LMFDB - Embedded newform 8039.2.a.a.1.10

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:	$1$
Dimension:	$279$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-2.64457 q^{2} -1.20149 q^{3} +4.99374 q^{4} -2.42138 q^{5} +3.17741 q^{6} +3.07885 q^{7} -7.91715 q^{8} -1.55643 q^{9} +O(q^{10})$	\(q-2.64457 q^{2} -1.20149 q^{3} +4.99374 q^{4} -2.42138 q^{5} +3.17741 q^{6} +3.07885 q^{7} -7.91715 q^{8} -1.55643 q^{9} +6.40351 q^{10} +1.20020 q^{11} -5.99992 q^{12} -2.23407 q^{13} -8.14224 q^{14} +2.90926 q^{15} +10.9500 q^{16} +2.35025 q^{17} +4.11608 q^{18} -4.78931 q^{19} -12.0918 q^{20} -3.69920 q^{21} -3.17401 q^{22} +4.75167 q^{23} +9.51236 q^{24} +0.863093 q^{25} +5.90815 q^{26} +5.47449 q^{27} +15.3750 q^{28} +6.94764 q^{29} -7.69374 q^{30} -7.95141 q^{31} -13.1236 q^{32} -1.44203 q^{33} -6.21540 q^{34} -7.45508 q^{35} -7.77240 q^{36} -11.9976 q^{37} +12.6656 q^{38} +2.68421 q^{39} +19.1705 q^{40} +6.99065 q^{41} +9.78280 q^{42} +2.56337 q^{43} +5.99349 q^{44} +3.76871 q^{45} -12.5661 q^{46} -3.90548 q^{47} -13.1562 q^{48} +2.47934 q^{49} -2.28251 q^{50} -2.82380 q^{51} -11.1564 q^{52} -0.351696 q^{53} -14.4777 q^{54} -2.90614 q^{55} -24.3758 q^{56} +5.75429 q^{57} -18.3735 q^{58} +13.3041 q^{59} +14.5281 q^{60} -5.15331 q^{61} +21.0281 q^{62} -4.79202 q^{63} +12.8064 q^{64} +5.40954 q^{65} +3.81353 q^{66} +4.18754 q^{67} +11.7365 q^{68} -5.70907 q^{69} +19.7155 q^{70} -2.44306 q^{71} +12.3225 q^{72} +1.32576 q^{73} +31.7285 q^{74} -1.03700 q^{75} -23.9166 q^{76} +3.69524 q^{77} -7.09857 q^{78} +9.23041 q^{79} -26.5141 q^{80} -1.90824 q^{81} -18.4872 q^{82} -4.38057 q^{83} -18.4729 q^{84} -5.69085 q^{85} -6.77901 q^{86} -8.34750 q^{87} -9.50217 q^{88} +11.3803 q^{89} -9.96661 q^{90} -6.87838 q^{91} +23.7286 q^{92} +9.55352 q^{93} +10.3283 q^{94} +11.5967 q^{95} +15.7679 q^{96} +6.14949 q^{97} -6.55679 q^{98} -1.86803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) $ 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * 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 $

$2$	−2.64457	−1.86999	−0.934996	−	0.354658i	$-0.884597\pi$
$2$	−2.64457	−1.86999	−0.934996	+	0.354658i	$0.884597\pi$
$3$	−1.20149	−0.693679	−0.346839	−	0.937925i	$-0.612745\pi$
$3$	−1.20149	−0.693679	−0.346839	+	0.937925i	$0.612745\pi$
$4$	4.99374	2.49687
$5$	−2.42138	−1.08288	−0.541438	−	0.840741i	$-0.682120\pi$
$5$	−2.42138	−1.08288	−0.541438	+	0.840741i	$0.682120\pi$
$6$	3.17741	1.29717
$7$	3.07885	1.16370	0.581849	−	0.813297i	$-0.302330\pi$
$7$	3.07885	1.16370	0.581849	+	0.813297i	$0.302330\pi$
$8$	−7.91715	−2.79914
$9$	−1.55643	−0.518810
$10$	6.40351	2.02497
$11$	1.20020	0.361874	0.180937	−	0.983495i	$-0.442087\pi$
$11$	1.20020	0.361874	0.180937	+	0.983495i	$0.442087\pi$
$12$	−5.99992	−1.73203
$13$	−2.23407	−0.619620	−0.309810	−	0.950799i	$-0.600265\pi$
$13$	−2.23407	−0.619620	−0.309810	+	0.950799i	$0.600265\pi$
$14$	−8.14224	−2.17610
$15$	2.90926	0.751168
$16$	10.9500	2.73749
$17$	2.35025	0.570019	0.285010	−	0.958525i	$-0.408003\pi$
$17$	2.35025	0.570019	0.285010	+	0.958525i	$0.408003\pi$
$18$	4.11608	0.970170
$19$	−4.78931	−1.09874	−0.549371	−	0.835578i	$-0.685133\pi$
$19$	−4.78931	−1.09874	−0.549371	+	0.835578i	$0.685133\pi$
$20$	−12.0918	−2.70380
$21$	−3.69920	−0.807232
$22$	−3.17401	−0.676702
$23$	4.75167	0.990792	0.495396	−	0.868667i	$-0.335023\pi$
$23$	4.75167	0.990792	0.495396	+	0.868667i	$0.335023\pi$
$24$	9.51236	1.94170
$25$	0.863093	0.172619
$26$	5.90815	1.15868
$27$	5.47449	1.05357
$28$	15.3750	2.90560
$29$	6.94764	1.29014	0.645072	−	0.764121i	$-0.276827\pi$
$29$	6.94764	1.29014	0.645072	+	0.764121i	$0.276827\pi$
$30$	−7.69374	−1.40468
$31$	−7.95141	−1.42812	−0.714058	−	0.700087i	$-0.753145\pi$
$31$	−7.95141	−1.42812	−0.714058	+	0.700087i	$0.753145\pi$
$32$	−13.1236	−2.31995
$33$	−1.44203	−0.251024
$34$	−6.21540	−1.06593
$35$	−7.45508	−1.26014
$36$	−7.77240	−1.29540
$37$	−11.9976	−1.97240	−0.986198	−	0.165572i	$-0.947053\pi$
$37$	−11.9976	−1.97240	−0.986198	+	0.165572i	$0.947053\pi$
$38$	12.6656	2.05464
$39$	2.68421	0.429817
$40$	19.1705	3.03111
$41$	6.99065	1.09176	0.545878	−	0.837865i	$-0.316196\pi$
$41$	6.99065	1.09176	0.545878	+	0.837865i	$0.316196\pi$
$42$	9.78280	1.50952
$43$	2.56337	0.390910	0.195455	−	0.980713i	$-0.437382\pi$
$43$	2.56337	0.390910	0.195455	+	0.980713i	$0.437382\pi$
$44$	5.99349	0.903553
$45$	3.76871	0.561806
$46$	−12.5661	−1.85277
$47$	−3.90548	−0.569672	−0.284836	−	0.958576i	$-0.591939\pi$
$47$	−3.90548	−0.569672	−0.284836	+	0.958576i	$0.591939\pi$
$48$	−13.1562	−1.89894
$49$	2.47934	0.354192
$50$	−2.28251	−0.322796
$51$	−2.82380	−0.395410
$52$	−11.1564	−1.54711
$53$	−0.351696	−0.0483091	−0.0241546	−	0.999708i	$-0.507689\pi$
$53$	−0.351696	−0.0483091	−0.0241546	+	0.999708i	$0.507689\pi$
$54$	−14.4777	−1.97016
$55$	−2.90614	−0.391864
$56$	−24.3758	−3.25735
$57$	5.75429	0.762174
$58$	−18.3735	−2.41256
$59$	13.3041	1.73204	0.866020	−	0.500009i	$-0.166670\pi$
$59$	13.3041	1.73204	0.866020	+	0.500009i	$0.166670\pi$
$60$	14.5281	1.87557
$61$	−5.15331	−0.659814	−0.329907	−	0.944013i	$-0.607017\pi$
$61$	−5.15331	−0.659814	−0.329907	+	0.944013i	$0.607017\pi$
$62$	21.0281	2.67057
$63$	−4.79202	−0.603737
$64$	12.8064	1.60080
$65$	5.40954	0.670971
$66$	3.81353	0.469414
$67$	4.18754	0.511590	0.255795	−	0.966731i	$-0.417663\pi$
$67$	4.18754	0.511590	0.255795	+	0.966731i	$0.417663\pi$
$68$	11.7365	1.42326
$69$	−5.70907	−0.687292
$70$	19.7155	2.35645
$71$	−2.44306	−0.289938	−0.144969	−	0.989436i	$-0.546308\pi$
