LMFDB - Embedded newform 8047.2.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-2.51491 q^{2} +3.41977 q^{3} +4.32478 q^{4} -1.23152 q^{5} -8.60042 q^{6} -3.67991 q^{7} -5.84661 q^{8} +8.69483 q^{9} +O(q^{10})$	\(q-2.51491 q^{2} +3.41977 q^{3} +4.32478 q^{4} -1.23152 q^{5} -8.60042 q^{6} -3.67991 q^{7} -5.84661 q^{8} +8.69483 q^{9} +3.09716 q^{10} -3.65258 q^{11} +14.7897 q^{12} +1.00000 q^{13} +9.25465 q^{14} -4.21151 q^{15} +6.05414 q^{16} +7.01735 q^{17} -21.8667 q^{18} +8.62252 q^{19} -5.32604 q^{20} -12.5845 q^{21} +9.18592 q^{22} +4.93337 q^{23} -19.9941 q^{24} -3.48336 q^{25} -2.51491 q^{26} +19.4750 q^{27} -15.9148 q^{28} -5.14343 q^{29} +10.5916 q^{30} +8.94218 q^{31} -3.53241 q^{32} -12.4910 q^{33} -17.6480 q^{34} +4.53188 q^{35} +37.6032 q^{36} -2.26585 q^{37} -21.6849 q^{38} +3.41977 q^{39} +7.20020 q^{40} +3.73997 q^{41} +31.6488 q^{42} -11.4301 q^{43} -15.7966 q^{44} -10.7078 q^{45} -12.4070 q^{46} -5.11982 q^{47} +20.7038 q^{48} +6.54176 q^{49} +8.76035 q^{50} +23.9977 q^{51} +4.32478 q^{52} +0.754064 q^{53} -48.9780 q^{54} +4.49822 q^{55} +21.5150 q^{56} +29.4870 q^{57} +12.9353 q^{58} +1.32726 q^{59} -18.2138 q^{60} -3.58069 q^{61} -22.4888 q^{62} -31.9962 q^{63} -3.22459 q^{64} -1.23152 q^{65} +31.4138 q^{66} -11.4041 q^{67} +30.3485 q^{68} +16.8710 q^{69} -11.3973 q^{70} +6.12978 q^{71} -50.8353 q^{72} -2.11389 q^{73} +5.69842 q^{74} -11.9123 q^{75} +37.2905 q^{76} +13.4412 q^{77} -8.60042 q^{78} -12.4578 q^{79} -7.45578 q^{80} +40.5156 q^{81} -9.40569 q^{82} +9.72427 q^{83} -54.4250 q^{84} -8.64200 q^{85} +28.7457 q^{86} -17.5894 q^{87} +21.3552 q^{88} +9.16831 q^{89} +26.9293 q^{90} -3.67991 q^{91} +21.3357 q^{92} +30.5802 q^{93} +12.8759 q^{94} -10.6188 q^{95} -12.0800 q^{96} +14.8408 q^{97} -16.4519 q^{98} -31.7586 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.51491	−1.77831	−0.889155	−	0.457606i	$-0.848707\pi$
$2$	−2.51491	−1.77831	−0.889155	+	0.457606i	$0.848707\pi$
$3$	3.41977	1.97441	0.987203	−	0.159469i	$-0.0509783\pi$
$3$	3.41977	1.97441	0.987203	+	0.159469i	$0.0509783\pi$
$4$	4.32478	2.16239
$5$	−1.23152	−0.550752	−0.275376	−	0.961337i	$-0.588802\pi$
$5$	−1.23152	−0.550752	−0.275376	+	0.961337i	$0.588802\pi$
$6$	−8.60042	−3.51111
$7$	−3.67991	−1.39088	−0.695438	−	0.718586i	$-0.744790\pi$
$7$	−3.67991	−1.39088	−0.695438	+	0.718586i	$0.744790\pi$
$8$	−5.84661	−2.06709
$9$	8.69483	2.89828
$10$	3.09716	0.979408
$11$	−3.65258	−1.10130	−0.550648	−	0.834738i	$-0.685619\pi$
$11$	−3.65258	−1.10130	−0.550648	+	0.834738i	$0.685619\pi$
$12$	14.7897	4.26943
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	9.25465	2.47341
$15$	−4.21151	−1.08741
$16$	6.05414	1.51353
$17$	7.01735	1.70196	0.850979	−	0.525200i	$-0.176009\pi$
$17$	7.01735	1.70196	0.850979	+	0.525200i	$0.176009\pi$
$18$	−21.8667	−5.15404
$19$	8.62252	1.97814	0.989071	−	0.147442i	$-0.0471041\pi$
$19$	8.62252	1.97814	0.989071	+	0.147442i	$0.0471041\pi$
$20$	−5.32604	−1.19094
$21$	−12.5845	−2.74615
$22$	9.18592	1.95845
$23$	4.93337	1.02868	0.514339	−	0.857587i	$-0.328037\pi$
$23$	4.93337	1.02868	0.514339	+	0.857587i	$0.328037\pi$
$24$	−19.9941	−4.08127
$25$	−3.48336	−0.696673
$26$	−2.51491	−0.493215
$27$	19.4750	3.74797
$28$	−15.9148	−3.00761
$29$	−5.14343	−0.955111	−0.477556	−	0.878601i	$-0.658477\pi$
$29$	−5.14343	−0.955111	−0.477556	+	0.878601i	$0.658477\pi$
$30$	10.5916	1.93375
$31$	8.94218	1.60606	0.803032	−	0.595936i	$-0.203219\pi$
$31$	8.94218	1.60606	0.803032	+	0.595936i	$0.203219\pi$
$32$	−3.53241	−0.624447
$33$	−12.4910	−2.17440
$34$	−17.6480	−3.02661
$35$	4.53188	0.766028
$36$	37.6032	6.26720
$37$	−2.26585	−0.372504	−0.186252	−	0.982502i	$-0.559634\pi$
$37$	−2.26585	−0.372504	−0.186252	+	0.982502i	$0.559634\pi$
$38$	−21.6849	−3.51775
$39$	3.41977	0.547602
$40$	7.20020	1.13845
$41$	3.73997	0.584085	0.292043	−	0.956405i	$-0.405665\pi$
$41$	3.73997	0.584085	0.292043	+	0.956405i	$0.405665\pi$
$42$	31.6488	4.88351
$43$	−11.4301	−1.74307	−0.871537	−	0.490330i	$-0.836876\pi$
$43$	−11.4301	−1.74307	−0.871537	+	0.490330i	$0.836876\pi$
$44$	−15.7966	−2.38143
$45$	−10.7078	−1.59623
$46$	−12.4070	−1.82931
$47$	−5.11982	−0.746802	−0.373401	−	0.927670i	$-0.621809\pi$
$47$	−5.11982	−0.746802	−0.373401	+	0.927670i	$0.621809\pi$
$48$	20.7038	2.98833
$49$	6.54176	0.934537
$50$	8.76035	1.23890
$51$	23.9977	3.36036
$52$	4.32478	0.599739
$53$	0.754064	0.103579	0.0517894	−	0.998658i	$-0.483508\pi$
$53$	0.754064	0.103579	0.0517894	+	0.998658i	$0.483508\pi$
$54$	−48.9780	−6.66506
$55$	4.49822	0.606540
$56$	21.5150	2.87506
$57$	29.4870	3.90565
$58$	12.9353	1.69848
$59$	1.32726	0.172794	0.0863972	−	0.996261i	$-0.472465\pi$
$59$	1.32726	0.172794	0.0863972	+	0.996261i	$0.472465\pi$
$60$	−18.2138	−2.35140
$61$	−3.58069	−0.458460	−0.229230	−	0.973372i	$-0.573621\pi$
$61$	−3.58069	−0.458460	−0.229230	+	0.973372i	$0.573621\pi$
$62$	−22.4888	−2.85608
$63$	−31.9962	−4.03115
$64$	−3.22459	−0.403074
$65$	−1.23152	−0.152751
$66$	31.4138	3.86677
$67$	−11.4041	−1.39323	−0.696614	−	0.717446i	$-0.745311\pi$
$67$	−11.4041	−1.39323	−0.696614	+	0.717446i	$0.745311\pi$
$68$	30.3485	3.68029
$69$	16.8710	2.03103
$70$	−11.3973	−1.36223
$71$	6.12978	0.727471	0.363735	−	0.931502i	$-0.381501\pi$
$71$	6.12978	0.727471	0.363735	+	0.931502i	$0.381501\pi$
$72$	−50.8353	−5.99099
$73$	−2.11389	−0.247412	−0.123706	−	0.992319i	$-0.539478\pi$
$73$	−2.11389	−0.247412	−0.123706	+	0.992319i	$0.539478\pi$
$74$	5.69842	0.662428
