LMFDB - Embedded newform 8042.2.a.a.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.36999 q^{3} +1.00000 q^{4} +1.57394 q^{5} -2.36999 q^{6} +1.59393 q^{7} +1.00000 q^{8} +2.61687 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.36999 q^{3} +1.00000 q^{4} +1.57394 q^{5} -2.36999 q^{6} +1.59393 q^{7} +1.00000 q^{8} +2.61687 q^{9} +1.57394 q^{10} -1.45522 q^{11} -2.36999 q^{12} +1.08958 q^{13} +1.59393 q^{14} -3.73022 q^{15} +1.00000 q^{16} -1.26543 q^{17} +2.61687 q^{18} +2.02440 q^{19} +1.57394 q^{20} -3.77760 q^{21} -1.45522 q^{22} -3.02412 q^{23} -2.36999 q^{24} -2.52272 q^{25} +1.08958 q^{26} +0.908022 q^{27} +1.59393 q^{28} -2.01834 q^{29} -3.73022 q^{30} -6.74478 q^{31} +1.00000 q^{32} +3.44885 q^{33} -1.26543 q^{34} +2.50874 q^{35} +2.61687 q^{36} +1.22529 q^{37} +2.02440 q^{38} -2.58230 q^{39} +1.57394 q^{40} +1.59789 q^{41} -3.77760 q^{42} +0.188669 q^{43} -1.45522 q^{44} +4.11878 q^{45} -3.02412 q^{46} -3.24332 q^{47} -2.36999 q^{48} -4.45939 q^{49} -2.52272 q^{50} +2.99907 q^{51} +1.08958 q^{52} -4.65195 q^{53} +0.908022 q^{54} -2.29042 q^{55} +1.59393 q^{56} -4.79781 q^{57} -2.01834 q^{58} -2.64062 q^{59} -3.73022 q^{60} -15.4282 q^{61} -6.74478 q^{62} +4.17110 q^{63} +1.00000 q^{64} +1.71493 q^{65} +3.44885 q^{66} -5.82549 q^{67} -1.26543 q^{68} +7.16714 q^{69} +2.50874 q^{70} +11.2978 q^{71} +2.61687 q^{72} -8.57257 q^{73} +1.22529 q^{74} +5.97884 q^{75} +2.02440 q^{76} -2.31951 q^{77} -2.58230 q^{78} +7.53323 q^{79} +1.57394 q^{80} -10.0026 q^{81} +1.59789 q^{82} -4.74304 q^{83} -3.77760 q^{84} -1.99171 q^{85} +0.188669 q^{86} +4.78344 q^{87} -1.45522 q^{88} -0.722253 q^{89} +4.11878 q^{90} +1.73672 q^{91} -3.02412 q^{92} +15.9851 q^{93} -3.24332 q^{94} +3.18627 q^{95} -2.36999 q^{96} +7.32169 q^{97} -4.45939 q^{98} -3.80810 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−2.36999	−1.36832	−0.684158	−	0.729334i	$-0.739830\pi$
$3$	−2.36999	−1.36832	−0.684158	+	0.729334i	$0.739830\pi$
$4$	1.00000	0.500000
$5$	1.57394	0.703886	0.351943	−	0.936021i	$-0.385521\pi$
$5$	1.57394	0.703886	0.351943	+	0.936021i	$0.385521\pi$
$6$	−2.36999	−0.967546
$7$	1.59393	0.602448	0.301224	−	0.953553i	$-0.402605\pi$
$7$	1.59393	0.602448	0.301224	+	0.953553i	$0.402605\pi$
$8$	1.00000	0.353553
$9$	2.61687	0.872289
$10$	1.57394	0.497722
$11$	−1.45522	−0.438764	−0.219382	−	0.975639i	$-0.570404\pi$
$11$	−1.45522	−0.438764	−0.219382	+	0.975639i	$0.570404\pi$
$12$	−2.36999	−0.684158
$13$	1.08958	0.302196	0.151098	−	0.988519i	$-0.451719\pi$
$13$	1.08958	0.302196	0.151098	+	0.988519i	$0.451719\pi$
$14$	1.59393	0.425995
$15$	−3.73022	−0.963138
$16$	1.00000	0.250000
$17$	−1.26543	−0.306913	−0.153456	−	0.988155i	$-0.549040\pi$
$17$	−1.26543	−0.306913	−0.153456	+	0.988155i	$0.549040\pi$
$18$	2.61687	0.616802
$19$	2.02440	0.464428	0.232214	−	0.972665i	$-0.425403\pi$
$19$	2.02440	0.464428	0.232214	+	0.972665i	$0.425403\pi$
$20$	1.57394	0.351943
$21$	−3.77760	−0.824340
$22$	−1.45522	−0.310253
$23$	−3.02412	−0.630572	−0.315286	−	0.948997i	$-0.602100\pi$
$23$	−3.02412	−0.630572	−0.315286	+	0.948997i	$0.602100\pi$
$24$	−2.36999	−0.483773
$25$	−2.52272	−0.504545
$26$	1.08958	0.213685
$27$	0.908022	0.174749
$28$	1.59393	0.301224
$29$	−2.01834	−0.374796	−0.187398	−	0.982284i	$-0.560005\pi$
$29$	−2.01834	−0.374796	−0.187398	+	0.982284i	$0.560005\pi$
$30$	−3.73022	−0.681042
$31$	−6.74478	−1.21140	−0.605699	−	0.795694i	$-0.707106\pi$
$31$	−6.74478	−1.21140	−0.605699	+	0.795694i	$0.707106\pi$
$32$	1.00000	0.176777
$33$	3.44885	0.600368
$34$	−1.26543	−0.217020
$35$	2.50874	0.424055
$36$	2.61687	0.436145
$37$	1.22529	0.201437	0.100718	−	0.994915i	$-0.467886\pi$
$37$	1.22529	0.201437	0.100718	+	0.994915i	$0.467886\pi$
$38$	2.02440	0.328400
$39$	−2.58230	−0.413500
$40$	1.57394	0.248861
$41$	1.59789	0.249549	0.124774	−	0.992185i	$-0.460179\pi$
$41$	1.59789	0.249549	0.124774	+	0.992185i	$0.460179\pi$
$42$	−3.77760	−0.582896
$43$	0.188669	0.0287718	0.0143859	−	0.999897i	$-0.495421\pi$
$43$	0.188669	0.0287718	0.0143859	+	0.999897i	$0.495421\pi$
$44$	−1.45522	−0.219382
$45$	4.11878	0.613992
$46$	−3.02412	−0.445882
$47$	−3.24332	−0.473086	−0.236543	−	0.971621i	$-0.576014\pi$
$47$	−3.24332	−0.473086	−0.236543	+	0.971621i	$0.576014\pi$
$48$	−2.36999	−0.342079
$49$	−4.45939	−0.637056
$50$	−2.52272	−0.356767
$51$	2.99907	0.419954
$52$	1.08958	0.151098
$53$	−4.65195	−0.638995	−0.319498	−	0.947587i	$-0.603514\pi$
$53$	−4.65195	−0.638995	−0.319498	+	0.947587i	$0.603514\pi$
$54$	0.908022	0.123566
$55$	−2.29042	−0.308840
$56$	1.59393	0.212998
$57$	−4.79781	−0.635485
$58$	−2.01834	−0.265021
$59$	−2.64062	−0.343780	−0.171890	−	0.985116i	$-0.554987\pi$
$59$	−2.64062	−0.343780	−0.171890	+	0.985116i	$0.554987\pi$
$60$	−3.73022	−0.481569
$61$	−15.4282	−1.97537	−0.987686	−	0.156448i	$-0.949996\pi$
$61$	−15.4282	−1.97537	−0.987686	+	0.156448i	$0.949996\pi$
$62$	−6.74478	−0.856587
$63$	4.17110	0.525509
$64$	1.00000	0.125000
$65$	1.71493	0.212711
$66$	3.44885	0.424524
$67$	−5.82549	−0.711696	−0.355848	−	0.934544i	$-0.615808\pi$
$67$	−5.82549	−0.711696	−0.355848	+	0.934544i	$0.615808\pi$
$68$	−1.26543	−0.153456
$69$	7.16714	0.862822
$70$	2.50874	0.299852
$71$	11.2978	1.34080	0.670400	−	0.742000i	$-0.266122\pi$
$71$	11.2978	1.34080	0.670400	+	0.742000i	$0.266122\pi$
$72$	2.61687	0.308401
$73$	−8.57257	−1.00334	−0.501672	−	0.865058i	$-0.667282\pi$
$73$	−8.57257	−1.00334	−0.501672	+	0.865058i	$0.667282\pi$
$74$	1.22529	0.142437
$75$	5.97884	0.690377
$76$	2.02440	0.232214
$77$	−2.31951	−0.264333
$78$	−2.58230	−0.292388
