LMFDB - Embedded newform 4009.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53259 q^{2} +1.29314 q^{3} +0.348820 q^{4} +3.28950 q^{5} -1.98185 q^{6} +4.48074 q^{7} +2.53058 q^{8} -1.32779 q^{9} +O(q^{10})$	\(q-1.53259 q^{2} +1.29314 q^{3} +0.348820 q^{4} +3.28950 q^{5} -1.98185 q^{6} +4.48074 q^{7} +2.53058 q^{8} -1.32779 q^{9} -5.04144 q^{10} +1.71386 q^{11} +0.451073 q^{12} -1.20289 q^{13} -6.86712 q^{14} +4.25378 q^{15} -4.57596 q^{16} +4.02684 q^{17} +2.03496 q^{18} +1.00000 q^{19} +1.14744 q^{20} +5.79422 q^{21} -2.62663 q^{22} -6.98173 q^{23} +3.27238 q^{24} +5.82081 q^{25} +1.84353 q^{26} -5.59644 q^{27} +1.56297 q^{28} +1.70211 q^{29} -6.51928 q^{30} +6.70143 q^{31} +1.95191 q^{32} +2.21625 q^{33} -6.17147 q^{34} +14.7394 q^{35} -0.463161 q^{36} -5.61232 q^{37} -1.53259 q^{38} -1.55550 q^{39} +8.32433 q^{40} +2.64972 q^{41} -8.88014 q^{42} -2.30921 q^{43} +0.597827 q^{44} -4.36777 q^{45} +10.7001 q^{46} +1.32013 q^{47} -5.91736 q^{48} +13.0770 q^{49} -8.92089 q^{50} +5.20726 q^{51} -0.419592 q^{52} -7.82267 q^{53} +8.57702 q^{54} +5.63773 q^{55} +11.3388 q^{56} +1.29314 q^{57} -2.60863 q^{58} +11.3788 q^{59} +1.48380 q^{60} +3.08462 q^{61} -10.2705 q^{62} -5.94949 q^{63} +6.16046 q^{64} -3.95690 q^{65} -3.39660 q^{66} +4.93026 q^{67} +1.40464 q^{68} -9.02835 q^{69} -22.5894 q^{70} +11.8709 q^{71} -3.36008 q^{72} +0.713471 q^{73} +8.60137 q^{74} +7.52712 q^{75} +0.348820 q^{76} +7.67934 q^{77} +2.38394 q^{78} +10.9688 q^{79} -15.0526 q^{80} -3.25359 q^{81} -4.06092 q^{82} -9.81972 q^{83} +2.02114 q^{84} +13.2463 q^{85} +3.53907 q^{86} +2.20106 q^{87} +4.33704 q^{88} +8.36655 q^{89} +6.69399 q^{90} -5.38983 q^{91} -2.43537 q^{92} +8.66588 q^{93} -2.02321 q^{94} +3.28950 q^{95} +2.52409 q^{96} +3.16042 q^{97} -20.0417 q^{98} -2.27564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * 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 $

$2$	−1.53259	−1.08370	−0.541851	−	0.840475i	$-0.682276\pi$
$2$	−1.53259	−1.08370	−0.541851	+	0.840475i	$0.682276\pi$
$3$	1.29314	0.746594	0.373297	−	0.927712i	$-0.378227\pi$
$3$	1.29314	0.746594	0.373297	+	0.927712i	$0.378227\pi$
$4$	0.348820	0.174410
$5$	3.28950	1.47111	0.735555	−	0.677465i	$-0.236922\pi$
$5$	3.28950	1.47111	0.735555	+	0.677465i	$0.236922\pi$
$6$	−1.98185	−0.809085
$7$	4.48074	1.69356	0.846780	−	0.531943i	$-0.178538\pi$
$7$	4.48074	1.69356	0.846780	+	0.531943i	$0.178538\pi$
$8$	2.53058	0.894694
$9$	−1.32779	−0.442597
$10$	−5.04144	−1.59424
$11$	1.71386	0.516747	0.258373	−	0.966045i	$-0.416813\pi$
$11$	1.71386	0.516747	0.258373	+	0.966045i	$0.416813\pi$
$12$	0.451073	0.130213
$13$	−1.20289	−0.333621	−0.166811	−	0.985989i	$-0.553347\pi$
$13$	−1.20289	−0.333621	−0.166811	+	0.985989i	$0.553347\pi$
$14$	−6.86712	−1.83531
$15$	4.25378	1.09832
$16$	−4.57596	−1.14399
$17$	4.02684	0.976651	0.488326	−	0.872662i	$-0.337608\pi$
$17$	4.02684	0.976651	0.488326	+	0.872662i	$0.337608\pi$
$18$	2.03496	0.479644
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.14744	0.256576
$21$	5.79422	1.26440
$22$	−2.62663	−0.560000
$23$	−6.98173	−1.45579	−0.727896	−	0.685688i	$-0.759502\pi$
$23$	−6.98173	−1.45579	−0.727896	+	0.685688i	$0.759502\pi$
$24$	3.27238	0.667973
$25$	5.82081	1.16416
$26$	1.84353	0.361546
$27$	−5.59644	−1.07703
$28$	1.56297	0.295374
$29$	1.70211	0.316074	0.158037	−	0.987433i	$-0.449483\pi$
$29$	1.70211	0.316074	0.158037	+	0.987433i	$0.449483\pi$
$30$	−6.51928	−1.19025
$31$	6.70143	1.20361	0.601806	−	0.798642i	$-0.294448\pi$
$31$	6.70143	1.20361	0.601806	+	0.798642i	$0.294448\pi$
$32$	1.95191	0.345052
$33$	2.21625	0.385800
$34$	−6.17147	−1.05840
$35$	14.7394	2.49141
$36$	−0.463161	−0.0771934
$37$	−5.61232	−0.922661	−0.461330	−	0.887228i	$-0.652628\pi$
$37$	−5.61232	−0.922661	−0.461330	+	0.887228i	$0.652628\pi$
$38$	−1.53259	−0.248618
$39$	−1.55550	−0.249080
$40$	8.32433	1.31619
$41$	2.64972	0.413816	0.206908	−	0.978360i	$-0.433660\pi$
$41$	2.64972	0.413816	0.206908	+	0.978360i	$0.433660\pi$
$42$	−8.88014	−1.37023
$43$	−2.30921	−0.352152	−0.176076	−	0.984377i	$-0.556340\pi$
$43$	−2.30921	−0.352152	−0.176076	+	0.984377i	$0.556340\pi$
$44$	0.597827	0.0901258
$45$	−4.36777	−0.651109
$46$	10.7001	1.57764
$47$	1.32013	0.192561	0.0962804	−	0.995354i	$-0.469305\pi$
$47$	1.32013	0.192561	0.0962804	+	0.995354i	$0.469305\pi$
$48$	−5.91736	−0.854097
$49$	13.0770	1.86815
$50$	−8.92089	−1.26160
$51$	5.20726	0.729162
$52$	−0.419592	−0.0581869
$53$	−7.82267	−1.07453	−0.537263	−	0.843415i	$-0.680542\pi$
$53$	−7.82267	−1.07453	−0.537263	+	0.843415i	$0.680542\pi$
$54$	8.57702	1.16718
$55$	5.63773	0.760191
$56$	11.3388	1.51522
$57$	1.29314	0.171280
$58$	−2.60863	−0.342530
$59$	11.3788	1.48139	0.740695	−	0.671842i	$-0.234496\pi$
$59$	11.3788	1.48139	0.740695	+	0.671842i	$0.234496\pi$
$60$	1.48380	0.191558
$61$	3.08462	0.394945	0.197473	−	0.980308i	$-0.436727\pi$
$61$	3.08462	0.394945	0.197473	+	0.980308i	$0.436727\pi$
$62$	−10.2705	−1.30436
$63$	−5.94949	−0.749565
$64$	6.16046	0.770058
$65$	−3.95690	−0.490793
$66$	−3.39660	−0.418092
$67$	4.93026	0.602328	0.301164	−	0.953572i	$-0.402625\pi$
$67$	4.93026	0.602328	0.301164	+	0.953572i	$0.402625\pi$
$68$	1.40464	0.170338
$69$	−9.02835	−1.08689
$70$	−22.5894	−2.69995
$71$	11.8709	1.40882	0.704411	−	0.709793i	$-0.251211\pi$
$71$	11.8709	1.40882	0.704411	+	0.709793i	$0.251211\pi$
$72$	−3.36008	−0.395989
$73$	0.713471	0.0835054	0.0417527	−	0.999128i	$-0.486706\pi$
$73$	0.713471	0.0835054	0.0417527	+	0.999128i	$0.486706\pi$
$74$	8.60137	0.999889
