LMFDB - Embedded newform 8006.2.a.a.1.5

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

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.86585 q^{3} +1.00000 q^{4} +2.47496 q^{5} -2.86585 q^{6} -3.03619 q^{7} +1.00000 q^{8} +5.21309 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.86585 q^{3} +1.00000 q^{4} +2.47496 q^{5} -2.86585 q^{6} -3.03619 q^{7} +1.00000 q^{8} +5.21309 q^{9} +2.47496 q^{10} -6.59575 q^{11} -2.86585 q^{12} +0.460614 q^{13} -3.03619 q^{14} -7.09286 q^{15} +1.00000 q^{16} -5.96668 q^{17} +5.21309 q^{18} +4.05967 q^{19} +2.47496 q^{20} +8.70125 q^{21} -6.59575 q^{22} +0.745760 q^{23} -2.86585 q^{24} +1.12542 q^{25} +0.460614 q^{26} -6.34238 q^{27} -3.03619 q^{28} +3.45675 q^{29} -7.09286 q^{30} +9.01512 q^{31} +1.00000 q^{32} +18.9024 q^{33} -5.96668 q^{34} -7.51443 q^{35} +5.21309 q^{36} +10.9665 q^{37} +4.05967 q^{38} -1.32005 q^{39} +2.47496 q^{40} +7.01127 q^{41} +8.70125 q^{42} +9.67372 q^{43} -6.59575 q^{44} +12.9022 q^{45} +0.745760 q^{46} +5.68647 q^{47} -2.86585 q^{48} +2.21842 q^{49} +1.12542 q^{50} +17.0996 q^{51} +0.460614 q^{52} -0.978448 q^{53} -6.34238 q^{54} -16.3242 q^{55} -3.03619 q^{56} -11.6344 q^{57} +3.45675 q^{58} -11.1609 q^{59} -7.09286 q^{60} -7.91319 q^{61} +9.01512 q^{62} -15.8279 q^{63} +1.00000 q^{64} +1.14000 q^{65} +18.9024 q^{66} -11.9292 q^{67} -5.96668 q^{68} -2.13723 q^{69} -7.51443 q^{70} -8.40463 q^{71} +5.21309 q^{72} +10.8013 q^{73} +10.9665 q^{74} -3.22528 q^{75} +4.05967 q^{76} +20.0259 q^{77} -1.32005 q^{78} -8.86674 q^{79} +2.47496 q^{80} +2.53703 q^{81} +7.01127 q^{82} -9.06171 q^{83} +8.70125 q^{84} -14.7673 q^{85} +9.67372 q^{86} -9.90652 q^{87} -6.59575 q^{88} -7.90264 q^{89} +12.9022 q^{90} -1.39851 q^{91} +0.745760 q^{92} -25.8360 q^{93} +5.68647 q^{94} +10.0475 q^{95} -2.86585 q^{96} -7.90053 q^{97} +2.21842 q^{98} -34.3843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * 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
$3$	−2.86585	−1.65460	−0.827299	−	0.561761i	$-0.810124\pi$
$3$	−2.86585	−1.65460	−0.827299	+	0.561761i	$0.810124\pi$
$4$	1.00000	0.500000
$5$	2.47496	1.10684	0.553418	−	0.832904i	$-0.313323\pi$
$5$	2.47496	1.10684	0.553418	+	0.832904i	$0.313323\pi$
$6$	−2.86585	−1.16998
$7$	−3.03619	−1.14757	−0.573785	−	0.819006i	$-0.694526\pi$
$7$	−3.03619	−1.14757	−0.573785	+	0.819006i	$0.694526\pi$
$8$	1.00000	0.353553
$9$	5.21309	1.73770
$10$	2.47496	0.782651
$11$	−6.59575	−1.98869	−0.994347	−	0.106175i	$-0.966140\pi$
$11$	−6.59575	−1.98869	−0.994347	+	0.106175i	$0.966140\pi$
$12$	−2.86585	−0.827299
$13$	0.460614	0.127751	0.0638756	−	0.997958i	$-0.479654\pi$
$13$	0.460614	0.127751	0.0638756	+	0.997958i	$0.479654\pi$
$14$	−3.03619	−0.811455
$15$	−7.09286	−1.83137
$16$	1.00000	0.250000
$17$	−5.96668	−1.44713	−0.723567	−	0.690254i	$-0.757499\pi$
$17$	−5.96668	−1.44713	−0.723567	+	0.690254i	$0.757499\pi$
$18$	5.21309	1.22874
$19$	4.05967	0.931352	0.465676	−	0.884955i	$-0.345811\pi$
$19$	4.05967	0.931352	0.465676	+	0.884955i	$0.345811\pi$
$20$	2.47496	0.553418
$21$	8.70125	1.89877
$22$	−6.59575	−1.40622
$23$	0.745760	0.155502	0.0777508	−	0.996973i	$-0.475226\pi$
$23$	0.745760	0.155502	0.0777508	+	0.996973i	$0.475226\pi$
$24$	−2.86585	−0.584989
$25$	1.12542	0.225084
$26$	0.460614	0.0903338
$27$	−6.34238	−1.22059
$28$	−3.03619	−0.573785
$29$	3.45675	0.641902	0.320951	−	0.947096i	$-0.395997\pi$
$29$	3.45675	0.641902	0.320951	+	0.947096i	$0.395997\pi$
$30$	−7.09286	−1.29497
$31$	9.01512	1.61916	0.809581	−	0.587008i	$-0.199694\pi$
$31$	9.01512	1.61916	0.809581	+	0.587008i	$0.199694\pi$
$32$	1.00000	0.176777
$33$	18.9024	3.29049
$34$	−5.96668	−1.02328
$35$	−7.51443	−1.27017
$36$	5.21309	0.868848
$37$	10.9665	1.80289	0.901443	−	0.432899i	$-0.142509\pi$
$37$	10.9665	1.80289	0.901443	+	0.432899i	$0.142509\pi$
$38$	4.05967	0.658565
$39$	−1.32005	−0.211377
$40$	2.47496	0.391325
$41$	7.01127	1.09498	0.547488	−	0.836814i	$-0.315584\pi$
$41$	7.01127	1.09498	0.547488	+	0.836814i	$0.315584\pi$
$42$	8.70125	1.34263
$43$	9.67372	1.47523	0.737614	−	0.675223i	$-0.235952\pi$
$43$	9.67372	1.47523	0.737614	+	0.675223i	$0.235952\pi$
$44$	−6.59575	−0.994347
$45$	12.9022	1.92334
$46$	0.745760	0.109956
$47$	5.68647	0.829456	0.414728	−	0.909945i	$-0.363877\pi$
$47$	5.68647	0.829456	0.414728	+	0.909945i	$0.363877\pi$
$48$	−2.86585	−0.413650
$49$	2.21842	0.316917
$50$	1.12542	0.159158
$51$	17.0996	2.39442
$52$	0.460614	0.0638756
$53$	−0.978448	−0.134400	−0.0672001	−	0.997740i	$-0.521407\pi$
$53$	−0.978448	−0.134400	−0.0672001	+	0.997740i	$0.521407\pi$
$54$	−6.34238	−0.863089
$55$	−16.3242	−2.20116
$56$	−3.03619	−0.405727
$57$	−11.6344	−1.54101
$58$	3.45675	0.453894
$59$	−11.1609	−1.45303	−0.726514	−	0.687152i	$-0.758861\pi$
$59$	−11.1609	−1.45303	−0.726514	+	0.687152i	$0.758861\pi$
$60$	−7.09286	−0.915684
$61$	−7.91319	−1.01318	−0.506590	−	0.862187i	$-0.669094\pi$
$61$	−7.91319	−1.01318	−0.506590	+	0.862187i	$0.669094\pi$
$62$	9.01512	1.14492
$63$	−15.8279	−1.99413
$64$	1.00000	0.125000
$65$	1.14000	0.141400
$66$	18.9024	2.32673
$67$	−11.9292	−1.45738	−0.728692	−	0.684841i	$-0.759871\pi$
$67$	−11.9292	−1.45738	−0.728692	+	0.684841i	$0.759871\pi$
$68$	−5.96668	−0.723567
$69$	−2.13723	−0.257293
$70$	−7.51443	−0.898147
$71$	−8.40463	−0.997446	−0.498723	−	0.866762i	$-0.666197\pi$
$71$	−8.40463	−0.997446	−0.498723	+	0.866762i	$0.666197\pi$
$72$	5.21309	0.614368
$73$	10.8013	1.26420	0.632100	−	0.774887i	$-0.282193\pi$
$73$	10.8013	1.26420	0.632100	+	0.774887i	$0.282193\pi$
