LMFDB - Embedded newform 8047.2.a.b.1.6

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			1.6
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)

$f(q)$	$=$	$q-2.65986 q^{2} -2.58294 q^{3} +5.07483 q^{4} +0.599360 q^{5} +6.87024 q^{6} -3.90786 q^{7} -8.17861 q^{8} +3.67156 q^{9} +O(q^{10})$	\(q-2.65986 q^{2} -2.58294 q^{3} +5.07483 q^{4} +0.599360 q^{5} +6.87024 q^{6} -3.90786 q^{7} -8.17861 q^{8} +3.67156 q^{9} -1.59421 q^{10} -2.24374 q^{11} -13.1080 q^{12} +1.00000 q^{13} +10.3944 q^{14} -1.54811 q^{15} +11.6043 q^{16} -5.38716 q^{17} -9.76583 q^{18} -1.97118 q^{19} +3.04165 q^{20} +10.0938 q^{21} +5.96803 q^{22} +6.80240 q^{23} +21.1248 q^{24} -4.64077 q^{25} -2.65986 q^{26} -1.73460 q^{27} -19.8317 q^{28} -10.4559 q^{29} +4.11774 q^{30} +1.00433 q^{31} -14.5084 q^{32} +5.79545 q^{33} +14.3291 q^{34} -2.34222 q^{35} +18.6326 q^{36} -0.135048 q^{37} +5.24305 q^{38} -2.58294 q^{39} -4.90193 q^{40} +10.9084 q^{41} -26.8479 q^{42} -2.44326 q^{43} -11.3866 q^{44} +2.20059 q^{45} -18.0934 q^{46} -7.37674 q^{47} -29.9731 q^{48} +8.27139 q^{49} +12.3438 q^{50} +13.9147 q^{51} +5.07483 q^{52} -13.0609 q^{53} +4.61379 q^{54} -1.34481 q^{55} +31.9609 q^{56} +5.09143 q^{57} +27.8112 q^{58} +5.66496 q^{59} -7.85639 q^{60} +14.3580 q^{61} -2.67138 q^{62} -14.3480 q^{63} +15.3818 q^{64} +0.599360 q^{65} -15.4151 q^{66} -8.28360 q^{67} -27.3390 q^{68} -17.5702 q^{69} +6.22995 q^{70} +5.77090 q^{71} -30.0283 q^{72} +11.5264 q^{73} +0.359209 q^{74} +11.9868 q^{75} -10.0034 q^{76} +8.76824 q^{77} +6.87024 q^{78} +2.74018 q^{79} +6.95512 q^{80} -6.53432 q^{81} -29.0147 q^{82} +9.98586 q^{83} +51.2241 q^{84} -3.22885 q^{85} +6.49872 q^{86} +27.0069 q^{87} +18.3507 q^{88} +0.0258145 q^{89} -5.85324 q^{90} -3.90786 q^{91} +34.5210 q^{92} -2.59413 q^{93} +19.6211 q^{94} -1.18144 q^{95} +37.4744 q^{96} -8.17149 q^{97} -22.0007 q^{98} -8.23805 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.65986	−1.88080	−0.940401	−	0.340068i	$-0.889550\pi$
$2$	−2.65986	−1.88080	−0.940401	+	0.340068i	$0.889550\pi$
$3$	−2.58294	−1.49126	−0.745630	−	0.666361i	$-0.767851\pi$
$3$	−2.58294	−1.49126	−0.745630	+	0.666361i	$0.767851\pi$
$4$	5.07483	2.53742
$5$	0.599360	0.268042	0.134021	−	0.990979i	$-0.457211\pi$
$5$	0.599360	0.268042	0.134021	+	0.990979i	$0.457211\pi$
$6$	6.87024	2.80476
$7$	−3.90786	−1.47703	−0.738517	−	0.674235i	$-0.764473\pi$
$7$	−3.90786	−1.47703	−0.738517	+	0.674235i	$0.764473\pi$
$8$	−8.17861	−2.89158
$9$	3.67156	1.22385
$10$	−1.59421	−0.504134
$11$	−2.24374	−0.676514	−0.338257	−	0.941054i	$-0.609837\pi$
$11$	−2.24374	−0.676514	−0.338257	+	0.941054i	$0.609837\pi$
$12$	−13.1080	−3.78395
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	10.3944	2.77801
$15$	−1.54811	−0.399720
$16$	11.6043	2.90106
$17$	−5.38716	−1.30658	−0.653290	−	0.757108i	$-0.726612\pi$
$17$	−5.38716	−1.30658	−0.653290	+	0.757108i	$0.726612\pi$
$18$	−9.76583	−2.30183
$19$	−1.97118	−0.452219	−0.226110	−	0.974102i	$-0.572601\pi$
$19$	−1.97118	−0.452219	−0.226110	+	0.974102i	$0.572601\pi$
$20$	3.04165	0.680134
$21$	10.0938	2.20264
$22$	5.96803	1.27239
$23$	6.80240	1.41840	0.709199	−	0.705008i	$-0.249057\pi$
$23$	6.80240	1.41840	0.709199	+	0.705008i	$0.249057\pi$
$24$	21.1248	4.31209
$25$	−4.64077	−0.928154
$26$	−2.65986	−0.521641
$27$	−1.73460	−0.333825
$28$	−19.8317	−3.74785
$29$	−10.4559	−1.94161	−0.970807	−	0.239863i	$-0.922897\pi$
$29$	−10.4559	−1.94161	−0.970807	+	0.239863i	$0.922897\pi$
$30$	4.11774	0.751794
$31$	1.00433	0.180383	0.0901917	−	0.995924i	$-0.471252\pi$
$31$	1.00433	0.180383	0.0901917	+	0.995924i	$0.471252\pi$
$32$	−14.5084	−2.56475
$33$	5.79545	1.00886
$34$	14.3291	2.45742
$35$	−2.34222	−0.395907
$36$	18.6326	3.10543
$37$	−0.135048	−0.0222018	−0.0111009	−	0.999938i	$-0.503534\pi$
$37$	−0.135048	−0.0222018	−0.0111009	+	0.999938i	$0.503534\pi$
$38$	5.24305	0.850535
$39$	−2.58294	−0.413601
$40$	−4.90193	−0.775063
$41$	10.9084	1.70360	0.851802	−	0.523863i	$-0.175510\pi$
$41$	10.9084	1.70360	0.851802	+	0.523863i	$0.175510\pi$
$42$	−26.8479	−4.14273
$43$	−2.44326	−0.372594	−0.186297	−	0.982493i	$-0.559649\pi$
$43$	−2.44326	−0.372594	−0.186297	+	0.982493i	$0.559649\pi$
$44$	−11.3866	−1.71660
$45$	2.20059	0.328044
$46$	−18.0934	−2.66773
$47$	−7.37674	−1.07601	−0.538004	−	0.842942i	$-0.680821\pi$
$47$	−7.37674	−1.07601	−0.538004	+	0.842942i	$0.680821\pi$
$48$	−29.9731	−4.32624
$49$	8.27139	1.18163
$50$	12.3438	1.74567
$51$	13.9147	1.94845
$52$	5.07483	0.703753
$53$	−13.0609	−1.79405	−0.897027	−	0.441976i	$-0.854278\pi$
$53$	−13.0609	−1.79405	−0.897027	+	0.441976i	$0.854278\pi$
$54$	4.61379	0.627858
$55$	−1.34481	−0.181334
$56$	31.9609	4.27095
$57$	5.09143	0.674376
$58$	27.8112	3.65179
$59$	5.66496	0.737514	0.368757	−	0.929526i	$-0.379783\pi$
$59$	5.66496	0.737514	0.368757	+	0.929526i	$0.379783\pi$
$60$	−7.85639	−1.01426
$61$	14.3580	1.83836	0.919178	−	0.393843i	$-0.128855\pi$
$61$	14.3580	1.83836	0.919178	+	0.393843i	$0.128855\pi$
$62$	−2.67138	−0.339265
$63$	−14.3480	−1.80767
$64$	15.3818	1.92273
$65$	0.599360	0.0743414
$66$	−15.4151	−1.89746
$67$	−8.28360	−1.01200	−0.506001	−	0.862533i	$-0.668877\pi$
$67$	−8.28360	−1.01200	−0.506001	+	0.862533i	$0.668877\pi$
$68$	−27.3390	−3.31533
$69$	−17.5702	−2.11520
$70$	6.22995	0.744622
$71$	5.77090	0.684880	0.342440	−	0.939540i	$-0.388747\pi$
$71$	5.77090	0.684880	0.342440	+	0.939540i	$0.388747\pi$
$72$	−30.0283	−3.53887
$73$	11.5264	1.34906	0.674532	−	0.738246i	$-0.264345\pi$
$73$	11.5264	1.34906	0.674532	+	0.738246i	$0.264345\pi$
