LMFDB - Embedded newform 8043.2.a.q.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.810573 q^{2} +1.00000 q^{3} -1.34297 q^{4} +0.143356 q^{5} -0.810573 q^{6} -1.00000 q^{7} +2.70972 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.810573 q^{2} +1.00000 q^{3} -1.34297 q^{4} +0.143356 q^{5} -0.810573 q^{6} -1.00000 q^{7} +2.70972 q^{8} +1.00000 q^{9} -0.116200 q^{10} -0.888944 q^{11} -1.34297 q^{12} -0.424212 q^{13} +0.810573 q^{14} +0.143356 q^{15} +0.489512 q^{16} -0.886689 q^{17} -0.810573 q^{18} -8.58099 q^{19} -0.192522 q^{20} -1.00000 q^{21} +0.720554 q^{22} +4.66414 q^{23} +2.70972 q^{24} -4.97945 q^{25} +0.343855 q^{26} +1.00000 q^{27} +1.34297 q^{28} +7.67733 q^{29} -0.116200 q^{30} +5.71573 q^{31} -5.81623 q^{32} -0.888944 q^{33} +0.718726 q^{34} -0.143356 q^{35} -1.34297 q^{36} -3.86654 q^{37} +6.95552 q^{38} -0.424212 q^{39} +0.388454 q^{40} +3.14537 q^{41} +0.810573 q^{42} +4.97952 q^{43} +1.19383 q^{44} +0.143356 q^{45} -3.78063 q^{46} +8.47457 q^{47} +0.489512 q^{48} +1.00000 q^{49} +4.03621 q^{50} -0.886689 q^{51} +0.569704 q^{52} +0.270384 q^{53} -0.810573 q^{54} -0.127435 q^{55} -2.70972 q^{56} -8.58099 q^{57} -6.22304 q^{58} +11.7274 q^{59} -0.192522 q^{60} -14.4988 q^{61} -4.63302 q^{62} -1.00000 q^{63} +3.73546 q^{64} -0.0608131 q^{65} +0.720554 q^{66} +8.57772 q^{67} +1.19080 q^{68} +4.66414 q^{69} +0.116200 q^{70} -13.6933 q^{71} +2.70972 q^{72} +3.15486 q^{73} +3.13411 q^{74} -4.97945 q^{75} +11.5240 q^{76} +0.888944 q^{77} +0.343855 q^{78} -8.70639 q^{79} +0.0701742 q^{80} +1.00000 q^{81} -2.54955 q^{82} -15.0147 q^{83} +1.34297 q^{84} -0.127112 q^{85} -4.03626 q^{86} +7.67733 q^{87} -2.40879 q^{88} +4.00183 q^{89} -0.116200 q^{90} +0.424212 q^{91} -6.26380 q^{92} +5.71573 q^{93} -6.86926 q^{94} -1.23013 q^{95} -5.81623 q^{96} -8.01650 q^{97} -0.810573 q^{98} -0.888944 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9}+O(q^{10}) $ 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9	\( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9} - 16 q^{10} - 2 q^{11} + 44 q^{12} - 34 q^{13} + 4 q^{14} - 16 q^{15} + 24 q^{16} - 4 q^{17} - 4 q^{18} - 22 q^{19} - 39 q^{20} - 44 q^{21} - 23 q^{22} - 56 q^{23} - 15 q^{24} + 32 q^{25} - 17 q^{26} + 44 q^{27} - 44 q^{28} - 33 q^{29} - 16 q^{30} - 32 q^{31} - 34 q^{32} - 2 q^{33} - 25 q^{34} + 16 q^{35} + 44 q^{36} - 47 q^{37} - 40 q^{38} - 34 q^{39} - 50 q^{40} + 2 q^{41} + 4 q^{42} - 12 q^{43} - 22 q^{44} - 16 q^{45} + 8 q^{46} - 27 q^{47} + 24 q^{48} + 44 q^{49} - 21 q^{50} - 4 q^{51} - 82 q^{52} - 114 q^{53} - 4 q^{54} - 29 q^{55} + 15 q^{56} - 22 q^{57} - 26 q^{58} - 40 q^{59} - 39 q^{60} - 47 q^{61} - 37 q^{62} - 44 q^{63} - 5 q^{64} - 20 q^{65} - 23 q^{66} - 14 q^{67} - 72 q^{68} - 56 q^{69} + 16 q^{70} - 65 q^{71} - 15 q^{72} - 21 q^{73} - 26 q^{74} + 32 q^{75} - 15 q^{76} + 2 q^{77} - 17 q^{78} + 6 q^{79} - 77 q^{80} + 44 q^{81} - 51 q^{82} - 30 q^{83} - 44 q^{84} - 26 q^{85} - 65 q^{86} - 33 q^{87} - 84 q^{88} - 32 q^{89} - 16 q^{90} + 34 q^{91} - 140 q^{92} - 32 q^{93} - 35 q^{94} - 50 q^{95} - 34 q^{96} - 83 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9 - 16 * q^10 - 2 * q^11 + 44 * q^12 - 34 * q^13 + 4 * q^14 - 16 * q^15 + 24 * q^16 - 4 * q^17 - 4 * q^18 - 22 * q^19 - 39 * q^20 - 44 * q^21 - 23 * q^22 - 56 * q^23 - 15 * q^24 + 32 * q^25 - 17 * q^26 + 44 * q^27 - 44 * q^28 - 33 * q^29 - 16 * q^30 - 32 * q^31 - 34 * q^32 - 2 * q^33 - 25 * q^34 + 16 * q^35 + 44 * q^36 - 47 * q^37 - 40 * q^38 - 34 * q^39 - 50 * q^40 + 2 * q^41 + 4 * q^42 - 12 * q^43 - 22 * q^44 - 16 * q^45 + 8 * q^46 - 27 * q^47 + 24 * q^48 + 44 * q^49 - 21 * q^50 - 4 * q^51 - 82 * q^52 - 114 * q^53 - 4 * q^54 - 29 * q^55 + 15 * q^56 - 22 * q^57 - 26 * q^58 - 40 * q^59 - 39 * q^60 - 47 * q^61 - 37 * q^62 - 44 * q^63 - 5 * q^64 - 20 * q^65 - 23 * q^66 - 14 * q^67 - 72 * q^68 - 56 * q^69 + 16 * q^70 - 65 * q^71 - 15 * q^72 - 21 * q^73 - 26 * q^74 + 32 * q^75 - 15 * q^76 + 2 * q^77 - 17 * q^78 + 6 * q^79 - 77 * q^80 + 44 * q^81 - 51 * q^82 - 30 * q^83 - 44 * q^84 - 26 * q^85 - 65 * q^86 - 33 * q^87 - 84 * q^88 - 32 * q^89 - 16 * q^90 + 34 * q^91 - 140 * q^92 - 32 * q^93 - 35 * q^94 - 50 * q^95 - 34 * q^96 - 83 * q^97 - 4 * q^98 - 2 * 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 $

$2$	−0.810573	−0.573162	−0.286581	−	0.958056i	$-0.592519\pi$
$2$	−0.810573	−0.573162	−0.286581	+	0.958056i	$0.592519\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.34297	−0.671485
$5$	0.143356	0.0641105	0.0320553	−	0.999486i	$-0.489795\pi$
$5$	0.143356	0.0641105	0.0320553	+	0.999486i	$0.489795\pi$
$6$	−0.810573	−0.330915
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.70972	0.958032
$9$	1.00000	0.333333
$10$	−0.116200	−0.0367457
$11$	−0.888944	−0.268027	−0.134013	−	0.990980i	$-0.542787\pi$
$11$	−0.888944	−0.268027	−0.134013	+	0.990980i	$0.542787\pi$
$12$	−1.34297	−0.387682
$13$	−0.424212	−0.117655	−0.0588276	−	0.998268i	$-0.518736\pi$
$13$	−0.424212	−0.117655	−0.0588276	+	0.998268i	$0.518736\pi$
$14$	0.810573	0.216635
$15$	0.143356	0.0370142
$16$	0.489512	0.122378
$17$	−0.886689	−0.215054	−0.107527	−	0.994202i	$-0.534293\pi$
$17$	−0.886689	−0.215054	−0.107527	+	0.994202i	$0.534293\pi$
$18$	−0.810573	−0.191054
$19$	−8.58099	−1.96861	−0.984306	−	0.176467i	$-0.943533\pi$
$19$	−8.58099	−1.96861	−0.984306	+	0.176467i	$0.943533\pi$
$20$	−0.192522	−0.0430493
$21$	−1.00000	−0.218218
$22$	0.720554	0.153623
$23$	4.66414	0.972540	0.486270	−	0.873809i	$-0.338357\pi$
$23$	4.66414	0.972540	0.486270	+	0.873809i	$0.338357\pi$
$24$	2.70972	0.553120
$25$	−4.97945	−0.995890
$26$	0.343855	0.0674355
$27$	1.00000	0.192450
$28$	1.34297	0.253798
$29$	7.67733	1.42564	0.712822	−	0.701345i	$-0.247417\pi$
$29$	7.67733	1.42564	0.712822	+	0.701345i	$0.247417\pi$
$30$	−0.116200	−0.0212152
$31$	5.71573	1.02658	0.513288	−	0.858217i	$-0.328427\pi$
$31$	5.71573	1.02658	0.513288	+	0.858217i	$0.328427\pi$
$32$	−5.81623	−1.02817
$33$	−0.888944	−0.154745
$34$	0.718726	0.123261
$35$	−0.143356	−0.0242315
$36$	−1.34297	−0.223828
$37$	−3.86654	−0.635655	−0.317827	−	0.948149i	$-0.602953\pi$
$37$	−3.86654	−0.635655	−0.317827	+	0.948149i	$0.602953\pi$
$38$	6.95552	1.12833
