LMFDB - Embedded newform 8047.2.a.c.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:	$1$
Dimension:	$151$
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)

$f(q)$	$=$	$q-2.53026 q^{2} -1.57653 q^{3} +4.40222 q^{4} -1.20027 q^{5} +3.98902 q^{6} +4.10699 q^{7} -6.07823 q^{8} -0.514564 q^{9} +O(q^{10})$	\(q-2.53026 q^{2} -1.57653 q^{3} +4.40222 q^{4} -1.20027 q^{5} +3.98902 q^{6} +4.10699 q^{7} -6.07823 q^{8} -0.514564 q^{9} +3.03700 q^{10} -0.0215764 q^{11} -6.94021 q^{12} -1.00000 q^{13} -10.3917 q^{14} +1.89226 q^{15} +6.57507 q^{16} -5.00346 q^{17} +1.30198 q^{18} -1.32964 q^{19} -5.28386 q^{20} -6.47477 q^{21} +0.0545938 q^{22} -6.39447 q^{23} +9.58249 q^{24} -3.55935 q^{25} +2.53026 q^{26} +5.54080 q^{27} +18.0798 q^{28} -9.28410 q^{29} -4.78791 q^{30} +6.53715 q^{31} -4.48019 q^{32} +0.0340157 q^{33} +12.6600 q^{34} -4.92950 q^{35} -2.26522 q^{36} +7.40338 q^{37} +3.36434 q^{38} +1.57653 q^{39} +7.29554 q^{40} +5.21804 q^{41} +16.3829 q^{42} +8.25645 q^{43} -0.0949838 q^{44} +0.617618 q^{45} +16.1797 q^{46} +7.68935 q^{47} -10.3658 q^{48} +9.86733 q^{49} +9.00607 q^{50} +7.88808 q^{51} -4.40222 q^{52} -5.22600 q^{53} -14.0197 q^{54} +0.0258975 q^{55} -24.9632 q^{56} +2.09622 q^{57} +23.4912 q^{58} +6.86456 q^{59} +8.33014 q^{60} +1.61811 q^{61} -16.5407 q^{62} -2.11331 q^{63} -1.81412 q^{64} +1.20027 q^{65} -0.0860686 q^{66} -1.86179 q^{67} -22.0263 q^{68} +10.0810 q^{69} +12.4729 q^{70} -5.39093 q^{71} +3.12764 q^{72} +2.16373 q^{73} -18.7325 q^{74} +5.61140 q^{75} -5.85337 q^{76} -0.0886138 q^{77} -3.98902 q^{78} -4.22642 q^{79} -7.89188 q^{80} -7.19153 q^{81} -13.2030 q^{82} -7.98133 q^{83} -28.5033 q^{84} +6.00551 q^{85} -20.8910 q^{86} +14.6366 q^{87} +0.131146 q^{88} +6.63813 q^{89} -1.56273 q^{90} -4.10699 q^{91} -28.1498 q^{92} -10.3060 q^{93} -19.4561 q^{94} +1.59593 q^{95} +7.06313 q^{96} +13.1193 q^{97} -24.9669 q^{98} +0.0111024 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) $ 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * 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 $

$2$	−2.53026	−1.78916	−0.894582	−	0.446904i	$-0.852527\pi$
$2$	−2.53026	−1.78916	−0.894582	+	0.446904i	$0.852527\pi$
$3$	−1.57653	−0.910208	−0.455104	−	0.890438i	$-0.650398\pi$
$3$	−1.57653	−0.910208	−0.455104	+	0.890438i	$0.650398\pi$
$4$	4.40222	2.20111
$5$	−1.20027	−0.536778	−0.268389	−	0.963311i	$-0.586491\pi$
$5$	−1.20027	−0.536778	−0.268389	+	0.963311i	$0.586491\pi$
$6$	3.98902	1.62851
$7$	4.10699	1.55229	0.776147	−	0.630552i	$-0.217171\pi$
$7$	4.10699	1.55229	0.776147	+	0.630552i	$0.217171\pi$
$8$	−6.07823	−2.14898
$9$	−0.514564	−0.171521
$10$	3.03700	0.960384
$11$	−0.0215764	−0.00650552	−0.00325276	−	0.999995i	$-0.501035\pi$
$11$	−0.0215764	−0.00650552	−0.00325276	+	0.999995i	$0.501035\pi$
$12$	−6.94021	−2.00347
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−10.3917	−2.77731
$15$	1.89226	0.488580
$16$	6.57507	1.64377
$17$	−5.00346	−1.21352	−0.606758	−	0.794887i	$-0.707530\pi$
$17$	−5.00346	−1.21352	−0.606758	+	0.794887i	$0.707530\pi$
$18$	1.30198	0.306880
$19$	−1.32964	−0.305041	−0.152520	−	0.988300i	$-0.548739\pi$
$19$	−1.32964	−0.305041	−0.152520	+	0.988300i	$0.548739\pi$
$20$	−5.28386	−1.18151
$21$	−6.47477	−1.41291
$22$	0.0545938	0.0116394
$23$	−6.39447	−1.33334	−0.666669	−	0.745354i	$-0.732281\pi$
$23$	−6.39447	−1.33334	−0.666669	+	0.745354i	$0.732281\pi$
$24$	9.58249	1.95602
$25$	−3.55935	−0.711869
$26$	2.53026	0.496225
$27$	5.54080	1.06633
$28$	18.0798	3.41677
$29$	−9.28410	−1.72401	−0.862007	−	0.506896i	$-0.830793\pi$
$29$	−9.28410	−1.72401	−0.862007	+	0.506896i	$0.830793\pi$
$30$	−4.78791	−0.874149
$31$	6.53715	1.17411	0.587054	−	0.809548i	$-0.300288\pi$
$31$	6.53715	1.17411	0.587054	+	0.809548i	$0.300288\pi$
$32$	−4.48019	−0.791992
$33$	0.0340157	0.00592137
$34$	12.6600	2.17118
$35$	−4.92950	−0.833238
$36$	−2.26522	−0.377537
$37$	7.40338	1.21711	0.608554	−	0.793513i	$-0.291750\pi$
$37$	7.40338	1.21711	0.608554	+	0.793513i	$0.291750\pi$
$38$	3.36434	0.545768
$39$	1.57653	0.252446
$40$	7.29554	1.15353
$41$	5.21804	0.814920	0.407460	−	0.913223i	$-0.366414\pi$
$41$	5.21804	0.814920	0.407460	+	0.913223i	$0.366414\pi$
$42$	16.3829	2.52793
$43$	8.25645	1.25910	0.629548	−	0.776961i	$-0.283240\pi$
$43$	8.25645	1.25910	0.629548	+	0.776961i	$0.283240\pi$
$44$	−0.0949838	−0.0143193
$45$	0.617618	0.0920690
$46$	16.1797	2.38556
$47$	7.68935	1.12161	0.560804	−	0.827949i	$-0.310492\pi$
$47$	7.68935	1.12161	0.560804	+	0.827949i	$0.310492\pi$
$48$	−10.3658	−1.49617
$49$	9.86733	1.40962
$50$	9.00607	1.27365
$51$	7.88808	1.10455
$52$	−4.40222	−0.610478
$53$	−5.22600	−0.717846	−0.358923	−	0.933367i	$-0.616856\pi$
$53$	−5.22600	−0.717846	−0.358923	+	0.933367i	$0.616856\pi$
$54$	−14.0197	−1.90784
$55$	0.0258975	0.00349202
$56$	−24.9632	−3.33585
$57$	2.09622	0.277650
$58$	23.4912	3.08454
$59$	6.86456	0.893689	0.446845	−	0.894612i	$-0.352548\pi$
$59$	6.86456	0.893689	0.446845	+	0.894612i	$0.352548\pi$
$60$	8.33014	1.07542
$61$	1.61811	0.207178	0.103589	−	0.994620i	$-0.466967\pi$
$61$	1.61811	0.207178	0.103589	+	0.994620i	$0.466967\pi$
$62$	−16.5407	−2.10067
$63$	−2.11331	−0.266252
$64$	−1.81412	−0.226764
$65$	1.20027	0.148876
$66$	−0.0860686	−0.0105943
$67$	−1.86179	−0.227454	−0.113727	−	0.993512i	$-0.536279\pi$
$67$	−1.86179	−0.227454	−0.113727	+	0.993512i	$0.536279\pi$
$68$	−22.0263	−2.67108
$69$	10.0810	1.21362
$70$	12.4729	1.49080
$71$	−5.39093	−0.639786	−0.319893	−	0.947454i	$-0.603647\pi$
$71$	−5.39093	−0.639786	−0.319893	+	0.947454i	$0.603647\pi$
$72$	3.12764	0.368596
$73$	2.16373	0.253245	0.126622	−	0.991951i	$-0.459586\pi$
$73$	2.16373	0.253245	0.126622	+	0.991951i	$0.459586\pi$
$74$	−18.7325	−2.17760
$75$	5.61140	0.647949
$76$	−5.85337	−0.671428
$77$	−0.0886138	−0.0100985
$78$	−3.98902	−0.451668
$79$	−4.22642	−0.475509	−0.237755	−	0.971325i	$-0.576411\pi$
