LMFDB - Embedded newform 8002.2.a.d.1.18

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:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.99879 q^{3} +1.00000 q^{4} +2.05838 q^{5} -1.99879 q^{6} +2.31041 q^{7} +1.00000 q^{8} +0.995149 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.99879 q^{3} +1.00000 q^{4} +2.05838 q^{5} -1.99879 q^{6} +2.31041 q^{7} +1.00000 q^{8} +0.995149 q^{9} +2.05838 q^{10} -3.05453 q^{11} -1.99879 q^{12} -1.17980 q^{13} +2.31041 q^{14} -4.11426 q^{15} +1.00000 q^{16} +0.133866 q^{17} +0.995149 q^{18} -2.51765 q^{19} +2.05838 q^{20} -4.61802 q^{21} -3.05453 q^{22} -1.56784 q^{23} -1.99879 q^{24} -0.763082 q^{25} -1.17980 q^{26} +4.00727 q^{27} +2.31041 q^{28} -6.46205 q^{29} -4.11426 q^{30} +3.37485 q^{31} +1.00000 q^{32} +6.10535 q^{33} +0.133866 q^{34} +4.75570 q^{35} +0.995149 q^{36} +5.00901 q^{37} -2.51765 q^{38} +2.35817 q^{39} +2.05838 q^{40} -0.906925 q^{41} -4.61802 q^{42} +11.9374 q^{43} -3.05453 q^{44} +2.04839 q^{45} -1.56784 q^{46} -3.39371 q^{47} -1.99879 q^{48} -1.66199 q^{49} -0.763082 q^{50} -0.267570 q^{51} -1.17980 q^{52} -11.2280 q^{53} +4.00727 q^{54} -6.28737 q^{55} +2.31041 q^{56} +5.03225 q^{57} -6.46205 q^{58} -13.9430 q^{59} -4.11426 q^{60} -0.114241 q^{61} +3.37485 q^{62} +2.29921 q^{63} +1.00000 q^{64} -2.42848 q^{65} +6.10535 q^{66} -10.6053 q^{67} +0.133866 q^{68} +3.13378 q^{69} +4.75570 q^{70} -14.0887 q^{71} +0.995149 q^{72} +10.2490 q^{73} +5.00901 q^{74} +1.52524 q^{75} -2.51765 q^{76} -7.05723 q^{77} +2.35817 q^{78} +2.04015 q^{79} +2.05838 q^{80} -10.9951 q^{81} -0.906925 q^{82} -11.3749 q^{83} -4.61802 q^{84} +0.275547 q^{85} +11.9374 q^{86} +12.9163 q^{87} -3.05453 q^{88} +0.534851 q^{89} +2.04839 q^{90} -2.72583 q^{91} -1.56784 q^{92} -6.74561 q^{93} -3.39371 q^{94} -5.18228 q^{95} -1.99879 q^{96} +2.81200 q^{97} -1.66199 q^{98} -3.03971 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.99879	−1.15400	−0.577000	−	0.816744i	$-0.695777\pi$
$3$	−1.99879	−1.15400	−0.577000	+	0.816744i	$0.695777\pi$
$4$	1.00000	0.500000
$5$	2.05838	0.920534	0.460267	−	0.887780i	$-0.347754\pi$
$5$	2.05838	0.920534	0.460267	+	0.887780i	$0.347754\pi$
$6$	−1.99879	−0.816001
$7$	2.31041	0.873254	0.436627	−	0.899643i	$-0.356173\pi$
$7$	2.31041	0.873254	0.436627	+	0.899643i	$0.356173\pi$
$8$	1.00000	0.353553
$9$	0.995149	0.331716
$10$	2.05838	0.650916
$11$	−3.05453	−0.920975	−0.460488	−	0.887666i	$-0.652325\pi$
$11$	−3.05453	−0.920975	−0.460488	+	0.887666i	$0.652325\pi$
$12$	−1.99879	−0.577000
$13$	−1.17980	−0.327218	−0.163609	−	0.986525i	$-0.552314\pi$
$13$	−1.17980	−0.327218	−0.163609	+	0.986525i	$0.552314\pi$
$14$	2.31041	0.617484
$15$	−4.11426	−1.06230
$16$	1.00000	0.250000
$17$	0.133866	0.0324673	0.0162336	−	0.999868i	$-0.494832\pi$
$17$	0.133866	0.0324673	0.0162336	+	0.999868i	$0.494832\pi$
$18$	0.995149	0.234559
$19$	−2.51765	−0.577589	−0.288795	−	0.957391i	$-0.593254\pi$
$19$	−2.51765	−0.577589	−0.288795	+	0.957391i	$0.593254\pi$
$20$	2.05838	0.460267
$21$	−4.61802	−1.00774
$22$	−3.05453	−0.651228
$23$	−1.56784	−0.326917	−0.163459	−	0.986550i	$-0.552265\pi$
$23$	−1.56784	−0.326917	−0.163459	+	0.986550i	$0.552265\pi$
$24$	−1.99879	−0.408001
$25$	−0.763082	−0.152616
$26$	−1.17980	−0.231378
$27$	4.00727	0.771199
$28$	2.31041	0.436627
$29$	−6.46205	−1.19997	−0.599986	−	0.800011i	$-0.704827\pi$
$29$	−6.46205	−1.19997	−0.599986	+	0.800011i	$0.704827\pi$
$30$	−4.11426	−0.751157
$31$	3.37485	0.606141	0.303070	−	0.952968i	$-0.401988\pi$
$31$	3.37485	0.606141	0.303070	+	0.952968i	$0.401988\pi$
$32$	1.00000	0.176777
$33$	6.10535	1.06281
$34$	0.133866	0.0229578
$35$	4.75570	0.803861
$36$	0.995149	0.165858
$37$	5.00901	0.823476	0.411738	−	0.911302i	$-0.364922\pi$
$37$	5.00901	0.823476	0.411738	+	0.911302i	$0.364922\pi$
$38$	−2.51765	−0.408417
$39$	2.35817	0.377610
$40$	2.05838	0.325458
$41$	−0.906925	−0.141638	−0.0708189	−	0.997489i	$-0.522561\pi$
$41$	−0.906925	−0.141638	−0.0708189	+	0.997489i	$0.522561\pi$
$42$	−4.61802	−0.712577
$43$	11.9374	1.82044	0.910219	−	0.414128i	$-0.135913\pi$
$43$	11.9374	1.82044	0.910219	+	0.414128i	$0.135913\pi$
$44$	−3.05453	−0.460488
$45$	2.04839	0.305356
$46$	−1.56784	−0.231165
$47$	−3.39371	−0.495024	−0.247512	−	0.968885i	$-0.579613\pi$
$47$	−3.39371	−0.495024	−0.247512	+	0.968885i	$0.579613\pi$
$48$	−1.99879	−0.288500
$49$	−1.66199	−0.237427
$50$	−0.763082	−0.107916
$51$	−0.267570	−0.0374673
$52$	−1.17980	−0.163609
$53$	−11.2280	−1.54228	−0.771141	−	0.636664i	$-0.780314\pi$
$53$	−11.2280	−1.54228	−0.771141	+	0.636664i	$0.780314\pi$
$54$	4.00727	0.545320
$55$	−6.28737	−0.847789
$56$	2.31041	0.308742
$57$	5.03225	0.666538
$58$	−6.46205	−0.848508
$59$	−13.9430	−1.81522	−0.907611	−	0.419812i	$-0.862096\pi$
$59$	−13.9430	−1.81522	−0.907611	+	0.419812i	$0.862096\pi$
$60$	−4.11426	−0.531148
$61$	−0.114241	−0.0146270	−0.00731351	−	0.999973i	$-0.502328\pi$
$61$	−0.114241	−0.0146270	−0.00731351	+	0.999973i	$0.502328\pi$
$62$	3.37485	0.428606
$63$	2.29921	0.289673
$64$	1.00000	0.125000
$65$	−2.42848	−0.301216
$66$	6.10535	0.751517
$67$	−10.6053	−1.29564	−0.647822	−	0.761792i	$-0.724320\pi$
$67$	−10.6053	−1.29564	−0.647822	+	0.761792i	$0.724320\pi$
$68$	0.133866	0.0162336
$69$	3.13378	0.377262
$70$	4.75570	0.568415
$71$	−14.0887	−1.67203	−0.836013	−	0.548710i	$-0.815119\pi$
$71$	−14.0887	−1.67203	−0.836013	+	0.548710i	$0.815119\pi$
$72$	0.995149	0.117279
$73$	10.2490	1.19956	0.599779	−	0.800166i	$-0.295255\pi$
$73$	10.2490	1.19956	0.599779	+	0.800166i	$0.295255\pi$
$74$	5.00901	0.582285
$75$	1.52524	0.176119