$71$	−2.44306	−0.289938	−0.144969	+	0.989436i	$0.546308\pi$
$72$	12.3225	1.45222
$73$	1.32576	0.155168	0.0775842	−	0.996986i	$-0.475279\pi$
$73$	1.32576	0.155168	0.0775842	+	0.996986i	$0.475279\pi$
$74$	31.7285	3.68836
$75$	−1.03700	−0.119742
$76$	−23.9166	−2.74342
$77$	3.69524	0.421112
$78$	−7.09857	−0.803755
$79$	9.23041	1.03850	0.519251	−	0.854622i	$-0.326211\pi$
$79$	9.23041	1.03850	0.519251	+	0.854622i	$0.326211\pi$
$80$	−26.5141	−2.96436
$81$	−1.90824	−0.212027
$82$	−18.4872	−2.04157
$83$	−4.38057	−0.480830	−0.240415	−	0.970670i	$-0.577284\pi$
$83$	−4.38057	−0.480830	−0.240415	+	0.970670i	$0.577284\pi$
$84$	−18.4729	−2.01555
$85$	−5.69085	−0.617260
$86$	−6.77901	−0.730999
$87$	−8.34750	−0.894946
$88$	−9.50217	−1.01293
$89$	11.3803	1.20631	0.603157	−	0.797623i	$-0.293909\pi$
$89$	11.3803	1.20631	0.603157	+	0.797623i	$0.293909\pi$
$90$	−9.96661	−1.05057
$91$	−6.87838	−0.721050
$92$	23.7286	2.47388
$93$	9.55352	0.990654
$94$	10.3283	1.06528
$95$	11.5967	1.18980
$96$	15.7679	1.60930
$97$	6.14949	0.624386	0.312193	−	0.950019i	$-0.398936\pi$
$97$	6.14949	0.624386	0.312193	+	0.950019i	$0.398936\pi$
$98$	−6.55679	−0.662335
$99$	−1.86803	−0.187744
$100$	4.31006	0.431006
$101$	−6.93059	−0.689620	−0.344810	−	0.938673i	$-0.612057\pi$
$101$	−6.93059	−0.689620	−0.344810	+	0.938673i	$0.612057\pi$
$102$	7.46772	0.739414
$103$	−1.41050	−0.138980	−0.0694902	−	0.997583i	$-0.522137\pi$
$103$	−1.41050	−0.138980	−0.0694902	+	0.997583i	$0.522137\pi$
$104$	17.6875	1.73440
$105$	8.95719	0.874132
$106$	0.930083	0.0903377
$107$	−0.285081	−0.0275598	−0.0137799	−	0.999905i	$-0.504386\pi$
$107$	−0.285081	−0.0275598	−0.0137799	+	0.999905i	$0.504386\pi$
$108$	27.3382	2.63062
$109$	−4.10616	−0.393299	−0.196650	−	0.980474i	$-0.563006\pi$
$109$	−4.10616	−0.393299	−0.196650	+	0.980474i	$0.563006\pi$
$110$	7.68550	0.732783
$111$	14.4150	1.36821
$112$	33.7133	3.18561
$113$	−18.7278	−1.76176	−0.880882	−	0.473336i	$-0.843050\pi$
$113$	−18.7278	−1.76176	−0.880882	+	0.473336i	$0.843050\pi$
$114$	−15.2176	−1.42526
$115$	−11.5056	−1.07290
$116$	34.6947	3.22132
$117$	3.47717	0.321465
$118$	−35.1835	−3.23890
$119$	7.23608	0.663330
$120$	−23.0331	−2.10262
$121$	−9.55952	−0.869047
$122$	13.6283	1.23385
$123$	−8.39917	−0.757328
$124$	−39.7073	−3.56582
$125$	10.0170	0.895951
$126$	12.6728	1.12898
$127$	8.72734	0.774426	0.387213	−	0.921990i	$-0.373438\pi$
$127$	8.72734	0.774426	0.387213	+	0.921990i	$0.373438\pi$
$128$	−7.62014	−0.673532
$129$	−3.07986	−0.271166
$130$	−14.3059	−1.25471
$131$	−14.4125	−1.25923	−0.629613	−	0.776909i	$-0.716786\pi$
$131$	−14.4125	−1.25923	−0.629613	+	0.776909i	$0.716786\pi$
$132$	−7.20110	−0.626775
$133$	−14.7456	−1.27860
$134$	−11.0742	−0.956669
$135$	−13.2558	−1.14088
$136$	−18.6073	−1.59556
$137$	−13.6081	−1.16262	−0.581311	−	0.813681i	$-0.697460\pi$
$137$	−13.6081	−1.16262	−0.581311	+	0.813681i	$0.697460\pi$
$138$	15.0980	1.28523
$139$	16.6117	1.40899	0.704493	−	0.709711i	$-0.251174\pi$
$139$	16.6117	1.40899	0.704493	+	0.709711i	$0.251174\pi$
$140$	−37.2288	−3.14640
$141$	4.69238	0.395170
$142$	6.46083	0.542181
$143$	−2.68133	−0.224224
$144$	−17.0428	−1.42024
$145$	−16.8229	−1.39707
$146$	−3.50606	−0.290164
$147$	−2.97890	−0.245695
$148$	−59.9130	−4.92482
$149$	−7.87647	−0.645265	−0.322633	−	0.946524i	$-0.604568\pi$
$149$	−7.87647	−0.645265	−0.322633	+	0.946524i	$0.604568\pi$
$150$	2.74241	0.223916
$151$	−20.5333	−1.67098	−0.835488	−	0.549509i	$-0.814815\pi$
$151$	−20.5333	−1.67098	−0.835488	+	0.549509i	$0.814815\pi$
$152$	37.9177	3.07553
$153$	−3.65800	−0.295732
$154$	−9.77232	−0.787476
$155$	19.2534	1.54647
$156$	13.4042	1.07320
$157$	13.0523	1.04168	0.520842	−	0.853653i	$-0.325618\pi$
$157$	13.0523	1.04168	0.520842	+	0.853653i	$0.325618\pi$
$158$	−24.4104	−1.94199
$159$	0.422558	0.0335110
$160$	31.7773	2.51222
$161$	14.6297	1.15298
$162$	5.04648	0.396489
$163$	−0.714370	−0.0559538	−0.0279769	−	0.999609i	$-0.508906\pi$
$163$	−0.714370	−0.0559538	−0.0279769	+	0.999609i	$0.508906\pi$
$164$	34.9095	2.72597
$165$	3.49170	0.271828
$166$	11.5847	0.899149
$167$	−1.93049	−0.149386	−0.0746930	−	0.997207i	$-0.523798\pi$
$167$	−1.93049	−0.149386	−0.0746930	+	0.997207i	$0.523798\pi$
$168$	29.2872	2.25955
$169$	−8.00893	−0.616071
$170$	15.0499	1.15427
$171$	7.45421	0.570038
$172$	12.8008	0.976052
$173$	19.2130	1.46073	0.730367	−	0.683055i	$-0.239349\pi$
$173$	19.2130	1.46073	0.730367	+	0.683055i	$0.239349\pi$
$174$	22.0755	1.67354
$175$	2.65734	0.200876
$176$	13.1422	0.990627
$177$	−15.9847	−1.20148
$178$	−30.0961	−2.25580
$179$	−4.34332	−0.324635	−0.162317	−	0.986739i	$-0.551897\pi$
$179$	−4.34332	−0.324635	−0.162317	+	0.986739i	$0.551897\pi$
$180$	18.8200	1.40276
$181$	14.6623	1.08984	0.544919	−	0.838489i	$-0.316560\pi$
$181$	14.6623	1.08984	0.544919	+	0.838489i	$0.316560\pi$
$182$	18.1903	1.34836
$183$	6.19164	0.457699
$184$	−37.6197	−2.77336
$185$	29.0508	2.13586
$186$	−25.2649	−1.85251
$187$	2.82077	0.206275
$188$	−19.5029	−1.42240
$189$	16.8552	1.22603
$190$	−30.6684	−2.22492
$191$	−7.36525	−0.532931	−0.266465	−	0.963844i	$-0.585856\pi$
$191$	−7.36525	−0.532931	−0.266465	+	0.963844i	$0.585856\pi$
$192$	−15.3867	−1.11044
$193$	27.0013	1.94360	0.971798	−	0.235815i	$-0.0757760\pi$
$193$	27.0013	1.94360	0.971798	+	0.235815i	$0.0757760\pi$
$194$	−16.2627	−1.16760
$195$	−6.49949	−0.465438
$196$	12.3812	0.884371
$197$	24.6674	1.75748	0.878741	−	0.477299i	$-0.158384\pi$
$197$	24.6674	1.75748	0.878741	+	0.477299i	$0.158384\pi$
$198$	4.94012	0.351079
$199$	5.79192	0.410578	0.205289	−	0.978701i	$-0.434186\pi$
