LMFDB - Embedded newform 8047.2.a.b.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-2.43267 q^{2} -0.839960 q^{3} +3.91786 q^{4} -3.64522 q^{5} +2.04334 q^{6} -2.93716 q^{7} -4.66552 q^{8} -2.29447 q^{9} +O(q^{10})$	\(q-2.43267 q^{2} -0.839960 q^{3} +3.91786 q^{4} -3.64522 q^{5} +2.04334 q^{6} -2.93716 q^{7} -4.66552 q^{8} -2.29447 q^{9} +8.86761 q^{10} -1.95237 q^{11} -3.29085 q^{12} +1.00000 q^{13} +7.14513 q^{14} +3.06184 q^{15} +3.51393 q^{16} -6.09230 q^{17} +5.58167 q^{18} +6.19342 q^{19} -14.2815 q^{20} +2.46710 q^{21} +4.74947 q^{22} -5.35888 q^{23} +3.91885 q^{24} +8.28765 q^{25} -2.43267 q^{26} +4.44714 q^{27} -11.5074 q^{28} -3.08816 q^{29} -7.44844 q^{30} -2.60935 q^{31} +0.782822 q^{32} +1.63992 q^{33} +14.8205 q^{34} +10.7066 q^{35} -8.98941 q^{36} +0.220676 q^{37} -15.0665 q^{38} -0.839960 q^{39} +17.0069 q^{40} -9.64711 q^{41} -6.00162 q^{42} -11.3459 q^{43} -7.64914 q^{44} +8.36384 q^{45} +13.0364 q^{46} -0.488560 q^{47} -2.95156 q^{48} +1.62691 q^{49} -20.1611 q^{50} +5.11729 q^{51} +3.91786 q^{52} -2.57418 q^{53} -10.8184 q^{54} +7.11684 q^{55} +13.7034 q^{56} -5.20223 q^{57} +7.51246 q^{58} +0.288421 q^{59} +11.9959 q^{60} +11.8875 q^{61} +6.34767 q^{62} +6.73922 q^{63} -8.93221 q^{64} -3.64522 q^{65} -3.98937 q^{66} -4.74662 q^{67} -23.8688 q^{68} +4.50124 q^{69} -26.0456 q^{70} +4.41906 q^{71} +10.7049 q^{72} -10.0232 q^{73} -0.536831 q^{74} -6.96129 q^{75} +24.2650 q^{76} +5.73444 q^{77} +2.04334 q^{78} +4.45139 q^{79} -12.8091 q^{80} +3.14798 q^{81} +23.4682 q^{82} +7.46556 q^{83} +9.66575 q^{84} +22.2078 q^{85} +27.6008 q^{86} +2.59393 q^{87} +9.10885 q^{88} +7.26752 q^{89} -20.3464 q^{90} -2.93716 q^{91} -20.9954 q^{92} +2.19175 q^{93} +1.18850 q^{94} -22.5764 q^{95} -0.657539 q^{96} +12.9408 q^{97} -3.95773 q^{98} +4.47966 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.43267	−1.72015	−0.860077	−	0.510164i	$-0.829585\pi$
$2$	−2.43267	−1.72015	−0.860077	+	0.510164i	$0.829585\pi$
$3$	−0.839960	−0.484951	−0.242476	−	0.970158i	$-0.577959\pi$
$3$	−0.839960	−0.484951	−0.242476	+	0.970158i	$0.577959\pi$
$4$	3.91786	1.95893
$5$	−3.64522	−1.63019	−0.815097	−	0.579325i	$-0.803316\pi$
$5$	−3.64522	−1.63019	−0.815097	+	0.579325i	$0.803316\pi$
$6$	2.04334	0.834191
$7$	−2.93716	−1.11014	−0.555071	−	0.831803i	$-0.687309\pi$
$7$	−2.93716	−1.11014	−0.555071	+	0.831803i	$0.687309\pi$
$8$	−4.66552	−1.64951
$9$	−2.29447	−0.764822
$10$	8.86761	2.80418
$11$	−1.95237	−0.588663	−0.294331	−	0.955703i	$-0.595097\pi$
$11$	−1.95237	−0.588663	−0.294331	+	0.955703i	$0.595097\pi$
$12$	−3.29085	−0.949986
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	7.14513	1.90962
$15$	3.06184	0.790564
$16$	3.51393	0.878483
$17$	−6.09230	−1.47760	−0.738800	−	0.673925i	$-0.764607\pi$
$17$	−6.09230	−1.47760	−0.738800	+	0.673925i	$0.764607\pi$
$18$	5.58167	1.31561
$19$	6.19342	1.42087	0.710434	−	0.703763i	$-0.248498\pi$
$19$	6.19342	1.42087	0.710434	+	0.703763i	$0.248498\pi$
$20$	−14.2815	−3.19344
$21$	2.46710	0.538365
$22$	4.74947	1.01259
$23$	−5.35888	−1.11740	−0.558702	−	0.829369i	$-0.688700\pi$
$23$	−5.35888	−1.11740	−0.558702	+	0.829369i	$0.688700\pi$
$24$	3.91885	0.799933
$25$	8.28765	1.65753
$26$	−2.43267	−0.477085
$27$	4.44714	0.855853
$28$	−11.5074	−2.17469
$29$	−3.08816	−0.573456	−0.286728	−	0.958012i	$-0.592568\pi$
$29$	−3.08816	−0.573456	−0.286728	+	0.958012i	$0.592568\pi$
$30$	−7.44844	−1.35989
$31$	−2.60935	−0.468652	−0.234326	−	0.972158i	$-0.575288\pi$
$31$	−2.60935	−0.468652	−0.234326	+	0.972158i	$0.575288\pi$
$32$	0.782822	0.138385
$33$	1.63992	0.285473
$34$	14.8205	2.54170
$35$	10.7066	1.80975
$36$	−8.98941	−1.49824
$37$	0.220676	0.0362789	0.0181395	−	0.999835i	$-0.494226\pi$
$37$	0.220676	0.0362789	0.0181395	+	0.999835i	$0.494226\pi$
$38$	−15.0665	−2.44411
$39$	−0.839960	−0.134501
$40$	17.0069	2.68902
$41$	−9.64711	−1.50663	−0.753313	−	0.657663i	$-0.771545\pi$
$41$	−9.64711	−1.50663	−0.753313	+	0.657663i	$0.771545\pi$
$42$	−6.00162	−0.926071
$43$	−11.3459	−1.73023	−0.865117	−	0.501570i	$-0.832756\pi$
$43$	−11.3459	−1.73023	−0.865117	+	0.501570i	$0.832756\pi$
$44$	−7.64914	−1.15315
$45$	8.36384	1.24681
$46$	13.0364	1.92211
$47$	−0.488560	−0.0712638	−0.0356319	−	0.999365i	$-0.511344\pi$
$47$	−0.488560	−0.0712638	−0.0356319	+	0.999365i	$0.511344\pi$
$48$	−2.95156	−0.426022
$49$	1.62691	0.232416
$50$	−20.1611	−2.85121
$51$	5.11729	0.716564
$52$	3.91786	0.543310
$53$	−2.57418	−0.353591	−0.176795	−	0.984248i	$-0.556573\pi$
$53$	−2.57418	−0.353591	−0.176795	+	0.984248i	$0.556573\pi$
$54$	−10.8184	−1.47220
$55$	7.11684	0.959634
$56$	13.7034	1.83119
$57$	−5.20223	−0.689052
$58$	7.51246	0.986434
$59$	0.288421	0.0375493	0.0187746	−	0.999824i	$-0.494023\pi$
$59$	0.288421	0.0375493	0.0187746	+	0.999824i	$0.494023\pi$
$60$	11.9959	1.54866
$61$	11.8875	1.52204	0.761021	−	0.648728i	$-0.224699\pi$
$61$	11.8875	1.52204	0.761021	+	0.648728i	$0.224699\pi$
$62$	6.34767	0.806154
$63$	6.73922	0.849062
$64$	−8.93221	−1.11653
$65$	−3.64522	−0.452134
$66$	−3.98937	−0.491057
$67$	−4.74662	−0.579892	−0.289946	−	0.957043i	$-0.593637\pi$
$67$	−4.74662	−0.579892	−0.289946	+	0.957043i	$0.593637\pi$
$68$	−23.8688	−2.89452
$69$	4.50124	0.541886
$70$	−26.0456	−3.11304
$71$	4.41906	0.524446	0.262223	−	0.965007i	$-0.415544\pi$
$71$	4.41906	0.524446	0.262223	+	0.965007i	$0.415544\pi$
$72$	10.7049	1.26158
$73$	−10.0232	−1.17313	−0.586564	−	0.809903i	$-0.699520\pi$
$73$	−10.0232	−1.17313	−0.586564	+	0.809903i	$0.699520\pi$
$74$	−0.536831	−0.0624053
