LMFDB - Embedded newform 2005.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2005,2,Mod(1,2005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2005, 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("2005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2005 = 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2005.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:	$16.0100056053$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	2005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34377 q^{2} +3.11940 q^{3} +3.49326 q^{4} -1.00000 q^{5} -7.31116 q^{6} -3.06432 q^{7} -3.49986 q^{8} +6.73066 q^{9} +O(q^{10})$	\(q-2.34377 q^{2} +3.11940 q^{3} +3.49326 q^{4} -1.00000 q^{5} -7.31116 q^{6} -3.06432 q^{7} -3.49986 q^{8} +6.73066 q^{9} +2.34377 q^{10} -2.76117 q^{11} +10.8969 q^{12} -1.77610 q^{13} +7.18207 q^{14} -3.11940 q^{15} +1.21636 q^{16} +3.21099 q^{17} -15.7751 q^{18} +6.14751 q^{19} -3.49326 q^{20} -9.55885 q^{21} +6.47156 q^{22} +1.43858 q^{23} -10.9175 q^{24} +1.00000 q^{25} +4.16278 q^{26} +11.6374 q^{27} -10.7045 q^{28} +6.32773 q^{29} +7.31116 q^{30} -2.57635 q^{31} +4.14887 q^{32} -8.61321 q^{33} -7.52582 q^{34} +3.06432 q^{35} +23.5119 q^{36} -6.01762 q^{37} -14.4084 q^{38} -5.54037 q^{39} +3.49986 q^{40} +3.64896 q^{41} +22.4038 q^{42} +0.966262 q^{43} -9.64550 q^{44} -6.73066 q^{45} -3.37171 q^{46} -5.91354 q^{47} +3.79430 q^{48} +2.39008 q^{49} -2.34377 q^{50} +10.0164 q^{51} -6.20439 q^{52} +6.67128 q^{53} -27.2754 q^{54} +2.76117 q^{55} +10.7247 q^{56} +19.1765 q^{57} -14.8308 q^{58} +4.61939 q^{59} -10.8969 q^{60} +2.30320 q^{61} +6.03837 q^{62} -20.6249 q^{63} -12.1567 q^{64} +1.77610 q^{65} +20.1874 q^{66} +4.68276 q^{67} +11.2168 q^{68} +4.48752 q^{69} -7.18207 q^{70} +10.0207 q^{71} -23.5564 q^{72} +12.4681 q^{73} +14.1039 q^{74} +3.11940 q^{75} +21.4749 q^{76} +8.46113 q^{77} +12.9854 q^{78} +17.2136 q^{79} -1.21636 q^{80} +16.1098 q^{81} -8.55233 q^{82} +4.20275 q^{83} -33.3916 q^{84} -3.21099 q^{85} -2.26470 q^{86} +19.7387 q^{87} +9.66373 q^{88} +5.37650 q^{89} +15.7751 q^{90} +5.44255 q^{91} +5.02535 q^{92} -8.03666 q^{93} +13.8600 q^{94} -6.14751 q^{95} +12.9420 q^{96} -16.0902 q^{97} -5.60180 q^{98} -18.5845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) $ 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * 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 $

$2$	−2.34377	−1.65730	−0.828648	−	0.559770i	$-0.810890\pi$
$2$	−2.34377	−1.65730	−0.828648	+	0.559770i	$0.810890\pi$
$3$	3.11940	1.80099	0.900493	−	0.434870i	$-0.143206\pi$
$3$	3.11940	1.80099	0.900493	+	0.434870i	$0.143206\pi$
$4$	3.49326	1.74663
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−7.31116	−2.98477
$7$	−3.06432	−1.15821	−0.579103	−	0.815255i	$-0.696597\pi$
$7$	−3.06432	−1.15821	−0.579103	+	0.815255i	$0.696597\pi$
$8$	−3.49986	−1.23739
$9$	6.73066	2.24355
$10$	2.34377	0.741165
$11$	−2.76117	−0.832525	−0.416263	−	0.909244i	$-0.636660\pi$
$11$	−2.76117	−0.832525	−0.416263	+	0.909244i	$0.636660\pi$
$12$	10.8969	3.14566
$13$	−1.77610	−0.492602	−0.246301	−	0.969193i	$-0.579215\pi$
$13$	−1.77610	−0.492602	−0.246301	+	0.969193i	$0.579215\pi$
$14$	7.18207	1.91949
$15$	−3.11940	−0.805426
$16$	1.21636	0.304089
$17$	3.21099	0.778779	0.389389	−	0.921073i	$-0.372686\pi$
$17$	3.21099	0.778779	0.389389	+	0.921073i	$0.372686\pi$
$18$	−15.7751	−3.71823
$19$	6.14751	1.41034	0.705168	−	0.709041i	$-0.250872\pi$
$19$	6.14751	1.41034	0.705168	+	0.709041i	$0.250872\pi$
$20$	−3.49326	−0.781117
$21$	−9.55885	−2.08591
$22$	6.47156	1.37974
$23$	1.43858	0.299965	0.149983	−	0.988689i	$-0.452078\pi$
$23$	1.43858	0.299965	0.149983	+	0.988689i	$0.452078\pi$
$24$	−10.9175	−2.22852
$25$	1.00000	0.200000
$26$	4.16278	0.816387
$27$	11.6374	2.23962
$28$	−10.7045	−2.02296
$29$	6.32773	1.17503	0.587515	−	0.809213i	$-0.300106\pi$
$29$	6.32773	1.17503	0.587515	+	0.809213i	$0.300106\pi$
$30$	7.31116	1.33483
$31$	−2.57635	−0.462726	−0.231363	−	0.972868i	$-0.574318\pi$
$31$	−2.57635	−0.462726	−0.231363	+	0.972868i	$0.574318\pi$
$32$	4.14887	0.733423
$33$	−8.61321	−1.49937
$34$	−7.52582	−1.29067
$35$	3.06432	0.517965
$36$	23.5119	3.91866
$37$	−6.01762	−0.989291	−0.494646	−	0.869095i	$-0.664702\pi$
$37$	−6.01762	−0.989291	−0.494646	+	0.869095i	$0.664702\pi$
$38$	−14.4084	−2.33734
$39$	−5.54037	−0.887169
$40$	3.49986	0.553377
$41$	3.64896	0.569872	0.284936	−	0.958547i	$-0.408028\pi$
$41$	3.64896	0.569872	0.284936	+	0.958547i	$0.408028\pi$
$42$	22.4038	3.45697
$43$	0.966262	0.147354	0.0736768	−	0.997282i	$-0.476527\pi$
$43$	0.966262	0.147354	0.0736768	+	0.997282i	$0.476527\pi$
$44$	−9.64550	−1.45411
$45$	−6.73066	−1.00335
$46$	−3.37171	−0.497131
$47$	−5.91354	−0.862578	−0.431289	−	0.902214i	$-0.641941\pi$
$47$	−5.91354	−0.862578	−0.431289	+	0.902214i	$0.641941\pi$
$48$	3.79430	0.547660
$49$	2.39008	0.341440
$50$	−2.34377	−0.331459
$51$	10.0164	1.40257
$52$	−6.20439	−0.860394
$53$	6.67128	0.916371	0.458185	−	0.888857i	$-0.348500\pi$
$53$	6.67128	0.916371	0.458185	+	0.888857i	$0.348500\pi$
$54$	−27.2754	−3.71171
$55$	2.76117	0.372317
$56$	10.7247	1.43315
$57$	19.1765	2.53999
$58$	−14.8308	−1.94737
$59$	4.61939	0.601393	0.300697	−	0.953720i	$-0.402781\pi$
$59$	4.61939	0.601393	0.300697	+	0.953720i	$0.402781\pi$
$60$	−10.8969	−1.40678
$61$	2.30320	0.294894	0.147447	−	0.989070i	$-0.452894\pi$
$61$	2.30320	0.294894	0.147447	+	0.989070i	$0.452894\pi$
$62$	6.03837	0.766874
$63$	−20.6249	−2.59849
$64$	−12.1567	−1.51959
$65$	1.77610	0.220298
$66$	20.1874	2.48489
$67$	4.68276	0.572090	0.286045	−	0.958216i	$-0.407659\pi$
$67$	4.68276	0.572090	0.286045	+	0.958216i	$0.407659\pi$
$68$	11.2168	1.36024
$69$	4.48752	0.540233
$70$	−7.18207	−0.858422
$71$	10.0207	1.18924	0.594622	−	0.804005i	$-0.297302\pi$
$71$	10.0207	1.18924	0.594622	+	0.804005i	$0.297302\pi$
$72$	−23.5564	−2.77615
