LMFDB - Embedded newform 8002.2.a.e.1.13

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			1.13
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.19428 q^{3} +1.00000 q^{4} +0.0495220 q^{5} +2.19428 q^{6} +1.71879 q^{7} -1.00000 q^{8} +1.81487 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.19428 q^{3} +1.00000 q^{4} +0.0495220 q^{5} +2.19428 q^{6} +1.71879 q^{7} -1.00000 q^{8} +1.81487 q^{9} -0.0495220 q^{10} -4.48853 q^{11} -2.19428 q^{12} +0.149594 q^{13} -1.71879 q^{14} -0.108665 q^{15} +1.00000 q^{16} -0.611407 q^{17} -1.81487 q^{18} -6.46347 q^{19} +0.0495220 q^{20} -3.77151 q^{21} +4.48853 q^{22} -3.90677 q^{23} +2.19428 q^{24} -4.99755 q^{25} -0.149594 q^{26} +2.60050 q^{27} +1.71879 q^{28} -4.90060 q^{29} +0.108665 q^{30} -6.63863 q^{31} -1.00000 q^{32} +9.84910 q^{33} +0.611407 q^{34} +0.0851178 q^{35} +1.81487 q^{36} -8.78763 q^{37} +6.46347 q^{38} -0.328251 q^{39} -0.0495220 q^{40} +6.85972 q^{41} +3.77151 q^{42} +2.76613 q^{43} -4.48853 q^{44} +0.0898762 q^{45} +3.90677 q^{46} -6.87998 q^{47} -2.19428 q^{48} -4.04577 q^{49} +4.99755 q^{50} +1.34160 q^{51} +0.149594 q^{52} -9.77518 q^{53} -2.60050 q^{54} -0.222281 q^{55} -1.71879 q^{56} +14.1827 q^{57} +4.90060 q^{58} +1.30319 q^{59} -0.108665 q^{60} -2.05964 q^{61} +6.63863 q^{62} +3.11938 q^{63} +1.00000 q^{64} +0.00740819 q^{65} -9.84910 q^{66} -6.84709 q^{67} -0.611407 q^{68} +8.57257 q^{69} -0.0851178 q^{70} +15.9726 q^{71} -1.81487 q^{72} -12.9027 q^{73} +8.78763 q^{74} +10.9660 q^{75} -6.46347 q^{76} -7.71483 q^{77} +0.328251 q^{78} +11.9686 q^{79} +0.0495220 q^{80} -11.1509 q^{81} -6.85972 q^{82} +4.22776 q^{83} -3.77151 q^{84} -0.0302781 q^{85} -2.76613 q^{86} +10.7533 q^{87} +4.48853 q^{88} +5.68357 q^{89} -0.0898762 q^{90} +0.257120 q^{91} -3.90677 q^{92} +14.5670 q^{93} +6.87998 q^{94} -0.320084 q^{95} +2.19428 q^{96} +1.41020 q^{97} +4.04577 q^{98} -8.14612 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.19428	−1.26687	−0.633435	−	0.773796i	$-0.718355\pi$
$3$	−2.19428	−1.26687	−0.633435	+	0.773796i	$0.718355\pi$
$4$	1.00000	0.500000
$5$	0.0495220	0.0221469	0.0110735	−	0.999939i	$-0.496475\pi$
$5$	0.0495220	0.0221469	0.0110735	+	0.999939i	$0.496475\pi$
$6$	2.19428	0.895812
$7$	1.71879	0.649641	0.324820	−	0.945776i	$-0.394696\pi$
$7$	1.71879	0.649641	0.324820	+	0.945776i	$0.394696\pi$
$8$	−1.00000	−0.353553
$9$	1.81487	0.604958
$10$	−0.0495220	−0.0156602
$11$	−4.48853	−1.35334	−0.676671	−	0.736285i	$-0.736578\pi$
$11$	−4.48853	−1.35334	−0.676671	+	0.736285i	$0.736578\pi$
$12$	−2.19428	−0.633435
$13$	0.149594	0.0414899	0.0207449	−	0.999785i	$-0.493396\pi$
$13$	0.149594	0.0414899	0.0207449	+	0.999785i	$0.493396\pi$
$14$	−1.71879	−0.459365
$15$	−0.108665	−0.0280572
$16$	1.00000	0.250000
$17$	−0.611407	−0.148288	−0.0741440	−	0.997248i	$-0.523622\pi$
$17$	−0.611407	−0.148288	−0.0741440	+	0.997248i	$0.523622\pi$
$18$	−1.81487	−0.427770
$19$	−6.46347	−1.48282	−0.741411	−	0.671052i	$-0.765843\pi$
$19$	−6.46347	−1.48282	−0.741411	+	0.671052i	$0.765843\pi$
$20$	0.0495220	0.0110735
$21$	−3.77151	−0.823010
$22$	4.48853	0.956958
$23$	−3.90677	−0.814619	−0.407309	−	0.913290i	$-0.633533\pi$
$23$	−3.90677	−0.814619	−0.407309	+	0.913290i	$0.633533\pi$
$24$	2.19428	0.447906
$25$	−4.99755	−0.999510
$26$	−0.149594	−0.0293378
$27$	2.60050	0.500467
$28$	1.71879	0.324820
$29$	−4.90060	−0.910018	−0.455009	−	0.890487i	$-0.650364\pi$
$29$	−4.90060	−0.910018	−0.455009	+	0.890487i	$0.650364\pi$
$30$	0.108665	0.0198395
$31$	−6.63863	−1.19233	−0.596167	−	0.802861i	$-0.703310\pi$
$31$	−6.63863	−1.19233	−0.596167	+	0.802861i	$0.703310\pi$
$32$	−1.00000	−0.176777
$33$	9.84910	1.71451
$34$	0.611407	0.104855
$35$	0.0851178	0.0143875
$36$	1.81487	0.302479
$37$	−8.78763	−1.44468	−0.722339	−	0.691539i	$-0.756933\pi$
$37$	−8.78763	−1.44468	−0.722339	+	0.691539i	$0.756933\pi$
$38$	6.46347	1.04851
$39$	−0.328251	−0.0525622
$40$	−0.0495220	−0.00783012
$41$	6.85972	1.07131	0.535654	−	0.844437i	$-0.320065\pi$
$41$	6.85972	1.07131	0.535654	+	0.844437i	$0.320065\pi$
$42$	3.77151	0.581956
$43$	2.76613	0.421832	0.210916	−	0.977504i	$-0.432355\pi$
$43$	2.76613	0.421832	0.210916	+	0.977504i	$0.432355\pi$
$44$	−4.48853	−0.676671
$45$	0.0898762	0.0133980
$46$	3.90677	0.576022
$47$	−6.87998	−1.00355	−0.501774	−	0.864999i	$-0.667319\pi$
$47$	−6.87998	−1.00355	−0.501774	+	0.864999i	$0.667319\pi$
$48$	−2.19428	−0.316717
$49$	−4.04577	−0.577967
$50$	4.99755	0.706760
$51$	1.34160	0.187862
$52$	0.149594	0.0207449
$53$	−9.77518	−1.34272	−0.671362	−	0.741129i	$-0.734290\pi$
$53$	−9.77518	−1.34272	−0.671362	+	0.741129i	$0.734290\pi$
$54$	−2.60050	−0.353883
$55$	−0.222281	−0.0299724
$56$	−1.71879	−0.229683
$57$	14.1827	1.87854
$58$	4.90060	0.643480
$59$	1.30319	0.169661	0.0848303	−	0.996395i	$-0.472965\pi$
$59$	1.30319	0.169661	0.0848303	+	0.996395i	$0.472965\pi$
$60$	−0.108665	−0.0140286
$61$	−2.05964	−0.263709	−0.131855	−	0.991269i	$-0.542093\pi$
$61$	−2.05964	−0.263709	−0.131855	+	0.991269i	$0.542093\pi$
$62$	6.63863	0.843107
$63$	3.11938	0.393005
$64$	1.00000	0.125000
$65$	0.00740819	0.000918872 0
$66$	−9.84910	−1.21234
$67$	−6.84709	−0.836506	−0.418253	−	0.908331i	$-0.637357\pi$
$67$	−6.84709	−0.836506	−0.418253	+	0.908331i	$0.637357\pi$
$68$	−0.611407	−0.0741440
$69$	8.57257	1.03202
$70$	−0.0851178	−0.0101735
$71$	15.9726	1.89560	0.947800	−	0.318864i	$-0.103301\pi$
$71$	15.9726	1.89560	0.947800	+	0.318864i	$0.103301\pi$
$72$	−1.81487	−0.213885
$73$	−12.9027	−1.51015	−0.755075	−	0.655639i	$-0.772400\pi$
$73$	−12.9027	−1.51015	−0.755075	+	0.655639i	$0.772400\pi$
$74$	8.78763	1.02154
$75$	10.9660	1.26625
