LMFDB - Embedded newform 8038.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.02523 q^{3} +1.00000 q^{4} +2.41251 q^{5} +3.02523 q^{6} +3.55505 q^{7} -1.00000 q^{8} +6.15201 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.02523 q^{3} +1.00000 q^{4} +2.41251 q^{5} +3.02523 q^{6} +3.55505 q^{7} -1.00000 q^{8} +6.15201 q^{9} -2.41251 q^{10} +3.30008 q^{11} -3.02523 q^{12} +6.51262 q^{13} -3.55505 q^{14} -7.29840 q^{15} +1.00000 q^{16} +0.572747 q^{17} -6.15201 q^{18} -7.21015 q^{19} +2.41251 q^{20} -10.7548 q^{21} -3.30008 q^{22} -1.04288 q^{23} +3.02523 q^{24} +0.820206 q^{25} -6.51262 q^{26} -9.53555 q^{27} +3.55505 q^{28} +3.45085 q^{29} +7.29840 q^{30} -4.32141 q^{31} -1.00000 q^{32} -9.98351 q^{33} -0.572747 q^{34} +8.57660 q^{35} +6.15201 q^{36} +5.23100 q^{37} +7.21015 q^{38} -19.7022 q^{39} -2.41251 q^{40} +6.02200 q^{41} +10.7548 q^{42} -9.95934 q^{43} +3.30008 q^{44} +14.8418 q^{45} +1.04288 q^{46} -7.80580 q^{47} -3.02523 q^{48} +5.63839 q^{49} -0.820206 q^{50} -1.73269 q^{51} +6.51262 q^{52} -4.22624 q^{53} +9.53555 q^{54} +7.96149 q^{55} -3.55505 q^{56} +21.8123 q^{57} -3.45085 q^{58} +1.48720 q^{59} -7.29840 q^{60} +8.78302 q^{61} +4.32141 q^{62} +21.8707 q^{63} +1.00000 q^{64} +15.7118 q^{65} +9.98351 q^{66} +0.963667 q^{67} +0.572747 q^{68} +3.15495 q^{69} -8.57660 q^{70} +2.36205 q^{71} -6.15201 q^{72} +6.22427 q^{73} -5.23100 q^{74} -2.48131 q^{75} -7.21015 q^{76} +11.7320 q^{77} +19.7022 q^{78} +12.0714 q^{79} +2.41251 q^{80} +10.3912 q^{81} -6.02200 q^{82} +8.92312 q^{83} -10.7548 q^{84} +1.38176 q^{85} +9.95934 q^{86} -10.4396 q^{87} -3.30008 q^{88} -0.840647 q^{89} -14.8418 q^{90} +23.1527 q^{91} -1.04288 q^{92} +13.0733 q^{93} +7.80580 q^{94} -17.3945 q^{95} +3.02523 q^{96} -1.87928 q^{97} -5.63839 q^{98} +20.3021 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * 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
$3$	−3.02523	−1.74662	−0.873308	−	0.487168i	$-0.838030\pi$
$3$	−3.02523	−1.74662	−0.873308	+	0.487168i	$0.838030\pi$
$4$	1.00000	0.500000
$5$	2.41251	1.07891	0.539454	−	0.842015i	$-0.318631\pi$
$5$	2.41251	1.07891	0.539454	+	0.842015i	$0.318631\pi$
$6$	3.02523	1.23504
$7$	3.55505	1.34368	0.671842	−	0.740695i	$-0.265504\pi$
$7$	3.55505	1.34368	0.671842	+	0.740695i	$0.265504\pi$
$8$	−1.00000	−0.353553
$9$	6.15201	2.05067
$10$	−2.41251	−0.762903
$11$	3.30008	0.995013	0.497506	−	0.867460i	$-0.334249\pi$
$11$	3.30008	0.995013	0.497506	+	0.867460i	$0.334249\pi$
$12$	−3.02523	−0.873308
$13$	6.51262	1.80628	0.903138	−	0.429350i	$-0.141257\pi$
$13$	6.51262	1.80628	0.903138	+	0.429350i	$0.141257\pi$
$14$	−3.55505	−0.950128
$15$	−7.29840	−1.88444
$16$	1.00000	0.250000
$17$	0.572747	0.138912	0.0694558	−	0.997585i	$-0.477874\pi$
$17$	0.572747	0.138912	0.0694558	+	0.997585i	$0.477874\pi$
$18$	−6.15201	−1.45004
$19$	−7.21015	−1.65412	−0.827060	−	0.562113i	$-0.809989\pi$
$19$	−7.21015	−1.65412	−0.827060	+	0.562113i	$0.809989\pi$
$20$	2.41251	0.539454
$21$	−10.7548	−2.34690
$22$	−3.30008	−0.703580
$23$	−1.04288	−0.217455	−0.108728	−	0.994072i	$-0.534678\pi$
$23$	−1.04288	−0.217455	−0.108728	+	0.994072i	$0.534678\pi$
$24$	3.02523	0.617522
$25$	0.820206	0.164041
$26$	−6.51262	−1.27723
$27$	−9.53555	−1.83512
$28$	3.55505	0.671842
$29$	3.45085	0.640807	0.320404	−	0.947281i	$-0.396181\pi$
$29$	3.45085	0.640807	0.320404	+	0.947281i	$0.396181\pi$
$30$	7.29840	1.33250
$31$	−4.32141	−0.776149	−0.388074	−	0.921628i	$-0.626860\pi$
$31$	−4.32141	−0.776149	−0.388074	+	0.921628i	$0.626860\pi$
$32$	−1.00000	−0.176777
$33$	−9.98351	−1.73791
$34$	−0.572747	−0.0982253
$35$	8.57660	1.44971
$36$	6.15201	1.02533
$37$	5.23100	0.859972	0.429986	−	0.902836i	$-0.358519\pi$
$37$	5.23100	0.859972	0.429986	+	0.902836i	$0.358519\pi$
$38$	7.21015	1.16964
$39$	−19.7022	−3.15487
$40$	−2.41251	−0.381451
$41$	6.02200	0.940479	0.470239	−	0.882539i	$-0.344168\pi$
$41$	6.02200	0.940479	0.470239	+	0.882539i	$0.344168\pi$
$42$	10.7548	1.65951
$43$	−9.95934	−1.51879	−0.759393	−	0.650633i	$-0.774504\pi$
$43$	−9.95934	−1.51879	−0.759393	+	0.650633i	$0.774504\pi$
$44$	3.30008	0.497506
$45$	14.8418	2.21248
$46$	1.04288	0.153764
$47$	−7.80580	−1.13859	−0.569296	−	0.822132i	$-0.692784\pi$
$47$	−7.80580	−1.13859	−0.569296	+	0.822132i	$0.692784\pi$
$48$	−3.02523	−0.436654
$49$	5.63839	0.805485
$50$	−0.820206	−0.115995
$51$	−1.73269	−0.242625
$52$	6.51262	0.903138
$53$	−4.22624	−0.580519	−0.290259	−	0.956948i	$-0.593742\pi$
$53$	−4.22624	−0.580519	−0.290259	+	0.956948i	$0.593742\pi$
$54$	9.53555	1.29762
$55$	7.96149	1.07353
$56$	−3.55505	−0.475064
$57$	21.8123	2.88911
$58$	−3.45085	−0.453119
$59$	1.48720	0.193617	0.0968085	−	0.995303i	$-0.469137\pi$
$59$	1.48720	0.193617	0.0968085	+	0.995303i	$0.469137\pi$
$60$	−7.29840	−0.942219
$61$	8.78302	1.12455	0.562275	−	0.826950i	$-0.309926\pi$
$61$	8.78302	1.12455	0.562275	+	0.826950i	$0.309926\pi$
$62$	4.32141	0.548820
$63$	21.8707	2.75545
$64$	1.00000	0.125000
$65$	15.7118	1.94880
$66$	9.98351	1.22888
$67$	0.963667	0.117731	0.0588654	−	0.998266i	$-0.481252\pi$
$67$	0.963667	0.117731	0.0588654	+	0.998266i	$0.481252\pi$
$68$	0.572747	0.0694558
$69$	3.15495	0.379811
$70$	−8.57660	−1.02510
$71$	2.36205	0.280324	0.140162	−	0.990129i	$-0.455238\pi$
$71$	2.36205	0.280324	0.140162	+	0.990129i	$0.455238\pi$
$72$	−6.15201	−0.725021
$73$	6.22427	0.728496	0.364248	−	0.931302i	$-0.381326\pi$
$73$	6.22427	0.728496	0.364248	+	0.931302i	$0.381326\pi$
$74$	−5.23100	−0.608092
$75$	−2.48131	−0.286517
$76$	−7.21015	−0.827060
