LMFDB - Embedded newform 8006.2.a.b.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, 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("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.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:	$63.9282318582$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.19035 q^{3} +1.00000 q^{4} +2.34635 q^{5} +2.19035 q^{6} -2.32589 q^{7} -1.00000 q^{8} +1.79762 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.19035 q^{3} +1.00000 q^{4} +2.34635 q^{5} +2.19035 q^{6} -2.32589 q^{7} -1.00000 q^{8} +1.79762 q^{9} -2.34635 q^{10} -1.82750 q^{11} -2.19035 q^{12} -2.80304 q^{13} +2.32589 q^{14} -5.13933 q^{15} +1.00000 q^{16} -6.34433 q^{17} -1.79762 q^{18} -2.41329 q^{19} +2.34635 q^{20} +5.09452 q^{21} +1.82750 q^{22} +5.57013 q^{23} +2.19035 q^{24} +0.505368 q^{25} +2.80304 q^{26} +2.63363 q^{27} -2.32589 q^{28} -3.05829 q^{29} +5.13933 q^{30} +0.969783 q^{31} -1.00000 q^{32} +4.00285 q^{33} +6.34433 q^{34} -5.45737 q^{35} +1.79762 q^{36} +10.2515 q^{37} +2.41329 q^{38} +6.13962 q^{39} -2.34635 q^{40} +1.93072 q^{41} -5.09452 q^{42} +5.74482 q^{43} -1.82750 q^{44} +4.21785 q^{45} -5.57013 q^{46} -2.61173 q^{47} -2.19035 q^{48} -1.59022 q^{49} -0.505368 q^{50} +13.8963 q^{51} -2.80304 q^{52} +13.9140 q^{53} -2.63363 q^{54} -4.28795 q^{55} +2.32589 q^{56} +5.28594 q^{57} +3.05829 q^{58} +10.3038 q^{59} -5.13933 q^{60} +9.16025 q^{61} -0.969783 q^{62} -4.18108 q^{63} +1.00000 q^{64} -6.57691 q^{65} -4.00285 q^{66} +8.23548 q^{67} -6.34433 q^{68} -12.2005 q^{69} +5.45737 q^{70} +5.77474 q^{71} -1.79762 q^{72} -1.11057 q^{73} -10.2515 q^{74} -1.10693 q^{75} -2.41329 q^{76} +4.25056 q^{77} -6.13962 q^{78} -10.2828 q^{79} +2.34635 q^{80} -11.1614 q^{81} -1.93072 q^{82} +7.02829 q^{83} +5.09452 q^{84} -14.8860 q^{85} -5.74482 q^{86} +6.69872 q^{87} +1.82750 q^{88} +3.90563 q^{89} -4.21785 q^{90} +6.51957 q^{91} +5.57013 q^{92} -2.12416 q^{93} +2.61173 q^{94} -5.66242 q^{95} +2.19035 q^{96} -10.1161 q^{97} +1.59022 q^{98} -3.28514 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9}+O(q^{10}) $ 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9	\( 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9} + 9 q^{10} - 5 q^{11} + q^{12} - 35 q^{13} + 8 q^{14} - 21 q^{15} + 75 q^{16} + 4 q^{17} - 66 q^{18} - 59 q^{19} - 9 q^{20} - 62 q^{21} + 5 q^{22} + 43 q^{23} - q^{24} + 44 q^{25} + 35 q^{26} + 4 q^{27} - 8 q^{28} - 38 q^{29} + 21 q^{30} - 51 q^{31} - 75 q^{32} - 19 q^{33} - 4 q^{34} + 14 q^{35} + 66 q^{36} - 63 q^{37} + 59 q^{38} - 34 q^{39} + 9 q^{40} - 27 q^{41} + 62 q^{42} - 39 q^{43} - 5 q^{44} - 52 q^{45} - 43 q^{46} + 40 q^{47} + q^{48} + 29 q^{49} - 44 q^{50} - 34 q^{51} - 35 q^{52} - 39 q^{53} - 4 q^{54} - 48 q^{55} + 8 q^{56} - 28 q^{57} + 38 q^{58} + 5 q^{59} - 21 q^{60} - 98 q^{61} + 51 q^{62} + 2 q^{63} + 75 q^{64} - q^{65} + 19 q^{66} - 59 q^{67} + 4 q^{68} - 69 q^{69} - 14 q^{70} - 9 q^{71} - 66 q^{72} - 51 q^{73} + 63 q^{74} - q^{75} - 59 q^{76} - 25 q^{77} + 34 q^{78} - 139 q^{79} - 9 q^{80} + 23 q^{81} + 27 q^{82} + 31 q^{83} - 62 q^{84} - 149 q^{85} + 39 q^{86} + q^{87} + 5 q^{88} - 39 q^{89} + 52 q^{90} - 93 q^{91} + 43 q^{92} - 83 q^{93} - 40 q^{94} + 2 q^{95} - q^{96} - 70 q^{97} - 29 q^{98} - 61 q^{99}+O(q^{100}) \) 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9 + 9 * q^10 - 5 * q^11 + q^12 - 35 * q^13 + 8 * q^14 - 21 * q^15 + 75 * q^16 + 4 * q^17 - 66 * q^18 - 59 * q^19 - 9 * q^20 - 62 * q^21 + 5 * q^22 + 43 * q^23 - q^24 + 44 * q^25 + 35 * q^26 + 4 * q^27 - 8 * q^28 - 38 * q^29 + 21 * q^30 - 51 * q^31 - 75 * q^32 - 19 * q^33 - 4 * q^34 + 14 * q^35 + 66 * q^36 - 63 * q^37 + 59 * q^38 - 34 * q^39 + 9 * q^40 - 27 * q^41 + 62 * q^42 - 39 * q^43 - 5 * q^44 - 52 * q^45 - 43 * q^46 + 40 * q^47 + q^48 + 29 * q^49 - 44 * q^50 - 34 * q^51 - 35 * q^52 - 39 * q^53 - 4 * q^54 - 48 * q^55 + 8 * q^56 - 28 * q^57 + 38 * q^58 + 5 * q^59 - 21 * q^60 - 98 * q^61 + 51 * q^62 + 2 * q^63 + 75 * q^64 - q^65 + 19 * q^66 - 59 * q^67 + 4 * q^68 - 69 * q^69 - 14 * q^70 - 9 * q^71 - 66 * q^72 - 51 * q^73 + 63 * q^74 - q^75 - 59 * q^76 - 25 * q^77 + 34 * q^78 - 139 * q^79 - 9 * q^80 + 23 * q^81 + 27 * q^82 + 31 * q^83 - 62 * q^84 - 149 * q^85 + 39 * q^86 + q^87 + 5 * q^88 - 39 * q^89 + 52 * q^90 - 93 * q^91 + 43 * q^92 - 83 * q^93 - 40 * q^94 + 2 * q^95 - q^96 - 70 * q^97 - 29 * q^98 - 61 * 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
$3$	−2.19035	−1.26460	−0.632299	−	0.774725i	$-0.717888\pi$
$3$	−2.19035	−1.26460	−0.632299	+	0.774725i	$0.717888\pi$
$4$	1.00000	0.500000
$5$	2.34635	1.04932	0.524660	−	0.851312i	$-0.324192\pi$
$5$	2.34635	1.04932	0.524660	+	0.851312i	$0.324192\pi$
$6$	2.19035	0.894205
$7$	−2.32589	−0.879105	−0.439553	−	0.898217i	$-0.644863\pi$
$7$	−2.32589	−0.879105	−0.439553	+	0.898217i	$0.644863\pi$
$8$	−1.00000	−0.353553
$9$	1.79762	0.599207
$10$	−2.34635	−0.741982
$11$	−1.82750	−0.551011	−0.275505	−	0.961300i	$-0.588845\pi$
$11$	−1.82750	−0.551011	−0.275505	+	0.961300i	$0.588845\pi$
$12$	−2.19035	−0.632299
$13$	−2.80304	−0.777422	−0.388711	−	0.921360i	$-0.627080\pi$
$13$	−2.80304	−0.777422	−0.388711	+	0.921360i	$0.627080\pi$
$14$	2.32589	0.621621
$15$	−5.13933	−1.32697
$16$	1.00000	0.250000
$17$	−6.34433	−1.53873	−0.769363	−	0.638812i	$-0.779426\pi$
$17$	−6.34433	−1.53873	−0.769363	+	0.638812i	$0.779426\pi$
$18$	−1.79762	−0.423703
$19$	−2.41329	−0.553646	−0.276823	−	0.960921i	$-0.589282\pi$
$19$	−2.41329	−0.553646	−0.276823	+	0.960921i	$0.589282\pi$
$20$	2.34635	0.524660
$21$	5.09452	1.11171
$22$	1.82750	0.389623
$23$	5.57013	1.16145	0.580726	−	0.814099i	$-0.302769\pi$
$23$	5.57013	1.16145	0.580726	+	0.814099i	$0.302769\pi$
$24$	2.19035	0.447103
$25$	0.505368	0.101074
$26$	2.80304	0.549721
$27$	2.63363	0.506842
$28$	−2.32589	−0.439553
$29$	−3.05829	−0.567910	−0.283955	−	0.958838i	$-0.591647\pi$
$29$	−3.05829	−0.567910	−0.283955	+	0.958838i	$0.591647\pi$
$30$	5.13933	0.938308
$31$	0.969783	0.174178	0.0870891	−	0.996201i	$-0.472244\pi$
$31$	0.969783	0.174178	0.0870891	+	0.996201i	$0.472244\pi$
$32$	−1.00000	−0.176777
$33$	4.00285	0.696807
$34$	6.34433	1.08804
$35$	−5.45737	−0.922463
