LMFDB - Embedded newform 8002.2.a.e.1.41

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.41
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.193143 q^{3} +1.00000 q^{4} -1.89295 q^{5} -0.193143 q^{6} +0.398672 q^{7} -1.00000 q^{8} -2.96270 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.193143 q^{3} +1.00000 q^{4} -1.89295 q^{5} -0.193143 q^{6} +0.398672 q^{7} -1.00000 q^{8} -2.96270 q^{9} +1.89295 q^{10} +0.0720742 q^{11} +0.193143 q^{12} +1.22384 q^{13} -0.398672 q^{14} -0.365611 q^{15} +1.00000 q^{16} +6.21131 q^{17} +2.96270 q^{18} +1.97304 q^{19} -1.89295 q^{20} +0.0770007 q^{21} -0.0720742 q^{22} -6.42095 q^{23} -0.193143 q^{24} -1.41672 q^{25} -1.22384 q^{26} -1.15165 q^{27} +0.398672 q^{28} +3.17503 q^{29} +0.365611 q^{30} -7.71025 q^{31} -1.00000 q^{32} +0.0139206 q^{33} -6.21131 q^{34} -0.754669 q^{35} -2.96270 q^{36} +3.20511 q^{37} -1.97304 q^{38} +0.236376 q^{39} +1.89295 q^{40} -5.43651 q^{41} -0.0770007 q^{42} +8.62281 q^{43} +0.0720742 q^{44} +5.60825 q^{45} +6.42095 q^{46} -1.45651 q^{47} +0.193143 q^{48} -6.84106 q^{49} +1.41672 q^{50} +1.19967 q^{51} +1.22384 q^{52} +2.67723 q^{53} +1.15165 q^{54} -0.136433 q^{55} -0.398672 q^{56} +0.381079 q^{57} -3.17503 q^{58} +6.28569 q^{59} -0.365611 q^{60} -1.22508 q^{61} +7.71025 q^{62} -1.18114 q^{63} +1.00000 q^{64} -2.31667 q^{65} -0.0139206 q^{66} +2.02486 q^{67} +6.21131 q^{68} -1.24016 q^{69} +0.754669 q^{70} +6.15138 q^{71} +2.96270 q^{72} -2.54741 q^{73} -3.20511 q^{74} -0.273630 q^{75} +1.97304 q^{76} +0.0287340 q^{77} -0.236376 q^{78} +2.70813 q^{79} -1.89295 q^{80} +8.66565 q^{81} +5.43651 q^{82} -1.54767 q^{83} +0.0770007 q^{84} -11.7577 q^{85} -8.62281 q^{86} +0.613234 q^{87} -0.0720742 q^{88} +1.44010 q^{89} -5.60825 q^{90} +0.487910 q^{91} -6.42095 q^{92} -1.48918 q^{93} +1.45651 q^{94} -3.73488 q^{95} -0.193143 q^{96} -15.4102 q^{97} +6.84106 q^{98} -0.213534 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * 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
$3$	0.193143	0.111511	0.0557556	−	0.998444i	$-0.482243\pi$
$3$	0.193143	0.111511	0.0557556	+	0.998444i	$0.482243\pi$
$4$	1.00000	0.500000
$5$	−1.89295	−0.846555	−0.423278	−	0.906000i	$-0.639120\pi$
$5$	−1.89295	−0.846555	−0.423278	+	0.906000i	$0.639120\pi$
$6$	−0.193143	−0.0788503
$7$	0.398672	0.150684	0.0753420	−	0.997158i	$-0.475995\pi$
$7$	0.398672	0.150684	0.0753420	+	0.997158i	$0.475995\pi$
$8$	−1.00000	−0.353553
$9$	−2.96270	−0.987565
$10$	1.89295	0.598605
$11$	0.0720742	0.0217312	0.0108656	−	0.999941i	$-0.496541\pi$
$11$	0.0720742	0.0217312	0.0108656	+	0.999941i	$0.496541\pi$
$12$	0.193143	0.0557556
$13$	1.22384	0.339432	0.169716	−	0.985493i	$-0.445715\pi$
$13$	1.22384	0.339432	0.169716	+	0.985493i	$0.445715\pi$
$14$	−0.398672	−0.106550
$15$	−0.365611	−0.0944003
$16$	1.00000	0.250000
$17$	6.21131	1.50646	0.753232	−	0.657755i	$-0.228494\pi$
$17$	6.21131	1.50646	0.753232	+	0.657755i	$0.228494\pi$
$18$	2.96270	0.698314
$19$	1.97304	0.452647	0.226323	−	0.974052i	$-0.427329\pi$
$19$	1.97304	0.452647	0.226323	+	0.974052i	$0.427329\pi$
$20$	−1.89295	−0.423278
$21$	0.0770007	0.0168029
$22$	−0.0720742	−0.0153663
$23$	−6.42095	−1.33886	−0.669430	−	0.742875i	$-0.733462\pi$
$23$	−6.42095	−1.33886	−0.669430	+	0.742875i	$0.733462\pi$
$24$	−0.193143	−0.0394251
$25$	−1.41672	−0.283345
$26$	−1.22384	−0.240014
$27$	−1.15165	−0.221636
$28$	0.398672	0.0753420
$29$	3.17503	0.589587	0.294794	−	0.955561i	$-0.404749\pi$
$29$	3.17503	0.589587	0.294794	+	0.955561i	$0.404749\pi$
$30$	0.365611	0.0667511
$31$	−7.71025	−1.38480	−0.692401	−	0.721513i	$-0.743447\pi$
$31$	−7.71025	−1.38480	−0.692401	+	0.721513i	$0.743447\pi$
$32$	−1.00000	−0.176777
$33$	0.0139206	0.00242327
$34$	−6.21131	−1.06523
$35$	−0.754669	−0.127562
$36$	−2.96270	−0.493783
$37$	3.20511	0.526917	0.263458	−	0.964671i	$-0.415137\pi$
$37$	3.20511	0.526917	0.263458	+	0.964671i	$0.415137\pi$
$38$	−1.97304	−0.320070
$39$	0.236376	0.0378504
$40$	1.89295	0.299302
$41$	−5.43651	−0.849041	−0.424520	−	0.905418i	$-0.639557\pi$
$41$	−5.43651	−0.849041	−0.424520	+	0.905418i	$0.639557\pi$
$42$	−0.0770007	−0.0118815
$43$	8.62281	1.31497	0.657483	−	0.753469i	$-0.271621\pi$
$43$	8.62281	1.31497	0.657483	+	0.753469i	$0.271621\pi$
$44$	0.0720742	0.0108656
$45$	5.60825	0.836028
$46$	6.42095	0.946717
$47$	−1.45651	−0.212454	−0.106227	−	0.994342i	$-0.533877\pi$
$47$	−1.45651	−0.212454	−0.106227	+	0.994342i	$0.533877\pi$
$48$	0.193143	0.0278778
$49$	−6.84106	−0.977294
$50$	1.41672	0.200355
$51$	1.19967	0.167988
$52$	1.22384	0.169716
$53$	2.67723	0.367746	0.183873	−	0.982950i	$-0.441136\pi$
$53$	2.67723	0.367746	0.183873	+	0.982950i	$0.441136\pi$
$54$	1.15165	0.156720
$55$	−0.136433	−0.0183967
$56$	−0.398672	−0.0532748
$57$	0.381079	0.0504751
$58$	−3.17503	−0.416901
$59$	6.28569	0.818328	0.409164	−	0.912461i	$-0.365820\pi$
$59$	6.28569	0.818328	0.409164	+	0.912461i	$0.365820\pi$
$60$	−0.365611	−0.0472001
$61$	−1.22508	−0.156855	−0.0784277	−	0.996920i	$-0.524990\pi$
$61$	−1.22508	−0.156855	−0.0784277	+	0.996920i	$0.524990\pi$
$62$	7.71025	0.979203
$63$	−1.18114	−0.148810
$64$	1.00000	0.125000
$65$	−2.31667	−0.287348
$66$	−0.0139206	−0.00171351
$67$	2.02486	0.247376	0.123688	−	0.992321i	$-0.460528\pi$
$67$	2.02486	0.247376	0.123688	+	0.992321i	$0.460528\pi$
$68$	6.21131	0.753232
$69$	−1.24016	−0.149298
$70$	0.754669	0.0902002
$71$	6.15138	0.730034	0.365017	−	0.931001i	$-0.381063\pi$
$71$	6.15138	0.730034	0.365017	+	0.931001i	$0.381063\pi$
$72$	2.96270	0.349157
$73$	−2.54741	−0.298152	−0.149076	−	0.988826i	$-0.547630\pi$
$73$	−2.54741	−0.298152	−0.149076	+	0.988826i	$0.547630\pi$
