LMFDB - Embedded newform 8002.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.22228 q^{3} +1.00000 q^{4} -3.24951 q^{5} -3.22228 q^{6} +4.15498 q^{7} +1.00000 q^{8} +7.38311 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.22228 q^{3} +1.00000 q^{4} -3.24951 q^{5} -3.22228 q^{6} +4.15498 q^{7} +1.00000 q^{8} +7.38311 q^{9} -3.24951 q^{10} +4.01228 q^{11} -3.22228 q^{12} -1.31798 q^{13} +4.15498 q^{14} +10.4708 q^{15} +1.00000 q^{16} -5.05365 q^{17} +7.38311 q^{18} -3.54118 q^{19} -3.24951 q^{20} -13.3885 q^{21} +4.01228 q^{22} +0.910439 q^{23} -3.22228 q^{24} +5.55929 q^{25} -1.31798 q^{26} -14.1236 q^{27} +4.15498 q^{28} +1.83498 q^{29} +10.4708 q^{30} +4.10189 q^{31} +1.00000 q^{32} -12.9287 q^{33} -5.05365 q^{34} -13.5016 q^{35} +7.38311 q^{36} +0.633925 q^{37} -3.54118 q^{38} +4.24691 q^{39} -3.24951 q^{40} -1.02441 q^{41} -13.3885 q^{42} -6.99386 q^{43} +4.01228 q^{44} -23.9915 q^{45} +0.910439 q^{46} -9.55466 q^{47} -3.22228 q^{48} +10.2638 q^{49} +5.55929 q^{50} +16.2843 q^{51} -1.31798 q^{52} -8.16514 q^{53} -14.1236 q^{54} -13.0379 q^{55} +4.15498 q^{56} +11.4107 q^{57} +1.83498 q^{58} -10.3152 q^{59} +10.4708 q^{60} +8.82410 q^{61} +4.10189 q^{62} +30.6767 q^{63} +1.00000 q^{64} +4.28279 q^{65} -12.9287 q^{66} -8.71303 q^{67} -5.05365 q^{68} -2.93369 q^{69} -13.5016 q^{70} +11.9175 q^{71} +7.38311 q^{72} -0.173148 q^{73} +0.633925 q^{74} -17.9136 q^{75} -3.54118 q^{76} +16.6709 q^{77} +4.24691 q^{78} +0.271615 q^{79} -3.24951 q^{80} +23.3610 q^{81} -1.02441 q^{82} +0.170942 q^{83} -13.3885 q^{84} +16.4219 q^{85} -6.99386 q^{86} -5.91282 q^{87} +4.01228 q^{88} +16.3950 q^{89} -23.9915 q^{90} -5.47619 q^{91} +0.910439 q^{92} -13.2175 q^{93} -9.55466 q^{94} +11.5071 q^{95} -3.22228 q^{96} -3.23701 q^{97} +10.2638 q^{98} +29.6231 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.22228	−1.86039	−0.930193	−	0.367071i	$-0.880361\pi$
$3$	−3.22228	−1.86039	−0.930193	+	0.367071i	$0.880361\pi$
$4$	1.00000	0.500000
$5$	−3.24951	−1.45322	−0.726612	−	0.687048i	$-0.758906\pi$
$5$	−3.24951	−1.45322	−0.726612	+	0.687048i	$0.758906\pi$
$6$	−3.22228	−1.31549
$7$	4.15498	1.57043	0.785217	−	0.619221i	$-0.212551\pi$
$7$	4.15498	1.57043	0.785217	+	0.619221i	$0.212551\pi$
$8$	1.00000	0.353553
$9$	7.38311	2.46104
$10$	−3.24951	−1.02758
$11$	4.01228	1.20975	0.604873	−	0.796322i	$-0.293224\pi$
$11$	4.01228	1.20975	0.604873	+	0.796322i	$0.293224\pi$
$12$	−3.22228	−0.930193
$13$	−1.31798	−0.365543	−0.182771	−	0.983155i	$-0.558507\pi$
$13$	−1.31798	−0.365543	−0.182771	+	0.983155i	$0.558507\pi$
$14$	4.15498	1.11046
$15$	10.4708	2.70356
$16$	1.00000	0.250000
$17$	−5.05365	−1.22569	−0.612845	−	0.790203i	$-0.709975\pi$
$17$	−5.05365	−1.22569	−0.612845	+	0.790203i	$0.709975\pi$
$18$	7.38311	1.74022
$19$	−3.54118	−0.812402	−0.406201	−	0.913784i	$-0.633147\pi$
$19$	−3.54118	−0.812402	−0.406201	+	0.913784i	$0.633147\pi$
$20$	−3.24951	−0.726612
$21$	−13.3885	−2.92161
$22$	4.01228	0.855420
$23$	0.910439	0.189840	0.0949198	−	0.995485i	$-0.469741\pi$
$23$	0.910439	0.189840	0.0949198	+	0.995485i	$0.469741\pi$
$24$	−3.22228	−0.657746
$25$	5.55929	1.11186
$26$	−1.31798	−0.258478
$27$	−14.1236	−2.71809
$28$	4.15498	0.785217
$29$	1.83498	0.340747	0.170373	−	0.985380i	$-0.445503\pi$
$29$	1.83498	0.340747	0.170373	+	0.985380i	$0.445503\pi$
$30$	10.4708	1.91170
$31$	4.10189	0.736721	0.368361	−	0.929683i	$-0.379919\pi$
$31$	4.10189	0.736721	0.368361	+	0.929683i	$0.379919\pi$
$32$	1.00000	0.176777
$33$	−12.9287	−2.25060
$34$	−5.05365	−0.866694
$35$	−13.5016	−2.28219
$36$	7.38311	1.23052
$37$	0.633925	0.104217	0.0521084	−	0.998641i	$-0.483406\pi$
$37$	0.633925	0.104217	0.0521084	+	0.998641i	$0.483406\pi$
$38$	−3.54118	−0.574455
$39$	4.24691	0.680050
$40$	−3.24951	−0.513792
$41$	−1.02441	−0.159986	−0.0799928	−	0.996795i	$-0.525490\pi$
$41$	−1.02441	−0.159986	−0.0799928	+	0.996795i	$0.525490\pi$
$42$	−13.3885	−2.06589
$43$	−6.99386	−1.06655	−0.533277	−	0.845941i	$-0.679040\pi$
$43$	−6.99386	−1.06655	−0.533277	+	0.845941i	$0.679040\pi$
$44$	4.01228	0.604873
$45$	−23.9915	−3.57644
$46$	0.910439	0.134237
$47$	−9.55466	−1.39369	−0.696845	−	0.717222i	$-0.745413\pi$
$47$	−9.55466	−1.39369	−0.696845	+	0.717222i	$0.745413\pi$
$48$	−3.22228	−0.465097
$49$	10.2638	1.46626
$50$	5.55929	0.786203
$51$	16.2843	2.28026
$52$	−1.31798	−0.182771
$53$	−8.16514	−1.12157	−0.560784	−	0.827962i	$-0.689500\pi$
$53$	−8.16514	−1.12157	−0.560784	+	0.827962i	$0.689500\pi$
$54$	−14.1236	−1.92198
$55$	−13.0379	−1.75803
$56$	4.15498	0.555232
$57$	11.4107	1.51138
$58$	1.83498	0.240944
$59$	−10.3152	−1.34293	−0.671465	−	0.741036i	$-0.734335\pi$
$59$	−10.3152	−1.34293	−0.671465	+	0.741036i	$0.734335\pi$
$60$	10.4708	1.35178
$61$	8.82410	1.12981	0.564905	−	0.825156i	$-0.308913\pi$
$61$	8.82410	1.12981	0.564905	+	0.825156i	$0.308913\pi$
$62$	4.10189	0.520941
$63$	30.6767	3.86490
$64$	1.00000	0.125000
$65$	4.28279	0.531215
$66$	−12.9287	−1.59141
$67$	−8.71303	−1.06447	−0.532233	−	0.846598i	$-0.678647\pi$
$67$	−8.71303	−1.06447	−0.532233	+	0.846598i	$0.678647\pi$
$68$	−5.05365	−0.612845
$69$	−2.93369	−0.353175
$70$	−13.5016	−1.61375
$71$	11.9175	1.41435	0.707173	−	0.707041i	$-0.249970\pi$
$71$	11.9175	1.41435	0.707173	+	0.707041i	$0.249970\pi$
$72$	7.38311	0.870108
$73$	−0.173148	−0.0202655	−0.0101327	−	0.999949i	$-0.503225\pi$
$73$	−0.173148	−0.0202655	−0.0101327	+	0.999949i	$0.503225\pi$
$74$	0.633925	0.0736923
$75$	−17.9136	−2.06849
$76$	−3.54118	−0.406201
$77$	16.6709	1.89983
$78$	4.24691	0.480868
