LMFDB - Embedded newform 8045.2.a.b.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11951 q^{2} +0.398701 q^{3} +2.49230 q^{4} -1.00000 q^{5} -0.845049 q^{6} -4.30325 q^{7} -1.04344 q^{8} -2.84104 q^{9} +O(q^{10})$	\(q-2.11951 q^{2} +0.398701 q^{3} +2.49230 q^{4} -1.00000 q^{5} -0.845049 q^{6} -4.30325 q^{7} -1.04344 q^{8} -2.84104 q^{9} +2.11951 q^{10} +2.97744 q^{11} +0.993685 q^{12} +0.514245 q^{13} +9.12077 q^{14} -0.398701 q^{15} -2.77303 q^{16} +4.01135 q^{17} +6.02160 q^{18} -8.36319 q^{19} -2.49230 q^{20} -1.71571 q^{21} -6.31069 q^{22} -7.24433 q^{23} -0.416022 q^{24} +1.00000 q^{25} -1.08995 q^{26} -2.32883 q^{27} -10.7250 q^{28} -2.25747 q^{29} +0.845049 q^{30} +9.54902 q^{31} +7.96433 q^{32} +1.18711 q^{33} -8.50208 q^{34} +4.30325 q^{35} -7.08073 q^{36} +9.26458 q^{37} +17.7258 q^{38} +0.205030 q^{39} +1.04344 q^{40} -6.74463 q^{41} +3.63646 q^{42} +8.73048 q^{43} +7.42068 q^{44} +2.84104 q^{45} +15.3544 q^{46} +7.19593 q^{47} -1.10561 q^{48} +11.5180 q^{49} -2.11951 q^{50} +1.59933 q^{51} +1.28166 q^{52} -12.2333 q^{53} +4.93596 q^{54} -2.97744 q^{55} +4.49020 q^{56} -3.33441 q^{57} +4.78471 q^{58} +12.4545 q^{59} -0.993685 q^{60} +9.08882 q^{61} -20.2392 q^{62} +12.2257 q^{63} -11.3344 q^{64} -0.514245 q^{65} -2.51608 q^{66} +0.180480 q^{67} +9.99751 q^{68} -2.88832 q^{69} -9.12077 q^{70} -6.37142 q^{71} +2.96446 q^{72} -13.1993 q^{73} -19.6363 q^{74} +0.398701 q^{75} -20.8436 q^{76} -12.8127 q^{77} -0.434563 q^{78} -3.25875 q^{79} +2.77303 q^{80} +7.59461 q^{81} +14.2953 q^{82} +6.64864 q^{83} -4.27608 q^{84} -4.01135 q^{85} -18.5043 q^{86} -0.900055 q^{87} -3.10678 q^{88} -4.82029 q^{89} -6.02160 q^{90} -2.21293 q^{91} -18.0551 q^{92} +3.80721 q^{93} -15.2518 q^{94} +8.36319 q^{95} +3.17539 q^{96} -0.486097 q^{97} -24.4124 q^{98} -8.45901 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) $ 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * 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 $

$2$	−2.11951	−1.49872	−0.749358	−	0.662165i	$-0.769638\pi$
$2$	−2.11951	−1.49872	−0.749358	+	0.662165i	$0.769638\pi$
$3$	0.398701	0.230190	0.115095	−	0.993354i	$-0.463283\pi$
$3$	0.398701	0.230190	0.115095	+	0.993354i	$0.463283\pi$
$4$	2.49230	1.24615
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.845049	−0.344990
$7$	−4.30325	−1.62648	−0.813238	−	0.581931i	$-0.802298\pi$
$7$	−4.30325	−1.62648	−0.813238	+	0.581931i	$0.802298\pi$
$8$	−1.04344	−0.368913
$9$	−2.84104	−0.947012
$10$	2.11951	0.670247
$11$	2.97744	0.897731	0.448865	−	0.893599i	$-0.351828\pi$
$11$	2.97744	0.897731	0.448865	+	0.893599i	$0.351828\pi$
$12$	0.993685	0.286852
$13$	0.514245	0.142626	0.0713130	−	0.997454i	$-0.477281\pi$
$13$	0.514245	0.142626	0.0713130	+	0.997454i	$0.477281\pi$
$14$	9.12077	2.43763
$15$	−0.398701	−0.102944
$16$	−2.77303	−0.693257
$17$	4.01135	0.972895	0.486448	−	0.873710i	$-0.338292\pi$
$17$	4.01135	0.972895	0.486448	+	0.873710i	$0.338292\pi$
$18$	6.02160	1.41930
$19$	−8.36319	−1.91865	−0.959323	−	0.282310i	$-0.908899\pi$
$19$	−8.36319	−1.91865	−0.959323	+	0.282310i	$0.908899\pi$
$20$	−2.49230	−0.557296
$21$	−1.71571	−0.374399
$22$	−6.31069	−1.34544
$23$	−7.24433	−1.51055	−0.755274	−	0.655409i	$-0.772496\pi$
$23$	−7.24433	−1.51055	−0.755274	+	0.655409i	$0.772496\pi$
$24$	−0.416022	−0.0849201
$25$	1.00000	0.200000
$26$	−1.08995	−0.213756
$27$	−2.32883	−0.448183
$28$	−10.7250	−2.02684
$29$	−2.25747	−0.419201	−0.209601	−	0.977787i	$-0.567216\pi$
$29$	−2.25747	−0.419201	−0.209601	+	0.977787i	$0.567216\pi$
$30$	0.845049	0.154284
$31$	9.54902	1.71506	0.857528	−	0.514438i	$-0.171999\pi$
$31$	9.54902	1.71506	0.857528	+	0.514438i	$0.171999\pi$
$32$	7.96433	1.40791
$33$	1.18711	0.206649
$34$	−8.50208	−1.45809
$35$	4.30325	0.727383
$36$	−7.08073	−1.18012
$37$	9.26458	1.52309	0.761544	−	0.648113i	$-0.224442\pi$
$37$	9.26458	1.52309	0.761544	+	0.648113i	$0.224442\pi$
$38$	17.7258	2.87551
$39$	0.205030	0.0328311
$40$	1.04344	0.164983
$41$	−6.74463	−1.05333	−0.526667	−	0.850071i	$-0.676559\pi$
$41$	−6.74463	−1.05333	−0.526667	+	0.850071i	$0.676559\pi$
$42$	3.63646	0.561118
$43$	8.73048	1.33139	0.665693	−	0.746226i	$-0.268136\pi$
$43$	8.73048	1.33139	0.665693	+	0.746226i	$0.268136\pi$
$44$	7.42068	1.11871
$45$	2.84104	0.423517
$46$	15.3544	2.26388
$47$	7.19593	1.04963	0.524817	−	0.851215i	$-0.324134\pi$
$47$	7.19593	1.04963	0.524817	+	0.851215i	$0.324134\pi$
$48$	−1.10561	−0.159581
$49$	11.5180	1.64543
$50$	−2.11951	−0.299743
$51$	1.59933	0.223951
$52$	1.28166	0.177734
$53$	−12.2333	−1.68038	−0.840190	−	0.542293i	$-0.817556\pi$
$53$	−12.2333	−1.68038	−0.840190	+	0.542293i	$0.817556\pi$
$54$	4.93596	0.671700
$55$	−2.97744	−0.401477
$56$	4.49020	0.600028
$57$	−3.33441	−0.441654
$58$	4.78471	0.628264
$59$	12.4545	1.62144	0.810719	−	0.585436i	$-0.199077\pi$
$59$	12.4545	1.62144	0.810719	+	0.585436i	$0.199077\pi$
$60$	−0.993685	−0.128284
$61$	9.08882	1.16370	0.581852	−	0.813295i	$-0.302328\pi$
$61$	9.08882	1.16370	0.581852	+	0.813295i	$0.302328\pi$
$62$	−20.2392	−2.57038
$63$	12.2257	1.54029
$64$	−11.3344	−1.41680
$65$	−0.514245	−0.0637843
$66$	−2.51608	−0.309708
$67$	0.180480	0.0220491	0.0110246	−	0.999939i	$-0.496491\pi$
$67$	0.180480	0.0220491	0.0110246	+	0.999939i	$0.496491\pi$
$68$	9.99751	1.21238
$69$	−2.88832	−0.347713
$70$	−9.12077	−1.09014
$71$	−6.37142	−0.756149	−0.378074	−	0.925775i	$-0.623414\pi$
$71$	−6.37142	−0.756149	−0.378074	+	0.925775i	$0.623414\pi$
$72$	2.96446	0.349365
$73$	−13.1993	−1.54486	−0.772430	−	0.635099i	$-0.780959\pi$
$73$	−13.1993	−1.54486	−0.772430	+	0.635099i	$0.780959\pi$
