LMFDB - Embedded newform 4022.2.a.d.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.0519256 q^{3} +1.00000 q^{4} -2.57858 q^{5} -0.0519256 q^{6} +3.17465 q^{7} -1.00000 q^{8} -2.99730 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.0519256 q^{3} +1.00000 q^{4} -2.57858 q^{5} -0.0519256 q^{6} +3.17465 q^{7} -1.00000 q^{8} -2.99730 q^{9} +2.57858 q^{10} +1.21936 q^{11} +0.0519256 q^{12} -6.52636 q^{13} -3.17465 q^{14} -0.133894 q^{15} +1.00000 q^{16} +0.682669 q^{17} +2.99730 q^{18} +2.63479 q^{19} -2.57858 q^{20} +0.164846 q^{21} -1.21936 q^{22} +2.04812 q^{23} -0.0519256 q^{24} +1.64905 q^{25} +6.52636 q^{26} -0.311414 q^{27} +3.17465 q^{28} +7.83712 q^{29} +0.133894 q^{30} +7.65134 q^{31} -1.00000 q^{32} +0.0633159 q^{33} -0.682669 q^{34} -8.18607 q^{35} -2.99730 q^{36} +0.366697 q^{37} -2.63479 q^{38} -0.338885 q^{39} +2.57858 q^{40} +1.87913 q^{41} -0.164846 q^{42} -3.14132 q^{43} +1.21936 q^{44} +7.72877 q^{45} -2.04812 q^{46} +1.74992 q^{47} +0.0519256 q^{48} +3.07838 q^{49} -1.64905 q^{50} +0.0354480 q^{51} -6.52636 q^{52} -0.658853 q^{53} +0.311414 q^{54} -3.14420 q^{55} -3.17465 q^{56} +0.136813 q^{57} -7.83712 q^{58} +1.65179 q^{59} -0.133894 q^{60} -13.5558 q^{61} -7.65134 q^{62} -9.51538 q^{63} +1.00000 q^{64} +16.8287 q^{65} -0.0633159 q^{66} -11.4620 q^{67} +0.682669 q^{68} +0.106350 q^{69} +8.18607 q^{70} +2.72321 q^{71} +2.99730 q^{72} -2.31887 q^{73} -0.366697 q^{74} +0.0856281 q^{75} +2.63479 q^{76} +3.87103 q^{77} +0.338885 q^{78} -12.4315 q^{79} -2.57858 q^{80} +8.97574 q^{81} -1.87913 q^{82} -2.41354 q^{83} +0.164846 q^{84} -1.76031 q^{85} +3.14132 q^{86} +0.406947 q^{87} -1.21936 q^{88} -0.424624 q^{89} -7.72877 q^{90} -20.7189 q^{91} +2.04812 q^{92} +0.397301 q^{93} -1.74992 q^{94} -6.79400 q^{95} -0.0519256 q^{96} -12.7283 q^{97} -3.07838 q^{98} -3.65478 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9}+O(q^{10}) $ 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9} + 13 q^{10} + 12 q^{11} - 5 q^{12} - 36 q^{13} + 22 q^{14} - 7 q^{15} + 37 q^{16} + 4 q^{17} - 32 q^{18} - 10 q^{19} - 13 q^{20} - 11 q^{21} - 12 q^{22} - q^{23} + 5 q^{24} + 8 q^{25} + 36 q^{26} - 20 q^{27} - 22 q^{28} - 6 q^{29} + 7 q^{30} - 23 q^{31} - 37 q^{32} - 27 q^{33} - 4 q^{34} + 24 q^{35} + 32 q^{36} - 46 q^{37} + 10 q^{38} + 5 q^{39} + 13 q^{40} + 11 q^{41} + 11 q^{42} - 22 q^{43} + 12 q^{44} - 57 q^{45} + q^{46} - 18 q^{47} - 5 q^{48} - q^{49} - 8 q^{50} + 18 q^{51} - 36 q^{52} - 25 q^{53} + 20 q^{54} - 25 q^{55} + 22 q^{56} - 25 q^{57} + 6 q^{58} + 24 q^{59} - 7 q^{60} - 40 q^{61} + 23 q^{62} - 38 q^{63} + 37 q^{64} + 14 q^{65} + 27 q^{66} - 49 q^{67} + 4 q^{68} - 19 q^{69} - 24 q^{70} + 27 q^{71} - 32 q^{72} - 87 q^{73} + 46 q^{74} - 5 q^{75} - 10 q^{76} - 50 q^{77} - 5 q^{78} + 11 q^{79} - 13 q^{80} + 5 q^{81} - 11 q^{82} - 9 q^{83} - 11 q^{84} - 68 q^{85} + 22 q^{86} - 12 q^{87} - 12 q^{88} - 3 q^{89} + 57 q^{90} - 19 q^{91} - q^{92} - 69 q^{93} + 18 q^{94} + 27 q^{95} + 5 q^{96} - 98 q^{97} + q^{98} + 33 q^{99}+O(q^{100}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9 + 13 * q^10 + 12 * q^11 - 5 * q^12 - 36 * q^13 + 22 * q^14 - 7 * q^15 + 37 * q^16 + 4 * q^17 - 32 * q^18 - 10 * q^19 - 13 * q^20 - 11 * q^21 - 12 * q^22 - q^23 + 5 * q^24 + 8 * q^25 + 36 * q^26 - 20 * q^27 - 22 * q^28 - 6 * q^29 + 7 * q^30 - 23 * q^31 - 37 * q^32 - 27 * q^33 - 4 * q^34 + 24 * q^35 + 32 * q^36 - 46 * q^37 + 10 * q^38 + 5 * q^39 + 13 * q^40 + 11 * q^41 + 11 * q^42 - 22 * q^43 + 12 * q^44 - 57 * q^45 + q^46 - 18 * q^47 - 5 * q^48 - q^49 - 8 * q^50 + 18 * q^51 - 36 * q^52 - 25 * q^53 + 20 * q^54 - 25 * q^55 + 22 * q^56 - 25 * q^57 + 6 * q^58 + 24 * q^59 - 7 * q^60 - 40 * q^61 + 23 * q^62 - 38 * q^63 + 37 * q^64 + 14 * q^65 + 27 * q^66 - 49 * q^67 + 4 * q^68 - 19 * q^69 - 24 * q^70 + 27 * q^71 - 32 * q^72 - 87 * q^73 + 46 * q^74 - 5 * q^75 - 10 * q^76 - 50 * q^77 - 5 * q^78 + 11 * q^79 - 13 * q^80 + 5 * q^81 - 11 * q^82 - 9 * q^83 - 11 * q^84 - 68 * q^85 + 22 * q^86 - 12 * q^87 - 12 * q^88 - 3 * q^89 + 57 * q^90 - 19 * q^91 - q^92 - 69 * q^93 + 18 * q^94 + 27 * q^95 + 5 * q^96 - 98 * q^97 + q^98 + 33 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.0519256	0.0299793	0.0149896	−	0.999888i	$-0.495228\pi$
$3$	0.0519256	0.0299793	0.0149896	+	0.999888i	$0.495228\pi$
$4$	1.00000	0.500000
$5$	−2.57858	−1.15317	−0.576587	−	0.817036i	$-0.695616\pi$
$5$	−2.57858	−1.15317	−0.576587	+	0.817036i	$0.695616\pi$
$6$	−0.0519256	−0.0211985
$7$	3.17465	1.19990	0.599952	−	0.800036i	$-0.295186\pi$
$7$	3.17465	1.19990	0.599952	+	0.800036i	$0.295186\pi$
$8$	−1.00000	−0.353553
$9$	−2.99730	−0.999101
$10$	2.57858	0.815417
$11$	1.21936	0.367650	0.183825	−	0.982959i	$-0.441152\pi$
$11$	1.21936	0.367650	0.183825	+	0.982959i	$0.441152\pi$
$12$	0.0519256	0.0149896
$13$	−6.52636	−1.81009	−0.905043	−	0.425321i	$-0.860161\pi$
$13$	−6.52636	−1.81009	−0.905043	+	0.425321i	$0.860161\pi$
$14$	−3.17465	−0.848460
$15$	−0.133894	−0.0345713
$16$	1.00000	0.250000
$17$	0.682669	0.165572	0.0827858	−	0.996567i	$-0.473618\pi$
$17$	0.682669	0.165572	0.0827858	+	0.996567i	$0.473618\pi$
$18$	2.99730	0.706471
$19$	2.63479	0.604461	0.302231	−	0.953235i	$-0.402269\pi$
$19$	2.63479	0.604461	0.302231	+	0.953235i	$0.402269\pi$
$20$	−2.57858	−0.576587
$21$	0.164846	0.0359722
$22$	−1.21936	−0.259968
$23$	2.04812	0.427062	0.213531	−	0.976936i	$-0.431504\pi$
$23$	2.04812	0.427062	0.213531	+	0.976936i	$0.431504\pi$
$24$	−0.0519256	−0.0105993
$25$	1.64905	0.329811
$26$	6.52636	1.27992
$27$	−0.311414	−0.0599316
$28$	3.17465	0.599952
$29$	7.83712	1.45532	0.727658	−	0.685940i	$-0.240609\pi$
$29$	7.83712	1.45532	0.727658	+	0.685940i	$0.240609\pi$
$30$	0.133894	0.0244456
$31$	7.65134	1.37422	0.687111	−	0.726553i	$-0.258879\pi$
$31$	7.65134	1.37422	0.687111	+	0.726553i	$0.258879\pi$
$32$	−1.00000	−0.176777
$33$	0.0633159	0.0110219
$34$	−0.682669	−0.117077
$35$	−8.18607	−1.38370
$36$	−2.99730	−0.499551
$37$	0.366697	0.0602846	0.0301423	−	0.999546i	$-0.490404\pi$
$37$	0.366697	0.0602846	0.0301423	+	0.999546i	$0.490404\pi$
$38$	−2.63479	−0.427419
$39$	−0.338885	−0.0542650
$40$	2.57858	0.407709
$41$	1.87913	0.293471	0.146736	−	0.989176i	$-0.453123\pi$
