LMFDB - Embedded newform 4027.2.a.b.1.20

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:	$1$
Dimension:	$159$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-2.29049 q^{2} +2.02403 q^{3} +3.24633 q^{4} -4.01248 q^{5} -4.63602 q^{6} +1.58118 q^{7} -2.85471 q^{8} +1.09671 q^{9} +O(q^{10})$	\(q-2.29049 q^{2} +2.02403 q^{3} +3.24633 q^{4} -4.01248 q^{5} -4.63602 q^{6} +1.58118 q^{7} -2.85471 q^{8} +1.09671 q^{9} +9.19054 q^{10} -2.14715 q^{11} +6.57068 q^{12} +4.41315 q^{13} -3.62167 q^{14} -8.12139 q^{15} +0.0460086 q^{16} -2.78337 q^{17} -2.51199 q^{18} -7.22508 q^{19} -13.0258 q^{20} +3.20036 q^{21} +4.91803 q^{22} +3.22321 q^{23} -5.77802 q^{24} +11.1000 q^{25} -10.1083 q^{26} -3.85233 q^{27} +5.13303 q^{28} +7.38099 q^{29} +18.6019 q^{30} +6.87606 q^{31} +5.60403 q^{32} -4.34591 q^{33} +6.37527 q^{34} -6.34445 q^{35} +3.56027 q^{36} +10.9746 q^{37} +16.5490 q^{38} +8.93236 q^{39} +11.4545 q^{40} -5.77644 q^{41} -7.33037 q^{42} -10.5071 q^{43} -6.97037 q^{44} -4.40051 q^{45} -7.38271 q^{46} -2.19319 q^{47} +0.0931229 q^{48} -4.49987 q^{49} -25.4244 q^{50} -5.63363 q^{51} +14.3266 q^{52} +3.69705 q^{53} +8.82371 q^{54} +8.61541 q^{55} -4.51380 q^{56} -14.6238 q^{57} -16.9061 q^{58} -2.21439 q^{59} -26.3647 q^{60} -2.77468 q^{61} -15.7495 q^{62} +1.73409 q^{63} -12.9280 q^{64} -17.7077 q^{65} +9.95425 q^{66} +9.94008 q^{67} -9.03574 q^{68} +6.52387 q^{69} +14.5319 q^{70} +6.00154 q^{71} -3.13078 q^{72} +11.1769 q^{73} -25.1371 q^{74} +22.4668 q^{75} -23.4550 q^{76} -3.39503 q^{77} -20.4595 q^{78} -14.2564 q^{79} -0.184609 q^{80} -11.0874 q^{81} +13.2309 q^{82} -12.7619 q^{83} +10.3894 q^{84} +11.1682 q^{85} +24.0665 q^{86} +14.9394 q^{87} +6.12950 q^{88} -6.32019 q^{89} +10.0793 q^{90} +6.97798 q^{91} +10.4636 q^{92} +13.9174 q^{93} +5.02347 q^{94} +28.9905 q^{95} +11.3427 q^{96} +2.82895 q^{97} +10.3069 q^{98} -2.35480 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9}+O(q^{10}) $ 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9} - 23 q^{10} - 33 q^{11} - 57 q^{12} - 90 q^{13} - 28 q^{14} - 22 q^{15} + 130 q^{16} - 145 q^{17} - 50 q^{18} - 28 q^{19} - 121 q^{20} - 69 q^{21} - 26 q^{22} - 79 q^{23} - 62 q^{24} + 123 q^{25} - 40 q^{26} - 70 q^{27} - 43 q^{28} - 109 q^{29} - 43 q^{30} - 21 q^{31} - 139 q^{32} - 83 q^{33} - 93 q^{35} + 75 q^{36} - 65 q^{37} - 122 q^{38} - 18 q^{39} - 43 q^{40} - 71 q^{41} - 88 q^{42} - 72 q^{43} - 79 q^{44} - 181 q^{45} - 11 q^{46} - 114 q^{47} - 118 q^{48} + 118 q^{49} - 77 q^{50} - 29 q^{51} - 169 q^{52} - 220 q^{53} - 80 q^{54} - 37 q^{55} - 72 q^{56} - 90 q^{57} - 8 q^{58} - 60 q^{59} - 42 q^{60} - 108 q^{61} - 152 q^{62} - 65 q^{63} + 114 q^{64} - 81 q^{65} - 40 q^{66} - 50 q^{67} - 319 q^{68} - 103 q^{69} + 4 q^{70} - 7 q^{71} - 129 q^{72} - 94 q^{73} - 79 q^{74} - 59 q^{75} - 46 q^{76} - 329 q^{77} + 8 q^{78} - 18 q^{79} - 190 q^{80} + 59 q^{81} - 56 q^{82} - 201 q^{83} - 71 q^{84} - 26 q^{85} - 52 q^{86} - 126 q^{87} - 66 q^{88} - 114 q^{89} - 33 q^{90} - 30 q^{91} - 204 q^{92} - 125 q^{93} + 9 q^{94} - 84 q^{95} - 88 q^{96} - 56 q^{97} - 110 q^{98} - 46 q^{99}+O(q^{100}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9 - 23 * q^10 - 33 * q^11 - 57 * q^12 - 90 * q^13 - 28 * q^14 - 22 * q^15 + 130 * q^16 - 145 * q^17 - 50 * q^18 - 28 * q^19 - 121 * q^20 - 69 * q^21 - 26 * q^22 - 79 * q^23 - 62 * q^24 + 123 * q^25 - 40 * q^26 - 70 * q^27 - 43 * q^28 - 109 * q^29 - 43 * q^30 - 21 * q^31 - 139 * q^32 - 83 * q^33 - 93 * q^35 + 75 * q^36 - 65 * q^37 - 122 * q^38 - 18 * q^39 - 43 * q^40 - 71 * q^41 - 88 * q^42 - 72 * q^43 - 79 * q^44 - 181 * q^45 - 11 * q^46 - 114 * q^47 - 118 * q^48 + 118 * q^49 - 77 * q^50 - 29 * q^51 - 169 * q^52 - 220 * q^53 - 80 * q^54 - 37 * q^55 - 72 * q^56 - 90 * q^57 - 8 * q^58 - 60 * q^59 - 42 * q^60 - 108 * q^61 - 152 * q^62 - 65 * q^63 + 114 * q^64 - 81 * q^65 - 40 * q^66 - 50 * q^67 - 319 * q^68 - 103 * q^69 + 4 * q^70 - 7 * q^71 - 129 * q^72 - 94 * q^73 - 79 * q^74 - 59 * q^75 - 46 * q^76 - 329 * q^77 + 8 * q^78 - 18 * q^79 - 190 * q^80 + 59 * q^81 - 56 * q^82 - 201 * q^83 - 71 * q^84 - 26 * q^85 - 52 * q^86 - 126 * q^87 - 66 * q^88 - 114 * q^89 - 33 * q^90 - 30 * q^91 - 204 * q^92 - 125 * q^93 + 9 * q^94 - 84 * q^95 - 88 * q^96 - 56 * q^97 - 110 * q^98 - 46 * 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 $

$2$	−2.29049	−1.61962	−0.809810	−	0.586693i	$-0.800430\pi$
$2$	−2.29049	−1.61962	−0.809810	+	0.586693i	$0.800430\pi$
$3$	2.02403	1.16858	0.584288	−	0.811547i	$-0.301374\pi$
$3$	2.02403	1.16858	0.584288	+	0.811547i	$0.301374\pi$
$4$	3.24633	1.62317
$5$	−4.01248	−1.79444	−0.897218	−	0.441588i	$-0.854415\pi$
$5$	−4.01248	−1.79444	−0.897218	+	0.441588i	$0.854415\pi$
$6$	−4.63602	−1.89265
$7$	1.58118	0.597629	0.298815	−	0.954311i	$-0.403409\pi$
$7$	1.58118	0.597629	0.298815	+	0.954311i	$0.403409\pi$
$8$	−2.85471	−1.00929
$9$	1.09671	0.365569
$10$	9.19054	2.90630
$11$	−2.14715	−0.647391	−0.323696	−	0.946161i	$-0.604925\pi$
$11$	−2.14715	−0.647391	−0.323696	+	0.946161i	$0.604925\pi$
$12$	6.57068	1.89679
$13$	4.41315	1.22399	0.611994	−	0.790862i	$-0.290368\pi$
$13$	4.41315	1.22399	0.611994	+	0.790862i	$0.290368\pi$
$14$	−3.62167	−0.967932
$15$	−8.12139	−2.09693
$16$	0.0460086	0.0115022
$17$	−2.78337	−0.675066	−0.337533	−	0.941314i	$-0.609592\pi$
$17$	−2.78337	−0.675066	−0.337533	+	0.941314i	$0.609592\pi$
$18$	−2.51199	−0.592082
$19$	−7.22508	−1.65755	−0.828774	−	0.559584i	$-0.810961\pi$
$19$	−7.22508	−1.65755	−0.828774	+	0.559584i	$0.810961\pi$
$20$	−13.0258	−2.91267
$21$	3.20036	0.698375
$22$	4.91803	1.04853
$23$	3.22321	0.672085	0.336042	−	0.941847i	$-0.390911\pi$
$23$	3.22321	0.672085	0.336042	+	0.941847i	$0.390911\pi$
$24$	−5.77802	−1.17943
$25$	11.1000	2.22000
$26$	−10.1083	−1.98239
$27$	−3.85233	−0.741381
$28$	5.13303	0.970052
$29$	7.38099	1.37062	0.685308	−	0.728253i	$-0.259668\pi$
$29$	7.38099	1.37062	0.685308	+	0.728253i	$0.259668\pi$
$30$	18.6019	3.39623
$31$	6.87606	1.23498	0.617489	−	0.786580i	$-0.288150\pi$
$31$	6.87606	1.23498	0.617489	+	0.786580i	$0.288150\pi$
$32$	5.60403	0.990663
$33$	−4.34591	−0.756525