$36$	1.79762	0.299603
$37$	10.2515	1.68534	0.842672	−	0.538428i	$-0.180982\pi$
$37$	10.2515	1.68534	0.842672	+	0.538428i	$0.180982\pi$
$38$	2.41329	0.391487
$39$	6.13962	0.983127
$40$	−2.34635	−0.370991
$41$	1.93072	0.301528	0.150764	−	0.988570i	$-0.451827\pi$
$41$	1.93072	0.301528	0.150764	+	0.988570i	$0.451827\pi$
$42$	−5.09452	−0.786101
$43$	5.74482	0.876078	0.438039	−	0.898956i	$-0.355673\pi$
$43$	5.74482	0.876078	0.438039	+	0.898956i	$0.355673\pi$
$44$	−1.82750	−0.275505
$45$	4.21785	0.628760
$46$	−5.57013	−0.821270
$47$	−2.61173	−0.380960	−0.190480	−	0.981691i	$-0.561005\pi$
$47$	−2.61173	−0.380960	−0.190480	+	0.981691i	$0.561005\pi$
$48$	−2.19035	−0.316149
$49$	−1.59022	−0.227174
$50$	−0.505368	−0.0714699
$51$	13.8963	1.94587
$52$	−2.80304	−0.388711
$53$	13.9140	1.91123	0.955615	−	0.294617i	$-0.0951921\pi$
$53$	13.9140	1.91123	0.955615	+	0.294617i	$0.0951921\pi$
$54$	−2.63363	−0.358391
$55$	−4.28795	−0.578187
$56$	2.32589	0.310811
$57$	5.28594	0.700139
$58$	3.05829	0.401573
$59$	10.3038	1.34144	0.670722	−	0.741709i	$-0.265984\pi$
$59$	10.3038	1.34144	0.670722	+	0.741709i	$0.265984\pi$
$60$	−5.13933	−0.663484
$61$	9.16025	1.17285	0.586425	−	0.810003i	$-0.300535\pi$
$61$	9.16025	1.17285	0.586425	+	0.810003i	$0.300535\pi$
$62$	−0.969783	−0.123163
$63$	−4.18108	−0.526766
$64$	1.00000	0.125000
$65$	−6.57691	−0.815765
$66$	−4.00285	−0.492717
$67$	8.23548	1.00612	0.503062	−	0.864250i	$-0.332207\pi$
$67$	8.23548	1.00612	0.503062	+	0.864250i	$0.332207\pi$
$68$	−6.34433	−0.769363
$69$	−12.2005	−1.46877
$70$	5.45737	0.652280
$71$	5.77474	0.685336	0.342668	−	0.939457i	$-0.388669\pi$
$71$	5.77474	0.685336	0.342668	+	0.939457i	$0.388669\pi$
$72$	−1.79762	−0.211852
$73$	−1.11057	−0.129982	−0.0649912	−	0.997886i	$-0.520702\pi$
$73$	−1.11057	−0.129982	−0.0649912	+	0.997886i	$0.520702\pi$
$74$	−10.2515	−1.19172
$75$	−1.10693	−0.127817
$76$	−2.41329	−0.276823
$77$	4.25056	0.484397
$78$	−6.13962	−0.695175
$79$	−10.2828	−1.15690	−0.578450	−	0.815718i	$-0.696342\pi$
$79$	−10.2828	−1.15690	−0.578450	+	0.815718i	$0.696342\pi$
$80$	2.34635	0.262330
$81$	−11.1614	−1.24016
$82$	−1.93072	−0.213213
$83$	7.02829	0.771455	0.385728	−	0.922613i	$-0.373950\pi$
$83$	7.02829	0.771455	0.385728	+	0.922613i	$0.373950\pi$
$84$	5.09452	0.555857
$85$	−14.8860	−1.61462
$86$	−5.74482	−0.619480
$87$	6.69872	0.718178
$88$	1.82750	0.194812
$89$	3.90563	0.413996	0.206998	−	0.978341i	$-0.433631\pi$
$89$	3.90563	0.413996	0.206998	+	0.978341i	$0.433631\pi$
$90$	−4.21785	−0.444601
$91$	6.51957	0.683436
$92$	5.57013	0.580726
$93$	−2.12416	−0.220265
$94$	2.61173	0.269380
$95$	−5.66242	−0.580952
$96$	2.19035	0.223551
$97$	−10.1161	−1.02713	−0.513565	−	0.858051i	$-0.671675\pi$
$97$	−10.1161	−1.02713	−0.513565	+	0.858051i	$0.671675\pi$
$98$	1.59022	0.160636
$99$	−3.28514	−0.330169
$100$	0.505368	0.0505368
$101$	−10.1441	−1.00937	−0.504686	−	0.863303i	$-0.668392\pi$
$101$	−10.1441	−1.00937	−0.504686	+	0.863303i	$0.668392\pi$
$102$	−13.8963	−1.37594
$103$	7.57681	0.746566	0.373283	−	0.927718i	$-0.378232\pi$
$103$	7.57681	0.746566	0.373283	+	0.927718i	$0.378232\pi$
$104$	2.80304	0.274860
$105$	11.9535	1.16654
$106$	−13.9140	−1.35144
$107$	3.56608	0.344746	0.172373	−	0.985032i	$-0.444857\pi$
$107$	3.56608	0.344746	0.172373	+	0.985032i	$0.444857\pi$
$108$	2.63363	0.253421
$109$	−5.34794	−0.512240	−0.256120	−	0.966645i	$-0.582444\pi$
$109$	−5.34794	−0.512240	−0.256120	+	0.966645i	$0.582444\pi$
$110$	4.28795	0.408840
$111$	−22.4544	−2.13128
$112$	−2.32589	−0.219776
$113$	−16.9274	−1.59239	−0.796197	−	0.605038i	$-0.793158\pi$
$113$	−16.9274	−1.59239	−0.796197	+	0.605038i	$0.793158\pi$
$114$	−5.28594	−0.495073
$115$	13.0695	1.21874
$116$	−3.05829	−0.283955
$117$	−5.03880	−0.465837
$118$	−10.3038	−0.948544
$119$	14.7562	1.35270
$120$	5.13933	0.469154
$121$	−7.66026	−0.696387
$122$	−9.16025	−0.829330
$123$	−4.22895	−0.381312
$124$	0.969783	0.0870891
$125$	−10.5460	−0.943262
$126$	4.18108	0.372480
$127$	13.2440	1.17521	0.587606	−	0.809147i	$-0.300070\pi$
$127$	13.2440	1.17521	0.587606	+	0.809147i	$0.300070\pi$
$128$	−1.00000	−0.0883883
$129$	−12.5832	−1.10789
$130$	6.57691	0.576833
$131$	−11.8650	−1.03665	−0.518327	−	0.855183i	$-0.673445\pi$
$131$	−11.8650	−1.03665	−0.518327	+	0.855183i	$0.673445\pi$
$132$	4.00285	0.348403
$133$	5.61305	0.486713
$134$	−8.23548	−0.711437
$135$	6.17942	0.531840
$136$	6.34433	0.544022
$137$	0.343159	0.0293181	0.0146590	−	0.999893i	$-0.495334\pi$
$137$	0.343159	0.0293181	0.0146590	+	0.999893i	$0.495334\pi$
$138$	12.2005	1.03858
$139$	20.6080	1.74794	0.873972	−	0.485976i	$-0.161536\pi$
$139$	20.6080	1.74794	0.873972	+	0.485976i	$0.161536\pi$
$140$	−5.45737	−0.461232
$141$	5.72060	0.481762
$142$	−5.77474	−0.484606
$143$	5.12254	0.428368
$144$	1.79762	0.149802
$145$	−7.17583	−0.595920
$146$	1.11057	0.0919115
$147$	3.48312	0.287283
$148$	10.2515	0.842672
$149$	0.0573712	0.00470003	0.00235002	−	0.999997i	$-0.499252\pi$
$149$	0.0573712	0.00470003	0.00235002	+	0.999997i	$0.499252\pi$
$150$	1.10693	0.0903806
$151$	−2.69000	−0.218909	−0.109454	−	0.993992i	$-0.534910\pi$
$151$	−2.69000	−0.218909	−0.109454	+	0.993992i	$0.534910\pi$
$152$	2.41329	0.195743
$153$	−11.4047	−0.922015
$154$	−4.25056	−0.342520
$155$	2.27545	0.182769
$156$	6.13962	0.491563
$157$	−14.2066	−1.13381	−0.566906	−	0.823783i	$-0.691860\pi$
$157$	−14.2066	−1.13381	−0.566906	+	0.823783i	$0.691860\pi$
$158$	10.2828	0.818052
$159$	−30.4764	−2.41694
$160$	−2.34635	−0.185495
$161$	−12.9555	−1.02104
$162$	11.1614	0.876924
$163$	−10.1200	−0.792659	−0.396329	−	0.918108i	$-0.629716\pi$
$163$	−10.1200	−0.792659	−0.396329	+	0.918108i	$0.629716\pi$
$164$	1.93072	0.150764
$165$	9.39210	0.731174
$166$	−7.02829	−0.545501
$167$	−1.74484	−0.135020	−0.0675099	−	0.997719i	$-0.521505\pi$
$167$	−1.74484	−0.135020	−0.0675099	+	0.997719i	$0.521505\pi$
$168$	−5.09452	−0.393050
$169$	−5.14299	−0.395614
$170$	14.8860	1.14171
$171$	−4.33817	−0.331748
$172$	5.74482	0.438039
$173$	−6.82361	−0.518789	−0.259395	−	0.965771i	$-0.583523\pi$
$173$	−6.82361	−0.518789	−0.259395	+	0.965771i	$0.583523\pi$
$174$	−6.69872	−0.507828
$175$	−1.17543	−0.0888544
