LMFDB - Embedded newform 169.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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([0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

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:	yes
Analytic conductor:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -18.1771 q^{7} +24.1771 q^{8} -13.4233 q^{9} +O(q^{10})$	\(q-2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -18.1771 q^{7} +24.1771 q^{8} -13.4233 q^{9} +1.43845 q^{10} -64.7386 q^{11} +5.30019 q^{12} +46.5616 q^{14} +2.06913 q^{15} -50.4233 q^{16} -25.5464 q^{17} +34.3845 q^{18} +107.970 q^{19} +0.807764 q^{20} +66.9763 q^{21} +165.831 q^{22} +73.2614 q^{23} -89.0843 q^{24} -124.685 q^{25} +148.946 q^{27} +26.1468 q^{28} +175.909 q^{29} -5.30019 q^{30} +113.093 q^{31} -64.2547 q^{32} +238.540 q^{33} +65.4384 q^{34} +10.2074 q^{35} +19.3087 q^{36} -114.808 q^{37} -276.570 q^{38} -13.5767 q^{40} +69.6458 q^{41} -171.563 q^{42} +438.302 q^{43} +93.1231 q^{44} +7.53789 q^{45} -187.663 q^{46} +31.9479 q^{47} +185.793 q^{48} -12.5937 q^{49} +319.386 q^{50} +94.1298 q^{51} +2.84658 q^{53} -381.533 q^{54} +36.3542 q^{55} -439.469 q^{56} -397.831 q^{57} -450.600 q^{58} -71.6325 q^{59} -2.97633 q^{60} -920.695 q^{61} -289.693 q^{62} +243.996 q^{63} +567.978 q^{64} -611.032 q^{66} +444.280 q^{67} +36.7471 q^{68} -269.943 q^{69} -26.1468 q^{70} +541.719 q^{71} -324.536 q^{72} -764.004 q^{73} +294.086 q^{74} +459.420 q^{75} -155.309 q^{76} +1176.76 q^{77} -421.538 q^{79} +28.3153 q^{80} -186.386 q^{81} -178.401 q^{82} -603.797 q^{83} -96.3419 q^{84} +14.3457 q^{85} -1122.73 q^{86} -648.165 q^{87} -1565.19 q^{88} +1159.88 q^{89} -19.3087 q^{90} -105.383 q^{92} -416.708 q^{93} -81.8362 q^{94} -60.6307 q^{95} +236.757 q^{96} -583.269 q^{97} +32.2595 q^{98} +869.006 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10}) $ 2 * q - q^2 + 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9	\( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 7 q^{10} - 80 q^{11} - 43 q^{12} + 89 q^{14} + 33 q^{15} - 39 q^{16} + 19 q^{17} + 110 q^{18} + 84 q^{19} - 19 q^{20} + 303 q^{21} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 335 q^{27} - 125 q^{28} - 44 q^{29} + 43 q^{30} + 86 q^{31} + 123 q^{32} + 106 q^{33} + 135 q^{34} + 107 q^{35} - 250 q^{36} - 209 q^{37} - 314 q^{38} - 89 q^{40} + 230 q^{41} + 197 q^{42} + 287 q^{43} + 178 q^{44} + 180 q^{45} + 4 q^{46} - 435 q^{47} + 285 q^{48} + 383 q^{49} + 144 q^{50} + 481 q^{51} - 118 q^{53} - 91 q^{54} - 18 q^{55} - 1015 q^{56} - 606 q^{57} - 794 q^{58} + 368 q^{59} - 175 q^{60} - 1058 q^{61} - 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} - 211 q^{68} + 796 q^{69} + 125 q^{70} + 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} - 516 q^{75} - 22 q^{76} + 762 q^{77} - 1008 q^{79} + 69 q^{80} + 122 q^{81} + 72 q^{82} - 1958 q^{83} - 1409 q^{84} + 173 q^{85} - 1359 q^{86} - 2558 q^{87} - 1242 q^{88} + 720 q^{89} + 250 q^{90} - 788 q^{92} - 652 q^{93} - 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} + 650 q^{98} + 130 q^{99}+O(q^{100}) \) 2 * q - q^2 + 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9 + 7 * q^10 - 80 * q^11 - 43 * q^12 + 89 * q^14 + 33 * q^15 - 39 * q^16 + 19 * q^17 + 110 * q^18 + 84 * q^19 - 19 * q^20 + 303 * q^21 + 142 * q^22 + 196 * q^23 - 273 * q^24 - 237 * q^25 + 335 * q^27 - 125 * q^28 - 44 * q^29 + 43 * q^30 + 86 * q^31 + 123 * q^32 + 106 * q^33 + 135 * q^34 + 107 * q^35 - 250 * q^36 - 209 * q^37 - 314 * q^38 - 89 * q^40 + 230 * q^41 + 197 * q^42 + 287 * q^43 + 178 * q^44 + 180 * q^45 + 4 * q^46 - 435 * q^47 + 285 * q^48 + 383 * q^49 + 144 * q^50 + 481 * q^51 - 118 * q^53 - 91 * q^54 - 18 * q^55 - 1015 * q^56 - 606 * q^57 - 794 * q^58 + 368 * q^59 - 175 * q^60 - 1058 * q^61 - 332 * q^62 + 1560 * q^63 + 769 * q^64 - 818 * q^66 - 68 * q^67 - 211 * q^68 + 796 * q^69 + 125 * q^70 + 131 * q^71 - 1350 * q^72 - 456 * q^73 + 147 * q^74 - 516 * q^75 - 22 * q^76 + 762 * q^77 - 1008 * q^79 + 69 * q^80 + 122 * q^81 + 72 * q^82 - 1958 * q^83 - 1409 * q^84 + 173 * q^85 - 1359 * q^86 - 2558 * q^87 - 1242 * q^88 + 720 * q^89 + 250 * q^90 - 788 * q^92 - 652 * q^93 - 811 * q^94 - 146 * q^95 + 1863 * q^96 + 928 * q^97 + 650 * q^98 + 130 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.56155	−0.905646	−0.452823	−	0.891601i	$-0.649583\pi$
$2$	−2.56155	−0.905646	−0.452823	+	0.891601i	$0.649583\pi$
$3$	−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
$5$	−0.561553	−0.0502268	−0.0251134	−	0.999685i	$-0.507995\pi$
$5$	−0.561553	−0.0502268	−0.0251134	+	0.999685i	$0.507995\pi$
$6$	9.43845	0.642205
$7$	−18.1771	−0.981470	−0.490735	−	0.871309i	$-0.663272\pi$
$7$	−18.1771	−0.981470	−0.490735	+	0.871309i	$0.663272\pi$
$8$	24.1771	1.06849
$9$	−13.4233	−0.497159
$10$	1.43845	0.0454877
$11$	−64.7386	−1.77449	−0.887247	−	0.461295i	$-0.847385\pi$
$11$	−64.7386	−1.77449	−0.887247	+	0.461295i	$0.847385\pi$
$12$	5.30019	0.127503
$13$	0	0
$13$	0	0
$14$	46.5616	0.888864
$15$	2.06913	0.0356165
$16$	−50.4233	−0.787864
$17$	−25.5464	−0.364465	−0.182233	−	0.983255i	$-0.558332\pi$
$17$	−25.5464	−0.364465	−0.182233	+	0.983255i	$0.558332\pi$
$18$	34.3845	0.450250
$19$	107.970	1.30368	0.651841	−	0.758356i	$-0.273997\pi$
$19$	107.970	1.30368	0.651841	+	0.758356i	$0.273997\pi$
$20$	0.807764	0.00903108
$21$	66.9763	0.695973
$22$	165.831	1.60706
$23$	73.2614	0.664176	0.332088	−	0.943248i	$-0.392247\pi$
$23$	73.2614	0.664176	0.332088	+	0.943248i	$0.392247\pi$
$24$	−89.0843	−0.757677
$25$	−124.685	−0.997477
$26$	0	0
$27$	148.946	1.06165
$28$	26.1468	0.176474
$29$	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
$31$	113.093	0.655228	0.327614	−	0.944812i	$-0.393755\pi$
$31$	113.093	0.655228	0.327614	+	0.944812i	$0.393755\pi$