$34$	6.37527	1.09335
$35$	−6.34445	−1.07241
$36$	3.56027	0.593379
$37$	10.9746	1.80421	0.902103	−	0.431520i	$-0.142023\pi$
$37$	10.9746	1.80421	0.902103	+	0.431520i	$0.142023\pi$
$38$	16.5490	2.68459
$39$	8.93236	1.43032
$40$	11.4545	1.81111
$41$	−5.77644	−0.902128	−0.451064	−	0.892492i	$-0.648955\pi$
$41$	−5.77644	−0.902128	−0.451064	+	0.892492i	$0.648955\pi$
$42$	−7.33037	−1.13110
$43$	−10.5071	−1.60232	−0.801162	−	0.598447i	$-0.795785\pi$
$43$	−10.5071	−1.60232	−0.801162	+	0.598447i	$0.795785\pi$
$44$	−6.97037	−1.05082
$45$	−4.40051	−0.655990
$46$	−7.38271	−1.08852
$47$	−2.19319	−0.319909	−0.159955	−	0.987124i	$-0.551135\pi$
$47$	−2.19319	−0.319909	−0.159955	+	0.987124i	$0.551135\pi$
$48$	0.0931229	0.0134411
$49$	−4.49987	−0.642839
$50$	−25.4244	−3.59556
$51$	−5.63363	−0.788866
$52$	14.3266	1.98674
$53$	3.69705	0.507829	0.253914	−	0.967227i	$-0.418282\pi$
$53$	3.69705	0.507829	0.253914	+	0.967227i	$0.418282\pi$
$54$	8.82371	1.20075
$55$	8.61541	1.16170
$56$	−4.51380	−0.603182
$57$	−14.6238	−1.93697
$58$	−16.9061	−2.21988
$59$	−2.21439	−0.288289	−0.144145	−	0.989557i	$-0.546043\pi$
$59$	−2.21439	−0.288289	−0.144145	+	0.989557i	$0.546043\pi$
$60$	−26.3647	−3.40367
$61$	−2.77468	−0.355261	−0.177630	−	0.984097i	$-0.556843\pi$
$61$	−2.77468	−0.355261	−0.177630	+	0.984097i	$0.556843\pi$
$62$	−15.7495	−2.00019
$63$	1.73409	0.218475
$64$	−12.9280	−1.61600
$65$	−17.7077	−2.19637
$66$	9.95425	1.22528
$67$	9.94008	1.21437	0.607187	−	0.794559i	$-0.292298\pi$
$67$	9.94008	1.21437	0.607187	+	0.794559i	$0.292298\pi$
$68$	−9.03574	−1.09574
$69$	6.52387	0.785382
$70$	14.5319	1.73689
$71$	6.00154	0.712252	0.356126	−	0.934438i	$-0.384097\pi$
$71$	6.00154	0.712252	0.356126	+	0.934438i	$0.384097\pi$
$72$	−3.13078	−0.368966
$73$	11.1769	1.30815	0.654077	−	0.756428i	$-0.273057\pi$
$73$	11.1769	1.30815	0.654077	+	0.756428i	$0.273057\pi$
$74$	−25.1371	−2.92213
$75$	22.4668	2.59424
$76$	−23.4550	−2.69047
$77$	−3.39503	−0.386900
$78$	−20.4595	−2.31658
$79$	−14.2564	−1.60397	−0.801986	−	0.597343i	$-0.796223\pi$
$79$	−14.2564	−1.60397	−0.801986	+	0.597343i	$0.796223\pi$
$80$	−0.184609	−0.0206399
$81$	−11.0874	−1.23193
$82$	13.2309	1.46110
$83$	−12.7619	−1.40080	−0.700402	−	0.713749i	$-0.746996\pi$
$83$	−12.7619	−1.40080	−0.700402	+	0.713749i	$0.746996\pi$
$84$	10.3894	1.13358
$85$	11.1682	1.21136
$86$	24.0665	2.59516
$87$	14.9394	1.60167
$88$	6.12950	0.653406
$89$	−6.32019	−0.669939	−0.334969	−	0.942229i	$-0.608726\pi$
$89$	−6.32019	−0.669939	−0.334969	+	0.942229i	$0.608726\pi$
$90$	10.0793	1.06245
$91$	6.97798	0.731491
$92$	10.4636	1.09091
$93$	13.9174	1.44316
$94$	5.02347	0.518131
$95$	28.9905	2.97436
$96$	11.3427	1.15766
$97$	2.82895	0.287236	0.143618	−	0.989633i	$-0.454126\pi$
$97$	2.82895	0.287236	0.143618	+	0.989633i	$0.454126\pi$
$98$	10.3069	1.04115
$99$	−2.35480	−0.236666
$100$	36.0343	3.60343
$101$	−5.82233	−0.579343	−0.289672	−	0.957126i	$-0.593546\pi$
$101$	−5.82233	−0.579343	−0.289672	+	0.957126i	$0.593546\pi$
$102$	12.9038	1.27766
$103$	9.46490	0.932604	0.466302	−	0.884626i	$-0.345586\pi$
$103$	9.46490	0.932604	0.466302	+	0.884626i	$0.345586\pi$
$104$	−12.5983	−1.23536
$105$	−12.8414	−1.25319
$106$	−8.46805	−0.822489
$107$	−3.55083	−0.343272	−0.171636	−	0.985160i	$-0.554905\pi$
$107$	−3.55083	−0.343272	−0.171636	+	0.985160i	$0.554905\pi$
$108$	−12.5059	−1.20338
$109$	−7.63999	−0.731778	−0.365889	−	0.930658i	$-0.619235\pi$
$109$	−7.63999	−0.731778	−0.365889	+	0.930658i	$0.619235\pi$
$110$	−19.7335	−1.88151
$111$	22.2129	2.10835
$112$	0.0727478	0.00687402
$113$	−7.00655	−0.659120	−0.329560	−	0.944135i	$-0.606900\pi$
$113$	−7.00655	−0.659120	−0.329560	+	0.944135i	$0.606900\pi$
$114$	33.4956	3.13715
$115$	−12.9331	−1.20601
$116$	23.9612	2.22474
$117$	4.83993	0.447452
$118$	5.07203	0.466919
$119$	−4.40100	−0.403439
$120$	23.1842	2.11642
$121$	−6.38973	−0.580885
$122$	6.35536	0.575387
$123$	−11.6917	−1.05420
$124$	22.3220	2.00457
$125$	−24.4762	−2.18921
$126$	−3.97191	−0.353846
$127$	−2.49492	−0.221388	−0.110694	−	0.993855i	$-0.535307\pi$
$127$	−2.49492	−0.221388	−0.110694	+	0.993855i	$0.535307\pi$
$128$	18.4033	1.62664
$129$	−21.2668	−1.87244
$130$	40.5592	3.55728
$131$	−21.9516	−1.91792	−0.958961	−	0.283539i	$-0.908491\pi$
$131$	−21.9516	−1.91792	−0.958961	+	0.283539i	$0.908491\pi$
$132$	−14.1083	−1.22797
$133$	−11.4241	−0.990599
$134$	−22.7676	−1.96682
$135$	15.4574	1.33036
$136$	7.94570	0.681339
$137$	2.53623	0.216685	0.108342	−	0.994114i	$-0.465446\pi$
$137$	2.53623	0.216685	0.108342	+	0.994114i	$0.465446\pi$
$138$	−14.9428	−1.27202
$139$	12.4691	1.05762	0.528808	−	0.848741i	$-0.322639\pi$
$139$	12.4691	1.05762	0.528808	+	0.848741i	$0.322639\pi$
$140$	−20.5962	−1.74070
$141$	−4.43908	−0.373838
$142$	−13.7465	−1.15358
$143$	−9.47571	−0.792399
$144$	0.0504579	0.00420483
$145$	−29.6161	−2.45948
$146$	−25.6005	−2.11871
$147$	−9.10789	−0.751206
$148$	35.6271	2.92853
$149$	−7.90538	−0.647634	−0.323817	−	0.946120i	$-0.604966\pi$
$149$	−7.90538	−0.647634	−0.323817	+	0.946120i	$0.604966\pi$
$150$	−51.4599	−4.20168
$151$	16.2885	1.32554	0.662768	−	0.748825i	$-0.269382\pi$
$151$	16.2885	1.32554	0.662768	+	0.748825i	$0.269382\pi$
$152$	20.6255	1.67295
$153$	−3.05254	−0.246783
$154$	7.77628	0.626630
$155$	−27.5901	−2.21609
$156$	28.9974	2.32165
$157$	−13.1541	−1.04981	−0.524907	−	0.851160i	$-0.675900\pi$
$157$	−13.1541	−1.04981	−0.524907	+	0.851160i	$0.675900\pi$
$158$	32.6541	2.59782
$159$	7.48295	0.593436
$160$	−22.4861	−1.77768
$161$	5.09646	0.401658
$162$	25.3954	1.99525
$163$	20.3301	1.59237	0.796187	−	0.605051i	$-0.206847\pi$
$163$	20.3301	1.59237	0.796187	+	0.605051i	$0.206847\pi$
$164$	−18.7522	−1.46430
$165$	17.4379	1.35754
$166$	29.2310	2.26877
$167$	22.0390	1.70543	0.852713	−	0.522380i	$-0.174956\pi$
$167$	22.0390	1.70543	0.852713	+	0.522380i	$0.174956\pi$
$168$	−9.13608	−0.704864
$169$	6.47591	0.498147
$170$	−25.5807	−1.96195
$171$	−7.92379	−0.605947
$172$	−34.1097	−2.60084
$173$	−0.978538	−0.0743969	−0.0371985	−	0.999308i	$-0.511843\pi$
$173$	−0.978538	−0.0743969	−0.0371985	+	0.999308i	$0.511843\pi$
$174$	−34.2184	−2.59409
$175$	17.5511	1.32674