$176$	−1.82750	−0.137753
$177$	−22.5690	−1.69639
$178$	−3.90563	−0.292740
$179$	−1.86074	−0.139079	−0.0695393	−	0.997579i	$-0.522153\pi$
$179$	−1.86074	−0.139079	−0.0695393	+	0.997579i	$0.522153\pi$
$180$	4.21785	0.314380
$181$	−0.749616	−0.0557185	−0.0278593	−	0.999612i	$-0.508869\pi$
$181$	−0.749616	−0.0557185	−0.0278593	+	0.999612i	$0.508869\pi$
$182$	−6.51957	−0.483262
$183$	−20.0641	−1.48318
$184$	−5.57013	−0.410635
$185$	24.0537	1.76847
$186$	2.12416	0.155751
$187$	11.5942	0.847855
$188$	−2.61173	−0.190480
$189$	−6.12554	−0.445568
$190$	5.66242	0.410795
$191$	−24.0268	−1.73852	−0.869260	−	0.494355i	$-0.835404\pi$
$191$	−24.0268	−1.73852	−0.869260	+	0.494355i	$0.835404\pi$
$192$	−2.19035	−0.158075
$193$	−22.1629	−1.59532	−0.797661	−	0.603107i	$-0.793929\pi$
$193$	−22.1629	−1.59532	−0.797661	+	0.603107i	$0.793929\pi$
$194$	10.1161	0.726290
$195$	14.4057	1.03161
$196$	−1.59022	−0.113587
$197$	4.38409	0.312353	0.156177	−	0.987729i	$-0.450083\pi$
$197$	4.38409	0.312353	0.156177	+	0.987729i	$0.450083\pi$
$198$	3.28514	0.233465
$199$	18.7385	1.32834	0.664168	−	0.747583i	$-0.268786\pi$
$199$	18.7385	1.32834	0.664168	+	0.747583i	$0.268786\pi$
$200$	−0.505368	−0.0357349
$201$	−18.0386	−1.27234
$202$	10.1441	0.713734
$203$	7.11326	0.499253
$204$	13.8963	0.972935
$205$	4.53015	0.316400
$206$	−7.57681	−0.527902
$207$	10.0130	0.695950
$208$	−2.80304	−0.194356
$209$	4.41027	0.305065
$210$	−11.9535	−0.824872
$211$	15.0275	1.03453	0.517267	−	0.855824i	$-0.326950\pi$
$211$	15.0275	1.03453	0.517267	+	0.855824i	$0.326950\pi$
$212$	13.9140	0.955615
$213$	−12.6487	−0.866674
$214$	−3.56608	−0.243772
$215$	13.4794	0.919286
$216$	−2.63363	−0.179196
$217$	−2.25561	−0.153121
$218$	5.34794	0.362208
$219$	2.43253	0.164375
$220$	−4.28795	−0.289093
$221$	17.7834	1.19624
$222$	22.4544	1.50704
$223$	−18.2612	−1.22286	−0.611430	−	0.791298i	$-0.709406\pi$
$223$	−18.2612	−1.22286	−0.611430	+	0.791298i	$0.709406\pi$
$224$	2.32589	0.155405
$225$	0.908460	0.0605640
$226$	16.9274	1.12599
$227$	12.2437	0.812645	0.406323	−	0.913730i	$-0.366811\pi$
$227$	12.2437	0.812645	0.406323	+	0.913730i	$0.366811\pi$
$228$	5.28594	0.350070
$229$	7.45508	0.492646	0.246323	−	0.969188i	$-0.420778\pi$
$229$	7.45508	0.492646	0.246323	+	0.969188i	$0.420778\pi$
$230$	−13.0695	−0.861776
$231$	−9.31021	−0.612567
$232$	3.05829	0.200787
$233$	−23.8002	−1.55920	−0.779601	−	0.626276i	$-0.784578\pi$
$233$	−23.8002	−1.55920	−0.779601	+	0.626276i	$0.784578\pi$
$234$	5.03880	0.329396
$235$	−6.12805	−0.399750
$236$	10.3038	0.670722
$237$	22.5228	1.46301
$238$	−14.7562	−0.956505
$239$	−25.6894	−1.66171	−0.830854	−	0.556490i	$-0.812148\pi$
$239$	−25.6894	−1.66171	−0.830854	+	0.556490i	$0.812148\pi$
$240$	−5.13933	−0.331742
$241$	−5.33928	−0.343933	−0.171967	−	0.985103i	$-0.555012\pi$
$241$	−5.33928	−0.343933	−0.171967	+	0.985103i	$0.555012\pi$
$242$	7.66026	0.492420
$243$	16.5465	1.06146
$244$	9.16025	0.586425
$245$	−3.73121	−0.238378
$246$	4.22895	0.269628
$247$	6.76453	0.430417
$248$	−0.969783	−0.0615813
$249$	−15.3944	−0.975580
$250$	10.5460	0.666987
$251$	−12.4706	−0.787135	−0.393567	−	0.919296i	$-0.628759\pi$
$251$	−12.4706	−0.787135	−0.393567	+	0.919296i	$0.628759\pi$
$252$	−4.18108	−0.263383
$253$	−10.1794	−0.639972
$254$	−13.2440	−0.831001
$255$	32.6056	2.04184
$256$	1.00000	0.0625000
$257$	9.79743	0.611147	0.305574	−	0.952168i	$-0.401152\pi$
$257$	9.79743	0.611147	0.305574	+	0.952168i	$0.401152\pi$
$258$	12.5832	0.783393
$259$	−23.8440	−1.48159
$260$	−6.57691	−0.407883
$261$	−5.49765	−0.340296
$262$	11.8650	0.733025
$263$	15.6100	0.962556	0.481278	−	0.876568i	$-0.340173\pi$
$263$	15.6100	0.962556	0.481278	+	0.876568i	$0.340173\pi$
$264$	−4.00285	−0.246358
$265$	32.6471	2.00549
$266$	−5.61305	−0.344158
$267$	−8.55469	−0.523539
$268$	8.23548	0.503062
$269$	25.9714	1.58350	0.791751	−	0.610843i	$-0.209169\pi$
$269$	25.9714	1.58350	0.791751	+	0.610843i	$0.209169\pi$
$270$	−6.17942	−0.376067
$271$	−2.75902	−0.167599	−0.0837994	−	0.996483i	$-0.526705\pi$
$271$	−2.75902	−0.167599	−0.0837994	+	0.996483i	$0.526705\pi$
$272$	−6.34433	−0.384682
$273$	−14.2801	−0.864272
$274$	−0.343159	−0.0207310
$275$	−0.923558	−0.0556927
$276$	−12.2005	−0.734384
$277$	7.15201	0.429722	0.214861	−	0.976645i	$-0.431070\pi$
$277$	7.15201	0.429722	0.214861	+	0.976645i	$0.431070\pi$
$278$	−20.6080	−1.23598
$279$	1.74330	0.104369
$280$	5.45737	0.326140
$281$	22.7085	1.35467	0.677337	−	0.735673i	$-0.263134\pi$
$281$	22.7085	1.35467	0.677337	+	0.735673i	$0.263134\pi$
$282$	−5.72060	−0.340657
$283$	1.54332	0.0917411	0.0458706	−	0.998947i	$-0.485394\pi$
$283$	1.54332	0.0917411	0.0458706	+	0.998947i	$0.485394\pi$
$284$	5.77474	0.342668
$285$	12.4027	0.734671
$286$	−5.12254	−0.302902
$287$	−4.49066	−0.265075
$288$	−1.79762	−0.105926
$289$	23.2505	1.36768
$290$	7.17583	0.421379
$291$	22.1577	1.29891
$292$	−1.11057	−0.0649912
$293$	−25.6389	−1.49784	−0.748919	−	0.662661i	$-0.769427\pi$
$293$	−25.6389	−1.49784	−0.748919	+	0.662661i	$0.769427\pi$
$294$	−3.48312	−0.203140
$295$	24.1764	1.40760
$296$	−10.2515	−0.595859
$297$	−4.81295	−0.279275
$298$	−0.0573712	−0.00332343
$299$	−15.6133	−0.902939
$300$	−1.10693	−0.0639087
$301$	−13.3619	−0.770165
$302$	2.69000	0.154792
$303$	22.2190	1.27645
$304$	−2.41329	−0.138411
$305$	21.4932	1.23070
$306$	11.4047	0.651963
$307$	−20.9526	−1.19583	−0.597914	−	0.801560i	$-0.704004\pi$
$307$	−20.9526	−1.19583	−0.597914	+	0.801560i	$0.704004\pi$
$308$	4.25056	0.242198
$309$	−16.5959	−0.944105
$310$	−2.27545	−0.129237
$311$	−18.4804	−1.04793	−0.523963	−	0.851741i	$-0.675547\pi$
$311$	−18.4804	−1.04793	−0.523963	+	0.851741i	$0.675547\pi$
$312$	−6.13962	−0.347588
$313$	−8.00993	−0.452748	−0.226374	−	0.974040i	$-0.572687\pi$
$313$	−8.00993	−0.452748	−0.226374	+	0.974040i	$0.572687\pi$
$314$	14.2066	0.801726
$315$	−9.81028	−0.552746
$316$	−10.2828	−0.578450
$317$	15.6206	0.877339	0.438670	−	0.898648i	$-0.355450\pi$
$317$	15.6206	0.877339	0.438670	+	0.898648i	$0.355450\pi$
$318$	30.4764	1.70903
$319$	5.58901	0.312925
$320$	2.34635	0.131165
$321$	−7.81096	−0.435965
$322$	12.9555	0.721983
$323$	15.3107	0.851909
$324$	−11.1614	−0.620079