$41$	1.87913	0.293471	0.146736	+	0.989176i	$0.453123\pi$
$42$	−0.164846	−0.0254362
$43$	−3.14132	−0.479047	−0.239524	−	0.970891i	$-0.576991\pi$
$43$	−3.14132	−0.479047	−0.239524	+	0.970891i	$0.576991\pi$
$44$	1.21936	0.183825
$45$	7.72877	1.15214
$46$	−2.04812	−0.301978
$47$	1.74992	0.255251	0.127626	−	0.991822i	$-0.459264\pi$
$47$	1.74992	0.255251	0.127626	+	0.991822i	$0.459264\pi$
$48$	0.0519256	0.00749482
$49$	3.07838	0.439769
$50$	−1.64905	−0.233211
$51$	0.0354480	0.00496371
$52$	−6.52636	−0.905043
$53$	−0.658853	−0.0905004	−0.0452502	−	0.998976i	$-0.514409\pi$
$53$	−0.658853	−0.0905004	−0.0452502	+	0.998976i	$0.514409\pi$
$54$	0.311414	0.0423780
$55$	−3.14420	−0.423964
$56$	−3.17465	−0.424230
$57$	0.136813	0.0181213
$58$	−7.83712	−1.02906
$59$	1.65179	0.215045	0.107523	−	0.994203i	$-0.465708\pi$
$59$	1.65179	0.215045	0.107523	+	0.994203i	$0.465708\pi$
$60$	−0.133894	−0.0172857
$61$	−13.5558	−1.73564	−0.867820	−	0.496878i	$-0.834480\pi$
$61$	−13.5558	−1.73564	−0.867820	+	0.496878i	$0.834480\pi$
$62$	−7.65134	−0.971721
$63$	−9.51538	−1.19883
$64$	1.00000	0.125000
$65$	16.8287	2.08734
$66$	−0.0633159	−0.00779364
$67$	−11.4620	−1.40030	−0.700151	−	0.713995i	$-0.746884\pi$
$67$	−11.4620	−1.40030	−0.700151	+	0.713995i	$0.746884\pi$
$68$	0.682669	0.0827858
$69$	0.106350	0.0128030
$70$	8.18607	0.978422
$71$	2.72321	0.323186	0.161593	−	0.986857i	$-0.448337\pi$
$71$	2.72321	0.323186	0.161593	+	0.986857i	$0.448337\pi$
$72$	2.99730	0.353236
$73$	−2.31887	−0.271404	−0.135702	−	0.990750i	$-0.543329\pi$
$73$	−2.31887	−0.271404	−0.135702	+	0.990750i	$0.543329\pi$
$74$	−0.366697	−0.0426276
$75$	0.0856281	0.00988748
$76$	2.63479	0.302231
$77$	3.87103	0.441144
$78$	0.338885	0.0383712
$79$	−12.4315	−1.39866	−0.699328	−	0.714800i	$-0.746518\pi$
$79$	−12.4315	−1.39866	−0.699328	+	0.714800i	$0.746518\pi$
$80$	−2.57858	−0.288294
$81$	8.97574	0.997305
$82$	−1.87913	−0.207516
$83$	−2.41354	−0.264920	−0.132460	−	0.991188i	$-0.542288\pi$
$83$	−2.41354	−0.264920	−0.132460	+	0.991188i	$0.542288\pi$
$84$	0.164846	0.0179861
$85$	−1.76031	−0.190933
$86$	3.14132	0.338737
$87$	0.406947	0.0436293
$88$	−1.21936	−0.129984
$89$	−0.424624	−0.0450100	−0.0225050	−	0.999747i	$-0.507164\pi$
$89$	−0.424624	−0.0450100	−0.0225050	+	0.999747i	$0.507164\pi$
$90$	−7.72877	−0.814684
$91$	−20.7189	−2.17193
$92$	2.04812	0.213531
$93$	0.397301	0.0411982
$94$	−1.74992	−0.180490
$95$	−6.79400	−0.697049
$96$	−0.0519256	−0.00529964
$97$	−12.7283	−1.29236	−0.646182	−	0.763184i	$-0.723635\pi$
$97$	−12.7283	−1.29236	−0.646182	+	0.763184i	$0.723635\pi$
$98$	−3.07838	−0.310964
$99$	−3.65478	−0.367319
$100$	1.64905	0.164905
$101$	14.6341	1.45615	0.728075	−	0.685498i	$-0.240415\pi$
$101$	14.6341	1.45615	0.728075	+	0.685498i	$0.240415\pi$
$102$	−0.0354480	−0.00350988
$103$	−14.0040	−1.37986	−0.689928	−	0.723878i	$-0.742358\pi$
$103$	−14.0040	−1.37986	−0.689928	+	0.723878i	$0.742358\pi$
$104$	6.52636	0.639962
$105$	−0.425067	−0.0414823
$106$	0.658853	0.0639935
$107$	−10.4605	−1.01125	−0.505627	−	0.862752i	$-0.668739\pi$
$107$	−10.4605	−1.01125	−0.505627	+	0.862752i	$0.668739\pi$
$108$	−0.311414	−0.0299658
$109$	4.37799	0.419335	0.209668	−	0.977773i	$-0.432762\pi$
$109$	4.37799	0.419335	0.209668	+	0.977773i	$0.432762\pi$
$110$	3.14420	0.299788
$111$	0.0190410	0.00180729
$112$	3.17465	0.299976
$113$	8.51250	0.800788	0.400394	−	0.916343i	$-0.368873\pi$
$113$	8.51250	0.800788	0.400394	+	0.916343i	$0.368873\pi$
$114$	−0.136813	−0.0128137
$115$	−5.28122	−0.492477
$116$	7.83712	0.727658
$117$	19.5615	1.80846
$118$	−1.65179	−0.152060
$119$	2.16723	0.198670
$120$	0.133894	0.0122228
$121$	−9.51317	−0.864834
$122$	13.5558	1.22728
$123$	0.0975752	0.00879806
$124$	7.65134	0.687111
$125$	8.64067	0.772845
$126$	9.51538	0.847697
$127$	10.6360	0.943797	0.471898	−	0.881653i	$-0.343569\pi$
$127$	10.6360	0.943797	0.471898	+	0.881653i	$0.343569\pi$
$128$	−1.00000	−0.0883883
$129$	−0.163115	−0.0143615
$130$	−16.8287	−1.47597
$131$	−1.82207	−0.159195	−0.0795976	−	0.996827i	$-0.525364\pi$
$131$	−1.82207	−0.159195	−0.0795976	+	0.996827i	$0.525364\pi$
$132$	0.0633159	0.00551094
$133$	8.36452	0.725296
$134$	11.4620	0.990163
$135$	0.803004	0.0691116
$136$	−0.682669	−0.0585384
$137$	3.76906	0.322012	0.161006	−	0.986953i	$-0.448526\pi$
$137$	3.76906	0.322012	0.161006	+	0.986953i	$0.448526\pi$
$138$	−0.106350	−0.00905309
$139$	8.86678	0.752071	0.376035	−	0.926605i	$-0.377287\pi$
$139$	8.86678	0.752071	0.376035	+	0.926605i	$0.377287\pi$
$140$	−8.18607	−0.691849
$141$	0.0908654	0.00765225
$142$	−2.72321	−0.228527
$143$	−7.95796	−0.665478
$144$	−2.99730	−0.249775
$145$	−20.2086	−1.67823
$146$	2.31887	0.191911
$147$	0.159847	0.0131840
$148$	0.366697	0.0301423
$149$	0.519198	0.0425344	0.0212672	−	0.999774i	$-0.493230\pi$
$149$	0.519198	0.0425344	0.0212672	+	0.999774i	$0.493230\pi$
$150$	−0.0856281	−0.00699150
$151$	5.72224	0.465669	0.232835	−	0.972516i	$-0.425200\pi$
$151$	5.72224	0.465669	0.232835	+	0.972516i	$0.425200\pi$
$152$	−2.63479	−0.213709
$153$	−2.04617	−0.165423
$154$	−3.87103	−0.311936
$155$	−19.7296	−1.58472
$156$	−0.338885	−0.0271325
$157$	−7.63001	−0.608941	−0.304471	−	0.952522i	$-0.598480\pi$
$157$	−7.63001	−0.608941	−0.304471	+	0.952522i	$0.598480\pi$
$158$	12.4315	0.989000
$159$	−0.0342114	−0.00271314
$160$	2.57858	0.203854
$161$	6.50205	0.512433
$162$	−8.97574	−0.705201
$163$	−5.93274	−0.464688	−0.232344	−	0.972634i	$-0.574640\pi$
$163$	−5.93274	−0.464688	−0.232344	+	0.972634i	$0.574640\pi$
$164$	1.87913	0.146736
$165$	−0.163265	−0.0127101
$166$	2.41354	0.187327
$167$	1.18016	0.0913232	0.0456616	−	0.998957i	$-0.485460\pi$
$167$	1.18016	0.0913232	0.0456616	+	0.998957i	$0.485460\pi$
$168$	−0.164846	−0.0127181
$169$	29.5933	2.27641
$170$	1.76031	0.135010
$171$	−7.89725	−0.603918
$172$	−3.14132	−0.239524
$173$	−22.6031	−1.71848	−0.859240	−	0.511572i	$-0.829063\pi$
$173$	−22.6031	−1.71848	−0.859240	+	0.511572i	$0.829063\pi$
$174$	−0.406947	−0.0308506
$175$	5.23516	0.395741
$176$	1.21936	0.0919125
$177$	0.0857705	0.00644690
$178$	0.424624	0.0318269
$179$	−14.2298	−1.06359	−0.531794	−	0.846874i	$-0.678482\pi$
$179$	−14.2298	−1.06359	−0.531794	+	0.846874i	$0.678482\pi$
