LMFDB - Embedded newform 4002.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.23252.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2 $ x^4 - x^3 - 6*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.565882$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.96842 q^{5} -1.00000 q^{6} +2.40254 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.96842 q^{5} -1.00000 q^{6} +2.40254 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.96842 q^{10} -2.43412 q^{11} +1.00000 q^{12} +2.67978 q^{13} -2.40254 q^{14} -3.96842 q^{15} +1.00000 q^{16} -2.40254 q^{17} -1.00000 q^{18} -0.957013 q^{19} -3.96842 q^{20} +2.40254 q^{21} +2.43412 q^{22} +1.00000 q^{23} -1.00000 q^{24} +10.7484 q^{25} -2.67978 q^{26} +1.00000 q^{27} +2.40254 q^{28} -1.00000 q^{29} +3.96842 q^{30} -7.81154 q^{31} -1.00000 q^{32} -2.43412 q^{33} +2.40254 q^{34} -9.53431 q^{35} +1.00000 q^{36} +11.1890 q^{37} +0.957013 q^{38} +2.67978 q^{39} +3.96842 q^{40} +4.38883 q^{41} -2.40254 q^{42} -7.33214 q^{43} -2.43412 q^{44} -3.96842 q^{45} -1.00000 q^{46} -2.08878 q^{47} +1.00000 q^{48} -1.22779 q^{49} -10.7484 q^{50} -2.40254 q^{51} +2.67978 q^{52} +11.1574 q^{53} -1.00000 q^{54} +9.65961 q^{55} -2.40254 q^{56} -0.957013 q^{57} +1.00000 q^{58} -5.54801 q^{59} -3.96842 q^{60} -3.56588 q^{61} +7.81154 q^{62} +2.40254 q^{63} +1.00000 q^{64} -10.6345 q^{65} +2.43412 q^{66} +13.5284 q^{67} -2.40254 q^{68} +1.00000 q^{69} +9.53431 q^{70} -14.3257 q^{71} -1.00000 q^{72} -15.4712 q^{73} -11.1890 q^{74} +10.7484 q^{75} -0.957013 q^{76} -5.84807 q^{77} -2.67978 q^{78} -10.6661 q^{79} -3.96842 q^{80} +1.00000 q^{81} -4.38883 q^{82} -6.81800 q^{83} +2.40254 q^{84} +9.53431 q^{85} +7.33214 q^{86} -1.00000 q^{87} +2.43412 q^{88} +10.5169 q^{89} +3.96842 q^{90} +6.43828 q^{91} +1.00000 q^{92} -7.81154 q^{93} +2.08878 q^{94} +3.79783 q^{95} -1.00000 q^{96} +14.5169 q^{97} +1.22779 q^{98} -2.43412 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 3 q^{10} - 11 q^{11} + 4 q^{12} - q^{13} + 2 q^{14} - 3 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 3 q^{20} - 2 q^{21} + 11 q^{22} + 4 q^{23} - 4 q^{24} + 3 q^{25} + q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} + 3 q^{30} - 17 q^{31} - 4 q^{32} - 11 q^{33} - 2 q^{34} - 24 q^{35} + 4 q^{36} + 15 q^{37} - 8 q^{38} - q^{39} + 3 q^{40} + q^{41} + 2 q^{42} + 4 q^{43} - 11 q^{44} - 3 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} + 16 q^{49} - 3 q^{50} + 2 q^{51} - q^{52} + 2 q^{53} - 4 q^{54} + 13 q^{55} + 2 q^{56} + 8 q^{57} + 4 q^{58} - 13 q^{59} - 3 q^{60} - 13 q^{61} + 17 q^{62} - 2 q^{63} + 4 q^{64} - 13 q^{65} + 11 q^{66} - 13 q^{67} + 2 q^{68} + 4 q^{69} + 24 q^{70} - 15 q^{71} - 4 q^{72} - 22 q^{73} - 15 q^{74} + 3 q^{75} + 8 q^{76} - 12 q^{77} + q^{78} - 26 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 22 q^{83} - 2 q^{84} + 24 q^{85} - 4 q^{86} - 4 q^{87} + 11 q^{88} - 24 q^{89} + 3 q^{90} + 30 q^{91} + 4 q^{92} - 17 q^{93} - 6 q^{94} - 4 q^{95} - 4 q^{96} - 8 q^{97} - 16 q^{98} - 11 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 3 * q^10 - 11 * q^11 + 4 * q^12 - q^13 + 2 * q^14 - 3 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 8 * q^19 - 3 * q^20 - 2 * q^21 + 11 * q^22 + 4 * q^23 - 4 * q^24 + 3 * q^25 + q^26 + 4 * q^27 - 2 * q^28 - 4 * q^29 + 3 * q^30 - 17 * q^31 - 4 * q^32 - 11 * q^33 - 2 * q^34 - 24 * q^35 + 4 * q^36 + 15 * q^37 - 8 * q^38 - q^39 + 3 * q^40 + q^41 + 2 * q^42 + 4 * q^43 - 11 * q^44 - 3 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 + 16 * q^49 - 3 * q^50 + 2 * q^51 - q^52 + 2 * q^53 - 4 * q^54 + 13 * q^55 + 2 * q^56 + 8 * q^57 + 4 * q^58 - 13 * q^59 - 3 * q^60 - 13 * q^61 + 17 * q^62 - 2 * q^63 + 4 * q^64 - 13 * q^65 + 11 * q^66 - 13 * q^67 + 2 * q^68 + 4 * q^69 + 24 * q^70 - 15 * q^71 - 4 * q^72 - 22 * q^73 - 15 * q^74 + 3 * q^75 + 8 * q^76 - 12 * q^77 + q^78 - 26 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 22 * q^83 - 2 * q^84 + 24 * q^85 - 4 * q^86 - 4 * q^87 + 11 * q^88 - 24 * q^89 + 3 * q^90 + 30 * q^91 + 4 * q^92 - 17 * q^93 - 6 * q^94 - 4 * q^95 - 4 * q^96 - 8 * q^97 - 16 * q^98 - 11 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.96842	−1.77473	−0.887367	−	0.461065i	$-0.847468\pi$
$5$	−3.96842	−1.77473	−0.887367	+	0.461065i	$0.847468\pi$
$6$	−1.00000	−0.408248
$7$	2.40254	0.908076	0.454038	−	0.890982i	$-0.349983\pi$
$7$	2.40254	0.908076	0.454038	+	0.890982i	$0.349983\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.96842	1.25493
$11$	−2.43412	−0.733914	−0.366957	−	0.930238i	$-0.619600\pi$
$11$	−2.43412	−0.733914	−0.366957	+	0.930238i	$0.619600\pi$
$12$	1.00000	0.288675
$13$	2.67978	0.743237	0.371618	−	0.928386i	$-0.378803\pi$
$13$	2.67978	0.743237	0.371618	+	0.928386i	$0.378803\pi$
$14$	−2.40254	−0.642106
$15$	−3.96842	−1.02464
$16$	1.00000	0.250000
$17$	−2.40254	−0.582702	−0.291351	−	0.956616i	$-0.594105\pi$
$17$	−2.40254	−0.582702	−0.291351	+	0.956616i	$0.594105\pi$
$18$	−1.00000	−0.235702
$19$	−0.957013	−0.219554	−0.109777	−	0.993956i	$-0.535014\pi$
$19$	−0.957013	−0.219554	−0.109777	+	0.993956i	$0.535014\pi$
$20$	−3.96842	−0.887367
$21$	2.40254	0.524278
$22$	2.43412	0.518956
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	10.7484	2.14968
$26$	−2.67978	−0.525548
$27$	1.00000	0.192450
$28$	2.40254	0.454038
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.96842	0.724532
$31$	−7.81154	−1.40299	−0.701497	−	0.712672i	$-0.747485\pi$
$31$	−7.81154	−1.40299	−0.701497	+	0.712672i	$0.747485\pi$
$32$	−1.00000	−0.176777
$33$	−2.43412	−0.423726
$34$	2.40254	0.412033
$35$	−9.53431	−1.61159
$36$	1.00000	0.166667
$37$	11.1890	1.83945	0.919727	−	0.392558i	$-0.128410\pi$
$37$	11.1890	1.83945	0.919727	+	0.392558i	$0.128410\pi$
$38$	0.957013	0.155248
$39$	2.67978	0.429108
$40$	3.96842	0.627463
$41$	4.38883	0.685421	0.342710	−	0.939441i	$-0.388655\pi$
$41$	4.38883	0.685421	0.342710	+	0.939441i	$0.388655\pi$
$42$	−2.40254	−0.370720
$43$	−7.33214	−1.11814	−0.559070	−	0.829120i	$-0.688842\pi$
$43$	−7.33214	−1.11814	−0.559070	+	0.829120i	$0.688842\pi$
