LMFDB - Embedded newform 8047.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$151$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73460 q^{2} +0.234090 q^{3} +5.47804 q^{4} -4.22814 q^{5} -0.640142 q^{6} +4.15879 q^{7} -9.51105 q^{8} -2.94520 q^{9} +O(q^{10})$	\(q-2.73460 q^{2} +0.234090 q^{3} +5.47804 q^{4} -4.22814 q^{5} -0.640142 q^{6} +4.15879 q^{7} -9.51105 q^{8} -2.94520 q^{9} +11.5623 q^{10} -3.89432 q^{11} +1.28235 q^{12} -1.00000 q^{13} -11.3726 q^{14} -0.989765 q^{15} +15.0529 q^{16} -6.50115 q^{17} +8.05395 q^{18} -4.84875 q^{19} -23.1619 q^{20} +0.973530 q^{21} +10.6494 q^{22} +2.23824 q^{23} -2.22644 q^{24} +12.8772 q^{25} +2.73460 q^{26} -1.39171 q^{27} +22.7820 q^{28} +8.02439 q^{29} +2.70661 q^{30} -4.47906 q^{31} -22.1414 q^{32} -0.911621 q^{33} +17.7781 q^{34} -17.5839 q^{35} -16.1339 q^{36} -4.17205 q^{37} +13.2594 q^{38} -0.234090 q^{39} +40.2141 q^{40} -2.71397 q^{41} -2.66222 q^{42} -1.51622 q^{43} -21.3333 q^{44} +12.4527 q^{45} -6.12068 q^{46} +10.2171 q^{47} +3.52372 q^{48} +10.2955 q^{49} -35.2139 q^{50} -1.52185 q^{51} -5.47804 q^{52} +8.58380 q^{53} +3.80578 q^{54} +16.4657 q^{55} -39.5545 q^{56} -1.13504 q^{57} -21.9435 q^{58} +4.14308 q^{59} -5.42197 q^{60} +9.64045 q^{61} +12.2484 q^{62} -12.2485 q^{63} +30.4423 q^{64} +4.22814 q^{65} +2.49292 q^{66} +8.00612 q^{67} -35.6136 q^{68} +0.523949 q^{69} +48.0851 q^{70} +11.7163 q^{71} +28.0120 q^{72} -3.84619 q^{73} +11.4089 q^{74} +3.01442 q^{75} -26.5617 q^{76} -16.1957 q^{77} +0.640142 q^{78} -0.883934 q^{79} -63.6456 q^{80} +8.50982 q^{81} +7.42163 q^{82} +15.0766 q^{83} +5.33304 q^{84} +27.4878 q^{85} +4.14625 q^{86} +1.87843 q^{87} +37.0391 q^{88} -12.4600 q^{89} -34.0532 q^{90} -4.15879 q^{91} +12.2612 q^{92} -1.04850 q^{93} -27.9398 q^{94} +20.5012 q^{95} -5.18309 q^{96} +3.16697 q^{97} -28.1541 q^{98} +11.4696 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) $ 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * 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 $

$2$	−2.73460	−1.93365	−0.966827	−	0.255431i	$-0.917783\pi$
$2$	−2.73460	−1.93365	−0.966827	+	0.255431i	$0.917783\pi$
$3$	0.234090	0.135152	0.0675759	−	0.997714i	$-0.478474\pi$
$3$	0.234090	0.135152	0.0675759	+	0.997714i	$0.478474\pi$
$4$	5.47804	2.73902
$5$	−4.22814	−1.89088	−0.945441	−	0.325793i	$-0.894369\pi$
$5$	−4.22814	−1.89088	−0.945441	+	0.325793i	$0.894369\pi$
$6$	−0.640142	−0.261337
$7$	4.15879	1.57187	0.785937	−	0.618307i	$-0.212181\pi$
$7$	4.15879	1.57187	0.785937	+	0.618307i	$0.212181\pi$
$8$	−9.51105	−3.36267
$9$	−2.94520	−0.981734
$10$	11.5623	3.65631
$11$	−3.89432	−1.17418	−0.587091	−	0.809521i	$-0.699727\pi$
$11$	−3.89432	−1.17418	−0.587091	+	0.809521i	$0.699727\pi$
$12$	1.28235	0.370184
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−11.3726	−3.03946
$15$	−0.989765	−0.255556
$16$	15.0529	3.76321
$17$	−6.50115	−1.57676	−0.788380	−	0.615188i	$-0.789080\pi$
$17$	−6.50115	−1.57676	−0.788380	+	0.615188i	$0.789080\pi$
$18$	8.05395	1.89833
$19$	−4.84875	−1.11238	−0.556190	−	0.831055i	$-0.687737\pi$
$19$	−4.84875	−1.11238	−0.556190	+	0.831055i	$0.687737\pi$
$20$	−23.1619	−5.17917
$21$	0.973530	0.212442
$22$	10.6494	2.27046
$23$	2.23824	0.466705	0.233352	−	0.972392i	$-0.425030\pi$
$23$	2.23824	0.466705	0.233352	+	0.972392i	$0.425030\pi$
$24$	−2.22644	−0.454471
$25$	12.8772	2.57544
$26$	2.73460	0.536299
$27$	−1.39171	−0.267835
$28$	22.7820	4.30539
$29$	8.02439	1.49009	0.745045	−	0.667014i	$-0.232428\pi$
$29$	8.02439	1.49009	0.745045	+	0.667014i	$0.232428\pi$
$30$	2.70661	0.494158
$31$	−4.47906	−0.804463	−0.402232	−	0.915538i	$-0.631765\pi$
$31$	−4.47906	−0.804463	−0.402232	+	0.915538i	$0.631765\pi$
$32$	−22.1414	−3.91409
$33$	−0.911621	−0.158693
$34$	17.7781	3.04891
$35$	−17.5839	−2.97223
$36$	−16.1339	−2.68899
$37$	−4.17205	−0.685881	−0.342940	−	0.939357i	$-0.611423\pi$
$37$	−4.17205	−0.685881	−0.342940	+	0.939357i	$0.611423\pi$
$38$	13.2594	2.15096
$39$	−0.234090	−0.0374844
$40$	40.2141	6.35840
$41$	−2.71397	−0.423851	−0.211925	−	0.977286i	$-0.567973\pi$
$41$	−2.71397	−0.423851	−0.211925	+	0.977286i	$0.567973\pi$
$42$	−2.66222	−0.410789
$43$	−1.51622	−0.231221	−0.115611	−	0.993295i	$-0.536882\pi$
$43$	−1.51622	−0.231221	−0.115611	+	0.993295i	$0.536882\pi$
$44$	−21.3333	−3.21611
$45$	12.4527	1.85634
$46$	−6.12068	−0.902446
$47$	10.2171	1.49032	0.745161	−	0.666884i	$-0.232372\pi$
$47$	10.2171	1.49032	0.745161	+	0.666884i	$0.232372\pi$
$48$	3.52372	0.508605
$49$	10.2955	1.47079
$50$	−35.2139	−4.98000
$51$	−1.52185	−0.213102
$52$	−5.47804	−0.759668
$53$	8.58380	1.17908	0.589538	−	0.807741i	$-0.299310\pi$
$53$	8.58380	1.17908	0.589538	+	0.807741i	$0.299310\pi$
$54$	3.80578	0.517901
$55$	16.4657	2.22024
$56$	−39.5545	−5.28569
$57$	−1.13504	−0.150340
$58$	−21.9435	−2.88132
$59$	4.14308	0.539383	0.269691	−	0.962947i	$-0.413078\pi$
$59$	4.14308	0.539383	0.269691	+	0.962947i	$0.413078\pi$
$60$	−5.42197	−0.699974
$61$	9.64045	1.23433	0.617167	−	0.786833i	$-0.288280\pi$
$61$	9.64045	1.23433	0.617167	+	0.786833i	$0.288280\pi$
$62$	12.2484	1.55555
$63$	−12.2485	−1.54316
$64$	30.4423	3.80528
$65$	4.22814	0.524436
$66$	2.49292	0.306857
$67$	8.00612	0.978104	0.489052	−	0.872255i	$-0.337343\pi$
$67$	8.00612	0.978104	0.489052	+	0.872255i	$0.337343\pi$
$68$	−35.6136	−4.31878
$69$	0.523949	0.0630760
$70$	48.0851	5.74726
$71$	11.7163	1.39047	0.695237	−	0.718780i	$-0.255299\pi$
$71$	11.7163	1.39047	0.695237	+	0.718780i	$0.255299\pi$
$72$	28.0120	3.30124
