LMFDB - Embedded newform 4002.2.a.y.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.654334$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.654334 q^{5} +1.00000 q^{6} -2.11309 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.654334 q^{5} +1.00000 q^{6} -2.11309 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.654334 q^{10} -0.654334 q^{11} -1.00000 q^{12} +2.65433 q^{13} +2.11309 q^{14} +0.654334 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.11309 q^{19} -0.654334 q^{20} +2.11309 q^{21} +0.654334 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.57185 q^{25} -2.65433 q^{26} -1.00000 q^{27} -2.11309 q^{28} -1.00000 q^{29} -0.654334 q^{30} +2.65433 q^{31} -1.00000 q^{32} +0.654334 q^{33} -4.00000 q^{34} +1.38267 q^{35} +1.00000 q^{36} +5.57185 q^{37} -6.11309 q^{38} -2.65433 q^{39} +0.654334 q^{40} -11.5718 q^{41} -2.11309 q^{42} -3.42176 q^{43} -0.654334 q^{44} -0.654334 q^{45} -1.00000 q^{46} -2.69133 q^{47} -1.00000 q^{48} -2.53485 q^{49} +4.57185 q^{50} -4.00000 q^{51} +2.65433 q^{52} -1.30867 q^{53} +1.00000 q^{54} +0.428152 q^{55} +2.11309 q^{56} -6.11309 q^{57} +1.00000 q^{58} +7.34567 q^{59} +0.654334 q^{60} -2.65433 q^{61} -2.65433 q^{62} -2.11309 q^{63} +1.00000 q^{64} -1.73682 q^{65} -0.654334 q^{66} -2.65433 q^{67} +4.00000 q^{68} -1.00000 q^{69} -1.38267 q^{70} -12.2632 q^{71} -1.00000 q^{72} -8.91751 q^{73} -5.57185 q^{74} +4.57185 q^{75} +6.11309 q^{76} +1.38267 q^{77} +2.65433 q^{78} +6.22618 q^{79} -0.654334 q^{80} +1.00000 q^{81} +11.5718 q^{82} +12.3393 q^{83} +2.11309 q^{84} -2.61733 q^{85} +3.42176 q^{86} +1.00000 q^{87} +0.654334 q^{88} -2.69133 q^{89} +0.654334 q^{90} -5.60885 q^{91} +1.00000 q^{92} -2.65433 q^{93} +2.69133 q^{94} -4.00000 q^{95} +1.00000 q^{96} +12.8044 q^{97} +2.53485 q^{98} -0.654334 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - q^5 + 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} + q^{10} - q^{11} - 3 q^{12} + 7 q^{13} - 6 q^{14} + q^{15} + 3 q^{16} + 12 q^{17} - 3 q^{18} + 6 q^{19} - q^{20} - 6 q^{21} + q^{22} + 3 q^{23} + 3 q^{24} + 10 q^{25} - 7 q^{26} - 3 q^{27} + 6 q^{28} - 3 q^{29} - q^{30} + 7 q^{31} - 3 q^{32} + q^{33} - 12 q^{34} + 8 q^{35} + 3 q^{36} - 7 q^{37} - 6 q^{38} - 7 q^{39} + q^{40} - 11 q^{41} + 6 q^{42} + 4 q^{43} - q^{44} - q^{45} - 3 q^{46} - 10 q^{47} - 3 q^{48} + 19 q^{49} - 10 q^{50} - 12 q^{51} + 7 q^{52} - 2 q^{53} + 3 q^{54} + 25 q^{55} - 6 q^{56} - 6 q^{57} + 3 q^{58} + 23 q^{59} + q^{60} - 7 q^{61} - 7 q^{62} + 6 q^{63} + 3 q^{64} - 27 q^{65} - q^{66} - 7 q^{67} + 12 q^{68} - 3 q^{69} - 8 q^{70} - 15 q^{71} - 3 q^{72} - 4 q^{73} + 7 q^{74} - 10 q^{75} + 6 q^{76} + 8 q^{77} + 7 q^{78} - 6 q^{79} - q^{80} + 3 q^{81} + 11 q^{82} - 6 q^{84} - 4 q^{85} - 4 q^{86} + 3 q^{87} + q^{88} - 10 q^{89} + q^{90} + 4 q^{91} + 3 q^{92} - 7 q^{93} + 10 q^{94} - 12 q^{95} + 3 q^{96} + 28 q^{97} - 19 q^{98} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - q^5 + 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 + q^10 - q^11 - 3 * q^12 + 7 * q^13 - 6 * q^14 + q^15 + 3 * q^16 + 12 * q^17 - 3 * q^18 + 6 * q^19 - q^20 - 6 * q^21 + q^22 + 3 * q^23 + 3 * q^24 + 10 * q^25 - 7 * q^26 - 3 * q^27 + 6 * q^28 - 3 * q^29 - q^30 + 7 * q^31 - 3 * q^32 + q^33 - 12 * q^34 + 8 * q^35 + 3 * q^36 - 7 * q^37 - 6 * q^38 - 7 * q^39 + q^40 - 11 * q^41 + 6 * q^42 + 4 * q^43 - q^44 - q^45 - 3 * q^46 - 10 * q^47 - 3 * q^48 + 19 * q^49 - 10 * q^50 - 12 * q^51 + 7 * q^52 - 2 * q^53 + 3 * q^54 + 25 * q^55 - 6 * q^56 - 6 * q^57 + 3 * q^58 + 23 * q^59 + q^60 - 7 * q^61 - 7 * q^62 + 6 * q^63 + 3 * q^64 - 27 * q^65 - q^66 - 7 * q^67 + 12 * q^68 - 3 * q^69 - 8 * q^70 - 15 * q^71 - 3 * q^72 - 4 * q^73 + 7 * q^74 - 10 * q^75 + 6 * q^76 + 8 * q^77 + 7 * q^78 - 6 * q^79 - q^80 + 3 * q^81 + 11 * q^82 - 6 * q^84 - 4 * q^85 - 4 * q^86 + 3 * q^87 + q^88 - 10 * q^89 + q^90 + 4 * q^91 + 3 * q^92 - 7 * q^93 + 10 * q^94 - 12 * q^95 + 3 * q^96 + 28 * q^97 - 19 * q^98 - 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
$5$	−0.654334	−0.292627	−0.146313	−	0.989238i	$-0.546741\pi$
$5$	−0.654334	−0.292627	−0.146313	+	0.989238i	$0.546741\pi$
$6$	1.00000	0.408248
$7$	−2.11309	−0.798673	−0.399337	−	0.916804i	$-0.630760\pi$
$7$	−2.11309	−0.798673	−0.399337	+	0.916804i	$0.630760\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.654334	0.206918
$11$	−0.654334	−0.197289	−0.0986445	−	0.995123i	$-0.531451\pi$
$11$	−0.654334	−0.197289	−0.0986445	+	0.995123i	$0.531451\pi$
$12$	−1.00000	−0.288675
$13$	2.65433	0.736180	0.368090	−	0.929790i	$-0.380012\pi$
$13$	2.65433	0.736180	0.368090	+	0.929790i	$0.380012\pi$
$14$	2.11309	0.564747
$15$	0.654334	0.168948
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	6.11309	1.40244	0.701220	−	0.712945i	$-0.252639\pi$
$19$	6.11309	1.40244	0.701220	+	0.712945i	$0.252639\pi$
$20$	−0.654334	−0.146313
$21$	2.11309	0.461114
$22$	0.654334	0.139504
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−4.57185	−0.914370
$26$	−2.65433	−0.520558
$27$	−1.00000	−0.192450
$28$	−2.11309	−0.399337
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−0.654334	−0.119464
$31$	2.65433	0.476732	0.238366	−	0.971175i	$-0.423388\pi$
$31$	2.65433	0.476732	0.238366	+	0.971175i	$0.423388\pi$
$32$	−1.00000	−0.176777
$33$	0.654334	0.113905
$34$	−4.00000	−0.685994
$35$	1.38267	0.233713
$36$	1.00000	0.166667
$37$	5.57185	0.916006	0.458003	−	0.888951i	$-0.348565\pi$
$37$	5.57185	0.916006	0.458003	+	0.888951i	$0.348565\pi$
$38$	−6.11309	−0.991674
$39$	−2.65433	−0.425034
$40$	0.654334	0.103459
$41$	−11.5718	−1.80722	−0.903609	−	0.428357i	$-0.859093\pi$
$41$	−11.5718	−1.80722	−0.903609	+	0.428357i	$0.859093\pi$
$42$	−2.11309	−0.326057
$43$	−3.42176	−0.521813	−0.260907	−	0.965364i	$-0.584021\pi$
$43$	−3.42176	−0.521813	−0.260907	+	0.965364i	$0.584021\pi$
$44$	−0.654334	−0.0986445
$45$	−0.654334	−0.0975423
$46$	−1.00000	−0.147442
$47$	−2.69133	−0.392571	−0.196286	−	0.980547i	$-0.562888\pi$