$32$	−64.2547	−0.354961
$33$	238.540	1.25832
$34$	65.4384	0.330077
$35$	10.2074	0.0492961
$36$	19.3087	0.0893921
$37$	−114.808	−0.510116	−0.255058	−	0.966926i	$-0.582095\pi$
$37$	−114.808	−0.510116	−0.255058	+	0.966926i	$0.582095\pi$
$38$	−276.570	−1.18067
$39$	0	0
$40$	−13.5767	−0.0536666
$41$	69.6458	0.265289	0.132645	−	0.991164i	$-0.457653\pi$
$41$	69.6458	0.265289	0.132645	+	0.991164i	$0.457653\pi$
$42$	−171.563	−0.630305
$43$	438.302	1.55443	0.777214	−	0.629236i	$-0.216632\pi$
$43$	438.302	1.55443	0.777214	+	0.629236i	$0.216632\pi$
$44$	93.1231	0.319064
$45$	7.53789	0.0249707
$46$	−187.663	−0.601508
$47$	31.9479	0.0991506	0.0495753	−	0.998770i	$-0.484213\pi$
$47$	31.9479	0.0991506	0.0495753	+	0.998770i	$0.484213\pi$
$48$	185.793	0.558684
$49$	−12.5937	−0.0367164
$50$	319.386	0.903361
$51$	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	$0.498826\pi$
$54$	−381.533	−0.961483
$55$	36.3542	0.0891272
$56$	−439.469	−1.04869
$57$	−397.831	−0.924457
$58$	−450.600	−1.02012
$59$	−71.6325	−0.158064	−0.0790319	−	0.996872i	$-0.525183\pi$
$59$	−71.6325	−0.158064	−0.0790319	+	0.996872i	$0.525183\pi$
$60$	−2.97633	−0.00640405
$61$	−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
$63$	243.996	0.487947
$64$	567.978	1.10933
$65$	0	0
$66$	−611.032	−1.13959
$67$	444.280	0.810112	0.405056	−	0.914292i	$-0.367252\pi$
$67$	444.280	0.810112	0.405056	+	0.914292i	$0.367252\pi$
$68$	36.7471	0.0655330
$69$	−269.943	−0.470976
$70$	−26.1468	−0.0446448
$71$	541.719	0.905496	0.452748	−	0.891639i	$-0.350444\pi$
$71$	541.719	0.905496	0.452748	+	0.891639i	$0.350444\pi$
$72$	−324.536	−0.531207
$73$	−764.004	−1.22493	−0.612465	−	0.790498i	$-0.709822\pi$
$73$	−764.004	−1.22493	−0.612465	+	0.790498i	$0.709822\pi$
$74$	294.086	0.461984
$75$	459.420	0.707324
$76$	−155.309	−0.234410
$77$	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	$0.597043\pi$
$80$	28.3153	0.0395719
$81$	−186.386	−0.255674
$82$	−178.401	−0.240258
$83$	−603.797	−0.798498	−0.399249	−	0.916842i	$-0.630729\pi$
$83$	−603.797	−0.798498	−0.399249	+	0.916842i	$0.630729\pi$
$84$	−96.3419	−0.125140
$85$	14.3457	0.0183059
$86$	−1122.73	−1.40776
$87$	−648.165	−0.798742
$88$	−1565.19	−1.89602
$89$	1159.88	1.38143	0.690715	−	0.723127i	$-0.257296\pi$
$89$	1159.88	1.38143	0.690715	+	0.723127i	$0.257296\pi$
$90$	−19.3087	−0.0226146
$91$	0	0
$92$	−105.383	−0.119423
$93$	−416.708	−0.464631
$94$	−81.8362	−0.0897953
$95$	−60.6307	−0.0654798
$96$	236.757	0.251707
$97$	−583.269	−0.610536	−0.305268	−	0.952267i	$-0.598746\pi$
$97$	−583.269	−0.610536	−0.305268	+	0.952267i	$0.598746\pi$
$98$	32.2595	0.0332521
$99$	869.006	0.882206
$100$	179.352	0.179352
$101$	921.740	0.908085	0.454043	−	0.890980i	$-0.349981\pi$
$101$	921.740	0.908085	0.454043	+	0.890980i	$0.349981\pi$
$102$	−241.118	−0.234061
$103$	−930.712	−0.890347	−0.445174	−	0.895444i	$-0.646858\pi$
$103$	−930.712	−0.890347	−0.445174	+	0.895444i	$0.646858\pi$
$104$	0	0
$105$	−37.6107	−0.0349565
$106$	−7.29168	−0.00668142
$107$	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
$109$	−671.853	−0.590384	−0.295192	−	0.955438i	$-0.595384\pi$
$109$	−671.853	−0.590384	−0.295192	+	0.955438i	$0.595384\pi$
$110$	−93.1231	−0.0807176
$111$	423.027	0.361730
$112$	916.548	0.773265
$113$	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
$115$	−41.1401	−0.0333594
$116$	−253.036	−0.202533
$117$	0	0
$118$	183.491	0.143150
$119$	464.359	0.357712
$120$	50.0255	0.0380557
$121$	2860.09	2.14883
$122$	2358.41	1.75017
$123$	−256.621	−0.188120
$124$	−162.678	−0.117814
$125$	140.211	0.100327
$126$	−625.009	−0.441907
$127$	−553.174	−0.386506	−0.193253	−	0.981149i	$-0.561904\pi$
$127$	−553.174	−0.386506	−0.193253	+	0.981149i	$0.561904\pi$
$128$	−940.868	−0.649702
$129$	−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	$0.259471\pi$
$132$	−343.127	−0.226253
$133$	−1962.57	−1.27952
$134$	−1138.05	−0.733674
$135$	−83.6411	−0.0533235
$136$	−617.637	−0.389426
$137$	1808.57	1.12786	0.563928	−	0.825824i	$-0.309290\pi$
$137$	1808.57	1.12786	0.563928	+	0.825824i	$0.309290\pi$
$138$	691.474	0.426537
$139$	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
$141$	−117.717	−0.0703090
$142$	−1387.64	−0.820058
$143$	0	0
$144$	676.847	0.391694
$145$	−98.7822	−0.0565753
$146$	1957.04	1.10935
$147$	46.4036	0.0260361
$148$	165.145	0.0917218
$149$	2759.02	1.51696	0.758482	−	0.651694i	$-0.225941\pi$
$149$	2759.02	1.51696	0.758482	+	0.651694i	$0.225941\pi$
$150$	−1176.83	−0.640585
$151$	976.355	0.526190	0.263095	−	0.964770i	$-0.415257\pi$
$151$	976.355	0.526190	0.263095	+	0.964770i	$0.415257\pi$
$152$	2610.39	1.39297
$153$	342.917	0.181197
$154$	−3014.33	−1.57728
$155$	−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	$0.545859\pi$
$158$	1079.79	0.543694
$159$	−10.4887	−0.00523149
$160$	36.0824	0.0178285
$161$	−1331.68	−0.651869
$162$	477.438	0.231550
$163$	−1508.53	−0.724892	−0.362446	−	0.932005i	$-0.618058\pi$
$163$	−1508.53	−0.724892	−0.362446	+	0.932005i	$0.618058\pi$
$164$	−100.182	−0.0477005
$165$	−133.953	−0.0632012
$166$	1546.66	0.723157
$167$	−592.521	−0.274555	−0.137277	−	0.990533i	$-0.543835\pi$
$167$	−592.521	−0.274555	−0.137277	+	0.990533i	$0.543835\pi$
$168$	1619.29	0.743638
$169$	0	0
$170$	−36.7471	−0.0165787
$171$	−1449.31	−0.648137
$172$	−630.474	−0.279495
$173$	−4495.57	−1.97568	−0.987838	−	0.155488i	$-0.950305\pi$
$173$	−4495.57	−1.97568	−0.987838	+	0.155488i	$0.950305\pi$
$174$	1660.31	0.723377
$175$	2266.40	0.978994
$176$	3264.34	1.39806
$177$	263.941	0.112085
$178$	−2971.10	−1.25109
$179$	−154.285	−0.0644235	−0.0322117	−	0.999481i	$-0.510255\pi$
$179$	−154.285	−0.0644235	−0.0322117	+	0.999481i	$0.510255\pi$
$180$	−10.8429	−0.00448988
$181$	1071.35	0.439959	0.219979	−	0.975505i	$-0.429401\pi$
$181$	1071.35	0.439959	0.219979	+	0.975505i	$0.429401\pi$