$74$	10.9665	1.27483
$75$	−3.22528	−0.372424
$76$	4.05967	0.465676
$77$	20.0259	2.28217
$78$	−1.32005	−0.149466
$79$	−8.86674	−0.997586	−0.498793	−	0.866721i	$-0.666223\pi$
$79$	−8.86674	−0.997586	−0.498793	+	0.866721i	$0.666223\pi$
$80$	2.47496	0.276709
$81$	2.53703	0.281892
$82$	7.01127	0.774265
$83$	−9.06171	−0.994652	−0.497326	−	0.867564i	$-0.665685\pi$
$83$	−9.06171	−0.994652	−0.497326	+	0.867564i	$0.665685\pi$
$84$	8.70125	0.949384
$85$	−14.7673	−1.60174
$86$	9.67372	1.04314
$87$	−9.90652	−1.06209
$88$	−6.59575	−0.703110
$89$	−7.90264	−0.837678	−0.418839	−	0.908061i	$-0.637563\pi$
$89$	−7.90264	−0.837678	−0.418839	+	0.908061i	$0.637563\pi$
$90$	12.9022	1.36001
$91$	−1.39851	−0.146604
$92$	0.745760	0.0777508
$93$	−25.8360	−2.67906
$94$	5.68647	0.586514
$95$	10.0475	1.03085
$96$	−2.86585	−0.292494
$97$	−7.90053	−0.802177	−0.401089	−	0.916039i	$-0.631368\pi$
$97$	−7.90053	−0.802177	−0.401089	+	0.916039i	$0.631368\pi$
$98$	2.21842	0.224094
$99$	−34.3843	−3.45575
$100$	1.12542	0.112542
$101$	15.0989	1.50240	0.751201	−	0.660074i	$-0.229475\pi$
$101$	15.0989	1.50240	0.751201	+	0.660074i	$0.229475\pi$
$102$	17.0996	1.69311
$103$	−14.2976	−1.40878	−0.704392	−	0.709811i	$-0.748780\pi$
$103$	−14.2976	−1.40878	−0.704392	+	0.709811i	$0.748780\pi$
$104$	0.460614	0.0451669
$105$	21.5352	2.10162
$106$	−0.978448	−0.0950353
$107$	5.04246	0.487473	0.243737	−	0.969841i	$-0.421627\pi$
$107$	5.04246	0.487473	0.243737	+	0.969841i	$0.421627\pi$
$108$	−6.34238	−0.610296
$109$	−7.31442	−0.700594	−0.350297	−	0.936639i	$-0.613919\pi$
$109$	−7.31442	−0.700594	−0.350297	+	0.936639i	$0.613919\pi$
$110$	−16.3242	−1.55645
$111$	−31.4284	−2.98305
$112$	−3.03619	−0.286893
$113$	4.13424	0.388917	0.194458	−	0.980911i	$-0.437705\pi$
$113$	4.13424	0.388917	0.194458	+	0.980911i	$0.437705\pi$
$114$	−11.6344	−1.08966
$115$	1.84572	0.172115
$116$	3.45675	0.320951
$117$	2.40122	0.221993
$118$	−11.1609	−1.02745
$119$	18.1160	1.66069
$120$	−7.09286	−0.647486
$121$	32.5040	2.95491
$122$	−7.91319	−0.716427
$123$	−20.0932	−1.81175
$124$	9.01512	0.809581
$125$	−9.58943	−0.857704
$126$	−15.8279	−1.41006
$127$	13.3895	1.18813	0.594065	−	0.804417i	$-0.297522\pi$
$127$	13.3895	1.18813	0.594065	+	0.804417i	$0.297522\pi$
$128$	1.00000	0.0883883
$129$	−27.7234	−2.44091
$130$	1.14000	0.0999846
$131$	−8.72786	−0.762557	−0.381279	−	0.924460i	$-0.624516\pi$
$131$	−8.72786	−0.762557	−0.381279	+	0.924460i	$0.624516\pi$
$132$	18.9024	1.64525
$133$	−12.3259	−1.06879
$134$	−11.9292	−1.03053
$135$	−15.6971	−1.35099
$136$	−5.96668	−0.511639
$137$	−15.4561	−1.32051	−0.660253	−	0.751043i	$-0.729551\pi$
$137$	−15.4561	−1.32051	−0.660253	+	0.751043i	$0.729551\pi$
$138$	−2.13723	−0.181934
$139$	−8.65975	−0.734510	−0.367255	−	0.930120i	$-0.619702\pi$
$139$	−8.65975	−0.734510	−0.367255	+	0.930120i	$0.619702\pi$
$140$	−7.51443	−0.635085
$141$	−16.2966	−1.37242
$142$	−8.40463	−0.705301
$143$	−3.03810	−0.254058
$144$	5.21309	0.434424
$145$	8.55531	0.710480
$146$	10.8013	0.893925
$147$	−6.35766	−0.524371
$148$	10.9665	0.901443
$149$	7.18075	0.588270	0.294135	−	0.955764i	$-0.404968\pi$
$149$	7.18075	0.588270	0.294135	+	0.955764i	$0.404968\pi$
$150$	−3.22528	−0.263343
$151$	−16.9642	−1.38053	−0.690264	−	0.723558i	$-0.742506\pi$
$151$	−16.9642	−1.38053	−0.690264	+	0.723558i	$0.742506\pi$
$152$	4.05967	0.329283
$153$	−31.1049	−2.51468
$154$	20.0259	1.61374
$155$	22.3120	1.79215
$156$	−1.32005	−0.105689
$157$	−8.86792	−0.707737	−0.353868	−	0.935295i	$-0.615134\pi$
$157$	−8.86792	−0.707737	−0.353868	+	0.935295i	$0.615134\pi$
$158$	−8.86674	−0.705400
$159$	2.80408	0.222378
$160$	2.47496	0.195663
$161$	−2.26426	−0.178449
$162$	2.53703	0.199328
$163$	−17.3827	−1.36152	−0.680758	−	0.732508i	$-0.738349\pi$
$163$	−17.3827	−1.36152	−0.680758	+	0.732508i	$0.738349\pi$
$164$	7.01127	0.547488
$165$	46.7827	3.64203
$166$	−9.06171	−0.703325
$167$	5.83608	0.451610	0.225805	−	0.974173i	$-0.427499\pi$
$167$	5.83608	0.451610	0.225805	+	0.974173i	$0.427499\pi$
$168$	8.70125	0.671316
$169$	−12.7878	−0.983680
$170$	−14.7673	−1.13260
$171$	21.1634	1.61841
$172$	9.67372	0.737614
$173$	13.7718	1.04705	0.523525	−	0.852010i	$-0.324617\pi$
$173$	13.7718	1.04705	0.523525	+	0.852010i	$0.324617\pi$
$174$	−9.90652	−0.751012
$175$	−3.41698	−0.258300
$176$	−6.59575	−0.497174
$177$	31.9855	2.40418
$178$	−7.90264	−0.592328
$179$	−4.40114	−0.328957	−0.164478	−	0.986381i	$-0.552594\pi$
$179$	−4.40114	−0.328957	−0.164478	+	0.986381i	$0.552594\pi$
$180$	12.9022	0.961672
$181$	−1.88849	−0.140370	−0.0701851	−	0.997534i	$-0.522359\pi$
$181$	−1.88849	−0.140370	−0.0701851	+	0.997534i	$0.522359\pi$
$182$	−1.39851	−0.103664
$183$	22.6780	1.67641
$184$	0.745760	0.0549781
$185$	27.1417	1.99550
$186$	−25.8360	−1.89438
$187$	39.3548	2.87791
$188$	5.68647	0.414728
$189$	19.2566	1.40071
$190$	10.0475	0.728923
$191$	9.26059	0.670073	0.335036	−	0.942205i	$-0.391251\pi$
$191$	9.26059	0.670073	0.335036	+	0.942205i	$0.391251\pi$
$192$	−2.86585	−0.206825
$193$	−3.52895	−0.254019	−0.127010	−	0.991901i	$-0.540538\pi$
$193$	−3.52895	−0.254019	−0.127010	+	0.991901i	$0.540538\pi$
$194$	−7.90053	−0.567225
$195$	−3.26707	−0.233960
$196$	2.21842	0.158459
$197$	2.04530	0.145722	0.0728608	−	0.997342i	$-0.476787\pi$
$197$	2.04530	0.145722	0.0728608	+	0.997342i	$0.476787\pi$
$198$	−34.3843	−2.44358
$199$	−13.6360	−0.966628	−0.483314	−	0.875447i	$-0.660567\pi$
$199$	−13.6360	−0.966628	−0.483314	+	0.875447i	$0.660567\pi$
$200$	1.12542	0.0795792
$201$	34.1873	2.41139
$202$	15.0989	1.06236
