LMFDB - Embedded newform 8042.2.a.a.1.12

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			1.12
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.22608 q^{3} +1.00000 q^{4} +1.44593 q^{5} -2.22608 q^{6} +1.32066 q^{7} +1.00000 q^{8} +1.95543 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.22608 q^{3} +1.00000 q^{4} +1.44593 q^{5} -2.22608 q^{6} +1.32066 q^{7} +1.00000 q^{8} +1.95543 q^{9} +1.44593 q^{10} -4.69336 q^{11} -2.22608 q^{12} +1.27986 q^{13} +1.32066 q^{14} -3.21876 q^{15} +1.00000 q^{16} -1.75842 q^{17} +1.95543 q^{18} -1.83542 q^{19} +1.44593 q^{20} -2.93990 q^{21} -4.69336 q^{22} +1.61507 q^{23} -2.22608 q^{24} -2.90928 q^{25} +1.27986 q^{26} +2.32529 q^{27} +1.32066 q^{28} +5.04102 q^{29} -3.21876 q^{30} +4.04230 q^{31} +1.00000 q^{32} +10.4478 q^{33} -1.75842 q^{34} +1.90959 q^{35} +1.95543 q^{36} -1.67445 q^{37} -1.83542 q^{38} -2.84906 q^{39} +1.44593 q^{40} -3.00389 q^{41} -2.93990 q^{42} -3.28239 q^{43} -4.69336 q^{44} +2.82742 q^{45} +1.61507 q^{46} +6.32528 q^{47} -2.22608 q^{48} -5.25585 q^{49} -2.90928 q^{50} +3.91438 q^{51} +1.27986 q^{52} -6.43256 q^{53} +2.32529 q^{54} -6.78628 q^{55} +1.32066 q^{56} +4.08579 q^{57} +5.04102 q^{58} +1.32811 q^{59} -3.21876 q^{60} -2.36183 q^{61} +4.04230 q^{62} +2.58247 q^{63} +1.00000 q^{64} +1.85059 q^{65} +10.4478 q^{66} +8.01836 q^{67} -1.75842 q^{68} -3.59527 q^{69} +1.90959 q^{70} -11.2422 q^{71} +1.95543 q^{72} -0.908233 q^{73} -1.67445 q^{74} +6.47629 q^{75} -1.83542 q^{76} -6.19835 q^{77} -2.84906 q^{78} -4.37488 q^{79} +1.44593 q^{80} -11.0426 q^{81} -3.00389 q^{82} +12.7989 q^{83} -2.93990 q^{84} -2.54256 q^{85} -3.28239 q^{86} -11.2217 q^{87} -4.69336 q^{88} -12.9009 q^{89} +2.82742 q^{90} +1.69026 q^{91} +1.61507 q^{92} -8.99849 q^{93} +6.32528 q^{94} -2.65389 q^{95} -2.22608 q^{96} -9.16110 q^{97} -5.25585 q^{98} -9.17755 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.22608	−1.28523	−0.642614	−	0.766190i	$-0.722150\pi$
$3$	−2.22608	−1.28523	−0.642614	+	0.766190i	$0.722150\pi$
$4$	1.00000	0.500000
$5$	1.44593	0.646641	0.323320	−	0.946290i	$-0.395201\pi$
$5$	1.44593	0.646641	0.323320	+	0.946290i	$0.395201\pi$
$6$	−2.22608	−0.908793
$7$	1.32066	0.499164	0.249582	−	0.968354i	$-0.419707\pi$
$7$	1.32066	0.499164	0.249582	+	0.968354i	$0.419707\pi$
$8$	1.00000	0.353553
$9$	1.95543	0.651811
$10$	1.44593	0.457244
$11$	−4.69336	−1.41510	−0.707551	−	0.706662i	$-0.750200\pi$
$11$	−4.69336	−1.41510	−0.707551	+	0.706662i	$0.750200\pi$
$12$	−2.22608	−0.642614
$13$	1.27986	0.354968	0.177484	−	0.984124i	$-0.443204\pi$
$13$	1.27986	0.354968	0.177484	+	0.984124i	$0.443204\pi$
$14$	1.32066	0.352962
$15$	−3.21876	−0.831080
$16$	1.00000	0.250000
$17$	−1.75842	−0.426480	−0.213240	−	0.977000i	$-0.568402\pi$
$17$	−1.75842	−0.426480	−0.213240	+	0.977000i	$0.568402\pi$
$18$	1.95543	0.460900
$19$	−1.83542	−0.421075	−0.210537	−	0.977586i	$-0.567521\pi$
$19$	−1.83542	−0.421075	−0.210537	+	0.977586i	$0.567521\pi$
$20$	1.44593	0.323320
$21$	−2.93990	−0.641539
$22$	−4.69336	−1.00063
$23$	1.61507	0.336765	0.168383	−	0.985722i	$-0.446146\pi$
$23$	1.61507	0.336765	0.168383	+	0.985722i	$0.446146\pi$
$24$	−2.22608	−0.454397
$25$	−2.90928	−0.581856
$26$	1.27986	0.251001
$27$	2.32529	0.447502
$28$	1.32066	0.249582
$29$	5.04102	0.936095	0.468047	−	0.883703i	$-0.344958\pi$
$29$	5.04102	0.936095	0.468047	+	0.883703i	$0.344958\pi$
$30$	−3.21876	−0.587663
$31$	4.04230	0.726019	0.363010	−	0.931785i	$-0.381749\pi$
$31$	4.04230	0.726019	0.363010	+	0.931785i	$0.381749\pi$
$32$	1.00000	0.176777
$33$	10.4478	1.81873
$34$	−1.75842	−0.301567
$35$	1.90959	0.322779
$36$	1.95543	0.325905
$37$	−1.67445	−0.275278	−0.137639	−	0.990482i	$-0.543951\pi$
$37$	−1.67445	−0.275278	−0.137639	+	0.990482i	$0.543951\pi$
$38$	−1.83542	−0.297745
$39$	−2.84906	−0.456215
$40$	1.44593	0.228622
$41$	−3.00389	−0.469130	−0.234565	−	0.972100i	$-0.575366\pi$
$41$	−3.00389	−0.469130	−0.234565	+	0.972100i	$0.575366\pi$
$42$	−2.93990	−0.453636
$43$	−3.28239	−0.500561	−0.250280	−	0.968173i	$-0.580523\pi$
$43$	−3.28239	−0.500561	−0.250280	+	0.968173i	$0.580523\pi$
$44$	−4.69336	−0.707551
$45$	2.82742	0.421487
$46$	1.61507	0.238129
$47$	6.32528	0.922637	0.461318	−	0.887235i	$-0.347377\pi$
$47$	6.32528	0.922637	0.461318	+	0.887235i	$0.347377\pi$
$48$	−2.22608	−0.321307
$49$	−5.25585	−0.750836
$50$	−2.90928	−0.411434
$51$	3.91438	0.548124
$52$	1.27986	0.177484
$53$	−6.43256	−0.883580	−0.441790	−	0.897119i	$-0.645656\pi$
$53$	−6.43256	−0.883580	−0.441790	+	0.897119i	$0.645656\pi$
$54$	2.32529	0.316432
$55$	−6.78628	−0.915062
$56$	1.32066	0.176481
$57$	4.08579	0.541177
$58$	5.04102	0.661919
$59$	1.32811	0.172905	0.0864524	−	0.996256i	$-0.472447\pi$
$59$	1.32811	0.172905	0.0864524	+	0.996256i	$0.472447\pi$
$60$	−3.21876	−0.415540
$61$	−2.36183	−0.302401	−0.151201	−	0.988503i	$-0.548314\pi$
$61$	−2.36183	−0.302401	−0.151201	+	0.988503i	$0.548314\pi$
$62$	4.04230	0.513373
$63$	2.58247	0.325360
$64$	1.00000	0.125000
$65$	1.85059	0.229537
$66$	10.4478	1.28604
$67$	8.01836	0.979599	0.489799	−	0.871835i	$-0.337070\pi$
$67$	8.01836	0.979599	0.489799	+	0.871835i	$0.337070\pi$
$68$	−1.75842	−0.213240
$69$	−3.59527	−0.432820
$70$	1.90959	0.228239
$71$	−11.2422	−1.33420	−0.667100	−	0.744968i	$-0.732465\pi$
$71$	−11.2422	−1.33420	−0.667100	+	0.744968i	$0.732465\pi$
$72$	1.95543	0.230450
$73$	−0.908233	−0.106301	−0.0531503	−	0.998587i	$-0.516926\pi$
$73$	−0.908233	−0.106301	−0.0531503	+	0.998587i	$0.516926\pi$
$74$	−1.67445	−0.194651
$75$	6.47629	0.747818
$76$	−1.83542	−0.210537