$75$	7.52712	0.869157
$76$	0.348820	0.0400124
$77$	7.67934	0.875142
$78$	2.38394	0.269928
$79$	10.9688	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$79$	10.9688	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$80$	−15.0526	−1.68294
$81$	−3.25359	−0.361510
$82$	−4.06092	−0.448454
$83$	−9.81972	−1.07785	−0.538927	−	0.842352i	$-0.681170\pi$
$83$	−9.81972	−1.07785	−0.538927	+	0.842352i	$0.681170\pi$
$84$	2.02114	0.220524
$85$	13.2463	1.43676
$86$	3.53907	0.381628
$87$	2.20106	0.235979
$88$	4.33704	0.462330
$89$	8.36655	0.886853	0.443426	−	0.896311i	$-0.353763\pi$
$89$	8.36655	0.886853	0.443426	+	0.896311i	$0.353763\pi$
$90$	6.69399	0.705608
$91$	−5.38983	−0.565008
$92$	−2.43537	−0.253905
$93$	8.66588	0.898610
$94$	−2.02321	−0.208678
$95$	3.28950	0.337496
$96$	2.52409	0.257614
$97$	3.16042	0.320892	0.160446	−	0.987045i	$-0.448707\pi$
$97$	3.16042	0.320892	0.160446	+	0.987045i	$0.448707\pi$
$98$	−20.0417	−2.02451
$99$	−2.27564	−0.228711
$100$	2.03042	0.203042
$101$	−6.39793	−0.636617	−0.318309	−	0.947987i	$-0.603115\pi$
$101$	−6.39793	−0.636617	−0.318309	+	0.947987i	$0.603115\pi$
$102$	−7.98057	−0.790194
$103$	−12.5519	−1.23678	−0.618389	−	0.785873i	$-0.712214\pi$
$103$	−12.5519	−1.23678	−0.618389	+	0.785873i	$0.712214\pi$
$104$	−3.04400	−0.298489
$105$	19.0601	1.86007
$106$	11.9889	1.16447
$107$	−7.33530	−0.709130	−0.354565	−	0.935031i	$-0.615371\pi$
$107$	−7.33530	−0.709130	−0.354565	+	0.935031i	$0.615371\pi$
$108$	−1.95215	−0.187846
$109$	20.5471	1.96806	0.984029	−	0.178009i	$-0.0569656\pi$
$109$	20.5471	1.96806	0.984029	+	0.178009i	$0.0569656\pi$
$110$	−8.64030	−0.823821
$111$	−7.25751	−0.688853
$112$	−20.5037	−1.93742
$113$	18.9253	1.78034	0.890172	−	0.455625i	$-0.150584\pi$
$113$	18.9253	1.78034	0.890172	+	0.455625i	$0.150584\pi$
$114$	−1.98185	−0.185617
$115$	−22.9664	−2.14163
$116$	0.593730	0.0551265
$117$	1.59719	0.147660
$118$	−17.4389	−1.60539
$119$	18.0432	1.65402
$120$	10.7645	0.982661
$121$	−8.06270	−0.732973
$122$	−4.72745	−0.428003
$123$	3.42645	0.308953
$124$	2.33759	0.209922
$125$	2.70006	0.241501
$126$	9.11811	0.812305
$127$	8.96527	0.795539	0.397770	−	0.917485i	$-0.369784\pi$
$127$	8.96527	0.795539	0.397770	+	0.917485i	$0.369784\pi$
$128$	−13.3453	−1.17956
$129$	−2.98613	−0.262914
$130$	6.06429	0.531874
$131$	−12.9719	−1.13336	−0.566680	−	0.823938i	$-0.691773\pi$
$131$	−12.9719	−1.13336	−0.566680	+	0.823938i	$0.691773\pi$
$132$	0.773073	0.0672874
$133$	4.48074	0.388529
$134$	−7.55605	−0.652744
$135$	−18.4095	−1.58444
$136$	10.1902	0.873803
$137$	11.1761	0.954838	0.477419	−	0.878676i	$-0.341572\pi$
$137$	11.1761	0.954838	0.477419	+	0.878676i	$0.341572\pi$
$138$	13.8367	1.17786
$139$	−10.7896	−0.915165	−0.457582	−	0.889167i	$-0.651284\pi$
$139$	−10.7896	−0.915165	−0.457582	+	0.889167i	$0.651284\pi$
$140$	5.14139	0.434527
$141$	1.70711	0.143765
$142$	−18.1932	−1.52674
$143$	−2.06158	−0.172398
$144$	6.07593	0.506328
$145$	5.59909	0.464979
$146$	−1.09346	−0.0904950
$147$	16.9104	1.39475
$148$	−1.95769	−0.160921
$149$	−3.50925	−0.287489	−0.143744	−	0.989615i	$-0.545914\pi$
$149$	−3.50925	−0.287489	−0.143744	+	0.989615i	$0.545914\pi$
$150$	−11.5360	−0.941907
$151$	−6.14810	−0.500325	−0.250163	−	0.968204i	$-0.580484\pi$
$151$	−6.14810	−0.500325	−0.250163	+	0.968204i	$0.580484\pi$
$152$	2.53058	0.205257
$153$	−5.34680	−0.432263
$154$	−11.7692	−0.948393
$155$	22.0444	1.77065
$156$	−0.542590	−0.0434420
$157$	−20.7226	−1.65385	−0.826923	−	0.562314i	$-0.809911\pi$
$157$	−20.7226	−1.65385	−0.826923	+	0.562314i	$0.809911\pi$
$158$	−16.8107	−1.33739
$159$	−10.1158	−0.802235
$160$	6.42080	0.507609
$161$	−31.2833	−2.46547
$162$	4.98641	0.391769
$163$	6.43331	0.503896	0.251948	−	0.967741i	$-0.418929\pi$
$163$	6.43331	0.503896	0.251948	+	0.967741i	$0.418929\pi$
$164$	0.924274	0.0721737
$165$	7.29036	0.567554
$166$	15.0496	1.16807
$167$	3.91467	0.302926	0.151463	−	0.988463i	$-0.451601\pi$
$167$	3.91467	0.302926	0.151463	+	0.988463i	$0.451601\pi$
$168$	14.6627	1.13125
$169$	−11.5531	−0.888697
$170$	−20.3011	−1.55702
$171$	−1.32779	−0.101539
$172$	−0.805500	−0.0614188
$173$	−19.3692	−1.47261	−0.736307	−	0.676648i	$-0.763432\pi$
$173$	−19.3692	−1.47261	−0.736307	+	0.676648i	$0.763432\pi$
$174$	−3.37332	−0.255731
$175$	26.0815	1.97158
$176$	−7.84254	−0.591154
$177$	14.7143	1.10600
$178$	−12.8225	−0.961084
$179$	−22.1406	−1.65487	−0.827434	−	0.561562i	$-0.810200\pi$
$179$	−22.1406	−1.65487	−0.827434	+	0.561562i	$0.810200\pi$
$180$	−1.52357	−0.113560
$181$	10.4286	0.775151	0.387576	−	0.921838i	$-0.373313\pi$
$181$	10.4286	0.775151	0.387576	+	0.921838i	$0.373313\pi$
$182$	8.26038	0.612300
$183$	3.98884	0.294864
$184$	−17.6678	−1.30249
$185$	−18.4617	−1.35733
$186$	−13.2812	−0.973826
$187$	6.90141	0.504681
$188$	0.460488	0.0335845
$189$	−25.0762	−1.82402
$190$	−5.04144	−0.365745
$191$	2.79169	0.201999	0.101000	−	0.994886i	$-0.467796\pi$
$191$	2.79169	0.201999	0.101000	+	0.994886i	$0.467796\pi$
$192$	7.96633	0.574920
$193$	−14.1495	−1.01850	−0.509250	−	0.860618i	$-0.670077\pi$
$193$	−14.1495	−1.01850	−0.509250	+	0.860618i	$0.670077\pi$
$194$	−4.84362	−0.347751
$195$	−5.11682	−0.366423
$196$	4.56153	0.325823
$197$	12.6690	0.902630	0.451315	−	0.892365i	$-0.350955\pi$
$197$	12.6690	0.902630	0.451315	+	0.892365i	$0.350955\pi$
$198$	3.48762	0.247854
$199$	−1.71644	−0.121675	−0.0608375	−	0.998148i	$-0.519377\pi$
$199$	−1.71644	−0.121675	−0.0608375	+	0.998148i	$0.519377\pi$
$200$	14.7300	1.04157
$201$	6.37552	0.449694
$202$	9.80537	0.689904
