LMFDB - Embedded newform 4002.2.a.ba.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.18398$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59819 q^{5} +1.00000 q^{6} +1.23023 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59819 q^{5} +1.00000 q^{6} +1.23023 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.59819 q^{10} +0.260186 q^{11} -1.00000 q^{12} +5.91704 q^{13} -1.23023 q^{14} -3.59819 q^{15} +1.00000 q^{16} -4.05866 q^{17} -1.00000 q^{18} +6.94699 q^{19} +3.59819 q^{20} -1.23023 q^{21} -0.260186 q^{22} +1.00000 q^{23} +1.00000 q^{24} +7.94699 q^{25} -5.91704 q^{26} -1.00000 q^{27} +1.23023 q^{28} -1.00000 q^{29} +3.59819 q^{30} +1.73981 q^{31} -1.00000 q^{32} -0.260186 q^{33} +4.05866 q^{34} +4.42662 q^{35} +1.00000 q^{36} +1.95089 q^{37} -6.94699 q^{38} -5.91704 q^{39} -3.59819 q^{40} +1.49042 q^{41} +1.23023 q^{42} -2.42662 q^{43} +0.260186 q^{44} +3.59819 q^{45} -1.00000 q^{46} +2.76977 q^{47} -1.00000 q^{48} -5.48652 q^{49} -7.94699 q^{50} +4.05866 q^{51} +5.91704 q^{52} -2.88833 q^{53} +1.00000 q^{54} +0.936200 q^{55} -1.23023 q^{56} -6.94699 q^{57} +1.00000 q^{58} +0.426620 q^{59} -3.59819 q^{60} -5.37361 q^{61} -1.73981 q^{62} +1.23023 q^{63} +1.00000 q^{64} +21.2907 q^{65} +0.260186 q^{66} +15.5643 q^{67} -4.05866 q^{68} -1.00000 q^{69} -4.42662 q^{70} +12.9362 q^{71} -1.00000 q^{72} +1.07906 q^{73} -1.95089 q^{74} -7.94699 q^{75} +6.94699 q^{76} +0.320090 q^{77} +5.91704 q^{78} +5.02871 q^{79} +3.59819 q^{80} +1.00000 q^{81} -1.49042 q^{82} +14.5452 q^{83} -1.23023 q^{84} -14.6038 q^{85} +2.42662 q^{86} +1.00000 q^{87} -0.260186 q^{88} -10.4279 q^{89} -3.59819 q^{90} +7.27935 q^{91} +1.00000 q^{92} -1.73981 q^{93} -2.76977 q^{94} +24.9966 q^{95} +1.00000 q^{96} -11.3488 q^{97} +5.48652 q^{98} +0.260186 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 - 4 * q^12 - 6 * q^14 - 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^19 + 2 * q^20 - 6 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^36 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^45 - 4 * q^46 + 10 * q^47 - 4 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^51 + 4 * q^53 + 4 * q^54 - 20 * q^55 - 6 * q^56 - 2 * q^57 + 4 * q^58 - 22 * q^59 - 2 * q^60 + 28 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 + 12 * q^65 + 24 * q^67 - 6 * q^68 - 4 * q^69 + 6 * q^70 + 28 * q^71 - 4 * q^72 - 10 * q^74 - 6 * q^75 + 2 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 4 * q^81 - 6 * q^82 + 20 * q^83 - 6 * q^84 - 10 * q^85 - 14 * q^86 + 4 * q^87 - 24 * q^89 - 2 * q^90 + 28 * q^91 + 4 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^95 + 4 * q^96 - 32 * q^97 - 6 * q^98

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
$5$	3.59819	1.60916	0.804580	−	0.593844i	$-0.202390\pi$
$5$	3.59819	1.60916	0.804580	+	0.593844i	$0.202390\pi$
$6$	1.00000	0.408248
$7$	1.23023	0.464985	0.232492	−	0.972598i	$-0.425312\pi$
$7$	1.23023	0.464985	0.232492	+	0.972598i	$0.425312\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.59819	−1.13785
$11$	0.260186	0.0784491	0.0392245	−	0.999230i	$-0.487511\pi$
$11$	0.260186	0.0784491	0.0392245	+	0.999230i	$0.487511\pi$
$12$	−1.00000	−0.288675
$13$	5.91704	1.64109	0.820546	−	0.571581i	$-0.193670\pi$
$13$	5.91704	1.64109	0.820546	+	0.571581i	$0.193670\pi$
$14$	−1.23023	−0.328794
$15$	−3.59819	−0.929049
$16$	1.00000	0.250000
$17$	−4.05866	−0.984370	−0.492185	−	0.870491i	$-0.663802\pi$
$17$	−4.05866	−0.984370	−0.492185	+	0.870491i	$0.663802\pi$
$18$	−1.00000	−0.235702
$19$	6.94699	1.59375	0.796875	−	0.604145i	$-0.206485\pi$
$19$	6.94699	1.59375	0.796875	+	0.604145i	$0.206485\pi$
$20$	3.59819	0.804580
$21$	−1.23023	−0.268459
$22$	−0.260186	−0.0554719
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	7.94699	1.58940
$26$	−5.91704	−1.16043
$27$	−1.00000	−0.192450
$28$	1.23023	0.232492
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.59819	0.656937
$31$	1.73981	0.312480	0.156240	−	0.987719i	$-0.450063\pi$
$31$	1.73981	0.312480	0.156240	+	0.987719i	$0.450063\pi$
$32$	−1.00000	−0.176777
$33$	−0.260186	−0.0452926
$34$	4.05866	0.696055
$35$	4.42662	0.748235
$36$	1.00000	0.166667
$37$	1.95089	0.320724	0.160362	−	0.987058i	$-0.448734\pi$
$37$	1.95089	0.320724	0.160362	+	0.987058i	$0.448734\pi$
$38$	−6.94699	−1.12695
$39$	−5.91704	−0.947485
$40$	−3.59819	−0.568924
$41$	1.49042	0.232765	0.116382	−	0.993204i	$-0.462870\pi$
$41$	1.49042	0.232765	0.116382	+	0.993204i	$0.462870\pi$
$42$	1.23023	0.189829
$43$	−2.42662	−0.370056	−0.185028	−	0.982733i	$-0.559238\pi$
$43$	−2.42662	−0.370056	−0.185028	+	0.982733i	$0.559238\pi$
$44$	0.260186	0.0392245
$45$	3.59819	0.536387
$46$	−1.00000	−0.147442
$47$	2.76977	0.404012	0.202006	−	0.979384i	$-0.435254\pi$
$47$	2.76977	0.404012	0.202006	+	0.979384i	$0.435254\pi$
$48$	−1.00000	−0.144338
$49$	−5.48652	−0.783789
$50$	−7.94699	−1.12387
$51$	4.05866	0.568326
$52$	5.91704	0.820546
$53$	−2.88833	−0.396743	−0.198371	−	0.980127i	$-0.563565\pi$
$53$	−2.88833	−0.396743	−0.198371	+	0.980127i	$0.563565\pi$
$54$	1.00000	0.136083
$55$	0.936200	0.126237
$56$	−1.23023	−0.164397
$57$	−6.94699	−0.920152
$58$	1.00000	0.131306
$59$	0.426620	0.0555412	0.0277706	−	0.999614i	$-0.491159\pi$
$59$	0.426620	0.0555412	0.0277706	+	0.999614i	$0.491159\pi$
$60$	−3.59819	−0.464525
$61$	−5.37361	−0.688021	−0.344010	−	0.938966i	$-0.611786\pi$
$61$	−5.37361	−0.688021	−0.344010	+	0.938966i	$0.611786\pi$
$62$	−1.73981	−0.220957
$63$	1.23023	0.154995
$64$	1.00000	0.125000
$65$	21.2907	2.64078
$66$	0.260186	0.0320267
$67$	15.5643	1.90149	0.950744	−	0.309978i	$-0.100322\pi$
$67$	15.5643	1.90149	0.950744	+	0.309978i	$0.100322\pi$
$68$	−4.05866	−0.492185
$69$	−1.00000	−0.120386