$76$	−6.46347	−0.741411
$77$	−7.71483	−0.879187
$78$	0.328251	0.0371671
$79$	11.9686	1.34657	0.673285	−	0.739383i	$-0.264883\pi$
$79$	11.9686	1.34657	0.673285	+	0.739383i	$0.264883\pi$
$80$	0.0495220	0.00553673
$81$	−11.1509	−1.23898
$82$	−6.85972	−0.757530
$83$	4.22776	0.464057	0.232029	−	0.972709i	$-0.425464\pi$
$83$	4.22776	0.464057	0.232029	+	0.972709i	$0.425464\pi$
$84$	−3.77151	−0.411505
$85$	−0.0302781	−0.00328412
$86$	−2.76613	−0.298280
$87$	10.7533	1.15287
$88$	4.48853	0.478479
$89$	5.68357	0.602457	0.301229	−	0.953552i	$-0.402603\pi$
$89$	5.68357	0.602457	0.301229	+	0.953552i	$0.402603\pi$
$90$	−0.0898762	−0.00947378
$91$	0.257120	0.0269535
$92$	−3.90677	−0.407309
$93$	14.5670	1.51053
$94$	6.87998	0.709616
$95$	−0.320084	−0.0328399
$96$	2.19428	0.223953
$97$	1.41020	0.143184	0.0715922	−	0.997434i	$-0.477192\pi$
$97$	1.41020	0.143184	0.0715922	+	0.997434i	$0.477192\pi$
$98$	4.04577	0.408684
$99$	−8.14612	−0.818716
$100$	−4.99755	−0.499755
$101$	−12.7606	−1.26973	−0.634864	−	0.772624i	$-0.718944\pi$
$101$	−12.7606	−1.26973	−0.634864	+	0.772624i	$0.718944\pi$
$102$	−1.34160	−0.132838
$103$	11.7694	1.15967	0.579836	−	0.814733i	$-0.303117\pi$
$103$	11.7694	1.15967	0.579836	+	0.814733i	$0.303117\pi$
$104$	−0.149594	−0.0146689
$105$	−0.186772	−0.0182271
$106$	9.77518	0.949450
$107$	−7.35234	−0.710778	−0.355389	−	0.934719i	$-0.615652\pi$
$107$	−7.35234	−0.710778	−0.355389	+	0.934719i	$0.615652\pi$
$108$	2.60050	0.250233
$109$	−0.814455	−0.0780106	−0.0390053	−	0.999239i	$-0.512419\pi$
$109$	−0.814455	−0.0780106	−0.0390053	+	0.999239i	$0.512419\pi$
$110$	0.222281	0.0211937
$111$	19.2825	1.83022
$112$	1.71879	0.162410
$113$	10.1353	0.953447	0.476724	−	0.879053i	$-0.341824\pi$
$113$	10.1353	0.953447	0.476724	+	0.879053i	$0.341824\pi$
$114$	−14.1827	−1.32833
$115$	−0.193471	−0.0180413
$116$	−4.90060	−0.455009
$117$	0.271494	0.0250996
$118$	−1.30319	−0.119968
$119$	−1.05088	−0.0963340
$120$	0.108665	0.00991973
$121$	9.14690	0.831536
$122$	2.05964	0.186471
$123$	−15.0522	−1.35721
$124$	−6.63863	−0.596167
$125$	−0.495099	−0.0442830
$126$	−3.11938	−0.277897
$127$	4.88085	0.433106	0.216553	−	0.976271i	$-0.430519\pi$
$127$	4.88085	0.433106	0.216553	+	0.976271i	$0.430519\pi$
$128$	−1.00000	−0.0883883
$129$	−6.06968	−0.534405
$130$	−0.00740819	−0.000649741 0
$131$	−0.0816734	−0.00713584	−0.00356792	−	0.999994i	$-0.501136\pi$
$131$	−0.0816734	−0.00713584	−0.00356792	+	0.999994i	$0.501136\pi$
$132$	9.84910	0.857254
$133$	−11.1093	−0.963301
$134$	6.84709	0.591499
$135$	0.128782	0.0110838
$136$	0.611407	0.0524277
$137$	4.22433	0.360909	0.180454	−	0.983583i	$-0.442243\pi$
$137$	4.22433	0.360909	0.180454	+	0.983583i	$0.442243\pi$
$138$	−8.57257	−0.729745
$139$	2.90677	0.246549	0.123274	−	0.992373i	$-0.460660\pi$
$139$	2.90677	0.246549	0.123274	+	0.992373i	$0.460660\pi$
$140$	0.0851178	0.00719377
$141$	15.0966	1.27136
$142$	−15.9726	−1.34039
$143$	−0.671456	−0.0561500
$144$	1.81487	0.151240
$145$	−0.242687	−0.0201541
$146$	12.9027	1.06784
$147$	8.87756	0.732209
$148$	−8.78763	−0.722339
$149$	16.1227	1.32082	0.660410	−	0.750905i	$-0.270383\pi$
$149$	16.1227	1.32082	0.660410	+	0.750905i	$0.270383\pi$
$150$	−10.9660	−0.895373
$151$	−9.26507	−0.753981	−0.376991	−	0.926217i	$-0.623041\pi$
$151$	−9.26507	−0.753981	−0.376991	+	0.926217i	$0.623041\pi$
$152$	6.46347	0.524256
$153$	−1.10963	−0.0897080
$154$	7.71483	0.621679
$155$	−0.328758	−0.0264065
$156$	−0.328251	−0.0262811
$157$	−9.73885	−0.777245	−0.388622	−	0.921397i	$-0.627049\pi$
$157$	−9.73885	−0.777245	−0.388622	+	0.921397i	$0.627049\pi$
$158$	−11.9686	−0.952168
$159$	21.4495	1.70106
$160$	−0.0495220	−0.00391506
$161$	−6.71492	−0.529210
$162$	11.1509	0.876094
$163$	2.70166	0.211610	0.105805	−	0.994387i	$-0.466258\pi$
$163$	2.70166	0.211610	0.105805	+	0.994387i	$0.466258\pi$
$164$	6.85972	0.535654
$165$	0.487747	0.0379711
$166$	−4.22776	−0.328138
$167$	20.6077	1.59467	0.797335	−	0.603538i	$-0.206243\pi$
$167$	20.6077	1.59467	0.797335	+	0.603538i	$0.206243\pi$
$168$	3.77151	0.290978
$169$	−12.9776	−0.998279
$170$	0.0302781	0.00232222
$171$	−11.7304	−0.897045
$172$	2.76613	0.210916
$173$	−19.6845	−1.49659	−0.748294	−	0.663368i	$-0.769127\pi$
$173$	−19.6845	−1.49659	−0.748294	+	0.663368i	$0.769127\pi$
$174$	−10.7533	−0.815205
$175$	−8.58972	−0.649322
$176$	−4.48853	−0.338336
$177$	−2.85956	−0.214938
$178$	−5.68357	−0.426002
$179$	−11.0404	−0.825200	−0.412600	−	0.910912i	$-0.635379\pi$
$179$	−11.0404	−0.825200	−0.412600	+	0.910912i	$0.635379\pi$
$180$	0.0898762	0.00669898
$181$	−7.43443	−0.552597	−0.276298	−	0.961072i	$-0.589108\pi$
$181$	−7.43443	−0.552597	−0.276298	+	0.961072i	$0.589108\pi$
$182$	−0.257120	−0.0190590
$183$	4.51942	0.334085
$184$	3.90677	0.288011
$185$	−0.435181	−0.0319951
$186$	−14.5670	−1.06811
$187$	2.74432	0.200685
$188$	−6.87998	−0.501774
$189$	4.46971	0.325123
$190$	0.320084	0.0232213
$191$	9.96951	0.721368	0.360684	−	0.932688i	$-0.382543\pi$
$191$	9.96951	0.721368	0.360684	+	0.932688i	$0.382543\pi$
$192$	−2.19428	−0.158359
$193$	−1.90500	−0.137125	−0.0685623	−	0.997647i	$-0.521841\pi$
$193$	−1.90500	−0.137125	−0.0685623	+	0.997647i	$0.521841\pi$
$194$	−1.41020	−0.101247
$195$	−0.0162556	−0.00116409
$196$	−4.04577	−0.288983
$197$	−8.29086	−0.590699	−0.295350	−	0.955389i	$-0.595436\pi$
$197$	−8.29086	−0.590699	−0.295350	+	0.955389i	$0.595436\pi$
$198$	8.14612	0.578919
$199$	−10.6771	−0.756877	−0.378439	−	0.925626i	$-0.623539\pi$
$199$	−10.6771	−0.756877	−0.378439	+	0.925626i	$0.623539\pi$
$200$	4.99755	0.353380
$201$	15.0245	1.05974
$202$	12.7606	0.897834
$203$	−8.42309	−0.591185