$75$	−6.96129	−0.803821
$76$	24.2650	2.78339
$77$	5.73444	0.653500
$78$	2.04334	0.231363
$79$	4.45139	0.500821	0.250410	−	0.968140i	$-0.419434\pi$
$79$	4.45139	0.500821	0.250410	+	0.968140i	$0.419434\pi$
$80$	−12.8091	−1.43210
$81$	3.14798	0.349776
$82$	23.4682	2.59163
$83$	7.46556	0.819452	0.409726	−	0.912209i	$-0.365624\pi$
$83$	7.46556	0.819452	0.409726	+	0.912209i	$0.365624\pi$
$84$	9.66575	1.05462
$85$	22.2078	2.40877
$86$	27.6008	2.97627
$87$	2.59393	0.278098
$88$	9.10885	0.971006
$89$	7.26752	0.770356	0.385178	−	0.922842i	$-0.374140\pi$
$89$	7.26752	0.770356	0.385178	+	0.922842i	$0.374140\pi$
$90$	−20.3464	−2.14470
$91$	−2.93716	−0.307898
$92$	−20.9954	−2.18892
$93$	2.19175	0.227273
$94$	1.18850	0.122585
$95$	−22.5764	−2.31629
$96$	−0.657539	−0.0671098
$97$	12.9408	1.31393	0.656967	−	0.753919i	$-0.271839\pi$
$97$	12.9408	1.31393	0.656967	+	0.753919i	$0.271839\pi$
$98$	−3.95773	−0.399791
$99$	4.47966	0.450223
$100$	32.4699	3.24699
$101$	−9.44266	−0.939580	−0.469790	−	0.882778i	$-0.655670\pi$
$101$	−9.44266	−0.939580	−0.469790	+	0.882778i	$0.655670\pi$
$102$	−12.4487	−1.23260
$103$	5.51149	0.543063	0.271532	−	0.962429i	$-0.412470\pi$
$103$	5.51149	0.543063	0.271532	+	0.962429i	$0.412470\pi$
$104$	−4.66552	−0.457492
$105$	−8.99312	−0.877639
$106$	6.26212	0.608230
$107$	−13.8940	−1.34318	−0.671591	−	0.740922i	$-0.734389\pi$
$107$	−13.8940	−1.34318	−0.671591	+	0.740922i	$0.734389\pi$
$108$	17.4233	1.67656
$109$	10.8139	1.03578	0.517892	−	0.855446i	$-0.326717\pi$
$109$	10.8139	1.03578	0.517892	+	0.855446i	$0.326717\pi$
$110$	−17.3129	−1.65072
$111$	−0.185359	−0.0175935
$112$	−10.3210	−0.975241
$113$	9.00074	0.846719	0.423359	−	0.905962i	$-0.360851\pi$
$113$	9.00074	0.846719	0.423359	+	0.905962i	$0.360851\pi$
$114$	12.6553	1.18528
$115$	19.5343	1.82158
$116$	−12.0990	−1.12336
$117$	−2.29447	−0.212124
$118$	−0.701633	−0.0645905
$119$	17.8941	1.64035
$120$	−14.2851	−1.30404
$121$	−7.18824	−0.653476
$122$	−28.9184	−2.61815
$123$	8.10319	0.730640
$124$	−10.2231	−0.918058
$125$	−11.9842	−1.07190
$126$	−16.3943	−1.46052
$127$	−0.678367	−0.0601954	−0.0300977	−	0.999547i	$-0.509582\pi$
$127$	−0.678367	−0.0601954	−0.0300977	+	0.999547i	$0.509582\pi$
$128$	20.1634	1.78221
$129$	9.53011	0.839079
$130$	8.86761	0.777741
$131$	11.6202	1.01526	0.507629	−	0.861576i	$-0.330522\pi$
$131$	11.6202	1.01526	0.507629	+	0.861576i	$0.330522\pi$
$132$	6.42497	0.559222
$133$	−18.1911	−1.57737
$134$	11.5469	0.997504
$135$	−16.2108	−1.39521
$136$	28.4238	2.43732
$137$	12.8243	1.09566	0.547828	−	0.836591i	$-0.315455\pi$
$137$	12.8243	1.09566	0.547828	+	0.836591i	$0.315455\pi$
$138$	−10.9500	−0.932128
$139$	3.89399	0.330284	0.165142	−	0.986270i	$-0.447192\pi$
$139$	3.89399	0.330284	0.165142	+	0.986270i	$0.447192\pi$
$140$	41.9470	3.54517
$141$	0.410371	0.0345595
$142$	−10.7501	−0.902128
$143$	−1.95237	−0.163266
$144$	−8.06260	−0.671884
$145$	11.2570	0.934845
$146$	24.3831	2.01796
$147$	−1.36654	−0.112710
$148$	0.864578	0.0710679
$149$	9.41585	0.771376	0.385688	−	0.922629i	$-0.373964\pi$
$149$	9.41585	0.771376	0.385688	+	0.922629i	$0.373964\pi$
$150$	16.9345	1.38270
$151$	22.2475	1.81048	0.905238	−	0.424906i	$-0.139693\pi$
$151$	22.2475	1.81048	0.905238	+	0.424906i	$0.139693\pi$
$152$	−28.8956	−2.34374
$153$	13.9786	1.13010
$154$	−13.9500	−1.12412
$155$	9.51165	0.763994
$156$	−3.29085	−0.263479
$157$	7.35290	0.586826	0.293413	−	0.955986i	$-0.405209\pi$
$157$	7.35290	0.586826	0.293413	+	0.955986i	$0.405209\pi$
$158$	−10.8287	−0.861489
$159$	2.16221	0.171474
$160$	−2.85356	−0.225594
$161$	15.7399	1.24048
$162$	−7.65799	−0.601668
$163$	−14.3471	−1.12375	−0.561877	−	0.827221i	$-0.689921\pi$
$163$	−14.3471	−1.12375	−0.561877	+	0.827221i	$0.689921\pi$
$164$	−37.7961	−2.95138
$165$	−5.97786	−0.465376
$166$	−18.1612	−1.40958
$167$	23.6284	1.82842	0.914210	−	0.405242i	$-0.132813\pi$
$167$	23.6284	1.82842	0.914210	+	0.405242i	$0.132813\pi$
$168$	−11.5103	−0.888039
$169$	1.00000	0.0769231
$170$	−54.0241	−4.14346
$171$	−14.2106	−1.08671
$172$	−44.4517	−3.38941
$173$	−10.0254	−0.762219	−0.381109	−	0.924530i	$-0.624458\pi$
$173$	−10.0254	−0.762219	−0.381109	+	0.924530i	$0.624458\pi$
$174$	−6.31016	−0.478372
$175$	−24.3422	−1.84009
$176$	−6.86051	−0.517131
$177$	−0.242262	−0.0182096
$178$	−17.6795	−1.32513
$179$	19.1672	1.43262	0.716310	−	0.697782i	$-0.245830\pi$
$179$	19.1672	1.43262	0.716310	+	0.697782i	$0.245830\pi$
$180$	32.7684	2.44241
$181$	4.06058	0.301820	0.150910	−	0.988547i	$-0.451780\pi$
$181$	4.06058	0.301820	0.150910	+	0.988547i	$0.451780\pi$
$182$	7.14513	0.529632
$183$	−9.98504	−0.738116
$184$	25.0020	1.84317
$185$	−0.804413	−0.0591416
$186$	−5.33179	−0.390946
$187$	11.8944	0.869808
$188$	−1.91411	−0.139601
$189$	−13.0620	−0.950118
$190$	54.9209	3.98438
$191$	−14.5472	−1.05260	−0.526300	−	0.850299i	$-0.676421\pi$
$191$	−14.5472	−1.05260	−0.526300	+	0.850299i	$0.676421\pi$
$192$	7.50270	0.541461
$193$	15.2869	1.10038	0.550189	−	0.835040i	$-0.314556\pi$
$193$	15.2869	1.10038	0.550189	+	0.835040i	$0.314556\pi$
$194$	−31.4805	−2.26017
$195$	3.06184	0.219263
$196$	6.37402	0.455287
$197$	−2.35092	−0.167496	−0.0837482	−	0.996487i	$-0.526689\pi$
$197$	−2.35092	−0.167496	−0.0837482	+	0.996487i	$0.526689\pi$
$198$	−10.8975	−0.774452
$199$	−4.48196	−0.317718	−0.158859	−	0.987301i	$-0.550782\pi$
$199$	−4.48196	−0.317718	−0.158859	+	0.987301i	$0.550782\pi$
$200$	−38.6662	−2.73411
$201$	3.98697	0.281219
$202$	22.9708	1.61622
$203$	9.07041	0.636618
$204$	20.0488	1.40370