$199$	5.79192	0.410578	0.205289	+	0.978701i	$0.434186\pi$
$200$	−6.83324	−0.483183
$201$	−5.03128	−0.354879
$202$	18.3284	1.28958
$203$	21.3908	1.50134
$204$	−14.1013	−0.987289
$205$	−16.9270	−1.18224
$206$	3.73015	0.259892
$207$	−7.39564	−0.514033
$208$	−24.4630	−1.69620
$209$	−5.74813	−0.397606
$210$	−23.6879	−1.63462
$211$	15.1905	1.04576	0.522878	−	0.852408i	$-0.324858\pi$
$211$	15.1905	1.04576	0.522878	+	0.852408i	$0.324858\pi$
$212$	−1.75628	−0.120622
$213$	2.93530	0.201124
$214$	0.753916	0.0515367
$215$	−6.20690	−0.423307
$216$	−43.3424	−2.94907
$217$	−24.4812	−1.66189
$218$	10.8590	0.735467
$219$	−1.59288	−0.107637
$220$	−14.5125	−0.978435
$221$	−5.25062	−0.353195
$222$	−38.1214	−2.55854
$223$	−18.4457	−1.23522	−0.617609	−	0.786486i	$-0.711898\pi$
$223$	−18.4457	−1.23522	−0.617609	+	0.786486i	$0.711898\pi$
$224$	−40.4057	−2.69972
$225$	−1.34334	−0.0895562
$226$	49.5270	3.29449
$227$	−20.7966	−1.38032	−0.690160	−	0.723657i	$-0.742460\pi$
$227$	−20.7966	−1.38032	−0.690160	+	0.723657i	$0.742460\pi$
$228$	28.7354	1.90305
$229$	−2.12194	−0.140222	−0.0701110	−	0.997539i	$-0.522335\pi$
$229$	−2.12194	−0.140222	−0.0701110	+	0.997539i	$0.522335\pi$
$230$	30.4274	2.00632
$231$	−4.43979	−0.292116
$232$	−55.0055	−3.61129
$233$	−16.3128	−1.06869	−0.534345	−	0.845267i	$-0.679442\pi$
$233$	−16.3128	−1.06869	−0.534345	+	0.845267i	$0.679442\pi$
$234$	−9.19562	−0.601136
$235$	9.45665	0.616884
$236$	66.4370	4.32468
$237$	−11.0902	−0.720387
$238$	−19.1363	−1.24042
$239$	0.228997	0.0148126	0.00740629	−	0.999973i	$-0.497642\pi$
$239$	0.228997	0.0148126	0.00740629	+	0.999973i	$0.497642\pi$
$240$	31.8563	2.05632
$241$	−21.2757	−1.37049	−0.685243	−	0.728315i	$-0.740304\pi$
$241$	−21.2757	−1.37049	−0.685243	+	0.728315i	$0.740304\pi$
$242$	25.2808	1.62511
$243$	−14.1307	−0.906487
$244$	−25.7343	−1.64747
$245$	−6.00343	−0.383545
$246$	22.2122	1.41620
$247$	10.6996	0.680802
$248$	62.9525	3.99749
$249$	5.26320	0.333542
$250$	−26.4907	−1.67542
$251$	13.5358	0.854369	0.427185	−	0.904164i	$-0.359505\pi$
$251$	13.5358	0.854369	0.427185	+	0.904164i	$0.359505\pi$
$252$	−23.9301	−1.50745
$253$	5.70296	0.358542
$254$	−23.0800	−1.44817
$255$	6.83749	0.428180
$256$	−5.46082	−0.341301
$257$	−9.93425	−0.619682	−0.309841	−	0.950788i	$-0.600276\pi$
$257$	−9.93425	−0.619682	−0.309841	+	0.950788i	$0.600276\pi$
$258$	8.14489	0.507079
$259$	−36.9389	−2.29527
$260$	27.0138	1.67533
$261$	−10.8135	−0.669340
$262$	38.1148	2.35474
$263$	17.0097	1.04886	0.524431	−	0.851453i	$-0.324278\pi$
$263$	17.0097	1.04886	0.524431	+	0.851453i	$0.324278\pi$
$264$	11.4167	0.702651
$265$	0.851590	0.0523127
$266$	38.9957	2.39098
$267$	−13.6733	−0.836795
$268$	20.9115	1.27737
$269$	−5.71692	−0.348567	−0.174283	−	0.984696i	$-0.555761\pi$
$269$	−5.71692	−0.348567	−0.174283	+	0.984696i	$0.555761\pi$
$270$	35.0560	2.13344
$271$	−21.8451	−1.32699	−0.663497	−	0.748179i	$-0.730928\pi$
$271$	−21.8451	−1.32699	−0.663497	+	0.748179i	$0.730928\pi$
$272$	25.7352	1.56042
$273$	8.26428	0.500177
$274$	35.9877	2.17410
$275$	1.03589	0.0624662
$276$	−28.5096	−1.71608
$277$	27.0042	1.62253	0.811264	−	0.584680i	$-0.198780\pi$
$277$	27.0042	1.62253	0.811264	+	0.584680i	$0.198780\pi$
$278$	−43.9308	−2.63479
$279$	12.3758	0.740920
$280$	59.0230	3.52730
$281$	14.7045	0.877199	0.438599	−	0.898683i	$-0.355475\pi$
$281$	14.7045	0.877199	0.438599	+	0.898683i	$0.355475\pi$
$282$	−12.4093	−0.738964
$283$	11.0404	0.656285	0.328143	−	0.944628i	$-0.393577\pi$
$283$	11.0404	0.656285	0.328143	+	0.944628i	$0.393577\pi$
$284$	−12.2000	−0.723937
$285$	−13.9333	−0.825340
$286$	7.09097	0.419298
$287$	21.5232	1.27047
$288$	20.4260	1.20361
$289$	−11.4763	−0.675078
$290$	44.4893	2.61250
$291$	−7.38853	−0.433123
$292$	6.62050	0.387435
$293$	−1.18881	−0.0694512	−0.0347256	−	0.999397i	$-0.511056\pi$
$293$	−1.18881	−0.0694512	−0.0347256	+	0.999397i	$0.511056\pi$
$294$	7.87789	0.459448
$295$	−32.2142	−1.87558
$296$	94.9869	5.52100
$297$	6.57049	0.381258
$298$	20.8299	1.20664
$299$	−10.6156	−0.613914
$300$	−5.17849	−0.298980
$301$	7.89224	0.454901
$302$	54.3017	3.12471
$303$	8.32702	0.478375
$304$	−52.4427	−3.00780
$305$	12.4781	0.714496
$306$	9.67382	0.553016
$307$	23.9047	1.36431	0.682156	−	0.731206i	$-0.261042\pi$
$307$	23.9047	1.36431	0.682156	+	0.731206i	$0.261042\pi$
$308$	18.4531	1.05146
$309$	1.69469	0.0964077
$310$	−50.9170	−2.89189
$311$	−5.22768	−0.296435	−0.148217	−	0.988955i	$-0.547354\pi$
$311$	−5.22768	−0.296435	−0.148217	+	0.988955i	$0.547354\pi$
$312$	−21.2513	−1.20312
$313$	24.0146	1.35739	0.678693	−	0.734422i	$-0.262547\pi$
$313$	24.0146	1.35739	0.678693	+	0.734422i	$0.262547\pi$
$314$	−34.5176	−1.94794
$315$	11.6033	0.653772
$316$	46.0943	2.59300
$317$	30.3540	1.70485	0.852426	−	0.522847i	$-0.175130\pi$
$317$	30.3540	1.70485	0.852426	+	0.522847i	$0.175130\pi$
$318$	−1.11748	−0.0626653
$319$	8.33856	0.466870
$320$	−31.0092	−1.73347
$321$	0.342521	0.0191177
$322$	−38.6893	−2.15607
$323$	−11.2561	−0.626304
$324$	−9.52927	−0.529404
$325$	−1.92821	−0.106958
$326$	1.88920	0.104633
$327$	4.93350	0.272823
$328$	−55.3460	−3.05597
$329$	−12.0244	−0.662926
$330$	−9.23403	−0.508316
$331$	−3.90517	−0.214648	−0.107324	−	0.994224i	$-0.534228\pi$
$331$	−3.90517	−0.214648	−0.107324	+	0.994224i	$0.534228\pi$
$332$	−21.8754	−1.20057
$333$	18.6734	1.02330
$334$	5.10532	0.279351
$335$	−10.1396	−0.553988
$336$	−40.5062	−2.20979
$337$	−25.2235	−1.37401	−0.687006	−	0.726651i	$-0.741076\pi$
$337$	−25.2235	−1.37401	−0.687006	+	0.726651i	$0.741076\pi$
$338$	21.1802	1.15205
