LMFDB - Embedded newform 4031.2.a.c.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23009 q^{2} +1.62667 q^{3} +2.97329 q^{4} +2.06861 q^{5} -3.62762 q^{6} +3.32810 q^{7} -2.17052 q^{8} -0.353941 q^{9} +O(q^{10})$	\(q-2.23009 q^{2} +1.62667 q^{3} +2.97329 q^{4} +2.06861 q^{5} -3.62762 q^{6} +3.32810 q^{7} -2.17052 q^{8} -0.353941 q^{9} -4.61317 q^{10} -2.00414 q^{11} +4.83656 q^{12} -4.84916 q^{13} -7.42196 q^{14} +3.36494 q^{15} -1.10613 q^{16} -1.62307 q^{17} +0.789318 q^{18} -7.82976 q^{19} +6.15056 q^{20} +5.41373 q^{21} +4.46940 q^{22} +1.63989 q^{23} -3.53072 q^{24} -0.720871 q^{25} +10.8140 q^{26} -5.45576 q^{27} +9.89541 q^{28} -1.00000 q^{29} -7.50411 q^{30} +6.32579 q^{31} +6.80781 q^{32} -3.26007 q^{33} +3.61959 q^{34} +6.88453 q^{35} -1.05237 q^{36} +5.26453 q^{37} +17.4610 q^{38} -7.88798 q^{39} -4.48995 q^{40} -10.6234 q^{41} -12.0731 q^{42} +2.54793 q^{43} -5.95888 q^{44} -0.732163 q^{45} -3.65711 q^{46} -0.424949 q^{47} -1.79931 q^{48} +4.07626 q^{49} +1.60761 q^{50} -2.64020 q^{51} -14.4179 q^{52} -6.57658 q^{53} +12.1668 q^{54} -4.14577 q^{55} -7.22371 q^{56} -12.7364 q^{57} +2.23009 q^{58} +11.3340 q^{59} +10.0049 q^{60} +8.32864 q^{61} -14.1071 q^{62} -1.17795 q^{63} -12.9697 q^{64} -10.0310 q^{65} +7.27025 q^{66} -5.61260 q^{67} -4.82586 q^{68} +2.66757 q^{69} -15.3531 q^{70} -12.4573 q^{71} +0.768235 q^{72} -13.5411 q^{73} -11.7404 q^{74} -1.17262 q^{75} -23.2801 q^{76} -6.66997 q^{77} +17.5909 q^{78} -1.51687 q^{79} -2.28815 q^{80} -7.81290 q^{81} +23.6911 q^{82} -8.94568 q^{83} +16.0966 q^{84} -3.35749 q^{85} -5.68210 q^{86} -1.62667 q^{87} +4.35002 q^{88} +9.67158 q^{89} +1.63279 q^{90} -16.1385 q^{91} +4.87588 q^{92} +10.2900 q^{93} +0.947674 q^{94} -16.1967 q^{95} +11.0741 q^{96} +15.2303 q^{97} -9.09041 q^{98} +0.709345 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	−2.23009	−1.57691	−0.788455	−	0.615093i	$-0.789119\pi$
$2$	−2.23009	−1.57691	−0.788455	+	0.615093i	$0.789119\pi$
$3$	1.62667	0.939159	0.469580	−	0.882890i	$-0.344406\pi$
$3$	1.62667	0.939159	0.469580	+	0.882890i	$0.344406\pi$
$4$	2.97329	1.48664
$5$	2.06861	0.925109	0.462554	−	0.886591i	$-0.346933\pi$
$5$	2.06861	0.925109	0.462554	+	0.886591i	$0.346933\pi$
$6$	−3.62762	−1.48097
$7$	3.32810	1.25790	0.628952	−	0.777444i	$-0.283484\pi$
$7$	3.32810	1.25790	0.628952	+	0.777444i	$0.283484\pi$
$8$	−2.17052	−0.767394
$9$	−0.353941	−0.117980
$10$	−4.61317	−1.45881
$11$	−2.00414	−0.604270	−0.302135	−	0.953265i	$-0.597699\pi$
$11$	−2.00414	−0.604270	−0.302135	+	0.953265i	$0.597699\pi$
$12$	4.83656	1.39620
$13$	−4.84916	−1.34491	−0.672457	−	0.740136i	$-0.734761\pi$
$13$	−4.84916	−1.34491	−0.672457	+	0.740136i	$0.734761\pi$
$14$	−7.42196	−1.98360
$15$	3.36494	0.868824
$16$	−1.10613	−0.276533
$17$	−1.62307	−0.393652	−0.196826	−	0.980438i	$-0.563063\pi$
$17$	−1.62307	−0.393652	−0.196826	+	0.980438i	$0.563063\pi$
$18$	0.789318	0.186044
$19$	−7.82976	−1.79627	−0.898135	−	0.439720i	$-0.855078\pi$
$19$	−7.82976	−1.79627	−0.898135	+	0.439720i	$0.855078\pi$
$20$	6.15056	1.37531
$21$	5.41373	1.18137
$22$	4.46940	0.952879
$23$	1.63989	0.341942	0.170971	−	0.985276i	$-0.445310\pi$
$23$	1.63989	0.341942	0.170971	+	0.985276i	$0.445310\pi$
$24$	−3.53072	−0.720706
$25$	−0.720871	−0.144174
$26$	10.8140	2.12081
$27$	−5.45576	−1.04996
$28$	9.89541	1.87006
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−7.50411	−1.37006
$31$	6.32579	1.13615	0.568073	−	0.822978i	$-0.307689\pi$
$31$	6.32579	1.13615	0.568073	+	0.822978i	$0.307689\pi$
$32$	6.80781	1.20346
$33$	−3.26007	−0.567506
$34$	3.61959	0.620754
$35$	6.88453	1.16370
$36$	−1.05237	−0.175395
$37$	5.26453	0.865483	0.432742	−	0.901518i	$-0.357546\pi$
$37$	5.26453	0.865483	0.432742	+	0.901518i	$0.357546\pi$
$38$	17.4610	2.83256
$39$	−7.88798	−1.26309
$40$	−4.48995	−0.709923
$41$	−10.6234	−1.65910	−0.829548	−	0.558436i	$-0.811402\pi$
$41$	−10.6234	−1.65910	−0.829548	+	0.558436i	$0.811402\pi$
$42$	−12.0731	−1.86292
$43$	2.54793	0.388556	0.194278	−	0.980947i	$-0.437764\pi$
$43$	2.54793	0.388556	0.194278	+	0.980947i	$0.437764\pi$
$44$	−5.95888	−0.898335
$45$	−0.732163	−0.109144
$46$	−3.65711	−0.539211
$47$	−0.424949	−0.0619852	−0.0309926	−	0.999520i	$-0.509867\pi$
$47$	−0.424949	−0.0619852	−0.0309926	+	0.999520i	$0.509867\pi$
$48$	−1.79931	−0.259708
$49$	4.07626	0.582323
$50$	1.60761	0.227350
$51$	−2.64020	−0.369702
$52$	−14.4179	−1.99941
$53$	−6.57658	−0.903363	−0.451682	−	0.892179i	$-0.649176\pi$
$53$	−6.57658	−0.903363	−0.451682	+	0.892179i	$0.649176\pi$
$54$	12.1668	1.65569
$55$	−4.14577	−0.559015
$56$	−7.22371	−0.965309
$57$	−12.7364	−1.68698
$58$	2.23009	0.292825
$59$	11.3340	1.47556	0.737778	−	0.675043i	$-0.235875\pi$
$59$	11.3340	1.47556	0.737778	+	0.675043i	$0.235875\pi$
$60$	10.0049	1.29163
$61$	8.32864	1.06637	0.533186	−	0.845998i	$-0.320994\pi$
$61$	8.32864	1.06637	0.533186	+	0.845998i	$0.320994\pi$
$62$	−14.1071	−1.79160
$63$	−1.17795	−0.148408
$64$	−12.9697	−1.62122
$65$	−10.0310	−1.24419
$66$	7.27025	0.894905
$67$	−5.61260	−0.685689	−0.342844	−	0.939392i	$-0.611390\pi$
$67$	−5.61260	−0.685689	−0.342844	+	0.939392i	$0.611390\pi$
$68$	−4.82586	−0.585221
$69$	2.66757	0.321138
$70$	−15.3531	−1.83505
$71$	−12.4573	−1.47841	−0.739206	−	0.673480i	$-0.764799\pi$
$71$	−12.4573	−1.47841	−0.739206	+	0.673480i	$0.764799\pi$
$72$	0.768235	0.0905373
$73$	−13.5411	−1.58487	−0.792433	−	0.609959i	$-0.791186\pi$
$73$	−13.5411	−1.58487	−0.792433	+	0.609959i	$0.791186\pi$
$74$	−11.7404	−1.36479