$44$	−2.43412	−0.366957
$45$	−3.96842	−0.591578
$46$	−1.00000	−0.147442
$47$	−2.08878	−0.304679	−0.152340	−	0.988328i	$-0.548681\pi$
$47$	−2.08878	−0.304679	−0.152340	+	0.988328i	$0.548681\pi$
$48$	1.00000	0.144338
$49$	−1.22779	−0.175399
$50$	−10.7484	−1.52005
$51$	−2.40254	−0.336423
$52$	2.67978	0.371618
$53$	11.1574	1.53259	0.766293	−	0.642492i	$-0.222099\pi$
$53$	11.1574	1.53259	0.766293	+	0.642492i	$0.222099\pi$
$54$	−1.00000	−0.136083
$55$	9.65961	1.30250
$56$	−2.40254	−0.321053
$57$	−0.957013	−0.126759
$58$	1.00000	0.131306
$59$	−5.54801	−0.722290	−0.361145	−	0.932510i	$-0.617614\pi$
$59$	−5.54801	−0.722290	−0.361145	+	0.932510i	$0.617614\pi$
$60$	−3.96842	−0.512321
$61$	−3.56588	−0.456564	−0.228282	−	0.973595i	$-0.573311\pi$
$61$	−3.56588	−0.456564	−0.228282	+	0.973595i	$0.573311\pi$
$62$	7.81154	0.992067
$63$	2.40254	0.302692
$64$	1.00000	0.125000
$65$	−10.6345	−1.31905
$66$	2.43412	0.299619
$67$	13.5284	1.65275	0.826376	−	0.563119i	$-0.190399\pi$
$67$	13.5284	1.65275	0.826376	+	0.563119i	$0.190399\pi$
$68$	−2.40254	−0.291351
$69$	1.00000	0.120386
$70$	9.53431	1.13957
$71$	−14.3257	−1.70015	−0.850073	−	0.526665i	$-0.823442\pi$
$71$	−14.3257	−1.70015	−0.850073	+	0.526665i	$0.823442\pi$
$72$	−1.00000	−0.117851
$73$	−15.4712	−1.81076	−0.905381	−	0.424600i	$-0.860415\pi$
$73$	−15.4712	−1.81076	−0.905381	+	0.424600i	$0.860415\pi$
$74$	−11.1890	−1.30069
$75$	10.7484	1.24112
$76$	−0.957013	−0.109777
$77$	−5.84807	−0.666450
$78$	−2.67978	−0.303425
$79$	−10.6661	−1.20003	−0.600013	−	0.799990i	$-0.704838\pi$
$79$	−10.6661	−1.20003	−0.600013	+	0.799990i	$0.704838\pi$
$80$	−3.96842	−0.443683
$81$	1.00000	0.111111
$82$	−4.38883	−0.484666
$83$	−6.81800	−0.748373	−0.374186	−	0.927354i	$-0.622078\pi$
$83$	−6.81800	−0.748373	−0.374186	+	0.927354i	$0.622078\pi$
$84$	2.40254	0.262139
$85$	9.53431	1.03414
$86$	7.33214	0.790645
$87$	−1.00000	−0.107211
$88$	2.43412	0.259478
$89$	10.5169	1.11479	0.557397	−	0.830246i	$-0.311800\pi$
$89$	10.5169	1.11479	0.557397	+	0.830246i	$0.311800\pi$
$90$	3.96842	0.418309
$91$	6.43828	0.674915
$92$	1.00000	0.104257
$93$	−7.81154	−0.810019
$94$	2.08878	0.215441
$95$	3.79783	0.389650
$96$	−1.00000	−0.102062
$97$	14.5169	1.47397	0.736986	−	0.675908i	$-0.236248\pi$
$97$	14.5169	1.47397	0.736986	+	0.675908i	$0.236248\pi$
$98$	1.22779	0.124026
$99$	−2.43412	−0.244638
$100$	10.7484	1.07484
$101$	7.17110	0.713551	0.356775	−	0.934190i	$-0.383876\pi$
$101$	7.17110	0.713551	0.356775	+	0.934190i	$0.383876\pi$
$102$	2.40254	0.237887
$103$	−3.87965	−0.382273	−0.191136	−	0.981563i	$-0.561217\pi$
$103$	−3.87965	−0.382273	−0.191136	+	0.981563i	$0.561217\pi$
$104$	−2.67978	−0.262774
$105$	−9.53431	−0.930453
$106$	−11.1574	−1.08370
$107$	−14.5801	−1.40951	−0.704756	−	0.709450i	$-0.748943\pi$
$107$	−14.5801	−1.40951	−0.704756	+	0.709450i	$0.748943\pi$
$108$	1.00000	0.0962250
$109$	−10.1776	−0.974833	−0.487416	−	0.873170i	$-0.662061\pi$
$109$	−10.1776	−0.974833	−0.487416	+	0.873170i	$0.662061\pi$
$110$	−9.65961	−0.921008
$111$	11.1890	1.06201
$112$	2.40254	0.227019
$113$	12.7419	1.19866	0.599330	−	0.800502i	$-0.295434\pi$
$113$	12.7419	1.19866	0.599330	+	0.800502i	$0.295434\pi$
$114$	0.957013	0.0896325
$115$	−3.96842	−0.370057
$116$	−1.00000	−0.0928477
$117$	2.67978	0.247746
$118$	5.54801	0.510736
$119$	−5.77221	−0.529138
$120$	3.96842	0.362266
$121$	−5.07507	−0.461370
$122$	3.56588	0.322840
$123$	4.38883	0.395728
$124$	−7.81154	−0.701497
$125$	−22.8120	−2.04037
$126$	−2.40254	−0.214035
$127$	1.17110	0.103918	0.0519590	−	0.998649i	$-0.483453\pi$
$127$	1.17110	0.103918	0.0519590	+	0.998649i	$0.483453\pi$
$128$	−1.00000	−0.0883883
$129$	−7.33214	−0.645559
$130$	10.6345	0.932707
$131$	2.33939	0.204393	0.102197	−	0.994764i	$-0.467413\pi$
$131$	2.33939	0.204393	0.102197	+	0.994764i	$0.467413\pi$
$132$	−2.43412	−0.211863
$133$	−2.29926	−0.199371
$134$	−13.5284	−1.16867
$135$	−3.96842	−0.341548
$136$	2.40254	0.206016
$137$	3.86099	0.329866	0.164933	−	0.986305i	$-0.447259\pi$
$137$	3.86099	0.329866	0.164933	+	0.986305i	$0.447259\pi$
$138$	−1.00000	−0.0851257
$139$	4.66607	0.395771	0.197885	−	0.980225i	$-0.436593\pi$
$139$	4.66607	0.395771	0.197885	+	0.980225i	$0.436593\pi$
$140$	−9.53431	−0.805796
$141$	−2.08878	−0.175907
$142$	14.3257	1.20218
$143$	−6.52290	−0.545472
$144$	1.00000	0.0833333
$145$	3.96842	0.329560
$146$	15.4712	1.28040
$147$	−1.22779	−0.101267
$148$	11.1890	0.919727
$149$	−6.43412	−0.527103	−0.263552	−	0.964645i	$-0.584894\pi$
$149$	−6.43412	−0.527103	−0.263552	+	0.964645i	$0.584894\pi$
$150$	−10.7484	−0.877602
$151$	−18.7191	−1.52334	−0.761670	−	0.647965i	$-0.775620\pi$
$151$	−18.7191	−1.52334	−0.761670	+	0.647965i	$0.775620\pi$
$152$	0.957013	0.0776240
$153$	−2.40254	−0.194234
$154$	5.84807	0.471251
$155$	30.9995	2.48994
$156$	2.67978	0.214554
$157$	−4.60292	−0.367353	−0.183676	−	0.982987i	$-0.558800\pi$
$157$	−4.60292	−0.367353	−0.183676	+	0.982987i	$0.558800\pi$
$158$	10.6661	0.848547
$159$	11.1574	0.884839
$160$	3.96842	0.313731
$161$	2.40254	0.189347
$162$	−1.00000	−0.0785674
$163$	−11.5077	−0.901351	−0.450676	−	0.892688i	$-0.648817\pi$
$163$	−11.5077	−0.901351	−0.450676	+	0.892688i	$0.648817\pi$
$164$	4.38883	0.342710
$165$	9.65961	0.752000
$166$	6.81800	0.529179
$167$	−5.03208	−0.389394	−0.194697	−	0.980863i	$-0.562372\pi$
$167$	−5.03208	−0.389394	−0.194697	+	0.980863i	$0.562372\pi$
$168$	−2.40254	−0.185360
$169$	−5.81879	−0.447599
$170$	−9.53431	−0.731248
$171$	−0.957013	−0.0731846
$172$	−7.33214	−0.559070
$173$	17.3578	1.31969	0.659843	−	0.751403i	$-0.270623\pi$
$173$	17.3578	1.31969	0.659843	+	0.751403i	$0.270623\pi$
$174$	1.00000	0.0758098
$175$	25.8235	1.95207
$176$	−2.43412	−0.183479
$177$	−5.54801	−0.417014
$178$	−10.5169	−0.788278
$179$	−7.68623	−0.574496	−0.287248	−	0.957856i	$-0.592740\pi$
$179$	−7.68623	−0.574496	−0.287248	+	0.957856i	$0.592740\pi$
$180$	−3.96842	−0.295789
$181$	−13.4336	−0.998514	−0.499257	−	0.866454i	$-0.666394\pi$
$181$	−13.4336	−0.998514	−0.499257	+	0.866454i	$0.666394\pi$
