LMFDB - Embedded newform 8047.2.a.e.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:	$0$
Dimension:	$168$
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)

$f(q)$	$=$	$q-2.69715 q^{2} +2.56324 q^{3} +5.27461 q^{4} +3.51333 q^{5} -6.91344 q^{6} -4.43417 q^{7} -8.83210 q^{8} +3.57021 q^{9} +O(q^{10})$	\(q-2.69715 q^{2} +2.56324 q^{3} +5.27461 q^{4} +3.51333 q^{5} -6.91344 q^{6} -4.43417 q^{7} -8.83210 q^{8} +3.57021 q^{9} -9.47597 q^{10} -4.75599 q^{11} +13.5201 q^{12} +1.00000 q^{13} +11.9596 q^{14} +9.00551 q^{15} +13.2723 q^{16} -0.810401 q^{17} -9.62937 q^{18} -3.93258 q^{19} +18.5314 q^{20} -11.3658 q^{21} +12.8276 q^{22} -5.29943 q^{23} -22.6388 q^{24} +7.34349 q^{25} -2.69715 q^{26} +1.46157 q^{27} -23.3885 q^{28} +3.45889 q^{29} -24.2892 q^{30} -2.75496 q^{31} -18.1330 q^{32} -12.1908 q^{33} +2.18577 q^{34} -15.5787 q^{35} +18.8314 q^{36} +6.94846 q^{37} +10.6067 q^{38} +2.56324 q^{39} -31.0301 q^{40} -0.642116 q^{41} +30.6553 q^{42} -9.53608 q^{43} -25.0860 q^{44} +12.5433 q^{45} +14.2934 q^{46} +10.7047 q^{47} +34.0200 q^{48} +12.6618 q^{49} -19.8065 q^{50} -2.07725 q^{51} +5.27461 q^{52} +10.5586 q^{53} -3.94208 q^{54} -16.7094 q^{55} +39.1630 q^{56} -10.0801 q^{57} -9.32913 q^{58} +0.223495 q^{59} +47.5005 q^{60} +8.54254 q^{61} +7.43053 q^{62} -15.8309 q^{63} +22.3630 q^{64} +3.51333 q^{65} +32.8803 q^{66} +2.66873 q^{67} -4.27454 q^{68} -13.5837 q^{69} +42.0180 q^{70} +10.2634 q^{71} -31.5324 q^{72} +8.73922 q^{73} -18.7410 q^{74} +18.8231 q^{75} -20.7428 q^{76} +21.0889 q^{77} -6.91344 q^{78} +16.0883 q^{79} +46.6298 q^{80} -6.96425 q^{81} +1.73188 q^{82} +10.9369 q^{83} -59.9503 q^{84} -2.84720 q^{85} +25.7202 q^{86} +8.86596 q^{87} +42.0054 q^{88} -16.8260 q^{89} -33.8312 q^{90} -4.43417 q^{91} -27.9524 q^{92} -7.06163 q^{93} -28.8723 q^{94} -13.8164 q^{95} -46.4794 q^{96} -9.97225 q^{97} -34.1508 q^{98} -16.9799 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * 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 $

$2$	−2.69715	−1.90717	−0.953586	−	0.301122i	$-0.902639\pi$
$2$	−2.69715	−1.90717	−0.953586	+	0.301122i	$0.902639\pi$
$3$	2.56324	1.47989	0.739944	−	0.672668i	$-0.234852\pi$
$3$	2.56324	1.47989	0.739944	+	0.672668i	$0.234852\pi$
$4$	5.27461	2.63730
$5$	3.51333	1.57121	0.785605	−	0.618729i	$-0.212352\pi$
$5$	3.51333	1.57121	0.785605	+	0.618729i	$0.212352\pi$
$6$	−6.91344	−2.82240
$7$	−4.43417	−1.67596	−0.837978	−	0.545703i	$-0.816263\pi$
$7$	−4.43417	−1.67596	−0.837978	+	0.545703i	$0.816263\pi$
$8$	−8.83210	−3.12262
$9$	3.57021	1.19007
$10$	−9.47597	−2.99657
$11$	−4.75599	−1.43399	−0.716993	−	0.697080i	$-0.754482\pi$
$11$	−4.75599	−1.43399	−0.716993	+	0.697080i	$0.754482\pi$
$12$	13.5201	3.90291
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	11.9596	3.19634
$15$	9.00551	2.32521
$16$	13.2723	3.31806
$17$	−0.810401	−0.196551	−0.0982755	−	0.995159i	$-0.531333\pi$
$17$	−0.810401	−0.196551	−0.0982755	+	0.995159i	$0.531333\pi$
$18$	−9.62937	−2.26966
$19$	−3.93258	−0.902195	−0.451097	−	0.892475i	$-0.648967\pi$
$19$	−3.93258	−0.902195	−0.451097	+	0.892475i	$0.648967\pi$
$20$	18.5314	4.14375
$21$	−11.3658	−2.48023
$22$	12.8276	2.73486
$23$	−5.29943	−1.10501	−0.552504	−	0.833510i	$-0.686328\pi$
$23$	−5.29943	−1.10501	−0.552504	+	0.833510i	$0.686328\pi$
$24$	−22.6388	−4.62112
$25$	7.34349	1.46870
$26$	−2.69715	−0.528954
$27$	1.46157	0.281280
$28$	−23.3885	−4.42001
$29$	3.45889	0.642299	0.321150	−	0.947028i	$-0.395931\pi$
$29$	3.45889	0.642299	0.321150	+	0.947028i	$0.395931\pi$
$30$	−24.2892	−4.43458
$31$	−2.75496	−0.494805	−0.247403	−	0.968913i	$-0.579577\pi$
$31$	−2.75496	−0.494805	−0.247403	+	0.968913i	$0.579577\pi$
$32$	−18.1330	−3.20550
$33$	−12.1908	−2.12214
$34$	2.18577	0.374856
$35$	−15.5787	−2.63328
$36$	18.8314	3.13857
$37$	6.94846	1.14232	0.571160	−	0.820839i	$-0.306494\pi$
$37$	6.94846	1.14232	0.571160	+	0.820839i	$0.306494\pi$
$38$	10.6067	1.72064
$39$	2.56324	0.410447
$40$	−31.0301	−4.90629
$41$	−0.642116	−0.100282	−0.0501409	−	0.998742i	$-0.515967\pi$
$41$	−0.642116	−0.100282	−0.0501409	+	0.998742i	$0.515967\pi$
$42$	30.6553	4.73022
$43$	−9.53608	−1.45424	−0.727119	−	0.686511i	$-0.759141\pi$
$43$	−9.53608	−1.45424	−0.727119	+	0.686511i	$0.759141\pi$
$44$	−25.0860	−3.78186
$45$	12.5433	1.86985
$46$	14.2934	2.10744
$47$	10.7047	1.56145	0.780724	−	0.624876i	$-0.214850\pi$
$47$	10.7047	1.56145	0.780724	+	0.624876i	$0.214850\pi$
$48$	34.0200	4.91036
$49$	12.6618	1.80883
$50$	−19.8065	−2.80106
$51$	−2.07725	−0.290873
$52$	5.27461	0.731456
$53$	10.5586	1.45034	0.725169	−	0.688571i	$-0.241762\pi$
$53$	10.5586	1.45034	0.725169	+	0.688571i	$0.241762\pi$
$54$	−3.94208	−0.536450
$55$	−16.7094	−2.25309
$56$	39.1630	5.23337
$57$	−10.0801	−1.33515
$58$	−9.32913	−1.22497
$59$	0.223495	0.0290966	0.0145483	−	0.999894i	$-0.495369\pi$
$59$	0.223495	0.0290966	0.0145483	+	0.999894i	$0.495369\pi$
$60$	47.5005	6.13229
$61$	8.54254	1.09376	0.546880	−	0.837211i	$-0.315815\pi$
$61$	8.54254	1.09376	0.546880	+	0.837211i	$0.315815\pi$
$62$	7.43053	0.943679
$63$	−15.8309	−1.99450
$64$	22.3630	2.79537
$65$	3.51333	0.435775
$66$	32.8803	4.04728
$67$	2.66873	0.326037	0.163018	−	0.986623i	$-0.447877\pi$
$67$	2.66873	0.326037	0.163018	+	0.986623i	$0.447877\pi$
$68$	−4.27454	−0.518365
$69$	−13.5837	−1.63529
$70$	42.0180	5.02211
$71$	10.2634	1.21805	0.609023	−	0.793153i	$-0.291562\pi$
$71$	10.2634	1.21805	0.609023	+	0.793153i	$0.291562\pi$
$72$	−31.5324	−3.71613