$203$	−10.4953	−0.736628
$204$	17.0996	1.19721
$205$	17.3526	1.21196
$206$	−14.2976	−0.996161
$207$	3.88771	0.270215
$208$	0.460614	0.0319378
$209$	−26.7766	−1.85218
$210$	21.5352	1.48607
$211$	−15.2818	−1.05204	−0.526022	−	0.850471i	$-0.676317\pi$
$211$	−15.2818	−1.05204	−0.526022	+	0.850471i	$0.676317\pi$
$212$	−0.978448	−0.0672001
$213$	24.0864	1.65037
$214$	5.04246	0.344696
$215$	23.9420	1.63283
$216$	−6.34238	−0.431544
$217$	−27.3716	−1.85810
$218$	−7.31442	−0.495395
$219$	−30.9550	−2.09174
$220$	−16.3242	−1.10058
$221$	−2.74834	−0.184873
$222$	−31.4284	−2.10934
$223$	−7.84028	−0.525024	−0.262512	−	0.964929i	$-0.584551\pi$
$223$	−7.84028	−0.525024	−0.262512	+	0.964929i	$0.584551\pi$
$224$	−3.03619	−0.202864
$225$	5.86691	0.391128
$226$	4.13424	0.275006
$227$	−4.49660	−0.298450	−0.149225	−	0.988803i	$-0.547678\pi$
$227$	−4.49660	−0.298450	−0.149225	+	0.988803i	$0.547678\pi$
$228$	−11.6344	−0.770507
$229$	−17.6798	−1.16831	−0.584157	−	0.811640i	$-0.698575\pi$
$229$	−17.6798	−1.16831	−0.584157	+	0.811640i	$0.698575\pi$
$230$	1.84572	0.121703
$231$	−57.3913	−3.77607
$232$	3.45675	0.226947
$233$	24.5478	1.60818	0.804090	−	0.594507i	$-0.202653\pi$
$233$	24.5478	1.60818	0.804090	+	0.594507i	$0.202653\pi$
$234$	2.40122	0.156973
$235$	14.0738	0.918071
$236$	−11.1609	−0.726514
$237$	25.4107	1.65060
$238$	18.1160	1.17428
$239$	4.37256	0.282838	0.141419	−	0.989950i	$-0.454834\pi$
$239$	4.37256	0.282838	0.141419	+	0.989950i	$0.454834\pi$
$240$	−7.09286	−0.457842
$241$	−26.2739	−1.69245	−0.846227	−	0.532823i	$-0.821131\pi$
$241$	−26.2739	−1.69245	−0.846227	+	0.532823i	$0.821131\pi$
$242$	32.5040	2.08944
$243$	11.7564	0.754173
$244$	−7.91319	−0.506590
$245$	5.49050	0.350775
$246$	−20.0932	−1.28110
$247$	1.86994	0.118981
$248$	9.01512	0.572461
$249$	25.9695	1.64575
$250$	−9.58943	−0.606489
$251$	−0.206223	−0.0130167	−0.00650835	−	0.999979i	$-0.502072\pi$
$251$	−0.206223	−0.0130167	−0.00650835	+	0.999979i	$0.502072\pi$
$252$	−15.8279	−0.997064
$253$	−4.91885	−0.309245
$254$	13.3895	0.840135
$255$	42.3208	2.65023
$256$	1.00000	0.0625000
$257$	17.9972	1.12263	0.561317	−	0.827601i	$-0.310295\pi$
$257$	17.9972	1.12263	0.561317	+	0.827601i	$0.310295\pi$
$258$	−27.7234	−1.72598
$259$	−33.2964	−2.06894
$260$	1.14000	0.0706998
$261$	18.0203	1.11543
$262$	−8.72786	−0.539209
$263$	18.5874	1.14615	0.573073	−	0.819505i	$-0.305751\pi$
$263$	18.5874	1.14615	0.573073	+	0.819505i	$0.305751\pi$
$264$	18.9024	1.16336
$265$	−2.42162	−0.148759
$266$	−12.3259	−0.755750
$267$	22.6478	1.38602
$268$	−11.9292	−0.728692
$269$	5.52662	0.336964	0.168482	−	0.985705i	$-0.446113\pi$
$269$	5.52662	0.336964	0.168482	+	0.985705i	$0.446113\pi$
$270$	−15.6971	−0.955297
$271$	−9.97466	−0.605917	−0.302959	−	0.953004i	$-0.597974\pi$
$271$	−9.97466	−0.605917	−0.302959	+	0.953004i	$0.597974\pi$
$272$	−5.96668	−0.361783
$273$	4.00791	0.242570
$274$	−15.4561	−0.933739
$275$	−7.42299	−0.447623
$276$	−2.13723	−0.128646
$277$	−28.4723	−1.71073	−0.855367	−	0.518023i	$-0.826668\pi$
$277$	−28.4723	−1.71073	−0.855367	+	0.518023i	$0.826668\pi$
$278$	−8.65975	−0.519377
$279$	46.9966	2.81361
$280$	−7.51443	−0.449073
$281$	−24.4047	−1.45586	−0.727931	−	0.685651i	$-0.759518\pi$
$281$	−24.4047	−1.45586	−0.727931	+	0.685651i	$0.759518\pi$
$282$	−16.2966	−0.970446
$283$	−15.6338	−0.929331	−0.464665	−	0.885486i	$-0.653825\pi$
$283$	−15.6338	−0.929331	−0.464665	+	0.885486i	$0.653825\pi$
$284$	−8.40463	−0.498723
$285$	−28.7947	−1.70565
$286$	−3.03810	−0.179646
$287$	−21.2875	−1.25656
$288$	5.21309	0.307184
$289$	18.6013	1.09420
$290$	8.55531	0.502385
$291$	22.6417	1.32728
$292$	10.8013	0.632100
$293$	−2.18118	−0.127426	−0.0637129	−	0.997968i	$-0.520294\pi$
$293$	−2.18118	−0.127426	−0.0637129	+	0.997968i	$0.520294\pi$
$294$	−6.35766	−0.370786
$295$	−27.6228	−1.60826
$296$	10.9665	0.637416
$297$	41.8328	2.42738
$298$	7.18075	0.415970
$299$	0.343507	0.0198655
$300$	−3.22528	−0.186212
$301$	−29.3712	−1.69293
$302$	−16.9642	−0.976181
$303$	−43.2713	−2.48587
$304$	4.05967	0.232838
$305$	−19.5848	−1.12142
$306$	−31.1049	−1.77815
$307$	−10.6098	−0.605531	−0.302765	−	0.953065i	$-0.597910\pi$
$307$	−10.6098	−0.605531	−0.302765	+	0.953065i	$0.597910\pi$
$308$	20.0259	1.14108
$309$	40.9748	2.33097
$310$	22.3120	1.26724
$311$	15.0157	0.851463	0.425732	−	0.904849i	$-0.360017\pi$
$311$	15.0157	0.851463	0.425732	+	0.904849i	$0.360017\pi$
$312$	−1.32005	−0.0747331
$313$	−5.00770	−0.283052	−0.141526	−	0.989935i	$-0.545201\pi$
$313$	−5.00770	−0.283052	−0.141526	+	0.989935i	$0.545201\pi$
$314$	−8.86792	−0.500446
$315$	−39.1734	−2.20717
$316$	−8.86674	−0.498793
$317$	1.03664	0.0582235	0.0291117	−	0.999576i	$-0.490732\pi$
$317$	1.03664	0.0582235	0.0291117	+	0.999576i	$0.490732\pi$
$318$	2.80408	0.157245
$319$	−22.7999	−1.27655
$320$	2.47496	0.138354
$321$	−14.4509	−0.806573
$322$	−2.26426	−0.126183
$323$	−24.2228	−1.34779
$324$	2.53703	0.140946
$325$	0.518384	0.0287548
$326$	−17.3827	−0.962737
$327$	20.9620	1.15920
$328$	7.01127	0.387132
$329$	−17.2652	−0.951859
$330$	46.7827	2.57531
$331$	−7.70069	−0.423268	−0.211634	−	0.977349i	$-0.567879\pi$
$331$	−7.70069	−0.423268	−0.211634	+	0.977349i	$0.567879\pi$
$332$	−9.06171	−0.497326
$333$	57.1695	3.13287
$334$	5.83608	0.319336
$335$	−29.5243	−1.61308
$336$	8.70125	0.474692
$337$	9.32450	0.507938	0.253969	−	0.967212i	$-0.418264\pi$
$337$	9.32450	0.507938	0.253969	+	0.967212i	$0.418264\pi$
$338$	−12.7878	−0.695567
$339$	−11.8481	−0.643501
$340$	−14.7673	−0.800869
$341$	−59.4615	−3.22002