$180$	7.72877	0.576069
$181$	−17.7655	−1.32050	−0.660250	−	0.751046i	$-0.729550\pi$
$181$	−17.7655	−1.32050	−0.660250	+	0.751046i	$0.729550\pi$
$182$	20.7189	1.53579
$183$	−0.703893	−0.0520332
$184$	−2.04812	−0.150989
$185$	−0.945555	−0.0695186
$186$	−0.397301	−0.0291315
$187$	0.832417	0.0608723
$188$	1.74992	0.127626
$189$	−0.988629	−0.0719121
$190$	6.79400	0.492888
$191$	−18.9021	−1.36770	−0.683852	−	0.729620i	$-0.739697\pi$
$191$	−18.9021	−1.36770	−0.683852	+	0.729620i	$0.739697\pi$
$192$	0.0519256	0.00374741
$193$	−3.96904	−0.285698	−0.142849	−	0.989745i	$-0.545626\pi$
$193$	−3.96904	−0.285698	−0.142849	+	0.989745i	$0.545626\pi$
$194$	12.7283	0.913839
$195$	0.873841	0.0625770
$196$	3.07838	0.219884
$197$	−12.4458	−0.886723	−0.443362	−	0.896343i	$-0.646214\pi$
$197$	−12.4458	−0.886723	−0.443362	+	0.896343i	$0.646214\pi$
$198$	3.65478	0.259734
$199$	13.8203	0.979692	0.489846	−	0.871809i	$-0.337053\pi$
$199$	13.8203	0.979692	0.489846	+	0.871809i	$0.337053\pi$
$200$	−1.64905	−0.116606
$201$	−0.595169	−0.0419800
$202$	−14.6341	−1.02965
$203$	24.8801	1.74624
$204$	0.0354480	0.00248186
$205$	−4.84549	−0.338424
$206$	14.0040	0.975706
$207$	−6.13883	−0.426678
$208$	−6.52636	−0.452521
$209$	3.21274	0.222230
$210$	0.425067	0.0293324
$211$	−1.79502	−0.123574	−0.0617872	−	0.998089i	$-0.519680\pi$
$211$	−1.79502	−0.123574	−0.0617872	+	0.998089i	$0.519680\pi$
$212$	−0.658853	−0.0452502
$213$	0.141405	0.00968888
$214$	10.4605	0.715064
$215$	8.10014	0.552425
$216$	0.311414	0.0211890
$217$	24.2903	1.64893
$218$	−4.37799	−0.296515
$219$	−0.120409	−0.00813648
$220$	−3.14420	−0.211982
$221$	−4.45534	−0.299699
$222$	−0.0190410	−0.00127795
$223$	−5.98752	−0.400954	−0.200477	−	0.979698i	$-0.564249\pi$
$223$	−5.98752	−0.400954	−0.200477	+	0.979698i	$0.564249\pi$
$224$	−3.17465	−0.212115
$225$	−4.94271	−0.329514
$226$	−8.51250	−0.566243
$227$	−12.5987	−0.836202	−0.418101	−	0.908401i	$-0.637304\pi$
$227$	−12.5987	−0.836202	−0.418101	+	0.908401i	$0.637304\pi$
$228$	0.136813	0.00906066
$229$	−14.7717	−0.976141	−0.488070	−	0.872804i	$-0.662299\pi$
$229$	−14.7717	−0.976141	−0.488070	+	0.872804i	$0.662299\pi$
$230$	5.28122	0.348234
$231$	0.201005	0.0132252
$232$	−7.83712	−0.514532
$233$	−3.46177	−0.226788	−0.113394	−	0.993550i	$-0.536172\pi$
$233$	−3.46177	−0.226788	−0.113394	+	0.993550i	$0.536172\pi$
$234$	−19.5615	−1.27877
$235$	−4.51229	−0.294349
$236$	1.65179	0.107523
$237$	−0.645515	−0.0419307
$238$	−2.16723	−0.140481
$239$	−9.57974	−0.619662	−0.309831	−	0.950792i	$-0.600272\pi$
$239$	−9.57974	−0.619662	−0.309831	+	0.950792i	$0.600272\pi$
$240$	−0.133894	−0.00864283
$241$	3.81441	0.245708	0.122854	−	0.992425i	$-0.460795\pi$
$241$	3.81441	0.245708	0.122854	+	0.992425i	$0.460795\pi$
$242$	9.51317	0.611530
$243$	1.40031	0.0898301
$244$	−13.5558	−0.867820
$245$	−7.93784	−0.507130
$246$	−0.0975752	−0.00622117
$247$	−17.1956	−1.09413
$248$	−7.65134	−0.485861
$249$	−0.125324	−0.00794211
$250$	−8.64067	−0.546484
$251$	−26.9935	−1.70381	−0.851906	−	0.523695i	$-0.824553\pi$
$251$	−26.9935	−1.70381	−0.851906	+	0.523695i	$0.824553\pi$
$252$	−9.51538	−0.599413
$253$	2.49738	0.157009
$254$	−10.6360	−0.667365
$255$	−0.0914054	−0.00572403
$256$	1.00000	0.0625000
$257$	5.49936	0.343041	0.171520	−	0.985181i	$-0.445132\pi$
$257$	5.49936	0.343041	0.171520	+	0.985181i	$0.445132\pi$
$258$	0.163115	0.0101551
$259$	1.16413	0.0723357
$260$	16.8287	1.04367
$261$	−23.4902	−1.45401
$262$	1.82207	0.112568
$263$	−17.8311	−1.09952	−0.549758	−	0.835324i	$-0.685280\pi$
$263$	−17.8311	−1.09952	−0.549758	+	0.835324i	$0.685280\pi$
$264$	−0.0633159	−0.00389682
$265$	1.69890	0.104363
$266$	−8.36452	−0.512861
$267$	−0.0220489	−0.00134937
$268$	−11.4620	−0.700151
$269$	−4.75480	−0.289905	−0.144953	−	0.989439i	$-0.546303\pi$
$269$	−4.75480	−0.289905	−0.144953	+	0.989439i	$0.546303\pi$
$270$	−0.803004	−0.0488693
$271$	−27.8709	−1.69304	−0.846519	−	0.532358i	$-0.821306\pi$
$271$	−27.8709	−1.69304	−0.846519	+	0.532358i	$0.821306\pi$
$272$	0.682669	0.0413929
$273$	−1.07584	−0.0651128
$274$	−3.76906	−0.227697
$275$	2.01078	0.121255
$276$	0.106350	0.00640150
$277$	23.6861	1.42316	0.711580	−	0.702605i	$-0.247980\pi$
$277$	23.6861	1.42316	0.711580	+	0.702605i	$0.247980\pi$
$278$	−8.86678	−0.531794
$279$	−22.9334	−1.37299
$280$	8.18607	0.489211
$281$	17.1977	1.02593	0.512965	−	0.858410i	$-0.328547\pi$
$281$	17.1977	1.02593	0.512965	+	0.858410i	$0.328547\pi$
$282$	−0.0908654	−0.00541096
$283$	−7.31309	−0.434718	−0.217359	−	0.976092i	$-0.569744\pi$
$283$	−7.31309	−0.434718	−0.217359	+	0.976092i	$0.569744\pi$
$284$	2.72321	0.161593
$285$	−0.352782	−0.0208970
$286$	7.95796	0.470564
$287$	5.96559	0.352138
$288$	2.99730	0.176618
$289$	−16.5340	−0.972586
$290$	20.2086	1.18669
$291$	−0.660925	−0.0387441
$292$	−2.31887	−0.135702
$293$	23.3594	1.36467	0.682337	−	0.731038i	$-0.260964\pi$
$293$	23.3594	1.36467	0.682337	+	0.731038i	$0.260964\pi$
$294$	−0.159847	−0.00932246
$295$	−4.25928	−0.247985
$296$	−0.366697	−0.0213138
$297$	−0.379724	−0.0220338
$298$	−0.519198	−0.0300763
$299$	−13.3667	−0.773018
$300$	0.0856281	0.00494374
$301$	−9.97259	−0.574810
$302$	−5.72224	−0.329278
$303$	0.759886	0.0436543
$304$	2.63479	0.151115
$305$	34.9546	2.00150
$306$	2.04617	0.116972
$307$	0.0440847	0.00251605	0.00125802	−	0.999999i	$-0.499600\pi$
$307$	0.0440847	0.00251605	0.00125802	+	0.999999i	$0.499600\pi$
$308$	3.87103	0.220572
$309$	−0.727167	−0.0413671
$310$	19.7296	1.12056
$311$	10.5281	0.596993	0.298496	−	0.954411i	$-0.403515\pi$
$311$	10.5281	0.596993	0.298496	+	0.954411i	$0.403515\pi$
$312$	0.338885	0.0191856
$313$	−16.8130	−0.950329	−0.475165	−	0.879897i	$-0.657612\pi$
$313$	−16.8130	−0.950329	−0.475165	+	0.879897i	$0.657612\pi$
$314$	7.63001	0.430586
$315$	24.5361	1.38245
$316$	−12.4315	−0.699328
$317$	−4.44349	−0.249571	−0.124786	−	0.992184i	$-0.539824\pi$
$317$	−4.44349	−0.249571	−0.124786	+	0.992184i	$0.539824\pi$
$318$	0.0342114	0.00191848
$319$	9.55625	0.535047
$320$	−2.57858	−0.144147
$321$	−0.543167	−0.0303166
$322$	−6.50205	−0.362345
$323$	1.79869	0.100082
$324$	8.97574	0.498652
$325$	−10.7623	−0.596985
$326$	5.93274	0.328584
$327$	0.227330	0.0125714
$328$	−1.87913	−0.103758
