LMFDB - Embedded newform 8039.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, 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("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1917381849$
Analytic rank:	$0$
Dimension:	$391$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79911 q^{2} -1.99039 q^{3} +5.83499 q^{4} -0.815598 q^{5} +5.57131 q^{6} -3.39799 q^{7} -10.7345 q^{8} +0.961647 q^{9} +O(q^{10})$	\(q-2.79911 q^{2} -1.99039 q^{3} +5.83499 q^{4} -0.815598 q^{5} +5.57131 q^{6} -3.39799 q^{7} -10.7345 q^{8} +0.961647 q^{9} +2.28295 q^{10} +3.98118 q^{11} -11.6139 q^{12} -0.110565 q^{13} +9.51134 q^{14} +1.62336 q^{15} +18.3772 q^{16} +0.842422 q^{17} -2.69175 q^{18} -5.36069 q^{19} -4.75901 q^{20} +6.76333 q^{21} -11.1437 q^{22} -5.15255 q^{23} +21.3659 q^{24} -4.33480 q^{25} +0.309484 q^{26} +4.05712 q^{27} -19.8273 q^{28} -0.805335 q^{29} -4.54395 q^{30} -4.20097 q^{31} -29.9705 q^{32} -7.92410 q^{33} -2.35803 q^{34} +2.77140 q^{35} +5.61120 q^{36} +9.71445 q^{37} +15.0051 q^{38} +0.220068 q^{39} +8.75508 q^{40} -7.05574 q^{41} -18.9313 q^{42} -9.63284 q^{43} +23.2302 q^{44} -0.784317 q^{45} +14.4225 q^{46} +3.06104 q^{47} -36.5777 q^{48} +4.54636 q^{49} +12.1336 q^{50} -1.67675 q^{51} -0.645147 q^{52} -8.74441 q^{53} -11.3563 q^{54} -3.24705 q^{55} +36.4759 q^{56} +10.6698 q^{57} +2.25422 q^{58} +10.9598 q^{59} +9.47228 q^{60} +6.66802 q^{61} +11.7590 q^{62} -3.26767 q^{63} +47.1363 q^{64} +0.0901767 q^{65} +22.1804 q^{66} -3.37484 q^{67} +4.91553 q^{68} +10.2556 q^{69} -7.75743 q^{70} -13.9044 q^{71} -10.3228 q^{72} +9.16446 q^{73} -27.1918 q^{74} +8.62794 q^{75} -31.2796 q^{76} -13.5280 q^{77} -0.615992 q^{78} +13.3181 q^{79} -14.9884 q^{80} -10.9602 q^{81} +19.7497 q^{82} -16.2285 q^{83} +39.4640 q^{84} -0.687078 q^{85} +26.9633 q^{86} +1.60293 q^{87} -42.7362 q^{88} -9.55595 q^{89} +2.19539 q^{90} +0.375700 q^{91} -30.0651 q^{92} +8.36157 q^{93} -8.56818 q^{94} +4.37217 q^{95} +59.6529 q^{96} -3.38277 q^{97} -12.7257 q^{98} +3.82849 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) $ 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * 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 $

$2$	−2.79911	−1.97927	−0.989633	−	0.143617i	$-0.954127\pi$
$2$	−2.79911	−1.97927	−0.989633	+	0.143617i	$0.954127\pi$
$3$	−1.99039	−1.14915	−0.574576	−	0.818451i	$-0.694833\pi$
$3$	−1.99039	−1.14915	−0.574576	+	0.818451i	$0.694833\pi$
$4$	5.83499	2.91750
$5$	−0.815598	−0.364747	−0.182373	−	0.983229i	$-0.558378\pi$
$5$	−0.815598	−0.364747	−0.182373	+	0.983229i	$0.558378\pi$
$6$	5.57131	2.27448
$7$	−3.39799	−1.28432	−0.642160	−	0.766570i	$-0.721962\pi$
$7$	−3.39799	−1.28432	−0.642160	+	0.766570i	$0.721962\pi$
$8$	−10.7345	−3.79524
$9$	0.961647	0.320549
$10$	2.28295	0.721931
$11$	3.98118	1.20037	0.600186	−	0.799861i	$-0.295093\pi$
$11$	3.98118	1.20037	0.600186	+	0.799861i	$0.295093\pi$
$12$	−11.6139	−3.35264
$13$	−0.110565	−0.0306653	−0.0153326	−	0.999882i	$-0.504881\pi$
$13$	−0.110565	−0.0306653	−0.0153326	+	0.999882i	$0.504881\pi$
$14$	9.51134	2.54201
$15$	1.62336	0.419149
$16$	18.3772	4.59429
$17$	0.842422	0.204317	0.102159	−	0.994768i	$-0.467425\pi$
$17$	0.842422	0.204317	0.102159	+	0.994768i	$0.467425\pi$
$18$	−2.69175	−0.634452
$19$	−5.36069	−1.22983	−0.614913	−	0.788595i	$-0.710809\pi$
$19$	−5.36069	−1.22983	−0.614913	+	0.788595i	$0.710809\pi$
$20$	−4.75901	−1.06415
$21$	6.76333	1.47588
$22$	−11.1437	−2.37586
$23$	−5.15255	−1.07438	−0.537191	−	0.843461i	$-0.680514\pi$
$23$	−5.15255	−1.07438	−0.537191	+	0.843461i	$0.680514\pi$
$24$	21.3659	4.36130
$25$	−4.33480	−0.866960
$26$	0.309484	0.0606947
$27$	4.05712	0.780792
$28$	−19.8273	−3.74700
$29$	−0.805335	−0.149547	−0.0747734	−	0.997201i	$-0.523823\pi$
$29$	−0.805335	−0.149547	−0.0747734	+	0.997201i	$0.523823\pi$
$30$	−4.54395	−0.829608
$31$	−4.20097	−0.754517	−0.377258	−	0.926108i	$-0.623133\pi$
$31$	−4.20097	−0.754517	−0.377258	+	0.926108i	$0.623133\pi$
$32$	−29.9705	−5.29809
$33$	−7.92410	−1.37941
$34$	−2.35803	−0.404399
$35$	2.77140	0.468452
$36$	5.61120	0.935200
$37$	9.71445	1.59705	0.798523	−	0.601964i	$-0.205615\pi$
$37$	9.71445	1.59705	0.798523	+	0.601964i	$0.205615\pi$
$38$	15.0051	2.43415
$39$	0.220068	0.0352390
$40$	8.75508	1.38430
$41$	−7.05574	−1.10192	−0.550960	−	0.834531i	$-0.685738\pi$
$41$	−7.05574	−1.10192	−0.550960	+	0.834531i	$0.685738\pi$
$42$	−18.9313	−2.92116
$43$	−9.63284	−1.46899	−0.734497	−	0.678612i	$-0.762582\pi$
$43$	−9.63284	−1.46899	−0.734497	+	0.678612i	$0.762582\pi$
$44$	23.2302	3.50208
$45$	−0.784317	−0.116919
$46$	14.4225	2.12649
$47$	3.06104	0.446499	0.223249	−	0.974761i	$-0.428334\pi$
$47$	3.06104	0.446499	0.223249	+	0.974761i	$0.428334\pi$
$48$	−36.5777	−5.27953
$49$	4.54636	0.649480
$50$	12.1336	1.71594
$51$	−1.67675	−0.234792
$52$	−0.645147	−0.0894658
$53$	−8.74441	−1.20114	−0.600569	−	0.799573i	$-0.705059\pi$
$53$	−8.74441	−1.20114	−0.600569	+	0.799573i	$0.705059\pi$
$54$	−11.3563	−1.54540
$55$	−3.24705	−0.437831
$56$	36.4759	4.87430
$57$	10.6698	1.41326
$58$	2.25422	0.295993
$59$	10.9598	1.42684	0.713421	−	0.700735i	$-0.247145\pi$
$59$	10.9598	1.42684	0.713421	+	0.700735i	$0.247145\pi$
$60$	9.47228	1.22287
$61$	6.66802	0.853752	0.426876	−	0.904310i	$-0.359614\pi$
$61$	6.66802	0.853752	0.426876	+	0.904310i	$0.359614\pi$
$62$	11.7590	1.49339
$63$	−3.26767	−0.411688
$64$	47.1363	5.89203
$65$	0.0901767	0.0111850
$66$	22.1804	2.73022
$67$	−3.37484	−0.412302	−0.206151	−	0.978520i	$-0.566094\pi$
$67$	−3.37484	−0.412302	−0.206151	+	0.978520i	$0.566094\pi$
$68$	4.91553	0.596095