$79$	−4.22642	−0.475509	−0.237755	+	0.971325i	$0.576411\pi$
$80$	−7.89188	−0.882339
$81$	−7.19153	−0.799059
$82$	−13.2030	−1.45803
$83$	−7.98133	−0.876065	−0.438032	−	0.898959i	$-0.644324\pi$
$83$	−7.98133	−0.876065	−0.438032	+	0.898959i	$0.644324\pi$
$84$	−28.5033	−3.10997
$85$	6.00551	0.651389
$86$	−20.8910	−2.25273
$87$	14.6366	1.56921
$88$	0.131146	0.0139802
$89$	6.63813	0.703640	0.351820	−	0.936068i	$-0.385563\pi$
$89$	6.63813	0.703640	0.351820	+	0.936068i	$0.385563\pi$
$90$	−1.56273	−0.164727
$91$	−4.10699	−0.430529
$92$	−28.1498	−2.93482
$93$	−10.3060	−1.06868
$94$	−19.4561	−2.00674
$95$	1.59593	0.163739
$96$	7.06313	0.720878
$97$	13.1193	1.33206	0.666032	−	0.745923i	$-0.267991\pi$
$97$	13.1193	1.33206	0.666032	+	0.745923i	$0.267991\pi$
$98$	−24.9669	−2.52204
$99$	0.0111024	0.00111584
$100$	−15.6690	−1.56690
$101$	15.5423	1.54652	0.773258	−	0.634092i	$-0.218626\pi$
$101$	15.5423	1.54652	0.773258	+	0.634092i	$0.218626\pi$
$102$	−19.9589	−1.97623
$103$	−9.48911	−0.934990	−0.467495	−	0.883996i	$-0.654843\pi$
$103$	−9.48911	−0.934990	−0.467495	+	0.883996i	$0.654843\pi$
$104$	6.07823	0.596020
$105$	7.77149	0.758420
$106$	13.2231	1.28434
$107$	−9.22072	−0.891400	−0.445700	−	0.895182i	$-0.647045\pi$
$107$	−9.22072	−0.891400	−0.445700	+	0.895182i	$0.647045\pi$
$108$	24.3918	2.34710
$109$	9.48828	0.908812	0.454406	−	0.890795i	$-0.349852\pi$
$109$	9.48828	0.908812	0.454406	+	0.890795i	$0.349852\pi$
$110$	−0.0655274	−0.00624780
$111$	−11.6716	−1.10782
$112$	27.0037	2.55161
$113$	7.09773	0.667698	0.333849	−	0.942627i	$-0.391652\pi$
$113$	7.09773	0.667698	0.333849	+	0.942627i	$0.391652\pi$
$114$	−5.30397	−0.496762
$115$	7.67510	0.715707
$116$	−40.8706	−3.79474
$117$	0.514564	0.0475715
$118$	−17.3691	−1.59896
$119$	−20.5491	−1.88373
$120$	−11.5016	−1.04995
$121$	−10.9995	−0.999958
$122$	−4.09425	−0.370676
$123$	−8.22637	−0.741747
$124$	28.7780	2.58434
$125$	10.2735	0.918894
$126$	5.34722	0.476368
$127$	−19.5186	−1.73200	−0.865999	−	0.500045i	$-0.833317\pi$
$127$	−19.5186	−1.73200	−0.865999	+	0.500045i	$0.833317\pi$
$128$	13.5506	1.19771
$129$	−13.0165	−1.14604
$130$	−3.03700	−0.266363
$131$	12.4012	1.08350	0.541749	−	0.840540i	$-0.317762\pi$
$131$	12.4012	1.08350	0.541749	+	0.840540i	$0.317762\pi$
$132$	0.149744	0.0130336
$133$	−5.46082	−0.473513
$134$	4.71081	0.406952
$135$	−6.65048	−0.572382
$136$	30.4122	2.60782
$137$	15.6093	1.33359	0.666797	−	0.745239i	$-0.267665\pi$
$137$	15.6093	1.33359	0.666797	+	0.745239i	$0.267665\pi$
$138$	−25.5077	−2.17136
$139$	16.1249	1.36770	0.683848	−	0.729624i	$-0.260305\pi$
$139$	16.1249	1.36770	0.683848	+	0.729624i	$0.260305\pi$
$140$	−21.7007	−1.83405
$141$	−12.1225	−1.02090
$142$	13.6405	1.14468
$143$	0.0215764	0.00180431
$144$	−3.38330	−0.281942
$145$	11.1435	0.925413
$146$	−5.47479	−0.453097
$147$	−15.5561	−1.28305
$148$	32.5913	2.67898
$149$	−1.53336	−0.125618	−0.0628088	−	0.998026i	$-0.520006\pi$
$149$	−1.53336	−0.125618	−0.0628088	+	0.998026i	$0.520006\pi$
$150$	−14.1983	−1.15929
$151$	−11.7558	−0.956675	−0.478338	−	0.878176i	$-0.658760\pi$
$151$	−11.7558	−0.956675	−0.478338	+	0.878176i	$0.658760\pi$
$152$	8.08187	0.655526
$153$	2.57460	0.208144
$154$	0.224216	0.0180678
$155$	−7.84637	−0.630235
$156$	6.94021	0.555662
$157$	−1.97978	−0.158004	−0.0790019	−	0.996874i	$-0.525173\pi$
$157$	−1.97978	−0.158004	−0.0790019	+	0.996874i	$0.525173\pi$
$158$	10.6939	0.850764
$159$	8.23892	0.653389
$160$	5.37744	0.425124
$161$	−26.2620	−2.06973
$162$	18.1964	1.42965
$163$	−13.1914	−1.03323	−0.516616	−	0.856217i	$-0.672808\pi$
$163$	−13.1914	−1.03323	−0.516616	+	0.856217i	$0.672808\pi$
$164$	22.9709	1.79373
$165$	−0.0408281	−0.00317846
$166$	20.1948	1.56742
$167$	−4.09386	−0.316793	−0.158396	−	0.987376i	$-0.550632\pi$
$167$	−4.09386	−0.316793	−0.158396	+	0.987376i	$0.550632\pi$
$168$	39.3552	3.03632
$169$	1.00000	0.0769231
$170$	−15.1955	−1.16544
$171$	0.684186	0.0523210
$172$	36.3467	2.77141
$173$	17.2203	1.30923	0.654617	−	0.755960i	$-0.272830\pi$
$173$	17.2203	1.30923	0.654617	+	0.755960i	$0.272830\pi$
$174$	−37.0345	−2.80758
$175$	−14.6182	−1.10503
$176$	−0.141866	−0.0106936
$177$	−10.8222	−0.813443
$178$	−16.7962	−1.25893
$179$	12.8494	0.960407	0.480204	−	0.877157i	$-0.340563\pi$
$179$	12.8494	0.960407	0.480204	+	0.877157i	$0.340563\pi$
$180$	2.71889	0.202654
$181$	−23.5542	−1.75077	−0.875385	−	0.483426i	$-0.839392\pi$
$181$	−23.5542	−1.75077	−0.875385	+	0.483426i	$0.839392\pi$
$182$	10.3917	0.770287
$183$	−2.55100	−0.188575
$184$	38.8671	2.86532
$185$	−8.88607	−0.653317
$186$	26.0769	1.91205
$187$	0.107956	0.00789455
$188$	33.8502	2.46878
$189$	22.7560	1.65526
$190$	−4.03812	−0.292956
$191$	−13.4261	−0.971475	−0.485738	−	0.874105i	$-0.661449\pi$
$191$	−13.4261	−0.971475	−0.485738	+	0.874105i	$0.661449\pi$
$192$	2.86000	0.206403
$193$	−3.11349	−0.224114	−0.112057	−	0.993702i	$-0.535744\pi$
$193$	−3.11349	−0.224114	−0.112057	+	0.993702i	$0.535744\pi$
$194$	−33.1953	−2.38328
$195$	−1.89226	−0.135508
$196$	43.4381	3.10272
$197$	−0.808148	−0.0575782	−0.0287891	−	0.999586i	$-0.509165\pi$
$197$	−0.808148	−0.0575782	−0.0287891	+	0.999586i	$0.509165\pi$
$198$	−0.0280920	−0.00199641
$199$	19.4651	1.37985	0.689924	−	0.723882i	$-0.257644\pi$
$199$	19.4651	1.37985	0.689924	+	0.723882i	$0.257644\pi$
$200$	21.6345	1.52979
$201$	2.93516	0.207030
$202$	−39.3260	−2.76697
$203$	−38.1297	−2.67618
$204$	34.7250	2.43124
$205$	−6.26307	−0.437432
$206$	24.0099	1.67285
$207$	3.29037	0.228696
$208$	−6.57507	−0.455899
$209$	0.0286888	0.00198445
$210$	−19.6639	−1.35694
$211$	1.79688	0.123702	0.0618510	−	0.998085i	$-0.480300\pi$
$211$	1.79688	0.123702	0.0618510	+	0.998085i	$0.480300\pi$
$212$	−23.0060	−1.58006
$213$	8.49894	0.582338
$214$	23.3308	1.59486
$215$	−9.90999	−0.675856
$216$	−33.6783	−2.29152
$217$	26.8480	1.82256
$218$	−24.0078	−1.62601
$219$	−3.41117	−0.230506