$73$	8.73922	1.02285	0.511424	−	0.859328i	$-0.329118\pi$
$73$	8.73922	1.02285	0.511424	+	0.859328i	$0.329118\pi$
$74$	−18.7410	−2.17860
$75$	18.8231	2.17351
$76$	−20.7428	−2.37936
$77$	21.0889	2.40330
$78$	−6.91344	−0.782793
$79$	16.0883	1.81008	0.905039	−	0.425329i	$-0.139842\pi$
$79$	16.0883	1.81008	0.905039	+	0.425329i	$0.139842\pi$
$80$	46.6298	5.21337
$81$	−6.96425	−0.773805
$82$	1.73188	0.191254
$83$	10.9369	1.20048	0.600242	−	0.799818i	$-0.295071\pi$
$83$	10.9369	1.20048	0.600242	+	0.799818i	$0.295071\pi$
$84$	−59.9503	−6.54111
$85$	−2.84720	−0.308823
$86$	25.7202	2.77348
$87$	8.86596	0.950531
$88$	42.0054	4.47779
$89$	−16.8260	−1.78356	−0.891778	−	0.452473i	$-0.850542\pi$
$89$	−16.8260	−1.78356	−0.891778	+	0.452473i	$0.850542\pi$
$90$	−33.8312	−3.56612
$91$	−4.43417	−0.464827
$92$	−27.9524	−2.91424
$93$	−7.06163	−0.732257
$94$	−28.8723	−2.97795
$95$	−13.8164	−1.41754
$96$	−46.4794	−4.74378
$97$	−9.97225	−1.01253	−0.506264	−	0.862378i	$-0.668974\pi$
$97$	−9.97225	−1.01253	−0.506264	+	0.862378i	$0.668974\pi$
$98$	−34.1508	−3.44975
$99$	−16.9799	−1.70654
$100$	38.7340	3.87340
$101$	10.7722	1.07187	0.535935	−	0.844259i	$-0.319959\pi$
$101$	10.7722	1.07187	0.535935	+	0.844259i	$0.319959\pi$
$102$	5.60266	0.554746
$103$	4.46053	0.439509	0.219754	−	0.975555i	$-0.429474\pi$
$103$	4.46053	0.439509	0.219754	+	0.975555i	$0.429474\pi$
$104$	−8.83210	−0.866058
$105$	−39.9319	−3.89696
$106$	−28.4782	−2.76604
$107$	7.66288	0.740799	0.370399	−	0.928873i	$-0.379221\pi$
$107$	7.66288	0.740799	0.370399	+	0.928873i	$0.379221\pi$
$108$	7.70923	0.741821
$109$	9.06538	0.868306	0.434153	−	0.900839i	$-0.357048\pi$
$109$	9.06538	0.868306	0.434153	+	0.900839i	$0.357048\pi$
$110$	45.0677	4.29703
$111$	17.8106	1.69051
$112$	−58.8514	−5.56093
$113$	19.8875	1.87086	0.935429	−	0.353514i	$-0.115013\pi$
$113$	19.8875	1.87086	0.935429	+	0.353514i	$0.115013\pi$
$114$	27.1876	2.54635
$115$	−18.6187	−1.73620
$116$	18.2443	1.69394
$117$	3.57021	0.330066
$118$	−0.602799	−0.0554921
$119$	3.59345	0.329411
$120$	−79.5376	−7.26075
$121$	11.6195	1.05632
$122$	−23.0405	−2.08599
$123$	−1.64590	−0.148406
$124$	−14.5313	−1.30495
$125$	8.23345	0.736422
$126$	42.6982	3.80386
$127$	19.0033	1.68627	0.843137	−	0.537699i	$-0.180706\pi$
$127$	19.0033	1.68627	0.843137	+	0.537699i	$0.180706\pi$
$128$	−24.0502	−2.12576
$129$	−24.4433	−2.15211
$130$	−9.47597	−0.831098
$131$	−7.79073	−0.680679	−0.340339	−	0.940303i	$-0.610542\pi$
$131$	−7.79073	−0.680679	−0.340339	+	0.940303i	$0.610542\pi$
$132$	−64.3015	−5.59672
$133$	17.4377	1.51204
$134$	−7.19795	−0.621808
$135$	5.13499	0.441950
$136$	7.15754	0.613754
$137$	−11.9693	−1.02260	−0.511302	−	0.859401i	$-0.670837\pi$
$137$	−11.9693	−1.02260	−0.511302	+	0.859401i	$0.670837\pi$
$138$	36.6373	3.11878
$139$	−18.0682	−1.53252	−0.766260	−	0.642530i	$-0.777885\pi$
$139$	−18.0682	−1.53252	−0.766260	+	0.642530i	$0.777885\pi$
$140$	−82.1714	−6.94475
$141$	27.4389	2.31077
$142$	−27.6820	−2.32302
$143$	−4.75599	−0.397716
$144$	47.3847	3.94872
$145$	12.1522	1.00919
$146$	−23.5710	−1.95075
$147$	32.4553	2.67687
$148$	36.6504	3.01264
$149$	4.55383	0.373064	0.186532	−	0.982449i	$-0.440275\pi$
$149$	4.55383	0.373064	0.186532	+	0.982449i	$0.440275\pi$
$150$	−50.7688	−4.14525
$151$	−12.0773	−0.982840	−0.491420	−	0.870923i	$-0.663522\pi$
$151$	−12.0773	−0.982840	−0.491420	+	0.870923i	$0.663522\pi$
$152$	34.7329	2.81721
$153$	−2.89330	−0.233909
$154$	−56.8798	−4.58350
$155$	−9.67908	−0.777443
$156$	13.5201	1.08247
$157$	−17.2426	−1.37611	−0.688055	−	0.725659i	$-0.741535\pi$
$157$	−17.2426	−1.37611	−0.688055	+	0.725659i	$0.741535\pi$
$158$	−43.3926	−3.45213
$159$	27.0643	2.14634
$160$	−63.7074	−5.03651
$161$	23.4986	1.85195
$162$	18.7836	1.47578
$163$	4.70748	0.368718	0.184359	−	0.982859i	$-0.440979\pi$
$163$	4.70748	0.368718	0.184359	+	0.982859i	$0.440979\pi$
$164$	−3.38691	−0.264473
$165$	−42.8302	−3.33432
$166$	−29.4985	−2.28953
$167$	−15.0767	−1.16667	−0.583335	−	0.812232i	$-0.698253\pi$
$167$	−15.0767	−1.16667	−0.583335	+	0.812232i	$0.698253\pi$
$168$	100.384	7.74481
$169$	1.00000	0.0769231
$170$	7.67933	0.588978
$171$	−14.0401	−1.07367
$172$	−50.2991	−3.83527
$173$	4.84945	0.368697	0.184349	−	0.982861i	$-0.440982\pi$
$173$	4.84945	0.368697	0.184349	+	0.982861i	$0.440982\pi$
$174$	−23.9128	−1.81283
$175$	−32.5622	−2.46147
$176$	−63.1228	−4.75806
$177$	0.572872	0.0430597
$178$	45.3823	3.40155
$179$	22.8950	1.71125	0.855626	−	0.517594i	$-0.173172\pi$
$179$	22.8950	1.71125	0.855626	+	0.517594i	$0.173172\pi$
$180$	66.1610	4.93135
$181$	20.0816	1.49265	0.746327	−	0.665579i	$-0.231815\pi$
$181$	20.0816	1.49265	0.746327	+	0.665579i	$0.231815\pi$
$182$	11.9596	0.886504
$183$	21.8966	1.61864
$184$	46.8051	3.45052
$185$	24.4122	1.79482
$186$	19.0463	1.39654
$187$	3.85426	0.281851
$188$	56.4633	4.11801
$189$	−6.48086	−0.471414
$190$	37.2650	2.70349
$191$	0.826856	0.0598292	0.0299146	−	0.999552i	$-0.490476\pi$
$191$	0.826856	0.0598292	0.0299146	+	0.999552i	$0.490476\pi$
$192$	57.3217	4.13684
$193$	20.1405	1.44974	0.724872	−	0.688884i	$-0.241899\pi$
$193$	20.1405	1.44974	0.724872	+	0.688884i	$0.241899\pi$
$194$	26.8966	1.93107
$195$	9.00551	0.644898
$196$	66.7861	4.77044
$197$	4.06066	0.289310	0.144655	−	0.989482i	$-0.453793\pi$
$197$	4.06066	0.289310	0.144655	+	0.989482i	$0.453793\pi$
$198$	45.7972	3.25467
$199$	1.64117	0.116339	0.0581697	−	0.998307i	$-0.481474\pi$
$199$	1.64117	0.116339	0.0581697	+	0.998307i	$0.481474\pi$
