 1.12717i
$709$	− 13624.9i	− 0.721712i	−0.932622	−	0.360856i	$-0.882485\pi$
$709$	− 13624.9i	− 0.721712i	0.932622	−	0.360856i	$-0.117515\pi$
$710$	2284.23i	0.120741i
$711$	−28398.4	−1.49792
$712$	−9315.43	−0.490324
$713$	3325.33i	0.174663i
$714$	−16418.2	−0.860553
$715$	0	0
$716$	193.069	0.0100773
$717$	− 4729.86i	− 0.246359i
$718$	4301.02	0.223555
$719$	−16235.8	−0.842131	−0.421066	−	0.907030i	$-0.638344\pi$
$719$	−16235.8	−0.842131	−0.421066	+	0.907030i	$0.638344\pi$
$720$	1970.09i	0.101973i
$721$	8770.37i	0.453018i
$722$	9813.51i	0.505846i
$723$	47123.8i	2.42400i
$724$	6837.08	0.350964
$725$	−24699.2	−1.26525
$726$	14891.8i	0.761277i
$727$	−24181.2	−1.23361	−0.616803	−	0.787118i	$-0.711572\pi$
$727$	−24181.2	−1.23361	−0.616803	+	0.787118i	$0.711572\pi$
$728$	0	0
$729$	−28693.9	−1.45780
$730$	1712.98i	0.0868496i
$731$	−6739.96	−0.341021
$732$	−6631.86	−0.334864
$733$	− 3053.70i	− 0.153876i	−0.997036	−	0.0769379i	$-0.975486\pi$
$733$	− 3053.70i	− 0.153876i	0.997036	−	0.0769379i	$-0.0245143\pi$
$734$	4747.41i	0.238733i
$735$	− 12236.1i	− 0.614060i
$736$	− 22983.4i	− 1.15106i
$737$	7818.10	0.390751
$738$	−12125.3	−0.604793
$739$	− 8033.62i	− 0.399894i	−0.979807	−	0.199947i	$-0.935923\pi$
$739$	− 8033.62i	− 0.399894i	0.979807	−	0.199947i	$-0.0640770\pi$
$740$	−1865.74	−0.0926836
$741$	0	0
$742$	−5128.54	−0.253739
$743$	16139.6i	0.796912i	0.917187	+	0.398456i	$0.130454\pi$
$743$	16139.6i	0.796912i	−0.917187	+	0.398456i	$0.869546\pi$
$744$	4982.79	0.245535
$745$	−6243.33	−0.307031
$746$	− 8408.53i	− 0.412678i
$747$	− 65574.9i	− 3.21186i
$748$	− 3780.96i	− 0.184820i
$749$	− 38988.7i	− 1.90202i
$750$	−11462.4	−0.558062
$751$	18491.1	0.898469	0.449235	−	0.893414i	$-0.351697\pi$
$751$	18491.1	0.898469	0.449235	+	0.893414i	$0.351697\pi$
$752$	5334.08i	0.258662i
$753$	45344.5	2.19448
$754$	0	0
$755$	9953.28	0.479784
$756$	− 28121.5i	− 1.35287i
$757$	160.630	0.00771227	0.00385613	−	0.999993i	$-0.498773\pi$
$757$	160.630	0.00771227	0.00385613	+	0.999993i	$0.498773\pi$
$758$	−5347.17	−0.256224
$759$	− 16267.7i	− 0.777973i
$760$	1807.87i	0.0862874i
$761$	26799.1i	1.27656i	0.769803	+	0.638282i	$0.220355\pi$
$761$	26799.1i	1.27656i	−0.769803	+	0.638282i	$0.779645\pi$
$762$	− 11733.1i	− 0.557803i
$763$	−23077.3	−1.09496
$764$	23807.9	1.12741
$765$	− 7682.57i	− 0.363090i
$766$	−598.052	−0.0282095
$767$	0	0
$768$	−30025.1	−1.41073
$769$	5145.82i	0.241304i	0.992695	+	0.120652i	$0.0384986\pi$
$769$	5145.82i	0.241304i	−0.992695	+	0.120652i	$0.961501\pi$
$770$	−2306.71	−0.107958
$771$	−5716.29	−0.267013
$772$	2626.83i	0.122463i
$773$	− 12810.6i	− 0.596072i	−0.954555	−	0.298036i	$-0.903668\pi$