$77$	11.7320	1.33698
$78$	19.7022	2.23083
$79$	12.0714	1.35813	0.679067	−	0.734076i	$-0.262385\pi$
$79$	12.0714	1.35813	0.679067	+	0.734076i	$0.262385\pi$
$80$	2.41251	0.269727
$81$	10.3912	1.15458
$82$	−6.02200	−0.665019
$83$	8.92312	0.979440	0.489720	−	0.871880i	$-0.337099\pi$
$83$	8.92312	0.979440	0.489720	+	0.871880i	$0.337099\pi$
$84$	−10.7548	−1.17345
$85$	1.38176	0.149873
$86$	9.95934	1.07394
$87$	−10.4396	−1.11924
$88$	−3.30008	−0.351790
$89$	−0.840647	−0.0891084	−0.0445542	−	0.999007i	$-0.514187\pi$
$89$	−0.840647	−0.0891084	−0.0445542	+	0.999007i	$0.514187\pi$
$90$	−14.8418	−1.56446
$91$	23.1527	2.42706
$92$	−1.04288	−0.108728
$93$	13.0733	1.35563
$94$	7.80580	0.805107
$95$	−17.3945	−1.78464
$96$	3.02523	0.308761
$97$	−1.87928	−0.190812	−0.0954058	−	0.995438i	$-0.530415\pi$
$97$	−1.87928	−0.190812	−0.0954058	+	0.995438i	$0.530415\pi$
$98$	−5.63839	−0.569564
$99$	20.3021	2.04044
$100$	0.820206	0.0820206
$101$	−2.19594	−0.218504	−0.109252	−	0.994014i	$-0.534846\pi$
$101$	−2.19594	−0.218504	−0.109252	+	0.994014i	$0.534846\pi$
$102$	1.73269	0.171562
$103$	3.95087	0.389291	0.194645	−	0.980874i	$-0.437644\pi$
$103$	3.95087	0.389291	0.194645	+	0.980874i	$0.437644\pi$
$104$	−6.51262	−0.638615
$105$	−25.9462	−2.53209
$106$	4.22624	0.410489
$107$	6.83769	0.661025	0.330512	−	0.943802i	$-0.392778\pi$
$107$	6.83769	0.661025	0.330512	+	0.943802i	$0.392778\pi$
$108$	−9.53555	−0.917558
$109$	4.60470	0.441051	0.220525	−	0.975381i	$-0.429223\pi$
$109$	4.60470	0.441051	0.220525	+	0.975381i	$0.429223\pi$
$110$	−7.96149	−0.759098
$111$	−15.8250	−1.50204
$112$	3.55505	0.335921
$113$	−10.8775	−1.02327	−0.511635	−	0.859203i	$-0.670960\pi$
$113$	−10.8775	−1.02327	−0.511635	+	0.859203i	$0.670960\pi$
$114$	−21.8123	−2.04291
$115$	−2.51596	−0.234614
$116$	3.45085	0.320404
$117$	40.0657	3.70408
$118$	−1.48720	−0.136908
$119$	2.03615	0.186653
$120$	7.29840	0.666249
$121$	−0.109449	−0.00994995
$122$	−8.78302	−0.795177
$123$	−18.2179	−1.64266
$124$	−4.32141	−0.388074
$125$	−10.0838	−0.901922
$126$	−21.8707	−1.94840
$127$	6.79621	0.603066	0.301533	−	0.953456i	$-0.402502\pi$
$127$	6.79621	0.603066	0.301533	+	0.953456i	$0.402502\pi$
$128$	−1.00000	−0.0883883
$129$	30.1293	2.65274
$130$	−15.7118	−1.37801
$131$	−3.13321	−0.273750	−0.136875	−	0.990588i	$-0.543706\pi$
$131$	−3.13321	−0.273750	−0.136875	+	0.990588i	$0.543706\pi$
$132$	−9.98351	−0.868953
$133$	−25.6324	−2.22261
$134$	−0.963667	−0.0832482
$135$	−23.0046	−1.97992
$136$	−0.572747	−0.0491127
$137$	15.3053	1.30762	0.653812	−	0.756657i	$-0.273169\pi$
$137$	15.3053	1.30762	0.653812	+	0.756657i	$0.273169\pi$
$138$	−3.15495	−0.268567
$139$	−9.32510	−0.790945	−0.395473	−	0.918478i	$-0.629419\pi$
$139$	−9.32510	−0.790945	−0.395473	+	0.918478i	$0.629419\pi$
$140$	8.57660	0.724855
$141$	23.6143	1.98868
$142$	−2.36205	−0.198219
$143$	21.4922	1.79727
$144$	6.15201	0.512667
$145$	8.32522	0.691372
$146$	−6.22427	−0.515125
$147$	−17.0574	−1.40687
$148$	5.23100	0.429986
$149$	−6.56817	−0.538086	−0.269043	−	0.963128i	$-0.586707\pi$
$149$	−6.56817	−0.538086	−0.269043	+	0.963128i	$0.586707\pi$
$150$	2.48131	0.202598
$151$	13.7345	1.11770	0.558850	−	0.829269i	$-0.311243\pi$
$151$	13.7345	1.11770	0.558850	+	0.829269i	$0.311243\pi$
$152$	7.21015	0.584820
$153$	3.52355	0.284862
$154$	−11.7320	−0.945389
$155$	−10.4255	−0.837393
$156$	−19.7022	−1.57744
$157$	20.5282	1.63833	0.819165	−	0.573558i	$-0.194437\pi$
$157$	20.5282	1.63833	0.819165	+	0.573558i	$0.194437\pi$
$158$	−12.0714	−0.960346
$159$	12.7853	1.01394
$160$	−2.41251	−0.190726
$161$	−3.70749	−0.292191
$162$	−10.3912	−0.816408
$163$	3.49194	0.273510	0.136755	−	0.990605i	$-0.456333\pi$
$163$	3.49194	0.273510	0.136755	+	0.990605i	$0.456333\pi$
$164$	6.02200	0.470239
$165$	−24.0853	−1.87504
$166$	−8.92312	−0.692569
$167$	19.8272	1.53427	0.767136	−	0.641485i	$-0.221681\pi$
$167$	19.8272	1.53427	0.767136	+	0.641485i	$0.221681\pi$
$168$	10.7548	0.829754
$169$	29.4142	2.26263
$170$	−1.38176	−0.105976
$171$	−44.3569	−3.39205
$172$	−9.95934	−0.759393
$173$	1.21940	0.0927092	0.0463546	−	0.998925i	$-0.485240\pi$
$173$	1.21940	0.0927092	0.0463546	+	0.998925i	$0.485240\pi$
$174$	10.4396	0.791426
$175$	2.91587	0.220419
$176$	3.30008	0.248753
$177$	−4.49912	−0.338175
$178$	0.840647	0.0630092
$179$	−11.3071	−0.845134	−0.422567	−	0.906332i	$-0.638871\pi$
$179$	−11.3071	−0.845134	−0.422567	+	0.906332i	$0.638871\pi$
$180$	14.8418	1.10624
$181$	−2.35665	−0.175169	−0.0875844	−	0.996157i	$-0.527915\pi$
$181$	−2.35665	−0.175169	−0.0875844	+	0.996157i	$0.527915\pi$
$182$	−23.1527	−1.71619
$183$	−26.5706	−1.96416
$184$	1.04288	0.0768821
$185$	12.6198	0.927830
$186$	−13.0733	−0.958578
$187$	1.89011	0.138219
$188$	−7.80580	−0.569296
$189$	−33.8994	−2.46582
$190$	17.3945	1.26193
$191$	18.2985	1.32403	0.662017	−	0.749489i	$-0.269701\pi$
$191$	18.2985	1.32403	0.662017	+	0.749489i	$0.269701\pi$
$192$	−3.02523	−0.218327
$193$	−17.7840	−1.28012	−0.640061	−	0.768324i	$-0.721091\pi$
$193$	−17.7840	−1.28012	−0.640061	+	0.768324i	$0.721091\pi$
$194$	1.87928	0.134924
$195$	−47.5317	−3.40381
$196$	5.63839	0.402742
$197$	−18.4631	−1.31544	−0.657719	−	0.753263i	$-0.728479\pi$
$197$	−18.4631	−1.31544	−0.657719	+	0.753263i	$0.728479\pi$
$198$	−20.3021	−1.44281
$199$	2.18599	0.154961	0.0774805	−	0.996994i	$-0.475312\pi$
$199$	2.18599	0.154961	0.0774805	+	0.996994i	$0.475312\pi$
$200$	−0.820206	−0.0579973
$201$	−2.91531	−0.205630
$202$	2.19594	0.154506
$203$	12.2680	0.861042
$204$	−1.73269	−0.121313