$76$	−2.51765	−0.288795
$77$	−7.05723	−0.804246
$78$	2.35817	0.267011
$79$	2.04015	0.229535	0.114768	−	0.993392i	$-0.463388\pi$
$79$	2.04015	0.229535	0.114768	+	0.993392i	$0.463388\pi$
$80$	2.05838	0.230134
$81$	−10.9951	−1.22168
$82$	−0.906925	−0.100153
$83$	−11.3749	−1.24856	−0.624280	−	0.781200i	$-0.714608\pi$
$83$	−11.3749	−1.24856	−0.624280	+	0.781200i	$0.714608\pi$
$84$	−4.61802	−0.503868
$85$	0.275547	0.0298873
$86$	11.9374	1.28724
$87$	12.9163	1.38477
$88$	−3.05453	−0.325614
$89$	0.534851	0.0566941	0.0283470	−	0.999598i	$-0.490976\pi$
$89$	0.534851	0.0566941	0.0283470	+	0.999598i	$0.490976\pi$
$90$	2.04839	0.215919
$91$	−2.72583	−0.285745
$92$	−1.56784	−0.163459
$93$	−6.74561	−0.699487
$94$	−3.39371	−0.350035
$95$	−5.18228	−0.531691
$96$	−1.99879	−0.204000
$97$	2.81200	0.285515	0.142758	−	0.989758i	$-0.454403\pi$
$97$	2.81200	0.285515	0.142758	+	0.989758i	$0.454403\pi$
$98$	−1.66199	−0.167886
$99$	−3.03971	−0.305503
$100$	−0.763082	−0.0763082
$101$	−6.93643	−0.690200	−0.345100	−	0.938566i	$-0.612155\pi$
$101$	−6.93643	−0.690200	−0.345100	+	0.938566i	$0.612155\pi$
$102$	−0.267570	−0.0264934
$103$	15.9364	1.57026	0.785132	−	0.619329i	$-0.212595\pi$
$103$	15.9364	1.57026	0.785132	+	0.619329i	$0.212595\pi$
$104$	−1.17980	−0.115689
$105$	−9.50564	−0.927655
$106$	−11.2280	−1.09056
$107$	7.05246	0.681787	0.340894	−	0.940102i	$-0.389270\pi$
$107$	7.05246	0.681787	0.340894	+	0.940102i	$0.389270\pi$
$108$	4.00727	0.385600
$109$	−1.17890	−0.112918	−0.0564592	−	0.998405i	$-0.517981\pi$
$109$	−1.17890	−0.112918	−0.0564592	+	0.998405i	$0.517981\pi$
$110$	−6.28737	−0.599478
$111$	−10.0119	−0.950291
$112$	2.31041	0.218314
$113$	0.694134	0.0652987	0.0326493	−	0.999467i	$-0.489606\pi$
$113$	0.694134	0.0652987	0.0326493	+	0.999467i	$0.489606\pi$
$114$	5.03225	0.471313
$115$	−3.22721	−0.300938
$116$	−6.46205	−0.599986
$117$	−1.17408	−0.108544
$118$	−13.9430	−1.28356
$119$	0.309286	0.0283522
$120$	−4.11426	−0.375579
$121$	−1.66985	−0.151805
$122$	−0.114241	−0.0103429
$123$	1.81275	0.163450
$124$	3.37485	0.303070
$125$	−11.8626	−1.06102
$126$	2.29921	0.204830
$127$	2.41691	0.214466	0.107233	−	0.994234i	$-0.465801\pi$
$127$	2.41691	0.214466	0.107233	+	0.994234i	$0.465801\pi$
$128$	1.00000	0.0883883
$129$	−23.8603	−2.10079
$130$	−2.42848	−0.212992
$131$	3.83531	0.335092	0.167546	−	0.985864i	$-0.446416\pi$
$131$	3.83531	0.335092	0.167546	+	0.985864i	$0.446416\pi$
$132$	6.10535	0.531403
$133$	−5.81682	−0.504382
$134$	−10.6053	−0.916158
$135$	8.24847	0.709916
$136$	0.133866	0.0114789
$137$	−10.9150	−0.932528	−0.466264	−	0.884646i	$-0.654400\pi$
$137$	−10.9150	−0.932528	−0.466264	+	0.884646i	$0.654400\pi$
$138$	3.13378	0.266765
$139$	3.28500	0.278630	0.139315	−	0.990248i	$-0.455510\pi$
$139$	3.28500	0.278630	0.139315	+	0.990248i	$0.455510\pi$
$140$	4.75570	0.401930
$141$	6.78331	0.571257
$142$	−14.0887	−1.18230
$143$	3.60374	0.301360
$144$	0.995149	0.0829291
$145$	−13.3013	−1.10462
$146$	10.2490	0.848216
$147$	3.32196	0.273991
$148$	5.00901	0.411738
$149$	−12.1043	−0.991621	−0.495810	−	0.868431i	$-0.665129\pi$
$149$	−12.1043	−0.991621	−0.495810	+	0.868431i	$0.665129\pi$
$150$	1.52524	0.124535
$151$	−9.04117	−0.735760	−0.367880	−	0.929873i	$-0.619916\pi$
$151$	−9.04117	−0.735760	−0.367880	+	0.929873i	$0.619916\pi$
$152$	−2.51765	−0.204209
$153$	0.133217	0.0107699
$154$	−7.05723	−0.568688
$155$	6.94671	0.557974
$156$	2.35817	0.188805
$157$	−2.02690	−0.161764	−0.0808821	−	0.996724i	$-0.525774\pi$
$157$	−2.02690	−0.161764	−0.0808821	+	0.996724i	$0.525774\pi$
$158$	2.04015	0.162306
$159$	22.4423	1.77979
$160$	2.05838	0.162729
$161$	−3.62236	−0.285482
$162$	−10.9951	−0.863859
$163$	11.6588	0.913187	0.456594	−	0.889675i	$-0.349069\pi$
$163$	11.6588	0.913187	0.456594	+	0.889675i	$0.349069\pi$
$164$	−0.906925	−0.0708189
$165$	12.5671	0.978349
$166$	−11.3749	−0.882866
$167$	10.3272	0.799143	0.399572	−	0.916702i	$-0.369159\pi$
$167$	10.3272	0.799143	0.399572	+	0.916702i	$0.369159\pi$
$168$	−4.61802	−0.356288
$169$	−11.6081	−0.892928
$170$	0.275547	0.0211335
$171$	−2.50544	−0.191596
$172$	11.9374	0.910219
$173$	−19.3822	−1.47360	−0.736802	−	0.676108i	$-0.763665\pi$
$173$	−19.3822	−1.47360	−0.736802	+	0.676108i	$0.763665\pi$
$174$	12.9163	0.979179
$175$	−1.76304	−0.133273
$176$	−3.05453	−0.230244
$177$	27.8691	2.09477
$178$	0.534851	0.0400887
$179$	−25.1615	−1.88066	−0.940330	−	0.340263i	$-0.889484\pi$
$179$	−25.1615	−1.88066	−0.940330	+	0.340263i	$0.889484\pi$
$180$	2.04839	0.152678
$181$	13.1748	0.979276	0.489638	−	0.871926i	$-0.337129\pi$
$181$	13.1748	0.979276	0.489638	+	0.871926i	$0.337129\pi$
$182$	−2.72583	−0.202052
$183$	0.228343	0.0168796
$184$	−1.56784	−0.115583
$185$	10.3104	0.758038
$186$	−6.74561	−0.494612
$187$	−0.408898	−0.0299016
$188$	−3.39371	−0.247512
$189$	9.25845	0.673453
$190$	−5.18228	−0.375962
$191$	8.61139	0.623098	0.311549	−	0.950230i	$-0.399152\pi$
$191$	8.61139	0.623098	0.311549	+	0.950230i	$0.399152\pi$
$192$	−1.99879	−0.144250
$193$	−1.18576	−0.0853530	−0.0426765	−	0.999089i	$-0.513588\pi$
$193$	−1.18576	−0.0853530	−0.0426765	+	0.999089i	$0.513588\pi$
$194$	2.81200	0.201890
$195$	4.85401	0.347603
$196$	−1.66199	−0.118713
$197$	21.8995	1.56028	0.780139	−	0.625606i	$-0.215148\pi$
$197$	21.8995	1.56028	0.780139	+	0.625606i	$0.215148\pi$
$198$	−3.03971	−0.216023
$199$	−14.8077	−1.04969	−0.524844	−	0.851198i	$-0.675876\pi$
$199$	−14.8077	−1.04969	−0.524844	+	0.851198i	$0.675876\pi$
$200$	−0.763082	−0.0539581
$201$	21.1977	1.49517
$202$	−6.93643	−0.488045
$203$	−14.9300	−1.04788
$204$	−0.267570	−0.0187336