$74$	−3.20511	−0.372586
$75$	−0.273630	−0.0315961
$76$	1.97304	0.226323
$77$	0.0287340	0.00327454
$78$	−0.236376	−0.0267643
$79$	2.70813	0.304688	0.152344	−	0.988328i	$-0.451318\pi$
$79$	2.70813	0.304688	0.152344	+	0.988328i	$0.451318\pi$
$80$	−1.89295	−0.211639
$81$	8.66565	0.962850
$82$	5.43651	0.600363
$83$	−1.54767	−0.169879	−0.0849396	−	0.996386i	$-0.527070\pi$
$83$	−1.54767	−0.169879	−0.0849396	+	0.996386i	$0.527070\pi$
$84$	0.0770007	0.00840147
$85$	−11.7577	−1.27530
$86$	−8.62281	−0.929821
$87$	0.613234	0.0657455
$88$	−0.0720742	−0.00768314
$89$	1.44010	0.152650	0.0763249	−	0.997083i	$-0.475681\pi$
$89$	1.44010	0.152650	0.0763249	+	0.997083i	$0.475681\pi$
$90$	−5.60825	−0.591161
$91$	0.487910	0.0511469
$92$	−6.42095	−0.669430
$93$	−1.48918	−0.154421
$94$	1.45651	0.150228
$95$	−3.73488	−0.383190
$96$	−0.193143	−0.0197126
$97$	−15.4102	−1.56467	−0.782335	−	0.622858i	$-0.785972\pi$
$97$	−15.4102	−1.56467	−0.782335	+	0.622858i	$0.785972\pi$
$98$	6.84106	0.691051
$99$	−0.213534	−0.0214610
$100$	−1.41672	−0.141672
$101$	7.75709	0.771860	0.385930	−	0.922528i	$-0.373881\pi$
$101$	7.75709	0.771860	0.385930	+	0.922528i	$0.373881\pi$
$102$	−1.19967	−0.118785
$103$	−9.09523	−0.896180	−0.448090	−	0.893988i	$-0.647896\pi$
$103$	−9.09523	−0.896180	−0.448090	+	0.893988i	$0.647896\pi$
$104$	−1.22384	−0.120007
$105$	−0.145759	−0.0142246
$106$	−2.67723	−0.260036
$107$	3.04206	0.294087	0.147043	−	0.989130i	$-0.453024\pi$
$107$	3.04206	0.294087	0.147043	+	0.989130i	$0.453024\pi$
$108$	−1.15165	−0.110818
$109$	−3.10864	−0.297754	−0.148877	−	0.988856i	$-0.547566\pi$
$109$	−3.10864	−0.297754	−0.148877	+	0.988856i	$0.547566\pi$
$110$	0.136433	0.0130084
$111$	0.619044	0.0587571
$112$	0.398672	0.0376710
$113$	2.98203	0.280526	0.140263	−	0.990114i	$-0.455205\pi$
$113$	2.98203	0.280526	0.140263	+	0.990114i	$0.455205\pi$
$114$	−0.381079	−0.0356913
$115$	12.1546	1.13342
$116$	3.17503	0.294794
$117$	−3.62586	−0.335211
$118$	−6.28569	−0.578645
$119$	2.47628	0.227000
$120$	0.365611	0.0333755
$121$	−10.9948	−0.999528
$122$	1.22508	0.110914
$123$	−1.05002	−0.0946775
$124$	−7.71025	−0.692401
$125$	12.1466	1.08642
$126$	1.18114	0.105225
$127$	−21.7266	−1.92793	−0.963964	−	0.266032i	$-0.914287\pi$
$127$	−21.7266	−1.92793	−0.963964	+	0.266032i	$0.914287\pi$
$128$	−1.00000	−0.0883883
$129$	1.66543	0.146633
$130$	2.31667	0.203185
$131$	18.1793	1.58833	0.794167	−	0.607700i	$-0.207908\pi$
$131$	18.1793	1.58833	0.794167	+	0.607700i	$0.207908\pi$
$132$	0.0139206	0.00121163
$133$	0.786597	0.0682066
$134$	−2.02486	−0.174921
$135$	2.18003	0.187627
$136$	−6.21131	−0.532616
$137$	−18.5408	−1.58405	−0.792025	−	0.610489i	$-0.790973\pi$
$137$	−18.5408	−1.58405	−0.792025	+	0.610489i	$0.790973\pi$
$138$	1.24016	0.105569
$139$	1.32936	0.112755	0.0563776	−	0.998410i	$-0.482045\pi$
$139$	1.32936	0.112755	0.0563776	+	0.998410i	$0.482045\pi$
$140$	−0.754669	−0.0637811
$141$	−0.281315	−0.0236910
$142$	−6.15138	−0.516212
$143$	0.0882072	0.00737625
$144$	−2.96270	−0.246891
$145$	−6.01018	−0.499118
$146$	2.54741	0.210825
$147$	−1.32130	−0.108979
$148$	3.20511	0.263458
$149$	−2.05178	−0.168088	−0.0840442	−	0.996462i	$-0.526784\pi$
$149$	−2.05178	−0.168088	−0.0840442	+	0.996462i	$0.526784\pi$
$150$	0.273630	0.0223418
$151$	−5.91990	−0.481755	−0.240877	−	0.970556i	$-0.577435\pi$
$151$	−5.91990	−0.481755	−0.240877	+	0.970556i	$0.577435\pi$
$152$	−1.97304	−0.160035
$153$	−18.4022	−1.48773
$154$	−0.0287340	−0.00231545
$155$	14.5952	1.17231
$156$	0.236376	0.0189252
$157$	7.10945	0.567396	0.283698	−	0.958914i	$-0.408439\pi$
$157$	7.10945	0.567396	0.283698	+	0.958914i	$0.408439\pi$
$158$	−2.70813	−0.215447
$159$	0.517088	0.0410078
$160$	1.89295	0.149651
$161$	−2.55985	−0.201745
$162$	−8.66565	−0.680838
$163$	−5.08181	−0.398038	−0.199019	−	0.979996i	$-0.563776\pi$
$163$	−5.08181	−0.398038	−0.199019	+	0.979996i	$0.563776\pi$
$164$	−5.43651	−0.424520
$165$	−0.0263511	−0.00205143
$166$	1.54767	0.120123
$167$	2.26797	0.175501	0.0877505	−	0.996142i	$-0.472032\pi$
$167$	2.26797	0.175501	0.0877505	+	0.996142i	$0.472032\pi$
$168$	−0.0770007	−0.00594074
$169$	−11.5022	−0.884786
$170$	11.7577	0.901777
$171$	−5.84552	−0.447018
$172$	8.62281	0.657483
$173$	19.9017	1.51310	0.756550	−	0.653935i	$-0.226883\pi$
$173$	19.9017	1.51310	0.756550	+	0.653935i	$0.226883\pi$
$174$	−0.613234	−0.0464891
$175$	−0.564808	−0.0426955
$176$	0.0720742	0.00543280
$177$	1.21404	0.0912526
$178$	−1.44010	−0.107940
$179$	9.70466	0.725360	0.362680	−	0.931914i	$-0.381862\pi$
$179$	9.70466	0.725360	0.362680	+	0.931914i	$0.381862\pi$
$180$	5.60825	0.418014
$181$	8.06824	0.599707	0.299854	−	0.953985i	$-0.403062\pi$
$181$	8.06824	0.599707	0.299854	+	0.953985i	$0.403062\pi$
$182$	−0.487910	−0.0361663
$183$	−0.236616	−0.0174911
$184$	6.42095	0.473358
$185$	−6.06713	−0.446064
$186$	1.48918	0.109192
$187$	0.447675	0.0327373
$188$	−1.45651	−0.106227
$189$	−0.459132	−0.0333969
$190$	3.73488	0.270957
$191$	8.07236	0.584095	0.292048	−	0.956404i	$-0.405663\pi$
$191$	8.07236	0.584095	0.292048	+	0.956404i	$0.405663\pi$
$192$	0.193143	0.0139389
$193$	7.01926	0.505258	0.252629	−	0.967563i	$-0.418705\pi$
$193$	7.01926	0.505258	0.252629	+	0.967563i	$0.418705\pi$
$194$	15.4102	1.10639
$195$	−0.447448	−0.0320425
$196$	−6.84106	−0.488647
$197$	11.2729	0.803161	0.401580	−	0.915824i	$-0.368461\pi$
$197$	11.2729	0.803161	0.401580	+	0.915824i	$0.368461\pi$
$198$	0.213534	0.0151752
$199$	5.75163	0.407722	0.203861	−	0.979000i	$-0.434651\pi$
$199$	5.75163	0.407722	0.203861	+	0.979000i	$0.434651\pi$
$200$	1.41672	0.100177
$201$	0.391087	0.0275851
$202$	−7.75709	−0.545787