$69$	10.2556	1.23463
$70$	−7.75743	−0.927191
$71$	−13.9044	−1.65015	−0.825073	−	0.565025i	$-0.808866\pi$
$71$	−13.9044	−1.65015	−0.825073	+	0.565025i	$0.808866\pi$
$72$	−10.3228	−1.21656
$73$	9.16446	1.07262	0.536310	−	0.844021i	$-0.319818\pi$
$73$	9.16446	1.07262	0.536310	+	0.844021i	$0.319818\pi$
$74$	−27.1918	−3.16098
$75$	8.62794	0.996268
$76$	−31.2796	−3.58801
$77$	−13.5280	−1.54166
$78$	−0.615992	−0.0697474
$79$	13.3181	1.49841	0.749203	−	0.662340i	$-0.230437\pi$
$79$	13.3181	1.49841	0.749203	+	0.662340i	$0.230437\pi$
$80$	−14.9884	−1.67575
$81$	−10.9602	−1.21780
$82$	19.7497	2.18099
$83$	−16.2285	−1.78131	−0.890655	−	0.454679i	$-0.849754\pi$
$83$	−16.2285	−1.78131	−0.890655	+	0.454679i	$0.849754\pi$
$84$	39.4640	4.30587
$85$	−0.687078	−0.0745241
$86$	26.9633	2.90753
$87$	1.60293	0.171852
$88$	−42.7362	−4.55569
$89$	−9.55595	−1.01293	−0.506465	−	0.862261i	$-0.669048\pi$
$89$	−9.55595	−1.01293	−0.506465	+	0.862261i	$0.669048\pi$
$90$	2.19539	0.231414
$91$	0.375700	0.0393840
$92$	−30.0651	−3.13450
$93$	8.36157	0.867054
$94$	−8.56818	−0.883740
$95$	4.37217	0.448575
$96$	59.6529	6.08830
$97$	−3.38277	−0.343468	−0.171734	−	0.985143i	$-0.554937\pi$
$97$	−3.38277	−0.343468	−0.171734	+	0.985143i	$0.554937\pi$
$98$	−12.7257	−1.28549
$99$	3.82849	0.384778
$100$	−25.2935	−2.52935
$101$	15.3819	1.53056	0.765279	−	0.643698i	$-0.222601\pi$
$101$	15.3819	1.53056	0.765279	+	0.643698i	$0.222601\pi$
$102$	4.69339	0.464715
$103$	−18.7143	−1.84398	−0.921988	−	0.387218i	$-0.873436\pi$
$103$	−18.7143	−1.84398	−0.921988	+	0.387218i	$0.873436\pi$
$104$	1.18687	0.116382
$105$	−5.51616	−0.538322
$106$	24.4765	2.37737
$107$	−12.5213	−1.21048	−0.605241	−	0.796042i	$-0.706923\pi$
$107$	−12.5213	−1.21048	−0.605241	+	0.796042i	$0.706923\pi$
$108$	23.6732	2.27796
$109$	4.33826	0.415530	0.207765	−	0.978179i	$-0.433381\pi$
$109$	4.33826	0.415530	0.207765	+	0.978179i	$0.433381\pi$
$110$	9.08882	0.866585
$111$	−19.3355	−1.83525
$112$	−62.4454	−5.90054
$113$	−1.46912	−0.138203	−0.0691017	−	0.997610i	$-0.522013\pi$
$113$	−1.46912	−0.138203	−0.0691017	+	0.997610i	$0.522013\pi$
$114$	−29.8660	−2.79721
$115$	4.20241	0.391877
$116$	−4.69912	−0.436302
$117$	−0.106325	−0.00982971
$118$	−30.6776	−2.82410
$119$	−2.86254	−0.262409
$120$	−17.4260	−1.59077
$121$	4.84981	0.440892
$122$	−18.6645	−1.68980
$123$	14.0437	1.26627
$124$	−24.5126	−2.20130
$125$	7.61345	0.680967
$126$	9.14655	0.814839
$127$	−11.5152	−1.02181	−0.510903	−	0.859639i	$-0.670689\pi$
$127$	−11.5152	−1.02181	−0.510903	+	0.859639i	$0.670689\pi$
$128$	−71.9984	−6.36382
$129$	19.1731	1.68810
$130$	−0.252414	−0.0221382
$131$	4.53910	0.396583	0.198291	−	0.980143i	$-0.436461\pi$
$131$	4.53910	0.396583	0.198291	+	0.980143i	$0.436461\pi$
$132$	−46.2371	−4.02442
$133$	18.2156	1.57949
$134$	9.44652	0.816055
$135$	−3.30898	−0.284791
$136$	−9.04302	−0.775433
$137$	−17.1376	−1.46416	−0.732082	−	0.681217i	$-0.761451\pi$
$137$	−17.1376	−1.46416	−0.732082	+	0.681217i	$0.761451\pi$
$138$	−28.7065	−2.44366
$139$	−3.34913	−0.284069	−0.142035	−	0.989862i	$-0.545364\pi$
$139$	−3.34913	−0.284069	−0.142035	+	0.989862i	$0.545364\pi$
$140$	16.1711	1.36671
$141$	−6.09266	−0.513095
$142$	38.9198	3.26608
$143$	−0.440180	−0.0368097
$144$	17.6723	1.47269
$145$	0.656830	0.0545467
$146$	−25.6523	−2.12300
$147$	−9.04902	−0.746350
$148$	56.6838	4.65938
$149$	3.23697	0.265183	0.132591	−	0.991171i	$-0.457670\pi$
$149$	3.23697	0.265183	0.132591	+	0.991171i	$0.457670\pi$
$150$	−24.1505	−1.97188
$151$	0.121490	0.00988674	0.00494337	−	0.999988i	$-0.498426\pi$
$151$	0.121490	0.00988674	0.00494337	+	0.999988i	$0.498426\pi$
$152$	57.5446	4.66748
$153$	0.810112	0.0654937
$154$	37.8664	3.05136
$155$	3.42631	0.275207
$156$	1.28409	0.102810
$157$	22.3784	1.78599	0.892994	−	0.450069i	$-0.148601\pi$
$157$	22.3784	1.78599	0.892994	+	0.450069i	$0.148601\pi$
$158$	−37.2788	−2.96575
$159$	17.4048	1.38029
$160$	24.4439	1.93246
$161$	17.5083	1.37985
$162$	30.6787	2.41035
$163$	6.24574	0.489205	0.244602	−	0.969624i	$-0.421343\pi$
$163$	6.24574	0.489205	0.244602	+	0.969624i	$0.421343\pi$
$164$	−41.1702	−3.21485
$165$	6.46288	0.503135
$166$	45.4253	3.52569
$167$	11.4996	0.889863	0.444931	−	0.895565i	$-0.353228\pi$
$167$	11.4996	0.889863	0.444931	+	0.895565i	$0.353228\pi$
$168$	−72.6013	−5.60131
$169$	−12.9878	−0.999060
$170$	1.92320	0.147503
$171$	−5.15509	−0.394219
$172$	−56.2075	−4.28578
$173$	4.24745	0.322928	0.161464	−	0.986879i	$-0.448378\pi$
$173$	4.24745	0.322928	0.161464	+	0.986879i	$0.448378\pi$
$174$	−4.48677	−0.340141
$175$	14.7296	1.11345
$176$	73.1628	5.51485
$177$	−21.8142	−1.63966
$178$	26.7481	2.00486
$179$	−17.5937	−1.31501	−0.657506	−	0.753449i	$-0.728389\pi$
$179$	−17.5937	−1.31501	−0.657506	+	0.753449i	$0.728389\pi$
$180$	−4.57649	−0.341111
$181$	−11.5159	−0.855971	−0.427986	−	0.903785i	$-0.640777\pi$
$181$	−11.5159	−0.855971	−0.427986	+	0.903785i	$0.640777\pi$
$182$	−1.05162	−0.0779515
$183$	−13.2719	−0.981091
$184$	55.3103	4.07753
$185$	−7.92309	−0.582517
$186$	−23.4049	−1.71613
$187$	3.35384	0.245257
$188$	17.8612	1.30266
$189$	−13.7860	−1.00279
$190$	−12.2382	−0.887849
$191$	5.06064	0.366175	0.183087	−	0.983097i	$-0.441391\pi$
$191$	5.06064	0.366175	0.183087	+	0.983097i	$0.441391\pi$
$192$	−93.8195	−6.77084
$193$	14.7791	1.06383	0.531913	−	0.846799i	$-0.321473\pi$
$193$	14.7791	1.06383	0.531913	+	0.846799i	$0.321473\pi$
$194$	9.46873	0.679815
$195$	−0.179487	−0.0128533
$196$	26.5280	1.89485
$197$	−24.4665	−1.74317	−0.871584	−	0.490246i	$-0.836907\pi$
