LMFDB - Embedded newform 4002.2.a.bb.1.4

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			1.4
Root			$-0.634868$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.51539 q^{5} -1.00000 q^{6} -1.88053 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.51539 q^{5} -1.00000 q^{6} -1.88053 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.51539 q^{10} -3.63487 q^{11} +1.00000 q^{12} +2.59694 q^{13} +1.88053 q^{14} +1.51539 q^{15} +1.00000 q^{16} +1.88053 q^{17} -1.00000 q^{18} -5.07441 q^{19} +1.51539 q^{20} -1.88053 q^{21} +3.63487 q^{22} +1.00000 q^{23} -1.00000 q^{24} -2.70358 q^{25} -2.59694 q^{26} +1.00000 q^{27} -1.88053 q^{28} -1.00000 q^{29} -1.51539 q^{30} -5.32721 q^{31} -1.00000 q^{32} -3.63487 q^{33} -1.88053 q^{34} -2.84974 q^{35} +1.00000 q^{36} +5.01955 q^{37} +5.07441 q^{38} +2.59694 q^{39} -1.51539 q^{40} -8.89747 q^{41} +1.88053 q^{42} +10.8400 q^{43} -3.63487 q^{44} +1.51539 q^{45} -1.00000 q^{46} -3.80468 q^{47} +1.00000 q^{48} -3.46362 q^{49} +2.70358 q^{50} +1.88053 q^{51} +2.59694 q^{52} -0.495848 q^{53} -1.00000 q^{54} -5.50826 q^{55} +1.88053 q^{56} -5.07441 q^{57} +1.00000 q^{58} -7.86668 q^{59} +1.51539 q^{60} -2.36513 q^{61} +5.32721 q^{62} -1.88053 q^{63} +1.00000 q^{64} +3.93539 q^{65} +3.63487 q^{66} -7.89177 q^{67} +1.88053 q^{68} +1.00000 q^{69} +2.84974 q^{70} +9.92825 q^{71} -1.00000 q^{72} +2.18105 q^{73} -5.01955 q^{74} -2.70358 q^{75} -5.07441 q^{76} +6.83546 q^{77} -2.59694 q^{78} -1.58000 q^{79} +1.51539 q^{80} +1.00000 q^{81} +8.89747 q^{82} -10.4155 q^{83} -1.88053 q^{84} +2.84974 q^{85} -10.8400 q^{86} -1.00000 q^{87} +3.63487 q^{88} -1.30196 q^{89} -1.51539 q^{90} -4.88362 q^{91} +1.00000 q^{92} -5.32721 q^{93} +3.80468 q^{94} -7.68973 q^{95} -1.00000 q^{96} +2.69804 q^{97} +3.46362 q^{98} -3.63487 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.51539	0.677705	0.338852	−	0.940840i	$-0.389961\pi$
$5$	1.51539	0.677705	0.338852	+	0.940840i	$0.389961\pi$
$6$	−1.00000	−0.408248
$7$	−1.88053	−0.710772	−0.355386	−	0.934720i	$-0.615651\pi$
$7$	−1.88053	−0.710772	−0.355386	+	0.934720i	$0.615651\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.51539	−0.479210
$11$	−3.63487	−1.09595	−0.547977	−	0.836493i	$-0.684602\pi$
$11$	−3.63487	−1.09595	−0.547977	+	0.836493i	$0.684602\pi$
$12$	1.00000	0.288675
$13$	2.59694	0.720262	0.360131	−	0.932902i	$-0.382732\pi$
$13$	2.59694	0.720262	0.360131	+	0.932902i	$0.382732\pi$
$14$	1.88053	0.502592
$15$	1.51539	0.391273
$16$	1.00000	0.250000
$17$	1.88053	0.456095	0.228047	−	0.973650i	$-0.426766\pi$
$17$	1.88053	0.456095	0.228047	+	0.973650i	$0.426766\pi$
$18$	−1.00000	−0.235702
$19$	−5.07441	−1.16415	−0.582075	−	0.813135i	$-0.697759\pi$
$19$	−5.07441	−1.16415	−0.582075	+	0.813135i	$0.697759\pi$
$20$	1.51539	0.338852
$21$	−1.88053	−0.410364
$22$	3.63487	0.774956
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−2.70358	−0.540716
$26$	−2.59694	−0.509302
$27$	1.00000	0.192450
$28$	−1.88053	−0.355386
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−1.51539	−0.276672
$31$	−5.32721	−0.956795	−0.478397	−	0.878143i	$-0.658782\pi$
$31$	−5.32721	−0.956795	−0.478397	+	0.878143i	$0.658782\pi$
$32$	−1.00000	−0.176777
$33$	−3.63487	−0.632749
$34$	−1.88053	−0.322508
$35$	−2.84974	−0.481694
$36$	1.00000	0.166667
$37$	5.01955	0.825208	0.412604	−	0.910910i	$-0.364619\pi$
$37$	5.01955	0.825208	0.412604	+	0.910910i	$0.364619\pi$
$38$	5.07441	0.823178
$39$	2.59694	0.415844
$40$	−1.51539	−0.239605
$41$	−8.89747	−1.38955	−0.694775	−	0.719227i	$-0.744496\pi$
$41$	−8.89747	−1.38955	−0.694775	+	0.719227i	$0.744496\pi$
$42$	1.88053	0.290171
$43$	10.8400	1.65308	0.826542	−	0.562875i	$-0.190305\pi$
$43$	10.8400	1.65308	0.826542	+	0.562875i	$0.190305\pi$
$44$	−3.63487	−0.547977
$45$	1.51539	0.225902
$46$	−1.00000	−0.147442
$47$	−3.80468	−0.554969	−0.277485	−	0.960730i	$-0.589501\pi$
$47$	−3.80468	−0.554969	−0.277485	+	0.960730i	$0.589501\pi$
$48$	1.00000	0.144338
$49$	−3.46362	−0.494803
$50$	2.70358	0.382344
$51$	1.88053	0.263326
$52$	2.59694	0.360131
$53$	−0.495848	−0.0681099	−0.0340550	−	0.999420i	$-0.510842\pi$
$53$	−0.495848	−0.0681099	−0.0340550	+	0.999420i	$0.510842\pi$
$54$	−1.00000	−0.136083
$55$	−5.50826	−0.742733
$56$	1.88053	0.251296
$57$	−5.07441	−0.672122
$58$	1.00000	0.131306
$59$	−7.86668	−1.02415	−0.512077	−	0.858939i	$-0.671124\pi$
$59$	−7.86668	−1.02415	−0.512077	+	0.858939i	$0.671124\pi$
$60$	1.51539	0.195637
$61$	−2.36513	−0.302824	−0.151412	−	0.988471i	$-0.548382\pi$
$61$	−2.36513	−0.302824	−0.151412	+	0.988471i	$0.548382\pi$
$62$	5.32721	0.676556
$63$	−1.88053	−0.236924
$64$	1.00000	0.125000
$65$	3.93539	0.488125
$66$	3.63487	0.447421
$67$	−7.89177	−0.964133	−0.482066	−	0.876135i	$-0.660114\pi$
$67$	−7.89177	−0.964133	−0.482066	+	0.876135i	$0.660114\pi$
$68$	1.88053	0.228047
$69$	1.00000	0.120386
$70$	2.84974	0.340609
$71$	9.92825	1.17827	0.589134	−	0.808036i	$-0.299469\pi$
$71$	9.92825	1.17827	0.589134	+	0.808036i	$0.299469\pi$
$72$	−1.00000	−0.117851
$73$	2.18105	0.255273	0.127636	−	0.991821i	$-0.459261\pi$
$73$	2.18105	0.255273	0.127636	+	0.991821i	$0.459261\pi$
$74$	−5.01955	−0.583510
$75$	−2.70358	−0.312183
$76$	−5.07441	−0.582075
$77$	6.83546	0.778973
$78$	−2.59694	−0.294046
$79$	−1.58000	−0.177764	−0.0888821	−	0.996042i	$-0.528329\pi$
$79$	−1.58000	−0.177764	−0.0888821	+	0.996042i	$0.528329\pi$
$80$	1.51539	0.169426
$81$	1.00000	0.111111
$82$	8.89747	0.982561
$83$	−10.4155	−1.14325	−0.571623	−	0.820516i	$-0.693686\pi$
