LMFDB - Embedded newform 402.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [402,2,Mod(1,402)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(402, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("402.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 402 = 2 \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	402.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:	$3.20998616126$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	402.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} +1.26795 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} +1.26795 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.46410 q^{10} -2.00000 q^{11} -1.00000 q^{12} +0.732051 q^{13} -1.26795 q^{14} +3.46410 q^{15} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{18} +2.00000 q^{19} -3.46410 q^{20} -1.26795 q^{21} +2.00000 q^{22} +5.26795 q^{23} +1.00000 q^{24} +7.00000 q^{25} -0.732051 q^{26} -1.00000 q^{27} +1.26795 q^{28} +7.66025 q^{29} -3.46410 q^{30} -0.196152 q^{31} -1.00000 q^{32} +2.00000 q^{33} -3.46410 q^{34} -4.39230 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} -0.732051 q^{39} +3.46410 q^{40} +1.46410 q^{41} +1.26795 q^{42} +10.9282 q^{43} -2.00000 q^{44} -3.46410 q^{45} -5.26795 q^{46} +9.66025 q^{47} -1.00000 q^{48} -5.39230 q^{49} -7.00000 q^{50} -3.46410 q^{51} +0.732051 q^{52} +0.535898 q^{53} +1.00000 q^{54} +6.92820 q^{55} -1.26795 q^{56} -2.00000 q^{57} -7.66025 q^{58} -9.46410 q^{59} +3.46410 q^{60} -4.73205 q^{61} +0.196152 q^{62} +1.26795 q^{63} +1.00000 q^{64} -2.53590 q^{65} -2.00000 q^{66} +1.00000 q^{67} +3.46410 q^{68} -5.26795 q^{69} +4.39230 q^{70} +8.19615 q^{71} -1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} -7.00000 q^{75} +2.00000 q^{76} -2.53590 q^{77} +0.732051 q^{78} +8.19615 q^{79} -3.46410 q^{80} +1.00000 q^{81} -1.46410 q^{82} +1.07180 q^{83} -1.26795 q^{84} -12.0000 q^{85} -10.9282 q^{86} -7.66025 q^{87} +2.00000 q^{88} -10.0000 q^{89} +3.46410 q^{90} +0.928203 q^{91} +5.26795 q^{92} +0.196152 q^{93} -9.66025 q^{94} -6.92820 q^{95} +1.00000 q^{96} -10.3923 q^{97} +5.39230 q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 6 q^{14} + 2 q^{16} - 2 q^{18} + 4 q^{19} - 6 q^{21} + 4 q^{22} + 14 q^{23} + 2 q^{24} + 14 q^{25} + 2 q^{26} - 2 q^{27} + 6 q^{28} - 2 q^{29} + 10 q^{31} - 2 q^{32} + 4 q^{33} + 12 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{38} + 2 q^{39} - 4 q^{41} + 6 q^{42} + 8 q^{43} - 4 q^{44} - 14 q^{46} + 2 q^{47} - 2 q^{48} + 10 q^{49} - 14 q^{50} - 2 q^{52} + 8 q^{53} + 2 q^{54} - 6 q^{56} - 4 q^{57} + 2 q^{58} - 12 q^{59} - 6 q^{61} - 10 q^{62} + 6 q^{63} + 2 q^{64} - 12 q^{65} - 4 q^{66} + 2 q^{67} - 14 q^{69} - 12 q^{70} + 6 q^{71} - 2 q^{72} + 12 q^{73} + 4 q^{74} - 14 q^{75} + 4 q^{76} - 12 q^{77} - 2 q^{78} + 6 q^{79} + 2 q^{81} + 4 q^{82} + 16 q^{83} - 6 q^{84} - 24 q^{85} - 8 q^{86} + 2 q^{87} + 4 q^{88} - 20 q^{89} - 12 q^{91} + 14 q^{92} - 10 q^{93} - 2 q^{94} + 2 q^{96} - 10 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 4 * q^11 - 2 * q^12 - 2 * q^13 - 6 * q^14 + 2 * q^16 - 2 * q^18 + 4 * q^19 - 6 * q^21 + 4 * q^22 + 14 * q^23 + 2 * q^24 + 14 * q^25 + 2 * q^26 - 2 * q^27 + 6 * q^28 - 2 * q^29 + 10 * q^31 - 2 * q^32 + 4 * q^33 + 12 * q^35 + 2 * q^36 - 4 * q^37 - 4 * q^38 + 2 * q^39 - 4 * q^41 + 6 * q^42 + 8 * q^43 - 4 * q^44 - 14 * q^46 + 2 * q^47 - 2 * q^48 + 10 * q^49 - 14 * q^50 - 2 * q^52 + 8 * q^53 + 2 * q^54 - 6 * q^56 - 4 * q^57 + 2 * q^58 - 12 * q^59 - 6 * q^61 - 10 * q^62 + 6 * q^63 + 2 * q^64 - 12 * q^65 - 4 * q^66 + 2 * q^67 - 14 * q^69 - 12 * q^70 + 6 * q^71 - 2 * q^72 + 12 * q^73 + 4 * q^74 - 14 * q^75 + 4 * q^76 - 12 * q^77 - 2 * q^78 + 6 * q^79 + 2 * q^81 + 4 * q^82 + 16 * q^83 - 6 * q^84 - 24 * q^85 - 8 * q^86 + 2 * q^87 + 4 * q^88 - 20 * q^89 - 12 * q^91 + 14 * q^92 - 10 * q^93 - 2 * q^94 + 2 * q^96 - 10 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	1.00000	0.408248
$7$	1.26795	0.479240	0.239620	−	0.970867i	$-0.422977\pi$
$7$	1.26795	0.479240	0.239620	+	0.970867i	$0.422977\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.46410	1.09545
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	−1.26795	−0.338874
$15$	3.46410	0.894427
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−3.46410	−0.774597
$21$	−1.26795	−0.276689
$22$	2.00000	0.426401
$23$	5.26795	1.09844	0.549222	−	0.835677i	$-0.314924\pi$
$23$	5.26795	1.09844	0.549222	+	0.835677i	$0.314924\pi$
$24$	1.00000	0.204124
$25$	7.00000	1.40000
$26$	−0.732051	−0.143567
$27$	−1.00000	−0.192450
$28$	1.26795	0.239620
$29$	7.66025	1.42247	0.711237	−	0.702953i	$-0.248135\pi$
$29$	7.66025	1.42247	0.711237	+	0.702953i	$0.248135\pi$
$30$	−3.46410	−0.632456
$31$	−0.196152	−0.0352300	−0.0176150	−	0.999845i	$-0.505607\pi$
$31$	−0.196152	−0.0352300	−0.0176150	+	0.999845i	$0.505607\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	−3.46410	−0.594089
$35$	−4.39230	−0.742435
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.00000	−0.324443
$39$	−0.732051	−0.117222
$40$	3.46410	0.547723
$41$	1.46410	0.228654	0.114327	−	0.993443i	$-0.463529\pi$
$41$	1.46410	0.228654	0.114327	+	0.993443i	$0.463529\pi$
$42$	1.26795	0.195649
$43$	10.9282	1.66654	0.833268	−	0.552870i	$-0.186467\pi$
$43$	10.9282	1.66654	0.833268	+	0.552870i	$0.186467\pi$
$44$	−2.00000	−0.301511
$45$	−3.46410	−0.516398
$46$	−5.26795	−0.776717
$47$	9.66025	1.40909	0.704546	−	0.709658i	$-0.251150\pi$
$47$	9.66025	1.40909	0.704546	+	0.709658i	$0.251150\pi$
$48$	−1.00000	−0.144338
$49$	−5.39230	−0.770329
$50$	−7.00000	−0.989949
$51$	−3.46410	−0.485071
$52$	0.732051	0.101517
$53$	0.535898	0.0736113	0.0368057	−	0.999322i	$-0.488282\pi$
$53$	0.535898	0.0736113	0.0368057	+	0.999322i	$0.488282\pi$
$54$	1.00000	0.136083
$55$	6.92820	0.934199
$56$	−1.26795	−0.169437