$73$	−3.84619	−0.450163	−0.225081	−	0.974340i	$-0.572265\pi$
$73$	−3.84619	−0.450163	−0.225081	+	0.974340i	$0.572265\pi$
$74$	11.4089	1.32626
$75$	3.01442	0.348075
$76$	−26.5617	−3.04683
$77$	−16.1957	−1.84567
$78$	0.640142	0.0724819
$79$	−0.883934	−0.0994503	−0.0497252	−	0.998763i	$-0.515835\pi$
$79$	−0.883934	−0.0994503	−0.0497252	+	0.998763i	$0.515835\pi$
$80$	−63.6456	−7.11579
$81$	8.50982	0.945536
$82$	7.42163	0.819581
$83$	15.0766	1.65487	0.827434	−	0.561563i	$-0.189800\pi$
$83$	15.0766	1.65487	0.827434	+	0.561563i	$0.189800\pi$
$84$	5.33304	0.581882
$85$	27.4878	2.98147
$86$	4.14625	0.447102
$87$	1.87843	0.201389
$88$	37.0391	3.94838
$89$	−12.4600	−1.32076	−0.660379	−	0.750932i	$-0.729604\pi$
$89$	−12.4600	−1.32076	−0.660379	+	0.750932i	$0.729604\pi$
$90$	−34.0532	−3.58953
$91$	−4.15879	−0.435959
$92$	12.2612	1.27831
$93$	−1.04850	−0.108725
$94$	−27.9398	−2.88177
$95$	20.5012	2.10338
$96$	−5.18309	−0.528997
$97$	3.16697	0.321557	0.160779	−	0.986991i	$-0.448600\pi$
$97$	3.16697	0.321557	0.160779	+	0.986991i	$0.448600\pi$
$98$	−28.1541	−2.84399
$99$	11.4696	1.15273
$100$	70.5417	7.05417
$101$	−1.32570	−0.131912	−0.0659561	−	0.997823i	$-0.521010\pi$
$101$	−1.32570	−0.131912	−0.0659561	+	0.997823i	$0.521010\pi$
$102$	4.16166	0.412066
$103$	−4.09674	−0.403664	−0.201832	−	0.979420i	$-0.564690\pi$
$103$	−4.09674	−0.403664	−0.201832	+	0.979420i	$0.564690\pi$
$104$	9.51105	0.932636
$105$	−4.11622	−0.401702
$106$	−23.4733	−2.27992
$107$	14.8043	1.43118	0.715592	−	0.698519i	$-0.246157\pi$
$107$	14.8043	1.43118	0.715592	+	0.698519i	$0.246157\pi$
$108$	−7.62385	−0.733606
$109$	−5.87672	−0.562887	−0.281444	−	0.959578i	$-0.590813\pi$
$109$	−5.87672	−0.562887	−0.281444	+	0.959578i	$0.590813\pi$
$110$	−45.0272	−4.29318
$111$	−0.976635	−0.0926981
$112$	62.6016	5.91530
$113$	1.91412	0.180065	0.0900327	−	0.995939i	$-0.471303\pi$
$113$	1.91412	0.180065	0.0900327	+	0.995939i	$0.471303\pi$
$114$	3.10389	0.290706
$115$	−9.46358	−0.882484
$116$	43.9579	4.08139
$117$	2.94520	0.272284
$118$	−11.3297	−1.04298
$119$	−27.0369	−2.47847
$120$	9.41371	0.859350
$121$	4.16574	0.378704
$122$	−26.3628	−2.38677
$123$	−0.635313	−0.0572843
$124$	−24.5365	−2.20344
$125$	−33.3058	−2.97896
$126$	33.4947	2.98394
$127$	−16.1463	−1.43276	−0.716378	−	0.697713i	$-0.754201\pi$
$127$	−16.1463	−1.43276	−0.716378	+	0.697713i	$0.754201\pi$
$128$	−38.9646	−3.44402
$129$	−0.354931	−0.0312500
$130$	−11.5623	−1.01408
$131$	−3.12904	−0.273386	−0.136693	−	0.990613i	$-0.543647\pi$
$131$	−3.12904	−0.273386	−0.136693	+	0.990613i	$0.543647\pi$
$132$	−4.99390	−0.434663
$133$	−20.1649	−1.74852
$134$	−21.8936	−1.89132
$135$	5.88435	0.506445
$136$	61.8328	5.30212
$137$	−10.7189	−0.915778	−0.457889	−	0.889009i	$-0.651394\pi$
$137$	−10.7189	−0.915778	−0.457889	+	0.889009i	$0.651394\pi$
$138$	−1.43279	−0.121967
$139$	19.1022	1.62022	0.810112	−	0.586275i	$-0.199406\pi$
$139$	19.1022	1.62022	0.810112	+	0.586275i	$0.199406\pi$
$140$	−96.3255	−8.14099
$141$	2.39173	0.201420
$142$	−32.0395	−2.68870
$143$	3.89432	0.325660
$144$	−44.3337	−3.69447
$145$	−33.9282	−2.81759
$146$	10.5178	0.870459
$147$	2.41008	0.198780
$148$	−22.8547	−1.87864
$149$	−1.78905	−0.146565	−0.0732823	−	0.997311i	$-0.523347\pi$
$149$	−1.78905	−0.146565	−0.0732823	+	0.997311i	$0.523347\pi$
$150$	−8.24323	−0.673057
$151$	−5.55287	−0.451886	−0.225943	−	0.974141i	$-0.572546\pi$
$151$	−5.55287	−0.451886	−0.225943	+	0.974141i	$0.572546\pi$
$152$	46.1167	3.74056
$153$	19.1472	1.54796
$154$	44.2886	3.56888
$155$	18.9381	1.52114
$156$	−1.28235	−0.102671
$157$	−12.6723	−1.01136	−0.505680	−	0.862721i	$-0.668759\pi$
$157$	−12.6723	−1.01136	−0.505680	+	0.862721i	$0.668759\pi$
$158$	2.41721	0.192303
$159$	2.00938	0.159354
$160$	93.6171	7.40108
$161$	9.30835	0.733601
$162$	−23.2710	−1.82834
$163$	−2.54104	−0.199030	−0.0995150	−	0.995036i	$-0.531729\pi$
$163$	−2.54104	−0.199030	−0.0995150	+	0.995036i	$0.531729\pi$
$164$	−14.8672	−1.16094
$165$	3.85446	0.300070
$166$	−41.2284	−3.19994
$167$	17.1829	1.32966	0.664828	−	0.746996i	$-0.268505\pi$
$167$	17.1829	1.32966	0.664828	+	0.746996i	$0.268505\pi$
$168$	−9.25930	−0.714370
$169$	1.00000	0.0769231
$170$	−75.1681	−5.76513
$171$	14.2806	1.09206
$172$	−8.30590	−0.633319
$173$	−1.48588	−0.112969	−0.0564847	−	0.998403i	$-0.517989\pi$
$173$	−1.48588	−0.112969	−0.0564847	+	0.998403i	$0.517989\pi$
$174$	−5.13675	−0.389416
$175$	53.5534	4.04826
$176$	−58.6206	−4.41870
$177$	0.969853	0.0728986
$178$	34.0731	2.55389
$179$	−13.4131	−1.00254	−0.501270	−	0.865291i	$-0.667134\pi$
$179$	−13.4131	−1.00254	−0.501270	+	0.865291i	$0.667134\pi$
$180$	68.2166	5.08456
$181$	15.6013	1.15964	0.579818	−	0.814746i	$-0.303124\pi$
$181$	15.6013	1.15964	0.579818	+	0.814746i	$0.303124\pi$
$182$	11.3726	0.842995
$183$	2.25673	0.166822
$184$	−21.2880	−1.56937
$185$	17.6400	1.29692
$186$	2.86724	0.210236
$187$	25.3176	1.85140
$188$	55.9699	4.08202
$189$	−5.78783	−0.421003
$190$	−56.0626	−4.06721
$191$	2.63680	0.190792	0.0953962	−	0.995439i	$-0.469588\pi$
$191$	2.63680	0.190792	0.0953962	+	0.995439i	$0.469588\pi$
$192$	7.12623	0.514291
$193$	20.4833	1.47442	0.737210	−	0.675664i	$-0.236143\pi$
$193$	20.4833	1.47442	0.737210	+	0.675664i	$0.236143\pi$
$194$	−8.66040	−0.621780
$195$	0.989765	0.0708786
$196$	56.3992	4.02852
$197$	−6.05693	−0.431539	−0.215769	−	0.976444i	$-0.569226\pi$
$197$	−6.05693	−0.431539	−0.215769	+	0.976444i	$0.569226\pi$
$198$	−31.3647	−2.22899
$199$	−23.7608	−1.68436	−0.842180	−	0.539196i	$-0.818728\pi$