$329$	5.55536	0.306277
$330$	0.163265	0.00898743
$331$	2.17908	0.119773	0.0598865	−	0.998205i	$-0.480926\pi$
$331$	2.17908	0.119773	0.0598865	+	0.998205i	$0.480926\pi$
$332$	−2.41354	−0.132460
$333$	−1.09910	−0.0602304
$334$	−1.18016	−0.0645753
$335$	29.5555	1.61479
$336$	0.164846	0.00899306
$337$	−8.75557	−0.476946	−0.238473	−	0.971149i	$-0.576647\pi$
$337$	−8.75557	−0.476946	−0.238473	+	0.971149i	$0.576647\pi$
$338$	−29.5933	−1.60966
$339$	0.442017	0.0240071
$340$	−1.76031	−0.0954664
$341$	9.32971	0.505232
$342$	7.89725	0.427035
$343$	−12.4498	−0.672223
$344$	3.14132	0.169369
$345$	−0.274231	−0.0147641
$346$	22.6031	1.21515
$347$	−11.3331	−0.608392	−0.304196	−	0.952609i	$-0.598388\pi$
$347$	−11.3331	−0.608392	−0.304196	+	0.952609i	$0.598388\pi$
$348$	0.406947	0.0218147
$349$	−16.0921	−0.861390	−0.430695	−	0.902497i	$-0.641732\pi$
$349$	−16.0921	−0.861390	−0.430695	+	0.902497i	$0.641732\pi$
$350$	−5.23516	−0.279831
$351$	2.03240	0.108481
$352$	−1.21936	−0.0649919
$353$	6.73910	0.358686	0.179343	−	0.983787i	$-0.442603\pi$
$353$	6.73910	0.358686	0.179343	+	0.983787i	$0.442603\pi$
$354$	−0.0857705	−0.00455865
$355$	−7.02201	−0.372690
$356$	−0.424624	−0.0225050
$357$	0.112535	0.00595598
$358$	14.2298	0.752070
$359$	25.7542	1.35925	0.679627	−	0.733558i	$-0.262142\pi$
$359$	25.7542	1.35925	0.679627	+	0.733558i	$0.262142\pi$
$360$	−7.72877	−0.407342
$361$	−12.0579	−0.634626
$362$	17.7655	0.933734
$363$	−0.493977	−0.0259271
$364$	−20.7189	−1.08596
$365$	5.97939	0.312976
$366$	0.703893	0.0367931
$367$	29.7654	1.55374	0.776870	−	0.629661i	$-0.216806\pi$
$367$	29.7654	1.55374	0.776870	+	0.629661i	$0.216806\pi$
$368$	2.04812	0.106765
$369$	−5.63234	−0.293208
$370$	0.945555	0.0491571
$371$	−2.09163	−0.108592
$372$	0.397301	0.0205991
$373$	−5.96350	−0.308778	−0.154389	−	0.988010i	$-0.549341\pi$
$373$	−5.96350	−0.308778	−0.154389	+	0.988010i	$0.549341\pi$
$374$	−0.832417	−0.0430433
$375$	0.448672	0.0231693
$376$	−1.74992	−0.0902450
$377$	−51.1478	−2.63425
$378$	0.988629	0.0508496
$379$	16.4134	0.843100	0.421550	−	0.906805i	$-0.361486\pi$
$379$	16.4134	0.843100	0.421550	+	0.906805i	$0.361486\pi$
$380$	−6.79400	−0.348525
$381$	0.552283	0.0282943
$382$	18.9021	0.967113
$383$	−9.53410	−0.487170	−0.243585	−	0.969880i	$-0.578323\pi$
$383$	−9.53410	−0.487170	−0.243585	+	0.969880i	$0.578323\pi$
$384$	−0.0519256	−0.00264982
$385$	−9.98174	−0.508716
$386$	3.96904	0.202019
$387$	9.41549	0.478616
$388$	−12.7283	−0.646182
$389$	−9.09973	−0.461375	−0.230687	−	0.973028i	$-0.574097\pi$
$389$	−9.09973	−0.461375	−0.230687	+	0.973028i	$0.574097\pi$
$390$	−0.873841	−0.0442486
$391$	1.39819	0.0707093
$392$	−3.07838	−0.155482
$393$	−0.0946122	−0.00477256
$394$	12.4458	0.627008
$395$	32.0557	1.61289
$396$	−3.65478	−0.183660
$397$	−24.4179	−1.22550	−0.612750	−	0.790277i	$-0.709937\pi$
$397$	−24.4179	−1.22550	−0.612750	+	0.790277i	$0.709937\pi$
$398$	−13.8203	−0.692747
$399$	0.434333	0.0217438
$400$	1.64905	0.0824526
$401$	26.1563	1.30618	0.653091	−	0.757280i	$-0.273472\pi$
$401$	26.1563	1.30618	0.653091	+	0.757280i	$0.273472\pi$
$402$	0.595169	0.0296844
$403$	−49.9354	−2.48746
$404$	14.6341	0.728075
$405$	−23.1446	−1.15007
$406$	−24.8801	−1.23478
$407$	0.447134	0.0221636
$408$	−0.0354480	−0.00175494
$409$	35.3718	1.74902	0.874511	−	0.485006i	$-0.161183\pi$
$409$	35.3718	1.74902	0.874511	+	0.485006i	$0.161183\pi$
$410$	4.84549	0.239302
$411$	0.195711	0.00965369
$412$	−14.0040	−0.689928
$413$	5.24386	0.258034
$414$	6.13883	0.301707
$415$	6.22349	0.305499
$416$	6.52636	0.319981
$417$	0.460413	0.0225465
$418$	−3.21274	−0.157140
$419$	21.5833	1.05441	0.527206	−	0.849737i	$-0.323240\pi$
$419$	21.5833	1.05441	0.527206	+	0.849737i	$0.323240\pi$
$420$	−0.425067	−0.0207411
$421$	−6.75573	−0.329254	−0.164627	−	0.986356i	$-0.552642\pi$
$421$	−6.75573	−0.329254	−0.164627	+	0.986356i	$0.552642\pi$
$422$	1.79502	0.0873803
$423$	−5.24503	−0.255022
$424$	0.658853	0.0319967
$425$	1.12576	0.0546072
$426$	−0.141405	−0.00685108
$427$	−43.0348	−2.08260
$428$	−10.4605	−0.505627
$429$	−0.413222	−0.0199505
$430$	−8.10014	−0.390623
$431$	−21.7971	−1.04993	−0.524965	−	0.851124i	$-0.675922\pi$
$431$	−21.7971	−1.04993	−0.524965	+	0.851124i	$0.675922\pi$
$432$	−0.311414	−0.0149829
$433$	−8.42128	−0.404701	−0.202351	−	0.979313i	$-0.564858\pi$
$433$	−8.42128	−0.404701	−0.202351	+	0.979313i	$0.564858\pi$
$434$	−24.2903	−1.16597
$435$	−1.04934	−0.0503122
$436$	4.37799	0.209668
$437$	5.39635	0.258142
$438$	0.120409	0.00575336
$439$	−22.1814	−1.05866	−0.529331	−	0.848416i	$-0.677557\pi$
$439$	−22.1814	−1.05866	−0.529331	+	0.848416i	$0.677557\pi$
$440$	3.14420	0.149894
$441$	−9.22685	−0.439374
$442$	4.45534	0.211919
$443$	−38.1514	−1.81263	−0.906314	−	0.422606i	$-0.861116\pi$
$443$	−38.1514	−1.81263	−0.906314	+	0.422606i	$0.861116\pi$
$444$	0.0190410	0.000903644 0
$445$	1.09492	0.0519044
$446$	5.98752	0.283517
$447$	0.0269597	0.00127515
$448$	3.17465	0.149988
$449$	4.13146	0.194975	0.0974877	−	0.995237i	$-0.468919\pi$
$449$	4.13146	0.194975	0.0974877	+	0.995237i	$0.468919\pi$
$450$	4.94271	0.233002
$451$	2.29133	0.107895
$452$	8.51250	0.400394
$453$	0.297131	0.0139604
$454$	12.5987	0.591284
$455$	53.4252	2.50461
$456$	−0.136813	−0.00640685
$457$	25.1342	1.17573	0.587865	−	0.808959i	$-0.299969\pi$
$457$	25.1342	1.17573	0.587865	+	0.808959i	$0.299969\pi$
$458$	14.7717	0.690236
$459$	−0.212592	−0.00992297
$460$	−5.28122	−0.246238
$461$	−31.9160	−1.48647	−0.743237	−	0.669028i	$-0.766711\pi$
$461$	−31.9160	−1.48647	−0.743237	+	0.669028i	$0.766711\pi$
$462$	−0.201005	−0.00935162
$463$	−3.54942	−0.164956	−0.0824778	−	0.996593i	$-0.526283\pi$
$463$	−3.54942	−0.164956	−0.0824778	+	0.996593i	$0.526283\pi$
$464$	7.83712	0.363829
$465$	−1.02447	−0.0475086
$466$	3.46177	0.160363
$467$	37.7439	1.74658	0.873291	−	0.487199i	$-0.161982\pi$
$467$	37.7439	1.74658	0.873291	+	0.487199i	$0.161982\pi$
$468$	19.5615	0.904229
$469$	−36.3877	−1.68023
$470$	4.51229	0.208136
$471$	−0.396193	−0.0182556
$472$	−1.65179	−0.0760300
$473$	−3.83039	−0.176122
$474$	0.645515	0.0296495
$475$	4.34490	0.199358
$476$	2.16723	0.0993349
$477$	1.97478	0.0904191
$478$	9.57974	0.438167
