LMFDB - Embedded newform 8042.2.a.a.1.57

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.57
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.93046 q^{3} +1.00000 q^{4} -3.49784 q^{5} +1.93046 q^{6} -2.88832 q^{7} +1.00000 q^{8} +0.726658 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.93046 q^{3} +1.00000 q^{4} -3.49784 q^{5} +1.93046 q^{6} -2.88832 q^{7} +1.00000 q^{8} +0.726658 q^{9} -3.49784 q^{10} +1.43362 q^{11} +1.93046 q^{12} +5.42179 q^{13} -2.88832 q^{14} -6.75242 q^{15} +1.00000 q^{16} +0.602820 q^{17} +0.726658 q^{18} -0.689583 q^{19} -3.49784 q^{20} -5.57577 q^{21} +1.43362 q^{22} -0.726438 q^{23} +1.93046 q^{24} +7.23488 q^{25} +5.42179 q^{26} -4.38859 q^{27} -2.88832 q^{28} -0.00220431 q^{29} -6.75242 q^{30} -8.02988 q^{31} +1.00000 q^{32} +2.76753 q^{33} +0.602820 q^{34} +10.1029 q^{35} +0.726658 q^{36} +8.02454 q^{37} -0.689583 q^{38} +10.4665 q^{39} -3.49784 q^{40} -8.22740 q^{41} -5.57577 q^{42} -3.37590 q^{43} +1.43362 q^{44} -2.54173 q^{45} -0.726438 q^{46} -3.51982 q^{47} +1.93046 q^{48} +1.34240 q^{49} +7.23488 q^{50} +1.16372 q^{51} +5.42179 q^{52} +6.67528 q^{53} -4.38859 q^{54} -5.01456 q^{55} -2.88832 q^{56} -1.33121 q^{57} -0.00220431 q^{58} -5.25720 q^{59} -6.75242 q^{60} -9.36485 q^{61} -8.02988 q^{62} -2.09882 q^{63} +1.00000 q^{64} -18.9645 q^{65} +2.76753 q^{66} -1.18335 q^{67} +0.602820 q^{68} -1.40236 q^{69} +10.1029 q^{70} +8.27572 q^{71} +0.726658 q^{72} +3.74956 q^{73} +8.02454 q^{74} +13.9666 q^{75} -0.689583 q^{76} -4.14074 q^{77} +10.4665 q^{78} -7.80026 q^{79} -3.49784 q^{80} -10.6519 q^{81} -8.22740 q^{82} +4.54253 q^{83} -5.57577 q^{84} -2.10857 q^{85} -3.37590 q^{86} -0.00425532 q^{87} +1.43362 q^{88} +10.4433 q^{89} -2.54173 q^{90} -15.6599 q^{91} -0.726438 q^{92} -15.5013 q^{93} -3.51982 q^{94} +2.41205 q^{95} +1.93046 q^{96} +1.18634 q^{97} +1.34240 q^{98} +1.04175 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	1.93046	1.11455	0.557274	−	0.830328i	$-0.311847\pi$
$3$	1.93046	1.11455	0.557274	+	0.830328i	$0.311847\pi$
$4$	1.00000	0.500000
$5$	−3.49784	−1.56428	−0.782141	−	0.623102i	$-0.785872\pi$
$5$	−3.49784	−1.56428	−0.782141	+	0.623102i	$0.785872\pi$
$6$	1.93046	0.788105
$7$	−2.88832	−1.09168	−0.545841	−	0.837889i	$-0.683790\pi$
$7$	−2.88832	−1.09168	−0.545841	+	0.837889i	$0.683790\pi$
$8$	1.00000	0.353553
$9$	0.726658	0.242219
$10$	−3.49784	−1.10611
$11$	1.43362	0.432251	0.216126	−	0.976366i	$-0.430658\pi$
$11$	1.43362	0.432251	0.216126	+	0.976366i	$0.430658\pi$
$12$	1.93046	0.557274
$13$	5.42179	1.50373	0.751867	−	0.659315i	$-0.229154\pi$