$204$	1.34160	0.0939308
$205$	0.339707	0.0237262
$206$	−11.7694	−0.820012
$207$	−7.09030	−0.492810
$208$	0.149594	0.0103725
$209$	29.0115	2.00677
$210$	0.186772	0.0128885
$211$	10.3373	0.711649	0.355825	−	0.934553i	$-0.384200\pi$
$211$	10.3373	0.711649	0.355825	+	0.934553i	$0.384200\pi$
$212$	−9.77518	−0.671362
$213$	−35.0484	−2.40148
$214$	7.35234	0.502596
$215$	0.136985	0.00934227
$216$	−2.60050	−0.176942
$217$	−11.4104	−0.774588
$218$	0.814455	0.0551619
$219$	28.3122	1.91316
$220$	−0.222281	−0.0149862
$221$	−0.0914627	−0.00615245
$222$	−19.2825	−1.29416
$223$	−26.1261	−1.74954	−0.874768	−	0.484543i	$-0.838986\pi$
$223$	−26.1261	−1.74954	−0.874768	+	0.484543i	$0.838986\pi$
$224$	−1.71879	−0.114841
$225$	−9.06992	−0.604661
$226$	−10.1353	−0.674189
$227$	4.18780	0.277954	0.138977	−	0.990296i	$-0.455619\pi$
$227$	4.18780	0.277954	0.138977	+	0.990296i	$0.455619\pi$
$228$	14.1827	0.939270
$229$	−5.14274	−0.339842	−0.169921	−	0.985458i	$-0.554351\pi$
$229$	−5.14274	−0.339842	−0.169921	+	0.985458i	$0.554351\pi$
$230$	0.193471	0.0127571
$231$	16.9285	1.11381
$232$	4.90060	0.321740
$233$	−17.6336	−1.15522	−0.577609	−	0.816313i	$-0.696014\pi$
$233$	−17.6336	−1.15522	−0.577609	+	0.816313i	$0.696014\pi$
$234$	−0.271494	−0.0177481
$235$	−0.340710	−0.0222255
$236$	1.30319	0.0848303
$237$	−26.2624	−1.70593
$238$	1.05088	0.0681184
$239$	2.86053	0.185032	0.0925160	−	0.995711i	$-0.470509\pi$
$239$	2.86053	0.185032	0.0925160	+	0.995711i	$0.470509\pi$
$240$	−0.108665	−0.00701431
$241$	−8.97528	−0.578149	−0.289074	−	0.957307i	$-0.593347\pi$
$241$	−8.97528	−0.578149	−0.289074	+	0.957307i	$0.593347\pi$
$242$	−9.14690	−0.587985
$243$	16.6666	1.06916
$244$	−2.05964	−0.131855
$245$	−0.200355	−0.0128002
$246$	15.0522	0.959691
$247$	−0.966895	−0.0615220
$248$	6.63863	0.421554
$249$	−9.27691	−0.587900
$250$	0.495099	0.0313128
$251$	11.6234	0.733665	0.366833	−	0.930287i	$-0.380442\pi$
$251$	11.6234	0.733665	0.366833	+	0.930287i	$0.380442\pi$
$252$	3.11938	0.196503
$253$	17.5357	1.10246
$254$	−4.88085	−0.306252
$255$	0.0664387	0.00416055
$256$	1.00000	0.0625000
$257$	−6.10648	−0.380912	−0.190456	−	0.981696i	$-0.560997\pi$
$257$	−6.10648	−0.380912	−0.190456	+	0.981696i	$0.560997\pi$
$258$	6.06968	0.377882
$259$	−15.1041	−0.938521
$260$	0.00740819	0.000459436 0
$261$	−8.89397	−0.550523
$262$	0.0816734	0.00504580
$263$	8.75820	0.540054	0.270027	−	0.962853i	$-0.412967\pi$
$263$	8.75820	0.540054	0.270027	+	0.962853i	$0.412967\pi$
$264$	−9.84910	−0.606170
$265$	−0.484086	−0.0297372
$266$	11.1093	0.681157
$267$	−12.4714	−0.763235
$268$	−6.84709	−0.418253
$269$	−8.49852	−0.518164	−0.259082	−	0.965855i	$-0.583420\pi$
$269$	−8.49852	−0.518164	−0.259082	+	0.965855i	$0.583420\pi$
$270$	−0.128782	−0.00783742
$271$	−7.03183	−0.427153	−0.213577	−	0.976926i	$-0.568511\pi$
$271$	−7.03183	−0.427153	−0.213577	+	0.976926i	$0.568511\pi$
$272$	−0.611407	−0.0370720
$273$	−0.564194	−0.0341466
$274$	−4.22433	−0.255201
$275$	22.4316	1.35268
$276$	8.57257	0.516008
$277$	27.0919	1.62779	0.813897	−	0.581009i	$-0.197342\pi$
$277$	27.0919	1.62779	0.813897	+	0.581009i	$0.197342\pi$
$278$	−2.90677	−0.174336
$279$	−12.0483	−0.721312
$280$	−0.0851178	−0.00508676
$281$	0.666362	0.0397518	0.0198759	−	0.999802i	$-0.493673\pi$
$281$	0.666362	0.0397518	0.0198759	+	0.999802i	$0.493673\pi$
$282$	−15.0966	−0.898991
$283$	−8.42508	−0.500819	−0.250409	−	0.968140i	$-0.580565\pi$
$283$	−8.42508	−0.500819	−0.250409	+	0.968140i	$0.580565\pi$
$284$	15.9726	0.947800
$285$	0.702354	0.0416039
$286$	0.671456	0.0397040
$287$	11.7904	0.695966
$288$	−1.81487	−0.106942
$289$	−16.6262	−0.978011
$290$	0.242687	0.0142511
$291$	−3.09438	−0.181396
$292$	−12.9027	−0.755075
$293$	20.1777	1.17879	0.589397	−	0.807844i	$-0.299365\pi$
$293$	20.1777	1.17879	0.589397	+	0.807844i	$0.299365\pi$
$294$	−8.87756	−0.517750
$295$	0.0645364	0.00375746
$296$	8.78763	0.510771
$297$	−11.6724	−0.677303
$298$	−16.1227	−0.933961
$299$	−0.584429	−0.0337984
$300$	10.9660	0.633124
$301$	4.75440	0.274039
$302$	9.26507	0.533145
$303$	28.0004	1.60858
$304$	−6.46347	−0.370705
$305$	−0.101997	−0.00584035
$306$	1.10963	0.0634332
$307$	−3.01484	−0.172066	−0.0860331	−	0.996292i	$-0.527419\pi$
$307$	−3.01484	−0.172066	−0.0860331	+	0.996292i	$0.527419\pi$
$308$	−7.71483	−0.439593
$309$	−25.8254	−1.46915
$310$	0.328758	0.0186722
$311$	18.4919	1.04858	0.524290	−	0.851540i	$-0.324331\pi$
$311$	18.4919	1.04858	0.524290	+	0.851540i	$0.324331\pi$
$312$	0.328251	0.0185836
$313$	10.2506	0.579401	0.289700	−	0.957117i	$-0.406444\pi$
$313$	10.2506	0.579401	0.289700	+	0.957117i	$0.406444\pi$
$314$	9.73885	0.549595
$315$	0.154478	0.00870386
$316$	11.9686	0.673285
$317$	−22.4731	−1.26222	−0.631108	−	0.775695i	$-0.717399\pi$
$317$	−22.4731	−1.26222	−0.631108	+	0.775695i	$0.717399\pi$
$318$	−21.4495	−1.20283
$319$	21.9965	1.23157
$320$	0.0495220	0.00276836
$321$	16.1331	0.900463
$322$	6.71492	0.374208
$323$	3.95181	0.219885
$324$	−11.1509	−0.619492
$325$	−0.747602	−0.0414695
$326$	−2.70166	−0.149631
$327$	1.78714	0.0988293
$328$	−6.85972	−0.378765
$329$	−11.8252	−0.651946
$330$	−0.487747	−0.0268496
$331$	−20.6999	−1.13777	−0.568883	−	0.822418i	$-0.692624\pi$
$331$	−20.6999	−1.13777	−0.568883	+	0.822418i	$0.692624\pi$
$332$	4.22776	0.232029
$333$	−15.9484	−0.873969
$334$	−20.6077	−1.12760
$335$	−0.339082	−0.0185260
$336$	−3.77151	−0.205753
$337$	−0.659518	−0.0359263	−0.0179631	−	0.999839i	$-0.505718\pi$
$337$	−0.659518	−0.0359263	−0.0179631	+	0.999839i	$0.505718\pi$
$338$	12.9776	0.705890
$339$	−22.2397	−1.20789
$340$	−0.0302781	−0.00164206
$341$	29.7977	1.61364
$342$	11.7304	0.634306