$70$	−4.42662	−0.529082
$71$	12.9362	1.53524	0.767622	−	0.640903i	$-0.221440\pi$
$71$	12.9362	1.53524	0.767622	+	0.640903i	$0.221440\pi$
$72$	−1.00000	−0.117851
$73$	1.07906	0.126295	0.0631474	−	0.998004i	$-0.479886\pi$
$73$	1.07906	0.126295	0.0631474	+	0.998004i	$0.479886\pi$
$74$	−1.95089	−0.226786
$75$	−7.94699	−0.917640
$76$	6.94699	0.796875
$77$	0.320090	0.0364776
$78$	5.91704	0.669973
$79$	5.02871	0.565774	0.282887	−	0.959153i	$-0.408708\pi$
$79$	5.02871	0.565774	0.282887	+	0.959153i	$0.408708\pi$
$80$	3.59819	0.402290
$81$	1.00000	0.111111
$82$	−1.49042	−0.164589
$83$	14.5452	1.59654	0.798271	−	0.602299i	$-0.205748\pi$
$83$	14.5452	1.59654	0.798271	+	0.602299i	$0.205748\pi$
$84$	−1.23023	−0.134230
$85$	−14.6038	−1.58401
$86$	2.42662	0.261669
$87$	1.00000	0.107211
$88$	−0.260186	−0.0277359
$89$	−10.4279	−1.10535	−0.552676	−	0.833396i	$-0.686393\pi$
$89$	−10.4279	−1.10535	−0.552676	+	0.833396i	$0.686393\pi$
$90$	−3.59819	−0.379283
$91$	7.27935	0.763083
$92$	1.00000	0.104257
$93$	−1.73981	−0.180410
$94$	−2.76977	−0.285680
$95$	24.9966	2.56460
$96$	1.00000	0.102062
$97$	−11.3488	−1.15230	−0.576148	−	0.817345i	$-0.695445\pi$
$97$	−11.3488	−1.15230	−0.576148	+	0.817345i	$0.695445\pi$
$98$	5.48652	0.554223
$99$	0.260186	0.0261497
$100$	7.94699	0.794699
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−4.05866	−0.401867
$103$	−8.54394	−0.841860	−0.420930	−	0.907093i	$-0.638296\pi$
$103$	−8.54394	−0.841860	−0.420930	+	0.907093i	$0.638296\pi$
$104$	−5.91704	−0.580214
$105$	−4.42662	−0.431994
$106$	2.88833	0.280539
$107$	1.16078	0.112217	0.0561084	−	0.998425i	$-0.482131\pi$
$107$	1.16078	0.112217	0.0561084	+	0.998425i	$0.482131\pi$
$108$	−1.00000	−0.0962250
$109$	−8.54519	−0.818480	−0.409240	−	0.912427i	$-0.634206\pi$
$109$	−8.54519	−0.818480	−0.409240	+	0.912427i	$0.634206\pi$
$110$	−0.936200	−0.0892631
$111$	−1.95089	−0.185170
$112$	1.23023	0.116246
$113$	−10.9718	−1.03214	−0.516070	−	0.856546i	$-0.672606\pi$
$113$	−10.9718	−1.03214	−0.516070	+	0.856546i	$0.672606\pi$
$114$	6.94699	0.650645
$115$	3.59819	0.335533
$116$	−1.00000	−0.0928477
$117$	5.91704	0.547031
$118$	−0.426620	−0.0392736
$119$	−4.99310	−0.457717
$120$	3.59819	0.328469
$121$	−10.9323	−0.993846
$122$	5.37361	0.486504
$123$	−1.49042	−0.134387
$124$	1.73981	0.156240
$125$	10.6038	0.948437
$126$	−1.23023	−0.109598
$127$	21.3736	1.89660	0.948301	−	0.317373i	$-0.102801\pi$
$127$	21.3736	1.89660	0.948301	+	0.317373i	$0.102801\pi$
$128$	−1.00000	−0.0883883
$129$	2.42662	0.213652
$130$	−21.2907	−1.86731
$131$	−20.1542	−1.76088	−0.880439	−	0.474159i	$-0.842752\pi$
$131$	−20.1542	−1.76088	−0.880439	+	0.474159i	$0.842752\pi$
$132$	−0.260186	−0.0226463
$133$	8.54643	0.741069
$134$	−15.5643	−1.34455
$135$	−3.59819	−0.309683
$136$	4.05866	0.348027
$137$	−16.9380	−1.44711	−0.723554	−	0.690268i	$-0.757493\pi$
$137$	−16.9380	−1.44711	−0.723554	+	0.690268i	$0.757493\pi$
$138$	1.00000	0.0851257
$139$	−9.53953	−0.809133	−0.404566	−	0.914509i	$-0.632577\pi$
$139$	−9.53953	−0.809133	−0.404566	+	0.914509i	$0.632577\pi$
$140$	4.42662	0.374118
$141$	−2.76977	−0.233256
$142$	−12.9362	−1.08558
$143$	1.53953	0.128742
$144$	1.00000	0.0833333
$145$	−3.59819	−0.298814
$146$	−1.07906	−0.0893039
$147$	5.48652	0.452521
$148$	1.95089	0.160362
$149$	8.75626	0.717341	0.358670	−	0.933464i	$-0.383230\pi$
$149$	8.75626	0.717341	0.358670	+	0.933464i	$0.383230\pi$
$150$	7.94699	0.648869
$151$	−13.5999	−1.10675	−0.553374	−	0.832933i	$-0.686660\pi$
$151$	−13.5999	−1.10675	−0.553374	+	0.832933i	$0.686660\pi$
$152$	−6.94699	−0.563475
$153$	−4.05866	−0.328123
$154$	−0.320090	−0.0257936
$155$	6.26019	0.502830
$156$	−5.91704	−0.473742
$157$	7.49042	0.597801	0.298900	−	0.954284i	$-0.403380\pi$
$157$	7.49042	0.597801	0.298900	+	0.954284i	$0.403380\pi$
$158$	−5.02871	−0.400063
$159$	2.88833	0.229060
$160$	−3.59819	−0.284462
$161$	1.23023	0.0969560
$162$	−1.00000	−0.0785674
$163$	9.66765	0.757229	0.378614	−	0.925555i	$-0.376401\pi$
$163$	9.66765	0.757229	0.378614	+	0.925555i	$0.376401\pi$
$164$	1.49042	0.116382
$165$	−0.936200	−0.0728831
$166$	−14.5452	−1.12893
$167$	−14.8149	−1.14641	−0.573207	−	0.819411i	$-0.694301\pi$
$167$	−14.8149	−1.14641	−0.573207	+	0.819411i	$0.694301\pi$
$168$	1.23023	0.0949146
$169$	22.0114	1.69318
$170$	14.6038	1.12006
$171$	6.94699	0.531250
$172$	−2.42662	−0.185028
$173$	14.3437	1.09053	0.545264	−	0.838264i	$-0.316429\pi$
$173$	14.3437	1.09053	0.545264	+	0.838264i	$0.316429\pi$
$174$	−1.00000	−0.0758098
$175$	9.77666	0.739046
$176$	0.260186	0.0196123
$177$	−0.426620	−0.0320667
$178$	10.4279	0.781601
$179$	−16.2371	−1.21362	−0.606810	−	0.794847i	$-0.707551\pi$
$179$	−16.2371	−1.21362	−0.606810	+	0.794847i	$0.707551\pi$
$180$	3.59819	0.268193
$181$	6.80537	0.505839	0.252920	−	0.967487i	$-0.418609\pi$
$181$	6.80537	0.505839	0.252920	+	0.967487i	$0.418609\pi$
$182$	−7.27935	−0.539581
$183$	5.37361	0.397229
$184$	−1.00000	−0.0737210
$185$	7.01967	0.516097
$186$	1.73981	0.127569
$187$	−1.05601	−0.0772229
$188$	2.76977	0.202006
$189$	−1.23023	−0.0894864
$190$	−24.9966	−1.81345
$191$	−10.0739	−0.728924	−0.364462	−	0.931218i	$-0.618747\pi$
$191$	−10.0739	−0.728924	−0.364462	+	0.931218i	$0.618747\pi$
$192$	−1.00000	−0.0721688
$193$	−8.10953	−0.583737	−0.291868	−	0.956458i	$-0.594277\pi$
$193$	−8.10953	−0.583737	−0.291868	+	0.956458i	$0.594277\pi$
$194$	11.3488	0.814796
$195$	−21.2907	−1.52466
$196$	−5.48652	−0.391895
$197$	11.0300	0.785852	0.392926	−	0.919570i	$-0.371463\pi$
$197$	11.0300	0.785852	0.392926	+	0.919570i	$0.371463\pi$
$198$	−0.260186	−0.0184906
$199$	−7.93230	−0.562306	−0.281153	−	0.959663i	$-0.590717\pi$
$199$	−7.93230	−0.562306	−0.281153	+	0.959663i	$0.590717\pi$
