LMFDB - Embedded newform 169.4.b.f.168.4

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			168.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	169.168
Dual form			169.4.b.f.168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{2} -3.68466 q^{3} +1.43845 q^{4} +0.561553i q^{5} -9.43845i q^{6} -18.1771i q^{7} +24.1771i q^{8} -13.4233 q^{9} +O(q^{10})$	\(q+2.56155i q^{2} -3.68466 q^{3} +1.43845 q^{4} +0.561553i q^{5} -9.43845i q^{6} -18.1771i q^{7} +24.1771i q^{8} -13.4233 q^{9} -1.43845 q^{10} -64.7386i q^{11} -5.30019 q^{12} +46.5616 q^{14} -2.06913i q^{15} -50.4233 q^{16} +25.5464 q^{17} -34.3845i q^{18} -107.970i q^{19} +0.807764i q^{20} +66.9763i q^{21} +165.831 q^{22} -73.2614 q^{23} -89.0843i q^{24} +124.685 q^{25} +148.946 q^{27} -26.1468i q^{28} +175.909 q^{29} +5.30019 q^{30} -113.093i q^{31} +64.2547i q^{32} +238.540i q^{33} +65.4384i q^{34} +10.2074 q^{35} -19.3087 q^{36} -114.808i q^{37} +276.570 q^{38} -13.5767 q^{40} -69.6458i q^{41} -171.563 q^{42} -438.302 q^{43} -93.1231i q^{44} -7.53789i q^{45} -187.663i q^{46} +31.9479i q^{47} +185.793 q^{48} +12.5937 q^{49} +319.386i q^{50} -94.1298 q^{51} +2.84658 q^{53} +381.533i q^{54} +36.3542 q^{55} +439.469 q^{56} +397.831i q^{57} +450.600i q^{58} -71.6325i q^{59} -2.97633i q^{60} -920.695 q^{61} +289.693 q^{62} +243.996i q^{63} -567.978 q^{64} -611.032 q^{66} -444.280i q^{67} +36.7471 q^{68} +269.943 q^{69} +26.1468i q^{70} -541.719i q^{71} -324.536i q^{72} -764.004i q^{73} +294.086 q^{74} -459.420 q^{75} -155.309i q^{76} -1176.76 q^{77} -421.538 q^{79} -28.3153i q^{80} -186.386 q^{81} +178.401 q^{82} +603.797i q^{83} +96.3419i q^{84} +14.3457i q^{85} -1122.73i q^{86} -648.165 q^{87} +1565.19 q^{88} +1159.88i q^{89} +19.3087 q^{90} -105.383 q^{92} +416.708i q^{93} -81.8362 q^{94} +60.6307 q^{95} -236.757i q^{96} +583.269i q^{97} +32.2595i q^{98} +869.006i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.56155i	0.905646i	0.891601	+	0.452823i	$0.149583\pi$
$2$	2.56155i	0.905646i	−0.891601	+	0.452823i	$0.850417\pi$
$3$	−3.68466	−0.709113	−0.354556	−	0.935035i	$-0.615368\pi$
$3$	−3.68466	−0.709113	−0.354556	+	0.935035i	$0.615368\pi$
$4$	1.43845	0.179806
$5$	0.561553i	0.0502268i	0.999685	+	0.0251134i	$0.00799469\pi$
$5$	0.561553i	0.0502268i	−0.999685	+	0.0251134i	$0.992005\pi$
$6$	− 9.43845i	− 0.642205i
$7$	− 18.1771i	− 0.981470i	−0.871309	−	0.490735i	$-0.836728\pi$
$7$	− 18.1771i	− 0.981470i	0.871309	−	0.490735i	$-0.163272\pi$
$8$	24.1771i	1.06849i
$9$	−13.4233	−0.497159
$10$	−1.43845	−0.0454877
$11$	− 64.7386i	− 1.77449i	−0.461295	−	0.887247i	$-0.652615\pi$
$11$	− 64.7386i	− 1.77449i	0.461295	−	0.887247i	$-0.347385\pi$
$12$	−5.30019	−0.127503
$13$	0	0
$13$	0	0
$14$	46.5616	0.888864
$15$	− 2.06913i	− 0.0356165i
$16$	−50.4233	−0.787864
$17$	25.5464	0.364465	0.182233	−	0.983255i	$-0.441668\pi$
$17$	25.5464	0.364465	0.182233	+	0.983255i	$0.441668\pi$
$18$	− 34.3845i	− 0.450250i
$19$	− 107.970i	− 1.30368i	−0.758356	−	0.651841i	$-0.773997\pi$
$19$	− 107.970i	− 1.30368i	0.758356	−	0.651841i	$-0.226003\pi$
$20$	0.807764i	0.00903108i
$21$	66.9763i	0.695973i
$22$	165.831	1.60706
$23$	−73.2614	−0.664176	−0.332088	−	0.943248i	$-0.607753\pi$
$23$	−73.2614	−0.664176	−0.332088	+	0.943248i	$0.607753\pi$
$24$	− 89.0843i	− 0.757677i
$25$	124.685	0.997477
$26$	0	0
$27$	148.946	1.06165
$28$	− 26.1468i	− 0.176474i
$29$	175.909	1.12640	0.563198	−	0.826322i	$-0.309571\pi$
$29$	175.909	1.12640	0.563198	+	0.826322i	$0.309571\pi$
$30$	5.30019	0.0322559
$31$	− 113.093i	− 0.655228i	−0.944812	−	0.327614i	$-0.893755\pi$
$31$	− 113.093i	− 0.655228i	0.944812	−	0.327614i	$-0.106245\pi$
$32$	64.2547i	0.354961i
$33$	238.540i	1.25832i
$34$	65.4384i	0.330077i
$35$	10.2074	0.0492961
$36$	−19.3087	−0.0893921
$37$	− 114.808i	− 0.510116i	−0.966926	−	0.255058i	$-0.917905\pi$
$37$	− 114.808i	− 0.510116i	0.966926	−	0.255058i	$-0.0820945\pi$
$38$	276.570	1.18067
$39$	0	0
$40$	−13.5767	−0.0536666
$41$	− 69.6458i	− 0.265289i	−0.991164	−	0.132645i	$-0.957653\pi$
$41$	− 69.6458i	− 0.265289i	0.991164	−	0.132645i	$-0.0423469\pi$
$42$	−171.563	−0.630305
$43$	−438.302	−1.55443	−0.777214	−	0.629236i	$-0.783368\pi$
$43$	−438.302	−1.55443	−0.777214	+	0.629236i	$0.783368\pi$
$44$	− 93.1231i	− 0.319064i
$45$	− 7.53789i	− 0.0249707i
$46$	− 187.663i	− 0.601508i
$47$	31.9479i	0.0991506i	0.998770	+	0.0495753i	$0.0157868\pi$
$47$	31.9479i	0.0991506i	−0.998770	+	0.0495753i	$0.984213\pi$
$48$	185.793	0.558684
$49$	12.5937	0.0367164
$50$	319.386i	0.903361i
$51$	−94.1298	−0.258447
$52$	0	0
$53$	2.84658	0.00737752	0.00368876	−	0.999993i	$-0.498826\pi$
$53$	2.84658	0.00737752	0.00368876	+	0.999993i	$0.498826\pi$
$54$	381.533i	0.961483i
$55$	36.3542	0.0891272
$56$	439.469	1.04869
$57$	397.831i	0.924457i
$58$	450.600i	1.02012i
$59$	− 71.6325i	− 0.158064i	−0.996872	−	0.0790319i	$-0.974817\pi$
$59$	− 71.6325i	− 0.158064i	0.996872	−	0.0790319i	$-0.0251829\pi$
$60$	− 2.97633i	− 0.00640405i
$61$	−920.695	−1.93251	−0.966253	−	0.257593i	$-0.917071\pi$
$61$	−920.695	−1.93251	−0.966253	+	0.257593i	$0.917071\pi$
$62$	289.693	0.593404
$63$	243.996i	0.487947i
$64$	−567.978	−1.10933
$65$	0	0
$66$	−611.032	−1.13959
$67$	− 444.280i	− 0.810112i	−0.914292	−	0.405056i	$-0.867252\pi$
$67$	− 444.280i	− 0.810112i	0.914292	−	0.405056i	$-0.132748\pi$
$68$	36.7471	0.0655330
$69$	269.943	0.470976
$70$	26.1468i	0.0446448i
$71$	− 541.719i	− 0.905496i	−0.891639	−	0.452748i	$-0.850444\pi$
$71$	− 541.719i	− 0.905496i	0.891639	−	0.452748i	$-0.149556\pi$
$72$	− 324.536i	− 0.531207i
$73$	− 764.004i	− 1.22493i	−0.790498	−	0.612465i	$-0.790178\pi$
$73$	− 764.004i	− 1.22493i	0.790498	−	0.612465i	$-0.209822\pi$
$74$	294.086	0.461984
$75$	−459.420	−0.707324
$76$	− 155.309i	− 0.234410i
$77$	−1176.76	−1.74161
$78$	0	0
$79$	−421.538	−0.600338	−0.300169	−	0.953886i	$-0.597043\pi$