$182$	−6.43828	−0.477237
$183$	−3.56588	−0.263598
$184$	−1.00000	−0.0737210
$185$	−44.4026	−3.26454
$186$	7.81154	0.572770
$187$	5.84807	0.427653
$188$	−2.08878	−0.152340
$189$	2.40254	0.174759
$190$	−3.79783	−0.275524
$191$	1.70490	0.123362	0.0616810	−	0.998096i	$-0.480354\pi$
$191$	1.70490	0.123362	0.0616810	+	0.998096i	$0.480354\pi$
$192$	1.00000	0.0721688
$193$	−18.0484	−1.29916	−0.649578	−	0.760295i	$-0.725054\pi$
$193$	−18.0484	−1.29916	−0.649578	+	0.760295i	$0.725054\pi$
$194$	−14.5169	−1.04226
$195$	−10.6345	−0.761552
$196$	−1.22779	−0.0876994
$197$	−26.7145	−1.90333	−0.951665	−	0.307137i	$-0.900629\pi$
$197$	−26.7145	−1.90333	−0.951665	+	0.307137i	$0.900629\pi$
$198$	2.43412	0.172985
$199$	−9.79367	−0.694255	−0.347128	−	0.937818i	$-0.612843\pi$
$199$	−9.79367	−0.694255	−0.347128	+	0.937818i	$0.612843\pi$
$200$	−10.7484	−0.760026
$201$	13.5284	0.954217
$202$	−7.17110	−0.504557
$203$	−2.40254	−0.168625
$204$	−2.40254	−0.168212
$205$	−17.4168	−1.21644
$206$	3.87965	0.270308
$207$	1.00000	0.0695048
$208$	2.67978	0.185809
$209$	2.32948	0.161134
$210$	9.53431	0.657930
$211$	−15.3083	−1.05387	−0.526934	−	0.849906i	$-0.676659\pi$
$211$	−15.3083	−1.05387	−0.526934	+	0.849906i	$0.676659\pi$
$212$	11.1574	0.766293
$213$	−14.3257	−0.981580
$214$	14.5801	0.996675
$215$	29.0970	1.98440
$216$	−1.00000	−0.0680414
$217$	−18.7676	−1.27402
$218$	10.1776	0.689311
$219$	−15.4712	−1.04544
$220$	9.65961	0.651251
$221$	−6.43828	−0.433085
$222$	−11.1890	−0.750954
$223$	14.2032	0.951115	0.475558	−	0.879685i	$-0.342246\pi$
$223$	14.2032	0.951115	0.475558	+	0.879685i	$0.342246\pi$
$224$	−2.40254	−0.160527
$225$	10.7484	0.716559
$226$	−12.7419	−0.847581
$227$	−23.8006	−1.57970	−0.789852	−	0.613298i	$-0.789843\pi$
$227$	−23.8006	−1.57970	−0.789852	+	0.613298i	$0.789843\pi$
$228$	−0.957013	−0.0633797
$229$	15.5055	1.02463	0.512317	−	0.858796i	$-0.328787\pi$
$229$	15.5055	1.02463	0.512317	+	0.858796i	$0.328787\pi$
$230$	3.96842	0.261670
$231$	−5.84807	−0.384775
$232$	1.00000	0.0656532
$233$	−11.5828	−0.758811	−0.379406	−	0.925230i	$-0.623872\pi$
$233$	−11.5828	−0.758811	−0.379406	+	0.925230i	$0.623872\pi$
$234$	−2.67978	−0.175183
$235$	8.28915	0.540725
$236$	−5.54801	−0.361145
$237$	−10.6661	−0.692836
$238$	5.77221	0.374157
$239$	−25.4602	−1.64689	−0.823443	−	0.567399i	$-0.807950\pi$
$239$	−25.4602	−1.64689	−0.823443	+	0.567399i	$0.807950\pi$
$240$	−3.96842	−0.256161
$241$	−14.8809	−0.958566	−0.479283	−	0.877660i	$-0.659103\pi$
$241$	−14.8809	−0.958566	−0.479283	+	0.877660i	$0.659103\pi$
$242$	5.07507	0.326238
$243$	1.00000	0.0641500
$244$	−3.56588	−0.228282
$245$	4.87240	0.311286
$246$	−4.38883	−0.279822
$247$	−2.56458	−0.163180
$248$	7.81154	0.496033
$249$	−6.81800	−0.432073
$250$	22.8120	1.44276
$251$	−11.1002	−0.700638	−0.350319	−	0.936631i	$-0.613927\pi$
$251$	−11.1002	−0.700638	−0.350319	+	0.936631i	$0.613927\pi$
$252$	2.40254	0.151346
$253$	−2.43412	−0.153032
$254$	−1.17110	−0.0734811
$255$	9.53431	0.597061
$256$	1.00000	0.0625000
$257$	12.5169	0.780786	0.390393	−	0.920648i	$-0.372339\pi$
$257$	12.5169	0.780786	0.390393	+	0.920648i	$0.372339\pi$
$258$	7.33214	0.456479
$259$	26.8820	1.67036
$260$	−10.6345	−0.659523
$261$	−1.00000	−0.0618984
$262$	−2.33939	−0.144528
$263$	9.18021	0.566076	0.283038	−	0.959109i	$-0.408658\pi$
$263$	9.18021	0.566076	0.283038	+	0.959109i	$0.408658\pi$
$264$	2.43412	0.149810
$265$	−44.2773	−2.71993
$266$	2.29926	0.140977
$267$	10.5169	0.643627
$268$	13.5284	0.826376
$269$	−0.841614	−0.0513141	−0.0256571	−	0.999671i	$-0.508168\pi$
$269$	−0.841614	−0.0513141	−0.0256571	+	0.999671i	$0.508168\pi$
$270$	3.96842	0.241511
$271$	5.86178	0.356078	0.178039	−	0.984023i	$-0.443025\pi$
$271$	5.86178	0.356078	0.178039	+	0.984023i	$0.443025\pi$
$272$	−2.40254	−0.145676
$273$	6.43828	0.389662
$274$	−3.86099	−0.233251
$275$	−26.1628	−1.57768
$276$	1.00000	0.0601929
$277$	−10.0896	−0.606224	−0.303112	−	0.952955i	$-0.598026\pi$
$277$	−10.0896	−0.606224	−0.303112	+	0.952955i	$0.598026\pi$
$278$	−4.66607	−0.279852
$279$	−7.81154	−0.467665
$280$	9.53431	0.569784
$281$	−6.17475	−0.368355	−0.184177	−	0.982893i	$-0.558962\pi$
$281$	−6.17475	−0.368355	−0.184177	+	0.982893i	$0.558962\pi$
$282$	2.08878	0.124385
$283$	22.0169	1.30877	0.654384	−	0.756163i	$-0.272928\pi$
$283$	22.0169	1.30877	0.654384	+	0.756163i	$0.272928\pi$
$284$	−14.3257	−0.850073
$285$	3.79783	0.224964
$286$	6.52290	0.385707
$287$	10.5444	0.622414
$288$	−1.00000	−0.0589256
$289$	−11.2278	−0.660458
$290$	−3.96842	−0.233034
$291$	14.5169	0.850998
$292$	−15.4712	−0.905381
$293$	−6.75485	−0.394622	−0.197311	−	0.980341i	$-0.563221\pi$
$293$	−6.75485	−0.394622	−0.197311	+	0.980341i	$0.563221\pi$
$294$	1.22779	0.0716062
$295$	22.0169	1.28187
$296$	−11.1890	−0.650345
$297$	−2.43412	−0.141242
$298$	6.43412	0.372718
$299$	2.67978	0.154976
$300$	10.7484	0.620559
$301$	−17.6158	−1.01536
$302$	18.7191	1.07716
$303$	7.17110	0.411969
$304$	−0.957013	−0.0548885
$305$	14.1509	0.810280
$306$	2.40254	0.137344
$307$	27.9990	1.59799	0.798994	−	0.601339i	$-0.205366\pi$
$307$	27.9990	1.59799	0.798994	+	0.601339i	$0.205366\pi$
$308$	−5.84807	−0.333225
$309$	−3.87965	−0.220705
$310$	−30.9995	−1.76065
$311$	15.7248	0.891670	0.445835	−	0.895115i	$-0.352907\pi$
$311$	15.7248	0.891670	0.445835	+	0.895115i	$0.352907\pi$
$312$	−2.67978	−0.151713
$313$	−14.1390	−0.799184	−0.399592	−	0.916693i	$-0.630848\pi$
$313$	−14.1390	−0.799184	−0.399592	+	0.916693i	$0.630848\pi$
$314$	4.60292	0.259758
$315$	−9.53431	−0.537197
$316$	−10.6661	−0.600013
$317$	12.9889	0.729532	0.364766	−	0.931099i	$-0.381149\pi$
$317$	12.9889	0.729532	0.364766	+	0.931099i	$0.381149\pi$
$318$	−11.1574	−0.625675
$319$	2.43412	0.136284
$320$	−3.96842	−0.221842
$321$	−14.5801	−0.813782
$322$	−2.40254	−0.133888
$323$	2.29926	0.127934
$324$	1.00000	0.0555556
$325$	28.8033	1.59772
$326$	11.5077	0.637352
$327$	−10.1776	−0.562820
$328$	−4.38883	−0.242333
$329$	−5.01837	−0.276672
$330$	−9.65961	−0.531744
$331$	13.1188	0.721077	0.360539	−	0.932744i	$-0.382593\pi$