$73$	12.4681	1.45929	0.729643	−	0.683829i	$-0.239686\pi$
$73$	12.4681	1.45929	0.729643	+	0.683829i	$0.239686\pi$
$74$	14.1039	1.63955
$75$	3.11940	0.360197
$76$	21.4749	2.46334
$77$	8.46113	0.964235
$78$	12.9854	1.47030
$79$	17.2136	1.93668	0.968340	−	0.249634i	$-0.0803103\pi$
$79$	17.2136	1.93668	0.968340	+	0.249634i	$0.0803103\pi$
$80$	−1.21636	−0.135993
$81$	16.1098	1.78997
$82$	−8.55233	−0.944447
$83$	4.20275	0.461312	0.230656	−	0.973035i	$-0.425913\pi$
$83$	4.20275	0.461312	0.230656	+	0.973035i	$0.425913\pi$
$84$	−33.3916	−3.64332
$85$	−3.21099	−0.348280
$86$	−2.26470	−0.244208
$87$	19.7387	2.11621
$88$	9.66373	1.03016
$89$	5.37650	0.569908	0.284954	−	0.958541i	$-0.408022\pi$
$89$	5.37650	0.569908	0.284954	+	0.958541i	$0.408022\pi$
$90$	15.7751	1.66284
$91$	5.44255	0.570534
$92$	5.02535	0.523929
$93$	−8.03666	−0.833363
$94$	13.8600	1.42955
$95$	−6.14751	−0.630721
$96$	12.9420	1.32089
$97$	−16.0902	−1.63371	−0.816854	−	0.576844i	$-0.804284\pi$
$97$	−16.0902	−1.63371	−0.816854	+	0.576844i	$0.804284\pi$
$98$	−5.60180	−0.565867
$99$	−18.5845	−1.86781
$100$	3.49326	0.349326
$101$	−2.30037	−0.228895	−0.114448	−	0.993429i	$-0.536510\pi$
$101$	−2.30037	−0.228895	−0.114448	+	0.993429i	$0.536510\pi$
$102$	−23.4760	−2.32447
$103$	−17.9038	−1.76411	−0.882057	−	0.471142i	$-0.843842\pi$
$103$	−17.9038	−1.76411	−0.882057	+	0.471142i	$0.843842\pi$
$104$	6.21611	0.609540
$105$	9.55885	0.932848
$106$	−15.6359	−1.51870
$107$	8.16296	0.789143	0.394572	−	0.918865i	$-0.370893\pi$
$107$	8.16296	0.789143	0.394572	+	0.918865i	$0.370893\pi$
$108$	40.6525	3.91179
$109$	14.6374	1.40201	0.701004	−	0.713158i	$-0.252736\pi$
$109$	14.6374	1.40201	0.701004	+	0.713158i	$0.252736\pi$
$110$	−6.47156	−0.617039
$111$	−18.7714	−1.78170
$112$	−3.72731	−0.352198
$113$	20.0300	1.88427	0.942134	−	0.335238i	$-0.108817\pi$
$113$	20.0300	1.88427	0.942134	+	0.335238i	$0.108817\pi$
$114$	−44.9454	−4.20952
$115$	−1.43858	−0.134149
$116$	22.1044	2.05234
$117$	−11.9543	−1.10518
$118$	−10.8268	−0.996686
$119$	−9.83950	−0.901986
$120$	10.9175	0.996625
$121$	−3.37592	−0.306902
$122$	−5.39817	−0.488727
$123$	11.3826	1.02633
$124$	−8.99986	−0.808211
$125$	−1.00000	−0.0894427
$126$	48.3401	4.30647
$127$	5.29233	0.469618	0.234809	−	0.972041i	$-0.424553\pi$
$127$	5.29233	0.469618	0.234809	+	0.972041i	$0.424553\pi$
$128$	20.1948	1.78499
$129$	3.01416	0.265382
$130$	−4.16278	−0.365100
$131$	8.81386	0.770071	0.385035	−	0.922902i	$-0.374189\pi$
$131$	8.81386	0.770071	0.385035	+	0.922902i	$0.374189\pi$
$132$	−30.0882	−2.61884
$133$	−18.8380	−1.63346
$134$	−10.9753	−0.948123
$135$	−11.6374	−1.00159
$136$	−11.2380	−0.963652
$137$	2.04973	0.175120	0.0875600	−	0.996159i	$-0.472093\pi$
$137$	2.04973	0.175120	0.0875600	+	0.996159i	$0.472093\pi$
$138$	−10.5177	−0.895327
$139$	−22.1975	−1.88276	−0.941382	−	0.337342i	$-0.890472\pi$
$139$	−22.1975	−1.88276	−0.941382	+	0.337342i	$0.890472\pi$
$140$	10.7045	0.904694
$141$	−18.4467	−1.55349
$142$	−23.4863	−1.97093
$143$	4.90413	0.410104
$144$	8.18688	0.682240
$145$	−6.32773	−0.525490
$146$	−29.2225	−2.41847
$147$	7.45561	0.614929
$148$	−21.0211	−1.72793
$149$	−7.85153	−0.643222	−0.321611	−	0.946872i	$-0.604224\pi$
$149$	−7.85153	−0.643222	−0.321611	+	0.946872i	$0.604224\pi$
$150$	−7.31116	−0.596954
$151$	6.97949	0.567983	0.283991	−	0.958827i	$-0.408341\pi$
$151$	6.97949	0.567983	0.283991	+	0.958827i	$0.408341\pi$
$152$	−21.5155	−1.74513
$153$	21.6120	1.74723
$154$	−19.8310	−1.59802
$155$	2.57635	0.206937
$156$	−19.3540	−1.54956
$157$	−2.59754	−0.207306	−0.103653	−	0.994613i	$-0.533053\pi$
$157$	−2.59754	−0.207306	−0.103653	+	0.994613i	$0.533053\pi$
$158$	−40.3447	−3.20965
$159$	20.8104	1.65037
$160$	−4.14887	−0.327997
$161$	−4.40828	−0.347421
$162$	−37.7576	−2.96652
$163$	16.7836	1.31459	0.657296	−	0.753633i	$-0.271700\pi$
$163$	16.7836	1.31459	0.657296	+	0.753633i	$0.271700\pi$
$164$	12.7468	0.995356
$165$	8.61321	0.670537
$166$	−9.85028	−0.764530
$167$	−4.44399	−0.343886	−0.171943	−	0.985107i	$-0.555005\pi$
$167$	−4.44399	−0.343886	−0.171943	+	0.985107i	$0.555005\pi$
$168$	33.4547	2.58108
$169$	−9.84546	−0.757343
$170$	7.52582	0.577204
$171$	41.3768	3.16416
$172$	3.37541	0.257372
$173$	−3.39265	−0.257939	−0.128969	−	0.991649i	$-0.541167\pi$
$173$	−3.39265	−0.257939	−0.128969	+	0.991649i	$0.541167\pi$
$174$	−46.2630	−3.50719
$175$	−3.06432	−0.231641
$176$	−3.35857	−0.253162
$177$	14.4097	1.08310
$178$	−12.6013	−0.944506
$179$	14.7066	1.09922	0.549611	−	0.835421i	$-0.314776\pi$
$179$	14.7066	1.09922	0.549611	+	0.835421i	$0.314776\pi$
$180$	−23.5119	−1.75248
$181$	−11.8016	−0.877209	−0.438604	−	0.898680i	$-0.644527\pi$
$181$	−11.8016	−0.877209	−0.438604	+	0.898680i	$0.644527\pi$
$182$	−12.7561	−0.945544
$183$	7.18459	0.531100
$184$	−5.03485	−0.371174
$185$	6.01762	0.442425
$186$	18.8361	1.38113
$187$	−8.86609	−0.648353
$188$	−20.6575	−1.50661
$189$	−35.6608	−2.59394
$190$	14.4084	1.04529
$191$	14.1767	1.02579	0.512894	−	0.858452i	$-0.328573\pi$
$191$	14.1767	1.02579	0.512894	+	0.858452i	$0.328573\pi$
$192$	−37.9216	−2.73676
$193$	−13.6828	−0.984911	−0.492456	−	0.870338i	$-0.663901\pi$
$193$	−13.6828	−0.984911	−0.492456	+	0.870338i	$0.663901\pi$
$194$	37.7116	2.70754
$195$	5.54037	0.396754
$196$	8.34918	0.596370
$197$	−22.1918	−1.58110	−0.790551	−	0.612396i	$-0.790206\pi$
$197$	−22.1918	−1.58110	−0.790551	+	0.612396i	$0.790206\pi$
$198$	43.5578	3.09552
$199$	4.76357	0.337681	0.168840	−	0.985643i	$-0.445998\pi$
$199$	4.76357	0.337681	0.168840	+	0.985643i	$0.445998\pi$
$200$	−3.49986	−0.247478
$201$	14.6074	1.03033
$202$	5.39154	0.379347
$203$	−19.3902	−1.36093
$204$	34.9897	2.44977
$205$	−3.64896	−0.254854
$206$	41.9624	2.92366
$207$	9.68261	0.672988
$208$	−2.16037	−0.149795
$209$	−16.9743	−1.17414
$210$	−22.4038	−1.54601
$211$	28.8574	1.98663	0.993313	−	0.115455i	$-0.0368328\pi$
