LMFDB - Embedded newform 8002.2.a.f.1.9

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:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.88148 q^{3} +1.00000 q^{4} +0.529257 q^{5} +2.88148 q^{6} -3.48565 q^{7} -1.00000 q^{8} +5.30294 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.88148 q^{3} +1.00000 q^{4} +0.529257 q^{5} +2.88148 q^{6} -3.48565 q^{7} -1.00000 q^{8} +5.30294 q^{9} -0.529257 q^{10} -3.60579 q^{11} -2.88148 q^{12} +0.327000 q^{13} +3.48565 q^{14} -1.52504 q^{15} +1.00000 q^{16} -4.64598 q^{17} -5.30294 q^{18} -1.57007 q^{19} +0.529257 q^{20} +10.0438 q^{21} +3.60579 q^{22} -5.65135 q^{23} +2.88148 q^{24} -4.71989 q^{25} -0.327000 q^{26} -6.63588 q^{27} -3.48565 q^{28} -0.0557094 q^{29} +1.52504 q^{30} +10.8737 q^{31} -1.00000 q^{32} +10.3900 q^{33} +4.64598 q^{34} -1.84481 q^{35} +5.30294 q^{36} +7.96720 q^{37} +1.57007 q^{38} -0.942245 q^{39} -0.529257 q^{40} +10.8596 q^{41} -10.0438 q^{42} -2.33391 q^{43} -3.60579 q^{44} +2.80662 q^{45} +5.65135 q^{46} +3.26957 q^{47} -2.88148 q^{48} +5.14977 q^{49} +4.71989 q^{50} +13.3873 q^{51} +0.327000 q^{52} -10.8570 q^{53} +6.63588 q^{54} -1.90839 q^{55} +3.48565 q^{56} +4.52412 q^{57} +0.0557094 q^{58} +1.55345 q^{59} -1.52504 q^{60} +11.9209 q^{61} -10.8737 q^{62} -18.4842 q^{63} +1.00000 q^{64} +0.173067 q^{65} -10.3900 q^{66} -6.85733 q^{67} -4.64598 q^{68} +16.2843 q^{69} +1.84481 q^{70} -9.45122 q^{71} -5.30294 q^{72} -4.32934 q^{73} -7.96720 q^{74} +13.6003 q^{75} -1.57007 q^{76} +12.5685 q^{77} +0.942245 q^{78} +2.30006 q^{79} +0.529257 q^{80} +3.21234 q^{81} -10.8596 q^{82} -3.64111 q^{83} +10.0438 q^{84} -2.45892 q^{85} +2.33391 q^{86} +0.160526 q^{87} +3.60579 q^{88} +2.34510 q^{89} -2.80662 q^{90} -1.13981 q^{91} -5.65135 q^{92} -31.3323 q^{93} -3.26957 q^{94} -0.830970 q^{95} +2.88148 q^{96} +11.7782 q^{97} -5.14977 q^{98} -19.1213 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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
$3$	−2.88148	−1.66362	−0.831812	−	0.555057i	$-0.812696\pi$
$3$	−2.88148	−1.66362	−0.831812	+	0.555057i	$0.812696\pi$
$4$	1.00000	0.500000
$5$	0.529257	0.236691	0.118345	−	0.992972i	$-0.462241\pi$
$5$	0.529257	0.236691	0.118345	+	0.992972i	$0.462241\pi$
$6$	2.88148	1.17636
$7$	−3.48565	−1.31745	−0.658726	−	0.752383i	$-0.728904\pi$
$7$	−3.48565	−1.31745	−0.658726	+	0.752383i	$0.728904\pi$
$8$	−1.00000	−0.353553
$9$	5.30294	1.76765
$10$	−0.529257	−0.167366
$11$	−3.60579	−1.08719	−0.543594	−	0.839349i	$-0.682937\pi$
$11$	−3.60579	−1.08719	−0.543594	+	0.839349i	$0.682937\pi$
$12$	−2.88148	−0.831812
$13$	0.327000	0.0906935	0.0453468	−	0.998971i	$-0.485561\pi$
$13$	0.327000	0.0906935	0.0453468	+	0.998971i	$0.485561\pi$
$14$	3.48565	0.931580
$15$	−1.52504	−0.393765
$16$	1.00000	0.250000
$17$	−4.64598	−1.12682	−0.563408	−	0.826179i	$-0.690510\pi$
$17$	−4.64598	−1.12682	−0.563408	+	0.826179i	$0.690510\pi$
$18$	−5.30294	−1.24991
$19$	−1.57007	−0.360198	−0.180099	−	0.983648i	$-0.557642\pi$
$19$	−1.57007	−0.360198	−0.180099	+	0.983648i	$0.557642\pi$
$20$	0.529257	0.118345
$21$	10.0438	2.19175
$22$	3.60579	0.768757
$23$	−5.65135	−1.17839	−0.589194	−	0.807992i	$-0.700555\pi$
$23$	−5.65135	−1.17839	−0.589194	+	0.807992i	$0.700555\pi$
$24$	2.88148	0.588180
$25$	−4.71989	−0.943977
$26$	−0.327000	−0.0641300
$27$	−6.63588	−1.27707
$28$	−3.48565	−0.658726
$29$	−0.0557094	−0.0103450	−0.00517248	−	0.999987i	$-0.501646\pi$
$29$	−0.0557094	−0.0103450	−0.00517248	+	0.999987i	$0.501646\pi$
$30$	1.52504	0.278434
$31$	10.8737	1.95297	0.976486	−	0.215582i	$-0.0691650\pi$
$31$	10.8737	1.95297	0.976486	+	0.215582i	$0.0691650\pi$
$32$	−1.00000	−0.176777
$33$	10.3900	1.80867
$34$	4.64598	0.796779
$35$	−1.84481	−0.311829
$36$	5.30294	0.883823
$37$	7.96720	1.30980	0.654900	−	0.755716i	$-0.272711\pi$
$37$	7.96720	1.30980	0.654900	+	0.755716i	$0.272711\pi$
$38$	1.57007	0.254699
$39$	−0.942245	−0.150880
$40$	−0.529257	−0.0836829
$41$	10.8596	1.69598	0.847992	−	0.530009i	$-0.177811\pi$
$41$	10.8596	1.69598	0.847992	+	0.530009i	$0.177811\pi$
$42$	−10.0438	−1.54980
$43$	−2.33391	−0.355918	−0.177959	−	0.984038i	$-0.556949\pi$
$43$	−2.33391	−0.355918	−0.177959	+	0.984038i	$0.556949\pi$
$44$	−3.60579	−0.543594
$45$	2.80662	0.418386
$46$	5.65135	0.833246
$47$	3.26957	0.476916	0.238458	−	0.971153i	$-0.423358\pi$
$47$	3.26957	0.476916	0.238458	+	0.971153i	$0.423358\pi$
$48$	−2.88148	−0.415906
$49$	5.14977	0.735682
$50$	4.71989	0.667493
$51$	13.3873	1.87460
$52$	0.327000	0.0453468
$53$	−10.8570	−1.49133	−0.745663	−	0.666323i	$-0.767867\pi$
$53$	−10.8570	−1.49133	−0.745663	+	0.666323i	$0.767867\pi$
$54$	6.63588	0.903028
$55$	−1.90839	−0.257327
$56$	3.48565	0.465790
$57$	4.52412	0.599235
$58$	0.0557094	0.00731500
$59$	1.55345	0.202242	0.101121	−	0.994874i	$-0.467757\pi$
$59$	1.55345	0.202242	0.101121	+	0.994874i	$0.467757\pi$
$60$	−1.52504	−0.196882
$61$	11.9209	1.52631	0.763154	−	0.646216i	$-0.223650\pi$
$61$	11.9209	1.52631	0.763154	+	0.646216i	$0.223650\pi$
$62$	−10.8737	−1.38096
$63$	−18.4842	−2.32879
$64$	1.00000	0.125000
$65$	0.173067	0.0214663
$66$	−10.3900	−1.27892
$67$	−6.85733	−0.837756	−0.418878	−	0.908043i	$-0.637576\pi$
$67$	−6.85733	−0.837756	−0.418878	+	0.908043i	$0.637576\pi$
$68$	−4.64598	−0.563408
$69$	16.2843	1.96039
$70$	1.84481	0.220496
$71$	−9.45122	−1.12165	−0.560827	−	0.827933i	$-0.689517\pi$
$71$	−9.45122	−1.12165	−0.560827	+	0.827933i	$0.689517\pi$
$72$	−5.30294	−0.624957
$73$	−4.32934	−0.506711	−0.253356	−	0.967373i	$-0.581534\pi$
$73$	−4.32934	−0.506711	−0.253356	+	0.967373i	$0.581534\pi$
