LMFDB - Embedded newform 8047.2.a.e.1.14

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			1.14
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)

$f(q)$	$=$	$q-2.50460 q^{2} -2.26782 q^{3} +4.27304 q^{4} +0.233077 q^{5} +5.68000 q^{6} -0.679959 q^{7} -5.69306 q^{8} +2.14302 q^{9} +O(q^{10})$	\(q-2.50460 q^{2} -2.26782 q^{3} +4.27304 q^{4} +0.233077 q^{5} +5.68000 q^{6} -0.679959 q^{7} -5.69306 q^{8} +2.14302 q^{9} -0.583765 q^{10} -1.61206 q^{11} -9.69050 q^{12} +1.00000 q^{13} +1.70303 q^{14} -0.528577 q^{15} +5.71278 q^{16} -2.21861 q^{17} -5.36742 q^{18} -3.42172 q^{19} +0.995946 q^{20} +1.54203 q^{21} +4.03757 q^{22} -0.266744 q^{23} +12.9109 q^{24} -4.94568 q^{25} -2.50460 q^{26} +1.94347 q^{27} -2.90549 q^{28} +10.0680 q^{29} +1.32388 q^{30} -8.93300 q^{31} -2.92213 q^{32} +3.65586 q^{33} +5.55673 q^{34} -0.158483 q^{35} +9.15722 q^{36} -3.09538 q^{37} +8.57006 q^{38} -2.26782 q^{39} -1.32692 q^{40} -10.2560 q^{41} -3.86216 q^{42} +7.01300 q^{43} -6.88839 q^{44} +0.499489 q^{45} +0.668089 q^{46} -9.11211 q^{47} -12.9556 q^{48} -6.53766 q^{49} +12.3870 q^{50} +5.03140 q^{51} +4.27304 q^{52} -0.459131 q^{53} -4.86763 q^{54} -0.375733 q^{55} +3.87105 q^{56} +7.75986 q^{57} -25.2165 q^{58} +0.580738 q^{59} -2.25863 q^{60} +8.73089 q^{61} +22.3736 q^{62} -1.45717 q^{63} -4.10678 q^{64} +0.233077 q^{65} -9.15649 q^{66} -11.1487 q^{67} -9.48019 q^{68} +0.604929 q^{69} +0.396936 q^{70} -15.8424 q^{71} -12.2004 q^{72} +2.72974 q^{73} +7.75269 q^{74} +11.2159 q^{75} -14.6212 q^{76} +1.09613 q^{77} +5.68000 q^{78} +4.43984 q^{79} +1.33152 q^{80} -10.8365 q^{81} +25.6871 q^{82} +1.25434 q^{83} +6.58914 q^{84} -0.517105 q^{85} -17.5648 q^{86} -22.8326 q^{87} +9.17755 q^{88} -10.1175 q^{89} -1.25102 q^{90} -0.679959 q^{91} -1.13981 q^{92} +20.2585 q^{93} +22.8222 q^{94} -0.797524 q^{95} +6.62688 q^{96} +11.6302 q^{97} +16.3742 q^{98} -3.45468 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.50460	−1.77102	−0.885511	−	0.464618i	$-0.846192\pi$
$2$	−2.50460	−1.77102	−0.885511	+	0.464618i	$0.846192\pi$
$3$	−2.26782	−1.30933	−0.654664	−	0.755920i	$-0.727190\pi$
$3$	−2.26782	−1.30933	−0.654664	+	0.755920i	$0.727190\pi$
$4$	4.27304	2.13652
$5$	0.233077	0.104235	0.0521175	−	0.998641i	$-0.483403\pi$
$5$	0.233077	0.104235	0.0521175	+	0.998641i	$0.483403\pi$
$6$	5.68000	2.31885
$7$	−0.679959	−0.257000	−0.128500	−	0.991709i	$-0.541016\pi$
$7$	−0.679959	−0.257000	−0.128500	+	0.991709i	$0.541016\pi$
$8$	−5.69306	−2.01280
$9$	2.14302	0.714341
$10$	−0.583765	−0.184603
$11$	−1.61206	−0.486054	−0.243027	−	0.970020i	$-0.578140\pi$
$11$	−1.61206	−0.486054	−0.243027	+	0.970020i	$0.578140\pi$
$12$	−9.69050	−2.79741
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.70303	0.455153
$15$	−0.528577	−0.136478
$16$	5.71278	1.42820
$17$	−2.21861	−0.538091	−0.269045	−	0.963127i	$-0.586708\pi$
$17$	−2.21861	−0.538091	−0.269045	+	0.963127i	$0.586708\pi$
$18$	−5.36742	−1.26511
$19$	−3.42172	−0.784997	−0.392498	−	0.919753i	$-0.628389\pi$
$19$	−3.42172	−0.784997	−0.392498	+	0.919753i	$0.628389\pi$
$20$	0.995946	0.222700
$21$	1.54203	0.336498
$22$	4.03757	0.860812
$23$	−0.266744	−0.0556201	−0.0278100	−	0.999613i	$-0.508853\pi$
$23$	−0.266744	−0.0556201	−0.0278100	+	0.999613i	$0.508853\pi$
$24$	12.9109	2.63542
$25$	−4.94568	−0.989135
$26$	−2.50460	−0.491193
$27$	1.94347	0.374022
$28$	−2.90549	−0.549086
$29$	10.0680	1.86959	0.934795	−	0.355188i	$-0.115583\pi$
$29$	10.0680	1.86959	0.934795	+	0.355188i	$0.115583\pi$
$30$	1.32388	0.241705
$31$	−8.93300	−1.60441	−0.802207	−	0.597046i	$-0.796341\pi$
$31$	−8.93300	−1.60441	−0.802207	+	0.597046i	$0.796341\pi$
$32$	−2.92213	−0.516565
$33$	3.65586	0.636404
$34$	5.55673	0.952971
$35$	−0.158483	−0.0267884
$36$	9.15722	1.52620
$37$	−3.09538	−0.508877	−0.254438	−	0.967089i	$-0.581891\pi$
$37$	−3.09538	−0.508877	−0.254438	+	0.967089i	$0.581891\pi$
$38$	8.57006	1.39025
$39$	−2.26782	−0.363142
$40$	−1.32692	−0.209804
$41$	−10.2560	−1.60171	−0.800857	−	0.598856i	$-0.795622\pi$
$41$	−10.2560	−1.60171	−0.800857	+	0.598856i	$0.795622\pi$
$42$	−3.86216	−0.595945
$43$	7.01300	1.06947	0.534736	−	0.845019i	$-0.320411\pi$
$43$	7.01300	1.06947	0.534736	+	0.845019i	$0.320411\pi$
$44$	−6.88839	−1.03846
$45$	0.499489	0.0744594
$46$	0.668089	0.0985044
$47$	−9.11211	−1.32914	−0.664569	−	0.747227i	$-0.731385\pi$
$47$	−9.11211	−1.32914	−0.664569	+	0.747227i	$0.731385\pi$
$48$	−12.9556	−1.86998
$49$	−6.53766	−0.933951
$50$	12.3870	1.75178
$51$	5.03140	0.704538
$52$	4.27304	0.592564
$53$	−0.459131	−0.0630664	−0.0315332	−	0.999503i	$-0.510039\pi$
$53$	−0.459131	−0.0630664	−0.0315332	+	0.999503i	$0.510039\pi$
$54$	−4.86763	−0.662401
$55$	−0.375733	−0.0506639
$56$	3.87105	0.517290
$57$	7.75986	1.02782
$58$	−25.2165	−3.31108
$59$	0.580738	0.0756057	0.0378028	−	0.999285i	$-0.487964\pi$
$59$	0.580738	0.0756057	0.0378028	+	0.999285i	$0.487964\pi$
$60$	−2.25863	−0.291588
$61$	8.73089	1.11788	0.558938	−	0.829210i	$-0.311209\pi$
$61$	8.73089	1.11788	0.558938	+	0.829210i	$0.311209\pi$
$62$	22.3736	2.84145
$63$	−1.45717	−0.183586
$64$	−4.10678	−0.513348
$65$	0.233077	0.0289096
$66$	−9.15649	−1.12709
$67$	−11.1487	−1.36203	−0.681017	−	0.732268i	$-0.738462\pi$
$67$	−11.1487	−1.36203	−0.681017	+	0.732268i	$0.738462\pi$
$68$	−9.48019	−1.14964
$69$	0.604929	0.0728249
$70$	0.396936	0.0474429
$71$	−15.8424	−1.88015	−0.940076	−	0.340964i	$-0.889247\pi$
$71$	−15.8424	−1.88015	−0.940076	+	0.340964i	$0.889247\pi$
$72$	−12.2004	−1.43783
$73$	2.72974	0.319491	0.159746	−	0.987158i	$-0.448933\pi$