$773$	− 12810.6i	− 0.596072i	0.954555	−	0.298036i	$-0.0963316\pi$
$774$	11440.8i	0.531306i
$775$	− 3042.94i	− 0.141039i
$776$	−32004.3	−1.48052
$777$	22231.6	1.02645
$778$	− 13411.1i	− 0.618011i
$779$	3843.64	0.176781
$780$	0	0
$781$	6268.13	0.287185
$782$	8537.89i	0.390428i
$783$	−40915.0	−1.86741
$784$	−4518.98	−0.205857
$785$	11556.8i	0.525452i
$786$	− 3816.23i	− 0.173181i
$787$	28073.0i	1.27153i	0.771883	+	0.635764i	$0.219315\pi$
$787$	28073.0i	1.27153i	−0.771883	+	0.635764i	$0.780685\pi$
$788$	24940.0i	1.12748i
$789$	28194.3	1.27217
$790$	3261.64	0.146891
$791$	− 43878.1i	− 1.97235i
$792$	−15650.0	−0.702144
$793$	0	0
$794$	11304.3	0.505259
$795$	− 3737.89i	− 0.166754i
$796$	2036.87	0.0906970
$797$	−30093.1	−1.33746	−0.668729	−	0.743507i	$-0.733161\pi$
$797$	−30093.1	−1.33746	−0.668729	+	0.743507i	$0.733161\pi$
$798$	− 8834.35i	− 0.391896i
$799$	− 20800.8i	− 0.921003i
$800$	21031.6i	0.929473i
$801$	21300.6i	0.939598i
$802$	−6667.24	−0.293552
$803$	4700.56	0.206574
$804$	− 24743.2i	− 1.08536i
$805$	−11880.2	−0.520151
$806$	0	0
$807$	−22456.8	−0.979575
$808$	− 7120.99i	− 0.310044i
$809$	−24337.1	−1.05766	−0.528831	−	0.848727i	$-0.677369\pi$
$809$	−24337.1	−1.05766	−0.528831	+	0.848727i	$0.677369\pi$
$810$	−1715.11	−0.0743984
$811$	− 19078.7i	− 0.826071i	−0.910715	−	0.413035i	$-0.864469\pi$
$811$	− 19078.7i	− 0.826071i	0.910715	−	0.413035i	$-0.135531\pi$
$812$	33238.5i	1.43651i
$813$	8588.54i	0.370496i
$814$	− 2244.74i	− 0.0966559i
$815$	−11687.1	−0.502309
$816$	−4419.33	−0.189593
$817$	− 3626.66i	− 0.155301i
$818$	21178.6	0.905248
$819$	0	0
$820$	−3176.26	−0.135268
$821$	− 2013.92i	− 0.0856104i	−0.999083	−	0.0428052i	$-0.986371\pi$
$821$	− 2013.92i	− 0.0856104i	0.999083	−	0.0428052i	$-0.0136295\pi$
$822$	35821.9	1.51999
$823$	7692.10	0.325795	0.162898	−	0.986643i	$-0.447916\pi$
$823$	7692.10	0.325795	0.162898	+	0.986643i	$0.447916\pi$
$824$	− 6834.10i	− 0.288929i
$825$	14886.2i	0.628209i
$826$	− 18657.3i	− 0.785922i
$827$	− 4762.76i	− 0.200263i	−0.994974	−	0.100131i	$-0.968074\pi$
$827$	− 4762.76i	− 0.200263i	0.994974	−	0.100131i	$-0.0319263\pi$
$828$	−33054.6	−1.38735
$829$	−19977.7	−0.836976	−0.418488	−	0.908222i	$-0.637440\pi$
$829$	−19977.7	−0.836976	−0.418488	+	0.908222i	$0.637440\pi$
$830$	7531.47i	0.314965i
$831$	−70714.0	−2.95191
$832$	0	0
$833$	17622.3	0.732984
$834$	− 27104.6i	− 1.12537i
$835$	11135.3	0.461499
$836$	2034.47	0.0841671
$837$	− 5040.72i	− 0.208164i
$838$	− 22762.6i	− 0.938330i
$839$	− 30615.8i	− 1.25980i	−0.776676	−	0.629901i	$-0.783095\pi$
$839$	− 30615.8i	− 1.25980i	0.776676	−	0.629901i	$-0.216905\pi$
$840$	17801.7i	0.731210i
$841$	23971.0	0.982861