$211$	28.8574	1.98663	0.993313	+	0.115455i	$0.0368328\pi$
$212$	23.3045	1.60056
$213$	31.2587	2.14181
$214$	−19.1321	−1.30784
$215$	−0.966262	−0.0658985
$216$	−40.7293	−2.77128
$217$	7.89476	0.535931
$218$	−34.3067	−2.32354
$219$	38.8931	2.62815
$220$	9.64550	0.650300
$221$	−5.70304	−0.383628
$222$	43.9958	2.95281
$223$	−5.73812	−0.384253	−0.192126	−	0.981370i	$-0.561538\pi$
$223$	−5.73812	−0.384253	−0.192126	+	0.981370i	$0.561538\pi$
$224$	−12.7135	−0.849455
$225$	6.73066	0.448710
$226$	−46.9458	−3.12279
$227$	−17.4984	−1.16141	−0.580704	−	0.814115i	$-0.697223\pi$
$227$	−17.4984	−1.16141	−0.580704	+	0.814115i	$0.697223\pi$
$228$	66.9887	4.43643
$229$	−8.92243	−0.589611	−0.294805	−	0.955557i	$-0.595255\pi$
$229$	−8.92243	−0.589611	−0.294805	+	0.955557i	$0.595255\pi$
$230$	3.37171	0.222324
$231$	26.3936	1.73657
$232$	−22.1462	−1.45397
$233$	16.1778	1.05985	0.529923	−	0.848046i	$-0.322221\pi$
$233$	16.1778	1.05985	0.529923	+	0.848046i	$0.322221\pi$
$234$	28.0182	1.83161
$235$	5.91354	0.385757
$236$	16.1367	1.05041
$237$	53.6961	3.48794
$238$	23.0615	1.49486
$239$	17.1281	1.10792	0.553962	−	0.832542i	$-0.313115\pi$
$239$	17.1281	1.10792	0.553962	+	0.832542i	$0.313115\pi$
$240$	−3.79430	−0.244921
$241$	7.87515	0.507283	0.253641	−	0.967298i	$-0.418372\pi$
$241$	7.87515	0.507283	0.253641	+	0.967298i	$0.418372\pi$
$242$	7.91238	0.508627
$243$	15.3405	0.984097
$244$	8.04567	0.515071
$245$	−2.39008	−0.152697
$246$	−26.6781	−1.70094
$247$	−10.9186	−0.694734
$248$	9.01687	0.572572
$249$	13.1101	0.830816
$250$	2.34377	0.148233
$251$	−7.89947	−0.498610	−0.249305	−	0.968425i	$-0.580202\pi$
$251$	−7.89947	−0.498610	−0.249305	+	0.968425i	$0.580202\pi$
$252$	−72.0482	−4.53861
$253$	−3.97218	−0.249729
$254$	−12.4040	−0.778297
$255$	−10.0164	−0.627248
$256$	−23.0186	−1.43866
$257$	−14.1119	−0.880278	−0.440139	−	0.897930i	$-0.645071\pi$
$257$	−14.1119	−0.880278	−0.440139	+	0.897930i	$0.645071\pi$
$258$	−7.06449	−0.439816
$259$	18.4399	1.14580
$260$	6.20439	0.384780
$261$	42.5898	2.63624
$262$	−20.6577	−1.27624
$263$	1.11513	0.0687617	0.0343808	−	0.999409i	$-0.489054\pi$
$263$	1.11513	0.0687617	0.0343808	+	0.999409i	$0.489054\pi$
$264$	30.1451	1.85530
$265$	−6.67128	−0.409813
$266$	44.1519	2.70712
$267$	16.7715	1.02640
$268$	16.3581	0.999231
$269$	8.18497	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$269$	8.18497	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$270$	27.2754	1.65993
$271$	23.8020	1.44587	0.722935	−	0.690916i	$-0.242793\pi$
$271$	23.8020	1.44587	0.722935	+	0.690916i	$0.242793\pi$
$272$	3.90570	0.236818
$273$	16.9775	1.02752
$274$	−4.80409	−0.290226
$275$	−2.76117	−0.166505
$276$	15.6761	0.943589
$277$	5.22659	0.314036	0.157018	−	0.987596i	$-0.449812\pi$
$277$	5.22659	0.314036	0.157018	+	0.987596i	$0.449812\pi$
$278$	52.0258	3.12030
$279$	−17.3405	−1.03815
$280$	−10.7247	−0.640924
$281$	16.4884	0.983617	0.491809	−	0.870703i	$-0.336336\pi$
$281$	16.4884	0.983617	0.491809	+	0.870703i	$0.336336\pi$
$282$	43.2348	2.57460
$283$	−27.4637	−1.63255	−0.816273	−	0.577667i	$-0.803963\pi$
$283$	−27.4637	−1.63255	−0.816273	+	0.577667i	$0.803963\pi$
$284$	35.0051	2.07717
$285$	−19.1765	−1.13592
$286$	−11.4941	−0.679663
$287$	−11.1816	−0.660029
$288$	27.9246	1.64547
$289$	−6.68956	−0.393504
$290$	14.8308	0.870892
$291$	−50.1916	−2.94229
$292$	43.5545	2.54883
$293$	−12.6304	−0.737878	−0.368939	−	0.929454i	$-0.620279\pi$
$293$	−12.6304	−0.737878	−0.368939	+	0.929454i	$0.620279\pi$
$294$	−17.4743	−1.01912
$295$	−4.61939	−0.268951
$296$	21.0609	1.22414
$297$	−32.1329	−1.86454
$298$	18.4022	1.06601
$299$	−2.55507	−0.147764
$300$	10.8969	0.629132
$301$	−2.96094	−0.170666
$302$	−16.3583	−0.941315
$303$	−7.17577	−0.412237
$304$	7.47756	0.428868
$305$	−2.30320	−0.131881
$306$	−50.6537	−2.89568
$307$	−13.2277	−0.754944	−0.377472	−	0.926021i	$-0.623206\pi$
$307$	−13.2277	−0.754944	−0.377472	+	0.926021i	$0.623206\pi$
$308$	29.5569	1.68416
$309$	−55.8491	−3.17715
$310$	−6.03837	−0.342956
$311$	−8.00761	−0.454070	−0.227035	−	0.973887i	$-0.572903\pi$
$311$	−8.00761	−0.454070	−0.227035	+	0.973887i	$0.572903\pi$
$312$	19.3905	1.09777
$313$	−20.4928	−1.15832	−0.579161	−	0.815213i	$-0.696620\pi$
$313$	−20.4928	−1.15832	−0.579161	+	0.815213i	$0.696620\pi$
$314$	6.08805	0.343568
$315$	20.6249	1.16208
$316$	60.1316	3.38267
$317$	23.0273	1.29334	0.646672	−	0.762768i	$-0.276160\pi$
$317$	23.0273	1.29334	0.646672	+	0.762768i	$0.276160\pi$
$318$	−48.7748	−2.73515
$319$	−17.4720	−0.978242
$320$	12.1567	0.679581
$321$	25.4635	1.42124
$322$	10.3320	0.575780
$323$	19.7396	1.09834
$324$	56.2756	3.12642
$325$	−1.77610	−0.0985204
$326$	−39.3369	−2.17867
$327$	45.6599	2.52500
$328$	−12.7709	−0.705153
$329$	18.1210	0.999043
$330$	−20.1874	−1.11128
$331$	−4.39056	−0.241327	−0.120663	−	0.992693i	$-0.538502\pi$
$331$	−4.39056	−0.241327	−0.120663	+	0.992693i	$0.538502\pi$
$332$	14.6813	0.805741
$333$	−40.5026	−2.21953
$334$	10.4157	0.569921
$335$	−4.68276	−0.255847
$336$	−11.6270	−0.634303
$337$	10.9165	0.594658	0.297329	−	0.954775i	$-0.403904\pi$
$337$	10.9165	0.594658	0.297329	+	0.954775i	$0.403904\pi$
$338$	23.0755	1.25514
$339$	62.4817	3.39354
$340$	−11.2168	−0.608317
$341$	7.11374	0.385231
$342$	−96.9777	−5.24395
$343$	14.1263	0.762748
$344$	−3.38179	−0.182334
$345$	−4.48752	−0.241600
$346$	7.95160	0.427481
$347$	2.46439	0.132296	0.0661478	−	0.997810i	$-0.478929\pi$
$347$	2.46439	0.132296	0.0661478	+	0.997810i	$0.478929\pi$
$348$	68.9525	3.69624
$349$	−23.6446	−1.26567	−0.632833	−	0.774288i	$-0.718108\pi$
$349$	−23.6446	−1.26567	−0.632833	+	0.774288i	$0.718108\pi$
$350$	7.18207	0.383898
$351$	−20.6692	−1.10324
$352$	−11.4557	−0.610593
$353$	−31.3932	−1.67089	−0.835446	−	0.549572i	$-0.814791\pi$
$353$	−31.3932	−1.67089	−0.835446	+	0.549572i	$0.814791\pi$
$354$	−33.7731	−1.79502
$355$	−10.0207	−0.531846
$356$	18.7815	0.995419