$83$	−10.4155	−1.14325	−0.571623	+	0.820516i	$0.693686\pi$
$84$	−1.88053	−0.205182
$85$	2.84974	0.309097
$86$	−10.8400	−1.16891
$87$	−1.00000	−0.107211
$88$	3.63487	0.387478
$89$	−1.30196	−0.138008	−0.0690039	−	0.997616i	$-0.521982\pi$
$89$	−1.30196	−0.138008	−0.0690039	+	0.997616i	$0.521982\pi$
$90$	−1.51539	−0.159737
$91$	−4.88362	−0.511942
$92$	1.00000	0.104257
$93$	−5.32721	−0.552406
$94$	3.80468	0.392422
$95$	−7.68973	−0.788950
$96$	−1.00000	−0.102062
$97$	2.69804	0.273944	0.136972	−	0.990575i	$-0.456263\pi$
$97$	2.69804	0.273944	0.136972	+	0.990575i	$0.456263\pi$
$98$	3.46362	0.349879
$99$	−3.63487	−0.365318
$100$	−2.70358	−0.270358
$101$	4.52109	0.449866	0.224933	−	0.974374i	$-0.427784\pi$
$101$	4.52109	0.449866	0.224933	+	0.974374i	$0.427784\pi$
$102$	−1.88053	−0.186200
$103$	3.32007	0.327136	0.163568	−	0.986532i	$-0.447700\pi$
$103$	3.32007	0.327136	0.163568	+	0.986532i	$0.447700\pi$
$104$	−2.59694	−0.254651
$105$	−2.84974	−0.278106
$106$	0.495848	0.0481610
$107$	−13.7288	−1.32722	−0.663608	−	0.748081i	$-0.730976\pi$
$107$	−13.7288	−1.32722	−0.663608	+	0.748081i	$0.730976\pi$
$108$	1.00000	0.0962250
$109$	−13.6094	−1.30354	−0.651770	−	0.758417i	$-0.725973\pi$
$109$	−13.6094	−1.30354	−0.651770	+	0.758417i	$0.725973\pi$
$110$	5.50826	0.525192
$111$	5.01955	0.476434
$112$	−1.88053	−0.177693
$113$	−6.79184	−0.638923	−0.319461	−	0.947599i	$-0.603502\pi$
$113$	−6.79184	−0.638923	−0.319461	+	0.947599i	$0.603502\pi$
$114$	5.07441	0.475262
$115$	1.51539	0.141311
$116$	−1.00000	−0.0928477
$117$	2.59694	0.240087
$118$	7.86668	0.724186
$119$	−3.53638	−0.324179
$120$	−1.51539	−0.138336
$121$	2.21226	0.201115
$122$	2.36513	0.214129
$123$	−8.89747	−0.802258
$124$	−5.32721	−0.478397
$125$	−11.6740	−1.04415
$126$	1.88053	0.167531
$127$	−1.47891	−0.131232	−0.0656159	−	0.997845i	$-0.520901\pi$
$127$	−1.47891	−0.131232	−0.0656159	+	0.997845i	$0.520901\pi$
$128$	−1.00000	−0.0883883
$129$	10.8400	0.954408
$130$	−3.93539	−0.345157
$131$	−12.9113	−1.12807	−0.564033	−	0.825752i	$-0.690751\pi$
$131$	−12.9113	−1.12807	−0.564033	+	0.825752i	$0.690751\pi$
$132$	−3.63487	−0.316375
$133$	9.54256	0.827445
$134$	7.89177	0.681745
$135$	1.51539	0.130424
$136$	−1.88053	−0.161254
$137$	3.34105	0.285446	0.142723	−	0.989763i	$-0.454414\pi$
$137$	3.34105	0.285446	0.142723	+	0.989763i	$0.454414\pi$
$138$	−1.00000	−0.0851257
$139$	−4.42000	−0.374899	−0.187450	−	0.982274i	$-0.560022\pi$
$139$	−4.42000	−0.374899	−0.187450	+	0.982274i	$0.560022\pi$
$140$	−2.84974	−0.240847
$141$	−3.80468	−0.320412
$142$	−9.92825	−0.833161
$143$	−9.43954	−0.789374
$144$	1.00000	0.0833333
$145$	−1.51539	−0.125847
$146$	−2.18105	−0.180505
$147$	−3.46362	−0.285675
$148$	5.01955	0.412604
$149$	−7.63487	−0.625473	−0.312736	−	0.949840i	$-0.601246\pi$
$149$	−7.63487	−0.625473	−0.312736	+	0.949840i	$0.601246\pi$
$150$	2.70358	0.220746
$151$	−18.3878	−1.49638	−0.748188	−	0.663487i	$-0.769076\pi$
$151$	−18.3878	−1.49638	−0.748188	+	0.663487i	$0.769076\pi$
$152$	5.07441	0.411589
$153$	1.88053	0.152032
$154$	−6.83546	−0.550817
$155$	−8.07282	−0.648424
$156$	2.59694	0.207922
$157$	15.4508	1.23311	0.616553	−	0.787313i	$-0.288528\pi$
$157$	15.4508	1.23311	0.616553	+	0.787313i	$0.288528\pi$
$158$	1.58000	0.125698
$159$	−0.495848	−0.0393233
$160$	−1.51539	−0.119802
$161$	−1.88053	−0.148206
$162$	−1.00000	−0.0785674
$163$	16.3437	1.28014	0.640070	−	0.768317i	$-0.278906\pi$
$163$	16.3437	1.28014	0.640070	+	0.768317i	$0.278906\pi$
$164$	−8.89747	−0.694775
$165$	−5.50826	−0.428817
$166$	10.4155	0.808397
$167$	−1.86215	−0.144097	−0.0720486	−	0.997401i	$-0.522954\pi$
$167$	−1.86215	−0.144097	−0.0720486	+	0.997401i	$0.522954\pi$
$168$	1.88053	0.145086
$169$	−6.25589	−0.481222
$170$	−2.84974	−0.218565
$171$	−5.07441	−0.388050
$172$	10.8400	0.826542
$173$	−10.0661	−0.765312	−0.382656	−	0.923891i	$-0.624991\pi$
$173$	−10.0661	−0.765312	−0.382656	+	0.923891i	$0.624991\pi$
$174$	1.00000	0.0758098
$175$	5.08415	0.384326
$176$	−3.63487	−0.273988
$177$	−7.86668	−0.591296
$178$	1.30196	0.0975862
$179$	−13.6852	−1.02288	−0.511440	−	0.859319i	$-0.670888\pi$
$179$	−13.6852	−1.02288	−0.511440	+	0.859319i	$0.670888\pi$
$180$	1.51539	0.112951
$181$	24.4379	1.81646	0.908229	−	0.418473i	$-0.137435\pi$
$181$	24.4379	1.81646	0.908229	+	0.418473i	$0.137435\pi$
$182$	4.88362	0.361998
$183$	−2.36513	−0.174836
$184$	−1.00000	−0.0737210
$185$	7.60659	0.559248
$186$	5.32721	0.390610
$187$	−6.83546	−0.499859
$188$	−3.80468	−0.277485
$189$	−1.88053	−0.136788
$190$	7.68973	0.557872
$191$	1.02408	0.0740996	0.0370498	−	0.999313i	$-0.488204\pi$
$191$	1.02408	0.0740996	0.0370498	+	0.999313i	$0.488204\pi$
$192$	1.00000	0.0721688
$193$	10.4057	0.749020	0.374510	−	0.927223i	$-0.377811\pi$
$193$	10.4057	0.749020	0.374510	+	0.927223i	$0.377811\pi$
$194$	−2.69804	−0.193708
$195$	3.93539	0.281819
$196$	−3.46362	−0.247402
$197$	10.8257	0.771301	0.385650	−	0.922645i	$-0.373977\pi$
$197$	10.8257	0.771301	0.385650	+	0.922645i	$0.373977\pi$
$198$	3.63487	0.258319
$199$	−10.8288	−0.767630	−0.383815	−	0.923410i	$-0.625390\pi$
$199$	−10.8288	−0.767630	−0.383815	+	0.923410i	$0.625390\pi$
$200$	2.70358	0.191172
$201$	−7.89177	−0.556642
$202$	−4.52109	−0.318103
$203$	1.88053	0.131987
$204$	1.88053	0.131663
$205$	−13.4832	−0.941705
$206$	−3.32007	−0.231320
$207$	1.00000	0.0695048
$208$	2.59694	0.180066
$209$	18.4448	1.27585
$210$	2.84974	0.196651
$211$	14.0800	0.969304	0.484652	−	0.874707i	$-0.338946\pi$
$211$	14.0800	0.969304	0.484652	+	0.874707i	$0.338946\pi$
$212$	−0.495848	−0.0340550