$74$	−7.96720	−0.926168
$75$	13.6003	1.57042
$76$	−1.57007	−0.180099
$77$	12.5685	1.43232
$78$	0.942245	0.106688
$79$	2.30006	0.258777	0.129388	−	0.991594i	$-0.458699\pi$
$79$	2.30006	0.258777	0.129388	+	0.991594i	$0.458699\pi$
$80$	0.529257	0.0591727
$81$	3.21234	0.356927
$82$	−10.8596	−1.19924
$83$	−3.64111	−0.399664	−0.199832	−	0.979830i	$-0.564040\pi$
$83$	−3.64111	−0.399664	−0.199832	+	0.979830i	$0.564040\pi$
$84$	10.0438	1.09587
$85$	−2.45892	−0.266707
$86$	2.33391	0.251672
$87$	0.160526	0.0172101
$88$	3.60579	0.384379
$89$	2.34510	0.248580	0.124290	−	0.992246i	$-0.460335\pi$
$89$	2.34510	0.248580	0.124290	+	0.992246i	$0.460335\pi$
$90$	−2.80662	−0.295843
$91$	−1.13981	−0.119484
$92$	−5.65135	−0.589194
$93$	−31.3323	−3.24901
$94$	−3.26957	−0.337231
$95$	−0.830970	−0.0852557
$96$	2.88148	0.294090
$97$	11.7782	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$97$	11.7782	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$98$	−5.14977	−0.520205
$99$	−19.1213	−1.92176
$100$	−4.71989	−0.471989
$101$	16.3907	1.63094	0.815469	−	0.578800i	$-0.196479\pi$
$101$	16.3907	1.63094	0.815469	+	0.578800i	$0.196479\pi$
$102$	−13.3873	−1.32554
$103$	−4.62577	−0.455790	−0.227895	−	0.973686i	$-0.573184\pi$
$103$	−4.62577	−0.455790	−0.227895	+	0.973686i	$0.573184\pi$
$104$	−0.327000	−0.0320650
$105$	5.31577	0.518766
$106$	10.8570	1.05453
$107$	6.22292	0.601592	0.300796	−	0.953688i	$-0.402748\pi$
$107$	6.22292	0.601592	0.300796	+	0.953688i	$0.402748\pi$
$108$	−6.63588	−0.638537
$109$	17.1142	1.63925	0.819624	−	0.572902i	$-0.194183\pi$
$109$	17.1142	1.63925	0.819624	+	0.572902i	$0.194183\pi$
$110$	1.90839	0.181958
$111$	−22.9573	−2.17901
$112$	−3.48565	−0.329363
$113$	−6.18176	−0.581531	−0.290766	−	0.956794i	$-0.593910\pi$
$113$	−6.18176	−0.581531	−0.290766	+	0.956794i	$0.593910\pi$
$114$	−4.52412	−0.423723
$115$	−2.99102	−0.278914
$116$	−0.0557094	−0.00517248
$117$	1.73406	0.160314
$118$	−1.55345	−0.143007
$119$	16.1943	1.48453
$120$	1.52504	0.139217
$121$	2.00173	0.181976
$122$	−11.9209	−1.07926
$123$	−31.2917	−2.82148
$124$	10.8737	0.976486
$125$	−5.14432	−0.460122
$126$	18.4842	1.64670
$127$	2.11956	0.188081	0.0940404	−	0.995568i	$-0.470022\pi$
$127$	2.11956	0.188081	0.0940404	+	0.995568i	$0.470022\pi$
$128$	−1.00000	−0.0883883
$129$	6.72512	0.592114
$130$	−0.173067	−0.0151790
$131$	−11.6270	−1.01586	−0.507928	−	0.861400i	$-0.669588\pi$
$131$	−11.6270	−1.01586	−0.507928	+	0.861400i	$0.669588\pi$
$132$	10.3900	0.904335
$133$	5.47271	0.474544
$134$	6.85733	0.592383
$135$	−3.51208	−0.302272
$136$	4.64598	0.398389
$137$	17.7101	1.51308	0.756539	−	0.653949i	$-0.226889\pi$
$137$	17.7101	1.51308	0.756539	+	0.653949i	$0.226889\pi$
$138$	−16.2843	−1.38621
$139$	5.19767	0.440861	0.220430	−	0.975403i	$-0.429254\pi$
$139$	5.19767	0.440861	0.220430	+	0.975403i	$0.429254\pi$
$140$	−1.84481	−0.155915
$141$	−9.42121	−0.793409
$142$	9.45122	0.793129
$143$	−1.17909	−0.0986008
$144$	5.30294	0.441912
$145$	−0.0294846	−0.00244856
$146$	4.32934	0.358299
$147$	−14.8390	−1.22390
$148$	7.96720	0.654900
$149$	−11.8127	−0.967731	−0.483865	−	0.875142i	$-0.660768\pi$
$149$	−11.8127	−0.967731	−0.483865	+	0.875142i	$0.660768\pi$
$150$	−13.6003	−1.11046
$151$	2.57670	0.209689	0.104844	−	0.994489i	$-0.466566\pi$
$151$	2.57670	0.209689	0.104844	+	0.994489i	$0.466566\pi$
$152$	1.57007	0.127349
$153$	−24.6373	−1.99181
$154$	−12.5685	−1.01280
$155$	5.75497	0.462250
$156$	−0.942245	−0.0754400
$157$	10.1025	0.806269	0.403135	−	0.915141i	$-0.367921\pi$
$157$	10.1025	0.806269	0.403135	+	0.915141i	$0.367921\pi$
$158$	−2.30006	−0.182983
$159$	31.2843	2.48101
$160$	−0.529257	−0.0418414
$161$	19.6986	1.55247
$162$	−3.21234	−0.252385
$163$	21.9261	1.71739	0.858693	−	0.512491i	$-0.171277\pi$
$163$	21.9261	1.71739	0.858693	+	0.512491i	$0.171277\pi$
$164$	10.8596	0.847992
$165$	5.49899	0.428096
$166$	3.64111	0.282605
$167$	7.00039	0.541706	0.270853	−	0.962621i	$-0.412694\pi$
$167$	7.00039	0.541706	0.270853	+	0.962621i	$0.412694\pi$
$168$	−10.0438	−0.774899
$169$	−12.8931	−0.991775
$170$	2.45892	0.188590
$171$	−8.32598	−0.636703
$172$	−2.33391	−0.177959
$173$	−19.6386	−1.49309	−0.746547	−	0.665333i	$-0.768290\pi$
$173$	−19.6386	−1.49309	−0.746547	+	0.665333i	$0.768290\pi$
$174$	−0.160526	−0.0121694
$175$	16.4519	1.24365
$176$	−3.60579	−0.271797
$177$	−4.47625	−0.336456
$178$	−2.34510	−0.175773
$179$	−5.00403	−0.374019	−0.187010	−	0.982358i	$-0.559880\pi$
$179$	−5.00403	−0.374019	−0.187010	+	0.982358i	$0.559880\pi$
$180$	2.80662	0.209193
$181$	−19.6641	−1.46162	−0.730809	−	0.682582i	$-0.760857\pi$
$181$	−19.6641	−1.46162	−0.730809	+	0.682582i	$0.760857\pi$
$182$	1.13981	0.0844882
$183$	−34.3497	−2.53920
$184$	5.65135	0.416623
$185$	4.21669	0.310018
$186$	31.3323	2.29740
$187$	16.7524	1.22506
$188$	3.26957	0.238458
$189$	23.1304	1.68249
$190$	0.830970	0.0602849
$191$	12.8381	0.928929	0.464465	−	0.885592i	$-0.346247\pi$
$191$	12.8381	0.928929	0.464465	+	0.885592i	$0.346247\pi$
$192$	−2.88148	−0.207953
$193$	4.98082	0.358527	0.179264	−	0.983801i	$-0.442628\pi$
$193$	4.98082	0.358527	0.179264	+	0.983801i	$0.442628\pi$
$194$	−11.7782	−0.845626
$195$	−0.498690	−0.0357119
$196$	5.14977	0.367841
$197$	7.83707	0.558368	0.279184	−	0.960238i	$-0.409936\pi$
$197$	7.83707	0.558368	0.279184	+	0.960238i	$0.409936\pi$
$198$	19.1213	1.35889
$199$	11.2366	0.796544	0.398272	−	0.917267i	$-0.369610\pi$
$199$	11.2366	0.796544	0.398272	+	0.917267i	$0.369610\pi$
$200$	4.71989	0.333746
$201$	19.7593	1.39371
$202$	−16.3907	−1.15325
$203$	0.194183	0.0136290