$200$	−7.94699	−0.561937
$201$	−15.5643	−1.09782
$202$	10.0000	0.703598
$203$	−1.23023	−0.0863455
$204$	4.05866	0.284163
$205$	5.36282	0.374556
$206$	8.54394	0.595285
$207$	1.00000	0.0695048
$208$	5.91704	0.410273
$209$	1.80751	0.125028
$210$	4.42662	0.305466
$211$	5.76587	0.396939	0.198469	−	0.980107i	$-0.436403\pi$
$211$	5.76587	0.396939	0.198469	+	0.980107i	$0.436403\pi$
$212$	−2.88833	−0.198371
$213$	−12.9362	−0.886374
$214$	−1.16078	−0.0793493
$215$	−8.73145	−0.595480
$216$	1.00000	0.0680414
$217$	2.14038	0.145298
$218$	8.54519	0.578753
$219$	−1.07906	−0.0729163
$220$	0.936200	0.0631186
$221$	−24.0153	−1.61544
$222$	1.95089	0.130935
$223$	−13.6569	−0.914531	−0.457265	−	0.889330i	$-0.651171\pi$
$223$	−13.6569	−0.914531	−0.457265	+	0.889330i	$0.651171\pi$
$224$	−1.23023	−0.0821985
$225$	7.94699	0.529799
$226$	10.9718	0.729834
$227$	4.14162	0.274889	0.137445	−	0.990509i	$-0.456111\pi$
$227$	4.14162	0.274889	0.137445	+	0.990509i	$0.456111\pi$
$228$	−6.94699	−0.460076
$229$	20.9623	1.38522	0.692612	−	0.721310i	$-0.256460\pi$
$229$	20.9623	1.38522	0.692612	+	0.721310i	$0.256460\pi$
$230$	−3.59819	−0.237258
$231$	−0.320090	−0.0210604
$232$	1.00000	0.0656532
$233$	11.9401	0.782222	0.391111	−	0.920344i	$-0.372091\pi$
$233$	11.9401	0.782222	0.391111	+	0.920344i	$0.372091\pi$
$234$	−5.91704	−0.386809
$235$	9.96615	0.650120
$236$	0.426620	0.0277706
$237$	−5.02871	−0.326650
$238$	4.99310	0.323655
$239$	7.88268	0.509888	0.254944	−	0.966956i	$-0.417943\pi$
$239$	7.88268	0.509888	0.254944	+	0.966956i	$0.417943\pi$
$240$	−3.59819	−0.232262
$241$	−2.22634	−0.143411	−0.0717055	−	0.997426i	$-0.522844\pi$
$241$	−2.22634	−0.143411	−0.0717055	+	0.997426i	$0.522844\pi$
$242$	10.9323	0.702755
$243$	−1.00000	−0.0641500
$244$	−5.37361	−0.344010
$245$	−19.7416	−1.26124
$246$	1.49042	0.0950257
$247$	41.1056	2.61549
$248$	−1.73981	−0.110478
$249$	−14.5452	−0.921764
$250$	−10.6038	−0.670646
$251$	24.1095	1.52178	0.760890	−	0.648881i	$-0.224763\pi$
$251$	24.1095	1.52178	0.760890	+	0.648881i	$0.224763\pi$
$252$	1.23023	0.0774975
$253$	0.260186	0.0163578
$254$	−21.3736	−1.34110
$255$	14.6038	0.914528
$256$	1.00000	0.0625000
$257$	−4.23860	−0.264397	−0.132198	−	0.991223i	$-0.542204\pi$
$257$	−4.23860	−0.264397	−0.132198	+	0.991223i	$0.542204\pi$
$258$	−2.42662	−0.151075
$259$	2.40005	0.149132
$260$	21.2907	1.32039
$261$	−1.00000	−0.0618984
$262$	20.1542	1.24513
$263$	−23.5649	−1.45307	−0.726536	−	0.687129i	$-0.758871\pi$
$263$	−23.5649	−1.45307	−0.726536	+	0.687129i	$0.758871\pi$
$264$	0.260186	0.0160133
$265$	−10.3928	−0.638423
$266$	−8.54643	−0.524015
$267$	10.4279	0.638175
$268$	15.5643	0.950744
$269$	10.4374	0.636380	0.318190	−	0.948027i	$-0.396925\pi$
$269$	10.4374	0.636380	0.318190	+	0.948027i	$0.396925\pi$
$270$	3.59819	0.218979
$271$	25.4718	1.54730	0.773652	−	0.633611i	$-0.218428\pi$
$271$	25.4718	1.54730	0.773652	+	0.633611i	$0.218428\pi$
$272$	−4.05866	−0.246092
$273$	−7.27935	−0.440566
$274$	16.9380	1.02326
$275$	2.06770	0.124687
$276$	−1.00000	−0.0601929
$277$	1.93863	0.116481	0.0582404	−	0.998303i	$-0.481451\pi$
$277$	1.93863	0.116481	0.0582404	+	0.998303i	$0.481451\pi$
$278$	9.53953	0.572143
$279$	1.73981	0.104160
$280$	−4.42662	−0.264541
$281$	−21.8341	−1.30251	−0.651256	−	0.758858i	$-0.725758\pi$
$281$	−21.8341	−1.30251	−0.651256	+	0.758858i	$0.725758\pi$
$282$	2.76977	0.164937
$283$	17.4240	1.03575	0.517874	−	0.855457i	$-0.326724\pi$
$283$	17.4240	1.03575	0.517874	+	0.855457i	$0.326724\pi$
$284$	12.9362	0.767622
$285$	−24.9966	−1.48067
$286$	−1.53953	−0.0910344
$287$	1.83357	0.108232
$288$	−1.00000	−0.0589256
$289$	−0.527268	−0.0310158
$290$	3.59819	0.211293
$291$	11.3488	0.665278
$292$	1.07906	0.0631474
$293$	−21.9436	−1.28196	−0.640980	−	0.767558i	$-0.721472\pi$
$293$	−21.9436	−1.28196	−0.640980	+	0.767558i	$0.721472\pi$
$294$	−5.48652	−0.319981
$295$	1.53506	0.0893747
$296$	−1.95089	−0.113393
$297$	−0.260186	−0.0150975
$298$	−8.75626	−0.507237
$299$	5.91704	0.342191
$300$	−7.94699	−0.458820
$301$	−2.98531	−0.172070
$302$	13.5999	0.782589
$303$	10.0000	0.574485
$304$	6.94699	0.398437
$305$	−19.3353	−1.10714
$306$	4.05866	0.232018
$307$	14.3928	0.821439	0.410719	−	0.911762i	$-0.365278\pi$
$307$	14.3928	0.821439	0.410719	+	0.911762i	$0.365278\pi$
$308$	0.320090	0.0182388
$309$	8.54394	0.486048
$310$	−6.26019	−0.355555
$311$	2.31709	0.131390	0.0656951	−	0.997840i	$-0.479074\pi$
$311$	2.31709	0.131390	0.0656951	+	0.997840i	$0.479074\pi$
$312$	5.91704	0.334986
$313$	0.452161	0.0255576	0.0127788	−	0.999918i	$-0.495932\pi$
$313$	0.452161	0.0255576	0.0127788	+	0.999918i	$0.495932\pi$
$314$	−7.49042	−0.422709
$315$	4.42662	0.249412
$316$	5.02871	0.282887
$317$	20.7625	1.16614	0.583069	−	0.812423i	$-0.301852\pi$
$317$	20.7625	1.16614	0.583069	+	0.812423i	$0.301852\pi$
$318$	−2.88833	−0.161970
$319$	−0.260186	−0.0145676
$320$	3.59819	0.201145
$321$	−1.16078	−0.0647884
$322$	−1.23023	−0.0685583
$323$	−28.1955	−1.56884
$324$	1.00000	0.0555556
$325$	47.0227	2.60835
$326$	−9.66765	−0.535441
$327$	8.54519	0.472550
$328$	−1.49042	−0.0822947
$329$	3.40746	0.187859
$330$	0.936200	0.0515361
$331$	29.3142	1.61126	0.805628	−	0.592422i	$-0.201828\pi$
$331$	29.3142	1.61126	0.805628	+	0.592422i	$0.201828\pi$
$332$	14.5452	0.798271
$333$	1.95089	0.106908
$334$	14.8149	0.810636
$335$	56.0035	3.05980
$336$	−1.23023	−0.0671148
$337$	21.6739	1.18065	0.590326	−	0.807165i	$-0.298999\pi$
$337$	21.6739	1.18065	0.590326	+	0.807165i	$0.298999\pi$
$338$	−22.0114	−1.19726
$339$	10.9718	0.595907
$340$	−14.6038	−0.792005
$341$	0.452675	0.0245137
$342$	−6.94699	−0.375650
$343$	−15.3613	−0.829435
$344$	2.42662	0.130835
$345$	−3.59819	−0.193720