$479$	−38.7264	−1.76946	−0.884728	−	0.466107i	$-0.845656\pi$
$479$	−38.7264	−1.76946	−0.884728	+	0.466107i	$0.845656\pi$
$480$	0.133894	0.00611140
$481$	−2.39319	−0.109120
$482$	−3.81441	−0.173742
$483$	0.337623	0.0153624
$484$	−9.51317	−0.432417
$485$	32.8209	1.49032
$486$	−1.40031	−0.0635194
$487$	22.4910	1.01917	0.509583	−	0.860422i	$-0.329800\pi$
$487$	22.4910	1.01917	0.509583	+	0.860422i	$0.329800\pi$
$488$	13.5558	0.613642
$489$	−0.308061	−0.0139310
$490$	7.93784	0.358595
$491$	−21.2004	−0.956759	−0.478380	−	0.878153i	$-0.658776\pi$
$491$	−21.2004	−0.956759	−0.478380	+	0.878153i	$0.658776\pi$
$492$	0.0975752	0.00439903
$493$	5.35016	0.240959
$494$	17.1956	0.773664
$495$	9.42413	0.423583
$496$	7.65134	0.343555
$497$	8.64524	0.387792
$498$	0.125324	0.00561592
$499$	18.3182	0.820033	0.410017	−	0.912078i	$-0.365523\pi$
$499$	18.3182	0.820033	0.410017	+	0.912078i	$0.365523\pi$
$500$	8.64067	0.386423
$501$	0.0612803	0.00273780
$502$	26.9935	1.20478
$503$	35.2798	1.57305	0.786524	−	0.617560i	$-0.211879\pi$
$503$	35.2798	1.57305	0.786524	+	0.617560i	$0.211879\pi$
$504$	9.51538	0.423849
$505$	−37.7352	−1.67919
$506$	−2.49738	−0.111022
$507$	1.53665	0.0682451
$508$	10.6360	0.471898
$509$	7.49236	0.332093	0.166047	−	0.986118i	$-0.446900\pi$
$509$	7.49236	0.332093	0.166047	+	0.986118i	$0.446900\pi$
$510$	0.0914054	0.00404750
$511$	−7.36160	−0.325658
$512$	−1.00000	−0.0441942
$513$	−0.820509	−0.0362263
$514$	−5.49936	−0.242566
$515$	36.1104	1.59121
$516$	−0.163115	−0.00718074
$517$	2.13377	0.0938431
$518$	−1.16413	−0.0511490
$519$	−1.17368	−0.0515188
$520$	−16.8287	−0.737987
$521$	−27.5770	−1.20817	−0.604085	−	0.796920i	$-0.706461\pi$
$521$	−27.5770	−1.20817	−0.604085	+	0.796920i	$0.706461\pi$
$522$	23.4902	1.02814
$523$	−41.8010	−1.82783	−0.913914	−	0.405907i	$-0.866956\pi$
$523$	−41.8010	−1.82783	−0.913914	+	0.405907i	$0.866956\pi$
$524$	−1.82207	−0.0795976
$525$	0.271839	0.0118640
$526$	17.8311	0.777475
$527$	5.22333	0.227532
$528$	0.0633159	0.00275547
$529$	−18.8052	−0.817618
$530$	−1.69890	−0.0737956
$531$	−4.95093	−0.214852
$532$	8.36452	0.362648
$533$	−12.2639	−0.531208
$534$	0.0220489	0.000954147 0
$535$	26.9732	1.16615
$536$	11.4620	0.495081
$537$	−0.738893	−0.0318856
$538$	4.75480	0.204994
$539$	3.75365	0.161681
$540$	0.803004	0.0345558
$541$	−21.0166	−0.903575	−0.451787	−	0.892126i	$-0.649213\pi$
$541$	−21.0166	−0.903575	−0.451787	+	0.892126i	$0.649213\pi$
$542$	27.8709	1.19716
$543$	−0.922485	−0.0395876
$544$	−0.682669	−0.0292692
$545$	−11.2890	−0.483567
$546$	1.07584	0.0460417
$547$	29.4924	1.26100	0.630501	−	0.776188i	$-0.282849\pi$
$547$	29.4924	1.26100	0.630501	+	0.776188i	$0.282849\pi$
$548$	3.76906	0.161006
$549$	40.6308	1.73408
$550$	−2.01078	−0.0857401
$551$	20.6491	0.879683
$552$	−0.106350	−0.00452655
$553$	−39.4657	−1.67825
$554$	−23.6861	−1.00633
$555$	−0.0490985	−0.00208412
$556$	8.86678	0.376035
$557$	−11.3674	−0.481654	−0.240827	−	0.970568i	$-0.577419\pi$
$557$	−11.3674	−0.481654	−0.240827	+	0.970568i	$0.577419\pi$
$558$	22.9334	0.970848
$559$	20.5014	0.867116
$560$	−8.18607	−0.345924
$561$	0.0432238	0.00182491
$562$	−17.1977	−0.725442
$563$	20.8797	0.879975	0.439988	−	0.898004i	$-0.354983\pi$
$563$	20.8797	0.879975	0.439988	+	0.898004i	$0.354983\pi$
$564$	0.0908654	0.00382613
$565$	−21.9501	−0.923448
$566$	7.31309	0.307392
$567$	28.4948	1.19667
$568$	−2.72321	−0.114264
$569$	1.99673	0.0837072	0.0418536	−	0.999124i	$-0.486674\pi$
$569$	1.99673	0.0837072	0.0418536	+	0.999124i	$0.486674\pi$
$570$	0.352782	0.0147764
$571$	23.0341	0.963947	0.481974	−	0.876186i	$-0.339920\pi$
$571$	23.0341	0.963947	0.481974	+	0.876186i	$0.339920\pi$
$572$	−7.95796	−0.332739
$573$	−0.981501	−0.0410028
$574$	−5.96559	−0.248999
$575$	3.37745	0.140850
$576$	−2.99730	−0.124888
$577$	29.5901	1.23185	0.615926	−	0.787804i	$-0.288782\pi$
$577$	29.5901	1.23185	0.615926	+	0.787804i	$0.288782\pi$
$578$	16.5340	0.687722
$579$	−0.206095	−0.00856501
$580$	−20.2086	−0.839117
$581$	−7.66213	−0.317879
$582$	0.660925	0.0273962
$583$	−0.803377	−0.0332725
$584$	2.31887	0.0959556
$585$	−50.4407	−2.08547
$586$	−23.3594	−0.964970
$587$	−12.4892	−0.515483	−0.257741	−	0.966214i	$-0.582978\pi$
$587$	−12.4892	−0.515483	−0.257741	+	0.966214i	$0.582978\pi$
$588$	0.159847	0.00659198
$589$	20.1596	0.830664
$590$	4.25928	0.175352
$591$	−0.646253	−0.0265833
$592$	0.366697	0.0150711
$593$	−38.3287	−1.57397	−0.786985	−	0.616972i	$-0.788359\pi$
$593$	−38.3287	−1.57397	−0.786985	+	0.616972i	$0.788359\pi$
$594$	0.379724	0.0155803
$595$	−5.58837	−0.229101
$596$	0.519198	0.0212672
$597$	0.717625	0.0293705
$598$	13.3667	0.546607
$599$	−6.73922	−0.275357	−0.137679	−	0.990477i	$-0.543964\pi$
$599$	−6.73922	−0.275357	−0.137679	+	0.990477i	$0.543964\pi$
$600$	−0.0856281	−0.00349575
$601$	0.325330	0.0132705	0.00663525	−	0.999978i	$-0.497888\pi$
$601$	0.325330	0.0132705	0.00663525	+	0.999978i	$0.497888\pi$
$602$	9.97259	0.406452
$603$	34.3550	1.39904
$604$	5.72224	0.232835
$605$	24.5304	0.997304
$606$	−0.759886	−0.0308682
$607$	0.666899	0.0270686	0.0135343	−	0.999908i	$-0.495692\pi$
$607$	0.666899	0.0270686	0.0135343	+	0.999908i	$0.495692\pi$
$608$	−2.63479	−0.106855
$609$	1.29191	0.0523510
$610$	−34.9546	−1.41527
$611$	−11.4206	−0.462027
$612$	−2.04617	−0.0827114
$613$	13.4705	0.544068	0.272034	−	0.962288i	$-0.412304\pi$
$613$	13.4705	0.544068	0.272034	+	0.962288i	$0.412304\pi$
$614$	−0.0440847	−0.00177912
$615$	−0.251605	−0.0101457
$616$	−3.87103	−0.155968
$617$	34.0740	1.37177	0.685885	−	0.727710i	$-0.259415\pi$
$617$	34.0740	1.37177	0.685885	+	0.727710i	$0.259415\pi$
$618$	0.727167	0.0292509
$619$	2.50120	0.100532	0.0502659	−	0.998736i	$-0.483993\pi$
$619$	2.50120	0.100532	0.0502659	+	0.998736i	$0.483993\pi$
$620$	−19.7296	−0.792358
$621$	−0.637812	−0.0255945
$622$	−10.5281	−0.422138
$623$	−1.34803	−0.0540077
$624$	−0.338885	−0.0135663
$625$	−30.5259	−1.22104
$626$	16.8130	0.671984
$627$	0.166824	0.00666230
$628$	−7.63001	−0.304471
$629$	0.250332	0.00998141
$630$	−24.5361	−0.977543
$631$	−29.7172	−1.18302	−0.591511	−	0.806297i	$-0.701468\pi$
$631$	−29.7172	−1.18302	−0.591511	+	0.806297i	$0.701468\pi$
$632$	12.4315	0.494500
$633$	−0.0932076	−0.00370467