$331$	13.1188	0.721077	0.360539	+	0.932744i	$0.382593\pi$
$332$	−6.81800	−0.374186
$333$	11.1890	0.613152
$334$	5.03208	0.275343
$335$	−53.6863	−2.93319
$336$	2.40254	0.131069
$337$	−23.8046	−1.29672	−0.648359	−	0.761334i	$-0.724545\pi$
$337$	−23.8046	−1.29672	−0.648359	+	0.761334i	$0.724545\pi$
$338$	5.81879	0.316501
$339$	12.7419	0.692047
$340$	9.53431	0.517070
$341$	19.0142	1.02968
$342$	0.957013	0.0517493
$343$	−19.7676	−1.06735
$344$	7.33214	0.395322
$345$	−3.96842	−0.213653
$346$	−17.3578	−0.933159
$347$	−20.9826	−1.12641	−0.563204	−	0.826318i	$-0.690431\pi$
$347$	−20.9826	−1.12641	−0.563204	+	0.826318i	$0.690431\pi$
$348$	−1.00000	−0.0536056
$349$	−17.3687	−0.929724	−0.464862	−	0.885383i	$-0.653896\pi$
$349$	−17.3687	−0.929724	−0.464862	+	0.885383i	$0.653896\pi$
$350$	−25.8235	−1.38032
$351$	2.67978	0.143036
$352$	2.43412	0.129739
$353$	−0.957013	−0.0509367	−0.0254683	−	0.999676i	$-0.508108\pi$
$353$	−0.957013	−0.0509367	−0.0254683	+	0.999676i	$0.508108\pi$
$354$	5.54801	0.294874
$355$	56.8504	3.01731
$356$	10.5169	0.557397
$357$	−5.77221	−0.305498
$358$	7.68623	0.406230
$359$	34.0723	1.79827	0.899133	−	0.437675i	$-0.144198\pi$
$359$	34.0723	1.79827	0.899133	+	0.437675i	$0.144198\pi$
$360$	3.96842	0.209154
$361$	−18.0841	−0.951796
$362$	13.4336	0.706056
$363$	−5.07507	−0.266372
$364$	6.43828	0.337457
$365$	61.3961	3.21362
$366$	3.56588	0.186392
$367$	11.9140	0.621907	0.310954	−	0.950425i	$-0.399352\pi$
$367$	11.9140	0.621907	0.310954	+	0.950425i	$0.399352\pi$
$368$	1.00000	0.0521286
$369$	4.38883	0.228474
$370$	44.4026	2.30838
$371$	26.8061	1.39170
$372$	−7.81154	−0.405010
$373$	−8.58010	−0.444261	−0.222130	−	0.975017i	$-0.571301\pi$
$373$	−8.58010	−0.444261	−0.222130	+	0.975017i	$0.571301\pi$
$374$	−5.84807	−0.302397
$375$	−22.8120	−1.17801
$376$	2.08878	0.107720
$377$	−2.67978	−0.138016
$378$	−2.40254	−0.123573
$379$	−28.1499	−1.44597	−0.722983	−	0.690866i	$-0.757229\pi$
$379$	−28.1499	−1.44597	−0.722983	+	0.690866i	$0.757229\pi$
$380$	3.79783	0.194825
$381$	1.17110	0.0599971
$382$	−1.70490	−0.0872301
$383$	−5.88381	−0.300649	−0.150324	−	0.988637i	$-0.548032\pi$
$383$	−5.88381	−0.300649	−0.150324	+	0.988637i	$0.548032\pi$
$384$	−1.00000	−0.0510310
$385$	23.2076	1.18277
$386$	18.0484	0.918642
$387$	−7.33214	−0.372714
$388$	14.5169	0.736986
$389$	30.8132	1.56229	0.781146	−	0.624349i	$-0.214636\pi$
$389$	30.8132	1.56229	0.781146	+	0.624349i	$0.214636\pi$
$390$	10.6345	0.538499
$391$	−2.40254	−0.121502
$392$	1.22779	0.0620128
$393$	2.33939	0.118007
$394$	26.7145	1.34586
$395$	42.3275	2.12973
$396$	−2.43412	−0.122319
$397$	2.49132	0.125036	0.0625179	−	0.998044i	$-0.480087\pi$
$397$	2.49132	0.125036	0.0625179	+	0.998044i	$0.480087\pi$
$398$	9.79367	0.490912
$399$	−2.29926	−0.115107
$400$	10.7484	0.537419
$401$	24.2302	1.21000	0.604998	−	0.796227i	$-0.293174\pi$
$401$	24.2302	1.21000	0.604998	+	0.796227i	$0.293174\pi$
$402$	−13.5284	−0.674733
$403$	−20.9332	−1.04276
$404$	7.17110	0.356775
$405$	−3.96842	−0.197193
$406$	2.40254	0.119236
$407$	−27.2353	−1.35000
$408$	2.40254	0.118944
$409$	19.4986	0.964142	0.482071	−	0.876132i	$-0.339885\pi$
$409$	19.4986	0.964142	0.482071	+	0.876132i	$0.339885\pi$
$410$	17.4168	0.860152
$411$	3.86099	0.190448
$412$	−3.87965	−0.191136
$413$	−13.3293	−0.655894
$414$	−1.00000	−0.0491473
$415$	27.0567	1.32816
$416$	−2.67978	−0.131387
$417$	4.66607	0.228498
$418$	−2.32948	−0.113939
$419$	29.3202	1.43239	0.716194	−	0.697902i	$-0.245883\pi$
$419$	29.3202	1.43239	0.716194	+	0.697902i	$0.245883\pi$
$420$	−9.53431	−0.465226
$421$	−5.71781	−0.278669	−0.139335	−	0.990245i	$-0.544496\pi$
$421$	−5.71781	−0.278669	−0.139335	+	0.990245i	$0.544496\pi$
$422$	15.3083	0.745197
$423$	−2.08878	−0.101560
$424$	−11.1574	−0.541851
$425$	−25.8235	−1.25262
$426$	14.3257	0.694082
$427$	−8.56718	−0.414595
$428$	−14.5801	−0.704756
$429$	−6.52290	−0.314928
$430$	−29.0970	−1.40318
$431$	−18.7401	−0.902681	−0.451340	−	0.892352i	$-0.649054\pi$
$431$	−18.7401	−0.902681	−0.451340	+	0.892352i	$0.649054\pi$
$432$	1.00000	0.0481125
$433$	28.8710	1.38745	0.693727	−	0.720238i	$-0.255968\pi$
$433$	28.8710	1.38745	0.693727	+	0.720238i	$0.255968\pi$
$434$	18.7676	0.900872
$435$	3.96842	0.190271
$436$	−10.1776	−0.487416
$437$	−0.957013	−0.0457801
$438$	15.4712	0.739240
$439$	−36.0238	−1.71932	−0.859662	−	0.510863i	$-0.829326\pi$
$439$	−36.0238	−1.71932	−0.859662	+	0.510863i	$0.829326\pi$
$440$	−9.65961	−0.460504
$441$	−1.22779	−0.0584663
$442$	6.43828	0.306238
$443$	−1.56172	−0.0741996	−0.0370998	−	0.999312i	$-0.511812\pi$
$443$	−1.56172	−0.0741996	−0.0370998	+	0.999312i	$0.511812\pi$
$444$	11.1890	0.531005
$445$	−41.7357	−1.97846
$446$	−14.2032	−0.672540
$447$	−6.43412	−0.304323
$448$	2.40254	0.113509
$449$	−36.4675	−1.72101	−0.860504	−	0.509444i	$-0.829851\pi$
$449$	−36.4675	−1.72101	−0.860504	+	0.509444i	$0.829851\pi$
$450$	−10.7484	−0.506684
$451$	−10.6829	−0.503040
$452$	12.7419	0.599330
$453$	−18.7191	−0.879501
$454$	23.8006	1.11702
$455$	−25.5498	−1.19779
$456$	0.957013	0.0448162
$457$	27.6697	1.29434	0.647168	−	0.762347i	$-0.275953\pi$
$457$	27.6697	1.29434	0.647168	+	0.762347i	$0.275953\pi$
$458$	−15.5055	−0.724526
$459$	−2.40254	−0.112141
$460$	−3.96842	−0.185029
$461$	−24.3229	−1.13283	−0.566415	−	0.824120i	$-0.691670\pi$
$461$	−24.3229	−1.13283	−0.566415	+	0.824120i	$0.691670\pi$
$462$	5.84807	0.272077
$463$	−17.3981	−0.808558	−0.404279	−	0.914636i	$-0.632478\pi$
$463$	−17.3981	−0.808558	−0.404279	+	0.914636i	$0.632478\pi$
$464$	−1.00000	−0.0464238
$465$	30.9995	1.43757
$466$	11.5828	0.536561
$467$	16.4625	0.761796	0.380898	−	0.924617i	$-0.375615\pi$
$467$	16.4625	0.761796	0.380898	+	0.924617i	$0.375615\pi$
$468$	2.67978	0.123873
$469$	32.5024	1.50082
$470$	−8.28915	−0.382350
$471$	−4.60292	−0.212091
$472$	5.54801	0.255368
$473$	17.8473	0.820619
$474$	10.6661	0.489909
$475$	−10.2863	−0.471970
$476$	−5.77221	−0.264569
$477$	11.1574	0.510862
$478$	25.4602	1.16452
$479$	−7.91998	−0.361873	−0.180936	−	0.983495i	$-0.557913\pi$