$13$	5.42179	1.50373	0.751867	+	0.659315i	$0.229154\pi$
$14$	−2.88832	−0.771936
$15$	−6.75242	−1.74347
$16$	1.00000	0.250000
$17$	0.602820	0.146205	0.0731027	−	0.997324i	$-0.476710\pi$
$17$	0.602820	0.146205	0.0731027	+	0.997324i	$0.476710\pi$
$18$	0.726658	0.171275
$19$	−0.689583	−0.158201	−0.0791006	−	0.996867i	$-0.525205\pi$
$19$	−0.689583	−0.158201	−0.0791006	+	0.996867i	$0.525205\pi$
$20$	−3.49784	−0.782141
$21$	−5.57577	−1.21673
$22$	1.43362	0.305648
$23$	−0.726438	−0.151473	−0.0757364	−	0.997128i	$-0.524131\pi$
$23$	−0.726438	−0.151473	−0.0757364	+	0.997128i	$0.524131\pi$
$24$	1.93046	0.394053
$25$	7.23488	1.44698
$26$	5.42179	1.06330
$27$	−4.38859	−0.844584
$28$	−2.88832	−0.545841
$29$	−0.00220431	−0.000409330 0	−0.000204665	−	1.00000i	$-0.500065\pi$
$29$	−0.00220431	−0.000409330 0	−0.000204665		1.00000i	$0.500065\pi$
$30$	−6.75242	−1.23282
$31$	−8.02988	−1.44221	−0.721105	−	0.692826i	$-0.756365\pi$
$31$	−8.02988	−1.44221	−0.721105	+	0.692826i	$0.756365\pi$
$32$	1.00000	0.176777
$33$	2.76753	0.481765
$34$	0.602820	0.103383
$35$	10.1029	1.70770
$36$	0.726658	0.121110
$37$	8.02454	1.31923	0.659613	−	0.751605i	$-0.270720\pi$
$37$	8.02454	1.31923	0.659613	+	0.751605i	$0.270720\pi$
$38$	−0.689583	−0.111865
$39$	10.4665	1.67599
$40$	−3.49784	−0.553057
$41$	−8.22740	−1.28490	−0.642452	−	0.766326i	$-0.722083\pi$
$41$	−8.22740	−1.28490	−0.642452	+	0.766326i	$0.722083\pi$
$42$	−5.57577	−0.860361
$43$	−3.37590	−0.514821	−0.257410	−	0.966302i	$-0.582869\pi$
$43$	−3.37590	−0.514821	−0.257410	+	0.966302i	$0.582869\pi$
$44$	1.43362	0.216126
$45$	−2.54173	−0.378899
$46$	−0.726438	−0.107107
$47$	−3.51982	−0.513419	−0.256709	−	0.966489i	$-0.582638\pi$
$47$	−3.51982	−0.513419	−0.256709	+	0.966489i	$0.582638\pi$
$48$	1.93046	0.278637
$49$	1.34240	0.191771
$50$	7.23488	1.02317
$51$	1.16372	0.162953
$52$	5.42179	0.751867
$53$	6.67528	0.916920	0.458460	−	0.888715i	$-0.348401\pi$
$53$	6.67528	0.916920	0.458460	+	0.888715i	$0.348401\pi$
$54$	−4.38859	−0.597211
$55$	−5.01456	−0.676163
$56$	−2.88832	−0.385968
$57$	−1.33121	−0.176323
$58$	−0.00220431	−0.000289440 0
$59$	−5.25720	−0.684429	−0.342215	−	0.939622i	$-0.611177\pi$
$59$	−5.25720	−0.684429	−0.342215	+	0.939622i	$0.611177\pi$
$60$	−6.75242	−0.871734
$61$	−9.36485	−1.19905	−0.599523	−	0.800358i	$-0.704643\pi$
$61$	−9.36485	−1.19905	−0.599523	+	0.800358i	$0.704643\pi$
$62$	−8.02988	−1.01980
$63$	−2.09882	−0.264427
$64$	1.00000	0.125000
$65$	−18.9645	−2.35226
$66$	2.76753	0.340660
$67$	−1.18335	−0.144570	−0.0722848	−	0.997384i	$-0.523029\pi$