$74$	0.359209	0.0417572
$75$	11.9868	1.38412
$76$	−10.0034	−1.14747
$77$	8.76824	0.999234
$78$	6.87024	0.777901
$79$	2.74018	0.308295	0.154147	−	0.988048i	$-0.450737\pi$
$79$	2.74018	0.308295	0.154147	+	0.988048i	$0.450737\pi$
$80$	6.95512	0.777607
$81$	−6.53432	−0.726035
$82$	−29.0147	−3.20414
$83$	9.98586	1.09609	0.548045	−	0.836449i	$-0.315372\pi$
$83$	9.98586	1.09609	0.548045	+	0.836449i	$0.315372\pi$
$84$	51.2241	5.58901
$85$	−3.22885	−0.350218
$86$	6.49872	0.700775
$87$	27.0069	2.89545
$88$	18.3507	1.95619
$89$	0.0258145	0.00273633	0.00136817	−	0.999999i	$-0.499564\pi$
$89$	0.0258145	0.00273633	0.00136817	+	0.999999i	$0.499564\pi$
$90$	−5.85324	−0.616986
$91$	−3.90786	−0.409655
$92$	34.5210	3.59907
$93$	−2.59413	−0.268998
$94$	19.6211	2.02376
$95$	−1.18144	−0.121214
$96$	37.4744	3.82471
$97$	−8.17149	−0.829689	−0.414844	−	0.909892i	$-0.636164\pi$
$97$	−8.17149	−0.829689	−0.414844	+	0.909892i	$0.636164\pi$
$98$	−22.0007	−2.22241
$99$	−8.23805	−0.827955
$100$	−23.5511	−2.35511
$101$	−7.92682	−0.788748	−0.394374	−	0.918950i	$-0.629039\pi$
$101$	−7.92682	−0.788748	−0.394374	+	0.918950i	$0.629039\pi$
$102$	−37.0111	−3.66465
$103$	11.7750	1.16023	0.580115	−	0.814535i	$-0.303008\pi$
$103$	11.7750	1.16023	0.580115	+	0.814535i	$0.303008\pi$
$104$	−8.17861	−0.801979
$105$	6.04979	0.590399
$106$	34.7401	3.37426
$107$	16.4407	1.58938	0.794691	−	0.607014i	$-0.207633\pi$
$107$	16.4407	1.58938	0.794691	+	0.607014i	$0.207633\pi$
$108$	−8.80282	−0.847052
$109$	5.97978	0.572759	0.286380	−	0.958116i	$-0.407548\pi$
$109$	5.97978	0.572759	0.286380	+	0.958116i	$0.407548\pi$
$110$	3.57700	0.341054
$111$	0.348821	0.0331087
$112$	−45.3478	−4.28497
$113$	−5.83160	−0.548591	−0.274295	−	0.961645i	$-0.588445\pi$
$113$	−5.83160	−0.548591	−0.274295	+	0.961645i	$0.588445\pi$
$114$	−13.5425	−1.26837
$115$	4.07708	0.380190
$116$	−53.0620	−4.92668
$117$	3.67156	0.339436
$118$	−15.0680	−1.38712
$119$	21.0523	1.92986
$120$	12.6614	1.15582
$121$	−5.96561	−0.542329
$122$	−38.1903	−3.45758
$123$	−28.1757	−2.54052
$124$	5.09682	0.457708
$125$	−5.77829	−0.516826
$126$	38.1635	3.39988
$127$	−16.6031	−1.47329	−0.736645	−	0.676280i	$-0.763591\pi$
$127$	−16.6031	−1.47329	−0.736645	+	0.676280i	$0.763591\pi$
$128$	−11.8966	−1.05152
$129$	6.31079	0.555634
$130$	−1.59421	−0.139821
$131$	16.7024	1.45930	0.729649	−	0.683822i	$-0.239684\pi$
$131$	16.7024	1.45930	0.729649	+	0.683822i	$0.239684\pi$
$132$	29.4109	2.55989
$133$	7.70309	0.667943
$134$	22.0332	1.90338
$135$	−1.03965	−0.0894789
$136$	44.0595	3.77807
$137$	−16.7026	−1.42700	−0.713501	−	0.700654i	$-0.752892\pi$
$137$	−16.7026	−1.42700	−0.713501	+	0.700654i	$0.752892\pi$
$138$	46.7341	3.97827
$139$	9.09521	0.771446	0.385723	−	0.922615i	$-0.373952\pi$
$139$	9.09521	0.771446	0.385723	+	0.922615i	$0.373952\pi$
$140$	−11.8863	−1.00458
$141$	19.0536	1.60461
$142$	−15.3498	−1.28812
$143$	−2.24374	−0.187631
$144$	42.6058	3.55048
$145$	−6.26685	−0.520434
$146$	−30.6586	−2.53732
$147$	−21.3645	−1.76211
$148$	−0.685348	−0.0563352
$149$	−1.73566	−0.142191	−0.0710954	−	0.997470i	$-0.522649\pi$
$149$	−1.73566	−0.142191	−0.0710954	+	0.997470i	$0.522649\pi$
$150$	−31.8832	−2.60325
$151$	23.5930	1.91997	0.959985	−	0.280051i	$-0.0903514\pi$
$151$	23.5930	1.91997	0.959985	+	0.280051i	$0.0903514\pi$
$152$	16.1215	1.30763
$153$	−19.7793	−1.59906
$154$	−23.3223	−1.87936
$155$	0.601956	0.0483503
$156$	−13.1080	−1.04948
$157$	−6.78825	−0.541761	−0.270881	−	0.962613i	$-0.587315\pi$
$157$	−6.78825	−0.541761	−0.270881	+	0.962613i	$0.587315\pi$
$158$	−7.28849	−0.579841
$159$	33.7355	2.67540
$160$	−8.69577	−0.687461
$161$	−26.5828	−2.09502
$162$	17.3803	1.36553
$163$	13.1082	1.02671	0.513357	−	0.858175i	$-0.328402\pi$
$163$	13.1082	1.02671	0.513357	+	0.858175i	$0.328402\pi$
$164$	55.3583	4.32275
$165$	3.47356	0.270416
$166$	−26.5609	−2.06153
$167$	−24.3678	−1.88564	−0.942818	−	0.333309i	$-0.891835\pi$
$167$	−24.3678	−1.88564	−0.942818	+	0.333309i	$0.891835\pi$
$168$	−82.5529	−6.36910
$169$	1.00000	0.0769231
$170$	8.58827	0.658690
$171$	−7.23730	−0.553451
$172$	−12.3991	−0.945425
$173$	−10.4507	−0.794550	−0.397275	−	0.917700i	$-0.630044\pi$
$173$	−10.4507	−0.794550	−0.397275	+	0.917700i	$0.630044\pi$
$174$	−71.8346	−5.44577
$175$	18.1355	1.37091
$176$	−26.0370	−1.96261
$177$	−14.6322	−1.09983
$178$	−0.0686629	−0.00514650
$179$	−17.1583	−1.28247	−0.641234	−	0.767345i	$-0.721577\pi$
$179$	−17.1583	−1.28247	−0.641234	+	0.767345i	$0.721577\pi$
$180$	11.1676	0.832384
$181$	−1.00854	−0.0749641	−0.0374820	−	0.999297i	$-0.511934\pi$
$181$	−1.00854	−0.0749641	−0.0374820	+	0.999297i	$0.511934\pi$
$182$	10.3944	0.770481
$183$	−37.0858	−2.74146
$184$	−55.6342	−4.10140
$185$	−0.0809425	−0.00595101
$186$	6.90000	0.505933
$187$	12.0874	0.883919
$188$	−37.4357	−2.73028
$189$	6.77859	0.493070
$190$	3.14247	0.227979
$191$	−5.66617	−0.409990	−0.204995	−	0.978763i	$-0.565718\pi$
$191$	−5.66617	−0.409990	−0.204995	+	0.978763i	$0.565718\pi$
$192$	−39.7303	−2.86729
$193$	14.9952	1.07938	0.539689	−	0.841864i	$-0.318542\pi$
$193$	14.9952	1.07938	0.539689	+	0.841864i	$0.318542\pi$
$194$	21.7350	1.56048
$195$	−1.54811	−0.110862
$196$	41.9759	2.99828
$197$	0.365077	0.0260107	0.0130053	−	0.999915i	$-0.495860\pi$
$197$	0.365077	0.0260107	0.0130053	+	0.999915i	$0.495860\pi$
$198$	21.9120	1.55722
$199$	5.47512	0.388121	0.194061	−	0.980990i	$-0.437834\pi$
$199$	5.47512	0.388121	0.194061	+	0.980990i	$0.437834\pi$
$200$	37.9550	2.68383
$201$	21.3960	1.50916