$479$	−7.91998	−0.361873	−0.180936	+	0.983495i	$0.557913\pi$
$480$	3.96842	0.181133
$481$	29.9839	1.36715
$482$	14.8809	0.677809
$483$	2.40254	0.109319
$484$	−5.07507	−0.230685
$485$	−57.6094	−2.61591
$486$	−1.00000	−0.0453609
$487$	−39.6715	−1.79769	−0.898844	−	0.438268i	$-0.855592\pi$
$487$	−39.6715	−1.79769	−0.898844	+	0.438268i	$0.855592\pi$
$488$	3.56588	0.161420
$489$	−11.5077	−0.520395
$490$	−4.87240	−0.220112
$491$	−24.2260	−1.09330	−0.546652	−	0.837360i	$-0.684098\pi$
$491$	−24.2260	−1.09330	−0.546652	+	0.837360i	$0.684098\pi$
$492$	4.38883	0.197864
$493$	2.40254	0.108205
$494$	2.56458	0.115386
$495$	9.65961	0.434167
$496$	−7.81154	−0.350749
$497$	−34.4181	−1.54386
$498$	6.81800	0.305522
$499$	−34.7429	−1.55531	−0.777654	−	0.628693i	$-0.783590\pi$
$499$	−34.7429	−1.55531	−0.777654	+	0.628693i	$0.783590\pi$
$500$	−22.8120	−1.02019
$501$	−5.03208	−0.224817
$502$	11.1002	0.495426
$503$	−18.4667	−0.823390	−0.411695	−	0.911322i	$-0.635063\pi$
$503$	−18.4667	−0.823390	−0.411695	+	0.911322i	$0.635063\pi$
$504$	−2.40254	−0.107018
$505$	−28.4580	−1.26636
$506$	2.43412	0.108210
$507$	−5.81879	−0.258422
$508$	1.17110	0.0519590
$509$	−17.4483	−0.773384	−0.386692	−	0.922209i	$-0.626382\pi$
$509$	−17.4483	−0.773384	−0.386692	+	0.922209i	$0.626382\pi$
$510$	−9.53431	−0.422186
$511$	−37.1701	−1.64431
$512$	−1.00000	−0.0441942
$513$	−0.957013	−0.0422532
$514$	−12.5169	−0.552099
$515$	15.3961	0.678433
$516$	−7.33214	−0.322779
$517$	5.08433	0.223609
$518$	−26.8820	−1.18113
$519$	17.3578	0.761921
$520$	10.6345	0.466353
$521$	2.17475	0.0952776	0.0476388	−	0.998865i	$-0.484830\pi$
$521$	2.17475	0.0952776	0.0476388	+	0.998865i	$0.484830\pi$
$522$	1.00000	0.0437688
$523$	−16.9739	−0.742216	−0.371108	−	0.928590i	$-0.621022\pi$
$523$	−16.9739	−0.742216	−0.371108	+	0.928590i	$0.621022\pi$
$524$	2.33939	0.102197
$525$	25.8235	1.12703
$526$	−9.18021	−0.400276
$527$	18.7676	0.817528
$528$	−2.43412	−0.105931
$529$	1.00000	0.0434783
$530$	44.2773	1.92328
$531$	−5.54801	−0.240763
$532$	−2.29926	−0.0996857
$533$	11.7611	0.509430
$534$	−10.5169	−0.455113
$535$	57.8600	2.50151
$536$	−13.5284	−0.584336
$537$	−7.68623	−0.331686
$538$	0.841614	0.0362846
$539$	2.98859	0.128728
$540$	−3.96842	−0.170774
$541$	14.5426	0.625234	0.312617	−	0.949879i	$-0.398794\pi$
$541$	14.5426	0.625234	0.312617	+	0.949879i	$0.398794\pi$
$542$	−5.86178	−0.251785
$543$	−13.4336	−0.576492
$544$	2.40254	0.103008
$545$	40.3889	1.73007
$546$	−6.43828	−0.275533
$547$	40.5525	1.73390	0.866949	−	0.498396i	$-0.166078\pi$
$547$	40.5525	1.73390	0.866949	+	0.498396i	$0.166078\pi$
$548$	3.86099	0.164933
$549$	−3.56588	−0.152188
$550$	26.1628	1.11559
$551$	0.957013	0.0407701
$552$	−1.00000	−0.0425628
$553$	−25.6257	−1.08971
$554$	10.0896	0.428665
$555$	−44.4026	−1.88478
$556$	4.66607	0.197885
$557$	−15.1458	−0.641749	−0.320875	−	0.947122i	$-0.603977\pi$
$557$	−15.1458	−0.641749	−0.320875	+	0.947122i	$0.603977\pi$
$558$	7.81154	0.330689
$559$	−19.6485	−0.831043
$560$	−9.53431	−0.402898
$561$	5.84807	0.246906
$562$	6.17475	0.260466
$563$	22.4468	0.946021	0.473011	−	0.881057i	$-0.343167\pi$
$563$	22.4468	0.946021	0.473011	+	0.881057i	$0.343167\pi$
$564$	−2.08878	−0.0879534
$565$	−50.5654	−2.12730
$566$	−22.0169	−0.925438
$567$	2.40254	0.100897
$568$	14.3257	0.601092
$569$	19.3339	0.810521	0.405260	−	0.914201i	$-0.367181\pi$
$569$	19.3339	0.810521	0.405260	+	0.914201i	$0.367181\pi$
$570$	−3.79783	−0.159074
$571$	33.7691	1.41319	0.706596	−	0.707618i	$-0.250230\pi$
$571$	33.7691	1.41319	0.706596	+	0.707618i	$0.250230\pi$
$572$	−6.52290	−0.272736
$573$	1.70490	0.0712231
$574$	−10.5444	−0.440113
$575$	10.7484	0.448239
$576$	1.00000	0.0416667
$577$	−14.6204	−0.608656	−0.304328	−	0.952567i	$-0.598432\pi$
$577$	−14.6204	−0.608656	−0.304328	+	0.952567i	$0.598432\pi$
$578$	11.2278	0.467015
$579$	−18.0484	−0.750068
$580$	3.96842	0.164780
$581$	−16.3805	−0.679579
$582$	−14.5169	−0.601747
$583$	−27.1584	−1.12479
$584$	15.4712	0.640201
$585$	−10.6345	−0.439682
$586$	6.75485	0.279040
$587$	0.678781	0.0280163	0.0140081	−	0.999902i	$-0.495541\pi$
$587$	0.678781	0.0280163	0.0140081	+	0.999902i	$0.495541\pi$
$588$	−1.22779	−0.0506333
$589$	7.47575	0.308033
$590$	−22.0169	−0.906420
$591$	−26.7145	−1.09889
$592$	11.1890	0.459864
$593$	21.7200	0.891933	0.445966	−	0.895050i	$-0.352860\pi$
$593$	21.7200	0.891933	0.445966	+	0.895050i	$0.352860\pi$
$594$	2.43412	0.0998731
$595$	22.9066	0.939078
$596$	−6.43412	−0.263552
$597$	−9.79367	−0.400828
$598$	−2.67978	−0.109584
$599$	−18.9669	−0.774967	−0.387484	−	0.921877i	$-0.626656\pi$
$599$	−18.9669	−0.774967	−0.387484	+	0.921877i	$0.626656\pi$
$600$	−10.7484	−0.438801
$601$	26.6085	1.08538	0.542692	−	0.839932i	$-0.317405\pi$
$601$	26.6085	1.08538	0.542692	+	0.839932i	$0.317405\pi$
$602$	17.6158	0.717965
$603$	13.5284	0.550917
$604$	−18.7191	−0.761670
$605$	20.1400	0.818809
$606$	−7.17110	−0.291306
$607$	11.6604	0.473281	0.236641	−	0.971597i	$-0.423954\pi$
$607$	11.6604	0.473281	0.236641	+	0.971597i	$0.423954\pi$
$608$	0.957013	0.0388120
$609$	−2.40254	−0.0973559
$610$	−14.1509	−0.572954
$611$	−5.59746	−0.226449
$612$	−2.40254	−0.0971170
$613$	−14.5012	−0.585699	−0.292850	−	0.956159i	$-0.594603\pi$
$613$	−14.5012	−0.585699	−0.292850	+	0.956159i	$0.594603\pi$
$614$	−27.9990	−1.12995
$615$	−17.4168	−0.702311
$616$	5.84807	0.235626
$617$	−33.5077	−1.34897	−0.674485	−	0.738289i	$-0.735634\pi$
$617$	−33.5077	−1.34897	−0.674485	+	0.738289i	$0.735634\pi$
$618$	3.87965	0.156062
$619$	16.4895	0.662770	0.331385	−	0.943496i	$-0.392484\pi$
$619$	16.4895	0.662770	0.331385	+	0.943496i	$0.392484\pi$
$620$	30.9995	1.24497
$621$	1.00000	0.0401286
$622$	−15.7248	−0.630506
$623$	25.2674	1.01232
$624$	2.67978	0.107277
$625$	36.7859	1.47144
$626$	14.1390	0.565109
$627$	2.32948	0.0930306
$628$	−4.60292	−0.183676
$629$	−26.8820	−1.07185
$630$	9.53431	0.379856
$631$	−12.9226	−0.514442	−0.257221	−	0.966353i	$-0.582807\pi$
$631$	−12.9226	−0.514442	−0.257221	+	0.966353i	$0.582807\pi$
$632$	10.6661	0.424273