$74$	−19.6363	−2.28268
$75$	0.398701	0.0460380
$76$	−20.8436	−2.39093
$77$	−12.8127	−1.46014
$78$	−0.434563	−0.0492045
$79$	−3.25875	−0.366638	−0.183319	−	0.983053i	$-0.558684\pi$
$79$	−3.25875	−0.366638	−0.183319	+	0.983053i	$0.558684\pi$
$80$	2.77303	0.310034
$81$	7.59461	0.843845
$82$	14.2953	1.57865
$83$	6.64864	0.729784	0.364892	−	0.931050i	$-0.381106\pi$
$83$	6.64864	0.729784	0.364892	+	0.931050i	$0.381106\pi$
$84$	−4.27608	−0.466558
$85$	−4.01135	−0.435092
$86$	−18.5043	−1.99537
$87$	−0.900055	−0.0964960
$88$	−3.10678	−0.331184
$89$	−4.82029	−0.510950	−0.255475	−	0.966816i	$-0.582232\pi$
$89$	−4.82029	−0.510950	−0.255475	+	0.966816i	$0.582232\pi$
$90$	−6.02160	−0.634732
$91$	−2.21293	−0.231978
$92$	−18.0551	−1.88237
$93$	3.80721	0.394789
$94$	−15.2518	−1.57311
$95$	8.36319	0.858045
$96$	3.17539	0.324087
$97$	−0.486097	−0.0493557	−0.0246778	−	0.999695i	$-0.507856\pi$
$97$	−0.486097	−0.0493557	−0.0246778	+	0.999695i	$0.507856\pi$
$98$	−24.4124	−2.46603
$99$	−8.45901	−0.850162
$100$	2.49230	0.249230
$101$	−0.319299	−0.0317714	−0.0158857	−	0.999874i	$-0.505057\pi$
$101$	−0.319299	−0.0317714	−0.0158857	+	0.999874i	$0.505057\pi$
$102$	−3.38979	−0.335639
$103$	−2.97654	−0.293287	−0.146644	−	0.989189i	$-0.546847\pi$
$103$	−2.97654	−0.293287	−0.146644	+	0.989189i	$0.546847\pi$
$104$	−0.536586	−0.0526166
$105$	1.71571	0.167436
$106$	25.9286	2.51841
$107$	13.0341	1.26006	0.630028	−	0.776572i	$-0.283043\pi$
$107$	13.0341	1.26006	0.630028	+	0.776572i	$0.283043\pi$
$108$	−5.80415	−0.558505
$109$	18.7773	1.79854	0.899271	−	0.437393i	$-0.144098\pi$
$109$	18.7773	1.79854	0.899271	+	0.437393i	$0.144098\pi$
$110$	6.31069	0.601701
$111$	3.69380	0.350600
$112$	11.9330	1.12757
$113$	4.60631	0.433325	0.216662	−	0.976247i	$-0.430483\pi$
$113$	4.60631	0.433325	0.216662	+	0.976247i	$0.430483\pi$
$114$	7.06731	0.661914
$115$	7.24433	0.675537
$116$	−5.62630	−0.522388
$117$	−1.46099	−0.135069
$118$	−26.3974	−2.43008
$119$	−17.2619	−1.58239
$120$	0.416022	0.0379774
$121$	−2.13488	−0.194080
$122$	−19.2638	−1.74406
$123$	−2.68909	−0.242467
$124$	23.7991	2.13722
$125$	−1.00000	−0.0894427
$126$	−25.9125	−2.30846
$127$	2.97448	0.263942	0.131971	−	0.991254i	$-0.457869\pi$
$127$	2.97448	0.263942	0.131971	+	0.991254i	$0.457869\pi$
$128$	8.09465	0.715473
$129$	3.48085	0.306472
$130$	1.08995	0.0955946
$131$	0.310578	0.0271354	0.0135677	−	0.999908i	$-0.495681\pi$
$131$	0.310578	0.0271354	0.0135677	+	0.999908i	$0.495681\pi$
$132$	2.95863	0.257516
$133$	35.9889	3.12063
$134$	−0.382528	−0.0330454
$135$	2.32883	0.200434
$136$	−4.18561	−0.358914
$137$	2.89159	0.247045	0.123522	−	0.992342i	$-0.460581\pi$
$137$	2.89159	0.247045	0.123522	+	0.992342i	$0.460581\pi$
$138$	6.12182	0.521124
$139$	0.184933	0.0156858	0.00784291	−	0.999969i	$-0.497503\pi$
$139$	0.184933	0.0156858	0.00784291	+	0.999969i	$0.497503\pi$
$140$	10.7250	0.906430
$141$	2.86903	0.241616
$142$	13.5043	1.13325
$143$	1.53113	0.128040
$144$	7.87827	0.656523
$145$	2.25747	0.187472
$146$	27.9760	2.31531
$147$	4.59224	0.378761
$148$	23.0902	1.89800
$149$	−21.7860	−1.78478	−0.892390	−	0.451265i	$-0.850973\pi$
$149$	−21.7860	−1.78478	−0.892390	+	0.451265i	$0.850973\pi$
$150$	−0.845049	−0.0689980
$151$	−17.0571	−1.38809	−0.694044	−	0.719933i	$-0.744173\pi$
$151$	−17.0571	−1.38809	−0.694044	+	0.719933i	$0.744173\pi$
$152$	8.72651	0.707813
$153$	−11.3964	−0.921344
$154$	27.1565	2.18833
$155$	−9.54902	−0.766996
$156$	0.510998	0.0409126
$157$	17.1553	1.36914	0.684571	−	0.728946i	$-0.259989\pi$
$157$	17.1553	1.36914	0.684571	+	0.728946i	$0.259989\pi$
$158$	6.90695	0.549487
$159$	−4.87745	−0.386807
$160$	−7.96433	−0.629636
$161$	31.1742	2.45687
$162$	−16.0968	−1.26468
$163$	−4.96576	−0.388948	−0.194474	−	0.980908i	$-0.562300\pi$
$163$	−4.96576	−0.388948	−0.194474	+	0.980908i	$0.562300\pi$
$164$	−16.8097	−1.31262
$165$	−1.18711	−0.0924161
$166$	−14.0918	−1.09374
$167$	19.0461	1.47383	0.736916	−	0.675984i	$-0.236281\pi$
$167$	19.0461	1.47383	0.736916	+	0.675984i	$0.236281\pi$
$168$	1.79025	0.138121
$169$	−12.7356	−0.979658
$170$	8.50208	0.652080
$171$	23.7601	1.81698
$172$	21.7590	1.65911
$173$	14.8878	1.13190	0.565948	−	0.824441i	$-0.308510\pi$
$173$	14.8878	1.13190	0.565948	+	0.824441i	$0.308510\pi$
$174$	1.90767	0.144620
$175$	−4.30325	−0.325295
$176$	−8.25651	−0.622358
$177$	4.96562	0.373239
$178$	10.2166	0.765769
$179$	−0.00165853	−0.000123964 0	−6.19821e−5	−	1.00000i	$-0.500020\pi$
$179$	−0.00165853	−0.000123964 0	−6.19821e−5		1.00000i	$0.500020\pi$
$180$	7.08073	0.527767
$181$	−0.556153	−0.0413385	−0.0206693	−	0.999786i	$-0.506580\pi$
$181$	−0.556153	−0.0413385	−0.0206693	+	0.999786i	$0.506580\pi$
$182$	4.69031	0.347669
$183$	3.62372	0.267873
$184$	7.55905	0.557260
$185$	−9.26458	−0.681146
$186$	−8.06940	−0.591677
$187$	11.9435	0.873398
$188$	17.9345	1.30800
$189$	10.0215	0.728960
$190$	−17.7258	−1.28597
$191$	−8.13478	−0.588612	−0.294306	−	0.955711i	$-0.595088\pi$
$191$	−8.13478	−0.588612	−0.294306	+	0.955711i	$0.595088\pi$
$192$	−4.51904	−0.326133
$193$	−2.63917	−0.189972	−0.0949858	−	0.995479i	$-0.530281\pi$
$193$	−2.63917	−0.189972	−0.0949858	+	0.995479i	$0.530281\pi$
$194$	1.03029	0.0739702
$195$	−0.205030	−0.0146825
$196$	28.7063	2.05045
$197$	10.0948	0.719224	0.359612	−	0.933102i	$-0.382909\pi$
$197$	10.0948	0.719224	0.359612	+	0.933102i	$0.382909\pi$
$198$	17.9289	1.27415
$199$	3.89756	0.276291	0.138145	−	0.990412i	$-0.455886\pi$
$199$	3.89756	0.276291	0.138145	+	0.990412i	$0.455886\pi$
$200$	−1.04344	−0.0737826
$201$	0.0719575	0.00507549