$79$	−421.538	−0.600338	−0.300169	+	0.953886i	$0.597043\pi$
$80$	− 28.3153i	− 0.0395719i
$81$	−186.386	−0.255674
$82$	178.401	0.240258
$83$	603.797i	0.798498i	0.916842	+	0.399249i	$0.130729\pi$
$83$	603.797i	0.798498i	−0.916842	+	0.399249i	$0.869271\pi$
$84$	96.3419i	0.125140i
$85$	14.3457i	0.0183059i
$86$	− 1122.73i	− 1.40776i
$87$	−648.165	−0.798742
$88$	1565.19	1.89602
$89$	1159.88i	1.38143i	0.723127	+	0.690715i	$0.242704\pi$
$89$	1159.88i	1.38143i	−0.723127	+	0.690715i	$0.757296\pi$
$90$	19.3087	0.0226146
$91$	0	0
$92$	−105.383	−0.119423
$93$	416.708i	0.464631i
$94$	−81.8362	−0.0897953
$95$	60.6307	0.0654798
$96$	− 236.757i	− 0.251707i
$97$	583.269i	0.610536i	0.952267	+	0.305268i	$0.0987460\pi$
$97$	583.269i	0.610536i	−0.952267	+	0.305268i	$0.901254\pi$
$98$	32.2595i	0.0332521i
$99$	869.006i	0.882206i
$100$	179.352	0.179352
$101$	−921.740	−0.908085	−0.454043	−	0.890980i	$-0.650019\pi$
$101$	−921.740	−0.908085	−0.454043	+	0.890980i	$0.650019\pi$
$102$	− 241.118i	− 0.234061i
$103$	930.712	0.890347	0.445174	−	0.895444i	$-0.353142\pi$
$103$	930.712	0.890347	0.445174	+	0.895444i	$0.353142\pi$
$104$	0	0
$105$	−37.6107	−0.0349565
$106$	7.29168i	0.00668142i
$107$	857.383	0.774638	0.387319	−	0.921946i	$-0.373401\pi$
$107$	857.383	0.774638	0.387319	+	0.921946i	$0.373401\pi$
$108$	214.251	0.190892
$109$	671.853i	0.590384i	0.955438	+	0.295192i	$0.0953836\pi$
$109$	671.853i	0.590384i	−0.955438	+	0.295192i	$0.904616\pi$
$110$	93.1231i	0.0807176i
$111$	423.027i	0.361730i
$112$	916.548i	0.773265i
$113$	641.474	0.534024	0.267012	−	0.963693i	$-0.413964\pi$
$113$	641.474	0.534024	0.267012	+	0.963693i	$0.413964\pi$
$114$	−1019.07	−0.837231
$115$	− 41.1401i	− 0.0333594i
$116$	253.036	0.202533
$117$	0	0
$118$	183.491	0.143150
$119$	− 464.359i	− 0.357712i
$120$	50.0255	0.0380557
$121$	−2860.09	−2.14883
$122$	− 2358.41i	− 1.75017i
$123$	256.621i	0.188120i
$124$	− 162.678i	− 0.117814i
$125$	140.211i	0.100327i
$126$	−625.009	−0.441907
$127$	553.174	0.386506	0.193253	−	0.981149i	$-0.438096\pi$
$127$	553.174	0.386506	0.193253	+	0.981149i	$0.438096\pi$
$128$	− 940.868i	− 0.649702i
$129$	1614.99	1.10227
$130$	0	0
$131$	2056.40	1.37152	0.685758	−	0.727830i	$-0.259471\pi$
$131$	2056.40	1.37152	0.685758	+	0.727830i	$0.259471\pi$
$132$	343.127i	0.226253i
$133$	−1962.57	−1.27952
$134$	1138.05	0.733674
$135$	83.6411i	0.0533235i
$136$	617.637i	0.389426i
$137$	1808.57i	1.12786i	0.825824	+	0.563928i	$0.190710\pi$
$137$	1808.57i	1.12786i	−0.825824	+	0.563928i	$0.809290\pi$
$138$	691.474i	0.426537i
$139$	1493.64	0.911428	0.455714	−	0.890126i	$-0.349384\pi$
$139$	1493.64	0.911428	0.455714	+	0.890126i	$0.349384\pi$
$140$	14.6828	0.00886373
$141$	− 117.717i	− 0.0703090i
$142$	1387.64	0.820058
$143$	0	0
$144$	676.847	0.391694
$145$	98.7822i	0.0565753i
$146$	1957.04	1.10935
$147$	−46.4036	−0.0260361
$148$	− 165.145i	− 0.0917218i
$149$	− 2759.02i	− 1.51696i	−0.651694	−	0.758482i	$-0.725941\pi$
$149$	− 2759.02i	− 1.51696i	0.651694	−	0.758482i	$-0.274059\pi$
$150$	− 1176.83i	− 0.640585i
$151$	976.355i	0.526190i	0.964770	+	0.263095i	$0.0847432\pi$
$151$	976.355i	0.526190i	−0.964770	+	0.263095i	$0.915257\pi$
$152$	2610.39	1.39297
$153$	−342.917	−0.181197
$154$	− 3014.33i	− 1.57728i
$155$	63.5076	0.0329100
$156$	0	0
$157$	−564.875	−0.287146	−0.143573	−	0.989640i	$-0.545859\pi$
$157$	−564.875	−0.287146	−0.143573	+	0.989640i	$0.545859\pi$
$158$	− 1079.79i	− 0.543694i
$159$	−10.4887	−0.00523149
$160$	−36.0824	−0.0178285
$161$	1331.68i	0.651869i
$162$	− 477.438i	− 0.231550i
$163$	− 1508.53i	− 0.724892i	−0.932005	−	0.362446i	$-0.881942\pi$
$163$	− 1508.53i	− 0.724892i	0.932005	−	0.362446i	$-0.118058\pi$
$164$	− 100.182i	− 0.0477005i
$165$	−133.953	−0.0632012
$166$	−1546.66	−0.723157
$167$	− 592.521i	− 0.274555i	−0.990533	−	0.137277i	$-0.956165\pi$
$167$	− 592.521i	− 0.274555i	0.990533	−	0.137277i	$-0.0438351\pi$
$168$	−1619.29	−0.743638
$169$	0	0
$170$	−36.7471	−0.0165787
$171$	1449.31i	0.648137i
$172$	−630.474	−0.279495
$173$	4495.57	1.97568	0.987838	−	0.155488i	$-0.0496952\pi$
$173$	4495.57	1.97568	0.987838	+	0.155488i	$0.0496952\pi$
$174$	− 1660.31i	− 0.723377i
$175$	− 2266.40i	− 0.978994i
$176$	3264.34i	1.39806i
$177$	263.941i	0.112085i
$178$	−2971.10	−1.25109
$179$	154.285	0.0644235	0.0322117	−	0.999481i	$-0.489745\pi$
$179$	154.285	0.0644235	0.0322117	+	0.999481i	$0.489745\pi$
$180$	− 10.8429i	− 0.00448988i
$181$	−1071.35	−0.439959	−0.219979	−	0.975505i	$-0.570599\pi$
$181$	−1071.35	−0.439959	−0.219979	+	0.975505i	$0.570599\pi$
$182$	0	0
$183$	3392.45	1.37037
$184$	− 1771.25i	− 0.709663i
$185$	64.4706	0.0256215
$186$	−1067.42	−0.420791
$187$	− 1653.84i	− 0.646742i
$188$	45.9554i	0.0178279i
$189$	− 2707.40i	− 1.04198i
$190$	155.309i	0.0593015i
$191$	677.203	0.256548	0.128274	−	0.991739i	$-0.459056\pi$
$191$	677.203	0.256548	0.128274	+	0.991739i	$0.459056\pi$
$192$	2092.81	0.786642
$193$	− 1321.68i	− 0.492936i	−0.969151	−	0.246468i	$-0.920730\pi$
$193$	− 1321.68i	− 0.492936i	0.969151	−	0.246468i	$-0.0792700\pi$
$194$	−1494.07	−0.552929
$195$	0	0
$196$	18.1154	0.00660183
$197$	1267.37i	0.458356i	0.973385	+	0.229178i	$0.0736038\pi$
$197$	1267.37i	0.458356i	−0.973385	+	0.229178i	$0.926396\pi$
$198$	−2226.00	−0.798966
$199$	−2396.24	−0.853593	−0.426796	−	0.904348i	$-0.640358\pi$
$199$	−2396.24	−0.853593	−0.426796	+	0.904348i	$0.640358\pi$
$200$	3014.51i	1.06579i
$201$	1637.02i	0.574460i
$202$	− 2361.09i	− 0.822403i
$203$	− 3197.51i	− 1.10552i
$204$	−135.401	−0.0464703
$205$	39.1098	0.0133246
$206$	2384.07i	0.806339i
$207$	983.409	0.330201
$208$	0	0
$209$	−6989.81	−2.31337
$210$	− 96.3419i	− 0.0316582i
$211$	−91.5539	−0.0298712	−0.0149356	−	0.999888i	$-0.504754\pi$
$211$	−91.5539	−0.0298712	−0.0149356	+	0.999888i	$0.504754\pi$
$212$	4.09466	0.00132652
$213$	1996.05i	0.642099i