$634$	4.44349	0.176473
$635$	−27.4259	−1.08836
$636$	−0.0342114	−0.00135657
$637$	−20.0906	−0.796019
$638$	−9.55625	−0.378335
$639$	−8.16230	−0.322896
$640$	2.57858	0.101927
$641$	2.35940	0.0931907	0.0465953	−	0.998914i	$-0.485163\pi$
$641$	2.35940	0.0931907	0.0465953	+	0.998914i	$0.485163\pi$
$642$	0.543167	0.0214371
$643$	18.2707	0.720526	0.360263	−	0.932851i	$-0.382687\pi$
$643$	18.2707	0.720526	0.360263	+	0.932851i	$0.382687\pi$
$644$	6.50205	0.256217
$645$	0.420605	0.0165613
$646$	−1.79869	−0.0707684
$647$	−27.7474	−1.09086	−0.545431	−	0.838156i	$-0.683634\pi$
$647$	−27.7474	−1.09086	−0.545431	+	0.838156i	$0.683634\pi$
$648$	−8.97574	−0.352600
$649$	2.01413	0.0790614
$650$	10.7623	0.422132
$651$	1.26129	0.0494338
$652$	−5.93274	−0.232344
$653$	21.1265	0.826745	0.413373	−	0.910562i	$-0.364351\pi$
$653$	21.1265	0.826745	0.413373	+	0.910562i	$0.364351\pi$
$654$	−0.227330	−0.00888930
$655$	4.69835	0.183580
$656$	1.87913	0.0733679
$657$	6.95037	0.271160
$658$	−5.55536	−0.216571
$659$	26.1945	1.02039	0.510197	−	0.860057i	$-0.329572\pi$
$659$	26.1945	1.02039	0.510197	+	0.860057i	$0.329572\pi$
$660$	−0.163265	−0.00635507
$661$	−42.8347	−1.66608	−0.833038	−	0.553215i	$-0.813401\pi$
$661$	−42.8347	−1.66608	−0.833038	+	0.553215i	$0.813401\pi$
$662$	−2.17908	−0.0846923
$663$	−0.231346	−0.00898475
$664$	2.41354	0.0936634
$665$	−21.5685	−0.836392
$666$	1.09910	0.0425893
$667$	16.0513	0.621510
$668$	1.18016	0.0456616
$669$	−0.310906	−0.0120203
$670$	−29.5555	−1.14183
$671$	−16.5293	−0.638108
$672$	−0.164846	−0.00635905
$673$	−19.1903	−0.739731	−0.369865	−	0.929085i	$-0.620596\pi$
$673$	−19.1903	−0.739731	−0.369865	+	0.929085i	$0.620596\pi$
$674$	8.75557	0.337252
$675$	−0.513538	−0.0197661
$676$	29.5933	1.13820
$677$	−3.38160	−0.129966	−0.0649828	−	0.997886i	$-0.520699\pi$
$677$	−3.38160	−0.129966	−0.0649828	+	0.997886i	$0.520699\pi$
$678$	−0.442017	−0.0169755
$679$	−40.4079	−1.55071
$680$	1.76031	0.0675049
$681$	−0.654193	−0.0250687
$682$	−9.32971	−0.357253
$683$	−43.3118	−1.65728	−0.828639	−	0.559783i	$-0.810884\pi$
$683$	−43.3118	−1.65728	−0.828639	+	0.559783i	$0.810884\pi$
$684$	−7.89725	−0.301959
$685$	−9.71880	−0.371336
$686$	12.4498	0.475334
$687$	−0.767029	−0.0292640
$688$	−3.14132	−0.119762
$689$	4.29991	0.163814
$690$	0.274231	0.0104398
$691$	−6.29761	−0.239572	−0.119786	−	0.992800i	$-0.538221\pi$
$691$	−6.29761	−0.239572	−0.119786	+	0.992800i	$0.538221\pi$
$692$	−22.6031	−0.859240
$693$	−11.6026	−0.440748
$694$	11.3331	0.430198
$695$	−22.8637	−0.867268
$696$	−0.406947	−0.0154253
$697$	1.28283	0.0485905
$698$	16.0921	0.609095
$699$	−0.179754	−0.00679894
$700$	5.23516	0.197870
$701$	17.3464	0.655166	0.327583	−	0.944822i	$-0.393766\pi$
$701$	17.3464	0.655166	0.327583	+	0.944822i	$0.393766\pi$
$702$	−2.03240	−0.0767079
$703$	0.966167	0.0364397
$704$	1.21936	0.0459562
$705$	−0.234303	−0.00882438
$706$	−6.73910	−0.253630
$707$	46.4582	1.74724
$708$	0.0857705	0.00322345
$709$	25.8544	0.970982	0.485491	−	0.874242i	$-0.338641\pi$
$709$	25.8544	0.970982	0.485491	+	0.874242i	$0.338641\pi$
$710$	7.02201	0.263532
$711$	37.2611	1.39740
$712$	0.424624	0.0159135
$713$	15.6708	0.586877
$714$	−0.112535	−0.00421151
$715$	20.5202	0.767412
$716$	−14.2298	−0.531794
$717$	−0.497434	−0.0185770
$718$	−25.7542	−0.961138
$719$	−0.398637	−0.0148667	−0.00743333	−	0.999972i	$-0.502366\pi$
$719$	−0.398637	−0.0148667	−0.00743333	+	0.999972i	$0.502366\pi$
$720$	7.72877	0.288034
$721$	−44.4578	−1.65569
$722$	12.0579	0.448749
$723$	0.198065	0.00736614
$724$	−17.7655	−0.660250
$725$	12.9238	0.479979
$726$	0.493977	0.0183332
$727$	17.5970	0.652637	0.326319	−	0.945260i	$-0.394192\pi$
$727$	17.5970	0.652637	0.326319	+	0.945260i	$0.394192\pi$
$728$	20.7189	0.767893
$729$	−26.8545	−0.994611
$730$	−5.97939	−0.221307
$731$	−2.14448	−0.0793165
$732$	−0.703893	−0.0260166
$733$	20.6500	0.762724	0.381362	−	0.924426i	$-0.375455\pi$
$733$	20.6500	0.762724	0.381362	+	0.924426i	$0.375455\pi$
$734$	−29.7654	−1.09866
$735$	−0.412177	−0.0152034
$736$	−2.04812	−0.0754946
$737$	−13.9762	−0.514821
$738$	5.63234	0.207329
$739$	−27.6569	−1.01738	−0.508688	−	0.860951i	$-0.669869\pi$
$739$	−27.6569	−1.01738	−0.508688	+	0.860951i	$0.669869\pi$
$740$	−0.945555	−0.0347593
$741$	−0.892890	−0.0328011
$742$	2.09163	0.0767860
$743$	22.3002	0.818116	0.409058	−	0.912508i	$-0.365857\pi$
$743$	22.3002	0.818116	0.409058	+	0.912508i	$0.365857\pi$
$744$	−0.397301	−0.0145657
$745$	−1.33879	−0.0490495
$746$	5.96350	0.218339
$747$	7.23410	0.264682
$748$	0.832417	0.0304362
$749$	−33.2083	−1.21341
$750$	−0.448672	−0.0163832
$751$	−13.6593	−0.498435	−0.249218	−	0.968448i	$-0.580173\pi$
$751$	−13.6593	−0.498435	−0.249218	+	0.968448i	$0.580173\pi$
$752$	1.74992	0.0638128
$753$	−1.40165	−0.0510790
$754$	51.1478	1.86269
$755$	−14.7552	−0.536998
$756$	−0.988629	−0.0359561
$757$	−33.1521	−1.20493	−0.602467	−	0.798144i	$-0.705816\pi$
$757$	−33.1521	−1.20493	−0.602467	+	0.798144i	$0.705816\pi$
$758$	−16.4134	−0.596161
$759$	0.129678	0.00470702
$760$	6.79400	0.246444
$761$	46.8186	1.69717	0.848587	−	0.529056i	$-0.177454\pi$
$761$	46.8186	1.69717	0.848587	+	0.529056i	$0.177454\pi$
$762$	−0.552283	−0.0200071
$763$	13.8986	0.503162
$764$	−18.9021	−0.683852
$765$	5.27619	0.190761
$766$	9.53410	0.344481
$767$	−10.7802	−0.389250
$768$	0.0519256	0.00187370
$769$	16.8566	0.607864	0.303932	−	0.952694i	$-0.401700\pi$
$769$	16.8566	0.607864	0.303932	+	0.952694i	$0.401700\pi$
$770$	9.98174	0.359717
$771$	0.285558	0.0102841
$772$	−3.96904	−0.142849
$773$	−28.0680	−1.00954	−0.504769	−	0.863255i	$-0.668422\pi$
$773$	−28.0680	−1.00954	−0.504769	+	0.863255i	$0.668422\pi$
$774$	−9.41549	−0.338433
$775$	12.6175	0.453233
$776$	12.7283	0.456919
$777$	0.0604483	0.00216857
$778$	9.09973	0.326241
$779$	4.95112	0.177392
$780$	0.873841	0.0312885
$781$	3.32057	0.118819
$782$	−1.39819	−0.0499990
$783$	−2.44059	−0.0872195
$784$	3.07838	0.109942
$785$	19.6746	0.702215
$786$	0.0946122	0.00337471
$787$	2.87696	0.102552	0.0512762	−	0.998685i	$-0.483671\pi$
$787$	2.87696	0.102552	0.0512762	+	0.998685i	$0.483671\pi$
$788$	−12.4458	−0.443362
$789$	−0.925894	−0.0329627
$790$	−32.0557	−1.14049
$791$	27.0242	0.960869