$75$	−1.17262	−0.135403
$76$	−23.2801	−2.67041
$77$	−6.66997	−0.760114
$78$	17.5909	1.99178
$79$	−1.51687	−0.170661	−0.0853306	−	0.996353i	$-0.527195\pi$
$79$	−1.51687	−0.170661	−0.0853306	+	0.996353i	$0.527195\pi$
$80$	−2.28815	−0.255823
$81$	−7.81290	−0.868100
$82$	23.6911	2.61624
$83$	−8.94568	−0.981916	−0.490958	−	0.871183i	$-0.663353\pi$
$83$	−8.94568	−0.981916	−0.490958	+	0.871183i	$0.663353\pi$
$84$	16.0966	1.75628
$85$	−3.35749	−0.364171
$86$	−5.68210	−0.612717
$87$	−1.62667	−0.174397
$88$	4.35002	0.463714
$89$	9.67158	1.02519	0.512593	−	0.858632i	$-0.328685\pi$
$89$	9.67158	1.02519	0.512593	+	0.858632i	$0.328685\pi$
$90$	1.63279	0.172111
$91$	−16.1385	−1.69177
$92$	4.87588	0.508346
$93$	10.2900	1.06702
$94$	0.947674	0.0977451
$95$	−16.1967	−1.66174
$96$	11.0741	1.13024
$97$	15.2303	1.54640	0.773202	−	0.634160i	$-0.218654\pi$
$97$	15.2303	1.54640	0.773202	+	0.634160i	$0.218654\pi$
$98$	−9.09041	−0.918270
$99$	0.709345	0.0712919
$100$	−2.14336	−0.214336
$101$	4.31572	0.429430	0.214715	−	0.976677i	$-0.431118\pi$
$101$	4.31572	0.429430	0.214715	+	0.976677i	$0.431118\pi$
$102$	5.88788	0.582987
$103$	−9.10379	−0.897023	−0.448511	−	0.893777i	$-0.648046\pi$
$103$	−9.10379	−0.897023	−0.448511	+	0.893777i	$0.648046\pi$
$104$	10.5252	1.03208
$105$	11.1989	1.09290
$106$	14.6664	1.42452
$107$	−5.45640	−0.527490	−0.263745	−	0.964592i	$-0.584958\pi$
$107$	−5.45640	−0.527490	−0.263745	+	0.964592i	$0.584958\pi$
$108$	−16.2215	−1.56092
$109$	−3.83694	−0.367512	−0.183756	−	0.982972i	$-0.558826\pi$
$109$	−3.83694	−0.367512	−0.183756	+	0.982972i	$0.558826\pi$
$110$	9.24543	0.881517
$111$	8.56366	0.812827
$112$	−3.68131	−0.347852
$113$	−0.301048	−0.0283202	−0.0141601	−	0.999900i	$-0.504507\pi$
$113$	−0.301048	−0.0283202	−0.0141601	+	0.999900i	$0.504507\pi$
$114$	28.4034	2.66022
$115$	3.39230	0.316333
$116$	−2.97329	−0.276063
$117$	1.71631	0.158673
$118$	−25.2757	−2.32682
$119$	−5.40174	−0.495177
$120$	−7.30367	−0.666731
$121$	−6.98343	−0.634858
$122$	−18.5736	−1.68157
$123$	−17.2808	−1.55815
$124$	18.8084	1.68904
$125$	−11.8342	−1.05849
$126$	2.62693	0.234026
$127$	−15.6010	−1.38436	−0.692181	−	0.721724i	$-0.743350\pi$
$127$	−15.6010	−1.38436	−0.692181	+	0.721724i	$0.743350\pi$
$128$	15.3080	1.35305
$129$	4.14464	0.364915
$130$	22.3700	1.96198
$131$	−2.18610	−0.191000	−0.0955002	−	0.995429i	$-0.530445\pi$
$131$	−2.18610	−0.191000	−0.0955002	+	0.995429i	$0.530445\pi$
$132$	−9.69314	−0.843679
$133$	−26.0582	−2.25953
$134$	12.5166	1.08127
$135$	−11.2858	−0.971328
$136$	3.52291	0.302087
$137$	6.34961	0.542484	0.271242	−	0.962511i	$-0.412566\pi$
$137$	6.34961	0.542484	0.271242	+	0.962511i	$0.412566\pi$
$138$	−5.94891	−0.506405
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	20.4697	1.73000
$141$	−0.691253	−0.0582140
$142$	27.7809	2.33132
$143$	9.71837	0.812691
$144$	0.391504	0.0326254
$145$	−2.06861	−0.171788
$146$	30.1978	2.49919
$147$	6.63073	0.546894
$148$	15.6530	1.28667
$149$	10.8391	0.887976	0.443988	−	0.896033i	$-0.353563\pi$
$149$	10.8391	0.887976	0.443988	+	0.896033i	$0.353563\pi$
$150$	2.61505	0.213518
$151$	−11.3195	−0.921169	−0.460585	−	0.887616i	$-0.652360\pi$
$151$	−11.3195	−0.921169	−0.460585	+	0.887616i	$0.652360\pi$
$152$	16.9946	1.37845
$153$	0.574471	0.0464432
$154$	14.8746	1.19863
$155$	13.0856	1.05106
$156$	−23.4533	−1.87776
$157$	3.51999	0.280926	0.140463	−	0.990086i	$-0.455141\pi$
$157$	3.51999	0.280926	0.140463	+	0.990086i	$0.455141\pi$
$158$	3.38275	0.269117
$159$	−10.6979	−0.848402
$160$	14.0827	1.11333
$161$	5.45774	0.430130
$162$	17.4235	1.36892
$163$	−20.4997	−1.60566	−0.802830	−	0.596208i	$-0.796673\pi$
$163$	−20.4997	−1.60566	−0.802830	+	0.596208i	$0.796673\pi$
$164$	−31.5864	−2.46649
$165$	−6.74380	−0.525004
$166$	19.9497	1.54839
$167$	3.20714	0.248176	0.124088	−	0.992271i	$-0.460400\pi$
$167$	3.20714	0.248176	0.124088	+	0.992271i	$0.460400\pi$
$168$	−11.7506	−0.906578
$169$	10.5143	0.808794
$170$	7.48750	0.574265
$171$	2.77127	0.211924
$172$	7.57573	0.577644
$173$	−9.90703	−0.753217	−0.376609	−	0.926372i	$-0.622910\pi$
$173$	−9.90703	−0.753217	−0.376609	+	0.926372i	$0.622910\pi$
$174$	3.62762	0.275009
$175$	−2.39913	−0.181357
$176$	2.21684	0.167100
$177$	18.4366	1.38578
$178$	−21.5685	−1.61662
$179$	21.6756	1.62011	0.810056	−	0.586352i	$-0.199436\pi$
$179$	21.6756	1.62011	0.810056	+	0.586352i	$0.199436\pi$
$180$	−2.17693	−0.162259
$181$	−11.3952	−0.846999	−0.423499	−	0.905896i	$-0.639198\pi$
$181$	−11.3952	−0.846999	−0.423499	+	0.905896i	$0.639198\pi$
$182$	35.9902	2.66777
$183$	13.5480	1.00149
$184$	−3.55942	−0.262404
$185$	10.8902	0.800666
$186$	−22.9476	−1.68260
$187$	3.25286	0.237872
$188$	−1.26350	−0.0921500
$189$	−18.1573	−1.32075
$190$	36.1200	2.62042
$191$	22.3658	1.61833	0.809166	−	0.587580i	$-0.199919\pi$
$191$	22.3658	1.61833	0.809166	+	0.587580i	$0.199919\pi$
$192$	−21.0975	−1.52258
$193$	0.808022	0.0581627	0.0290814	−	0.999577i	$-0.490742\pi$
$193$	0.808022	0.0581627	0.0290814	+	0.999577i	$0.490742\pi$
$194$	−33.9649	−2.43854
$195$	−16.3171	−1.16849
$196$	12.1199	0.865707
$197$	0.534542	0.0380845	0.0190423	−	0.999819i	$-0.493938\pi$
$197$	0.534542	0.0380845	0.0190423	+	0.999819i	$0.493938\pi$
$198$	−1.58190	−0.112421
$199$	−15.5310	−1.10096	−0.550482	−	0.834847i	$-0.685556\pi$
$199$	−15.5310	−1.10096	−0.550482	+	0.834847i	$0.685556\pi$
$200$	1.56466	0.110638
$201$	−9.12986	−0.643971
$202$	−9.62443	−0.677173
$203$	−3.32810	−0.233587
$204$	−7.85008	−0.549616