$357$	−30.6933	−1.62446
$358$	−34.4689	−1.82174
$359$	−22.1744	−1.17032	−0.585160	−	0.810918i	$-0.698968\pi$
$359$	−22.1744	−1.17032	−0.585160	+	0.810918i	$0.698968\pi$
$360$	23.5564	1.24153
$361$	18.7919	0.989046
$362$	27.6603	1.45379
$363$	−10.5308	−0.552726
$364$	19.0123	0.996513
$365$	−12.4681	−0.652612
$366$	−16.8390	−0.880191
$367$	11.4390	0.597111	0.298555	−	0.954392i	$-0.403495\pi$
$367$	11.4390	0.597111	0.298555	+	0.954392i	$0.403495\pi$
$368$	1.74983	0.0912162
$369$	24.5599	1.27854
$370$	−14.1039	−0.733229
$371$	−20.4430	−1.06135
$372$	−28.0742	−1.45558
$373$	6.61125	0.342318	0.171159	−	0.985243i	$-0.445249\pi$
$373$	6.61125	0.342318	0.171159	+	0.985243i	$0.445249\pi$
$374$	20.7801	1.07451
$375$	−3.11940	−0.161085
$376$	20.6966	1.06734
$377$	−11.2387	−0.578822
$378$	83.5807	4.29893
$379$	29.5996	1.52043	0.760215	−	0.649671i	$-0.225093\pi$
$379$	29.5996	1.52043	0.760215	+	0.649671i	$0.225093\pi$
$380$	−21.4749	−1.10164
$381$	16.5089	0.845776
$382$	−33.2269	−1.70004
$383$	16.0822	0.821762	0.410881	−	0.911689i	$-0.365221\pi$
$383$	16.0822	0.821762	0.410881	+	0.911689i	$0.365221\pi$
$384$	62.9957	3.21473
$385$	−8.46113	−0.431219
$386$	32.0694	1.63229
$387$	6.50358	0.330595
$388$	−56.2071	−2.85349
$389$	0.825395	0.0418492	0.0209246	−	0.999781i	$-0.493339\pi$
$389$	0.825395	0.0418492	0.0209246	+	0.999781i	$0.493339\pi$
$390$	−12.9854	−0.657539
$391$	4.61927	0.233607
$392$	−8.36496	−0.422494
$393$	27.4940	1.38689
$394$	52.0125	2.62035
$395$	−17.2136	−0.866110
$396$	−64.9206	−3.26238
$397$	−39.6694	−1.99095	−0.995476	−	0.0950176i	$-0.969709\pi$
$397$	−39.6694	−1.99095	−0.995476	+	0.0950176i	$0.969709\pi$
$398$	−11.1647	−0.559637
$399$	−58.7631	−2.94184
$400$	1.21636	0.0608178
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−34.2364	−1.70756
$403$	4.57585	0.227940
$404$	−8.03579	−0.399796
$405$	−16.1098	−0.800500
$406$	45.4462	2.25546
$407$	16.6157	0.823610
$408$	−35.0559	−1.73552
$409$	28.6012	1.41424	0.707120	−	0.707094i	$-0.249994\pi$
$409$	28.6012	1.41424	0.707120	+	0.707094i	$0.249994\pi$
$410$	8.55233	0.422369
$411$	6.39392	0.315389
$412$	−62.5427	−3.08126
$413$	−14.1553	−0.696537
$414$	−22.6938	−1.11534
$415$	−4.20275	−0.206305
$416$	−7.36881	−0.361286
$417$	−69.2428	−3.39083
$418$	39.7840	1.94590
$419$	34.0425	1.66309	0.831544	−	0.555459i	$-0.187458\pi$
$419$	34.0425	1.66309	0.831544	+	0.555459i	$0.187458\pi$
$420$	33.3916	1.62934
$421$	0.461640	0.0224990	0.0112495	−	0.999937i	$-0.496419\pi$
$421$	0.461640	0.0224990	0.0112495	+	0.999937i	$0.496419\pi$
$422$	−67.6351	−3.29243
$423$	−39.8020	−1.93524
$424$	−23.3486	−1.13391
$425$	3.21099	0.155756
$426$	−73.2633	−3.54962
$427$	−7.05774	−0.341548
$428$	28.5154	1.37834
$429$	15.2979	0.738591
$430$	2.26470	0.109213
$431$	−13.1890	−0.635293	−0.317646	−	0.948209i	$-0.602892\pi$
$431$	−13.1890	−0.635293	−0.317646	+	0.948209i	$0.602892\pi$
$432$	14.1552	0.681044
$433$	2.32755	0.111855	0.0559275	−	0.998435i	$-0.482188\pi$
$433$	2.32755	0.111855	0.0559275	+	0.998435i	$0.482188\pi$
$434$	−18.5035	−0.888197
$435$	−19.7387	−0.946399
$436$	51.1322	2.44879
$437$	8.84370	0.423052
$438$	−91.1565	−4.35563
$439$	12.1346	0.579151	0.289576	−	0.957155i	$-0.406486\pi$
$439$	12.1346	0.579151	0.289576	+	0.957155i	$0.406486\pi$
$440$	−9.66373	−0.460700
$441$	16.0868	0.766038
$442$	13.3666	0.635785
$443$	−19.4810	−0.925570	−0.462785	−	0.886471i	$-0.653150\pi$
$443$	−19.4810	−0.925570	−0.462785	+	0.886471i	$0.653150\pi$
$444$	−65.5733	−3.11197
$445$	−5.37650	−0.254871
$446$	13.4488	0.636821
$447$	−24.4921	−1.15843
$448$	37.2521	1.76000
$449$	−4.26017	−0.201050	−0.100525	−	0.994935i	$-0.532052\pi$
$449$	−4.26017	−0.201050	−0.100525	+	0.994935i	$0.532052\pi$
$450$	−15.7751	−0.743646
$451$	−10.0754	−0.474433
$452$	69.9702	3.29112
$453$	21.7718	1.02293
$454$	41.0122	1.92480
$455$	−5.44255	−0.255151
$456$	−67.1153	−3.14296
$457$	−5.50296	−0.257418	−0.128709	−	0.991682i	$-0.541083\pi$
$457$	−5.50296	−0.257418	−0.128709	+	0.991682i	$0.541083\pi$
$458$	20.9121	0.977160
$459$	37.3676	1.74417
$460$	−5.02535	−0.234308
$461$	−26.5935	−1.23858	−0.619291	−	0.785161i	$-0.712580\pi$
$461$	−26.5935	−1.23858	−0.619291	+	0.785161i	$0.712580\pi$
$462$	−61.8607	−2.87802
$463$	−37.5778	−1.74639	−0.873195	−	0.487370i	$-0.837956\pi$
$463$	−37.5778	−1.74639	−0.873195	+	0.487370i	$0.837956\pi$
$464$	7.69678	0.357314
$465$	8.03666	0.372691
$466$	−37.9171	−1.75648
$467$	39.5875	1.83189	0.915945	−	0.401305i	$-0.131443\pi$
$467$	39.5875	1.83189	0.915945	+	0.401305i	$0.131443\pi$
$468$	−41.7596	−1.93034
$469$	−14.3495	−0.662598
$470$	−13.8600	−0.639313
$471$	−8.10277	−0.373356
$472$	−16.1672	−0.744157
$473$	−2.66802	−0.122676
$474$	−125.851	−5.78054
$475$	6.14751	0.282067
$476$	−34.3720	−1.57544
$477$	44.9021	2.05592
$478$	−40.1443	−1.83616
$479$	6.10223	0.278818	0.139409	−	0.990235i	$-0.455480\pi$
$479$	6.10223	0.278818	0.139409	+	0.990235i	$0.455480\pi$
$480$	−12.9420	−0.590718
$481$	10.6879	0.487327
$482$	−18.4575	−0.840718
$483$	−13.7512	−0.625701
$484$	−11.7930	−0.536044
$485$	16.0902	0.730616
$486$	−35.9547	−1.63094
$487$	−0.318041	−0.0144118	−0.00720591	−	0.999974i	$-0.502294\pi$
$487$	−0.318041	−0.0144118	−0.00720591	+	0.999974i	$0.502294\pi$
$488$	−8.06088	−0.364899
$489$	52.3547	2.36756
$490$	5.60180	0.253063
$491$	−30.8518	−1.39232	−0.696160	−	0.717887i	$-0.745110\pi$
$491$	−30.8518	−1.39232	−0.696160	+	0.717887i	$0.745110\pi$
$492$	39.7623	1.79262
$493$	20.3183	0.915089
$494$	25.5907	1.15138
$495$	18.5845	0.835312
$496$	−3.13376	−0.140710
$497$	−30.7068	−1.37739
$498$	−30.7270	−1.37691
$499$	−22.0336	−0.986360	−0.493180	−	0.869927i	$-0.664166\pi$
$499$	−22.0336	−0.986360	−0.493180	+	0.869927i	$0.664166\pi$
$500$	−3.49326	−0.156223
$501$	−13.8626	−0.619334
$502$	18.5145	0.826344
$503$	−39.0883	−1.74286	−0.871430	−	0.490520i	$-0.836807\pi$