$67$	−1.18335	−0.144570	−0.0722848	+	0.997384i	$0.523029\pi$
$68$	0.602820	0.0731027
$69$	−1.40236	−0.168824
$70$	10.1029	1.20753
$71$	8.27572	0.982147	0.491073	−	0.871118i	$-0.336605\pi$
$71$	8.27572	0.982147	0.491073	+	0.871118i	$0.336605\pi$
$72$	0.726658	0.0856375
$73$	3.74956	0.438853	0.219426	−	0.975629i	$-0.429581\pi$
$73$	3.74956	0.438853	0.219426	+	0.975629i	$0.429581\pi$
$74$	8.02454	0.932834
$75$	13.9666	1.61272
$76$	−0.689583	−0.0791006
$77$	−4.14074	−0.471881
$78$	10.4665	1.18510
$79$	−7.80026	−0.877597	−0.438799	−	0.898585i	$-0.644596\pi$
$79$	−7.80026	−0.877597	−0.438799	+	0.898585i	$0.644596\pi$
$80$	−3.49784	−0.391070
$81$	−10.6519	−1.18355
$82$	−8.22740	−0.908565
$83$	4.54253	0.498608	0.249304	−	0.968425i	$-0.419798\pi$
$83$	4.54253	0.498608	0.249304	+	0.968425i	$0.419798\pi$
$84$	−5.57577	−0.608367
$85$	−2.10857	−0.228706
$86$	−3.37590	−0.364033
$87$	−0.00425532	−0.000456218 0
$88$	1.43362	0.152824
$89$	10.4433	1.10699	0.553494	−	0.832853i	$-0.313294\pi$
$89$	10.4433	1.10699	0.553494	+	0.832853i	$0.313294\pi$
$90$	−2.54173	−0.267922
$91$	−15.6599	−1.64160
$92$	−0.726438	−0.0757364
$93$	−15.5013	−1.60741
$94$	−3.51982	−0.363042
$95$	2.41205	0.247471
$96$	1.93046	0.197026
$97$	1.18634	0.120455	0.0602275	−	0.998185i	$-0.480817\pi$
$97$	1.18634	0.120455	0.0602275	+	0.998185i	$0.480817\pi$
$98$	1.34240	0.135603
$99$	1.04175	0.104700
$100$	7.23488	0.723488
$101$	−15.7449	−1.56667	−0.783336	−	0.621598i	$-0.786484\pi$
$101$	−15.7449	−1.56667	−0.783336	+	0.621598i	$0.786484\pi$
$102$	1.16372	0.115225
$103$	10.3826	1.02303	0.511514	−	0.859275i	$-0.329085\pi$
$103$	10.3826	1.02303	0.511514	+	0.859275i	$0.329085\pi$
$104$	5.42179	0.531650
$105$	19.5032	1.90331
$106$	6.67528	0.648361
$107$	−15.3995	−1.48873	−0.744363	−	0.667775i	$-0.767247\pi$
$107$	−15.3995	−1.48873	−0.744363	+	0.667775i	$0.767247\pi$
$108$	−4.38859	−0.422292
$109$	−10.9609	−1.04987	−0.524933	−	0.851143i	$-0.675910\pi$
$109$	−10.9609	−1.04987	−0.524933	+	0.851143i	$0.675910\pi$
$110$	−5.01456	−0.478119
$111$	15.4910	1.47034
$112$	−2.88832	−0.272921
$113$	−10.6739	−1.00412	−0.502058	−	0.864834i	$-0.667424\pi$
$113$	−10.6739	−1.00412	−0.502058	+	0.864834i	$0.667424\pi$
$114$	−1.33121	−0.124679
$115$	2.54096	0.236946
$116$	−0.00220431	−0.000204665 0
$117$	3.93979	0.364233
$118$	−5.25720	−0.483965
$119$	−1.74114	−0.159610
$120$	−6.75242	−0.616409
$121$	−8.94475	−0.813159
$122$	−9.36485	−0.847854
$123$	−15.8826	−1.43209
$124$	−8.02988	−0.721105
$125$	−7.81723	−0.699195
$126$	−2.09882	−0.186978