$75$	−11.9123	−1.37551
$76$	37.2905	4.27751
$77$	13.4412	1.53177
$78$	−8.60042	−0.973806
$79$	−12.4578	−1.40161	−0.700805	−	0.713353i	$-0.747176\pi$
$79$	−12.4578	−1.40161	−0.700805	+	0.713353i	$0.747176\pi$
$80$	−7.45578	−0.833582
$81$	40.5156	4.50174
$82$	−9.40569	−1.03868
$83$	9.72427	1.06738	0.533689	−	0.845681i	$-0.320805\pi$
$83$	9.72427	1.06738	0.533689	+	0.845681i	$0.320805\pi$
$84$	−54.4250	−5.93825
$85$	−8.64200	−0.937356
$86$	28.7457	3.09973
$87$	−17.5894	−1.88578
$88$	21.3552	2.27647
$89$	9.16831	0.971839	0.485920	−	0.874003i	$-0.338485\pi$
$89$	9.16831	0.971839	0.485920	+	0.874003i	$0.338485\pi$
$90$	26.9293	2.83860
$91$	−3.67991	−0.385760
$92$	21.3357	2.22440
$93$	30.5802	3.17102
$94$	12.8759	1.32805
$95$	−10.6188	−1.08946
$96$	−12.0800	−1.23291
$97$	14.8408	1.50686	0.753428	−	0.657531i	$-0.228399\pi$
$97$	14.8408	1.50686	0.753428	+	0.657531i	$0.228399\pi$
$98$	−16.4519	−1.66190
$99$	−31.7586	−3.19186
$100$	−15.0648	−1.50648
$101$	9.12911	0.908381	0.454190	−	0.890905i	$-0.349929\pi$
$101$	9.12911	0.908381	0.454190	+	0.890905i	$0.349929\pi$
$102$	−60.3522	−5.97576
$103$	0.585984	0.0577387	0.0288694	−	0.999583i	$-0.490809\pi$
$103$	0.585984	0.0577387	0.0288694	+	0.999583i	$0.490809\pi$
$104$	−5.84661	−0.573307
$105$	15.4980	1.51245
$106$	−1.89640	−0.184195
$107$	14.5457	1.40618	0.703092	−	0.711099i	$-0.251802\pi$
$107$	14.5457	1.40618	0.703092	+	0.711099i	$0.251802\pi$
$108$	84.2251	8.10457
$109$	−14.7569	−1.41346	−0.706729	−	0.707484i	$-0.749830\pi$
$109$	−14.7569	−1.41346	−0.706729	+	0.707484i	$0.749830\pi$
$110$	−11.3126	−1.07862
$111$	−7.74870	−0.735474
$112$	−22.2787	−2.10514
$113$	0.0440581	0.00414464	0.00207232	−	0.999998i	$-0.499340\pi$
$113$	0.0440581	0.00414464	0.00207232	+	0.999998i	$0.499340\pi$
$114$	−74.1573	−6.94546
$115$	−6.07554	−0.566547
$116$	−22.2442	−2.06532
$117$	8.69483	0.803838
$118$	−3.33794	−0.307282
$119$	−25.8232	−2.36721
$120$	24.6230	2.24777
$121$	2.34137	0.212852
$122$	9.00511	0.815285
$123$	12.7898	1.15322
$124$	38.6729	3.47293
$125$	10.4474	0.934445
$126$	80.4677	7.16863
$127$	8.08119	0.717090	0.358545	−	0.933512i	$-0.383273\pi$
$127$	8.08119	0.717090	0.358545	+	0.933512i	$0.383273\pi$
$128$	15.1744	1.34124
$129$	−39.0883	−3.44154
$130$	3.09716	0.271639
$131$	−13.9757	−1.22106	−0.610531	−	0.791992i	$-0.709044\pi$
$131$	−13.9757	−1.22106	−0.610531	+	0.791992i	$0.709044\pi$
$132$	−54.0208	−4.70191
$133$	−31.7301	−2.75135
$134$	28.6802	2.47759
$135$	−23.9839	−2.06420
$136$	−41.0277	−3.51810
$137$	−9.33113	−0.797212	−0.398606	−	0.917122i	$-0.630506\pi$
$137$	−9.33113	−0.797212	−0.398606	+	0.917122i	$0.630506\pi$
$138$	−42.4291	−3.61180
$139$	3.96183	0.336038	0.168019	−	0.985784i	$-0.446263\pi$
$139$	3.96183	0.336038	0.168019	+	0.985784i	$0.446263\pi$
$140$	19.5994	1.65645
$141$	−17.5086	−1.47449
$142$	−15.4158	−1.29367
$143$	−3.65258	−0.305444
$144$	52.6397	4.38664
$145$	6.33423	0.526029
$146$	5.31624	0.439975
$147$	22.3713	1.84516
$148$	−9.79931	−0.805499
$149$	−6.10042	−0.499766	−0.249883	−	0.968276i	$-0.580392\pi$
$149$	−6.10042	−0.499766	−0.249883	+	0.968276i	$0.580392\pi$
$150$	29.9584	2.44609
$151$	−4.78084	−0.389059	−0.194530	−	0.980897i	$-0.562318\pi$
$151$	−4.78084	−0.389059	−0.194530	+	0.980897i	$0.562318\pi$
$152$	−50.4125	−4.08899
$153$	61.0147	4.93275
$154$	−33.8034	−2.72396
$155$	−11.0125	−0.884542
$156$	14.7897	1.18413
$157$	18.7456	1.49606	0.748030	−	0.663665i	$-0.231000\pi$
$157$	18.7456	1.49606	0.748030	+	0.663665i	$0.231000\pi$
$158$	31.3302	2.49250
$159$	2.57873	0.204506
$160$	4.35022	0.343915
$161$	−18.1544	−1.43077
$162$	−101.893	−8.00549
$163$	7.56485	0.592525	0.296262	−	0.955107i	$-0.404260\pi$
$163$	7.56485	0.592525	0.296262	+	0.955107i	$0.404260\pi$
$164$	16.1745	1.26302
$165$	15.3829	1.19756
$166$	−24.4557	−1.89813
$167$	13.1185	1.01514	0.507569	−	0.861611i	$-0.330544\pi$
$167$	13.1185	1.01514	0.507569	+	0.861611i	$0.330544\pi$
$168$	73.5764	5.67654
$169$	1.00000	0.0769231
$170$	21.7339	1.66691
$171$	74.9714	5.73320
$172$	−49.4326	−3.76920
$173$	−4.53282	−0.344624	−0.172312	−	0.985042i	$-0.555124\pi$
$173$	−4.53282	−0.344624	−0.172312	+	0.985042i	$0.555124\pi$
$174$	44.2357	3.35350
$175$	12.8185	0.968985
$176$	−22.1132	−1.66685
$177$	4.53892	0.341166
$178$	−23.0575	−1.72823
$179$	14.5430	1.08700	0.543499	−	0.839410i	$-0.317099\pi$
$179$	14.5430	1.08700	0.543499	+	0.839410i	$0.317099\pi$
$180$	−46.3090	−3.45167
$181$	−15.3291	−1.13940	−0.569701	−	0.821852i	$-0.692941\pi$
$181$	−15.3291	−1.13940	−0.569701	+	0.821852i	$0.692941\pi$
$182$	9.25465	0.686001
$183$	−12.2451	−0.905187
$184$	−28.8435	−2.12637
$185$	2.79044	0.205157
$186$	−76.9065	−5.63906
$187$	−25.6315	−1.87436
$188$	−22.1421	−1.61488
$189$	−71.6664	−5.21296
$190$	26.7053	1.93741
$191$	−17.7374	−1.28343	−0.641716	−	0.766943i	$-0.721777\pi$
$191$	−17.7374	−1.28343	−0.641716	+	0.766943i	$0.721777\pi$
$192$	−11.0274	−0.795831
$193$	−0.600077	−0.0431945	−0.0215972	−	0.999767i	$-0.506875\pi$
$193$	−0.600077	−0.0431945	−0.0215972	+	0.999767i	$0.506875\pi$
$194$	−37.3233	−2.67966
$195$	−4.21151	−0.301593
$196$	28.2916	2.02083
$197$	−0.853709	−0.0608242	−0.0304121	−	0.999537i	$-0.509682\pi$
$197$	−0.853709	−0.0608242	−0.0304121	+	0.999537i	$0.509682\pi$
$198$	79.8701	5.67612
$199$	−11.3831	−0.806923	−0.403462	−	0.914997i	$-0.632193\pi$
$199$	−11.3831	−0.806923	−0.403462	+	0.914997i	$0.632193\pi$
$200$	20.3658	1.44008
$201$	−38.9993	−2.75080
$202$	−22.9589	−1.61538