$205$	−1.86679	−0.130383
$206$	15.9364	1.11034
$207$	−1.56023	−0.108444
$208$	−1.17980	−0.0818046
$209$	7.69024	0.531945
$210$	−9.50564	−0.655951
$211$	12.9449	0.891164	0.445582	−	0.895241i	$-0.352997\pi$
$211$	12.9449	0.891164	0.445582	+	0.895241i	$0.352997\pi$
$212$	−11.2280	−0.771141
$213$	28.1604	1.92952
$214$	7.05246	0.482096
$215$	24.5717	1.67578
$216$	4.00727	0.272660
$217$	7.79730	0.529315
$218$	−1.17890	−0.0798454
$219$	−20.4856	−1.38429
$220$	−6.28737	−0.423895
$221$	−0.157936	−0.0106239
$222$	−10.0119	−0.671957
$223$	14.4248	0.965956	0.482978	−	0.875632i	$-0.339555\pi$
$223$	14.4248	0.965956	0.482978	+	0.875632i	$0.339555\pi$
$224$	2.31041	0.154371
$225$	−0.759381	−0.0506254
$226$	0.694134	0.0461731
$227$	11.5492	0.766547	0.383273	−	0.923635i	$-0.374797\pi$
$227$	11.5492	0.766547	0.383273	+	0.923635i	$0.374797\pi$
$228$	5.03225	0.333269
$229$	−2.62170	−0.173247	−0.0866234	−	0.996241i	$-0.527608\pi$
$229$	−2.62170	−0.173247	−0.0866234	+	0.996241i	$0.527608\pi$
$230$	−3.22721	−0.212796
$231$	14.1059	0.928100
$232$	−6.46205	−0.424254
$233$	27.3261	1.79019	0.895097	−	0.445872i	$-0.147106\pi$
$233$	27.3261	1.79019	0.895097	+	0.445872i	$0.147106\pi$
$234$	−1.17408	−0.0767520
$235$	−6.98554	−0.455686
$236$	−13.9430	−0.907611
$237$	−4.07783	−0.264884
$238$	0.309286	0.0200480
$239$	−7.37212	−0.476863	−0.238431	−	0.971159i	$-0.576633\pi$
$239$	−7.37212	−0.476863	−0.238431	+	0.971159i	$0.576633\pi$
$240$	−4.11426	−0.265574
$241$	−1.77515	−0.114348	−0.0571739	−	0.998364i	$-0.518209\pi$
$241$	−1.77515	−0.114348	−0.0571739	+	0.998364i	$0.518209\pi$
$242$	−1.66985	−0.107342
$243$	9.95510	0.638620
$244$	−0.114241	−0.00731351
$245$	−3.42100	−0.218560
$246$	1.81275	0.115577
$247$	2.97033	0.188998
$248$	3.37485	0.214303
$249$	22.7361	1.44084
$250$	−11.8626	−0.750257
$251$	−15.2230	−0.960865	−0.480432	−	0.877032i	$-0.659520\pi$
$251$	−15.2230	−0.960865	−0.480432	+	0.877032i	$0.659520\pi$
$252$	2.29921	0.144836
$253$	4.78901	0.301083
$254$	2.41691	0.151650
$255$	−0.550760	−0.0344899
$256$	1.00000	0.0625000
$257$	−18.5499	−1.15711	−0.578555	−	0.815643i	$-0.696383\pi$
$257$	−18.5499	−1.15711	−0.578555	+	0.815643i	$0.696383\pi$
$258$	−23.8603	−1.48548
$259$	11.5729	0.719104
$260$	−2.42848	−0.150608
$261$	−6.43070	−0.398050
$262$	3.83531	0.236946
$263$	1.58811	0.0979273	0.0489636	−	0.998801i	$-0.484408\pi$
$263$	1.58811	0.0979273	0.0489636	+	0.998801i	$0.484408\pi$
$264$	6.10535	0.375759
$265$	−23.1114	−1.41972
$266$	−5.81682	−0.356652
$267$	−1.06905	−0.0654249
$268$	−10.6053	−0.647822
$269$	4.93143	0.300675	0.150337	−	0.988635i	$-0.451964\pi$
$269$	4.93143	0.300675	0.150337	+	0.988635i	$0.451964\pi$
$270$	8.24847	0.501986
$271$	−8.46366	−0.514131	−0.257065	−	0.966394i	$-0.582756\pi$
$271$	−8.46366	−0.514131	−0.257065	+	0.966394i	$0.582756\pi$
$272$	0.133866	0.00811682
$273$	5.44836	0.329750
$274$	−10.9150	−0.659397
$275$	2.33086	0.140556
$276$	3.13378	0.188631
$277$	−8.86433	−0.532606	−0.266303	−	0.963889i	$-0.585802\pi$
$277$	−8.86433	−0.532606	−0.266303	+	0.963889i	$0.585802\pi$
$278$	3.28500	0.197021
$279$	3.35848	0.201067
$280$	4.75570	0.284208
$281$	1.22720	0.0732089	0.0366044	−	0.999330i	$-0.488346\pi$
$281$	1.22720	0.0732089	0.0366044	+	0.999330i	$0.488346\pi$
$282$	6.78331	0.403940
$283$	−3.26756	−0.194236	−0.0971181	−	0.995273i	$-0.530962\pi$
$283$	−3.26756	−0.194236	−0.0971181	+	0.995273i	$0.530962\pi$
$284$	−14.0887	−0.836013
$285$	10.3583	0.613571
$286$	3.60374	0.213094
$287$	−2.09537	−0.123686
$288$	0.995149	0.0586397
$289$	−16.9821	−0.998946
$290$	−13.3013	−0.781081
$291$	−5.62059	−0.329485
$292$	10.2490	0.599779
$293$	1.98823	0.116154	0.0580768	−	0.998312i	$-0.481503\pi$
$293$	1.98823	0.116154	0.0580768	+	0.998312i	$0.481503\pi$
$294$	3.32196	0.193741
$295$	−28.6999	−1.67097
$296$	5.00901	0.291143
$297$	−12.2403	−0.710256
$298$	−12.1043	−0.701182
$299$	1.84974	0.106973
$300$	1.52524	0.0880597
$301$	27.5804	1.58971
$302$	−9.04117	−0.520261
$303$	13.8644	0.796491
$304$	−2.51765	−0.144397
$305$	−0.235150	−0.0134647
$306$	0.133217	0.00761549
$307$	−12.3062	−0.702351	−0.351176	−	0.936310i	$-0.614218\pi$
$307$	−12.3062	−0.702351	−0.351176	+	0.936310i	$0.614218\pi$
$308$	−7.05723	−0.402123
$309$	−31.8535	−1.81208
$310$	6.94671	0.394547
$311$	15.4280	0.874839	0.437419	−	0.899258i	$-0.355893\pi$
$311$	15.4280	0.874839	0.437419	+	0.899258i	$0.355893\pi$
$312$	2.35817	0.133505
$313$	−19.7519	−1.11644	−0.558222	−	0.829691i	$-0.688516\pi$
$313$	−19.7519	−1.11644	−0.558222	+	0.829691i	$0.688516\pi$
$314$	−2.02690	−0.114385
$315$	4.73263	0.266654
$316$	2.04015	0.114768
$317$	−9.83219	−0.552231	−0.276116	−	0.961124i	$-0.589047\pi$
$317$	−9.83219	−0.552231	−0.276116	+	0.961124i	$0.589047\pi$
$318$	22.4423	1.25850
$319$	19.7385	1.10514
$320$	2.05838	0.115067
$321$	−14.0964	−0.786782
$322$	−3.62236	−0.201866
$323$	−0.337028	−0.0187528
$324$	−10.9951	−0.610840
$325$	0.900287	0.0499389
$326$	11.6588	0.645721
$327$	2.35638	0.130308
$328$	−0.906925	−0.0500766
$329$	−7.84088	−0.432282
$330$	12.5671	0.691797
$331$	6.83200	0.375520	0.187760	−	0.982215i	$-0.439877\pi$
$331$	6.83200	0.375520	0.187760	+	0.982215i	$0.439877\pi$
$332$	−11.3749	−0.624280
$333$	4.98471	0.273160
$334$	10.3272	0.565080
$335$	−21.8297	−1.19268
$336$	−4.61802	−0.251934
$337$	−11.9388	−0.650347	−0.325174	−	0.945654i	$-0.605423\pi$
$337$	−11.9388	−0.650347	−0.325174	+	0.945654i	$0.605423\pi$
$338$	−11.6081	−0.631396
$339$	−1.38743	−0.0753547
$340$	0.275547	0.0149436
$341$	−10.3086	−0.558241
$342$	−2.50544	−0.135479
$343$	−20.0128	−1.08059
