LMFDB - Embedded newform 4022.2.a.d.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, 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("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.16293 q^{3} +1.00000 q^{4} +3.05784 q^{5} +1.16293 q^{6} +2.13691 q^{7} -1.00000 q^{8} -1.64760 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.16293 q^{3} +1.00000 q^{4} +3.05784 q^{5} +1.16293 q^{6} +2.13691 q^{7} -1.00000 q^{8} -1.64760 q^{9} -3.05784 q^{10} +2.96503 q^{11} -1.16293 q^{12} -2.55607 q^{13} -2.13691 q^{14} -3.55604 q^{15} +1.00000 q^{16} +0.966746 q^{17} +1.64760 q^{18} -1.03195 q^{19} +3.05784 q^{20} -2.48507 q^{21} -2.96503 q^{22} -7.18065 q^{23} +1.16293 q^{24} +4.35037 q^{25} +2.55607 q^{26} +5.40482 q^{27} +2.13691 q^{28} -3.72364 q^{29} +3.55604 q^{30} -5.13329 q^{31} -1.00000 q^{32} -3.44811 q^{33} -0.966746 q^{34} +6.53433 q^{35} -1.64760 q^{36} +3.04420 q^{37} +1.03195 q^{38} +2.97252 q^{39} -3.05784 q^{40} -9.21011 q^{41} +2.48507 q^{42} -8.73092 q^{43} +2.96503 q^{44} -5.03810 q^{45} +7.18065 q^{46} -1.66879 q^{47} -1.16293 q^{48} -2.43361 q^{49} -4.35037 q^{50} -1.12425 q^{51} -2.55607 q^{52} +1.21637 q^{53} -5.40482 q^{54} +9.06658 q^{55} -2.13691 q^{56} +1.20008 q^{57} +3.72364 q^{58} -10.4656 q^{59} -3.55604 q^{60} -4.34071 q^{61} +5.13329 q^{62} -3.52079 q^{63} +1.00000 q^{64} -7.81605 q^{65} +3.44811 q^{66} +6.10476 q^{67} +0.966746 q^{68} +8.35056 q^{69} -6.53433 q^{70} -1.70705 q^{71} +1.64760 q^{72} -15.5689 q^{73} -3.04420 q^{74} -5.05916 q^{75} -1.03195 q^{76} +6.33601 q^{77} -2.97252 q^{78} -5.88804 q^{79} +3.05784 q^{80} -1.34259 q^{81} +9.21011 q^{82} -8.56847 q^{83} -2.48507 q^{84} +2.95615 q^{85} +8.73092 q^{86} +4.33031 q^{87} -2.96503 q^{88} -0.694381 q^{89} +5.03810 q^{90} -5.46210 q^{91} -7.18065 q^{92} +5.96964 q^{93} +1.66879 q^{94} -3.15552 q^{95} +1.16293 q^{96} +3.16886 q^{97} +2.43361 q^{98} -4.88520 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9}+O(q^{10}) $ 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9} + 13 q^{10} + 12 q^{11} - 5 q^{12} - 36 q^{13} + 22 q^{14} - 7 q^{15} + 37 q^{16} + 4 q^{17} - 32 q^{18} - 10 q^{19} - 13 q^{20} - 11 q^{21} - 12 q^{22} - q^{23} + 5 q^{24} + 8 q^{25} + 36 q^{26} - 20 q^{27} - 22 q^{28} - 6 q^{29} + 7 q^{30} - 23 q^{31} - 37 q^{32} - 27 q^{33} - 4 q^{34} + 24 q^{35} + 32 q^{36} - 46 q^{37} + 10 q^{38} + 5 q^{39} + 13 q^{40} + 11 q^{41} + 11 q^{42} - 22 q^{43} + 12 q^{44} - 57 q^{45} + q^{46} - 18 q^{47} - 5 q^{48} - q^{49} - 8 q^{50} + 18 q^{51} - 36 q^{52} - 25 q^{53} + 20 q^{54} - 25 q^{55} + 22 q^{56} - 25 q^{57} + 6 q^{58} + 24 q^{59} - 7 q^{60} - 40 q^{61} + 23 q^{62} - 38 q^{63} + 37 q^{64} + 14 q^{65} + 27 q^{66} - 49 q^{67} + 4 q^{68} - 19 q^{69} - 24 q^{70} + 27 q^{71} - 32 q^{72} - 87 q^{73} + 46 q^{74} - 5 q^{75} - 10 q^{76} - 50 q^{77} - 5 q^{78} + 11 q^{79} - 13 q^{80} + 5 q^{81} - 11 q^{82} - 9 q^{83} - 11 q^{84} - 68 q^{85} + 22 q^{86} - 12 q^{87} - 12 q^{88} - 3 q^{89} + 57 q^{90} - 19 q^{91} - q^{92} - 69 q^{93} + 18 q^{94} + 27 q^{95} + 5 q^{96} - 98 q^{97} + q^{98} + 33 q^{99}+O(q^{100}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9 + 13 * q^10 + 12 * q^11 - 5 * q^12 - 36 * q^13 + 22 * q^14 - 7 * q^15 + 37 * q^16 + 4 * q^17 - 32 * q^18 - 10 * q^19 - 13 * q^20 - 11 * q^21 - 12 * q^22 - q^23 + 5 * q^24 + 8 * q^25 + 36 * q^26 - 20 * q^27 - 22 * q^28 - 6 * q^29 + 7 * q^30 - 23 * q^31 - 37 * q^32 - 27 * q^33 - 4 * q^34 + 24 * q^35 + 32 * q^36 - 46 * q^37 + 10 * q^38 + 5 * q^39 + 13 * q^40 + 11 * q^41 + 11 * q^42 - 22 * q^43 + 12 * q^44 - 57 * q^45 + q^46 - 18 * q^47 - 5 * q^48 - q^49 - 8 * q^50 + 18 * q^51 - 36 * q^52 - 25 * q^53 + 20 * q^54 - 25 * q^55 + 22 * q^56 - 25 * q^57 + 6 * q^58 + 24 * q^59 - 7 * q^60 - 40 * q^61 + 23 * q^62 - 38 * q^63 + 37 * q^64 + 14 * q^65 + 27 * q^66 - 49 * q^67 + 4 * q^68 - 19 * q^69 - 24 * q^70 + 27 * q^71 - 32 * q^72 - 87 * q^73 + 46 * q^74 - 5 * q^75 - 10 * q^76 - 50 * q^77 - 5 * q^78 + 11 * q^79 - 13 * q^80 + 5 * q^81 - 11 * q^82 - 9 * q^83 - 11 * q^84 - 68 * q^85 + 22 * q^86 - 12 * q^87 - 12 * q^88 - 3 * q^89 + 57 * q^90 - 19 * q^91 - q^92 - 69 * q^93 + 18 * q^94 + 27 * q^95 + 5 * q^96 - 98 * q^97 + q^98 + 33 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.16293	−0.671415	−0.335708	−	0.941966i	$-0.608975\pi$
$3$	−1.16293	−0.671415	−0.335708	+	0.941966i	$0.608975\pi$
$4$	1.00000	0.500000
$5$	3.05784	1.36751	0.683753	−	0.729713i	$-0.260346\pi$
$5$	3.05784	1.36751	0.683753	+	0.729713i	$0.260346\pi$
$6$	1.16293	0.474762
$7$	2.13691	0.807677	0.403838	−	0.914830i	$-0.367676\pi$
$7$	2.13691	0.807677	0.403838	+	0.914830i	$0.367676\pi$
$8$	−1.00000	−0.353553
$9$	−1.64760	−0.549201
$10$	−3.05784	−0.966973
$11$	2.96503	0.893991	0.446995	−	0.894536i	$-0.352494\pi$
$11$	2.96503	0.893991	0.446995	+	0.894536i	$0.352494\pi$
$12$	−1.16293	−0.335708
$13$	−2.55607	−0.708927	−0.354464	−	0.935070i	$-0.615336\pi$
$13$	−2.55607	−0.708927	−0.354464	+	0.935070i	$0.615336\pi$
$14$	−2.13691	−0.571114
$15$	−3.55604	−0.918165
$16$	1.00000	0.250000
$17$	0.966746	0.234470	0.117235	−	0.993104i	$-0.462597\pi$
$17$	0.966746	0.234470	0.117235	+	0.993104i	$0.462597\pi$
$18$	1.64760	0.388344
$19$	−1.03195	−0.236745	−0.118372	−	0.992969i	$-0.537768\pi$
$19$	−1.03195	−0.236745	−0.118372	+	0.992969i	$0.537768\pi$
$20$	3.05784	0.683753
$21$	−2.48507	−0.542287
$22$	−2.96503	−0.632147
$23$	−7.18065	−1.49727	−0.748634	−	0.662983i	$-0.769290\pi$
$23$	−7.18065	−1.49727	−0.748634	+	0.662983i	$0.769290\pi$
$24$	1.16293	0.237381
$25$	4.35037	0.870074
$26$	2.55607	0.501287
$27$	5.40482	1.04016
$28$	2.13691	0.403838
$29$	−3.72364	−0.691462	−0.345731	−	0.938334i	$-0.612369\pi$
$29$	−3.72364	−0.691462	−0.345731	+	0.938334i	$0.612369\pi$
$30$	3.55604	0.649241
$31$	−5.13329	−0.921967	−0.460983	−	0.887409i	$-0.652503\pi$
$31$	−5.13329	−0.921967	−0.460983	+	0.887409i	$0.652503\pi$
$32$	−1.00000	−0.176777
$33$	−3.44811	−0.600239
$34$	−0.966746	−0.165796
$35$	6.53433	1.10450
$36$	−1.64760	−0.274601
$37$	3.04420	0.500463	0.250232	−	0.968186i	$-0.419493\pi$
$37$	3.04420	0.500463	0.250232	+	0.968186i	$0.419493\pi$