$205$	14.5281	1.01469
$206$	−3.95087	−0.275270
$207$	−6.41580	−0.445929
$208$	6.51262	0.451569
$209$	−23.7941	−1.64587
$210$	25.9462	1.79046
$211$	−4.37274	−0.301032	−0.150516	−	0.988608i	$-0.548093\pi$
$211$	−4.37274	−0.301032	−0.150516	+	0.988608i	$0.548093\pi$
$212$	−4.22624	−0.290259
$213$	−7.14574	−0.489618
$214$	−6.83769	−0.467415
$215$	−24.0270	−1.63863
$216$	9.53555	0.648812
$217$	−15.3629	−1.04290
$218$	−4.60470	−0.311870
$219$	−18.8299	−1.27240
$220$	7.96149	0.536763
$221$	3.73009	0.250913
$222$	15.8250	1.06210
$223$	−11.8687	−0.794788	−0.397394	−	0.917648i	$-0.630085\pi$
$223$	−11.8687	−0.794788	−0.397394	+	0.917648i	$0.630085\pi$
$224$	−3.55505	−0.237532
$225$	5.04591	0.336394
$226$	10.8775	0.723561
$227$	29.8441	1.98082	0.990411	−	0.138154i	$-0.0441170\pi$
$227$	29.8441	1.98082	0.990411	+	0.138154i	$0.0441170\pi$
$228$	21.8123	1.44456
$229$	−3.85892	−0.255005	−0.127502	−	0.991838i	$-0.540696\pi$
$229$	−3.85892	−0.255005	−0.127502	+	0.991838i	$0.540696\pi$
$230$	2.51596	0.165897
$231$	−35.4919	−2.33519
$232$	−3.45085	−0.226560
$233$	29.2555	1.91659	0.958295	−	0.285781i	$-0.0922532\pi$
$233$	29.2555	1.91659	0.958295	+	0.285781i	$0.0922532\pi$
$234$	−40.0657	−2.61918
$235$	−18.8316	−1.22844
$236$	1.48720	0.0968085
$237$	−36.5186	−2.37214
$238$	−2.03615	−0.131984
$239$	−0.607549	−0.0392991	−0.0196495	−	0.999807i	$-0.506255\pi$
$239$	−0.607549	−0.0392991	−0.0196495	+	0.999807i	$0.506255\pi$
$240$	−7.29840	−0.471109
$241$	−5.32172	−0.342802	−0.171401	−	0.985201i	$-0.554829\pi$
$241$	−5.32172	−0.342802	−0.171401	+	0.985201i	$0.554829\pi$
$242$	0.109449	0.00703568
$243$	−2.82906	−0.181484
$244$	8.78302	0.562275
$245$	13.6027	0.869043
$246$	18.2179	1.16153
$247$	−46.9570	−2.98780
$248$	4.32141	0.274410
$249$	−26.9945	−1.71071
$250$	10.0838	0.637755
$251$	−20.6029	−1.30044	−0.650221	−	0.759745i	$-0.725324\pi$
$251$	−20.6029	−1.30044	−0.650221	+	0.759745i	$0.725324\pi$
$252$	21.8707	1.37773
$253$	−3.44159	−0.216371
$254$	−6.79621	−0.426432
$255$	−4.18014	−0.261770
$256$	1.00000	0.0625000
$257$	−1.88657	−0.117681	−0.0588403	−	0.998267i	$-0.518740\pi$
$257$	−1.88657	−0.117681	−0.0588403	+	0.998267i	$0.518740\pi$
$258$	−30.1293	−1.87577
$259$	18.5965	1.15553
$260$	15.7118	0.974402
$261$	21.2297	1.31408
$262$	3.13321	0.193571
$263$	1.83934	0.113419	0.0567094	−	0.998391i	$-0.481939\pi$
$263$	1.83934	0.113419	0.0567094	+	0.998391i	$0.481939\pi$
$264$	9.98351	0.614442
$265$	−10.1958	−0.626326
$266$	25.6324	1.57163
$267$	2.54315	0.155638
$268$	0.963667	0.0588654
$269$	24.1443	1.47210	0.736052	−	0.676925i	$-0.236688\pi$
$269$	24.1443	1.47210	0.736052	+	0.676925i	$0.236688\pi$
$270$	23.0046	1.40002
$271$	−3.94379	−0.239568	−0.119784	−	0.992800i	$-0.538220\pi$
$271$	−3.94379	−0.239568	−0.119784	+	0.992800i	$0.538220\pi$
$272$	0.572747	0.0347279
$273$	−70.0422	−4.23915
$274$	−15.3053	−0.924629
$275$	2.70675	0.163223
$276$	3.15495	0.189906
$277$	−14.6667	−0.881239	−0.440619	−	0.897694i	$-0.645241\pi$
$277$	−14.6667	−0.881239	−0.440619	+	0.897694i	$0.645241\pi$
$278$	9.32510	0.559283
$279$	−26.5854	−1.59162
$280$	−8.57660	−0.512550
$281$	13.4398	0.801751	0.400875	−	0.916133i	$-0.368706\pi$
$281$	13.4398	0.801751	0.400875	+	0.916133i	$0.368706\pi$
$282$	−23.6143	−1.40621
$283$	23.1482	1.37602	0.688010	−	0.725701i	$-0.258485\pi$
$283$	23.1482	1.37602	0.688010	+	0.725701i	$0.258485\pi$
$284$	2.36205	0.140162
$285$	52.6225	3.11709
$286$	−21.4922	−1.27086
$287$	21.4085	1.26371
$288$	−6.15201	−0.362511
$289$	−16.6720	−0.980704
$290$	−8.32522	−0.488874
$291$	5.68524	0.333275
$292$	6.22427	0.364248
$293$	31.9093	1.86416	0.932082	−	0.362248i	$-0.117991\pi$
$293$	31.9093	1.86416	0.932082	+	0.362248i	$0.117991\pi$
$294$	17.0574	0.994809
$295$	3.58789	0.208895
$296$	−5.23100	−0.304046
$297$	−31.4681	−1.82596
$298$	6.56817	0.380484
$299$	−6.79188	−0.392785
$300$	−2.48131	−0.143258
$301$	−35.4060	−2.04077
$302$	−13.7345	−0.790334
$303$	6.64323	0.381644
$304$	−7.21015	−0.413530
$305$	21.1891	1.21329
$306$	−3.52355	−0.201428
$307$	−8.96667	−0.511755	−0.255877	−	0.966709i	$-0.582364\pi$
$307$	−8.96667	−0.511755	−0.255877	+	0.966709i	$0.582364\pi$
$308$	11.7320	0.668491
$309$	−11.9523	−0.679942
$310$	10.4255	0.592126
$311$	−12.5292	−0.710468	−0.355234	−	0.934777i	$-0.615599\pi$
$311$	−12.5292	−0.710468	−0.355234	+	0.934777i	$0.615599\pi$
$312$	19.7022	1.11542
$313$	−2.96374	−0.167521	−0.0837604	−	0.996486i	$-0.526693\pi$
$313$	−2.96374	−0.167521	−0.0837604	+	0.996486i	$0.526693\pi$
$314$	−20.5282	−1.15847
$315$	52.7633	2.97288
$316$	12.0714	0.679067
$317$	−9.25485	−0.519804	−0.259902	−	0.965635i	$-0.583690\pi$
$317$	−9.25485	−0.519804	−0.259902	+	0.965635i	$0.583690\pi$
$318$	−12.7853	−0.716966
$319$	11.3881	0.637611
$320$	2.41251	0.134863
$321$	−20.6856	−1.15456
$322$	3.70749	0.206610
$323$	−4.12959	−0.229777
$324$	10.3912	0.577288
$325$	5.34169	0.296304
$326$	−3.49194	−0.193401
$327$	−13.9303	−0.770346
$328$	−6.02200	−0.332509
$329$	−27.7500	−1.52991
$330$	24.0853	1.32585
$331$	−7.92232	−0.435450	−0.217725	−	0.976010i	$-0.569864\pi$
$331$	−7.92232	−0.435450	−0.217725	+	0.976010i	$0.569864\pi$
$332$	8.92312	0.489720
$333$	32.1812	1.76352
$334$	−19.8272	−1.08489
$335$	2.32486	0.127021
$336$	−10.7548	−0.586725
$337$	−29.8578	−1.62646	−0.813230	−	0.581942i	$-0.802293\pi$
$337$	−29.8578	−1.62646	−0.813230	+	0.581942i	$0.802293\pi$
$338$	−29.4142	−1.59992
$339$	32.9069	1.78726
$340$	1.38176	0.0749364
$341$	−14.2610	−0.772278
$342$	44.3569	2.39855
$343$	−4.84058	−0.261367