$205$	35.1659	2.45609
$206$	−13.4076	−0.934153
$207$	12.2958	0.854615
$208$	3.51393	0.243647
$209$	−12.0919	−0.836413
$210$	21.8773	1.50967
$211$	0.549258	0.0378125	0.0189062	−	0.999821i	$-0.493982\pi$
$211$	0.549258	0.0378125	0.0189062	+	0.999821i	$0.493982\pi$
$212$	−10.0853	−0.692660
$213$	−3.71183	−0.254331
$214$	33.7994	2.31048
$215$	41.3584	2.82062
$216$	−20.7482	−1.41174
$217$	7.66407	0.520271
$218$	−26.3066	−1.78171
$219$	8.41910	0.568910
$220$	27.8828	1.87986
$221$	−6.09230	−0.409812
$222$	0.450917	0.0302635
$223$	0.0405410	0.00271482	0.00135741	−	0.999999i	$-0.499568\pi$
$223$	0.0405410	0.00271482	0.00135741	+	0.999999i	$0.499568\pi$
$224$	−2.29927	−0.153627
$225$	−19.0157	−1.26772
$226$	−21.8958	−1.45649
$227$	−4.64935	−0.308588	−0.154294	−	0.988025i	$-0.549310\pi$
$227$	−4.64935	−0.308588	−0.154294	+	0.988025i	$0.549310\pi$
$228$	−20.3816	−1.34981
$229$	−18.9317	−1.25104	−0.625520	−	0.780209i	$-0.715113\pi$
$229$	−18.9317	−1.25104	−0.625520	+	0.780209i	$0.715113\pi$
$230$	−47.5204	−3.13341
$231$	−4.81670	−0.316915
$232$	14.4079	0.945923
$233$	16.0613	1.05221	0.526104	−	0.850420i	$-0.323652\pi$
$233$	16.0613	1.05221	0.526104	+	0.850420i	$0.323652\pi$
$234$	5.58167	0.364885
$235$	1.78091	0.116174
$236$	1.13000	0.0735565
$237$	−3.73899	−0.242874
$238$	−43.5303	−2.82165
$239$	−21.3159	−1.37881	−0.689405	−	0.724376i	$-0.742128\pi$
$239$	−21.3159	−1.37881	−0.689405	+	0.724376i	$0.742128\pi$
$240$	10.7591	0.694497
$241$	20.4739	1.31884	0.659419	−	0.751776i	$-0.270803\pi$
$241$	20.4739	1.31884	0.659419	+	0.751776i	$0.270803\pi$
$242$	17.4866	1.12408
$243$	−15.9856	−1.02548
$244$	46.5737	2.98158
$245$	−5.93045	−0.378883
$246$	−19.7123	−1.25681
$247$	6.19342	0.394078
$248$	12.1740	0.773047
$249$	−6.27077	−0.397394
$250$	29.1536	1.84383
$251$	−9.23733	−0.583055	−0.291527	−	0.956562i	$-0.594163\pi$
$251$	−9.23733	−0.583055	−0.291527	+	0.956562i	$0.594163\pi$
$252$	26.4033	1.66325
$253$	10.4625	0.657774
$254$	1.65024	0.103545
$255$	−18.6537	−1.16814
$256$	−31.1865	−1.94916
$257$	−6.66713	−0.415884	−0.207942	−	0.978141i	$-0.566677\pi$
$257$	−6.66713	−0.415884	−0.207942	+	0.978141i	$0.566677\pi$
$258$	−23.1836	−1.44335
$259$	−0.648161	−0.0402747
$260$	−14.2815	−0.885700
$261$	7.08567	0.438592
$262$	−28.2680	−1.74640
$263$	0.311831	0.0192283	0.00961417	−	0.999954i	$-0.496940\pi$
$263$	0.311831	0.0192283	0.00961417	+	0.999954i	$0.496940\pi$
$264$	−7.65107	−0.470891
$265$	9.38345	0.576421
$266$	44.2528	2.71331
$267$	−6.10443	−0.373585
$268$	−18.5966	−1.13597
$269$	8.93944	0.545047	0.272524	−	0.962149i	$-0.412142\pi$
$269$	8.93944	0.545047	0.272524	+	0.962149i	$0.412142\pi$
$270$	39.4355	2.39997
$271$	1.73849	0.105606	0.0528029	−	0.998605i	$-0.483184\pi$
$271$	1.73849	0.105606	0.0528029	+	0.998605i	$0.483184\pi$
$272$	−21.4079	−1.29805
$273$	2.46710	0.149316
$274$	−31.1973	−1.88470
$275$	−16.1806	−0.975726
$276$	17.6353	1.06152
$277$	−17.5446	−1.05415	−0.527077	−	0.849818i	$-0.676712\pi$
$277$	−17.5446	−1.05415	−0.527077	+	0.849818i	$0.676712\pi$
$278$	−9.47277	−0.568139
$279$	5.98706	0.358436
$280$	−49.9519	−2.98520
$281$	14.6425	0.873496	0.436748	−	0.899584i	$-0.356130\pi$
$281$	14.6425	0.873496	0.436748	+	0.899584i	$0.356130\pi$
$282$	−0.998295	−0.0594476
$283$	−19.9903	−1.18830	−0.594150	−	0.804355i	$-0.702511\pi$
$283$	−19.9903	−1.18830	−0.594150	+	0.804355i	$0.702511\pi$
$284$	17.3133	1.02735
$285$	18.9633	1.12329
$286$	4.74947	0.280842
$287$	28.3351	1.67257
$288$	−1.79616	−0.105840
$289$	20.1161	1.18330
$290$	−27.3846	−1.60808
$291$	−10.8697	−0.637194
$292$	−39.2696	−2.29808
$293$	−30.9640	−1.80894	−0.904469	−	0.426540i	$-0.859732\pi$
$293$	−30.9640	−1.80894	−0.904469	+	0.426540i	$0.859732\pi$
$294$	3.32434	0.193879
$295$	−1.05136	−0.0612125
$296$	−1.02957	−0.0598425
$297$	−8.68248	−0.503809
$298$	−22.9056	−1.32689
$299$	−5.35888	−0.309912
$300$	−27.2734	−1.57463
$301$	33.3247	1.92081
$302$	−54.1207	−3.11430
$303$	7.93146	0.455650
$304$	21.7633	1.24821
$305$	−43.3327	−2.48122
$306$	−34.0052	−1.94395
$307$	21.9702	1.25390	0.626952	−	0.779058i	$-0.284302\pi$
$307$	21.9702	1.25390	0.626952	+	0.779058i	$0.284302\pi$
$308$	22.4667	1.28016
$309$	−4.62943	−0.263359
$310$	−23.1387	−1.31419
$311$	5.77549	0.327498	0.163749	−	0.986502i	$-0.447641\pi$
$311$	5.77549	0.327498	0.163749	+	0.986502i	$0.447641\pi$
$312$	3.91885	0.221861
$313$	−21.5018	−1.21535	−0.607675	−	0.794185i	$-0.707898\pi$
$313$	−21.5018	−1.21535	−0.607675	+	0.794185i	$0.707898\pi$
$314$	−17.8872	−1.00943
$315$	−24.5659	−1.38413
$316$	17.4399	0.981074
$317$	−22.8103	−1.28116	−0.640578	−	0.767893i	$-0.721305\pi$
$317$	−22.8103	−1.28116	−0.640578	+	0.767893i	$0.721305\pi$
$318$	−5.25993	−0.294962
$319$	6.02924	0.337573
$320$	32.5599	1.82015
$321$	11.6704	0.651378
$322$	−38.2899	−2.13381
$323$	−37.7322	−2.09947
$324$	12.3334	0.685187
$325$	8.28765	0.459716
$326$	34.9018	1.93303
$327$	−9.08324	−0.502304
$328$	45.0088	2.48520
$329$	1.43498	0.0791130
$330$	14.5421	0.800518
$331$	−20.9138	−1.14952	−0.574762	−	0.818320i	$-0.694906\pi$
$331$	−20.9138	−1.14952	−0.574762	+	0.818320i	$0.694906\pi$
$332$	29.2491	1.60525
$333$	−0.506334	−0.0277469
$334$	−57.4800	−3.14516
$335$	17.3025	0.945336
$336$	8.66922	0.472944
$337$	−3.13546	−0.170799	−0.0853996	−	0.996347i	$-0.527217\pi$
$337$	−3.13546	−0.170799	−0.0853996	+	0.996347i	$0.527217\pi$
$338$	−2.43267	−0.132320
$339$	−7.56026	−0.410617
$340$	87.0071	4.71862
$341$	5.09442	0.275878
$342$	34.5697	1.86931
$343$	15.7816	0.852128