$203$	1.26579	0.0888414
$204$	1.19967	0.0839938
$205$	10.2911	0.718760
$206$	9.09523	0.633695
$207$	19.0233	1.32221
$208$	1.22384	0.0848579
$209$	0.142205	0.00983655
$210$	0.145759	0.0100583
$211$	−0.352380	−0.0242588	−0.0121294	−	0.999926i	$-0.503861\pi$
$211$	−0.352380	−0.0242588	−0.0121294	+	0.999926i	$0.503861\pi$
$212$	2.67723	0.183873
$213$	1.18810	0.0814069
$214$	−3.04206	−0.207951
$215$	−16.3226	−1.11319
$216$	1.15165	0.0783600
$217$	−3.07386	−0.208667
$218$	3.10864	0.210544
$219$	−0.492014	−0.0332473
$220$	−0.136433	−0.00919833
$221$	7.60164	0.511342
$222$	−0.619044	−0.0415475
$223$	6.01462	0.402769	0.201384	−	0.979512i	$-0.435456\pi$
$223$	6.01462	0.402769	0.201384	+	0.979512i	$0.435456\pi$
$224$	−0.398672	−0.0266374
$225$	4.19732	0.279821
$226$	−2.98203	−0.198362
$227$	1.29928	0.0862365	0.0431183	−	0.999070i	$-0.486271\pi$
$227$	1.29928	0.0862365	0.0431183	+	0.999070i	$0.486271\pi$
$228$	0.381079	0.0252376
$229$	27.5795	1.82250	0.911252	−	0.411848i	$-0.135117\pi$
$229$	27.5795	1.82250	0.911252	+	0.411848i	$0.135117\pi$
$230$	−12.1546	−0.801448
$231$	0.00554977	0.000365148 0
$232$	−3.17503	−0.208451
$233$	−12.2937	−0.805384	−0.402692	−	0.915335i	$-0.631925\pi$
$233$	−12.2937	−0.805384	−0.402692	+	0.915335i	$0.631925\pi$
$234$	3.62586	0.237030
$235$	2.75711	0.179854
$236$	6.28569	0.409164
$237$	0.523056	0.0339761
$238$	−2.47628	−0.160513
$239$	20.9876	1.35758	0.678789	−	0.734334i	$-0.262505\pi$
$239$	20.9876	1.35758	0.678789	+	0.734334i	$0.262505\pi$
$240$	−0.365611	−0.0236001
$241$	23.3613	1.50483	0.752415	−	0.658689i	$-0.228889\pi$
$241$	23.3613	1.50483	0.752415	+	0.658689i	$0.228889\pi$
$242$	10.9948	0.706773
$243$	5.12867	0.329004
$244$	−1.22508	−0.0784277
$245$	12.9498	0.827333
$246$	1.05002	0.0669471
$247$	2.41468	0.153643
$248$	7.71025	0.489601
$249$	−0.298922	−0.0189434
$250$	−12.1466	−0.768216
$251$	−14.9746	−0.945187	−0.472593	−	0.881281i	$-0.656682\pi$
$251$	−14.9746	−0.945187	−0.472593	+	0.881281i	$0.656682\pi$
$252$	−1.18114	−0.0744051
$253$	−0.462785	−0.0290950
$254$	21.7266	1.36325
$255$	−2.27092	−0.142211
$256$	1.00000	0.0625000
$257$	−16.0019	−0.998170	−0.499085	−	0.866553i	$-0.666331\pi$
$257$	−16.0019	−0.998170	−0.499085	+	0.866553i	$0.666331\pi$
$258$	−1.66543	−0.103685
$259$	1.27779	0.0793979
$260$	−2.31667	−0.143674
$261$	−9.40663	−0.582256
$262$	−18.1793	−1.12312
$263$	9.69832	0.598024	0.299012	−	0.954249i	$-0.403343\pi$
$263$	9.69832	0.598024	0.299012	+	0.954249i	$0.403343\pi$
$264$	−0.0139206	−0.000856755 0
$265$	−5.06788	−0.311317
$266$	−0.786597	−0.0482294
$267$	0.278144	0.0170222
$268$	2.02486	0.123688
$269$	−28.2199	−1.72060	−0.860300	−	0.509788i	$-0.829724\pi$
$269$	−28.2199	−1.72060	−0.860300	+	0.509788i	$0.829724\pi$
$270$	−2.18003	−0.132672
$271$	7.84250	0.476398	0.238199	−	0.971216i	$-0.423443\pi$
$271$	7.84250	0.476398	0.238199	+	0.971216i	$0.423443\pi$
$272$	6.21131	0.376616
$273$	0.0942364	0.00570345
$274$	18.5408	1.12009
$275$	−0.102109	−0.00615741
$276$	−1.24016	−0.0746489
$277$	1.11751	0.0671449	0.0335724	−	0.999436i	$-0.489312\pi$
$277$	1.11751	0.0671449	0.0335724	+	0.999436i	$0.489312\pi$
$278$	−1.32936	−0.0797300
$279$	22.8431	1.36758
$280$	0.754669	0.0451001
$281$	20.6275	1.23053	0.615266	−	0.788319i	$-0.289048\pi$
$281$	20.6275	1.23053	0.615266	+	0.788319i	$0.289048\pi$
$282$	0.281315	0.0167521
$283$	6.79028	0.403640	0.201820	−	0.979423i	$-0.435314\pi$
$283$	6.79028	0.403640	0.201820	+	0.979423i	$0.435314\pi$
$284$	6.15138	0.365017
$285$	−0.721365	−0.0427300
$286$	−0.0882072	−0.00521580
$287$	−2.16739	−0.127937
$288$	2.96270	0.174579
$289$	21.5804	1.26943
$290$	6.01018	0.352930
$291$	−2.97637	−0.174478
$292$	−2.54741	−0.149076
$293$	−19.6711	−1.14920	−0.574598	−	0.818436i	$-0.694842\pi$
$293$	−19.6711	−1.14920	−0.574598	+	0.818436i	$0.694842\pi$
$294$	1.32130	0.0770599
$295$	−11.8985	−0.692760
$296$	−3.20511	−0.186293
$297$	−0.0830044	−0.00481641
$298$	2.05178	0.118856
$299$	−7.85820	−0.454451
$300$	−0.273630	−0.0157980
$301$	3.43768	0.198144
$302$	5.91990	0.340652
$303$	1.49823	0.0860709
$304$	1.97304	0.113162
$305$	2.31902	0.132787
$306$	18.4022	1.05199
$307$	−7.23223	−0.412765	−0.206383	−	0.978471i	$-0.566169\pi$
$307$	−7.23223	−0.412765	−0.206383	+	0.978471i	$0.566169\pi$
$308$	0.0287340	0.00163727
$309$	−1.75668	−0.0999340
$310$	−14.5952	−0.828949
$311$	1.33111	0.0754801	0.0377400	−	0.999288i	$-0.487984\pi$
$311$	1.33111	0.0754801	0.0377400	+	0.999288i	$0.487984\pi$
$312$	−0.236376	−0.0133821
$313$	16.5452	0.935191	0.467596	−	0.883942i	$-0.345120\pi$
$313$	16.5452	0.935191	0.467596	+	0.883942i	$0.345120\pi$
$314$	−7.10945	−0.401210
$315$	2.23585	0.125976
$316$	2.70813	0.152344
$317$	24.7716	1.39131	0.695655	−	0.718376i	$-0.255114\pi$
$317$	24.7716	1.39131	0.695655	+	0.718376i	$0.255114\pi$
$318$	−0.517088	−0.0289969
$319$	0.228837	0.0128124
$320$	−1.89295	−0.105819
$321$	0.587552	0.0327939
$322$	2.55985	0.142655
$323$	12.2552	0.681896
$324$	8.66565	0.481425
$325$	−1.73384	−0.0961761
$326$	5.08181	0.281456
$327$	−0.600413	−0.0332029
$328$	5.43651	0.300181
$329$	−0.580672	−0.0320135
$330$	0.0263511	0.00145058
$331$	31.6706	1.74078	0.870388	−	0.492367i	$-0.163868\pi$
$331$	31.6706	1.74078	0.870388	+	0.492367i	$0.163868\pi$
$332$	−1.54767	−0.0849396
$333$	−9.49576	−0.520365
$334$	−2.26797	−0.124098
$335$	−3.83296	−0.209417
$336$	0.0770007	0.00420073
$337$	4.18090	0.227748	0.113874	−	0.993495i	$-0.463674\pi$
$337$	4.18090	0.227748	0.113874	+	0.993495i	$0.463674\pi$
$338$	11.5022	0.625638
$339$	0.575958	0.0312817
$340$	−11.7577	−0.637652
$341$	−0.555710	−0.0300934
$342$	5.84552	0.316090
$343$	−5.51805	−0.297947
