LMFDB - Embedded newform 8047.2.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.51095 q^{2} -2.93259 q^{3} +4.30485 q^{4} -3.06715 q^{5} +7.36357 q^{6} +3.82085 q^{7} -5.78735 q^{8} +5.60007 q^{9} +O(q^{10})$	\(q-2.51095 q^{2} -2.93259 q^{3} +4.30485 q^{4} -3.06715 q^{5} +7.36357 q^{6} +3.82085 q^{7} -5.78735 q^{8} +5.60007 q^{9} +7.70146 q^{10} -0.749081 q^{11} -12.6243 q^{12} +1.00000 q^{13} -9.59395 q^{14} +8.99470 q^{15} +5.92201 q^{16} -6.57187 q^{17} -14.0615 q^{18} +5.95299 q^{19} -13.2036 q^{20} -11.2050 q^{21} +1.88090 q^{22} +7.17254 q^{23} +16.9719 q^{24} +4.40744 q^{25} -2.51095 q^{26} -7.62494 q^{27} +16.4482 q^{28} +0.412040 q^{29} -22.5852 q^{30} +0.475285 q^{31} -3.29516 q^{32} +2.19675 q^{33} +16.5016 q^{34} -11.7191 q^{35} +24.1075 q^{36} +7.83017 q^{37} -14.9476 q^{38} -2.93259 q^{39} +17.7507 q^{40} -6.09709 q^{41} +28.1351 q^{42} +3.56425 q^{43} -3.22468 q^{44} -17.1763 q^{45} -18.0099 q^{46} -7.17897 q^{47} -17.3668 q^{48} +7.59890 q^{49} -11.0668 q^{50} +19.2726 q^{51} +4.30485 q^{52} +11.7259 q^{53} +19.1458 q^{54} +2.29755 q^{55} -22.1126 q^{56} -17.4577 q^{57} -1.03461 q^{58} -8.83797 q^{59} +38.7208 q^{60} -4.24712 q^{61} -1.19342 q^{62} +21.3970 q^{63} -3.57005 q^{64} -3.06715 q^{65} -5.51591 q^{66} -11.6021 q^{67} -28.2909 q^{68} -21.0341 q^{69} +29.4261 q^{70} +12.4021 q^{71} -32.4095 q^{72} +2.64223 q^{73} -19.6611 q^{74} -12.9252 q^{75} +25.6267 q^{76} -2.86213 q^{77} +7.36357 q^{78} -4.45566 q^{79} -18.1637 q^{80} +5.56059 q^{81} +15.3095 q^{82} -13.0176 q^{83} -48.2357 q^{84} +20.1569 q^{85} -8.94963 q^{86} -1.20834 q^{87} +4.33519 q^{88} +18.6311 q^{89} +43.1287 q^{90} +3.82085 q^{91} +30.8767 q^{92} -1.39382 q^{93} +18.0260 q^{94} -18.2588 q^{95} +9.66336 q^{96} -17.0653 q^{97} -19.0804 q^{98} -4.19491 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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 $

$2$	−2.51095	−1.77551	−0.887753	−	0.460320i	$-0.847735\pi$
$2$	−2.51095	−1.77551	−0.887753	+	0.460320i	$0.847735\pi$
$3$	−2.93259	−1.69313	−0.846565	−	0.532285i	$-0.821333\pi$
$3$	−2.93259	−1.69313	−0.846565	+	0.532285i	$0.821333\pi$
$4$	4.30485	2.15242
$5$	−3.06715	−1.37167	−0.685837	−	0.727755i	$-0.740564\pi$
$5$	−3.06715	−1.37167	−0.685837	+	0.727755i	$0.740564\pi$
$6$	7.36357	3.00616
$7$	3.82085	1.44415	0.722073	−	0.691817i	$-0.243189\pi$
$7$	3.82085	1.44415	0.722073	+	0.691817i	$0.243189\pi$
$8$	−5.78735	−2.04614
$9$	5.60007	1.86669
$10$	7.70146	2.43542
$11$	−0.749081	−0.225856	−0.112928	−	0.993603i	$-0.536023\pi$
$11$	−0.749081	−0.225856	−0.112928	+	0.993603i	$0.536023\pi$
$12$	−12.6243	−3.64433
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−9.59395	−2.56409
$15$	8.99470	2.32242
$16$	5.92201	1.48050
$17$	−6.57187	−1.59391	−0.796956	−	0.604037i	$-0.793558\pi$
$17$	−6.57187	−1.59391	−0.796956	+	0.604037i	$0.793558\pi$
$18$	−14.0615	−3.31432
$19$	5.95299	1.36571	0.682855	−	0.730554i	$-0.260738\pi$
$19$	5.95299	1.36571	0.682855	+	0.730554i	$0.260738\pi$
$20$	−13.2036	−2.95242
$21$	−11.2050	−2.44513
$22$	1.88090	0.401009
$23$	7.17254	1.49558	0.747789	−	0.663937i	$-0.231116\pi$
$23$	7.17254	1.49558	0.747789	+	0.663937i	$0.231116\pi$
$24$	16.9719	3.46437
$25$	4.40744	0.881488
$26$	−2.51095	−0.492437
$27$	−7.62494	−1.46742
$28$	16.4482	3.10841
$29$	0.412040	0.0765140	0.0382570	−	0.999268i	$-0.487819\pi$
$29$	0.412040	0.0765140	0.0382570	+	0.999268i	$0.487819\pi$
$30$	−22.5852	−4.12348
$31$	0.475285	0.0853638	0.0426819	−	0.999089i	$-0.486410\pi$
$31$	0.475285	0.0853638	0.0426819	+	0.999089i	$0.486410\pi$
$32$	−3.29516	−0.582508
$33$	2.19675	0.382404
$34$	16.5016	2.83000
$35$	−11.7191	−1.98090
$36$	24.1075	4.01791
$37$	7.83017	1.28727	0.643636	−	0.765332i	$-0.277425\pi$
$37$	7.83017	1.28727	0.643636	+	0.765332i	$0.277425\pi$
$38$	−14.9476	−2.42483
$39$	−2.93259	−0.469590
$40$	17.7507	2.80663
$41$	−6.09709	−0.952206	−0.476103	−	0.879390i	$-0.657951\pi$
$41$	−6.09709	−0.952206	−0.476103	+	0.879390i	$0.657951\pi$
$42$	28.1351	4.34134
$43$	3.56425	0.543542	0.271771	−	0.962362i	$-0.412391\pi$
$43$	3.56425	0.543542	0.271771	+	0.962362i	$0.412391\pi$
$44$	−3.22468	−0.486138
$45$	−17.1763	−2.56049
$46$	−18.0099	−2.65541
$47$	−7.17897	−1.04716	−0.523580	−	0.851976i	$-0.675404\pi$
$47$	−7.17897	−1.04716	−0.523580	+	0.851976i	$0.675404\pi$
$48$	−17.3668	−2.50669
$49$	7.59890	1.08556
$50$	−11.0668	−1.56509
$51$	19.2726	2.69870
$52$	4.30485	0.596975
$53$	11.7259	1.61068	0.805338	−	0.592816i	$-0.201984\pi$
$53$	11.7259	1.61068	0.805338	+	0.592816i	$0.201984\pi$
$54$	19.1458	2.60541
$55$	2.29755	0.309801
$56$	−22.1126	−2.95492
$57$	−17.4577	−2.31233
$58$	−1.03461	−0.135851
$59$	−8.83797	−1.15061	−0.575303	−	0.817940i	$-0.695116\pi$
$59$	−8.83797	−1.15061	−0.575303	+	0.817940i	$0.695116\pi$
$60$	38.7208	4.99884
$61$	−4.24712	−0.543788	−0.271894	−	0.962327i	$-0.587650\pi$
$61$	−4.24712	−0.543788	−0.271894	+	0.962327i	$0.587650\pi$
$62$	−1.19342	−0.151564
$63$	21.3970	2.69577
$64$	−3.57005	−0.446256
$65$	−3.06715	−0.380434
$66$	−5.51591	−0.678961
$67$	−11.6021	−1.41743	−0.708713	−	0.705497i	$-0.750724\pi$
$67$	−11.6021	−1.41743	−0.708713	+	0.705497i	$0.750724\pi$
$68$	−28.2909	−3.43078
$69$	−21.0341	−2.53221
$70$	29.4261	3.51709
$71$	12.4021	1.47185	0.735927	−	0.677061i	$-0.236747\pi$
$71$	12.4021	1.47185	0.735927	+	0.677061i	$0.236747\pi$
$72$	−32.4095	−3.81950
$73$	2.64223	0.309250	0.154625	−	0.987973i	$-0.450583\pi$
$73$	2.64223	0.309250	0.154625	+	0.987973i	$0.450583\pi$