$203$	7.62671	0.535290
$204$	1.81640	0.127173
$205$	8.71625	0.608769
$206$	19.2369	1.34030
$207$	9.27029	0.644330
$208$	5.50438	0.381660
$209$	1.71386	0.118550
$210$	−29.2112	−2.01576
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−2.72870	−0.187408
$213$	15.3508	1.05182
$214$	11.2420	0.768486
$215$	−7.59616	−0.518054
$216$	−14.1622	−0.963616
$217$	30.0274	2.03839
$218$	−31.4902	−2.13279
$219$	0.922617	0.0623446
$220$	1.96655	0.132585
$221$	−4.84384	−0.325832
$222$	11.1228	0.746511
$223$	21.0707	1.41100	0.705498	−	0.708712i	$-0.250724\pi$
$223$	21.0707	1.41100	0.705498	+	0.708712i	$0.250724\pi$
$224$	8.74599	0.584366
$225$	−7.72883	−0.515255
$226$	−29.0047	−1.92936
$227$	−29.0305	−1.92682	−0.963411	−	0.268028i	$-0.913628\pi$
$227$	−29.0305	−1.92682	−0.963411	+	0.268028i	$0.913628\pi$
$228$	0.451073	0.0298730
$229$	7.32167	0.483829	0.241915	−	0.970298i	$-0.422225\pi$
$229$	7.32167	0.483829	0.241915	+	0.970298i	$0.422225\pi$
$230$	35.1980	2.32089
$231$	9.93045	0.653376
$232$	4.30732	0.282789
$233$	−8.05581	−0.527754	−0.263877	−	0.964556i	$-0.585001\pi$
$233$	−8.05581	−0.527754	−0.263877	+	0.964556i	$0.585001\pi$
$234$	−2.44783	−0.160019
$235$	4.34257	0.283278
$236$	3.96914	0.258369
$237$	14.1842	0.921364
$238$	−27.6528	−1.79246
$239$	0.923587	0.0597418	0.0298709	−	0.999554i	$-0.490490\pi$
$239$	0.923587	0.0597418	0.0298709	+	0.999554i	$0.490490\pi$
$240$	−19.4651	−1.25647
$241$	13.8987	0.895295	0.447648	−	0.894210i	$-0.352262\pi$
$241$	13.8987	0.895295	0.447648	+	0.894210i	$0.352262\pi$
$242$	12.3568	0.794324
$243$	12.5820	0.807133
$244$	1.07598	0.0688824
$245$	43.0169	2.74825
$246$	−5.25133	−0.334813
$247$	−1.20289	−0.0765380
$248$	16.9585	1.07686
$249$	−12.6983	−0.804719
$250$	−4.13807	−0.261715
$251$	14.5822	0.920423	0.460212	−	0.887809i	$-0.347774\pi$
$251$	14.5822	0.920423	0.460212	+	0.887809i	$0.347774\pi$
$252$	−2.07530	−0.130732
$253$	−11.9657	−0.752276
$254$	−13.7401	−0.862128
$255$	17.1293	1.07268
$256$	8.13183	0.508239
$257$	−18.0809	−1.12785	−0.563927	−	0.825824i	$-0.690710\pi$
$257$	−18.0809	−1.12785	−0.563927	+	0.825824i	$0.690710\pi$
$258$	4.57651	0.284921
$259$	−25.1474	−1.56258
$260$	−1.38025	−0.0855993
$261$	−2.26005	−0.139894
$262$	19.8806	1.22823
$263$	19.3859	1.19538	0.597692	−	0.801726i	$-0.296085\pi$
$263$	19.3859	1.19538	0.597692	+	0.801726i	$0.296085\pi$
$264$	5.60840	0.345173
$265$	−25.7327	−1.58075
$266$	−6.86712	−0.421050
$267$	10.8191	0.662119
$268$	1.71977	0.105052
$269$	−5.27925	−0.321881	−0.160941	−	0.986964i	$-0.551453\pi$
$269$	−5.27925	−0.321881	−0.160941	+	0.986964i	$0.551453\pi$
$270$	28.2141	1.71706
$271$	−31.5323	−1.91545	−0.957724	−	0.287688i	$-0.907113\pi$
$271$	−31.5323	−1.91545	−0.957724	+	0.287688i	$0.907113\pi$
$272$	−18.4267	−1.11728
$273$	−6.96980	−0.421831
$274$	−17.1283	−1.03476
$275$	9.97603	0.601577
$276$	−3.14927	−0.189564
$277$	−0.873104	−0.0524597	−0.0262299	−	0.999656i	$-0.508350\pi$
$277$	−0.873104	−0.0524597	−0.0262299	+	0.999656i	$0.508350\pi$
$278$	16.5360	0.991766
$279$	−8.89811	−0.532716
$280$	37.2991	2.22905
$281$	27.5686	1.64460	0.822302	−	0.569052i	$-0.192690\pi$
$281$	27.5686	1.64460	0.822302	+	0.569052i	$0.192690\pi$
$282$	−2.61629	−0.155798
$283$	−32.9272	−1.95732	−0.978660	−	0.205488i	$-0.934122\pi$
$283$	−32.9272	−1.95732	−0.978660	+	0.205488i	$0.934122\pi$
$284$	4.14082	0.245713
$285$	4.25378	0.251972
$286$	3.15955	0.186828
$287$	11.8727	0.700823
$288$	−2.59173	−0.152719
$289$	−0.784596	−0.0461527
$290$	−8.58109	−0.503899
$291$	4.08686	0.239576
$292$	0.248873	0.0145642
$293$	15.8965	0.928684	0.464342	−	0.885656i	$-0.346291\pi$
$293$	15.8965	0.928684	0.464342	+	0.885656i	$0.346291\pi$
$294$	−25.9166	−1.51149
$295$	37.4305	2.17929
$296$	−14.2024	−0.825498
$297$	−9.59148	−0.556554
$298$	5.37822	0.311552
$299$	8.39825	0.485683
$300$	2.62561	0.151590
$301$	−10.3470	−0.596390
$302$	9.42249	0.542204
$303$	−8.27341	−0.475295
$304$	−4.57596	−0.262450
$305$	10.1469	0.581008
$306$	8.19443	0.468445
$307$	12.4364	0.709786	0.354893	−	0.934907i	$-0.384517\pi$
$307$	12.4364	0.709786	0.354893	+	0.934907i	$0.384517\pi$
$308$	2.67871	0.152633
$309$	−16.2314	−0.923370
$310$	−33.7849	−1.91885
$311$	18.1132	1.02711	0.513553	−	0.858058i	$-0.328329\pi$
$311$	18.1132	1.02711	0.513553	+	0.858058i	$0.328329\pi$
$312$	−3.93632	−0.222850
$313$	−7.34815	−0.415342	−0.207671	−	0.978199i	$-0.566588\pi$
$313$	−7.34815	−0.415342	−0.207671	+	0.978199i	$0.566588\pi$
$314$	31.7592	1.79228
$315$	−19.5708	−1.10269
$316$	3.82615	0.215238
$317$	−6.00935	−0.337519	−0.168759	−	0.985657i	$-0.553976\pi$
$317$	−6.00935	−0.337519	−0.168759	+	0.985657i	$0.553976\pi$
$318$	15.5033	0.869384
$319$	2.91717	0.163330
$320$	20.2648	1.13284
$321$	−9.48556	−0.529432
$322$	47.9444	2.67183
$323$	4.02684	0.224059
$324$	−1.13492	−0.0630510
$325$	−7.00179	−0.388389
$326$	−9.85960	−0.546073
$327$	26.5703	1.46934
$328$	6.70531	0.370239
$329$	5.91516	0.326113
$330$	−11.1731	−0.615060
$331$	−22.0167	−1.21015	−0.605073	−	0.796170i	$-0.706856\pi$
$331$	−22.0167	−1.21015	−0.605073	+	0.796170i	$0.706856\pi$
$332$	−3.42531	−0.187989
$333$	7.45200	0.408367
$334$	−5.99957	−0.328282
$335$	16.2181	0.886090
$336$	−26.5141	−1.44646
$337$	29.4562	1.60458	0.802290	−	0.596935i	$-0.203615\pi$
$337$	29.4562	1.60458	0.802290	+	0.596935i	$0.203615\pi$
$338$	17.7061	0.963082
$339$	24.4731	1.32919
$340$	4.62057	0.250585
$341$	11.4853	0.621963
$342$	2.03496	0.110038
$343$	27.2295	1.47026
$344$	−5.84364	−0.315068
$345$	−29.6987	−1.59893
$346$	29.6850	1.59587