$57$	−2.00000	−0.264906
$58$	−7.66025	−1.00584
$59$	−9.46410	−1.23212	−0.616061	−	0.787699i	$-0.711272\pi$
$59$	−9.46410	−1.23212	−0.616061	+	0.787699i	$0.711272\pi$
$60$	3.46410	0.447214
$61$	−4.73205	−0.605877	−0.302939	−	0.953010i	$-0.597968\pi$
$61$	−4.73205	−0.605877	−0.302939	+	0.953010i	$0.597968\pi$
$62$	0.196152	0.0249114
$63$	1.26795	0.159747
$64$	1.00000	0.125000
$65$	−2.53590	−0.314539
$66$	−2.00000	−0.246183
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	3.46410	0.420084
$69$	−5.26795	−0.634187
$70$	4.39230	0.524981
$71$	8.19615	0.972704	0.486352	−	0.873763i	$-0.338327\pi$
$71$	8.19615	0.972704	0.486352	+	0.873763i	$0.338327\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	2.00000	0.232495
$75$	−7.00000	−0.808290
$76$	2.00000	0.229416
$77$	−2.53590	−0.288992
$78$	0.732051	0.0828884
$79$	8.19615	0.922139	0.461070	−	0.887364i	$-0.347466\pi$
$79$	8.19615	0.922139	0.461070	+	0.887364i	$0.347466\pi$
$80$	−3.46410	−0.387298
$81$	1.00000	0.111111
$82$	−1.46410	−0.161683
$83$	1.07180	0.117645	0.0588225	−	0.998268i	$-0.481265\pi$
$83$	1.07180	0.117645	0.0588225	+	0.998268i	$0.481265\pi$
$84$	−1.26795	−0.138345
$85$	−12.0000	−1.30158
$86$	−10.9282	−1.17842
$87$	−7.66025	−0.821265
$88$	2.00000	0.213201
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	3.46410	0.365148
$91$	0.928203	0.0973021
$92$	5.26795	0.549222
$93$	0.196152	0.0203401
$94$	−9.66025	−0.996379
$95$	−6.92820	−0.710819
$96$	1.00000	0.102062
$97$	−10.3923	−1.05518	−0.527589	−	0.849500i	$-0.676904\pi$
$97$	−10.3923	−1.05518	−0.527589	+	0.849500i	$0.676904\pi$
$98$	5.39230	0.544705
$99$	−2.00000	−0.201008
$100$	7.00000	0.700000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	3.46410	0.342997
$103$	−6.53590	−0.644001	−0.322001	−	0.946739i	$-0.604355\pi$
$103$	−6.53590	−0.644001	−0.322001	+	0.946739i	$0.604355\pi$
$104$	−0.732051	−0.0717835
$105$	4.39230	0.428645
$106$	−0.535898	−0.0520511
$107$	−12.3923	−1.19801	−0.599005	−	0.800746i	$-0.704437\pi$
$107$	−12.3923	−1.19801	−0.599005	+	0.800746i	$0.704437\pi$
$108$	−1.00000	−0.0962250
$109$	−7.26795	−0.696143	−0.348072	−	0.937468i	$-0.613163\pi$
$109$	−7.26795	−0.696143	−0.348072	+	0.937468i	$0.613163\pi$
$110$	−6.92820	−0.660578
$111$	2.00000	0.189832
$112$	1.26795	0.119810
$113$	6.53590	0.614846	0.307423	−	0.951573i	$-0.400533\pi$
$113$	6.53590	0.614846	0.307423	+	0.951573i	$0.400533\pi$
$114$	2.00000	0.187317
$115$	−18.2487	−1.70170
$116$	7.66025	0.711237
$117$	0.732051	0.0676781
$118$	9.46410	0.871241
$119$	4.39230	0.402642
$120$	−3.46410	−0.316228
$121$	−7.00000	−0.636364
$122$	4.73205	0.428420
$123$	−1.46410	−0.132014
$124$	−0.196152	−0.0176150
$125$	−6.92820	−0.619677
$126$	−1.26795	−0.112958
$127$	15.3205	1.35948	0.679738	−	0.733455i	$-0.262094\pi$
$127$	15.3205	1.35948	0.679738	+	0.733455i	$0.262094\pi$
$128$	−1.00000	−0.0883883
$129$	−10.9282	−0.962175
$130$	2.53590	0.222413
$131$	17.8564	1.56012	0.780061	−	0.625704i	$-0.215188\pi$
$131$	17.8564	1.56012	0.780061	+	0.625704i	$0.215188\pi$
$132$	2.00000	0.174078
$133$	2.53590	0.219890
$134$	−1.00000	−0.0863868
$135$	3.46410	0.298142
$136$	−3.46410	−0.297044
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	5.26795	0.448438
$139$	19.3205	1.63874	0.819372	−	0.573262i	$-0.194322\pi$
$139$	19.3205	1.63874	0.819372	+	0.573262i	$0.194322\pi$
$140$	−4.39230	−0.371218
$141$	−9.66025	−0.813540
$142$	−8.19615	−0.687806
$143$	−1.46410	−0.122434
$144$	1.00000	0.0833333
$145$	−26.5359	−2.20369
$146$	−6.00000	−0.496564
$147$	5.39230	0.444750
$148$	−2.00000	−0.164399
$149$	−22.5885	−1.85052	−0.925259	−	0.379335i	$-0.876153\pi$
$149$	−22.5885	−1.85052	−0.925259	+	0.379335i	$0.876153\pi$
$150$	7.00000	0.571548
$151$	17.8564	1.45313	0.726567	−	0.687096i	$-0.241115\pi$
$151$	17.8564	1.45313	0.726567	+	0.687096i	$0.241115\pi$
$152$	−2.00000	−0.162221
$153$	3.46410	0.280056
$154$	2.53590	0.204349
$155$	0.679492	0.0545781
$156$	−0.732051	−0.0586110
$157$	14.3923	1.14863	0.574315	−	0.818634i	$-0.305268\pi$
$157$	14.3923	1.14863	0.574315	+	0.818634i	$0.305268\pi$
$158$	−8.19615	−0.652051
$159$	−0.535898	−0.0424995
$160$	3.46410	0.273861
$161$	6.67949	0.526418
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	1.46410	0.114327
$165$	−6.92820	−0.539360
$166$	−1.07180	−0.0831876
$167$	17.6603	1.36659	0.683296	−	0.730142i	$-0.260546\pi$
$167$	17.6603	1.36659	0.683296	+	0.730142i	$0.260546\pi$
$168$	1.26795	0.0978244
$169$	−12.4641	−0.958777
$170$	12.0000	0.920358
$171$	2.00000	0.152944
$172$	10.9282	0.833268
$173$	20.7321	1.57623	0.788114	−	0.615529i	$-0.211058\pi$
$173$	20.7321	1.57623	0.788114	+	0.615529i	$0.211058\pi$
$174$	7.66025	0.580722
$175$	8.87564	0.670936
$176$	−2.00000	−0.150756
$177$	9.46410	0.711365
$178$	10.0000	0.749532
$179$	−7.07180	−0.528571	−0.264285	−	0.964445i	$-0.585136\pi$
$179$	−7.07180	−0.528571	−0.264285	+	0.964445i	$0.585136\pi$
$180$	−3.46410	−0.258199
$181$	2.39230	0.177819	0.0889093	−	0.996040i	$-0.471662\pi$
$181$	2.39230	0.177819	0.0889093	+	0.996040i	$0.471662\pi$
$182$	−0.928203	−0.0688030
$183$	4.73205	0.349803
$184$	−5.26795	−0.388358
$185$	6.92820	0.509372
$186$	−0.196152	−0.0143826
$187$	−6.92820	−0.506640
$188$	9.66025	0.704546
$189$	−1.26795	−0.0922297
$190$	6.92820	0.502625
$191$	10.5359	0.762351	0.381175	−	0.924503i	$-0.375519\pi$
$191$	10.5359	0.762351	0.381175	+	0.924503i	$0.375519\pi$
$192$	−1.00000	−0.0721688
$193$	−17.4641	−1.25709	−0.628547	−	0.777772i	$-0.716350\pi$
$193$	−17.4641	−1.25709	−0.628547	+	0.777772i	$0.716350\pi$
$194$	10.3923	0.746124
$195$	2.53590	0.181599
$196$	−5.39230	−0.385165
$197$	26.7846	1.90832	0.954162	−	0.299290i	$-0.0967498\pi$
$197$	26.7846	1.90832	0.954162	+	0.299290i	$0.0967498\pi$
$198$	2.00000	0.142134
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−7.00000	−0.494975