$339$	22.5012	1.22210
$340$	−28.4187	−1.54122
$341$	−9.54329	−0.516798
$342$	−19.7132	−1.06597
$343$	−13.9184	−0.751526
$344$	−20.2946	−1.09421
$345$	13.8239	0.744251
$346$	−50.8100	−2.73156
$347$	25.6832	1.37875	0.689374	−	0.724406i	$-0.257886\pi$
$347$	25.6832	1.37875	0.689374	+	0.724406i	$0.257886\pi$
$348$	−41.6853	−2.23456
$349$	14.4882	0.775537	0.387768	−	0.921757i	$-0.373246\pi$
$349$	14.4882	0.775537	0.387768	+	0.921757i	$0.373246\pi$
$350$	−7.02751	−0.375636
$351$	−12.2304	−0.652810
$352$	−15.7510	−0.839530
$353$	13.5434	0.720841	0.360420	−	0.932790i	$-0.382633\pi$
$353$	13.5434	0.720841	0.360420	+	0.932790i	$0.382633\pi$
$354$	42.2725	2.24676
$355$	5.91558	0.313966
$356$	56.8305	3.01201
$357$	−8.69405	−0.460138
$358$	11.4862	0.607064
$359$	16.3207	0.861372	0.430686	−	0.902502i	$-0.358272\pi$
$359$	16.3207	0.861372	0.430686	+	0.902502i	$0.358272\pi$
$360$	−29.8374	−1.57257
$361$	3.93745	0.207234
$362$	−38.7754	−2.03799
$363$	11.4856	0.602840
$364$	−34.3488	−1.80037
$365$	−3.21017	−0.168028
$366$	−16.3742	−0.855894
$367$	−17.8075	−0.929545	−0.464773	−	0.885430i	$-0.653864\pi$
$367$	−17.8075	−0.929545	−0.464773	+	0.885430i	$0.653864\pi$
$368$	52.0307	2.71229
$369$	−10.8804	−0.566413
$370$	−76.8268	−3.99404
$371$	−1.08282	−0.0562172
$372$	47.7078	2.47353
$373$	35.6397	1.84535	0.922677	−	0.385574i	$-0.125997\pi$
$373$	35.6397	1.84535	0.922677	+	0.385574i	$0.125997\pi$
$374$	−7.45972	−0.385733
$375$	−12.0353	−0.621502
$376$	30.9202	1.59459
$377$	−15.5215	−0.799399
$378$	−44.5746	−2.29267
$379$	−26.1416	−1.34280	−0.671401	−	0.741094i	$-0.734307\pi$
$379$	−26.1416	−1.34280	−0.671401	+	0.741094i	$0.734307\pi$
$380$	57.9111	2.97078
$381$	−10.4858	−0.537203
$382$	19.4779	0.996576
$383$	−37.2449	−1.90313	−0.951563	−	0.307455i	$-0.900523\pi$
$383$	−37.2449	−1.90313	−0.951563	+	0.307455i	$0.900523\pi$
$384$	9.15550	0.467215
$385$	−8.94759	−0.456012
$386$	−71.4068	−3.63451
$387$	−3.98970	−0.202808
$388$	30.7089	1.55901
$389$	−20.9538	−1.06240	−0.531201	−	0.847246i	$-0.678259\pi$
$389$	−20.9538	−1.06240	−0.531201	+	0.847246i	$0.678259\pi$
$390$	17.1883	0.870366
$391$	11.1676	0.564771
$392$	−19.6293	−0.991430
$393$	17.3164	0.873498
$394$	−65.2347	−3.28648
$395$	−22.3503	−1.12457
$396$	−9.32844	−0.468772
$397$	16.8831	0.847338	0.423669	−	0.905817i	$-0.360742\pi$
$397$	16.8831	0.847338	0.423669	+	0.905817i	$0.360742\pi$
$398$	−15.3171	−0.767779
$399$	17.7166	0.886940
$400$	9.45084	0.472542
$401$	−24.7215	−1.23453	−0.617266	−	0.786755i	$-0.711760\pi$
$401$	−24.7215	−1.23453	−0.617266	+	0.786755i	$0.711760\pi$
$402$	13.3056	0.663621
$403$	17.7640	0.884889
$404$	−34.6096	−1.72189
$405$	4.62059	0.229599
$406$	−56.5694	−2.80749
$407$	−14.3995	−0.713759
$408$	22.3564	1.10681
$409$	5.79490	0.286539	0.143270	−	0.989684i	$-0.454238\pi$
$409$	5.79490	0.286539	0.143270	+	0.989684i	$0.454238\pi$
$410$	44.7647	2.21077
$411$	16.3500	0.806487
$412$	−7.04366	−0.347016
$413$	40.9612	2.01557
$414$	19.5583	0.961237
$415$	10.6070	0.520679
$416$	29.3191	1.43749
$417$	−19.9587	−0.977384
$418$	15.2013	0.743521
$419$	14.6891	0.717608	0.358804	−	0.933413i	$-0.383185\pi$
$419$	14.6891	0.717608	0.358804	+	0.933413i	$0.383185\pi$
$420$	44.7299	2.18259
$421$	36.6490	1.78616	0.893081	−	0.449896i	$-0.148539\pi$
$421$	36.6490	1.78616	0.893081	+	0.449896i	$0.148539\pi$
$422$	−40.1722	−1.95555
$423$	6.07859	0.295551
$424$	2.78443	0.135224
$425$	2.02849	0.0983960
$426$	−7.76261	−0.376100
$427$	−15.8663	−0.767824
$428$	−1.42362	−0.0688133
$429$	3.22159	0.155540
$430$	16.4146	0.791581
$431$	37.3225	1.79776	0.898881	−	0.438192i	$-0.144381\pi$
$431$	37.3225	1.79776	0.898881	+	0.438192i	$0.144381\pi$
$432$	59.9455	2.88413
$433$	28.7690	1.38255	0.691276	−	0.722591i	$-0.257049\pi$
$433$	28.7690	1.38255	0.691276	+	0.722591i	$0.257049\pi$
$434$	64.7423	3.10773
$435$	20.2125	0.969115
$436$	−20.5051	−0.982017
$437$	−22.7572	−1.08863
$438$	4.21249	0.201280
$439$	−22.7056	−1.08368	−0.541839	−	0.840482i	$-0.682272\pi$
$439$	−22.7056	−1.08368	−0.541839	+	0.840482i	$0.682272\pi$
$440$	23.0084	1.09688
$441$	−3.85892	−0.183758
$442$	13.8856	0.660472
$443$	13.1587	0.625187	0.312593	−	0.949887i	$-0.398802\pi$
$443$	13.1587	0.625187	0.312593	+	0.949887i	$0.398802\pi$
$444$	71.9847	3.41624
$445$	−27.5562	−1.30629
$446$	48.7810	2.30985
$447$	9.46348	0.447607
$448$	39.4290	1.86285
$449$	17.8870	0.844142	0.422071	−	0.906563i	$-0.361303\pi$
$449$	17.8870	0.844142	0.422071	+	0.906563i	$0.361303\pi$
$450$	3.55256	0.167469
$451$	8.39018	0.395078
$452$	−93.5218	−4.39890
$453$	24.6705	1.15912
$454$	54.9980	2.58119
$455$	16.6552	0.780807
$456$	−45.5576	−2.13343
$457$	−34.7622	−1.62611	−0.813054	−	0.582188i	$-0.802197\pi$
$457$	−34.7622	−1.62611	−0.813054	+	0.582188i	$0.802197\pi$
$458$	5.61163	0.262214
$459$	12.8664	0.600553
$460$	−57.4561	−2.67890
$461$	35.2045	1.63964	0.819818	−	0.572624i	$-0.194074\pi$
$461$	35.2045	1.63964	0.819818	+	0.572624i	$0.194074\pi$
$462$	11.7413	0.546255
$463$	−8.48650	−0.394401	−0.197200	−	0.980363i	$-0.563185\pi$
$463$	−8.48650	−0.394401	−0.197200	+	0.980363i	$0.563185\pi$
$464$	76.0765	3.53176
$465$	−23.1327	−1.07275
$466$	43.1404	1.99844
$467$	31.9472	1.47834	0.739170	−	0.673519i	$-0.235218\pi$
$467$	31.9472	1.47834	0.739170	+	0.673519i	$0.235218\pi$
$468$	17.3641	0.802656
$469$	12.8928	0.595336
$470$	−25.0088	−1.15357
$471$	−15.6821	−0.722594
$472$	−105.330	−4.84822
$473$	3.07656	0.141460
$474$	29.3288	1.34712
$475$	−4.13362	−0.189663
$476$	36.1351	1.65625
$477$	0.547389	0.0250632