$79$	0.271615	0.0305591	0.0152795	−	0.999883i	$-0.495136\pi$
$79$	0.271615	0.0305591	0.0152795	+	0.999883i	$0.495136\pi$
$80$	−3.24951	−0.363306
$81$	23.3610	2.59567
$82$	−1.02441	−0.113127
$83$	0.170942	0.0187633	0.00938166	−	0.999956i	$-0.497014\pi$
$83$	0.170942	0.0187633	0.00938166	+	0.999956i	$0.497014\pi$
$84$	−13.3885	−1.46081
$85$	16.4219	1.78120
$86$	−6.99386	−0.754167
$87$	−5.91282	−0.633921
$88$	4.01228	0.427710
$89$	16.3950	1.73786	0.868932	−	0.494931i	$-0.164807\pi$
$89$	16.3950	1.73786	0.868932	+	0.494931i	$0.164807\pi$
$90$	−23.9915	−2.52892
$91$	−5.47619	−0.574060
$92$	0.910439	0.0949198
$93$	−13.2175	−1.37059
$94$	−9.55466	−0.985488
$95$	11.5071	1.18060
$96$	−3.22228	−0.328873
$97$	−3.23701	−0.328669	−0.164334	−	0.986405i	$-0.552548\pi$
$97$	−3.23701	−0.328669	−0.164334	+	0.986405i	$0.552548\pi$
$98$	10.2638	1.03680
$99$	29.6231	2.97723
$100$	5.55929	0.555929
$101$	−12.5159	−1.24538	−0.622691	−	0.782468i	$-0.713961\pi$
$101$	−12.5159	−1.24538	−0.622691	+	0.782468i	$0.713961\pi$
$102$	16.2843	1.61239
$103$	4.18616	0.412475	0.206237	−	0.978502i	$-0.433878\pi$
$103$	4.18616	0.412475	0.206237	+	0.978502i	$0.433878\pi$
$104$	−1.31798	−0.129239
$105$	43.5061	4.24576
$106$	−8.16514	−0.793069
$107$	−2.30019	−0.222368	−0.111184	−	0.993800i	$-0.535464\pi$
$107$	−2.30019	−0.222368	−0.111184	+	0.993800i	$0.535464\pi$
$108$	−14.1236	−1.35905
$109$	12.3776	1.18556	0.592780	−	0.805364i	$-0.298030\pi$
$109$	12.3776	1.18556	0.592780	+	0.805364i	$0.298030\pi$
$110$	−13.0379	−1.24312
$111$	−2.04269	−0.193883
$112$	4.15498	0.392608
$113$	−2.00849	−0.188943	−0.0944716	−	0.995528i	$-0.530116\pi$
$113$	−2.00849	−0.188943	−0.0944716	+	0.995528i	$0.530116\pi$
$114$	11.4107	1.06871
$115$	−2.95848	−0.275880
$116$	1.83498	0.170373
$117$	−9.73081	−0.899614
$118$	−10.3152	−0.949595
$119$	−20.9978	−1.92487
$120$	10.4708	0.955852
$121$	5.09836	0.463487
$122$	8.82410	0.798896
$123$	3.30093	0.297635
$124$	4.10189	0.368361
$125$	−1.81743	−0.162556
$126$	30.6767	2.73289
$127$	8.53198	0.757091	0.378545	−	0.925583i	$-0.376424\pi$
$127$	8.53198	0.757091	0.378545	+	0.925583i	$0.376424\pi$
$128$	1.00000	0.0883883
$129$	22.5362	1.98420
$130$	4.28279	0.375626
$131$	−1.51561	−0.132420	−0.0662098	−	0.997806i	$-0.521091\pi$
$131$	−1.51561	−0.132420	−0.0662098	+	0.997806i	$0.521091\pi$
$132$	−12.9287	−1.12530
$133$	−14.7135	−1.27582
$134$	−8.71303	−0.752691
$135$	45.8948	3.95000
$136$	−5.05365	−0.433347
$137$	−2.17483	−0.185808	−0.0929039	−	0.995675i	$-0.529615\pi$
$137$	−2.17483	−0.185808	−0.0929039	+	0.995675i	$0.529615\pi$
$138$	−2.93369	−0.249733
$139$	15.1915	1.28853	0.644263	−	0.764804i	$-0.277164\pi$
$139$	15.1915	1.28853	0.644263	+	0.764804i	$0.277164\pi$
$140$	−13.5016	−1.14110
$141$	30.7878	2.59280
$142$	11.9175	1.00009
$143$	−5.28811	−0.442214
$144$	7.38311	0.615259
$145$	−5.96277	−0.495181
$146$	−0.173148	−0.0143299
$147$	−33.0730	−2.72781
$148$	0.633925	0.0521084
$149$	8.77423	0.718813	0.359407	−	0.933181i	$-0.382979\pi$
$149$	8.77423	0.718813	0.359407	+	0.933181i	$0.382979\pi$
$150$	−17.9136	−1.46264
$151$	−8.85813	−0.720865	−0.360432	−	0.932785i	$-0.617371\pi$
$151$	−8.85813	−0.720865	−0.360432	+	0.932785i	$0.617371\pi$
$152$	−3.54118	−0.287228
$153$	−37.3117	−3.01647
$154$	16.6709	1.34338
$155$	−13.3291	−1.07062
$156$	4.24691	0.340025
$157$	−3.20786	−0.256015	−0.128008	−	0.991773i	$-0.540858\pi$
$157$	−3.20786	−0.256015	−0.128008	+	0.991773i	$0.540858\pi$
$158$	0.271615	0.0216085
$159$	26.3104	2.08655
$160$	−3.24951	−0.256896
$161$	3.78285	0.298131
$162$	23.3610	1.83541
$163$	4.69438	0.367692	0.183846	−	0.982955i	$-0.441145\pi$
$163$	4.69438	0.367692	0.183846	+	0.982955i	$0.441145\pi$
$164$	−1.02441	−0.0799928
$165$	42.0119	3.27062
$166$	0.170942	0.0132677
$167$	−20.6282	−1.59625	−0.798127	−	0.602489i	$-0.794176\pi$
$167$	−20.6282	−1.59625	−0.798127	+	0.602489i	$0.794176\pi$
$168$	−13.3885	−1.03295
$169$	−11.2629	−0.866379
$170$	16.4219	1.25950
$171$	−26.1449	−1.99935
$172$	−6.99386	−0.533277
$173$	−6.66495	−0.506727	−0.253363	−	0.967371i	$-0.581537\pi$
$173$	−6.66495	−0.506727	−0.253363	+	0.967371i	$0.581537\pi$
$174$	−5.91282	−0.448250
$175$	23.0987	1.74610
$176$	4.01228	0.302437
$177$	33.2386	2.49837
$178$	16.3950	1.22886
$179$	0.610096	0.0456007	0.0228004	−	0.999740i	$-0.492742\pi$
$179$	0.610096	0.0456007	0.0228004	+	0.999740i	$0.492742\pi$
$180$	−23.9915	−1.78822
$181$	4.37792	0.325408	0.162704	−	0.986675i	$-0.447978\pi$
$181$	4.37792	0.325408	0.162704	+	0.986675i	$0.447978\pi$
$182$	−5.47619	−0.405922
$183$	−28.4337	−2.10188
$184$	0.910439	0.0671185
$185$	−2.05995	−0.151450
$186$	−13.2175	−0.969151
$187$	−20.2766	−1.48277
$188$	−9.55466	−0.696845
$189$	−58.6833	−4.26858
$190$	11.5071	0.834812
$191$	−4.79662	−0.347071	−0.173536	−	0.984828i	$-0.555519\pi$
$191$	−4.79662	−0.347071	−0.173536	+	0.984828i	$0.555519\pi$
$192$	−3.22228	−0.232548
$193$	−7.60851	−0.547673	−0.273836	−	0.961776i	$-0.588293\pi$
$193$	−7.60851	−0.547673	−0.273836	+	0.961776i	$0.588293\pi$
$194$	−3.23701	−0.232404
$195$	−13.8004	−0.988265
$196$	10.2638	0.733131
$197$	−21.7451	−1.54927	−0.774636	−	0.632407i	$-0.782067\pi$
$197$	−21.7451	−1.54927	−0.774636	+	0.632407i	$0.782067\pi$
$198$	29.6231	2.10522
$199$	0.904665	0.0641300	0.0320650	−	0.999486i	$-0.489792\pi$
$199$	0.904665	0.0641300	0.0320650	+	0.999486i	$0.489792\pi$
$200$	5.55929	0.393102
$201$	28.0759	1.98032
$202$	−12.5159	−0.880618
$203$	7.62429	0.535120
$204$	16.2843	1.14013
$205$	3.32882	0.232495