$79$	7.53323	0.847554	0.423777	−	0.905767i	$-0.360704\pi$
$79$	7.53323	0.847554	0.423777	+	0.905767i	$0.360704\pi$
$80$	1.57394	0.175971
$81$	−10.0026	−1.11140
$82$	1.59789	0.176458
$83$	−4.74304	−0.520616	−0.260308	−	0.965526i	$-0.583824\pi$
$83$	−4.74304	−0.520616	−0.260308	+	0.965526i	$0.583824\pi$
$84$	−3.77760	−0.412170
$85$	−1.99171	−0.216031
$86$	0.188669	0.0203447
$87$	4.78344	0.512839
$88$	−1.45522	−0.155126
$89$	−0.722253	−0.0765586	−0.0382793	−	0.999267i	$-0.512188\pi$
$89$	−0.722253	−0.0765586	−0.0382793	+	0.999267i	$0.512188\pi$
$90$	4.11878	0.434158
$91$	1.73672	0.182057
$92$	−3.02412	−0.315286
$93$	15.9851	1.65757
$94$	−3.24332	−0.334522
$95$	3.18627	0.326905
$96$	−2.36999	−0.241886
$97$	7.32169	0.743405	0.371703	−	0.928352i	$-0.378774\pi$
$97$	7.32169	0.743405	0.371703	+	0.928352i	$0.378774\pi$
$98$	−4.45939	−0.450467
$99$	−3.80810	−0.382729
$100$	−2.52272	−0.252272
$101$	8.46910	0.842707	0.421353	−	0.906897i	$-0.361555\pi$
$101$	8.46910	0.842707	0.421353	+	0.906897i	$0.361555\pi$
$102$	2.99907	0.296952
$103$	12.6629	1.24772	0.623858	−	0.781538i	$-0.285564\pi$
$103$	12.6629	1.24772	0.623858	+	0.781538i	$0.285564\pi$
$104$	1.08958	0.106842
$105$	−5.94570	−0.580241
$106$	−4.65195	−0.451838
$107$	−9.20169	−0.889561	−0.444781	−	0.895640i	$-0.646718\pi$
$107$	−9.20169	−0.889561	−0.444781	+	0.895640i	$0.646718\pi$
$108$	0.908022	0.0873745
$109$	5.30425	0.508055	0.254027	−	0.967197i	$-0.418245\pi$
$109$	5.30425	0.508055	0.254027	+	0.967197i	$0.418245\pi$
$110$	−2.29042	−0.218383
$111$	−2.90393	−0.275629
$112$	1.59393	0.150612
$113$	−6.07270	−0.571272	−0.285636	−	0.958338i	$-0.592205\pi$
$113$	−6.07270	−0.571272	−0.285636	+	0.958338i	$0.592205\pi$
$114$	−4.79781	−0.449356
$115$	−4.75977	−0.443851
$116$	−2.01834	−0.187398
$117$	2.85129	0.263602
$118$	−2.64062	−0.243089
$119$	−2.01701	−0.184899
$120$	−3.73022	−0.340521
$121$	−8.88235	−0.807486
$122$	−15.4282	−1.39680
$123$	−3.78699	−0.341461
$124$	−6.74478	−0.605699
$125$	−11.8403	−1.05903
$126$	4.17110	0.371591
$127$	13.7667	1.22160	0.610798	−	0.791787i	$-0.290849\pi$
$127$	13.7667	1.22160	0.610798	+	0.791787i	$0.290849\pi$
$128$	1.00000	0.0883883
$129$	−0.447145	−0.0393689
$130$	1.71493	0.150410
$131$	−4.95814	−0.433195	−0.216597	−	0.976261i	$-0.569496\pi$
$131$	−4.95814	−0.433195	−0.216597	+	0.976261i	$0.569496\pi$
$132$	3.44885	0.300184
$133$	3.22674	0.279794
$134$	−5.82549	−0.503245
$135$	1.42917	0.123003
$136$	−1.26543	−0.108510
$137$	−9.81564	−0.838607	−0.419303	−	0.907846i	$-0.637726\pi$
$137$	−9.81564	−0.838607	−0.419303	+	0.907846i	$0.637726\pi$
$138$	7.16714	0.610107
$139$	1.27239	0.107923	0.0539613	−	0.998543i	$-0.482815\pi$
$139$	1.27239	0.107923	0.0539613	+	0.998543i	$0.482815\pi$
$140$	2.50874	0.212027
$141$	7.68664	0.647331
$142$	11.2978	0.948089
$143$	−1.58558	−0.132593
$144$	2.61687	0.218072
$145$	−3.17673	−0.263813
$146$	−8.57257	−0.709471
$147$	10.5687	0.871694
$148$	1.22529	0.100718
$149$	−16.1054	−1.31940	−0.659702	−	0.751528i	$-0.729317\pi$
$149$	−16.1054	−1.31940	−0.659702	+	0.751528i	$0.729317\pi$
$150$	5.97884	0.488170
$151$	9.18990	0.747863	0.373932	−	0.927456i	$-0.378009\pi$
$151$	9.18990	0.747863	0.373932	+	0.927456i	$0.378009\pi$
$152$	2.02440	0.164200
$153$	−3.31147	−0.267717
$154$	−2.31951	−0.186911
$155$	−10.6158	−0.852685
$156$	−2.58230	−0.206750
$157$	−11.1585	−0.890543	−0.445271	−	0.895396i	$-0.646893\pi$
$157$	−11.1585	−0.890543	−0.445271	+	0.895396i	$0.646893\pi$
$158$	7.53323	0.599311
$159$	11.0251	0.874347
$160$	1.57394	0.124431
$161$	−4.82023	−0.379887
$162$	−10.0026	−0.785879
$163$	−6.20270	−0.485833	−0.242916	−	0.970047i	$-0.578104\pi$
$163$	−6.20270	−0.485833	−0.242916	+	0.970047i	$0.578104\pi$
$164$	1.59789	0.124774
$165$	5.42827	0.422590
$166$	−4.74304	−0.368131
$167$	−23.0755	−1.78564	−0.892818	−	0.450418i	$-0.851275\pi$
$167$	−23.0755	−1.78564	−0.892818	+	0.450418i	$0.851275\pi$
$168$	−3.77760	−0.291448
$169$	−11.8128	−0.908678
$170$	−1.99171	−0.152757
$171$	5.29758	0.405116
$172$	0.188669	0.0143859
$173$	−2.50660	−0.190573	−0.0952866	−	0.995450i	$-0.530377\pi$
$173$	−2.50660	−0.190573	−0.0952866	+	0.995450i	$0.530377\pi$
$174$	4.78344	0.362632
$175$	−4.02104	−0.303962
$176$	−1.45522	−0.109691
$177$	6.25826	0.470400
$178$	−0.722253	−0.0541351
$179$	14.6526	1.09519	0.547593	−	0.836745i	$-0.315544\pi$
$179$	14.6526	1.09519	0.547593	+	0.836745i	$0.315544\pi$
$180$	4.11878	0.306996
$181$	17.2563	1.28265	0.641327	−	0.767268i	$-0.278384\pi$
$181$	17.2563	1.28265	0.641327	+	0.767268i	$0.278384\pi$
$182$	1.73672	0.128734
$183$	36.5646	2.70293
$184$	−3.02412	−0.222941
$185$	1.92853	0.141788
$186$	15.9851	1.17208
$187$	1.84148	0.134662
$188$	−3.24332	−0.236543
$189$	1.44732	0.105277
$190$	3.18627	0.231156
$191$	18.3904	1.33068	0.665342	−	0.746539i	$-0.268286\pi$
$191$	18.3904	1.33068	0.665342	+	0.746539i	$0.268286\pi$
$192$	−2.36999	−0.171040
$193$	−6.51055	−0.468639	−0.234320	−	0.972160i	$-0.575286\pi$
$193$	−6.51055	−0.468639	−0.234320	+	0.972160i	$0.575286\pi$
$194$	7.32169	0.525667
$195$	−4.06438	−0.291056
$196$	−4.45939	−0.318528
$197$	13.5903	0.968266	0.484133	−	0.874994i	$-0.339135\pi$
$197$	13.5903	0.968266	0.484133	+	0.874994i	$0.339135\pi$
$198$	−3.80810	−0.270630
$199$	−13.7842	−0.977135	−0.488568	−	0.872526i	$-0.662480\pi$
$199$	−13.7842	−0.977135	−0.488568	+	0.872526i	$0.662480\pi$
$200$	−2.52272	−0.178384
$201$	13.8064	0.973826
$202$	8.46910	0.595884
$203$	−3.21709	−0.225795
$204$	2.99907	0.209977
$205$	2.51498	0.175654
$206$	12.6629	0.882269