$204$	13.3873	0.937299
$205$	5.74752	0.401424
$206$	4.62577	0.322292
$207$	−29.9688	−2.08297
$208$	0.327000	0.0226734
$209$	5.66134	0.391603
$210$	−5.31577	−0.366823
$211$	−26.0860	−1.79584	−0.897918	−	0.440163i	$-0.854921\pi$
$211$	−26.0860	−1.79584	−0.897918	+	0.440163i	$0.854921\pi$
$212$	−10.8570	−0.745663
$213$	27.2335	1.86601
$214$	−6.22292	−0.425390
$215$	−1.23524	−0.0842425
$216$	6.63588	0.451514
$217$	−37.9019	−2.57295
$218$	−17.1142	−1.15912
$219$	12.4749	0.842977
$220$	−1.90839	−0.128664
$221$	−1.51924	−0.102195
$222$	22.9573	1.54080
$223$	15.3589	1.02851	0.514253	−	0.857639i	$-0.328069\pi$
$223$	15.3589	1.02851	0.514253	+	0.857639i	$0.328069\pi$
$224$	3.48565	0.232895
$225$	−25.0293	−1.66862
$226$	6.18176	0.411205
$227$	11.2655	0.747720	0.373860	−	0.927485i	$-0.378034\pi$
$227$	11.2655	0.747720	0.373860	+	0.927485i	$0.378034\pi$
$228$	4.52412	0.299617
$229$	18.6218	1.23056	0.615282	−	0.788307i	$-0.289042\pi$
$229$	18.6218	1.23056	0.615282	+	0.788307i	$0.289042\pi$
$230$	2.99102	0.197222
$231$	−36.2160	−2.38284
$232$	0.0557094	0.00365750
$233$	26.3662	1.72731	0.863655	−	0.504083i	$-0.168170\pi$
$233$	26.3662	1.72731	0.863655	+	0.504083i	$0.168170\pi$
$234$	−1.73406	−0.113359
$235$	1.73044	0.112882
$236$	1.55345	0.101121
$237$	−6.62758	−0.430507
$238$	−16.1943	−1.04972
$239$	−4.74230	−0.306754	−0.153377	−	0.988168i	$-0.549015\pi$
$239$	−4.74230	−0.306754	−0.153377	+	0.988168i	$0.549015\pi$
$240$	−1.52504	−0.0984412
$241$	−0.602244	−0.0387939	−0.0193970	−	0.999812i	$-0.506175\pi$
$241$	−0.602244	−0.0387939	−0.0193970	+	0.999812i	$0.506175\pi$
$242$	−2.00173	−0.128676
$243$	10.6513	0.683283
$244$	11.9209	0.763154
$245$	2.72555	0.174129
$246$	31.2917	1.99509
$247$	−0.513413	−0.0326677
$248$	−10.8737	−0.690480
$249$	10.4918	0.664891
$250$	5.14432	0.325355
$251$	18.3509	1.15830	0.579151	−	0.815221i	$-0.303384\pi$
$251$	18.3509	1.15830	0.579151	+	0.815221i	$0.303384\pi$
$252$	−18.4842	−1.16440
$253$	20.3776	1.28113
$254$	−2.11956	−0.132993
$255$	7.08532	0.443700
$256$	1.00000	0.0625000
$257$	−3.51878	−0.219496	−0.109748	−	0.993959i	$-0.535004\pi$
$257$	−3.51878	−0.219496	−0.109748	+	0.993959i	$0.535004\pi$
$258$	−6.72512	−0.418688
$259$	−27.7709	−1.72560
$260$	0.173067	0.0107332
$261$	−0.295423	−0.0182862
$262$	11.6270	0.718319
$263$	−23.1779	−1.42921	−0.714604	−	0.699529i	$-0.753393\pi$
$263$	−23.1779	−1.42921	−0.714604	+	0.699529i	$0.753393\pi$
$264$	−10.3900	−0.639462
$265$	−5.74615	−0.352983
$266$	−5.47271	−0.335554
$267$	−6.75736	−0.413544
$268$	−6.85733	−0.418878
$269$	−0.0862323	−0.00525768	−0.00262884	−	0.999997i	$-0.500837\pi$
$269$	−0.0862323	−0.00525768	−0.00262884	+	0.999997i	$0.500837\pi$
$270$	3.51208	0.213739
$271$	−7.17299	−0.435728	−0.217864	−	0.975979i	$-0.569909\pi$
$271$	−7.17299	−0.435728	−0.217864	+	0.975979i	$0.569909\pi$
$272$	−4.64598	−0.281704
$273$	3.28434	0.198777
$274$	−17.7101	−1.06991
$275$	17.0189	1.02628
$276$	16.2843	0.980197
$277$	−3.84964	−0.231302	−0.115651	−	0.993290i	$-0.536895\pi$
$277$	−3.84964	−0.231302	−0.115651	+	0.993290i	$0.536895\pi$
$278$	−5.19767	−0.311736
$279$	57.6625	3.45216
$280$	1.84481	0.110248
$281$	−12.3515	−0.736827	−0.368414	−	0.929662i	$-0.620099\pi$
$281$	−12.3515	−0.736827	−0.368414	+	0.929662i	$0.620099\pi$
$282$	9.42121	0.561025
$283$	−16.8890	−1.00394	−0.501972	−	0.864884i	$-0.667392\pi$
$283$	−16.8890	−1.00394	−0.501972	+	0.864884i	$0.667392\pi$
$284$	−9.45122	−0.560827
$285$	2.39442	0.141833
$286$	1.17909	0.0697213
$287$	−37.8528	−2.23438
$288$	−5.30294	−0.312479
$289$	4.58511	0.269712
$290$	0.0294846	0.00173139
$291$	−33.9387	−1.98952
$292$	−4.32934	−0.253356
$293$	3.09466	0.180792	0.0903960	−	0.995906i	$-0.471187\pi$
$293$	3.09466	0.180792	0.0903960	+	0.995906i	$0.471187\pi$
$294$	14.8390	0.865426
$295$	0.822176	0.0478689
$296$	−7.96720	−0.463084
$297$	23.9276	1.38842
$298$	11.8127	0.684289
$299$	−1.84799	−0.106872
$300$	13.6003	0.785212
$301$	8.13520	0.468905
$302$	−2.57670	−0.148272
$303$	−47.2296	−2.71327
$304$	−1.57007	−0.0900496
$305$	6.30919	0.361263
$306$	24.6373	1.40842
$307$	−5.69351	−0.324946	−0.162473	−	0.986713i	$-0.551947\pi$
$307$	−5.69351	−0.324946	−0.162473	+	0.986713i	$0.551947\pi$
$308$	12.5685	0.716159
$309$	13.3291	0.758264
$310$	−5.75497	−0.326860
$311$	0.306373	0.0173728	0.00868640	−	0.999962i	$-0.497235\pi$
$311$	0.306373	0.0173728	0.00868640	+	0.999962i	$0.497235\pi$
$312$	0.942245	0.0533441
$313$	6.30858	0.356582	0.178291	−	0.983978i	$-0.442943\pi$
$313$	6.30858	0.356582	0.178291	+	0.983978i	$0.442943\pi$
$314$	−10.1025	−0.570118
$315$	−9.78289	−0.551203
$316$	2.30006	0.129388
$317$	−2.86247	−0.160772	−0.0803861	−	0.996764i	$-0.525615\pi$
$317$	−2.86247	−0.160772	−0.0803861	+	0.996764i	$0.525615\pi$
$318$	−31.2843	−1.75434
$319$	0.200876	0.0112469
$320$	0.529257	0.0295864
$321$	−17.9312	−1.00082
$322$	−19.6986	−1.09776
$323$	7.29450	0.405877
$324$	3.21234	0.178463
$325$	−1.54340	−0.0856126
$326$	−21.9261	−1.21437
$327$	−49.3144	−2.72709
$328$	−10.8596	−0.599621
$329$	−11.3966	−0.628314
$330$	−5.49899	−0.302709
$331$	−3.10822	−0.170843	−0.0854216	−	0.996345i	$-0.527224\pi$
$331$	−3.10822	−0.170843	−0.0854216	+	0.996345i	$0.527224\pi$
$332$	−3.64111	−0.199832
$333$	42.2496	2.31526
$334$	−7.00039	−0.383044
$335$	−3.62929	−0.198289
$336$	10.0438	0.547937
$337$	−0.691820	−0.0376858	−0.0188429	−	0.999822i	$-0.505998\pi$
$337$	−0.691820	−0.0376858	−0.0188429	+	0.999822i	$0.505998\pi$
$338$	12.8931	0.701291
$339$	17.8126	0.967450
$340$	−2.45892	−0.133353
$341$	−39.2082	−2.12325
$342$	8.32598	0.450217