$220$	0.114006	0.00768631
$221$	5.00346	0.336569
$222$	29.5322	1.98207
$223$	−1.78221	−0.119345	−0.0596727	−	0.998218i	$-0.519006\pi$
$223$	−1.78221	−0.119345	−0.0596727	+	0.998218i	$0.519006\pi$
$224$	−18.4001	−1.22941
$225$	1.83151	0.122101
$226$	−17.9591	−1.19462
$227$	−28.9894	−1.92409	−0.962045	−	0.272889i	$-0.912021\pi$
$227$	−28.9894	−1.92409	−0.962045	+	0.272889i	$0.912021\pi$
$228$	9.22799	0.611139
$229$	−16.2040	−1.07079	−0.535396	−	0.844601i	$-0.679838\pi$
$229$	−16.2040	−1.07079	−0.535396	+	0.844601i	$0.679838\pi$
$230$	−19.4200	−1.28052
$231$	0.139702	0.00919171
$232$	56.4309	3.70487
$233$	−18.3672	−1.20327	−0.601637	−	0.798770i	$-0.705485\pi$
$233$	−18.3672	−1.20327	−0.601637	+	0.798770i	$0.705485\pi$
$234$	−1.30198	−0.0851132
$235$	−9.22932	−0.602054
$236$	30.2193	1.96711
$237$	6.66306	0.432812
$238$	51.9946	3.37031
$239$	9.28794	0.600787	0.300394	−	0.953815i	$-0.402882\pi$
$239$	9.28794	0.600787	0.300394	+	0.953815i	$0.402882\pi$
$240$	12.4418	0.803112
$241$	−13.9362	−0.897708	−0.448854	−	0.893605i	$-0.648168\pi$
$241$	−13.9362	−0.897708	−0.448854	+	0.893605i	$0.648168\pi$
$242$	27.8317	1.78909
$243$	−5.28477	−0.339018
$244$	7.12328	0.456021
$245$	−11.8435	−0.756653
$246$	20.8149	1.32711
$247$	1.32964	0.0846031
$248$	−39.7343	−2.52313
$249$	12.5828	0.797401
$250$	−25.9947	−1.64405
$251$	−19.5822	−1.23602	−0.618009	−	0.786171i	$-0.712060\pi$
$251$	−19.5822	−1.23602	−0.618009	+	0.786171i	$0.712060\pi$
$252$	−9.30324	−0.586049
$253$	0.137969	0.00867406
$254$	49.3872	3.09883
$255$	−9.46785	−0.592900
$256$	−30.6582	−1.91614
$257$	−10.1810	−0.635070	−0.317535	−	0.948247i	$-0.602855\pi$
$257$	−10.1810	−0.635070	−0.317535	+	0.948247i	$0.602855\pi$
$258$	32.9352	2.05045
$259$	30.4056	1.88931
$260$	5.28386	0.327691
$261$	4.77727	0.295705
$262$	−31.3783	−1.93856
$263$	30.7401	1.89551	0.947757	−	0.318993i	$-0.103345\pi$
$263$	30.7401	1.89551	0.947757	+	0.318993i	$0.103345\pi$
$264$	−0.206755	−0.0127249
$265$	6.27262	0.385324
$266$	13.8173	0.847193
$267$	−10.4652	−0.640459
$268$	−8.19599	−0.500650
$269$	15.7357	0.959425	0.479713	−	0.877426i	$-0.340741\pi$
$269$	15.7357	0.959425	0.479713	+	0.877426i	$0.340741\pi$
$270$	16.8274	1.02408
$271$	23.2916	1.41487	0.707433	−	0.706781i	$-0.249853\pi$
$271$	23.2916	1.41487	0.707433	+	0.706781i	$0.249853\pi$
$272$	−32.8981	−1.99474
$273$	6.47477	0.391871
$274$	−39.4956	−2.38602
$275$	0.0767977	0.00463108
$276$	44.3789	2.67130
$277$	−6.51093	−0.391204	−0.195602	−	0.980683i	$-0.562666\pi$
$277$	−6.51093	−0.391204	−0.195602	+	0.980683i	$0.562666\pi$
$278$	−40.8002	−2.44703
$279$	−3.36379	−0.201385
$280$	29.9627	1.79061
$281$	27.1050	1.61695	0.808476	−	0.588530i	$-0.200293\pi$
$281$	27.1050	1.61695	0.808476	+	0.588530i	$0.200293\pi$
$282$	30.6730	1.82655
$283$	−17.0672	−1.01454	−0.507269	−	0.861788i	$-0.669345\pi$
$283$	−17.0672	−1.01454	−0.507269	+	0.861788i	$0.669345\pi$
$284$	−23.7320	−1.40824
$285$	−2.51603	−0.149037
$286$	−0.0545938	−0.00322820
$287$	21.4304	1.26500
$288$	2.30534	0.135844
$289$	8.03457	0.472622
$290$	−28.1958	−1.65572
$291$	−20.6829	−1.21246
$292$	9.52519	0.557420
$293$	30.1727	1.76271	0.881353	−	0.472459i	$-0.156634\pi$
$293$	30.1727	1.76271	0.881353	+	0.472459i	$0.156634\pi$
$294$	39.3610	2.29558
$295$	−8.23934	−0.479713
$296$	−44.9994	−2.61554
$297$	−0.119550	−0.00693702
$298$	3.87979	0.224750
$299$	6.39447	0.369802
$300$	24.7026	1.42621
$301$	33.9091	1.95449
$302$	29.7453	1.71165
$303$	−24.5028	−1.40765
$304$	−8.74249	−0.501416
$305$	−1.94218	−0.111209
$306$	−6.51441	−0.372404
$307$	−4.89698	−0.279485	−0.139743	−	0.990188i	$-0.544628\pi$
$307$	−4.89698	−0.279485	−0.139743	+	0.990188i	$0.544628\pi$
$308$	−0.390097	−0.0222278
$309$	14.9598	0.851035
$310$	19.8534	1.12759
$311$	−12.1501	−0.688967	−0.344483	−	0.938792i	$-0.611946\pi$
$311$	−12.1501	−0.688967	−0.344483	+	0.938792i	$0.611946\pi$
$312$	−9.58249	−0.542502
$313$	16.7686	0.947819	0.473910	−	0.880574i	$-0.342842\pi$
$313$	16.7686	0.947819	0.473910	+	0.880574i	$0.342842\pi$
$314$	5.00936	0.282695
$315$	2.53655	0.142918
$316$	−18.6056	−1.04665
$317$	−31.7109	−1.78106	−0.890530	−	0.454924i	$-0.849666\pi$
$317$	−31.7109	−1.78106	−0.890530	+	0.454924i	$0.849666\pi$
$318$	−20.8466	−1.16902
$319$	0.200317	0.0112156
$320$	2.17743	0.121722
$321$	14.5367	0.811360
$322$	66.4496	3.70309
$323$	6.65280	0.370172
$324$	−31.6587	−1.75881
$325$	3.55935	0.197437
$326$	33.3777	1.84862
$327$	−14.9585	−0.827208
$328$	−31.7164	−1.75125
$329$	31.5801	1.74106
$330$	0.103306	0.00568679
$331$	−23.5583	−1.29488	−0.647440	−	0.762117i	$-0.724160\pi$
$331$	−23.5583	−1.29488	−0.647440	+	0.762117i	$0.724160\pi$
$332$	−35.1355	−1.92831
$333$	−3.80951	−0.208760
$334$	10.3585	0.566794
$335$	2.23465	0.122092
$336$	−42.5721	−2.32250
$337$	−34.3221	−1.86964	−0.934822	−	0.355116i	$-0.884441\pi$
$337$	−34.3221	−1.86964	−0.934822	+	0.355116i	$0.884441\pi$
$338$	−2.53026	−0.137628
$339$	−11.1898	−0.607744
$340$	26.4376	1.43378
$341$	−0.141048	−0.00763818
$342$	−1.73117	−0.0936109
$343$	11.7761	0.635849
$344$	−50.1846	−2.70577
$345$	−12.1000	−0.651442
$346$	−43.5718	−2.34244
$347$	−27.9141	−1.49851	−0.749254	−	0.662283i	$-0.769588\pi$
$347$	−27.9141	−1.49851	−0.749254	+	0.662283i	$0.769588\pi$
$348$	64.4336	3.45400
$349$	26.6149	1.42466	0.712332	−	0.701842i	$-0.247639\pi$
$349$	26.6149	1.42466	0.712332	+	0.701842i	$0.247639\pi$
$350$	36.9878	1.97708
$351$	−5.54080	−0.295746
$352$	0.0966661	0.00515232
$353$	4.30457	0.229109	0.114555	−	0.993417i	$-0.463456\pi$
$353$	4.30457	0.229109	0.114555	+	0.993417i	$0.463456\pi$
$354$	27.3829	1.45538
$355$	6.47059	0.343423
$356$	29.2225	1.54879
$357$	32.3962	1.71459
$358$	−32.5123	−1.71833
$359$	21.3646	1.12758	0.563790	−	0.825918i	$-0.309343\pi$
$359$	21.3646	1.12758	0.563790	+	0.825918i	$0.309343\pi$
$360$	−3.75402	−0.197854
$361$	−17.2321	−0.906950