$213$	9.92825	0.680273
$214$	13.7288	0.938483
$215$	16.4269	1.12030
$216$	−1.00000	−0.0680414
$217$	10.0180	0.680063
$218$	13.6094	0.921742
$219$	2.18105	0.147382
$220$	−5.50826	−0.371367
$221$	4.88362	0.328508
$222$	−5.01955	−0.336890
$223$	8.38324	0.561383	0.280692	−	0.959798i	$-0.409436\pi$
$223$	8.38324	0.561383	0.280692	+	0.959798i	$0.409436\pi$
$224$	1.88053	0.125648
$225$	−2.70358	−0.180239
$226$	6.79184	0.451787
$227$	−22.2638	−1.47770	−0.738849	−	0.673871i	$-0.764630\pi$
$227$	−22.2638	−1.47770	−0.738849	+	0.673871i	$0.764630\pi$
$228$	−5.07441	−0.336061
$229$	13.2878	0.878085	0.439043	−	0.898466i	$-0.355318\pi$
$229$	13.2878	0.878085	0.439043	+	0.898466i	$0.355318\pi$
$230$	−1.51539	−0.0999221
$231$	6.83546	0.449741
$232$	1.00000	0.0656532
$233$	23.5560	1.54320	0.771602	−	0.636106i	$-0.219456\pi$
$233$	23.5560	1.54320	0.771602	+	0.636106i	$0.219456\pi$
$234$	−2.59694	−0.169767
$235$	−5.76558	−0.376105
$236$	−7.86668	−0.512077
$237$	−1.58000	−0.102632
$238$	3.53638	0.229229
$239$	−8.75551	−0.566347	−0.283173	−	0.959069i	$-0.591387\pi$
$239$	−8.75551	−0.566347	−0.283173	+	0.959069i	$0.591387\pi$
$240$	1.51539	0.0978183
$241$	4.13290	0.266223	0.133112	−	0.991101i	$-0.457503\pi$
$241$	4.13290	0.266223	0.133112	+	0.991101i	$0.457503\pi$
$242$	−2.21226	−0.142210
$243$	1.00000	0.0641500
$244$	−2.36513	−0.151412
$245$	−5.24875	−0.335330
$246$	8.89747	0.567282
$247$	−13.1780	−0.838493
$248$	5.32721	0.338278
$249$	−10.4155	−0.660053
$250$	11.6740	0.738326
$251$	−3.21487	−0.202921	−0.101460	−	0.994840i	$-0.532352\pi$
$251$	−3.21487	−0.202921	−0.101460	+	0.994840i	$0.532352\pi$
$252$	−1.88053	−0.118462
$253$	−3.63487	−0.228522
$254$	1.47891	0.0927949
$255$	2.84974	0.178458
$256$	1.00000	0.0625000
$257$	0.698038	0.0435424	0.0217712	−	0.999763i	$-0.493069\pi$
$257$	0.698038	0.0435424	0.0217712	+	0.999763i	$0.493069\pi$
$258$	−10.8400	−0.674869
$259$	−9.43939	−0.586535
$260$	3.93539	0.244063
$261$	−1.00000	−0.0618984
$262$	12.9113	0.797664
$263$	−21.6755	−1.33657	−0.668283	−	0.743907i	$-0.732970\pi$
$263$	−21.6755	−1.33657	−0.668283	+	0.743907i	$0.732970\pi$
$264$	3.63487	0.223711
$265$	−0.751405	−0.0461584
$266$	−9.54256	−0.585092
$267$	−1.30196	−0.0796788
$268$	−7.89177	−0.482066
$269$	17.9237	1.09283	0.546414	−	0.837515i	$-0.315992\pi$
$269$	17.9237	1.09283	0.546414	+	0.837515i	$0.315992\pi$
$270$	−1.51539	−0.0922239
$271$	2.18148	0.132515	0.0662576	−	0.997803i	$-0.478894\pi$
$271$	2.18148	0.132515	0.0662576	+	0.997803i	$0.478894\pi$
$272$	1.88053	0.114024
$273$	−4.88362	−0.295570
$274$	−3.34105	−0.201840
$275$	9.82716	0.592600
$276$	1.00000	0.0601929
$277$	−8.64510	−0.519434	−0.259717	−	0.965685i	$-0.583629\pi$
$277$	−8.64510	−0.519434	−0.259717	+	0.965685i	$0.583629\pi$
$278$	4.42000	0.265094
$279$	−5.32721	−0.318932
$280$	2.84974	0.170304
$281$	0.344147	0.0205301	0.0102651	−	0.999947i	$-0.496732\pi$
$281$	0.344147	0.0205301	0.0102651	+	0.999947i	$0.496732\pi$
$282$	3.80468	0.226565
$283$	−11.9211	−0.708637	−0.354318	−	0.935125i	$-0.615287\pi$
$283$	−11.9211	−0.708637	−0.354318	+	0.935125i	$0.615287\pi$
$284$	9.92825	0.589134
$285$	−7.68973	−0.455500
$286$	9.43954	0.558172
$287$	16.7319	0.987654
$288$	−1.00000	−0.0589256
$289$	−13.4636	−0.791978
$290$	1.51539	0.0889870
$291$	2.69804	0.158162
$292$	2.18105	0.127636
$293$	0.615322	0.0359475	0.0179737	−	0.999838i	$-0.494278\pi$
$293$	0.615322	0.0359475	0.0179737	+	0.999838i	$0.494278\pi$
$294$	3.46362	0.202002
$295$	−11.9211	−0.694074
$296$	−5.01955	−0.291755
$297$	−3.63487	−0.210916
$298$	7.63487	0.442276
$299$	2.59694	0.150185
$300$	−2.70358	−0.156091
$301$	−20.3849	−1.17497
$302$	18.3878	1.05810
$303$	4.52109	0.259730
$304$	−5.07441	−0.291037
$305$	−3.58411	−0.205225
$306$	−1.88053	−0.107503
$307$	−2.41957	−0.138092	−0.0690461	−	0.997613i	$-0.521996\pi$
$307$	−2.41957	−0.138092	−0.0690461	+	0.997613i	$0.521996\pi$
$308$	6.83546	0.389487
$309$	3.32007	0.188872
$310$	8.07282	0.458505
$311$	24.6356	1.39696	0.698479	−	0.715631i	$-0.253861\pi$
$311$	24.6356	1.39696	0.698479	+	0.715631i	$0.253861\pi$
$312$	−2.59694	−0.147023
$313$	−14.6589	−0.828573	−0.414286	−	0.910147i	$-0.635969\pi$
$313$	−14.6589	−0.828573	−0.414286	+	0.910147i	$0.635969\pi$
$314$	−15.4508	−0.871938
$315$	−2.84974	−0.160565
$316$	−1.58000	−0.0888821
$317$	−30.3018	−1.70192	−0.850958	−	0.525234i	$-0.823978\pi$
$317$	−30.3018	−1.70192	−0.850958	+	0.525234i	$0.823978\pi$
$318$	0.495848	0.0278058
$319$	3.63487	0.203514
$320$	1.51539	0.0847131
$321$	−13.7288	−0.766268
$322$	1.88053	0.104798
$323$	−9.54256	−0.530962
$324$	1.00000	0.0555556
$325$	−7.02105	−0.389458
$326$	−16.3437	−0.905195
$327$	−13.6094	−0.752599
$328$	8.89747	0.491280
$329$	7.15479	0.394457
$330$	5.50826	0.303220
$331$	−1.44625	−0.0794933	−0.0397467	−	0.999210i	$-0.512655\pi$
$331$	−1.44625	−0.0794933	−0.0397467	+	0.999210i	$0.512655\pi$
$332$	−10.4155	−0.571623
$333$	5.01955	0.275069
$334$	1.86215	0.101892
$335$	−11.9591	−0.653398
$336$	−1.88053	−0.102591
$337$	23.8339	1.29831	0.649157	−	0.760655i	$-0.275122\pi$
$337$	23.8339	1.29831	0.649157	+	0.760655i	$0.275122\pi$
$338$	6.25589	0.340275
$339$	−6.79184	−0.368882
$340$	2.84974	0.154549
$341$	19.3637	1.04860
$342$	5.07441	0.274393
$343$	19.6771	1.06246
$344$	−10.8400	−0.584453
$345$	1.51539	0.0815861
$346$	10.0661	0.541157
$347$	−15.8483	−0.850781	−0.425391	−	0.905010i	$-0.639863\pi$
$347$	−15.8483	−0.850781	−0.425391	+	0.905010i	$0.639863\pi$
$348$	−1.00000	−0.0536056
$349$	11.0027	0.588959	0.294480	−	0.955658i	$-0.404854\pi$