$203$	18.9274	1.32844
$204$	103.785	7.26639
$205$	−4.60584	−0.321686
$206$	−1.47370	−0.102677
$207$	42.8948	2.98140
$208$	6.05414	0.419779
$209$	−31.4945	−2.17852
$210$	−38.9761	−2.68960
$211$	−2.16098	−0.148768	−0.0743841	−	0.997230i	$-0.523699\pi$
$211$	−2.16098	−0.148768	−0.0743841	+	0.997230i	$0.523699\pi$
$212$	3.26116	0.223977
$213$	20.9624	1.43632
$214$	−36.5811	−2.50063
$215$	14.0764	0.960001
$216$	−113.863	−7.74738
$217$	−32.9065	−2.23384
$218$	37.1124	2.51357
$219$	−7.22902	−0.488492
$220$	19.4538	1.31158
$221$	7.01735	0.472038
$222$	19.4873	1.30790
$223$	22.1147	1.48091	0.740456	−	0.672105i	$-0.234610\pi$
$223$	22.1147	1.48091	0.740456	+	0.672105i	$0.234610\pi$
$224$	12.9989	0.868529
$225$	−30.2873	−2.01915
$226$	−0.110802	−0.00737046
$227$	−15.0790	−1.00083	−0.500415	−	0.865786i	$-0.666819\pi$
$227$	−15.0790	−1.00083	−0.500415	+	0.865786i	$0.666819\pi$
$228$	127.525	8.44554
$229$	23.1135	1.52738	0.763692	−	0.645581i	$-0.223385\pi$
$229$	23.1135	1.52738	0.763692	+	0.645581i	$0.223385\pi$
$230$	15.2794	1.00750
$231$	45.9658	3.02433
$232$	30.0716	1.97430
$233$	16.3261	1.06956	0.534779	−	0.844992i	$-0.320395\pi$
$233$	16.3261	1.06956	0.534779	+	0.844992i	$0.320395\pi$
$234$	−21.8667	−1.42947
$235$	6.30515	0.411303
$236$	5.74010	0.373648
$237$	−42.6028	−2.76735
$238$	64.9432	4.20964
$239$	−12.0265	−0.777927	−0.388963	−	0.921253i	$-0.627167\pi$
$239$	−12.0265	−0.777927	−0.388963	+	0.921253i	$0.627167\pi$
$240$	−25.4971	−1.64583
$241$	11.2420	0.724159	0.362079	−	0.932147i	$-0.382067\pi$
$241$	11.2420	0.724159	0.362079	+	0.932147i	$0.382067\pi$
$242$	−5.88833	−0.378516
$243$	80.1291	5.14028
$244$	−15.4857	−0.991369
$245$	−8.05630	−0.514698
$246$	−32.1653	−2.05079
$247$	8.62252	0.548638
$248$	−52.2814	−3.31987
$249$	33.2548	2.10744
$250$	−26.2743	−1.66173
$251$	3.26361	0.205997	0.102999	−	0.994681i	$-0.467156\pi$
$251$	3.26361	0.205997	0.102999	+	0.994681i	$0.467156\pi$
$252$	−138.377	−8.71690
$253$	−18.0195	−1.13288
$254$	−20.3235	−1.27521
$255$	−29.5537	−1.85072
$256$	−31.7130	−1.98206
$257$	18.8198	1.17395	0.586973	−	0.809607i	$-0.300319\pi$
$257$	18.8198	1.17395	0.586973	+	0.809607i	$0.300319\pi$
$258$	98.3037	6.12012
$259$	8.33815	0.518107
$260$	−5.32604	−0.330307
$261$	−44.7213	−2.76818
$262$	35.1476	2.17143
$263$	10.8232	0.667386	0.333693	−	0.942682i	$-0.391705\pi$
$263$	10.8232	0.667386	0.333693	+	0.942682i	$0.391705\pi$
$264$	73.0299	4.49468
$265$	−0.928644	−0.0570462
$266$	79.7984	4.89275
$267$	31.3535	1.91881
$268$	−49.3200	−3.01270
$269$	18.9372	1.15462	0.577312	−	0.816524i	$-0.304102\pi$
$269$	18.9372	1.15462	0.577312	+	0.816524i	$0.304102\pi$
$270$	60.3172	3.67079
$271$	−14.1422	−0.859076	−0.429538	−	0.903049i	$-0.641324\pi$
$271$	−14.1422	−0.859076	−0.429538	+	0.903049i	$0.641324\pi$
$272$	42.4840	2.57597
$273$	−12.5845	−0.761646
$274$	23.4670	1.41769
$275$	12.7233	0.767242
$276$	72.9633	4.39187
$277$	−4.64798	−0.279270	−0.139635	−	0.990203i	$-0.544593\pi$
$277$	−4.64798	−0.279270	−0.139635	+	0.990203i	$0.544593\pi$
$278$	−9.96366	−0.597580
$279$	77.7508	4.65482
$280$	−26.4961	−1.58345
$281$	15.0412	0.897281	0.448640	−	0.893712i	$-0.351908\pi$
$281$	15.0412	0.897281	0.448640	+	0.893712i	$0.351908\pi$
$282$	44.0326	2.62210
$283$	3.73246	0.221872	0.110936	−	0.993828i	$-0.464615\pi$
$283$	3.73246	0.221872	0.110936	+	0.993828i	$0.464615\pi$
$284$	26.5099	1.57307
$285$	−36.3138	−2.15105
$286$	9.18592	0.543175
$287$	−13.7628	−0.812390
$288$	−30.7137	−1.80982
$289$	32.2432	1.89666
$290$	−15.9300	−0.935443
$291$	50.7521	2.97514
$292$	−9.14210	−0.535001
$293$	2.82710	0.165161	0.0825804	−	0.996584i	$-0.473684\pi$
$293$	2.82710	0.165161	0.0825804	+	0.996584i	$0.473684\pi$
$294$	−56.2619	−3.28126
$295$	−1.63454	−0.0951668
$296$	13.2476	0.769999
$297$	−71.1342	−4.12762
$298$	15.3420	0.888739
$299$	4.93337	0.285304
$300$	−51.5180	−2.97440
$301$	42.0618	2.42440
$302$	12.0234	0.691868
$303$	31.2195	1.79351
$304$	52.2019	2.99398
$305$	4.40968	0.252498
$306$	−153.447	−8.77196
$307$	6.73546	0.384413	0.192207	−	0.981354i	$-0.438436\pi$
$307$	6.73546	0.384413	0.192207	+	0.981354i	$0.438436\pi$
$308$	58.1301	3.31227
$309$	2.00393	0.114000
$310$	27.6954	1.57299
$311$	32.5901	1.84801	0.924007	−	0.382376i	$-0.124894\pi$
$311$	32.5901	1.84801	0.924007	+	0.382376i	$0.124894\pi$
$312$	−19.9941	−1.13194
$313$	8.32635	0.470633	0.235317	−	0.971919i	$-0.424387\pi$
$313$	8.32635	0.470633	0.235317	+	0.971919i	$0.424387\pi$
$314$	−47.1435	−2.66046
$315$	39.4039	2.22016
$316$	−53.8771	−3.03082
$317$	1.74053	0.0977580	0.0488790	−	0.998805i	$-0.484435\pi$
$317$	1.74053	0.0977580	0.0488790	+	0.998805i	$0.484435\pi$
$318$	−6.48527	−0.363676
$319$	18.7868	1.05186
$320$	3.97114	0.221994
$321$	49.7429	2.77638
$322$	45.6566	2.54434
$323$	60.5073	3.36671
$324$	175.221	9.73450
$325$	−3.48336	−0.193222
$326$	−19.0249	−1.05369
$327$	−50.4653	−2.79074
$328$	−21.8661	−1.20735
$329$	18.8405	1.03871
$330$	−38.6866	−2.12963
$331$	27.2657	1.49866	0.749329	−	0.662198i	$-0.230376\pi$
$331$	27.2657	1.49866	0.749329	+	0.662198i	$0.230376\pi$
$332$	42.0553	2.30808
$333$	−19.7012	−1.07962
$334$	−32.9918	−1.80523
$335$	14.0443	0.767322
$336$	−76.1881	−4.15640
$337$	−0.614271	−0.0334615	−0.0167307	−	0.999860i	$-0.505326\pi$
$337$	−0.614271	−0.0334615	−0.0167307	+	0.999860i	$0.505326\pi$
$338$	−2.51491	−0.136793
$339$	0.150669	0.00818320
$340$	−37.3747	−2.02693
$341$	−32.6621	−1.76875
$342$	−188.546	−10.1954
$343$	1.68628	0.0910508
