LMFDB - Embedded newform 8031.2.a.b.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$102$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93543 q^{2} -1.00000 q^{3} +1.74588 q^{4} -1.47227 q^{5} +1.93543 q^{6} +2.60220 q^{7} +0.491826 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.93543 q^{2} -1.00000 q^{3} +1.74588 q^{4} -1.47227 q^{5} +1.93543 q^{6} +2.60220 q^{7} +0.491826 q^{8} +1.00000 q^{9} +2.84947 q^{10} -2.40787 q^{11} -1.74588 q^{12} +3.47825 q^{13} -5.03637 q^{14} +1.47227 q^{15} -4.44366 q^{16} +1.32542 q^{17} -1.93543 q^{18} +3.60647 q^{19} -2.57041 q^{20} -2.60220 q^{21} +4.66025 q^{22} +4.21996 q^{23} -0.491826 q^{24} -2.83243 q^{25} -6.73191 q^{26} -1.00000 q^{27} +4.54314 q^{28} -1.74461 q^{29} -2.84947 q^{30} +3.17126 q^{31} +7.61673 q^{32} +2.40787 q^{33} -2.56526 q^{34} -3.83113 q^{35} +1.74588 q^{36} -0.558331 q^{37} -6.98006 q^{38} -3.47825 q^{39} -0.724099 q^{40} +0.406250 q^{41} +5.03637 q^{42} -6.59414 q^{43} -4.20385 q^{44} -1.47227 q^{45} -8.16742 q^{46} +0.962569 q^{47} +4.44366 q^{48} -0.228550 q^{49} +5.48196 q^{50} -1.32542 q^{51} +6.07262 q^{52} -12.5376 q^{53} +1.93543 q^{54} +3.54502 q^{55} +1.27983 q^{56} -3.60647 q^{57} +3.37656 q^{58} -5.16945 q^{59} +2.57041 q^{60} +5.36889 q^{61} -6.13775 q^{62} +2.60220 q^{63} -5.85432 q^{64} -5.12092 q^{65} -4.66025 q^{66} +9.22787 q^{67} +2.31403 q^{68} -4.21996 q^{69} +7.41489 q^{70} -4.18089 q^{71} +0.491826 q^{72} +2.75633 q^{73} +1.08061 q^{74} +2.83243 q^{75} +6.29647 q^{76} -6.26575 q^{77} +6.73191 q^{78} -5.44566 q^{79} +6.54225 q^{80} +1.00000 q^{81} -0.786267 q^{82} -15.7493 q^{83} -4.54314 q^{84} -1.95138 q^{85} +12.7625 q^{86} +1.74461 q^{87} -1.18425 q^{88} -5.81491 q^{89} +2.84947 q^{90} +9.05112 q^{91} +7.36755 q^{92} -3.17126 q^{93} -1.86298 q^{94} -5.30968 q^{95} -7.61673 q^{96} +1.88961 q^{97} +0.442342 q^{98} -2.40787 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) $ 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * 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 $

$2$	−1.93543	−1.36855	−0.684277	−	0.729222i	$-0.739882\pi$
$2$	−1.93543	−1.36855	−0.684277	+	0.729222i	$0.739882\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.74588	0.872941
$5$	−1.47227	−0.658418	−0.329209	−	0.944257i	$-0.606782\pi$
$5$	−1.47227	−0.658418	−0.329209	+	0.944257i	$0.606782\pi$
$6$	1.93543	0.790135
$7$	2.60220	0.983540	0.491770	−	0.870725i	$-0.336350\pi$
$7$	2.60220	0.983540	0.491770	+	0.870725i	$0.336350\pi$
$8$	0.491826	0.173887
$9$	1.00000	0.333333
$10$	2.84947	0.901081
$11$	−2.40787	−0.725999	−0.362999	−	0.931789i	$-0.618247\pi$
$11$	−2.40787	−0.725999	−0.362999	+	0.931789i	$0.618247\pi$
$12$	−1.74588	−0.503993
$13$	3.47825	0.964694	0.482347	−	0.875980i	$-0.339784\pi$
$13$	3.47825	0.964694	0.482347	+	0.875980i	$0.339784\pi$
$14$	−5.03637	−1.34603
$15$	1.47227	0.380138
$16$	−4.44366	−1.11091
$17$	1.32542	0.321462	0.160731	−	0.986998i	$-0.448615\pi$
$17$	1.32542	0.321462	0.160731	+	0.986998i	$0.448615\pi$
$18$	−1.93543	−0.456185
$19$	3.60647	0.827381	0.413690	−	0.910418i	$-0.364240\pi$
$19$	3.60647	0.827381	0.413690	+	0.910418i	$0.364240\pi$
$20$	−2.57041	−0.574760
$21$	−2.60220	−0.567847
$22$	4.66025	0.993569
$23$	4.21996	0.879922	0.439961	−	0.898017i	$-0.354992\pi$
$23$	4.21996	0.879922	0.439961	+	0.898017i	$0.354992\pi$
$24$	−0.491826	−0.100394
$25$	−2.83243	−0.566486
$26$	−6.73191	−1.32024
$27$	−1.00000	−0.192450
$28$	4.54314	0.858572
$29$	−1.74461	−0.323965	−0.161983	−	0.986794i	$-0.551789\pi$
$29$	−1.74461	−0.323965	−0.161983	+	0.986794i	$0.551789\pi$
$30$	−2.84947	−0.520239
$31$	3.17126	0.569576	0.284788	−	0.958591i	$-0.408077\pi$
$31$	3.17126	0.569576	0.284788	+	0.958591i	$0.408077\pi$
$32$	7.61673	1.34646
$33$	2.40787	0.419156
$34$	−2.56526	−0.439939
$35$	−3.83113	−0.647580
$36$	1.74588	0.290980
$37$	−0.558331	−0.0917890	−0.0458945	−	0.998946i	$-0.514614\pi$
$37$	−0.558331	−0.0917890	−0.0458945	+	0.998946i	$0.514614\pi$
$38$	−6.98006	−1.13232
$39$	−3.47825	−0.556966
$40$	−0.724099	−0.114490
$41$	0.406250	0.0634456	0.0317228	−	0.999497i	$-0.489901\pi$
$41$	0.406250	0.0634456	0.0317228	+	0.999497i	$0.489901\pi$
$42$	5.03637	0.777129
$43$	−6.59414	−1.00560	−0.502798	−	0.864404i	$-0.667696\pi$
$43$	−6.59414	−1.00560	−0.502798	+	0.864404i	$0.667696\pi$
$44$	−4.20385	−0.633754
$45$	−1.47227	−0.219473
$46$	−8.16742	−1.20422
$47$	0.962569	0.140405	0.0702025	−	0.997533i	$-0.477635\pi$
$47$	0.962569	0.140405	0.0702025	+	0.997533i	$0.477635\pi$
$48$	4.44366	0.641387
$49$	−0.228550	−0.0326500
$50$	5.48196	0.775267
$51$	−1.32542	−0.185596
$52$	6.07262	0.842121
$53$	−12.5376	−1.72217	−0.861085	−	0.508461i	$-0.830214\pi$
$53$	−12.5376	−1.72217	−0.861085	+	0.508461i	$0.830214\pi$
$54$	1.93543	0.263378
$55$	3.54502	0.478010
$56$	1.27983	0.171024
$57$	−3.60647	−0.477688
$58$	3.37656	0.443364
$59$	−5.16945	−0.673005	−0.336502	−	0.941683i	$-0.609244\pi$
$59$	−5.16945	−0.673005	−0.336502	+	0.941683i	$0.609244\pi$
$60$	2.57041	0.331838
$61$	5.36889	0.687416	0.343708	−	0.939077i	$-0.388317\pi$
$61$	5.36889	0.687416	0.343708	+	0.939077i	$0.388317\pi$
$62$	−6.13775	−0.779495
$63$	2.60220	0.327847
$64$	−5.85432	−0.731790
$65$	−5.12092	−0.635172
$66$	−4.66025	−0.573637
$67$	9.22787	1.12736	0.563682	−	0.825992i	$-0.309384\pi$
$67$	9.22787	1.12736	0.563682	+	0.825992i	$0.309384\pi$
$68$	2.31403	0.280618
$69$	−4.21996	−0.508023
$70$	7.41489	0.886248
$71$	−4.18089	−0.496180	−0.248090	−	0.968737i	$-0.579803\pi$
$71$	−4.18089	−0.496180	−0.248090	+	0.968737i	$0.579803\pi$
$72$	0.491826	0.0579622