$349$	11.0027	0.588959	0.294480	+	0.955658i	$0.404854\pi$
$350$	−5.08415	−0.271760
$351$	2.59694	0.138615
$352$	3.63487	0.193739
$353$	−5.07441	−0.270084	−0.135042	−	0.990840i	$-0.543117\pi$
$353$	−5.07441	−0.270084	−0.135042	+	0.990840i	$0.543117\pi$
$354$	7.86668	0.418109
$355$	15.0452	0.798517
$356$	−1.30196	−0.0690039
$357$	−3.53638	−0.187165
$358$	13.6852	0.723285
$359$	−30.8918	−1.63041	−0.815204	−	0.579174i	$-0.803375\pi$
$359$	−30.8918	−1.63041	−0.815204	+	0.579174i	$0.803375\pi$
$360$	−1.51539	−0.0798683
$361$	6.74965	0.355245
$362$	−24.4379	−1.28443
$363$	2.21226	0.116114
$364$	−4.88362	−0.255971
$365$	3.30515	0.172999
$366$	2.36513	0.123627
$367$	20.1488	1.05176	0.525880	−	0.850559i	$-0.323736\pi$
$367$	20.1488	1.05176	0.525880	+	0.850559i	$0.323736\pi$
$368$	1.00000	0.0521286
$369$	−8.89747	−0.463184
$370$	−7.60659	−0.395448
$371$	0.932455	0.0484106
$372$	−5.32721	−0.276203
$373$	−7.72883	−0.400183	−0.200092	−	0.979777i	$-0.564124\pi$
$373$	−7.72883	−0.400183	−0.200092	+	0.979777i	$0.564124\pi$
$374$	6.83546	0.353453
$375$	−11.6740	−0.602841
$376$	3.80468	0.196211
$377$	−2.59694	−0.133749
$378$	1.88053	0.0967238
$379$	20.0037	1.02752	0.513760	−	0.857934i	$-0.328252\pi$
$379$	20.0037	1.02752	0.513760	+	0.857934i	$0.328252\pi$
$380$	−7.68973	−0.394475
$381$	−1.47891	−0.0757667
$382$	−1.02408	−0.0523964
$383$	13.8386	0.707117	0.353559	−	0.935412i	$-0.384971\pi$
$383$	13.8386	0.707117	0.353559	+	0.935412i	$0.384971\pi$
$384$	−1.00000	−0.0510310
$385$	10.3584	0.527914
$386$	−10.4057	−0.529637
$387$	10.8400	0.551028
$388$	2.69804	0.136972
$389$	−36.3772	−1.84440	−0.922198	−	0.386718i	$-0.873609\pi$
$389$	−36.3772	−1.84440	−0.922198	+	0.386718i	$0.873609\pi$
$390$	−3.93539	−0.199276
$391$	1.88053	0.0951023
$392$	3.46362	0.174939
$393$	−12.9113	−0.651290
$394$	−10.8257	−0.545392
$395$	−2.39433	−0.120472
$396$	−3.63487	−0.182659
$397$	−0.0758501	−0.00380681	−0.00190340	−	0.999998i	$-0.500606\pi$
$397$	−0.0758501	−0.00380681	−0.00190340	+	0.999998i	$0.500606\pi$
$398$	10.8288	0.542796
$399$	9.54256	0.477726
$400$	−2.70358	−0.135179
$401$	−13.3149	−0.664912	−0.332456	−	0.943119i	$-0.607877\pi$
$401$	−13.3149	−0.664912	−0.332456	+	0.943119i	$0.607877\pi$
$402$	7.89177	0.393606
$403$	−13.8345	−0.689143
$404$	4.52109	0.224933
$405$	1.51539	0.0753005
$406$	−1.88053	−0.0933289
$407$	−18.2454	−0.904390
$408$	−1.88053	−0.0930999
$409$	19.8528	0.981659	0.490830	−	0.871256i	$-0.336694\pi$
$409$	19.8528	0.981659	0.490830	+	0.871256i	$0.336694\pi$
$410$	13.4832	0.665886
$411$	3.34105	0.164802
$412$	3.32007	0.163568
$413$	14.7935	0.727940
$414$	−1.00000	−0.0491473
$415$	−15.7835	−0.774783
$416$	−2.59694	−0.127326
$417$	−4.42000	−0.216448
$418$	−18.4448	−0.902165
$419$	−18.3230	−0.895137	−0.447569	−	0.894250i	$-0.647710\pi$
$419$	−18.3230	−0.895137	−0.447569	+	0.894250i	$0.647710\pi$
$420$	−2.84974	−0.139053
$421$	−17.2006	−0.838306	−0.419153	−	0.907916i	$-0.637673\pi$
$421$	−17.2006	−0.838306	−0.419153	+	0.907916i	$0.637673\pi$
$422$	−14.0800	−0.685401
$423$	−3.80468	−0.184990
$424$	0.495848	0.0240805
$425$	−5.08415	−0.246618
$426$	−9.92825	−0.481026
$427$	4.44769	0.215239
$428$	−13.7288	−0.663608
$429$	−9.43954	−0.455745
$430$	−16.4269	−0.792174
$431$	28.0518	1.35121	0.675605	−	0.737264i	$-0.263883\pi$
$431$	28.0518	1.35121	0.675605	+	0.737264i	$0.263883\pi$
$432$	1.00000	0.0481125
$433$	41.2232	1.98106	0.990531	−	0.137288i	$-0.0438387\pi$
$433$	41.2232	1.98106	0.990531	+	0.137288i	$0.0438387\pi$
$434$	−10.0180	−0.480877
$435$	−1.51539	−0.0726576
$436$	−13.6094	−0.651770
$437$	−5.07441	−0.242742
$438$	−2.18105	−0.104215
$439$	0.486105	0.0232005	0.0116003	−	0.999933i	$-0.496307\pi$
$439$	0.486105	0.0232005	0.0116003	+	0.999933i	$0.496307\pi$
$440$	5.50826	0.262596
$441$	−3.46362	−0.164934
$442$	−4.88362	−0.232290
$443$	−12.8836	−0.612119	−0.306060	−	0.952012i	$-0.599011\pi$
$443$	−12.8836	−0.612119	−0.306060	+	0.952012i	$0.599011\pi$
$444$	5.01955	0.238217
$445$	−1.97299	−0.0935285
$446$	−8.38324	−0.396958
$447$	−7.63487	−0.361117
$448$	−1.88053	−0.0888465
$449$	−22.6842	−1.07053	−0.535267	−	0.844683i	$-0.679789\pi$
$449$	−22.6842	−1.07053	−0.535267	+	0.844683i	$0.679789\pi$
$450$	2.70358	0.127448
$451$	32.3411	1.52288
$452$	−6.79184	−0.319461
$453$	−18.3878	−0.863933
$454$	22.2638	1.04489
$455$	−7.40061	−0.346946
$456$	5.07441	0.237631
$457$	−33.0113	−1.54420	−0.772102	−	0.635499i	$-0.780794\pi$
$457$	−33.0113	−1.54420	−0.772102	+	0.635499i	$0.780794\pi$
$458$	−13.2878	−0.620900
$459$	1.88053	0.0877754
$460$	1.51539	0.0706556
$461$	9.88175	0.460239	0.230120	−	0.973162i	$-0.426088\pi$
$461$	9.88175	0.460239	0.230120	+	0.973162i	$0.426088\pi$
$462$	−6.83546	−0.318015
$463$	−20.1443	−0.936185	−0.468092	−	0.883680i	$-0.655059\pi$
$463$	−20.1443	−0.936185	−0.468092	+	0.883680i	$0.655059\pi$
$464$	−1.00000	−0.0464238
$465$	−8.07282	−0.374368
$466$	−23.5560	−1.09121
$467$	18.3623	0.849704	0.424852	−	0.905263i	$-0.360326\pi$
$467$	18.3623	0.849704	0.424852	+	0.905263i	$0.360326\pi$
$468$	2.59694	0.120044
$469$	14.8407	0.685279
$470$	5.76558	0.265947
$471$	15.4508	0.711935
$472$	7.86668	0.362093
$473$	−39.4020	−1.81170
$474$	1.58000	0.0725719
$475$	13.7191	0.629475
$476$	−3.53638	−0.162090
$477$	−0.495848	−0.0227033
$478$	8.75551	0.400468
$479$	−30.8903	−1.41142	−0.705708	−	0.708503i	$-0.749371\pi$
$479$	−30.8903	−1.41142	−0.705708	+	0.708503i	$0.749371\pi$
$480$	−1.51539	−0.0691680
$481$	13.0355	0.594366
$482$	−4.13290	−0.188248
$483$	−1.88053	−0.0855669
