LMFDB - Embedded newform 8013.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8013,2,Mod(1,8013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8013, 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("8013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8013 = 3 \cdot 2671 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8013.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:	$63.9841271397$
Analytic rank:	$0$
Dimension:	$129$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69326 q^{2} +1.00000 q^{3} +5.25363 q^{4} -3.14797 q^{5} -2.69326 q^{6} +0.0430508 q^{7} -8.76286 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.69326 q^{2} +1.00000 q^{3} +5.25363 q^{4} -3.14797 q^{5} -2.69326 q^{6} +0.0430508 q^{7} -8.76286 q^{8} +1.00000 q^{9} +8.47829 q^{10} -3.97005 q^{11} +5.25363 q^{12} +0.212754 q^{13} -0.115947 q^{14} -3.14797 q^{15} +13.0934 q^{16} -6.24792 q^{17} -2.69326 q^{18} -1.70064 q^{19} -16.5383 q^{20} +0.0430508 q^{21} +10.6924 q^{22} -1.61766 q^{23} -8.76286 q^{24} +4.90972 q^{25} -0.573002 q^{26} +1.00000 q^{27} +0.226173 q^{28} -8.54219 q^{29} +8.47829 q^{30} -6.57126 q^{31} -17.7381 q^{32} -3.97005 q^{33} +16.8273 q^{34} -0.135523 q^{35} +5.25363 q^{36} -7.20692 q^{37} +4.58026 q^{38} +0.212754 q^{39} +27.5852 q^{40} +6.68846 q^{41} -0.115947 q^{42} -8.46296 q^{43} -20.8572 q^{44} -3.14797 q^{45} +4.35677 q^{46} +1.17838 q^{47} +13.0934 q^{48} -6.99815 q^{49} -13.2231 q^{50} -6.24792 q^{51} +1.11773 q^{52} -8.24058 q^{53} -2.69326 q^{54} +12.4976 q^{55} -0.377248 q^{56} -1.70064 q^{57} +23.0063 q^{58} +6.26247 q^{59} -16.5383 q^{60} -1.14722 q^{61} +17.6981 q^{62} +0.0430508 q^{63} +21.5865 q^{64} -0.669744 q^{65} +10.6924 q^{66} -0.662651 q^{67} -32.8243 q^{68} -1.61766 q^{69} +0.364997 q^{70} -10.3048 q^{71} -8.76286 q^{72} +13.1083 q^{73} +19.4101 q^{74} +4.90972 q^{75} -8.93454 q^{76} -0.170914 q^{77} -0.573002 q^{78} -14.5271 q^{79} -41.2176 q^{80} +1.00000 q^{81} -18.0137 q^{82} +10.4232 q^{83} +0.226173 q^{84} +19.6683 q^{85} +22.7929 q^{86} -8.54219 q^{87} +34.7890 q^{88} -1.48612 q^{89} +8.47829 q^{90} +0.00915924 q^{91} -8.49858 q^{92} -6.57126 q^{93} -3.17369 q^{94} +5.35357 q^{95} -17.7381 q^{96} +0.764689 q^{97} +18.8478 q^{98} -3.97005 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 129 q + 15 q^{2} + 129 q^{3} + 151 q^{4} + 16 q^{5} + 15 q^{6} + 61 q^{7} + 42 q^{8} + 129 q^{9}+O(q^{10}) $ 129 * q + 15 * q^2 + 129 * q^3 + 151 * q^4 + 16 * q^5 + 15 * q^6 + 61 * q^7 + 42 * q^8 + 129 * q^9	\( 129 q + 15 q^{2} + 129 q^{3} + 151 q^{4} + 16 q^{5} + 15 q^{6} + 61 q^{7} + 42 q^{8} + 129 q^{9} + 41 q^{10} + 51 q^{11} + 151 q^{12} + 56 q^{13} + 5 q^{14} + 16 q^{15} + 195 q^{16} + 18 q^{17} + 15 q^{18} + 93 q^{19} + 44 q^{20} + 61 q^{21} + 46 q^{22} + 50 q^{23} + 42 q^{24} + 193 q^{25} + q^{26} + 129 q^{27} + 145 q^{28} + 24 q^{29} + 41 q^{30} + 67 q^{31} + 89 q^{32} + 51 q^{33} + 73 q^{34} + 56 q^{35} + 151 q^{36} + 95 q^{37} + 9 q^{38} + 56 q^{39} + 103 q^{40} + 7 q^{41} + 5 q^{42} + 150 q^{43} + 69 q^{44} + 16 q^{45} + 72 q^{46} + 53 q^{47} + 195 q^{48} + 240 q^{49} + 17 q^{50} + 18 q^{51} + 124 q^{52} + 34 q^{53} + 15 q^{54} + 66 q^{55} - 17 q^{56} + 93 q^{57} + 57 q^{58} + 49 q^{59} + 44 q^{60} + 113 q^{61} + 27 q^{62} + 61 q^{63} + 262 q^{64} + 22 q^{65} + 46 q^{66} + 185 q^{67} + 2 q^{68} + 50 q^{69} + 25 q^{70} + 41 q^{71} + 42 q^{72} + 153 q^{73} - q^{74} + 193 q^{75} + 190 q^{76} + 39 q^{77} + q^{78} + 101 q^{79} + 48 q^{80} + 129 q^{81} + 15 q^{82} + 162 q^{83} + 145 q^{84} + 99 q^{85} + 13 q^{86} + 24 q^{87} + 86 q^{88} - 4 q^{89} + 41 q^{90} + 117 q^{91} + 56 q^{92} + 67 q^{93} + 49 q^{94} + 71 q^{95} + 89 q^{96} + 159 q^{97} + 40 q^{98} + 51 q^{99}+O(q^{100}) \) 129 * q + 15 * q^2 + 129 * q^3 + 151 * q^4 + 16 * q^5 + 15 * q^6 + 61 * q^7 + 42 * q^8 + 129 * q^9 + 41 * q^10 + 51 * q^11 + 151 * q^12 + 56 * q^13 + 5 * q^14 + 16 * q^15 + 195 * q^16 + 18 * q^17 + 15 * q^18 + 93 * q^19 + 44 * q^20 + 61 * q^21 + 46 * q^22 + 50 * q^23 + 42 * q^24 + 193 * q^25 + q^26 + 129 * q^27 + 145 * q^28 + 24 * q^29 + 41 * q^30 + 67 * q^31 + 89 * q^32 + 51 * q^33 + 73 * q^34 + 56 * q^35 + 151 * q^36 + 95 * q^37 + 9 * q^38 + 56 * q^39 + 103 * q^40 + 7 * q^41 + 5 * q^42 + 150 * q^43 + 69 * q^44 + 16 * q^45 + 72 * q^46 + 53 * q^47 + 195 * q^48 + 240 * q^49 + 17 * q^50 + 18 * q^51 + 124 * q^52 + 34 * q^53 + 15 * q^54 + 66 * q^55 - 17 * q^56 + 93 * q^57 + 57 * q^58 + 49 * q^59 + 44 * q^60 + 113 * q^61 + 27 * q^62 + 61 * q^63 + 262 * q^64 + 22 * q^65 + 46 * q^66 + 185 * q^67 + 2 * q^68 + 50 * q^69 + 25 * q^70 + 41 * q^71 + 42 * q^72 + 153 * q^73 - q^74 + 193 * q^75 + 190 * q^76 + 39 * q^77 + q^78 + 101 * q^79 + 48 * q^80 + 129 * q^81 + 15 * q^82 + 162 * q^83 + 145 * q^84 + 99 * q^85 + 13 * q^86 + 24 * q^87 + 86 * q^88 - 4 * q^89 + 41 * q^90 + 117 * q^91 + 56 * q^92 + 67 * q^93 + 49 * q^94 + 71 * q^95 + 89 * q^96 + 159 * q^97 + 40 * q^98 + 51 * 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 $

$2$	−2.69326	−1.90442	−0.952210	−	0.305444i	$-0.901195\pi$
$2$	−2.69326	−1.90442	−0.952210	+	0.305444i	$0.901195\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.25363	2.62682
$5$	−3.14797	−1.40782	−0.703908	−	0.710292i	$-0.748563\pi$
$5$	−3.14797	−1.40782	−0.703908	+	0.710292i	$0.748563\pi$
$6$	−2.69326	−1.09952
$7$	0.0430508	0.0162717	0.00813583	−	0.999967i	$-0.497410\pi$
$7$	0.0430508	0.0162717	0.00813583	+	0.999967i	$0.497410\pi$
$8$	−8.76286	−3.09814
$9$	1.00000	0.333333
$10$	8.47829	2.68107
$11$	−3.97005	−1.19702	−0.598508	−	0.801117i	$-0.704240\pi$
$11$	−3.97005	−1.19702	−0.598508	+	0.801117i	$0.704240\pi$
$12$	5.25363	1.51659
$13$	0.212754	0.0590074	0.0295037	−	0.999565i	$-0.490607\pi$
$13$	0.212754	0.0590074	0.0295037	+	0.999565i	$0.490607\pi$
$14$	−0.115947	−0.0309881
$15$	−3.14797	−0.812803
$16$	13.0934	3.27335
$17$	−6.24792	−1.51534	−0.757672	−	0.652636i	$-0.773663\pi$
$17$	−6.24792	−1.51534	−0.757672	+	0.652636i	$0.773663\pi$
$18$	−2.69326	−0.634807
$19$	−1.70064	−0.390154	−0.195077	−	0.980788i	$-0.562496\pi$
$19$	−1.70064	−0.390154	−0.195077	+	0.980788i	$0.562496\pi$
$20$	−16.5383	−3.69807
$21$	0.0430508	0.00939445
$22$	10.6924	2.27962
$23$	−1.61766	−0.337305	−0.168653	−	0.985676i	$-0.553942\pi$
$23$	−1.61766	−0.337305	−0.168653	+	0.985676i	$0.553942\pi$