$197$	−24.4665	−1.74317	−0.871584	+	0.490246i	$0.836907\pi$
$198$	−10.7163	−0.761578
$199$	−17.9884	−1.27516	−0.637581	−	0.770383i	$-0.720065\pi$
$199$	−17.9884	−1.27516	−0.637581	+	0.770383i	$0.720065\pi$
$200$	46.5321	3.29032
$201$	6.71724	0.473797
$202$	−43.0556	−3.02938
$203$	2.73652	0.192066
$204$	−9.78381	−0.685004
$205$	5.75465	0.401922
$206$	52.3833	3.64972
$207$	−4.95493	−0.344392
$208$	−2.03187	−0.140885
$209$	−21.3419	−1.47625
$210$	15.4403	1.06548
$211$	−3.85797	−0.265594	−0.132797	−	0.991143i	$-0.542396\pi$
$211$	−3.85797	−0.265594	−0.132797	+	0.991143i	$0.542396\pi$
$212$	−51.0236	−3.50431
$213$	27.6751	1.89627
$214$	35.0485	2.39587
$215$	7.85653	0.535811
$216$	−43.5513	−2.96329
$217$	14.2749	0.969041
$218$	−12.1432	−0.822444
$219$	−18.2408	−1.23260
$220$	−18.9465	−1.27737
$221$	−0.0931425	−0.00626544
$222$	54.1222	3.63244
$223$	−0.888951	−0.0595285	−0.0297643	−	0.999557i	$-0.509476\pi$
$223$	−0.888951	−0.0595285	−0.0297643	+	0.999557i	$0.509476\pi$
$224$	101.840	6.80444
$225$	−4.16855	−0.277903
$226$	4.11223	0.273541
$227$	9.32837	0.619145	0.309573	−	0.950876i	$-0.399814\pi$
$227$	9.32837	0.619145	0.309573	+	0.950876i	$0.399814\pi$
$228$	62.2585	4.12317
$229$	−10.4664	−0.691640	−0.345820	−	0.938301i	$-0.612399\pi$
$229$	−10.4664	−0.691640	−0.345820	+	0.938301i	$0.612399\pi$
$230$	−11.7630	−0.775629
$231$	26.9260	1.77160
$232$	8.64490	0.567566
$233$	−19.7964	−1.29691	−0.648453	−	0.761254i	$-0.724584\pi$
$233$	−19.7964	−1.29691	−0.648453	+	0.761254i	$0.724584\pi$
$234$	0.297614	0.0194556
$235$	−2.49658	−0.162859
$236$	63.9503	4.16281
$237$	−26.5083	−1.72190
$238$	8.01256	0.519377
$239$	10.0173	0.647968	0.323984	−	0.946063i	$-0.394978\pi$
$239$	10.0173	0.647968	0.323984	+	0.946063i	$0.394978\pi$
$240$	29.8327	1.92569
$241$	−8.42904	−0.542962	−0.271481	−	0.962444i	$-0.587513\pi$
$241$	−8.42904	−0.542962	−0.271481	+	0.962444i	$0.587513\pi$
$242$	−13.5751	−0.872642
$243$	9.64366	0.618641
$244$	38.9078	2.49082
$245$	−3.70800	−0.236895
$246$	−39.3097	−2.50629
$247$	0.592705	0.0377129
$248$	45.0955	2.86357
$249$	32.3010	2.04700
$250$	−21.3108	−1.34782
$251$	−14.8383	−0.936588	−0.468294	−	0.883573i	$-0.655131\pi$
$251$	−14.8383	−0.936588	−0.468294	+	0.883573i	$0.655131\pi$
$252$	−19.0668	−1.20110
$253$	−20.5132	−1.28966
$254$	32.2322	2.02243
$255$	1.36755	0.0856395
$256$	107.259	6.70367
$257$	−25.7004	−1.60315	−0.801574	−	0.597896i	$-0.796004\pi$
$257$	−25.7004	−1.60315	−0.801574	+	0.597896i	$0.796004\pi$
$258$	−53.6675	−3.34119
$259$	−33.0096	−2.05112
$260$	0.526181	0.0326323
$261$	−0.774447	−0.0479371
$262$	−12.7054	−0.784943
$263$	8.40297	0.518149	0.259075	−	0.965857i	$-0.416582\pi$
$263$	8.40297	0.518149	0.259075	+	0.965857i	$0.416582\pi$
$264$	85.0616	5.23518
$265$	7.13193	0.438111
$266$	−50.9873	−3.12623
$267$	19.0201	1.16401
$268$	−19.6921	−1.20289
$269$	3.66124	0.223230	0.111615	−	0.993752i	$-0.464398\pi$
$269$	3.66124	0.223230	0.111615	+	0.993752i	$0.464398\pi$
$270$	9.26217	0.563678
$271$	−9.02354	−0.548141	−0.274070	−	0.961710i	$-0.588370\pi$
$271$	−9.02354	−0.548141	−0.274070	+	0.961710i	$0.588370\pi$
$272$	15.4813	0.938693
$273$	−0.747788	−0.0452582
$274$	47.9699	2.89797
$275$	−17.2576	−1.04067
$276$	59.8412	3.60202
$277$	28.2070	1.69479	0.847396	−	0.530961i	$-0.178169\pi$
$277$	28.2070	1.69479	0.847396	+	0.530961i	$0.178169\pi$
$278$	9.37456	0.562249
$279$	−4.03985	−0.241860
$280$	−29.7497	−1.77788
$281$	−22.3310	−1.33215	−0.666077	−	0.745883i	$-0.732028\pi$
$281$	−22.3310	−1.33215	−0.666077	+	0.745883i	$0.732028\pi$
$282$	17.0540	1.01555
$283$	5.95249	0.353839	0.176919	−	0.984225i	$-0.443387\pi$
$283$	5.95249	0.353839	0.176919	+	0.984225i	$0.443387\pi$
$284$	−81.1320	−4.81430
$285$	−8.70231	−0.515480
$286$	1.23211	0.0728562
$287$	23.9753	1.41522
$288$	−28.8210	−1.69830
$289$	−16.2903	−0.958254
$290$	−1.83854	−0.107963
$291$	6.73302	0.394697
$292$	53.4746	3.12936
$293$	−8.43838	−0.492976	−0.246488	−	0.969146i	$-0.579277\pi$
$293$	−8.43838	−0.492976	−0.246488	+	0.969146i	$0.579277\pi$
$294$	25.3292	1.47723
$295$	−8.93879	−0.520436
$296$	−104.280	−6.06117
$297$	16.1521	0.937241
$298$	−9.06062	−0.524867
$299$	0.569693	0.0329462
$300$	50.3439	2.90661
$301$	32.7323	1.88666
$302$	−0.340064	−0.0195685
$303$	−30.6160	−1.75884
$304$	−98.5142	−5.65017
$305$	−5.43842	−0.311403
$306$	−2.26759	−0.129630
$307$	6.41500	0.366123	0.183062	−	0.983101i	$-0.441399\pi$
$307$	6.41500	0.366123	0.183062	+	0.983101i	$0.441399\pi$
$308$	−78.9359	−4.49779
$309$	37.2488	2.11901
$310$	−9.59059	−0.544709
$311$	−0.0948551	−0.00537874	−0.00268937	−	0.999996i	$-0.500856\pi$
$311$	−0.0948551	−0.00537874	−0.00268937	+	0.999996i	$0.500856\pi$
$312$	−2.36233	−0.133740
$313$	−26.5916	−1.50305	−0.751524	−	0.659706i	$-0.770681\pi$
$313$	−26.5916	−1.50305	−0.751524	+	0.659706i	$0.770681\pi$
$314$	−62.6394	−3.53494
$315$	2.66511	0.150162
$316$	77.7112	4.37160
$317$	−31.9726	−1.79576	−0.897880	−	0.440241i	$-0.854893\pi$
$317$	−31.9726	−1.79576	−0.897880	+	0.440241i	$0.854893\pi$
$318$	−48.7178	−2.73196
$319$	−3.20618	−0.179512
$320$	−38.4443	−2.14910
$321$	24.9223	1.39103
$322$	−49.0077	−2.73109
$323$	−4.51596	−0.251275
$324$	−63.9525	−3.55292
$325$	0.479278	0.0265855
$326$	−17.4825	−0.968266
$327$	−8.63482	−0.477506
$328$	75.7401	4.18205
$329$	−10.4014	−0.573448
$330$	−18.0903	−0.995838
$331$	28.9496	1.59122	0.795608	−	0.605812i	$-0.207152\pi$
$331$	28.9496	1.59122	0.795608	+	0.605812i	$0.207152\pi$
$332$	−94.6932	−5.19697
$333$	9.34187	0.511931
$334$	−32.1885	−1.76128
$335$	2.75251	0.150386
$336$	124.291	6.78061