$201$	−1.00000	−0.0705346
$202$	10.0000	0.703598
$203$	9.71281	0.681706
$204$	−3.46410	−0.242536
$205$	−5.07180	−0.354230
$206$	6.53590	0.455378
$207$	5.26795	0.366148
$208$	0.732051	0.0507586
$209$	−4.00000	−0.276686
$210$	−4.39230	−0.303098
$211$	−7.85641	−0.540857	−0.270429	−	0.962740i	$-0.587165\pi$
$211$	−7.85641	−0.540857	−0.270429	+	0.962740i	$0.587165\pi$
$212$	0.535898	0.0368057
$213$	−8.19615	−0.561591
$214$	12.3923	0.847121
$215$	−37.8564	−2.58179
$216$	1.00000	0.0680414
$217$	−0.248711	−0.0168836
$218$	7.26795	0.492248
$219$	−6.00000	−0.405442
$220$	6.92820	0.467099
$221$	2.53590	0.170583
$222$	−2.00000	−0.134231
$223$	1.07180	0.0717728	0.0358864	−	0.999356i	$-0.488575\pi$
$223$	1.07180	0.0717728	0.0358864	+	0.999356i	$0.488575\pi$
$224$	−1.26795	−0.0847184
$225$	7.00000	0.466667
$226$	−6.53590	−0.434761
$227$	5.46410	0.362665	0.181333	−	0.983422i	$-0.441959\pi$
$227$	5.46410	0.362665	0.181333	+	0.983422i	$0.441959\pi$
$228$	−2.00000	−0.132453
$229$	−12.7321	−0.841358	−0.420679	−	0.907210i	$-0.638208\pi$
$229$	−12.7321	−0.841358	−0.420679	+	0.907210i	$0.638208\pi$
$230$	18.2487	1.20328
$231$	2.53590	0.166850
$232$	−7.66025	−0.502920
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	−0.732051	−0.0478557
$235$	−33.4641	−2.18296
$236$	−9.46410	−0.616061
$237$	−8.19615	−0.532397
$238$	−4.39230	−0.284711
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	3.46410	0.223607
$241$	10.5359	0.678677	0.339338	−	0.940664i	$-0.389797\pi$
$241$	10.5359	0.678677	0.339338	+	0.940664i	$0.389797\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	−4.73205	−0.302939
$245$	18.6795	1.19339
$246$	1.46410	0.0933477
$247$	1.46410	0.0931586
$248$	0.196152	0.0124557
$249$	−1.07180	−0.0679224
$250$	6.92820	0.438178
$251$	−17.8564	−1.12709	−0.563543	−	0.826087i	$-0.690562\pi$
$251$	−17.8564	−1.12709	−0.563543	+	0.826087i	$0.690562\pi$
$252$	1.26795	0.0798733
$253$	−10.5359	−0.662386
$254$	−15.3205	−0.961294
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	−30.7846	−1.92029	−0.960146	−	0.279500i	$-0.909831\pi$
$257$	−30.7846	−1.92029	−0.960146	+	0.279500i	$0.909831\pi$
$258$	10.9282	0.680360
$259$	−2.53590	−0.157573
$260$	−2.53590	−0.157270
$261$	7.66025	0.474158
$262$	−17.8564	−1.10317
$263$	18.0526	1.11317	0.556584	−	0.830791i	$-0.312112\pi$
$263$	18.0526	1.11317	0.556584	+	0.830791i	$0.312112\pi$
$264$	−2.00000	−0.123091
$265$	−1.85641	−0.114038
$266$	−2.53590	−0.155486
$267$	10.0000	0.611990
$268$	1.00000	0.0610847
$269$	24.0526	1.46651	0.733255	−	0.679954i	$-0.238000\pi$
$269$	24.0526	1.46651	0.733255	+	0.679954i	$0.238000\pi$
$270$	−3.46410	−0.210819
$271$	−20.5885	−1.25066	−0.625330	−	0.780361i	$-0.715036\pi$
$271$	−20.5885	−1.25066	−0.625330	+	0.780361i	$0.715036\pi$
$272$	3.46410	0.210042
$273$	−0.928203	−0.0561774
$274$	−12.9282	−0.781021
$275$	−14.0000	−0.844232
$276$	−5.26795	−0.317093
$277$	−0.928203	−0.0557703	−0.0278852	−	0.999611i	$-0.508877\pi$
$277$	−0.928203	−0.0557703	−0.0278852	+	0.999611i	$0.508877\pi$
$278$	−19.3205	−1.15877
$279$	−0.196152	−0.0117433
$280$	4.39230	0.262490
$281$	23.8564	1.42315	0.711577	−	0.702608i	$-0.247981\pi$
$281$	23.8564	1.42315	0.711577	+	0.702608i	$0.247981\pi$
$282$	9.66025	0.575260
$283$	−14.9282	−0.887390	−0.443695	−	0.896178i	$-0.646333\pi$
$283$	−14.9282	−0.887390	−0.443695	+	0.896178i	$0.646333\pi$
$284$	8.19615	0.486352
$285$	6.92820	0.410391
$286$	1.46410	0.0865741
$287$	1.85641	0.109580
$288$	−1.00000	−0.0589256
$289$	−5.00000	−0.294118
$290$	26.5359	1.55824
$291$	10.3923	0.609208
$292$	6.00000	0.351123
$293$	−28.0526	−1.63885	−0.819424	−	0.573188i	$-0.805707\pi$
$293$	−28.0526	−1.63885	−0.819424	+	0.573188i	$0.805707\pi$
$294$	−5.39230	−0.314486
$295$	32.7846	1.90879
$296$	2.00000	0.116248
$297$	2.00000	0.116052
$298$	22.5885	1.30851
$299$	3.85641	0.223022
$300$	−7.00000	−0.404145
$301$	13.8564	0.798670
$302$	−17.8564	−1.02752
$303$	10.0000	0.574485
$304$	2.00000	0.114708
$305$	16.3923	0.938621
$306$	−3.46410	−0.198030
$307$	−3.85641	−0.220097	−0.110048	−	0.993926i	$-0.535101\pi$
$307$	−3.85641	−0.220097	−0.110048	+	0.993926i	$0.535101\pi$
$308$	−2.53590	−0.144496
$309$	6.53590	0.371814
$310$	−0.679492	−0.0385925
$311$	6.92820	0.392862	0.196431	−	0.980518i	$-0.437065\pi$
$311$	6.92820	0.392862	0.196431	+	0.980518i	$0.437065\pi$
$312$	0.732051	0.0414442
$313$	5.32051	0.300733	0.150366	−	0.988630i	$-0.451955\pi$
$313$	5.32051	0.300733	0.150366	+	0.988630i	$0.451955\pi$
$314$	−14.3923	−0.812205
$315$	−4.39230	−0.247478
$316$	8.19615	0.461070
$317$	−20.0526	−1.12626	−0.563132	−	0.826367i	$-0.690404\pi$
$317$	−20.0526	−1.12626	−0.563132	+	0.826367i	$0.690404\pi$
$318$	0.535898	0.0300517
$319$	−15.3205	−0.857784
$320$	−3.46410	−0.193649
$321$	12.3923	0.691671
$322$	−6.67949	−0.372234
$323$	6.92820	0.385496
$324$	1.00000	0.0555556
$325$	5.12436	0.284248
$326$	−4.00000	−0.221540
$327$	7.26795	0.401919
$328$	−1.46410	−0.0808415
$329$	12.2487	0.675293
$330$	6.92820	0.381385
$331$	18.9282	1.04039	0.520194	−	0.854048i	$-0.325860\pi$
$331$	18.9282	1.04039	0.520194	+	0.854048i	$0.325860\pi$
$332$	1.07180	0.0588225
$333$	−2.00000	−0.109599
$334$	−17.6603	−0.966326
$335$	−3.46410	−0.189264
$336$	−1.26795	−0.0691723
$337$	15.8564	0.863753	0.431877	−	0.901933i	$-0.357852\pi$
$337$	15.8564	0.863753	0.431877	+	0.901933i	$0.357852\pi$
$338$	12.4641	0.677958
$339$	−6.53590	−0.354981
$340$	−12.0000	−0.650791
$341$	0.392305	0.0212445
$342$	−2.00000	−0.108148
$343$	−15.7128	−0.848412
$344$	−10.9282	−0.589209
$345$	18.2487	0.982478
$346$	−20.7321	−1.11456
$347$	−20.9282	−1.12348	−0.561742	−	0.827312i	$-0.689869\pi$
$347$	−20.9282	−1.12348	−0.561742	+	0.827312i	$0.689869\pi$
$348$	−7.66025	−0.410633
$349$	2.39230	0.128057	0.0640286	−	0.997948i	$-0.479605\pi$
$349$	2.39230	0.128057	0.0640286	+	0.997948i	$0.479605\pi$
