LMFDB - Embedded newform 3971.2.a.s.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3971,2,Mod(1,3971)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3971, 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("3971.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3971 = 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3971.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:	$31.7085946427$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	3971.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.68098 q^{2} -1.09464 q^{3} +0.825695 q^{4} -0.508118 q^{5} -1.84007 q^{6} -0.0790567 q^{7} -1.97398 q^{8} -1.80176 q^{9} +O(q^{10})$	\(q+1.68098 q^{2} -1.09464 q^{3} +0.825695 q^{4} -0.508118 q^{5} -1.84007 q^{6} -0.0790567 q^{7} -1.97398 q^{8} -1.80176 q^{9} -0.854136 q^{10} +1.00000 q^{11} -0.903841 q^{12} +3.65860 q^{13} -0.132893 q^{14} +0.556208 q^{15} -4.96962 q^{16} +3.58996 q^{17} -3.02872 q^{18} -0.419550 q^{20} +0.0865388 q^{21} +1.68098 q^{22} +5.54840 q^{23} +2.16081 q^{24} -4.74182 q^{25} +6.15003 q^{26} +5.25621 q^{27} -0.0652767 q^{28} +4.03987 q^{29} +0.934974 q^{30} -6.49633 q^{31} -4.40586 q^{32} -1.09464 q^{33} +6.03466 q^{34} +0.0401701 q^{35} -1.48770 q^{36} +1.71356 q^{37} -4.00486 q^{39} +1.00302 q^{40} -11.0598 q^{41} +0.145470 q^{42} -4.63138 q^{43} +0.825695 q^{44} +0.915505 q^{45} +9.32675 q^{46} -6.93061 q^{47} +5.43996 q^{48} -6.99375 q^{49} -7.97090 q^{50} -3.92973 q^{51} +3.02089 q^{52} -3.10045 q^{53} +8.83558 q^{54} -0.508118 q^{55} +0.156057 q^{56} +6.79094 q^{58} +4.14146 q^{59} +0.459258 q^{60} +8.91839 q^{61} -10.9202 q^{62} +0.142441 q^{63} +2.53307 q^{64} -1.85900 q^{65} -1.84007 q^{66} -2.96229 q^{67} +2.96422 q^{68} -6.07352 q^{69} +0.0675252 q^{70} -12.8263 q^{71} +3.55664 q^{72} -9.18768 q^{73} +2.88046 q^{74} +5.19060 q^{75} -0.0790567 q^{77} -6.73209 q^{78} -8.28030 q^{79} +2.52515 q^{80} -0.348401 q^{81} -18.5912 q^{82} -2.76429 q^{83} +0.0714547 q^{84} -1.82413 q^{85} -7.78526 q^{86} -4.42222 q^{87} -1.97398 q^{88} +3.58632 q^{89} +1.53895 q^{90} -0.289237 q^{91} +4.58128 q^{92} +7.11116 q^{93} -11.6502 q^{94} +4.82285 q^{96} -15.3181 q^{97} -11.7564 q^{98} -1.80176 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 6 q^{2} - 15 q^{3} + 18 q^{4} + 6 q^{5} - 6 q^{6} + 6 q^{7} - 15 q^{8} + 18 q^{9}+O(q^{10}) $ 21 * q - 6 * q^2 - 15 * q^3 + 18 * q^4 + 6 * q^5 - 6 * q^6 + 6 * q^7 - 15 * q^8 + 18 * q^9	\( 21 q - 6 q^{2} - 15 q^{3} + 18 q^{4} + 6 q^{5} - 6 q^{6} + 6 q^{7} - 15 q^{8} + 18 q^{9} + 6 q^{10} + 21 q^{11} - 30 q^{12} - 36 q^{13} - 12 q^{14} - 12 q^{15} + 12 q^{16} + 21 q^{17} - 6 q^{18} + 15 q^{20} + 3 q^{21} - 6 q^{22} + 6 q^{23} - 18 q^{24} + 3 q^{25} + 6 q^{26} - 24 q^{27} - 27 q^{29} - 30 q^{30} - 54 q^{31} - 21 q^{32} - 15 q^{33} - 39 q^{34} - 33 q^{35} + 54 q^{36} - 57 q^{37} + 27 q^{40} - 24 q^{41} + 6 q^{42} + 12 q^{43} + 18 q^{44} - 3 q^{45} - 15 q^{46} + 12 q^{47} + 39 q^{48} + 21 q^{49} - 33 q^{50} + 18 q^{51} + 18 q^{52} - 39 q^{53} + 30 q^{54} + 6 q^{55} - 57 q^{56} + 30 q^{58} - 27 q^{59} - 12 q^{60} - 30 q^{61} + 21 q^{62} - 18 q^{63} - 15 q^{64} + 24 q^{65} - 6 q^{66} - 27 q^{67} - 48 q^{68} - 54 q^{69} - 51 q^{70} - 42 q^{71} - 15 q^{72} + 6 q^{73} + 84 q^{74} - 27 q^{75} + 6 q^{77} + 48 q^{78} - 39 q^{79} - 27 q^{80} - 27 q^{81} - 15 q^{82} + 12 q^{83} + 48 q^{84} - 42 q^{85} + 45 q^{86} + 75 q^{87} - 15 q^{88} - 75 q^{89} + 87 q^{90} - 45 q^{91} + 12 q^{92} + 66 q^{93} - 42 q^{94} - 3 q^{96} - 39 q^{97} + 45 q^{98} + 18 q^{99}+O(q^{100}) \) 21 * q - 6 * q^2 - 15 * q^3 + 18 * q^4 + 6 * q^5 - 6 * q^6 + 6 * q^7 - 15 * q^8 + 18 * q^9 + 6 * q^10 + 21 * q^11 - 30 * q^12 - 36 * q^13 - 12 * q^14 - 12 * q^15 + 12 * q^16 + 21 * q^17 - 6 * q^18 + 15 * q^20 + 3 * q^21 - 6 * q^22 + 6 * q^23 - 18 * q^24 + 3 * q^25 + 6 * q^26 - 24 * q^27 - 27 * q^29 - 30 * q^30 - 54 * q^31 - 21 * q^32 - 15 * q^33 - 39 * q^34 - 33 * q^35 + 54 * q^36 - 57 * q^37 + 27 * q^40 - 24 * q^41 + 6 * q^42 + 12 * q^43 + 18 * q^44 - 3 * q^45 - 15 * q^46 + 12 * q^47 + 39 * q^48 + 21 * q^49 - 33 * q^50 + 18 * q^51 + 18 * q^52 - 39 * q^53 + 30 * q^54 + 6 * q^55 - 57 * q^56 + 30 * q^58 - 27 * q^59 - 12 * q^60 - 30 * q^61 + 21 * q^62 - 18 * q^63 - 15 * q^64 + 24 * q^65 - 6 * q^66 - 27 * q^67 - 48 * q^68 - 54 * q^69 - 51 * q^70 - 42 * q^71 - 15 * q^72 + 6 * q^73 + 84 * q^74 - 27 * q^75 + 6 * q^77 + 48 * q^78 - 39 * q^79 - 27 * q^80 - 27 * q^81 - 15 * q^82 + 12 * q^83 + 48 * q^84 - 42 * q^85 + 45 * q^86 + 75 * q^87 - 15 * q^88 - 75 * q^89 + 87 * q^90 - 45 * q^91 + 12 * q^92 + 66 * q^93 - 42 * q^94 - 3 * q^96 - 39 * q^97 + 45 * q^98 + 18 * 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 $

$2$	1.68098	1.18863	0.594316	−	0.804231i	$-0.297423\pi$
$2$	1.68098	1.18863	0.594316	+	0.804231i	$0.297423\pi$
$3$	−1.09464	−0.631992	−0.315996	−	0.948760i	$-0.602339\pi$
$3$	−1.09464	−0.631992	−0.315996	+	0.948760i	$0.602339\pi$
$4$	0.825695	0.412847
$5$	−0.508118	−0.227237	−0.113619	−	0.993524i	$-0.536244\pi$
$5$	−0.508118	−0.227237	−0.113619	+	0.993524i	$0.536244\pi$
$6$	−1.84007	−0.751207
$7$	−0.0790567	−0.0298806	−0.0149403	−	0.999888i	$-0.504756\pi$
$7$	−0.0790567	−0.0298806	−0.0149403	+	0.999888i	$0.504756\pi$
$8$	−1.97398	−0.697909
$9$	−1.80176	−0.600586
$10$	−0.854136	−0.270102
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.903841	−0.260916
$13$	3.65860	1.01471	0.507356	−	0.861736i	$-0.330623\pi$
$13$	3.65860	1.01471	0.507356	+	0.861736i	$0.330623\pi$
$14$	−0.132893	−0.0355171
$15$	0.556208	0.143612
$16$	−4.96962	−1.24240
$17$	3.58996	0.870694	0.435347	−	0.900263i	$-0.356626\pi$
$17$	3.58996	0.870694	0.435347	+	0.900263i	$0.356626\pi$
$18$	−3.02872	−0.713876
$19$	0	0
$19$	0	0
$20$	−0.419550	−0.0938143
$21$	0.0865388	0.0188843
$22$	1.68098	0.358386
$23$	5.54840	1.15692	0.578461	−	0.815710i	$-0.303654\pi$
$23$	5.54840	1.15692	0.578461	+	0.815710i	$0.303654\pi$
$24$	2.16081	0.441073
$25$	−4.74182	−0.948363
$26$	6.15003	1.20612
$27$	5.25621	1.01156
$28$	−0.0652767	−0.0123361
$29$	4.03987	0.750185	0.375093	−	0.926987i	$-0.377611\pi$
$29$	4.03987	0.750185	0.375093	+	0.926987i	$0.377611\pi$
$30$	0.934974	0.170702