$337$	−16.3202	−0.889019	−0.444510	−	0.895774i	$-0.646622\pi$
$337$	−16.3202	−0.889019	−0.444510	+	0.895774i	$0.646622\pi$
$338$	36.3542	1.97741
$339$	2.92412	0.158817
$340$	−4.00910	−0.217424
$341$	−16.7248	−0.905700
$342$	14.4296	0.780265
$343$	8.33746	0.450181
$344$	103.404	5.57518
$345$	−8.36443	−0.450326
$346$	−11.8891	−0.639160
$347$	−13.2417	−0.710854	−0.355427	−	0.934704i	$-0.615665\pi$
$347$	−13.2417	−0.710854	−0.355427	+	0.934704i	$0.615665\pi$
$348$	9.35308	0.501378
$349$	−5.32025	−0.284786	−0.142393	−	0.989810i	$-0.545480\pi$
$349$	−5.32025	−0.284786	−0.142393	+	0.989810i	$0.545480\pi$
$350$	−41.2298	−2.20382
$351$	−0.448576	−0.0239432
$352$	−119.318	−6.35967
$353$	−13.9939	−0.744818	−0.372409	−	0.928069i	$-0.621468\pi$
$353$	−13.9939	−0.744818	−0.372409	+	0.928069i	$0.621468\pi$
$354$	61.0604	3.24532
$355$	11.3404	0.601886
$356$	−55.7589	−2.95522
$357$	5.69758	0.301548
$358$	49.2465	2.60276
$359$	−1.00226	−0.0528972	−0.0264486	−	0.999650i	$-0.508420\pi$
$359$	−1.00226	−0.0528972	−0.0264486	+	0.999650i	$0.508420\pi$
$360$	8.41929	0.443736
$361$	9.73696	0.512472
$362$	32.2343	1.69420
$363$	−9.65300	−0.506651
$364$	2.19220	0.114903
$365$	−7.47452	−0.391234
$366$	37.1496	1.94184
$367$	15.4336	0.805629	0.402814	−	0.915282i	$-0.368032\pi$
$367$	15.4336	0.805629	0.402814	+	0.915282i	$0.368032\pi$
$368$	−94.6892	−4.93602
$369$	−6.78512	−0.353219
$370$	22.1776	1.15296
$371$	29.7134	1.54265
$372$	48.7897	2.52963
$373$	−7.52425	−0.389591	−0.194796	−	0.980844i	$-0.562404\pi$
$373$	−7.52425	−0.389591	−0.194796	+	0.980844i	$0.562404\pi$
$374$	−9.38774	−0.485428
$375$	−15.1537	−0.782535
$376$	−32.8589	−1.69457
$377$	0.0890419	0.00458589
$378$	38.5886	1.98478
$379$	−7.03105	−0.361161	−0.180580	−	0.983560i	$-0.557798\pi$
$379$	−7.03105	−0.361161	−0.180580	+	0.983560i	$0.557798\pi$
$380$	25.5116	1.30872
$381$	22.9196	1.17421
$382$	−14.1653	−0.724758
$383$	−27.8905	−1.42514	−0.712570	−	0.701601i	$-0.752469\pi$
$383$	−27.8905	−1.42514	−0.712570	+	0.701601i	$0.752469\pi$
$384$	143.305	7.31299
$385$	11.0334	0.562316
$386$	−41.3684	−2.10560
$387$	−9.26339	−0.470884
$388$	−19.7384	−1.00207
$389$	−33.3006	−1.68841	−0.844204	−	0.536022i	$-0.819926\pi$
$389$	−33.3006	−1.68841	−0.844204	+	0.536022i	$0.819926\pi$
$390$	0.502402	0.0254401
$391$	−4.34062	−0.219515
$392$	−48.8031	−2.46493
$393$	−9.03457	−0.455734
$394$	68.4844	3.45020
$395$	−10.8622	−0.546539
$396$	22.3392	1.12259
$397$	24.0538	1.20723	0.603613	−	0.797278i	$-0.293727\pi$
$397$	24.0538	1.20723	0.603613	+	0.797278i	$0.293727\pi$
$398$	50.3514	2.52389
$399$	−36.2561	−1.81507
$400$	−79.6613	−3.98306
$401$	5.99123	0.299188	0.149594	−	0.988748i	$-0.452203\pi$
$401$	5.99123	0.299188	0.149594	+	0.988748i	$0.452203\pi$
$402$	−18.8023	−0.937771
$403$	0.464481	0.0231374
$404$	89.7534	4.46540
$405$	8.93910	0.444188
$406$	−7.65981	−0.380150
$407$	38.6750	1.91705
$408$	17.9991	0.891090
$409$	16.0492	0.793582	0.396791	−	0.917909i	$-0.370124\pi$
$409$	16.0492	0.793582	0.396791	+	0.917909i	$0.370124\pi$
$410$	−16.1079	−0.795511
$411$	34.1105	1.68255
$412$	−109.198	−5.37979
$413$	−37.2413	−1.83252
$414$	13.8694	0.681643
$415$	13.2359	0.649727
$416$	3.31369	0.162467
$417$	6.66607	0.326439
$418$	59.7381	2.92189
$419$	2.55247	0.124696	0.0623481	−	0.998054i	$-0.480141\pi$
$419$	2.55247	0.124696	0.0623481	+	0.998054i	$0.480141\pi$
$420$	−32.1867	−1.57055
$421$	14.0731	0.685882	0.342941	−	0.939357i	$-0.388577\pi$
$421$	14.0731	0.685882	0.342941	+	0.939357i	$0.388577\pi$
$422$	10.7989	0.525681
$423$	2.94364	0.143125
$424$	93.8673	4.55860
$425$	−3.65173	−0.177135
$426$	−77.4656	−3.75322
$427$	−22.6579	−1.09649
$428$	−73.0618	−3.53158
$429$	0.876129	0.0422999
$430$	−21.9912	−1.06051
$431$	−40.1900	−1.93588	−0.967941	−	0.251176i	$-0.919183\pi$
$431$	−40.1900	−1.93588	−0.967941	+	0.251176i	$0.919183\pi$
$432$	74.5582	3.58718
$433$	34.5281	1.65932	0.829658	−	0.558273i	$-0.188536\pi$
$433$	34.5281	1.65932	0.829658	+	0.558273i	$0.188536\pi$
$434$	−39.9569	−1.91799
$435$	−1.30735	−0.0626824
$436$	25.3137	1.21231
$437$	27.6212	1.32130
$438$	51.0580	2.43965
$439$	−33.8064	−1.61349	−0.806746	−	0.590898i	$-0.798774\pi$
$439$	−33.8064	−1.61349	−0.806746	+	0.590898i	$0.798774\pi$
$440$	34.8556	1.66167
$441$	4.37199	0.208190
$442$	0.260716	0.0124010
$443$	0.328488	0.0156069	0.00780347	−	0.999970i	$-0.497516\pi$
$443$	0.328488	0.0156069	0.00780347	+	0.999970i	$0.497516\pi$
$444$	−112.823	−5.35433
$445$	7.79382	0.369463
$446$	2.48827	0.117823
$447$	−6.44283	−0.304735
$448$	−160.169	−7.56726
$449$	31.9255	1.50666	0.753329	−	0.657644i	$-0.228447\pi$
$449$	31.9255	1.50666	0.753329	+	0.657644i	$0.228447\pi$
$450$	11.6682	0.550044
$451$	−28.0902	−1.32271
$452$	−8.57231	−0.403208
$453$	−0.241813	−0.0113614
$454$	−26.1111	−1.22545
$455$	−0.306420	−0.0143652
$456$	−114.536	−5.36364
$457$	4.21781	0.197301	0.0986505	−	0.995122i	$-0.468547\pi$
$457$	4.21781	0.197301	0.0986505	+	0.995122i	$0.468547\pi$
$458$	29.2966	1.36894
$459$	3.41780	0.159529
$460$	24.5210	1.14330
$461$	−7.55466	−0.351856	−0.175928	−	0.984403i	$-0.556293\pi$
$461$	−7.55466	−0.351856	−0.175928	+	0.984403i	$0.556293\pi$
$462$	−75.3688	−3.50647
$463$	−19.4907	−0.905808	−0.452904	−	0.891559i	$-0.649612\pi$
$463$	−19.4907	−0.905808	−0.452904	+	0.891559i	$0.649612\pi$
$464$	−14.7998	−0.687061
$465$	−6.81968	−0.316255
$466$	55.4123	2.56692
$467$	−30.1344	−1.39445	−0.697227	−	0.716850i	$-0.745583\pi$
$467$	−30.1344	−1.39445	−0.697227	+	0.716850i	$0.745583\pi$
$468$	−0.620403	−0.0286782
$469$	11.4677	0.529528
$470$	6.98819	0.322341