$127$	−16.3458	−1.45046	−0.725228	−	0.688508i	$-0.758266\pi$
$127$	−16.3458	−1.45046	−0.725228	+	0.688508i	$0.758266\pi$
$128$	1.00000	0.0883883
$129$	−6.51703	−0.573793
$130$	−18.9645	−1.66330
$131$	16.8693	1.47388	0.736939	−	0.675960i	$-0.236271\pi$
$131$	16.8693	1.47388	0.736939	+	0.675960i	$0.236271\pi$
$132$	2.76753	0.240883
$133$	1.99174	0.172706
$134$	−1.18335	−0.102226
$135$	15.3506	1.32117
$136$	0.602820	0.0516914
$137$	−18.9766	−1.62128	−0.810641	−	0.585543i	$-0.800881\pi$
$137$	−18.9766	−1.62128	−0.810641	+	0.585543i	$0.800881\pi$
$138$	−1.40236	−0.119377
$139$	−13.4422	−1.14015	−0.570076	−	0.821592i	$-0.693086\pi$
$139$	−13.4422	−1.14015	−0.570076	+	0.821592i	$0.693086\pi$
$140$	10.1029	0.853849
$141$	−6.79486	−0.572230
$142$	8.27572	0.694483
$143$	7.77276	0.649991
$144$	0.726658	0.0605548
$145$	0.00771031	0.000640307 0
$146$	3.74956	0.310316
$147$	2.59144	0.213738
$148$	8.02454	0.659613
$149$	−23.3604	−1.91376	−0.956880	−	0.290484i	$-0.906184\pi$
$149$	−23.3604	−1.91376	−0.956880	+	0.290484i	$0.906184\pi$
$150$	13.9666	1.14037
$151$	−24.3618	−1.98254	−0.991268	−	0.131865i	$-0.957903\pi$
$151$	−24.3618	−1.98254	−0.991268	+	0.131865i	$0.957903\pi$
$152$	−0.689583	−0.0559326
$153$	0.438044	0.0354138
$154$	−4.14074	−0.333671
$155$	28.0872	2.25602
$156$	10.4665	0.837993
$157$	−8.79082	−0.701584	−0.350792	−	0.936453i	$-0.614088\pi$
$157$	−8.79082	−0.701584	−0.350792	+	0.936453i	$0.614088\pi$
$158$	−7.80026	−0.620555
$159$	12.8863	1.02195
$160$	−3.49784	−0.276528
$161$	2.09819	0.165360
$162$	−10.6519	−0.836896
$163$	12.5357	0.981868	0.490934	−	0.871197i	$-0.336656\pi$
$163$	12.5357	0.981868	0.490934	+	0.871197i	$0.336656\pi$
$164$	−8.22740	−0.642452
$165$	−9.68038	−0.753616
$166$	4.54253	0.352569
$167$	1.32841	0.102796	0.0513978	−	0.998678i	$-0.483632\pi$
$167$	1.32841	0.102796	0.0513978	+	0.998678i	$0.483632\pi$
$168$	−5.57577	−0.430180
$169$	16.3958	1.26122
$170$	−2.10857	−0.161720
$171$	−0.501091	−0.0383194
$172$	−3.37590	−0.257410
$173$	22.1477	1.68386	0.841928	−	0.539589i	$-0.181420\pi$
$173$	22.1477	1.68386	0.841928	+	0.539589i	$0.181420\pi$
$174$	−0.00425532	−0.000322595 0
$175$	−20.8966	−1.57964
$176$	1.43362	0.108063
$177$	−10.1488	−0.762830
$178$	10.4433	0.782758
$179$	−1.29998	−0.0971650	−0.0485825	−	0.998819i	$-0.515470\pi$
$179$	−1.29998	−0.0971650	−0.0485825	+	0.998819i	$0.515470\pi$
$180$	−2.54173	−0.189450
$181$	−12.6475	−0.940079	−0.470040	−	0.882645i	$-0.655760\pi$
$181$	−12.6475	−0.940079	−0.470040	+	0.882645i	$0.655760\pi$
$182$	−15.6599	−1.16079
$183$	−18.0784	−1.33640
$184$	−0.726438	−0.0535537