$176$	−0.0987875	−0.00744639
$177$	−4.48200	−0.336888
$178$	14.4763	1.08505
$179$	−20.2281	−1.51192	−0.755959	−	0.654619i	$-0.772829\pi$
$179$	−20.2281	−1.51192	−0.755959	+	0.654619i	$0.772829\pi$
$180$	−14.2855	−1.06478
$181$	1.20264	0.0893912	0.0446956	−	0.999001i	$-0.485768\pi$
$181$	1.20264	0.0893912	0.0446956	+	0.999001i	$0.485768\pi$
$182$	−15.9830	−1.18474
$183$	−5.61603	−0.415149
$184$	−9.20131	−0.678330
$185$	−44.0352	−3.23753
$186$	−31.8776	−2.33738
$187$	5.97632	0.437032
$188$	−7.11981	−0.519266
$189$	−6.09122	−0.443071
$190$	−66.4024	−4.81733
$191$	−10.7126	−0.775135	−0.387567	−	0.921841i	$-0.626685\pi$
$191$	−10.7126	−0.775135	−0.387567	+	0.921841i	$0.626685\pi$
$192$	−26.1667	−1.88842
$193$	5.83801	0.420229	0.210114	−	0.977677i	$-0.432616\pi$
$193$	5.83801	0.420229	0.210114	+	0.977677i	$0.432616\pi$
$194$	−6.47967	−0.465213
$195$	−35.8409	−2.56662
$196$	−14.6081	−1.04343
$197$	−8.79299	−0.626474	−0.313237	−	0.949675i	$-0.601413\pi$
$197$	−8.79299	−0.626474	−0.313237	+	0.949675i	$0.601413\pi$
$198$	5.39363	0.383309
$199$	−13.5260	−0.958836	−0.479418	−	0.877587i	$-0.659152\pi$
$199$	−13.5260	−0.958836	−0.479418	+	0.877587i	$0.659152\pi$
$200$	−31.6873	−2.24063
$201$	20.1191	1.41909
$202$	13.3360	0.938316
$203$	11.6707	0.819120
$204$	−18.2886	−1.28046
$205$	23.1778	1.61881
$206$	−21.6792	−1.51046
$207$	3.53491	0.245693
$208$	0.203043	0.0140785
$209$	15.5134	1.07308
$210$	29.4130	2.02969
$211$	−19.9452	−1.37309	−0.686544	−	0.727088i	$-0.740873\pi$
$211$	−19.9452	−1.37309	−0.686544	+	0.727088i	$0.740873\pi$
$212$	12.0019	0.824291
$213$	12.1473	0.832320
$214$	8.13314	0.555970
$215$	42.1597	2.87527
$216$	10.9973	0.748270
$217$	10.8723	0.738058
$218$	17.4993	1.18520
$219$	22.6224	1.52868
$220$	27.9685	1.88564
$221$	−12.2834	−0.826273
$222$	−50.8783	−3.41473
$223$	−10.2750	−0.688065	−0.344032	−	0.938958i	$-0.611793\pi$
$223$	−10.2750	−0.688065	−0.344032	+	0.938958i	$0.611793\pi$
$224$	8.86098	0.592049
$225$	12.1734	0.811563
$226$	16.0484	1.06752
$227$	24.5340	1.62838	0.814191	−	0.580597i	$-0.197181\pi$
$227$	24.5340	1.62838	0.814191	+	0.580597i	$0.197181\pi$
$228$	−47.4737	−3.14402
$229$	−19.4456	−1.28500	−0.642500	−	0.766286i	$-0.722103\pi$
$229$	−19.4456	−1.28500	−0.642500	+	0.766286i	$0.722103\pi$
$230$	29.6230	1.95328
$231$	−6.87165	−0.452122
$232$	−21.0706	−1.38335
$233$	−11.7294	−0.768416	−0.384208	−	0.923246i	$-0.625526\pi$
$233$	−11.7294	−0.768416	−0.384208	+	0.923246i	$0.625526\pi$
$234$	−11.0858	−0.724702
$235$	8.80012	0.574057
$236$	−7.18865	−0.467941
$237$	−28.8554	−1.87436
$238$	10.0804	0.653418
$239$	−16.8469	−1.08974	−0.544868	−	0.838522i	$-0.683420\pi$
$239$	−16.8469	−1.08974	−0.544868	+	0.838522i	$0.683420\pi$
$240$	−0.373654	−0.0241193
$241$	−16.7632	−1.07981	−0.539905	−	0.841726i	$-0.681540\pi$
$241$	−16.7632	−1.07981	−0.539905	+	0.841726i	$0.681540\pi$
$242$	14.6356	0.940812
$243$	−10.8842	−0.698220
$244$	−9.00752	−0.576647
$245$	18.0557	1.15353
$246$	26.7797	1.70741
$247$	−31.8854	−2.02882
$248$	−19.6292	−1.24645
$249$	−25.8305	−1.63694
$250$	56.0623	3.54569
$251$	3.81191	0.240606	0.120303	−	0.992737i	$-0.461613\pi$
$251$	3.81191	0.240606	0.120303	+	0.992737i	$0.461613\pi$
$252$	5.62943	0.354621
$253$	−6.92072	−0.435102
$254$	5.71457	0.358564
$255$	22.6048	1.41557
$256$	−16.2966	−1.01854
$257$	−1.49498	−0.0932542	−0.0466271	−	0.998912i	$-0.514847\pi$
$257$	−1.49498	−0.0932542	−0.0466271	+	0.998912i	$0.514847\pi$
$258$	48.7113	3.03264
$259$	17.3527	1.07825
$260$	−57.4850	−3.56507
$261$	8.09478	0.501054
$262$	50.2799	3.10630
$263$	−1.88880	−0.116469	−0.0582343	−	0.998303i	$-0.518547\pi$
$263$	−1.88880	−0.116469	−0.0582343	+	0.998303i	$0.518547\pi$
$264$	12.4063	0.763555
$265$	−14.8343	−0.911267
$266$	26.1668	1.60439
$267$	−12.7923	−0.782874
$268$	32.2688	1.97113
$269$	5.96939	0.363960	0.181980	−	0.983302i	$-0.441749\pi$
$269$	5.96939	0.363960	0.181980	+	0.983302i	$0.441749\pi$
$270$	−35.4050	−2.15468
$271$	9.34446	0.567635	0.283818	−	0.958878i	$-0.408399\pi$
$271$	9.34446	0.567635	0.283818	+	0.958878i	$0.408399\pi$
$272$	−0.128059	−0.00776471
$273$	14.1237	0.854803
$274$	−5.80920	−0.350947
$275$	−23.8334	−1.43721
$276$	21.1787	1.27481
$277$	−25.4384	−1.52845	−0.764224	−	0.644951i	$-0.776878\pi$
$277$	−25.4384	−1.52845	−0.764224	+	0.644951i	$0.776878\pi$
$278$	−28.5603	−1.71294
$279$	7.54102	0.451469
$280$	18.1116	1.08237
$281$	−16.3323	−0.974301	−0.487150	−	0.873318i	$-0.661964\pi$
$281$	−16.3323	−0.974301	−0.487150	+	0.873318i	$0.661964\pi$
$282$	10.1677	0.605475
$283$	−21.2978	−1.26603	−0.633013	−	0.774142i	$-0.718182\pi$
$283$	−21.2978	−1.26603	−0.633013	+	0.774142i	$0.718182\pi$
$284$	19.4830	1.15610
$285$	58.6777	3.47577
$286$	21.7040	1.28338
$287$	−9.13358	−0.539138
$288$	6.14598	0.362155
$289$	−9.25286	−0.544286
$290$	67.8353	3.98343
$291$	5.72588	0.335657
$292$	36.2839	2.12335
$293$	11.3793	0.664788	0.332394	−	0.943141i	$-0.392144\pi$
$293$	11.3793	0.664788	0.332394	+	0.943141i	$0.392144\pi$
$294$	20.8615	1.21667
$295$	8.88520	0.517317
$296$	−31.3292	−1.82097
$297$	8.27154	0.479963
$298$	18.1072	1.04892
$299$	14.2245	0.822624
$300$	72.9346	4.21088
$301$	−16.6137	−0.957596
$302$	−37.3085	−2.14686
$303$	−11.7846	−0.677006
$304$	−0.332416	−0.0190654
$305$	11.1333	0.637493
$306$	6.99180	0.399695
$307$	11.6867	0.666997	0.333499	−	0.942751i	$-0.391771\pi$
$307$	11.6867	0.666997	0.333499	+	0.942751i	$0.391771\pi$
$308$	−11.0214	−0.628003
$309$	19.1573	1.08982
$310$	63.1947	3.58922
$311$	2.63743	0.149555	0.0747776	−	0.997200i	$-0.476175\pi$
$311$	2.63743	0.149555	0.0747776	+	0.997200i	$0.476175\pi$
$312$	−25.4993	−1.44361
$313$	8.07492	0.456421	0.228211	−	0.973612i	$-0.426713\pi$
$313$	8.07492	0.456421	0.228211	+	0.973612i	$0.426713\pi$
$314$	30.1294	1.70030
$315$	−6.95800	−0.392039
$316$	−46.2810	−2.60351
$317$	10.7356	0.602969	0.301484	−	0.953471i	$-0.402518\pi$
$317$	10.7356	0.602969	0.301484	+	0.953471i	$0.402518\pi$
$318$	−17.1396	−0.961141
$319$	−15.8481	−0.887325
$320$	51.8733	2.89981
$321$	−7.18700	−0.401139
$322$	−11.6734	−0.650532
$323$	20.1101	1.11895
$324$	−35.9932	−1.99962