$344$	66.8273	3.60309
$345$	−20.7769	−1.11859
$346$	11.3996	0.612849
$347$	2.62114	0.140710	0.0703550	−	0.997522i	$-0.477587\pi$
$347$	2.62114	0.140710	0.0703550	+	0.997522i	$0.477587\pi$
$348$	−76.0700	−4.07778
$349$	22.0308	1.17928	0.589642	−	0.807665i	$-0.299269\pi$
$349$	22.0308	1.17928	0.589642	+	0.807665i	$0.299269\pi$
$350$	−32.2373	−1.72316
$351$	19.4750	1.03950
$352$	12.9024	0.687701
$353$	−6.20196	−0.330097	−0.165049	−	0.986285i	$-0.552778\pi$
$353$	−6.20196	−0.330097	−0.165049	+	0.986285i	$0.552778\pi$
$354$	−11.4150	−0.606699
$355$	−7.54894	−0.400656
$356$	39.6509	2.10149
$357$	−88.3096	−4.67384
$358$	−36.5745	−1.93302
$359$	−23.4243	−1.23629	−0.618143	−	0.786066i	$-0.712115\pi$
$359$	−23.4243	−1.23629	−0.618143	+	0.786066i	$0.712115\pi$
$360$	62.6045	3.29955
$361$	55.3478	2.91304
$362$	38.5513	2.02621
$363$	8.00694	0.420256
$364$	−15.9148	−0.834162
$365$	2.60329	0.136263
$366$	30.7954	1.60970
$367$	−4.86539	−0.253971	−0.126986	−	0.991905i	$-0.540530\pi$
$367$	−4.86539	−0.253971	−0.126986	+	0.991905i	$0.540530\pi$
$368$	29.8673	1.55694
$369$	32.5184	1.69284
$370$	−7.01771	−0.364833
$371$	−2.77489	−0.144065
$372$	132.253	6.85698
$373$	−3.49126	−0.180771	−0.0903854	−	0.995907i	$-0.528810\pi$
$373$	−3.49126	−0.180771	−0.0903854	+	0.995907i	$0.528810\pi$
$374$	64.4609	3.33319
$375$	35.7278	1.84497
$376$	29.9336	1.54371
$377$	−5.14343	−0.264900
$378$	180.235	9.27027
$379$	23.6296	1.21377	0.606886	−	0.794789i	$-0.292418\pi$
$379$	23.6296	1.21377	0.606886	+	0.794789i	$0.292418\pi$
$380$	−45.9239	−2.35585
$381$	27.6358	1.41583
$382$	44.6079	2.28234
$383$	11.4097	0.583007	0.291503	−	0.956570i	$-0.405845\pi$
$383$	11.4097	0.583007	0.291503	+	0.956570i	$0.405845\pi$
$384$	51.8929	2.64815
$385$	−16.5531	−0.843623
$386$	1.50914	0.0768132
$387$	−99.3828	−5.05191
$388$	64.1831	3.25841
$389$	10.6781	0.541401	0.270701	−	0.962664i	$-0.412745\pi$
$389$	10.6781	0.541401	0.270701	+	0.962664i	$0.412745\pi$
$390$	10.5916	0.536325
$391$	34.6192	1.75077
$392$	−38.2471	−1.93177
$393$	−47.7937	−2.41087
$394$	2.14700	0.108164
$395$	15.3420	0.771939
$396$	−137.349	−6.90204
$397$	13.8741	0.696321	0.348160	−	0.937435i	$-0.386806\pi$
$397$	13.8741	0.696321	0.348160	+	0.937435i	$0.386806\pi$
$398$	28.6274	1.43496
$399$	−108.510	−5.43228
$400$	−21.0888	−1.05444
$401$	−30.4265	−1.51943	−0.759713	−	0.650259i	$-0.774660\pi$
$401$	−30.4265	−1.51943	−0.759713	+	0.650259i	$0.774660\pi$
$402$	98.0797	4.89177
$403$	8.94218	0.445442
$404$	39.4814	1.96427
$405$	−49.8957	−2.47934
$406$	−47.6007	−2.36238
$407$	8.27622	0.410237
$408$	−140.305	−6.94615
$409$	−27.8507	−1.37713	−0.688564	−	0.725176i	$-0.741759\pi$
$409$	−27.8507	−1.37713	−0.688564	+	0.725176i	$0.741759\pi$
$410$	11.5833	0.572057
$411$	−31.9103	−1.57402
$412$	2.53425	0.124853
$413$	−4.88420	−0.240336
$414$	−107.877	−5.30185
$415$	−11.9756	−0.587860
$416$	−3.53241	−0.173190
$417$	13.5486	0.663476
$418$	79.2058	3.87408
$419$	−30.5700	−1.49344	−0.746721	−	0.665138i	$-0.768373\pi$
$419$	−30.5700	−1.49344	−0.746721	+	0.665138i	$0.768373\pi$
$420$	67.0253	3.27050
$421$	−20.8364	−1.01551	−0.507753	−	0.861503i	$-0.669524\pi$
$421$	−20.8364	−1.01551	−0.507753	+	0.861503i	$0.669524\pi$
$422$	5.43468	0.264556
$423$	−44.5160	−2.16444
$424$	−4.40872	−0.214106
$425$	−24.4440	−1.18571
$426$	−52.7187	−2.55423
$427$	13.1766	0.637662
$428$	62.9068	3.04071
$429$	−12.4910	−0.603071
$430$	−35.4008	−1.70718
$431$	12.2381	0.589491	0.294745	−	0.955576i	$-0.404765\pi$
$431$	12.2381	0.589491	0.294745	+	0.955576i	$0.404765\pi$
$432$	117.905	5.67268
$433$	−20.7625	−0.997780	−0.498890	−	0.866665i	$-0.666259\pi$
$433$	−20.7625	−0.997780	−0.498890	+	0.866665i	$0.666259\pi$
$434$	82.7568	3.97245
$435$	21.6616	1.03860
$436$	−63.8205	−3.05645
$437$	42.5381	2.03487
$438$	18.1803	0.868690
$439$	12.1912	0.581852	0.290926	−	0.956746i	$-0.406037\pi$
$439$	12.1912	0.581852	0.290926	+	0.956746i	$0.406037\pi$
$440$	−26.2993	−1.25377
$441$	56.8795	2.70855
$442$	−17.6480	−0.839431
$443$	22.7114	1.07905	0.539526	−	0.841969i	$-0.318604\pi$
$443$	22.7114	1.07905	0.539526	+	0.841969i	$0.318604\pi$
$444$	−33.5114	−1.59038
$445$	−11.2909	−0.535242
$446$	−55.6166	−2.63352
$447$	−20.8620	−0.986741
$448$	11.8662	0.560626
$449$	−1.50263	−0.0709136	−0.0354568	−	0.999371i	$-0.511289\pi$
$449$	−1.50263	−0.0709136	−0.0354568	+	0.999371i	$0.511289\pi$
$450$	76.1698	3.59068
$451$	−13.6606	−0.643250
$452$	0.190542	0.00896232
$453$	−16.3494	−0.768161
$454$	37.9224	1.77979
$455$	4.53188	0.212458
$456$	−172.399	−8.07333
$457$	4.49754	0.210386	0.105193	−	0.994452i	$-0.466454\pi$
$457$	4.49754	0.210386	0.105193	+	0.994452i	$0.466454\pi$
$458$	−58.1284	−2.71616
$459$	136.663	6.37889
$460$	−26.2753	−1.22509
$461$	13.6659	0.636485	0.318243	−	0.948009i	$-0.396907\pi$
$461$	13.6659	0.636485	0.318243	+	0.948009i	$0.396907\pi$
$462$	−115.600	−5.37819
$463$	−28.2195	−1.31147	−0.655736	−	0.754990i	$-0.727642\pi$
$463$	−28.2195	−1.31147	−0.655736	+	0.754990i	$0.727642\pi$
$464$	−31.1390	−1.44559
$465$	−37.6601	−1.74645
$466$	−41.0587	−1.90201
$467$	−0.237557	−0.0109928	−0.00549642	−	0.999985i	$-0.501750\pi$
$467$	−0.237557	−0.0109928	−0.00549642	+	0.999985i	$0.501750\pi$
$468$	37.6032	1.73821
$469$	41.9659	1.93781
$470$	−15.8569	−0.731424
$471$	64.1056	2.95383
$472$	−7.75996	−0.357181
$473$	41.7494	1.91964
$474$	107.142	4.92120
$475$	−30.0354	−1.37812
$476$	−111.680	−5.11883
$477$	6.55646	0.300200
$478$	30.2455	1.38340