$792$	3.65478	0.129867
$793$	88.4699	3.14166
$794$	24.4179	0.866560
$795$	0.0882166	0.00312872
$796$	13.8203	0.489846
$797$	−15.1768	−0.537589	−0.268794	−	0.963198i	$-0.586625\pi$
$797$	−15.1768	−0.537589	−0.268794	+	0.963198i	$0.586625\pi$
$798$	−0.434333	−0.0153752
$799$	1.19461	0.0422624
$800$	−1.64905	−0.0583028
$801$	1.27273	0.0449696
$802$	−26.1563	−0.923610
$803$	−2.82753	−0.0997815
$804$	−0.595169	−0.0209900
$805$	−16.7660	−0.590925
$806$	49.9354	1.75890
$807$	−0.246896	−0.00869114
$808$	−14.6341	−0.514826
$809$	25.8558	0.909041	0.454521	−	0.890736i	$-0.349811\pi$
$809$	25.8558	0.909041	0.454521	+	0.890736i	$0.349811\pi$
$810$	23.1446	0.813219
$811$	30.4947	1.07081	0.535407	−	0.844594i	$-0.320158\pi$
$811$	30.4947	1.07081	0.535407	+	0.844594i	$0.320158\pi$
$812$	24.8801	0.873120
$813$	−1.44722	−0.0507561
$814$	−0.447134	−0.0156720
$815$	15.2980	0.535866
$816$	0.0354480	0.00124093
$817$	−8.27671	−0.289565
$818$	−35.3718	−1.23674
$819$	62.1008	2.16998
$820$	−4.84549	−0.169212
$821$	−9.41713	−0.328660	−0.164330	−	0.986405i	$-0.552546\pi$
$821$	−9.41713	−0.328660	−0.164330	+	0.986405i	$0.552546\pi$
$822$	−0.195711	−0.00682619
$823$	10.5640	0.368239	0.184119	−	0.982904i	$-0.441057\pi$
$823$	10.5640	0.368239	0.184119	+	0.982904i	$0.441057\pi$
$824$	14.0040	0.487853
$825$	0.104411	0.00363513
$826$	−5.24386	−0.182457
$827$	−9.13808	−0.317762	−0.158881	−	0.987298i	$-0.550789\pi$
$827$	−9.13808	−0.317762	−0.158881	+	0.987298i	$0.550789\pi$
$828$	−6.13883	−0.213339
$829$	15.8231	0.549559	0.274779	−	0.961507i	$-0.411395\pi$
$829$	15.8231	0.549559	0.274779	+	0.961507i	$0.411395\pi$
$830$	−6.22349	−0.216020
$831$	1.22992	0.0426653
$832$	−6.52636	−0.226261
$833$	2.10152	0.0728132
$834$	−0.460413	−0.0159428
$835$	−3.04312	−0.105312
$836$	3.21274	0.111115
$837$	−2.38273	−0.0823593
$838$	−21.5833	−0.745582
$839$	−41.2293	−1.42339	−0.711696	−	0.702487i	$-0.752073\pi$
$839$	−41.2293	−1.42339	−0.711696	+	0.702487i	$0.752073\pi$
$840$	0.425067	0.0146662
$841$	32.4205	1.11795
$842$	6.75573	0.232818
$843$	0.893002	0.0307566
$844$	−1.79502	−0.0617872
$845$	−76.3086	−2.62510
$846$	5.24503	0.180328
$847$	−30.2010	−1.03772
$848$	−0.658853	−0.0226251
$849$	−0.379737	−0.0130325
$850$	−1.12576	−0.0386132
$851$	0.751038	0.0257452
$852$	0.141405	0.00484444
$853$	−30.3110	−1.03783	−0.518914	−	0.854827i	$-0.673663\pi$
$853$	−30.3110	−1.03783	−0.518914	+	0.854827i	$0.673663\pi$
$854$	43.0348	1.47262
$855$	20.3637	0.696423
$856$	10.4605	0.357532
$857$	−18.5261	−0.632841	−0.316421	−	0.948619i	$-0.602481\pi$
$857$	−18.5261	−0.632841	−0.316421	+	0.948619i	$0.602481\pi$
$858$	0.413222	0.0141072
$859$	35.8237	1.22229	0.611145	−	0.791519i	$-0.290709\pi$
$859$	35.8237	1.22229	0.611145	+	0.791519i	$0.290709\pi$
$860$	8.10014	0.276212
$861$	0.309767	0.0105568
$862$	21.7971	0.742413
$863$	−7.35047	−0.250213	−0.125107	−	0.992143i	$-0.539927\pi$
$863$	−7.35047	−0.250213	−0.125107	+	0.992143i	$0.539927\pi$
$864$	0.311414	0.0105945
$865$	58.2838	1.98171
$866$	8.42128	0.286167
$867$	−0.858536	−0.0291574
$868$	24.2903	0.824467
$869$	−15.1585	−0.514216
$870$	1.04934	0.0355761
$871$	74.8048	2.53466
$872$	−4.37799	−0.148257
$873$	38.1506	1.29120
$874$	−5.39635	−0.182534
$875$	27.4311	0.927340
$876$	−0.120409	−0.00406824
$877$	4.97139	0.167872	0.0839359	−	0.996471i	$-0.473251\pi$
$877$	4.97139	0.167872	0.0839359	+	0.996471i	$0.473251\pi$
$878$	22.1814	0.748586
$879$	1.21295	0.0409119
$880$	−3.14420	−0.105991
$881$	−11.4542	−0.385903	−0.192951	−	0.981208i	$-0.561806\pi$
$881$	−11.4542	−0.385903	−0.192951	+	0.981208i	$0.561806\pi$
$882$	9.22685	0.310684
$883$	−16.0195	−0.539098	−0.269549	−	0.962987i	$-0.586875\pi$
$883$	−16.0195	−0.539098	−0.269549	+	0.962987i	$0.586875\pi$
$884$	−4.45534	−0.149849
$885$	−0.221166	−0.00743440
$886$	38.1514	1.28172
$887$	−20.2778	−0.680862	−0.340431	−	0.940269i	$-0.610573\pi$
$887$	−20.2778	−0.680862	−0.340431	+	0.940269i	$0.610573\pi$
$888$	−0.0190410	−0.000638973 0
$889$	33.7657	1.13247
$890$	−1.09492	−0.0367020
$891$	10.9446	0.366659
$892$	−5.98752	−0.200477
$893$	4.61065	0.154290
$894$	−0.0269597	−0.000901666 0
$895$	36.6927	1.22650
$896$	−3.17465	−0.106058
$897$	−0.694076	−0.0231745
$898$	−4.13146	−0.137868
$899$	59.9645	1.99993
$900$	−4.94271	−0.164757
$901$	−0.449779	−0.0149843
$902$	−2.29133	−0.0762931
$903$	−0.517833	−0.0172324
$904$	−8.51250	−0.283121
$905$	45.8097	1.52277
$906$	−0.297131	−0.00987151
$907$	30.4213	1.01012	0.505061	−	0.863083i	$-0.331470\pi$
$907$	30.4213	1.01012	0.505061	+	0.863083i	$0.331470\pi$
$908$	−12.5987	−0.418101
$909$	−43.8629	−1.45484
$910$	−53.4252	−1.77103
$911$	4.21617	0.139688	0.0698439	−	0.997558i	$-0.477750\pi$
$911$	4.21617	0.139688	0.0698439	+	0.997558i	$0.477750\pi$
$912$	0.136813	0.00453033
$913$	−2.94296	−0.0973978
$914$	−25.1342	−0.831366
$915$	1.81504	0.0600034
$916$	−14.7717	−0.488070
$917$	−5.78444	−0.191019
$918$	0.212592	0.00701660
$919$	32.8632	1.08406	0.542029	−	0.840360i	$-0.317656\pi$
$919$	32.8632	1.08406	0.542029	+	0.840360i	$0.317656\pi$
$920$	5.28122	0.174117
$921$	0.00228913	7.54293e−5 0
$922$	31.9160	1.05110
$923$	−17.7727	−0.584994
$924$	0.201005	0.00661259
$925$	0.604702	0.0198825
$926$	3.54942	0.116641
$927$	41.9743	1.37862
$928$	−7.83712	−0.257266
$929$	10.8100	0.354664	0.177332	−	0.984151i	$-0.443253\pi$
$929$	10.8100	0.354664	0.177332	+	0.984151i	$0.443253\pi$
$930$	1.02447	0.0335937
$931$	8.11088	0.265823
$932$	−3.46177	−0.113394
$933$	0.546677	0.0178974
$934$	−37.7439	−1.23502
$935$	−2.14645	−0.0701964
$936$	−19.5615	−0.639387
$937$	−24.4548	−0.798904	−0.399452	−	0.916754i	$-0.630800\pi$
$937$	−24.4548	−0.798904	−0.399452	+	0.916754i	$0.630800\pi$
$938$	36.3877	1.18810
$939$	−0.873028	−0.0284902
$940$	−4.51229	−0.147175
$941$	−32.2373	−1.05091	−0.525454	−	0.850822i	$-0.676104\pi$
$941$	−32.2373	−1.05091	−0.525454	+	0.850822i	$0.676104\pi$
$942$	0.396193	0.0129087
$943$	3.84869	0.125330
$944$	1.65179	0.0537613
$945$	2.54925	0.0829272
$946$	3.83039	0.124537
$947$	23.5330	0.764719	0.382359	−	0.924014i	$-0.375112\pi$
$947$	23.5330	0.764719	0.382359	+	0.924014i	$0.375112\pi$
$948$	−0.645515	−0.0209654
$949$	15.1338	0.491264
$950$	−4.34490	−0.140967