$185$	−28.0685	−2.06364
$186$	−15.5013	−1.13661
$187$	0.864213	0.0631975
$188$	−3.51982	−0.256709
$189$	12.6756	0.922017
$190$	2.41205	0.174989
$191$	26.8770	1.94475	0.972376	−	0.233420i	$-0.0749917\pi$
$191$	26.8770	1.94475	0.972376	+	0.233420i	$0.0749917\pi$
$192$	1.93046	0.139319
$193$	4.87673	0.351035	0.175517	−	0.984476i	$-0.443840\pi$
$193$	4.87673	0.351035	0.175517	+	0.984476i	$0.443840\pi$
$194$	1.18634	0.0851745
$195$	−36.6102	−2.62171
$196$	1.34240	0.0958856
$197$	−11.1725	−0.796004	−0.398002	−	0.917384i	$-0.630296\pi$
$197$	−11.1725	−0.796004	−0.398002	+	0.917384i	$0.630296\pi$
$198$	1.04175	0.0740338
$199$	−15.9700	−1.13208	−0.566040	−	0.824378i	$-0.691525\pi$
$199$	−15.9700	−1.13208	−0.566040	+	0.824378i	$0.691525\pi$
$200$	7.23488	0.511583
$201$	−2.28441	−0.161130
$202$	−15.7449	−1.10781
$203$	0.00636675	0.000446858 0
$204$	1.16372	0.0814765
$205$	28.7781	2.00995
$206$	10.3826	0.723391
$207$	−0.527872	−0.0366896
$208$	5.42179	0.375934
$209$	−0.988597	−0.0683827
$210$	19.5032	1.34585
$211$	21.8399	1.50352	0.751759	−	0.659438i	$-0.229206\pi$
$211$	21.8399	1.50352	0.751759	+	0.659438i	$0.229206\pi$
$212$	6.67528	0.458460
$213$	15.9759	1.09465
$214$	−15.3995	−1.05269
$215$	11.8084	0.805324
$216$	−4.38859	−0.298605
$217$	23.1929	1.57443
$218$	−10.9609	−0.742368
$219$	7.23835	0.489123
$220$	−5.01456	−0.338081
$221$	3.26837	0.219854
$222$	15.4910	1.03969
$223$	19.4803	1.30450	0.652250	−	0.758004i	$-0.273825\pi$
$223$	19.4803	1.30450	0.652250	+	0.758004i	$0.273825\pi$
$224$	−2.88832	−0.192984
$225$	5.25728	0.350485
$226$	−10.6739	−0.710017
$227$	22.8084	1.51384	0.756922	−	0.653505i	$-0.226702\pi$
$227$	22.8084	1.51384	0.756922	+	0.653505i	$0.226702\pi$
$228$	−1.33121	−0.0881615
$229$	−24.9537	−1.64899	−0.824493	−	0.565872i	$-0.808540\pi$
$229$	−24.9537	−1.64899	−0.824493	+	0.565872i	$0.808540\pi$
$230$	2.54096	0.167546
$231$	−7.99352	−0.525935
$232$	−0.00220431	−0.000144720 0
$233$	18.5593	1.21586	0.607929	−	0.793991i	$-0.292000\pi$
$233$	18.5593	1.21586	0.607929	+	0.793991i	$0.292000\pi$
$234$	3.93979	0.257552
$235$	12.3118	0.803131
$236$	−5.25720	−0.342215
$237$	−15.0580	−0.978125
$238$	−1.74114	−0.112861
$239$	−26.8834	−1.73894	−0.869472	−	0.493982i	$-0.835541\pi$
$239$	−26.8834	−1.73894	−0.869472	+	0.493982i	$0.835541\pi$
$240$	−6.75242	−0.435867
$241$	19.9119	1.28264	0.641319	−	0.767275i	$-0.278388\pi$
$241$	19.9119	1.28264	0.641319	+	0.767275i	$0.278388\pi$
$242$	−8.94475	−0.574990
$243$	−7.39734	−0.474540
$244$	−9.36485	−0.599523
$245$	−4.69549	−0.299984
$246$	−15.8826	−1.01264