$206$	4.18616	0.291664
$207$	6.72187	0.467202
$208$	−1.31798	−0.0913856
$209$	−14.2082	−0.982801
$210$	43.5061	3.00220
$211$	−12.4416	−0.856512	−0.428256	−	0.903657i	$-0.640872\pi$
$211$	−12.4416	−0.856512	−0.428256	+	0.903657i	$0.640872\pi$
$212$	−8.16514	−0.560784
$213$	−38.4015	−2.63123
$214$	−2.30019	−0.157238
$215$	22.7266	1.54994
$216$	−14.1236	−0.960991
$217$	17.0433	1.15697
$218$	12.3776	0.838318
$219$	0.557933	0.0377016
$220$	−13.0379	−0.879016
$221$	6.66062	0.448042
$222$	−2.04269	−0.137096
$223$	−5.36978	−0.359587	−0.179794	−	0.983704i	$-0.557543\pi$
$223$	−5.36978	−0.359587	−0.179794	+	0.983704i	$0.557543\pi$
$224$	4.15498	0.277616
$225$	41.0449	2.73633
$226$	−2.00849	−0.133603
$227$	2.64894	0.175816	0.0879080	−	0.996129i	$-0.471982\pi$
$227$	2.64894	0.175816	0.0879080	+	0.996129i	$0.471982\pi$
$228$	11.4107	0.755691
$229$	−27.2277	−1.79926	−0.899630	−	0.436653i	$-0.856164\pi$
$229$	−27.2277	−1.79926	−0.899630	+	0.436653i	$0.856164\pi$
$230$	−2.95848	−0.195076
$231$	−53.7184	−3.53441
$232$	1.83498	0.120472
$233$	7.53866	0.493874	0.246937	−	0.969032i	$-0.420576\pi$
$233$	7.53866	0.493874	0.246937	+	0.969032i	$0.420576\pi$
$234$	−9.73081	−0.636123
$235$	31.0479	2.02534
$236$	−10.3152	−0.671465
$237$	−0.875220	−0.0568517
$238$	−20.9978	−1.36109
$239$	−14.9478	−0.966892	−0.483446	−	0.875374i	$-0.660615\pi$
$239$	−14.9478	−0.966892	−0.483446	+	0.875374i	$0.660615\pi$
$240$	10.4708	0.675889
$241$	23.6181	1.52137	0.760687	−	0.649119i	$-0.224862\pi$
$241$	23.6181	1.52137	0.760687	+	0.649119i	$0.224862\pi$
$242$	5.09836	0.327735
$243$	−32.9049	−2.11085
$244$	8.82410	0.564905
$245$	−33.3524	−2.13081
$246$	3.30093	0.210460
$247$	4.66721	0.296968
$248$	4.10189	0.260470
$249$	−0.550823	−0.0349070
$250$	−1.81743	−0.114945
$251$	−18.4620	−1.16531	−0.582656	−	0.812719i	$-0.697987\pi$
$251$	−18.4620	−1.16531	−0.582656	+	0.812719i	$0.697987\pi$
$252$	30.6767	1.93245
$253$	3.65293	0.229658
$254$	8.53198	0.535344
$255$	−52.9159	−3.31372
$256$	1.00000	0.0625000
$257$	−6.70260	−0.418096	−0.209048	−	0.977905i	$-0.567037\pi$
$257$	−6.70260	−0.418096	−0.209048	+	0.977905i	$0.567037\pi$
$258$	22.5362	1.40304
$259$	2.63395	0.163665
$260$	4.28279	0.265608
$261$	13.5478	0.838591
$262$	−1.51561	−0.0936348
$263$	−21.4031	−1.31977	−0.659887	−	0.751365i	$-0.729396\pi$
$263$	−21.4031	−1.31977	−0.659887	+	0.751365i	$0.729396\pi$
$264$	−12.9287	−0.795706
$265$	26.5327	1.62989
$266$	−14.7135	−0.902144
$267$	−52.8293	−3.23310
$268$	−8.71303	−0.532233
$269$	23.4274	1.42839	0.714196	−	0.699945i	$-0.246792\pi$
$269$	23.4274	1.42839	0.714196	+	0.699945i	$0.246792\pi$
$270$	45.8948	2.79307
$271$	25.6175	1.55615	0.778075	−	0.628171i	$-0.216196\pi$
$271$	25.6175	1.55615	0.778075	+	0.628171i	$0.216196\pi$
$272$	−5.05365	−0.306423
$273$	17.6458	1.06797
$274$	−2.17483	−0.131386
$275$	22.3054	1.34507
$276$	−2.93369	−0.176588
$277$	−16.8074	−1.00986	−0.504928	−	0.863161i	$-0.668481\pi$
$277$	−16.8074	−1.00986	−0.504928	+	0.863161i	$0.668481\pi$
$278$	15.1915	0.911125
$279$	30.2847	1.81310
$280$	−13.5016	−0.806876
$281$	−12.7635	−0.761408	−0.380704	−	0.924697i	$-0.624318\pi$
$281$	−12.7635	−0.761408	−0.380704	+	0.924697i	$0.624318\pi$
$282$	30.7878	1.83339
$283$	26.3964	1.56910	0.784552	−	0.620063i	$-0.212893\pi$
$283$	26.3964	1.56910	0.784552	+	0.620063i	$0.212893\pi$
$284$	11.9175	0.707173
$285$	−37.0791	−2.19638
$286$	−5.28811	−0.312692
$287$	−4.25639	−0.251247
$288$	7.38311	0.435054
$289$	8.53938	0.502316
$290$	−5.96277	−0.350146
$291$	10.4306	0.611451
$292$	−0.173148	−0.0101327
$293$	30.3087	1.77065	0.885327	−	0.464968i	$-0.153934\pi$
$293$	30.3087	1.77065	0.885327	+	0.464968i	$0.153934\pi$
$294$	−33.0730	−1.92886
$295$	33.5195	1.95158
$296$	0.633925	0.0368462
$297$	−56.6679	−3.28820
$298$	8.77423	0.508278
$299$	−1.19994	−0.0693945
$300$	−17.9136	−1.03424
$301$	−29.0593	−1.67495
$302$	−8.85813	−0.509728
$303$	40.3299	2.31689
$304$	−3.54118	−0.203101
$305$	−28.6740	−1.64187
$306$	−37.3117	−2.13297
$307$	−17.4097	−0.993625	−0.496812	−	0.867858i	$-0.665496\pi$
$307$	−17.4097	−0.993625	−0.496812	+	0.867858i	$0.665496\pi$
$308$	16.6709	0.949914
$309$	−13.4890	−0.767362
$310$	−13.3291	−0.757043
$311$	−28.8678	−1.63694	−0.818470	−	0.574549i	$-0.805177\pi$
$311$	−28.8678	−1.63694	−0.818470	+	0.574549i	$0.805177\pi$
$312$	4.24691	0.240434
$313$	10.8881	0.615432	0.307716	−	0.951478i	$-0.400435\pi$
$313$	10.8881	0.615432	0.307716	+	0.951478i	$0.400435\pi$
$314$	−3.20786	−0.181030
$315$	−99.6840	−5.61656
$316$	0.271615	0.0152795
$317$	−34.1716	−1.91927	−0.959635	−	0.281248i	$-0.909252\pi$
$317$	−34.1716	−1.91927	−0.959635	+	0.281248i	$0.909252\pi$
$318$	26.3104	1.47541
$319$	7.36244	0.412217
$320$	−3.24951	−0.181653
$321$	7.41188	0.413691
$322$	3.78285	0.210810
$323$	17.8959	0.995754
$324$	23.3610	1.29783
$325$	−7.32705	−0.406432
$326$	4.69438	0.259998
$327$	−39.8842	−2.20560
$328$	−1.02441	−0.0565635
$329$	−39.6994	−2.18870
$330$	42.0119	2.31268
$331$	−18.2828	−1.00491	−0.502456	−	0.864603i	$-0.667570\pi$
$331$	−18.2828	−1.00491	−0.502456	+	0.864603i	$0.667570\pi$
$332$	0.170942	0.00938166
$333$	4.68034	0.256481
$334$	−20.6282	−1.12872
$335$	28.3131	1.54691
$336$	−13.3885	−0.730403
$337$	−25.1174	−1.36823	−0.684115	−	0.729374i	$-0.739812\pi$
$337$	−25.1174	−1.36823	−0.684115	+	0.729374i	$0.739812\pi$
$338$	−11.2629	−0.612622
$339$	6.47194	0.351507
$340$	16.4219	0.890601
$341$	16.4579	0.891246
$342$	−26.1449	−1.41376
$343$	13.5612	0.732233
$344$	−6.99386	−0.377084