$202$	0.676755	0.0476163
$203$	9.71445	0.681821
$204$	3.98602	0.279077
$205$	6.74463	0.471066
$206$	6.30880	0.439555
$207$	20.5814	1.43051
$208$	−1.42602	−0.0988764
$209$	−24.9009	−1.72243
$210$	−3.63646	−0.250940
$211$	−12.2270	−0.841742	−0.420871	−	0.907121i	$-0.638275\pi$
$211$	−12.2270	−0.841742	−0.420871	+	0.907121i	$0.638275\pi$
$212$	−30.4892	−2.09401
$213$	−2.54029	−0.174058
$214$	−27.6259	−1.88847
$215$	−8.73048	−0.595414
$216$	2.43000	0.165341
$217$	−41.0919	−2.78950
$218$	−39.7986	−2.69550
$219$	−5.26257	−0.355612
$220$	−7.42068	−0.500302
$221$	2.06282	0.138760
$222$	−7.82903	−0.525450
$223$	−5.89963	−0.395068	−0.197534	−	0.980296i	$-0.563293\pi$
$223$	−5.89963	−0.395068	−0.197534	+	0.980296i	$0.563293\pi$
$224$	−34.2725	−2.28993
$225$	−2.84104	−0.189402
$226$	−9.76310	−0.649431
$227$	4.08252	0.270966	0.135483	−	0.990780i	$-0.456741\pi$
$227$	4.08252	0.270966	0.135483	+	0.990780i	$0.456741\pi$
$228$	−8.31037	−0.550368
$229$	21.4333	1.41635	0.708177	−	0.706035i	$-0.249518\pi$
$229$	21.4333	1.41635	0.708177	+	0.706035i	$0.249518\pi$
$230$	−15.3544	−1.01244
$231$	−5.10842	−0.336109
$232$	2.35554	0.154649
$233$	−14.2799	−0.935507	−0.467753	−	0.883859i	$-0.654936\pi$
$233$	−14.2799	−0.935507	−0.467753	+	0.883859i	$0.654936\pi$
$234$	3.09658	0.202430
$235$	−7.19593	−0.469411
$236$	31.0404	2.02056
$237$	−1.29927	−0.0843965
$238$	36.5866	2.37156
$239$	−21.8254	−1.41177	−0.705883	−	0.708329i	$-0.749449\pi$
$239$	−21.8254	−1.41177	−0.705883	+	0.708329i	$0.749449\pi$
$240$	1.10561	0.0713667
$241$	−28.8587	−1.85896	−0.929478	−	0.368879i	$-0.879742\pi$
$241$	−28.8587	−1.85896	−0.929478	+	0.368879i	$0.879742\pi$
$242$	4.52488	0.290870
$243$	10.0145	0.642428
$244$	22.6521	1.45015
$245$	−11.5180	−0.735857
$246$	5.69955	0.363390
$247$	−4.30073	−0.273649
$248$	−9.96386	−0.632706
$249$	2.65082	0.167989
$250$	2.11951	0.134049
$251$	15.9044	1.00388	0.501938	−	0.864903i	$-0.332620\pi$
$251$	15.9044	1.00388	0.501938	+	0.864903i	$0.332620\pi$
$252$	30.4702	1.91944
$253$	−21.5695	−1.35606
$254$	−6.30443	−0.395575
$255$	−1.59933	−0.100154
$256$	5.51213	0.344508
$257$	1.02339	0.0638375	0.0319187	−	0.999490i	$-0.489838\pi$
$257$	1.02339	0.0638375	0.0319187	+	0.999490i	$0.489838\pi$
$258$	−7.37768	−0.459315
$259$	−39.8678	−2.47727
$260$	−1.28166	−0.0794849
$261$	6.41355	0.396989
$262$	−0.658273	−0.0406682
$263$	14.7839	0.911612	0.455806	−	0.890079i	$-0.349351\pi$
$263$	14.7839	0.911612	0.455806	+	0.890079i	$0.349351\pi$
$264$	−1.23868	−0.0762354
$265$	12.2333	0.751489
$266$	−76.2787	−4.67695
$267$	−1.92186	−0.117616
$268$	0.449810	0.0274765
$269$	−19.2258	−1.17222	−0.586109	−	0.810232i	$-0.699341\pi$
$269$	−19.2258	−1.17222	−0.586109	+	0.810232i	$0.699341\pi$
$270$	−4.93596	−0.300393
$271$	−14.1330	−0.858519	−0.429259	−	0.903181i	$-0.641225\pi$
$271$	−14.1330	−0.858519	−0.429259	+	0.903181i	$0.641225\pi$
$272$	−11.1236	−0.674466
$273$	−0.882297	−0.0533990
$274$	−6.12874	−0.370250
$275$	2.97744	0.179546
$276$	−7.19858	−0.433304
$277$	−12.3574	−0.742483	−0.371242	−	0.928536i	$-0.621068\pi$
$277$	−12.3574	−0.742483	−0.371242	+	0.928536i	$0.621068\pi$
$278$	−0.391967	−0.0235086
$279$	−27.1291	−1.62418
$280$	−4.49020	−0.268341
$281$	29.7074	1.77220	0.886099	−	0.463496i	$-0.153405\pi$
$281$	29.7074	1.77220	0.886099	+	0.463496i	$0.153405\pi$
$282$	−6.08092	−0.362113
$283$	17.0426	1.01308	0.506539	−	0.862217i	$-0.330925\pi$
$283$	17.0426	1.01308	0.506539	+	0.862217i	$0.330925\pi$
$284$	−15.8795	−0.942276
$285$	3.33441	0.197514
$286$	−3.24524	−0.191895
$287$	29.0239	1.71322
$288$	−22.6270	−1.33331
$289$	−0.909073	−0.0534749
$290$	−4.78471	−0.280968
$291$	−0.193807	−0.0113612
$292$	−32.8967	−1.92513
$293$	19.4444	1.13595	0.567977	−	0.823045i	$-0.307726\pi$
$293$	19.4444	1.13595	0.567977	+	0.823045i	$0.307726\pi$
$294$	−9.73327	−0.567656
$295$	−12.4545	−0.725129
$296$	−9.66706	−0.561887
$297$	−6.93394	−0.402348
$298$	46.1756	2.67488
$299$	−3.72536	−0.215443
$300$	0.993685	0.0573704
$301$	−37.5695	−2.16547
$302$	36.1526	2.08035
$303$	−0.127305	−0.00731346
$304$	23.1913	1.33011
$305$	−9.08882	−0.520424
$306$	24.1547	1.38083
$307$	−10.4593	−0.596945	−0.298473	−	0.954418i	$-0.596477\pi$
$307$	−10.4593	−0.596945	−0.298473	+	0.954418i	$0.596477\pi$
$308$	−31.9331	−1.81955
$309$	−1.18675	−0.0675119
$310$	20.2392	1.14951
$311$	−18.5664	−1.05280	−0.526402	−	0.850236i	$-0.676459\pi$
$311$	−18.5664	−1.05280	−0.526402	+	0.850236i	$0.676459\pi$
$312$	−0.213937	−0.0121118
$313$	22.0493	1.24630	0.623151	−	0.782102i	$-0.285852\pi$
$313$	22.0493	1.24630	0.623151	+	0.782102i	$0.285852\pi$
$314$	−36.3608	−2.05196
$315$	−12.2257	−0.688840
$316$	−8.12181	−0.456887
$317$	0.843415	0.0473709	0.0236855	−	0.999719i	$-0.492460\pi$
$317$	0.843415	0.0473709	0.0236855	+	0.999719i	$0.492460\pi$
$318$	10.3378	0.579714
$319$	−6.72146	−0.376330
$320$	11.3344	0.633612
$321$	5.19672	0.290053
$322$	−66.0739	−3.68215
$323$	−33.5477	−1.86664
$324$	18.9281	1.05156
$325$	0.514245	0.0285252
$326$	10.5250	0.582923
$327$	7.48654	0.414006
$328$	7.03764	0.388589
$329$	−30.9659	−1.70721
$330$	2.51608	0.138506
$331$	13.1011	0.720099	0.360050	−	0.932933i	$-0.382760\pi$
$331$	13.1011	0.720099	0.360050	+	0.932933i	$0.382760\pi$
$332$	16.5704	0.909422
$333$	−26.3210	−1.44238
$334$	−40.3683	−2.20886
$335$	−0.180480	−0.00986066
$336$	4.75771	0.259555
$337$	−14.3386	−0.781074	−0.390537	−	0.920587i	$-0.627711\pi$
$337$	−14.3386	−0.781074	−0.390537	+	0.920587i	$0.627711\pi$
$338$	26.9931	1.46823
$339$	1.83654	0.0997472
$340$	−9.99751	−0.542191
$341$	28.4316	1.53966