$31$	−6.49633	−1.16677	−0.583387	−	0.812194i	$-0.698273\pi$
$31$	−6.49633	−1.16677	−0.583387	+	0.812194i	$0.698273\pi$
$32$	−4.40586	−0.778854
$33$	−1.09464	−0.190553
$34$	6.03466	1.03494
$35$	0.0401701	0.00678999
$36$	−1.48770	−0.247950
$37$	1.71356	0.281707	0.140854	−	0.990030i	$-0.455015\pi$
$37$	1.71356	0.281707	0.140854	+	0.990030i	$0.455015\pi$
$38$	0	0
$39$	−4.00486	−0.641291
$40$	1.00302	0.158591
$41$	−11.0598	−1.72724	−0.863622	−	0.504139i	$-0.831810\pi$
$41$	−11.0598	−1.72724	−0.863622	+	0.504139i	$0.831810\pi$
$42$	0.145470	0.0224465
$43$	−4.63138	−0.706279	−0.353139	−	0.935571i	$-0.614886\pi$
$43$	−4.63138	−0.706279	−0.353139	+	0.935571i	$0.614886\pi$
$44$	0.825695	0.124478
$45$	0.915505	0.136475
$46$	9.32675	1.37515
$47$	−6.93061	−1.01093	−0.505467	−	0.862846i	$-0.668680\pi$
$47$	−6.93061	−1.01093	−0.505467	+	0.862846i	$0.668680\pi$
$48$	5.43996	0.785190
$49$	−6.99375	−0.999107
$50$	−7.97090	−1.12726
$51$	−3.92973	−0.550272
$52$	3.02089	0.418922
$53$	−3.10045	−0.425880	−0.212940	−	0.977065i	$-0.568304\pi$
$53$	−3.10045	−0.425880	−0.212940	+	0.977065i	$0.568304\pi$
$54$	8.83558	1.20237
$55$	−0.508118	−0.0685146
$56$	0.156057	0.0208539
$57$	0	0
$58$	6.79094	0.891694
$59$	4.14146	0.539172	0.269586	−	0.962976i	$-0.413113\pi$
$59$	4.14146	0.539172	0.269586	+	0.962976i	$0.413113\pi$
$60$	0.459258	0.0592899
$61$	8.91839	1.14188	0.570941	−	0.820991i	$-0.306578\pi$
$61$	8.91839	1.14188	0.570941	+	0.820991i	$0.306578\pi$
$62$	−10.9202	−1.38687
$63$	0.142441	0.0179459
$64$	2.53307	0.316633
$65$	−1.85900	−0.230581
$66$	−1.84007	−0.226497
$67$	−2.96229	−0.361902	−0.180951	−	0.983492i	$-0.557917\pi$
$67$	−2.96229	−0.361902	−0.180951	+	0.983492i	$0.557917\pi$
$68$	2.96422	0.359464
$69$	−6.07352	−0.731165
$70$	0.0675252	0.00807080
$71$	−12.8263	−1.52220	−0.761099	−	0.648636i	$-0.775340\pi$
$71$	−12.8263	−1.52220	−0.761099	+	0.648636i	$0.775340\pi$
$72$	3.55664	0.419154
$73$	−9.18768	−1.07534	−0.537668	−	0.843157i	$-0.680695\pi$
$73$	−9.18768	−1.07534	−0.537668	+	0.843157i	$0.680695\pi$
$74$	2.88046	0.334846
$75$	5.19060	0.599358
$76$	0	0
$77$	−0.0790567	−0.00900934
$78$	−6.73209	−0.762259
$79$	−8.28030	−0.931607	−0.465803	−	0.884888i	$-0.654235\pi$
$79$	−8.28030	−0.931607	−0.465803	+	0.884888i	$0.654235\pi$
$80$	2.52515	0.282321
$81$	−0.348401	−0.0387112
$82$	−18.5912	−2.05306
$83$	−2.76429	−0.303420	−0.151710	−	0.988425i	$-0.548478\pi$
$83$	−2.76429	−0.303420	−0.151710	+	0.988425i	$0.548478\pi$
$84$	0.0714547	0.00779634
$85$	−1.82413	−0.197854
$86$	−7.78526	−0.839506
$87$	−4.42222	−0.474111
$88$	−1.97398	−0.210427
$89$	3.58632	0.380149	0.190075	−	0.981770i	$-0.439127\pi$
$89$	3.58632	0.380149	0.190075	+	0.981770i	$0.439127\pi$
$90$	1.53895	0.162219
$91$	−0.289237	−0.0303202
$92$	4.58128	0.477632
$93$	7.11116	0.737393
$94$	−11.6502	−1.20163
$95$	0	0
$96$	4.82285	0.492230
$97$	−15.3181	−1.55532	−0.777661	−	0.628684i	$-0.783594\pi$
$97$	−15.3181	−1.55532	−0.777661	+	0.628684i	$0.783594\pi$
$98$	−11.7564	−1.18757
$99$	−1.80176	−0.181083
$100$	−3.91529	−0.391529
$101$	−10.7885	−1.07350	−0.536748	−	0.843743i	$-0.680347\pi$
$101$	−10.7885	−1.07350	−0.536748	+	0.843743i	$0.680347\pi$
$102$	−6.60580	−0.654071
$103$	3.07575	0.303062	0.151531	−	0.988452i	$-0.451580\pi$
$103$	3.07575	0.303062	0.151531	+	0.988452i	$0.451580\pi$
$104$	−7.22202	−0.708177
$105$	−0.0439719	−0.00429122
$106$	−5.21180	−0.506215
$107$	−13.8336	−1.33735	−0.668673	−	0.743557i	$-0.733137\pi$
$107$	−13.8336	−1.33735	−0.668673	+	0.743557i	$0.733137\pi$
$108$	4.34003	0.417619
$109$	3.28010	0.314177	0.157088	−	0.987585i	$-0.449789\pi$
$109$	3.28010	0.314177	0.157088	+	0.987585i	$0.449789\pi$
$110$	−0.854136	−0.0814387
$111$	−1.87573	−0.178037
$112$	0.392881	0.0371238
$113$	−13.7861	−1.29688	−0.648441	−	0.761265i	$-0.724579\pi$
$113$	−13.7861	−1.29688	−0.648441	+	0.761265i	$0.724579\pi$
$114$	0	0
$115$	−2.81924	−0.262896
$116$	3.33570	0.309712
$117$	−6.59191	−0.609422
$118$	6.96171	0.640878
$119$	−0.283811	−0.0260169
$120$	−1.09795	−0.100228
$121$	1.00000	0.0909091
$122$	14.9916	1.35728
$123$	12.1065	1.09161
$124$	−5.36398	−0.481700
$125$	4.94999	0.442741
$126$	0.239440	0.0213310
$127$	5.66489	0.502678	0.251339	−	0.967899i	$-0.419129\pi$
$127$	5.66489	0.502678	0.251339	+	0.967899i	$0.419129\pi$
$128$	13.0698	1.15521
$129$	5.06971	0.446363
$130$	−3.12494	−0.274076
$131$	3.26324	0.285111	0.142555	−	0.989787i	$-0.454468\pi$
$131$	3.26324	0.285111	0.142555	+	0.989787i	$0.454468\pi$
$132$	−0.903841	−0.0786693
$133$	0	0
$134$	−4.97955	−0.430168
$135$	−2.67077	−0.229864
$136$	−7.08653	−0.607665
$137$	−10.4802	−0.895387	−0.447694	−	0.894187i	$-0.647754\pi$
$137$	−10.4802	−0.895387	−0.447694	+	0.894187i	$0.647754\pi$
$138$	−10.2095	−0.869087
$139$	16.3722	1.38867	0.694335	−	0.719652i	$-0.255699\pi$
$139$	16.3722	1.38867	0.694335	+	0.719652i	$0.255699\pi$
$140$	0.0331683	0.00280323
$141$	7.58655	0.638903
$142$	−21.5607	−1.80933
$143$	3.65860	0.305947
$144$	8.95404	0.746170
$145$	−2.05273	−0.170470
$146$	−15.4443	−1.27818
$147$	7.65566	0.631428
$148$	1.41488	0.116302
$149$	−17.4690	−1.43112	−0.715558	−	0.698554i	$-0.753827\pi$
$149$	−17.4690	−1.43112	−0.715558	+	0.698554i	$0.753827\pi$
$150$	8.72529	0.712417
$151$	3.64299	0.296463	0.148231	−	0.988953i	$-0.452642\pi$
$151$	3.64299	0.296463	0.148231	+	0.988953i	$0.452642\pi$
$152$	0	0
$153$	−6.46824	−0.522927
$154$	−0.132893	−0.0107088
$155$	3.30090	0.265135
$156$	−3.30679	−0.264755
$157$	10.6010	0.846055	0.423027	−	0.906117i	$-0.360967\pi$
$157$	10.6010	0.846055	0.423027	+	0.906117i	$0.360967\pi$
$158$	−13.9190	−1.10734
$159$	3.39389	0.269153
$160$	2.23870	0.176985
$161$	−0.438638	−0.0345695
$162$	−0.585656	−0.0460135
$163$	19.8522	1.55494	0.777471	−	0.628919i	$-0.216502\pi$
$163$	19.8522	1.55494	0.777471	+	0.628919i	$0.216502\pi$
$164$	−9.13199	−0.713089
$165$	0.556208	0.0433007
$166$	−4.64671	−0.360655
$167$	−6.51610	−0.504231	−0.252115	−	0.967697i	$-0.581126\pi$
$167$	−6.51610	−0.504231	−0.252115	+	0.967697i	$0.581126\pi$
$168$	−0.170826	−0.0131795
$169$	0.385351	0.0296424
$170$	−3.06632	−0.235176
$171$	0	0
$172$	−3.82411	−0.291585