$202$	21.0842	1.48348
$203$	40.8602	2.86783
$204$	70.6148	4.94402
$205$	6.53805	0.456637
$206$	−31.3199	−2.18216
$207$	24.9754	1.73591
$208$	11.6043	0.804611
$209$	4.42282	0.305933
$210$	−16.0916	−1.11042
$211$	2.26944	0.156235	0.0781174	−	0.996944i	$-0.475109\pi$
$211$	2.26944	0.156235	0.0781174	+	0.996944i	$0.475109\pi$
$212$	−66.2819	−4.55226
$213$	−14.9059	−1.02133
$214$	−43.7299	−2.98931
$215$	−1.46439	−0.0998707
$216$	14.1866	0.965279
$217$	−3.92479	−0.266432
$218$	−15.9054	−1.07725
$219$	−29.7720	−2.01180
$220$	−6.82468	−0.460120
$221$	−5.38716	−0.362380
$222$	−0.927814	−0.0622708
$223$	17.1933	1.15135	0.575674	−	0.817679i	$-0.304740\pi$
$223$	17.1933	1.15135	0.575674	+	0.817679i	$0.304740\pi$
$224$	56.6970	3.78822
$225$	−17.0389	−1.13592
$226$	15.5112	1.03179
$227$	20.2222	1.34219	0.671097	−	0.741370i	$-0.265824\pi$
$227$	20.2222	1.34219	0.671097	+	0.741370i	$0.265824\pi$
$228$	25.8382	1.71117
$229$	2.22853	0.147265	0.0736327	−	0.997285i	$-0.476541\pi$
$229$	2.22853	0.147265	0.0736327	+	0.997285i	$0.476541\pi$
$230$	−10.8445	−0.715062
$231$	−22.6478	−1.49012
$232$	85.5148	5.61432
$233$	5.14335	0.336952	0.168476	−	0.985706i	$-0.446115\pi$
$233$	5.14335	0.336952	0.168476	+	0.985706i	$0.446115\pi$
$234$	−9.76583	−0.638412
$235$	−4.42132	−0.288415
$236$	28.7487	1.87138
$237$	−7.07772	−0.459747
$238$	−55.9961	−3.62969
$239$	3.19761	0.206836	0.103418	−	0.994638i	$-0.467022\pi$
$239$	3.19761	0.206836	0.103418	+	0.994638i	$0.467022\pi$
$240$	−17.9646	−1.15961
$241$	−23.7720	−1.53129	−0.765644	−	0.643265i	$-0.777579\pi$
$241$	−23.7720	−1.53129	−0.765644	+	0.643265i	$0.777579\pi$
$242$	15.8677	1.02001
$243$	22.0815	1.41653
$244$	72.8645	4.66467
$245$	4.95754	0.316725
$246$	74.9433	4.77821
$247$	−1.97118	−0.125423
$248$	−8.21404	−0.521592
$249$	−25.7928	−1.63456
$250$	15.3694	0.972047
$251$	10.7050	0.675694	0.337847	−	0.941201i	$-0.390301\pi$
$251$	10.7050	0.675694	0.337847	+	0.941201i	$0.390301\pi$
$252$	−72.8135	−4.58682
$253$	−15.2628	−0.959566
$254$	44.1619	2.77097
$255$	8.33991	0.522266
$256$	0.879483	0.0549677
$257$	−26.2906	−1.63996	−0.819981	−	0.572390i	$-0.806016\pi$
$257$	−26.2906	−1.63996	−0.819981	+	0.572390i	$0.806016\pi$
$258$	−16.7858	−1.04504
$259$	0.527750	0.0327928
$260$	3.04165	0.188635
$261$	−38.3895	−2.37625
$262$	−44.4260	−2.74465
$263$	−21.2644	−1.31122	−0.655609	−	0.755101i	$-0.727588\pi$
$263$	−21.2644	−1.31122	−0.655609	+	0.755101i	$0.727588\pi$
$264$	−47.3987	−2.91719
$265$	−7.82818	−0.480881
$266$	−20.4891	−1.25627
$267$	−0.0666773	−0.00408058
$268$	−42.0379	−2.56787
$269$	13.8465	0.844236	0.422118	−	0.906541i	$-0.361287\pi$
$269$	13.8465	0.844236	0.422118	+	0.906541i	$0.361287\pi$
$270$	2.76532	0.168292
$271$	3.69372	0.224377	0.112189	−	0.993687i	$-0.464214\pi$
$271$	3.69372	0.224377	0.112189	+	0.993687i	$0.464214\pi$
$272$	−62.5140	−3.79047
$273$	10.0938	0.610902
$274$	44.4266	2.68391
$275$	10.4127	0.627909
$276$	−89.1656	−5.36714
$277$	14.7473	0.886080	0.443040	−	0.896502i	$-0.353900\pi$
$277$	14.7473	0.886080	0.443040	+	0.896502i	$0.353900\pi$
$278$	−24.1919	−1.45094
$279$	3.68747	0.220763
$280$	19.1561	1.14479
$281$	23.3895	1.39530	0.697651	−	0.716438i	$-0.254229\pi$
$281$	23.3895	1.39530	0.697651	+	0.716438i	$0.254229\pi$
$282$	−50.6799	−3.01795
$283$	−6.22626	−0.370113	−0.185056	−	0.982728i	$-0.559247\pi$
$283$	−6.22626	−0.370113	−0.185056	+	0.982728i	$0.559247\pi$
$284$	29.2864	1.73783
$285$	3.05160	0.180761
$286$	5.96803	0.352897
$287$	−42.6285	−2.51628
$288$	−53.2686	−3.13888
$289$	12.0215	0.707149
$290$	16.6689	0.978832
$291$	21.1064	1.23728
$292$	58.4946	3.42314
$293$	6.32713	0.369635	0.184818	−	0.982773i	$-0.440831\pi$
$293$	6.32713	0.369635	0.184818	+	0.982773i	$0.440831\pi$
$294$	56.8264	3.31418
$295$	3.39535	0.197685
$296$	1.10451	0.0641982
$297$	3.89201	0.225837
$298$	4.61661	0.267433
$299$	6.80240	0.393393
$300$	60.8311	3.51208
$301$	9.54793	0.550333
$302$	−62.7539	−3.61108
$303$	20.4745	1.17623
$304$	−22.8741	−1.31192
$305$	8.60562	0.492756
$306$	52.6101	3.00752
$307$	29.7130	1.69581	0.847906	−	0.530147i	$-0.177863\pi$
$307$	29.7130	1.69581	0.847906	+	0.530147i	$0.177863\pi$
$308$	44.4974	2.53547
$309$	−30.4142	−1.73020
$310$	−1.60112	−0.0909373
$311$	30.0465	1.70378	0.851891	−	0.523719i	$-0.175456\pi$
$311$	30.0465	1.70378	0.851891	+	0.523719i	$0.175456\pi$
$312$	21.1248	1.19596
$313$	23.2483	1.31407	0.657037	−	0.753858i	$-0.271809\pi$
$313$	23.2483	1.31407	0.657037	+	0.753858i	$0.271809\pi$
$314$	18.0558	1.01895
$315$	−8.59959	−0.484532
$316$	13.9060	0.782272
$317$	14.6310	0.821761	0.410880	−	0.911689i	$-0.365221\pi$
$317$	14.6310	0.821761	0.410880	+	0.911689i	$0.365221\pi$
$318$	−89.7316	−5.03190
$319$	23.4604	1.31353
$320$	9.21924	0.515371
$321$	−42.4653	−2.37018
$322$	70.7065	3.94032
$323$	10.6191	0.590860
$324$	−33.1606	−1.84225
$325$	−4.64077	−0.257423
$326$	−34.8659	−1.93105
$327$	−15.4454	−0.854133
$328$	−89.2155	−4.92610
$329$	28.8273	1.58930
$330$	−9.23916	−0.508599
$331$	−12.1120	−0.665734	−0.332867	−	0.942974i	$-0.608016\pi$
$331$	−12.1120	−0.665734	−0.332867	+	0.942974i	$0.608016\pi$
$332$	50.6766	2.78124
$333$	−0.495838	−0.0271718
$334$	64.8148	3.54651
$335$	−4.96486	−0.271259
$336$	117.131	6.39000
$337$	17.8256	0.971021	0.485511	−	0.874231i	$-0.338634\pi$
$337$	17.8256	0.971021	0.485511	+	0.874231i	$0.338634\pi$
$338$	−2.65986	−0.144677
$339$	15.0627	0.818091
$340$	−16.3859	−0.888648
$341$	−2.25346	−0.122032
$342$	19.2502	1.04093
$343$	−4.96841	−0.268269
$344$	19.9825	1.07738