$342$	21.1634	1.14439
$343$	14.5178	0.783885
$344$	9.67372	0.521572
$345$	−5.28957	−0.284781
$346$	13.7718	0.740376
$347$	16.2574	0.872745	0.436372	−	0.899766i	$-0.356263\pi$
$347$	16.2574	0.872745	0.436372	+	0.899766i	$0.356263\pi$
$348$	−9.90652	−0.531045
$349$	−7.88230	−0.421930	−0.210965	−	0.977494i	$-0.567661\pi$
$349$	−7.88230	−0.421930	−0.210965	+	0.977494i	$0.567661\pi$
$350$	−3.41698	−0.182645
$351$	−2.92139	−0.155932
$352$	−6.59575	−0.351555
$353$	−23.3718	−1.24396	−0.621978	−	0.783035i	$-0.713671\pi$
$353$	−23.3718	−1.24396	−0.621978	+	0.783035i	$0.713671\pi$
$354$	31.9855	1.70001
$355$	−20.8011	−1.10401
$356$	−7.90264	−0.418839
$357$	−51.9176	−2.74777
$358$	−4.40114	−0.232607
$359$	−28.2644	−1.49174	−0.745868	−	0.666093i	$-0.767965\pi$
$359$	−28.2644	−1.49174	−0.745868	+	0.666093i	$0.767965\pi$
$360$	12.9022	0.680005
$361$	−2.51908	−0.132583
$362$	−1.88849	−0.0992567
$363$	−93.1515	−4.88919
$364$	−1.39851	−0.0733018
$365$	26.7328	1.39926
$366$	22.6780	1.18540
$367$	5.61548	0.293126	0.146563	−	0.989201i	$-0.453179\pi$
$367$	5.61548	0.293126	0.146563	+	0.989201i	$0.453179\pi$
$368$	0.745760	0.0388754
$369$	36.5504	1.90274
$370$	27.1417	1.41103
$371$	2.97075	0.154234
$372$	−25.8360	−1.33953
$373$	30.4929	1.57886	0.789430	−	0.613840i	$-0.210376\pi$
$373$	30.4929	1.57886	0.789430	+	0.613840i	$0.210376\pi$
$374$	39.3548	2.03499
$375$	27.4818	1.41916
$376$	5.68647	0.293257
$377$	1.59223	0.0820038
$378$	19.2566	0.990455
$379$	31.7108	1.62887	0.814436	−	0.580253i	$-0.197047\pi$
$379$	31.7108	1.62887	0.814436	+	0.580253i	$0.197047\pi$
$380$	10.0475	0.515427
$381$	−38.3724	−1.96588
$382$	9.26059	0.473813
$383$	2.11021	0.107827	0.0539134	−	0.998546i	$-0.482831\pi$
$383$	2.11021	0.107827	0.0539134	+	0.998546i	$0.482831\pi$
$384$	−2.86585	−0.146247
$385$	49.5634	2.52598
$386$	−3.52895	−0.179619
$387$	50.4299	2.56350
$388$	−7.90053	−0.401089
$389$	21.2357	1.07669	0.538346	−	0.842724i	$-0.319049\pi$
$389$	21.2357	1.07669	0.538346	+	0.842724i	$0.319049\pi$
$390$	−3.26707	−0.165434
$391$	−4.44971	−0.225032
$392$	2.21842	0.112047
$393$	25.0127	1.26173
$394$	2.04530	0.103041
$395$	−21.9448	−1.10416
$396$	−34.3843	−1.72787
$397$	19.1503	0.961125	0.480563	−	0.876960i	$-0.340432\pi$
$397$	19.1503	0.961125	0.480563	+	0.876960i	$0.340432\pi$
$398$	−13.6360	−0.683509
$399$	35.3242	1.76842
$400$	1.12542	0.0562710
$401$	13.8212	0.690199	0.345100	−	0.938566i	$-0.387845\pi$
$401$	13.8212	0.690199	0.345100	+	0.938566i	$0.387845\pi$
$402$	34.1873	1.70511
$403$	4.15249	0.206850
$404$	15.0989	0.751201
$405$	6.27905	0.312009
$406$	−10.4953	−0.520875
$407$	−72.3325	−3.58539
$408$	17.0996	0.846557
$409$	11.7677	0.581873	0.290936	−	0.956742i	$-0.406033\pi$
$409$	11.7677	0.581873	0.290936	+	0.956742i	$0.406033\pi$
$410$	17.3526	0.856983
$411$	44.2949	2.18491
$412$	−14.2976	−0.704392
$413$	33.8866	1.66745
$414$	3.88771	0.191071
$415$	−22.4274	−1.10092
$416$	0.460614	0.0225834
$417$	24.8175	1.21532
$418$	−26.7766	−1.30969
$419$	13.8631	0.677257	0.338628	−	0.940920i	$-0.390037\pi$
$419$	13.8631	0.677257	0.338628	+	0.940920i	$0.390037\pi$
$420$	21.5352	1.05081
$421$	8.87403	0.432494	0.216247	−	0.976339i	$-0.430618\pi$
$421$	8.87403	0.432494	0.216247	+	0.976339i	$0.430618\pi$
$422$	−15.2818	−0.743908
$423$	29.6441	1.44134
$424$	−0.978448	−0.0475176
$425$	−6.71502	−0.325727
$426$	24.0864	1.16699
$427$	24.0259	1.16270
$428$	5.04246	0.243737
$429$	8.70672	0.420365
$430$	23.9420	1.15459
$431$	3.22135	0.155167	0.0775834	−	0.996986i	$-0.475280\pi$
$431$	3.22135	0.155167	0.0775834	+	0.996986i	$0.475280\pi$
$432$	−6.34238	−0.305148
$433$	−6.32566	−0.303992	−0.151996	−	0.988381i	$-0.548570\pi$
$433$	−6.32566	−0.303992	−0.151996	+	0.988381i	$0.548570\pi$
$434$	−27.3716	−1.31388
$435$	−24.5182	−1.17556
$436$	−7.31442	−0.350297
$437$	3.02754	0.144827
$438$	−30.9550	−1.47909
$439$	−25.9743	−1.23969	−0.619843	−	0.784725i	$-0.712804\pi$
$439$	−25.9743	−1.23969	−0.619843	+	0.784725i	$0.712804\pi$
$440$	−16.3242	−0.778227
$441$	11.5648	0.550706
$442$	−2.74834	−0.130725
$443$	−12.6907	−0.602951	−0.301476	−	0.953474i	$-0.597479\pi$
$443$	−12.6907	−0.602951	−0.301476	+	0.953474i	$0.597479\pi$
$444$	−31.4284	−1.49153
$445$	−19.5587	−0.927171
$446$	−7.84028	−0.371248
$447$	−20.5789	−0.973351
$448$	−3.03619	−0.143446
$449$	−4.72237	−0.222863	−0.111431	−	0.993772i	$-0.535544\pi$
$449$	−4.72237	−0.222863	−0.111431	+	0.993772i	$0.535544\pi$
$450$	5.86691	0.276569
$451$	−46.2446	−2.17757
$452$	4.13424	0.194458
$453$	48.6169	2.28422
$454$	−4.49660	−0.211036
$455$	−3.46125	−0.162266
$456$	−11.6344	−0.544831
$457$	6.36898	0.297928	0.148964	−	0.988843i	$-0.452406\pi$
$457$	6.36898	0.297928	0.148964	+	0.988843i	$0.452406\pi$
$458$	−17.6798	−0.826123
$459$	37.8430	1.76636
$460$	1.84572	0.0860574
$461$	−30.6362	−1.42687	−0.713435	−	0.700721i	$-0.752862\pi$
$461$	−30.6362	−1.42687	−0.713435	+	0.700721i	$0.752862\pi$
$462$	−57.3913	−2.67008
$463$	−32.8235	−1.52544	−0.762719	−	0.646730i	$-0.776136\pi$
$463$	−32.8235	−1.52544	−0.762719	+	0.646730i	$0.776136\pi$
$464$	3.45675	0.160476
$465$	−63.9429	−2.96528
$466$	24.5478	1.13716
$467$	16.1110	0.745526	0.372763	−	0.927927i	$-0.378410\pi$
$467$	16.1110	0.745526	0.372763	+	0.927927i	$0.378410\pi$
$468$	2.40122	0.110996
$469$	36.2193	1.67245
$470$	14.0738	0.649175
$471$	25.4141	1.17102
$472$	−11.1609	−0.513723
$473$	−63.8055	−2.93378
$474$	25.4107	1.16715
$475$	4.56883	0.209632
$476$	18.1160	0.830344
$477$	−5.10074	−0.233547
$478$	4.37256	0.199996