$350$	−8.87564	−0.474423
$351$	−0.732051	−0.0390740
$352$	2.00000	0.106600
$353$	−23.3205	−1.24123	−0.620613	−	0.784117i	$-0.713116\pi$
$353$	−23.3205	−1.24123	−0.620613	+	0.784117i	$0.713116\pi$
$354$	−9.46410	−0.503011
$355$	−28.3923	−1.50691
$356$	−10.0000	−0.529999
$357$	−4.39230	−0.232465
$358$	7.07180	0.373756
$359$	−26.4449	−1.39571	−0.697853	−	0.716241i	$-0.745861\pi$
$359$	−26.4449	−1.39571	−0.697853	+	0.716241i	$0.745861\pi$
$360$	3.46410	0.182574
$361$	−15.0000	−0.789474
$362$	−2.39230	−0.125737
$363$	7.00000	0.367405
$364$	0.928203	0.0486511
$365$	−20.7846	−1.08792
$366$	−4.73205	−0.247348
$367$	−14.7321	−0.769007	−0.384503	−	0.923124i	$-0.625627\pi$
$367$	−14.7321	−0.769007	−0.384503	+	0.923124i	$0.625627\pi$
$368$	5.26795	0.274611
$369$	1.46410	0.0762181
$370$	−6.92820	−0.360180
$371$	0.679492	0.0352775
$372$	0.196152	0.0101700
$373$	25.5167	1.32120	0.660601	−	0.750737i	$-0.270301\pi$
$373$	25.5167	1.32120	0.660601	+	0.750737i	$0.270301\pi$
$374$	6.92820	0.358249
$375$	6.92820	0.357771
$376$	−9.66025	−0.498190
$377$	5.60770	0.288811
$378$	1.26795	0.0652163
$379$	14.2487	0.731907	0.365954	−	0.930633i	$-0.380743\pi$
$379$	14.2487	0.731907	0.365954	+	0.930633i	$0.380743\pi$
$380$	−6.92820	−0.355409
$381$	−15.3205	−0.784893
$382$	−10.5359	−0.539063
$383$	−22.5359	−1.15153	−0.575765	−	0.817615i	$-0.695296\pi$
$383$	−22.5359	−1.15153	−0.575765	+	0.817615i	$0.695296\pi$
$384$	1.00000	0.0510310
$385$	8.78461	0.447705
$386$	17.4641	0.888899
$387$	10.9282	0.555512
$388$	−10.3923	−0.527589
$389$	−8.73205	−0.442733	−0.221366	−	0.975191i	$-0.571052\pi$
$389$	−8.73205	−0.442733	−0.221366	+	0.975191i	$0.571052\pi$
$390$	−2.53590	−0.128410
$391$	18.2487	0.922877
$392$	5.39230	0.272353
$393$	−17.8564	−0.900737
$394$	−26.7846	−1.34939
$395$	−28.3923	−1.42857
$396$	−2.00000	−0.100504
$397$	4.53590	0.227650	0.113825	−	0.993501i	$-0.463690\pi$
$397$	4.53590	0.227650	0.113825	+	0.993501i	$0.463690\pi$
$398$	0	0
$399$	−2.53590	−0.126954
$400$	7.00000	0.350000
$401$	−4.14359	−0.206921	−0.103461	−	0.994634i	$-0.532992\pi$
$401$	−4.14359	−0.206921	−0.103461	+	0.994634i	$0.532992\pi$
$402$	1.00000	0.0498755
$403$	−0.143594	−0.00715290
$404$	−10.0000	−0.497519
$405$	−3.46410	−0.172133
$406$	−9.71281	−0.482039
$407$	4.00000	0.198273
$408$	3.46410	0.171499
$409$	−15.4641	−0.764651	−0.382325	−	0.924028i	$-0.624877\pi$
$409$	−15.4641	−0.764651	−0.382325	+	0.924028i	$0.624877\pi$
$410$	5.07180	0.250478
$411$	−12.9282	−0.637701
$412$	−6.53590	−0.322001
$413$	−12.0000	−0.590481
$414$	−5.26795	−0.258906
$415$	−3.71281	−0.182255
$416$	−0.732051	−0.0358917
$417$	−19.3205	−0.946129
$418$	4.00000	0.195646
$419$	−25.8564	−1.26317	−0.631584	−	0.775307i	$-0.717595\pi$
$419$	−25.8564	−1.26317	−0.631584	+	0.775307i	$0.717595\pi$
$420$	4.39230	0.214323
$421$	24.5359	1.19581	0.597903	−	0.801568i	$-0.296001\pi$
$421$	24.5359	1.19581	0.597903	+	0.801568i	$0.296001\pi$
$422$	7.85641	0.382444
$423$	9.66025	0.469698
$424$	−0.535898	−0.0260255
$425$	24.2487	1.17624
$426$	8.19615	0.397105
$427$	−6.00000	−0.290360
$428$	−12.3923	−0.599005
$429$	1.46410	0.0706875
$430$	37.8564	1.82560
$431$	23.8038	1.14659	0.573295	−	0.819349i	$-0.305665\pi$
$431$	23.8038	1.14659	0.573295	+	0.819349i	$0.305665\pi$
$432$	−1.00000	−0.0481125
$433$	−25.3205	−1.21683	−0.608413	−	0.793621i	$-0.708194\pi$
$433$	−25.3205	−1.21683	−0.608413	+	0.793621i	$0.708194\pi$
$434$	0.248711	0.0119385
$435$	26.5359	1.27230
$436$	−7.26795	−0.348072
$437$	10.5359	0.504000
$438$	6.00000	0.286691
$439$	−19.3205	−0.922118	−0.461059	−	0.887370i	$-0.652530\pi$
$439$	−19.3205	−0.922118	−0.461059	+	0.887370i	$0.652530\pi$
$440$	−6.92820	−0.330289
$441$	−5.39230	−0.256776
$442$	−2.53590	−0.120620
$443$	17.0718	0.811106	0.405553	−	0.914072i	$-0.367079\pi$
$443$	17.0718	0.811106	0.405553	+	0.914072i	$0.367079\pi$
$444$	2.00000	0.0949158
$445$	34.6410	1.64214
$446$	−1.07180	−0.0507510
$447$	22.5885	1.06840
$448$	1.26795	0.0599050
$449$	−22.3923	−1.05676	−0.528379	−	0.849009i	$-0.677200\pi$
$449$	−22.3923	−1.05676	−0.528379	+	0.849009i	$0.677200\pi$
$450$	−7.00000	−0.329983
$451$	−2.92820	−0.137884
$452$	6.53590	0.307423
$453$	−17.8564	−0.838967
$454$	−5.46410	−0.256443
$455$	−3.21539	−0.150740
$456$	2.00000	0.0936586
$457$	26.5359	1.24130	0.620648	−	0.784089i	$-0.286869\pi$
$457$	26.5359	1.24130	0.620648	+	0.784089i	$0.286869\pi$
$458$	12.7321	0.594930
$459$	−3.46410	−0.161690
$460$	−18.2487	−0.850851
$461$	26.1962	1.22008	0.610038	−	0.792372i	$-0.291154\pi$
$461$	26.1962	1.22008	0.610038	+	0.792372i	$0.291154\pi$
$462$	−2.53590	−0.117981
$463$	42.7321	1.98593	0.992963	−	0.118422i	$-0.0377834\pi$
$463$	42.7321	1.98593	0.992963	+	0.118422i	$0.0377834\pi$
$464$	7.66025	0.355618
$465$	−0.679492	−0.0315107
$466$	−19.8564	−0.919830
$467$	−5.07180	−0.234695	−0.117347	−	0.993091i	$-0.537439\pi$
$467$	−5.07180	−0.234695	−0.117347	+	0.993091i	$0.537439\pi$
$468$	0.732051	0.0338391
$469$	1.26795	0.0585485
$470$	33.4641	1.54358
$471$	−14.3923	−0.663162
$472$	9.46410	0.435621
$473$	−21.8564	−1.00496
$474$	8.19615	0.376462
$475$	14.0000	0.642364
$476$	4.39230	0.201321
$477$	0.535898	0.0245371
$478$	6.92820	0.316889
$479$	−5.66025	−0.258624	−0.129312	−	0.991604i	$-0.541277\pi$
$479$	−5.66025	−0.258624	−0.129312	+	0.991604i	$0.541277\pi$
$480$	−3.46410	−0.158114
$481$	−1.46410	−0.0667573
$482$	−10.5359	−0.479897
$483$	−6.67949	−0.303927
$484$	−7.00000	−0.318182
$485$	36.0000	1.63468
$486$	1.00000	0.0453609
$487$	−24.5885	−1.11421	−0.557105	−	0.830442i	$-0.688088\pi$
$487$	−24.5885	−1.11421	−0.557105	+	0.830442i	$0.688088\pi$
$488$	4.73205	0.214210
$489$	−4.00000	−0.180886
$490$	−18.6795	−0.843853
$491$	17.8564	0.805848	0.402924	−	0.915233i	$-0.367994\pi$
$491$	17.8564	0.805848	0.402924	+	0.915233i	$0.367994\pi$