$24$	−8.76286	−1.78871
$25$	4.90972	0.981944
$26$	−0.573002	−0.112375
$27$	1.00000	0.192450
$28$	0.226173	0.0427427
$29$	−8.54219	−1.58625	−0.793123	−	0.609062i	$-0.791546\pi$
$29$	−8.54219	−1.58625	−0.793123	+	0.609062i	$0.791546\pi$
$30$	8.47829	1.54792
$31$	−6.57126	−1.18023	−0.590116	−	0.807318i	$-0.700918\pi$
$31$	−6.57126	−1.18023	−0.590116	+	0.807318i	$0.700918\pi$
$32$	−17.7381	−3.13568
$33$	−3.97005	−0.691097
$34$	16.8273	2.88585
$35$	−0.135523	−0.0229075
$36$	5.25363	0.875605
$37$	−7.20692	−1.18481	−0.592405	−	0.805640i	$-0.701821\pi$
$37$	−7.20692	−1.18481	−0.592405	+	0.805640i	$0.701821\pi$
$38$	4.58026	0.743017
$39$	0.212754	0.0340680
$40$	27.5852	4.36161
$41$	6.68846	1.04456	0.522281	−	0.852774i	$-0.325081\pi$
$41$	6.68846	1.04456	0.522281	+	0.852774i	$0.325081\pi$
$42$	−0.115947	−0.0178910
$43$	−8.46296	−1.29059	−0.645295	−	0.763934i	$-0.723265\pi$
$43$	−8.46296	−1.29059	−0.645295	+	0.763934i	$0.723265\pi$
$44$	−20.8572	−3.14434
$45$	−3.14797	−0.469272
$46$	4.35677	0.642371
$47$	1.17838	0.171885	0.0859425	−	0.996300i	$-0.472610\pi$
$47$	1.17838	0.171885	0.0859425	+	0.996300i	$0.472610\pi$
$48$	13.0934	1.88987
$49$	−6.99815	−0.999735
$50$	−13.2231	−1.87003
$51$	−6.24792	−0.874884
$52$	1.11773	0.155002
$53$	−8.24058	−1.13193	−0.565965	−	0.824429i	$-0.691496\pi$
$53$	−8.24058	−1.13193	−0.565965	+	0.824429i	$0.691496\pi$
$54$	−2.69326	−0.366506
$55$	12.4976	1.68518
$56$	−0.377248	−0.0504119
$57$	−1.70064	−0.225255
$58$	23.0063	3.02088
$59$	6.26247	0.815304	0.407652	−	0.913137i	$-0.366348\pi$
$59$	6.26247	0.815304	0.407652	+	0.913137i	$0.366348\pi$
$60$	−16.5383	−2.13508
$61$	−1.14722	−0.146887	−0.0734435	−	0.997299i	$-0.523399\pi$
$61$	−1.14722	−0.146887	−0.0734435	+	0.997299i	$0.523399\pi$
$62$	17.6981	2.24766
$63$	0.0430508	0.00542389
$64$	21.5865	2.69831
$65$	−0.669744	−0.0830716
$66$	10.6924	1.31614
$67$	−0.662651	−0.0809557	−0.0404779	−	0.999180i	$-0.512888\pi$
$67$	−0.662651	−0.0809557	−0.0404779	+	0.999180i	$0.512888\pi$
$68$	−32.8243	−3.98053
$69$	−1.61766	−0.194743
$70$	0.364997	0.0436255
$71$	−10.3048	−1.22296	−0.611479	−	0.791261i	$-0.709425\pi$
$71$	−10.3048	−1.22296	−0.611479	+	0.791261i	$0.709425\pi$
$72$	−8.76286	−1.03271
$73$	13.1083	1.53422	0.767108	−	0.641518i	$-0.221695\pi$
$73$	13.1083	1.53422	0.767108	+	0.641518i	$0.221695\pi$
$74$	19.4101	2.25638
$75$	4.90972	0.566926
$76$	−8.93454	−1.02486
$77$	−0.170914	−0.0194774
$78$	−0.573002	−0.0648797
$79$	−14.5271	−1.63443	−0.817214	−	0.576334i	$-0.804483\pi$
$79$	−14.5271	−1.63443	−0.817214	+	0.576334i	$0.804483\pi$
$80$	−41.2176	−4.60827
$81$	1.00000	0.111111
$82$	−18.0137	−1.98928
$83$	10.4232	1.14410	0.572049	−	0.820220i	$-0.306149\pi$
$83$	10.4232	1.14410	0.572049	+	0.820220i	$0.306149\pi$
$84$	0.226173	0.0246775
$85$	19.6683	2.13332
$86$	22.7929	2.45782
$87$	−8.54219	−0.915819
$88$	34.7890	3.70852
$89$	−1.48612	−0.157529	−0.0787643	−	0.996893i	$-0.525097\pi$
$89$	−1.48612	−0.157529	−0.0787643	+	0.996893i	$0.525097\pi$
$90$	8.47829	0.893691
$91$	0.00915924	0.000960149 0
$92$	−8.49858	−0.886038
$93$	−6.57126	−0.681408
$94$	−3.17369	−0.327341
$95$	5.35357	0.549265
$96$	−17.7381	−1.81039
$97$	0.764689	0.0776424	0.0388212	−	0.999246i	$-0.487640\pi$
$97$	0.764689	0.0776424	0.0388212	+	0.999246i	$0.487640\pi$
$98$	18.8478	1.90392
$99$	−3.97005	−0.399005
$100$	25.7939	2.57939
$101$	−6.48839	−0.645618	−0.322809	−	0.946464i	$-0.604627\pi$
$101$	−6.48839	−0.645618	−0.322809	+	0.946464i	$0.604627\pi$
$102$	16.8273	1.66615
$103$	0.608752	0.0599821	0.0299911	−	0.999550i	$-0.490452\pi$
$103$	0.608752	0.0599821	0.0299911	+	0.999550i	$0.490452\pi$
$104$	−1.86434	−0.182813
$105$	−0.135523	−0.0132256
$106$	22.1940	2.15567
$107$	10.1347	0.979754	0.489877	−	0.871792i	$-0.337042\pi$
$107$	10.1347	0.979754	0.489877	+	0.871792i	$0.337042\pi$
$108$	5.25363	0.505531
$109$	−2.02593	−0.194049	−0.0970245	−	0.995282i	$-0.530933\pi$
$109$	−2.02593	−0.194049	−0.0970245	+	0.995282i	$0.530933\pi$
$110$	−33.6593	−3.20929
$111$	−7.20692	−0.684050
$112$	0.563680	0.0532628
$113$	−14.2966	−1.34491	−0.672455	−	0.740138i	$-0.734760\pi$
$113$	−14.2966	−1.34491	−0.672455	+	0.740138i	$0.734760\pi$
$114$	4.58026	0.428981
$115$	5.09234	0.474863
$116$	−44.8775	−4.16677
$117$	0.212754	0.0196691
$118$	−16.8664	−1.55268
$119$	−0.268978	−0.0246571
$120$	27.5852	2.51818
$121$	4.76132	0.432847
$122$	3.08977	0.279735
$123$	6.68846	0.603078
$124$	−34.5230	−3.10025
$125$	0.284200	0.0254197
$126$	−0.115947	−0.0103294
$127$	−21.4040	−1.89930	−0.949648	−	0.313320i	$-0.898559\pi$
$127$	−21.4040	−1.89930	−0.949648	+	0.313320i	$0.898559\pi$
$128$	−22.6618	−2.00304
$129$	−8.46296	−0.745122
$130$	1.80379	0.158203
$131$	−1.98638	−0.173550	−0.0867752	−	0.996228i	$-0.527656\pi$
$131$	−1.98638	−0.173550	−0.0867752	+	0.996228i	$0.527656\pi$
$132$	−20.8572	−1.81539
$133$	−0.0732139	−0.00634845
$134$	1.78469	0.154174
$135$	−3.14797	−0.270934
$136$	54.7497	4.69475
$137$	6.48064	0.553679	0.276839	−	0.960916i	$-0.410713\pi$
$137$	6.48064	0.553679	0.276839	+	0.960916i	$0.410713\pi$
$138$	4.35677	0.370873
$139$	10.8856	0.923301	0.461651	−	0.887062i	$-0.347257\pi$
$139$	10.8856	0.923301	0.461651	+	0.887062i	$0.347257\pi$
$140$	−0.711986	−0.0601738
$141$	1.17838	0.0992378
$142$	27.7535	2.32903
$143$	−0.844646	−0.0706328
$144$	13.0934	1.09112
$145$	26.8906	2.23314
$146$	−35.3041	−2.92179
$147$	−6.99815	−0.577197
$148$	−37.8625	−3.11228
$149$	9.48025	0.776653	0.388326	−	0.921522i	$-0.373053\pi$
$149$	9.48025	0.776653	0.388326	+	0.921522i	$0.373053\pi$
$150$	−13.2231	−1.07966
$151$	−16.2086	−1.31904	−0.659519	−	0.751688i	$-0.729240\pi$
$151$	−16.2086	−1.31904	−0.659519	+	0.751688i	$0.729240\pi$
$152$	14.9025	1.20875
$153$	−6.24792	−0.505114
$154$	0.460315	0.0370932
$155$	20.6861	1.66155
$156$	1.11773	0.0894902
$157$	8.08975	0.645632	0.322816	−	0.946462i	$-0.395371\pi$
$157$	8.08975	0.645632	0.322816	+	0.946462i	$0.395371\pi$
$158$	39.1252	3.11264
$159$	−8.24058	−0.653521
$160$	55.8390	4.41446
$161$	−0.0696414	−0.00548851
$162$	−2.69326	−0.211602
$163$	12.8992	1.01034	0.505172	−	0.863019i	$-0.331429\pi$
$163$	12.8992	1.01034	0.505172	+	0.863019i	$0.331429\pi$
$164$	35.1387	2.74387
$165$	12.4976	0.972938
$166$	−28.0724	−2.17884