$73$	2.72974	0.319491	0.159746	+	0.987158i	$0.448933\pi$
$74$	7.75269	0.901232
$75$	11.2159	1.29510
$76$	−14.6212	−1.67716
$77$	1.09613	0.124916
$78$	5.68000	0.643133
$79$	4.43984	0.499521	0.249761	−	0.968308i	$-0.419648\pi$
$79$	4.43984	0.499521	0.249761	+	0.968308i	$0.419648\pi$
$80$	1.33152	0.148868
$81$	−10.8365	−1.20406
$82$	25.6871	2.83667
$83$	1.25434	0.137682	0.0688409	−	0.997628i	$-0.478070\pi$
$83$	1.25434	0.137682	0.0688409	+	0.997628i	$0.478070\pi$
$84$	6.58914	0.718934
$85$	−0.517105	−0.0560879
$86$	−17.5648	−1.89406
$87$	−22.8326	−2.44791
$88$	9.17755	0.978330
$89$	−10.1175	−1.07245	−0.536225	−	0.844075i	$-0.680150\pi$
$89$	−10.1175	−1.07245	−0.536225	+	0.844075i	$0.680150\pi$
$90$	−1.25102	−0.131869
$91$	−0.679959	−0.0712790
$92$	−1.13981	−0.118833
$93$	20.2585	2.10070
$94$	22.8222	2.35393
$95$	−0.797524	−0.0818242
$96$	6.62688	0.676353
$97$	11.6302	1.18087	0.590436	−	0.807085i	$-0.298956\pi$
$97$	11.6302	1.18087	0.590436	+	0.807085i	$0.298956\pi$
$98$	16.3742	1.65405
$99$	−3.45468	−0.347208
$100$	−21.1331	−2.11331
$101$	12.3960	1.23345	0.616726	−	0.787178i	$-0.288459\pi$
$101$	12.3960	1.23345	0.616726	+	0.787178i	$0.288459\pi$
$102$	−12.6017	−1.24775
$103$	−0.925030	−0.0911460	−0.0455730	−	0.998961i	$-0.514511\pi$
$103$	−0.925030	−0.0911460	−0.0455730	+	0.998961i	$0.514511\pi$
$104$	−5.69306	−0.558251
$105$	0.359410	0.0350749
$106$	1.14994	0.111692
$107$	6.47972	0.626418	0.313209	−	0.949684i	$-0.398596\pi$
$107$	6.47972	0.626418	0.313209	+	0.949684i	$0.398596\pi$
$108$	8.30454	0.799104
$109$	2.32370	0.222570	0.111285	−	0.993789i	$-0.464503\pi$
$109$	2.32370	0.222570	0.111285	+	0.993789i	$0.464503\pi$
$110$	0.941063	0.0897268
$111$	7.01976	0.666287
$112$	−3.88446	−0.367047
$113$	−1.21538	−0.114333	−0.0571665	−	0.998365i	$-0.518207\pi$
$113$	−1.21538	−0.114333	−0.0571665	+	0.998365i	$0.518207\pi$
$114$	−19.4354	−1.82029
$115$	−0.0621719	−0.00579756
$116$	43.0212	3.99441
$117$	2.14302	0.198123
$118$	−1.45452	−0.133899
$119$	1.50856	0.138289
$120$	3.00922	0.274703
$121$	−8.40127	−0.763752
$122$	−21.8674	−1.97978
$123$	23.2587	2.09717
$124$	−38.1710	−3.42786
$125$	−2.31810	−0.207338
$126$	3.64962	0.325134
$127$	−5.47018	−0.485400	−0.242700	−	0.970101i	$-0.578033\pi$
$127$	−5.47018	−0.485400	−0.242700	+	0.970101i	$0.578033\pi$
$128$	16.1301	1.42571
$129$	−15.9043	−1.40029
$130$	−0.583765	−0.0511995
$131$	−12.6703	−1.10701	−0.553505	−	0.832846i	$-0.686710\pi$
$131$	−12.6703	−1.10701	−0.553505	+	0.832846i	$0.686710\pi$
$132$	15.6216	1.35969
$133$	2.32663	0.201744
$134$	27.9231	2.41219
$135$	0.452978	0.0389862
$136$	12.6307	1.08307
$137$	13.7424	1.17409	0.587044	−	0.809555i	$-0.300291\pi$
$137$	13.7424	1.17409	0.587044	+	0.809555i	$0.300291\pi$
$138$	−1.51511	−0.128975
$139$	−6.79142	−0.576041	−0.288021	−	0.957624i	$-0.592997\pi$
$139$	−6.79142	−0.576041	−0.288021	+	0.957624i	$0.592997\pi$
$140$	−0.677202	−0.0572340
$141$	20.6647	1.74028
$142$	39.6791	3.32979
$143$	−1.61206	−0.134807
$144$	12.2426	1.02022
$145$	2.34663	0.194877
$146$	−6.83691	−0.565826
$147$	14.8262	1.22285
$148$	−13.2267	−1.08722
$149$	0.120244	0.00985081	0.00492540	−	0.999988i	$-0.498432\pi$
$149$	0.120244	0.00985081	0.00492540	+	0.999988i	$0.498432\pi$
$150$	−28.0914	−2.29366
$151$	−11.1413	−0.906668	−0.453334	−	0.891341i	$-0.649766\pi$
$151$	−11.1413	−0.906668	−0.453334	+	0.891341i	$0.649766\pi$
$152$	19.4801	1.58004
$153$	−4.75452	−0.384380
$154$	−2.74538	−0.221229
$155$	−2.08207	−0.167236
$156$	−9.69050	−0.775861
$157$	−20.4053	−1.62852	−0.814258	−	0.580503i	$-0.802856\pi$
$157$	−20.4053	−1.62852	−0.814258	+	0.580503i	$0.802856\pi$
$158$	−11.1200	−0.884663
$159$	1.04123	0.0825747
$160$	−0.681081	−0.0538442
$161$	0.181375	0.0142944
$162$	27.1412	2.13241
$163$	−9.64489	−0.755446	−0.377723	−	0.925919i	$-0.623293\pi$
$163$	−9.64489	−0.755446	−0.377723	+	0.925919i	$0.623293\pi$
$164$	−43.8242	−3.42209
$165$	0.852097	0.0663356
$166$	−3.14163	−0.243838
$167$	23.8346	1.84438	0.922190	−	0.386737i	$-0.126398\pi$
$167$	23.8346	1.84438	0.922190	+	0.386737i	$0.126398\pi$
$168$	−8.77885	−0.677303
$169$	1.00000	0.0769231
$170$	1.29514	0.0993329
$171$	−7.33283	−0.560755
$172$	29.9668	2.28495
$173$	4.90094	0.372611	0.186306	−	0.982492i	$-0.440349\pi$
$173$	4.90094	0.372611	0.186306	+	0.982492i	$0.440349\pi$
$174$	57.1865	4.33530
$175$	3.36285	0.254208
$176$	−9.20934	−0.694180
$177$	−1.31701	−0.0989927
$178$	25.3402	1.89933
$179$	14.3935	1.07582	0.537910	−	0.843002i	$-0.319214\pi$
$179$	14.3935	1.07582	0.537910	+	0.843002i	$0.319214\pi$
$180$	2.13433	0.159084
$181$	6.60251	0.490761	0.245380	−	0.969427i	$-0.421087\pi$
$181$	6.60251	0.490761	0.245380	+	0.969427i	$0.421087\pi$
$182$	1.70303	0.126237
$183$	−19.8001	−1.46367
$184$	1.51859	0.111952
$185$	−0.721460	−0.0530428
$186$	−50.7394	−3.72039
$187$	3.57652	0.261541
$188$	−38.9364	−2.83973
$189$	−1.32148	−0.0961236
$190$	1.99748	0.144912
$191$	−17.3169	−1.25301	−0.626503	−	0.779419i	$-0.715514\pi$
$191$	−17.3169	−1.25301	−0.626503	+	0.779419i	$0.715514\pi$
$192$	9.31345	0.672141
$193$	5.10973	0.367807	0.183903	−	0.982944i	$-0.441127\pi$
$193$	5.10973	0.367807	0.183903	+	0.982944i	$0.441127\pi$
$194$	−29.1291	−2.09135
$195$	−0.528577	−0.0378522
$196$	−27.9357	−1.99540
$197$	12.9398	0.921923	0.460962	−	0.887420i	$-0.347505\pi$
$197$	12.9398	0.921923	0.460962	+	0.887420i	$0.347505\pi$
$198$	8.65260	0.614913
$199$	12.8562	0.911354	0.455677	−	0.890145i	$-0.349397\pi$
$199$	12.8562	0.911354	0.455677	+	0.890145i	$0.349397\pi$
$200$	28.1560	1.99093
$201$	25.2833	1.78335