$345$	9.53306	0.513242
$346$	−6.66495	−0.358310
$347$	16.8830	0.906325	0.453162	−	0.891428i	$-0.350296\pi$
$347$	16.8830	0.906325	0.453162	+	0.891428i	$0.350296\pi$
$348$	−5.91282	−0.316960
$349$	−8.99134	−0.481295	−0.240648	−	0.970613i	$-0.577360\pi$
$349$	−8.99134	−0.481295	−0.240648	+	0.970613i	$0.577360\pi$
$350$	23.0987	1.23468
$351$	18.6147	0.993579
$352$	4.01228	0.213855
$353$	30.3918	1.61759	0.808795	−	0.588091i	$-0.200120\pi$
$353$	30.3918	1.61759	0.808795	+	0.588091i	$0.200120\pi$
$354$	33.2386	1.76661
$355$	−38.7260	−2.05536
$356$	16.3950	0.868932
$357$	67.6609	3.58099
$358$	0.610096	0.0322446
$359$	−17.5605	−0.926805	−0.463403	−	0.886148i	$-0.653372\pi$
$359$	−17.5605	−0.926805	−0.463403	+	0.886148i	$0.653372\pi$
$360$	−23.9915	−1.26446
$361$	−6.46004	−0.340002
$362$	4.37792	0.230098
$363$	−16.4284	−0.862265
$364$	−5.47619	−0.287030
$365$	0.562647	0.0294503
$366$	−28.4337	−1.48626
$367$	7.02086	0.366486	0.183243	−	0.983068i	$-0.441340\pi$
$367$	7.02086	0.366486	0.183243	+	0.983068i	$0.441340\pi$
$368$	0.910439	0.0474599
$369$	−7.56332	−0.393731
$370$	−2.05995	−0.107091
$371$	−33.9260	−1.76135
$372$	−13.2175	−0.685293
$373$	11.6776	0.604645	0.302323	−	0.953206i	$-0.402238\pi$
$373$	11.6776	0.604645	0.302323	+	0.953206i	$0.402238\pi$
$374$	−20.2766	−1.04848
$375$	5.85628	0.302417
$376$	−9.55466	−0.492744
$377$	−2.41847	−0.124557
$378$	−58.6833	−3.01834
$379$	−3.27561	−0.168257	−0.0841285	−	0.996455i	$-0.526811\pi$
$379$	−3.27561	−0.168257	−0.0841285	+	0.996455i	$0.526811\pi$
$380$	11.5071	0.590301
$381$	−27.4925	−1.40848
$382$	−4.79662	−0.245417
$383$	−29.6332	−1.51419	−0.757093	−	0.653307i	$-0.773381\pi$
$383$	−29.6332	−1.51419	−0.757093	+	0.653307i	$0.773381\pi$
$384$	−3.22228	−0.164436
$385$	−54.1723	−2.76087
$386$	−7.60851	−0.387263
$387$	−51.6364	−2.62483
$388$	−3.23701	−0.164334
$389$	−24.3437	−1.23427	−0.617136	−	0.786856i	$-0.711707\pi$
$389$	−24.3437	−1.23427	−0.617136	+	0.786856i	$0.711707\pi$
$390$	−13.8004	−0.698809
$391$	−4.60104	−0.232685
$392$	10.2638	0.518402
$393$	4.88373	0.246352
$394$	−21.7451	−1.09550
$395$	−0.882614	−0.0444091
$396$	29.6231	1.48862
$397$	−37.5789	−1.88603	−0.943015	−	0.332750i	$-0.892023\pi$
$397$	−37.5789	−1.88603	−0.943015	+	0.332750i	$0.892023\pi$
$398$	0.904665	0.0453468
$399$	47.4111	2.37353
$400$	5.55929	0.277965
$401$	−9.45682	−0.472251	−0.236126	−	0.971723i	$-0.575878\pi$
$401$	−9.45682	−0.472251	−0.236126	+	0.971723i	$0.575878\pi$
$402$	28.0759	1.40030
$403$	−5.40622	−0.269303
$404$	−12.5159	−0.622691
$405$	−75.9117	−3.77208
$406$	7.62429	0.378387
$407$	2.54348	0.126076
$408$	16.2843	0.806193
$409$	−32.7404	−1.61891	−0.809453	−	0.587184i	$-0.800236\pi$
$409$	−32.7404	−1.61891	−0.809453	+	0.587184i	$0.800236\pi$
$410$	3.32882	0.164399
$411$	7.00790	0.345674
$412$	4.18616	0.206237
$413$	−42.8596	−2.10898
$414$	6.72187	0.330362
$415$	−0.555477	−0.0272673
$416$	−1.31798	−0.0646194
$417$	−48.9513	−2.39715
$418$	−14.2082	−0.694945
$419$	−5.40318	−0.263962	−0.131981	−	0.991252i	$-0.542134\pi$
$419$	−5.40318	−0.263962	−0.131981	+	0.991252i	$0.542134\pi$
$420$	43.5061	2.12288
$421$	−20.5428	−1.00119	−0.500597	−	0.865680i	$-0.666886\pi$
$421$	−20.5428	−1.00119	−0.500597	+	0.865680i	$0.666886\pi$
$422$	−12.4416	−0.605645
$423$	−70.5431	−3.42992
$424$	−8.16514	−0.396534
$425$	−28.0947	−1.36279
$426$	−38.4015	−1.86056
$427$	36.6639	1.77429
$428$	−2.30019	−0.111184
$429$	17.0398	0.822689
$430$	22.7266	1.09597
$431$	21.5927	1.04008	0.520041	−	0.854141i	$-0.325917\pi$
$431$	21.5927	1.04008	0.520041	+	0.854141i	$0.325917\pi$
$432$	−14.1236	−0.679523
$433$	28.4300	1.36626	0.683130	−	0.730297i	$-0.260618\pi$
$433$	28.4300	1.36626	0.683130	+	0.730297i	$0.260618\pi$
$434$	17.0433	0.818103
$435$	19.2137	0.921229
$436$	12.3776	0.592780
$437$	−3.22403	−0.154226
$438$	0.557933	0.0266591
$439$	−8.47822	−0.404644	−0.202322	−	0.979319i	$-0.564849\pi$
$439$	−8.47822	−0.404644	−0.202322	+	0.979319i	$0.564849\pi$
$440$	−13.0379	−0.621558
$441$	75.7790	3.60852
$442$	6.66062	0.316813
$443$	22.8282	1.08460	0.542300	−	0.840185i	$-0.317554\pi$
$443$	22.8282	1.08460	0.542300	+	0.840185i	$0.317554\pi$
$444$	−2.04269	−0.0969417
$445$	−53.2756	−2.52551
$446$	−5.36978	−0.254266
$447$	−28.2731	−1.33727
$448$	4.15498	0.196304
$449$	−20.7064	−0.977194	−0.488597	−	0.872510i	$-0.662491\pi$
$449$	−20.7064	−0.977194	−0.488597	+	0.872510i	$0.662491\pi$
$450$	41.0449	1.93487
$451$	−4.11021	−0.193542
$452$	−2.00849	−0.0944716
$453$	28.5434	1.34109
$454$	2.64894	0.124321
$455$	17.7949	0.834238
$456$	11.4107	0.534354
$457$	31.9285	1.49355	0.746775	−	0.665077i	$-0.231601\pi$
$457$	31.9285	1.49355	0.746775	+	0.665077i	$0.231601\pi$
$458$	−27.2277	−1.27227
$459$	71.3758	3.33154
$460$	−2.95848	−0.137940
$461$	23.3573	1.08786	0.543928	−	0.839132i	$-0.316936\pi$
$461$	23.3573	1.08786	0.543928	+	0.839132i	$0.316936\pi$
$462$	−53.7184	−2.49921
$463$	12.0630	0.560617	0.280308	−	0.959910i	$-0.409563\pi$
$463$	12.0630	0.560617	0.280308	+	0.959910i	$0.409563\pi$
$464$	1.83498	0.0851867
$465$	42.9502	1.99177
$466$	7.53866	0.349222
$467$	7.60503	0.351919	0.175959	−	0.984397i	$-0.443697\pi$
$467$	7.60503	0.351919	0.175959	+	0.984397i	$0.443697\pi$
$468$	−9.73081	−0.449807
$469$	−36.2024	−1.67167
$470$	31.0479	1.43213
$471$	10.3366	0.476287
$472$	−10.3152	−0.474798
$473$	−28.0613	−1.29026
$474$	−0.875220	−0.0402002
$475$	−19.6865	−0.903277
$476$	−20.9978	−0.962433
$477$	−60.2841	−2.76022
$478$	−14.9478	−0.683696
$479$	−24.2770	−1.10925	−0.554623	−	0.832102i	$-0.687137\pi$