$362$	59.5983	3.13242
$363$	17.3411	0.910169
$364$	−18.0798	−0.947641
$365$	−2.59706	−0.135936
$366$	6.45469	0.337392
$367$	28.5502	1.49031	0.745155	−	0.666892i	$-0.232376\pi$
$367$	28.5502	1.49031	0.745155	+	0.666892i	$0.232376\pi$
$368$	−42.0441	−2.19170
$369$	−2.68502	−0.139776
$370$	22.4841	1.16889
$371$	−21.4631	−1.11431
$372$	−45.3692	−2.35228
$373$	29.7115	1.53840	0.769201	−	0.639007i	$-0.220655\pi$
$373$	29.7115	1.53840	0.769201	+	0.639007i	$0.220655\pi$
$374$	−0.273158	−0.0141246
$375$	−16.1965	−0.836385
$376$	−46.7377	−2.41031
$377$	9.28410	0.478156
$378$	−57.5786	−2.96152
$379$	−21.5422	−1.10655	−0.553274	−	0.832999i	$-0.686622\pi$
$379$	−21.5422	−1.10655	−0.553274	+	0.832999i	$0.686622\pi$
$380$	7.02564	0.360408
$381$	30.7716	1.57648
$382$	33.9714	1.73813
$383$	26.5507	1.35668	0.678338	−	0.734750i	$-0.262701\pi$
$383$	26.5507	1.35668	0.678338	+	0.734750i	$0.262701\pi$
$384$	−21.3628	−1.09017
$385$	0.106361	0.00542064
$386$	7.87793	0.400976
$387$	−4.24847	−0.215962
$388$	57.7541	2.93202
$389$	−14.7253	−0.746602	−0.373301	−	0.927710i	$-0.621774\pi$
$389$	−14.7253	−0.746602	−0.373301	+	0.927710i	$0.621774\pi$
$390$	4.78791	0.242445
$391$	31.9944	1.61803
$392$	−59.9759	−3.02924
$393$	−19.5508	−0.986209
$394$	2.04482	0.103017
$395$	5.07285	0.255243
$396$	0.0488753	0.00245608
$397$	1.12757	0.0565910	0.0282955	−	0.999600i	$-0.490992\pi$
$397$	1.12757	0.0565910	0.0282955	+	0.999600i	$0.490992\pi$
$398$	−49.2519	−2.46877
$399$	8.60913	0.430995
$400$	−23.4030	−1.17015
$401$	−7.32981	−0.366033	−0.183017	−	0.983110i	$-0.558586\pi$
$401$	−7.32981	−0.366033	−0.183017	+	0.983110i	$0.558586\pi$
$402$	−7.42671	−0.370411
$403$	−6.53715	−0.325639
$404$	68.4205	3.40405
$405$	8.63180	0.428917
$406$	96.4780	4.78812
$407$	−0.159738	−0.00791791
$408$	−47.9456	−2.37366
$409$	−11.5893	−0.573056	−0.286528	−	0.958072i	$-0.592501\pi$
$409$	−11.5893	−0.573056	−0.286528	+	0.958072i	$0.592501\pi$
$410$	15.8472	0.782637
$411$	−24.6085	−1.21385
$412$	−41.7731	−2.05801
$413$	28.1926	1.38727
$414$	−8.32548	−0.409175
$415$	9.57977	0.470252
$416$	4.48019	0.219659
$417$	−25.4213	−1.24489
$418$	−0.0725902	−0.00355050
$419$	37.8938	1.85123	0.925616	−	0.378463i	$-0.123547\pi$
$419$	37.8938	1.85123	0.925616	+	0.378463i	$0.123547\pi$
$420$	34.2118	1.66936
$421$	4.85674	0.236703	0.118352	−	0.992972i	$-0.462239\pi$
$421$	4.85674	0.236703	0.118352	+	0.992972i	$0.462239\pi$
$422$	−4.54656	−0.221323
$423$	−3.95667	−0.192380
$424$	31.7648	1.54264
$425$	17.8090	0.863865
$426$	−21.5045	−1.04190
$427$	6.64557	0.321602
$428$	−40.5916	−1.96207
$429$	−0.0340157	−0.00164229
$430$	25.0749	1.20922
$431$	13.6030	0.655236	0.327618	−	0.944810i	$-0.393754\pi$
$431$	13.6030	0.655236	0.327618	+	0.944810i	$0.393754\pi$
$432$	36.4312	1.75280
$433$	−6.20636	−0.298259	−0.149129	−	0.988818i	$-0.547647\pi$
$433$	−6.20636	−0.298259	−0.149129	+	0.988818i	$0.547647\pi$
$434$	−67.9324	−3.26086
$435$	−17.5679	−0.842319
$436$	41.7694	2.00039
$437$	8.50235	0.406723
$438$	8.63115	0.412412
$439$	−2.15817	−0.103004	−0.0515018	−	0.998673i	$-0.516401\pi$
$439$	−2.15817	−0.103004	−0.0515018	+	0.998673i	$0.516401\pi$
$440$	−0.157411	−0.00750428
$441$	−5.07738	−0.241780
$442$	−12.6600	−0.602177
$443$	−12.5138	−0.594547	−0.297273	−	0.954792i	$-0.596077\pi$
$443$	−12.5138	−0.594547	−0.297273	+	0.954792i	$0.596077\pi$
$444$	−51.3810	−2.43843
$445$	−7.96756	−0.377699
$446$	4.50945	0.213529
$447$	2.41738	0.114338
$448$	−7.45054	−0.352005
$449$	36.4803	1.72161	0.860807	−	0.508931i	$-0.169959\pi$
$449$	36.4803	1.72161	0.860807	+	0.508931i	$0.169959\pi$
$450$	−4.63420	−0.218458
$451$	−0.112586	−0.00530148
$452$	31.2457	1.46968
$453$	18.5334	0.870773
$454$	73.3506	3.44251
$455$	4.92950	0.231099
$456$	−12.7413	−0.596665
$457$	−26.0889	−1.22039	−0.610193	−	0.792253i	$-0.708908\pi$
$457$	−26.0889	−1.22039	−0.610193	+	0.792253i	$0.708908\pi$
$458$	41.0004	1.91582
$459$	−27.7232	−1.29401
$460$	33.7875	1.57535
$461$	−32.1448	−1.49713	−0.748567	−	0.663059i	$-0.769258\pi$
$461$	−32.1448	−1.49713	−0.748567	+	0.663059i	$0.769258\pi$
$462$	−0.353482	−0.0164455
$463$	−21.8693	−1.01635	−0.508175	−	0.861254i	$-0.669680\pi$
$463$	−21.8693	−1.01635	−0.508175	+	0.861254i	$0.669680\pi$
$464$	−61.0437	−2.83388
$465$	12.3700	0.573645
$466$	46.4737	2.15285
$467$	−21.8783	−1.01241	−0.506203	−	0.862414i	$-0.668951\pi$
$467$	−21.8783	−1.01241	−0.506203	+	0.862414i	$0.668951\pi$
$468$	2.26522	0.104710
$469$	−7.64633	−0.353075
$470$	23.3526	1.07717
$471$	3.12118	0.143816
$472$	−41.7244	−1.92052
$473$	−0.178144	−0.00819107
$474$	−16.8593	−0.774372
$475$	4.73265	0.217149
$476$	−90.4617	−4.14630
$477$	2.68911	0.123126
$478$	−23.5009	−1.07491
$479$	−20.8747	−0.953788	−0.476894	−	0.878961i	$-0.658238\pi$
$479$	−20.8747	−0.953788	−0.476894	+	0.878961i	$0.658238\pi$
$480$	−8.47768	−0.386951
$481$	−7.40338	−0.337565
$482$	35.2622	1.60615
$483$	41.4027	1.88389
$484$	−48.4223	−2.20101
$485$	−15.7468	−0.715023
$486$	13.3719	0.606560
$487$	5.08040	0.230215	0.115107	−	0.993353i	$-0.463279\pi$
$487$	5.08040	0.230215	0.115107	+	0.993353i	$0.463279\pi$
$488$	−9.83527	−0.445222
$489$	20.7966	0.940455
$490$	29.9671	1.35378
$491$	24.7862	1.11859	0.559294	−	0.828970i	$-0.311072\pi$
$491$	24.7862	1.11859	0.559294	+	0.828970i	$0.311072\pi$
$492$	−36.2143	−1.63267
$493$	46.4526	2.09212
$494$	−3.36434	−0.151369
$495$	−0.0133259	−0.000598956 0
$496$	42.9823	1.92996
$497$	−22.1405	−0.993136
$498$	−31.8377	−1.42668
$499$	43.0815	1.92859	0.964296	−	0.264826i	$-0.0853145\pi$
$499$	43.0815	1.92859	0.964296	+	0.264826i	$0.0853145\pi$
$500$	45.2264	2.02259
$501$	6.45409	0.288347
$502$	49.5481	2.21144
$503$	−21.5611	−0.961362	−0.480681	−	0.876896i	$-0.659610\pi$
$503$	−21.5611	−0.961362	−0.480681	+	0.876896i	$0.659610\pi$
$504$	12.8452	0.572170
$505$	−18.6550	−0.830136
$506$	−0.349098	−0.0155193