$842$	24748.1	1.01292
$843$	13324.1i	0.544372i
$844$	11804.7	0.481439
$845$	0	0
$846$	−35308.5	−1.43491
$847$	− 29842.9i	− 1.21064i
$848$	−1380.47	−0.0559026
$849$	60488.6	2.44519
$850$	− 7812.83i	− 0.315268i
$851$	− 11561.0i	− 0.465696i
$852$	− 19837.8i	− 0.797690i
$853$	5660.88i	0.227227i	0.993525	+	0.113614i	$0.0362426\pi$
$853$	5660.88i	0.227227i	−0.993525	+	0.113614i	$0.963757\pi$
$854$	−5827.01	−0.233485
$855$	4133.86	0.165351
$856$	30381.0i	1.21309i
$857$	−41346.1	−1.64802	−0.824012	−	0.566572i	$-0.808269\pi$
$857$	−41346.1	−1.64802	−0.824012	+	0.566572i	$0.808269\pi$
$858$	0	0
$859$	−34810.5	−1.38268	−0.691339	−	0.722530i	$-0.742979\pi$
$859$	−34810.5	−1.38268	−0.691339	+	0.722530i	$0.742979\pi$
$860$	2996.96i	0.118832i
$861$	37847.4	1.49807
$862$	16700.4	0.659880
$863$	8360.51i	0.329774i	0.986312	+	0.164887i	$0.0527260\pi$
$863$	8360.51i	0.329774i	−0.986312	+	0.164887i	$0.947274\pi$
$864$	34839.5i	1.37183i
$865$	− 347.500i	− 0.0136594i
$866$	25108.2i	0.985233i
$867$	−25434.1	−0.996293
$868$	−4094.99	−0.160130
$869$	− 8950.21i	− 0.349385i
$870$	10621.6	0.413917
$871$	0	0
$872$	17982.4	0.698352
$873$	73180.6i	2.83710i
$874$	−4594.10	−0.177801
$875$	22970.4	0.887475
$876$	− 14876.6i	− 0.573784i
$877$	40579.3i	1.56245i	0.624251	+	0.781223i	$0.285404\pi$
$877$	40579.3i	1.56245i	−0.624251	+	0.781223i	$0.714596\pi$
$878$	− 9425.22i	− 0.362285i
$879$	5558.43i	0.213289i
$880$	−620.903	−0.0237848
$881$	−10445.2	−0.399442	−0.199721	−	0.979853i	$-0.564004\pi$
$881$	−10445.2	−0.399442	−0.199721	+	0.979853i	$0.564004\pi$
$882$	− 29913.0i	− 1.14198i
$883$	−18227.6	−0.694685	−0.347343	−	0.937738i	$-0.612916\pi$
$883$	−18227.6	−0.694685	−0.347343	+	0.937738i	$0.612916\pi$
$884$	0	0
$885$	13598.2	0.516496
$886$	15929.9i	0.604036i
$887$	−23517.7	−0.890245	−0.445122	−	0.895470i	$-0.646840\pi$
$887$	−23517.7	−0.890245	−0.445122	+	0.895470i	$0.646840\pi$
$888$	−17323.4	−0.654658
$889$	23512.9i	0.887061i
$890$	− 2446.43i	− 0.0921399i
$891$	4706.40i	0.176959i
$892$	32960.1i	1.23720i
$893$	11192.6	0.419424
$894$	−23773.1	−0.889366
$895$	123.639i	0.00461766i
$896$	32181.2	1.19989
$897$	0	0
$898$	9092.21	0.337874
$899$	5957.95i	0.221033i
$900$	30247.5	1.12028
$901$	5383.28	0.199049
$902$	− 3821.47i	− 0.141065i
$903$	− 35710.9i	− 1.31604i
$904$	34191.0i	1.25794i
$905$	4378.38i	0.160820i
$906$	37899.7	1.38977
$907$	30564.6	1.11894	0.559471	−	0.828850i	$-0.311004\pi$
$907$	30564.6	1.11894	0.559471	+	0.828850i	$0.311004\pi$
$908$	4981.26i	0.182058i
$909$	−16282.8	−0.594132
$910$	0	0
$911$	−32766.5	−1.19166	−0.595831	−	0.803110i	$-0.703177\pi$
$911$	−32766.5	−1.19166	−0.595831	+	0.803110i	$0.703177\pi$