$344$	−8.62281	−0.464911
$345$	2.34757	0.126389
$346$	−19.9017	−1.06992
$347$	−17.6118	−0.945449	−0.472725	−	0.881210i	$-0.656729\pi$
$347$	−17.6118	−0.945449	−0.472725	+	0.881210i	$0.656729\pi$
$348$	0.613234	0.0328728
$349$	14.0874	0.754080	0.377040	−	0.926197i	$-0.376942\pi$
$349$	14.0874	0.754080	0.377040	+	0.926197i	$0.376942\pi$
$350$	0.564808	0.0301903
$351$	−1.40944	−0.0752301
$352$	−0.0720742	−0.00384157
$353$	24.8100	1.32050	0.660251	−	0.751045i	$-0.270450\pi$
$353$	24.8100	1.32050	0.660251	+	0.751045i	$0.270450\pi$
$354$	−1.21404	−0.0645254
$355$	−11.6443	−0.618014
$356$	1.44010	0.0763249
$357$	0.478276	0.0253130
$358$	−9.70466	−0.512907
$359$	20.7624	1.09580	0.547898	−	0.836545i	$-0.315428\pi$
$359$	20.7624	1.09580	0.547898	+	0.836545i	$0.315428\pi$
$360$	−5.60825	−0.295581
$361$	−15.1071	−0.795111
$362$	−8.06824	−0.424057
$363$	−2.12357	−0.111458
$364$	0.487910	0.0255735
$365$	4.82213	0.252402
$366$	0.236616	0.0123681
$367$	5.16918	0.269829	0.134914	−	0.990857i	$-0.456924\pi$
$367$	5.16918	0.269829	0.134914	+	0.990857i	$0.456924\pi$
$368$	−6.42095	−0.334715
$369$	16.1067	0.838483
$370$	6.06713	0.315415
$371$	1.06734	0.0554134
$372$	−1.48918	−0.0772104
$373$	10.5803	0.547826	0.273913	−	0.961755i	$-0.411682\pi$
$373$	10.5803	0.547826	0.273913	+	0.961755i	$0.411682\pi$
$374$	−0.447675	−0.0231487
$375$	2.34602	0.121148
$376$	1.45651	0.0751139
$377$	3.88572	0.200125
$378$	0.459132	0.0236152
$379$	−10.4336	−0.535939	−0.267969	−	0.963427i	$-0.586353\pi$
$379$	−10.4336	−0.535939	−0.267969	+	0.963427i	$0.586353\pi$
$380$	−3.73488	−0.191595
$381$	−4.19635	−0.214985
$382$	−8.07236	−0.413018
$383$	8.51000	0.434840	0.217420	−	0.976078i	$-0.430236\pi$
$383$	8.51000	0.434840	0.217420	+	0.976078i	$0.430236\pi$
$384$	−0.193143	−0.00985628
$385$	−0.0543921	−0.00277208
$386$	−7.01926	−0.357271
$387$	−25.5468	−1.29861
$388$	−15.4102	−0.782335
$389$	14.9168	0.756310	0.378155	−	0.925742i	$-0.376559\pi$
$389$	14.9168	0.756310	0.378155	+	0.925742i	$0.376559\pi$
$390$	0.447448	0.0226574
$391$	−39.8825	−2.01694
$392$	6.84106	0.345526
$393$	3.51121	0.177117
$394$	−11.2729	−0.567920
$395$	−5.12636	−0.257935
$396$	−0.213534	−0.0107305
$397$	29.3302	1.47204	0.736020	−	0.676959i	$-0.236703\pi$
$397$	29.3302	1.47204	0.736020	+	0.676959i	$0.236703\pi$
$398$	−5.75163	−0.288303
$399$	0.151926	0.00760580
$400$	−1.41672	−0.0708361
$401$	−5.06935	−0.253151	−0.126576	−	0.991957i	$-0.540399\pi$
$401$	−5.06935	−0.253151	−0.126576	+	0.991957i	$0.540399\pi$
$402$	−0.391087	−0.0195056
$403$	−9.43610	−0.470046
$404$	7.75709	0.385930
$405$	−16.4037	−0.815106
$406$	−1.26579	−0.0628203
$407$	0.231006	0.0114505
$408$	−1.19967	−0.0593926
$409$	9.78754	0.483963	0.241981	−	0.970281i	$-0.422203\pi$
$409$	9.78754	0.483963	0.241981	+	0.970281i	$0.422203\pi$
$410$	−10.2911	−0.508240
$411$	−3.58103	−0.176639
$412$	−9.09523	−0.448090
$413$	2.50593	0.123309
$414$	−19.0233	−0.934945
$415$	2.92968	0.143812
$416$	−1.22384	−0.0600036
$417$	0.256757	0.0125735
$418$	−0.142205	−0.00695549
$419$	−25.0626	−1.22439	−0.612193	−	0.790708i	$-0.709712\pi$
$419$	−25.0626	−1.22439	−0.612193	+	0.790708i	$0.709712\pi$
$420$	−0.145759	−0.00711231
$421$	1.03241	0.0503166	0.0251583	−	0.999683i	$-0.491991\pi$
$421$	1.03241	0.0503166	0.0251583	+	0.999683i	$0.491991\pi$
$422$	0.352380	0.0171536
$423$	4.31521	0.209812
$424$	−2.67723	−0.130018
$425$	−8.79971	−0.426848
$426$	−1.18810	−0.0575634
$427$	−0.488406	−0.0236356
$428$	3.04206	0.147043
$429$	0.0170366	0.000822534 0
$430$	16.3226	0.787145
$431$	10.6462	0.512807	0.256404	−	0.966570i	$-0.417462\pi$
$431$	10.6462	0.512807	0.256404	+	0.966570i	$0.417462\pi$
$432$	−1.15165	−0.0554089
$433$	−39.9969	−1.92213	−0.961063	−	0.276328i	$-0.910882\pi$
$433$	−39.9969	−1.92213	−0.961063	+	0.276328i	$0.910882\pi$
$434$	3.07386	0.147550
$435$	−1.16082	−0.0556572
$436$	−3.10864	−0.148877
$437$	−12.6688	−0.606031
$438$	0.492014	0.0235094
$439$	−14.3081	−0.682887	−0.341443	−	0.939902i	$-0.610916\pi$
$439$	−14.3081	−0.682887	−0.341443	+	0.939902i	$0.610916\pi$
$440$	0.136433	0.00650420
$441$	20.2680	0.965142
$442$	−7.60164	−0.361573
$443$	25.2432	1.19934	0.599670	−	0.800247i	$-0.295298\pi$
$443$	25.2432	1.19934	0.599670	+	0.800247i	$0.295298\pi$
$444$	0.619044	0.0293785
$445$	−2.72604	−0.129226
$446$	−6.01462	−0.284800
$447$	−0.396287	−0.0187437
$448$	0.398672	0.0188355
$449$	24.9990	1.17977	0.589887	−	0.807486i	$-0.299172\pi$
$449$	24.9990	1.17977	0.589887	+	0.807486i	$0.299172\pi$
$450$	−4.19732	−0.197863
$451$	−0.391832	−0.0184507
$452$	2.98203	0.140263
$453$	−1.14339	−0.0537210
$454$	−1.29928	−0.0609784
$455$	−0.923592	−0.0432987
$456$	−0.381079	−0.0178457
$457$	−22.3620	−1.04605	−0.523025	−	0.852317i	$-0.675197\pi$
$457$	−22.3620	−1.04605	−0.523025	+	0.852317i	$0.675197\pi$
$458$	−27.5795	−1.28871
$459$	−7.15327	−0.333886
$460$	12.1546	0.566709
$461$	16.7120	0.778358	0.389179	−	0.921162i	$-0.372759\pi$
$461$	16.7120	0.778358	0.389179	+	0.921162i	$0.372759\pi$
$462$	−0.00554977	−0.000258199 0
$463$	−8.98307	−0.417478	−0.208739	−	0.977971i	$-0.566936\pi$
$463$	−8.98307	−0.417478	−0.208739	+	0.977971i	$0.566936\pi$
$464$	3.17503	0.147397
$465$	2.81895	0.130726
$466$	12.2937	0.569493
$467$	35.3400	1.63534	0.817669	−	0.575689i	$-0.195266\pi$
$467$	35.3400	1.63534	0.817669	+	0.575689i	$0.195266\pi$
$468$	−3.62586	−0.167605
$469$	0.807254	0.0372755
$470$	−2.75711	−0.127176
$471$	1.37314	0.0632710
$472$	−6.28569	−0.289323
$473$	0.621482	0.0285758
$474$	−0.523056	−0.0240247
$475$	−2.79525	−0.128255
$476$	2.47628	0.113500
$477$	−7.93182	−0.363173
$478$	−20.9876	−0.959952