$507$	−1.57653	−0.0700160
$508$	−85.9252	−3.81232
$509$	−18.6581	−0.827006	−0.413503	−	0.910503i	$-0.635695\pi$
$509$	−18.6581	−0.827006	−0.413503	+	0.910503i	$0.635695\pi$
$510$	23.9561	1.06079
$511$	8.88639	0.393111
$512$	50.4721	2.23057
$513$	−7.36728	−0.325274
$514$	25.7605	1.13624
$515$	11.3895	0.501882
$516$	−57.3015	−2.52256
$517$	−0.165908	−0.00729663
$518$	−76.9340	−3.38028
$519$	−27.1483	−1.19168
$520$	−7.29554	−0.319930
$521$	−26.1581	−1.14601	−0.573004	−	0.819553i	$-0.694222\pi$
$521$	−26.1581	−1.14601	−0.573004	+	0.819553i	$0.694222\pi$
$522$	−12.0877	−0.529066
$523$	8.17397	0.357423	0.178711	−	0.983902i	$-0.442807\pi$
$523$	8.17397	0.357423	0.178711	+	0.983902i	$0.442807\pi$
$524$	54.5928	2.38490
$525$	23.0459	1.00581
$526$	−77.7804	−3.39139
$527$	−32.7084	−1.42480
$528$	0.223656	0.00973337
$529$	17.8892	0.777792
$530$	−15.8714	−0.689408
$531$	−3.53226	−0.153287
$532$	−24.0397	−1.04225
$533$	−5.21804	−0.226018
$534$	26.4796	1.14589
$535$	11.0674	0.478484
$536$	11.3164	0.488793
$537$	−20.2574	−0.874170
$538$	−39.8155	−1.71657
$539$	−0.212901	−0.00917030
$540$	−29.2768	−1.25987
$541$	−6.33309	−0.272281	−0.136140	−	0.990690i	$-0.543470\pi$
$541$	−6.33309	−0.272281	−0.136140	+	0.990690i	$0.543470\pi$
$542$	−58.9339	−2.53143
$543$	37.1338	1.59357
$544$	22.4164	0.961096
$545$	−11.3885	−0.487831
$546$	−16.3829	−0.701121
$547$	−19.0205	−0.813257	−0.406629	−	0.913594i	$-0.633296\pi$
$547$	−19.0205	−0.813257	−0.406629	+	0.913594i	$0.633296\pi$
$548$	68.7156	2.93538
$549$	−0.832623	−0.0355355
$550$	−0.194318	−0.00828575
$551$	12.3445	0.525895
$552$	−61.2749	−2.60803
$553$	−17.3578	−0.738130
$554$	16.4743	0.699928
$555$	14.0091	0.594654
$556$	70.9853	3.01045
$557$	−13.1957	−0.559119	−0.279559	−	0.960128i	$-0.590188\pi$
$557$	−13.1957	−0.559119	−0.279559	+	0.960128i	$0.590188\pi$
$558$	8.51126	0.360310
$559$	−8.25645	−0.349211
$560$	−32.4118	−1.36965
$561$	−0.170196	−0.00718568
$562$	−68.5828	−2.89299
$563$	−29.9920	−1.26401	−0.632005	−	0.774964i	$-0.717768\pi$
$563$	−29.9920	−1.26401	−0.632005	+	0.774964i	$0.717768\pi$
$564$	−53.3657	−2.24710
$565$	−8.51921	−0.358406
$566$	43.1844	1.81518
$567$	−29.5355	−1.24037
$568$	32.7673	1.37489
$569$	−6.69659	−0.280735	−0.140368	−	0.990099i	$-0.544828\pi$
$569$	−6.69659	−0.280735	−0.140368	+	0.990099i	$0.544828\pi$
$570$	6.36621	0.266651
$571$	6.04386	0.252928	0.126464	−	0.991971i	$-0.459637\pi$
$571$	6.04386	0.252928	0.126464	+	0.991971i	$0.459637\pi$
$572$	0.0949838	0.00397147
$573$	21.1665	0.884244
$574$	−54.2245	−2.26329
$575$	22.7601	0.949163
$576$	0.933479	0.0388950
$577$	12.1914	0.507535	0.253767	−	0.967265i	$-0.418330\pi$
$577$	12.1914	0.507535	0.253767	+	0.967265i	$0.418330\pi$
$578$	−20.3295	−0.845598
$579$	4.90849	0.203990
$580$	49.0559	2.03693
$581$	−32.7792	−1.35991
$582$	52.3332	2.16928
$583$	0.112758	0.00466996
$584$	−13.1516	−0.544218
$585$	−0.617618	−0.0255353
$586$	−76.3447	−3.15377
$587$	−45.1080	−1.86180	−0.930902	−	0.365268i	$-0.880977\pi$
$587$	−45.1080	−1.86180	−0.930902	+	0.365268i	$0.880977\pi$
$588$	−68.4813	−2.82412
$589$	−8.69207	−0.358151
$590$	20.8477	0.858285
$591$	1.27407	0.0524081
$592$	48.6777	2.00064
$593$	−30.6167	−1.25728	−0.628639	−	0.777697i	$-0.716388\pi$
$593$	−30.6167	−1.25728	−0.628639	+	0.777697i	$0.716388\pi$
$594$	0.302493	0.0124115
$595$	24.6645	1.01115
$596$	−6.75017	−0.276498
$597$	−30.6873	−1.25595
$598$	−16.1797	−0.661636
$599$	−44.5337	−1.81960	−0.909800	−	0.415047i	$-0.863765\pi$
$599$	−44.5337	−1.81960	−0.909800	+	0.415047i	$0.863765\pi$
$600$	−34.1074	−1.39243
$601$	8.52235	0.347634	0.173817	−	0.984778i	$-0.444390\pi$
$601$	8.52235	0.347634	0.173817	+	0.984778i	$0.444390\pi$
$602$	−85.7989	−3.49690
$603$	0.958009	0.0390132
$604$	−51.7517	−2.10575
$605$	13.2024	0.536756
$606$	61.9985	2.51852
$607$	−15.1676	−0.615635	−0.307818	−	0.951445i	$-0.599599\pi$
$607$	−15.1676	−0.615635	−0.307818	+	0.951445i	$0.599599\pi$
$608$	5.95704	0.241590
$609$	60.1124	2.43588
$610$	4.91421	0.198971
$611$	−7.68935	−0.311078
$612$	11.3339	0.458148
$613$	−9.64184	−0.389430	−0.194715	−	0.980860i	$-0.562378\pi$
$613$	−9.64184	−0.389430	−0.194715	+	0.980860i	$0.562378\pi$
$614$	12.3906	0.500045
$615$	9.87389	0.398154
$616$	0.538615	0.0217014
$617$	−21.6550	−0.871796	−0.435898	−	0.899996i	$-0.643569\pi$
$617$	−21.6550	−0.871796	−0.435898	+	0.899996i	$0.643569\pi$
$618$	−37.8523	−1.52264
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	−34.5414	−1.38722
$621$	−35.4305	−1.42178
$622$	30.7428	1.23267
$623$	27.2627	1.09226
$624$	10.3658	0.414963
$625$	5.46567	0.218627
$626$	−42.4290	−1.69580
$627$	−0.0452287	−0.00180626
$628$	−8.71542	−0.347783
$629$	−37.0425	−1.47698
$630$	−6.41812	−0.255704
$631$	−10.7101	−0.426361	−0.213181	−	0.977013i	$-0.568382\pi$
$631$	−10.7101	−0.426361	−0.213181	+	0.977013i	$0.568382\pi$
$632$	25.6891	1.02186
$633$	−2.83282	−0.112595
$634$	80.2368	3.18661
$635$	23.4277	0.929699
$636$	36.2695	1.43818
$637$	−9.86733	−0.390958
$638$	−0.506854	−0.0200666
$639$	2.77398	0.109737
$640$	−16.2644	−0.642905
$641$	6.94812	0.274434	0.137217	−	0.990541i	$-0.456184\pi$
$641$	6.94812	0.274434	0.137217	+	0.990541i	$0.456184\pi$
$642$	−36.7816	−1.45166
$643$	−2.57038	−0.101366	−0.0506830	−	0.998715i	$-0.516140\pi$
$643$	−2.57038	−0.101366	−0.0506830	+	0.998715i	$0.516140\pi$
$644$	−115.611	−4.55571
$645$	15.6234	0.615169
$646$	−16.8333	−0.662298
$647$	−6.74814	−0.265297	−0.132648	−	0.991163i	$-0.542348\pi$
$647$	−6.74814	−0.265297	−0.132648	+	0.991163i	$0.542348\pi$
$648$	43.7118	1.71716
$649$	−0.148112	−0.00581391
$650$	−9.00607	−0.353247
$651$	−42.3266	−1.65891
$652$	−58.0715	−2.27425
$653$	−25.1054	−0.982449	−0.491224	−	0.871033i	$-0.663451\pi$
$653$	−25.1054	−0.982449	−0.491224	+	0.871033i	$0.663451\pi$
$654$	37.8489	1.48001
$655$	−14.8848	−0.581598
$656$	34.3090	1.33954
$657$	−1.11338	−0.0434370