$479$	−24.2770	−1.10925	−0.554623	+	0.832102i	$0.687137\pi$
$480$	10.4708	0.477926
$481$	−0.835503	−0.0380956
$482$	23.6181	1.07577
$483$	−12.1894	−0.554638
$484$	5.09836	0.231744
$485$	10.5187	0.477629
$486$	−32.9049	−1.49259
$487$	28.9530	1.31199	0.655994	−	0.754766i	$-0.272250\pi$
$487$	28.9530	1.31199	0.655994	+	0.754766i	$0.272250\pi$
$488$	8.82410	0.399448
$489$	−15.1266	−0.684050
$490$	−33.3524	−1.50671
$491$	25.3134	1.14238	0.571189	−	0.820819i	$-0.306483\pi$
$491$	25.3134	1.14238	0.571189	+	0.820819i	$0.306483\pi$
$492$	3.30093	0.148818
$493$	−9.27334	−0.417650
$494$	4.66721	0.209988
$495$	−96.2604	−4.32658
$496$	4.10189	0.184180
$497$	49.5169	2.22114
$498$	−0.550823	−0.0246830
$499$	−27.7596	−1.24269	−0.621345	−	0.783537i	$-0.713413\pi$
$499$	−27.7596	−1.24269	−0.621345	+	0.783537i	$0.713413\pi$
$500$	−1.81743	−0.0812780
$501$	66.4698	2.96965
$502$	−18.4620	−0.824000
$503$	8.99347	0.400999	0.200499	−	0.979694i	$-0.435744\pi$
$503$	8.99347	0.400999	0.200499	+	0.979694i	$0.435744\pi$
$504$	30.6767	1.36645
$505$	40.6706	1.80982
$506$	3.65293	0.162393
$507$	36.2923	1.61180
$508$	8.53198	0.378545
$509$	−1.97858	−0.0876990	−0.0438495	−	0.999038i	$-0.513962\pi$
$509$	−1.97858	−0.0876990	−0.0438495	+	0.999038i	$0.513962\pi$
$510$	−52.9159	−2.34316
$511$	−0.719427	−0.0318256
$512$	1.00000	0.0441942
$513$	50.0143	2.20819
$514$	−6.70260	−0.295639
$515$	−13.6030	−0.599418
$516$	22.5362	0.992101
$517$	−38.3359	−1.68601
$518$	2.63395	0.115729
$519$	21.4764	0.942708
$520$	4.28279	0.187813
$521$	−12.9049	−0.565376	−0.282688	−	0.959212i	$-0.591226\pi$
$521$	−12.9049	−0.565376	−0.282688	+	0.959212i	$0.591226\pi$
$522$	13.5478	0.592973
$523$	−17.9372	−0.784339	−0.392169	−	0.919893i	$-0.628275\pi$
$523$	−17.9372	−0.784339	−0.392169	+	0.919893i	$0.628275\pi$
$524$	−1.51561	−0.0662098
$525$	−74.4307	−3.24842
$526$	−21.4031	−0.933221
$527$	−20.7295	−0.902992
$528$	−12.9287	−0.562649
$529$	−22.1711	−0.963961
$530$	26.5327	1.15251
$531$	−76.1586	−3.30500
$532$	−14.7135	−0.637912
$533$	1.35015	0.0584816
$534$	−52.8293	−2.28615
$535$	7.47450	0.323151
$536$	−8.71303	−0.376346
$537$	−1.96590	−0.0848350
$538$	23.4274	1.01003
$539$	41.1813	1.77381
$540$	45.8948	1.97500
$541$	−31.4229	−1.35098	−0.675488	−	0.737371i	$-0.736067\pi$
$541$	−31.4229	−1.35098	−0.675488	+	0.737371i	$0.736067\pi$
$542$	25.6175	1.10036
$543$	−14.1069	−0.605385
$544$	−5.05365	−0.216673
$545$	−40.2211	−1.72288
$546$	17.6458	0.755172
$547$	30.5503	1.30623	0.653117	−	0.757257i	$-0.273461\pi$
$547$	30.5503	1.30623	0.653117	+	0.757257i	$0.273461\pi$
$548$	−2.17483	−0.0929039
$549$	65.1493	2.78050
$550$	22.3054	0.951107
$551$	−6.49799	−0.276824
$552$	−2.93369	−0.124866
$553$	1.12855	0.0479910
$554$	−16.8074	−0.714077
$555$	6.63773	0.281756
$556$	15.1915	0.644263
$557$	5.22241	0.221281	0.110640	−	0.993861i	$-0.464710\pi$
$557$	5.22241	0.221281	0.110640	+	0.993861i	$0.464710\pi$
$558$	30.2847	1.28205
$559$	9.21778	0.389871
$560$	−13.5016	−0.570548
$561$	65.3371	2.75853
$562$	−12.7635	−0.538397
$563$	−3.35396	−0.141353	−0.0706764	−	0.997499i	$-0.522516\pi$
$563$	−3.35396	−0.141353	−0.0706764	+	0.997499i	$0.522516\pi$
$564$	30.7878	1.29640
$565$	6.52662	0.274577
$566$	26.3964	1.10952
$567$	97.0644	4.07632
$568$	11.9175	0.500047
$569$	6.37811	0.267384	0.133692	−	0.991023i	$-0.457317\pi$
$569$	6.37811	0.267384	0.133692	+	0.991023i	$0.457317\pi$
$570$	−37.0791	−1.55307
$571$	13.6188	0.569929	0.284964	−	0.958538i	$-0.408018\pi$
$571$	13.6188	0.569929	0.284964	+	0.958538i	$0.408018\pi$
$572$	−5.28811	−0.221107
$573$	15.4561	0.645687
$574$	−4.25639	−0.177658
$575$	5.06140	0.211075
$576$	7.38311	0.307630
$577$	32.1498	1.33841	0.669207	−	0.743076i	$-0.266634\pi$
$577$	32.1498	1.33841	0.669207	+	0.743076i	$0.266634\pi$
$578$	8.53938	0.355191
$579$	24.5168	1.01888
$580$	−5.96277	−0.247591
$581$	0.710260	0.0294665
$582$	10.4306	0.432361
$583$	−32.7608	−1.35681
$584$	−0.173148	−0.00716493
$585$	31.6203	1.30734
$586$	30.3087	1.25204
$587$	−39.9900	−1.65057	−0.825283	−	0.564719i	$-0.808984\pi$
$587$	−39.9900	−1.65057	−0.825283	+	0.564719i	$0.808984\pi$
$588$	−33.0730	−1.36391
$589$	−14.5255	−0.598514
$590$	33.5195	1.37997
$591$	70.0688	2.88224
$592$	0.633925	0.0260542
$593$	16.2840	0.668705	0.334353	−	0.942448i	$-0.391482\pi$
$593$	16.2840	0.668705	0.334353	+	0.942448i	$0.391482\pi$
$594$	−56.6679	−2.32511
$595$	68.2325	2.79726
$596$	8.77423	0.359407
$597$	−2.91509	−0.119307
$598$	−1.19994	−0.0490693
$599$	17.2849	0.706240	0.353120	−	0.935578i	$-0.385121\pi$
$599$	17.2849	0.706240	0.353120	+	0.935578i	$0.385121\pi$
$600$	−17.9136	−0.731321
$601$	12.8816	0.525450	0.262725	−	0.964871i	$-0.415379\pi$
$601$	12.8816	0.525450	0.262725	+	0.964871i	$0.415379\pi$
$602$	−29.0593	−1.18437
$603$	−64.3293	−2.61969
$604$	−8.85813	−0.360432
$605$	−16.5672	−0.673551
$606$	40.3299	1.63829
$607$	9.22519	0.374439	0.187219	−	0.982318i	$-0.440052\pi$
$607$	9.22519	0.374439	0.187219	+	0.982318i	$0.440052\pi$
$608$	−3.54118	−0.143614
$609$	−24.5676	−0.995530
$610$	−28.6740	−1.16097
$611$	12.5929	0.509453
$612$	−37.3117	−1.50823
$613$	−25.4547	−1.02810	−0.514052	−	0.857759i	$-0.671856\pi$
$613$	−25.4547	−1.02810	−0.514052	+	0.857759i	$0.671856\pi$
$614$	−17.4097	−0.702599
$615$	−10.7264	−0.432530
$616$	16.6709	0.671690
$617$	−7.92234	−0.318942	−0.159471	−	0.987203i	$-0.550979\pi$
$617$	−7.92234	−0.318942	−0.159471	+	0.987203i	$0.550979\pi$
$618$	−13.4890	−0.542607