$343$	−18.9853	−1.02511
$344$	−2.76613	−0.149140
$345$	0.424531	0.0228560
$346$	19.6845	1.05825
$347$	−26.6562	−1.43098	−0.715491	−	0.698622i	$-0.753797\pi$
$347$	−26.6562	−1.43098	−0.715491	+	0.698622i	$0.753797\pi$
$348$	10.7533	0.576437
$349$	−35.1290	−1.88041	−0.940206	−	0.340606i	$-0.889368\pi$
$349$	−35.1290	−1.88041	−0.940206	+	0.340606i	$0.889368\pi$
$350$	8.58972	0.459140
$351$	0.389019	0.0207643
$352$	4.48853	0.239239
$353$	9.85262	0.524402	0.262201	−	0.965013i	$-0.415552\pi$
$353$	9.85262	0.524402	0.262201	+	0.965013i	$0.415552\pi$
$354$	2.85956	0.151984
$355$	0.790996	0.0419817
$356$	5.68357	0.301229
$357$	2.30593	0.122043
$358$	11.0404	0.583505
$359$	24.5956	1.29810	0.649052	−	0.760744i	$-0.275166\pi$
$359$	24.5956	1.29810	0.649052	+	0.760744i	$0.275166\pi$
$360$	−0.0898762	−0.00473689
$361$	22.7764	1.19876
$362$	7.43443	0.390745
$363$	−20.0709	−1.05345
$364$	0.257120	0.0134768
$365$	−0.638969	−0.0334452
$366$	−4.51942	−0.236234
$367$	−20.3170	−1.06054	−0.530268	−	0.847830i	$-0.677909\pi$
$367$	−20.3170	−1.06054	−0.530268	+	0.847830i	$0.677909\pi$
$368$	−3.90677	−0.203655
$369$	12.4495	0.648097
$370$	0.435181	0.0226240
$371$	−16.8015	−0.872289
$372$	14.5670	0.755265
$373$	32.0593	1.65997	0.829984	−	0.557787i	$-0.188349\pi$
$373$	32.0593	1.65997	0.829984	+	0.557787i	$0.188349\pi$
$374$	−2.74432	−0.141905
$375$	1.08639	0.0561007
$376$	6.87998	0.354808
$377$	−0.733099	−0.0377565
$378$	−4.46971	−0.229897
$379$	16.0863	0.826296	0.413148	−	0.910664i	$-0.364429\pi$
$379$	16.0863	0.826296	0.413148	+	0.910664i	$0.364429\pi$
$380$	−0.320084	−0.0164200
$381$	−10.7100	−0.548689
$382$	−9.96951	−0.510084
$383$	23.8546	1.21891	0.609456	−	0.792820i	$-0.291388\pi$
$383$	23.8546	1.21891	0.609456	+	0.792820i	$0.291388\pi$
$384$	2.19428	0.111976
$385$	−0.382054	−0.0194713
$386$	1.90500	0.0969617
$387$	5.02019	0.255190
$388$	1.41020	0.0715922
$389$	26.6943	1.35346	0.676728	−	0.736233i	$-0.263397\pi$
$389$	26.6943	1.35346	0.676728	+	0.736233i	$0.263397\pi$
$390$	0.0162556	0.000823137 0
$391$	2.38863	0.120798
$392$	4.04577	0.204342
$393$	0.179214	0.00904017
$394$	8.29086	0.417687
$395$	0.592707	0.0298223
$396$	−8.14612	−0.409358
$397$	33.2828	1.67042	0.835209	−	0.549933i	$-0.185347\pi$
$397$	33.2828	1.67042	0.835209	+	0.549933i	$0.185347\pi$
$398$	10.6771	0.535193
$399$	24.3770	1.22038
$400$	−4.99755	−0.249877
$401$	−17.2486	−0.861355	−0.430678	−	0.902506i	$-0.641725\pi$
$401$	−17.2486	−0.861355	−0.430678	+	0.902506i	$0.641725\pi$
$402$	−15.0245	−0.749352
$403$	−0.993098	−0.0494697
$404$	−12.7606	−0.634864
$405$	−0.552213	−0.0274397
$406$	8.42309	0.418031
$407$	39.4435	1.95514
$408$	−1.34160	−0.0664191
$409$	−17.6415	−0.872316	−0.436158	−	0.899870i	$-0.643661\pi$
$409$	−17.6415	−0.872316	−0.436158	+	0.899870i	$0.643661\pi$
$410$	−0.339707	−0.0167769
$411$	−9.26938	−0.457225
$412$	11.7694	0.579836
$413$	2.23990	0.110218
$414$	7.09030	0.348469
$415$	0.209367	0.0102774
$416$	−0.149594	−0.00733444
$417$	−6.37827	−0.312345
$418$	−29.0115	−1.41900
$419$	−25.6130	−1.25128	−0.625638	−	0.780113i	$-0.715161\pi$
$419$	−25.6130	−1.25128	−0.625638	+	0.780113i	$0.715161\pi$
$420$	−0.186772	−0.00911356
$421$	−1.23490	−0.0601851	−0.0300926	−	0.999547i	$-0.509580\pi$
$421$	−1.23490	−0.0601851	−0.0300926	+	0.999547i	$0.509580\pi$
$422$	−10.3373	−0.503212
$423$	−12.4863	−0.607105
$424$	9.77518	0.474725
$425$	3.05554	0.148215
$426$	35.0484	1.69810
$427$	−3.54008	−0.171316
$428$	−7.35234	−0.355389
$429$	1.47336	0.0711347
$430$	−0.136985	−0.00660598
$431$	3.10156	0.149397	0.0746985	−	0.997206i	$-0.476201\pi$
$431$	3.10156	0.149397	0.0746985	+	0.997206i	$0.476201\pi$
$432$	2.60050	0.125117
$433$	19.4689	0.935616	0.467808	−	0.883830i	$-0.345044\pi$
$433$	19.4689	0.935616	0.467808	+	0.883830i	$0.345044\pi$
$434$	11.4104	0.547717
$435$	0.532525	0.0255326
$436$	−0.814455	−0.0390053
$437$	25.2513	1.20793
$438$	−28.3122	−1.35281
$439$	−39.9161	−1.90509	−0.952546	−	0.304393i	$-0.901546\pi$
$439$	−39.9161	−1.90509	−0.952546	+	0.304393i	$0.901546\pi$
$440$	0.222281	0.0105968
$441$	−7.34256	−0.349646
$442$	0.0914627	0.00435044
$443$	31.8463	1.51306	0.756531	−	0.653958i	$-0.226893\pi$
$443$	31.8463	1.51306	0.756531	+	0.653958i	$0.226893\pi$
$444$	19.2825	0.915109
$445$	0.281462	0.0133426
$446$	26.1261	1.23711
$447$	−35.3777	−1.67331
$448$	1.71879	0.0812051
$449$	22.1570	1.04565	0.522827	−	0.852439i	$-0.324877\pi$
$449$	22.1570	1.04565	0.522827	+	0.852439i	$0.324877\pi$
$450$	9.06992	0.427560
$451$	−30.7901	−1.44985
$452$	10.1353	0.476724
$453$	20.3302	0.955196
$454$	−4.18780	−0.196543
$455$	0.0127331	0.000596937 0
$456$	−14.1827	−0.664164
$457$	−34.7269	−1.62446	−0.812228	−	0.583340i	$-0.801745\pi$
$457$	−34.7269	−1.62446	−0.812228	+	0.583340i	$0.801745\pi$
$458$	5.14274	0.240304
$459$	−1.58996	−0.0742132
$460$	−0.193471	−0.00902064
$461$	−5.36635	−0.249936	−0.124968	−	0.992161i	$-0.539883\pi$
$461$	−5.36635	−0.249936	−0.124968	+	0.992161i	$0.539883\pi$
$462$	−16.9285	−0.787586
$463$	−8.49012	−0.394569	−0.197285	−	0.980346i	$-0.563212\pi$
$463$	−8.49012	−0.394569	−0.197285	+	0.980346i	$0.563212\pi$
$464$	−4.90060	−0.227505
$465$	0.721388	0.0334536
$466$	17.6336	0.816863
$467$	16.9438	0.784066	0.392033	−	0.919951i	$-0.371772\pi$
$467$	16.9438	0.784066	0.392033	+	0.919951i	$0.371772\pi$
$468$	0.271494	0.0125498
$469$	−11.7687	−0.543428
$470$	0.340710	0.0157158
$471$	21.3698	0.984668
$472$	−1.30319	−0.0599841
$473$	−12.4159	−0.570883
$474$	26.2624	1.20627
$475$	32.3015	1.48209
$476$	−1.05088	−0.0481670
$477$	−17.7407	−0.812292
$478$	−2.86053	−0.130837