$346$	−14.3437	−0.771120
$347$	−9.91653	−0.532347	−0.266173	−	0.963925i	$-0.585759\pi$
$347$	−9.91653	−0.532347	−0.266173	+	0.963925i	$0.585759\pi$
$348$	1.00000	0.0536056
$349$	−11.6107	−0.621509	−0.310754	−	0.950490i	$-0.600582\pi$
$349$	−11.6107	−0.621509	−0.310754	+	0.950490i	$0.600582\pi$
$350$	−9.77666	−0.522585
$351$	−5.91704	−0.315828
$352$	−0.260186	−0.0138680
$353$	3.08686	0.164297	0.0821484	−	0.996620i	$-0.473822\pi$
$353$	3.08686	0.164297	0.0821484	+	0.996620i	$0.473822\pi$
$354$	0.426620	0.0226746
$355$	46.5469	2.47046
$356$	−10.4279	−0.552676
$357$	4.99310	0.264263
$358$	16.2371	0.858159
$359$	−18.1451	−0.957664	−0.478832	−	0.877907i	$-0.658940\pi$
$359$	−18.1451	−0.957664	−0.478832	+	0.877907i	$0.658940\pi$
$360$	−3.59819	−0.189641
$361$	29.2607	1.54004
$362$	−6.80537	−0.357682
$363$	10.9323	0.573797
$364$	7.27935	0.381541
$365$	3.88268	0.203229
$366$	−5.37361	−0.280883
$367$	−26.8359	−1.40082	−0.700411	−	0.713740i	$-0.747000\pi$
$367$	−26.8359	−1.40082	−0.700411	+	0.713740i	$0.747000\pi$
$368$	1.00000	0.0521286
$369$	1.49042	0.0775882
$370$	−7.01967	−0.364935
$371$	−3.55332	−0.184479
$372$	−1.73981	−0.0902051
$373$	−17.9404	−0.928918	−0.464459	−	0.885595i	$-0.653751\pi$
$373$	−17.9404	−0.928918	−0.464459	+	0.885595i	$0.653751\pi$
$374$	1.05601	0.0546048
$375$	−10.6038	−0.547580
$376$	−2.76977	−0.142840
$377$	−5.91704	−0.304743
$378$	1.23023	0.0632764
$379$	0.913144	0.0469050	0.0234525	−	0.999725i	$-0.492534\pi$
$379$	0.913144	0.0469050	0.0234525	+	0.999725i	$0.492534\pi$
$380$	24.9966	1.28230
$381$	−21.3736	−1.09500
$382$	10.0739	0.515427
$383$	−27.2829	−1.39409	−0.697044	−	0.717028i	$-0.745502\pi$
$383$	−27.2829	−1.39409	−0.697044	+	0.717028i	$0.745502\pi$
$384$	1.00000	0.0510310
$385$	1.15174	0.0586984
$386$	8.10953	0.412764
$387$	−2.42662	−0.123352
$388$	−11.3488	−0.576148
$389$	24.9054	1.26275	0.631376	−	0.775477i	$-0.282491\pi$
$389$	24.9054	1.26275	0.631376	+	0.775477i	$0.282491\pi$
$390$	21.2907	1.07809
$391$	−4.05866	−0.205255
$392$	5.48652	0.277111
$393$	20.1542	1.01664
$394$	−11.0300	−0.555681
$395$	18.0943	0.910421
$396$	0.260186	0.0130748
$397$	36.5966	1.83673	0.918365	−	0.395735i	$-0.129510\pi$
$397$	36.5966	1.83673	0.918365	+	0.395735i	$0.129510\pi$
$398$	7.93230	0.397610
$399$	−8.54643	−0.427857
$400$	7.94699	0.397350
$401$	−6.97056	−0.348093	−0.174047	−	0.984737i	$-0.555684\pi$
$401$	−6.97056	−0.348093	−0.174047	+	0.984737i	$0.555684\pi$
$402$	15.5643	0.776279
$403$	10.2945	0.512808
$404$	−10.0000	−0.497519
$405$	3.59819	0.178796
$406$	1.23023	0.0610555
$407$	0.507594	0.0251605
$408$	−4.05866	−0.200934
$409$	−22.9578	−1.13519	−0.567595	−	0.823308i	$-0.692126\pi$
$409$	−22.9578	−1.13519	−0.567595	+	0.823308i	$0.692126\pi$
$410$	−5.36282	−0.264851
$411$	16.9380	0.835488
$412$	−8.54394	−0.420930
$413$	0.524843	0.0258258
$414$	−1.00000	−0.0491473
$415$	52.3364	2.56909
$416$	−5.91704	−0.290107
$417$	9.53953	0.467153
$418$	−1.80751	−0.0884082
$419$	18.2512	0.891627	0.445814	−	0.895126i	$-0.352914\pi$
$419$	18.2512	0.891627	0.445814	+	0.895126i	$0.352914\pi$
$420$	−4.42662	−0.215997
$421$	−25.9897	−1.26666	−0.633331	−	0.773881i	$-0.718313\pi$
$421$	−25.9897	−1.26666	−0.633331	+	0.773881i	$0.718313\pi$
$422$	−5.76587	−0.280678
$423$	2.76977	0.134671
$424$	2.88833	0.140270
$425$	−32.2542	−1.56456
$426$	12.9362	0.626761
$427$	−6.61080	−0.319919
$428$	1.16078	0.0561084
$429$	−1.53953	−0.0743293
$430$	8.73145	0.421068
$431$	4.08296	0.196669	0.0983346	−	0.995153i	$-0.468648\pi$
$431$	4.08296	0.196669	0.0983346	+	0.995153i	$0.468648\pi$
$432$	−1.00000	−0.0481125
$433$	8.66646	0.416484	0.208242	−	0.978077i	$-0.433226\pi$
$433$	8.66646	0.416484	0.208242	+	0.978077i	$0.433226\pi$
$434$	−2.14038	−0.102741
$435$	3.59819	0.172520
$436$	−8.54519	−0.409240
$437$	6.94699	0.332320
$438$	1.07906	0.0515596
$439$	17.8336	0.851150	0.425575	−	0.904923i	$-0.360072\pi$
$439$	17.8336	0.851150	0.425575	+	0.904923i	$0.360072\pi$
$440$	−0.936200	−0.0446316
$441$	−5.48652	−0.261263
$442$	24.0153	1.14229
$443$	26.3467	1.25177	0.625884	−	0.779916i	$-0.284738\pi$
$443$	26.3467	1.25177	0.625884	+	0.779916i	$0.284738\pi$
$444$	−1.95089	−0.0925851
$445$	−37.5215	−1.77869
$446$	13.6569	0.646671
$447$	−8.75626	−0.414157
$448$	1.23023	0.0581231
$449$	9.77718	0.461413	0.230707	−	0.973023i	$-0.425896\pi$
$449$	9.77718	0.461413	0.230707	+	0.973023i	$0.425896\pi$
$450$	−7.94699	−0.374625
$451$	0.387787	0.0182602
$452$	−10.9718	−0.516070
$453$	13.5999	0.638981
$454$	−4.14162	−0.194376
$455$	26.1925	1.22792
$456$	6.94699	0.325323
$457$	35.8371	1.67639	0.838194	−	0.545372i	$-0.183612\pi$
$457$	35.8371	1.67639	0.838194	+	0.545372i	$0.183612\pi$
$458$	−20.9623	−0.979502
$459$	4.05866	0.189442
$460$	3.59819	0.167767
$461$	−12.3775	−0.576478	−0.288239	−	0.957558i	$-0.593070\pi$
$461$	−12.3775	−0.576478	−0.288239	+	0.957558i	$0.593070\pi$
$462$	0.320090	0.0148919
$463$	−7.31619	−0.340012	−0.170006	−	0.985443i	$-0.554379\pi$
$463$	−7.31619	−0.340012	−0.170006	+	0.985443i	$0.554379\pi$
$464$	−1.00000	−0.0464238
$465$	−6.26019	−0.290309
$466$	−11.9401	−0.553114
$467$	8.42713	0.389961	0.194981	−	0.980807i	$-0.437536\pi$
$467$	8.42713	0.389961	0.194981	+	0.980807i	$0.437536\pi$
$468$	5.91704	0.273515
$469$	19.1478	0.884163
$470$	−9.96615	−0.459704
$471$	−7.49042	−0.345140
$472$	−0.426620	−0.0196368
$473$	−0.631373	−0.0290306
$474$	5.02871	0.230976
$475$	55.2077	2.53310
$476$	−4.99310	−0.228859
$477$	−2.88833	−0.132248
$478$	−7.88268	−0.360545
$479$	2.15241	0.0983463	0.0491731	−	0.998790i	$-0.484341\pi$
$479$	2.15241	0.0983463	0.0491731	+	0.998790i	$0.484341\pi$
$480$	3.59819	0.164234
$481$	11.5435	0.526338
$482$	2.22634	0.101407
$483$	−1.23023	−0.0559776