$345$	−10.5308	−0.566962
$346$	27.7973	1.49439
$347$	−18.2641	−0.980467	−0.490234	−	0.871591i	$-0.663089\pi$
$347$	−18.2641	−0.980467	−0.490234	+	0.871591i	$0.663089\pi$
$348$	137.056	7.34696
$349$	20.9162	1.11962	0.559810	−	0.828621i	$-0.310874\pi$
$349$	20.9162	1.11962	0.559810	+	0.828621i	$0.310874\pi$
$350$	−48.2378	−2.57842
$351$	−1.73460	−0.0925863
$352$	32.5532	1.73509
$353$	19.4957	1.03765	0.518825	−	0.854881i	$-0.326370\pi$
$353$	19.4957	1.03765	0.518825	+	0.854881i	$0.326370\pi$
$354$	38.9196	2.06855
$355$	3.45885	0.183576
$356$	0.131004	0.00694322
$357$	−54.3767	−2.87792
$358$	45.6385	2.41207
$359$	14.4743	0.763923	0.381961	−	0.924178i	$-0.375249\pi$
$359$	14.4743	0.763923	0.381961	+	0.924178i	$0.375249\pi$
$360$	−17.9977	−0.948564
$361$	−15.1145	−0.795498
$362$	2.68257	0.140993
$363$	15.4088	0.808752
$364$	−19.8317	−1.03947
$365$	6.90847	0.361606
$366$	98.6430	5.15615
$367$	14.8002	0.772565	0.386282	−	0.922381i	$-0.373759\pi$
$367$	14.8002	0.772565	0.386282	+	0.922381i	$0.373759\pi$
$368$	78.9368	4.11486
$369$	40.0508	2.08496
$370$	0.215295	0.0111927
$371$	51.0402	2.64988
$372$	−13.1648	−0.682561
$373$	27.0044	1.39823	0.699117	−	0.715008i	$-0.253577\pi$
$373$	27.0044	1.39823	0.699117	+	0.715008i	$0.253577\pi$
$374$	−32.1508	−1.66248
$375$	14.9250	0.770721
$376$	60.3314	3.11136
$377$	−10.4559	−0.538507
$378$	−18.0301	−0.927367
$379$	−12.6254	−0.648524	−0.324262	−	0.945967i	$-0.605116\pi$
$379$	−12.6254	−0.648524	−0.324262	+	0.945967i	$0.605116\pi$
$380$	−5.99563	−0.307570
$381$	42.8848	2.19706
$382$	15.0712	0.771109
$383$	11.9853	0.612420	0.306210	−	0.951964i	$-0.400939\pi$
$383$	11.9853	0.612420	0.306210	+	0.951964i	$0.400939\pi$
$384$	30.7281	1.56808
$385$	5.25533	0.267836
$386$	−39.8851	−2.03010
$387$	−8.97058	−0.456000
$388$	−41.4689	−2.10527
$389$	−30.9272	−1.56807	−0.784036	−	0.620716i	$-0.786842\pi$
$389$	−30.9272	−1.56807	−0.784036	+	0.620716i	$0.786842\pi$
$390$	4.11774	0.208510
$391$	−36.6456	−1.85325
$392$	−67.6485	−3.41676
$393$	−43.1413	−2.17619
$394$	−0.971053	−0.0489209
$395$	1.64236	0.0826359
$396$	−41.8067	−2.10087
$397$	13.0082	0.652864	0.326432	−	0.945221i	$-0.394154\pi$
$397$	13.0082	0.652864	0.326432	+	0.945221i	$0.394154\pi$
$398$	−14.5630	−0.729979
$399$	−19.8966	−0.996076
$400$	−53.8527	−2.69263
$401$	33.5218	1.67400	0.836999	−	0.547205i	$-0.184308\pi$
$401$	33.5218	1.67400	0.836999	+	0.547205i	$0.184308\pi$
$402$	−56.9103	−2.83843
$403$	1.00433	0.0500293
$404$	−40.2273	−2.00138
$405$	−3.91641	−0.194608
$406$	−108.682	−5.39382
$407$	0.303014	0.0150198
$408$	−113.803	−5.63408
$409$	−6.59118	−0.325913	−0.162956	−	0.986633i	$-0.552103\pi$
$409$	−6.59118	−0.325913	−0.162956	+	0.986633i	$0.552103\pi$
$410$	−17.3903	−0.858844
$411$	43.1419	2.12803
$412$	59.7564	2.94399
$413$	−22.1379	−1.08933
$414$	−66.4310	−3.26491
$415$	5.98512	0.293798
$416$	−14.5084	−0.711334
$417$	−23.4923	−1.15043
$418$	−11.7641	−0.575399
$419$	−33.5443	−1.63875	−0.819373	−	0.573261i	$-0.805678\pi$
$419$	−33.5443	−1.63875	−0.819373	+	0.573261i	$0.805678\pi$
$420$	30.7017	1.49809
$421$	16.4992	0.804123	0.402062	−	0.915613i	$-0.368294\pi$
$421$	16.4992	0.804123	0.402062	+	0.915613i	$0.368294\pi$
$422$	−6.03639	−0.293847
$423$	−27.0841	−1.31688
$424$	106.820	5.18764
$425$	25.0006	1.21271
$426$	39.6475	1.92093
$427$	−56.1092	−2.71531
$428$	83.4338	4.03292
$429$	5.79545	0.279807
$430$	3.89507	0.187837
$431$	−2.79993	−0.134868	−0.0674340	−	0.997724i	$-0.521481\pi$
$431$	−2.79993	−0.134868	−0.0674340	+	0.997724i	$0.521481\pi$
$432$	−20.1288	−0.968447
$433$	−32.2715	−1.55087	−0.775434	−	0.631429i	$-0.782469\pi$
$433$	−32.2715	−1.55087	−0.775434	+	0.631429i	$0.782469\pi$
$434$	10.4394	0.501106
$435$	16.1869	0.776101
$436$	30.3464	1.45333
$437$	−13.4087	−0.641427
$438$	79.1892	3.78381
$439$	34.1561	1.63018	0.815091	−	0.579333i	$-0.196687\pi$
$439$	34.1561	1.63018	0.815091	+	0.579333i	$0.196687\pi$
$440$	10.9987	0.524341
$441$	30.3689	1.44614
$442$	14.3291	0.681565
$443$	−6.70246	−0.318443	−0.159222	−	0.987243i	$-0.550898\pi$
$443$	−6.70246	−0.318443	−0.159222	+	0.987243i	$0.550898\pi$
$444$	1.77021	0.0840104
$445$	0.0154722	0.000733452 0
$446$	−45.7317	−2.16546
$447$	4.48310	0.212043
$448$	−60.1100	−2.83993
$449$	−36.1352	−1.70533	−0.852663	−	0.522462i	$-0.825014\pi$
$449$	−36.1352	−1.70533	−0.852663	+	0.522462i	$0.825014\pi$
$450$	45.3209	2.13645
$451$	−24.4756	−1.15251
$452$	−29.5944	−1.39200
$453$	−60.9392	−2.86317
$454$	−53.7881	−2.52440
$455$	−2.34222	−0.109805
$456$	−41.6408	−1.95001
$457$	−34.9102	−1.63303	−0.816516	−	0.577323i	$-0.804097\pi$
$457$	−34.9102	−1.63303	−0.816516	+	0.577323i	$0.804097\pi$
$458$	−5.92757	−0.276977
$459$	9.34459	0.436168
$460$	20.6905	0.964700
$461$	3.16348	0.147338	0.0736689	−	0.997283i	$-0.476529\pi$
$461$	3.16348	0.147338	0.0736689	+	0.997283i	$0.476529\pi$
$462$	60.2399	2.80261
$463$	6.63613	0.308407	0.154204	−	0.988039i	$-0.450719\pi$
$463$	6.63613	0.308407	0.154204	+	0.988039i	$0.450719\pi$
$464$	−121.333	−5.63275
$465$	−1.55481	−0.0721028
$466$	−13.6806	−0.633740
$467$	38.9260	1.80128	0.900640	−	0.434566i	$-0.143098\pi$
$467$	38.9260	1.80128	0.900640	+	0.434566i	$0.143098\pi$
$468$	18.6326	0.861291
$469$	32.3712	1.49476
$470$	11.7601	0.542451
$471$	17.5336	0.807907
$472$	−46.3315	−2.13258
$473$	5.48205	0.252065
$474$	18.8257	0.864694
$475$	9.14778	0.419729
$476$	106.837	4.89686
$477$	−47.9539	−2.19566
$478$	−8.50518	−0.389018
$479$	2.57323	0.117574	0.0587869	−	0.998271i	$-0.481277\pi$