$619$	−21.6602	−0.870597	−0.435298	−	0.900286i	$-0.643357\pi$
$619$	−21.6602	−0.870597	−0.435298	+	0.900286i	$0.643357\pi$
$620$	−13.3291	−0.535310
$621$	−12.8587	−0.516002
$622$	−28.8678	−1.15749
$623$	68.1208	2.72920
$624$	4.24691	0.170013
$625$	−21.8907	−0.875629
$626$	10.8881	0.435176
$627$	45.7828	1.82839
$628$	−3.20786	−0.128008
$629$	−3.20364	−0.127737
$630$	−99.6840	−3.97151
$631$	−28.7491	−1.14448	−0.572242	−	0.820085i	$-0.693926\pi$
$631$	−28.7491	−1.14448	−0.572242	+	0.820085i	$0.693926\pi$
$632$	0.271615	0.0108043
$633$	40.0902	1.59344
$634$	−34.1716	−1.35713
$635$	−27.7247	−1.10022
$636$	26.3104	1.04328
$637$	−13.5276	−0.535981
$638$	7.36244	0.291482
$639$	87.9881	3.48076
$640$	−3.24951	−0.128448
$641$	−24.1732	−0.954785	−0.477392	−	0.878690i	$-0.658418\pi$
$641$	−24.1732	−0.954785	−0.477392	+	0.878690i	$0.658418\pi$
$642$	7.41188	0.292524
$643$	−27.3609	−1.07901	−0.539505	−	0.841983i	$-0.681388\pi$
$643$	−27.3609	−1.07901	−0.539505	+	0.841983i	$0.681388\pi$
$644$	3.78285	0.149065
$645$	−73.2315	−2.88349
$646$	17.8959	0.704104
$647$	−3.86104	−0.151793	−0.0758965	−	0.997116i	$-0.524182\pi$
$647$	−3.86104	−0.151793	−0.0758965	+	0.997116i	$0.524182\pi$
$648$	23.3610	0.917706
$649$	−41.3876	−1.62461
$650$	−7.32705	−0.287391
$651$	−54.9182	−2.15241
$652$	4.69438	0.183846
$653$	−39.2519	−1.53605	−0.768023	−	0.640422i	$-0.778759\pi$
$653$	−39.2519	−1.53605	−0.768023	+	0.640422i	$0.778759\pi$
$654$	−39.8842	−1.55959
$655$	4.92499	0.192435
$656$	−1.02441	−0.0399964
$657$	−1.27837	−0.0498741
$658$	−39.6994	−1.54764
$659$	47.5482	1.85221	0.926107	−	0.377260i	$-0.123134\pi$
$659$	47.5482	1.85221	0.926107	+	0.377260i	$0.123134\pi$
$660$	42.0119	1.63531
$661$	22.1418	0.861218	0.430609	−	0.902539i	$-0.358299\pi$
$661$	22.1418	0.861218	0.430609	+	0.902539i	$0.358299\pi$
$662$	−18.2828	−0.710581
$663$	−21.4624	−0.833531
$664$	0.170942	0.00663383
$665$	47.8117	1.85406
$666$	4.68034	0.181360
$667$	1.67064	0.0646873
$668$	−20.6282	−0.798127
$669$	17.3030	0.668971
$670$	28.3131	1.09383
$671$	35.4047	1.36678
$672$	−13.3885	−0.516473
$673$	−5.53511	−0.213363	−0.106681	−	0.994293i	$-0.534022\pi$
$673$	−5.53511	−0.213363	−0.106681	+	0.994293i	$0.534022\pi$
$674$	−25.1174	−0.967485
$675$	−78.5174	−3.02214
$676$	−11.2629	−0.433189
$677$	0.342279	0.0131549	0.00657743	−	0.999978i	$-0.497906\pi$
$677$	0.342279	0.0131549	0.00657743	+	0.999978i	$0.497906\pi$
$678$	6.47194	0.248553
$679$	−13.4497	−0.516153
$680$	16.4219	0.629750
$681$	−8.53562	−0.327086
$682$	16.4579	0.630206
$683$	−48.9943	−1.87471	−0.937357	−	0.348371i	$-0.886735\pi$
$683$	−48.9943	−1.87471	−0.937357	+	0.348371i	$0.886735\pi$
$684$	−26.1449	−0.999676
$685$	7.06711	0.270020
$686$	13.5612	0.517767
$687$	87.7355	3.34732
$688$	−6.99386	−0.266638
$689$	10.7615	0.409981
$690$	9.53306	0.362917
$691$	48.6212	1.84964	0.924819	−	0.380407i	$-0.124216\pi$
$691$	48.6212	1.84964	0.924819	+	0.380407i	$0.124216\pi$
$692$	−6.66495	−0.253363
$693$	123.083	4.67554
$694$	16.8830	0.640868
$695$	−49.3648	−1.87252
$696$	−5.91282	−0.224125
$697$	5.17700	0.196093
$698$	−8.99134	−0.340327
$699$	−24.2917	−0.918797
$700$	23.0987	0.873050
$701$	−0.929734	−0.0351156	−0.0175578	−	0.999846i	$-0.505589\pi$
$701$	−0.929734	−0.0351156	−0.0175578	+	0.999846i	$0.505589\pi$
$702$	18.6147	0.702566
$703$	−2.24484	−0.0846659
$704$	4.01228	0.151218
$705$	−100.045	−3.76792
$706$	30.3918	1.14381
$707$	−52.0034	−1.95579
$708$	33.2386	1.24918
$709$	−44.4503	−1.66937	−0.834684	−	0.550729i	$-0.814350\pi$
$709$	−44.4503	−1.66937	−0.834684	+	0.550729i	$0.814350\pi$
$710$	−38.7260	−1.45336
$711$	2.00536	0.0752070
$712$	16.3950	0.614428
$713$	3.73452	0.139859
$714$	67.6609	2.53214
$715$	17.1837	0.642636
$716$	0.610096	0.0228004
$717$	48.1660	1.79879
$718$	−17.5605	−0.655350
$719$	−19.0275	−0.709607	−0.354803	−	0.934941i	$-0.615452\pi$
$719$	−19.0275	−0.709607	−0.354803	+	0.934941i	$0.615452\pi$
$720$	−23.9915	−0.894109
$721$	17.3934	0.647764
$722$	−6.46004	−0.240418
$723$	−76.1042	−2.83034
$724$	4.37792	0.162704
$725$	10.2012	0.378862
$726$	−16.4284	−0.609714
$727$	31.3357	1.16218	0.581088	−	0.813841i	$-0.302627\pi$
$727$	31.3357	1.16218	0.581088	+	0.813841i	$0.302627\pi$
$728$	−5.47619	−0.202961
$729$	35.9458	1.33133
$730$	0.562647	0.0208245
$731$	35.3445	1.30726
$732$	−28.4337	−1.05094
$733$	20.6537	0.762863	0.381431	−	0.924397i	$-0.375431\pi$
$733$	20.6537	0.762863	0.381431	+	0.924397i	$0.375431\pi$
$734$	7.02086	0.259145
$735$	107.471	3.96412
$736$	0.910439	0.0335592
$737$	−34.9591	−1.28773
$738$	−7.56332	−0.278410
$739$	35.6158	1.31015	0.655073	−	0.755565i	$-0.272638\pi$
$739$	35.6158	1.31015	0.655073	+	0.755565i	$0.272638\pi$
$740$	−2.05995	−0.0757251
$741$	−15.0391	−0.552475
$742$	−33.9260	−1.24546
$743$	−38.1440	−1.39937	−0.699685	−	0.714452i	$-0.746676\pi$
$743$	−38.1440	−1.39937	−0.699685	+	0.714452i	$0.746676\pi$
$744$	−13.2175	−0.484575
$745$	−28.5119	−1.04460
$746$	11.6776	0.427549
$747$	1.26208	0.0461772
$748$	−20.2766	−0.741387
$749$	−9.55726	−0.349215
$750$	5.85628	0.213841
$751$	17.3697	0.633829	0.316915	−	0.948454i	$-0.397353\pi$
$751$	17.3697	0.633829	0.316915	+	0.948454i	$0.397353\pi$
$752$	−9.55466	−0.348422
$753$	59.4899	2.16793
$754$	−2.41847	−0.0880754
$755$	28.7846	1.04758
$756$	−58.6833	−2.13429
$757$	−29.3394	−1.06636	−0.533179	−	0.846003i	$-0.679003\pi$
$757$	−29.3394	−1.06636	−0.533179	+	0.846003i	$0.679003\pi$
$758$	−3.27561	−0.118976
$759$	−11.7708	−0.427252
$760$	11.5071	0.417406
$761$	51.0998	1.85237	0.926183	−	0.377076i	$-0.123070\pi$