$325$	−1.41657	−0.0785769
$326$	10.1200	0.560494
$327$	11.7139	0.647777
$328$	−1.93072	−0.106606
$329$	6.07462	0.334904
$330$	−9.39210	−0.517018
$331$	−23.3453	−1.28317	−0.641586	−	0.767052i	$-0.721723\pi$
$331$	−23.3453	−1.28317	−0.641586	+	0.767052i	$0.721723\pi$
$332$	7.02829	0.385728
$333$	18.4284	1.00987
$334$	1.74484	0.0954734
$335$	19.3233	1.05575
$336$	5.09452	0.277929
$337$	21.0030	1.14411	0.572054	−	0.820216i	$-0.306147\pi$
$337$	21.0030	1.14411	0.572054	+	0.820216i	$0.306147\pi$
$338$	5.14299	0.279742
$339$	37.0768	2.01374
$340$	−14.8860	−0.807308
$341$	−1.77228	−0.0959741
$342$	4.33817	0.234582
$343$	19.9799	1.07882
$344$	−5.74482	−0.309740
$345$	−28.6267	−1.54121
$346$	6.82361	0.366840
$347$	15.3784	0.825554	0.412777	−	0.910832i	$-0.364559\pi$
$347$	15.3784	0.825554	0.412777	+	0.910832i	$0.364559\pi$
$348$	6.69872	0.359089
$349$	18.6949	1.00071	0.500356	−	0.865820i	$-0.333202\pi$
$349$	18.6949	1.00071	0.500356	+	0.865820i	$0.333202\pi$
$350$	1.17543	0.0628295
$351$	−7.38216	−0.394030
$352$	1.82750	0.0974059
$353$	14.6667	0.780628	0.390314	−	0.920682i	$-0.372366\pi$
$353$	14.6667	0.780628	0.390314	+	0.920682i	$0.372366\pi$
$354$	22.5690	1.19953
$355$	13.5496	0.719137
$356$	3.90563	0.206998
$357$	−32.3213	−1.71062
$358$	1.86074	0.0983434
$359$	−17.1296	−0.904068	−0.452034	−	0.892001i	$-0.649301\pi$
$359$	−17.1296	−0.904068	−0.452034	+	0.892001i	$0.649301\pi$
$360$	−4.21785	−0.222300
$361$	−13.1760	−0.693476
$362$	0.749616	0.0393989
$363$	16.7786	0.880649
$364$	6.51957	0.341718
$365$	−2.60579	−0.136393
$366$	20.0641	1.04877
$367$	36.6351	1.91234	0.956169	−	0.292814i	$-0.0945917\pi$
$367$	36.6351	1.91234	0.956169	+	0.292814i	$0.0945917\pi$
$368$	5.57013	0.290363
$369$	3.47071	0.180678
$370$	−24.0537	−1.25049
$371$	−32.3624	−1.68017
$372$	−2.12416	−0.110133
$373$	−7.74263	−0.400898	−0.200449	−	0.979704i	$-0.564240\pi$
$373$	−7.74263	−0.400898	−0.200449	+	0.979704i	$0.564240\pi$
$374$	−11.5942	−0.599524
$375$	23.0994	1.19285
$376$	2.61173	0.134690
$377$	8.57250	0.441506
$378$	6.12554	0.315064
$379$	−36.1066	−1.85467	−0.927335	−	0.374232i	$-0.877906\pi$
$379$	−36.1066	−1.85467	−0.927335	+	0.374232i	$0.877906\pi$
$380$	−5.66242	−0.290476
$381$	−29.0089	−1.48617
$382$	24.0268	1.22932
$383$	25.6407	1.31018	0.655090	−	0.755551i	$-0.272631\pi$
$383$	25.6407	1.31018	0.655090	+	0.755551i	$0.272631\pi$
$384$	2.19035	0.111776
$385$	9.97332	0.508287
$386$	22.1629	1.12806
$387$	10.3270	0.524952
$388$	−10.1161	−0.513565
$389$	−28.6418	−1.45220	−0.726098	−	0.687591i	$-0.758668\pi$
$389$	−28.6418	−1.45220	−0.726098	+	0.687591i	$0.758668\pi$
$390$	−14.4057	−0.729462
$391$	−35.3387	−1.78716
$392$	1.59022	0.0803180
$393$	25.9886	1.31095
$394$	−4.38409	−0.220867
$395$	−24.1270	−1.21396
$396$	−3.28514	−0.165085
$397$	−20.8607	−1.04697	−0.523483	−	0.852036i	$-0.675368\pi$
$397$	−20.8607	−1.04697	−0.523483	+	0.852036i	$0.675368\pi$
$398$	−18.7385	−0.939275
$399$	−12.2945	−0.615496
$400$	0.505368	0.0252684
$401$	−30.0405	−1.50015	−0.750075	−	0.661353i	$-0.769983\pi$
$401$	−30.0405	−1.50015	−0.750075	+	0.661353i	$0.769983\pi$
$402$	18.0386	0.899682
$403$	−2.71834	−0.135410
$404$	−10.1441	−0.504686
$405$	−26.1886	−1.30132
$406$	−7.11326	−0.353025
$407$	−18.7347	−0.928642
$408$	−13.8963	−0.687969
$409$	22.9368	1.13415	0.567075	−	0.823666i	$-0.308075\pi$
$409$	22.9368	1.13415	0.567075	+	0.823666i	$0.308075\pi$
$410$	−4.53015	−0.223728
$411$	−0.751638	−0.0370756
$412$	7.57681	0.373283
$413$	−23.9656	−1.17927
$414$	−10.0130	−0.492111
$415$	16.4908	0.809504
$416$	2.80304	0.137430
$417$	−45.1386	−2.21045
$418$	−4.41027	−0.215713
$419$	5.52564	0.269945	0.134973	−	0.990849i	$-0.456905\pi$
$419$	5.52564	0.269945	0.134973	+	0.990849i	$0.456905\pi$
$420$	11.9535	0.583272
$421$	−2.38955	−0.116459	−0.0582297	−	0.998303i	$-0.518546\pi$
$421$	−2.38955	−0.116459	−0.0582297	+	0.998303i	$0.518546\pi$
$422$	−15.0275	−0.731526
$423$	−4.69491	−0.228274
$424$	−13.9140	−0.675722
$425$	−3.20622	−0.155525
$426$	12.6487	0.612831
$427$	−21.3058	−1.03106
$428$	3.56608	0.172373
$429$	−11.2201	−0.541713
$430$	−13.4794	−0.650034
$431$	41.0490	1.97726	0.988631	−	0.150365i	$-0.0480448\pi$
$431$	41.0490	1.97726	0.988631	+	0.150365i	$0.0480448\pi$
$432$	2.63363	0.126710
$433$	22.7514	1.09336	0.546681	−	0.837341i	$-0.315891\pi$
$433$	22.7514	1.09336	0.546681	+	0.837341i	$0.315891\pi$
$434$	2.25561	0.108273
$435$	15.7175	0.753599
$436$	−5.34794	−0.256120
$437$	−13.4423	−0.643033
$438$	−2.43253	−0.116231
$439$	−29.5338	−1.40957	−0.704785	−	0.709421i	$-0.748957\pi$
$439$	−29.5338	−1.40957	−0.704785	+	0.709421i	$0.748957\pi$
$440$	4.28795	0.204420
$441$	−2.85860	−0.136124
$442$	−17.7834	−0.845870
$443$	−27.6804	−1.31514	−0.657568	−	0.753395i	$-0.728415\pi$
$443$	−27.6804	−1.31514	−0.657568	+	0.753395i	$0.728415\pi$
$444$	−22.4544	−1.06564
$445$	9.16399	0.434415
$446$	18.2612	0.864693
$447$	−0.125663	−0.00594365
$448$	−2.32589	−0.109888
$449$	−8.57134	−0.404506	−0.202253	−	0.979333i	$-0.564826\pi$
$449$	−8.57134	−0.404506	−0.202253	+	0.979333i	$0.564826\pi$
$450$	−0.908460	−0.0428252
$451$	−3.52839	−0.166145
$452$	−16.9274	−0.796197
$453$	5.89203	0.276832
$454$	−12.2437	−0.574627
$455$	15.2972	0.717144
$456$	−5.28594	−0.247537
$457$	0.612194	0.0286372	0.0143186	−	0.999897i	$-0.495442\pi$
$457$	0.612194	0.0286372	0.0143186	+	0.999897i	$0.495442\pi$
$458$	−7.45508	−0.348353
$459$	−16.7086	−0.779891
$460$	13.0695	0.609368
$461$	22.6558	1.05518	0.527592	−	0.849498i	$-0.323095\pi$
$461$	22.6558	1.05518	0.527592	+	0.849498i	$0.323095\pi$
$462$	9.31021	0.433150
$463$	7.72115	0.358832	0.179416	−	0.983773i	$-0.442579\pi$
$463$	7.72115	0.358832	0.179416	+	0.983773i	$0.442579\pi$
$464$	−3.05829	−0.141978
$465$	−4.98403	−0.231129
$466$	23.8002	1.10252
$467$	19.6007	0.907013	0.453507	−	0.891253i	$-0.350173\pi$
$467$	19.6007	0.907013	0.453507	+	0.891253i	$0.350173\pi$
$468$	−5.03880	−0.232918
$469$	−19.1549	−0.884489
$470$	6.12805	0.282666
$471$	31.1174	1.43382
$472$	−10.3038	−0.474272
$473$	−10.4986	−0.482728
$474$	−22.5228	−1.03451
$475$	−1.21960	−0.0559590
$476$	14.7562	0.676351
$477$	25.0120	1.14522
$478$	25.6894	1.17501