$347$	11.4519	0.614771	0.307386	−	0.951585i	$-0.400546\pi$
$347$	11.4519	0.614771	0.307386	+	0.951585i	$0.400546\pi$
$348$	0.767775	0.0411571
$349$	−1.44914	−0.0775709	−0.0387854	−	0.999248i	$-0.512349\pi$
$349$	−1.44914	−0.0775709	−0.0387854	+	0.999248i	$0.512349\pi$
$350$	−39.9722	−2.13660
$351$	6.73189	0.359322
$352$	3.34529	0.178305
$353$	5.52109	0.293858	0.146929	−	0.989147i	$-0.453061\pi$
$353$	5.52109	0.293858	0.146929	+	0.989147i	$0.453061\pi$
$354$	−22.5510	−1.19857
$355$	39.0495	2.07253
$356$	2.91842	0.154676
$357$	23.3324	1.23488
$358$	33.9324	1.79338
$359$	33.5795	1.77226	0.886130	−	0.463436i	$-0.153384\pi$
$359$	33.5795	1.77226	0.886130	+	0.463436i	$0.153384\pi$
$360$	−11.0530	−0.582543
$361$	1.00000	0.0526316
$362$	−15.9827	−0.840033
$363$	−10.4262	−0.547233
$364$	−1.88008	−0.0985430
$365$	2.34696	0.122846
$366$	−6.11324	−0.319544
$367$	7.34259	0.383280	0.191640	−	0.981465i	$-0.438619\pi$
$367$	7.34259	0.383280	0.191640	+	0.981465i	$0.438619\pi$
$368$	31.9482	1.66541
$369$	−3.51827	−0.183154
$370$	28.2942	1.47095
$371$	−35.0513	−1.81978
$372$	3.02283	0.156727
$373$	−30.7066	−1.58993	−0.794963	−	0.606658i	$-0.792510\pi$
$373$	−30.7066	−1.58993	−0.794963	+	0.606658i	$0.792510\pi$
$374$	−10.5770	−0.546924
$375$	3.49155	0.180303
$376$	3.34069	0.172283
$377$	−2.04745	−0.105449
$378$	38.4314	1.97670
$379$	−4.13800	−0.212555	−0.106277	−	0.994337i	$-0.533893\pi$
$379$	−4.13800	−0.212555	−0.106277	+	0.994337i	$0.533893\pi$
$380$	1.14744	0.0588626
$381$	11.5933	0.593945
$382$	−4.27850	−0.218907
$383$	4.50527	0.230209	0.115104	−	0.993353i	$-0.463280\pi$
$383$	4.50527	0.230209	0.115104	+	0.993353i	$0.463280\pi$
$384$	−17.2573	−0.880656
$385$	25.2612	1.28743
$386$	21.6853	1.10375
$387$	3.06616	0.155862
$388$	1.10242	0.0559668
$389$	−27.4349	−1.39100	−0.695502	−	0.718524i	$-0.744818\pi$
$389$	−27.4349	−1.39100	−0.695502	+	0.718524i	$0.744818\pi$
$390$	7.84197	0.397094
$391$	−28.1143	−1.42180
$392$	33.0924	1.67142
$393$	−16.7745	−0.846160
$394$	−19.4163	−0.978182
$395$	36.0820	1.81548
$396$	−0.793790	−0.0398895
$397$	−18.7082	−0.938939	−0.469469	−	0.882949i	$-0.655555\pi$
$397$	−18.7082	−0.938939	−0.469469	+	0.882949i	$0.655555\pi$
$398$	2.63059	0.131859
$399$	5.79422	0.290074
$400$	−26.6358	−1.33179
$401$	−16.0659	−0.802291	−0.401146	−	0.916014i	$-0.631388\pi$
$401$	−16.0659	−0.802291	−0.401146	+	0.916014i	$0.631388\pi$
$402$	−9.77103	−0.487335
$403$	−8.06108	−0.401551
$404$	−2.23172	−0.111032
$405$	−10.7027	−0.531821
$406$	−11.6886	−0.580095
$407$	−9.61872	−0.476782
$408$	13.1774	0.652376
$409$	26.6869	1.31958	0.659791	−	0.751449i	$-0.270645\pi$
$409$	26.6869	1.31958	0.659791	+	0.751449i	$0.270645\pi$
$410$	−13.3584	−0.659724
$411$	14.4522	0.712877
$412$	−4.37836	−0.215706
$413$	50.9853	2.50882
$414$	−14.2075	−0.698261
$415$	−32.3020	−1.58564
$416$	−2.34793	−0.115117
$417$	−13.9525	−0.683256
$418$	−2.62663	−0.128473
$419$	13.8544	0.676833	0.338417	−	0.940996i	$-0.390109\pi$
$419$	13.8544	0.676833	0.338417	+	0.940996i	$0.390109\pi$
$420$	6.64854	0.324415
$421$	−3.02860	−0.147605	−0.0738025	−	0.997273i	$-0.523513\pi$
$421$	−3.02860	−0.147605	−0.0738025	+	0.997273i	$0.523513\pi$
$422$	1.53259	0.0746051
$423$	−1.75286	−0.0852269
$424$	−19.7959	−0.961372
$425$	23.4394	1.13698
$426$	−23.5264	−1.13986
$427$	13.8214	0.668863
$428$	−2.55870	−0.123679
$429$	−2.66591	−0.128711
$430$	11.6418	0.561416
$431$	−26.3474	−1.26911	−0.634555	−	0.772878i	$-0.718817\pi$
$431$	−26.3474	−1.26911	−0.634555	+	0.772878i	$0.718817\pi$
$432$	25.6091	1.23212
$433$	−15.3051	−0.735518	−0.367759	−	0.929921i	$-0.619875\pi$
$433$	−15.3051	−0.735518	−0.367759	+	0.929921i	$0.619875\pi$
$434$	−46.0195	−2.20901
$435$	7.24040	0.347151
$436$	7.16725	0.343249
$437$	−6.98173	−0.333981
$438$	−1.41399	−0.0675630
$439$	11.1814	0.533660	0.266830	−	0.963744i	$-0.414024\pi$
$439$	11.1814	0.533660	0.266830	+	0.963744i	$0.414024\pi$
$440$	14.2667	0.680138
$441$	−17.3636	−0.826836
$442$	7.42359	0.353104
$443$	−1.73369	−0.0823703	−0.0411851	−	0.999152i	$-0.513113\pi$
$443$	−1.73369	−0.0823703	−0.0411851	+	0.999152i	$0.513113\pi$
$444$	−2.53157	−0.120143
$445$	27.5218	1.30466
$446$	−32.2926	−1.52910
$447$	−4.53794	−0.214637
$448$	27.6034	1.30414
$449$	11.5499	0.545074	0.272537	−	0.962145i	$-0.412137\pi$
$449$	11.5499	0.545074	0.272537	+	0.962145i	$0.412137\pi$
$450$	11.8451	0.558383
$451$	4.54123	0.213838
$452$	6.60153	0.310510
$453$	−7.95035	−0.373540
$454$	44.4917	2.08810
$455$	−17.7298	−0.831188
$456$	3.27238	0.153243
$457$	−30.1431	−1.41004	−0.705018	−	0.709190i	$-0.749061\pi$
$457$	−30.1431	−1.41004	−0.705018	+	0.709190i	$0.749061\pi$
$458$	−11.2211	−0.524327
$459$	−22.5359	−1.05189
$460$	−8.01114	−0.373521
$461$	−40.6648	−1.89395	−0.946974	−	0.321311i	$-0.895877\pi$
$461$	−40.6648	−1.89395	−0.946974	+	0.321311i	$0.895877\pi$
$462$	−15.2193	−0.708065
$463$	−35.0819	−1.63040	−0.815198	−	0.579182i	$-0.803372\pi$
$463$	−35.0819	−1.63040	−0.815198	+	0.579182i	$0.803372\pi$
$464$	−7.78880	−0.361586
$465$	28.5064	1.32195
$466$	12.3462	0.571928
$467$	−3.82168	−0.176846	−0.0884232	−	0.996083i	$-0.528183\pi$
$467$	−3.82168	−0.176846	−0.0884232	+	0.996083i	$0.528183\pi$
$468$	0.557131	0.0257534
$469$	22.0912	1.02008
$470$	−6.65536	−0.306989
$471$	−26.7973	−1.23475
$472$	28.7948	1.32539
$473$	−3.95766	−0.181973
$474$	−21.7385	−0.998484
$475$	5.82081	0.267077
$476$	6.29383	0.288477
$477$	10.3869	0.475583
$478$	−1.41548	−0.0647424
$479$	−2.76166	−0.126184	−0.0630918	−	0.998008i	$-0.520096\pi$