$247$	−3.73878	−0.237893
$248$	−8.02988	−0.509898
$249$	8.76916	0.555723
$250$	−7.81723	−0.494405
$251$	14.8311	0.936132	0.468066	−	0.883694i	$-0.344951\pi$
$251$	14.8311	0.936132	0.468066	+	0.883694i	$0.344951\pi$
$252$	−2.09882	−0.132213
$253$	−1.04143	−0.0654744
$254$	−16.3458	−1.02563
$255$	−4.07050	−0.254904
$256$	1.00000	0.0625000
$257$	−25.0530	−1.56276	−0.781382	−	0.624053i	$-0.785485\pi$
$257$	−25.0530	−1.56276	−0.781382	+	0.624053i	$0.785485\pi$
$258$	−6.51703	−0.405733
$259$	−23.1774	−1.44018
$260$	−18.9645	−1.17613
$261$	−0.00160178	−9.91475e−5 0
$262$	16.8693	1.04219
$263$	−17.8987	−1.10368	−0.551841	−	0.833949i	$-0.686075\pi$
$263$	−17.8987	−1.10368	−0.551841	+	0.833949i	$0.686075\pi$
$264$	2.76753	0.170330
$265$	−23.3491	−1.43432
$266$	1.99174	0.122121
$267$	20.1603	1.23379
$268$	−1.18335	−0.0722848
$269$	−10.6461	−0.649106	−0.324553	−	0.945868i	$-0.605214\pi$
$269$	−10.6461	−0.649106	−0.324553	+	0.945868i	$0.605214\pi$
$270$	15.3506	0.934206
$271$	−18.7684	−1.14010	−0.570049	−	0.821611i	$-0.693076\pi$
$271$	−18.7684	−1.14010	−0.570049	+	0.821611i	$0.693076\pi$
$272$	0.602820	0.0365513
$273$	−30.2307	−1.82964
$274$	−18.9766	−1.14642
$275$	10.3720	0.625457
$276$	−1.40236	−0.0844119
$277$	−17.4142	−1.04632	−0.523158	−	0.852236i	$-0.675246\pi$
$277$	−17.4142	−1.04632	−0.523158	+	0.852236i	$0.675246\pi$
$278$	−13.4422	−0.806209
$279$	−5.83498	−0.349331
$280$	10.1029	0.603763
$281$	−8.58728	−0.512274	−0.256137	−	0.966640i	$-0.582450\pi$
$281$	−8.58728	−0.512274	−0.256137	+	0.966640i	$0.582450\pi$
$282$	−6.79486	−0.404628
$283$	13.6714	0.812680	0.406340	−	0.913722i	$-0.366805\pi$
$283$	13.6714	0.812680	0.406340	+	0.913722i	$0.366805\pi$
$284$	8.27572	0.491073
$285$	4.65636	0.275819
$286$	7.77276	0.459613
$287$	23.7634	1.40271
$288$	0.726658	0.0428187
$289$	−16.6366	−0.978624
$290$	0.00771031	0.000452765 0
$291$	2.29018	0.134253
$292$	3.74956	0.219426
$293$	−24.0618	−1.40571	−0.702853	−	0.711336i	$-0.748091\pi$
$293$	−24.0618	−1.40571	−0.702853	+	0.711336i	$0.748091\pi$
$294$	2.59144	0.151136
$295$	18.3888	1.07064
$296$	8.02454	0.466417
$297$	−6.29155	−0.365072
$298$	−23.3604	−1.35323
$299$	−3.93860	−0.227775
$300$	13.9666	0.806362
$301$	9.75070	0.562021
$302$	−24.3618	−1.40186
$303$	−30.3948	−1.74613
$304$	−0.689583	−0.0395503
$305$	32.7567	1.87564
$306$	0.438044	0.0250413
$307$	−28.3216	−1.61640	−0.808199	−	0.588909i	$-0.799557\pi$
$307$	−28.3216	−1.61640	−0.808199	+	0.588909i	$0.799557\pi$
$308$	−4.14074	−0.235941
$309$	20.0432	1.14022
$310$	28.0872	1.59525