$478$	−0.605598	−0.0276994
$479$	−28.2153	−1.28919	−0.644594	−	0.764525i	$-0.722974\pi$
$479$	−28.2153	−1.28919	−0.644594	+	0.764525i	$0.722974\pi$
$480$	−38.1800	−1.74267
$481$	26.8035	1.22214
$482$	56.2649	2.56280
$483$	−17.5774	−0.799800
$484$	−47.7378	−2.16990
$485$	−14.8903	−0.676132
$486$	37.3697	1.69512
$487$	−32.2212	−1.46008	−0.730040	−	0.683404i	$-0.760499\pi$
$487$	−32.2212	−1.46008	−0.730040	+	0.683404i	$0.760499\pi$
$488$	40.7996	1.84691
$489$	0.858306	0.0388139
$490$	15.8765	0.717227
$491$	−21.0243	−0.948815	−0.474407	−	0.880305i	$-0.657338\pi$
$491$	−21.0243	−0.948815	−0.474407	+	0.880305i	$0.657338\pi$
$492$	−41.9433	−1.89095
$493$	16.3287	0.735408
$494$	−28.2959	−1.27309
$495$	4.52321	0.203303
$496$	−87.0677	−3.90946
$497$	−7.52182	−0.337400
$498$	−13.9189	−0.623720
$499$	−29.5984	−1.32501	−0.662503	−	0.749059i	$-0.730506\pi$
$499$	−29.5984	−1.32501	−0.662503	+	0.749059i	$0.730506\pi$
$500$	50.0225	2.23707
$501$	2.31946	0.103626
$502$	−35.7962	−1.59766
$503$	16.5726	0.738935	0.369468	−	0.929244i	$-0.379540\pi$
$503$	16.5726	0.738935	0.369468	+	0.929244i	$0.379540\pi$
$504$	37.9391	1.68994
$505$	16.7816	0.746772
$506$	−15.0819	−0.670471
$507$	9.62262	0.427356
$508$	43.5821	1.93364
$509$	−29.5410	−1.30938	−0.654692	−	0.755896i	$-0.727202\pi$
$509$	−29.5410	−1.30938	−0.654692	+	0.755896i	$0.727202\pi$
$510$	−18.0822	−0.800693
$511$	4.08182	0.180569
$512$	29.6818	1.31176
$513$	−26.2190	−1.15760
$514$	26.2718	1.15880
$515$	3.41535	0.150498
$516$	−15.3800	−0.677067
$517$	−4.68735	−0.206150
$518$	97.6874	4.29214
$519$	−23.0841	−1.01328
$520$	−42.8281	−1.87814
$521$	−38.0991	−1.66915	−0.834577	−	0.550891i	$-0.814288\pi$
$521$	−38.0991	−1.66915	−0.834577	+	0.550891i	$0.814288\pi$
$522$	28.5971	1.25166
$523$	−1.41125	−0.0617099	−0.0308549	−	0.999524i	$-0.509823\pi$
$523$	−1.41125	−0.0617099	−0.0308549	+	0.999524i	$0.509823\pi$
$524$	−71.9723	−3.14412
$525$	−3.19276	−0.139343
$526$	−44.9832	−1.96136
$527$	−18.6878	−0.814054
$528$	−15.7901	−0.687177
$529$	−0.421605	−0.0183307
$530$	−2.25209	−0.0978244
$531$	−20.7068	−0.898599
$532$	−73.6356	−3.19251
$533$	−15.6176	−0.676473
$534$	36.1601	1.56480
$535$	0.690290	0.0298439
$536$	−33.1534	−1.43201
$537$	5.21844	0.225192
$538$	15.1188	0.651817
$539$	2.97571	0.128173
$540$	−66.1962	−2.84863
$541$	23.6189	1.01545	0.507727	−	0.861518i	$-0.330486\pi$
$541$	23.6189	1.01545	0.507727	+	0.861518i	$0.330486\pi$
$542$	57.7708	2.48147
$543$	−17.6165	−0.755998
$544$	−30.8438	−1.32242
$545$	9.94260	0.425894
$546$	−21.8555	−0.935327
$547$	6.10678	0.261107	0.130553	−	0.991441i	$-0.458325\pi$
$547$	6.10678	0.261107	0.130553	+	0.991441i	$0.458325\pi$
$548$	−67.9556	−2.90292
$549$	8.02076	0.342318
$550$	−2.73947	−0.116811
$551$	−33.2744	−1.41754
$552$	45.1996	1.92382
$553$	28.4191	1.20850
$554$	−71.4145	−3.03411
$555$	−34.9042	−1.48160
$556$	82.9545	3.51806
$557$	9.03751	0.382932	0.191466	−	0.981499i	$-0.438676\pi$
$557$	9.03751	0.382932	0.191466	+	0.981499i	$0.438676\pi$
$558$	−32.7287	−1.38551
$559$	−5.72675	−0.242216
$560$	−81.6329	−3.44962
$561$	−3.38912	−0.143089
$562$	−38.8871	−1.64035
$563$	34.2678	1.44421	0.722107	−	0.691782i	$-0.243174\pi$
$563$	34.2678	1.44421	0.722107	+	0.691782i	$0.243174\pi$
$564$	23.4325	0.986687
$565$	45.3472	1.90777
$566$	−29.1972	−1.22725
$567$	−5.87520	−0.246735
$568$	19.3421	0.811575
$569$	−11.6373	−0.487860	−0.243930	−	0.969793i	$-0.578437\pi$
$569$	−11.6373	−0.487860	−0.243930	+	0.969793i	$0.578437\pi$
$570$	36.8477	1.54338
$571$	0.186681	0.00781236	0.00390618	−	0.999992i	$-0.498757\pi$
$571$	0.186681	0.00781236	0.00390618	+	0.999992i	$0.498757\pi$
$572$	−13.3899	−0.559859
$573$	8.84925	0.369683
$574$	−56.9195	−2.37578
$575$	4.10114	0.171029
$576$	−19.9323	−0.830510
$577$	37.0936	1.54423	0.772113	−	0.635486i	$-0.219200\pi$
$577$	37.0936	1.54423	0.772113	+	0.635486i	$0.219200\pi$
$578$	30.3499	1.26239
$579$	−32.4417	−1.34823
$580$	−84.0092	−3.48829
$581$	−13.4871	−0.559541
$582$	19.5395	0.809937
$583$	−0.422105	−0.0174818
$584$	−10.4962	−0.434337
$585$	−8.41956	−0.348106
$586$	3.14390	0.129873
$587$	−13.6888	−0.564996	−0.282498	−	0.959268i	$-0.591163\pi$
$587$	−13.6888	−0.564996	−0.282498	+	0.959268i	$0.591163\pi$
$588$	−14.8758	−0.613469
$589$	38.0817	1.56913
$590$	85.1927	3.50733
$591$	−29.6376	−1.21913
$592$	−131.373	−5.39942
$593$	−26.2988	−1.07996	−0.539982	−	0.841677i	$-0.681569\pi$
$593$	−26.2988	−1.07996	−0.539982	+	0.841677i	$0.681569\pi$
$594$	−17.3761	−0.712950
$595$	−17.5213	−0.718304
$596$	−39.3330	−1.61114
$597$	−6.95892	−0.284810
$598$	28.0736	1.14802
$599$	30.0899	1.22944	0.614721	−	0.788745i	$-0.289269\pi$
$599$	30.0899	1.22944	0.614721	+	0.788745i	$0.289269\pi$
$600$	8.21005	0.335174
$601$	−3.11378	−0.127014	−0.0635069	−	0.997981i	$-0.520228\pi$
$601$	−3.11378	−0.127014	−0.0635069	+	0.997981i	$0.520228\pi$
$602$	−20.8716	−0.850662
$603$	−6.51761	−0.265418
$604$	−102.538	−4.17221
$605$	23.1473	0.941070
$606$	−22.0214	−0.894557
$607$	−9.35656	−0.379771	−0.189886	−	0.981806i	$-0.560812\pi$
$607$	−9.35656	−0.379771	−0.189886	+	0.981806i	$0.560812\pi$
$608$	62.8531	2.54903
$609$	−25.7007	−1.04145
$610$	−32.9993	−1.33610
$611$	8.72511	0.352980
$612$	−18.2671	−0.738403
$613$	−25.7596	−1.04042	−0.520210	−	0.854038i	$-0.674146\pi$
$613$	−25.7596	−1.04042	−0.520210	+	0.854038i	$0.674146\pi$
$614$	−63.2176	−2.55125
$615$	20.3376	0.820092
$616$	−29.2558	−1.17875
$617$	−7.84163	−0.315692	−0.157846	−	0.987464i	$-0.550455\pi$
$617$	−7.84163	−0.315692	−0.157846	+	0.987464i	$0.550455\pi$