$39$	−0.424212	−0.0679283
$40$	0.388454	0.0614199
$41$	3.14537	0.491224	0.245612	−	0.969368i	$-0.421011\pi$
$41$	3.14537	0.491224	0.245612	+	0.969368i	$0.421011\pi$
$42$	0.810573	0.125074
$43$	4.97952	0.759369	0.379685	−	0.925116i	$-0.376033\pi$
$43$	4.97952	0.759369	0.379685	+	0.925116i	$0.376033\pi$
$44$	1.19383	0.179976
$45$	0.143356	0.0213702
$46$	−3.78063	−0.557423
$47$	8.47457	1.23614	0.618071	−	0.786122i	$-0.287914\pi$
$47$	8.47457	1.23614	0.618071	+	0.786122i	$0.287914\pi$
$48$	0.489512	0.0706550
$49$	1.00000	0.142857
$50$	4.03621	0.570806
$51$	−0.886689	−0.124161
$52$	0.569704	0.0790038
$53$	0.270384	0.0371401	0.0185701	−	0.999828i	$-0.494089\pi$
$53$	0.270384	0.0371401	0.0185701	+	0.999828i	$0.494089\pi$
$54$	−0.810573	−0.110305
$55$	−0.127435	−0.0171833
$56$	−2.70972	−0.362102
$57$	−8.58099	−1.13658
$58$	−6.22304	−0.817125
$59$	11.7274	1.52678	0.763388	−	0.645940i	$-0.223534\pi$
$59$	11.7274	1.52678	0.763388	+	0.645940i	$0.223534\pi$
$60$	−0.192522	−0.0248545
$61$	−14.4988	−1.85638	−0.928188	−	0.372113i	$-0.878633\pi$
$61$	−14.4988	−1.85638	−0.928188	+	0.372113i	$0.878633\pi$
$62$	−4.63302	−0.588394
$63$	−1.00000	−0.125988
$64$	3.73546	0.466932
$65$	−0.0608131	−0.00754294
$66$	0.720554	0.0886941
$67$	8.57772	1.04794	0.523968	−	0.851738i	$-0.324451\pi$
$67$	8.57772	1.04794	0.523968	+	0.851738i	$0.324451\pi$
$68$	1.19080	0.144405
$69$	4.66414	0.561496
$70$	0.116200	0.0138886
$71$	−13.6933	−1.62509	−0.812546	−	0.582897i	$-0.801919\pi$
$71$	−13.6933	−1.62509	−0.812546	+	0.582897i	$0.801919\pi$
$72$	2.70972	0.319344
$73$	3.15486	0.369248	0.184624	−	0.982809i	$-0.440893\pi$
$73$	3.15486	0.369248	0.184624	+	0.982809i	$0.440893\pi$
$74$	3.13411	0.364333
$75$	−4.97945	−0.574977
$76$	11.5240	1.32189
$77$	0.888944	0.101305
$78$	0.343855	0.0389339
$79$	−8.70639	−0.979546	−0.489773	−	0.871850i	$-0.662920\pi$
$79$	−8.70639	−0.979546	−0.489773	+	0.871850i	$0.662920\pi$
$80$	0.0701742	0.00784572
$81$	1.00000	0.111111
$82$	−2.54955	−0.281551
$83$	−15.0147	−1.64808	−0.824038	−	0.566535i	$-0.808284\pi$
$83$	−15.0147	−1.64808	−0.824038	+	0.566535i	$0.808284\pi$
$84$	1.34297	0.146530
$85$	−0.127112	−0.0137872
$86$	−4.03626	−0.435242
$87$	7.67733	0.823096
$88$	−2.40879	−0.256778
$89$	4.00183	0.424194	0.212097	−	0.977249i	$-0.431971\pi$
$89$	4.00183	0.424194	0.212097	+	0.977249i	$0.431971\pi$
$90$	−0.116200	−0.0122486
$91$	0.424212	0.0444695
$92$	−6.26380	−0.653047
$93$	5.71573	0.592694
$94$	−6.86926	−0.708510
$95$	−1.23013	−0.126209
$96$	−5.81623	−0.593617
$97$	−8.01650	−0.813952	−0.406976	−	0.913439i	$-0.633417\pi$
$97$	−8.01650	−0.813952	−0.406976	+	0.913439i	$0.633417\pi$
$98$	−0.810573	−0.0818803
$99$	−0.888944	−0.0893422
$100$	6.68725	0.668725
$101$	5.29355	0.526728	0.263364	−	0.964697i	$-0.415168\pi$
$101$	5.29355	0.526728	0.263364	+	0.964697i	$0.415168\pi$
$102$	0.718726	0.0711645
$103$	−14.0558	−1.38495	−0.692477	−	0.721440i	$-0.743481\pi$
$103$	−14.0558	−1.38495	−0.692477	+	0.721440i	$0.743481\pi$
$104$	−1.14950	−0.112717
$105$	−0.143356	−0.0139901
$106$	−0.219166	−0.0212873
$107$	2.47676	0.239437	0.119719	−	0.992808i	$-0.461801\pi$
$107$	2.47676	0.239437	0.119719	+	0.992808i	$0.461801\pi$
$108$	−1.34297	−0.129227
$109$	4.33520	0.415237	0.207618	−	0.978210i	$-0.433429\pi$
$109$	4.33520	0.415237	0.207618	+	0.978210i	$0.433429\pi$
$110$	0.103295	0.00984883
$111$	−3.86654	−0.366995
$112$	−0.489512	−0.0462545
$113$	−20.5499	−1.93318	−0.966588	−	0.256334i	$-0.917485\pi$
$113$	−20.5499	−1.93318	−0.966588	+	0.256334i	$0.917485\pi$
$114$	6.95552	0.651444
$115$	0.668630	0.0623501
$116$	−10.3104	−0.957299
$117$	−0.424212	−0.0392184
$118$	−9.50591	−0.875090
$119$	0.886689	0.0812826
$120$	0.388454	0.0354608
$121$	−10.2098	−0.928162
$122$	11.7523	1.06400
$123$	3.14537	0.283609
$124$	−7.67606	−0.689330
$125$	−1.43061	−0.127958
$126$	0.810573	0.0722116
$127$	8.19208	0.726930	0.363465	−	0.931608i	$-0.381594\pi$
$127$	8.19208	0.726930	0.363465	+	0.931608i	$0.381594\pi$
$128$	8.60460	0.760546
$129$	4.97952	0.438422
$130$	0.0492935	0.00432333
$131$	3.78047	0.330301	0.165150	−	0.986268i	$-0.447189\pi$
$131$	3.78047	0.330301	0.165150	+	0.986268i	$0.447189\pi$
$132$	1.19383	0.103909
$133$	8.58099	0.744066
$134$	−6.95287	−0.600637
$135$	0.143356	0.0123381
$136$	−2.40268	−0.206028
$137$	4.90427	0.419000	0.209500	−	0.977809i	$-0.432816\pi$
$137$	4.90427	0.419000	0.209500	+	0.977809i	$0.432816\pi$
$138$	−3.78063	−0.321828
$139$	−13.0960	−1.11079	−0.555396	−	0.831586i	$-0.687433\pi$
$139$	−13.0960	−1.11079	−0.555396	+	0.831586i	$0.687433\pi$
$140$	0.192522	0.0162711
$141$	8.47457	0.713687
$142$	11.0994	0.931441
$143$	0.377101	0.0315347
$144$	0.489512	0.0407927
$145$	1.10059	0.0913988
$146$	−2.55725	−0.211639
$147$	1.00000	0.0824786
$148$	5.19264	0.426833
$149$	6.70758	0.549506	0.274753	−	0.961515i	$-0.411404\pi$
$149$	6.70758	0.549506	0.274753	+	0.961515i	$0.411404\pi$
$150$	4.03621	0.329555
$151$	−7.41638	−0.603537	−0.301768	−	0.953381i	$-0.597577\pi$
$151$	−7.41638	−0.603537	−0.301768	+	0.953381i	$0.597577\pi$
$152$	−23.2521	−1.88599
$153$	−0.886689	−0.0716845
$154$	−0.720554	−0.0580639
$155$	0.819381	0.0658143
$156$	0.569704	0.0456128
$157$	24.8568	1.98379	0.991894	−	0.127069i	$-0.0405571\pi$
$157$	24.8568	1.98379	0.991894	+	0.127069i	$0.0405571\pi$
$158$	7.05717	0.561438
$159$	0.270384	0.0214429
$160$	−0.833789	−0.0659168
$161$	−4.66414	−0.367586
$162$	−0.810573	−0.0636847
$163$	−14.8777	−1.16531	−0.582654	−	0.812720i	$-0.697986\pi$
$163$	−14.8777	−1.16531	−0.582654	+	0.812720i	$0.697986\pi$
$164$	−4.22414	−0.329850
$165$	−0.127435	−0.00992080
$166$	12.1705	0.944614
$167$	−13.0226	−1.00772	−0.503860	−	0.863785i	$-0.668087\pi$
$167$	−13.0226	−1.00772	−0.503860	+	0.863785i	$0.668087\pi$
$168$	−2.70972	−0.209060
$169$	−12.8200	−0.986157
$170$	0.103033	0.00790230
$171$	−8.58099	−0.656204
$172$	−6.68735	−0.509905
$173$	−6.52257	−0.495901	−0.247951	−	0.968773i	$-0.579757\pi$
$173$	−6.52257	−0.495901	−0.247951	+	0.968773i	$0.579757\pi$
$174$	−6.22304	−0.471767
$175$	4.97945	0.376411
$176$	−0.435149	−0.0328006
$177$	11.7274	0.881485
$178$	−3.24378	−0.243132
$179$	−12.0548	−0.901017	−0.450508	−	0.892772i	$-0.648757\pi$