$484$	−10.9323	−0.496923
$485$	−40.8352	−1.85423
$486$	1.00000	0.0453609
$487$	−33.4837	−1.51729	−0.758645	−	0.651505i	$-0.774138\pi$
$487$	−33.4837	−1.51729	−0.758645	+	0.651505i	$0.774138\pi$
$488$	5.37361	0.243252
$489$	−9.66765	−0.437186
$490$	19.7416	0.891833
$491$	−0.652958	−0.0294676	−0.0147338	−	0.999891i	$-0.504690\pi$
$491$	−0.652958	−0.0294676	−0.0147338	+	0.999891i	$0.504690\pi$
$492$	−1.49042	−0.0671933
$493$	4.05866	0.182793
$494$	−41.1056	−1.84943
$495$	0.936200	0.0420790
$496$	1.73981	0.0781200
$497$	15.9146	0.713865
$498$	14.5452	0.651785
$499$	2.14286	0.0959277	0.0479639	−	0.998849i	$-0.484727\pi$
$499$	2.14286	0.0959277	0.0479639	+	0.998849i	$0.484727\pi$
$500$	10.6038	0.474218
$501$	14.8149	0.661882
$502$	−24.1095	−1.07606
$503$	0.272736	0.0121607	0.00608034	−	0.999982i	$-0.498065\pi$
$503$	0.272736	0.0121607	0.00608034	+	0.999982i	$0.498065\pi$
$504$	−1.23023	−0.0547990
$505$	−35.9819	−1.60117
$506$	−0.260186	−0.0115667
$507$	−22.0114	−0.977559
$508$	21.3736	0.948301
$509$	10.0413	0.445074	0.222537	−	0.974924i	$-0.428566\pi$
$509$	10.0413	0.445074	0.222537	+	0.974924i	$0.428566\pi$
$510$	−14.6038	−0.646669
$511$	1.32750	0.0587252
$512$	−1.00000	−0.0441942
$513$	−6.94699	−0.306717
$514$	4.23860	0.186957
$515$	−30.7428	−1.35469
$516$	2.42662	0.106826
$517$	0.720655	0.0316943
$518$	−2.40005	−0.105452
$519$	−14.3437	−0.629617
$520$	−21.2907	−0.933657
$521$	27.9308	1.22367	0.611836	−	0.790985i	$-0.290431\pi$
$521$	27.9308	1.22367	0.611836	+	0.790985i	$0.290431\pi$
$522$	1.00000	0.0437688
$523$	37.3385	1.63270	0.816350	−	0.577558i	$-0.195994\pi$
$523$	37.3385	1.63270	0.816350	+	0.577558i	$0.195994\pi$
$524$	−20.1542	−0.880439
$525$	−9.77666	−0.426688
$526$	23.5649	1.02748
$527$	−7.06132	−0.307596
$528$	−0.260186	−0.0113231
$529$	1.00000	0.0434783
$530$	10.3928	0.451433
$531$	0.426620	0.0185137
$532$	8.54643	0.370535
$533$	8.81888	0.381988
$534$	−10.4279	−0.451258
$535$	4.17671	0.180575
$536$	−15.5643	−0.672277
$537$	16.2371	0.700684
$538$	−10.4374	−0.449989
$539$	−1.42752	−0.0614875
$540$	−3.59819	−0.154842
$541$	34.0757	1.46503	0.732514	−	0.680752i	$-0.238347\pi$
$541$	34.0757	1.46503	0.732514	+	0.680752i	$0.238347\pi$
$542$	−25.4718	−1.09411
$543$	−6.80537	−0.292046
$544$	4.05866	0.174014
$545$	−30.7472	−1.31707
$546$	7.27935	0.311527
$547$	−22.2945	−0.953246	−0.476623	−	0.879108i	$-0.658139\pi$
$547$	−22.2945	−0.953246	−0.476623	+	0.879108i	$0.658139\pi$
$548$	−16.9380	−0.723554
$549$	−5.37361	−0.229340
$550$	−2.06770	−0.0881669
$551$	−6.94699	−0.295952
$552$	1.00000	0.0425628
$553$	6.18649	0.263076
$554$	−1.93863	−0.0823643
$555$	−7.01967	−0.297969
$556$	−9.53953	−0.404566
$557$	30.6071	1.29686	0.648432	−	0.761273i	$-0.275425\pi$
$557$	30.6071	1.29686	0.648432	+	0.761273i	$0.275425\pi$
$558$	−1.73981	−0.0736522
$559$	−14.3584	−0.607296
$560$	4.42662	0.187059
$561$	1.05601	0.0445847
$562$	21.8341	0.921015
$563$	21.0982	0.889182	0.444591	−	0.895734i	$-0.353349\pi$
$563$	21.0982	0.889182	0.444591	+	0.895734i	$0.353349\pi$
$564$	−2.76977	−0.116628
$565$	−39.4787	−1.66088
$566$	−17.4240	−0.732384
$567$	1.23023	0.0516650
$568$	−12.9362	−0.542791
$569$	6.02034	0.252386	0.126193	−	0.992006i	$-0.459724\pi$
$569$	6.02034	0.252386	0.126193	+	0.992006i	$0.459724\pi$
$570$	24.9966	1.04699
$571$	−12.7071	−0.531778	−0.265889	−	0.964004i	$-0.585665\pi$
$571$	−12.7071	−0.531778	−0.265889	+	0.964004i	$0.585665\pi$
$572$	1.53953	0.0643710
$573$	10.0739	0.420844
$574$	−1.83357	−0.0765316
$575$	7.94699	0.331412
$576$	1.00000	0.0416667
$577$	4.66075	0.194030	0.0970148	−	0.995283i	$-0.469071\pi$
$577$	4.66075	0.194030	0.0970148	+	0.995283i	$0.469071\pi$
$578$	0.527268	0.0219315
$579$	8.10953	0.337021
$580$	−3.59819	−0.149407
$581$	17.8940	0.742368
$582$	−11.3488	−0.470423
$583$	−0.751504	−0.0311241
$584$	−1.07906	−0.0446520
$585$	21.2907	0.880260
$586$	21.9436	0.906482
$587$	10.8871	0.449358	0.224679	−	0.974433i	$-0.427867\pi$
$587$	10.8871	0.449358	0.224679	+	0.974433i	$0.427867\pi$
$588$	5.48652	0.226260
$589$	12.0865	0.498014
$590$	−1.53506	−0.0631975
$591$	−11.0300	−0.453712
$592$	1.95089	0.0801810
$593$	16.2945	0.669137	0.334568	−	0.942371i	$-0.391409\pi$
$593$	16.2945	0.669137	0.334568	+	0.942371i	$0.391409\pi$
$594$	0.260186	0.0106756
$595$	−17.9662	−0.736540
$596$	8.75626	0.358670
$597$	7.93230	0.324648
$598$	−5.91704	−0.241966
$599$	−17.4693	−0.713778	−0.356889	−	0.934147i	$-0.616163\pi$
$599$	−17.4693	−0.713778	−0.356889	+	0.934147i	$0.616163\pi$
$600$	7.94699	0.324435
$601$	2.30234	0.0939145	0.0469572	−	0.998897i	$-0.485048\pi$
$601$	2.30234	0.0939145	0.0469572	+	0.998897i	$0.485048\pi$
$602$	2.98531	0.121672
$603$	15.5643	0.633829
$604$	−13.5999	−0.553374
$605$	−39.3365	−1.59926
$606$	−10.0000	−0.406222
$607$	26.0241	1.05629	0.528143	−	0.849156i	$-0.322889\pi$
$607$	26.0241	1.05629	0.528143	+	0.849156i	$0.322889\pi$
$608$	−6.94699	−0.281738
$609$	1.23023	0.0498516
$610$	19.3353	0.782863
$611$	16.3888	0.663021
$612$	−4.05866	−0.164062
$613$	14.2697	0.576349	0.288175	−	0.957578i	$-0.406952\pi$
$613$	14.2697	0.576349	0.288175	+	0.957578i	$0.406952\pi$
$614$	−14.3928	−0.580845
$615$	−5.36282	−0.216250
$616$	−0.320090	−0.0128968
$617$	−43.8250	−1.76433	−0.882165	−	0.470941i	$-0.843915\pi$
$617$	−43.8250	−1.76433	−0.882165	+	0.470941i	$0.843915\pi$
$618$	−8.54394	−0.343688
$619$	6.40752	0.257540	0.128770	−	0.991674i	$-0.458897\pi$
$619$	6.40752	0.257540	0.128770	+	0.991674i	$0.458897\pi$
$620$	6.26019	0.251415
$621$	−1.00000	−0.0401286
$622$	−2.31709	−0.0929069
$623$	−12.8287	−0.513972
$624$	−5.91704	−0.236871
$625$	−1.58028	−0.0632110
$626$	−0.452161	−0.0180720
$627$	−1.80751	−0.0721850
$628$	7.49042	0.298900