$479$	−17.2598	−0.788619	−0.394310	−	0.918978i	$-0.629016\pi$
$479$	−17.2598	−0.788619	−0.394310	+	0.918978i	$0.629016\pi$
$480$	0.108665	0.00495987
$481$	−1.31458	−0.0599395
$482$	8.97528	0.408813
$483$	14.7344	0.670439
$484$	9.14690	0.415768
$485$	0.0698360	0.00317109
$486$	−16.6666	−0.756013
$487$	27.6962	1.25504	0.627518	−	0.778602i	$-0.284071\pi$
$487$	27.6962	1.25504	0.627518	+	0.778602i	$0.284071\pi$
$488$	2.05964	0.0932354
$489$	−5.92820	−0.268083
$490$	0.200355	0.00905109
$491$	−15.1407	−0.683289	−0.341645	−	0.939829i	$-0.610984\pi$
$491$	−15.1407	−0.683289	−0.341645	+	0.939829i	$0.610984\pi$
$492$	−15.0522	−0.678604
$493$	2.99626	0.134945
$494$	0.966895	0.0435027
$495$	−0.403412	−0.0181320
$496$	−6.63863	−0.298083
$497$	27.4535	1.23146
$498$	9.27691	0.415708
$499$	−3.04613	−0.136364	−0.0681818	−	0.997673i	$-0.521720\pi$
$499$	−3.04613	−0.136364	−0.0681818	+	0.997673i	$0.521720\pi$
$500$	−0.495099	−0.0221415
$501$	−45.2190	−2.02024
$502$	−11.6234	−0.518780
$503$	−26.2002	−1.16821	−0.584104	−	0.811679i	$-0.698554\pi$
$503$	−26.2002	−1.16821	−0.584104	+	0.811679i	$0.698554\pi$
$504$	−3.11938	−0.138948
$505$	−0.631931	−0.0281206
$506$	−17.5357	−0.779556
$507$	28.4766	1.26469
$508$	4.88085	0.216553
$509$	−26.7347	−1.18499	−0.592496	−	0.805573i	$-0.701858\pi$
$509$	−26.7347	−1.18499	−0.592496	+	0.805573i	$0.701858\pi$
$510$	−0.0664387	−0.00294196
$511$	−22.1770	−0.981055
$512$	−1.00000	−0.0441942
$513$	−16.8082	−0.742102
$514$	6.10648	0.269345
$515$	0.582844	0.0256832
$516$	−6.06968	−0.267203
$517$	30.8810	1.35814
$518$	15.1041	0.663635
$519$	43.1934	1.89598
$520$	−0.00740819	−0.000324870 0
$521$	15.5473	0.681138	0.340569	−	0.940220i	$-0.389380\pi$
$521$	15.5473	0.681138	0.340569	+	0.940220i	$0.389380\pi$
$522$	8.89397	0.389278
$523$	22.2301	0.972055	0.486027	−	0.873944i	$-0.338446\pi$
$523$	22.2301	0.972055	0.486027	+	0.873944i	$0.338446\pi$
$524$	−0.0816734	−0.00356792
$525$	18.8483	0.822606
$526$	−8.75820	−0.381876
$527$	4.05891	0.176809
$528$	9.84910	0.428627
$529$	−7.73712	−0.336396
$530$	0.484086	0.0210274
$531$	2.36512	0.102638
$532$	−11.1093	−0.481651
$533$	1.02617	0.0444485
$534$	12.4714	0.539689
$535$	−0.364103	−0.0157415
$536$	6.84709	0.295749
$537$	24.2258	1.04542
$538$	8.49852	0.366397
$539$	18.1595	0.782187
$540$	0.128782	0.00554189
$541$	−23.8470	−1.02526	−0.512631	−	0.858609i	$-0.671329\pi$
$541$	−23.8470	−1.02526	−0.512631	+	0.858609i	$0.671329\pi$
$542$	7.03183	0.302043
$543$	16.3132	0.700068
$544$	0.611407	0.0262139
$545$	−0.0403334	−0.00172769
$546$	0.564194	0.0241453
$547$	−12.6750	−0.541946	−0.270973	−	0.962587i	$-0.587345\pi$
$547$	−12.6750	−0.541946	−0.270973	+	0.962587i	$0.587345\pi$
$548$	4.22433	0.180454
$549$	−3.73798	−0.159533
$550$	−22.4316	−0.956488
$551$	31.6749	1.34939
$552$	−8.57257	−0.364873
$553$	20.5714	0.874786
$554$	−27.0919	−1.15102
$555$	0.954910	0.0405337
$556$	2.90677	0.123274
$557$	34.7629	1.47295	0.736476	−	0.676464i	$-0.236489\pi$
$557$	34.7629	1.47295	0.736476	+	0.676464i	$0.236489\pi$
$558$	12.0483	0.510044
$559$	0.413797	0.0175017
$560$	0.0851178	0.00359688
$561$	−6.02181	−0.254241
$562$	−0.666362	−0.0281088
$563$	27.8562	1.17400	0.586999	−	0.809588i	$-0.300309\pi$
$563$	27.8562	1.17400	0.586999	+	0.809588i	$0.300309\pi$
$564$	15.0966	0.635682
$565$	0.501919	0.0211159
$566$	8.42508	0.354132
$567$	−19.1660	−0.804894
$568$	−15.9726	−0.670196
$569$	29.2247	1.22517	0.612583	−	0.790407i	$-0.290131\pi$
$569$	29.2247	1.22517	0.612583	+	0.790407i	$0.290131\pi$
$570$	−0.702354	−0.0294184
$571$	−2.25160	−0.0942264	−0.0471132	−	0.998890i	$-0.515002\pi$
$571$	−2.25160	−0.0942264	−0.0471132	+	0.998890i	$0.515002\pi$
$572$	−0.671456	−0.0280750
$573$	−21.8759	−0.913879
$574$	−11.7904	−0.492122
$575$	19.5243	0.814219
$576$	1.81487	0.0756198
$577$	16.4483	0.684753	0.342376	−	0.939563i	$-0.388768\pi$
$577$	16.4483	0.684753	0.342376	+	0.939563i	$0.388768\pi$
$578$	16.6262	0.691558
$579$	4.18010	0.173719
$580$	−0.242687	−0.0100770
$581$	7.26663	0.301471
$582$	3.09438	0.128266
$583$	43.8762	1.81717
$584$	12.9027	0.533919
$585$	0.0134449	0.000555879 0
$586$	−20.1777	−0.833533
$587$	0.383184	0.0158157	0.00790786	−	0.999969i	$-0.497483\pi$
$587$	0.383184	0.0158157	0.00790786	+	0.999969i	$0.497483\pi$
$588$	8.87756	0.366104
$589$	42.9086	1.76802
$590$	−0.0645364	−0.00265692
$591$	18.1925	0.748339
$592$	−8.78763	−0.361169
$593$	−5.88116	−0.241510	−0.120755	−	0.992682i	$-0.538532\pi$
$593$	−5.88116	−0.241510	−0.120755	+	0.992682i	$0.538532\pi$
$594$	11.6724	0.478925
$595$	−0.0520416	−0.00213350
$596$	16.1227	0.660410
$597$	23.4285	0.958865
$598$	0.584429	0.0238991
$599$	11.2660	0.460317	0.230159	−	0.973153i	$-0.426076\pi$
$599$	11.2660	0.460317	0.230159	+	0.973153i	$0.426076\pi$
$600$	−10.9660	−0.447686
$601$	3.99074	0.162786	0.0813929	−	0.996682i	$-0.474063\pi$
$601$	3.99074	0.162786	0.0813929	+	0.996682i	$0.474063\pi$
$602$	−4.75440	−0.193775
$603$	−12.4266	−0.506051
$604$	−9.26507	−0.376991
$605$	0.452973	0.0184160
$606$	−28.0004	−1.13744
$607$	15.3100	0.621416	0.310708	−	0.950505i	$-0.399434\pi$
$607$	15.3100	0.621416	0.310708	+	0.950505i	$0.399434\pi$
$608$	6.46347	0.262128
$609$	18.4826	0.748954
$610$	0.101997	0.00412975
$611$	−1.02920	−0.0416371
$612$	−1.10963	−0.0448540
$613$	41.6194	1.68099	0.840496	−	0.541818i	$-0.182264\pi$
$613$	41.6194	1.68099	0.840496	+	0.541818i	$0.182264\pi$
$614$	3.01484	0.121669
$615$	−0.745413	−0.0300580
$616$	7.71483	0.310839
$617$	−24.6308	−0.991600	−0.495800	−	0.868437i	$-0.665125\pi$
$617$	−24.6308	−0.991600	−0.495800	+	0.868437i	$0.665125\pi$
$618$	25.8254	1.03885
$619$	0.383970	0.0154331	0.00771654	−	0.999970i	$-0.497544\pi$