$503$	−39.0883	−1.74286	−0.871430	+	0.490520i	$0.836807\pi$
$504$	72.1844	3.21535
$505$	2.30037	0.102365
$506$	9.30988	0.413874
$507$	−30.7119	−1.36396
$508$	18.4875	0.820250
$509$	16.1084	0.713993	0.356997	−	0.934106i	$-0.383801\pi$
$509$	16.1084	0.713993	0.356997	+	0.934106i	$0.383801\pi$
$510$	23.4760	1.03954
$511$	−38.2064	−1.69015
$512$	13.5607	0.599303
$513$	71.5411	3.15861
$514$	33.0751	1.45888
$515$	17.9038	0.788936
$516$	10.5292	0.463524
$517$	16.3283	0.718118
$518$	−43.2190	−1.89893
$519$	−10.5830	−0.464544
$520$	−6.21611	−0.272595
$521$	22.8375	1.00053	0.500265	−	0.865873i	$-0.333236\pi$
$521$	22.8375	1.00053	0.500265	+	0.865873i	$0.333236\pi$
$522$	−99.8207	−4.36903
$523$	−20.1905	−0.882870	−0.441435	−	0.897293i	$-0.645530\pi$
$523$	−20.1905	−0.882870	−0.441435	+	0.897293i	$0.645530\pi$
$524$	30.7891	1.34503
$525$	−9.55885	−0.417182
$526$	−2.61360	−0.113958
$527$	−8.27262	−0.360361
$528$	−10.4767	−0.455941
$529$	−20.9305	−0.910021
$530$	15.6359	0.679182
$531$	31.0915	1.34926
$532$	−65.8059	−2.85305
$533$	−6.48092	−0.280720
$534$	−39.3084	−1.70104
$535$	−8.16296	−0.352915
$536$	−16.3890	−0.707898
$537$	45.8757	1.97968
$538$	−19.1837	−0.827068
$539$	−6.59943	−0.284257
$540$	−40.6525	−1.74941
$541$	−7.51226	−0.322977	−0.161489	−	0.986875i	$-0.551630\pi$
$541$	−7.51226	−0.322977	−0.161489	+	0.986875i	$0.551630\pi$
$542$	−55.7865	−2.39623
$543$	−36.8140	−1.57984
$544$	13.3220	0.571174
$545$	−14.6374	−0.626997
$546$	−39.7913	−1.70291
$547$	27.2925	1.16694	0.583471	−	0.812134i	$-0.301694\pi$
$547$	27.2925	1.16694	0.583471	+	0.812134i	$0.301694\pi$
$548$	7.16023	0.305870
$549$	15.5020	0.661610
$550$	6.47156	0.275948
$551$	38.8998	1.65719
$552$	−15.7057	−0.668479
$553$	−52.7480	−2.24307
$554$	−12.2499	−0.520450
$555$	18.7714	0.796801
$556$	−77.5416	−3.28849
$557$	25.8692	1.09611	0.548057	−	0.836441i	$-0.315368\pi$
$557$	25.8692	1.09611	0.548057	+	0.836441i	$0.315368\pi$
$558$	40.6422	1.72052
$559$	−1.71618	−0.0725866
$560$	3.72731	0.157508
$561$	−27.6569	−1.16767
$562$	−38.6451	−1.63015
$563$	−0.747413	−0.0314997	−0.0157498	−	0.999876i	$-0.505014\pi$
$563$	−0.747413	−0.0314997	−0.0157498	+	0.999876i	$0.505014\pi$
$564$	−64.4391	−2.71338
$565$	−20.0300	−0.842670
$566$	64.3685	2.70561
$567$	−49.3655	−2.07316
$568$	−35.0713	−1.47156
$569$	17.2721	0.724083	0.362042	−	0.932162i	$-0.382080\pi$
$569$	17.2721	0.724083	0.362042	+	0.932162i	$0.382080\pi$
$570$	44.9454	1.88256
$571$	−9.25832	−0.387449	−0.193724	−	0.981056i	$-0.562057\pi$
$571$	−9.25832	−0.387449	−0.193724	+	0.981056i	$0.562057\pi$
$572$	17.1314	0.716300
$573$	44.2227	1.84743
$574$	26.2071	1.09386
$575$	1.43858	0.0599931
$576$	−81.8226	−3.40928
$577$	−21.4924	−0.894740	−0.447370	−	0.894349i	$-0.647639\pi$
$577$	−21.4924	−0.894740	−0.447370	+	0.894349i	$0.647639\pi$
$578$	15.6788	0.652152
$579$	−42.6822	−1.77381
$580$	−22.1044	−0.917836
$581$	−12.8786	−0.534294
$582$	117.638	4.87624
$583$	−18.4206	−0.762902
$584$	−43.6368	−1.80570
$585$	11.9543	0.494251
$586$	29.6028	1.22288
$587$	−9.39098	−0.387607	−0.193804	−	0.981040i	$-0.562082\pi$
$587$	−9.39098	−0.387607	−0.193804	+	0.981040i	$0.562082\pi$
$588$	26.0444	1.07405
$589$	−15.8381	−0.652598
$590$	10.8268	0.445732
$591$	−69.2252	−2.84754
$592$	−7.31958	−0.300833
$593$	21.7825	0.894500	0.447250	−	0.894409i	$-0.352403\pi$
$593$	21.7825	0.894500	0.447250	+	0.894409i	$0.352403\pi$
$594$	75.3122	3.09010
$595$	9.83950	0.403380
$596$	−27.4274	−1.12347
$597$	14.8595	0.608158
$598$	5.98850	0.244888
$599$	−38.6192	−1.57794	−0.788970	−	0.614432i	$-0.789385\pi$
$599$	−38.6192	−1.57794	−0.788970	+	0.614432i	$0.789385\pi$
$600$	−10.9175	−0.445704
$601$	−19.2712	−0.786090	−0.393045	−	0.919519i	$-0.628578\pi$
$601$	−19.2712	−0.786090	−0.393045	+	0.919519i	$0.628578\pi$
$602$	6.93976	0.282844
$603$	31.5181	1.28351
$604$	24.3812	0.992056
$605$	3.37592	0.137251
$606$	16.8184	0.683199
$607$	−24.7506	−1.00460	−0.502298	−	0.864695i	$-0.667512\pi$
$607$	−24.7506	−1.00460	−0.502298	+	0.864695i	$0.667512\pi$
$608$	25.5052	1.03437
$609$	−60.4858	−2.45101
$610$	5.39817	0.218565
$611$	10.5030	0.424908
$612$	75.4965	3.05177
$613$	−17.9455	−0.724812	−0.362406	−	0.932020i	$-0.618045\pi$
$613$	−17.9455	−0.724812	−0.362406	+	0.932020i	$0.618045\pi$
$614$	31.0027	1.25117
$615$	−11.3826	−0.458989
$616$	−29.6128	−1.19313
$617$	4.02949	0.162221	0.0811105	−	0.996705i	$-0.474153\pi$
$617$	4.02949	0.162221	0.0811105	+	0.996705i	$0.474153\pi$
$618$	130.898	5.26547
$619$	−16.8855	−0.678686	−0.339343	−	0.940663i	$-0.610205\pi$
$619$	−16.8855	−0.678686	−0.339343	+	0.940663i	$0.610205\pi$
$620$	8.99986	0.361443
$621$	16.7414	0.671808
$622$	18.7680	0.752528
$623$	−16.4753	−0.660070
$624$	−6.73907	−0.269779
$625$	1.00000	0.0400000
$626$	48.0304	1.91968
$627$	−52.9498	−2.11461
$628$	−9.07390	−0.362088
$629$	−19.3225	−0.770439
$630$	−48.3401	−1.92591
$631$	−17.4317	−0.693945	−0.346973	−	0.937875i	$-0.612790\pi$
$631$	−17.4317	−0.693945	−0.346973	+	0.937875i	$0.612790\pi$
$632$	−60.2452	−2.39643
$633$	90.0178	3.57789
$634$	−53.9708	−2.14346
$635$	−5.29233	−0.210020
$636$	72.6961	2.88259
$637$	−4.24502	−0.168194
$638$	40.9503	1.62124
$639$	67.4462	2.66813
$640$	−20.1948	−0.798270
$641$	−9.77928	−0.386258	−0.193129	−	0.981173i	$-0.561864\pi$
$641$	−9.77928	−0.386258	−0.193129	+	0.981173i	$0.561864\pi$
$642$	−59.6807	−2.35541
$643$	6.68731	0.263722	0.131861	−	0.991268i	$-0.457905\pi$
$643$	6.68731	0.263722	0.131861	+	0.991268i	$0.457905\pi$
$644$	−15.3993	−0.606817
$645$	−3.01416	−0.118682
$646$	−46.2650	−1.82027
$647$	29.6436	1.16541	0.582706	−	0.812683i	$-0.301994\pi$
$647$	29.6436	1.16541	0.582706	+	0.812683i	$0.301994\pi$
$648$	−56.3820	−2.21489
$649$	−12.7549	−0.500675
$650$	4.16278	0.163277
$651$	24.6269	0.965205
$652$	58.6294	2.29611
$653$	−0.801959	−0.0313831	−0.0156915	−	0.999877i	$-0.504995\pi$