$199$	−23.7608	−1.68436	−0.842180	+	0.539196i	$0.818728\pi$
$200$	−122.476	−8.66033
$201$	1.87415	0.132193
$202$	3.62526	0.255073
$203$	33.3717	2.34223
$204$	−8.33678	−0.583691
$205$	11.4751	0.801452
$206$	11.2030	0.780547
$207$	−6.59206	−0.458180
$208$	−15.0529	−1.04373
$209$	18.8826	1.30614
$210$	11.2562	0.776753
$211$	−5.00277	−0.344405	−0.172203	−	0.985062i	$-0.555088\pi$
$211$	−5.00277	−0.344405	−0.172203	+	0.985062i	$0.555088\pi$
$212$	47.0224	3.22951
$213$	2.74268	0.187925
$214$	−40.4838	−2.76741
$215$	6.41078	0.437212
$216$	13.2366	0.900640
$217$	−18.6275	−1.26451
$218$	16.0705	1.08843
$219$	−0.900354	−0.0608403
$220$	90.2000	6.08128
$221$	6.50115	0.437315
$222$	2.67071	0.179246
$223$	−29.4110	−1.96950	−0.984752	−	0.173964i	$-0.944342\pi$
$223$	−29.4110	−1.96950	−0.984752	+	0.173964i	$0.944342\pi$
$224$	−92.0815	−6.15246
$225$	−37.9259	−2.52839
$226$	−5.23436	−0.348184
$227$	12.2851	0.815393	0.407696	−	0.913118i	$-0.366332\pi$
$227$	12.2851	0.815393	0.407696	+	0.913118i	$0.366332\pi$
$228$	−6.21782	−0.411785
$229$	8.90938	0.588749	0.294374	−	0.955690i	$-0.404889\pi$
$229$	8.90938	0.588749	0.294374	+	0.955690i	$0.404889\pi$
$230$	25.8791	1.70642
$231$	−3.79124	−0.249445
$232$	−76.3204	−5.01068
$233$	−23.9620	−1.56980	−0.784902	−	0.619620i	$-0.787287\pi$
$233$	−23.9620	−1.56980	−0.784902	+	0.619620i	$0.787287\pi$
$234$	−8.05395	−0.526503
$235$	−43.1995	−2.81802
$236$	22.6960	1.47738
$237$	−0.206920	−0.0134409
$238$	73.9351	4.79250
$239$	−2.34230	−0.151511	−0.0757553	−	0.997126i	$-0.524137\pi$
$239$	−2.34230	−0.151511	−0.0757553	+	0.997126i	$0.524137\pi$
$240$	−14.8988	−0.961713
$241$	−24.8698	−1.60201	−0.801003	−	0.598660i	$-0.795700\pi$
$241$	−24.8698	−1.60201	−0.801003	+	0.598660i	$0.795700\pi$
$242$	−11.3916	−0.732282
$243$	6.16720	0.395626
$244$	52.8108	3.38086
$245$	−43.5309	−2.78109
$246$	1.73733	0.110768
$247$	4.84875	0.308519
$248$	42.6006	2.70514
$249$	3.52927	0.223659
$250$	91.0781	5.76029
$251$	−4.43079	−0.279669	−0.139835	−	0.990175i	$-0.544657\pi$
$251$	−4.43079	−0.279669	−0.139835	+	0.990175i	$0.544657\pi$
$252$	−67.0976	−4.22675
$253$	−8.71641	−0.547996
$254$	44.1538	2.77045
$255$	6.43461	0.402951
$256$	45.6681	2.85426
$257$	−3.00546	−0.187475	−0.0937377	−	0.995597i	$-0.529881\pi$
$257$	−3.00546	−0.187475	−0.0937377	+	0.995597i	$0.529881\pi$
$258$	0.970595	0.0604266
$259$	−17.3507	−1.07812
$260$	23.1619	1.43644
$261$	−23.6334	−1.46287
$262$	8.55668	0.528633
$263$	−17.6697	−1.08956	−0.544779	−	0.838580i	$-0.683387\pi$
$263$	−17.6697	−1.08956	−0.544779	+	0.838580i	$0.683387\pi$
$264$	8.67048	0.533631
$265$	−36.2935	−2.22949
$266$	55.1430	3.38104
$267$	−2.91676	−0.178503
$268$	43.8579	2.67905
$269$	−18.4722	−1.12627	−0.563136	−	0.826364i	$-0.690405\pi$
$269$	−18.4722	−1.12627	−0.563136	+	0.826364i	$0.690405\pi$
$270$	−16.0914	−0.979289
$271$	7.04794	0.428132	0.214066	−	0.976819i	$-0.431329\pi$
$271$	7.04794	0.428132	0.214066	+	0.976819i	$0.431329\pi$
$272$	−97.8609	−5.93369
$273$	−0.973530	−0.0589207
$274$	29.3119	1.77080
$275$	−50.1479	−3.02403
$276$	2.87021	0.172766
$277$	−16.5894	−0.996762	−0.498381	−	0.866958i	$-0.666072\pi$
$277$	−16.5894	−0.996762	−0.498381	+	0.866958i	$0.666072\pi$
$278$	−52.2368	−3.13296
$279$	13.1917	0.789769
$280$	167.242	9.99461
$281$	−15.8660	−0.946488	−0.473244	−	0.880931i	$-0.656917\pi$
$281$	−15.8660	−0.946488	−0.473244	+	0.880931i	$0.656917\pi$
$282$	−6.54042	−0.389477
$283$	26.9656	1.60294	0.801469	−	0.598036i	$-0.204052\pi$
$283$	26.9656	1.60294	0.801469	+	0.598036i	$0.204052\pi$
$284$	64.1826	3.80854
$285$	4.79913	0.284276
$286$	−10.6494	−0.629713
$287$	−11.2868	−0.666240
$288$	65.2110	3.84259
$289$	25.2650	1.48618
$290$	92.7802	5.44824
$291$	0.741356	0.0434590
$292$	−21.0696	−1.23300
$293$	12.5180	0.731307	0.365653	−	0.930751i	$-0.380846\pi$
$293$	12.5180	0.731307	0.365653	+	0.930751i	$0.380846\pi$
$294$	−6.59059	−0.384371
$295$	−17.5175	−1.01991
$296$	39.6806	2.30639
$297$	5.41977	0.314487
$298$	4.89233	0.283405
$299$	−2.23824	−0.129441
$300$	16.5131	0.953384
$301$	−6.30563	−0.363450
$302$	15.1849	0.873792
$303$	−0.310333	−0.0178282
$304$	−72.9876	−4.18612
$305$	−40.7612	−2.33398
$306$	−52.3600	−2.99322
$307$	−10.9743	−0.626334	−0.313167	−	0.949698i	$-0.601390\pi$
$307$	−10.9743	−0.626334	−0.313167	+	0.949698i	$0.601390\pi$
$308$	−88.7205	−5.05532
$309$	−0.959006	−0.0545560
$310$	−51.7881	−2.94137
$311$	13.4107	0.760448	0.380224	−	0.924894i	$-0.375847\pi$
$311$	13.4107	0.760448	0.380224	+	0.924894i	$0.375847\pi$
$312$	2.22644	0.126047
$313$	20.4776	1.15746	0.578732	−	0.815518i	$-0.303548\pi$
$313$	20.4776	1.15746	0.578732	+	0.815518i	$0.303548\pi$
$314$	34.6537	1.95562
$315$	51.7883	2.91794
$316$	−4.84223	−0.272396
$317$	3.98598	0.223875	0.111938	−	0.993715i	$-0.464294\pi$
$317$	3.98598	0.223875	0.111938	+	0.993715i	$0.464294\pi$
$318$	−5.49485	−0.308136
$319$	−31.2495	−1.74964
$320$	−128.714	−7.19535
$321$	3.46553	0.193427
$322$	−25.4546	−1.41853
$323$	31.5225	1.75396
$324$	46.6171	2.58984
$325$	−12.8772	−0.714297
$326$	6.94874	0.384855
$327$	−1.37568	−0.0760753
$328$	25.8127	1.42527
$329$	42.4909	2.34260
$330$	−10.5404	−0.580231
$331$	−29.2045	−1.60523	−0.802613	−	0.596500i	$-0.796558\pi$
$331$	−29.2045	−1.60523	−0.802613	+	0.596500i	$0.796558\pi$
$332$	82.5901	4.53272
$333$	12.2875	0.673352
$334$	−46.9885	−2.57110
$335$	−33.8510	−1.84948
$336$	14.6544	0.799463
$337$	31.7813	1.73124	0.865619	−	0.500703i	$-0.166925\pi$
$337$	31.7813	1.73124	0.865619	+	0.500703i	$0.166925\pi$
$338$	−2.73460	−0.148743