$343$	6.44925	0.348227
$344$	2.33391	0.125836
$345$	8.61856	0.464008
$346$	19.6386	1.05578
$347$	12.8886	0.691899	0.345949	−	0.938253i	$-0.387557\pi$
$347$	12.8886	0.691899	0.345949	+	0.938253i	$0.387557\pi$
$348$	0.160526	0.00860507
$349$	27.6326	1.47914	0.739569	−	0.673081i	$-0.235029\pi$
$349$	27.6326	1.47914	0.739569	+	0.673081i	$0.235029\pi$
$350$	−16.4519	−0.879390
$351$	−2.16993	−0.115822
$352$	3.60579	0.192189
$353$	14.9512	0.795774	0.397887	−	0.917435i	$-0.369744\pi$
$353$	14.9512	0.795774	0.397887	+	0.917435i	$0.369744\pi$
$354$	4.47625	0.237910
$355$	−5.00212	−0.265485
$356$	2.34510	0.124290
$357$	−46.6635	−2.46969
$358$	5.00403	0.264471
$359$	3.45432	0.182312	0.0911561	−	0.995837i	$-0.470944\pi$
$359$	3.45432	0.182312	0.0911561	+	0.995837i	$0.470944\pi$
$360$	−2.80662	−0.147922
$361$	−16.5349	−0.870257
$362$	19.6641	1.03352
$363$	−5.76795	−0.302739
$364$	−1.13981	−0.0597422
$365$	−2.29133	−0.119934
$366$	34.3497	1.79549
$367$	15.0774	0.787033	0.393516	−	0.919318i	$-0.371258\pi$
$367$	15.0774	0.787033	0.393516	+	0.919318i	$0.371258\pi$
$368$	−5.65135	−0.294597
$369$	57.5878	2.99790
$370$	−4.21669	−0.219215
$371$	37.8438	1.96475
$372$	−31.3323	−1.62451
$373$	21.5316	1.11486	0.557432	−	0.830223i	$-0.311787\pi$
$373$	21.5316	1.11486	0.557432	+	0.830223i	$0.311787\pi$
$374$	−16.7524	−0.866247
$375$	14.8233	0.765470
$376$	−3.26957	−0.168615
$377$	−0.0182170	−0.000938222 0
$378$	−23.1304	−1.18970
$379$	−13.1690	−0.676445	−0.338223	−	0.941066i	$-0.609826\pi$
$379$	−13.1690	−0.676445	−0.338223	+	0.941066i	$0.609826\pi$
$380$	−0.830970	−0.0426278
$381$	−6.10748	−0.312896
$382$	−12.8381	−0.656852
$383$	−27.8840	−1.42481	−0.712403	−	0.701770i	$-0.752393\pi$
$383$	−27.8840	−1.42481	−0.712403	+	0.701770i	$0.752393\pi$
$384$	2.88148	0.147045
$385$	6.65198	0.339016
$386$	−4.98082	−0.253517
$387$	−12.3766	−0.629137
$388$	11.7782	0.597948
$389$	10.5612	0.535476	0.267738	−	0.963492i	$-0.413724\pi$
$389$	10.5612	0.535476	0.267738	+	0.963492i	$0.413724\pi$
$390$	0.498690	0.0252521
$391$	26.2560	1.32783
$392$	−5.14977	−0.260103
$393$	33.5030	1.69000
$394$	−7.83707	−0.394826
$395$	1.21732	0.0612501
$396$	−19.1213	−0.960881
$397$	−10.4931	−0.526635	−0.263318	−	0.964709i	$-0.584817\pi$
$397$	−10.4931	−0.526635	−0.263318	+	0.964709i	$0.584817\pi$
$398$	−11.2366	−0.563242
$399$	−15.7695	−0.789464
$400$	−4.71989	−0.235994
$401$	5.19810	0.259581	0.129790	−	0.991541i	$-0.458570\pi$
$401$	5.19810	0.259581	0.129790	+	0.991541i	$0.458570\pi$
$402$	−19.7593	−0.985502
$403$	3.55570	0.177122
$404$	16.3907	0.815469
$405$	1.70015	0.0844813
$406$	−0.194183	−0.00963716
$407$	−28.7281	−1.42400
$408$	−13.3873	−0.662770
$409$	11.3570	0.561568	0.280784	−	0.959771i	$-0.409406\pi$
$409$	11.3570	0.561568	0.280784	+	0.959771i	$0.409406\pi$
$410$	−5.74752	−0.283850
$411$	−51.0314	−2.51719
$412$	−4.62577	−0.227895
$413$	−5.41480	−0.266445
$414$	29.9688	1.47288
$415$	−1.92708	−0.0945968
$416$	−0.327000	−0.0160325
$417$	−14.9770	−0.733426
$418$	−5.66134	−0.276905
$419$	−32.8796	−1.60627	−0.803137	−	0.595795i	$-0.796837\pi$
$419$	−32.8796	−1.60627	−0.803137	+	0.595795i	$0.796837\pi$
$420$	5.31577	0.259383
$421$	14.9284	0.727566	0.363783	−	0.931484i	$-0.381485\pi$
$421$	14.9284	0.727566	0.363783	+	0.931484i	$0.381485\pi$
$422$	26.0860	1.26985
$423$	17.3383	0.843019
$424$	10.8570	0.527263
$425$	21.9285	1.06369
$426$	−27.2335	−1.31947
$427$	−41.5520	−2.01084
$428$	6.22292	0.300796
$429$	3.39754	0.164035
$430$	1.23524	0.0595685
$431$	22.2475	1.07163	0.535813	−	0.844337i	$-0.320005\pi$
$431$	22.2475	1.07163	0.535813	+	0.844337i	$0.320005\pi$
$432$	−6.63588	−0.319269
$433$	−23.6056	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$433$	−23.6056	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$434$	37.9019	1.81935
$435$	0.0849592	0.00407348
$436$	17.1142	0.819624
$437$	8.87301	0.424454
$438$	−12.4749	−0.596075
$439$	−23.9189	−1.14159	−0.570793	−	0.821094i	$-0.693364\pi$
$439$	−23.9189	−1.14159	−0.570793	+	0.821094i	$0.693364\pi$
$440$	1.90839	0.0909789
$441$	27.3089	1.30042
$442$	1.51924	0.0722627
$443$	31.9070	1.51595	0.757974	−	0.652285i	$-0.226189\pi$
$443$	31.9070	1.51595	0.757974	+	0.652285i	$0.226189\pi$
$444$	−22.9573	−1.08951
$445$	1.24116	0.0588366
$446$	−15.3589	−0.727263
$447$	34.0380	1.60994
$448$	−3.48565	−0.164682
$449$	−14.0302	−0.662128	−0.331064	−	0.943608i	$-0.607408\pi$
$449$	−14.0302	−0.662128	−0.331064	+	0.943608i	$0.607408\pi$
$450$	25.0293	1.17989
$451$	−39.1574	−1.84385
$452$	−6.18176	−0.290766
$453$	−7.42471	−0.348843
$454$	−11.2655	−0.528718
$455$	−0.603252	−0.0282809
$456$	−4.52412	−0.211862
$457$	−17.3962	−0.813761	−0.406881	−	0.913481i	$-0.633384\pi$
$457$	−17.3962	−0.813761	−0.406881	+	0.913481i	$0.633384\pi$
$458$	−18.6218	−0.870140
$459$	30.8301	1.43903
$460$	−2.99102	−0.139457
$461$	4.16782	0.194115	0.0970573	−	0.995279i	$-0.469057\pi$
$461$	4.16782	0.194115	0.0970573	+	0.995279i	$0.469057\pi$
$462$	36.2160	1.68492
$463$	−17.9818	−0.835687	−0.417844	−	0.908519i	$-0.637214\pi$
$463$	−17.9818	−0.835687	−0.417844	+	0.908519i	$0.637214\pi$
$464$	−0.0557094	−0.00258624
$465$	−16.5828	−0.769011
$466$	−26.3662	−1.22139
$467$	31.4672	1.45613	0.728063	−	0.685510i	$-0.240421\pi$
$467$	31.4672	1.45613	0.728063	+	0.685510i	$0.240421\pi$
$468$	1.73406	0.0801570
$469$	23.9023	1.10370
$470$	−1.73044	−0.0798194
$471$	−29.1102	−1.34133
$472$	−1.55345	−0.0715035
$473$	8.41559	0.386949
$474$	6.62758	0.304415
$475$	7.41055	0.340019
$476$	16.1943	0.742263
$477$	−57.5741	−2.63614
$478$	4.74230	0.216908