$74$	−19.6611	−2.28556
$75$	−12.9252	−1.49247
$76$	25.6267	2.93959
$77$	−2.86213	−0.326169
$78$	7.36357	0.833760
$79$	−4.45566	−0.501300	−0.250650	−	0.968078i	$-0.580644\pi$
$79$	−4.45566	−0.501300	−0.250650	+	0.968078i	$0.580644\pi$
$80$	−18.1637	−2.03077
$81$	5.56059	0.617843
$82$	15.3095	1.69065
$83$	−13.0176	−1.42887	−0.714434	−	0.699703i	$-0.753316\pi$
$83$	−13.0176	−1.42887	−0.714434	+	0.699703i	$0.753316\pi$
$84$	−48.2357	−5.26295
$85$	20.1569	2.18633
$86$	−8.94963	−0.965063
$87$	−1.20834	−0.129548
$88$	4.33519	0.462133
$89$	18.6311	1.97490	0.987448	−	0.157943i	$-0.0504864\pi$
$89$	18.6311	1.97490	0.987448	+	0.157943i	$0.0504864\pi$
$90$	43.1287	4.54617
$91$	3.82085	0.400534
$92$	30.8767	3.21912
$93$	−1.39382	−0.144532
$94$	18.0260	1.85924
$95$	−18.2588	−1.87331
$96$	9.66336	0.986263
$97$	−17.0653	−1.73272	−0.866359	−	0.499421i	$-0.833546\pi$
$97$	−17.0653	−1.73272	−0.866359	+	0.499421i	$0.833546\pi$
$98$	−19.0804	−1.92741
$99$	−4.19491	−0.421604
$100$	18.9734	1.89734
$101$	8.21460	0.817383	0.408691	−	0.912673i	$-0.365985\pi$
$101$	8.21460	0.817383	0.408691	+	0.912673i	$0.365985\pi$
$102$	−48.3924	−4.79156
$103$	−12.6003	−1.24155	−0.620774	−	0.783989i	$-0.713182\pi$
$103$	−12.6003	−1.24155	−0.620774	+	0.783989i	$0.713182\pi$
$104$	−5.78735	−0.567496
$105$	34.3674	3.35392
$106$	−29.4431	−2.85976
$107$	−4.80628	−0.464641	−0.232320	−	0.972639i	$-0.574632\pi$
$107$	−4.80628	−0.464641	−0.232320	+	0.972639i	$0.574632\pi$
$108$	−32.8242	−3.15851
$109$	0.0196813	0.00188513	0.000942564	−	1.00000i	$-0.499700\pi$
$109$	0.0196813	0.00188513	0.000942564		1.00000i	$0.499700\pi$
$110$	−5.76901	−0.550054
$111$	−22.9627	−2.17952
$112$	22.6271	2.13806
$113$	−17.5678	−1.65264	−0.826318	−	0.563203i	$-0.809569\pi$
$113$	−17.5678	−1.65264	−0.826318	+	0.563203i	$0.809569\pi$
$114$	43.8353	4.10555
$115$	−21.9993	−2.05144
$116$	1.77377	0.164691
$117$	5.60007	0.517727
$118$	22.1917	2.04291
$119$	−25.1101	−2.30184
$120$	−52.0554	−4.75199
$121$	−10.4389	−0.948989
$122$	10.6643	0.965498
$123$	17.8803	1.61221
$124$	2.04603	0.183739
$125$	1.81748	0.162560
$126$	−53.7268	−4.78636
$127$	−18.2482	−1.61926	−0.809631	−	0.586939i	$-0.800333\pi$
$127$	−18.2482	−1.61926	−0.809631	+	0.586939i	$0.800333\pi$
$128$	15.5545	1.37484
$129$	−10.4525	−0.920288
$130$	7.70146	0.675463
$131$	−2.35837	−0.206052	−0.103026	−	0.994679i	$-0.532853\pi$
$131$	−2.35837	−0.206052	−0.103026	+	0.994679i	$0.532853\pi$
$132$	9.45665	0.823096
$133$	22.7455	1.97228
$134$	29.1323	2.51665
$135$	23.3869	2.01282
$136$	38.0337	3.26136
$137$	−23.0178	−1.96654	−0.983271	−	0.182149i	$-0.941695\pi$
$137$	−23.0178	−1.96654	−0.983271	+	0.182149i	$0.941695\pi$
$138$	52.8155	4.49595
$139$	1.43869	0.122028	0.0610139	−	0.998137i	$-0.480567\pi$
$139$	1.43869	0.122028	0.0610139	+	0.998137i	$0.480567\pi$
$140$	−50.4491	−4.26373
$141$	21.0530	1.77298
$142$	−31.1409	−2.61329
$143$	−0.749081	−0.0626413
$144$	33.1637	2.76364
$145$	−1.26379	−0.104952
$146$	−6.63450	−0.549075
$147$	−22.2844	−1.83799
$148$	33.7077	2.77075
$149$	4.54911	0.372678	0.186339	−	0.982486i	$-0.440338\pi$
$149$	4.54911	0.372678	0.186339	+	0.982486i	$0.440338\pi$
$150$	32.4545	2.64990
$151$	−10.0100	−0.814600	−0.407300	−	0.913294i	$-0.633530\pi$
$151$	−10.0100	−0.814600	−0.407300	+	0.913294i	$0.633530\pi$
$152$	−34.4520	−2.79443
$153$	−36.8029	−2.97534
$154$	7.18664	0.579116
$155$	−1.45777	−0.117091
$156$	−12.6243	−1.01076
$157$	14.8366	1.18409	0.592043	−	0.805906i	$-0.298321\pi$
$157$	14.8366	1.18409	0.592043	+	0.805906i	$0.298321\pi$
$158$	11.1879	0.890062
$159$	−34.3872	−2.72708
$160$	10.1068	0.799011
$161$	27.4052	2.15983
$162$	−13.9623	−1.09698
$163$	15.7145	1.23086	0.615428	−	0.788193i	$-0.288983\pi$
$163$	15.7145	1.23086	0.615428	+	0.788193i	$0.288983\pi$
$164$	−26.2470	−2.04955
$165$	−6.73776	−0.524534
$166$	32.6865	2.53696
$167$	2.36711	0.183172	0.0915861	−	0.995797i	$-0.470806\pi$
$167$	2.36711	0.183172	0.0915861	+	0.995797i	$0.470806\pi$
$168$	64.8471	5.00306
$169$	1.00000	0.0769231
$170$	−50.6130	−3.88184
$171$	33.3372	2.54936
$172$	15.3435	1.16993
$173$	−15.5318	−1.18086	−0.590432	−	0.807087i	$-0.701043\pi$
$173$	−15.5318	−1.18086	−0.590432	+	0.807087i	$0.701043\pi$
$174$	3.03409	0.230014
$175$	16.8402	1.27300
$176$	−4.43607	−0.334381
$177$	25.9181	1.94813
$178$	−46.7818	−3.50644
$179$	11.5547	0.863635	0.431818	−	0.901961i	$-0.357872\pi$
$179$	11.5547	0.863635	0.431818	+	0.901961i	$0.357872\pi$
$180$	−73.9413	−5.51126
$181$	18.7197	1.39142	0.695712	−	0.718321i	$-0.255089\pi$
$181$	18.7197	1.39142	0.695712	+	0.718321i	$0.255089\pi$
$182$	−9.59395	−0.711151
$183$	12.4550	0.920703
$184$	−41.5100	−3.06015
$185$	−24.0163	−1.76572
$186$	3.49980	0.256617
$187$	4.92286	0.359995
$188$	−30.9044	−2.25393
$189$	−29.1338	−2.11917
$190$	45.8467	3.32607
$191$	22.8428	1.65285	0.826423	−	0.563050i	$-0.190372\pi$
$191$	22.8428	1.65285	0.826423	+	0.563050i	$0.190372\pi$
$192$	10.4695	0.755570
$193$	−0.741921	−0.0534046	−0.0267023	−	0.999643i	$-0.508501\pi$
$193$	−0.741921	−0.0534046	−0.0267023	+	0.999643i	$0.508501\pi$
$194$	42.8500	3.07645
$195$	8.99470	0.644124
$196$	32.7121	2.33658
$197$	7.14733	0.509226	0.254613	−	0.967043i	$-0.418052\pi$
$197$	7.14733	0.509226	0.254613	+	0.967043i	$0.418052\pi$
$198$	10.5332	0.748560
$199$	1.89945	0.134649	0.0673244	−	0.997731i	$-0.478554\pi$
$199$	1.89945	0.134649	0.0673244	+	0.997731i	$0.478554\pi$
$200$	−25.5074	−1.80364
$201$	34.0243	2.39989
$202$	−20.6264	−1.45127