$342$	−50.3597	−2.72314
$343$	−19.4421	−1.04977
$344$	−9.10976	−0.491165
$345$	2.88832	0.155502
$346$	−31.5547	−1.69639
$347$	−17.1263	−0.919390	−0.459695	−	0.888077i	$-0.652041\pi$
$347$	−17.1263	−0.919390	−0.459695	+	0.888077i	$0.652041\pi$
$348$	−2.24321	−0.120249
$349$	−13.9144	−0.744821	−0.372411	−	0.928068i	$-0.621469\pi$
$349$	−13.9144	−0.744821	−0.372411	+	0.928068i	$0.621469\pi$
$350$	9.12077	0.487526
$351$	−1.19759	−0.0639226
$352$	23.7133	1.26392
$353$	8.07188	0.429623	0.214811	−	0.976656i	$-0.431086\pi$
$353$	8.07188	0.429623	0.214811	+	0.976656i	$0.431086\pi$
$354$	−10.5247	−0.559380
$355$	6.37142	0.338160
$356$	−12.0136	−0.636721
$357$	−6.88232	−0.364251
$358$	0.00351526	0.000185787 0
$359$	2.53064	0.133562	0.0667810	−	0.997768i	$-0.478727\pi$
$359$	2.53064	0.133562	0.0667810	+	0.997768i	$0.478727\pi$
$360$	−2.96446	−0.156241
$361$	50.9429	2.68121
$362$	1.17877	0.0619547
$363$	−0.851177	−0.0446752
$364$	−5.51529	−0.289080
$365$	13.1993	0.690883
$366$	−7.68050	−0.401466
$367$	10.3920	0.542456	0.271228	−	0.962515i	$-0.412570\pi$
$367$	10.3920	0.542456	0.271228	+	0.962515i	$0.412570\pi$
$368$	20.0887	1.04720
$369$	19.1618	0.997521
$370$	19.6363	1.02084
$371$	52.6432	2.73310
$372$	9.48872	0.491967
$373$	18.5975	0.962941	0.481470	−	0.876462i	$-0.340103\pi$
$373$	18.5975	0.962941	0.481470	+	0.876462i	$0.340103\pi$
$374$	−25.3144	−1.30898
$375$	−0.398701	−0.0205888
$376$	−7.50855	−0.387224
$377$	−1.16089	−0.0597890
$378$	−21.2407	−1.09250
$379$	−25.0458	−1.28651	−0.643257	−	0.765650i	$-0.722417\pi$
$379$	−25.0458	−1.28651	−0.643257	+	0.765650i	$0.722417\pi$
$380$	20.8436	1.06925
$381$	1.18593	0.0607569
$382$	17.2417	0.882162
$383$	11.6996	0.597819	0.298910	−	0.954281i	$-0.403377\pi$
$383$	11.6996	0.597819	0.298910	+	0.954281i	$0.403377\pi$
$384$	3.22735	0.164695
$385$	12.8127	0.652994
$386$	5.59374	0.284714
$387$	−24.8036	−1.26084
$388$	−1.21150	−0.0615047
$389$	1.59666	0.0809540	0.0404770	−	0.999180i	$-0.487112\pi$
$389$	1.59666	0.0809540	0.0404770	+	0.999180i	$0.487112\pi$
$390$	0.434563	0.0220049
$391$	−29.0595	−1.46960
$392$	−12.0184	−0.607019
$393$	0.123828	0.00624629
$394$	−21.3960	−1.07791
$395$	3.25875	0.163966
$396$	−21.0824	−1.05943
$397$	23.3527	1.17204	0.586020	−	0.810296i	$-0.300694\pi$
$397$	23.3527	1.17204	0.586020	+	0.810296i	$0.300694\pi$
$398$	−8.26090	−0.414081
$399$	14.3488	0.718339
$400$	−2.77303	−0.138651
$401$	20.7024	1.03383	0.516913	−	0.856038i	$-0.327081\pi$
$401$	20.7024	1.03383	0.516913	+	0.856038i	$0.327081\pi$
$402$	−0.152514	−0.00760672
$403$	4.91054	0.244611
$404$	−0.795789	−0.0395920
$405$	−7.59461	−0.377379
$406$	−20.5898	−1.02186
$407$	27.5847	1.36732
$408$	−1.66881	−0.0826184
$409$	−27.8812	−1.37864	−0.689319	−	0.724458i	$-0.742090\pi$
$409$	−27.8812	−1.37864	−0.689319	+	0.724458i	$0.742090\pi$
$410$	−14.2953	−0.705994
$411$	1.15288	0.0568673
$412$	−7.41845	−0.365481
$413$	−53.5949	−2.63723
$414$	−43.6224	−2.14393
$415$	−6.64864	−0.326369
$416$	4.09562	0.200804
$417$	0.0737330	0.00361072
$418$	52.7775	2.58143
$419$	−25.2032	−1.23126	−0.615629	−	0.788036i	$-0.711098\pi$
$419$	−25.2032	−1.23126	−0.615629	+	0.788036i	$0.711098\pi$
$420$	4.27608	0.208651
$421$	−2.25887	−0.110091	−0.0550453	−	0.998484i	$-0.517530\pi$
$421$	−2.25887	−0.110091	−0.0550453	+	0.998484i	$0.517530\pi$
$422$	25.9152	1.26153
$423$	−20.4439	−0.994017
$424$	12.7648	0.619914
$425$	4.01135	0.194579
$426$	5.38416	0.260864
$427$	−39.1115	−1.89274
$428$	32.4850	1.57022
$429$	0.610464	0.0294735
$430$	18.5043	0.892357
$431$	5.02868	0.242223	0.121112	−	0.992639i	$-0.461354\pi$
$431$	5.02868	0.242223	0.121112	+	0.992639i	$0.461354\pi$
$432$	6.45790	0.310706
$433$	−2.35693	−0.113267	−0.0566335	−	0.998395i	$-0.518037\pi$
$433$	−2.35693	−0.113267	−0.0566335	+	0.998395i	$0.518037\pi$
$434$	87.0945	4.18067
$435$	0.900055	0.0431543
$436$	46.7988	2.24126
$437$	60.5857	2.89821
$438$	11.1541	0.532961
$439$	−17.7155	−0.845517	−0.422758	−	0.906242i	$-0.638938\pi$
$439$	−17.7155	−0.845517	−0.422758	+	0.906242i	$0.638938\pi$
$440$	3.10678	0.148110
$441$	−32.7230	−1.55824
$442$	−4.37215	−0.207962
$443$	−36.1564	−1.71784	−0.858922	−	0.512107i	$-0.828865\pi$
$443$	−36.1564	−1.71784	−0.858922	+	0.512107i	$0.828865\pi$
$444$	9.20607	0.436901
$445$	4.82029	0.228504
$446$	12.5043	0.592095
$447$	−8.68611	−0.410839
$448$	48.7748	2.30439
$449$	−3.51789	−0.166020	−0.0830098	−	0.996549i	$-0.526453\pi$
$449$	−3.51789	−0.166020	−0.0830098	+	0.996549i	$0.526453\pi$
$450$	6.02160	0.283861
$451$	−20.0817	−0.945611
$452$	11.4803	0.539989
$453$	−6.80069	−0.319524
$454$	−8.65292	−0.406102
$455$	2.21293	0.103744
$456$	3.47927	0.162932
$457$	2.98318	0.139547	0.0697736	−	0.997563i	$-0.477772\pi$
$457$	2.98318	0.139547	0.0697736	+	0.997563i	$0.477772\pi$
$458$	−45.4281	−2.12271
$459$	−9.34174	−0.436035
$460$	18.0551	0.841823
$461$	2.60707	0.121423	0.0607116	−	0.998155i	$-0.480663\pi$
$461$	2.60707	0.121423	0.0607116	+	0.998155i	$0.480663\pi$
$462$	10.8273	0.503733
$463$	−37.8063	−1.75701	−0.878504	−	0.477736i	$-0.841458\pi$
$463$	−37.8063	−1.75701	−0.878504	+	0.477736i	$0.841458\pi$
$464$	6.26001	0.290614
$465$	−3.80721	−0.176555
$466$	30.2663	1.40206
$467$	−7.22233	−0.334210	−0.167105	−	0.985939i	$-0.553442\pi$
$467$	−7.22233	−0.334210	−0.167105	+	0.985939i	$0.553442\pi$
$468$	−3.64123	−0.168316
$469$	−0.776650	−0.0358624
$470$	15.2518	0.703514
$471$	6.83984	0.315163
$472$	−12.9956	−0.598169
$473$	25.9944	1.19523
$474$	2.75381	0.126487
$475$	−8.36319	−0.383729
$476$	−43.0218	−1.97190
$477$	34.7554	1.59134