$77$	−6.19835	−0.706367
$78$	−2.84906	−0.322593
$79$	−4.37488	−0.492212	−0.246106	−	0.969243i	$-0.579151\pi$
$79$	−4.37488	−0.492212	−0.246106	+	0.969243i	$0.579151\pi$
$80$	1.44593	0.161660
$81$	−11.0426	−1.22695
$82$	−3.00389	−0.331725
$83$	12.7989	1.40486	0.702429	−	0.711753i	$-0.252099\pi$
$83$	12.7989	1.40486	0.702429	+	0.711753i	$0.252099\pi$
$84$	−2.93990	−0.320769
$85$	−2.54256	−0.275779
$86$	−3.28239	−0.353950
$87$	−11.2217	−1.20309
$88$	−4.69336	−0.500314
$89$	−12.9009	−1.36749	−0.683745	−	0.729721i	$-0.739650\pi$
$89$	−12.9009	−1.36749	−0.683745	+	0.729721i	$0.739650\pi$
$90$	2.82742	0.298037
$91$	1.69026	0.177187
$92$	1.61507	0.168383
$93$	−8.99849	−0.933100
$94$	6.32528	0.652403
$95$	−2.65389	−0.272284
$96$	−2.22608	−0.227198
$97$	−9.16110	−0.930169	−0.465085	−	0.885266i	$-0.653976\pi$
$97$	−9.16110	−0.930169	−0.465085	+	0.885266i	$0.653976\pi$
$98$	−5.25585	−0.530921
$99$	−9.17755	−0.922379
$100$	−2.90928	−0.290928
$101$	−9.75684	−0.970842	−0.485421	−	0.874281i	$-0.661334\pi$
$101$	−9.75684	−0.970842	−0.485421	+	0.874281i	$0.661334\pi$
$102$	3.91438	0.387582
$103$	−13.9265	−1.37222	−0.686112	−	0.727496i	$-0.740684\pi$
$103$	−13.9265	−1.37222	−0.686112	+	0.727496i	$0.740684\pi$
$104$	1.27986	0.125500
$105$	−4.25090	−0.414845
$106$	−6.43256	−0.624785
$107$	0.0699806	0.00676528	0.00338264	−	0.999994i	$-0.498923\pi$
$107$	0.0699806	0.00676528	0.00338264	+	0.999994i	$0.498923\pi$
$108$	2.32529	0.223751
$109$	7.14893	0.684743	0.342372	−	0.939565i	$-0.388770\pi$
$109$	7.14893	0.684743	0.342372	+	0.939565i	$0.388770\pi$
$110$	−6.78628	−0.647047
$111$	3.72746	0.353795
$112$	1.32066	0.124791
$113$	18.1278	1.70532	0.852662	−	0.522463i	$-0.174987\pi$
$113$	18.1278	1.70532	0.852662	+	0.522463i	$0.174987\pi$
$114$	4.08579	0.382670
$115$	2.33528	0.217766
$116$	5.04102	0.468047
$117$	2.50267	0.231372
$118$	1.32811	0.122262
$119$	−2.32228	−0.212883
$120$	−3.21876	−0.293831
$121$	11.0277	1.00251
$122$	−2.36183	−0.213830
$123$	6.68691	0.602938
$124$	4.04230	0.363010
$125$	−11.4363	−1.02289
$126$	2.58247	0.230064
$127$	15.1331	1.34285	0.671425	−	0.741073i	$-0.265683\pi$
$127$	15.1331	1.34285	0.671425	+	0.741073i	$0.265683\pi$
$128$	1.00000	0.0883883
$129$	7.30687	0.643334
$130$	1.85059	0.162307
$131$	−5.41604	−0.473201	−0.236601	−	0.971607i	$-0.576033\pi$
$131$	−5.41604	−0.473201	−0.236601	+	0.971607i	$0.576033\pi$
$132$	10.4478	0.909364
$133$	−2.42397	−0.210185
$134$	8.01836	0.692681
$135$	3.36221	0.289373
$136$	−1.75842	−0.150783
$137$	−16.3466	−1.39658	−0.698291	−	0.715814i	$-0.746056\pi$
$137$	−16.3466	−1.39658	−0.698291	+	0.715814i	$0.746056\pi$
$138$	−3.59527	−0.306050
$139$	−9.63523	−0.817250	−0.408625	−	0.912702i	$-0.633992\pi$
$139$	−9.63523	−0.817250	−0.408625	+	0.912702i	$0.633992\pi$
$140$	1.90959	0.161390
$141$	−14.0806	−1.18580
$142$	−11.2422	−0.943422
$143$	−6.00683	−0.502316
$144$	1.95543	0.162953
$145$	7.28898	0.605317
$146$	−0.908233	−0.0751659
$147$	11.6999	0.964995
$148$	−1.67445	−0.137639
$149$	3.55096	0.290906	0.145453	−	0.989365i	$-0.453536\pi$
$149$	3.55096	0.290906	0.145453	+	0.989365i	$0.453536\pi$
$150$	6.47629	0.528787
$151$	8.65736	0.704526	0.352263	−	0.935901i	$-0.385412\pi$
$151$	8.65736	0.704526	0.352263	+	0.935901i	$0.385412\pi$
$152$	−1.83542	−0.148872
$153$	−3.43847	−0.277984
$154$	−6.19835	−0.499477
$155$	5.84490	0.469474
$156$	−2.84906	−0.228108
$157$	−18.1654	−1.44975	−0.724877	−	0.688878i	$-0.758103\pi$
$157$	−18.1654	−1.44975	−0.724877	+	0.688878i	$0.758103\pi$
$158$	−4.37488	−0.348047
$159$	14.3194	1.13560
$160$	1.44593	0.114311
$161$	2.13296	0.168101
$162$	−11.0426	−0.867587
$163$	−14.1560	−1.10879	−0.554393	−	0.832255i	$-0.687050\pi$
$163$	−14.1560	−1.10879	−0.554393	+	0.832255i	$0.687050\pi$
$164$	−3.00389	−0.234565
$165$	15.1068	1.17606
$166$	12.7989	0.993385
$167$	8.75105	0.677176	0.338588	−	0.940935i	$-0.390051\pi$
$167$	8.75105	0.677176	0.338588	+	0.940935i	$0.390051\pi$
$168$	−2.93990	−0.226818
$169$	−11.3620	−0.873997
$170$	−2.54256	−0.195005
$171$	−3.58904	−0.274461
$172$	−3.28239	−0.250280
$173$	−20.5300	−1.56087	−0.780433	−	0.625239i	$-0.785001\pi$
$173$	−20.5300	−1.56087	−0.780433	+	0.625239i	$0.785001\pi$
$174$	−11.2217	−0.850717
$175$	−3.84218	−0.290441
$176$	−4.69336	−0.353776
$177$	−2.95647	−0.222222
$178$	−12.9009	−0.966961
$179$	−7.14274	−0.533874	−0.266937	−	0.963714i	$-0.586012\pi$
$179$	−7.14274	−0.533874	−0.266937	+	0.963714i	$0.586012\pi$
$180$	2.82742	0.210744
$181$	−4.22490	−0.314034	−0.157017	−	0.987596i	$-0.550188\pi$
$181$	−4.22490	−0.314034	−0.157017	+	0.987596i	$0.550188\pi$
$182$	1.69026	0.125290
$183$	5.25762	0.388655
$184$	1.61507	0.119065
$185$	−2.42114	−0.178006
$186$	−8.99849	−0.659802
$187$	8.25291	0.603512
$188$	6.32528	0.461318
$189$	3.07092	0.223377
$190$	−2.65389	−0.192534
$191$	5.89653	0.426658	0.213329	−	0.976980i	$-0.431569\pi$
$191$	5.89653	0.426658	0.213329	+	0.976980i	$0.431569\pi$
$192$	−2.22608	−0.160653
$193$	22.3976	1.61222	0.806108	−	0.591768i	$-0.201570\pi$
$193$	22.3976	1.61222	0.806108	+	0.591768i	$0.201570\pi$
$194$	−9.16110	−0.657729
$195$	−4.11955	−0.295007
$196$	−5.25585	−0.375418
$197$	−11.9420	−0.850833	−0.425417	−	0.904998i	$-0.639872\pi$
$197$	−11.9420	−0.850833	−0.425417	+	0.904998i	$0.639872\pi$
$198$	−9.17755	−0.652220
$199$	−2.13617	−0.151429	−0.0757147	−	0.997130i	$-0.524124\pi$
$199$	−2.13617	−0.151429	−0.0757147	+	0.997130i	$0.524124\pi$
$200$	−2.90928	−0.205717
$201$	−17.8495	−1.25901
$202$	−9.75684	−0.686489
$203$	6.65749	0.467264
$204$	3.91438	0.274062