$38$	1.03195	0.167404
$39$	2.97252	0.475985
$40$	−3.05784	−0.483487
$41$	−9.21011	−1.43838	−0.719189	−	0.694815i	$-0.755486\pi$
$41$	−9.21011	−1.43838	−0.719189	+	0.694815i	$0.755486\pi$
$42$	2.48507	0.383455
$43$	−8.73092	−1.33145	−0.665726	−	0.746196i	$-0.731878\pi$
$43$	−8.73092	−1.33145	−0.665726	+	0.746196i	$0.731878\pi$
$44$	2.96503	0.446995
$45$	−5.03810	−0.751036
$46$	7.18065	1.05873
$47$	−1.66879	−0.243419	−0.121709	−	0.992566i	$-0.538838\pi$
$47$	−1.66879	−0.243419	−0.121709	+	0.992566i	$0.538838\pi$
$48$	−1.16293	−0.167854
$49$	−2.43361	−0.347658
$50$	−4.35037	−0.615235
$51$	−1.12425	−0.157427
$52$	−2.55607	−0.354464
$53$	1.21637	0.167082	0.0835408	−	0.996504i	$-0.473377\pi$
$53$	1.21637	0.167082	0.0835408	+	0.996504i	$0.473377\pi$
$54$	−5.40482	−0.735503
$55$	9.06658	1.22254
$56$	−2.13691	−0.285557
$57$	1.20008	0.158954
$58$	3.72364	0.488937
$59$	−10.4656	−1.36251	−0.681253	−	0.732048i	$-0.738565\pi$
$59$	−10.4656	−1.36251	−0.681253	+	0.732048i	$0.738565\pi$
$60$	−3.55604	−0.459082
$61$	−4.34071	−0.555771	−0.277885	−	0.960614i	$-0.589634\pi$
$61$	−4.34071	−0.555771	−0.277885	+	0.960614i	$0.589634\pi$
$62$	5.13329	0.651929
$63$	−3.52079	−0.443577
$64$	1.00000	0.125000
$65$	−7.81605	−0.969462
$66$	3.44811	0.424433
$67$	6.10476	0.745815	0.372908	−	0.927868i	$-0.378361\pi$
$67$	6.10476	0.745815	0.372908	+	0.927868i	$0.378361\pi$
$68$	0.966746	0.117235
$69$	8.35056	1.00529
$70$	−6.53433	−0.781002
$71$	−1.70705	−0.202589	−0.101295	−	0.994856i	$-0.532298\pi$
$71$	−1.70705	−0.202589	−0.101295	+	0.994856i	$0.532298\pi$
$72$	1.64760	0.194172
$73$	−15.5689	−1.82220	−0.911102	−	0.412181i	$-0.864767\pi$
$73$	−15.5689	−1.82220	−0.911102	+	0.412181i	$0.864767\pi$
$74$	−3.04420	−0.353881
$75$	−5.05916	−0.584181
$76$	−1.03195	−0.118372
$77$	6.33601	0.722056
$78$	−2.97252	−0.336572
$79$	−5.88804	−0.662456	−0.331228	−	0.943551i	$-0.607463\pi$
$79$	−5.88804	−0.662456	−0.331228	+	0.943551i	$0.607463\pi$
$80$	3.05784	0.341877
$81$	−1.34259	−0.149177
$82$	9.21011	1.01709
$83$	−8.56847	−0.940511	−0.470256	−	0.882530i	$-0.655838\pi$
$83$	−8.56847	−0.940511	−0.470256	+	0.882530i	$0.655838\pi$
$84$	−2.48507	−0.271143
$85$	2.95615	0.320640
$86$	8.73092	0.941479
$87$	4.33031	0.464258
$88$	−2.96503	−0.316073
$89$	−0.694381	−0.0736042	−0.0368021	−	0.999323i	$-0.511717\pi$
$89$	−0.694381	−0.0736042	−0.0368021	+	0.999323i	$0.511717\pi$
$90$	5.03810	0.531063
$91$	−5.46210	−0.572584
$92$	−7.18065	−0.748634
$93$	5.96964	0.619023
$94$	1.66879	0.172123
$95$	−3.15552	−0.323750
$96$	1.16293	0.118691
$97$	3.16886	0.321749	0.160875	−	0.986975i	$-0.448569\pi$
$97$	3.16886	0.321749	0.160875	+	0.986975i	$0.448569\pi$
$98$	2.43361	0.245831
$99$	−4.88520	−0.490981
$100$	4.35037	0.435037
$101$	2.38671	0.237486	0.118743	−	0.992925i	$-0.462114\pi$
$101$	2.38671	0.237486	0.118743	+	0.992925i	$0.462114\pi$
$102$	1.12425	0.111318
$103$	−17.4313	−1.71755	−0.858777	−	0.512349i	$-0.828775\pi$
$103$	−17.4313	−1.71755	−0.858777	+	0.512349i	$0.828775\pi$
$104$	2.55607	0.250644
$105$	−7.59894	−0.741581
$106$	−1.21637	−0.118145
$107$	15.8714	1.53435	0.767173	−	0.641440i	$-0.221663\pi$
$107$	15.8714	1.53435	0.767173	+	0.641440i	$0.221663\pi$
$108$	5.40482	0.520079
$109$	15.0679	1.44325	0.721623	−	0.692286i	$-0.243396\pi$
$109$	15.0679	1.44325	0.721623	+	0.692286i	$0.243396\pi$
$110$	−9.06658	−0.864465
$111$	−3.54018	−0.336019
$112$	2.13691	0.201919
$113$	7.14717	0.672349	0.336175	−	0.941800i	$-0.390867\pi$
$113$	7.14717	0.672349	0.336175	+	0.941800i	$0.390867\pi$
$114$	−1.20008	−0.112397
$115$	−21.9573	−2.04752
$116$	−3.72364	−0.345731
$117$	4.21140	0.389344
$118$	10.4656	0.963437
$119$	2.06585	0.189376
$120$	3.55604	0.324620
$121$	−2.20859	−0.200781
$122$	4.34071	0.392989
$123$	10.7107	0.965749
$124$	−5.13329	−0.460983
$125$	−1.98647	−0.177675
$126$	3.52079	0.313656
$127$	4.19037	0.371835	0.185917	−	0.982565i	$-0.440474\pi$
$127$	4.19037	0.371835	0.185917	+	0.982565i	$0.440474\pi$
$128$	−1.00000	−0.0883883
$129$	10.1534	0.893958
$130$	7.81605	0.685513
$131$	8.08351	0.706259	0.353130	−	0.935574i	$-0.385117\pi$
$131$	8.08351	0.706259	0.353130	+	0.935574i	$0.385117\pi$
$132$	−3.44811	−0.300120
$133$	−2.20518	−0.191213
$134$	−6.10476	−0.527371
$135$	16.5271	1.42242
$136$	−0.966746	−0.0828978
$137$	−4.81781	−0.411613	−0.205807	−	0.978593i	$-0.565982\pi$
$137$	−4.81781	−0.411613	−0.205807	+	0.978593i	$0.565982\pi$
$138$	−8.35056	−0.710847
$139$	14.0581	1.19239	0.596195	−	0.802840i	$-0.296678\pi$
$139$	14.0581	1.19239	0.596195	+	0.802840i	$0.296678\pi$
$140$	6.53433	0.552252
$141$	1.94068	0.163435
$142$	1.70705	0.143252
$143$	−7.57884	−0.633774
$144$	−1.64760	−0.137300
$145$	−11.3863	−0.945578
$146$	15.5689	1.28849
$147$	2.83010	0.233423
$148$	3.04420	0.250232
$149$	−4.17264	−0.341836	−0.170918	−	0.985285i	$-0.554673\pi$
$149$	−4.17264	−0.341836	−0.170918	+	0.985285i	$0.554673\pi$
$150$	5.05916	0.413078
$151$	21.7967	1.77379	0.886896	−	0.461969i	$-0.152857\pi$
$151$	21.7967	1.77379	0.886896	+	0.461969i	$0.152857\pi$
$152$	1.03195	0.0837018
$153$	−1.59282	−0.128771
$154$	−6.33601	−0.510570
$155$	−15.6968	−1.26080
$156$	2.97252	0.237992
$157$	−13.7140	−1.09449	−0.547247	−	0.836971i	$-0.684324\pi$
$157$	−13.7140	−1.09449	−0.547247	+	0.836971i	$0.684324\pi$
$158$	5.88804	0.468427
$159$	−1.41455	−0.112181
$160$	−3.05784	−0.241743
$161$	−15.3444	−1.20931
$162$	1.34259	0.105484
$163$	25.2066	1.97433	0.987166	−	0.159695i	$-0.0510511\pi$
$163$	25.2066	1.97433	0.987166	+	0.159695i	$0.0510511\pi$
$164$	−9.21011	−0.719189
$165$	−10.5438	−0.820831
$166$	8.56847	0.665042
$167$	8.62120	0.667128	0.333564	−	0.942727i	$-0.391749\pi$
$167$	8.62120	0.667128	0.333564	+	0.942727i	$0.391749\pi$
$168$	2.48507	0.191727
$169$	−6.46649	−0.497423
$170$	−2.95615	−0.226727
$171$	1.70024	0.130020
$172$	−8.73092	−0.665726
$173$	−8.84315	−0.672332	−0.336166	−	0.941803i	$-0.609130\pi$
$173$	−8.84315	−0.672332	−0.336166	+	0.941803i	$0.609130\pi$
$174$	−4.33031	−0.328280
$175$	9.29636	0.702739
$176$	2.96503	0.223498
$177$	12.1707	0.914807
$178$	0.694381	0.0520461