$658$	−79.9058	−3.11505
$659$	40.7796	1.58855	0.794273	−	0.607561i	$-0.207852\pi$
$659$	40.7796	1.58855	0.794273	+	0.607561i	$0.207852\pi$
$660$	−0.179734	−0.00699614
$661$	40.1917	1.56328	0.781638	−	0.623733i	$-0.214385\pi$
$661$	40.1917	1.56328	0.781638	+	0.623733i	$0.214385\pi$
$662$	59.6086	2.31675
$663$	−7.88808	−0.306348
$664$	48.5124	1.88264
$665$	6.55447	0.254172
$666$	9.63906	0.373506
$667$	59.3669	2.29869
$668$	−18.0221	−0.697295
$669$	2.80970	0.108629
$670$	−5.65425	−0.218443
$671$	−0.0349130	−0.00134780
$672$	29.0082	1.11901
$673$	35.6010	1.37232	0.686159	−	0.727452i	$-0.259295\pi$
$673$	35.6010	1.37232	0.686159	+	0.727452i	$0.259295\pi$
$674$	86.8439	3.34510
$675$	−19.7216	−0.759086
$676$	4.40222	0.169316
$677$	35.6729	1.37102	0.685511	−	0.728062i	$-0.259579\pi$
$677$	35.6729	1.37102	0.685511	+	0.728062i	$0.259579\pi$
$678$	28.3130	1.08735
$679$	53.8808	2.06776
$680$	−36.5029	−1.39982
$681$	45.7025	1.75132
$682$	0.356888	0.0136659
$683$	1.84420	0.0705662	0.0352831	−	0.999377i	$-0.488767\pi$
$683$	1.84420	0.0705662	0.0352831	+	0.999377i	$0.488767\pi$
$684$	3.01194	0.115164
$685$	−18.7354	−0.715844
$686$	−29.7966	−1.13764
$687$	25.5461	0.974644
$688$	54.2868	2.06966
$689$	5.22600	0.199095
$690$	30.6162	1.16554
$691$	37.7973	1.43788	0.718938	−	0.695074i	$-0.244628\pi$
$691$	37.7973	1.43788	0.718938	+	0.695074i	$0.244628\pi$
$692$	75.8075	2.88177
$693$	0.0455975	0.00173211
$694$	70.6300	2.68108
$695$	−19.3543	−0.734150
$696$	−88.9648	−3.37220
$697$	−26.1082	−0.988919
$698$	−67.3427	−2.54896
$699$	28.9563	1.09523
$700$	−64.3524	−2.43229
$701$	−48.7355	−1.84071	−0.920357	−	0.391078i	$-0.872102\pi$
$701$	−48.7355	−1.84071	−0.920357	+	0.391078i	$0.872102\pi$
$702$	14.0197	0.529139
$703$	−9.84384	−0.371267
$704$	0.0391420	0.00147522
$705$	14.5503	0.547995
$706$	−10.8917	−0.409914
$707$	63.8320	2.40065
$708$	−47.6415	−1.79048
$709$	−36.5255	−1.37174	−0.685872	−	0.727722i	$-0.740579\pi$
$709$	−36.5255	−1.37174	−0.685872	+	0.727722i	$0.740579\pi$
$710$	−16.3723	−0.614440
$711$	2.17476	0.0815600
$712$	−40.3481	−1.51211
$713$	−41.8016	−1.56548
$714$	−81.9709	−3.06768
$715$	−0.0258975	−0.000968512 0
$716$	56.5657	2.11396
$717$	−14.6427	−0.546841
$718$	−54.0580	−2.01743
$719$	41.6752	1.55422	0.777112	−	0.629362i	$-0.216684\pi$
$719$	41.6752	1.55422	0.777112	+	0.629362i	$0.216684\pi$
$720$	4.06088	0.151340
$721$	−38.9716	−1.45138
$722$	43.6016	1.62268
$723$	21.9708	0.817101
$724$	−103.691	−3.85363
$725$	33.0453	1.22727
$726$	−43.8774	−1.62844
$727$	−43.2344	−1.60347	−0.801737	−	0.597677i	$-0.796091\pi$
$727$	−43.2344	−1.60347	−0.801737	+	0.597677i	$0.796091\pi$
$728$	24.9632	0.925198
$729$	29.9062	1.10764
$730$	6.57124	0.243213
$731$	−41.3108	−1.52793
$732$	−11.2300	−0.415074
$733$	0.152421	0.00562979	0.00281489	−	0.999996i	$-0.499104\pi$
$733$	0.152421	0.00562979	0.00281489	+	0.999996i	$0.499104\pi$
$734$	−72.2395	−2.66641
$735$	18.6716	0.688711
$736$	28.6484	1.05599
$737$	0.0401706	0.00147970
$738$	6.79379	0.250083
$739$	26.3949	0.970952	0.485476	−	0.874250i	$-0.338646\pi$
$739$	26.3949	0.970952	0.485476	+	0.874250i	$0.338646\pi$
$740$	−39.1184	−1.43802
$741$	−2.09622	−0.0770064
$742$	54.3072	1.99368
$743$	−16.2918	−0.597689	−0.298845	−	0.954302i	$-0.596601\pi$
$743$	−16.2918	−0.597689	−0.298845	+	0.954302i	$0.596601\pi$
$744$	62.6422	2.29658
$745$	1.84045	0.0674288
$746$	−75.1777	−2.75245
$747$	4.10691	0.150264
$748$	0.475247	0.0173768
$749$	−37.8693	−1.38372
$750$	40.9814	1.49643
$751$	−29.4059	−1.07304	−0.536518	−	0.843889i	$-0.680261\pi$
$751$	−29.4059	−1.07304	−0.536518	+	0.843889i	$0.680261\pi$
$752$	50.5581	1.84366
$753$	30.8719	1.12503
$754$	−23.4912	−0.855499
$755$	14.1102	0.513522
$756$	100.177	3.64340
$757$	−32.9147	−1.19631	−0.598153	−	0.801382i	$-0.704099\pi$
$757$	−32.9147	−1.19631	−0.598153	+	0.801382i	$0.704099\pi$
$758$	54.5074	1.97980
$759$	−0.217512	−0.00789519
$760$	−9.70045	−0.351872
$761$	−6.87999	−0.249399	−0.124700	−	0.992195i	$-0.539797\pi$
$761$	−6.87999	−0.249399	−0.124700	+	0.992195i	$0.539797\pi$
$762$	−77.8602	−2.82058
$763$	38.9682	1.41074
$764$	−59.1044	−2.13832
$765$	−3.09022	−0.111727
$766$	−67.1801	−2.42732
$767$	−6.86456	−0.247865
$768$	48.3335	1.74408
$769$	32.5288	1.17302	0.586510	−	0.809942i	$-0.300501\pi$
$769$	32.5288	1.17302	0.586510	+	0.809942i	$0.300501\pi$
$770$	−0.269120	−0.00969842
$771$	16.0505	0.578046
$772$	−13.7062	−0.493298
$773$	5.75741	0.207079	0.103540	−	0.994625i	$-0.466983\pi$
$773$	5.75741	0.207079	0.103540	+	0.994625i	$0.466983\pi$
$774$	10.7497	0.386392
$775$	−23.2680	−0.835811
$776$	−79.7422	−2.86258
$777$	−47.9352	−1.71966
$778$	37.2588	1.33579
$779$	−6.93812	−0.248584
$780$	−8.33014	−0.298267
$781$	0.116317	0.00416214
$782$	−80.9542	−2.89492
$783$	−51.4414	−1.83837
$784$	64.8784	2.31709
$785$	2.37628	0.0848130
$786$	49.4687	1.76449
$787$	15.1720	0.540823	0.270411	−	0.962745i	$-0.412840\pi$
$787$	15.1720	0.540823	0.270411	+	0.962745i	$0.412840\pi$
$788$	−3.55764	−0.126736
$789$	−48.4625	−1.72531
$790$	−12.8356	−0.456672
$791$	29.1503	1.03646
$792$	−0.0674831	−0.00239791
$793$	−1.61811	−0.0574609
$794$	−2.85304	−0.101251
$795$	−9.88895	−0.350725
$796$	85.6898	3.03719
$797$	5.35729	0.189765	0.0948826	−	0.995488i	$-0.469752\pi$
$797$	5.35729	0.189765	0.0948826	+	0.995488i	$0.469752\pi$
$798$	−21.7833	−0.771121
$799$	−38.4733	−1.36109
$800$	15.9465	0.563795
$801$	−3.41574	−0.120689
$802$	18.5463	0.654893
$803$	−0.0466853	−0.00164749
$804$	12.9212	0.455695
$805$	31.5215	1.11099
$806$	16.5407	0.582621
$807$	−24.8078	−0.873277
$808$	−94.4696	−3.32343
$809$	9.76284	0.343243	0.171622	−	0.985163i	$-0.445099\pi$
$809$	9.76284	0.343243	0.171622	+	0.985163i	$0.445099\pi$
$810$	−21.8407	−0.767404
$811$	9.18529	0.322539	0.161270	−	0.986910i	$-0.448441\pi$
$811$	9.18529	0.322539	0.161270	+	0.986910i	$0.448441\pi$
$812$	−167.855	−5.89056
