LMFDB - Embedded newform 8002.2.a.d.1.69

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.69
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +3.43798 q^{3} +1.00000 q^{4} -3.02837 q^{5} +3.43798 q^{6} -1.10688 q^{7} +1.00000 q^{8} +8.81968 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.43798 q^{3} +1.00000 q^{4} -3.02837 q^{5} +3.43798 q^{6} -1.10688 q^{7} +1.00000 q^{8} +8.81968 q^{9} -3.02837 q^{10} -3.55811 q^{11} +3.43798 q^{12} -3.39792 q^{13} -1.10688 q^{14} -10.4115 q^{15} +1.00000 q^{16} -3.28878 q^{17} +8.81968 q^{18} -3.76773 q^{19} -3.02837 q^{20} -3.80543 q^{21} -3.55811 q^{22} +2.60254 q^{23} +3.43798 q^{24} +4.17102 q^{25} -3.39792 q^{26} +20.0079 q^{27} -1.10688 q^{28} -3.31720 q^{29} -10.4115 q^{30} -0.0680546 q^{31} +1.00000 q^{32} -12.2327 q^{33} -3.28878 q^{34} +3.35204 q^{35} +8.81968 q^{36} +6.93670 q^{37} -3.76773 q^{38} -11.6820 q^{39} -3.02837 q^{40} -12.2150 q^{41} -3.80543 q^{42} -6.57947 q^{43} -3.55811 q^{44} -26.7092 q^{45} +2.60254 q^{46} -4.85573 q^{47} +3.43798 q^{48} -5.77482 q^{49} +4.17102 q^{50} -11.3068 q^{51} -3.39792 q^{52} +7.41215 q^{53} +20.0079 q^{54} +10.7753 q^{55} -1.10688 q^{56} -12.9534 q^{57} -3.31720 q^{58} -11.4666 q^{59} -10.4115 q^{60} -10.3409 q^{61} -0.0680546 q^{62} -9.76232 q^{63} +1.00000 q^{64} +10.2902 q^{65} -12.2327 q^{66} +3.88492 q^{67} -3.28878 q^{68} +8.94748 q^{69} +3.35204 q^{70} -2.65842 q^{71} +8.81968 q^{72} -12.7369 q^{73} +6.93670 q^{74} +14.3399 q^{75} -3.76773 q^{76} +3.93840 q^{77} -11.6820 q^{78} +6.47742 q^{79} -3.02837 q^{80} +42.3277 q^{81} -12.2150 q^{82} +0.889778 q^{83} -3.80543 q^{84} +9.95965 q^{85} -6.57947 q^{86} -11.4044 q^{87} -3.55811 q^{88} +3.27105 q^{89} -26.7092 q^{90} +3.76109 q^{91} +2.60254 q^{92} -0.233970 q^{93} -4.85573 q^{94} +11.4101 q^{95} +3.43798 q^{96} +10.6617 q^{97} -5.77482 q^{98} -31.3814 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	3.43798	1.98492	0.992458	−	0.122584i	$-0.0391181\pi$
$3$	3.43798	1.98492	0.992458	+	0.122584i	$0.0391181\pi$
$4$	1.00000	0.500000
$5$	−3.02837	−1.35433	−0.677164	−	0.735832i	$-0.736791\pi$
$5$	−3.02837	−1.35433	−0.677164	+	0.735832i	$0.736791\pi$
$6$	3.43798	1.40355
$7$	−1.10688	−0.418361	−0.209181	−	0.977877i	$-0.567080\pi$
$7$	−1.10688	−0.418361	−0.209181	+	0.977877i	$0.567080\pi$
$8$	1.00000	0.353553
$9$	8.81968	2.93989
$10$	−3.02837	−0.957654
$11$	−3.55811	−1.07281	−0.536406	−	0.843960i	$-0.680218\pi$
$11$	−3.55811	−1.07281	−0.536406	+	0.843960i	$0.680218\pi$
$12$	3.43798	0.992458
$13$	−3.39792	−0.942413	−0.471207	−	0.882023i	$-0.656181\pi$
$13$	−3.39792	−0.942413	−0.471207	+	0.882023i	$0.656181\pi$
$14$	−1.10688	−0.295826
$15$	−10.4115	−2.68823
$16$	1.00000	0.250000
$17$	−3.28878	−0.797647	−0.398823	−	0.917028i	$-0.630581\pi$
$17$	−3.28878	−0.797647	−0.398823	+	0.917028i	$0.630581\pi$
$18$	8.81968	2.07882
$19$	−3.76773	−0.864376	−0.432188	−	0.901783i	$-0.642258\pi$
$19$	−3.76773	−0.864376	−0.432188	+	0.901783i	$0.642258\pi$
$20$	−3.02837	−0.677164
$21$	−3.80543	−0.830412
$22$	−3.55811	−0.758592
$23$	2.60254	0.542668	0.271334	−	0.962485i	$-0.412535\pi$
$23$	2.60254	0.542668	0.271334	+	0.962485i	$0.412535\pi$
$24$	3.43798	0.701774
$25$	4.17102	0.834204
$26$	−3.39792	−0.666387
$27$	20.0079	3.85052
$28$	−1.10688	−0.209181
$29$	−3.31720	−0.615988	−0.307994	−	0.951388i	$-0.599658\pi$
$29$	−3.31720	−0.615988	−0.307994	+	0.951388i	$0.599658\pi$
$30$	−10.4115	−1.90086
$31$	−0.0680546	−0.0122230	−0.00611149	−	0.999981i	$-0.501945\pi$
$31$	−0.0680546	−0.0122230	−0.00611149	+	0.999981i	$0.501945\pi$
$32$	1.00000	0.176777
$33$	−12.2327	−2.12944
$34$	−3.28878	−0.564021
$35$	3.35204	0.566598
$36$	8.81968	1.46995
$37$	6.93670	1.14039	0.570194	−	0.821510i	$-0.306868\pi$
$37$	6.93670	1.14039	0.570194	+	0.821510i	$0.306868\pi$
$38$	−3.76773	−0.611206
$39$	−11.6820	−1.87061
$40$	−3.02837	−0.478827
$41$	−12.2150	−1.90767	−0.953833	−	0.300338i	$-0.902901\pi$
$41$	−12.2150	−1.90767	−0.953833	+	0.300338i	$0.902901\pi$
$42$	−3.80543	−0.587190
$43$	−6.57947	−1.00336	−0.501680	−	0.865053i	$-0.667284\pi$
$43$	−6.57947	−1.00336	−0.501680	+	0.865053i	$0.667284\pi$
$44$	−3.55811	−0.536406
$45$	−26.7092	−3.98158
$46$	2.60254	0.383724
$47$	−4.85573	−0.708281	−0.354141	−	0.935192i	$-0.615227\pi$
$47$	−4.85573	−0.708281	−0.354141	+	0.935192i	$0.615227\pi$
$48$	3.43798	0.496229
$49$	−5.77482	−0.824974
$50$	4.17102	0.589871
$51$	−11.3068	−1.58326
$52$	−3.39792	−0.471207
$53$	7.41215	1.01814	0.509069	−	0.860726i	$-0.329990\pi$
$53$	7.41215	1.01814	0.509069	+	0.860726i	$0.329990\pi$
$54$	20.0079	2.72273
$55$	10.7753	1.45294
$56$	−1.10688	−0.147913
$57$	−12.9534	−1.71571
$58$	−3.31720	−0.435569
$59$	−11.4666	−1.49283	−0.746415	−	0.665480i	$-0.768227\pi$
$59$	−11.4666	−1.49283	−0.746415	+	0.665480i	$0.768227\pi$
$60$	−10.4115	−1.34411
$61$	−10.3409	−1.32401	−0.662007	−	0.749498i	$-0.730295\pi$
$61$	−10.3409	−1.32401	−0.662007	+	0.749498i	$0.730295\pi$
$62$	−0.0680546	−0.00864295
$63$	−9.76232	−1.22994
$64$	1.00000	0.125000
$65$	10.2902	1.27634
$66$	−12.2327	−1.50574
$67$	3.88492	0.474619	0.237309	−	0.971434i	$-0.423734\pi$
$67$	3.88492	0.474619	0.237309	+	0.971434i	$0.423734\pi$
$68$	−3.28878	−0.398823
$69$	8.94748	1.07715
$70$	3.35204	0.400646
$71$	−2.65842	−0.315496	−0.157748	−	0.987479i	$-0.550423\pi$
$71$	−2.65842	−0.315496	−0.157748	+	0.987479i	$0.550423\pi$
$72$	8.81968	1.03941
$73$	−12.7369	−1.49075	−0.745374	−	0.666647i	$-0.767729\pi$
$73$	−12.7369	−1.49075	−0.745374	+	0.666647i	$0.767729\pi$