$619$	0.383970	0.0154331	0.00771654	+	0.999970i	$0.497544\pi$
$620$	−0.328758	−0.0132032
$621$	−10.1596	−0.407689
$622$	−18.4919	−0.741458
$623$	9.76886	0.391381
$624$	−0.328251	−0.0131406
$625$	24.9632	0.998529
$626$	−10.2506	−0.409698
$627$	−63.6593	−2.54231
$628$	−9.73885	−0.388622
$629$	5.37282	0.214228
$630$	−0.154478	−0.00615456
$631$	11.8514	0.471795	0.235898	−	0.971778i	$-0.424197\pi$
$631$	11.8514	0.471795	0.235898	+	0.971778i	$0.424197\pi$
$632$	−11.9686	−0.476084
$633$	−22.6830	−0.901566
$634$	22.4731	0.892521
$635$	0.241710	0.00959196
$636$	21.4495	0.850528
$637$	−0.605222	−0.0239798
$638$	−21.9965	−0.870849
$639$	28.9883	1.14676
$640$	−0.0495220	−0.00195753
$641$	−39.9262	−1.57699	−0.788495	−	0.615041i	$-0.789140\pi$
$641$	−39.9262	−1.57699	−0.788495	+	0.615041i	$0.789140\pi$
$642$	−16.1331	−0.636723
$643$	−3.87221	−0.152705	−0.0763526	−	0.997081i	$-0.524327\pi$
$643$	−3.87221	−0.152705	−0.0763526	+	0.997081i	$0.524327\pi$
$644$	−6.71492	−0.264605
$645$	−0.300583	−0.0118354
$646$	−3.95181	−0.155482
$647$	4.47846	0.176066	0.0880331	−	0.996118i	$-0.471942\pi$
$647$	4.47846	0.176066	0.0880331	+	0.996118i	$0.471942\pi$
$648$	11.1509	0.438047
$649$	−5.84940	−0.229609
$650$	0.747602	0.0293234
$651$	25.0376	0.981302
$652$	2.70166	0.105805
$653$	−2.35056	−0.0919844	−0.0459922	−	0.998942i	$-0.514645\pi$
$653$	−2.35056	−0.0919844	−0.0459922	+	0.998942i	$0.514645\pi$
$654$	−1.78714	−0.0698829
$655$	−0.00404463	−0.000158037 0
$656$	6.85972	0.267827
$657$	−23.4168	−0.913577
$658$	11.8252	0.460995
$659$	25.5062	0.993582	0.496791	−	0.867870i	$-0.334512\pi$
$659$	25.5062	0.993582	0.496791	+	0.867870i	$0.334512\pi$
$660$	0.487747	0.0189855
$661$	−3.68444	−0.143308	−0.0716541	−	0.997430i	$-0.522828\pi$
$661$	−3.68444	−0.143308	−0.0716541	+	0.997430i	$0.522828\pi$
$662$	20.6999	0.804523
$663$	0.200695	0.00779435
$664$	−4.22776	−0.164069
$665$	−0.550156	−0.0213341
$666$	15.9484	0.617990
$667$	19.1455	0.741318
$668$	20.6077	0.797335
$669$	57.3281	2.21643
$670$	0.339082	0.0130999
$671$	9.24474	0.356889
$672$	3.77151	0.145489
$673$	7.47582	0.288172	0.144086	−	0.989565i	$-0.453976\pi$
$673$	7.47582	0.288172	0.144086	+	0.989565i	$0.453976\pi$
$674$	0.659518	0.0254037
$675$	−12.9961	−0.500221
$676$	−12.9776	−0.499139
$677$	14.1155	0.542504	0.271252	−	0.962508i	$-0.412562\pi$
$677$	14.1155	0.542504	0.271252	+	0.962508i	$0.412562\pi$
$678$	22.2397	0.854109
$679$	2.42384	0.0930184
$680$	0.0302781	0.00116111
$681$	−9.18922	−0.352132
$682$	−29.7977	−1.14101
$683$	−14.3026	−0.547272	−0.273636	−	0.961833i	$-0.588226\pi$
$683$	−14.3026	−0.547272	−0.273636	+	0.961833i	$0.588226\pi$
$684$	−11.7304	−0.448522
$685$	0.209197	0.00799302
$686$	18.9853	0.724863
$687$	11.2846	0.430535
$688$	2.76613	0.105458
$689$	−1.46231	−0.0557095
$690$	−0.424531	−0.0161616
$691$	42.6761	1.62347	0.811737	−	0.584023i	$-0.198522\pi$
$691$	42.6761	1.62347	0.811737	+	0.584023i	$0.198522\pi$
$692$	−19.6845	−0.748294
$693$	−14.0014	−0.531871
$694$	26.6562	1.01186
$695$	0.143949	0.00546030
$696$	−10.7533	−0.407603
$697$	−4.19408	−0.158862
$698$	35.1290	1.32965
$699$	38.6932	1.46351
$700$	−8.58972	−0.324661
$701$	−36.5514	−1.38053	−0.690263	−	0.723558i	$-0.742505\pi$
$701$	−36.5514	−1.38053	−0.690263	+	0.723558i	$0.742505\pi$
$702$	−0.389019	−0.0146826
$703$	56.7986	2.14220
$704$	−4.48853	−0.169168
$705$	0.747615	0.0281568
$706$	−9.85262	−0.370808
$707$	−21.9328	−0.824868
$708$	−2.85956	−0.107469
$709$	25.8383	0.970376	0.485188	−	0.874410i	$-0.338751\pi$
$709$	25.8383	0.970376	0.485188	+	0.874410i	$0.338751\pi$
$710$	−0.790996	−0.0296855
$711$	21.7214	0.814618
$712$	−5.68357	−0.213001
$713$	25.9356	0.971297
$714$	−2.30593	−0.0862971
$715$	−0.0332519	−0.00124355
$716$	−11.0404	−0.412600
$717$	−6.27680	−0.234411
$718$	−24.5956	−0.917899
$719$	−1.22318	−0.0456168	−0.0228084	−	0.999740i	$-0.507261\pi$
$719$	−1.22318	−0.0456168	−0.0228084	+	0.999740i	$0.507261\pi$
$720$	0.0898762	0.00334949
$721$	20.2291	0.753370
$722$	−22.7764	−0.847650
$723$	19.6943	0.732439
$724$	−7.43443	−0.276298
$725$	24.4910	0.909572
$726$	20.0709	0.744900
$727$	−8.79628	−0.326236	−0.163118	−	0.986607i	$-0.552155\pi$
$727$	−8.79628	−0.326236	−0.163118	+	0.986607i	$0.552155\pi$
$728$	−0.257120	−0.00952950
$729$	−3.11870	−0.115508
$730$	0.638969	0.0236493
$731$	−1.69123	−0.0625526
$732$	4.51942	0.167043
$733$	15.8151	0.584142	0.292071	−	0.956397i	$-0.405656\pi$
$733$	15.8151	0.584142	0.292071	+	0.956397i	$0.405656\pi$
$734$	20.3170	0.749913
$735$	0.439634	0.0162162
$736$	3.90677	0.144006
$737$	30.7334	1.13208
$738$	−12.4495	−0.458274
$739$	−1.33529	−0.0491196	−0.0245598	−	0.999698i	$-0.507818\pi$
$739$	−1.33529	−0.0491196	−0.0245598	+	0.999698i	$0.507818\pi$
$740$	−0.435181	−0.0159976
$741$	2.12164	0.0779404
$742$	16.8015	0.616801
$743$	33.0831	1.21370	0.606850	−	0.794816i	$-0.292433\pi$
$743$	33.0831	1.21370	0.606850	+	0.794816i	$0.292433\pi$
$744$	−14.5670	−0.534053
$745$	0.798426	0.0292521
$746$	−32.0593	−1.17377
$747$	7.67286	0.280735
$748$	2.74432	0.100342
$749$	−12.6371	−0.461750
$750$	−1.08639	−0.0396692
$751$	41.3968	1.51059	0.755296	−	0.655384i	$-0.227493\pi$
$751$	41.3968	1.51059	0.755296	+	0.655384i	$0.227493\pi$
$752$	−6.87998	−0.250887
$753$	−25.5051	−0.929458
$754$	0.733099	0.0266979
$755$	−0.458825	−0.0166984
$756$	4.46971	0.162562
$757$	−18.4458	−0.670423	−0.335212	−	0.942143i	$-0.608808\pi$
$757$	−18.4458	−0.670423	−0.335212	+	0.942143i	$0.608808\pi$
$758$	−16.0863	−0.584280
$759$	−38.4782	−1.39667
$760$	0.320084	0.0116107
$761$	−0.286940	−0.0104016	−0.00520078	−	0.999986i	$-0.501655\pi$