$182$	0	0
$183$	3392.45	1.37037
$184$	1771.25	0.709663
$185$	64.4706	0.0256215
$186$	1067.42	0.420791
$187$	1653.84	0.646742
$188$	−45.9554	−0.0178279
$189$	−2707.40	−1.04198
$190$	155.309	0.0593015
$191$	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
$193$	−1321.68	−0.492936	−0.246468	−	0.969151i	$-0.579270\pi$
$193$	−1321.68	−0.492936	−0.246468	+	0.969151i	$0.579270\pi$
$194$	1494.07	0.552929
$195$	0	0
$196$	18.1154	0.00660183
$197$	−1267.37	−0.458356	−0.229178	−	0.973385i	$-0.573604\pi$
$197$	−1267.37	−0.458356	−0.229178	+	0.973385i	$0.573604\pi$
$198$	−2226.00	−0.798966
$199$	2396.24	0.853593	0.426796	−	0.904348i	$-0.359642\pi$
$199$	2396.24	0.853593	0.426796	+	0.904348i	$0.359642\pi$
$200$	−3014.51	−1.06579
$201$	−1637.02	−0.574460
$202$	−2361.09	−0.822403
$203$	−3197.51	−1.10552
$204$	−135.401	−0.0464703
$205$	−39.1098	−0.0133246
$206$	2384.07	0.806339
$207$	−983.409	−0.330201
$208$	0	0
$209$	−6989.81	−2.31337
$210$	96.3419	0.0316582
$211$	−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
$213$	−1996.05	−0.642099
$214$	−2196.23	−0.701548
$215$	−246.130	−0.0780740
$216$	3601.08	1.13436
$217$	−2055.70	−0.643087
$218$	1720.99	0.534679
$219$	2815.09	0.868613
$220$	−52.2935	−0.0160256
$221$	0	0
$222$	−1083.61	−0.327599
$223$	−1235.42	−0.370985	−0.185493	−	0.982646i	$-0.559388\pi$
$223$	−1235.42	−0.370985	−0.185493	+	0.982646i	$0.559388\pi$
$224$	1167.96	0.348383
$225$	1673.68	0.495905
$226$	−1643.17	−0.483637
$227$	−3301.66	−0.965370	−0.482685	−	0.875794i	$-0.660338\pi$
$227$	−3301.66	−0.965370	−0.482685	+	0.875794i	$0.660338\pi$
$228$	572.260	0.166223
$229$	−211.283	−0.0609694	−0.0304847	−	0.999535i	$-0.509705\pi$
$229$	−211.283	−0.0609694	−0.0304847	+	0.999535i	$0.509705\pi$
$230$	105.383	0.0302118
$231$	−4335.96	−1.23500
$232$	4252.97	1.20354
$233$	−256.724	−0.0721827	−0.0360913	−	0.999348i	$-0.511491\pi$
$233$	−256.724	−0.0721827	−0.0360913	+	0.999348i	$0.511491\pi$
$234$	0	0
$235$	−17.9404	−0.00498002
$236$	103.040	0.0284208
$237$	1553.22	0.425708
$238$	−1189.48	−0.323960
$239$	3549.62	0.960694	0.480347	−	0.877078i	$-0.340511\pi$
$239$	3549.62	0.960694	0.480347	+	0.877078i	$0.340511\pi$
$240$	−104.332	−0.0280609
$241$	5030.10	1.34447	0.672235	−	0.740338i	$-0.265335\pi$
$241$	5030.10	1.34447	0.672235	+	0.740338i	$0.265335\pi$
$242$	−7326.27	−1.94608
$243$	−3334.77	−0.880353
$244$	1324.37	0.347476
$245$	7.07204	0.00184415
$246$	657.349	0.170370
$247$	0	0
$248$	2734.25	0.700102
$249$	2224.79	0.566226
$250$	−359.158	−0.0908606
$251$	−718.784	−0.180754	−0.0903770	−	0.995908i	$-0.528807\pi$
$251$	−718.784	−0.180754	−0.0903770	+	0.995908i	$0.528807\pi$
$252$	−350.976	−0.0877357
$253$	−4742.84	−1.17858
$254$	1416.98	0.350038
$255$	−52.8588	−0.0129810
$256$	−2133.74	−0.520933
$257$	1280.79	0.310871	0.155435	−	0.987846i	$-0.450322\pi$
$257$	1280.79	0.310871	0.155435	+	0.987846i	$0.450322\pi$
$258$	4136.89	0.998262
$259$	2086.87	0.500663
$260$	0	0
$261$	−2361.28	−0.559998
$262$	−5267.58	−1.24211
$263$	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
$265$	−1.59851	−0.000370549 0
$266$	5027.24	1.15880
$267$	−4273.77	−0.979590
$268$	−639.074	−0.145663
$269$	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
$271$	3929.93	0.880909	0.440455	−	0.897775i	$-0.354817\pi$
$271$	3929.93	0.880909	0.440455	+	0.897775i	$0.354817\pi$
$272$	1288.13	0.287149
$273$	0	0
$274$	−4632.74	−1.02144
$275$	8071.91	1.77002
$276$	388.299	0.0846842
$277$	−5884.40	−1.27639	−0.638194	−	0.769876i	$-0.720318\pi$
$277$	−5884.40	−1.27639	−0.638194	+	0.769876i	$0.720318\pi$
$278$	−3826.03	−0.825431
$279$	−1518.08	−0.325752
$280$	246.785	0.0526722
$281$	−3529.79	−0.749358	−0.374679	−	0.927155i	$-0.622247\pi$
$281$	−3529.79	−0.749358	−0.374679	+	0.927155i	$0.622247\pi$
$282$	301.538	0.0636750
$283$	−2611.00	−0.548438	−0.274219	−	0.961667i	$-0.588419\pi$
$283$	−2611.00	−0.548438	−0.274219	+	0.961667i	$0.588419\pi$
$284$	−779.234	−0.162813
$285$	223.403	0.0464325
$286$	0	0
$287$	−1265.96	−0.260373
$288$	862.510	0.176472
$289$	−4260.38	−0.867165
$290$	253.036	0.0512372
$291$	2149.15	0.432939
$292$	1098.98	0.220250
$293$	5491.03	1.09484	0.547422	−	0.836857i	$-0.315609\pi$
$293$	5491.03	1.09484	0.547422	+	0.836857i	$0.315609\pi$
$294$	−118.865	−0.0235795
$295$	40.2255	0.00793904
$296$	−2775.72	−0.545052
$297$	−9642.56	−1.88390
$298$	−7067.37	−1.37383
$299$	0	0
$300$	−660.852	−0.127181
$301$	−7967.05	−1.52563
$302$	−2500.99	−0.476542
$303$	−3396.30	−0.643935
$304$	−5444.19	−1.02712
$305$	517.019	0.0970637
$306$	−878.399	−0.164100
$307$	7307.59	1.35852	0.679261	−	0.733897i	$-0.262300\pi$
$307$	7307.59	1.35852	0.679261	+	0.733897i	$0.262300\pi$
$308$	−1692.71	−0.313152
$309$	3429.36	0.631357
$310$	162.678	0.0298048
$311$	7904.92	1.44131	0.720654	−	0.693295i	$-0.243842\pi$
$311$	7904.92	1.44131	0.720654	+	0.693295i	$0.243842\pi$
$312$	0	0
$313$	10002.4	1.80629	0.903145	−	0.429336i	$-0.141252\pi$
$313$	10002.4	1.80629	0.903145	+	0.429336i	$0.141252\pi$
$314$	1446.96	0.260053
$315$	−137.017	−0.0245080
$316$	606.360	0.107944
$317$	6230.81	1.10397	0.551983	−	0.833856i	$-0.313871\pi$
$317$	6230.81	1.10397	0.551983	+	0.833856i	$0.313871\pi$
$318$	26.8673	0.00473788
$319$	−11388.1	−1.99878
$320$	−318.950	−0.0557182
$321$	−3159.16	−0.549306
$322$	3411.16	0.590362
$323$	−2758.24	−0.475147
$324$	268.107	0.0459717
$325$	0	0
$326$	3864.19	0.656495
$327$	2475.55	0.418649
$328$	1683.83	0.283458
$329$	−580.719	−0.0973134
$330$	343.127	0.0572379
$331$	4634.51	0.769594	0.384797	−	0.923001i	$-0.374271\pi$
$331$	4634.51	0.769594	0.384797	+	0.923001i	$0.374271\pi$
$332$	868.531	0.143575
$333$	1541.10	0.253609
$334$	1517.77	0.248649
$335$	−249.487	−0.0406893
$336$	−3377.17	−0.548332
$337$	3029.82	0.489747	0.244874	−	0.969555i	$-0.421254\pi$
