LMFDB - Embedded newform 4027.2.a.c.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4027,2,Mod(1,4027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4027, 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("4027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4027 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4027.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:	$32.1557568940$
Analytic rank:	$0$
Dimension:	$174$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41774 q^{2} -1.12112 q^{3} +3.84547 q^{4} +1.14031 q^{5} +2.71058 q^{6} +1.36358 q^{7} -4.46187 q^{8} -1.74309 q^{9} +O(q^{10})$	\(q-2.41774 q^{2} -1.12112 q^{3} +3.84547 q^{4} +1.14031 q^{5} +2.71058 q^{6} +1.36358 q^{7} -4.46187 q^{8} -1.74309 q^{9} -2.75699 q^{10} +0.0870099 q^{11} -4.31124 q^{12} +6.43997 q^{13} -3.29679 q^{14} -1.27843 q^{15} +3.09671 q^{16} +2.91130 q^{17} +4.21433 q^{18} +3.65638 q^{19} +4.38505 q^{20} -1.52874 q^{21} -0.210367 q^{22} -0.109143 q^{23} +5.00230 q^{24} -3.69968 q^{25} -15.5702 q^{26} +5.31758 q^{27} +5.24362 q^{28} +6.93773 q^{29} +3.09092 q^{30} +4.75193 q^{31} +1.43670 q^{32} -0.0975486 q^{33} -7.03877 q^{34} +1.55491 q^{35} -6.70299 q^{36} -8.84338 q^{37} -8.84018 q^{38} -7.21998 q^{39} -5.08794 q^{40} +8.26020 q^{41} +3.69610 q^{42} -4.98099 q^{43} +0.334594 q^{44} -1.98767 q^{45} +0.263881 q^{46} +5.18867 q^{47} -3.47179 q^{48} -5.14064 q^{49} +8.94487 q^{50} -3.26392 q^{51} +24.7647 q^{52} +4.14206 q^{53} -12.8565 q^{54} +0.0992186 q^{55} -6.08413 q^{56} -4.09924 q^{57} -16.7736 q^{58} +11.6476 q^{59} -4.91617 q^{60} +3.54812 q^{61} -11.4889 q^{62} -2.37684 q^{63} -9.66699 q^{64} +7.34359 q^{65} +0.235847 q^{66} +1.69075 q^{67} +11.1953 q^{68} +0.122363 q^{69} -3.75938 q^{70} +7.48161 q^{71} +7.77743 q^{72} -10.0462 q^{73} +21.3810 q^{74} +4.14779 q^{75} +14.0605 q^{76} +0.118645 q^{77} +17.4561 q^{78} -12.0541 q^{79} +3.53123 q^{80} -0.732387 q^{81} -19.9710 q^{82} -2.74298 q^{83} -5.87873 q^{84} +3.31980 q^{85} +12.0427 q^{86} -7.77804 q^{87} -0.388227 q^{88} +2.53675 q^{89} +4.80567 q^{90} +8.78142 q^{91} -0.419708 q^{92} -5.32749 q^{93} -12.5449 q^{94} +4.16942 q^{95} -1.61071 q^{96} +2.88915 q^{97} +12.4287 q^{98} -0.151666 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9}+O(q^{10}) $ 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9} + 20 q^{10} + 35 q^{11} + 23 q^{12} + 91 q^{13} + 18 q^{14} + 16 q^{15} + 201 q^{16} + 148 q^{17} + 39 q^{18} + 36 q^{19} + 128 q^{20} + 57 q^{21} + 17 q^{22} + 96 q^{23} + 24 q^{24} + 226 q^{25} + 44 q^{26} + 62 q^{27} + 32 q^{28} + 122 q^{29} + 25 q^{30} + 23 q^{31} + 104 q^{32} + 91 q^{33} + 6 q^{34} + 80 q^{35} + 222 q^{36} + 71 q^{37} + 125 q^{38} + 16 q^{39} + 53 q^{40} + 97 q^{41} + 14 q^{42} + 38 q^{43} + 70 q^{44} + 185 q^{45} - 23 q^{46} + 110 q^{47} + 36 q^{48} + 210 q^{49} + 51 q^{50} + 33 q^{51} + 118 q^{52} + 214 q^{53} + 8 q^{54} + 37 q^{55} + 41 q^{56} + 76 q^{57} + 2 q^{58} + 66 q^{59} - 12 q^{60} + 114 q^{61} + 175 q^{62} + 62 q^{63} + 190 q^{64} + 128 q^{65} + 12 q^{66} - 6 q^{67} + 348 q^{68} + 115 q^{69} - 38 q^{70} + 54 q^{71} + 101 q^{72} + 107 q^{73} + 71 q^{74} - q^{75} + 31 q^{76} + 368 q^{77} - 14 q^{78} - 14 q^{79} + 205 q^{80} + 222 q^{81} + 26 q^{82} + 246 q^{83} + 41 q^{84} + 87 q^{85} + 33 q^{86} + 100 q^{87} - 6 q^{88} + 147 q^{89} + 50 q^{90} - 23 q^{91} + 189 q^{92} + 117 q^{93} + 23 q^{94} + 42 q^{95} + 38 q^{96} + 52 q^{97} + 148 q^{98} + 38 q^{99}+O(q^{100}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9 + 20 * q^10 + 35 * q^11 + 23 * q^12 + 91 * q^13 + 18 * q^14 + 16 * q^15 + 201 * q^16 + 148 * q^17 + 39 * q^18 + 36 * q^19 + 128 * q^20 + 57 * q^21 + 17 * q^22 + 96 * q^23 + 24 * q^24 + 226 * q^25 + 44 * q^26 + 62 * q^27 + 32 * q^28 + 122 * q^29 + 25 * q^30 + 23 * q^31 + 104 * q^32 + 91 * q^33 + 6 * q^34 + 80 * q^35 + 222 * q^36 + 71 * q^37 + 125 * q^38 + 16 * q^39 + 53 * q^40 + 97 * q^41 + 14 * q^42 + 38 * q^43 + 70 * q^44 + 185 * q^45 - 23 * q^46 + 110 * q^47 + 36 * q^48 + 210 * q^49 + 51 * q^50 + 33 * q^51 + 118 * q^52 + 214 * q^53 + 8 * q^54 + 37 * q^55 + 41 * q^56 + 76 * q^57 + 2 * q^58 + 66 * q^59 - 12 * q^60 + 114 * q^61 + 175 * q^62 + 62 * q^63 + 190 * q^64 + 128 * q^65 + 12 * q^66 - 6 * q^67 + 348 * q^68 + 115 * q^69 - 38 * q^70 + 54 * q^71 + 101 * q^72 + 107 * q^73 + 71 * q^74 - q^75 + 31 * q^76 + 368 * q^77 - 14 * q^78 - 14 * q^79 + 205 * q^80 + 222 * q^81 + 26 * q^82 + 246 * q^83 + 41 * q^84 + 87 * q^85 + 33 * q^86 + 100 * q^87 - 6 * q^88 + 147 * q^89 + 50 * q^90 - 23 * q^91 + 189 * q^92 + 117 * q^93 + 23 * q^94 + 42 * q^95 + 38 * q^96 + 52 * q^97 + 148 * q^98 + 38 * 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 $

$2$	−2.41774	−1.70960	−0.854801	−	0.518957i	$-0.826321\pi$
$2$	−2.41774	−1.70960	−0.854801	+	0.518957i	$0.826321\pi$
$3$	−1.12112	−0.647280	−0.323640	−	0.946180i	$-0.604907\pi$
$3$	−1.12112	−0.647280	−0.323640	+	0.946180i	$0.604907\pi$
$4$	3.84547	1.92274
$5$	1.14031	0.509964	0.254982	−	0.966946i	$-0.417930\pi$
$5$	1.14031	0.509964	0.254982	+	0.966946i	$0.417930\pi$
$6$	2.71058	1.10659
$7$	1.36358	0.515385	0.257693	−	0.966227i	$-0.417038\pi$
$7$	1.36358	0.515385	0.257693	+	0.966227i	$0.417038\pi$
$8$	−4.46187	−1.57751
$9$	−1.74309	−0.581029
$10$	−2.75699	−0.871836
$11$	0.0870099	0.0262345	0.0131172	−	0.999914i	$-0.495825\pi$
$11$	0.0870099	0.0262345	0.0131172	+	0.999914i	$0.495825\pi$
$12$	−4.31124	−1.24455
$13$	6.43997	1.78613	0.893063	−	0.449932i	$-0.148552\pi$
$13$	6.43997	1.78613	0.893063	+	0.449932i	$0.148552\pi$
$14$	−3.29679	−0.881104
$15$	−1.27843	−0.330090
$16$	3.09671	0.774178
$17$	2.91130	0.706094	0.353047	−	0.935606i	$-0.385146\pi$
$17$	2.91130	0.706094	0.353047	+	0.935606i	$0.385146\pi$
$18$	4.21433	0.993328
$19$	3.65638	0.838831	0.419415	−	0.907794i	$-0.362235\pi$
$19$	3.65638	0.838831	0.419415	+	0.907794i	$0.362235\pi$
$20$	4.38505	0.980527
$21$	−1.52874	−0.333599
$22$	−0.210367	−0.0448505
$23$	−0.109143	−0.0227580	−0.0113790	−	0.999935i	$-0.503622\pi$
$23$	−0.109143	−0.0227580	−0.0113790	+	0.999935i	$0.503622\pi$
$24$	5.00230	1.02109
$25$	−3.69968	−0.739936
$26$	−15.5702	−3.05356
$27$	5.31758	1.02337
$28$	5.24362	0.990950
$29$	6.93773	1.28830	0.644152	−	0.764897i	$-0.277210\pi$