$479$	35.7260	1.63236	0.816182	−	0.577795i	$-0.196087\pi$
$479$	35.7260	1.63236	0.816182	+	0.577795i	$0.196087\pi$
$480$	−7.09286	−0.323743
$481$	5.05133	0.230321
$482$	−26.2739	−1.19675
$483$	6.48904	0.295262
$484$	32.5040	1.47745
$485$	−19.5535	−0.887878
$486$	11.7564	0.533281
$487$	10.3524	0.469113	0.234556	−	0.972103i	$-0.424636\pi$
$487$	10.3524	0.469113	0.234556	+	0.972103i	$0.424636\pi$
$488$	−7.91319	−0.358213
$489$	49.8161	2.25276
$490$	5.49050	0.248036
$491$	0.148822	0.00671625	0.00335812	−	0.999994i	$-0.498931\pi$
$491$	0.148822	0.00671625	0.00335812	+	0.999994i	$0.498931\pi$
$492$	−20.0932	−0.905873
$493$	−20.6253	−0.928918
$494$	1.86994	0.0841326
$495$	−85.0996	−3.82494
$496$	9.01512	0.404791
$497$	25.5180	1.14464
$498$	25.9695	1.16372
$499$	−13.6067	−0.609118	−0.304559	−	0.952494i	$-0.598509\pi$
$499$	−13.6067	−0.609118	−0.304559	+	0.952494i	$0.598509\pi$
$500$	−9.58943	−0.428852
$501$	−16.7253	−0.747233
$502$	−0.206223	−0.00920419
$503$	−26.7448	−1.19249	−0.596246	−	0.802802i	$-0.703342\pi$
$503$	−26.7448	−1.19249	−0.596246	+	0.802802i	$0.703342\pi$
$504$	−15.8279	−0.705031
$505$	37.3693	1.66291
$506$	−4.91885	−0.218669
$507$	36.6480	1.62759
$508$	13.3895	0.594065
$509$	−6.58416	−0.291838	−0.145919	−	0.989297i	$-0.546614\pi$
$509$	−6.58416	−0.291838	−0.145919	+	0.989297i	$0.546614\pi$
$510$	42.3208	1.87400
$511$	−32.7948	−1.45076
$512$	1.00000	0.0441942
$513$	−25.7480	−1.13680
$514$	17.9972	0.793822
$515$	−35.3860	−1.55929
$516$	−27.7234	−1.22045
$517$	−37.5065	−1.64954
$518$	−33.2964	−1.46296
$519$	−39.4679	−1.73245
$520$	1.14000	0.0499923
$521$	31.6859	1.38818	0.694092	−	0.719886i	$-0.255806\pi$
$521$	31.6859	1.38818	0.694092	+	0.719886i	$0.255806\pi$
$522$	18.0203	0.788729
$523$	−6.21216	−0.271639	−0.135819	−	0.990734i	$-0.543367\pi$
$523$	−6.21216	−0.271639	−0.135819	+	0.990734i	$0.543367\pi$
$524$	−8.72786	−0.381279
$525$	9.79256	0.427382
$526$	18.5874	0.810447
$527$	−53.7904	−2.34314
$528$	18.9024	0.822623
$529$	−22.4438	−0.975819
$530$	−2.42162	−0.105188
$531$	−58.1829	−2.52492
$532$	−12.3259	−0.534396
$533$	3.22949	0.139885
$534$	22.6478	0.980065
$535$	12.4799	0.539552
$536$	−11.9292	−0.515263
$537$	12.6130	0.544291
$538$	5.52662	0.238270
$539$	−14.6322	−0.630252
$540$	−15.6971	−0.675497
$541$	−16.5704	−0.712418	−0.356209	−	0.934406i	$-0.615931\pi$
$541$	−16.5704	−0.712418	−0.356209	+	0.934406i	$0.615931\pi$
$542$	−9.97466	−0.428448
$543$	5.41212	0.232256
$544$	−5.96668	−0.255819
$545$	−18.1029	−0.775442
$546$	4.00791	0.171523
$547$	−17.1902	−0.735002	−0.367501	−	0.930023i	$-0.619786\pi$
$547$	−17.1902	−0.735002	−0.367501	+	0.930023i	$0.619786\pi$
$548$	−15.4561	−0.660253
$549$	−41.2522	−1.76060
$550$	−7.42299	−0.316517
$551$	14.0333	0.597837
$552$	−2.13723	−0.0909668
$553$	26.9211	1.14480
$554$	−28.4723	−1.20967
$555$	−77.7840	−3.30175
$556$	−8.65975	−0.367255
$557$	−33.2237	−1.40773	−0.703866	−	0.710332i	$-0.748545\pi$
$557$	−33.2237	−1.40773	−0.703866	+	0.710332i	$0.748545\pi$
$558$	46.9966	1.98953
$559$	4.45585	0.188462
$560$	−7.51443	−0.317543
$561$	−112.785	−4.76178
$562$	−24.4047	−1.02945
$563$	30.8095	1.29847	0.649233	−	0.760590i	$-0.275090\pi$
$563$	30.8095	1.29847	0.649233	+	0.760590i	$0.275090\pi$
$564$	−16.2966	−0.686209
$565$	10.2321	0.430467
$566$	−15.6338	−0.657136
$567$	−7.70290	−0.323491
$568$	−8.40463	−0.352650
$569$	26.1141	1.09476	0.547379	−	0.836885i	$-0.315626\pi$
$569$	26.1141	1.09476	0.547379	+	0.836885i	$0.315626\pi$
$570$	−28.7947	−1.20608
$571$	−26.7061	−1.11762	−0.558808	−	0.829297i	$-0.688741\pi$
$571$	−26.7061	−1.11762	−0.558808	+	0.829297i	$0.688741\pi$
$572$	−3.03810	−0.127029
$573$	−26.5394	−1.10870
$574$	−21.2875	−0.888523
$575$	0.839293	0.0350009
$576$	5.21309	0.217212
$577$	−22.2291	−0.925408	−0.462704	−	0.886513i	$-0.653121\pi$
$577$	−22.2291	−0.925408	−0.462704	+	0.886513i	$0.653121\pi$
$578$	18.6013	0.773713
$579$	10.1134	0.420300
$580$	8.55531	0.355240
$581$	27.5130	1.14143
$582$	22.6417	0.938529
$583$	6.45360	0.267281
$584$	10.8013	0.446962
$585$	5.94292	0.245710
$586$	−2.18118	−0.0901037
$587$	24.1671	0.997481	0.498740	−	0.866751i	$-0.333796\pi$
$587$	24.1671	0.997481	0.498740	+	0.866751i	$0.333796\pi$
$588$	−6.35766	−0.262186
$589$	36.5984	1.50801
$590$	−27.6228	−1.13721
$591$	−5.86152	−0.241111
$592$	10.9665	0.450721
$593$	−0.632539	−0.0259752	−0.0129876	−	0.999916i	$-0.504134\pi$
$593$	−0.632539	−0.0259752	−0.0129876	+	0.999916i	$0.504134\pi$
$594$	41.8328	1.71642
$595$	44.8362	1.83811
$596$	7.18075	0.294135
$597$	39.0786	1.59938
$598$	0.343507	0.0140471
$599$	−6.83850	−0.279414	−0.139707	−	0.990193i	$-0.544616\pi$
$599$	−6.83850	−0.279414	−0.139707	+	0.990193i	$0.544616\pi$
$600$	−3.22528	−0.131672
$601$	0.553980	0.0225973	0.0112986	−	0.999936i	$-0.496403\pi$
$601$	0.553980	0.0225973	0.0112986	+	0.999936i	$0.496403\pi$
$602$	−29.3712	−1.19708
$603$	−62.1880	−2.53249
$604$	−16.9642	−0.690264
$605$	80.4460	3.27060
$606$	−43.2713	−1.75778
$607$	18.1993	0.738689	0.369344	−	0.929293i	$-0.379582\pi$
$607$	18.1993	0.738689	0.369344	+	0.929293i	$0.379582\pi$
$608$	4.05967	0.164641
$609$	30.0780	1.21882
$610$	−19.5848	−0.792966
$611$	2.61926	0.105964
$612$	−31.1049	−1.25734
$613$	−33.2885	−1.34451	−0.672255	−	0.740320i	$-0.734674\pi$
$613$	−33.2885	−1.34451	−0.672255	+	0.740320i	$0.734674\pi$
$614$	−10.6098	−0.428175
$615$	−49.7299	−2.00530
$616$	20.0259	0.806868
$617$	−0.494258	−0.0198981	−0.00994903	−	0.999951i	$-0.503167\pi$
$617$	−0.494258	−0.0198981	−0.00994903	+	0.999951i	$0.503167\pi$