$337$	3029.82	0.489747	0.244874	+	0.969555i	$0.421254\pi$
$338$	0	0
$339$	−2363.61	−0.378684
$340$	−20.6355	−0.00329151
$341$	−7321.47	−1.16270
$342$	3712.48	0.586982
$343$	6463.66	1.01751
$344$	10596.9	1.66089
$345$	151.587	0.0236556
$346$	11515.6	1.78926
$347$	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
$349$	−7565.68	−1.16040	−0.580202	−	0.814472i	$-0.697027\pi$
$349$	−7565.68	−1.16040	−0.580202	+	0.814472i	$0.697027\pi$
$350$	−5805.51	−0.886622
$351$	0	0
$352$	4159.76	0.629875
$353$	2339.44	0.352736	0.176368	−	0.984324i	$-0.443565\pi$
$353$	2339.44	0.352736	0.176368	+	0.984324i	$0.443565\pi$
$354$	−676.100	−0.101509
$355$	−304.204	−0.0454802
$356$	−1668.43	−0.248389
$357$	−1711.00	−0.253658
$358$	395.209	0.0583449
$359$	2531.68	0.372192	0.186096	−	0.982532i	$-0.440417\pi$
$359$	2531.68	0.372192	0.186096	+	0.982532i	$0.440417\pi$
$360$	182.244	0.0266809
$361$	4798.45	0.699585
$362$	−2744.31	−0.398447
$363$	−10538.5	−1.52376
$364$	0	0
$365$	429.028	0.0615243
$366$	−8689.93	−1.24107
$367$	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
$369$	−934.876	−0.131891
$370$	−165.145	−0.0232040
$371$	−51.7426	−0.00724081
$372$	599.413	0.0835433
$373$	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
$375$	−516.630	−0.0711431
$376$	772.407	0.105941
$377$	0	0
$378$	6935.16	0.943667
$379$	1865.73	0.252866	0.126433	−	0.991975i	$-0.459647\pi$
$379$	1865.73	0.252866	0.126433	+	0.991975i	$0.459647\pi$
$380$	87.2140	0.0117736
$381$	2038.26	0.274076
$382$	−1734.69	−0.232342
$383$	−10836.0	−1.44567	−0.722837	−	0.691019i	$-0.757162\pi$
$383$	−10836.0	−1.44567	−0.722837	+	0.691019i	$0.757162\pi$
$384$	3466.78	0.460712
$385$	−660.813	−0.0874757
$386$	3385.55	0.446425
$387$	−5883.46	−0.772798
$388$	839.001	0.109778
$389$	−9520.34	−1.24088	−0.620438	−	0.784256i	$-0.713045\pi$
$389$	−9520.34	−1.24088	−0.620438	+	0.784256i	$0.713045\pi$
$390$	0	0
$391$	−1871.56	−0.242069
$392$	−304.480	−0.0392310
$393$	−7577.13	−0.972559
$394$	3246.43	0.415108
$395$	236.716	0.0301531
$396$	−1250.02	−0.158626
$397$	10108.8	1.27796	0.638978	−	0.769225i	$-0.279358\pi$
$397$	10108.8	1.27796	0.638978	+	0.769225i	$0.279358\pi$
$398$	−6138.10	−0.773053
$399$	7231.41	0.907327
$400$	6287.01	0.785876
$401$	−2084.38	−0.259573	−0.129787	−	0.991542i	$-0.541429\pi$
$401$	−2084.38	−0.259573	−0.129787	+	0.991542i	$0.541429\pi$
$402$	4193.32	0.520258
$403$	0	0
$404$	−1325.88	−0.163279
$405$	104.666	0.0128417
$406$	8190.60	1.00121
$407$	7432.50	0.905197
$408$	2275.78	0.276147
$409$	9716.53	1.17470	0.587349	−	0.809334i	$-0.300172\pi$
$409$	9716.53	1.17470	0.587349	+	0.809334i	$0.300172\pi$
$410$	100.182	0.0120674
$411$	−6663.95	−0.799777
$412$	1338.78	0.160090
$413$	1302.07	0.155135
$414$	2519.05	0.299045
$415$	339.064	0.0401060
$416$	0	0
$417$	−5503.54	−0.646305
$418$	17904.8	2.09510
$419$	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
$421$	9463.37	1.09553	0.547763	−	0.836633i	$-0.315479\pi$
$421$	9463.37	1.09553	0.547763	+	0.836633i	$0.315479\pi$
$422$	234.520	0.0270527
$423$	−428.846	−0.0492936
$424$	68.8221	0.00788278
$425$	3185.24	0.363546
$426$	5112.98	0.581514
$427$	16735.5	1.89670
$428$	−1233.30	−0.139285
$429$	0	0
$430$	630.474	0.0707074
$431$	−4852.28	−0.542288	−0.271144	−	0.962539i	$-0.587402\pi$
$431$	−4852.28	−0.542288	−0.271144	+	0.962539i	$0.587402\pi$
$432$	−7510.35	−0.836439
$433$	−8208.00	−0.910973	−0.455486	−	0.890243i	$-0.650535\pi$
$433$	−8208.00	−0.910973	−0.455486	+	0.890243i	$0.650535\pi$
$434$	5265.78	0.582409
$435$	363.979	0.0401183
$436$	966.425	0.106155
$437$	7910.01	0.865874
$438$	−7211.01	−0.786656
$439$	−2993.80	−0.325481	−0.162741	−	0.986669i	$-0.552033\pi$
$439$	−2993.80	−0.325481	−0.162741	+	0.986669i	$0.552033\pi$
$440$	878.938	0.0952311
$441$	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	$0.324999\pi$
$444$	−608.503	−0.0650411
$445$	−651.335	−0.0693848
$446$	3164.59	0.335981
$447$	−10166.0	−1.07570
$448$	−10324.2	−1.08878
$449$	561.459	0.0590131	0.0295065	−	0.999565i	$-0.490606\pi$
$449$	561.459	0.0590131	0.0295065	+	0.999565i	$0.490606\pi$
$450$	−4287.22	−0.449114
$451$	−4508.78	−0.470754
$452$	−922.726	−0.0960207
$453$	−3597.54	−0.373128
$454$	8457.38	0.874283
$455$	0	0
$456$	−9618.40	−0.987770
$457$	−13758.4	−1.40830	−0.704148	−	0.710054i	$-0.748671\pi$
$457$	−13758.4	−1.40830	−0.704148	+	0.710054i	$0.748671\pi$
$458$	541.213	0.0552166
$459$	−3805.03	−0.386936
$460$	59.1779	0.00599823
$461$	−12009.2	−1.21329	−0.606644	−	0.794974i	$-0.707485\pi$
$461$	−12009.2	−1.21329	−0.606644	+	0.794974i	$0.707485\pi$
$462$	11106.8	1.11847
$463$	−13635.7	−1.36870	−0.684348	−	0.729156i	$-0.739913\pi$
$463$	−13635.7	−1.36870	−0.684348	+	0.729156i	$0.739913\pi$
$464$	−8869.91	−0.887447
$465$	234.004	0.0233369
$466$	657.613	0.0653719
$467$	8821.95	0.874157	0.437079	−	0.899423i	$-0.356013\pi$
$467$	8821.95	0.874157	0.437079	+	0.899423i	$0.356013\pi$
$468$	0	0
$469$	−8075.72	−0.795100
$470$	45.9554	0.00451013
$471$	2081.37	0.203619
$472$	−1731.87	−0.168889
$473$	−28375.1	−2.75832
$474$	−3978.66	−0.385540
$475$	−13462.2	−1.30039
$476$	−667.956	−0.0643187
$477$	−38.2105	−0.00366780
$478$	−9092.54	−0.870049
$479$	14620.0	1.39459	0.697293	−	0.716786i	$-0.254388\pi$
$479$	14620.0	1.39459	0.697293	+	0.716786i	$0.254388\pi$
$480$	−132.951	−0.0126424
$481$	0	0
$482$	−12884.9	−1.21761
$483$	4906.78	0.462249
$484$	−4114.09	−0.386372
$485$	327.536	0.0306653
$486$	8542.20	0.797288
$487$	9798.86	0.911763	0.455882	−	0.890040i	$-0.349324\pi$
$487$	9798.86	0.911763	0.455882	+	0.890040i	$0.349324\pi$
$488$	−22259.7	−2.06486
$489$	5558.43	0.514030
$490$	−18.1154	−0.00167014
$491$	−10836.1	−0.995977	−0.497989	−	0.867184i	$-0.665928\pi$
$491$	−10836.1	−0.995977	−0.497989	+	0.867184i	$0.665928\pi$
$492$	369.136	0.0338251