$214$	2196.23i	0.701548i
$215$	− 246.130i	− 0.0780740i
$216$	3601.08i	1.13436i
$217$	−2055.70	−0.643087
$218$	−1720.99	−0.534679
$219$	2815.09i	0.868613i
$220$	52.2935	0.0160256
$221$	0	0
$222$	−1083.61	−0.327599
$223$	1235.42i	0.370985i	0.982646	+	0.185493i	$0.0593880\pi$
$223$	1235.42i	0.370985i	−0.982646	+	0.185493i	$0.940612\pi$
$224$	1167.96	0.348383
$225$	−1673.68	−0.495905
$226$	1643.17i	0.483637i
$227$	3301.66i	0.965370i	0.875794	+	0.482685i	$0.160338\pi$
$227$	3301.66i	0.965370i	−0.875794	+	0.482685i	$0.839662\pi$
$228$	572.260i	0.166223i
$229$	− 211.283i	− 0.0609694i	−0.999535	−	0.0304847i	$-0.990295\pi$
$229$	− 211.283i	− 0.0609694i	0.999535	−	0.0304847i	$-0.00970508\pi$
$230$	105.383	0.0302118
$231$	4335.96	1.23500
$232$	4252.97i	1.20354i
$233$	256.724	0.0721827	0.0360913	−	0.999348i	$-0.488509\pi$
$233$	256.724	0.0721827	0.0360913	+	0.999348i	$0.488509\pi$
$234$	0	0
$235$	−17.9404	−0.00498002
$236$	− 103.040i	− 0.0284208i
$237$	1553.22	0.425708
$238$	1189.48	0.323960
$239$	− 3549.62i	− 0.960694i	−0.877078	−	0.480347i	$-0.840511\pi$
$239$	− 3549.62i	− 0.960694i	0.877078	−	0.480347i	$-0.159489\pi$
$240$	104.332i	0.0280609i
$241$	5030.10i	1.34447i	0.740338	+	0.672235i	$0.234665\pi$
$241$	5030.10i	1.34447i	−0.740338	+	0.672235i	$0.765335\pi$
$242$	− 7326.27i	− 1.94608i
$243$	−3334.77	−0.880353
$244$	−1324.37	−0.347476
$245$	7.07204i	0.00184415i
$246$	−657.349	−0.170370
$247$	0	0
$248$	2734.25	0.700102
$249$	− 2224.79i	− 0.566226i
$250$	−359.158	−0.0908606
$251$	718.784	0.180754	0.0903770	−	0.995908i	$-0.471193\pi$
$251$	718.784	0.180754	0.0903770	+	0.995908i	$0.471193\pi$
$252$	350.976i	0.0877357i
$253$	4742.84i	1.17858i
$254$	1416.98i	0.350038i
$255$	− 52.8588i	− 0.0129810i
$256$	−2133.74	−0.520933
$257$	−1280.79	−0.310871	−0.155435	−	0.987846i	$-0.549678\pi$
$257$	−1280.79	−0.310871	−0.155435	+	0.987846i	$0.549678\pi$
$258$	4136.89i	0.998262i
$259$	−2086.87	−0.500663
$260$	0	0
$261$	−2361.28	−0.559998
$262$	5267.58i	1.24211i
$263$	5225.55	1.22517	0.612587	−	0.790403i	$-0.290129\pi$
$263$	5225.55	1.22517	0.612587	+	0.790403i	$0.290129\pi$
$264$	−5767.19	−1.34449
$265$	1.59851i	0 0.000370549i
$266$	− 5027.24i	− 1.15880i
$267$	− 4273.77i	− 0.979590i
$268$	− 639.074i	− 0.145663i
$269$	6443.80	1.46054	0.730270	−	0.683158i	$-0.239394\pi$
$269$	6443.80	1.46054	0.730270	+	0.683158i	$0.239394\pi$
$270$	−214.251	−0.0482922
$271$	3929.93i	0.880909i	0.897775	+	0.440455i	$0.145183\pi$
$271$	3929.93i	0.880909i	−0.897775	+	0.440455i	$0.854817\pi$
$272$	−1288.13	−0.287149
$273$	0	0
$274$	−4632.74	−1.02144
$275$	− 8071.91i	− 1.77002i
$276$	388.299	0.0846842
$277$	5884.40	1.27639	0.638194	−	0.769876i	$-0.279682\pi$
$277$	5884.40	1.27639	0.638194	+	0.769876i	$0.279682\pi$
$278$	3826.03i	0.825431i
$279$	1518.08i	0.325752i
$280$	246.785i	0.0526722i
$281$	− 3529.79i	− 0.749358i	−0.927155	−	0.374679i	$-0.877753\pi$
$281$	− 3529.79i	− 0.749358i	0.927155	−	0.374679i	$-0.122247\pi$
$282$	301.538	0.0636750
$283$	2611.00	0.548438	0.274219	−	0.961667i	$-0.411581\pi$
$283$	2611.00	0.548438	0.274219	+	0.961667i	$0.411581\pi$
$284$	− 779.234i	− 0.162813i
$285$	−223.403	−0.0464325
$286$	0	0
$287$	−1265.96	−0.260373
$288$	− 862.510i	− 0.176472i
$289$	−4260.38	−0.867165
$290$	−253.036	−0.0512372
$291$	− 2149.15i	− 0.432939i
$292$	− 1098.98i	− 0.220250i
$293$	5491.03i	1.09484i	0.836857	+	0.547422i	$0.184391\pi$
$293$	5491.03i	1.09484i	−0.836857	+	0.547422i	$0.815609\pi$
$294$	− 118.865i	− 0.0235795i
$295$	40.2255	0.00793904
$296$	2775.72	0.545052
$297$	− 9642.56i	− 1.88390i
$298$	7067.37	1.37383
$299$	0	0
$300$	−660.852	−0.127181
$301$	7967.05i	1.52563i
$302$	−2500.99	−0.476542
$303$	3396.30	0.643935
$304$	5444.19i	1.02712i
$305$	− 517.019i	− 0.0970637i
$306$	− 878.399i	− 0.164100i
$307$	7307.59i	1.35852i	0.733897	+	0.679261i	$0.237700\pi$
$307$	7307.59i	1.35852i	−0.733897	+	0.679261i	$0.762300\pi$
$308$	−1692.71	−0.313152
$309$	−3429.36	−0.631357
$310$	162.678i	0.0298048i
$311$	−7904.92	−1.44131	−0.720654	−	0.693295i	$-0.756158\pi$
$311$	−7904.92	−1.44131	−0.720654	+	0.693295i	$0.756158\pi$
$312$	0	0
$313$	10002.4	1.80629	0.903145	−	0.429336i	$-0.141252\pi$
$313$	10002.4	1.80629	0.903145	+	0.429336i	$0.141252\pi$
$314$	− 1446.96i	− 0.260053i
$315$	−137.017	−0.0245080
$316$	−606.360	−0.107944
$317$	− 6230.81i	− 1.10397i	−0.833856	−	0.551983i	$-0.813871\pi$
$317$	− 6230.81i	− 1.10397i	0.833856	−	0.551983i	$-0.186129\pi$
$318$	− 26.8673i	− 0.00473788i
$319$	− 11388.1i	− 1.99878i
$320$	− 318.950i	− 0.0557182i
$321$	−3159.16	−0.549306
$322$	−3411.16	−0.590362
$323$	− 2758.24i	− 0.475147i
$324$	−268.107	−0.0459717
$325$	0	0
$326$	3864.19	0.656495
$327$	− 2475.55i	− 0.418649i
$328$	1683.83	0.283458
$329$	580.719	0.0973134
$330$	− 343.127i	− 0.0572379i
$331$	− 4634.51i	− 0.769594i	−0.923001	−	0.384797i	$-0.874271\pi$
$331$	− 4634.51i	− 0.769594i	0.923001	−	0.384797i	$-0.125729\pi$
$332$	868.531i	0.143575i
$333$	1541.10i	0.253609i
$334$	1517.77	0.248649
$335$	249.487	0.0406893
$336$	− 3377.17i	− 0.548332i
$337$	−3029.82	−0.489747	−0.244874	−	0.969555i	$-0.578746\pi$
$337$	−3029.82	−0.489747	−0.244874	+	0.969555i	$0.578746\pi$
$338$	0	0
$339$	−2363.61	−0.378684
$340$	20.6355i	0.00329151i
$341$	−7321.47	−1.16270
$342$	−3712.48	−0.586982
$343$	− 6463.66i	− 1.01751i
$344$	− 10596.9i	− 1.66089i
$345$	151.587i	0.0236556i
$346$	11515.6i	1.78926i
$347$	2841.60	0.439611	0.219805	−	0.975544i	$-0.429458\pi$
$347$	2841.60	0.439611	0.219805	+	0.975544i	$0.429458\pi$
$348$	−932.351	−0.143619
$349$	− 7565.68i	− 1.16040i	−0.814472	−	0.580202i	$-0.802973\pi$
$349$	− 7565.68i	− 1.16040i	0.814472	−	0.580202i	$-0.197027\pi$
$350$	5805.51	0.886622
$351$	0	0
$352$	4159.76	0.629875
$353$	− 2339.44i	− 0.352736i	−0.984324	−	0.176368i	$-0.943565\pi$