$207$	−7.91371	−0.550041
$208$	1.08958	0.0755490
$209$	−2.94593	−0.203774
$210$	−5.94570	−0.410292
$211$	−9.53853	−0.656659	−0.328330	−	0.944563i	$-0.606486\pi$
$211$	−9.53853	−0.656659	−0.328330	+	0.944563i	$0.606486\pi$
$212$	−4.65195	−0.319498
$213$	−26.7757	−1.83464
$214$	−9.20169	−0.629015
$215$	0.296954	0.0202521
$216$	0.908022	0.0617831
$217$	−10.7507	−0.729804
$218$	5.30425	0.359249
$219$	20.3169	1.37289
$220$	−2.29042	−0.154420
$221$	−1.37879	−0.0927478
$222$	−2.90393	−0.194899
$223$	12.7731	0.855347	0.427673	−	0.903933i	$-0.359333\pi$
$223$	12.7731	0.855347	0.427673	+	0.903933i	$0.359333\pi$
$224$	1.59393	0.106499
$225$	−6.60164	−0.440109
$226$	−6.07270	−0.403950
$227$	19.9844	1.32641	0.663206	−	0.748437i	$-0.269195\pi$
$227$	19.9844	1.32641	0.663206	+	0.748437i	$0.269195\pi$
$228$	−4.79781	−0.317742
$229$	−11.6113	−0.767294	−0.383647	−	0.923480i	$-0.625332\pi$
$229$	−11.6113	−0.767294	−0.383647	+	0.923480i	$0.625332\pi$
$230$	−4.75977	−0.313850
$231$	5.49722	0.361691
$232$	−2.01834	−0.132510
$233$	24.2822	1.59078	0.795392	−	0.606096i	$-0.207265\pi$
$233$	24.2822	1.59078	0.795392	+	0.606096i	$0.207265\pi$
$234$	2.85129	0.186395
$235$	−5.10477	−0.332999
$236$	−2.64062	−0.171890
$237$	−17.8537	−1.15972
$238$	−2.01701	−0.130743
$239$	−19.6722	−1.27249	−0.636245	−	0.771487i	$-0.719513\pi$
$239$	−19.6722	−1.27249	−0.636245	+	0.771487i	$0.719513\pi$
$240$	−3.73022	−0.240785
$241$	8.85103	0.570145	0.285072	−	0.958506i	$-0.407982\pi$
$241$	8.85103	0.570145	0.285072	+	0.958506i	$0.407982\pi$
$242$	−8.88235	−0.570979
$243$	20.9820	1.34600
$244$	−15.4282	−0.987686
$245$	−7.01880	−0.448415
$246$	−3.78699	−0.241450
$247$	2.20575	0.140348
$248$	−6.74478	−0.428294
$249$	11.2410	0.712367
$250$	−11.8403	−0.748846
$251$	0.989008	0.0624256	0.0312128	−	0.999513i	$-0.490063\pi$
$251$	0.989008	0.0624256	0.0312128	+	0.999513i	$0.490063\pi$
$252$	4.17110	0.262755
$253$	4.40074	0.276672
$254$	13.7667	0.863798
$255$	4.72034	0.295599
$256$	1.00000	0.0625000
$257$	14.9999	0.935669	0.467834	−	0.883816i	$-0.345034\pi$
$257$	14.9999	0.935669	0.467834	+	0.883816i	$0.345034\pi$
$258$	−0.447145	−0.0278380
$259$	1.95303	0.121355
$260$	1.71493	0.106356
$261$	−5.28172	−0.326930
$262$	−4.95814	−0.306315
$263$	18.3854	1.13369	0.566846	−	0.823824i	$-0.308164\pi$
$263$	18.3854	1.13369	0.566846	+	0.823824i	$0.308164\pi$
$264$	3.44885	0.212262
$265$	−7.32188	−0.449779
$266$	3.22674	0.197844
$267$	1.71173	0.104756
$268$	−5.82549	−0.355848
$269$	13.0813	0.797582	0.398791	−	0.917042i	$-0.369430\pi$
$269$	13.0813	0.797582	0.398791	+	0.917042i	$0.369430\pi$
$270$	1.42917	0.0869765
$271$	−9.28426	−0.563979	−0.281989	−	0.959418i	$-0.590994\pi$
$271$	−9.28426	−0.563979	−0.281989	+	0.959418i	$0.590994\pi$
$272$	−1.26543	−0.0767282
$273$	−4.11601	−0.249112
$274$	−9.81564	−0.592984
$275$	3.67111	0.221376
$276$	7.16714	0.431411
$277$	−17.5260	−1.05304	−0.526518	−	0.850164i	$-0.676503\pi$
$277$	−17.5260	−1.05304	−0.526518	+	0.850164i	$0.676503\pi$
$278$	1.27239	0.0763129
$279$	−17.6502	−1.05669
$280$	2.50874	0.149926
$281$	−28.0768	−1.67492	−0.837461	−	0.546498i	$-0.815961\pi$
$281$	−28.0768	−1.67492	−0.837461	+	0.546498i	$0.815961\pi$
$282$	7.68664	0.457732
$283$	−16.4120	−0.975593	−0.487796	−	0.872957i	$-0.662199\pi$
$283$	−16.4120	−0.975593	−0.487796	+	0.872957i	$0.662199\pi$
$284$	11.2978	0.670400
$285$	−7.55144	−0.447309
$286$	−1.58558	−0.0937572
$287$	2.54692	0.150340
$288$	2.61687	0.154200
$289$	−15.3987	−0.905805
$290$	−3.17673	−0.186544
$291$	−17.3524	−1.01721
$292$	−8.57257	−0.501672
$293$	−9.15821	−0.535029	−0.267514	−	0.963554i	$-0.586202\pi$
$293$	−9.15821	−0.535029	−0.267514	+	0.963554i	$0.586202\pi$
$294$	10.5687	0.616381
$295$	−4.15617	−0.241982
$296$	1.22529	0.0712186
$297$	−1.32137	−0.0766735
$298$	−16.1054	−0.932959
$299$	−3.29503	−0.190556
$300$	5.97884	0.345188
$301$	0.300726	0.0173335
$302$	9.18990	0.528819
$303$	−20.0717	−1.15309
$304$	2.02440	0.116107
$305$	−24.2829	−1.39044
$306$	−3.31147	−0.189304
$307$	14.8303	0.846409	0.423204	−	0.906034i	$-0.360905\pi$
$307$	14.8303	0.846409	0.423204	+	0.906034i	$0.360905\pi$
$308$	−2.31951	−0.132166
$309$	−30.0111	−1.70727
$310$	−10.6158	−0.602940
$311$	25.6635	1.45524	0.727622	−	0.685978i	$-0.240625\pi$
$311$	25.6635	1.45524	0.727622	+	0.685978i	$0.240625\pi$
$312$	−2.58230	−0.146194
$313$	15.9469	0.901372	0.450686	−	0.892683i	$-0.351179\pi$
$313$	15.9469	0.901372	0.450686	+	0.892683i	$0.351179\pi$
$314$	−11.1585	−0.629709
$315$	6.56504	0.369898
$316$	7.53323	0.423777
$317$	−3.76227	−0.211310	−0.105655	−	0.994403i	$-0.533694\pi$
$317$	−3.76227	−0.211310	−0.105655	+	0.994403i	$0.533694\pi$
$318$	11.0251	0.618257
$319$	2.93711	0.164447
$320$	1.57394	0.0879857
$321$	21.8079	1.21720
$322$	−4.82023	−0.268621
$323$	−2.56174	−0.142539
$324$	−10.0026	−0.555700
$325$	−2.74872	−0.152471
$326$	−6.20270	−0.343536
$327$	−12.5710	−0.695180
$328$	1.59789	0.0882288
$329$	−5.16961	−0.285010
$330$	5.42827	0.298816
$331$	−15.4315	−0.848194	−0.424097	−	0.905617i	$-0.639408\pi$
$331$	−15.4315	−0.848194	−0.424097	+	0.905617i	$0.639408\pi$
$332$	−4.74304	−0.260308
$333$	3.20642	0.175711
$334$	−23.0755	−1.26264
$335$	−9.16894	−0.500953
$336$	−3.77760	−0.206085
$337$	−17.5762	−0.957438	−0.478719	−	0.877968i	$-0.658899\pi$
$337$	−17.5762	−0.957438	−0.478719	+	0.877968i	$0.658899\pi$
$338$	−11.8128	−0.642532
$339$	14.3923	0.781681
$340$	−1.99171	−0.108016
$341$	9.81510	0.531517
$342$	5.29758	0.286460
$343$	−18.2655	−0.986242
$344$	0.188669	0.0101724
$345$	11.2806	0.607328