$653$	−0.801959	−0.0313831	−0.0156915	+	0.999877i	$0.504995\pi$
$654$	−107.016	−4.18467
$655$	−8.81386	−0.344386
$656$	4.43844	0.173292
$657$	83.9187	3.27398
$658$	−42.4715	−1.65571
$659$	−46.8421	−1.82471	−0.912355	−	0.409399i	$-0.865738\pi$
$659$	−46.8421	−1.82471	−0.912355	+	0.409399i	$0.865738\pi$
$660$	30.0882	1.17118
$661$	35.9749	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$661$	35.9749	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$662$	10.2905	0.399950
$663$	−17.7901	−0.690909
$664$	−14.7091	−0.570822
$665$	18.8380	0.730505
$666$	94.9287	3.67841
$667$	9.10297	0.352468
$668$	−15.5240	−0.600642
$669$	−17.8995	−0.692034
$670$	10.9753	0.424014
$671$	−6.35953	−0.245507
$672$	−39.6584	−1.52986
$673$	−10.3891	−0.400470	−0.200235	−	0.979748i	$-0.564171\pi$
$673$	−10.3891	−0.400470	−0.200235	+	0.979748i	$0.564171\pi$
$674$	−25.5857	−0.985524
$675$	11.6374	0.447924
$676$	−34.3928	−1.32280
$677$	37.9387	1.45810	0.729051	−	0.684460i	$-0.239962\pi$
$677$	37.9387	1.45810	0.729051	+	0.684460i	$0.239962\pi$
$678$	−146.443	−5.62410
$679$	49.3055	1.89217
$680$	11.2380	0.430958
$681$	−54.5844	−2.09168
$682$	−16.6730	−0.638442
$683$	−40.2928	−1.54176	−0.770880	−	0.636980i	$-0.780183\pi$
$683$	−40.2928	−1.54176	−0.770880	+	0.636980i	$0.780183\pi$
$684$	144.540	5.52662
$685$	−2.04973	−0.0783160
$686$	−33.1088	−1.26410
$687$	−27.8326	−1.06188
$688$	1.17532	0.0448086
$689$	−11.8489	−0.451406
$690$	10.5177	0.400402
$691$	34.8164	1.32448	0.662238	−	0.749293i	$-0.269607\pi$
$691$	34.8164	1.32448	0.662238	+	0.749293i	$0.269607\pi$
$692$	−11.8514	−0.450523
$693$	56.9490	2.16331
$694$	−5.77597	−0.219253
$695$	22.1975	0.841998
$696$	−69.0829	−2.61858
$697$	11.7168	0.443804
$698$	55.4175	2.09758
$699$	50.4651	1.90877
$700$	−10.7045	−0.404592
$701$	19.6023	0.740370	0.370185	−	0.928958i	$-0.379294\pi$
$701$	19.6023	0.740370	0.370185	+	0.928958i	$0.379294\pi$
$702$	48.4439	1.82840
$703$	−36.9934	−1.39523
$704$	33.5668	1.26510
$705$	18.4467	0.694743
$706$	73.5785	2.76916
$707$	7.04908	0.265108
$708$	50.3369	1.89178
$709$	−0.223632	−0.00839867	−0.00419933	−	0.999991i	$-0.501337\pi$
$709$	−0.223632	−0.00839867	−0.00419933	+	0.999991i	$0.501337\pi$
$710$	23.4863	0.881427
$711$	115.859	4.34504
$712$	−18.8170	−0.705198
$713$	−3.70629	−0.138802
$714$	71.9382	2.69222
$715$	−4.90413	−0.183404
$716$	51.3740	1.91994
$717$	53.4293	1.99535
$718$	51.9717	1.93957
$719$	−41.4574	−1.54610	−0.773051	−	0.634344i	$-0.781270\pi$
$719$	−41.4574	−1.54610	−0.773051	+	0.634344i	$0.781270\pi$
$720$	−8.18688	−0.305107
$721$	54.8631	2.04321
$722$	−44.0438	−1.63914
$723$	24.5657	0.913609
$724$	−41.2262	−1.53216
$725$	6.32773	0.235006
$726$	24.6819	0.916030
$727$	20.2651	0.751593	0.375796	−	0.926702i	$-0.377369\pi$
$727$	20.2651	0.751593	0.375796	+	0.926702i	$0.377369\pi$
$728$	−19.0482	−0.705973
$729$	−0.475959	−0.0176281
$730$	29.2225	1.08157
$731$	3.10265	0.114756
$732$	25.0977	0.927636
$733$	43.4350	1.60431	0.802154	−	0.597118i	$-0.203687\pi$
$733$	43.4350	1.60431	0.802154	+	0.597118i	$0.203687\pi$
$734$	−26.8104	−0.989590
$735$	−7.45561	−0.275004
$736$	5.96849	0.220002
$737$	−12.9299	−0.476280
$738$	−57.5628	−2.11891
$739$	26.1777	0.962961	0.481481	−	0.876457i	$-0.340099\pi$
$739$	26.1777	0.962961	0.481481	+	0.876457i	$0.340099\pi$
$740$	21.0211	0.772753
$741$	−34.0595	−1.25121
$742$	47.9136	1.75896
$743$	−19.7246	−0.723624	−0.361812	−	0.932251i	$-0.617842\pi$
$743$	−19.7246	−0.723624	−0.361812	+	0.932251i	$0.617842\pi$
$744$	28.1272	1.03119
$745$	7.85153	0.287658
$746$	−15.4953	−0.567322
$747$	28.2873	1.03498
$748$	−30.9716	−1.13243
$749$	−25.0139	−0.913990
$750$	7.31116	0.266966
$751$	50.9822	1.86037	0.930184	−	0.367095i	$-0.119648\pi$
$751$	50.9822	1.86037	0.930184	+	0.367095i	$0.119648\pi$
$752$	−7.19297	−0.262301
$753$	−24.6416	−0.897990
$754$	26.3409	0.959280
$755$	−6.97949	−0.254009
$756$	−124.572	−4.53066
$757$	−12.8658	−0.467616	−0.233808	−	0.972283i	$-0.575119\pi$
$757$	−12.8658	−0.467616	−0.233808	+	0.972283i	$0.575119\pi$
$758$	−69.3747	−2.51980
$759$	−12.3908	−0.449758
$760$	21.5155	0.780447
$761$	2.44406	0.0885971	0.0442986	−	0.999018i	$-0.485895\pi$
$761$	2.44406	0.0885971	0.0442986	+	0.999018i	$0.485895\pi$
$762$	−38.6931	−1.40170
$763$	−44.8537	−1.62381
$764$	49.5229	1.79167
$765$	−21.6120	−0.781385
$766$	−37.6930	−1.36190
$767$	−8.20450	−0.296247
$768$	−71.8042	−2.59101
$769$	9.77166	0.352375	0.176188	−	0.984357i	$-0.443623\pi$
$769$	9.77166	0.352375	0.176188	+	0.984357i	$0.443623\pi$
$770$	19.8310	0.714658
$771$	−44.0207	−1.58537
$772$	−47.7977	−1.72028
$773$	34.8354	1.25294	0.626471	−	0.779445i	$-0.284499\pi$
$773$	34.8354	1.25294	0.626471	+	0.779445i	$0.284499\pi$
$774$	−15.2429	−0.547894
$775$	−2.57635	−0.0925451
$776$	56.3134	2.02153
$777$	57.5216	2.06358
$778$	−1.93454	−0.0693565
$779$	22.4320	0.803711
$780$	19.3540	0.692983
$781$	−27.6690	−0.990076
$782$	−10.8265	−0.387155
$783$	73.6384	2.63162
$784$	2.90719	0.103828
$785$	2.59754	0.0927103
$786$	−64.4395	−2.29848
$787$	−11.4733	−0.408980	−0.204490	−	0.978869i	$-0.565554\pi$
$787$	−11.4733	−0.408980	−0.204490	+	0.978869i	$0.565554\pi$
$788$	−77.5218	−2.76160
$789$	3.47853	0.123839
$790$	40.3447	1.43540
$791$	−61.3785	−2.18237
$792$	65.0433	2.31121
$793$	−4.09071	−0.145265
$794$	92.9760	3.29960
$795$	−20.8104	−0.738068
$796$	16.6404	0.589803
$797$	−21.6283	−0.766112	−0.383056	−	0.923725i	$-0.625128\pi$
$797$	−21.6283	−0.766112	−0.383056	+	0.923725i	$0.625128\pi$
$798$	137.727	4.87549
$799$	−18.9883	−0.671757
$800$	4.14887	0.146685
$801$	36.1874	1.27862
$802$	−2.34377	−0.0827614
$803$	−34.4267	−1.21489
$804$	51.0275	1.79960
$805$	4.40828	0.155372
$806$	−10.7248	−0.377763
$807$	25.5322	0.898775
$808$	8.05098	0.283233
$809$	−20.3222	−0.714490	−0.357245	−	0.934011i	$-0.616284\pi$
$809$	−20.3222	−0.714490	−0.357245	+	0.934011i	$0.616284\pi$