$202$	−31.0471	−2.18447
$203$	−6.84586	−0.480485
$204$	21.4994	1.50526
$205$	−2.39043	−0.166955
$206$	2.31683	0.161421
$207$	−0.571639	−0.0397317
$208$	5.71278	0.396110
$209$	5.51602	0.381551
$210$	−0.900180	−0.0621183
$211$	−13.1451	−0.904949	−0.452475	−	0.891777i	$-0.649459\pi$
$211$	−13.1451	−0.904949	−0.452475	+	0.891777i	$0.649459\pi$
$212$	−1.96188	−0.134743
$213$	35.9279	2.46174
$214$	−16.2291	−1.10940
$215$	1.63457	0.111477
$216$	−11.0643	−0.752831
$217$	6.07407	0.412335
$218$	−5.81994	−0.394176
$219$	−6.19056	−0.418319
$220$	−1.60552	−0.108244
$221$	−2.21861	−0.149240
$222$	−17.5817	−1.18001
$223$	−22.1561	−1.48368	−0.741841	−	0.670575i	$-0.766047\pi$
$223$	−22.1561	−1.48368	−0.741841	+	0.670575i	$0.766047\pi$
$224$	1.98693	0.132757
$225$	−10.5987	−0.706580
$226$	3.04403	0.202486
$227$	−3.58155	−0.237716	−0.118858	−	0.992911i	$-0.537923\pi$
$227$	−3.58155	−0.237716	−0.118858	+	0.992911i	$0.537923\pi$
$228$	33.1582	2.19595
$229$	−2.41028	−0.159276	−0.0796378	−	0.996824i	$-0.525376\pi$
$229$	−2.41028	−0.159276	−0.0796378	+	0.996824i	$0.525376\pi$
$230$	0.155716	0.0102676
$231$	−2.48584	−0.163556
$232$	−57.3180	−3.76311
$233$	−13.0550	−0.855264	−0.427632	−	0.903953i	$-0.640652\pi$
$233$	−13.0550	−0.855264	−0.427632	+	0.903953i	$0.640652\pi$
$234$	−5.36742	−0.350879
$235$	−2.12382	−0.138543
$236$	2.48152	0.161533
$237$	−10.0688	−0.654037
$238$	−3.77834	−0.244914
$239$	17.1869	1.11173	0.555865	−	0.831272i	$-0.312387\pi$
$239$	17.1869	1.11173	0.555865	+	0.831272i	$0.312387\pi$
$240$	−3.01964	−0.194917
$241$	−25.3167	−1.63079	−0.815397	−	0.578902i	$-0.803481\pi$
$241$	−25.3167	−1.63079	−0.815397	+	0.578902i	$0.803481\pi$
$242$	21.0418	1.35262
$243$	18.7449	1.20249
$244$	37.3074	2.38836
$245$	−1.52378	−0.0973504
$246$	−58.2539	−3.71413
$247$	−3.42172	−0.217719
$248$	50.8561	3.22936
$249$	−2.84463	−0.180271
$250$	5.80593	0.367199
$251$	13.8073	0.871510	0.435755	−	0.900065i	$-0.356481\pi$
$251$	13.8073	0.871510	0.435755	+	0.900065i	$0.356481\pi$
$252$	−6.22653	−0.392235
$253$	0.430008	0.0270344
$254$	13.7006	0.859655
$255$	1.17270	0.0734375
$256$	−32.1860	−2.01162
$257$	−7.57609	−0.472584	−0.236292	−	0.971682i	$-0.575932\pi$
$257$	−7.57609	−0.472584	−0.236292	+	0.971682i	$0.575932\pi$
$258$	39.8338	2.47995
$259$	2.10473	0.130781
$260$	0.995946	0.0617659
$261$	21.5761	1.33552
$262$	31.7341	1.96054
$263$	23.3062	1.43712	0.718562	−	0.695463i	$-0.244801\pi$
$263$	23.3062	1.43712	0.718562	+	0.695463i	$0.244801\pi$
$264$	−20.8131	−1.28095
$265$	−0.107013	−0.00657373
$266$	−5.82728	−0.357294
$267$	22.9446	1.40419
$268$	−47.6390	−2.91001
$269$	−9.21802	−0.562032	−0.281016	−	0.959703i	$-0.590671\pi$
$269$	−9.21802	−0.562032	−0.281016	+	0.959703i	$0.590671\pi$
$270$	−1.13453	−0.0690454
$271$	−22.2992	−1.35458	−0.677290	−	0.735717i	$-0.736846\pi$
$271$	−22.2992	−1.35458	−0.677290	+	0.735717i	$0.736846\pi$
$272$	−12.6744	−0.768499
$273$	1.54203	0.0933277
$274$	−34.4192	−2.07934
$275$	7.97272	0.480773
$276$	2.58489	0.155592
$277$	−13.6457	−0.819890	−0.409945	−	0.912110i	$-0.634452\pi$
$277$	−13.6457	−0.819890	−0.409945	+	0.912110i	$0.634452\pi$
$278$	17.0098	1.02018
$279$	−19.1436	−1.14610
$280$	0.902250	0.0539198
$281$	−11.8648	−0.707797	−0.353898	−	0.935284i	$-0.615144\pi$
$281$	−11.8648	−0.707797	−0.353898	+	0.935284i	$0.615144\pi$
$282$	−51.7568	−3.08207
$283$	5.29145	0.314544	0.157272	−	0.987555i	$-0.449730\pi$
$283$	5.29145	0.314544	0.157272	+	0.987555i	$0.449730\pi$
$284$	−67.6954	−4.01698
$285$	1.80864	0.107135
$286$	4.03757	0.238746
$287$	6.97364	0.411641
$288$	−6.26220	−0.369003
$289$	−12.0778	−0.710458
$290$	−5.87737	−0.345131
$291$	−26.3753	−1.54615
$292$	11.6643	0.682600
$293$	26.7768	1.56431	0.782157	−	0.623081i	$-0.214119\pi$
$293$	26.7768	1.56431	0.782157	+	0.623081i	$0.214119\pi$
$294$	−37.1339	−2.16569
$295$	0.135357	0.00788076
$296$	17.6222	1.02427
$297$	−3.13299	−0.181795
$298$	−0.301165	−0.0174460
$299$	−0.266744	−0.0154262
$300$	47.9260	2.76701
$301$	−4.76855	−0.274855
$302$	27.9046	1.60573
$303$	−28.1120	−1.61499
$304$	−19.5476	−1.12113
$305$	2.03497	0.116522
$306$	11.9082	0.680746
$307$	−0.0172484	−0.000984416 0	−0.000492208	−	1.00000i	$-0.500157\pi$
$307$	−0.0172484	−0.000984416 0	−0.000492208		1.00000i	$0.500157\pi$
$308$	4.68382	0.266885
$309$	2.09781	0.119340
$310$	5.21477	0.296179
$311$	−12.6990	−0.720095	−0.360048	−	0.932934i	$-0.617240\pi$
$311$	−12.6990	−0.720095	−0.360048	+	0.932934i	$0.617240\pi$
$312$	12.9109	0.730933
$313$	−29.6175	−1.67408	−0.837039	−	0.547143i	$-0.815715\pi$
$313$	−29.6175	−1.67408	−0.837039	+	0.547143i	$0.815715\pi$
$314$	51.1071	2.88414
$315$	−0.339632	−0.0191361
$316$	18.9716	1.06724
$317$	−0.597982	−0.0335860	−0.0167930	−	0.999859i	$-0.505346\pi$
$317$	−0.597982	−0.0335860	−0.0167930	+	0.999859i	$0.505346\pi$
$318$	−2.60786	−0.146242
$319$	−16.2303	−0.908721
$320$	−0.957195	−0.0535088
$321$	−14.6949	−0.820187
$322$	−0.454273	−0.0253156
$323$	7.59145	0.422400
$324$	−46.3049	−2.57249
$325$	−4.94568	−0.274337
$326$	24.1566	1.33791
$327$	−5.26974	−0.291417
$328$	58.3879	3.22393
$329$	6.19586	0.341589
$330$	−2.13416	−0.117482
$331$	−19.7855	−1.08751	−0.543755	−	0.839244i	$-0.682998\pi$
$331$	−19.7855	−1.08751	−0.543755	+	0.839244i	$0.682998\pi$
$332$	5.35985	0.294160
$333$	−6.63346	−0.363511
$334$	−59.6963	−3.26644
$335$	−2.59851	−0.141972
$336$	8.80926	0.480584
$337$	−10.6548	−0.580405	−0.290203	−	0.956965i	$-0.593723\pi$
$337$	−10.6548	−0.580405	−0.290203	+	0.956965i	$0.593723\pi$
$338$	−2.50460	−0.136232