$633$	−15.3083	−0.608451
$634$	−12.9889	−0.515857
$635$	−4.64741	−0.184427
$636$	11.1574	0.442419
$637$	−3.29021	−0.130363
$638$	−2.43412	−0.0963677
$639$	−14.3257	−0.566715
$640$	3.96842	0.156866
$641$	10.3997	0.410765	0.205382	−	0.978682i	$-0.434156\pi$
$641$	10.3997	0.410765	0.205382	+	0.978682i	$0.434156\pi$
$642$	14.5801	0.575430
$643$	30.2762	1.19398	0.596989	−	0.802249i	$-0.296364\pi$
$643$	30.2762	1.19398	0.596989	+	0.802249i	$0.296364\pi$
$644$	2.40254	0.0946734
$645$	29.0970	1.14569
$646$	−2.29926	−0.0904633
$647$	0.236906	0.00931372	0.00465686	−	0.999989i	$-0.498518\pi$
$647$	0.236906	0.00931372	0.00465686	+	0.999989i	$0.498518\pi$
$648$	−1.00000	−0.0392837
$649$	13.5045	0.530099
$650$	−28.8033	−1.12976
$651$	−18.7676	−0.735559
$652$	−11.5077	−0.450676
$653$	17.3055	0.677217	0.338609	−	0.940927i	$-0.390044\pi$
$653$	17.3055	0.677217	0.338609	+	0.940927i	$0.390044\pi$
$654$	10.1776	0.397974
$655$	−9.28369	−0.362744
$656$	4.38883	0.171355
$657$	−15.4712	−0.603587
$658$	5.01837	0.195637
$659$	−18.5771	−0.723661	−0.361830	−	0.932244i	$-0.617848\pi$
$659$	−18.5771	−0.723661	−0.361830	+	0.932244i	$0.617848\pi$
$660$	9.65961	0.376000
$661$	25.1374	0.977730	0.488865	−	0.872359i	$-0.337411\pi$
$661$	25.1374	0.977730	0.488865	+	0.872359i	$0.337411\pi$
$662$	−13.1188	−0.509879
$663$	−6.43828	−0.250042
$664$	6.81800	0.264590
$665$	9.12446	0.353831
$666$	−11.1890	−0.433564
$667$	−1.00000	−0.0387202
$668$	−5.03208	−0.194697
$669$	14.2032	0.549127
$670$	53.6863	2.07408
$671$	8.67978	0.335079
$672$	−2.40254	−0.0926801
$673$	33.3979	1.28739	0.643697	−	0.765280i	$-0.277400\pi$
$673$	33.3979	1.28739	0.643697	+	0.765280i	$0.277400\pi$
$674$	23.8046	0.916919
$675$	10.7484	0.413706
$676$	−5.81879	−0.223800
$677$	22.2635	0.855657	0.427828	−	0.903860i	$-0.359279\pi$
$677$	22.2635	0.855657	0.427828	+	0.903860i	$0.359279\pi$
$678$	−12.7419	−0.489351
$679$	34.8776	1.33848
$680$	−9.53431	−0.365624
$681$	−23.8006	−0.912042
$682$	−19.0142	−0.728092
$683$	−33.1893	−1.26995	−0.634977	−	0.772531i	$-0.718990\pi$
$683$	−33.1893	−1.26995	−0.634977	+	0.772531i	$0.718990\pi$
$684$	−0.957013	−0.0365923
$685$	−15.3220	−0.585425
$686$	19.7676	0.754731
$687$	15.5055	0.591573
$688$	−7.33214	−0.279535
$689$	29.8993	1.13907
$690$	3.96842	0.151075
$691$	41.8965	1.59382	0.796909	−	0.604099i	$-0.206467\pi$
$691$	41.8965	1.59382	0.796909	+	0.604099i	$0.206467\pi$
$692$	17.3578	0.659843
$693$	−5.84807	−0.222150
$694$	20.9826	0.796490
$695$	−18.5169	−0.702388
$696$	1.00000	0.0379049
$697$	−10.5444	−0.399396
$698$	17.3687	0.657414
$699$	−11.5828	−0.438100
$700$	25.8235	0.976035
$701$	19.3024	0.729040	0.364520	−	0.931196i	$-0.381233\pi$
$701$	19.3024	0.729040	0.364520	+	0.931196i	$0.381233\pi$
$702$	−2.67978	−0.101142
$703$	−10.7080	−0.403859
$704$	−2.43412	−0.0917393
$705$	8.28915	0.312188
$706$	0.957013	0.0360177
$707$	17.2289	0.647958
$708$	−5.54801	−0.208507
$709$	−12.9534	−0.486476	−0.243238	−	0.969967i	$-0.578210\pi$
$709$	−12.9534	−0.486476	−0.243238	+	0.969967i	$0.578210\pi$
$710$	−56.8504	−2.13356
$711$	−10.6661	−0.400009
$712$	−10.5169	−0.394139
$713$	−7.81154	−0.292545
$714$	5.77221	0.216019
$715$	25.8856	0.968067
$716$	−7.68623	−0.287248
$717$	−25.4602	−0.950830
$718$	−34.0723	−1.27157
$719$	19.4803	0.726491	0.363246	−	0.931693i	$-0.381669\pi$
$719$	19.4803	0.726491	0.363246	+	0.931693i	$0.381669\pi$
$720$	−3.96842	−0.147894
$721$	−9.32102	−0.347133
$722$	18.0841	0.673021
$723$	−14.8809	−0.553428
$724$	−13.4336	−0.499257
$725$	−10.7484	−0.399185
$726$	5.07507	0.188353
$727$	10.2133	0.378790	0.189395	−	0.981901i	$-0.439347\pi$
$727$	10.2133	0.378790	0.189395	+	0.981901i	$0.439347\pi$
$728$	−6.43828	−0.238618
$729$	1.00000	0.0370370
$730$	−61.3961	−2.27237
$731$	17.6158	0.651543
$732$	−3.56588	−0.131799
$733$	13.6821	0.505359	0.252679	−	0.967550i	$-0.418688\pi$
$733$	13.6821	0.505359	0.252679	+	0.967550i	$0.418688\pi$
$734$	−11.9140	−0.439755
$735$	4.87240	0.179721
$736$	−1.00000	−0.0368605
$737$	−32.9296	−1.21298
$738$	−4.38883	−0.161555
$739$	11.8517	0.435971	0.217985	−	0.975952i	$-0.430052\pi$
$739$	11.8517	0.435971	0.217985	+	0.975952i	$0.430052\pi$
$740$	−44.4026	−1.63227
$741$	−2.56458	−0.0942123
$742$	−26.8061	−0.984083
$743$	3.44137	0.126252	0.0631258	−	0.998006i	$-0.479893\pi$
$743$	3.44137	0.126252	0.0631258	+	0.998006i	$0.479893\pi$
$744$	7.81154	0.286385
$745$	25.5333	0.935468
$746$	8.58010	0.314140
$747$	−6.81800	−0.249458
$748$	5.84807	0.213827
$749$	−35.0293	−1.27994
$750$	22.8120	0.832978
$751$	−9.39350	−0.342774	−0.171387	−	0.985204i	$-0.554825\pi$
$751$	−9.39350	−0.342774	−0.171387	+	0.985204i	$0.554825\pi$
$752$	−2.08878	−0.0761699
$753$	−11.1002	−0.404513
$754$	2.67978	0.0975917
$755$	74.2854	2.70352
$756$	2.40254	0.0873796
$757$	5.37541	0.195373	0.0976864	−	0.995217i	$-0.468856\pi$
$757$	5.37541	0.195373	0.0976864	+	0.995217i	$0.468856\pi$
$758$	28.1499	1.02245
$759$	−2.43412	−0.0883529
$760$	−3.79783	−0.137762
$761$	0.227791	0.00825743	0.00412871	−	0.999991i	$-0.498686\pi$
$761$	0.227791	0.00825743	0.00412871	+	0.999991i	$0.498686\pi$
$762$	−1.17110	−0.0424244
$763$	−24.4520	−0.885222
$764$	1.70490	0.0616810
$765$	9.53431	0.344714
$766$	5.88381	0.212591
$767$	−14.8674	−0.536832
$768$	1.00000	0.0360844
$769$	−16.4095	−0.591742	−0.295871	−	0.955228i	$-0.595610\pi$
$769$	−16.4095	−0.591742	−0.295871	+	0.955228i	$0.595610\pi$
$770$	−23.2076	−0.836345
$771$	12.5169	0.450787
$772$	−18.0484	−0.649578
$773$	−8.79518	−0.316341	−0.158170	−	0.987412i	$-0.550559\pi$
$773$	−8.79518	−0.316341	−0.158170	+	0.987412i	$0.550559\pi$
$774$	7.33214	0.263548
$775$	−83.9615	−3.01599
$776$	−14.5169	−0.521128
$777$	26.8820	0.964385
$778$	−30.8132	−1.10471
$779$	−4.20017	−0.150487
$780$	−10.6345	−0.380776
$781$	34.8704	1.24776
$782$	2.40254	0.0859147
$783$	−1.00000	−0.0357371
$784$	−1.22779	−0.0438497
$785$	18.2663	0.651953
$786$	−2.33939	−0.0834433
$787$	15.9912	0.570026	0.285013	−	0.958524i	$-0.408002\pi$
$787$	15.9912	0.570026	0.285013	+	0.958524i	$0.408002\pi$
$788$	−26.7145	−0.951665