$74$	6.93670	0.806375
$75$	14.3399	1.65583
$76$	−3.76773	−0.432188
$77$	3.93840	0.448823
$78$	−11.6820	−1.32272
$79$	6.47742	0.728767	0.364384	−	0.931249i	$-0.381280\pi$
$79$	6.47742	0.728767	0.364384	+	0.931249i	$0.381280\pi$
$80$	−3.02837	−0.338582
$81$	42.3277	4.70308
$82$	−12.2150	−1.34892
$83$	0.889778	0.0976658	0.0488329	−	0.998807i	$-0.484450\pi$
$83$	0.889778	0.0976658	0.0488329	+	0.998807i	$0.484450\pi$
$84$	−3.80543	−0.415206
$85$	9.95965	1.08028
$86$	−6.57947	−0.709482
$87$	−11.4044	−1.22268
$88$	−3.55811	−0.379296
$89$	3.27105	0.346731	0.173365	−	0.984858i	$-0.444536\pi$
$89$	3.27105	0.346731	0.173365	+	0.984858i	$0.444536\pi$
$90$	−26.7092	−2.81540
$91$	3.76109	0.394269
$92$	2.60254	0.271334
$93$	−0.233970	−0.0242616
$94$	−4.85573	−0.500830
$95$	11.4101	1.17065
$96$	3.43798	0.350887
$97$	10.6617	1.08253	0.541263	−	0.840853i	$-0.317946\pi$
$97$	10.6617	1.08253	0.541263	+	0.840853i	$0.317946\pi$
$98$	−5.77482	−0.583345
$99$	−31.3814	−3.15395
$100$	4.17102	0.417102
$101$	1.25060	0.124439	0.0622195	−	0.998062i	$-0.480182\pi$
$101$	1.25060	0.124439	0.0622195	+	0.998062i	$0.480182\pi$
$102$	−11.3068	−1.11954
$103$	−5.74436	−0.566009	−0.283004	−	0.959119i	$-0.591331\pi$
$103$	−5.74436	−0.566009	−0.283004	+	0.959119i	$0.591331\pi$
$104$	−3.39792	−0.333193
$105$	11.5242	1.12465
$106$	7.41215	0.719932
$107$	−3.99596	−0.386304	−0.193152	−	0.981169i	$-0.561871\pi$
$107$	−3.99596	−0.386304	−0.193152	+	0.981169i	$0.561871\pi$
$108$	20.0079	1.92526
$109$	17.3709	1.66383	0.831917	−	0.554901i	$-0.187244\pi$
$109$	17.3709	1.66383	0.831917	+	0.554901i	$0.187244\pi$
$110$	10.7753	1.02738
$111$	23.8482	2.26357
$112$	−1.10688	−0.104590
$113$	4.53896	0.426989	0.213495	−	0.976944i	$-0.431515\pi$
$113$	4.53896	0.426989	0.213495	+	0.976944i	$0.431515\pi$
$114$	−12.9534	−1.21319
$115$	−7.88147	−0.734950
$116$	−3.31720	−0.307994
$117$	−29.9685	−2.77059
$118$	−11.4666	−1.05559
$119$	3.64029	0.333705
$120$	−10.4115	−0.950432
$121$	1.66018	0.150925
$122$	−10.3409	−0.936219
$123$	−41.9950	−3.78656
$124$	−0.0680546	−0.00611149
$125$	2.51046	0.224542
$126$	−9.76232	−0.869697
$127$	1.79138	0.158960	0.0794798	−	0.996836i	$-0.474674\pi$
$127$	1.79138	0.158960	0.0794798	+	0.996836i	$0.474674\pi$
$128$	1.00000	0.0883883
$129$	−22.6201	−1.99158
$130$	10.2902	0.902506
$131$	21.7566	1.90089	0.950443	−	0.310899i	$-0.100630\pi$
$131$	21.7566	1.90089	0.950443	+	0.310899i	$0.100630\pi$
$132$	−12.2327	−1.06472
$133$	4.17042	0.361622
$134$	3.88492	0.335606
$135$	−60.5913	−5.21487
$136$	−3.28878	−0.282011
$137$	−10.0062	−0.854891	−0.427446	−	0.904041i	$-0.640586\pi$
$137$	−10.0062	−0.854891	−0.427446	+	0.904041i	$0.640586\pi$
$138$	8.94748	0.761660
$139$	−6.83235	−0.579512	−0.289756	−	0.957101i	$-0.593574\pi$
$139$	−6.83235	−0.579512	−0.289756	+	0.957101i	$0.593574\pi$
$140$	3.35204	0.283299
$141$	−16.6939	−1.40588
$142$	−2.65842	−0.223089
$143$	12.0902	1.01103
$144$	8.81968	0.734973
$145$	10.0457	0.834250
$146$	−12.7369	−1.05412
$147$	−19.8537	−1.63750
$148$	6.93670	0.570194
$149$	12.5019	1.02419	0.512097	−	0.858928i	$-0.328869\pi$
$149$	12.5019	1.02419	0.512097	+	0.858928i	$0.328869\pi$
$150$	14.3399	1.17085
$151$	−1.25092	−0.101799	−0.0508994	−	0.998704i	$-0.516209\pi$
$151$	−1.25092	−0.101799	−0.0508994	+	0.998704i	$0.516209\pi$
$152$	−3.76773	−0.305603
$153$	−29.0060	−2.34500
$154$	3.93840	0.317366
$155$	0.206095	0.0165539
$156$	−11.6820	−0.935305
$157$	16.2356	1.29574	0.647872	−	0.761749i	$-0.275659\pi$
$157$	16.2356	1.29574	0.647872	+	0.761749i	$0.275659\pi$
$158$	6.47742	0.515316
$159$	25.4828	2.02092
$160$	−3.02837	−0.239414
$161$	−2.88070	−0.227031
$162$	42.3277	3.32558
$163$	−16.9850	−1.33037	−0.665185	−	0.746679i	$-0.731647\pi$
$163$	−16.9850	−1.33037	−0.665185	+	0.746679i	$0.731647\pi$
$164$	−12.2150	−0.953833
$165$	37.0452	2.88396
$166$	0.889778	0.0690602
$167$	−18.9280	−1.46470	−0.732348	−	0.680930i	$-0.761576\pi$
$167$	−18.9280	−1.46470	−0.732348	+	0.680930i	$0.761576\pi$
$168$	−3.80543	−0.293595
$169$	−1.45415	−0.111858
$170$	9.95965	0.763870
$171$	−33.2302	−2.54117
$172$	−6.57947	−0.501680
$173$	−23.3465	−1.77500	−0.887502	−	0.460804i	$-0.847561\pi$
$173$	−23.3465	−1.77500	−0.887502	+	0.460804i	$0.847561\pi$
$174$	−11.4044	−0.864568
$175$	−4.61682	−0.348999
$176$	−3.55811	−0.268203
$177$	−39.4221	−2.96314
$178$	3.27105	0.245176
$179$	−23.3419	−1.74465	−0.872327	−	0.488923i	$-0.837390\pi$
$179$	−23.3419	−1.74465	−0.872327	+	0.488923i	$0.837390\pi$
$180$	−26.7092	−1.99079
$181$	−12.2632	−0.911516	−0.455758	−	0.890104i	$-0.650632\pi$
$181$	−12.2632	−0.911516	−0.455758	+	0.890104i	$0.650632\pi$
$182$	3.76109	0.278790
$183$	−35.5517	−2.62806
$184$	2.60254	0.191862
$185$	−21.0069	−1.54446
$186$	−0.233970	−0.0171555
$187$	11.7019	0.855725
$188$	−4.85573	−0.354141
$189$	−22.1464	−1.61091
$190$	11.4101	0.827774
$191$	14.7586	1.06789	0.533946	−	0.845519i	$-0.320709\pi$
$191$	14.7586	1.06789	0.533946	+	0.845519i	$0.320709\pi$
$192$	3.43798	0.248115
$193$	4.26628	0.307093	0.153547	−	0.988141i	$-0.450930\pi$
$193$	4.26628	0.307093	0.153547	+	0.988141i	$0.450930\pi$
$194$	10.6617	0.765462
$195$	35.3773	2.53342
$196$	−5.77482	−0.412487
$197$	−18.6566	−1.32923	−0.664615	−	0.747186i	$-0.731404\pi$
$197$	−18.6566	−1.32923	−0.664615	+	0.747186i	$0.731404\pi$
$198$	−31.3814	−2.23018
$199$	19.1054	1.35435	0.677174	−	0.735823i	$-0.263204\pi$
$199$	19.1054	1.35435	0.677174	+	0.735823i	$0.263204\pi$
$200$	4.17102	0.294936
$201$	13.3563	0.942078
$202$	1.25060	0.0879916