$912$	− 2377.97i	− 0.0863404i
$913$	20667.0	0.749154
$914$	7216.87	0.261174
$915$	− 4246.96i	− 0.153443i
$916$	3491.08i	0.125926i
$917$	7647.64i	0.275406i
$918$	− 12942.2i	− 0.465312i
$919$	20686.7	0.742538	0.371269	−	0.928525i	$-0.378923\pi$
$919$	20686.7	0.742538	0.371269	+	0.928525i	$0.378923\pi$
$920$	9257.35	0.331745
$921$	− 871.989i	− 0.0311976i
$922$	8007.28	0.286015
$923$	0	0
$924$	20033.0	0.713242
$925$	10579.2i	0.376047i
$926$	−10128.6	−0.359447
$927$	−15626.8	−0.553669
$928$	− 41179.0i	− 1.45665i
$929$	− 45632.2i	− 1.61156i	−0.592212	−	0.805782i	$-0.701745\pi$
$929$	− 45632.2i	− 1.61156i	0.592212	−	0.805782i	$-0.298255\pi$
$930$	1308.59i	0.0461401i
$931$	9482.26i	0.333801i
$932$	−12812.3	−0.450301
$933$	33687.1	1.18206
$934$	− 20265.9i	− 0.709979i
$935$	2421.28	0.0846892
$936$	0	0
$937$	−17761.4	−0.619253	−0.309626	−	0.950858i	$-0.600204\pi$
$937$	−17761.4	−0.619253	−0.309626	+	0.950858i	$0.600204\pi$
$938$	− 21740.4i	− 0.756768i
$939$	−32909.6	−1.14373
$940$	−9249.19	−0.320931
$941$	44888.3i	1.55507i	0.628841	+	0.777534i	$0.283530\pi$
$941$	44888.3i	1.55507i	−0.628841	+	0.777534i	$0.716470\pi$
$942$	44005.5i	1.52206i
$943$	− 19681.7i	− 0.679664i
$944$	− 5022.05i	− 0.173150i
$945$	18008.6	0.619917
$946$	−3605.74	−0.123925
$947$	16069.6i	0.551415i	0.961242	+	0.275708i	$0.0889122\pi$
$947$	16069.6i	0.551415i	−0.961242	+	0.275708i	$0.911088\pi$
$948$	−28326.2	−0.970457
$949$	0	0
$950$	4203.96	0.143573
$951$	− 38271.6i	− 1.30499i
$952$	−25637.8	−0.872823
$953$	−3512.03	−0.119377	−0.0596883	−	0.998217i	$-0.519011\pi$
$953$	−3512.03	−0.119377	−0.0596883	+	0.998217i	$0.519011\pi$
$954$	− 9137.88i	− 0.310115i
$955$	15246.3i	0.516605i
$956$	− 3028.94i	− 0.102472i
$957$	− 29146.7i	− 0.984513i
$958$	9071.00	0.305919
$959$	−71786.5	−2.41721
$960$	− 6217.78i	− 0.209040i
$961$	29057.0	0.975361
$962$	0	0
$963$	69468.9	2.32461
$964$	30177.5i	1.00825i
$965$	−1682.19	−0.0561158
$966$	−45237.0	−1.50670
$967$	37011.9i	1.23084i	0.788199	+	0.615421i	$0.211014\pi$
$967$	37011.9i	1.23084i	−0.788199	+	0.615421i	$0.788986\pi$
$968$	23254.4i	0.772132i
$969$	9273.16i	0.307427i
$970$	− 8405.00i	− 0.278215i
$971$	19532.3	0.645542	0.322771	−	0.946477i	$-0.395386\pi$
$971$	19532.3	0.645542	0.322771	+	0.946477i	$0.395386\pi$
$972$	−13043.1	−0.430409
$973$	54317.1i	1.78965i
$974$	−8412.30	−0.276743
$975$	0	0
$976$	−1568.47	−0.0514402
$977$	− 30201.2i	− 0.988970i	−0.869186	−	0.494485i	$-0.835357\pi$
$977$	− 30201.2i	− 0.988970i	0.869186	−	0.494485i	$-0.164643\pi$
$978$	−44501.8	−1.45502
$979$	−6713.21	−0.219157
$980$	− 7835.83i	− 0.255415i
$981$	− 41118.5i	− 1.33824i
$982$	− 23827.8i	− 0.774315i