$200$	−64.8584	−4.58618
$201$	6.84059	0.482498
$202$	−29.0541	−2.04424
$203$	−15.3373	−1.07647
$204$	−10.9567	−0.767122
$205$	−2.25597	−0.157564
$206$	−12.0307	−0.838219
$207$	−18.9201	−1.31504
$208$	13.2723	0.920265
$209$	18.7033	1.29373
$210$	107.702	7.43217
$211$	12.3398	0.849505	0.424753	−	0.905309i	$-0.360361\pi$
$211$	12.3398	0.849505	0.424753	+	0.905309i	$0.360361\pi$
$212$	55.6926	3.82498
$213$	26.3077	1.80257
$214$	−20.6679	−1.41283
$215$	−33.5034	−2.28491
$216$	−12.9088	−0.878331
$217$	12.2159	0.829273
$218$	−24.4507	−1.65601
$219$	22.4007	1.51370
$220$	−88.1354	−5.94209
$221$	−0.810401	−0.0545134
$222$	−48.0378	−3.22408
$223$	22.8200	1.52814	0.764070	−	0.645133i	$-0.223198\pi$
$223$	22.8200	1.52814	0.764070	+	0.645133i	$0.223198\pi$
$224$	80.4049	5.37228
$225$	26.2178	1.74785
$226$	−53.6395	−3.56805
$227$	−1.41334	−0.0938067	−0.0469033	−	0.998899i	$-0.514935\pi$
$227$	−1.41334	−0.0938067	−0.0469033	+	0.998899i	$0.514935\pi$
$228$	−53.1688	−3.52119
$229$	2.94128	0.194365	0.0971827	−	0.995267i	$-0.469017\pi$
$229$	2.94128	0.194365	0.0971827	+	0.995267i	$0.469017\pi$
$230$	50.2173	3.31123
$231$	54.0558	3.55661
$232$	−30.5492	−2.00566
$233$	−6.03642	−0.395459	−0.197730	−	0.980257i	$-0.563357\pi$
$233$	−6.03642	−0.395459	−0.197730	+	0.980257i	$0.563357\pi$
$234$	−9.62937	−0.629492
$235$	37.6093	2.45336
$236$	1.17885	0.0767365
$237$	41.2383	2.67871
$238$	−9.69207	−0.628243
$239$	−10.7650	−0.696327	−0.348164	−	0.937434i	$-0.613195\pi$
$239$	−10.7650	−0.696327	−0.348164	+	0.937434i	$0.613195\pi$
$240$	119.523	7.71521
$241$	−21.4370	−1.38088	−0.690441	−	0.723389i	$-0.742583\pi$
$241$	−21.4370	−1.38088	−0.690441	+	0.723389i	$0.742583\pi$
$242$	−31.3395	−2.01458
$243$	−22.2358	−1.42643
$244$	45.0586	2.88458
$245$	44.4852	2.84205
$246$	4.43923	0.283035
$247$	−3.93258	−0.250224
$248$	24.3321	1.54509
$249$	28.0340	1.77658
$250$	−22.2068	−1.40448
$251$	5.09122	0.321355	0.160678	−	0.987007i	$-0.448632\pi$
$251$	5.09122	0.321355	0.160678	+	0.987007i	$0.448632\pi$
$252$	−83.5017	−5.26011
$253$	25.2041	1.58457
$254$	−51.2548	−3.21601
$255$	−7.29807	−0.457023
$256$	20.1410	1.25881
$257$	−0.931014	−0.0580750	−0.0290375	−	0.999578i	$-0.509244\pi$
$257$	−0.931014	−0.0580750	−0.0290375	+	0.999578i	$0.509244\pi$
$258$	65.9271	4.10444
$259$	−30.8106	−1.91448
$260$	18.5314	1.14927
$261$	12.3489	0.764380
$262$	21.0127	1.29817
$263$	5.71536	0.352424	0.176212	−	0.984352i	$-0.443616\pi$
$263$	5.71536	0.352424	0.176212	+	0.984352i	$0.443616\pi$
$264$	107.670	6.62663
$265$	37.0959	2.27878
$266$	−47.0320	−2.88372
$267$	−43.1292	−2.63946
$268$	14.0765	0.859858
$269$	3.40373	0.207529	0.103765	−	0.994602i	$-0.466911\pi$
$269$	3.40373	0.207529	0.103765	+	0.994602i	$0.466911\pi$
$270$	−13.8498	−0.842874
$271$	−6.20548	−0.376956	−0.188478	−	0.982077i	$-0.560355\pi$
$271$	−6.20548	−0.376956	−0.188478	+	0.982077i	$0.560355\pi$
$272$	−10.7558	−0.652169
$273$	−11.3658	−0.687892
$274$	32.2829	1.95028
$275$	−34.9256	−2.10609
$276$	−71.6488	−4.31275
$277$	4.25925	0.255914	0.127957	−	0.991780i	$-0.459158\pi$
$277$	4.25925	0.255914	0.127957	+	0.991780i	$0.459158\pi$
$278$	48.7325	2.92278
$279$	−9.83577	−0.588852
$280$	137.592	8.22272
$281$	−15.6608	−0.934242	−0.467121	−	0.884193i	$-0.654709\pi$
$281$	−15.6608	−0.934242	−0.467121	+	0.884193i	$0.654709\pi$
$282$	−74.0066	−4.40703
$283$	−14.7577	−0.877256	−0.438628	−	0.898669i	$-0.644535\pi$
$283$	−14.7577	−0.877256	−0.438628	+	0.898669i	$0.644535\pi$
$284$	54.1356	3.21236
$285$	−35.4149	−2.09780
$286$	12.8276	0.758513
$287$	2.84725	0.168068
$288$	−64.7387	−3.81476
$289$	−16.3433	−0.961368
$290$	−32.7763	−1.92469
$291$	−25.5613	−1.49843
$292$	46.0960	2.69756
$293$	23.5494	1.37577	0.687886	−	0.725819i	$-0.258539\pi$
$293$	23.5494	1.37577	0.687886	+	0.725819i	$0.258539\pi$
$294$	−87.5367	−5.10525
$295$	0.785212	0.0457168
$296$	−61.3695	−3.56703
$297$	−6.95124	−0.403352
$298$	−12.2823	−0.711497
$299$	−5.29943	−0.306474
$300$	99.2846	5.73220
$301$	42.2845	2.43724
$302$	32.5744	1.87444
$303$	27.6117	1.58625
$304$	−52.1942	−2.99354
$305$	30.0128	1.71853
$306$	7.80365	0.446105
$307$	21.3989	1.22130	0.610649	−	0.791901i	$-0.290909\pi$
$307$	21.3989	1.22130	0.610649	+	0.791901i	$0.290909\pi$
$308$	111.235	6.33823
$309$	11.4334	0.650424
$310$	26.1059	1.48272
$311$	4.65855	0.264162	0.132081	−	0.991239i	$-0.457834\pi$
$311$	4.65855	0.264162	0.132081	+	0.991239i	$0.457834\pi$
$312$	−22.6388	−1.28167
$313$	13.6519	0.771652	0.385826	−	0.922572i	$-0.373916\pi$
$313$	13.6519	0.771652	0.385826	+	0.922572i	$0.373916\pi$
$314$	46.5058	2.62448
$315$	−55.6191	−3.13378
$316$	84.8596	4.77372
$317$	6.23044	0.349936	0.174968	−	0.984574i	$-0.444018\pi$
$317$	6.23044	0.349936	0.174968	+	0.984574i	$0.444018\pi$
$318$	−72.9964	−4.09343
$319$	−16.4504	−0.921048
$320$	78.5686	4.39212
$321$	19.6418	1.09630
$322$	−63.3791	−3.53198
$323$	3.18696	0.177327
$324$	−36.7337	−2.04076
$325$	7.34349	0.407344
$326$	−12.6968	−0.703209
$327$	23.2368	1.28500
$328$	5.67123	0.313141
$329$	−47.4666	−2.61692
$330$	115.519	6.35913
$331$	4.55570	0.250404	0.125202	−	0.992131i	$-0.460042\pi$
$331$	4.55570	0.250404	0.125202	+	0.992131i	$0.460042\pi$
$332$	57.6880	3.16604
$333$	24.8074	1.35944
$334$	40.6641	2.22504
$335$	9.37611	0.512272
$336$	−150.850	−8.22956
$337$	−11.6019	−0.631997	−0.315998	−	0.948760i	$-0.602339\pi$
$337$	−11.6019	−0.631997	−0.315998	+	0.948760i	$0.602339\pi$
$338$	−2.69715	−0.146705
$339$	50.9765	2.76866
$340$	−15.0179	−0.814459