$167$	11.3998	0.882141	0.441070	−	0.897473i	$-0.354599\pi$
$167$	11.3998	0.882141	0.441070	+	0.897473i	$0.354599\pi$
$168$	−0.377248	−0.0291053
$169$	−12.9547	−0.996518
$170$	−52.9717	−4.06274
$171$	−1.70064	−0.130051
$172$	−44.4613	−3.39014
$173$	−8.14401	−0.619177	−0.309589	−	0.950871i	$-0.600191\pi$
$173$	−8.14401	−0.619177	−0.309589	+	0.950871i	$0.600191\pi$
$174$	23.0063	1.74410
$175$	0.211367	0.0159779
$176$	−51.9814	−3.91825
$177$	6.26247	0.470716
$178$	4.00251	0.300001
$179$	−21.9234	−1.63864	−0.819318	−	0.573340i	$-0.805647\pi$
$179$	−21.9234	−1.63864	−0.819318	+	0.573340i	$0.805647\pi$
$180$	−16.5383	−1.23269
$181$	0.291204	0.0216450	0.0108225	−	0.999941i	$-0.496555\pi$
$181$	0.291204	0.0216450	0.0108225	+	0.999941i	$0.496555\pi$
$182$	−0.0246682	−0.00182853
$183$	−1.14722	−0.0848053
$184$	14.1753	1.04502
$185$	22.6872	1.66799
$186$	17.6981	1.29769
$187$	24.8046	1.81389
$188$	6.19080	0.451510
$189$	0.0430508	0.00313148
$190$	−14.4185	−1.04603
$191$	6.48551	0.469275	0.234638	−	0.972083i	$-0.424610\pi$
$191$	6.48551	0.469275	0.234638	+	0.972083i	$0.424610\pi$
$192$	21.5865	1.55787
$193$	−21.6045	−1.55513	−0.777564	−	0.628804i	$-0.783545\pi$
$193$	−21.6045	−1.55513	−0.777564	+	0.628804i	$0.783545\pi$
$194$	−2.05950	−0.147864
$195$	−0.669744	−0.0479614
$196$	−36.7657	−2.62612
$197$	−16.1402	−1.14994	−0.574971	−	0.818174i	$-0.694987\pi$
$197$	−16.1402	−1.14994	−0.574971	+	0.818174i	$0.694987\pi$
$198$	10.6924	0.759874
$199$	−2.42626	−0.171993	−0.0859965	−	0.996295i	$-0.527407\pi$
$199$	−2.42626	−0.171993	−0.0859965	+	0.996295i	$0.527407\pi$
$200$	−43.0232	−3.04220
$201$	−0.662651	−0.0467398
$202$	17.4749	1.22953
$203$	−0.367748	−0.0258109
$204$	−32.8243	−2.29816
$205$	−21.0551	−1.47055
$206$	−1.63953	−0.114231
$207$	−1.61766	−0.112435
$208$	2.78567	0.193152
$209$	6.75164	0.467020
$210$	0.364997	0.0251872
$211$	−19.3488	−1.33203	−0.666015	−	0.745939i	$-0.732001\pi$
$211$	−19.3488	−1.33203	−0.666015	+	0.745939i	$0.732001\pi$
$212$	−43.2930	−2.97337
$213$	−10.3048	−0.706075
$214$	−27.2952	−1.86586
$215$	26.6412	1.81691
$216$	−8.76286	−0.596237
$217$	−0.282898	−0.0192043
$218$	5.45635	0.369551
$219$	13.1083	0.885780
$220$	65.6578	4.42665
$221$	−1.32927	−0.0894165
$222$	19.4101	1.30272
$223$	−19.6443	−1.31548	−0.657741	−	0.753245i	$-0.728488\pi$
$223$	−19.6443	−1.31548	−0.657741	+	0.753245i	$0.728488\pi$
$224$	−0.763639	−0.0510228
$225$	4.90972	0.327315
$226$	38.5044	2.56127
$227$	−17.1918	−1.14106	−0.570529	−	0.821278i	$-0.693262\pi$
$227$	−17.1918	−1.14106	−0.570529	+	0.821278i	$0.693262\pi$
$228$	−8.93454	−0.591705
$229$	−0.447614	−0.0295791	−0.0147896	−	0.999891i	$-0.504708\pi$
$229$	−0.447614	−0.0295791	−0.0147896	+	0.999891i	$0.504708\pi$
$230$	−13.7150	−0.904339
$231$	−0.170914	−0.0112453
$232$	74.8541	4.91441
$233$	4.91670	0.322104	0.161052	−	0.986946i	$-0.448511\pi$
$233$	4.91670	0.322104	0.161052	+	0.986946i	$0.448511\pi$
$234$	−0.573002	−0.0374583
$235$	−3.70952	−0.241982
$236$	32.9007	2.14165
$237$	−14.5271	−0.943637
$238$	0.724426	0.0469576
$239$	19.3251	1.25004	0.625019	−	0.780610i	$-0.285091\pi$
$239$	19.3251	1.25004	0.625019	+	0.780610i	$0.285091\pi$
$240$	−41.2176	−2.66058
$241$	5.97867	0.385120	0.192560	−	0.981285i	$-0.438321\pi$
$241$	5.97867	0.385120	0.192560	+	0.981285i	$0.438321\pi$
$242$	−12.8235	−0.824323
$243$	1.00000	0.0641500
$244$	−6.02710	−0.385845
$245$	22.0300	1.40744
$246$	−18.0137	−1.14851
$247$	−0.361819	−0.0230220
$248$	57.5830	3.65653
$249$	10.4232	0.660545
$250$	−0.765425	−0.0484097
$251$	13.7369	0.867065	0.433532	−	0.901138i	$-0.357267\pi$
$251$	13.7369	0.867065	0.433532	+	0.901138i	$0.357267\pi$
$252$	0.226173	0.0142476
$253$	6.42219	0.403760
$254$	57.6464	3.61706
$255$	19.6683	1.23167
$256$	17.8610	1.11631
$257$	10.2362	0.638514	0.319257	−	0.947668i	$-0.396567\pi$
$257$	10.2362	0.638514	0.319257	+	0.947668i	$0.396567\pi$
$258$	22.7929	1.41903
$259$	−0.310263	−0.0192788
$260$	−3.51859	−0.218214
$261$	−8.54219	−0.528749
$262$	5.34982	0.330513
$263$	24.6857	1.52219	0.761093	−	0.648642i	$-0.224663\pi$
$263$	24.6857	1.52219	0.761093	+	0.648642i	$0.224663\pi$
$264$	34.7890	2.14112
$265$	25.9411	1.59355
$266$	0.197184	0.0120901
$267$	−1.48612	−0.0909492
$268$	−3.48132	−0.212656
$269$	1.46734	0.0894652	0.0447326	−	0.998999i	$-0.485756\pi$
$269$	1.46734	0.0894652	0.0447326	+	0.998999i	$0.485756\pi$
$270$	8.47829	0.515972
$271$	−19.9447	−1.21156	−0.605778	−	0.795634i	$-0.707138\pi$
$271$	−19.9447	−1.21156	−0.605778	+	0.795634i	$0.707138\pi$
$272$	−81.8064	−4.96024
$273$	0.00915924	0.000554342 0
$274$	−17.4540	−1.05444
$275$	−19.4918	−1.17540
$276$	−8.49858	−0.511554
$277$	15.5404	0.933734	0.466867	−	0.884327i	$-0.345383\pi$
$277$	15.5404	0.933734	0.466867	+	0.884327i	$0.345383\pi$
$278$	−29.3176	−1.75835
$279$	−6.57126	−0.393411
$280$	1.18757	0.0709706
$281$	9.32142	0.556069	0.278035	−	0.960571i	$-0.410317\pi$
$281$	9.32142	0.556069	0.278035	+	0.960571i	$0.410317\pi$
$282$	−3.17369	−0.188991
$283$	16.2950	0.968636	0.484318	−	0.874892i	$-0.339068\pi$
$283$	16.2950	0.968636	0.484318	+	0.874892i	$0.339068\pi$
$284$	−54.1378	−3.21248
$285$	5.35357	0.317118
$286$	2.27485	0.134515
$287$	0.287943	0.0169967
$288$	−17.7381	−1.04523
$289$	22.0365	1.29626
$290$	−72.4232	−4.25284
$291$	0.764689	0.0448269
$292$	68.8664	4.03010
$293$	7.15965	0.418271	0.209136	−	0.977887i	$-0.432935\pi$
$293$	7.15965	0.418271	0.209136	+	0.977887i	$0.432935\pi$
$294$	18.8478	1.09923
$295$	−19.7141	−1.14780
$296$	63.1532	3.67071
$297$	−3.97005	−0.230366
$298$	−25.5328	−1.47907
$299$	−0.344164	−0.0199035
$300$	25.7939	1.48921
$301$	−0.364337	−0.0210000
$302$	43.6540	2.51200
$303$	−6.48839	−0.372748
$304$	−22.2671	−1.27711
$305$	3.61143	0.206790
$306$	16.8273	0.961950
$307$	21.1676	1.20810	0.604049	−	0.796947i	$-0.293553\pi$
$307$	21.1676	1.20810	0.604049	+	0.796947i	$0.293553\pi$
$308$	−0.897918	−0.0511636
$309$	0.608752	0.0346307
$310$	−55.7130	−3.16429
$311$	5.06551	0.287239	0.143619	−	0.989633i	$-0.454126\pi$
$311$	5.06551	0.287239	0.143619	+	0.989633i	$0.454126\pi$
$312$	−1.86434	−0.105547
$313$	0.834190	0.0471512	0.0235756	−	0.999722i	$-0.492495\pi$
$313$	0.834190	0.0471512	0.0235756	+	0.999722i	$0.492495\pi$
$314$	−21.7878	−1.22955
$315$	−0.135523	−0.00763583
$316$	−76.3201	−4.29334