$478$	46.2590	2.11584
$479$	−35.4918	−1.62166	−0.810831	−	0.585281i	$-0.800984\pi$
$479$	−35.4918	−1.62166	−0.810831	+	0.585281i	$0.800984\pi$
$480$	−3.17539	−0.144936
$481$	4.76427	0.217232
$482$	61.1663	2.78605
$483$	12.4292	0.565547
$484$	−5.32076	−0.241853
$485$	0.486097	0.0220725
$486$	−21.2257	−0.962818
$487$	−22.9084	−1.03808	−0.519040	−	0.854750i	$-0.673710\pi$
$487$	−22.9084	−1.03808	−0.519040	+	0.854750i	$0.673710\pi$
$488$	−9.48366	−0.429305
$489$	−1.97985	−0.0895321
$490$	24.4124	1.10284
$491$	−0.688062	−0.0310518	−0.0155259	−	0.999879i	$-0.504942\pi$
$491$	−0.688062	−0.0310518	−0.0155259	+	0.999879i	$0.504942\pi$
$492$	−6.70204	−0.302151
$493$	−9.05549	−0.407839
$494$	9.11542	0.410122
$495$	8.45901	0.380204
$496$	−26.4797	−1.18897
$497$	27.4178	1.22986
$498$	−5.61843	−0.251768
$499$	16.6745	0.746455	0.373227	−	0.927740i	$-0.378251\pi$
$499$	16.6745	0.746455	0.373227	+	0.927740i	$0.378251\pi$
$500$	−2.49230	−0.111459
$501$	7.59370	0.339262
$502$	−33.7095	−1.50453
$503$	12.8407	0.572538	0.286269	−	0.958149i	$-0.407585\pi$
$503$	12.8407	0.572538	0.286269	+	0.958149i	$0.407585\pi$
$504$	−12.7568	−0.568234
$505$	0.319299	0.0142086
$506$	45.7167	2.03236
$507$	−5.07768	−0.225508
$508$	7.41331	0.328912
$509$	12.9690	0.574843	0.287421	−	0.957804i	$-0.407202\pi$
$509$	12.9690	0.574843	0.287421	+	0.957804i	$0.407202\pi$
$510$	3.38979	0.150102
$511$	56.7999	2.51268
$512$	−27.8723	−1.23179
$513$	19.4764	0.859905
$514$	−2.16909	−0.0956743
$515$	2.97654	0.131162
$516$	8.67534	0.381911
$517$	21.4254	0.942289
$518$	84.5001	3.71272
$519$	5.93577	0.260551
$520$	0.536586	0.0235308
$521$	21.2399	0.930536	0.465268	−	0.885170i	$-0.345958\pi$
$521$	21.2399	0.930536	0.465268	+	0.885170i	$0.345958\pi$
$522$	−13.5936	−0.594974
$523$	−31.7788	−1.38959	−0.694795	−	0.719208i	$-0.744505\pi$
$523$	−31.7788	−1.38959	−0.694795	+	0.719208i	$0.744505\pi$
$524$	0.774056	0.0338148
$525$	−1.71571	−0.0748798
$526$	−31.3345	−1.36625
$527$	38.3045	1.66857
$528$	−3.29188	−0.143261
$529$	29.4803	1.28175
$530$	−25.9286	−1.12627
$531$	−35.3837	−1.53552
$532$	89.6953	3.88879
$533$	−3.46840	−0.150233
$534$	4.07338	0.176273
$535$	−13.0341	−0.563514
$536$	−0.188320	−0.00813420
$537$	−0.000661257 0	−2.85354e−5 0
$538$	40.7492	1.75682
$539$	34.2941	1.47715
$540$	5.80415	0.249771
$541$	4.22434	0.181619	0.0908093	−	0.995868i	$-0.471055\pi$
$541$	4.22434	0.181619	0.0908093	+	0.995868i	$0.471055\pi$
$542$	29.9550	1.28668
$543$	−0.221739	−0.00951572
$544$	31.9477	1.36975
$545$	−18.7773	−0.804332
$546$	1.87003	0.0800300
$547$	29.2877	1.25225	0.626126	−	0.779722i	$-0.284640\pi$
$547$	29.2877	1.25225	0.626126	+	0.779722i	$0.284640\pi$
$548$	7.20672	0.307856
$549$	−25.8217	−1.10204
$550$	−6.31069	−0.269089
$551$	18.8796	0.804299
$552$	3.01380	0.128276
$553$	14.0232	0.596329
$554$	26.1915	1.11277
$555$	−3.69380	−0.156793
$556$	0.460910	0.0195469
$557$	5.86724	0.248603	0.124301	−	0.992244i	$-0.460331\pi$
$557$	5.86724	0.248603	0.124301	+	0.992244i	$0.460331\pi$
$558$	57.5004	2.43418
$559$	4.48961	0.189890
$560$	−11.9330	−0.504263
$561$	4.76190	0.201048
$562$	−62.9651	−2.65602
$563$	−26.8040	−1.12966	−0.564828	−	0.825209i	$-0.691057\pi$
$563$	−26.8040	−1.12966	−0.564828	+	0.825209i	$0.691057\pi$
$564$	7.15049	0.301090
$565$	−4.60631	−0.193789
$566$	−36.1219	−1.51832
$567$	−32.6815	−1.37249
$568$	6.64822	0.278953
$569$	−16.8386	−0.705911	−0.352955	−	0.935640i	$-0.614823\pi$
$569$	−16.8386	−0.705911	−0.352955	+	0.935640i	$0.614823\pi$
$570$	−7.06731	−0.296017
$571$	−36.5651	−1.53020	−0.765100	−	0.643911i	$-0.777311\pi$
$571$	−36.5651	−1.53020	−0.765100	+	0.643911i	$0.777311\pi$
$572$	3.81605	0.159557
$573$	−3.24334	−0.135493
$574$	−61.5163	−2.56764
$575$	−7.24433	−0.302110
$576$	32.2014	1.34173
$577$	−24.7590	−1.03073	−0.515366	−	0.856970i	$-0.672344\pi$
$577$	−24.7590	−1.03073	−0.515366	+	0.856970i	$0.672344\pi$
$578$	1.92679	0.0801437
$579$	−1.05224	−0.0437296
$580$	5.62630	0.233619
$581$	−28.6108	−1.18698
$582$	0.410776	0.0170272
$583$	−36.4240	−1.50853
$584$	13.7727	0.569919
$585$	1.46099	0.0604045
$586$	−41.2125	−1.70247
$587$	−32.0198	−1.32160	−0.660800	−	0.750562i	$-0.729783\pi$
$587$	−32.0198	−1.32160	−0.660800	+	0.750562i	$0.729783\pi$
$588$	11.4452	0.471994
$589$	−79.8603	−3.29059
$590$	26.3974	1.08676
$591$	4.02481	0.165558
$592$	−25.6909	−1.05589
$593$	6.79758	0.279143	0.139571	−	0.990212i	$-0.455427\pi$
$593$	6.79758	0.279143	0.139571	+	0.990212i	$0.455427\pi$
$594$	14.6965	0.603005
$595$	17.2619	0.707667
$596$	−54.2974	−2.22411
$597$	1.55396	0.0635994
$598$	7.89593	0.322889
$599$	−29.6893	−1.21307	−0.606537	−	0.795055i	$-0.707442\pi$
$599$	−29.6893	−1.21307	−0.606537	+	0.795055i	$0.707442\pi$
$600$	−0.416022	−0.0169840
$601$	9.21440	0.375863	0.187932	−	0.982182i	$-0.439822\pi$
$601$	9.21440	0.375863	0.187932	+	0.982182i	$0.439822\pi$
$602$	79.6287	3.24542
$603$	−0.512750	−0.0208808
$604$	−42.5115	−1.72977
$605$	2.13488	0.0867950
$606$	0.269823	0.0109608
$607$	−25.8348	−1.04860	−0.524300	−	0.851533i	$-0.675673\pi$
$607$	−25.8348	−1.04860	−0.524300	+	0.851533i	$0.675673\pi$
$608$	−66.6072	−2.70128
$609$	3.87316	0.156948
$610$	19.2638	0.779968
$611$	3.70047	0.149705
$612$	−28.4033	−1.14813
$613$	41.1470	1.66191	0.830955	−	0.556339i	$-0.187795\pi$
$613$	41.1470	1.66191	0.830955	+	0.556339i	$0.187795\pi$
$614$	22.1686	0.894652
$615$	2.68909	0.108435
$616$	13.3693	0.538664
$617$	−16.3659	−0.658868	−0.329434	−	0.944179i	$-0.606858\pi$
$617$	−16.3659	−0.658868	−0.329434	+	0.944179i	$0.606858\pi$
$618$	2.51532	0.101181