$344$	9.95934	0.536972
$345$	7.61135	0.409781
$346$	−1.21940	−0.0655553
$347$	21.7842	1.16944	0.584718	−	0.811237i	$-0.301205\pi$
$347$	21.7842	1.16944	0.584718	+	0.811237i	$0.301205\pi$
$348$	−10.4396	−0.559622
$349$	1.13202	0.0605956	0.0302978	−	0.999541i	$-0.490354\pi$
$349$	1.13202	0.0605956	0.0302978	+	0.999541i	$0.490354\pi$
$350$	−2.91587	−0.155860
$351$	−62.1014	−3.31473
$352$	−3.30008	−0.175895
$353$	−25.0792	−1.33483	−0.667414	−	0.744687i	$-0.732599\pi$
$353$	−25.0792	−1.33483	−0.667414	+	0.744687i	$0.732599\pi$
$354$	4.49912	0.239126
$355$	5.69847	0.302443
$356$	−0.840647	−0.0445542
$357$	−6.15981	−0.326012
$358$	11.3071	0.597600
$359$	27.0704	1.42872	0.714360	−	0.699779i	$-0.246718\pi$
$359$	27.0704	1.42872	0.714360	+	0.699779i	$0.246718\pi$
$360$	−14.8418	−0.782231
$361$	32.9862	1.73612
$362$	2.35665	0.123863
$363$	0.331110	0.0173788
$364$	23.1527	1.21353
$365$	15.0161	0.785980
$366$	26.5706	1.38887
$367$	0.784121	0.0409308	0.0204654	−	0.999791i	$-0.493485\pi$
$367$	0.784121	0.0409308	0.0204654	+	0.999791i	$0.493485\pi$
$368$	−1.04288	−0.0543639
$369$	37.0474	1.92861
$370$	−12.6198	−0.656075
$371$	−15.0245	−0.780033
$372$	13.0733	0.677817
$373$	13.7136	0.710063	0.355032	−	0.934854i	$-0.384470\pi$
$373$	13.7136	0.710063	0.355032	+	0.934854i	$0.384470\pi$
$374$	−1.89011	−0.0977355
$375$	30.5058	1.57531
$376$	7.80580	0.402553
$377$	22.4741	1.15748
$378$	33.8994	1.74359
$379$	1.88498	0.0968249	0.0484124	−	0.998827i	$-0.484584\pi$
$379$	1.88498	0.0968249	0.0484124	+	0.998827i	$0.484584\pi$
$380$	−17.3945	−0.892322
$381$	−20.5601	−1.05333
$382$	−18.2985	−0.936234
$383$	22.1930	1.13401	0.567004	−	0.823715i	$-0.308102\pi$
$383$	22.1930	1.13401	0.567004	+	0.823715i	$0.308102\pi$
$384$	3.02523	0.154381
$385$	28.3035	1.44248
$386$	17.7840	0.905183
$387$	−61.2699	−3.11453
$388$	−1.87928	−0.0954058
$389$	−34.3354	−1.74088	−0.870438	−	0.492279i	$-0.836164\pi$
$389$	−34.3354	−1.74088	−0.870438	+	0.492279i	$0.836164\pi$
$390$	47.5317	2.40686
$391$	−0.597306	−0.0302071
$392$	−5.63839	−0.284782
$393$	9.47869	0.478136
$394$	18.4631	0.930156
$395$	29.1223	1.46530
$396$	20.3021	1.02022
$397$	22.1026	1.10930	0.554650	−	0.832084i	$-0.312852\pi$
$397$	22.1026	1.10930	0.554650	+	0.832084i	$0.312852\pi$
$398$	−2.18599	−0.109574
$399$	77.5440	3.88206
$400$	0.820206	0.0410103
$401$	31.2685	1.56148	0.780738	−	0.624858i	$-0.214843\pi$
$401$	31.2685	1.56148	0.780738	+	0.624858i	$0.214843\pi$
$402$	2.91531	0.145403
$403$	−28.1437	−1.40194
$404$	−2.19594	−0.109252
$405$	25.0688	1.24568
$406$	−12.2680	−0.608849
$407$	17.2627	0.855683
$408$	1.73269	0.0857810
$409$	−15.0074	−0.742068	−0.371034	−	0.928619i	$-0.620997\pi$
$409$	−15.0074	−0.742068	−0.371034	+	0.928619i	$0.620997\pi$
$410$	−14.5281	−0.717494
$411$	−46.3022	−2.28392
$412$	3.95087	0.194645
$413$	5.28708	0.260160
$414$	6.41580	0.315320
$415$	21.5271	1.05672
$416$	−6.51262	−0.319308
$417$	28.2106	1.38148
$418$	23.7941	1.16381
$419$	−20.0539	−0.979696	−0.489848	−	0.871808i	$-0.662948\pi$
$419$	−20.0539	−0.979696	−0.489848	+	0.871808i	$0.662948\pi$
$420$	−25.9462	−1.26604
$421$	16.9008	0.823694	0.411847	−	0.911253i	$-0.364884\pi$
$421$	16.9008	0.823694	0.411847	+	0.911253i	$0.364884\pi$
$422$	4.37274	0.212862
$423$	−48.0213	−2.33488
$424$	4.22624	0.205244
$425$	0.469771	0.0227872
$426$	7.14574	0.346212
$427$	31.2241	1.51104
$428$	6.83769	0.330512
$429$	−65.0188	−3.13914
$430$	24.0270	1.15869
$431$	−8.45395	−0.407212	−0.203606	−	0.979053i	$-0.565266\pi$
$431$	−8.45395	−0.407212	−0.203606	+	0.979053i	$0.565266\pi$
$432$	−9.53555	−0.458779
$433$	−10.2531	−0.492731	−0.246365	−	0.969177i	$-0.579236\pi$
$433$	−10.2531	−0.492731	−0.246365	+	0.969177i	$0.579236\pi$
$434$	15.3629	0.737441
$435$	−25.1857	−1.20756
$436$	4.60470	0.220525
$437$	7.51931	0.359698
$438$	18.8299	0.899725
$439$	−3.58408	−0.171059	−0.0855293	−	0.996336i	$-0.527258\pi$
$439$	−3.58408	−0.171059	−0.0855293	+	0.996336i	$0.527258\pi$
$440$	−7.96149	−0.379549
$441$	34.6874	1.65178
$442$	−3.73009	−0.177422
$443$	1.53474	0.0729176	0.0364588	−	0.999335i	$-0.488392\pi$
$443$	1.53474	0.0729176	0.0364588	+	0.999335i	$0.488392\pi$
$444$	−15.8250	−0.751020
$445$	−2.02807	−0.0961398
$446$	11.8687	0.562000
$447$	19.8702	0.939830
$448$	3.55505	0.167960
$449$	5.51659	0.260344	0.130172	−	0.991491i	$-0.458447\pi$
$449$	5.51659	0.260344	0.130172	+	0.991491i	$0.458447\pi$
$450$	−5.04591	−0.237867
$451$	19.8731	0.935788
$452$	−10.8775	−0.511635
$453$	−41.5501	−1.95219
$454$	−29.8441	−1.40065
$455$	55.8561	2.61858
$456$	−21.8123	−1.02146
$457$	8.25167	0.385997	0.192998	−	0.981199i	$-0.438179\pi$
$457$	8.25167	0.385997	0.192998	+	0.981199i	$0.438179\pi$
$458$	3.85892	0.180316
$459$	−5.46146	−0.254919
$460$	−2.51596	−0.117307
$461$	11.9841	0.558155	0.279077	−	0.960269i	$-0.409971\pi$
$461$	11.9841	0.558155	0.279077	+	0.960269i	$0.409971\pi$
$462$	35.4919	1.65123
$463$	11.5986	0.539033	0.269517	−	0.962996i	$-0.413136\pi$
$463$	11.5986	0.539033	0.269517	+	0.962996i	$0.413136\pi$
$464$	3.45085	0.160202
$465$	31.5394	1.46260
$466$	−29.2555	−1.35523
$467$	29.3969	1.36033	0.680163	−	0.733061i	$-0.261909\pi$
$467$	29.3969	1.36033	0.680163	+	0.733061i	$0.261909\pi$
$468$	40.0657	1.85204
$469$	3.42589	0.158193
$470$	18.8316	0.868635
$471$	−62.1025	−2.86153
$472$	−1.48720	−0.0684540
$473$	−32.8667	−1.51121
$474$	36.5186	1.67736
$475$	−5.91380	−0.271344
$476$	2.03615	0.0933266
$477$	−25.9999	−1.19045
$478$	0.607549	0.0277887
$479$	17.7499	0.811012	0.405506	−	0.914092i	$-0.367095\pi$