$479$	2.57323	0.117574	0.0587869	+	0.998271i	$0.481277\pi$
$480$	22.4606	1.02518
$481$	−0.135048	−0.00615767
$482$	63.2300	2.88005
$483$	68.6618	3.12422
$484$	−30.2745	−1.37611
$485$	−4.89766	−0.222391
$486$	−58.7337	−2.66421
$487$	27.1382	1.22975	0.614875	−	0.788624i	$-0.289206\pi$
$487$	27.1382	1.22975	0.614875	+	0.788624i	$0.289206\pi$
$488$	−117.429	−5.31574
$489$	−33.8577	−1.53110
$490$	−13.1863	−0.595698
$491$	−8.56114	−0.386359	−0.193179	−	0.981163i	$-0.561880\pi$
$491$	−8.56114	−0.386359	−0.193179	+	0.981163i	$0.561880\pi$
$492$	−142.987	−6.44635
$493$	56.3277	2.53687
$494$	5.24305	0.235896
$495$	−4.93755	−0.221926
$496$	11.6545	0.523304
$497$	−22.5519	−1.01159
$498$	68.6052	3.07427
$499$	−16.5623	−0.741430	−0.370715	−	0.928747i	$-0.620887\pi$
$499$	−16.5623	−0.741430	−0.370715	+	0.928747i	$0.620887\pi$
$500$	−29.3238	−1.31140
$501$	62.9404	2.81197
$502$	−28.4738	−1.27085
$503$	9.73394	0.434015	0.217007	−	0.976170i	$-0.430370\pi$
$503$	9.73394	0.434015	0.217007	+	0.976170i	$0.430370\pi$
$504$	117.346	5.22702
$505$	−4.75102	−0.211417
$506$	40.5969	1.80475
$507$	−2.58294	−0.114712
$508$	−84.2581	−3.73835
$509$	−1.08851	−0.0482473	−0.0241237	−	0.999709i	$-0.507680\pi$
$509$	−1.08851	−0.0482473	−0.0241237	+	0.999709i	$0.507680\pi$
$510$	−22.1830	−0.982278
$511$	−45.0436	−1.99261
$512$	21.4538	0.948134
$513$	3.41921	0.150962
$514$	69.9292	3.08444
$515$	7.05749	0.310990
$516$	32.0262	1.40987
$517$	16.5515	0.727934
$518$	−1.40374	−0.0616768
$519$	26.9934	1.18488
$520$	−4.90193	−0.214964
$521$	−21.7020	−0.950780	−0.475390	−	0.879775i	$-0.657693\pi$
$521$	−21.7020	−0.950780	−0.475390	+	0.879775i	$0.657693\pi$
$522$	102.111	4.46926
$523$	−23.9664	−1.04798	−0.523988	−	0.851725i	$-0.675556\pi$
$523$	−23.9664	−1.04798	−0.523988	+	0.851725i	$0.675556\pi$
$524$	84.7620	3.70284
$525$	−46.8428	−2.04439
$526$	56.5602	2.46614
$527$	−5.41050	−0.235685
$528$	67.2519	2.92676
$529$	23.2726	1.01185
$530$	20.8218	0.904443
$531$	20.7992	0.902610
$532$	39.0919	1.69485
$533$	10.9084	0.472495
$534$	0.177352	0.00767477
$535$	9.85389	0.426021
$536$	67.7483	2.92628
$537$	44.3187	1.91249
$538$	−36.8297	−1.58784
$539$	−18.5589	−0.799387
$540$	−5.27606	−0.227045
$541$	−14.2889	−0.614330	−0.307165	−	0.951656i	$-0.599380\pi$
$541$	−14.2889	−0.614330	−0.307165	+	0.951656i	$0.599380\pi$
$542$	−9.82475	−0.422009
$543$	2.60499	0.111791
$544$	78.1593	3.35105
$545$	3.58404	0.153523
$546$	−26.8479	−1.14899
$547$	1.35973	0.0581377	0.0290689	−	0.999577i	$-0.490746\pi$
$547$	1.35973	0.0581377	0.0290689	+	0.999577i	$0.490746\pi$
$548$	−84.7631	−3.62090
$549$	52.7164	2.24988
$550$	−27.6963	−1.18097
$551$	20.6105	0.878035
$552$	143.700	6.11626
$553$	−10.7083	−0.455362
$554$	−39.2257	−1.66654
$555$	0.209069	0.00887450
$556$	46.1567	1.95748
$557$	15.3855	0.651907	0.325953	−	0.945386i	$-0.394315\pi$
$557$	15.3855	0.651907	0.325953	+	0.945386i	$0.394315\pi$
$558$	−9.80813	−0.415211
$559$	−2.44326	−0.103339
$560$	−27.1797	−1.14855
$561$	−31.2210	−1.31815
$562$	−62.2127	−2.62429
$563$	4.88977	0.206079	0.103040	−	0.994677i	$-0.467143\pi$
$563$	4.88977	0.206079	0.103040	+	0.994677i	$0.467143\pi$
$564$	96.6940	4.07155
$565$	−3.49523	−0.147045
$566$	16.5610	0.696109
$567$	25.5352	1.07238
$568$	−47.1980	−1.98038
$569$	−14.4521	−0.605862	−0.302931	−	0.953012i	$-0.597965\pi$
$569$	−14.4521	−0.605862	−0.302931	+	0.953012i	$0.597965\pi$
$570$	−8.11681	−0.339976
$571$	−0.483688	−0.0202417	−0.0101209	−	0.999949i	$-0.503222\pi$
$571$	−0.483688	−0.0202417	−0.0101209	+	0.999949i	$0.503222\pi$
$572$	−11.3866	−0.476099
$573$	14.6354	0.611401
$574$	113.386	4.73263
$575$	−31.5683	−1.31649
$576$	56.4753	2.35314
$577$	−10.1132	−0.421017	−0.210508	−	0.977592i	$-0.567512\pi$
$577$	−10.1132	−0.421017	−0.210508	+	0.977592i	$0.567512\pi$
$578$	−31.9755	−1.33001
$579$	−38.7316	−1.60963
$580$	−31.8032	−1.32056
$581$	−39.0234	−1.61896
$582$	−56.1401	−2.32708
$583$	29.3053	1.21370
$584$	−94.2700	−3.90092
$585$	2.20059	0.0909831
$586$	−16.8293	−0.695210
$587$	−29.4474	−1.21542	−0.607711	−	0.794158i	$-0.707912\pi$
$587$	−29.4474	−1.21542	−0.607711	+	0.794158i	$0.707912\pi$
$588$	−108.421	−4.47121
$589$	−1.97972	−0.0815728
$590$	−9.03113	−0.371806
$591$	−0.942972	−0.0387887
$592$	−1.56714	−0.0644089
$593$	31.1972	1.28112	0.640559	−	0.767909i	$-0.278703\pi$
$593$	31.1972	1.28112	0.640559	+	0.767909i	$0.278703\pi$
$594$	−10.3522	−0.424755
$595$	12.6179	0.517283
$596$	−8.80819	−0.360797
$597$	−14.1419	−0.578789
$598$	−18.0934	−0.739894
$599$	−18.6954	−0.763872	−0.381936	−	0.924189i	$-0.624742\pi$
$599$	−18.6954	−0.763872	−0.381936	+	0.924189i	$0.624742\pi$
$600$	−98.0355	−4.00228
$601$	20.8606	0.850920	0.425460	−	0.904977i	$-0.360112\pi$
$601$	20.8606	0.850920	0.425460	+	0.904977i	$0.360112\pi$
$602$	−25.3961	−1.03507
$603$	−30.4138	−1.23854
$604$	119.730	4.87176
$605$	−3.57555	−0.145367
$606$	−54.4591	−2.21225
$607$	−39.1901	−1.59068	−0.795339	−	0.606165i	$-0.792707\pi$
$607$	−39.1901	−1.59068	−0.795339	+	0.606165i	$0.792707\pi$
$608$	28.5987	1.15983
$609$	−105.539	−4.27667
$610$	−22.8897	−0.926777
$611$	−7.37674	−0.298431
$612$	−100.377	−4.05749
$613$	−16.5368	−0.667916	−0.333958	−	0.942588i	$-0.608384\pi$
$613$	−16.5368	−0.667916	−0.333958	+	0.942588i	$0.608384\pi$
$614$	−79.0323	−3.18948
$615$	−16.8874	−0.680964
$616$	−71.7120	−2.88936
$617$	−39.8494	−1.60428	−0.802138	−	0.597139i	$-0.796304\pi$
$617$	−39.8494	−1.60428	−0.802138	+	0.597139i	$0.796304\pi$
$618$	80.8974	3.25417