$173$	−3.77346	−0.286891	−0.143445	−	0.989658i	$-0.545818\pi$
$173$	−3.77346	−0.286891	−0.143445	+	0.989658i	$0.545818\pi$
$174$	−7.43366	−0.563544
$175$	0.374872	0.0283377
$176$	−4.96962	−0.374599
$177$	−4.53342	−0.340753
$178$	6.02853	0.451858
$179$	−3.12439	−0.233528	−0.116764	−	0.993160i	$-0.537252\pi$
$179$	−3.12439	−0.233528	−0.116764	+	0.993160i	$0.537252\pi$
$180$	0.755928	0.0563435
$181$	12.2063	0.907285	0.453643	−	0.891184i	$-0.350124\pi$
$181$	12.2063	0.907285	0.453643	+	0.891184i	$0.350124\pi$
$182$	−0.486201	−0.0360396
$183$	−9.76245	−0.721661
$184$	−10.9524	−0.807425
$185$	−0.870689	−0.0640143
$186$	11.9537	0.876489
$187$	3.58996	0.262524
$188$	−5.72257	−0.417362
$189$	−0.415538	−0.0302260
$190$	0	0
$191$	1.89191	0.136894	0.0684469	−	0.997655i	$-0.478196\pi$
$191$	1.89191	0.136894	0.0684469	+	0.997655i	$0.478196\pi$
$192$	−2.77280	−0.200110
$193$	0.966731	0.0695868	0.0347934	−	0.999395i	$-0.488923\pi$
$193$	0.966731	0.0695868	0.0347934	+	0.999395i	$0.488923\pi$
$194$	−25.7495	−1.84871
$195$	2.03494	0.145725
$196$	−5.77470	−0.412479
$197$	19.7115	1.40438	0.702192	−	0.711988i	$-0.252205\pi$
$197$	19.7115	1.40438	0.702192	+	0.711988i	$0.252205\pi$
$198$	−3.02872	−0.215242
$199$	22.1538	1.57044	0.785219	−	0.619218i	$-0.212550\pi$
$199$	22.1538	1.57044	0.785219	+	0.619218i	$0.212550\pi$
$200$	9.36027	0.661871
$201$	3.24265	0.228719
$202$	−18.1353	−1.27599
$203$	−0.319379	−0.0224160
$204$	−3.24476	−0.227178
$205$	5.61966	0.392494
$206$	5.17027	0.360230
$207$	−9.99687	−0.694830
$208$	−18.1818	−1.26068
$209$	0	0
$210$	−0.0739160	−0.00510069
$211$	−11.7734	−0.810514	−0.405257	−	0.914203i	$-0.632818\pi$
$211$	−11.7734	−0.810514	−0.405257	+	0.914203i	$0.632818\pi$
$212$	−2.56003	−0.175823
$213$	14.0402	0.962017
$214$	−23.2540	−1.58961
$215$	2.35329	0.160493
$216$	−10.3757	−0.705975
$217$	0.513578	0.0348639
$218$	5.51379	0.373441
$219$	10.0572	0.679604
$220$	−0.419550	−0.0282861
$221$	13.1342	0.883505
$222$	−3.15307	−0.211620
$223$	−20.1775	−1.35118	−0.675592	−	0.737276i	$-0.736112\pi$
$223$	−20.1775	−1.35118	−0.675592	+	0.737276i	$0.736112\pi$
$224$	0.348313	0.0232726
$225$	8.54360	0.569573
$226$	−23.1741	−1.54152
$227$	−23.7410	−1.57574	−0.787872	−	0.615839i	$-0.788817\pi$
$227$	−23.7410	−1.57574	−0.787872	+	0.615839i	$0.788817\pi$
$228$	0	0
$229$	−2.98452	−0.197222	−0.0986112	−	0.995126i	$-0.531440\pi$
$229$	−2.98452	−0.197222	−0.0986112	+	0.995126i	$0.531440\pi$
$230$	−4.73909	−0.312486
$231$	0.0865388	0.00569384
$232$	−7.97464	−0.523561
$233$	−10.2948	−0.674435	−0.337218	−	0.941427i	$-0.609486\pi$
$233$	−10.2948	−0.674435	−0.337218	+	0.941427i	$0.609486\pi$
$234$	−11.0809	−0.724379
$235$	3.52157	0.229722
$236$	3.41958	0.222596
$237$	9.06397	0.588768
$238$	−0.477080	−0.0309245
$239$	−20.2631	−1.31071	−0.655356	−	0.755320i	$-0.727482\pi$
$239$	−20.2631	−1.31071	−0.655356	+	0.755320i	$0.727482\pi$
$240$	−2.76414	−0.178424
$241$	17.4961	1.12702	0.563510	−	0.826109i	$-0.309450\pi$
$241$	17.4961	1.12702	0.563510	+	0.826109i	$0.309450\pi$
$242$	1.68098	0.108058
$243$	−15.3873	−0.987093
$244$	7.36387	0.471423
$245$	3.55365	0.227034
$246$	20.3508	1.29752
$247$	0	0
$248$	12.8236	0.814302
$249$	3.02591	0.191759
$250$	8.32084	0.526256
$251$	20.6121	1.30103	0.650513	−	0.759495i	$-0.274554\pi$
$251$	20.6121	1.30103	0.650513	+	0.759495i	$0.274554\pi$
$252$	0.117613	0.00740891
$253$	5.54840	0.348825
$254$	9.52257	0.597499
$255$	1.99677	0.125042
$256$	16.9039	1.05649
$257$	−5.08444	−0.317159	−0.158579	−	0.987346i	$-0.550691\pi$
$257$	−5.08444	−0.317159	−0.158579	+	0.987346i	$0.550691\pi$
$258$	8.52208	0.530561
$259$	−0.135468	−0.00841758
$260$	−1.53497	−0.0951946
$261$	−7.27886	−0.450550
$262$	5.48544	0.338892
$263$	−16.0804	−0.991559	−0.495779	−	0.868449i	$-0.665118\pi$
$263$	−16.0804	−0.991559	−0.495779	+	0.868449i	$0.665118\pi$
$264$	2.16081	0.132988
$265$	1.57540	0.0967757
$266$	0	0
$267$	−3.92574	−0.240251
$268$	−2.44595	−0.149410
$269$	2.16204	0.131822	0.0659109	−	0.997826i	$-0.479005\pi$
$269$	2.16204	0.131822	0.0659109	+	0.997826i	$0.479005\pi$
$270$	−4.48952	−0.273223
$271$	−6.85773	−0.416577	−0.208289	−	0.978067i	$-0.566789\pi$
$271$	−6.85773	−0.416577	−0.208289	+	0.978067i	$0.566789\pi$
$272$	−17.8408	−1.08175
$273$	0.316611	0.0191622
$274$	−17.6171	−1.06429
$275$	−4.74182	−0.285942
$276$	−5.01487	−0.301860
$277$	26.6091	1.59879	0.799394	−	0.600807i	$-0.205154\pi$
$277$	26.6091	1.59879	0.799394	+	0.600807i	$0.205154\pi$
$278$	27.5213	1.65062
$279$	11.7048	0.700748
$280$	−0.0792952	−0.00473879
$281$	13.2563	0.790806	0.395403	−	0.918508i	$-0.370605\pi$
$281$	13.2563	0.790806	0.395403	+	0.918508i	$0.370605\pi$
$282$	12.7528	0.759420
$283$	8.74456	0.519810	0.259905	−	0.965634i	$-0.416309\pi$
$283$	8.74456	0.519810	0.259905	+	0.965634i	$0.416309\pi$
$284$	−10.5906	−0.628435
$285$	0	0
$286$	6.15003	0.363659
$287$	0.874348	0.0516111
$288$	7.93829	0.467768
$289$	−4.11215	−0.241891
$290$	−3.45060	−0.202626
$291$	16.7679	0.982952
$292$	−7.58622	−0.443950
$293$	9.05697	0.529114	0.264557	−	0.964370i	$-0.414774\pi$
$293$	9.05697	0.529114	0.264557	+	0.964370i	$0.414774\pi$
$294$	12.8690	0.750536
$295$	−2.10435	−0.122520
$296$	−3.38253	−0.196606
$297$	5.25621	0.304996
$298$	−29.3650	−1.70107
$299$	20.2994	1.17394
$300$	4.28585	0.247444
$301$	0.366141	0.0211040
$302$	6.12380	0.352385
$303$	11.8096	0.678441
$304$	0	0
$305$	−4.53159	−0.259478
$306$	−10.8730	−0.621568
$307$	−11.8871	−0.678433	−0.339217	−	0.940708i	$-0.610162\pi$
$307$	−11.8871	−0.678433	−0.339217	+	0.940708i	$0.610162\pi$
$308$	−0.0652767	−0.00371948
$309$	−3.36684	−0.191533
$310$	5.54875	0.315148
$311$	30.2784	1.71693	0.858466	−	0.512870i	$-0.171418\pi$
$311$	30.2784	1.71693	0.858466	+	0.512870i	$0.171418\pi$
$312$	7.90553	0.447562
$313$	−17.4650	−0.987178	−0.493589	−	0.869695i	$-0.664315\pi$
$313$	−17.4650	−0.987178	−0.493589	+	0.869695i	$0.664315\pi$
$314$	17.8201	1.00565
$315$	−0.0723768	−0.00407797
$316$	−6.83700	−0.384611
$317$	−14.8243	−0.832616	−0.416308	−	0.909224i	$-0.636676\pi$
$317$	−14.8243	−0.832616	−0.416308	+	0.909224i	$0.636676\pi$
$318$	5.70506	0.319924
$319$	4.03987	0.226189
$320$	−1.28710	−0.0719509