$484$	2.21226	0.100557
$485$	4.08859	0.185653
$486$	−1.00000	−0.0453609
$487$	−6.24869	−0.283155	−0.141578	−	0.989927i	$-0.545217\pi$
$487$	−6.24869	−0.283155	−0.141578	+	0.989927i	$0.545217\pi$
$488$	2.36513	0.107065
$489$	16.3437	0.739089
$490$	5.24875	0.237114
$491$	0.796371	0.0359397	0.0179699	−	0.999839i	$-0.494280\pi$
$491$	0.796371	0.0359397	0.0179699	+	0.999839i	$0.494280\pi$
$492$	−8.89747	−0.401129
$493$	−1.88053	−0.0846946
$494$	13.1780	0.592904
$495$	−5.50826	−0.247578
$496$	−5.32721	−0.239199
$497$	−18.6703	−0.837479
$498$	10.4155	0.466728
$499$	2.09833	0.0939343	0.0469672	−	0.998896i	$-0.485044\pi$
$499$	2.09833	0.0939343	0.0469672	+	0.998896i	$0.485044\pi$
$500$	−11.6740	−0.522075
$501$	−1.86215	−0.0831946
$502$	3.21487	0.143487
$503$	−7.84377	−0.349736	−0.174868	−	0.984592i	$-0.555950\pi$
$503$	−7.84377	−0.349736	−0.174868	+	0.984592i	$0.555950\pi$
$504$	1.88053	0.0837653
$505$	6.85124	0.304876
$506$	3.63487	0.161590
$507$	−6.25589	−0.277834
$508$	−1.47891	−0.0656159
$509$	−18.9986	−0.842096	−0.421048	−	0.907038i	$-0.638338\pi$
$509$	−18.9986	−0.842096	−0.421048	+	0.907038i	$0.638338\pi$
$510$	−2.84974	−0.126189
$511$	−4.10152	−0.181441
$512$	−1.00000	−0.0441942
$513$	−5.07441	−0.224041
$514$	−0.698038	−0.0307891
$515$	5.03121	0.221702
$516$	10.8400	0.477204
$517$	13.8295	0.608220
$518$	9.43939	0.414743
$519$	−10.0661	−0.441853
$520$	−3.93539	−0.172578
$521$	−4.34415	−0.190321	−0.0951603	−	0.995462i	$-0.530336\pi$
$521$	−4.34415	−0.190321	−0.0951603	+	0.995462i	$0.530336\pi$
$522$	1.00000	0.0437688
$523$	12.8467	0.561747	0.280874	−	0.959745i	$-0.409376\pi$
$523$	12.8467	0.561747	0.280874	+	0.959745i	$0.409376\pi$
$524$	−12.9113	−0.564033
$525$	5.08415	0.221891
$526$	21.6755	0.945095
$527$	−10.0180	−0.436389
$528$	−3.63487	−0.158187
$529$	1.00000	0.0434783
$530$	0.751405	0.0326389
$531$	−7.86668	−0.341385
$532$	9.54256	0.413723
$533$	−23.1062	−1.00084
$534$	1.30196	0.0563414
$535$	−20.8046	−0.899460
$536$	7.89177	0.340872
$537$	−13.6852	−0.590560
$538$	−17.9237	−0.772747
$539$	12.5898	0.542281
$540$	1.51539	0.0652122
$541$	−6.52807	−0.280664	−0.140332	−	0.990105i	$-0.544817\pi$
$541$	−6.52807	−0.280664	−0.140332	+	0.990105i	$0.544817\pi$
$542$	−2.18148	−0.0937024
$543$	24.4379	1.04873
$544$	−1.88053	−0.0806269
$545$	−20.6235	−0.883415
$546$	4.88362	0.209000
$547$	−11.8842	−0.508132	−0.254066	−	0.967187i	$-0.581768\pi$
$547$	−11.8842	−0.508132	−0.254066	+	0.967187i	$0.581768\pi$
$548$	3.34105	0.142723
$549$	−2.36513	−0.100941
$550$	−9.82716	−0.419031
$551$	5.07441	0.216177
$552$	−1.00000	−0.0425628
$553$	2.97124	0.126350
$554$	8.64510	0.367295
$555$	7.60659	0.322882
$556$	−4.42000	−0.187450
$557$	31.1444	1.31963	0.659814	−	0.751429i	$-0.270635\pi$
$557$	31.1444	1.31963	0.659814	+	0.751429i	$0.270635\pi$
$558$	5.32721	0.225519
$559$	28.1508	1.19065
$560$	−2.84974	−0.120423
$561$	−6.83546	−0.288594
$562$	−0.344147	−0.0145170
$563$	2.23224	0.0940776	0.0470388	−	0.998893i	$-0.485022\pi$
$563$	2.23224	0.0940776	0.0470388	+	0.998893i	$0.485022\pi$
$564$	−3.80468	−0.160206
$565$	−10.2923	−0.433001
$566$	11.9211	0.501082
$567$	−1.88053	−0.0789747
$568$	−9.92825	−0.416580
$569$	28.4200	1.19143	0.595714	−	0.803197i	$-0.296869\pi$
$569$	28.4200	1.19143	0.595714	+	0.803197i	$0.296869\pi$
$570$	7.68973	0.322087
$571$	26.7484	1.11938	0.559692	−	0.828701i	$-0.310919\pi$
$571$	26.7484	1.11938	0.559692	+	0.828701i	$0.310919\pi$
$572$	−9.43954	−0.394687
$573$	1.02408	0.0427814
$574$	−16.7319	−0.698377
$575$	−2.70358	−0.112747
$576$	1.00000	0.0416667
$577$	−43.9392	−1.82921	−0.914607	−	0.404344i	$-0.867500\pi$
$577$	−43.9392	−1.82921	−0.914607	+	0.404344i	$0.867500\pi$
$578$	13.4636	0.560013
$579$	10.4057	0.432447
$580$	−1.51539	−0.0629233
$581$	19.5866	0.812587
$582$	−2.69804	−0.111837
$583$	1.80234	0.0746453
$584$	−2.18105	−0.0902525
$585$	3.93539	0.162708
$586$	−0.615322	−0.0254187
$587$	−29.8226	−1.23091	−0.615456	−	0.788171i	$-0.711028\pi$
$587$	−29.8226	−1.23091	−0.615456	+	0.788171i	$0.711028\pi$
$588$	−3.46362	−0.142837
$589$	27.0324	1.11385
$590$	11.9211	0.490785
$591$	10.8257	0.445311
$592$	5.01955	0.206302
$593$	−40.1570	−1.64905	−0.824526	−	0.565824i	$-0.808558\pi$
$593$	−40.1570	−1.64905	−0.824526	+	0.565824i	$0.808558\pi$
$594$	3.63487	0.149140
$595$	−5.35901	−0.219698
$596$	−7.63487	−0.312736
$597$	−10.8288	−0.443191
$598$	−2.59694	−0.106197
$599$	8.28172	0.338382	0.169191	−	0.985583i	$-0.445885\pi$
$599$	8.28172	0.338382	0.169191	+	0.985583i	$0.445885\pi$
$600$	2.70358	0.110373
$601$	26.4562	1.07917	0.539586	−	0.841931i	$-0.318581\pi$
$601$	26.4562	1.07917	0.539586	+	0.841931i	$0.318581\pi$
$602$	20.3849	0.830826
$603$	−7.89177	−0.321378
$604$	−18.3878	−0.748188
$605$	3.35245	0.136297
$606$	−4.52109	−0.183657
$607$	−6.66784	−0.270639	−0.135320	−	0.990802i	$-0.543206\pi$
$607$	−6.66784	−0.270639	−0.135320	+	0.990802i	$0.543206\pi$
$608$	5.07441	0.205795
$609$	1.88053	0.0762028
$610$	3.58411	0.145116
$611$	−9.88053	−0.399723
$612$	1.88053	0.0760158
$613$	19.4320	0.784850	0.392425	−	0.919784i	$-0.371636\pi$
$613$	19.4320	0.784850	0.392425	+	0.919784i	$0.371636\pi$
$614$	2.41957	0.0976459
$615$	−13.4832	−0.543694
$616$	−6.83546	−0.275409
$617$	42.0698	1.69367	0.846833	−	0.531859i	$-0.178506\pi$
$617$	42.0698	1.69367	0.846833	+	0.531859i	$0.178506\pi$
$618$	−3.32007	−0.133553
$619$	−13.3358	−0.536013	−0.268006	−	0.963417i	$-0.586365\pi$
$619$	−13.3358	−0.536013	−0.268006	+	0.963417i	$0.586365\pi$
$620$	−8.07282	−0.324212
$621$	1.00000	0.0401286