$346$	−2.50660	−0.134756
$347$	−3.30225	−0.177274	−0.0886371	−	0.996064i	$-0.528251\pi$
$347$	−3.30225	−0.177274	−0.0886371	+	0.996064i	$0.528251\pi$
$348$	4.78344	0.256420
$349$	−3.24409	−0.173652	−0.0868261	−	0.996223i	$-0.527672\pi$
$349$	−3.24409	−0.173652	−0.0868261	+	0.996223i	$0.527672\pi$
$350$	−4.02104	−0.214934
$351$	0.989366	0.0528084
$352$	−1.45522	−0.0775632
$353$	−6.94052	−0.369407	−0.184703	−	0.982794i	$-0.559132\pi$
$353$	−6.94052	−0.369407	−0.184703	+	0.982794i	$0.559132\pi$
$354$	6.25826	0.332623
$355$	17.7820	0.943770
$356$	−0.722253	−0.0382793
$357$	4.78030	0.253000
$358$	14.6526	0.774413
$359$	−25.3192	−1.33630	−0.668148	−	0.744028i	$-0.732913\pi$
$359$	−25.3192	−1.33630	−0.668148	+	0.744028i	$0.732913\pi$
$360$	4.11878	0.217079
$361$	−14.9018	−0.784306
$362$	17.2563	0.906974
$363$	21.0511	1.10490
$364$	1.73672	0.0910287
$365$	−13.4927	−0.706239
$366$	36.5646	1.91126
$367$	−29.5013	−1.53995	−0.769977	−	0.638072i	$-0.779732\pi$
$367$	−29.5013	−1.53995	−0.769977	+	0.638072i	$0.779732\pi$
$368$	−3.02412	−0.157643
$369$	4.18147	0.217679
$370$	1.92853	0.100260
$371$	−7.41488	−0.384962
$372$	15.9851	0.828787
$373$	−29.0522	−1.50427	−0.752134	−	0.659011i	$-0.770975\pi$
$373$	−29.0522	−1.50427	−0.752134	+	0.659011i	$0.770975\pi$
$374$	1.84148	0.0952206
$375$	28.0614	1.44908
$376$	−3.24332	−0.167261
$377$	−2.19915	−0.113262
$378$	1.44732	0.0744422
$379$	3.19572	0.164153	0.0820767	−	0.996626i	$-0.473845\pi$
$379$	3.19572	0.164153	0.0820767	+	0.996626i	$0.473845\pi$
$380$	3.18627	0.163452
$381$	−32.6269	−1.67153
$382$	18.3904	0.940935
$383$	5.19416	0.265409	0.132705	−	0.991156i	$-0.457634\pi$
$383$	5.19416	0.265409	0.132705	+	0.991156i	$0.457634\pi$
$384$	−2.36999	−0.120943
$385$	−3.65076	−0.186060
$386$	−6.51055	−0.331378
$387$	0.493723	0.0250973
$388$	7.32169	0.371703
$389$	37.6167	1.90724	0.953620	−	0.301012i	$-0.0973245\pi$
$389$	37.6167	1.90724	0.953620	+	0.301012i	$0.0973245\pi$
$390$	−4.06438	−0.205808
$391$	3.82682	0.193531
$392$	−4.45939	−0.225233
$393$	11.7508	0.592747
$394$	13.5903	0.684668
$395$	11.8568	0.596581
$396$	−3.80810	−0.191364
$397$	−27.9068	−1.40060	−0.700302	−	0.713847i	$-0.746951\pi$
$397$	−27.9068	−1.40060	−0.700302	+	0.713847i	$0.746951\pi$
$398$	−13.7842	−0.690939
$399$	−7.64736	−0.382847
$400$	−2.52272	−0.126136
$401$	17.7271	0.885249	0.442624	−	0.896707i	$-0.354048\pi$
$401$	17.7271	0.885249	0.442624	+	0.896707i	$0.354048\pi$
$402$	13.8064	0.688599
$403$	−7.34899	−0.366079
$404$	8.46910	0.421353
$405$	−15.7435	−0.782299
$406$	−3.21709	−0.159661
$407$	−1.78306	−0.0883831
$408$	2.99907	0.148476
$409$	−9.57326	−0.473367	−0.236684	−	0.971587i	$-0.576061\pi$
$409$	−9.57326	−0.473367	−0.236684	+	0.971587i	$0.576061\pi$
$410$	2.51498	0.124206
$411$	23.2630	1.14748
$412$	12.6629	0.623858
$413$	−4.20897	−0.207110
$414$	−7.91371	−0.388938
$415$	−7.46524	−0.366454
$416$	1.08958	0.0534212
$417$	−3.01555	−0.147672
$418$	−2.94593	−0.144090
$419$	−15.7393	−0.768916	−0.384458	−	0.923142i	$-0.625612\pi$
$419$	−15.7393	−0.768916	−0.384458	+	0.923142i	$0.625612\pi$
$420$	−5.94570	−0.290121
$421$	−34.8963	−1.70074	−0.850370	−	0.526185i	$-0.823622\pi$
$421$	−34.8963	−1.70074	−0.850370	+	0.526185i	$0.823622\pi$
$422$	−9.53853	−0.464328
$423$	−8.48733	−0.412668
$424$	−4.65195	−0.225919
$425$	3.19234	0.154851
$426$	−26.7757	−1.29729
$427$	−24.5914	−1.19006
$428$	−9.20169	−0.444781
$429$	3.75781	0.181429
$430$	0.296954	0.0143204
$431$	−32.7475	−1.57739	−0.788696	−	0.614783i	$-0.789244\pi$
$431$	−32.7475	−1.57739	−0.788696	+	0.614783i	$0.789244\pi$
$432$	0.908022	0.0436872
$433$	36.1285	1.73623	0.868113	−	0.496367i	$-0.165333\pi$
$433$	36.1285	1.73623	0.868113	+	0.496367i	$0.165333\pi$
$434$	−10.7507	−0.516050
$435$	7.52884	0.360980
$436$	5.30425	0.254027
$437$	−6.12201	−0.292856
$438$	20.3169	0.970781
$439$	−1.93567	−0.0923846	−0.0461923	−	0.998933i	$-0.514709\pi$
$439$	−1.93567	−0.0923846	−0.0461923	+	0.998933i	$0.514709\pi$
$440$	−2.29042	−0.109191
$441$	−11.6696	−0.555697
$442$	−1.37879	−0.0655826
$443$	11.9595	0.568213	0.284106	−	0.958793i	$-0.408303\pi$
$443$	11.9595	0.568213	0.284106	+	0.958793i	$0.408303\pi$
$444$	−2.90393	−0.137814
$445$	−1.13678	−0.0538885
$446$	12.7731	0.604821
$447$	38.1696	1.80536
$448$	1.59393	0.0753061
$449$	−26.6005	−1.25536	−0.627678	−	0.778473i	$-0.715994\pi$
$449$	−26.6005	−1.25536	−0.627678	+	0.778473i	$0.715994\pi$
$450$	−6.60164	−0.311204
$451$	−2.32527	−0.109493
$452$	−6.07270	−0.285636
$453$	−21.7800	−1.02331
$454$	19.9844	0.937914
$455$	2.73348	0.128148
$456$	−4.79781	−0.224678
$457$	4.75305	0.222338	0.111169	−	0.993802i	$-0.464540\pi$
$457$	4.75305	0.222338	0.111169	+	0.993802i	$0.464540\pi$
$458$	−11.6113	−0.542559
$459$	−1.14904	−0.0536327
$460$	−4.75977	−0.221925
$461$	−4.08028	−0.190038	−0.0950188	−	0.995475i	$-0.530291\pi$
$461$	−4.08028	−0.190038	−0.0950188	+	0.995475i	$0.530291\pi$
$462$	5.49722	0.255754
$463$	4.31122	0.200359	0.100180	−	0.994969i	$-0.468058\pi$
$463$	4.31122	0.200359	0.100180	+	0.994969i	$0.468058\pi$
$464$	−2.01834	−0.0936989
$465$	25.1595	1.16674
$466$	24.2822	1.12485
$467$	−32.5024	−1.50403	−0.752017	−	0.659144i	$-0.770919\pi$
$467$	−32.5024	−1.50403	−0.752017	+	0.659144i	$0.770919\pi$
$468$	2.85129	0.131801
$469$	−9.28541	−0.428760
$470$	−5.10477	−0.235466
$471$	26.4455	1.21854
$472$	−2.64062	−0.121545
$473$	−0.274555	−0.0126240
$474$	−17.8537	−0.820047
$475$	−5.10700	−0.234325
$476$	−2.01701	−0.0924495
$477$	−12.1735	−0.557388
$478$	−19.6722	−0.899786