$479$	17.7499	0.811012	0.405506	+	0.914092i	$0.367095\pi$
$480$	7.29840	0.333125
$481$	34.0675	1.55335
$482$	5.32172	0.242398
$483$	11.2160	0.510346
$484$	−0.109449	−0.00497498
$485$	−4.53377	−0.205868
$486$	2.82906	0.128329
$487$	17.9303	0.812499	0.406249	−	0.913762i	$-0.366836\pi$
$487$	17.9303	0.812499	0.406249	+	0.913762i	$0.366836\pi$
$488$	−8.78302	−0.397588
$489$	−10.5639	−0.477717
$490$	−13.6027	−0.614507
$491$	−5.30759	−0.239528	−0.119764	−	0.992802i	$-0.538214\pi$
$491$	−5.30759	−0.239528	−0.119764	+	0.992802i	$0.538214\pi$
$492$	−18.2179	−0.821328
$493$	1.97647	0.0890156
$494$	46.9570	2.11269
$495$	48.9791	2.20145
$496$	−4.32141	−0.194037
$497$	8.39720	0.376666
$498$	26.9945	1.20965
$499$	−38.9432	−1.74334	−0.871668	−	0.490097i	$-0.836961\pi$
$499$	−38.9432	−1.74334	−0.871668	+	0.490097i	$0.836961\pi$
$500$	−10.0838	−0.450961
$501$	−59.9817	−2.67978
$502$	20.6029	0.919551
$503$	−5.84374	−0.260560	−0.130280	−	0.991477i	$-0.541588\pi$
$503$	−5.84374	−0.260560	−0.130280	+	0.991477i	$0.541588\pi$
$504$	−21.8707	−0.974199
$505$	−5.29773	−0.235746
$506$	3.44159	0.152997
$507$	−88.9848	−3.95195
$508$	6.79621	0.301533
$509$	24.5903	1.08994	0.544972	−	0.838454i	$-0.316540\pi$
$509$	24.5903	1.08994	0.544972	+	0.838454i	$0.316540\pi$
$510$	4.18014	0.185100
$511$	22.1276	0.978868
$512$	−1.00000	−0.0441942
$513$	68.7527	3.03550
$514$	1.88657	0.0832128
$515$	9.53151	0.420009
$516$	30.1293	1.32637
$517$	−25.7598	−1.13291
$518$	−18.5965	−0.817083
$519$	−3.68896	−0.161927
$520$	−15.7118	−0.689007
$521$	−9.55013	−0.418399	−0.209199	−	0.977873i	$-0.567086\pi$
$521$	−9.55013	−0.418399	−0.209199	+	0.977873i	$0.567086\pi$
$522$	−21.2297	−0.929198
$523$	−31.5960	−1.38160	−0.690798	−	0.723048i	$-0.742740\pi$
$523$	−31.5960	−1.38160	−0.690798	+	0.723048i	$0.742740\pi$
$524$	−3.13321	−0.136875
$525$	−8.82118	−0.384988
$526$	−1.83934	−0.0801992
$527$	−2.47508	−0.107816
$528$	−9.98351	−0.434476
$529$	−21.9124	−0.952713
$530$	10.1958	0.442879
$531$	9.14927	0.397045
$532$	−25.6324	−1.11131
$533$	39.2190	1.69876
$534$	−2.54315	−0.110053
$535$	16.4960	0.713184
$536$	−0.963667	−0.0416241
$537$	34.2066	1.47613
$538$	−24.1443	−1.04094
$539$	18.6072	0.801467
$540$	−23.0046	−0.989960
$541$	43.9916	1.89135	0.945674	−	0.325116i	$-0.105403\pi$
$541$	43.9916	1.89135	0.945674	+	0.325116i	$0.105403\pi$
$542$	3.94379	0.169400
$543$	7.12942	0.305953
$544$	−0.572747	−0.0245563
$545$	11.1089	0.475853
$546$	70.0422	2.99753
$547$	30.5442	1.30598	0.652988	−	0.757369i	$-0.273515\pi$
$547$	30.5442	1.30598	0.652988	+	0.757369i	$0.273515\pi$
$548$	15.3053	0.653812
$549$	54.0332	2.30608
$550$	−2.70675	−0.115416
$551$	−24.8812	−1.05997
$552$	−3.15495	−0.134284
$553$	42.9143	1.82490
$554$	14.6667	0.623130
$555$	−38.1779	−1.62056
$556$	−9.32510	−0.395473
$557$	12.3605	0.523732	0.261866	−	0.965104i	$-0.415662\pi$
$557$	12.3605	0.523732	0.261866	+	0.965104i	$0.415662\pi$
$558$	26.5854	1.12545
$559$	−64.8614	−2.74335
$560$	8.57660	0.362427
$561$	−5.71803	−0.241415
$562$	−13.4398	−0.566923
$563$	−6.83572	−0.288091	−0.144046	−	0.989571i	$-0.546011\pi$
$563$	−6.83572	−0.288091	−0.144046	+	0.989571i	$0.546011\pi$
$564$	23.6143	0.994342
$565$	−26.2421	−1.10401
$566$	−23.1482	−0.972993
$567$	36.9412	1.55138
$568$	−2.36205	−0.0991094
$569$	−29.3327	−1.22969	−0.614845	−	0.788648i	$-0.710781\pi$
$569$	−29.3327	−1.22969	−0.614845	+	0.788648i	$0.710781\pi$
$570$	−52.6225	−2.20411
$571$	−8.59221	−0.359573	−0.179786	−	0.983706i	$-0.557541\pi$
$571$	−8.59221	−0.359573	−0.179786	+	0.983706i	$0.557541\pi$
$572$	21.4922	0.898634
$573$	−55.3572	−2.31258
$574$	−21.4085	−0.893575
$575$	−0.855376	−0.0356716
$576$	6.15201	0.256334
$577$	−20.0331	−0.833990	−0.416995	−	0.908909i	$-0.636917\pi$
$577$	−20.0331	−0.833990	−0.416995	+	0.908909i	$0.636917\pi$
$578$	16.6720	0.693462
$579$	53.8007	2.23588
$580$	8.32522	0.345686
$581$	31.7222	1.31606
$582$	−5.68524	−0.235661
$583$	−13.9469	−0.577623
$584$	−6.22427	−0.257562
$585$	96.6589	3.99635
$586$	−31.9093	−1.31816
$587$	1.85468	0.0765507	0.0382753	−	0.999267i	$-0.487814\pi$
$587$	1.85468	0.0765507	0.0382753	+	0.999267i	$0.487814\pi$
$588$	−17.0574	−0.703437
$589$	31.1580	1.28384
$590$	−3.58789	−0.147711
$591$	55.8550	2.29757
$592$	5.23100	0.214993
$593$	45.6341	1.87397	0.936983	−	0.349375i	$-0.113606\pi$
$593$	45.6341	1.87397	0.936983	+	0.349375i	$0.113606\pi$
$594$	31.4681	1.29115
$595$	4.91222	0.201382
$596$	−6.56817	−0.269043
$597$	−6.61313	−0.270657
$598$	6.79188	0.277741
$599$	−27.8791	−1.13911	−0.569554	−	0.821954i	$-0.692884\pi$
$599$	−27.8791	−1.13911	−0.569554	+	0.821954i	$0.692884\pi$
$600$	2.48131	0.101299
$601$	15.8647	0.647135	0.323568	−	0.946205i	$-0.395118\pi$
$601$	15.8647	0.647135	0.323568	+	0.946205i	$0.395118\pi$
$602$	35.4060	1.44304
$603$	5.92849	0.241427
$604$	13.7345	0.558850
$605$	−0.264048	−0.0107351
$606$	−6.64323	−0.269863
$607$	−48.6510	−1.97468	−0.987341	−	0.158612i	$-0.949298\pi$
$607$	−48.6510	−1.97468	−0.987341	+	0.158612i	$0.949298\pi$
$608$	7.21015	0.292410
$609$	−37.1134	−1.50391
$610$	−21.1891	−0.857922
$611$	−50.8362	−2.05661
$612$	3.52355	0.142431
$613$	−34.7506	−1.40356	−0.701782	−	0.712392i	$-0.747612\pi$
$613$	−34.7506	−1.40356	−0.701782	+	0.712392i	$0.747612\pi$
$614$	8.96667	0.361865
$615$	−43.9510	−1.77227
$616$	−11.7320	−0.472694
$617$	−34.9615	−1.40750	−0.703749	−	0.710448i	$-0.748492\pi$
$617$	−34.9615	−1.40750	−0.703749	+	0.710448i	$0.748492\pi$
$618$	11.9523	0.480791