$622$	−24.6356	−0.987798
$623$	2.44837	0.0980920
$624$	2.59694	0.103961
$625$	−4.17275	−0.166910
$626$	14.6589	0.585889
$627$	18.4448	0.736615
$628$	15.4508	0.616553
$629$	9.43939	0.376373
$630$	2.84974	0.113536
$631$	−1.60552	−0.0639147	−0.0319573	−	0.999489i	$-0.510174\pi$
$631$	−1.60552	−0.0639147	−0.0319573	+	0.999489i	$0.510174\pi$
$632$	1.58000	0.0628491
$633$	14.0800	0.559628
$634$	30.3018	1.20344
$635$	−2.24113	−0.0889364
$636$	−0.495848	−0.0196616
$637$	−8.99483	−0.356388
$638$	−3.63487	−0.143906
$639$	9.92825	0.392756
$640$	−1.51539	−0.0599012
$641$	−3.83403	−0.151435	−0.0757175	−	0.997129i	$-0.524125\pi$
$641$	−3.83403	−0.151435	−0.0757175	+	0.997129i	$0.524125\pi$
$642$	13.7288	0.541834
$643$	4.05790	0.160028	0.0800139	−	0.996794i	$-0.474504\pi$
$643$	4.05790	0.160028	0.0800139	+	0.996794i	$0.474504\pi$
$644$	−1.88053	−0.0741031
$645$	16.4269	0.646807
$646$	9.54256	0.375447
$647$	−25.7329	−1.01167	−0.505833	−	0.862631i	$-0.668815\pi$
$647$	−25.7329	−1.01167	−0.505833	+	0.862631i	$0.668815\pi$
$648$	−1.00000	−0.0392837
$649$	28.5943	1.12243
$650$	7.02105	0.275388
$651$	10.0180	0.392635
$652$	16.3437	0.640070
$653$	−22.0335	−0.862236	−0.431118	−	0.902296i	$-0.641881\pi$
$653$	−22.0335	−0.862236	−0.431118	+	0.902296i	$0.641881\pi$
$654$	13.6094	0.532168
$655$	−19.5657	−0.764496
$656$	−8.89747	−0.347388
$657$	2.18105	0.0850909
$658$	−7.15479	−0.278923
$659$	25.8038	1.00517	0.502587	−	0.864527i	$-0.332382\pi$
$659$	25.8038	1.00517	0.502587	+	0.864527i	$0.332382\pi$
$660$	−5.50826	−0.214409
$661$	42.6373	1.65840	0.829199	−	0.558954i	$-0.188797\pi$
$661$	42.6373	1.65840	0.829199	+	0.558954i	$0.188797\pi$
$662$	1.44625	0.0562103
$663$	4.88362	0.189664
$664$	10.4155	0.404198
$665$	14.4607	0.560764
$666$	−5.01955	−0.194503
$667$	−1.00000	−0.0387202
$668$	−1.86215	−0.0720486
$669$	8.38324	0.324115
$670$	11.9591	0.462022
$671$	8.59694	0.331881
$672$	1.88053	0.0725429
$673$	2.56514	0.0988790	0.0494395	−	0.998777i	$-0.484257\pi$
$673$	2.56514	0.0988790	0.0494395	+	0.998777i	$0.484257\pi$
$674$	−23.8339	−0.918046
$675$	−2.70358	−0.104061
$676$	−6.25589	−0.240611
$677$	17.4605	0.671063	0.335531	−	0.942029i	$-0.391084\pi$
$677$	17.4605	0.671063	0.335531	+	0.942029i	$0.391084\pi$
$678$	6.79184	0.260839
$679$	−5.07373	−0.194712
$680$	−2.84974	−0.109282
$681$	−22.2638	−0.853149
$682$	−19.3637	−0.741474
$683$	25.8720	0.989965	0.494983	−	0.868903i	$-0.335174\pi$
$683$	25.8720	0.989965	0.494983	+	0.868903i	$0.335174\pi$
$684$	−5.07441	−0.194025
$685$	5.06301	0.193448
$686$	−19.6771	−0.751276
$687$	13.2878	0.506963
$688$	10.8400	0.413271
$689$	−1.28769	−0.0490570
$690$	−1.51539	−0.0576901
$691$	0.758812	0.0288666	0.0144333	−	0.999896i	$-0.495406\pi$
$691$	0.758812	0.0288666	0.0144333	+	0.999896i	$0.495406\pi$
$692$	−10.0661	−0.382656
$693$	6.83546	0.259658
$694$	15.8483	0.601593
$695$	−6.69804	−0.254071
$696$	1.00000	0.0379049
$697$	−16.7319	−0.633767
$698$	−11.0027	−0.416457
$699$	23.5560	0.890969
$700$	5.08415	0.192163
$701$	22.9046	0.865095	0.432547	−	0.901611i	$-0.357615\pi$
$701$	22.9046	0.865095	0.432547	+	0.901611i	$0.357615\pi$
$702$	−2.59694	−0.0980153
$703$	−25.4712	−0.960666
$704$	−3.63487	−0.136994
$705$	−5.76558	−0.217144
$706$	5.07441	0.190978
$707$	−8.50203	−0.319752
$708$	−7.86668	−0.295648
$709$	37.4456	1.40630	0.703149	−	0.711043i	$-0.251777\pi$
$709$	37.4456	1.40630	0.703149	+	0.711043i	$0.251777\pi$
$710$	−15.0452	−0.564637
$711$	−1.58000	−0.0592547
$712$	1.30196	0.0487931
$713$	−5.32721	−0.199505
$714$	3.53638	0.132346
$715$	−14.3046	−0.534963
$716$	−13.6852	−0.511440
$717$	−8.75551	−0.326980
$718$	30.8918	1.15287
$719$	−26.3776	−0.983719	−0.491859	−	0.870675i	$-0.663683\pi$
$719$	−26.3776	−0.983719	−0.491859	+	0.870675i	$0.663683\pi$
$720$	1.51539	0.0564754
$721$	−6.24348	−0.232519
$722$	−6.74965	−0.251196
$723$	4.13290	0.153704
$724$	24.4379	0.908229
$725$	2.70358	0.100408
$726$	−2.21226	−0.0821048
$727$	6.60626	0.245013	0.122506	−	0.992468i	$-0.460907\pi$
$727$	6.60626	0.245013	0.122506	+	0.992468i	$0.460907\pi$
$728$	4.88362	0.180999
$729$	1.00000	0.0370370
$730$	−3.30515	−0.122329
$731$	20.3849	0.753963
$732$	−2.36513	−0.0874178
$733$	32.2037	1.18947	0.594735	−	0.803922i	$-0.297257\pi$
$733$	32.2037	1.18947	0.594735	+	0.803922i	$0.297257\pi$
$734$	−20.1488	−0.743707
$735$	−5.24875	−0.193603
$736$	−1.00000	−0.0368605
$737$	28.6855	1.05665
$738$	8.89747	0.327520
$739$	5.95846	0.219185	0.109593	−	0.993977i	$-0.465045\pi$
$739$	5.95846	0.219185	0.109593	+	0.993977i	$0.465045\pi$
$740$	7.60659	0.279624
$741$	−13.1780	−0.484104
$742$	−0.932455	−0.0342315
$743$	7.56355	0.277480	0.138740	−	0.990329i	$-0.455695\pi$
$743$	7.56355	0.277480	0.138740	+	0.990329i	$0.455695\pi$
$744$	5.32721	0.195305
$745$	−11.5698	−0.423886
$746$	7.72883	0.282972
$747$	−10.4155	−0.381082
$748$	−6.83546	−0.249929
$749$	25.8174	0.943348
$750$	11.6740	0.426273
$751$	25.0692	0.914788	0.457394	−	0.889264i	$-0.348783\pi$
$751$	25.0692	0.914788	0.457394	+	0.889264i	$0.348783\pi$
$752$	−3.80468	−0.138742
$753$	−3.21487	−0.117156
$754$	2.59694	0.0945751
$755$	−27.8647	−1.01410
$756$	−1.88053	−0.0683941
$757$	−11.4207	−0.415094	−0.207547	−	0.978225i	$-0.566548\pi$
$757$	−11.4207	−0.415094	−0.207547	+	0.978225i	$0.566548\pi$
$758$	−20.0037	−0.726566
$759$	−3.63487	−0.131937
$760$	7.68973	0.278936
$761$	2.46362	0.0893062	0.0446531	−	0.999003i	$-0.485782\pi$
$761$	2.46362	0.0893062	0.0446531	+	0.999003i	$0.485782\pi$
$762$	1.47891	0.0535752
$763$	25.5927	0.926520