$479$	−16.9784	−0.775764	−0.387882	−	0.921709i	$-0.626793\pi$
$479$	−16.9784	−0.775764	−0.387882	+	0.921709i	$0.626793\pi$
$480$	5.13933	0.234577
$481$	−28.7354	−1.31022
$482$	5.33928	0.243197
$483$	28.3771	1.29120
$484$	−7.66026	−0.348194
$485$	−23.7358	−1.07779
$486$	−16.5465	−0.750565
$487$	−4.86546	−0.220475	−0.110238	−	0.993905i	$-0.535161\pi$
$487$	−4.86546	−0.220475	−0.110238	+	0.993905i	$0.535161\pi$
$488$	−9.16025	−0.414665
$489$	22.1663	1.00239
$490$	3.73121	0.168559
$491$	−28.0324	−1.26509	−0.632543	−	0.774525i	$-0.717989\pi$
$491$	−28.0324	−1.26509	−0.632543	+	0.774525i	$0.717989\pi$
$492$	−4.22895	−0.190656
$493$	19.4028	0.873858
$494$	−6.76453	−0.304351
$495$	−7.70811	−0.346454
$496$	0.969783	0.0435446
$497$	−13.4314	−0.602482
$498$	15.3944	0.689839
$499$	−26.6649	−1.19368	−0.596842	−	0.802359i	$-0.703578\pi$
$499$	−26.6649	−1.19368	−0.596842	+	0.802359i	$0.703578\pi$
$500$	−10.5460	−0.471631
$501$	3.82181	0.170746
$502$	12.4706	0.556588
$503$	−15.5463	−0.693177	−0.346589	−	0.938017i	$-0.612660\pi$
$503$	−15.5463	−0.693177	−0.346589	+	0.938017i	$0.612660\pi$
$504$	4.18108	0.186240
$505$	−23.8016	−1.05916
$506$	10.1794	0.452529
$507$	11.2649	0.500293
$508$	13.2440	0.587606
$509$	−6.52951	−0.289415	−0.144708	−	0.989474i	$-0.546224\pi$
$509$	−6.52951	−0.289415	−0.144708	+	0.989474i	$0.546224\pi$
$510$	−32.6056	−1.44380
$511$	2.58307	0.114268
$512$	−1.00000	−0.0441942
$513$	−6.35570	−0.280611
$514$	−9.79743	−0.432146
$515$	17.7779	0.783387
$516$	−12.5832	−0.553943
$517$	4.77293	0.209913
$518$	23.8440	1.04765
$519$	14.9461	0.656060
$520$	6.57691	0.288417
$521$	−9.17439	−0.401937	−0.200969	−	0.979598i	$-0.564409\pi$
$521$	−9.17439	−0.401937	−0.200969	+	0.979598i	$0.564409\pi$
$522$	5.49765	0.240625
$523$	−0.111915	−0.00489369	−0.00244685	−	0.999997i	$-0.500779\pi$
$523$	−0.111915	−0.00489369	−0.00244685	+	0.999997i	$0.500779\pi$
$524$	−11.8650	−0.518327
$525$	2.57461	0.112365
$526$	−15.6100	−0.680630
$527$	−6.15263	−0.268013
$528$	4.00285	0.174202
$529$	8.02631	0.348970
$530$	−32.6471	−1.41810
$531$	18.5224	0.803803
$532$	5.61305	0.243357
$533$	−5.41189	−0.234415
$534$	8.55469	0.370198
$535$	8.36729	0.361749
$536$	−8.23548	−0.355719
$537$	4.07567	0.175878
$538$	−25.9714	−1.11971
$539$	2.90611	0.125175
$540$	6.17942	0.265920
$541$	10.8405	0.466068	0.233034	−	0.972469i	$-0.425135\pi$
$541$	10.8405	0.466068	0.233034	+	0.972469i	$0.425135\pi$
$542$	2.75902	0.118510
$543$	1.64192	0.0704615
$544$	6.34433	0.272011
$545$	−12.5482	−0.537504
$546$	14.2801	0.611133
$547$	25.4169	1.08675	0.543374	−	0.839490i	$-0.317146\pi$
$547$	25.4169	1.08675	0.543374	+	0.839490i	$0.317146\pi$
$548$	0.343159	0.0146590
$549$	16.4667	0.702780
$550$	0.923558	0.0393807
$551$	7.38053	0.314421
$552$	12.2005	0.519288
$553$	23.9166	1.01704
$554$	−7.15201	−0.303860
$555$	−52.6860	−2.23640
$556$	20.6080	0.873972
$557$	10.1308	0.429255	0.214627	−	0.976696i	$-0.431146\pi$
$557$	10.1308	0.429255	0.214627	+	0.976696i	$0.431146\pi$
$558$	−1.74330	−0.0737999
$559$	−16.1030	−0.681082
$560$	−5.45737	−0.230616
$561$	−25.3954	−1.07219
$562$	−22.7085	−0.957899
$563$	−41.1153	−1.73280	−0.866402	−	0.499348i	$-0.833573\pi$
$563$	−41.1153	−1.73280	−0.866402	+	0.499348i	$0.833573\pi$
$564$	5.72060	0.240881
$565$	−39.7176	−1.67093
$566$	−1.54332	−0.0648708
$567$	25.9603	1.09023
$568$	−5.77474	−0.242303
$569$	−42.3185	−1.77408	−0.887042	−	0.461689i	$-0.847244\pi$
$569$	−42.3185	−1.77408	−0.887042	+	0.461689i	$0.847244\pi$
$570$	−12.4027	−0.519491
$571$	−29.9641	−1.25396	−0.626979	−	0.779036i	$-0.715709\pi$
$571$	−29.9641	−1.25396	−0.626979	+	0.779036i	$0.715709\pi$
$572$	5.12254	0.214184
$573$	52.6271	2.19853
$574$	4.49066	0.187436
$575$	2.81497	0.117392
$576$	1.79762	0.0749009
$577$	−2.38992	−0.0994935	−0.0497467	−	0.998762i	$-0.515841\pi$
$577$	−2.38992	−0.0994935	−0.0497467	+	0.998762i	$0.515841\pi$
$578$	−23.2505	−0.967094
$579$	48.5445	2.01744
$580$	−7.17583	−0.297960
$581$	−16.3471	−0.678190
$582$	−22.1577	−0.918465
$583$	−25.4277	−1.05311
$584$	1.11057	0.0459557
$585$	−11.8228	−0.488812
$586$	25.6389	1.05913
$587$	26.8153	1.10679	0.553393	−	0.832920i	$-0.313333\pi$
$587$	26.8153	1.10679	0.553393	+	0.832920i	$0.313333\pi$
$588$	3.48312	0.143642
$589$	−2.34037	−0.0964331
$590$	−24.1764	−0.995327
$591$	−9.60267	−0.395001
$592$	10.2515	0.421336
$593$	−11.6809	−0.479677	−0.239838	−	0.970813i	$-0.577094\pi$
$593$	−11.6809	−0.479677	−0.239838	+	0.970813i	$0.577094\pi$
$594$	4.81295	0.197478
$595$	34.6233	1.41942
$596$	0.0573712	0.00235002
$597$	−41.0438	−1.67981
$598$	15.6133	0.638474
$599$	−7.82159	−0.319581	−0.159791	−	0.987151i	$-0.551082\pi$
$599$	−7.82159	−0.319581	−0.159791	+	0.987151i	$0.551082\pi$
$600$	1.10693	0.0451903
$601$	−39.8561	−1.62577	−0.812883	−	0.582427i	$-0.802103\pi$
$601$	−39.8561	−1.62577	−0.812883	+	0.582427i	$0.802103\pi$
$602$	13.3619	0.544589
$603$	14.8043	0.602876
$604$	−2.69000	−0.109454
$605$	−17.9737	−0.730733
$606$	−22.2190	−0.902587
$607$	30.1328	1.22305	0.611527	−	0.791223i	$-0.290556\pi$
$607$	30.1328	1.22305	0.611527	+	0.791223i	$0.290556\pi$
$608$	2.41329	0.0978717
$609$	−15.5805	−0.631354
$610$	−21.4932	−0.870233
$611$	7.32078	0.296167
$612$	−11.4047	−0.461008
$613$	39.8111	1.60795	0.803977	−	0.594661i	$-0.202714\pi$
$613$	39.8111	1.60795	0.803977	+	0.594661i	$0.202714\pi$
$614$	20.9526	0.845579
$615$	−9.92261	−0.400118
$616$	−4.25056	−0.171260
$617$	−38.6736	−1.55694	−0.778471	−	0.627681i	$-0.784004\pi$
$617$	−38.6736	−1.55694	−0.778471	+	0.627681i	$0.784004\pi$
$618$	16.5959	0.667583
$619$	−40.6713	−1.63472	−0.817360	−	0.576128i	$-0.804563\pi$
$619$	−40.6713	−1.63472	−0.817360	+	0.576128i	$0.804563\pi$
$620$	2.27545	0.0913844
$621$	14.6696	0.588672
$622$	18.4804	0.740995
$623$	−9.08409	−0.363946
$624$	6.13962	0.245782
$625$	−27.2714	−1.09086
$626$	8.00993	0.320141
$627$	−9.66003	−0.385784
$628$	−14.2066	−0.566906
$629$	−65.0392	−2.59328
$630$	9.81028	0.390851
$631$	−15.0868	−0.600596	−0.300298	−	0.953845i	$-0.597086\pi$
$631$	−15.0868	−0.600596	−0.300298	+	0.953845i	$0.597086\pi$
$632$	10.2828	0.409026
$633$	−32.9154	−1.30827
$634$	−15.6206	−0.620372