$618$	−4.48173	−0.180282
$619$	−19.8294	−0.797011	−0.398505	−	0.917166i	$-0.630471\pi$
$619$	−19.8294	−0.797011	−0.398505	+	0.917166i	$0.630471\pi$
$620$	96.1465	3.86134
$621$	26.0130	1.04387
$622$	13.8250	0.554330
$623$	35.0384	1.40378
$624$	29.3920	1.17662
$625$	−28.5705	−1.14282
$626$	−63.5083	−2.53830
$627$	6.90630	0.275811
$628$	65.1796	2.60095
$629$	−28.1974	−1.12430
$630$	−30.6857	−1.22255
$631$	−6.65841	−0.265067	−0.132534	−	0.991179i	$-0.542311\pi$
$631$	−6.65841	−0.265067	−0.132534	+	0.991179i	$0.542311\pi$
$632$	−73.0785	−2.90691
$633$	−18.2512	−0.725418
$634$	−80.2733	−3.18806
$635$	−21.1322	−0.838607
$636$	2.11014	0.0836727
$637$	−5.53902	−0.219464
$638$	−22.0519	−0.873043
$639$	3.80245	0.150422
$640$	18.4513	0.729351
$641$	−23.2406	−0.917949	−0.458975	−	0.888449i	$-0.651783\pi$
$641$	−23.2406	−0.917949	−0.458975	+	0.888449i	$0.651783\pi$
$642$	−0.905821	−0.0357499
$643$	−44.9322	−1.77195	−0.885976	−	0.463731i	$-0.846510\pi$
$643$	−44.9322	−1.77195	−0.885976	+	0.463731i	$0.846510\pi$
$644$	73.0570	2.87885
$645$	7.45751	0.293639
$646$	29.7674	1.17118
$647$	0.580499	0.0228218	0.0114109	−	0.999935i	$-0.496368\pi$
$647$	0.580499	0.0228218	0.0114109	+	0.999935i	$0.496368\pi$
$648$	15.1079	0.593492
$649$	15.9675	0.626781
$650$	5.09929	0.200010
$651$	29.4139	1.15282
$652$	−3.56738	−0.139709
$653$	17.4462	0.682724	0.341362	−	0.939932i	$-0.389112\pi$
$653$	17.4462	0.682724	0.341362	+	0.939932i	$0.389112\pi$
$654$	−13.0470	−0.510178
$655$	34.8982	1.36358
$656$	76.5474	2.98867
$657$	−2.06345	−0.0805028
$658$	31.7993	1.23967
$659$	−25.7099	−1.00152	−0.500758	−	0.865587i	$-0.666945\pi$
$659$	−25.7099	−1.00152	−0.500758	+	0.865587i	$0.666945\pi$
$660$	17.4366	0.678720
$661$	−20.7336	−0.806444	−0.403222	−	0.915102i	$-0.632110\pi$
$661$	−20.7336	−0.806444	−0.403222	+	0.915102i	$0.632110\pi$
$662$	10.3275	0.401390
$663$	6.30856	0.245004
$664$	34.6817	1.34591
$665$	35.7047	1.38457
$666$	−49.3832	−1.91356
$667$	33.0129	1.27827
$668$	−9.64038	−0.372998
$669$	22.1623	0.856844
$670$	26.8150	1.03595
$671$	−6.18501	−0.238770
$672$	48.5470	1.87274
$673$	5.00510	0.192933	0.0964663	−	0.995336i	$-0.469246\pi$
$673$	5.00510	0.192933	0.0964663	+	0.995336i	$0.469246\pi$
$674$	66.7053	2.56939
$675$	4.72500	0.181865
$676$	−39.9945	−1.53825
$677$	29.0563	1.11672	0.558362	−	0.829598i	$-0.311430\pi$
$677$	29.0563	1.11672	0.558362	+	0.829598i	$0.311430\pi$
$678$	−59.5060	−2.28532
$679$	18.9334	0.726596
$680$	45.0554	1.72779
$681$	24.9869	0.957498
$682$	25.2379	0.966408
$683$	−11.4055	−0.436419	−0.218210	−	0.975902i	$-0.570022\pi$
$683$	−11.4055	−0.436419	−0.218210	+	0.975902i	$0.570022\pi$
$684$	37.2244	1.42331
$685$	32.9505	1.25898
$686$	36.8083	1.40535
$687$	2.54949	0.0972691
$688$	28.0688	1.07011
$689$	0.785713	0.0299333
$690$	−36.5581	−1.39174
$691$	−5.71597	−0.217446	−0.108723	−	0.994072i	$-0.534676\pi$
$691$	−5.71597	−0.217446	−0.108723	+	0.994072i	$0.534676\pi$
$692$	95.9445	3.64726
$693$	−5.75138	−0.218477
$694$	−67.9210	−2.57825
$695$	−40.2233	−1.52576
$696$	66.0885	2.50508
$697$	16.4298	0.622322
$698$	−38.3151	−1.45025
$699$	19.5997	0.741327
$700$	13.2701	0.501561
$701$	−22.2660	−0.840974	−0.420487	−	0.907299i	$-0.638141\pi$
$701$	−22.2660	−0.840974	−0.420487	+	0.907299i	$0.638141\pi$
$702$	32.3441	1.22075
$703$	57.4602	2.16715
$704$	15.3702	0.579288
$705$	−11.3620	−0.427919
$706$	−35.8164	−1.34797
$707$	−21.3383	−0.802509
$708$	−79.8232	−2.99994
$709$	−20.1189	−0.755580	−0.377790	−	0.925891i	$-0.623316\pi$
$709$	−20.1189	−0.755580	−0.377790	+	0.925891i	$0.623316\pi$
$710$	−15.6441	−0.587114
$711$	−14.3665	−0.538785
$712$	−90.0999	−3.37664
$713$	−37.7825	−1.41497
$714$	22.9920	0.860455
$715$	6.49253	0.242807
$716$	−21.6894	−0.810571
$717$	−0.275137	−0.0102752
$718$	−43.1611	−1.61076
$719$	−8.00004	−0.298351	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00004	−0.298351	−0.149175	+	0.988811i	$0.547662\pi$
$720$	41.2672	1.53794
$721$	−4.34271	−0.161731
$722$	−10.4129	−0.387527
$723$	25.5624	0.950677
$724$	73.2196	2.72118
$725$	5.99646	0.222703
$726$	−30.3746	−1.12731
$727$	20.1445	0.747116	0.373558	−	0.927607i	$-0.378138\pi$
$727$	20.1445	0.747116	0.373558	+	0.927607i	$0.378138\pi$
$728$	54.4572	2.01832
$729$	22.7026	0.840838
$730$	8.48951	0.314211
$731$	6.02456	0.222826
$732$	30.9194	1.14282
$733$	20.6490	0.762688	0.381344	−	0.924433i	$-0.375461\pi$
$733$	20.6490	0.762688	0.381344	+	0.924433i	$0.375461\pi$
$734$	47.0932	1.73824
$735$	7.21305	0.266057
$736$	−62.3592	−2.29859
$737$	5.02589	0.185131
$738$	28.7741	1.05919
$739$	−52.2834	−1.92328	−0.961638	−	0.274322i	$-0.911547\pi$
$739$	−52.2834	−1.92328	−0.961638	+	0.274322i	$0.911547\pi$
$740$	145.072	5.33296
$741$	−12.8555	−0.472258
$742$	2.86359	0.105126
$743$	16.1191	0.591352	0.295676	−	0.955288i	$-0.404455\pi$
$743$	16.1191	0.591352	0.295676	+	0.955288i	$0.404455\pi$
$744$	−75.6367	−2.77297
$745$	19.0719	0.698742
$746$	−94.2516	−3.45080
$747$	6.81805	0.249459
$748$	14.0862	0.515043
$749$	−0.877723	−0.0320713
$750$	31.8283	1.16220
$751$	−28.3806	−1.03562	−0.517811	−	0.855495i	$-0.673253\pi$
$751$	−28.3806	−1.03562	−0.517811	+	0.855495i	$0.673253\pi$
$752$	−42.7648	−1.55947
$753$	−16.2630	−0.592658
$754$	41.0477	1.49487
$755$	49.7189	1.80946
$756$	84.1703	3.06124
$757$	15.3818	0.559062	0.279531	−	0.960137i	$-0.409821\pi$
$757$	15.3818	0.559062	0.279531	+	0.960137i	$0.409821\pi$
$758$	69.1332	2.51103
$759$	−6.85203	−0.248713
$760$	−91.8132	−3.33041
$761$	−18.3163	−0.663965	−0.331982	−	0.943286i	$-0.607717\pi$