$789$	9.18021	0.326824
$790$	−42.3275	−1.50594
$791$	30.6130	1.08847
$792$	2.43412	0.0864926
$793$	−9.55577	−0.339335
$794$	−2.49132	−0.0884136
$795$	−44.2773	−1.57035
$796$	−9.79367	−0.347128
$797$	−55.8590	−1.97863	−0.989313	−	0.145804i	$-0.953423\pi$
$797$	−55.8590	−1.97863	−0.989313	+	0.145804i	$0.953423\pi$
$798$	2.29926	0.0813931
$799$	5.01837	0.177537
$800$	−10.7484	−0.380013
$801$	10.5169	0.371598
$802$	−24.2302	−0.855597
$803$	37.6586	1.32894
$804$	13.5284	0.477108
$805$	−9.53431	−0.336040
$806$	20.9332	0.737340
$807$	−0.841614	−0.0296262
$808$	−7.17110	−0.252278
$809$	−34.1162	−1.19946	−0.599731	−	0.800202i	$-0.704726\pi$
$809$	−34.1162	−1.19946	−0.599731	+	0.800202i	$0.704726\pi$
$810$	3.96842	0.139436
$811$	−12.3440	−0.433456	−0.216728	−	0.976232i	$-0.569538\pi$
$811$	−12.3440	−0.433456	−0.216728	+	0.976232i	$0.569538\pi$
$812$	−2.40254	−0.0843127
$813$	5.86178	0.205582
$814$	27.2353	0.954595
$815$	45.6674	1.59966
$816$	−2.40254	−0.0841058
$817$	7.01695	0.245492
$818$	−19.4986	−0.681751
$819$	6.43828	0.224972
$820$	−17.4168	−0.608220
$821$	−44.9451	−1.56860	−0.784298	−	0.620385i	$-0.786976\pi$
$821$	−44.9451	−1.56860	−0.784298	+	0.620385i	$0.786976\pi$
$822$	−3.86099	−0.134667
$823$	22.1748	0.772964	0.386482	−	0.922297i	$-0.373690\pi$
$823$	22.1748	0.772964	0.386482	+	0.922297i	$0.373690\pi$
$824$	3.87965	0.135154
$825$	−26.1628	−0.910873
$826$	13.3293	0.463787
$827$	32.5483	1.13182	0.565908	−	0.824468i	$-0.308526\pi$
$827$	32.5483	1.13182	0.565908	+	0.824468i	$0.308526\pi$
$828$	1.00000	0.0347524
$829$	46.2260	1.60550	0.802748	−	0.596319i	$-0.203371\pi$
$829$	46.2260	1.60550	0.802748	+	0.596319i	$0.203371\pi$
$830$	−27.0567	−0.939152
$831$	−10.0896	−0.350003
$832$	2.67978	0.0929046
$833$	2.94982	0.102205
$834$	−4.66607	−0.161573
$835$	19.9694	0.691071
$836$	2.32948	0.0805668
$837$	−7.81154	−0.270006
$838$	−29.3202	−1.01285
$839$	−7.00130	−0.241712	−0.120856	−	0.992670i	$-0.538564\pi$
$839$	−7.00130	−0.241712	−0.120856	+	0.992670i	$0.538564\pi$
$840$	9.53431	0.328965
$841$	1.00000	0.0344828
$842$	5.71781	0.197049
$843$	−6.17475	−0.212670
$844$	−15.3083	−0.526934
$845$	23.0914	0.794369
$846$	2.08878	0.0718136
$847$	−12.1931	−0.418959
$848$	11.1574	0.383146
$849$	22.0169	0.755617
$850$	25.8235	0.885737
$851$	11.1890	0.383553
$852$	−14.3257	−0.490790
$853$	11.8263	0.404924	0.202462	−	0.979290i	$-0.435106\pi$
$853$	11.8263	0.404924	0.202462	+	0.979290i	$0.435106\pi$
$854$	8.56718	0.293163
$855$	3.79783	0.129883
$856$	14.5801	0.498337
$857$	−45.3788	−1.55011	−0.775055	−	0.631894i	$-0.782278\pi$
$857$	−45.3788	−1.55011	−0.775055	+	0.631894i	$0.782278\pi$
$858$	6.52290	0.222688
$859$	38.8061	1.32405	0.662023	−	0.749483i	$-0.269698\pi$
$859$	38.8061	1.32405	0.662023	+	0.749483i	$0.269698\pi$
$860$	29.0970	0.992201
$861$	10.5444	0.359351
$862$	18.7401	0.638292
$863$	29.6882	1.01060	0.505300	−	0.862944i	$-0.331382\pi$
$863$	29.6882	1.01060	0.505300	+	0.862944i	$0.331382\pi$
$864$	−1.00000	−0.0340207
$865$	−68.8830	−2.34209
$866$	−28.8710	−0.981078
$867$	−11.2278	−0.381316
$868$	−18.7676	−0.637012
$869$	25.9625	0.880717
$870$	−3.96842	−0.134542
$871$	36.2530	1.22839
$872$	10.1776	0.344655
$873$	14.5169	0.491324
$874$	0.957013	0.0323715
$875$	−54.8069	−1.85281
$876$	−15.4712	−0.522722
$877$	46.7757	1.57950	0.789751	−	0.613427i	$-0.210210\pi$
$877$	46.7757	1.57950	0.789751	+	0.613427i	$0.210210\pi$
$878$	36.0238	1.21575
$879$	−6.75485	−0.227835
$880$	9.65961	0.325625
$881$	29.0886	0.980021	0.490010	−	0.871717i	$-0.336993\pi$
$881$	29.0886	0.980021	0.490010	+	0.871717i	$0.336993\pi$
$882$	1.22779	0.0413419
$883$	51.3220	1.72712	0.863562	−	0.504243i	$-0.168228\pi$
$883$	51.3220	1.72712	0.863562	+	0.504243i	$0.168228\pi$
$884$	−6.43828	−0.216543
$885$	22.0169	0.740089
$886$	1.56172	0.0524671
$887$	−24.4365	−0.820497	−0.410248	−	0.911974i	$-0.634558\pi$
$887$	−24.4365	−0.820497	−0.410248	+	0.911974i	$0.634558\pi$
$888$	−11.1890	−0.375477
$889$	2.81361	0.0943654
$890$	41.7357	1.39898
$891$	−2.43412	−0.0815460
$892$	14.2032	0.475558
$893$	1.99899	0.0668935
$894$	6.43412	0.215189
$895$	30.5022	1.01958
$896$	−2.40254	−0.0802633
$897$	2.67978	0.0894752
$898$	36.4675	1.21694
$899$	7.81154	0.260529
$900$	10.7484	0.358280
$901$	−26.8061	−0.893041
$902$	10.6829	0.355703
$903$	−17.6158	−0.586216
$904$	−12.7419	−0.423790
$905$	53.3103	1.77210
$906$	18.7191	0.621901
$907$	22.4655	0.745954	0.372977	−	0.927841i	$-0.378337\pi$
$907$	22.4655	0.745954	0.372977	+	0.927841i	$0.378337\pi$
$908$	−23.8006	−0.789852
$909$	7.17110	0.237850
$910$	25.5498	0.846968
$911$	−36.6941	−1.21573	−0.607864	−	0.794041i	$-0.707974\pi$
$911$	−36.6941	−1.21573	−0.607864	+	0.794041i	$0.707974\pi$
$912$	−0.957013	−0.0316899
$913$	16.5958	0.549241
$914$	−27.6697	−0.915234
$915$	14.1509	0.467815
$916$	15.5055	0.512317
$917$	5.62048	0.185605
$918$	2.40254	0.0792957
$919$	−16.2931	−0.537460	−0.268730	−	0.963216i	$-0.586604\pi$
$919$	−16.2931	−0.537460	−0.268730	+	0.963216i	$0.586604\pi$
$920$	3.96842	0.130835
$921$	27.9990	0.922599
$922$	24.3229	0.801031
$923$	−38.3896	−1.26361
$924$	−5.84807	−0.192387
$925$	120.263	3.95423
$926$	17.3981	0.571737
$927$	−3.87965	−0.127424
$928$	1.00000	0.0328266
$929$	54.6598	1.79333	0.896665	−	0.442711i	$-0.145983\pi$
$929$	54.6598	1.79333	0.896665	+	0.442711i	$0.145983\pi$
$930$	−30.9995	−1.01651
$931$	1.17501	0.0385095
$932$	−11.5828	−0.379406
$933$	15.7248	0.514806
$934$	−16.4625	−0.538671
$935$	−23.2076	−0.758971
$936$	−2.67978	−0.0875913
$937$	7.19313	0.234989	0.117495	−	0.993074i	$-0.462514\pi$
$937$	7.19313	0.234989	0.117495	+	0.993074i	$0.462514\pi$
$938$	−32.5024	−1.06124
$939$	−14.1390	−0.461409
$940$	8.28915	0.270362
$941$	11.9152	0.388424	0.194212	−	0.980960i	$-0.437785\pi$
$941$	11.9152	0.388424	0.194212	+	0.980960i	$0.437785\pi$
$942$	4.60292	0.149971
$943$	4.38883	0.142920
$944$	−5.54801	−0.180572
$945$	−9.53431	−0.310151
$946$	−17.8473	−0.580266
$947$	26.2032	0.851489	0.425744	−	0.904843i	$-0.360012\pi$