$317$	−22.2444	−1.24937	−0.624685	−	0.780877i	$-0.714773\pi$
$317$	−22.2444	−1.24937	−0.624685	+	0.780877i	$0.714773\pi$
$318$	22.1940	1.24458
$319$	33.9130	1.89876
$320$	−67.9537	−3.79873
$321$	10.1347	0.565661
$322$	0.187562	0.0104524
$323$	10.6255	0.591217
$324$	5.25363	0.291868
$325$	1.04456	0.0579420
$326$	−34.7409	−1.92412
$327$	−2.02593	−0.112034
$328$	−58.6100	−3.23620
$329$	0.0507303	0.00279685
$330$	−33.6593	−1.85288
$331$	10.7556	0.591181	0.295591	−	0.955315i	$-0.404484\pi$
$331$	10.7556	0.591181	0.295591	+	0.955315i	$0.404484\pi$
$332$	54.7598	3.00533
$333$	−7.20692	−0.394937
$334$	−30.7025	−1.67997
$335$	2.08601	0.113971
$336$	0.563680	0.0307513
$337$	32.0241	1.74446	0.872232	−	0.489093i	$-0.162672\pi$
$337$	32.0241	1.74446	0.872232	+	0.489093i	$0.162672\pi$
$338$	34.8904	1.89779
$339$	−14.2966	−0.776484
$340$	103.330	5.60385
$341$	26.0882	1.41276
$342$	4.58026	0.247672
$343$	−0.602631	−0.0325390
$344$	74.1598	3.99843
$345$	5.09234	0.274162
$346$	21.9339	1.17917
$347$	−20.3762	−1.09385	−0.546926	−	0.837181i	$-0.684202\pi$
$347$	−20.3762	−1.09385	−0.546926	+	0.837181i	$0.684202\pi$
$348$	−44.8775	−2.40569
$349$	12.6317	0.676161	0.338081	−	0.941117i	$-0.390222\pi$
$349$	12.6317	0.676161	0.338081	+	0.941117i	$0.390222\pi$
$350$	−0.569266	−0.0304286
$351$	0.212754	0.0113560
$352$	70.4212	3.75346
$353$	8.98285	0.478109	0.239054	−	0.971006i	$-0.423163\pi$
$353$	8.98285	0.478109	0.239054	+	0.971006i	$0.423163\pi$
$354$	−16.8664	−0.896441
$355$	32.4393	1.72170
$356$	−7.80754	−0.413799
$357$	−0.268978	−0.0142358
$358$	59.0455	3.12065
$359$	9.13891	0.482333	0.241167	−	0.970484i	$-0.422470\pi$
$359$	9.13891	0.482333	0.241167	+	0.970484i	$0.422470\pi$
$360$	27.5852	1.45387
$361$	−16.1078	−0.847780
$362$	−0.784286	−0.0412211
$363$	4.76132	0.249904
$364$	0.0481193	0.00252213
$365$	−41.2647	−2.15989
$366$	3.08977	0.161505
$367$	−20.1263	−1.05058	−0.525291	−	0.850922i	$-0.676044\pi$
$367$	−20.1263	−1.05058	−0.525291	+	0.850922i	$0.676044\pi$
$368$	−21.1806	−1.10412
$369$	6.68846	0.348187
$370$	−61.1024	−3.17656
$371$	−0.354763	−0.0184184
$372$	−34.5230	−1.78993
$373$	1.96957	0.101981	0.0509903	−	0.998699i	$-0.483762\pi$
$373$	1.96957	0.101981	0.0509903	+	0.998699i	$0.483762\pi$
$374$	−66.8051	−3.45441
$375$	0.284200	0.0146760
$376$	−10.3260	−0.532524
$377$	−1.81739	−0.0936003
$378$	−0.115947	−0.00596366
$379$	−4.66433	−0.239591	−0.119795	−	0.992799i	$-0.538224\pi$
$379$	−4.66433	−0.239591	−0.119795	+	0.992799i	$0.538224\pi$
$380$	28.1257	1.44282
$381$	−21.4040	−1.09656
$382$	−17.4671	−0.893697
$383$	−8.15198	−0.416547	−0.208273	−	0.978071i	$-0.566784\pi$
$383$	−8.15198	−0.416547	−0.208273	+	0.978071i	$0.566784\pi$
$384$	−22.6618	−1.15645
$385$	0.538032	0.0274206
$386$	58.1865	2.96162
$387$	−8.46296	−0.430197
$388$	4.01740	0.203952
$389$	22.3259	1.13197	0.565984	−	0.824416i	$-0.308496\pi$
$389$	22.3259	1.13197	0.565984	+	0.824416i	$0.308496\pi$
$390$	1.80379	0.0913386
$391$	10.1070	0.511133
$392$	61.3238	3.09732
$393$	−1.98638	−0.100199
$394$	43.4697	2.18997
$395$	45.7309	2.30097
$396$	−20.8572	−1.04811
$397$	−4.17190	−0.209382	−0.104691	−	0.994505i	$-0.533385\pi$
$397$	−4.17190	−0.209382	−0.104691	+	0.994505i	$0.533385\pi$
$398$	6.53454	0.327547
$399$	−0.0732139	−0.00366528
$400$	64.2848	3.21424
$401$	−28.8507	−1.44074	−0.720368	−	0.693592i	$-0.756027\pi$
$401$	−28.8507	−1.44074	−0.720368	+	0.693592i	$0.756027\pi$
$402$	1.78469	0.0890122
$403$	−1.39806	−0.0696425
$404$	−34.0876	−1.69592
$405$	−3.14797	−0.156424
$406$	0.990440	0.0491547
$407$	28.6118	1.41824
$408$	54.7497	2.71051
$409$	−9.32567	−0.461125	−0.230562	−	0.973058i	$-0.574057\pi$
$409$	−9.32567	−0.461125	−0.230562	+	0.973058i	$0.574057\pi$
$410$	56.7067	2.80054
$411$	6.48064	0.319667
$412$	3.19816	0.157562
$413$	0.269604	0.0132664
$414$	4.35677	0.214124
$415$	−32.8120	−1.61068
$416$	−3.77386	−0.185029
$417$	10.8856	0.533068
$418$	−18.1839	−0.889403
$419$	−32.8357	−1.60413	−0.802064	−	0.597238i	$-0.796265\pi$
$419$	−32.8357	−1.60413	−0.802064	+	0.597238i	$0.796265\pi$
$420$	−0.711986	−0.0347413
$421$	−32.2409	−1.57133	−0.785663	−	0.618655i	$-0.787678\pi$
$421$	−32.2409	−1.57133	−0.785663	+	0.618655i	$0.787678\pi$
$422$	52.1114	2.53674
$423$	1.17838	0.0572950
$424$	72.2111	3.50688
$425$	−30.6755	−1.48798
$426$	27.7535	1.34466
$427$	−0.0493889	−0.00239010
$428$	53.2437	2.57363
$429$	−0.844646	−0.0407799
$430$	−71.7515	−3.46016
$431$	10.2569	0.494057	0.247028	−	0.969008i	$-0.420546\pi$
$431$	10.2569	0.494057	0.247028	+	0.969008i	$0.420546\pi$
$432$	13.0934	0.629956
$433$	1.65518	0.0795430	0.0397715	−	0.999209i	$-0.487337\pi$
$433$	1.65518	0.0795430	0.0397715	+	0.999209i	$0.487337\pi$
$434$	0.761916	0.0365731
$435$	26.8906	1.28930
$436$	−10.6435	−0.509731
$437$	2.75106	0.131601
$438$	−35.3041	−1.68690
$439$	−7.25909	−0.346457	−0.173229	−	0.984882i	$-0.555420\pi$
$439$	−7.25909	−0.346457	−0.173229	+	0.984882i	$0.555420\pi$
$440$	−109.515	−5.22092
$441$	−6.99815	−0.333245
$442$	3.58007	0.170287
$443$	13.0817	0.621529	0.310764	−	0.950487i	$-0.399415\pi$
$443$	13.0817	0.621529	0.310764	+	0.950487i	$0.399415\pi$
$444$	−37.8625	−1.79687
$445$	4.67827	0.221771
$446$	52.9072	2.50523
$447$	9.48025	0.448401
$448$	0.929316	0.0439060
$449$	−30.2153	−1.42595	−0.712973	−	0.701191i	$-0.752652\pi$
$449$	−30.2153	−1.42595	−0.712973	+	0.701191i	$0.752652\pi$
$450$	−13.2231	−0.623345
$451$	−26.5535	−1.25036
$452$	−75.1090	−3.53283
$453$	−16.2086	−0.761547
$454$	46.3018	2.17305
$455$	−0.0288330	−0.00135171
$456$	14.9025	0.697873
$457$	13.7250	0.642030	0.321015	−	0.947074i	$-0.395976\pi$
$457$	13.7250	0.642030	0.321015	+	0.947074i	$0.395976\pi$
$458$	1.20554	0.0563311
$459$	−6.24792	−0.291628
$460$	26.7533	1.24738
$461$	−24.7390	−1.15221	−0.576105	−	0.817376i	$-0.695428\pi$
$461$	−24.7390	−1.15221	−0.576105	+	0.817376i	$0.695428\pi$
$462$	0.460315	0.0214158
$463$	5.41303	0.251565	0.125782	−	0.992058i	$-0.459856\pi$
$463$	5.41303	0.251565	0.125782	+	0.992058i	$0.459856\pi$
$464$	−111.846	−5.19233
$465$	20.6861	0.959296
$466$	−13.2419	−0.613421
$467$	−12.7700	−0.590925	−0.295463	−	0.955354i	$-0.595474\pi$
$467$	−12.7700	−0.590925	−0.295463	+	0.955354i	$0.595474\pi$
$468$	1.11773	0.0516672
$469$	−0.0285276	−0.00131728
$470$	9.99069	0.460836
$471$	8.08975	0.372756