$761$	−18.3163	−0.663965	−0.331982	+	0.943286i	$0.607717\pi$
$762$	27.7304	1.00457
$763$	−12.6423	−0.457681
$764$	−36.7801	−1.33066
$765$	8.85741	0.320240
$766$	98.4967	3.55883
$767$	−29.7222	−1.07321
$768$	6.56111	0.236754
$769$	−40.9745	−1.47758	−0.738790	−	0.673936i	$-0.764602\pi$
$769$	−40.9745	−1.47758	−0.738790	+	0.673936i	$0.764602\pi$
$770$	23.6625	0.852738
$771$	11.9359	0.429860
$772$	134.837	4.85291
$773$	−14.8247	−0.533208	−0.266604	−	0.963806i	$-0.585902\pi$
$773$	−14.8247	−0.533208	−0.266604	+	0.963806i	$0.585902\pi$
$774$	10.5510	0.379249
$775$	−6.86281	−0.246519
$776$	−48.6864	−1.74774
$777$	44.3816	1.59218
$778$	55.4139	1.98668
$779$	−33.4804	−1.19956
$780$	−32.4568	−1.16214
$781$	−2.93216	−0.104921
$782$	−29.5335	−1.05612
$783$	38.0348	1.35925
$784$	27.1487	0.969597
$785$	−31.6045	−1.12801
$786$	−45.7945	−1.63343
$787$	9.25926	0.330057	0.165028	−	0.986289i	$-0.447228\pi$
$787$	9.25926	0.330057	0.165028	+	0.986289i	$0.447228\pi$
$788$	123.183	4.38820
$789$	−20.4369	−0.727573
$790$	59.1070	2.10293
$791$	−57.6602	−2.05016
$792$	14.7895	0.525520
$793$	11.5129	0.408834
$794$	−44.6485	−1.58452
$795$	−1.02317	−0.0362882
$796$	28.9234	1.02516
$797$	−43.0874	−1.52623	−0.763116	−	0.646261i	$-0.776332\pi$
$797$	−43.0874	−1.52623	−0.763116	+	0.646261i	$0.776332\pi$
$798$	−46.8528	−1.65857
$799$	−9.17884	−0.324724
$800$	−11.3269	−0.400467
$801$	−17.7127	−0.625847
$802$	65.3776	2.30856
$803$	1.59118	0.0561514
$804$	−25.1249	−0.886087
$805$	−35.4241	−1.24854
$806$	−46.9782	−1.65473
$807$	6.86881	0.241793
$808$	54.8706	1.93034
$809$	−23.3182	−0.819824	−0.409912	−	0.912125i	$-0.634441\pi$
$809$	−23.3182	−0.819824	−0.409912	+	0.912125i	$0.634441\pi$
$810$	−12.2195	−0.429348
$811$	−38.2639	−1.34363	−0.671814	−	0.740720i	$-0.734484\pi$
$811$	−38.2639	−1.34363	−0.671814	+	0.740720i	$0.734484\pi$
$812$	106.820	3.74865
$813$	26.2466	0.920507
$814$	38.0806	1.33472
$815$	1.72976	0.0605909
$816$	−30.9205	−1.08243
$817$	−12.2768	−0.429510
$818$	−15.3250	−0.535826
$819$	10.7057	0.374088
$820$	−84.5292	−2.95189
$821$	17.6521	0.616063	0.308031	−	0.951376i	$-0.400330\pi$
$821$	17.6521	0.616063	0.308031	+	0.951376i	$0.400330\pi$
$822$	−43.2387	−1.50812
$823$	−32.9503	−1.14858	−0.574288	−	0.818653i	$-0.694721\pi$
$823$	−32.9503	−1.14858	−0.574288	+	0.818653i	$0.694721\pi$
$824$	11.1671	0.389025
$825$	−1.24460	−0.0433315
$826$	−108.325	−3.76910
$827$	−2.28102	−0.0793188	−0.0396594	−	0.999213i	$-0.512627\pi$
$827$	−2.28102	−0.0793188	−0.0396594	+	0.999213i	$0.512627\pi$
$828$	−36.9319	−1.28347
$829$	40.4311	1.40423	0.702116	−	0.712063i	$-0.252239\pi$
$829$	40.4311	1.40423	0.702116	+	0.712063i	$0.252239\pi$
$830$	−28.0510	−0.973666
$831$	−32.4452	−1.12551
$832$	−28.6104	−0.991887
$833$	5.82707	0.201896
$834$	52.7823	1.82770
$835$	4.67446	0.161766
$836$	−28.7047	−0.992771
$837$	−43.5299	−1.50461
$838$	−38.8463	−1.34192
$839$	2.75936	0.0952637	0.0476318	−	0.998865i	$-0.484833\pi$
$839$	2.75936	0.0952637	0.0476318	+	0.998865i	$0.484833\pi$
$840$	−70.9154	−2.44681
$841$	19.2697	0.664474
$842$	−96.9207	−3.34011
$843$	−17.6673	−0.608494
$844$	75.8573	2.61112
$845$	19.3927	0.667128
$846$	−16.0753	−0.552679
$847$	−29.4324	−1.01131
$848$	−3.85106	−0.132246
$849$	−13.2649	−0.455251
$850$	−5.36447	−0.184000
$851$	−57.0087	−1.95423
$852$	14.6581	0.502180
$853$	−33.9149	−1.16122	−0.580611	−	0.814181i	$-0.697186\pi$
$853$	−33.9149	−1.16122	−0.580611	+	0.814181i	$0.697186\pi$
$854$	41.9595	1.43582
$855$	−18.0495	−0.617280
$856$	2.25703	0.0771437
$857$	−3.42254	−0.116912	−0.0584558	−	0.998290i	$-0.518618\pi$
$857$	−3.42254	−0.116912	−0.0584558	+	0.998290i	$0.518618\pi$
$858$	−8.51971	−0.290858
$859$	−49.9581	−1.70455	−0.852274	−	0.523095i	$-0.824777\pi$
$859$	−49.9581	−1.70455	−0.852274	+	0.523095i	$0.824777\pi$
$860$	−30.9956	−1.05694
$861$	−25.8598	−0.881301
$862$	−98.7020	−3.36180
$863$	−45.6509	−1.55397	−0.776987	−	0.629517i	$-0.783253\pi$
$863$	−45.6509	−1.55397	−0.776987	+	0.629517i	$0.783253\pi$
$864$	−71.8452	−2.44422
$865$	−46.5219	−1.58179
$866$	−76.0817	−2.58536
$867$	13.7887	0.468287
$868$	−122.253	−4.14954
$869$	11.0783	0.375807
$870$	−53.4533	−1.81224
$871$	−9.35527	−0.316991
$872$	32.5091	1.10090
$873$	−9.57124	−0.323937
$874$	60.1830	2.03572
$875$	30.8410	1.04262
$876$	−7.95444	−0.268756
$877$	−23.1234	−0.780822	−0.390411	−	0.920641i	$-0.627667\pi$
$877$	−23.1234	−0.780822	−0.390411	+	0.920641i	$0.627667\pi$
$878$	60.0465	2.02647
$879$	1.42834	0.0481768
$880$	−31.8222	−1.07273
$881$	−22.7052	−0.764959	−0.382479	−	0.923964i	$-0.624930\pi$
$881$	−22.7052	−0.764959	−0.382479	+	0.923964i	$0.624930\pi$
$882$	10.2052	0.343626
$883$	−26.4642	−0.890593	−0.445296	−	0.895383i	$-0.646902\pi$
$883$	−26.4642	−0.890593	−0.445296	+	0.895383i	$0.646902\pi$
$884$	−26.2203	−0.881883
$885$	38.7050	1.30105
$886$	−34.7990	−1.16909
$887$	−15.5374	−0.521695	−0.260848	−	0.965380i	$-0.584002\pi$
$887$	−15.5374	−0.521695	−0.260848	+	0.965380i	$0.584002\pi$
$888$	−114.126	−3.82980
$889$	26.8702	0.901198
$890$	72.8742	2.44275
$891$	−2.29027	−0.0767271
$892$	−92.1132	−3.08418
$893$	18.7045	0.625923
$894$	−25.0268	−0.837022
$895$	10.5168	0.351539
$896$	−23.4613	−0.783787
$897$	12.7545	0.425859
$898$	−47.3035	−1.57854
$899$	−55.2436	−1.84248
$900$	−6.70831	−0.223610
$901$	−0.826573	−0.0275371
$902$	−22.1884	−0.738793
$903$	−9.48243	−0.315555
$904$	148.271	4.93142
$905$	−35.5030	−1.18016
$906$	−65.2428	−2.16755
$907$	−29.2826	−0.972313	−0.486156	−	0.873872i	$-0.661601\pi$
$907$	−29.2826	−0.972313	−0.486156	+	0.873872i	$0.661601\pi$