$479$	−12.7050	−0.580505	−0.290253	−	0.956950i	$-0.593739\pi$
$479$	−12.7050	−0.580505	−0.290253	+	0.956950i	$0.593739\pi$
$480$	1.52504	0.0696084
$481$	2.60527	0.118790
$482$	0.602244	0.0274314
$483$	−56.7613	−2.58273
$484$	2.00173	0.0909878
$485$	6.23369	0.283058
$486$	−10.6513	−0.483154
$487$	15.4411	0.699705	0.349852	−	0.936805i	$-0.386232\pi$
$487$	15.4411	0.699705	0.349852	+	0.936805i	$0.386232\pi$
$488$	−11.9209	−0.539632
$489$	−63.1797	−2.85708
$490$	−2.72555	−0.123128
$491$	−26.0356	−1.17497	−0.587486	−	0.809234i	$-0.699882\pi$
$491$	−26.0356	−1.17497	−0.587486	+	0.809234i	$0.699882\pi$
$492$	−31.2917	−1.41074
$493$	0.258824	0.0116569
$494$	0.513413	0.0230995
$495$	−10.1201	−0.454863
$496$	10.8737	0.488243
$497$	32.9437	1.47773
$498$	−10.4918	−0.470149
$499$	−33.3964	−1.49503	−0.747514	−	0.664246i	$-0.768753\pi$
$499$	−33.3964	−1.49503	−0.747514	+	0.664246i	$0.768753\pi$
$500$	−5.14432	−0.230061
$501$	−20.1715	−0.901196
$502$	−18.3509	−0.819043
$503$	−16.8450	−0.751083	−0.375541	−	0.926806i	$-0.622543\pi$
$503$	−16.8450	−0.751083	−0.375541	+	0.926806i	$0.622543\pi$
$504$	18.4842	0.823352
$505$	8.67491	0.386028
$506$	−20.3776	−0.905894
$507$	37.1512	1.64994
$508$	2.11956	0.0940404
$509$	4.55917	0.202082	0.101041	−	0.994882i	$-0.467783\pi$
$509$	4.55917	0.202082	0.101041	+	0.994882i	$0.467783\pi$
$510$	−7.08532	−0.313743
$511$	15.0906	0.667568
$512$	−1.00000	−0.0441942
$513$	10.4188	0.460000
$514$	3.51878	0.155207
$515$	−2.44822	−0.107881
$516$	6.72512	0.296057
$517$	−11.7894	−0.518497
$518$	27.7709	1.22018
$519$	56.5882	2.48395
$520$	−0.173067	−0.00758949
$521$	17.8409	0.781626	0.390813	−	0.920470i	$-0.372194\pi$
$521$	17.8409	0.781626	0.390813	+	0.920470i	$0.372194\pi$
$522$	0.295423	0.0129303
$523$	−2.38852	−0.104443	−0.0522213	−	0.998636i	$-0.516630\pi$
$523$	−2.38852	−0.104443	−0.0522213	+	0.998636i	$0.516630\pi$
$524$	−11.6270	−0.507928
$525$	−47.4058	−2.06896
$526$	23.1779	1.01060
$527$	−50.5189	−2.20064
$528$	10.3900	0.452168
$529$	8.93776	0.388598
$530$	5.74615	0.249597
$531$	8.23787	0.357493
$532$	5.47271	0.237272
$533$	3.55109	0.153815
$534$	6.75736	0.292420
$535$	3.29352	0.142391
$536$	6.85733	0.296191
$537$	14.4190	0.622227
$538$	0.0862323	0.00371774
$539$	−18.5690	−0.799823
$540$	−3.51208	−0.151136
$541$	−18.6159	−0.800361	−0.400181	−	0.916436i	$-0.631053\pi$
$541$	−18.6159	−0.800361	−0.400181	+	0.916436i	$0.631053\pi$
$542$	7.17299	0.308106
$543$	56.6616	2.43158
$544$	4.64598	0.199195
$545$	9.05783	0.387995
$546$	−3.28434	−0.140557
$547$	−31.9940	−1.36796	−0.683982	−	0.729499i	$-0.739753\pi$
$547$	−31.9940	−1.36796	−0.683982	+	0.729499i	$0.739753\pi$
$548$	17.7101	0.756539
$549$	63.2156	2.69797
$550$	−17.0189	−0.725690
$551$	0.0874675	0.00372624
$552$	−16.2843	−0.693104
$553$	−8.01720	−0.340926
$554$	3.84964	0.163555
$555$	−12.1503	−0.515753
$556$	5.19767	0.220430
$557$	4.93948	0.209292	0.104646	−	0.994510i	$-0.466629\pi$
$557$	4.93948	0.209292	0.104646	+	0.994510i	$0.466629\pi$
$558$	−57.6625	−2.44105
$559$	−0.763189	−0.0322794
$560$	−1.84481	−0.0779573
$561$	−48.2718	−2.03804
$562$	12.3515	0.521016
$563$	−22.0530	−0.929425	−0.464712	−	0.885462i	$-0.653842\pi$
$563$	−22.0530	−0.929425	−0.464712	+	0.885462i	$0.653842\pi$
$564$	−9.42121	−0.396705
$565$	−3.27174	−0.137643
$566$	16.8890	0.709896
$567$	−11.1971	−0.470234
$568$	9.45122	0.396564
$569$	−16.8993	−0.708456	−0.354228	−	0.935159i	$-0.615256\pi$
$569$	−16.8993	−0.708456	−0.354228	+	0.935159i	$0.615256\pi$
$570$	−2.39442	−0.100291
$571$	4.89339	0.204782	0.102391	−	0.994744i	$-0.467351\pi$
$571$	4.89339	0.204782	0.102391	+	0.994744i	$0.467351\pi$
$572$	−1.17909	−0.0493004
$573$	−36.9926	−1.54539
$574$	37.8528	1.57994
$575$	26.6737	1.11237
$576$	5.30294	0.220956
$577$	−7.19850	−0.299677	−0.149839	−	0.988710i	$-0.547875\pi$
$577$	−7.19850	−0.299677	−0.149839	+	0.988710i	$0.547875\pi$
$578$	−4.58511	−0.190715
$579$	−14.3521	−0.596455
$580$	−0.0294846	−0.00122428
$581$	12.6917	0.526538
$582$	33.9387	1.40680
$583$	39.1481	1.62135
$584$	4.32934	0.179150
$585$	0.917764	0.0379449
$586$	−3.09466	−0.127839
$587$	−20.2753	−0.836852	−0.418426	−	0.908251i	$-0.637418\pi$
$587$	−20.2753	−0.836852	−0.418426	+	0.908251i	$0.637418\pi$
$588$	−14.8390	−0.611949
$589$	−17.0724	−0.703457
$590$	−0.822176	−0.0338485
$591$	−22.5824	−0.928915
$592$	7.96720	0.327450
$593$	−0.643631	−0.0264307	−0.0132154	−	0.999913i	$-0.504207\pi$
$593$	−0.643631	−0.0264307	−0.0132154	+	0.999913i	$0.504207\pi$
$594$	−23.9276	−0.981761
$595$	8.57092	0.351374
$596$	−11.8127	−0.483865
$597$	−32.3782	−1.32515
$598$	1.84799	0.0755700
$599$	−11.1889	−0.457165	−0.228582	−	0.973525i	$-0.573409\pi$
$599$	−11.1889	−0.457165	−0.228582	+	0.973525i	$0.573409\pi$
$600$	−13.6003	−0.555229
$601$	−37.3874	−1.52506	−0.762532	−	0.646950i	$-0.776044\pi$
$601$	−37.3874	−1.52506	−0.762532	+	0.646950i	$0.776044\pi$
$602$	−8.13520	−0.331566
$603$	−36.3640	−1.48086
$604$	2.57670	0.104844
$605$	1.05943	0.0430720
$606$	47.2296	1.91857
$607$	29.8956	1.21342	0.606712	−	0.794922i	$-0.292488\pi$
$607$	29.8956	1.21342	0.606712	+	0.794922i	$0.292488\pi$
$608$	1.57007	0.0636747
$609$	−0.559536	−0.0226735
$610$	−6.30919	−0.255452
$611$	1.06915	0.0432532
$612$	−24.6373	−0.995905
$613$	−8.12772	−0.328276	−0.164138	−	0.986437i	$-0.552484\pi$
$613$	−8.12772	−0.328276	−0.164138	+	0.986437i	$0.552484\pi$
$614$	5.69351	0.229771
$615$	−16.5614	−0.667819
$616$	−12.5685	−0.506401
$617$	−16.3215	−0.657080	−0.328540	−	0.944490i	$-0.606557\pi$
$617$	−16.3215	−0.657080	−0.328540	+	0.944490i	$0.606557\pi$