$179$	−12.0548	−0.901017	−0.450508	+	0.892772i	$0.648757\pi$
$180$	−0.192522	−0.0143498
$181$	8.92294	0.663237	0.331618	−	0.943414i	$-0.392405\pi$
$181$	8.92294	0.663237	0.331618	+	0.943414i	$0.392405\pi$
$182$	−0.343855	−0.0254882
$183$	−14.4988	−1.07178
$184$	12.6385	0.931725
$185$	−0.554289	−0.0407522
$186$	−4.63302	−0.339709
$187$	0.788217	0.0576401
$188$	−11.3811	−0.830052
$189$	−1.00000	−0.0727393
$190$	0.997112	0.0723381
$191$	15.6690	1.13377	0.566883	−	0.823798i	$-0.308149\pi$
$191$	15.6690	1.13377	0.566883	+	0.823798i	$0.308149\pi$
$192$	3.73546	0.269584
$193$	17.1874	1.23718	0.618589	−	0.785715i	$-0.287705\pi$
$193$	17.1874	1.23718	0.618589	+	0.785715i	$0.287705\pi$
$194$	6.49796	0.466526
$195$	−0.0608131	−0.00435492
$196$	−1.34297	−0.0959265
$197$	18.3736	1.30906	0.654531	−	0.756035i	$-0.272866\pi$
$197$	18.3736	1.30906	0.654531	+	0.756035i	$0.272866\pi$
$198$	0.720554	0.0512076
$199$	−7.48545	−0.530629	−0.265315	−	0.964162i	$-0.585476\pi$
$199$	−7.48545	−0.530629	−0.265315	+	0.964162i	$0.585476\pi$
$200$	−13.4929	−0.954094
$201$	8.57772	0.605026
$202$	−4.29081	−0.301900
$203$	−7.67733	−0.538843
$204$	1.19080	0.0833725
$205$	0.450906	0.0314927
$206$	11.3932	0.793803
$207$	4.66414	0.324180
$208$	−0.207657	−0.0143984
$209$	7.62802	0.527641
$210$	0.116200	0.00801857
$211$	2.73926	0.188578	0.0942891	−	0.995545i	$-0.469942\pi$
$211$	2.73926	0.188578	0.0942891	+	0.995545i	$0.469942\pi$
$212$	−0.363118	−0.0249391
$213$	−13.6933	−0.938247
$214$	−2.00759	−0.137236
$215$	0.713841	0.0486836
$216$	2.70972	0.184373
$217$	−5.71573	−0.388009
$218$	−3.51400	−0.237998
$219$	3.15486	0.213186
$220$	0.171142	0.0115384
$221$	0.376144	0.0253022
$222$	3.13411	0.210348
$223$	3.49272	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$223$	3.49272	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$224$	5.81623	0.388613
$225$	−4.97945	−0.331963
$226$	16.6572	1.10802
$227$	1.57627	0.104621	0.0523103	−	0.998631i	$-0.483342\pi$
$227$	1.57627	0.104621	0.0523103	+	0.998631i	$0.483342\pi$
$228$	11.5240	0.763196
$229$	9.96316	0.658384	0.329192	−	0.944263i	$-0.393224\pi$
$229$	9.96316	0.658384	0.329192	+	0.944263i	$0.393224\pi$
$230$	−0.541974	−0.0357367
$231$	0.888944	0.0584882
$232$	20.8034	1.36581
$233$	6.59409	0.431993	0.215997	−	0.976394i	$-0.430700\pi$
$233$	6.59409	0.431993	0.215997	+	0.976394i	$0.430700\pi$
$234$	0.343855	0.0224785
$235$	1.21488	0.0792498
$236$	−15.7495	−1.02521
$237$	−8.70639	−0.565541
$238$	−0.718726	−0.0465881
$239$	7.62877	0.493464	0.246732	−	0.969084i	$-0.420643\pi$
$239$	7.62877	0.493464	0.246732	+	0.969084i	$0.420643\pi$
$240$	0.0701742	0.00452973
$241$	−15.6026	−1.00505	−0.502527	−	0.864562i	$-0.667596\pi$
$241$	−15.6026	−1.00505	−0.502527	+	0.864562i	$0.667596\pi$
$242$	8.27577	0.531987
$243$	1.00000	0.0641500
$244$	19.4714	1.24653
$245$	0.143356	0.00915865
$246$	−2.54955	−0.162554
$247$	3.64016	0.231618
$248$	15.4880	0.983492
$249$	−15.0147	−0.951517
$250$	1.15961	0.0733404
$251$	0.775745	0.0489646	0.0244823	−	0.999700i	$-0.492206\pi$
$251$	0.775745	0.0489646	0.0244823	+	0.999700i	$0.492206\pi$
$252$	1.34297	0.0845992
$253$	−4.14616	−0.260667
$254$	−6.64029	−0.416649
$255$	−0.127112	−0.00796004
$256$	−14.4456	−0.902849
$257$	−25.4902	−1.59003	−0.795016	−	0.606588i	$-0.792538\pi$
$257$	−25.4902	−1.59003	−0.795016	+	0.606588i	$0.792538\pi$
$258$	−4.03626	−0.251287
$259$	3.86654	0.240255
$260$	0.0816703	0.00506497
$261$	7.67733	0.475215
$262$	−3.06435	−0.189316
$263$	−7.06795	−0.435828	−0.217914	−	0.975968i	$-0.569925\pi$
$263$	−7.06795	−0.435828	−0.217914	+	0.975968i	$0.569925\pi$
$264$	−2.40879	−0.148251
$265$	0.0387611	0.00238107
$266$	−6.95552	−0.426470
$267$	4.00183	0.244908
$268$	−11.5196	−0.703674
$269$	4.05657	0.247333	0.123667	−	0.992324i	$-0.460535\pi$
$269$	4.05657	0.247333	0.123667	+	0.992324i	$0.460535\pi$
$270$	−0.116200	−0.00707172
$271$	−27.0355	−1.64229	−0.821145	−	0.570719i	$-0.806665\pi$
$271$	−27.0355	−1.64229	−0.821145	+	0.570719i	$0.806665\pi$
$272$	−0.434045	−0.0263178
$273$	0.424212	0.0256745
$274$	−3.97527	−0.240155
$275$	4.42645	0.266925
$276$	−6.26380	−0.377037
$277$	26.7417	1.60675	0.803376	−	0.595472i	$-0.203035\pi$
$277$	26.7417	1.60675	0.803376	+	0.595472i	$0.203035\pi$
$278$	10.6153	0.636663
$279$	5.71573	0.342192
$280$	−0.388454	−0.0232146
$281$	16.2635	0.970200	0.485100	−	0.874459i	$-0.338783\pi$
$281$	16.2635	0.970200	0.485100	+	0.874459i	$0.338783\pi$
$282$	−6.86926	−0.409058
$283$	−5.39124	−0.320476	−0.160238	−	0.987078i	$-0.551226\pi$
$283$	−5.39124	−0.320476	−0.160238	+	0.987078i	$0.551226\pi$
$284$	18.3897	1.09123
$285$	−1.23013	−0.0728667
$286$	−0.305668	−0.0180745
$287$	−3.14537	−0.185665
$288$	−5.81623	−0.342725
$289$	−16.2138	−0.953752
$290$	−0.892107	−0.0523863
$291$	−8.01650	−0.469935
$292$	−4.23688	−0.247945
$293$	−24.1437	−1.41049	−0.705246	−	0.708963i	$-0.749163\pi$
$293$	−24.1437	−1.41049	−0.705246	+	0.708963i	$0.749163\pi$
$294$	−0.810573	−0.0472736
$295$	1.68119	0.0978825
$296$	−10.4772	−0.608977
$297$	−0.888944	−0.0515818
$298$	−5.43698	−0.314956
$299$	−1.97858	−0.114424
$300$	6.68725	0.386089
$301$	−4.97952	−0.287015
$302$	6.01152	0.345924
$303$	5.29355	0.304106
$304$	−4.20049	−0.240915
$305$	−2.07848	−0.119013
$306$	0.718726	0.0410869
$307$	1.44454	0.0824444	0.0412222	−	0.999150i	$-0.486875\pi$
$307$	1.44454	0.0824444	0.0412222	+	0.999150i	$0.486875\pi$
$308$	−1.19383	−0.0680245
$309$	−14.0558	−0.799604
$310$	−0.664169	−0.0377222
$311$	4.31443	0.244649	0.122325	−	0.992490i	$-0.460965\pi$
$311$	4.31443	0.244649	0.122325	+	0.992490i	$0.460965\pi$
$312$	−1.14950	−0.0650775
$313$	−29.8052	−1.68469	−0.842346	−	0.538937i	$-0.818826\pi$
$313$	−29.8052	−1.68469	−0.842346	+	0.538937i	$0.818826\pi$
$314$	−20.1483	−1.13703
$315$	−0.143356	−0.00807717
$316$	11.6924	0.657751
$317$	−21.6497	−1.21597	−0.607985	−	0.793949i	$-0.708022\pi$
$317$	−21.6497	−1.21597	−0.607985	+	0.793949i	$0.708022\pi$
$318$	−0.219166	−0.0122902
$319$	−6.82471	−0.382111
$320$	0.535499	0.0299353
$321$	2.47676	0.138239
$322$	3.78063	0.210686
$323$	7.60866	0.423357
$324$	−1.34297	−0.0746095
$325$	2.11234	0.117172
$326$	12.0594	0.667911
$327$	4.33520	0.239737
$328$	8.52308	0.470609
$329$	−8.47457	−0.467218