$179$	12.5877	0.940849	0.470425	−	0.882440i	$-0.344101\pi$
$179$	12.5877	0.940849	0.470425	+	0.882440i	$0.344101\pi$
$180$	−5.03810	−0.375518
$181$	−1.67433	−0.124452	−0.0622260	−	0.998062i	$-0.519820\pi$
$181$	−1.67433	−0.124452	−0.0622260	+	0.998062i	$0.519820\pi$
$182$	5.46210	0.404878
$183$	5.04792	0.373153
$184$	7.18065	0.529364
$185$	9.30867	0.684387
$186$	−5.96964	−0.437715
$187$	2.86643	0.209614
$188$	−1.66879	−0.121709
$189$	11.5496	0.840111
$190$	3.15552	0.228926
$191$	−3.12124	−0.225845	−0.112923	−	0.993604i	$-0.536021\pi$
$191$	−3.12124	−0.225845	−0.112923	+	0.993604i	$0.536021\pi$
$192$	−1.16293	−0.0839269
$193$	7.66190	0.551516	0.275758	−	0.961227i	$-0.411071\pi$
$193$	7.66190	0.551516	0.275758	+	0.961227i	$0.411071\pi$
$194$	−3.16886	−0.227511
$195$	9.08949	0.650912
$196$	−2.43361	−0.173829
$197$	−20.3129	−1.44723	−0.723616	−	0.690202i	$-0.757522\pi$
$197$	−20.3129	−1.44723	−0.723616	+	0.690202i	$0.757522\pi$
$198$	4.88520	0.347176
$199$	1.25313	0.0888320	0.0444160	−	0.999013i	$-0.485857\pi$
$199$	1.25313	0.0888320	0.0444160	+	0.999013i	$0.485857\pi$
$200$	−4.35037	−0.307618
$201$	−7.09938	−0.500752
$202$	−2.38671	−0.167928
$203$	−7.95708	−0.558478
$204$	−1.12425	−0.0787135
$205$	−28.1630	−1.96699
$206$	17.4313	1.21449
$207$	11.8309	0.822302
$208$	−2.55607	−0.177232
$209$	−3.05975	−0.211647
$210$	7.59894	0.524377
$211$	25.8028	1.77634	0.888168	−	0.459520i	$-0.151978\pi$
$211$	25.8028	1.77634	0.888168	+	0.459520i	$0.151978\pi$
$212$	1.21637	0.0835408
$213$	1.98517	0.136022
$214$	−15.8714	−1.08495
$215$	−26.6977	−1.82077
$216$	−5.40482	−0.367751
$217$	−10.9694	−0.744651
$218$	−15.0679	−1.02053
$219$	18.1055	1.22346
$220$	9.06658	0.611269
$221$	−2.47107	−0.166222
$222$	3.54018	0.237601
$223$	−20.7896	−1.39218	−0.696089	−	0.717956i	$-0.745078\pi$
$223$	−20.7896	−1.39218	−0.696089	+	0.717956i	$0.745078\pi$
$224$	−2.13691	−0.142778
$225$	−7.16768	−0.477846
$226$	−7.14717	−0.475423
$227$	−24.7747	−1.64435	−0.822176	−	0.569234i	$-0.807240\pi$
$227$	−24.7747	−1.64435	−0.822176	+	0.569234i	$0.807240\pi$
$228$	1.20008	0.0794770
$229$	14.9930	0.990768	0.495384	−	0.868674i	$-0.335027\pi$
$229$	14.9930	0.990768	0.495384	+	0.868674i	$0.335027\pi$
$230$	21.9573	1.44782
$231$	−7.36831	−0.484799
$232$	3.72364	0.244469
$233$	−13.2104	−0.865444	−0.432722	−	0.901527i	$-0.642447\pi$
$233$	−13.2104	−0.865444	−0.432722	+	0.901527i	$0.642447\pi$
$234$	−4.21140	−0.275308
$235$	−5.10290	−0.332876
$236$	−10.4656	−0.681253
$237$	6.84735	0.444783
$238$	−2.06585	−0.133909
$239$	11.5531	0.747308	0.373654	−	0.927568i	$-0.378105\pi$
$239$	11.5531	0.747308	0.373654	+	0.927568i	$0.378105\pi$
$240$	−3.55604	−0.229541
$241$	24.1694	1.55689	0.778443	−	0.627716i	$-0.216010\pi$
$241$	24.1694	1.55689	0.778443	+	0.627716i	$0.216010\pi$
$242$	2.20859	0.141973
$243$	−14.6531	−0.939998
$244$	−4.34071	−0.277885
$245$	−7.44157	−0.475424
$246$	−10.7107	−0.682888
$247$	2.63773	0.167835
$248$	5.13329	0.325964
$249$	9.96449	0.631474
$250$	1.98647	0.125635
$251$	11.7945	0.744461	0.372230	−	0.928140i	$-0.378593\pi$
$251$	11.7945	0.744461	0.372230	+	0.928140i	$0.378593\pi$
$252$	−3.52079	−0.221789
$253$	−21.2909	−1.33854
$254$	−4.19037	−0.262927
$255$	−3.43779	−0.215283
$256$	1.00000	0.0625000
$257$	25.9820	1.62072	0.810358	−	0.585936i	$-0.199273\pi$
$257$	25.9820	1.62072	0.810358	+	0.585936i	$0.199273\pi$
$258$	−10.1534	−0.632124
$259$	6.50519	0.404213
$260$	−7.81605	−0.484731
$261$	6.13508	0.379752
$262$	−8.08351	−0.499401
$263$	−1.84525	−0.113783	−0.0568915	−	0.998380i	$-0.518119\pi$
$263$	−1.84525	−0.113783	−0.0568915	+	0.998380i	$0.518119\pi$
$264$	3.44811	0.212217
$265$	3.71947	0.228485
$266$	2.20518	0.135208
$267$	0.807513	0.0494190
$268$	6.10476	0.372908
$269$	25.6329	1.56287	0.781434	−	0.623988i	$-0.214489\pi$
$269$	25.6329	1.56287	0.781434	+	0.623988i	$0.214489\pi$
$270$	−16.5271	−1.00580
$271$	−30.0547	−1.82569	−0.912845	−	0.408306i	$-0.866120\pi$
$271$	−30.0547	−1.82569	−0.912845	+	0.408306i	$0.866120\pi$
$272$	0.966746	0.0586176
$273$	6.35202	0.384442
$274$	4.81781	0.291054
$275$	12.8990	0.777838
$276$	8.35056	0.502645
$277$	−19.0276	−1.14326	−0.571629	−	0.820512i	$-0.693688\pi$
$277$	−19.0276	−1.14326	−0.571629	+	0.820512i	$0.693688\pi$
$278$	−14.0581	−0.843147
$279$	8.45763	0.506345
$280$	−6.53433	−0.390501
$281$	7.15806	0.427014	0.213507	−	0.976942i	$-0.431511\pi$
$281$	7.15806	0.427014	0.213507	+	0.976942i	$0.431511\pi$
$282$	−1.94068	−0.115566
$283$	−1.24476	−0.0739933	−0.0369966	−	0.999315i	$-0.511779\pi$
$283$	−1.24476	−0.0739933	−0.0369966	+	0.999315i	$0.511779\pi$
$284$	−1.70705	−0.101295
$285$	3.66964	0.217371
$286$	7.57884	0.448146
$287$	−19.6812	−1.16174
$288$	1.64760	0.0970860
$289$	−16.0654	−0.945024
$290$	11.3863	0.668625
$291$	−3.68515	−0.216027
$292$	−15.5689	−0.911102
$293$	−2.10734	−0.123112	−0.0615561	−	0.998104i	$-0.519606\pi$
$293$	−2.10734	−0.123112	−0.0615561	+	0.998104i	$0.519606\pi$
$294$	−2.83010	−0.165055
$295$	−32.0021	−1.86324
$296$	−3.04420	−0.176940
$297$	16.0255	0.929891
$298$	4.17264	0.241714
$299$	18.3543	1.06145
$300$	−5.05916	−0.292090
$301$	−18.6572	−1.07538
$302$	−21.7967	−1.25426
$303$	−2.77556	−0.159452
$304$	−1.03195	−0.0591861
$305$	−13.2732	−0.760020
$306$	1.59282	0.0910552
$307$	−6.02806	−0.344040	−0.172020	−	0.985093i	$-0.555029\pi$
$307$	−6.02806	−0.344040	−0.172020	+	0.985093i	$0.555029\pi$
$308$	6.33601	0.361028
$309$	20.2713	1.15319
$310$	15.6968	0.891517
$311$	8.47349	0.480488	0.240244	−	0.970713i	$-0.422773\pi$
$311$	8.47349	0.480488	0.240244	+	0.970713i	$0.422773\pi$
$312$	−2.97252	−0.168286
$313$	−13.3825	−0.756424	−0.378212	−	0.925719i	$-0.623461\pi$
$313$	−13.3825	−0.756424	−0.378212	+	0.925719i	$0.623461\pi$
$314$	13.7140	0.773924
$315$	−10.7660	−0.606595
$316$	−5.88804	−0.331228
$317$	−26.4469	−1.48541	−0.742704	−	0.669620i	$-0.766457\pi$
$317$	−26.4469	−1.48541	−0.742704	+	0.669620i	$0.766457\pi$
$318$	1.41455	0.0793241
$319$	−11.0407	−0.618160
$320$	3.05784	0.170938
$321$	−18.4573	−1.03018
$322$	15.3444	0.855111
$323$	−0.997630	−0.0555096
$324$	−1.34259	−0.0745883
$325$	−11.1199	−0.616819