$341$	13.1026	0.709544
$342$	37.8682	2.04768
$343$	−25.1055	−1.35557
$344$	84.2236	4.54103
$345$	−47.7241	−2.56938
$346$	−13.0797	−0.703169
$347$	28.6125	1.53600	0.768000	−	0.640450i	$-0.221252\pi$
$347$	28.6125	1.53600	0.768000	+	0.640450i	$0.221252\pi$
$348$	46.7645	2.50684
$349$	−28.4300	−1.52182	−0.760910	−	0.648857i	$-0.775247\pi$
$349$	−28.4300	−1.52182	−0.760910	+	0.648857i	$0.775247\pi$
$350$	87.8252	4.69445
$351$	1.46157	0.0780131
$352$	86.2407	4.59664
$353$	20.9877	1.11706	0.558530	−	0.829484i	$-0.311366\pi$
$353$	20.9877	1.11706	0.558530	+	0.829484i	$0.311366\pi$
$354$	−1.54512	−0.0821222
$355$	36.0588	1.91380
$356$	−88.7507	−4.70378
$357$	9.21088	0.487491
$358$	−61.7512	−3.26365
$359$	−20.6896	−1.09196	−0.545978	−	0.837800i	$-0.683842\pi$
$359$	−20.6896	−1.09196	−0.545978	+	0.837800i	$0.683842\pi$
$360$	−110.784	−5.83882
$361$	−3.53485	−0.186045
$362$	−54.1631	−2.84675
$363$	29.7835	1.56323
$364$	−23.3885	−1.22589
$365$	30.7038	1.60711
$366$	−59.0584	−3.08703
$367$	−29.4162	−1.53551	−0.767756	−	0.640742i	$-0.778627\pi$
$367$	−29.4162	−1.53551	−0.767756	+	0.640742i	$0.778627\pi$
$368$	−70.3354	−3.66649
$369$	−2.29249	−0.119342
$370$	−65.8434	−3.42304
$371$	−46.8187	−2.43070
$372$	−37.2473	−1.93118
$373$	7.33708	0.379900	0.189950	−	0.981794i	$-0.439167\pi$
$373$	7.33708	0.379900	0.189950	+	0.981794i	$0.439167\pi$
$374$	−10.3955	−0.537539
$375$	21.1043	1.08982
$376$	−94.5454	−4.87581
$377$	3.45889	0.178142
$378$	17.4798	0.899066
$379$	4.55679	0.234066	0.117033	−	0.993128i	$-0.462662\pi$
$379$	4.55679	0.234066	0.117033	+	0.993128i	$0.462662\pi$
$380$	−72.8763	−3.73847
$381$	48.7101	2.49550
$382$	−2.23015	−0.114105
$383$	26.2496	1.34129	0.670647	−	0.741777i	$-0.266017\pi$
$383$	26.2496	1.34129	0.670647	+	0.741777i	$0.266017\pi$
$384$	−61.6465	−3.14588
$385$	74.0921	3.77609
$386$	−54.3219	−2.76491
$387$	−34.0458	−1.73064
$388$	−52.5997	−2.67034
$389$	−11.7604	−0.596274	−0.298137	−	0.954523i	$-0.596365\pi$
$389$	−11.7604	−0.596274	−0.298137	+	0.954523i	$0.596365\pi$
$390$	−24.2892	−1.22993
$391$	4.29466	0.217190
$392$	−111.830	−5.64829
$393$	−19.9695	−1.00733
$394$	−10.9522	−0.551763
$395$	56.5236	2.84401
$396$	−89.5622	−4.50067
$397$	−37.1277	−1.86339	−0.931693	−	0.363246i	$-0.881669\pi$
$397$	−37.1277	−1.86339	−0.931693	+	0.363246i	$0.881669\pi$
$398$	−4.42648	−0.221879
$399$	44.6970	2.23765
$400$	97.4647	4.87323
$401$	−18.6705	−0.932360	−0.466180	−	0.884690i	$-0.654370\pi$
$401$	−18.6705	−0.932360	−0.466180	+	0.884690i	$0.654370\pi$
$402$	−18.4501	−0.920206
$403$	−2.75496	−0.137234
$404$	56.8189	2.82685
$405$	−24.4677	−1.21581
$406$	41.3669	2.05301
$407$	−33.0468	−1.63807
$408$	18.3465	0.908287
$409$	−17.8141	−0.880850	−0.440425	−	0.897789i	$-0.645172\pi$
$409$	−17.8141	−0.880850	−0.440425	+	0.897789i	$0.645172\pi$
$410$	6.08468	0.300501
$411$	−30.6801	−1.51334
$412$	23.5275	1.15912
$413$	−0.991014	−0.0487646
$414$	51.0302	2.50800
$415$	38.4251	1.88621
$416$	−18.1330	−0.889046
$417$	−46.3130	−2.26796
$418$	−50.4456	−2.46737
$419$	−21.3132	−1.04122	−0.520610	−	0.853795i	$-0.674295\pi$
$419$	−21.3132	−1.04122	−0.520610	+	0.853795i	$0.674295\pi$
$420$	−210.625	−10.2775
$421$	32.5008	1.58399	0.791997	−	0.610525i	$-0.209042\pi$
$421$	32.5008	1.58399	0.791997	+	0.610525i	$0.209042\pi$
$422$	−33.2822	−1.62015
$423$	38.2182	1.85823
$424$	−93.2548	−4.52885
$425$	−5.95117	−0.288674
$426$	−70.9556	−3.43781
$427$	−37.8791	−1.83310
$428$	40.4187	1.95371
$429$	−12.1908	−0.588575
$430$	90.3636	4.35772
$431$	27.6717	1.33290	0.666450	−	0.745549i	$-0.267813\pi$
$431$	27.6717	1.33290	0.666450	+	0.745549i	$0.267813\pi$
$432$	19.3984	0.933306
$433$	−22.2683	−1.07014	−0.535072	−	0.844806i	$-0.679716\pi$
$433$	−22.2683	−1.07014	−0.535072	+	0.844806i	$0.679716\pi$
$434$	−32.9482	−1.58157
$435$	31.1491	1.49348
$436$	47.8163	2.28999
$437$	20.8404	0.996933
$438$	−60.4181	−2.88689
$439$	−6.64179	−0.316996	−0.158498	−	0.987359i	$-0.550665\pi$
$439$	−6.64179	−0.316996	−0.158498	+	0.987359i	$0.550665\pi$
$440$	147.579	7.03555
$441$	45.2053	2.15263
$442$	2.18577	0.103966
$443$	−23.4368	−1.11352	−0.556758	−	0.830675i	$-0.687955\pi$
$443$	−23.4368	−1.11352	−0.556758	+	0.830675i	$0.687955\pi$
$444$	93.9438	4.45837
$445$	−59.1154	−2.80234
$446$	−61.5489	−2.91443
$447$	11.6726	0.552093
$448$	−99.1612	−4.68493
$449$	7.30338	0.344668	0.172334	−	0.985039i	$-0.444869\pi$
$449$	7.30338	0.344668	0.172334	+	0.985039i	$0.444869\pi$
$450$	−70.7132	−3.33345
$451$	3.05390	0.143803
$452$	104.899	4.93402
$453$	−30.9571	−1.45449
$454$	3.81199	0.178905
$455$	−15.5787	−0.730340
$456$	89.0288	4.16915
$457$	−7.37875	−0.345163	−0.172582	−	0.984995i	$-0.555211\pi$
$457$	−7.37875	−0.345163	−0.172582	+	0.984995i	$0.555211\pi$
$458$	−7.93307	−0.370688
$459$	−1.18446	−0.0552859
$460$	−98.2061	−4.57888
$461$	9.68840	0.451234	0.225617	−	0.974216i	$-0.427560\pi$
$461$	9.68840	0.451234	0.225617	+	0.974216i	$0.427560\pi$
$462$	−145.797	−6.78307
$463$	2.19231	0.101885	0.0509427	−	0.998702i	$-0.483777\pi$
$463$	2.19231	0.101885	0.0509427	+	0.998702i	$0.483777\pi$
$464$	45.9072	2.13119
$465$	−24.8098	−1.15053
$466$	16.2811	0.754208
$467$	29.5710	1.36838	0.684192	−	0.729302i	$-0.260155\pi$
$467$	29.5710	1.36838	0.684192	+	0.729302i	$0.260155\pi$
$468$	18.8314	0.870483
$469$	−11.8336	−0.546423
$470$	−101.438	−4.67898
$471$	−44.1969	−2.03649
$472$	−1.97393	−0.0908575
$473$	45.3535	2.08536
$474$	−111.226	−5.10876
$475$	−28.8788	−1.32505