$325$	48.9860	2.71726
$326$	−46.5657	−2.57904
$327$	−15.4636	−0.855138
$328$	16.4900	0.910510
$329$	−3.46782	−0.191187
$330$	−39.9412	−2.19869
$331$	2.82029	0.155017	0.0775086	−	0.996992i	$-0.475303\pi$
$331$	2.82029	0.155017	0.0775086	+	0.996992i	$0.475303\pi$
$332$	−41.4294	−2.27374
$333$	12.0359	0.659562
$334$	−50.4799	−2.76214
$335$	−39.8844	−2.17912
$336$	0.147244	0.00803282
$337$	8.93967	0.486975	0.243488	−	0.969904i	$-0.421708\pi$
$337$	8.93967	0.486975	0.243488	+	0.969904i	$0.421708\pi$
$338$	−14.8330	−0.806808
$339$	−14.1815	−0.770232
$340$	36.2557	1.96624
$341$	−14.7640	−0.799513
$342$	18.1493	0.981404
$343$	−18.1834	−0.981809
$344$	29.9948	1.61721
$345$	−26.1769	−1.40932
$346$	2.24133	0.120495
$347$	−3.44323	−0.184842	−0.0924210	−	0.995720i	$-0.529461\pi$
$347$	−3.44323	−0.184842	−0.0924210	+	0.995720i	$0.529461\pi$
$348$	48.4982	2.59977
$349$	−9.13125	−0.488785	−0.244392	−	0.969676i	$-0.578588\pi$
$349$	−9.13125	−0.488785	−0.244392	+	0.969676i	$0.578588\pi$
$350$	−40.2005	−2.14881
$351$	−17.0009	−0.907441
$352$	−12.0327	−0.641346
$353$	−19.7582	−1.05162	−0.525811	−	0.850602i	$-0.676238\pi$
$353$	−19.7582	−1.05162	−0.525811	+	0.850602i	$0.676238\pi$
$354$	10.2660	0.545630
$355$	−24.0811	−1.27809
$356$	−20.5174	−1.08742
$357$	−8.90777	−0.471449
$358$	46.3322	2.44873
$359$	5.48173	0.289315	0.144657	−	0.989482i	$-0.453792\pi$
$359$	5.48173	0.289315	0.144657	+	0.989482i	$0.453792\pi$
$360$	12.5622	0.662085
$361$	33.2018	1.74746
$362$	−2.75462	−0.144780
$363$	−12.9330	−0.678808
$364$	22.6528	1.18733
$365$	−44.8470	−2.34740
$366$	12.8635	0.672383
$367$	28.7999	1.50334	0.751672	−	0.659537i	$-0.229248\pi$
$367$	28.7999	1.50334	0.751672	+	0.659537i	$0.229248\pi$
$368$	0.148295	0.00773042
$369$	−6.33506	−0.329790
$370$	100.862	5.24357
$371$	5.84570	0.303493
$372$	45.1804	2.34250
$373$	−1.84793	−0.0956822	−0.0478411	−	0.998855i	$-0.515234\pi$
$373$	−1.84793	−0.0956822	−0.0478411	+	0.998855i	$0.515234\pi$
$374$	−13.6887	−0.707825
$375$	−49.5405	−2.55826
$376$	6.26091	0.322882
$377$	32.5734	1.67762
$378$	13.9519	0.717606
$379$	−21.6524	−1.11221	−0.556105	−	0.831112i	$-0.687705\pi$
$379$	−21.6524	−1.11221	−0.556105	+	0.831112i	$0.687705\pi$
$380$	94.1128	4.82788
$381$	−5.04979	−0.258709
$382$	24.5370	1.25542
$383$	−0.0570891	−0.00291712	−0.00145856	−	0.999999i	$-0.500464\pi$
$383$	−0.0570891	−0.00291712	−0.00145856	+	0.999999i	$0.500464\pi$
$384$	37.2489	1.90085
$385$	13.6225	0.694267
$386$	−13.3719	−0.680611
$387$	−11.5233	−0.585760
$388$	9.18370	0.466232
$389$	4.96016	0.251490	0.125745	−	0.992063i	$-0.459868\pi$
$389$	4.96016	0.251490	0.125745	+	0.992063i	$0.459868\pi$
$390$	82.0932	4.15695
$391$	−8.97137	−0.453702
$392$	12.8458	0.648812
$393$	−44.4308	−2.24124
$394$	20.1402	1.01465
$395$	57.2036	2.87822
$396$	−7.64445	−0.384148
$397$	−20.3244	−1.02005	−0.510027	−	0.860158i	$-0.670365\pi$
$397$	−20.3244	−1.02005	−0.510027	+	0.860158i	$0.670365\pi$
$398$	30.9812	1.55295
$399$	−23.1228	−1.15759
$400$	0.510696	0.0255348
$401$	10.2326	0.510991	0.255496	−	0.966810i	$-0.417761\pi$
$401$	10.2326	0.510991	0.255496	+	0.966810i	$0.417761\pi$
$402$	−46.0824	−2.29838
$403$	30.3451	1.51160
$404$	−18.9012	−0.940370
$405$	44.4878	2.21062
$406$	−26.7315	−1.32666
$407$	−23.5641	−1.16803
$408$	16.0824	0.796196
$409$	−18.0457	−0.892300	−0.446150	−	0.894958i	$-0.647205\pi$
$409$	−18.0457	−0.892300	−0.446150	+	0.894958i	$0.647205\pi$
$410$	−53.0886	−2.62186
$411$	5.13341	0.253212
$412$	30.7262	1.51377
$413$	−3.50135	−0.172290
$414$	−8.09667	−0.397930
$415$	51.2070	2.51365
$416$	24.7315	1.21256
$417$	25.2379	1.23590
$418$	−35.5331	−1.73798
$419$	−8.69399	−0.424729	−0.212365	−	0.977190i	$-0.568116\pi$
$419$	−8.69399	−0.424729	−0.212365	+	0.977190i	$0.568116\pi$
$420$	−41.6873	−2.03413
$421$	−3.73488	−0.182027	−0.0910135	−	0.995850i	$-0.529011\pi$
$421$	−3.73488	−0.182027	−0.0910135	+	0.995850i	$0.529011\pi$
$422$	45.6843	2.22388
$423$	−2.40528	−0.116949
$424$	−10.5540	−0.512548
$425$	−30.8954	−1.49865
$426$	−27.8233	−1.34804
$427$	−4.38726	−0.212314
$428$	−11.5272	−0.557188
$429$	−19.1791	−0.925978
$430$	−96.5663	−4.65684
$431$	25.1939	1.21355	0.606774	−	0.794874i	$-0.292463\pi$
$431$	25.1939	1.21355	0.606774	+	0.794874i	$0.292463\pi$
$432$	−0.177240	−0.00852748
$433$	−34.5901	−1.66229	−0.831146	−	0.556054i	$-0.812315\pi$
$433$	−34.5901	−1.66229	−0.831146	+	0.556054i	$0.812315\pi$
$434$	−24.9028	−1.19537
$435$	−59.9439	−2.87409
$436$	−24.8019	−1.18780
$437$	−23.2879	−1.11401
$438$	−51.8162	−2.47588
$439$	−27.9582	−1.33437	−0.667187	−	0.744890i	$-0.732502\pi$
$439$	−27.9582	−1.33437	−0.667187	+	0.744890i	$0.732502\pi$
$440$	−24.5945	−1.17250
$441$	−4.93504	−0.235002
$442$	28.1350	1.33825
$443$	−35.5441	−1.68875	−0.844375	−	0.535752i	$-0.820028\pi$
$443$	−35.5441	−1.68875	−0.844375	+	0.535752i	$0.820028\pi$
$444$	72.1103	3.42220
$445$	25.3596	1.20216
$446$	23.5348	1.11440
$447$	−16.0007	−0.756809
$448$	−20.4415	−0.965768
$449$	−26.4462	−1.24807	−0.624037	−	0.781395i	$-0.714509\pi$
$449$	−26.4462	−1.24807	−0.624037	+	0.781395i	$0.714509\pi$
$450$	−27.8831	−1.31442
$451$	12.4029	0.584030
$452$	−22.7456	−1.06986
$453$	32.9684	1.54899
$454$	−56.1949	−2.63736
$455$	−27.9990	−1.31261
$456$	41.7467	1.95497
$457$	33.0930	1.54803	0.774014	−	0.633169i	$-0.218246\pi$
$457$	33.0930	1.54803	0.774014	+	0.633169i	$0.218246\pi$
$458$	44.5398	2.08121
$459$	10.7224	0.500481
$460$	−41.9850	−1.95756
$461$	−10.3898	−0.483902	−0.241951	−	0.970289i	$-0.577787\pi$
$461$	−10.3898	−0.483902	−0.241951	+	0.970289i	$0.577787\pi$
$462$	15.7394	0.732265
$463$	23.1827	1.07739	0.538697	−	0.842500i	$-0.318917\pi$
$463$	23.1827	1.07739	0.538697	+	0.842500i	$0.318917\pi$
$464$	0.339589	0.0157650
$465$	−55.8432	−2.58967
$466$	26.8660	1.24454
$467$	22.8318	1.05653	0.528266	−	0.849079i	$-0.322843\pi$
$467$	22.8318	1.05653	0.528266	+	0.849079i	$0.322843\pi$
$468$	15.7120	0.726289
$469$	15.7170	0.725746
$470$	−20.1566	−0.929753
$471$	−26.6244	−1.22679
$472$	6.32144	0.290968
$473$	22.5604	1.03733
$474$	66.0930	3.03575
$475$	−80.1984	−3.67976
$476$	−14.2871	−0.654849
$477$	4.05458	0.185646