$29$	6.93773	1.28830	0.644152	+	0.764897i	$0.277210\pi$
$30$	3.09092	0.564321
$31$	4.75193	0.853472	0.426736	−	0.904376i	$-0.359663\pi$
$31$	4.75193	0.853472	0.426736	+	0.904376i	$0.359663\pi$
$32$	1.43670	0.253975
$33$	−0.0975486	−0.0169810
$34$	−7.03877	−1.20714
$35$	1.55491	0.262828
$36$	−6.70299	−1.11717
$37$	−8.84338	−1.45384	−0.726921	−	0.686721i	$-0.759049\pi$
$37$	−8.84338	−1.45384	−0.726921	+	0.686721i	$0.759049\pi$
$38$	−8.84018	−1.43407
$39$	−7.21998	−1.15612
$40$	−5.08794	−0.804474
$41$	8.26020	1.29003	0.645014	−	0.764171i	$-0.276852\pi$
$41$	8.26020	1.29003	0.645014	+	0.764171i	$0.276852\pi$
$42$	3.69610	0.570320
$43$	−4.98099	−0.759594	−0.379797	−	0.925070i	$-0.624006\pi$
$43$	−4.98099	−0.759594	−0.379797	+	0.925070i	$0.624006\pi$
$44$	0.334594	0.0504419
$45$	−1.98767	−0.296304
$46$	0.263881	0.0389071
$47$	5.18867	0.756846	0.378423	−	0.925633i	$-0.376466\pi$
$47$	5.18867	0.756846	0.378423	+	0.925633i	$0.376466\pi$
$48$	−3.47179	−0.501110
$49$	−5.14064	−0.734378
$50$	8.94487	1.26500
$51$	−3.26392	−0.457040
$52$	24.7647	3.43425
$53$	4.14206	0.568956	0.284478	−	0.958683i	$-0.408180\pi$
$53$	4.14206	0.568956	0.284478	+	0.958683i	$0.408180\pi$
$54$	−12.8565	−1.74955
$55$	0.0992186	0.0133786
$56$	−6.08413	−0.813026
$57$	−4.09924	−0.542958
$58$	−16.7736	−2.20249
$59$	11.6476	1.51639	0.758197	−	0.652026i	$-0.226081\pi$
$59$	11.6476	1.51639	0.758197	+	0.652026i	$0.226081\pi$
$60$	−4.91617	−0.634675
$61$	3.54812	0.454291	0.227145	−	0.973861i	$-0.427061\pi$
$61$	3.54812	0.454291	0.227145	+	0.973861i	$0.427061\pi$
$62$	−11.4889	−1.45910
$63$	−2.37684	−0.299454
$64$	−9.66699	−1.20837
$65$	7.34359	0.910860
$66$	0.235847	0.0290308
$67$	1.69075	0.206558	0.103279	−	0.994652i	$-0.467067\pi$
$67$	1.69075	0.206558	0.103279	+	0.994652i	$0.467067\pi$
$68$	11.1953	1.35763
$69$	0.122363	0.0147308
$70$	−3.75938	−0.449331
$71$	7.48161	0.887904	0.443952	−	0.896051i	$-0.353576\pi$
$71$	7.48161	0.887904	0.443952	+	0.896051i	$0.353576\pi$
$72$	7.77743	0.916579
$73$	−10.0462	−1.17582	−0.587910	−	0.808926i	$-0.700049\pi$
$73$	−10.0462	−1.17582	−0.587910	+	0.808926i	$0.700049\pi$
$74$	21.3810	2.48549
$75$	4.14779	0.478946
$76$	14.0605	1.61285
$77$	0.118645	0.0135209
$78$	17.4561	1.97651
$79$	−12.0541	−1.35620	−0.678099	−	0.734971i	$-0.737196\pi$
$79$	−12.0541	−1.35620	−0.678099	+	0.734971i	$0.737196\pi$
$80$	3.53123	0.394803
$81$	−0.732387	−0.0813764
$82$	−19.9710	−2.20543
$83$	−2.74298	−0.301081	−0.150540	−	0.988604i	$-0.548101\pi$
$83$	−2.74298	−0.301081	−0.150540	+	0.988604i	$0.548101\pi$
$84$	−5.87873	−0.641422
$85$	3.31980	0.360083
$86$	12.0427	1.29860
$87$	−7.77804	−0.833893
$88$	−0.388227	−0.0413851
$89$	2.53675	0.268895	0.134447	−	0.990921i	$-0.457074\pi$
$89$	2.53675	0.268895	0.134447	+	0.990921i	$0.457074\pi$
$90$	4.80567	0.506562
$91$	8.78142	0.920543
$92$	−0.419708	−0.0437576
$93$	−5.32749	−0.552435
$94$	−12.5449	−1.29390
$95$	4.16942	0.427774
$96$	−1.61071	−0.164393
$97$	2.88915	0.293348	0.146674	−	0.989185i	$-0.453143\pi$
$97$	2.88915	0.293348	0.146674	+	0.989185i	$0.453143\pi$
$98$	12.4287	1.25549
$99$	−0.151666	−0.0152430
$100$	−14.2270	−1.42270
$101$	7.17439	0.713879	0.356939	−	0.934128i	$-0.383820\pi$
$101$	7.17439	0.713879	0.356939	+	0.934128i	$0.383820\pi$
$102$	7.89132	0.781357
$103$	19.0756	1.87957	0.939785	−	0.341766i	$-0.111025\pi$
$103$	19.0756	1.87957	0.939785	+	0.341766i	$0.111025\pi$
$104$	−28.7343	−2.81763
$105$	−1.74325	−0.170123
$106$	−10.0144	−0.972688
$107$	5.73582	0.554503	0.277251	−	0.960797i	$-0.410577\pi$
$107$	5.73582	0.554503	0.277251	+	0.960797i	$0.410577\pi$
$108$	20.4486	1.96767
$109$	11.2233	1.07499	0.537496	−	0.843266i	$-0.319370\pi$
$109$	11.2233	1.07499	0.537496	+	0.843266i	$0.319370\pi$
$110$	−0.239885	−0.0228721
$111$	9.91450	0.941043
$112$	4.22262	0.399000
$113$	6.04302	0.568480	0.284240	−	0.958753i	$-0.408259\pi$
$113$	6.04302	0.568480	0.284240	+	0.958753i	$0.408259\pi$
$114$	9.91091	0.928242
$115$	−0.124458	−0.0116058
$116$	26.6789	2.47707
$117$	−11.2254	−1.03779
$118$	−28.1610	−2.59243
$119$	3.96980	0.363911
$120$	5.70420	0.520720
$121$	−10.9924	−0.999312
$122$	−8.57845	−0.776656
$123$	−9.26069	−0.835008
$124$	18.2734	1.64100
$125$	−9.92038	−0.887305
$126$	5.74659	0.511947
$127$	−10.9045	−0.967622	−0.483811	−	0.875173i	$-0.660748\pi$
$127$	−10.9045	−0.967622	−0.483811	+	0.875173i	$0.660748\pi$
$128$	20.4989	1.81186
$129$	5.58430	0.491670
$130$	−17.7549	−1.55721
$131$	−16.4200	−1.43462	−0.717312	−	0.696752i	$-0.754628\pi$
$131$	−16.4200	−1.43462	−0.717312	+	0.696752i	$0.754628\pi$
$132$	−0.375120	−0.0326500
$133$	4.98577	0.432321
$134$	−4.08780	−0.353132
$135$	6.06371	0.521881
$136$	−12.9899	−1.11387
$137$	2.08476	0.178113	0.0890565	−	0.996027i	$-0.471615\pi$
$137$	2.08476	0.178113	0.0890565	+	0.996027i	$0.471615\pi$
$138$	−0.295842	−0.0251838
$139$	−11.5106	−0.976320	−0.488160	−	0.872754i	$-0.662332\pi$
$139$	−11.5106	−0.976320	−0.488160	+	0.872754i	$0.662332\pi$
$140$	5.97937	0.505349
$141$	−5.81713	−0.489891
$142$	−18.0886	−1.51796
$143$	0.560341	0.0468580
$144$	−5.39784	−0.449820
$145$	7.91120	0.656989
$146$	24.2892	2.01018
$147$	5.76329	0.475348
$148$	−34.0070	−2.79535
$149$	−6.63520	−0.543577	−0.271789	−	0.962357i	$-0.587615\pi$
$149$	−6.63520	−0.543577	−0.271789	+	0.962357i	$0.587615\pi$
$150$	−10.0283	−0.818806
$151$	−8.84094	−0.719465	−0.359733	−	0.933055i	$-0.617132\pi$
$151$	−8.84094	−0.719465	−0.359733	+	0.933055i	$0.617132\pi$
$152$	−16.3143	−1.32326
$153$	−5.07465	−0.410261
$154$	−0.286853	−0.0231153
$155$	5.41870	0.435241
$156$	−27.7642	−2.22292
$157$	14.0256	1.11936	0.559682	−	0.828707i	$-0.310923\pi$
$157$	14.0256	1.11936	0.559682	+	0.828707i	$0.310923\pi$
$158$	29.1438	2.31856
$159$	−4.64375	−0.368274
$160$	1.63829	0.129518
$161$	−0.148826	−0.0117291
$162$	1.77072	0.139121
$163$	−3.79103	−0.296937	−0.148468	−	0.988917i	$-0.547434\pi$
$163$	−3.79103	−0.296937	−0.148468	+	0.988917i	$0.547434\pi$
$164$	31.7644	2.48038
$165$	−0.111236	−0.00865972
$166$	6.63181	0.514728
$167$	−10.9434	−0.846828	−0.423414	−	0.905936i	$-0.639168\pi$
$167$	−10.9434	−0.846828	−0.423414	+	0.905936i	$0.639168\pi$
$168$	6.82105	0.526255
$169$	28.4732	2.19024
$170$	−8.02642	−0.615598