$761$	−0.286940	−0.0104016	−0.00520078	+	0.999986i	$0.501655\pi$
$762$	10.7100	0.387981
$763$	−1.39988	−0.0506789
$764$	9.96951	0.360684
$765$	−0.0549510	−0.00198676
$766$	−23.8546	−0.861901
$767$	0.194949	0.00703919
$768$	−2.19428	−0.0791793
$769$	−23.2997	−0.840207	−0.420104	−	0.907476i	$-0.638006\pi$
$769$	−23.2997	−0.840207	−0.420104	+	0.907476i	$0.638006\pi$
$770$	0.382054	0.0137683
$771$	13.3993	0.482566
$772$	−1.90500	−0.0685623
$773$	21.7576	0.782566	0.391283	−	0.920270i	$-0.372031\pi$
$773$	21.7576	0.782566	0.391283	+	0.920270i	$0.372031\pi$
$774$	−5.02019	−0.180447
$775$	33.1769	1.19175
$776$	−1.41020	−0.0506233
$777$	33.1426	1.18898
$778$	−26.6943	−0.957038
$779$	−44.3376	−1.58856
$780$	−0.0162556	−0.000582046 0
$781$	−71.6936	−2.56540
$782$	−2.38863	−0.0854172
$783$	−12.7440	−0.455434
$784$	−4.04577	−0.144492
$785$	−0.482287	−0.0172136
$786$	−0.179214	−0.00639237
$787$	37.5681	1.33916	0.669580	−	0.742740i	$-0.266474\pi$
$787$	37.5681	1.33916	0.669580	+	0.742740i	$0.266474\pi$
$788$	−8.29086	−0.295350
$789$	−19.2180	−0.684178
$790$	−0.592707	−0.0210876
$791$	17.4204	0.619398
$792$	8.14612	0.289460
$793$	−0.308109	−0.0109413
$794$	−33.2828	−1.18116
$795$	1.06222	0.0376731
$796$	−10.6771	−0.378439
$797$	−6.13569	−0.217337	−0.108669	−	0.994078i	$-0.534659\pi$
$797$	−6.13569	−0.217337	−0.108669	+	0.994078i	$0.534659\pi$
$798$	−24.3770	−0.862937
$799$	4.20647	0.148814
$800$	4.99755	0.176690
$801$	10.3150	0.364462
$802$	17.2486	0.609070
$803$	57.9143	2.04375
$804$	15.0245	0.529872
$805$	−0.332536	−0.0117204
$806$	0.993098	0.0349804
$807$	18.6482	0.656446
$808$	12.7606	0.448917
$809$	7.78281	0.273629	0.136814	−	0.990597i	$-0.456314\pi$
$809$	7.78281	0.273629	0.136814	+	0.990597i	$0.456314\pi$
$810$	0.552213	0.0194028
$811$	−39.6378	−1.39187	−0.695936	−	0.718104i	$-0.745010\pi$
$811$	−39.6378	−1.39187	−0.695936	+	0.718104i	$0.745010\pi$
$812$	−8.42309	−0.295592
$813$	15.4298	0.541147
$814$	−39.4435	−1.38250
$815$	0.133792	0.00468651
$816$	1.34160	0.0469654
$817$	−17.8788	−0.625501
$818$	17.6415	0.616820
$819$	0.466641	0.0163057
$820$	0.339707	0.0118631
$821$	−5.71077	−0.199307	−0.0996537	−	0.995022i	$-0.531773\pi$
$821$	−5.71077	−0.199307	−0.0996537	+	0.995022i	$0.531773\pi$
$822$	9.26938	0.323307
$823$	28.6113	0.997326	0.498663	−	0.866796i	$-0.333825\pi$
$823$	28.6113	0.997326	0.498663	+	0.866796i	$0.333825\pi$
$824$	−11.7694	−0.410006
$825$	−49.2213	−1.71367
$826$	−2.23990	−0.0779362
$827$	−19.3506	−0.672887	−0.336444	−	0.941704i	$-0.609224\pi$
$827$	−19.3506	−0.672887	−0.336444	+	0.941704i	$0.609224\pi$
$828$	−7.09030	−0.246405
$829$	18.0246	0.626021	0.313011	−	0.949750i	$-0.398662\pi$
$829$	18.0246	0.626021	0.313011	+	0.949750i	$0.398662\pi$
$830$	−0.209367	−0.00726725
$831$	−59.4473	−2.06220
$832$	0.149594	0.00518623
$833$	2.47361	0.0857056
$834$	6.37827	0.220861
$835$	1.02053	0.0353170
$836$	29.0115	1.00338
$837$	−17.2638	−0.596723
$838$	25.6130	0.884786
$839$	30.3266	1.04699	0.523494	−	0.852029i	$-0.324628\pi$
$839$	30.3266	1.04699	0.523494	+	0.852029i	$0.324628\pi$
$840$	0.186772	0.00644426
$841$	−4.98415	−0.171867
$842$	1.23490	0.0425573
$843$	−1.46219	−0.0503604
$844$	10.3373	0.355825
$845$	−0.642678	−0.0221088
$846$	12.4863	0.429288
$847$	15.7216	0.540200
$848$	−9.77518	−0.335681
$849$	18.4870	0.634472
$850$	−3.05554	−0.104804
$851$	34.3313	1.17686
$852$	−35.0484	−1.20074
$853$	14.5564	0.498403	0.249202	−	0.968452i	$-0.419832\pi$
$853$	14.5564	0.498403	0.249202	+	0.968452i	$0.419832\pi$
$854$	3.54008	0.121139
$855$	−0.580912	−0.0198668
$856$	7.35234	0.251298
$857$	−12.0512	−0.411661	−0.205831	−	0.978588i	$-0.565990\pi$
$857$	−12.0512	−0.411661	−0.205831	+	0.978588i	$0.565990\pi$
$858$	−1.47336	−0.0502998
$859$	13.5353	0.461820	0.230910	−	0.972975i	$-0.425830\pi$
$859$	13.5353	0.461820	0.230910	+	0.972975i	$0.425830\pi$
$860$	0.136985	0.00467113
$861$	−25.8715	−0.881698
$862$	−3.10156	−0.105640
$863$	−42.6120	−1.45053	−0.725265	−	0.688470i	$-0.758283\pi$
$863$	−42.6120	−1.45053	−0.725265	+	0.688470i	$0.758283\pi$
$864$	−2.60050	−0.0884708
$865$	−0.974817	−0.0331448
$866$	−19.4689	−0.661581
$867$	36.4825	1.23901
$868$	−11.4104	−0.387294
$869$	−53.7213	−1.82237
$870$	−0.532525	−0.0180543
$871$	−1.02428	−0.0347065
$872$	0.814455	0.0275809
$873$	2.55934	0.0866205
$874$	−25.2513	−0.854138
$875$	−0.850969	−0.0287680
$876$	28.3122	0.956581
$877$	−0.806782	−0.0272431	−0.0136216	−	0.999907i	$-0.504336\pi$
$877$	−0.806782	−0.0272431	−0.0136216	+	0.999907i	$0.504336\pi$
$878$	39.9161	1.34710
$879$	−44.2756	−1.49338
$880$	−0.222281	−0.00749309
$881$	51.7530	1.74360	0.871802	−	0.489858i	$-0.162951\pi$
$881$	51.7530	1.74360	0.871802	+	0.489858i	$0.162951\pi$
$882$	7.34256	0.247237
$883$	4.21607	0.141882	0.0709410	−	0.997481i	$-0.477400\pi$
$883$	4.21607	0.141882	0.0709410	+	0.997481i	$0.477400\pi$
$884$	−0.0914627	−0.00307623
$885$	−0.141611	−0.00476021
$886$	−31.8463	−1.06990
$887$	−11.1549	−0.374546	−0.187273	−	0.982308i	$-0.559965\pi$
$887$	−11.1549	−0.374546	−0.187273	+	0.982308i	$0.559965\pi$
$888$	−19.2825	−0.647080
$889$	8.38915	0.281363
$890$	−0.281462	−0.00943462
$891$	50.0509	1.67677
$892$	−26.1261	−0.874768
$893$	44.4685	1.48808
$894$	35.3777	1.18321
$895$	−0.546744	−0.0182756
$896$	−1.71879	−0.0574207
$897$	1.28240	0.0428182
$898$	−22.1570	−0.739389
$899$	32.5333	1.08504
$900$	−9.06992	−0.302331
$901$	5.97662	0.199110
$902$	30.7901	1.02520
$903$	−10.4325	−0.347172
$904$	−10.1353	−0.337095
$905$	−0.368168	−0.0122383
$906$	−20.3302	−0.675425
$907$	18.1152	0.601506	0.300753	−	0.953702i	$-0.402762\pi$