$478$	38.5876	1.76496
$479$	−20.7238	−0.946895	−0.473448	−	0.880822i	$-0.656991\pi$
$479$	−20.7238	−0.946895	−0.473448	+	0.880822i	$0.656991\pi$
$480$	−45.5126	−2.07735
$481$	48.4324	2.20833
$482$	38.3958	1.74888
$483$	10.3154	0.469367
$484$	−20.7432	−0.942873
$485$	−11.3511	−0.515427
$486$	24.9301	1.13085
$487$	−29.8289	−1.35167	−0.675837	−	0.737051i	$-0.736218\pi$
$487$	−29.8289	−1.35167	−0.675837	+	0.737051i	$0.736218\pi$
$488$	7.92089	0.358562
$489$	41.1487	1.86081
$490$	−41.3563	−1.86829
$491$	34.9970	1.57939	0.789695	−	0.613499i	$-0.210239\pi$
$491$	34.9970	1.57939	0.789695	+	0.613499i	$0.210239\pi$
$492$	−37.9551	−1.71115
$493$	−20.5440	−0.925256
$494$	73.0330	3.28591
$495$	9.44858	0.424682
$496$	0.316358	0.0142049
$497$	9.48951	0.425663
$498$	59.1645	2.65123
$499$	−37.6833	−1.68694	−0.843468	−	0.537179i	$-0.819490\pi$
$499$	−37.6833	−1.68694	−0.843468	+	0.537179i	$0.819490\pi$
$500$	−79.4578	−3.55346
$501$	44.6075	1.99292
$502$	−8.73114	−0.389690
$503$	−28.2686	−1.26044	−0.630218	−	0.776418i	$-0.717034\pi$
$503$	−28.2686	−1.26044	−0.630218	+	0.776418i	$0.717034\pi$
$504$	−4.95032	−0.220505
$505$	23.3620	1.03959
$506$	15.8518	0.704699
$507$	13.1074	0.582122
$508$	−8.09933	−0.359350
$509$	−22.5492	−0.999474	−0.499737	−	0.866177i	$-0.666570\pi$
$509$	−22.5492	−0.999474	−0.499737	+	0.866177i	$0.666570\pi$
$510$	−51.7761	−2.29268
$511$	17.6726	0.781792
$512$	0.520516	0.0230038
$513$	27.8334	1.22887
$514$	3.42423	0.151036
$515$	−37.9777	−1.67350
$516$	−69.0391	−3.03928
$517$	4.70911	0.207106
$518$	−39.7462	−1.74635
$519$	−1.98059	−0.0869384
$520$	50.5503	2.21678
$521$	−9.62134	−0.421518	−0.210759	−	0.977538i	$-0.567594\pi$
$521$	−9.62134	−0.421518	−0.210759	+	0.977538i	$0.567594\pi$
$522$	−18.5410	−0.811517
$523$	−37.4694	−1.63842	−0.819211	−	0.573492i	$-0.805589\pi$
$523$	−37.4694	−1.63842	−0.819211	+	0.573492i	$0.805589\pi$
$524$	−71.2622	−3.11310
$525$	35.5240	1.55039
$526$	4.32628	0.188635
$527$	−19.1386	−0.833691
$528$	−0.199949	−0.00870167
$529$	−12.6109	−0.548302
$530$	33.9779	1.47590
$531$	−2.42854	−0.105390
$532$	−37.0866	−1.60791
$533$	−25.4923	−1.10419
$534$	29.3005	1.26796
$535$	14.2477	0.615980
$536$	−28.3760	−1.22566
$537$	−40.9423	−1.76679
$538$	−13.6728	−0.589477
$539$	9.66192	0.416168
$540$	50.1798	2.15940
$541$	23.8828	1.02680	0.513401	−	0.858149i	$-0.328385\pi$
$541$	23.8828	1.02680	0.513401	+	0.858149i	$0.328385\pi$
$542$	−21.4034	−0.919353
$543$	2.43417	0.104460
$544$	−15.5981	−0.668763
$545$	30.6553	1.31313
$546$	−32.3501	−1.38445
$547$	−0.460471	−0.0196883	−0.00984416	−	0.999952i	$-0.503134\pi$
$547$	−0.460471	−0.0196883	−0.00984416	+	0.999952i	$0.503134\pi$
$548$	8.23344	0.351715
$549$	−3.04300	−0.129872
$550$	54.5901	2.32773
$551$	−53.3283	−2.27186
$552$	−18.6238	−0.792680
$553$	−22.5419	−0.958580
$554$	58.2664	2.47550
$555$	−89.1287	−3.78330
$556$	40.4789	1.71669
$557$	−39.2841	−1.66452	−0.832260	−	0.554386i	$-0.812953\pi$
$557$	−39.2841	−1.66452	−0.832260	+	0.554386i	$0.812953\pi$
$558$	−17.2726	−0.731208
$559$	−46.3696	−1.96123
$560$	−0.291899	−0.0123350
$561$	12.0963	0.510705
$562$	37.4088	1.57800
$563$	10.2770	0.433123	0.216562	−	0.976269i	$-0.430516\pi$
$563$	10.2770	0.433123	0.216562	+	0.976269i	$0.430516\pi$
$564$	−14.4107	−0.606801
$565$	28.1136	1.18275
$566$	48.7824	2.05048
$567$	−17.5311	−0.736236
$568$	−17.1327	−0.718870
$569$	33.8331	1.41836	0.709179	−	0.705028i	$-0.249066\pi$
$569$	33.8331	1.41836	0.709179	+	0.705028i	$0.249066\pi$
$570$	−134.401	−5.62942
$571$	32.4752	1.35905	0.679523	−	0.733654i	$-0.262187\pi$
$571$	32.4752	1.35905	0.679523	+	0.733654i	$0.262187\pi$
$572$	−30.7613	−1.28620
$573$	−21.6826	−0.905803
$574$	20.9203	0.873198
$575$	35.7776	1.49203
$576$	−14.1782	−0.590759
$577$	−34.1746	−1.42271	−0.711353	−	0.702835i	$-0.751917\pi$
$577$	−34.1746	−1.42271	−0.711353	+	0.702835i	$0.751917\pi$
$578$	21.1936	0.881536
$579$	11.8163	0.491069
$580$	−96.1437	−3.99215
$581$	−20.1789	−0.837161
$582$	−13.1151	−0.543636
$583$	−7.93813	−0.328764
$584$	−31.9067	−1.32031
$585$	−19.4201	−0.802924
$586$	−26.0642	−1.07670
$587$	29.0302	1.19820	0.599102	−	0.800673i	$-0.295524\pi$
$587$	29.0302	1.19820	0.599102	+	0.800673i	$0.295524\pi$
$588$	−29.5672	−1.21933
$589$	−49.6801	−2.04703
$590$	−20.3514	−0.837856
$591$	−17.7973	−0.732083
$592$	0.504924	0.0207523
$593$	11.1470	0.457754	0.228877	−	0.973455i	$-0.426495\pi$
$593$	11.1470	0.457754	0.228877	+	0.973455i	$0.426495\pi$
$594$	−18.9459	−0.777358
$595$	17.6589	0.723946
$596$	−25.6635	−1.05122
$597$	−27.3772	−1.12047
$598$	−32.5810	−1.33234
$599$	47.2353	1.92998	0.964992	−	0.262280i	$-0.0844742\pi$
$599$	47.2353	1.92998	0.964992	+	0.262280i	$0.0844742\pi$
$600$	−64.1361	−2.61834
$601$	−16.3067	−0.665164	−0.332582	−	0.943074i	$-0.607920\pi$
$601$	−16.3067	−0.665164	−0.332582	+	0.943074i	$0.607920\pi$
$602$	38.0534	1.55094
$603$	10.9014	0.443937
$604$	52.8777	2.15156
$605$	25.6387	1.04236
$606$	26.9924	1.09649
$607$	−28.5529	−1.15893	−0.579463	−	0.814998i	$-0.696738\pi$
$607$	−28.5529	−1.15893	−0.579463	+	0.814998i	$0.696738\pi$
$608$	−40.4896	−1.64207
$609$	23.6218	0.957204
$610$	−25.5008	−1.03250
$611$	−9.67887	−0.391565
$612$	−9.90955	−0.400570
$613$	23.9745	0.968322	0.484161	−	0.874979i	$-0.339125\pi$
$613$	23.9745	0.968322	0.484161	+	0.874979i	$0.339125\pi$
$614$	−26.7683	−1.08028
$615$	46.9127	1.89170
$616$	9.69183	0.390495
$617$	−7.06161	−0.284290	−0.142145	−	0.989846i	$-0.545400\pi$
$617$	−7.06161	−0.284290	−0.142145	+	0.989846i	$0.545400\pi$
$618$	−43.8795	−1.76509
$619$	−33.4615	−1.34493	−0.672465	−	0.740129i	$-0.734764\pi$
$619$	−33.4615	−1.34493	−0.672465	+	0.740129i	$0.734764\pi$
$620$	−89.5665	−3.59708
$621$	−12.4168	−0.498271
$622$	−6.04101	−0.242222
$623$	−9.99335	−0.400375
$624$	0.410966	0.0164518
$625$	42.7101	1.70840
$626$	−18.4955	−0.739229
$627$	31.3995	1.25398
$628$	−42.7027	−1.70402
$629$	−30.5462	−1.21796
$630$	15.9372	0.634953
$631$	−27.9594	−1.11305	−0.556523	−	0.830832i	$-0.687865\pi$
$631$	−27.9594	−1.11305	−0.556523	+	0.830832i	$0.687865\pi$
$632$	40.6979	1.61887
$633$	−40.3698	−1.60456
$634$	−24.5897	−0.976580