$476$	18.9540	0.868757
$477$	37.6965	1.72600
$478$	29.0347	1.32802
$479$	29.9489	1.36840	0.684201	−	0.729294i	$-0.260151\pi$
$479$	29.9489	1.36840	0.684201	+	0.729294i	$0.260151\pi$
$480$	−163.297	−7.45347
$481$	6.94846	0.316822
$482$	57.8189	2.63358
$483$	60.2325	2.74067
$484$	61.2882	2.78583
$485$	−35.0358	−1.59089
$486$	59.9732	2.72044
$487$	−15.6538	−0.709343	−0.354671	−	0.934991i	$-0.615407\pi$
$487$	−15.6538	−0.709343	−0.354671	+	0.934991i	$0.615407\pi$
$488$	−75.4486	−3.41540
$489$	12.0664	0.545662
$490$	−119.983	−5.42028
$491$	4.45739	0.201159	0.100580	−	0.994929i	$-0.467930\pi$
$491$	4.45739	0.201159	0.100580	+	0.994929i	$0.467930\pi$
$492$	−8.68147	−0.391391
$493$	−2.80308	−0.126245
$494$	10.6067	0.477220
$495$	−59.6559	−2.68133
$496$	−36.5645	−1.64180
$497$	−45.5098	−2.04139
$498$	−75.6118	−3.38825
$499$	−20.8285	−0.932412	−0.466206	−	0.884676i	$-0.654379\pi$
$499$	−20.8285	−0.932412	−0.466206	+	0.884676i	$0.654379\pi$
$500$	43.4282	1.94217
$501$	−38.6452	−1.72654
$502$	−13.7318	−0.612880
$503$	−5.95760	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$503$	−5.95760	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$504$	139.820	6.22807
$505$	37.8462	1.68413
$506$	−67.9791	−3.02204
$507$	2.56324	0.113838
$508$	100.235	4.44721
$509$	14.8961	0.660257	0.330128	−	0.943936i	$-0.392908\pi$
$509$	14.8961	0.660257	0.330128	+	0.943936i	$0.392908\pi$
$510$	19.6840	0.871621
$511$	−38.7512	−1.71425
$512$	−6.22275	−0.275009
$513$	−5.74775	−0.253770
$514$	2.51108	0.110759
$515$	15.6713	0.690560
$516$	−128.929	−5.67577
$517$	−50.9117	−2.23909
$518$	83.1008	3.65124
$519$	12.4303	0.545630
$520$	−31.0301	−1.36076
$521$	−20.1690	−0.883619	−0.441810	−	0.897109i	$-0.645663\pi$
$521$	−20.1690	−0.883619	−0.441810	+	0.897109i	$0.645663\pi$
$522$	−33.3069	−1.45780
$523$	19.2927	0.843610	0.421805	−	0.906687i	$-0.361397\pi$
$523$	19.2927	0.843610	0.421805	+	0.906687i	$0.361397\pi$
$524$	−41.0930	−1.79516
$525$	−83.4649	−3.64271
$526$	−15.4152	−0.672133
$527$	2.23262	0.0972545
$528$	−161.799	−7.04139
$529$	5.08399	0.221043
$530$	−100.053	−4.34603
$531$	0.797923	0.0346269
$532$	91.9770	3.98771
$533$	−0.642116	−0.0278131
$534$	116.326	5.03391
$535$	26.9222	1.16395
$536$	−23.5704	−1.01809
$537$	58.6854	2.53246
$538$	−9.18036	−0.395794
$539$	−60.2195	−2.59384
$540$	27.0851	1.16556
$541$	40.2765	1.73162	0.865811	−	0.500372i	$-0.166803\pi$
$541$	40.2765	1.73162	0.865811	+	0.500372i	$0.166803\pi$
$542$	16.7371	0.718920
$543$	51.4740	2.20896
$544$	14.6950	0.630044
$545$	31.8497	1.36429
$546$	30.6553	1.31193
$547$	−20.1229	−0.860393	−0.430197	−	0.902735i	$-0.641556\pi$
$547$	−20.1229	−0.860393	−0.430197	+	0.902735i	$0.641556\pi$
$548$	−63.1332	−2.69692
$549$	30.4986	1.30165
$550$	94.1995	4.01668
$551$	−13.6023	−0.579479
$552$	119.973	5.10638
$553$	−71.3383	−3.03361
$554$	−11.4878	−0.488071
$555$	62.5744	2.65614
$556$	−95.3024	−4.04172
$557$	−17.7961	−0.754046	−0.377023	−	0.926204i	$-0.623052\pi$
$557$	−17.7961	−0.754046	−0.377023	+	0.926204i	$0.623052\pi$
$558$	26.5285	1.12304
$559$	−9.53608	−0.403333
$560$	−206.764	−8.73739
$561$	9.87940	0.417109
$562$	42.2394	1.78176
$563$	10.2399	0.431560	0.215780	−	0.976442i	$-0.430771\pi$
$563$	10.2399	0.431560	0.215780	+	0.976442i	$0.430771\pi$
$564$	144.729	6.09420
$565$	69.8713	2.93951
$566$	39.8038	1.67308
$567$	30.8806	1.29686
$568$	−90.6477	−3.80349
$569$	−27.5884	−1.15657	−0.578283	−	0.815836i	$-0.696277\pi$
$569$	−27.5884	−1.15657	−0.578283	+	0.815836i	$0.696277\pi$
$570$	95.5191	4.00086
$571$	−8.32982	−0.348592	−0.174296	−	0.984693i	$-0.555765\pi$
$571$	−8.32982	−0.348592	−0.174296	+	0.984693i	$0.555765\pi$
$572$	−25.0860	−1.04890
$573$	2.11943	0.0885405
$574$	−7.67945	−0.320534
$575$	−38.9163	−1.62292
$576$	79.8405	3.32669
$577$	−17.4929	−0.728239	−0.364120	−	0.931352i	$-0.618630\pi$
$577$	−17.4929	−0.728239	−0.364120	+	0.931352i	$0.618630\pi$
$578$	44.0802	1.83349
$579$	51.6249	2.14546
$580$	64.0981	2.66153
$581$	−48.4962	−2.01196
$582$	68.9425	2.85776
$583$	−50.2167	−2.07976
$584$	−77.1857	−3.19397
$585$	12.5433	0.518602
$586$	−63.5162	−2.62383
$587$	−0.801730	−0.0330909	−0.0165455	−	0.999863i	$-0.505267\pi$
$587$	−0.801730	−0.0330909	−0.0165455	+	0.999863i	$0.505267\pi$
$588$	171.189	7.05971
$589$	10.8341	0.446411
$590$	−2.11783	−0.0871898
$591$	10.4084	0.428146
$592$	92.2217	3.79029
$593$	32.8297	1.34815	0.674077	−	0.738661i	$-0.264542\pi$
$593$	32.8297	1.34815	0.674077	+	0.738661i	$0.264542\pi$
$594$	18.7485	0.769261
$595$	12.6250	0.517574
$596$	24.0196	0.983883
$597$	4.20672	0.172169
$598$	14.2934	0.584499
$599$	24.4970	1.00092	0.500461	−	0.865759i	$-0.333164\pi$
$599$	24.4970	1.00092	0.500461	+	0.865759i	$0.333164\pi$
$600$	−166.248	−6.78704
$601$	−30.0913	−1.22745	−0.613724	−	0.789520i	$-0.710329\pi$
$601$	−30.0913	−1.22745	−0.613724	+	0.789520i	$0.710329\pi$
$602$	−114.048	−4.64824
$603$	9.52790	0.388006
$604$	−63.7032	−2.59205
$605$	40.8231	1.65969
$606$	−74.4727	−3.02525
$607$	−19.1767	−0.778360	−0.389180	−	0.921162i	$-0.627242\pi$
$607$	−19.1767	−0.778360	−0.389180	+	0.921162i	$0.627242\pi$
$608$	71.3096	2.89199
$609$	−39.3131	−1.59305
$610$	−80.9489	−3.27753
$611$	10.7047	0.433068
$612$	−15.2610	−0.616889
$613$	−11.5023	−0.464573	−0.232286	−	0.972647i	$-0.574621\pi$
$613$	−11.5023	−0.464573	−0.232286	+	0.972647i	$0.574621\pi$
$614$	−57.7159	−2.32923
$615$	−5.78259	−0.233176
$616$	−186.259	−7.50458
$617$	35.4244	1.42613	0.713066	−	0.701097i	$-0.247306\pi$