$353$	− 2339.44i	− 0.352736i	0.984324	−	0.176368i	$-0.0564348\pi$
$354$	−676.100	−0.101509
$355$	304.204	0.0454802
$356$	1668.43i	0.248389i
$357$	1711.00i	0.253658i
$358$	395.209i	0.0583449i
$359$	2531.68i	0.372192i	0.982532	+	0.186096i	$0.0595835\pi$
$359$	2531.68i	0.372192i	−0.982532	+	0.186096i	$0.940417\pi$
$360$	182.244	0.0266809
$361$	−4798.45	−0.699585
$362$	− 2744.31i	− 0.398447i
$363$	10538.5	1.52376
$364$	0	0
$365$	429.028	0.0615243
$366$	8689.93i	1.24107i
$367$	6577.81	0.935583	0.467792	−	0.883839i	$-0.345050\pi$
$367$	6577.81	0.935583	0.467792	+	0.883839i	$0.345050\pi$
$368$	3694.08	0.523280
$369$	934.876i	0.131891i
$370$	165.145i	0.0232040i
$371$	− 51.7426i	− 0.00724081i
$372$	599.413i	0.0835433i
$373$	2902.72	0.402942	0.201471	−	0.979495i	$-0.435428\pi$
$373$	2902.72	0.402942	0.201471	+	0.979495i	$0.435428\pi$
$374$	4236.40	0.585719
$375$	− 516.630i	− 0.0711431i
$376$	−772.407	−0.105941
$377$	0	0
$378$	6935.16	0.943667
$379$	− 1865.73i	− 0.252866i	−0.991975	−	0.126433i	$-0.959647\pi$
$379$	− 1865.73i	− 0.252866i	0.991975	−	0.126433i	$-0.0403529\pi$
$380$	87.2140	0.0117736
$381$	−2038.26	−0.274076
$382$	1734.69i	0.232342i
$383$	10836.0i	1.44567i	0.691019	+	0.722837i	$0.257162\pi$
$383$	10836.0i	1.44567i	−0.691019	+	0.722837i	$0.742838\pi$
$384$	3466.78i	0.460712i
$385$	− 660.813i	− 0.0874757i
$386$	3385.55	0.446425
$387$	5883.46	0.772798
$388$	839.001i	0.109778i
$389$	9520.34	1.24088	0.620438	−	0.784256i	$-0.286955\pi$
$389$	9520.34	1.24088	0.620438	+	0.784256i	$0.286955\pi$
$390$	0	0
$391$	−1871.56	−0.242069
$392$	304.480i	0.0392310i
$393$	−7577.13	−0.972559
$394$	−3246.43	−0.415108
$395$	− 236.716i	− 0.0301531i
$396$	1250.02i	0.158626i
$397$	10108.8i	1.27796i	0.769225	+	0.638978i	$0.220642\pi$
$397$	10108.8i	1.27796i	−0.769225	+	0.638978i	$0.779358\pi$
$398$	− 6138.10i	− 0.773053i
$399$	7231.41	0.907327
$400$	−6287.01	−0.785876
$401$	− 2084.38i	− 0.259573i	−0.991542	−	0.129787i	$-0.958571\pi$
$401$	− 2084.38i	− 0.259573i	0.991542	−	0.129787i	$-0.0414292\pi$
$402$	−4193.32	−0.520258
$403$	0	0
$404$	−1325.88	−0.163279
$405$	− 104.666i	− 0.0128417i
$406$	8190.60	1.00121
$407$	−7432.50	−0.905197
$408$	− 2275.78i	− 0.276147i
$409$	− 9716.53i	− 1.17470i	−0.809334	−	0.587349i	$-0.800172\pi$
$409$	− 9716.53i	− 1.17470i	0.809334	−	0.587349i	$-0.199828\pi$
$410$	100.182i	0.0120674i
$411$	− 6663.95i	− 0.799777i
$412$	1338.78	0.160090
$413$	−1302.07	−0.155135
$414$	2519.05i	0.299045i
$415$	−339.064	−0.0401060
$416$	0	0
$417$	−5503.54	−0.646305
$418$	− 17904.8i	− 2.09510i
$419$	13381.9	1.56026	0.780129	−	0.625619i	$-0.215153\pi$
$419$	13381.9	1.56026	0.780129	+	0.625619i	$0.215153\pi$
$420$	−54.1011	−0.00628539
$421$	− 9463.37i	− 1.09553i	−0.836633	−	0.547763i	$-0.815479\pi$
$421$	− 9463.37i	− 1.09553i	0.836633	−	0.547763i	$-0.184521\pi$
$422$	− 234.520i	− 0.0270527i
$423$	− 428.846i	− 0.0492936i
$424$	68.8221i	0.00788278i
$425$	3185.24	0.363546
$426$	−5112.98	−0.581514
$427$	16735.5i	1.89670i
$428$	1233.30	0.139285
$429$	0	0
$430$	630.474	0.0707074
$431$	4852.28i	0.542288i	0.962539	+	0.271144i	$0.0874019\pi$
$431$	4852.28i	0.542288i	−0.962539	+	0.271144i	$0.912598\pi$
$432$	−7510.35	−0.836439
$433$	8208.00	0.910973	0.455486	−	0.890243i	$-0.349465\pi$
$433$	8208.00	0.910973	0.455486	+	0.890243i	$0.349465\pi$
$434$	− 5265.78i	− 0.582409i
$435$	− 363.979i	− 0.0401183i
$436$	966.425i	0.106155i
$437$	7910.01i	0.865874i
$438$	−7211.01	−0.786656
$439$	2993.80	0.325481	0.162741	−	0.986669i	$-0.447967\pi$
$439$	2993.80	0.325481	0.162741	+	0.986669i	$0.447967\pi$
$440$	878.938i	0.0952311i
$441$	−169.049	−0.0182539
$442$	0	0
$443$	9743.67	1.04500	0.522501	−	0.852639i	$-0.324999\pi$
$443$	9743.67	1.04500	0.522501	+	0.852639i	$0.324999\pi$
$444$	608.503i	0.0650411i
$445$	−651.335	−0.0693848
$446$	−3164.59	−0.335981
$447$	10166.0i	1.07570i
$448$	10324.2i	1.08878i
$449$	561.459i	0.0590131i	0.999565	+	0.0295065i	$0.00939359\pi$
$449$	561.459i	0.0590131i	−0.999565	+	0.0295065i	$0.990606\pi$
$450$	− 4287.22i	− 0.449114i
$451$	−4508.78	−0.470754
$452$	922.726	0.0960207
$453$	− 3597.54i	− 0.373128i
$454$	−8457.38	−0.874283
$455$	0	0
$456$	−9618.40	−0.987770
$457$	13758.4i	1.40830i	0.710054	+	0.704148i	$0.248671\pi$
$457$	13758.4i	1.40830i	−0.710054	+	0.704148i	$0.751329\pi$
$458$	541.213	0.0552166
$459$	3805.03	0.386936
$460$	− 59.1779i	− 0.00599823i
$461$	12009.2i	1.21329i	0.794974	+	0.606644i	$0.207485\pi$
$461$	12009.2i	1.21329i	−0.794974	+	0.606644i	$0.792515\pi$
$462$	11106.8i	1.11847i
$463$	− 13635.7i	− 1.36870i	−0.729156	−	0.684348i	$-0.760087\pi$
$463$	− 13635.7i	− 1.36870i	0.729156	−	0.684348i	$-0.239913\pi$
$464$	−8869.91	−0.887447
$465$	−234.004	−0.0233369
$466$	657.613i	0.0653719i
$467$	−8821.95	−0.874157	−0.437079	−	0.899423i	$-0.643987\pi$
$467$	−8821.95	−0.874157	−0.437079	+	0.899423i	$0.643987\pi$
$468$	0	0
$469$	−8075.72	−0.795100
$470$	− 45.9554i	− 0.00451013i
$471$	2081.37	0.203619
$472$	1731.87	0.168889
$473$	28375.1i	2.75832i
$474$	3978.66i	0.385540i
$475$	− 13462.2i	− 1.30039i
$476$	− 667.956i	− 0.0643187i
$477$	−38.2105	−0.00366780
$478$	9092.54	0.870049
$479$	14620.0i	1.39459i	0.716786	+	0.697293i	$0.245612\pi$
$479$	14620.0i	1.39459i	−0.716786	+	0.697293i	$0.754388\pi$
$480$	132.951	0.0126424
$481$	0	0
$482$	−12884.9	−1.21761
$483$	− 4906.78i	− 0.462249i
$484$	−4114.09	−0.386372
$485$	−327.536	−0.0306653
$486$	− 8542.20i	− 0.797288i
$487$	− 9798.86i	− 0.911763i	−0.890040	−	0.455882i	$-0.849324\pi$
$487$	− 9798.86i	− 0.911763i	0.890040	−	0.455882i	$-0.150676\pi$
$488$	− 22259.7i	− 2.06486i
$489$	5558.43i	0.514030i
$490$	−18.1154	−0.00167014
$491$	10836.1	0.995977	0.497989	−	0.867184i	$-0.334072\pi$
$491$	10836.1	0.995977	0.497989	+	0.867184i	$0.334072\pi$
$492$	369.136i	0.0338251i
$493$	4493.84	0.410532
$494$	0	0
$495$	−487.993	−0.0443104