$635$	10.0108	0.397267
$636$	24.2921	0.963246
$637$	−19.8586	−0.786828
$638$	36.2999	1.43713
$639$	6.58193	0.260377
$640$	−73.8430	−2.91890
$641$	22.1323	0.874175	0.437087	−	0.899419i	$-0.356010\pi$
$641$	22.1323	0.874175	0.437087	+	0.899419i	$0.356010\pi$
$642$	16.4617	0.649693
$643$	−38.2987	−1.51035	−0.755176	−	0.655522i	$-0.772449\pi$
$643$	−38.2987	−1.51035	−0.755176	+	0.655522i	$0.772449\pi$
$644$	16.5448	0.651957
$645$	85.3326	3.35997
$646$	−46.0618	−1.81228
$647$	44.5000	1.74947	0.874737	−	0.484598i	$-0.161034\pi$
$647$	44.5000	1.74947	0.874737	+	0.484598i	$0.161034\pi$
$648$	31.6512	1.24338
$649$	4.75464	0.186636
$650$	−112.202	−4.40092
$651$	22.0058	0.862477
$652$	65.9981	2.58469
$653$	−18.2087	−0.712562	−0.356281	−	0.934379i	$-0.615955\pi$
$653$	−18.2087	−0.712562	−0.356281	+	0.934379i	$0.615955\pi$
$654$	35.4191	1.38500
$655$	88.0804	3.44159
$656$	−0.265766	−0.0103764
$657$	12.2578	0.478221
$658$	7.94300	0.309650
$659$	28.5074	1.11049	0.555246	−	0.831686i	$-0.312624\pi$
$659$	28.5074	1.11049	0.555246	+	0.831686i	$0.312624\pi$
$660$	56.6091	2.20351
$661$	23.1114	0.898929	0.449464	−	0.893298i	$-0.351615\pi$
$661$	23.1114	0.898929	0.449464	+	0.893298i	$0.351615\pi$
$662$	−6.45984	−0.251069
$663$	−24.8621	−0.965562
$664$	36.4316	1.41382
$665$	45.8391	1.77757
$666$	−27.5680	−1.06824
$667$	23.7905	0.921170
$668$	71.5458	2.76819
$669$	−20.7969	−0.804056
$670$	91.3547	3.52934
$671$	5.95765	0.229993
$672$	17.9349	0.691854
$673$	23.4533	0.904060	0.452030	−	0.892003i	$-0.350700\pi$
$673$	23.4533	0.904060	0.452030	+	0.892003i	$0.350700\pi$
$674$	−20.4762	−0.788714
$675$	−42.7609	−1.64587
$676$	21.0230	0.808575
$677$	−28.8984	−1.11066	−0.555329	−	0.831631i	$-0.687407\pi$
$677$	−28.8984	−1.11066	−0.555329	+	0.831631i	$0.687407\pi$
$678$	32.4825	1.24748
$679$	4.47307	0.171661
$680$	−31.8820	−1.22262
$681$	49.6577	1.90289
$682$	33.8167	1.29491
$683$	37.3283	1.42833	0.714163	−	0.699979i	$-0.246807\pi$
$683$	37.3283	1.42833	0.714163	+	0.699979i	$0.246807\pi$
$684$	−25.7233	−0.983553
$685$	−10.1766	−0.388827
$686$	41.6487	1.59016
$687$	−39.3585	−1.50162
$688$	−0.483419	−0.0184302
$689$	16.3156	0.621577
$690$	59.9579	2.28256
$691$	−44.4423	−1.69066	−0.845332	−	0.534241i	$-0.820597\pi$
$691$	−44.4423	−1.69066	−0.845332	+	0.534241i	$0.820597\pi$
$692$	−3.17666	−0.120759
$693$	−3.72335	−0.141438
$694$	7.88667	0.299374
$695$	−50.0321	−1.89782
$696$	−42.6475	−1.61655
$697$	16.0780	0.608996
$698$	20.9150	0.791645
$699$	−23.7406	−0.897953
$700$	56.9767	2.15352
$701$	−2.95840	−0.111737	−0.0558685	−	0.998438i	$-0.517793\pi$
$701$	−2.95840	−0.111737	−0.0558685	+	0.998438i	$0.517793\pi$
$702$	38.9404	1.46971
$703$	−79.2921	−2.99056
$704$	27.7584	1.04618
$705$	17.8117	0.670829
$706$	45.2558	1.70323
$707$	−9.20614	−0.346233
$708$	−14.5501	−0.546825
$709$	−5.16985	−0.194158	−0.0970790	−	0.995277i	$-0.530950\pi$
$709$	−5.16985	−0.194158	−0.0970790	+	0.995277i	$0.530950\pi$
$710$	55.1574	2.07002
$711$	−15.6351	−0.586362
$712$	18.0423	0.676164
$713$	22.1630	0.830010
$714$	20.4031	0.763568
$715$	38.0211	1.42191
$716$	−65.6671	−2.45409
$717$	−34.0987	−1.27344
$718$	−12.5558	−0.468580
$719$	7.37184	0.274923	0.137462	−	0.990507i	$-0.456106\pi$
$719$	7.37184	0.274923	0.137462	+	0.990507i	$0.456106\pi$
$720$	−0.202462	−0.00754530
$721$	14.9657	0.557352
$722$	−76.0483	−2.83022
$723$	−33.9292	−1.26184
$724$	3.90415	0.145097
$725$	81.9291	3.04277
$726$	29.6229	1.09941
$727$	27.9384	1.03618	0.518089	−	0.855327i	$-0.326644\pi$
$727$	27.9384	1.03618	0.518089	+	0.855327i	$0.326644\pi$
$728$	−19.9201	−0.738288
$729$	11.2321	0.416005
$730$	102.722	3.80189
$731$	29.2453	1.08167
$732$	−18.2315	−0.673856
$733$	41.8116	1.54435	0.772174	−	0.635411i	$-0.219170\pi$
$733$	41.8116	1.54435	0.772174	+	0.635411i	$0.219170\pi$
$734$	−65.9659	−2.43485
$735$	36.5452	1.34799
$736$	18.0630	0.665809
$737$	−21.3429	−0.786175
$738$	14.5104	0.534134
$739$	−17.3395	−0.637842	−0.318921	−	0.947781i	$-0.603321\pi$
$739$	−17.3395	−0.637842	−0.318921	+	0.947781i	$0.603321\pi$
$740$	−142.953	−5.25505
$741$	−64.5370	−2.37083
$742$	−13.3895	−0.491544
$743$	−5.31186	−0.194873	−0.0974367	−	0.995242i	$-0.531064\pi$
$743$	−5.31186	−0.194873	−0.0974367	+	0.995242i	$0.531064\pi$
$744$	−39.7300	−1.45657
$745$	31.7202	1.16214
$746$	4.23266	0.154969
$747$	−13.9961	−0.512090
$748$	19.4011	0.709375
$749$	−5.61450	−0.205149
$750$	113.472	4.14341
$751$	−34.9317	−1.27468	−0.637338	−	0.770584i	$-0.719965\pi$
$751$	−34.9317	−1.27468	−0.637338	+	0.770584i	$0.719965\pi$
$752$	−0.100905	−0.00367964
$753$	7.71543	0.281166
$754$	−74.6091	−2.71710
$755$	−65.3571	−2.37859
$756$	−19.7741	−0.719178
$757$	39.5031	1.43576	0.717882	−	0.696165i	$-0.245112\pi$
$757$	39.5031	1.43576	0.717882	+	0.696165i	$0.245112\pi$
$758$	49.5946	1.80136
$759$	−14.0078	−0.508449
$760$	−82.7594	−3.00200
$761$	29.1792	1.05775	0.528873	−	0.848701i	$-0.322615\pi$
$761$	29.1792	1.05775	0.528873	+	0.848701i	$0.322615\pi$
$762$	11.5665	0.419009
$763$	−12.0802	−0.437332
$764$	−34.7766	−1.25817
$765$	12.2483	0.442836
$766$	0.130762	0.00472462
$767$	−9.77244	−0.352863
$768$	−32.9848	−1.19024
$769$	−15.8690	−0.572249	−0.286125	−	0.958192i	$-0.592367\pi$
$769$	−15.8690	−0.572249	−0.286125	+	0.958192i	$0.592367\pi$
$770$	−31.2022	−1.12445
$771$	−3.02588	−0.108975
$772$	18.9521	0.682101
$773$	42.6885	1.53540	0.767699	−	0.640811i	$-0.221402\pi$
$773$	42.6885	1.53540	0.767699	+	0.640811i	$0.221402\pi$
$774$	26.3939	0.948708
$775$	76.3243	2.74165
$776$	−8.07582	−0.289905
$777$	35.1225	1.26001
$778$	−11.3612	−0.407319
$779$	41.7352	1.49532
$780$	−116.352	−4.16605
$781$	−12.8862	−0.461106
$782$	20.5488	0.734824
$783$	−28.4340	−1.01615
$784$	−0.207033	−0.00739404
$785$	52.7807	1.88382
$786$	101.768	3.62995
$787$	−49.0728	−1.74926	−0.874629	−	0.484793i	$-0.838895\pi$
$787$	−49.0728	−1.74926	−0.874629	+	0.484793i	$0.838895\pi$
$788$	−28.5450	−1.01687
$789$	−3.82300	−0.136102
$790$	−131.024	−4.66163
$791$	−11.0786	−0.393910
$792$	6.72226	0.238865
$793$	−12.2451	−0.434835
$794$	46.5529	1.65210
$795$	−30.0252	−1.06488
$796$	−43.9101	−1.55635