$339$	0.448077	0.0243362
$340$	150.579	8.16631
$341$	17.4429	0.944586
$342$	−39.0516	−2.11167
$343$	13.7053	0.740018
$344$	14.4208	0.777519
$345$	−2.21533	−0.119269
$346$	4.06329	0.218444
$347$	0.550483	0.0295515	0.0147757	−	0.999891i	$-0.495297\pi$
$347$	0.550483	0.0295515	0.0147757	+	0.999891i	$0.495297\pi$
$348$	10.2901	0.551607
$349$	−4.95430	−0.265197	−0.132599	−	0.991170i	$-0.542332\pi$
$349$	−4.95430	−0.265197	−0.132599	+	0.991170i	$0.542332\pi$
$350$	−146.447	−7.82794
$351$	1.39171	0.0742841
$352$	86.2259	4.59585
$353$	−29.6478	−1.57799	−0.788996	−	0.614398i	$-0.789399\pi$
$353$	−29.6478	−1.57799	−0.788996	+	0.614398i	$0.789399\pi$
$354$	−2.65216	−0.140961
$355$	−49.5384	−2.62922
$356$	−68.2564	−3.61758
$357$	−6.32907	−0.334970
$358$	36.6794	1.93857
$359$	−23.3468	−1.23220	−0.616099	−	0.787669i	$-0.711288\pi$
$359$	−23.3468	−1.23220	−0.616099	+	0.787669i	$0.711288\pi$
$360$	−118.439	−6.24226
$361$	4.51040	0.237389
$362$	−42.6633	−2.24233
$363$	0.975158	0.0511825
$364$	−22.7820	−1.19410
$365$	16.2622	0.851204
$366$	−6.17126	−0.322577
$367$	14.5411	0.759039	0.379519	−	0.925184i	$-0.376089\pi$
$367$	14.5411	0.759039	0.379519	+	0.925184i	$0.376089\pi$
$368$	33.6919	1.75631
$369$	7.99319	0.416109
$370$	−48.2384	−2.50780
$371$	35.6982	1.85336
$372$	−5.74374	−0.297799
$373$	32.1364	1.66396	0.831981	−	0.554804i	$-0.187207\pi$
$373$	32.1364	1.66396	0.831981	+	0.554804i	$0.187207\pi$
$374$	−69.2335	−3.57998
$375$	−7.79656	−0.402612
$376$	−97.1757	−5.01146
$377$	−8.02439	−0.413277
$378$	15.8274	0.814074
$379$	−6.99860	−0.359494	−0.179747	−	0.983713i	$-0.557528\pi$
$379$	−6.99860	−0.359494	−0.179747	+	0.983713i	$0.557528\pi$
$380$	112.306	5.76120
$381$	−3.77969	−0.193640
$382$	−7.21060	−0.368927
$383$	−29.0115	−1.48242	−0.741209	−	0.671274i	$-0.765747\pi$
$383$	−29.0115	−1.48242	−0.741209	+	0.671274i	$0.765747\pi$
$384$	−9.12122	−0.465465
$385$	68.4775	3.48994
$386$	−56.0136	−2.85102
$387$	4.46557	0.226998
$388$	17.3488	0.880751
$389$	28.2875	1.43423	0.717117	−	0.696953i	$-0.245461\pi$
$389$	28.2875	1.43423	0.717117	+	0.696953i	$0.245461\pi$
$390$	−2.70661	−0.137055
$391$	−14.5511	−0.735882
$392$	−97.9212	−4.94577
$393$	−0.732477	−0.0369486
$394$	16.5633	0.834447
$395$	3.73740	0.188049
$396$	62.8307	3.15736
$397$	0.937514	0.0470525	0.0235262	−	0.999723i	$-0.492511\pi$
$397$	0.937514	0.0470525	0.0235262	+	0.999723i	$0.492511\pi$
$398$	64.9764	3.25697
$399$	−4.72041	−0.236316
$400$	193.838	9.69191
$401$	1.09027	0.0544453	0.0272227	−	0.999629i	$-0.491334\pi$
$401$	1.09027	0.0544453	0.0272227	+	0.999629i	$0.491334\pi$
$402$	−5.12506	−0.255615
$403$	4.47906	0.223118
$404$	−7.26225	−0.361310
$405$	−35.9807	−1.78790
$406$	−91.2583	−4.52907
$407$	16.2473	0.805349
$408$	14.4744	0.716591
$409$	10.7848	0.533275	0.266638	−	0.963797i	$-0.414087\pi$
$409$	10.7848	0.533275	0.266638	+	0.963797i	$0.414087\pi$
$410$	−31.3797	−1.54973
$411$	−2.50919	−0.123769
$412$	−22.4421	−1.10564
$413$	17.2302	0.847842
$414$	18.0267	0.885962
$415$	−63.7459	−3.12916
$416$	22.1414	1.08557
$417$	4.47162	0.218976
$418$	−51.6364	−2.52562
$419$	22.7266	1.11027	0.555133	−	0.831761i	$-0.312667\pi$
$419$	22.7266	1.11027	0.555133	+	0.831761i	$0.312667\pi$
$420$	−22.5488	−1.10027
$421$	26.6331	1.29802	0.649009	−	0.760781i	$-0.275184\pi$
$421$	26.6331	1.29802	0.649009	+	0.760781i	$0.275184\pi$
$422$	13.6806	0.665961
$423$	−30.0915	−1.46310
$424$	−81.6410	−3.96484
$425$	−83.7165	−4.06085
$426$	−7.50013	−0.363382
$427$	40.0926	1.94022
$428$	81.0984	3.92004
$429$	0.911621	0.0440135
$430$	−17.5309	−0.845417
$431$	11.2987	0.544238	0.272119	−	0.962264i	$-0.412276\pi$
$431$	11.2987	0.544238	0.272119	+	0.962264i	$0.412276\pi$
$432$	−20.9492	−1.00792
$433$	31.1927	1.49903	0.749513	−	0.661989i	$-0.230288\pi$
$433$	31.1927	1.49903	0.749513	+	0.661989i	$0.230288\pi$
$434$	50.9387	2.44513
$435$	−7.94226	−0.380802
$436$	−32.1929	−1.54176
$437$	−10.8527	−0.519153
$438$	2.46211	0.117644
$439$	29.1484	1.39118	0.695589	−	0.718440i	$-0.255144\pi$
$439$	29.1484	1.39118	0.695589	+	0.718440i	$0.255144\pi$
$440$	−156.607	−7.46592
$441$	−30.3224	−1.44392
$442$	−17.7781	−0.845616
$443$	−3.90194	−0.185387	−0.0926934	−	0.995695i	$-0.529548\pi$
$443$	−3.90194	−0.185387	−0.0926934	+	0.995695i	$0.529548\pi$
$444$	−5.35005	−0.253902
$445$	52.6827	2.49740
$446$	80.4273	3.80834
$447$	−0.418798	−0.0198085
$448$	126.603	5.98143
$449$	16.2914	0.768840	0.384420	−	0.923158i	$-0.374402\pi$
$449$	16.2914	0.768840	0.384420	+	0.923158i	$0.374402\pi$
$450$	103.712	4.88904
$451$	10.5691	0.497678
$452$	10.4856	0.493203
$453$	−1.29987	−0.0610732
$454$	−33.5949	−1.57669
$455$	17.5839	0.824348
$456$	10.7955	0.505544
$457$	−25.7817	−1.20602	−0.603008	−	0.797735i	$-0.706031\pi$
$457$	−25.7817	−1.20602	−0.603008	+	0.797735i	$0.706031\pi$
$458$	−24.3636	−1.13844
$459$	9.04773	0.422312
$460$	−51.8419	−2.41714
$461$	17.1728	0.799816	0.399908	−	0.916555i	$-0.369042\pi$
$461$	17.1728	0.799816	0.399908	+	0.916555i	$0.369042\pi$
$462$	10.3675	0.482341
$463$	1.53809	0.0714809	0.0357405	−	0.999361i	$-0.488621\pi$
$463$	1.53809	0.0714809	0.0357405	+	0.999361i	$0.488621\pi$
$464$	120.790	5.60753
$465$	4.43322	0.205586
$466$	65.5266	3.03546
$467$	−38.6250	−1.78735	−0.893676	−	0.448714i	$-0.851882\pi$
$467$	−38.6250	−1.78735	−0.893676	+	0.448714i	$0.851882\pi$
$468$	16.1339	0.745792
$469$	33.2958	1.53746
$470$	118.133	5.44909
$471$	−2.96646	−0.136687
$472$	−39.4050	−1.81376
$473$	5.90464	0.271496
$474$	0.565843	0.0259901
$475$	−62.4382	−2.86486