$629$	−7.91800	−0.315711
$630$	−4.42662	−0.176361
$631$	−21.9176	−0.872524	−0.436262	−	0.899820i	$-0.643698\pi$
$631$	−21.9176	−0.872524	−0.436262	+	0.899820i	$0.643698\pi$
$632$	−5.02871	−0.200031
$633$	−5.76587	−0.229173
$634$	−20.7625	−0.824584
$635$	76.9064	3.05194
$636$	2.88833	0.114530
$637$	−32.4640	−1.28627
$638$	0.260186	0.0103009
$639$	12.9362	0.511748
$640$	−3.59819	−0.142231
$641$	−16.9605	−0.669899	−0.334950	−	0.942236i	$-0.608719\pi$
$641$	−16.9605	−0.669899	−0.334950	+	0.942236i	$0.608719\pi$
$642$	1.16078	0.0458123
$643$	30.3474	1.19678	0.598392	−	0.801203i	$-0.295807\pi$
$643$	30.3474	1.19678	0.598392	+	0.801203i	$0.295807\pi$
$644$	1.23023	0.0484780
$645$	8.73145	0.343800
$646$	28.1955	1.10934
$647$	25.7398	1.01194	0.505968	−	0.862552i	$-0.331135\pi$
$647$	25.7398	1.01194	0.505968	+	0.862552i	$0.331135\pi$
$648$	−1.00000	−0.0392837
$649$	0.111001	0.00435715
$650$	−47.0227	−1.84438
$651$	−2.14038	−0.0838881
$652$	9.66765	0.378614
$653$	−32.7663	−1.28225	−0.641123	−	0.767438i	$-0.721531\pi$
$653$	−32.7663	−1.28225	−0.641123	+	0.767438i	$0.721531\pi$
$654$	−8.54519	−0.334143
$655$	−72.5186	−2.83354
$656$	1.49042	0.0581911
$657$	1.07906	0.0420983
$658$	−3.40746	−0.132837
$659$	−31.4041	−1.22333	−0.611666	−	0.791116i	$-0.709500\pi$
$659$	−31.4041	−1.22333	−0.611666	+	0.791116i	$0.709500\pi$
$660$	−0.936200	−0.0364415
$661$	14.3218	0.557055	0.278528	−	0.960428i	$-0.410154\pi$
$661$	14.3218	0.557055	0.278528	+	0.960428i	$0.410154\pi$
$662$	−29.3142	−1.13933
$663$	24.0153	0.932676
$664$	−14.5452	−0.564463
$665$	30.7517	1.19250
$666$	−1.95089	−0.0755954
$667$	−1.00000	−0.0387202
$668$	−14.8149	−0.573207
$669$	13.6569	0.528004
$670$	−56.0035	−2.16360
$671$	−1.39814	−0.0539746
$672$	1.23023	0.0474573
$673$	0.669058	0.0257903	0.0128952	−	0.999917i	$-0.495895\pi$
$673$	0.669058	0.0257903	0.0128952	+	0.999917i	$0.495895\pi$
$674$	−21.6739	−0.834846
$675$	−7.94699	−0.305880
$676$	22.0114	0.846591
$677$	−25.5892	−0.983471	−0.491736	−	0.870745i	$-0.663637\pi$
$677$	−25.5892	−0.983471	−0.491736	+	0.870745i	$0.663637\pi$
$678$	−10.9718	−0.421370
$679$	−13.9617	−0.535800
$680$	14.6038	0.560032
$681$	−4.14162	−0.158707
$682$	−0.452675	−0.0173338
$683$	−19.3182	−0.739190	−0.369595	−	0.929193i	$-0.620503\pi$
$683$	−19.3182	−0.739190	−0.369595	+	0.929193i	$0.620503\pi$
$684$	6.94699	0.265625
$685$	−60.9460	−2.32863
$686$	15.3613	0.586499
$687$	−20.9623	−0.799760
$688$	−2.42662	−0.0925140
$689$	−17.0904	−0.651091
$690$	3.59819	0.136981
$691$	−39.2116	−1.49168	−0.745841	−	0.666124i	$-0.767952\pi$
$691$	−39.2116	−1.49168	−0.745841	+	0.666124i	$0.767952\pi$
$692$	14.3437	0.545264
$693$	0.320090	0.0121592
$694$	9.91653	0.376426
$695$	−34.3251	−1.30202
$696$	−1.00000	−0.0379049
$697$	−6.04911	−0.229126
$698$	11.6107	0.439473
$699$	−11.9401	−0.451616
$700$	9.77666	0.369523
$701$	−25.5634	−0.965516	−0.482758	−	0.875754i	$-0.660365\pi$
$701$	−25.5634	−0.965516	−0.482758	+	0.875754i	$0.660365\pi$
$702$	5.91704	0.223324
$703$	13.5528	0.511154
$704$	0.260186	0.00980613
$705$	−9.96615	−0.375347
$706$	−3.08686	−0.116175
$707$	−12.3023	−0.462677
$708$	−0.426620	−0.0160334
$709$	−37.0999	−1.39332	−0.696658	−	0.717403i	$-0.745331\pi$
$709$	−37.0999	−1.39332	−0.696658	+	0.717403i	$0.745331\pi$
$710$	−46.5469	−1.74688
$711$	5.02871	0.188591
$712$	10.4279	0.390801
$713$	1.73981	0.0651565
$714$	−4.99310	−0.186862
$715$	5.53953	0.207167
$716$	−16.2371	−0.606810
$717$	−7.88268	−0.294384
$718$	18.1451	0.677170
$719$	17.2907	0.644833	0.322416	−	0.946598i	$-0.395505\pi$
$719$	17.2907	0.644833	0.322416	+	0.946598i	$0.395505\pi$
$720$	3.59819	0.134097
$721$	−10.5111	−0.391452
$722$	−29.2607	−1.08897
$723$	2.22634	0.0827984
$724$	6.80537	0.252920
$725$	−7.94699	−0.295144
$726$	−10.9323	−0.405736
$727$	−19.3410	−0.717318	−0.358659	−	0.933469i	$-0.616766\pi$
$727$	−19.3410	−0.717318	−0.358659	+	0.933469i	$0.616766\pi$
$728$	−7.27935	−0.269790
$729$	1.00000	0.0370370
$730$	−3.88268	−0.143704
$731$	9.84883	0.364272
$732$	5.37361	0.198614
$733$	1.32150	0.0488108	0.0244054	−	0.999702i	$-0.492231\pi$
$733$	1.32150	0.0488108	0.0244054	+	0.999702i	$0.492231\pi$
$734$	26.8359	0.990531
$735$	19.7416	0.728179
$736$	−1.00000	−0.0368605
$737$	4.04963	0.149170
$738$	−1.49042	−0.0548631
$739$	−40.9741	−1.50726	−0.753628	−	0.657301i	$-0.771698\pi$
$739$	−40.9741	−1.50726	−0.753628	+	0.657301i	$0.771698\pi$
$740$	7.01967	0.258048
$741$	−41.1056	−1.51005
$742$	3.55332	0.130447
$743$	−33.7691	−1.23887	−0.619434	−	0.785049i	$-0.712638\pi$
$743$	−33.7691	−1.23887	−0.619434	+	0.785049i	$0.712638\pi$
$744$	1.73981	0.0637847
$745$	31.5067	1.15432
$746$	17.9404	0.656844
$747$	14.5452	0.532180
$748$	−1.05601	−0.0386114
$749$	1.42803	0.0521791
$750$	10.6038	0.387198
$751$	0.658994	0.0240470	0.0120235	−	0.999928i	$-0.496173\pi$
$751$	0.658994	0.0240470	0.0120235	+	0.999928i	$0.496173\pi$
$752$	2.76977	0.101003
$753$	−24.1095	−0.878600
$754$	5.91704	0.215486
$755$	−48.9352	−1.78094
$756$	−1.23023	−0.0447432
$757$	28.5833	1.03888	0.519438	−	0.854508i	$-0.326141\pi$
$757$	28.5833	1.03888	0.519438	+	0.854508i	$0.326141\pi$
$758$	−0.913144	−0.0331669
$759$	−0.260186	−0.00944416
$760$	−24.9966	−0.906723
$761$	−24.1159	−0.874199	−0.437099	−	0.899413i	$-0.643994\pi$
$761$	−24.1159	−0.874199	−0.437099	+	0.899413i	$0.643994\pi$
$762$	21.3736	0.774284
$763$	−10.5126	−0.380581
$764$	−10.0739	−0.364462
$765$	−14.6038	−0.528003
$766$	27.2829	0.985770
$767$	2.52433	0.0911482
$768$	−1.00000	−0.0360844
$769$	−23.1676	−0.835446	−0.417723	−	0.908575i	$-0.637172\pi$
$769$	−23.1676	−0.835446	−0.417723	+	0.908575i	$0.637172\pi$
$770$	−1.15174	−0.0415060
$771$	4.23860	0.152649