$326$	−25.2066	−1.39606
$327$	−17.5229	−0.969018
$328$	9.21011	0.508543
$329$	−3.56607	−0.196604
$330$	10.5438	0.580415
$331$	−26.5639	−1.46009	−0.730043	−	0.683402i	$-0.760500\pi$
$331$	−26.5639	−1.46009	−0.730043	+	0.683402i	$0.760500\pi$
$332$	−8.56847	−0.470256
$333$	−5.01563	−0.274855
$334$	−8.62120	−0.471731
$335$	18.6674	1.01991
$336$	−2.48507	−0.135572
$337$	15.3211	0.834596	0.417298	−	0.908770i	$-0.362977\pi$
$337$	15.3211	0.834596	0.417298	+	0.908770i	$0.362977\pi$
$338$	6.46649	0.351731
$339$	−8.31163	−0.451426
$340$	2.95615	0.160320
$341$	−15.2204	−0.824230
$342$	−1.70024	−0.0919383
$343$	−20.1588	−1.08847
$344$	8.73092	0.470740
$345$	25.5347	1.37474
$346$	8.84315	0.475410
$347$	11.2901	0.606086	0.303043	−	0.952977i	$-0.401997\pi$
$347$	11.2901	0.606086	0.303043	+	0.952977i	$0.401997\pi$
$348$	4.33031	0.232129
$349$	10.3245	0.552656	0.276328	−	0.961063i	$-0.410882\pi$
$349$	10.3245	0.552656	0.276328	+	0.961063i	$0.410882\pi$
$350$	−9.29636	−0.496911
$351$	−13.8151	−0.737396
$352$	−2.96503	−0.158037
$353$	7.39201	0.393437	0.196718	−	0.980460i	$-0.436972\pi$
$353$	7.39201	0.393437	0.196718	+	0.980460i	$0.436972\pi$
$354$	−12.1707	−0.646866
$355$	−5.21987	−0.277042
$356$	−0.694381	−0.0368021
$357$	−2.40243	−0.127150
$358$	−12.5877	−0.665281
$359$	−5.47904	−0.289173	−0.144586	−	0.989492i	$-0.546185\pi$
$359$	−5.47904	−0.289173	−0.144586	+	0.989492i	$0.546185\pi$
$360$	5.03810	0.265531
$361$	−17.9351	−0.943952
$362$	1.67433	0.0880008
$363$	2.56842	0.134807
$364$	−5.46210	−0.286292
$365$	−47.6072	−2.49188
$366$	−5.04792	−0.263859
$367$	−0.707477	−0.0369300	−0.0184650	−	0.999830i	$-0.505878\pi$
$367$	−0.707477	−0.0369300	−0.0184650	+	0.999830i	$0.505878\pi$
$368$	−7.18065	−0.374317
$369$	15.1746	0.789959
$370$	−9.30867	−0.483934
$371$	2.59928	0.134948
$372$	5.96964	0.309511
$373$	−4.98270	−0.257995	−0.128997	−	0.991645i	$-0.541176\pi$
$373$	−4.98270	−0.257995	−0.128997	+	0.991645i	$0.541176\pi$
$374$	−2.86643	−0.148220
$375$	2.31011	0.119294
$376$	1.66879	0.0860615
$377$	9.51788	0.490196
$378$	−11.5496	−0.594048
$379$	−27.1450	−1.39434	−0.697171	−	0.716904i	$-0.745558\pi$
$379$	−27.1450	−1.39434	−0.697171	+	0.716904i	$0.745558\pi$
$380$	−3.15552	−0.161875
$381$	−4.87309	−0.249656
$382$	3.12124	0.159697
$383$	20.4650	1.04571	0.522857	−	0.852420i	$-0.324866\pi$
$383$	20.4650	1.04571	0.522857	+	0.852420i	$0.324866\pi$
$384$	1.16293	0.0593453
$385$	19.3745	0.987416
$386$	−7.66190	−0.389980
$387$	14.3851	0.731235
$388$	3.16886	0.160875
$389$	9.63349	0.488437	0.244219	−	0.969720i	$-0.421469\pi$
$389$	9.63349	0.488437	0.244219	+	0.969720i	$0.421469\pi$
$390$	−9.08949	−0.460264
$391$	−6.94187	−0.351065
$392$	2.43361	0.122916
$393$	−9.40052	−0.474193
$394$	20.3129	1.02335
$395$	−18.0047	−0.905913
$396$	−4.88520	−0.245490
$397$	−13.0461	−0.654767	−0.327384	−	0.944891i	$-0.606167\pi$
$397$	−13.0461	−0.654767	−0.327384	+	0.944891i	$0.606167\pi$
$398$	−1.25313	−0.0628137
$399$	2.56446	0.128383
$400$	4.35037	0.217518
$401$	−37.7923	−1.88726	−0.943628	−	0.331007i	$-0.892611\pi$
$401$	−37.7923	−1.88726	−0.943628	+	0.331007i	$0.892611\pi$
$402$	7.09938	0.354085
$403$	13.1211	0.653607
$404$	2.38671	0.118743
$405$	−4.10542	−0.204000
$406$	7.95708	0.394903
$407$	9.02615	0.447410
$408$	1.12425	0.0556589
$409$	−24.6539	−1.21906	−0.609528	−	0.792764i	$-0.708641\pi$
$409$	−24.6539	−1.21906	−0.609528	+	0.792764i	$0.708641\pi$
$410$	28.1630	1.39087
$411$	5.60275	0.276363
$412$	−17.4313	−0.858777
$413$	−22.3641	−1.10046
$414$	−11.8309	−0.581455
$415$	−26.2010	−1.28616
$416$	2.55607	0.125322
$417$	−16.3485	−0.800589
$418$	3.05975	0.149657
$419$	−6.81166	−0.332772	−0.166386	−	0.986061i	$-0.553210\pi$
$419$	−6.81166	−0.332772	−0.166386	+	0.986061i	$0.553210\pi$
$420$	−7.59894	−0.370790
$421$	−10.3996	−0.506848	−0.253424	−	0.967355i	$-0.581557\pi$
$421$	−10.3996	−0.506848	−0.253424	+	0.967355i	$0.581557\pi$
$422$	−25.8028	−1.25606
$423$	2.74951	0.133686
$424$	−1.21637	−0.0590723
$425$	4.20570	0.204007
$426$	−1.98517	−0.0961818
$427$	−9.27571	−0.448883
$428$	15.8714	0.767173
$429$	8.81362	0.425526
$430$	26.6977	1.28748
$431$	37.9459	1.82779	0.913896	−	0.405948i	$-0.133059\pi$
$431$	37.9459	1.82779	0.913896	+	0.405948i	$0.133059\pi$
$432$	5.40482	0.260039
$433$	−7.58288	−0.364410	−0.182205	−	0.983261i	$-0.558323\pi$
$433$	−7.58288	−0.364410	−0.182205	+	0.983261i	$0.558323\pi$
$434$	10.9694	0.526548
$435$	13.2414	0.634876
$436$	15.0679	0.721623
$437$	7.41004	0.354470
$438$	−18.1055	−0.865114
$439$	−10.5983	−0.505829	−0.252915	−	0.967489i	$-0.581389\pi$
$439$	−10.5983	−0.505829	−0.252915	+	0.967489i	$0.581389\pi$
$440$	−9.06658	−0.432232
$441$	4.00962	0.190934
$442$	2.47107	0.117537
$443$	28.7867	1.36770	0.683848	−	0.729624i	$-0.260305\pi$
$443$	28.7867	1.36770	0.683848	+	0.729624i	$0.260305\pi$
$444$	−3.54018	−0.168009
$445$	−2.12330	−0.100654
$446$	20.7896	0.984418
$447$	4.85247	0.229514
$448$	2.13691	0.100960
$449$	−26.5599	−1.25344	−0.626720	−	0.779244i	$-0.715603\pi$
$449$	−26.5599	−1.25344	−0.626720	+	0.779244i	$0.715603\pi$
$450$	7.16768	0.337888
$451$	−27.3083	−1.28590
$452$	7.14717	0.336175
$453$	−25.3480	−1.19095
$454$	24.7747	1.16273
$455$	−16.7022	−0.783012
$456$	−1.20008	−0.0561987
$457$	−23.1639	−1.08356	−0.541780	−	0.840520i	$-0.682249\pi$
$457$	−23.1639	−1.08356	−0.541780	+	0.840520i	$0.682249\pi$
$458$	−14.9930	−0.700579
$459$	5.22509	0.243886
$460$	−21.9573	−1.02376
$461$	−30.0670	−1.40036	−0.700180	−	0.713966i	$-0.746897\pi$
$461$	−30.0670	−1.40036	−0.700180	+	0.713966i	$0.746897\pi$
$462$	7.36831	0.342805
$463$	−22.1273	−1.02834	−0.514171	−	0.857688i	$-0.671900\pi$
$463$	−22.1273	−1.02834	−0.514171	+	0.857688i	$0.671900\pi$
$464$	−3.72364	−0.172865
$465$	18.2542	0.846518
$466$	13.2104	0.611961
$467$	16.6884	0.772249	0.386124	−	0.922447i	$-0.373814\pi$
$467$	16.6884	0.772249	0.386124	+	0.922447i	$0.373814\pi$
$468$	4.21140	0.194672
$469$	13.0453	0.602378
$470$	5.10290	0.235379
$471$	15.9483	0.734860
$472$	10.4656	0.481718
$473$	−25.8874	−1.19031
$474$	−6.84735	−0.314509
$475$	−4.48934	−0.205985
$476$	2.06585	0.0946882