$171$	−6.37339	−0.487385
$172$	−19.1543	−1.46050
$173$	−19.6040	−1.49047	−0.745233	−	0.666804i	$-0.767662\pi$
$173$	−19.6040	−1.49047	−0.745233	+	0.666804i	$0.767662\pi$
$174$	18.8053	1.42562
$175$	−5.04482	−0.381352
$176$	0.269444	0.0203101
$177$	−13.0584	−0.981531
$178$	−6.13320	−0.459703
$179$	−20.8510	−1.55848	−0.779240	−	0.626726i	$-0.784395\pi$
$179$	−20.8510	−1.55848	−0.779240	+	0.626726i	$0.784395\pi$
$180$	−7.64352	−0.569714
$181$	−18.2169	−1.35405	−0.677027	−	0.735958i	$-0.736732\pi$
$181$	−18.2169	−1.35405	−0.677027	+	0.735958i	$0.736732\pi$
$182$	−21.2312	−1.57376
$183$	−3.97788	−0.294053
$184$	0.486984	0.0359010
$185$	−10.0842	−0.741408
$186$	12.8805	0.944444
$187$	0.253312	0.0185240
$188$	19.9529	1.45522
$189$	7.25095	0.527429
$190$	−10.0806	−0.731322
$191$	−12.0557	−0.872320	−0.436160	−	0.899869i	$-0.643662\pi$
$191$	−12.0557	−0.872320	−0.436160	+	0.899869i	$0.643662\pi$
$192$	10.8379	0.782156
$193$	21.6413	1.55778	0.778888	−	0.627163i	$-0.215784\pi$
$193$	21.6413	1.55778	0.778888	+	0.627163i	$0.215784\pi$
$194$	−6.98521	−0.501509
$195$	−8.23305	−0.589581
$196$	−19.7682	−1.41201
$197$	19.3181	1.37636	0.688179	−	0.725541i	$-0.258410\pi$
$197$	19.3181	1.37636	0.688179	+	0.725541i	$0.258410\pi$
$198$	0.366688	0.0260594
$199$	−4.89929	−0.347301	−0.173651	−	0.984807i	$-0.555556\pi$
$199$	−4.89929	−0.347301	−0.173651	+	0.984807i	$0.555556\pi$
$200$	16.5075	1.16726
$201$	−1.89554	−0.133701
$202$	−17.3458	−1.22045
$203$	9.46016	0.663973
$204$	−12.5513	−0.878768
$205$	9.41923	0.657868
$206$	−46.1198	−3.21332
$207$	0.190247	0.0132231
$208$	19.9427	1.38278
$209$	0.318141	0.0220063
$210$	4.21472	0.290843
$211$	−17.0224	−1.17187	−0.585934	−	0.810359i	$-0.699272\pi$
$211$	−17.0224	−1.17187	−0.585934	+	0.810359i	$0.699272\pi$
$212$	15.9282	1.09395
$213$	−8.38779	−0.574722
$214$	−13.8677	−0.947978
$215$	−5.67990	−0.387366
$216$	−23.7264	−1.61437
$217$	6.47965	0.439867
$218$	−27.1349	−1.83781
$219$	11.2630	0.761085
$220$	0.381542	0.0257236
$221$	18.7487	1.26117
$222$	−23.9707	−1.60881
$223$	0.795044	0.0532401	0.0266201	−	0.999646i	$-0.491526\pi$
$223$	0.795044	0.0532401	0.0266201	+	0.999646i	$0.491526\pi$
$224$	1.95906	0.130895
$225$	6.44887	0.429924
$226$	−14.6105	−0.971874
$227$	−6.10083	−0.404926	−0.202463	−	0.979290i	$-0.564895\pi$
$227$	−6.10083	−0.404926	−0.202463	+	0.979290i	$0.564895\pi$
$228$	−15.7635	−1.04397
$229$	−10.9860	−0.725972	−0.362986	−	0.931795i	$-0.618243\pi$
$229$	−10.9860	−0.725972	−0.362986	+	0.931795i	$0.618243\pi$
$230$	0.300907	0.0198412
$231$	−0.133015	−0.00875178
$232$	−30.9553	−2.03231
$233$	−13.8203	−0.905401	−0.452700	−	0.891663i	$-0.649539\pi$
$233$	−13.8203	−0.905401	−0.452700	+	0.891663i	$0.649539\pi$
$234$	27.1402	1.77421
$235$	5.91672	0.385964
$236$	44.7907	2.91562
$237$	13.5142	0.877839
$238$	−9.59794	−0.622142
$239$	9.16542	0.592862	0.296431	−	0.955054i	$-0.404204\pi$
$239$	9.16542	0.592862	0.296431	+	0.955054i	$0.404204\pi$
$240$	−3.95893	−0.255548
$241$	8.76857	0.564833	0.282417	−	0.959292i	$-0.408864\pi$
$241$	8.76857	0.564833	0.282417	+	0.959292i	$0.408864\pi$
$242$	26.5768	1.70842
$243$	−15.1316	−0.970695
$244$	13.6442	0.873481
$245$	−5.86195	−0.374506
$246$	22.3900	1.42753
$247$	23.5470	1.49826
$248$	−21.2025	−1.34636
$249$	3.07521	0.194884
$250$	23.9849	1.51694
$251$	−20.4788	−1.29261	−0.646304	−	0.763080i	$-0.723686\pi$
$251$	−20.4788	−1.29261	−0.646304	+	0.763080i	$0.723686\pi$
$252$	−9.14008	−0.575771
$253$	−0.00949656	−0.000597044 0
$254$	26.3644	1.65425
$255$	−3.72190	−0.233074
$256$	−30.2270	−1.88919
$257$	22.3861	1.39641	0.698204	−	0.715899i	$-0.253983\pi$
$257$	22.3861	1.39641	0.698204	+	0.715899i	$0.253983\pi$
$258$	−13.5014	−0.840559
$259$	−12.0587	−0.749289
$260$	28.2396	1.75134
$261$	−12.0931	−0.748542
$262$	39.6994	2.45264
$263$	15.1711	0.935492	0.467746	−	0.883863i	$-0.345066\pi$
$263$	15.1711	0.935492	0.467746	+	0.883863i	$0.345066\pi$
$264$	0.435250	0.0267878
$265$	4.72325	0.290147
$266$	−12.0543	−0.739097
$267$	−2.84400	−0.174050
$268$	6.50174	0.397157
$269$	−9.22904	−0.562705	−0.281352	−	0.959605i	$-0.590783\pi$
$269$	−9.22904	−0.562705	−0.281352	+	0.959605i	$0.590783\pi$
$270$	−14.6605	−0.892209
$271$	11.4607	0.696191	0.348095	−	0.937459i	$-0.386829\pi$
$271$	11.4607	0.696191	0.348095	+	0.937459i	$0.386829\pi$
$272$	9.01546	0.546643
$273$	−9.84504	−0.595849
$274$	−5.04041	−0.304502
$275$	−0.321909	−0.0194118
$276$	0.470544	0.0283234
$277$	16.4137	0.986205	0.493103	−	0.869971i	$-0.335863\pi$
$277$	16.4137	0.986205	0.493103	+	0.869971i	$0.335863\pi$
$278$	27.8298	1.66912
$279$	−8.28303	−0.495892
$280$	−6.93782	−0.414614
$281$	8.91042	0.531551	0.265776	−	0.964035i	$-0.414372\pi$
$281$	8.91042	0.531551	0.265776	+	0.964035i	$0.414372\pi$
$282$	14.0643	0.837518
$283$	9.10448	0.541205	0.270602	−	0.962691i	$-0.412777\pi$
$283$	9.10448	0.541205	0.270602	+	0.962691i	$0.412777\pi$
$284$	28.7703	1.70720
$285$	−4.67443	−0.276889
$286$	−1.35476	−0.0801085
$287$	11.2635	0.664861
$288$	−2.50429	−0.147567
$289$	−8.52433	−0.501431
$290$	−19.1272	−1.12319
$291$	−3.23908	−0.189878
$292$	−38.6324	−2.26079
$293$	−33.1169	−1.93471	−0.967356	−	0.253422i	$-0.918444\pi$
$293$	−33.1169	−1.93471	−0.967356	+	0.253422i	$0.918444\pi$
$294$	−13.9341	−0.812655
$295$	13.2820	0.773306
$296$	39.4580	2.29345
$297$	0.462682	0.0268475
$298$	16.0422	0.929300
$299$	−0.702880	−0.0406486
$300$	15.9502	0.920886
$301$	−6.79199	−0.391484
$302$	21.3751	1.23000
$303$	−8.04336	−0.462079
$304$	11.3228	0.649404
$305$	4.04598	0.231672
$306$	12.2692	0.701383
$307$	12.5643	0.717083	0.358542	−	0.933514i	$-0.383274\pi$
$307$	12.5643	0.717083	0.358542	+	0.933514i	$0.383274\pi$
$308$	0.456246	0.0259970
$309$	−21.3860	−1.21661
$310$	−13.1010	−0.744088
$311$	18.4434	1.04583	0.522913	−	0.852386i	$-0.324845\pi$
$311$	18.4434	1.04583	0.522913	+	0.852386i	$0.324845\pi$
$312$	32.2147	1.82380
$313$	20.7952	1.17541	0.587707	−	0.809074i	$-0.300031\pi$
$313$	20.7952	1.17541	0.587707	+	0.809074i	$0.300031\pi$
$314$	−33.9102	−1.91367
$315$	−2.71035	−0.152711
$316$	−46.3539	−2.60761
$317$	−1.02881	−0.0577839	−0.0288920	−	0.999583i	$-0.509198\pi$
$317$	−1.02881	−0.0577839	−0.0288920	+	0.999583i	$0.509198\pi$
$318$	11.2274	0.629601