$321$	15.1429	0.845192
$322$	−0.737342	−0.0410905
$323$	0	0
$324$	−0.287673	−0.0159818
$325$	−17.3484	−0.962316
$326$	33.3711	1.84825
$327$	−3.59054	−0.198557
$328$	21.8318	1.20546
$329$	0.547911	0.0302073
$330$	0.934974	0.0514686
$331$	−29.1161	−1.60037	−0.800184	−	0.599755i	$-0.795265\pi$
$331$	−29.1161	−1.60037	−0.800184	+	0.599755i	$0.795265\pi$
$332$	−2.28246	−0.125266
$333$	−3.08741	−0.169189
$334$	−10.9534	−0.599345
$335$	1.50519	0.0822375
$336$	−0.430065	−0.0234620
$337$	−35.8346	−1.95204	−0.976018	−	0.217691i	$-0.930147\pi$
$337$	−35.8346	−1.95204	−0.976018	+	0.217691i	$0.930147\pi$
$338$	0.647767	0.0352339
$339$	15.0908	0.819620
$340$	−1.50617	−0.0816836
$341$	−6.49633	−0.351796
$342$	0	0
$343$	1.10630	0.0597345
$344$	9.14227	0.492918
$345$	3.08606	0.166148
$346$	−6.34311	−0.341008
$347$	−14.2896	−0.767108	−0.383554	−	0.923518i	$-0.625300\pi$
$347$	−14.2896	−0.767108	−0.383554	+	0.923518i	$0.625300\pi$
$348$	−3.65140	−0.195736
$349$	−21.7678	−1.16520	−0.582602	−	0.812758i	$-0.697965\pi$
$349$	−21.7678	−1.16520	−0.582602	+	0.812758i	$0.697965\pi$
$350$	0.630153	0.0336831
$351$	19.2304	1.02644
$352$	−4.40586	−0.234833
$353$	11.1964	0.595922	0.297961	−	0.954578i	$-0.403693\pi$
$353$	11.1964	0.595922	0.297961	+	0.954578i	$0.403693\pi$
$354$	−7.62059	−0.405030
$355$	6.51725	0.345900
$356$	2.96121	0.156944
$357$	0.310671	0.0164425
$358$	−5.25204	−0.277579
$359$	3.96451	0.209239	0.104620	−	0.994512i	$-0.466638\pi$
$359$	3.96451	0.209239	0.104620	+	0.994512i	$0.466638\pi$
$360$	−1.80719	−0.0952474
$361$	0	0
$362$	20.5185	1.07843
$363$	−1.09464	−0.0574539
$364$	−0.238821	−0.0125176
$365$	4.66842	0.244356
$366$	−16.4105	−0.857790
$367$	2.35416	0.122886	0.0614431	−	0.998111i	$-0.480430\pi$
$367$	2.35416	0.122886	0.0614431	+	0.998111i	$0.480430\pi$
$368$	−27.5734	−1.43736
$369$	19.9270	1.03736
$370$	−1.46361	−0.0760895
$371$	0.245111	0.0127255
$372$	5.87165	0.304431
$373$	8.84850	0.458158	0.229079	−	0.973408i	$-0.426429\pi$
$373$	8.84850	0.458158	0.229079	+	0.973408i	$0.426429\pi$
$374$	6.03466	0.312045
$375$	−5.41847	−0.279809
$376$	13.6809	0.705540
$377$	14.7803	0.761223
$378$	−0.698512	−0.0359276
$379$	−19.4443	−0.998788	−0.499394	−	0.866375i	$-0.666444\pi$
$379$	−19.4443	−0.998788	−0.499394	+	0.866375i	$0.666444\pi$
$380$	0	0
$381$	−6.20103	−0.317688
$382$	3.18026	0.162716
$383$	31.3751	1.60319	0.801596	−	0.597866i	$-0.203985\pi$
$383$	31.3751	1.60319	0.801596	+	0.597866i	$0.203985\pi$
$384$	−14.3067	−0.730087
$385$	0.0401701	0.00204726
$386$	1.62506	0.0827132
$387$	8.34462	0.424181
$388$	−12.6481	−0.642111
$389$	28.0046	1.41989	0.709945	−	0.704257i	$-0.248720\pi$
$389$	28.0046	1.41989	0.709945	+	0.704257i	$0.248720\pi$
$390$	3.42070	0.173214
$391$	19.9186	1.00732
$392$	13.8055	0.697286
$393$	−3.57208	−0.180188
$394$	33.1346	1.66930
$395$	4.20737	0.211696
$396$	−1.48770	−0.0747598
$397$	−19.4844	−0.977895	−0.488947	−	0.872313i	$-0.662619\pi$
$397$	−19.4844	−0.977895	−0.488947	+	0.872313i	$0.662619\pi$
$398$	37.2400	1.86667
$399$	0	0
$400$	23.5650	1.17825
$401$	29.3906	1.46770	0.733849	−	0.679312i	$-0.237722\pi$
$401$	29.3906	1.46770	0.733849	+	0.679312i	$0.237722\pi$
$402$	5.45083	0.271863
$403$	−23.7675	−1.18394
$404$	−8.90801	−0.443190
$405$	0.177029	0.00879664
$406$	−0.536869	−0.0266444
$407$	1.71356	0.0849379
$408$	7.75722	0.384040
$409$	−33.0180	−1.63263	−0.816317	−	0.577605i	$-0.803987\pi$
$409$	−33.0180	−1.63263	−0.816317	+	0.577605i	$0.803987\pi$
$410$	9.44655	0.466532
$411$	11.4721	0.565878
$412$	2.53963	0.125118
$413$	−0.327410	−0.0161108
$414$	−16.8045	−0.825898
$415$	1.40458	0.0689483
$416$	−16.1193	−0.790313
$417$	−17.9217	−0.877629
$418$	0	0
$419$	21.9296	1.07133	0.535664	−	0.844431i	$-0.320061\pi$
$419$	21.9296	1.07133	0.535664	+	0.844431i	$0.320061\pi$
$420$	−0.0363074	−0.00177162
$421$	−27.9157	−1.36053	−0.680264	−	0.732967i	$-0.738135\pi$
$421$	−27.9157	−1.36053	−0.680264	+	0.732967i	$0.738135\pi$
$422$	−19.7908	−0.963403
$423$	12.4873	0.607152
$424$	6.12024	0.297225
$425$	−17.0230	−0.825735
$426$	23.6013	1.14348
$427$	−0.705058	−0.0341201
$428$	−11.4223	−0.552120
$429$	−4.00486	−0.193356
$430$	3.95583	0.190767
$431$	−21.4052	−1.03105	−0.515526	−	0.856874i	$-0.672403\pi$
$431$	−21.4052	−1.03105	−0.515526	+	0.856874i	$0.672403\pi$
$432$	−26.1214	−1.25676
$433$	−35.7468	−1.71788	−0.858942	−	0.512073i	$-0.828878\pi$
$433$	−35.7468	−1.71788	−0.858942	+	0.512073i	$0.828878\pi$
$434$	0.863315	0.0414404
$435$	2.24701	0.107736
$436$	2.70836	0.129707
$437$	0	0
$438$	16.9060	0.807800
$439$	3.84638	0.183577	0.0917887	−	0.995779i	$-0.470742\pi$
$439$	3.84638	0.183577	0.0917887	+	0.995779i	$0.470742\pi$
$440$	1.00302	0.0478169
$441$	12.6010	0.600049
$442$	22.0784	1.05016
$443$	1.86405	0.0885635	0.0442817	−	0.999019i	$-0.485900\pi$
$443$	1.86405	0.0885635	0.0442817	+	0.999019i	$0.485900\pi$
$444$	−1.54878	−0.0735020
$445$	−1.82227	−0.0863840
$446$	−33.9179	−1.60606
$447$	19.1223	0.904454
$448$	−0.200256	−0.00946120
$449$	14.1925	0.669785	0.334893	−	0.942256i	$-0.391300\pi$
$449$	14.1925	0.669785	0.334893	+	0.942256i	$0.391300\pi$
$450$	14.3616	0.677013
$451$	−11.0598	−0.520784
$452$	−11.3831	−0.535415
$453$	−3.98778	−0.187362
$454$	−39.9081	−1.87298
$455$	0.146966	0.00688989
$456$	0	0
$457$	−23.3764	−1.09350	−0.546752	−	0.837295i	$-0.684136\pi$
$457$	−23.3764	−1.09350	−0.546752	+	0.837295i	$0.684136\pi$
$458$	−5.01691	−0.234425
$459$	18.8696	0.880758
$460$	−2.32783	−0.108536
$461$	−4.34005	−0.202136	−0.101068	−	0.994880i	$-0.532226\pi$
$461$	−4.34005	−0.202136	−0.101068	+	0.994880i	$0.532226\pi$
$462$	0.145470	0.00676788
$463$	17.8770	0.830817	0.415408	−	0.909635i	$-0.363639\pi$
$463$	17.8770	0.830817	0.415408	+	0.909635i	$0.363639\pi$
$464$	−20.0766	−0.932033
$465$	−3.61331	−0.167563
$466$	−17.3054	−0.801655
$467$	−20.8465	−0.964662	−0.482331	−	0.875989i	$-0.660210\pi$
$467$	−20.8465	−0.964662	−0.482331	+	0.875989i	$0.660210\pi$
$468$	−5.44290	−0.251598
$469$	0.234189	0.0108138
$470$	5.91969	0.273055
$471$	−11.6044	−0.534700
$472$	−8.17518	−0.376293
$473$	−4.63138	−0.212951
$474$	15.2364	0.699829
$475$	0	0
$476$	−0.234341	−0.0107410