$617$	35.4244	1.42613	0.713066	+	0.701097i	$0.247306\pi$
$618$	−30.8376	−1.24047
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−51.0534	−2.05035
$621$	−7.74552	−0.310817
$622$	−12.5648	−0.503803
$623$	74.6094	2.98916
$624$	34.0200	1.36189
$625$	−7.79061	−0.311624
$626$	−36.8212	−1.47167
$627$	47.9411	1.91458
$628$	−90.9479	−3.62922
$629$	−5.63104	−0.224524
$630$	150.013	5.97666
$631$	33.6504	1.33960	0.669800	−	0.742542i	$-0.266380\pi$
$631$	33.6504	1.33960	0.669800	+	0.742542i	$0.266380\pi$
$632$	−142.094	−5.65218
$633$	31.6298	1.25717
$634$	−16.8044	−0.667388
$635$	66.7650	2.64949
$636$	142.753	5.66054
$637$	12.6618	0.501680
$638$	44.3693	1.75660
$639$	36.6426	1.44956
$640$	−84.4963	−3.34001
$641$	40.4595	1.59806	0.799028	−	0.601294i	$-0.205348\pi$
$641$	40.4595	1.59806	0.799028	+	0.601294i	$0.205348\pi$
$642$	−52.9769	−2.09083
$643$	−6.88093	−0.271357	−0.135679	−	0.990753i	$-0.543321\pi$
$643$	−6.88093	−0.271357	−0.135679	+	0.990753i	$0.543321\pi$
$644$	123.946	4.88414
$645$	−85.8773	−3.38141
$646$	−8.59571	−0.338194
$647$	−18.2720	−0.718346	−0.359173	−	0.933271i	$-0.616941\pi$
$647$	−18.2720	−0.718346	−0.359173	+	0.933271i	$0.616941\pi$
$648$	61.5089	2.41630
$649$	−1.06294	−0.0417241
$650$	−19.8065	−0.776874
$651$	31.3124	1.22723
$652$	24.8301	0.972422
$653$	−2.13855	−0.0836880	−0.0418440	−	0.999124i	$-0.513323\pi$
$653$	−2.13855	−0.0836880	−0.0418440	+	0.999124i	$0.513323\pi$
$654$	−62.6730	−2.45071
$655$	−27.3714	−1.06949
$656$	−8.52233	−0.332741
$657$	31.2008	1.21726
$658$	128.025	4.99091
$659$	−21.0740	−0.820927	−0.410464	−	0.911877i	$-0.634633\pi$
$659$	−21.0740	−0.820927	−0.410464	+	0.911877i	$0.634633\pi$
$660$	−225.912	−8.79362
$661$	16.4005	0.637904	0.318952	−	0.947771i	$-0.396669\pi$
$661$	16.4005	0.637904	0.318952	+	0.947771i	$0.396669\pi$
$662$	−12.2874	−0.477563
$663$	−2.07725	−0.0806738
$664$	−96.5960	−3.74865
$665$	61.2644	2.37573
$666$	−66.9093	−2.59268
$667$	−18.3301	−0.709746
$668$	−79.5237	−3.07686
$669$	58.4932	2.26148
$670$	−25.2888	−0.976990
$671$	−40.6283	−1.56844
$672$	206.097	7.95037
$673$	−25.9564	−1.00055	−0.500273	−	0.865868i	$-0.666767\pi$
$673$	−25.9564	−1.00055	−0.500273	+	0.865868i	$0.666767\pi$
$674$	31.2921	1.20533
$675$	10.7331	0.413116
$676$	5.27461	0.202869
$677$	5.82421	0.223843	0.111921	−	0.993717i	$-0.464300\pi$
$677$	5.82421	0.223843	0.111921	+	0.993717i	$0.464300\pi$
$678$	−137.491	−5.28031
$679$	44.2186	1.69695
$680$	25.1468	0.964335
$681$	−3.62273	−0.138823
$682$	−35.3396	−1.35322
$683$	18.4445	0.705760	0.352880	−	0.935669i	$-0.385202\pi$
$683$	18.4445	0.705760	0.352880	+	0.935669i	$0.385202\pi$
$684$	−74.0560	−2.83160
$685$	−42.0520	−1.60672
$686$	67.7131	2.58530
$687$	7.53921	0.287639
$688$	−126.565	−4.82526
$689$	10.5586	0.402251
$690$	128.719	4.90025
$691$	−28.2657	−1.07528	−0.537640	−	0.843175i	$-0.680684\pi$
$691$	−28.2657	−1.07528	−0.537640	+	0.843175i	$0.680684\pi$
$692$	25.5790	0.972366
$693$	75.2916	2.86009
$694$	−77.1722	−2.92942
$695$	−63.4794	−2.40791
$696$	−78.3050	−2.96815
$697$	0.520371	0.0197105
$698$	76.6798	2.90237
$699$	−15.4728	−0.585235
$700$	−171.753	−6.49165
$701$	47.9440	1.81082	0.905411	−	0.424537i	$-0.139563\pi$
$701$	47.9440	1.81082	0.905411	+	0.424537i	$0.139563\pi$
$702$	−3.94208	−0.148784
$703$	−27.3253	−1.03059
$704$	−106.358	−4.00853
$705$	96.4018	3.63070
$706$	−56.6068	−2.13043
$707$	−47.7655	−1.79641
$708$	3.02167	0.113561
$709$	9.41346	0.353530	0.176765	−	0.984253i	$-0.443437\pi$
$709$	9.41346	0.353530	0.176765	+	0.984253i	$0.443437\pi$
$710$	−97.2560	−3.64995
$711$	57.4386	2.15412
$712$	148.609	5.56936
$713$	14.5997	0.546764
$714$	−24.8431	−0.929730
$715$	−16.7094	−0.624895
$716$	120.762	4.51309
$717$	−27.5932	−1.03049
$718$	55.8029	2.08255
$719$	−18.6801	−0.696650	−0.348325	−	0.937374i	$-0.613249\pi$
$719$	−18.6801	−0.696650	−0.348325	+	0.937374i	$0.613249\pi$
$720$	166.478	6.20427
$721$	−19.7787	−0.736598
$722$	9.53401	0.354819
$723$	−54.9483	−2.04355
$724$	105.923	3.93658
$725$	25.4003	0.943344
$726$	−80.3306	−2.98135
$727$	−21.4098	−0.794047	−0.397024	−	0.917808i	$-0.629957\pi$
$727$	−21.4098	−0.794047	−0.397024	+	0.917808i	$0.629957\pi$
$728$	39.1630	1.45148
$729$	−36.1029	−1.33714
$730$	−82.8126	−3.06503
$731$	7.72804	0.285832
$732$	115.496	4.26885
$733$	−4.36620	−0.161269	−0.0806346	−	0.996744i	$-0.525695\pi$
$733$	−4.36620	−0.161269	−0.0806346	+	0.996744i	$0.525695\pi$
$734$	79.3398	2.92848
$735$	114.026	4.20592
$736$	96.0949	3.54210
$737$	−12.6924	−0.467532
$738$	6.18318	0.227606
$739$	10.7027	0.393706	0.196853	−	0.980433i	$-0.436928\pi$
$739$	10.7027	0.393706	0.196853	+	0.980433i	$0.436928\pi$
$740$	128.765	4.73349
$741$	−10.0801	−0.370303
$742$	126.277	4.63577
$743$	46.5889	1.70918	0.854590	−	0.519303i	$-0.173809\pi$
$743$	46.5889	1.70918	0.854590	+	0.519303i	$0.173809\pi$
$744$	62.3690	2.28656
$745$	15.9991	0.586162
$746$	−19.7892	−0.724534
$747$	39.0471	1.42866
$748$	20.3297	0.743328
$749$	−33.9785	−1.24155
$750$	−56.9215	−2.07848
$751$	23.7551	0.866837	0.433419	−	0.901193i	$-0.357307\pi$
$751$	23.7551	0.866837	0.433419	+	0.901193i	$0.357307\pi$
$752$	142.076	5.18099
$753$	13.0500	0.475570
$754$	−9.32913	−0.339747
$755$	−42.4317	−1.54425
$756$	−34.1840	−1.24326
$757$	26.6869	0.969952	0.484976	−	0.874528i	$-0.338828\pi$
$757$	26.6869	0.969952	0.484976	+	0.874528i	$0.338828\pi$
$758$	−12.2903	−0.446405
$759$	64.6041	2.34498
$760$	122.028	4.42642
$761$	25.9285	0.939906	0.469953	−	0.882691i	$-0.344271\pi$