$619$	−20.2655	−0.814538	−0.407269	−	0.913308i	$-0.633519\pi$
$619$	−20.2655	−0.814538	−0.407269	+	0.913308i	$0.633519\pi$
$620$	−10.4255	−0.418696
$621$	9.94442	0.399056
$622$	12.5292	0.502377
$623$	−2.98855	−0.119734
$624$	−19.7022	−0.788718
$625$	−28.4283	−1.13713
$626$	2.96374	0.118455
$627$	71.9825	2.87471
$628$	20.5282	0.819165
$629$	2.99604	0.119460
$630$	−52.7633	−2.10214
$631$	−27.4667	−1.09343	−0.546717	−	0.837318i	$-0.684123\pi$
$631$	−27.4667	−1.09343	−0.546717	+	0.837318i	$0.684123\pi$
$632$	−12.0714	−0.480173
$633$	13.2285	0.525787
$634$	9.25485	0.367557
$635$	16.3959	0.650652
$636$	12.7853	0.506972
$637$	36.7207	1.45493
$638$	−11.3881	−0.450859
$639$	14.5313	0.574851
$640$	−2.41251	−0.0953628
$641$	9.17388	0.362346	0.181173	−	0.983451i	$-0.442011\pi$
$641$	9.17388	0.362346	0.181173	+	0.983451i	$0.442011\pi$
$642$	20.6856	0.816395
$643$	32.6285	1.28674	0.643372	−	0.765554i	$-0.277535\pi$
$643$	32.6285	1.28674	0.643372	+	0.765554i	$0.277535\pi$
$644$	−3.70749	−0.146096
$645$	72.6872	2.86206
$646$	4.12959	0.162477
$647$	40.2855	1.58379	0.791894	−	0.610659i	$-0.209095\pi$
$647$	40.2855	1.58379	0.791894	+	0.610659i	$0.209095\pi$
$648$	−10.3912	−0.408204
$649$	4.90789	0.192651
$650$	−5.34169	−0.209518
$651$	46.4761	1.82154
$652$	3.49194	0.136755
$653$	5.96323	0.233359	0.116680	−	0.993170i	$-0.462775\pi$
$653$	5.96323	0.233359	0.116680	+	0.993170i	$0.462775\pi$
$654$	13.9303	0.544717
$655$	−7.55891	−0.295351
$656$	6.02200	0.235120
$657$	38.2918	1.49390
$658$	27.7500	1.08181
$659$	13.9941	0.545131	0.272566	−	0.962137i	$-0.412128\pi$
$659$	13.9941	0.545131	0.272566	+	0.962137i	$0.412128\pi$
$660$	−24.0853	−0.937520
$661$	28.5822	1.11172	0.555859	−	0.831277i	$-0.312390\pi$
$661$	28.5822	1.11172	0.555859	+	0.831277i	$0.312390\pi$
$662$	7.92232	0.307910
$663$	−11.2844	−0.438248
$664$	−8.92312	−0.346284
$665$	−61.8385	−2.39800
$666$	−32.1812	−1.24700
$667$	−3.59882	−0.139347
$668$	19.8272	0.767136
$669$	35.9056	1.38819
$670$	−2.32486	−0.0898171
$671$	28.9847	1.11894
$672$	10.7548	0.414877
$673$	31.3428	1.20818	0.604088	−	0.796918i	$-0.293538\pi$
$673$	31.3428	1.20818	0.604088	+	0.796918i	$0.293538\pi$
$674$	29.8578	1.15008
$675$	−7.82111	−0.301035
$676$	29.4142	1.13132
$677$	9.64782	0.370796	0.185398	−	0.982664i	$-0.440643\pi$
$677$	9.64782	0.370796	0.185398	+	0.982664i	$0.440643\pi$
$678$	−32.9069	−1.26378
$679$	−6.68092	−0.256390
$680$	−1.38176	−0.0529880
$681$	−90.2852	−3.45974
$682$	14.2610	0.546083
$683$	33.9665	1.29969	0.649845	−	0.760066i	$-0.274834\pi$
$683$	33.9665	1.29969	0.649845	+	0.760066i	$0.274834\pi$
$684$	−44.3569	−1.69603
$685$	36.9243	1.41080
$686$	4.84058	0.184814
$687$	11.6741	0.445395
$688$	−9.95934	−0.379696
$689$	−27.5239	−1.04858
$690$	−7.61135	−0.289759
$691$	22.9483	0.872993	0.436497	−	0.899706i	$-0.356219\pi$
$691$	22.9483	0.872993	0.436497	+	0.899706i	$0.356219\pi$
$692$	1.21940	0.0463546
$693$	72.1752	2.74171
$694$	−21.7842	−0.826916
$695$	−22.4969	−0.853356
$696$	10.4396	0.395713
$697$	3.44909	0.130643
$698$	−1.13202	−0.0428476
$699$	−88.5045	−3.34755
$700$	2.91587	0.110210
$701$	−5.94975	−0.224719	−0.112360	−	0.993668i	$-0.535841\pi$
$701$	−5.94975	−0.224719	−0.112360	+	0.993668i	$0.535841\pi$
$702$	62.1014	2.34387
$703$	−37.7163	−1.42250
$704$	3.30008	0.124377
$705$	56.9698	2.14561
$706$	25.0792	0.943866
$707$	−7.80669	−0.293601
$708$	−4.49912	−0.169087
$709$	−41.0611	−1.54208	−0.771042	−	0.636784i	$-0.780264\pi$
$709$	−41.0611	−1.54208	−0.771042	+	0.636784i	$0.780264\pi$
$710$	−5.69847	−0.213860
$711$	74.2631	2.78508
$712$	0.840647	0.0315046
$713$	4.50671	0.168778
$714$	6.15981	0.230525
$715$	51.8501	1.93909
$716$	−11.3071	−0.422567
$717$	1.83798	0.0686404
$718$	−27.0704	−1.01026
$719$	−24.3259	−0.907205	−0.453602	−	0.891204i	$-0.649861\pi$
$719$	−24.3259	−0.907205	−0.453602	+	0.891204i	$0.649861\pi$
$720$	14.8418	0.553121
$721$	14.0455	0.523083
$722$	−32.9862	−1.22762
$723$	16.0994	0.598744
$724$	−2.35665	−0.0875844
$725$	2.83041	0.105119
$726$	−0.331110	−0.0122886
$727$	−8.39833	−0.311477	−0.155738	−	0.987798i	$-0.549776\pi$
$727$	−8.39833	−0.311477	−0.155738	+	0.987798i	$0.549776\pi$
$728$	−23.1527	−0.858096
$729$	−22.6150	−0.837592
$730$	−15.0161	−0.555772
$731$	−5.70418	−0.210977
$732$	−26.5706	−0.982079
$733$	15.7858	0.583063	0.291531	−	0.956561i	$-0.405835\pi$
$733$	15.7858	0.583063	0.291531	+	0.956561i	$0.405835\pi$
$734$	−0.784121	−0.0289424
$735$	−41.1512	−1.51789
$736$	1.04288	0.0384410
$737$	3.18018	0.117144
$738$	−37.0474	−1.36373
$739$	−2.56298	−0.0942807	−0.0471403	−	0.998888i	$-0.515011\pi$
$739$	−2.56298	−0.0942807	−0.0471403	+	0.998888i	$0.515011\pi$
$740$	12.6198	0.463915
$741$	142.056	5.21854
$742$	15.0245	0.551567
$743$	−43.3674	−1.59100	−0.795498	−	0.605956i	$-0.792791\pi$
$743$	−43.3674	−1.59100	−0.795498	+	0.605956i	$0.792791\pi$
$744$	−13.0733	−0.479289
$745$	−15.8458	−0.580545
$746$	−13.7136	−0.502090
$747$	54.8951	2.00851
$748$	1.89011	0.0691094
$749$	24.3084	0.888208
$750$	−30.5058	−1.11391
$751$	−14.0800	−0.513786	−0.256893	−	0.966440i	$-0.582699\pi$
$751$	−14.0800	−0.513786	−0.256893	+	0.966440i	$0.582699\pi$
$752$	−7.80580	−0.284648
$753$	62.3284	2.27137
$754$	−22.4741	−0.818459
$755$	33.1347	1.20590
$756$	−33.8994	−1.23291
$757$	−25.4046	−0.923347	−0.461674	−	0.887050i	$-0.652751\pi$
$757$	−25.4046	−0.923347	−0.461674	+	0.887050i	$0.652751\pi$
$758$	−1.88498	−0.0684655
$759$	10.4116	0.377917
$760$	17.3945	0.630967
$761$	−0.887035	−0.0321550	−0.0160775	−	0.999871i	$-0.505118\pi$