$330$	0.103295	0.00568623
$331$	7.58968	0.417166	0.208583	−	0.978005i	$-0.433115\pi$
$331$	7.58968	0.417166	0.208583	+	0.978005i	$0.433115\pi$
$332$	20.1643	1.10666
$333$	−3.86654	−0.211885
$334$	10.5558	0.577587
$335$	1.22966	0.0671837
$336$	−0.489512	−0.0267051
$337$	−1.28609	−0.0700576	−0.0350288	−	0.999386i	$-0.511152\pi$
$337$	−1.28609	−0.0700576	−0.0350288	+	0.999386i	$0.511152\pi$
$338$	10.3916	0.565228
$339$	−20.5499	−1.11612
$340$	0.170707	0.00925790
$341$	−5.08096	−0.275150
$342$	6.95552	0.376111
$343$	−1.00000	−0.0539949
$344$	13.4931	0.727500
$345$	0.668630	0.0359978
$346$	5.28702	0.284232
$347$	2.93230	0.157414	0.0787071	−	0.996898i	$-0.474921\pi$
$347$	2.93230	0.157414	0.0787071	+	0.996898i	$0.474921\pi$
$348$	−10.3104	−0.552697
$349$	−10.9454	−0.585894	−0.292947	−	0.956129i	$-0.594636\pi$
$349$	−10.9454	−0.585894	−0.292947	+	0.956129i	$0.594636\pi$
$350$	−4.03621	−0.215744
$351$	−0.424212	−0.0226428
$352$	5.17030	0.275578
$353$	−26.0310	−1.38549	−0.692744	−	0.721184i	$-0.743598\pi$
$353$	−26.0310	−1.38549	−0.692744	+	0.721184i	$0.743598\pi$
$354$	−9.50591	−0.505234
$355$	−1.96300	−0.104185
$356$	−5.37435	−0.284840
$357$	0.886689	0.0469285
$358$	9.77129	0.516429
$359$	−14.4552	−0.762915	−0.381457	−	0.924386i	$-0.624578\pi$
$359$	−14.4552	−0.762915	−0.381457	+	0.924386i	$0.624578\pi$
$360$	0.388454	0.0204733
$361$	54.6333	2.87544
$362$	−7.23269	−0.380142
$363$	−10.2098	−0.535874
$364$	−0.569704	−0.0298606
$365$	0.452266	0.0236727
$366$	11.7523	0.614303
$367$	−14.1617	−0.739233	−0.369617	−	0.929184i	$-0.620511\pi$
$367$	−14.1617	−0.739233	−0.369617	+	0.929184i	$0.620511\pi$
$368$	2.28315	0.119018
$369$	3.14537	0.163741
$370$	0.449292	0.0233576
$371$	−0.270384	−0.0140377
$372$	−7.67606	−0.397985
$373$	−8.06187	−0.417428	−0.208714	−	0.977977i	$-0.566928\pi$
$373$	−8.06187	−0.417428	−0.208714	+	0.977977i	$0.566928\pi$
$374$	−0.638908	−0.0330371
$375$	−1.43061	−0.0738763
$376$	22.9637	1.18426
$377$	−3.25681	−0.167734
$378$	0.810573	0.0416914
$379$	−36.3610	−1.86774	−0.933869	−	0.357615i	$-0.883590\pi$
$379$	−36.3610	−1.86774	−0.933869	+	0.357615i	$0.883590\pi$
$380$	1.65203	0.0847474
$381$	8.19208	0.419693
$382$	−12.7008	−0.649832
$383$	1.00000	0.0510976
$383$	1.00000	0.0510976
$384$	8.60460	0.439102
$385$	0.127435	0.00649469
$386$	−13.9317	−0.709104
$387$	4.97952	0.253123
$388$	10.7659	0.546557
$389$	−30.0008	−1.52110	−0.760550	−	0.649279i	$-0.775071\pi$
$389$	−30.0008	−1.52110	−0.760550	+	0.649279i	$0.775071\pi$
$390$	0.0492935	0.00249607
$391$	−4.13564	−0.209148
$392$	2.70972	0.136862
$393$	3.78047	0.190699
$394$	−14.8931	−0.750305
$395$	−1.24811	−0.0627992
$396$	1.19383	0.0599920
$397$	34.9611	1.75465	0.877325	−	0.479897i	$-0.159326\pi$
$397$	34.9611	1.75465	0.877325	+	0.479897i	$0.159326\pi$
$398$	6.06750	0.304137
$399$	8.58099	0.429587
$400$	−2.43750	−0.121875
$401$	−15.3380	−0.765945	−0.382972	−	0.923760i	$-0.625100\pi$
$401$	−15.3380	−0.765945	−0.382972	+	0.923760i	$0.625100\pi$
$402$	−6.95287	−0.346778
$403$	−2.42468	−0.120782
$404$	−7.10908	−0.353690
$405$	0.143356	0.00712339
$406$	6.22304	0.308844
$407$	3.43713	0.170372
$408$	−2.40268	−0.118950
$409$	−26.4333	−1.30704	−0.653522	−	0.756907i	$-0.726709\pi$
$409$	−26.4333	−1.30704	−0.653522	+	0.756907i	$0.726709\pi$
$410$	−0.365493	−0.0180504
$411$	4.90427	0.241910
$412$	18.8765	0.929977
$413$	−11.7274	−0.577067
$414$	−3.78063	−0.185808
$415$	−2.15244	−0.105659
$416$	2.46732	0.120970
$417$	−13.0960	−0.641316
$418$	−6.18307	−0.302424
$419$	21.0401	1.02788	0.513939	−	0.857827i	$-0.328186\pi$
$419$	21.0401	1.02788	0.513939	+	0.857827i	$0.328186\pi$
$420$	0.192522	0.00939412
$421$	−34.3988	−1.67650	−0.838248	−	0.545289i	$-0.816420\pi$
$421$	−34.3988	−1.67650	−0.838248	+	0.545289i	$0.816420\pi$
$422$	−2.22037	−0.108086
$423$	8.47457	0.412048
$424$	0.732667	0.0355814
$425$	4.41522	0.214170
$426$	11.0994	0.537767
$427$	14.4988	0.701644
$428$	−3.32621	−0.160779
$429$	0.377101	0.0182066
$430$	−0.578621	−0.0279036
$431$	0.702192	0.0338234	0.0169117	−	0.999857i	$-0.494617\pi$
$431$	0.702192	0.0338234	0.0169117	+	0.999857i	$0.494617\pi$
$432$	0.489512	0.0235517
$433$	−35.8069	−1.72077	−0.860385	−	0.509644i	$-0.829777\pi$
$433$	−35.8069	−1.72077	−0.860385	+	0.509644i	$0.829777\pi$
$434$	4.63302	0.222392
$435$	1.10059	0.0527691
$436$	−5.82204	−0.278825
$437$	−40.0229	−1.91456
$438$	−2.55725	−0.122190
$439$	19.3752	0.924729	0.462365	−	0.886690i	$-0.347001\pi$
$439$	19.3752	0.924729	0.462365	+	0.886690i	$0.347001\pi$
$440$	−0.345314	−0.0164622
$441$	1.00000	0.0476190
$442$	−0.304892	−0.0145023
$443$	−34.1503	−1.62253	−0.811264	−	0.584680i	$-0.801220\pi$
$443$	−34.1503	−1.62253	−0.811264	+	0.584680i	$0.801220\pi$
$444$	5.19264	0.246432
$445$	0.573685	0.0271953
$446$	−2.83111	−0.134057
$447$	6.70758	0.317257
$448$	−3.73546	−0.176484
$449$	−32.5305	−1.53521	−0.767604	−	0.640925i	$-0.778551\pi$
$449$	−32.5305	−1.53521	−0.767604	+	0.640925i	$0.778551\pi$
$450$	4.03621	0.190269
$451$	−2.79606	−0.131661
$452$	27.5980	1.29810
$453$	−7.41638	−0.348452
$454$	−1.27768	−0.0599646
$455$	0.0608131	0.00285096
$456$	−23.2521	−1.08888
$457$	24.8959	1.16458	0.582292	−	0.812980i	$-0.302156\pi$
$457$	24.8959	1.16458	0.582292	+	0.812980i	$0.302156\pi$
$458$	−8.07587	−0.377361
$459$	−0.886689	−0.0413871
$460$	−0.897951	−0.0418672
$461$	−17.9370	−0.835412	−0.417706	−	0.908582i	$-0.637166\pi$
$461$	−17.9370	−0.835412	−0.417706	+	0.908582i	$0.637166\pi$
$462$	−0.720554	−0.0335232
$463$	1.82916	0.0850084	0.0425042	−	0.999096i	$-0.486466\pi$
$463$	1.82916	0.0850084	0.0425042	+	0.999096i	$0.486466\pi$
$464$	3.75814	0.174467
$465$	0.819381	0.0379979
$466$	−5.34500	−0.247602
$467$	−38.7435	−1.79284	−0.896418	−	0.443209i	$-0.853840\pi$
$467$	−38.7435	−1.79284	−0.896418	+	0.443209i	$0.853840\pi$
$468$	0.569704	0.0263346
$469$	−8.57772	−0.396082
$470$	−0.984746	−0.0454230
$471$	24.8568	1.14534
$472$	31.7780	1.46270
$473$	−4.42651	−0.203531
$474$	7.05717	0.324147
$475$	42.7286	1.96052
$476$	−1.19080	−0.0545801
$477$	0.270384	0.0123800
$478$	−6.18368	−0.282835
$479$	−14.7469	−0.673804	−0.336902	−	0.941540i	$-0.609379\pi$
$479$	−14.7469	−0.673804	−0.336902	+	0.941540i	$0.609379\pi$