$477$	−2.00410	−0.0917614
$478$	−11.5531	−0.528427
$479$	−3.82413	−0.174729	−0.0873645	−	0.996176i	$-0.527844\pi$
$479$	−3.82413	−0.174729	−0.0873645	+	0.996176i	$0.527844\pi$
$480$	3.55604	0.162310
$481$	−7.78119	−0.354792
$482$	−24.1694	−1.10088
$483$	17.8444	0.811949
$484$	−2.20859	−0.100390
$485$	9.68987	0.439994
$486$	14.6531	0.664679
$487$	15.8914	0.720108	0.360054	−	0.932931i	$-0.382758\pi$
$487$	15.8914	0.720108	0.360054	+	0.932931i	$0.382758\pi$
$488$	4.34071	0.196495
$489$	−29.3134	−1.32560
$490$	7.44157	0.336176
$491$	29.9313	1.35078	0.675390	−	0.737461i	$-0.263975\pi$
$491$	29.9313	1.35078	0.675390	+	0.737461i	$0.263975\pi$
$492$	10.7107	0.482874
$493$	−3.59981	−0.162127
$494$	−2.63773	−0.118677
$495$	−14.9381	−0.671419
$496$	−5.13329	−0.230492
$497$	−3.64781	−0.163627
$498$	−9.96449	−0.446519
$499$	19.9866	0.894724	0.447362	−	0.894353i	$-0.352364\pi$
$499$	19.9866	0.894724	0.447362	+	0.894353i	$0.352364\pi$
$500$	−1.98647	−0.0888375
$501$	−10.0258	−0.447920
$502$	−11.7945	−0.526413
$503$	26.0223	1.16028	0.580140	−	0.814517i	$-0.302998\pi$
$503$	26.0223	1.16028	0.580140	+	0.814517i	$0.302998\pi$
$504$	3.52079	0.156828
$505$	7.29816	0.324764
$506$	21.2909	0.946494
$507$	7.52005	0.333977
$508$	4.19037	0.185917
$509$	−3.67980	−0.163104	−0.0815522	−	0.996669i	$-0.525988\pi$
$509$	−3.67980	−0.163104	−0.0815522	+	0.996669i	$0.525988\pi$
$510$	3.43779	0.152228
$511$	−33.2694	−1.47175
$512$	−1.00000	−0.0441942
$513$	−5.57748	−0.246252
$514$	−25.9820	−1.14602
$515$	−53.3020	−2.34877
$516$	10.1534	0.446979
$517$	−4.94803	−0.217614
$518$	−6.50519	−0.285821
$519$	10.2839	0.451414
$520$	7.81605	0.342757
$521$	37.4566	1.64100	0.820502	−	0.571643i	$-0.193694\pi$
$521$	37.4566	1.64100	0.820502	+	0.571643i	$0.193694\pi$
$522$	−6.13508	−0.268525
$523$	18.0802	0.790593	0.395297	−	0.918554i	$-0.370642\pi$
$523$	18.0802	0.790593	0.395297	+	0.918554i	$0.370642\pi$
$524$	8.08351	0.353130
$525$	−10.8110	−0.471830
$526$	1.84525	0.0804568
$527$	−4.96259	−0.216174
$528$	−3.44811	−0.150060
$529$	28.5617	1.24181
$530$	−3.71947	−0.161563
$531$	17.2432	0.748290
$532$	−2.20518	−0.0956066
$533$	23.5417	1.01970
$534$	−0.807513	−0.0349445
$535$	48.5322	2.09823
$536$	−6.10476	−0.263686
$537$	−14.6386	−0.631701
$538$	−25.6329	−1.10511
$539$	−7.21572	−0.310803
$540$	16.5271	0.711211
$541$	−7.74381	−0.332932	−0.166466	−	0.986047i	$-0.553236\pi$
$541$	−7.74381	−0.332932	−0.166466	+	0.986047i	$0.553236\pi$
$542$	30.0547	1.29096
$543$	1.94712	0.0835589
$544$	−0.966746	−0.0414489
$545$	46.0753	1.97365
$546$	−6.35202	−0.271841
$547$	−28.7823	−1.23064	−0.615321	−	0.788276i	$-0.710974\pi$
$547$	−28.7823	−1.23064	−0.615321	+	0.788276i	$0.710974\pi$
$548$	−4.81781	−0.205807
$549$	7.15177	0.305230
$550$	−12.8990	−0.550014
$551$	3.84259	0.163700
$552$	−8.35056	−0.355423
$553$	−12.5822	−0.535050
$554$	19.0276	0.808405
$555$	−10.8253	−0.459508
$556$	14.0581	0.596195
$557$	2.56068	0.108499	0.0542496	−	0.998527i	$-0.482723\pi$
$557$	2.56068	0.108499	0.0542496	+	0.998527i	$0.482723\pi$
$558$	−8.45763	−0.358040
$559$	22.3169	0.943903
$560$	6.53433	0.276126
$561$	−3.33345	−0.140738
$562$	−7.15806	−0.301944
$563$	−23.9761	−1.01047	−0.505236	−	0.862981i	$-0.668595\pi$
$563$	−23.9761	−1.01047	−0.505236	+	0.862981i	$0.668595\pi$
$564$	1.94068	0.0817175
$565$	21.8549	0.919442
$566$	1.24476	0.0523212
$567$	−2.86900	−0.120487
$568$	1.70705	0.0716261
$569$	13.1721	0.552202	0.276101	−	0.961129i	$-0.410958\pi$
$569$	13.1721	0.552202	0.276101	+	0.961129i	$0.410958\pi$
$570$	−3.66964	−0.153704
$571$	31.0341	1.29874	0.649368	−	0.760474i	$-0.275034\pi$
$571$	31.0341	1.29874	0.649368	+	0.760474i	$0.275034\pi$
$572$	−7.57884	−0.316887
$573$	3.62977	0.151636
$574$	19.6812	0.821477
$575$	−31.2385	−1.30273
$576$	−1.64760	−0.0686502
$577$	−7.23730	−0.301293	−0.150646	−	0.988588i	$-0.548136\pi$
$577$	−7.23730	−0.301293	−0.150646	+	0.988588i	$0.548136\pi$
$578$	16.0654	0.668233
$579$	−8.91022	−0.370296
$580$	−11.3863	−0.472789
$581$	−18.3101	−0.759629
$582$	3.68515	0.152754
$583$	3.60658	0.149369
$584$	15.5689	0.644246
$585$	12.8778	0.532430
$586$	2.10734	0.0870535
$587$	43.8355	1.80928	0.904641	−	0.426174i	$-0.140139\pi$
$587$	43.8355	1.80928	0.904641	+	0.426174i	$0.140139\pi$
$588$	2.83010	0.116711
$589$	5.29728	0.218271
$590$	32.0021	1.31751
$591$	23.6224	0.971695
$592$	3.04420	0.125116
$593$	2.69031	0.110478	0.0552390	−	0.998473i	$-0.482408\pi$
$593$	2.69031	0.110478	0.0552390	+	0.998473i	$0.482408\pi$
$594$	−16.0255	−0.657532
$595$	6.31704	0.258973
$596$	−4.17264	−0.170918
$597$	−1.45730	−0.0596432
$598$	−18.3543	−0.750561
$599$	−9.25838	−0.378287	−0.189144	−	0.981949i	$-0.560571\pi$
$599$	−9.25838	−0.378287	−0.189144	+	0.981949i	$0.560571\pi$
$600$	5.05916	0.206539
$601$	−20.4789	−0.835352	−0.417676	−	0.908596i	$-0.637155\pi$
$601$	−20.4789	−0.835352	−0.417676	+	0.908596i	$0.637155\pi$
$602$	18.6572	0.760411
$603$	−10.0582	−0.409603
$604$	21.7967	0.886896
$605$	−6.75350	−0.274569
$606$	2.77556	0.112749
$607$	4.34394	0.176315	0.0881576	−	0.996107i	$-0.471902\pi$
$607$	4.34394	0.176315	0.0881576	+	0.996107i	$0.471902\pi$
$608$	1.03195	0.0418509
$609$	9.25350	0.374971
$610$	13.2732	0.537415
$611$	4.26556	0.172566
$612$	−1.59282	−0.0643857
$613$	−30.2346	−1.22116	−0.610582	−	0.791953i	$-0.709064\pi$
$613$	−30.2346	−1.22116	−0.610582	+	0.791953i	$0.709064\pi$
$614$	6.02806	0.243273
$615$	32.7515	1.32067
$616$	−6.33601	−0.255285
$617$	−25.2324	−1.01582	−0.507910	−	0.861410i	$-0.669582\pi$
$617$	−25.2324	−1.01582	−0.507910	+	0.861410i	$0.669582\pi$
$618$	−20.2713	−0.815430
$619$	27.4419	1.10298	0.551491	−	0.834181i	$-0.314059\pi$
$619$	27.4419	1.10298	0.551491	+	0.834181i	$0.314059\pi$
$620$	−15.6968	−0.630398
$621$	−38.8101	−1.55740
$622$	−8.47349	−0.339756
$623$	−1.48383	−0.0594484
$624$	2.97252	0.118996
$625$	−27.8261	−1.11305
$626$	13.3825	0.534872
$627$	3.55826	0.142103
$628$	−13.7140	−0.547247
$629$	2.94297	0.117344
$630$	10.7660	0.428927
$631$	38.8244	1.54558	0.772788	−	0.634665i	$-0.218862\pi$
$631$	38.8244	1.54558	0.772788	+	0.634665i	$0.218862\pi$
$632$	5.88804	0.234213