$493$	−4493.84	−0.410532
$494$	0	0
$495$	−487.993	−0.0443104
$496$	−5702.51	−0.516230
$497$	−9846.86	−0.888717
$498$	−5698.91	−0.512800
$499$	−2589.96	−0.232349	−0.116175	−	0.993229i	$-0.537063\pi$
$499$	−2589.96	−0.232349	−0.116175	+	0.993229i	$0.537063\pi$
$500$	−201.686	−0.0180394
$501$	2183.24	0.194690
$502$	1841.20	0.163699
$503$	−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
$505$	−517.606	−0.0456102
$506$	12149.0	1.06737
$507$	0	0
$508$	795.712	0.0694961
$509$	1012.89	0.0882038	0.0441019	−	0.999027i	$-0.485957\pi$
$509$	1012.89	0.0882038	0.0441019	+	0.999027i	$0.485957\pi$
$510$	135.401	0.0117562
$511$	13887.4	1.20223
$512$	12992.6	1.12148
$513$	16081.7	1.38406
$514$	−3280.82	−0.281539
$515$	522.644	0.0447193
$516$	2323.08	0.198194
$517$	−2068.26	−0.175942
$518$	−5345.63	−0.453424
$519$	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	$0.706463\pi$
$522$	6048.54	0.507160
$523$	−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
$525$	−8350.92	−0.694217
$526$	−13385.5	−1.10957
$527$	−2889.11	−0.238808
$528$	−12028.0	−0.991382
$529$	−6799.77	−0.558870
$530$	4.09466	0.000335586 0
$531$	961.545	0.0785828
$532$	2823.06	0.230066
$533$	0	0
$534$	10947.5	0.887161
$535$	−481.466	−0.0389076
$536$	10741.4	0.865593
$537$	568.488	0.0456835
$538$	−16506.1	−1.32273
$539$	815.301	0.0651530
$540$	120.313	0.00958788
$541$	−17592.2	−1.39806	−0.699029	−	0.715094i	$-0.746384\pi$
$541$	−17592.2	−1.39806	−0.699029	+	0.715094i	$0.746384\pi$
$542$	−10066.7	−0.797792
$543$	−3947.55	−0.311980
$544$	1641.48	0.129371
$545$	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	$0.365336\pi$
$548$	−2601.53	−0.202795
$549$	12358.8	0.960763
$550$	−20676.6	−1.60301
$551$	18992.8	1.46846
$552$	−6526.44	−0.503231
$553$	7662.33	0.589214
$554$	15073.2	1.15596
$555$	−237.552	−0.0181685
$556$	−2148.52	−0.163880
$557$	507.558	0.0386102	0.0193051	−	0.999814i	$-0.493855\pi$
$557$	507.558	0.0386102	0.0193051	+	0.999814i	$0.493855\pi$
$558$	3888.64	0.295016
$559$	0	0
$560$	−514.690	−0.0388386
$561$	−6093.83	−0.458613
$562$	9041.75	0.678653
$563$	−3443.14	−0.257746	−0.128873	−	0.991661i	$-0.541136\pi$
$563$	−3443.14	−0.257746	−0.128873	+	0.991661i	$0.541136\pi$
$564$	169.330	0.0126420
$565$	−360.221	−0.0268223
$566$	6688.21	0.496690
$567$	3387.96	0.250936
$568$	13097.2	0.967509
$569$	23972.2	1.76620	0.883098	−	0.469189i	$-0.155454\pi$
$569$	23972.2	1.76620	0.883098	+	0.469189i	$0.155454\pi$
$570$	−572.260	−0.0420514
$571$	−7458.32	−0.546622	−0.273311	−	0.961926i	$-0.588119\pi$
$571$	−7458.32	−0.546622	−0.273311	+	0.961926i	$0.588119\pi$
$572$	0	0
$573$	−2495.26	−0.181922
$574$	3242.82	0.235806
$575$	−9134.57	−0.662501
$576$	−7624.14	−0.551515
$577$	−5669.57	−0.409059	−0.204530	−	0.978860i	$-0.565566\pi$
$577$	−5669.57	−0.409059	−0.204530	+	0.978860i	$0.565566\pi$
$578$	10913.2	0.785344
$579$	4869.94	0.349547
$580$	142.093	0.0101726
$581$	10975.3	0.783702
$582$	−5505.15	−0.392089
$583$	−184.284	−0.0130914
$584$	−18471.4	−1.30882
$585$	0	0
$586$	−14065.6	−0.991541
$587$	−1017.39	−0.0715371	−0.0357685	−	0.999360i	$-0.511388\pi$
$587$	−1017.39	−0.0715371	−0.0357685	+	0.999360i	$0.511388\pi$
$588$	−66.7491	−0.00468144
$589$	12210.6	0.854208
$590$	−103.040	−0.00718996
$591$	4669.81	0.325026
$592$	5788.99	0.401902
$593$	10198.2	0.706221	0.353111	−	0.935582i	$-0.385124\pi$
$593$	10198.2	0.706221	0.353111	+	0.935582i	$0.385124\pi$
$594$	24699.9	1.70615
$595$	−260.762	−0.0179667
$596$	−3968.70	−0.272759
$597$	−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	$0.359612\pi$
$600$	11107.4	0.755766
$601$	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
$603$	−5963.70	−0.402754
$604$	−1404.44	−0.0946120
$605$	−1606.09	−0.107929
$606$	8699.80	0.583177
$607$	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
$609$	11781.7	0.783942
$610$	−1324.37	−0.0879053
$611$	0	0
$612$	−493.268	−0.0325803
$613$	23085.4	1.52106	0.760530	−	0.649302i	$-0.224939\pi$
$613$	23085.4	1.52106	0.760530	+	0.649302i	$0.224939\pi$
$614$	−18718.8	−1.23034
$615$	144.106	0.00944866
$616$	28450.6	1.86089
$617$	−3049.24	−0.198959	−0.0994796	−	0.995040i	$-0.531718\pi$
$617$	−3049.24	−0.198959	−0.0994796	+	0.995040i	$0.531718\pi$
$618$	−8784.48	−0.571786
$619$	−7296.58	−0.473787	−0.236894	−	0.971536i	$-0.576129\pi$
$619$	−7296.58	−0.473787	−0.236894	+	0.971536i	$0.576129\pi$
$620$	91.3523	0.00591741
$621$	10912.0	0.705126
$622$	−20248.9	−1.30531
$623$	−21083.3	−1.35583
$624$	0	0
$625$	15506.8	0.992438
$626$	−25621.7	−1.63586
$627$	25755.1	1.64044
$628$	812.543	0.0516305
$629$	2932.92	0.185920
$630$	350.976	0.0221956
$631$	23829.5	1.50339	0.751694	−	0.659512i	$-0.229237\pi$
$631$	23829.5	1.50339	0.751694	+	0.659512i	$0.229237\pi$
$632$	−10191.6	−0.641453
$633$	337.345	0.0211821
$634$	−15960.5	−0.999801
$635$	310.637	0.0194130
$636$	15.0874	0.000940653 0
$637$	0	0
$638$	29171.3	1.81019
$639$	−7271.65	−0.450175
$640$	528.347	0.0326324
$641$	13405.3	0.826016	0.413008	−	0.910727i	$-0.364478\pi$
$641$	13405.3	0.826016	0.413008	+	0.910727i	$0.364478\pi$
$642$	8092.36	0.497477
$643$	−5251.51	−0.322083	−0.161042	−	0.986948i	$-0.551485\pi$
$643$	−5251.51	−0.322083	−0.161042	+	0.986948i	$0.551485\pi$
$644$	1915.55	0.117210
$645$	906.904	0.0553633
$646$	7065.37	0.430315
$647$	21611.4	1.31319	0.656595	−	0.754244i	$-0.271996\pi$
$647$	21611.4	1.31319	0.656595	+	0.754244i	$0.271996\pi$
$648$	−4506.28	−0.273184
$649$	4637.39	0.280483
$650$	0	0
$651$	7574.54	0.456021
$652$	2169.94	0.130340
$653$	−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
$655$	−1154.78	−0.0688869
$656$	−3511.77	−0.209012
$657$	10255.4	0.608985
$658$	1487.54	0.0881314
$659$	−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
$661$	−26981.1	−1.58766	−0.793831	−	0.608139i	$-0.791916\pi$