$797$	−24.1208	−0.854403	−0.427201	−	0.904157i	$-0.640500\pi$
$797$	−24.1208	−0.854403	−0.427201	+	0.904157i	$0.640500\pi$
$798$	52.9625	1.87485
$799$	6.10445	0.215960
$800$	62.2048	2.19927
$801$	−6.93139	−0.244909
$802$	−23.4376	−0.827611
$803$	−23.9985	−0.846888
$804$	65.3131	2.30342
$805$	−20.4495	−0.720749
$806$	−69.5051	−2.44821
$807$	12.0822	0.425315
$808$	16.6210	0.584726
$809$	−6.65644	−0.234028	−0.117014	−	0.993130i	$-0.537332\pi$
$809$	−6.65644	−0.234028	−0.117014	+	0.993130i	$0.537332\pi$
$810$	−101.899	−3.58036
$811$	24.0680	0.845141	0.422570	−	0.906330i	$-0.361128\pi$
$811$	24.0680	0.845141	0.422570	+	0.906330i	$0.361128\pi$
$812$	37.8869	1.32957
$813$	18.9135	0.663325
$814$	53.9732	1.89176
$815$	−81.5740	−2.85741
$816$	−0.259195	−0.00907365
$817$	75.9150	2.65593
$818$	41.3333	1.44519
$819$	7.65279	0.267410
$820$	75.2430	2.62760
$821$	29.9586	1.04556	0.522782	−	0.852466i	$-0.324894\pi$
$821$	29.9586	1.04556	0.522782	+	0.852466i	$0.324894\pi$
$822$	−11.7580	−0.410108
$823$	−8.48611	−0.295807	−0.147904	−	0.989002i	$-0.547253\pi$
$823$	−8.48611	−0.295807	−0.147904	+	0.989002i	$0.547253\pi$
$824$	−27.0195	−0.941270
$825$	−48.2396	−1.67949
$826$	8.01979	0.279044
$827$	−8.28547	−0.288114	−0.144057	−	0.989569i	$-0.546015\pi$
$827$	−8.28547	−0.288114	−0.144057	+	0.989569i	$0.546015\pi$
$828$	11.4755	0.398801
$829$	28.9340	1.00492	0.502460	−	0.864601i	$-0.332429\pi$
$829$	28.9340	1.00492	0.502460	+	0.864601i	$0.332429\pi$
$830$	−117.289	−4.07116
$831$	−51.4882	−1.78611
$832$	−57.0532	−1.97796
$833$	12.5248	0.433959
$834$	−57.8071	−2.00169
$835$	−88.4309	−3.06028
$836$	50.3615	1.74179
$837$	−26.4888	−0.915588
$838$	19.9135	0.687900
$839$	46.3523	1.60026	0.800129	−	0.599828i	$-0.204764\pi$
$839$	46.3523	1.60026	0.800129	+	0.599828i	$0.204764\pi$
$840$	36.6584	1.26483
$841$	25.4791	0.878588
$842$	8.55470	0.294814
$843$	−33.0570	−1.13854
$844$	−64.7489	−2.22875
$845$	−25.9845	−0.893893
$846$	5.50927	0.189413
$847$	−10.1033	−0.347154
$848$	0.170096	0.00584113
$849$	−43.1075	−1.47945
$850$	70.7655	2.42724
$851$	35.3733	1.21258
$852$	39.4342	1.35099
$853$	−16.8208	−0.575932	−0.287966	−	0.957641i	$-0.592979\pi$
$853$	−16.8208	−0.575932	−0.287966	+	0.957641i	$0.592979\pi$
$854$	10.0490	0.343868
$855$	31.7941	1.08733
$856$	10.1366	0.346462
$857$	7.73056	0.264071	0.132035	−	0.991245i	$-0.457849\pi$
$857$	7.73056	0.264071	0.132035	+	0.991245i	$0.457849\pi$
$858$	43.9296	1.49973
$859$	−13.1395	−0.448315	−0.224157	−	0.974553i	$-0.571963\pi$
$859$	−13.1395	−0.448315	−0.224157	+	0.974553i	$0.571963\pi$
$860$	136.864	4.66704
$861$	−18.4867	−0.630024
$862$	−57.7064	−1.96549
$863$	36.5876	1.24546	0.622728	−	0.782438i	$-0.286024\pi$
$863$	36.5876	1.24546	0.622728	+	0.782438i	$0.286024\pi$
$864$	−21.5886	−0.734458
$865$	3.92637	0.133500
$866$	79.2281	2.69228
$867$	−18.7281	−0.636039
$868$	35.2950	1.19799
$869$	30.6107	1.03840
$870$	137.301	4.65493
$871$	43.8671	1.48638
$872$	21.8099	0.738578
$873$	3.10252	0.105005
$874$	53.3407	1.80428
$875$	−38.7012	−1.30834
$876$	73.4397	2.48130
$877$	1.14605	0.0386995	0.0193498	−	0.999813i	$-0.493840\pi$
$877$	1.14605	0.0386995	0.0193498	+	0.999813i	$0.493840\pi$
$878$	64.0380	2.16118
$879$	23.0322	0.776855
$880$	0.396383	0.0133621
$881$	−48.9932	−1.65062	−0.825312	−	0.564677i	$-0.809001\pi$
$881$	−48.9932	−1.65062	−0.825312	+	0.564677i	$0.809001\pi$
$882$	11.3036	0.380614
$883$	22.6771	0.763145	0.381573	−	0.924339i	$-0.375383\pi$
$883$	22.6771	0.763145	0.381573	+	0.924339i	$0.375383\pi$
$884$	−39.8761	−1.34118
$885$	17.9839	0.604523
$886$	81.4133	2.73513
$887$	25.7542	0.864741	0.432370	−	0.901696i	$-0.357677\pi$
$887$	25.7542	0.864741	0.432370	+	0.901696i	$0.357677\pi$
$888$	−63.4112	−2.12794
$889$	−3.94491	−0.132308
$890$	−58.0859	−1.94705
$891$	23.8062	0.797539
$892$	−33.3561	−1.11684
$893$	15.8460	0.530265
$894$	36.6495	1.22574
$895$	81.1648	2.71304
$896$	29.0989	0.972127
$897$	28.7908	0.961298
$898$	60.5747	2.02141
$899$	50.7522	1.69268
$900$	39.5191	1.31730
$901$	−10.2903	−0.342818
$902$	−28.4087	−0.945905
$903$	−33.6266	−1.11902
$904$	20.0016	0.665245
$905$	−4.82555	−0.160407
$906$	−75.5136	−2.50877
$907$	−42.4779	−1.41046	−0.705228	−	0.708981i	$-0.749155\pi$
$907$	−42.4779	−1.41046	−0.705228	+	0.708981i	$0.749155\pi$
$908$	79.6457	2.64313
$909$	−6.38538	−0.211790
$910$	64.1314	2.12593
$911$	52.1346	1.72730	0.863649	−	0.504094i	$-0.168173\pi$
$911$	52.1346	1.72730	0.863649	+	0.504094i	$0.168173\pi$
$912$	−0.672821	−0.0222793
$913$	27.4018	0.906867
$914$	−75.7992	−2.50721
$915$	22.5342	0.744959
$916$	−63.1268	−2.08577
$917$	−34.7094	−1.14621
$918$	−24.5596	−0.810589
$919$	11.5193	0.379988	0.189994	−	0.981785i	$-0.439153\pi$
$919$	11.5193	0.379988	0.189994	+	0.981785i	$0.439153\pi$
$920$	36.9201	1.21722
$921$	23.6543	0.779437
$922$	23.7977	0.783736
$923$	26.4857	0.871788
$924$	−22.3077	−0.733869
$925$	121.818	4.00534
$926$	−53.0998	−1.74497
$927$	10.3802	0.340931
$928$	41.3633	1.35782
$929$	7.21605	0.236751	0.118375	−	0.992969i	$-0.462231\pi$
$929$	7.21605	0.236751	0.118375	+	0.992969i	$0.462231\pi$
$930$	127.908	4.19427
$931$	32.5120	1.06554
$932$	−38.0774	−1.24727
$933$	5.33825	0.174766
$934$	−52.2960	−1.71118
$935$	−23.9799	−0.784225
$936$	−13.8166	−0.451609
$937$	−47.7693	−1.56056	−0.780278	−	0.625432i	$-0.784923\pi$
$937$	−47.7693	−1.56056	−0.780278	+	0.625432i	$0.784923\pi$
$938$	−35.9997	−1.17543
$939$	16.3439	0.533363
$940$	28.5681	0.931789
$941$	−47.4346	−1.54632	−0.773162	−	0.634208i	$-0.781326\pi$
$941$	−47.4346	−1.54632	−0.773162	+	0.634208i	$0.781326\pi$
$942$	60.9828	1.98693
$943$	−18.6186	−0.606307
$944$	−0.101881	−0.00331595
$945$	24.4409	0.795062
$946$	−51.6744	−1.68008
$947$	10.4742	0.340365	0.170183	−	0.985413i	$-0.445564\pi$
$947$	10.4742	0.340365	0.170183	+	0.985413i	$0.445564\pi$
$948$	−93.6743	−3.04240
$949$	49.3253	1.60117
$950$	183.694	5.95980
$951$	21.7291	0.704615
$952$	12.5636	0.407188
$953$	−41.4296	−1.34204	−0.671018	−	0.741441i	$-0.734143\pi$
$953$	−41.4296	−1.34204	−0.671018	+	0.741441i	$0.734143\pi$
$954$	−9.28696	−0.300676
$955$	42.9840	1.39093
$956$	−54.6907	−1.76882