$761$	51.0998	1.85237	0.926183	+	0.377076i	$0.123070\pi$
$762$	−27.4925	−0.995947
$763$	51.4287	1.86184
$764$	−4.79662	−0.173536
$765$	121.244	4.38360
$766$	−29.6332	−1.07069
$767$	13.5953	0.490898
$768$	−3.22228	−0.116274
$769$	7.38549	0.266327	0.133164	−	0.991094i	$-0.457486\pi$
$769$	7.38549	0.266327	0.133164	+	0.991094i	$0.457486\pi$
$770$	−54.1723	−1.95223
$771$	21.5977	0.777821
$772$	−7.60851	−0.273836
$773$	−45.1563	−1.62416	−0.812080	−	0.583546i	$-0.801665\pi$
$773$	−45.1563	−1.62416	−0.812080	+	0.583546i	$0.801665\pi$
$774$	−51.6364	−1.85603
$775$	22.8036	0.819130
$776$	−3.23701	−0.116202
$777$	−8.48732	−0.304481
$778$	−24.3437	−0.872762
$779$	3.62761	0.129973
$780$	−13.8004	−0.494133
$781$	47.8163	1.71100
$782$	−4.60104	−0.164533
$783$	−25.9165	−0.926181
$784$	10.2638	0.366565
$785$	10.4240	0.372047
$786$	4.88373	0.174197
$787$	0.674944	0.0240591	0.0120296	−	0.999928i	$-0.496171\pi$
$787$	0.674944	0.0240591	0.0120296	+	0.999928i	$0.496171\pi$
$788$	−21.7451	−0.774636
$789$	68.9670	2.45529
$790$	−0.882614	−0.0314020
$791$	−8.34525	−0.296723
$792$	29.6231	1.05261
$793$	−11.6300	−0.412994
$794$	−37.5789	−1.33362
$795$	−85.4958	−3.03222
$796$	0.904665	0.0320650
$797$	−18.3650	−0.650520	−0.325260	−	0.945625i	$-0.605452\pi$
$797$	−18.3650	−0.650520	−0.325260	+	0.945625i	$0.605452\pi$
$798$	47.4111	1.67834
$799$	48.2859	1.70823
$800$	5.55929	0.196551
$801$	121.046	4.27695
$802$	−9.45682	−0.333932
$803$	−0.694719	−0.0245161
$804$	28.0759	0.990159
$805$	−12.2924	−0.433251
$806$	−5.40622	−0.190426
$807$	−75.4897	−2.65736
$808$	−12.5159	−0.440309
$809$	−18.1098	−0.636706	−0.318353	−	0.947972i	$-0.603130\pi$
$809$	−18.1098	−0.636706	−0.318353	+	0.947972i	$0.603130\pi$
$810$	−75.9117	−2.66726
$811$	50.7246	1.78118	0.890590	−	0.454807i	$-0.150292\pi$
$811$	50.7246	1.78118	0.890590	+	0.454807i	$0.150292\pi$
$812$	7.62429	0.267560
$813$	−82.5467	−2.89504
$814$	2.54348	0.0891491
$815$	−15.2544	−0.534339
$816$	16.2843	0.570064
$817$	24.7665	0.866471
$818$	−32.7404	−1.14474
$819$	−40.4313	−1.41278
$820$	3.32882	0.116247
$821$	49.2698	1.71953	0.859764	−	0.510691i	$-0.170610\pi$
$821$	49.2698	1.71953	0.859764	+	0.510691i	$0.170610\pi$
$822$	7.00790	0.244429
$823$	0.986461	0.0343859	0.0171929	−	0.999852i	$-0.494527\pi$
$823$	0.986461	0.0343859	0.0171929	+	0.999852i	$0.494527\pi$
$824$	4.18616	0.145832
$825$	−71.8744	−2.50235
$826$	−42.8596	−1.49128
$827$	3.26917	0.113680	0.0568400	−	0.998383i	$-0.481898\pi$
$827$	3.26917	0.113680	0.0568400	+	0.998383i	$0.481898\pi$
$828$	6.72187	0.233601
$829$	12.2143	0.424220	0.212110	−	0.977246i	$-0.431966\pi$
$829$	12.2143	0.424220	0.212110	+	0.977246i	$0.431966\pi$
$830$	−0.555477	−0.0192809
$831$	54.1581	1.87872
$832$	−1.31798	−0.0456928
$833$	−51.8698	−1.79718
$834$	−48.9513	−1.69504
$835$	67.0313	2.31971
$836$	−14.2082	−0.491401
$837$	−57.9336	−2.00248
$838$	−5.40318	−0.186650
$839$	19.9491	0.688720	0.344360	−	0.938838i	$-0.388096\pi$
$839$	19.9491	0.688720	0.344360	+	0.938838i	$0.388096\pi$
$840$	43.5061	1.50110
$841$	−25.6329	−0.883892
$842$	−20.5428	−0.707951
$843$	41.1277	1.41651
$844$	−12.4416	−0.428256
$845$	36.5989	1.25904
$846$	−70.5431	−2.42532
$847$	21.1836	0.727876
$848$	−8.16514	−0.280392
$849$	−85.0567	−2.91914
$850$	−28.0947	−0.963641
$851$	0.577151	0.0197845
$852$	−38.4015	−1.31561
$853$	26.2853	0.899993	0.449997	−	0.893030i	$-0.351425\pi$
$853$	26.2853	0.899993	0.449997	+	0.893030i	$0.351425\pi$
$854$	36.6639	1.25461
$855$	84.9581	2.90551
$856$	−2.30019	−0.0786190
$857$	28.3725	0.969185	0.484592	−	0.874740i	$-0.338968\pi$
$857$	28.3725	0.969185	0.484592	+	0.874740i	$0.338968\pi$
$858$	17.0398	0.581729
$859$	47.5443	1.62219	0.811095	−	0.584915i	$-0.198872\pi$
$859$	47.5443	1.62219	0.811095	+	0.584915i	$0.198872\pi$
$860$	22.7266	0.774971
$861$	13.7153	0.467416
$862$	21.5927	0.735449
$863$	19.8348	0.675185	0.337592	−	0.941292i	$-0.390387\pi$
$863$	19.8348	0.675185	0.337592	+	0.941292i	$0.390387\pi$
$864$	−14.1236	−0.480495
$865$	21.6578	0.736387
$866$	28.4300	0.966091
$867$	−27.5163	−0.934502
$868$	17.0433	0.578486
$869$	1.08979	0.0369687
$870$	19.2137	0.651407
$871$	11.4836	0.389108
$872$	12.3776	0.419159
$873$	−23.8992	−0.808866
$874$	−3.22403	−0.109054
$875$	−7.55139	−0.255284
$876$	0.557933	0.0188508
$877$	−45.7623	−1.54528	−0.772641	−	0.634843i	$-0.781065\pi$
$877$	−45.7623	−1.54528	−0.772641	+	0.634843i	$0.781065\pi$
$878$	−8.47822	−0.286126
$879$	−97.6633	−3.29410
$880$	−13.0379	−0.439508
$881$	−46.7616	−1.57544	−0.787720	−	0.616034i	$-0.788738\pi$
$881$	−46.7616	−1.57544	−0.787720	+	0.616034i	$0.788738\pi$
$882$	75.7790	2.55161
$883$	43.2751	1.45632	0.728162	−	0.685405i	$-0.240375\pi$
$883$	43.2751	1.45632	0.728162	+	0.685405i	$0.240375\pi$
$884$	6.66062	0.224021
$885$	−108.009	−3.63069
$886$	22.8282	0.766928
$887$	−33.7021	−1.13161	−0.565803	−	0.824540i	$-0.691434\pi$
$887$	−33.7021	−1.13161	−0.565803	+	0.824540i	$0.691434\pi$
$888$	−2.04269	−0.0685481
$889$	35.4502	1.18896
$890$	−53.2756	−1.78580
$891$	93.7307	3.14010
$892$	−5.36978	−0.179794
$893$	33.8348	1.13224
$894$	−28.2731	−0.945593
$895$	−1.98251	−0.0662681
$896$	4.15498	0.138808
$897$	3.86656	0.129101
$898$	−20.7064	−0.690980
$899$	7.52688	0.251035
$900$	41.0449	1.36816
$901$	41.2638	1.37470
$902$	−4.11021	−0.136855
$903$	93.6374	3.11606
$904$	−2.00849	−0.0668015
$905$	−14.2261	−0.472891
$906$	28.5434	0.948292
$907$	−6.72653	−0.223351	−0.111675	−	0.993745i	$-0.535622\pi$
$907$	−6.72653	−0.223351	−0.111675	+	0.993745i	$0.535622\pi$