$661$	−26981.1	−1.58766	−0.793831	+	0.608139i	$0.791916\pi$
$662$	−11871.5	−0.696980
$663$	0	0
$664$	−14598.1	−0.853185
$665$	1102.09	0.0642664
$666$	−3947.60	−0.229680
$667$	12887.3	0.748126
$668$	852.310	0.0493665
$669$	4552.09	0.263070
$670$	639.074	0.0368501
$671$	59604.5	3.42922
$672$	−4303.55	−0.247043
$673$	11149.2	0.638591	0.319296	−	0.947655i	$-0.396554\pi$
$673$	11149.2	0.638591	0.319296	+	0.947655i	$0.396554\pi$
$674$	−7761.04	−0.443537
$675$	−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	$0.470010\pi$
$678$	6054.51	0.342953
$679$	10602.1	0.599223
$680$	346.836	0.0195596
$681$	12165.5	0.684556
$682$	18754.3	1.05299
$683$	−24505.2	−1.37287	−0.686433	−	0.727193i	$-0.740824\pi$
$683$	−24505.2	−1.37287	−0.686433	+	0.727193i	$0.740824\pi$
$684$	2084.75	0.116539
$685$	−1015.61	−0.0566486
$686$	−16557.0	−0.921500
$687$	778.506	0.0432342
$688$	−22100.6	−1.22468
$689$	0	0
$690$	−388.299	−0.0214236
$691$	21752.8	1.19756	0.598782	−	0.800912i	$-0.295652\pi$
$691$	21752.8	1.19756	0.598782	+	0.800912i	$0.295652\pi$
$692$	6466.64	0.355238
$693$	−15796.0	−0.865858
$694$	−7278.90	−0.398132
$695$	−838.755	−0.0457781
$696$	−15670.7	−0.853445
$697$	−1779.20	−0.0966887
$698$	19379.9	1.05092
$699$	945.941	0.0511857
$700$	−3260.10	−0.176029
$701$	34250.9	1.84542	0.922709	−	0.385496i	$-0.125970\pi$
$701$	34250.9	1.84542	0.922709	+	0.385496i	$0.125970\pi$
$702$	0	0
$703$	−12395.8	−0.665028
$704$	−36770.1	−1.96850
$705$	66.1043	0.00353140
$706$	−5992.59	−0.319454
$707$	−16754.6	−0.891259
$708$	−379.666	−0.0201536
$709$	5527.11	0.292771	0.146386	−	0.989228i	$-0.453236\pi$
$709$	5527.11	0.292771	0.146386	+	0.989228i	$0.453236\pi$
$710$	779.234	0.0411889
$711$	5658.43	0.298464
$712$	28042.6	1.47604
$713$	8285.33	0.435187
$714$	4382.83	0.229724
$715$	0	0
$716$	221.931	0.0115837
$717$	−13079.1	−0.681241
$718$	−6485.02	−0.337074
$719$	−3777.78	−0.195949	−0.0979745	−	0.995189i	$-0.531236\pi$
$719$	−3777.78	−0.195949	−0.0979745	+	0.995189i	$0.531236\pi$
$720$	−380.085	−0.0196735
$721$	16917.6	0.873849
$722$	−12291.5	−0.633576
$723$	−18534.2	−0.953380
$724$	−1541.08	−0.0791072
$725$	−21933.2	−1.12355
$726$	26994.8	1.37999
$727$	19076.8	0.973204	0.486602	−	0.873624i	$-0.338236\pi$
$727$	19076.8	0.973204	0.486602	+	0.873624i	$0.338236\pi$
$728$	0	0
$729$	17319.9	0.879944
$730$	−1098.98	−0.0557192
$731$	−11197.0	−0.566535
$732$	−4879.86	−0.246400
$733$	−7997.30	−0.402984	−0.201492	−	0.979490i	$-0.564579\pi$
$733$	−7997.30	−0.402984	−0.201492	+	0.979490i	$0.564579\pi$
$734$	−16849.4	−0.847307
$735$	−26.0581	−0.00130771
$736$	−4707.39	−0.235756
$737$	−28762.1	−1.43754
$738$	2394.74	0.119446
$739$	−28983.6	−1.44273	−0.721367	−	0.692553i	$-0.756486\pi$
$739$	−28983.6	−1.44273	−0.721367	+	0.692553i	$0.756486\pi$
$740$	−92.7376	−0.00460689
$741$	0	0
$742$	132.541	0.00655761
$743$	19145.4	0.945324	0.472662	−	0.881244i	$-0.343293\pi$
$743$	19145.4	0.945324	0.472662	+	0.881244i	$0.343293\pi$
$744$	−10074.8	−0.496451
$745$	−1549.33	−0.0761923
$746$	−7435.47	−0.364922
$747$	8104.95	0.396981
$748$	−2378.96	−0.116288
$749$	−15584.7	−0.760284
$750$	1323.38	0.0644304
$751$	−25516.9	−1.23985	−0.619923	−	0.784663i	$-0.712836\pi$
$751$	−25516.9	−1.23985	−0.619923	+	0.784663i	$0.712836\pi$
$752$	−1610.92	−0.0781172
$753$	2648.47	0.128175
$754$	0	0
$755$	−548.275	−0.0264288
$756$	3894.46	0.187355
$757$	−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
$759$	17475.7	0.835744
$760$	−1465.87	−0.0699642
$761$	2343.06	0.111611	0.0558053	−	0.998442i	$-0.482227\pi$
$761$	2343.06	0.111611	0.0558053	+	0.998442i	$0.482227\pi$
$762$	−5221.11	−0.248216
$763$	12212.3	0.579444
$764$	−974.121	−0.0461289
$765$	−192.566	−0.00910096
$766$	27756.9	1.30927
$767$	0	0
$768$	7862.11	0.369400
$769$	7100.18	0.332950	0.166475	−	0.986046i	$-0.446761\pi$
$769$	7100.18	0.332950	0.166475	+	0.986046i	$0.446761\pi$
$770$	1692.71	0.0792219
$771$	−4719.29	−0.220442
$772$	1901.17	0.0886328
$773$	−12270.4	−0.570940	−0.285470	−	0.958388i	$-0.592150\pi$
$773$	−12270.4	−0.570940	−0.285470	+	0.958388i	$0.592150\pi$
$774$	15070.8	0.699881
$775$	−14100.9	−0.653575
$776$	−14101.7	−0.652349
$777$	−7689.40	−0.355027
$778$	24386.9	1.12379
$779$	7519.64	0.345852
$780$	0	0
$781$	−35070.1	−1.60680
$782$	4794.11	0.219229
$783$	26201.0	1.19584
$784$	635.017	0.0289275
$785$	317.207	0.0144224
$786$	19409.2	0.880794
$787$	−3425.04	−0.155133	−0.0775663	−	0.996987i	$-0.524715\pi$
$787$	−3425.04	−0.155133	−0.0775663	+	0.996987i	$0.524715\pi$
$788$	1823.04	0.0824151
$789$	−19254.3	−0.868787
$790$	−606.360	−0.0273080
$791$	−11660.1	−0.524129
$792$	21010.0	0.942624
$793$	0	0
$794$	−25894.3	−1.15737
$795$	5.88995	0.000262761 0
$796$	−3446.87	−0.153481
$797$	−11781.1	−0.523600	−0.261800	−	0.965122i	$-0.584316\pi$
$797$	−11781.1	−0.523600	−0.261800	+	0.965122i	$0.584316\pi$
$798$	−18523.6	−0.821717
$799$	−816.154	−0.0361370
$800$	8011.58	0.354065
$801$	−15569.4	−0.686790
$802$	5339.24	0.235081
$803$	49460.6	2.17363
$804$	2354.77	0.103291
$805$	747.807	0.0327413
$806$	0	0
$807$	−23743.2	−1.03569
$808$	22285.0	0.970276
$809$	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
$811$	−12803.3	−0.554359	−0.277180	−	0.960818i	$-0.589400\pi$
$811$	−12803.3	−0.554359	−0.277180	+	0.960818i	$0.589400\pi$
$812$	4599.45	0.198780
$813$	−14480.5	−0.624664
$814$	−19038.7	−0.819788
$815$	847.121	0.0364090
$816$	−4746.33	−0.203621
$817$	47323.3	2.02648
$818$	−24889.4	−1.06386
$819$	0	0
$820$	56.2574	0.00239585
$821$	−19335.1	−0.821923	−0.410962	−	0.911653i	$-0.634807\pi$
$821$	−19335.1	−0.821923	−0.410962	+	0.911653i	$0.634807\pi$
$822$	17070.1	0.724315
$823$	−2125.90	−0.0900417	−0.0450209	−	0.998986i	$-0.514335\pi$
$823$	−2125.90	−0.0900417	−0.0450209	+	0.998986i	$0.514335\pi$
$824$	−22501.9	−0.951324