$633$	−30.0067	−1.19266
$634$	26.4469	1.05034
$635$	12.8135	0.508487
$636$	−1.41455	−0.0560906
$637$	6.22047	0.246464
$638$	11.0407	0.437105
$639$	2.81254	0.111262
$640$	−3.05784	−0.120872
$641$	−11.0983	−0.438358	−0.219179	−	0.975685i	$-0.570338\pi$
$641$	−11.0983	−0.438358	−0.219179	+	0.975685i	$0.570338\pi$
$642$	18.4573	0.728450
$643$	−24.2937	−0.958048	−0.479024	−	0.877802i	$-0.659009\pi$
$643$	−24.2937	−0.958048	−0.479024	+	0.877802i	$0.659009\pi$
$644$	−15.3444	−0.604655
$645$	31.0475	1.22249
$646$	0.997630	0.0392512
$647$	−25.4103	−0.998981	−0.499491	−	0.866319i	$-0.666479\pi$
$647$	−25.4103	−0.998981	−0.499491	+	0.866319i	$0.666479\pi$
$648$	1.34259	0.0527419
$649$	−31.0308	−1.21807
$650$	11.1199	0.436157
$651$	12.7566	0.499970
$652$	25.2066	0.987166
$653$	15.3219	0.599593	0.299796	−	0.954003i	$-0.403081\pi$
$653$	15.3219	0.599593	0.299796	+	0.954003i	$0.403081\pi$
$654$	17.5229	0.685199
$655$	24.7180	0.965814
$656$	−9.21011	−0.359594
$657$	25.6514	1.00076
$658$	3.56607	0.139020
$659$	−1.30908	−0.0509946	−0.0254973	−	0.999675i	$-0.508117\pi$
$659$	−1.30908	−0.0509946	−0.0254973	+	0.999675i	$0.508117\pi$
$660$	−10.5438	−0.410415
$661$	33.4020	1.29919	0.649594	−	0.760282i	$-0.274939\pi$
$661$	33.4020	1.29919	0.649594	+	0.760282i	$0.274939\pi$
$662$	26.5639	1.03244
$663$	2.87368	0.111604
$664$	8.56847	0.332521
$665$	−6.74307	−0.261485
$666$	5.01563	0.194352
$667$	26.7381	1.03530
$668$	8.62120	0.333564
$669$	24.1768	0.934729
$670$	−18.6674	−0.721183
$671$	−12.8703	−0.496854
$672$	2.48507	0.0958637
$673$	−34.9715	−1.34805	−0.674026	−	0.738708i	$-0.735436\pi$
$673$	−34.9715	−1.34805	−0.674026	+	0.738708i	$0.735436\pi$
$674$	−15.3211	−0.590148
$675$	23.5130	0.905014
$676$	−6.46649	−0.248711
$677$	9.03644	0.347299	0.173649	−	0.984808i	$-0.444444\pi$
$677$	9.03644	0.347299	0.173649	+	0.984808i	$0.444444\pi$
$678$	8.31163	0.319206
$679$	6.77158	0.259869
$680$	−2.95615	−0.113363
$681$	28.8111	1.10404
$682$	15.2204	0.582818
$683$	−38.0711	−1.45675	−0.728376	−	0.685178i	$-0.759724\pi$
$683$	−38.0711	−1.45675	−0.728376	+	0.685178i	$0.759724\pi$
$684$	1.70024	0.0650102
$685$	−14.7321	−0.562884
$686$	20.1588	0.769666
$687$	−17.4358	−0.665217
$688$	−8.73092	−0.332863
$689$	−3.10914	−0.118449
$690$	−25.5347	−0.972088
$691$	−12.6697	−0.481980	−0.240990	−	0.970528i	$-0.577472\pi$
$691$	−12.6697	−0.481980	−0.240990	+	0.970528i	$0.577472\pi$
$692$	−8.84315	−0.336166
$693$	−10.4392	−0.396554
$694$	−11.2901	−0.428568
$695$	42.9873	1.63060
$696$	−4.33031	−0.164140
$697$	−8.90384	−0.337257
$698$	−10.3245	−0.390787
$699$	15.3628	0.581073
$700$	9.29636	0.351369
$701$	29.5053	1.11440	0.557199	−	0.830379i	$-0.311876\pi$
$701$	29.5053	1.11440	0.557199	+	0.830379i	$0.311876\pi$
$702$	13.8151	0.521418
$703$	−3.14145	−0.118482
$704$	2.96503	0.111749
$705$	5.93429	0.223498
$706$	−7.39201	−0.278202
$707$	5.10018	0.191812
$708$	12.1707	0.457404
$709$	−9.93366	−0.373067	−0.186533	−	0.982449i	$-0.559725\pi$
$709$	−9.93366	−0.373067	−0.186533	+	0.982449i	$0.559725\pi$
$710$	5.21987	0.195898
$711$	9.70115	0.363822
$712$	0.694381	0.0260230
$713$	36.8604	1.38043
$714$	2.40243	0.0899088
$715$	−23.1748	−0.866690
$716$	12.5877	0.470425
$717$	−13.4354	−0.501754
$718$	5.47904	0.204476
$719$	−38.2083	−1.42493	−0.712465	−	0.701708i	$-0.752421\pi$
$719$	−38.2083	−1.42493	−0.712465	+	0.701708i	$0.752421\pi$
$720$	−5.03810	−0.187759
$721$	−37.2491	−1.38723
$722$	17.9351	0.667475
$723$	−28.1072	−1.04532
$724$	−1.67433	−0.0622260
$725$	−16.1992	−0.601623
$726$	−2.56842	−0.0953231
$727$	−20.5084	−0.760615	−0.380307	−	0.924860i	$-0.624182\pi$
$727$	−20.5084	−0.760615	−0.380307	+	0.924860i	$0.624182\pi$
$728$	5.46210	0.202439
$729$	21.0683	0.780306
$730$	47.6072	1.76202
$731$	−8.44058	−0.312186
$732$	5.04792	0.186576
$733$	26.7743	0.988931	0.494465	−	0.869197i	$-0.335364\pi$
$733$	26.7743	0.988931	0.494465	+	0.869197i	$0.335364\pi$
$734$	0.707477	0.0261134
$735$	8.65399	0.319207
$736$	7.18065	0.264682
$737$	18.1008	0.666752
$738$	−15.1746	−0.558585
$739$	33.4202	1.22938	0.614691	−	0.788768i	$-0.289281\pi$
$739$	33.4202	1.22938	0.614691	+	0.788768i	$0.289281\pi$
$740$	9.30867	0.342193
$741$	−3.06748	−0.112687
$742$	−2.59928	−0.0954226
$743$	52.8798	1.93997	0.969986	−	0.243162i	$-0.0781845\pi$
$743$	52.8798	1.93997	0.969986	+	0.243162i	$0.0781845\pi$
$744$	−5.96964	−0.218858
$745$	−12.7592	−0.467463
$746$	4.98270	0.182430
$747$	14.1174	0.516530
$748$	2.86643	0.104807
$749$	33.9158	1.23926
$750$	−2.31011	−0.0843534
$751$	−5.89876	−0.215249	−0.107624	−	0.994192i	$-0.534324\pi$
$751$	−5.89876	−0.215249	−0.107624	+	0.994192i	$0.534324\pi$
$752$	−1.66879	−0.0608546
$753$	−13.7161	−0.499842
$754$	−9.51788	−0.346621
$755$	66.6508	2.42567
$756$	11.5496	0.420056
$757$	−13.1314	−0.477268	−0.238634	−	0.971110i	$-0.576700\pi$
$757$	−13.1314	−0.477268	−0.238634	+	0.971110i	$0.576700\pi$
$758$	27.1450	0.985949
$759$	24.7597	0.898719
$760$	3.15552	0.114463
$761$	−16.8014	−0.609050	−0.304525	−	0.952504i	$-0.598498\pi$
$761$	−16.8014	−0.609050	−0.304525	+	0.952504i	$0.598498\pi$
$762$	4.87309	0.176533
$763$	32.1989	1.16568
$764$	−3.12124	−0.112923
$765$	−4.87057	−0.176096
$766$	−20.4650	−0.739432
$767$	26.7508	0.965917
$768$	−1.16293	−0.0419635
$769$	−15.0235	−0.541760	−0.270880	−	0.962613i	$-0.587315\pi$
$769$	−15.0235	−0.541760	−0.270880	+	0.962613i	$0.587315\pi$
$770$	−19.3745	−0.698208
$771$	−30.2152	−1.08817
$772$	7.66190	0.275758
$773$	−8.54464	−0.307329	−0.153665	−	0.988123i	$-0.549108\pi$
$773$	−8.54464	−0.307329	−0.153665	+	0.988123i	$0.549108\pi$
$774$	−14.3851	−0.517062
$775$	−22.3317	−0.802179
$776$	−3.16886	−0.113756
$777$	−7.56505	−0.271395
$778$	−9.63349	−0.345377
$779$	9.50433	0.340528
$780$	9.08949	0.325456
$781$	−5.06145	−0.181113
$782$	6.94187	0.248241
$783$	−20.1256	−0.719229
$784$	−2.43361	−0.0869145
$785$	−41.9351	−1.49673
$786$	9.40052	0.335305
$787$	−7.02819	−0.250528	−0.125264	−	0.992123i	$-0.539978\pi$
$787$	−7.02819	−0.250528	−0.125264	+	0.992123i	$0.539978\pi$
$788$	−20.3129	−0.723616
$789$	2.14589	0.0763957
$790$	18.0047	0.640577
$791$	15.2729	0.543041