$477$	5.58626	0.255777
$478$	−34.0619	−1.55796
$479$	−2.20290	−0.100653	−0.0503266	−	0.998733i	$-0.516026\pi$
$479$	−2.20290	−0.100653	−0.0503266	+	0.998733i	$0.516026\pi$
$480$	−2.45057	−0.111853
$481$	6.26922	0.285852
$482$	29.4105	1.33961
$483$	0.480152	0.0218477
$484$	0.825695	0.0375316
$485$	7.78343	0.353427
$486$	−25.8657	−1.17329
$487$	20.2803	0.918988	0.459494	−	0.888181i	$-0.348031\pi$
$487$	20.2803	0.918988	0.459494	+	0.888181i	$0.348031\pi$
$488$	−17.6048	−0.796930
$489$	−21.7310	−0.982711
$490$	5.97362	0.269860
$491$	20.1305	0.908477	0.454238	−	0.890880i	$-0.349911\pi$
$491$	20.1305	0.908477	0.454238	+	0.890880i	$0.349911\pi$
$492$	9.99627	0.450667
$493$	14.5030	0.653182
$494$	0	0
$495$	0.915505	0.0411489
$496$	32.2843	1.44961
$497$	1.01400	0.0454842
$498$	5.08649	0.227931
$499$	26.4006	1.18185	0.590927	−	0.806725i	$-0.298762\pi$
$499$	26.4006	1.18185	0.590927	+	0.806725i	$0.298762\pi$
$500$	4.08718	0.182784
$501$	7.13280	0.318670
$502$	34.6486	1.54644
$503$	−8.37294	−0.373331	−0.186665	−	0.982424i	$-0.559768\pi$
$503$	−8.37294	−0.373331	−0.186665	+	0.982424i	$0.559768\pi$
$504$	−0.281176	−0.0125246
$505$	5.48183	0.243938
$506$	9.32675	0.414625
$507$	−0.421822	−0.0187338
$508$	4.67747	0.207529
$509$	−37.7849	−1.67478	−0.837392	−	0.546603i	$-0.815921\pi$
$509$	−37.7849	−1.67478	−0.837392	+	0.546603i	$0.815921\pi$
$510$	3.35652	0.148629
$511$	0.726347	0.0321317
$512$	2.27556	0.100567
$513$	0	0
$514$	−8.54685	−0.376985
$515$	−1.56284	−0.0688670
$516$	4.18603	0.184280
$517$	−6.93061	−0.304808
$518$	−0.227719	−0.0100054
$519$	4.13059	0.181313
$520$	3.66964	0.160924
$521$	−20.4850	−0.897464	−0.448732	−	0.893666i	$-0.648124\pi$
$521$	−20.4850	−0.897464	−0.448732	+	0.893666i	$0.648124\pi$
$522$	−12.2356	−0.535539
$523$	30.2943	1.32468	0.662339	−	0.749204i	$-0.269564\pi$
$523$	30.2943	1.32468	0.662339	+	0.749204i	$0.269564\pi$
$524$	2.69444	0.117707
$525$	−0.410351	−0.0179092
$526$	−27.0308	−1.17860
$527$	−23.3216	−1.01590
$528$	5.43996	0.236744
$529$	7.78473	0.338467
$530$	2.64821	0.115031
$531$	−7.46191	−0.323819
$532$	0	0
$533$	−40.4632	−1.75266
$534$	−6.59909	−0.285571
$535$	7.02911	0.303895
$536$	5.84752	0.252574
$537$	3.42009	0.147588
$538$	3.63434	0.156688
$539$	−6.99375	−0.301242
$540$	−2.20524	−0.0948986
$541$	44.4071	1.90921	0.954605	−	0.297874i	$-0.0962775\pi$
$541$	44.4071	1.90921	0.954605	+	0.297874i	$0.0962775\pi$
$542$	−11.5277	−0.495157
$543$	−13.3615	−0.573397
$544$	−15.8169	−0.678144
$545$	−1.66668	−0.0713927
$546$	0.532217	0.0227768
$547$	−43.4320	−1.85702	−0.928509	−	0.371311i	$-0.878908\pi$
$547$	−43.4320	−1.85702	−0.928509	+	0.371311i	$0.878908\pi$
$548$	−8.65348	−0.369658
$549$	−16.0688	−0.685798
$550$	−7.97090	−0.339880
$551$	0	0
$552$	11.9890	0.510287
$553$	0.654613	0.0278370
$554$	44.7294	1.90037
$555$	0.953094	0.0404566
$556$	13.5184	0.573309
$557$	44.5753	1.88871	0.944357	−	0.328921i	$-0.106685\pi$
$557$	44.5753	1.88871	0.944357	+	0.328921i	$0.106685\pi$
$558$	19.6755	0.832932
$559$	−16.9444	−0.716670
$560$	−0.199630	−0.00843591
$561$	−3.92973	−0.165913
$562$	22.2836	0.939978
$563$	7.72130	0.325414	0.162707	−	0.986674i	$-0.447977\pi$
$563$	7.72130	0.325414	0.162707	+	0.986674i	$0.447977\pi$
$564$	6.26417	0.263769
$565$	7.00494	0.294700
$566$	14.6994	0.617864
$567$	0.0275434	0.00115672
$568$	25.3188	1.06235
$569$	11.1299	0.466589	0.233294	−	0.972406i	$-0.425049\pi$
$569$	11.1299	0.466589	0.233294	+	0.972406i	$0.425049\pi$
$570$	0	0
$571$	9.06319	0.379283	0.189641	−	0.981853i	$-0.439267\pi$
$571$	9.06319	0.379283	0.189641	+	0.981853i	$0.439267\pi$
$572$	3.02089	0.126310
$573$	−2.07097	−0.0865158
$574$	1.46976	0.0613467
$575$	−26.3095	−1.09718
$576$	−4.56397	−0.190166
$577$	−30.8928	−1.28608	−0.643042	−	0.765831i	$-0.722328\pi$
$577$	−30.8928	−1.28608	−0.643042	+	0.765831i	$0.722328\pi$
$578$	−6.91245	−0.287520
$579$	−1.05823	−0.0439783
$580$	−1.69493	−0.0703781
$581$	0.218535	0.00906637
$582$	28.1865	1.16837
$583$	−3.10045	−0.128408
$584$	18.1363	0.750486
$585$	3.34947	0.138483
$586$	15.2246	0.628922
$587$	39.6253	1.63551	0.817756	−	0.575565i	$-0.195218\pi$
$587$	39.6253	1.63551	0.817756	+	0.575565i	$0.195218\pi$
$588$	6.32124	0.260683
$589$	0	0
$590$	−3.53737	−0.145631
$591$	−21.5770	−0.887560
$592$	−8.51572	−0.349994
$593$	−23.0757	−0.947604	−0.473802	−	0.880632i	$-0.657119\pi$
$593$	−23.0757	−0.947604	−0.473802	+	0.880632i	$0.657119\pi$
$594$	8.83558	0.362528
$595$	0.144209	0.00591201
$596$	−14.4241	−0.590832
$597$	−24.2504	−0.992505
$598$	34.1228	1.39539
$599$	−17.8031	−0.727415	−0.363707	−	0.931513i	$-0.618489\pi$
$599$	−17.8031	−0.727415	−0.363707	+	0.931513i	$0.618489\pi$
$600$	−10.2462	−0.418297
$601$	3.56274	0.145327	0.0726636	−	0.997357i	$-0.476850\pi$
$601$	3.56274	0.145327	0.0726636	+	0.997357i	$0.476850\pi$
$602$	0.615476	0.0250849
$603$	5.33733	0.217353
$604$	3.00800	0.122394
$605$	−0.508118	−0.0206579
$606$	19.8516	0.806417
$607$	−3.06484	−0.124398	−0.0621990	−	0.998064i	$-0.519811\pi$
$607$	−3.06484	−0.124398	−0.0621990	+	0.998064i	$0.519811\pi$
$608$	0	0
$609$	0.349606	0.0141667
$610$	−7.61752	−0.308424
$611$	−25.3563	−1.02581
$612$	−5.34080	−0.215889
$613$	19.5820	0.790909	0.395455	−	0.918486i	$-0.370587\pi$
$613$	19.5820	0.790909	0.395455	+	0.918486i	$0.370587\pi$
$614$	−19.9820	−0.806408
$615$	−6.15153	−0.248053
$616$	0.156057	0.00628770
$617$	18.5725	0.747701	0.373850	−	0.927489i	$-0.378037\pi$
$617$	18.5725	0.747701	0.373850	+	0.927489i	$0.378037\pi$
$618$	−5.65960	−0.227662
$619$	−11.8209	−0.475123	−0.237561	−	0.971373i	$-0.576348\pi$
$619$	−11.8209	−0.475123	−0.237561	+	0.971373i	$0.576348\pi$
$620$	2.72554	0.109460
$621$	29.1635	1.17029
$622$	50.8974	2.04080
$623$	−0.283522	−0.0113591
$624$	19.9026	0.796743
$625$	21.1939	0.847756
$626$	−29.3582	−1.17339
$627$	0	0
$628$	8.75322	0.349292
$629$	6.15161	0.245281
$630$	−0.121664	−0.00484721
$631$	30.8436	1.22786	0.613932	−	0.789359i	$-0.289587\pi$
$631$	30.8436	1.22786	0.613932	+	0.789359i	$0.289587\pi$
$632$	16.3452	0.650176
$633$	12.8877	0.512239
$634$	−24.9194	−0.989675
$635$	−2.87843	−0.114227
$636$	2.80232	0.111119
$637$	−25.5873	−1.01381