$772$	−8.10953	−0.291868
$773$	−10.0241	−0.360541	−0.180271	−	0.983617i	$-0.557697\pi$
$773$	−10.0241	−0.360541	−0.180271	+	0.983617i	$0.557697\pi$
$774$	2.42662	0.0872231
$775$	13.8263	0.496655
$776$	11.3488	0.407398
$777$	−2.40005	−0.0861013
$778$	−24.9054	−0.892900
$779$	10.3539	0.370968
$780$	−21.2907	−0.762328
$781$	3.36582	0.120438
$782$	4.05866	0.145137
$783$	1.00000	0.0357371
$784$	−5.48652	−0.195947
$785$	26.9520	0.961957
$786$	−20.1542	−0.718875
$787$	2.86279	0.102047	0.0510237	−	0.998697i	$-0.483752\pi$
$787$	2.86279	0.102047	0.0510237	+	0.998697i	$0.483752\pi$
$788$	11.0300	0.392926
$789$	23.5649	0.838931
$790$	−18.0943	−0.643765
$791$	−13.4979	−0.479930
$792$	−0.260186	−0.00924531
$793$	−31.7959	−1.12910
$794$	−36.5966	−1.29876
$795$	10.3928	0.368594
$796$	−7.93230	−0.281153
$797$	−53.3899	−1.89117	−0.945584	−	0.325379i	$-0.894508\pi$
$797$	−53.3899	−1.89117	−0.945584	+	0.325379i	$0.894508\pi$
$798$	8.54643	0.302540
$799$	−11.2415	−0.397697
$800$	−7.94699	−0.280969
$801$	−10.4279	−0.368450
$802$	6.97056	0.246139
$803$	0.280757	0.00990771
$804$	−15.5643	−0.548912
$805$	4.42662	0.156018
$806$	−10.2945	−0.362610
$807$	−10.4374	−0.367414
$808$	10.0000	0.351799
$809$	−8.89404	−0.312698	−0.156349	−	0.987702i	$-0.549972\pi$
$809$	−8.89404	−0.312698	−0.156349	+	0.987702i	$0.549972\pi$
$810$	−3.59819	−0.126428
$811$	−36.1424	−1.26913	−0.634566	−	0.772869i	$-0.718821\pi$
$811$	−36.1424	−1.26913	−0.634566	+	0.772869i	$0.718821\pi$
$812$	−1.23023	−0.0431728
$813$	−25.4718	−0.893336
$814$	−0.507594	−0.0177912
$815$	34.7861	1.21850
$816$	4.05866	0.142082
$817$	−16.8577	−0.589777
$818$	22.9578	0.802700
$819$	7.27935	0.254361
$820$	5.36282	0.187278
$821$	−17.6491	−0.615956	−0.307978	−	0.951393i	$-0.599652\pi$
$821$	−17.6491	−0.615956	−0.307978	+	0.951393i	$0.599652\pi$
$822$	−16.9380	−0.590779
$823$	−11.0177	−0.384052	−0.192026	−	0.981390i	$-0.561506\pi$
$823$	−11.0177	−0.384052	−0.192026	+	0.981390i	$0.561506\pi$
$824$	8.54394	0.297642
$825$	−2.06770	−0.0719880
$826$	−0.524843	−0.0182616
$827$	21.4807	0.746956	0.373478	−	0.927639i	$-0.378165\pi$
$827$	21.4807	0.746956	0.373478	+	0.927639i	$0.378165\pi$
$828$	1.00000	0.0347524
$829$	−38.3284	−1.33120	−0.665600	−	0.746309i	$-0.731824\pi$
$829$	−38.3284	−1.33120	−0.665600	+	0.746309i	$0.731824\pi$
$830$	−52.3364	−1.81662
$831$	−1.93863	−0.0672502
$832$	5.91704	0.205136
$833$	22.2679	0.771538
$834$	−9.53953	−0.330327
$835$	−53.3069	−1.84476
$836$	1.80751	0.0625141
$837$	−1.73981	−0.0601368
$838$	−18.2512	−0.630476
$839$	−32.5560	−1.12396	−0.561979	−	0.827152i	$-0.689960\pi$
$839$	−32.5560	−1.12396	−0.561979	+	0.827152i	$0.689960\pi$
$840$	4.42662	0.152733
$841$	1.00000	0.0344828
$842$	25.9897	0.895665
$843$	21.8341	0.752006
$844$	5.76587	0.198469
$845$	79.2011	2.72460
$846$	−2.76977	−0.0952265
$847$	−13.4493	−0.462123
$848$	−2.88833	−0.0991857
$849$	−17.4240	−0.597989
$850$	32.2542	1.10631
$851$	1.95089	0.0668756
$852$	−12.9362	−0.443187
$853$	12.3858	0.424082	0.212041	−	0.977261i	$-0.431989\pi$
$853$	12.3858	0.424082	0.212041	+	0.977261i	$0.431989\pi$
$854$	6.61080	0.226217
$855$	24.9966	0.854866
$856$	−1.16078	−0.0396747
$857$	5.65833	0.193285	0.0966424	−	0.995319i	$-0.469190\pi$
$857$	5.65833	0.193285	0.0966424	+	0.995319i	$0.469190\pi$
$858$	1.53953	0.0525587
$859$	−48.8675	−1.66734	−0.833670	−	0.552264i	$-0.813764\pi$
$859$	−48.8675	−1.66734	−0.833670	+	0.552264i	$0.813764\pi$
$860$	−8.73145	−0.297740
$861$	−1.83357	−0.0624878
$862$	−4.08296	−0.139066
$863$	37.6019	1.27998	0.639992	−	0.768381i	$-0.278937\pi$
$863$	37.6019	1.27998	0.639992	+	0.768381i	$0.278937\pi$
$864$	1.00000	0.0340207
$865$	51.6113	1.75484
$866$	−8.66646	−0.294498
$867$	0.527268	0.0179070
$868$	2.14038	0.0726492
$869$	1.30840	0.0443844
$870$	−3.59819	−0.121990
$871$	92.0949	3.12052
$872$	8.54519	0.289376
$873$	−11.3488	−0.384099
$874$	−6.94699	−0.234986
$875$	13.0452	0.441009
$876$	−1.07906	−0.0364582
$877$	20.8111	0.702740	0.351370	−	0.936237i	$-0.385716\pi$
$877$	20.8111	0.702740	0.351370	+	0.936237i	$0.385716\pi$
$878$	−17.8336	−0.601854
$879$	21.9436	0.740140
$880$	0.936200	0.0315593
$881$	39.5870	1.33372	0.666859	−	0.745184i	$-0.267638\pi$
$881$	39.5870	1.33372	0.666859	+	0.745184i	$0.267638\pi$
$882$	5.48652	0.184741
$883$	15.8518	0.533456	0.266728	−	0.963772i	$-0.414057\pi$
$883$	15.8518	0.533456	0.266728	+	0.963772i	$0.414057\pi$
$884$	−24.0153	−0.807721
$885$	−1.53506	−0.0516005
$886$	−26.3467	−0.885133
$887$	16.0471	0.538810	0.269405	−	0.963027i	$-0.413173\pi$
$887$	16.0471	0.538810	0.269405	+	0.963027i	$0.413173\pi$
$888$	1.95089	0.0654675
$889$	26.2945	0.881891
$890$	37.5215	1.25772
$891$	0.260186	0.00871656
$892$	−13.6569	−0.457265
$893$	19.2415	0.643894
$894$	8.75626	0.292853
$895$	−58.4243	−1.95291
$896$	−1.23023	−0.0410992
$897$	−5.91704	−0.197564
$898$	−9.77718	−0.326269
$899$	−1.73981	−0.0580260
$900$	7.94699	0.264900
$901$	11.7228	0.390542
$902$	−0.387787	−0.0129119
$903$	2.98531	0.0993449
$904$	10.9718	0.364917
$905$	24.4870	0.813977
$906$	−13.5999	−0.451828
$907$	−50.3850	−1.67301	−0.836503	−	0.547963i	$-0.815404\pi$
$907$	−50.3850	−1.67301	−0.836503	+	0.547963i	$0.815404\pi$
$908$	4.14162	0.137445
$909$	−10.0000	−0.331679
$910$	−26.1925	−0.868273
$911$	27.7391	0.919037	0.459518	−	0.888168i	$-0.348022\pi$
$911$	27.7391	0.919037	0.459518	+	0.888168i	$0.348022\pi$
$912$	−6.94699	−0.230038
$913$	3.78445	0.125247
$914$	−35.8371	−1.18539
$915$	19.3353	0.639205
$916$	20.9623	0.692612
$917$	−24.7944	−0.818782
$918$	−4.05866	−0.133956
$919$	48.3840	1.59604	0.798021	−	0.602630i	$-0.205880\pi$
$919$	48.3840	1.59604	0.798021	+	0.602630i	$0.205880\pi$
$920$	−3.59819	−0.118629
$921$	−14.3928	−0.474258