$476$	−148.109	−6.78858
$477$	−25.2810	−1.15754
$478$	6.40525	0.292969
$479$	41.8898	1.91400	0.956998	−	0.290096i	$-0.0936873\pi$
$479$	41.8898	1.91400	0.956998	+	0.290096i	$0.0936873\pi$
$480$	21.9148	1.00027
$481$	4.17205	0.190229
$482$	68.0091	3.09773
$483$	2.17899	0.0991475
$484$	22.8201	1.03728
$485$	−13.3904	−0.608026
$486$	−16.8648	−0.765004
$487$	12.8425	0.581950	0.290975	−	0.956731i	$-0.406020\pi$
$487$	12.8425	0.581950	0.290975	+	0.956731i	$0.406020\pi$
$488$	−91.6908	−4.15065
$489$	−0.594833	−0.0268993
$490$	119.040	5.37766
$491$	−16.4250	−0.741248	−0.370624	−	0.928783i	$-0.620856\pi$
$491$	−16.4250	−0.741248	−0.370624	+	0.928783i	$0.620856\pi$
$492$	−3.48027	−0.156903
$493$	−52.1677	−2.34952
$494$	−13.2594	−0.596569
$495$	−48.4949	−2.17969
$496$	−67.4226	−3.02737
$497$	48.7258	2.18565
$498$	−9.65115	−0.432478
$499$	27.8091	1.24491	0.622453	−	0.782657i	$-0.286136\pi$
$499$	27.8091	1.24491	0.622453	+	0.782657i	$0.286136\pi$
$500$	−182.451	−8.15944
$501$	4.02235	0.179706
$502$	12.1164	0.540784
$503$	18.4235	0.821463	0.410731	−	0.911756i	$-0.365273\pi$
$503$	18.4235	0.821463	0.410731	+	0.911756i	$0.365273\pi$
$504$	116.496	5.18914
$505$	5.60525	0.249430
$506$	23.8359	1.05964
$507$	0.234090	0.0103963
$508$	−88.4503	−3.92435
$509$	−30.6007	−1.35635	−0.678175	−	0.734900i	$-0.737229\pi$
$509$	−30.6007	−1.35635	−0.678175	+	0.734900i	$0.737229\pi$
$510$	−17.5961	−0.779168
$511$	−15.9955	−0.707599
$512$	−46.9548	−2.07513
$513$	6.74807	0.297934
$514$	8.21873	0.362513
$515$	17.3216	0.763281
$516$	−1.94433	−0.0855943
$517$	−39.7888	−1.74991
$518$	47.4472	2.08471
$519$	−0.347829	−0.0152680
$520$	−40.2141	−1.76350
$521$	−15.0762	−0.660498	−0.330249	−	0.943894i	$-0.607133\pi$
$521$	−15.0762	−0.660498	−0.330249	+	0.943894i	$0.607133\pi$
$522$	64.6280	2.82869
$523$	−37.8627	−1.65562	−0.827811	−	0.561007i	$-0.810414\pi$
$523$	−37.8627	−1.65562	−0.827811	+	0.561007i	$0.810414\pi$
$524$	−17.1410	−0.748809
$525$	12.5363	0.547130
$526$	48.3195	2.10683
$527$	29.1191	1.26845
$528$	−13.7225	−0.597195
$529$	−17.9903	−0.782187
$530$	99.2483	4.31107
$531$	−12.2022	−0.529530
$532$	−110.464	−4.78924
$533$	2.71397	0.117555
$534$	7.97618	0.345163
$535$	−62.5946	−2.70620
$536$	−76.1467	−3.28904
$537$	−3.13986	−0.135495
$538$	50.5142	2.17782
$539$	−40.0940	−1.72697
$540$	32.2347	1.38716
$541$	−3.64894	−0.156880	−0.0784402	−	0.996919i	$-0.524994\pi$
$541$	−3.64894	−0.156880	−0.0784402	+	0.996919i	$0.524994\pi$
$542$	−19.2733	−0.827859
$543$	3.65211	0.156727
$544$	143.945	6.17158
$545$	24.8476	1.06435
$546$	2.66222	0.113932
$547$	−19.0430	−0.814221	−0.407111	−	0.913379i	$-0.633464\pi$
$547$	−19.0430	−0.814221	−0.407111	+	0.913379i	$0.633464\pi$
$548$	−58.7186	−2.50834
$549$	−28.3931	−1.21179
$550$	137.134	5.84743
$551$	−38.9083	−1.65755
$552$	−4.98330	−0.212104
$553$	−3.67609	−0.156323
$554$	45.3655	1.92739
$555$	4.12935	0.175281
$556$	104.642	4.43783
$557$	−21.1231	−0.895016	−0.447508	−	0.894280i	$-0.647688\pi$
$557$	−21.1231	−0.895016	−0.447508	+	0.894280i	$0.647688\pi$
$558$	−36.0741	−1.52714
$559$	1.51622	0.0641292
$560$	−264.688	−11.1851
$561$	5.92659	0.250221
$562$	43.3873	1.83018
$563$	−17.0397	−0.718139	−0.359070	−	0.933311i	$-0.616906\pi$
$563$	−17.0397	−0.718139	−0.359070	+	0.933311i	$0.616906\pi$
$564$	13.1020	0.551693
$565$	−8.09318	−0.340483
$566$	−73.7401	−3.09953
$567$	35.3905	1.48626
$568$	−111.435	−4.67570
$569$	1.77535	0.0744265	0.0372132	−	0.999307i	$-0.488152\pi$
$569$	1.77535	0.0744265	0.0372132	+	0.999307i	$0.488152\pi$
$570$	−13.1237	−0.549691
$571$	37.8775	1.58512	0.792562	−	0.609792i	$-0.208747\pi$
$571$	37.8775	1.58512	0.792562	+	0.609792i	$0.208747\pi$
$572$	21.3333	0.891988
$573$	0.617249	0.0257859
$574$	30.8650	1.28828
$575$	28.8222	1.20197
$576$	−89.6587	−3.73578
$577$	−40.2914	−1.67735	−0.838676	−	0.544631i	$-0.816670\pi$
$577$	−40.2914	−1.67735	−0.838676	+	0.544631i	$0.816670\pi$
$578$	−69.0896	−2.87375
$579$	4.79493	0.199271
$580$	−185.860	−7.71743
$581$	62.7003	2.60124
$582$	−2.02731	−0.0840348
$583$	−33.4281	−1.38445
$584$	36.5813	1.51375
$585$	−12.4527	−0.514857
$586$	−34.2316	−1.41409
$587$	41.2492	1.70254	0.851269	−	0.524730i	$-0.175834\pi$
$587$	41.2492	1.70254	0.851269	+	0.524730i	$0.175834\pi$
$588$	13.2025	0.544462
$589$	21.7179	0.894869
$590$	47.9034	1.97215
$591$	−1.41787	−0.0583233
$592$	−62.8013	−2.58112
$593$	23.3336	0.958197	0.479098	−	0.877761i	$-0.340964\pi$
$593$	23.3336	0.958197	0.479098	+	0.877761i	$0.340964\pi$
$594$	−14.8209	−0.608110
$595$	114.316	4.68649
$596$	−9.80048	−0.401443
$597$	−5.56217	−0.227644
$598$	6.12068	0.250293
$599$	−12.4723	−0.509605	−0.254802	−	0.966993i	$-0.582010\pi$
$599$	−12.4723	−0.509605	−0.254802	+	0.966993i	$0.582010\pi$
$600$	−28.6703	−1.17046
$601$	20.8526	0.850597	0.425299	−	0.905053i	$-0.360169\pi$
$601$	20.8526	0.850597	0.425299	+	0.905053i	$0.360169\pi$
$602$	17.2434	0.702787
$603$	−23.5797	−0.960238
$604$	−30.4188	−1.23773
$605$	−17.6133	−0.716084
$606$	0.848638	0.0344735
$607$	−41.9779	−1.70383	−0.851916	−	0.523679i	$-0.824559\pi$
$607$	−41.9779	−1.70383	−0.851916	+	0.523679i	$0.824559\pi$
$608$	107.358	4.35396
$609$	7.81198	0.316557
$610$	111.466	4.51311
$611$	−10.2171	−0.413341
$612$	104.889	4.23989
$613$	−20.9299	−0.845349	−0.422674	−	0.906282i	$-0.638909\pi$
$613$	−20.9299	−0.845349	−0.422674	+	0.906282i	$0.638909\pi$
$614$	30.0102	1.21111
$615$	2.68619	0.108318
$616$	154.038	6.20636
$617$	4.44521	0.178957	0.0894787	−	0.995989i	$-0.471480\pi$