$638$	6.79094	0.268856
$639$	23.1098	0.914210
$640$	−6.64098	−0.262508
$641$	24.1406	0.953497	0.476749	−	0.879040i	$-0.341815\pi$
$641$	24.1406	0.953497	0.476749	+	0.879040i	$0.341815\pi$
$642$	25.4549	1.00462
$643$	−6.26410	−0.247032	−0.123516	−	0.992343i	$-0.539417\pi$
$643$	−6.26410	−0.247032	−0.123516	+	0.992343i	$0.539417\pi$
$644$	−0.362181	−0.0142719
$645$	−2.57601	−0.101430
$646$	0	0
$647$	29.4783	1.15891	0.579455	−	0.815004i	$-0.303265\pi$
$647$	29.4783	1.15891	0.579455	+	0.815004i	$0.303265\pi$
$648$	0.687738	0.0270169
$649$	4.14146	0.162567
$650$	−29.1623	−1.14384
$651$	−0.562185	−0.0220337
$652$	16.3918	0.641954
$653$	17.3120	0.677471	0.338735	−	0.940882i	$-0.390001\pi$
$653$	17.3120	0.677471	0.338735	+	0.940882i	$0.390001\pi$
$654$	−6.03563	−0.236012
$655$	−1.65811	−0.0647878
$656$	54.9628	2.14594
$657$	16.5540	0.645831
$658$	0.921028	0.0359054
$659$	37.1379	1.44669	0.723343	−	0.690488i	$-0.242604\pi$
$659$	37.1379	1.44669	0.723343	+	0.690488i	$0.242604\pi$
$660$	0.459258	0.0178766
$661$	−2.69780	−0.104932	−0.0524661	−	0.998623i	$-0.516708\pi$
$661$	−2.69780	−0.104932	−0.0524661	+	0.998623i	$0.516708\pi$
$662$	−48.9437	−1.90225
$663$	−14.3773	−0.558368
$664$	5.45666	0.211759
$665$	0	0
$666$	−5.18988	−0.201104
$667$	22.4148	0.867905
$668$	−5.38031	−0.208170
$669$	22.0871	0.853937
$670$	2.53020	0.0977502
$671$	8.91839	0.344291
$672$	−0.381278	−0.0147081
$673$	25.6388	0.988302	0.494151	−	0.869376i	$-0.335479\pi$
$673$	25.6388	0.988302	0.494151	+	0.869376i	$0.335479\pi$
$674$	−60.2373	−2.32025
$675$	−24.9240	−0.959324
$676$	0.318182	0.0122378
$677$	0.329487	0.0126632	0.00633161	−	0.999980i	$-0.497985\pi$
$677$	0.329487	0.0126632	0.00633161	+	0.999980i	$0.497985\pi$
$678$	25.3673	0.974227
$679$	1.21100	0.0464740
$680$	3.60079	0.138084
$681$	25.9879	0.995859
$682$	−10.9202	−0.418156
$683$	21.8760	0.837061	0.418531	−	0.908203i	$-0.362545\pi$
$683$	21.8760	0.837061	0.418531	+	0.908203i	$0.362545\pi$
$684$	0	0
$685$	5.32520	0.203465
$686$	1.85967	0.0710024
$687$	3.26698	0.124643
$688$	23.0162	0.877484
$689$	−11.3433	−0.432146
$690$	5.18761	0.197489
$691$	−21.1929	−0.806218	−0.403109	−	0.915152i	$-0.632070\pi$
$691$	−21.1929	−0.806218	−0.403109	+	0.915152i	$0.632070\pi$
$692$	−3.11572	−0.118442
$693$	0.142441	0.00541088
$694$	−24.0206	−0.911810
$695$	−8.31900	−0.315557
$696$	8.72938	0.330886
$697$	−39.7042	−1.50390
$698$	−36.5912	−1.38500
$699$	11.2691	0.426238
$700$	0.309530	0.0116991
$701$	30.1959	1.14048	0.570242	−	0.821476i	$-0.306849\pi$
$701$	30.1959	1.14048	0.570242	+	0.821476i	$0.306849\pi$
$702$	32.3259	1.22006
$703$	0	0
$704$	2.53307	0.0954686
$705$	−3.85486	−0.145182
$706$	18.8209	0.708333
$707$	0.852903	0.0320767
$708$	−3.74322	−0.140679
$709$	37.8617	1.42193	0.710963	−	0.703229i	$-0.248259\pi$
$709$	37.8617	1.42193	0.710963	+	0.703229i	$0.248259\pi$
$710$	10.9554	0.411148
$711$	14.9191	0.559510
$712$	−7.07934	−0.265309
$713$	−36.0442	−1.34987
$714$	0.522232	0.0195441
$715$	−1.85900	−0.0695227
$716$	−2.57979	−0.0964114
$717$	22.1809	0.828360
$718$	6.66427	0.248708
$719$	−38.1719	−1.42357	−0.711786	−	0.702396i	$-0.752114\pi$
$719$	−38.1719	−1.42357	−0.711786	+	0.702396i	$0.752114\pi$
$720$	−4.54971	−0.169558
$721$	−0.243158	−0.00905569
$722$	0	0
$723$	−19.1519	−0.712269
$724$	10.0787	0.374570
$725$	−19.1563	−0.711448
$726$	−1.84007	−0.0682915
$727$	22.3720	0.829731	0.414865	−	0.909883i	$-0.363829\pi$
$727$	22.3720	0.829731	0.414865	+	0.909883i	$0.363829\pi$
$728$	0.570949	0.0211608
$729$	17.8888	0.662546
$730$	7.84753	0.290450
$731$	−16.6265	−0.614953
$732$	−8.06080	−0.297936
$733$	−33.4848	−1.23679	−0.618394	−	0.785868i	$-0.712216\pi$
$733$	−33.4848	−1.23679	−0.618394	+	0.785868i	$0.712216\pi$
$734$	3.95730	0.146067
$735$	−3.88998	−0.143484
$736$	−24.4455	−0.901072
$737$	−2.96229	−0.109117
$738$	33.4969	1.23304
$739$	−8.70066	−0.320059	−0.160029	−	0.987112i	$-0.551159\pi$
$739$	−8.70066	−0.320059	−0.160029	+	0.987112i	$0.551159\pi$
$740$	−0.718924	−0.0264282
$741$	0	0
$742$	0.412027	0.0151260
$743$	31.5966	1.15917	0.579584	−	0.814912i	$-0.303215\pi$
$743$	31.5966	1.15917	0.579584	+	0.814912i	$0.303215\pi$
$744$	−14.0373	−0.514633
$745$	8.87631	0.325203
$746$	14.8742	0.544582
$747$	4.98057	0.182230
$748$	2.96422	0.108382
$749$	1.09364	0.0399607
$750$	−9.10835	−0.332590
$751$	−30.6511	−1.11848	−0.559238	−	0.829007i	$-0.688906\pi$
$751$	−30.6511	−1.11848	−0.559238	+	0.829007i	$0.688906\pi$
$752$	34.4425	1.25599
$753$	−22.5629	−0.822239
$754$	24.8453	0.904814
$755$	−1.85107	−0.0673674
$756$	−0.343108	−0.0124787
$757$	−42.4313	−1.54219	−0.771095	−	0.636720i	$-0.780291\pi$
$757$	−42.4313	−1.54219	−0.771095	+	0.636720i	$0.780291\pi$
$758$	−32.6855	−1.18719
$759$	−6.07352	−0.220455
$760$	0	0
$761$	−33.9440	−1.23047	−0.615234	−	0.788344i	$-0.710939\pi$
$761$	−33.9440	−1.23047	−0.615234	+	0.788344i	$0.710939\pi$
$762$	−10.4238	−0.377615
$763$	−0.259314	−0.00938780
$764$	1.56214	0.0565162
$765$	3.28663	0.118828
$766$	52.7409	1.90561
$767$	15.1519	0.547105
$768$	−18.5037	−0.667695
$769$	−48.2113	−1.73854	−0.869272	−	0.494333i	$-0.835412\pi$
$769$	−48.2113	−1.73854	−0.869272	+	0.494333i	$0.835412\pi$
$770$	0.0675252	0.00243344
$771$	5.56565	0.200442
$772$	0.798225	0.0287287
$773$	−4.58368	−0.164864	−0.0824318	−	0.996597i	$-0.526269\pi$
$773$	−4.58368	−0.164864	−0.0824318	+	0.996597i	$0.526269\pi$
$774$	14.0271	0.504195
$775$	30.8044	1.10653
$776$	30.2378	1.08547
$777$	0.148289	0.00531985
$778$	47.0752	1.68773
$779$	0	0
$780$	1.68024	0.0601623
$781$	−12.8263	−0.458960
$782$	33.4827	1.19734
$783$	21.2344	0.758856
$784$	34.7563	1.24130
$785$	−5.38658	−0.192255
$786$	−6.00460	−0.214177
$787$	−18.8649	−0.672463	−0.336231	−	0.941779i	$-0.609152\pi$
$787$	−18.8649	−0.672463	−0.336231	+	0.941779i	$0.609152\pi$
$788$	16.2757	0.579796
$789$	17.6023	0.626658
$790$	7.07251	0.251629
$791$	1.08988	0.0387517
$792$	3.55664	0.126380
$793$	32.6288	1.15868
$794$	−32.7529	−1.16236
$795$	−1.72450	−0.0611615
$796$	18.2922	0.648351
$797$	−29.7563	−1.05402	−0.527012	−	0.849858i	$-0.676688\pi$
$797$	−29.7563	−1.05402	−0.527012	+	0.849858i	$0.676688\pi$
$798$	0	0
$799$	−24.8807	−0.880215