$480$	−0.833789	−0.0380571
$481$	1.64023	0.0747881
$482$	12.6471	0.576059
$483$	−4.66414	−0.212226
$484$	13.7114	0.623247
$485$	−1.14921	−0.0521829
$486$	−0.810573	−0.0367684
$487$	−14.2882	−0.647459	−0.323730	−	0.946150i	$-0.604937\pi$
$487$	−14.2882	−0.647459	−0.323730	+	0.946150i	$0.604937\pi$
$488$	−39.2876	−1.77847
$489$	−14.8777	−0.672791
$490$	−0.116200	−0.00524939
$491$	14.6110	0.659384	0.329692	−	0.944089i	$-0.393055\pi$
$491$	14.6110	0.659384	0.329692	+	0.944089i	$0.393055\pi$
$492$	−4.22414	−0.190439
$493$	−6.80740	−0.306590
$494$	−2.95061	−0.132754
$495$	−0.127435	−0.00572778
$496$	2.79792	0.125630
$497$	13.6933	0.614227
$498$	12.1705	0.545373
$499$	−17.9092	−0.801728	−0.400864	−	0.916138i	$-0.631290\pi$
$499$	−17.9092	−0.801728	−0.400864	+	0.916138i	$0.631290\pi$
$500$	1.92127	0.0859216
$501$	−13.0226	−0.581807
$502$	−0.628798	−0.0280646
$503$	20.9433	0.933815	0.466907	−	0.884306i	$-0.345368\pi$
$503$	20.9433	0.933815	0.466907	+	0.884306i	$0.345368\pi$
$504$	−2.70972	−0.120701
$505$	0.758859	0.0337688
$506$	3.36077	0.149404
$507$	−12.8200	−0.569358
$508$	−11.0017	−0.488123
$509$	−5.83668	−0.258706	−0.129353	−	0.991599i	$-0.541290\pi$
$509$	−5.83668	−0.258706	−0.129353	+	0.991599i	$0.541290\pi$
$510$	0.103033	0.00456239
$511$	−3.15486	−0.139563
$512$	−5.50000	−0.243068
$513$	−8.58099	−0.378860
$514$	20.6616	0.911346
$515$	−2.01497	−0.0887902
$516$	−6.68735	−0.294394
$517$	−7.53342	−0.331319
$518$	−3.13411	−0.137705
$519$	−6.52257	−0.286309
$520$	−0.164787	−0.00722638
$521$	−26.7670	−1.17268	−0.586342	−	0.810064i	$-0.699432\pi$
$521$	−26.7670	−1.17268	−0.586342	+	0.810064i	$0.699432\pi$
$522$	−6.22304	−0.272375
$523$	10.3938	0.454487	0.227244	−	0.973838i	$-0.427029\pi$
$523$	10.3938	0.454487	0.227244	+	0.973838i	$0.427029\pi$
$524$	−5.07706	−0.221792
$525$	4.97945	0.217321
$526$	5.72909	0.249800
$527$	−5.06807	−0.220769
$528$	−0.435149	−0.0189374
$529$	−1.24580	−0.0541653
$530$	−0.0314187	−0.00136474
$531$	11.7274	0.508926
$532$	−11.5240	−0.499629
$533$	−1.33430	−0.0577951
$534$	−3.24378	−0.140372
$535$	0.355057	0.0153504
$536$	23.2433	1.00396
$537$	−12.0548	−0.520202
$538$	−3.28815	−0.141762
$539$	−0.888944	−0.0382895
$540$	−0.192522	−0.00828484
$541$	−6.70658	−0.288339	−0.144169	−	0.989553i	$-0.546051\pi$
$541$	−6.70658	−0.288339	−0.144169	+	0.989553i	$0.546051\pi$
$542$	21.9143	0.941299
$543$	8.92294	0.382920
$544$	5.15719	0.221113
$545$	0.621474	0.0266210
$546$	−0.343855	−0.0147156
$547$	14.3245	0.612473	0.306237	−	0.951955i	$-0.400930\pi$
$547$	14.3245	0.612473	0.306237	+	0.951955i	$0.400930\pi$
$548$	−6.58629	−0.281352
$549$	−14.4988	−0.618792
$550$	−3.58796	−0.152991
$551$	−65.8790	−2.80654
$552$	12.6385	0.537931
$553$	8.70639	0.370233
$554$	−21.6761	−0.920929
$555$	−0.554289	−0.0235283
$556$	17.5876	0.745880
$557$	19.0952	0.809092	0.404546	−	0.914518i	$-0.367430\pi$
$557$	19.0952	0.809092	0.404546	+	0.914518i	$0.367430\pi$
$558$	−4.63302	−0.196131
$559$	−2.11237	−0.0893438
$560$	−0.0701742	−0.00296540
$561$	0.788217	0.0332785
$562$	−13.1828	−0.556082
$563$	0.731423	0.0308258	0.0154129	−	0.999881i	$-0.495094\pi$
$563$	0.731423	0.0308258	0.0154129	+	0.999881i	$0.495094\pi$
$564$	−11.3811	−0.479231
$565$	−2.94595	−0.123937
$566$	4.36999	0.183684
$567$	−1.00000	−0.0419961
$568$	−37.1050	−1.55689
$569$	24.6545	1.03357	0.516786	−	0.856115i	$-0.327128\pi$
$569$	24.6545	1.03357	0.516786	+	0.856115i	$0.327128\pi$
$570$	0.997112	0.0417644
$571$	−10.7185	−0.448556	−0.224278	−	0.974525i	$-0.572002\pi$
$571$	−10.7185	−0.448556	−0.224278	+	0.974525i	$0.572002\pi$
$572$	−0.506435	−0.0211751
$573$	15.6690	0.654580
$574$	2.54955	0.106416
$575$	−23.2248	−0.968543
$576$	3.73546	0.155644
$577$	2.19342	0.0913135	0.0456567	−	0.998957i	$-0.485462\pi$
$577$	2.19342	0.0913135	0.0456567	+	0.998957i	$0.485462\pi$
$578$	13.1425	0.546654
$579$	17.1874	0.714285
$580$	−1.47806	−0.0613729
$581$	15.0147	0.622914
$582$	6.49796	0.269349
$583$	−0.240357	−0.00995455
$584$	8.54880	0.353752
$585$	−0.0608131	−0.00251431
$586$	19.5703	0.808440
$587$	−30.7353	−1.26858	−0.634290	−	0.773095i	$-0.718707\pi$
$587$	−30.7353	−1.26858	−0.634290	+	0.773095i	$0.718707\pi$
$588$	−1.34297	−0.0553832
$589$	−49.0466	−2.02093
$590$	−1.36273	−0.0561025
$591$	18.3736	0.755787
$592$	−1.89272	−0.0777901
$593$	6.18101	0.253824	0.126912	−	0.991914i	$-0.459493\pi$
$593$	6.18101	0.253824	0.126912	+	0.991914i	$0.459493\pi$
$594$	0.720554	0.0295647
$595$	0.127112	0.00521107
$596$	−9.00808	−0.368985
$597$	−7.48545	−0.306359
$598$	1.60379	0.0655837
$599$	9.38206	0.383341	0.191670	−	0.981459i	$-0.438610\pi$
$599$	9.38206	0.383341	0.191670	+	0.981459i	$0.438610\pi$
$600$	−13.4929	−0.550847
$601$	25.2561	1.03022	0.515109	−	0.857125i	$-0.327752\pi$
$601$	25.2561	1.03022	0.515109	+	0.857125i	$0.327752\pi$
$602$	4.03626	0.164506
$603$	8.57772	0.349312
$604$	9.95998	0.405266
$605$	−1.46363	−0.0595049
$606$	−4.29081	−0.174302
$607$	−23.5190	−0.954605	−0.477302	−	0.878739i	$-0.658385\pi$
$607$	−23.5190	−0.954605	−0.477302	+	0.878739i	$0.658385\pi$
$608$	49.9090	2.02408
$609$	−7.67733	−0.311101
$610$	1.68476	0.0682138
$611$	−3.59501	−0.145439
$612$	1.19080	0.0481351
$613$	−25.4819	−1.02920	−0.514601	−	0.857429i	$-0.672060\pi$
$613$	−25.4819	−1.02920	−0.514601	+	0.857429i	$0.672060\pi$
$614$	−1.17091	−0.0472540
$615$	0.450906	0.0181823
$616$	2.40879	0.0970530
$617$	−18.1745	−0.731679	−0.365840	−	0.930678i	$-0.619218\pi$
$617$	−18.1745	−0.731679	−0.365840	+	0.930678i	$0.619218\pi$
$618$	11.3932	0.458303
$619$	−3.92060	−0.157582	−0.0787912	−	0.996891i	$-0.525106\pi$
$619$	−3.92060	−0.157582	−0.0787912	+	0.996891i	$0.525106\pi$
$620$	−1.10041	−0.0441933
$621$	4.66414	0.187165
$622$	−3.49716	−0.140224
$623$	−4.00183	−0.160330
$624$	−0.207657	−0.00831293
$625$	24.6922	0.987686
$626$	24.1593	0.965601
$627$	7.62802	0.304634
$628$	−33.3819	−1.33208
$629$	3.42841	0.136700
$630$	0.116200	0.00462953
$631$	16.6747	0.663809	0.331905	−	0.943313i	$-0.392309\pi$
$631$	16.6747	0.663809	0.331905	+	0.943313i	$0.392309\pi$
$632$	−23.5919	−0.938436
$633$	2.73926	0.108876
$634$	17.5487	0.696947
$635$	1.17438	0.0466039
$636$	−0.363118	−0.0143986
$637$	−0.424212	−0.0168079