$344$	11.9374	0.643622
$345$	6.45050	0.347283
$346$	−19.3822	−1.04200
$347$	−11.7684	−0.631761	−0.315881	−	0.948799i	$-0.602300\pi$
$347$	−11.7684	−0.631761	−0.315881	+	0.948799i	$0.602300\pi$
$348$	12.9163	0.692384
$349$	−11.2253	−0.600877	−0.300438	−	0.953801i	$-0.597133\pi$
$349$	−11.2253	−0.600877	−0.300438	+	0.953801i	$0.597133\pi$
$350$	−1.76304	−0.0942382
$351$	−4.72779	−0.252351
$352$	−3.05453	−0.162807
$353$	−11.7271	−0.624172	−0.312086	−	0.950054i	$-0.601028\pi$
$353$	−11.7271	−0.624172	−0.312086	+	0.950054i	$0.601028\pi$
$354$	27.8691	1.48122
$355$	−28.9999	−1.53916
$356$	0.534851	0.0283470
$357$	−0.618197	−0.0327184
$358$	−25.1615	−1.32983
$359$	7.88693	0.416256	0.208128	−	0.978102i	$-0.433263\pi$
$359$	7.88693	0.416256	0.208128	+	0.978102i	$0.433263\pi$
$360$	2.04839	0.107960
$361$	−12.6614	−0.666391
$362$	13.1748	0.692453
$363$	3.33768	0.175183
$364$	−2.72583	−0.142872
$365$	21.0964	1.10423
$366$	0.228343	0.0119357
$367$	−1.79646	−0.0937747	−0.0468873	−	0.998900i	$-0.514930\pi$
$367$	−1.79646	−0.0937747	−0.0468873	+	0.998900i	$0.514930\pi$
$368$	−1.56784	−0.0817293
$369$	−0.902525	−0.0469836
$370$	10.3104	0.536014
$371$	−25.9413	−1.34680
$372$	−6.74561	−0.349743
$373$	13.8286	0.716016	0.358008	−	0.933719i	$-0.383456\pi$
$373$	13.8286	0.716016	0.358008	+	0.933719i	$0.383456\pi$
$374$	−0.408898	−0.0211436
$375$	23.7108	1.22442
$376$	−3.39371	−0.175017
$377$	7.62394	0.392653
$378$	9.25845	0.476203
$379$	−3.65081	−0.187530	−0.0937648	−	0.995594i	$-0.529890\pi$
$379$	−3.65081	−0.187530	−0.0937648	+	0.995594i	$0.529890\pi$
$380$	−5.18228	−0.265845
$381$	−4.83088	−0.247494
$382$	8.61139	0.440597
$383$	13.4424	0.686873	0.343437	−	0.939176i	$-0.388409\pi$
$383$	13.4424	0.686873	0.343437	+	0.939176i	$0.388409\pi$
$384$	−1.99879	−0.102000
$385$	−14.5264	−0.740336
$386$	−1.18576	−0.0603537
$387$	11.8795	0.603869
$388$	2.81200	0.142758
$389$	−31.4414	−1.59414	−0.797072	−	0.603884i	$-0.793619\pi$
$389$	−31.4414	−1.59414	−0.797072	+	0.603884i	$0.793619\pi$
$390$	4.85401	0.245793
$391$	−0.209881	−0.0106141
$392$	−1.66199	−0.0839431
$393$	−7.66596	−0.386697
$394$	21.8995	1.10328
$395$	4.19941	0.211295
$396$	−3.03971	−0.152751
$397$	−20.6448	−1.03613	−0.518066	−	0.855340i	$-0.673348\pi$
$397$	−20.6448	−1.03613	−0.518066	+	0.855340i	$0.673348\pi$
$398$	−14.8077	−0.742242
$399$	11.6266	0.582057
$400$	−0.763082	−0.0381541
$401$	−23.1779	−1.15745	−0.578724	−	0.815524i	$-0.696449\pi$
$401$	−23.1779	−1.15745	−0.578724	+	0.815524i	$0.696449\pi$
$402$	21.1977	1.05725
$403$	−3.98166	−0.198340
$404$	−6.93643	−0.345100
$405$	−22.6321	−1.12460
$406$	−14.9300	−0.740964
$407$	−15.3002	−0.758401
$408$	−0.267570	−0.0132467
$409$	12.5721	0.621652	0.310826	−	0.950467i	$-0.399394\pi$
$409$	12.5721	0.621652	0.310826	+	0.950467i	$0.399394\pi$
$410$	−1.86679	−0.0921944
$411$	21.8167	1.07614
$412$	15.9364	0.785132
$413$	−32.2141	−1.58515
$414$	−1.56023	−0.0766813
$415$	−23.4139	−1.14934
$416$	−1.17980	−0.0578446
$417$	−6.56601	−0.321539
$418$	7.69024	0.376142
$419$	12.3554	0.603599	0.301799	−	0.953371i	$-0.402413\pi$
$419$	12.3554	0.603599	0.301799	+	0.953371i	$0.402413\pi$
$420$	−9.50564	−0.463828
$421$	−21.8116	−1.06303	−0.531516	−	0.847048i	$-0.678377\pi$
$421$	−21.8116	−1.06303	−0.531516	+	0.847048i	$0.678377\pi$
$422$	12.9449	0.630148
$423$	−3.37725	−0.164207
$424$	−11.2280	−0.545279
$425$	−0.102151	−0.00495504
$426$	28.1604	1.36438
$427$	−0.263943	−0.0127731
$428$	7.05246	0.340894
$429$	−7.20311	−0.347770
$430$	24.5717	1.18495
$431$	14.8941	0.717425	0.358713	−	0.933448i	$-0.383216\pi$
$431$	14.8941	0.717425	0.358713	+	0.933448i	$0.383216\pi$
$432$	4.00727	0.192800
$433$	6.86545	0.329932	0.164966	−	0.986299i	$-0.447248\pi$
$433$	6.86545	0.329932	0.164966	+	0.986299i	$0.447248\pi$
$434$	7.79730	0.374282
$435$	26.5865	1.27473
$436$	−1.17890	−0.0564592
$437$	3.94727	0.188824
$438$	−20.4856	−0.978841
$439$	11.9953	0.572506	0.286253	−	0.958154i	$-0.407590\pi$
$439$	11.9953	0.572506	0.286253	+	0.958154i	$0.407590\pi$
$440$	−6.28737	−0.299739
$441$	−1.65393	−0.0787584
$442$	−0.157936	−0.00751223
$443$	34.4921	1.63877	0.819385	−	0.573244i	$-0.194315\pi$
$443$	34.4921	1.63877	0.819385	+	0.573244i	$0.194315\pi$
$444$	−10.0119	−0.475146
$445$	1.10092	0.0521888
$446$	14.4248	0.683034
$447$	24.1939	1.14433
$448$	2.31041	0.109157
$449$	3.34325	0.157778	0.0788888	−	0.996883i	$-0.474863\pi$
$449$	3.34325	0.157778	0.0788888	+	0.996883i	$0.474863\pi$
$450$	−0.759381	−0.0357975
$451$	2.77023	0.130445
$452$	0.694134	0.0326493
$453$	18.0714	0.849067
$454$	11.5492	0.542030
$455$	−5.61079	−0.263038
$456$	5.03225	0.235657
$457$	−4.44436	−0.207898	−0.103949	−	0.994583i	$-0.533148\pi$
$457$	−4.44436	−0.207898	−0.103949	+	0.994583i	$0.533148\pi$
$458$	−2.62170	−0.122504
$459$	0.536437	0.0250388
$460$	−3.22721	−0.150469
$461$	−1.34714	−0.0627425	−0.0313713	−	0.999508i	$-0.509987\pi$
$461$	−1.34714	−0.0627425	−0.0313713	+	0.999508i	$0.509987\pi$
$462$	14.1059	0.656266
$463$	−3.14173	−0.146008	−0.0730042	−	0.997332i	$-0.523259\pi$
$463$	−3.14173	−0.146008	−0.0730042	+	0.997332i	$0.523259\pi$
$464$	−6.46205	−0.299993
$465$	−13.8850	−0.643902
$466$	27.3261	1.26586
$467$	−17.2444	−0.797976	−0.398988	−	0.916956i	$-0.630638\pi$
$467$	−17.2444	−0.797976	−0.398988	+	0.916956i	$0.630638\pi$
$468$	−1.17408	−0.0542718
$469$	−24.5026	−1.13143
$470$	−6.98554	−0.322219
$471$	4.05134	0.186676
$472$	−13.9430	−0.641778
$473$	−36.4632	−1.67658
$474$	−4.07783	−0.187301
$475$	1.92118	0.0881496
$476$	0.309286	0.0141761
$477$	−11.1735	−0.511600
$478$	−7.37212	−0.337193