$761$	25.9285	0.939906	0.469953	+	0.882691i	$0.344271\pi$
$762$	−131.378	−4.75934
$763$	−40.1974	−1.45524
$764$	4.36134	0.157788
$765$	−10.1651	−0.367520
$766$	−70.7992	−2.55808
$767$	0.223495	0.00806994
$768$	51.6261	1.86290
$769$	−41.9434	−1.51252	−0.756259	−	0.654273i	$-0.772975\pi$
$769$	−41.9434	−1.51252	−0.756259	+	0.654273i	$0.772975\pi$
$770$	−199.837	−7.20164
$771$	−2.38641	−0.0859446
$772$	106.233	3.82341
$773$	18.1158	0.651581	0.325791	−	0.945442i	$-0.394370\pi$
$773$	18.1158	0.651581	0.325791	+	0.945442i	$0.394370\pi$
$774$	91.8264	3.30063
$775$	−20.2310	−0.726720
$776$	88.0759	3.16174
$777$	−78.9750	−2.83321
$778$	31.7194	1.13720
$779$	2.52517	0.0904736
$780$	47.5005	1.70079
$781$	−48.8128	−1.74666
$782$	−11.5833	−0.414219
$783$	5.05542	0.180666
$784$	168.051	6.00182
$785$	−60.5789	−2.16216
$786$	53.8607	1.92115
$787$	−13.8712	−0.494454	−0.247227	−	0.968958i	$-0.579519\pi$
$787$	−13.8712	−0.494454	−0.247227	+	0.968958i	$0.579519\pi$
$788$	21.4184	0.762997
$789$	14.6498	0.521548
$790$	−152.452	−5.42402
$791$	−88.1845	−3.13548
$792$	149.968	5.32888
$793$	8.54254	0.303355
$794$	100.139	3.55380
$795$	95.0858	3.37235
$796$	8.65653	0.306822
$797$	−17.8694	−0.632966	−0.316483	−	0.948598i	$-0.602502\pi$
$797$	−17.8694	−0.632966	−0.316483	+	0.948598i	$0.602502\pi$
$798$	−120.554	−4.26758
$799$	−8.67514	−0.306904
$800$	−133.160	−4.70791
$801$	−60.0724	−2.12255
$802$	50.3571	1.77817
$803$	−41.5637	−1.46675
$804$	36.0814	1.27249
$805$	82.5582	2.90979
$806$	7.43053	0.261729
$807$	8.72458	0.307120
$808$	−95.1408	−3.34704
$809$	36.5895	1.28642	0.643208	−	0.765691i	$-0.277603\pi$
$809$	36.5895	1.28642	0.643208	+	0.765691i	$0.277603\pi$
$810$	65.9930	2.31876
$811$	8.66987	0.304440	0.152220	−	0.988347i	$-0.451358\pi$
$811$	8.66987	0.304440	0.152220	+	0.988347i	$0.451358\pi$
$812$	−80.8981	−2.83897
$813$	−15.9062	−0.557853
$814$	89.1322	3.12408
$815$	16.5389	0.579333
$816$	−27.5698	−0.965137
$817$	37.5013	1.31201
$818$	48.0472	1.67993
$819$	−15.8309	−0.553176
$820$	−11.8993	−0.415543
$821$	−23.0167	−0.803289	−0.401645	−	0.915796i	$-0.631561\pi$
$821$	−23.0167	−0.803289	−0.401645	+	0.915796i	$0.631561\pi$
$822$	82.7489	2.88620
$823$	−44.5753	−1.55380	−0.776899	−	0.629625i	$-0.783209\pi$
$823$	−44.5753	−1.55380	−0.776899	+	0.629625i	$0.783209\pi$
$824$	−39.3958	−1.37242
$825$	−89.5227	−3.11678
$826$	2.67291	0.0930024
$827$	−17.7206	−0.616206	−0.308103	−	0.951353i	$-0.599694\pi$
$827$	−17.7206	−0.616206	−0.308103	+	0.951353i	$0.599694\pi$
$828$	−99.7959	−3.46815
$829$	29.3850	1.02058	0.510292	−	0.860001i	$-0.329537\pi$
$829$	29.3850	1.02058	0.510292	+	0.860001i	$0.329537\pi$
$830$	−103.638	−3.59733
$831$	10.9175	0.378724
$832$	22.3630	0.775297
$833$	−10.2611	−0.355528
$834$	124.913	4.32539
$835$	−52.9694	−1.83308
$836$	98.6526	3.41197
$837$	−4.02658	−0.139179
$838$	57.4849	1.98578
$839$	−55.9794	−1.93262	−0.966311	−	0.257376i	$-0.917142\pi$
$839$	−55.9794	−1.93262	−0.966311	+	0.257376i	$0.917142\pi$
$840$	352.683	12.1687
$841$	−17.0361	−0.587452
$842$	−87.6595	−3.02095
$843$	−40.1423	−1.38257
$844$	65.0875	2.24040
$845$	3.51333	0.120862
$846$	−103.080	−3.54396
$847$	−51.5227	−1.77034
$848$	140.137	4.81232
$849$	−37.8276	−1.29824
$850$	16.0512	0.550551
$851$	−36.8229	−1.26227
$852$	138.763	4.75393
$853$	43.5354	1.49062	0.745311	−	0.666717i	$-0.232301\pi$
$853$	43.5354	1.49062	0.745311	+	0.666717i	$0.232301\pi$
$854$	102.165	3.49603
$855$	−49.3275	−1.68697
$856$	−67.6793	−2.31323
$857$	3.20928	0.109627	0.0548134	−	0.998497i	$-0.482544\pi$
$857$	3.20928	0.109627	0.0548134	+	0.998497i	$0.482544\pi$
$858$	32.8803	1.12251
$859$	−31.9669	−1.09070	−0.545348	−	0.838210i	$-0.683602\pi$
$859$	−31.9669	−1.09070	−0.545348	+	0.838210i	$0.683602\pi$
$860$	−176.717	−6.02601
$861$	7.29819	0.248722
$862$	−74.6348	−2.54207
$863$	19.7108	0.670965	0.335482	−	0.942046i	$-0.391101\pi$
$863$	19.7108	0.670965	0.335482	+	0.942046i	$0.391101\pi$
$864$	−26.5028	−0.901644
$865$	17.0377	0.579300
$866$	60.0608	2.04095
$867$	−41.8917	−1.42272
$868$	64.4343	2.18704
$869$	−76.5160	−2.59563
$870$	−84.0136	−2.84833
$871$	2.66873	0.0904263
$872$	−80.0663	−2.71139
$873$	−35.6030	−1.20498
$874$	−56.2097	−1.90132
$875$	−36.5085	−1.23421
$876$	118.155	3.99209
$877$	−52.2319	−1.76374	−0.881872	−	0.471488i	$-0.843717\pi$
$877$	−52.2319	−1.76374	−0.881872	+	0.471488i	$0.843717\pi$
$878$	17.9139	0.604565
$879$	60.3628	2.03599
$880$	−221.771	−7.47590
$881$	27.2360	0.917603	0.458802	−	0.888539i	$-0.348279\pi$
$881$	27.2360	0.917603	0.458802	+	0.888539i	$0.348279\pi$
$882$	−121.925	−4.10544
$883$	−13.1361	−0.442065	−0.221032	−	0.975266i	$-0.570943\pi$
$883$	−13.1361	−0.442065	−0.221032	+	0.975266i	$0.570943\pi$
$884$	−4.27454	−0.143768
$885$	2.01269	0.0676557
$886$	63.2125	2.12367
$887$	−36.7200	−1.23294	−0.616468	−	0.787380i	$-0.711437\pi$
$887$	−36.7200	−1.23294	−0.616468	+	0.787380i	$0.711437\pi$
$888$	−157.305	−5.27880
$889$	−84.2639	−2.82612
$890$	159.443	5.34454
$891$	33.1219	1.10963
$892$	120.367	4.03017
$893$	−42.0972	−1.40873
$894$	−31.4826	−1.05294
$895$	80.4377	2.68873
$896$	106.643	3.56268
$897$	−13.5837	−0.453547
$898$	−19.6983	−0.657340
$899$	−9.52910	−0.317813
$900$	138.288	4.60961
$901$	−8.55671	−0.285065
$902$	−8.23682	−0.274256
$903$	108.385	3.60684
$904$	−175.648	−5.84198
$905$	70.5533	2.34527
$906$	83.4960	2.77397
$907$	−10.0524	−0.333783	−0.166892	−	0.985975i	$-0.553373\pi$