$800$	20.8918	0.738636
$801$	−6.46168	−0.228312
$802$	49.4051	1.74455
$803$	−9.18768	−0.324226
$804$	2.67744	0.0944261
$805$	0.222880	0.00785548
$806$	−39.9526	−1.40727
$807$	−2.36666	−0.0833104
$808$	21.2963	0.749202
$809$	−28.1923	−0.991190	−0.495595	−	0.868554i	$-0.665050\pi$
$809$	−28.1923	−0.991190	−0.495595	+	0.868554i	$0.665050\pi$
$810$	0.297582	0.0104560
$811$	36.9805	1.29856	0.649280	−	0.760549i	$-0.275070\pi$
$811$	36.9805	1.29856	0.649280	+	0.760549i	$0.275070\pi$
$812$	−0.263709	−0.00925438
$813$	7.50676	0.263274
$814$	2.88046	0.100960
$815$	−10.0872	−0.353341
$816$	19.5293	0.683661
$817$	0	0
$818$	−55.5025	−1.94060
$819$	0.521134	0.0182099
$820$	4.64013	0.162040
$821$	−43.5919	−1.52137	−0.760684	−	0.649122i	$-0.775136\pi$
$821$	−43.5919	−1.52137	−0.760684	+	0.649122i	$0.775136\pi$
$822$	19.2844	0.672621
$823$	−14.9002	−0.519388	−0.259694	−	0.965691i	$-0.583622\pi$
$823$	−14.9002	−0.519388	−0.259694	+	0.965691i	$0.583622\pi$
$824$	−6.07147	−0.211510
$825$	5.19060	0.180713
$826$	−0.550370	−0.0191498
$827$	26.4327	0.919153	0.459577	−	0.888138i	$-0.348001\pi$
$827$	26.4327	0.919153	0.459577	+	0.888138i	$0.348001\pi$
$828$	−8.25436	−0.286859
$829$	−15.9510	−0.554001	−0.277000	−	0.960870i	$-0.589340\pi$
$829$	−15.9510	−0.554001	−0.277000	+	0.960870i	$0.589340\pi$
$830$	2.36108	0.0819542
$831$	−29.1275	−1.01042
$832$	9.26748	0.321292
$833$	−25.1073	−0.869917
$834$	−30.1260	−1.04318
$835$	3.31095	0.114580
$836$	0	0
$837$	−34.1461	−1.18026
$838$	36.8631	1.27342
$839$	31.7128	1.09485	0.547424	−	0.836856i	$-0.315609\pi$
$839$	31.7128	1.09485	0.547424	+	0.836856i	$0.315609\pi$
$840$	0.0867999	0.00299488
$841$	−12.6794	−0.437222
$842$	−46.9257	−1.61717
$843$	−14.5109	−0.499784
$844$	−9.72123	−0.334619
$845$	−0.195804	−0.00673585
$846$	20.9909	0.721681
$847$	−0.0790567	−0.00271642
$848$	15.4081	0.529115
$849$	−9.57217	−0.328516
$850$	−28.6152	−0.981495
$851$	9.50750	0.325913
$852$	11.5929	0.397166
$853$	13.7535	0.470912	0.235456	−	0.971885i	$-0.424342\pi$
$853$	13.7535	0.470912	0.235456	+	0.971885i	$0.424342\pi$
$854$	−1.18519	−0.0405563
$855$	0	0
$856$	27.3073	0.933345
$857$	47.5712	1.62500	0.812500	−	0.582961i	$-0.198106\pi$
$857$	47.5712	1.62500	0.812500	+	0.582961i	$0.198106\pi$
$858$	−6.73209	−0.229830
$859$	37.8586	1.29172	0.645859	−	0.763457i	$-0.276499\pi$
$859$	37.8586	1.29172	0.645859	+	0.763457i	$0.276499\pi$
$860$	1.94310	0.0662590
$861$	−0.957099	−0.0326178
$862$	−35.9817	−1.22554
$863$	4.04675	0.137753	0.0688765	−	0.997625i	$-0.478059\pi$
$863$	4.04675	0.137753	0.0688765	+	0.997625i	$0.478059\pi$
$864$	−23.1581	−0.787856
$865$	1.91736	0.0651923
$866$	−60.0897	−2.04193
$867$	4.50134	0.152873
$868$	0.424059	0.0143935
$869$	−8.28030	−0.280890
$870$	3.77717	0.128058
$871$	−10.8378	−0.367226
$872$	−6.47487	−0.219267
$873$	27.5996	0.934104
$874$	0	0
$875$	−0.391330	−0.0132294
$876$	8.30420	0.280573
$877$	26.1977	0.884633	0.442316	−	0.896859i	$-0.354157\pi$
$877$	26.1977	0.884633	0.442316	+	0.896859i	$0.354157\pi$
$878$	6.46568	0.218206
$879$	−9.91415	−0.334396
$880$	2.52515	0.0851229
$881$	10.5495	0.355422	0.177711	−	0.984083i	$-0.443131\pi$
$881$	10.5495	0.355422	0.177711	+	0.984083i	$0.443131\pi$
$882$	21.1821	0.713238
$883$	0.302694	0.0101865	0.00509323	−	0.999987i	$-0.498379\pi$
$883$	0.302694	0.0101865	0.00509323	+	0.999987i	$0.498379\pi$
$884$	10.8449	0.364753
$885$	2.30351	0.0774317
$886$	3.13342	0.105269
$887$	37.2880	1.25201	0.626004	−	0.779820i	$-0.284690\pi$
$887$	37.2880	1.25201	0.626004	+	0.779820i	$0.284690\pi$
$888$	3.70267	0.124253
$889$	−0.447847	−0.0150203
$890$	−3.06321	−0.102679
$891$	−0.348401	−0.0116719
$892$	−16.6604	−0.557832
$893$	0	0
$894$	32.1442	1.07506
$895$	1.58756	0.0530662
$896$	−1.03325	−0.0345185
$897$	−22.2206	−0.741923
$898$	23.8573	0.796129
$899$	−26.2443	−0.875297
$900$	7.05441	0.235147
$901$	−11.1305	−0.370811
$902$	−18.5912	−0.619021
$903$	−0.400794	−0.0133376
$904$	27.2134	0.905106
$905$	−6.20223	−0.206169
$906$	−6.70338	−0.222705
$907$	17.6040	0.584533	0.292266	−	0.956337i	$-0.405591\pi$
$907$	17.6040	0.584533	0.292266	+	0.956337i	$0.405591\pi$
$908$	−19.6028	−0.650542
$909$	19.4383	0.644726
$910$	0.247048	0.00818955
$911$	57.8952	1.91815	0.959076	−	0.283148i	$-0.0913789\pi$
$911$	57.8952	1.91815	0.959076	+	0.283148i	$0.0913789\pi$
$912$	0	0
$913$	−2.76429	−0.0914845
$914$	−39.2953	−1.29977
$915$	4.96048	0.163988
$916$	−2.46430	−0.0814227
$917$	−0.257981	−0.00851928
$918$	31.7194	1.04690
$919$	−12.8718	−0.424600	−0.212300	−	0.977205i	$-0.568095\pi$
$919$	−12.8718	−0.424600	−0.212300	+	0.977205i	$0.568095\pi$
$920$	5.56514	0.183477
$921$	13.0121	0.428765
$922$	−7.29553	−0.240266
$923$	−46.9262	−1.54459
$924$	0.0714547	0.00235069
$925$	−8.12537	−0.267161
$926$	30.0510	0.987536
$927$	−5.54175	−0.182015
$928$	−17.7991	−0.584284
$929$	18.9629	0.622153	0.311077	−	0.950385i	$-0.399310\pi$
$929$	18.9629	0.622153	0.311077	+	0.950385i	$0.399310\pi$
$930$	−6.07390	−0.199171
$931$	0	0
$932$	−8.50037	−0.278439
$933$	−33.1441	−1.08509
$934$	−35.0426	−1.14663
$935$	−1.82413	−0.0596553
$936$	13.0123	0.425321
$937$	−47.0256	−1.53626	−0.768130	−	0.640294i	$-0.778812\pi$
$937$	−47.0256	−1.53626	−0.768130	+	0.640294i	$0.778812\pi$
$938$	0.393667	0.0128537
$939$	19.1179	0.623889
$940$	2.90774	0.0948401
$941$	−18.3303	−0.597550	−0.298775	−	0.954324i	$-0.596578\pi$
$941$	−18.3303	−0.597550	−0.298775	+	0.954324i	$0.596578\pi$
$942$	−19.5067	−0.635562
$943$	−61.3640	−1.99829
$944$	−20.5815	−0.669870
$945$	0.211143	0.00686847
$946$	−7.78526	−0.253121
$947$	−43.9083	−1.42683	−0.713415	−	0.700742i	$-0.752852\pi$
$947$	−43.9083	−1.42683	−0.713415	+	0.700742i	$0.752852\pi$
$948$	7.48408	0.243072
$949$	−33.6140	−1.09116
$950$	0	0
$951$	16.2273	0.526207
$952$	0.560238	0.0181574
$953$	36.1391	1.17066	0.585329	−	0.810796i	$-0.300965\pi$
$953$	36.1391	1.17066	0.585329	+	0.810796i	$0.300965\pi$
$954$	9.39039	0.304025
$955$	−0.961313	−0.0311074
$956$	−16.7312	−0.541124
$957$	−4.42222	−0.142950
$958$	−3.70303	−0.119640
$959$	0.828533	0.0267547
$960$	1.40891	0.0454724
$961$	11.2023	0.361363
$962$	10.5384	0.339773
$963$	24.9248	0.803191