$761$	−0.887035	−0.0321550	−0.0160775	+	0.999871i	$0.505118\pi$
$762$	20.5601	0.744813
$763$	16.3700	0.592632
$764$	18.2985	0.662017
$765$	8.50059	0.307339
$766$	−22.1930	−0.801865
$767$	9.68558	0.349726
$768$	−3.02523	−0.109164
$769$	33.7041	1.21540	0.607701	−	0.794166i	$-0.292092\pi$
$769$	33.7041	1.21540	0.607701	+	0.794166i	$0.292092\pi$
$770$	−28.3035	−1.01999
$771$	5.70729	0.205543
$772$	−17.7840	−0.640061
$773$	1.95508	0.0703194	0.0351597	−	0.999382i	$-0.488806\pi$
$773$	1.95508	0.0703194	0.0351597	+	0.999382i	$0.488806\pi$
$774$	61.2699	2.20230
$775$	−3.54445	−0.127320
$776$	1.87928	0.0674621
$777$	−56.2586	−2.01827
$778$	34.3354	1.23098
$779$	−43.4195	−1.55567
$780$	−47.5317	−1.70191
$781$	7.79496	0.278925
$782$	0.597306	0.0213596
$783$	−32.9058	−1.17596
$784$	5.63839	0.201371
$785$	49.5245	1.76761
$786$	−9.47869	−0.338094
$787$	−40.0974	−1.42932	−0.714660	−	0.699472i	$-0.753418\pi$
$787$	−40.0974	−1.42932	−0.714660	+	0.699472i	$0.753418\pi$
$788$	−18.4631	−0.657719
$789$	−5.56444	−0.198099
$790$	−29.1223	−1.03612
$791$	−38.6701	−1.37495
$792$	−20.3021	−0.721405
$793$	57.2005	2.03125
$794$	−22.1026	−0.784393
$795$	30.8448	1.09395
$796$	2.18599	0.0774805
$797$	26.6847	0.945219	0.472609	−	0.881272i	$-0.343312\pi$
$797$	26.6847	0.945219	0.472609	+	0.881272i	$0.343312\pi$
$798$	−77.5440	−2.74503
$799$	−4.47075	−0.158164
$800$	−0.820206	−0.0289987
$801$	−5.17167	−0.182732
$802$	−31.2685	−1.10413
$803$	20.5406	0.724863
$804$	−2.91531	−0.102815
$805$	−8.94436	−0.315247
$806$	28.1437	0.991321
$807$	−73.0421	−2.57120
$808$	2.19594	0.0772530
$809$	20.4086	0.717527	0.358763	−	0.933429i	$-0.383199\pi$
$809$	20.4086	0.717527	0.358763	+	0.933429i	$0.383199\pi$
$810$	−25.0688	−0.880829
$811$	50.2472	1.76442	0.882209	−	0.470858i	$-0.156056\pi$
$811$	50.2472	1.76442	0.882209	+	0.470858i	$0.156056\pi$
$812$	12.2680	0.430521
$813$	11.9309	0.418434
$814$	−17.2627	−0.605059
$815$	8.42434	0.295092
$816$	−1.73269	−0.0606563
$817$	71.8083	2.51225
$818$	15.0074	0.524721
$819$	142.436	4.97710
$820$	14.5281	0.507345
$821$	18.3785	0.641415	0.320707	−	0.947178i	$-0.396079\pi$
$821$	18.3785	0.641415	0.320707	+	0.947178i	$0.396079\pi$
$822$	46.3022	1.61497
$823$	−46.0744	−1.60605	−0.803027	−	0.595943i	$-0.796779\pi$
$823$	−46.0744	−1.60605	−0.803027	+	0.595943i	$0.796779\pi$
$824$	−3.95087	−0.137635
$825$	−8.18853	−0.285088
$826$	−5.28708	−0.183961
$827$	12.1286	0.421753	0.210876	−	0.977513i	$-0.432368\pi$
$827$	12.1286	0.421753	0.210876	+	0.977513i	$0.432368\pi$
$828$	−6.41580	−0.222965
$829$	17.5636	0.610008	0.305004	−	0.952351i	$-0.401342\pi$
$829$	17.5636	0.610008	0.305004	+	0.952351i	$0.401342\pi$
$830$	−21.5271	−0.747217
$831$	44.3702	1.53919
$832$	6.51262	0.225785
$833$	3.22937	0.111891
$834$	−28.2106	−0.976852
$835$	47.8332	1.65534
$836$	−23.7941	−0.822936
$837$	41.2070	1.42432
$838$	20.0539	0.692750
$839$	−5.76947	−0.199184	−0.0995920	−	0.995028i	$-0.531754\pi$
$839$	−5.76947	−0.199184	−0.0995920	+	0.995028i	$0.531754\pi$
$840$	25.9462	0.895228
$841$	−17.0916	−0.589366
$842$	−16.9008	−0.582439
$843$	−40.6584	−1.40035
$844$	−4.37274	−0.150516
$845$	70.9622	2.44117
$846$	48.0213	1.65101
$847$	−0.389099	−0.0133696
$848$	−4.22624	−0.145130
$849$	−70.0287	−2.40338
$850$	−0.469771	−0.0161130
$851$	−5.45531	−0.187005
$852$	−7.14574	−0.244809
$853$	−20.2923	−0.694796	−0.347398	−	0.937718i	$-0.612935\pi$
$853$	−20.2923	−0.694796	−0.347398	+	0.937718i	$0.612935\pi$
$854$	−31.2241	−1.06847
$855$	−107.011	−3.65971
$856$	−6.83769	−0.233708
$857$	−45.8366	−1.56575	−0.782874	−	0.622180i	$-0.786247\pi$
$857$	−45.8366	−1.56575	−0.782874	+	0.622180i	$0.786247\pi$
$858$	65.0188	2.21971
$859$	−41.8770	−1.42883	−0.714413	−	0.699725i	$-0.753306\pi$
$859$	−41.8770	−1.42883	−0.714413	+	0.699725i	$0.753306\pi$
$860$	−24.0270	−0.819314
$861$	−64.7657	−2.20721
$862$	8.45395	0.287943
$863$	−4.34158	−0.147789	−0.0738946	−	0.997266i	$-0.523543\pi$
$863$	−4.34158	−0.147789	−0.0738946	+	0.997266i	$0.523543\pi$
$864$	9.53555	0.324406
$865$	2.94181	0.100025
$866$	10.2531	0.348413
$867$	50.4365	1.71291
$868$	−15.3629	−0.521449
$869$	39.8365	1.35136
$870$	25.1857	0.853875
$871$	6.27600	0.212654
$872$	−4.60470	−0.155935
$873$	−11.5613	−0.391291
$874$	−7.51931	−0.254345
$875$	−35.8484	−1.21190
$876$	−18.8299	−0.636202
$877$	49.0126	1.65504	0.827519	−	0.561438i	$-0.189752\pi$
$877$	49.0126	1.65504	0.827519	+	0.561438i	$0.189752\pi$
$878$	3.58408	0.120957
$879$	−96.5330	−3.25598
$880$	7.96149	0.268382
$881$	−45.7993	−1.54302	−0.771510	−	0.636218i	$-0.780498\pi$
$881$	−45.7993	−1.54302	−0.771510	+	0.636218i	$0.780498\pi$
$882$	−34.6874	−1.16799
$883$	−5.03118	−0.169313	−0.0846563	−	0.996410i	$-0.526979\pi$
$883$	−5.03118	−0.169313	−0.0846563	+	0.996410i	$0.526979\pi$
$884$	3.73009	0.125456
$885$	−10.8542	−0.364859
$886$	−1.53474	−0.0515606
$887$	52.0287	1.74695	0.873476	−	0.486867i	$-0.161860\pi$
$887$	52.0287	1.74695	0.873476	+	0.486867i	$0.161860\pi$
$888$	15.8250	0.531052
$889$	24.1609	0.810330
$890$	2.02807	0.0679811
$891$	34.2918	1.14882
$892$	−11.8687	−0.397394
$893$	56.2809	1.88337
$894$	−19.8702	−0.664560
$895$	−27.2786	−0.911822
$896$	−3.55505	−0.118766
$897$	20.5470	0.686044
$898$	−5.51659	−0.184091
$899$	−14.9126	−0.497362
$900$	5.04591	0.168197
$901$	−2.42057	−0.0806408
$902$	−19.8731	−0.661702
$903$	107.111	3.56444
$904$	10.8775	0.361780
$905$	−5.68545	−0.188991
$906$	41.5501	1.38041
$907$	8.90054	0.295538	0.147769	−	0.989022i	$-0.452791\pi$