$47$	−2.69133	−0.392571	−0.196286	+	0.980547i	$0.562888\pi$
$48$	−1.00000	−0.144338
$49$	−2.53485	−0.362121
$50$	4.57185	0.646557
$51$	−4.00000	−0.560112
$52$	2.65433	0.368090
$53$	−1.30867	−0.179759	−0.0898796	−	0.995953i	$-0.528648\pi$
$53$	−1.30867	−0.179759	−0.0898796	+	0.995953i	$0.528648\pi$
$54$	1.00000	0.136083
$55$	0.428152	0.0577320
$56$	2.11309	0.282374
$57$	−6.11309	−0.809699
$58$	1.00000	0.131306
$59$	7.34567	0.956324	0.478162	−	0.878272i	$-0.341303\pi$
$59$	7.34567	0.956324	0.478162	+	0.878272i	$0.341303\pi$
$60$	0.654334	0.0844741
$61$	−2.65433	−0.339853	−0.169926	−	0.985457i	$-0.554353\pi$
$61$	−2.65433	−0.339853	−0.169926	+	0.985457i	$0.554353\pi$
$62$	−2.65433	−0.337101
$63$	−2.11309	−0.266224
$64$	1.00000	0.125000
$65$	−1.73682	−0.215426
$66$	−0.654334	−0.0805429
$67$	−2.65433	−0.324278	−0.162139	−	0.986768i	$-0.551839\pi$
$67$	−2.65433	−0.324278	−0.162139	+	0.986768i	$0.551839\pi$
$68$	4.00000	0.485071
$69$	−1.00000	−0.120386
$70$	−1.38267	−0.165260
$71$	−12.2632	−1.45537	−0.727686	−	0.685911i	$-0.759404\pi$
$71$	−12.2632	−1.45537	−0.727686	+	0.685911i	$0.759404\pi$
$72$	−1.00000	−0.117851
$73$	−8.91751	−1.04372	−0.521858	−	0.853032i	$-0.674761\pi$
$73$	−8.91751	−1.04372	−0.521858	+	0.853032i	$0.674761\pi$
$74$	−5.57185	−0.647714
$75$	4.57185	0.527911
$76$	6.11309	0.701220
$77$	1.38267	0.157569
$78$	2.65433	0.300544
$79$	6.22618	0.700500	0.350250	−	0.936656i	$-0.386097\pi$
$79$	6.22618	0.700500	0.350250	+	0.936656i	$0.386097\pi$
$80$	−0.654334	−0.0731567
$81$	1.00000	0.111111
$82$	11.5718	1.27790
$83$	12.3393	1.35441	0.677206	−	0.735794i	$-0.263191\pi$
$83$	12.3393	1.35441	0.677206	+	0.735794i	$0.263191\pi$
$84$	2.11309	0.230557
$85$	−2.61733	−0.283890
$86$	3.42176	0.368978
$87$	1.00000	0.107211
$88$	0.654334	0.0697522
$89$	−2.69133	−0.285281	−0.142640	−	0.989775i	$-0.545559\pi$
$89$	−2.69133	−0.285281	−0.142640	+	0.989775i	$0.545559\pi$
$90$	0.654334	0.0689728
$91$	−5.60885	−0.587967
$92$	1.00000	0.104257
$93$	−2.65433	−0.275242
$94$	2.69133	0.277590
$95$	−4.00000	−0.410391
$96$	1.00000	0.102062
$97$	12.8044	1.30009	0.650046	−	0.759895i	$-0.274750\pi$
$97$	12.8044	1.30009	0.650046	+	0.759895i	$0.274750\pi$
$98$	2.53485	0.256058
$99$	−0.654334	−0.0657630
$100$	−4.57185	−0.457185
$101$	−5.79803	−0.576925	−0.288463	−	0.957491i	$-0.593144\pi$
$101$	−5.79803	−0.576925	−0.288463	+	0.957491i	$0.593144\pi$
$102$	4.00000	0.396059
$103$	9.45876	0.931999	0.466000	−	0.884785i	$-0.345695\pi$
$103$	9.45876	0.931999	0.466000	+	0.884785i	$0.345695\pi$
$104$	−2.65433	−0.260279
$105$	−1.38267	−0.134934
$106$	1.30867	0.127109
$107$	−6.80442	−0.657808	−0.328904	−	0.944363i	$-0.606679\pi$
$107$	−6.80442	−0.657808	−0.328904	+	0.944363i	$0.606679\pi$
$108$	−1.00000	−0.0962250
$109$	17.0306	1.63124	0.815618	−	0.578591i	$-0.196397\pi$
$109$	17.0306	1.63124	0.815618	+	0.578591i	$0.196397\pi$
$110$	−0.428152	−0.0408227
$111$	−5.57185	−0.528856
$112$	−2.11309	−0.199668
$113$	19.3699	1.82216	0.911082	−	0.412224i	$-0.135248\pi$
$113$	19.3699	1.82216	0.911082	+	0.412224i	$0.135248\pi$
$114$	6.11309	0.572543
$115$	−0.654334	−0.0610169
$116$	−1.00000	−0.0928477
$117$	2.65433	0.245393
$118$	−7.34567	−0.676223
$119$	−8.45236	−0.774827
$120$	−0.654334	−0.0597322
$121$	−10.5718	−0.961077
$122$	2.65433	0.240312
$123$	11.5718	1.04340
$124$	2.65433	0.238366
$125$	6.26318	0.560196
$126$	2.11309	0.188249
$127$	11.5718	1.02684	0.513418	−	0.858139i	$-0.328379\pi$
$127$	11.5718	1.02684	0.513418	+	0.858139i	$0.328379\pi$
$128$	−1.00000	−0.0883883
$129$	3.42176	0.301269
$130$	1.73682	0.152329
$131$	15.7610	1.37705	0.688524	−	0.725214i	$-0.258259\pi$
$131$	15.7610	1.37705	0.688524	+	0.725214i	$0.258259\pi$
$132$	0.654334	0.0569524
$133$	−12.9175	−1.12009
$134$	2.65433	0.229299
$135$	0.654334	0.0563161
$136$	−4.00000	−0.342997
$137$	−13.6088	−1.16268	−0.581341	−	0.813660i	$-0.697472\pi$
$137$	−13.6088	−1.16268	−0.581341	+	0.813660i	$0.697472\pi$
$138$	1.00000	0.0851257
$139$	14.2262	1.20665	0.603324	−	0.797496i	$-0.293842\pi$
$139$	14.2262	1.20665	0.603324	+	0.797496i	$0.293842\pi$
$140$	1.38267	0.116857
$141$	2.69133	0.226651
$142$	12.2632	1.02910
$143$	−1.73682	−0.145240
$144$	1.00000	0.0833333
$145$	0.654334	0.0543394
$146$	8.91751	0.738019
$147$	2.53485	0.209071
$148$	5.57185	0.458003
$149$	−12.9545	−1.06128	−0.530638	−	0.847599i	$-0.678047\pi$
$149$	−12.9545	−1.06128	−0.530638	+	0.847599i	$0.678047\pi$
$150$	−4.57185	−0.373290
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−6.11309	−0.495837
$153$	4.00000	0.323381
$154$	−1.38267	−0.111418
$155$	−1.73682	−0.139505
$156$	−2.65433	−0.212517
$157$	7.83503	0.625303	0.312652	−	0.949868i	$-0.398783\pi$
$157$	7.83503	0.625303	0.312652	+	0.949868i	$0.398783\pi$
$158$	−6.22618	−0.495328
$159$	1.30867	0.103784
$160$	0.654334	0.0517296
$161$	−2.11309	−0.166535
$162$	−1.00000	−0.0785674
$163$	15.7980	1.23740	0.618699	−	0.785628i	$-0.287660\pi$
$163$	15.7980	1.23740	0.618699	+	0.785628i	$0.287660\pi$
$164$	−11.5718	−0.903609
$165$	−0.428152	−0.0333316
$166$	−12.3393	−0.957713
$167$	10.1892	0.788463	0.394231	−	0.919011i	$-0.371011\pi$
$167$	10.1892	0.788463	0.394231	+	0.919011i	$0.371011\pi$
$168$	−2.11309	−0.163028
$169$	−5.95451	−0.458040
$170$	2.61733	0.200740
$171$	6.11309	0.467480
$172$	−3.42176	−0.260907
$173$	−6.69133	−0.508733	−0.254366	−	0.967108i	$-0.581867\pi$
$173$	−6.69133	−0.508733	−0.254366	+	0.967108i	$0.581867\pi$
$174$	−1.00000	−0.0758098
$175$	9.66073	0.730282
$176$	−0.654334	−0.0493222
$177$	−7.34567	−0.552134
$178$	2.69133	0.201724
$179$	7.77382	0.581042	0.290521	−	0.956869i	$-0.406171\pi$
$179$	7.77382	0.581042	0.290521	+	0.956869i	$0.406171\pi$
$180$	−0.654334	−0.0487711
$181$	0.804424	0.0597923	0.0298962	−	0.999553i	$-0.490482\pi$
$181$	0.804424	0.0597923	0.0298962	+	0.999553i	$0.490482\pi$
$182$	5.60885	0.415755
$183$	2.65433	0.196214
$184$	−1.00000	−0.0737210
$185$	−3.64585	−0.268048
$186$	2.65433	0.194625
$187$	−2.61733	−0.191398
$188$	−2.69133	−0.196286