$339$	2.75626	0.149699
$340$	−2.20961	−0.119833
$341$	14.4005	0.779832
$342$	18.3658	0.993110
$343$	9.20505	0.497026
$344$	−39.9254	−2.15264
$345$	0.140995	0.00759091
$346$	−12.2749	−0.659903
$347$	−31.0845	−1.66870	−0.834351	−	0.551233i	$-0.814157\pi$
$347$	−31.0845	−1.66870	−0.834351	+	0.551233i	$0.814157\pi$
$348$	−97.5644	−5.23000
$349$	22.6243	1.21105	0.605525	−	0.795826i	$-0.292963\pi$
$349$	22.6243	1.21105	0.605525	+	0.795826i	$0.292963\pi$
$350$	−8.42262	−0.450208
$351$	1.94347	0.103735
$352$	4.71065	0.251078
$353$	27.3830	1.45745	0.728725	−	0.684806i	$-0.240113\pi$
$353$	27.3830	1.45745	0.728725	+	0.684806i	$0.240113\pi$
$354$	3.29859	0.175318
$355$	−3.69251	−0.195978
$356$	−43.2323	−2.29131
$357$	−3.42115	−0.181066
$358$	−36.0500	−1.90530
$359$	−31.1011	−1.64145	−0.820726	−	0.571321i	$-0.806431\pi$
$359$	−31.1011	−1.64145	−0.820726	+	0.571321i	$0.806431\pi$
$360$	−2.84362	−0.149872
$361$	−7.29182	−0.383780
$362$	−16.5367	−0.869148
$363$	19.0526	1.00000
$364$	−2.90549	−0.152289
$365$	0.636238	0.0333022
$366$	49.5914	2.59219
$367$	−15.6106	−0.814866	−0.407433	−	0.913235i	$-0.633576\pi$
$367$	−15.6106	−0.814866	−0.407433	+	0.913235i	$0.633576\pi$
$368$	−1.52385	−0.0794363
$369$	−21.9788	−1.14417
$370$	1.80697	0.0939399
$371$	0.312190	0.0162081
$372$	86.5652	4.48819
$373$	−7.50065	−0.388369	−0.194185	−	0.980965i	$-0.562206\pi$
$373$	−7.50065	−0.388369	−0.194185	+	0.980965i	$0.562206\pi$
$374$	−8.95777	−0.463195
$375$	5.25705	0.271473
$376$	51.8758	2.67529
$377$	10.0680	0.518531
$378$	3.30979	0.170237
$379$	19.6220	1.00791	0.503957	−	0.863729i	$-0.331877\pi$
$379$	19.6220	1.00791	0.503957	+	0.863729i	$0.331877\pi$
$380$	−3.40785	−0.174819
$381$	12.4054	0.635549
$382$	43.3719	2.21910
$383$	−5.70394	−0.291458	−0.145729	−	0.989325i	$-0.546553\pi$
$383$	−5.70394	−0.291458	−0.145729	+	0.989325i	$0.546553\pi$
$384$	−36.5803	−1.86673
$385$	0.255483	0.0130206
$386$	−12.7979	−0.651394
$387$	15.0290	0.763968
$388$	49.6965	2.52296
$389$	−15.4384	−0.782758	−0.391379	−	0.920230i	$-0.628002\pi$
$389$	−15.4384	−0.782758	−0.391379	+	0.920230i	$0.628002\pi$
$390$	1.32388	0.0670370
$391$	0.591801	0.0299286
$392$	37.2193	1.87986
$393$	28.7340	1.44944
$394$	−32.4091	−1.63275
$395$	1.03482	0.0520676
$396$	−14.7620	−0.741817
$397$	6.32003	0.317193	0.158597	−	0.987343i	$-0.449303\pi$
$397$	6.32003	0.317193	0.158597	+	0.987343i	$0.449303\pi$
$398$	−32.1998	−1.61403
$399$	−5.27639	−0.264150
$400$	−28.2536	−1.41268
$401$	3.84745	0.192133	0.0960663	−	0.995375i	$-0.469374\pi$
$401$	3.84745	0.192133	0.0960663	+	0.995375i	$0.469374\pi$
$402$	−63.3248	−3.15835
$403$	−8.93300	−0.444984
$404$	52.9687	2.63529
$405$	−2.52574	−0.125505
$406$	17.1462	0.850949
$407$	4.98993	0.247341
$408$	−28.6441	−1.41809
$409$	8.02130	0.396628	0.198314	−	0.980139i	$-0.436453\pi$
$409$	8.02130	0.396628	0.198314	+	0.980139i	$0.436453\pi$
$410$	5.98707	0.295680
$411$	−31.1652	−1.53727
$412$	−3.95269	−0.194735
$413$	−0.394878	−0.0194307
$414$	1.43173	0.0703657
$415$	0.292358	0.0143513
$416$	−2.92213	−0.143269
$417$	15.4018	0.754227
$418$	−13.8154	−0.675735
$419$	3.50476	0.171219	0.0856093	−	0.996329i	$-0.472716\pi$
$419$	3.50476	0.171219	0.0856093	+	0.996329i	$0.472716\pi$
$420$	1.53577	0.0749381
$421$	−26.9213	−1.31207	−0.656033	−	0.754732i	$-0.727767\pi$
$421$	−26.9213	−1.31207	−0.656033	+	0.754732i	$0.727767\pi$
$422$	32.9234	1.60268
$423$	−19.5275	−0.949457
$424$	2.61386	0.126940
$425$	10.9725	0.532244
$426$	−89.9851	−4.35979
$427$	−5.93664	−0.287294
$428$	27.6881	1.33835
$429$	3.65586	0.176507
$430$	−4.09394	−0.197427
$431$	−21.0700	−1.01491	−0.507453	−	0.861680i	$-0.669413\pi$
$431$	−21.0700	−1.01491	−0.507453	+	0.861680i	$0.669413\pi$
$432$	11.1026	0.534176
$433$	−3.54795	−0.170504	−0.0852519	−	0.996359i	$-0.527170\pi$
$433$	−3.54795	−0.170504	−0.0852519	+	0.996359i	$0.527170\pi$
$434$	−15.2131	−0.730254
$435$	−5.32174	−0.255158
$436$	9.92925	0.475525
$437$	0.912725	0.0436616
$438$	15.5049	0.740853
$439$	32.8459	1.56765	0.783825	−	0.620982i	$-0.213266\pi$
$439$	32.8459	1.56765	0.783825	+	0.620982i	$0.213266\pi$
$440$	2.13907	0.101976
$441$	−14.0103	−0.667159
$442$	5.55673	0.264307
$443$	3.56178	0.169225	0.0846125	−	0.996414i	$-0.473035\pi$
$443$	3.56178	0.169225	0.0846125	+	0.996414i	$0.473035\pi$
$444$	29.9957	1.42353
$445$	−2.35814	−0.111787
$446$	55.4923	2.62764
$447$	−0.272693	−0.0128979
$448$	2.79244	0.131930
$449$	−14.9196	−0.704097	−0.352049	−	0.935982i	$-0.614515\pi$
$449$	−14.9196	−0.704097	−0.352049	+	0.935982i	$0.614515\pi$
$450$	26.5455	1.25137
$451$	16.5332	0.778519
$452$	−5.19335	−0.244275
$453$	25.2666	1.18713
$454$	8.97037	0.421000
$455$	−0.158483	−0.00742977
$456$	−44.1774	−2.06879
$457$	6.37388	0.298158	0.149079	−	0.988825i	$-0.452369\pi$
$457$	6.37388	0.298158	0.149079	+	0.988825i	$0.452369\pi$
$458$	6.03679	0.282081
$459$	−4.31180	−0.201258
$460$	−0.265663	−0.0123866
$461$	30.0801	1.40097	0.700486	−	0.713666i	$-0.252967\pi$
$461$	30.0801	1.40097	0.700486	+	0.713666i	$0.252967\pi$
$462$	6.22603	0.289661
$463$	−5.71386	−0.265546	−0.132773	−	0.991146i	$-0.542388\pi$
$463$	−5.71386	−0.265546	−0.132773	+	0.991146i	$0.542388\pi$
$464$	57.5166	2.67014
$465$	4.72177	0.218967
$466$	32.6977	1.51469
$467$	22.4919	1.04080	0.520401	−	0.853922i	$-0.325783\pi$
$467$	22.4919	1.04080	0.520401	+	0.853922i	$0.325783\pi$
$468$	9.15722	0.423293
$469$	7.58068	0.350043
$470$	5.31933	0.245362
$471$	46.2755	2.13226
$472$	−3.30618	−0.152179
$473$	−11.3054	−0.519822
$474$	25.2183	1.15831
$475$	16.9227	0.776468