$635$	31.0750	1.23317
$636$	−30.4764	−1.20847
$637$	4.45743	0.176610
$638$	−5.58901	−0.221271
$639$	10.3808	0.410658
$640$	−2.34635	−0.0927477
$641$	−23.8540	−0.942177	−0.471089	−	0.882086i	$-0.656139\pi$
$641$	−23.8540	−0.942177	−0.471089	+	0.882086i	$0.656139\pi$
$642$	7.81096	0.308274
$643$	44.5198	1.75569	0.877845	−	0.478945i	$-0.158981\pi$
$643$	44.5198	1.75569	0.877845	+	0.478945i	$0.158981\pi$
$644$	−12.9555	−0.510519
$645$	−29.5245	−1.16253
$646$	−15.3107	−0.602391
$647$	36.4895	1.43455	0.717276	−	0.696789i	$-0.245389\pi$
$647$	36.4895	1.43455	0.717276	+	0.696789i	$0.245389\pi$
$648$	11.1614	0.438462
$649$	−18.8302	−0.739150
$650$	1.41657	0.0555623
$651$	4.94058	0.193636
$652$	−10.1200	−0.396329
$653$	12.1476	0.475370	0.237685	−	0.971342i	$-0.423611\pi$
$653$	12.1476	0.475370	0.237685	+	0.971342i	$0.423611\pi$
$654$	−11.7139	−0.458048
$655$	−27.8396	−1.08778
$656$	1.93072	0.0753820
$657$	−1.99638	−0.0778864
$658$	−6.07462	−0.236813
$659$	2.30942	0.0899621	0.0449811	−	0.998988i	$-0.485677\pi$
$659$	2.30942	0.0899621	0.0449811	+	0.998988i	$0.485677\pi$
$660$	9.39210	0.365587
$661$	18.0835	0.703365	0.351683	−	0.936119i	$-0.385610\pi$
$661$	18.0835	0.703365	0.351683	+	0.936119i	$0.385610\pi$
$662$	23.3453	0.907339
$663$	−38.9518	−1.51276
$664$	−7.02829	−0.272751
$665$	13.1702	0.510718
$666$	−18.4284	−0.714085
$667$	−17.0351	−0.659600
$668$	−1.74484	−0.0675099
$669$	39.9984	1.54643
$670$	−19.3233	−0.746526
$671$	−16.7403	−0.646253
$672$	−5.09452	−0.196525
$673$	−29.4818	−1.13644	−0.568221	−	0.822876i	$-0.692368\pi$
$673$	−29.4818	−1.13644	−0.568221	+	0.822876i	$0.692368\pi$
$674$	−21.0030	−0.809007
$675$	1.33095	0.0512284
$676$	−5.14299	−0.197807
$677$	−17.4494	−0.670636	−0.335318	−	0.942105i	$-0.608844\pi$
$677$	−17.4494	−0.670636	−0.335318	+	0.942105i	$0.608844\pi$
$678$	−37.0768	−1.42393
$679$	23.5289	0.902955
$680$	14.8860	0.570853
$681$	−26.8180	−1.02767
$682$	1.77228	0.0678639
$683$	19.5132	0.746651	0.373325	−	0.927700i	$-0.378218\pi$
$683$	19.5132	0.746651	0.373325	+	0.927700i	$0.378218\pi$
$684$	−4.33817	−0.165874
$685$	0.805172	0.0307640
$686$	−19.9799	−0.762837
$687$	−16.3292	−0.622998
$688$	5.74482	0.219019
$689$	−39.0014	−1.48583
$690$	28.6267	1.08980
$691$	−16.9466	−0.644681	−0.322340	−	0.946624i	$-0.604470\pi$
$691$	−16.9466	−0.644681	−0.322340	+	0.946624i	$0.604470\pi$
$692$	−6.82361	−0.259395
$693$	7.64090	0.290254
$694$	−15.3784	−0.583755
$695$	48.3535	1.83415
$696$	−6.69872	−0.253914
$697$	−12.2491	−0.463969
$698$	−18.6949	−0.707611
$699$	52.1307	1.97176
$700$	−1.17543	−0.0444272
$701$	21.7600	0.821864	0.410932	−	0.911666i	$-0.365203\pi$
$701$	21.7600	0.821864	0.410932	+	0.911666i	$0.365203\pi$
$702$	7.38216	0.278622
$703$	−24.7399	−0.933083
$704$	−1.82750	−0.0688764
$705$	13.4225	0.505522
$706$	−14.6667	−0.551987
$707$	23.5940	0.887345
$708$	−22.5690	−0.848194
$709$	−1.52764	−0.0573716	−0.0286858	−	0.999588i	$-0.509132\pi$
$709$	−1.52764	−0.0573716	−0.0286858	+	0.999588i	$0.509132\pi$
$710$	−13.5496	−0.508507
$711$	−18.4845	−0.693223
$712$	−3.90563	−0.146370
$713$	5.40182	0.202300
$714$	32.3213	1.20959
$715$	12.0193	0.449496
$716$	−1.86074	−0.0695393
$717$	56.2687	2.10139
$718$	17.1296	0.639272
$719$	25.9143	0.966440	0.483220	−	0.875499i	$-0.339467\pi$
$719$	25.9143	0.966440	0.483220	+	0.875499i	$0.339467\pi$
$720$	4.21785	0.157190
$721$	−17.6229	−0.656310
$722$	13.1760	0.490362
$723$	11.6949	0.434937
$724$	−0.749616	−0.0278593
$725$	−1.54556	−0.0574008
$726$	−16.7786	−0.622713
$727$	2.27437	0.0843517	0.0421759	−	0.999110i	$-0.486571\pi$
$727$	2.27437	0.0843517	0.0421759	+	0.999110i	$0.486571\pi$
$728$	−6.51957	−0.241631
$729$	−2.75832	−0.102160
$730$	2.60579	0.0964446
$731$	−36.4471	−1.34804
$732$	−20.0641	−0.741592
$733$	−25.6131	−0.946040	−0.473020	−	0.881052i	$-0.656836\pi$
$733$	−25.6131	−0.946040	−0.473020	+	0.881052i	$0.656836\pi$
$734$	−36.6351	−1.35223
$735$	8.17264	0.301452
$736$	−5.57013	−0.205318
$737$	−15.0503	−0.554385
$738$	−3.47071	−0.127758
$739$	−39.7827	−1.46343	−0.731716	−	0.681610i	$-0.761280\pi$
$739$	−39.7827	−1.46343	−0.731716	+	0.681610i	$0.761280\pi$
$740$	24.0537	0.884233
$741$	−14.8167	−0.544304
$742$	32.3624	1.18806
$743$	45.1460	1.65625	0.828123	−	0.560547i	$-0.189409\pi$
$743$	45.1460	1.65625	0.828123	+	0.560547i	$0.189409\pi$
$744$	2.12416	0.0778756
$745$	0.134613	0.00493184
$746$	7.74263	0.283478
$747$	12.6342	0.462261
$748$	11.5942	0.423927
$749$	−8.29433	−0.303068
$750$	−23.0994	−0.843470
$751$	11.3641	0.414682	0.207341	−	0.978269i	$-0.433519\pi$
$751$	11.3641	0.414682	0.207341	+	0.978269i	$0.433519\pi$
$752$	−2.61173	−0.0952401
$753$	27.3149	0.995409
$754$	−8.57250	−0.312192
$755$	−6.31168	−0.229706
$756$	−6.12554	−0.222784
$757$	−15.2431	−0.554019	−0.277009	−	0.960867i	$-0.589343\pi$
$757$	−15.2431	−0.554019	−0.277009	+	0.960867i	$0.589343\pi$
$758$	36.1066	1.31145
$759$	22.2964	0.809308
$760$	5.66242	0.205398
$761$	18.3336	0.664591	0.332295	−	0.943175i	$-0.392177\pi$
$761$	18.3336	0.664591	0.332295	+	0.943175i	$0.392177\pi$
$762$	29.0089	1.05088
$763$	12.4387	0.450313
$764$	−24.0268	−0.869260
$765$	−26.7594	−0.967490
$766$	−25.6407	−0.926437
$767$	−28.8820	−1.04287
$768$	−2.19035	−0.0790373
$769$	9.78309	0.352787	0.176394	−	0.984320i	$-0.443557\pi$
$769$	9.78309	0.352787	0.176394	+	0.984320i	$0.443557\pi$
$770$	−9.97332	−0.359413
$771$	−21.4598	−0.772855
$772$	−22.1629	−0.797661
$773$	45.4208	1.63367	0.816837	−	0.576869i	$-0.195726\pi$
$773$	45.4208	1.63367	0.816837	+	0.576869i	$0.195726\pi$
$774$	−10.3270	−0.371197
$775$	0.490098	0.0176048
$776$	10.1161	0.363145
$777$	52.2266	1.87362
$778$	28.6418	1.02686
$779$	−4.65939	−0.166940
$780$	14.4057	0.515807
$781$	−10.5533	−0.377627
$782$	35.3387	1.26371
$783$	−8.05440	−0.287841
$784$	−1.59022	−0.0567934
$785$	−33.3337	−1.18973
$786$	−25.9886	−0.926981
$787$	−3.37804	−0.120414	−0.0602071	−	0.998186i	$-0.519176\pi$
$787$	−3.37804	−0.120414	−0.0602071	+	0.998186i	$0.519176\pi$
$788$	4.38409	0.156177
$789$	−34.1914	−1.21725
$790$	24.1270	0.858399
$791$	39.3713	1.39988
$792$	3.28514	0.116733
$793$	−25.6765	−0.911800
$794$	20.8607	0.740317
$795$	−71.5084	−2.53614