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	3.05483	0.122685
$621$	−11.7995	−0.473496
$622$	−79.9194	−3.20448
$623$	−0.100880	−0.00404165
$624$	−29.9731	−1.19988
$625$	19.7406	0.789623
$626$	−61.8372	−2.47151
$627$	−11.4239	−0.456225
$628$	−34.4492	−1.37467
$629$	0.727528	0.0290084
$630$	22.8737	0.911309
$631$	27.2691	1.08557	0.542784	−	0.839873i	$-0.317370\pi$
$631$	27.2691	1.08557	0.542784	+	0.839873i	$0.317370\pi$
$632$	−22.4109	−0.891457
$633$	−5.86183	−0.232987
$634$	−38.9164	−1.54557
$635$	−9.95125	−0.394903
$636$	171.202	6.78860
$637$	8.27139	0.327724
$638$	−62.4012	−2.47049
$639$	21.1882	0.838193
$640$	−7.13032	−0.281850
$641$	45.4628	1.79567	0.897836	−	0.440331i	$-0.145139\pi$
$641$	45.4628	1.79567	0.897836	+	0.440331i	$0.145139\pi$
$642$	112.952	4.45784
$643$	15.6912	0.618802	0.309401	−	0.950932i	$-0.399872\pi$
$643$	15.6912	0.618802	0.309401	+	0.950932i	$0.399872\pi$
$644$	−134.903	−5.31594
$645$	3.78243	0.148933
$646$	−28.2452	−1.11129
$647$	14.0326	0.551680	0.275840	−	0.961204i	$-0.411044\pi$
$647$	14.0326	0.551680	0.275840	+	0.961204i	$0.411044\pi$
$648$	53.4416	2.09939
$649$	−12.7107	−0.498939
$650$	12.3438	0.484163
$651$	10.1375	0.397319
$652$	66.5219	2.60520
$653$	2.57443	0.100745	0.0503725	−	0.998730i	$-0.483959\pi$
$653$	2.57443	0.100745	0.0503725	+	0.998730i	$0.483959\pi$
$654$	41.0825	1.60645
$655$	10.0108	0.391153
$656$	126.584	4.94227
$657$	42.3199	1.65106
$658$	−76.6764	−2.98916
$659$	7.70514	0.300150	0.150075	−	0.988675i	$-0.452049\pi$
$659$	7.70514	0.300150	0.150075	+	0.988675i	$0.452049\pi$
$660$	17.6277	0.686158
$661$	30.0611	1.16924	0.584621	−	0.811306i	$-0.301243\pi$
$661$	30.0611	1.16924	0.584621	+	0.811306i	$0.301243\pi$
$662$	32.2161	1.25211
$663$	13.9147	0.540402
$664$	−81.6705	−3.16943
$665$	4.61692	0.179037
$666$	1.31886	0.0511047
$667$	−71.1252	−2.75398
$668$	−123.662	−4.78464
$669$	−44.4092	−1.71696
$670$	13.2058	0.510185
$671$	−32.2157	−1.24367
$672$	−146.445	−5.64923
$673$	36.4508	1.40507	0.702537	−	0.711647i	$-0.252051\pi$
$673$	36.4508	1.40507	0.702537	+	0.711647i	$0.252051\pi$
$674$	−47.4135	−1.82630
$675$	8.04989	0.309840
$676$	5.07483	0.195186
$677$	−47.0194	−1.80710	−0.903551	−	0.428480i	$-0.859049\pi$
$677$	−47.0194	−1.80710	−0.903551	+	0.428480i	$0.859049\pi$
$678$	−40.0645	−1.53867
$679$	31.9331	1.22548
$680$	26.4075	1.01268
$681$	−52.2326	−2.00156
$682$	5.99389	0.229518
$683$	22.2040	0.849614	0.424807	−	0.905284i	$-0.360342\pi$
$683$	22.2040	0.849614	0.424807	+	0.905284i	$0.360342\pi$
$684$	−36.7281	−1.40433
$685$	−10.0109	−0.382496
$686$	13.2153	0.504561
$687$	−5.75615	−0.219611
$688$	−28.3522	−1.08092
$689$	−13.0609	−0.497581
$690$	28.0105	1.06634
$691$	−12.2979	−0.467836	−0.233918	−	0.972256i	$-0.575155\pi$
$691$	−12.2979	−0.467836	−0.233918	+	0.972256i	$0.575155\pi$
$692$	−53.0354	−2.01610
$693$	32.1931	1.22292
$694$	48.5798	1.84406
$695$	5.45130	0.206780
$696$	−220.879	−8.37241
$697$	−58.7653	−2.22589
$698$	−55.6341	−2.10578
$699$	−13.2849	−0.502482
$700$	92.0345	3.47858
$701$	−18.0585	−0.682060	−0.341030	−	0.940052i	$-0.610776\pi$
$701$	−18.0585	−0.682060	−0.341030	+	0.940052i	$0.610776\pi$
$702$	4.61379	0.174136
$703$	0.266204	0.0100401
$704$	−34.5129	−1.30075
$705$	11.4200	0.430101
$706$	−51.8556	−1.95161
$707$	30.9769	1.16501
$708$	−74.2561	−2.79071
$709$	29.0464	1.09086	0.545430	−	0.838156i	$-0.316366\pi$
$709$	29.0464	1.09086	0.545430	+	0.838156i	$0.316366\pi$
$710$	−9.20003	−0.345271
$711$	10.0608	0.377308
$712$	−0.211127	−0.00791231
$713$	6.83187	0.255855
$714$	144.634	5.41280
$715$	−1.34481	−0.0502930
$716$	−87.0752	−3.25415
$717$	−8.25922	−0.308446
$718$	−38.4995	−1.43679
$719$	−32.5691	−1.21462	−0.607312	−	0.794464i	$-0.707752\pi$
$719$	−32.5691	−1.21462	−0.607312	+	0.794464i	$0.707752\pi$
$720$	25.5362	0.951677
$721$	−46.0153	−1.71370
$722$	40.2023	1.49617
$723$	61.4015	2.28355
$724$	−5.11816	−0.190215
$725$	48.5234	1.80212
$726$	−40.9852	−1.52110
$727$	−26.1060	−0.968217	−0.484108	−	0.875008i	$-0.660856\pi$
$727$	−26.1060	−0.968217	−0.484108	+	0.875008i	$0.660856\pi$
$728$	31.9609	1.18455
$729$	−37.4323	−1.38638
$730$	−18.3755	−0.680109
$731$	13.1622	0.486823
$732$	−188.204	−6.95624
$733$	4.64473	0.171557	0.0857785	−	0.996314i	$-0.472662\pi$
$733$	4.64473	0.171557	0.0857785	+	0.996314i	$0.472662\pi$
$734$	−39.3664	−1.45304
$735$	−12.8050	−0.472320
$736$	−98.6921	−3.63784
$737$	18.5863	0.684634
$738$	−106.529	−3.92140
$739$	−19.5355	−0.718623	−0.359312	−	0.933218i	$-0.616988\pi$
$739$	−19.5355	−0.718623	−0.359312	+	0.933218i	$0.616988\pi$
$740$	−0.410770	−0.0151002
$741$	5.09143	0.187038
$742$	−135.760	−4.98389
$743$	−52.2786	−1.91792	−0.958958	−	0.283549i	$-0.908488\pi$
$743$	−52.2786	−1.91792	−0.958958	+	0.283549i	$0.908488\pi$
$744$	21.2163	0.777829
$745$	−1.04028	−0.0381131
$746$	−71.8277	−2.62980
$747$	36.6637	1.34145
$748$	61.3416	2.24287
$749$	−64.2480	−2.34757
$750$	−39.6982	−1.44957
$751$	−28.9954	−1.05806	−0.529029	−	0.848604i	$-0.677444\pi$
$751$	−28.9954	−1.05806	−0.529029	+	0.848604i	$0.677444\pi$
$752$	−85.6015	−3.12157
$753$	−27.6504	−1.00764
$754$	27.8112	1.01282
$755$	14.1407	0.514632
$756$	34.4002	1.25112
$757$	45.4148	1.65063	0.825314	−	0.564675i	$-0.190998\pi$
$757$	45.4148	1.65063	0.825314	+	0.564675i	$0.190998\pi$
$758$	33.5818	1.21975
$759$	39.4229	1.43096
$760$	9.66258	0.350499
$761$	−14.7985	−0.536446	−0.268223	−	0.963357i	$-0.586436\pi$
$761$	−14.7985	−0.536446	−0.268223	+	0.963357i	$0.586436\pi$
$762$	−114.067	−4.13223
$763$	−23.3682	−0.845985