$479$	−2.76166	−0.126184	−0.0630918	+	0.998008i	$0.520096\pi$
$480$	8.30299	0.378978
$481$	6.75100	0.307819
$482$	−21.3010	−0.970233
$483$	−40.4537	−1.84071
$484$	−2.81243	−0.127838
$485$	10.3962	0.472067
$486$	−19.2829	−0.874692
$487$	13.0934	0.593317	0.296658	−	0.954984i	$-0.404128\pi$
$487$	13.0934	0.593317	0.296658	+	0.954984i	$0.404128\pi$
$488$	7.80587	0.353355
$489$	8.31916	0.376205
$490$	−65.9270	−2.97828
$491$	29.4477	1.32896	0.664478	−	0.747307i	$-0.268654\pi$
$491$	29.4477	1.32896	0.664478	+	0.747307i	$0.268654\pi$
$492$	1.19521	0.0538845
$493$	6.85412	0.308694
$494$	1.84353	0.0829444
$495$	−7.48573	−0.336459
$496$	−30.6655	−1.37692
$497$	53.1906	2.38592
$498$	19.4612	0.872076
$499$	24.3487	1.09000	0.544998	−	0.838437i	$-0.316530\pi$
$499$	24.3487	1.09000	0.544998	+	0.838437i	$0.316530\pi$
$500$	0.941835	0.0421201
$501$	5.06222	0.226163
$502$	−22.3485	−0.997464
$503$	−11.8122	−0.526678	−0.263339	−	0.964703i	$-0.584824\pi$
$503$	−11.8122	−0.526678	−0.263339	+	0.964703i	$0.584824\pi$
$504$	−15.0556	−0.670631
$505$	−21.0460	−0.936534
$506$	18.3384	0.815243
$507$	−14.9397	−0.663496
$508$	3.12727	0.138750
$509$	23.5098	1.04205	0.521026	−	0.853541i	$-0.325549\pi$
$509$	23.5098	1.04205	0.521026	+	0.853541i	$0.325549\pi$
$510$	−26.2521	−1.16246
$511$	3.19688	0.141421
$512$	14.2278	0.628785
$513$	−5.59644	−0.247089
$514$	27.7105	1.22226
$515$	−41.2895	−1.81943
$516$	−1.04162	−0.0458549
$517$	2.26251	0.0995052
$518$	38.5405	1.69337
$519$	−25.0471	−1.09944
$520$	−10.0132	−0.439110
$521$	42.0844	1.84375	0.921875	−	0.387487i	$-0.126657\pi$
$521$	42.0844	1.84375	0.921875	+	0.387487i	$0.126657\pi$
$522$	3.46372	0.151603
$523$	−36.7849	−1.60849	−0.804247	−	0.594296i	$-0.797431\pi$
$523$	−36.7849	−1.60849	−0.804247	+	0.594296i	$0.797431\pi$
$524$	−4.52486	−0.197669
$525$	33.7270	1.47197
$526$	−29.7105	−1.29544
$527$	26.9856	1.17551
$528$	−10.1415	−0.441352
$529$	25.7446	1.11933
$530$	39.4375	1.71306
$531$	−15.1086	−0.655659
$532$	1.56297	0.0677634
$533$	−3.18732	−0.138058
$534$	−16.5812	−0.717539
$535$	−24.1295	−1.04321
$536$	12.4764	0.538899
$537$	−28.6309	−1.23551
$538$	8.09090	0.348823
$539$	22.4121	0.965358
$540$	−6.42159	−0.276341
$541$	2.21426	0.0951985	0.0475992	−	0.998867i	$-0.484843\pi$
$541$	2.21426	0.0951985	0.0475992	+	0.998867i	$0.484843\pi$
$542$	48.3259	2.07578
$543$	13.4856	0.578723
$544$	7.86001	0.336995
$545$	67.5898	2.89523
$546$	10.6818	0.457140
$547$	−27.6879	−1.18385	−0.591925	−	0.805993i	$-0.701632\pi$
$547$	−27.6879	−1.18385	−0.591925	+	0.805993i	$0.701632\pi$
$548$	3.89845	0.166533
$549$	−4.09574	−0.174802
$550$	−15.2891	−0.651930
$551$	1.70211	0.0725123
$552$	−22.8469	−0.972429
$553$	49.1485	2.09001
$554$	1.33811	0.0568507
$555$	−23.8736	−1.01338
$556$	−3.76364	−0.159614
$557$	−16.1351	−0.683666	−0.341833	−	0.939761i	$-0.611048\pi$
$557$	−16.1351	−0.683666	−0.341833	+	0.939761i	$0.611048\pi$
$558$	13.6371	0.577305
$559$	2.77773	0.117485
$560$	−67.4469	−2.85015
$561$	8.92449	0.376792
$562$	−42.2512	−1.78226
$563$	5.59506	0.235804	0.117902	−	0.993025i	$-0.462383\pi$
$563$	5.59506	0.235804	0.117902	+	0.993025i	$0.462383\pi$
$564$	0.595474	0.0250740
$565$	62.2548	2.61908
$566$	50.4638	2.12115
$567$	−14.5785	−0.612239
$568$	30.0403	1.26046
$569$	44.2110	1.85342	0.926711	−	0.375775i	$-0.122624\pi$
$569$	44.2110	1.85342	0.926711	+	0.375775i	$0.122624\pi$
$570$	−6.51928	−0.273063
$571$	−30.2322	−1.26518	−0.632588	−	0.774488i	$-0.718007\pi$
$571$	−30.2322	−1.26518	−0.632588	+	0.774488i	$0.718007\pi$
$572$	−0.719120	−0.0300679
$573$	3.61004	0.150812
$574$	−18.1959	−0.759483
$575$	−40.6393	−1.69478
$576$	−8.17981	−0.340826
$577$	−15.4104	−0.641545	−0.320772	−	0.947156i	$-0.603942\pi$
$577$	−15.4104	−0.641545	−0.320772	+	0.947156i	$0.603942\pi$
$578$	1.20246	0.0500158
$579$	−18.2972	−0.760406
$580$	1.95308	0.0810970
$581$	−43.9996	−1.82541
$582$	−6.26347	−0.259629
$583$	−13.4069	−0.555258
$584$	1.80549	0.0747118
$585$	5.25395	0.217224
$586$	−24.3628	−1.00642
$587$	−29.5522	−1.21975	−0.609876	−	0.792497i	$-0.708781\pi$
$587$	−29.5522	−1.21975	−0.609876	+	0.792497i	$0.708781\pi$
$588$	5.89868	0.243258
$589$	6.70143	0.276128
$590$	−57.3654	−2.36170
$591$	16.3828	0.673898
$592$	25.6818	1.05552
$593$	−17.6722	−0.725709	−0.362855	−	0.931846i	$-0.618198\pi$
$593$	−17.6722	−0.725709	−0.362855	+	0.931846i	$0.618198\pi$
$594$	14.6998	0.603139
$595$	59.3531	2.43324
$596$	−1.22410	−0.0501409
$597$	−2.21959	−0.0908418
$598$	−12.8710	−0.526336
$599$	−11.0371	−0.450963	−0.225481	−	0.974247i	$-0.572395\pi$
$599$	−11.0371	−0.450963	−0.225481	+	0.974247i	$0.572395\pi$
$600$	19.0479	0.777629
$601$	−9.30034	−0.379369	−0.189684	−	0.981845i	$-0.560747\pi$
$601$	−9.30034	−0.379369	−0.189684	+	0.981845i	$0.560747\pi$
$602$	15.8576	0.646309
$603$	−6.54637	−0.266589
$604$	−2.14458	−0.0872617
$605$	−26.5222	−1.07828
$606$	12.6797	0.515078
$607$	−23.0022	−0.933631	−0.466815	−	0.884355i	$-0.654599\pi$
$607$	−23.0022	−0.933631	−0.466815	+	0.884355i	$0.654599\pi$
$608$	1.95191	0.0791603
$609$	9.86240	0.399644
$610$	−15.5509	−0.629639
$611$	−1.58797	−0.0642424
$612$	−1.86507	−0.0753910
$613$	−9.57265	−0.386636	−0.193318	−	0.981136i	$-0.561925\pi$
$613$	−9.57265	−0.386636	−0.193318	+	0.981136i	$0.561925\pi$
$614$	−19.0599	−0.769196
$615$	11.2713	0.454503
$616$	19.4331	0.782984
$617$	−7.65788	−0.308295	−0.154147	−	0.988048i	$-0.549263\pi$
$617$	−7.65788	−0.308295	−0.154147	+	0.988048i	$0.549263\pi$
$618$	24.8760	1.00066
$619$	16.3560	0.657403	0.328701	−	0.944434i	$-0.393389\pi$