$813$	−36.7199	−1.28782
$814$	0.404178	0.0141664
$815$	15.8333	0.554616
$816$	51.8647	1.81563
$817$	−10.9781	−0.384076
$818$	29.3240	1.02529
$819$	2.11331	0.0738450
$820$	−27.5714	−0.962834
$821$	−32.9896	−1.15134	−0.575672	−	0.817681i	$-0.695260\pi$
$821$	−32.9896	−1.15134	−0.575672	+	0.817681i	$0.695260\pi$
$822$	62.2659	2.17177
$823$	−2.19653	−0.0765662	−0.0382831	−	0.999267i	$-0.512189\pi$
$823$	−2.19653	−0.0765662	−0.0382831	+	0.999267i	$0.512189\pi$
$824$	57.6770	2.00927
$825$	−0.121074	−0.00421524
$826$	−71.3347	−2.48205
$827$	15.7387	0.547288	0.273644	−	0.961831i	$-0.411771\pi$
$827$	15.7387	0.547288	0.273644	+	0.961831i	$0.411771\pi$
$828$	14.4849	0.503385
$829$	33.5836	1.16641	0.583204	−	0.812326i	$-0.301799\pi$
$829$	33.5836	1.16641	0.583204	+	0.812326i	$0.301799\pi$
$830$	−24.2393	−0.841359
$831$	10.2647	0.356077
$832$	1.81412	0.0628931
$833$	−49.3708	−1.71060
$834$	64.3226	2.22731
$835$	4.91375	0.170047
$836$	0.126294	0.00436798
$837$	36.2211	1.25198
$838$	−95.8812	−3.31216
$839$	−47.0500	−1.62435	−0.812173	−	0.583417i	$-0.801715\pi$
$839$	−47.0500	−1.62435	−0.812173	+	0.583417i	$0.801715\pi$
$840$	−47.2369	−1.62983
$841$	57.1945	1.97223
$842$	−12.2888	−0.423501
$843$	−42.7318	−1.47176
$844$	7.91023	0.272282
$845$	−1.20027	−0.0412906
$846$	10.0114	0.344199
$847$	−45.1749	−1.55223
$848$	−34.3613	−1.17997
$849$	26.9069	0.923441
$850$	−45.0615	−1.54560
$851$	−47.3406	−1.62282
$852$	37.4142	1.28179
$853$	19.1359	0.655201	0.327600	−	0.944816i	$-0.393760\pi$
$853$	19.1359	0.655201	0.327600	+	0.944816i	$0.393760\pi$
$854$	−16.8150	−0.575398
$855$	−0.821210	−0.0280848
$856$	56.0456	1.91560
$857$	−33.9729	−1.16049	−0.580245	−	0.814442i	$-0.697043\pi$
$857$	−33.9729	−1.16049	−0.580245	+	0.814442i	$0.697043\pi$
$858$	0.0860686	0.00293833
$859$	14.1369	0.482345	0.241172	−	0.970482i	$-0.422468\pi$
$859$	14.1369	0.482345	0.241172	+	0.970482i	$0.422468\pi$
$860$	−43.6259	−1.48763
$861$	−33.7856	−1.15141
$862$	−34.4192	−1.17232
$863$	−3.02965	−0.103130	−0.0515652	−	0.998670i	$-0.516421\pi$
$863$	−3.02965	−0.103130	−0.0515652	+	0.998670i	$0.516421\pi$
$864$	−24.8238	−0.844524
$865$	−20.6691	−0.702769
$866$	15.7037	0.533634
$867$	−12.6667	−0.430184
$868$	118.191	4.01165
$869$	0.0911907	0.00309343
$870$	44.4515	1.50705
$871$	1.86179	0.0630843
$872$	−57.6719	−1.95302
$873$	−6.75073	−0.228478
$874$	−21.5132	−0.727693
$875$	42.1933	1.42639
$876$	−15.0167	−0.507368
$877$	14.7088	0.496681	0.248341	−	0.968673i	$-0.420115\pi$
$877$	14.7088	0.496681	0.248341	+	0.968673i	$0.420115\pi$
$878$	5.46072	0.184290
$879$	−47.5680	−1.60443
$880$	0.170278	0.00574007
$881$	−53.4277	−1.80002	−0.900012	−	0.435866i	$-0.856442\pi$
$881$	−53.4277	−1.80002	−0.900012	+	0.435866i	$0.856442\pi$
$882$	12.8471	0.432584
$883$	−39.3750	−1.32507	−0.662537	−	0.749029i	$-0.730520\pi$
$883$	−39.3750	−1.32507	−0.662537	+	0.749029i	$0.730520\pi$
$884$	22.0263	0.740824
$885$	12.9895	0.436639
$886$	31.6631	1.06374
$887$	−41.1711	−1.38239	−0.691195	−	0.722668i	$-0.742916\pi$
$887$	−41.1711	−1.38239	−0.691195	+	0.722668i	$0.742916\pi$
$888$	70.9428	2.38068
$889$	−80.1627	−2.68857
$890$	20.1600	0.675765
$891$	0.155167	0.00519829
$892$	−7.84566	−0.262692
$893$	−10.2241	−0.342136
$894$	−6.11660	−0.204570
$895$	−15.4228	−0.515526
$896$	55.6519	1.85920
$897$	−10.0810	−0.336596
$898$	−92.3048	−3.08025
$899$	−60.6916	−2.02418
$900$	8.06271	0.268757
$901$	26.1480	0.871118
$902$	0.284872	0.00948521
$903$	−53.4586	−1.77899
$904$	−43.1416	−1.43487
$905$	28.2715	0.939775
$906$	−46.8942	−1.55796
$907$	58.0474	1.92743	0.963716	−	0.266930i	$-0.0860093\pi$
$907$	58.0474	1.92743	0.963716	+	0.266930i	$0.0860093\pi$
$908$	−127.617	−4.23513
$909$	−7.99751	−0.265261
$910$	−12.4729	−0.413473
$911$	3.35040	0.111004	0.0555018	−	0.998459i	$-0.482324\pi$
$911$	3.35040	0.111004	0.0555018	+	0.998459i	$0.482324\pi$
$912$	13.7828	0.456393
$913$	0.172208	0.00569925
$914$	66.0116	2.18347
$915$	3.06189	0.101223
$916$	−71.3336	−2.35693
$917$	50.9316	1.68191
$918$	70.1468	2.31519
$919$	−26.7215	−0.881461	−0.440730	−	0.897640i	$-0.645281\pi$
$919$	−26.7215	−0.881461	−0.440730	+	0.897640i	$0.645281\pi$
$920$	−46.6511	−1.53804
$921$	7.72021	0.254390
$922$	81.3348	2.67862
$923$	5.39093	0.177445
$924$	0.614998	0.0202320
$925$	−26.3512	−0.866421
$926$	55.3349	1.81842
$927$	4.88276	0.160371
$928$	41.5945	1.36541
$929$	−38.1431	−1.25143	−0.625717	−	0.780050i	$-0.715194\pi$
$929$	−38.1431	−1.25143	−0.625717	+	0.780050i	$0.715194\pi$
$930$	−31.2993	−1.02635
$931$	−13.1200	−0.429991
$932$	−80.8563	−2.64854
$933$	19.1549	0.627103
$934$	55.3577	1.81136
$935$	−0.129577	−0.00423762
$936$	−3.12764	−0.102230
$937$	−45.3922	−1.48290	−0.741450	−	0.671008i	$-0.765862\pi$
$937$	−45.3922	−1.48290	−0.741450	+	0.671008i	$0.765862\pi$
$938$	19.3472	0.631709
$939$	−26.4362	−0.862713
$940$	−40.6295	−1.32519
$941$	18.6768	0.608846	0.304423	−	0.952537i	$-0.401536\pi$
$941$	18.6768	0.608846	0.304423	+	0.952537i	$0.401536\pi$
$942$	−7.89739	−0.257311
$943$	−33.3666	−1.08656
$944$	45.1350	1.46902
$945$	−27.3134	−0.888505
$946$	0.450751	0.0146552
$947$	34.1385	1.10935	0.554676	−	0.832066i	$-0.312842\pi$
$947$	34.1385	1.10935	0.554676	+	0.832066i	$0.312842\pi$
$948$	29.3322	0.952667
$949$	−2.16373	−0.0702375
$950$	−11.9748	−0.388515
$951$	49.9930	1.62114
$952$	124.902	4.04811
$953$	−16.7155	−0.541469	−0.270734	−	0.962654i	$-0.587267\pi$
$953$	−16.7155	−0.541469	−0.270734	+	0.962654i	$0.587267\pi$
$954$	−6.80415	−0.220293
$955$	16.1149	0.521467
$956$	40.8875	1.32240
$957$	−0.315805	−0.0102085
$958$	52.8184	1.70648
$959$	64.1072	2.07013
$960$	−3.43278	−0.110793
$961$	11.7344	0.378529
$962$	18.7325	0.603959
$963$	4.74465	0.152894
$964$	−61.3501	−1.97595
$965$	3.73703	0.120299
$966$	−104.760	−3.37059
$967$	−24.3577	−0.783290	−0.391645	−	0.920117i	$-0.628094\pi$