$205$	−21.9756	−1.53484
$206$	20.3022	1.41452
$207$	−0.580425	−0.0403423
$208$	5.36380	0.371913
$209$	15.6919	1.08543
$210$	−24.9744	−1.72340
$211$	12.2123	0.840727	0.420364	−	0.907356i	$-0.361902\pi$
$211$	12.2123	0.840727	0.420364	+	0.907356i	$0.361902\pi$
$212$	−19.5541	−1.34298
$213$	−20.2640	−1.38846
$214$	12.1683	0.831805
$215$	5.27066	0.359456
$216$	11.8418	0.805734
$217$	21.0529	1.42916
$218$	8.55670	0.579533
$219$	−22.0269	−1.48844
$220$	−12.3266	−0.831057
$221$	7.87052	0.529429
$222$	−19.0977	−1.28175
$223$	−22.0995	−1.47989	−0.739945	−	0.672667i	$-0.765149\pi$
$223$	−22.0995	−1.47989	−0.739945	+	0.672667i	$0.765149\pi$
$224$	22.6571	1.51384
$225$	0.255145	0.0170097
$226$	0.671362	0.0446583
$227$	−20.8932	−1.38673	−0.693365	−	0.720586i	$-0.743873\pi$
$227$	−20.8932	−1.38673	−0.693365	+	0.720586i	$0.743873\pi$
$228$	−37.8691	−2.50794
$229$	−15.1793	−1.00307	−0.501537	−	0.865136i	$-0.667232\pi$
$229$	−15.1793	−1.00307	−0.501537	+	0.865136i	$0.667232\pi$
$230$	−7.56511	−0.498829
$231$	−10.8499	−0.713868
$232$	2.17052	0.142502
$233$	16.1529	1.05821	0.529106	−	0.848556i	$-0.322527\pi$
$233$	16.1529	1.05821	0.529106	+	0.848556i	$0.322527\pi$
$234$	−3.82753	−0.250213
$235$	−0.879053	−0.0573431
$236$	33.6991	2.19363
$237$	−2.46745	−0.160278
$238$	12.0464	0.780849
$239$	8.09766	0.523794	0.261897	−	0.965096i	$-0.415652\pi$
$239$	8.09766	0.523794	0.261897	+	0.965096i	$0.415652\pi$
$240$	−3.72206	−0.240258
$241$	−25.6664	−1.65332	−0.826660	−	0.562702i	$-0.809762\pi$
$241$	−25.6664	−1.65332	−0.826660	+	0.562702i	$0.809762\pi$
$242$	15.5737	1.00111
$243$	3.65825	0.234677
$244$	24.7634	1.58532
$245$	8.43217	0.538712
$246$	38.5376	2.45707
$247$	37.9677	2.41583
$248$	−13.7302	−0.871872
$249$	−14.5517	−0.922176
$250$	26.3914	1.66914
$251$	−11.8184	−0.745971	−0.372985	−	0.927837i	$-0.621666\pi$
$251$	−11.8184	−0.745971	−0.372985	+	0.927837i	$0.621666\pi$
$252$	−3.50239	−0.220630
$253$	−3.28657	−0.206625
$254$	34.7915	2.18301
$255$	−5.46154	−0.342015
$256$	−8.19878	−0.512424
$257$	7.95527	0.496236	0.248118	−	0.968730i	$-0.420188\pi$
$257$	7.95527	0.496236	0.248118	+	0.968730i	$0.420188\pi$
$258$	−9.24292	−0.575439
$259$	17.5209	1.08869
$260$	−29.8250	−1.84967
$261$	0.353941	0.0219084
$262$	4.87519	0.301190
$263$	19.0798	1.17651	0.588254	−	0.808676i	$-0.299815\pi$
$263$	19.0798	1.17651	0.588254	+	0.808676i	$0.299815\pi$
$264$	7.07605	0.435501
$265$	−13.6044	−0.835709
$266$	58.1121	3.56308
$267$	15.7325	0.962812
$268$	−16.6879	−1.01938
$269$	20.3650	1.24168	0.620839	−	0.783938i	$-0.286792\pi$
$269$	20.3650	1.24168	0.620839	+	0.783938i	$0.286792\pi$
$270$	25.1683	1.53170
$271$	−12.0740	−0.733442	−0.366721	−	0.930331i	$-0.619520\pi$
$271$	−12.0740	−0.733442	−0.366721	+	0.930331i	$0.619520\pi$
$272$	1.79533	0.108858
$273$	−26.2520	−1.58884
$274$	−14.1602	−0.855448
$275$	1.44472	0.0871202
$276$	7.93146	0.477418
$277$	−30.9603	−1.86022	−0.930111	−	0.367279i	$-0.880289\pi$
$277$	−30.9603	−1.86022	−0.930111	+	0.367279i	$0.880289\pi$
$278$	2.23009	0.133752
$279$	−2.23895	−0.134043
$280$	−14.9430	−0.893015
$281$	17.0485	1.01703	0.508513	−	0.861054i	$-0.330195\pi$
$281$	17.0485	1.01703	0.508513	+	0.861054i	$0.330195\pi$
$282$	1.54155	0.0917982
$283$	11.4998	0.683595	0.341798	−	0.939774i	$-0.388964\pi$
$283$	11.4998	0.683595	0.341798	+	0.939774i	$0.388964\pi$
$284$	−37.0392	−2.19787
$285$	−26.3467	−1.56064
$286$	−21.6728	−1.28154
$287$	−35.3557	−2.08698
$288$	−2.40956	−0.141985
$289$	−14.3656	−0.845038
$290$	4.61317	0.270895
$291$	24.7747	1.45232
$292$	−40.2616	−2.35613
$293$	23.1786	1.35411	0.677055	−	0.735933i	$-0.263256\pi$
$293$	23.1786	1.35411	0.677055	+	0.735933i	$0.263256\pi$
$294$	−14.7871	−0.862402
$295$	23.4455	1.36505
$296$	−11.4268	−0.664167
$297$	10.9341	0.634460
$298$	−24.1722	−1.40026
$299$	−7.95211	−0.459882
$300$	−3.48654	−0.201295
$301$	8.47977	0.488766
$302$	25.2435	1.45260
$303$	7.02026	0.403303
$304$	8.66073	0.496727
$305$	17.2287	0.986511
$306$	−1.28112	−0.0732367
$307$	18.6235	1.06290	0.531451	−	0.847089i	$-0.321647\pi$
$307$	18.6235	1.06290	0.531451	+	0.847089i	$0.321647\pi$
$308$	−19.8318	−1.13002
$309$	−14.8089	−0.842447
$310$	−29.1819	−1.65742
$311$	−10.2082	−0.578854	−0.289427	−	0.957200i	$-0.593465\pi$
$311$	−10.2082	−0.578854	−0.289427	+	0.957200i	$0.593465\pi$
$312$	17.1210	0.969287
$313$	27.0218	1.52736	0.763682	−	0.645593i	$-0.223390\pi$
$313$	27.0218	1.52736	0.763682	+	0.645593i	$0.223390\pi$
$314$	−7.84989	−0.442995
$315$	−2.43671	−0.137293
$316$	−4.51009	−0.253713
$317$	−0.589377	−0.0331027	−0.0165514	−	0.999863i	$-0.505269\pi$
$317$	−0.589377	−0.0331027	−0.0165514	+	0.999863i	$0.505269\pi$
$318$	23.8573	1.33785
$319$	2.00414	0.112210
$320$	−26.8293	−1.49980
$321$	−8.87577	−0.495397
$322$	−12.1712	−0.678276
$323$	12.7083	0.707106
$324$	−23.2300	−1.29056
$325$	3.49562	0.193902
$326$	45.7161	2.53198
$327$	−6.24143	−0.345152
$328$	23.0583	1.27318
$329$	−1.41427	−0.0779715
$330$	15.0393	0.827885
$331$	3.15968	0.173672	0.0868359	−	0.996223i	$-0.472324\pi$
$331$	3.15968	0.173672	0.0868359	+	0.996223i	$0.472324\pi$
$332$	−26.5981	−1.45976
$333$	−1.86333	−0.102110
$334$	−7.15220	−0.391351
$335$	−11.6103	−0.634336
$336$	−5.98829	−0.326688
$337$	−23.3601	−1.27251	−0.636253	−	0.771480i	$-0.719517\pi$
$337$	−23.3601	−1.27251	−0.636253	+	0.771480i	$0.719517\pi$
$338$	−23.4479	−1.27540
$339$	−0.489705	−0.0265971
$340$	−9.98280	−0.541393
$341$	−12.6777	−0.686539
$342$	−6.18017	−0.334185
$343$	−9.73051	−0.525398