$618$	−13.3291	−0.536174
$619$	−13.1770	−0.529626	−0.264813	−	0.964300i	$-0.585310\pi$
$619$	−13.1770	−0.529626	−0.264813	+	0.964300i	$0.585310\pi$
$620$	5.75497	0.231125
$621$	37.5017	1.50489
$622$	−0.306373	−0.0122844
$623$	−8.17420	−0.327493
$624$	−0.942245	−0.0377200
$625$	20.8768	0.835071
$626$	−6.30858	−0.252141
$627$	−16.3131	−0.651480
$628$	10.1025	0.403135
$629$	−37.0154	−1.47590
$630$	9.78289	0.389760
$631$	28.5679	1.13727	0.568635	−	0.822590i	$-0.307472\pi$
$631$	28.5679	1.13727	0.568635	+	0.822590i	$0.307472\pi$
$632$	−2.30006	−0.0914914
$633$	75.1664	2.98760
$634$	2.86247	0.113683
$635$	1.12179	0.0445170
$636$	31.2843	1.24050
$637$	1.68398	0.0667216
$638$	−0.200876	−0.00795277
$639$	−50.1192	−1.98269
$640$	−0.529257	−0.0209207
$641$	−26.1215	−1.03174	−0.515869	−	0.856667i	$-0.672531\pi$
$641$	−26.1215	−1.03174	−0.515869	+	0.856667i	$0.672531\pi$
$642$	17.9312	0.707689
$643$	−21.4575	−0.846202	−0.423101	−	0.906083i	$-0.639058\pi$
$643$	−21.4575	−0.846202	−0.423101	+	0.906083i	$0.639058\pi$
$644$	19.6986	0.776235
$645$	3.55932	0.140148
$646$	−7.29450	−0.286998
$647$	−9.53958	−0.375040	−0.187520	−	0.982261i	$-0.560045\pi$
$647$	−9.53958	−0.375040	−0.187520	+	0.982261i	$0.560045\pi$
$648$	−3.21234	−0.126193
$649$	−5.60143	−0.219875
$650$	1.54340	0.0605373
$651$	109.214	4.28042
$652$	21.9261	0.858693
$653$	−15.6138	−0.611015	−0.305508	−	0.952190i	$-0.598826\pi$
$653$	−15.6138	−0.611015	−0.305508	+	0.952190i	$0.598826\pi$
$654$	49.3144	1.92834
$655$	−6.15367	−0.240444
$656$	10.8596	0.423996
$657$	−22.9582	−0.895686
$658$	11.3966	0.444285
$659$	−31.5433	−1.22875	−0.614377	−	0.789013i	$-0.710592\pi$
$659$	−31.5433	−1.22875	−0.614377	+	0.789013i	$0.710592\pi$
$660$	5.49899	0.214048
$661$	10.8532	0.422140	0.211070	−	0.977471i	$-0.432305\pi$
$661$	10.8532	0.422140	0.211070	+	0.977471i	$0.432305\pi$
$662$	3.10822	0.120804
$663$	4.37765	0.170014
$664$	3.64111	0.141303
$665$	2.89647	0.112320
$666$	−42.2496	−1.63714
$667$	0.314833	0.0121904
$668$	7.00039	0.270853
$669$	−44.2563	−1.71105
$670$	3.62929	0.140212
$671$	−42.9841	−1.65938
$672$	−10.0438	−0.387450
$673$	−41.5889	−1.60313	−0.801567	−	0.597904i	$-0.796000\pi$
$673$	−41.5889	−1.60313	−0.801567	+	0.597904i	$0.796000\pi$
$674$	0.691820	0.0266479
$675$	31.3206	1.20553
$676$	−12.8931	−0.495887
$677$	34.5604	1.32826	0.664132	−	0.747615i	$-0.268801\pi$
$677$	34.5604	1.32826	0.664132	+	0.747615i	$0.268801\pi$
$678$	−17.8126	−0.684090
$679$	−41.0547	−1.57554
$680$	2.45892	0.0942951
$681$	−32.4614	−1.24393
$682$	39.2082	1.50136
$683$	−0.801180	−0.0306563	−0.0153281	−	0.999883i	$-0.504879\pi$
$683$	−0.801180	−0.0306563	−0.0153281	+	0.999883i	$0.504879\pi$
$684$	−8.32598	−0.318352
$685$	9.37320	0.358132
$686$	−6.44925	−0.246234
$687$	−53.6584	−2.04719
$688$	−2.33391	−0.0889795
$689$	−3.55025	−0.135254
$690$	−8.61856	−0.328103
$691$	1.38922	0.0528485	0.0264242	−	0.999651i	$-0.491588\pi$
$691$	1.38922	0.0528485	0.0264242	+	0.999651i	$0.491588\pi$
$692$	−19.6386	−0.746547
$693$	66.6502	2.53183
$694$	−12.8886	−0.489246
$695$	2.75090	0.104348
$696$	−0.160526	−0.00608470
$697$	−50.4534	−1.91106
$698$	−27.6326	−1.04591
$699$	−75.9739	−2.87360
$700$	16.4519	0.621823
$701$	40.6688	1.53604	0.768020	−	0.640426i	$-0.221242\pi$
$701$	40.6688	1.53604	0.768020	+	0.640426i	$0.221242\pi$
$702$	2.16993	0.0818988
$703$	−12.5090	−0.471788
$704$	−3.60579	−0.135898
$705$	−4.98624	−0.187793
$706$	−14.9512	−0.562697
$707$	−57.1324	−2.14868
$708$	−4.47625	−0.168228
$709$	10.0598	0.377803	0.188902	−	0.981996i	$-0.439507\pi$
$709$	10.0598	0.377803	0.188902	+	0.981996i	$0.439507\pi$
$710$	5.00212	0.187726
$711$	12.1971	0.457426
$712$	−2.34510	−0.0878864
$713$	−61.4510	−2.30136
$714$	46.6635	1.74634
$715$	−0.624044	−0.0233379
$716$	−5.00403	−0.187010
$717$	13.6649	0.510324
$718$	−3.45432	−0.128914
$719$	−12.1884	−0.454551	−0.227276	−	0.973830i	$-0.572982\pi$
$719$	−12.1884	−0.454551	−0.227276	+	0.973830i	$0.572982\pi$
$720$	2.80662	0.104596
$721$	16.1238	0.600482
$722$	16.5349	0.615365
$723$	1.73535	0.0645385
$724$	−19.6641	−0.730809
$725$	0.262942	0.00976542
$726$	5.76795	0.214069
$727$	−10.7223	−0.397669	−0.198835	−	0.980033i	$-0.563716\pi$
$727$	−10.7223	−0.397669	−0.198835	+	0.980033i	$0.563716\pi$
$728$	1.13981	0.0422441
$729$	−40.3286	−1.49365
$730$	2.29133	0.0848061
$731$	10.8433	0.401054
$732$	−34.3497	−1.26960
$733$	−11.8322	−0.437033	−0.218516	−	0.975833i	$-0.570122\pi$
$733$	−11.8322	−0.437033	−0.218516	+	0.975833i	$0.570122\pi$
$734$	−15.0774	−0.556516
$735$	−7.85363	−0.289685
$736$	5.65135	0.208312
$737$	24.7261	0.910797
$738$	−57.5878	−2.11984
$739$	14.9596	0.550299	0.275150	−	0.961401i	$-0.411273\pi$
$739$	14.9596	0.550299	0.275150	+	0.961401i	$0.411273\pi$
$740$	4.21669	0.155009
$741$	1.47939	0.0543467
$742$	−37.8438	−1.38929
$743$	−23.3658	−0.857208	−0.428604	−	0.903492i	$-0.640994\pi$
$743$	−23.3658	−0.857208	−0.428604	+	0.903492i	$0.640994\pi$
$744$	31.3323	1.14870
$745$	−6.25193	−0.229053
$746$	−21.5316	−0.788327
$747$	−19.3086	−0.706465
$748$	16.7524	0.612529
$749$	−21.6909	−0.792569
$750$	−14.8233	−0.541269
$751$	−2.10661	−0.0768712	−0.0384356	−	0.999261i	$-0.512237\pi$
$751$	−2.10661	−0.0768712	−0.0384356	+	0.999261i	$0.512237\pi$
$752$	3.26957	0.119229
$753$	−52.8779	−1.92698
$754$	0.0182170	0.000663423 0
$755$	1.36373	0.0496314
$756$	23.1304	0.841243
$757$	−15.1566	−0.550876	−0.275438	−	0.961319i	$-0.588823\pi$
$757$	−15.1566	−0.550876	−0.275438	+	0.961319i	$0.588823\pi$
$758$	13.1690	0.478319
$759$	−58.7177	−2.13132
$760$	0.830970	0.0301424