$957$	−32.0771	−1.03691
$958$	47.4676	1.53361
$959$	4.01023	0.129497
$960$	104.993	3.38864
$961$	16.2802	0.525169
$962$	−110.934	−3.57665
$963$	−3.89422	−0.125490
$964$	−54.4188	−1.75271
$965$	−23.4249	−0.754074
$966$	−23.6273	−0.760196
$967$	−1.19534	−0.0384396	−0.0192198	−	0.999815i	$-0.506118\pi$
$967$	−1.19534	−0.0384396	−0.0192198	+	0.999815i	$0.506118\pi$
$968$	18.2408	0.586282
$969$	40.7034	1.30758
$970$	25.9995	0.834795
$971$	9.61827	0.308665	0.154332	−	0.988019i	$-0.450677\pi$
$971$	9.61827	0.308665	0.154332	+	0.988019i	$0.450677\pi$
$972$	−35.3337	−1.13333
$973$	19.7159	0.632062
$974$	68.3226	2.18920
$975$	99.1493	3.17532
$976$	−0.127659	−0.00408626
$977$	−7.08804	−0.226767	−0.113383	−	0.993551i	$-0.536169\pi$
$977$	−7.08804	−0.226767	−0.113383	+	0.993551i	$0.536169\pi$
$978$	−94.2506	−3.01380
$979$	13.5704	0.433712
$980$	58.6147	1.87238
$981$	−8.37882	−0.267515
$982$	−80.1601	−2.55801
$983$	−6.44833	−0.205670	−0.102835	−	0.994698i	$-0.532791\pi$
$983$	−6.44833	−0.205670	−0.102835	+	0.994698i	$0.532791\pi$
$984$	33.3764	1.06400
$985$	35.2817	1.12417
$986$	47.0558	1.49856
$987$	−7.01898	−0.223417
$988$	−103.511	−3.29311
$989$	−33.8667	−1.07690
$990$	−21.6418	−0.687823
$991$	20.3685	0.647028	0.323514	−	0.946223i	$-0.395136\pi$
$991$	20.3685	0.647028	0.323514	+	0.946223i	$0.395136\pi$
$992$	38.5337	1.22345
$993$	5.70836	0.181149
$994$	−21.7356	−0.689411
$995$	54.2730	1.72057
$996$	−83.8545	−2.65703
$997$	−52.6413	−1.66717	−0.833584	−	0.552393i	$-0.813715\pi$
$997$	−52.6413	−1.66717	−0.833584	+	0.552393i	$0.813715\pi$
$998$	86.3132	2.73220
$999$	−42.2776	−1.33760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4027.2.a.b.1.20	✓	159

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4027.2.a.b.1.20	✓	159	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)

\(f(q)\)	\(=\)	\(q-2.29049 q^{2} +2.02403 q^{3} +3.24633 q^{4} -4.01248 q^{5} -4.63602 q^{6} +1.58118 q^{7} -2.85471 q^{8} +1.09671 q^{9} +O(q^{10})\)	\(q-2.29049 q^{2} +2.02403 q^{3} +3.24633 q^{4} -4.01248 q^{5} -4.63602 q^{6} +1.58118 q^{7} -2.85471 q^{8} +1.09671 q^{9} +9.19054 q^{10} -2.14715 q^{11} +6.57068 q^{12} +4.41315 q^{13} -3.62167 q^{14} -8.12139 q^{15} +0.0460086 q^{16} -2.78337 q^{17} -2.51199 q^{18} -7.22508 q^{19} -13.0258 q^{20} +3.20036 q^{21} +4.91803 q^{22} +3.22321 q^{23} -5.77802 q^{24} +11.1000 q^{25} -10.1083 q^{26} -3.85233 q^{27} +5.13303 q^{28} +7.38099 q^{29} +18.6019 q^{30} +6.87606 q^{31} +5.60403 q^{32} -4.34591 q^{33} +6.37527 q^{34} -6.34445 q^{35} +3.56027 q^{36} +10.9746 q^{37} +16.5490 q^{38} +8.93236 q^{39} +11.4545 q^{40} -5.77644 q^{41} -7.33037 q^{42} -10.5071 q^{43} -6.97037 q^{44} -4.40051 q^{45} -7.38271 q^{46} -2.19319 q^{47} +0.0931229 q^{48} -4.49987 q^{49} -25.4244 q^{50} -5.63363 q^{51} +14.3266 q^{52} +3.69705 q^{53} +8.82371 q^{54} +8.61541 q^{55} -4.51380 q^{56} -14.6238 q^{57} -16.9061 q^{58} -2.21439 q^{59} -26.3647 q^{60} -2.77468 q^{61} -15.7495 q^{62} +1.73409 q^{63} -12.9280 q^{64} -17.7077 q^{65} +9.95425 q^{66} +9.94008 q^{67} -9.03574 q^{68} +6.52387 q^{69} +14.5319 q^{70} +6.00154 q^{71} -3.13078 q^{72} +11.1769 q^{73} -25.1371 q^{74} +22.4668 q^{75} -23.4550 q^{76} -3.39503 q^{77} -20.4595 q^{78} -14.2564 q^{79} -0.184609 q^{80} -11.0874 q^{81} +13.2309 q^{82} -12.7619 q^{83} +10.3894 q^{84} +11.1682 q^{85} +24.0665 q^{86} +14.9394 q^{87} +6.12950 q^{88} -6.32019 q^{89} +10.0793 q^{90} +6.97798 q^{91} +10.4636 q^{92} +13.9174 q^{93} +5.02347 q^{94} +28.9905 q^{95} +11.3427 q^{96} +2.82895 q^{97} +10.3069 q^{98} -2.35480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9}+O(q^{10}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9	\( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9} - 23 q^{10} - 33 q^{11} - 57 q^{12} - 90 q^{13} - 28 q^{14} - 22 q^{15} + 130 q^{16} - 145 q^{17} - 50 q^{18} - 28 q^{19} - 121 q^{20} - 69 q^{21} - 26 q^{22} - 79 q^{23} - 62 q^{24} + 123 q^{25} - 40 q^{26} - 70 q^{27} - 43 q^{28} - 109 q^{29} - 43 q^{30} - 21 q^{31} - 139 q^{32} - 83 q^{33} - 93 q^{35} + 75 q^{36} - 65 q^{37} - 122 q^{38} - 18 q^{39} - 43 q^{40} - 71 q^{41} - 88 q^{42} - 72 q^{43} - 79 q^{44} - 181 q^{45} - 11 q^{46} - 114 q^{47} - 118 q^{48} + 118 q^{49} - 77 q^{50} - 29 q^{51} - 169 q^{52} - 220 q^{53} - 80 q^{54} - 37 q^{55} - 72 q^{56} - 90 q^{57} - 8 q^{58} - 60 q^{59} - 42 q^{60} - 108 q^{61} - 152 q^{62} - 65 q^{63} + 114 q^{64} - 81 q^{65} - 40 q^{66} - 50 q^{67} - 319 q^{68} - 103 q^{69} + 4 q^{70} - 7 q^{71} - 129 q^{72} - 94 q^{73} - 79 q^{74} - 59 q^{75} - 46 q^{76} - 329 q^{77} + 8 q^{78} - 18 q^{79} - 190 q^{80} + 59 q^{81} - 56 q^{82} - 201 q^{83} - 71 q^{84} - 26 q^{85} - 52 q^{86} - 126 q^{87} - 66 q^{88} - 114 q^{89} - 33 q^{90} - 30 q^{91} - 204 q^{92} - 125 q^{93} + 9 q^{94} - 84 q^{95} - 88 q^{96} - 56 q^{97} - 110 q^{98} - 46 q^{99}+O(q^{100}) \) 159 * q - 22 * q^2 - 19 * q^3 + 148 * q^4 - 70 * q^5 - 23 * q^6 - 19 * q^7 - 66 * q^8 + 126 * q^9 - 23 * q^10 - 33 * q^11 - 57 * q^12 - 90 * q^13 - 28 * q^14 - 22 * q^15 + 130 * q^16 - 145 * q^17 - 50 * q^18 - 28 * q^19 - 121 * q^20 - 69 * q^21 - 26 * q^22 - 79 * q^23 - 62 * q^24 + 123 * q^25 - 40 * q^26 - 70 * q^27 - 43 * q^28 - 109 * q^29 - 43 * q^30 - 21 * q^31 - 139 * q^32 - 83 * q^33 - 93 * q^35 + 75 * q^36 - 65 * q^37 - 122 * q^38 - 18 * q^39 - 43 * q^40 - 71 * q^41 - 88 * q^42 - 72 * q^43 - 79 * q^44 - 181 * q^45 - 11 * q^46 - 114 * q^47 - 118 * q^48 + 118 * q^49 - 77 * q^50 - 29 * q^51 - 169 * q^52 - 220 * q^53 - 80 * q^54 - 37 * q^55 - 72 * q^56 - 90 * q^57 - 8 * q^58 - 60 * q^59 - 42 * q^60 - 108 * q^61 - 152 * q^62 - 65 * q^63 + 114 * q^64 - 81 * q^65 - 40 * q^66 - 50 * q^67 - 319 * q^68 - 103 * q^69 + 4 * q^70 - 7 * q^71 - 129 * q^72 - 94 * q^73 - 79 * q^74 - 59 * q^75 - 46 * q^76 - 329 * q^77 + 8 * q^78 - 18 * q^79 - 190 * q^80 + 59 * q^81 - 56 * q^82 - 201 * q^83 - 71 * q^84 - 26 * q^85 - 52 * q^86 - 126 * q^87 - 66 * q^88 - 114 * q^89 - 33 * q^90 - 30 * q^91 - 204 * q^92 - 125 * q^93 + 9 * q^94 - 84 * q^95 - 88 * q^96 - 56 * q^97 - 110 * q^98 - 46 * q^99

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