$344$	52.9346	2.85404
$345$	−16.4080	−0.883379
$346$	24.3885	1.31113
$347$	28.6562	1.53835	0.769173	−	0.639041i	$-0.220668\pi$
$347$	28.6562	1.53835	0.769173	+	0.639041i	$0.220668\pi$
$348$	10.1627	0.544776
$349$	−16.8790	−0.903514	−0.451757	−	0.892141i	$-0.649203\pi$
$349$	−16.8790	−0.903514	−0.451757	+	0.892141i	$0.649203\pi$
$350$	59.2163	3.16525
$351$	4.44714	0.237371
$352$	−1.52836	−0.0814619
$353$	28.1451	1.49801	0.749006	−	0.662563i	$-0.230531\pi$
$353$	28.1451	1.49801	0.749006	+	0.662563i	$0.230531\pi$
$354$	0.589344	0.0313233
$355$	−16.1085	−0.854948
$356$	28.4732	1.50907
$357$	−15.0303	−0.795487
$358$	−46.6273	−2.46433
$359$	−18.7833	−0.991344	−0.495672	−	0.868510i	$-0.665078\pi$
$359$	−18.7833	−0.991344	−0.495672	+	0.868510i	$0.665078\pi$
$360$	−39.0217	−2.05662
$361$	19.3585	1.01887
$362$	−9.87803	−0.519178
$363$	6.03783	0.316904
$364$	−11.5074	−0.603151
$365$	36.5368	1.91243
$366$	24.2903	1.26967
$367$	−9.81686	−0.512436	−0.256218	−	0.966619i	$-0.582476\pi$
$367$	−9.81686	−0.512436	−0.256218	+	0.966619i	$0.582476\pi$
$368$	−18.8307	−0.981620
$369$	22.1350	1.15230
$370$	1.95687	0.101733
$371$	7.56077	0.392536
$372$	8.58696	0.445213
$373$	17.6484	0.913799	0.456899	−	0.889518i	$-0.348960\pi$
$373$	17.6484	0.913799	0.456899	+	0.889518i	$0.348960\pi$
$374$	−28.9352	−1.49620
$375$	10.0663	0.519819
$376$	2.27939	0.117550
$377$	−3.08816	−0.159048
$378$	31.7754	1.63435
$379$	−15.6693	−0.804879	−0.402439	−	0.915447i	$-0.631838\pi$
$379$	−15.6693	−0.804879	−0.402439	+	0.915447i	$0.631838\pi$
$380$	−88.4513	−4.53746
$381$	0.569802	0.0291918
$382$	35.3885	1.81064
$383$	0.0432659	0.00221078	0.00110539	−	0.999999i	$-0.499648\pi$
$383$	0.0432659	0.00221078	0.00110539	+	0.999999i	$0.499648\pi$
$384$	−16.9365	−0.864286
$385$	−20.9033	−1.06533
$386$	−37.1880	−1.89282
$387$	26.0328	1.32332
$388$	50.7001	2.57391
$389$	39.1737	1.98619	0.993094	−	0.117325i	$-0.0374318\pi$
$389$	39.1737	1.98619	0.993094	+	0.117325i	$0.0374318\pi$
$390$	−7.44844	−0.377166
$391$	32.6479	1.65107
$392$	−7.59039	−0.383373
$393$	−9.76047	−0.492351
$394$	5.71901	0.288120
$395$	−16.2263	−0.816434
$396$	17.5507	0.881956
$397$	−25.7581	−1.29276	−0.646381	−	0.763015i	$-0.723718\pi$
$397$	−25.7581	−1.29276	−0.646381	+	0.763015i	$0.723718\pi$
$398$	10.9031	0.546524
$399$	15.2798	0.764946
$400$	29.1222	1.45611
$401$	2.56358	0.128019	0.0640095	−	0.997949i	$-0.479611\pi$
$401$	2.56358	0.128019	0.0640095	+	0.997949i	$0.479611\pi$
$402$	−9.69897	−0.483741
$403$	−2.60935	−0.129981
$404$	−36.9951	−1.84057
$405$	−11.4751	−0.570202
$406$	−22.0653	−1.09508
$407$	−0.430842	−0.0213560
$408$	−23.8748	−1.18198
$409$	26.2935	1.30013	0.650066	−	0.759878i	$-0.274741\pi$
$409$	26.2935	1.30013	0.650066	+	0.759878i	$0.274741\pi$
$410$	−85.5468	−4.22486
$411$	−10.7719	−0.531340
$412$	21.5933	1.06382
$413$	−0.847140	−0.0416850
$414$	−29.9115	−1.47007
$415$	−27.2136	−1.33586
$416$	0.782822	0.0383810
$417$	−3.27079	−0.160171
$418$	29.4155	1.43876
$419$	12.1058	0.591406	0.295703	−	0.955280i	$-0.404446\pi$
$419$	12.1058	0.591406	0.295703	+	0.955280i	$0.404446\pi$
$420$	−35.2338	−1.71923
$421$	14.5550	0.709366	0.354683	−	0.934987i	$-0.384589\pi$
$421$	14.5550	0.709366	0.354683	+	0.934987i	$0.384589\pi$
$422$	−1.33616	−0.0650433
$423$	1.12098	0.0545041
$424$	12.0099	0.583252
$425$	−50.4908	−2.44916
$426$	9.02965	0.437488
$427$	−34.9156	−1.68968
$428$	−54.4347	−2.63120
$429$	1.63992	0.0791759
$430$	−100.611	−4.85190
$431$	20.9584	1.00953	0.504764	−	0.863257i	$-0.331579\pi$
$431$	20.9584	1.00953	0.504764	+	0.863257i	$0.331579\pi$
$432$	15.6270	0.751852
$433$	4.52672	0.217540	0.108770	−	0.994067i	$-0.465309\pi$
$433$	4.52672	0.217540	0.108770	+	0.994067i	$0.465309\pi$
$434$	−18.6441	−0.894946
$435$	−9.45545	−0.453354
$436$	42.3674	2.02903
$437$	−33.1898	−1.58768
$438$	−20.4809	−0.978613
$439$	−36.8148	−1.75707	−0.878537	−	0.477675i	$-0.841480\pi$
$439$	−36.8148	−1.75707	−0.878537	+	0.477675i	$0.841480\pi$
$440$	−33.2038	−1.58293
$441$	−3.73289	−0.177757
$442$	14.8205	0.704941
$443$	−28.7694	−1.36688	−0.683438	−	0.730008i	$-0.739516\pi$
$443$	−28.7694	−1.36688	−0.683438	+	0.730008i	$0.739516\pi$
$444$	−0.726211	−0.0344645
$445$	−26.4917	−1.25583
$446$	−0.0986226	−0.00466992
$447$	−7.90893	−0.374080
$448$	26.2353	1.23950
$449$	2.50012	0.117988	0.0589940	−	0.998258i	$-0.481211\pi$
$449$	2.50012	0.117988	0.0589940	+	0.998258i	$0.481211\pi$
$450$	46.2589	2.18067
$451$	18.8348	0.886894
$452$	35.2637	1.65866
$453$	−18.6870	−0.877992
$454$	11.3103	0.530819
$455$	10.7066	0.501933
$456$	24.2711	1.13660
$457$	35.5183	1.66148	0.830739	−	0.556663i	$-0.187918\pi$
$457$	35.5183	1.66148	0.830739	+	0.556663i	$0.187918\pi$
$458$	46.0544	2.15198
$459$	−27.0933	−1.26461
$460$	76.5328	3.56836
$461$	21.5051	1.00159	0.500795	−	0.865566i	$-0.333041\pi$
$461$	21.5051	1.00159	0.500795	+	0.865566i	$0.333041\pi$
$462$	11.7174	0.545143
$463$	−31.1781	−1.44897	−0.724485	−	0.689290i	$-0.757923\pi$
$463$	−31.1781	−1.44897	−0.724485	+	0.689290i	$0.757923\pi$
$464$	−10.8516	−0.503772
$465$	−7.98940	−0.370500
$466$	−39.0717	−1.80996
$467$	13.4678	0.623215	0.311608	−	0.950211i	$-0.399133\pi$
$467$	13.4678	0.623215	0.311608	+	0.950211i	$0.399133\pi$
$468$	−8.98941	−0.415536
$469$	13.9416	0.643763
$470$	−4.33236	−0.199837
$471$	−6.17615	−0.284582
$472$	−1.34564	−0.0619379
$473$	22.1515	1.01852
$474$	9.09572	0.417780
$475$	51.3289	2.35513
$476$	70.1065	3.21333
$477$	5.90637	0.270434
$478$	51.8544	2.37177