$619$	16.3560	0.657403	0.328701	+	0.944434i	$0.393389\pi$
$620$	7.68952	0.308818
$621$	39.0728	1.56794
$622$	−27.7601	−1.11308
$623$	37.4883	1.50194
$624$	7.11792	0.284945
$625$	−20.2222	−0.808888
$626$	11.2617	0.450107
$627$	2.21625	0.0885086
$628$	−7.22847	−0.288447
$629$	−22.5999	−0.901117
$630$	29.9940	1.19499
$631$	−27.3351	−1.08819	−0.544096	−	0.839023i	$-0.683127\pi$
$631$	−27.3351	−1.08819	−0.544096	+	0.839023i	$0.683127\pi$
$632$	27.7575	1.10413
$633$	−1.29314	−0.0513976
$634$	9.20985	0.365770
$635$	29.4913	1.17033
$636$	−3.52859	−0.139918
$637$	−15.7302	−0.623253
$638$	−4.47082	−0.177001
$639$	−15.7621	−0.623541
$640$	−43.8992	−1.73527
$641$	18.1905	0.718481	0.359241	−	0.933245i	$-0.383036\pi$
$641$	18.1905	0.718481	0.359241	+	0.933245i	$0.383036\pi$
$642$	14.5374	0.573747
$643$	36.6960	1.44715	0.723575	−	0.690245i	$-0.242497\pi$
$643$	36.6960	1.44715	0.723575	+	0.690245i	$0.242497\pi$
$644$	−10.9122	−0.430003
$645$	−9.82289	−0.386776
$646$	−6.17147	−0.242813
$647$	21.2182	0.834173	0.417086	−	0.908867i	$-0.363051\pi$
$647$	21.2182	0.834173	0.417086	+	0.908867i	$0.363051\pi$
$648$	−8.23346	−0.323441
$649$	19.5016	0.765504
$650$	10.7308	0.420898
$651$	38.8296	1.52185
$652$	2.24407	0.0878844
$653$	24.1995	0.947000	0.473500	−	0.880794i	$-0.342990\pi$
$653$	24.1995	0.947000	0.473500	+	0.880794i	$0.342990\pi$
$654$	−40.7212	−1.59233
$655$	−42.6711	−1.66730
$656$	−12.1250	−0.473402
$657$	−0.947341	−0.0369593
$658$	−9.06549	−0.353409
$659$	−6.02676	−0.234769	−0.117385	−	0.993087i	$-0.537451\pi$
$659$	−6.02676	−0.234769	−0.117385	+	0.993087i	$0.537451\pi$
$660$	2.54303	0.0989871
$661$	16.2277	0.631183	0.315592	−	0.948895i	$-0.397797\pi$
$661$	16.2277	0.631183	0.315592	+	0.948895i	$0.397797\pi$
$662$	33.7425	1.31144
$663$	−6.26375	−0.243264
$664$	−24.8495	−0.964349
$665$	14.7394	0.571569
$666$	−11.4208	−0.442548
$667$	−11.8837	−0.460138
$668$	1.36552	0.0528334
$669$	27.2473	1.05344
$670$	−24.8556	−0.960257
$671$	5.28660	0.204087
$672$	11.3098	0.436284
$673$	11.2545	0.433830	0.216915	−	0.976190i	$-0.430401\pi$
$673$	11.2545	0.433830	0.216915	+	0.976190i	$0.430401\pi$
$674$	−45.1441	−1.73889
$675$	−32.5758	−1.25384
$676$	−4.02994	−0.154998
$677$	−22.9150	−0.880694	−0.440347	−	0.897828i	$-0.645145\pi$
$677$	−22.9150	−0.880694	−0.440347	+	0.897828i	$0.645145\pi$
$678$	−37.5071	−1.44045
$679$	14.1610	0.543450
$680$	33.5207	1.28546
$681$	−37.5405	−1.43855
$682$	−17.6022	−0.674023
$683$	23.7623	0.909238	0.454619	−	0.890686i	$-0.349775\pi$
$683$	23.7623	0.909238	0.454619	+	0.890686i	$0.349775\pi$
$684$	−0.463161	−0.0177094
$685$	36.7638	1.40467
$686$	−41.7316	−1.59332
$687$	9.46793	0.361224
$688$	10.5669	0.402859
$689$	9.40980	0.358485
$690$	45.5159	1.73276
$691$	−23.3944	−0.889967	−0.444983	−	0.895539i	$-0.646790\pi$
$691$	−23.3944	−0.889967	−0.444983	+	0.895539i	$0.646790\pi$
$692$	−6.75636	−0.256838
$693$	−10.1966	−0.387336
$694$	−17.5511	−0.666229
$695$	−35.4925	−1.34631
$696$	5.56996	0.211129
$697$	10.6700	0.404154
$698$	2.22094	0.0840637
$699$	−10.4173	−0.394018
$700$	9.09776	0.343863
$701$	51.8439	1.95812	0.979059	−	0.203578i	$-0.0652572\pi$
$701$	51.8439	1.95812	0.979059	+	0.203578i	$0.0652572\pi$
$702$	−10.3172	−0.389398
$703$	−5.61232	−0.211673
$704$	10.5581	0.397925
$705$	5.61554	0.211494
$706$	−8.46154	−0.318454
$707$	−28.6674	−1.07815
$708$	5.13265	0.192897
$709$	45.2442	1.69918	0.849591	−	0.527442i	$-0.176849\pi$
$709$	45.2442	1.69918	0.849591	+	0.527442i	$0.176849\pi$
$710$	−59.8467	−2.24601
$711$	−14.5643	−0.546205
$712$	21.1722	0.793461
$713$	−46.7876	−1.75221
$714$	−35.7588	−1.33824
$715$	−6.78156	−0.253616
$716$	−7.72310	−0.288626
$717$	1.19433	0.0446029
$718$	−51.4635	−1.92060
$719$	−1.24302	−0.0463567	−0.0231784	−	0.999731i	$-0.507379\pi$
$719$	−1.24302	−0.0463567	−0.0231784	+	0.999731i	$0.507379\pi$
$720$	19.9868	0.744863
$721$	−56.2419	−2.09456
$722$	−1.53259	−0.0570369
$723$	17.9730	0.668422
$724$	3.63770	0.135194
$725$	9.90766	0.367961
$726$	15.9790	0.593037
$727$	17.3718	0.644285	0.322142	−	0.946691i	$-0.395597\pi$
$727$	17.3718	0.644285	0.322142	+	0.946691i	$0.395597\pi$
$728$	−13.6394	−0.505509
$729$	26.0310	0.964111
$730$	−3.59692	−0.133128
$731$	−9.29883	−0.343930
$732$	1.39139	0.0514272
$733$	14.9509	0.552226	0.276113	−	0.961125i	$-0.410954\pi$
$733$	14.9509	0.552226	0.276113	+	0.961125i	$0.410954\pi$
$734$	−11.2531	−0.415361
$735$	55.6268	2.05182
$736$	−13.6277	−0.502324
$737$	8.44976	0.311251
$738$	5.39206	0.198484
$739$	20.4838	0.753508	0.376754	−	0.926313i	$-0.377040\pi$
$739$	20.4838	0.753508	0.376754	+	0.926313i	$0.377040\pi$
$740$	−6.43983	−0.236733
$741$	−1.55550	−0.0571428
$742$	53.7192	1.97209
$743$	−16.3926	−0.601385	−0.300692	−	0.953721i	$-0.597218\pi$
$743$	−16.3926	−0.601385	−0.300692	+	0.953721i	$0.597218\pi$
$744$	21.9297	0.803981
$745$	−11.5437	−0.422927
$746$	47.0605	1.72301
$747$	13.0385	0.477055
$748$	2.40735	0.0880215
$749$	−32.8676	−1.20095
$750$	−5.35110	−0.195395
$751$	−14.9563	−0.545764	−0.272882	−	0.962048i	$-0.587977\pi$
$751$	−14.9563	−0.545764	−0.272882	+	0.962048i	$0.587977\pi$
$752$	−6.04087	−0.220288
$753$	18.8569	0.687182
$754$	3.13789	0.114275
$755$	−20.2242	−0.736033
$756$	−8.74707	−0.318128
$757$	7.64817	0.277977	0.138989	−	0.990294i	$-0.455615\pi$
$757$	7.64817	0.277977	0.138989	+	0.990294i	$0.455615\pi$
$758$	6.34184	0.230346
$759$	−15.4733	−0.561645
$760$	8.32433	0.301955
$761$	39.8487	1.44451	0.722256	−	0.691625i	$-0.243105\pi$