$203$	1.57435	0.110497
$204$	82.9655	5.80875
$205$	18.7007	1.30612
$206$	31.6388	2.20438
$207$	40.1667	2.79178
$208$	5.92201	0.410618
$209$	−4.45927	−0.308454
$210$	−86.2947	−5.95490
$211$	−0.153142	−0.0105427	−0.00527136	−	0.999986i	$-0.501678\pi$
$211$	−0.153142	−0.0105427	−0.00527136	+	0.999986i	$0.501678\pi$
$212$	50.4782	3.46686
$213$	−36.3701	−2.49204
$214$	12.0683	0.824972
$215$	−10.9321	−0.745563
$216$	44.1282	3.00254
$217$	1.81599	0.123278
$218$	−0.0494187	−0.00334706
$219$	−7.74857	−0.523600
$220$	9.89059	0.666823
$221$	−6.57187	−0.442072
$222$	57.6580	3.86975
$223$	−23.4689	−1.57159	−0.785797	−	0.618485i	$-0.787747\pi$
$223$	−23.4689	−1.57159	−0.785797	+	0.618485i	$0.787747\pi$
$224$	−12.5903	−0.841227
$225$	24.6820	1.64547
$226$	44.1117	2.93427
$227$	−2.07557	−0.137760	−0.0688802	−	0.997625i	$-0.521943\pi$
$227$	−2.07557	−0.137760	−0.0688802	+	0.997625i	$0.521943\pi$
$228$	−75.1526	−4.97710
$229$	−19.0655	−1.25988	−0.629942	−	0.776642i	$-0.716921\pi$
$229$	−19.0655	−1.25988	−0.629942	+	0.776642i	$0.716921\pi$
$230$	55.2390	3.64235
$231$	8.39343	0.552247
$232$	−2.38462	−0.156558
$233$	9.65407	0.632459	0.316230	−	0.948683i	$-0.397583\pi$
$233$	9.65407	0.632459	0.316230	+	0.948683i	$0.397583\pi$
$234$	−14.0615	−0.919227
$235$	22.0190	1.43636
$236$	−38.0461	−2.47659
$237$	13.0666	0.848767
$238$	63.0502	4.08694
$239$	5.68936	0.368014	0.184007	−	0.982925i	$-0.441093\pi$
$239$	5.68936	0.368014	0.184007	+	0.982925i	$0.441093\pi$
$240$	53.2668	3.43835
$241$	14.6817	0.945732	0.472866	−	0.881134i	$-0.343219\pi$
$241$	14.6817	0.945732	0.472866	+	0.881134i	$0.343219\pi$
$242$	26.2115	1.68494
$243$	6.56790	0.421331
$244$	−18.2832	−1.17046
$245$	−23.3070	−1.48903
$246$	−44.8964	−2.86249
$247$	5.95299	0.378780
$248$	−2.75064	−0.174666
$249$	38.1753	2.41926
$250$	−4.56358	−0.288626
$251$	8.85729	0.559067	0.279534	−	0.960136i	$-0.409820\pi$
$251$	8.85729	0.559067	0.279534	+	0.960136i	$0.409820\pi$
$252$	92.1110	5.80245
$253$	−5.37281	−0.337786
$254$	45.8201	2.87501
$255$	−59.1120	−3.70174
$256$	−31.9165	−1.99478
$257$	−3.33415	−0.207979	−0.103989	−	0.994578i	$-0.533161\pi$
$257$	−3.33415	−0.207979	−0.103989	+	0.994578i	$0.533161\pi$
$258$	26.2456	1.63398
$259$	29.9179	1.85901
$260$	−13.2036	−0.818855
$261$	2.30746	0.142828
$262$	5.92175	0.365847
$263$	−11.3542	−0.700130	−0.350065	−	0.936725i	$-0.613840\pi$
$263$	−11.3542	−0.700130	−0.350065	+	0.936725i	$0.613840\pi$
$264$	−12.7133	−0.782451
$265$	−35.9651	−2.20932
$266$	−57.1127	−3.50180
$267$	−54.6374	−3.34376
$268$	−49.9454	−3.05090
$269$	0.573796	0.0349850	0.0174925	−	0.999847i	$-0.494432\pi$
$269$	0.573796	0.0349850	0.0174925	+	0.999847i	$0.494432\pi$
$270$	−58.7231	−3.57378
$271$	−25.0626	−1.52245	−0.761223	−	0.648490i	$-0.775401\pi$
$271$	−25.0626	−1.52245	−0.761223	+	0.648490i	$0.775401\pi$
$272$	−38.9187	−2.35979
$273$	−11.2050	−0.678156
$274$	57.7964	3.49161
$275$	−3.30153	−0.199090
$276$	−90.5486	−5.45038
$277$	−5.48033	−0.329281	−0.164641	−	0.986354i	$-0.552646\pi$
$277$	−5.48033	−0.329281	−0.164641	+	0.986354i	$0.552646\pi$
$278$	−3.61247	−0.216661
$279$	2.66163	0.159348
$280$	67.8227	4.05318
$281$	31.4182	1.87425	0.937126	−	0.348992i	$-0.113476\pi$
$281$	31.4182	1.87425	0.937126	+	0.348992i	$0.113476\pi$
$282$	−52.8628	−3.14794
$283$	−5.70825	−0.339320	−0.169660	−	0.985503i	$-0.554267\pi$
$283$	−5.70825	−0.339320	−0.169660	+	0.985503i	$0.554267\pi$
$284$	53.3890	3.16805
$285$	53.5454	3.17176
$286$	1.88090	0.111220
$287$	−23.2961	−1.37512
$288$	−18.4532	−1.08736
$289$	26.1895	1.54056
$290$	3.17331	0.186343
$291$	50.0455	2.93372
$292$	11.3744	0.665636
$293$	18.3929	1.07453	0.537263	−	0.843415i	$-0.319458\pi$
$293$	18.3929	1.07453	0.537263	+	0.843415i	$0.319458\pi$
$294$	55.9550	3.26336
$295$	27.1074	1.57826
$296$	−45.3159	−2.63393
$297$	5.71169	0.331426
$298$	−11.4226	−0.661692
$299$	7.17254	0.414799
$300$	−55.6410	−3.21244
$301$	13.6185	0.784954
$302$	25.1345	1.44633
$303$	−24.0900	−1.38394
$304$	35.2537	2.02194
$305$	13.0266	0.745899
$306$	92.4102	5.28274
$307$	−13.6422	−0.778599	−0.389299	−	0.921111i	$-0.627283\pi$
$307$	−13.6422	−0.778599	−0.389299	+	0.921111i	$0.627283\pi$
$308$	−12.3210	−0.702055
$309$	36.9516	2.10210
$310$	3.66039	0.207896
$311$	10.8019	0.612518	0.306259	−	0.951948i	$-0.400923\pi$
$311$	10.8019	0.612518	0.306259	+	0.951948i	$0.400923\pi$
$312$	16.9719	0.960845
$313$	−10.5925	−0.598722	−0.299361	−	0.954140i	$-0.596774\pi$
$313$	−10.5925	−0.598722	−0.299361	+	0.954140i	$0.596774\pi$
$314$	−37.2538	−2.10235
$315$	−65.6280	−3.69772
$316$	−19.1809	−1.07901
$317$	22.1119	1.24193	0.620963	−	0.783840i	$-0.286742\pi$
$317$	22.1119	1.24193	0.620963	+	0.783840i	$0.286742\pi$
$318$	86.3444	4.84195
$319$	−0.308652	−0.0172812
$320$	10.9499	0.612118
$321$	14.0948	0.786697
$322$	−68.8130	−3.83480
$323$	−39.1223	−2.17682
$324$	23.9375	1.32986
$325$	4.40744	0.244481
$326$	−39.4583	−2.18539
$327$	−0.0577172	−0.00319177
$328$	35.2860	1.94834
$329$	−27.4298	−1.51225
$330$	16.9181	0.931313
$331$	20.2468	1.11287	0.556433	−	0.830893i	$-0.312170\pi$
$331$	20.2468	1.11287	0.556433	+	0.830893i	$0.312170\pi$
$332$	−56.0388	−3.07553
$333$	43.8495	2.40294
$334$	−5.94368	−0.325224
$335$	35.5855	1.94425
$336$	−66.3561	−3.62002
$337$	3.44942	0.187902	0.0939510	−	0.995577i	$-0.470050\pi$
$337$	3.44942	0.187902	0.0939510	+	0.995577i	$0.470050\pi$
$338$	−2.51095	−0.136577
$339$	51.5190	2.79813
$340$	86.7726	4.70590
$341$	−0.356027	−0.0192799
$342$	−83.7079	−4.52640