$951$	−0.230731	−0.00748196
$952$	−2.16723	−0.0702404
$953$	1.32309	0.0428590	0.0214295	−	0.999770i	$-0.493178\pi$
$953$	1.32309	0.0428590	0.0214295	+	0.999770i	$0.493178\pi$
$954$	−1.97478	−0.0639360
$955$	48.7404	1.57720
$956$	−9.57974	−0.309831
$957$	0.496214	0.0160403
$958$	38.7264	1.25119
$959$	11.9654	0.386384
$960$	−0.133894	−0.00432142
$961$	27.5430	0.888484
$962$	2.39319	0.0771596
$963$	31.3532	1.01034
$964$	3.81441	0.122854
$965$	10.2345	0.329459
$966$	−0.337623	−0.0108628
$967$	−57.6364	−1.85346	−0.926730	−	0.375727i	$-0.877393\pi$
$967$	−57.6364	−1.85346	−0.926730	+	0.375727i	$0.877393\pi$
$968$	9.51317	0.305765
$969$	0.0933979	0.00300037
$970$	−32.8209	−1.05382
$971$	−35.2159	−1.13013	−0.565066	−	0.825046i	$-0.691149\pi$
$971$	−35.2159	−1.13013	−0.565066	+	0.825046i	$0.691149\pi$
$972$	1.40031	0.0449150
$973$	28.1489	0.902412
$974$	−22.4910	−0.720659
$975$	−0.558839	−0.0178972
$976$	−13.5558	−0.433910
$977$	52.1142	1.66728	0.833640	−	0.552308i	$-0.186253\pi$
$977$	52.1142	1.66728	0.833640	+	0.552308i	$0.186253\pi$
$978$	0.308061	0.00985071
$979$	−0.517768	−0.0165479
$980$	−7.93784	−0.253565
$981$	−13.1222	−0.418958
$982$	21.2004	0.676531
$983$	16.2625	0.518694	0.259347	−	0.965784i	$-0.416493\pi$
$983$	16.2625	0.518694	0.259347	+	0.965784i	$0.416493\pi$
$984$	−0.0975752	−0.00311058
$985$	32.0923	1.02255
$986$	−5.35016	−0.170384
$987$	0.288466	0.00918196
$988$	−17.1956	−0.547063
$989$	−6.43379	−0.204583
$990$	−9.42413	−0.299519
$991$	55.1129	1.75072	0.875360	−	0.483472i	$-0.160625\pi$
$991$	55.1129	1.75072	0.875360	+	0.483472i	$0.160625\pi$
$992$	−7.65134	−0.242930
$993$	0.113150	0.00359071
$994$	−8.64524	−0.274210
$995$	−35.6366	−1.12976
$996$	−0.125324	−0.00397106
$997$	23.9074	0.757154	0.378577	−	0.925570i	$-0.376414\pi$
$997$	23.9074	0.757154	0.378577	+	0.925570i	$0.376414\pi$
$998$	−18.3182	−0.579851
$999$	−0.114194	−0.00361295

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.d.1.20	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.d.1.20	✓	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.0519256 q^{3} +1.00000 q^{4} -2.57858 q^{5} -0.0519256 q^{6} +3.17465 q^{7} -1.00000 q^{8} -2.99730 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.0519256 q^{3} +1.00000 q^{4} -2.57858 q^{5} -0.0519256 q^{6} +3.17465 q^{7} -1.00000 q^{8} -2.99730 q^{9} +2.57858 q^{10} +1.21936 q^{11} +0.0519256 q^{12} -6.52636 q^{13} -3.17465 q^{14} -0.133894 q^{15} +1.00000 q^{16} +0.682669 q^{17} +2.99730 q^{18} +2.63479 q^{19} -2.57858 q^{20} +0.164846 q^{21} -1.21936 q^{22} +2.04812 q^{23} -0.0519256 q^{24} +1.64905 q^{25} +6.52636 q^{26} -0.311414 q^{27} +3.17465 q^{28} +7.83712 q^{29} +0.133894 q^{30} +7.65134 q^{31} -1.00000 q^{32} +0.0633159 q^{33} -0.682669 q^{34} -8.18607 q^{35} -2.99730 q^{36} +0.366697 q^{37} -2.63479 q^{38} -0.338885 q^{39} +2.57858 q^{40} +1.87913 q^{41} -0.164846 q^{42} -3.14132 q^{43} +1.21936 q^{44} +7.72877 q^{45} -2.04812 q^{46} +1.74992 q^{47} +0.0519256 q^{48} +3.07838 q^{49} -1.64905 q^{50} +0.0354480 q^{51} -6.52636 q^{52} -0.658853 q^{53} +0.311414 q^{54} -3.14420 q^{55} -3.17465 q^{56} +0.136813 q^{57} -7.83712 q^{58} +1.65179 q^{59} -0.133894 q^{60} -13.5558 q^{61} -7.65134 q^{62} -9.51538 q^{63} +1.00000 q^{64} +16.8287 q^{65} -0.0633159 q^{66} -11.4620 q^{67} +0.682669 q^{68} +0.106350 q^{69} +8.18607 q^{70} +2.72321 q^{71} +2.99730 q^{72} -2.31887 q^{73} -0.366697 q^{74} +0.0856281 q^{75} +2.63479 q^{76} +3.87103 q^{77} +0.338885 q^{78} -12.4315 q^{79} -2.57858 q^{80} +8.97574 q^{81} -1.87913 q^{82} -2.41354 q^{83} +0.164846 q^{84} -1.76031 q^{85} +3.14132 q^{86} +0.406947 q^{87} -1.21936 q^{88} -0.424624 q^{89} -7.72877 q^{90} -20.7189 q^{91} +2.04812 q^{92} +0.397301 q^{93} -1.74992 q^{94} -6.79400 q^{95} -0.0519256 q^{96} -12.7283 q^{97} -3.07838 q^{98} -3.65478 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9}+O(q^{10}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9} + 13 q^{10} + 12 q^{11} - 5 q^{12} - 36 q^{13} + 22 q^{14} - 7 q^{15} + 37 q^{16} + 4 q^{17} - 32 q^{18} - 10 q^{19} - 13 q^{20} - 11 q^{21} - 12 q^{22} - q^{23} + 5 q^{24} + 8 q^{25} + 36 q^{26} - 20 q^{27} - 22 q^{28} - 6 q^{29} + 7 q^{30} - 23 q^{31} - 37 q^{32} - 27 q^{33} - 4 q^{34} + 24 q^{35} + 32 q^{36} - 46 q^{37} + 10 q^{38} + 5 q^{39} + 13 q^{40} + 11 q^{41} + 11 q^{42} - 22 q^{43} + 12 q^{44} - 57 q^{45} + q^{46} - 18 q^{47} - 5 q^{48} - q^{49} - 8 q^{50} + 18 q^{51} - 36 q^{52} - 25 q^{53} + 20 q^{54} - 25 q^{55} + 22 q^{56} - 25 q^{57} + 6 q^{58} + 24 q^{59} - 7 q^{60} - 40 q^{61} + 23 q^{62} - 38 q^{63} + 37 q^{64} + 14 q^{65} + 27 q^{66} - 49 q^{67} + 4 q^{68} - 19 q^{69} - 24 q^{70} + 27 q^{71} - 32 q^{72} - 87 q^{73} + 46 q^{74} - 5 q^{75} - 10 q^{76} - 50 q^{77} - 5 q^{78} + 11 q^{79} - 13 q^{80} + 5 q^{81} - 11 q^{82} - 9 q^{83} - 11 q^{84} - 68 q^{85} + 22 q^{86} - 12 q^{87} - 12 q^{88} - 3 q^{89} + 57 q^{90} - 19 q^{91} - q^{92} - 69 q^{93} + 18 q^{94} + 27 q^{95} + 5 q^{96} - 98 q^{97} + q^{98} + 33 q^{99}+O(q^{100}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9 + 13 * q^10 + 12 * q^11 - 5 * q^12 - 36 * q^13 + 22 * q^14 - 7 * q^15 + 37 * q^16 + 4 * q^17 - 32 * q^18 - 10 * q^19 - 13 * q^20 - 11 * q^21 - 12 * q^22 - q^23 + 5 * q^24 + 8 * q^25 + 36 * q^26 - 20 * q^27 - 22 * q^28 - 6 * q^29 + 7 * q^30 - 23 * q^31 - 37 * q^32 - 27 * q^33 - 4 * q^34 + 24 * q^35 + 32 * q^36 - 46 * q^37 + 10 * q^38 + 5 * q^39 + 13 * q^40 + 11 * q^41 + 11 * q^42 - 22 * q^43 + 12 * q^44 - 57 * q^45 + q^46 - 18 * q^47 - 5 * q^48 - q^49 - 8 * q^50 + 18 * q^51 - 36 * q^52 - 25 * q^53 + 20 * q^54 - 25 * q^55 + 22 * q^56 - 25 * q^57 + 6 * q^58 + 24 * q^59 - 7 * q^60 - 40 * q^61 + 23 * q^62 - 38 * q^63 + 37 * q^64 + 14 * q^65 + 27 * q^66 - 49 * q^67 + 4 * q^68 - 19 * q^69 - 24 * q^70 + 27 * q^71 - 32 * q^72 - 87 * q^73 + 46 * q^74 - 5 * q^75 - 10 * q^76 - 50 * q^77 - 5 * q^78 + 11 * q^79 - 13 * q^80 + 5 * q^81 - 11 * q^82 - 9 * q^83 - 11 * q^84 - 68 * q^85 + 22 * q^86 - 12 * q^87 - 12 * q^88 - 3 * q^89 + 57 * q^90 - 19 * q^91 - q^92 - 69 * q^93 + 18 * q^94 + 27 * q^95 + 5 * q^96 - 98 * q^97 + q^98 + 33 * q^99

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