$638$	5.53193	0.219011
$639$	−13.6933	−0.541697
$640$	1.23352	0.0487590
$641$	−26.7643	−1.05713	−0.528563	−	0.848894i	$-0.677269\pi$
$641$	−26.7643	−1.05713	−0.528563	+	0.848894i	$0.677269\pi$
$642$	−2.00759	−0.0792334
$643$	0.924905	0.0364747	0.0182373	−	0.999834i	$-0.494195\pi$
$643$	0.924905	0.0364747	0.0182373	+	0.999834i	$0.494195\pi$
$644$	6.26380	0.246828
$645$	0.713841	0.0281075
$646$	−6.16738	−0.242652
$647$	41.0449	1.61364	0.806821	−	0.590796i	$-0.201186\pi$
$647$	41.0449	1.61364	0.806821	+	0.590796i	$0.201186\pi$
$648$	2.70972	0.106448
$649$	−10.4250	−0.409217
$650$	−1.71221	−0.0671583
$651$	−5.71573	−0.224017
$652$	19.9803	0.782488
$653$	6.99632	0.273787	0.136894	−	0.990586i	$-0.456288\pi$
$653$	6.99632	0.273787	0.136894	+	0.990586i	$0.456288\pi$
$654$	−3.51400	−0.137408
$655$	0.541951	0.0211758
$656$	1.53970	0.0601150
$657$	3.15486	0.123083
$658$	6.86926	0.267792
$659$	14.3451	0.558807	0.279404	−	0.960174i	$-0.409863\pi$
$659$	14.3451	0.558807	0.279404	+	0.960174i	$0.409863\pi$
$660$	0.171142	0.00666167
$661$	−26.7975	−1.04230	−0.521152	−	0.853464i	$-0.674497\pi$
$661$	−26.7975	−1.04230	−0.521152	+	0.853464i	$0.674497\pi$
$662$	−6.15199	−0.239104
$663$	0.376144	0.0146082
$664$	−40.6856	−1.57891
$665$	1.23013	0.0477025
$666$	3.13411	0.121444
$667$	35.8081	1.38650
$668$	17.4890	0.676669
$669$	3.49272	0.135036
$670$	−0.996733	−0.0385072
$671$	12.8886	0.497558
$672$	5.81623	0.224366
$673$	−9.97120	−0.384361	−0.192181	−	0.981360i	$-0.561556\pi$
$673$	−9.97120	−0.384361	−0.192181	+	0.981360i	$0.561556\pi$
$674$	1.04247	0.0401543
$675$	−4.97945	−0.191659
$676$	17.2169	0.662190
$677$	3.07701	0.118259	0.0591295	−	0.998250i	$-0.481168\pi$
$677$	3.07701	0.118259	0.0591295	+	0.998250i	$0.481168\pi$
$678$	16.6572	0.639717
$679$	8.01650	0.307645
$680$	−0.344438	−0.0132086
$681$	1.57627	0.0604027
$682$	4.11849	0.157705
$683$	18.1050	0.692767	0.346383	−	0.938093i	$-0.387410\pi$
$683$	18.1050	0.692767	0.346383	+	0.938093i	$0.387410\pi$
$684$	11.5240	0.440632
$685$	0.703054	0.0268623
$686$	0.810573	0.0309478
$687$	9.96316	0.380118
$688$	2.43753	0.0929301
$689$	−0.114700	−0.00436973
$690$	−0.541974	−0.0206326
$691$	44.3131	1.68575	0.842876	−	0.538108i	$-0.180861\pi$
$691$	44.3131	1.68575	0.842876	+	0.538108i	$0.180861\pi$
$692$	8.75961	0.332991
$693$	0.888944	0.0337682
$694$	−2.37685	−0.0902238
$695$	−1.87739	−0.0712134
$696$	20.8034	0.788552
$697$	−2.78896	−0.105640
$698$	8.87206	0.335812
$699$	6.59409	0.249411
$700$	−6.68725	−0.252754
$701$	−8.28300	−0.312845	−0.156422	−	0.987690i	$-0.549996\pi$
$701$	−8.28300	−0.312845	−0.156422	+	0.987690i	$0.549996\pi$
$702$	0.343855	0.0129780
$703$	33.1787	1.25136
$704$	−3.32061	−0.125150
$705$	1.21488	0.0457549
$706$	21.1000	0.794109
$707$	−5.29355	−0.199084
$708$	−15.7495	−0.591904
$709$	15.6010	0.585907	0.292953	−	0.956127i	$-0.405362\pi$
$709$	15.6010	0.585907	0.292953	+	0.956127i	$0.405362\pi$
$710$	1.59116	0.0597152
$711$	−8.70639	−0.326515
$712$	10.8439	0.406391
$713$	26.6590	0.998386
$714$	−0.718726	−0.0268977
$715$	0.0540595	0.00202171
$716$	16.1892	0.605020
$717$	7.62877	0.284902
$718$	11.7170	0.437274
$719$	−24.5682	−0.916237	−0.458119	−	0.888891i	$-0.651477\pi$
$719$	−24.5682	−0.916237	−0.458119	+	0.888891i	$0.651477\pi$
$720$	0.0701742	0.00261524
$721$	14.0558	0.523464
$722$	−44.2843	−1.64809
$723$	−15.6026	−0.580268
$724$	−11.9832	−0.445354
$725$	−38.2289	−1.41978
$726$	8.27577	0.307143
$727$	26.9345	0.998945	0.499472	−	0.866330i	$-0.333527\pi$
$727$	26.9345	0.998945	0.499472	+	0.866330i	$0.333527\pi$
$728$	1.14950	0.0426032
$729$	1.00000	0.0370370
$730$	−0.366595	−0.0135683
$731$	−4.41528	−0.163305
$732$	19.4714	0.719684
$733$	39.5434	1.46057	0.730284	−	0.683144i	$-0.239388\pi$
$733$	39.5434	1.46057	0.730284	+	0.683144i	$0.239388\pi$
$734$	11.4791	0.423700
$735$	0.143356	0.00528775
$736$	−27.1277	−0.999941
$737$	−7.62512	−0.280875
$738$	−2.54955	−0.0938504
$739$	−14.5050	−0.533575	−0.266788	−	0.963755i	$-0.585962\pi$
$739$	−14.5050	−0.533575	−0.266788	+	0.963755i	$0.585962\pi$
$740$	0.744394	0.0273645
$741$	3.64016	0.133725
$742$	0.219166	0.00804585
$743$	−3.33447	−0.122330	−0.0611650	−	0.998128i	$-0.519482\pi$
$743$	−3.33447	−0.122330	−0.0611650	+	0.998128i	$0.519482\pi$
$744$	15.4880	0.567819
$745$	0.961568	0.0352291
$746$	6.53474	0.239254
$747$	−15.0147	−0.549359
$748$	−1.05855	−0.0387045
$749$	−2.47676	−0.0904987
$750$	1.15961	0.0423431
$751$	26.7448	0.975932	0.487966	−	0.872863i	$-0.337739\pi$
$751$	26.7448	0.975932	0.487966	+	0.872863i	$0.337739\pi$
$752$	4.14840	0.151277
$753$	0.775745	0.0282697
$754$	2.63989	0.0961390
$755$	−1.06318	−0.0386930
$756$	1.34297	0.0488434
$757$	−15.6152	−0.567544	−0.283772	−	0.958892i	$-0.591586\pi$
$757$	−15.6152	−0.567544	−0.283772	+	0.958892i	$0.591586\pi$
$758$	29.4732	1.07052
$759$	−4.14616	−0.150496
$760$	−3.33332	−0.120912
$761$	23.6289	0.856547	0.428274	−	0.903649i	$-0.359122\pi$
$761$	23.6289	0.856547	0.428274	+	0.903649i	$0.359122\pi$
$762$	−6.64029	−0.240552
$763$	−4.33520	−0.156945
$764$	−21.0430	−0.761307
$765$	−0.127112	−0.00459573
$766$	−0.810573	−0.0292872
$767$	−4.97490	−0.179633
$768$	−14.4456	−0.521260
$769$	−20.1151	−0.725368	−0.362684	−	0.931912i	$-0.618140\pi$
$769$	−20.1151	−0.725368	−0.362684	+	0.931912i	$0.618140\pi$
$770$	−0.103295	−0.00372251
$771$	−25.4902	−0.918006
$772$	−23.0822	−0.830747
$773$	4.96166	0.178458	0.0892292	−	0.996011i	$-0.471560\pi$
$773$	4.96166	0.178458	0.0892292	+	0.996011i	$0.471560\pi$
$774$	−4.03626	−0.145081
$775$	−28.4612	−1.02236
$776$	−21.7225	−0.779792
$777$	3.86654	0.138711
$778$	24.3178	0.871837
$779$	−26.9904	−0.967031
$780$	0.0816703	0.00292426
$781$	12.1725	0.435568
$782$	3.35224	0.119876
$783$	7.67733	0.274365
$784$	0.489512	0.0174826
$785$	3.56336	0.127182
$786$	−3.06435	−0.109302
$787$	−8.61157	−0.306969	−0.153485	−	0.988151i	$-0.549050\pi$
$787$	−8.61157	−0.306969	−0.153485	+	0.988151i	$0.549050\pi$
$788$	−24.6752	−0.879016
$789$	−7.06795	−0.251626
$790$	1.01168	0.0359941
$791$	20.5499	0.730672
$792$	−2.40879	−0.0855927
$793$	6.15054	0.218412
$794$	−28.3386	−1.00570
$795$	0.0387611	0.00137471
$796$	10.0527	0.356310
$797$	−45.9942	−1.62920	−0.814600	−	0.580024i	$-0.803043\pi$