$479$	3.89369	0.177907	0.0889536	−	0.996036i	$-0.471648\pi$
$479$	3.89369	0.177907	0.0889536	+	0.996036i	$0.471648\pi$
$480$	2.39688	0.109402
$481$	0.220676	0.0100620
$482$	−49.8061	−2.26860
$483$	−13.2209	−0.601571
$484$	−28.1625	−1.28012
$485$	−47.1719	−2.14197
$486$	38.8876	1.76398
$487$	36.8745	1.67094	0.835472	−	0.549534i	$-0.185195\pi$
$487$	36.8745	1.67094	0.835472	+	0.549534i	$0.185195\pi$
$488$	−55.4615	−2.51063
$489$	12.0510	0.544966
$490$	14.4268	0.651737
$491$	−18.3643	−0.828769	−0.414384	−	0.910102i	$-0.636003\pi$
$491$	−18.3643	−0.828769	−0.414384	+	0.910102i	$0.636003\pi$
$492$	31.7472	1.43127
$493$	18.8140	0.847339
$494$	−15.0665	−0.677875
$495$	−16.3294	−0.733950
$496$	−9.16907	−0.411703
$497$	−12.9795	−0.582210
$498$	15.2547	0.683579
$499$	−9.08841	−0.406853	−0.203427	−	0.979090i	$-0.565208\pi$
$499$	−9.08841	−0.406853	−0.203427	+	0.979090i	$0.565208\pi$
$500$	−46.9525	−2.09978
$501$	−19.8469	−0.886694
$502$	22.4713	1.00294
$503$	−1.07688	−0.0480157	−0.0240078	−	0.999712i	$-0.507643\pi$
$503$	−1.07688	−0.0480157	−0.0240078	+	0.999712i	$0.507643\pi$
$504$	−31.4420	−1.40054
$505$	34.4206	1.53170
$506$	−25.4519	−1.13147
$507$	−0.839960	−0.0373039
$508$	−2.65775	−0.117919
$509$	40.4809	1.79429	0.897143	−	0.441740i	$-0.145639\pi$
$509$	40.4809	1.79429	0.897143	+	0.441740i	$0.145639\pi$
$510$	45.3781	2.00938
$511$	29.4398	1.30234
$512$	35.5395	1.57064
$513$	27.5430	1.21605
$514$	16.2189	0.715385
$515$	−20.0906	−0.885298
$516$	37.3377	1.64370
$517$	0.953852	0.0419504
$518$	1.57676	0.0692788
$519$	8.42096	0.369639
$520$	17.0069	0.745801
$521$	−7.01078	−0.307148	−0.153574	−	0.988137i	$-0.549078\pi$
$521$	−7.01078	−0.307148	−0.153574	+	0.988137i	$0.549078\pi$
$522$	−17.2371	−0.754447
$523$	29.3279	1.28242	0.641210	−	0.767365i	$-0.278433\pi$
$523$	29.3279	1.28242	0.641210	+	0.767365i	$0.278433\pi$
$524$	45.5262	1.98882
$525$	20.4464	0.892356
$526$	−0.758582	−0.0330757
$527$	15.8969	0.692480
$528$	5.76256	0.250783
$529$	5.71759	0.248591
$530$	−22.8268	−0.991533
$531$	−0.661773	−0.0287185
$532$	−71.2702	−3.08995
$533$	−9.64711	−0.417863
$534$	14.8500	0.642624
$535$	50.6467	2.18965
$536$	22.1455	0.956539
$537$	−16.0996	−0.694751
$538$	−21.7467	−0.937566
$539$	−3.17634	−0.136815
$540$	−63.5118	−2.73311
$541$	33.6492	1.44669	0.723346	−	0.690485i	$-0.242603\pi$
$541$	33.6492	1.44669	0.723346	+	0.690485i	$0.242603\pi$
$542$	−4.22917	−0.181658
$543$	−3.41072	−0.146368
$544$	−4.76918	−0.204477
$545$	−39.4191	−1.68853
$546$	−6.00162	−0.256846
$547$	−23.9739	−1.02505	−0.512525	−	0.858672i	$-0.671290\pi$
$547$	−23.9739	−1.02505	−0.512525	+	0.858672i	$0.671290\pi$
$548$	50.2440	2.14632
$549$	−27.2755	−1.16409
$550$	39.3620	1.67840
$551$	−19.1263	−0.814806
$552$	−21.0007	−0.893848
$553$	−13.0745	−0.555982
$554$	42.6802	1.81331
$555$	0.675675	0.0286808
$556$	15.2561	0.647003
$557$	−12.7715	−0.541145	−0.270572	−	0.962700i	$-0.587213\pi$
$557$	−12.7715	−0.541145	−0.270572	+	0.962700i	$0.587213\pi$
$558$	−14.5645	−0.616565
$559$	−11.3459	−0.479881
$560$	37.6223	1.58983
$561$	−9.99086	−0.421814
$562$	−35.6202	−1.50255
$563$	23.8832	1.00656	0.503278	−	0.864125i	$-0.332127\pi$
$563$	23.8832	1.00656	0.503278	+	0.864125i	$0.332127\pi$
$564$	1.60778	0.0676996
$565$	−32.8097	−1.38031
$566$	48.6297	2.04406
$567$	−9.24612	−0.388301
$568$	−20.6172	−0.865080
$569$	−30.4046	−1.27463	−0.637314	−	0.770605i	$-0.719954\pi$
$569$	−30.4046	−1.27463	−0.637314	+	0.770605i	$0.719954\pi$
$570$	−46.1313	−1.93223
$571$	12.0930	0.506075	0.253037	−	0.967457i	$-0.418570\pi$
$571$	12.0930	0.506075	0.253037	+	0.967457i	$0.418570\pi$
$572$	−7.64914	−0.319826
$573$	12.2191	0.510460
$574$	−68.9298	−2.87708
$575$	−44.4125	−1.85213
$576$	20.4947	0.853944
$577$	33.6188	1.39957	0.699785	−	0.714354i	$-0.253279\pi$
$577$	33.6188	1.39957	0.699785	+	0.714354i	$0.253279\pi$
$578$	−48.9358	−2.03546
$579$	−12.8404	−0.533630
$580$	44.1035	1.83130
$581$	−21.9275	−0.909708
$582$	26.4424	1.09607
$583$	5.02576	0.208146
$584$	46.7635	1.93509
$585$	8.36384	0.345802
$586$	75.3251	3.11165
$587$	−0.476412	−0.0196636	−0.00983181	−	0.999952i	$-0.503130\pi$
$587$	−0.476412	−0.0196636	−0.00983181	+	0.999952i	$0.503130\pi$
$588$	−5.35392	−0.220792
$589$	−16.1608	−0.665893
$590$	2.55761	0.105295
$591$	1.97468	0.0812275
$592$	0.775441	0.0318704
$593$	10.3245	0.423978	0.211989	−	0.977272i	$-0.432006\pi$
$593$	10.3245	0.423978	0.211989	+	0.977272i	$0.432006\pi$
$594$	21.1216	0.866629
$595$	−65.2278	−2.67408
$596$	36.8900	1.51107
$597$	3.76467	0.154078
$598$	13.0364	0.533097
$599$	21.0777	0.861212	0.430606	−	0.902540i	$-0.358300\pi$
$599$	21.0777	0.861212	0.430606	+	0.902540i	$0.358300\pi$
$600$	32.4781	1.32591
$601$	−14.2327	−0.580564	−0.290282	−	0.956941i	$-0.593749\pi$
$601$	−14.2327	−0.580564	−0.290282	+	0.956941i	$0.593749\pi$
$602$	−81.0680	−3.30408
$603$	10.8910	0.443514
$604$	87.1627	3.54660
$605$	26.2027	1.06529
$606$	−19.2946	−0.783789
$607$	−33.3191	−1.35238	−0.676190	−	0.736727i	$-0.736370\pi$
$607$	−33.3191	−1.35238	−0.676190	+	0.736727i	$0.736370\pi$
$608$	4.84835	0.196626
$609$	−7.61878	−0.308729
$610$	105.414	4.26808
$611$	−0.488560	−0.0197650
$612$	54.7662	2.21379
$613$	−47.3358	−1.91187	−0.955937	−	0.293570i	$-0.905156\pi$
$613$	−47.3358	−1.91187	−0.955937	+	0.293570i	$0.905156\pi$
$614$	−53.4461	−2.15691
$615$	−29.5379	−1.19108
$616$	−26.7541	−1.07796
$617$	−0.451673	−0.0181837	−0.00909184	−	0.999959i	$-0.502894\pi$
$617$	−0.451673	−0.0181837	−0.00909184	+	0.999959i	$0.502894\pi$
$618$	11.2619	0.453019