$344$	−5.53033	−0.298175
$345$	5.51815	0.297087
$346$	22.0935	1.18776
$347$	19.1623	1.02869	0.514343	−	0.857585i	$-0.328036\pi$
$347$	19.1623	1.02869	0.514343	+	0.857585i	$0.328036\pi$
$348$	−4.83656	−0.259267
$349$	−3.23030	−0.172914	−0.0864569	−	0.996256i	$-0.527554\pi$
$349$	−3.23030	−0.172914	−0.0864569	+	0.996256i	$0.527554\pi$
$350$	5.35027	0.285984
$351$	26.4558	1.41211
$352$	−13.6438	−0.727216
$353$	−2.26308	−0.120452	−0.0602259	−	0.998185i	$-0.519182\pi$
$353$	−2.26308	−0.120452	−0.0602259	+	0.998185i	$0.519182\pi$
$354$	−41.1153	−2.18525
$355$	−25.7693	−1.36769
$356$	28.7564	1.52409
$357$	−8.78686	−0.465050
$358$	−48.3385	−2.55477
$359$	−2.32848	−0.122893	−0.0614463	−	0.998110i	$-0.519571\pi$
$359$	−2.32848	−0.122893	−0.0614463	+	0.998110i	$0.519571\pi$
$360$	1.58917	0.0837569
$361$	42.3051	2.22658
$362$	25.4123	1.33564
$363$	−11.3598	−0.596232
$364$	−47.9844	−2.51506
$365$	−28.0112	−1.46617
$366$	−30.2131	−1.57927
$367$	28.0257	1.46293	0.731465	−	0.681879i	$-0.238837\pi$
$367$	28.0257	1.46293	0.731465	+	0.681879i	$0.238837\pi$
$368$	−1.81394	−0.0945580
$369$	3.76005	0.195740
$370$	−24.2862	−1.26258
$371$	−21.8875	−1.13634
$372$	30.5951	1.58628
$373$	14.2166	0.736110	0.368055	−	0.929804i	$-0.380024\pi$
$373$	14.2166	0.736110	0.368055	+	0.929804i	$0.380024\pi$
$374$	−7.25415	−0.375103
$375$	−19.2504	−0.994086
$376$	0.922361	0.0475671
$377$	4.84916	0.249744
$378$	40.4924	2.08270
$379$	7.51950	0.386251	0.193125	−	0.981174i	$-0.438138\pi$
$379$	7.51950	0.386251	0.193125	+	0.981174i	$0.438138\pi$
$380$	−48.1574	−2.47042
$381$	−25.3776	−1.30014
$382$	−49.8777	−2.55196
$383$	16.9232	0.864733	0.432367	−	0.901698i	$-0.357679\pi$
$383$	16.9232	0.864733	0.432367	+	0.901698i	$0.357679\pi$
$384$	24.9011	1.27073
$385$	−13.7975	−0.703188
$386$	−1.80196	−0.0917174
$387$	−0.901815	−0.0458419
$388$	45.2841	2.29895
$389$	−25.4599	−1.29087	−0.645433	−	0.763817i	$-0.723323\pi$
$389$	−25.4599	−1.29087	−0.645433	+	0.763817i	$0.723323\pi$
$390$	36.3886	1.84261
$391$	−2.66166	−0.134606
$392$	−8.84760	−0.446871
$393$	−3.55607	−0.179380
$394$	−1.19207	−0.0600559
$395$	−3.13781	−0.157880
$396$	2.10909	0.105986
$397$	−31.1844	−1.56510	−0.782549	−	0.622588i	$-0.786081\pi$
$397$	−31.1844	−1.56510	−0.782549	+	0.622588i	$0.786081\pi$
$398$	34.6355	1.73612
$399$	−42.3882	−2.12206
$400$	0.797377	0.0398689
$401$	16.1080	0.804397	0.402199	−	0.915552i	$-0.368246\pi$
$401$	16.1080	0.804397	0.402199	+	0.915552i	$0.368246\pi$
$402$	20.3604	1.01548
$403$	−30.6747	−1.52802
$404$	12.8319	0.638410
$405$	−16.1618	−0.803087
$406$	7.42196	0.368345
$407$	−10.5508	−0.522986
$408$	5.73061	0.283708
$409$	13.3481	0.660023	0.330011	−	0.943977i	$-0.392947\pi$
$409$	13.3481	0.660023	0.330011	+	0.943977i	$0.392947\pi$
$410$	49.0075	2.42031
$411$	10.3287	0.509479
$412$	−27.0682	−1.33355
$413$	37.7206	1.85611
$414$	1.29440	0.0636162
$415$	−18.5051	−0.908379
$416$	−33.0121	−1.61855
$417$	−1.62667	−0.0796584
$418$	−34.9943	−1.71163
$419$	11.9738	0.584960	0.292480	−	0.956272i	$-0.405520\pi$
$419$	11.9738	0.584960	0.292480	+	0.956272i	$0.405520\pi$
$420$	33.2975	1.62475
$421$	9.39326	0.457799	0.228900	−	0.973450i	$-0.426487\pi$
$421$	9.39326	0.457799	0.228900	+	0.973450i	$0.426487\pi$
$422$	−27.2344	−1.32575
$423$	0.150407	0.00731303
$424$	14.2746	0.693236
$425$	1.17002	0.0567545
$426$	45.1904	2.18948
$427$	27.7186	1.34139
$428$	−16.2235	−0.784191
$429$	15.8086	0.763246
$430$	−11.7540	−0.566830
$431$	−17.4723	−0.841613	−0.420806	−	0.907150i	$-0.638253\pi$
$431$	−17.4723	−0.841613	−0.420806	+	0.907150i	$0.638253\pi$
$432$	6.03478	0.290349
$433$	−17.6263	−0.847065	−0.423533	−	0.905881i	$-0.639210\pi$
$433$	−17.6263	−0.847065	−0.423533	+	0.905881i	$0.639210\pi$
$434$	−46.9497	−2.25366
$435$	−3.36494	−0.161337
$436$	−11.4083	−0.546360
$437$	−12.8400	−0.614219
$438$	49.1220	2.34714
$439$	23.7987	1.13585	0.567925	−	0.823080i	$-0.307746\pi$
$439$	23.7987	1.13585	0.567925	+	0.823080i	$0.307746\pi$
$440$	8.99847	0.428985
$441$	−1.44275	−0.0687025
$442$	−17.5520	−0.834861
$443$	−3.55883	−0.169085	−0.0845426	−	0.996420i	$-0.526943\pi$
$443$	−3.55883	−0.169085	−0.0845426	+	0.996420i	$0.526943\pi$
$444$	25.4622	1.20838
$445$	20.0067	0.948407
$446$	49.2838	2.33365
$447$	17.6317	0.833951
$448$	−43.1646	−2.03934
$449$	21.7192	1.02499	0.512496	−	0.858689i	$-0.328721\pi$
$449$	21.7192	1.02499	0.512496	+	0.858689i	$0.328721\pi$
$450$	−0.568997	−0.0268228
$451$	21.2907	1.00254
$452$	−0.895101	−0.0421020
$453$	−18.4131	−0.865125
$454$	46.5937	2.18675
$455$	−33.3842	−1.56507
$456$	27.6447	1.29458
$457$	−29.8607	−1.39683	−0.698413	−	0.715695i	$-0.746110\pi$
$457$	−29.8607	−1.39683	−0.698413	+	0.715695i	$0.746110\pi$
$458$	33.8511	1.58176
$459$	8.85508	0.413320
$460$	10.0863	0.470275
$461$	−36.6909	−1.70886	−0.854432	−	0.519562i	$-0.826095\pi$
$461$	−36.6909	−1.70886	−0.854432	+	0.519562i	$0.826095\pi$
$462$	24.1961	1.12571
$463$	−6.64217	−0.308688	−0.154344	−	0.988017i	$-0.549326\pi$
$463$	−6.64217	−0.308688	−0.154344	+	0.988017i	$0.549326\pi$
$464$	1.10613	0.0513508
$465$	21.2859	0.987110
$466$	−36.0224	−1.66870
$467$	−25.5035	−1.18016	−0.590079	−	0.807345i	$-0.700904\pi$
$467$	−25.5035	−1.18016	−0.590079	+	0.807345i	$0.700904\pi$
$468$	5.10310	0.235891
$469$	−18.6793	−0.862531
$470$	1.96036	0.0904248
$471$	5.72587	0.263834
$472$	−24.6006	−1.13233
$473$	−5.10640	−0.234792
$474$	5.50263	0.252744
$475$	5.64425	0.258976
$476$	−16.0609	−0.736152
$477$	2.32772	0.106579
$478$	−18.0585	−0.825976