$907$	18.1152	0.601506	0.300753	+	0.953702i	$0.402762\pi$
$908$	4.18780	0.138977
$909$	−23.1589	−0.768133
$910$	−0.0127331	−0.000422098 0
$911$	46.4176	1.53788	0.768942	−	0.639319i	$-0.220784\pi$
$911$	46.4176	1.53788	0.768942	+	0.639319i	$0.220784\pi$
$912$	14.1827	0.469635
$913$	−18.9764	−0.628029
$914$	34.7269	1.14866
$915$	0.223811	0.00739896
$916$	−5.14274	−0.169921
$917$	−0.140379	−0.00463573
$918$	1.58996	0.0524767
$919$	−4.97111	−0.163982	−0.0819909	−	0.996633i	$-0.526128\pi$
$919$	−4.97111	−0.163982	−0.0819909	+	0.996633i	$0.526128\pi$
$920$	0.193471	0.00637856
$921$	6.61542	0.217985
$922$	5.36635	0.176731
$923$	2.38940	0.0786482
$924$	16.9285	0.556907
$925$	43.9166	1.44397
$926$	8.49012	0.279003
$927$	21.3600	0.701553
$928$	4.90060	0.160870
$929$	26.8589	0.881211	0.440606	−	0.897701i	$-0.354764\pi$
$929$	26.8589	0.881211	0.440606	+	0.897701i	$0.354764\pi$
$930$	−0.721388	−0.0236553
$931$	26.1497	0.857022
$932$	−17.6336	−0.577609
$933$	−40.5765	−1.32841
$934$	−16.9438	−0.554418
$935$	0.135904	0.00444454
$936$	−0.271494	−0.00887406
$937$	19.1563	0.625809	0.312905	−	0.949785i	$-0.398698\pi$
$937$	19.1563	0.625809	0.312905	+	0.949785i	$0.398698\pi$
$938$	11.7687	0.384262
$939$	−22.4928	−0.734025
$940$	−0.340710	−0.0111127
$941$	−29.7348	−0.969325	−0.484663	−	0.874701i	$-0.661058\pi$
$941$	−29.7348	−0.969325	−0.484663	+	0.874701i	$0.661058\pi$
$942$	−21.3698	−0.696265
$943$	−26.7994	−0.872708
$944$	1.30319	0.0424151
$945$	0.221349	0.00720048
$946$	12.4159	0.403675
$947$	−46.5827	−1.51373	−0.756867	−	0.653569i	$-0.773271\pi$
$947$	−46.5827	−1.51373	−0.756867	+	0.653569i	$0.773271\pi$
$948$	−26.2624	−0.852964
$949$	−1.93017	−0.0626559
$950$	−32.3015	−1.04800
$951$	49.3123	1.59906
$952$	1.05088	0.0340592
$953$	19.9156	0.645129	0.322564	−	0.946548i	$-0.395455\pi$
$953$	19.9156	0.645129	0.322564	+	0.946548i	$0.395455\pi$
$954$	17.7407	0.574377
$955$	0.493710	0.0159761
$956$	2.86053	0.0925160
$957$	−48.2665	−1.56023
$958$	17.2598	0.557638
$959$	7.26073	0.234461
$960$	−0.108665	−0.00350716
$961$	13.0714	0.421659
$962$	1.31458	0.0423836
$963$	−13.3436	−0.429991
$964$	−8.97528	−0.289074
$965$	−0.0943392	−0.00303689
$966$	−14.7344	−0.474072
$967$	−46.8596	−1.50690	−0.753452	−	0.657503i	$-0.771613\pi$
$967$	−46.8596	−1.50690	−0.753452	+	0.657503i	$0.771613\pi$
$968$	−9.14690	−0.293992
$969$	−8.67139	−0.278565
$970$	−0.0698360	−0.00224230
$971$	−42.6905	−1.37000	−0.685002	−	0.728541i	$-0.740199\pi$
$971$	−42.6905	−1.37000	−0.685002	+	0.728541i	$0.740199\pi$
$972$	16.6666	0.534582
$973$	4.99612	0.160168
$974$	−27.6962	−0.887444
$975$	1.64045	0.0525365
$976$	−2.05964	−0.0659274
$977$	10.1283	0.324034	0.162017	−	0.986788i	$-0.448200\pi$
$977$	10.1283	0.324034	0.162017	+	0.986788i	$0.448200\pi$
$978$	5.92820	0.189563
$979$	−25.5109	−0.815331
$980$	−0.200355	−0.00640009
$981$	−1.47813	−0.0471932
$982$	15.1407	0.483159
$983$	27.3605	0.872666	0.436333	−	0.899785i	$-0.356277\pi$
$983$	27.3605	0.872666	0.436333	+	0.899785i	$0.356277\pi$
$984$	15.0522	0.479846
$985$	−0.410580	−0.0130822
$986$	−2.99626	−0.0954204
$987$	25.9479	0.825930
$988$	−0.966895	−0.0307610
$989$	−10.8067	−0.343632
$990$	0.403412	0.0128213
$991$	−43.0139	−1.36638	−0.683191	−	0.730239i	$-0.739409\pi$
$991$	−43.0139	−1.36638	−0.683191	+	0.730239i	$0.739409\pi$
$992$	6.63863	0.210777
$993$	45.4213	1.44140
$994$	−27.4535	−0.870773
$995$	−0.528750	−0.0167625
$996$	−9.27691	−0.293950
$997$	−39.7241	−1.25807	−0.629037	−	0.777375i	$-0.716551\pi$
$997$	−39.7241	−1.25807	−0.629037	+	0.777375i	$0.716551\pi$
$998$	3.04613	0.0964237
$999$	−22.8522	−0.723013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.13	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.19428 q^{3} +1.00000 q^{4} +0.0495220 q^{5} +2.19428 q^{6} +1.71879 q^{7} -1.00000 q^{8} +1.81487 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.19428 q^{3} +1.00000 q^{4} +0.0495220 q^{5} +2.19428 q^{6} +1.71879 q^{7} -1.00000 q^{8} +1.81487 q^{9} -0.0495220 q^{10} -4.48853 q^{11} -2.19428 q^{12} +0.149594 q^{13} -1.71879 q^{14} -0.108665 q^{15} +1.00000 q^{16} -0.611407 q^{17} -1.81487 q^{18} -6.46347 q^{19} +0.0495220 q^{20} -3.77151 q^{21} +4.48853 q^{22} -3.90677 q^{23} +2.19428 q^{24} -4.99755 q^{25} -0.149594 q^{26} +2.60050 q^{27} +1.71879 q^{28} -4.90060 q^{29} +0.108665 q^{30} -6.63863 q^{31} -1.00000 q^{32} +9.84910 q^{33} +0.611407 q^{34} +0.0851178 q^{35} +1.81487 q^{36} -8.78763 q^{37} +6.46347 q^{38} -0.328251 q^{39} -0.0495220 q^{40} +6.85972 q^{41} +3.77151 q^{42} +2.76613 q^{43} -4.48853 q^{44} +0.0898762 q^{45} +3.90677 q^{46} -6.87998 q^{47} -2.19428 q^{48} -4.04577 q^{49} +4.99755 q^{50} +1.34160 q^{51} +0.149594 q^{52} -9.77518 q^{53} -2.60050 q^{54} -0.222281 q^{55} -1.71879 q^{56} +14.1827 q^{57} +4.90060 q^{58} +1.30319 q^{59} -0.108665 q^{60} -2.05964 q^{61} +6.63863 q^{62} +3.11938 q^{63} +1.00000 q^{64} +0.00740819 q^{65} -9.84910 q^{66} -6.84709 q^{67} -0.611407 q^{68} +8.57257 q^{69} -0.0851178 q^{70} +15.9726 q^{71} -1.81487 q^{72} -12.9027 q^{73} +8.78763 q^{74} +10.9660 q^{75} -6.46347 q^{76} -7.71483 q^{77} +0.328251 q^{78} +11.9686 q^{79} +0.0495220 q^{80} -11.1509 q^{81} -6.85972 q^{82} +4.22776 q^{83} -3.77151 q^{84} -0.0302781 q^{85} -2.76613 q^{86} +10.7533 q^{87} +4.48853 q^{88} +5.68357 q^{89} -0.0898762 q^{90} +0.257120 q^{91} -3.90677 q^{92} +14.5670 q^{93} +6.87998 q^{94} -0.320084 q^{95} +2.19428 q^{96} +1.41020 q^{97} +4.04577 q^{98} -8.14612 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 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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