$479$	34.3493	1.56946	0.784731	−	0.619837i	$-0.212801\pi$
$479$	34.3493	1.56946	0.784731	+	0.619837i	$0.212801\pi$
$480$	14.8768	0.679028
$481$	−2.26585	−0.103314
$482$	−28.2725	−1.28778
$483$	−62.0838	−2.82491
$484$	10.1259	0.460268
$485$	−18.2767	−0.829903
$486$	−201.518	−9.14102
$487$	−21.4126	−0.970296	−0.485148	−	0.874432i	$-0.661234\pi$
$487$	−21.4126	−0.970296	−0.485148	+	0.874432i	$0.661234\pi$
$488$	20.9349	0.947677
$489$	25.8700	1.16988
$490$	20.2609	0.915293
$491$	1.17946	0.0532284	0.0266142	−	0.999646i	$-0.491527\pi$
$491$	1.17946	0.0532284	0.0266142	+	0.999646i	$0.491527\pi$
$492$	55.3132	2.49371
$493$	−36.0933	−1.62556
$494$	−21.6849	−0.975648
$495$	39.1113	1.75792
$496$	54.1372	2.43083
$497$	−22.5571	−1.01182
$498$	−83.6328	−3.74768
$499$	−19.0027	−0.850677	−0.425338	−	0.905034i	$-0.639845\pi$
$499$	−19.0027	−0.850677	−0.425338	+	0.905034i	$0.639845\pi$
$500$	45.1827	2.02063
$501$	44.8622	2.00429
$502$	−8.20770	−0.366327
$503$	−20.5683	−0.917096	−0.458548	−	0.888670i	$-0.651630\pi$
$503$	−20.5683	−0.917096	−0.458548	+	0.888670i	$0.651630\pi$
$504$	187.069	8.33273
$505$	−11.2427	−0.500292
$506$	45.3176	2.01461
$507$	3.41977	0.151877
$508$	34.9494	1.55063
$509$	−36.2968	−1.60883	−0.804413	−	0.594071i	$-0.797520\pi$
$509$	−36.2968	−1.60883	−0.804413	+	0.594071i	$0.797520\pi$
$510$	74.3248	3.29116
$511$	7.77893	0.344120
$512$	49.4066	2.18349
$513$	167.924	7.41402
$514$	−47.3301	−2.08764
$515$	−0.721650	−0.0317997
$516$	−169.048	−7.44194
$517$	18.7006	0.822450
$518$	−20.9697	−0.921356
$519$	−15.5012	−0.680428
$520$	7.20020	0.315750
$521$	40.2833	1.76484	0.882421	−	0.470460i	$-0.155912\pi$
$521$	40.2833	1.76484	0.882421	+	0.470460i	$0.155912\pi$
$522$	112.470	4.92268
$523$	−7.46349	−0.326356	−0.163178	−	0.986597i	$-0.552174\pi$
$523$	−7.46349	−0.326356	−0.163178	+	0.986597i	$0.552174\pi$
$524$	−60.4417	−2.64041
$525$	43.8362	1.91317
$526$	−27.2193	−1.18682
$527$	62.7505	2.73345
$528$	−75.6222	−3.29104
$529$	1.33815	0.0581803
$530$	2.33546	0.101446
$531$	11.5403	0.500806
$532$	−137.226	−5.94949
$533$	3.73997	0.161996
$534$	−78.8513	−3.41223
$535$	−17.9133	−0.774458
$536$	66.6750	2.87992
$537$	49.7339	2.14618
$538$	−47.6254	−2.05328
$539$	−23.8943	−1.02920
$540$	−103.725	−4.46360
$541$	−11.2592	−0.484070	−0.242035	−	0.970268i	$-0.577815\pi$
$541$	−11.2592	−0.484070	−0.242035	+	0.970268i	$0.577815\pi$
$542$	35.5663	1.52770
$543$	−52.4220	−2.24964
$544$	−24.7881	−1.06278
$545$	18.1734	0.778465
$546$	31.6488	1.35444
$547$	−6.26671	−0.267945	−0.133973	−	0.990985i	$-0.542773\pi$
$547$	−6.26671	−0.267945	−0.133973	+	0.990985i	$0.542773\pi$
$548$	−40.3550	−1.72388
$549$	−31.1335	−1.32875
$550$	−31.9979	−1.36440
$551$	−44.3493	−1.88935
$552$	−98.6381	−4.19831
$553$	45.8435	1.94947
$554$	11.6893	0.496629
$555$	9.54267	0.405064
$556$	17.1340	0.726645
$557$	−6.40671	−0.271461	−0.135730	−	0.990746i	$-0.543338\pi$
$557$	−6.40671	−0.271461	−0.135730	+	0.990746i	$0.543338\pi$
$558$	−195.536	−8.27771
$559$	−11.4301	−0.483442
$560$	27.4366	1.15941
$561$	−87.6538	−3.70074
$562$	−37.8272	−1.59564
$563$	24.7133	1.04154	0.520770	−	0.853697i	$-0.325645\pi$
$563$	24.7133	1.04154	0.520770	+	0.853697i	$0.325645\pi$
$564$	−75.7208	−3.18842
$565$	−0.0542584	−0.00228267
$566$	−9.38681	−0.394557
$567$	−149.094	−6.26136
$568$	−35.8384	−1.50375
$569$	−31.8776	−1.33638	−0.668190	−	0.743991i	$-0.732931\pi$
$569$	−31.8776	−1.33638	−0.668190	+	0.743991i	$0.732931\pi$
$570$	91.3260	3.82523
$571$	10.6747	0.446724	0.223362	−	0.974736i	$-0.428297\pi$
$571$	10.6747	0.446724	0.223362	+	0.974736i	$0.428297\pi$
$572$	−15.7966	−0.660489
$573$	−60.6578	−2.53401
$574$	34.6121	1.44468
$575$	−17.1847	−0.716652
$576$	−28.0373	−1.16822
$577$	14.3053	0.595539	0.297770	−	0.954638i	$-0.403757\pi$
$577$	14.3053	0.595539	0.297770	+	0.954638i	$0.403757\pi$
$578$	−81.0889	−3.37285
$579$	−2.05212	−0.0852834
$580$	27.3941	1.13748
$581$	−35.7845	−1.48459
$582$	−127.637	−5.29073
$583$	−2.75428	−0.114071
$584$	12.3591	0.511422
$585$	−10.7078	−0.442715
$586$	−7.10989	−0.293707
$587$	14.7634	0.609349	0.304674	−	0.952457i	$-0.401452\pi$
$587$	14.7634	0.609349	0.304674	+	0.952457i	$0.401452\pi$
$588$	96.7510	3.98994
$589$	77.1041	3.17702
$590$	4.11073	0.169236
$591$	−2.91949	−0.120092
$592$	−13.7178	−0.563798
$593$	−9.62542	−0.395269	−0.197634	−	0.980276i	$-0.563326\pi$
$593$	−9.62542	−0.395269	−0.197634	+	0.980276i	$0.563326\pi$
$594$	178.896	7.34020
$595$	31.8018	1.30375
$596$	−26.3830	−1.08069
$597$	−38.9274	−1.59319
$598$	−12.4070	−0.507359
$599$	43.3034	1.76933	0.884665	−	0.466227i	$-0.154387\pi$
$599$	43.3034	1.76933	0.884665	+	0.466227i	$0.154387\pi$
$600$	69.6465	2.84331
$601$	13.8356	0.564364	0.282182	−	0.959361i	$-0.408942\pi$
$601$	13.8356	0.564364	0.282182	+	0.959361i	$0.408942\pi$
$602$	−105.782	−4.31134
$603$	−99.1564	−4.03796
$604$	−20.6761	−0.841297
$605$	−2.88344	−0.117228
$606$	−78.5142	−3.18942
$607$	11.6961	0.474731	0.237366	−	0.971420i	$-0.423716\pi$
$607$	11.6961	0.474731	0.237366	+	0.971420i	$0.423716\pi$
$608$	−30.4582	−1.23524
$609$	64.7273	2.62288
$610$	−11.0900	−0.449019
$611$	−5.11982	−0.207126
$612$	263.875	10.6665
$613$	−15.7156	−0.634746	−0.317373	−	0.948301i	$-0.602801\pi$
$613$	−15.7156	−0.634746	−0.317373	+	0.948301i	$0.602801\pi$
$614$	−16.9391	−0.683606
$615$	−15.7509	−0.635138
$616$	−78.5853	−3.16629
$617$	1.70050	0.0684597	0.0342298	−	0.999414i	$-0.489102\pi$
$617$	1.70050	0.0684597	0.0342298	+	0.999414i	$0.489102\pi$