$73$	2.75633	0.322604	0.161302	−	0.986905i	$-0.448431\pi$
$73$	2.75633	0.322604	0.161302	+	0.986905i	$0.448431\pi$
$74$	1.08061	0.125618
$75$	2.83243	0.327061
$76$	6.29647	0.722255
$77$	−6.26575	−0.714048
$78$	6.73191	0.762239
$79$	−5.44566	−0.612685	−0.306343	−	0.951921i	$-0.599105\pi$
$79$	−5.44566	−0.612685	−0.306343	+	0.951921i	$0.599105\pi$
$80$	6.54225	0.731446
$81$	1.00000	0.111111
$82$	−0.786267	−0.0868287
$83$	−15.7493	−1.72871	−0.864356	−	0.502880i	$-0.832274\pi$
$83$	−15.7493	−1.72871	−0.864356	+	0.502880i	$0.832274\pi$
$84$	−4.54314	−0.495697
$85$	−1.95138	−0.211656
$86$	12.7625	1.37621
$87$	1.74461	0.187042
$88$	−1.18425	−0.126242
$89$	−5.81491	−0.616379	−0.308190	−	0.951325i	$-0.599723\pi$
$89$	−5.81491	−0.616379	−0.308190	+	0.951325i	$0.599723\pi$
$90$	2.84947	0.300360
$91$	9.05112	0.948815
$92$	7.36755	0.768120
$93$	−3.17126	−0.328845
$94$	−1.86298	−0.192152
$95$	−5.30968	−0.544762
$96$	−7.61673	−0.777379
$97$	1.88961	0.191861	0.0959303	−	0.995388i	$-0.469417\pi$
$97$	1.88961	0.191861	0.0959303	+	0.995388i	$0.469417\pi$
$98$	0.442342	0.0446832
$99$	−2.40787	−0.242000
$100$	−4.94509	−0.494509
$101$	−1.68701	−0.167864	−0.0839318	−	0.996472i	$-0.526748\pi$
$101$	−1.68701	−0.167864	−0.0839318	+	0.996472i	$0.526748\pi$
$102$	2.56526	0.253999
$103$	−12.0554	−1.18785	−0.593927	−	0.804519i	$-0.702423\pi$
$103$	−12.0554	−1.18785	−0.593927	+	0.804519i	$0.702423\pi$
$104$	1.71070	0.167748
$105$	3.83113	0.373880
$106$	24.2656	2.35688
$107$	−13.6493	−1.31952	−0.659762	−	0.751475i	$-0.729343\pi$
$107$	−13.6493	−1.31952	−0.659762	+	0.751475i	$0.729343\pi$
$108$	−1.74588	−0.167998
$109$	10.9339	1.04728	0.523638	−	0.851941i	$-0.324574\pi$
$109$	10.9339	1.04728	0.523638	+	0.851941i	$0.324574\pi$
$110$	−6.86113	−0.654183
$111$	0.558331	0.0529944
$112$	−11.5633	−1.09263
$113$	−1.34525	−0.126550	−0.0632750	−	0.997996i	$-0.520155\pi$
$113$	−1.34525	−0.126550	−0.0632750	+	0.997996i	$0.520155\pi$
$114$	6.98006	0.653743
$115$	−6.21290	−0.579356
$116$	−3.04588	−0.282803
$117$	3.47825	0.321565
$118$	10.0051	0.921044
$119$	3.44902	0.316171
$120$	0.724099	0.0661009
$121$	−5.20219	−0.472926
$122$	−10.3911	−0.940766
$123$	−0.406250	−0.0366303
$124$	5.53665	0.497206
$125$	11.5314	1.03140
$126$	−5.03637	−0.448676
$127$	3.43360	0.304683	0.152341	−	0.988328i	$-0.451319\pi$
$127$	3.43360	0.304683	0.152341	+	0.988328i	$0.451319\pi$
$128$	−3.90285	−0.344966
$129$	6.59414	0.580581
$130$	9.91117	0.869267
$131$	13.9966	1.22289	0.611445	−	0.791287i	$-0.290588\pi$
$131$	13.9966	1.22289	0.611445	+	0.791287i	$0.290588\pi$
$132$	4.20385	0.365898
$133$	9.38476	0.813762
$134$	−17.8599	−1.54286
$135$	1.47227	0.126713
$136$	0.651877	0.0558980
$137$	−11.4060	−0.974484	−0.487242	−	0.873267i	$-0.661997\pi$
$137$	−11.4060	−0.974484	−0.487242	+	0.873267i	$0.661997\pi$
$138$	8.16742	0.695257
$139$	18.0682	1.53253	0.766263	−	0.642527i	$-0.222114\pi$
$139$	18.0682	1.53253	0.766263	+	0.642527i	$0.222114\pi$
$140$	−6.68871	−0.565299
$141$	−0.962569	−0.0810629
$142$	8.09180	0.679049
$143$	−8.37517	−0.700367
$144$	−4.44366	−0.370305
$145$	2.56853	0.213305
$146$	−5.33467	−0.441501
$147$	0.228550	0.0188505
$148$	−0.974780	−0.0801264
$149$	−5.53288	−0.453271	−0.226636	−	0.973980i	$-0.572773\pi$
$149$	−5.53288	−0.453271	−0.226636	+	0.973980i	$0.572773\pi$
$150$	−5.48196	−0.447601
$151$	−11.8835	−0.967065	−0.483533	−	0.875326i	$-0.660647\pi$
$151$	−11.8835	−0.967065	−0.483533	+	0.875326i	$0.660647\pi$
$152$	1.77375	0.143871
$153$	1.32542	0.107154
$154$	12.1269	0.977214
$155$	−4.66895	−0.375019
$156$	−6.07262	−0.486199
$157$	−3.16027	−0.252217	−0.126108	−	0.992016i	$-0.540249\pi$
$157$	−3.16027	−0.252217	−0.126108	+	0.992016i	$0.540249\pi$
$158$	10.5397	0.838493
$159$	12.5376	0.994295
$160$	−11.2139	−0.886534
$161$	10.9812	0.865438
$162$	−1.93543	−0.152062
$163$	−15.6178	−1.22328	−0.611639	−	0.791137i	$-0.709490\pi$
$163$	−15.6178	−1.22328	−0.611639	+	0.791137i	$0.709490\pi$
$164$	0.709264	0.0553842
$165$	−3.54502	−0.275979
$166$	30.4817	2.36584
$167$	−22.4580	−1.73785	−0.868925	−	0.494944i	$-0.835188\pi$
$167$	−22.4580	−1.73785	−0.868925	+	0.494944i	$0.835188\pi$
$168$	−1.27983	−0.0987410
$169$	−0.901748	−0.0693652
$170$	3.77675	0.289663
$171$	3.60647	0.275794
$172$	−11.5126	−0.877827
$173$	0.262869	0.0199856	0.00999278	−	0.999950i	$-0.496819\pi$
$173$	0.262869	0.0199856	0.00999278	+	0.999950i	$0.496819\pi$
$174$	−3.37656	−0.255977
$175$	−7.37055	−0.557161
$176$	10.6997	0.806523
$177$	5.16945	0.388559
$178$	11.2543	0.843549
$179$	−2.58745	−0.193395	−0.0966975	−	0.995314i	$-0.530828\pi$
$179$	−2.58745	−0.193395	−0.0966975	+	0.995314i	$0.530828\pi$
$180$	−2.57041	−0.191587
$181$	17.4719	1.29867	0.649337	−	0.760501i	$-0.275046\pi$
$181$	17.4719	1.29867	0.649337	+	0.760501i	$0.275046\pi$
$182$	−17.5178	−1.29850
$183$	−5.36889	−0.396880
$184$	2.07548	0.153007
$185$	0.822012	0.0604355
$186$	6.13775	0.450042
$187$	−3.19144	−0.233381
$188$	1.68053	0.122565
$189$	−2.60220	−0.189282
$190$	10.2765	0.745537
$191$	−8.78795	−0.635873	−0.317937	−	0.948112i	$-0.602990\pi$
$191$	−8.78795	−0.635873	−0.317937	+	0.948112i	$0.602990\pi$
$192$	5.85432	0.422499
$193$	−17.6540	−1.27076	−0.635380	−	0.772200i	$-0.719156\pi$
$193$	−17.6540	−1.27076	−0.635380	+	0.772200i	$0.719156\pi$
$194$	−3.65720	−0.262572
$195$	5.12092	0.366717
$196$	−0.399021	−0.0285015
$197$	2.86455	0.204090	0.102045	−	0.994780i	$-0.467461\pi$
$197$	2.86455	0.204090	0.102045	+	0.994780i	$0.467461\pi$
$198$	4.66025	0.331190
$199$	−23.9740	−1.69947	−0.849735	−	0.527209i	$-0.823238\pi$
$199$	−23.9740	−1.69947	−0.849735	+	0.527209i	$0.823238\pi$