$476$	6.44613	0.295458
$477$	−0.983927	−0.0450509
$478$	−43.0465	−1.96890
$479$	16.0152	0.731751	0.365876	−	0.930664i	$-0.380770\pi$
$479$	16.0152	0.731751	0.365876	+	0.930664i	$0.380770\pi$
$480$	1.54457	0.0704997
$481$	−3.09538	−0.141137
$482$	63.4084	2.88817
$483$	−0.411327	−0.0187160
$484$	−35.8989	−1.63177
$485$	2.71074	0.123088
$486$	−46.9485	−2.12963
$487$	−39.5946	−1.79420	−0.897102	−	0.441824i	$-0.854332\pi$
$487$	−39.5946	−1.79420	−0.897102	+	0.441824i	$0.854332\pi$
$488$	−49.7055	−2.25006
$489$	21.8729	0.989127
$490$	3.81645	0.172410
$491$	−34.4912	−1.55657	−0.778283	−	0.627914i	$-0.783909\pi$
$491$	−34.4912	−1.55657	−0.778283	+	0.627914i	$0.783909\pi$
$492$	99.3855	4.48064
$493$	−22.3370	−1.00601
$494$	8.57006	0.385585
$495$	−0.805205	−0.0361913
$496$	−51.0323	−2.29142
$497$	10.7722	0.483200
$498$	7.12466	0.319264
$499$	−7.28157	−0.325968	−0.162984	−	0.986629i	$-0.552112\pi$
$499$	−7.28157	−0.325968	−0.162984	+	0.986629i	$0.552112\pi$
$500$	−9.90535	−0.442981
$501$	−54.0528	−2.41490
$502$	−34.5818	−1.54346
$503$	−13.3606	−0.595721	−0.297860	−	0.954609i	$-0.596273\pi$
$503$	−13.3606	−0.595721	−0.297860	+	0.954609i	$0.596273\pi$
$504$	8.29574	0.369522
$505$	2.88923	0.128569
$506$	−1.07700	−0.0478784
$507$	−2.26782	−0.100718
$508$	−23.3743	−1.03707
$509$	−34.4099	−1.52519	−0.762595	−	0.646876i	$-0.776075\pi$
$509$	−34.4099	−1.52519	−0.762595	+	0.646876i	$0.776075\pi$
$510$	−2.93716	−0.130059
$511$	−1.85611	−0.0821094
$512$	48.3529	2.13692
$513$	−6.65003	−0.293606
$514$	18.9751	0.836956
$515$	−0.215603	−0.00950060
$516$	−67.9595	−2.99175
$517$	14.6893	0.646033
$518$	−5.27151	−0.231617
$519$	−11.1145	−0.487871
$520$	−1.32692	−0.0581893
$521$	−18.9349	−0.829551	−0.414776	−	0.909924i	$-0.636140\pi$
$521$	−18.9349	−0.829551	−0.414776	+	0.909924i	$0.636140\pi$
$522$	−54.0395	−2.36524
$523$	−20.3435	−0.889560	−0.444780	−	0.895640i	$-0.646718\pi$
$523$	−20.3435	−0.889560	−0.444780	+	0.895640i	$0.646718\pi$
$524$	−54.1407	−2.36515
$525$	−7.62636	−0.332842
$526$	−58.3728	−2.54518
$527$	19.8188	0.863320
$528$	20.8852	0.908910
$529$	−22.9288	−0.996906
$530$	0.268024	0.0116422
$531$	1.24454	0.0540082
$532$	9.94178	0.431031
$533$	−10.2560	−0.444235
$534$	−57.4672	−2.48685
$535$	1.51027	0.0652947
$536$	63.4704	2.74150
$537$	−32.6419	−1.40860
$538$	23.0875	0.995372
$539$	10.5391	0.453951
$540$	1.93559	0.0832947
$541$	14.0355	0.603432	0.301716	−	0.953398i	$-0.402440\pi$
$541$	14.0355	0.603432	0.301716	+	0.953398i	$0.402440\pi$
$542$	55.8506	2.39899
$543$	−14.9733	−0.642567
$544$	6.48306	0.277959
$545$	0.541600	0.0231996
$546$	−3.86216	−0.165285
$547$	−18.3360	−0.783990	−0.391995	−	0.919967i	$-0.628215\pi$
$547$	−18.3360	−0.783990	−0.391995	+	0.919967i	$0.628215\pi$
$548$	58.7216	2.50846
$549$	18.7105	0.798544
$550$	−19.9685	−0.851460
$551$	−34.4501	−1.46762
$552$	−3.44390	−0.146582
$553$	−3.01891	−0.128377
$554$	34.1770	1.45204
$555$	1.63614	0.0694504
$556$	−29.0200	−1.23072
$557$	25.9777	1.10071	0.550355	−	0.834931i	$-0.314492\pi$
$557$	25.9777	1.10071	0.550355	+	0.834931i	$0.314492\pi$
$558$	47.9472	2.02977
$559$	7.01300	0.296618
$560$	−0.905376	−0.0382591
$561$	−8.11092	−0.342443
$562$	29.7167	1.25352
$563$	33.3190	1.40423	0.702113	−	0.712065i	$-0.252240\pi$
$563$	33.3190	1.40423	0.702113	+	0.712065i	$0.252240\pi$
$564$	88.3009	3.71814
$565$	−0.283276	−0.0119175
$566$	−13.2530	−0.557065
$567$	7.36839	0.309443
$568$	90.1920	3.78437
$569$	40.0741	1.67999	0.839997	−	0.542591i	$-0.182557\pi$
$569$	40.0741	1.67999	0.839997	+	0.542591i	$0.182557\pi$
$570$	−4.52993	−0.189738
$571$	16.9453	0.709139	0.354570	−	0.935030i	$-0.384627\pi$
$571$	16.9453	0.709139	0.354570	+	0.935030i	$0.384627\pi$
$572$	−6.88839	−0.288018
$573$	39.2716	1.64060
$574$	−17.4662	−0.729025
$575$	1.31923	0.0550158
$576$	−8.80092	−0.366705
$577$	14.7277	0.613124	0.306562	−	0.951851i	$-0.400821\pi$
$577$	14.7277	0.613124	0.306562	+	0.951851i	$0.400821\pi$
$578$	30.2501	1.25824
$579$	−11.5880	−0.481580
$580$	10.0272	0.416358
$581$	−0.852901	−0.0353843
$582$	66.0597	2.73826
$583$	0.740146	0.0306537
$584$	−15.5406	−0.643073
$585$	0.499489	0.0206513
$586$	−67.0651	−2.77044
$587$	0.985212	0.0406640	0.0203320	−	0.999793i	$-0.493528\pi$
$587$	0.985212	0.0406640	0.0203320	+	0.999793i	$0.493528\pi$
$588$	63.3531	2.61264
$589$	30.5662	1.25946
$590$	−0.339014	−0.0139570
$591$	−29.3452	−1.20710
$592$	−17.6832	−0.726775
$593$	−26.6718	−1.09528	−0.547641	−	0.836714i	$-0.684474\pi$
$593$	−26.6718	−1.09528	−0.547641	+	0.836714i	$0.684474\pi$
$594$	7.84691	0.321962
$595$	0.351610	0.0144146
$596$	0.513809	0.0210464
$597$	−29.1557	−1.19326
$598$	0.668089	0.0273202
$599$	22.7314	0.928778	0.464389	−	0.885631i	$-0.346274\pi$
$599$	22.7314	0.928778	0.464389	+	0.885631i	$0.346274\pi$
$600$	−63.8529	−2.60678
$601$	16.1431	0.658489	0.329245	−	0.944245i	$-0.393206\pi$
$601$	16.1431	0.658489	0.329245	+	0.944245i	$0.393206\pi$
$602$	11.9433	0.486774
$603$	−23.8920	−0.972957
$604$	−47.6073	−1.93711
$605$	−1.95814	−0.0796097
$606$	70.4094	2.86019
$607$	−0.587187	−0.0238332	−0.0119166	−	0.999929i	$-0.503793\pi$
$607$	−0.587187	−0.0238332	−0.0119166	+	0.999929i	$0.503793\pi$
$608$	9.99873	0.405502
$609$	15.5252	0.629113
$610$	−5.09678	−0.206363
$611$	−9.11211	−0.368636
$612$	−20.3163	−0.821236
$613$	24.7782	1.00078	0.500390	−	0.865800i	$-0.333190\pi$
$613$	24.7782	1.00078	0.500390	+	0.865800i	$0.333190\pi$
$614$	0.0432003	0.00174342
$615$	5.42107	0.218599
$616$	−6.24035	−0.251431
$617$	−21.2092	−0.853848	−0.426924	−	0.904287i	$-0.640403\pi$