$797$	−45.9942	−1.62920	−0.814600	+	0.580024i	$0.803043\pi$
$798$	−6.95552	−0.246223
$799$	−7.51430	−0.265837
$800$	28.9616	1.02395
$801$	4.00183	0.141398
$802$	12.4326	0.439010
$803$	−2.80449	−0.0989684
$804$	−11.5196	−0.406266
$805$	−0.668630	−0.0235661
$806$	1.96538	0.0692276
$807$	4.05657	0.142798
$808$	14.3441	0.504622
$809$	−23.3965	−0.822577	−0.411289	−	0.911505i	$-0.634921\pi$
$809$	−23.3965	−0.822577	−0.411289	+	0.911505i	$0.634921\pi$
$810$	−0.116200	−0.00408286
$811$	−24.7502	−0.869097	−0.434549	−	0.900648i	$-0.643092\pi$
$811$	−24.7502	−0.869097	−0.434549	+	0.900648i	$0.643092\pi$
$812$	10.3104	0.361825
$813$	−27.0355	−0.948177
$814$	−2.78605	−0.0976510
$815$	−2.13280	−0.0747086
$816$	−0.434045	−0.0151946
$817$	−42.7292	−1.49490
$818$	21.4262	0.749148
$819$	0.424212	0.0148232
$820$	−0.605554	−0.0211469
$821$	43.1476	1.50586	0.752931	−	0.658100i	$-0.228639\pi$
$821$	43.1476	1.50586	0.752931	+	0.658100i	$0.228639\pi$
$822$	−3.97527	−0.138653
$823$	28.4239	0.990794	0.495397	−	0.868667i	$-0.335023\pi$
$823$	28.4239	0.990794	0.495397	+	0.868667i	$0.335023\pi$
$824$	−38.0872	−1.32683
$825$	4.42645	0.154109
$826$	9.50591	0.330753
$827$	−45.8266	−1.59355	−0.796773	−	0.604279i	$-0.793461\pi$
$827$	−45.8266	−1.59355	−0.796773	+	0.604279i	$0.793461\pi$
$828$	−6.26380	−0.217682
$829$	25.2739	0.877797	0.438899	−	0.898537i	$-0.355369\pi$
$829$	25.2739	0.877797	0.438899	+	0.898537i	$0.355369\pi$
$830$	1.74471	0.0605597
$831$	26.7417	0.927659
$832$	−1.58463	−0.0549370
$833$	−0.886689	−0.0307219
$834$	10.6153	0.367578
$835$	−1.86686	−0.0646055
$836$	−10.2442	−0.354303
$837$	5.71573	0.197565
$838$	−17.0546	−0.589141
$839$	29.4421	1.01646	0.508228	−	0.861223i	$-0.330301\pi$
$839$	29.4421	1.01646	0.508228	+	0.861223i	$0.330301\pi$
$840$	−0.388454	−0.0134029
$841$	29.9413	1.03246
$842$	27.8828	0.960904
$843$	16.2635	0.560145
$844$	−3.67874	−0.126628
$845$	−1.83782	−0.0632231
$846$	−6.86926	−0.236170
$847$	10.2098	0.350812
$848$	0.132356	0.00454514
$849$	−5.39124	−0.185027
$850$	−3.57886	−0.122754
$851$	−18.0341	−0.618200
$852$	18.3897	0.630019
$853$	13.9301	0.476958	0.238479	−	0.971148i	$-0.423351\pi$
$853$	13.9301	0.476958	0.238479	+	0.971148i	$0.423351\pi$
$854$	−11.7523	−0.402156
$855$	−1.23013	−0.0420696
$856$	6.71133	0.229388
$857$	15.2018	0.519282	0.259641	−	0.965705i	$-0.416396\pi$
$857$	15.2018	0.519282	0.259641	+	0.965705i	$0.416396\pi$
$858$	−0.305668	−0.0104353
$859$	−53.7411	−1.83362	−0.916811	−	0.399322i	$-0.869246\pi$
$859$	−53.7411	−1.83362	−0.916811	+	0.399322i	$0.869246\pi$
$860$	−0.958668	−0.0326903
$861$	−3.14537	−0.107194
$862$	−0.569178	−0.0193863
$863$	−29.8386	−1.01572	−0.507858	−	0.861441i	$-0.669563\pi$
$863$	−29.8386	−1.01572	−0.507858	+	0.861441i	$0.669563\pi$
$864$	−5.81623	−0.197872
$865$	−0.935046	−0.0317925
$866$	29.0241	0.986280
$867$	−16.2138	−0.550649
$868$	7.67606	0.260542
$869$	7.73950	0.262544
$870$	−0.892107	−0.0302452
$871$	−3.63877	−0.123295
$872$	11.7472	0.397810
$873$	−8.01650	−0.271317
$874$	32.4415	1.09735
$875$	1.43061	0.0483634
$876$	−4.23688	−0.143151
$877$	−23.1390	−0.781350	−0.390675	−	0.920529i	$-0.627758\pi$
$877$	−23.1390	−0.781350	−0.390675	+	0.920529i	$0.627758\pi$
$878$	−15.7050	−0.530020
$879$	−24.1437	−0.814348
$880$	−0.0623810	−0.00210286
$881$	−32.2296	−1.08584	−0.542921	−	0.839784i	$-0.682682\pi$
$881$	−32.2296	−1.08584	−0.542921	+	0.839784i	$0.682682\pi$
$882$	−0.810573	−0.0272934
$883$	−21.0663	−0.708939	−0.354469	−	0.935068i	$-0.615339\pi$
$883$	−21.0663	−0.708939	−0.354469	+	0.935068i	$0.615339\pi$
$884$	−0.505150	−0.0169900
$885$	1.68119	0.0565125
$886$	27.6813	0.929971
$887$	3.40795	0.114428	0.0572138	−	0.998362i	$-0.481778\pi$
$887$	3.40795	0.114428	0.0572138	+	0.998362i	$0.481778\pi$
$888$	−10.4772	−0.351593
$889$	−8.19208	−0.274754
$890$	−0.465014	−0.0155873
$891$	−0.888944	−0.0297807
$892$	−4.69062	−0.157054
$893$	−72.7201	−2.43349
$894$	−5.43698	−0.181840
$895$	−1.72812	−0.0577647
$896$	−8.60460	−0.287460
$897$	−1.97858	−0.0660630
$898$	26.3683	0.879923
$899$	43.8815	1.46353
$900$	6.68725	0.222908
$901$	−0.239747	−0.00798712
$902$	2.26641	0.0754632
$903$	−4.97952	−0.165708
$904$	−55.6847	−1.85204
$905$	1.27915	0.0425204
$906$	6.01152	0.199719
$907$	56.5374	1.87729	0.938647	−	0.344880i	$-0.112081\pi$
$907$	56.5374	1.87729	0.938647	+	0.344880i	$0.112081\pi$
$908$	−2.11688	−0.0702512
$909$	5.29355	0.175576
$910$	−0.0492935	−0.00163406
$911$	−38.5940	−1.27868	−0.639338	−	0.768926i	$-0.720791\pi$
$911$	−38.5940	−1.27868	−0.639338	+	0.768926i	$0.720791\pi$
$912$	−4.20049	−0.139092
$913$	13.3472	0.441728
$914$	−20.1800	−0.667495
$915$	−2.07848	−0.0687123
$916$	−13.3802	−0.442095
$917$	−3.78047	−0.124842
$918$	0.718726	0.0237215
$919$	−1.15162	−0.0379886	−0.0189943	−	0.999820i	$-0.506046\pi$
$919$	−1.15162	−0.0379886	−0.0189943	+	0.999820i	$0.506046\pi$
$920$	1.81180	0.0597334
$921$	1.44454	0.0475993
$922$	14.5393	0.478826
$923$	5.80885	0.191201
$924$	−1.19383	−0.0392740
$925$	19.2532	0.633042
$926$	−1.48267	−0.0487236
$927$	−14.0558	−0.461652
$928$	−44.6531	−1.46581
$929$	−14.8520	−0.487279	−0.243639	−	0.969866i	$-0.578341\pi$
$929$	−14.8520	−0.487279	−0.243639	+	0.969866i	$0.578341\pi$
$930$	−0.664169	−0.0217789
$931$	−8.58099	−0.281230
$932$	−8.85567	−0.290077
$933$	4.31443	0.141248
$934$	31.4045	1.02759
$935$	0.112995	0.00369534
$936$	−1.14950	−0.0375725
$937$	43.7206	1.42829	0.714145	−	0.699998i	$-0.246816\pi$
$937$	43.7206	1.42829	0.714145	+	0.699998i	$0.246816\pi$
$938$	6.95287	0.227019
$939$	−29.8052	−0.972657
$940$	−1.63154	−0.0532151
$941$	−59.0761	−1.92583	−0.962913	−	0.269812i	$-0.913038\pi$
$941$	−59.0761	−1.92583	−0.962913	+	0.269812i	$0.913038\pi$
$942$	−20.1483	−0.656465
$943$	14.6704	0.477735
$944$	5.74070	0.186844
$945$	−0.143356	−0.00466336
$946$	3.58801	0.116656
$947$	8.92651	0.290073	0.145036	−	0.989426i	$-0.453670\pi$
$947$	8.92651	0.290073	0.145036	+	0.989426i	$0.453670\pi$
$948$	11.6924	0.379752
$949$	−1.33833	−0.0434440
$950$	−34.6346	−1.12370
$951$	−21.6497	−0.702040
$952$	2.40268	0.0778713
$953$	−18.3821	−0.595453	−0.297727	−	0.954651i	$-0.596228\pi$
$953$	−18.3821	−0.595453	−0.297727	+	0.954651i	$0.596228\pi$