$922$	12.3775	0.407632
$923$	76.5440	2.51948
$924$	−0.320090	−0.0105302
$925$	15.5037	0.509758
$926$	7.31619	0.240425
$927$	−8.54394	−0.280620
$928$	1.00000	0.0328266
$929$	−26.9976	−0.885762	−0.442881	−	0.896580i	$-0.646044\pi$
$929$	−26.9976	−0.885762	−0.442881	+	0.896580i	$0.646044\pi$
$930$	6.26019	0.205280
$931$	−38.1148	−1.24916
$932$	11.9401	0.391111
$933$	−2.31709	−0.0758581
$934$	−8.42713	−0.275744
$935$	−3.79972	−0.124264
$936$	−5.91704	−0.193405
$937$	40.6191	1.32697	0.663484	−	0.748190i	$-0.269077\pi$
$937$	40.6191	1.32697	0.663484	+	0.748190i	$0.269077\pi$
$938$	−19.1478	−0.625197
$939$	−0.452161	−0.0147557
$940$	9.96615	0.325060
$941$	17.9404	0.584840	0.292420	−	0.956290i	$-0.405540\pi$
$941$	17.9404	0.584840	0.292420	+	0.956290i	$0.405540\pi$
$942$	7.49042	0.244051
$943$	1.49042	0.0485348
$944$	0.426620	0.0138853
$945$	−4.42662	−0.143998
$946$	0.631373	0.0205277
$947$	17.6721	0.574267	0.287133	−	0.957891i	$-0.407298\pi$
$947$	17.6721	0.574267	0.287133	+	0.957891i	$0.407298\pi$
$948$	−5.02871	−0.163325
$949$	6.38486	0.207261
$950$	−55.2077	−1.79117
$951$	−20.7625	−0.673270
$952$	4.99310	0.161827
$953$	13.3623	0.432848	0.216424	−	0.976300i	$-0.430561\pi$
$953$	13.3623	0.432848	0.216424	+	0.976300i	$0.430561\pi$
$954$	2.88833	0.0935132
$955$	−36.2479	−1.17296
$956$	7.88268	0.254944
$957$	0.260186	0.00841062
$958$	−2.15241	−0.0695413
$959$	−20.8377	−0.672883
$960$	−3.59819	−0.116131
$961$	−27.9730	−0.902356
$962$	−11.5435	−0.372177
$963$	1.16078	0.0374056
$964$	−2.22634	−0.0717055
$965$	−29.1797	−0.939326
$966$	1.23023	0.0395821
$967$	−15.7511	−0.506522	−0.253261	−	0.967398i	$-0.581503\pi$
$967$	−15.7511	−0.506522	−0.253261	+	0.967398i	$0.581503\pi$
$968$	10.9323	0.351378
$969$	28.1955	0.905770
$970$	40.8352	1.31114
$971$	−21.5165	−0.690497	−0.345248	−	0.938511i	$-0.612205\pi$
$971$	−21.5165	−0.690497	−0.345248	+	0.938511i	$0.612205\pi$
$972$	−1.00000	−0.0320750
$973$	−11.7359	−0.376234
$974$	33.4837	1.07289
$975$	−47.0227	−1.50593
$976$	−5.37361	−0.172005
$977$	−5.18112	−0.165759	−0.0828794	−	0.996560i	$-0.526412\pi$
$977$	−5.18112	−0.165759	−0.0828794	+	0.996560i	$0.526412\pi$
$978$	9.66765	0.309137
$979$	−2.71318	−0.0867138
$980$	−19.7416	−0.630621
$981$	−8.54519	−0.272827
$982$	0.652958	0.0208367
$983$	4.54270	0.144890	0.0724448	−	0.997372i	$-0.476920\pi$
$983$	4.54270	0.144890	0.0724448	+	0.997372i	$0.476920\pi$
$984$	1.49042	0.0475129
$985$	39.6879	1.26456
$986$	−4.05866	−0.129254
$987$	−3.40746	−0.108461
$988$	41.1056	1.30774
$989$	−2.42662	−0.0771620
$990$	−0.936200	−0.0297544
$991$	−23.7556	−0.754621	−0.377311	−	0.926087i	$-0.623151\pi$
$991$	−23.7556	−0.754621	−0.377311	+	0.926087i	$0.623151\pi$
$992$	−1.73981	−0.0552391
$993$	−29.3142	−0.930259
$994$	−15.9146	−0.504779
$995$	−28.5420	−0.904841
$996$	−14.5452	−0.460882
$997$	25.9278	0.821143	0.410571	−	0.911828i	$-0.365329\pi$
$997$	25.9278	0.821143	0.410571	+	0.911828i	$0.365329\pi$
$998$	−2.14286	−0.0678312
$999$	−1.95089	−0.0617234

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.ba.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59819 q^{5} +1.00000 q^{6} +1.23023 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59819 q^{5} +1.00000 q^{6} +1.23023 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.59819 q^{10} +0.260186 q^{11} -1.00000 q^{12} +5.91704 q^{13} -1.23023 q^{14} -3.59819 q^{15} +1.00000 q^{16} -4.05866 q^{17} -1.00000 q^{18} +6.94699 q^{19} +3.59819 q^{20} -1.23023 q^{21} -0.260186 q^{22} +1.00000 q^{23} +1.00000 q^{24} +7.94699 q^{25} -5.91704 q^{26} -1.00000 q^{27} +1.23023 q^{28} -1.00000 q^{29} +3.59819 q^{30} +1.73981 q^{31} -1.00000 q^{32} -0.260186 q^{33} +4.05866 q^{34} +4.42662 q^{35} +1.00000 q^{36} +1.95089 q^{37} -6.94699 q^{38} -5.91704 q^{39} -3.59819 q^{40} +1.49042 q^{41} +1.23023 q^{42} -2.42662 q^{43} +0.260186 q^{44} +3.59819 q^{45} -1.00000 q^{46} +2.76977 q^{47} -1.00000 q^{48} -5.48652 q^{49} -7.94699 q^{50} +4.05866 q^{51} +5.91704 q^{52} -2.88833 q^{53} +1.00000 q^{54} +0.936200 q^{55} -1.23023 q^{56} -6.94699 q^{57} +1.00000 q^{58} +0.426620 q^{59} -3.59819 q^{60} -5.37361 q^{61} -1.73981 q^{62} +1.23023 q^{63} +1.00000 q^{64} +21.2907 q^{65} +0.260186 q^{66} +15.5643 q^{67} -4.05866 q^{68} -1.00000 q^{69} -4.42662 q^{70} +12.9362 q^{71} -1.00000 q^{72} +1.07906 q^{73} -1.95089 q^{74} -7.94699 q^{75} +6.94699 q^{76} +0.320090 q^{77} +5.91704 q^{78} +5.02871 q^{79} +3.59819 q^{80} +1.00000 q^{81} -1.49042 q^{82} +14.5452 q^{83} -1.23023 q^{84} -14.6038 q^{85} +2.42662 q^{86} +1.00000 q^{87} -0.260186 q^{88} -10.4279 q^{89} -3.59819 q^{90} +7.27935 q^{91} +1.00000 q^{92} -1.73981 q^{93} -2.76977 q^{94} +24.9966 q^{95} +1.00000 q^{96} -11.3488 q^{97} +5.48652 q^{98} +0.260186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 - 4 * q^12 - 6 * q^14 - 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^19 + 2 * q^20 - 6 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^36 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^45 - 4 * q^46 + 10 * q^47 - 4 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^51 + 4 * q^53 + 4 * q^54 - 20 * q^55 - 6 * q^56 - 2 * q^57 + 4 * q^58 - 22 * q^59 - 2 * q^60 + 28 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 + 12 * q^65 + 24 * q^67 - 6 * q^68 - 4 * q^69 + 6 * q^70 + 28 * q^71 - 4 * q^72 - 10 * q^74 - 6 * q^75 + 2 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 4 * q^81 - 6 * q^82 + 20 * q^83 - 6 * q^84 - 10 * q^85 - 14 * q^86 + 4 * q^87 - 24 * q^89 - 2 * q^90 + 28 * q^91 + 4 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^95 + 4 * q^96 - 32 * q^97 - 6 * q^98

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