$472$	−54.8772	−2.52593
$473$	33.5984	1.54486
$474$	39.1252	1.79708
$475$	−8.34967	−0.383109
$476$	−1.41311	−0.0647698
$477$	−8.24058	−0.377310
$478$	−52.0475	−2.38060
$479$	−27.3049	−1.24759	−0.623797	−	0.781587i	$-0.714411\pi$
$479$	−27.3049	−1.24759	−0.623797	+	0.781587i	$0.714411\pi$
$480$	55.8390	2.54869
$481$	−1.53330	−0.0699126
$482$	−16.1021	−0.733430
$483$	−0.0696414	−0.00316880
$484$	25.0142	1.13701
$485$	−2.40722	−0.109306
$486$	−2.69326	−0.122169
$487$	32.2183	1.45995	0.729974	−	0.683475i	$-0.239532\pi$
$487$	32.2183	1.45995	0.729974	+	0.683475i	$0.239532\pi$
$488$	10.0530	0.455077
$489$	12.8992	0.583323
$490$	−59.3323	−2.68036
$491$	−36.8943	−1.66502	−0.832508	−	0.554013i	$-0.813096\pi$
$491$	−36.8943	−1.66502	−0.832508	+	0.554013i	$0.813096\pi$
$492$	35.1387	1.58417
$493$	53.3709	2.40371
$494$	0.974471	0.0438435
$495$	12.4976	0.561726
$496$	−86.0400	−3.86331
$497$	−0.443631	−0.0198996
$498$	−28.0724	−1.25796
$499$	−29.1658	−1.30564	−0.652820	−	0.757513i	$-0.726414\pi$
$499$	−29.1658	−1.30564	−0.652820	+	0.757513i	$0.726414\pi$
$500$	1.49308	0.0667727
$501$	11.3998	0.509304
$502$	−36.9970	−1.65126
$503$	−21.5005	−0.958660	−0.479330	−	0.877635i	$-0.659120\pi$
$503$	−21.5005	−0.958660	−0.479330	+	0.877635i	$0.659120\pi$
$504$	−0.377248	−0.0168040
$505$	20.4252	0.908912
$506$	−17.2966	−0.768928
$507$	−12.9547	−0.575340
$508$	−112.449	−4.98910
$509$	7.84723	0.347822	0.173911	−	0.984761i	$-0.444359\pi$
$509$	7.84723	0.347822	0.173911	+	0.984761i	$0.444359\pi$
$510$	−52.9717	−2.34563
$511$	0.564324	0.0249642
$512$	−2.78071	−0.122891
$513$	−1.70064	−0.0750851
$514$	−27.5686	−1.21600
$515$	−1.91633	−0.0844438
$516$	−44.4613	−1.95730
$517$	−4.67825	−0.205749
$518$	0.835619	0.0367150
$519$	−8.14401	−0.357482
$520$	5.86888	0.257367
$521$	−9.57773	−0.419608	−0.209804	−	0.977743i	$-0.567283\pi$
$521$	−9.57773	−0.419608	−0.209804	+	0.977743i	$0.567283\pi$
$522$	23.0063	1.00696
$523$	−16.3655	−0.715611	−0.357806	−	0.933796i	$-0.616475\pi$
$523$	−16.3655	−0.715611	−0.357806	+	0.933796i	$0.616475\pi$
$524$	−10.4357	−0.455885
$525$	0.211367	0.00922482
$526$	−66.4850	−2.89888
$527$	41.0567	1.78846
$528$	−51.9814	−2.26220
$529$	−20.3832	−0.886225
$530$	−69.8661	−3.03479
$531$	6.26247	0.271768
$532$	−0.384639	−0.0166762
$533$	1.42300	0.0616369
$534$	4.00251	0.173206
$535$	−31.9036	−1.37931
$536$	5.80672	0.250812
$537$	−21.9234	−0.946066
$538$	−3.95192	−0.170379
$539$	27.7830	1.19670
$540$	−16.5383	−0.711694
$541$	21.6362	0.930212	0.465106	−	0.885255i	$-0.346016\pi$
$541$	21.6362	0.930212	0.465106	+	0.885255i	$0.346016\pi$
$542$	53.7162	2.30731
$543$	0.291204	0.0124967
$544$	110.826	4.75164
$545$	6.37757	0.273185
$546$	−0.0246682	−0.00105570
$547$	−17.0895	−0.730694	−0.365347	−	0.930871i	$-0.619050\pi$
$547$	−17.0895	−0.730694	−0.365347	+	0.930871i	$0.619050\pi$
$548$	34.0469	1.45441
$549$	−1.14722	−0.0489624
$550$	52.4965	2.23846
$551$	14.5272	0.618880
$552$	14.1753	0.603342
$553$	−0.625403	−0.0265949
$554$	−41.8544	−1.77822
$555$	22.6872	0.963017
$556$	57.1887	2.42534
$557$	−29.8407	−1.26439	−0.632196	−	0.774808i	$-0.717846\pi$
$557$	−29.8407	−1.26439	−0.632196	+	0.774808i	$0.717846\pi$
$558$	17.6981	0.749219
$559$	−1.80053	−0.0761544
$560$	−1.77445	−0.0749841
$561$	24.8046	1.04725
$562$	−25.1050	−1.05899
$563$	−43.5907	−1.83713	−0.918564	−	0.395272i	$-0.870650\pi$
$563$	−43.5907	−1.83713	−0.918564	+	0.395272i	$0.870650\pi$
$564$	6.19080	0.260680
$565$	45.0053	1.89339
$566$	−43.8866	−1.84469
$567$	0.0430508	0.00180796
$568$	90.2998	3.78889
$569$	−30.5079	−1.27896	−0.639480	−	0.768808i	$-0.720850\pi$
$569$	−30.5079	−1.27896	−0.639480	+	0.768808i	$0.720850\pi$
$570$	−14.4185	−0.603926
$571$	34.2415	1.43296	0.716480	−	0.697607i	$-0.245752\pi$
$571$	34.2415	1.43296	0.716480	+	0.697607i	$0.245752\pi$
$572$	−4.43746	−0.185539
$573$	6.48551	0.270936
$574$	−0.775505	−0.0323689
$575$	−7.94225	−0.331215
$576$	21.5865	0.899438
$577$	−21.3609	−0.889266	−0.444633	−	0.895713i	$-0.646666\pi$
$577$	−21.3609	−0.889266	−0.444633	+	0.895713i	$0.646666\pi$
$578$	−59.3499	−2.46863
$579$	−21.6045	−0.897853
$580$	141.273	5.86605
$581$	0.448728	0.0186164
$582$	−2.05950	−0.0853692
$583$	32.7155	1.35494
$584$	−114.867	−4.75322
$585$	−0.669744	−0.0276905
$586$	−19.2828	−0.796564
$587$	29.5178	1.21833	0.609165	−	0.793043i	$-0.291505\pi$
$587$	29.5178	1.21833	0.609165	+	0.793043i	$0.291505\pi$
$588$	−36.7657	−1.51619
$589$	11.1753	0.460472
$590$	53.0951	2.18589
$591$	−16.1402	−0.663919
$592$	−94.3629	−3.87829
$593$	−3.23084	−0.132675	−0.0663373	−	0.997797i	$-0.521131\pi$
$593$	−3.23084	−0.132675	−0.0663373	+	0.997797i	$0.521131\pi$
$594$	10.6924	0.438713
$595$	0.846734	0.0347127
$596$	49.8058	2.04012
$597$	−2.42626	−0.0993002
$598$	0.926922	0.0379046
$599$	−47.5239	−1.94178	−0.970888	−	0.239534i	$-0.923005\pi$
$599$	−47.5239	−1.94178	−0.970888	+	0.239534i	$0.923005\pi$
$600$	−43.0232	−1.75642
$601$	35.1770	1.43490	0.717449	−	0.696611i	$-0.245309\pi$
$601$	35.1770	1.43490	0.717449	+	0.696611i	$0.245309\pi$
$602$	0.981253	0.0399929
$603$	−0.662651	−0.0269852
$604$	−85.1541	−3.46487
$605$	−14.9885	−0.609369
$606$	17.4749	0.709869
$607$	47.6240	1.93300	0.966499	−	0.256670i	$-0.0826254\pi$
$607$	47.6240	1.93300	0.966499	+	0.256670i	$0.0826254\pi$
$608$	30.1662	1.22340
$609$	−0.367748	−0.0149019
$610$	−9.72651	−0.393815
$611$	0.250706	0.0101425
$612$	−32.8243	−1.32684
$613$	−23.0412	−0.930626	−0.465313	−	0.885146i	$-0.654058\pi$
$613$	−23.0412	−0.930626	−0.465313	+	0.885146i	$0.654058\pi$
$614$	−57.0097	−2.30072
$615$	−21.0551	−0.849022
$616$	1.49769	0.0603438
$617$	−3.66906	−0.147711	−0.0738553	−	0.997269i	$-0.523530\pi$
$617$	−3.66906	−0.147711	−0.0738553	+	0.997269i	$0.523530\pi$
$618$	−1.63953	−0.0659514
$619$	20.9193	0.840817	0.420409	−	0.907335i	$-0.361887\pi$
$619$	20.9193	0.840817	0.420409	+	0.907335i	$0.361887\pi$
$620$	108.677	4.36458
$621$	−1.61766	−0.0649144
$622$	−13.6427	−0.547023
$623$	−0.0639787	−0.00256325
$624$	2.78567	0.111516
$625$	−25.4433	−1.01773
$626$	−2.24669	−0.0897957
$627$	6.75164	0.269634
$628$	42.5005	1.69596
$629$	45.0282	1.79539
$630$	0.364997	0.0145418
$631$	10.3667	0.412690	0.206345	−	0.978479i	$-0.433843\pi$
$631$	10.3667	0.412690	0.206345	+	0.978479i	$0.433843\pi$
$632$	127.299	5.06369