$761$	−34.8920	−1.26483	−0.632417	−	0.774628i	$-0.717937\pi$
$761$	−34.8920	−1.26483	−0.632417	+	0.774628i	$0.717937\pi$
$762$	6.10748	0.221251
$763$	−59.6543	−2.15963
$764$	12.8381	0.464465
$765$	−13.0395	−0.471443
$766$	27.8840	1.00749
$767$	0.507980	0.0183421
$768$	−2.88148	−0.103977
$769$	29.5631	1.06607	0.533037	−	0.846092i	$-0.321051\pi$
$769$	29.5631	1.06607	0.533037	+	0.846092i	$0.321051\pi$
$770$	−6.65198	−0.239721
$771$	10.1393	0.365158
$772$	4.98082	0.179264
$773$	17.8152	0.640768	0.320384	−	0.947288i	$-0.396188\pi$
$773$	17.8152	0.640768	0.320384	+	0.947288i	$0.396188\pi$
$774$	12.3766	0.444867
$775$	−51.3226	−1.84356
$776$	−11.7782	−0.422813
$777$	80.0213	2.87075
$778$	−10.5612	−0.378639
$779$	−17.0503	−0.610891
$780$	−0.498690	−0.0178560
$781$	34.0791	1.21945
$782$	−26.2560	−0.938914
$783$	0.369680	0.0132113
$784$	5.14977	0.183920
$785$	5.34683	0.190837
$786$	−33.5030	−1.19501
$787$	−2.42009	−0.0862668	−0.0431334	−	0.999069i	$-0.513734\pi$
$787$	−2.42009	−0.0862668	−0.0431334	+	0.999069i	$0.513734\pi$
$788$	7.83707	0.279184
$789$	66.7866	2.37767
$790$	−1.21732	−0.0433104
$791$	21.5475	0.766140
$792$	19.1213	0.679445
$793$	3.89812	0.138426
$794$	10.4931	0.372387
$795$	16.5574	0.587231
$796$	11.2366	0.398272
$797$	−45.2275	−1.60204	−0.801021	−	0.598637i	$-0.795709\pi$
$797$	−45.2275	−1.60204	−0.801021	+	0.598637i	$0.795709\pi$
$798$	15.7695	0.558235
$799$	−15.1904	−0.537396
$800$	4.71989	0.166873
$801$	12.4359	0.439402
$802$	−5.19810	−0.183551
$803$	15.6107	0.550890
$804$	19.7593	0.696855
$805$	10.4256	0.367456
$806$	−3.55570	−0.125244
$807$	0.248477	0.00874680
$808$	−16.3907	−0.576624
$809$	−39.3658	−1.38403	−0.692014	−	0.721884i	$-0.743276\pi$
$809$	−39.3658	−1.38403	−0.692014	+	0.721884i	$0.743276\pi$
$810$	−1.70015	−0.0597373
$811$	13.7830	0.483988	0.241994	−	0.970278i	$-0.422199\pi$
$811$	13.7830	0.483988	0.241994	+	0.970278i	$0.422199\pi$
$812$	0.194183	0.00681450
$813$	20.6688	0.724888
$814$	28.7281	1.00692
$815$	11.6045	0.406489
$816$	13.3873	0.468649
$817$	3.66440	0.128201
$818$	−11.3570	−0.397089
$819$	−6.04434	−0.211206
$820$	5.74752	0.200712
$821$	−15.7944	−0.551227	−0.275614	−	0.961269i	$-0.588881\pi$
$821$	−15.7944	−0.551227	−0.275614	+	0.961269i	$0.588881\pi$
$822$	51.0314	1.77992
$823$	11.4107	0.397751	0.198875	−	0.980025i	$-0.436271\pi$
$823$	11.4107	0.397751	0.198875	+	0.980025i	$0.436271\pi$
$824$	4.62577	0.161146
$825$	−49.0397	−1.70734
$826$	5.41480	0.188405
$827$	13.6740	0.475492	0.237746	−	0.971327i	$-0.423591\pi$
$827$	13.6740	0.475492	0.237746	+	0.971327i	$0.423591\pi$
$828$	−29.9688	−1.04149
$829$	20.0594	0.696693	0.348347	−	0.937366i	$-0.386743\pi$
$829$	20.0594	0.696693	0.348347	+	0.937366i	$0.386743\pi$
$830$	1.92708	0.0668901
$831$	11.0927	0.384800
$832$	0.327000	0.0113367
$833$	−23.9257	−0.828977
$834$	14.9770	0.518611
$835$	3.70500	0.128217
$836$	5.66134	0.195802
$837$	−72.1564	−2.49409
$838$	32.8796	1.13581
$839$	52.2904	1.80526	0.902632	−	0.430414i	$-0.141632\pi$
$839$	52.2904	1.80526	0.902632	+	0.430414i	$0.141632\pi$
$840$	−5.31577	−0.183412
$841$	−28.9969	−0.999893
$842$	−14.9284	−0.514467
$843$	35.5906	1.22580
$844$	−26.0860	−0.897918
$845$	−6.82375	−0.234744
$846$	−17.3383	−0.596104
$847$	−6.97734	−0.239744
$848$	−10.8570	−0.372831
$849$	48.6652	1.67019
$850$	−21.9285	−0.752141
$851$	−45.0254	−1.54345
$852$	27.2335	0.933005
$853$	52.9767	1.81389	0.906944	−	0.421252i	$-0.138409\pi$
$853$	52.9767	1.81389	0.906944	+	0.421252i	$0.138409\pi$
$854$	41.5520	1.42188
$855$	−4.40658	−0.150702
$856$	−6.22292	−0.212695
$857$	−14.3448	−0.490008	−0.245004	−	0.969522i	$-0.578789\pi$
$857$	−14.3448	−0.490008	−0.245004	+	0.969522i	$0.578789\pi$
$858$	−3.39754	−0.115990
$859$	−2.26370	−0.0772365	−0.0386183	−	0.999254i	$-0.512296\pi$
$859$	−2.26370	−0.0772365	−0.0386183	+	0.999254i	$0.512296\pi$
$860$	−1.23524	−0.0421213
$861$	109.072	3.71717
$862$	−22.2475	−0.757753
$863$	−9.70125	−0.330234	−0.165117	−	0.986274i	$-0.552800\pi$
$863$	−9.70125	−0.330234	−0.165117	+	0.986274i	$0.552800\pi$
$864$	6.63588	0.225757
$865$	−10.3938	−0.353401
$866$	23.6056	0.802152
$867$	−13.2119	−0.448700
$868$	−37.9019	−1.28647
$869$	−8.29353	−0.281339
$870$	−0.0849592	−0.00288039
$871$	−2.24235	−0.0759790
$872$	−17.1142	−0.579561
$873$	62.4591	2.11392
$874$	−8.87301	−0.300134
$875$	17.9313	0.606189
$876$	12.4749	0.421489
$877$	−9.96145	−0.336374	−0.168187	−	0.985755i	$-0.553791\pi$
$877$	−9.96145	−0.336374	−0.168187	+	0.985755i	$0.553791\pi$
$878$	23.9189	0.807224
$879$	−8.91721	−0.300770
$880$	−1.90839	−0.0643318
$881$	−21.5003	−0.724363	−0.362182	−	0.932108i	$-0.617968\pi$
$881$	−21.5003	−0.724363	−0.362182	+	0.932108i	$0.617968\pi$
$882$	−27.3089	−0.919539
$883$	28.1453	0.947166	0.473583	−	0.880749i	$-0.342960\pi$
$883$	28.1453	0.947166	0.473583	+	0.880749i	$0.342960\pi$
$884$	−1.51924	−0.0510974
$885$	−2.36909	−0.0796359
$886$	−31.9070	−1.07194
$887$	4.41708	0.148311	0.0741555	−	0.997247i	$-0.476374\pi$
$887$	4.41708	0.148311	0.0741555	+	0.997247i	$0.476374\pi$
$888$	22.9573	0.770398
$889$	−7.38806	−0.247787
$890$	−1.24116	−0.0416038
$891$	−11.5830	−0.388046
$892$	15.3589	0.514253
$893$	−5.13345	−0.171784
$894$	−34.0380	−1.13840
$895$	−2.64842	−0.0885269
$896$	3.48565	0.116447
$897$	5.32496	0.177795
$898$	14.0302	0.468195
$899$	−0.605766	−0.0202034
$900$	−25.0293	−0.834309
$901$	50.4415	1.68045
$902$	39.1574	1.30380
$903$	−23.4414	−0.780082
$904$	6.18176	0.205602
$905$	−10.4073	−0.345952
$906$	7.42471	0.246669
$907$	−15.8293	−0.525603	−0.262801	−	0.964850i	$-0.584646\pi$