$200$	−1.39306	−0.0985044
$201$	−9.22787	−0.650884
$202$	3.26508	0.229731
$203$	−4.53982	−0.318633
$204$	−2.31403	−0.162015
$205$	−0.598108	−0.0417737
$206$	23.3324	1.62564
$207$	4.21996	0.293307
$208$	−15.4562	−1.07169
$209$	−8.68389	−0.600677
$210$	−7.41489	−0.511676
$211$	26.8883	1.85107	0.925535	−	0.378663i	$-0.123616\pi$
$211$	26.8883	1.85107	0.925535	+	0.378663i	$0.123616\pi$
$212$	−21.8892	−1.50335
$213$	4.18089	0.286470
$214$	26.4172	1.80584
$215$	9.70833	0.662103
$216$	−0.491826	−0.0334645
$217$	8.25226	0.560200
$218$	−21.1618	−1.43326
$219$	−2.75633	−0.186255
$220$	6.18919	0.417275
$221$	4.61016	0.310113
$222$	−1.08061	−0.0725257
$223$	24.3924	1.63343	0.816717	−	0.577038i	$-0.195792\pi$
$223$	24.3924	1.63343	0.816717	+	0.577038i	$0.195792\pi$
$224$	19.8203	1.32430
$225$	−2.83243	−0.188829
$226$	2.60363	0.173191
$227$	16.2983	1.08176	0.540879	−	0.841100i	$-0.318092\pi$
$227$	16.2983	1.08176	0.540879	+	0.841100i	$0.318092\pi$
$228$	−6.29647	−0.416994
$229$	12.6886	0.838489	0.419245	−	0.907873i	$-0.362295\pi$
$229$	12.6886	0.838489	0.419245	+	0.907873i	$0.362295\pi$
$230$	12.0246	0.792880
$231$	6.26575	0.412256
$232$	−0.858043	−0.0563333
$233$	21.1850	1.38788	0.693939	−	0.720033i	$-0.255874\pi$
$233$	21.1850	1.38788	0.693939	+	0.720033i	$0.255874\pi$
$234$	−6.73191	−0.440079
$235$	−1.41716	−0.0924452
$236$	−9.02525	−0.587494
$237$	5.44566	0.353734
$238$	−6.67532	−0.432697
$239$	16.6877	1.07944	0.539720	−	0.841844i	$-0.318530\pi$
$239$	16.6877	1.07944	0.539720	+	0.841844i	$0.318530\pi$
$240$	−6.54225	−0.422301
$241$	24.5264	1.57988	0.789942	−	0.613182i	$-0.210111\pi$
$241$	24.5264	1.57988	0.789942	+	0.613182i	$0.210111\pi$
$242$	10.0685	0.647225
$243$	−1.00000	−0.0641500
$244$	9.37345	0.600074
$245$	0.336486	0.0214973
$246$	0.786267	0.0501306
$247$	12.5442	0.798169
$248$	1.55971	0.0990417
$249$	15.7493	0.998073
$250$	−22.3183	−1.41153
$251$	−19.2046	−1.21218	−0.606092	−	0.795395i	$-0.707264\pi$
$251$	−19.2046	−1.21218	−0.606092	+	0.795395i	$0.707264\pi$
$252$	4.54314	0.286191
$253$	−10.1611	−0.638822
$254$	−6.64549	−0.416975
$255$	1.95138	0.122200
$256$	19.2623	1.20390
$257$	−12.2874	−0.766468	−0.383234	−	0.923651i	$-0.625190\pi$
$257$	−12.2874	−0.766468	−0.383234	+	0.923651i	$0.625190\pi$
$258$	−12.7625	−0.794557
$259$	−1.45289	−0.0902781
$260$	−8.94052	−0.554468
$261$	−1.74461	−0.107988
$262$	−27.0895	−1.67359
$263$	−14.0212	−0.864583	−0.432292	−	0.901734i	$-0.642295\pi$
$263$	−14.0212	−0.864583	−0.432292	+	0.901734i	$0.642295\pi$
$264$	1.18425	0.0728856
$265$	18.4587	1.13391
$266$	−18.1635	−1.11368
$267$	5.81491	0.355867
$268$	16.1108	0.984123
$269$	18.4234	1.12330	0.561649	−	0.827376i	$-0.310167\pi$
$269$	18.4234	1.12330	0.561649	+	0.827376i	$0.310167\pi$
$270$	−2.84947	−0.173413
$271$	−5.10610	−0.310174	−0.155087	−	0.987901i	$-0.549566\pi$
$271$	−5.10610	−0.310174	−0.155087	+	0.987901i	$0.549566\pi$
$272$	−5.88973	−0.357117
$273$	−9.05112	−0.547799
$274$	22.0756	1.33363
$275$	6.82011	0.411268
$276$	−7.36755	−0.443474
$277$	1.65651	0.0995300	0.0497650	−	0.998761i	$-0.484153\pi$
$277$	1.65651	0.0995300	0.0497650	+	0.998761i	$0.484153\pi$
$278$	−34.9697	−2.09735
$279$	3.17126	0.189859
$280$	−1.88425	−0.112606
$281$	−9.64760	−0.575527	−0.287764	−	0.957701i	$-0.592912\pi$
$281$	−9.64760	−0.575527	−0.287764	+	0.957701i	$0.592912\pi$
$282$	1.86298	0.110939
$283$	7.82497	0.465146	0.232573	−	0.972579i	$-0.425286\pi$
$283$	7.82497	0.465146	0.232573	+	0.972579i	$0.425286\pi$
$284$	−7.29934	−0.433136
$285$	5.30968	0.314519
$286$	16.2095	0.958490
$287$	1.05714	0.0624012
$288$	7.61673	0.448820
$289$	−15.2433	−0.896662
$290$	−4.97120	−0.291919
$291$	−1.88961	−0.110771
$292$	4.81222	0.281614
$293$	28.0752	1.64017	0.820084	−	0.572242i	$-0.193926\pi$
$293$	28.0752	1.64017	0.820084	+	0.572242i	$0.193926\pi$
$294$	−0.442342	−0.0257979
$295$	7.61081	0.443118
$296$	−0.274601	−0.0159609
$297$	2.40787	0.139719
$298$	10.7085	0.620327
$299$	14.6781	0.848855
$300$	4.94509	0.285505
$301$	−17.1593	−0.989044
$302$	22.9997	1.32348
$303$	1.68701	0.0969161
$304$	−16.0259	−0.919149
$305$	−7.90444	−0.452607
$306$	−2.56526	−0.146646
$307$	3.05248	0.174214	0.0871072	−	0.996199i	$-0.472238\pi$
$307$	3.05248	0.174214	0.0871072	+	0.996199i	$0.472238\pi$
$308$	−10.9393	−0.623322
$309$	12.0554	0.685808
$310$	9.03641	0.513234
$311$	15.5048	0.879194	0.439597	−	0.898195i	$-0.355121\pi$
$311$	15.5048	0.879194	0.439597	+	0.898195i	$0.355121\pi$
$312$	−1.71070	−0.0968491
$313$	32.6797	1.84716	0.923582	−	0.383401i	$-0.125247\pi$
$313$	32.6797	1.84716	0.923582	+	0.383401i	$0.125247\pi$
$314$	6.11647	0.345173
$315$	−3.83113	−0.215860
$316$	−9.50749	−0.534838
$317$	4.82995	0.271277	0.135639	−	0.990758i	$-0.456691\pi$
$317$	4.82995	0.271277	0.135639	+	0.990758i	$0.456691\pi$
$318$	−24.2656	−1.36075
$319$	4.20078	0.235198
$320$	8.61912	0.481824
$321$	13.6493	0.761828
$322$	−21.2533	−1.18440
$323$	4.78010	0.265972
$324$	1.74588	0.0969935
$325$	−9.85191	−0.546486
$326$	30.2271	1.67412
$327$	−10.9339	−0.604646
$328$	0.199804	0.0110323
$329$	2.50480	0.138094
$330$	6.86113	0.377693
$331$	−1.38826	−0.0763056	−0.0381528	−	0.999272i	$-0.512147\pi$
$331$	−1.38826	−0.0763056	−0.0381528	+	0.999272i	$0.512147\pi$
$332$	−27.4965	−1.50906
$333$	−0.558331	−0.0305963
$334$	43.4658	2.37834
$335$	−13.5859	−0.742277
$336$	11.5633	0.630829
$337$	−15.6513	−0.852581	−0.426290	−	0.904586i	$-0.640180\pi$
$337$	−15.6513	−0.852581	−0.426290	+	0.904586i	$0.640180\pi$
$338$	1.74527	0.0949301
$339$	1.34525	0.0730637
$340$	−3.40687	−0.184764