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	37.2653	1.49661
$621$	−23.8317	−0.956333
$622$	−14.0498	−0.563348
$623$	−21.3459	−0.855205
$624$	−2.95156	−0.118157
$625$	2.24688	0.0898750
$626$	52.3066	2.09059
$627$	10.1567	0.405619
$628$	28.8077	1.14955
$629$	−1.34442	−0.0536057
$630$	59.7608	2.38093
$631$	16.4564	0.655118	0.327559	−	0.944831i	$-0.393774\pi$
$631$	16.4564	0.655118	0.327559	+	0.944831i	$0.393774\pi$
$632$	−20.7681	−0.826110
$633$	−0.461355	−0.0183372
$634$	55.4899	2.20379
$635$	2.47280	0.0981301
$636$	8.47123	0.335906
$637$	1.62691	0.0644605
$638$	−14.6671	−0.580677
$639$	−10.1394	−0.401108
$640$	−73.5002	−2.90535
$641$	−40.9811	−1.61866	−0.809328	−	0.587358i	$-0.800168\pi$
$641$	−40.9811	−1.61866	−0.809328	+	0.587358i	$0.800168\pi$
$642$	−28.3902	−1.12047
$643$	35.0821	1.38350	0.691752	−	0.722135i	$-0.256839\pi$
$643$	35.0821	1.38350	0.691752	+	0.722135i	$0.256839\pi$
$644$	61.6667	2.43001
$645$	−34.7394	−1.36786
$646$	91.7898	3.61142
$647$	−9.70406	−0.381506	−0.190753	−	0.981638i	$-0.561093\pi$
$647$	−9.70406	−0.381506	−0.190753	+	0.981638i	$0.561093\pi$
$648$	−14.6870	−0.576959
$649$	−0.563106	−0.0221039
$650$	−20.1611	−0.790783
$651$	−6.43751	−0.252306
$652$	−56.2102	−2.20136
$653$	−28.0300	−1.09690	−0.548450	−	0.836184i	$-0.684782\pi$
$653$	−28.0300	−1.09690	−0.548450	+	0.836184i	$0.684782\pi$
$654$	22.0965	0.864041
$655$	−42.3581	−1.65507
$656$	−33.8993	−1.32355
$657$	22.9979	0.897235
$658$	−3.49082	−0.136087
$659$	−4.00968	−0.156195	−0.0780975	−	0.996946i	$-0.524885\pi$
$659$	−4.00968	−0.156195	−0.0780975	+	0.996946i	$0.524885\pi$
$660$	−23.4204	−0.911640
$661$	−15.9496	−0.620367	−0.310183	−	0.950677i	$-0.600390\pi$
$661$	−15.9496	−0.620367	−0.310183	+	0.950677i	$0.600390\pi$
$662$	50.8762	1.97736
$663$	5.11729	0.198739
$664$	−34.8307	−1.35170
$665$	66.3105	2.57141
$666$	1.23174	0.0477290
$667$	16.5491	0.640782
$668$	92.5728	3.58175
$669$	−0.0340528	−0.00131656
$670$	−42.0912	−1.62612
$671$	−23.2089	−0.895969
$672$	1.93130	0.0745014
$673$	8.33607	0.321332	0.160666	−	0.987009i	$-0.448636\pi$
$673$	8.33607	0.321332	0.160666	+	0.987009i	$0.448636\pi$
$674$	7.62752	0.293801
$675$	36.8563	1.41860
$676$	3.91786	0.150687
$677$	−40.8130	−1.56857	−0.784286	−	0.620399i	$-0.786971\pi$
$677$	−40.8130	−1.56857	−0.784286	+	0.620399i	$0.786971\pi$
$678$	18.3916	0.706325
$679$	−38.0091	−1.45865
$680$	−103.611	−3.97330
$681$	3.90526	0.149650
$682$	−12.3930	−0.474553
$683$	35.1141	1.34361	0.671803	−	0.740730i	$-0.265520\pi$
$683$	35.1141	1.34361	0.671803	+	0.740730i	$0.265520\pi$
$684$	−55.6752	−2.12880
$685$	−46.7476	−1.78613
$686$	−38.3914	−1.46579
$687$	15.9018	0.606693
$688$	−39.8688	−1.51998
$689$	−2.57418	−0.0980684
$690$	39.9153	1.51955
$691$	3.16458	0.120386	0.0601932	−	0.998187i	$-0.480828\pi$
$691$	3.16458	0.120386	0.0601932	+	0.998187i	$0.480828\pi$
$692$	−39.2783	−1.49314
$693$	−13.1575	−0.499811
$694$	−69.7110	−2.64619
$695$	−14.1944	−0.538426
$696$	−12.1020	−0.458726
$697$	58.7731	2.22619
$698$	41.0610	1.55418
$699$	−13.4908	−0.510269
$700$	−95.3693	−3.60462
$701$	−17.4612	−0.659500	−0.329750	−	0.944068i	$-0.606965\pi$
$701$	−17.4612	−0.659500	−0.329750	+	0.944068i	$0.606965\pi$
$702$	−10.8184	−0.408315
$703$	1.36674	0.0515476
$704$	17.4390	0.657258
$705$	−1.49589	−0.0563386
$706$	−68.4676	−2.57681
$707$	27.7346	1.04307
$708$	−0.949151	−0.0356713
$709$	−29.7660	−1.11789	−0.558943	−	0.829206i	$-0.688793\pi$
$709$	−29.7660	−1.11789	−0.558943	+	0.829206i	$0.688793\pi$
$710$	39.1865	1.47064
$711$	−10.2136	−0.383039
$712$	−33.9068	−1.27071
$713$	13.9832	0.523674
$714$	36.5637	1.36836
$715$	7.11684	0.266155
$716$	75.0943	2.80641
$717$	17.9045	0.668656
$718$	45.6934	1.70526
$719$	40.4446	1.50833	0.754164	−	0.656686i	$-0.228042\pi$
$719$	40.4446	1.50833	0.754164	+	0.656686i	$0.228042\pi$
$720$	29.3900	1.09530
$721$	−16.1881	−0.602878
$722$	−47.0928	−1.75261
$723$	−17.1972	−0.639572
$724$	15.9088	0.591246
$725$	−25.5936	−0.950521
$726$	−14.6880	−0.545124
$727$	−47.9207	−1.77728	−0.888641	−	0.458603i	$-0.848350\pi$
$727$	−47.9207	−1.77728	−0.888641	+	0.458603i	$0.848350\pi$
$728$	13.7034	0.507881
$729$	3.98332	0.147531
$730$	−88.8820	−3.28967
$731$	69.1227	2.55659
$732$	−39.1201	−1.44592
$733$	0.795034	0.0293652	0.0146826	−	0.999892i	$-0.495326\pi$
$733$	0.795034	0.0293652	0.0146826	+	0.999892i	$0.495326\pi$
$734$	23.8811	0.881469
$735$	4.98134	0.183740
$736$	−4.19505	−0.154631
$737$	9.26718	0.341361
$738$	−53.8470	−1.98214
$739$	−33.3608	−1.22720	−0.613599	−	0.789618i	$-0.710279\pi$
$739$	−33.3608	−1.22720	−0.613599	+	0.789618i	$0.710279\pi$
$740$	−3.15158	−0.115854
$741$	−5.20223	−0.191109
$742$	−18.3928	−0.675222
$743$	−5.17500	−0.189852	−0.0949262	−	0.995484i	$-0.530262\pi$
$743$	−5.17500	−0.189852	−0.0949262	+	0.995484i	$0.530262\pi$
$744$	−10.2256	−0.374890
$745$	−34.3229	−1.25749
$746$	−42.9326	−1.57188
$747$	−17.1295	−0.626735
$748$	46.6008	1.70389
$749$	40.8089	1.49112
$750$	−24.4878	−0.894170
$751$	−34.1669	−1.24677	−0.623383	−	0.781916i	$-0.714242\pi$
$751$	−34.1669	−1.24677	−0.623383	+	0.781916i	$0.714242\pi$
$752$	−1.71677	−0.0626041
$753$	7.75899	0.282753
$754$	7.51246	0.273587
$755$	−81.0971	−2.95142
$756$	−51.1750	−1.86122
$757$	33.8610	1.23070	0.615350	−	0.788254i	$-0.289015\pi$
$757$	33.8610	1.23070	0.615350	+	0.788254i	$0.289015\pi$
$758$	38.1182	1.38452
$759$	−8.78811	−0.318988
$760$	105.331	3.82075
$761$	14.4804	0.524914	0.262457	−	0.964944i	$-0.415467\pi$