$761$	39.8487	1.44451	0.722256	+	0.691625i	$0.243105\pi$
$762$	−17.7678	−0.643659
$763$	92.0663	3.33302
$764$	0.973797	0.0352307
$765$	−17.5883	−0.635906
$766$	−6.90472	−0.249478
$767$	−13.6874	−0.494223
$768$	10.5156	0.379448
$769$	−46.2457	−1.66766	−0.833830	−	0.552021i	$-0.813857\pi$
$769$	−46.2457	−1.66766	−0.833830	+	0.552021i	$0.813857\pi$
$770$	−38.7149	−1.39519
$771$	−23.3811	−0.842049
$772$	−4.93562	−0.177637
$773$	−19.5738	−0.704022	−0.352011	−	0.935996i	$-0.614502\pi$
$773$	−19.5738	−0.704022	−0.352011	+	0.935996i	$0.614502\pi$
$774$	−4.69915	−0.168907
$775$	39.0078	1.40120
$776$	7.99768	0.287100
$777$	−32.5190	−1.16661
$778$	42.0463	1.50743
$779$	2.64972	0.0949360
$780$	−1.78485	−0.0639079
$781$	20.3451	0.728004
$782$	43.0876	1.54081
$783$	−9.52575	−0.340423
$784$	−59.8400	−2.13714
$785$	−68.1671	−2.43299
$786$	25.7083	0.916986
$787$	2.42023	0.0862718	0.0431359	−	0.999069i	$-0.486265\pi$
$787$	2.42023	0.0862718	0.0431359	+	0.999069i	$0.486265\pi$
$788$	4.41920	0.157428
$789$	25.0686	0.892466
$790$	−55.2987	−1.96744
$791$	84.7994	3.01512
$792$	−5.75869	−0.204626
$793$	−3.71046	−0.131762
$794$	28.6720	1.01753
$795$	−33.2759	−1.18018
$796$	−0.598727	−0.0212213
$797$	20.5088	0.726461	0.363230	−	0.931699i	$-0.381674\pi$
$797$	20.5088	0.726461	0.363230	+	0.931699i	$0.381674\pi$
$798$	−8.88014	−0.314353
$799$	5.31595	0.188065
$800$	11.3617	0.401696
$801$	−11.1090	−0.392519
$802$	24.6223	0.869444
$803$	1.22279	0.0431512
$804$	2.22391	0.0784312
$805$	−102.906	−3.62698
$806$	12.3543	0.435162
$807$	−6.82680	−0.240315
$808$	−16.1904	−0.569577
$809$	−10.2006	−0.358634	−0.179317	−	0.983791i	$-0.557389\pi$
$809$	−10.2006	−0.358634	−0.179317	+	0.983791i	$0.557389\pi$
$810$	16.4028	0.576335
$811$	−17.3564	−0.609465	−0.304733	−	0.952438i	$-0.598567\pi$
$811$	−17.3564	−0.609465	−0.304733	+	0.952438i	$0.598567\pi$
$812$	2.66035	0.0933600
$813$	−40.7756	−1.43006
$814$	14.7415	0.516690
$815$	21.1624	0.741285
$816$	−23.8282	−0.834155
$817$	−2.30921	−0.0807892
$818$	−40.9000	−1.43003
$819$	7.15658	0.250071
$820$	3.04040	0.106175
$821$	5.33961	0.186354	0.0931768	−	0.995650i	$-0.470298\pi$
$821$	5.33961	0.186354	0.0931768	+	0.995650i	$0.470298\pi$
$822$	−22.1493	−0.772546
$823$	26.1794	0.912558	0.456279	−	0.889837i	$-0.349182\pi$
$823$	26.1794	0.912558	0.456279	+	0.889837i	$0.349182\pi$
$824$	−31.7636	−1.10654
$825$	12.9004	0.449134
$826$	−78.1394	−2.71882
$827$	−34.6542	−1.20505	−0.602523	−	0.798102i	$-0.705838\pi$
$827$	−34.6542	−1.20505	−0.602523	+	0.798102i	$0.705838\pi$
$828$	3.23366	0.112378
$829$	−20.5213	−0.712734	−0.356367	−	0.934346i	$-0.615985\pi$
$829$	−20.5213	−0.712734	−0.356367	+	0.934346i	$0.615985\pi$
$830$	49.5055	1.71836
$831$	−1.12904	−0.0391661
$832$	−7.41035	−0.256908
$833$	52.6590	1.82453
$834$	21.3834	0.740446
$835$	12.8773	0.445638
$836$	0.597827	0.0206763
$837$	−37.5041	−1.29633
$838$	−21.2331	−0.733486
$839$	−22.3454	−0.771447	−0.385724	−	0.922614i	$-0.626048\pi$
$839$	−22.3454	−0.771447	−0.385724	+	0.922614i	$0.626048\pi$
$840$	48.2330	1.66420
$841$	−26.1028	−0.900097
$842$	4.64159	0.159960
$843$	35.6500	1.22785
$844$	−0.348820	−0.0120069
$845$	−38.0038	−1.30737
$846$	2.68641	0.0923605
$847$	−36.1268	−1.24133
$848$	35.7963	1.22925
$849$	−42.5794	−1.46132
$850$	−35.9230	−1.23215
$851$	39.1837	1.34320
$852$	5.35466	0.183448
$853$	−3.88273	−0.132942	−0.0664711	−	0.997788i	$-0.521174\pi$
$853$	−3.88273	−0.132942	−0.0664711	+	0.997788i	$0.521174\pi$
$854$	−21.1825	−0.724849
$855$	−4.36777	−0.149375
$856$	−18.5625	−0.634454
$857$	24.1487	0.824905	0.412453	−	0.910979i	$-0.364672\pi$
$857$	24.1487	0.824905	0.412453	+	0.910979i	$0.364672\pi$
$858$	4.08573	0.139485
$859$	−18.6252	−0.635484	−0.317742	−	0.948177i	$-0.602925\pi$
$859$	−18.6252	−0.635484	−0.317742	+	0.948177i	$0.602925\pi$
$860$	−2.64969	−0.0903538
$861$	15.3530	0.523230
$862$	40.3797	1.37534
$863$	−34.9373	−1.18928	−0.594640	−	0.803992i	$-0.702705\pi$
$863$	−34.9373	−1.18928	−0.594640	+	0.803992i	$0.702705\pi$
$864$	−10.9237	−0.371633
$865$	−63.7150	−2.16637
$866$	23.4564	0.797082
$867$	−1.01459	−0.0344573
$868$	10.4741	0.355516
$869$	18.7990	0.637712
$870$	−11.0965	−0.376208
$871$	−5.93056	−0.200949
$872$	51.9961	1.76081
$873$	−4.19638	−0.142026
$874$	10.7001	0.361936
$875$	12.0983	0.408996
$876$	0.321827	0.0108735
$877$	−47.9333	−1.61859	−0.809296	−	0.587401i	$-0.800151\pi$
$877$	−47.9333	−1.61859	−0.809296	+	0.587401i	$0.800151\pi$
$878$	−17.1365	−0.578328
$879$	20.5564	0.693350
$880$	−25.7980	−0.869652
$881$	−19.3298	−0.651236	−0.325618	−	0.945501i	$-0.605572\pi$
$881$	−19.3298	−0.651236	−0.325618	+	0.945501i	$0.605572\pi$
$882$	26.6112	0.896044
$883$	−0.535428	−0.0180186	−0.00900929	−	0.999959i	$-0.502868\pi$
$883$	−0.535428	−0.0180186	−0.00900929	+	0.999959i	$0.502868\pi$
$884$	−1.68963	−0.0568283
$885$	48.4028	1.62704
$886$	2.65704	0.0892649
$887$	33.8741	1.13738	0.568691	−	0.822551i	$-0.307450\pi$
$887$	33.8741	1.13738	0.568691	+	0.822551i	$0.307450\pi$
$888$	−18.3657	−0.616312
$889$	40.1711	1.34729
$890$	−42.1795	−1.41386
$891$	−5.57618	−0.186809
$892$	7.34987	0.246092
$893$	1.32013	0.0441765
$894$	6.95479	0.232603
$895$	−72.8316	−2.43449
$896$	−59.7966	−1.99766
$897$	10.8601	0.362608
$898$	−17.7012	−0.590697
$899$	11.4066	0.380431
$900$	−2.69597	−0.0898657
$901$	−31.5006	−1.04944
$902$	−6.95983	−0.231737
$903$	−13.3801	−0.445261
$904$	47.8919	1.59286
$905$	34.3049	1.14033
$906$	12.1846	0.404806
$907$	−13.1975	−0.438217	−0.219108	−	0.975701i	$-0.570315\pi$