$311$	3.99269	0.226405	0.113202	−	0.993572i	$-0.463889\pi$
$311$	3.99269	0.226405	0.113202	+	0.993572i	$0.463889\pi$
$312$	10.4665	0.592550
$313$	16.9204	0.956398	0.478199	−	0.878252i	$-0.341290\pi$
$313$	16.9204	0.956398	0.478199	+	0.878252i	$0.341290\pi$
$314$	−8.79082	−0.496095
$315$	7.34134	0.413638
$316$	−7.80026	−0.438799
$317$	−0.439446	−0.0246818	−0.0123409	−	0.999924i	$-0.503928\pi$
$317$	−0.439446	−0.0246818	−0.0123409	+	0.999924i	$0.503928\pi$
$318$	12.8863	0.722630
$319$	−0.00316013	−0.000176933 0
$320$	−3.49784	−0.195535
$321$	−29.7281	−1.65926
$322$	2.09819	0.116927
$323$	−0.415695	−0.0231299
$324$	−10.6519	−0.591775
$325$	39.2260	2.17587
$326$	12.5357	0.694286
$327$	−21.1596	−1.17013
$328$	−8.22740	−0.454282
$329$	10.1664	0.560490
$330$	−9.68038	−0.532887
$331$	−14.3632	−0.789475	−0.394737	−	0.918794i	$-0.629164\pi$
$331$	−14.3632	−0.789475	−0.394737	+	0.918794i	$0.629164\pi$
$332$	4.54253	0.249304
$333$	5.83110	0.319542
$334$	1.32841	0.0726874
$335$	4.13918	0.226148
$336$	−5.57577	−0.304183
$337$	2.24411	0.122244	0.0611222	−	0.998130i	$-0.480532\pi$
$337$	2.24411	0.122244	0.0611222	+	0.998130i	$0.480532\pi$
$338$	16.3958	0.891815
$339$	−20.6055	−1.11914
$340$	−2.10857	−0.114353
$341$	−11.5118	−0.623397
$342$	−0.501091	−0.0270959
$343$	16.3410	0.882329
$344$	−3.37590	−0.182017
$345$	4.90522	0.264088
$346$	22.1477	1.19067
$347$	21.8444	1.17267	0.586333	−	0.810070i	$-0.300571\pi$
$347$	21.8444	1.17267	0.586333	+	0.810070i	$0.300571\pi$
$348$	−0.00425532	−0.000228109 0
$349$	33.9036	1.81482	0.907408	−	0.420250i	$-0.138058\pi$
$349$	33.9036	1.81482	0.907408	+	0.420250i	$0.138058\pi$
$350$	−20.8966	−1.11697
$351$	−23.7940	−1.27003
$352$	1.43362	0.0764120
$353$	28.4989	1.51684	0.758421	−	0.651765i	$-0.225971\pi$
$353$	28.4989	1.51684	0.758421	+	0.651765i	$0.225971\pi$
$354$	−10.1488	−0.539402
$355$	−28.9471	−1.53635
$356$	10.4433	0.553494
$357$	−3.36119	−0.177893
$358$	−1.29998	−0.0687060
$359$	−11.5177	−0.607882	−0.303941	−	0.952691i	$-0.598303\pi$
$359$	−11.5177	−0.607882	−0.303941	+	0.952691i	$0.598303\pi$
$360$	−2.54173	−0.133961
$361$	−18.5245	−0.974972
$362$	−12.6475	−0.664737
$363$	−17.2674	−0.906305
$364$	−15.6599	−0.820800
$365$	−13.1153	−0.686489
$366$	−18.0784	−0.944974
$367$	17.6374	0.920666	0.460333	−	0.887746i	$-0.347730\pi$
$367$	17.6374	0.920666	0.460333	+	0.887746i	$0.347730\pi$
$368$	−0.726438	−0.0378682
$369$	−5.97851	−0.311229
$370$	−28.0685	−1.45921
$371$	−19.2804	−1.00099
$372$	−15.5013	−0.803706
$373$	−32.9489	−1.70603	−0.853015	−	0.521886i	$-0.825229\pi$