$343$	2.28831	0.123557
$344$	−20.6275	−1.11216
$345$	64.5148	3.47336
$346$	38.9996	2.09663
$347$	−19.7336	−1.05935	−0.529677	−	0.848199i	$-0.677687\pi$
$347$	−19.7336	−1.05935	−0.529677	+	0.848199i	$0.677687\pi$
$348$	−5.20174	−0.278843
$349$	−10.1289	−0.542189	−0.271094	−	0.962553i	$-0.587386\pi$
$349$	−10.1289	−0.542189	−0.271094	+	0.962553i	$0.587386\pi$
$350$	−42.2847	−2.26021
$351$	−7.62494	−0.406989
$352$	2.46834	0.131563
$353$	17.2160	0.916313	0.458157	−	0.888871i	$-0.348510\pi$
$353$	17.2160	0.916313	0.458157	+	0.888871i	$0.348510\pi$
$354$	−65.0790	−3.45891
$355$	−38.0390	−2.01890
$356$	80.2042	4.25081
$357$	73.6377	3.89732
$358$	−29.0131	−1.53339
$359$	−29.4540	−1.55452	−0.777261	−	0.629178i	$-0.783392\pi$
$359$	−29.4540	−1.55452	−0.777261	+	0.629178i	$0.783392\pi$
$360$	99.4051	5.23911
$361$	16.4381	0.865164
$362$	−47.0041	−2.47048
$363$	30.6129	1.60676
$364$	16.4482	0.862119
$365$	−8.10413	−0.424190
$366$	−31.2739	−1.63471
$367$	−35.7815	−1.86778	−0.933891	−	0.357558i	$-0.883609\pi$
$367$	−35.7815	−1.86778	−0.933891	+	0.357558i	$0.883609\pi$
$368$	42.4759	2.21421
$369$	−34.1442	−1.77747
$370$	60.3037	3.13504
$371$	44.8029	2.32605
$372$	−6.00016	−0.311094
$373$	−1.08646	−0.0562545	−0.0281273	−	0.999604i	$-0.508954\pi$
$373$	−1.08646	−0.0562545	−0.0281273	+	0.999604i	$0.508954\pi$
$374$	−12.3610	−0.639174
$375$	−5.32991	−0.275235
$376$	41.5472	2.14263
$377$	0.412040	0.0212212
$378$	73.1533	3.76260
$379$	7.08196	0.363776	0.181888	−	0.983319i	$-0.441779\pi$
$379$	7.08196	0.363776	0.181888	+	0.983319i	$0.441779\pi$
$380$	−78.6011	−4.03215
$381$	53.5143	2.74162
$382$	−57.3570	−2.93464
$383$	16.7244	0.854578	0.427289	−	0.904115i	$-0.359469\pi$
$383$	16.7244	0.854578	0.427289	+	0.904115i	$0.359469\pi$
$384$	−45.6150	−2.32778
$385$	8.77858	0.447398
$386$	1.86292	0.0948203
$387$	19.9600	1.01463
$388$	−73.4635	−3.72955
$389$	−19.5849	−0.992992	−0.496496	−	0.868039i	$-0.665380\pi$
$389$	−19.5849	−0.992992	−0.496496	+	0.868039i	$0.665380\pi$
$390$	−22.5852	−1.14365
$391$	−47.1370	−2.38382
$392$	−43.9775	−2.22120
$393$	6.91614	0.348873
$394$	−17.9466	−0.904134
$395$	13.6662	0.687621
$396$	−18.0584	−0.907470
$397$	−16.6132	−0.833792	−0.416896	−	0.908954i	$-0.636882\pi$
$397$	−16.6132	−0.833792	−0.416896	+	0.908954i	$0.636882\pi$
$398$	−4.76943	−0.239070
$399$	−66.7032	−3.33934
$400$	26.1009	1.30505
$401$	17.1393	0.855894	0.427947	−	0.903804i	$-0.359237\pi$
$401$	17.1393	0.855894	0.427947	+	0.903804i	$0.359237\pi$
$402$	−85.4331	−4.26101
$403$	0.475285	0.0236756
$404$	35.3626	1.75935
$405$	−17.0552	−0.847479
$406$	−3.95309	−0.196189
$407$	−5.86543	−0.290738
$408$	−111.537	−5.52191
$409$	16.8137	0.831382	0.415691	−	0.909506i	$-0.363540\pi$
$409$	16.8137	0.831382	0.415691	+	0.909506i	$0.363540\pi$
$410$	−46.9565	−2.31902
$411$	67.5017	3.32961
$412$	−54.2425	−2.67234
$413$	−33.7686	−1.66164
$414$	−100.856	−4.95682
$415$	39.9270	1.95994
$416$	−3.29516	−0.161559
$417$	−4.21908	−0.206609
$418$	11.1970	0.547663
$419$	12.5347	0.612360	0.306180	−	0.951974i	$-0.400949\pi$
$419$	12.5347	0.612360	0.306180	+	0.951974i	$0.400949\pi$
$420$	147.946	7.21905
$421$	−33.0625	−1.61137	−0.805685	−	0.592344i	$-0.798203\pi$
$421$	−33.0625	−1.61137	−0.805685	+	0.592344i	$0.798203\pi$
$422$	0.384531	0.0187187
$423$	−40.2027	−1.95472
$424$	−67.8618	−3.29566
$425$	−28.9651	−1.40501
$426$	91.3234	4.42463
$427$	−16.2276	−0.785309
$428$	−20.6903	−1.00010
$429$	2.19675	0.106060
$430$	27.4499	1.32375
$431$	36.8853	1.77670	0.888350	−	0.459167i	$-0.151852\pi$
$431$	36.8853	1.77670	0.888350	+	0.459167i	$0.151852\pi$
$432$	−45.1550	−2.17252
$433$	15.1503	0.728075	0.364038	−	0.931384i	$-0.381398\pi$
$433$	15.1503	0.728075	0.364038	+	0.931384i	$0.381398\pi$
$434$	−4.55986	−0.218880
$435$	3.70618	0.177698
$436$	0.0847250	0.00405759
$437$	42.6981	2.04253
$438$	19.4562	0.929655
$439$	27.5223	1.31357	0.656784	−	0.754078i	$-0.271916\pi$
$439$	27.5223	1.31357	0.656784	+	0.754078i	$0.271916\pi$
$440$	−13.2967	−0.633895
$441$	42.5544	2.02640
$442$	16.5016	0.784901
$443$	11.6361	0.552849	0.276425	−	0.961036i	$-0.410850\pi$
$443$	11.6361	0.552849	0.276425	+	0.961036i	$0.410850\pi$
$444$	−98.8507	−4.69125
$445$	−57.1446	−2.70891
$446$	58.9291	2.79037
$447$	−13.3407	−0.630992
$448$	−13.6406	−0.644459
$449$	−24.4068	−1.15183	−0.575914	−	0.817510i	$-0.695354\pi$
$449$	−24.4068	−1.15183	−0.575914	+	0.817510i	$0.695354\pi$
$450$	−61.9751	−2.92153
$451$	4.56721	0.215062
$452$	−75.6266	−3.55717
$453$	29.3551	1.37922
$454$	5.21164	0.244594
$455$	−11.7191	−0.549402
$456$	101.034	4.73133
$457$	8.27890	0.387270	0.193635	−	0.981074i	$-0.437972\pi$
$457$	8.27890	0.387270	0.193635	+	0.981074i	$0.437972\pi$
$458$	47.8724	2.23693
$459$	50.1101	2.33894
$460$	−94.7036	−4.41558
$461$	−6.89724	−0.321236	−0.160618	−	0.987017i	$-0.551349\pi$
$461$	−6.89724	−0.321236	−0.160618	+	0.987017i	$0.551349\pi$
$462$	−21.0755	−0.980519
$463$	9.61766	0.446970	0.223485	−	0.974707i	$-0.428257\pi$
$463$	9.61766	0.446970	0.223485	+	0.974707i	$0.428257\pi$
$464$	2.44011	0.113279
$465$	4.27505	0.198251
$466$	−24.2408	−1.12294
$467$	−34.1652	−1.58098	−0.790488	−	0.612477i	$-0.790173\pi$
$467$	−34.1652	−1.58098	−0.790488	+	0.612477i	$0.790173\pi$
$468$	24.1075	1.11437
$469$	−44.3300	−2.04697
$470$	−55.2885	−2.55027
$471$	−43.5095	−2.00481
$472$	51.1484	2.35430
$473$	−2.66991	−0.122762
$474$	−32.8095	−1.50699
$475$	26.2375	1.20386
$476$	−108.095	−4.95454
$477$	65.6658	3.00663
$478$	−14.2857	−0.653411