$479$	2.85060	0.130247	0.0651236	−	0.997877i	$-0.479256\pi$
$479$	2.85060	0.130247	0.0651236	+	0.997877i	$0.479256\pi$
$480$	22.9079	1.04560
$481$	−25.5285	−1.16400
$482$	57.2384	2.60714
$483$	8.87794	0.403960
$484$	−20.7638	−0.943808
$485$	31.5055	1.43059
$486$	−8.15821	−0.370064
$487$	−13.9850	−0.633720	−0.316860	−	0.948472i	$-0.602629\pi$
$487$	−13.9850	−0.633720	−0.316860	+	0.948472i	$0.602629\pi$
$488$	−18.0775	−0.818329
$489$	−33.3463	−1.50797
$490$	−18.8045	−0.849500
$491$	−10.1823	−0.459520	−0.229760	−	0.973247i	$-0.573794\pi$
$491$	−10.1823	−0.459520	−0.229760	+	0.973247i	$0.573794\pi$
$492$	−51.3807	−2.31642
$493$	1.62307	0.0730994
$494$	−84.6713	−3.80954
$495$	1.46736	0.0659527
$496$	−6.99715	−0.314181
$497$	−41.4592	−1.85970
$498$	32.4515	1.45419
$499$	25.7520	1.15282	0.576409	−	0.817161i	$-0.304453\pi$
$499$	25.7520	1.15282	0.576409	+	0.817161i	$0.304453\pi$
$500$	−35.1866	−1.57359
$501$	5.21696	0.233077
$502$	26.3561	1.17633
$503$	−39.8631	−1.77741	−0.888703	−	0.458484i	$-0.848393\pi$
$503$	−39.8631	−1.77741	−0.888703	+	0.458484i	$0.848393\pi$
$504$	2.55676	0.113887
$505$	8.92752	0.397270
$506$	7.32935	0.325829
$507$	17.1033	0.759586
$508$	−46.3862	−2.05805
$509$	−23.0811	−1.02305	−0.511526	−	0.859268i	$-0.670920\pi$
$509$	−23.0811	−1.02305	−0.511526	+	0.859268i	$0.670920\pi$
$510$	12.1797	0.539326
$511$	−45.0662	−1.99361
$512$	−12.3321	−0.545006
$513$	42.7173	1.88601
$514$	−17.7410	−0.782520
$515$	−18.8321	−0.829843
$516$	12.3232	0.542500
$517$	0.851657	0.0374558
$518$	−39.0731	−1.71677
$519$	−16.1155	−0.707391
$520$	21.7725	0.954786
$521$	6.04798	0.264967	0.132483	−	0.991185i	$-0.457705\pi$
$521$	6.04798	0.264967	0.132483	+	0.991185i	$0.457705\pi$
$522$	−0.789318	−0.0345475
$523$	12.7479	0.557429	0.278714	−	0.960374i	$-0.410092\pi$
$523$	12.7479	0.557429	0.278714	+	0.960374i	$0.410092\pi$
$524$	−6.49991	−0.283950
$525$	−3.90260	−0.170323
$526$	−42.5495	−1.85525
$527$	−10.2672	−0.447246
$528$	3.60607	0.156934
$529$	−20.3107	−0.883076
$530$	30.3389	1.31784
$531$	−4.01155	−0.174086
$532$	−77.4786	−3.35912
$533$	51.5145	2.23134
$534$	−35.0848	−1.51827
$535$	−11.2871	−0.487986
$536$	12.1823	0.526194
$537$	35.2591	1.52154
$538$	−45.4158	−1.95801
$539$	−8.16938	−0.351880
$540$	−33.5560	−1.44402
$541$	17.3732	0.746934	0.373467	−	0.927643i	$-0.378169\pi$
$541$	17.3732	0.746934	0.373467	+	0.927643i	$0.378169\pi$
$542$	26.9260	1.15657
$543$	−18.5362	−0.795466
$544$	−11.0496	−0.473746
$545$	−7.93711	−0.339988
$546$	58.5443	2.50546
$547$	−16.9118	−0.723094	−0.361547	−	0.932354i	$-0.617751\pi$
$547$	−16.9118	−0.723094	−0.361547	+	0.932354i	$0.617751\pi$
$548$	18.8792	0.806481
$549$	−2.94784	−0.125811
$550$	−3.22186	−0.137381
$551$	7.82976	0.333559
$552$	−5.79001	−0.246439
$553$	−5.04830	−0.214675
$554$	69.0441	2.93340
$555$	17.7148	0.751953
$556$	−2.97329	−0.126096
$557$	34.1912	1.44873	0.724363	−	0.689419i	$-0.242134\pi$
$557$	34.1912	1.44873	0.724363	+	0.689419i	$0.242134\pi$
$558$	4.99306	0.211373
$559$	−12.3553	−0.522574
$560$	−7.61519	−0.321800
$561$	5.29133	0.223400
$562$	−38.0195	−1.60376
$563$	5.85329	0.246687	0.123343	−	0.992364i	$-0.460638\pi$
$563$	5.85329	0.246687	0.123343	+	0.992364i	$0.460638\pi$
$564$	−2.05529	−0.0865435
$565$	−0.622749	−0.0261992
$566$	−25.6457	−1.07797
$567$	−26.0021	−1.09199
$568$	27.0388	1.13452
$569$	−14.2454	−0.597200	−0.298600	−	0.954378i	$-0.596520\pi$
$569$	−14.2454	−0.597200	−0.298600	+	0.954378i	$0.596520\pi$
$570$	58.7554	2.46099
$571$	−28.8133	−1.20580	−0.602900	−	0.797816i	$-0.705988\pi$
$571$	−28.8133	−1.20580	−0.602900	+	0.797816i	$0.705988\pi$
$572$	28.8955	1.20818
$573$	36.3818	1.51987
$574$	78.8464	3.29098
$575$	−1.18215	−0.0492992
$576$	4.59052	0.191272
$577$	9.18311	0.382298	0.191149	−	0.981561i	$-0.438779\pi$
$577$	9.18311	0.382298	0.191149	+	0.981561i	$0.438779\pi$
$578$	32.0366	1.33255
$579$	1.31439	0.0546241
$580$	−6.15056	−0.255388
$581$	−29.7721	−1.23516
$582$	−55.2497	−2.29018
$583$	13.1804	0.545875
$584$	29.3912	1.21622
$585$	3.55038	0.146790
$586$	−51.6903	−2.13531
$587$	13.7807	0.568789	0.284394	−	0.958707i	$-0.408208\pi$
$587$	13.7807	0.568789	0.284394	+	0.958707i	$0.408208\pi$
$588$	19.7151	0.813036
$589$	−49.5294	−2.04082
$590$	−52.2855	−2.15256
$591$	0.869524	0.0357674
$592$	−5.82326	−0.239334
$593$	40.9084	1.67991	0.839953	−	0.542659i	$-0.182582\pi$
$593$	40.9084	1.67991	0.839953	+	0.542659i	$0.182582\pi$
$594$	−24.3840	−1.00049
$595$	−11.1741	−0.458092
$596$	32.2279	1.32011
$597$	−25.2639	−1.03398
$598$	17.7339	0.725193
$599$	2.14220	0.0875280	0.0437640	−	0.999042i	$-0.486065\pi$
$599$	2.14220	0.0875280	0.0437640	+	0.999042i	$0.486065\pi$
$600$	2.54519	0.103907
$601$	31.7908	1.29678	0.648388	−	0.761310i	$-0.275444\pi$
$601$	31.7908	1.29678	0.648388	+	0.761310i	$0.275444\pi$
$602$	−18.9106	−0.770739
$603$	1.98653	0.0808977
$604$	−33.6562	−1.36945
$605$	−14.4460	−0.587312
$606$	−15.6558	−0.635973
$607$	−22.6788	−0.920506	−0.460253	−	0.887788i	$-0.652241\pi$
$607$	−22.6788	−0.920506	−0.460253	+	0.887788i	$0.652241\pi$
$608$	−53.3035	−2.16174
$609$	−5.41373	−0.219375
$610$	−38.4214	−1.55564
$611$	2.06065	0.0833648
$612$	1.70807	0.0690445
$613$	41.9692	1.69512	0.847559	−	0.530701i	$-0.178071\pi$
$613$	41.9692	1.69512	0.847559	+	0.530701i	$0.178071\pi$
$614$	−41.5321	−1.67610
$615$	−35.7471	−1.44146
$616$	14.4773	0.583307
$617$	0.831162	0.0334613	0.0167307	−	0.999860i	$-0.494674\pi$
$617$	0.831162	0.0334613	0.0167307	+	0.999860i	$0.494674\pi$