$319$	0.603651	0.0337980
$320$	−11.0234	−0.616228
$321$	−6.43055	−0.358918
$322$	0.359823	0.0200521
$323$	10.6448	0.592294
$324$	−2.81638	−0.156465
$325$	−23.8258	−1.32162
$326$	9.16573	0.507643
$327$	−12.5826	−0.695821
$328$	−36.8560	−2.03503
$329$	7.07518	0.390067
$330$	0.268940	0.0148047
$331$	−17.1569	−0.943028	−0.471514	−	0.881859i	$-0.656292\pi$
$331$	−17.1569	−0.943028	−0.471514	+	0.881859i	$0.656292\pi$
$332$	−10.5480	−0.578899
$333$	15.4148	0.844724
$334$	26.4584	1.44774
$335$	1.92799	0.105337
$336$	−4.73407	−0.258265
$337$	8.10201	0.441344	0.220672	−	0.975348i	$-0.429175\pi$
$337$	8.10201	0.441344	0.220672	+	0.975348i	$0.429175\pi$
$338$	−68.8407	−3.74444
$339$	−6.77496	−0.367966
$340$	12.7662	0.692344
$341$	0.413465	0.0223904
$342$	15.4092	0.833234
$343$	−16.5548	−0.893873
$344$	22.2246	1.19827
$345$	0.139532	0.00751217
$346$	47.3975	2.54810
$347$	21.7315	1.16661	0.583305	−	0.812253i	$-0.301759\pi$
$347$	21.7315	1.16661	0.583305	+	0.812253i	$0.301759\pi$
$348$	−29.9102	−1.60336
$349$	−2.97203	−0.159089	−0.0795445	−	0.996831i	$-0.525347\pi$
$349$	−2.97203	−0.159089	−0.0795445	+	0.996831i	$0.525347\pi$
$350$	12.1971	0.651961
$351$	34.2450	1.82786
$352$	0.125007	0.00666290
$353$	12.4442	0.662339	0.331170	−	0.943571i	$-0.392557\pi$
$353$	12.4442	0.662339	0.331170	+	0.943571i	$0.392557\pi$
$354$	31.5719	1.67803
$355$	8.53139	0.452799
$356$	9.75500	0.517014
$357$	−4.45062	−0.235552
$358$	50.4124	2.66438
$359$	0.846392	0.0446709	0.0223354	−	0.999751i	$-0.492890\pi$
$359$	0.846392	0.0446709	0.0223354	+	0.999751i	$0.492890\pi$
$360$	8.86872	0.467423
$361$	−5.63090	−0.296363
$362$	44.0438	2.31489
$363$	12.3238	0.646834
$364$	33.7687	1.76996
$365$	−11.4559	−0.599627
$366$	9.61748	0.502714
$367$	−9.12744	−0.476448	−0.238224	−	0.971210i	$-0.576565\pi$
$367$	−9.12744	−0.476448	−0.238224	+	0.971210i	$0.576565\pi$
$368$	−0.337986	−0.0176187
$369$	−14.3983	−0.749543
$370$	24.3811	1.26751
$371$	5.64804	0.293232
$372$	−20.4867	−1.06219
$373$	32.8144	1.69907	0.849534	−	0.527534i	$-0.176883\pi$
$373$	32.8144	1.69907	0.849534	+	0.527534i	$0.176883\pi$
$374$	−0.612443	−0.0316687
$375$	11.1219	0.574335
$376$	−23.1512	−1.19393
$377$	44.6788	2.30107
$378$	−17.5309	−0.901693
$379$	30.2612	1.55441	0.777206	−	0.629247i	$-0.216636\pi$
$379$	30.2612	1.55441	0.777206	+	0.629247i	$0.216636\pi$
$380$	16.0334	0.822496
$381$	12.2253	0.626322
$382$	29.1476	1.49132
$383$	16.6575	0.851156	0.425578	−	0.904922i	$-0.360071\pi$
$383$	16.6575	0.851156	0.425578	+	0.904922i	$0.360071\pi$
$384$	−22.9817	−1.17278
$385$	0.135293	0.00689516
$386$	−52.3231	−2.66318
$387$	8.68230	0.441346
$388$	11.1101	0.564032
$389$	33.2405	1.68536	0.842680	−	0.538414i	$-0.180976\pi$
$389$	33.2405	1.68536	0.842680	+	0.538414i	$0.180976\pi$
$390$	19.9054	1.00795
$391$	−0.317750	−0.0160693
$392$	22.9369	1.15849
$393$	18.4088	0.928603
$394$	−46.7062	−2.35302
$395$	−13.7455	−0.691612
$396$	−0.583226	−0.0293082
$397$	30.8250	1.54706	0.773531	−	0.633758i	$-0.218489\pi$
$397$	30.8250	1.54706	0.773531	+	0.633758i	$0.218489\pi$
$398$	11.8452	0.593746
$399$	−5.58965	−0.279833
$400$	−11.4569	−0.572843
$401$	35.2234	1.75897	0.879486	−	0.475925i	$-0.157887\pi$
$401$	35.2234	1.75897	0.879486	+	0.475925i	$0.157887\pi$
$402$	4.58292	0.228575
$403$	30.6023	1.52441
$404$	27.5889	1.37260
$405$	−0.835152	−0.0414991
$406$	−22.8722	−1.13513
$407$	−0.769461	−0.0381408
$408$	14.5632	0.720986
$409$	−7.12867	−0.352490	−0.176245	−	0.984346i	$-0.556395\pi$
$409$	−7.12867	−0.352490	−0.176245	+	0.984346i	$0.556395\pi$
$410$	−22.7733	−1.12469
$411$	−2.33727	−0.115289
$412$	73.3545	3.61392
$413$	15.8825	0.781527
$414$	−0.459967	−0.0226061
$415$	−3.12786	−0.153541
$416$	9.25230	0.453631
$417$	12.9048	0.631952
$418$	−0.769182	−0.0376219
$419$	−17.4140	−0.850728	−0.425364	−	0.905022i	$-0.639854\pi$
$419$	−17.4140	−0.850728	−0.425364	+	0.905022i	$0.639854\pi$
$420$	−6.70360	−0.327102
$421$	2.64487	0.128903	0.0644516	−	0.997921i	$-0.479470\pi$
$421$	2.64487	0.128903	0.0644516	+	0.997921i	$0.479470\pi$
$422$	41.1557	2.00343
$423$	−9.04431	−0.439749
$424$	−18.4814	−0.897534
$425$	−10.7709	−0.522465
$426$	20.2795	0.982545
$427$	4.83816	0.234135
$428$	22.0569	1.06616
$429$	−0.628210	−0.0303303
$430$	13.7325	0.662241
$431$	5.47008	0.263484	0.131742	−	0.991284i	$-0.457943\pi$
$431$	5.47008	0.263484	0.131742	+	0.991284i	$0.457943\pi$
$432$	16.4670	0.792269
$433$	29.0482	1.39597	0.697984	−	0.716114i	$-0.254081\pi$
$433$	29.0482	1.39597	0.697984	+	0.716114i	$0.254081\pi$
$434$	−15.6661	−0.751998
$435$	−8.86941	−0.425256
$436$	43.1587	2.06693
$437$	−0.399070	−0.0190901
$438$	−27.2311	−1.30115
$439$	16.5909	0.791840	0.395920	−	0.918285i	$-0.370426\pi$
$439$	16.5909	0.791840	0.395920	+	0.918285i	$0.370426\pi$
$440$	−0.442701	−0.0211049
$441$	8.96059	0.426695
$442$	−45.3295	−2.15610
$443$	12.2823	0.583549	0.291774	−	0.956487i	$-0.405754\pi$
$443$	12.2823	0.583549	0.291774	+	0.956487i	$0.405754\pi$
$444$	38.1259	1.80938
$445$	2.89269	0.137127
$446$	−1.92221	−0.0910194
$447$	7.43887	0.351846
$448$	−13.1817	−0.622778
$449$	10.3039	0.486271	0.243136	−	0.969992i	$-0.421824\pi$
$449$	10.3039	0.486271	0.243136	+	0.969992i	$0.421824\pi$
$450$	−15.5917	−0.734999
$451$	0.718719	0.0338432
$452$	23.2383	1.09304
$453$	9.91177	0.465695
$454$	14.7502	0.692263
$455$	10.0136	0.469444
$456$	18.2903	0.856522
$457$	−20.7743	−0.971780	−0.485890	−	0.874020i	$-0.661504\pi$
$457$	−20.7743	−0.971780	−0.485890	+	0.874020i	$0.661504\pi$
$458$	26.5612	1.24112
$459$	15.4811	0.722594
$460$	−0.478600	−0.0223148
$461$	−3.64989	−0.169993	−0.0849963	−	0.996381i	$-0.527088\pi$
$461$	−3.64989	−0.169993	−0.0849963	+	0.996381i	$0.527088\pi$
$462$	0.321597	0.0149620
$463$	−2.20595	−0.102519	−0.0512595	−	0.998685i	$-0.516324\pi$
$463$	−2.20595	−0.102519	−0.0512595	+	0.998685i	$0.516324\pi$
$464$	21.4842	0.997377
$465$	−6.07502	−0.281722
$466$	33.4140	1.54787
$467$	−2.58229	−0.119494	−0.0597471	−	0.998214i	$-0.519029\pi$
$467$	−2.58229	−0.119494	−0.0597471	+	0.998214i	$0.519029\pi$
$468$	−43.1670	−1.99540
$469$	2.30548	0.106457
$470$	−14.3051	−0.659845
$471$	−15.7244	−0.724542
$472$	−51.9703	−2.39213
$473$	−0.433395	−0.0199275
$474$	−32.6738	−1.50076