$617$	4.44521	0.178957	0.0894787	+	0.995989i	$0.471480\pi$
$618$	2.62250	0.105492
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	103.744	4.16645
$621$	−3.11498	−0.125000
$622$	−36.6728	−1.47044
$623$	−51.8185	−2.07607
$624$	−3.52372	−0.141062
$625$	76.4358	3.05743
$626$	−55.9981	−2.23814
$627$	4.42023	0.176527
$628$	−69.4195	−2.77014
$629$	27.1231	1.08147
$630$	−141.620	−5.64228
$631$	14.1381	0.562830	0.281415	−	0.959586i	$-0.409196\pi$
$631$	14.1381	0.562830	0.281415	+	0.959586i	$0.409196\pi$
$632$	8.40714	0.334418
$633$	−1.17110	−0.0465470
$634$	−10.9001	−0.432897
$635$	68.2690	2.70917
$636$	11.0075	0.436475
$637$	−10.2955	−0.407923
$638$	85.4550	3.38320
$639$	−34.5070	−1.36508
$640$	164.748	6.51223
$641$	35.7155	1.41068	0.705339	−	0.708870i	$-0.250795\pi$
$641$	35.7155	1.41068	0.705339	+	0.708870i	$0.250795\pi$
$642$	−9.47684	−0.374021
$643$	−25.9908	−1.02498	−0.512488	−	0.858694i	$-0.671276\pi$
$643$	−25.9908	−1.02498	−0.512488	+	0.858694i	$0.671276\pi$
$644$	50.9915	2.00935
$645$	1.50070	0.0590900
$646$	−86.2014	−3.39155
$647$	−18.3979	−0.723296	−0.361648	−	0.932315i	$-0.617786\pi$
$647$	−18.3979	−0.723296	−0.361648	+	0.932315i	$0.617786\pi$
$648$	−80.9374	−3.17952
$649$	−16.1345	−0.633334
$650$	35.2139	1.38120
$651$	−4.36050	−0.170901
$652$	−13.9199	−0.545147
$653$	−23.3867	−0.915193	−0.457596	−	0.889160i	$-0.651290\pi$
$653$	−23.3867	−0.915193	−0.457596	+	0.889160i	$0.651290\pi$
$654$	3.76193	0.147103
$655$	13.2300	0.516940
$656$	−40.8530	−1.59504
$657$	11.3278	0.441940
$658$	−116.196	−4.52978
$659$	9.54687	0.371893	0.185947	−	0.982560i	$-0.440465\pi$
$659$	9.54687	0.371893	0.185947	+	0.982560i	$0.440465\pi$
$660$	21.1149	0.821897
$661$	38.0422	1.47967	0.739836	−	0.672788i	$-0.234903\pi$
$661$	38.0422	1.47967	0.739836	+	0.672788i	$0.234903\pi$
$662$	79.8628	3.10395
$663$	1.52185	0.0591039
$664$	−143.394	−5.56477
$665$	85.2602	3.30625
$666$	−33.6015	−1.30203
$667$	17.9605	0.695432
$668$	94.1289	3.64196
$669$	−6.88481	−0.266182
$670$	92.5690	3.57625
$671$	−37.5430	−1.44933
$672$	−21.5554	−0.831516
$673$	37.0459	1.42801	0.714007	−	0.700139i	$-0.246879\pi$
$673$	37.0459	1.42801	0.714007	+	0.700139i	$0.246879\pi$
$674$	−86.9092	−3.34762
$675$	−17.9213	−0.689792
$676$	5.47804	0.210694
$677$	30.7011	1.17994	0.589970	−	0.807425i	$-0.299139\pi$
$677$	30.7011	1.17994	0.589970	+	0.807425i	$0.299139\pi$
$678$	−1.22531	−0.0470578
$679$	13.1708	0.505447
$680$	−261.438	−10.0257
$681$	2.87582	0.110202
$682$	−47.6994	−1.82650
$683$	−14.9283	−0.571217	−0.285609	−	0.958346i	$-0.592196\pi$
$683$	−14.9283	−0.571217	−0.285609	+	0.958346i	$0.592196\pi$
$684$	78.2295	2.99118
$685$	45.3211	1.73163
$686$	−37.4786	−1.43094
$687$	2.08560	0.0795705
$688$	−22.8234	−0.870134
$689$	−8.58380	−0.327017
$690$	6.05804	0.230626
$691$	21.8981	0.833043	0.416522	−	0.909126i	$-0.363249\pi$
$691$	21.8981	0.833043	0.416522	+	0.909126i	$0.363249\pi$
$692$	−8.13971	−0.309425
$693$	47.6995	1.81195
$694$	−1.50535	−0.0571423
$695$	−80.7667	−3.06365
$696$	−17.8658	−0.677202
$697$	17.6439	0.668312
$698$	13.5480	0.512800
$699$	−5.60927	−0.212162
$700$	293.368	11.0883
$701$	−10.6288	−0.401446	−0.200723	−	0.979648i	$-0.564329\pi$
$701$	−10.6288	−0.401446	−0.200723	+	0.979648i	$0.564329\pi$
$702$	−3.80578	−0.143640
$703$	20.2292	0.762960
$704$	−118.552	−4.46810
$705$	−10.1126	−0.380861
$706$	81.0748	3.05129
$707$	−5.51331	−0.207349
$708$	5.31289	0.199671
$709$	9.69830	0.364227	0.182114	−	0.983277i	$-0.441706\pi$
$709$	9.69830	0.364227	0.182114	+	0.983277i	$0.441706\pi$
$710$	135.468	5.08401
$711$	2.60336	0.0976337
$712$	118.508	4.44127
$713$	−10.0252	−0.375447
$714$	17.3075	0.647716
$715$	−16.4657	−0.615784
$716$	−73.4773	−2.74598
$717$	−0.548308	−0.0204769
$718$	63.8442	2.38264
$719$	−26.5395	−0.989756	−0.494878	−	0.868963i	$-0.664787\pi$
$719$	−26.5395	−0.989756	−0.494878	+	0.868963i	$0.664787\pi$
$720$	187.449	6.98582
$721$	−17.0375	−0.634509
$722$	−12.3341	−0.459029
$723$	−5.82178	−0.216514
$724$	85.4646	3.17627
$725$	103.331	3.83763
$726$	−2.66667	−0.0989693
$727$	−28.9314	−1.07301	−0.536503	−	0.843898i	$-0.680255\pi$
$727$	−28.9314	−1.07301	−0.536503	+	0.843898i	$0.680255\pi$
$728$	39.5545	1.46599
$729$	−24.0858	−0.892066
$730$	−44.4707	−1.64594
$731$	9.85716	0.364580
$732$	12.3625	0.456930
$733$	−41.6162	−1.53713	−0.768566	−	0.639771i	$-0.779029\pi$
$733$	−41.6162	−1.53713	−0.768566	+	0.639771i	$0.779029\pi$
$734$	−39.7641	−1.46772
$735$	−10.1901	−0.375869
$736$	−49.5578	−1.82672
$737$	−31.1784	−1.14847
$738$	−21.8582	−0.804611
$739$	24.3745	0.896630	0.448315	−	0.893876i	$-0.352024\pi$
$739$	24.3745	0.896630	0.448315	+	0.893876i	$0.352024\pi$
$740$	96.6327	3.55229
$741$	1.13504	0.0416969
$742$	−97.6203	−3.58375
$743$	−17.9667	−0.659134	−0.329567	−	0.944132i	$-0.606903\pi$
$743$	−17.9667	−0.659134	−0.329567	+	0.944132i	$0.606903\pi$
$744$	9.97237	0.365605
$745$	7.56435	0.277136
$746$	−87.8803	−3.21753
$747$	−44.4035	−1.62464
$748$	138.691	5.07103
$749$	61.5678	2.24964
$750$	21.3205	0.778513
$751$	−47.4587	−1.73179	−0.865897	−	0.500223i	$-0.833251\pi$
$751$	−47.4587	−1.73179	−0.865897	+	0.500223i	$0.833251\pi$
$752$	153.797	5.60840
$753$	−1.03720	−0.0377978
$754$	21.9435	0.799135
$755$	23.4783	0.854463
$756$	−31.7060	−1.15314
$757$	29.7081	1.07976	0.539879	−	0.841743i	$-0.318470\pi$
$757$	29.7081	1.07976	0.539879	+	0.841743i	$0.318470\pi$
$758$	19.1384	0.695137
$759$	−2.04042	−0.0740627
$760$	−194.988	−7.07296
$761$	4.79960	0.173985	0.0869927	−	0.996209i	$-0.472274\pi$