$205$	−4.34343	−0.303358
$206$	−13.9265	−0.970308
$207$	3.15816	0.219507
$208$	1.27986	0.0887421
$209$	8.61430	0.595864
$210$	−4.25090	−0.293340
$211$	9.83082	0.676781	0.338391	−	0.941006i	$-0.390117\pi$
$211$	9.83082	0.676781	0.338391	+	0.941006i	$0.390117\pi$
$212$	−6.43256	−0.441790
$213$	25.0260	1.71475
$214$	0.0699806	0.00478377
$215$	−4.74612	−0.323683
$216$	2.32529	0.158216
$217$	5.33852	0.362402
$218$	7.14893	0.484187
$219$	2.02180	0.136621
$220$	−6.78628	−0.457531
$221$	−2.25053	−0.151387
$222$	3.72746	0.250171
$223$	−17.5616	−1.17601	−0.588005	−	0.808857i	$-0.700087\pi$
$223$	−17.5616	−1.17601	−0.588005	+	0.808857i	$0.700087\pi$
$224$	1.32066	0.0882405
$225$	−5.68890	−0.379260
$226$	18.1278	1.20585
$227$	−4.98834	−0.331088	−0.165544	−	0.986202i	$-0.552938\pi$
$227$	−4.98834	−0.331088	−0.165544	+	0.986202i	$0.552938\pi$
$228$	4.08579	0.270588
$229$	−4.96024	−0.327782	−0.163891	−	0.986478i	$-0.552405\pi$
$229$	−4.96024	−0.327782	−0.163891	+	0.986478i	$0.552405\pi$
$230$	2.33528	0.153984
$231$	13.7980	0.907843
$232$	5.04102	0.330959
$233$	3.88242	0.254346	0.127173	−	0.991881i	$-0.459410\pi$
$233$	3.88242	0.254346	0.127173	+	0.991881i	$0.459410\pi$
$234$	2.50267	0.163605
$235$	9.14593	0.596614
$236$	1.32811	0.0864524
$237$	9.73883	0.632605
$238$	−2.32228	−0.150531
$239$	19.0246	1.23060	0.615299	−	0.788294i	$-0.289035\pi$
$239$	19.0246	1.23060	0.615299	+	0.788294i	$0.289035\pi$
$240$	−3.21876	−0.207770
$241$	8.14953	0.524957	0.262479	−	0.964938i	$-0.415460\pi$
$241$	8.14953	0.524957	0.262479	+	0.964938i	$0.415460\pi$
$242$	11.0277	0.708885
$243$	17.6058	1.12941
$244$	−2.36183	−0.151201
$245$	−7.59960	−0.485521
$246$	6.68691	0.426342
$247$	−2.34908	−0.149468
$248$	4.04230	0.256687
$249$	−28.4913	−1.80556
$250$	−11.4363	−0.723294
$251$	1.16166	0.0733236	0.0366618	−	0.999328i	$-0.488328\pi$
$251$	1.16166	0.0733236	0.0366618	+	0.999328i	$0.488328\pi$
$252$	2.58247	0.162680
$253$	−7.58011	−0.476557
$254$	15.1331	0.949538
$255$	5.65994	0.354439
$256$	1.00000	0.0625000
$257$	25.1089	1.56625	0.783126	−	0.621863i	$-0.213624\pi$
$257$	25.1089	1.56625	0.783126	+	0.621863i	$0.213624\pi$
$258$	7.30687	0.454906
$259$	−2.21138	−0.137409
$260$	1.85059	0.114768
$261$	9.85738	0.610157
$262$	−5.41604	−0.334604
$263$	6.73112	0.415058	0.207529	−	0.978229i	$-0.433458\pi$
$263$	6.73112	0.415058	0.207529	+	0.978229i	$0.433458\pi$
$264$	10.4478	0.643018
$265$	−9.30104	−0.571359
$266$	−2.42397	−0.148623
$267$	28.7184	1.75754
$268$	8.01836	0.489799
$269$	−6.39523	−0.389924	−0.194962	−	0.980811i	$-0.562458\pi$
$269$	−6.39523	−0.389924	−0.194962	+	0.980811i	$0.562458\pi$
$270$	3.36221	0.204618
$271$	9.50505	0.577390	0.288695	−	0.957421i	$-0.406779\pi$
$271$	9.50505	0.577390	0.288695	+	0.957421i	$0.406779\pi$
$272$	−1.75842	−0.106620
$273$	−3.76265	−0.227726
$274$	−16.3466	−0.987533
$275$	13.6543	0.823386
$276$	−3.59527	−0.216410
$277$	−22.0202	−1.32307	−0.661533	−	0.749916i	$-0.730094\pi$
$277$	−22.0202	−1.32307	−0.661533	+	0.749916i	$0.730094\pi$
$278$	−9.63523	−0.577883
$279$	7.90445	0.473227
$280$	1.90959	0.114120
$281$	−30.9873	−1.84855	−0.924275	−	0.381728i	$-0.875329\pi$
$281$	−30.9873	−1.84855	−0.924275	+	0.381728i	$0.875329\pi$
$282$	−14.0806	−0.838486
$283$	16.6965	0.992505	0.496253	−	0.868178i	$-0.334709\pi$
$283$	16.6965	0.992505	0.496253	+	0.868178i	$0.334709\pi$
$284$	−11.2422	−0.667100
$285$	5.90778	0.349947
$286$	−6.00683	−0.355191
$287$	−3.96713	−0.234172
$288$	1.95543	0.115225
$289$	−13.9080	−0.818115
$290$	7.28898	0.428024
$291$	20.3933	1.19548
$292$	−0.908233	−0.0531503
$293$	6.92641	0.404645	0.202322	−	0.979319i	$-0.435151\pi$
$293$	6.92641	0.404645	0.202322	+	0.979319i	$0.435151\pi$
$294$	11.6999	0.682355
$295$	1.92035	0.111807
$296$	−1.67445	−0.0973254
$297$	−10.9134	−0.633262
$298$	3.55096	0.205702
$299$	2.06706	0.119541
$300$	6.47629	0.373909
$301$	−4.33494	−0.249862
$302$	8.65736	0.498175
$303$	21.7195	1.24775
$304$	−1.83542	−0.105269
$305$	−3.41505	−0.195545
$306$	−3.43847	−0.196564
$307$	6.50377	0.371190	0.185595	−	0.982626i	$-0.440579\pi$
$307$	6.50377	0.371190	0.185595	+	0.982626i	$0.440579\pi$
$308$	−6.19835	−0.353184
$309$	31.0016	1.76362
$310$	5.84490	0.331968
$311$	−29.7435	−1.68660	−0.843300	−	0.537444i	$-0.819390\pi$
$311$	−29.7435	−1.68660	−0.843300	+	0.537444i	$0.819390\pi$
$312$	−2.84906	−0.161296
$313$	−10.7975	−0.610313	−0.305156	−	0.952302i	$-0.598709\pi$
$313$	−10.7975	−0.610313	−0.305156	+	0.952302i	$0.598709\pi$
$314$	−18.1654	−1.02513
$315$	3.73407	0.210391
$316$	−4.37488	−0.246106
$317$	−2.34579	−0.131753	−0.0658765	−	0.997828i	$-0.520984\pi$
$317$	−2.34579	−0.131753	−0.0658765	+	0.997828i	$0.520984\pi$
$318$	14.3194	0.802992
$319$	−23.6594	−1.32467
$320$	1.44593	0.0808301
$321$	−0.155782	−0.00869492
$322$	2.13296	0.118865
$323$	3.22744	0.179580
$324$	−11.0426	−0.613477
$325$	−3.72346	−0.206540
$326$	−14.1560	−0.784030
$327$	−15.9141	−0.880051
$328$	−3.00389	−0.165862
$329$	8.35356	0.460547
$330$	15.1068	0.831603
$331$	0.327186	0.0179837	0.00899187	−	0.999960i	$-0.497138\pi$
$331$	0.327186	0.0179837	0.00899187	+	0.999960i	$0.497138\pi$
$332$	12.7989	0.702429
$333$	−3.27427	−0.179429
$334$	8.75105	0.478836
$335$	11.5940	0.633448
$336$	−2.93990	−0.160385
$337$	−6.08028	−0.331214	−0.165607	−	0.986192i	$-0.552958\pi$
$337$	−6.08028	−0.331214	−0.165607	+	0.986192i	$0.552958\pi$
$338$	−11.3620	−0.618010
$339$	−40.3540	−2.19173
$340$	−2.54256	−0.137890
$341$	−18.9720	−1.02739
$342$	−3.58904	−0.194073
$343$	−16.1858	−0.873953
$344$	−3.28239	−0.176975