$907$	8.90054	0.295538	0.147769	+	0.989022i	$0.452791\pi$
$908$	29.8441	0.990411
$909$	−13.5095	−0.448080
$910$	−55.8561	−1.85161
$911$	−26.3404	−0.872697	−0.436349	−	0.899778i	$-0.643729\pi$
$911$	−26.3404	−0.872697	−0.436349	+	0.899778i	$0.643729\pi$
$912$	21.8123	0.722279
$913$	29.4470	0.974555
$914$	−8.25167	−0.272941
$915$	−64.1019	−2.11914
$916$	−3.85892	−0.127502
$917$	−11.1387	−0.367833
$918$	5.46146	0.180255
$919$	46.4292	1.53156	0.765780	−	0.643103i	$-0.222353\pi$
$919$	46.4292	1.53156	0.765780	+	0.643103i	$0.222353\pi$
$920$	2.51596	0.0829487
$921$	27.1262	0.893840
$922$	−11.9841	−0.394675
$923$	15.3831	0.506342
$924$	−35.4919	−1.16760
$925$	4.29050	0.141071
$926$	−11.5986	−0.381154
$927$	24.3058	0.798307
$928$	−3.45085	−0.113280
$929$	−13.8012	−0.452803	−0.226402	−	0.974034i	$-0.572696\pi$
$929$	−13.8012	−0.452803	−0.226402	+	0.974034i	$0.572696\pi$
$930$	−31.5394	−1.03422
$931$	−40.6536	−1.33237
$932$	29.2555	0.958295
$933$	37.9038	1.24092
$934$	−29.3969	−0.961895
$935$	4.55992	0.149125
$936$	−40.0657	−1.30959
$937$	−5.80962	−0.189792	−0.0948961	−	0.995487i	$-0.530252\pi$
$937$	−5.80962	−0.189792	−0.0948961	+	0.995487i	$0.530252\pi$
$938$	−3.42589	−0.111859
$939$	8.96600	0.292594
$940$	−18.8316	−0.614218
$941$	27.3925	0.892969	0.446485	−	0.894791i	$-0.352676\pi$
$941$	27.3925	0.892969	0.446485	+	0.894791i	$0.352676\pi$
$942$	62.1025	2.02341
$943$	−6.28022	−0.204512
$944$	1.48720	0.0484043
$945$	−81.7825	−2.66039
$946$	32.8667	1.06859
$947$	32.1095	1.04342	0.521709	−	0.853124i	$-0.325295\pi$
$947$	32.1095	1.04342	0.521709	+	0.853124i	$0.325295\pi$
$948$	−36.5186	−1.18607
$949$	40.5363	1.31587
$950$	5.91380	0.191869
$951$	27.9980	0.907899
$952$	−2.03615	−0.0659919
$953$	41.8551	1.35582	0.677910	−	0.735145i	$-0.262886\pi$
$953$	41.8551	1.35582	0.677910	+	0.735145i	$0.262886\pi$
$954$	25.9999	0.841777
$955$	44.1454	1.42851
$956$	−0.607549	−0.0196495
$957$	−34.4516	−1.11366
$958$	−17.7499	−0.573472
$959$	54.4113	1.75703
$960$	−7.29840	−0.235555
$961$	−12.3254	−0.397593
$962$	−34.0675	−1.09838
$963$	42.0655	1.35554
$964$	−5.32172	−0.171401
$965$	−42.9041	−1.38113
$966$	−11.2160	−0.360869
$967$	41.6439	1.33918	0.669588	−	0.742733i	$-0.266471\pi$
$967$	41.6439	1.33918	0.669588	+	0.742733i	$0.266471\pi$
$968$	0.109449	0.00351784
$969$	12.4930	0.401332
$970$	4.53377	0.145571
$971$	−32.5449	−1.04442	−0.522208	−	0.852818i	$-0.674891\pi$
$971$	−32.5449	−1.04442	−0.522208	+	0.852818i	$0.674891\pi$
$972$	−2.82906	−0.0907422
$973$	−33.1512	−1.06278
$974$	−17.9303	−0.574523
$975$	−16.1598	−0.517529
$976$	8.78302	0.281138
$977$	−38.4727	−1.23085	−0.615425	−	0.788196i	$-0.711016\pi$
$977$	−38.4727	−1.23085	−0.615425	+	0.788196i	$0.711016\pi$
$978$	10.5639	0.337797
$979$	−2.77421	−0.0886640
$980$	13.6027	0.434522
$981$	28.3282	0.904449
$982$	5.30759	0.169372
$983$	−54.4020	−1.73515	−0.867577	−	0.497303i	$-0.834324\pi$
$983$	−54.4020	−1.73515	−0.867577	+	0.497303i	$0.834324\pi$
$984$	18.2179	0.580767
$985$	−44.5423	−1.41924
$986$	−1.97647	−0.0629435
$987$	83.9501	2.67216
$988$	−46.9570	−1.49390
$989$	10.3864	0.330268
$990$	−48.9791	−1.55666
$991$	−46.1941	−1.46740	−0.733702	−	0.679471i	$-0.762209\pi$
$991$	−46.1941	−1.46740	−0.733702	+	0.679471i	$0.762209\pi$
$992$	4.32141	0.137205
$993$	23.9668	0.760564
$994$	−8.39720	−0.266343
$995$	5.27373	0.167189
$996$	−26.9945	−0.855353
$997$	39.8239	1.26123	0.630617	−	0.776094i	$-0.282802\pi$
$997$	39.8239	1.26123	0.630617	+	0.776094i	$0.282802\pi$
$998$	38.9432	1.23272
$999$	−49.8805	−1.57815

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.3	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.3	✓	83	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.02523 q^{3} +1.00000 q^{4} +2.41251 q^{5} +3.02523 q^{6} +3.55505 q^{7} -1.00000 q^{8} +6.15201 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.02523 q^{3} +1.00000 q^{4} +2.41251 q^{5} +3.02523 q^{6} +3.55505 q^{7} -1.00000 q^{8} +6.15201 q^{9} -2.41251 q^{10} +3.30008 q^{11} -3.02523 q^{12} +6.51262 q^{13} -3.55505 q^{14} -7.29840 q^{15} +1.00000 q^{16} +0.572747 q^{17} -6.15201 q^{18} -7.21015 q^{19} +2.41251 q^{20} -10.7548 q^{21} -3.30008 q^{22} -1.04288 q^{23} +3.02523 q^{24} +0.820206 q^{25} -6.51262 q^{26} -9.53555 q^{27} +3.55505 q^{28} +3.45085 q^{29} +7.29840 q^{30} -4.32141 q^{31} -1.00000 q^{32} -9.98351 q^{33} -0.572747 q^{34} +8.57660 q^{35} +6.15201 q^{36} +5.23100 q^{37} +7.21015 q^{38} -19.7022 q^{39} -2.41251 q^{40} +6.02200 q^{41} +10.7548 q^{42} -9.95934 q^{43} +3.30008 q^{44} +14.8418 q^{45} +1.04288 q^{46} -7.80580 q^{47} -3.02523 q^{48} +5.63839 q^{49} -0.820206 q^{50} -1.73269 q^{51} +6.51262 q^{52} -4.22624 q^{53} +9.53555 q^{54} +7.96149 q^{55} -3.55505 q^{56} +21.8123 q^{57} -3.45085 q^{58} +1.48720 q^{59} -7.29840 q^{60} +8.78302 q^{61} +4.32141 q^{62} +21.8707 q^{63} +1.00000 q^{64} +15.7118 q^{65} +9.98351 q^{66} +0.963667 q^{67} +0.572747 q^{68} +3.15495 q^{69} -8.57660 q^{70} +2.36205 q^{71} -6.15201 q^{72} +6.22427 q^{73} -5.23100 q^{74} -2.48131 q^{75} -7.21015 q^{76} +11.7320 q^{77} +19.7022 q^{78} +12.0714 q^{79} +2.41251 q^{80} +10.3912 q^{81} -6.02200 q^{82} +8.92312 q^{83} -10.7548 q^{84} +1.38176 q^{85} +9.95934 q^{86} -10.4396 q^{87} -3.30008 q^{88} -0.840647 q^{89} -14.8418 q^{90} +23.1527 q^{91} -1.04288 q^{92} +13.0733 q^{93} +7.80580 q^{94} -17.3945 q^{95} +3.02523 q^{96} -1.87928 q^{97} -5.63839 q^{98} +20.3021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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