$496$	5702.51i	0.516230i
$497$	−9846.86	−0.888717
$498$	5698.91	0.512800
$499$	2589.96i	0.232349i	0.993229	+	0.116175i	$0.0370633\pi$
$499$	2589.96i	0.232349i	−0.993229	+	0.116175i	$0.962937\pi$
$500$	201.686i	0.0180394i
$501$	2183.24i	0.194690i
$502$	1841.20i	0.163699i
$503$	−17067.5	−1.51292	−0.756462	−	0.654038i	$-0.773074\pi$
$503$	−17067.5	−1.51292	−0.756462	+	0.654038i	$0.773074\pi$
$504$	−5899.12	−0.521364
$505$	− 517.606i	− 0.0456102i
$506$	−12149.0	−1.06737
$507$	0	0
$508$	795.712	0.0694961
$509$	− 1012.89i	− 0.0882038i	−0.999027	−	0.0441019i	$-0.985957\pi$
$509$	− 1012.89i	− 0.0882038i	0.999027	−	0.0441019i	$-0.0140426\pi$
$510$	135.401	0.0117562
$511$	−13887.4	−1.20223
$512$	− 12992.6i	− 1.12148i
$513$	− 16081.7i	− 1.38406i
$514$	− 3280.82i	− 0.281539i
$515$	522.644i	0.0447193i
$516$	2323.08	0.198194
$517$	2068.26	0.175942
$518$	− 5345.63i	− 0.453424i
$519$	−16564.6	−1.40098
$520$	0	0
$521$	−14367.7	−1.20818	−0.604089	−	0.796917i	$-0.706463\pi$
$521$	−14367.7	−1.20818	−0.604089	+	0.796917i	$0.706463\pi$
$522$	− 6048.54i	− 0.507160i
$523$	−16219.9	−1.35611	−0.678057	−	0.735010i	$-0.737178\pi$
$523$	−16219.9	−1.35611	−0.678057	+	0.735010i	$0.737178\pi$
$524$	2958.02	0.246607
$525$	8350.92i	0.694217i
$526$	13385.5i	1.10957i
$527$	− 2889.11i	− 0.238808i
$528$	− 12028.0i	− 0.991382i
$529$	−6799.77	−0.558870
$530$	−4.09466	−0.000335586 0
$531$	961.545i	0.0785828i
$532$	−2823.06	−0.230066
$533$	0	0
$534$	10947.5	0.887161
$535$	481.466i	0.0389076i
$536$	10741.4	0.865593
$537$	−568.488	−0.0456835
$538$	16506.1i	1.32273i
$539$	− 815.301i	− 0.0651530i
$540$	120.313i	0.00958788i
$541$	− 17592.2i	− 1.39806i	−0.715094	−	0.699029i	$-0.753616\pi$
$541$	− 17592.2i	− 1.39806i	0.715094	−	0.699029i	$-0.246384\pi$
$542$	−10066.7	−0.797792
$543$	3947.55	0.311980
$544$	1641.48i	0.129371i
$545$	−377.281	−0.0296531
$546$	0	0
$547$	10504.6	0.821103	0.410552	−	0.911837i	$-0.365336\pi$
$547$	10504.6	0.821103	0.410552	+	0.911837i	$0.365336\pi$
$548$	2601.53i	0.202795i
$549$	12358.8	0.960763
$550$	20676.6	1.60301
$551$	− 18992.8i	− 1.46846i
$552$	6526.44i	0.503231i
$553$	7662.33i	0.589214i
$554$	15073.2i	1.15596i
$555$	−237.552	−0.0181685
$556$	2148.52	0.163880
$557$	507.558i	0.0386102i	0.999814	+	0.0193051i	$0.00614539\pi$
$557$	507.558i	0.0386102i	−0.999814	+	0.0193051i	$0.993855\pi$
$558$	−3888.64	−0.295016
$559$	0	0
$560$	−514.690	−0.0388386
$561$	6093.83i	0.458613i
$562$	9041.75	0.678653
$563$	3443.14	0.257746	0.128873	−	0.991661i	$-0.458864\pi$
$563$	3443.14	0.257746	0.128873	+	0.991661i	$0.458864\pi$
$564$	− 169.330i	− 0.0126420i
$565$	360.221i	0.0268223i
$566$	6688.21i	0.496690i
$567$	3387.96i	0.250936i
$568$	13097.2	0.967509
$569$	−23972.2	−1.76620	−0.883098	−	0.469189i	$-0.844546\pi$
$569$	−23972.2	−1.76620	−0.883098	+	0.469189i	$0.844546\pi$
$570$	− 572.260i	− 0.0420514i
$571$	7458.32	0.546622	0.273311	−	0.961926i	$-0.411881\pi$
$571$	7458.32	0.546622	0.273311	+	0.961926i	$0.411881\pi$
$572$	0	0
$573$	−2495.26	−0.181922
$574$	− 3242.82i	− 0.235806i
$575$	−9134.57	−0.662501
$576$	7624.14	0.551515
$577$	5669.57i	0.409059i	0.978860	+	0.204530i	$0.0655665\pi$
$577$	5669.57i	0.409059i	−0.978860	+	0.204530i	$0.934434\pi$
$578$	− 10913.2i	− 0.785344i
$579$	4869.94i	0.349547i
$580$	142.093i	0.0101726i
$581$	10975.3	0.783702
$582$	5505.15	0.392089
$583$	− 184.284i	− 0.0130914i
$584$	18471.4	1.30882
$585$	0	0
$586$	−14065.6	−0.991541
$587$	1017.39i	0.0715371i	0.999360	+	0.0357685i	$0.0113879\pi$
$587$	1017.39i	0.0715371i	−0.999360	+	0.0357685i	$0.988612\pi$
$588$	−66.7491	−0.00468144
$589$	−12210.6	−0.854208
$590$	103.040i	0.00718996i
$591$	− 4669.81i	− 0.325026i
$592$	5788.99i	0.401902i
$593$	10198.2i	0.706221i	0.935582	+	0.353111i	$0.114876\pi$
$593$	10198.2i	0.706221i	−0.935582	+	0.353111i	$0.885124\pi$
$594$	24699.9	1.70615
$595$	260.762	0.0179667
$596$	− 3968.70i	− 0.272759i
$597$	8829.33	0.605294
$598$	0	0
$599$	12516.3	0.853763	0.426881	−	0.904308i	$-0.359612\pi$
$599$	12516.3	0.853763	0.426881	+	0.904308i	$0.359612\pi$
$600$	− 11107.4i	− 0.755766i
$601$	9627.46	0.653431	0.326716	−	0.945123i	$-0.394058\pi$
$601$	9627.46	0.653431	0.326716	+	0.945123i	$0.394058\pi$
$602$	−20408.0	−1.38168
$603$	5963.70i	0.402754i
$604$	1404.44i	0.0946120i
$605$	− 1606.09i	− 0.107929i
$606$	8699.80i	0.583177i
$607$	6667.20	0.445821	0.222910	−	0.974839i	$-0.428444\pi$
$607$	6667.20	0.445821	0.222910	+	0.974839i	$0.428444\pi$
$608$	6937.56	0.462755
$609$	11781.7i	0.783942i
$610$	1324.37	0.0879053
$611$	0	0
$612$	−493.268	−0.0325803
$613$	− 23085.4i	− 1.52106i	−0.649302	−	0.760530i	$-0.724939\pi$
$613$	− 23085.4i	− 1.52106i	0.649302	−	0.760530i	$-0.275061\pi$
$614$	−18718.8	−1.23034
$615$	−144.106	−0.00944866
$616$	− 28450.6i	− 1.86089i
$617$	3049.24i	0.198959i	0.995040	+	0.0994796i	$0.0317178\pi$
$617$	3049.24i	0.198959i	−0.995040	+	0.0994796i	$0.968282\pi$
$618$	− 8784.48i	− 0.571786i
$619$	− 7296.58i	− 0.473787i	−0.971536	−	0.236894i	$-0.923871\pi$
$619$	− 7296.58i	− 0.473787i	0.971536	−	0.236894i	$-0.0761293\pi$
$620$	91.3523	0.00591741
$621$	−10912.0	−0.705126
$622$	− 20248.9i	− 1.30531i
$623$	21083.3	1.35583
$624$	0	0
$625$	15506.8	0.992438
$626$	25621.7i	1.63586i
$627$	25755.1	1.64044
$628$	−812.543	−0.0516305
$629$	− 2932.92i	− 0.185920i
$630$	− 350.976i	− 0.0221956i
$631$	23829.5i	1.50339i	0.659512	+	0.751694i	$0.270763\pi$
$631$	23829.5i	1.50339i	−0.659512	+	0.751694i	$0.729237\pi$
$632$	− 10191.6i	− 0.641453i
$633$	337.345	0.0211821
$634$	15960.5	0.999801
$635$	310.637i	0.0194130i
$636$	−15.0874	−0.000940653 0
$637$	0	0
$638$	29171.3	1.81019
$639$	7271.65i	0.450175i
$640$	528.347	0.0326324
$641$	−13405.3	−0.826016	−0.413008	−	0.910727i	$-0.635522\pi$
$641$	−13405.3	−0.826016	−0.413008	+	0.910727i	$0.635522\pi$
$642$	− 8092.36i	− 0.497477i