$203$	3.67174	0.257705
$204$	−11.3068	−0.791631
$205$	36.9916	2.58361
$206$	−5.74436	−0.400228
$207$	22.9536	1.59539
$208$	−3.39792	−0.235603
$209$	13.4060	0.927313
$210$	11.5242	0.795248
$211$	20.4400	1.40715	0.703574	−	0.710622i	$-0.251586\pi$
$211$	20.4400	1.40715	0.703574	+	0.710622i	$0.251586\pi$
$212$	7.41215	0.509069
$213$	−9.13957	−0.626233
$214$	−3.99596	−0.273158
$215$	19.9251	1.35888
$216$	20.0079	1.36137
$217$	0.0753283	0.00511362
$218$	17.3709	1.17651
$219$	−43.7893	−2.95901
$220$	10.7753	0.726469
$221$	11.1750	0.751713
$222$	23.8482	1.60059
$223$	2.88661	0.193302	0.0966510	−	0.995318i	$-0.469187\pi$
$223$	2.88661	0.193302	0.0966510	+	0.995318i	$0.469187\pi$
$224$	−1.10688	−0.0739565
$225$	36.7871	2.45247
$226$	4.53896	0.301927
$227$	−20.8776	−1.38570	−0.692849	−	0.721083i	$-0.743645\pi$
$227$	−20.8776	−1.38570	−0.692849	+	0.721083i	$0.743645\pi$
$228$	−12.9534	−0.857857
$229$	−4.74199	−0.313360	−0.156680	−	0.987649i	$-0.550079\pi$
$229$	−4.74199	−0.313360	−0.156680	+	0.987649i	$0.550079\pi$
$230$	−7.88147	−0.519688
$231$	13.5401	0.890876
$232$	−3.31720	−0.217785
$233$	−17.0081	−1.11424	−0.557119	−	0.830432i	$-0.688093\pi$
$233$	−17.0081	−1.11424	−0.557119	+	0.830432i	$0.688093\pi$
$234$	−29.9685	−1.95911
$235$	14.7049	0.959245
$236$	−11.4666	−0.746415
$237$	22.2692	1.44654
$238$	3.64029	0.235965
$239$	−11.4421	−0.740128	−0.370064	−	0.929006i	$-0.620664\pi$
$239$	−11.4421	−0.740128	−0.370064	+	0.929006i	$0.620664\pi$
$240$	−10.4115	−0.672057
$241$	−29.4736	−1.89856	−0.949282	−	0.314426i	$-0.898188\pi$
$241$	−29.4736	−1.89856	−0.949282	+	0.314426i	$0.898188\pi$
$242$	1.66018	0.106720
$243$	85.4978	5.48469
$244$	−10.3409	−0.662007
$245$	17.4883	1.11729
$246$	−41.9950	−2.67750
$247$	12.8024	0.814599
$248$	−0.0680546	−0.00432147
$249$	3.05904	0.193858
$250$	2.51046	0.158775
$251$	5.64597	0.356370	0.178185	−	0.983997i	$-0.442977\pi$
$251$	5.64597	0.356370	0.178185	+	0.983997i	$0.442977\pi$
$252$	−9.76232	−0.614969
$253$	−9.26015	−0.582181
$254$	1.79138	0.112401
$255$	34.2410	2.14426
$256$	1.00000	0.0625000
$257$	9.95819	0.621175	0.310588	−	0.950545i	$-0.399474\pi$
$257$	9.95819	0.621175	0.310588	+	0.950545i	$0.399474\pi$
$258$	−22.6201	−1.40826
$259$	−7.67810	−0.477094
$260$	10.2902	0.638168
$261$	−29.2566	−1.81094
$262$	21.7566	1.34413
$263$	20.1663	1.24351	0.621755	−	0.783212i	$-0.286420\pi$
$263$	20.1663	1.24351	0.621755	+	0.783212i	$0.286420\pi$
$264$	−12.2327	−0.752871
$265$	−22.4467	−1.37889
$266$	4.17042	0.255705
$267$	11.2458	0.688231
$268$	3.88492	0.237309
$269$	−27.3605	−1.66820	−0.834099	−	0.551615i	$-0.814012\pi$
$269$	−27.3605	−1.66820	−0.834099	+	0.551615i	$0.814012\pi$
$270$	−60.5913	−3.68747
$271$	1.99574	0.121232	0.0606162	−	0.998161i	$-0.480693\pi$
$271$	1.99574	0.121232	0.0606162	+	0.998161i	$0.480693\pi$
$272$	−3.28878	−0.199412
$273$	12.9305	0.782591
$274$	−10.0062	−0.604499
$275$	−14.8410	−0.894944
$276$	8.94748	0.538575
$277$	19.7122	1.18439	0.592196	−	0.805794i	$-0.298261\pi$
$277$	19.7122	1.18439	0.592196	+	0.805794i	$0.298261\pi$
$278$	−6.83235	−0.409777
$279$	−0.600220	−0.0359342
$280$	3.35204	0.200323
$281$	15.6704	0.934819	0.467410	−	0.884041i	$-0.345187\pi$
$281$	15.6704	0.934819	0.467410	+	0.884041i	$0.345187\pi$
$282$	−16.6939	−0.994106
$283$	7.60392	0.452006	0.226003	−	0.974127i	$-0.427434\pi$
$283$	7.60392	0.452006	0.226003	+	0.974127i	$0.427434\pi$
$284$	−2.65842	−0.157748
$285$	39.2276	2.32364
$286$	12.0902	0.714907
$287$	13.5206	0.798094
$288$	8.81968	0.519704
$289$	−6.18391	−0.363760
$290$	10.0457	0.589904
$291$	36.6545	2.14873
$292$	−12.7369	−0.745374
$293$	13.7236	0.801741	0.400870	−	0.916135i	$-0.368708\pi$
$293$	13.7236	0.801741	0.400870	+	0.916135i	$0.368708\pi$
$294$	−19.8537	−1.15789
$295$	34.7252	2.02178
$296$	6.93670	0.403188
$297$	−71.1904	−4.13089
$298$	12.5019	0.724214
$299$	−8.84323	−0.511417
$300$	14.3399	0.827913
$301$	7.28268	0.419767
$302$	−1.25092	−0.0719826
$303$	4.29952	0.247001
$304$	−3.76773	−0.216094
$305$	31.3160	1.79315
$306$	−29.0060	−1.65816
$307$	−15.7064	−0.896413	−0.448206	−	0.893930i	$-0.647937\pi$
$307$	−15.7064	−0.896413	−0.448206	+	0.893930i	$0.647937\pi$
$308$	3.93840	0.224411
$309$	−19.7490	−1.12348
$310$	0.206095	0.0117054
$311$	9.59119	0.543867	0.271933	−	0.962316i	$-0.412337\pi$
$311$	9.59119	0.543867	0.271933	+	0.962316i	$0.412337\pi$
$312$	−11.6820	−0.661361
$313$	6.02135	0.340347	0.170173	−	0.985414i	$-0.445567\pi$
$313$	6.02135	0.340347	0.170173	+	0.985414i	$0.445567\pi$
$314$	16.2356	0.916230
$315$	29.5639	1.66574
$316$	6.47742	0.364384
$317$	22.5522	1.26666	0.633330	−	0.773882i	$-0.281688\pi$
$317$	22.5522	1.26666	0.633330	+	0.773882i	$0.281688\pi$
$318$	25.4828	1.42900
$319$	11.8030	0.660839
$320$	−3.02837	−0.169291
$321$	−13.7380	−0.766781
$322$	−2.88070	−0.160535
$323$	12.3912	0.689467
$324$	42.3277	2.35154
$325$	−14.1728	−0.786165
$326$	−16.9850	−0.940713
$327$	59.7208	3.30257
$328$	−12.2150	−0.674462
$329$	5.37471	0.296317
$330$	37.0452	2.03927
$331$	32.9979	1.81373	0.906865	−	0.421421i	$-0.138469\pi$
$331$	32.9979	1.81373	0.906865	+	0.421421i	$0.138469\pi$
$332$	0.889778	0.0488329
$333$	61.1795	3.35262
$334$	−18.9280	−1.03570
$335$	−11.7650	−0.642789
$336$	−3.80543	−0.207603
$337$	16.5450	0.901263	0.450632	−	0.892710i	$-0.351199\pi$
$337$	16.5450	0.901263	0.450632	+	0.892710i	$0.351199\pi$
$338$	−1.45415	−0.0790953
$339$	15.6048	0.847538
$340$	9.95965	0.540138
$341$	0.242146	0.0131129
$342$	−33.2302	−1.79688
$343$	14.1402	0.763498
$344$	−6.57947	−0.354741