$964$	14.4464	0.465288
$965$	−0.491213	−0.0158127
$966$	0.807126	0.0259689
$967$	−32.5993	−1.04832	−0.524161	−	0.851619i	$-0.675621\pi$
$967$	−32.5993	−1.04832	−0.524161	+	0.851619i	$0.675621\pi$
$968$	−1.97398	−0.0634462
$969$	0	0
$970$	13.0838	0.420095
$971$	−24.9854	−0.801818	−0.400909	−	0.916118i	$-0.631306\pi$
$971$	−24.9854	−0.801818	−0.400909	+	0.916118i	$0.631306\pi$
$972$	−12.7052	−0.407519
$973$	−1.29433	−0.0414943
$974$	34.0908	1.09234
$975$	18.9903	0.608177
$976$	−44.3210	−1.41868
$977$	−22.6053	−0.723207	−0.361604	−	0.932332i	$-0.617771\pi$
$977$	−22.6053	−0.723207	−0.361604	+	0.932332i	$0.617771\pi$
$978$	−36.5294	−1.16808
$979$	3.58632	0.114619
$980$	2.93423	0.0937306
$981$	−5.90995	−0.188690
$982$	33.8390	1.07984
$983$	41.6836	1.32950	0.664751	−	0.747065i	$-0.268538\pi$
$983$	41.6836	1.32950	0.664751	+	0.747065i	$0.268538\pi$
$984$	−23.8980	−0.761841
$985$	−10.0157	−0.319128
$986$	24.3792	0.776393
$987$	−0.599767	−0.0190908
$988$	0	0
$989$	−25.6967	−0.817109
$990$	1.53895	0.0489109
$991$	21.1058	0.670448	0.335224	−	0.942138i	$-0.391188\pi$
$991$	21.1058	0.670448	0.335224	+	0.942138i	$0.391188\pi$
$992$	28.6219	0.908747
$993$	31.8718	1.01142
$994$	1.70452	0.0540640
$995$	−11.2567	−0.356862
$996$	2.49848	0.0791672
$997$	−0.667646	−0.0211446	−0.0105723	−	0.999944i	$-0.503365\pi$
$997$	−0.667646	−0.0211446	−0.0105723	+	0.999944i	$0.503365\pi$
$998$	44.3789	1.40479
$999$	9.00681	0.284963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3971.2.a.s.1.17		21
19.14	odd	18		209.2.j.b.177.2	yes	42
19.15	odd	18		209.2.j.b.111.2	✓	42
19.18	odd	2		3971.2.a.t.1.5		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.j.b.111.2	✓	42	19.15	odd	18
209.2.j.b.177.2	yes	42	19.14	odd	18
3971.2.a.s.1.17		21	1.1	even	1	trivial
3971.2.a.t.1.5		21	19.18	odd	2

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:	\( N \)	\(=\)	\( 3971 = 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3971.a (trivial)

\(f(q)\)	\(=\)	\(q+1.68098 q^{2} -1.09464 q^{3} +0.825695 q^{4} -0.508118 q^{5} -1.84007 q^{6} -0.0790567 q^{7} -1.97398 q^{8} -1.80176 q^{9} +O(q^{10})\)	\(q+1.68098 q^{2} -1.09464 q^{3} +0.825695 q^{4} -0.508118 q^{5} -1.84007 q^{6} -0.0790567 q^{7} -1.97398 q^{8} -1.80176 q^{9} -0.854136 q^{10} +1.00000 q^{11} -0.903841 q^{12} +3.65860 q^{13} -0.132893 q^{14} +0.556208 q^{15} -4.96962 q^{16} +3.58996 q^{17} -3.02872 q^{18} -0.419550 q^{20} +0.0865388 q^{21} +1.68098 q^{22} +5.54840 q^{23} +2.16081 q^{24} -4.74182 q^{25} +6.15003 q^{26} +5.25621 q^{27} -0.0652767 q^{28} +4.03987 q^{29} +0.934974 q^{30} -6.49633 q^{31} -4.40586 q^{32} -1.09464 q^{33} +6.03466 q^{34} +0.0401701 q^{35} -1.48770 q^{36} +1.71356 q^{37} -4.00486 q^{39} +1.00302 q^{40} -11.0598 q^{41} +0.145470 q^{42} -4.63138 q^{43} +0.825695 q^{44} +0.915505 q^{45} +9.32675 q^{46} -6.93061 q^{47} +5.43996 q^{48} -6.99375 q^{49} -7.97090 q^{50} -3.92973 q^{51} +3.02089 q^{52} -3.10045 q^{53} +8.83558 q^{54} -0.508118 q^{55} +0.156057 q^{56} +6.79094 q^{58} +4.14146 q^{59} +0.459258 q^{60} +8.91839 q^{61} -10.9202 q^{62} +0.142441 q^{63} +2.53307 q^{64} -1.85900 q^{65} -1.84007 q^{66} -2.96229 q^{67} +2.96422 q^{68} -6.07352 q^{69} +0.0675252 q^{70} -12.8263 q^{71} +3.55664 q^{72} -9.18768 q^{73} +2.88046 q^{74} +5.19060 q^{75} -0.0790567 q^{77} -6.73209 q^{78} -8.28030 q^{79} +2.52515 q^{80} -0.348401 q^{81} -18.5912 q^{82} -2.76429 q^{83} +0.0714547 q^{84} -1.82413 q^{85} -7.78526 q^{86} -4.42222 q^{87} -1.97398 q^{88} +3.58632 q^{89} +1.53895 q^{90} -0.289237 q^{91} +4.58128 q^{92} +7.11116 q^{93} -11.6502 q^{94} +4.82285 q^{96} -15.3181 q^{97} -11.7564 q^{98} -1.80176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 6 q^{2} - 15 q^{3} + 18 q^{4} + 6 q^{5} - 6 q^{6} + 6 q^{7} - 15 q^{8} + 18 q^{9}+O(q^{10}) \) 21 * q - 6 * q^2 - 15 * q^3 + 18 * q^4 + 6 * q^5 - 6 * q^6 + 6 * q^7 - 15 * q^8 + 18 * q^9	\( 21 q - 6 q^{2} - 15 q^{3} + 18 q^{4} + 6 q^{5} - 6 q^{6} + 6 q^{7} - 15 q^{8} + 18 q^{9} + 6 q^{10} + 21 q^{11} - 30 q^{12} - 36 q^{13} - 12 q^{14} - 12 q^{15} + 12 q^{16} + 21 q^{17} - 6 q^{18} + 15 q^{20} + 3 q^{21} - 6 q^{22} + 6 q^{23} - 18 q^{24} + 3 q^{25} + 6 q^{26} - 24 q^{27} - 27 q^{29} - 30 q^{30} - 54 q^{31} - 21 q^{32} - 15 q^{33} - 39 q^{34} - 33 q^{35} + 54 q^{36} - 57 q^{37} + 27 q^{40} - 24 q^{41} + 6 q^{42} + 12 q^{43} + 18 q^{44} - 3 q^{45} - 15 q^{46} + 12 q^{47} + 39 q^{48} + 21 q^{49} - 33 q^{50} + 18 q^{51} + 18 q^{52} - 39 q^{53} + 30 q^{54} + 6 q^{55} - 57 q^{56} + 30 q^{58} - 27 q^{59} - 12 q^{60} - 30 q^{61} + 21 q^{62} - 18 q^{63} - 15 q^{64} + 24 q^{65} - 6 q^{66} - 27 q^{67} - 48 q^{68} - 54 q^{69} - 51 q^{70} - 42 q^{71} - 15 q^{72} + 6 q^{73} + 84 q^{74} - 27 q^{75} + 6 q^{77} + 48 q^{78} - 39 q^{79} - 27 q^{80} - 27 q^{81} - 15 q^{82} + 12 q^{83} + 48 q^{84} - 42 q^{85} + 45 q^{86} + 75 q^{87} - 15 q^{88} - 75 q^{89} + 87 q^{90} - 45 q^{91} + 12 q^{92} + 66 q^{93} - 42 q^{94} - 3 q^{96} - 39 q^{97} + 45 q^{98} + 18 q^{99}+O(q^{100}) \) 21 * q - 6 * q^2 - 15 * q^3 + 18 * q^4 + 6 * q^5 - 6 * q^6 + 6 * q^7 - 15 * q^8 + 18 * q^9 + 6 * q^10 + 21 * q^11 - 30 * q^12 - 36 * q^13 - 12 * q^14 - 12 * q^15 + 12 * q^16 + 21 * q^17 - 6 * q^18 + 15 * q^20 + 3 * q^21 - 6 * q^22 + 6 * q^23 - 18 * q^24 + 3 * q^25 + 6 * q^26 - 24 * q^27 - 27 * q^29 - 30 * q^30 - 54 * q^31 - 21 * q^32 - 15 * q^33 - 39 * q^34 - 33 * q^35 + 54 * q^36 - 57 * q^37 + 27 * q^40 - 24 * q^41 + 6 * q^42 + 12 * q^43 + 18 * q^44 - 3 * q^45 - 15 * q^46 + 12 * q^47 + 39 * q^48 + 21 * q^49 - 33 * q^50 + 18 * q^51 + 18 * q^52 - 39 * q^53 + 30 * q^54 + 6 * q^55 - 57 * q^56 + 30 * q^58 - 27 * q^59 - 12 * q^60 - 30 * q^61 + 21 * q^62 - 18 * q^63 - 15 * q^64 + 24 * q^65 - 6 * q^66 - 27 * q^67 - 48 * q^68 - 54 * q^69 - 51 * q^70 - 42 * q^71 - 15 * q^72 + 6 * q^73 + 84 * q^74 - 27 * q^75 + 6 * q^77 + 48 * q^78 - 39 * q^79 - 27 * q^80 - 27 * q^81 - 15 * q^82 + 12 * q^83 + 48 * q^84 - 42 * q^85 + 45 * q^86 + 75 * q^87 - 15 * q^88 - 75 * q^89 + 87 * q^90 - 45 * q^91 + 12 * q^92 + 66 * q^93 - 42 * q^94 - 3 * q^96 - 39 * q^97 + 45 * q^98 + 18 * q^99

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