$189$	2.11309	0.153705
$190$	4.00000	0.290191
$191$	−6.07609	−0.439651	−0.219825	−	0.975539i	$-0.570549\pi$
$191$	−6.07609	−0.439651	−0.219825	+	0.975539i	$0.570549\pi$
$192$	−1.00000	−0.0721688
$193$	−0.691333	−0.0497632	−0.0248816	−	0.999690i	$-0.507921\pi$
$193$	−0.691333	−0.0497632	−0.0248816	+	0.999690i	$0.507921\pi$
$194$	−12.8044	−0.919304
$195$	1.73682	0.124376
$196$	−2.53485	−0.181061
$197$	−15.8350	−1.12820	−0.564100	−	0.825707i	$-0.690777\pi$
$197$	−15.8350	−1.12820	−0.564100	+	0.825707i	$0.690777\pi$
$198$	0.654334	0.0465015
$199$	1.23258	0.0873750	0.0436875	−	0.999045i	$-0.486089\pi$
$199$	1.23258	0.0873750	0.0436875	+	0.999045i	$0.486089\pi$
$200$	4.57185	0.323278
$201$	2.65433	0.187222
$202$	5.79803	0.407948
$203$	2.11309	0.148310
$204$	−4.00000	−0.280056
$205$	7.57185	0.528841
$206$	−9.45876	−0.659023
$207$	1.00000	0.0695048
$208$	2.65433	0.184045
$209$	−4.00000	−0.276686
$210$	1.38267	0.0954130
$211$	4.65433	0.320418	0.160209	−	0.987083i	$-0.448783\pi$
$211$	4.65433	0.320418	0.160209	+	0.987083i	$0.448783\pi$
$212$	−1.30867	−0.0898796
$213$	12.2632	0.840259
$214$	6.80442	0.465141
$215$	2.23897	0.152697
$216$	1.00000	0.0680414
$217$	−5.60885	−0.380753
$218$	−17.0306	−1.15346
$219$	8.91751	0.602590
$220$	0.428152	0.0288660
$221$	10.6173	0.714199
$222$	5.57185	0.373958
$223$	13.3827	0.896170	0.448085	−	0.893991i	$-0.352106\pi$
$223$	13.3827	0.896170	0.448085	+	0.893991i	$0.352106\pi$
$224$	2.11309	0.141187
$225$	−4.57185	−0.304790
$226$	−19.3699	−1.28847
$227$	−12.3393	−0.818986	−0.409493	−	0.912313i	$-0.634294\pi$
$227$	−12.3393	−0.818986	−0.409493	+	0.912313i	$0.634294\pi$
$228$	−6.11309	−0.404849
$229$	6.58033	0.434841	0.217420	−	0.976078i	$-0.430236\pi$
$229$	6.58033	0.434841	0.217420	+	0.976078i	$0.430236\pi$
$230$	0.654334	0.0431455
$231$	−1.38267	−0.0909727
$232$	1.00000	0.0656532
$233$	7.08249	0.463989	0.231995	−	0.972717i	$-0.425475\pi$
$233$	7.08249	0.463989	0.231995	+	0.972717i	$0.425475\pi$
$234$	−2.65433	−0.173519
$235$	1.76103	0.114877
$236$	7.34567	0.478162
$237$	−6.22618	−0.404434
$238$	8.45236	0.547885
$239$	6.95451	0.449850	0.224925	−	0.974376i	$-0.427786\pi$
$239$	6.95451	0.449850	0.224925	+	0.974376i	$0.427786\pi$
$240$	0.654334	0.0422370
$241$	15.5348	1.00069	0.500344	−	0.865827i	$-0.333207\pi$
$241$	15.5348	1.00069	0.500344	+	0.865827i	$0.333207\pi$
$242$	10.5718	0.679584
$243$	−1.00000	−0.0641500
$244$	−2.65433	−0.169926
$245$	1.65864	0.105966
$246$	−11.5718	−0.737794
$247$	16.2262	1.03245
$248$	−2.65433	−0.168550
$249$	−12.3393	−0.781970
$250$	−6.26318	−0.396118
$251$	1.96300	0.123903	0.0619517	−	0.998079i	$-0.480268\pi$
$251$	1.96300	0.123903	0.0619517	+	0.998079i	$0.480268\pi$
$252$	−2.11309	−0.133112
$253$	−0.654334	−0.0411376
$254$	−11.5718	−0.726082
$255$	2.61733	0.163904
$256$	1.00000	0.0625000
$257$	30.6785	1.91367	0.956837	−	0.290624i	$-0.0938627\pi$
$257$	30.6785	1.91367	0.956837	+	0.290624i	$0.0938627\pi$
$258$	−3.42176	−0.213029
$259$	−11.7738	−0.731590
$260$	−1.73682	−0.107713
$261$	−1.00000	−0.0618984
$262$	−15.7610	−0.973720
$263$	−11.2568	−0.694123	−0.347062	−	0.937842i	$-0.612821\pi$
$263$	−11.2568	−0.694123	−0.347062	+	0.937842i	$0.612821\pi$
$264$	−0.654334	−0.0402714
$265$	0.856305	0.0526024
$266$	12.9175	0.792024
$267$	2.69133	0.164707
$268$	−2.65433	−0.162139
$269$	−29.8720	−1.82133	−0.910665	−	0.413146i	$-0.864430\pi$
$269$	−29.8720	−1.82133	−0.910665	+	0.413146i	$0.864430\pi$
$270$	−0.654334	−0.0398215
$271$	−5.73682	−0.348487	−0.174243	−	0.984703i	$-0.555748\pi$
$271$	−5.73682	−0.348487	−0.174243	+	0.984703i	$0.555748\pi$
$272$	4.00000	0.242536
$273$	5.60885	0.339463
$274$	13.6088	0.822141
$275$	2.99151	0.180395
$276$	−1.00000	−0.0601929
$277$	−11.1807	−0.671783	−0.335891	−	0.941901i	$-0.609038\pi$
$277$	−11.1807	−0.671783	−0.335891	+	0.941901i	$0.609038\pi$
$278$	−14.2262	−0.853230
$279$	2.65433	0.158911
$280$	−1.38267	−0.0826301
$281$	−6.50424	−0.388011	−0.194005	−	0.981000i	$-0.562148\pi$
$281$	−6.50424	−0.388011	−0.194005	+	0.981000i	$0.562148\pi$
$282$	−2.69133	−0.160267
$283$	3.51064	0.208686	0.104343	−	0.994541i	$-0.466726\pi$
$283$	3.51064	0.208686	0.104343	+	0.994541i	$0.466726\pi$
$284$	−12.2632	−0.727686
$285$	4.00000	0.236940
$286$	1.73682	0.102700
$287$	24.4524	1.44338
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−0.654334	−0.0384238
$291$	−12.8044	−0.750609
$292$	−8.91751	−0.521858
$293$	13.6479	0.797321	0.398661	−	0.917099i	$-0.369475\pi$
$293$	13.6479	0.797321	0.398661	+	0.917099i	$0.369475\pi$
$294$	−2.53485	−0.147835
$295$	−4.80652	−0.279846
$296$	−5.57185	−0.323857
$297$	0.654334	0.0379683
$298$	12.9545	0.750435
$299$	2.65433	0.153504
$300$	4.57185	0.263956
$301$	7.23048	0.416758
$302$	−4.00000	−0.230174
$303$	5.79803	0.333088
$304$	6.11309	0.350610
$305$	1.73682	0.0994500
$306$	−4.00000	−0.228665
$307$	13.1067	0.748039	0.374019	−	0.927421i	$-0.377979\pi$
$307$	13.1067	0.748039	0.374019	+	0.927421i	$0.377979\pi$
$308$	1.38267	0.0787847
$309$	−9.45876	−0.538090
$310$	1.73682	0.0986447
$311$	−10.9175	−0.619075	−0.309538	−	0.950887i	$-0.600174\pi$
$311$	−10.9175	−0.619075	−0.309538	+	0.950887i	$0.600174\pi$
$312$	2.65433	0.150272
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−7.83503	−0.442156
$315$	1.38267	0.0779044
$316$	6.22618	0.350250
$317$	25.8720	1.45312	0.726559	−	0.687104i	$-0.241118\pi$
$317$	25.8720	1.45312	0.726559	+	0.687104i	$0.241118\pi$
$318$	−1.30867	−0.0733864
$319$	0.654334	0.0366356
$320$	−0.654334	−0.0365784
$321$	6.80442	0.379786
$322$	2.11309	0.117758
$323$	24.4524	1.36057
$324$	1.00000	0.0555556
$325$	−12.1352	−0.673140
$326$	−15.7980	−0.874972
$327$	−17.0306	−0.941795
$328$	11.5718	0.638948
$329$	5.68703	0.313536
$330$	0.428152	0.0235690
$331$	−6.84352	−0.376154	−0.188077	−	0.982154i	$-0.560225\pi$
$331$	−6.84352	−0.376154	−0.188077	+	0.982154i	$0.560225\pi$
$332$	12.3393	0.677206
$333$	5.57185	0.305335
$334$	−10.1892	−0.557527
$335$	1.73682	0.0948926
$336$	2.11309	0.115279