$479$	17.9948	0.822202	0.411101	−	0.911590i	$-0.365144\pi$
$479$	17.9948	0.822202	0.411101	+	0.911590i	$0.365144\pi$
$480$	0.365611	0.0166878
$481$	3.92254	0.178852
$482$	−23.3613	−1.06408
$483$	−0.494418	−0.0224968
$484$	−10.9948	−0.499764
$485$	29.1708	1.32458
$486$	−5.12867	−0.232641
$487$	39.6892	1.79849	0.899245	−	0.437445i	$-0.144116\pi$
$487$	39.6892	1.79849	0.899245	+	0.437445i	$0.144116\pi$
$488$	1.22508	0.0554568
$489$	−0.981516	−0.0443857
$490$	−12.9498	−0.585013
$491$	−10.2844	−0.464130	−0.232065	−	0.972700i	$-0.574548\pi$
$491$	−10.2844	−0.464130	−0.232065	+	0.972700i	$0.574548\pi$
$492$	−1.05002	−0.0473387
$493$	19.7211	0.888192
$494$	−2.41468	−0.108642
$495$	0.404210	0.0181679
$496$	−7.71025	−0.346200
$497$	2.45238	0.110004
$498$	0.298922	0.0133950
$499$	−30.2073	−1.35226	−0.676132	−	0.736780i	$-0.736345\pi$
$499$	−30.2073	−1.35226	−0.676132	+	0.736780i	$0.736345\pi$
$500$	12.1466	0.543211
$501$	0.438043	0.0195703
$502$	14.9746	0.668348
$503$	28.0954	1.25271	0.626357	−	0.779536i	$-0.284545\pi$
$503$	28.0954	1.25271	0.626357	+	0.779536i	$0.284545\pi$
$504$	1.18114	0.0526124
$505$	−14.6838	−0.653422
$506$	0.462785	0.0205733
$507$	−2.22157	−0.0986635
$508$	−21.7266	−0.963964
$509$	−12.3736	−0.548449	−0.274224	−	0.961666i	$-0.588421\pi$
$509$	−12.3736	−0.548449	−0.274224	+	0.961666i	$0.588421\pi$
$510$	2.27092	0.100558
$511$	−1.01558	−0.0449267
$512$	−1.00000	−0.0441942
$513$	−2.27226	−0.100323
$514$	16.0019	0.705813
$515$	17.2169	0.758666
$516$	1.66543	0.0733167
$517$	−0.104977	−0.00461689
$518$	−1.27779	−0.0561428
$519$	3.84388	0.168728
$520$	2.31667	0.101593
$521$	−17.4719	−0.765458	−0.382729	−	0.923861i	$-0.625016\pi$
$521$	−17.4719	−0.765458	−0.382729	+	0.923861i	$0.625016\pi$
$522$	9.40663	0.411717
$523$	33.4627	1.46322	0.731610	−	0.681723i	$-0.238769\pi$
$523$	33.4627	1.46322	0.731610	+	0.681723i	$0.238769\pi$
$524$	18.1793	0.794167
$525$	−0.109089	−0.00476102
$526$	−9.69832	−0.422867
$527$	−47.8908	−2.08615
$528$	0.0139206	0.000605817 0
$529$	18.2286	0.792546
$530$	5.06788	0.220135
$531$	−18.6226	−0.808152
$532$	0.786597	0.0341033
$533$	−6.65341	−0.288191
$534$	−0.278144	−0.0120365
$535$	−5.75847	−0.248961
$536$	−2.02486	−0.0874605
$537$	1.87439	0.0808858
$538$	28.2199	1.21665
$539$	−0.493064	−0.0212378
$540$	2.18003	0.0938134
$541$	16.8392	0.723973	0.361986	−	0.932183i	$-0.382099\pi$
$541$	16.8392	0.723973	0.361986	+	0.932183i	$0.382099\pi$
$542$	−7.84250	−0.336864
$543$	1.55832	0.0668740
$544$	−6.21131	−0.266308
$545$	5.88452	0.252065
$546$	−0.0942364	−0.00403295
$547$	−3.21339	−0.137394	−0.0686972	−	0.997638i	$-0.521884\pi$
$547$	−3.21339	−0.137394	−0.0686972	+	0.997638i	$0.521884\pi$
$548$	−18.5408	−0.792025
$549$	3.62954	0.154905
$550$	0.102109	0.00435395
$551$	6.26446	0.266875
$552$	1.24016	0.0527847
$553$	1.07966	0.0459116
$554$	−1.11751	−0.0474786
$555$	−1.17182	−0.0497411
$556$	1.32936	0.0563776
$557$	25.8383	1.09480	0.547401	−	0.836870i	$-0.315617\pi$
$557$	25.8383	1.09480	0.547401	+	0.836870i	$0.315617\pi$
$558$	−22.8431	−0.967027
$559$	10.5529	0.446341
$560$	−0.754669	−0.0318906
$561$	0.0864653	0.00365057
$562$	−20.6275	−0.870118
$563$	36.3052	1.53008	0.765041	−	0.643981i	$-0.222719\pi$
$563$	36.3052	1.53008	0.765041	+	0.643981i	$0.222719\pi$
$564$	−0.281315	−0.0118455
$565$	−5.64484	−0.237480
$566$	−6.79028	−0.285417
$567$	3.45476	0.145086
$568$	−6.15138	−0.258106
$569$	23.9374	1.00351	0.501755	−	0.865010i	$-0.332688\pi$
$569$	23.9374	1.00351	0.501755	+	0.865010i	$0.332688\pi$
$570$	0.721365	0.0302147
$571$	−31.9259	−1.33606	−0.668029	−	0.744135i	$-0.732862\pi$
$571$	−31.9259	−1.33606	−0.668029	+	0.744135i	$0.732862\pi$
$572$	0.0882072	0.00368813
$573$	1.55912	0.0651331
$574$	2.16739	0.0904650
$575$	9.09670	0.379359
$576$	−2.96270	−0.123446
$577$	37.3286	1.55401	0.777004	−	0.629495i	$-0.216738\pi$
$577$	37.3286	1.55401	0.777004	+	0.629495i	$0.216738\pi$
$578$	−21.5804	−0.897626
$579$	1.35572	0.0563419
$580$	−6.01018	−0.249559
$581$	−0.617015	−0.0255981
$582$	2.97637	0.123375
$583$	0.192959	0.00799156
$584$	2.54741	0.105413
$585$	6.86359	0.283775
$586$	19.6711	0.812604
$587$	−5.22887	−0.215819	−0.107909	−	0.994161i	$-0.534416\pi$
$587$	−5.22887	−0.215819	−0.107909	+	0.994161i	$0.534416\pi$
$588$	−1.32130	−0.0544896
$589$	−15.2126	−0.626826
$590$	11.8985	0.489855
$591$	2.17728	0.0895613
$592$	3.20511	0.131729
$593$	4.23156	0.173769	0.0868847	−	0.996218i	$-0.472309\pi$
$593$	4.23156	0.173769	0.0868847	+	0.996218i	$0.472309\pi$
$594$	0.0830044	0.00340571
$595$	−4.68748	−0.192168
$596$	−2.05178	−0.0840442
$597$	1.11089	0.0454656
$598$	7.85820	0.321346
$599$	4.19513	0.171408	0.0857041	−	0.996321i	$-0.472686\pi$
$599$	4.19513	0.171408	0.0857041	+	0.996321i	$0.472686\pi$
$600$	0.273630	0.0111709
$601$	29.2857	1.19459	0.597294	−	0.802022i	$-0.296242\pi$
$601$	29.2857	1.19459	0.597294	+	0.802022i	$0.296242\pi$
$602$	−3.43768	−0.140109
$603$	−5.99903	−0.244300
$604$	−5.91990	−0.240877
$605$	20.8127	0.846155
$606$	−1.49823	−0.0608614
$607$	−11.3294	−0.459846	−0.229923	−	0.973209i	$-0.573847\pi$
$607$	−11.3294	−0.459846	−0.229923	+	0.973209i	$0.573847\pi$
$608$	−1.97304	−0.0800174
$609$	0.244479	0.00990680
$610$	−2.31902	−0.0938944
$611$	−1.78254	−0.0721137
$612$	−18.4022	−0.743866
$613$	2.39189	0.0966075	0.0483037	−	0.998833i	$-0.484618\pi$
$613$	2.39189	0.0966075	0.0483037	+	0.998833i	$0.484618\pi$
$614$	7.23223	0.291869
$615$	1.98765	0.0801497
$616$	−0.0287340	−0.00115773
$617$	36.7536	1.47965	0.739823	−	0.672802i	$-0.234909\pi$
$617$	36.7536	1.47965	0.739823	+	0.672802i	$0.234909\pi$
$618$	1.75668	0.0706640
$619$	−33.7874	−1.35803	−0.679016	−	0.734123i	$-0.737593\pi$