$761$	4.79960	0.173985	0.0869927	+	0.996209i	$0.472274\pi$
$762$	10.3360	0.374432
$763$	−24.4400	−0.884788
$764$	14.4445	0.522584
$765$	−80.9571	−2.92701
$766$	79.3349	2.86649
$767$	−4.14308	−0.149598
$768$	10.6904	0.385758
$769$	−36.6778	−1.32264	−0.661318	−	0.750106i	$-0.730003\pi$
$769$	−36.6778	−1.32264	−0.661318	+	0.750106i	$0.730003\pi$
$770$	−187.259	−6.74833
$771$	−0.703547	−0.0253376
$772$	112.208	4.03847
$773$	−4.02799	−0.144877	−0.0724383	−	0.997373i	$-0.523078\pi$
$773$	−4.02799	−0.144877	−0.0724383	+	0.997373i	$0.523078\pi$
$774$	−12.2115	−0.438935
$775$	−57.6777	−2.07184
$776$	−30.1212	−1.08129
$777$	−4.06162	−0.145710
$778$	−77.3550	−2.77331
$779$	13.1594	0.471483
$780$	5.42197	0.194138
$781$	−45.6272	−1.63267
$782$	39.7915	1.42294
$783$	−11.1676	−0.399099
$784$	154.977	5.53489
$785$	53.5803	1.91236
$786$	2.00303	0.0714458
$787$	−4.84997	−0.172883	−0.0864414	−	0.996257i	$-0.527550\pi$
$787$	−4.84997	−0.172883	−0.0864414	+	0.996257i	$0.527550\pi$
$788$	−33.1801	−1.18199
$789$	−4.13629	−0.147256
$790$	−10.2203	−0.363621
$791$	7.96043	0.283040
$792$	−109.088	−3.87626
$793$	−9.64045	−0.342342
$794$	−2.56373	−0.0909832
$795$	−8.49595	−0.301320
$796$	−130.163	−4.61350
$797$	−28.7918	−1.01986	−0.509929	−	0.860217i	$-0.670328\pi$
$797$	−28.7918	−1.01986	−0.509929	+	0.860217i	$0.670328\pi$
$798$	12.9084	0.456953
$799$	−66.4232	−2.34988
$800$	−285.119	−10.0805
$801$	36.6972	1.29663
$802$	−2.98144	−0.105278
$803$	14.9783	0.528573
$804$	10.2667	0.362078
$805$	−39.3570	−1.38715
$806$	−12.2484	−0.431433
$807$	−4.32416	−0.152218
$808$	12.6088	0.443577
$809$	27.8828	0.980307	0.490153	−	0.871636i	$-0.336941\pi$
$809$	27.8828	0.980307	0.490153	+	0.871636i	$0.336941\pi$
$810$	98.3929	3.45717
$811$	−9.44965	−0.331822	−0.165911	−	0.986141i	$-0.553056\pi$
$811$	−9.44965	−0.331822	−0.165911	+	0.986141i	$0.553056\pi$
$812$	182.812	6.41543
$813$	1.64985	0.0578628
$814$	−44.4299	−1.55727
$815$	10.7439	0.376342
$816$	−22.9082	−0.801949
$817$	7.35176	0.257206
$818$	−29.4922	−1.03117
$819$	12.2485	0.427996
$820$	62.8608	2.19519
$821$	36.6260	1.27826	0.639128	−	0.769101i	$-0.279296\pi$
$821$	36.6260	1.27826	0.639128	+	0.769101i	$0.279296\pi$
$822$	6.86163	0.239327
$823$	35.4892	1.23708	0.618538	−	0.785755i	$-0.287725\pi$
$823$	35.4892	1.23708	0.618538	+	0.785755i	$0.287725\pi$
$824$	38.9644	1.35739
$825$	−11.7391	−0.408703
$826$	−47.1177	−1.63943
$827$	35.4305	1.23204	0.616020	−	0.787731i	$-0.288744\pi$
$827$	35.4305	1.23204	0.616020	+	0.787731i	$0.288744\pi$
$828$	−36.1116	−1.25496
$829$	41.3616	1.43655	0.718273	−	0.695761i	$-0.244933\pi$
$829$	41.3616	1.43655	0.718273	+	0.695761i	$0.244933\pi$
$830$	174.319	6.05072
$831$	−3.88342	−0.134714
$832$	−30.4423	−1.05540
$833$	−66.9327	−2.31908
$834$	−12.2281	−0.423425
$835$	−72.6519	−2.51422
$836$	103.440	3.57754
$837$	6.23356	0.215463
$838$	−62.1481	−2.14687
$839$	7.65849	0.264400	0.132200	−	0.991223i	$-0.457796\pi$
$839$	7.65849	0.264400	0.132200	+	0.991223i	$0.457796\pi$
$840$	39.1496	1.35079
$841$	35.3908	1.22037
$842$	−72.8309	−2.50992
$843$	−3.71408	−0.127920
$844$	−27.4054	−0.943333
$845$	−4.22814	−0.145452
$846$	82.2883	2.82913
$847$	17.3244	0.595274
$848$	129.211	4.43711
$849$	6.31237	0.216640
$850$	228.931	7.85227
$851$	−9.33804	−0.320104
$852$	15.0245	0.514731
$853$	−30.2477	−1.03566	−0.517831	−	0.855483i	$-0.673260\pi$
$853$	−30.2477	−1.03566	−0.517831	+	0.855483i	$0.673260\pi$
$854$	−109.637	−3.75171
$855$	−60.3802	−2.06496
$856$	−140.804	−4.81259
$857$	1.64372	0.0561485	0.0280742	−	0.999606i	$-0.491063\pi$
$857$	1.64372	0.0561485	0.0280742	+	0.999606i	$0.491063\pi$
$858$	−2.49292	−0.0851069
$859$	−34.4304	−1.17475	−0.587375	−	0.809315i	$-0.699838\pi$
$859$	−34.4304	−1.17475	−0.587375	+	0.809315i	$0.699838\pi$
$860$	35.1185	1.19753
$861$	−2.64213	−0.0900436
$862$	−30.8974	−1.05237
$863$	−40.0732	−1.36411	−0.682054	−	0.731302i	$-0.738913\pi$
$863$	−40.0732	−1.36411	−0.682054	+	0.731302i	$0.738913\pi$
$864$	30.8145	1.04833
$865$	6.28251	0.213612
$866$	−85.2997	−2.89860
$867$	5.91428	0.200859
$868$	−102.042	−3.46353
$869$	3.44232	0.116773
$870$	21.7189	0.736340
$871$	−8.00612	−0.271277
$872$	55.8938	1.89280
$873$	−9.32736	−0.315683
$874$	29.6777	1.00386
$875$	−138.512	−4.68255
$876$	−4.93218	−0.166643
$877$	−3.27454	−0.110573	−0.0552866	−	0.998471i	$-0.517607\pi$
$877$	−3.27454	−0.110573	−0.0552866	+	0.998471i	$0.517607\pi$
$878$	−79.7092	−2.69006
$879$	2.93033	0.0988375
$880$	247.856	8.35524
$881$	−35.4077	−1.19292	−0.596459	−	0.802644i	$-0.703426\pi$
$881$	−35.4077	−1.19292	−0.596459	+	0.802644i	$0.703426\pi$
$882$	82.9196	2.79205
$883$	−3.01915	−0.101602	−0.0508012	−	0.998709i	$-0.516177\pi$
$883$	−3.01915	−0.101602	−0.0508012	+	0.998709i	$0.516177\pi$
$884$	35.6136	1.19781
$885$	−4.10067	−0.137843
$886$	10.6703	0.358474
$887$	−38.5357	−1.29390	−0.646952	−	0.762531i	$-0.723956\pi$
$887$	−38.5357	−1.29390	−0.646952	+	0.762531i	$0.723956\pi$
$888$	9.28883	0.311713
$889$	−67.1492	−2.25211
$890$	−144.066	−4.82911
$891$	−33.1400	−1.11023
$892$	−161.114	−5.39451
$893$	−49.5404	−1.65781
$894$	1.14525	0.0383028
$895$	56.7123	1.89568
$896$	−162.046	−5.41356
$897$	−0.523949	−0.0174941
$898$	−44.5505	−1.48667
$899$	−35.9417	−1.19872
$900$	−207.760	−6.92532
$901$	−55.8046	−1.85912
$902$	−28.9022	−0.962338
$903$	−1.47608	−0.0491210
$904$	−18.2053	−0.605500
$905$	−65.9645	−2.19273
$906$	3.55463	0.118095
$907$	−13.5666	−0.450472	−0.225236	−	0.974304i	$-0.572315\pi$
$907$	−13.5666	−0.450472	−0.225236	+	0.974304i	$0.572315\pi$