$345$	−27.0963	−1.45882
$346$	−23.3465	−1.25512
$347$	−8.55827	−0.459432	−0.229716	−	0.973258i	$-0.573780\pi$
$347$	−8.55827	−0.459432	−0.229716	+	0.973258i	$0.573780\pi$
$348$	−11.4044	−0.611342
$349$	−13.6373	−0.729987	−0.364993	−	0.931010i	$-0.618929\pi$
$349$	−13.6373	−0.729987	−0.364993	+	0.931010i	$0.618929\pi$
$350$	−4.61682	−0.246779
$351$	−67.9853	−3.62878
$352$	−3.55811	−0.189648
$353$	3.63509	0.193476	0.0967382	−	0.995310i	$-0.469159\pi$
$353$	3.63509	0.193476	0.0967382	+	0.995310i	$0.469159\pi$
$354$	−39.4221	−2.09526
$355$	8.05066	0.427285
$356$	3.27105	0.173365
$357$	12.5152	0.662376
$358$	−23.3419	−1.23366
$359$	30.0878	1.58797	0.793986	−	0.607936i	$-0.208002\pi$
$359$	30.0878	1.58797	0.793986	+	0.607936i	$0.208002\pi$
$360$	−26.7092	−1.40770
$361$	−4.80422	−0.252854
$362$	−12.2632	−0.644539
$363$	5.70764	0.299574
$364$	3.76109	0.197135
$365$	38.5722	2.01896
$366$	−35.5517	−1.85832
$367$	−11.0103	−0.574734	−0.287367	−	0.957821i	$-0.592780\pi$
$367$	−11.0103	−0.574734	−0.287367	+	0.957821i	$0.592780\pi$
$368$	2.60254	0.135667
$369$	−107.733	−5.60833
$370$	−21.0069	−1.09210
$371$	−8.20436	−0.425949
$372$	−0.233970	−0.0121308
$373$	−6.52618	−0.337913	−0.168956	−	0.985624i	$-0.554040\pi$
$373$	−6.52618	−0.337913	−0.168956	+	0.985624i	$0.554040\pi$
$374$	11.7019	0.605089
$375$	8.63089	0.445697
$376$	−4.85573	−0.250415
$377$	11.2716	0.580515
$378$	−22.1464	−1.13909
$379$	−0.544847	−0.0279869	−0.0139935	−	0.999902i	$-0.504454\pi$
$379$	−0.544847	−0.0279869	−0.0139935	+	0.999902i	$0.504454\pi$
$380$	11.4101	0.585324
$381$	6.15874	0.315522
$382$	14.7586	0.755114
$383$	−18.0197	−0.920764	−0.460382	−	0.887721i	$-0.652288\pi$
$383$	−18.0197	−0.920764	−0.460382	+	0.887721i	$0.652288\pi$
$384$	3.43798	0.175443
$385$	−11.9269	−0.607853
$386$	4.26628	0.217148
$387$	−58.0288	−2.94977
$388$	10.6617	0.541263
$389$	6.95139	0.352450	0.176225	−	0.984350i	$-0.443611\pi$
$389$	6.95139	0.352450	0.176225	+	0.984350i	$0.443611\pi$
$390$	35.3773	1.79140
$391$	−8.55920	−0.432857
$392$	−5.77482	−0.291672
$393$	74.7988	3.77310
$394$	−18.6566	−0.939907
$395$	−19.6160	−0.986990
$396$	−31.3814	−1.57698
$397$	12.3712	0.620892	0.310446	−	0.950591i	$-0.399522\pi$
$397$	12.3712	0.620892	0.310446	+	0.950591i	$0.399522\pi$
$398$	19.1054	0.957669
$399$	14.3378	0.717789
$400$	4.17102	0.208551
$401$	3.96628	0.198066	0.0990332	−	0.995084i	$-0.468425\pi$
$401$	3.96628	0.198066	0.0990332	+	0.995084i	$0.468425\pi$
$402$	13.3563	0.666150
$403$	0.231244	0.0115191
$404$	1.25060	0.0622195
$405$	−128.184	−6.36951
$406$	3.67174	0.182225
$407$	−24.6816	−1.22342
$408$	−11.3068	−0.559768
$409$	5.10460	0.252406	0.126203	−	0.992004i	$-0.459721\pi$
$409$	5.10460	0.252406	0.126203	+	0.992004i	$0.459721\pi$
$410$	36.9916	1.82688
$411$	−34.4012	−1.69689
$412$	−5.74436	−0.283004
$413$	12.6922	0.624543
$414$	22.9536	1.12811
$415$	−2.69458	−0.132272
$416$	−3.39792	−0.166597
$417$	−23.4894	−1.15028
$418$	13.4060	0.655709
$419$	−14.2452	−0.695925	−0.347963	−	0.937508i	$-0.613126\pi$
$419$	−14.2452	−0.695925	−0.347963	+	0.937508i	$0.613126\pi$
$420$	11.5242	0.562325
$421$	−2.08365	−0.101551	−0.0507754	−	0.998710i	$-0.516169\pi$
$421$	−2.08365	−0.101551	−0.0507754	+	0.998710i	$0.516169\pi$
$422$	20.4400	0.995004
$423$	−42.8260	−2.08227
$424$	7.41215	0.359966
$425$	−13.7176	−0.665400
$426$	−9.13957	−0.442814
$427$	11.4461	0.553916
$428$	−3.99596	−0.193152
$429$	41.5658	2.00681
$430$	19.9251	0.960872
$431$	−32.3361	−1.55758	−0.778788	−	0.627287i	$-0.784165\pi$
$431$	−32.3361	−1.55758	−0.778788	+	0.627287i	$0.784165\pi$
$432$	20.0079	0.962631
$433$	−29.3043	−1.40828	−0.704138	−	0.710063i	$-0.748666\pi$
$433$	−29.3043	−1.40828	−0.704138	+	0.710063i	$0.748666\pi$
$434$	0.0753283	0.00361587
$435$	34.5369	1.65592
$436$	17.3709	0.831917
$437$	−9.80568	−0.469069
$438$	−43.7893	−2.09233
$439$	−6.52153	−0.311256	−0.155628	−	0.987816i	$-0.549740\pi$
$439$	−6.52153	−0.311256	−0.155628	+	0.987816i	$0.549740\pi$
$440$	10.7753	0.513691
$441$	−50.9320	−2.42533
$442$	11.1750	0.531541
$443$	−4.94546	−0.234966	−0.117483	−	0.993075i	$-0.537483\pi$
$443$	−4.94546	−0.234966	−0.117483	+	0.993075i	$0.537483\pi$
$444$	23.8482	1.13179
$445$	−9.90595	−0.469587
$446$	2.88661	0.136685
$447$	42.9812	2.03294
$448$	−1.10688	−0.0522952
$449$	−24.4368	−1.15324	−0.576622	−	0.817011i	$-0.695630\pi$
$449$	−24.4368	−1.15324	−0.576622	+	0.817011i	$0.695630\pi$
$450$	36.7871	1.73416
$451$	43.4624	2.04657
$452$	4.53896	0.213495
$453$	−4.30065	−0.202062
$454$	−20.8776	−0.979836
$455$	−11.3900	−0.533970
$456$	−12.9534	−0.606597
$457$	20.4506	0.956639	0.478319	−	0.878186i	$-0.341246\pi$
$457$	20.4506	0.956639	0.478319	+	0.878186i	$0.341246\pi$
$458$	−4.74199	−0.221579
$459$	−65.8017	−3.07136
$460$	−7.88147	−0.367475
$461$	−6.06096	−0.282287	−0.141144	−	0.989989i	$-0.545078\pi$
$461$	−6.06096	−0.282287	−0.141144	+	0.989989i	$0.545078\pi$
$462$	13.5401	0.629944
$463$	39.8186	1.85053	0.925264	−	0.379324i	$-0.123844\pi$
$463$	39.8186	1.85053	0.925264	+	0.379324i	$0.123844\pi$
$464$	−3.31720	−0.153997
$465$	0.708548	0.0328581
$466$	−17.0081	−0.787886
$467$	8.37003	0.387319	0.193659	−	0.981069i	$-0.437964\pi$
$467$	8.37003	0.387319	0.193659	+	0.981069i	$0.437964\pi$
$468$	−29.9685	−1.38530
$469$	−4.30014	−0.198562
$470$	14.7049	0.678289
$471$	55.8177	2.57194
$472$	−11.4666	−0.527795
$473$	23.4105	1.07642
$474$	22.2692	1.02286
$475$	−15.7153	−0.721066
$476$	3.64029	0.166852
$477$	65.3728	2.99322
$478$	−11.4421	−0.523350