$619$	−33.7874	−1.35803	−0.679016	+	0.734123i	$0.737593\pi$
$620$	14.5952	0.586156
$621$	7.39470	0.296739
$622$	−1.33111	−0.0533725
$623$	0.574126	0.0230019
$624$	0.236376	0.00946260
$625$	−15.9093	−0.636371
$626$	−16.5452	−0.661280
$627$	0.0274660	0.00109689
$628$	7.10945	0.283698
$629$	19.9079	0.793781
$630$	−2.23585	−0.0890785
$631$	31.9821	1.27319	0.636594	−	0.771199i	$-0.280343\pi$
$631$	31.9821	1.27319	0.636594	+	0.771199i	$0.280343\pi$
$632$	−2.70813	−0.107724
$633$	−0.0680597	−0.00270513
$634$	−24.7716	−0.983804
$635$	41.1276	1.63210
$636$	0.517088	0.0205039
$637$	−8.37235	−0.331725
$638$	−0.228837	−0.00905976
$639$	−18.2247	−0.720957
$640$	1.89295	0.0748256
$641$	28.2971	1.11767	0.558835	−	0.829279i	$-0.311249\pi$
$641$	28.2971	1.11767	0.558835	+	0.829279i	$0.311249\pi$
$642$	−0.587552	−0.0231888
$643$	−13.5804	−0.535557	−0.267778	−	0.963481i	$-0.586290\pi$
$643$	−13.5804	−0.535557	−0.267778	+	0.963481i	$0.586290\pi$
$644$	−2.55985	−0.100872
$645$	−3.15259	−0.124133
$646$	−12.2552	−0.482173
$647$	−15.9887	−0.628579	−0.314289	−	0.949327i	$-0.601766\pi$
$647$	−15.9887	−0.628579	−0.314289	+	0.949327i	$0.601766\pi$
$648$	−8.66565	−0.340419
$649$	0.453037	0.0177832
$650$	1.73384	0.0680068
$651$	−0.593695	−0.0232687
$652$	−5.08181	−0.199019
$653$	−18.8516	−0.737720	−0.368860	−	0.929485i	$-0.620252\pi$
$653$	−18.8516	−0.737720	−0.368860	+	0.929485i	$0.620252\pi$
$654$	0.600413	0.0234780
$655$	−34.4126	−1.34461
$656$	−5.43651	−0.212260
$657$	7.54721	0.294445
$658$	0.580672	0.0226369
$659$	0.302937	0.0118007	0.00590037	−	0.999983i	$-0.498122\pi$
$659$	0.302937	0.0118007	0.00590037	+	0.999983i	$0.498122\pi$
$660$	−0.0263511	−0.00102572
$661$	7.77666	0.302477	0.151238	−	0.988497i	$-0.451674\pi$
$661$	7.77666	0.302477	0.151238	+	0.988497i	$0.451674\pi$
$662$	−31.6706	−1.23091
$663$	1.46820	0.0570203
$664$	1.54767	0.0600614
$665$	−1.48899	−0.0577407
$666$	9.49576	0.367953
$667$	−20.3867	−0.789375
$668$	2.26797	0.0877505
$669$	1.16168	0.0449132
$670$	3.83296	0.148080
$671$	−0.0882967	−0.00340866
$672$	−0.0770007	−0.00297037
$673$	23.7356	0.914940	0.457470	−	0.889225i	$-0.348756\pi$
$673$	23.7356	0.914940	0.457470	+	0.889225i	$0.348756\pi$
$674$	−4.18090	−0.161042
$675$	1.63157	0.0627992
$676$	−11.5022	−0.442393
$677$	−13.1042	−0.503637	−0.251818	−	0.967775i	$-0.581029\pi$
$677$	−13.1042	−0.503637	−0.251818	+	0.967775i	$0.581029\pi$
$678$	−0.575958	−0.0221195
$679$	−6.14363	−0.235771
$680$	11.7577	0.450888
$681$	0.250947	0.00961633
$682$	0.555710	0.0212792
$683$	39.6769	1.51819	0.759097	−	0.650978i	$-0.225641\pi$
$683$	39.6769	1.51819	0.759097	+	0.650978i	$0.225641\pi$
$684$	−5.84552	−0.223509
$685$	35.0969	1.34099
$686$	5.51805	0.210680
$687$	5.32679	0.203230
$688$	8.62281	0.328742
$689$	3.27650	0.124825
$690$	−2.34757	−0.0893704
$691$	−15.9397	−0.606374	−0.303187	−	0.952931i	$-0.598051\pi$
$691$	−15.9397	−0.606374	−0.303187	+	0.952931i	$0.598051\pi$
$692$	19.9017	0.756550
$693$	−0.0851301	−0.00323382
$694$	17.6118	0.668533
$695$	−2.51643	−0.0954535
$696$	−0.613234	−0.0232446
$697$	−33.7679	−1.27905
$698$	−14.0874	−0.533215
$699$	−2.37443	−0.0898093
$700$	−0.564808	−0.0213477
$701$	−22.3323	−0.843481	−0.421740	−	0.906717i	$-0.638581\pi$
$701$	−22.3323	−0.843481	−0.421740	+	0.906717i	$0.638581\pi$
$702$	1.40944	0.0531957
$703$	6.32381	0.238507
$704$	0.0720742	0.00271640
$705$	0.532517	0.0200557
$706$	−24.8100	−0.933735
$707$	3.09254	0.116307
$708$	1.21404	0.0456263
$709$	46.1510	1.73324	0.866619	−	0.498970i	$-0.166288\pi$
$709$	46.1510	1.73324	0.866619	+	0.498970i	$0.166288\pi$
$710$	11.6443	0.437002
$711$	−8.02336	−0.300899
$712$	−1.44010	−0.0539699
$713$	49.5071	1.85406
$714$	−0.478276	−0.0178990
$715$	−0.166972	−0.00624441
$716$	9.70466	0.362680
$717$	4.05361	0.151385
$718$	−20.7624	−0.774845
$719$	−33.2689	−1.24072	−0.620361	−	0.784316i	$-0.713014\pi$
$719$	−33.2689	−1.24072	−0.620361	+	0.784316i	$0.713014\pi$
$720$	5.60825	0.209007
$721$	−3.62602	−0.135040
$722$	15.1071	0.562228
$723$	4.51206	0.167805
$724$	8.06824	0.299854
$725$	−4.49813	−0.167056
$726$	2.12357	0.0788130
$727$	−38.6935	−1.43506	−0.717531	−	0.696526i	$-0.754728\pi$
$727$	−38.6935	−1.43506	−0.717531	+	0.696526i	$0.754728\pi$
$728$	−0.487910	−0.0180832
$729$	−25.0064	−0.926163
$730$	−4.82213	−0.178475
$731$	53.5590	1.98095
$732$	−0.236616	−0.00874556
$733$	−4.69220	−0.173310	−0.0866551	−	0.996238i	$-0.527618\pi$
$733$	−4.69220	−0.173310	−0.0866551	+	0.996238i	$0.527618\pi$
$734$	−5.16918	−0.190798
$735$	2.50117	0.0922569
$736$	6.42095	0.236679
$737$	0.145940	0.00537577
$738$	−16.1067	−0.592897
$739$	1.49200	0.0548842	0.0274421	−	0.999623i	$-0.491264\pi$
$739$	1.49200	0.0548842	0.0274421	+	0.999623i	$0.491264\pi$
$740$	−6.06713	−0.223032
$741$	0.466379	0.0171329
$742$	−1.06734	−0.0391832
$743$	−20.3300	−0.745835	−0.372917	−	0.927865i	$-0.621643\pi$
$743$	−20.3300	−0.745835	−0.372917	+	0.927865i	$0.621643\pi$
$744$	1.48918	0.0545960
$745$	3.88393	0.142296
$746$	−10.5803	−0.387371
$747$	4.58529	0.167767
$748$	0.447675	0.0163686
$749$	1.21278	0.0443141
$750$	−2.34602	−0.0856646
$751$	22.7893	0.831595	0.415797	−	0.909457i	$-0.363503\pi$
$751$	22.7893	0.831595	0.415797	+	0.909457i	$0.363503\pi$
$752$	−1.45651	−0.0531136
$753$	−2.89223	−0.105399
$754$	−3.88572	−0.141509
$755$	11.2061	0.407832
$756$	−0.459132	−0.0166985
$757$	32.9397	1.19721	0.598607	−	0.801043i	$-0.295721\pi$
$757$	32.9397	1.19721	0.598607	+	0.801043i	$0.295721\pi$
$758$	10.4336	0.378966
$759$	−0.0893836	−0.00324442
$760$	3.73488	0.135478
$761$	−3.87141	−0.140339	−0.0701693	−	0.997535i	$-0.522354\pi$