$619$	19.7285	0.792956	0.396478	−	0.918044i	$-0.370232\pi$
$619$	19.7285	0.792956	0.396478	+	0.918044i	$0.370232\pi$
$620$	−23.7991	−0.955794
$621$	16.8708	0.677002
$622$	39.3516	1.57785
$623$	20.7429	0.831048
$624$	−0.568554	−0.0227604
$625$	1.00000	0.0400000
$626$	−46.7337	−1.86785
$627$	−9.92800	−0.396486
$628$	42.7563	1.70616
$629$	37.1635	1.48180
$630$	25.9125	1.03238
$631$	−10.2062	−0.406301	−0.203150	−	0.979148i	$-0.565118\pi$
$631$	−10.2062	−0.406301	−0.203150	+	0.979148i	$0.565118\pi$
$632$	3.40032	0.135258
$633$	−4.87492	−0.193761
$634$	−1.78762	−0.0709956
$635$	−2.97448	−0.118039
$636$	−12.1561	−0.482020
$637$	5.92307	0.234681
$638$	14.2462	0.564012
$639$	18.1014	0.716082
$640$	−8.09465	−0.319969
$641$	−28.8232	−1.13845	−0.569224	−	0.822182i	$-0.692756\pi$
$641$	−28.8232	−1.13845	−0.569224	+	0.822182i	$0.692756\pi$
$642$	−11.0145	−0.434707
$643$	−17.4187	−0.686928	−0.343464	−	0.939166i	$-0.611600\pi$
$643$	−17.4187	−0.686928	−0.343464	+	0.939166i	$0.611600\pi$
$644$	77.6956	3.06164
$645$	−3.48085	−0.137058
$646$	71.1045	2.79757
$647$	43.1838	1.69773	0.848865	−	0.528610i	$-0.177286\pi$
$647$	43.1838	1.69773	0.848865	+	0.528610i	$0.177286\pi$
$648$	−7.92454	−0.311305
$649$	37.0825	1.45561
$650$	−1.08995	−0.0427512
$651$	−16.3834	−0.642115
$652$	−12.3762	−0.484689
$653$	16.1806	0.633195	0.316598	−	0.948560i	$-0.397459\pi$
$653$	16.1806	0.633195	0.316598	+	0.948560i	$0.397459\pi$
$654$	−15.8678	−0.620479
$655$	−0.310578	−0.0121353
$656$	18.7030	0.730231
$657$	37.4997	1.46300
$658$	65.6324	2.55862
$659$	−16.8066	−0.654691	−0.327345	−	0.944905i	$-0.606154\pi$
$659$	−16.8066	−0.654691	−0.327345	+	0.944905i	$0.606154\pi$
$660$	−2.95863	−0.115165
$661$	37.0318	1.44037	0.720185	−	0.693782i	$-0.244057\pi$
$661$	37.0318	1.44037	0.720185	+	0.693782i	$0.244057\pi$
$662$	−27.7678	−1.07922
$663$	0.822448	0.0319412
$664$	−6.93748	−0.269227
$665$	−35.9889	−1.39559
$666$	55.7876	2.16172
$667$	16.3538	0.633223
$668$	47.4687	1.83662
$669$	−2.35219	−0.0909408
$670$	0.382528	0.0147783
$671$	27.0614	1.04469
$672$	−13.6645	−0.527119
$673$	−25.6665	−0.989372	−0.494686	−	0.869072i	$-0.664717\pi$
$673$	−25.6665	−0.989372	−0.494686	+	0.869072i	$0.664717\pi$
$674$	30.3908	1.17061
$675$	−2.32883	−0.0896366
$676$	−31.7409	−1.22080
$677$	0.589825	0.0226688	0.0113344	−	0.999936i	$-0.496392\pi$
$677$	0.589825	0.0226688	0.0113344	+	0.999936i	$0.496392\pi$
$678$	−3.89256	−0.149493
$679$	2.09180	0.0802759
$680$	4.18561	0.160511
$681$	1.62770	0.0623738
$682$	−60.2610	−2.30751
$683$	−44.3929	−1.69865	−0.849323	−	0.527873i	$-0.822990\pi$
$683$	−44.3929	−1.69865	−0.849323	+	0.527873i	$0.822990\pi$
$684$	59.2175	2.26424
$685$	−2.89159	−0.110482
$686$	41.2075	1.57331
$687$	8.54549	0.326031
$688$	−24.2098	−0.922992
$689$	−6.29094	−0.239666
$690$	−6.12182	−0.233054
$691$	−29.0534	−1.10524	−0.552622	−	0.833432i	$-0.686373\pi$
$691$	−29.0534	−1.10524	−0.552622	+	0.833432i	$0.686373\pi$
$692$	37.1048	1.41051
$693$	36.4012	1.38277
$694$	36.2994	1.37791
$695$	−0.184933	−0.00701491
$696$	0.939156	0.0355986
$697$	−27.0551	−1.02478
$698$	29.4917	1.11628
$699$	−5.69341	−0.215344
$700$	−10.7250	−0.405368
$701$	−11.4628	−0.432943	−0.216471	−	0.976289i	$-0.569455\pi$
$701$	−11.4628	−0.432943	−0.216471	+	0.976289i	$0.569455\pi$
$702$	2.53830	0.0958018
$703$	−77.4814	−2.92227
$704$	−33.7474	−1.27190
$705$	−2.86903	−0.108054
$706$	−17.1084	−0.643883
$707$	1.37402	0.0516754
$708$	12.3758	0.465113
$709$	−25.4479	−0.955716	−0.477858	−	0.878437i	$-0.658587\pi$
$709$	−25.4479	−0.955716	−0.477858	+	0.878437i	$0.658587\pi$
$710$	−13.5043	−0.506806
$711$	9.25824	0.347211
$712$	5.02970	0.188496
$713$	−69.1763	−2.59067
$714$	14.5871	0.545909
$715$	−1.53113	−0.0572611
$716$	−0.00413356	−0.000154478 0
$717$	−8.70180	−0.324975
$718$	−5.36370	−0.200172
$719$	43.6068	1.62626	0.813130	−	0.582082i	$-0.197762\pi$
$719$	43.6068	1.62626	0.813130	+	0.582082i	$0.197762\pi$
$720$	−7.87827	−0.293606
$721$	12.8088	0.477025
$722$	−107.974	−4.01837
$723$	−11.5060	−0.427913
$724$	−1.38610	−0.0515141
$725$	−2.25747	−0.0838402
$726$	1.80408	0.0669555
$727$	−10.4475	−0.387478	−0.193739	−	0.981053i	$-0.562061\pi$
$727$	−10.4475	−0.387478	−0.193739	+	0.981053i	$0.562061\pi$
$728$	2.30906	0.0855796
$729$	−18.7910	−0.695964
$730$	−27.9760	−1.03544
$731$	35.0210	1.29530
$732$	9.03142	0.333811
$733$	−26.7135	−0.986687	−0.493344	−	0.869834i	$-0.664225\pi$
$733$	−26.7135	−0.986687	−0.493344	+	0.869834i	$0.664225\pi$
$734$	−22.0258	−0.812987
$735$	−4.59224	−0.169387
$736$	−57.6963	−2.12671
$737$	0.537367	0.0197942
$738$	−40.6135	−1.49500
$739$	48.3600	1.77895	0.889475	−	0.456983i	$-0.151070\pi$
$739$	48.3600	1.77895	0.889475	+	0.456983i	$0.151070\pi$
$740$	−23.0902	−0.848811
$741$	−1.71471	−0.0629913
$742$	−111.578	−4.09614
$743$	−10.2845	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$743$	−10.2845	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$744$	−3.97260	−0.145643
$745$	21.7860	0.798178
$746$	−39.4175	−1.44318
$747$	−18.8890	−0.691114
$748$	29.7669	1.08839
$749$	−56.0891	−2.04945
$750$	0.845049	0.0308568
$751$	−12.1561	−0.443581	−0.221790	−	0.975094i	$-0.571190\pi$
$751$	−12.1561	−0.443581	−0.221790	+	0.975094i	$0.571190\pi$
$752$	−19.9545	−0.727666
$753$	6.34110	0.231083
$754$	2.46052	0.0896067
$755$	17.0571	0.620772
$756$	24.9767	0.908395
$757$	30.9949	1.12653	0.563264	−	0.826277i	$-0.309545\pi$
$757$	30.9949	1.12653	0.563264	+	0.826277i	$0.309545\pi$
$758$	53.0846	1.92812
$759$	−8.59980	−0.312153
$760$	−8.72651	−0.316544
$761$	40.6347	1.47301	0.736503	−	0.676434i	$-0.236476\pi$