$618$	40.9748	1.64825
$619$	−11.8474	−0.476185	−0.238093	−	0.971242i	$-0.576522\pi$
$619$	−11.8474	−0.476185	−0.238093	+	0.971242i	$0.576522\pi$
$620$	22.3120	0.896073
$621$	−4.72989	−0.189804
$622$	15.0157	0.602076
$623$	23.9939	0.961294
$624$	−1.32005	−0.0528443
$625$	−29.3605	−1.17442
$626$	−5.00770	−0.200148
$627$	76.7377	3.06461
$628$	−8.86792	−0.353868
$629$	−65.4338	−2.60902
$630$	−39.1734	−1.56071
$631$	−31.1470	−1.23994	−0.619972	−	0.784624i	$-0.712856\pi$
$631$	−31.1470	−1.23994	−0.619972	+	0.784624i	$0.712856\pi$
$632$	−8.86674	−0.352700
$633$	43.7954	1.74071
$634$	1.03664	0.0411702
$635$	33.1386	1.31506
$636$	2.80408	0.111189
$637$	1.02184	0.0404866
$638$	−22.7999	−0.902656
$639$	−43.8141	−1.73326
$640$	2.47496	0.0978313
$641$	−3.95936	−0.156385	−0.0781927	−	0.996938i	$-0.524915\pi$
$641$	−3.95936	−0.156385	−0.0781927	+	0.996938i	$0.524915\pi$
$642$	−14.4509	−0.570333
$643$	36.7901	1.45086	0.725430	−	0.688296i	$-0.241641\pi$
$643$	36.7901	1.45086	0.725430	+	0.688296i	$0.241641\pi$
$644$	−2.26426	−0.0892245
$645$	−68.6143	−2.70168
$646$	−24.2228	−0.953032
$647$	14.7910	0.581496	0.290748	−	0.956800i	$-0.406096\pi$
$647$	14.7910	0.581496	0.290748	+	0.956800i	$0.406096\pi$
$648$	2.53703	0.0996640
$649$	73.6147	2.88963
$650$	0.518384	0.0203327
$651$	78.4428	3.07441
$652$	−17.3827	−0.680758
$653$	−9.75690	−0.381817	−0.190908	−	0.981608i	$-0.561143\pi$
$653$	−9.75690	−0.381817	−0.190908	+	0.981608i	$0.561143\pi$
$654$	20.9620	0.819680
$655$	−21.6011	−0.844025
$656$	7.01127	0.273744
$657$	56.3083	2.19680
$658$	−17.2652	−0.673066
$659$	32.1743	1.25333	0.626666	−	0.779288i	$-0.284419\pi$
$659$	32.1743	1.25333	0.626666	+	0.779288i	$0.284419\pi$
$660$	46.7827	1.82102
$661$	−30.2842	−1.17792	−0.588959	−	0.808163i	$-0.700462\pi$
$661$	−30.2842	−1.17792	−0.588959	+	0.808163i	$0.700462\pi$
$662$	−7.70069	−0.299296
$663$	7.87632	0.305891
$664$	−9.06171	−0.351663
$665$	−30.5061	−1.18298
$666$	57.1695	2.21527
$667$	2.57791	0.0998169
$668$	5.83608	0.225805
$669$	22.4691	0.868704
$670$	−29.5243	−1.14062
$671$	52.1935	2.01491
$672$	8.70125	0.335658
$673$	21.3654	0.823576	0.411788	−	0.911280i	$-0.364904\pi$
$673$	21.3654	0.823576	0.411788	+	0.911280i	$0.364904\pi$
$674$	9.32450	0.359166
$675$	−7.13784	−0.274736
$676$	−12.7878	−0.491840
$677$	21.8331	0.839115	0.419558	−	0.907729i	$-0.362185\pi$
$677$	21.8331	0.839115	0.419558	+	0.907729i	$0.362185\pi$
$678$	−11.8481	−0.455024
$679$	23.9875	0.920555
$680$	−14.7673	−0.566300
$681$	12.8866	0.493815
$682$	−59.4615	−2.27690
$683$	5.51685	0.211096	0.105548	−	0.994414i	$-0.466340\pi$
$683$	5.51685	0.211096	0.105548	+	0.994414i	$0.466340\pi$
$684$	21.1634	0.809204
$685$	−38.2533	−1.46158
$686$	14.5178	0.554291
$687$	50.6677	1.93309
$688$	9.67372	0.368807
$689$	−0.450687	−0.0171698
$690$	−5.28957	−0.201370
$691$	−37.6660	−1.43288	−0.716442	−	0.697647i	$-0.754230\pi$
$691$	−37.6660	−1.43288	−0.716442	+	0.697647i	$0.754230\pi$
$692$	13.7718	0.523525
$693$	104.397	3.96571
$694$	16.2574	0.617124
$695$	−21.4325	−0.812982
$696$	−9.90652	−0.375506
$697$	−41.8340	−1.58458
$698$	−7.88230	−0.298350
$699$	−70.3503	−2.66089
$700$	−3.41698	−0.129150
$701$	1.54674	0.0584197	0.0292098	−	0.999573i	$-0.490701\pi$
$701$	1.54674	0.0584197	0.0292098	+	0.999573i	$0.490701\pi$
$702$	−2.92139	−0.110261
$703$	44.5205	1.67912
$704$	−6.59575	−0.248587
$705$	−40.3333	−1.51904
$706$	−23.3718	−0.879610
$707$	−45.8432	−1.72411
$708$	31.9855	1.20209
$709$	−49.8792	−1.87325	−0.936627	−	0.350329i	$-0.886070\pi$
$709$	−49.8792	−1.87325	−0.936627	+	0.350329i	$0.886070\pi$
$710$	−20.8011	−0.780651
$711$	−46.2231	−1.73350
$712$	−7.90264	−0.296164
$713$	6.72311	0.251783
$714$	−51.9176	−1.94297
$715$	−7.51916	−0.281201
$716$	−4.40114	−0.164478
$717$	−12.5311	−0.467983
$718$	−28.2644	−1.05482
$719$	35.3575	1.31861	0.659305	−	0.751875i	$-0.270850\pi$
$719$	35.3575	1.31861	0.659305	+	0.751875i	$0.270850\pi$
$720$	12.9022	0.480836
$721$	43.4102	1.61668
$722$	−2.51908	−0.0937504
$723$	75.2972	2.80033
$724$	−1.88849	−0.0701851
$725$	3.89029	0.144482
$726$	−93.1515	−3.45718
$727$	39.3522	1.45949	0.729746	−	0.683718i	$-0.239638\pi$
$727$	39.3522	1.45949	0.729746	+	0.683718i	$0.239638\pi$
$728$	−1.39851	−0.0518322
$729$	−41.3031	−1.52975
$730$	26.7328	0.989427
$731$	−57.7200	−2.13485
$732$	22.6780	0.838203
$733$	−9.99869	−0.369310	−0.184655	−	0.982803i	$-0.559117\pi$
$733$	−9.99869	−0.369310	−0.184655	+	0.982803i	$0.559117\pi$
$734$	5.61548	0.207271
$735$	−15.7349	−0.580392
$736$	0.745760	0.0274891
$737$	78.6821	2.89829
$738$	36.5504	1.34544
$739$	2.61104	0.0960485	0.0480242	−	0.998846i	$-0.484708\pi$
$739$	2.61104	0.0960485	0.0480242	+	0.998846i	$0.484708\pi$
$740$	27.1417	0.997748
$741$	−5.35897	−0.196867
$742$	2.97075	0.109060
$743$	32.2325	1.18250	0.591248	−	0.806490i	$-0.298635\pi$
$743$	32.2325	1.18250	0.591248	+	0.806490i	$0.298635\pi$
$744$	−25.8360	−0.947192
$745$	17.7721	0.651118
$746$	30.4929	1.11642
$747$	−47.2395	−1.72840
$748$	39.3548	1.43895
$749$	−15.3099	−0.559410
$750$	27.4818	1.00350
$751$	6.61938	0.241545	0.120772	−	0.992680i	$-0.461463\pi$
$751$	6.61938	0.241545	0.120772	+	0.992680i	$0.461463\pi$
$752$	5.68647	0.207364
$753$	0.591005	0.0215374
$754$	1.59223	0.0579855
$755$	−41.9857	−1.52802
$756$	19.2566	0.700357
$757$	−1.42843	−0.0519171	−0.0259585	−	0.999663i	$-0.508264\pi$
$757$	−1.42843	−0.0519171	−0.0259585	+	0.999663i	$0.508264\pi$
$758$	31.7108	1.15179
$759$	14.0967	0.511677
$760$	10.0475	0.364462
$761$	−45.0450	−1.63288	−0.816440	−	0.577430i	$-0.804056\pi$