$764$	1.02408	0.0370498
$765$	2.84974	0.103032
$766$	−13.8386	−0.500007
$767$	−20.4293	−0.737660
$768$	1.00000	0.0360844
$769$	−9.55449	−0.344544	−0.172272	−	0.985049i	$-0.555111\pi$
$769$	−9.55449	−0.344544	−0.172272	+	0.985049i	$0.555111\pi$
$770$	−10.3584	−0.373292
$771$	0.698038	0.0251392
$772$	10.4057	0.374510
$773$	−31.5951	−1.13640	−0.568198	−	0.822892i	$-0.692359\pi$
$773$	−31.5951	−1.13640	−0.568198	+	0.822892i	$0.692359\pi$
$774$	−10.8400	−0.389636
$775$	14.4025	0.517354
$776$	−2.69804	−0.0968539
$777$	−9.43939	−0.338636
$778$	36.3772	1.30418
$779$	45.1494	1.61765
$780$	3.93539	0.140910
$781$	−36.0879	−1.29133
$782$	−1.88053	−0.0672475
$783$	−1.00000	−0.0357371
$784$	−3.46362	−0.123701
$785$	23.4140	0.835683
$786$	12.9113	0.460531
$787$	−8.69501	−0.309943	−0.154972	−	0.987919i	$-0.549529\pi$
$787$	−8.69501	−0.309943	−0.154972	+	0.987919i	$0.549529\pi$
$788$	10.8257	0.385650
$789$	−21.6755	−0.771667
$790$	2.39433	0.0851863
$791$	12.7722	0.454128
$792$	3.63487	0.129159
$793$	−6.14211	−0.218113
$794$	0.0758501	0.00269182
$795$	−0.751405	−0.0266496
$796$	−10.8288	−0.383815
$797$	5.49809	0.194752	0.0973761	−	0.995248i	$-0.468955\pi$
$797$	5.49809	0.194752	0.0973761	+	0.995248i	$0.468955\pi$
$798$	−9.54256	−0.337803
$799$	−7.15479	−0.253118
$800$	2.70358	0.0955860
$801$	−1.30196	−0.0460026
$802$	13.3149	0.470164
$803$	−7.92783	−0.279767
$804$	−7.89177	−0.278321
$805$	−2.84974	−0.100440
$806$	13.8345	0.487298
$807$	17.9237	0.630945
$808$	−4.52109	−0.159051
$809$	−6.11249	−0.214904	−0.107452	−	0.994210i	$-0.534269\pi$
$809$	−6.11249	−0.214904	−0.107452	+	0.994210i	$0.534269\pi$
$810$	−1.51539	−0.0532455
$811$	−34.3022	−1.20451	−0.602256	−	0.798303i	$-0.705731\pi$
$811$	−34.3022	−1.20451	−0.602256	+	0.798303i	$0.705731\pi$
$812$	1.88053	0.0659935
$813$	2.18148	0.0765077
$814$	18.2454	0.639501
$815$	24.7672	0.867556
$816$	1.88053	0.0658316
$817$	−55.0066	−1.92444
$818$	−19.8528	−0.694138
$819$	−4.88362	−0.170647
$820$	−13.4832	−0.470853
$821$	−19.5914	−0.683745	−0.341872	−	0.939746i	$-0.611061\pi$
$821$	−19.5914	−0.683745	−0.341872	+	0.939746i	$0.611061\pi$
$822$	−3.34105	−0.116533
$823$	−32.0702	−1.11790	−0.558949	−	0.829202i	$-0.688795\pi$
$823$	−32.0702	−1.11790	−0.558949	+	0.829202i	$0.688795\pi$
$824$	−3.32007	−0.115660
$825$	9.82716	0.342138
$826$	−14.7935	−0.514732
$827$	−7.36572	−0.256131	−0.128066	−	0.991766i	$-0.540877\pi$
$827$	−7.36572	−0.256131	−0.128066	+	0.991766i	$0.540877\pi$
$828$	1.00000	0.0347524
$829$	21.2036	0.736432	0.368216	−	0.929740i	$-0.379969\pi$
$829$	21.2036	0.736432	0.368216	+	0.929740i	$0.379969\pi$
$830$	15.7835	0.547855
$831$	−8.64510	−0.299895
$832$	2.59694	0.0900328
$833$	−6.51343	−0.225677
$834$	4.42000	0.153052
$835$	−2.82189	−0.0976554
$836$	18.4448	0.637927
$837$	−5.32721	−0.184135
$838$	18.3230	0.632958
$839$	4.81283	0.166157	0.0830786	−	0.996543i	$-0.473525\pi$
$839$	4.81283	0.166157	0.0830786	+	0.996543i	$0.473525\pi$
$840$	2.84974	0.0983253
$841$	1.00000	0.0344828
$842$	17.2006	0.592772
$843$	0.344147	0.0118531
$844$	14.0800	0.484652
$845$	−9.48014	−0.326127
$846$	3.80468	0.130807
$847$	−4.16022	−0.142947
$848$	−0.495848	−0.0170275
$849$	−11.9211	−0.409132
$850$	5.08415	0.174385
$851$	5.01955	0.172068
$852$	9.92825	0.340136
$853$	1.03765	0.0355286	0.0177643	−	0.999842i	$-0.494345\pi$
$853$	1.03765	0.0355286	0.0177643	+	0.999842i	$0.494345\pi$
$854$	−4.44769	−0.152197
$855$	−7.68973	−0.262983
$856$	13.7288	0.469242
$857$	28.5057	0.973737	0.486868	−	0.873475i	$-0.338139\pi$
$857$	28.5057	0.973737	0.486868	+	0.873475i	$0.338139\pi$
$858$	9.43954	0.322261
$859$	12.9325	0.441250	0.220625	−	0.975359i	$-0.429190\pi$
$859$	12.9325	0.441250	0.220625	+	0.975359i	$0.429190\pi$
$860$	16.4269	0.560151
$861$	16.7319	0.570222
$862$	−28.0518	−0.955449
$863$	48.7983	1.66111	0.830556	−	0.556935i	$-0.188023\pi$
$863$	48.7983	1.66111	0.830556	+	0.556935i	$0.188023\pi$
$864$	−1.00000	−0.0340207
$865$	−15.2541	−0.518656
$866$	−41.2232	−1.40082
$867$	−13.4636	−0.457249
$868$	10.0180	0.340031
$869$	5.74310	0.194821
$870$	1.51539	0.0513767
$871$	−20.4945	−0.694429
$872$	13.6094	0.460871
$873$	2.69804	0.0913147
$874$	5.07441	0.171645
$875$	21.9532	0.742153
$876$	2.18105	0.0736908
$877$	−40.6341	−1.37212	−0.686058	−	0.727547i	$-0.740660\pi$
$877$	−40.6341	−1.37212	−0.686058	+	0.727547i	$0.740660\pi$
$878$	−0.486105	−0.0164052
$879$	0.615322	0.0207543
$880$	−5.50826	−0.185683
$881$	−13.4336	−0.452591	−0.226295	−	0.974059i	$-0.572661\pi$
$881$	−13.4336	−0.452591	−0.226295	+	0.974059i	$0.572661\pi$
$882$	3.46362	0.116626
$883$	30.9370	1.04111	0.520556	−	0.853827i	$-0.325725\pi$
$883$	30.9370	1.04111	0.520556	+	0.853827i	$0.325725\pi$
$884$	4.88362	0.164254
$885$	−11.9211	−0.400724
$886$	12.8836	0.432834
$887$	14.1436	0.474896	0.237448	−	0.971400i	$-0.423689\pi$
$887$	14.1436	0.474896	0.237448	+	0.971400i	$0.423689\pi$
$888$	−5.01955	−0.168445
$889$	2.78112	0.0932759
$890$	1.97299	0.0661346
$891$	−3.63487	−0.121773
$892$	8.38324	0.280692
$893$	19.3065	0.646067
$894$	7.63487	0.255348
$895$	−20.7385	−0.693211
$896$	1.88053	0.0628240
$897$	2.59694	0.0867094
$898$	22.6842	0.756981
$899$	5.32721	0.177672
$900$	−2.70358	−0.0901194
$901$	−0.932455	−0.0310646
$902$	−32.3411	−1.07684
$903$	−20.3849	−0.678367
$904$	6.79184	0.225893
$905$	37.0331	1.23102
$906$	18.3878	0.610893
$907$	−4.42889	−0.147059	−0.0735294	−	0.997293i	$-0.523426\pi$
$907$	−4.42889	−0.147059	−0.0735294	+	0.997293i	$0.523426\pi$
$908$	−22.2638	−0.738849
$909$	4.52109	0.149955