$618$	−5.03971	−0.202727
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−47.6264	−1.91272
$621$	96.0775	3.85546
$622$	−81.9611	−3.28634
$623$	−33.7386	−1.35171
$624$	20.7038	0.828814
$625$	4.55063	0.182025
$626$	−20.9400	−0.836932
$627$	−107.704	−4.30128
$628$	81.0704	3.23506
$629$	−15.9003	−0.633986
$630$	−99.0974	−3.94814
$631$	−33.3863	−1.32909	−0.664544	−	0.747249i	$-0.731374\pi$
$631$	−33.3863	−1.32909	−0.664544	+	0.747249i	$0.731374\pi$
$632$	72.8357	2.89725
$633$	−7.39007	−0.293729
$634$	−4.37728	−0.173844
$635$	−9.95214	−0.394939
$636$	11.1524	0.442222
$637$	6.54176	0.259194
$638$	−47.2472	−1.87053
$639$	53.2974	2.10841
$640$	−18.6875	−0.738689
$641$	44.9698	1.77620	0.888100	−	0.459651i	$-0.152025\pi$
$641$	44.9698	1.77620	0.888100	+	0.459651i	$0.152025\pi$
$642$	−125.099	−4.93726
$643$	13.8616	0.546646	0.273323	−	0.961922i	$-0.411877\pi$
$643$	13.8616	0.546646	0.273323	+	0.961922i	$0.411877\pi$
$644$	−78.5136	−3.09387
$645$	48.1380	1.89543
$646$	−152.170	−5.98706
$647$	28.5990	1.12434	0.562172	−	0.827020i	$-0.309966\pi$
$647$	28.5990	1.12434	0.562172	+	0.827020i	$0.309966\pi$
$648$	−236.879	−9.30548
$649$	−4.84792	−0.190298
$650$	8.76035	0.343609
$651$	−112.533	−4.41050
$652$	32.7163	1.28127
$653$	−7.32597	−0.286687	−0.143344	−	0.989673i	$-0.545785\pi$
$653$	−7.32597	−0.286687	−0.143344	+	0.989673i	$0.545785\pi$
$654$	126.916	4.96280
$655$	17.2113	0.672502
$656$	22.6423	0.884033
$657$	−18.3799	−0.717069
$658$	−47.3822	−1.84715
$659$	−13.3919	−0.521675	−0.260837	−	0.965383i	$-0.583999\pi$
$659$	−13.3919	−0.521675	−0.260837	+	0.965383i	$0.583999\pi$
$660$	66.5276	2.58958
$661$	11.5256	0.448295	0.224148	−	0.974555i	$-0.428040\pi$
$661$	11.5256	0.448295	0.224148	+	0.974555i	$0.428040\pi$
$662$	−68.5708	−2.66508
$663$	23.9977	0.931995
$664$	−56.8540	−2.20636
$665$	39.0762	1.51531
$666$	49.5468	1.91990
$667$	−25.3745	−0.982503
$668$	56.7345	2.19512
$669$	75.6273	2.92392
$670$	−35.3202	−1.36454
$671$	13.0788	0.504900
$672$	44.4534	1.71483
$673$	−39.4518	−1.52075	−0.760377	−	0.649482i	$-0.774986\pi$
$673$	−39.4518	−1.52075	−0.760377	+	0.649482i	$0.774986\pi$
$674$	1.54484	0.0595049
$675$	−67.8386	−2.61111
$676$	4.32478	0.166338
$677$	9.32072	0.358224	0.179112	−	0.983829i	$-0.442677\pi$
$677$	9.32072	0.358224	0.179112	+	0.983829i	$0.442677\pi$
$678$	−0.378918	−0.0145523
$679$	−54.6129	−2.09585
$680$	50.5263	1.93760
$681$	−51.5668	−1.97604
$682$	82.1422	3.14539
$683$	35.6818	1.36533	0.682663	−	0.730733i	$-0.260822\pi$
$683$	35.6818	1.36533	0.682663	+	0.730733i	$0.260822\pi$
$684$	324.234	12.3974
$685$	11.4915	0.439066
$686$	−4.24085	−0.161917
$687$	79.0429	3.01568
$688$	−69.1994	−2.63820
$689$	0.754064	0.0287276
$690$	52.2522	1.98921
$691$	35.5531	1.35250	0.676252	−	0.736670i	$-0.263603\pi$
$691$	35.5531	1.35250	0.676252	+	0.736670i	$0.263603\pi$
$692$	−19.6034	−0.745211
$693$	116.869	4.43948
$694$	−6.59193	−0.250226
$695$	−4.87907	−0.185074
$696$	102.838	3.89807
$697$	26.2447	0.994089
$698$	−55.4056	−2.09713
$699$	55.8315	2.11174
$700$	55.4370	2.09532
$701$	3.41184	0.128863	0.0644317	−	0.997922i	$-0.479477\pi$
$701$	3.41184	0.128863	0.0644317	+	0.997922i	$0.479477\pi$
$702$	−48.9780	−1.84855
$703$	−19.5374	−0.736866
$704$	11.7781	0.443903
$705$	21.5622	0.812078
$706$	15.5974	0.587015
$707$	−33.5943	−1.26345
$708$	19.6298	0.737734
$709$	12.0648	0.453105	0.226552	−	0.973999i	$-0.427255\pi$
$709$	12.0648	0.453105	0.226552	+	0.973999i	$0.427255\pi$
$710$	18.9849	0.712491
$711$	−108.318	−4.06225
$712$	−53.6035	−2.00888
$713$	44.1151	1.65212
$714$	222.091	8.31154
$715$	4.49822	0.168224
$716$	62.8954	2.35051
$717$	−41.1277	−1.53594
$718$	58.9100	2.19850
$719$	−39.0723	−1.45715	−0.728576	−	0.684965i	$-0.759818\pi$
$719$	−39.0723	−1.45715	−0.728576	+	0.684965i	$0.759818\pi$
$720$	−64.8268	−2.41595
$721$	−2.15637	−0.0803074
$722$	−139.195	−5.18029
$723$	38.4449	1.42978
$724$	−66.2949	−2.46383
$725$	17.9164	0.665400
$726$	−20.1367	−0.747345
$727$	40.6463	1.50749	0.753744	−	0.657168i	$-0.228246\pi$
$727$	40.6463	1.50749	0.753744	+	0.657168i	$0.228246\pi$
$728$	21.5150	0.797399
$729$	152.476	5.64727
$730$	−6.54705	−0.242317
$731$	−80.2090	−2.96664
$732$	−52.9575	−1.95736
$733$	−34.5652	−1.27670	−0.638348	−	0.769748i	$-0.720382\pi$
$733$	−34.5652	−1.27670	−0.638348	+	0.769748i	$0.720382\pi$
$734$	12.2360	0.451640
$735$	−27.5507	−1.01622
$736$	−17.4267	−0.642356
$737$	41.6543	1.53436
$738$	−81.7809	−3.01040
$739$	−37.4667	−1.37823	−0.689117	−	0.724650i	$-0.742001\pi$
$739$	−37.4667	−1.37823	−0.689117	+	0.724650i	$0.742001\pi$
$740$	12.0680	0.443630
$741$	29.4870	1.08323
$742$	6.97860	0.256193
$743$	4.06302	0.149058	0.0745289	−	0.997219i	$-0.476255\pi$
$743$	4.06302	0.149058	0.0745289	+	0.997219i	$0.476255\pi$
$744$	−178.790	−6.55478
$745$	7.51278	0.275247
$746$	8.78022	0.321467
$747$	84.5509	3.09356
$748$	−110.850	−4.05309
$749$	−53.5268	−1.95583
$750$	−89.8522	−3.28094
$751$	−6.09921	−0.222563	−0.111282	−	0.993789i	$-0.535496\pi$
$751$	−6.09921	−0.222563	−0.111282	+	0.993789i	$0.535496\pi$
$752$	−30.9961	−1.13031
$753$	11.1608	0.406723
$754$	12.9353	0.471075
$755$	5.88769	0.214275
$756$	−309.941	−11.2725
$757$	−26.5955	−0.966630	−0.483315	−	0.875447i	$-0.660567\pi$
$757$	−26.5955	−0.966630	−0.483315	+	0.875447i	$0.660567\pi$
$758$	−59.4264	−2.15847
$759$	−61.6227	−2.23676
$760$	62.0839	2.25202
$761$	9.65559	0.350015	0.175007	−	0.984567i	$-0.444005\pi$