$475$	−13.5274	−0.620681
$476$	15.2657	0.699704
$477$	−7.21997	−0.330580
$478$	−22.1596	−1.01356
$479$	−13.8358	−0.632176	−0.316088	−	0.948730i	$-0.602369\pi$
$479$	−13.8358	−0.632176	−0.316088	+	0.948730i	$0.602369\pi$
$480$	−1.83672	−0.0838345
$481$	−56.9510	−2.59674
$482$	−21.2001	−0.965639
$483$	0.166852	0.00759203
$484$	−42.2711	−1.92141
$485$	3.29454	0.149597
$486$	36.5844	1.65950
$487$	−3.79210	−0.171836	−0.0859181	−	0.996302i	$-0.527382\pi$
$487$	−3.79210	−0.171836	−0.0859181	+	0.996302i	$0.527382\pi$
$488$	−15.8313	−0.716649
$489$	4.25021	0.192201
$490$	14.1727	0.640257
$491$	11.2189	0.506300	0.253150	−	0.967427i	$-0.418533\pi$
$491$	11.2189	0.506300	0.253150	+	0.967427i	$0.418533\pi$
$492$	−35.6117	−1.60550
$493$	20.1978	0.909664
$494$	−56.9304	−2.56142
$495$	−0.172947	−0.00777338
$496$	14.7154	0.660740
$497$	10.2018	0.457613
$498$	−7.43506	−0.333173
$499$	−38.5875	−1.72742	−0.863708	−	0.503993i	$-0.831864\pi$
$499$	−38.5875	−1.72742	−0.863708	+	0.503993i	$0.831864\pi$
$500$	−38.1485	−1.70605
$501$	12.2689	0.548135
$502$	49.5124	2.20984
$503$	−32.4265	−1.44583	−0.722914	−	0.690938i	$-0.757198\pi$
$503$	−32.4265	−1.44583	−0.722914	+	0.690938i	$0.757198\pi$
$504$	10.6052	0.472392
$505$	8.18106	0.364053
$506$	0.0229602	0.00102071
$507$	−31.9219	−1.41770
$508$	−41.9331	−1.86048
$509$	−0.971433	−0.0430580	−0.0215290	−	0.999768i	$-0.506853\pi$
$509$	−0.971433	−0.0430580	−0.0215290	+	0.999768i	$0.506853\pi$
$510$	8.99859	0.398464
$511$	−13.6988	−0.606001
$512$	32.0833	1.41790
$513$	19.4431	0.858433
$514$	−54.1238	−2.38730
$515$	21.7521	0.958514
$516$	21.4743	0.945351
$517$	0.451466	0.0198554
$518$	29.1547	1.28099
$519$	21.9785	0.964749
$520$	−32.7662	−1.43689
$521$	−39.6649	−1.73775	−0.868876	−	0.495030i	$-0.835157\pi$
$521$	−39.6649	−1.73775	−0.868876	+	0.495030i	$0.835157\pi$
$522$	29.2379	1.27971
$523$	−19.7760	−0.864742	−0.432371	−	0.901696i	$-0.642323\pi$
$523$	−19.7760	−0.864742	−0.432371	+	0.901696i	$0.642323\pi$
$524$	−63.1427	−2.75840
$525$	5.65585	0.246842
$526$	−36.6799	−1.59932
$527$	13.8343	0.602632
$528$	−0.302080	−0.0131463
$529$	−22.9881	−0.999482
$530$	−11.4196	−0.496036
$531$	−20.3028	−0.881068
$532$	19.1726	0.831240
$533$	53.1954	2.30415
$534$	6.87607	0.297557
$535$	6.54064	0.282777
$536$	−7.54392	−0.325848
$537$	23.3765	1.00877
$538$	22.3134	0.962000
$539$	−0.447287	−0.0192660
$540$	23.3178	1.00344
$541$	26.1412	1.12390	0.561948	−	0.827172i	$-0.310052\pi$
$541$	26.1412	1.12390	0.561948	+	0.827172i	$0.310052\pi$
$542$	−27.7091	−1.19021
$543$	20.4234	0.876452
$544$	4.18267	0.179330
$545$	12.7980	0.548208
$546$	23.8028	1.01866
$547$	−24.2586	−1.03722	−0.518612	−	0.855010i	$-0.673551\pi$
$547$	−24.2586	−1.03722	−0.518612	+	0.855010i	$0.673551\pi$
$548$	8.01688	0.342464
$549$	−6.18469	−0.263956
$550$	0.778292	0.0331865
$551$	25.3670	1.08067
$552$	−0.545969	−0.0232380
$553$	−16.4368	−0.698965
$554$	−39.6841	−1.68602
$555$	11.3056	0.479898
$556$	−44.2639	−1.87721
$557$	7.12398	0.301853	0.150926	−	0.988545i	$-0.451774\pi$
$557$	7.12398	0.301853	0.150926	+	0.988545i	$0.451774\pi$
$558$	20.0262	0.847778
$559$	−32.0774	−1.35673
$560$	4.81512	0.203476
$561$	−0.283993	−0.0119902
$562$	−21.5431	−0.908741
$563$	2.07416	0.0874155	0.0437077	−	0.999044i	$-0.486083\pi$
$563$	2.07416	0.0874155	0.0437077	+	0.999044i	$0.486083\pi$
$564$	−22.3696	−0.941931
$565$	6.89095	0.289904
$566$	−22.0123	−0.925244
$567$	−0.998670	−0.0419402
$568$	−33.3820	−1.40068
$569$	30.8822	1.29465	0.647325	−	0.762214i	$-0.275888\pi$
$569$	30.8822	1.29465	0.647325	+	0.762214i	$0.275888\pi$
$570$	11.3016	0.473370
$571$	24.1494	1.01062	0.505311	−	0.862937i	$-0.331378\pi$
$571$	24.1494	1.01062	0.505311	+	0.862937i	$0.331378\pi$
$572$	2.15477	0.0900956
$573$	13.5159	0.564635
$574$	−27.2321	−1.13665
$575$	0.403796	0.0168395
$576$	16.8504	0.702100
$577$	20.9884	0.873759	0.436879	−	0.899520i	$-0.356084\pi$
$577$	20.9884	0.873759	0.436879	+	0.899520i	$0.356084\pi$
$578$	20.6096	0.857247
$579$	−24.2625	−1.00832
$580$	30.4223	1.26322
$581$	−3.74027	−0.155173
$582$	7.83127	0.324616
$583$	0.360400	0.0149262
$584$	44.8250	1.85487
$585$	−12.8005	−0.529236
$586$	80.0682	3.30759
$587$	36.8633	1.52151	0.760755	−	0.649039i	$-0.224829\pi$
$587$	36.8633	1.52151	0.760755	+	0.649039i	$0.224829\pi$
$588$	22.1626	0.913969
$589$	17.3749	0.715919
$590$	−32.1124	−1.32205
$591$	−21.6579	−0.890889
$592$	−27.3854	−1.12553
$593$	−14.3409	−0.588910	−0.294455	−	0.955665i	$-0.595138\pi$
$593$	−14.3409	−0.588910	−0.294455	+	0.955665i	$0.595138\pi$
$594$	−1.11864	−0.0458985
$595$	4.52682	0.185581
$596$	−25.5155	−1.04516
$597$	5.49269	0.224801
$598$	1.69938	0.0694929
$599$	2.81822	0.115149	0.0575746	−	0.998341i	$-0.481663\pi$
$599$	2.81822	0.115149	0.0575746	+	0.998341i	$0.481663\pi$
$600$	−18.5069	−0.755542
$601$	−36.6676	−1.49570	−0.747851	−	0.663867i	$-0.768914\pi$
$601$	−36.6676	−1.49570	−0.747851	+	0.663867i	$0.768914\pi$
$602$	16.4213	0.669281
$603$	−2.94713	−0.120016
$604$	−33.9976	−1.38334
$605$	−12.5348	−0.509613
$606$	19.4468	0.789971
$607$	−18.6646	−0.757573	−0.378786	−	0.925484i	$-0.623658\pi$
$607$	−18.6646	−0.757573	−0.378786	+	0.925484i	$0.623658\pi$
$608$	5.25312	0.213042
$609$	−10.6060	−0.429776
$610$	−9.78213	−0.396067
$611$	33.4149	1.35182
$612$	−19.5144	−0.788824
$613$	24.1854	0.976839	0.488420	−	0.872609i	$-0.337574\pi$
$613$	24.1854	0.976839	0.488420	+	0.872609i	$0.337574\pi$
$614$	−30.3773	−1.22593
$615$	−10.5601	−0.425825
$616$	−0.529379	−0.0213293
$617$	−2.83779	−0.114245	−0.0571225	−	0.998367i	$-0.518193\pi$
$617$	−2.83779	−0.114245	−0.0571225	+	0.998367i	$0.518193\pi$
$618$	51.7058	2.07991
$619$	11.4424	0.459910	0.229955	−	0.973201i	$-0.426142\pi$
$619$	11.4424	0.459910	0.229955	+	0.973201i	$0.426142\pi$
$620$	20.8375	0.836853
$621$	−0.580379	−0.0232898
$622$	−44.5913	−1.78795
$623$	3.45907	0.138585
$624$	−22.3582	−0.895045
$625$	7.18606	0.287442
$626$	−50.2774	−2.00949
$627$	−0.356675	−0.0142442
$628$	53.9350	2.15224
$629$	−25.7457	−1.02655
$630$	6.55292	0.261075
$631$	15.8419	0.630655	0.315327	−	0.948983i	$-0.397886\pi$
$631$	15.8419	0.630655	0.315327	+	0.948983i	$0.397886\pi$
$632$	53.7841	2.13942
$633$	19.0841	0.758527
$634$	2.48740	0.0987875
$635$	−12.4346	−0.493453