$345$	−5.19852	−0.279879
$346$	−20.5300	−1.10370
$347$	−11.3965	−0.611794	−0.305897	−	0.952065i	$-0.598956\pi$
$347$	−11.3965	−0.611794	−0.305897	+	0.952065i	$0.598956\pi$
$348$	−11.2217	−0.601547
$349$	10.3551	0.554293	0.277147	−	0.960828i	$-0.410611\pi$
$349$	10.3551	0.554293	0.277147	+	0.960828i	$0.410611\pi$
$350$	−3.84218	−0.205373
$351$	2.97604	0.158849
$352$	−4.69336	−0.250157
$353$	−11.2397	−0.598230	−0.299115	−	0.954217i	$-0.596691\pi$
$353$	−11.2397	−0.598230	−0.299115	+	0.954217i	$0.596691\pi$
$354$	−2.95647	−0.157135
$355$	−16.2554	−0.862748
$356$	−12.9009	−0.683745
$357$	5.16958	0.273603
$358$	−7.14274	−0.377506
$359$	−5.87584	−0.310115	−0.155057	−	0.987905i	$-0.549556\pi$
$359$	−5.87584	−0.310115	−0.155057	+	0.987905i	$0.549556\pi$
$360$	2.82742	0.149018
$361$	−15.6312	−0.822696
$362$	−4.22490	−0.222056
$363$	−24.5484	−1.28846
$364$	1.69026	0.0885936
$365$	−1.31324	−0.0687383
$366$	5.25762	0.274820
$367$	−13.3605	−0.697415	−0.348707	−	0.937232i	$-0.613379\pi$
$367$	−13.3605	−0.697415	−0.348707	+	0.937232i	$0.613379\pi$
$368$	1.61507	0.0841913
$369$	−5.87391	−0.305784
$370$	−2.42114	−0.125869
$371$	−8.49524	−0.441051
$372$	−8.99849	−0.466550
$373$	34.0479	1.76293	0.881467	−	0.472245i	$-0.156556\pi$
$373$	34.0479	1.76293	0.881467	+	0.472245i	$0.156556\pi$
$374$	8.25291	0.426748
$375$	25.4581	1.31465
$376$	6.32528	0.326201
$377$	6.45179	0.332284
$378$	3.07092	0.157951
$379$	−25.2972	−1.29943	−0.649715	−	0.760178i	$-0.725112\pi$
$379$	−25.2972	−1.29943	−0.649715	+	0.760178i	$0.725112\pi$
$380$	−2.65389	−0.136142
$381$	−33.6876	−1.72587
$382$	5.89653	0.301693
$383$	17.0415	0.870782	0.435391	−	0.900241i	$-0.356610\pi$
$383$	17.0415	0.870782	0.435391	+	0.900241i	$0.356610\pi$
$384$	−2.22608	−0.113599
$385$	−8.96239	−0.456766
$386$	22.3976	1.14001
$387$	−6.41850	−0.326271
$388$	−9.16110	−0.465085
$389$	18.9159	0.959074	0.479537	−	0.877522i	$-0.340805\pi$
$389$	18.9159	0.959074	0.479537	+	0.877522i	$0.340805\pi$
$390$	−4.11955	−0.208602
$391$	−2.83997	−0.143624
$392$	−5.25585	−0.265461
$393$	12.0565	0.608172
$394$	−11.9420	−0.601630
$395$	−6.32578	−0.318284
$396$	−9.17755	−0.461189
$397$	−8.33504	−0.418323	−0.209162	−	0.977881i	$-0.567074\pi$
$397$	−8.33504	−0.418323	−0.209162	+	0.977881i	$0.567074\pi$
$398$	−2.13617	−0.107077
$399$	5.39596	0.270136
$400$	−2.90928	−0.145464
$401$	−15.2801	−0.763054	−0.381527	−	0.924358i	$-0.624602\pi$
$401$	−15.2801	−0.763054	−0.381527	+	0.924358i	$0.624602\pi$
$402$	−17.8495	−0.890253
$403$	5.17357	0.257714
$404$	−9.75684	−0.485421
$405$	−15.9668	−0.793398
$406$	6.65749	0.330406
$407$	7.85880	0.389546
$408$	3.91438	0.193791
$409$	−12.4078	−0.613527	−0.306764	−	0.951786i	$-0.599246\pi$
$409$	−12.4078	−0.613527	−0.306764	+	0.951786i	$0.599246\pi$
$410$	−4.34343	−0.214507
$411$	36.3888	1.79493
$412$	−13.9265	−0.686112
$413$	1.75398	0.0863078
$414$	3.15816	0.155215
$415$	18.5063	0.908439
$416$	1.27986	0.0627501
$417$	21.4488	1.05035
$418$	8.61430	0.421339
$419$	−19.6693	−0.960907	−0.480453	−	0.877020i	$-0.659528\pi$
$419$	−19.6693	−0.960907	−0.480453	+	0.877020i	$0.659528\pi$
$420$	−4.25090	−0.207423
$421$	−14.0490	−0.684704	−0.342352	−	0.939572i	$-0.611224\pi$
$421$	−14.0490	−0.684704	−0.342352	+	0.939572i	$0.611224\pi$
$422$	9.83082	0.478557
$423$	12.3687	0.601385
$424$	−6.43256	−0.312393
$425$	5.11574	0.248150
$426$	25.0260	1.21251
$427$	−3.11918	−0.150948
$428$	0.0699806	0.00338264
$429$	13.3717	0.645591
$430$	−4.74612	−0.228878
$431$	15.8254	0.762283	0.381142	−	0.924517i	$-0.375531\pi$
$431$	15.8254	0.762283	0.381142	+	0.924517i	$0.375531\pi$
$432$	2.32529	0.111876
$433$	−3.72748	−0.179131	−0.0895657	−	0.995981i	$-0.528548\pi$
$433$	−3.72748	−0.179131	−0.0895657	+	0.995981i	$0.528548\pi$
$434$	5.33852	0.256257
$435$	−16.2258	−0.777970
$436$	7.14893	0.342372
$437$	−2.96433	−0.141803
$438$	2.02180	0.0966053
$439$	2.55266	0.121832	0.0609160	−	0.998143i	$-0.480598\pi$
$439$	2.55266	0.121832	0.0609160	+	0.998143i	$0.480598\pi$
$440$	−6.78628	−0.323523
$441$	−10.2775	−0.489403
$442$	−2.25053	−0.107047
$443$	−7.30889	−0.347256	−0.173628	−	0.984811i	$-0.555549\pi$
$443$	−7.30889	−0.347256	−0.173628	+	0.984811i	$0.555549\pi$
$444$	3.72746	0.176897
$445$	−18.6538	−0.884274
$446$	−17.5616	−0.831565
$447$	−7.90473	−0.373881
$448$	1.32066	0.0623954
$449$	5.87689	0.277347	0.138674	−	0.990338i	$-0.455716\pi$
$449$	5.87689	0.277347	0.138674	+	0.990338i	$0.455716\pi$
$450$	−5.68890	−0.268177
$451$	14.0984	0.663866
$452$	18.1278	0.852662
$453$	−19.2720	−0.905476
$454$	−4.98834	−0.234114
$455$	2.44400	0.114576
$456$	4.08579	0.191335
$457$	−28.3939	−1.32821	−0.664105	−	0.747639i	$-0.731187\pi$
$457$	−28.3939	−1.32821	−0.664105	+	0.747639i	$0.731187\pi$
$458$	−4.96024	−0.231777
$459$	−4.08884	−0.190851
$460$	2.33528	0.108883
$461$	−26.7478	−1.24577	−0.622885	−	0.782313i	$-0.714040\pi$
$461$	−26.7478	−1.24577	−0.622885	+	0.782313i	$0.714040\pi$
$462$	13.7980	0.641942
$463$	−36.7289	−1.70694	−0.853468	−	0.521145i	$-0.825505\pi$
$463$	−36.7289	−1.70694	−0.853468	+	0.521145i	$0.825505\pi$
$464$	5.04102	0.234024
$465$	−13.0112	−0.603381
$466$	3.88242	0.179850
$467$	−7.38286	−0.341638	−0.170819	−	0.985302i	$-0.554641\pi$
$467$	−7.38286	−0.341638	−0.170819	+	0.985302i	$0.554641\pi$
$468$	2.50267	0.115686
$469$	10.5896	0.488980
$470$	9.14593	0.421870
$471$	40.4376	1.86326
$472$	1.32811	0.0611311
$473$	15.4055	0.708344
$474$	9.73883	0.447319
$475$	5.33975	0.245005
$476$	−2.32228	−0.106442
$477$	−12.5784	−0.575927
$478$	19.0246	0.870164