$910$	7.40061	0.245328
$911$	21.4246	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$911$	21.4246	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$912$	−5.07441	−0.168031
$913$	37.8588	1.25294
$914$	33.0113	1.09192
$915$	−3.58411	−0.118487
$916$	13.2878	0.439043
$917$	24.2801	0.801798
$918$	−1.88053	−0.0620666
$919$	43.8632	1.44691	0.723457	−	0.690369i	$-0.242552\pi$
$919$	43.8632	1.44691	0.723457	+	0.690369i	$0.242552\pi$
$920$	−1.51539	−0.0499611
$921$	−2.41957	−0.0797276
$922$	−9.88175	−0.325438
$923$	25.7831	0.848661
$924$	6.83546	0.224870
$925$	−13.5707	−0.446204
$926$	20.1443	0.661983
$927$	3.32007	0.109045
$928$	1.00000	0.0328266
$929$	36.0040	1.18125	0.590626	−	0.806945i	$-0.298881\pi$
$929$	36.0040	1.18125	0.590626	+	0.806945i	$0.298881\pi$
$930$	8.07282	0.264718
$931$	17.5758	0.576025
$932$	23.5560	0.771602
$933$	24.6356	0.806534
$934$	−18.3623	−0.600831
$935$	−10.3584	−0.338757
$936$	−2.59694	−0.0848837
$937$	−11.4989	−0.375654	−0.187827	−	0.982202i	$-0.560144\pi$
$937$	−11.4989	−0.375654	−0.187827	+	0.982202i	$0.560144\pi$
$938$	−14.8407	−0.484565
$939$	−14.6589	−0.478377
$940$	−5.76558	−0.188053
$941$	−35.9023	−1.17038	−0.585191	−	0.810896i	$-0.698980\pi$
$941$	−35.9023	−1.17038	−0.585191	+	0.810896i	$0.698980\pi$
$942$	−15.4508	−0.503414
$943$	−8.89747	−0.289741
$944$	−7.86668	−0.256039
$945$	−2.84974	−0.0927020
$946$	39.4020	1.28107
$947$	20.3832	0.662366	0.331183	−	0.943566i	$-0.392552\pi$
$947$	20.3832	0.662366	0.331183	+	0.943566i	$0.392552\pi$
$948$	−1.58000	−0.0513161
$949$	5.66406	0.183863
$950$	−13.7191	−0.445106
$951$	−30.3018	−0.982602
$952$	3.53638	0.114615
$953$	17.6016	0.570173	0.285086	−	0.958502i	$-0.407978\pi$
$953$	17.6016	0.570173	0.285086	+	0.958502i	$0.407978\pi$
$954$	0.495848	0.0160537
$955$	1.55188	0.0502177
$956$	−8.75551	−0.283173
$957$	3.63487	0.117499
$958$	30.8903	0.998021
$959$	−6.28294	−0.202887
$960$	1.51539	0.0489091
$961$	−2.62086	−0.0845440
$962$	−13.0355	−0.420281
$963$	−13.7288	−0.442405
$964$	4.13290	0.133112
$965$	15.7688	0.507615
$966$	1.88053	0.0605049
$967$	22.2953	0.716969	0.358484	−	0.933536i	$-0.383294\pi$
$967$	22.2953	0.716969	0.358484	+	0.933536i	$0.383294\pi$
$968$	−2.21226	−0.0711049
$969$	−9.54256	−0.306551
$970$	−4.08859	−0.131277
$971$	−6.47033	−0.207643	−0.103821	−	0.994596i	$-0.533107\pi$
$971$	−6.47033	−0.207643	−0.103821	+	0.994596i	$0.533107\pi$
$972$	1.00000	0.0320750
$973$	8.31192	0.266468
$974$	6.24869	0.200221
$975$	−7.02105	−0.224853
$976$	−2.36513	−0.0757060
$977$	45.0880	1.44249	0.721246	−	0.692679i	$-0.243570\pi$
$977$	45.0880	1.44249	0.721246	+	0.692679i	$0.243570\pi$
$978$	−16.3437	−0.522615
$979$	4.73246	0.151250
$980$	−5.24875	−0.167665
$981$	−13.6094	−0.434513
$982$	−0.796371	−0.0254132
$983$	8.60542	0.274470	0.137235	−	0.990538i	$-0.456178\pi$
$983$	8.60542	0.274470	0.137235	+	0.990538i	$0.456178\pi$
$984$	8.89747	0.283641
$985$	16.4052	0.522714
$986$	1.88053	0.0598882
$987$	7.15479	0.227740
$988$	−13.1780	−0.419247
$989$	10.8400	0.344692
$990$	5.50826	0.175064
$991$	0.196126	0.00623014	0.00311507	−	0.999995i	$-0.499008\pi$
$991$	0.196126	0.00623014	0.00311507	+	0.999995i	$0.499008\pi$
$992$	5.32721	0.169139
$993$	−1.44625	−0.0458955
$994$	18.6703	0.592187
$995$	−16.4098	−0.520227
$996$	−10.4155	−0.330027
$997$	8.35757	0.264687	0.132343	−	0.991204i	$-0.457750\pi$
$997$	8.35757	0.264687	0.132343	+	0.991204i	$0.457750\pi$
$998$	−2.09833	−0.0664216
$999$	5.01955	0.158811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bb.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bb.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.51539 q^{5} -1.00000 q^{6} -1.88053 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.51539 q^{5} -1.00000 q^{6} -1.88053 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.51539 q^{10} -3.63487 q^{11} +1.00000 q^{12} +2.59694 q^{13} +1.88053 q^{14} +1.51539 q^{15} +1.00000 q^{16} +1.88053 q^{17} -1.00000 q^{18} -5.07441 q^{19} +1.51539 q^{20} -1.88053 q^{21} +3.63487 q^{22} +1.00000 q^{23} -1.00000 q^{24} -2.70358 q^{25} -2.59694 q^{26} +1.00000 q^{27} -1.88053 q^{28} -1.00000 q^{29} -1.51539 q^{30} -5.32721 q^{31} -1.00000 q^{32} -3.63487 q^{33} -1.88053 q^{34} -2.84974 q^{35} +1.00000 q^{36} +5.01955 q^{37} +5.07441 q^{38} +2.59694 q^{39} -1.51539 q^{40} -8.89747 q^{41} +1.88053 q^{42} +10.8400 q^{43} -3.63487 q^{44} +1.51539 q^{45} -1.00000 q^{46} -3.80468 q^{47} +1.00000 q^{48} -3.46362 q^{49} +2.70358 q^{50} +1.88053 q^{51} +2.59694 q^{52} -0.495848 q^{53} -1.00000 q^{54} -5.50826 q^{55} +1.88053 q^{56} -5.07441 q^{57} +1.00000 q^{58} -7.86668 q^{59} +1.51539 q^{60} -2.36513 q^{61} +5.32721 q^{62} -1.88053 q^{63} +1.00000 q^{64} +3.93539 q^{65} +3.63487 q^{66} -7.89177 q^{67} +1.88053 q^{68} +1.00000 q^{69} +2.84974 q^{70} +9.92825 q^{71} -1.00000 q^{72} +2.18105 q^{73} -5.01955 q^{74} -2.70358 q^{75} -5.07441 q^{76} +6.83546 q^{77} -2.59694 q^{78} -1.58000 q^{79} +1.51539 q^{80} +1.00000 q^{81} +8.89747 q^{82} -10.4155 q^{83} -1.88053 q^{84} +2.84974 q^{85} -10.8400 q^{86} -1.00000 q^{87} +3.63487 q^{88} -1.30196 q^{89} -1.51539 q^{90} -4.88362 q^{91} +1.00000 q^{92} -5.32721 q^{93} +3.80468 q^{94} -7.68973 q^{95} -1.00000 q^{96} +2.69804 q^{97} +3.46362 q^{98} -3.63487 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 3 q^{5} - 4 q^{6} - 2 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 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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