$761$	40.6347	1.47301	0.736503	+	0.676434i	$0.236476\pi$
$762$	−2.51358	−0.0910575
$763$	−80.8036	−2.92529
$764$	−20.2743	−0.733500
$765$	11.3964	0.412038
$766$	−24.7973	−0.895962
$767$	6.40467	0.231259
$768$	2.19769	0.0793024
$769$	−43.7760	−1.57860	−0.789301	−	0.614006i	$-0.789557\pi$
$769$	−43.7760	−1.57860	−0.789301	+	0.614006i	$0.789557\pi$
$770$	−27.1565	−0.978653
$771$	0.408028	0.0146948
$772$	−6.57762	−0.236734
$773$	27.5709	0.991656	0.495828	−	0.868421i	$-0.334865\pi$
$773$	27.5709	0.991656	0.495828	+	0.868421i	$0.334865\pi$
$774$	52.5714	1.88964
$775$	9.54902	0.343011
$776$	0.507215	0.0182079
$777$	−15.8954	−0.570243
$778$	−3.38414	−0.121327
$779$	56.4066	2.02098
$780$	−0.510998	−0.0182967
$781$	−18.9705	−0.678818
$782$	61.5919	2.20252
$783$	5.25725	0.187879
$784$	−31.9397	−1.14070
$785$	−17.1553	−0.612299
$786$	−0.262454	−0.00936142
$787$	12.8776	0.459037	0.229519	−	0.973304i	$-0.426285\pi$
$787$	12.8776	0.459037	0.229519	+	0.973304i	$0.426285\pi$
$788$	25.1593	0.896263
$789$	5.89434	0.209844
$790$	−6.90695	−0.245738
$791$	−19.8221	−0.704793
$792$	8.82649	0.313636
$793$	4.67388	0.165974
$794$	−49.4963	−1.75656
$795$	4.87745	0.172985
$796$	9.71391	0.344300
$797$	−43.4639	−1.53957	−0.769784	−	0.638304i	$-0.779636\pi$
$797$	−43.4639	−1.53957	−0.769784	+	0.638304i	$0.779636\pi$
$798$	−30.4124	−1.07659
$799$	28.8654	1.02118
$800$	7.96433	0.281582
$801$	13.6946	0.483876
$802$	−43.8788	−1.54941
$803$	−39.3001	−1.38687
$804$	0.179340	0.00632483
$805$	−31.1742	−1.09875
$806$	−10.4079	−0.366603
$807$	−7.66535	−0.269833
$808$	0.333170	0.0117209
$809$	−16.4215	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$809$	−16.4215	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$810$	16.0968	0.565584
$811$	−46.7625	−1.64205	−0.821027	−	0.570890i	$-0.806598\pi$
$811$	−46.7625	−1.64205	−0.821027	+	0.570890i	$0.806598\pi$
$812$	24.2114	0.849653
$813$	−5.63484	−0.197623
$814$	−58.4659	−2.04923
$815$	4.96576	0.173943
$816$	−4.43498	−0.155255
$817$	−73.0146	−2.55446
$818$	59.0944	2.06619
$819$	6.28701	0.219686
$820$	16.8097	0.587020
$821$	−27.2183	−0.949925	−0.474962	−	0.880006i	$-0.657538\pi$
$821$	−27.2183	−0.949925	−0.474962	+	0.880006i	$0.657538\pi$
$822$	−2.44353	−0.0852280
$823$	−37.0398	−1.29112	−0.645562	−	0.763707i	$-0.723377\pi$
$823$	−37.0398	−1.29112	−0.645562	+	0.763707i	$0.723377\pi$
$824$	3.10585	0.108197
$825$	1.18711	0.0413298
$826$	113.595	3.95246
$827$	54.6797	1.90140	0.950701	−	0.310110i	$-0.100366\pi$
$827$	54.6797	1.90140	0.950701	+	0.310110i	$0.100366\pi$
$828$	51.2952	1.78263
$829$	−20.8311	−0.723495	−0.361748	−	0.932276i	$-0.617820\pi$
$829$	−20.8311	−0.723495	−0.361748	+	0.932276i	$0.617820\pi$
$830$	14.0918	0.489135
$831$	−4.92690	−0.170912
$832$	−5.82866	−0.202072
$833$	46.2027	1.60083
$834$	−0.156278	−0.00541145
$835$	−19.0461	−0.659118
$836$	−62.0605	−2.14641
$837$	−22.2380	−0.768659
$838$	53.4184	1.84531
$839$	40.4091	1.39508	0.697538	−	0.716548i	$-0.254279\pi$
$839$	40.4091	1.39508	0.697538	+	0.716548i	$0.254279\pi$
$840$	−1.79025	−0.0617694
$841$	−23.9038	−0.824270
$842$	4.78769	0.164995
$843$	11.8444	0.407942
$844$	−30.4734	−1.04894
$845$	12.7356	0.438116
$846$	43.3310	1.48975
$847$	9.18691	0.315666
$848$	33.9234	1.16493
$849$	6.79491	0.233201
$850$	−8.50208	−0.291619
$851$	−67.1157	−2.30070
$852$	−6.33118	−0.216903
$853$	21.1766	0.725074	0.362537	−	0.931969i	$-0.381911\pi$
$853$	21.1766	0.725074	0.362537	+	0.931969i	$0.381911\pi$
$854$	82.8970	2.83668
$855$	−23.7601	−0.812579
$856$	−13.6004	−0.464851
$857$	−12.7855	−0.436746	−0.218373	−	0.975865i	$-0.570075\pi$
$857$	−12.7855	−0.436746	−0.218373	+	0.975865i	$0.570075\pi$
$858$	−1.29388	−0.0441724
$859$	−47.8836	−1.63377	−0.816884	−	0.576802i	$-0.804300\pi$
$859$	−47.8836	−1.63377	−0.816884	+	0.576802i	$0.804300\pi$
$860$	−21.7590	−0.741976
$861$	11.5718	0.394368
$862$	−10.6583	−0.363024
$863$	3.19453	0.108743	0.0543716	−	0.998521i	$-0.482684\pi$
$863$	3.19453	0.108743	0.0543716	+	0.998521i	$0.482684\pi$
$864$	−18.5476	−0.631001
$865$	−14.8878	−0.506199
$866$	4.99553	0.169755
$867$	−0.362448	−0.0123094
$868$	−102.413	−3.47614
$869$	−9.70273	−0.329142
$870$	−1.90767	−0.0646761
$871$	0.0928108	0.00314478
$872$	−19.5931	−0.663505
$873$	1.38102	0.0467405
$874$	−128.412	−4.34359
$875$	4.30325	0.145477
$876$	−13.1159	−0.443147
$877$	−48.4805	−1.63707	−0.818536	−	0.574456i	$-0.805214\pi$
$877$	−48.4805	−1.63707	−0.818536	+	0.574456i	$0.805214\pi$
$878$	37.5482	1.26719
$879$	7.75250	0.261485
$880$	8.25651	0.278327
$881$	−13.3511	−0.449811	−0.224906	−	0.974381i	$-0.572207\pi$
$881$	−13.3511	−0.449811	−0.224906	+	0.974381i	$0.572207\pi$
$882$	69.3567	2.33536
$883$	−6.13493	−0.206457	−0.103228	−	0.994658i	$-0.532917\pi$
$883$	−6.13493	−0.206457	−0.103228	+	0.994658i	$0.532917\pi$
$884$	5.14117	0.172916
$885$	−4.96562	−0.166918
$886$	76.6337	2.57456
$887$	28.3462	0.951771	0.475886	−	0.879507i	$-0.342128\pi$
$887$	28.3462	0.951771	0.475886	+	0.879507i	$0.342128\pi$
$888$	−3.85427	−0.129341
$889$	−12.7999	−0.429296
$890$	−10.2166	−0.342462
$891$	22.6125	0.757546
$892$	−14.7037	−0.492315
$893$	−60.1809	−2.01388
$894$	18.4103	0.615731
$895$	0.00165853	5.54385e−5 0
$896$	−34.8333	−1.16370
$897$	−1.48531	−0.0495929
$898$	7.45619	0.248816
$899$	−21.5566	−0.718953
$900$	−7.08073	−0.236024
$901$	−49.0722	−1.63483
$902$	42.5633	1.41720
$903$	−14.9790	−0.498469
$904$	−4.80642	−0.159859
$905$	0.556153	0.0184871
$906$	14.4141	0.478876
$907$	15.2790	0.507331	0.253665	−	0.967292i	$-0.418364\pi$