$479$	−11.6682	−0.533132	−0.266566	−	0.963817i	$-0.585889\pi$
$479$	−11.6682	−0.533132	−0.266566	+	0.963817i	$0.585889\pi$
$480$	−3.21876	−0.146916
$481$	−2.14306	−0.0977149
$482$	8.14953	0.371201
$483$	−4.74814	−0.216048
$484$	11.0277	0.501257
$485$	−13.2463	−0.601485
$486$	17.6058	0.798615
$487$	−19.5412	−0.885495	−0.442748	−	0.896646i	$-0.645996\pi$
$487$	−19.5412	−0.885495	−0.442748	+	0.896646i	$0.645996\pi$
$488$	−2.36183	−0.106915
$489$	31.5125	1.42504
$490$	−7.59960	−0.343315
$491$	1.52840	0.0689757	0.0344879	−	0.999405i	$-0.489020\pi$
$491$	1.52840	0.0689757	0.0344879	+	0.999405i	$0.489020\pi$
$492$	6.68691	0.301469
$493$	−8.86424	−0.399225
$494$	−2.34908	−0.105690
$495$	−13.2701	−0.596448
$496$	4.04230	0.181505
$497$	−14.8471	−0.665984
$498$	−28.4913	−1.27673
$499$	−42.1288	−1.88595	−0.942973	−	0.332870i	$-0.891983\pi$
$499$	−42.1288	−1.88595	−0.942973	+	0.332870i	$0.891983\pi$
$500$	−11.4363	−0.511446
$501$	−19.4805	−0.870326
$502$	1.16166	0.0518476
$503$	12.3148	0.549088	0.274544	−	0.961575i	$-0.411473\pi$
$503$	12.3148	0.549088	0.274544	+	0.961575i	$0.411473\pi$
$504$	2.58247	0.115032
$505$	−14.1077	−0.627786
$506$	−7.58011	−0.336977
$507$	25.2926	1.12329
$508$	15.1331	0.671425
$509$	12.1132	0.536909	0.268455	−	0.963292i	$-0.413487\pi$
$509$	12.1132	0.536909	0.268455	+	0.963292i	$0.413487\pi$
$510$	5.65994	0.250626
$511$	−1.19947	−0.0530614
$512$	1.00000	0.0441942
$513$	−4.26789	−0.188432
$514$	25.1089	1.10751
$515$	−20.1368	−0.887335
$516$	7.30687	0.321667
$517$	−29.6868	−1.30563
$518$	−2.21138	−0.0971626
$519$	45.7014	2.00607
$520$	1.85059	0.0811536
$521$	−26.4763	−1.15995	−0.579974	−	0.814635i	$-0.696937\pi$
$521$	−26.4763	−1.15995	−0.579974	+	0.814635i	$0.696937\pi$
$522$	9.85738	0.431446
$523$	10.3520	0.452660	0.226330	−	0.974051i	$-0.427327\pi$
$523$	10.3520	0.452660	0.226330	+	0.974051i	$0.427327\pi$
$524$	−5.41604	−0.236601
$525$	8.55299	0.373283
$526$	6.73112	0.293491
$527$	−7.10807	−0.309632
$528$	10.4478	0.454682
$529$	−20.3916	−0.886589
$530$	−9.30104	−0.404012
$531$	2.59702	0.112701
$532$	−2.42397	−0.105093
$533$	−3.84455	−0.166526
$534$	28.7184	1.24277
$535$	0.101187	0.00437470
$536$	8.01836	0.346341
$537$	15.9003	0.686149
$538$	−6.39523	−0.275718
$539$	24.6676	1.06251
$540$	3.36221	0.144687
$541$	27.3982	1.17794	0.588971	−	0.808154i	$-0.299533\pi$
$541$	27.3982	1.17794	0.588971	+	0.808154i	$0.299533\pi$
$542$	9.50505	0.408277
$543$	9.40496	0.403606
$544$	−1.75842	−0.0753917
$545$	10.3369	0.442783
$546$	−3.76265	−0.161027
$547$	−30.1829	−1.29053	−0.645264	−	0.763959i	$-0.723253\pi$
$547$	−30.1829	−1.29053	−0.645264	+	0.763959i	$0.723253\pi$
$548$	−16.3466	−0.698291
$549$	−4.61840	−0.197108
$550$	13.6543	0.582222
$551$	−9.25240	−0.394166
$552$	−3.59527	−0.153025
$553$	−5.77774	−0.245694
$554$	−22.0202	−0.935549
$555$	5.38965	0.228778
$556$	−9.63523	−0.408625
$557$	7.68484	0.325617	0.162809	−	0.986658i	$-0.447945\pi$
$557$	7.68484	0.325617	0.162809	+	0.986658i	$0.447945\pi$
$558$	7.90445	0.334622
$559$	−4.20099	−0.177683
$560$	1.90959	0.0806948
$561$	−18.3716	−0.775651
$562$	−30.9873	−1.30712
$563$	27.6489	1.16526	0.582631	−	0.812737i	$-0.302023\pi$
$563$	27.6489	1.16526	0.582631	+	0.812737i	$0.302023\pi$
$564$	−14.0806	−0.592899
$565$	26.2116	1.10273
$566$	16.6965	0.701807
$567$	−14.5835	−0.612450
$568$	−11.2422	−0.471711
$569$	−29.8901	−1.25306	−0.626530	−	0.779398i	$-0.715525\pi$
$569$	−29.8901	−1.25306	−0.626530	+	0.779398i	$0.715525\pi$
$570$	5.90778	0.247450
$571$	−3.17758	−0.132978	−0.0664888	−	0.997787i	$-0.521180\pi$
$571$	−3.17758	−0.132978	−0.0664888	+	0.997787i	$0.521180\pi$
$572$	−6.00683	−0.251158
$573$	−13.1262	−0.548353
$574$	−3.96713	−0.165585
$575$	−4.69869	−0.195949
$576$	1.95543	0.0814763
$577$	−15.8826	−0.661202	−0.330601	−	0.943771i	$-0.607251\pi$
$577$	−15.8826	−0.661202	−0.330601	+	0.943771i	$0.607251\pi$
$578$	−13.9080	−0.578495
$579$	−49.8589	−2.07207
$580$	7.28898	0.302658
$581$	16.9030	0.701254
$582$	20.3933	0.845332
$583$	30.1903	1.25036
$584$	−0.908233	−0.0375829
$585$	3.61870	0.149615
$586$	6.92641	0.286127
$587$	18.2326	0.752540	0.376270	−	0.926510i	$-0.377207\pi$
$587$	18.2326	0.752540	0.376270	+	0.926510i	$0.377207\pi$
$588$	11.6999	0.482498
$589$	−7.41933	−0.305708
$590$	1.92035	0.0790597
$591$	26.5839	1.09351
$592$	−1.67445	−0.0688195
$593$	8.27513	0.339819	0.169909	−	0.985460i	$-0.445652\pi$
$593$	8.27513	0.339819	0.169909	+	0.985460i	$0.445652\pi$
$594$	−10.9134	−0.447784
$595$	−3.35786	−0.137659
$596$	3.55096	0.145453
$597$	4.75529	0.194621
$598$	2.06706	0.0845283
$599$	−5.06788	−0.207068	−0.103534	−	0.994626i	$-0.533015\pi$
$599$	−5.06788	−0.207068	−0.103534	+	0.994626i	$0.533015\pi$
$600$	6.47629	0.264393
$601$	−23.3877	−0.954004	−0.477002	−	0.878902i	$-0.658277\pi$
$601$	−23.3877	−0.954004	−0.477002	+	0.878902i	$0.658277\pi$
$602$	−4.33494	−0.176679
$603$	15.6794	0.638513
$604$	8.65736	0.352263
$605$	15.9452	0.648266
$606$	21.7195	0.882295
$607$	20.6447	0.837941	0.418971	−	0.908000i	$-0.362391\pi$
$607$	20.6447	0.837941	0.418971	+	0.908000i	$0.362391\pi$
$608$	−1.83542	−0.0744362
$609$	−14.8201	−0.600541
$610$	−3.41505	−0.138271
$611$	8.09545	0.327507
$612$	−3.43847	−0.138992
$613$	−11.4334	−0.461790	−0.230895	−	0.972979i	$-0.574165\pi$
$613$	−11.4334	−0.461790	−0.230895	+	0.972979i	$0.574165\pi$
$614$	6.50377	0.262471
$615$	9.66882	0.389884
$616$	−6.19835	−0.249739
$617$	2.24546	0.0903986	0.0451993	−	0.998978i	$-0.485608\pi$
$617$	2.24546	0.0903986	0.0451993	+	0.998978i	$0.485608\pi$
$618$	31.0016	1.24707