$643$	5251.51i	0.322083i	0.986948	+	0.161042i	$0.0514853\pi$
$643$	5251.51i	0.322083i	−0.986948	+	0.161042i	$0.948515\pi$
$644$	1915.55i	0.117210i
$645$	906.904i	0.0553633i
$646$	7065.37	0.430315
$647$	−21611.4	−1.31319	−0.656595	−	0.754244i	$-0.728004\pi$
$647$	−21611.4	−1.31319	−0.656595	+	0.754244i	$0.728004\pi$
$648$	− 4506.28i	− 0.273184i
$649$	−4637.39	−0.280483
$650$	0	0
$651$	7574.54	0.456021
$652$	− 2169.94i	− 0.130340i
$653$	−21595.8	−1.29420	−0.647099	−	0.762406i	$-0.724018\pi$
$653$	−21595.8	−1.29420	−0.647099	+	0.762406i	$0.724018\pi$
$654$	6341.25	0.379148
$655$	1154.78i	0.0688869i
$656$	3511.77i	0.209012i
$657$	10255.4i	0.608985i
$658$	1487.54i	0.0881314i
$659$	−16642.6	−0.983768	−0.491884	−	0.870661i	$-0.663692\pi$
$659$	−16642.6	−0.983768	−0.491884	+	0.870661i	$0.663692\pi$
$660$	−192.684	−0.0113640
$661$	− 26981.1i	− 1.58766i	−0.608139	−	0.793831i	$-0.708084\pi$
$661$	− 26981.1i	− 1.58766i	0.608139	−	0.793831i	$-0.291916\pi$
$662$	11871.5	0.696980
$663$	0	0
$664$	−14598.1	−0.853185
$665$	− 1102.09i	− 0.0642664i
$666$	−3947.60	−0.229680
$667$	−12887.3	−0.748126
$668$	− 852.310i	− 0.0493665i
$669$	− 4552.09i	− 0.263070i
$670$	639.074i	0.0368501i
$671$	59604.5i	3.42922i
$672$	−4303.55	−0.247043
$673$	−11149.2	−0.638591	−0.319296	−	0.947655i	$-0.603446\pi$
$673$	−11149.2	−0.638591	−0.319296	+	0.947655i	$0.603446\pi$
$674$	− 7761.04i	− 0.443537i
$675$	18571.3	1.05898
$676$	0	0
$677$	3314.33	0.188154	0.0940769	−	0.995565i	$-0.470010\pi$
$677$	3314.33	0.188154	0.0940769	+	0.995565i	$0.470010\pi$
$678$	− 6054.51i	− 0.342953i
$679$	10602.1	0.599223
$680$	−346.836	−0.0195596
$681$	− 12165.5i	− 0.684556i
$682$	− 18754.3i	− 1.05299i
$683$	− 24505.2i	− 1.37287i	−0.727193	−	0.686433i	$-0.759176\pi$
$683$	− 24505.2i	− 1.37287i	0.727193	−	0.686433i	$-0.240824\pi$
$684$	2084.75i	0.116539i
$685$	−1015.61	−0.0566486
$686$	16557.0	0.921500
$687$	778.506i	0.0432342i
$688$	22100.6	1.22468
$689$	0	0
$690$	−388.299	−0.0214236
$691$	− 21752.8i	− 1.19756i	−0.800912	−	0.598782i	$-0.795652\pi$
$691$	− 21752.8i	− 1.19756i	0.800912	−	0.598782i	$-0.204348\pi$
$692$	6466.64	0.355238
$693$	15796.0	0.865858
$694$	7278.90i	0.398132i
$695$	838.755i	0.0457781i
$696$	− 15670.7i	− 0.853445i
$697$	− 1779.20i	− 0.0966887i
$698$	19379.9	1.05092
$699$	−945.941	−0.0511857
$700$	− 3260.10i	− 0.176029i
$701$	−34250.9	−1.84542	−0.922709	−	0.385496i	$-0.874030\pi$
$701$	−34250.9	−1.84542	−0.922709	+	0.385496i	$0.874030\pi$
$702$	0	0
$703$	−12395.8	−0.665028
$704$	36770.1i	1.96850i
$705$	66.1043	0.00353140
$706$	5992.59	0.319454
$707$	16754.6i	0.891259i
$708$	379.666i	0.0201536i
$709$	5527.11i	0.292771i	0.989228	+	0.146386i	$0.0467641\pi$
$709$	5527.11i	0.292771i	−0.989228	+	0.146386i	$0.953236\pi$
$710$	779.234i	0.0411889i
$711$	5658.43	0.298464
$712$	−28042.6	−1.47604
$713$	8285.33i	0.435187i
$714$	−4382.83	−0.229724
$715$	0	0
$716$	221.931	0.0115837
$717$	13079.1i	0.681241i
$718$	−6485.02	−0.337074
$719$	3777.78	0.195949	0.0979745	−	0.995189i	$-0.468764\pi$
$719$	3777.78	0.195949	0.0979745	+	0.995189i	$0.468764\pi$
$720$	380.085i	0.0196735i
$721$	− 16917.6i	− 0.873849i
$722$	− 12291.5i	− 0.633576i
$723$	− 18534.2i	− 0.953380i
$724$	−1541.08	−0.0791072
$725$	21933.2	1.12355
$726$	26994.8i	1.37999i
$727$	−19076.8	−0.973204	−0.486602	−	0.873624i	$-0.661764\pi$
$727$	−19076.8	−0.973204	−0.486602	+	0.873624i	$0.661764\pi$
$728$	0	0
$729$	17319.9	0.879944
$730$	1098.98i	0.0557192i
$731$	−11197.0	−0.566535
$732$	4879.86	0.246400
$733$	7997.30i	0.402984i	0.979490	+	0.201492i	$0.0645790\pi$
$733$	7997.30i	0.402984i	−0.979490	+	0.201492i	$0.935421\pi$
$734$	16849.4i	0.847307i
$735$	− 26.0581i	− 0.00130771i
$736$	− 4707.39i	− 0.235756i
$737$	−28762.1	−1.43754
$738$	−2394.74	−0.119446
$739$	− 28983.6i	− 1.44273i	−0.692553	−	0.721367i	$-0.743514\pi$
$739$	− 28983.6i	− 1.44273i	0.692553	−	0.721367i	$-0.256486\pi$
$740$	92.7376	0.00460689
$741$	0	0
$742$	132.541	0.00655761
$743$	− 19145.4i	− 0.945324i	−0.881244	−	0.472662i	$-0.843293\pi$
$743$	− 19145.4i	− 0.945324i	0.881244	−	0.472662i	$-0.156707\pi$
$744$	−10074.8	−0.496451
$745$	1549.33	0.0761923
$746$	7435.47i	0.364922i
$747$	− 8104.95i	− 0.396981i
$748$	− 2378.96i	− 0.116288i
$749$	− 15584.7i	− 0.760284i
$750$	1323.38	0.0644304
$751$	25516.9	1.23985	0.619923	−	0.784663i	$-0.287164\pi$
$751$	25516.9	1.23985	0.619923	+	0.784663i	$0.287164\pi$
$752$	− 1610.92i	− 0.0781172i
$753$	−2648.47	−0.128175
$754$	0	0
$755$	−548.275	−0.0264288
$756$	− 3894.46i	− 0.187355i
$757$	−17230.6	−0.827289	−0.413645	−	0.910438i	$-0.635744\pi$
$757$	−17230.6	−0.827289	−0.413645	+	0.910438i	$0.635744\pi$
$758$	4779.17	0.229007
$759$	− 17475.7i	− 0.835744i
$760$	1465.87i	0.0699642i
$761$	2343.06i	0.111611i	0.998442	+	0.0558053i	$0.0177726\pi$
$761$	2343.06i	0.111611i	−0.998442	+	0.0558053i	$0.982227\pi$
$762$	− 5221.11i	− 0.248216i
$763$	12212.3	0.579444
$764$	974.121	0.0461289
$765$	− 192.566i	− 0.00910096i
$766$	−27756.9	−1.30927
$767$	0	0
$768$	7862.11	0.369400
$769$	− 7100.18i	− 0.332950i	−0.986046	−	0.166475i	$-0.946761\pi$
$769$	− 7100.18i	− 0.332950i	0.986046	−	0.166475i	$-0.0532386\pi$
$770$	1692.71	0.0792219
$771$	4719.29	0.220442
$772$	− 1901.17i	− 0.0886328i
$773$	12270.4i	0.570940i	0.958388	+	0.285470i	$0.0921498\pi$
$773$	12270.4i	0.570940i	−0.958388	+	0.285470i	$0.907850\pi$
$774$	15070.8i	0.699881i
$775$	− 14100.9i	− 0.653575i
$776$	−14101.7	−0.652349
$777$	7689.40	0.355027
$778$	24386.9i	1.12379i
$779$	−7519.64	−0.345852
$780$	0	0
$781$	−35070.1	−1.60680
$782$	− 4794.11i	− 0.219229i
$783$	26201.0	1.19584
$784$	−635.017	−0.0289275
$785$	− 317.207i	− 0.0144224i
$786$	− 19409.2i	− 0.880794i
$787$	− 3425.04i	− 0.155133i	−0.996987	−	0.0775663i	$-0.975285\pi$
$787$	− 3425.04i	− 0.155133i	0.996987	−	0.0775663i	$-0.0247150\pi$
$788$	1823.04i	0.0824151i