$792$	4.88520	0.173588
$793$	11.0952	0.394001
$794$	13.0461	0.462990
$795$	−4.32547	−0.153408
$796$	1.25313	0.0444160
$797$	26.8855	0.952333	0.476166	−	0.879355i	$-0.342026\pi$
$797$	26.8855	0.952333	0.476166	+	0.879355i	$0.342026\pi$
$798$	−2.56446	−0.0907808
$799$	−1.61330	−0.0570745
$800$	−4.35037	−0.153809
$801$	1.14406	0.0404235
$802$	37.7923	1.33449
$803$	−46.1623	−1.62903
$804$	−7.09938	−0.250376
$805$	−46.9207	−1.65374
$806$	−13.1211	−0.462170
$807$	−29.8092	−1.04933
$808$	−2.38671	−0.0839640
$809$	26.9905	0.948937	0.474468	−	0.880273i	$-0.342640\pi$
$809$	26.9905	0.948937	0.474468	+	0.880273i	$0.342640\pi$
$810$	4.10542	0.144250
$811$	5.46033	0.191738	0.0958690	−	0.995394i	$-0.469437\pi$
$811$	5.46033	0.191738	0.0958690	+	0.995394i	$0.469437\pi$
$812$	−7.95708	−0.279239
$813$	34.9513	1.22580
$814$	−9.02615	−0.316366
$815$	77.0777	2.69991
$816$	−1.12425	−0.0393568
$817$	9.00983	0.315214
$818$	24.6539	0.862003
$819$	8.99938	0.314464
$820$	−28.1630	−0.983495
$821$	24.1731	0.843647	0.421823	−	0.906678i	$-0.361390\pi$
$821$	24.1731	0.843647	0.421823	+	0.906678i	$0.361390\pi$
$822$	−5.60275	−0.195418
$823$	16.9114	0.589495	0.294748	−	0.955575i	$-0.404764\pi$
$823$	16.9114	0.589495	0.294748	+	0.955575i	$0.404764\pi$
$824$	17.4313	0.607247
$825$	−15.0006	−0.522252
$826$	22.3641	0.778146
$827$	47.4277	1.64922	0.824611	−	0.565700i	$-0.191394\pi$
$827$	47.4277	1.64922	0.824611	+	0.565700i	$0.191394\pi$
$828$	11.8309	0.411151
$829$	5.77234	0.200482	0.100241	−	0.994963i	$-0.468039\pi$
$829$	5.77234	0.200482	0.100241	+	0.994963i	$0.468039\pi$
$830$	26.2010	0.909449
$831$	22.1277	0.767601
$832$	−2.55607	−0.0886159
$833$	−2.35268	−0.0815155
$834$	16.3485	0.566102
$835$	26.3622	0.912302
$836$	−3.05975	−0.105824
$837$	−27.7445	−0.958991
$838$	6.81166	0.235305
$839$	−14.2497	−0.491956	−0.245978	−	0.969275i	$-0.579109\pi$
$839$	−14.2497	−0.491956	−0.245978	+	0.969275i	$0.579109\pi$
$840$	7.59894	0.262188
$841$	−15.1345	−0.521881
$842$	10.3996	0.358395
$843$	−8.32429	−0.286704
$844$	25.8028	0.888168
$845$	−19.7735	−0.680228
$846$	−2.74951	−0.0945301
$847$	−4.71955	−0.162166
$848$	1.21637	0.0417704
$849$	1.44756	0.0496802
$850$	−4.20570	−0.144254
$851$	−21.8593	−0.749328
$852$	1.98517	0.0680108
$853$	−22.5597	−0.772430	−0.386215	−	0.922409i	$-0.626218\pi$
$853$	−22.5597	−0.772430	−0.386215	+	0.922409i	$0.626218\pi$
$854$	9.27571	0.317408
$855$	5.19905	0.177804
$856$	−15.8714	−0.542473
$857$	−4.55168	−0.155483	−0.0777413	−	0.996974i	$-0.524771\pi$
$857$	−4.55168	−0.155483	−0.0777413	+	0.996974i	$0.524771\pi$
$858$	−8.81362	−0.300892
$859$	21.6324	0.738089	0.369045	−	0.929412i	$-0.379685\pi$
$859$	21.6324	0.738089	0.369045	+	0.929412i	$0.379685\pi$
$860$	−26.6977	−0.910385
$861$	22.8878	0.780013
$862$	−37.9459	−1.29244
$863$	11.0714	0.376874	0.188437	−	0.982085i	$-0.439658\pi$
$863$	11.0714	0.376874	0.188437	+	0.982085i	$0.439658\pi$
$864$	−5.40482	−0.183876
$865$	−27.0409	−0.919418
$866$	7.58288	0.257677
$867$	18.6829	0.634503
$868$	−10.9694	−0.372326
$869$	−17.4582	−0.592229
$870$	−13.2414	−0.448925
$871$	−15.6042	−0.528729
$872$	−15.0679	−0.510265
$873$	−5.22103	−0.176705
$874$	−7.41004	−0.250648
$875$	−4.24490	−0.143504
$876$	18.1055	0.611728
$877$	−2.58246	−0.0872033	−0.0436017	−	0.999049i	$-0.513883\pi$
$877$	−2.58246	−0.0872033	−0.0436017	+	0.999049i	$0.513883\pi$
$878$	10.5983	0.357675
$879$	2.45068	0.0826595
$880$	9.06658	0.305635
$881$	18.5582	0.625243	0.312621	−	0.949878i	$-0.398793\pi$
$881$	18.5582	0.625243	0.312621	+	0.949878i	$0.398793\pi$
$882$	−4.00962	−0.135011
$883$	8.84074	0.297515	0.148757	−	0.988874i	$-0.452473\pi$
$883$	8.84074	0.297515	0.148757	+	0.988874i	$0.452473\pi$
$884$	−2.47107	−0.0831112
$885$	37.2161	1.25100
$886$	−28.7867	−0.967108
$887$	23.2145	0.779468	0.389734	−	0.920928i	$-0.372567\pi$
$887$	23.2145	0.779468	0.389734	+	0.920928i	$0.372567\pi$
$888$	3.54018	0.118801
$889$	8.95445	0.300322
$890$	2.12330	0.0711733
$891$	−3.98082	−0.133363
$892$	−20.7896	−0.696089
$893$	1.72210	0.0576280
$894$	−4.85247	−0.162291
$895$	38.4911	1.28662
$896$	−2.13691	−0.0713892
$897$	−21.3446	−0.712677
$898$	26.5599	0.886316
$899$	19.1145	0.637505
$900$	−7.16768	−0.238923
$901$	1.17592	0.0391757
$902$	27.3083	0.909266
$903$	21.6969	0.722029
$904$	−7.14717	−0.237711
$905$	−5.11983	−0.170189
$906$	25.3480	0.842130
$907$	16.8608	0.559852	0.279926	−	0.960022i	$-0.409690\pi$
$907$	16.8608	0.559852	0.279926	+	0.960022i	$0.409690\pi$
$908$	−24.7747	−0.822176
$909$	−3.93235	−0.130428
$910$	16.7022	0.553673
$911$	28.4955	0.944099	0.472050	−	0.881572i	$-0.343514\pi$
$911$	28.4955	0.944099	0.472050	+	0.881572i	$0.343514\pi$
$912$	1.20008	0.0397385
$913$	−25.4058	−0.840808
$914$	23.1639	0.766193
$915$	15.4357	0.510289
$916$	14.9930	0.495384
$917$	17.2737	0.570429
$918$	−5.22509	−0.172454
$919$	32.1253	1.05972	0.529858	−	0.848087i	$-0.322245\pi$
$919$	32.1253	1.05972	0.529858	+	0.848087i	$0.322245\pi$
$920$	21.9573	0.723909
$921$	7.01019	0.230993
$922$	30.0670	0.990205
$923$	4.36334	0.143621
$924$	−7.36831	−0.242400
$925$	13.2434	0.435440
$926$	22.1273	0.727147
$927$	28.7198	0.943283
$928$	3.72364	0.122234
$929$	−8.07648	−0.264981	−0.132490	−	0.991184i	$-0.542297\pi$
$929$	−8.07648	−0.264981	−0.132490	+	0.991184i	$0.542297\pi$
$930$	−18.2542	−0.598578
$931$	2.51135	0.0823061
$932$	−13.2104	−0.432722
$933$	−9.85404	−0.322607
$934$	−16.6884	−0.546062
$935$	8.76509	0.286649
$936$	−4.21140	−0.137654
$937$	−0.759730	−0.0248193	−0.0124097	−	0.999923i	$-0.503950\pi$
$937$	−0.759730	−0.0248193	−0.0124097	+	0.999923i	$0.503950\pi$
$938$	−13.0453	−0.425945
$939$	15.5629	0.507875
$940$	−5.10290	−0.166438
$941$	13.2384	0.431561	0.215780	−	0.976442i	$-0.430770\pi$
$941$	13.2384	0.431561	0.215780	+	0.976442i	$0.430770\pi$
$942$	−15.9483	−0.519625
$943$	66.1346	2.15364
$944$	−10.4656	−0.340626
$945$	35.3169	1.14886
$946$	25.8874	0.841674
$947$	−26.1776	−0.850659	−0.425329	−	0.905039i	$-0.639842\pi$
$947$	−26.1776	−0.850659	−0.425329	+	0.905039i	$0.639842\pi$
$948$	6.84735	0.222392
$949$	39.7953	1.29181
$950$	4.48934	0.145654
$951$	30.7558	0.997326