$796$	18.7385	0.664168
$797$	−52.2579	−1.85107	−0.925535	−	0.378663i	$-0.876384\pi$
$797$	−52.2579	−1.85107	−0.925535	+	0.378663i	$0.876384\pi$
$798$	12.2945	0.435222
$799$	16.5697	0.586194
$800$	−0.505368	−0.0178675
$801$	7.02085	0.248069
$802$	30.0405	1.06077
$803$	2.02956	0.0716217
$804$	−18.0386	−0.636171
$805$	−30.3982	−1.07140
$806$	2.71834	0.0957494
$807$	−56.8863	−2.00249
$808$	10.1441	0.356867
$809$	−5.76137	−0.202559	−0.101280	−	0.994858i	$-0.532294\pi$
$809$	−5.76137	−0.202559	−0.101280	+	0.994858i	$0.532294\pi$
$810$	26.1886	0.920175
$811$	−28.7264	−1.00872	−0.504361	−	0.863493i	$-0.668272\pi$
$811$	−28.7264	−1.00872	−0.504361	+	0.863493i	$0.668272\pi$
$812$	7.11326	0.249626
$813$	6.04322	0.211945
$814$	18.7347	0.656649
$815$	−23.7451	−0.831753
$816$	13.8963	0.486467
$817$	−13.8639	−0.485037
$818$	−22.9368	−0.801965
$819$	11.7197	0.409520
$820$	4.53015	0.158200
$821$	−6.26989	−0.218821	−0.109410	−	0.993997i	$-0.534896\pi$
$821$	−6.26989	−0.218821	−0.109410	+	0.993997i	$0.534896\pi$
$822$	0.751638	0.0262164
$823$	15.6789	0.546533	0.273266	−	0.961938i	$-0.411896\pi$
$823$	15.6789	0.546533	0.273266	+	0.961938i	$0.411896\pi$
$824$	−7.57681	−0.263951
$825$	2.02291	0.0704288
$826$	23.9656	0.833870
$827$	21.4423	0.745620	0.372810	−	0.927908i	$-0.378394\pi$
$827$	21.4423	0.745620	0.372810	+	0.927908i	$0.378394\pi$
$828$	10.0130	0.347975
$829$	33.4727	1.16256	0.581278	−	0.813705i	$-0.302553\pi$
$829$	33.4727	1.16256	0.581278	+	0.813705i	$0.302553\pi$
$830$	−16.4908	−0.572406
$831$	−15.6654	−0.543426
$832$	−2.80304	−0.0971778
$833$	10.0889	0.349558
$834$	45.1386	1.56302
$835$	−4.09401	−0.141679
$836$	4.41027	0.152532
$837$	2.55405	0.0882808
$838$	−5.52564	−0.190880
$839$	50.4091	1.74032	0.870158	−	0.492772i	$-0.164016\pi$
$839$	50.4091	1.74032	0.870158	+	0.492772i	$0.164016\pi$
$840$	−11.9535	−0.412436
$841$	−19.6469	−0.677478
$842$	2.38955	0.0823492
$843$	−49.7394	−1.71312
$844$	15.0275	0.517267
$845$	−12.0673	−0.415126
$846$	4.69491	0.161414
$847$	17.8170	0.612198
$848$	13.9140	0.477808
$849$	−3.38042	−0.116016
$850$	3.20622	0.109973
$851$	57.1024	1.95744
$852$	−12.6487	−0.433337
$853$	−4.52921	−0.155077	−0.0775386	−	0.996989i	$-0.524706\pi$
$853$	−4.52921	−0.155077	−0.0775386	+	0.996989i	$0.524706\pi$
$854$	21.3058	0.729069
$855$	−10.1789	−0.348110
$856$	−3.56608	−0.121886
$857$	9.35651	0.319612	0.159806	−	0.987148i	$-0.448913\pi$
$857$	9.35651	0.319612	0.159806	+	0.987148i	$0.448913\pi$
$858$	11.2201	0.383049
$859$	48.0339	1.63890	0.819448	−	0.573153i	$-0.194280\pi$
$859$	48.0339	1.63890	0.819448	+	0.573153i	$0.194280\pi$
$860$	13.4794	0.459643
$861$	9.83610	0.335213
$862$	−41.0490	−1.39813
$863$	−45.8190	−1.55970	−0.779848	−	0.625969i	$-0.784704\pi$
$863$	−45.8190	−1.55970	−0.779848	+	0.625969i	$0.784704\pi$
$864$	−2.63363	−0.0895978
$865$	−16.0106	−0.544376
$866$	−22.7514	−0.773124
$867$	−50.9267	−1.72956
$868$	−2.25561	−0.0765605
$869$	18.7917	0.637465
$870$	−15.7175	−0.532875
$871$	−23.0844	−0.782183
$872$	5.34794	0.181104
$873$	−18.1848	−0.615463
$874$	13.4423	0.454693
$875$	24.5289	0.829227
$876$	2.43253	0.0821877
$877$	−40.8123	−1.37813	−0.689066	−	0.724698i	$-0.741979\pi$
$877$	−40.8123	−1.37813	−0.689066	+	0.724698i	$0.741979\pi$
$878$	29.5338	0.996717
$879$	56.1580	1.89416
$880$	−4.28795	−0.144547
$881$	45.3403	1.52755	0.763777	−	0.645480i	$-0.223343\pi$
$881$	45.3403	1.52755	0.763777	+	0.645480i	$0.223343\pi$
$882$	2.85860	0.0962542
$883$	−34.5810	−1.16374	−0.581872	−	0.813280i	$-0.697680\pi$
$883$	−34.5810	−1.16374	−0.581872	+	0.813280i	$0.697680\pi$
$884$	17.7834	0.598120
$885$	−52.9547	−1.78005
$886$	27.6804	0.929941
$887$	0.433356	0.0145507	0.00727533	−	0.999974i	$-0.497684\pi$
$887$	0.433356	0.0145507	0.00727533	+	0.999974i	$0.497684\pi$
$888$	22.4544	0.753522
$889$	−30.8041	−1.03314
$890$	−9.16399	−0.307178
$891$	20.3975	0.683340
$892$	−18.2612	−0.611430
$893$	6.30286	0.210917
$894$	0.125663	0.00420280
$895$	−4.36596	−0.145938
$896$	2.32589	0.0777027
$897$	34.1985	1.14185
$898$	8.57134	0.286029
$899$	−2.96588	−0.0989176
$900$	0.908460	0.0302820
$901$	−88.2748	−2.94086
$902$	3.52839	0.117482
$903$	29.2671	0.973948
$904$	16.9274	0.562996
$905$	−1.75886	−0.0584666
$906$	−5.89203	−0.195750
$907$	−8.95401	−0.297313	−0.148657	−	0.988889i	$-0.547495\pi$
$907$	−8.95401	−0.297313	−0.148657	+	0.988889i	$0.547495\pi$
$908$	12.2437	0.406323
$909$	−18.2352	−0.604823
$910$	−15.2972	−0.507097
$911$	−5.38712	−0.178483	−0.0892416	−	0.996010i	$-0.528444\pi$
$911$	−5.38712	−0.178483	−0.0892416	+	0.996010i	$0.528444\pi$
$912$	5.28594	0.175035
$913$	−12.8442	−0.425080
$914$	−0.612194	−0.0202496
$915$	−47.0775	−1.55633
$916$	7.45508	0.246323
$917$	27.5968	0.911328
$918$	16.7086	0.551466
$919$	−8.99731	−0.296794	−0.148397	−	0.988928i	$-0.547411\pi$
$919$	−8.99731	−0.296794	−0.148397	+	0.988928i	$0.547411\pi$
$920$	−13.0695	−0.430888
$921$	45.8935	1.51224
$922$	−22.6558	−0.746128
$923$	−16.1868	−0.532796
$924$	−9.31021	−0.306283
$925$	5.18080	0.170344
$926$	−7.72115	−0.253733
$927$	13.6202	0.447347
$928$	3.05829	0.100393
$929$	−4.96634	−0.162940	−0.0814701	−	0.996676i	$-0.525962\pi$
$929$	−4.96634	−0.162940	−0.0814701	+	0.996676i	$0.525962\pi$
$930$	4.98403	0.163433
$931$	3.83765	0.125774
$932$	−23.8002	−0.779601
$933$	40.4784	1.32520
$934$	−19.6007	−0.641355
$935$	27.2042	0.889671
$936$	5.03880	0.164698
$937$	53.4269	1.74538	0.872691	−	0.488274i	$-0.162373\pi$
$937$	53.4269	1.74538	0.872691	+	0.488274i	$0.162373\pi$
$938$	19.1549	0.625428
$939$	17.5445	0.572544
$940$	−6.12805	−0.199875
$941$	26.4586	0.862527	0.431264	−	0.902226i	$-0.358068\pi$
$941$	26.4586	0.862527	0.431264	+	0.902226i	$0.358068\pi$
$942$	−31.1174	−1.01386
$943$	10.7544	0.350210
$944$	10.3038	0.335361
$945$	−14.3727	−0.467543
$946$	10.4986	0.341340
$947$	23.8556	0.775202	0.387601	−	0.921827i	$-0.373304\pi$
$947$	23.8556	0.775202	0.387601	+	0.921827i	$0.373304\pi$
$948$	22.5228	0.731507
$949$	3.11297	0.101051
$950$	1.21960	0.0395690
$951$	−34.2145	−1.10948
$952$	−14.7562	−0.478253
$953$	6.81199	0.220662	0.110331	−	0.993895i	$-0.464809\pi$
$953$	6.81199	0.220662	0.110331	+	0.993895i	$0.464809\pi$