$471$	−44.5416	−2.05237
$472$	−117.648	−5.41521
$473$	−38.3501	−1.76334
$474$	74.1994	3.40809
$475$	23.2375	1.06621
$476$	−16.7029	−0.765577
$477$	−8.40903	−0.385023
$478$	−28.0396	−1.28250
$479$	−35.5728	−1.62536	−0.812682	−	0.582708i	$-0.801993\pi$
$479$	−35.5728	−1.62536	−0.812682	+	0.582708i	$0.801993\pi$
$480$	−48.6528	−2.22069
$481$	−1.07408	−0.0489738
$482$	23.5938	1.07467
$483$	−34.8484	−1.58566
$484$	28.2986	1.28630
$485$	2.75898	0.125279
$486$	−26.9936	−1.22446
$487$	−35.6018	−1.61327	−0.806636	−	0.591049i	$-0.798714\pi$
$487$	−35.6018	−1.61327	−0.806636	+	0.591049i	$0.798714\pi$
$488$	−71.5782	−3.24019
$489$	−12.4315	−0.562170
$490$	10.3791	0.468879
$491$	−2.77354	−0.125168	−0.0625841	−	0.998040i	$-0.519934\pi$
$491$	−2.77354	−0.125168	−0.0625841	+	0.998040i	$0.519934\pi$
$492$	81.9446	3.69435
$493$	−0.678432	−0.0305550
$494$	−1.65904	−0.0746439
$495$	−3.12251	−0.140346
$496$	−77.2019	−3.46647
$497$	47.2470	2.11932
$498$	−90.4140	−4.05155
$499$	25.9912	1.16353	0.581763	−	0.813358i	$-0.302363\pi$
$499$	25.9912	1.16353	0.581763	+	0.813358i	$0.302363\pi$
$500$	44.4244	1.98672
$501$	−22.8886	−1.02259
$502$	41.5341	1.85376
$503$	−8.90427	−0.397022	−0.198511	−	0.980099i	$-0.563611\pi$
$503$	−8.90427	−0.397022	−0.198511	+	0.980099i	$0.563611\pi$
$504$	35.0770	1.56245
$505$	−12.5455	−0.558266
$506$	57.4187	2.55257
$507$	25.8507	1.14807
$508$	−67.1909	−2.98111
$509$	29.9678	1.32830	0.664150	−	0.747599i	$-0.268793\pi$
$509$	29.9678	1.32830	0.664150	+	0.747599i	$0.268793\pi$
$510$	−3.82792	−0.169503
$511$	−31.1408	−1.37759
$512$	−156.231	−6.90452
$513$	−21.7489	−0.960238
$514$	71.9382	3.17306
$515$	15.2634	0.672584
$516$	111.875	4.92502
$517$	12.1866	0.535964
$518$	92.3975	4.05971
$519$	−8.45408	−0.371093
$520$	−0.968007	−0.0424499
$521$	−7.87702	−0.345099	−0.172549	−	0.985001i	$-0.555200\pi$
$521$	−7.87702	−0.345099	−0.172549	+	0.985001i	$0.555200\pi$
$522$	2.16776	0.0948803
$523$	−4.88072	−0.213419	−0.106710	−	0.994290i	$-0.534031\pi$
$523$	−4.88072	−0.213419	−0.106710	+	0.994290i	$0.534031\pi$
$524$	26.4856	1.15703
$525$	−29.3177	−1.27953
$526$	−23.5208	−1.02556
$527$	−3.53899	−0.154161
$528$	−145.622	−6.33740
$529$	3.54879	0.154295
$530$	−19.9630	−0.867138
$531$	10.5394	0.457373
$532$	106.288	4.60816
$533$	0.780118	0.0337907
$534$	−53.2392	−2.30388
$535$	10.2124	0.441519
$536$	36.2274	1.56478
$537$	35.0182	1.51115
$538$	−10.2482	−0.441831
$539$	18.0999	0.779617
$540$	−19.3079	−0.830878
$541$	14.2523	0.612753	0.306376	−	0.951910i	$-0.400883\pi$
$541$	14.2523	0.612753	0.306376	+	0.951910i	$0.400883\pi$
$542$	25.2578	1.08492
$543$	22.9212	0.983641
$544$	−25.2478	−1.08249
$545$	−3.53827	−0.151563
$546$	2.09314	0.0895780
$547$	34.4267	1.47198	0.735989	−	0.676993i	$-0.236717\pi$
$547$	34.4267	1.47198	0.735989	+	0.676993i	$0.236717\pi$
$548$	−99.9977	−4.27169
$549$	6.41228	0.273669
$550$	48.3059	2.05977
$551$	4.31715	0.183917
$552$	−110.089	−4.68570
$553$	−45.2549	−1.92443
$554$	−78.9543	−3.35445
$555$	15.7700	0.669401
$556$	−19.5421	−0.828771
$557$	32.1632	1.36280	0.681400	−	0.731911i	$-0.261371\pi$
$557$	32.1632	1.36280	0.681400	+	0.731911i	$0.261371\pi$
$558$	11.3080	0.478704
$559$	1.06506	0.0450471
$560$	50.9304	2.15220
$561$	−6.67544	−0.281837
$562$	62.5068	2.63669
$563$	−6.38184	−0.268963	−0.134481	−	0.990916i	$-0.542937\pi$
$563$	−6.38184	−0.268963	−0.134481	+	0.990916i	$0.542937\pi$
$564$	−35.5506	−1.49695
$565$	1.19821	0.0504092
$566$	−16.6617	−0.700342
$567$	37.2426	1.56404
$568$	149.257	6.26270
$569$	4.84422	0.203080	0.101540	−	0.994831i	$-0.467623\pi$
$569$	4.84422	0.203080	0.101540	+	0.994831i	$0.467623\pi$
$570$	24.3587	1.02027
$571$	15.7071	0.657320	0.328660	−	0.944448i	$-0.393403\pi$
$571$	15.7071	0.657320	0.328660	+	0.944448i	$0.393403\pi$
$572$	−2.56845	−0.107392
$573$	−10.0726	−0.420790
$574$	−67.1095	−2.80110
$575$	22.3353	0.931445
$576$	45.3284	1.88869
$577$	−14.8473	−0.618101	−0.309051	−	0.951046i	$-0.600011\pi$
$577$	−14.8473	−0.618101	−0.309051	+	0.951046i	$0.600011\pi$
$578$	45.5983	1.89664
$579$	−29.4162	−1.22250
$580$	3.83260	0.159140
$581$	55.1444	2.28777
$582$	−18.8464	−0.781210
$583$	−34.8131	−1.44181
$584$	−98.3764	−4.07084
$585$	0.0867182	0.00358536
$586$	23.6199	0.975730
$587$	9.93434	0.410034	0.205017	−	0.978758i	$-0.434275\pi$
$587$	9.93434	0.410034	0.205017	+	0.978758i	$0.434275\pi$
$588$	−52.8009	−2.17747
$589$	22.5201	0.927924
$590$	25.0206	1.03008
$591$	48.6979	2.00316
$592$	178.524	7.33729
$593$	10.5249	0.432204	0.216102	−	0.976371i	$-0.430666\pi$
$593$	10.5249	0.432204	0.216102	+	0.976371i	$0.430666\pi$
$594$	−45.2115	−1.85505
$595$	2.33469	0.0957128
$596$	18.8877	0.773670
$597$	35.8039	1.46535
$598$	−1.59463	−0.0652093
$599$	−30.2808	−1.23724	−0.618620	−	0.785690i	$-0.712308\pi$
$599$	−30.2808	−1.23724	−0.618620	+	0.785690i	$0.712308\pi$
$600$	−92.6170	−3.78107
$601$	−22.0343	−0.898799	−0.449399	−	0.893331i	$-0.648362\pi$
$601$	−22.0343	−0.898799	−0.449399	+	0.893331i	$0.648362\pi$
$602$	−91.6212	−3.73420
$603$	−3.24540	−0.132163
$604$	0.708895	0.0288445
$605$	−3.95549	−0.160814
$606$	85.6975	3.48122
$607$	25.8248	1.04820	0.524098	−	0.851658i	$-0.324403\pi$
$607$	25.8248	1.04820	0.524098	+	0.851658i	$0.324403\pi$
$608$	160.662	6.51572
$609$	−5.44674	−0.220713
$610$	15.2227	0.616350
$611$	−0.338444	−0.0136920
$612$	4.72700	0.191078
$613$	2.39214	0.0966177	0.0483088	−	0.998832i	$-0.484617\pi$
$613$	2.39214	0.0966177	0.0483088	+	0.998832i	$0.484617\pi$
$614$	−17.9563	−0.724656
$615$	−11.4540	−0.461869
$616$	145.217	5.85097