$908$	2.64894	0.0879080
$909$	−92.4065	−3.06493
$910$	17.7949	0.589895
$911$	−49.5477	−1.64159	−0.820794	−	0.571225i	$-0.806468\pi$
$911$	−49.5477	−1.64159	−0.820794	+	0.571225i	$0.806468\pi$
$912$	11.4107	0.377846
$913$	0.685866	0.0226989
$914$	31.9285	1.05610
$915$	92.3956	3.05451
$916$	−27.2277	−0.899630
$917$	−6.29733	−0.207956
$918$	71.3758	2.35575
$919$	−15.4865	−0.510852	−0.255426	−	0.966829i	$-0.582216\pi$
$919$	−15.4865	−0.510852	−0.255426	+	0.966829i	$0.582216\pi$
$920$	−2.95848	−0.0975381
$921$	56.0990	1.84853
$922$	23.3573	0.769231
$923$	−15.7070	−0.517004
$924$	−53.7184	−1.76721
$925$	3.52418	0.115874
$926$	12.0630	0.396416
$927$	30.9069	1.01512
$928$	1.83498	0.0602361
$929$	−20.8896	−0.685367	−0.342683	−	0.939451i	$-0.611336\pi$
$929$	−20.8896	−0.685367	−0.342683	+	0.939451i	$0.611336\pi$
$930$	42.9502	1.40839
$931$	−36.3461	−1.19119
$932$	7.53866	0.246937
$933$	93.0201	3.04534
$934$	7.60503	0.248844
$935$	65.8891	2.15480
$936$	−9.73081	−0.318061
$937$	−57.7379	−1.88622	−0.943108	−	0.332487i	$-0.892112\pi$
$937$	−57.7379	−1.88622	−0.943108	+	0.332487i	$0.892112\pi$
$938$	−36.2024	−1.18205
$939$	−35.0846	−1.14494
$940$	31.0479	1.01267
$941$	−36.7870	−1.19922	−0.599611	−	0.800292i	$-0.704678\pi$
$941$	−36.7870	−1.19922	−0.599611	+	0.800292i	$0.704678\pi$
$942$	10.3366	0.336786
$943$	−0.932661	−0.0303716
$944$	−10.3152	−0.335733
$945$	190.692	6.20321
$946$	−28.0613	−0.912352
$947$	16.8676	0.548123	0.274062	−	0.961712i	$-0.411633\pi$
$947$	16.8676	0.548123	0.274062	+	0.961712i	$0.411633\pi$
$948$	−0.875220	−0.0284258
$949$	0.228206	0.00740789
$950$	−19.6865	−0.638713
$951$	110.111	3.57058
$952$	−20.9978	−0.680543
$953$	53.7987	1.74271	0.871355	−	0.490653i	$-0.163242\pi$
$953$	53.7987	1.74271	0.871355	+	0.490653i	$0.163242\pi$
$954$	−60.2841	−1.95177
$955$	15.5867	0.504372
$956$	−14.9478	−0.483446
$957$	−23.7239	−0.766884
$958$	−24.2770	−0.784355
$959$	−9.03635	−0.291799
$960$	10.4708	0.337945
$961$	−14.1745	−0.457242
$962$	−0.835503	−0.0269377
$963$	−16.9826	−0.547256
$964$	23.6181	0.760687
$965$	24.7239	0.795891
$966$	−12.1894	−0.392188
$967$	−16.8046	−0.540399	−0.270200	−	0.962804i	$-0.587090\pi$
$967$	−16.8046	−0.540399	−0.270200	+	0.962804i	$0.587090\pi$
$968$	5.09836	0.163867
$969$	−57.6656	−1.85249
$970$	10.5187	0.337735
$971$	−45.3507	−1.45537	−0.727687	−	0.685909i	$-0.759405\pi$
$971$	−45.3507	−1.45537	−0.727687	+	0.685909i	$0.759405\pi$
$972$	−32.9049	−1.05542
$973$	63.1203	2.02354
$974$	28.9530	0.927715
$975$	23.6098	0.756120
$976$	8.82410	0.282452
$977$	29.1634	0.933021	0.466510	−	0.884516i	$-0.345511\pi$
$977$	29.1634	0.933021	0.466510	+	0.884516i	$0.345511\pi$
$978$	−15.1266	−0.483696
$979$	65.7812	2.10238
$980$	−33.3524	−1.06540
$981$	91.3853	2.91771
$982$	25.3134	0.807783
$983$	14.1756	0.452131	0.226066	−	0.974112i	$-0.427414\pi$
$983$	14.1756	0.452131	0.226066	+	0.974112i	$0.427414\pi$
$984$	3.30093	0.105230
$985$	70.6608	2.25144
$986$	−9.27334	−0.295323
$987$	127.923	4.07182
$988$	4.66721	0.148484
$989$	−6.36748	−0.202474
$990$	−96.2604	−3.05936
$991$	−45.5764	−1.44778	−0.723891	−	0.689914i	$-0.757648\pi$
$991$	−45.5764	−1.44778	−0.723891	+	0.689914i	$0.757648\pi$
$992$	4.10189	0.130235
$993$	58.9123	1.86953
$994$	49.5169	1.57058
$995$	−2.93971	−0.0931952
$996$	−0.550823	−0.0174535
$997$	15.9140	0.504002	0.252001	−	0.967727i	$-0.418911\pi$
$997$	15.9140	0.504002	0.252001	+	0.967727i	$0.418911\pi$
$998$	−27.7596	−0.878714
$999$	−8.95333	−0.283271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.3	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.3	✓	69	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.22228 q^{3} +1.00000 q^{4} -3.24951 q^{5} -3.22228 q^{6} +4.15498 q^{7} +1.00000 q^{8} +7.38311 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.22228 q^{3} +1.00000 q^{4} -3.24951 q^{5} -3.22228 q^{6} +4.15498 q^{7} +1.00000 q^{8} +7.38311 q^{9} -3.24951 q^{10} +4.01228 q^{11} -3.22228 q^{12} -1.31798 q^{13} +4.15498 q^{14} +10.4708 q^{15} +1.00000 q^{16} -5.05365 q^{17} +7.38311 q^{18} -3.54118 q^{19} -3.24951 q^{20} -13.3885 q^{21} +4.01228 q^{22} +0.910439 q^{23} -3.22228 q^{24} +5.55929 q^{25} -1.31798 q^{26} -14.1236 q^{27} +4.15498 q^{28} +1.83498 q^{29} +10.4708 q^{30} +4.10189 q^{31} +1.00000 q^{32} -12.9287 q^{33} -5.05365 q^{34} -13.5016 q^{35} +7.38311 q^{36} +0.633925 q^{37} -3.54118 q^{38} +4.24691 q^{39} -3.24951 q^{40} -1.02441 q^{41} -13.3885 q^{42} -6.99386 q^{43} +4.01228 q^{44} -23.9915 q^{45} +0.910439 q^{46} -9.55466 q^{47} -3.22228 q^{48} +10.2638 q^{49} +5.55929 q^{50} +16.2843 q^{51} -1.31798 q^{52} -8.16514 q^{53} -14.1236 q^{54} -13.0379 q^{55} +4.15498 q^{56} +11.4107 q^{57} +1.83498 q^{58} -10.3152 q^{59} +10.4708 q^{60} +8.82410 q^{61} +4.10189 q^{62} +30.6767 q^{63} +1.00000 q^{64} +4.28279 q^{65} -12.9287 q^{66} -8.71303 q^{67} -5.05365 q^{68} -2.93369 q^{69} -13.5016 q^{70} +11.9175 q^{71} +7.38311 q^{72} -0.173148 q^{73} +0.633925 q^{74} -17.9136 q^{75} -3.54118 q^{76} +16.6709 q^{77} +4.24691 q^{78} +0.271615 q^{79} -3.24951 q^{80} +23.3610 q^{81} -1.02441 q^{82} +0.170942 q^{83} -13.3885 q^{84} +16.4219 q^{85} -6.99386 q^{86} -5.91282 q^{87} +4.01228 q^{88} +16.3950 q^{89} -23.9915 q^{90} -5.47619 q^{91} +0.910439 q^{92} -13.2175 q^{93} -9.55466 q^{94} +11.5071 q^{95} -3.22228 q^{96} -3.23701 q^{97} +10.2638 q^{98} +29.6231 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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