$479$	38.4726	1.75786	0.878930	−	0.476951i	$-0.158258\pi$
$479$	38.4726	1.75786	0.878930	+	0.476951i	$0.158258\pi$
$480$	−4.11426	−0.187789
$481$	−5.90964	−0.269456
$482$	−1.77515	−0.0808561
$483$	7.24032	0.329446
$484$	−1.66985	−0.0759023
$485$	5.78816	0.262827
$486$	9.95510	0.451573
$487$	−16.0693	−0.728169	−0.364084	−	0.931366i	$-0.618618\pi$
$487$	−16.0693	−0.728169	−0.364084	+	0.931366i	$0.618618\pi$
$488$	−0.114241	−0.00517143
$489$	−23.3034	−1.05382
$490$	−3.42100	−0.154545
$491$	−19.2444	−0.868488	−0.434244	−	0.900795i	$-0.642984\pi$
$491$	−19.2444	−0.868488	−0.434244	+	0.900795i	$0.642984\pi$
$492$	1.81275	0.0817251
$493$	−0.865049	−0.0389598
$494$	2.97033	0.133642
$495$	−6.25687	−0.281226
$496$	3.37485	0.151535
$497$	−32.5508	−1.46010
$498$	22.7361	1.01883
$499$	10.1059	0.452403	0.226202	−	0.974080i	$-0.427369\pi$
$499$	10.1059	0.452403	0.226202	+	0.974080i	$0.427369\pi$
$500$	−11.8626	−0.530512
$501$	−20.6419	−0.922212
$502$	−15.2230	−0.679434
$503$	30.3869	1.35489	0.677443	−	0.735576i	$-0.263088\pi$
$503$	30.3869	1.35489	0.677443	+	0.735576i	$0.263088\pi$
$504$	2.29921	0.102415
$505$	−14.2778	−0.635353
$506$	4.78901	0.212898
$507$	23.2020	1.03044
$508$	2.41691	0.107233
$509$	−21.5598	−0.955621	−0.477810	−	0.878463i	$-0.658569\pi$
$509$	−21.5598	−0.955621	−0.477810	+	0.878463i	$0.658569\pi$
$510$	−0.550760	−0.0243880
$511$	23.6795	1.04752
$512$	1.00000	0.0441942
$513$	−10.0889	−0.445436
$514$	−18.5499	−0.818200
$515$	32.8032	1.44548
$516$	−23.8603	−1.05039
$517$	10.3662	0.455905
$518$	11.5729	0.508483
$519$	38.7410	1.70054
$520$	−2.42848	−0.106496
$521$	−25.2085	−1.10440	−0.552201	−	0.833711i	$-0.686212\pi$
$521$	−25.2085	−1.10440	−0.552201	+	0.833711i	$0.686212\pi$
$522$	−6.43070	−0.281464
$523$	−4.26413	−0.186457	−0.0932287	−	0.995645i	$-0.529719\pi$
$523$	−4.26413	−0.186457	−0.0932287	+	0.995645i	$0.529719\pi$
$524$	3.83531	0.167546
$525$	3.52393	0.153797
$526$	1.58811	0.0692450
$527$	0.451778	0.0196798
$528$	6.10535	0.265701
$529$	−20.5419	−0.893125
$530$	−23.1114	−1.00390
$531$	−13.8754	−0.602139
$532$	−5.81682	−0.252191
$533$	1.06999	0.0463465
$534$	−1.06905	−0.0462624
$535$	14.5166	0.627608
$536$	−10.6053	−0.458079
$537$	50.2925	2.17028
$538$	4.93143	0.212609
$539$	5.07659	0.218664
$540$	8.24847	0.354958
$541$	−15.1448	−0.651128	−0.325564	−	0.945520i	$-0.605554\pi$
$541$	−15.1448	−0.651128	−0.325564	+	0.945520i	$0.605554\pi$
$542$	−8.46366	−0.363545
$543$	−26.3336	−1.13008
$544$	0.133866	0.00573946
$545$	−2.42663	−0.103945
$546$	5.44836	0.233168
$547$	22.4040	0.957927	0.478963	−	0.877835i	$-0.341013\pi$
$547$	22.4040	0.957927	0.478963	+	0.877835i	$0.341013\pi$
$548$	−10.9150	−0.466264
$549$	−0.113686	−0.00485202
$550$	2.33086	0.0993881
$551$	16.2692	0.693091
$552$	3.13378	0.133382
$553$	4.71360	0.200443
$554$	−8.86433	−0.376609
$555$	−20.6083	−0.874776
$556$	3.28500	0.139315
$557$	3.07380	0.130241	0.0651206	−	0.997877i	$-0.479257\pi$
$557$	3.07380	0.130241	0.0651206	+	0.997877i	$0.479257\pi$
$558$	3.35848	0.142176
$559$	−14.0838	−0.595681
$560$	4.75570	0.200965
$561$	0.817300	0.0345064
$562$	1.22720	0.0517665
$563$	−14.5243	−0.612127	−0.306063	−	0.952011i	$-0.599012\pi$
$563$	−14.5243	−0.612127	−0.306063	+	0.952011i	$0.599012\pi$
$564$	6.78331	0.285629
$565$	1.42879	0.0601097
$566$	−3.26756	−0.137346
$567$	−25.4033	−1.06684
$568$	−14.0887	−0.591150
$569$	−10.8718	−0.455769	−0.227884	−	0.973688i	$-0.573181\pi$
$569$	−10.8718	−0.455769	−0.227884	+	0.973688i	$0.573181\pi$
$570$	10.3583	0.433860
$571$	−5.45313	−0.228206	−0.114103	−	0.993469i	$-0.536399\pi$
$571$	−5.45313	−0.228206	−0.114103	+	0.993469i	$0.536399\pi$
$572$	3.60374	0.150680
$573$	−17.2123	−0.719056
$574$	−2.09537	−0.0874591
$575$	1.19639	0.0498929
$576$	0.995149	0.0414645
$577$	−28.8939	−1.20287	−0.601435	−	0.798922i	$-0.705404\pi$
$577$	−28.8939	−1.20287	−0.601435	+	0.798922i	$0.705404\pi$
$578$	−16.9821	−0.706361
$579$	2.37009	0.0984974
$580$	−13.3013	−0.552308
$581$	−26.2808	−1.09031
$582$	−5.62059	−0.232981
$583$	34.2962	1.42040
$584$	10.2490	0.424108
$585$	−2.41670	−0.0999182
$586$	1.98823	0.0821330
$587$	2.18944	0.0903677	0.0451839	−	0.998979i	$-0.485613\pi$
$587$	2.18944	0.0903677	0.0451839	+	0.998979i	$0.485613\pi$
$588$	3.32196	0.136995
$589$	−8.49670	−0.350100
$590$	−28.6999	−1.18156
$591$	−43.7725	−1.80056
$592$	5.00901	0.205869
$593$	−24.9483	−1.02451	−0.512253	−	0.858835i	$-0.671189\pi$
$593$	−24.9483	−1.02451	−0.512253	+	0.858835i	$0.671189\pi$
$594$	−12.2403	−0.502227
$595$	0.636627	0.0260992
$596$	−12.1043	−0.495810
$597$	29.5974	1.21134
$598$	1.84974	0.0756416
$599$	−2.89609	−0.118331	−0.0591655	−	0.998248i	$-0.518844\pi$
$599$	−2.89609	−0.118331	−0.0591655	+	0.998248i	$0.518844\pi$
$600$	1.52524	0.0622676
$601$	31.9769	1.30436	0.652182	−	0.758062i	$-0.273854\pi$
$601$	31.9769	1.30436	0.652182	+	0.758062i	$0.273854\pi$
$602$	27.5804	1.12409
$603$	−10.5539	−0.429786
$604$	−9.04117	−0.367880
$605$	−3.43718	−0.139741
$606$	13.8644	0.563204
$607$	−22.3191	−0.905905	−0.452953	−	0.891535i	$-0.649629\pi$
$607$	−22.3191	−0.905905	−0.452953	+	0.891535i	$0.649629\pi$
$608$	−2.51765	−0.102104
$609$	29.8419	1.20925
$610$	−0.235150	−0.00952096
$611$	4.00391	0.161981
$612$	0.133217	0.00538496
$613$	−11.1096	−0.448711	−0.224356	−	0.974507i	$-0.572028\pi$
$613$	−11.1096	−0.448711	−0.224356	+	0.974507i	$0.572028\pi$
$614$	−12.3062	−0.496637
$615$	3.73132	0.150461
$616$	−7.05723	−0.284344
$617$	22.8667	0.920578	0.460289	−	0.887769i	$-0.347746\pi$
$617$	22.8667	0.920578	0.460289	+	0.887769i	$0.347746\pi$
$618$	−31.8535	−1.28134