$810$	37.7576	1.32667
$811$	−48.8594	−1.71568	−0.857842	−	0.513913i	$-0.828195\pi$
$811$	−48.8594	−1.71568	−0.857842	+	0.513913i	$0.828195\pi$
$812$	−67.7351	−2.37704
$813$	74.2480	2.60399
$814$	−38.9434	−1.36497
$815$	−16.7836	−0.587903
$816$	12.1835	0.426506
$817$	5.94010	0.207818
$818$	−67.0347	−2.34381
$819$	36.6319	1.28002
$820$	−12.7468	−0.445137
$821$	−17.3372	−0.605073	−0.302537	−	0.953138i	$-0.597833\pi$
$821$	−17.3372	−0.605073	−0.302537	+	0.953138i	$0.597833\pi$
$822$	−14.9859	−0.522693
$823$	−47.5127	−1.65619	−0.828094	−	0.560589i	$-0.810575\pi$
$823$	−47.5127	−1.65619	−0.828094	+	0.560589i	$0.810575\pi$
$824$	62.6609	2.18290
$825$	−8.61321	−0.299873
$826$	33.1768	1.15437
$827$	2.45690	0.0854349	0.0427174	−	0.999087i	$-0.486398\pi$
$827$	2.45690	0.0854349	0.0427174	+	0.999087i	$0.486398\pi$
$828$	33.8239	1.17546
$829$	15.8901	0.551884	0.275942	−	0.961174i	$-0.411010\pi$
$829$	15.8901	0.551884	0.275942	+	0.961174i	$0.411010\pi$
$830$	9.85028	0.341908
$831$	16.3038	0.565574
$832$	21.5916	0.748552
$833$	7.67451	0.265906
$834$	162.289	5.61962
$835$	4.44399	0.153791
$836$	−59.2958	−2.05079
$837$	−29.9820	−1.03633
$838$	−79.7879	−2.75623
$839$	33.8464	1.16851	0.584253	−	0.811571i	$-0.301387\pi$
$839$	33.8464	1.16851	0.584253	+	0.811571i	$0.301387\pi$
$840$	−33.4547	−1.15430
$841$	11.0402	0.380696
$842$	−1.08198	−0.0372875
$843$	51.4340	1.77148
$844$	100.806	3.46990
$845$	9.84546	0.338694
$846$	93.2867	3.20726
$847$	10.3449	0.355455
$848$	8.11465	0.278658
$849$	−85.6702	−2.94019
$850$	−7.52582	−0.258133
$851$	−8.65685	−0.296753
$852$	109.195	3.74096
$853$	38.2944	1.31118	0.655588	−	0.755119i	$-0.272421\pi$
$853$	38.2944	1.31118	0.655588	+	0.755119i	$0.272421\pi$
$854$	16.5417	0.566046
$855$	−41.3768	−1.41506
$856$	−28.5693	−0.976477
$857$	−32.2492	−1.10161	−0.550806	−	0.834634i	$-0.685679\pi$
$857$	−32.2492	−1.10161	−0.550806	+	0.834634i	$0.685679\pi$
$858$	−35.8548	−1.22406
$859$	48.4844	1.65427	0.827133	−	0.562006i	$-0.189970\pi$
$859$	48.4844	1.65427	0.827133	+	0.562006i	$0.189970\pi$
$860$	−3.37541	−0.115100
$861$	−34.8799	−1.18870
$862$	30.9120	1.05287
$863$	45.9796	1.56516	0.782582	−	0.622547i	$-0.213902\pi$
$863$	45.9796	1.56516	0.782582	+	0.622547i	$0.213902\pi$
$864$	48.2821	1.64259
$865$	3.39265	0.115354
$866$	−5.45525	−0.185377
$867$	−20.8674	−0.708695
$868$	27.5785	0.936074
$869$	−47.5297	−1.61234
$870$	46.2630	1.56846
$871$	−8.31706	−0.281813
$872$	−51.2289	−1.73483
$873$	−108.297	−3.66531
$874$	−20.7276	−0.701122
$875$	3.06432	0.103593
$876$	135.864	4.59041
$877$	−11.7804	−0.397796	−0.198898	−	0.980020i	$-0.563736\pi$
$877$	−11.7804	−0.397796	−0.198898	+	0.980020i	$0.563736\pi$
$878$	−28.4407	−0.959825
$879$	−39.3994	−1.32891
$880$	3.35857	0.113217
$881$	−39.8805	−1.34361	−0.671805	−	0.740728i	$-0.734481\pi$
$881$	−39.8805	−1.34361	−0.671805	+	0.740728i	$0.734481\pi$
$882$	−37.7038	−1.26955
$883$	5.33594	0.179569	0.0897843	−	0.995961i	$-0.471382\pi$
$883$	5.33594	0.179569	0.0897843	+	0.995961i	$0.471382\pi$
$884$	−19.9222	−0.670056
$885$	−14.4097	−0.484377
$886$	45.6590	1.53394
$887$	−19.9477	−0.669778	−0.334889	−	0.942258i	$-0.608699\pi$
$887$	−19.9477	−0.669778	−0.334889	+	0.942258i	$0.608699\pi$
$888$	65.6973	2.20466
$889$	−16.2174	−0.543914
$890$	12.6013	0.422396
$891$	−44.4818	−1.49020
$892$	−20.0448	−0.671148
$893$	−36.3535	−1.21652
$894$	57.4038	1.91987
$895$	−14.7066	−0.491587
$896$	−61.8834	−2.06738
$897$	−7.97028	−0.266120
$898$	9.98486	0.333199
$899$	−16.3024	−0.543717
$900$	23.5119	0.783731
$901$	21.4214	0.713650
$902$	23.6145	0.786276
$903$	−9.23635	−0.307367
$904$	−70.1024	−2.33157
$905$	11.8016	0.392300
$906$	−51.0281	−1.69530
$907$	15.8567	0.526512	0.263256	−	0.964726i	$-0.415204\pi$
$907$	15.8567	0.526512	0.263256	+	0.964726i	$0.415204\pi$
$908$	−61.1264	−2.02855
$909$	−15.4830	−0.513538
$910$	12.7561	0.422860
$911$	−26.7564	−0.886478	−0.443239	−	0.896403i	$-0.646171\pi$
$911$	−26.7564	−0.886478	−0.443239	+	0.896403i	$0.646171\pi$
$912$	23.3255	0.772385
$913$	−11.6045	−0.384054
$914$	12.8977	0.426618
$915$	−7.18459	−0.237515
$916$	−31.1684	−1.02983
$917$	−27.0085	−0.891900
$918$	−87.5810	−2.89060
$919$	29.9127	0.986729	0.493364	−	0.869823i	$-0.335767\pi$
$919$	29.9127	0.986729	0.493364	+	0.869823i	$0.335767\pi$
$920$	5.03485	0.165994
$921$	−41.2624	−1.35964
$922$	62.3290	2.05270
$923$	−17.7979	−0.585824
$924$	92.1999	3.03316
$925$	−6.01762	−0.197858
$926$	88.0739	2.89429
$927$	−120.504	−3.95788
$928$	26.2529	0.861794
$929$	−47.9092	−1.57185	−0.785925	−	0.618322i	$-0.787813\pi$
$929$	−47.9092	−1.57185	−0.785925	+	0.618322i	$0.787813\pi$
$930$	−18.8361	−0.617660
$931$	14.6930	0.481545
$932$	56.5134	1.85116
$933$	−24.9789	−0.817774
$934$	−92.7839	−3.03598
$935$	8.86609	0.289952
$936$	41.8385	1.36754
$937$	21.4850	0.701885	0.350943	−	0.936397i	$-0.385861\pi$
$937$	21.4850	0.701885	0.350943	+	0.936397i	$0.385861\pi$
$938$	33.6319	1.09812
$939$	−63.9252	−2.08612
$940$	20.6575	0.673775
$941$	−23.5290	−0.767023	−0.383511	−	0.923536i	$-0.625285\pi$
$941$	−23.5290	−0.767023	−0.383511	+	0.923536i	$0.625285\pi$
$942$	18.9910	0.618762
$943$	5.24933	0.170942
$944$	5.61882	0.182877
$945$	35.6608	1.16005
$946$	6.25322	0.203310
$947$	56.6756	1.84171	0.920856	−	0.389904i	$-0.127492\pi$
$947$	56.6756	1.84171	0.920856	+	0.389904i	$0.127492\pi$
$948$	187.574	6.09214
$949$	−22.1447	−0.718847
$950$	−14.4084	−0.467469
$951$	71.8315	2.32930
$952$	34.4369	1.11611
$953$	−13.4951	−0.437150	−0.218575	−	0.975820i	$-0.570141\pi$
$953$	−13.4951	−0.437150	−0.218575	+	0.975820i	$0.570141\pi$
$954$	−105.240	−3.40728
$955$	−14.1767	−0.458747
$956$	59.8329	1.93513
$957$	−54.5021	−1.76180
$958$	−14.3022	−0.462084
$959$	−6.28103	−0.202825
$960$	37.9216	1.22392
$961$	−24.3624	−0.785885
$962$	−25.0500	−0.807645
$963$	54.9421	1.77048
$964$	27.5100	0.886036
$965$	13.6828	0.440466