$908$	−103.853	−3.44648
$909$	10.7870	0.357781
$910$	−44.0458	−1.46010
$911$	−39.4076	−1.30563	−0.652816	−	0.757517i	$-0.726412\pi$
$911$	−39.4076	−1.30563	−0.652816	+	0.757517i	$0.726412\pi$
$912$	63.0093	2.08645
$913$	−5.25756	−0.174000
$914$	91.9310	3.04081
$915$	−14.9923	−0.495631
$916$	−10.5964	−0.350116
$917$	−44.3740	−1.46536
$918$	−34.0261	−1.12303
$919$	32.6058	1.07557	0.537784	−	0.843083i	$-0.319262\pi$
$919$	32.6058	1.07557	0.537784	+	0.843083i	$0.319262\pi$
$920$	91.0917	3.00321
$921$	−28.7212	−0.946395
$922$	−93.1007	−3.06611
$923$	5.45796	0.179651
$924$	−22.1711	−0.729377
$925$	−10.3551	−0.340472
$926$	22.4431	0.737527
$927$	2.19534	0.0721043
$928$	−91.1783	−2.99307
$929$	−33.6850	−1.10517	−0.552585	−	0.833457i	$-0.686358\pi$
$929$	−33.6850	−1.10517	−0.552585	+	0.833457i	$0.686358\pi$
$930$	61.1761	2.00604
$931$	−11.8743	−0.389165
$932$	−81.4621	−2.66838
$933$	6.28099	0.205630
$934$	−84.4865	−2.76448
$935$	−6.83017	−0.223370
$936$	−27.5293	−0.899823
$937$	−37.3626	−1.22058	−0.610291	−	0.792177i	$-0.708948\pi$
$937$	−37.3626	−1.22058	−0.610291	+	0.792177i	$0.708948\pi$
$938$	−34.0960	−1.11327
$939$	−28.8532	−0.941590
$940$	47.2241	1.54028
$941$	18.0045	0.586930	0.293465	−	0.955970i	$-0.405192\pi$
$941$	18.0045	0.586930	0.293465	+	0.955970i	$0.405192\pi$
$942$	41.4724	1.35124
$943$	33.2173	1.08170
$944$	145.679	4.74145
$945$	−40.8128	−1.32764
$946$	−8.13617	−0.264530
$947$	−18.1699	−0.590443	−0.295221	−	0.955429i	$-0.595393\pi$
$947$	−18.1699	−0.590443	−0.295221	+	0.955429i	$0.595393\pi$
$948$	−55.3817	−1.79871
$949$	−2.96184	−0.0961454
$950$	10.9316	0.354669
$951$	−36.4700	−1.18262
$952$	−57.2891	−1.85675
$953$	56.2145	1.82097	0.910483	−	0.413546i	$-0.135710\pi$
$953$	56.2145	1.82097	0.910483	+	0.413546i	$0.135710\pi$
$954$	−1.44761	−0.0468680
$955$	17.8341	0.577097
$956$	1.14355	0.0369851
$957$	−10.0187	−0.323858
$958$	74.6172	2.41077
$959$	−41.8975	−1.35294
$960$	37.2571	1.20247
$961$	32.2250	1.03951
$962$	−70.8837	−2.28538
$963$	0.443708	0.0142983
$964$	−106.245	−3.42193
$965$	−65.3805	−2.10467
$966$	46.4846	1.49562
$967$	−10.4954	−0.337508	−0.168754	−	0.985658i	$-0.553974\pi$
$967$	−10.4954	−0.337508	−0.168754	+	0.985658i	$0.553974\pi$
$968$	75.6842	2.43258
$969$	13.5240	0.434454
$970$	39.3783	1.26436
$971$	−7.10764	−0.228095	−0.114047	−	0.993475i	$-0.536382\pi$
$971$	−7.10764	−0.228095	−0.114047	+	0.993475i	$0.536382\pi$
$972$	−70.5653	−2.26338
$973$	51.1450	1.63963
$974$	85.2111	2.73034
$975$	2.31672	0.0741945
$976$	−56.4286	−1.80624
$977$	−6.36315	−0.203575	−0.101788	−	0.994806i	$-0.532456\pi$
$977$	−6.36315	−0.203575	−0.101788	+	0.994806i	$0.532456\pi$
$978$	−2.26985	−0.0725818
$979$	13.6587	0.436534
$980$	−29.9796	−0.957663
$981$	6.39095	0.204047
$982$	55.6003	1.77428
$983$	61.8334	1.97218	0.986089	−	0.166217i	$-0.0531551\pi$
$983$	61.8334	1.97218	0.986089	+	0.166217i	$0.0531551\pi$
$984$	66.4975	2.11986
$985$	−59.7293	−1.90313
$986$	−43.1824	−1.37521
$987$	14.4471	0.459858
$988$	53.4313	1.69988
$989$	12.1803	0.387311
$990$	−11.9619	−0.380175
$991$	26.3601	0.837357	0.418679	−	0.908134i	$-0.362493\pi$
$991$	26.3601	0.837357	0.418679	+	0.908134i	$0.362493\pi$
$992$	104.351	3.31316
$993$	4.69202	0.148897
$994$	19.8920	0.630935
$995$	−14.0245	−0.444605
$996$	26.2831	0.832811
$997$	−48.2750	−1.52888	−0.764442	−	0.644692i	$-0.776986\pi$
$997$	−48.2750	−1.52888	−0.764442	+	0.644692i	$0.776986\pi$
$998$	78.2749	2.47775
$999$	−65.6808	−2.07805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.a.1.10	✓	279

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.a.1.10	✓	279	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)

\(f(q)\)	\(=\)	\(q-2.64457 q^{2} -1.20149 q^{3} +4.99374 q^{4} -2.42138 q^{5} +3.17741 q^{6} +3.07885 q^{7} -7.91715 q^{8} -1.55643 q^{9} +O(q^{10})\)	\(q-2.64457 q^{2} -1.20149 q^{3} +4.99374 q^{4} -2.42138 q^{5} +3.17741 q^{6} +3.07885 q^{7} -7.91715 q^{8} -1.55643 q^{9} +6.40351 q^{10} +1.20020 q^{11} -5.99992 q^{12} -2.23407 q^{13} -8.14224 q^{14} +2.90926 q^{15} +10.9500 q^{16} +2.35025 q^{17} +4.11608 q^{18} -4.78931 q^{19} -12.0918 q^{20} -3.69920 q^{21} -3.17401 q^{22} +4.75167 q^{23} +9.51236 q^{24} +0.863093 q^{25} +5.90815 q^{26} +5.47449 q^{27} +15.3750 q^{28} +6.94764 q^{29} -7.69374 q^{30} -7.95141 q^{31} -13.1236 q^{32} -1.44203 q^{33} -6.21540 q^{34} -7.45508 q^{35} -7.77240 q^{36} -11.9976 q^{37} +12.6656 q^{38} +2.68421 q^{39} +19.1705 q^{40} +6.99065 q^{41} +9.78280 q^{42} +2.56337 q^{43} +5.99349 q^{44} +3.76871 q^{45} -12.5661 q^{46} -3.90548 q^{47} -13.1562 q^{48} +2.47934 q^{49} -2.28251 q^{50} -2.82380 q^{51} -11.1564 q^{52} -0.351696 q^{53} -14.4777 q^{54} -2.90614 q^{55} -24.3758 q^{56} +5.75429 q^{57} -18.3735 q^{58} +13.3041 q^{59} +14.5281 q^{60} -5.15331 q^{61} +21.0281 q^{62} -4.79202 q^{63} +12.8064 q^{64} +5.40954 q^{65} +3.81353 q^{66} +4.18754 q^{67} +11.7365 q^{68} -5.70907 q^{69} +19.7155 q^{70} -2.44306 q^{71} +12.3225 q^{72} +1.32576 q^{73} +31.7285 q^{74} -1.03700 q^{75} -23.9166 q^{76} +3.69524 q^{77} -7.09857 q^{78} +9.23041 q^{79} -26.5141 q^{80} -1.90824 q^{81} -18.4872 q^{82} -4.38057 q^{83} -18.4729 q^{84} -5.69085 q^{85} -6.77901 q^{86} -8.34750 q^{87} -9.50217 q^{88} +11.3803 q^{89} -9.96661 q^{90} -6.87838 q^{91} +23.7286 q^{92} +9.55352 q^{93} +10.3283 q^{94} +11.5967 q^{95} +15.7679 q^{96} +6.14949 q^{97} -6.55679 q^{98} -1.86803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * q^99

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