$619$	−27.6211	−1.11019	−0.555093	−	0.831788i	$-0.687318\pi$
$619$	−27.6211	−1.11019	−0.555093	+	0.831788i	$0.687318\pi$
$620$	6.94671	0.278987
$621$	−6.28276	−0.252118
$622$	15.4280	0.618604
$623$	1.23573	0.0495083
$624$	2.35817	0.0944025
$625$	−20.6023	−0.824092
$626$	−19.7519	−0.789445
$627$	−15.3712	−0.613865
$628$	−2.02690	−0.0808821
$629$	0.670536	0.0267360
$630$	4.73263	0.188553
$631$	31.2957	1.24586	0.622930	−	0.782277i	$-0.285942\pi$
$631$	31.2957	1.24586	0.622930	+	0.782277i	$0.285942\pi$
$632$	2.04015	0.0811529
$633$	−25.8741	−1.02840
$634$	−9.83219	−0.390486
$635$	4.97491	0.197423
$636$	22.4423	0.889897
$637$	1.96082	0.0776904
$638$	19.7385	0.781455
$639$	−14.0204	−0.554638
$640$	2.05838	0.0813645
$641$	36.8362	1.45494	0.727471	−	0.686138i	$-0.240696\pi$
$641$	36.8362	1.45494	0.727471	+	0.686138i	$0.240696\pi$
$642$	−14.0964	−0.556339
$643$	30.8316	1.21588	0.607940	−	0.793983i	$-0.291996\pi$
$643$	30.8316	1.21588	0.607940	+	0.793983i	$0.291996\pi$
$644$	−3.62236	−0.142741
$645$	−49.1136	−1.93384
$646$	−0.337028	−0.0132602
$647$	−10.2856	−0.404368	−0.202184	−	0.979348i	$-0.564804\pi$
$647$	−10.2856	−0.404368	−0.202184	+	0.979348i	$0.564804\pi$
$648$	−10.9951	−0.431929
$649$	42.5893	1.67177
$650$	0.900287	0.0353122
$651$	−15.5851	−0.610830
$652$	11.6588	0.456594
$653$	15.7195	0.615152	0.307576	−	0.951524i	$-0.400482\pi$
$653$	15.7195	0.615152	0.307576	+	0.951524i	$0.400482\pi$
$654$	2.35638	0.0921416
$655$	7.89451	0.308464
$656$	−0.906925	−0.0354095
$657$	10.1993	0.397913
$658$	−7.84088	−0.305669
$659$	−11.1967	−0.436161	−0.218080	−	0.975931i	$-0.569979\pi$
$659$	−11.1967	−0.436161	−0.218080	+	0.975931i	$0.569979\pi$
$660$	12.5671	0.489175
$661$	−15.4849	−0.602294	−0.301147	−	0.953578i	$-0.597370\pi$
$661$	−15.4849	−0.602294	−0.301147	+	0.953578i	$0.597370\pi$
$662$	6.83200	0.265533
$663$	0.315680	0.0122600
$664$	−11.3749	−0.441433
$665$	−11.9732	−0.464301
$666$	4.98471	0.193154
$667$	10.1315	0.392291
$668$	10.3272	0.399572
$669$	−28.8321	−1.11471
$670$	−21.8297	−0.843355
$671$	0.348951	0.0134711
$672$	−4.61802	−0.178144
$673$	27.6145	1.06446	0.532230	−	0.846600i	$-0.321354\pi$
$673$	27.6145	1.06446	0.532230	+	0.846600i	$0.321354\pi$
$674$	−11.9388	−0.459865
$675$	−3.05788	−0.117698
$676$	−11.6081	−0.446464
$677$	36.0282	1.38468	0.692339	−	0.721573i	$-0.256580\pi$
$677$	36.0282	1.38468	0.692339	+	0.721573i	$0.256580\pi$
$678$	−1.38743	−0.0532838
$679$	6.49688	0.249327
$680$	0.275547	0.0105667
$681$	−23.0844	−0.884595
$682$	−10.3086	−0.394736
$683$	−36.8052	−1.40831	−0.704156	−	0.710046i	$-0.748674\pi$
$683$	−36.8052	−1.40831	−0.704156	+	0.710046i	$0.748674\pi$
$684$	−2.50544	−0.0957979
$685$	−22.4671	−0.858424
$686$	−20.0128	−0.764091
$687$	5.24022	0.199927
$688$	11.9374	0.455109
$689$	13.2468	0.504663
$690$	6.45050	0.245566
$691$	10.3573	0.394009	0.197004	−	0.980403i	$-0.436879\pi$
$691$	10.3573	0.394009	0.197004	+	0.980403i	$0.436879\pi$
$692$	−19.3822	−0.736802
$693$	−7.02299	−0.266781
$694$	−11.7684	−0.446723
$695$	6.76177	0.256488
$696$	12.9163	0.489589
$697$	−0.121406	−0.00459860
$698$	−11.2253	−0.424884
$699$	−54.6191	−2.06588
$700$	−1.76304	−0.0666365
$701$	35.9180	1.35660	0.678302	−	0.734783i	$-0.262716\pi$
$701$	35.9180	1.35660	0.678302	+	0.734783i	$0.262716\pi$
$702$	−4.72779	−0.178439
$703$	−12.6109	−0.475631
$704$	−3.05453	−0.115122
$705$	13.9626	0.525862
$706$	−11.7271	−0.441357
$707$	−16.0260	−0.602720
$708$	27.8691	1.04738
$709$	−19.6049	−0.736278	−0.368139	−	0.929771i	$-0.620005\pi$
$709$	−19.6049	−0.736278	−0.368139	+	0.929771i	$0.620005\pi$
$710$	−28.9999	−1.08835
$711$	2.03026	0.0761406
$712$	0.534851	0.0200444
$713$	−5.29122	−0.198158
$714$	−0.618197	−0.0231354
$715$	7.41786	0.277412
$716$	−25.1615	−0.940330
$717$	14.7353	0.550300
$718$	7.88693	0.294338
$719$	−24.1943	−0.902297	−0.451148	−	0.892449i	$-0.648986\pi$
$719$	−24.1943	−0.902297	−0.451148	+	0.892449i	$0.648986\pi$
$720$	2.04839	0.0763391
$721$	36.8198	1.37124
$722$	−12.6614	−0.471210
$723$	3.54816	0.131957
$724$	13.1748	0.489638
$725$	4.93107	0.183135
$726$	3.33768	0.123873
$727$	37.8695	1.40450	0.702252	−	0.711929i	$-0.252178\pi$
$727$	37.8695	1.40450	0.702252	+	0.711929i	$0.252178\pi$
$728$	−2.72583	−0.101026
$729$	13.0872	0.484713
$730$	21.0964	0.780812
$731$	1.59801	0.0591047
$732$	0.228343	0.00843979
$733$	2.80019	0.103427	0.0517136	−	0.998662i	$-0.483532\pi$
$733$	2.80019	0.103427	0.0517136	+	0.998662i	$0.483532\pi$
$734$	−1.79646	−0.0663087
$735$	6.83785	0.252218
$736$	−1.56784	−0.0577913
$737$	32.3942	1.19326
$738$	−0.902525	−0.0332224
$739$	7.65527	0.281604	0.140802	−	0.990038i	$-0.455032\pi$
$739$	7.65527	0.281604	0.140802	+	0.990038i	$0.455032\pi$
$740$	10.3104	0.379019
$741$	−5.93706	−0.218103
$742$	−25.9413	−0.952335
$743$	−13.2583	−0.486400	−0.243200	−	0.969976i	$-0.578197\pi$
$743$	−13.2583	−0.486400	−0.243200	+	0.969976i	$0.578197\pi$
$744$	−6.74561	−0.247306
$745$	−24.9152	−0.912821
$746$	13.8286	0.506299
$747$	−11.3197	−0.414168
$748$	−0.408898	−0.0149508
$749$	16.2941	0.595373
$750$	23.7108	0.865796
$751$	−2.79614	−0.102033	−0.0510163	−	0.998698i	$-0.516246\pi$
$751$	−2.79614	−0.102033	−0.0510163	+	0.998698i	$0.516246\pi$
$752$	−3.39371	−0.123756
$753$	30.4274	1.10884
$754$	7.62394	0.277648
$755$	−18.6101	−0.677292
$756$	9.25845	0.336727
$757$	−27.3903	−0.995519	−0.497759	−	0.867315i	$-0.665844\pi$
$757$	−27.3903	−0.995519	−0.497759	+	0.867315i	$0.665844\pi$
$758$	−3.65081	−0.132604
$759$	−9.57221	−0.347449
$760$	−5.18228	−0.187981
$761$	0.387886	0.0140608	0.00703042	−	0.999975i	$-0.497762\pi$