$825$	−29742.2	−1.25514
$826$	−3335.32	−0.140497
$827$	6989.24	0.293881	0.146941	−	0.989145i	$-0.453057\pi$
$827$	6989.24	0.293881	0.146941	+	0.989145i	$0.453057\pi$
$828$	1414.58	0.0593721
$829$	−32649.7	−1.36788	−0.683938	−	0.729540i	$-0.739734\pi$
$829$	−32649.7	−1.36788	−0.683938	+	0.729540i	$0.739734\pi$
$830$	−868.531	−0.0363219
$831$	21682.0	0.905103
$832$	0	0
$833$	321.724	0.0133819
$834$	14097.6	0.585324
$835$	332.732	0.0137900
$836$	10054.5	0.415958
$837$	16844.7	0.695626
$838$	−34278.4	−1.41304
$839$	4038.23	0.166168	0.0830841	−	0.996543i	$-0.473523\pi$
$839$	4038.23	0.166168	0.0830841	+	0.996543i	$0.473523\pi$
$840$	−909.318	−0.0373505
$841$	6555.00	0.268769
$842$	−24240.9	−0.992159
$843$	13006.1	0.531380
$844$	131.695	0.00537102
$845$	0	0
$846$	1098.51	0.0446426
$847$	−51988.1	−2.10901
$848$	−143.534	−0.00581248
$849$	9620.64	0.388904
$850$	−8159.17	−0.329244
$851$	−8410.97	−0.338807
$852$	2871.21	0.115453
$853$	−8114.12	−0.325700	−0.162850	−	0.986651i	$-0.552069\pi$
$853$	−8114.12	−0.325700	−0.162850	+	0.986651i	$0.552069\pi$
$854$	−42869.0	−1.71774
$855$	813.863	0.0325538
$856$	20729.0	0.827690
$857$	−22298.1	−0.888786	−0.444393	−	0.895832i	$-0.646581\pi$
$857$	−22298.1	−0.888786	−0.444393	+	0.895832i	$0.646581\pi$
$858$	0	0
$859$	33550.5	1.33263	0.666315	−	0.745670i	$-0.267870\pi$
$859$	33550.5	1.33263	0.666315	+	0.745670i	$0.267870\pi$
$860$	354.045	0.0140382
$861$	4664.62	0.184634
$862$	12429.4	0.491121
$863$	14120.5	0.556972	0.278486	−	0.960440i	$-0.410167\pi$
$863$	14120.5	0.556972	0.278486	+	0.960440i	$0.410167\pi$
$864$	−9570.49	−0.376846
$865$	2524.50	0.0992319
$866$	21025.2	0.825018
$867$	15698.1	0.614918
$868$	2957.01	0.115631
$869$	27289.8	1.06530
$870$	−932.351	−0.0363329
$871$	0	0
$872$	−16243.4	−0.630817
$873$	7829.39	0.303533
$874$	−20261.9	−0.784175
$875$	−2548.63	−0.0984679
$876$	−4049.36	−0.156182
$877$	1941.69	0.0747619	0.0373809	−	0.999301i	$-0.488099\pi$
$877$	1941.69	0.0747619	0.0373809	+	0.999301i	$0.488099\pi$
$878$	7668.78	0.294771
$879$	−20232.6	−0.776368
$880$	−1833.10	−0.0702201
$881$	−790.231	−0.0302197	−0.0151099	−	0.999886i	$-0.504810\pi$
$881$	−790.231	−0.0302197	−0.0151099	+	0.999886i	$0.504810\pi$
$882$	−433.029	−0.0165316
$883$	−36638.6	−1.39636	−0.698180	−	0.715922i	$-0.746007\pi$
$883$	−36638.6	−1.39636	−0.698180	+	0.715922i	$0.746007\pi$
$884$	0	0
$885$	−148.217	−0.00562968
$886$	−24958.9	−0.946401
$887$	−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
$889$	10055.1	0.379344
$890$	1668.43	0.0628381
$891$	12066.4	0.453692
$892$	1777.08	0.0667053
$893$	3449.40	0.129261
$894$	26040.9	0.974202
$895$	86.6392	0.00323579
$896$	17102.2	0.637663
$897$	0	0
$898$	−1438.21	−0.0534449
$899$	19894.0	0.738046
$900$	−2407.50	−0.0891666
$901$	−72.7200	−0.00268885
$902$	11549.5	0.426336
$903$	29355.9	1.08184
$904$	15509.0	0.570598
$905$	−601.618	−0.0220977
$906$	9215.28	0.337922
$907$	−10464.4	−0.383093	−0.191547	−	0.981484i	$-0.561350\pi$
$907$	−10464.4	−0.383093	−0.191547	+	0.981484i	$0.561350\pi$
$908$	4749.26	0.173579
$909$	−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	$0.724212\pi$
$912$	20060.0	0.728346
$913$	39089.0	1.41693
$914$	35242.9	1.27542
$915$	−1905.04	−0.0688291
$916$	303.920	0.0109627
$917$	−37379.4	−1.34610
$918$	9746.80	0.350427
$919$	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
$921$	−26926.0	−0.963346
$922$	30762.3	1.09881
$923$	0	0
$924$	6237.04	0.222060
$925$	14314.8	0.508829
$926$	34928.6	1.23955
$927$	12493.2	0.442644
$928$	−11303.0	−0.399826
$929$	−55733.8	−1.96832	−0.984159	−	0.177290i	$-0.943267\pi$
$929$	−55733.8	−1.96832	−0.984159	+	0.177290i	$0.943267\pi$
$930$	−599.413	−0.0211350
$931$	−1359.74	−0.0478665
$932$	369.284	0.0129789
$933$	−29126.9	−1.02205
$934$	−22597.9	−0.791677
$935$	−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	$0.517758\pi$
$938$	20686.4	0.720079
$939$	−36855.4	−1.28086
$940$	25.8064	0.000895437 0
$941$	−8823.35	−0.305667	−0.152834	−	0.988252i	$-0.548840\pi$
$941$	−8823.35	−0.305667	−0.152834	+	0.988252i	$0.548840\pi$
$942$	−5331.54	−0.184407
$943$	5102.35	0.176199
$944$	3611.95	0.124533
$945$	1520.35	0.0523354
$946$	72684.3	2.49806
$947$	−28290.4	−0.970766	−0.485383	−	0.874301i	$-0.661320\pi$
$947$	−28290.4	−0.970766	−0.485383	+	0.874301i	$0.661320\pi$
$948$	−2234.23	−0.0765447
$949$	0	0
$950$	34484.0	1.17769
$951$	−22958.4	−0.782836
$952$	11226.8	0.382210
$953$	−12399.0	−0.421452	−0.210726	−	0.977545i	$-0.567583\pi$
$953$	−12399.0	−0.421452	−0.210726	+	0.977545i	$0.567583\pi$
$954$	97.8783	0.00332173
$955$	−380.285	−0.0128856
$956$	−5105.94	−0.172739
$957$	41961.3	1.41736
$958$	−37450.0	−1.26300
$959$	−32874.5	−1.10696
$960$	1175.22	0.0395105
$961$	−17001.0	−0.570676
$962$	0	0
$963$	−11508.9	−0.385118
$964$	−7235.53	−0.241744
$965$	742.193	0.0247586
$966$	−12569.0	−0.418634
$967$	26667.1	0.886820	0.443410	−	0.896319i	$-0.353769\pi$
$967$	26667.1	0.886820	0.443410	+	0.896319i	$0.353769\pi$
$968$	69148.6	2.29599
$969$	10163.2	0.336933
$970$	−839.001	−0.0277719
$971$	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
$973$	−27149.9	−0.894539
$974$	−25100.3	−0.825735
$975$	0	0
$976$	46424.5	1.52255
$977$	−778.759	−0.0255012	−0.0127506	−	0.999919i	$-0.504059\pi$
$977$	−778.759	−0.0255012	−0.0127506	+	0.999919i	$0.504059\pi$
$978$	−14238.2	−0.465529
$979$	−75089.2	−2.45134
$980$	−10.1728	−0.000331589 0
$981$	9018.48	0.293515
$982$	27757.2	0.902003
$983$	−5997.90	−0.194612	−0.0973059	−	0.995255i	$-0.531023\pi$
$983$	−5997.90	−0.194612	−0.0973059	+	0.995255i	$0.531023\pi$
$984$	−6204.35	−0.201004
$985$	711.693	0.0230218
$986$	11511.2	0.371797
$987$	2139.75	0.0690062
$988$	0	0
$989$	32110.6	1.03241
$990$	1250.02	0.0401295
$991$	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