$966$	32.2297	1.03697
$967$	−20.7808	−0.668266	−0.334133	−	0.942526i	$-0.608444\pi$
$967$	−20.7808	−0.668266	−0.334133	+	0.942526i	$0.608444\pi$
$968$	11.8153	0.379757
$969$	61.5756	1.97809
$970$	−37.7116	−1.21085
$971$	−37.8248	−1.21385	−0.606927	−	0.794757i	$-0.707598\pi$
$971$	−37.8248	−1.21385	−0.606927	+	0.794757i	$0.707598\pi$
$972$	53.5886	1.71885
$973$	68.0202	2.18063
$974$	0.745415	0.0238847
$975$	−5.54037	−0.177434
$976$	2.80151	0.0896741
$977$	59.8627	1.91518	0.957589	−	0.288137i	$-0.0930357\pi$
$977$	59.8627	1.91518	0.957589	+	0.288137i	$0.0930357\pi$
$978$	−122.707	−3.92375
$979$	−14.8455	−0.474463
$980$	−8.34918	−0.266705
$981$	98.5192	3.14548
$982$	72.3094	2.30749
$983$	−26.9063	−0.858179	−0.429089	−	0.903262i	$-0.641165\pi$
$983$	−26.9063	−0.858179	−0.429089	+	0.903262i	$0.641165\pi$
$984$	−39.8374	−1.26997
$985$	22.1918	0.707090
$986$	−47.6214	−1.51657
$987$	56.5266	1.79926
$988$	−38.1415	−1.21344
$989$	1.39005	0.0442010
$990$	−43.5578	−1.38436
$991$	−26.8385	−0.852552	−0.426276	−	0.904593i	$-0.640175\pi$
$991$	−26.8385	−0.852552	−0.426276	+	0.904593i	$0.640175\pi$
$992$	−10.6889	−0.339374
$993$	−13.6959	−0.434627
$994$	71.9697	2.28274
$995$	−4.76357	−0.151015
$996$	45.7969	1.45113
$997$	−11.4182	−0.361617	−0.180809	−	0.983518i	$-0.557871\pi$
$997$	−11.4182	−0.361617	−0.180809	+	0.983518i	$0.557871\pi$
$998$	51.6417	1.63469
$999$	−70.0295	−2.21564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2005.2.a.f.1.4	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2005.2.a.f.1.4	✓	37	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:	\( N \)	\(=\)	\( 2005 = 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34377 q^{2} +3.11940 q^{3} +3.49326 q^{4} -1.00000 q^{5} -7.31116 q^{6} -3.06432 q^{7} -3.49986 q^{8} +6.73066 q^{9} +O(q^{10})\)	\(q-2.34377 q^{2} +3.11940 q^{3} +3.49326 q^{4} -1.00000 q^{5} -7.31116 q^{6} -3.06432 q^{7} -3.49986 q^{8} +6.73066 q^{9} +2.34377 q^{10} -2.76117 q^{11} +10.8969 q^{12} -1.77610 q^{13} +7.18207 q^{14} -3.11940 q^{15} +1.21636 q^{16} +3.21099 q^{17} -15.7751 q^{18} +6.14751 q^{19} -3.49326 q^{20} -9.55885 q^{21} +6.47156 q^{22} +1.43858 q^{23} -10.9175 q^{24} +1.00000 q^{25} +4.16278 q^{26} +11.6374 q^{27} -10.7045 q^{28} +6.32773 q^{29} +7.31116 q^{30} -2.57635 q^{31} +4.14887 q^{32} -8.61321 q^{33} -7.52582 q^{34} +3.06432 q^{35} +23.5119 q^{36} -6.01762 q^{37} -14.4084 q^{38} -5.54037 q^{39} +3.49986 q^{40} +3.64896 q^{41} +22.4038 q^{42} +0.966262 q^{43} -9.64550 q^{44} -6.73066 q^{45} -3.37171 q^{46} -5.91354 q^{47} +3.79430 q^{48} +2.39008 q^{49} -2.34377 q^{50} +10.0164 q^{51} -6.20439 q^{52} +6.67128 q^{53} -27.2754 q^{54} +2.76117 q^{55} +10.7247 q^{56} +19.1765 q^{57} -14.8308 q^{58} +4.61939 q^{59} -10.8969 q^{60} +2.30320 q^{61} +6.03837 q^{62} -20.6249 q^{63} -12.1567 q^{64} +1.77610 q^{65} +20.1874 q^{66} +4.68276 q^{67} +11.2168 q^{68} +4.48752 q^{69} -7.18207 q^{70} +10.0207 q^{71} -23.5564 q^{72} +12.4681 q^{73} +14.1039 q^{74} +3.11940 q^{75} +21.4749 q^{76} +8.46113 q^{77} +12.9854 q^{78} +17.2136 q^{79} -1.21636 q^{80} +16.1098 q^{81} -8.55233 q^{82} +4.20275 q^{83} -33.3916 q^{84} -3.21099 q^{85} -2.26470 q^{86} +19.7387 q^{87} +9.66373 q^{88} +5.37650 q^{89} +15.7751 q^{90} +5.44255 q^{91} +5.02535 q^{92} -8.03666 q^{93} +13.8600 q^{94} -6.14751 q^{95} +12.9420 q^{96} -16.0902 q^{97} -5.60180 q^{98} -18.5845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9}+O(q^{10}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9	\( 37 q + 7 q^{2} + 3 q^{3} + 43 q^{4} - 37 q^{5} + 8 q^{6} - 16 q^{7} + 21 q^{8} + 54 q^{9} - 7 q^{10} + 42 q^{11} - 13 q^{13} + 14 q^{14} - 3 q^{15} + 63 q^{16} + 18 q^{17} + 22 q^{18} + 22 q^{19} - 43 q^{20} + 16 q^{21} - 10 q^{22} + 23 q^{23} + 23 q^{24} + 37 q^{25} + 21 q^{26} + 3 q^{27} - 18 q^{28} + 33 q^{29} - 8 q^{30} + 11 q^{31} + 54 q^{32} + 2 q^{33} + 8 q^{34} + 16 q^{35} + 91 q^{36} - 11 q^{37} + 29 q^{38} + 25 q^{39} - 21 q^{40} + 24 q^{41} + 4 q^{42} + 25 q^{43} + 84 q^{44} - 54 q^{45} + 31 q^{46} + 7 q^{47} + 4 q^{48} + 45 q^{49} + 7 q^{50} + 94 q^{51} - 43 q^{52} + 49 q^{53} + 38 q^{54} - 42 q^{55} + 46 q^{56} + 6 q^{57} + 15 q^{58} + 69 q^{59} + 9 q^{61} + 17 q^{62} - 38 q^{63} + 107 q^{64} + 13 q^{65} + 74 q^{66} + 13 q^{67} + 86 q^{68} - 14 q^{70} + 51 q^{71} + 81 q^{72} - 47 q^{73} + 79 q^{74} + 3 q^{75} + 59 q^{76} + 2 q^{77} + 20 q^{78} + 67 q^{79} - 63 q^{80} + 125 q^{81} - 24 q^{82} + 80 q^{83} + 50 q^{84} - 18 q^{85} + 69 q^{86} - 32 q^{87} - 12 q^{88} + 34 q^{89} - 22 q^{90} + 39 q^{91} + 85 q^{92} + q^{93} + 12 q^{94} - 22 q^{95} + 77 q^{96} - 14 q^{97} + 40 q^{98} + 133 q^{99}+O(q^{100}) \) 37 * q + 7 * q^2 + 3 * q^3 + 43 * q^4 - 37 * q^5 + 8 * q^6 - 16 * q^7 + 21 * q^8 + 54 * q^9 - 7 * q^10 + 42 * q^11 - 13 * q^13 + 14 * q^14 - 3 * q^15 + 63 * q^16 + 18 * q^17 + 22 * q^18 + 22 * q^19 - 43 * q^20 + 16 * q^21 - 10 * q^22 + 23 * q^23 + 23 * q^24 + 37 * q^25 + 21 * q^26 + 3 * q^27 - 18 * q^28 + 33 * q^29 - 8 * q^30 + 11 * q^31 + 54 * q^32 + 2 * q^33 + 8 * q^34 + 16 * q^35 + 91 * q^36 - 11 * q^37 + 29 * q^38 + 25 * q^39 - 21 * q^40 + 24 * q^41 + 4 * q^42 + 25 * q^43 + 84 * q^44 - 54 * q^45 + 31 * q^46 + 7 * q^47 + 4 * q^48 + 45 * q^49 + 7 * q^50 + 94 * q^51 - 43 * q^52 + 49 * q^53 + 38 * q^54 - 42 * q^55 + 46 * q^56 + 6 * q^57 + 15 * q^58 + 69 * q^59 + 9 * q^61 + 17 * q^62 - 38 * q^63 + 107 * q^64 + 13 * q^65 + 74 * q^66 + 13 * q^67 + 86 * q^68 - 14 * q^70 + 51 * q^71 + 81 * q^72 - 47 * q^73 + 79 * q^74 + 3 * q^75 + 59 * q^76 + 2 * q^77 + 20 * q^78 + 67 * q^79 - 63 * q^80 + 125 * q^81 - 24 * q^82 + 80 * q^83 + 50 * q^84 - 18 * q^85 + 69 * q^86 - 32 * q^87 - 12 * q^88 + 34 * q^89 - 22 * q^90 + 39 * q^91 + 85 * q^92 + q^93 + 12 * q^94 - 22 * q^95 + 77 * q^96 - 14 * q^97 + 40 * q^98 + 133 * q^99

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