$761$	9.65559	0.350015	0.175007	+	0.984567i	$0.444005\pi$
$762$	−69.5017	−2.51778
$763$	54.3042	1.96595
$764$	−76.7102	−2.77528
$765$	−75.1407	−2.71672
$766$	−28.6943	−1.03677
$767$	1.32726	0.0479245
$768$	−108.451	−3.91340
$769$	40.3177	1.45389	0.726947	−	0.686693i	$-0.240939\pi$
$769$	40.3177	1.45389	0.726947	+	0.686693i	$0.240939\pi$
$770$	41.6295	1.50022
$771$	64.3594	2.31785
$772$	−2.59520	−0.0934032
$773$	32.4100	1.16571	0.582854	−	0.812577i	$-0.301936\pi$
$773$	32.4100	1.16571	0.582854	+	0.812577i	$0.301936\pi$
$774$	249.939	8.98387
$775$	−31.1489	−1.11890
$776$	−86.7683	−3.11480
$777$	28.5146	1.02295
$778$	−26.8545	−0.962779
$779$	32.2480	1.15540
$780$	−18.2138	−0.652160
$781$	−22.3895	−0.801160
$782$	−87.0642	−3.11341
$783$	−100.168	−3.57973
$784$	39.6047	1.41445
$785$	−23.0855	−0.823958
$786$	120.197	4.28728
$787$	−3.31612	−0.118207	−0.0591035	−	0.998252i	$-0.518824\pi$
$787$	−3.31612	−0.118207	−0.0591035	+	0.998252i	$0.518824\pi$
$788$	−3.69210	−0.131526
$789$	37.0128	1.31769
$790$	−38.5837	−1.37275
$791$	−0.162130	−0.00576468
$792$	185.680	6.59785
$793$	−3.58069	−0.127154
$794$	−34.8921	−1.23827
$795$	−3.17575	−0.112632
$796$	−49.2291	−1.74488
$797$	−0.858310	−0.0304029	−0.0152014	−	0.999884i	$-0.504839\pi$
$797$	−0.858310	−0.0304029	−0.0152014	+	0.999884i	$0.504839\pi$
$798$	272.892	9.66028
$799$	−35.9276	−1.27103
$800$	12.3047	0.435035
$801$	79.7170	2.81666
$802$	76.5199	2.70201
$803$	7.72116	0.272474
$804$	−168.663	−5.94829
$805$	22.3574	0.787996
$806$	−22.4888	−0.792134
$807$	64.7610	2.27969
$808$	−53.3743	−1.87770
$809$	−8.00886	−0.281576	−0.140788	−	0.990040i	$-0.544964\pi$
$809$	−8.00886	−0.281576	−0.140788	+	0.990040i	$0.544964\pi$
$810$	125.483	4.40904
$811$	−33.6266	−1.18079	−0.590395	−	0.807114i	$-0.701028\pi$
$811$	−33.6266	−1.18079	−0.590395	+	0.807114i	$0.701028\pi$
$812$	81.8567	2.87261
$813$	−48.3630	−1.69616
$814$	−20.8140	−0.729529
$815$	−9.31625	−0.326334
$816$	145.286	5.08601
$817$	−98.5562	−3.44805
$818$	70.0420	2.44896
$819$	−31.9962	−1.11804
$820$	−19.9192	−0.695610
$821$	−29.4906	−1.02923	−0.514614	−	0.857422i	$-0.672065\pi$
$821$	−29.4906	−1.02923	−0.514614	+	0.857422i	$0.672065\pi$
$822$	80.2516	2.79910
$823$	−46.1402	−1.60835	−0.804173	−	0.594395i	$-0.797392\pi$
$823$	−46.1402	−1.60835	−0.804173	+	0.594395i	$0.797392\pi$
$824$	−3.42602	−0.119351
$825$	43.5107	1.51485
$826$	12.2833	0.427391
$827$	−31.6981	−1.10225	−0.551125	−	0.834423i	$-0.685801\pi$
$827$	−31.6981	−1.10225	−0.551125	+	0.834423i	$0.685801\pi$
$828$	185.511	6.44694
$829$	−40.2112	−1.39659	−0.698297	−	0.715808i	$-0.746058\pi$
$829$	−40.2112	−1.39659	−0.698297	+	0.715808i	$0.746058\pi$
$830$	30.1176	1.04540
$831$	−15.8950	−0.551393
$832$	−3.22459	−0.111793
$833$	45.9058	1.59054
$834$	−34.0734	−1.17987
$835$	−16.1556	−0.559089
$836$	−136.207	−4.71080
$837$	174.149	6.01948
$838$	76.8808	2.65580
$839$	48.4845	1.67387	0.836935	−	0.547302i	$-0.184345\pi$
$839$	48.4845	1.67387	0.836935	+	0.547302i	$0.184345\pi$
$840$	−90.6106	−3.12636
$841$	−2.54511	−0.0877623
$842$	52.4017	1.80588
$843$	51.4373	1.77160
$844$	−9.34577	−0.321695
$845$	−1.23152	−0.0423655
$846$	111.954	3.84905
$847$	−8.61603	−0.296050
$848$	4.56521	0.156770
$849$	12.7642	0.438065
$850$	61.4744	2.10856
$851$	−11.1783	−0.383187
$852$	90.6579	3.10589
$853$	9.43678	0.323109	0.161555	−	0.986864i	$-0.448349\pi$
$853$	9.43678	0.323109	0.161555	+	0.986864i	$0.448349\pi$
$854$	−33.1380	−1.13396
$855$	−92.3286	−3.15757
$856$	−85.0428	−2.90670
$857$	14.6872	0.501707	0.250853	−	0.968025i	$-0.419289\pi$
$857$	14.6872	0.501707	0.250853	+	0.968025i	$0.419289\pi$
$858$	31.4138	1.07245
$859$	−7.71669	−0.263290	−0.131645	−	0.991297i	$-0.542026\pi$
$859$	−7.71669	−0.263290	−0.131645	+	0.991297i	$0.542026\pi$
$860$	60.8772	2.07589
$861$	−47.0655	−1.60399
$862$	−30.7779	−1.04830
$863$	15.9057	0.541435	0.270717	−	0.962659i	$-0.412739\pi$
$863$	15.9057	0.541435	0.270717	+	0.962659i	$0.412739\pi$
$864$	−68.7937	−2.34041
$865$	5.58225	0.189802
$866$	52.2157	1.77436
$867$	110.265	3.74478
$868$	−142.313	−4.83042
$869$	45.5031	1.54359
$870$	−54.4770	−1.84694
$871$	−11.4041	−0.386412
$872$	86.2780	2.92174
$873$	129.038	4.36728
$874$	−106.979	−3.61863
$875$	−38.4456	−1.29970
$876$	−31.2639	−1.05631
$877$	−13.9315	−0.470434	−0.235217	−	0.971943i	$-0.575580\pi$
$877$	−13.9315	−0.470434	−0.235217	+	0.971943i	$0.575580\pi$
$878$	−30.6597	−1.03471
$879$	9.66802	0.326094
$880$	27.2329	0.918020
$881$	−31.6949	−1.06783	−0.533915	−	0.845538i	$-0.679280\pi$
$881$	−31.6949	−1.06783	−0.533915	+	0.845538i	$0.679280\pi$
$882$	−143.047	−4.81664
$883$	−42.9493	−1.44536	−0.722679	−	0.691184i	$-0.757090\pi$
$883$	−42.9493	−1.44536	−0.722679	+	0.691184i	$0.757090\pi$
$884$	30.3485	1.02073
$885$	−5.58976	−0.187898
$886$	−57.1172	−1.91889
$887$	−10.0192	−0.336412	−0.168206	−	0.985752i	$-0.553797\pi$
$887$	−10.0192	−0.336412	−0.168206	+	0.985752i	$0.553797\pi$
$888$	45.3036	1.52029
$889$	−29.7381	−0.997384
$890$	28.3957	0.951827
$891$	−147.987	−4.95774
$892$	95.6413	3.20231
$893$	−44.1457	−1.47728
$894$	52.4662	1.75473
$895$	−17.9100	−0.598666
$896$	−55.8403	−1.86550
$897$	16.8710	0.563306
$898$	3.77899	0.126106
$899$	−45.9935	−1.53397
$900$	−130.986	−4.36619
$901$	5.29154	0.176287
$902$	34.3551	1.14390
$903$	143.842	4.78675
$904$	−0.257590	−0.00856733
$905$	18.8781	0.627528
$906$	41.1172	1.36603
$907$	−14.0298	−0.465853	−0.232926	−	0.972494i	$-0.574830\pi$