$947$	26.2032	0.851489	0.425744	+	0.904843i	$0.360012\pi$
$948$	−10.6661	−0.346418
$949$	−41.4593	−1.34582
$950$	10.2863	0.333733
$951$	12.9889	0.421196
$952$	5.77221	0.187078
$953$	−50.2514	−1.62780	−0.813902	−	0.581003i	$-0.802661\pi$
$953$	−50.2514	−1.62780	−0.813902	+	0.581003i	$0.802661\pi$
$954$	−11.1574	−0.361234
$955$	−6.76575	−0.218935
$956$	−25.4602	−0.823443
$957$	2.43412	0.0786839
$958$	7.91998	0.255883
$959$	9.27618	0.299543
$960$	−3.96842	−0.128080
$961$	30.0202	0.968393
$962$	−29.9839	−0.966721
$963$	−14.5801	−0.469837
$964$	−14.8809	−0.479283
$965$	71.6239	2.30565
$966$	−2.40254	−0.0773005
$967$	−21.4191	−0.688793	−0.344396	−	0.938824i	$-0.611916\pi$
$967$	−21.4191	−0.688793	−0.344396	+	0.938824i	$0.611916\pi$
$968$	5.07507	0.163119
$969$	2.29926	0.0738630
$970$	57.6094	1.84973
$971$	7.41395	0.237925	0.118963	−	0.992899i	$-0.462043\pi$
$971$	7.41395	0.237925	0.118963	+	0.992899i	$0.462043\pi$
$972$	1.00000	0.0320750
$973$	11.2104	0.359390
$974$	39.6715	1.27116
$975$	28.8033	0.922444
$976$	−3.56588	−0.114141
$977$	51.0784	1.63414	0.817071	−	0.576537i	$-0.195596\pi$
$977$	51.0784	1.63414	0.817071	+	0.576537i	$0.195596\pi$
$978$	11.5077	0.367975
$979$	−25.5995	−0.818163
$980$	4.87240	0.155643
$981$	−10.1776	−0.324944
$982$	24.2260	0.773083
$983$	−57.0261	−1.81885	−0.909426	−	0.415866i	$-0.863478\pi$
$983$	−57.0261	−1.81885	−0.909426	+	0.415866i	$0.863478\pi$
$984$	−4.38883	−0.139911
$985$	106.015	3.37790
$986$	−2.40254	−0.0765125
$987$	−5.01837	−0.159737
$988$	−2.56458	−0.0815902
$989$	−7.33214	−0.233148
$990$	−9.65961	−0.307003
$991$	−49.3420	−1.56740	−0.783701	−	0.621138i	$-0.786670\pi$
$991$	−49.3420	−1.56740	−0.783701	+	0.621138i	$0.786670\pi$
$992$	7.81154	0.248017
$993$	13.1188	0.416314
$994$	34.4181	1.09167
$995$	38.8654	1.23212
$996$	−6.81800	−0.216037
$997$	−21.4582	−0.679589	−0.339795	−	0.940500i	$-0.610358\pi$
$997$	−21.4582	−0.679589	−0.339795	+	0.940500i	$0.610358\pi$
$998$	34.7429	1.09977
$999$	11.1890	0.354003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bb.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bb.1.1	✓	4	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.96842 q^{5} -1.00000 q^{6} +2.40254 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.96842 q^{5} -1.00000 q^{6} +2.40254 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.96842 q^{10} -2.43412 q^{11} +1.00000 q^{12} +2.67978 q^{13} -2.40254 q^{14} -3.96842 q^{15} +1.00000 q^{16} -2.40254 q^{17} -1.00000 q^{18} -0.957013 q^{19} -3.96842 q^{20} +2.40254 q^{21} +2.43412 q^{22} +1.00000 q^{23} -1.00000 q^{24} +10.7484 q^{25} -2.67978 q^{26} +1.00000 q^{27} +2.40254 q^{28} -1.00000 q^{29} +3.96842 q^{30} -7.81154 q^{31} -1.00000 q^{32} -2.43412 q^{33} +2.40254 q^{34} -9.53431 q^{35} +1.00000 q^{36} +11.1890 q^{37} +0.957013 q^{38} +2.67978 q^{39} +3.96842 q^{40} +4.38883 q^{41} -2.40254 q^{42} -7.33214 q^{43} -2.43412 q^{44} -3.96842 q^{45} -1.00000 q^{46} -2.08878 q^{47} +1.00000 q^{48} -1.22779 q^{49} -10.7484 q^{50} -2.40254 q^{51} +2.67978 q^{52} +11.1574 q^{53} -1.00000 q^{54} +9.65961 q^{55} -2.40254 q^{56} -0.957013 q^{57} +1.00000 q^{58} -5.54801 q^{59} -3.96842 q^{60} -3.56588 q^{61} +7.81154 q^{62} +2.40254 q^{63} +1.00000 q^{64} -10.6345 q^{65} +2.43412 q^{66} +13.5284 q^{67} -2.40254 q^{68} +1.00000 q^{69} +9.53431 q^{70} -14.3257 q^{71} -1.00000 q^{72} -15.4712 q^{73} -11.1890 q^{74} +10.7484 q^{75} -0.957013 q^{76} -5.84807 q^{77} -2.67978 q^{78} -10.6661 q^{79} -3.96842 q^{80} +1.00000 q^{81} -4.38883 q^{82} -6.81800 q^{83} +2.40254 q^{84} +9.53431 q^{85} +7.33214 q^{86} -1.00000 q^{87} +2.43412 q^{88} +10.5169 q^{89} +3.96842 q^{90} +6.43828 q^{91} +1.00000 q^{92} -7.81154 q^{93} +2.08878 q^{94} +3.79783 q^{95} -1.00000 q^{96} +14.5169 q^{97} +1.22779 q^{98} -2.43412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9} + 3 q^{10} - 11 q^{11} + 4 q^{12} - q^{13} + 2 q^{14} - 3 q^{15} + 4 q^{16} + 2 q^{17} - 4 q^{18} + 8 q^{19} - 3 q^{20} - 2 q^{21} + 11 q^{22} + 4 q^{23} - 4 q^{24} + 3 q^{25} + q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} + 3 q^{30} - 17 q^{31} - 4 q^{32} - 11 q^{33} - 2 q^{34} - 24 q^{35} + 4 q^{36} + 15 q^{37} - 8 q^{38} - q^{39} + 3 q^{40} + q^{41} + 2 q^{42} + 4 q^{43} - 11 q^{44} - 3 q^{45} - 4 q^{46} + 6 q^{47} + 4 q^{48} + 16 q^{49} - 3 q^{50} + 2 q^{51} - q^{52} + 2 q^{53} - 4 q^{54} + 13 q^{55} + 2 q^{56} + 8 q^{57} + 4 q^{58} - 13 q^{59} - 3 q^{60} - 13 q^{61} + 17 q^{62} - 2 q^{63} + 4 q^{64} - 13 q^{65} + 11 q^{66} - 13 q^{67} + 2 q^{68} + 4 q^{69} + 24 q^{70} - 15 q^{71} - 4 q^{72} - 22 q^{73} - 15 q^{74} + 3 q^{75} + 8 q^{76} - 12 q^{77} + q^{78} - 26 q^{79} - 3 q^{80} + 4 q^{81} - q^{82} - 22 q^{83} - 2 q^{84} + 24 q^{85} - 4 q^{86} - 4 q^{87} + 11 q^{88} - 24 q^{89} + 3 q^{90} + 30 q^{91} + 4 q^{92} - 17 q^{93} - 6 q^{94} - 4 q^{95} - 4 q^{96} - 8 q^{97} - 16 q^{98} - 11 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 3 * q^5 - 4 * q^6 - 2 * q^7 - 4 * q^8 + 4 * q^9 + 3 * q^10 - 11 * q^11 + 4 * q^12 - q^13 + 2 * q^14 - 3 * q^15 + 4 * q^16 + 2 * q^17 - 4 * q^18 + 8 * q^19 - 3 * q^20 - 2 * q^21 + 11 * q^22 + 4 * q^23 - 4 * q^24 + 3 * q^25 + q^26 + 4 * q^27 - 2 * q^28 - 4 * q^29 + 3 * q^30 - 17 * q^31 - 4 * q^32 - 11 * q^33 - 2 * q^34 - 24 * q^35 + 4 * q^36 + 15 * q^37 - 8 * q^38 - q^39 + 3 * q^40 + q^41 + 2 * q^42 + 4 * q^43 - 11 * q^44 - 3 * q^45 - 4 * q^46 + 6 * q^47 + 4 * q^48 + 16 * q^49 - 3 * q^50 + 2 * q^51 - q^52 + 2 * q^53 - 4 * q^54 + 13 * q^55 + 2 * q^56 + 8 * q^57 + 4 * q^58 - 13 * q^59 - 3 * q^60 - 13 * q^61 + 17 * q^62 - 2 * q^63 + 4 * q^64 - 13 * q^65 + 11 * q^66 - 13 * q^67 + 2 * q^68 + 4 * q^69 + 24 * q^70 - 15 * q^71 - 4 * q^72 - 22 * q^73 - 15 * q^74 + 3 * q^75 + 8 * q^76 - 12 * q^77 + q^78 - 26 * q^79 - 3 * q^80 + 4 * q^81 - q^82 - 22 * q^83 - 2 * q^84 + 24 * q^85 - 4 * q^86 - 4 * q^87 + 11 * q^88 - 24 * q^89 + 3 * q^90 + 30 * q^91 + 4 * q^92 - 17 * q^93 - 6 * q^94 - 4 * q^95 - 4 * q^96 - 8 * q^97 - 16 * q^98 - 11 * q^99

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