$617$	−21.2092	−0.853848	−0.426924	+	0.904287i	$0.640403\pi$
$618$	−5.25417	−0.211354
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−8.89678	−0.357303
$621$	−0.518411	−0.0208031
$622$	31.8060	1.27530
$623$	6.87946	0.275620
$624$	−12.9556	−0.518638
$625$	24.1881	0.967523
$626$	74.1800	2.96483
$627$	−12.5094	−0.499575
$628$	−87.1924	−3.47936
$629$	6.86742	0.273822
$630$	0.850642	0.0338904
$631$	−30.9805	−1.23332	−0.616658	−	0.787231i	$-0.711514\pi$
$631$	−30.9805	−1.23332	−0.616658	+	0.787231i	$0.711514\pi$
$632$	−25.2763	−1.00544
$633$	29.8109	1.18488
$634$	1.49771	0.0594816
$635$	−1.27497	−0.0505957
$636$	4.44920	0.176422
$637$	−6.53766	−0.259031
$638$	40.6504	1.60937
$639$	−33.9507	−1.34307
$640$	3.75956	0.148609
$641$	34.9526	1.38054	0.690272	−	0.723550i	$-0.257491\pi$
$641$	34.9526	1.38054	0.690272	+	0.723550i	$0.257491\pi$
$642$	36.8048	1.45257
$643$	−19.4582	−0.767357	−0.383679	−	0.923467i	$-0.625343\pi$
$643$	−19.4582	−0.767357	−0.383679	+	0.923467i	$0.625343\pi$
$644$	0.775023	0.0305402
$645$	−3.70691	−0.145959
$646$	−19.0136	−0.748079
$647$	20.5162	0.806575	0.403288	−	0.915073i	$-0.367867\pi$
$647$	20.5162	0.806575	0.403288	+	0.915073i	$0.367867\pi$
$648$	61.6930	2.42353
$649$	−0.936184	−0.0367484
$650$	12.3870	0.485856
$651$	−13.7749	−0.539881
$652$	−41.2130	−1.61403
$653$	16.8685	0.660114	0.330057	−	0.943961i	$-0.392932\pi$
$653$	16.8685	0.660114	0.330057	+	0.943961i	$0.392932\pi$
$654$	13.1986	0.516106
$655$	−2.95316	−0.115389
$656$	−58.5901	−2.28756
$657$	5.84989	0.228226
$658$	−15.5182	−0.604961
$659$	41.1965	1.60479	0.802394	−	0.596794i	$-0.203559\pi$
$659$	41.1965	1.60479	0.802394	+	0.596794i	$0.203559\pi$
$660$	3.64104	0.141727
$661$	21.1595	0.823008	0.411504	−	0.911408i	$-0.365004\pi$
$661$	21.1595	0.823008	0.411504	+	0.911408i	$0.365004\pi$
$662$	49.5548	1.92600
$663$	5.03140	0.195404
$664$	−7.14104	−0.277126
$665$	0.542283	0.0210288
$666$	16.6142	0.643787
$667$	−2.68560	−0.103987
$668$	101.846	3.94055
$669$	50.2461	1.94263
$670$	6.50823	0.251435
$671$	−14.0747	−0.543348
$672$	−4.50600	−0.173823
$673$	17.9768	0.692955	0.346477	−	0.938058i	$-0.387378\pi$
$673$	17.9768	0.692955	0.346477	+	0.938058i	$0.387378\pi$
$674$	26.6861	1.02791
$675$	−9.61179	−0.369958
$676$	4.27304	0.164348
$677$	31.7744	1.22119	0.610595	−	0.791943i	$-0.290930\pi$
$677$	31.7744	1.22119	0.610595	+	0.791943i	$0.290930\pi$
$678$	−6.90333	−0.265121
$679$	−7.90808	−0.303484
$680$	2.94391	0.112894
$681$	8.12233	0.311248
$682$	−36.0676	−1.38110
$683$	19.7171	0.754455	0.377227	−	0.926121i	$-0.376878\pi$
$683$	19.7171	0.754455	0.377227	+	0.926121i	$0.376878\pi$
$684$	−31.3335	−1.19806
$685$	3.20302	0.122381
$686$	−23.0550	−0.880244
$687$	5.46608	0.208544
$688$	40.0638	1.52742
$689$	−0.459131	−0.0174915
$690$	−0.353136	−0.0134437
$691$	−42.0264	−1.59876	−0.799381	−	0.600825i	$-0.794839\pi$
$691$	−42.0264	−1.59876	−0.799381	+	0.600825i	$0.794839\pi$
$692$	20.9419	0.796092
$693$	2.34904	0.0892326
$694$	77.8543	2.95531
$695$	−1.58292	−0.0600437
$696$	129.987	4.92715
$697$	22.7540	0.861867
$698$	−56.6649	−2.14480
$699$	29.6065	1.11982
$700$	14.3696	0.543120
$701$	−10.3285	−0.390101	−0.195050	−	0.980793i	$-0.562487\pi$
$701$	−10.3285	−0.390101	−0.195050	+	0.980793i	$0.562487\pi$
$702$	−4.86763	−0.183717
$703$	10.5915	0.399467
$704$	6.62037	0.249515
$705$	4.81645	0.181398
$706$	−68.5836	−2.58118
$707$	−8.42879	−0.316997
$708$	−5.62764	−0.211500
$709$	−39.7615	−1.49327	−0.746636	−	0.665232i	$-0.768332\pi$
$709$	−39.7615	−1.49327	−0.746636	+	0.665232i	$0.768332\pi$
$710$	9.24826	0.347081
$711$	9.51468	0.356828
$712$	57.5993	2.15863
$713$	2.38283	0.0892376
$714$	8.56862	0.320672
$715$	−0.375733	−0.0140516
$716$	61.5040	2.29851
$717$	−38.9769	−1.45562
$718$	77.8959	2.90705
$719$	−22.6854	−0.846022	−0.423011	−	0.906125i	$-0.639027\pi$
$719$	−22.6854	−0.846022	−0.423011	+	0.906125i	$0.639027\pi$
$720$	2.85347	0.106343
$721$	0.628982	0.0234245
$722$	18.2631	0.679683
$723$	57.4139	2.13525
$724$	28.2128	1.04852
$725$	−49.7933	−1.84928
$726$	−47.7192	−1.77102
$727$	6.97996	0.258872	0.129436	−	0.991588i	$-0.458683\pi$
$727$	6.97996	0.258872	0.129436	+	0.991588i	$0.458683\pi$
$728$	3.87105	0.143471
$729$	−10.0005	−0.370391
$730$	−1.59352	−0.0589789
$731$	−15.5591	−0.575473
$732$	−84.6066	−3.12715
$733$	41.1073	1.51833	0.759166	−	0.650897i	$-0.225607\pi$
$733$	41.1073	1.51833	0.759166	+	0.650897i	$0.225607\pi$
$734$	39.0983	1.44315
$735$	3.45565	0.127464
$736$	0.779463	0.0287314
$737$	17.9724	0.662022
$738$	55.0481	2.02635
$739$	14.2608	0.524591	0.262295	−	0.964988i	$-0.415521\pi$
$739$	14.2608	0.524591	0.262295	+	0.964988i	$0.415521\pi$
$740$	−3.08283	−0.113327
$741$	7.75986	0.285066
$742$	−0.781912	−0.0287049
$743$	25.5043	0.935661	0.467830	−	0.883818i	$-0.345036\pi$
$743$	25.5043	0.935661	0.467830	+	0.883818i	$0.345036\pi$
$744$	−115.333	−4.22830
$745$	0.0280262	0.00102680
$746$	18.7862	0.687810
$747$	2.68808	0.0983518
$748$	15.2826	0.558788
$749$	−4.40594	−0.160990
$750$	−13.1668	−0.480785
$751$	38.9682	1.42197	0.710985	−	0.703207i	$-0.248249\pi$
$751$	38.9682	1.42197	0.710985	+	0.703207i	$0.248249\pi$
$752$	−52.0555	−1.89827
$753$	−31.3125	−1.14109
$754$	−25.2165	−0.918330
$755$	−2.59678	−0.0945066
$756$	−5.64674	−0.205370
$757$	46.7254	1.69826	0.849131	−	0.528182i	$-0.177126\pi$
$757$	46.7254	1.69826	0.849131	+	0.528182i	$0.177126\pi$
$758$	−49.1453	−1.78504
$759$	−0.975181	−0.0353968
$760$	4.54035	0.164696
$761$	35.9588	1.30350	0.651752	−	0.758432i	$-0.274034\pi$
$761$	35.9588	1.30350	0.651752	+	0.758432i	$0.274034\pi$