$479$	11.7102	0.535055	0.267527	−	0.963550i	$-0.413793\pi$
$479$	11.7102	0.535055	0.267527	+	0.963550i	$0.413793\pi$
$480$	−3.73022	−0.170260
$481$	1.33506	0.0608733
$482$	8.85103	0.403153
$483$	11.4239	0.519806
$484$	−8.88235	−0.403743
$485$	11.5239	0.523272
$486$	20.9820	0.951765
$487$	−26.8841	−1.21824	−0.609118	−	0.793080i	$-0.708476\pi$
$487$	−26.8841	−1.21824	−0.609118	+	0.793080i	$0.708476\pi$
$488$	−15.4282	−0.698400
$489$	14.7004	0.664773
$490$	−7.01880	−0.317077
$491$	−20.7525	−0.936545	−0.468273	−	0.883584i	$-0.655123\pi$
$491$	−20.7525	−0.936545	−0.468273	+	0.883584i	$0.655123\pi$
$492$	−3.78699	−0.170731
$493$	2.55407	0.115030
$494$	2.20575	0.0992413
$495$	−5.99371	−0.269397
$496$	−6.74478	−0.302849
$497$	18.0079	0.807763
$498$	11.2410	0.503720
$499$	15.4767	0.692831	0.346415	−	0.938081i	$-0.387399\pi$
$499$	15.4767	0.692831	0.346415	+	0.938081i	$0.387399\pi$
$500$	−11.8403	−0.529514
$501$	54.6888	2.44331
$502$	0.989008	0.0441416
$503$	−19.8746	−0.886164	−0.443082	−	0.896481i	$-0.646115\pi$
$503$	−19.8746	−0.886164	−0.443082	+	0.896481i	$0.646115\pi$
$504$	4.17110	0.185796
$505$	13.3298	0.593169
$506$	4.40074	0.195637
$507$	27.9963	1.24336
$508$	13.7667	0.610798
$509$	−16.7878	−0.744105	−0.372052	−	0.928212i	$-0.621346\pi$
$509$	−16.7878	−0.744105	−0.372052	+	0.928212i	$0.621346\pi$
$510$	4.72034	0.209020
$511$	−13.6641	−0.604463
$512$	1.00000	0.0441942
$513$	1.83820	0.0811584
$514$	14.9999	0.661618
$515$	19.9307	0.878250
$516$	−0.447145	−0.0196845
$517$	4.71972	0.207573
$518$	1.95303	0.0858111
$519$	5.94062	0.260764
$520$	1.71493	0.0752048
$521$	−2.48079	−0.108685	−0.0543426	−	0.998522i	$-0.517306\pi$
$521$	−2.48079	−0.108685	−0.0543426	+	0.998522i	$0.517306\pi$
$522$	−5.28172	−0.231175
$523$	32.2891	1.41190	0.705952	−	0.708260i	$-0.250519\pi$
$523$	32.2891	1.41190	0.705952	+	0.708260i	$0.250519\pi$
$524$	−4.95814	−0.216597
$525$	9.52984	0.415917
$526$	18.3854	0.801641
$527$	8.53507	0.371793
$528$	3.44885	0.150092
$529$	−13.8547	−0.602379
$530$	−7.32188	−0.318042
$531$	−6.91016	−0.299875
$532$	3.22674	0.139897
$533$	1.74103	0.0754126
$534$	1.71173	0.0740740
$535$	−14.4829	−0.626149
$536$	−5.82549	−0.251623
$537$	−34.7265	−1.49856
$538$	13.0813	0.563976
$539$	6.48937	0.279517
$540$	1.42917	0.0615016
$541$	−7.29027	−0.313433	−0.156717	−	0.987644i	$-0.550091\pi$
$541$	−7.29027	−0.313433	−0.156717	+	0.987644i	$0.550091\pi$
$542$	−9.28426	−0.398793
$543$	−40.8974	−1.75508
$544$	−1.26543	−0.0542550
$545$	8.34855	0.357613
$546$	−4.11601	−0.176149
$547$	11.3262	0.484275	0.242138	−	0.970242i	$-0.422151\pi$
$547$	11.3262	0.484275	0.242138	+	0.970242i	$0.422151\pi$
$548$	−9.81564	−0.419303
$549$	−40.3734	−1.72310
$550$	3.67111	0.156537
$551$	−4.08591	−0.174066
$552$	7.16714	0.305054
$553$	12.0074	0.510608
$554$	−17.5260	−0.744608
$555$	−4.57060	−0.194011
$556$	1.27239	0.0539613
$557$	−19.2760	−0.816752	−0.408376	−	0.912814i	$-0.633905\pi$
$557$	−19.2760	−0.816752	−0.408376	+	0.912814i	$0.633905\pi$
$558$	−17.6502	−0.747192
$559$	0.205571	0.00869473
$560$	2.50874	0.106014
$561$	−4.36429	−0.184260
$562$	−28.0768	−1.18435
$563$	−22.3783	−0.943131	−0.471566	−	0.881831i	$-0.656311\pi$
$563$	−22.3783	−0.943131	−0.471566	+	0.881831i	$0.656311\pi$
$564$	7.68664	0.323666
$565$	−9.55805	−0.402110
$566$	−16.4120	−0.689848
$567$	−15.9434	−0.669562
$568$	11.2978	0.474045
$569$	40.9996	1.71879	0.859397	−	0.511309i	$-0.170839\pi$
$569$	40.9996	1.71879	0.859397	+	0.511309i	$0.170839\pi$
$570$	−7.55144	−0.316295
$571$	28.2362	1.18165	0.590824	−	0.806800i	$-0.298803\pi$
$571$	28.2362	1.18165	0.590824	+	0.806800i	$0.298803\pi$
$572$	−1.58558	−0.0662963
$573$	−43.5851	−1.82080
$574$	2.54692	0.106307
$575$	7.62902	0.318152
$576$	2.61687	0.109036
$577$	−31.6393	−1.31716	−0.658581	−	0.752510i	$-0.728843\pi$
$577$	−31.6393	−1.31716	−0.658581	+	0.752510i	$0.728843\pi$
$578$	−15.3987	−0.640501
$579$	15.4300	0.641247
$580$	−3.17673	−0.131907
$581$	−7.56006	−0.313644
$582$	−17.3524	−0.719278
$583$	6.76959	0.280368
$584$	−8.57257	−0.354736
$585$	4.48775	0.185546
$586$	−9.15821	−0.378322
$587$	−26.8560	−1.10846	−0.554232	−	0.832362i	$-0.686988\pi$
$587$	−26.8560	−1.10846	−0.554232	+	0.832362i	$0.686988\pi$
$588$	10.5687	0.435847
$589$	−13.6541	−0.562607
$590$	−4.15617	−0.171107
$591$	−32.2088	−1.32489
$592$	1.22529	0.0503592
$593$	−20.2072	−0.829810	−0.414905	−	0.909865i	$-0.636185\pi$
$593$	−20.2072	−0.829810	−0.414905	+	0.909865i	$0.636185\pi$
$594$	−1.32137	−0.0542164
$595$	−3.17465	−0.130148
$596$	−16.1054	−0.659702
$597$	32.6684	1.33703
$598$	−3.29503	−0.134744
$599$	−36.4067	−1.48754	−0.743769	−	0.668437i	$-0.766964\pi$
$599$	−36.4067	−1.48754	−0.743769	+	0.668437i	$0.766964\pi$
$600$	5.97884	0.244085
$601$	−20.0267	−0.816907	−0.408453	−	0.912779i	$-0.633932\pi$
$601$	−20.0267	−0.816907	−0.408453	+	0.912779i	$0.633932\pi$
$602$	0.300726	0.0122567
$603$	−15.2445	−0.620805
$604$	9.18990	0.373932
$605$	−13.9803	−0.568378
$606$	−20.0717	−0.815357
$607$	−7.65184	−0.310579	−0.155289	−	0.987869i	$-0.549631\pi$
$607$	−7.65184	−0.310579	−0.155289	+	0.987869i	$0.549631\pi$
$608$	2.02440	0.0821001
$609$	7.62447	0.308959
$610$	−24.2829	−0.983187
$611$	−3.53386	−0.142965
$612$	−3.31147	−0.133858
$613$	22.0456	0.890414	0.445207	−	0.895428i	$-0.353130\pi$
$613$	22.0456	0.890414	0.445207	+	0.895428i	$0.353130\pi$
$614$	14.8303	0.598501
$615$	−5.96048	−0.240350
$616$	−2.31951	−0.0934557
$617$	−38.3564	−1.54417	−0.772086	−	0.635518i	$-0.780786\pi$
$617$	−38.3564	−1.54417	−0.772086	+	0.635518i	$0.780786\pi$
$618$	−30.0111	−1.20722