$761$	0.387886	0.0140608	0.00703042	+	0.999975i	$0.497762\pi$
$762$	−4.83088	−0.175004
$763$	−2.72375	−0.0986065
$764$	8.61139	0.311549
$765$	0.274210	0.00991409
$766$	13.4424	0.485693
$767$	16.4500	0.593974
$768$	−1.99879	−0.0721250
$769$	12.9538	0.467127	0.233563	−	0.972342i	$-0.424961\pi$
$769$	12.9538	0.467127	0.233563	+	0.972342i	$0.424961\pi$
$770$	−14.5264	−0.523496
$771$	37.0773	1.33530
$772$	−1.18576	−0.0426765
$773$	21.4322	0.770864	0.385432	−	0.922736i	$-0.374053\pi$
$773$	21.4322	0.770864	0.385432	+	0.922736i	$0.374053\pi$
$774$	11.8795	0.427000
$775$	−2.57529	−0.0925071
$776$	2.81200	0.100945
$777$	−23.1317	−0.829846
$778$	−31.4414	−1.12723
$779$	2.28332	0.0818085
$780$	4.85401	0.173802
$781$	43.0345	1.53989
$782$	−0.209881	−0.00750531
$783$	−25.8952	−0.925418
$784$	−1.66199	−0.0593567
$785$	−4.17213	−0.148910
$786$	−7.66596	−0.273436
$787$	−24.1541	−0.861000	−0.430500	−	0.902591i	$-0.641663\pi$
$787$	−24.1541	−0.861000	−0.430500	+	0.902591i	$0.641663\pi$
$788$	21.8995	0.780139
$789$	−3.17430	−0.113008
$790$	4.19941	0.149408
$791$	1.60374	0.0570223
$792$	−3.03971	−0.108011
$793$	0.134781	0.00478623
$794$	−20.6448	−0.732656
$795$	46.1948	1.63836
$796$	−14.8077	−0.524844
$797$	35.3332	1.25156	0.625782	−	0.779998i	$-0.284780\pi$
$797$	35.3332	1.25156	0.625782	+	0.779998i	$0.284780\pi$
$798$	11.6266	0.411576
$799$	−0.454303	−0.0160721
$800$	−0.763082	−0.0269790
$801$	0.532256	0.0188063
$802$	−23.1779	−0.818439
$803$	−31.3060	−1.10476
$804$	21.1977	0.747586
$805$	−7.45618	−0.262796
$806$	−3.98166	−0.140248
$807$	−9.85688	−0.346979
$808$	−6.93643	−0.244023
$809$	−29.0498	−1.02134	−0.510668	−	0.859778i	$-0.670602\pi$
$809$	−29.0498	−1.02134	−0.510668	+	0.859778i	$0.670602\pi$
$810$	−22.6321	−0.795212
$811$	48.0953	1.68886	0.844428	−	0.535669i	$-0.179941\pi$
$811$	48.0953	1.68886	0.844428	+	0.535669i	$0.179941\pi$
$812$	−14.9300	−0.523940
$813$	16.9170	0.593307
$814$	−15.3002	−0.536270
$815$	23.9982	0.840620
$816$	−0.267570	−0.00936681
$817$	−30.0542	−1.05146
$818$	12.5721	0.439574
$819$	−2.71261	−0.0947863
$820$	−1.86679	−0.0651913
$821$	−34.9974	−1.22142	−0.610709	−	0.791855i	$-0.709115\pi$
$821$	−34.9974	−1.22142	−0.610709	+	0.791855i	$0.709115\pi$
$822$	21.8167	0.760944
$823$	6.38339	0.222511	0.111256	−	0.993792i	$-0.464513\pi$
$823$	6.38339	0.222511	0.111256	+	0.993792i	$0.464513\pi$
$824$	15.9364	0.555172
$825$	−4.65889	−0.162202
$826$	−32.2141	−1.12087
$827$	−24.4635	−0.850680	−0.425340	−	0.905034i	$-0.639845\pi$
$827$	−24.4635	−0.850680	−0.425340	+	0.905034i	$0.639845\pi$
$828$	−1.56023	−0.0542219
$829$	−6.55621	−0.227706	−0.113853	−	0.993498i	$-0.536319\pi$
$829$	−6.55621	−0.227706	−0.113853	+	0.993498i	$0.536319\pi$
$830$	−23.4139	−0.812708
$831$	17.7179	0.614627
$832$	−1.17980	−0.0409023
$833$	−0.222484	−0.00770861
$834$	−6.56601	−0.227362
$835$	21.2573	0.735639
$836$	7.69024	0.265973
$837$	13.5239	0.467456
$838$	12.3554	0.426809
$839$	2.50184	0.0863731	0.0431865	−	0.999067i	$-0.486249\pi$
$839$	2.50184	0.0863731	0.0431865	+	0.999067i	$0.486249\pi$
$840$	−9.50564	−0.327976
$841$	12.7580	0.439932
$842$	−21.8116	−0.751677
$843$	−2.45292	−0.0844831
$844$	12.9449	0.445582
$845$	−23.8938	−0.821971
$846$	−3.37725	−0.116112
$847$	−3.85805	−0.132564
$848$	−11.2280	−0.385571
$849$	6.53115	0.224149
$850$	−0.102151	−0.00350374
$851$	−7.85332	−0.269208
$852$	28.1604	0.964759
$853$	14.5627	0.498617	0.249309	−	0.968424i	$-0.419797\pi$
$853$	14.5627	0.498617	0.249309	+	0.968424i	$0.419797\pi$
$854$	−0.263943	−0.00903195
$855$	−5.15714	−0.176370
$856$	7.05246	0.241048
$857$	−26.9155	−0.919417	−0.459709	−	0.888070i	$-0.652046\pi$
$857$	−26.9155	−0.919417	−0.459709	+	0.888070i	$0.652046\pi$
$858$	−7.20311	−0.245910
$859$	5.76283	0.196625	0.0983127	−	0.995156i	$-0.468655\pi$
$859$	5.76283	0.196625	0.0983127	+	0.995156i	$0.468655\pi$
$860$	24.5717	0.837888
$861$	4.18820	0.142734
$862$	14.8941	0.507296
$863$	−11.1285	−0.378820	−0.189410	−	0.981898i	$-0.560657\pi$
$863$	−11.1285	−0.378820	−0.189410	+	0.981898i	$0.560657\pi$
$864$	4.00727	0.136330
$865$	−39.8960	−1.35650
$866$	6.86545	0.233297
$867$	33.9436	1.15278
$868$	7.79730	0.264658
$869$	−6.23171	−0.211396
$870$	26.5865	0.901368
$871$	12.5122	0.423958
$872$	−1.17890	−0.0399227
$873$	2.79836	0.0947101
$874$	3.94727	0.133519
$875$	−27.4075	−0.926543
$876$	−20.4856	−0.692145
$877$	39.6363	1.33842	0.669211	−	0.743073i	$-0.266632\pi$
$877$	39.6363	1.33842	0.669211	+	0.743073i	$0.266632\pi$
$878$	11.9953	0.404823
$879$	−3.97405	−0.134041
$880$	−6.28737	−0.211947
$881$	−5.63151	−0.189730	−0.0948651	−	0.995490i	$-0.530242\pi$
$881$	−5.63151	−0.189730	−0.0948651	+	0.995490i	$0.530242\pi$
$882$	−1.65393	−0.0556906
$883$	4.80069	0.161556	0.0807781	−	0.996732i	$-0.474259\pi$
$883$	4.80069	0.161556	0.0807781	+	0.996732i	$0.474259\pi$
$884$	−0.157936	−0.00531195
$885$	57.3650	1.92831
$886$	34.4921	1.15878
$887$	31.1995	1.04758	0.523789	−	0.851848i	$-0.324518\pi$
$887$	31.1995	1.04758	0.523789	+	0.851848i	$0.324518\pi$
$888$	−10.0119	−0.335979
$889$	5.58406	0.187283
$890$	1.10092	0.0369031
$891$	33.5849	1.12514
$892$	14.4248	0.482978
$893$	8.54418	0.285920
$894$	24.1939	0.809164
$895$	−51.7919	−1.73121
$896$	2.31041	0.0771855
$897$	−3.69724	−0.123447
$898$	3.34325	0.111566
$899$	−21.8084	−0.727352
$900$	−0.759381	−0.0253127
$901$	−1.50305	−0.0500737
$902$	2.77023	0.0922385
$903$	−55.1272	−1.83452
$904$	0.694134	0.0230866
$905$	27.1187	0.901457
$906$	18.0714	0.600381
$907$	20.8151	0.691153	0.345577	−	0.938391i	$-0.387683\pi$