$337$	11.9239	0.649537	0.324768	−	0.945794i	$-0.394714\pi$
$337$	11.9239	0.649537	0.324768	+	0.945794i	$0.394714\pi$
$338$	5.95451	0.323883
$339$	−19.3699	−1.05203
$340$	−2.61733	−0.141945
$341$	−1.73682	−0.0940540
$342$	−6.11309	−0.330558
$343$	20.1480	1.08789
$344$	3.42176	0.184489
$345$	0.654334	0.0352281
$346$	6.69133	0.359728
$347$	35.0697	1.88264	0.941320	−	0.337515i	$-0.109587\pi$
$347$	35.0697	1.88264	0.941320	+	0.337515i	$0.109587\pi$
$348$	1.00000	0.0536056
$349$	31.4069	1.68117	0.840586	−	0.541678i	$-0.182211\pi$
$349$	31.4069	1.68117	0.840586	+	0.541678i	$0.182211\pi$
$350$	−9.66073	−0.516388
$351$	−2.65433	−0.141678
$352$	0.654334	0.0348761
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	7.34567	0.390418
$355$	8.02421	0.425881
$356$	−2.69133	−0.142640
$357$	8.45236	0.447346
$358$	−7.77382	−0.410859
$359$	12.1871	0.643210	0.321605	−	0.946874i	$-0.395778\pi$
$359$	12.1871	0.643210	0.321605	+	0.946874i	$0.395778\pi$
$360$	0.654334	0.0344864
$361$	18.3699	0.966836
$362$	−0.804424	−0.0422796
$363$	10.5718	0.554878
$364$	−5.60885	−0.293983
$365$	5.83503	0.305419
$366$	−2.65433	−0.138744
$367$	30.2262	1.57779	0.788897	−	0.614526i	$-0.210653\pi$
$367$	30.2262	1.57779	0.788897	+	0.614526i	$0.210653\pi$
$368$	1.00000	0.0521286
$369$	−11.5718	−0.602406
$370$	3.64585	0.189539
$371$	2.76533	0.143569
$372$	−2.65433	−0.137621
$373$	2.03909	0.105580	0.0527901	−	0.998606i	$-0.483189\pi$
$373$	2.03909	0.105580	0.0527901	+	0.998606i	$0.483189\pi$
$374$	2.61733	0.135339
$375$	−6.26318	−0.323429
$376$	2.69133	0.138795
$377$	−2.65433	−0.136705
$378$	−2.11309	−0.108686
$379$	23.6479	1.21471	0.607357	−	0.794429i	$-0.292230\pi$
$379$	23.6479	1.21471	0.607357	+	0.794429i	$0.292230\pi$
$380$	−4.00000	−0.205196
$381$	−11.5718	−0.592844
$382$	6.07609	0.310880
$383$	8.85630	0.452536	0.226268	−	0.974065i	$-0.427347\pi$
$383$	8.85630	0.452536	0.226268	+	0.974065i	$0.427347\pi$
$384$	1.00000	0.0510310
$385$	−0.904725	−0.0461090
$386$	0.691333	0.0351879
$387$	−3.42176	−0.173938
$388$	12.8044	0.650046
$389$	−16.4133	−0.832186	−0.416093	−	0.909322i	$-0.636601\pi$
$389$	−16.4133	−0.832186	−0.416093	+	0.909322i	$0.636601\pi$
$390$	−1.73682	−0.0879473
$391$	4.00000	0.202289
$392$	2.53485	0.128029
$393$	−15.7610	−0.795039
$394$	15.8350	0.797757
$395$	−4.07400	−0.204985
$396$	−0.654334	−0.0328815
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	−1.23258	−0.0617834
$399$	12.9175	0.646685
$400$	−4.57185	−0.228592
$401$	14.0761	0.702926	0.351463	−	0.936202i	$-0.385684\pi$
$401$	14.0761	0.702926	0.351463	+	0.936202i	$0.385684\pi$
$402$	−2.65433	−0.132386
$403$	7.04549	0.350961
$404$	−5.79803	−0.288463
$405$	−0.654334	−0.0325141
$406$	−2.11309	−0.104871
$407$	−3.64585	−0.180718
$408$	4.00000	0.198030
$409$	9.21769	0.455786	0.227893	−	0.973686i	$-0.426816\pi$
$409$	9.21769	0.455786	0.227893	+	0.973686i	$0.426816\pi$
$410$	−7.57185	−0.373947
$411$	13.6088	0.671275
$412$	9.45876	0.466000
$413$	−15.5221	−0.763791
$414$	−1.00000	−0.0491473
$415$	−8.07400	−0.396337
$416$	−2.65433	−0.130139
$417$	−14.2262	−0.696659
$418$	4.00000	0.195646
$419$	−17.4958	−0.854724	−0.427362	−	0.904081i	$-0.640557\pi$
$419$	−17.4958	−0.854724	−0.427362	+	0.904081i	$0.640557\pi$
$420$	−1.38267	−0.0674672
$421$	5.87203	0.286185	0.143093	−	0.989709i	$-0.454295\pi$
$421$	5.87203	0.286185	0.143093	+	0.989709i	$0.454295\pi$
$422$	−4.65433	−0.226569
$423$	−2.69133	−0.130857
$424$	1.30867	0.0635545
$425$	−18.2874	−0.887069
$426$	−12.2632	−0.594153
$427$	5.60885	0.271431
$428$	−6.80442	−0.328904
$429$	1.73682	0.0838544
$430$	−2.23897	−0.107973
$431$	−0.300180	−0.0144592	−0.00722958	−	0.999974i	$-0.502301\pi$
$431$	−0.300180	−0.0144592	−0.00722958	+	0.999974i	$0.502301\pi$
$432$	−1.00000	−0.0481125
$433$	−5.25679	−0.252625	−0.126313	−	0.991990i	$-0.540314\pi$
$433$	−5.25679	−0.252625	−0.126313	+	0.991990i	$0.540314\pi$
$434$	5.60885	0.269233
$435$	−0.654334	−0.0313729
$436$	17.0306	0.815618
$437$	6.11309	0.292429
$438$	−8.91751	−0.426095
$439$	−0.526361	−0.0251219	−0.0125609	−	0.999921i	$-0.503998\pi$
$439$	−0.526361	−0.0251219	−0.0125609	+	0.999921i	$0.503998\pi$
$440$	−0.428152	−0.0204114
$441$	−2.53485	−0.120707
$442$	−10.6173	−0.505015
$443$	−23.6870	−1.12540	−0.562702	−	0.826660i	$-0.690238\pi$
$443$	−23.6870	−1.12540	−0.562702	+	0.826660i	$0.690238\pi$
$444$	−5.57185	−0.264428
$445$	1.76103	0.0834808
$446$	−13.3827	−0.633688
$447$	12.9545	0.612727
$448$	−2.11309	−0.0998341
$449$	−25.4069	−1.19902	−0.599512	−	0.800366i	$-0.704639\pi$
$449$	−25.4069	−1.19902	−0.599512	+	0.800366i	$0.704639\pi$
$450$	4.57185	0.215519
$451$	7.57185	0.356544
$452$	19.3699	0.911082
$453$	−4.00000	−0.187936
$454$	12.3393	0.579111
$455$	3.67006	0.172055
$456$	6.11309	0.286272
$457$	−29.1437	−1.36328	−0.681642	−	0.731686i	$-0.738734\pi$
$457$	−29.1437	−1.36328	−0.681642	+	0.731686i	$0.738734\pi$
$458$	−6.58033	−0.307479
$459$	−4.00000	−0.186704
$460$	−0.654334	−0.0305085
$461$	−0.110998	−0.00516971	−0.00258486	−	0.999997i	$-0.500823\pi$
$461$	−0.110998	−0.00516971	−0.00258486	+	0.999997i	$0.500823\pi$
$462$	1.38267	0.0643274
$463$	−14.2874	−0.663991	−0.331996	−	0.943281i	$-0.607722\pi$
$463$	−14.2874	−0.663991	−0.331996	+	0.943281i	$0.607722\pi$
$464$	−1.00000	−0.0464238
$465$	1.73682	0.0805431
$466$	−7.08249	−0.328090
$467$	4.80652	0.222419	0.111210	−	0.993797i	$-0.464528\pi$
$467$	4.80652	0.222419	0.111210	+	0.993797i	$0.464528\pi$
$468$	2.65433	0.122697
$469$	5.60885	0.258992
$470$	−1.76103	−0.0812302
$471$	−7.83503	−0.361019
$472$	−7.34567	−0.338112
$473$	2.23897	0.102948
$474$	6.22618	0.285978
$475$	−27.9481	−1.28235
$476$	−8.45236	−0.387413
$477$	−1.30867	−0.0599197
$478$	−6.95451	−0.318092
$479$	17.5200	0.800508	0.400254	−	0.916404i	$-0.368922\pi$
$479$	17.5200	0.800508	0.400254	+	0.916404i	$0.368922\pi$
$480$	−0.654334	−0.0298661
$481$	14.7895	0.674345
$482$	−15.5348	−0.707593
$483$	2.11309	0.0961490
$484$	−10.5718	−0.480539
$485$	−8.37836	−0.380442