$492$	−1.46410	−0.0660068
$493$	26.5359	1.19512
$494$	−1.46410	−0.0658730
$495$	6.92820	0.311400
$496$	−0.196152	−0.00880750
$497$	10.3923	0.466159
$498$	1.07180	0.0480284
$499$	23.3205	1.04397	0.521985	−	0.852955i	$-0.325192\pi$
$499$	23.3205	1.04397	0.521985	+	0.852955i	$0.325192\pi$
$500$	−6.92820	−0.309839
$501$	−17.6603	−0.789002
$502$	17.8564	0.796970
$503$	−17.4641	−0.778686	−0.389343	−	0.921093i	$-0.627298\pi$
$503$	−17.4641	−0.778686	−0.389343	+	0.921093i	$0.627298\pi$
$504$	−1.26795	−0.0564789
$505$	34.6410	1.54150
$506$	10.5359	0.468378
$507$	12.4641	0.553550
$508$	15.3205	0.679738
$509$	−30.5885	−1.35581	−0.677905	−	0.735150i	$-0.737112\pi$
$509$	−30.5885	−1.35581	−0.677905	+	0.735150i	$0.737112\pi$
$510$	−12.0000	−0.531369
$511$	7.60770	0.336545
$512$	−1.00000	−0.0441942
$513$	−2.00000	−0.0883022
$514$	30.7846	1.35785
$515$	22.6410	0.997682
$516$	−10.9282	−0.481087
$517$	−19.3205	−0.849715
$518$	2.53590	0.111421
$519$	−20.7321	−0.910036
$520$	2.53590	0.111207
$521$	−12.3923	−0.542917	−0.271458	−	0.962450i	$-0.587506\pi$
$521$	−12.3923	−0.542917	−0.271458	+	0.962450i	$0.587506\pi$
$522$	−7.66025	−0.335280
$523$	−9.85641	−0.430991	−0.215495	−	0.976505i	$-0.569137\pi$
$523$	−9.85641	−0.430991	−0.215495	+	0.976505i	$0.569137\pi$
$524$	17.8564	0.780061
$525$	−8.87564	−0.387365
$526$	−18.0526	−0.787129
$527$	−0.679492	−0.0295991
$528$	2.00000	0.0870388
$529$	4.75129	0.206578
$530$	1.85641	0.0806371
$531$	−9.46410	−0.410707
$532$	2.53590	0.109945
$533$	1.07180	0.0464247
$534$	−10.0000	−0.432742
$535$	42.9282	1.85595
$536$	−1.00000	−0.0431934
$537$	7.07180	0.305171
$538$	−24.0526	−1.03698
$539$	10.7846	0.464526
$540$	3.46410	0.149071
$541$	−29.1244	−1.25215	−0.626077	−	0.779761i	$-0.715340\pi$
$541$	−29.1244	−1.25215	−0.626077	+	0.779761i	$0.715340\pi$
$542$	20.5885	0.884350
$543$	−2.39230	−0.102664
$544$	−3.46410	−0.148522
$545$	25.1769	1.07846
$546$	0.928203	0.0397234
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	12.9282	0.552265
$549$	−4.73205	−0.201959
$550$	14.0000	0.596962
$551$	15.3205	0.652676
$552$	5.26795	0.224219
$553$	10.3923	0.441926
$554$	0.928203	0.0394356
$555$	−6.92820	−0.294086
$556$	19.3205	0.819372
$557$	34.5885	1.46556	0.732780	−	0.680466i	$-0.238222\pi$
$557$	34.5885	1.46556	0.732780	+	0.680466i	$0.238222\pi$
$558$	0.196152	0.00830379
$559$	8.00000	0.338364
$560$	−4.39230	−0.185609
$561$	6.92820	0.292509
$562$	−23.8564	−1.00632
$563$	−9.85641	−0.415398	−0.207699	−	0.978193i	$-0.566597\pi$
$563$	−9.85641	−0.415398	−0.207699	+	0.978193i	$0.566597\pi$
$564$	−9.66025	−0.406770
$565$	−22.6410	−0.952515
$566$	14.9282	0.627479
$567$	1.26795	0.0532489
$568$	−8.19615	−0.343903
$569$	1.60770	0.0673981	0.0336990	−	0.999432i	$-0.489271\pi$
$569$	1.60770	0.0673981	0.0336990	+	0.999432i	$0.489271\pi$
$570$	−6.92820	−0.290191
$571$	33.0718	1.38401	0.692006	−	0.721892i	$-0.256727\pi$
$571$	33.0718	1.38401	0.692006	+	0.721892i	$0.256727\pi$
$572$	−1.46410	−0.0612172
$573$	−10.5359	−0.440143
$574$	−1.85641	−0.0774849
$575$	36.8756	1.53782
$576$	1.00000	0.0416667
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	5.00000	0.207973
$579$	17.4641	0.725783
$580$	−26.5359	−1.10184
$581$	1.35898	0.0563802
$582$	−10.3923	−0.430775
$583$	−1.07180	−0.0443893
$584$	−6.00000	−0.248282
$585$	−2.53590	−0.104846
$586$	28.0526	1.15884
$587$	−12.9282	−0.533604	−0.266802	−	0.963751i	$-0.585967\pi$
$587$	−12.9282	−0.533604	−0.266802	+	0.963751i	$0.585967\pi$
$588$	5.39230	0.222375
$589$	−0.392305	−0.0161646
$590$	−32.7846	−1.34972
$591$	−26.7846	−1.10177
$592$	−2.00000	−0.0821995
$593$	−4.14359	−0.170157	−0.0850785	−	0.996374i	$-0.527114\pi$
$593$	−4.14359	−0.170157	−0.0850785	+	0.996374i	$0.527114\pi$
$594$	−2.00000	−0.0820610
$595$	−15.2154	−0.623770
$596$	−22.5885	−0.925259
$597$	0	0
$598$	−3.85641	−0.157700
$599$	−26.6410	−1.08852	−0.544261	−	0.838916i	$-0.683190\pi$
$599$	−26.6410	−1.08852	−0.544261	+	0.838916i	$0.683190\pi$
$600$	7.00000	0.285774
$601$	−6.53590	−0.266605	−0.133302	−	0.991075i	$-0.542558\pi$
$601$	−6.53590	−0.266605	−0.133302	+	0.991075i	$0.542558\pi$
$602$	−13.8564	−0.564745
$603$	1.00000	0.0407231
$604$	17.8564	0.726567
$605$	24.2487	0.985850
$606$	−10.0000	−0.406222
$607$	38.2487	1.55247	0.776234	−	0.630445i	$-0.217128\pi$
$607$	38.2487	1.55247	0.776234	+	0.630445i	$0.217128\pi$
$608$	−2.00000	−0.0811107
$609$	−9.71281	−0.393583
$610$	−16.3923	−0.663705
$611$	7.07180	0.286094
$612$	3.46410	0.140028
$613$	−30.7846	−1.24338	−0.621689	−	0.783264i	$-0.713553\pi$
$613$	−30.7846	−1.24338	−0.621689	+	0.783264i	$0.713553\pi$
$614$	3.85641	0.155632
$615$	5.07180	0.204515
$616$	2.53590	0.102174
$617$	−29.7128	−1.19619	−0.598096	−	0.801424i	$-0.704076\pi$
$617$	−29.7128	−1.19619	−0.598096	+	0.801424i	$0.704076\pi$
$618$	−6.53590	−0.262912
$619$	33.7128	1.35503	0.677516	−	0.735508i	$-0.263056\pi$
$619$	33.7128	1.35503	0.677516	+	0.735508i	$0.263056\pi$
$620$	0.679492	0.0272891
$621$	−5.26795	−0.211396
$622$	−6.92820	−0.277796
$623$	−12.6795	−0.507993
$624$	−0.732051	−0.0293055
$625$	−11.0000	−0.440000
$626$	−5.32051	−0.212650
$627$	4.00000	0.159745
$628$	14.3923	0.574315
$629$	−6.92820	−0.276246
$630$	4.39230	0.174994
$631$	11.4115	0.454286	0.227143	−	0.973861i	$-0.427061\pi$
$631$	11.4115	0.454286	0.227143	+	0.973861i	$0.427061\pi$
$632$	−8.19615	−0.326025
$633$	7.85641	0.312264
$634$	20.0526	0.796389
$635$	−53.0718	−2.10609
$636$	−0.535898	−0.0212498
$637$	−3.94744	−0.156403
$638$	15.3205	0.606545
$639$	8.19615	0.324235
$640$	3.46410	0.136931
$641$	−0.928203	−0.0366618	−0.0183309	−	0.999832i	$-0.505835\pi$
$641$	−0.928203	−0.0366618	−0.0183309	+	0.999832i	$0.505835\pi$
$642$	−12.3923	−0.489085
$643$	19.8564	0.783060	0.391530	−	0.920165i	$-0.371946\pi$
$643$	19.8564	0.783060	0.391530	+	0.920165i	$0.371946\pi$