$983$	− 38774.9i	− 1.25812i	−0.777359	−	0.629058i	$-0.783441\pi$
$983$	− 38774.9i	− 1.25812i	0.777359	−	0.629058i	$-0.216559\pi$
$984$	−29491.7	−0.955447
$985$	−15971.3	−0.516638
$986$	15297.2i	0.494080i
$987$	110211.	3.55425
$988$	0	0
$989$	−18570.6	−0.597079
$990$	− 4110.02i	− 0.131944i
$991$	−27728.9	−0.888838	−0.444419	−	0.895819i	$-0.646590\pi$
$991$	−27728.9	−0.888838	−0.444419	+	0.895819i	$0.646590\pi$
$992$	5073.25	0.162375
$993$	35880.6i	1.14666i
$994$	− 17430.3i	− 0.556192i
$995$	1304.38i	0.0415596i
$996$	− 65408.2i	− 2.08086i
$997$	−48918.2	−1.55392	−0.776958	−	0.629552i	$-0.783239\pi$
$997$	−48918.2	−1.55392	−0.776958	+	0.629552i	$0.783239\pi$
$998$	2898.31	0.0919283
$999$	17524.8i	0.555016i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.f.168.3		4
13.2	odd	12		169.4.c.j.22.1		4
13.3	even	3		169.4.e.f.147.2		8
13.4	even	6		169.4.e.f.23.2		8
13.5	odd	4		169.4.a.g.1.2		2
13.6	odd	12		169.4.c.j.146.1		4
13.7	odd	12		169.4.c.g.146.2		4
13.8	odd	4		13.4.a.b.1.1	✓	2
13.9	even	3		169.4.e.f.23.3		8
13.10	even	6		169.4.e.f.147.3		8
13.11	odd	12		169.4.c.g.22.2		4
13.12	even	2	inner	169.4.b.f.168.2		4
39.5	even	4		1521.4.a.r.1.1		2
39.8	even	4		117.4.a.d.1.2		2
52.47	even	4		208.4.a.h.1.1		2
65.8	even	4		325.4.b.e.274.3		4
65.34	odd	4		325.4.a.f.1.2		2
65.47	even	4		325.4.b.e.274.2		4
91.34	even	4		637.4.a.b.1.1		2
104.21	odd	4		832.4.a.s.1.1		2
104.99	even	4		832.4.a.z.1.2		2
143.21	even	4		1573.4.a.b.1.2		2
156.47	odd	4		1872.4.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.1	✓	2	13.8	odd	4
117.4.a.d.1.2		2	39.8	even	4
169.4.a.g.1.2		2	13.5	odd	4
169.4.b.f.168.2		4	13.12	even	2	inner
169.4.b.f.168.3		4	1.1	even	1	trivial
169.4.c.g.22.2		4	13.11	odd	12
169.4.c.g.146.2		4	13.7	odd	12
169.4.c.j.22.1		4	13.2	odd	12
169.4.c.j.146.1		4	13.6	odd	12
169.4.e.f.23.2		8	13.4	even	6
169.4.e.f.23.3		8	13.9	even	3
169.4.e.f.147.2		8	13.3	even	3
169.4.e.f.147.3		8	13.10	even	6
208.4.a.h.1.1		2	52.47	even	4
325.4.a.f.1.2		2	65.34	odd	4
325.4.b.e.274.2		4	65.47	even	4
325.4.b.e.274.3		4	65.8	even	4
637.4.a.b.1.1		2	91.34	even	4
832.4.a.s.1.1		2	104.21	odd	4
832.4.a.z.1.2		2	104.99	even	4
1521.4.a.r.1.1		2	39.5	even	4
1573.4.a.b.1.2		2	143.21	even	4
1872.4.a.bb.1.2		2	156.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{2} +8.68466 q^{3} +5.56155 q^{4} +3.56155i q^{5} +13.5616i q^{6} -27.1771i q^{7} +21.1771i q^{8} +48.4233 q^{9} +O(q^{10})\)	\(q+1.56155i q^{2} +8.68466 q^{3} +5.56155 q^{4} +3.56155i q^{5} +13.5616i q^{6} -27.1771i q^{7} +21.1771i q^{8} +48.4233 q^{9} -5.56155 q^{10} +15.2614i q^{11} +48.3002 q^{12} +42.4384 q^{14} +30.9309i q^{15} +11.4233 