$617$	17.7895	0.716179	0.358089	−	0.933687i	$-0.383428\pi$
$617$	17.7895	0.716179	0.358089	+	0.933687i	$0.383428\pi$
$618$	−104.263	−4.19408
$619$	−36.7992	−1.47909	−0.739543	−	0.673109i	$-0.764958\pi$
$619$	−36.7992	−1.47909	−0.739543	+	0.673109i	$0.764958\pi$
$620$	19.9925	0.802917
$621$	−20.9045	−0.838868
$622$	0.265510	0.0106460
$623$	32.4711	1.30093
$624$	4.04422	0.161898
$625$	15.4645	0.618579
$626$	74.4327	2.97493
$627$	42.4786	1.69643
$628$	130.578	5.21061
$629$	8.18367	0.326304
$630$	−7.45991	−0.297210
$631$	−23.1523	−0.921679	−0.460840	−	0.887483i	$-0.652452\pi$
$631$	−23.1523	−0.921679	−0.460840	+	0.887483i	$0.652452\pi$
$632$	−142.964	−5.68681
$633$	7.67887	0.305207
$634$	89.4946	3.55429
$635$	9.39175	0.372700
$636$	101.557	4.02699
$637$	−0.502669	−0.0199165
$638$	8.97445	0.355302
$639$	−13.3711	−0.528953
$640$	58.7218	2.32118
$641$	−34.7972	−1.37441	−0.687203	−	0.726466i	$-0.741162\pi$
$641$	−34.7972	−1.37441	−0.687203	+	0.726466i	$0.741162\pi$
$642$	−69.7602	−2.75321
$643$	−4.93636	−0.194671	−0.0973355	−	0.995252i	$-0.531032\pi$
$643$	−4.93636	−0.194671	−0.0973355	+	0.995252i	$0.531032\pi$
$644$	102.161	4.02571
$645$	−15.6375	−0.615728
$646$	12.6407	0.497340
$647$	−2.60542	−0.102430	−0.0512148	−	0.998688i	$-0.516309\pi$
$647$	−2.60542	−0.102430	−0.0512148	+	0.998688i	$0.516309\pi$
$648$	117.653	4.62183
$649$	43.6329	1.71274
$650$	−1.34155	−0.0526199
$651$	−28.4125	−1.11358
$652$	36.4439	1.42725
$653$	−31.6862	−1.23998	−0.619989	−	0.784611i	$-0.712863\pi$
$653$	−31.6862	−1.23998	−0.619989	+	0.784611i	$0.712863\pi$
$654$	24.1698	0.945112
$655$	−3.70208	−0.144652
$656$	−129.664	−5.06254
$657$	8.81297	0.343827
$658$	29.1146	1.13501
$659$	33.7387	1.31427	0.657136	−	0.753772i	$-0.271768\pi$
$659$	33.7387	1.31427	0.657136	+	0.753772i	$0.271768\pi$
$660$	37.7109	1.46789
$661$	21.4787	0.835423	0.417712	−	0.908580i	$-0.362832\pi$
$661$	21.4787	0.835423	0.417712	+	0.908580i	$0.362832\pi$
$662$	−81.0331	−3.14944
$663$	0.185390	0.00719994
$664$	174.206	6.76049
$665$	−14.8566	−0.576114
$666$	−26.1489	−1.01325
$667$	4.14953	0.160670
$668$	67.0999	2.59617
$669$	1.76936	0.0684073
$670$	−7.70457	−0.297654
$671$	26.5466	1.02482
$672$	−202.700	−7.81933
$673$	9.81596	0.378378	0.189189	−	0.981941i	$-0.439414\pi$
$673$	9.81596	0.378378	0.189189	+	0.981941i	$0.439414\pi$
$674$	45.6820	1.75961
$675$	−17.5868	−0.676915
$676$	−75.7836	−2.91475
$677$	18.8515	0.724522	0.362261	−	0.932077i	$-0.382005\pi$
$677$	18.8515	0.724522	0.362261	+	0.932077i	$0.382005\pi$
$678$	−8.18493	−0.314340
$679$	11.4946	0.441123
$680$	7.37547	0.282837
$681$	−18.5671	−0.711492
$682$	46.8146	1.79262
$683$	1.99010	0.0761489	0.0380745	−	0.999275i	$-0.487878\pi$
$683$	1.99010	0.0761489	0.0380745	+	0.999275i	$0.487878\pi$
$684$	−30.0799	−1.15013
$685$	13.9774	0.534049
$686$	−23.3374	−0.891028
$687$	20.8322	0.794799
$688$	−177.024	−6.74898
$689$	0.966827	0.0368332
$690$	23.4129	0.891315
$691$	4.73273	0.180041	0.0900207	−	0.995940i	$-0.471307\pi$
$691$	4.73273	0.180041	0.0900207	+	0.995940i	$0.471307\pi$
$692$	24.7839	0.942141
$693$	−13.0092	−0.494178
$694$	37.0651	1.40697
$695$	2.73154	0.103613
$696$	−17.2067	−0.652219
$697$	−5.94391	−0.225142
$698$	14.8919	0.563668
$699$	39.4026	1.49034
$700$	85.9472	3.24850
$701$	14.6558	0.553543	0.276771	−	0.960936i	$-0.410736\pi$
$701$	14.6558	0.553543	0.276771	+	0.960936i	$0.410736\pi$
$702$	1.25561	0.0473900
$703$	−52.0761	−1.96409
$704$	187.658	7.07263
$705$	4.96917	0.187150
$706$	39.1703	1.47419
$707$	−52.2677	−1.96573
$708$	−127.286	−4.78370
$709$	47.0214	1.76592	0.882962	−	0.469444i	$-0.155545\pi$
$709$	47.0214	1.76592	0.882962	+	0.469444i	$0.155545\pi$
$710$	−31.7430	−1.19129
$711$	12.8073	0.480313
$712$	102.579	3.84431
$713$	21.6457	0.810639
$714$	−15.9481	−0.596843
$715$	0.359010	0.0134262
$716$	−102.659	−3.83654
$717$	−19.9384	−0.744614
$718$	2.80543	0.104698
$719$	−47.0798	−1.75578	−0.877891	−	0.478861i	$-0.841050\pi$
$719$	−47.0798	−1.75578	−0.877891	+	0.478861i	$0.841050\pi$
$720$	−14.4135	−0.537160
$721$	63.5911	2.36826
$722$	−27.2548	−1.01432
$723$	16.7771	0.623946
$724$	−67.1953	−2.49729
$725$	3.49096	0.129651
$726$	27.0198	1.00280
$727$	−10.3697	−0.384591	−0.192296	−	0.981337i	$-0.561593\pi$
$727$	−10.3697	−0.384591	−0.192296	+	0.981337i	$0.561593\pi$
$728$	−4.03297	−0.149472
$729$	13.6859	0.506885
$730$	20.9220	0.774357
$731$	−8.11492	−0.300141
$732$	−77.4417	−2.86233
$733$	16.4810	0.608738	0.304369	−	0.952554i	$-0.401554\pi$
$733$	16.4810	0.608738	0.304369	+	0.952554i	$0.401554\pi$
$734$	−43.2003	−1.59455
$735$	7.38036	0.272229
$736$	154.425	5.69216
$737$	−13.4358	−0.494915
$738$	18.9923	0.699116
$739$	24.8937	0.915728	0.457864	−	0.889022i	$-0.348615\pi$
$739$	24.8937	0.915728	0.457864	+	0.889022i	$0.348615\pi$
$740$	−46.2312	−1.69949
$741$	−1.17971	−0.0433379
$742$	−83.1711	−3.05331
$743$	51.9011	1.90407	0.952034	−	0.305993i	$-0.0989885\pi$
$743$	51.9011	1.90407	0.952034	+	0.305993i	$0.0989885\pi$
$744$	−89.7576	−3.29067
$745$	−2.64007	−0.0967245
$746$	21.0612	0.771105
$747$	−15.6061	−0.570997
$748$	19.5696	0.715536
$749$	42.5474	1.55465
$750$	42.4169	1.54884
$751$	16.9022	0.616769	0.308384	−	0.951262i	$-0.400212\pi$
$751$	16.9022	0.616769	0.308384	+	0.951262i	$0.400212\pi$
$752$	56.2532	2.05134
$753$	29.5341	1.07628
$754$	−0.249238	−0.00907670
$755$	−0.0990872	−0.00360615
$756$	−80.4415	−2.92563
$757$	35.9964	1.30831	0.654155	−	0.756360i	$-0.273024\pi$
$757$	35.9964	1.30831	0.654155	+	0.756360i	$0.273024\pi$
$758$	19.6807	0.714833
$759$	40.8293	1.48201
$760$	−46.9332	−1.70245