$764$	−28.7549	−1.04031
$765$	−11.8549	−0.428616
$766$	−31.8792	−1.15184
$767$	5.66496	0.204550
$768$	−2.27165	−0.0819711
$769$	−36.0435	−1.29976	−0.649881	−	0.760036i	$-0.725181\pi$
$769$	−36.0435	−1.29976	−0.649881	+	0.760036i	$0.725181\pi$
$770$	−13.9784	−0.503747
$771$	67.9069	2.44561
$772$	76.0981	2.73883
$773$	−19.8286	−0.713186	−0.356593	−	0.934260i	$-0.616062\pi$
$773$	−19.8286	−0.713186	−0.356593	+	0.934260i	$0.616062\pi$
$774$	23.8605	0.857647
$775$	−4.66087	−0.167423
$776$	66.8314	2.39911
$777$	−1.36315	−0.0489026
$778$	82.2619	2.94923
$779$	−21.5024	−0.770403
$780$	−7.85639	−0.281304
$781$	−12.9484	−0.463331
$782$	97.4721	3.48559
$783$	18.1369	0.648158
$784$	95.9833	3.42798
$785$	−4.06860	−0.145215
$786$	114.750	4.09298
$787$	−9.95588	−0.354889	−0.177444	−	0.984131i	$-0.556783\pi$
$787$	−9.95588	−0.354889	−0.177444	+	0.984131i	$0.556783\pi$
$788$	1.85271	0.0659999
$789$	54.9246	1.95537
$790$	−4.36843	−0.155422
$791$	22.7891	0.810287
$792$	67.3758	2.39409
$793$	14.3580	0.509868
$794$	−34.6000	−1.22791
$795$	20.2197	0.717119
$796$	27.7853	0.984825
$797$	0.0542137	0.00192035	0.000960175	−	1.00000i	$-0.499694\pi$
$797$	0.0542137	0.00192035	0.000960175		1.00000i	$0.499694\pi$
$798$	52.9221	1.87342
$799$	39.7397	1.40589
$800$	67.3303	2.38048
$801$	0.0947796	0.00334887
$802$	−89.1631	−3.14846
$803$	−25.8623	−0.912661
$804$	108.581	3.82936
$805$	−15.9327	−0.561553
$806$	−2.67138	−0.0940953
$807$	−35.7646	−1.25897
$808$	64.8304	2.28072
$809$	−53.3467	−1.87557	−0.937784	−	0.347218i	$-0.887126\pi$
$809$	−53.3467	−1.87557	−0.937784	+	0.347218i	$0.887126\pi$
$810$	10.4171	0.366019
$811$	20.8040	0.730526	0.365263	−	0.930904i	$-0.380979\pi$
$811$	20.8040	0.730526	0.365263	+	0.930904i	$0.380979\pi$
$812$	207.359	7.27687
$813$	−9.54064	−0.334605
$814$	−0.805973	−0.0282493
$815$	7.85653	0.275202
$816$	161.470	5.65257
$817$	4.81610	0.168494
$818$	17.5316	0.612977
$819$	−14.3480	−0.501358
$820$	33.1795	1.15868
$821$	−33.7272	−1.17709	−0.588544	−	0.808465i	$-0.700299\pi$
$821$	−33.7272	−1.17709	−0.588544	+	0.808465i	$0.700299\pi$
$822$	−114.751	−4.00241
$823$	−39.1297	−1.36397	−0.681987	−	0.731364i	$-0.738884\pi$
$823$	−39.1297	−1.36397	−0.681987	+	0.731364i	$0.738884\pi$
$824$	−96.3035	−3.35489
$825$	−26.8953	−0.936375
$826$	58.8835	2.04882
$827$	21.9711	0.764010	0.382005	−	0.924160i	$-0.375234\pi$
$827$	21.9711	0.764010	0.382005	+	0.924160i	$0.375234\pi$
$828$	126.746	4.40473
$829$	29.7441	1.03306	0.516528	−	0.856270i	$-0.327224\pi$
$829$	29.7441	1.03306	0.516528	+	0.856270i	$0.327224\pi$
$830$	−15.9196	−0.552576
$831$	−38.0914	−1.32138
$832$	15.3818	0.533269
$833$	−44.5593	−1.54389
$834$	62.4863	2.16372
$835$	−14.6051	−0.505429
$836$	22.4451	0.776279
$837$	−1.74212	−0.0602164
$838$	89.2230	3.08216
$839$	−32.1524	−1.11002	−0.555011	−	0.831843i	$-0.687286\pi$
$839$	−32.1524	−1.11002	−0.555011	+	0.831843i	$0.687286\pi$
$840$	−49.4789	−1.70718
$841$	80.3260	2.76986
$842$	−43.8856	−1.51240
$843$	−60.4136	−2.08076
$844$	11.5170	0.396433
$845$	0.599360	0.0206186
$846$	72.0399	2.47678
$847$	23.3128	0.801037
$848$	−151.562	−5.20467
$849$	16.0820	0.551934
$850$	−66.4979	−2.28086
$851$	−0.918653	−0.0314910
$852$	−75.6448	−2.59155
$853$	−11.2807	−0.386245	−0.193122	−	0.981175i	$-0.561861\pi$
$853$	−11.2807	−0.386245	−0.193122	+	0.981175i	$0.561861\pi$
$854$	149.242	5.10696
$855$	−4.33775	−0.148348
$856$	−134.462	−4.59582
$857$	−8.06981	−0.275659	−0.137830	−	0.990456i	$-0.544013\pi$
$857$	−8.06981	−0.275659	−0.137830	+	0.990456i	$0.544013\pi$
$858$	−15.4151	−0.526261
$859$	−12.3178	−0.420277	−0.210138	−	0.977672i	$-0.567391\pi$
$859$	−12.3178	−0.420277	−0.210138	+	0.977672i	$0.567391\pi$
$860$	−7.43154	−0.253414
$861$	110.107	3.75243
$862$	7.44742	0.253660
$863$	−3.32543	−0.113199	−0.0565995	−	0.998397i	$-0.518026\pi$
$863$	−3.32543	−0.113199	−0.0565995	+	0.998397i	$0.518026\pi$
$864$	25.1664	0.856177
$865$	−6.26371	−0.212973
$866$	85.8374	2.91687
$867$	−31.0509	−1.05454
$868$	−19.9177	−0.676049
$869$	−6.14827	−0.208566
$870$	−43.0548	−1.45969
$871$	−8.28360	−0.280679
$872$	−48.9063	−1.65618
$873$	−30.0021	−1.01542
$874$	35.6653	1.20640
$875$	22.5808	0.763369
$876$	−151.088	−5.10479
$877$	28.5152	0.962892	0.481446	−	0.876476i	$-0.340112\pi$
$877$	28.5152	0.962892	0.481446	+	0.876476i	$0.340112\pi$
$878$	−90.8503	−3.06605
$879$	−16.3426	−0.551222
$880$	−15.6055	−0.526062
$881$	−3.36251	−0.113286	−0.0566429	−	0.998395i	$-0.518040\pi$
$881$	−3.36251	−0.113286	−0.0566429	+	0.998395i	$0.518040\pi$
$882$	−80.7769	−2.71990
$883$	−32.4508	−1.09206	−0.546029	−	0.837766i	$-0.683861\pi$
$883$	−32.4508	−1.09206	−0.546029	+	0.837766i	$0.683861\pi$
$884$	−27.3390	−0.919508
$885$	−8.76996	−0.294799
$886$	17.8276	0.598929
$887$	−7.75431	−0.260364	−0.130182	−	0.991490i	$-0.541556\pi$
$887$	−7.75431	−0.260364	−0.130182	+	0.991490i	$0.541556\pi$
$888$	−2.85287	−0.0957362
$889$	64.8827	2.17610
$890$	−0.0411538	−0.00137948
$891$	14.6613	0.491173
$892$	87.2531	2.92145
$893$	14.5409	0.486591
$894$	−11.9244	−0.398812
$895$	−10.2840	−0.343755
$896$	46.4901	1.55313
$897$	−17.5702	−0.586651
$898$	96.1144	3.20738
$899$	−10.5012	−0.350235
$900$	−86.4694	−2.88231
$901$	70.3612	2.34407
$902$	65.1017	2.16765
$903$	−24.6617	−0.820690
$904$	47.6944	1.58629
$905$	−0.604477	−0.0200935
$906$	162.089	5.38506
$907$	17.4546	0.579571	0.289786	−	0.957092i	$-0.406416\pi$
$907$	17.4546	0.579571	0.289786	+	0.957092i	$0.406416\pi$
$908$	102.624	3.40570
$909$	−29.1038	−0.965312
$910$	6.22995	0.206521