$761$	14.4804	0.524914	0.262457	+	0.964944i	$0.415467\pi$
$762$	−1.38614	−0.0502144
$763$	−31.7622	−1.14987
$764$	−56.9941	−2.06197
$765$	−50.9550	−1.84228
$766$	−0.105251	−0.00380289
$767$	0.288421	0.0104143
$768$	26.1954	0.945246
$769$	−34.3195	−1.23759	−0.618797	−	0.785551i	$-0.712380\pi$
$769$	−34.3195	−1.23759	−0.618797	+	0.785551i	$0.712380\pi$
$770$	50.8507	1.83253
$771$	5.60012	0.201683
$772$	59.8922	2.15557
$773$	6.88095	0.247490	0.123745	−	0.992314i	$-0.460509\pi$
$773$	6.88095	0.247490	0.123745	+	0.992314i	$0.460509\pi$
$774$	−63.3291	−2.27632
$775$	−21.6253	−0.776805
$776$	−60.3754	−2.16735
$777$	0.544429	0.0195313
$778$	−95.2966	−3.41655
$779$	−59.7486	−2.14072
$780$	11.9959	0.429521
$781$	−8.62766	−0.308722
$782$	−79.4214	−2.84010
$783$	−13.7335	−0.490794
$784$	5.71686	0.204173
$785$	−26.8030	−0.956639
$786$	23.7440	0.846919
$787$	−17.4441	−0.621814	−0.310907	−	0.950440i	$-0.600633\pi$
$787$	−17.4441	−0.621814	−0.310907	+	0.950440i	$0.600633\pi$
$788$	−9.21060	−0.328114
$789$	−0.261926	−0.00932481
$790$	39.4732	1.40439
$791$	−26.4366	−0.939978
$792$	−20.9000	−0.742647
$793$	11.8875	0.422138
$794$	62.6609	2.22375
$795$	−7.88173	−0.279536
$796$	−17.5597	−0.622388
$797$	−36.1084	−1.27903	−0.639513	−	0.768780i	$-0.720864\pi$
$797$	−36.1084	−1.27903	−0.639513	+	0.768780i	$0.720864\pi$
$798$	−37.1706	−1.31583
$799$	2.97645	0.105299
$800$	6.48775	0.229377
$801$	−16.6751	−0.589185
$802$	−6.23633	−0.220212
$803$	19.5691	0.690577
$804$	15.6204	0.550890
$805$	−57.3754	−2.02222
$806$	6.34767	0.223587
$807$	−7.50877	−0.264321
$808$	44.0550	1.54985
$809$	45.2575	1.59117	0.795585	−	0.605842i	$-0.207164\pi$
$809$	45.2575	1.59117	0.795585	+	0.605842i	$0.207164\pi$
$810$	27.9151	0.980835
$811$	50.4390	1.77115	0.885576	−	0.464494i	$-0.153764\pi$
$811$	50.4390	1.77115	0.885576	+	0.464494i	$0.153764\pi$
$812$	35.5366	1.24709
$813$	−1.46026	−0.0512137
$814$	1.04809	0.0367357
$815$	52.2985	1.83194
$816$	17.9818	0.629489
$817$	−70.2700	−2.45844
$818$	−63.9634	−2.23643
$819$	6.73922	0.235487
$820$	137.775	4.81131
$821$	−19.5226	−0.681343	−0.340671	−	0.940182i	$-0.610654\pi$
$821$	−19.5226	−0.681343	−0.340671	+	0.940182i	$0.610654\pi$
$822$	26.2045	0.913987
$823$	46.8485	1.63304	0.816518	−	0.577321i	$-0.195902\pi$
$823$	46.8485	1.63304	0.816518	+	0.577321i	$0.195902\pi$
$824$	−25.7140	−0.895790
$825$	13.5910	0.473180
$826$	2.06081	0.0717047
$827$	−52.0032	−1.80833	−0.904164	−	0.427185i	$-0.859505\pi$
$827$	−52.0032	−1.80833	−0.904164	+	0.427185i	$0.859505\pi$
$828$	48.1732	1.67413
$829$	41.9539	1.45712	0.728560	−	0.684982i	$-0.240190\pi$
$829$	41.9539	1.45712	0.728560	+	0.684982i	$0.240190\pi$
$830$	66.2017	2.29789
$831$	14.7368	0.511213
$832$	−8.93221	−0.309669
$833$	−9.91162	−0.343417
$834$	7.95674	0.275520
$835$	−86.1307	−2.98068
$836$	−47.3743	−1.63848
$837$	−11.6041	−0.401097
$838$	−29.4493	−1.01731
$839$	43.9044	1.51575	0.757875	−	0.652400i	$-0.226238\pi$
$839$	43.9044	1.51575	0.757875	+	0.652400i	$0.226238\pi$
$840$	41.9576	1.44768
$841$	−19.4633	−0.671148
$842$	−35.4074	−1.22022
$843$	−12.2991	−0.423603
$844$	2.15192	0.0740721
$845$	−3.64522	−0.125399
$846$	−2.72698	−0.0937556
$847$	21.1130	0.725451
$848$	−9.04549	−0.310623
$849$	16.7910	0.576267
$850$	122.827	4.21294
$851$	−1.18258	−0.0405382
$852$	−14.5425	−0.498217
$853$	2.05638	0.0704091	0.0352046	−	0.999380i	$-0.488792\pi$
$853$	2.05638	0.0704091	0.0352046	+	0.999380i	$0.488792\pi$
$854$	84.9379	2.90652
$855$	51.8008	1.77155
$856$	64.8227	2.21559
$857$	−40.5263	−1.38435	−0.692176	−	0.721729i	$-0.743348\pi$
$857$	−40.5263	−1.38435	−0.692176	+	0.721729i	$0.743348\pi$
$858$	−3.98937	−0.136195
$859$	43.4286	1.48177	0.740883	−	0.671635i	$-0.234408\pi$
$859$	43.4286	1.48177	0.740883	+	0.671635i	$0.234408\pi$
$860$	162.036	5.52540
$861$	−23.8004	−0.811114
$862$	−50.9847	−1.73654
$863$	−45.4429	−1.54690	−0.773448	−	0.633860i	$-0.781469\pi$
$863$	−45.4429	−1.54690	−0.773448	+	0.633860i	$0.781469\pi$
$864$	3.48132	0.118437
$865$	36.5449	1.24256
$866$	−11.0120	−0.374203
$867$	−16.8967	−0.573843
$868$	30.0268	1.01918
$869$	−8.69078	−0.294815
$870$	23.0019	0.779839
$871$	−4.74662	−0.160833
$872$	−50.4525	−1.70854
$873$	−29.6921	−1.00493
$874$	80.7397	2.73106
$875$	35.1995	1.18996
$876$	32.9849	1.11446
$877$	−48.2621	−1.62970	−0.814848	−	0.579675i	$-0.803180\pi$
$877$	−48.2621	−1.62970	−0.814848	+	0.579675i	$0.803180\pi$
$878$	89.5581	3.02244
$879$	26.0085	0.877246
$880$	25.0081	0.843023
$881$	−32.7464	−1.10326	−0.551628	−	0.834090i	$-0.685993\pi$
$881$	−32.7464	−1.10326	−0.551628	+	0.834090i	$0.685993\pi$
$882$	9.08088	0.305769
$883$	−1.19651	−0.0402658	−0.0201329	−	0.999797i	$-0.506409\pi$
$883$	−1.19651	−0.0402658	−0.0201329	+	0.999797i	$0.506409\pi$
$884$	−23.8688	−0.802795
$885$	0.883100	0.0296851
$886$	69.9864	2.35124
$887$	−25.5300	−0.857214	−0.428607	−	0.903491i	$-0.640995\pi$
$887$	−25.5300	−0.857214	−0.428607	+	0.903491i	$0.640995\pi$
$888$	0.864797	0.0290207
$889$	1.99247	0.0668254
$890$	64.4456	2.16022
$891$	−6.14604	−0.205900
$892$	0.158834	0.00531816
$893$	−3.02586	−0.101257
$894$	19.2398	0.643475
$895$	−69.8685	−2.33545
$896$	−59.2233	−1.97851
$897$	4.50124	0.150292
$898$	−6.08196	−0.202958
$899$	8.05807	0.268752
$900$	−74.5011	−2.48337
$901$	15.6827	0.522465
$902$	−45.8187	−1.52560
$903$	−27.9915	−0.931497
$904$	−41.9932	−1.39667
$905$	−14.8017	−0.492025
$906$	45.4592	1.51028
$907$	−7.60563	−0.252541	−0.126270	−	0.991996i	$-0.540301\pi$
$907$	−7.60563	−0.252541	−0.126270	+	0.991996i	$0.540301\pi$