$636$	−17.8574	−0.708093
$637$	−33.1056	−1.31169
$638$	−1.45947	−0.0577810
$639$	−13.0411	−0.515898
$640$	23.3752	0.923985
$641$	−15.5611	−0.614626	−0.307313	−	0.951608i	$-0.599430\pi$
$641$	−15.5611	−0.614626	−0.307313	+	0.951608i	$0.599430\pi$
$642$	15.5474	0.613607
$643$	−15.0313	−0.592778	−0.296389	−	0.955067i	$-0.595782\pi$
$643$	−15.0313	−0.592778	−0.296389	+	0.955067i	$0.595782\pi$
$644$	−0.572306	−0.0225520
$645$	6.36786	0.250734
$646$	−25.7364	−1.01259
$647$	−1.26205	−0.0496163	−0.0248082	−	0.999692i	$-0.507897\pi$
$647$	−1.26205	−0.0496163	−0.0248082	+	0.999692i	$0.507897\pi$
$648$	3.26782	0.128372
$649$	1.01346	0.0397818
$650$	57.6047	2.25944
$651$	−7.26447	−0.284717
$652$	−14.5783	−0.570931
$653$	−42.9048	−1.67899	−0.839496	−	0.543365i	$-0.817150\pi$
$653$	−42.9048	−1.67899	−0.839496	+	0.543365i	$0.817150\pi$
$654$	30.4215	1.18958
$655$	−18.7240	−0.731607
$656$	25.5795	0.998711
$657$	17.5114	0.683186
$658$	−17.1060	−0.666860
$659$	29.4462	1.14706	0.573530	−	0.819185i	$-0.305574\pi$
$659$	29.4462	1.14706	0.573530	+	0.819185i	$0.305574\pi$
$660$	−0.427755	−0.0166504
$661$	−4.19473	−0.163156	−0.0815780	−	0.996667i	$-0.525996\pi$
$661$	−4.19473	−0.163156	−0.0815780	+	0.996667i	$0.525996\pi$
$662$	41.4809	1.61220
$663$	−21.0195	−0.816331
$664$	12.2388	0.474958
$665$	5.68535	0.220468
$666$	−37.2689	−1.44414
$667$	−0.757208	−0.0293192
$668$	−42.0827	−1.62823
$669$	−0.891341	−0.0344612
$670$	−4.66138	−0.180085
$671$	0.308722	0.0119181
$672$	−2.19634	−0.0847257
$673$	44.3742	1.71050	0.855250	−	0.518216i	$-0.173404\pi$
$673$	44.3742	1.71050	0.855250	+	0.518216i	$0.173404\pi$
$674$	−19.5886	−0.754523
$675$	−19.6733	−0.757227
$676$	109.493	4.21126
$677$	22.3922	0.860603	0.430302	−	0.902685i	$-0.358407\pi$
$677$	22.3922	0.860603	0.430302	+	0.902685i	$0.358407\pi$
$678$	16.3801	0.629074
$679$	3.93959	0.151188
$680$	−14.8125	−0.568035
$681$	6.83977	0.262101
$682$	−0.999651	−0.0382786
$683$	35.1837	1.34627	0.673133	−	0.739522i	$-0.264948\pi$
$683$	35.1837	1.34627	0.673133	+	0.739522i	$0.264948\pi$
$684$	−24.5087	−0.937113
$685$	2.37728	0.0908313
$686$	40.0251	1.52817
$687$	12.3166	0.469907
$688$	−15.4247	−0.588061
$689$	26.6747	1.01623
$690$	−0.337353	−0.0128428
$691$	4.56256	0.173568	0.0867839	−	0.996227i	$-0.472341\pi$
$691$	4.56256	0.173568	0.0867839	+	0.996227i	$0.472341\pi$
$692$	−75.3867	−2.86577
$693$	−0.206809	−0.00785601
$694$	−52.5412	−1.99444
$695$	−13.1258	−0.497889
$696$	34.7046	1.31548
$697$	24.0479	0.910881
$698$	7.18560	0.271979
$699$	15.4943	0.586047
$700$	−19.3997	−0.733240
$701$	−40.8926	−1.54449	−0.772245	−	0.635325i	$-0.780866\pi$
$701$	−40.8926	−1.54449	−0.772245	+	0.635325i	$0.780866\pi$
$702$	−82.7956	−3.12492
$703$	−32.3347	−1.21953
$704$	−0.841124	−0.0317010
$705$	−6.63336	−0.249827
$706$	−30.0869	−1.13234
$707$	9.78287	0.367923
$708$	−50.2158	−1.88722
$709$	−50.6931	−1.90382	−0.951910	−	0.306378i	$-0.900883\pi$
$709$	−50.6931	−1.90382	−0.951910	+	0.306378i	$0.900883\pi$
$710$	−20.6267	−0.774106
$711$	21.0114	0.787990
$712$	−11.3187	−0.424185
$713$	−0.518643	−0.0194233
$714$	10.7605	0.402700
$715$	0.638965	0.0238959
$716$	−80.1821	−2.99655
$717$	−10.2756	−0.383747
$718$	−2.04636	−0.0763694
$719$	46.2645	1.72537	0.862687	−	0.505739i	$-0.168780\pi$
$719$	46.2645	1.72537	0.862687	+	0.505739i	$0.168780\pi$
$720$	−6.15524	−0.229392
$721$	26.0111	0.968703
$722$	13.6140	0.506662
$723$	−9.83063	−0.365605
$724$	−70.0527	−2.60349
$725$	−25.6674	−0.953263
$726$	−29.7959	−1.10583
$727$	−42.6200	−1.58069	−0.790344	−	0.612663i	$-0.790098\pi$
$727$	−42.6200	−1.58069	−0.790344	+	0.612663i	$0.790098\pi$
$728$	−39.1816	−1.45217
$729$	19.1616	0.709687
$730$	27.6973	1.02512
$731$	−14.5012	−0.536345
$732$	−15.2968	−0.565387
$733$	18.7431	0.692294	0.346147	−	0.938180i	$-0.387490\pi$
$733$	18.7431	0.692294	0.346147	+	0.938180i	$0.387490\pi$
$734$	22.0678	0.814537
$735$	6.57196	0.242410
$736$	−0.156806	−0.00577996
$737$	0.147112	0.00541894
$738$	34.8113	1.28142
$739$	1.78903	0.0658106	0.0329053	−	0.999458i	$-0.489524\pi$
$739$	1.78903	0.0658106	0.0329053	+	0.999458i	$0.489524\pi$
$740$	−38.7786	−1.42553
$741$	−26.3990	−0.969791
$742$	−13.6555	−0.501309
$743$	−6.38946	−0.234407	−0.117203	−	0.993108i	$-0.537393\pi$
$743$	−6.38946	−0.234407	−0.117203	+	0.993108i	$0.537393\pi$
$744$	23.7706	0.871473
$745$	−7.56622	−0.277205
$746$	−79.3368	−2.90473
$747$	4.78125	0.174937
$748$	0.974104	0.0356168
$749$	7.82126	0.285783
$750$	−26.8900	−0.981884
$751$	13.7345	0.501180	0.250590	−	0.968093i	$-0.419375\pi$
$751$	13.7345	0.501180	0.250590	+	0.968093i	$0.419375\pi$
$752$	16.0678	0.585934
$753$	22.9592	0.836679
$754$	−108.022	−3.93392
$755$	−10.0815	−0.366902
$756$	27.8833	1.01411
$757$	49.9638	1.81597	0.907983	−	0.419007i	$-0.137622\pi$
$757$	49.9638	1.81597	0.907983	+	0.419007i	$0.137622\pi$
$758$	−73.1636	−2.65742
$759$	0.0106468	0.000386454 0
$760$	−18.6034	−0.674818
$761$	26.7674	0.970318	0.485159	−	0.874426i	$-0.338762\pi$
$761$	26.7674	0.970318	0.485159	+	0.874426i	$0.338762\pi$
$762$	−29.5577	−1.07076
$763$	15.3038	0.554035
$764$	−46.3599	−1.67724
$765$	−5.78670	−0.209219
$766$	−40.2734	−1.45514
$767$	75.0104	2.70847
$768$	33.8882	1.22283
$769$	50.7200	1.82901	0.914505	−	0.404575i	$-0.132580\pi$
$769$	50.7200	1.82901	0.914505	+	0.404575i	$0.132580\pi$
$770$	−0.327103	−0.0117880
$771$	−25.0976	−0.903867
$772$	83.2211	2.99519
$773$	6.03803	0.217173	0.108587	−	0.994087i	$-0.465368\pi$
$773$	6.03803	0.217173	0.108587	+	0.994087i	$0.465368\pi$
$774$	−20.9916	−0.754526
$775$	−17.5806	−0.631515
$776$	−12.8910	−0.462760
$777$	13.5192	0.485000
$778$	−80.3669	−2.88129
$779$	30.2024	1.08211
$780$	−31.6600	−1.13361
$781$	0.650974	0.0232937
$782$	0.768236	0.0274721
$783$	36.8919	1.31841
$784$	−15.9191	−0.568539
$785$	15.9936	0.570836
$786$	−44.5078	−1.58754
$787$	20.6142	0.734817	0.367408	−	0.930060i	$-0.380245\pi$
$787$	20.6142	0.734817	0.367408	+	0.930060i	$0.380245\pi$
$788$	74.2872	2.64637
$789$	−17.0087	−0.605525
$790$	33.2331	1.18238
$791$	8.24016	0.292986
$792$	0.676713	0.0240460
$793$	22.8498	0.811420
$794$	−74.5269	−2.64486
$795$	−5.29534	−0.187806
$796$	−18.8401	−0.667768
$797$	42.8669	1.51842	0.759212	−	0.650843i	$-0.225584\pi$