$911$	−10.0043	−0.331456	−0.165728	−	0.986171i	$-0.552997\pi$
$911$	−10.0043	−0.331456	−0.165728	+	0.986171i	$0.552997\pi$
$912$	59.0823	1.95641
$913$	−22.4057	−0.741521
$914$	92.8562	3.07141
$915$	−22.2278	−0.734827
$916$	11.3094	0.373674
$917$	−65.2708	−2.15543
$918$	−24.8553	−0.820346
$919$	32.5243	1.07288	0.536439	−	0.843939i	$-0.319769\pi$
$919$	32.5243	1.07288	0.536439	+	0.843939i	$0.319769\pi$
$920$	−33.3449	−1.09935
$921$	−76.7468	−2.52889
$922$	−8.41439	−0.277113
$923$	5.77090	0.189952
$924$	−114.934	−3.78105
$925$	0.626728	0.0206067
$926$	−17.6511	−0.580053
$927$	43.2328	1.41995
$928$	151.699	4.97976
$929$	−29.1706	−0.957057	−0.478529	−	0.878072i	$-0.658830\pi$
$929$	−29.1706	−0.957057	−0.478529	+	0.878072i	$0.658830\pi$
$930$	4.13558	0.135611
$931$	−16.3044	−0.534355
$932$	26.1016	0.854987
$933$	−77.6083	−2.54078
$934$	−103.538	−3.38785
$935$	7.24471	0.236927
$936$	−30.0283	−0.981505
$937$	48.3898	1.58083	0.790413	−	0.612574i	$-0.209866\pi$
$937$	48.3898	1.58083	0.790413	+	0.612574i	$0.209866\pi$
$938$	−86.1026	−2.81135
$939$	−60.0490	−1.95963
$940$	−22.4374	−0.731829
$941$	−40.9168	−1.33385	−0.666925	−	0.745125i	$-0.732390\pi$
$941$	−40.9168	−1.33385	−0.666925	+	0.745125i	$0.732390\pi$
$942$	−46.6369	−1.51951
$943$	74.2032	2.41639
$944$	65.7376	2.13958
$945$	4.06281	0.132163
$946$	−14.5815	−0.474084
$947$	−2.95800	−0.0961221	−0.0480611	−	0.998844i	$-0.515304\pi$
$947$	−2.95800	−0.0961221	−0.0480611	+	0.998844i	$0.515304\pi$
$948$	−35.9182	−1.16657
$949$	11.5264	0.374163
$950$	−24.3318	−0.789427
$951$	−37.7910	−1.22546
$952$	−172.179	−5.58034
$953$	−21.5717	−0.698777	−0.349389	−	0.936978i	$-0.613611\pi$
$953$	−21.5717	−0.698777	−0.349389	+	0.936978i	$0.613611\pi$
$954$	127.551	4.12960
$955$	−3.39607	−0.109894
$956$	16.2273	0.524829
$957$	−60.5967	−1.95881
$958$	−6.84442	−0.221133
$959$	65.2716	2.10773
$960$	−23.8127	−0.768552
$961$	−29.9913	−0.967462
$962$	0.359209	0.0115814
$963$	60.3631	1.94517
$964$	−120.639	−3.88551
$965$	8.98752	0.289318
$966$	−182.630	−5.87604
$967$	−3.61422	−0.116226	−0.0581128	−	0.998310i	$-0.518508\pi$
$967$	−3.61422	−0.116226	−0.0581128	+	0.998310i	$0.518508\pi$
$968$	48.7904	1.56818
$969$	−27.4284	−0.881126
$970$	13.0271	0.418274
$971$	−15.9826	−0.512905	−0.256453	−	0.966557i	$-0.582554\pi$
$971$	−15.9826	−0.512905	−0.256453	+	0.966557i	$0.582554\pi$
$972$	112.060	3.59433
$973$	−35.5428	−1.13945
$974$	−72.1838	−2.31292
$975$	11.9868	0.383885
$976$	166.614	5.33319
$977$	9.87777	0.316018	0.158009	−	0.987438i	$-0.449493\pi$
$977$	9.87777	0.316018	0.158009	+	0.987438i	$0.449493\pi$
$978$	90.0565	2.87969
$979$	−0.0579212	−0.00185117
$980$	25.1587	0.803664
$981$	21.9552	0.700974
$982$	22.7714	0.726664
$983$	−11.8157	−0.376861	−0.188431	−	0.982087i	$-0.560340\pi$
$983$	−11.8157	−0.376861	−0.188431	+	0.982087i	$0.560340\pi$
$984$	230.438	7.34609
$985$	0.218813	0.00697195
$986$	−149.824	−4.77135
$987$	−74.4590	−2.37006
$988$	−10.0034	−0.318251
$989$	−16.6200	−0.528486
$990$	13.1332	0.417400
$991$	26.3314	0.836446	0.418223	−	0.908344i	$-0.362653\pi$
$991$	26.3314	0.836446	0.418223	+	0.908344i	$0.362653\pi$
$992$	−14.5713	−0.462639
$993$	31.2845	0.992782
$994$	59.9848	1.90260
$995$	3.28157	0.104033
$996$	−130.894	−4.14755
$997$	−0.944094	−0.0298997	−0.0149499	−	0.999888i	$-0.504759\pi$
$997$	−0.944094	−0.0298997	−0.0149499	+	0.999888i	$0.504759\pi$
$998$	44.0533	1.39448
$999$	0.234255	0.00741151

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.6	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.6	✓	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)

\(f(q)\)	\(=\)	\(q-2.65986 q^{2} -2.58294 q^{3} +5.07483 q^{4} +0.599360 q^{5} +6.87024 q^{6} -3.90786 q^{7} -8.17861 q^{8} +3.67156 q^{9} +O(q^{10})\)	\(q-2.65986 q^{2} -2.58294 q^{3} +5.07483 q^{4} +0.599360 q^{5} +6.87024 q^{6} -3.90786 q^{7} -8.17861 q^{8} +3.67156 q^{9} -1.59421 q^{10} -2.24374 q^{11} -13.1080 q^{12} +1.00000 q^{13} +10.3944 q^{14} -1.54811 q^{15} +11.6043 q^{16} -5.38716 q^{17} -9.76583 q^{18} -1.97118 q^{19} +3.04165 q^{20} +10.0938 q^{21} +5.96803 q^{22} +6.80240 q^{23} +21.1248 q^{24} -4.64077 q^{25} -2.65986 q^{26} -1.73460 q^{27} -19.8317 q^{28} -10.4559 q^{29} +4.11774 q^{30} +1.00433 q^{31} -14.5084 q^{32} +5.79545 q^{33} +14.3291 q^{34} -2.34222 q^{35} +18.6326 q^{36} -0.135048 q^{37} +5.24305 q^{38} -2.58294 q^{39} -4.90193 q^{40} +10.9084 q^{41} -26.8479 q^{42} -2.44326 q^{43} -11.3866 q^{44} +2.20059 q^{45} -18.0934 q^{46} -7.37674 q^{47} -29.9731 q^{48} +8.27139 q^{49} +12.3438 q^{50} +13.9147 q^{51} +5.07483 q^{52} -13.0609 q^{53} +4.61379 q^{54} -1.34481 q^{55} +31.9609 q^{56} +5.09143 q^{57} +27.8112 q^{58} +5.66496 q^{59} -7.85639 q^{60} +14.3580 q^{61} -2.67138 q^{62} -14.3480 q^{63} +15.3818 q^{64} +0.599360 q^{65} -15.4151 q^{66} -8.28360 q^{67} -27.3390 q^{68} -17.5702 q^{69} +6.22995 q^{70} +5.77090 q^{71} -30.0283 q^{72} +11.5264 q^{73} +0.359209 q^{74} +11.9868 q^{75} -10.0034 q^{76} +8.76824 q^{77} +6.87024 q^{78} +2.74018 q^{79} +6.95512 q^{80} -6.53432 q^{81} -29.0147 q^{82} +9.98586 q^{83} +51.2241 q^{84} -3.22885 q^{85} +6.49872 q^{86} +27.0069 q^{87} +18.3507 q^{88} +0.0258145 q^{89} -5.85324 q^{90} -3.90786 q^{91} +34.5210 q^{92} -2.59413 q^{93} +19.6211 q^{94} -1.18144 q^{95} +37.4744 q^{96} -8.17149 q^{97} -22.0007 q^{98} -8.23805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 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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