$762$	−31.0706	−1.12557
$763$	−1.58002	−0.0572005
$764$	−73.9957	−2.67707
$765$	−1.10817	−0.0400659
$766$	14.2861	0.516178
$767$	0.580738	0.0209692
$768$	72.9922	2.63388
$769$	−44.3631	−1.59978	−0.799888	−	0.600149i	$-0.795108\pi$
$769$	−44.3631	−1.59978	−0.799888	+	0.600149i	$0.795108\pi$
$770$	−0.639884	−0.0230598
$771$	17.1812	0.618767
$772$	21.8341	0.785826
$773$	−3.59759	−0.129396	−0.0646982	−	0.997905i	$-0.520608\pi$
$773$	−3.59759	−0.129396	−0.0646982	+	0.997905i	$0.520608\pi$
$774$	−37.6417	−1.35300
$775$	44.1797	1.58698
$776$	−66.2117	−2.37686
$777$	−4.77315	−0.171236
$778$	38.6671	1.38628
$779$	35.0931	1.25734
$780$	−2.25863	−0.0808719
$781$	25.5390	0.913856
$782$	−1.48223	−0.0530043
$783$	19.5670	0.699267
$784$	−37.3482	−1.33386
$785$	−4.75599	−0.169749
$786$	−71.9674	−2.56699
$787$	36.6998	1.30821	0.654104	−	0.756405i	$-0.273046\pi$
$787$	36.6998	1.30821	0.654104	+	0.756405i	$0.273046\pi$
$788$	55.2923	1.96971
$789$	−52.8544	−1.88167
$790$	−2.59182	−0.0922129
$791$	0.826405	0.0293836
$792$	19.6677	0.698861
$793$	8.73089	0.310043
$794$	−15.8292	−0.561756
$795$	0.242686	0.00860718
$796$	54.9352	1.94713
$797$	48.2567	1.70934	0.854669	−	0.519173i	$-0.173760\pi$
$797$	48.2567	1.70934	0.854669	+	0.519173i	$0.173760\pi$
$798$	13.2153	0.467815
$799$	20.2162	0.715197
$800$	14.4519	0.510952
$801$	−21.6820	−0.766094
$802$	−9.63634	−0.340271
$803$	−4.40049	−0.155290
$804$	108.037	3.81016
$805$	0.0422743	0.00148997
$806$	22.3736	0.788077
$807$	20.9048	0.735885
$808$	−70.5714	−2.48269
$809$	−12.9612	−0.455693	−0.227846	−	0.973697i	$-0.573168\pi$
$809$	−12.9612	−0.455693	−0.227846	+	0.973697i	$0.573168\pi$
$810$	6.32598	0.222272
$811$	21.7778	0.764722	0.382361	−	0.924013i	$-0.375111\pi$
$811$	21.7778	0.764722	0.382361	+	0.924013i	$0.375111\pi$
$812$	−29.2526	−1.02657
$813$	50.5706	1.77359
$814$	−12.4978	−0.438047
$815$	−2.24800	−0.0787440
$816$	28.7433	1.00622
$817$	−23.9965	−0.839533
$818$	−20.0902	−0.702437
$819$	−1.45717	−0.0509175
$820$	−10.2144	−0.356702
$821$	12.7265	0.444159	0.222079	−	0.975029i	$-0.428716\pi$
$821$	12.7265	0.444159	0.222079	+	0.975029i	$0.428716\pi$
$822$	78.0566	2.72254
$823$	11.9318	0.415918	0.207959	−	0.978138i	$-0.433318\pi$
$823$	11.9318	0.415918	0.207959	+	0.978138i	$0.433318\pi$
$824$	5.26625	0.183459
$825$	−18.0807	−0.629490
$826$	0.989013	0.0344122
$827$	−29.5707	−1.02828	−0.514138	−	0.857708i	$-0.671888\pi$
$827$	−29.5707	−1.02828	−0.514138	+	0.857708i	$0.671888\pi$
$828$	−2.44264	−0.0848875
$829$	42.5660	1.47838	0.739189	−	0.673498i	$-0.235209\pi$
$829$	42.5660	1.47838	0.739189	+	0.673498i	$0.235209\pi$
$830$	−0.732240	−0.0254164
$831$	30.9460	1.07351
$832$	−4.10678	−0.142377
$833$	14.5045	0.502550
$834$	−38.5753	−1.33575
$835$	5.55530	0.192249
$836$	23.5702	0.815191
$837$	−17.3610	−0.600085
$838$	−8.77803	−0.303232
$839$	−31.7522	−1.09621	−0.548103	−	0.836411i	$-0.684650\pi$
$839$	−31.7522	−1.09621	−0.548103	+	0.836411i	$0.684650\pi$
$840$	−2.04614	−0.0705987
$841$	72.3656	2.49537
$842$	67.4273	2.32370
$843$	26.9073	0.926738
$844$	−56.1697	−1.93344
$845$	0.233077	0.00801808
$846$	48.9085	1.68151
$847$	5.71251	0.196284
$848$	−2.62291	−0.0900712
$849$	−12.0001	−0.411842
$850$	−27.4818	−0.942617
$851$	0.825674	0.0283037
$852$	153.521	5.25955
$853$	43.5753	1.49199	0.745995	−	0.665951i	$-0.231974\pi$
$853$	43.5753	1.49199	0.745995	+	0.665951i	$0.231974\pi$
$854$	14.8689	0.508805
$855$	−1.70911	−0.0584504
$856$	−36.8894	−1.26085
$857$	8.27122	0.282540	0.141270	−	0.989971i	$-0.454881\pi$
$857$	8.27122	0.282540	0.141270	+	0.989971i	$0.454881\pi$
$858$	−9.15649	−0.312597
$859$	28.9893	0.989103	0.494551	−	0.869148i	$-0.335332\pi$
$859$	28.9893	0.989103	0.494551	+	0.869148i	$0.335332\pi$
$860$	6.98457	0.238172
$861$	−15.8150	−0.538973
$862$	52.7719	1.79742
$863$	9.32704	0.317496	0.158748	−	0.987319i	$-0.449254\pi$
$863$	9.32704	0.317496	0.158748	+	0.987319i	$0.449254\pi$
$864$	−5.67909	−0.193206
$865$	1.14229	0.0388392
$866$	8.88622	0.301966
$867$	27.3903	0.930223
$868$	25.9547	0.880961
$869$	−7.15728	−0.242794
$870$	13.3288	0.451890
$871$	−11.1487	−0.377760
$872$	−13.2290	−0.447989
$873$	24.9239	0.843545
$874$	−2.28602	−0.0773256
$875$	1.57622	0.0532858
$876$	−26.4525	−0.893747
$877$	−2.20112	−0.0743266	−0.0371633	−	0.999309i	$-0.511832\pi$
$877$	−2.20112	−0.0743266	−0.0371633	+	0.999309i	$0.511832\pi$
$878$	−82.2660	−2.77634
$879$	−60.7249	−2.04820
$880$	−2.14648	−0.0723579
$881$	25.8796	0.871906	0.435953	−	0.899970i	$-0.356411\pi$
$881$	25.8796	0.871906	0.435953	+	0.899970i	$0.356411\pi$
$882$	35.0904	1.18155
$883$	20.2271	0.680696	0.340348	−	0.940300i	$-0.389455\pi$
$883$	20.2271	0.680696	0.340348	+	0.940300i	$0.389455\pi$
$884$	−9.48019	−0.318853
$885$	−0.306965	−0.0103185
$886$	−8.92084	−0.299701
$887$	−28.7007	−0.963675	−0.481838	−	0.876260i	$-0.660031\pi$
$887$	−28.7007	−0.963675	−0.481838	+	0.876260i	$0.660031\pi$
$888$	−39.9639	−1.34110
$889$	3.71950	0.124748
$890$	5.90622	0.197977
$891$	17.4691	0.585237
$892$	−94.6739	−3.16992
$893$	31.1791	1.04337
$894$	0.682988	0.0228425
$895$	3.35479	0.112138
$896$	−10.9678	−0.366409
$897$	0.604929	0.0201980
$898$	37.3676	1.24697
$899$	−89.9378	−2.99959
$900$	−45.2886	−1.50962
$901$	1.01863	0.0339355
$902$	−41.4092	−1.37877
$903$	10.8142	0.359875
$904$	6.91921	0.230129
$905$	1.53889	0.0511545
$906$	−63.2827	−2.10243
$907$	−43.9553	−1.45951	−0.729757	−	0.683707i	$-0.760367\pi$
$907$	−43.9553	−1.45951	−0.729757	+	0.683707i	$0.760367\pi$
$908$	−15.3041	−0.507885
$909$	26.5650	0.881105