$633$	−19.3488	−0.769047
$634$	59.9099	2.37933
$635$	67.3791	2.67386
$636$	−43.2930	−1.71668
$637$	−1.48889	−0.0589918
$638$	−91.3363	−3.61604
$639$	−10.3048	−0.407653
$640$	71.3387	2.81991
$641$	12.1861	0.481322	0.240661	−	0.970609i	$-0.422636\pi$
$641$	12.1861	0.481322	0.240661	+	0.970609i	$0.422636\pi$
$642$	−27.2952	−1.07726
$643$	27.4309	1.08177	0.540884	−	0.841097i	$-0.318090\pi$
$643$	27.4309	1.08177	0.540884	+	0.841097i	$0.318090\pi$
$644$	−0.365870	−0.0144173
$645$	26.6412	1.04899
$646$	−28.6171	−1.12593
$647$	24.1269	0.948525	0.474263	−	0.880383i	$-0.342715\pi$
$647$	24.1269	0.948525	0.474263	+	0.880383i	$0.342715\pi$
$648$	−8.76286	−0.344238
$649$	−24.8623	−0.975932
$650$	−2.81328	−0.110346
$651$	−0.282898	−0.0110876
$652$	67.7677	2.65399
$653$	24.6003	0.962683	0.481342	−	0.876533i	$-0.340150\pi$
$653$	24.6003	0.962683	0.481342	+	0.876533i	$0.340150\pi$
$654$	5.45635	0.213360
$655$	6.25305	0.244327
$656$	87.5745	3.41921
$657$	13.1083	0.511405
$658$	−0.136630	−0.00532639
$659$	35.7993	1.39454	0.697271	−	0.716807i	$-0.254397\pi$
$659$	35.7993	1.39454	0.697271	+	0.716807i	$0.254397\pi$
$660$	65.6578	2.55573
$661$	47.1408	1.83357	0.916783	−	0.399386i	$-0.130777\pi$
$661$	47.1408	1.83357	0.916783	+	0.399386i	$0.130777\pi$
$662$	−28.9676	−1.12586
$663$	−1.32927	−0.0516246
$664$	−91.3373	−3.54457
$665$	0.230475	0.00893745
$666$	19.4101	0.752125
$667$	13.8184	0.535049
$668$	59.8902	2.31722
$669$	−19.6443	−0.759493
$670$	−5.61815	−0.217048
$671$	4.55454	0.175826
$672$	−0.763639	−0.0294580
$673$	−45.2055	−1.74254	−0.871272	−	0.490801i	$-0.836704\pi$
$673$	−45.2055	−1.74254	−0.871272	+	0.490801i	$0.836704\pi$
$674$	−86.2491	−3.32219
$675$	4.90972	0.188975
$676$	−68.0594	−2.61767
$677$	15.9940	0.614700	0.307350	−	0.951597i	$-0.400558\pi$
$677$	15.9940	0.614700	0.307350	+	0.951597i	$0.400558\pi$
$678$	38.5044	1.47875
$679$	0.0329205	0.00126337
$680$	−172.350	−6.60933
$681$	−17.1918	−0.658790
$682$	−70.2623	−2.69048
$683$	−13.4672	−0.515308	−0.257654	−	0.966237i	$-0.582949\pi$
$683$	−13.4672	−0.515308	−0.257654	+	0.966237i	$0.582949\pi$
$684$	−8.93454	−0.341621
$685$	−20.4009	−0.779478
$686$	1.62304	0.0619680
$687$	−0.447614	−0.0170775
$688$	−110.809	−4.22455
$689$	−1.75322	−0.0667923
$690$	−13.7150	−0.522120
$691$	36.2228	1.37798	0.688991	−	0.724770i	$-0.258054\pi$
$691$	36.2228	1.37798	0.688991	+	0.724770i	$0.258054\pi$
$692$	−42.7856	−1.62646
$693$	−0.170914	−0.00649248
$694$	54.8783	2.08315
$695$	−34.2674	−1.29984
$696$	74.8541	2.83734
$697$	−41.7889	−1.58287
$698$	−34.0205	−1.28769
$699$	4.91670	0.185967
$700$	1.11045	0.0419709
$701$	29.0413	1.09688	0.548438	−	0.836191i	$-0.315223\pi$
$701$	29.0413	1.09688	0.548438	+	0.836191i	$0.315223\pi$
$702$	−0.573002	−0.0216266
$703$	12.2564	0.462258
$704$	−85.6996	−3.22992
$705$	−3.70952	−0.139709
$706$	−24.1931	−0.910520
$707$	−0.279330	−0.0105053
$708$	32.9007	1.23648
$709$	0.0797127	0.00299367	0.00149684	−	0.999999i	$-0.499524\pi$
$709$	0.0797127	0.00299367	0.00149684	+	0.999999i	$0.499524\pi$
$710$	−87.3673	−3.27884
$711$	−14.5271	−0.544809
$712$	13.0227	0.488046
$713$	10.6300	0.398098
$714$	0.724426	0.0271110
$715$	2.65892	0.0994380
$716$	−115.178	−4.30439
$717$	19.3251	0.721709
$718$	−24.6134	−0.918565
$719$	−5.28187	−0.196980	−0.0984902	−	0.995138i	$-0.531401\pi$
$719$	−5.28187	−0.196980	−0.0984902	+	0.995138i	$0.531401\pi$
$720$	−41.2176	−1.53609
$721$	0.0262073	0.000976009 0
$722$	43.3825	1.61453
$723$	5.97867	0.222349
$724$	1.52988	0.0568574
$725$	−41.9398	−1.55760
$726$	−12.8235	−0.475923
$727$	25.8097	0.957228	0.478614	−	0.878025i	$-0.341139\pi$
$727$	25.8097	0.957228	0.478614	+	0.878025i	$0.341139\pi$
$728$	−0.0802612	−0.00297468
$729$	1.00000	0.0370370
$730$	111.136	4.11334
$731$	52.8759	1.95569
$732$	−6.02710	−0.222768
$733$	19.3541	0.714860	0.357430	−	0.933940i	$-0.383653\pi$
$733$	19.3541	0.714860	0.357430	+	0.933940i	$0.383653\pi$
$734$	54.2052	2.00075
$735$	22.0300	0.812587
$736$	28.6942	1.05768
$737$	2.63076	0.0969053
$738$	−18.0137	−0.663094
$739$	3.59517	0.132250	0.0661252	−	0.997811i	$-0.478936\pi$
$739$	3.59517	0.132250	0.0661252	+	0.997811i	$0.478936\pi$
$740$	119.190	4.38151
$741$	−0.361819	−0.0132917
$742$	0.955469	0.0350764
$743$	14.5914	0.535307	0.267653	−	0.963515i	$-0.413752\pi$
$743$	14.5914	0.535307	0.267653	+	0.963515i	$0.413752\pi$
$744$	57.5830	2.11110
$745$	−29.8436	−1.09338
$746$	−5.30456	−0.194214
$747$	10.4232	0.381366
$748$	130.314	4.76475
$749$	0.436305	0.0159422
$750$	−0.765425	−0.0279494
$751$	−44.1301	−1.61033	−0.805165	−	0.593051i	$-0.797923\pi$
$751$	−44.1301	−1.61033	−0.805165	+	0.593051i	$0.797923\pi$
$752$	15.4290	0.562639
$753$	13.7369	0.500600
$754$	4.89469	0.178254
$755$	51.0243	1.85696
$756$	0.226173	0.00822583
$757$	−33.7818	−1.22782	−0.613911	−	0.789375i	$-0.710405\pi$
$757$	−33.7818	−1.22782	−0.613911	+	0.789375i	$0.710405\pi$
$758$	12.5622	0.456281
$759$	6.42219	0.233111
$760$	−46.9126	−1.70170
$761$	−30.1995	−1.09473	−0.547365	−	0.836894i	$-0.684369\pi$
$761$	−30.1995	−1.09473	−0.547365	+	0.836894i	$0.684369\pi$
$762$	57.6464	2.08831
$763$	−0.0872179	−0.00315750
$764$	34.0725	1.23270
$765$	19.6683	0.711108
$766$	21.9554	0.793280
$767$	1.33237	0.0481090
$768$	17.8610	0.644504
$769$	33.5653	1.21040	0.605199	−	0.796075i	$-0.293094\pi$
$769$	33.5653	1.21040	0.605199	+	0.796075i	$0.293094\pi$
$770$	−1.44906	−0.0522204
$771$	10.2362	0.368646
$772$	−113.502	−4.08503
$773$	−32.9732	−1.18596	−0.592982	−	0.805216i	$-0.702049\pi$
$773$	−32.9732	−1.18596	−0.592982	+	0.805216i	$0.702049\pi$
$774$	22.7929	0.819275
$775$	−32.2630	−1.15892
$776$	−6.70087	−0.240547
$777$	−0.310263	−0.0111306
$778$	−60.1294	−2.15574
$779$	−11.3747	−0.407540
$780$	−3.51859	−0.125986
$781$	40.9107	1.46390
$782$	−27.2207	−0.973412
$783$	−8.54219	−0.305273
$784$	−91.6294	−3.27248
$785$	−25.4663	−0.908931
$786$	5.34982	0.190822
$787$	21.9281	0.781653	0.390827	−	0.920464i	$-0.372189\pi$
$787$	21.9281	0.781653	0.390827	+	0.920464i	$0.372189\pi$
$788$	−84.7947	−3.02068
$789$	24.6857	0.878835
$790$	−123.165	−4.38202
$791$	−0.615479	−0.0218839
$792$	34.7890	1.23617
$793$	−0.244077	−0.00866743
$794$	11.2360	0.398750
$795$	25.9411	0.920036
$796$	−12.7467	−0.451794
$797$	−6.14979	−0.217837	−0.108918	−	0.994051i	$-0.534739\pi$