$619$	−36.7028	−1.47521	−0.737606	−	0.675232i	$-0.764044\pi$
$619$	−36.7028	−1.47521	−0.737606	+	0.675232i	$0.764044\pi$
$620$	−10.6158	−0.426343
$621$	−2.74597	−0.110192
$622$	25.6635	1.02901
$623$	−1.15122	−0.0461226
$624$	−2.58230	−0.103375
$625$	−6.02224	−0.240889
$626$	15.9469	0.637366
$627$	6.98184	0.278828
$628$	−11.1585	−0.445271
$629$	−1.55052	−0.0618235
$630$	6.56504	0.261558
$631$	17.5818	0.699919	0.349960	−	0.936765i	$-0.386195\pi$
$631$	17.5818	0.699919	0.349960	+	0.936765i	$0.386195\pi$
$632$	7.53323	0.299656
$633$	22.6062	0.898517
$634$	−3.76227	−0.149419
$635$	21.6679	0.859863
$636$	11.0251	0.437174
$637$	−4.85888	−0.192516
$638$	2.93711	0.116281
$639$	29.5648	1.16957
$640$	1.57394	0.0622153
$641$	23.2445	0.918101	0.459050	−	0.888410i	$-0.348190\pi$
$641$	23.2445	0.918101	0.459050	+	0.888410i	$0.348190\pi$
$642$	21.8079	0.860691
$643$	−12.4072	−0.489291	−0.244645	−	0.969613i	$-0.578672\pi$
$643$	−12.4072	−0.489291	−0.244645	+	0.969613i	$0.578672\pi$
$644$	−4.82023	−0.189944
$645$	−0.703778	−0.0277112
$646$	−2.56174	−0.100790
$647$	41.4121	1.62808	0.814039	−	0.580811i	$-0.197264\pi$
$647$	41.4121	1.62808	0.814039	+	0.580811i	$0.197264\pi$
$648$	−10.0026	−0.392940
$649$	3.84268	0.150838
$650$	−2.74872	−0.107814
$651$	25.4791	0.998603
$652$	−6.20270	−0.242916
$653$	32.3478	1.26587	0.632933	−	0.774206i	$-0.281851\pi$
$653$	32.3478	1.26587	0.632933	+	0.774206i	$0.281851\pi$
$654$	−12.5710	−0.491566
$655$	−7.80380	−0.304920
$656$	1.59789	0.0623872
$657$	−22.4333	−0.875206
$658$	−5.16961	−0.201533
$659$	42.9015	1.67120	0.835602	−	0.549335i	$-0.185119\pi$
$659$	42.9015	1.67120	0.835602	+	0.549335i	$0.185119\pi$
$660$	5.42827	0.211295
$661$	−23.9138	−0.930140	−0.465070	−	0.885274i	$-0.653971\pi$
$661$	−23.9138	−0.930140	−0.465070	+	0.885274i	$0.653971\pi$
$662$	−15.4315	−0.599764
$663$	3.26773	0.126908
$664$	−4.74304	−0.184066
$665$	5.07869	0.196943
$666$	3.20642	0.124246
$667$	6.10369	0.236336
$668$	−23.0755	−0.892818
$669$	−30.2720	−1.17038
$670$	−9.16894	−0.354227
$671$	22.4513	0.866722
$672$	−3.77760	−0.145724
$673$	38.1793	1.47170	0.735852	−	0.677142i	$-0.236782\pi$
$673$	38.1793	1.47170	0.735852	+	0.677142i	$0.236782\pi$
$674$	−17.5762	−0.677011
$675$	−2.29069	−0.0881687
$676$	−11.8128	−0.454339
$677$	−28.3519	−1.08965	−0.544825	−	0.838550i	$-0.683404\pi$
$677$	−28.3519	−1.08965	−0.544825	+	0.838550i	$0.683404\pi$
$678$	14.3923	0.552732
$679$	11.6703	0.447863
$680$	−1.99171	−0.0763787
$681$	−47.3629	−1.81495
$682$	9.81510	0.375840
$683$	40.9899	1.56844	0.784218	−	0.620485i	$-0.213064\pi$
$683$	40.9899	1.56844	0.784218	+	0.620485i	$0.213064\pi$
$684$	5.29758	0.202558
$685$	−15.4492	−0.590283
$686$	−18.2655	−0.697378
$687$	27.5186	1.04990
$688$	0.188669	0.00719295
$689$	−5.06869	−0.193102
$690$	11.2806	0.429446
$691$	9.83019	0.373958	0.186979	−	0.982364i	$-0.440130\pi$
$691$	9.83019	0.373958	0.186979	+	0.982364i	$0.440130\pi$
$692$	−2.50660	−0.0952866
$693$	−6.06985	−0.230574
$694$	−3.30225	−0.125352
$695$	2.00266	0.0759652
$696$	4.78344	0.181316
$697$	−2.02202	−0.0765896
$698$	−3.24409	−0.122791
$699$	−57.5488	−2.17669
$700$	−4.02104	−0.151981
$701$	51.0308	1.92741	0.963704	−	0.266973i	$-0.0860236\pi$
$701$	51.0308	1.92741	0.963704	+	0.266973i	$0.0860236\pi$
$702$	0.989366	0.0373412
$703$	2.48047	0.0935529
$704$	−1.45522	−0.0548455
$705$	12.0983	0.455647
$706$	−6.94052	−0.261210
$707$	13.4991	0.507687
$708$	6.25826	0.235200
$709$	24.2127	0.909326	0.454663	−	0.890663i	$-0.349760\pi$
$709$	24.2127	0.909326	0.454663	+	0.890663i	$0.349760\pi$
$710$	17.7820	0.667346
$711$	19.7135	0.739312
$712$	−0.722253	−0.0270676
$713$	20.3970	0.763873
$714$	4.78030	0.178898
$715$	−2.49560	−0.0933301
$716$	14.6526	0.547593
$717$	46.6230	1.74117
$718$	−25.3192	−0.944904
$719$	7.56756	0.282222	0.141111	−	0.989994i	$-0.454933\pi$
$719$	7.56756	0.282222	0.141111	+	0.989994i	$0.454933\pi$
$720$	4.11878	0.153498
$721$	20.1838	0.751685
$722$	−14.9018	−0.554588
$723$	−20.9769	−0.780139
$724$	17.2563	0.641327
$725$	5.09171	0.189101
$726$	21.0511	0.781280
$727$	−18.4701	−0.685019	−0.342509	−	0.939514i	$-0.611277\pi$
$727$	−18.4701	−0.685019	−0.342509	+	0.939514i	$0.611277\pi$
$728$	1.73672	0.0643670
$729$	−19.7195	−0.730351
$730$	−13.4927	−0.499387
$731$	−0.238749	−0.00883044
$732$	36.5646	1.35147
$733$	−29.2011	−1.07857	−0.539284	−	0.842124i	$-0.681305\pi$
$733$	−29.2011	−1.07857	−0.539284	+	0.842124i	$0.681305\pi$
$734$	−29.5013	−1.08891
$735$	16.6345	0.613573
$736$	−3.02412	−0.111470
$737$	8.47734	0.312267
$738$	4.18147	0.153922
$739$	5.37525	0.197732	0.0988659	−	0.995101i	$-0.468479\pi$
$739$	5.37525	0.197732	0.0988659	+	0.995101i	$0.468479\pi$
$740$	1.92853	0.0708942
$741$	−5.22761	−0.192041
$742$	−7.41488	−0.272209
$743$	−12.9288	−0.474312	−0.237156	−	0.971472i	$-0.576215\pi$
$743$	−12.9288	−0.474312	−0.237156	+	0.971472i	$0.576215\pi$
$744$	15.9851	0.586041
$745$	−25.3488	−0.928709
$746$	−29.0522	−1.06368
$747$	−12.4119	−0.454128
$748$	1.84148	0.0673311
$749$	−14.6668	−0.535915
$750$	28.0614	1.02466
$751$	−8.67100	−0.316409	−0.158205	−	0.987406i	$-0.550571\pi$
$751$	−8.67100	−0.316409	−0.158205	+	0.987406i	$0.550571\pi$
$752$	−3.24332	−0.118272
$753$	−2.34394	−0.0854180
$754$	−2.19915	−0.0800882
$755$	14.4643	0.526410
$756$	1.44732	0.0526386
$757$	39.6710	1.44187	0.720933	−	0.693005i	$-0.243713\pi$
$757$	39.6710	1.44187	0.720933	+	0.693005i	$0.243713\pi$
$758$	3.19572	0.116074
$759$	−10.4297	−0.378575
$760$	3.18627	0.115578
$761$	38.1915	1.38444	0.692220	−	0.721686i	$-0.256633\pi$