$486$	1.00000	0.0453609
$487$	26.2874	1.19120	0.595598	−	0.803283i	$-0.296915\pi$
$487$	26.2874	1.19120	0.595598	+	0.803283i	$0.296915\pi$
$488$	2.65433	0.120156
$489$	−15.7980	−0.714412
$490$	−1.65864	−0.0749295
$491$	−12.6173	−0.569412	−0.284706	−	0.958615i	$-0.591896\pi$
$491$	−12.6173	−0.569412	−0.284706	+	0.958615i	$0.591896\pi$
$492$	11.5718	0.521699
$493$	−4.00000	−0.180151
$494$	−16.2262	−0.730050
$495$	0.428152	0.0192440
$496$	2.65433	0.119183
$497$	25.9132	1.16237
$498$	12.3393	0.552936
$499$	15.4608	0.692123	0.346061	−	0.938212i	$-0.387519\pi$
$499$	15.4608	0.692123	0.346061	+	0.938212i	$0.387519\pi$
$500$	6.26318	0.280098
$501$	−10.1892	−0.455219
$502$	−1.96300	−0.0876130
$503$	14.5782	0.650012	0.325006	−	0.945712i	$-0.394634\pi$
$503$	14.5782	0.650012	0.325006	+	0.945712i	$0.394634\pi$
$504$	2.11309	0.0941245
$505$	3.79384	0.168824
$506$	0.654334	0.0290887
$507$	5.95451	0.264449
$508$	11.5718	0.513418
$509$	28.5264	1.26441	0.632204	−	0.774802i	$-0.282150\pi$
$509$	28.5264	1.26441	0.632204	+	0.774802i	$0.282150\pi$
$510$	−2.61733	−0.115897
$511$	18.8435	0.833588
$512$	−1.00000	−0.0441942
$513$	−6.11309	−0.269900
$514$	−30.6785	−1.35317
$515$	−6.18918	−0.272728
$516$	3.42176	0.150634
$517$	1.76103	0.0774500
$518$	11.7738	0.517312
$519$	6.69133	0.293717
$520$	1.73682	0.0761646
$521$	−3.66073	−0.160379	−0.0801897	−	0.996780i	$-0.525553\pi$
$521$	−3.66073	−0.160379	−0.0801897	+	0.996780i	$0.525553\pi$
$522$	1.00000	0.0437688
$523$	11.1807	0.488898	0.244449	−	0.969662i	$-0.421393\pi$
$523$	11.1807	0.488898	0.244449	+	0.969662i	$0.421393\pi$
$524$	15.7610	0.688524
$525$	−9.66073	−0.421629
$526$	11.2568	0.490819
$527$	10.6173	0.462498
$528$	0.654334	0.0284762
$529$	1.00000	0.0434783
$530$	−0.856305	−0.0371955
$531$	7.34567	0.318775
$532$	−12.9175	−0.560045
$533$	−30.7155	−1.33044
$534$	−2.69133	−0.116465
$535$	4.45236	0.192492
$536$	2.65433	0.114650
$537$	−7.77382	−0.335465
$538$	29.8720	1.28787
$539$	1.65864	0.0714425
$540$	0.654334	0.0281580
$541$	25.5348	1.09783	0.548914	−	0.835879i	$-0.315041\pi$
$541$	25.5348	1.09783	0.548914	+	0.835879i	$0.315041\pi$
$542$	5.73682	0.246417
$543$	−0.804424	−0.0345211
$544$	−4.00000	−0.171499
$545$	−11.1437	−0.477343
$546$	−5.60885	−0.240036
$547$	0.317154	0.0135605	0.00678026	−	0.999977i	$-0.497842\pi$
$547$	0.317154	0.0135605	0.00678026	+	0.999977i	$0.497842\pi$
$548$	−13.6088	−0.581341
$549$	−2.65433	−0.113284
$550$	−2.99151	−0.127559
$551$	−6.11309	−0.260426
$552$	1.00000	0.0425628
$553$	−13.1565	−0.559471
$554$	11.1807	0.475022
$555$	3.64585	0.154758
$556$	14.2262	0.603324
$557$	26.9417	1.14156	0.570779	−	0.821104i	$-0.306641\pi$
$557$	26.9417	1.14156	0.570779	+	0.821104i	$0.306641\pi$
$558$	−2.65433	−0.112367
$559$	−9.08249	−0.384148
$560$	1.38267	0.0584283
$561$	2.61733	0.110504
$562$	6.50424	0.274365
$563$	−41.6331	−1.75462	−0.877312	−	0.479920i	$-0.840666\pi$
$563$	−41.6331	−1.75462	−0.877312	+	0.479920i	$0.840666\pi$
$564$	2.69133	0.113326
$565$	−12.6744	−0.533214
$566$	−3.51064	−0.147563
$567$	−2.11309	−0.0887415
$568$	12.2632	0.514552
$569$	10.0612	0.421788	0.210894	−	0.977509i	$-0.432363\pi$
$569$	10.0612	0.421788	0.210894	+	0.977509i	$0.432363\pi$
$570$	−4.00000	−0.167542
$571$	−12.7895	−0.535226	−0.267613	−	0.963527i	$-0.586235\pi$
$571$	−12.7895	−0.535226	−0.267613	+	0.963527i	$0.586235\pi$
$572$	−1.73682	−0.0726201
$573$	6.07609	0.253832
$574$	−24.4524	−1.02062
$575$	−4.57185	−0.190659
$576$	1.00000	0.0416667
$577$	12.6913	0.528347	0.264174	−	0.964475i	$-0.414901\pi$
$577$	12.6913	0.528347	0.264174	+	0.964475i	$0.414901\pi$
$578$	1.00000	0.0415945
$579$	0.691333	0.0287308
$580$	0.654334	0.0271697
$581$	−26.0740	−1.08173
$582$	12.8044	0.530760
$583$	0.856305	0.0354645
$584$	8.91751	0.369009
$585$	−1.73682	−0.0718086
$586$	−13.6479	−0.563791
$587$	−30.6173	−1.26371	−0.631856	−	0.775086i	$-0.717707\pi$
$587$	−30.6173	−1.26371	−0.631856	+	0.775086i	$0.717707\pi$
$588$	2.53485	0.104535
$589$	16.2262	0.668588
$590$	4.80652	0.197881
$591$	15.8350	0.651366
$592$	5.57185	0.229002
$593$	6.52636	0.268006	0.134003	−	0.990981i	$-0.457217\pi$
$593$	6.52636	0.268006	0.134003	+	0.990981i	$0.457217\pi$
$594$	−0.654334	−0.0268476
$595$	5.53066	0.226735
$596$	−12.9545	−0.530638
$597$	−1.23258	−0.0504460
$598$	−2.65433	−0.108544
$599$	15.4439	0.631020	0.315510	−	0.948922i	$-0.397824\pi$
$599$	15.4439	0.631020	0.315510	+	0.948922i	$0.397824\pi$
$600$	−4.57185	−0.186645
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	−7.23048	−0.294693
$603$	−2.65433	−0.108093
$604$	4.00000	0.162758
$605$	6.91751	0.281237
$606$	−5.79803	−0.235529
$607$	−27.5348	−1.11761	−0.558803	−	0.829301i	$-0.688739\pi$
$607$	−27.5348	−1.11761	−0.558803	+	0.829301i	$0.688739\pi$
$608$	−6.11309	−0.247919
$609$	−2.11309	−0.0856268
$610$	−1.73682	−0.0703218
$611$	−7.14370	−0.289003
$612$	4.00000	0.161690
$613$	42.4133	1.71306	0.856528	−	0.516101i	$-0.172617\pi$
$613$	42.4133	1.71306	0.856528	+	0.516101i	$0.172617\pi$
$614$	−13.1067	−0.528943
$615$	−7.57185	−0.305326
$616$	−1.38267	−0.0557092
$617$	16.3002	0.656221	0.328110	−	0.944639i	$-0.393588\pi$
$617$	16.3002	0.656221	0.328110	+	0.944639i	$0.393588\pi$
$618$	9.45876	0.380487
$619$	−21.2568	−0.854382	−0.427191	−	0.904161i	$-0.640497\pi$
$619$	−21.2568	−0.854382	−0.427191	+	0.904161i	$0.640497\pi$
$620$	−1.73682	−0.0697523
$621$	−1.00000	−0.0401286
$622$	10.9175	0.437752
$623$	5.68703	0.227846
$624$	−2.65433	−0.106258
$625$	18.7610	0.750441
$626$	−14.0000	−0.559553
$627$	4.00000	0.159745
$628$	7.83503	0.312652
$629$	22.2874	0.888656
$630$	−1.38267	−0.0550867
$631$	13.5328	0.538731	0.269365	−	0.963038i	$-0.413186\pi$
$631$	13.5328	0.538731	0.269365	+	0.963038i	$0.413186\pi$
$632$	−6.22618	−0.247664
$633$	−4.65433	−0.184993
$634$	−25.8720	−1.02751
$635$	−7.57185	−0.300480
$636$	1.30867	0.0518920
$637$	−6.72833	−0.266586
$638$	−0.654334	−0.0259053
$639$	−12.2632	−0.485124
$640$	0.654334	0.0258648