$618$	33.0251	1.32846
$619$	40.0922	1.61144	0.805720	−	0.592296i	$-0.201778\pi$
$619$	40.0922	1.61144	0.805720	+	0.592296i	$0.201778\pi$
$620$	38.9072	1.56255
$621$	−8.94687	−0.359026
$622$	22.7652	0.912800
$623$	32.1880	1.28958
$624$	8.72514	0.349285
$625$	−20.8760	−0.835040
$626$	−60.2610	−2.40852
$627$	25.5256	1.01939
$628$	10.4660	0.417637
$629$	−8.54470	−0.340700
$630$	5.43408	0.216499
$631$	−30.6732	−1.22108	−0.610540	−	0.791985i	$-0.709048\pi$
$631$	−30.6732	−1.22108	−0.610540	+	0.791985i	$0.709048\pi$
$632$	3.29240	0.130964
$633$	19.8653	0.789577
$634$	1.31436	0.0522000
$635$	−32.2722	−1.28068
$636$	−31.8081	−1.26127
$637$	−19.7664	−0.783174
$638$	−4.46940	−0.176945
$639$	4.40915	0.174423
$640$	31.6663	1.25172
$641$	2.81583	0.111219	0.0556093	−	0.998453i	$-0.482290\pi$
$641$	2.81583	0.111219	0.0556093	+	0.998453i	$0.482290\pi$
$642$	19.7938	0.781197
$643$	13.6795	0.539467	0.269734	−	0.962935i	$-0.413064\pi$
$643$	13.6795	0.539467	0.269734	+	0.962935i	$0.413064\pi$
$644$	16.2274	0.639450
$645$	8.57363	0.337586
$646$	−28.3405	−1.11504
$647$	13.8958	0.546301	0.273151	−	0.961971i	$-0.411934\pi$
$647$	13.8958	0.546301	0.273151	+	0.961971i	$0.411934\pi$
$648$	16.9581	0.666176
$649$	−22.7148	−0.891634
$650$	−7.79553	−0.305766
$651$	34.2461	1.34221
$652$	−60.9515	−2.38704
$653$	7.37217	0.288495	0.144248	−	0.989542i	$-0.453924\pi$
$653$	7.37217	0.288495	0.144248	+	0.989542i	$0.453924\pi$
$654$	13.9189	0.544274
$655$	−4.52218	−0.176696
$656$	11.7509	0.458794
$657$	4.79275	0.186983
$658$	3.15396	0.122954
$659$	−33.1597	−1.29172	−0.645859	−	0.763456i	$-0.723501\pi$
$659$	−33.1597	−1.29172	−0.645859	+	0.763456i	$0.723501\pi$
$660$	−20.0513	−0.780495
$661$	25.8127	1.00400	0.502000	−	0.864868i	$-0.332598\pi$
$661$	25.8127	1.00400	0.502000	+	0.864868i	$0.332598\pi$
$662$	−7.04636	−0.273865
$663$	12.8028	0.497218
$664$	19.4168	0.753517
$665$	−53.9042	−2.09031
$666$	4.15539	0.161018
$667$	−1.63989	−0.0634970
$668$	9.53575	0.368949
$669$	−35.9486	−1.38985
$670$	25.8919	1.00029
$671$	−16.6917	−0.644377
$672$	36.8556	1.42174
$673$	12.9902	0.500736	0.250368	−	0.968151i	$-0.419448\pi$
$673$	12.9902	0.500736	0.250368	+	0.968151i	$0.419448\pi$
$674$	52.0951	2.00663
$675$	3.93290	0.151377
$676$	31.2621	1.20239
$677$	12.7421	0.489718	0.244859	−	0.969559i	$-0.421258\pi$
$677$	12.7421	0.489718	0.244859	+	0.969559i	$0.421258\pi$
$678$	1.09209	0.0419413
$679$	50.6880	1.94523
$680$	7.28750	0.279463
$681$	−33.9864	−1.30236
$682$	28.2725	1.08261
$683$	−35.9900	−1.37712	−0.688559	−	0.725181i	$-0.741756\pi$
$683$	−35.9900	−1.37712	−0.688559	+	0.725181i	$0.741756\pi$
$684$	8.23978	0.315056
$685$	13.1348	0.501856
$686$	21.6999	0.828506
$687$	−24.6917	−0.942047
$688$	−2.81834	−0.107448
$689$	31.8909	1.21495
$690$	−12.3060	−0.468480
$691$	−46.8937	−1.78392	−0.891960	−	0.452115i	$-0.850670\pi$
$691$	−46.8937	−1.78392	−0.891960	+	0.452115i	$0.850670\pi$
$692$	−29.4565	−1.11977
$693$	2.36077	0.0896784
$694$	−42.7336	−1.62214
$695$	−2.06861	−0.0784667
$696$	3.53072	0.133832
$697$	17.2425	0.653107
$698$	7.20384	0.272670
$699$	26.2755	0.993829
$700$	−7.13331	−0.269614
$701$	27.2766	1.03022	0.515112	−	0.857123i	$-0.327750\pi$
$701$	27.2766	1.03022	0.515112	+	0.857123i	$0.327750\pi$
$702$	−58.9988	−2.22677
$703$	−41.2200	−1.55464
$704$	25.9931	0.979653
$705$	−1.42993	−0.0538543
$706$	5.04687	0.189941
$707$	14.3632	0.540182
$708$	54.8174	2.06017
$709$	30.7105	1.15336	0.576678	−	0.816971i	$-0.304349\pi$
$709$	30.7105	1.15336	0.576678	+	0.816971i	$0.304349\pi$
$710$	57.4677	2.15673
$711$	0.536882	0.0201346
$712$	−20.9923	−0.786721
$713$	10.3736	0.388495
$714$	19.5955	0.733342
$715$	20.1035	0.751828
$716$	64.4479	2.40853
$717$	13.1722	0.491926
$718$	5.19272	0.193791
$719$	7.48553	0.279163	0.139582	−	0.990211i	$-0.455424\pi$
$719$	7.48553	0.279163	0.139582	+	0.990211i	$0.455424\pi$
$720$	0.809868	0.0301820
$721$	−30.2983	−1.12837
$722$	−94.3441	−3.51112
$723$	−41.7508	−1.55273
$724$	−33.8812	−1.25919
$725$	0.720871	0.0267725
$726$	25.3332	0.940205
$727$	−22.8517	−0.847522	−0.423761	−	0.905774i	$-0.639290\pi$
$727$	−22.8517	−0.847522	−0.423761	+	0.905774i	$0.639290\pi$
$728$	35.0289	1.29826
$729$	29.3895	1.08850
$730$	62.4674	2.31202
$731$	−4.13547	−0.152956
$732$	40.2820	1.48887
$733$	27.2790	1.00757	0.503787	−	0.863828i	$-0.331940\pi$
$733$	27.2790	1.00757	0.503787	+	0.863828i	$0.331940\pi$
$734$	−62.4998	−2.30691
$735$	13.7164	0.505936
$736$	11.1641	0.411514
$737$	11.2484	0.414341
$738$	−8.38524	−0.308665
$739$	−44.8365	−1.64934	−0.824668	−	0.565617i	$-0.808638\pi$
$739$	−44.8365	−1.64934	−0.824668	+	0.565617i	$0.808638\pi$
$740$	32.3798	1.19031
$741$	61.7610	2.26885
$742$	48.8111	1.79191
$743$	−25.3567	−0.930246	−0.465123	−	0.885246i	$-0.653990\pi$
$743$	−25.3567	−0.930246	−0.465123	+	0.885246i	$0.653990\pi$
$744$	−22.3346	−0.818826
$745$	22.4219	0.821474
$746$	−31.7043	−1.16078
$747$	3.16624	0.115847
$748$	9.67168	0.353632
$749$	−18.1595	−0.663532
$750$	42.9301	1.56758
$751$	42.1518	1.53814	0.769071	−	0.639163i	$-0.220719\pi$
$751$	42.1518	1.53814	0.769071	+	0.639163i	$0.220719\pi$
$752$	0.470049	0.0171409
$753$	−19.2247	−0.700585
$754$	−10.8140	−0.393824
$755$	−23.4156	−0.852182
$756$	−53.9870	−1.96349
$757$	−11.0969	−0.403323	−0.201662	−	0.979455i	$-0.564634\pi$
$757$	−11.0969	−0.403323	−0.201662	+	0.979455i	$0.564634\pi$
$758$	−16.7691	−0.609083
$759$	−5.34618	−0.194054
$760$	35.1552	1.27521
$761$	−47.0563	−1.70579	−0.852895	−	0.522082i	$-0.825156\pi$