$797$	42.8669	1.51842	0.759212	+	0.650843i	$0.225584\pi$
$798$	13.5143	0.478402
$799$	15.1058	0.534405
$800$	−5.31533	−0.187925
$801$	−4.42178	−0.156236
$802$	−85.1610	−3.00714
$803$	−0.874120	−0.0308470
$804$	−7.28924	−0.257072
$805$	−0.169709	−0.00598144
$806$	−73.9884	−2.60613
$807$	10.3469	0.364227
$808$	−32.0112	−1.12615
$809$	−9.34123	−0.328420	−0.164210	−	0.986425i	$-0.552508\pi$
$809$	−9.34123	−0.328420	−0.164210	+	0.986425i	$0.552508\pi$
$810$	2.01918	0.0709468
$811$	17.9739	0.631150	0.315575	−	0.948901i	$-0.397803\pi$
$811$	17.9739	0.631150	0.315575	+	0.948901i	$0.397803\pi$
$812$	36.3788	1.27665
$813$	−12.8489	−0.450630
$814$	1.86036	0.0652055
$815$	−4.32297	−0.151427
$816$	−10.1074	−0.353831
$817$	−18.2124	−0.637171
$818$	17.2353	0.602617
$819$	−15.3068	−0.534862
$820$	36.2214	1.26491
$821$	−7.10180	−0.247854	−0.123927	−	0.992291i	$-0.539549\pi$
$821$	−7.10180	−0.247854	−0.123927	+	0.992291i	$0.539549\pi$
$822$	5.65091	0.197098
$823$	0.388681	0.0135486	0.00677428	−	0.999977i	$-0.497844\pi$
$823$	0.388681	0.0135486	0.00677428	+	0.999977i	$0.497844\pi$
$824$	−85.1127	−2.96504
$825$	0.360899	0.0125649
$826$	−38.3998	−1.33610
$827$	−15.0709	−0.524068	−0.262034	−	0.965059i	$-0.584393\pi$
$827$	−15.0709	−0.524068	−0.262034	+	0.965059i	$0.584393\pi$
$828$	0.731588	0.0254244
$829$	27.4237	0.952463	0.476232	−	0.879320i	$-0.342002\pi$
$829$	27.4237	0.952463	0.476232	+	0.879320i	$0.342002\pi$
$830$	7.56235	0.262493
$831$	−18.4018	−0.638351
$832$	−62.2551	−2.15831
$833$	−14.9660	−0.518540
$834$	−31.2005	−1.08039
$835$	−12.4790	−0.431852
$836$	1.22340	0.0423123
$837$	25.2688	0.873416
$838$	42.1025	1.45441
$839$	21.2984	0.735304	0.367652	−	0.929963i	$-0.380162\pi$
$839$	21.2984	0.735304	0.367652	+	0.929963i	$0.380162\pi$
$840$	7.77814	0.268371
$841$	19.1321	0.659728
$842$	−6.39461	−0.220373
$843$	−9.98967	−0.344062
$844$	−65.4591	−2.25319
$845$	32.4684	1.11695
$846$	21.8668	0.751796
$847$	−14.9891	−0.515031
$848$	12.8268	0.440473
$849$	−10.2072	−0.350311
$850$	26.0412	0.893206
$851$	0.965197	0.0330865
$852$	−32.2550	−1.10504
$853$	−20.7052	−0.708933	−0.354467	−	0.935069i	$-0.615338\pi$
$853$	−20.7052	−0.708933	−0.354467	+	0.935069i	$0.615338\pi$
$854$	−11.6974	−0.400277
$855$	−7.26767	−0.248549
$856$	−25.5925	−0.874734
$857$	−15.8628	−0.541861	−0.270931	−	0.962599i	$-0.587331\pi$
$857$	−15.8628	−0.541861	−0.270931	+	0.962599i	$0.587331\pi$
$858$	1.51885	0.0518526
$859$	−34.3885	−1.17332	−0.586660	−	0.809833i	$-0.699558\pi$
$859$	−34.3885	−1.17332	−0.586660	+	0.809833i	$0.699558\pi$
$860$	−21.8419	−0.744802
$861$	−12.6277	−0.430351
$862$	−13.2252	−0.450453
$863$	−38.0561	−1.29545	−0.647723	−	0.761876i	$-0.724278\pi$
$863$	−38.0561	−1.29545	−0.647723	+	0.761876i	$0.724278\pi$
$864$	7.63976	0.259910
$865$	−22.3548	−0.760085
$866$	−70.2310	−2.38655
$867$	9.55680	0.324566
$868$	24.9173	0.845749
$869$	−1.04883	−0.0355791
$870$	21.4439	0.727018
$871$	10.8884	0.368939
$872$	−50.0767	−1.69581
$873$	−5.03603	−0.170444
$874$	0.964848	0.0326365
$875$	−13.5272	−0.457304
$876$	43.3117	1.46337
$877$	−14.5169	−0.490202	−0.245101	−	0.969498i	$-0.578821\pi$
$877$	−14.5169	−0.490202	−0.245101	+	0.969498i	$0.578821\pi$
$878$	−40.1125	−1.35373
$879$	37.1281	1.25230
$880$	0.307252	0.0103574
$881$	26.5788	0.895463	0.447732	−	0.894168i	$-0.352232\pi$
$881$	26.5788	0.895463	0.447732	+	0.894168i	$0.352232\pi$
$882$	−21.6644	−0.729478
$883$	−24.1050	−0.811198	−0.405599	−	0.914051i	$-0.632937\pi$
$883$	−24.1050	−0.811198	−0.405599	+	0.914051i	$0.632937\pi$
$884$	72.0975	2.42490
$885$	−14.8907	−0.500546
$886$	−29.6954	−0.997635
$887$	−30.6039	−1.02758	−0.513789	−	0.857917i	$-0.671758\pi$
$887$	−30.6039	−1.02758	−0.513789	+	0.857917i	$0.671758\pi$
$888$	−44.2372	−1.48450
$889$	−14.8692	−0.498698
$890$	−6.99378	−0.234432
$891$	−0.0637249	−0.00213487
$892$	3.05732	0.102367
$893$	18.9718	0.634866
$894$	−17.9853	−0.601517
$895$	−23.7767	−0.794769
$896$	27.9519	0.933808
$897$	0.788014	0.0263110
$898$	−24.9122	−0.831330
$899$	32.9676	1.09953
$900$	24.7989	0.826631
$901$	12.0588	0.401736
$902$	−1.73768	−0.0578583
$903$	7.61464	0.253400
$904$	−26.9632	−0.896783
$905$	−20.7730	−0.690520
$906$	−23.9641	−0.796153
$907$	17.9419	0.595751	0.297875	−	0.954605i	$-0.403722\pi$
$907$	17.9419	0.595751	0.297875	+	0.954605i	$0.403722\pi$
$908$	−23.4606	−0.778567
$909$	−12.5056	−0.414784
$910$	−24.2103	−0.802562
$911$	−26.2901	−0.871031	−0.435515	−	0.900181i	$-0.643434\pi$
$911$	−26.2901	−0.871031	−0.435515	+	0.900181i	$0.643434\pi$
$912$	−12.6942	−0.420346
$913$	−0.238666	−0.00789870
$914$	50.2268	1.66136
$915$	−4.53603	−0.149957
$916$	−42.2462	−1.39585
$917$	−22.3900	−0.739384
$918$	−37.4292	−1.23535
$919$	−20.4143	−0.673407	−0.336703	−	0.941611i	$-0.609312\pi$
$919$	−20.4143	−0.673407	−0.336703	+	0.941611i	$0.609312\pi$
$920$	0.555316	0.0183082
$921$	−14.0861	−0.464153
$922$	8.82450	0.290619
$923$	48.1813	1.58591
$924$	−0.511507	−0.0168274
$925$	32.7177	1.07575
$926$	5.33341	0.175267
$927$	−33.2503	−1.09208
$928$	9.96744	0.327197
$929$	−11.6955	−0.383716	−0.191858	−	0.981423i	$-0.561451\pi$
$929$	−11.6955	−0.383716	−0.191858	+	0.981423i	$0.561451\pi$
$930$	14.6878	0.481633
$931$	−18.7961	−0.616019
$932$	−53.1457	−1.74085
$933$	−20.6772	−0.676942
$934$	6.24331	0.204287
$935$	0.288855	0.00944658
$936$	50.0864	1.63713
$937$	52.6629	1.72042	0.860211	−	0.509938i	$-0.170332\pi$
$937$	52.6629	1.72042	0.860211	+	0.509938i	$0.170332\pi$
$938$	−5.57405	−0.181999
$939$	−23.3139	−0.760821
$940$	22.7526	0.742108
$941$	3.60812	0.117621	0.0588107	−	0.998269i	$-0.481269\pi$
$941$	3.60812	0.117621	0.0588107	+	0.998269i	$0.481269\pi$
$942$	38.0175	1.23868
$943$	−0.901547	−0.0293584
$944$	36.0694	1.17396
$945$	8.26837	0.268970
$946$	1.04784	0.0340681
$947$	51.9572	1.68838	0.844191	−	0.536042i	$-0.180081\pi$
$947$	51.9572	1.68838	0.844191	+	0.536042i	$0.180081\pi$
$948$	51.9683	1.68785
$949$	−64.6973	−2.10016
$950$	32.7058	1.06112
$951$	1.15342	0.0374024
$952$	−17.7127	−0.574073
$953$	−11.5914	−0.375481	−0.187740	−	0.982219i	$-0.560116\pi$
$953$	−11.5914	−0.375481	−0.187740	+	0.982219i	$0.560116\pi$
$954$	17.4560	0.565160
$955$	−13.7473	−0.444852
$956$	35.2454	1.13992
$957$	−0.676766	−0.0218767