$341$	−7.63597	−0.413511
$342$	−6.98006	−0.377438
$343$	−18.8101	−1.01565
$344$	−3.24317	−0.174860
$345$	6.21290	0.334491
$346$	−0.508764	−0.0273513
$347$	9.76498	0.524212	0.262106	−	0.965039i	$-0.415583\pi$
$347$	9.76498	0.524212	0.262106	+	0.965039i	$0.415583\pi$
$348$	3.04588	0.163276
$349$	−27.5930	−1.47702	−0.738510	−	0.674243i	$-0.764470\pi$
$349$	−27.5930	−1.47702	−0.738510	+	0.674243i	$0.764470\pi$
$350$	14.2652	0.762506
$351$	−3.47825	−0.185655
$352$	−18.3401	−0.977529
$353$	−20.1447	−1.07219	−0.536097	−	0.844156i	$-0.680102\pi$
$353$	−20.1447	−1.07219	−0.536097	+	0.844156i	$0.680102\pi$
$354$	−10.0051	−0.531765
$355$	6.15538	0.326694
$356$	−10.1522	−0.538063
$357$	−3.44902	−0.182541
$358$	5.00782	0.264672
$359$	10.0066	0.528130	0.264065	−	0.964505i	$-0.414937\pi$
$359$	10.0066	0.528130	0.264065	+	0.964505i	$0.414937\pi$
$360$	−0.724099	−0.0381634
$361$	−5.99338	−0.315441
$362$	−33.8156	−1.77731
$363$	5.20219	0.273044
$364$	15.8022	0.828260
$365$	−4.05805	−0.212408
$366$	10.3911	0.543151
$367$	−25.3455	−1.32302	−0.661511	−	0.749935i	$-0.730085\pi$
$367$	−25.3455	−1.32302	−0.661511	+	0.749935i	$0.730085\pi$
$368$	−18.7520	−0.977518
$369$	0.406250	0.0211485
$370$	−1.59094	−0.0827093
$371$	−32.6253	−1.69382
$372$	−5.53665	−0.287062
$373$	−16.7655	−0.868082	−0.434041	−	0.900893i	$-0.642913\pi$
$373$	−16.7655	−0.868082	−0.434041	+	0.900893i	$0.642913\pi$
$374$	6.17680	0.319395
$375$	−11.5314	−0.595480
$376$	0.473416	0.0244146
$377$	−6.06819	−0.312528
$378$	5.03637	0.259043
$379$	−23.2090	−1.19217	−0.596083	−	0.802923i	$-0.703277\pi$
$379$	−23.2090	−1.19217	−0.596083	+	0.802923i	$0.703277\pi$
$380$	−9.27009	−0.475545
$381$	−3.43360	−0.175909
$382$	17.0084	0.870227
$383$	32.5732	1.66441	0.832206	−	0.554467i	$-0.187078\pi$
$383$	32.5732	1.66441	0.832206	+	0.554467i	$0.187078\pi$
$384$	3.90285	0.199166
$385$	9.22486	0.470142
$386$	34.1680	1.73910
$387$	−6.59414	−0.335199
$388$	3.29903	0.167483
$389$	18.9253	0.959552	0.479776	−	0.877391i	$-0.340718\pi$
$389$	18.9253	0.959552	0.479776	+	0.877391i	$0.340718\pi$
$390$	−9.91117	−0.501872
$391$	5.59323	0.282862
$392$	−0.112407	−0.00567739
$393$	−13.9966	−0.706036
$394$	−5.54412	−0.279309
$395$	8.01747	0.403403
$396$	−4.20385	−0.211251
$397$	−2.71020	−0.136021	−0.0680105	−	0.997685i	$-0.521665\pi$
$397$	−2.71020	−0.136021	−0.0680105	+	0.997685i	$0.521665\pi$
$398$	46.3999	2.32582
$399$	−9.38476	−0.469825
$400$	12.5864	0.629318
$401$	2.97160	0.148395	0.0741974	−	0.997244i	$-0.476361\pi$
$401$	2.97160	0.148395	0.0741974	+	0.997244i	$0.476361\pi$
$402$	17.8599	0.890770
$403$	11.0305	0.549466
$404$	−2.94532	−0.146535
$405$	−1.47227	−0.0731575
$406$	8.78650	0.436066
$407$	1.34438	0.0666387
$408$	−0.651877	−0.0322727
$409$	−14.8469	−0.734132	−0.367066	−	0.930195i	$-0.619638\pi$
$409$	−14.8469	−0.734132	−0.367066	+	0.930195i	$0.619638\pi$
$410$	1.15760	0.0571696
$411$	11.4060	0.562618
$412$	−21.0473	−1.03693
$413$	−13.4519	−0.661927
$414$	−8.16742	−0.401407
$415$	23.1872	1.13822
$416$	26.4929	1.29892
$417$	−18.0682	−0.884804
$418$	16.8070	0.822059
$419$	11.7282	0.572962	0.286481	−	0.958086i	$-0.407515\pi$
$419$	11.7282	0.572962	0.286481	+	0.958086i	$0.407515\pi$
$420$	6.68871	0.326376
$421$	−36.6230	−1.78490	−0.892449	−	0.451149i	$-0.851014\pi$
$421$	−36.6230	−1.78490	−0.892449	+	0.451149i	$0.851014\pi$
$422$	−52.0405	−2.53329
$423$	0.962569	0.0468017
$424$	−6.16631	−0.299462
$425$	−3.75417	−0.182104
$426$	−8.09180	−0.392049
$427$	13.9709	0.676101
$428$	−23.8300	−1.15187
$429$	8.37517	0.404357
$430$	−18.7898	−0.906123
$431$	−36.3836	−1.75254	−0.876268	−	0.481823i	$-0.839975\pi$
$431$	−36.3836	−1.75254	−0.876268	+	0.481823i	$0.839975\pi$
$432$	4.44366	0.213796
$433$	33.8235	1.62545	0.812726	−	0.582646i	$-0.197983\pi$
$433$	33.8235	1.62545	0.812726	+	0.582646i	$0.197983\pi$
$434$	−15.9717	−0.766665
$435$	−2.56853	−0.123151
$436$	19.0893	0.914211
$437$	15.2191	0.728030
$438$	5.33467	0.254901
$439$	7.14928	0.341217	0.170608	−	0.985339i	$-0.445427\pi$
$439$	7.14928	0.341217	0.170608	+	0.985339i	$0.445427\pi$
$440$	1.74353	0.0831197
$441$	−0.228550	−0.0108833
$442$	−8.92263	−0.424406
$443$	24.3650	1.15762	0.578809	−	0.815463i	$-0.303518\pi$
$443$	24.3650	1.15762	0.578809	+	0.815463i	$0.303518\pi$
$444$	0.974780	0.0462610
$445$	8.56110	0.405835
$446$	−47.2097	−2.23544
$447$	5.53288	0.261696
$448$	−15.2341	−0.719744
$449$	−25.9171	−1.22310	−0.611552	−	0.791204i	$-0.709455\pi$
$449$	−25.9171	−1.22310	−0.611552	+	0.791204i	$0.709455\pi$
$450$	5.48196	0.258422
$451$	−0.978195	−0.0460614
$452$	−2.34864	−0.110471
$453$	11.8835	0.558335
$454$	−31.5442	−1.48044
$455$	−13.3257	−0.624717
$456$	−1.77375	−0.0830637
$457$	−6.82586	−0.319300	−0.159650	−	0.987174i	$-0.551037\pi$
$457$	−6.82586	−0.319300	−0.159650	+	0.987174i	$0.551037\pi$
$458$	−24.5580	−1.14752
$459$	−1.32542	−0.0618654
$460$	−10.8470	−0.505744
$461$	−16.1251	−0.751022	−0.375511	−	0.926818i	$-0.622533\pi$
$461$	−16.1251	−0.751022	−0.375511	+	0.926818i	$0.622533\pi$
$462$	−12.1269	−0.564195
$463$	22.5852	1.04962	0.524812	−	0.851218i	$-0.324135\pi$
$463$	22.5852	1.04962	0.524812	+	0.851218i	$0.324135\pi$
$464$	7.75244	0.359898
$465$	4.66895	0.216517
$466$	−41.0021	−1.89939
$467$	20.1978	0.934645	0.467322	−	0.884087i	$-0.345219\pi$
$467$	20.1978	0.934645	0.467322	+	0.884087i	$0.345219\pi$
$468$	6.07262	0.280707
$469$	24.0128	1.10881
$470$	2.74281	0.126516
$471$	3.16027	0.145617
$472$	−2.54247	−0.117027
$473$	15.8778	0.730062
$474$	−10.5397	−0.484104
$475$	−10.2151	−0.468700
$476$	6.02158	0.275999
$477$	−12.5376	−0.574057