$641$	−13.6088	−0.537517	−0.268759	−	0.963208i	$-0.586613\pi$
$641$	−13.6088	−0.537517	−0.268759	+	0.963208i	$0.586613\pi$
$642$	−6.80442	−0.268549
$643$	−28.1352	−1.10954	−0.554772	−	0.832002i	$-0.687195\pi$
$643$	−28.1352	−1.10954	−0.554772	+	0.832002i	$0.687195\pi$
$644$	−2.11309	−0.0832674
$645$	−2.23897	−0.0881594
$646$	−24.4524	−0.962065
$647$	35.8592	1.40977	0.704886	−	0.709321i	$-0.250998\pi$
$647$	35.8592	1.40977	0.704886	+	0.709321i	$0.250998\pi$
$648$	−1.00000	−0.0392837
$649$	−4.80652	−0.188672
$650$	12.1352	0.475982
$651$	5.60885	0.219828
$652$	15.7980	0.618699
$653$	20.6415	0.807766	0.403883	−	0.914811i	$-0.367660\pi$
$653$	20.6415	0.807766	0.403883	+	0.914811i	$0.367660\pi$
$654$	17.0306	0.665949
$655$	−10.3130	−0.402961
$656$	−11.5718	−0.451805
$657$	−8.91751	−0.347905
$658$	−5.68703	−0.221704
$659$	−30.2874	−1.17983	−0.589915	−	0.807466i	$-0.700839\pi$
$659$	−30.2874	−1.17983	−0.589915	+	0.807466i	$0.700839\pi$
$660$	−0.428152	−0.0166658
$661$	−2.56545	−0.0997846	−0.0498923	−	0.998755i	$-0.515888\pi$
$661$	−2.56545	−0.0997846	−0.0498923	+	0.998755i	$0.515888\pi$
$662$	6.84352	0.265981
$663$	−10.6173	−0.412343
$664$	−12.3393	−0.478857
$665$	8.45236	0.327769
$666$	−5.57185	−0.215905
$667$	−1.00000	−0.0387202
$668$	10.1892	0.394231
$669$	−13.3827	−0.517404
$670$	−1.73682	−0.0670992
$671$	1.73682	0.0670492
$672$	−2.11309	−0.0815142
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−11.9239	−0.459292
$675$	4.57185	0.175970
$676$	−5.95451	−0.229020
$677$	44.9438	1.72733	0.863666	−	0.504065i	$-0.168163\pi$
$677$	44.9438	1.72733	0.863666	+	0.504065i	$0.168163\pi$
$678$	19.3699	0.743896
$679$	−27.0569	−1.03835
$680$	2.61733	0.100370
$681$	12.3393	0.472842
$682$	1.73682	0.0665063
$683$	35.5718	1.36112	0.680559	−	0.732693i	$-0.261737\pi$
$683$	35.5718	1.36112	0.680559	+	0.732693i	$0.261737\pi$
$684$	6.11309	0.233740
$685$	8.90472	0.340232
$686$	−20.1480	−0.769254
$687$	−6.58033	−0.251055
$688$	−3.42176	−0.130453
$689$	−3.47364	−0.132335
$690$	−0.654334	−0.0249101
$691$	−7.59606	−0.288968	−0.144484	−	0.989507i	$-0.546152\pi$
$691$	−7.59606	−0.288968	−0.144484	+	0.989507i	$0.546152\pi$
$692$	−6.69133	−0.254366
$693$	1.38267	0.0525231
$694$	−35.0697	−1.33123
$695$	−9.30867	−0.353098
$696$	−1.00000	−0.0379049
$697$	−46.2874	−1.75326
$698$	−31.4069	−1.18877
$699$	−7.08249	−0.267884
$700$	9.66073	0.365141
$701$	2.48936	0.0940219	0.0470109	−	0.998894i	$-0.485030\pi$
$701$	2.48936	0.0940219	0.0470109	+	0.998894i	$0.485030\pi$
$702$	2.65433	0.100181
$703$	34.0612	1.28464
$704$	−0.654334	−0.0246611
$705$	−1.76103	−0.0663242
$706$	−2.00000	−0.0752710
$707$	12.2518	0.460775
$708$	−7.34567	−0.276067
$709$	−50.4617	−1.89513	−0.947564	−	0.319565i	$-0.896463\pi$
$709$	−50.4617	−1.89513	−0.947564	+	0.319565i	$0.896463\pi$
$710$	−8.02421	−0.301143
$711$	6.22618	0.233500
$712$	2.69133	0.100862
$713$	2.65433	0.0994056
$714$	−8.45236	−0.316322
$715$	1.13646	0.0425012
$716$	7.77382	0.290521
$717$	−6.95451	−0.259721
$718$	−12.1871	−0.454818
$719$	35.2419	1.31430	0.657151	−	0.753759i	$-0.271762\pi$
$719$	35.2419	1.31430	0.657151	+	0.753759i	$0.271762\pi$
$720$	−0.654334	−0.0243856
$721$	−19.9872	−0.744363
$722$	−18.3699	−0.683656
$723$	−15.5348	−0.577747
$724$	0.804424	0.0298962
$725$	4.57185	0.169794
$726$	−10.5718	−0.392358
$727$	24.8435	0.921395	0.460697	−	0.887557i	$-0.347599\pi$
$727$	24.8435	0.921395	0.460697	+	0.887557i	$0.347599\pi$
$728$	5.60885	0.207878
$729$	1.00000	0.0370370
$730$	−5.83503	−0.215964
$731$	−13.6870	−0.506233
$732$	2.65433	0.0981070
$733$	−46.4026	−1.71392	−0.856959	−	0.515384i	$-0.827649\pi$
$733$	−46.4026	−1.71392	−0.856959	+	0.515384i	$0.827649\pi$
$734$	−30.2262	−1.11567
$735$	−1.65864	−0.0611797
$736$	−1.00000	−0.0368605
$737$	1.73682	0.0639766
$738$	11.5718	0.425966
$739$	21.3329	0.784743	0.392371	−	0.919807i	$-0.371655\pi$
$739$	21.3329	0.784743	0.392371	+	0.919807i	$0.371655\pi$
$740$	−3.64585	−0.134024
$741$	−16.2262	−0.596084
$742$	−2.76533	−0.101519
$743$	−29.2938	−1.07468	−0.537342	−	0.843364i	$-0.680572\pi$
$743$	−29.2938	−1.07468	−0.537342	+	0.843364i	$0.680572\pi$
$744$	2.65433	0.0973126
$745$	8.47657	0.310558
$746$	−2.03909	−0.0746564
$747$	12.3393	0.451470
$748$	−2.61733	−0.0956992
$749$	14.3784	0.525374
$750$	6.26318	0.228699
$751$	24.9175	0.909253	0.454627	−	0.890682i	$-0.349773\pi$
$751$	24.9175	0.909253	0.454627	+	0.890682i	$0.349773\pi$
$752$	−2.69133	−0.0981428
$753$	−1.96300	−0.0715357
$754$	2.65433	0.0966651
$755$	−2.61733	−0.0952545
$756$	2.11309	0.0768524
$757$	−29.7240	−1.08034	−0.540169	−	0.841556i	$-0.681640\pi$
$757$	−29.7240	−1.08034	−0.540169	+	0.841556i	$0.681640\pi$
$758$	−23.6479	−0.858932
$759$	0.654334	0.0237508
$760$	4.00000	0.145095
$761$	53.5221	1.94017	0.970087	−	0.242759i	$-0.0780525\pi$
$761$	53.5221	1.94017	0.970087	+	0.242759i	$0.0780525\pi$
$762$	11.5718	0.419204
$763$	−35.9872	−1.30282
$764$	−6.07609	−0.219825
$765$	−2.61733	−0.0946299
$766$	−8.85630	−0.319991
$767$	19.4978	0.704027
$768$	−1.00000	−0.0360844
$769$	−43.9723	−1.58568	−0.792841	−	0.609428i	$-0.791399\pi$
$769$	−43.9723	−1.58568	−0.792841	+	0.609428i	$0.791399\pi$
$770$	0.904725	0.0326040
$771$	−30.6785	−1.10486
$772$	−0.691333	−0.0248816
$773$	−49.5442	−1.78198	−0.890990	−	0.454023i	$-0.849988\pi$
$773$	−49.5442	−1.78198	−0.890990	+	0.454023i	$0.849988\pi$
$774$	3.42176	0.122993
$775$	−12.1352	−0.435910
$776$	−12.8044	−0.459652
$777$	11.7738	0.422383
$778$	16.4133	0.588444
$779$	−70.7398	−2.53451
$780$	1.73682	0.0621881
$781$	8.02421	0.287129
$782$	−4.00000	−0.143040
$783$	1.00000	0.0357371
$784$	−2.53485	−0.0905303
$785$	−5.12672	−0.182981
$786$	15.7610	0.562177
$787$	8.81931	0.314374	0.157187	−	0.987569i	$-0.449757\pi$
$787$	8.81931	0.314374	0.157187	+	0.987569i	$0.449757\pi$
$788$	−15.8350	−0.564100
$789$	11.2568	0.400752
$790$	4.07400	0.144946
$791$	−40.9303	−1.45531
$792$	0.654334	0.0232507
$793$	−7.04549	−0.250193
$794$	10.0000	0.354887
$795$	−0.856305	−0.0303700