$954$	−0.219166	−0.00709577
$955$	2.24623	0.0726864
$956$	−10.2452	−0.331354
$957$	−6.82471	−0.220612
$958$	11.9535	0.386199
$959$	−4.90427	−0.158367
$960$	0.535499	0.0172831
$961$	1.66956	0.0538568
$962$	−1.32953	−0.0428657
$963$	2.47676	0.0798124
$964$	20.9539	0.674879
$965$	2.46391	0.0793162
$966$	3.78063	0.121640
$967$	56.0604	1.80278	0.901391	−	0.433006i	$-0.142547\pi$
$967$	56.0604	1.80278	0.901391	+	0.433006i	$0.142547\pi$
$968$	−27.6657	−0.889208
$969$	7.60866	0.244425
$970$	0.931518	0.0299092
$971$	31.0501	0.996446	0.498223	−	0.867049i	$-0.333986\pi$
$971$	31.0501	0.996446	0.498223	+	0.867049i	$0.333986\pi$
$972$	−1.34297	−0.0430758
$973$	13.0960	0.419840
$974$	11.5816	0.371099
$975$	2.11234	0.0676491
$976$	−7.09731	−0.227179
$977$	−10.9400	−0.350002	−0.175001	−	0.984568i	$-0.555993\pi$
$977$	−10.9400	−0.350002	−0.175001	+	0.984568i	$0.555993\pi$
$978$	12.0594	0.385618
$979$	−3.55741	−0.113695
$980$	−0.192522	−0.00614990
$981$	4.33520	0.138412
$982$	−11.8433	−0.377934
$983$	53.3451	1.70144	0.850722	−	0.525616i	$-0.176165\pi$
$983$	53.3451	1.70144	0.850722	+	0.525616i	$0.176165\pi$
$984$	8.52308	0.271706
$985$	2.63395	0.0839247
$986$	5.51790	0.175726
$987$	−8.47457	−0.269748
$988$	−4.88862	−0.155528
$989$	23.2252	0.738517
$990$	0.103295	0.00328294
$991$	50.8510	1.61534	0.807668	−	0.589638i	$-0.200730\pi$
$991$	50.8510	1.61534	0.807668	+	0.589638i	$0.200730\pi$
$992$	−33.2440	−1.05550
$993$	7.58968	0.240851
$994$	−11.0994	−0.352051
$995$	−1.07308	−0.0340189
$996$	20.1643	0.638930
$997$	14.1360	0.447691	0.223846	−	0.974625i	$-0.428139\pi$
$997$	14.1360	0.447691	0.223846	+	0.974625i	$0.428139\pi$
$998$	14.5168	0.459520
$999$	−3.86654	−0.122332

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.q.1.18	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.q.1.18	✓	44	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-0.810573 q^{2} +1.00000 q^{3} -1.34297 q^{4} +0.143356 q^{5} -0.810573 q^{6} -1.00000 q^{7} +2.70972 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.810573 q^{2} +1.00000 q^{3} -1.34297 q^{4} +0.143356 q^{5} -0.810573 q^{6} -1.00000 q^{7} +2.70972 q^{8} +1.00000 q^{9} -0.116200 q^{10} -0.888944 q^{11} -1.34297 q^{12} -0.424212 q^{13} +0.810573 q^{14} +0.143356 q^{15} +0.489512 q^{16} -0.886689 q^{17} -0.810573 q^{18} -8.58099 q^{19} -0.192522 q^{20} -1.00000 q^{21} +0.720554 q^{22} +4.66414 q^{23} +2.70972 q^{24} -4.97945 q^{25} +0.343855 q^{26} +1.00000 q^{27} +1.34297 q^{28} +7.67733 q^{29} -0.116200 q^{30} +5.71573 q^{31} -5.81623 q^{32} -0.888944 q^{33} +0.718726 q^{34} -0.143356 q^{35} -1.34297 q^{36} -3.86654 q^{37} +6.95552 q^{38} -0.424212 q^{39} +0.388454 q^{40} +3.14537 q^{41} +0.810573 q^{42} +4.97952 q^{43} +1.19383 q^{44} +0.143356 q^{45} -3.78063 q^{46} +8.47457 q^{47} +0.489512 q^{48} +1.00000 q^{49} +4.03621 q^{50} -0.886689 q^{51} +0.569704 q^{52} +0.270384 q^{53} -0.810573 q^{54} -0.127435 q^{55} -2.70972 q^{56} -8.58099 q^{57} -6.22304 q^{58} +11.7274 q^{59} -0.192522 q^{60} -14.4988 q^{61} -4.63302 q^{62} -1.00000 q^{63} +3.73546 q^{64} -0.0608131 q^{65} +0.720554 q^{66} +8.57772 q^{67} +1.19080 q^{68} +4.66414 q^{69} +0.116200 q^{70} -13.6933 q^{71} +2.70972 q^{72} +3.15486 q^{73} +3.13411 q^{74} -4.97945 q^{75} +11.5240 q^{76} +0.888944 q^{77} +0.343855 q^{78} -8.70639 q^{79} +0.0701742 q^{80} +1.00000 q^{81} -2.54955 q^{82} -15.0147 q^{83} +1.34297 q^{84} -0.127112 q^{85} -4.03626 q^{86} +7.67733 q^{87} -2.40879 q^{88} +4.00183 q^{89} -0.116200 q^{90} +0.424212 q^{91} -6.26380 q^{92} +5.71573 q^{93} -6.86926 q^{94} -1.23013 q^{95} -5.81623 q^{96} -8.01650 q^{97} -0.810573 q^{98} -0.888944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9}+O(q^{10}) \) 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9	\( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9} - 16 q^{10} - 2 q^{11} + 44 q^{12} - 34 q^{13} + 4 q^{14} - 16 q^{15} + 24 q^{16} - 4 q^{17} - 4 q^{18} - 22 q^{19} - 39 q^{20} - 44 q^{21} - 23 q^{22} - 56 q^{23} - 15 q^{24} + 32 q^{25} - 17 q^{26} + 44 q^{27} - 44 q^{28} - 33 q^{29} - 16 q^{30} - 32 q^{31} - 34 q^{32} - 2 q^{33} - 25 q^{34} + 16 q^{35} + 44 q^{36} - 47 q^{37} - 40 q^{38} - 34 q^{39} - 50 q^{40} + 2 q^{41} + 4 q^{42} - 12 q^{43} - 22 q^{44} - 16 q^{45} + 8 q^{46} - 27 q^{47} + 24 q^{48} + 44 q^{49} - 21 q^{50} - 4 q^{51} - 82 q^{52} - 114 q^{53} - 4 q^{54} - 29 q^{55} + 15 q^{56} - 22 q^{57} - 26 q^{58} - 40 q^{59} - 39 q^{60} - 47 q^{61} - 37 q^{62} - 44 q^{63} - 5 q^{64} - 20 q^{65} - 23 q^{66} - 14 q^{67} - 72 q^{68} - 56 q^{69} + 16 q^{70} - 65 q^{71} - 15 q^{72} - 21 q^{73} - 26 q^{74} + 32 q^{75} - 15 q^{76} + 2 q^{77} - 17 q^{78} + 6 q^{79} - 77 q^{80} + 44 q^{81} - 51 q^{82} - 30 q^{83} - 44 q^{84} - 26 q^{85} - 65 q^{86} - 33 q^{87} - 84 q^{88} - 32 q^{89} - 16 q^{90} + 34 q^{91} - 140 q^{92} - 32 q^{93} - 35 q^{94} - 50 q^{95} - 34 q^{96} - 83 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 44 * q - 4 * q^2 + 44 * q^3 + 44 * q^4 - 16 * q^5 - 4 * q^6 - 44 * q^7 - 15 * q^8 + 44 * q^9 - 16 * q^10 - 2 * q^11 + 44 * q^12 - 34 * q^13 + 4 * q^14 - 16 * q^15 + 24 * q^16 - 4 * q^17 - 4 * q^18 - 22 * q^19 - 39 * q^20 - 44 * q^21 - 23 * q^22 - 56 * q^23 - 15 * q^24 + 32 * q^25 - 17 * q^26 + 44 * q^27 - 44 * q^28 - 33 * q^29 - 16 * q^30 - 32 * q^31 - 34 * q^32 - 2 * q^33 - 25 * q^34 + 16 * q^35 + 44 * q^36 - 47 * q^37 - 40 * q^38 - 34 * q^39 - 50 * q^40 + 2 * q^41 + 4 * q^42 - 12 * q^43 - 22 * q^44 - 16 * q^45 + 8 * q^46 - 27 * q^47 + 24 * q^48 + 44 * q^49 - 21 * q^50 - 4 * q^51 - 82 * q^52 - 114 * q^53 - 4 * q^54 - 29 * q^55 + 15 * q^56 - 22 * q^57 - 26 * q^58 - 40 * q^59 - 39 * q^60 - 47 * q^61 - 37 * q^62 - 44 * q^63 - 5 * q^64 - 20 * q^65 - 23 * q^66 - 14 * q^67 - 72 * q^68 - 56 * q^69 + 16 * q^70 - 65 * q^71 - 15 * q^72 - 21 * q^73 - 26 * q^74 + 32 * q^75 - 15 * q^76 + 2 * q^77 - 17 * q^78 + 6 * q^79 - 77 * q^80 + 44 * q^81 - 51 * q^82 - 30 * q^83 - 44 * q^84 - 26 * q^85 - 65 * q^86 - 33 * q^87 - 84 * q^88 - 32 * q^89 - 16 * q^90 + 34 * q^91 - 140 * q^92 - 32 * q^93 - 35 * q^94 - 50 * q^95 - 34 * q^96 - 83 * q^97 - 4 * q^98 - 2 * q^99

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