$952$	−2.06585	−0.0669547
$953$	43.0860	1.39569	0.697846	−	0.716247i	$-0.254142\pi$
$953$	43.0860	1.39569	0.697846	+	0.716247i	$0.254142\pi$
$954$	2.00410	0.0648851
$955$	−9.54425	−0.308845
$956$	11.5531	0.373654
$957$	12.8395	0.415042
$958$	3.82413	0.123552
$959$	−10.2952	−0.332450
$960$	−3.55604	−0.114771
$961$	−4.64929	−0.149977
$962$	7.78119	0.250876
$963$	−26.1498	−0.842665
$964$	24.1694	0.778443
$965$	23.4288	0.754201
$966$	−17.8444	−0.574135
$967$	−13.2214	−0.425172	−0.212586	−	0.977142i	$-0.568189\pi$
$967$	−13.2214	−0.425172	−0.212586	+	0.977142i	$0.568189\pi$
$968$	2.20859	0.0709866
$969$	1.16017	0.0372700
$970$	−9.68987	−0.311123
$971$	−39.0441	−1.25298	−0.626492	−	0.779427i	$-0.715510\pi$
$971$	−39.0441	−1.25298	−0.626492	+	0.779427i	$0.715510\pi$
$972$	−14.6531	−0.469999
$973$	30.0409	0.963066
$974$	−15.8914	−0.509194
$975$	12.9316	0.414142
$976$	−4.34071	−0.138943
$977$	−42.2650	−1.35218	−0.676088	−	0.736820i	$-0.736326\pi$
$977$	−42.2650	−1.35218	−0.676088	+	0.736820i	$0.736326\pi$
$978$	29.3134	0.937339
$979$	−2.05886	−0.0658015
$980$	−7.44157	−0.237712
$981$	−24.8260	−0.792633
$982$	−29.9313	−0.955145
$983$	14.6723	0.467974	0.233987	−	0.972240i	$-0.424823\pi$
$983$	14.6723	0.467974	0.233987	+	0.972240i	$0.424823\pi$
$984$	−10.7107	−0.341444
$985$	−62.1135	−1.97910
$986$	3.59981	0.114641
$987$	4.14707	0.132003
$988$	2.63773	0.0839173
$989$	62.6936	1.99354
$990$	14.9381	0.474765
$991$	−21.8084	−0.692766	−0.346383	−	0.938093i	$-0.612590\pi$
$991$	−21.8084	−0.692766	−0.346383	+	0.938093i	$0.612590\pi$
$992$	5.13329	0.162982
$993$	30.8919	0.980324
$994$	3.64781	0.115702
$995$	3.83187	0.121478
$996$	9.96449	0.315737
$997$	−38.7176	−1.22620	−0.613100	−	0.790005i	$-0.710078\pi$
$997$	−38.7176	−1.22620	−0.613100	+	0.790005i	$0.710078\pi$
$998$	−19.9866	−0.632665
$999$	16.4533	0.520561

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.d.1.15	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.d.1.15	✓	37	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.16293 q^{3} +1.00000 q^{4} +3.05784 q^{5} +1.16293 q^{6} +2.13691 q^{7} -1.00000 q^{8} -1.64760 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.16293 q^{3} +1.00000 q^{4} +3.05784 q^{5} +1.16293 q^{6} +2.13691 q^{7} -1.00000 q^{8} -1.64760 q^{9} -3.05784 q^{10} +2.96503 q^{11} -1.16293 q^{12} -2.55607 q^{13} -2.13691 q^{14} -3.55604 q^{15} +1.00000 q^{16} +0.966746 q^{17} +1.64760 q^{18} -1.03195 q^{19} +3.05784 q^{20} -2.48507 q^{21} -2.96503 q^{22} -7.18065 q^{23} +1.16293 q^{24} +4.35037 q^{25} +2.55607 q^{26} +5.40482 q^{27} +2.13691 q^{28} -3.72364 q^{29} +3.55604 q^{30} -5.13329 q^{31} -1.00000 q^{32} -3.44811 q^{33} -0.966746 q^{34} +6.53433 q^{35} -1.64760 q^{36} +3.04420 q^{37} +1.03195 q^{38} +2.97252 q^{39} -3.05784 q^{40} -9.21011 q^{41} +2.48507 q^{42} -8.73092 q^{43} +2.96503 q^{44} -5.03810 q^{45} +7.18065 q^{46} -1.66879 q^{47} -1.16293 q^{48} -2.43361 q^{49} -4.35037 q^{50} -1.12425 q^{51} -2.55607 q^{52} +1.21637 q^{53} -5.40482 q^{54} +9.06658 q^{55} -2.13691 q^{56} +1.20008 q^{57} +3.72364 q^{58} -10.4656 q^{59} -3.55604 q^{60} -4.34071 q^{61} +5.13329 q^{62} -3.52079 q^{63} +1.00000 q^{64} -7.81605 q^{65} +3.44811 q^{66} +6.10476 q^{67} +0.966746 q^{68} +8.35056 q^{69} -6.53433 q^{70} -1.70705 q^{71} +1.64760 q^{72} -15.5689 q^{73} -3.04420 q^{74} -5.05916 q^{75} -1.03195 q^{76} +6.33601 q^{77} -2.97252 q^{78} -5.88804 q^{79} +3.05784 q^{80} -1.34259 q^{81} +9.21011 q^{82} -8.56847 q^{83} -2.48507 q^{84} +2.95615 q^{85} +8.73092 q^{86} +4.33031 q^{87} -2.96503 q^{88} -0.694381 q^{89} +5.03810 q^{90} -5.46210 q^{91} -7.18065 q^{92} +5.96964 q^{93} +1.66879 q^{94} -3.15552 q^{95} +1.16293 q^{96} +3.16886 q^{97} +2.43361 q^{98} -4.88520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9}+O(q^{10}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9} + 13 q^{10} + 12 q^{11} - 5 q^{12} - 36 q^{13} + 22 q^{14} - 7 q^{15} + 37 q^{16} + 4 q^{17} - 32 q^{18} - 10 q^{19} - 13 q^{20} - 11 q^{21} - 12 q^{22} - q^{23} + 5 q^{24} + 8 q^{25} + 36 q^{26} - 20 q^{27} - 22 q^{28} - 6 q^{29} + 7 q^{30} - 23 q^{31} - 37 q^{32} - 27 q^{33} - 4 q^{34} + 24 q^{35} + 32 q^{36} - 46 q^{37} + 10 q^{38} + 5 q^{39} + 13 q^{40} + 11 q^{41} + 11 q^{42} - 22 q^{43} + 12 q^{44} - 57 q^{45} + q^{46} - 18 q^{47} - 5 q^{48} - q^{49} - 8 q^{50} + 18 q^{51} - 36 q^{52} - 25 q^{53} + 20 q^{54} - 25 q^{55} + 22 q^{56} - 25 q^{57} + 6 q^{58} + 24 q^{59} - 7 q^{60} - 40 q^{61} + 23 q^{62} - 38 q^{63} + 37 q^{64} + 14 q^{65} + 27 q^{66} - 49 q^{67} + 4 q^{68} - 19 q^{69} - 24 q^{70} + 27 q^{71} - 32 q^{72} - 87 q^{73} + 46 q^{74} - 5 q^{75} - 10 q^{76} - 50 q^{77} - 5 q^{78} + 11 q^{79} - 13 q^{80} + 5 q^{81} - 11 q^{82} - 9 q^{83} - 11 q^{84} - 68 q^{85} + 22 q^{86} - 12 q^{87} - 12 q^{88} - 3 q^{89} + 57 q^{90} - 19 q^{91} - q^{92} - 69 q^{93} + 18 q^{94} + 27 q^{95} + 5 q^{96} - 98 q^{97} + q^{98} + 33 q^{99}+O(q^{100}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9 + 13 * q^10 + 12 * q^11 - 5 * q^12 - 36 * q^13 + 22 * q^14 - 7 * q^15 + 37 * q^16 + 4 * q^17 - 32 * q^18 - 10 * q^19 - 13 * q^20 - 11 * q^21 - 12 * q^22 - q^23 + 5 * q^24 + 8 * q^25 + 36 * q^26 - 20 * q^27 - 22 * q^28 - 6 * q^29 + 7 * q^30 - 23 * q^31 - 37 * q^32 - 27 * q^33 - 4 * q^34 + 24 * q^35 + 32 * q^36 - 46 * q^37 + 10 * q^38 + 5 * q^39 + 13 * q^40 + 11 * q^41 + 11 * q^42 - 22 * q^43 + 12 * q^44 - 57 * q^45 + q^46 - 18 * q^47 - 5 * q^48 - q^49 - 8 * q^50 + 18 * q^51 - 36 * q^52 - 25 * q^53 + 20 * q^54 - 25 * q^55 + 22 * q^56 - 25 * q^57 + 6 * q^58 + 24 * q^59 - 7 * q^60 - 40 * q^61 + 23 * q^62 - 38 * q^63 + 37 * q^64 + 14 * q^65 + 27 * q^66 - 49 * q^67 + 4 * q^68 - 19 * q^69 - 24 * q^70 + 27 * q^71 - 32 * q^72 - 87 * q^73 + 46 * q^74 - 5 * q^75 - 10 * q^76 - 50 * q^77 - 5 * q^78 + 11 * q^79 - 13 * q^80 + 5 * q^81 - 11 * q^82 - 9 * q^83 - 11 * q^84 - 68 * q^85 + 22 * q^86 - 12 * q^87 - 12 * q^88 - 3 * q^89 + 57 * q^90 - 19 * q^91 - q^92 - 69 * q^93 + 18 * q^94 + 27 * q^95 + 5 * q^96 - 98 * q^97 + q^98 + 33 * q^99

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