$796$	1.23258	0.0436875
$797$	−35.1046	−1.24347	−0.621734	−	0.783228i	$-0.713572\pi$
$797$	−35.1046	−1.24347	−0.621734	+	0.783228i	$0.713572\pi$
$798$	−12.9175	−0.457275
$799$	−10.7653	−0.380850
$800$	4.57185	0.161639
$801$	−2.69133	−0.0950936
$802$	−14.0761	−0.497044
$803$	5.83503	0.205914
$804$	2.65433	0.0936111
$805$	1.38267	0.0487326
$806$	−7.04549	−0.248167
$807$	29.8720	1.05155
$808$	5.79803	0.203974
$809$	−1.81082	−0.0636650	−0.0318325	−	0.999493i	$-0.510134\pi$
$809$	−1.81082	−0.0636650	−0.0318325	+	0.999493i	$0.510134\pi$
$810$	0.654334	0.0229909
$811$	−52.5876	−1.84660	−0.923300	−	0.384080i	$-0.874519\pi$
$811$	−52.5876	−1.84660	−0.923300	+	0.384080i	$0.874519\pi$
$812$	2.11309	0.0741549
$813$	5.73682	0.201199
$814$	3.64585	0.127787
$815$	−10.3372	−0.362096
$816$	−4.00000	−0.140028
$817$	−20.9175	−0.731811
$818$	−9.21769	−0.322289
$819$	−5.60885	−0.195989
$820$	7.57185	0.264420
$821$	−9.83503	−0.343245	−0.171622	−	0.985163i	$-0.554901\pi$
$821$	−9.83503	−0.343245	−0.171622	+	0.985163i	$0.554901\pi$
$822$	−13.6088	−0.474663
$823$	10.9715	0.382442	0.191221	−	0.981547i	$-0.438755\pi$
$823$	10.9715	0.382442	0.191221	+	0.981547i	$0.438755\pi$
$824$	−9.45876	−0.329511
$825$	−2.99151	−0.104151
$826$	15.5221	0.540081
$827$	−16.8065	−0.584420	−0.292210	−	0.956354i	$-0.594391\pi$
$827$	−16.8065	−0.584420	−0.292210	+	0.956354i	$0.594391\pi$
$828$	1.00000	0.0347524
$829$	18.3784	0.638307	0.319153	−	0.947703i	$-0.396601\pi$
$829$	18.3784	0.638307	0.319153	+	0.947703i	$0.396601\pi$
$830$	8.07400	0.280253
$831$	11.1807	0.387854
$832$	2.65433	0.0920225
$833$	−10.1394	−0.351309
$834$	14.2262	0.492612
$835$	−6.66712	−0.230725
$836$	−4.00000	−0.138343
$837$	−2.65433	−0.0917472
$838$	17.4958	0.604381
$839$	−3.61094	−0.124663	−0.0623317	−	0.998055i	$-0.519854\pi$
$839$	−3.61094	−0.124663	−0.0623317	+	0.998055i	$0.519854\pi$
$840$	1.38267	0.0477065
$841$	1.00000	0.0344828
$842$	−5.87203	−0.202363
$843$	6.50424	0.224018
$844$	4.65433	0.160209
$845$	3.89624	0.134035
$846$	2.69133	0.0925299
$847$	22.3393	0.767586
$848$	−1.30867	−0.0449398
$849$	−3.51064	−0.120485
$850$	18.2874	0.627252
$851$	5.57185	0.191000
$852$	12.2632	0.420130
$853$	−16.2390	−0.556012	−0.278006	−	0.960579i	$-0.589673\pi$
$853$	−16.2390	−0.556012	−0.278006	+	0.960579i	$0.589673\pi$
$854$	−5.60885	−0.191931
$855$	−4.00000	−0.136797
$856$	6.80442	0.232570
$857$	32.0612	1.09519	0.547595	−	0.836744i	$-0.315543\pi$
$857$	32.0612	1.09519	0.547595	+	0.836744i	$0.315543\pi$
$858$	−1.73682	−0.0592940
$859$	22.7398	0.775870	0.387935	−	0.921687i	$-0.373188\pi$
$859$	22.7398	0.775870	0.387935	+	0.921687i	$0.373188\pi$
$860$	2.23897	0.0763483
$861$	−24.4524	−0.833334
$862$	0.300180	0.0102242
$863$	10.6415	0.362242	0.181121	−	0.983461i	$-0.442027\pi$
$863$	10.6415	0.362242	0.181121	+	0.983461i	$0.442027\pi$
$864$	1.00000	0.0340207
$865$	4.37836	0.148869
$866$	5.25679	0.178633
$867$	1.00000	0.0339618
$868$	−5.60885	−0.190377
$869$	−4.07400	−0.138201
$870$	0.654334	0.0221840
$871$	−7.04549	−0.238727
$872$	−17.0306	−0.576729
$873$	12.8044	0.433364
$874$	−6.11309	−0.206778
$875$	−13.2347	−0.447413
$876$	8.91751	0.301295
$877$	−0.0612095	−0.00206690	−0.00103345	−	0.999999i	$-0.500329\pi$
$877$	−0.0612095	−0.00206690	−0.00103345	+	0.999999i	$0.500329\pi$
$878$	0.526361	0.0177638
$879$	−13.6479	−0.460334
$880$	0.428152	0.0144330
$881$	−10.9957	−0.370454	−0.185227	−	0.982696i	$-0.559302\pi$
$881$	−10.9957	−0.370454	−0.185227	+	0.982696i	$0.559302\pi$
$882$	2.53485	0.0853528
$883$	41.5051	1.39676	0.698379	−	0.715728i	$-0.253905\pi$
$883$	41.5051	1.39676	0.698379	+	0.715728i	$0.253905\pi$
$884$	10.6173	0.357100
$885$	4.80652	0.161569
$886$	23.6870	0.795781
$887$	13.1565	0.441752	0.220876	−	0.975302i	$-0.429108\pi$
$887$	13.1565	0.441752	0.220876	+	0.975302i	$0.429108\pi$
$888$	5.57185	0.186979
$889$	−24.4524	−0.820106
$890$	−1.76103	−0.0590298
$891$	−0.654334	−0.0219210
$892$	13.3827	0.448085
$893$	−16.4524	−0.550557
$894$	−12.9545	−0.433264
$895$	−5.08667	−0.170029
$896$	2.11309	0.0705934
$897$	−2.65433	−0.0886256
$898$	25.4069	0.847838
$899$	−2.65433	−0.0885270
$900$	−4.57185	−0.152395
$901$	−5.23467	−0.174392
$902$	−7.57185	−0.252115
$903$	−7.23048	−0.240615
$904$	−19.3699	−0.644233
$905$	−0.526361	−0.0174968
$906$	4.00000	0.132891
$907$	44.2483	1.46924	0.734620	−	0.678478i	$-0.237360\pi$
$907$	44.2483	1.46924	0.734620	+	0.678478i	$0.237360\pi$
$908$	−12.3393	−0.409493
$909$	−5.79803	−0.192308
$910$	−3.67006	−0.121661
$911$	−37.3720	−1.23819	−0.619094	−	0.785317i	$-0.712500\pi$
$911$	−37.3720	−1.23819	−0.619094	+	0.785317i	$0.712500\pi$
$912$	−6.11309	−0.202425
$913$	−8.07400	−0.267210
$914$	29.1437	0.963988
$915$	−1.73682	−0.0574175
$916$	6.58033	0.217420
$917$	−33.3045	−1.09981
$918$	4.00000	0.132020
$919$	−14.5412	−0.479671	−0.239836	−	0.970813i	$-0.577094\pi$
$919$	−14.5412	−0.479671	−0.239836	+	0.970813i	$0.577094\pi$
$920$	0.654334	0.0215727
$921$	−13.1067	−0.431880
$922$	0.110998	0.00365554
$923$	−32.5506	−1.07142
$924$	−1.38267	−0.0454864
$925$	−25.4736	−0.837568
$926$	14.2874	0.469513
$927$	9.45876	0.310666
$928$	1.00000	0.0328266
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	−1.73682	−0.0569526
$931$	−15.4958	−0.507853
$932$	7.08249	0.231995
$933$	10.9175	0.357423
$934$	−4.80652	−0.157274
$935$	1.71261	0.0560083
$936$	−2.65433	−0.0867596
$937$	−31.6870	−1.03517	−0.517585	−	0.855632i	$-0.673169\pi$
$937$	−31.6870	−1.03517	−0.517585	+	0.855632i	$0.673169\pi$
$938$	−5.60885	−0.183135
$939$	−14.0000	−0.456873
$940$	1.76103	0.0574384
$941$	37.7811	1.23163	0.615814	−	0.787892i	$-0.288827\pi$
$941$	37.7811	1.23163	0.615814	+	0.787892i	$0.288827\pi$
$942$	7.83503	0.255279
$943$	−11.5718	−0.376831
$944$	7.34567	0.239081
$945$	−1.38267	−0.0449781
$946$	−2.23897	−0.0727952
$947$	20.5433	0.667569	0.333784	−	0.942649i	$-0.391674\pi$
$947$	20.5433	0.667569	0.333784	+	0.942649i	$0.391674\pi$
$948$	−6.22618	−0.202217
$949$	−23.6701	−0.768363