$644$	6.67949	0.263209
$645$	37.8564	1.49059
$646$	−6.92820	−0.272587
$647$	−16.3923	−0.644448	−0.322224	−	0.946663i	$-0.604430\pi$
$647$	−16.3923	−0.644448	−0.322224	+	0.946663i	$0.604430\pi$
$648$	−1.00000	−0.0392837
$649$	18.9282	0.742997
$650$	−5.12436	−0.200994
$651$	0.248711	0.00974776
$652$	4.00000	0.156652
$653$	0.928203	0.0363234	0.0181617	−	0.999835i	$-0.494219\pi$
$653$	0.928203	0.0363234	0.0181617	+	0.999835i	$0.494219\pi$
$654$	−7.26795	−0.284199
$655$	−61.8564	−2.41693
$656$	1.46410	0.0571636
$657$	6.00000	0.234082
$658$	−12.2487	−0.477504
$659$	−23.3205	−0.908438	−0.454219	−	0.890890i	$-0.650082\pi$
$659$	−23.3205	−0.908438	−0.454219	+	0.890890i	$0.650082\pi$
$660$	−6.92820	−0.269680
$661$	36.4449	1.41754	0.708770	−	0.705439i	$-0.249250\pi$
$661$	36.4449	1.41754	0.708770	+	0.705439i	$0.249250\pi$
$662$	−18.9282	−0.735666
$663$	−2.53590	−0.0984861
$664$	−1.07180	−0.0415938
$665$	−8.78461	−0.340653
$666$	2.00000	0.0774984
$667$	40.3538	1.56251
$668$	17.6603	0.683296
$669$	−1.07180	−0.0414381
$670$	3.46410	0.133830
$671$	9.46410	0.365358
$672$	1.26795	0.0489122
$673$	−15.1769	−0.585027	−0.292514	−	0.956261i	$-0.594492\pi$
$673$	−15.1769	−0.585027	−0.292514	+	0.956261i	$0.594492\pi$
$674$	−15.8564	−0.610766
$675$	−7.00000	−0.269430
$676$	−12.4641	−0.479389
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	6.53590	0.251010
$679$	−13.1769	−0.505684
$680$	12.0000	0.460179
$681$	−5.46410	−0.209385
$682$	−0.392305	−0.0150221
$683$	41.8564	1.60159	0.800795	−	0.598938i	$-0.204410\pi$
$683$	41.8564	1.60159	0.800795	+	0.598938i	$0.204410\pi$
$684$	2.00000	0.0764719
$685$	−44.7846	−1.71113
$686$	15.7128	0.599918
$687$	12.7321	0.485758
$688$	10.9282	0.416634
$689$	0.392305	0.0149456
$690$	−18.2487	−0.694717
$691$	−9.07180	−0.345107	−0.172554	−	0.985000i	$-0.555202\pi$
$691$	−9.07180	−0.345107	−0.172554	+	0.985000i	$0.555202\pi$
$692$	20.7321	0.788114
$693$	−2.53590	−0.0963308
$694$	20.9282	0.794424
$695$	−66.9282	−2.53873
$696$	7.66025	0.290361
$697$	5.07180	0.192108
$698$	−2.39230	−0.0905501
$699$	−19.8564	−0.751038
$700$	8.87564	0.335468
$701$	40.6410	1.53499	0.767495	−	0.641055i	$-0.221503\pi$
$701$	40.6410	1.53499	0.767495	+	0.641055i	$0.221503\pi$
$702$	0.732051	0.0276295
$703$	−4.00000	−0.150863
$704$	−2.00000	−0.0753778
$705$	33.4641	1.26033
$706$	23.3205	0.877679
$707$	−12.6795	−0.476861
$708$	9.46410	0.355683
$709$	40.9282	1.53709	0.768545	−	0.639795i	$-0.220981\pi$
$709$	40.9282	1.53709	0.768545	+	0.639795i	$0.220981\pi$
$710$	28.3923	1.06554
$711$	8.19615	0.307380
$712$	10.0000	0.374766
$713$	−1.03332	−0.0386982
$714$	4.39230	0.164378
$715$	5.07180	0.189674
$716$	−7.07180	−0.264285
$717$	6.92820	0.258738
$718$	26.4449	0.986914
$719$	12.9808	0.484101	0.242050	−	0.970264i	$-0.422180\pi$
$719$	12.9808	0.484101	0.242050	+	0.970264i	$0.422180\pi$
$720$	−3.46410	−0.129099
$721$	−8.28719	−0.308631
$722$	15.0000	0.558242
$723$	−10.5359	−0.391834
$724$	2.39230	0.0889093
$725$	53.6218	1.99146
$726$	−7.00000	−0.259794
$727$	−19.4115	−0.719934	−0.359967	−	0.932965i	$-0.617212\pi$
$727$	−19.4115	−0.719934	−0.359967	+	0.932965i	$0.617212\pi$
$728$	−0.928203	−0.0344015
$729$	1.00000	0.0370370
$730$	20.7846	0.769273
$731$	37.8564	1.40017
$732$	4.73205	0.174902
$733$	−12.7321	−0.470269	−0.235135	−	0.971963i	$-0.575553\pi$
$733$	−12.7321	−0.470269	−0.235135	+	0.971963i	$0.575553\pi$
$734$	14.7321	0.543770
$735$	−18.6795	−0.689003
$736$	−5.26795	−0.194179
$737$	−2.00000	−0.0736709
$738$	−1.46410	−0.0538943
$739$	−37.1769	−1.36757	−0.683787	−	0.729681i	$-0.739668\pi$
$739$	−37.1769	−1.36757	−0.683787	+	0.729681i	$0.739668\pi$
$740$	6.92820	0.254686
$741$	−1.46410	−0.0537851
$742$	−0.679492	−0.0249449
$743$	17.6603	0.647892	0.323946	−	0.946076i	$-0.394990\pi$
$743$	17.6603	0.647892	0.323946	+	0.946076i	$0.394990\pi$
$744$	−0.196152	−0.00719130
$745$	78.2487	2.86681
$746$	−25.5167	−0.934231
$747$	1.07180	0.0392150
$748$	−6.92820	−0.253320
$749$	−15.7128	−0.574134
$750$	−6.92820	−0.252982
$751$	0.679492	0.0247950	0.0123975	−	0.999923i	$-0.496054\pi$
$751$	0.679492	0.0247950	0.0123975	+	0.999923i	$0.496054\pi$
$752$	9.66025	0.352273
$753$	17.8564	0.650724
$754$	−5.60770	−0.204220
$755$	−61.8564	−2.25119
$756$	−1.26795	−0.0461149
$757$	−26.1962	−0.952115	−0.476058	−	0.879414i	$-0.657935\pi$
$757$	−26.1962	−0.952115	−0.476058	+	0.879414i	$0.657935\pi$
$758$	−14.2487	−0.517537
$759$	10.5359	0.382429
$760$	6.92820	0.251312
$761$	0.535898	0.0194263	0.00971315	−	0.999953i	$-0.496908\pi$
$761$	0.535898	0.0194263	0.00971315	+	0.999953i	$0.496908\pi$
$762$	15.3205	0.555003
$763$	−9.21539	−0.333620
$764$	10.5359	0.381175
$765$	−12.0000	−0.433861
$766$	22.5359	0.814255
$767$	−6.92820	−0.250163
$768$	−1.00000	−0.0360844
$769$	−11.1769	−0.403050	−0.201525	−	0.979483i	$-0.564590\pi$
$769$	−11.1769	−0.403050	−0.201525	+	0.979483i	$0.564590\pi$
$770$	−8.78461	−0.316575
$771$	30.7846	1.10868
$772$	−17.4641	−0.628547
$773$	1.80385	0.0648799	0.0324399	−	0.999474i	$-0.489672\pi$
$773$	1.80385	0.0648799	0.0324399	+	0.999474i	$0.489672\pi$
$774$	−10.9282	−0.392806
$775$	−1.37307	−0.0493220
$776$	10.3923	0.373062
$777$	2.53590	0.0909748
$778$	8.73205	0.313059
$779$	2.92820	0.104914
$780$	2.53590	0.0907997
$781$	−16.3923	−0.586563
$782$	−18.2487	−0.652573
$783$	−7.66025	−0.273755
$784$	−5.39230	−0.192582
$785$	−49.8564	−1.77945
$786$	17.8564	0.636917
$787$	−34.2487	−1.22083	−0.610417	−	0.792080i	$-0.708998\pi$
$787$	−34.2487	−1.22083	−0.610417	+	0.792080i	$0.708998\pi$
$788$	26.7846	0.954162
$789$	−18.0526	−0.642688
$790$	28.3923	1.01015
$791$	8.28719	0.294658
$792$	2.00000	0.0710669
$793$	−3.46410	−0.123014
$794$	−4.53590	−0.160973
$795$	1.85641	0.0658400
$796$	0	0
$797$	−38.9808	−1.38077	−0.690385	−	0.723442i	$-0.742559\pi$