$907$	20.8151	0.691153	0.345577	+	0.938391i	$0.387683\pi$
$908$	11.5492	0.383273
$909$	−6.90278	−0.228951
$910$	−5.61079	−0.185996
$911$	13.5482	0.448871	0.224436	−	0.974489i	$-0.427946\pi$
$911$	13.5482	0.448871	0.224436	+	0.974489i	$0.427946\pi$
$912$	5.03225	0.166634
$913$	34.7451	1.14989
$914$	−4.44436	−0.147006
$915$	0.470015	0.0155382
$916$	−2.62170	−0.0866234
$917$	8.86115	0.292621
$918$	0.536437	0.0177051
$919$	−0.485829	−0.0160260	−0.00801301	−	0.999968i	$-0.502551\pi$
$919$	−0.485829	−0.0160260	−0.00801301	+	0.999968i	$0.502551\pi$
$920$	−3.22721	−0.106398
$921$	24.5974	0.810513
$922$	−1.34714	−0.0443657
$923$	16.6219	0.547118
$924$	14.1059	0.464050
$925$	−3.82229	−0.125676
$926$	−3.14173	−0.103244
$927$	15.8591	0.520882
$928$	−6.46205	−0.212127
$929$	−45.4186	−1.49014	−0.745068	−	0.666989i	$-0.767583\pi$
$929$	−45.4186	−1.49014	−0.745068	+	0.666989i	$0.767583\pi$
$930$	−13.8850	−0.455307
$931$	4.18431	0.137135
$932$	27.3261	0.895097
$933$	−30.8372	−1.00956
$934$	−17.2444	−0.564254
$935$	−0.841666	−0.0275254
$936$	−1.17408	−0.0383760
$937$	−48.0337	−1.56919	−0.784596	−	0.620007i	$-0.787130\pi$
$937$	−48.0337	−1.56919	−0.784596	+	0.620007i	$0.787130\pi$
$938$	−24.5026	−0.800039
$939$	39.4799	1.28838
$940$	−6.98554	−0.227843
$941$	−58.5051	−1.90721	−0.953605	−	0.301059i	$-0.902660\pi$
$941$	−58.5051	−1.90721	−0.953605	+	0.301059i	$0.902660\pi$
$942$	4.05134	0.132000
$943$	1.42191	0.0463039
$944$	−13.9430	−0.453806
$945$	19.0574	0.619937
$946$	−36.4632	−1.18552
$947$	35.1193	1.14122	0.570611	−	0.821220i	$-0.306706\pi$
$947$	35.1193	1.14122	0.570611	+	0.821220i	$0.306706\pi$
$948$	−4.07783	−0.132442
$949$	−12.0918	−0.392518
$950$	1.92118	0.0623312
$951$	19.6525	0.637275
$952$	0.309286	0.0100240
$953$	56.5572	1.83207	0.916033	−	0.401102i	$-0.131373\pi$
$953$	56.5572	1.83207	0.916033	+	0.401102i	$0.131373\pi$
$954$	−11.1735	−0.361756
$955$	17.7255	0.573583
$956$	−7.37212	−0.238431
$957$	−39.4531	−1.27534
$958$	38.4726	1.24299
$959$	−25.2181	−0.814334
$960$	−4.11426	−0.132787
$961$	−19.6104	−0.632593
$962$	−5.90964	−0.190534
$963$	7.01825	0.226160
$964$	−1.77515	−0.0571739
$965$	−2.44075	−0.0785704
$966$	7.24032	0.232954
$967$	−13.4723	−0.433240	−0.216620	−	0.976256i	$-0.569503\pi$
$967$	−13.4723	−0.433240	−0.216620	+	0.976256i	$0.569503\pi$
$968$	−1.66985	−0.0536710
$969$	0.673648	0.0216407
$970$	5.78816	0.185847
$971$	60.1256	1.92952	0.964761	−	0.263128i	$-0.0847542\pi$
$971$	60.1256	1.92952	0.964761	+	0.263128i	$0.0847542\pi$
$972$	9.95510	0.319310
$973$	7.58971	0.243315
$974$	−16.0693	−0.514893
$975$	−1.79948	−0.0576295
$976$	−0.114241	−0.00365675
$977$	52.1944	1.66985	0.834923	−	0.550367i	$-0.185512\pi$
$977$	52.1944	1.66985	0.834923	+	0.550367i	$0.185512\pi$
$978$	−23.3034	−0.745162
$979$	−1.63372	−0.0522138
$980$	−3.42100	−0.109280
$981$	−1.17318	−0.0374569
$982$	−19.2444	−0.614114
$983$	25.2508	0.805375	0.402687	−	0.915338i	$-0.368076\pi$
$983$	25.2508	0.805375	0.402687	+	0.915338i	$0.368076\pi$
$984$	1.81275	0.0577884
$985$	45.0775	1.43629
$986$	−0.865049	−0.0275488
$987$	15.6722	0.498853
$988$	2.97033	0.0944989
$989$	−18.7159	−0.595132
$990$	−6.25687	−0.198857
$991$	−3.89647	−0.123775	−0.0618877	−	0.998083i	$-0.519712\pi$
$991$	−3.89647	−0.123775	−0.0618877	+	0.998083i	$0.519712\pi$
$992$	3.37485	0.107152
$993$	−13.6557	−0.433351
$994$	−32.5508	−1.03245
$995$	−30.4798	−0.966274
$996$	22.7361	0.720420
$997$	56.7553	1.79746	0.898729	−	0.438505i	$-0.144492\pi$
$997$	56.7553	1.79746	0.898729	+	0.438505i	$0.144492\pi$
$998$	10.1059	0.319898
$999$	20.0724	0.635064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.18	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.18	✓	69	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.99879 q^{3} +1.00000 q^{4} +2.05838 q^{5} -1.99879 q^{6} +2.31041 q^{7} +1.00000 q^{8} +0.995149 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.99879 q^{3} +1.00000 q^{4} +2.05838 q^{5} -1.99879 q^{6} +2.31041 q^{7} +1.00000 q^{8} +0.995149 q^{9} +2.05838 q^{10} -3.05453 q^{11} -1.99879 q^{12} -1.17980 q^{13} +2.31041 q^{14} -4.11426 q^{15} +1.00000 q^{16} +0.133866 q^{17} +0.995149 q^{18} -2.51765 q^{19} +2.05838 q^{20} -4.61802 q^{21} -3.05453 q^{22} -1.56784 q^{23} -1.99879 q^{24} -0.763082 q^{25} -1.17980 q^{26} +4.00727 q^{27} +2.31041 q^{28} -6.46205 q^{29} -4.11426 q^{30} +3.37485 q^{31} +1.00000 q^{32} +6.10535 q^{33} +0.133866 q^{34} +4.75570 q^{35} +0.995149 q^{36} +5.00901 q^{37} -2.51765 q^{38} +2.35817 q^{39} +2.05838 q^{40} -0.906925 q^{41} -4.61802 q^{42} +11.9374 q^{43} -3.05453 q^{44} +2.04839 q^{45} -1.56784 q^{46} -3.39371 q^{47} -1.99879 q^{48} -1.66199 q^{49} -0.763082 q^{50} -0.267570 q^{51} -1.17980 q^{52} -11.2280 q^{53} +4.00727 q^{54} -6.28737 q^{55} +2.31041 q^{56} +5.03225 q^{57} -6.46205 q^{58} -13.9430 q^{59} -4.11426 q^{60} -0.114241 q^{61} +3.37485 q^{62} +2.29921 q^{63} +1.00000 q^{64} -2.42848 q^{65} +6.10535 q^{66} -10.6053 q^{67} +0.133866 q^{68} +3.13378 q^{69} +4.75570 q^{70} -14.0887 q^{71} +0.995149 q^{72} +10.2490 q^{73} +5.00901 q^{74} +1.52524 q^{75} -2.51765 q^{76} -7.05723 q^{77} +2.35817 q^{78} +2.04015 q^{79} +2.05838 q^{80} -10.9951 q^{81} -0.906925 q^{82} -11.3749 q^{83} -4.61802 q^{84} +0.275547 q^{85} +11.9374 q^{86} +12.9163 q^{87} -3.05453 q^{88} +0.534851 q^{89} +2.04839 q^{90} -2.72583 q^{91} -1.56784 q^{92} -6.74561 q^{93} -3.39371 q^{94} -5.18228 q^{95} -1.99879 q^{96} +2.81200 q^{97} -1.66199 q^{98} -3.03971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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