$479$	8.25406	0.377138	0.188569	−	0.982060i	$-0.439615\pi$
$479$	8.25406	0.377138	0.188569	+	0.982060i	$0.439615\pi$
$480$	−10.4115	−0.475216
$481$	−23.5704	−1.07472
$482$	−29.4736	−1.34249
$483$	−9.90379	−0.450638
$484$	1.66018	0.0754625
$485$	−32.2874	−1.46610
$486$	85.4978	3.87826
$487$	−2.64483	−0.119848	−0.0599242	−	0.998203i	$-0.519086\pi$
$487$	−2.64483	−0.119848	−0.0599242	+	0.998203i	$0.519086\pi$
$488$	−10.3409	−0.468109
$489$	−58.3941	−2.64067
$490$	17.4883	0.790040
$491$	−17.2133	−0.776826	−0.388413	−	0.921485i	$-0.626977\pi$
$491$	−17.2133	−0.776826	−0.388413	+	0.921485i	$0.626977\pi$
$492$	−41.9950	−1.89328
$493$	10.9095	0.491341
$494$	12.8024	0.576009
$495$	95.0345	4.27148
$496$	−0.0680546	−0.00305574
$497$	2.94255	0.131991
$498$	3.05904	0.137079
$499$	31.7780	1.42258	0.711290	−	0.702898i	$-0.248111\pi$
$499$	31.7780	1.42258	0.711290	+	0.702898i	$0.248111\pi$
$500$	2.51046	0.112271
$501$	−65.0742	−2.90730
$502$	5.64597	0.251992
$503$	−28.5592	−1.27339	−0.636695	−	0.771116i	$-0.719699\pi$
$503$	−28.5592	−1.27339	−0.636695	+	0.771116i	$0.719699\pi$
$504$	−9.76232	−0.434848
$505$	−3.78727	−0.168531
$506$	−9.26015	−0.411664
$507$	−4.99933	−0.222028
$508$	1.79138	0.0794798
$509$	37.5294	1.66346	0.831731	−	0.555179i	$-0.187350\pi$
$509$	37.5294	1.66346	0.831731	+	0.555179i	$0.187350\pi$
$510$	34.2410	1.51622
$511$	14.0983	0.623671
$512$	1.00000	0.0441942
$513$	−75.3844	−3.32830
$514$	9.95819	0.439237
$515$	17.3960	0.766561
$516$	−22.6201	−0.995792
$517$	17.2772	0.759852
$518$	−7.67810	−0.337356
$519$	−80.2648	−3.52323
$520$	10.2902	0.451253
$521$	16.3605	0.716764	0.358382	−	0.933575i	$-0.383328\pi$
$521$	16.3605	0.716764	0.358382	+	0.933575i	$0.383328\pi$
$522$	−29.2566	−1.28053
$523$	−2.18763	−0.0956582	−0.0478291	−	0.998856i	$-0.515230\pi$
$523$	−2.18763	−0.0956582	−0.0478291	+	0.998856i	$0.515230\pi$
$524$	21.7566	0.950443
$525$	−15.8725	−0.692733
$526$	20.1663	0.879294
$527$	0.223817	0.00974961
$528$	−12.2327	−0.532360
$529$	−16.2268	−0.705511
$530$	−22.4467	−0.975024
$531$	−101.132	−4.38876
$532$	4.17042	0.180811
$533$	41.5057	1.79781
$534$	11.2458	0.486653
$535$	12.1012	0.523182
$536$	3.88492	0.167803
$537$	−80.2488	−3.46299
$538$	−27.3605	−1.17959
$539$	20.5475	0.885042
$540$	−60.5913	−2.60744
$541$	1.75587	0.0754907	0.0377454	−	0.999287i	$-0.487982\pi$
$541$	1.75587	0.0754907	0.0377454	+	0.999287i	$0.487982\pi$
$542$	1.99574	0.0857243
$543$	−42.1606	−1.80928
$544$	−3.28878	−0.141005
$545$	−52.6056	−2.25338
$546$	12.9305	0.553376
$547$	−16.6701	−0.712762	−0.356381	−	0.934341i	$-0.615989\pi$
$547$	−16.6701	−0.712762	−0.356381	+	0.934341i	$0.615989\pi$
$548$	−10.0062	−0.427446
$549$	−91.2032	−3.89246
$550$	−14.8410	−0.632821
$551$	12.4983	0.532445
$552$	8.94748	0.380830
$553$	−7.16973	−0.304888
$554$	19.7122	0.837491
$555$	−72.2212	−3.06562
$556$	−6.83235	−0.289756
$557$	24.2787	1.02872	0.514361	−	0.857574i	$-0.328029\pi$
$557$	24.2787	1.02872	0.514361	+	0.857574i	$0.328029\pi$
$558$	−0.600220	−0.0254093
$559$	22.3565	0.945579
$560$	3.35204	0.141650
$561$	40.2307	1.69854
$562$	15.6704	0.661017
$563$	10.5274	0.443677	0.221839	−	0.975083i	$-0.428794\pi$
$563$	10.5274	0.443677	0.221839	+	0.975083i	$0.428794\pi$
$564$	−16.6939	−0.702939
$565$	−13.7456	−0.578283
$566$	7.60392	0.319616
$567$	−46.8517	−1.96758
$568$	−2.65842	−0.111545
$569$	3.86713	0.162119	0.0810593	−	0.996709i	$-0.474170\pi$
$569$	3.86713	0.162119	0.0810593	+	0.996709i	$0.474170\pi$
$570$	39.2276	1.64306
$571$	41.7024	1.74519	0.872595	−	0.488444i	$-0.162436\pi$
$571$	41.7024	1.74519	0.872595	+	0.488444i	$0.162436\pi$
$572$	12.0902	0.505516
$573$	50.7396	2.11968
$574$	13.5206	0.564337
$575$	10.8553	0.452696
$576$	8.81968	0.367487
$577$	21.6281	0.900391	0.450196	−	0.892930i	$-0.351354\pi$
$577$	21.6281	0.900391	0.450196	+	0.892930i	$0.351354\pi$
$578$	−6.18391	−0.257217
$579$	14.6674	0.609555
$580$	10.0457	0.417125
$581$	−0.984877	−0.0408596
$582$	36.6545	1.51938
$583$	−26.3733	−1.09227
$584$	−12.7369	−0.527059
$585$	90.7558	3.75229
$586$	13.7236	0.566916
$587$	15.0833	0.622554	0.311277	−	0.950319i	$-0.399243\pi$
$587$	15.0833	0.622554	0.311277	+	0.950319i	$0.399243\pi$
$588$	−19.8537	−0.818752
$589$	0.256411	0.0105652
$590$	34.7252	1.42962
$591$	−64.1410	−2.63841
$592$	6.93670	0.285097
$593$	11.5085	0.472597	0.236299	−	0.971680i	$-0.424066\pi$
$593$	11.5085	0.472597	0.236299	+	0.971680i	$0.424066\pi$
$594$	−71.1904	−2.92098
$595$	−11.0241	−0.451945
$596$	12.5019	0.512097
$597$	65.6840	2.68827
$598$	−8.84323	−0.361627
$599$	30.3556	1.24030	0.620149	−	0.784484i	$-0.287072\pi$
$599$	30.3556	1.24030	0.620149	+	0.784484i	$0.287072\pi$
$600$	14.3399	0.585423
$601$	43.8864	1.79016	0.895082	−	0.445902i	$-0.147117\pi$
$601$	43.8864	1.79016	0.895082	+	0.445902i	$0.147117\pi$
$602$	7.28268	0.296820
$603$	34.2638	1.39533
$604$	−1.25092	−0.0508994
$605$	−5.02762	−0.204402
$606$	4.29952	0.174656
$607$	31.0841	1.26167	0.630833	−	0.775919i	$-0.282713\pi$
$607$	31.0841	1.26167	0.630833	+	0.775919i	$0.282713\pi$
$608$	−3.76773	−0.152802
$609$	12.6233	0.511524
$610$	31.3160	1.26795
$611$	16.4994	0.667493
$612$	−29.0060	−1.17250
$613$	3.91015	0.157929	0.0789647	−	0.996877i	$-0.474839\pi$
$613$	3.91015	0.157929	0.0789647	+	0.996877i	$0.474839\pi$
$614$	−15.7064	−0.633860
$615$	127.176	5.12824
$616$	3.93840	0.158683
$617$	−14.7367	−0.593277	−0.296638	−	0.954990i	$-0.595866\pi$
$617$	−14.7367	−0.593277	−0.296638	+	0.954990i	$0.595866\pi$
$618$	−19.7490	−0.794420