$619$	15.7379	0.632562	0.316281	−	0.948666i	$-0.397566\pi$
$619$	15.7379	0.632562	0.316281	+	0.948666i	$0.397566\pi$
$620$	5.84490	0.234737
$621$	3.75551	0.150703
$622$	−29.7435	−1.19261
$623$	−17.0377	−0.682601
$624$	−2.84906	−0.114054
$625$	−1.98969	−0.0795876
$626$	−10.7975	−0.431556
$627$	−19.1761	−0.765820
$628$	−18.1654	−0.724877
$629$	2.94439	0.117400
$630$	3.73407	0.148769
$631$	−27.0675	−1.07754	−0.538770	−	0.842453i	$-0.681111\pi$
$631$	−27.0675	−1.07754	−0.538770	+	0.842453i	$0.681111\pi$
$632$	−4.37488	−0.174023
$633$	−21.8842	−0.869818
$634$	−2.34579	−0.0931634
$635$	21.8815	0.868341
$636$	14.3194	0.567801
$637$	−6.72673	−0.266523
$638$	−23.6594	−0.936683
$639$	−21.9833	−0.869646
$640$	1.44593	0.0571555
$641$	18.1861	0.718309	0.359154	−	0.933278i	$-0.383065\pi$
$641$	18.1861	0.718309	0.359154	+	0.933278i	$0.383065\pi$
$642$	−0.155782	−0.00614824
$643$	25.3873	1.00118	0.500588	−	0.865686i	$-0.333117\pi$
$643$	25.3873	1.00118	0.500588	+	0.865686i	$0.333117\pi$
$644$	2.13296	0.0840505
$645$	10.5652	0.416006
$646$	3.22744	0.126982
$647$	−25.7397	−1.01193	−0.505967	−	0.862553i	$-0.668864\pi$
$647$	−25.7397	−1.01193	−0.505967	+	0.862553i	$0.668864\pi$
$648$	−11.0426	−0.433794
$649$	−6.23329	−0.244678
$650$	−3.72346	−0.146046
$651$	−11.8840	−0.465770
$652$	−14.1560	−0.554393
$653$	16.3795	0.640981	0.320491	−	0.947252i	$-0.396152\pi$
$653$	16.3795	0.640981	0.320491	+	0.947252i	$0.396152\pi$
$654$	−15.9141	−0.622290
$655$	−7.83122	−0.305991
$656$	−3.00389	−0.117282
$657$	−1.77599	−0.0692879
$658$	8.35356	0.325656
$659$	33.4879	1.30450	0.652251	−	0.758003i	$-0.273825\pi$
$659$	33.4879	1.30450	0.652251	+	0.758003i	$0.273825\pi$
$660$	15.1068	0.588032
$661$	14.0937	0.548180	0.274090	−	0.961704i	$-0.411623\pi$
$661$	14.0937	0.548180	0.274090	+	0.961704i	$0.411623\pi$
$662$	0.327186	0.0127164
$663$	5.00985	0.194566
$664$	12.7989	0.496693
$665$	−3.50490	−0.135914
$666$	−3.27427	−0.126876
$667$	8.14160	0.315244
$668$	8.75105	0.338588
$669$	39.0935	1.51144
$670$	11.5940	0.447916
$671$	11.0849	0.427929
$672$	−2.93990	−0.113409
$673$	−16.8891	−0.651025	−0.325513	−	0.945538i	$-0.605537\pi$
$673$	−16.8891	−0.651025	−0.325513	+	0.945538i	$0.605537\pi$
$674$	−6.08028	−0.234204
$675$	−6.76492	−0.260382
$676$	−11.3620	−0.436999
$677$	49.3306	1.89593	0.947965	−	0.318374i	$-0.103137\pi$
$677$	49.3306	1.89593	0.947965	+	0.318374i	$0.103137\pi$
$678$	−40.3540	−1.54979
$679$	−12.0987	−0.464306
$680$	−2.54256	−0.0975026
$681$	11.1044	0.425523
$682$	−18.9720	−0.726476
$683$	−44.9875	−1.72140	−0.860700	−	0.509112i	$-0.829974\pi$
$683$	−44.9875	−1.72140	−0.860700	+	0.509112i	$0.829974\pi$
$684$	−3.58904	−0.137230
$685$	−23.6360	−0.903087
$686$	−16.1858	−0.617978
$687$	11.0419	0.421275
$688$	−3.28239	−0.125140
$689$	−8.23275	−0.313643
$690$	−5.19852	−0.197904
$691$	14.5597	0.553876	0.276938	−	0.960888i	$-0.410680\pi$
$691$	14.5597	0.553876	0.276938	+	0.960888i	$0.410680\pi$
$692$	−20.5300	−0.780433
$693$	−12.1205	−0.460418
$694$	−11.3965	−0.432604
$695$	−13.9319	−0.528467
$696$	−11.2217	−0.425358
$697$	5.28211	0.200074
$698$	10.3551	0.391945
$699$	−8.64258	−0.326892
$700$	−3.84218	−0.145221
$701$	32.9733	1.24538	0.622692	−	0.782467i	$-0.286039\pi$
$701$	32.9733	1.24538	0.622692	+	0.782467i	$0.286039\pi$
$702$	2.97604	0.112323
$703$	3.07332	0.115912
$704$	−4.69336	−0.176888
$705$	−20.3596	−0.766786
$706$	−11.2397	−0.423013
$707$	−12.8855	−0.484609
$708$	−2.95647	−0.111111
$709$	−29.3855	−1.10360	−0.551798	−	0.833978i	$-0.686058\pi$
$709$	−29.3855	−1.10360	−0.551798	+	0.833978i	$0.686058\pi$
$710$	−16.2554	−0.610055
$711$	−8.55478	−0.320829
$712$	−12.9009	−0.483481
$713$	6.52860	0.244498
$714$	5.16958	0.193467
$715$	−8.68547	−0.324818
$716$	−7.14274	−0.266937
$717$	−42.3502	−1.58160
$718$	−5.87584	−0.219284
$719$	37.5592	1.40072	0.700362	−	0.713788i	$-0.253022\pi$
$719$	37.5592	1.40072	0.700362	+	0.713788i	$0.253022\pi$
$720$	2.82742	0.105372
$721$	−18.3923	−0.684964
$722$	−15.6312	−0.581734
$723$	−18.1415	−0.674690
$724$	−4.22490	−0.157017
$725$	−14.6657	−0.544672
$726$	−24.5484	−0.911078
$727$	−52.2273	−1.93700	−0.968501	−	0.249009i	$-0.919895\pi$
$727$	−52.2273	−1.93700	−0.968501	+	0.249009i	$0.919895\pi$
$728$	1.69026	0.0626451
$729$	−6.06417	−0.224599
$730$	−1.31324	−0.0486053
$731$	5.77183	0.213479
$732$	5.25762	0.194327
$733$	44.4111	1.64036	0.820181	−	0.572104i	$-0.193873\pi$
$733$	44.4111	1.64036	0.820181	+	0.572104i	$0.193873\pi$
$734$	−13.3605	−0.493147
$735$	16.9173	0.624005
$736$	1.61507	0.0595323
$737$	−37.6331	−1.38623
$738$	−5.87391	−0.216222
$739$	34.6183	1.27346	0.636728	−	0.771088i	$-0.280287\pi$
$739$	34.6183	1.27346	0.636728	+	0.771088i	$0.280287\pi$
$740$	−2.42114	−0.0890029
$741$	5.22923	0.192101
$742$	−8.49524	−0.311870
$743$	−15.8354	−0.580945	−0.290473	−	0.956883i	$-0.593813\pi$
$743$	−15.8354	−0.580945	−0.290473	+	0.956883i	$0.593813\pi$
$744$	−8.99849	−0.329901
$745$	5.13445	0.188112
$746$	34.0479	1.24658
$747$	25.0273	0.915702
$748$	8.25291	0.301756
$749$	0.0924207	0.00337698
$750$	25.4581	0.929598
$751$	10.9417	0.399268	0.199634	−	0.979871i	$-0.436025\pi$
$751$	10.9417	0.399268	0.199634	+	0.979871i	$0.436025\pi$
$752$	6.32528	0.230659
$753$	−2.58596	−0.0942375
$754$	6.45179	0.234960
$755$	12.5180	0.455575
$756$	3.07092	0.111688
$757$	−5.70389	−0.207311	−0.103656	−	0.994613i	$-0.533054\pi$
$757$	−5.70389	−0.207311	−0.103656	+	0.994613i	$0.533054\pi$
$758$	−25.2972	−0.918836
$759$	16.8739	0.612485
$760$	−2.65389	−0.0962669
$761$	50.8699	1.84403	0.922016	−	0.387151i	$-0.126541\pi$