$761$	−47.0563	−1.70579	−0.852895	+	0.522082i	$0.825156\pi$
$762$	56.5943	2.05020
$763$	−12.7697	−0.462295
$764$	66.4999	2.40588
$765$	1.18835	0.0429650
$766$	−37.7401	−1.36361
$767$	−54.9602	−1.98450
$768$	−13.3367	−0.481248
$769$	−31.5357	−1.13721	−0.568603	−	0.822612i	$-0.692516\pi$
$769$	−31.5357	−1.13721	−0.568603	+	0.822612i	$0.692516\pi$
$770$	30.7697	1.10886
$771$	12.9406	0.466045
$772$	2.40248	0.0864673
$773$	30.7894	1.10742	0.553710	−	0.832710i	$-0.313212\pi$
$773$	30.7894	1.10742	0.553710	+	0.832710i	$0.313212\pi$
$774$	2.01113	0.0722885
$775$	−4.56008	−0.163803
$776$	−33.0577	−1.18670
$777$	28.5007	1.02246
$778$	56.7777	2.03558
$779$	83.1786	2.98018
$780$	−48.5155	−1.73713
$781$	24.9662	0.893360
$782$	5.93575	0.212262
$783$	5.45576	0.194973
$784$	−4.50887	−0.161031
$785$	7.28148	0.259887
$786$	7.93034	0.282866
$787$	26.6385	0.949559	0.474780	−	0.880105i	$-0.342528\pi$
$787$	26.6385	0.949559	0.474780	+	0.880105i	$0.342528\pi$
$788$	1.58935	0.0566182
$789$	31.0365	1.10493
$790$	6.99758	0.248963
$791$	−1.00192	−0.0356241
$792$	−1.53965	−0.0547090
$793$	−40.3869	−1.43418
$794$	69.5439	2.46802
$795$	−22.1298	−0.784864
$796$	−46.1782	−1.63674
$797$	25.1024	0.889171	0.444586	−	0.895736i	$-0.353351\pi$
$797$	25.1024	0.889171	0.444586	+	0.895736i	$0.353351\pi$
$798$	94.5293	3.34630
$799$	0.689723	0.0244006
$800$	−4.90755	−0.173508
$801$	−3.42316	−0.120952
$802$	−35.9224	−1.26846
$803$	27.1382	0.957687
$804$	−27.1457	−0.957356
$805$	11.2899	0.397917
$806$	68.4074	2.40955
$807$	33.1272	1.16613
$808$	−9.36736	−0.329542
$809$	2.94582	0.103570	0.0517848	−	0.998658i	$-0.483509\pi$
$809$	2.94582	0.103570	0.0517848	+	0.998658i	$0.483509\pi$
$810$	36.0423	1.26640
$811$	33.7456	1.18497	0.592485	−	0.805582i	$-0.298147\pi$
$811$	33.7456	1.18497	0.592485	+	0.805582i	$0.298147\pi$
$812$	−9.89541	−0.347261
$813$	−19.6404	−0.688818
$814$	23.5293	0.824701
$815$	−42.4058	−1.48541
$816$	2.92041	0.102235
$817$	−19.9497	−0.697951
$818$	−29.7675	−1.04080
$819$	5.71206	0.199596
$820$	−65.3398	−2.28177
$821$	−40.4922	−1.41319	−0.706594	−	0.707619i	$-0.749769\pi$
$821$	−40.4922	−1.41319	−0.706594	+	0.707619i	$0.749769\pi$
$822$	−23.0340	−0.803402
$823$	−30.9083	−1.07739	−0.538697	−	0.842500i	$-0.681083\pi$
$823$	−30.9083	−1.07739	−0.538697	+	0.842500i	$0.681083\pi$
$824$	19.7599	0.688370
$825$	2.35009	0.0818197
$826$	−84.1202	−2.92691
$827$	47.7083	1.65898	0.829489	−	0.558522i	$-0.188632\pi$
$827$	47.7083	1.65898	0.829489	+	0.558522i	$0.188632\pi$
$828$	−1.72577	−0.0599747
$829$	12.1639	0.422469	0.211234	−	0.977435i	$-0.432252\pi$
$829$	12.1639	0.422469	0.211234	+	0.977435i	$0.432252\pi$
$830$	41.2680	1.43243
$831$	−50.3622	−1.74704
$832$	62.8923	2.18040
$833$	−6.61605	−0.229233
$834$	3.62762	0.125614
$835$	6.63431	0.229590
$836$	46.6566	1.61365
$837$	−34.5120	−1.19291
$838$	−26.7027	−0.922428
$839$	33.8683	1.16926	0.584632	−	0.811299i	$-0.301239\pi$
$839$	33.8683	1.16926	0.584632	+	0.811299i	$0.301239\pi$
$840$	−24.3074	−0.838683
$841$	1.00000	0.0344828
$842$	−20.9478	−0.721908
$843$	27.7322	0.955149
$844$	36.3106	1.24986
$845$	21.7500	0.748222
$846$	−0.335420	−0.0115320
$847$	−23.2416	−0.798590
$848$	7.27456	0.249809
$849$	18.7065	0.642005
$850$	−2.60926	−0.0894968
$851$	8.63327	0.295945
$852$	−60.2506	−2.06415
$853$	45.8445	1.56969	0.784843	−	0.619695i	$-0.212744\pi$
$853$	45.8445	1.56969	0.784843	+	0.619695i	$0.212744\pi$
$854$	−61.8148	−2.11526
$855$	5.73266	0.196053
$856$	11.8432	0.404793
$857$	−22.8100	−0.779173	−0.389587	−	0.920990i	$-0.627382\pi$
$857$	−22.8100	−0.779173	−0.389587	+	0.920990i	$0.627382\pi$
$858$	−35.2546	−1.20357
$859$	39.6239	1.35195	0.675974	−	0.736925i	$-0.263723\pi$
$859$	39.6239	1.35195	0.675974	+	0.736925i	$0.263723\pi$
$860$	15.6712	0.534383
$861$	−57.5122	−1.96001
$862$	38.9648	1.32715
$863$	−3.44833	−0.117382	−0.0586912	−	0.998276i	$-0.518693\pi$
$863$	−3.44833	−0.117382	−0.0586912	+	0.998276i	$0.518693\pi$
$864$	−37.1417	−1.26359
$865$	−20.4937	−0.696808
$866$	39.3081	1.33575
$867$	−23.3682	−0.793625
$868$	62.5963	2.12466
$869$	3.04002	0.103125
$870$	7.50411	0.254413
$871$	27.2164	0.922192
$872$	8.32815	0.282027
$873$	−5.39062	−0.182445
$874$	28.6343	0.968569
$875$	−39.3855	−1.33147
$876$	−65.4924	−2.21278
$877$	−44.8728	−1.51525	−0.757624	−	0.652692i	$-0.773640\pi$
$877$	−44.8728	−1.51525	−0.757624	+	0.652692i	$0.773640\pi$
$878$	−53.0732	−1.79113
$879$	37.7040	1.27172
$880$	4.58576	0.154586
$881$	−38.1710	−1.28601	−0.643007	−	0.765861i	$-0.722313\pi$
$881$	−38.1710	−1.28601	−0.643007	+	0.765861i	$0.722313\pi$
$882$	3.21746	0.108338
$883$	12.1408	0.408571	0.204286	−	0.978911i	$-0.434513\pi$
$883$	12.1408	0.408571	0.204286	+	0.978911i	$0.434513\pi$
$884$	23.4013	0.787072
$885$	38.1381	1.28200
$886$	7.93651	0.266632
$887$	−27.0924	−0.909674	−0.454837	−	0.890575i	$-0.650302\pi$
$887$	−27.0924	−0.909674	−0.454837	+	0.890575i	$0.650302\pi$
$888$	−18.5876	−0.623759
$889$	−51.9216	−1.74139
$890$	−44.6166	−1.49555
$891$	15.6581	0.524567
$892$	−65.7081	−2.20007
$893$	3.32725	0.111342
$894$	−39.3202	−1.31507
$895$	44.8383	1.49878
$896$	50.9467	1.70201
$897$	−12.9355	−0.431903
$898$	−48.4357	−1.61632
$899$	−6.32579	−0.210977
$900$	0.758621	0.0252874
$901$	10.6743	0.355611
$902$	−47.4802	−1.58092
$903$	13.7938	0.459029
$904$	0.653429	0.0217327
$905$	−23.5722	−0.783566
$906$	41.0629	1.36422
$907$	1.50185	0.0498680	0.0249340	−	0.999689i	$-0.492062\pi$