$619$	−2.49664	−0.100349	−0.0501743	−	0.998740i	$-0.515978\pi$
$619$	−2.49664	−0.100349	−0.0501743	+	0.998740i	$0.515978\pi$
$620$	0.206095	0.00827696
$621$	52.0715	2.08956
$622$	9.59119	0.384572
$623$	−3.62066	−0.145059
$624$	−11.6820	−0.467653
$625$	−28.4577	−1.13831
$626$	6.02135	0.240662
$627$	46.0895	1.84064
$628$	16.2356	0.647872
$629$	−22.8133	−0.909626
$630$	29.5639	1.17785
$631$	−40.5084	−1.61261	−0.806306	−	0.591498i	$-0.798537\pi$
$631$	−40.5084	−1.61261	−0.806306	+	0.591498i	$0.798537\pi$
$632$	6.47742	0.257658
$633$	70.2723	2.79307
$634$	22.5522	0.895664
$635$	−5.42497	−0.215283
$636$	25.4828	1.01046
$637$	19.6224	0.777466
$638$	11.8030	0.467284
$639$	−23.4464	−0.927524
$640$	−3.02837	−0.119707
$641$	−12.0795	−0.477113	−0.238556	−	0.971129i	$-0.576674\pi$
$641$	−12.0795	−0.477113	−0.238556	+	0.971129i	$0.576674\pi$
$642$	−13.7380	−0.542196
$643$	−14.8849	−0.587003	−0.293502	−	0.955959i	$-0.594821\pi$
$643$	−14.8849	−0.587003	−0.293502	+	0.955959i	$0.594821\pi$
$644$	−2.88070	−0.113516
$645$	68.5019	2.69726
$646$	12.3912	0.487527
$647$	−24.0069	−0.943808	−0.471904	−	0.881650i	$-0.656433\pi$
$647$	−24.0069	−0.943808	−0.471904	+	0.881650i	$0.656433\pi$
$648$	42.3277	1.66279
$649$	40.7996	1.60153
$650$	−14.1728	−0.555902
$651$	0.258977	0.0101501
$652$	−16.9850	−0.665185
$653$	30.9321	1.21046	0.605232	−	0.796049i	$-0.293080\pi$
$653$	30.9321	1.21046	0.605232	+	0.796049i	$0.293080\pi$
$654$	59.7208	2.33527
$655$	−65.8871	−2.57442
$656$	−12.2150	−0.476916
$657$	−112.336	−4.38264
$658$	5.37471	0.209528
$659$	−49.3848	−1.92376	−0.961878	−	0.273477i	$-0.911826\pi$
$659$	−49.3848	−1.92376	−0.961878	+	0.273477i	$0.911826\pi$
$660$	37.0452	1.44198
$661$	−8.16827	−0.317709	−0.158854	−	0.987302i	$-0.550780\pi$
$661$	−8.16827	−0.317709	−0.158854	+	0.987302i	$0.550780\pi$
$662$	32.9979	1.28250
$663$	38.4194	1.49209
$664$	0.889778	0.0345301
$665$	−12.6296	−0.489754
$666$	61.1795	2.37066
$667$	−8.63315	−0.334277
$668$	−18.9280	−0.732348
$669$	9.92411	0.383688
$670$	−11.7650	−0.454521
$671$	36.7940	1.42042
$672$	−3.80543	−0.146798
$673$	43.4340	1.67426	0.837129	−	0.547006i	$-0.184232\pi$
$673$	43.4340	1.67426	0.837129	+	0.547006i	$0.184232\pi$
$674$	16.5450	0.637289
$675$	83.4534	3.21212
$676$	−1.45415	−0.0559288
$677$	−8.49919	−0.326650	−0.163325	−	0.986572i	$-0.552222\pi$
$677$	−8.49919	−0.326650	−0.163325	+	0.986572i	$0.552222\pi$
$678$	15.6048	0.599300
$679$	−11.8012	−0.452887
$680$	9.95965	0.381935
$681$	−71.7768	−2.75049
$682$	0.242146	0.00927225
$683$	7.32278	0.280198	0.140099	−	0.990137i	$-0.455258\pi$
$683$	7.32278	0.280198	0.140099	+	0.990137i	$0.455258\pi$
$684$	−33.2302	−1.27059
$685$	30.3026	1.15780
$686$	14.1402	0.539875
$687$	−16.3029	−0.621993
$688$	−6.57947	−0.250840
$689$	−25.1859	−0.959506
$690$	−27.0963	−1.03154
$691$	−43.2215	−1.64422	−0.822111	−	0.569327i	$-0.807204\pi$
$691$	−43.2215	−1.64422	−0.822111	+	0.569327i	$0.807204\pi$
$692$	−23.3465	−0.887502
$693$	34.7355	1.31949
$694$	−8.55827	−0.324868
$695$	20.6909	0.784849
$696$	−11.4044	−0.432284
$697$	40.1725	1.52164
$698$	−13.6373	−0.516179
$699$	−58.4735	−2.21167
$700$	−4.61682	−0.174499
$701$	−32.9547	−1.24468	−0.622341	−	0.782746i	$-0.713818\pi$
$701$	−32.9547	−1.24468	−0.622341	+	0.782746i	$0.713818\pi$
$702$	−67.9853	−2.56594
$703$	−26.1356	−0.985723
$704$	−3.55811	−0.134101
$705$	50.5553	1.90402
$706$	3.63509	0.136808
$707$	−1.38426	−0.0520604
$708$	−39.4221	−1.48157
$709$	−39.1080	−1.46873	−0.734366	−	0.678754i	$-0.762520\pi$
$709$	−39.1080	−1.46873	−0.734366	+	0.678754i	$0.762520\pi$
$710$	8.05066	0.302136
$711$	57.1288	2.14250
$712$	3.27105	0.122588
$713$	−0.177115	−0.00663302
$714$	12.5152	0.468370
$715$	−36.6135	−1.36927
$716$	−23.3419	−0.872327
$717$	−39.3377	−1.46909
$718$	30.0878	1.12287
$719$	33.0430	1.23229	0.616147	−	0.787631i	$-0.288693\pi$
$719$	33.0430	1.23229	0.616147	+	0.787631i	$0.288693\pi$
$720$	−26.7092	−0.995395
$721$	6.35832	0.236796
$722$	−4.80422	−0.178795
$723$	−101.330	−3.76849
$724$	−12.2632	−0.455758
$725$	−13.8361	−0.513860
$726$	5.70764	0.211831
$727$	9.20275	0.341311	0.170656	−	0.985331i	$-0.445411\pi$
$727$	9.20275	0.341311	0.170656	+	0.985331i	$0.445411\pi$
$728$	3.76109	0.139395
$729$	166.956	6.18357
$730$	38.5722	1.42762
$731$	21.6384	0.800327
$732$	−35.5517	−1.31403
$733$	18.7385	0.692123	0.346062	−	0.938212i	$-0.387519\pi$
$733$	18.7385	0.692123	0.346062	+	0.938212i	$0.387519\pi$
$734$	−11.0103	−0.406398
$735$	60.1243	2.21772
$736$	2.60254	0.0959311
$737$	−13.8230	−0.509177
$738$	−107.733	−3.96569
$739$	−37.1679	−1.36724	−0.683622	−	0.729836i	$-0.739596\pi$
$739$	−37.1679	−1.36724	−0.683622	+	0.729836i	$0.739596\pi$
$740$	−21.0069	−0.772229
$741$	44.0145	1.61691
$742$	−8.20436	−0.301192
$743$	−17.8928	−0.656422	−0.328211	−	0.944604i	$-0.606446\pi$
$743$	−17.8928	−0.656422	−0.328211	+	0.944604i	$0.606446\pi$
$744$	−0.233970	−0.00857776
$745$	−37.8603	−1.38709
$746$	−6.52618	−0.238941
$747$	7.84756	0.287127
$748$	11.7019	0.427862
$749$	4.42305	0.161615
$750$	8.63089	0.315155
$751$	−1.64746	−0.0601165	−0.0300582	−	0.999548i	$-0.509569\pi$
$751$	−1.64746	−0.0601165	−0.0300582	+	0.999548i	$0.509569\pi$
$752$	−4.85573	−0.177070
$753$	19.4107	0.707365
$754$	11.2716	0.410486
$755$	3.78826	0.137869
$756$	−22.1464	−0.805455
$757$	−1.42479	−0.0517848	−0.0258924	−	0.999665i	$-0.508243\pi$
$757$	−1.42479	−0.0517848	−0.0258924	+	0.999665i	$0.508243\pi$
$758$	−0.544847	−0.0197897
$759$	−31.8362	−1.15558
$760$	11.4101	0.413887
$761$	12.1917	0.441950	0.220975	−	0.975279i	$-0.429076\pi$