$797$	−6.14979	−0.217837	−0.108918	+	0.994051i	$0.534739\pi$
$798$	0.197184	0.00698023
$799$	−7.36245	−0.260465
$800$	−87.0891	−3.07907
$801$	−1.48612	−0.0525096
$802$	77.7024	2.74377
$803$	−52.0408	−1.83648
$804$	−3.48132	−0.122777
$805$	0.219229	0.00772681
$806$	3.76534	0.132629
$807$	1.46734	0.0516527
$808$	56.8568	2.00022
$809$	12.5388	0.440840	0.220420	−	0.975405i	$-0.429257\pi$
$809$	12.5388	0.440840	0.220420	+	0.975405i	$0.429257\pi$
$810$	8.47829	0.297897
$811$	44.9269	1.57760	0.788798	−	0.614653i	$-0.210704\pi$
$811$	44.9269	1.57760	0.788798	+	0.614653i	$0.210704\pi$
$812$	−1.93201	−0.0678003
$813$	−19.9447	−0.699492
$814$	−77.0590	−2.70092
$815$	−40.6064	−1.42238
$816$	−81.8064	−2.86380
$817$	14.3925	0.503529
$818$	25.1164	0.878175
$819$	0.00915924	0.000320050 0
$820$	−110.616	−3.86286
$821$	26.8584	0.937364	0.468682	−	0.883367i	$-0.344729\pi$
$821$	26.8584	0.937364	0.468682	+	0.883367i	$0.344729\pi$
$822$	−17.4540	−0.608780
$823$	−18.6761	−0.651009	−0.325505	−	0.945540i	$-0.605534\pi$
$823$	−18.6761	−0.651009	−0.325505	+	0.945540i	$0.605534\pi$
$824$	−5.33441	−0.185833
$825$	−19.4918	−0.678619
$826$	−0.726113	−0.0252647
$827$	−28.7854	−1.00097	−0.500484	−	0.865746i	$-0.666845\pi$
$827$	−28.7854	−1.00097	−0.500484	+	0.865746i	$0.666845\pi$
$828$	−8.49858	−0.295346
$829$	−37.8459	−1.31444	−0.657220	−	0.753698i	$-0.728268\pi$
$829$	−37.8459	−1.31444	−0.657220	+	0.753698i	$0.728268\pi$
$830$	88.3712	3.06741
$831$	15.5404	0.539092
$832$	4.59262	0.159221
$833$	43.7239	1.51494
$834$	−29.3176	−1.01519
$835$	−35.8861	−1.24189
$836$	35.4706	1.22678
$837$	−6.57126	−0.227136
$838$	88.4349	3.05493
$839$	36.1131	1.24676	0.623380	−	0.781919i	$-0.285759\pi$
$839$	36.1131	1.24676	0.623380	+	0.781919i	$0.285759\pi$
$840$	1.18757	0.0409749
$841$	43.9691	1.51617
$842$	86.8330	2.99246
$843$	9.32142	0.321047
$844$	−101.652	−3.49900
$845$	40.7811	1.40291
$846$	−3.17369	−0.109114
$847$	0.204978	0.00704314
$848$	−107.897	−3.70520
$849$	16.2950	0.559242
$850$	82.6171	2.83374
$851$	11.6583	0.399642
$852$	−54.1378	−1.85473
$853$	36.3723	1.24536	0.622682	−	0.782475i	$-0.286043\pi$
$853$	36.3723	1.24536	0.622682	+	0.782475i	$0.286043\pi$
$854$	0.133017	0.00455175
$855$	5.35357	0.183088
$856$	−88.8086	−3.03542
$857$	−3.82716	−0.130733	−0.0653666	−	0.997861i	$-0.520822\pi$
$857$	−3.82716	−0.130733	−0.0653666	+	0.997861i	$0.520822\pi$
$858$	2.27485	0.0776620
$859$	−4.60465	−0.157109	−0.0785544	−	0.996910i	$-0.525030\pi$
$859$	−4.60465	−0.157109	−0.0785544	+	0.996910i	$0.525030\pi$
$860$	139.963	4.77269
$861$	0.287943	0.00981308
$862$	−27.6244	−0.940892
$863$	−46.6427	−1.58774	−0.793868	−	0.608090i	$-0.791936\pi$
$863$	−46.6427	−1.58774	−0.793868	+	0.608090i	$0.791936\pi$
$864$	−17.7381	−0.603463
$865$	25.6371	0.871687
$866$	−4.45783	−0.151483
$867$	22.0365	0.748399
$868$	−1.48624	−0.0504463
$869$	57.6734	1.95644
$870$	−72.4232	−2.45538
$871$	−0.140982	−0.00477699
$872$	17.7530	0.601191
$873$	0.764689	0.0258808
$874$	−7.40930	−0.250623
$875$	0.0122350	0.000413620 0
$876$	68.8664	2.32678
$877$	8.13899	0.274834	0.137417	−	0.990513i	$-0.456120\pi$
$877$	8.13899	0.274834	0.137417	+	0.990513i	$0.456120\pi$
$878$	19.5506	0.659801
$879$	7.15965	0.241489
$880$	163.636	5.51617
$881$	49.3414	1.66235	0.831177	−	0.556008i	$-0.187668\pi$
$881$	49.3414	1.66235	0.831177	+	0.556008i	$0.187668\pi$
$882$	18.8478	0.634639
$883$	34.2071	1.15116	0.575581	−	0.817745i	$-0.304776\pi$
$883$	34.2071	1.15116	0.575581	+	0.817745i	$0.304776\pi$
$884$	−6.98350	−0.234881
$885$	−19.7141	−0.662681
$886$	−35.2323	−1.18365
$887$	1.71402	0.0575510	0.0287755	−	0.999586i	$-0.490839\pi$
$887$	1.71402	0.0575510	0.0287755	+	0.999586i	$0.490839\pi$
$888$	63.1532	2.11928
$889$	−0.921458	−0.0309047
$890$	−12.5998	−0.422346
$891$	−3.97005	−0.133002
$892$	−103.204	−3.45553
$893$	−2.00401	−0.0670616
$894$	−25.5328	−0.853943
$895$	69.0144	2.30690
$896$	−0.975608	−0.0325928
$897$	−0.344164	−0.0114913
$898$	81.3774	2.71560
$899$	56.1329	1.87214
$900$	25.7939	0.859795
$901$	51.4865	1.71526
$902$	71.5154	2.38120
$903$	−0.364337	−0.0121244
$904$	125.279	4.16672
$905$	−0.916700	−0.0304721
$906$	43.6540	1.45031
$907$	11.3809	0.377897	0.188949	−	0.981987i	$-0.439492\pi$
$907$	11.3809	0.377897	0.188949	+	0.981987i	$0.439492\pi$
$908$	−90.3192	−2.99735
$909$	−6.48839	−0.215206
$910$	0.0776547	0.00257423
$911$	5.12921	0.169938	0.0849691	−	0.996384i	$-0.472921\pi$
$911$	5.12921	0.169938	0.0849691	+	0.996384i	$0.472921\pi$
$912$	−22.2671	−0.737339
$913$	−41.3808	−1.36950
$914$	−36.9650	−1.22269
$915$	3.61143	0.119390
$916$	−2.35160	−0.0776990
$917$	−0.0855150	−0.00282395
$918$	16.8273	0.555382
$919$	28.4185	0.937440	0.468720	−	0.883347i	$-0.344715\pi$
$919$	28.4185	0.937440	0.468720	+	0.883347i	$0.344715\pi$
$920$	−44.6235	−1.47119
$921$	21.1676	0.697495
$922$	66.6285	2.19429
$923$	−2.19240	−0.0721636
$924$	−0.897918	−0.0295393
$925$	−35.3839	−1.16342
$926$	−14.5787	−0.479085
$927$	0.608752	0.0199940
$928$	151.522	4.97396
$929$	−17.1314	−0.562062	−0.281031	−	0.959699i	$-0.590676\pi$
$929$	−17.1314	−0.562062	−0.281031	+	0.959699i	$0.590676\pi$
$930$	−55.7130	−1.82690
$931$	11.9013	0.390051
$932$	25.8306	0.846108
$933$	5.06551	0.165837
$934$	34.3929	1.12537
$935$	−78.0841	−2.55362
$936$	−1.86434	−0.0609378
$937$	58.9549	1.92597	0.962986	−	0.269552i	$-0.0868757\pi$
$937$	58.9549	1.92597	0.962986	+	0.269552i	$0.0868757\pi$
$938$	0.0768323	0.00250866
$939$	0.834190	0.0272228
$940$	−19.4884	−0.635643
$941$	4.06419	0.132489	0.0662444	−	0.997803i	$-0.478898\pi$
$941$	4.06419	0.132489	0.0662444	+	0.997803i	$0.478898\pi$
$942$	−21.7878	−0.709884
$943$	−10.8196	−0.352336
$944$	81.9969	2.66877
$945$	−0.135523	−0.00440855
$946$	−90.4891	−2.94206
$947$	43.0426	1.39870	0.699349	−	0.714780i	$-0.253473\pi$
$947$	43.0426	1.39870	0.699349	+	0.714780i	$0.253473\pi$
$948$	−76.3201	−2.47876
$949$	2.78886	0.0905301
$950$	22.4878	0.729601
$951$	−22.2444	−0.721324
$952$	2.35702	0.0763913
$953$	8.56721	0.277519	0.138760	−	0.990326i	$-0.455688\pi$
$953$	8.56721	0.277519	0.138760	+	0.990326i	$0.455688\pi$
$954$	22.1940	0.718557
$955$	−20.4162	−0.660653
$956$	101.527	3.28362
$957$	33.9130	1.09625
$958$	73.5391	2.37594
$959$	0.278997	0.00900928
$960$	−67.9537	−2.19320
$961$	12.1814	0.392949