$789$	−19254.3	−0.868787
$790$	606.360	0.0273080
$791$	− 11660.1i	− 0.524129i
$792$	−21010.0	−0.942624
$793$	0	0
$794$	−25894.3	−1.15737
$795$	− 5.88995i	0 0.000262761i
$796$	−3446.87	−0.153481
$797$	11781.1	0.523600	0.261800	−	0.965122i	$-0.415684\pi$
$797$	11781.1	0.523600	0.261800	+	0.965122i	$0.415684\pi$
$798$	18523.6i	0.821717i
$799$	816.154i	0.0361370i
$800$	8011.58i	0.354065i
$801$	− 15569.4i	− 0.686790i
$802$	5339.24	0.235081
$803$	−49460.6	−2.17363
$804$	2354.77i	0.103291i
$805$	−747.807	−0.0327413
$806$	0	0
$807$	−23743.2	−1.03569
$808$	− 22285.0i	− 0.970276i
$809$	18910.1	0.821810	0.410905	−	0.911678i	$-0.365213\pi$
$809$	18910.1	0.821810	0.410905	+	0.911678i	$0.365213\pi$
$810$	268.107	0.0116300
$811$	12803.3i	0.554359i	0.960818	+	0.277180i	$0.0893997\pi$
$811$	12803.3i	0.554359i	−0.960818	+	0.277180i	$0.910600\pi$
$812$	− 4599.45i	− 0.198780i
$813$	− 14480.5i	− 0.624664i
$814$	− 19038.7i	− 0.819788i
$815$	847.121	0.0364090
$816$	4746.33	0.203621
$817$	47323.3i	2.02648i
$818$	24889.4	1.06386
$819$	0	0
$820$	56.2574	0.00239585
$821$	19335.1i	0.821923i	0.911653	+	0.410962i	$0.134807\pi$
$821$	19335.1i	0.821923i	−0.911653	+	0.410962i	$0.865193\pi$
$822$	17070.1	0.724315
$823$	2125.90	0.0900417	0.0450209	−	0.998986i	$-0.485665\pi$
$823$	2125.90	0.0900417	0.0450209	+	0.998986i	$0.485665\pi$
$824$	22501.9i	0.951324i
$825$	29742.2i	1.25514i
$826$	− 3335.32i	− 0.140497i
$827$	6989.24i	0.293881i	0.989145	+	0.146941i	$0.0469426\pi$
$827$	6989.24i	0.293881i	−0.989145	+	0.146941i	$0.953057\pi$
$828$	1414.58	0.0593721
$829$	32649.7	1.36788	0.683938	−	0.729540i	$-0.260266\pi$
$829$	32649.7	1.36788	0.683938	+	0.729540i	$0.260266\pi$
$830$	− 868.531i	− 0.0363219i
$831$	−21682.0	−0.905103
$832$	0	0
$833$	321.724	0.0133819
$834$	− 14097.6i	− 0.585324i
$835$	332.732	0.0137900
$836$	−10054.5	−0.415958
$837$	− 16844.7i	− 0.695626i
$838$	34278.4i	1.41304i
$839$	4038.23i	0.166168i	0.996543	+	0.0830841i	$0.0264770\pi$
$839$	4038.23i	0.166168i	−0.996543	+	0.0830841i	$0.973523\pi$
$840$	− 909.318i	− 0.0373505i
$841$	6555.00	0.268769
$842$	24240.9	0.992159
$843$	13006.1i	0.531380i
$844$	−131.695	−0.00537102
$845$	0	0
$846$	1098.51	0.0446426
$847$	51988.1i	2.10901i
$848$	−143.534	−0.00581248
$849$	−9620.64	−0.388904
$850$	8159.17i	0.329244i
$851$	8410.97i	0.338807i
$852$	2871.21i	0.115453i
$853$	− 8114.12i	− 0.325700i	−0.986651	−	0.162850i	$-0.947931\pi$
$853$	− 8114.12i	− 0.325700i	0.986651	−	0.162850i	$-0.0520687\pi$
$854$	−42869.0	−1.71774
$855$	−813.863	−0.0325538
$856$	20729.0i	0.827690i
$857$	22298.1	0.888786	0.444393	−	0.895832i	$-0.353419\pi$
$857$	22298.1	0.888786	0.444393	+	0.895832i	$0.353419\pi$
$858$	0	0
$859$	33550.5	1.33263	0.666315	−	0.745670i	$-0.267870\pi$
$859$	33550.5	1.33263	0.666315	+	0.745670i	$0.267870\pi$
$860$	− 354.045i	− 0.0140382i
$861$	4664.62	0.184634
$862$	−12429.4	−0.491121
$863$	− 14120.5i	− 0.556972i	−0.960440	−	0.278486i	$-0.910167\pi$
$863$	− 14120.5i	− 0.556972i	0.960440	−	0.278486i	$-0.0898326\pi$
$864$	9570.49i	0.376846i
$865$	2524.50i	0.0992319i
$866$	21025.2i	0.825018i
$867$	15698.1	0.614918
$868$	−2957.01	−0.115631
$869$	27289.8i	1.06530i
$870$	932.351	0.0363329
$871$	0	0
$872$	−16243.4	−0.630817
$873$	− 7829.39i	− 0.303533i
$874$	−20261.9	−0.784175
$875$	2548.63	0.0984679
$876$	4049.36i	0.156182i
$877$	− 1941.69i	− 0.0747619i	−0.999301	−	0.0373809i	$-0.988099\pi$
$877$	− 1941.69i	− 0.0747619i	0.999301	−	0.0373809i	$-0.0119015\pi$
$878$	7668.78i	0.294771i
$879$	− 20232.6i	− 0.776368i
$880$	−1833.10	−0.0702201
$881$	790.231	0.0302197	0.0151099	−	0.999886i	$-0.495190\pi$
$881$	790.231	0.0302197	0.0151099	+	0.999886i	$0.495190\pi$
$882$	− 433.029i	− 0.0165316i
$883$	36638.6	1.39636	0.698180	−	0.715922i	$-0.253993\pi$
$883$	36638.6	1.39636	0.698180	+	0.715922i	$0.253993\pi$
$884$	0	0
$885$	−148.217	−0.00562968
$886$	24958.9i	0.946401i
$887$	−40686.3	−1.54015	−0.770075	−	0.637954i	$-0.779781\pi$
$887$	−40686.3	−1.54015	−0.770075	+	0.637954i	$0.779781\pi$
$888$	−10227.6	−0.386503
$889$	− 10055.1i	− 0.379344i
$890$	− 1668.43i	− 0.0628381i
$891$	12066.4i	0.453692i
$892$	1777.08i	0.0667053i
$893$	3449.40	0.129261
$894$	−26040.9	−0.974202
$895$	86.6392i	0.00323579i
$896$	−17102.2	−0.637663
$897$	0	0
$898$	−1438.21	−0.0534449
$899$	− 19894.0i	− 0.738046i
$900$	−2407.50	−0.0891666
$901$	72.7200	0.00268885
$902$	− 11549.5i	− 0.426336i
$903$	− 29355.9i	− 1.08184i
$904$	15509.0i	0.570598i
$905$	− 601.618i	− 0.0220977i
$906$	9215.28	0.337922
$907$	10464.4	0.383093	0.191547	−	0.981484i	$-0.438650\pi$
$907$	10464.4	0.383093	0.191547	+	0.981484i	$0.438650\pi$
$908$	4749.26i	0.173579i
$909$	12372.8	0.451463
$910$	0	0
$911$	−35611.5	−1.29513	−0.647563	−	0.762011i	$-0.724212\pi$
$911$	−35611.5	−1.29513	−0.647563	+	0.762011i	$0.724212\pi$
$912$	− 20060.0i	− 0.728346i
$913$	39089.0	1.41693
$914$	−35242.9	−1.27542
$915$	1905.04i	0.0688291i
$916$	− 303.920i	− 0.0109627i
$917$	− 37379.4i	− 1.34610i
$918$	9746.80i	0.350427i
$919$	1077.25	0.0386674	0.0193337	−	0.999813i	$-0.493846\pi$
$919$	1077.25	0.0386674	0.0193337	+	0.999813i	$0.493846\pi$
$920$	994.648	0.0356441
$921$	− 26926.0i	− 0.963346i
$922$	−30762.3	−1.09881
$923$	0	0
$924$	6237.04	0.222060
$925$	− 14314.8i	− 0.508829i
$926$	34928.6	1.23955
$927$	−12493.2	−0.442644
$928$	11303.0i	0.399826i
$929$	55733.8i	1.96832i	0.177290	+	0.984159i	$0.443267\pi$
$929$	55733.8i	1.96832i	−0.177290	+	0.984159i	$0.556733\pi$
$930$	− 599.413i	− 0.0211350i
$931$	− 1359.74i	− 0.0478665i
$932$	369.284	0.0129789
$933$	29126.9	1.02205
$934$	− 22597.9i	− 0.791677i
$935$	928.718	0.0324838
$936$	0	0
$937$	−3198.60	−0.111519	−0.0557596	−	0.998444i	$-0.517758\pi$
$937$	−3198.60	−0.111519	−0.0557596	+	0.998444i	$0.517758\pi$
$938$	− 20686.4i	− 0.720079i
$939$	−36855.4	−1.28086
$940$	−25.8064	−0.000895437 0