$993$	−17076.6	−0.545729
$994$	25223.3	0.804863
$995$	−1345.62	−0.0428732
$996$	−3200.24	−0.101811
$997$	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
$999$	−17100.2	−0.541567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.g.1.1		2
3.2	odd	2		1521.4.a.r.1.2		2
13.2	odd	12		169.4.e.f.147.1		8
13.3	even	3		169.4.c.j.22.2		4
13.4	even	6		169.4.c.g.146.1		4
13.5	odd	4		169.4.b.f.168.4		4
13.6	odd	12		169.4.e.f.23.4		8
13.7	odd	12		169.4.e.f.23.1		8
13.8	odd	4		169.4.b.f.168.1		4
13.9	even	3		169.4.c.j.146.2		4
13.10	even	6		169.4.c.g.22.1		4
13.11	odd	12		169.4.e.f.147.4		8
13.12	even	2		13.4.a.b.1.2	✓	2
39.38	odd	2		117.4.a.d.1.1		2
52.51	odd	2		208.4.a.h.1.2		2
65.12	odd	4		325.4.b.e.274.4		4
65.38	odd	4		325.4.b.e.274.1		4
65.64	even	2		325.4.a.f.1.1		2
91.90	odd	2		637.4.a.b.1.2		2
104.51	odd	2		832.4.a.z.1.1		2
104.77	even	2		832.4.a.s.1.2		2
143.142	odd	2		1573.4.a.b.1.1		2
156.155	even	2		1872.4.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.2	✓	2	13.12	even	2
117.4.a.d.1.1		2	39.38	odd	2
169.4.a.g.1.1		2	1.1	even	1	trivial
169.4.b.f.168.1		4	13.8	odd	4
169.4.b.f.168.4		4	13.5	odd	4
169.4.c.g.22.1		4	13.10	even	6
169.4.c.g.146.1		4	13.4	even	6
169.4.c.j.22.2		4	13.3	even	3
169.4.c.j.146.2		4	13.9	even	3
169.4.e.f.23.1		8	13.7	odd	12
169.4.e.f.23.4		8	13.6	odd	12
169.4.e.f.147.1		8	13.2	odd	12
169.4.e.f.147.4		8	13.11	odd	12
208.4.a.h.1.2		2	52.51	odd	2
325.4.a.f.1.1		2	65.64	even	2
325.4.b.e.274.1		4	65.38	odd	4
325.4.b.e.274.4		4	65.12	odd	4
637.4.a.b.1.2		2	91.90	odd	2
832.4.a.s.1.2		2	104.77	even	2
832.4.a.z.1.1		2	104.51	odd	2
1521.4.a.r.1.2		2	3.2	odd	2
1573.4.a.b.1.1		2	143.142	odd	2
1872.4.a.bb.1.1		2	156.155	even	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -18.1771 q^{7} +24.1771 q^{8} -13.4233 q^{9} +O(q^{10})\)	\(q-2.56155 q^{2} -3.68466 q^{3} -1.43845 q^{4} -0.561553 q^{5} +9.43845 q^{6} -18.1771 q^{7} +24.1771 q^{8} -13.4233 q^{9} +1.43845 q^{10} -64.7386 q^{11} +5.30019 q^{12} +46.5616 q^{14} +2.06913 q^{15} -50.4233 q^{16} -25.5464 q^{17} +34.3845 q^{18} +107.970 q^{19} +0.807764 q^{20} +66.9763 q^{21} +165.831 q^{22} +73.2614 q^{23} -89.0843 q^{24} -124.685 q^{25} +148.946 q^{27} +26.1468 q^{28} +175.909 q^{29} -5.30019 q^{30} +113.093 q^{31} -64.2547 q^{32} +238.540 q^{33} +65.4384 q^{34} +10.2074 q^{35} +19.3087 q^{36} -114.808 q^{37} -276.570 q^{38} -13.5767 q^{40} +69.6458 q^{41} -171.563 q^{42} +438.302 q^{43} +93.1231 q^{44} +7.53789 q^{45} -187.663 q^{46} +31.9479 q^{47} +185.793 q^{48} -12.5937 q^{49} +319.386 q^{50} +94.1298 q^{51} +2.84658 q^{53} -381.533 q^{54} +36.3542 q^{55} -439.469 q^{56} -397.831 q^{57} -450.600 q^{58} -71.6325 q^{59} -2.97633 q^{60} -920.695 q^{61} -289.693 q^{62} +243.996 q^{63} +567.978 q^{64} -611.032 q^{66} +444.280 q^{67} +36.7471 q^{68} -269.943 q^{69} -26.1468 q^{70} +541.719 q^{71} -324.536 q^{72} -764.004 q^{73} +294.086 q^{74} +459.420 q^{75} -155.309 q^{76} +1176.76 q^{77} -421.538 q^{79} +28.3153 q^{80} -186.386 q^{81} -178.401 q^{82} -603.797 q^{83} -96.3419 q^{84} +14.3457 q^{85} -1122.73 q^{86} -648.165 q^{87} -1565.19 q^{88} +1159.88 q^{89} -19.3087 q^{90} -105.383 q^{92} -416.708 q^{93} -81.8362 q^{94} -60.6307 q^{95} +236.757 q^{96} -583.269 q^{97} +32.2595 q^{98} +869.006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10}) \) 2 * q - q^2 + 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9	\( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9} + 7 q^{10} - 80 q^{11} - 43 q^{12} + 89 q^{14} + 33 q^{15} - 39 q^{16} + 19 q^{17} + 110 q^{18} + 84 q^{19} - 19 q^{20} + 303 q^{21} + 142 q^{22} + 196 q^{23} - 273 q^{24} - 237 q^{25} + 335 q^{27} - 125 q^{28} - 44 q^{29} + 43 q^{30} + 86 q^{31} + 123 q^{32} + 106 q^{33} + 135 q^{34} + 107 q^{35} - 250 q^{36} - 209 q^{37} - 314 q^{38} - 89 q^{40} + 230 q^{41} + 197 q^{42} + 287 q^{43} + 178 q^{44} + 180 q^{45} + 4 q^{46} - 435 q^{47} + 285 q^{48} + 383 q^{49} + 144 q^{50} + 481 q^{51} - 118 q^{53} - 91 q^{54} - 18 q^{55} - 1015 q^{56} - 606 q^{57} - 794 q^{58} + 368 q^{59} - 175 q^{60} - 1058 q^{61} - 332 q^{62} + 1560 q^{63} + 769 q^{64} - 818 q^{66} - 68 q^{67} - 211 q^{68} + 796 q^{69} + 125 q^{70} + 131 q^{71} - 1350 q^{72} - 456 q^{73} + 147 q^{74} - 516 q^{75} - 22 q^{76} + 762 q^{77} - 1008 q^{79} + 69 q^{80} + 122 q^{81} + 72 q^{82} - 1958 q^{83} - 1409 q^{84} + 173 q^{85} - 1359 q^{86} - 2558 q^{87} - 1242 q^{88} + 720 q^{89} + 250 q^{90} - 788 q^{92} - 652 q^{93} - 811 q^{94} - 146 q^{95} + 1863 q^{96} + 928 q^{97} + 650 q^{98} + 130 q^{99}+O(q^{100}) \) 2 * q - q^2 + 5 * q^3 - 7 * q^4 + 3 * q^5 + 23 * q^6 + 9 * q^7 + 3 * q^8 + 35 * q^9 + 7 * q^10 - 80 * q^11 - 43 * q^12 + 89 * q^14 + 33 * q^15 - 39 * q^16 + 19 * q^17 + 110 * q^18 + 84 * q^19 - 19 * q^20 + 303 * q^21 + 142 * q^22 + 196 * q^23 - 273 * q^24 - 237 * q^25 + 335 * q^27 - 125 * q^28 - 44 * q^29 + 43 * q^30 + 86 * q^31 + 123 * q^32 + 106 * q^33 + 135 * q^34 + 107 * q^35 - 250 * q^36 - 209 * q^37 - 314 * q^38 - 89 * q^40 + 230 * q^41 + 197 * q^42 + 287 * q^43 + 178 * q^44 + 180 * q^45 + 4 * q^46 - 435 * q^47 + 285 * q^48 + 383 * q^49 + 144 * q^50 + 481 * q^51 - 118 * q^53 - 91 * q^54 - 18 * q^55 - 1015 * q^56 - 606 * q^57 - 794 * q^58 + 368 * q^59 - 175 * q^60 - 1058 * q^61 - 332 * q^62 + 1560 * q^63 + 769 * q^64 - 818 * q^66 - 68 * q^67 - 211 * q^68 + 796 * q^69 + 125 * q^70 + 131 * q^71 - 1350 * q^72 - 456 * q^73 + 147 * q^74 - 516 * q^75 - 22 * q^76 + 762 * q^77 - 1008 * q^79 + 69 * q^80 + 122 * q^81 + 72 * q^82 - 1958 * q^83 - 1409 * q^84 + 173 * q^85 - 1359 * q^86 - 2558 * q^87 - 1242 * q^88 + 720 * q^89 + 250 * q^90 - 788 * q^92 - 652 * q^93 - 811 * q^94 - 146 * q^95 + 1863 * q^96 + 928 * q^97 + 650 * q^98 + 130 * q^99

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