$761$	5.91408	0.214385	0.107193	−	0.994238i	$-0.465814\pi$
$761$	5.91408	0.214385	0.107193	+	0.994238i	$0.465814\pi$
$762$	−64.1545	−2.32407
$763$	−14.7414	−0.533673
$764$	29.5288	1.06831
$765$	−0.660726	−0.0238886
$766$	78.0685	2.82073
$767$	−1.21177	−0.0437545
$768$	−213.486	−7.70353
$769$	−30.6653	−1.10582	−0.552909	−	0.833242i	$-0.686482\pi$
$769$	−30.6653	−1.10582	−0.552909	+	0.833242i	$0.686482\pi$
$770$	−30.8838	−1.11297
$771$	51.1538	1.84226
$772$	86.2362	3.10371
$773$	−9.66649	−0.347680	−0.173840	−	0.984774i	$-0.555617\pi$
$773$	−9.66649	−0.347680	−0.173840	+	0.984774i	$0.555617\pi$
$774$	25.9292	0.932006
$775$	18.2104	0.654136
$776$	36.3125	1.30354
$777$	65.7020	2.35705
$778$	93.2119	3.34181
$779$	37.8236	1.35517
$780$	−1.04730	−0.0374995
$781$	−55.3559	−1.98079
$782$	12.1499	0.434478
$783$	−3.26734	−0.116765
$784$	83.5491	2.98390
$785$	−18.2517	−0.651433
$786$	25.2887	0.902019
$787$	−37.7582	−1.34593	−0.672967	−	0.739673i	$-0.734980\pi$
$787$	−37.7582	−1.34593	−0.672967	+	0.739673i	$0.734980\pi$
$788$	−142.762	−5.08569
$789$	−16.7252	−0.595432
$790$	30.4046	1.08175
$791$	4.99206	0.177497
$792$	−41.0971	−1.46032
$793$	−0.737250	−0.0261805
$794$	−67.3291	−2.38942
$795$	−14.1953	−0.503456
$796$	−104.962	−3.72028
$797$	−10.1526	−0.359623	−0.179811	−	0.983701i	$-0.557549\pi$
$797$	−10.1526	−0.359623	−0.179811	+	0.983701i	$0.557549\pi$
$798$	101.485	3.59252
$799$	2.57869	0.0912275
$800$	129.916	4.59323
$801$	−9.18945	−0.324693
$802$	−16.7701	−0.592172
$803$	36.4854	1.28754
$804$	39.1950	1.38230
$805$	−14.2798	−0.503296
$806$	−1.30013	−0.0457952
$807$	−7.28728	−0.256525
$808$	−165.118	−5.80883
$809$	48.9695	1.72168	0.860838	−	0.508879i	$-0.169940\pi$
$809$	48.9695	1.72168	0.860838	+	0.508879i	$0.169940\pi$
$810$	−25.0215	−0.879166
$811$	47.3189	1.66159	0.830796	−	0.556576i	$-0.187885\pi$
$811$	47.3189	1.66159	0.830796	+	0.556576i	$0.187885\pi$
$812$	15.9676	0.560352
$813$	17.9603	0.629897
$814$	−108.255	−3.79435
$815$	−5.09402	−0.178436
$816$	−30.8138	−1.07870
$817$	51.6386	1.80661
$818$	−44.9234	−1.57071
$819$	0.361290	0.0126245
$820$	33.5783	1.17261
$821$	28.8423	1.00660	0.503302	−	0.864111i	$-0.332118\pi$
$821$	28.8423	1.00660	0.503302	+	0.864111i	$0.332118\pi$
$822$	−95.4788	−3.33021
$823$	36.8013	1.28281	0.641407	−	0.767201i	$-0.278351\pi$
$823$	36.8013	1.28281	0.641407	+	0.767201i	$0.278351\pi$
$824$	200.890	6.99833
$825$	34.3494	1.19589
$826$	104.242	3.62705
$827$	4.17453	0.145163	0.0725814	−	0.997362i	$-0.476876\pi$
$827$	4.17453	0.145163	0.0725814	+	0.997362i	$0.476876\pi$
$828$	−28.9120	−1.00476
$829$	−37.2895	−1.29512	−0.647559	−	0.762015i	$-0.724210\pi$
$829$	−37.2895	−1.29512	−0.647559	+	0.762015i	$0.724210\pi$
$830$	−37.0488	−1.28598
$831$	−56.1428	−1.94757
$832$	−5.21163	−0.180681
$833$	3.82995	0.132700
$834$	−18.6590	−0.646109
$835$	−9.37902	−0.324574
$836$	−124.530	−4.30695
$837$	−17.0438	−0.589121
$838$	−7.14462	−0.246807
$839$	−2.63861	−0.0910950	−0.0455475	−	0.998962i	$-0.514503\pi$
$839$	−2.63861	−0.0910950	−0.0455475	+	0.998962i	$0.514503\pi$
$840$	59.2135	2.04306
$841$	−28.3514	−0.977636
$842$	−39.3922	−1.35754
$843$	44.4473	1.53085
$844$	−22.5112	−0.774869
$845$	10.5928	0.364404
$846$	−8.23956	−0.283282
$847$	−16.4796	−0.566246
$848$	−160.697	−5.51837
$849$	−11.8478	−0.406615
$850$	10.2216	0.350597
$851$	−50.0542	−1.71584
$852$	161.484	5.53236
$853$	42.5797	1.45790	0.728951	−	0.684566i	$-0.240008\pi$
$853$	42.5797	1.45790	0.728951	+	0.684566i	$0.240008\pi$
$854$	63.4218	2.17025
$855$	4.20448	0.143790
$856$	134.411	4.59407
$857$	10.5662	0.360936	0.180468	−	0.983581i	$-0.442239\pi$
$857$	10.5662	0.360936	0.180468	+	0.983581i	$0.442239\pi$
$858$	−2.45238	−0.0837228
$859$	−26.1261	−0.891412	−0.445706	−	0.895180i	$-0.647047\pi$
$859$	−26.1261	−0.891412	−0.445706	+	0.895180i	$0.647047\pi$
$860$	45.8428	1.56323
$861$	−47.7202	−1.62630
$862$	112.496	3.83163
$863$	37.3748	1.27225	0.636126	−	0.771585i	$-0.280536\pi$
$863$	37.3748	1.27225	0.636126	+	0.771585i	$0.280536\pi$
$864$	−121.594	−4.13670
$865$	−3.46422	−0.117787
$866$	−96.6478	−3.28423
$867$	32.4241	1.10118
$868$	83.2938	2.82717
$869$	53.0219	1.79864
$870$	3.65940	0.124065
$871$	0.373139	0.0126433
$872$	−46.5692	−1.57703
$873$	−3.25303	−0.110098
$874$	−77.3147	−2.61521
$875$	−25.8704	−0.874580
$876$	−106.435	−3.59611
$877$	3.87918	0.130990	0.0654952	−	0.997853i	$-0.479137\pi$
$877$	3.87918	0.130990	0.0654952	+	0.997853i	$0.479137\pi$
$878$	94.6277	3.19353
$879$	16.7957	0.566504
$880$	−59.6714	−2.01152
$881$	−14.4302	−0.486165	−0.243082	−	0.970006i	$-0.578159\pi$
$881$	−14.4302	−0.486165	−0.243082	+	0.970006i	$0.578159\pi$
$882$	−12.2377	−0.412063
$883$	−39.7137	−1.33647	−0.668236	−	0.743949i	$-0.732950\pi$
$883$	−39.7137	−1.33647	−0.668236	+	0.743949i	$0.732950\pi$
$884$	−0.543486	−0.0182794
$885$	17.7917	0.598060
$886$	−0.919472	−0.0308903
$887$	−54.1425	−1.81793	−0.908964	−	0.416875i	$-0.863125\pi$
$887$	−54.1425	−1.81793	−0.908964	+	0.416875i	$0.863125\pi$
$888$	207.558	6.96520
$889$	39.1284	1.31233
$890$	−21.8157	−0.731265
$891$	−43.6345	−1.46181
$892$	−5.18702	−0.173674
$893$	−16.4093	−0.549116
$894$	18.0341	0.603152
$895$	14.3494	0.479646
$896$	244.650	8.17319
$897$	−1.13391	−0.0378601
$898$	−89.3629	−2.98208
$899$	3.38319	0.112836
$900$	−24.3234	−0.810781
$901$	−7.36649	−0.245413
$902$	78.6273	2.61800
$903$	−65.1500	−2.16806
$904$	15.7704	0.524514
$905$	9.39236	0.312213
$906$	0.676860	0.0224872
$907$	45.2784	1.50344	0.751722	−	0.659480i	$-0.229224\pi$
$907$	45.2784	1.50344	0.751722	+	0.659480i	$0.229224\pi$