$907$	15.2790	0.507331	0.253665	+	0.967292i	$0.418364\pi$
$908$	10.1749	0.337665
$909$	0.907139	0.0300879
$910$	−4.69031	−0.155482
$911$	35.9277	1.19034	0.595169	−	0.803600i	$-0.297085\pi$
$911$	35.9277	1.19034	0.595169	+	0.803600i	$0.297085\pi$
$912$	9.24641	0.306179
$913$	19.7959	0.655149
$914$	−6.32286	−0.209142
$915$	−3.62372	−0.119797
$916$	53.4184	1.76499
$917$	−1.33650	−0.0441350
$918$	19.7999	0.653493
$919$	−21.5834	−0.711972	−0.355986	−	0.934491i	$-0.615855\pi$
$919$	−21.5834	−0.711972	−0.355986	+	0.934491i	$0.615855\pi$
$920$	−7.55905	−0.249214
$921$	−4.17015	−0.137411
$922$	−5.52569	−0.181979
$923$	−3.27647	−0.107846
$924$	−12.7317	−0.418844
$925$	9.26458	0.304618
$926$	80.1306	2.63326
$927$	8.45647	0.277747
$928$	−17.9792	−0.590197
$929$	−6.33909	−0.207979	−0.103989	−	0.994578i	$-0.533161\pi$
$929$	−6.33909	−0.207979	−0.103989	+	0.994578i	$0.533161\pi$
$930$	8.06940	0.264606
$931$	−96.3271	−3.15699
$932$	−35.5898	−1.16578
$933$	−7.40244	−0.242345
$934$	15.3078	0.500886
$935$	−11.9435	−0.390595
$936$	1.52446	0.0498285
$937$	−27.2663	−0.890750	−0.445375	−	0.895344i	$-0.646930\pi$
$937$	−27.2663	−0.890750	−0.445375	+	0.895344i	$0.646930\pi$
$938$	1.64611	0.0537475
$939$	8.79109	0.286886
$940$	−17.9345	−0.584957
$941$	27.0650	0.882294	0.441147	−	0.897435i	$-0.354572\pi$
$941$	27.0650	0.882294	0.441147	+	0.897435i	$0.354572\pi$
$942$	−14.4971	−0.472340
$943$	48.8604	1.59111
$944$	−34.5367	−1.12407
$945$	−10.0215	−0.326001
$946$	−55.0954	−1.79130
$947$	−14.5782	−0.473727	−0.236864	−	0.971543i	$-0.576119\pi$
$947$	−14.5782	−0.473727	−0.236864	+	0.971543i	$0.576119\pi$
$948$	−3.23817	−0.105171
$949$	−6.78768	−0.220337
$950$	17.7258	0.575102
$951$	0.336271	0.0109043
$952$	18.0118	0.583765
$953$	31.2915	1.01363	0.506816	−	0.862054i	$-0.330822\pi$
$953$	31.2915	1.01363	0.506816	+	0.862054i	$0.330822\pi$
$954$	−73.6643	−2.38497
$955$	8.13478	0.263235
$956$	−54.3955	−1.75927
$957$	−2.67985	−0.0866274
$958$	75.2251	2.43041
$959$	−12.4432	−0.401813
$960$	4.51904	0.145851
$961$	60.1839	1.94141
$962$	−10.0979	−0.325569
$963$	−37.0304	−1.19329
$964$	−71.9248	−2.31654
$965$	2.63917	0.0849579
$966$	−26.3437	−0.847596
$967$	25.3826	0.816249	0.408125	−	0.912926i	$-0.366183\pi$
$967$	25.3826	0.816249	0.408125	+	0.912926i	$0.366183\pi$
$968$	2.22762	0.0715985
$969$	−13.3755	−0.429683
$970$	−1.03029	−0.0330805
$971$	43.2157	1.38686	0.693429	−	0.720525i	$-0.256099\pi$
$971$	43.2157	1.38686	0.693429	+	0.720525i	$0.256099\pi$
$972$	24.9591	0.800563
$973$	−0.795814	−0.0255126
$974$	48.5545	1.55579
$975$	0.205030	0.00656622
$976$	−25.2035	−0.806745
$977$	−7.79760	−0.249467	−0.124734	−	0.992190i	$-0.539808\pi$
$977$	−7.79760	−0.249467	−0.124734	+	0.992190i	$0.539808\pi$
$978$	4.19631	0.134183
$979$	−14.3521	−0.458695
$980$	−28.7063	−0.916990
$981$	−53.3471	−1.70324
$982$	1.45835	0.0465379
$983$	2.85657	0.0911106	0.0455553	−	0.998962i	$-0.485494\pi$
$983$	2.85657	0.0911106	0.0455553	+	0.998962i	$0.485494\pi$
$984$	2.80592	0.0894493
$985$	−10.0948	−0.321647
$986$	19.1932	0.611235
$987$	−12.3461	−0.392982
$988$	−10.7187	−0.341008
$989$	−63.2465	−2.01112
$990$	−17.9289	−0.569818
$991$	−35.5353	−1.12882	−0.564409	−	0.825496i	$-0.690896\pi$
$991$	−35.5353	−1.12882	−0.564409	+	0.825496i	$0.690896\pi$
$992$	76.0516	2.41464
$993$	5.22341	0.165760
$994$	−58.1123	−1.84321
$995$	−3.89756	−0.123561
$996$	6.60666	0.209340
$997$	10.2080	0.323291	0.161645	−	0.986849i	$-0.448320\pi$
$997$	10.2080	0.323291	0.161645	+	0.986849i	$0.448320\pi$
$998$	−35.3418	−1.11872
$999$	−21.5756	−0.682622

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.b.1.17	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.b.1.17	✓	126	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11951 q^{2} +0.398701 q^{3} +2.49230 q^{4} -1.00000 q^{5} -0.845049 q^{6} -4.30325 q^{7} -1.04344 q^{8} -2.84104 q^{9} +O(q^{10})\)	\(q-2.11951 q^{2} +0.398701 q^{3} +2.49230 q^{4} -1.00000 q^{5} -0.845049 q^{6} -4.30325 q^{7} -1.04344 q^{8} -2.84104 q^{9} +2.11951 q^{10} +2.97744 q^{11} +0.993685 q^{12} +0.514245 q^{13} +9.12077 q^{14} -0.398701 q^{15} -2.77303 q^{16} +4.01135 q^{17} +6.02160 q^{18} -8.36319 q^{19} -2.49230 q^{20} -1.71571 q^{21} -6.31069 q^{22} -7.24433 q^{23} -0.416022 q^{24} +1.00000 q^{25} -1.08995 q^{26} -2.32883 q^{27} -10.7250 q^{28} -2.25747 q^{29} +0.845049 q^{30} +9.54902 q^{31} +7.96433 q^{32} +1.18711 q^{33} -8.50208 q^{34} +4.30325 q^{35} -7.08073 q^{36} +9.26458 q^{37} +17.7258 q^{38} +0.205030 q^{39} +1.04344 q^{40} -6.74463 q^{41} +3.63646 q^{42} +8.73048 q^{43} +7.42068 q^{44} +2.84104 q^{45} +15.3544 q^{46} +7.19593 q^{47} -1.10561 q^{48} +11.5180 q^{49} -2.11951 q^{50} +1.59933 q^{51} +1.28166 q^{52} -12.2333 q^{53} +4.93596 q^{54} -2.97744 q^{55} +4.49020 q^{56} -3.33441 q^{57} +4.78471 q^{58} +12.4545 q^{59} -0.993685 q^{60} +9.08882 q^{61} -20.2392 q^{62} +12.2257 q^{63} -11.3344 q^{64} -0.514245 q^{65} -2.51608 q^{66} +0.180480 q^{67} +9.99751 q^{68} -2.88832 q^{69} -9.12077 q^{70} -6.37142 q^{71} +2.96446 q^{72} -13.1993 q^{73} -19.6363 q^{74} +0.398701 q^{75} -20.8436 q^{76} -12.8127 q^{77} -0.434563 q^{78} -3.25875 q^{79} +2.77303 q^{80} +7.59461 q^{81} +14.2953 q^{82} +6.64864 q^{83} -4.27608 q^{84} -4.01135 q^{85} -18.5043 q^{86} -0.900055 q^{87} -3.10678 q^{88} -4.82029 q^{89} -6.02160 q^{90} -2.21293 q^{91} -18.0551 q^{92} +3.80721 q^{93} -15.2518 q^{94} +8.36319 q^{95} +3.17539 q^{96} -0.486097 q^{97} -24.4124 q^{98} -8.45901 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

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