$478$	−32.2979	−1.47727
$479$	−20.6513	−0.943581	−0.471790	−	0.881711i	$-0.656392\pi$
$479$	−20.6513	−0.943581	−0.471790	+	0.881711i	$0.656392\pi$
$480$	11.2139	0.511840
$481$	−1.94202	−0.0885483
$482$	−47.4691	−2.16216
$483$	−10.9812	−0.499661
$484$	−9.08241	−0.412837
$485$	−2.78201	−0.126324
$486$	1.93543	0.0877928
$487$	−13.8071	−0.625662	−0.312831	−	0.949809i	$-0.601277\pi$
$487$	−13.8071	−0.625662	−0.312831	+	0.949809i	$0.601277\pi$
$488$	2.64056	0.119532
$489$	15.6178	0.706260
$490$	−0.651245	−0.0294202
$491$	−4.43150	−0.199991	−0.0999954	−	0.994988i	$-0.531883\pi$
$491$	−4.43150	−0.199991	−0.0999954	+	0.994988i	$0.531883\pi$
$492$	−0.709264	−0.0319761
$493$	−2.31234	−0.104143
$494$	−24.2784	−1.09234
$495$	3.54502	0.159337
$496$	−14.0920	−0.632750
$497$	−10.8795	−0.488013
$498$	−30.4817	−1.36592
$499$	−37.9321	−1.69808	−0.849038	−	0.528332i	$-0.822818\pi$
$499$	−37.9321	−1.69808	−0.849038	+	0.528332i	$0.822818\pi$
$500$	20.1325	0.900354
$501$	22.4580	1.00335
$502$	37.1691	1.65894
$503$	−34.4167	−1.53457	−0.767283	−	0.641309i	$-0.778392\pi$
$503$	−34.4167	−1.53457	−0.767283	+	0.641309i	$0.778392\pi$
$504$	1.27983	0.0570082
$505$	2.48373	0.110524
$506$	19.6661	0.874263
$507$	0.901748	0.0400480
$508$	5.99466	0.265970
$509$	−39.9977	−1.77287	−0.886434	−	0.462855i	$-0.846825\pi$
$509$	−39.9977	−1.77287	−0.886434	+	0.462855i	$0.846825\pi$
$510$	−3.77675	−0.167237
$511$	7.17252	0.317294
$512$	−29.4751	−1.30263
$513$	−3.60647	−0.159229
$514$	23.7814	1.04895
$515$	17.7488	0.782104
$516$	11.5126	0.506813
$517$	−2.31774	−0.101934
$518$	2.81196	0.123550
$519$	−0.262869	−0.0115387
$520$	−2.51860	−0.110448
$521$	−10.7631	−0.471539	−0.235769	−	0.971809i	$-0.575761\pi$
$521$	−10.7631	−0.471539	−0.235769	+	0.971809i	$0.575761\pi$
$522$	3.37656	0.147788
$523$	23.6148	1.03260	0.516302	−	0.856406i	$-0.327308\pi$
$523$	23.6148	1.03260	0.516302	+	0.856406i	$0.327308\pi$
$524$	24.4365	1.06751
$525$	7.37055	0.321677
$526$	27.1370	1.18323
$527$	4.20327	0.183097
$528$	−10.6997	−0.465646
$529$	−5.19197	−0.225738
$530$	−35.7254	−1.55181
$531$	−5.16945	−0.224335
$532$	16.3847	0.710366
$533$	1.41304	0.0612056
$534$	−11.2543	−0.487023
$535$	20.0954	0.868798
$536$	4.53851	0.196034
$537$	2.58745	0.111657
$538$	−35.6573	−1.53729
$539$	0.550317	0.0237038
$540$	2.57041	0.110613
$541$	6.38845	0.274661	0.137330	−	0.990525i	$-0.456148\pi$
$541$	6.38845	0.274661	0.137330	+	0.990525i	$0.456148\pi$
$542$	9.88250	0.424490
$543$	−17.4719	−0.749790
$544$	10.0954	0.432836
$545$	−16.0976	−0.689546
$546$	17.5178	0.749692
$547$	11.3468	0.485154	0.242577	−	0.970132i	$-0.422007\pi$
$547$	11.3468	0.485154	0.242577	+	0.970132i	$0.422007\pi$
$548$	−19.9136	−0.850667
$549$	5.36889	0.229139
$550$	−13.1998	−0.562843
$551$	−6.29187	−0.268043
$552$	−2.07548	−0.0883385
$553$	−14.1707	−0.602600
$554$	−3.20606	−0.136212
$555$	−0.822012	−0.0348924
$556$	31.5450	1.33781
$557$	27.5538	1.16749	0.583746	−	0.811936i	$-0.301586\pi$
$557$	27.5538	1.16749	0.583746	+	0.811936i	$0.301586\pi$
$558$	−6.13775	−0.259832
$559$	−22.9361	−0.970093
$560$	17.0243	0.719406
$561$	3.19144	0.134743
$562$	18.6722	0.787641
$563$	11.7062	0.493356	0.246678	−	0.969097i	$-0.420661\pi$
$563$	11.7062	0.493356	0.246678	+	0.969097i	$0.420661\pi$
$564$	−1.68053	−0.0707632
$565$	1.98056	0.0833228
$566$	−15.1447	−0.636578
$567$	2.60220	0.109282
$568$	−2.05627	−0.0862791
$569$	−7.75157	−0.324963	−0.162481	−	0.986712i	$-0.551950\pi$
$569$	−7.75157	−0.324963	−0.162481	+	0.986712i	$0.551950\pi$
$570$	−10.2765	−0.430436
$571$	−3.76060	−0.157376	−0.0786880	−	0.996899i	$-0.525073\pi$
$571$	−3.76060	−0.157376	−0.0786880	+	0.996899i	$0.525073\pi$
$572$	−14.6221	−0.611379
$573$	8.78795	0.367122
$574$	−2.04603	−0.0853995
$575$	−11.9527	−0.498463
$576$	−5.85432	−0.243930
$577$	−15.1695	−0.631515	−0.315757	−	0.948840i	$-0.602259\pi$
$577$	−15.1695	−0.631515	−0.315757	+	0.948840i	$0.602259\pi$
$578$	29.5022	1.22713
$579$	17.6540	0.733673
$580$	4.48435	0.186202
$581$	−40.9829	−1.70026
$582$	3.65720	0.151596
$583$	30.1888	1.25029
$584$	1.35563	0.0560965
$585$	−5.12092	−0.211724
$586$	−54.3375	−2.24466
$587$	−26.4965	−1.09363	−0.546814	−	0.837254i	$-0.684159\pi$
$587$	−26.4965	−1.09363	−0.546814	+	0.837254i	$0.684159\pi$
$588$	0.399021	0.0164553
$589$	11.4371	0.471256
$590$	−14.7302	−0.606432
$591$	−2.86455	−0.117832
$592$	2.48103	0.101970
$593$	−28.3354	−1.16359	−0.581797	−	0.813334i	$-0.697650\pi$
$593$	−28.3354	−1.16359	−0.581797	+	0.813334i	$0.697650\pi$
$594$	−4.66025	−0.191212
$595$	−5.07787	−0.208173
$596$	−9.65976	−0.395679
$597$	23.9740	0.981190
$598$	−28.4084	−1.16170
$599$	−18.7937	−0.767891	−0.383946	−	0.923356i	$-0.625435\pi$
$599$	−18.7937	−0.767891	−0.383946	+	0.923356i	$0.625435\pi$
$600$	1.39306	0.0568715
$601$	25.3351	1.03344	0.516720	−	0.856154i	$-0.327153\pi$
$601$	25.3351	1.03344	0.516720	+	0.856154i	$0.327153\pi$
$602$	33.2105	1.35356
$603$	9.22787	0.375788
$604$	−20.7472	−0.844191
$605$	7.65901	0.311383
$606$	−3.26508	−0.132635
$607$	−22.9143	−0.930063	−0.465031	−	0.885294i	$-0.653957\pi$
$607$	−22.9143	−0.930063	−0.465031	+	0.885294i	$0.653957\pi$
$608$	27.4695	1.11404
$609$	4.53982	0.183963
$610$	15.2985	0.619417
$611$	3.34806	0.135448
$612$	2.31403	0.0935392
$613$	18.9857	0.766826	0.383413	−	0.923577i	$-0.374749\pi$
$613$	18.9857	0.766826	0.383413	+	0.923577i	$0.374749\pi$
$614$	−5.90786	−0.238422
$615$	0.598108	0.0241180
$616$	−3.08166	−0.124164
$617$	−6.14706	−0.247471	−0.123736	−	0.992315i	$-0.539487\pi$
$617$	−6.14706	−0.247471	−0.123736	+	0.992315i	$0.539487\pi$