q^{16} -44.5464 q^{17} +75.6155i q^{18} -23.9697i q^{19} +19.8078i q^{20} -236.024i q^{21} -23.8314 q^{22} -122.739 q^{23} +183.916i q^{24} +112.315 q^{25} +186.054 q^{27} -151.147i q^{28} -219.909 q^{29} -48.3002 q^{30} -27.0928i q^{31} +187.255i q^{32} +132.540i q^{33} -69.5616i q^{34} +96.7926 q^{35} +269.309 q^{36} +94.1922i q^{37} +37.4299 q^{38} -75.4233 q^{40} +160.354i q^{41} +368.563 q^{42} +151.302 q^{43} +84.8769i q^{44} +172.462i q^{45} -191.663i q^{46} +466.948i q^{47} +99.2074 q^{48} -395.594 q^{49} +175.386i q^{50} -386.870 q^{51} -120.847 q^{53} +290.533i q^{54} -54.3542 q^{55} +575.531 q^{56} -208.169i q^{57} -343.400i q^{58} -439.633i q^{59} +172.024i q^{60} -137.305 q^{61} +42.3068 q^{62} -1316.00i q^{63} -201.022 q^{64} -206.968 q^{66} -512.280i q^{67} -247.747 q^{68} -1065.94 q^{69} +151.147i q^{70} -410.719i q^{71} +1025.46i q^{72} -308.004i q^{73} -147.086 q^{74} +975.420 q^{75} -133.309i q^{76} +414.759 q^{77} -586.462 q^{79} +40.6847i q^{80} +308.386 q^{81} -250.401 q^{82} -1354.20i q^{83} -1312.66i q^{84} -158.654i q^{85} +236.266i q^{86} -1909.84 q^{87} -323.191 q^{88} +439.882i q^{89} -269.309 q^{90} -682.617 q^{92} -235.292i q^{93} -729.164 q^{94} +85.3693 q^{95} +1626.24i q^{96} +1511.27i q^{97} -617.740i q^{98} +739.006i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9}+O(q^{10}) \) 4 * q + 10 * q^3 + 14 * q^4 + 70 * q^9	\( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9} - 14 q^{10} + 86 q^{12} + 178 q^{14} - 78 q^{16} - 38 q^{17} + 284 q^{22} - 392 q^{23} + 474 q^{25} + 670 q^{27} - 88 q^{29} - 86 q^{30} + 214 q^{35} + 500 q^{36} + 628 q^{38} - 178 q^{40} + 394 q^{42} - 574 q^{43} + 570 q^{48} - 766 q^{49} - 962 q^{51} - 236 q^{53} - 36 q^{55} + 2030 q^{56} - 2116 q^{61} + 664 q^{62} - 1538 q^{64} - 1636 q^{66} - 422 q^{68} - 1592 q^{69} + 294 q^{74} + 1032 q^{75} - 1524 q^{77} - 2016 q^{79} + 244 q^{81} - 144 q^{82} - 5116 q^{87} + 2484 q^{88} - 500 q^{90} - 1576 q^{92} - 1622 q^{94} + 292 q^{95}+O(q^{100}) \) 4 * q + 10 * q^3 + 14 * q^4 + 70 * q^9 - 14 * q^10 + 86 * q^12 + 178 * q^14 - 78 * q^16 - 38 * q^17 + 284 * q^22 - 392 * q^23 + 474 * q^25 + 670 * q^27 - 88 * q^29 - 86 * q^30 + 214 * q^35 + 500 * q^36 + 628 * q^38 - 178 * q^40 + 394 * q^42 - 574 * q^43 + 570 * q^48 - 766 * q^49 - 962 * q^51 - 236 * q^53 - 36 * q^55 + 2030 * q^56 - 2116 * q^61 + 664 * q^62 - 1538 * q^64 - 1636 * q^66 - 422 * q^68 - 1592 * q^69 + 294 * q^74 + 1032 * q^75 - 1524 * q^77 - 2016 * q^79 + 244 * q^81 - 144 * q^82 - 5116 * q^87 + 2484 * q^88 - 500 * q^90 - 1576 * q^92 - 1622 * q^94 + 292 * q^95

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