$907$	−14.0298	−0.465853	−0.232926	+	0.972494i	$0.574830\pi$
$908$	−65.2134	−2.16418
$909$	79.3761	2.63274
$910$	−11.3973	−0.377816
$911$	−10.6514	−0.352896	−0.176448	−	0.984310i	$-0.556461\pi$
$911$	−10.6514	−0.352896	−0.176448	+	0.984310i	$0.556461\pi$
$912$	178.519	5.91134
$913$	−35.5187	−1.17550
$914$	−11.3109	−0.374131
$915$	15.0801	0.498533
$916$	99.9608	3.30280
$917$	51.4293	1.69835
$918$	−343.696	−11.3436
$919$	−46.6730	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$919$	−46.6730	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$920$	35.5213	1.17110
$921$	23.0337	0.758988
$922$	−34.3686	−1.13187
$923$	6.12978	0.201764
$924$	198.792	6.53977
$925$	7.89279	0.259513
$926$	70.9696	2.33221
$927$	5.09503	0.167343
$928$	18.1687	0.596416
$929$	32.2637	1.05854	0.529270	−	0.848454i	$-0.322466\pi$
$929$	32.2637	1.05854	0.529270	+	0.848454i	$0.322466\pi$
$930$	94.7118	3.10572
$931$	56.4064	1.84865
$932$	70.6068	2.31280
$933$	111.451	3.64873
$934$	0.597436	0.0195487
$935$	31.5656	1.03231
$936$	−50.8353	−1.66160
$937$	15.3930	0.502868	0.251434	−	0.967874i	$-0.419098\pi$
$937$	15.3930	0.502868	0.251434	+	0.967874i	$0.419098\pi$
$938$	−105.541	−3.44602
$939$	28.4742	0.929221
$940$	27.2684	0.889396
$941$	15.5975	0.508463	0.254231	−	0.967143i	$-0.418178\pi$
$941$	15.5975	0.508463	0.254231	+	0.967143i	$0.418178\pi$
$942$	−161.220	−5.25283
$943$	18.4507	0.600836
$944$	8.03541	0.261530
$945$	88.2585	2.87105
$946$	−104.996	−3.41371
$947$	5.63661	0.183165	0.0915827	−	0.995797i	$-0.470807\pi$
$947$	5.63661	0.183165	0.0915827	+	0.995797i	$0.470807\pi$
$948$	−184.247	−5.98408
$949$	−2.11389	−0.0686197
$950$	75.5362	2.45072
$951$	5.95222	0.193014
$952$	150.978	4.89324
$953$	44.2140	1.43223	0.716115	−	0.697982i	$-0.245918\pi$
$953$	44.2140	1.43223	0.716115	+	0.697982i	$0.245918\pi$
$954$	−16.4889	−0.533849
$955$	21.8439	0.706852
$956$	−52.0117	−1.68218
$957$	64.2466	2.07680
$958$	−86.3855	−2.79099
$959$	34.3377	1.10882
$960$	13.5804	0.438305
$961$	48.9626	1.57944
$962$	5.69842	0.183724
$963$	126.472	4.07551
$964$	48.6190	1.56591
$965$	0.739005	0.0237894
$966$	156.135	5.02357
$967$	−9.32019	−0.299717	−0.149859	−	0.988707i	$-0.547882\pi$
$967$	−9.32019	−0.299717	−0.149859	+	0.988707i	$0.547882\pi$
$968$	−13.6891	−0.439983
$969$	206.921	6.64726
$970$	45.9643	1.47583
$971$	32.3878	1.03937	0.519686	−	0.854357i	$-0.326049\pi$
$971$	32.3878	1.03937	0.519686	+	0.854357i	$0.326049\pi$
$972$	346.540	11.1153
$973$	−14.5792	−0.467388
$974$	53.8507	1.72549
$975$	−11.9123	−0.381499
$976$	−21.6780	−0.693895
$977$	−33.6386	−1.07619	−0.538097	−	0.842883i	$-0.680856\pi$
$977$	−33.6386	−1.07619	−0.538097	+	0.842883i	$0.680856\pi$
$978$	−65.0609	−2.08042
$979$	−33.4880	−1.07028
$980$	−34.8417	−1.11298
$981$	−128.309	−4.09659
$982$	−2.96625	−0.0946567
$983$	44.8544	1.43063	0.715316	−	0.698801i	$-0.246283\pi$
$983$	44.8544	1.43063	0.715316	+	0.698801i	$0.246283\pi$
$984$	−74.7772	−2.38381
$985$	1.05136	0.0334991
$986$	90.7714	2.89075
$987$	64.4302	2.05083
$988$	37.2905	1.18637
$989$	−56.3889	−1.79306
$990$	−98.3614	−3.12613
$991$	7.72935	0.245531	0.122765	−	0.992436i	$-0.460824\pi$
$991$	7.72935	0.245531	0.122765	+	0.992436i	$0.460824\pi$
$992$	−31.5874	−1.00290
$993$	93.2424	2.95896
$994$	56.7290	1.79933
$995$	14.0184	0.444414
$996$	143.820	4.55710
$997$	15.5965	0.493947	0.246973	−	0.969022i	$-0.420564\pi$
$997$	15.5965	0.493947	0.246973	+	0.969022i	$0.420564\pi$
$998$	47.7900	1.51277
$999$	−44.1276	−1.39613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.13	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.13	✓	168	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.51491 q^{2} +3.41977 q^{3} +4.32478 q^{4} -1.23152 q^{5} -8.60042 q^{6} -3.67991 q^{7} -5.84661 q^{8} +8.69483 q^{9} +O(q^{10})\)	\(q-2.51491 q^{2} +3.41977 q^{3} +4.32478 q^{4} -1.23152 q^{5} -8.60042 q^{6} -3.67991 q^{7} -5.84661 q^{8} +8.69483 q^{9} +3.09716 q^{10} -3.65258 q^{11} +14.7897 q^{12} +1.00000 q^{13} +9.25465 q^{14} -4.21151 q^{15} +6.05414 q^{16} +7.01735 q^{17} -21.8667 q^{18} +8.62252 q^{19} -5.32604 q^{20} -12.5845 q^{21} +9.18592 q^{22} +4.93337 q^{23} -19.9941 q^{24} -3.48336 q^{25} -2.51491 q^{26} +19.4750 q^{27} -15.9148 q^{28} -5.14343 q^{29} +10.5916 q^{30} +8.94218 q^{31} -3.53241 q^{32} -12.4910 q^{33} -17.6480 q^{34} +4.53188 q^{35} +37.6032 q^{36} -2.26585 q^{37} -21.6849 q^{38} +3.41977 q^{39} +7.20020 q^{40} +3.73997 q^{41} +31.6488 q^{42} -11.4301 q^{43} -15.7966 q^{44} -10.7078 q^{45} -12.4070 q^{46} -5.11982 q^{47} +20.7038 q^{48} +6.54176 q^{49} +8.76035 q^{50} +23.9977 q^{51} +4.32478 q^{52} +0.754064 q^{53} -48.9780 q^{54} +4.49822 q^{55} +21.5150 q^{56} +29.4870 q^{57} +12.9353 q^{58} +1.32726 q^{59} -18.2138 q^{60} -3.58069 q^{61} -22.4888 q^{62} -31.9962 q^{63} -3.22459 q^{64} -1.23152 q^{65} +31.4138 q^{66} -11.4041 q^{67} +30.3485 q^{68} +16.8710 q^{69} -11.3973 q^{70} +6.12978 q^{71} -50.8353 q^{72} -2.11389 q^{73} +5.69842 q^{74} -11.9123 q^{75} +37.2905 q^{76} +13.4412 q^{77} -8.60042 q^{78} -12.4578 q^{79} -7.45578 q^{80} +40.5156 q^{81} -9.40569 q^{82} +9.72427 q^{83} -54.4250 q^{84} -8.64200 q^{85} +28.7457 q^{86} -17.5894 q^{87} +21.3552 q^{88} +9.16831 q^{89} +26.9293 q^{90} -3.67991 q^{91} +21.3357 q^{92} +30.5802 q^{93} +12.8759 q^{94} -10.6188 q^{95} -12.0800 q^{96} +14.8408 q^{97} -16.4519 q^{98} -31.7586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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