$761$	50.8699	1.84403	0.922016	+	0.387151i	$0.126541\pi$
$762$	−33.6876	−1.22037
$763$	9.44132	0.341799
$764$	5.89653	0.213329
$765$	−4.97180	−0.179756
$766$	17.0415	0.615736
$767$	1.69979	0.0613757
$768$	−2.22608	−0.0803267
$769$	−12.7091	−0.458301	−0.229150	−	0.973391i	$-0.573595\pi$
$769$	−12.7091	−0.458301	−0.229150	+	0.973391i	$0.573595\pi$
$770$	−8.96239	−0.322982
$771$	−55.8945	−2.01299
$772$	22.3976	0.806108
$773$	17.6765	0.635781	0.317890	−	0.948127i	$-0.397026\pi$
$773$	17.6765	0.635781	0.317890	+	0.948127i	$0.397026\pi$
$774$	−6.41850	−0.230708
$775$	−11.7602	−0.422439
$776$	−9.16110	−0.328864
$777$	4.92272	0.176601
$778$	18.9159	0.678168
$779$	5.51341	0.197538
$780$	−4.11955	−0.147504
$781$	52.7636	1.88803
$782$	−2.83997	−0.101557
$783$	11.7218	0.418905
$784$	−5.25585	−0.187709
$785$	−26.2659	−0.937470
$786$	12.0565	0.430042
$787$	9.34658	0.333170	0.166585	−	0.986027i	$-0.446726\pi$
$787$	9.34658	0.333170	0.166585	+	0.986027i	$0.446726\pi$
$788$	−11.9420	−0.425417
$789$	−14.9840	−0.533445
$790$	−6.32578	−0.225061
$791$	23.9408	0.851236
$792$	−9.17755	−0.326110
$793$	−3.02280	−0.107343
$794$	−8.33504	−0.295799
$795$	20.7049	0.734326
$796$	−2.13617	−0.0757147
$797$	26.9195	0.953539	0.476769	−	0.879028i	$-0.341808\pi$
$797$	26.9195	0.953539	0.476769	+	0.879028i	$0.341808\pi$
$798$	5.39596	0.191015
$799$	−11.1225	−0.393486
$800$	−2.90928	−0.102859
$801$	−25.2268	−0.891345
$802$	−15.2801	−0.539561
$803$	4.26267	0.150426
$804$	−17.8495	−0.629504
$805$	3.08412	0.108701
$806$	5.17357	0.182231
$807$	14.2363	0.501141
$808$	−9.75684	−0.343244
$809$	−5.77202	−0.202934	−0.101467	−	0.994839i	$-0.532354\pi$
$809$	−5.77202	−0.202934	−0.101467	+	0.994839i	$0.532354\pi$
$810$	−15.9668	−0.561017
$811$	9.12700	0.320492	0.160246	−	0.987077i	$-0.448771\pi$
$811$	9.12700	0.320492	0.160246	+	0.987077i	$0.448771\pi$
$812$	6.65749	0.233632
$813$	−21.1590	−0.742078
$814$	7.85880	0.275451
$815$	−20.4687	−0.716986
$816$	3.91438	0.137031
$817$	6.02458	0.210773
$818$	−12.4078	−0.433829
$819$	3.30519	0.115493
$820$	−4.34343	−0.151679
$821$	19.2026	0.670174	0.335087	−	0.942187i	$-0.391234\pi$
$821$	19.2026	0.670174	0.335087	+	0.942187i	$0.391234\pi$
$822$	36.3888	1.26920
$823$	−9.99020	−0.348236	−0.174118	−	0.984725i	$-0.555707\pi$
$823$	−9.99020	−0.348236	−0.174118	+	0.984725i	$0.555707\pi$
$824$	−13.9265	−0.485154
$825$	−30.3956	−1.05824
$826$	1.75398	0.0610288
$827$	33.1679	1.15336	0.576680	−	0.816970i	$-0.304348\pi$
$827$	33.1679	1.15336	0.576680	+	0.816970i	$0.304348\pi$
$828$	3.15816	0.109754
$829$	−11.5001	−0.399416	−0.199708	−	0.979855i	$-0.563999\pi$
$829$	−11.5001	−0.399416	−0.199708	+	0.979855i	$0.563999\pi$
$830$	18.5063	0.642363
$831$	49.0187	1.70044
$832$	1.27986	0.0443710
$833$	9.24200	0.320216
$834$	21.4488	0.742711
$835$	12.6534	0.437890
$836$	8.61430	0.297932
$837$	9.39954	0.324895
$838$	−19.6693	−0.679464
$839$	−30.4146	−1.05003	−0.525015	−	0.851093i	$-0.675940\pi$
$839$	−30.4146	−1.05003	−0.525015	+	0.851093i	$0.675940\pi$
$840$	−4.25090	−0.146670
$841$	−3.58808	−0.123727
$842$	−14.0490	−0.484159
$843$	68.9803	2.37581
$844$	9.83082	0.338391
$845$	−16.4286	−0.565162
$846$	12.3687	0.425243
$847$	14.5638	0.500419
$848$	−6.43256	−0.220895
$849$	−37.1678	−1.27560
$850$	5.11574	0.175468
$851$	−2.70435	−0.0927040
$852$	25.0260	0.857375
$853$	23.9854	0.821246	0.410623	−	0.911805i	$-0.365311\pi$
$853$	23.9854	0.821246	0.410623	+	0.911805i	$0.365311\pi$
$854$	−3.11918	−0.106736
$855$	−5.18951	−0.177478
$856$	0.0699806	0.00239189
$857$	−12.2807	−0.419501	−0.209750	−	0.977755i	$-0.567265\pi$
$857$	−12.2807	−0.419501	−0.209750	+	0.977755i	$0.567265\pi$
$858$	13.3717	0.456502
$859$	−28.9811	−0.988822	−0.494411	−	0.869228i	$-0.664616\pi$
$859$	−28.9811	−0.988822	−0.494411	+	0.869228i	$0.664616\pi$
$860$	−4.74612	−0.161841
$861$	8.83115	0.300965
$862$	15.8254	0.539016
$863$	26.6918	0.908598	0.454299	−	0.890849i	$-0.349890\pi$
$863$	26.6918	0.908598	0.454299	+	0.890849i	$0.349890\pi$
$864$	2.32529	0.0791080
$865$	−29.6850	−1.00932
$866$	−3.72748	−0.126665
$867$	30.9602	1.05146
$868$	5.33852	0.181201
$869$	20.5329	0.696531
$870$	−16.2258	−0.550108
$871$	10.2624	0.347727
$872$	7.14893	0.242093
$873$	−17.9139	−0.606294
$874$	−2.96433	−0.100270
$875$	−15.1035	−0.510590
$876$	2.02180	0.0683103
$877$	16.4113	0.554171	0.277086	−	0.960845i	$-0.410631\pi$
$877$	16.4113	0.554171	0.277086	+	0.960845i	$0.410631\pi$
$878$	2.55266	0.0861482
$879$	−15.4187	−0.520061
$880$	−6.78628	−0.228766
$881$	−30.3579	−1.02278	−0.511391	−	0.859348i	$-0.670870\pi$
$881$	−30.3579	−1.02278	−0.511391	+	0.859348i	$0.670870\pi$
$882$	−10.2775	−0.346060
$883$	36.0715	1.21390	0.606951	−	0.794739i	$-0.292392\pi$
$883$	36.0715	1.21390	0.606951	+	0.794739i	$0.292392\pi$
$884$	−2.25053	−0.0756934
$885$	−4.27486	−0.143698
$886$	−7.30889	−0.245547
$887$	−16.2250	−0.544782	−0.272391	−	0.962187i	$-0.587814\pi$
$887$	−16.2250	−0.544782	−0.272391	+	0.962187i	$0.587814\pi$
$888$	3.72746	0.125085
$889$	19.9858	0.670301
$890$	−18.6538	−0.625276
$891$	51.8268	1.73626
$892$	−17.5616	−0.588005
$893$	−11.6096	−0.388499
$894$	−7.90473	−0.264374
$895$	−10.3279	−0.345224
$896$	1.32066	0.0441202
$897$	−4.60144	−0.153637
$898$	5.87689	0.196114
$899$	20.3774	0.679623
$900$	−5.68890	−0.189630
$901$	11.3111	0.376829
$902$	14.0984	0.469424
$903$	9.64991	0.321129
$904$	18.1278	0.602923
$905$	−6.10892	−0.203067
$906$	−19.2720	−0.640268
$907$	−43.8409	−1.45571	−0.727857	−	0.685729i	$-0.759484\pi$
$907$	−43.8409	−1.45571	−0.727857	+	0.685729i	$0.759484\pi$