$908$	54.4310	1.80635
$909$	14.7920	0.490619
$910$	0.857702	0.0284325
$911$	13.2597	0.439314	0.219657	−	0.975577i	$-0.429506\pi$
$911$	13.2597	0.439314	0.219657	+	0.975577i	$0.429506\pi$
$912$	196.081	6.49291
$913$	−64.6087	−2.13823
$914$	−11.8061	−0.390511
$915$	10.8246	0.357850
$916$	−61.0714	−2.01786
$917$	−15.4238	−0.509340
$918$	−9.56679	−0.315751
$919$	−21.9968	−0.725607	−0.362804	−	0.931866i	$-0.618180\pi$
$919$	−21.9968	−0.725607	−0.362804	+	0.931866i	$0.618180\pi$
$920$	−45.1110	−1.48727
$921$	−12.7683	−0.420731
$922$	21.1463	0.696416
$923$	1.53734	0.0506022
$924$	157.113	5.16864
$925$	−42.1102	−1.38457
$926$	54.5564	1.79284
$927$	−17.9966	−0.591085
$928$	24.1363	0.792312
$929$	−7.13080	−0.233954	−0.116977	−	0.993135i	$-0.537320\pi$
$929$	−7.13080	−0.233954	−0.116977	+	0.993135i	$0.537320\pi$
$930$	19.0890	0.625953
$931$	−24.3716	−0.798747
$932$	−115.512	−3.78372
$933$	0.188799	0.00618099
$934$	84.3494	2.76000
$935$	−2.73538	−0.0894566
$936$	1.14135	0.0373061
$937$	46.3122	1.51295	0.756477	−	0.654020i	$-0.226919\pi$
$937$	46.3122	1.51295	0.756477	+	0.654020i	$0.226919\pi$
$938$	−32.0992	−1.04808
$939$	52.9277	1.72723
$940$	−14.5675	−0.475140
$941$	28.0375	0.913995	0.456997	−	0.889468i	$-0.348925\pi$
$941$	28.0375	0.913995	0.456997	+	0.889468i	$0.348925\pi$
$942$	124.677	4.06219
$943$	36.3550	1.18388
$944$	201.410	6.55533
$945$	11.2439	0.365763
$946$	107.346	3.49012
$947$	41.4013	1.34536	0.672681	−	0.739932i	$-0.265143\pi$
$947$	41.4013	1.34536	0.672681	+	0.739932i	$0.265143\pi$
$948$	−154.675	−5.02362
$949$	−1.01327	−0.0328921
$950$	−65.0442	−2.11031
$951$	63.6379	2.06360
$952$	30.7281	0.995904
$953$	38.5840	1.24986	0.624929	−	0.780681i	$-0.285128\pi$
$953$	38.5840	1.24986	0.624929	+	0.780681i	$0.285128\pi$
$954$	23.5378	0.762064
$955$	−4.12745	−0.133561
$956$	58.4511	1.89044
$957$	6.38155	0.206286
$958$	99.5721	3.21703
$959$	58.2334	1.88045
$960$	76.5190	2.46964
$961$	−13.3518	−0.430704
$962$	3.00646	0.0969323
$963$	−12.0411	−0.388019
$964$	−49.1834	−1.58409
$965$	−12.0538	−0.388027
$966$	97.5443	3.13844
$967$	−25.2387	−0.811621	−0.405810	−	0.913957i	$-0.633011\pi$
$967$	−25.2387	−0.811621	−0.405810	+	0.913957i	$0.633011\pi$
$968$	−52.0605	−1.67329
$969$	8.98852	0.288753
$970$	−7.72268	−0.247960
$971$	−17.4192	−0.559009	−0.279504	−	0.960144i	$-0.590170\pi$
$971$	−17.4192	−0.559009	−0.279504	+	0.960144i	$0.590170\pi$
$972$	56.2707	1.80488
$973$	11.3803	0.364836
$974$	99.6532	3.19310
$975$	−0.953949	−0.0305508
$976$	122.539	3.92238
$977$	34.4743	1.10293	0.551466	−	0.834198i	$-0.314069\pi$
$977$	34.4743	1.10293	0.551466	+	0.834198i	$0.314069\pi$
$978$	34.7970	1.11268
$979$	−38.0440	−1.21589
$980$	−21.6362	−0.691142
$981$	4.17187	0.133198
$982$	7.76344	0.247741
$983$	−35.3480	−1.12742	−0.563712	−	0.825971i	$-0.690627\pi$
$983$	−35.3480	−1.12742	−0.563712	+	0.825971i	$0.690627\pi$
$984$	−150.752	−4.80581
$985$	19.9549	0.635815
$986$	1.89900	0.0604765
$987$	20.7028	0.658978
$988$	3.45843	0.110027
$989$	49.6337	1.57826
$990$	8.74024	0.277783
$991$	−42.2796	−1.34306	−0.671528	−	0.740979i	$-0.734362\pi$
$991$	−42.2796	−1.34306	−0.671528	+	0.740979i	$0.734362\pi$
$992$	125.905	3.99749
$993$	−57.6210	−1.82855
$994$	−132.249	−4.19469
$995$	14.6713	0.465111
$996$	188.476	5.97210
$997$	42.7368	1.35349	0.676744	−	0.736219i	$-0.263391\pi$
$997$	42.7368	1.35349	0.676744	+	0.736219i	$0.263391\pi$
$998$	−72.7521	−2.30293
$999$	39.4127	1.24696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.b.1.3	✓	391

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.b.1.3	✓	391	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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79911 q^{2} -1.99039 q^{3} +5.83499 q^{4} -0.815598 q^{5} +5.57131 q^{6} -3.39799 q^{7} -10.7345 q^{8} +0.961647 q^{9} +O(q^{10})\)	\(q-2.79911 q^{2} -1.99039 q^{3} +5.83499 q^{4} -0.815598 q^{5} +5.57131 q^{6} -3.39799 q^{7} -10.7345 q^{8} +0.961647 q^{9} +2.28295 q^{10} +3.98118 q^{11} -11.6139 q^{12} -0.110565 q^{13} +9.51134 q^{14} +1.62336 q^{15} +18.3772 q^{16} +0.842422 q^{17} -2.69175 q^{18} -5.36069 q^{19} -4.75901 q^{20} +6.76333 q^{21} -11.1437 q^{22} -5.15255 q^{23} +21.3659 q^{24} -4.33480 q^{25} +0.309484 q^{26} +4.05712 q^{27} -19.8273 q^{28} -0.805335 q^{29} -4.54395 q^{30} -4.20097 q^{31} -29.9705 q^{32} -7.92410 q^{33} -2.35803 q^{34} +2.77140 q^{35} +5.61120 q^{36} +9.71445 q^{37} +15.0051 q^{38} +0.220068 q^{39} +8.75508 q^{40} -7.05574 q^{41} -18.9313 q^{42} -9.63284 q^{43} +23.2302 q^{44} -0.784317 q^{45} +14.4225 q^{46} +3.06104 q^{47} -36.5777 q^{48} +4.54636 q^{49} +12.1336 q^{50} -1.67675 q^{51} -0.645147 q^{52} -8.74441 q^{53} -11.3563 q^{54} -3.24705 q^{55} +36.4759 q^{56} +10.6698 q^{57} +2.25422 q^{58} +10.9598 q^{59} +9.47228 q^{60} +6.66802 q^{61} +11.7590 q^{62} -3.26767 q^{63} +47.1363 q^{64} +0.0901767 q^{65} +22.1804 q^{66} -3.37484 q^{67} +4.91553 q^{68} +10.2556 q^{69} -7.75743 q^{70} -13.9044 q^{71} -10.3228 q^{72} +9.16446 q^{73} -27.1918 q^{74} +8.62794 q^{75} -31.2796 q^{76} -13.5280 q^{77} -0.615992 q^{78} +13.3181 q^{79} -14.9884 q^{80} -10.9602 q^{81} +19.7497 q^{82} -16.2285 q^{83} +39.4640 q^{84} -0.687078 q^{85} +26.9633 q^{86} +1.60293 q^{87} -42.7362 q^{88} -9.55595 q^{89} +2.19539 q^{90} +0.375700 q^{91} -30.0651 q^{92} +8.36157 q^{93} -8.56818 q^{94} +4.37217 q^{95} +59.6529 q^{96} -3.38277 q^{97} -12.7257 q^{98} +3.82849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * q^99

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