$950$	27.9481	0.906757
$951$	−25.8720	−0.838958
$952$	8.45236	0.273943
$953$	−5.82570	−0.188713	−0.0943565	−	0.995538i	$-0.530079\pi$
$953$	−5.82570	−0.188713	−0.0943565	+	0.995538i	$0.530079\pi$
$954$	1.30867	0.0423697
$955$	3.97579	0.128654
$956$	6.95451	0.224925
$957$	−0.654334	−0.0211516
$958$	−17.5200	−0.566044
$959$	28.7567	0.928603
$960$	0.654334	0.0211185
$961$	−23.9545	−0.772726
$962$	−14.7895	−0.476834
$963$	−6.80442	−0.219269
$964$	15.5348	0.500344
$965$	0.452362	0.0145621
$966$	−2.11309	−0.0679876
$967$	−42.7398	−1.37442	−0.687209	−	0.726460i	$-0.741164\pi$
$967$	−42.7398	−1.37442	−0.687209	+	0.726460i	$0.741164\pi$
$968$	10.5718	0.339792
$969$	−24.4524	−0.785523
$970$	8.37836	0.269013
$971$	−25.0327	−0.803337	−0.401669	−	0.915785i	$-0.631570\pi$
$971$	−25.0327	−0.803337	−0.401669	+	0.915785i	$0.631570\pi$
$972$	−1.00000	−0.0320750
$973$	−30.0612	−0.963718
$974$	−26.2874	−0.842302
$975$	12.1352	0.388638
$976$	−2.65433	−0.0849631
$977$	−33.5940	−1.07477	−0.537383	−	0.843338i	$-0.680587\pi$
$977$	−33.5940	−1.07477	−0.537383	+	0.843338i	$0.680587\pi$
$978$	15.7980	0.505165
$979$	1.76103	0.0562827
$980$	1.65864	0.0529832
$981$	17.0306	0.543745
$982$	12.6173	0.402635
$983$	46.3763	1.47917	0.739587	−	0.673061i	$-0.235021\pi$
$983$	46.3763	1.47917	0.739587	+	0.673061i	$0.235021\pi$
$984$	−11.5718	−0.368897
$985$	10.3614	0.330141
$986$	4.00000	0.127386
$987$	−5.68703	−0.181020
$988$	16.2262	0.516224
$989$	−3.42176	−0.108806
$990$	−0.428152	−0.0136076
$991$	1.00849	0.0320356	0.0160178	−	0.999872i	$-0.494901\pi$
$991$	1.00849	0.0320356	0.0160178	+	0.999872i	$0.494901\pi$
$992$	−2.65433	−0.0842752
$993$	6.84352	0.217172
$994$	−25.9132	−0.821917
$995$	−0.806516	−0.0255683
$996$	−12.3393	−0.390985
$997$	−19.2347	−0.609168	−0.304584	−	0.952485i	$-0.598517\pi$
$997$	−19.2347	−0.609168	−0.304584	+	0.952485i	$0.598517\pi$
$998$	−15.4608	−0.489405
$999$	−5.57185	−0.176285

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.y.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.y.1.2	✓	3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.654334 q^{5} +1.00000 q^{6} -2.11309 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.654334 q^{5} +1.00000 q^{6} -2.11309 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.654334 q^{10} -0.654334 q^{11} -1.00000 q^{12} +2.65433 q^{13} +2.11309 q^{14} +0.654334 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +6.11309 q^{19} -0.654334 q^{20} +2.11309 q^{21} +0.654334 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.57185 q^{25} -2.65433 q^{26} -1.00000 q^{27} -2.11309 q^{28} -1.00000 q^{29} -0.654334 q^{30} +2.65433 q^{31} -1.00000 q^{32} +0.654334 q^{33} -4.00000 q^{34} +1.38267 q^{35} +1.00000 q^{36} +5.57185 q^{37} -6.11309 q^{38} -2.65433 q^{39} +0.654334 q^{40} -11.5718 q^{41} -2.11309 q^{42} -3.42176 q^{43} -0.654334 q^{44} -0.654334 q^{45} -1.00000 q^{46} -2.69133 q^{47} -1.00000 q^{48} -2.53485 q^{49} +4.57185 q^{50} -4.00000 q^{51} +2.65433 q^{52} -1.30867 q^{53} +1.00000 q^{54} +0.428152 q^{55} +2.11309 q^{56} -6.11309 q^{57} +1.00000 q^{58} +7.34567 q^{59} +0.654334 q^{60} -2.65433 q^{61} -2.65433 q^{62} -2.11309 q^{63} +1.00000 q^{64} -1.73682 q^{65} -0.654334 q^{66} -2.65433 q^{67} +4.00000 q^{68} -1.00000 q^{69} -1.38267 q^{70} -12.2632 q^{71} -1.00000 q^{72} -8.91751 q^{73} -5.57185 q^{74} +4.57185 q^{75} +6.11309 q^{76} +1.38267 q^{77} +2.65433 q^{78} +6.22618 q^{79} -0.654334 q^{80} +1.00000 q^{81} +11.5718 q^{82} +12.3393 q^{83} +2.11309 q^{84} -2.61733 q^{85} +3.42176 q^{86} +1.00000 q^{87} +0.654334 q^{88} -2.69133 q^{89} +0.654334 q^{90} -5.60885 q^{91} +1.00000 q^{92} -2.65433 q^{93} +2.69133 q^{94} -4.00000 q^{95} +1.00000 q^{96} +12.8044 q^{97} +2.53485 q^{98} -0.654334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - q^5 + 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 6 q^{7} - 3 q^{8} + 3 q^{9} + q^{10} - q^{11} - 3 q^{12} + 7 q^{13} - 6 q^{14} + q^{15} + 3 q^{16} + 12 q^{17} - 3 q^{18} + 6 q^{19} - q^{20} - 6 q^{21} + q^{22} + 3 q^{23} + 3 q^{24} + 10 q^{25} - 7 q^{26} - 3 q^{27} + 6 q^{28} - 3 q^{29} - q^{30} + 7 q^{31} - 3 q^{32} + q^{33} - 12 q^{34} + 8 q^{35} + 3 q^{36} - 7 q^{37} - 6 q^{38} - 7 q^{39} + q^{40} - 11 q^{41} + 6 q^{42} + 4 q^{43} - q^{44} - q^{45} - 3 q^{46} - 10 q^{47} - 3 q^{48} + 19 q^{49} - 10 q^{50} - 12 q^{51} + 7 q^{52} - 2 q^{53} + 3 q^{54} + 25 q^{55} - 6 q^{56} - 6 q^{57} + 3 q^{58} + 23 q^{59} + q^{60} - 7 q^{61} - 7 q^{62} + 6 q^{63} + 3 q^{64} - 27 q^{65} - q^{66} - 7 q^{67} + 12 q^{68} - 3 q^{69} - 8 q^{70} - 15 q^{71} - 3 q^{72} - 4 q^{73} + 7 q^{74} - 10 q^{75} + 6 q^{76} + 8 q^{77} + 7 q^{78} - 6 q^{79} - q^{80} + 3 q^{81} + 11 q^{82} - 6 q^{84} - 4 q^{85} - 4 q^{86} + 3 q^{87} + q^{88} - 10 q^{89} + q^{90} + 4 q^{91} + 3 q^{92} - 7 q^{93} + 10 q^{94} - 12 q^{95} + 3 q^{96} + 28 q^{97} - 19 q^{98} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 - q^5 + 3 * q^6 + 6 * q^7 - 3 * q^8 + 3 * q^9 + q^10 - q^11 - 3 * q^12 + 7 * q^13 - 6 * q^14 + q^15 + 3 * q^16 + 12 * q^17 - 3 * q^18 + 6 * q^19 - q^20 - 6 * q^21 + q^22 + 3 * q^23 + 3 * q^24 + 10 * q^25 - 7 * q^26 - 3 * q^27 + 6 * q^28 - 3 * q^29 - q^30 + 7 * q^31 - 3 * q^32 + q^33 - 12 * q^34 + 8 * q^35 + 3 * q^36 - 7 * q^37 - 6 * q^38 - 7 * q^39 + q^40 - 11 * q^41 + 6 * q^42 + 4 * q^43 - q^44 - q^45 - 3 * q^46 - 10 * q^47 - 3 * q^48 + 19 * q^49 - 10 * q^50 - 12 * q^51 + 7 * q^52 - 2 * q^53 + 3 * q^54 + 25 * q^55 - 6 * q^56 - 6 * q^57 + 3 * q^58 + 23 * q^59 + q^60 - 7 * q^61 - 7 * q^62 + 6 * q^63 + 3 * q^64 - 27 * q^65 - q^66 - 7 * q^67 + 12 * q^68 - 3 * q^69 - 8 * q^70 - 15 * q^71 - 3 * q^72 - 4 * q^73 + 7 * q^74 - 10 * q^75 + 6 * q^76 + 8 * q^77 + 7 * q^78 - 6 * q^79 - q^80 + 3 * q^81 + 11 * q^82 - 6 * q^84 - 4 * q^85 - 4 * q^86 + 3 * q^87 + q^88 - 10 * q^89 + q^90 + 4 * q^91 + 3 * q^92 - 7 * q^93 + 10 * q^94 - 12 * q^95 + 3 * q^96 + 28 * q^97 - 19 * q^98 - q^99

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