$908$	−18.2155	−0.604503
$909$	21.6659	0.718612
$910$	−26.0456	−0.863403
$911$	39.0561	1.29399	0.646993	−	0.762496i	$-0.276026\pi$
$911$	39.0561	1.29399	0.646993	+	0.762496i	$0.276026\pi$
$912$	−18.2803	−0.605321
$913$	−14.5756	−0.482381
$914$	−86.4042	−2.85800
$915$	36.3977	1.20327
$916$	−74.1717	−2.45070
$917$	−34.1303	−1.12708
$918$	65.9090	2.17532
$919$	−15.9975	−0.527710	−0.263855	−	0.964562i	$-0.584994\pi$
$919$	−15.9975	−0.527710	−0.263855	+	0.964562i	$0.584994\pi$
$920$	−91.1378	−3.00472
$921$	−18.4541	−0.608082
$922$	−52.3146	−1.72289
$923$	4.41906	0.145455
$924$	−18.8712	−0.620816
$925$	1.82888	0.0601334
$926$	75.8460	2.49245
$927$	−12.6459	−0.415347
$928$	−2.41748	−0.0793575
$929$	36.3506	1.19263	0.596313	−	0.802752i	$-0.296632\pi$
$929$	36.3506	1.19263	0.596313	+	0.802752i	$0.296632\pi$
$930$	19.4355	0.637317
$931$	10.0761	0.330232
$932$	62.9258	2.06120
$933$	−4.85118	−0.158821
$934$	−32.7626	−1.07203
$935$	−43.3579	−1.41796
$936$	10.7049	0.349900
$937$	−43.0018	−1.40481	−0.702404	−	0.711779i	$-0.747890\pi$
$937$	−43.0018	−1.40481	−0.702404	+	0.711779i	$0.747890\pi$
$938$	−33.9152	−1.10737
$939$	18.0606	0.589386
$940$	6.97736	0.227577
$941$	−7.30276	−0.238063	−0.119032	−	0.992890i	$-0.537979\pi$
$941$	−7.30276	−0.238063	−0.119032	+	0.992890i	$0.537979\pi$
$942$	15.0245	0.489525
$943$	51.6977	1.68351
$944$	1.01349	0.0329864
$945$	47.6138	1.54888
$946$	−53.8871	−1.75202
$947$	−52.9038	−1.71914	−0.859571	−	0.511016i	$-0.829269\pi$
$947$	−52.9038	−1.71914	−0.859571	+	0.511016i	$0.829269\pi$
$948$	−14.6489	−0.475773
$949$	−10.0232	−0.325367
$950$	−124.866	−4.05119
$951$	19.1598	0.621298
$952$	−83.4852	−2.70577
$953$	−7.50053	−0.242966	−0.121483	−	0.992594i	$-0.538765\pi$
$953$	−7.50053	−0.242966	−0.121483	+	0.992594i	$0.538765\pi$
$954$	−14.3682	−0.465188
$955$	53.0279	1.71594
$956$	−83.5128	−2.70100
$957$	−5.06432	−0.163706
$958$	−9.47204	−0.306028
$959$	−37.6671	−1.21633
$960$	−27.3490	−0.882686
$961$	−24.1913	−0.780365
$962$	−0.536831	−0.0173081
$963$	31.8793	1.02730
$964$	80.2138	2.58351
$965$	−55.7243	−1.79383
$966$	32.1620	1.03479
$967$	−13.8994	−0.446973	−0.223487	−	0.974707i	$-0.571744\pi$
$967$	−13.8994	−0.446973	−0.223487	+	0.974707i	$0.571744\pi$
$968$	33.5369	1.07792
$969$	31.6935	1.01814
$970$	114.754	3.68451
$971$	15.3058	0.491187	0.245593	−	0.969373i	$-0.421017\pi$
$971$	15.3058	0.491187	0.245593	+	0.969373i	$0.421017\pi$
$972$	−62.6294	−2.00884
$973$	−11.4373	−0.366662
$974$	−89.7033	−2.87428
$975$	−6.96129	−0.222940
$976$	41.7720	1.33709
$977$	−7.75540	−0.248117	−0.124059	−	0.992275i	$-0.539591\pi$
$977$	−7.75540	−0.248117	−0.124059	+	0.992275i	$0.539591\pi$
$978$	−29.3161	−0.937426
$979$	−14.1889	−0.453480
$980$	−23.2347	−0.742205
$981$	−24.8121	−0.792190
$982$	44.6742	1.42561
$983$	−13.0950	−0.417666	−0.208833	−	0.977951i	$-0.566966\pi$
$983$	−13.0950	−0.417666	−0.208833	+	0.977951i	$0.566966\pi$
$984$	−37.8056	−1.20520
$985$	8.56964	0.273051
$986$	−45.7681	−1.45755
$987$	−1.20533	−0.0383659
$988$	24.2650	0.771972
$989$	60.8013	1.93337
$990$	39.7239	1.26251
$991$	−29.3422	−0.932087	−0.466043	−	0.884762i	$-0.654321\pi$
$991$	−29.3422	−0.932087	−0.466043	+	0.884762i	$0.654321\pi$
$992$	−2.04265	−0.0648543
$993$	17.5667	0.557463
$994$	31.5748	1.00149
$995$	16.3378	0.517942
$996$	−24.5680	−0.778468
$997$	35.6671	1.12959	0.564794	−	0.825232i	$-0.308956\pi$
$997$	35.6671	1.12959	0.564794	+	0.825232i	$0.308956\pi$
$998$	22.1091	0.699850
$999$	0.981377	0.0310494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.15	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.15	✓	142	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.43267 q^{2} -0.839960 q^{3} +3.91786 q^{4} -3.64522 q^{5} +2.04334 q^{6} -2.93716 q^{7} -4.66552 q^{8} -2.29447 q^{9} +O(q^{10})\)	\(q-2.43267 q^{2} -0.839960 q^{3} +3.91786 q^{4} -3.64522 q^{5} +2.04334 q^{6} -2.93716 q^{7} -4.66552 q^{8} -2.29447 q^{9} +8.86761 q^{10} -1.95237 q^{11} -3.29085 q^{12} +1.00000 q^{13} +7.14513 q^{14} +3.06184 q^{15} +3.51393 q^{16} -6.09230 q^{17} +5.58167 q^{18} +6.19342 q^{19} -14.2815 q^{20} +2.46710 q^{21} +4.74947 q^{22} -5.35888 q^{23} +3.91885 q^{24} +8.28765 q^{25} -2.43267 q^{26} +4.44714 q^{27} -11.5074 q^{28} -3.08816 q^{29} -7.44844 q^{30} -2.60935 q^{31} +0.782822 q^{32} +1.63992 q^{33} +14.8205 q^{34} +10.7066 q^{35} -8.98941 q^{36} +0.220676 q^{37} -15.0665 q^{38} -0.839960 q^{39} +17.0069 q^{40} -9.64711 q^{41} -6.00162 q^{42} -11.3459 q^{43} -7.64914 q^{44} +8.36384 q^{45} +13.0364 q^{46} -0.488560 q^{47} -2.95156 q^{48} +1.62691 q^{49} -20.1611 q^{50} +5.11729 q^{51} +3.91786 q^{52} -2.57418 q^{53} -10.8184 q^{54} +7.11684 q^{55} +13.7034 q^{56} -5.20223 q^{57} +7.51246 q^{58} +0.288421 q^{59} +11.9959 q^{60} +11.8875 q^{61} +6.34767 q^{62} +6.73922 q^{63} -8.93221 q^{64} -3.64522 q^{65} -3.98937 q^{66} -4.74662 q^{67} -23.8688 q^{68} +4.50124 q^{69} -26.0456 q^{70} +4.41906 q^{71} +10.7049 q^{72} -10.0232 q^{73} -0.536831 q^{74} -6.96129 q^{75} +24.2650 q^{76} +5.73444 q^{77} +2.04334 q^{78} +4.45139 q^{79} -12.8091 q^{80} +3.14798 q^{81} +23.4682 q^{82} +7.46556 q^{83} +9.66575 q^{84} +22.2078 q^{85} +27.6008 q^{86} +2.59393 q^{87} +9.10885 q^{88} +7.26752 q^{89} -20.3464 q^{90} -2.93716 q^{91} -20.9954 q^{92} +2.19175 q^{93} +1.18850 q^{94} -22.5764 q^{95} -0.657539 q^{96} +12.9408 q^{97} -3.95773 q^{98} +4.47966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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