$479$	18.1797	0.830654	0.415327	−	0.909672i	$-0.363667\pi$
$479$	18.1797	0.830654	0.415327	+	0.909672i	$0.363667\pi$
$480$	−29.6390	−1.35283
$481$	7.83017	0.357025
$482$	−36.8650	−1.67915
$483$	−80.3681	−3.65688
$484$	−44.9378	−2.04263
$485$	52.3419	2.37672
$486$	−16.4917	−0.748076
$487$	3.03173	0.137381	0.0686904	−	0.997638i	$-0.478118\pi$
$487$	3.03173	0.137381	0.0686904	+	0.997638i	$0.478118\pi$
$488$	24.5795	1.11266
$489$	−46.0842	−2.08400
$490$	58.5226	2.64378
$491$	30.2964	1.36726	0.683629	−	0.729829i	$-0.260401\pi$
$491$	30.2964	1.36726	0.683629	+	0.729829i	$0.260401\pi$
$492$	76.9718	3.47016
$493$	−2.70788	−0.121957
$494$	−14.9476	−0.672526
$495$	12.8664	0.578303
$496$	2.81465	0.126381
$497$	47.3864	2.12557
$498$	−95.8560	−4.29541
$499$	15.0802	0.675082	0.337541	−	0.941311i	$-0.390405\pi$
$499$	15.0802	0.675082	0.337541	+	0.941311i	$0.390405\pi$
$500$	7.82395	0.349898
$501$	−6.94175	−0.310135
$502$	−22.2402	−0.992627
$503$	−40.5273	−1.80702	−0.903512	−	0.428562i	$-0.859020\pi$
$503$	−40.5273	−1.80702	−0.903512	+	0.428562i	$0.859020\pi$
$504$	−123.832	−5.51592
$505$	−25.1954	−1.12118
$506$	13.4908	0.599741
$507$	−2.93259	−0.130241
$508$	−78.5555	−3.48534
$509$	−24.9313	−1.10506	−0.552531	−	0.833492i	$-0.686338\pi$
$509$	−24.9313	−1.10506	−0.552531	+	0.833492i	$0.686338\pi$
$510$	148.427	6.57246
$511$	10.0956	0.446602
$512$	49.0315	2.16691
$513$	−45.3912	−2.00407
$514$	8.37187	0.369267
$515$	38.6472	1.70300
$516$	−44.9963	−1.98085
$517$	5.37763	0.236508
$518$	−75.1222	−3.30068
$519$	45.5485	1.99936
$520$	17.7507	0.778419
$521$	−11.7291	−0.513862	−0.256931	−	0.966430i	$-0.582711\pi$
$521$	−11.7291	−0.513862	−0.256931	+	0.966430i	$0.582711\pi$
$522$	−5.79390	−0.253592
$523$	31.7559	1.38859	0.694294	−	0.719692i	$-0.255717\pi$
$523$	31.7559	1.38859	0.694294	+	0.719692i	$0.255717\pi$
$524$	−10.1524	−0.443512
$525$	−49.3853	−2.15535
$526$	28.5098	1.24308
$527$	−3.12351	−0.136062
$528$	13.0092	0.566151
$529$	28.4453	1.23675
$530$	90.3065	3.92266
$531$	−49.4933	−2.14783
$532$	97.9159	4.24519
$533$	−6.09709	−0.264094
$534$	137.192	5.93686
$535$	14.7416	0.637335
$536$	67.1455	2.90025
$537$	−33.8850	−1.46225
$538$	−1.44077	−0.0621160
$539$	−5.69219	−0.245180
$540$	100.677	4.33244
$541$	−1.98254	−0.0852361	−0.0426181	−	0.999091i	$-0.513570\pi$
$541$	−1.98254	−0.0852361	−0.0426181	+	0.999091i	$0.513570\pi$
$542$	62.9309	2.70311
$543$	−54.8971	−2.35586
$544$	21.6554	0.928467
$545$	−0.0603656	−0.00258578
$546$	28.1351	1.20407
$547$	5.77163	0.246777	0.123389	−	0.992358i	$-0.460624\pi$
$547$	5.77163	0.246777	0.123389	+	0.992358i	$0.460624\pi$
$548$	−99.0880	−4.23283
$549$	−23.7842	−1.01508
$550$	8.28996	0.353485
$551$	2.45287	0.104496
$552$	121.732	5.18124
$553$	−17.0244	−0.723951
$554$	13.7608	0.584641
$555$	70.4300	2.98959
$556$	6.19333	0.262656
$557$	17.1828	0.728059	0.364029	−	0.931387i	$-0.381401\pi$
$557$	17.1828	0.728059	0.364029	+	0.931387i	$0.381401\pi$
$558$	−6.68321	−0.282923
$559$	3.56425	0.150752
$560$	−69.4009	−2.93272
$561$	−14.4367	−0.609519
$562$	−78.8893	−3.32775
$563$	21.7127	0.915081	0.457540	−	0.889189i	$-0.348731\pi$
$563$	21.7127	0.915081	0.457540	+	0.889189i	$0.348731\pi$
$564$	90.6298	3.81620
$565$	53.8831	2.26688
$566$	14.3331	0.602465
$567$	21.2462	0.892255
$568$	−71.7750	−3.01161
$569$	−46.7764	−1.96097	−0.980484	−	0.196598i	$-0.937011\pi$
$569$	−46.7764	−1.96097	−0.980484	+	0.196598i	$0.937011\pi$
$570$	−134.450	−5.63147
$571$	36.8554	1.54235	0.771176	−	0.636622i	$-0.219669\pi$
$571$	36.8554	1.54235	0.771176	+	0.636622i	$0.219669\pi$
$572$	−3.22468	−0.134831
$573$	−66.9884	−2.79848
$574$	58.4952	2.44154
$575$	31.6125	1.31833
$576$	−19.9925	−0.833022
$577$	4.67849	0.194768	0.0973840	−	0.995247i	$-0.468953\pi$
$577$	4.67849	0.194768	0.0973840	+	0.995247i	$0.468953\pi$
$578$	−65.7604	−2.73527
$579$	2.17575	0.0904210
$580$	−5.44043	−0.225902
$581$	−49.7383	−2.06349
$582$	−125.662	−5.20884
$583$	−8.78364	−0.363781
$584$	−15.2915	−0.632767
$585$	−17.1763	−0.710152
$586$	−46.1836	−1.90783
$587$	−6.23092	−0.257178	−0.128589	−	0.991698i	$-0.541045\pi$
$587$	−6.23092	−0.257178	−0.128589	+	0.991698i	$0.541045\pi$
$588$	−95.9311	−3.95613
$589$	2.82937	0.116582
$590$	−68.0653	−2.80220
$591$	−20.9602	−0.862186
$592$	46.3704	1.90581
$593$	2.84182	0.116699	0.0583497	−	0.998296i	$-0.481416\pi$
$593$	2.84182	0.116699	0.0583497	+	0.998296i	$0.481416\pi$
$594$	−14.3418	−0.588449
$595$	77.0167	3.15738
$596$	19.5832	0.802160
$597$	−5.57032	−0.227978
$598$	−18.0099	−0.736478
$599$	−45.8994	−1.87540	−0.937699	−	0.347449i	$-0.887048\pi$
$599$	−45.8994	−1.87540	−0.937699	+	0.347449i	$0.887048\pi$
$600$	74.8026	3.05380
$601$	−12.4576	−0.508155	−0.254077	−	0.967184i	$-0.581772\pi$
$601$	−12.4576	−0.508155	−0.254077	+	0.967184i	$0.581772\pi$
$602$	−34.1952	−1.39369
$603$	−64.9728	−2.64590
$604$	−43.0914	−1.75336
$605$	32.0177	1.30170
$606$	60.4887	2.45719
$607$	−2.81382	−0.114209	−0.0571046	−	0.998368i	$-0.518187\pi$
$607$	−2.81382	−0.114209	−0.0571046	+	0.998368i	$0.518187\pi$
$608$	−19.6161	−0.795538
$609$	−4.61691	−0.187086
$610$	−32.7090	−1.32435
$611$	−7.17897	−0.290430
$612$	−158.431	−6.40420
$613$	31.3727	1.26713	0.633566	−	0.773689i	$-0.281591\pi$
$613$	31.3727	1.26713	0.633566	+	0.773689i	$0.281591\pi$
$614$	34.2547	1.38241
$615$	−54.8415	−2.21142
$616$	16.5641	0.667387
$617$	−16.2240	−0.653152	−0.326576	−	0.945171i	$-0.605895\pi$
$617$	−16.2240	−0.653152	−0.326576	+	0.945171i	$0.605895\pi$
$618$	−92.7835	−3.73230