$797$	−38.9808	−1.38077	−0.690385	+	0.723442i	$0.742559\pi$
$798$	2.53590	0.0897698
$799$	33.4641	1.18387
$800$	−7.00000	−0.247487
$801$	−10.0000	−0.353333
$802$	4.14359	0.146315
$803$	−12.0000	−0.423471
$804$	−1.00000	−0.0352673
$805$	−23.1384	−0.815523
$806$	0.143594	0.00505787
$807$	−24.0526	−0.846690
$808$	10.0000	0.351799
$809$	−35.8564	−1.26064	−0.630322	−	0.776334i	$-0.717077\pi$
$809$	−35.8564	−1.26064	−0.630322	+	0.776334i	$0.717077\pi$
$810$	3.46410	0.121716
$811$	−7.60770	−0.267142	−0.133571	−	0.991039i	$-0.542644\pi$
$811$	−7.60770	−0.267142	−0.133571	+	0.991039i	$0.542644\pi$
$812$	9.71281	0.340853
$813$	20.5885	0.722069
$814$	−4.00000	−0.140200
$815$	−13.8564	−0.485369
$816$	−3.46410	−0.121268
$817$	21.8564	0.764659
$818$	15.4641	0.540690
$819$	0.928203	0.0324340
$820$	−5.07180	−0.177115
$821$	−19.2679	−0.672456	−0.336228	−	0.941781i	$-0.609151\pi$
$821$	−19.2679	−0.672456	−0.336228	+	0.941781i	$0.609151\pi$
$822$	12.9282	0.450923
$823$	30.2487	1.05440	0.527202	−	0.849740i	$-0.323241\pi$
$823$	30.2487	1.05440	0.527202	+	0.849740i	$0.323241\pi$
$824$	6.53590	0.227689
$825$	14.0000	0.487417
$826$	12.0000	0.417533
$827$	−40.3923	−1.40458	−0.702289	−	0.711892i	$-0.747839\pi$
$827$	−40.3923	−1.40458	−0.702289	+	0.711892i	$0.747839\pi$
$828$	5.26795	0.183074
$829$	44.6410	1.55045	0.775223	−	0.631687i	$-0.217637\pi$
$829$	44.6410	1.55045	0.775223	+	0.631687i	$0.217637\pi$
$830$	3.71281	0.128874
$831$	0.928203	0.0321990
$832$	0.732051	0.0253793
$833$	−18.6795	−0.647206
$834$	19.3205	0.669014
$835$	−61.1769	−2.11711
$836$	−4.00000	−0.138343
$837$	0.196152	0.00678002
$838$	25.8564	0.893195
$839$	−43.1244	−1.48882	−0.744409	−	0.667724i	$-0.767269\pi$
$839$	−43.1244	−1.48882	−0.744409	+	0.667724i	$0.767269\pi$
$840$	−4.39230	−0.151549
$841$	29.6795	1.02343
$842$	−24.5359	−0.845563
$843$	−23.8564	−0.821658
$844$	−7.85641	−0.270429
$845$	43.1769	1.48533
$846$	−9.66025	−0.332126
$847$	−8.87564	−0.304971
$848$	0.535898	0.0184028
$849$	14.9282	0.512335
$850$	−24.2487	−0.831724
$851$	−10.5359	−0.361166
$852$	−8.19615	−0.280796
$853$	−13.6077	−0.465919	−0.232959	−	0.972486i	$-0.574841\pi$
$853$	−13.6077	−0.465919	−0.232959	+	0.972486i	$0.574841\pi$
$854$	6.00000	0.205316
$855$	−6.92820	−0.236940
$856$	12.3923	0.423560
$857$	35.3205	1.20653	0.603263	−	0.797542i	$-0.293867\pi$
$857$	35.3205	1.20653	0.603263	+	0.797542i	$0.293867\pi$
$858$	−1.46410	−0.0499836
$859$	−2.00000	−0.0682391	−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000	−0.0682391	−0.0341196	+	0.999418i	$0.510863\pi$
$860$	−37.8564	−1.29089
$861$	−1.85641	−0.0632662
$862$	−23.8038	−0.810762
$863$	−4.98076	−0.169547	−0.0847736	−	0.996400i	$-0.527017\pi$
$863$	−4.98076	−0.169547	−0.0847736	+	0.996400i	$0.527017\pi$
$864$	1.00000	0.0340207
$865$	−71.8179	−2.44188
$866$	25.3205	0.860426
$867$	5.00000	0.169809
$868$	−0.248711	−0.00844181
$869$	−16.3923	−0.556071
$870$	−26.5359	−0.899651
$871$	0.732051	0.0248046
$872$	7.26795	0.246124
$873$	−10.3923	−0.351726
$874$	−10.5359	−0.356382
$875$	−8.78461	−0.296974
$876$	−6.00000	−0.202721
$877$	9.32051	0.314731	0.157366	−	0.987540i	$-0.449700\pi$
$877$	9.32051	0.314731	0.157366	+	0.987540i	$0.449700\pi$
$878$	19.3205	0.652036
$879$	28.0526	0.946189
$880$	6.92820	0.233550
$881$	−48.9282	−1.64843	−0.824217	−	0.566275i	$-0.808384\pi$
$881$	−48.9282	−1.64843	−0.824217	+	0.566275i	$0.808384\pi$
$882$	5.39230	0.181568
$883$	44.3923	1.49392	0.746960	−	0.664869i	$-0.231513\pi$
$883$	44.3923	1.49392	0.746960	+	0.664869i	$0.231513\pi$
$884$	2.53590	0.0852915
$885$	−32.7846	−1.10204
$886$	−17.0718	−0.573538
$887$	−3.12436	−0.104906	−0.0524528	−	0.998623i	$-0.516704\pi$
$887$	−3.12436	−0.104906	−0.0524528	+	0.998623i	$0.516704\pi$
$888$	−2.00000	−0.0671156
$889$	19.4256	0.651515
$890$	−34.6410	−1.16117
$891$	−2.00000	−0.0670025
$892$	1.07180	0.0358864
$893$	19.3205	0.646536
$894$	−22.5885	−0.755471
$895$	24.4974	0.818859
$896$	−1.26795	−0.0423592
$897$	−3.85641	−0.128762
$898$	22.3923	0.747241
$899$	−1.50258	−0.0501138
$900$	7.00000	0.233333
$901$	1.85641	0.0618459
$902$	2.92820	0.0974985
$903$	−13.8564	−0.461112
$904$	−6.53590	−0.217381
$905$	−8.28719	−0.275475
$906$	17.8564	0.593239
$907$	−53.4256	−1.77397	−0.886984	−	0.461799i	$-0.847204\pi$
$907$	−53.4256	−1.77397	−0.886984	+	0.461799i	$0.847204\pi$
$908$	5.46410	0.181333
$909$	−10.0000	−0.331679
$910$	3.21539	0.106589
$911$	43.8038	1.45129	0.725643	−	0.688071i	$-0.241542\pi$
$911$	43.8038	1.45129	0.725643	+	0.688071i	$0.241542\pi$
$912$	−2.00000	−0.0662266
$913$	−2.14359	−0.0709426
$914$	−26.5359	−0.877730
$915$	−16.3923	−0.541913
$916$	−12.7321	−0.420679
$917$	22.6410	0.747672
$918$	3.46410	0.114332
$919$	13.3731	0.441137	0.220568	−	0.975372i	$-0.429209\pi$
$919$	13.3731	0.441137	0.220568	+	0.975372i	$0.429209\pi$
$920$	18.2487	0.601642
$921$	3.85641	0.127073
$922$	−26.1962	−0.862724
$923$	6.00000	0.197492
$924$	2.53590	0.0834249
$925$	−14.0000	−0.460317
$926$	−42.7321	−1.40426
$927$	−6.53590	−0.214667
$928$	−7.66025	−0.251460
$929$	−52.3923	−1.71894	−0.859468	−	0.511190i	$-0.829205\pi$
$929$	−52.3923	−1.71894	−0.859468	+	0.511190i	$0.829205\pi$
$930$	0.679492	0.0222814
$931$	−10.7846	−0.353451
$932$	19.8564	0.650418
$933$	−6.92820	−0.226819
$934$	5.07180	0.165954
$935$	24.0000	0.784884
$936$	−0.732051	−0.0239278
$937$	41.7128	1.36270	0.681349	−	0.731959i	$-0.261394\pi$
$937$	41.7128	1.36270	0.681349	+	0.731959i	$0.261394\pi$
$938$	−1.26795	−0.0414000
$939$	−5.32051	−0.173628
$940$	−33.4641	−1.09148
$941$	−8.92820	−0.291051	−0.145526	−	0.989354i	$-0.546487\pi$
$941$	−8.92820	−0.291051	−0.145526	+	0.989354i	$0.546487\pi$
$942$	14.3923	0.468927
$943$	7.71281	0.251164
$944$	−9.46410	−0.308030
$945$	4.39230	0.142882
$946$	21.8564	0.710613
$947$	36.7846	1.19534	0.597670	−	0.801743i	$-0.296093\pi$