$958$	33.4515	1.08077
$959$	2.84274	0.0917969
$960$	12.3586	0.398872
$961$	−8.41913	−0.271585
$962$	137.693	4.43940
$963$	−9.99803	−0.322182
$964$	33.7193	1.08602
$965$	24.6779	0.794410
$966$	−0.403405	−0.0129793
$967$	−49.3372	−1.58658	−0.793289	−	0.608846i	$-0.791633\pi$
$967$	−49.3372	−1.58658	−0.793289	+	0.608846i	$0.791633\pi$
$968$	49.0468	1.57642
$969$	−11.9341	−0.383380
$970$	−7.96534	−0.255752
$971$	−22.8586	−0.733568	−0.366784	−	0.930306i	$-0.619541\pi$
$971$	−22.8586	−0.733568	−0.366784	+	0.930306i	$0.619541\pi$
$972$	−58.1883	−1.86639
$973$	−15.6957	−0.503181
$974$	9.16831	0.293771
$975$	26.7116	0.855457
$976$	10.9875	0.351702
$977$	14.6817	0.469709	0.234854	−	0.972031i	$-0.424539\pi$
$977$	14.6817	0.469709	0.234854	+	0.972031i	$0.424539\pi$
$978$	−10.2759	−0.328587
$979$	0.220722	0.00705431
$980$	−22.5420	−0.720077
$981$	−19.5631	−0.624602
$982$	−27.1243	−0.865571
$983$	35.0352	1.11745	0.558725	−	0.829353i	$-0.311291\pi$
$983$	35.0352	1.11745	0.558725	+	0.829353i	$0.311291\pi$
$984$	41.3200	1.31723
$985$	22.0287	0.701894
$986$	−48.8331	−1.55516
$987$	−7.93214	−0.252483
$988$	90.5492	2.88075
$989$	0.543643	0.0172868
$990$	0.418140	0.0132894
$991$	−35.7011	−1.13408	−0.567041	−	0.823689i	$-0.691912\pi$
$991$	−35.7011	−1.13408	−0.567041	+	0.823689i	$0.691912\pi$
$992$	6.82710	0.216761
$993$	19.2350	0.610403
$994$	−24.6653	−0.782335
$995$	−5.58673	−0.177111
$996$	11.8256	0.374710
$997$	−11.7568	−0.372342	−0.186171	−	0.982517i	$-0.559608\pi$
$997$	−11.7568	−0.372342	−0.186171	+	0.982517i	$0.559608\pi$
$998$	93.2947	2.95319
$999$	−47.0253	−1.48782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.c.1.15	✓	174

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.c.1.15	✓	174	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:	\( N \)	\(=\)	\( 4027 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41774 q^{2} -1.12112 q^{3} +3.84547 q^{4} +1.14031 q^{5} +2.71058 q^{6} +1.36358 q^{7} -4.46187 q^{8} -1.74309 q^{9} +O(q^{10})\)	\(q-2.41774 q^{2} -1.12112 q^{3} +3.84547 q^{4} +1.14031 q^{5} +2.71058 q^{6} +1.36358 q^{7} -4.46187 q^{8} -1.74309 q^{9} -2.75699 q^{10} +0.0870099 q^{11} -4.31124 q^{12} +6.43997 q^{13} -3.29679 q^{14} -1.27843 q^{15} +3.09671 q^{16} +2.91130 q^{17} +4.21433 q^{18} +3.65638 q^{19} +4.38505 q^{20} -1.52874 q^{21} -0.210367 q^{22} -0.109143 q^{23} +5.00230 q^{24} -3.69968 q^{25} -15.5702 q^{26} +5.31758 q^{27} +5.24362 q^{28} +6.93773 q^{29} +3.09092 q^{30} +4.75193 q^{31} +1.43670 q^{32} -0.0975486 q^{33} -7.03877 q^{34} +1.55491 q^{35} -6.70299 q^{36} -8.84338 q^{37} -8.84018 q^{38} -7.21998 q^{39} -5.08794 q^{40} +8.26020 q^{41} +3.69610 q^{42} -4.98099 q^{43} +0.334594 q^{44} -1.98767 q^{45} +0.263881 q^{46} +5.18867 q^{47} -3.47179 q^{48} -5.14064 q^{49} +8.94487 q^{50} -3.26392 q^{51} +24.7647 q^{52} +4.14206 q^{53} -12.8565 q^{54} +0.0992186 q^{55} -6.08413 q^{56} -4.09924 q^{57} -16.7736 q^{58} +11.6476 q^{59} -4.91617 q^{60} +3.54812 q^{61} -11.4889 q^{62} -2.37684 q^{63} -9.66699 q^{64} +7.34359 q^{65} +0.235847 q^{66} +1.69075 q^{67} +11.1953 q^{68} +0.122363 q^{69} -3.75938 q^{70} +7.48161 q^{71} +7.77743 q^{72} -10.0462 q^{73} +21.3810 q^{74} +4.14779 q^{75} +14.0605 q^{76} +0.118645 q^{77} +17.4561 q^{78} -12.0541 q^{79} +3.53123 q^{80} -0.732387 q^{81} -19.9710 q^{82} -2.74298 q^{83} -5.87873 q^{84} +3.31980 q^{85} +12.0427 q^{86} -7.77804 q^{87} -0.388227 q^{88} +2.53675 q^{89} +4.80567 q^{90} +8.78142 q^{91} -0.419708 q^{92} -5.32749 q^{93} -12.5449 q^{94} +4.16942 q^{95} -1.61071 q^{96} +2.88915 q^{97} +12.4287 q^{98} -0.151666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9}+O(q^{10}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9	\( 174 q + 21 q^{2} + 17 q^{3} + 187 q^{4} + 72 q^{5} + 21 q^{6} + 24 q^{7} + 54 q^{8} + 197 q^{9} + 20 q^{10} + 35 q^{11} + 23 q^{12} + 91 q^{13} + 18 q^{14} + 16 q^{15} + 201 q^{16} + 148 q^{17} + 39 q^{18} + 36 q^{19} + 128 q^{20} + 57 q^{21} + 17 q^{22} + 96 q^{23} + 24 q^{24} + 226 q^{25} + 44 q^{26} + 62 q^{27} + 32 q^{28} + 122 q^{29} + 25 q^{30} + 23 q^{31} + 104 q^{32} + 91 q^{33} + 6 q^{34} + 80 q^{35} + 222 q^{36} + 71 q^{37} + 125 q^{38} + 16 q^{39} + 53 q^{40} + 97 q^{41} + 14 q^{42} + 38 q^{43} + 70 q^{44} + 185 q^{45} - 23 q^{46} + 110 q^{47} + 36 q^{48} + 210 q^{49} + 51 q^{50} + 33 q^{51} + 118 q^{52} + 214 q^{53} + 8 q^{54} + 37 q^{55} + 41 q^{56} + 76 q^{57} + 2 q^{58} + 66 q^{59} - 12 q^{60} + 114 q^{61} + 175 q^{62} + 62 q^{63} + 190 q^{64} + 128 q^{65} + 12 q^{66} - 6 q^{67} + 348 q^{68} + 115 q^{69} - 38 q^{70} + 54 q^{71} + 101 q^{72} + 107 q^{73} + 71 q^{74} - q^{75} + 31 q^{76} + 368 q^{77} - 14 q^{78} - 14 q^{79} + 205 q^{80} + 222 q^{81} + 26 q^{82} + 246 q^{83} + 41 q^{84} + 87 q^{85} + 33 q^{86} + 100 q^{87} - 6 q^{88} + 147 q^{89} + 50 q^{90} - 23 q^{91} + 189 q^{92} + 117 q^{93} + 23 q^{94} + 42 q^{95} + 38 q^{96} + 52 q^{97} + 148 q^{98} + 38 q^{99}+O(q^{100}) \) 174 * q + 21 * q^2 + 17 * q^3 + 187 * q^4 + 72 * q^5 + 21 * q^6 + 24 * q^7 + 54 * q^8 + 197 * q^9 + 20 * q^10 + 35 * q^11 + 23 * q^12 + 91 * q^13 + 18 * q^14 + 16 * q^15 + 201 * q^16 + 148 * q^17 + 39 * q^18 + 36 * q^19 + 128 * q^20 + 57 * q^21 + 17 * q^22 + 96 * q^23 + 24 * q^24 + 226 * q^25 + 44 * q^26 + 62 * q^27 + 32 * q^28 + 122 * q^29 + 25 * q^30 + 23 * q^31 + 104 * q^32 + 91 * q^33 + 6 * q^34 + 80 * q^35 + 222 * q^36 + 71 * q^37 + 125 * q^38 + 16 * q^39 + 53 * q^40 + 97 * q^41 + 14 * q^42 + 38 * q^43 + 70 * q^44 + 185 * q^45 - 23 * q^46 + 110 * q^47 + 36 * q^48 + 210 * q^49 + 51 * q^50 + 33 * q^51 + 118 * q^52 + 214 * q^53 + 8 * q^54 + 37 * q^55 + 41 * q^56 + 76 * q^57 + 2 * q^58 + 66 * q^59 - 12 * q^60 + 114 * q^61 + 175 * q^62 + 62 * q^63 + 190 * q^64 + 128 * q^65 + 12 * q^66 - 6 * q^67 + 348 * q^68 + 115 * q^69 - 38 * q^70 + 54 * q^71 + 101 * q^72 + 107 * q^73 + 71 * q^74 - q^75 + 31 * q^76 + 368 * q^77 - 14 * q^78 - 14 * q^79 + 205 * q^80 + 222 * q^81 + 26 * q^82 + 246 * q^83 + 41 * q^84 + 87 * q^85 + 33 * q^86 + 100 * q^87 - 6 * q^88 + 147 * q^89 + 50 * q^90 - 23 * q^91 + 189 * q^92 + 117 * q^93 + 23 * q^94 + 42 * q^95 + 38 * q^96 + 52 * q^97 + 148 * q^98 + 38 * q^99

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