$967$	−24.3577	−0.783290	−0.391645	+	0.920117i	$0.628094\pi$
$968$	66.8577	2.14889
$969$	−10.4883	−0.336933
$970$	39.8434	1.27929
$971$	−30.7709	−0.987485	−0.493742	−	0.869608i	$-0.664371\pi$
$971$	−30.7709	−0.987485	−0.493742	+	0.869608i	$0.664371\pi$
$972$	−23.2647	−0.746216
$973$	66.2248	2.12307
$974$	−12.8547	−0.411892
$975$	−5.61140	−0.179709
$976$	10.6392	0.340553
$977$	51.8543	1.65897	0.829483	−	0.558532i	$-0.188635\pi$
$977$	51.8543	1.65897	0.829483	+	0.558532i	$0.188635\pi$
$978$	−52.6208	−1.68263
$979$	−0.143227	−0.00457754
$980$	−52.1376	−1.66547
$981$	−4.88233	−0.155881
$982$	−62.7156	−2.00134
$983$	20.1982	0.644222	0.322111	−	0.946702i	$-0.395608\pi$
$983$	20.1982	0.644222	0.322111	+	0.946702i	$0.395608\pi$
$984$	50.0018	1.59400
$985$	0.969998	0.0309067
$986$	−117.537	−3.74314
$987$	−49.7868	−1.58473
$988$	5.85337	0.186221
$989$	−52.7956	−1.67880
$990$	0.0337181	0.00107163
$991$	−19.9698	−0.634361	−0.317180	−	0.948365i	$-0.602736\pi$
$991$	−19.9698	−0.634361	−0.317180	+	0.948365i	$0.602736\pi$
$992$	−29.2877	−0.929884
$993$	37.1402	1.17861
$994$	56.0212	1.77688
$995$	−23.3635	−0.740672
$996$	55.3921	1.75517
$997$	20.9983	0.665023	0.332511	−	0.943099i	$-0.392104\pi$
$997$	20.9983	0.665023	0.332511	+	0.943099i	$0.392104\pi$
$998$	−109.007	−3.45057
$999$	41.0206	1.29784

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.c.1.13	✓	151

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.c.1.13	✓	151	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)

\(f(q)\)	\(=\)	\(q-2.53026 q^{2} -1.57653 q^{3} +4.40222 q^{4} -1.20027 q^{5} +3.98902 q^{6} +4.10699 q^{7} -6.07823 q^{8} -0.514564 q^{9} +O(q^{10})\)	\(q-2.53026 q^{2} -1.57653 q^{3} +4.40222 q^{4} -1.20027 q^{5} +3.98902 q^{6} +4.10699 q^{7} -6.07823 q^{8} -0.514564 q^{9} +3.03700 q^{10} -0.0215764 q^{11} -6.94021 q^{12} -1.00000 q^{13} -10.3917 q^{14} +1.89226 q^{15} +6.57507 q^{16} -5.00346 q^{17} +1.30198 q^{18} -1.32964 q^{19} -5.28386 q^{20} -6.47477 q^{21} +0.0545938 q^{22} -6.39447 q^{23} +9.58249 q^{24} -3.55935 q^{25} +2.53026 q^{26} +5.54080 q^{27} +18.0798 q^{28} -9.28410 q^{29} -4.78791 q^{30} +6.53715 q^{31} -4.48019 q^{32} +0.0340157 q^{33} +12.6600 q^{34} -4.92950 q^{35} -2.26522 q^{36} +7.40338 q^{37} +3.36434 q^{38} +1.57653 q^{39} +7.29554 q^{40} +5.21804 q^{41} +16.3829 q^{42} +8.25645 q^{43} -0.0949838 q^{44} +0.617618 q^{45} +16.1797 q^{46} +7.68935 q^{47} -10.3658 q^{48} +9.86733 q^{49} +9.00607 q^{50} +7.88808 q^{51} -4.40222 q^{52} -5.22600 q^{53} -14.0197 q^{54} +0.0258975 q^{55} -24.9632 q^{56} +2.09622 q^{57} +23.4912 q^{58} +6.86456 q^{59} +8.33014 q^{60} +1.61811 q^{61} -16.5407 q^{62} -2.11331 q^{63} -1.81412 q^{64} +1.20027 q^{65} -0.0860686 q^{66} -1.86179 q^{67} -22.0263 q^{68} +10.0810 q^{69} +12.4729 q^{70} -5.39093 q^{71} +3.12764 q^{72} +2.16373 q^{73} -18.7325 q^{74} +5.61140 q^{75} -5.85337 q^{76} -0.0886138 q^{77} -3.98902 q^{78} -4.22642 q^{79} -7.89188 q^{80} -7.19153 q^{81} -13.2030 q^{82} -7.98133 q^{83} -28.5033 q^{84} +6.00551 q^{85} -20.8910 q^{86} +14.6366 q^{87} +0.131146 q^{88} +6.63813 q^{89} -1.56273 q^{90} -4.10699 q^{91} -28.1498 q^{92} -10.3060 q^{93} -19.4561 q^{94} +1.59593 q^{95} +7.06313 q^{96} +13.1193 q^{97} -24.9669 q^{98} +0.0111024 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9}+O(q^{10}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9	\( 151 q - 13 q^{2} - 16 q^{3} + 151 q^{4} - 43 q^{5} - 17 q^{6} - 18 q^{7} - 39 q^{8} + 135 q^{9} - 3 q^{10} - 27 q^{11} - 52 q^{12} - 151 q^{13} - 9 q^{14} - 14 q^{15} + 143 q^{16} - 111 q^{17} - 37 q^{18} - 17 q^{19} - 107 q^{20} - 29 q^{21} - 16 q^{22} - 47 q^{23} - 46 q^{24} + 122 q^{25} + 13 q^{26} - 55 q^{27} - 44 q^{28} + 37 q^{29} - 14 q^{30} - 27 q^{31} - 86 q^{32} - 94 q^{33} - 10 q^{34} - 47 q^{35} + 124 q^{36} - 59 q^{37} - 80 q^{38} + 16 q^{39} + 5 q^{40} - 129 q^{41} - 77 q^{42} - 11 q^{43} - 99 q^{44} - 122 q^{45} - 17 q^{46} - 130 q^{47} - 111 q^{48} + 99 q^{49} - 72 q^{50} + 15 q^{51} - 151 q^{52} - 43 q^{53} - 49 q^{54} - 40 q^{55} - 50 q^{56} - 85 q^{57} - 73 q^{58} - 74 q^{59} - 43 q^{60} - 7 q^{61} - 110 q^{62} - 70 q^{63} + 141 q^{64} + 43 q^{65} - 16 q^{66} - 39 q^{67} - 222 q^{68} + 19 q^{69} - 52 q^{70} - 72 q^{71} - 106 q^{72} - 143 q^{73} + 20 q^{74} - 73 q^{75} - 88 q^{76} - 86 q^{77} + 17 q^{78} + 10 q^{79} - 239 q^{80} + 103 q^{81} - 96 q^{82} - 96 q^{83} - 75 q^{84} - 24 q^{85} - 109 q^{86} - 65 q^{87} - 45 q^{88} - 237 q^{89} - 79 q^{90} + 18 q^{91} - 153 q^{92} - 137 q^{93} - 23 q^{94} + 10 q^{95} - 109 q^{96} - 160 q^{97} - 119 q^{98} - 86 q^{99}+O(q^{100}) \) 151 * q - 13 * q^2 - 16 * q^3 + 151 * q^4 - 43 * q^5 - 17 * q^6 - 18 * q^7 - 39 * q^8 + 135 * q^9 - 3 * q^10 - 27 * q^11 - 52 * q^12 - 151 * q^13 - 9 * q^14 - 14 * q^15 + 143 * q^16 - 111 * q^17 - 37 * q^18 - 17 * q^19 - 107 * q^20 - 29 * q^21 - 16 * q^22 - 47 * q^23 - 46 * q^24 + 122 * q^25 + 13 * q^26 - 55 * q^27 - 44 * q^28 + 37 * q^29 - 14 * q^30 - 27 * q^31 - 86 * q^32 - 94 * q^33 - 10 * q^34 - 47 * q^35 + 124 * q^36 - 59 * q^37 - 80 * q^38 + 16 * q^39 + 5 * q^40 - 129 * q^41 - 77 * q^42 - 11 * q^43 - 99 * q^44 - 122 * q^45 - 17 * q^46 - 130 * q^47 - 111 * q^48 + 99 * q^49 - 72 * q^50 + 15 * q^51 - 151 * q^52 - 43 * q^53 - 49 * q^54 - 40 * q^55 - 50 * q^56 - 85 * q^57 - 73 * q^58 - 74 * q^59 - 43 * q^60 - 7 * q^61 - 110 * q^62 - 70 * q^63 + 141 * q^64 + 43 * q^65 - 16 * q^66 - 39 * q^67 - 222 * q^68 + 19 * q^69 - 52 * q^70 - 72 * q^71 - 106 * q^72 - 143 * q^73 + 20 * q^74 - 73 * q^75 - 88 * q^76 - 86 * q^77 + 17 * q^78 + 10 * q^79 - 239 * q^80 + 103 * q^81 - 96 * q^82 - 96 * q^83 - 75 * q^84 - 24 * q^85 - 109 * q^86 - 65 * q^87 - 45 * q^88 - 237 * q^89 - 79 * q^90 + 18 * q^91 - 153 * q^92 - 137 * q^93 - 23 * q^94 + 10 * q^95 - 109 * q^96 - 160 * q^97 - 119 * q^98 - 86 * q^99

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