$761$	38.1915	1.38444	0.692220	+	0.721686i	$0.256633\pi$
$762$	−32.6269	−1.18195
$763$	8.45460	0.306077
$764$	18.3904	0.665342
$765$	−5.21205	−0.188442
$766$	5.19416	0.187673
$767$	−2.87718	−0.103889
$768$	−2.36999	−0.0855198
$769$	−25.9183	−0.934637	−0.467318	−	0.884089i	$-0.654780\pi$
$769$	−25.9183	−0.934637	−0.467318	+	0.884089i	$0.654780\pi$
$770$	−3.65076	−0.131564
$771$	−35.5497	−1.28029
$772$	−6.51055	−0.234320
$773$	−15.6243	−0.561966	−0.280983	−	0.959713i	$-0.590660\pi$
$773$	−15.6243	−0.561966	−0.280983	+	0.959713i	$0.590660\pi$
$774$	0.493723	0.0177465
$775$	17.0152	0.611204
$776$	7.32169	0.262833
$777$	−4.62866	−0.166052
$778$	37.6167	1.34862
$779$	3.23476	0.115897
$780$	−4.06438	−0.145528
$781$	−16.4407	−0.588295
$782$	3.82682	0.136847
$783$	−1.83269	−0.0654952
$784$	−4.45939	−0.159264
$785$	−17.5627	−0.626840
$786$	11.7508	0.419136
$787$	36.3232	1.29478	0.647392	−	0.762157i	$-0.275860\pi$
$787$	36.3232	1.29478	0.647392	+	0.762157i	$0.275860\pi$
$788$	13.5903	0.484133
$789$	−43.5732	−1.55125
$790$	11.8568	0.421847
$791$	−9.67946	−0.344162
$792$	−3.80810	−0.135315
$793$	−16.8103	−0.596950
$794$	−27.9068	−0.990376
$795$	17.3528	0.615441
$796$	−13.7842	−0.488568
$797$	11.7487	0.416160	0.208080	−	0.978112i	$-0.433279\pi$
$797$	11.7487	0.416160	0.208080	+	0.978112i	$0.433279\pi$
$798$	−7.64736	−0.270714
$799$	4.10420	0.145196
$800$	−2.52272	−0.0891918
$801$	−1.89004	−0.0667812
$802$	17.7271	0.625965
$803$	12.4749	0.440231
$804$	13.8064	0.486913
$805$	−7.58673	−0.267397
$806$	−7.34899	−0.258857
$807$	−31.0026	−1.09134
$808$	8.46910	0.297942
$809$	−32.4310	−1.14021	−0.570106	−	0.821571i	$-0.693098\pi$
$809$	−32.4310	−1.14021	−0.570106	+	0.821571i	$0.693098\pi$
$810$	−15.7435	−0.553169
$811$	−14.5762	−0.511840	−0.255920	−	0.966698i	$-0.582378\pi$
$811$	−14.5762	−0.511840	−0.255920	+	0.966698i	$0.582378\pi$
$812$	−3.21709	−0.112898
$813$	22.0036	0.771701
$814$	−1.78306	−0.0624963
$815$	−9.76265	−0.341971
$816$	2.99907	0.104988
$817$	0.381942	0.0133624
$818$	−9.57326	−0.334721
$819$	4.54476	0.158807
$820$	2.51498	0.0878268
$821$	1.87768	0.0655314	0.0327657	−	0.999463i	$-0.489568\pi$
$821$	1.87768	0.0655314	0.0327657	+	0.999463i	$0.489568\pi$
$822$	23.2630	0.811390
$823$	21.5499	0.751182	0.375591	−	0.926785i	$-0.377440\pi$
$823$	21.5499	0.751182	0.375591	+	0.926785i	$0.377440\pi$
$824$	12.6629	0.441134
$825$	−8.70050	−0.302913
$826$	−4.20897	−0.146449
$827$	−19.2726	−0.670175	−0.335088	−	0.942187i	$-0.608766\pi$
$827$	−19.2726	−0.670175	−0.335088	+	0.942187i	$0.608766\pi$
$828$	−7.91371	−0.275021
$829$	−23.9777	−0.832778	−0.416389	−	0.909186i	$-0.636705\pi$
$829$	−23.9777	−0.832778	−0.416389	+	0.909186i	$0.636705\pi$
$830$	−7.46524	−0.259122
$831$	41.5365	1.44089
$832$	1.08958	0.0377745
$833$	5.64306	0.195521
$834$	−3.01555	−0.104420
$835$	−36.3194	−1.25688
$836$	−2.94593	−0.101887
$837$	−6.12441	−0.211690
$838$	−15.7393	−0.543706
$839$	6.37702	0.220159	0.110080	−	0.993923i	$-0.464889\pi$
$839$	6.37702	0.220159	0.110080	+	0.993923i	$0.464889\pi$
$840$	−5.94570	−0.205146
$841$	−24.9263	−0.859528
$842$	−34.8963	−1.20261
$843$	66.5418	2.29182
$844$	−9.53853	−0.328330
$845$	−18.5926	−0.639605
$846$	−8.48733	−0.291800
$847$	−14.1578	−0.486469
$848$	−4.65195	−0.159749
$849$	38.8964	1.33492
$850$	3.19234	0.109496
$851$	−3.70542	−0.127020
$852$	−26.7757	−0.917319
$853$	18.7988	0.643659	0.321829	−	0.946798i	$-0.395702\pi$
$853$	18.7988	0.643659	0.321829	+	0.946798i	$0.395702\pi$
$854$	−24.5914	−0.841499
$855$	8.33805	0.285155
$856$	−9.20169	−0.314507
$857$	−1.74194	−0.0595035	−0.0297518	−	0.999557i	$-0.509472\pi$
$857$	−1.74194	−0.0595035	−0.0297518	+	0.999557i	$0.509472\pi$
$858$	3.75781	0.128289
$859$	47.8082	1.63120	0.815598	−	0.578619i	$-0.196408\pi$
$859$	47.8082	1.63120	0.815598	+	0.578619i	$0.196408\pi$
$860$	0.296954	0.0101260
$861$	−6.03619	−0.205713
$862$	−32.7475	−1.11539
$863$	−4.25556	−0.144861	−0.0724305	−	0.997373i	$-0.523076\pi$
$863$	−4.25556	−0.144861	−0.0724305	+	0.997373i	$0.523076\pi$
$864$	0.908022	0.0308915
$865$	−3.94523	−0.134142
$866$	36.1285	1.22770
$867$	36.4948	1.23943
$868$	−10.7507	−0.364902
$869$	−10.9625	−0.371876
$870$	7.52884	0.255251
$871$	−6.34735	−0.215072
$872$	5.30425	0.179625
$873$	19.1599	0.648464
$874$	−6.12201	−0.207080
$875$	−18.8726	−0.638010
$876$	20.3169	0.686446
$877$	20.0559	0.677240	0.338620	−	0.940923i	$-0.390040\pi$
$877$	20.0559	0.677240	0.338620	+	0.940923i	$0.390040\pi$
$878$	−1.93567	−0.0653258
$879$	21.7049	0.732088
$880$	−2.29042	−0.0772099
$881$	28.4515	0.958555	0.479278	−	0.877663i	$-0.340899\pi$
$881$	28.4515	0.958555	0.479278	+	0.877663i	$0.340899\pi$
$882$	−11.6696	−0.392937
$883$	35.8629	1.20688	0.603441	−	0.797408i	$-0.293796\pi$
$883$	35.8629	1.20688	0.603441	+	0.797408i	$0.293796\pi$
$884$	−1.37879	−0.0463739
$885$	9.85010	0.331108
$886$	11.9595	0.401787
$887$	33.4678	1.12374	0.561869	−	0.827226i	$-0.310082\pi$
$887$	33.4678	1.12374	0.561869	+	0.827226i	$0.310082\pi$
$888$	−2.90393	−0.0974496
$889$	21.9431	0.735948
$890$	−1.13678	−0.0381049
$891$	14.5559	0.487643
$892$	12.7731	0.427673
$893$	−6.56576	−0.219715
$894$	38.1696	1.27658
$895$	23.0622	0.770885
$896$	1.59393	0.0532494
$897$	7.80919	0.260741
$898$	−26.6005	−0.887670
$899$	13.6132	0.454027
$900$	−6.60164	−0.220055
$901$	5.88674	0.196116
$902$	−2.32527	−0.0774232
$903$	−0.712718	−0.0237178
$904$	−6.07270	−0.201975
$905$	27.1604	0.902842
$906$	−21.7800	−0.723592
$907$	20.4980	0.680623	0.340312	−	0.940313i	$-0.389467\pi$