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−6.27549	−0.252030
$621$	−54.6902	−2.19464
$622$	−27.1229	−1.08753
$623$	71.1868	2.85204
$624$	−17.3668	−0.695230
$625$	−27.6117	−1.10447
$626$	26.5971	1.06304
$627$	13.0772	0.522253
$628$	63.8691	2.54866
$629$	−51.4588	−2.05180
$630$	164.788	6.56533
$631$	7.61067	0.302976	0.151488	−	0.988459i	$-0.451594\pi$
$631$	7.61067	0.302976	0.151488	+	0.988459i	$0.451594\pi$
$632$	25.7864	1.02573
$633$	0.449102	0.0178502
$634$	−55.5217	−2.20505
$635$	55.9699	2.22110
$636$	−148.032	−5.86984
$637$	7.59890	0.301079
$638$	0.775007	0.0306828
$639$	69.4524	2.74749
$640$	−47.7082	−1.88583
$641$	−12.0456	−0.475773	−0.237887	−	0.971293i	$-0.576455\pi$
$641$	−12.0456	−0.475773	−0.237887	+	0.971293i	$0.576455\pi$
$642$	−35.3914	−1.39679
$643$	−8.31163	−0.327779	−0.163889	−	0.986479i	$-0.552404\pi$
$643$	−8.31163	−0.327779	−0.163889	+	0.986479i	$0.552404\pi$
$644$	117.975	4.64887
$645$	32.0593	1.26233
$646$	98.2340	3.86496
$647$	−32.3214	−1.27069	−0.635343	−	0.772230i	$-0.719141\pi$
$647$	−32.3214	−1.27069	−0.635343	+	0.772230i	$0.719141\pi$
$648$	−32.1810	−1.26419
$649$	6.62036	0.259872
$650$	−11.0668	−0.434077
$651$	−5.32556	−0.208725
$652$	67.6486	2.64932
$653$	−19.4110	−0.759612	−0.379806	−	0.925066i	$-0.624009\pi$
$653$	−19.4110	−0.759612	−0.379806	+	0.925066i	$0.624009\pi$
$654$	0.144925	0.00566700
$655$	7.23350	0.282636
$656$	−36.1071	−1.40974
$657$	14.7967	0.577274
$658$	68.8747	2.68501
$659$	−18.5649	−0.723185	−0.361592	−	0.932336i	$-0.617767\pi$
$659$	−18.5649	−0.723185	−0.361592	+	0.932336i	$0.617767\pi$
$660$	−29.0050	−1.12902
$661$	−44.1347	−1.71664	−0.858320	−	0.513114i	$-0.828492\pi$
$661$	−44.1347	−1.71664	−0.858320	+	0.513114i	$0.828492\pi$
$662$	−50.8387	−1.97590
$663$	19.2726	0.748485
$664$	75.3374	2.92366
$665$	−69.7640	−2.70533
$666$	−110.104	−4.26643
$667$	2.95538	0.114433
$668$	10.1900	0.394264
$669$	68.8246	2.66091
$670$	−89.3533	−3.45202
$671$	3.18143	0.122818
$672$	36.9223	1.42431
$673$	−26.0350	−1.00357	−0.501787	−	0.864991i	$-0.667324\pi$
$673$	−26.0350	−1.00357	−0.501787	+	0.864991i	$0.667324\pi$
$674$	−8.66131	−0.333621
$675$	−33.6065	−1.29351
$676$	4.30485	0.165571
$677$	−36.4331	−1.40024	−0.700118	−	0.714027i	$-0.746869\pi$
$677$	−36.4331	−1.40024	−0.700118	+	0.714027i	$0.746869\pi$
$678$	−129.361	−4.96810
$679$	−65.2040	−2.50230
$680$	−116.655	−4.47352
$681$	6.08679	0.233246
$682$	0.893964	0.0342317
$683$	10.3166	0.394754	0.197377	−	0.980328i	$-0.436758\pi$
$683$	10.3166	0.394754	0.197377	+	0.980328i	$0.436758\pi$
$684$	143.511	5.48730
$685$	70.5991	2.69745
$686$	−5.74582	−0.219376
$687$	55.9112	2.13315
$688$	21.1075	0.804716
$689$	11.7259	0.446721
$690$	−161.993	−6.16698
$691$	10.4846	0.398853	0.199427	−	0.979913i	$-0.436092\pi$
$691$	10.4846	0.398853	0.199427	+	0.979913i	$0.436092\pi$
$692$	−66.8622	−2.54172
$693$	−16.0281	−0.608857
$694$	49.5500	1.88089
$695$	−4.41268	−0.167382
$696$	6.99311	0.265073
$697$	40.0693	1.51773
$698$	25.4332	0.962660
$699$	−28.3114	−1.07084
$700$	72.4943	2.74003
$701$	2.50436	0.0945885	0.0472943	−	0.998881i	$-0.484940\pi$
$701$	2.50436	0.0945885	0.0472943	+	0.998881i	$0.484940\pi$
$702$	19.1458	0.722612
$703$	46.6129	1.75804
$704$	2.67426	0.100790
$705$	−64.5727	−2.43195
$706$	−43.2283	−1.62692
$707$	31.3867	1.18042
$708$	111.574	4.19319
$709$	−5.59934	−0.210288	−0.105144	−	0.994457i	$-0.533530\pi$
$709$	−5.59934	−0.210288	−0.105144	+	0.994457i	$0.533530\pi$
$710$	95.5139	3.58457
$711$	−24.9520	−0.935773
$712$	−107.825	−4.04091
$713$	3.40900	0.127668
$714$	−184.900	−6.91972
$715$	2.29755	0.0859234
$716$	49.7410	1.85891
$717$	−16.6845	−0.623096
$718$	73.9574	2.76006
$719$	−5.25914	−0.196133	−0.0980665	−	0.995180i	$-0.531266\pi$
$719$	−5.25914	−0.196133	−0.0980665	+	0.995180i	$0.531266\pi$
$720$	−101.718	−3.79081
$721$	−48.1440	−1.79298
$722$	−41.2752	−1.53610
$723$	−43.0554	−1.60125
$724$	80.5854	2.99493
$725$	1.81604	0.0674462
$726$	−76.8674	−2.85282
$727$	19.0262	0.705643	0.352821	−	0.935691i	$-0.385222\pi$
$727$	19.0262	0.705643	0.352821	+	0.935691i	$0.385222\pi$
$728$	−22.1126	−0.819547
$729$	−35.9427	−1.33121
$730$	20.3490	0.753151
$731$	−23.4238	−0.866359
$732$	53.6171	1.98174
$733$	37.8883	1.39944	0.699718	−	0.714419i	$-0.253309\pi$
$733$	37.8883	1.39944	0.699718	+	0.714419i	$0.253309\pi$
$734$	89.8455	3.31626
$735$	68.3498	2.52112
$736$	−23.6347	−0.871186
$737$	8.69093	0.320135
$738$	85.7341	3.15592
$739$	21.4831	0.790268	0.395134	−	0.918623i	$-0.370698\pi$
$739$	21.4831	0.790268	0.395134	+	0.918623i	$0.370698\pi$
$740$	−103.387	−3.80057
$741$	−17.4577	−0.641324
$742$	−112.498	−4.12992
$743$	49.7677	1.82580	0.912900	−	0.408184i	$-0.133838\pi$
$743$	49.7677	1.82580	0.912900	+	0.408184i	$0.133838\pi$
$744$	8.06649	0.295732
$745$	−13.9528	−0.511192
$746$	2.72803	0.0998803
$747$	−72.8995	−2.66725
$748$	21.1922	0.774862
$749$	−18.3641	−0.671009
$750$	13.3831	0.488682
$751$	−49.1593	−1.79385	−0.896924	−	0.442186i	$-0.854203\pi$
$751$	−49.1593	−1.79385	−0.896924	+	0.442186i	$0.854203\pi$
$752$	−42.5140	−1.55032
$753$	−25.9748	−0.946574
$754$	−1.03461	−0.0376783
$755$	30.7021	1.11736
$756$	−125.416	−4.56135
$757$	20.9112	0.760032	0.380016	−	0.924980i	$-0.375918\pi$
$757$	20.9112	0.760032	0.380016	+	0.924980i	$0.375918\pi$
$758$	−17.7824	−0.645886
$759$	15.7562	0.571915
$760$	105.670	3.83304
$761$	−8.72024	−0.316108	−0.158054	−	0.987430i	$-0.550522\pi$
$761$	−8.72024	−0.316108	−0.158054	+	0.987430i	$0.550522\pi$