$907$	−13.1975	−0.438217	−0.219108	+	0.975701i	$0.570315\pi$
$908$	−10.1264	−0.336057
$909$	8.49512	0.281765
$910$	27.1725	0.900760
$911$	−33.8070	−1.12008	−0.560038	−	0.828467i	$-0.689213\pi$
$911$	−33.8070	−1.12008	−0.560038	+	0.828467i	$0.689213\pi$
$912$	−5.91736	−0.195943
$913$	−16.8296	−0.556978
$914$	46.1969	1.52806
$915$	13.1213	0.433777
$916$	2.55394	0.0843847
$917$	−58.1237	−1.91941
$918$	34.5382	1.13993
$919$	50.6466	1.67068	0.835339	−	0.549735i	$-0.185271\pi$
$919$	50.6466	1.67068	0.835339	+	0.549735i	$0.185271\pi$
$920$	−58.1182	−1.91610
$921$	16.0821	0.529922
$922$	62.3223	2.05247
$923$	−14.2794	−0.470013
$924$	3.46394	0.113955
$925$	−32.6683	−1.07413
$926$	53.7661	1.76686
$927$	16.6663	0.547394
$928$	3.32236	0.109062
$929$	−48.1689	−1.58037	−0.790186	−	0.612868i	$-0.790016\pi$
$929$	−48.1689	−1.58037	−0.790186	+	0.612868i	$0.790016\pi$
$930$	−43.6885	−1.43260
$931$	13.0770	0.428582
$932$	−2.81003	−0.0920455
$933$	23.4229	0.766832
$934$	5.85706	0.191649
$935$	22.7022	0.742442
$936$	4.04180	0.132110
$937$	−14.8612	−0.485494	−0.242747	−	0.970090i	$-0.578048\pi$
$937$	−14.8612	−0.485494	−0.242747	+	0.970090i	$0.578048\pi$
$938$	−33.8567	−1.10546
$939$	−9.50218	−0.310092
$940$	1.51477	0.0494065
$941$	−22.1416	−0.721795	−0.360897	−	0.932606i	$-0.617530\pi$
$941$	−22.1416	−0.721795	−0.360897	+	0.932606i	$0.617530\pi$
$942$	41.0691	1.33810
$943$	−18.4996	−0.602430
$944$	−52.0689	−1.69470
$945$	−82.4880	−2.68334
$946$	6.06546	0.197205
$947$	−15.8259	−0.514272	−0.257136	−	0.966375i	$-0.582779\pi$
$947$	−15.8259	−0.514272	−0.257136	+	0.966375i	$0.582779\pi$
$948$	4.94774	0.160695
$949$	−0.858226	−0.0278592
$950$	−8.92089	−0.289432
$951$	−7.77093	−0.251990
$952$	45.6597	1.47984
$953$	37.7868	1.22403	0.612016	−	0.790845i	$-0.290359\pi$
$953$	37.7868	1.22403	0.612016	+	0.790845i	$0.290359\pi$
$954$	−15.9188	−0.515390
$955$	9.18326	0.297163
$956$	0.322165	0.0104196
$957$	3.77231	0.121941
$958$	4.23248	0.136745
$959$	50.0772	1.61708
$960$	26.2052	0.845771
$961$	13.9092	0.448684
$962$	−10.3465	−0.333584
$963$	9.73975	0.313859
$964$	4.84815	0.156148
$965$	−46.5447	−1.49833
$966$	61.9987	1.99478
$967$	33.3657	1.07297	0.536484	−	0.843910i	$-0.319752\pi$
$967$	33.3657	1.07297	0.536484	+	0.843910i	$0.319752\pi$
$968$	−20.4033	−0.655786
$969$	5.20726	0.167281
$970$	−15.9331	−0.511580
$971$	5.62153	0.180403	0.0902017	−	0.995924i	$-0.471249\pi$
$971$	5.62153	0.180403	0.0902017	+	0.995924i	$0.471249\pi$
$972$	4.38884	0.140772
$973$	−48.3455	−1.54989
$974$	−20.0667	−0.642979
$975$	−9.05428	−0.289969
$976$	−14.1151	−0.451814
$977$	3.67879	0.117695	0.0588474	−	0.998267i	$-0.481257\pi$
$977$	3.67879	0.117695	0.0588474	+	0.998267i	$0.481257\pi$
$978$	−12.7498	−0.407694
$979$	14.3391	0.458278
$980$	15.0051	0.479322
$981$	−27.2823	−0.871057
$982$	−45.1312	−1.44019
$983$	−44.0997	−1.40656	−0.703282	−	0.710911i	$-0.748283\pi$
$983$	−44.0997	−1.40656	−0.703282	+	0.710911i	$0.748283\pi$
$984$	8.67090	0.276418
$985$	41.6747	1.32787
$986$	−10.5045	−0.334532
$987$	7.64912	0.243474
$988$	−0.419592	−0.0133490
$989$	16.1223	0.512660
$990$	11.4725	0.364621
$991$	2.45638	0.0780294	0.0390147	−	0.999239i	$-0.487578\pi$
$991$	2.45638	0.0780294	0.0390147	+	0.999239i	$0.487578\pi$
$992$	13.0806	0.415309
$993$	−28.4706	−0.903488
$994$	−81.5191	−2.58563
$995$	−5.64622	−0.178997
$996$	−4.42941	−0.140351
$997$	−56.0592	−1.77541	−0.887706	−	0.460410i	$-0.847702\pi$
$997$	−56.0592	−1.77541	−0.887706	+	0.460410i	$0.847702\pi$
$998$	−37.3164	−1.18123
$999$	31.4090	0.993737

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.19	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.19	✓	82	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53259 q^{2} +1.29314 q^{3} +0.348820 q^{4} +3.28950 q^{5} -1.98185 q^{6} +4.48074 q^{7} +2.53058 q^{8} -1.32779 q^{9} +O(q^{10})\)	\(q-1.53259 q^{2} +1.29314 q^{3} +0.348820 q^{4} +3.28950 q^{5} -1.98185 q^{6} +4.48074 q^{7} +2.53058 q^{8} -1.32779 q^{9} -5.04144 q^{10} +1.71386 q^{11} +0.451073 q^{12} -1.20289 q^{13} -6.86712 q^{14} +4.25378 q^{15} -4.57596 q^{16} +4.02684 q^{17} +2.03496 q^{18} +1.00000 q^{19} +1.14744 q^{20} +5.79422 q^{21} -2.62663 q^{22} -6.98173 q^{23} +3.27238 q^{24} +5.82081 q^{25} +1.84353 q^{26} -5.59644 q^{27} +1.56297 q^{28} +1.70211 q^{29} -6.51928 q^{30} +6.70143 q^{31} +1.95191 q^{32} +2.21625 q^{33} -6.17147 q^{34} +14.7394 q^{35} -0.463161 q^{36} -5.61232 q^{37} -1.53259 q^{38} -1.55550 q^{39} +8.32433 q^{40} +2.64972 q^{41} -8.88014 q^{42} -2.30921 q^{43} +0.597827 q^{44} -4.36777 q^{45} +10.7001 q^{46} +1.32013 q^{47} -5.91736 q^{48} +13.0770 q^{49} -8.92089 q^{50} +5.20726 q^{51} -0.419592 q^{52} -7.82267 q^{53} +8.57702 q^{54} +5.63773 q^{55} +11.3388 q^{56} +1.29314 q^{57} -2.60863 q^{58} +11.3788 q^{59} +1.48380 q^{60} +3.08462 q^{61} -10.2705 q^{62} -5.94949 q^{63} +6.16046 q^{64} -3.95690 q^{65} -3.39660 q^{66} +4.93026 q^{67} +1.40464 q^{68} -9.02835 q^{69} -22.5894 q^{70} +11.8709 q^{71} -3.36008 q^{72} +0.713471 q^{73} +8.60137 q^{74} +7.52712 q^{75} +0.348820 q^{76} +7.67934 q^{77} +2.38394 q^{78} +10.9688 q^{79} -15.0526 q^{80} -3.25359 q^{81} -4.06092 q^{82} -9.81972 q^{83} +2.02114 q^{84} +13.2463 q^{85} +3.53907 q^{86} +2.20106 q^{87} +4.33704 q^{88} +8.36655 q^{89} +6.69399 q^{90} -5.38983 q^{91} -2.43537 q^{92} +8.66588 q^{93} -2.02321 q^{94} +3.28950 q^{95} +2.52409 q^{96} +3.16042 q^{97} -20.0417 q^{98} -2.27564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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