$947$	36.7846	1.19534	0.597670	+	0.801743i	$0.296093\pi$
$948$	−8.19615	−0.266199
$949$	4.39230	0.142580
$950$	−14.0000	−0.454220
$951$	20.0526	0.650249
$952$	−4.39230	−0.142355
$953$	53.3205	1.72722	0.863610	−	0.504160i	$-0.168198\pi$
$953$	53.3205	1.72722	0.863610	+	0.504160i	$0.168198\pi$
$954$	−0.535898	−0.0173504
$955$	−36.4974	−1.18103
$956$	−6.92820	−0.224074
$957$	15.3205	0.495242
$958$	5.66025	0.182875
$959$	16.3923	0.529335
$960$	3.46410	0.111803
$961$	−30.9615	−0.998759
$962$	1.46410	0.0472045
$963$	−12.3923	−0.399336
$964$	10.5359	0.339338
$965$	60.4974	1.94748
$966$	6.67949	0.214909
$967$	−52.7846	−1.69744	−0.848719	−	0.528844i	$-0.822626\pi$
$967$	−52.7846	−1.69744	−0.848719	+	0.528844i	$0.822626\pi$
$968$	7.00000	0.224989
$969$	−6.92820	−0.222566
$970$	−36.0000	−1.15589
$971$	−23.3205	−0.748391	−0.374195	−	0.927350i	$-0.622081\pi$
$971$	−23.3205	−0.748391	−0.374195	+	0.927350i	$0.622081\pi$
$972$	−1.00000	−0.0320750
$973$	24.4974	0.785351
$974$	24.5885	0.787865
$975$	−5.12436	−0.164111
$976$	−4.73205	−0.151469
$977$	44.5359	1.42483	0.712415	−	0.701759i	$-0.247601\pi$
$977$	44.5359	1.42483	0.712415	+	0.701759i	$0.247601\pi$
$978$	4.00000	0.127906
$979$	20.0000	0.639203
$980$	18.6795	0.596694
$981$	−7.26795	−0.232048
$982$	−17.8564	−0.569821
$983$	−1.07180	−0.0341850	−0.0170925	−	0.999854i	$-0.505441\pi$
$983$	−1.07180	−0.0341850	−0.0170925	+	0.999854i	$0.505441\pi$
$984$	1.46410	0.0466739
$985$	−92.7846	−2.95636
$986$	−26.5359	−0.845075
$987$	−12.2487	−0.389881
$988$	1.46410	0.0465793
$989$	57.5692	1.83059
$990$	−6.92820	−0.220193
$991$	44.5885	1.41640	0.708200	−	0.706012i	$-0.249508\pi$
$991$	44.5885	1.41640	0.708200	+	0.706012i	$0.249508\pi$
$992$	0.196152	0.00622785
$993$	−18.9282	−0.600668
$994$	−10.3923	−0.329624
$995$	0	0
$996$	−1.07180	−0.0339612
$997$	−32.5359	−1.03042	−0.515211	−	0.857063i	$-0.672286\pi$
$997$	−32.5359	−1.03042	−0.515211	+	0.857063i	$0.672286\pi$
$998$	−23.3205	−0.738198
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.2.a.e.1.1	✓	2
3.2	odd	2		1206.2.a.k.1.2		2
4.3	odd	2		3216.2.a.o.1.1		2
12.11	even	2		9648.2.a.ba.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.2.a.e.1.1	✓	2	1.1	even	1	trivial
1206.2.a.k.1.2		2	3.2	odd	2
3216.2.a.o.1.1		2	4.3	odd	2
9648.2.a.ba.1.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	402.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} +1.26795 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} +1.26795 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.46410 q^{10} -2.00000 q^{11} -1.00000 q^{12} +0.732051 q^{13} -1.26795 q^{14} +3.46410 q^{15} +1.00000 q^{16} +3.46410 q^{17} -1.00000 q^{18} +2.00000 q^{19} -3.46410 q^{20} -1.26795 q^{21} +2.00000 q^{22} +5.26795 q^{23} +1.00000 q^{24} +7.00000 q^{25} -0.732051 q^{26} -1.00000 q^{27} +1.26795 q^{28} +7.66025 q^{29} -3.46410 q^{30} -0.196152 q^{31} -1.00000 q^{32} +2.00000 q^{33} -3.46410 q^{34} -4.39230 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{38} -0.732051 q^{39} +3.46410 q^{40} +1.46410 q^{41} +1.26795 q^{42} +10.9282 q^{43} -2.00000 q^{44} -3.46410 q^{45} -5.26795 q^{46} +9.66025 q^{47} -1.00000 q^{48} -5.39230 q^{49} -7.00000 q^{50} -3.46410 q^{51} +0.732051 q^{52} +0.535898 q^{53} +1.00000 q^{54} +6.92820 q^{55} -1.26795 q^{56} -2.00000 q^{57} -7.66025 q^{58} -9.46410 q^{59} +3.46410 q^{60} -4.73205 q^{61} +0.196152 q^{62} +1.26795 q^{63} +1.00000 q^{64} -2.53590 q^{65} -2.00000 q^{66} +1.00000 q^{67} +3.46410 q^{68} -5.26795 q^{69} +4.39230 q^{70} +8.19615 q^{71} -1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} -7.00000 q^{75} +2.00000 q^{76} -2.53590 q^{77} +0.732051 q^{78} +8.19615 q^{79} -3.46410 q^{80} +1.00000 q^{81} -1.46410 q^{82} +1.07180 q^{83} -1.26795 q^{84} -12.0000 q^{85} -10.9282 q^{86} -7.66025 q^{87} +2.00000 q^{88} -10.0000 q^{89} +3.46410 q^{90} +0.928203 q^{91} +5.26795 q^{92} +0.196152 q^{93} -9.66025 q^{94} -6.92820 q^{95} +1.00000 q^{96} -10.3923 q^{97} +5.39230 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 6 q^{14} + 2 q^{16} - 2 q^{18} + 4 q^{19} - 6 q^{21} + 4 q^{22} + 14 q^{23} + 2 q^{24} + 14 q^{25} + 2 q^{26} - 2 q^{27} + 6 q^{28} - 2 q^{29} + 10 q^{31} - 2 q^{32} + 4 q^{33} + 12 q^{35} + 2 q^{36} - 4 q^{37} - 4 q^{38} + 2 q^{39} - 4 q^{41} + 6 q^{42} + 8 q^{43} - 4 q^{44} - 14 q^{46} + 2 q^{47} - 2 q^{48} + 10 q^{49} - 14 q^{50} - 2 q^{52} + 8 q^{53} + 2 q^{54} - 6 q^{56} - 4 q^{57} + 2 q^{58} - 12 q^{59} - 6 q^{61} - 10 q^{62} + 6 q^{63} + 2 q^{64} - 12 q^{65} - 4 q^{66} + 2 q^{67} - 14 q^{69} - 12 q^{70} + 6 q^{71} - 2 q^{72} + 12 q^{73} + 4 q^{74} - 14 q^{75} + 4 q^{76} - 12 q^{77} - 2 q^{78} + 6 q^{79} + 2 q^{81} + 4 q^{82} + 16 q^{83} - 6 q^{84} - 24 q^{85} - 8 q^{86} + 2 q^{87} + 4 q^{88} - 20 q^{89} - 12 q^{91} + 14 q^{92} - 10 q^{93} - 2 q^{94} + 2 q^{96} - 10 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 4 * q^11 - 2 * q^12 - 2 * q^13 - 6 * q^14 + 2 * q^16 - 2 * q^18 + 4 * q^19 - 6 * q^21 + 4 * q^22 + 14 * q^23 + 2 * q^24 + 14 * q^25 + 2 * q^26 - 2 * q^27 + 6 * q^28 - 2 * q^29 + 10 * q^31 - 2 * q^32 + 4 * q^33 + 12 * q^35 + 2 * q^36 - 4 * q^37 - 4 * q^38 + 2 * q^39 - 4 * q^41 + 6 * q^42 + 8 * q^43 - 4 * q^44 - 14 * q^46 + 2 * q^47 - 2 * q^48 + 10 * q^49 - 14 * q^50 - 2 * q^52 + 8 * q^53 + 2 * q^54 - 6 * q^56 - 4 * q^57 + 2 * q^58 - 12 * q^59 - 6 * q^61 - 10 * q^62 + 6 * q^63 + 2 * q^64 - 12 * q^65 - 4 * q^66 + 2 * q^67 - 14 * q^69 - 12 * q^70 + 6 * q^71 - 2 * q^72 + 12 * q^73 + 4 * q^74 - 14 * q^75 + 4 * q^76 - 12 * q^77 - 2 * q^78 + 6 * q^79 + 2 * q^81 + 4 * q^82 + 16 * q^83 - 6 * q^84 - 24 * q^85 - 8 * q^86 + 2 * q^87 + 4 * q^88 - 20 * q^89 - 12 * q^91 + 14 * q^92 - 10 * q^93 - 2 * q^94 + 2 * q^96 - 10 * q^98 - 4 * q^99

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