$618$	−23.3324	−0.938565
$619$	40.5099	1.62823	0.814115	−	0.580703i	$-0.197222\pi$
$619$	40.5099	1.62823	0.814115	+	0.580703i	$0.197222\pi$
$620$	−8.15143	−0.327369
$621$	−4.21996	−0.169341
$622$	−30.0083	−1.20322
$623$	−15.1316	−0.606234
$624$	15.4562	0.618742
$625$	−2.81519	−0.112608
$626$	−63.2492	−2.52794
$627$	8.68389	0.346801
$628$	−5.51746	−0.220171
$629$	−0.740024	−0.0295067
$630$	7.41489	0.295416
$631$	22.7189	0.904426	0.452213	−	0.891910i	$-0.350635\pi$
$631$	22.7189	0.904426	0.452213	+	0.891910i	$0.350635\pi$
$632$	−2.67832	−0.106538
$633$	−26.8883	−1.06872
$634$	−9.34803	−0.371258
$635$	−5.05518	−0.200609
$636$	21.8892	0.867961
$637$	−0.794954	−0.0314972
$638$	−8.13031	−0.321882
$639$	−4.18089	−0.165393
$640$	5.74603	0.227132
$641$	−19.9071	−0.786283	−0.393141	−	0.919478i	$-0.628612\pi$
$641$	−19.9071	−0.786283	−0.393141	+	0.919478i	$0.628612\pi$
$642$	−26.4172	−1.04260
$643$	37.4569	1.47716	0.738578	−	0.674168i	$-0.235498\pi$
$643$	37.4569	1.47716	0.738578	+	0.674168i	$0.235498\pi$
$644$	19.1718	0.755476
$645$	−9.70833	−0.382265
$646$	−9.25153	−0.363997
$647$	−1.74875	−0.0687503	−0.0343752	−	0.999409i	$-0.510944\pi$
$647$	−1.74875	−0.0687503	−0.0343752	+	0.999409i	$0.510944\pi$
$648$	0.491826	0.0193207
$649$	12.4473	0.488601
$650$	19.0677	0.747895
$651$	−8.25226	−0.323432
$652$	−27.2668	−1.06785
$653$	−32.2070	−1.26036	−0.630179	−	0.776450i	$-0.717018\pi$
$653$	−32.2070	−1.26036	−0.630179	+	0.776450i	$0.717018\pi$
$654$	21.1618	0.827490
$655$	−20.6068	−0.805173
$656$	−1.80524	−0.0704826
$657$	2.75633	0.107535
$658$	−4.84785	−0.188989
$659$	10.4622	0.407548	0.203774	−	0.979018i	$-0.434679\pi$
$659$	10.4622	0.407548	0.203774	+	0.979018i	$0.434679\pi$
$660$	−6.18919	−0.240914
$661$	−31.7349	−1.23434	−0.617171	−	0.786829i	$-0.711721\pi$
$661$	−31.7349	−1.23434	−0.617171	+	0.786829i	$0.711721\pi$
$662$	2.68687	0.104428
$663$	−4.61016	−0.179044
$664$	−7.74593	−0.300600
$665$	−13.8169	−0.535795
$666$	1.08061	0.0418727
$667$	−7.36217	−0.285064
$668$	−39.2090	−1.51704
$669$	−24.3924	−0.943064
$670$	26.2945	1.01585
$671$	−12.9276	−0.499063
$672$	−19.8203	−0.764583
$673$	−43.1352	−1.66274	−0.831370	−	0.555719i	$-0.812443\pi$
$673$	−43.1352	−1.66274	−0.831370	+	0.555719i	$0.812443\pi$
$674$	30.2920	1.16680
$675$	2.83243	0.109020
$676$	−1.57435	−0.0605518
$677$	45.7907	1.75988	0.879940	−	0.475084i	$-0.157582\pi$
$677$	45.7907	1.75988	0.879940	+	0.475084i	$0.157582\pi$
$678$	−2.60363	−0.0999916
$679$	4.91714	0.188702
$680$	−0.959738	−0.0368043
$681$	−16.2983	−0.624553
$682$	14.7789	0.565913
$683$	0.138630	0.00530453	0.00265227	−	0.999996i	$-0.499156\pi$
$683$	0.138630	0.00530453	0.00265227	+	0.999996i	$0.499156\pi$
$684$	6.29647	0.240752
$685$	16.7927	0.641617
$686$	36.4057	1.38998
$687$	−12.6886	−0.484102
$688$	29.3021	1.11713
$689$	−43.6089	−1.66137
$690$	−12.0246	−0.457770
$691$	−23.2566	−0.884725	−0.442362	−	0.896836i	$-0.645859\pi$
$691$	−23.2566	−0.884725	−0.442362	+	0.896836i	$0.645859\pi$
$692$	0.458938	0.0174462
$693$	−6.26575	−0.238016
$694$	−18.8994	−0.717412
$695$	−26.6012	−1.00904
$696$	0.858043	0.0325240
$697$	0.538453	0.0203954
$698$	53.4043	2.02138
$699$	−21.1850	−0.801292
$700$	−12.8681	−0.486369
$701$	8.38091	0.316542	0.158271	−	0.987396i	$-0.449408\pi$
$701$	8.38091	0.316542	0.158271	+	0.987396i	$0.449408\pi$
$702$	6.73191	0.254080
$703$	−2.01360	−0.0759444
$704$	14.0964	0.531278
$705$	1.41716	0.0533733
$706$	38.9886	1.46736
$707$	−4.38994	−0.165101
$708$	9.02525	0.339190
$709$	30.6545	1.15126	0.575628	−	0.817712i	$-0.304758\pi$
$709$	30.6545	1.15126	0.575628	+	0.817712i	$0.304758\pi$
$710$	−11.9133	−0.447098
$711$	−5.44566	−0.204228
$712$	−2.85992	−0.107180
$713$	13.3826	0.501182
$714$	6.67532	0.249818
$715$	12.3305	0.461134
$716$	−4.51738	−0.168822
$717$	−16.6877	−0.623215
$718$	−19.3671	−0.722775
$719$	−6.81886	−0.254301	−0.127150	−	0.991883i	$-0.540583\pi$
$719$	−6.81886	−0.254301	−0.127150	+	0.991883i	$0.540583\pi$
$720$	6.54225	0.243815
$721$	−31.3706	−1.16830
$722$	11.5998	0.431699
$723$	−24.5264	−0.912146
$724$	30.5038	1.13367
$725$	4.94148	0.183522
$726$	−10.0685	−0.373676
$727$	9.17351	0.340227	0.170113	−	0.985425i	$-0.445587\pi$
$727$	9.17351	0.340227	0.170113	+	0.985425i	$0.445587\pi$
$728$	4.45157	0.164986
$729$	1.00000	0.0370370
$730$	7.85406	0.290692
$731$	−8.74002	−0.323261
$732$	−9.37345	−0.346453
$733$	−33.2394	−1.22773	−0.613863	−	0.789412i	$-0.710385\pi$
$733$	−33.2394	−1.22773	−0.613863	+	0.789412i	$0.710385\pi$
$734$	49.0543	1.81063
$735$	−0.336486	−0.0124115
$736$	32.1423	1.18478
$737$	−22.2195	−0.818465
$738$	−0.786267	−0.0289429
$739$	−44.2054	−1.62612	−0.813060	−	0.582180i	$-0.802200\pi$
$739$	−44.2054	−1.62612	−0.813060	+	0.582180i	$0.802200\pi$
$740$	1.43514	0.0527566
$741$	−12.5442	−0.460823
$742$	63.1440	2.31809
$743$	−0.488039	−0.0179044	−0.00895220	−	0.999960i	$-0.502850\pi$
$743$	−0.488039	−0.0179044	−0.00895220	+	0.999960i	$0.502850\pi$
$744$	−1.55971	−0.0571817
$745$	8.14588	0.298442
$746$	32.4483	1.18802
$747$	−15.7493	−0.576238
$748$	−5.57188	−0.203728
$749$	−35.5181	−1.29780
$750$	22.3183	0.814947
$751$	−45.0230	−1.64291	−0.821457	−	0.570271i	$-0.806838\pi$
$751$	−45.0230	−1.64291	−0.821457	+	0.570271i	$0.806838\pi$
$752$	−4.27733	−0.155978
$753$	19.2046	0.699855
$754$	11.7445	0.427711
$755$	17.4957	0.636733
$756$	−4.54314	−0.165232
$757$	−13.9696	−0.507735	−0.253868	−	0.967239i	$-0.581703\pi$
$757$	−13.9696	−0.507735	−0.253868	+	0.967239i	$0.581703\pi$
$758$	44.9194	1.63154
$759$	10.1611	0.368824
$760$	−2.61144	−0.0947269