$908$	67.2984	2.23338
$909$	3.90446	0.129503
$910$	−48.0851	−1.59400
$911$	6.73419	0.223114	0.111557	−	0.993758i	$-0.464416\pi$
$911$	6.73419	0.223114	0.111557	+	0.993758i	$0.464416\pi$
$912$	−17.0857	−0.565762
$913$	−58.7130	−1.94312
$914$	70.5026	2.33202
$915$	−9.54178	−0.315442
$916$	48.8060	1.61259
$917$	−13.0130	−0.429728
$918$	−24.7419	−0.816605
$919$	−16.3504	−0.539351	−0.269675	−	0.962951i	$-0.586916\pi$
$919$	−16.3504	−0.539351	−0.269675	+	0.962951i	$0.586916\pi$
$920$	90.0086	2.96750
$921$	−2.56896	−0.0846502
$922$	−46.9607	−1.54657
$923$	−11.7163	−0.385648
$924$	−20.7686	−0.683236
$925$	−53.7242	−1.76644
$926$	−4.20605	−0.138219
$927$	12.0657	0.396291
$928$	−177.671	−5.83235
$929$	42.6245	1.39846	0.699232	−	0.714895i	$-0.253525\pi$
$929$	42.6245	1.39846	0.699232	+	0.714895i	$0.253525\pi$
$930$	−12.1231	−0.397532
$931$	−49.9204	−1.63607
$932$	−131.265	−4.29973
$933$	3.13930	0.102776
$934$	105.624	3.45612
$935$	−107.046	−3.50079
$936$	−28.0120	−0.915600
$937$	−33.2835	−1.08732	−0.543662	−	0.839304i	$-0.682963\pi$
$937$	−33.2835	−1.08732	−0.543662	+	0.839304i	$0.682963\pi$
$938$	−91.0506	−2.97291
$939$	4.79361	0.156433
$940$	−236.649	−7.71863
$941$	−56.7738	−1.85077	−0.925387	−	0.379024i	$-0.876260\pi$
$941$	−56.7738	−1.85077	−0.925387	+	0.379024i	$0.876260\pi$
$942$	8.11209	0.264306
$943$	−6.07451	−0.197813
$944$	62.3651	2.02981
$945$	24.4718	0.796067
$946$	−16.1468	−0.524979
$947$	33.2327	1.07992	0.539959	−	0.841691i	$-0.318440\pi$
$947$	33.2327	1.07992	0.539959	+	0.841691i	$0.318440\pi$
$948$	−1.13352	−0.0368149
$949$	3.84619	0.124853
$950$	170.744	5.53966
$951$	0.933078	0.0302571
$952$	257.149	8.33426
$953$	−1.71633	−0.0555973	−0.0277986	−	0.999614i	$-0.508850\pi$
$953$	−1.71633	−0.0555973	−0.0277986	+	0.999614i	$0.508850\pi$
$954$	69.1335	2.23828
$955$	−11.1488	−0.360766
$956$	−12.8312	−0.414991
$957$	−7.31520	−0.236467
$958$	−114.552	−3.70101
$959$	−44.5777	−1.43949
$960$	−30.1307	−0.972464
$961$	−10.9380	−0.352839
$962$	−11.4089	−0.367837
$963$	−43.6016	−1.40504
$964$	−136.238	−4.38793
$965$	−86.6063	−2.78795
$966$	−5.95867	−0.191717
$967$	−11.4948	−0.369649	−0.184824	−	0.982772i	$-0.559172\pi$
$967$	−11.4948	−0.369649	−0.184824	+	0.982772i	$0.559172\pi$
$968$	−39.6206	−1.27345
$969$	7.37909	0.237051
$970$	36.6174	1.17571
$971$	20.8700	0.669750	0.334875	−	0.942263i	$-0.391306\pi$
$971$	20.8700	0.669750	0.334875	+	0.942263i	$0.391306\pi$
$972$	33.7842	1.08363
$973$	79.4419	2.54679
$974$	−35.1191	−1.12529
$975$	−3.01442	−0.0965386
$976$	145.116	4.64506
$977$	45.8012	1.46531	0.732655	−	0.680600i	$-0.238281\pi$
$977$	45.8012	1.46531	0.732655	+	0.680600i	$0.238281\pi$
$978$	1.62663	0.0520139
$979$	48.5233	1.55081
$980$	−238.464	−7.61745
$981$	17.3081	0.552606
$982$	44.9157	1.43332
$983$	−50.2693	−1.60334	−0.801671	−	0.597766i	$-0.796055\pi$
$983$	−50.2693	−1.60334	−0.801671	+	0.597766i	$0.796055\pi$
$984$	6.04250	0.192628
$985$	25.6096	0.815989
$986$	142.658	4.54315
$987$	9.94669	0.316607
$988$	26.5617	0.845039
$989$	−3.39365	−0.107912
$990$	132.614	4.21476
$991$	5.99103	0.190311	0.0951557	−	0.995462i	$-0.469665\pi$
$991$	5.99103	0.190311	0.0951557	+	0.995462i	$0.469665\pi$
$992$	99.1728	3.14874
$993$	−6.83649	−0.216949
$994$	−133.246	−4.22629
$995$	100.464	3.18493
$996$	19.3335	0.612605
$997$	−7.19490	−0.227865	−0.113932	−	0.993489i	$-0.536345\pi$
$997$	−7.19490	−0.227865	−0.113932	+	0.993489i	$0.536345\pi$
$998$	−76.0468	−2.40722
$999$	5.80629	0.183703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.c.1.4	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.c.1.4	✓	151	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73460 q^{2} +0.234090 q^{3} +5.47804 q^{4} -4.22814 q^{5} -0.640142 q^{6} +4.15879 q^{7} -9.51105 q^{8} -2.94520 q^{9} +O(q^{10})\)	\(q-2.73460 q^{2} +0.234090 q^{3} +5.47804 q^{4} -4.22814 q^{5} -0.640142 q^{6} +4.15879 q^{7} -9.51105 q^{8} -2.94520 q^{9} +11.5623 q^{10} -3.89432 q^{11} +1.28235 q^{12} -1.00000 q^{13} -11.3726 q^{14} -0.989765 q^{15} +15.0529 q^{16} -6.50115 q^{17} +8.05395 q^{18} -4.84875 q^{19} -23.1619 q^{20} +0.973530 q^{21} +10.6494 q^{22} +2.23824 q^{23} -2.22644 q^{24} +12.8772 q^{25} +2.73460 q^{26} -1.39171 q^{27} +22.7820 q^{28} +8.02439 q^{29} +2.70661 q^{30} -4.47906 q^{31} -22.1414 q^{32} -0.911621 q^{33} +17.7781 q^{34} -17.5839 q^{35} -16.1339 q^{36} -4.17205 q^{37} +13.2594 q^{38} -0.234090 q^{39} +40.2141 q^{40} -2.71397 q^{41} -2.66222 q^{42} -1.51622 q^{43} -21.3333 q^{44} +12.4527 q^{45} -6.12068 q^{46} +10.2171 q^{47} +3.52372 q^{48} +10.2955 q^{49} -35.2139 q^{50} -1.52185 q^{51} -5.47804 q^{52} +8.58380 q^{53} +3.80578 q^{54} +16.4657 q^{55} -39.5545 q^{56} -1.13504 q^{57} -21.9435 q^{58} +4.14308 q^{59} -5.42197 q^{60} +9.64045 q^{61} +12.2484 q^{62} -12.2485 q^{63} +30.4423 q^{64} +4.22814 q^{65} +2.49292 q^{66} +8.00612 q^{67} -35.6136 q^{68} +0.523949 q^{69} +48.0851 q^{70} +11.7163 q^{71} +28.0120 q^{72} -3.84619 q^{73} +11.4089 q^{74} +3.01442 q^{75} -26.5617 q^{76} -16.1957 q^{77} +0.640142 q^{78} -0.883934 q^{79} -63.6456 q^{80} +8.50982 q^{81} +7.42163 q^{82} +15.0766 q^{83} +5.33304 q^{84} +27.4878 q^{85} +4.14625 q^{86} +1.87843 q^{87} +37.0391 q^{88} -12.4600 q^{89} -34.0532 q^{90} -4.15879 q^{91} +12.2612 q^{92} -1.04850 q^{93} -27.9398 q^{94} +20.5012 q^{95} -5.18309 q^{96} +3.16697 q^{97} -28.1541 q^{98} +11.4696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * q^99

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