$761$	−3.87141	−0.140339	−0.0701693	+	0.997535i	$0.522354\pi$
$762$	4.19635	0.152018
$763$	−1.23933	−0.0448668
$764$	8.07236	0.292048
$765$	34.8346	1.25945
$766$	−8.51000	−0.307479
$767$	7.69267	0.277766
$768$	0.193143	0.00696944
$769$	−11.6982	−0.421848	−0.210924	−	0.977502i	$-0.567647\pi$
$769$	−11.6982	−0.421848	−0.210924	+	0.977502i	$0.567647\pi$
$770$	0.0543921	0.00196016
$771$	−3.09065	−0.111307
$772$	7.01926	0.252629
$773$	15.3991	0.553866	0.276933	−	0.960889i	$-0.410682\pi$
$773$	15.3991	0.553866	0.276933	+	0.960889i	$0.410682\pi$
$774$	25.5468	0.918259
$775$	10.9233	0.392376
$776$	15.4102	0.553194
$777$	0.246796	0.00885375
$778$	−14.9168	−0.534792
$779$	−10.7265	−0.384316
$780$	−0.447448	−0.0160212
$781$	0.443356	0.0158645
$782$	39.8825	1.42620
$783$	−3.65653	−0.130674
$784$	−6.84106	−0.244324
$785$	−13.4579	−0.480332
$786$	−3.51121	−0.125241
$787$	28.1486	1.00339	0.501695	−	0.865045i	$-0.332710\pi$
$787$	28.1486	1.00339	0.501695	+	0.865045i	$0.332710\pi$
$788$	11.2729	0.401580
$789$	1.87316	0.0666864
$790$	5.12636	0.182388
$791$	1.18885	0.0422707
$792$	0.213534	0.00758760
$793$	−1.49930	−0.0532417
$794$	−29.3302	−1.04089
$795$	−0.978825	−0.0347153
$796$	5.75163	0.203861
$797$	−0.144914	−0.00513312	−0.00256656	−	0.999997i	$-0.500817\pi$
$797$	−0.144914	−0.00513312	−0.00256656	+	0.999997i	$0.500817\pi$
$798$	−0.151926	−0.00537811
$799$	−9.04686	−0.320055
$800$	1.41672	0.0500887
$801$	−4.26657	−0.150752
$802$	5.06935	0.179005
$803$	−0.183603	−0.00647920
$804$	0.391087	0.0137926
$805$	4.84569	0.170788
$806$	9.43610	0.332372
$807$	−5.45048	−0.191866
$808$	−7.75709	−0.272894
$809$	−10.7224	−0.376981	−0.188490	−	0.982075i	$-0.560359\pi$
$809$	−10.7224	−0.376981	−0.188490	+	0.982075i	$0.560359\pi$
$810$	16.4037	0.576367
$811$	−4.98745	−0.175133	−0.0875665	−	0.996159i	$-0.527909\pi$
$811$	−4.98745	−0.175133	−0.0875665	+	0.996159i	$0.527909\pi$
$812$	1.26579	0.0444207
$813$	1.51472	0.0531237
$814$	−0.231006	−0.00809675
$815$	9.61964	0.336961
$816$	1.19967	0.0419969
$817$	17.0132	0.595215
$818$	−9.78754	−0.342213
$819$	−1.44553	−0.0505109
$820$	10.2911	0.359380
$821$	−28.2116	−0.984591	−0.492295	−	0.870428i	$-0.663842\pi$
$821$	−28.2116	−0.984591	−0.492295	+	0.870428i	$0.663842\pi$
$822$	3.58103	0.124903
$823$	28.8120	1.00432	0.502161	−	0.864774i	$-0.332538\pi$
$823$	28.8120	1.00432	0.502161	+	0.864774i	$0.332538\pi$
$824$	9.09523	0.316847
$825$	−0.0197217	−0.000686620 0
$826$	−2.50593	−0.0871925
$827$	9.38090	0.326206	0.163103	−	0.986609i	$-0.447850\pi$
$827$	9.38090	0.326206	0.163103	+	0.986609i	$0.447850\pi$
$828$	19.0233	0.661106
$829$	−42.9819	−1.49282	−0.746412	−	0.665484i	$-0.768225\pi$
$829$	−42.9819	−1.49282	−0.746412	+	0.665484i	$0.768225\pi$
$830$	−2.92968	−0.101691
$831$	0.215840	0.00748740
$832$	1.22384	0.0424290
$833$	−42.4920	−1.47226
$834$	−0.256757	−0.00889078
$835$	−4.29317	−0.148571
$836$	0.142205	0.00491828
$837$	8.87953	0.306921
$838$	25.0626	0.865772
$839$	28.2605	0.975662	0.487831	−	0.872938i	$-0.337788\pi$
$839$	28.2605	0.975662	0.487831	+	0.872938i	$0.337788\pi$
$840$	0.145759	0.00502916
$841$	−18.9192	−0.652387
$842$	−1.03241	−0.0355792
$843$	3.98405	0.137218
$844$	−0.352380	−0.0121294
$845$	21.7732	0.749020
$846$	−4.31521	−0.148360
$847$	−4.38332	−0.150613
$848$	2.67723	0.0919365
$849$	1.31149	0.0450104
$850$	8.79971	0.301827
$851$	−20.5798	−0.705468
$852$	1.18810	0.0407035
$853$	−48.0424	−1.64494	−0.822470	−	0.568809i	$-0.807404\pi$
$853$	−48.0424	−1.64494	−0.822470	+	0.568809i	$0.807404\pi$
$854$	0.488406	0.0167129
$855$	11.0653	0.378426
$856$	−3.04206	−0.103975
$857$	−48.4878	−1.65631	−0.828155	−	0.560499i	$-0.810609\pi$
$857$	−48.4878	−1.65631	−0.828155	+	0.560499i	$0.810609\pi$
$858$	−0.0170366	−0.000581620 0
$859$	30.4583	1.03922	0.519612	−	0.854402i	$-0.326076\pi$
$859$	30.4583	1.03922	0.519612	+	0.854402i	$0.326076\pi$
$860$	−16.3226	−0.556596
$861$	−0.418616	−0.0142664
$862$	−10.6462	−0.362609
$863$	−32.9197	−1.12060	−0.560299	−	0.828290i	$-0.689314\pi$
$863$	−32.9197	−1.12060	−0.560299	+	0.828290i	$0.689314\pi$
$864$	1.15165	0.0391800
$865$	−37.6731	−1.28092
$866$	39.9969	1.35915
$867$	4.16810	0.141556
$868$	−3.07386	−0.104334
$869$	0.195186	0.00662124
$870$	1.16082	0.0393556
$871$	2.47810	0.0839671
$872$	3.10864	0.105272
$873$	45.6558	1.54521
$874$	12.6688	0.428528
$875$	4.84250	0.163706
$876$	−0.492014	−0.0166236
$877$	36.1376	1.22028	0.610140	−	0.792294i	$-0.291113\pi$
$877$	36.1376	1.22028	0.610140	+	0.792294i	$0.291113\pi$
$878$	14.3081	0.482874
$879$	−3.79933	−0.128148
$880$	−0.136433	−0.00459916
$881$	57.5645	1.93940	0.969699	−	0.244304i	$-0.0785593\pi$
$881$	57.5645	1.93940	0.969699	+	0.244304i	$0.0785593\pi$
$882$	−20.2680	−0.682458
$883$	32.4035	1.09046	0.545232	−	0.838285i	$-0.316441\pi$
$883$	32.4035	1.09046	0.545232	+	0.838285i	$0.316441\pi$
$884$	7.60164	0.255671
$885$	−2.29812	−0.0772504
$886$	−25.2432	−0.848062
$887$	11.9832	0.402357	0.201178	−	0.979555i	$-0.435523\pi$
$887$	11.9832	0.402357	0.201178	+	0.979555i	$0.435523\pi$
$888$	−0.619044	−0.0207738
$889$	−8.66181	−0.290508
$890$	2.72604	0.0913769
$891$	0.624570	0.0209239
$892$	6.01462	0.201384
$893$	−2.87376	−0.0961667
$894$	0.396287	0.0132538
$895$	−18.3705	−0.614058
$896$	−0.398672	−0.0133187
$897$	−1.51776	−0.0506764
$898$	−24.9990	−0.834226
$899$	−24.4802	−0.816462
$900$	4.19732	0.139911
$901$	16.6291	0.553996
$902$	0.391832	0.0130466
$903$	0.663963	0.0220953
$904$	−2.98203	−0.0991808
$905$	−15.2728	−0.507685
$906$	1.14339	0.0379865
$907$	−7.23261	−0.240155	−0.120077	−	0.992765i	$-0.538314\pi$
$907$	−7.23261	−0.240155	−0.120077	+	0.992765i	$0.538314\pi$