$761$	−46.9958	−1.70359	−0.851797	−	0.523871i	$-0.824487\pi$
$761$	−46.9958	−1.70359	−0.851797	+	0.523871i	$0.824487\pi$
$762$	6.64549	0.240741
$763$	28.4522	1.03004
$764$	−15.3427	−0.555080
$765$	−1.95138	−0.0705522
$766$	−63.0431	−2.27784
$767$	−17.9807	−0.649244
$768$	−19.2623	−0.695069
$769$	41.1824	1.48508	0.742538	−	0.669804i	$-0.233622\pi$
$769$	41.1824	1.48508	0.742538	+	0.669804i	$0.233622\pi$
$770$	−17.8540	−0.643415
$771$	12.2874	0.442521
$772$	−30.8217	−1.10930
$773$	−49.7939	−1.79096	−0.895481	−	0.445099i	$-0.853169\pi$
$773$	−49.7939	−1.79096	−0.895481	+	0.445099i	$0.853169\pi$
$774$	12.7625	0.458738
$775$	−8.98238	−0.322657
$776$	0.929358	0.0333620
$777$	1.45289	0.0521221
$778$	−36.6286	−1.31320
$779$	1.46513	0.0524936
$780$	8.94052	0.320122
$781$	10.0670	0.360226
$782$	−10.8253	−0.387112
$783$	1.74461	0.0623472
$784$	1.01560	0.0362713
$785$	4.65276	0.166064
$786$	27.0895	0.966249
$787$	−34.3827	−1.22561	−0.612805	−	0.790234i	$-0.709959\pi$
$787$	−34.3827	−1.22561	−0.612805	+	0.790234i	$0.709959\pi$
$788$	5.00116	0.178159
$789$	14.0212	0.499167
$790$	−15.5172	−0.552079
$791$	−3.50060	−0.124467
$792$	−1.18425	−0.0420805
$793$	18.6744	0.663146
$794$	5.24540	0.186152
$795$	−18.4587	−0.654662
$796$	−41.8558	−1.48354
$797$	37.7058	1.33561	0.667803	−	0.744338i	$-0.267235\pi$
$797$	37.7058	1.33561	0.667803	+	0.744338i	$0.267235\pi$
$798$	18.1635	0.642982
$799$	1.27581	0.0451349
$800$	−21.5739	−0.762751
$801$	−5.81491	−0.205460
$802$	−5.75133	−0.203086
$803$	−6.63686	−0.234210
$804$	−16.1108	−0.568183
$805$	−16.1672	−0.569820
$806$	−21.3487	−0.751975
$807$	−18.4234	−0.648536
$808$	−0.829715	−0.0291893
$809$	−16.9052	−0.594354	−0.297177	−	0.954822i	$-0.596045\pi$
$809$	−16.9052	−0.594354	−0.297177	+	0.954822i	$0.596045\pi$
$810$	2.84947	0.100120
$811$	−14.8247	−0.520564	−0.260282	−	0.965533i	$-0.583816\pi$
$811$	−14.8247	−0.520564	−0.260282	+	0.965533i	$0.583816\pi$
$812$	−7.92599	−0.278148
$813$	5.10610	0.179079
$814$	−2.60196	−0.0911986
$815$	22.9935	0.805428
$816$	5.88973	0.206182
$817$	−23.7815	−0.832011
$818$	28.7351	1.00470
$819$	9.05112	0.316272
$820$	−1.04423	−0.0364660
$821$	1.10715	0.0386397	0.0193199	−	0.999813i	$-0.493850\pi$
$821$	1.10715	0.0386397	0.0193199	+	0.999813i	$0.493850\pi$
$822$	−22.0756	−0.769974
$823$	9.21174	0.321101	0.160551	−	0.987028i	$-0.448673\pi$
$823$	9.21174	0.321101	0.160551	+	0.987028i	$0.448673\pi$
$824$	−5.92916	−0.206552
$825$	−6.82011	−0.237446
$826$	26.0353	0.905883
$827$	36.2408	1.26022	0.630109	−	0.776507i	$-0.283010\pi$
$827$	36.2408	1.26022	0.630109	+	0.776507i	$0.283010\pi$
$828$	7.36755	0.256040
$829$	−0.314438	−0.0109209	−0.00546043	−	0.999985i	$-0.501738\pi$
$829$	−0.314438	−0.0109209	−0.00546043	+	0.999985i	$0.501738\pi$
$830$	−44.8772	−1.55771
$831$	−1.65651	−0.0574637
$832$	−20.3628	−0.705953
$833$	−0.302925	−0.0104957
$834$	34.9697	1.21090
$835$	33.0641	1.14423
$836$	−15.1611	−0.524356
$837$	−3.17126	−0.109615
$838$	−22.6992	−0.784129
$839$	−5.31306	−0.183427	−0.0917136	−	0.995785i	$-0.529234\pi$
$839$	−5.31306	−0.183427	−0.0917136	+	0.995785i	$0.529234\pi$
$840$	1.88425	0.0650129
$841$	−25.9563	−0.895046
$842$	70.8812	2.44273
$843$	9.64760	0.332281
$844$	46.9439	1.61588
$845$	1.32761	0.0456713
$846$	−1.86298	−0.0640507
$847$	−13.5371	−0.465141
$848$	55.7128	1.91318
$849$	−7.82497	−0.268552
$850$	7.26592	0.249219
$851$	−2.35613	−0.0807671
$852$	7.29934	0.250071
$853$	−41.8526	−1.43301	−0.716503	−	0.697584i	$-0.754258\pi$
$853$	−41.8526	−1.43301	−0.716503	+	0.697584i	$0.754258\pi$
$854$	−27.0397	−0.925280
$855$	−5.30968	−0.181587
$856$	−6.71306	−0.229448
$857$	29.6431	1.01259	0.506295	−	0.862360i	$-0.331015\pi$
$857$	29.6431	1.01259	0.506295	+	0.862360i	$0.331015\pi$
$858$	−16.2095	−0.553384
$859$	28.4008	0.969022	0.484511	−	0.874785i	$-0.338998\pi$
$859$	28.4008	0.969022	0.484511	+	0.874785i	$0.338998\pi$
$860$	16.9496	0.577977
$861$	−1.05714	−0.0360274
$862$	70.4179	2.39844
$863$	−19.3752	−0.659538	−0.329769	−	0.944062i	$-0.606971\pi$
$863$	−19.3752	−0.659538	−0.329769	+	0.944062i	$0.606971\pi$
$864$	−7.61673	−0.259126
$865$	−0.387013	−0.0131588
$866$	−65.4629	−2.22452
$867$	15.2433	0.517688
$868$	14.4075	0.489022
$869$	13.1124	0.444809
$870$	4.97120	0.168540
$871$	32.0969	1.08756
$872$	5.37757	0.182108
$873$	1.88961	0.0639535
$874$	−29.4556	−0.996349
$875$	30.0071	1.01442
$876$	−4.81222	−0.162590
$877$	38.3709	1.29569	0.647847	−	0.761770i	$-0.275670\pi$
$877$	38.3709	1.29569	0.647847	+	0.761770i	$0.275670\pi$
$878$	−13.8369	−0.466974
$879$	−28.0752	−0.946952
$880$	−15.7529	−0.531029
$881$	12.0706	0.406668	0.203334	−	0.979109i	$-0.434822\pi$
$881$	12.0706	0.406668	0.203334	+	0.979109i	$0.434822\pi$
$882$	0.442342	0.0148944
$883$	14.2430	0.479316	0.239658	−	0.970857i	$-0.422965\pi$
$883$	14.2430	0.479316	0.239658	+	0.970857i	$0.422965\pi$
$884$	8.04879	0.270710
$885$	−7.61081	−0.255834
$886$	−47.1568	−1.58426
$887$	−43.0626	−1.44590	−0.722950	−	0.690900i	$-0.757214\pi$
$887$	−43.0626	−1.44590	−0.722950	+	0.690900i	$0.757214\pi$
$888$	0.274601	0.00921502
$889$	8.93492	0.299668
$890$	−16.5694	−0.555408
$891$	−2.40787	−0.0806665
$892$	42.5862	1.42589
$893$	3.47147	0.116168
$894$	−10.7085	−0.358146
$895$	3.80941	0.127335
$896$	−10.1560	−0.339288
$897$	−14.6781	−0.490087
$898$	50.1607	1.67389
$899$	−5.53261	−0.184523
$900$	−4.94509	−0.164836
$901$	−16.6176	−0.553613
$902$	1.89323	0.0630375
$903$	17.1593	0.571025
$904$	−0.661627	−0.0220054
$905$	−25.7233	−0.855070
$906$	−22.9997	−0.764112
$907$	−41.7760	−1.38715	−0.693575	−	0.720385i	$-0.743965\pi$