$954$	−25.0120	−0.809795
$955$	−56.3754	−1.82426
$956$	−25.6894	−0.830854
$957$	−12.2419	−0.395724
$958$	16.9784	0.548548
$959$	−0.798152	−0.0257737
$960$	−5.13933	−0.165871
$961$	−30.0595	−0.969662
$962$	28.7354	0.926468
$963$	6.41046	0.206574
$964$	−5.33928	−0.171967
$965$	−52.0020	−1.67400
$966$	−28.3771	−0.913018
$967$	−25.4575	−0.818657	−0.409328	−	0.912387i	$-0.634237\pi$
$967$	−25.4575	−0.818657	−0.409328	+	0.912387i	$0.634237\pi$
$968$	7.66026	0.246210
$969$	−33.5357	−1.07732
$970$	23.7358	0.762111
$971$	46.1404	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$971$	46.1404	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$972$	16.5465	0.530729
$973$	−47.9319	−1.53663
$974$	4.86546	0.155899
$975$	3.10277	0.0993682
$976$	9.16025	0.293213
$977$	−17.7896	−0.569141	−0.284570	−	0.958655i	$-0.591851\pi$
$977$	−17.7896	−0.569141	−0.284570	+	0.958655i	$0.591851\pi$
$978$	−22.1663	−0.708800
$979$	−7.13753	−0.228116
$980$	−3.73121	−0.119189
$981$	−9.61357	−0.306938
$982$	28.0324	0.894551
$983$	−4.39628	−0.140219	−0.0701097	−	0.997539i	$-0.522335\pi$
$983$	−4.39628	−0.140219	−0.0701097	+	0.997539i	$0.522335\pi$
$984$	4.22895	0.134814
$985$	10.2866	0.327759
$986$	−19.4028	−0.617911
$987$	−13.3055	−0.423519
$988$	6.76453	0.215208
$989$	31.9994	1.01752
$990$	7.70811	0.244980
$991$	−43.4988	−1.38179	−0.690893	−	0.722957i	$-0.742783\pi$
$991$	−43.4988	−1.38179	−0.690893	+	0.722957i	$0.742783\pi$
$992$	−0.969783	−0.0307907
$993$	51.1342	1.62269
$994$	13.4314	0.426019
$995$	43.9671	1.39385
$996$	−15.3944	−0.487790
$997$	−16.0506	−0.508329	−0.254165	−	0.967161i	$-0.581800\pi$
$997$	−16.0506	−0.508329	−0.254165	+	0.967161i	$0.581800\pi$
$998$	26.6649	0.844063
$999$	26.9987	0.854203

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.b.1.15	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.b.1.15	✓	75	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:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.19035 q^{3} +1.00000 q^{4} +2.34635 q^{5} +2.19035 q^{6} -2.32589 q^{7} -1.00000 q^{8} +1.79762 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.19035 q^{3} +1.00000 q^{4} +2.34635 q^{5} +2.19035 q^{6} -2.32589 q^{7} -1.00000 q^{8} +1.79762 q^{9} -2.34635 q^{10} -1.82750 q^{11} -2.19035 q^{12} -2.80304 q^{13} +2.32589 q^{14} -5.13933 q^{15} +1.00000 q^{16} -6.34433 q^{17} -1.79762 q^{18} -2.41329 q^{19} +2.34635 q^{20} +5.09452 q^{21} +1.82750 q^{22} +5.57013 q^{23} +2.19035 q^{24} +0.505368 q^{25} +2.80304 q^{26} +2.63363 q^{27} -2.32589 q^{28} -3.05829 q^{29} +5.13933 q^{30} +0.969783 q^{31} -1.00000 q^{32} +4.00285 q^{33} +6.34433 q^{34} -5.45737 q^{35} +1.79762 q^{36} +10.2515 q^{37} +2.41329 q^{38} +6.13962 q^{39} -2.34635 q^{40} +1.93072 q^{41} -5.09452 q^{42} +5.74482 q^{43} -1.82750 q^{44} +4.21785 q^{45} -5.57013 q^{46} -2.61173 q^{47} -2.19035 q^{48} -1.59022 q^{49} -0.505368 q^{50} +13.8963 q^{51} -2.80304 q^{52} +13.9140 q^{53} -2.63363 q^{54} -4.28795 q^{55} +2.32589 q^{56} +5.28594 q^{57} +3.05829 q^{58} +10.3038 q^{59} -5.13933 q^{60} +9.16025 q^{61} -0.969783 q^{62} -4.18108 q^{63} +1.00000 q^{64} -6.57691 q^{65} -4.00285 q^{66} +8.23548 q^{67} -6.34433 q^{68} -12.2005 q^{69} +5.45737 q^{70} +5.77474 q^{71} -1.79762 q^{72} -1.11057 q^{73} -10.2515 q^{74} -1.10693 q^{75} -2.41329 q^{76} +4.25056 q^{77} -6.13962 q^{78} -10.2828 q^{79} +2.34635 q^{80} -11.1614 q^{81} -1.93072 q^{82} +7.02829 q^{83} +5.09452 q^{84} -14.8860 q^{85} -5.74482 q^{86} +6.69872 q^{87} +1.82750 q^{88} +3.90563 q^{89} -4.21785 q^{90} +6.51957 q^{91} +5.57013 q^{92} -2.12416 q^{93} +2.61173 q^{94} -5.66242 q^{95} +2.19035 q^{96} -10.1161 q^{97} +1.59022 q^{98} -3.28514 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9}+O(q^{10}) \) 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9	\( 75 q - 75 q^{2} + q^{3} + 75 q^{4} - 9 q^{5} - q^{6} - 8 q^{7} - 75 q^{8} + 66 q^{9} + 9 q^{10} - 5 q^{11} + q^{12} - 35 q^{13} + 8 q^{14} - 21 q^{15} + 75 q^{16} + 4 q^{17} - 66 q^{18} - 59 q^{19} - 9 q^{20} - 62 q^{21} + 5 q^{22} + 43 q^{23} - q^{24} + 44 q^{25} + 35 q^{26} + 4 q^{27} - 8 q^{28} - 38 q^{29} + 21 q^{30} - 51 q^{31} - 75 q^{32} - 19 q^{33} - 4 q^{34} + 14 q^{35} + 66 q^{36} - 63 q^{37} + 59 q^{38} - 34 q^{39} + 9 q^{40} - 27 q^{41} + 62 q^{42} - 39 q^{43} - 5 q^{44} - 52 q^{45} - 43 q^{46} + 40 q^{47} + q^{48} + 29 q^{49} - 44 q^{50} - 34 q^{51} - 35 q^{52} - 39 q^{53} - 4 q^{54} - 48 q^{55} + 8 q^{56} - 28 q^{57} + 38 q^{58} + 5 q^{59} - 21 q^{60} - 98 q^{61} + 51 q^{62} + 2 q^{63} + 75 q^{64} - q^{65} + 19 q^{66} - 59 q^{67} + 4 q^{68} - 69 q^{69} - 14 q^{70} - 9 q^{71} - 66 q^{72} - 51 q^{73} + 63 q^{74} - q^{75} - 59 q^{76} - 25 q^{77} + 34 q^{78} - 139 q^{79} - 9 q^{80} + 23 q^{81} + 27 q^{82} + 31 q^{83} - 62 q^{84} - 149 q^{85} + 39 q^{86} + q^{87} + 5 q^{88} - 39 q^{89} + 52 q^{90} - 93 q^{91} + 43 q^{92} - 83 q^{93} - 40 q^{94} + 2 q^{95} - q^{96} - 70 q^{97} - 29 q^{98} - 61 q^{99}+O(q^{100}) \) 75 * q - 75 * q^2 + q^3 + 75 * q^4 - 9 * q^5 - q^6 - 8 * q^7 - 75 * q^8 + 66 * q^9 + 9 * q^10 - 5 * q^11 + q^12 - 35 * q^13 + 8 * q^14 - 21 * q^15 + 75 * q^16 + 4 * q^17 - 66 * q^18 - 59 * q^19 - 9 * q^20 - 62 * q^21 + 5 * q^22 + 43 * q^23 - q^24 + 44 * q^25 + 35 * q^26 + 4 * q^27 - 8 * q^28 - 38 * q^29 + 21 * q^30 - 51 * q^31 - 75 * q^32 - 19 * q^33 - 4 * q^34 + 14 * q^35 + 66 * q^36 - 63 * q^37 + 59 * q^38 - 34 * q^39 + 9 * q^40 - 27 * q^41 + 62 * q^42 - 39 * q^43 - 5 * q^44 - 52 * q^45 - 43 * q^46 + 40 * q^47 + q^48 + 29 * q^49 - 44 * q^50 - 34 * q^51 - 35 * q^52 - 39 * q^53 - 4 * q^54 - 48 * q^55 + 8 * q^56 - 28 * q^57 + 38 * q^58 + 5 * q^59 - 21 * q^60 - 98 * q^61 + 51 * q^62 + 2 * q^63 + 75 * q^64 - q^65 + 19 * q^66 - 59 * q^67 + 4 * q^68 - 69 * q^69 - 14 * q^70 - 9 * q^71 - 66 * q^72 - 51 * q^73 + 63 * q^74 - q^75 - 59 * q^76 - 25 * q^77 + 34 * q^78 - 139 * q^79 - 9 * q^80 + 23 * q^81 + 27 * q^82 + 31 * q^83 - 62 * q^84 - 149 * q^85 + 39 * q^86 + q^87 + 5 * q^88 - 39 * q^89 + 52 * q^90 - 93 * q^91 + 43 * q^92 - 83 * q^93 - 40 * q^94 + 2 * q^95 - q^96 - 70 * q^97 - 29 * q^98 - 61 * q^99

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