$941$	8823.35i	0.305667i	0.988252	+	0.152834i	$0.0488399\pi$
$941$	8823.35i	0.305667i	−0.988252	+	0.152834i	$0.951160\pi$
$942$	5331.54i	0.184407i
$943$	5102.35i	0.176199i
$944$	3611.95i	0.124533i
$945$	1520.35	0.0523354
$946$	−72684.3	−2.49806
$947$	− 28290.4i	− 0.970766i	−0.874301	−	0.485383i	$-0.838680\pi$
$947$	− 28290.4i	− 0.970766i	0.874301	−	0.485383i	$-0.161320\pi$
$948$	2234.23	0.0765447
$949$	0	0
$950$	34484.0	1.17769
$951$	22958.4i	0.782836i
$952$	11226.8	0.382210
$953$	12399.0	0.421452	0.210726	−	0.977545i	$-0.432417\pi$
$953$	12399.0	0.421452	0.210726	+	0.977545i	$0.432417\pi$
$954$	− 97.8783i	− 0.00332173i
$955$	380.285i	0.0128856i
$956$	− 5105.94i	− 0.172739i
$957$	41961.3i	1.41736i
$958$	−37450.0	−1.26300
$959$	32874.5	1.10696
$960$	1175.22i	0.0395105i
$961$	17001.0	0.570676
$962$	0	0
$963$	−11508.9	−0.385118
$964$	7235.53i	0.241744i
$965$	742.193	0.0247586
$966$	12569.0	0.418634
$967$	− 26667.1i	− 0.886820i	−0.896319	−	0.443410i	$-0.853769\pi$
$967$	− 26667.1i	− 0.886820i	0.896319	−	0.443410i	$-0.146231\pi$
$968$	− 69148.6i	− 2.29599i
$969$	10163.2i	0.336933i
$970$	− 839.001i	− 0.0277719i
$971$	49420.7	1.63335	0.816676	−	0.577096i	$-0.195814\pi$
$971$	49420.7	1.63335	0.816676	+	0.577096i	$0.195814\pi$
$972$	−4796.89	−0.158293
$973$	− 27149.9i	− 0.894539i
$974$	25100.3	0.825735
$975$	0	0
$976$	46424.5	1.52255
$977$	778.759i	0.0255012i	0.999919	+	0.0127506i	$0.00405876\pi$
$977$	778.759i	0.0255012i	−0.999919	+	0.0127506i	$0.995941\pi$
$978$	−14238.2	−0.465529
$979$	75089.2	2.45134
$980$	10.1728i	0 0.000331589i
$981$	− 9018.48i	− 0.293515i
$982$	27757.2i	0.902003i
$983$	− 5997.90i	− 0.194612i	−0.995255	−	0.0973059i	$-0.968977\pi$
$983$	− 5997.90i	− 0.194612i	0.995255	−	0.0973059i	$-0.0310225\pi$
$984$	−6204.35	−0.201004
$985$	−711.693	−0.0230218
$986$	11511.2i	0.371797i
$987$	−2139.75	−0.0690062
$988$	0	0
$989$	32110.6	1.03241
$990$	− 1250.02i	− 0.0401295i
$991$	8974.94	0.287688	0.143844	−	0.989600i	$-0.454054\pi$
$991$	8974.94	0.287688	0.143844	+	0.989600i	$0.454054\pi$
$992$	7266.75	0.232580
$993$	17076.6i	0.545729i
$994$	− 25223.3i	− 0.804863i
$995$	− 1345.62i	− 0.0428732i
$996$	− 3200.24i	− 0.101811i
$997$	28530.2	0.906280	0.453140	−	0.891439i	$-0.350304\pi$
$997$	28530.2	0.906280	0.453140	+	0.891439i	$0.350304\pi$
$998$	−6634.31	−0.210426
$999$	− 17100.2i	− 0.541567i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.b.f.168.4		4
13.2	odd	12		169.4.c.g.22.1		4
13.3	even	3		169.4.e.f.147.1		8
13.4	even	6		169.4.e.f.23.1		8
13.5	odd	4		13.4.a.b.1.2	✓	2
13.6	odd	12		169.4.c.g.146.1		4
13.7	odd	12		169.4.c.j.146.2		4
13.8	odd	4		169.4.a.g.1.1		2
13.9	even	3		169.4.e.f.23.4		8
13.10	even	6		169.4.e.f.147.4		8
13.11	odd	12		169.4.c.j.22.2		4
13.12	even	2	inner	169.4.b.f.168.1		4
39.5	even	4		117.4.a.d.1.1		2
39.8	even	4		1521.4.a.r.1.2		2
52.31	even	4		208.4.a.h.1.2		2
65.18	even	4		325.4.b.e.274.1		4
65.44	odd	4		325.4.a.f.1.1		2
65.57	even	4		325.4.b.e.274.4		4
91.83	even	4		637.4.a.b.1.2		2
104.5	odd	4		832.4.a.s.1.2		2
104.83	even	4		832.4.a.z.1.1		2
143.109	even	4		1573.4.a.b.1.1		2
156.83	odd	4		1872.4.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.2	✓	2	13.5	odd	4
117.4.a.d.1.1		2	39.5	even	4
169.4.a.g.1.1		2	13.8	odd	4
169.4.b.f.168.1		4	13.12	even	2	inner
169.4.b.f.168.4		4	1.1	even	1	trivial
169.4.c.g.22.1		4	13.2	odd	12
169.4.c.g.146.1		4	13.6	odd	12
169.4.c.j.22.2		4	13.11	odd	12
169.4.c.j.146.2		4	13.7	odd	12
169.4.e.f.23.1		8	13.4	even	6
169.4.e.f.23.4		8	13.9	even	3
169.4.e.f.147.1		8	13.3	even	3
169.4.e.f.147.4		8	13.10	even	6
208.4.a.h.1.2		2	52.31	even	4
325.4.a.f.1.1		2	65.44	odd	4
325.4.b.e.274.1		4	65.18	even	4
325.4.b.e.274.4		4	65.57	even	4
637.4.a.b.1.2		2	91.83	even	4
832.4.a.s.1.2		2	104.5	odd	4
832.4.a.z.1.1		2	104.83	even	4
1521.4.a.r.1.2		2	39.8	even	4
1573.4.a.b.1.1		2	143.109	even	4
1872.4.a.bb.1.1		2	156.83	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)

\(f(q)\)	\(=\)	\(q+2.56155i q^{2} -3.68466 q^{3} +1.43845 q^{4} +0.561553i q^{5} -9.43845i q^{6} -18.1771i q^{7} +24.1771i q^{8} -13.4233 q^{9} +O(q^{10})\)	\(q+2.56155i q^{2} -3.68466 q^{3} +1.43845 q^{4} +0.561553i q^{5} -9.43845i q^{6} -18.1771i q^{7} +24.1771i q^{8} -13.4233 q^{9} -1.43845 q^{10} -64.7386i q^{11} -5.30019 q^{12} +46.5616 q^{14} -2.06913i q^{15} -50.4233 q^{16} +25.5464 q^{17} -34.3845i q^{18} -107.970i q^{19} +0.807764i q^{20} +66.9763i q^{21} +165.831 q^{22} -73.2614 q^{23} -89.0843i q^{24} +124.685 q^{25} +148.946 q^{27} -26.1468i q^{28} +175.909 q^{29} +5.30019 q^{30} -113.093i q^{31} +64.2547i q^{32} +238.540i q^{33} +65.4384i q^{34} +10.2074 q^{35} -19.3087 q^{36} -114.808i q^{37} +276.570 q^{38} -13.5767 q^{40} -69.6458i q^{41} -171.563 q^{42} -438.302 q^{43} -93.1231i q^{44} -7.53789i q^{45} -187.663i q^{46} +31.9479i q^{47} +185.793 q^{48} +12.5937 q^{49} +319.386i q^{50} -94.1298 q^{51} +2.84658 q^{53} +381.533i q^{54} +36.3542 q^{55} +439.469 q^{56} +397.831i q^{57} +450.600i q^{58} -71.6325i q^{59} -2.97633i q^{60} -920.695 q^{61} +289.693 q^{62} +243.996i q^{63} -567.978 q^{64} -611.032 q^{66} -444.280i q^{67} +36.7471 q^{68} +269.943 q^{69} +26.1468i q^{70} -541.719i q^{71} -324.536i q^{72} -764.004i q^{73} +294.086 q^{74} -459.420 q^{75} -155.309i q^{76} -1176.76 q^{77} -421.538 q^{79} -28.3153i q^{80} -186.386 q^{81} +178.401 q^{82} +603.797i q^{83} +96.3419i q^{84} +14.3457i q^{85} -1122.73i q^{86} -648.165 q^{87} +1565.19 q^{88} +1159.88i q^{89} +19.3087 q^{90} -105.383 q^{92} +416.708i q^{93} -81.8362 q^{94} +60.6307 q^{95} -236.757i q^{96} +583.269i q^{97} +32.2595i q^{98} +869.006i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{3} + 14 q^{4} + 70 q^{9}+O(q^{10}) \) 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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