$907$	−41.7760	−1.38715	−0.693575	+	0.720385i	$0.743965\pi$
$908$	28.4550	0.944311
$909$	−1.68701	−0.0559545
$910$	25.7909	0.854959
$911$	12.6680	0.419710	0.209855	−	0.977733i	$-0.432701\pi$
$911$	12.6680	0.419710	0.209855	+	0.977733i	$0.432701\pi$
$912$	16.0259	0.530671
$913$	37.9222	1.25504
$914$	13.2110	0.436980
$915$	7.90444	0.261313
$916$	22.1529	0.731952
$917$	36.4220	1.20276
$918$	2.56526	0.0846662
$919$	−30.1733	−0.995325	−0.497662	−	0.867371i	$-0.665808\pi$
$919$	−30.1733	−0.995325	−0.497662	+	0.867371i	$0.665808\pi$
$920$	−3.05567	−0.100742
$921$	−3.05248	−0.100583
$922$	31.2090	1.02781
$923$	−14.5422	−0.478662
$924$	10.9393	0.359875
$925$	1.58143	0.0519972
$926$	−43.7121	−1.43647
$927$	−12.0554	−0.395951
$928$	−13.2882	−0.436207
$929$	32.7923	1.07588	0.537940	−	0.842983i	$-0.319203\pi$
$929$	32.7923	1.07588	0.537940	+	0.842983i	$0.319203\pi$
$930$	−9.03641	−0.296316
$931$	−0.824257	−0.0270139
$932$	36.9866	1.21154
$933$	−15.5048	−0.507603
$934$	−39.0915	−1.27911
$935$	4.69865	0.153662
$936$	1.71070	0.0559158
$937$	14.9553	0.488569	0.244285	−	0.969704i	$-0.421447\pi$
$937$	14.9553	0.488569	0.244285	+	0.969704i	$0.421447\pi$
$938$	−46.4750	−1.51746
$939$	−32.6797	−1.06646
$940$	−2.47419	−0.0806992
$941$	−5.72089	−0.186496	−0.0932479	−	0.995643i	$-0.529725\pi$
$941$	−5.72089	−0.186496	−0.0932479	+	0.995643i	$0.529725\pi$
$942$	−6.11647	−0.199285
$943$	1.71436	0.0558271
$944$	22.9713	0.747651
$945$	3.83113	0.124627
$946$	−30.7303	−0.999129
$947$	23.6005	0.766913	0.383457	−	0.923559i	$-0.374734\pi$
$947$	23.6005	0.766913	0.383457	+	0.923559i	$0.374734\pi$
$948$	9.50749	0.308789
$949$	9.58721	0.311214
$950$	19.7705	0.641441
$951$	−4.82995	−0.156622
$952$	1.69632	0.0549779
$953$	−23.0658	−0.747174	−0.373587	−	0.927595i	$-0.621872\pi$
$953$	−23.0658	−0.747174	−0.373587	+	0.927595i	$0.621872\pi$
$954$	24.2656	0.785628
$955$	12.9382	0.418670
$956$	29.1348	0.942288
$957$	−4.20078	−0.135792
$958$	39.9690	1.29134
$959$	−29.6808	−0.958443
$960$	−8.61912	−0.278181
$961$	−20.9431	−0.675584
$962$	3.75863	0.121183
$963$	−13.6493	−0.439841
$964$	42.8202	1.37915
$965$	25.9913	0.836691
$966$	21.2533	0.683813
$967$	25.9368	0.834073	0.417036	−	0.908890i	$-0.363069\pi$
$967$	25.9368	0.834073	0.417036	+	0.908890i	$0.363069\pi$
$968$	−2.55857	−0.0822356
$969$	−4.78010	−0.153559
$970$	5.38437	0.172882
$971$	−13.0232	−0.417935	−0.208968	−	0.977923i	$-0.567010\pi$
$971$	−13.0232	−0.417935	−0.208968	+	0.977923i	$0.567010\pi$
$972$	−1.74588	−0.0559992
$973$	47.0171	1.50730
$974$	26.7227	0.856252
$975$	9.85191	0.315514
$976$	−23.8575	−0.763660
$977$	−37.0401	−1.18502	−0.592509	−	0.805564i	$-0.701863\pi$
$977$	−37.0401	−1.18502	−0.592509	+	0.805564i	$0.701863\pi$
$978$	−30.2271	−0.966555
$979$	14.0015	0.447491
$980$	0.587465	0.0187659
$981$	10.9339	0.349092
$982$	8.57685	0.273698
$983$	−7.18678	−0.229223	−0.114611	−	0.993410i	$-0.536562\pi$
$983$	−7.18678	−0.229223	−0.114611	+	0.993410i	$0.536562\pi$
$984$	−0.199804	−0.00636952
$985$	−4.21738	−0.134377
$986$	4.47537	0.142525
$987$	−2.50480	−0.0797286
$988$	21.9007	0.696755
$989$	−27.8270	−0.884846
$990$	−6.86113	−0.218061
$991$	−27.6254	−0.877551	−0.438775	−	0.898597i	$-0.644588\pi$
$991$	−27.6254	−0.877551	−0.438775	+	0.898597i	$0.644588\pi$
$992$	24.1547	0.766911
$993$	1.38826	0.0440550
$994$	21.0565	0.667872
$995$	35.2961	1.11896
$996$	27.4965	0.871259
$997$	0.539614	0.0170897	0.00854487	−	0.999963i	$-0.497280\pi$
$997$	0.539614	0.0170897	0.00854487	+	0.999963i	$0.497280\pi$
$998$	73.4149	2.32391
$999$	0.558331	0.0176648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.b.1.20	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.b.1.20	✓	102	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93543 q^{2} -1.00000 q^{3} +1.74588 q^{4} -1.47227 q^{5} +1.93543 q^{6} +2.60220 q^{7} +0.491826 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.93543 q^{2} -1.00000 q^{3} +1.74588 q^{4} -1.47227 q^{5} +1.93543 q^{6} +2.60220 q^{7} +0.491826 q^{8} +1.00000 q^{9} +2.84947 q^{10} -2.40787 q^{11} -1.74588 q^{12} +3.47825 q^{13} -5.03637 q^{14} +1.47227 q^{15} -4.44366 q^{16} +1.32542 q^{17} -1.93543 q^{18} +3.60647 q^{19} -2.57041 q^{20} -2.60220 q^{21} +4.66025 q^{22} +4.21996 q^{23} -0.491826 q^{24} -2.83243 q^{25} -6.73191 q^{26} -1.00000 q^{27} +4.54314 q^{28} -1.74461 q^{29} -2.84947 q^{30} +3.17126 q^{31} +7.61673 q^{32} +2.40787 q^{33} -2.56526 q^{34} -3.83113 q^{35} +1.74588 q^{36} -0.558331 q^{37} -6.98006 q^{38} -3.47825 q^{39} -0.724099 q^{40} +0.406250 q^{41} +5.03637 q^{42} -6.59414 q^{43} -4.20385 q^{44} -1.47227 q^{45} -8.16742 q^{46} +0.962569 q^{47} +4.44366 q^{48} -0.228550 q^{49} +5.48196 q^{50} -1.32542 q^{51} +6.07262 q^{52} -12.5376 q^{53} +1.93543 q^{54} +3.54502 q^{55} +1.27983 q^{56} -3.60647 q^{57} +3.37656 q^{58} -5.16945 q^{59} +2.57041 q^{60} +5.36889 q^{61} -6.13775 q^{62} +2.60220 q^{63} -5.85432 q^{64} -5.12092 q^{65} -4.66025 q^{66} +9.22787 q^{67} +2.31403 q^{68} -4.21996 q^{69} +7.41489 q^{70} -4.18089 q^{71} +0.491826 q^{72} +2.75633 q^{73} +1.08061 q^{74} +2.83243 q^{75} +6.29647 q^{76} -6.26575 q^{77} +6.73191 q^{78} -5.44566 q^{79} +6.54225 q^{80} +1.00000 q^{81} -0.786267 q^{82} -15.7493 q^{83} -4.54314 q^{84} -1.95138 q^{85} +12.7625 q^{86} +1.74461 q^{87} -1.18425 q^{88} -5.81491 q^{89} +2.84947 q^{90} +9.05112 q^{91} +7.36755 q^{92} -3.17126 q^{93} -1.86298 q^{94} -5.30968 q^{95} -7.61673 q^{96} +1.88961 q^{97} +0.442342 q^{98} -2.40787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * q^99

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