$761$	12.1917	0.441950	0.220975	+	0.975279i	$0.429076\pi$
$762$	6.15874	0.223107
$763$	−19.2275	−0.696083
$764$	14.7586	0.533946
$765$	87.8409	3.17589
$766$	−18.0197	−0.651079
$767$	38.9627	1.40686
$768$	3.43798	0.124057
$769$	4.02194	0.145035	0.0725174	−	0.997367i	$-0.476897\pi$
$769$	4.02194	0.145035	0.0725174	+	0.997367i	$0.476897\pi$
$770$	−11.9269	−0.429817
$771$	34.2360	1.23298
$772$	4.26628	0.153547
$773$	−28.9481	−1.04119	−0.520595	−	0.853804i	$-0.674290\pi$
$773$	−28.9481	−1.04119	−0.520595	+	0.853804i	$0.674290\pi$
$774$	−58.0288	−2.08580
$775$	−0.283857	−0.0101965
$776$	10.6617	0.382731
$777$	−26.3971	−0.946991
$778$	6.95139	0.249220
$779$	46.0229	1.64894
$780$	35.3773	1.26671
$781$	9.45895	0.338468
$782$	−8.55920	−0.306076
$783$	−66.3702	−2.37188
$784$	−5.77482	−0.206243
$785$	−49.1675	−1.75486
$786$	74.7988	2.66798
$787$	39.3738	1.40352	0.701762	−	0.712412i	$-0.252397\pi$
$787$	39.3738	1.40352	0.701762	+	0.712412i	$0.252397\pi$
$788$	−18.6566	−0.664615
$789$	69.3314	2.46826
$790$	−19.6160	−0.697907
$791$	−5.02408	−0.178636
$792$	−31.3814	−1.11509
$793$	35.1374	1.24777
$794$	12.3712	0.439037
$795$	−77.1713	−2.73699
$796$	19.1054	0.677174
$797$	−23.8361	−0.844318	−0.422159	−	0.906522i	$-0.638728\pi$
$797$	−23.8361	−0.844318	−0.422159	+	0.906522i	$0.638728\pi$
$798$	14.3378	0.507553
$799$	15.9694	0.564958
$800$	4.17102	0.147468
$801$	28.8496	1.01935
$802$	3.96628	0.140054
$803$	45.3195	1.59929
$804$	13.3563	0.471039
$805$	8.72384	0.307475
$806$	0.231244	0.00814523
$807$	−94.0646	−3.31123
$808$	1.25060	0.0439958
$809$	16.6446	0.585194	0.292597	−	0.956236i	$-0.405481\pi$
$809$	16.6446	0.585194	0.292597	+	0.956236i	$0.405481\pi$
$810$	−128.184	−4.50392
$811$	13.2833	0.466441	0.233220	−	0.972424i	$-0.425074\pi$
$811$	13.2833	0.466441	0.233220	+	0.972424i	$0.425074\pi$
$812$	3.67174	0.128853
$813$	6.86130	0.240636
$814$	−24.6816	−0.865089
$815$	51.4369	1.80176
$816$	−11.3068	−0.395816
$817$	24.7897	0.867280
$818$	5.10460	0.178478
$819$	33.1716	1.15911
$820$	36.9916	1.29180
$821$	−36.1236	−1.26072	−0.630361	−	0.776302i	$-0.717093\pi$
$821$	−36.1236	−1.26072	−0.630361	+	0.776302i	$0.717093\pi$
$822$	−34.4012	−1.19988
$823$	−15.1357	−0.527597	−0.263798	−	0.964578i	$-0.584975\pi$
$823$	−15.1357	−0.527597	−0.263798	+	0.964578i	$0.584975\pi$
$824$	−5.74436	−0.200114
$825$	−51.0229	−1.77639
$826$	12.6922	0.441618
$827$	−0.735973	−0.0255923	−0.0127961	−	0.999918i	$-0.504073\pi$
$827$	−0.735973	−0.0255923	−0.0127961	+	0.999918i	$0.504073\pi$
$828$	22.9536	0.797693
$829$	24.2725	0.843018	0.421509	−	0.906824i	$-0.361501\pi$
$829$	24.2725	0.843018	0.421509	+	0.906824i	$0.361501\pi$
$830$	−2.69458	−0.0935301
$831$	67.7701	2.35092
$832$	−3.39792	−0.117802
$833$	18.9921	0.658038
$834$	−23.4894	−0.813373
$835$	57.3211	1.98368
$836$	13.4060	0.463656
$837$	−1.36163	−0.0470648
$838$	−14.2452	−0.492093
$839$	−39.8963	−1.37737	−0.688686	−	0.725059i	$-0.741812\pi$
$839$	−39.8963	−1.37737	−0.688686	+	0.725059i	$0.741812\pi$
$840$	11.5242	0.397624
$841$	−17.9962	−0.620559
$842$	−2.08365	−0.0718073
$843$	53.8745	1.85554
$844$	20.4400	0.703574
$845$	4.40370	0.151492
$846$	−42.8260	−1.47239
$847$	−1.83761	−0.0631412
$848$	7.41215	0.254534
$849$	26.1421	0.897194
$850$	−13.7176	−0.470509
$851$	18.0531	0.618852
$852$	−9.13957	−0.313116
$853$	8.79970	0.301296	0.150648	−	0.988587i	$-0.451864\pi$
$853$	8.79970	0.301296	0.150648	+	0.988587i	$0.451864\pi$
$854$	11.4461	0.391678
$855$	100.633	3.44158
$856$	−3.99596	−0.136579
$857$	−4.40766	−0.150563	−0.0752814	−	0.997162i	$-0.523986\pi$
$857$	−4.40766	−0.150563	−0.0752814	+	0.997162i	$0.523986\pi$
$858$	41.5658	1.41903
$859$	−43.9012	−1.49789	−0.748945	−	0.662632i	$-0.769439\pi$
$859$	−43.9012	−1.49789	−0.748945	+	0.662632i	$0.769439\pi$
$860$	19.9251	0.679439
$861$	46.4834	1.58415
$862$	−32.3361	−1.10137
$863$	7.97488	0.271468	0.135734	−	0.990745i	$-0.456661\pi$
$863$	7.97488	0.271468	0.135734	+	0.990745i	$0.456661\pi$
$864$	20.0079	0.680683
$865$	70.7019	2.40394
$866$	−29.3043	−0.995802
$867$	−21.2601	−0.722032
$868$	0.0753283	0.00255681
$869$	−23.0474	−0.781830
$870$	34.5369	1.17091
$871$	−13.2006	−0.447287
$872$	17.3709	0.588254
$873$	94.0324	3.18251
$874$	−9.80568	−0.331682
$875$	−2.77877	−0.0939397
$876$	−43.7893	−1.47950
$877$	−23.4807	−0.792888	−0.396444	−	0.918059i	$-0.629756\pi$
$877$	−23.4807	−0.792888	−0.396444	+	0.918059i	$0.629756\pi$
$878$	−6.52153	−0.220091
$879$	47.1814	1.59139
$880$	10.7753	0.363235
$881$	−20.8774	−0.703379	−0.351689	−	0.936117i	$-0.614393\pi$
$881$	−20.8774	−0.703379	−0.351689	+	0.936117i	$0.614393\pi$
$882$	−50.9320	−1.71497
$883$	22.5264	0.758074	0.379037	−	0.925381i	$-0.376255\pi$
$883$	22.5264	0.758074	0.379037	+	0.925381i	$0.376255\pi$
$884$	11.1750	0.375856
$885$	119.385	4.01307
$886$	−4.94546	−0.166146
$887$	13.5862	0.456181	0.228091	−	0.973640i	$-0.426752\pi$
$887$	13.5862	0.456181	0.228091	+	0.973640i	$0.426752\pi$
$888$	23.8482	0.800294
$889$	−1.98285	−0.0665026
$890$	−9.90595	−0.332048
$891$	−150.607	−5.04552
$892$	2.88661	0.0966510
$893$	18.2951	0.612221
$894$	42.9812	1.43750
$895$	70.6878	2.36283
$896$	−1.10688	−0.0369783
$897$	−30.4028	−1.01512
$898$	−24.4368	−0.815467
$899$	0.225751	0.00752920
$900$	36.7871	1.22624
$901$	−24.3770	−0.812114
$902$	43.4624	1.44714
$903$	25.0377	0.833202
$904$	4.53896	0.150963
$905$	37.1375	1.23449
$906$	−4.30065	−0.142880
$907$	46.2650	1.53620	0.768102	−	0.640327i	$-0.221201\pi$