$907$	1.50185	0.0498680	0.0249340	+	0.999689i	$0.492062\pi$
$908$	−62.1215	−2.06158
$909$	−1.52751	−0.0506643
$910$	74.4496	2.46798
$911$	51.8367	1.71743	0.858714	−	0.512456i	$-0.171264\pi$
$911$	51.8367	1.71743	0.858714	+	0.512456i	$0.171264\pi$
$912$	14.0882	0.466506
$913$	17.9284	0.593343
$914$	66.5920	2.20267
$915$	28.0254	0.926490
$916$	−45.1324	−1.49122
$917$	−7.27556	−0.240260
$918$	−19.7476	−0.651768
$919$	51.5656	1.70099	0.850496	−	0.525981i	$-0.176302\pi$
$919$	51.5656	1.70099	0.850496	+	0.525981i	$0.176302\pi$
$920$	−7.36304	−0.242752
$921$	30.2944	0.998234
$922$	81.8239	2.69473
$923$	60.4075	1.98834
$924$	−32.2597	−1.06127
$925$	−3.79505	−0.124780
$926$	14.8126	0.486773
$927$	3.22220	0.105831
$928$	−6.80781	−0.223477
$929$	29.9546	0.982777	0.491389	−	0.870940i	$-0.336489\pi$
$929$	29.9546	0.982777	0.491389	+	0.870940i	$0.336489\pi$
$930$	−47.4694	−1.55658
$931$	−31.9161	−1.04601
$932$	48.0272	1.57318
$933$	−16.6054	−0.543636
$934$	56.8749	1.86100
$935$	6.72888	0.220058
$936$	−3.72529	−0.121765
$937$	23.0102	0.751711	0.375855	−	0.926678i	$-0.377349\pi$
$937$	23.0102	0.751711	0.375855	+	0.926678i	$0.377349\pi$
$938$	41.6565	1.36013
$939$	43.9556	1.43444
$940$	−2.61368	−0.0852488
$941$	−17.5020	−0.570550	−0.285275	−	0.958446i	$-0.592085\pi$
$941$	−17.5020	−0.570550	−0.285275	+	0.958446i	$0.592085\pi$
$942$	−12.7692	−0.416043
$943$	−17.4212	−0.567314
$944$	−12.5368	−0.408039
$945$	−37.5603	−1.22184
$946$	11.3877	0.370247
$947$	36.8926	1.19885	0.599424	−	0.800432i	$-0.295396\pi$
$947$	36.8926	1.19885	0.599424	+	0.800432i	$0.295396\pi$
$948$	−7.33644	−0.238276
$949$	65.6629	2.13151
$950$	−12.5872	−0.408381
$951$	−0.958723	−0.0310887
$952$	11.7246	0.379996
$953$	33.8136	1.09533	0.547666	−	0.836697i	$-0.315517\pi$
$953$	33.8136	1.09533	0.547666	+	0.836697i	$0.315517\pi$
$954$	−5.19102	−0.168065
$955$	46.2660	1.49713
$956$	24.0767	0.778696
$957$	3.26007	0.105383
$958$	−6.35709	−0.205388
$959$	21.1321	0.682393
$960$	−43.6424	−1.40855
$961$	9.01561	0.290826
$962$	56.9308	1.83552
$963$	1.93124	0.0622334
$964$	−76.3137	−2.45790
$965$	1.67148	0.0538068
$966$	−19.7986	−0.637009
$967$	−9.92616	−0.319204	−0.159602	−	0.987181i	$-0.551021\pi$
$967$	−9.92616	−0.319204	−0.159602	+	0.987181i	$0.551021\pi$
$968$	15.1577	0.487186
$969$	20.6721	0.664085
$970$	−70.2600	−2.25591
$971$	5.11391	0.164113	0.0820566	−	0.996628i	$-0.473851\pi$
$971$	5.11391	0.164113	0.0820566	+	0.996628i	$0.473851\pi$
$972$	10.8770	0.348881
$973$	−3.32810	−0.106694
$974$	31.1877	0.999320
$975$	5.68622	0.182105
$976$	−9.21256	−0.294887
$977$	−44.3943	−1.42030	−0.710150	−	0.704051i	$-0.751373\pi$
$977$	−44.3943	−1.42030	−0.710150	+	0.704051i	$0.751373\pi$
$978$	74.3651	2.37793
$979$	−19.3832	−0.619489
$980$	25.0713	0.800873
$981$	1.35805	0.0433591
$982$	22.7074	0.724622
$983$	11.0923	0.353790	0.176895	−	0.984230i	$-0.443395\pi$
$983$	11.0923	0.353790	0.176895	+	0.984230i	$0.443395\pi$
$984$	37.5082	1.19572
$985$	1.10576	0.0352323
$986$	−3.61959	−0.115271
$987$	−2.30056	−0.0732276
$988$	112.889	3.59148
$989$	4.17834	0.132863
$990$	−3.27233	−0.104002
$991$	−38.6015	−1.22622	−0.613108	−	0.789999i	$-0.710081\pi$
$991$	−38.6015	−1.22622	−0.613108	+	0.789999i	$0.710081\pi$
$992$	43.0647	1.36731
$993$	5.13976	0.163105
$994$	92.4577	2.93258
$995$	−32.1275	−1.01851
$996$	−43.2664	−1.37095
$997$	25.3592	0.803135	0.401568	−	0.915829i	$-0.368465\pi$
$997$	25.3592	0.803135	0.401568	+	0.915829i	$0.368465\pi$
$998$	−57.4293	−1.81789
$999$	−28.7220	−0.908724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.8	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.8	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23009 q^{2} +1.62667 q^{3} +2.97329 q^{4} +2.06861 q^{5} -3.62762 q^{6} +3.32810 q^{7} -2.17052 q^{8} -0.353941 q^{9} +O(q^{10})\)	\(q-2.23009 q^{2} +1.62667 q^{3} +2.97329 q^{4} +2.06861 q^{5} -3.62762 q^{6} +3.32810 q^{7} -2.17052 q^{8} -0.353941 q^{9} -4.61317 q^{10} -2.00414 q^{11} +4.83656 q^{12} -4.84916 q^{13} -7.42196 q^{14} +3.36494 q^{15} -1.10613 q^{16} -1.62307 q^{17} +0.789318 q^{18} -7.82976 q^{19} +6.15056 q^{20} +5.41373 q^{21} +4.46940 q^{22} +1.63989 q^{23} -3.53072 q^{24} -0.720871 q^{25} +10.8140 q^{26} -5.45576 q^{27} +9.89541 q^{28} -1.00000 q^{29} -7.50411 q^{30} +6.32579 q^{31} +6.80781 q^{32} -3.26007 q^{33} +3.61959 q^{34} +6.88453 q^{35} -1.05237 q^{36} +5.26453 q^{37} +17.4610 q^{38} -7.88798 q^{39} -4.48995 q^{40} -10.6234 q^{41} -12.0731 q^{42} +2.54793 q^{43} -5.95888 q^{44} -0.732163 q^{45} -3.65711 q^{46} -0.424949 q^{47} -1.79931 q^{48} +4.07626 q^{49} +1.60761 q^{50} -2.64020 q^{51} -14.4179 q^{52} -6.57658 q^{53} +12.1668 q^{54} -4.14577 q^{55} -7.22371 q^{56} -12.7364 q^{57} +2.23009 q^{58} +11.3340 q^{59} +10.0049 q^{60} +8.32864 q^{61} -14.1071 q^{62} -1.17795 q^{63} -12.9697 q^{64} -10.0310 q^{65} +7.27025 q^{66} -5.61260 q^{67} -4.82586 q^{68} +2.66757 q^{69} -15.3531 q^{70} -12.4573 q^{71} +0.768235 q^{72} -13.5411 q^{73} -11.7404 q^{74} -1.17262 q^{75} -23.2801 q^{76} -6.66997 q^{77} +17.5909 q^{78} -1.51687 q^{79} -2.28815 q^{80} -7.81290 q^{81} +23.6911 q^{82} -8.94568 q^{83} +16.0966 q^{84} -3.35749 q^{85} -5.68210 q^{86} -1.62667 q^{87} +4.35002 q^{88} +9.67158 q^{89} +1.63279 q^{90} -16.1385 q^{91} +4.87588 q^{92} +10.2900 q^{93} +0.947674 q^{94} -16.1967 q^{95} +11.0741 q^{96} +15.2303 q^{97} -9.09041 q^{98} +0.709345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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