$907$	20.4980	0.680623	0.340312	+	0.940313i	$0.389467\pi$
$908$	19.9844	0.663206
$909$	22.1625	0.735084
$910$	2.73348	0.0906141
$911$	47.6928	1.58013	0.790067	−	0.613021i	$-0.210046\pi$
$911$	47.6928	1.58013	0.790067	+	0.613021i	$0.210046\pi$
$912$	−4.79781	−0.158871
$913$	6.90214	0.228427
$914$	4.75305	0.157217
$915$	57.5504	1.90256
$916$	−11.6113	−0.383647
$917$	−7.90292	−0.260978
$918$	−1.14904	−0.0379240
$919$	−15.1264	−0.498975	−0.249487	−	0.968378i	$-0.580262\pi$
$919$	−15.1264	−0.498975	−0.249487	+	0.968378i	$0.580262\pi$
$920$	−4.75977	−0.156925
$921$	−35.1477	−1.15815
$922$	−4.08028	−0.134377
$923$	12.3099	0.405184
$924$	5.49722	0.180845
$925$	−3.09107	−0.101634
$926$	4.31122	0.141675
$927$	33.1372	1.08837
$928$	−2.01834	−0.0662552
$929$	−6.35714	−0.208571	−0.104285	−	0.994547i	$-0.533256\pi$
$929$	−6.35714	−0.208571	−0.104285	+	0.994547i	$0.533256\pi$
$930$	25.1595	0.825012
$931$	−9.02758	−0.295867
$932$	24.2822	0.795392
$933$	−60.8224	−1.99124
$934$	−32.5024	−1.06351
$935$	2.89837	0.0947868
$936$	2.85129	0.0931975
$937$	−32.8472	−1.07307	−0.536536	−	0.843877i	$-0.680267\pi$
$937$	−32.8472	−1.07307	−0.536536	+	0.843877i	$0.680267\pi$
$938$	−9.28541	−0.303179
$939$	−37.7940	−1.23336
$940$	−5.10477	−0.166499
$941$	41.6656	1.35826	0.679130	−	0.734018i	$-0.262357\pi$
$941$	41.6656	1.35826	0.679130	+	0.734018i	$0.262357\pi$
$942$	26.4455	0.861641
$943$	−4.83221	−0.157358
$944$	−2.64062	−0.0859450
$945$	2.27799	0.0741031
$946$	−0.274555	−0.00892654
$947$	48.8264	1.58665	0.793323	−	0.608801i	$-0.208349\pi$
$947$	48.8264	1.58665	0.793323	+	0.608801i	$0.208349\pi$
$948$	−17.8537	−0.579861
$949$	−9.34053	−0.303206
$950$	−5.10700	−0.165693
$951$	8.91655	0.289139
$952$	−2.01701	−0.0653717
$953$	−36.6982	−1.18877	−0.594386	−	0.804180i	$-0.702605\pi$
$953$	−36.6982	−1.18877	−0.594386	+	0.804180i	$0.702605\pi$
$954$	−12.1735	−0.394133
$955$	28.9453	0.936649
$956$	−19.6722	−0.636245
$957$	−6.96094	−0.225015
$958$	11.7102	0.378341
$959$	−15.6454	−0.505217
$960$	−3.73022	−0.120392
$961$	14.4920	0.467484
$962$	1.33506	0.0430439
$963$	−24.0796	−0.775954
$964$	8.85103	0.285072
$965$	−10.2472	−0.329869
$966$	11.4239	0.367558
$967$	24.8447	0.798952	0.399476	−	0.916744i	$-0.369192\pi$
$967$	24.8447	0.798952	0.399476	+	0.916744i	$0.369192\pi$
$968$	−8.88235	−0.285489
$969$	6.07130	0.195038
$970$	11.5239	0.370009
$971$	47.3468	1.51943	0.759715	−	0.650256i	$-0.225339\pi$
$971$	47.3468	1.51943	0.759715	+	0.650256i	$0.225339\pi$
$972$	20.9820	0.672999
$973$	2.02810	0.0650178
$974$	−26.8841	−0.861423
$975$	6.51444	0.208629
$976$	−15.4282	−0.493843
$977$	−56.3957	−1.80426	−0.902129	−	0.431467i	$-0.857996\pi$
$977$	−56.3957	−1.80426	−0.902129	+	0.431467i	$0.857996\pi$
$978$	14.7004	0.470066
$979$	1.05103	0.0335912
$980$	−7.01880	−0.224207
$981$	13.8805	0.443171
$982$	−20.7525	−0.662237
$983$	−11.0927	−0.353804	−0.176902	−	0.984228i	$-0.556608\pi$
$983$	−11.0927	−0.353804	−0.176902	+	0.984228i	$0.556608\pi$
$984$	−3.78699	−0.120725
$985$	21.3902	0.681549
$986$	2.55407	0.0813382
$987$	12.2519	0.389984
$988$	2.20575	0.0701742
$989$	−0.570559	−0.0181427
$990$	−5.99371	−0.190493
$991$	24.1386	0.766787	0.383393	−	0.923585i	$-0.374755\pi$
$991$	24.1386	0.766787	0.383393	+	0.923585i	$0.374755\pi$
$992$	−6.74478	−0.214147
$993$	36.5726	1.16060
$994$	18.0079	0.571175
$995$	−21.6954	−0.687791
$996$	11.2410	0.356184
$997$	21.7831	0.689878	0.344939	−	0.938625i	$-0.387900\pi$
$997$	21.7831	0.689878	0.344939	+	0.938625i	$0.387900\pi$
$998$	15.4767	0.489905
$999$	1.11259	0.0352008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.10	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.10	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.36999 q^{3} +1.00000 q^{4} +1.57394 q^{5} -2.36999 q^{6} +1.59393 q^{7} +1.00000 q^{8} +2.61687 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.36999 q^{3} +1.00000 q^{4} +1.57394 q^{5} -2.36999 q^{6} +1.59393 q^{7} +1.00000 q^{8} +2.61687 q^{9} +1.57394 q^{10} -1.45522 q^{11} -2.36999 q^{12} +1.08958 q^{13} +1.59393 q^{14} -3.73022 q^{15} +1.00000 q^{16} -1.26543 q^{17} +2.61687 q^{18} +2.02440 q^{19} +1.57394 q^{20} -3.77760 q^{21} -1.45522 q^{22} -3.02412 q^{23} -2.36999 q^{24} -2.52272 q^{25} +1.08958 q^{26} +0.908022 q^{27} +1.59393 q^{28} -2.01834 q^{29} -3.73022 q^{30} -6.74478 q^{31} +1.00000 q^{32} +3.44885 q^{33} -1.26543 q^{34} +2.50874 q^{35} +2.61687 q^{36} +1.22529 q^{37} +2.02440 q^{38} -2.58230 q^{39} +1.57394 q^{40} +1.59789 q^{41} -3.77760 q^{42} +0.188669 q^{43} -1.45522 q^{44} +4.11878 q^{45} -3.02412 q^{46} -3.24332 q^{47} -2.36999 q^{48} -4.45939 q^{49} -2.52272 q^{50} +2.99907 q^{51} +1.08958 q^{52} -4.65195 q^{53} +0.908022 q^{54} -2.29042 q^{55} +1.59393 q^{56} -4.79781 q^{57} -2.01834 q^{58} -2.64062 q^{59} -3.73022 q^{60} -15.4282 q^{61} -6.74478 q^{62} +4.17110 q^{63} +1.00000 q^{64} +1.71493 q^{65} +3.44885 q^{66} -5.82549 q^{67} -1.26543 q^{68} +7.16714 q^{69} +2.50874 q^{70} +11.2978 q^{71} +2.61687 q^{72} -8.57257 q^{73} +1.22529 q^{74} +5.97884 q^{75} +2.02440 q^{76} -2.31951 q^{77} -2.58230 q^{78} +7.53323 q^{79} +1.57394 q^{80} -10.0026 q^{81} +1.59789 q^{82} -4.74304 q^{83} -3.77760 q^{84} -1.99171 q^{85} +0.188669 q^{86} +4.78344 q^{87} -1.45522 q^{88} -0.722253 q^{89} +4.11878 q^{90} +1.73672 q^{91} -3.02412 q^{92} +15.9851 q^{93} -3.24332 q^{94} +3.18627 q^{95} -2.36999 q^{96} +7.32169 q^{97} -4.45939 q^{98} -3.80810 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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