$910$	0.396936	0.0131583
$911$	7.26103	0.240569	0.120284	−	0.992739i	$-0.461619\pi$
$911$	7.26103	0.240569	0.120284	+	0.992739i	$0.461619\pi$
$912$	44.3304	1.46793
$913$	−2.02207	−0.0669208
$914$	−15.9640	−0.528044
$915$	−4.61494	−0.152565
$916$	−10.2992	−0.340295
$917$	8.61529	0.284502
$918$	10.7993	0.356432
$919$	−12.2843	−0.405222	−0.202611	−	0.979259i	$-0.564943\pi$
$919$	−12.2843	−0.405222	−0.202611	+	0.979259i	$0.564943\pi$
$920$	0.353948	0.0116693
$921$	0.0391163	0.00128892
$922$	−75.3388	−2.48115
$923$	−15.8424	−0.521461
$924$	−10.6221	−0.349441
$925$	15.3087	0.503348
$926$	14.3110	0.470287
$927$	−1.98236	−0.0651093
$928$	−29.4202	−0.965764
$929$	29.3965	0.964468	0.482234	−	0.876043i	$-0.339826\pi$
$929$	29.3965	0.964468	0.482234	+	0.876043i	$0.339826\pi$
$930$	−11.8262	−0.387795
$931$	22.3700	0.733149
$932$	−55.7847	−1.82729
$933$	28.7991	0.942841
$934$	−56.3333	−1.84328
$935$	0.833604	0.0272618
$936$	−12.2004	−0.398781
$937$	17.2301	0.562882	0.281441	−	0.959579i	$-0.409188\pi$
$937$	17.2301	0.562882	0.281441	+	0.959579i	$0.409188\pi$
$938$	−18.9866	−0.619934
$939$	67.1672	2.19192
$940$	−9.07516	−0.295999
$941$	22.8351	0.744403	0.372201	−	0.928152i	$-0.378603\pi$
$941$	22.8351	0.744403	0.372201	+	0.928152i	$0.378603\pi$
$942$	−115.902	−3.77629
$943$	2.73572	0.0890874
$944$	3.31763	0.107980
$945$	−0.308007	−0.0100195
$946$	28.3155	0.920615
$947$	−36.4412	−1.18418	−0.592089	−	0.805872i	$-0.701697\pi$
$947$	−36.4412	−1.18418	−0.592089	+	0.805872i	$0.701697\pi$
$948$	−43.0243	−1.39736
$949$	2.72974	0.0886110
$950$	−42.3847	−1.37514
$951$	1.35612	0.0439751
$952$	−8.58832	−0.278349
$953$	9.54493	0.309191	0.154595	−	0.987978i	$-0.450593\pi$
$953$	9.54493	0.309191	0.154595	+	0.987978i	$0.450593\pi$
$954$	2.46435	0.0797862
$955$	−4.03616	−0.130607
$956$	73.4404	2.37523
$957$	36.8074	1.18981
$958$	−40.1116	−1.29595
$959$	−9.34424	−0.301741
$960$	2.17075	0.0700606
$961$	48.7984	1.57414
$962$	7.75269	0.249957
$963$	13.8862	0.447476
$964$	−108.179	−3.48422
$965$	1.19096	0.0383383
$966$	1.03021	0.0331465
$967$	−21.3973	−0.688091	−0.344046	−	0.938953i	$-0.611798\pi$
$967$	−21.3973	−0.688091	−0.344046	+	0.938953i	$0.611798\pi$
$968$	47.8289	1.53728
$969$	−17.2161	−0.553060
$970$	−6.78932	−0.217992
$971$	25.3420	0.813263	0.406631	−	0.913592i	$-0.366703\pi$
$971$	25.3420	0.813263	0.406631	+	0.913592i	$0.366703\pi$
$972$	80.0977	2.56913
$973$	4.61789	0.148043
$974$	99.1688	3.17757
$975$	11.2159	0.359197
$976$	49.8777	1.59655
$977$	45.3944	1.45230	0.726148	−	0.687539i	$-0.241309\pi$
$977$	45.3944	1.45230	0.726148	+	0.687539i	$0.241309\pi$
$978$	−54.7830	−1.75177
$979$	16.3099	0.521268
$980$	−6.51115	−0.207991
$981$	4.97974	0.158991
$982$	86.3867	2.75671
$983$	−15.3138	−0.488435	−0.244218	−	0.969720i	$-0.578531\pi$
$983$	−15.3138	−0.488435	−0.244218	+	0.969720i	$0.578531\pi$
$984$	−132.413	−4.22118
$985$	3.01597	0.0960967
$986$	55.9454	1.78166
$987$	−14.0511	−0.447252
$988$	−14.6212	−0.465161
$989$	−1.87068	−0.0594841
$990$	2.01672	0.0640955
$991$	−34.3678	−1.09173	−0.545865	−	0.837873i	$-0.683799\pi$
$991$	−34.3678	−1.09173	−0.545865	+	0.837873i	$0.683799\pi$
$992$	26.1034	0.828784
$993$	44.8700	1.42391
$994$	−26.9801	−0.855757
$995$	2.99649	0.0949951
$996$	−12.1552	−0.385152
$997$	−3.53020	−0.111803	−0.0559013	−	0.998436i	$-0.517803\pi$
$997$	−3.53020	−0.111803	−0.0559013	+	0.998436i	$0.517803\pi$
$998$	18.2375	0.577297
$999$	−6.01578	−0.190331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.14	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.14	✓	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)

\(f(q)\)	\(=\)	\(q-2.50460 q^{2} -2.26782 q^{3} +4.27304 q^{4} +0.233077 q^{5} +5.68000 q^{6} -0.679959 q^{7} -5.69306 q^{8} +2.14302 q^{9} +O(q^{10})\)	\(q-2.50460 q^{2} -2.26782 q^{3} +4.27304 q^{4} +0.233077 q^{5} +5.68000 q^{6} -0.679959 q^{7} -5.69306 q^{8} +2.14302 q^{9} -0.583765 q^{10} -1.61206 q^{11} -9.69050 q^{12} +1.00000 q^{13} +1.70303 q^{14} -0.528577 q^{15} +5.71278 q^{16} -2.21861 q^{17} -5.36742 q^{18} -3.42172 q^{19} +0.995946 q^{20} +1.54203 q^{21} +4.03757 q^{22} -0.266744 q^{23} +12.9109 q^{24} -4.94568 q^{25} -2.50460 q^{26} +1.94347 q^{27} -2.90549 q^{28} +10.0680 q^{29} +1.32388 q^{30} -8.93300 q^{31} -2.92213 q^{32} +3.65586 q^{33} +5.55673 q^{34} -0.158483 q^{35} +9.15722 q^{36} -3.09538 q^{37} +8.57006 q^{38} -2.26782 q^{39} -1.32692 q^{40} -10.2560 q^{41} -3.86216 q^{42} +7.01300 q^{43} -6.88839 q^{44} +0.499489 q^{45} +0.668089 q^{46} -9.11211 q^{47} -12.9556 q^{48} -6.53766 q^{49} +12.3870 q^{50} +5.03140 q^{51} +4.27304 q^{52} -0.459131 q^{53} -4.86763 q^{54} -0.375733 q^{55} +3.87105 q^{56} +7.75986 q^{57} -25.2165 q^{58} +0.580738 q^{59} -2.25863 q^{60} +8.73089 q^{61} +22.3736 q^{62} -1.45717 q^{63} -4.10678 q^{64} +0.233077 q^{65} -9.15649 q^{66} -11.1487 q^{67} -9.48019 q^{68} +0.604929 q^{69} +0.396936 q^{70} -15.8424 q^{71} -12.2004 q^{72} +2.72974 q^{73} +7.75269 q^{74} +11.2159 q^{75} -14.6212 q^{76} +1.09613 q^{77} +5.68000 q^{78} +4.43984 q^{79} +1.33152 q^{80} -10.8365 q^{81} +25.6871 q^{82} +1.25434 q^{83} +6.58914 q^{84} -0.517105 q^{85} -17.5648 q^{86} -22.8326 q^{87} +9.17755 q^{88} -10.1175 q^{89} -1.25102 q^{90} -0.679959 q^{91} -1.13981 q^{92} +20.2585 q^{93} +22.8222 q^{94} -0.797524 q^{95} +6.62688 q^{96} +11.6302 q^{97} +16.3742 q^{98} -3.45468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 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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