$761$	−45.0450	−1.63288	−0.816440	+	0.577430i	$0.804056\pi$
$762$	−38.3724	−1.39009
$763$	22.2079	0.803981
$764$	9.26059	0.335036
$765$	−76.9832	−2.78333
$766$	2.11021	0.0762451
$767$	−5.14087	−0.185626
$768$	−2.86585	−0.103412
$769$	34.4117	1.24092	0.620459	−	0.784239i	$-0.286946\pi$
$769$	34.4117	1.24092	0.620459	+	0.784239i	$0.286946\pi$
$770$	49.5634	1.78614
$771$	−51.5772	−1.85751
$772$	−3.52895	−0.127010
$773$	−9.92328	−0.356916	−0.178458	−	0.983948i	$-0.557111\pi$
$773$	−9.92328	−0.356916	−0.178458	+	0.983948i	$0.557111\pi$
$774$	50.4299	1.81267
$775$	10.1458	0.364448
$776$	−7.90053	−0.283612
$777$	95.4224	3.42326
$778$	21.2357	0.761337
$779$	28.4634	1.01981
$780$	−3.26707	−0.116980
$781$	55.4349	1.98361
$782$	−4.44971	−0.159121
$783$	−21.9240	−0.783501
$784$	2.21842	0.0792293
$785$	−21.9477	−0.783348
$786$	25.0127	0.892175
$787$	46.4567	1.65600	0.828002	−	0.560725i	$-0.189478\pi$
$787$	46.4567	1.65600	0.828002	+	0.560725i	$0.189478\pi$
$788$	2.04530	0.0728608
$789$	−53.2685	−1.89641
$790$	−21.9448	−0.780761
$791$	−12.5523	−0.446309
$792$	−34.3843	−1.22179
$793$	−3.64492	−0.129435
$794$	19.1503	0.679618
$795$	6.93999	0.246136
$796$	−13.6360	−0.483314
$797$	14.9293	0.528823	0.264411	−	0.964410i	$-0.414822\pi$
$797$	14.9293	0.528823	0.264411	+	0.964410i	$0.414822\pi$
$798$	35.3242	1.25046
$799$	−33.9293	−1.20033
$800$	1.12542	0.0397896
$801$	−41.1972	−1.45563
$802$	13.8212	0.488045
$803$	−71.2429	−2.51411
$804$	34.1873	1.20569
$805$	−5.60396	−0.197514
$806$	4.15249	0.146265
$807$	−15.8385	−0.557540
$808$	15.0989	0.531179
$809$	−23.0630	−0.810852	−0.405426	−	0.914128i	$-0.632877\pi$
$809$	−23.0630	−0.810852	−0.405426	+	0.914128i	$0.632877\pi$
$810$	6.27905	0.220623
$811$	31.0027	1.08865	0.544326	−	0.838874i	$-0.316785\pi$
$811$	31.0027	1.08865	0.544326	+	0.838874i	$0.316785\pi$
$812$	−10.4953	−0.368314
$813$	28.5859	1.00255
$814$	−72.3325	−2.53525
$815$	−43.0214	−1.50697
$816$	17.0996	0.598606
$817$	39.2721	1.37396
$818$	11.7677	0.411446
$819$	−7.29055	−0.254752
$820$	17.3526	0.605979
$821$	−1.51527	−0.0528834	−0.0264417	−	0.999650i	$-0.508418\pi$
$821$	−1.51527	−0.0528834	−0.0264417	+	0.999650i	$0.508418\pi$
$822$	44.2949	1.54496
$823$	7.64274	0.266409	0.133205	−	0.991089i	$-0.457473\pi$
$823$	7.64274	0.266409	0.133205	+	0.991089i	$0.457473\pi$
$824$	−14.2976	−0.498080
$825$	21.2732	0.740637
$826$	33.8866	1.17907
$827$	−52.7380	−1.83388	−0.916940	−	0.399025i	$-0.869349\pi$
$827$	−52.7380	−1.83388	−0.916940	+	0.399025i	$0.869349\pi$
$828$	3.88771	0.135107
$829$	36.8774	1.28080	0.640402	−	0.768040i	$-0.278768\pi$
$829$	36.8774	1.28080	0.640402	+	0.768040i	$0.278768\pi$
$830$	−22.4274	−0.778465
$831$	81.5973	2.83058
$832$	0.460614	0.0159689
$833$	−13.2366	−0.458622
$834$	24.8175	0.859361
$835$	14.4441	0.499857
$836$	−26.7766	−0.926088
$837$	−57.1773	−1.97634
$838$	13.8631	0.478893
$839$	−17.8047	−0.614687	−0.307343	−	0.951599i	$-0.599440\pi$
$839$	−17.8047	−0.614687	−0.307343	+	0.951599i	$0.599440\pi$
$840$	21.5352	0.743036
$841$	−17.0509	−0.587961
$842$	8.87403	0.305819
$843$	69.9401	2.40887
$844$	−15.2818	−0.526022
$845$	−31.6494	−1.08877
$846$	29.6441	1.01918
$847$	−98.6881	−3.39096
$848$	−0.978448	−0.0336000
$849$	44.8040	1.53767
$850$	−6.71502	−0.230323
$851$	8.17839	0.280352
$852$	24.0864	0.825186
$853$	−14.2989	−0.489585	−0.244793	−	0.969575i	$-0.578720\pi$
$853$	−14.2989	−0.489585	−0.244793	+	0.969575i	$0.578720\pi$
$854$	24.0259	0.822150
$855$	52.3786	1.79131
$856$	5.04246	0.172348
$857$	1.67435	0.0571947	0.0285973	−	0.999591i	$-0.490896\pi$
$857$	1.67435	0.0571947	0.0285973	+	0.999591i	$0.490896\pi$
$858$	8.70672	0.297243
$859$	47.8373	1.63219	0.816094	−	0.577919i	$-0.196135\pi$
$859$	47.8373	1.63219	0.816094	+	0.577919i	$0.196135\pi$
$860$	23.9420	0.816417
$861$	61.0068	2.07910
$862$	3.22135	0.109719
$863$	−26.4724	−0.901130	−0.450565	−	0.892744i	$-0.648777\pi$
$863$	−26.4724	−0.901130	−0.450565	+	0.892744i	$0.648777\pi$
$864$	−6.34238	−0.215772
$865$	34.0846	1.15891
$866$	−6.32566	−0.214955
$867$	−53.3086	−1.81045
$868$	−27.3716	−0.929052
$869$	58.4828	1.98389
$870$	−24.5182	−0.831246
$871$	−5.49476	−0.186183
$872$	−7.31442	−0.247697
$873$	−41.1862	−1.39394
$874$	3.02754	0.102408
$875$	29.1153	0.984276
$876$	−30.9550	−1.04587
$877$	49.1659	1.66022	0.830108	−	0.557603i	$-0.188279\pi$
$877$	49.1659	1.66022	0.830108	+	0.557603i	$0.188279\pi$
$878$	−25.9743	−0.876591
$879$	6.25093	0.210839
$880$	−16.3242	−0.550289
$881$	11.5351	0.388628	0.194314	−	0.980939i	$-0.437752\pi$
$881$	11.5351	0.388628	0.194314	+	0.980939i	$0.437752\pi$
$882$	11.5648	0.389408
$883$	−48.0872	−1.61826	−0.809132	−	0.587627i	$-0.800062\pi$
$883$	−48.0872	−1.61826	−0.809132	+	0.587627i	$0.800062\pi$
$884$	−2.74834	−0.0924366
$885$	79.1628	2.66103
$886$	−12.6907	−0.426351
$887$	−22.8569	−0.767458	−0.383729	−	0.923446i	$-0.625360\pi$
$887$	−22.8569	−0.767458	−0.383729	+	0.923446i	$0.625360\pi$
$888$	−31.4284	−1.05467
$889$	−40.6532	−1.36346
$890$	−19.5587	−0.655609
$891$	−16.7336	−0.560598
$892$	−7.84028	−0.262512
$893$	23.0852	0.772516
$894$	−20.5789	−0.688263
$895$	−10.8926	−0.364101
$896$	−3.03619	−0.101432
$897$	−0.984440	−0.0328695
$898$	−4.72237	−0.157588
$899$	31.1630	1.03934
$900$	5.86691	0.195564
$901$	5.83809	0.194495
$902$	−46.2446	−1.53978
$903$	84.1734	2.80112
$904$	4.13424	0.137503
$905$	−4.67393	−0.155367
$906$	48.6169	1.61519
$907$	−39.5111	−1.31194	−0.655971	−	0.754786i	$-0.727741\pi$
$907$	−39.5111	−1.31194	−0.655971	+	0.754786i	$0.727741\pi$