$762$	−134.372	−4.86777
$763$	0.0751994	0.00272240
$764$	98.3347	3.55762
$765$	112.880	4.08120
$766$	−41.9941	−1.51731
$767$	−8.83797	−0.319121
$768$	93.5979	3.37742
$769$	28.6009	1.03137	0.515687	−	0.856777i	$-0.327537\pi$
$769$	28.6009	1.03137	0.515687	+	0.856777i	$0.327537\pi$
$770$	−22.0425	−0.794358
$771$	9.77769	0.352135
$772$	−3.19386	−0.114949
$773$	−9.93121	−0.357201	−0.178600	−	0.983922i	$-0.557157\pi$
$773$	−9.93121	−0.357201	−0.178600	+	0.983922i	$0.557157\pi$
$774$	−50.1185	−1.80147
$775$	2.09479	0.0752471
$776$	98.7628	3.54538
$777$	−87.7369	−3.14754
$778$	49.1765	1.76306
$779$	−36.2959	−1.30044
$780$	38.7208	1.38643
$781$	−9.29014	−0.332427
$782$	118.358	4.23249
$783$	−3.14178	−0.112278
$784$	45.0008	1.60717
$785$	−45.5060	−1.62418
$786$	−17.3661	−0.619427
$787$	1.23055	0.0438644	0.0219322	−	0.999759i	$-0.493018\pi$
$787$	1.23055	0.0438644	0.0219322	+	0.999759i	$0.493018\pi$
$788$	30.7682	1.09607
$789$	33.2972	1.18541
$790$	−34.3151	−1.22087
$791$	−67.1238	−2.38665
$792$	24.2774	0.862659
$793$	−4.24712	−0.150820
$794$	41.7148	1.48040
$795$	105.471	3.74067
$796$	8.17686	0.289821
$797$	18.0324	0.638741	0.319370	−	0.947630i	$-0.396529\pi$
$797$	18.0324	0.638741	0.319370	+	0.947630i	$0.396529\pi$
$798$	167.488	5.92901
$799$	47.1793	1.66908
$800$	−14.5232	−0.513474
$801$	104.336	3.68652
$802$	−43.0358	−1.51965
$803$	−1.97924	−0.0698460
$804$	146.469	5.16557
$805$	−84.0560	−2.96258
$806$	−1.19342	−0.0420363
$807$	−1.68271	−0.0592341
$808$	−47.5407	−1.67248
$809$	−18.2396	−0.641271	−0.320636	−	0.947203i	$-0.603896\pi$
$809$	−18.2396	−0.641271	−0.320636	+	0.947203i	$0.603896\pi$
$810$	42.8246	1.50470
$811$	5.62309	0.197453	0.0987266	−	0.995115i	$-0.468523\pi$
$811$	5.62309	0.197453	0.0987266	+	0.995115i	$0.468523\pi$
$812$	6.77732	0.237837
$813$	73.4984	2.57770
$814$	14.7278	0.516208
$815$	−48.1988	−1.68833
$816$	114.133	3.99544
$817$	21.2179	0.742321
$818$	−42.2182	−1.47612
$819$	21.3970	0.747673
$820$	80.5038	2.81131
$821$	−7.54530	−0.263333	−0.131666	−	0.991294i	$-0.542033\pi$
$821$	−7.54530	−0.263333	−0.131666	+	0.991294i	$0.542033\pi$
$822$	−169.493	−5.91175
$823$	−2.50448	−0.0873006	−0.0436503	−	0.999047i	$-0.513899\pi$
$823$	−2.50448	−0.0873006	−0.0436503	+	0.999047i	$0.513899\pi$
$824$	72.9225	2.54038
$825$	9.68202	0.337085
$826$	84.7911	2.95026
$827$	−2.94367	−0.102361	−0.0511807	−	0.998689i	$-0.516298\pi$
$827$	−2.94367	−0.102361	−0.0511807	+	0.998689i	$0.516298\pi$
$828$	172.912	6.00909
$829$	−31.1995	−1.08360	−0.541801	−	0.840507i	$-0.682257\pi$
$829$	−31.1995	−1.08360	−0.541801	+	0.840507i	$0.682257\pi$
$830$	−100.255	−3.47988
$831$	16.0715	0.557516
$832$	−3.57005	−0.123769
$833$	−49.9390	−1.73028
$834$	10.5939	0.366836
$835$	−7.26029	−0.251253
$836$	−19.1965	−0.663924
$837$	−3.62402	−0.125264
$838$	−31.4739	−1.08725
$839$	1.50613	0.0519975	0.0259987	−	0.999662i	$-0.491723\pi$
$839$	1.50613	0.0519975	0.0259987	+	0.999662i	$0.491723\pi$
$840$	−198.896	−6.86257
$841$	−28.8302	−0.994146
$842$	83.0183	2.86100
$843$	−92.1366	−3.17335
$844$	−0.659252	−0.0226924
$845$	−3.06715	−0.105513
$846$	100.947	3.47063
$847$	−39.8854	−1.37048
$848$	69.4409	2.38461
$849$	16.7399	0.574514
$850$	72.7298	2.49461
$851$	56.1622	1.92521
$852$	−156.568	−5.36393
$853$	−28.3662	−0.971241	−0.485620	−	0.874170i	$-0.661406\pi$
$853$	−28.3662	−0.971241	−0.485620	+	0.874170i	$0.661406\pi$
$854$	40.7466	1.39432
$855$	−102.250	−3.49689
$856$	27.8156	0.950718
$857$	17.7051	0.604796	0.302398	−	0.953182i	$-0.402213\pi$
$857$	17.7051	0.604796	0.302398	+	0.953182i	$0.402213\pi$
$858$	−5.51591	−0.188310
$859$	15.2851	0.521520	0.260760	−	0.965404i	$-0.416027\pi$
$859$	15.2851	0.521520	0.260760	+	0.965404i	$0.416027\pi$
$860$	−47.0610	−1.60477
$861$	68.3178	2.32826
$862$	−92.6169	−3.15454
$863$	−8.30837	−0.282820	−0.141410	−	0.989951i	$-0.545164\pi$
$863$	−8.30837	−0.282820	−0.141410	+	0.989951i	$0.545164\pi$
$864$	25.1254	0.854784
$865$	47.6386	1.61976
$866$	−38.0415	−1.29270
$867$	−76.8030	−2.60837
$868$	7.81758	0.265346
$869$	3.33765	0.113222
$870$	−9.30602	−0.315504
$871$	−11.6021	−0.393123
$872$	−0.113903	−0.00385723
$873$	−95.5669	−3.23445
$874$	−107.213	−3.62652
$875$	6.94430	0.234760
$876$	−33.3564	−1.12701
$877$	34.6992	1.17171	0.585854	−	0.810417i	$-0.300759\pi$
$877$	34.6992	1.17171	0.585854	+	0.810417i	$0.300759\pi$
$878$	−69.1070	−2.33225
$879$	−53.9388	−1.81931
$880$	13.6061	0.458662
$881$	29.0239	0.977841	0.488920	−	0.872328i	$-0.337391\pi$
$881$	29.0239	0.977841	0.488920	+	0.872328i	$0.337391\pi$
$882$	−106.852	−3.59789
$883$	−26.6122	−0.895571	−0.447785	−	0.894141i	$-0.647787\pi$
$883$	−26.6122	−0.895571	−0.447785	+	0.894141i	$0.647787\pi$
$884$	−28.2909	−0.951526
$885$	−79.4949	−2.67219
$886$	−29.2177	−0.981587
$887$	53.8416	1.80782	0.903911	−	0.427720i	$-0.140683\pi$
$887$	53.8416	1.80782	0.903911	+	0.427720i	$0.140683\pi$
$888$	132.893	4.45959
$889$	−69.7235	−2.33845
$890$	143.487	4.80969
$891$	−4.16533	−0.139544
$892$	−101.030	−3.38273
$893$	−42.7363	−1.43012
$894$	33.4977	1.12033
$895$	−35.4399	−1.18463
$896$	59.4315	1.98547
$897$	−21.0341	−0.702308
$898$	61.2842	2.04508
$899$	0.195837	0.00653152
$900$	106.252	3.54174
$901$	−77.0611	−2.56728
$902$	−11.4680	−0.381843
$903$	−39.9373	−1.32903
$904$	101.671	3.38152
$905$	−57.4162	−1.90858
$906$	−73.7091	−2.44882
$907$	17.0271	0.565375	0.282688	−	0.959212i	$-0.408774\pi$
$907$	17.0271	0.565375	0.282688	+	0.959212i	$0.408774\pi$
$908$	−8.93501	−0.296519