$907$	−15.8293	−0.525603	−0.262801	+	0.964850i	$0.584646\pi$
$908$	11.2655	0.373860
$909$	86.9190	2.88292
$910$	0.603252	0.0199976
$911$	34.8621	1.15503	0.577517	−	0.816379i	$-0.304022\pi$
$911$	34.8621	1.15503	0.577517	+	0.816379i	$0.304022\pi$
$912$	4.52412	0.149809
$913$	13.1291	0.434510
$914$	17.3962	0.575416
$915$	−18.1798	−0.601007
$916$	18.6218	0.615282
$917$	40.5277	1.33834
$918$	−30.8301	−1.01755
$919$	38.3036	1.26352	0.631760	−	0.775164i	$-0.282333\pi$
$919$	38.3036	1.26352	0.631760	+	0.775164i	$0.282333\pi$
$920$	2.99102	0.0986109
$921$	16.4057	0.540588
$922$	−4.16782	−0.137260
$923$	−3.09055	−0.101727
$924$	−36.2160	−1.19142
$925$	−37.6043	−1.23642
$926$	17.9818	0.590920
$927$	−24.5302	−0.805676
$928$	0.0557094	0.00182875
$929$	−37.6528	−1.23535	−0.617675	−	0.786434i	$-0.711925\pi$
$929$	−37.6528	−1.23535	−0.617675	+	0.786434i	$0.711925\pi$
$930$	16.5828	0.543773
$931$	−8.08549	−0.264991
$932$	26.3662	0.863655
$933$	−0.882807	−0.0289018
$934$	−31.4672	−1.02964
$935$	8.86634	0.289960
$936$	−1.73406	−0.0566796
$937$	37.7831	1.23432	0.617161	−	0.786837i	$-0.288283\pi$
$937$	37.7831	1.23432	0.617161	+	0.786837i	$0.288283\pi$
$938$	−23.9023	−0.780436
$939$	−18.1781	−0.593218
$940$	1.73044	0.0564408
$941$	−41.8825	−1.36533	−0.682666	−	0.730731i	$-0.739179\pi$
$941$	−41.8825	−1.36533	−0.682666	+	0.730731i	$0.739179\pi$
$942$	29.1102	0.948463
$943$	−61.3714	−1.99853
$944$	1.55345	0.0505606
$945$	12.2419	0.398229
$946$	−8.41559	−0.273614
$947$	38.0514	1.23651	0.618253	−	0.785979i	$-0.287841\pi$
$947$	38.0514	1.23651	0.618253	+	0.785979i	$0.287841\pi$
$948$	−6.62758	−0.215254
$949$	−1.41570	−0.0459554
$950$	−7.41055	−0.240430
$951$	8.24815	0.267464
$952$	−16.1943	−0.524859
$953$	−30.6704	−0.993511	−0.496755	−	0.867891i	$-0.665475\pi$
$953$	−30.6704	−0.993511	−0.496755	+	0.867891i	$0.665475\pi$
$954$	57.5741	1.86403
$955$	6.79463	0.219869
$956$	−4.74230	−0.153377
$957$	−0.578821	−0.0187106
$958$	12.7050	0.410479
$959$	−61.7313	−1.99341
$960$	−1.52504	−0.0492206
$961$	87.2370	2.81410
$962$	−2.60527	−0.0839974
$963$	32.9997	1.06340
$964$	−0.602244	−0.0193970
$965$	2.63613	0.0848602
$966$	56.7613	1.82626
$967$	59.3082	1.90722	0.953611	−	0.301040i	$-0.0973339\pi$
$967$	59.3082	1.90722	0.953611	+	0.301040i	$0.0973339\pi$
$968$	−2.00173	−0.0643381
$969$	−21.0190	−0.675227
$970$	−6.23369	−0.200152
$971$	−2.25405	−0.0723361	−0.0361680	−	0.999346i	$-0.511515\pi$
$971$	−2.25405	−0.0723361	−0.0361680	+	0.999346i	$0.511515\pi$
$972$	10.6513	0.341641
$973$	−18.1173	−0.580813
$974$	−15.4411	−0.494766
$975$	4.44729	0.142427
$976$	11.9209	0.381577
$977$	−10.2829	−0.328979	−0.164490	−	0.986379i	$-0.552598\pi$
$977$	−10.2829	−0.328979	−0.164490	+	0.986379i	$0.552598\pi$
$978$	63.1797	2.02026
$979$	−8.45594	−0.270253
$980$	2.72555	0.0870646
$981$	90.7558	2.89761
$982$	26.0356	0.830830
$983$	−50.2243	−1.60191	−0.800953	−	0.598727i	$-0.795674\pi$
$983$	−50.2243	−1.60191	−0.800953	+	0.598727i	$0.795674\pi$
$984$	31.2917	0.997544
$985$	4.14783	0.132161
$986$	−0.258824	−0.00824265
$987$	32.8391	1.04528
$988$	−0.513413	−0.0163338
$989$	13.1897	0.419409
$990$	10.1201	0.321637
$991$	7.20541	0.228887	0.114444	−	0.993430i	$-0.463491\pi$
$991$	7.20541	0.228887	0.114444	+	0.993430i	$0.463491\pi$
$992$	−10.8737	−0.345240
$993$	8.95628	0.284219
$994$	−32.9437	−1.04491
$995$	5.94707	0.188535
$996$	10.4918	0.332445
$997$	−33.5664	−1.06306	−0.531529	−	0.847040i	$-0.678382\pi$
$997$	−33.5664	−1.06306	−0.531529	+	0.847040i	$0.678382\pi$
$998$	33.3964	1.05714
$999$	−52.8693	−1.67271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.9	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.9	✓	89	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.88148 q^{3} +1.00000 q^{4} +0.529257 q^{5} +2.88148 q^{6} -3.48565 q^{7} -1.00000 q^{8} +5.30294 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.88148 q^{3} +1.00000 q^{4} +0.529257 q^{5} +2.88148 q^{6} -3.48565 q^{7} -1.00000 q^{8} +5.30294 q^{9} -0.529257 q^{10} -3.60579 q^{11} -2.88148 q^{12} +0.327000 q^{13} +3.48565 q^{14} -1.52504 q^{15} +1.00000 q^{16} -4.64598 q^{17} -5.30294 q^{18} -1.57007 q^{19} +0.529257 q^{20} +10.0438 q^{21} +3.60579 q^{22} -5.65135 q^{23} +2.88148 q^{24} -4.71989 q^{25} -0.327000 q^{26} -6.63588 q^{27} -3.48565 q^{28} -0.0557094 q^{29} +1.52504 q^{30} +10.8737 q^{31} -1.00000 q^{32} +10.3900 q^{33} +4.64598 q^{34} -1.84481 q^{35} +5.30294 q^{36} +7.96720 q^{37} +1.57007 q^{38} -0.942245 q^{39} -0.529257 q^{40} +10.8596 q^{41} -10.0438 q^{42} -2.33391 q^{43} -3.60579 q^{44} +2.80662 q^{45} +5.65135 q^{46} +3.26957 q^{47} -2.88148 q^{48} +5.14977 q^{49} +4.71989 q^{50} +13.3873 q^{51} +0.327000 q^{52} -10.8570 q^{53} +6.63588 q^{54} -1.90839 q^{55} +3.48565 q^{56} +4.52412 q^{57} +0.0557094 q^{58} +1.55345 q^{59} -1.52504 q^{60} +11.9209 q^{61} -10.8737 q^{62} -18.4842 q^{63} +1.00000 q^{64} +0.173067 q^{65} -10.3900 q^{66} -6.85733 q^{67} -4.64598 q^{68} +16.2843 q^{69} +1.84481 q^{70} -9.45122 q^{71} -5.30294 q^{72} -4.32934 q^{73} -7.96720 q^{74} +13.6003 q^{75} -1.57007 q^{76} +12.5685 q^{77} +0.942245 q^{78} +2.30006 q^{79} +0.529257 q^{80} +3.21234 q^{81} -10.8596 q^{82} -3.64111 q^{83} +10.0438 q^{84} -2.45892 q^{85} +2.33391 q^{86} +0.160526 q^{87} +3.60579 q^{88} +2.34510 q^{89} -2.80662 q^{90} -1.13981 q^{91} -5.65135 q^{92} -31.3323 q^{93} -3.26957 q^{94} -0.830970 q^{95} +2.88148 q^{96} +11.7782 q^{97} -5.14977 q^{98} -19.1213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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