$962$	4.12958	0.133143
$963$	10.1347	0.326585
$964$	31.4097	1.01164
$965$	68.0104	2.18933
$966$	0.187562	0.00603472
$967$	24.1298	0.775963	0.387982	−	0.921667i	$-0.373172\pi$
$967$	24.1298	0.775963	0.387982	+	0.921667i	$0.373172\pi$
$968$	−41.7228	−1.34102
$969$	10.6255	0.341339
$970$	6.48326	0.208165
$971$	−31.5150	−1.01136	−0.505681	−	0.862720i	$-0.668759\pi$
$971$	−31.5150	−1.01136	−0.505681	+	0.862720i	$0.668759\pi$
$972$	5.25363	0.168510
$973$	0.468632	0.0150236
$974$	−86.7720	−2.78036
$975$	1.04456	0.0334528
$976$	−15.0211	−0.480812
$977$	−22.3005	−0.713455	−0.356727	−	0.934208i	$-0.616108\pi$
$977$	−22.3005	−0.713455	−0.356727	+	0.934208i	$0.616108\pi$
$978$	−34.7409	−1.11089
$979$	5.89998	0.188564
$980$	115.737	3.69709
$981$	−2.02593	−0.0646830
$982$	99.3658	3.17089
$983$	−4.38519	−0.139866	−0.0699330	−	0.997552i	$-0.522279\pi$
$983$	−4.38519	−0.139866	−0.0699330	+	0.997552i	$0.522279\pi$
$984$	−58.6100	−1.86842
$985$	50.8089	1.61891
$986$	−143.742	−4.57767
$987$	0.0507303	0.00161476
$988$	−1.90086	−0.0604745
$989$	13.6902	0.435322
$990$	−33.6593	−1.06976
$991$	−53.6221	−1.70336	−0.851681	−	0.524060i	$-0.824417\pi$
$991$	−53.6221	−1.70336	−0.851681	+	0.524060i	$0.824417\pi$
$992$	116.562	3.70084
$993$	10.7556	0.341319
$994$	1.19481	0.0378971
$995$	7.63779	0.242134
$996$	54.7598	1.73513
$997$	13.8753	0.439435	0.219717	−	0.975564i	$-0.429486\pi$
$997$	13.8753	0.439435	0.219717	+	0.975564i	$0.429486\pi$
$998$	78.5510	2.48649
$999$	−7.20692	−0.228017

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8013.2.a.d.1.4	✓	129

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8013.2.a.d.1.4	✓	129	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:	\( N \)	\(=\)	\( 8013 = 3 \cdot 2671 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69326 q^{2} +1.00000 q^{3} +5.25363 q^{4} -3.14797 q^{5} -2.69326 q^{6} +0.0430508 q^{7} -8.76286 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.69326 q^{2} +1.00000 q^{3} +5.25363 q^{4} -3.14797 q^{5} -2.69326 q^{6} +0.0430508 q^{7} -8.76286 q^{8} +1.00000 q^{9} +8.47829 q^{10} -3.97005 q^{11} +5.25363 q^{12} +0.212754 q^{13} -0.115947 q^{14} -3.14797 q^{15} +13.0934 q^{16} -6.24792 q^{17} -2.69326 q^{18} -1.70064 q^{19} -16.5383 q^{20} +0.0430508 q^{21} +10.6924 q^{22} -1.61766 q^{23} -8.76286 q^{24} +4.90972 q^{25} -0.573002 q^{26} +1.00000 q^{27} +0.226173 q^{28} -8.54219 q^{29} +8.47829 q^{30} -6.57126 q^{31} -17.7381 q^{32} -3.97005 q^{33} +16.8273 q^{34} -0.135523 q^{35} +5.25363 q^{36} -7.20692 q^{37} +4.58026 q^{38} +0.212754 q^{39} +27.5852 q^{40} +6.68846 q^{41} -0.115947 q^{42} -8.46296 q^{43} -20.8572 q^{44} -3.14797 q^{45} +4.35677 q^{46} +1.17838 q^{47} +13.0934 q^{48} -6.99815 q^{49} -13.2231 q^{50} -6.24792 q^{51} +1.11773 q^{52} -8.24058 q^{53} -2.69326 q^{54} +12.4976 q^{55} -0.377248 q^{56} -1.70064 q^{57} +23.0063 q^{58} +6.26247 q^{59} -16.5383 q^{60} -1.14722 q^{61} +17.6981 q^{62} +0.0430508 q^{63} +21.5865 q^{64} -0.669744 q^{65} +10.6924 q^{66} -0.662651 q^{67} -32.8243 q^{68} -1.61766 q^{69} +0.364997 q^{70} -10.3048 q^{71} -8.76286 q^{72} +13.1083 q^{73} +19.4101 q^{74} +4.90972 q^{75} -8.93454 q^{76} -0.170914 q^{77} -0.573002 q^{78} -14.5271 q^{79} -41.2176 q^{80} +1.00000 q^{81} -18.0137 q^{82} +10.4232 q^{83} +0.226173 q^{84} +19.6683 q^{85} +22.7929 q^{86} -8.54219 q^{87} +34.7890 q^{88} -1.48612 q^{89} +8.47829 q^{90} +0.00915924 q^{91} -8.49858 q^{92} -6.57126 q^{93} -3.17369 q^{94} +5.35357 q^{95} -17.7381 q^{96} +0.764689 q^{97} +18.8478 q^{98} -3.97005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 129 q + 15 q^{2} + 129 q^{3} + 151 q^{4} + 16 q^{5} + 15 q^{6} + 61 q^{7} + 42 q^{8} + 129 q^{9}+O(q^{10}) \) 129 * q + 15 * q^2 + 129 * q^3 + 151 * q^4 + 16 * q^5 + 15 * q^6 + 61 * q^7 + 42 * q^8 + 129 * q^9	\( 129 q + 15 q^{2} + 129 q^{3} + 151 q^{4} + 16 q^{5} + 15 q^{6} + 61 q^{7} + 42 q^{8} + 129 q^{9} + 41 q^{10} + 51 q^{11} + 151 q^{12} + 56 q^{13} + 5 q^{14} + 16 q^{15} + 195 q^{16} + 18 q^{17} + 15 q^{18} + 93 q^{19} + 44 q^{20} + 61 q^{21} + 46 q^{22} + 50 q^{23} + 42 q^{24} + 193 q^{25} + q^{26} + 129 q^{27} + 145 q^{28} + 24 q^{29} + 41 q^{30} + 67 q^{31} + 89 q^{32} + 51 q^{33} + 73 q^{34} + 56 q^{35} + 151 q^{36} + 95 q^{37} + 9 q^{38} + 56 q^{39} + 103 q^{40} + 7 q^{41} + 5 q^{42} + 150 q^{43} + 69 q^{44} + 16 q^{45} + 72 q^{46} + 53 q^{47} + 195 q^{48} + 240 q^{49} + 17 q^{50} + 18 q^{51} + 124 q^{52} + 34 q^{53} + 15 q^{54} + 66 q^{55} - 17 q^{56} + 93 q^{57} + 57 q^{58} + 49 q^{59} + 44 q^{60} + 113 q^{61} + 27 q^{62} + 61 q^{63} + 262 q^{64} + 22 q^{65} + 46 q^{66} + 185 q^{67} + 2 q^{68} + 50 q^{69} + 25 q^{70} + 41 q^{71} + 42 q^{72} + 153 q^{73} - q^{74} + 193 q^{75} + 190 q^{76} + 39 q^{77} + q^{78} + 101 q^{79} + 48 q^{80} + 129 q^{81} + 15 q^{82} + 162 q^{83} + 145 q^{84} + 99 q^{85} + 13 q^{86} + 24 q^{87} + 86 q^{88} - 4 q^{89} + 41 q^{90} + 117 q^{91} + 56 q^{92} + 67 q^{93} + 49 q^{94} + 71 q^{95} + 89 q^{96} + 159 q^{97} + 40 q^{98} + 51 q^{99}+O(q^{100}) \) 129 * q + 15 * q^2 + 129 * q^3 + 151 * q^4 + 16 * q^5 + 15 * q^6 + 61 * q^7 + 42 * q^8 + 129 * q^9 + 41 * q^10 + 51 * q^11 + 151 * q^12 + 56 * q^13 + 5 * q^14 + 16 * q^15 + 195 * q^16 + 18 * q^17 + 15 * q^18 + 93 * q^19 + 44 * q^20 + 61 * q^21 + 46 * q^22 + 50 * q^23 + 42 * q^24 + 193 * q^25 + q^26 + 129 * q^27 + 145 * q^28 + 24 * q^29 + 41 * q^30 + 67 * q^31 + 89 * q^32 + 51 * q^33 + 73 * q^34 + 56 * q^35 + 151 * q^36 + 95 * q^37 + 9 * q^38 + 56 * q^39 + 103 * q^40 + 7 * q^41 + 5 * q^42 + 150 * q^43 + 69 * q^44 + 16 * q^45 + 72 * q^46 + 53 * q^47 + 195 * q^48 + 240 * q^49 + 17 * q^50 + 18 * q^51 + 124 * q^52 + 34 * q^53 + 15 * q^54 + 66 * q^55 - 17 * q^56 + 93 * q^57 + 57 * q^58 + 49 * q^59 + 44 * q^60 + 113 * q^61 + 27 * q^62 + 61 * q^63 + 262 * q^64 + 22 * q^65 + 46 * q^66 + 185 * q^67 + 2 * q^68 + 50 * q^69 + 25 * q^70 + 41 * q^71 + 42 * q^72 + 153 * q^73 - q^74 + 193 * q^75 + 190 * q^76 + 39 * q^77 + q^78 + 101 * q^79 + 48 * q^80 + 129 * q^81 + 15 * q^82 + 162 * q^83 + 145 * q^84 + 99 * q^85 + 13 * q^86 + 24 * q^87 + 86 * q^88 - 4 * q^89 + 41 * q^90 + 117 * q^91 + 56 * q^92 + 67 * q^93 + 49 * q^94 + 71 * q^95 + 89 * q^96 + 159 * q^97 + 40 * q^98 + 51 * q^99

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