$907$	46.2650	1.53620	0.768102	+	0.640327i	$0.221201\pi$
$908$	−20.8776	−0.692849
$909$	11.0299	0.365837
$910$	−11.3900	−0.377574
$911$	−8.34215	−0.276388	−0.138194	−	0.990405i	$-0.544130\pi$
$911$	−8.34215	−0.276388	−0.138194	+	0.990405i	$0.544130\pi$
$912$	−12.9534	−0.428929
$913$	−3.16593	−0.104777
$914$	20.4506	0.676446
$915$	107.664	3.55925
$916$	−4.74199	−0.156680
$917$	−24.0820	−0.795257
$918$	−65.8017	−2.17178
$919$	−36.0211	−1.18823	−0.594113	−	0.804381i	$-0.702497\pi$
$919$	−36.0211	−1.18823	−0.594113	+	0.804381i	$0.702497\pi$
$920$	−7.88147	−0.259844
$921$	−53.9983	−1.77930
$922$	−6.06096	−0.199607
$923$	9.03308	0.297327
$924$	13.5401	0.445438
$925$	28.9331	0.951316
$926$	39.8186	1.30852
$927$	−50.6634	−1.66400
$928$	−3.31720	−0.108892
$929$	−24.3499	−0.798893	−0.399447	−	0.916756i	$-0.630798\pi$
$929$	−24.3499	−0.798893	−0.399447	+	0.916756i	$0.630798\pi$
$930$	0.708548	0.0232342
$931$	21.7579	0.713088
$932$	−17.0081	−0.557119
$933$	32.9743	1.07953
$934$	8.37003	0.273876
$935$	−35.4376	−1.15893
$936$	−29.9685	−0.979553
$937$	−49.4009	−1.61386	−0.806929	−	0.590648i	$-0.798872\pi$
$937$	−49.4009	−1.61386	−0.806929	+	0.590648i	$0.798872\pi$
$938$	−4.30014	−0.140405
$939$	20.7013	0.675560
$940$	14.7049	0.479623
$941$	−50.8487	−1.65762	−0.828810	−	0.559531i	$-0.810981\pi$
$941$	−50.8487	−1.65762	−0.828810	+	0.559531i	$0.810981\pi$
$942$	55.8177	1.81864
$943$	−31.7901	−1.03523
$944$	−11.4666	−0.373208
$945$	67.0673	2.18170
$946$	23.4105	0.761141
$947$	−49.0269	−1.59316	−0.796579	−	0.604534i	$-0.793359\pi$
$947$	−49.0269	−1.59316	−0.796579	+	0.604534i	$0.793359\pi$
$948$	22.2692	0.723271
$949$	43.2791	1.40490
$950$	−15.7153	−0.509871
$951$	77.5340	2.51421
$952$	3.64029	0.117982
$953$	56.0042	1.81415	0.907077	−	0.420964i	$-0.138308\pi$
$953$	56.0042	1.81415	0.907077	+	0.420964i	$0.138308\pi$
$954$	65.3728	2.11652
$955$	−44.6944	−1.44628
$956$	−11.4421	−0.370064
$957$	40.5783	1.31171
$958$	8.25406	0.266677
$959$	11.0757	0.357653
$960$	−10.4115	−0.336028
$961$	−30.9954	−0.999851
$962$	−23.5704	−0.759939
$963$	−35.2431	−1.13569
$964$	−29.4736	−0.949282
$965$	−12.9199	−0.415905
$966$	−9.90379	−0.318649
$967$	−34.7373	−1.11708	−0.558539	−	0.829478i	$-0.688638\pi$
$967$	−34.7373	−1.11708	−0.558539	+	0.829478i	$0.688638\pi$
$968$	1.66018	0.0533601
$969$	42.6008	1.36853
$970$	−32.2874	−1.03669
$971$	−39.4288	−1.26533	−0.632665	−	0.774425i	$-0.718039\pi$
$971$	−39.4288	−1.26533	−0.632665	+	0.774425i	$0.718039\pi$
$972$	85.4978	2.74234
$973$	7.56259	0.242445
$974$	−2.64483	−0.0847457
$975$	−48.7257	−1.56047
$976$	−10.3409	−0.331003
$977$	15.0785	0.482404	0.241202	−	0.970475i	$-0.422458\pi$
$977$	15.0785	0.482404	0.241202	+	0.970475i	$0.422458\pi$
$978$	−58.3941	−1.86724
$979$	−11.6388	−0.371977
$980$	17.4883	0.558643
$981$	153.206	4.89149
$982$	−17.2133	−0.549299
$983$	10.4102	0.332034	0.166017	−	0.986123i	$-0.446909\pi$
$983$	10.4102	0.332034	0.166017	+	0.986123i	$0.446909\pi$
$984$	−41.9950	−1.33875
$985$	56.4991	1.80021
$986$	10.9095	0.347430
$987$	18.4781	0.588165
$988$	12.8024	0.407300
$989$	−17.1234	−0.544491
$990$	95.0345	3.02040
$991$	−37.4664	−1.19016	−0.595080	−	0.803667i	$-0.702880\pi$
$991$	−37.4664	−1.19016	−0.595080	+	0.803667i	$0.702880\pi$
$992$	−0.0680546	−0.00216074
$993$	113.446	3.60010
$994$	2.94255	0.0933319
$995$	−57.8583	−1.83423
$996$	3.05904	0.0969292
$997$	30.8986	0.978569	0.489285	−	0.872124i	$-0.337258\pi$
$997$	30.8986	0.978569	0.489285	+	0.872124i	$0.337258\pi$
$998$	31.7780	1.00592
$999$	138.789	4.39109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.69	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.69	✓	69	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.43798 q^{3} +1.00000 q^{4} -3.02837 q^{5} +3.43798 q^{6} -1.10688 q^{7} +1.00000 q^{8} +8.81968 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.43798 q^{3} +1.00000 q^{4} -3.02837 q^{5} +3.43798 q^{6} -1.10688 q^{7} +1.00000 q^{8} +8.81968 q^{9} -3.02837 q^{10} -3.55811 q^{11} +3.43798 q^{12} -3.39792 q^{13} -1.10688 q^{14} -10.4115 q^{15} +1.00000 q^{16} -3.28878 q^{17} +8.81968 q^{18} -3.76773 q^{19} -3.02837 q^{20} -3.80543 q^{21} -3.55811 q^{22} +2.60254 q^{23} +3.43798 q^{24} +4.17102 q^{25} -3.39792 q^{26} +20.0079 q^{27} -1.10688 q^{28} -3.31720 q^{29} -10.4115 q^{30} -0.0680546 q^{31} +1.00000 q^{32} -12.2327 q^{33} -3.28878 q^{34} +3.35204 q^{35} +8.81968 q^{36} +6.93670 q^{37} -3.76773 q^{38} -11.6820 q^{39} -3.02837 q^{40} -12.2150 q^{41} -3.80543 q^{42} -6.57947 q^{43} -3.55811 q^{44} -26.7092 q^{45} +2.60254 q^{46} -4.85573 q^{47} +3.43798 q^{48} -5.77482 q^{49} +4.17102 q^{50} -11.3068 q^{51} -3.39792 q^{52} +7.41215 q^{53} +20.0079 q^{54} +10.7753 q^{55} -1.10688 q^{56} -12.9534 q^{57} -3.31720 q^{58} -11.4666 q^{59} -10.4115 q^{60} -10.3409 q^{61} -0.0680546 q^{62} -9.76232 q^{63} +1.00000 q^{64} +10.2902 q^{65} -12.2327 q^{66} +3.88492 q^{67} -3.28878 q^{68} +8.94748 q^{69} +3.35204 q^{70} -2.65842 q^{71} +8.81968 q^{72} -12.7369 q^{73} +6.93670 q^{74} +14.3399 q^{75} -3.76773 q^{76} +3.93840 q^{77} -11.6820 q^{78} +6.47742 q^{79} -3.02837 q^{80} +42.3277 q^{81} -12.2150 q^{82} +0.889778 q^{83} -3.80543 q^{84} +9.95965 q^{85} -6.57947 q^{86} -11.4044 q^{87} -3.55811 q^{88} +3.27105 q^{89} -26.7092 q^{90} +3.76109 q^{91} +2.60254 q^{92} -0.233970 q^{93} -4.85573 q^{94} +11.4101 q^{95} +3.43798 q^{96} +10.6617 q^{97} -5.77482 q^{98} -31.3814 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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