$908$	−4.49660	−0.149225
$909$	78.7122	2.61072
$910$	−3.46125	−0.114739
$911$	−3.35442	−0.111137	−0.0555684	−	0.998455i	$-0.517697\pi$
$911$	−3.35442	−0.111137	−0.0555684	+	0.998455i	$0.517697\pi$
$912$	−11.6344	−0.385254
$913$	59.7688	1.97806
$914$	6.36898	0.210667
$915$	56.1271	1.85551
$916$	−17.6798	−0.584157
$917$	26.4994	0.875088
$918$	37.8430	1.24900
$919$	−57.3542	−1.89194	−0.945970	−	0.324253i	$-0.894887\pi$
$919$	−57.3542	−1.89194	−0.945970	+	0.324253i	$0.894887\pi$
$920$	1.84572	0.0608517
$921$	30.4060	1.00191
$922$	−30.6362	−1.00895
$923$	−3.87129	−0.127425
$924$	−57.3913	−1.88804
$925$	12.3419	0.405801
$926$	−32.8235	−1.07865
$927$	−74.5347	−2.44804
$928$	3.45675	0.113473
$929$	−1.69163	−0.0555007	−0.0277503	−	0.999615i	$-0.508834\pi$
$929$	−1.69163	−0.0555007	−0.0277503	+	0.999615i	$0.508834\pi$
$930$	−63.9429	−2.09677
$931$	9.00606	0.295162
$932$	24.5478	0.804090
$933$	−43.0328	−1.40883
$934$	16.1110	0.527167
$935$	97.4015	3.18537
$936$	2.40122	0.0784864
$937$	−43.0469	−1.40628	−0.703140	−	0.711051i	$-0.748219\pi$
$937$	−43.0469	−1.40628	−0.703140	+	0.711051i	$0.748219\pi$
$938$	36.2193	1.18260
$939$	14.3513	0.468337
$940$	14.0738	0.459036
$941$	−22.2073	−0.723936	−0.361968	−	0.932190i	$-0.617895\pi$
$941$	−22.2073	−0.723936	−0.361968	+	0.932190i	$0.617895\pi$
$942$	25.4141	0.828036
$943$	5.22872	0.170271
$944$	−11.1609	−0.363257
$945$	47.6594	1.55036
$946$	−63.8055	−2.07449
$947$	−39.0901	−1.27026	−0.635129	−	0.772406i	$-0.719053\pi$
$947$	−39.0901	−1.27026	−0.635129	+	0.772406i	$0.719053\pi$
$948$	25.4107	0.825302
$949$	4.97524	0.161503
$950$	4.56883	0.148233
$951$	−2.97085	−0.0963365
$952$	18.1160	0.587142
$953$	−5.55333	−0.179890	−0.0899450	−	0.995947i	$-0.528669\pi$
$953$	−5.55333	−0.179890	−0.0899450	+	0.995947i	$0.528669\pi$
$954$	−5.10074	−0.165142
$955$	22.9196	0.741660
$956$	4.37256	0.141419
$957$	65.3410	2.11217
$958$	35.7260	1.15426
$959$	46.9277	1.51537
$960$	−7.09286	−0.228921
$961$	50.2723	1.62169
$962$	5.05133	0.162861
$963$	26.2868	0.847080
$964$	−26.2739	−0.846227
$965$	−8.73400	−0.281157
$966$	6.48904	0.208781
$967$	−24.8846	−0.800233	−0.400117	−	0.916464i	$-0.631030\pi$
$967$	−24.8846	−0.800233	−0.400117	+	0.916464i	$0.631030\pi$
$968$	32.5040	1.04472
$969$	69.4188	2.23005
$970$	−19.5535	−0.627824
$971$	27.2248	0.873684	0.436842	−	0.899538i	$-0.356097\pi$
$971$	27.2248	0.873684	0.436842	+	0.899538i	$0.356097\pi$
$972$	11.7564	0.377086
$973$	26.2926	0.842902
$974$	10.3524	0.331713
$975$	−1.48561	−0.0475776
$976$	−7.91319	−0.253295
$977$	43.9128	1.40489	0.702447	−	0.711736i	$-0.252091\pi$
$977$	43.9128	1.40489	0.702447	+	0.711736i	$0.252091\pi$
$978$	49.8161	1.59294
$979$	52.1239	1.66589
$980$	5.49050	0.175388
$981$	−38.1307	−1.21742
$982$	0.148822	0.00474910
$983$	−57.8391	−1.84478	−0.922391	−	0.386258i	$-0.873767\pi$
$983$	−57.8391	−1.84478	−0.922391	+	0.386258i	$0.873767\pi$
$984$	−20.0932	−0.640549
$985$	5.06203	0.161290
$986$	−20.6253	−0.656845
$987$	49.4794	1.57495
$988$	1.86994	0.0594907
$989$	7.21427	0.229400
$990$	−85.0996	−2.70464
$991$	−7.05603	−0.224142	−0.112071	−	0.993700i	$-0.535748\pi$
$991$	−7.05603	−0.224142	−0.112071	+	0.993700i	$0.535748\pi$
$992$	9.01512	0.286230
$993$	22.0690	0.700339
$994$	25.5180	0.809382
$995$	−33.7485	−1.06990
$996$	25.9695	0.822875
$997$	−1.77139	−0.0561006	−0.0280503	−	0.999607i	$-0.508930\pi$
$997$	−1.77139	−0.0561006	−0.0280503	+	0.999607i	$0.508930\pi$
$998$	−13.6067	−0.430711
$999$	−69.5539	−2.20059

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.a.1.5	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.a.1.5	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.86585 q^{3} +1.00000 q^{4} +2.47496 q^{5} -2.86585 q^{6} -3.03619 q^{7} +1.00000 q^{8} +5.21309 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.86585 q^{3} +1.00000 q^{4} +2.47496 q^{5} -2.86585 q^{6} -3.03619 q^{7} +1.00000 q^{8} +5.21309 q^{9} +2.47496 q^{10} -6.59575 q^{11} -2.86585 q^{12} +0.460614 q^{13} -3.03619 q^{14} -7.09286 q^{15} +1.00000 q^{16} -5.96668 q^{17} +5.21309 q^{18} +4.05967 q^{19} +2.47496 q^{20} +8.70125 q^{21} -6.59575 q^{22} +0.745760 q^{23} -2.86585 q^{24} +1.12542 q^{25} +0.460614 q^{26} -6.34238 q^{27} -3.03619 q^{28} +3.45675 q^{29} -7.09286 q^{30} +9.01512 q^{31} +1.00000 q^{32} +18.9024 q^{33} -5.96668 q^{34} -7.51443 q^{35} +5.21309 q^{36} +10.9665 q^{37} +4.05967 q^{38} -1.32005 q^{39} +2.47496 q^{40} +7.01127 q^{41} +8.70125 q^{42} +9.67372 q^{43} -6.59575 q^{44} +12.9022 q^{45} +0.745760 q^{46} +5.68647 q^{47} -2.86585 q^{48} +2.21842 q^{49} +1.12542 q^{50} +17.0996 q^{51} +0.460614 q^{52} -0.978448 q^{53} -6.34238 q^{54} -16.3242 q^{55} -3.03619 q^{56} -11.6344 q^{57} +3.45675 q^{58} -11.1609 q^{59} -7.09286 q^{60} -7.91319 q^{61} +9.01512 q^{62} -15.8279 q^{63} +1.00000 q^{64} +1.14000 q^{65} +18.9024 q^{66} -11.9292 q^{67} -5.96668 q^{68} -2.13723 q^{69} -7.51443 q^{70} -8.40463 q^{71} +5.21309 q^{72} +10.8013 q^{73} +10.9665 q^{74} -3.22528 q^{75} +4.05967 q^{76} +20.0259 q^{77} -1.32005 q^{78} -8.86674 q^{79} +2.47496 q^{80} +2.53703 q^{81} +7.01127 q^{82} -9.06171 q^{83} +8.70125 q^{84} -14.7673 q^{85} +9.67372 q^{86} -9.90652 q^{87} -6.59575 q^{88} -7.90264 q^{89} +12.9022 q^{90} -1.39851 q^{91} +0.745760 q^{92} -25.8360 q^{93} +5.68647 q^{94} +10.0475 q^{95} -2.86585 q^{96} -7.90053 q^{97} +2.21842 q^{98} -34.3843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * q^99

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