$909$	46.0023	1.52580
$910$	29.4261	0.975467
$911$	−43.1106	−1.42832	−0.714160	−	0.699983i	$-0.753191\pi$
$911$	−43.1106	−1.42832	−0.714160	+	0.699983i	$0.753191\pi$
$912$	−103.385	−3.42341
$913$	9.75123	0.322719
$914$	−20.7879	−0.687601
$915$	−38.2015	−1.26290
$916$	−82.0740	−2.71180
$917$	−9.01100	−0.297569
$918$	−125.824	−4.15280
$919$	−29.3492	−0.968141	−0.484071	−	0.875029i	$-0.660842\pi$
$919$	−29.3492	−0.968141	−0.484071	+	0.875029i	$0.660842\pi$
$920$	127.317	4.19753
$921$	40.0068	1.31827
$922$	17.3186	0.570357
$923$	12.4021	0.408219
$924$	36.1325	1.18867
$925$	34.5110	1.13471
$926$	−24.1494	−0.793599
$927$	−70.5628	−2.31759
$928$	−1.35774	−0.0445700
$929$	−10.4799	−0.343834	−0.171917	−	0.985111i	$-0.554996\pi$
$929$	−10.4799	−0.343834	−0.171917	+	0.985111i	$0.554996\pi$
$930$	−10.7344	−0.351995
$931$	45.2362	1.48256
$932$	41.5593	1.36132
$933$	−31.6774	−1.03707
$934$	85.7869	2.80703
$935$	−15.0992	−0.493796
$936$	−32.4095	−1.05934
$937$	−25.8119	−0.843240	−0.421620	−	0.906773i	$-0.638538\pi$
$937$	−25.8119	−0.843240	−0.421620	+	0.906773i	$0.638538\pi$
$938$	111.310	3.63441
$939$	31.0634	1.01371
$940$	94.7885	3.09166
$941$	4.84424	0.157918	0.0789588	−	0.996878i	$-0.474840\pi$
$941$	4.84424	0.157918	0.0789588	+	0.996878i	$0.474840\pi$
$942$	109.250	3.55956
$943$	−43.7316	−1.42410
$944$	−52.3386	−1.70348
$945$	89.3577	2.90681
$946$	6.70399	0.217966
$947$	−1.27499	−0.0414316	−0.0207158	−	0.999785i	$-0.506595\pi$
$947$	−1.27499	−0.0414316	−0.0207158	+	0.999785i	$0.506595\pi$
$948$	56.2497	1.82691
$949$	2.64223	0.0857704
$950$	−65.8808	−2.13746
$951$	−64.8450	−2.10274
$952$	145.321	4.70988
$953$	−16.6521	−0.539414	−0.269707	−	0.962942i	$-0.586927\pi$
$953$	−16.6521	−0.539414	−0.269707	+	0.962942i	$0.586927\pi$
$954$	−164.883	−5.33830
$955$	−70.0623	−2.26716
$956$	24.4918	0.792122
$957$	0.905148	0.0292593
$958$	−45.6484	−1.47483
$959$	−87.9475	−2.83997
$960$	−32.1115	−1.03640
$961$	−30.7741	−0.992713
$962$	−19.6611	−0.633900
$963$	−26.9155	−0.867340
$964$	63.2025	2.03562
$965$	2.27559	0.0732537
$966$	201.800	6.49281
$967$	−41.3705	−1.33039	−0.665193	−	0.746671i	$-0.731651\pi$
$967$	−41.3705	−1.33039	−0.665193	+	0.746671i	$0.731651\pi$
$968$	60.4134	1.94176
$969$	114.730	3.68564
$970$	−131.428	−4.21989
$971$	−49.9053	−1.60154	−0.800769	−	0.598973i	$-0.795575\pi$
$971$	−49.9053	−1.60154	−0.800769	+	0.598973i	$0.795575\pi$
$972$	28.2738	0.906883
$973$	5.49701	0.176226
$974$	−7.61251	−0.243921
$975$	−12.9252	−0.413938
$976$	−25.1515	−0.805079
$977$	−17.7932	−0.569254	−0.284627	−	0.958638i	$-0.591870\pi$
$977$	−17.7932	−0.569254	−0.284627	+	0.958638i	$0.591870\pi$
$978$	115.715	3.70015
$979$	−13.9562	−0.446043
$980$	−100.333	−3.20502
$981$	0.110217	0.00351895
$982$	−76.0727	−2.42758
$983$	12.9381	0.412660	0.206330	−	0.978482i	$-0.433848\pi$
$983$	12.9381	0.412660	0.206330	+	0.978482i	$0.433848\pi$
$984$	−103.479	−3.29880
$985$	−21.9220	−0.698492
$986$	6.79933	0.216535
$987$	80.4402	2.56044
$988$	25.6267	0.815295
$989$	25.5647	0.812910
$990$	−32.3069	−1.02678
$991$	27.9144	0.886729	0.443365	−	0.896341i	$-0.353785\pi$
$991$	27.9144	0.886729	0.443365	+	0.896341i	$0.353785\pi$
$992$	−1.56614	−0.0497251
$993$	−59.3756	−1.88423
$994$	−118.985	−3.77397
$995$	−5.82592	−0.184694
$996$	164.339	5.20727
$997$	−28.1399	−0.891198	−0.445599	−	0.895233i	$-0.647009\pi$
$997$	−28.1399	−0.891198	−0.445599	+	0.895233i	$0.647009\pi$
$998$	−37.8655	−1.19861
$999$	−59.7045	−1.88897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.12	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.12	✓	142	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)

\(f(q)\)	\(=\)	\(q-2.51095 q^{2} -2.93259 q^{3} +4.30485 q^{4} -3.06715 q^{5} +7.36357 q^{6} +3.82085 q^{7} -5.78735 q^{8} +5.60007 q^{9} +O(q^{10})\)	\(q-2.51095 q^{2} -2.93259 q^{3} +4.30485 q^{4} -3.06715 q^{5} +7.36357 q^{6} +3.82085 q^{7} -5.78735 q^{8} +5.60007 q^{9} +7.70146 q^{10} -0.749081 q^{11} -12.6243 q^{12} +1.00000 q^{13} -9.59395 q^{14} +8.99470 q^{15} +5.92201 q^{16} -6.57187 q^{17} -14.0615 q^{18} +5.95299 q^{19} -13.2036 q^{20} -11.2050 q^{21} +1.88090 q^{22} +7.17254 q^{23} +16.9719 q^{24} +4.40744 q^{25} -2.51095 q^{26} -7.62494 q^{27} +16.4482 q^{28} +0.412040 q^{29} -22.5852 q^{30} +0.475285 q^{31} -3.29516 q^{32} +2.19675 q^{33} +16.5016 q^{34} -11.7191 q^{35} +24.1075 q^{36} +7.83017 q^{37} -14.9476 q^{38} -2.93259 q^{39} +17.7507 q^{40} -6.09709 q^{41} +28.1351 q^{42} +3.56425 q^{43} -3.22468 q^{44} -17.1763 q^{45} -18.0099 q^{46} -7.17897 q^{47} -17.3668 q^{48} +7.59890 q^{49} -11.0668 q^{50} +19.2726 q^{51} +4.30485 q^{52} +11.7259 q^{53} +19.1458 q^{54} +2.29755 q^{55} -22.1126 q^{56} -17.4577 q^{57} -1.03461 q^{58} -8.83797 q^{59} +38.7208 q^{60} -4.24712 q^{61} -1.19342 q^{62} +21.3970 q^{63} -3.57005 q^{64} -3.06715 q^{65} -5.51591 q^{66} -11.6021 q^{67} -28.2909 q^{68} -21.0341 q^{69} +29.4261 q^{70} +12.4021 q^{71} -32.4095 q^{72} +2.64223 q^{73} -19.6611 q^{74} -12.9252 q^{75} +25.6267 q^{76} -2.86213 q^{77} +7.36357 q^{78} -4.45566 q^{79} -18.1637 q^{80} +5.56059 q^{81} +15.3095 q^{82} -13.0176 q^{83} -48.2357 q^{84} +20.1569 q^{85} -8.94963 q^{86} -1.20834 q^{87} +4.33519 q^{88} +18.6311 q^{89} +43.1287 q^{90} +3.82085 q^{91} +30.8767 q^{92} -1.39382 q^{93} +18.0260 q^{94} -18.2588 q^{95} +9.66336 q^{96} -17.0653 q^{97} -19.0804 q^{98} -4.19491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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