$907$	−10.0524	−0.333783	−0.166892	+	0.985975i	$0.553373\pi$
$908$	−7.45481	−0.247397
$909$	38.4588	1.27560
$910$	42.0180	1.39288
$911$	21.0620	0.697814	0.348907	−	0.937157i	$-0.386553\pi$
$911$	21.0620	0.697814	0.348907	+	0.937157i	$0.386553\pi$
$912$	−133.786	−4.43010
$913$	−52.0160	−1.72148
$914$	19.9016	0.658286
$915$	76.9300	2.54323
$916$	15.5141	0.512600
$917$	34.5454	1.14079
$918$	3.19467	0.105440
$919$	−29.2587	−0.965155	−0.482577	−	0.875853i	$-0.660299\pi$
$919$	−29.2587	−0.965155	−0.482577	+	0.875853i	$0.660299\pi$
$920$	164.442	5.42148
$921$	54.8505	1.80739
$922$	−26.1311	−0.860581
$923$	10.2634	0.337825
$924$	285.123	9.37987
$925$	51.0259	1.67772
$926$	−5.91299	−0.194313
$927$	15.9250	0.523046
$928$	−62.7202	−2.05889
$929$	15.9720	0.524025	0.262013	−	0.965064i	$-0.415614\pi$
$929$	15.9720	0.524025	0.262013	+	0.965064i	$0.415614\pi$
$930$	66.9158	2.19425
$931$	−49.7936	−1.63192
$932$	−31.8397	−1.04295
$933$	11.9410	0.390931
$934$	−79.7575	−2.60974
$935$	13.5413	0.442848
$936$	−31.5324	−1.03067
$937$	−39.7984	−1.30016	−0.650079	−	0.759867i	$-0.725264\pi$
$937$	−39.7984	−1.30016	−0.650079	+	0.759867i	$0.725264\pi$
$938$	31.9169	1.04212
$939$	34.9932	1.14196
$940$	198.374	6.47026
$941$	22.8410	0.744597	0.372298	−	0.928113i	$-0.378570\pi$
$941$	22.8410	0.744597	0.372298	+	0.928113i	$0.378570\pi$
$942$	119.206	3.88393
$943$	3.40285	0.110812
$944$	2.96628	0.0965443
$945$	−22.7694	−0.740689
$946$	−122.325	−3.97713
$947$	6.85162	0.222648	0.111324	−	0.993784i	$-0.464491\pi$
$947$	6.85162	0.222648	0.111324	+	0.993784i	$0.464491\pi$
$948$	217.516	7.06458
$949$	8.73922	0.283687
$950$	77.8905	2.52710
$951$	15.9701	0.517866
$952$	−31.7377	−1.02862
$953$	33.6880	1.09126	0.545630	−	0.838026i	$-0.316290\pi$
$953$	33.6880	1.09126	0.545630	+	0.838026i	$0.316290\pi$
$954$	−101.673	−3.29178
$955$	2.90502	0.0940042
$956$	−56.7809	−1.83643
$957$	−42.1665	−1.36305
$958$	−80.7767	−2.60978
$959$	53.0737	1.71384
$960$	201.390	6.49984
$961$	−23.4102	−0.755168
$962$	−18.7410	−0.604235
$963$	27.3581	0.881601
$964$	−113.072	−3.64180
$965$	70.7602	2.27785
$966$	−162.456	−5.22693
$967$	39.5997	1.27344	0.636721	−	0.771095i	$-0.280291\pi$
$967$	39.5997	1.27344	0.636721	+	0.771095i	$0.280291\pi$
$968$	−102.624	−3.29847
$969$	8.16895	0.262425
$970$	94.4967	3.03411
$971$	4.19772	0.134711	0.0673556	−	0.997729i	$-0.478544\pi$
$971$	4.19772	0.134711	0.0673556	+	0.997729i	$0.478544\pi$
$972$	−117.285	−3.76192
$973$	80.1172	2.56844
$974$	42.2207	1.35284
$975$	18.8231	0.602823
$976$	113.379	3.62917
$977$	22.7554	0.728011	0.364005	−	0.931397i	$-0.381409\pi$
$977$	22.7554	0.728011	0.364005	+	0.931397i	$0.381409\pi$
$978$	−32.5449	−1.04067
$979$	80.0245	2.55760
$980$	234.642	7.49535
$981$	32.3653	1.03334
$982$	−12.0222	−0.383645
$983$	−2.15324	−0.0686778	−0.0343389	−	0.999410i	$-0.510933\pi$
$983$	−2.15324	−0.0686778	−0.0343389	+	0.999410i	$0.510933\pi$
$984$	14.5367	0.463414
$985$	14.2664	0.454566
$986$	7.56033	0.240770
$987$	−121.668	−3.87275
$988$	−20.7428	−0.659916
$989$	50.5358	1.60694
$990$	160.901	5.11376
$991$	−43.4452	−1.38008	−0.690041	−	0.723771i	$-0.742407\pi$
$991$	−43.4452	−1.38008	−0.690041	+	0.723771i	$0.742407\pi$
$992$	49.9558	1.58610
$993$	11.6774	0.370570
$994$	122.747	3.89328
$995$	5.76597	0.182794
$996$	147.868	4.68539
$997$	4.56182	0.144474	0.0722372	−	0.997387i	$-0.476986\pi$
$997$	4.56182	0.144474	0.0722372	+	0.997387i	$0.476986\pi$
$998$	56.1776	1.77827
$999$	10.1557	0.321312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.4	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.4	✓	168	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)

\(f(q)\)	\(=\)	\(q-2.69715 q^{2} +2.56324 q^{3} +5.27461 q^{4} +3.51333 q^{5} -6.91344 q^{6} -4.43417 q^{7} -8.83210 q^{8} +3.57021 q^{9} +O(q^{10})\)	\(q-2.69715 q^{2} +2.56324 q^{3} +5.27461 q^{4} +3.51333 q^{5} -6.91344 q^{6} -4.43417 q^{7} -8.83210 q^{8} +3.57021 q^{9} -9.47597 q^{10} -4.75599 q^{11} +13.5201 q^{12} +1.00000 q^{13} +11.9596 q^{14} +9.00551 q^{15} +13.2723 q^{16} -0.810401 q^{17} -9.62937 q^{18} -3.93258 q^{19} +18.5314 q^{20} -11.3658 q^{21} +12.8276 q^{22} -5.29943 q^{23} -22.6388 q^{24} +7.34349 q^{25} -2.69715 q^{26} +1.46157 q^{27} -23.3885 q^{28} +3.45889 q^{29} -24.2892 q^{30} -2.75496 q^{31} -18.1330 q^{32} -12.1908 q^{33} +2.18577 q^{34} -15.5787 q^{35} +18.8314 q^{36} +6.94846 q^{37} +10.6067 q^{38} +2.56324 q^{39} -31.0301 q^{40} -0.642116 q^{41} +30.6553 q^{42} -9.53608 q^{43} -25.0860 q^{44} +12.5433 q^{45} +14.2934 q^{46} +10.7047 q^{47} +34.0200 q^{48} +12.6618 q^{49} -19.8065 q^{50} -2.07725 q^{51} +5.27461 q^{52} +10.5586 q^{53} -3.94208 q^{54} -16.7094 q^{55} +39.1630 q^{56} -10.0801 q^{57} -9.32913 q^{58} +0.223495 q^{59} +47.5005 q^{60} +8.54254 q^{61} +7.43053 q^{62} -15.8309 q^{63} +22.3630 q^{64} +3.51333 q^{65} +32.8803 q^{66} +2.66873 q^{67} -4.27454 q^{68} -13.5837 q^{69} +42.0180 q^{70} +10.2634 q^{71} -31.5324 q^{72} +8.73922 q^{73} -18.7410 q^{74} +18.8231 q^{75} -20.7428 q^{76} +21.0889 q^{77} -6.91344 q^{78} +16.0883 q^{79} +46.6298 q^{80} -6.96425 q^{81} +1.73188 q^{82} +10.9369 q^{83} -59.9503 q^{84} -2.84720 q^{85} +25.7202 q^{86} +8.86596 q^{87} +42.0054 q^{88} -16.8260 q^{89} -33.8312 q^{90} -4.43417 q^{91} -27.9524 q^{92} -7.06163 q^{93} -28.8723 q^{94} -13.8164 q^{95} -46.4794 q^{96} -9.97225 q^{97} -34.1508 q^{98} -16.9799 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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