$908$	−4.98834	−0.165544
$909$	−19.0788	−0.632805
$910$	2.44400	0.0810178
$911$	44.3590	1.46968	0.734840	−	0.678241i	$-0.237257\pi$
$911$	44.3590	1.46968	0.734840	+	0.678241i	$0.237257\pi$
$912$	4.08579	0.135294
$913$	−60.0698	−1.98802
$914$	−28.3939	−0.939187
$915$	7.60217	0.251320
$916$	−4.96024	−0.163891
$917$	−7.15276	−0.236205
$918$	−4.08884	−0.134952
$919$	52.9228	1.74576	0.872882	−	0.487932i	$-0.162249\pi$
$919$	52.9228	1.74576	0.872882	+	0.487932i	$0.162249\pi$
$920$	2.33528	0.0769919
$921$	−14.4779	−0.477064
$922$	−26.7478	−0.880893
$923$	−14.3884	−0.473599
$924$	13.7980	0.453922
$925$	4.87144	0.160172
$926$	−36.7289	−1.20699
$927$	−27.2324	−0.894430
$928$	5.04102	0.165480
$929$	−33.9073	−1.11246	−0.556232	−	0.831027i	$-0.687753\pi$
$929$	−33.9073	−1.11246	−0.556232	+	0.831027i	$0.687753\pi$
$930$	−13.0112	−0.426654
$931$	9.64670	0.316158
$932$	3.88242	0.127173
$933$	66.2114	2.16766
$934$	−7.38286	−0.241575
$935$	11.9331	0.390256
$936$	2.50267	0.0818024
$937$	31.5868	1.03190	0.515948	−	0.856620i	$-0.327440\pi$
$937$	31.5868	1.03190	0.515948	+	0.856620i	$0.327440\pi$
$938$	10.5896	0.345761
$939$	24.0362	0.784391
$940$	9.14593	0.298307
$941$	6.89053	0.224625	0.112312	−	0.993673i	$-0.464174\pi$
$941$	6.89053	0.224625	0.112312	+	0.993673i	$0.464174\pi$
$942$	40.4376	1.31753
$943$	−4.85150	−0.157987
$944$	1.32811	0.0432262
$945$	4.44035	0.144445
$946$	15.4055	0.500875
$947$	10.0686	0.327184	0.163592	−	0.986528i	$-0.447692\pi$
$947$	10.0686	0.327184	0.163592	+	0.986528i	$0.447692\pi$
$948$	9.73883	0.316302
$949$	−1.16241	−0.0377334
$950$	5.33975	0.173245
$951$	5.22193	0.169333
$952$	−2.32228	−0.0752655
$953$	−6.39412	−0.207126	−0.103563	−	0.994623i	$-0.533024\pi$
$953$	−6.39412	−0.207126	−0.103563	+	0.994623i	$0.533024\pi$
$954$	−12.5784	−0.407242
$955$	8.52599	0.275895
$956$	19.0246	0.615299
$957$	52.6676	1.70250
$958$	−11.6682	−0.376981
$959$	−21.5883	−0.697123
$960$	−3.21876	−0.103885
$961$	−14.6598	−0.472896
$962$	−2.14306	−0.0690949
$963$	0.136842	0.00440968
$964$	8.14953	0.262479
$965$	32.3854	1.04252
$966$	−4.74814	−0.152769
$967$	−32.0285	−1.02997	−0.514984	−	0.857200i	$-0.672202\pi$
$967$	−32.0285	−1.02997	−0.514984	+	0.857200i	$0.672202\pi$
$968$	11.0277	0.354442
$969$	−7.18455	−0.230801
$970$	−13.2463	−0.425314
$971$	37.6530	1.20834	0.604171	−	0.796854i	$-0.293504\pi$
$971$	37.6530	1.20834	0.604171	+	0.796854i	$0.293504\pi$
$972$	17.6058	0.564706
$973$	−12.7249	−0.407941
$974$	−19.5412	−0.626140
$975$	8.28872	0.265452
$976$	−2.36183	−0.0756003
$977$	−59.4357	−1.90152	−0.950759	−	0.309931i	$-0.899694\pi$
$977$	−59.4357	−1.90152	−0.950759	+	0.309931i	$0.899694\pi$
$978$	31.5125	1.00766
$979$	60.5485	1.93514
$980$	−7.59960	−0.242760
$981$	13.9792	0.446323
$982$	1.52840	0.0487732
$983$	51.6959	1.64884	0.824422	−	0.565975i	$-0.191500\pi$
$983$	51.6959	1.64884	0.824422	+	0.565975i	$0.191500\pi$
$984$	6.68691	0.213171
$985$	−17.2673	−0.550183
$986$	−8.86424	−0.282295
$987$	−18.5957	−0.591907
$988$	−2.34908	−0.0747341
$989$	−5.30130	−0.168571
$990$	−13.2701	−0.421752
$991$	30.2304	0.960299	0.480149	−	0.877187i	$-0.340582\pi$
$991$	30.2304	0.960299	0.480149	+	0.877187i	$0.340582\pi$
$992$	4.04230	0.128343
$993$	−0.728341	−0.0231132
$994$	−14.8471	−0.470922
$995$	−3.08876	−0.0979204
$996$	−28.4913	−0.902782
$997$	−6.31333	−0.199945	−0.0999725	−	0.994990i	$-0.531875\pi$
$997$	−6.31333	−0.199945	−0.0999725	+	0.994990i	$0.531875\pi$
$998$	−42.1288	−1.33356
$999$	−3.89358	−0.123188

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.12	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.12	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.22608 q^{3} +1.00000 q^{4} +1.44593 q^{5} -2.22608 q^{6} +1.32066 q^{7} +1.00000 q^{8} +1.95543 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.22608 q^{3} +1.00000 q^{4} +1.44593 q^{5} -2.22608 q^{6} +1.32066 q^{7} +1.00000 q^{8} +1.95543 q^{9} +1.44593 q^{10} -4.69336 q^{11} -2.22608 q^{12} +1.27986 q^{13} +1.32066 q^{14} -3.21876 q^{15} +1.00000 q^{16} -1.75842 q^{17} +1.95543 q^{18} -1.83542 q^{19} +1.44593 q^{20} -2.93990 q^{21} -4.69336 q^{22} +1.61507 q^{23} -2.22608 q^{24} -2.90928 q^{25} +1.27986 q^{26} +2.32529 q^{27} +1.32066 q^{28} +5.04102 q^{29} -3.21876 q^{30} +4.04230 q^{31} +1.00000 q^{32} +10.4478 q^{33} -1.75842 q^{34} +1.90959 q^{35} +1.95543 q^{36} -1.67445 q^{37} -1.83542 q^{38} -2.84906 q^{39} +1.44593 q^{40} -3.00389 q^{41} -2.93990 q^{42} -3.28239 q^{43} -4.69336 q^{44} +2.82742 q^{45} +1.61507 q^{46} +6.32528 q^{47} -2.22608 q^{48} -5.25585 q^{49} -2.90928 q^{50} +3.91438 q^{51} +1.27986 q^{52} -6.43256 q^{53} +2.32529 q^{54} -6.78628 q^{55} +1.32066 q^{56} +4.08579 q^{57} +5.04102 q^{58} +1.32811 q^{59} -3.21876 q^{60} -2.36183 q^{61} +4.04230 q^{62} +2.58247 q^{63} +1.00000 q^{64} +1.85059 q^{65} +10.4478 q^{66} +8.01836 q^{67} -1.75842 q^{68} -3.59527 q^{69} +1.90959 q^{70} -11.2422 q^{71} +1.95543 q^{72} -0.908233 q^{73} -1.67445 q^{74} +6.47629 q^{75} -1.83542 q^{76} -6.19835 q^{77} -2.84906 q^{78} -4.37488 q^{79} +1.44593 q^{80} -11.0426 q^{81} -3.00389 q^{82} +12.7989 q^{83} -2.93990 q^{84} -2.54256 q^{85} -3.28239 q^{86} -11.2217 q^{87} -4.69336 q^{88} -12.9009 q^{89} +2.82742 q^{90} +1.69026 q^{91} +1.61507 q^{92} -8.99849 q^{93} +6.32528 q^{94} -2.65389 q^{95} -2.22608 q^{96} -9.16110 q^{97} -5.25585 q^{98} -9.17755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 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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