$908$	1.29928	0.0431183
$909$	−22.9819	−0.762262
$910$	0.923592	0.0306168
$911$	6.05510	0.200614	0.100307	−	0.994957i	$-0.468017\pi$
$911$	6.05510	0.200614	0.100307	+	0.994957i	$0.468017\pi$
$912$	0.381079	0.0126188
$913$	−0.111547	−0.00369168
$914$	22.3620	0.739669
$915$	0.447903	0.0148072
$916$	27.5795	0.911252
$917$	7.24759	0.239336
$918$	7.15327	0.236093
$919$	−9.51324	−0.313813	−0.156906	−	0.987613i	$-0.550152\pi$
$919$	−9.51324	−0.313813	−0.156906	+	0.987613i	$0.550152\pi$
$920$	−12.1546	−0.400724
$921$	−1.39685	−0.0460279
$922$	−16.7120	−0.550382
$923$	7.52829	0.247797
$924$	0.00554977	0.000182574 0
$925$	−4.54075	−0.149299
$926$	8.98307	0.295202
$927$	26.9464	0.885036
$928$	−3.17503	−0.104225
$929$	−13.1805	−0.432438	−0.216219	−	0.976345i	$-0.569372\pi$
$929$	−13.1805	−0.432438	−0.216219	+	0.976345i	$0.569372\pi$
$930$	−2.81895	−0.0924370
$931$	−13.4977	−0.442369
$932$	−12.2937	−0.402692
$933$	0.257094	0.00841687
$934$	−35.3400	−1.15636
$935$	−0.847429	−0.0277139
$936$	3.62586	0.118515
$937$	−2.79275	−0.0912350	−0.0456175	−	0.998959i	$-0.514526\pi$
$937$	−2.79275	−0.0912350	−0.0456175	+	0.998959i	$0.514526\pi$
$938$	−0.807254	−0.0263578
$939$	3.19559	0.104284
$940$	2.75711	0.0899271
$941$	0.720590	0.0234906	0.0117453	−	0.999931i	$-0.496261\pi$
$941$	0.720590	0.0234906	0.0117453	+	0.999931i	$0.496261\pi$
$942$	−1.37314	−0.0447393
$943$	34.9076	1.13675
$944$	6.28569	0.204582
$945$	0.869116	0.0282723
$946$	−0.621482	−0.0202061
$947$	56.1592	1.82493	0.912465	−	0.409154i	$-0.134176\pi$
$947$	56.1592	1.82493	0.912465	+	0.409154i	$0.134176\pi$
$948$	0.523056	0.0169881
$949$	−3.11762	−0.101202
$950$	2.79525	0.0906900
$951$	4.78445	0.155146
$952$	−2.47628	−0.0802566
$953$	−0.137614	−0.00445777	−0.00222888	−	0.999998i	$-0.500709\pi$
$953$	−0.137614	−0.00445777	−0.00222888	+	0.999998i	$0.500709\pi$
$954$	7.93182	0.256802
$955$	−15.2806	−0.494469
$956$	20.9876	0.678789
$957$	0.0441983	0.00142873
$958$	−17.9948	−0.581384
$959$	−7.39171	−0.238691
$960$	−0.365611	−0.0118000
$961$	28.4480	0.917677
$962$	−3.92254	−0.126468
$963$	−9.01269	−0.290430
$964$	23.3613	0.752415
$965$	−13.2871	−0.427728
$966$	0.494418	0.0159076
$967$	−2.32812	−0.0748674	−0.0374337	−	0.999299i	$-0.511918\pi$
$967$	−2.32812	−0.0748674	−0.0374337	+	0.999299i	$0.511918\pi$
$968$	10.9948	0.353386
$969$	2.36700	0.0760390
$970$	−29.1708	−0.936619
$971$	19.2804	0.618737	0.309368	−	0.950942i	$-0.399882\pi$
$971$	19.2804	0.618737	0.309368	+	0.950942i	$0.399882\pi$
$972$	5.12867	0.164502
$973$	0.529981	0.0169904
$974$	−39.6892	−1.27172
$975$	−0.334879	−0.0107247
$976$	−1.22508	−0.0392139
$977$	−19.3899	−0.620338	−0.310169	−	0.950681i	$-0.600386\pi$
$977$	−19.3899	−0.620338	−0.310169	+	0.950681i	$0.600386\pi$
$978$	0.981516	0.0313854
$979$	0.103794	0.00331726
$980$	12.9498	0.413667
$981$	9.20997	0.294052
$982$	10.2844	0.328190
$983$	12.3527	0.393991	0.196995	−	0.980404i	$-0.436882\pi$
$983$	12.3527	0.393991	0.196995	+	0.980404i	$0.436882\pi$
$984$	1.05002	0.0334735
$985$	−21.3391	−0.679920
$986$	−19.7211	−0.628047
$987$	−0.112153	−0.00356986
$988$	2.41468	0.0768213
$989$	−55.3666	−1.76056
$990$	−0.404210	−0.0128466
$991$	−28.9110	−0.918388	−0.459194	−	0.888336i	$-0.651862\pi$
$991$	−28.9110	−0.918388	−0.459194	+	0.888336i	$0.651862\pi$
$992$	7.71025	0.244801
$993$	6.11696	0.194116
$994$	−2.45238	−0.0777849
$995$	−10.8876	−0.345159
$996$	−0.298922	−0.00947171
$997$	39.0315	1.23614	0.618070	−	0.786123i	$-0.287915\pi$
$997$	39.0315	1.23614	0.618070	+	0.786123i	$0.287915\pi$
$998$	30.2073	0.956195
$999$	−3.69117	−0.116784

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.41	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.41	✓	77	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.193143 q^{3} +1.00000 q^{4} -1.89295 q^{5} -0.193143 q^{6} +0.398672 q^{7} -1.00000 q^{8} -2.96270 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.193143 q^{3} +1.00000 q^{4} -1.89295 q^{5} -0.193143 q^{6} +0.398672 q^{7} -1.00000 q^{8} -2.96270 q^{9} +1.89295 q^{10} +0.0720742 q^{11} +0.193143 q^{12} +1.22384 q^{13} -0.398672 q^{14} -0.365611 q^{15} +1.00000 q^{16} +6.21131 q^{17} +2.96270 q^{18} +1.97304 q^{19} -1.89295 q^{20} +0.0770007 q^{21} -0.0720742 q^{22} -6.42095 q^{23} -0.193143 q^{24} -1.41672 q^{25} -1.22384 q^{26} -1.15165 q^{27} +0.398672 q^{28} +3.17503 q^{29} +0.365611 q^{30} -7.71025 q^{31} -1.00000 q^{32} +0.0139206 q^{33} -6.21131 q^{34} -0.754669 q^{35} -2.96270 q^{36} +3.20511 q^{37} -1.97304 q^{38} +0.236376 q^{39} +1.89295 q^{40} -5.43651 q^{41} -0.0770007 q^{42} +8.62281 q^{43} +0.0720742 q^{44} +5.60825 q^{45} +6.42095 q^{46} -1.45651 q^{47} +0.193143 q^{48} -6.84106 q^{49} +1.41672 q^{50} +1.19967 q^{51} +1.22384 q^{52} +2.67723 q^{53} +1.15165 q^{54} -0.136433 q^{55} -0.398672 q^{56} +0.381079 q^{57} -3.17503 q^{58} +6.28569 q^{59} -0.365611 q^{60} -1.22508 q^{61} +7.71025 q^{62} -1.18114 q^{63} +1.00000 q^{64} -2.31667 q^{65} -0.0139206 q^{66} +2.02486 q^{67} +6.21131 q^{68} -1.24016 q^{69} +0.754669 q^{70} +6.15138 q^{71} +2.96270 q^{72} -2.54741 q^{73} -3.20511 q^{74} -0.273630 q^{75} +1.97304 q^{76} +0.0287340 q^{77} -0.236376 q^{78} +2.70813 q^{79} -1.89295 q^{80} +8.66565 q^{81} +5.43651 q^{82} -1.54767 q^{83} +0.0770007 q^{84} -11.7577 q^{85} -8.62281 q^{86} +0.613234 q^{87} -0.0720742 q^{88} +1.44010 q^{89} -5.60825 q^{90} +0.487910 q^{91} -6.42095 q^{92} -1.48918 q^{93} +1.45651 q^{94} -3.73488 q^{95} -0.193143 q^{96} -15.4102 q^{97} +6.84106 q^{98} -0.213534 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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