$373$	−32.9489	−1.70603	−0.853015	+	0.521886i	$0.825229\pi$
$374$	0.864213	0.0446874
$375$	−15.0908	−0.779287
$376$	−3.51982	−0.181521
$377$	−0.0119513	−0.000615523 0
$378$	12.6756	0.651965
$379$	8.57770	0.440607	0.220304	−	0.975431i	$-0.429295\pi$
$379$	8.57770	0.440607	0.220304	+	0.975431i	$0.429295\pi$
$380$	2.41205	0.123736
$381$	−31.5549	−1.61660
$382$	26.8770	1.37515
$383$	25.6327	1.30977	0.654885	−	0.755729i	$-0.272717\pi$
$383$	25.6327	1.30977	0.654885	+	0.755729i	$0.272717\pi$
$384$	1.93046	0.0985131
$385$	14.4837	0.738155
$386$	4.87673	0.248219
$387$	−2.45313	−0.124700
$388$	1.18634	0.0602275
$389$	21.1460	1.07214	0.536072	−	0.844172i	$-0.319908\pi$
$389$	21.1460	1.07214	0.536072	+	0.844172i	$0.319908\pi$
$390$	−36.6102	−1.85383
$391$	−0.437912	−0.0221461
$392$	1.34240	0.0678014
$393$	32.5654	1.64271
$394$	−11.1725	−0.562860
$395$	27.2840	1.37281
$396$	1.04175	0.0523498
$397$	−26.1637	−1.31312	−0.656559	−	0.754275i	$-0.727989\pi$
$397$	−26.1637	−1.31312	−0.656559	+	0.754275i	$0.727989\pi$
$398$	−15.9700	−0.800502
$399$	3.84496	0.192489
$400$	7.23488	0.361744
$401$	−34.9966	−1.74765	−0.873825	−	0.486241i	$-0.838368\pi$
$401$	−34.9966	−1.74765	−0.873825	+	0.486241i	$0.838368\pi$
$402$	−2.28441	−0.113936
$403$	−43.5363	−2.16870
$404$	−15.7449	−0.783336
$405$	37.2588	1.85140
$406$	0.00636675	0.000315976 0
$407$	11.5041	0.570237
$408$	1.16372	0.0576126
$409$	−22.3525	−1.10526	−0.552631	−	0.833426i	$-0.686376\pi$
$409$	−22.3525	−1.10526	−0.552631	+	0.833426i	$0.686376\pi$
$410$	28.7781	1.42125
$411$	−36.6335	−1.80700
$412$	10.3826	0.511514
$413$	15.1845	0.747180
$414$	−0.527872	−0.0259435
$415$	−15.8891	−0.779963
$416$	5.42179	0.265825
$417$	−25.9495	−1.27075
$418$	−0.988597	−0.0483539
$419$	−4.20461	−0.205409	−0.102704	−	0.994712i	$-0.532750\pi$
$419$	−4.20461	−0.205409	−0.102704	+	0.994712i	$0.532750\pi$
$420$	19.5032	0.951657
$421$	5.01054	0.244199	0.122099	−	0.992518i	$-0.461037\pi$
$421$	5.01054	0.244199	0.122099	+	0.992518i	$0.461037\pi$
$422$	21.8399	1.06315
$423$	−2.55771	−0.124360
$424$	6.67528	0.324180
$425$	4.36133	0.211556
$426$	15.9759	0.774035
$427$	27.0487	1.30898
$428$	−15.3995	−0.744363
$429$	15.0050	0.724447
$430$	11.8084	0.569450
$431$	31.0104	1.49372	0.746859	−	0.664982i	$-0.231561\pi$
$431$	31.0104	1.49372	0.746859	+	0.664982i	$0.231561\pi$
$432$	−4.38859	−0.211146
$433$	21.6898	1.04235	0.521173	−	0.853451i	$-0.325495\pi$
$433$	21.6898	1.04235	0.521173	+	0.853451i	$0.325495\pi$
$434$	23.1929	1.11329
$435$	0.0148844	0.000713653 0
$436$	−10.9609	−0.524933
$437$	0.500940	0.0239632
$438$	7.23835	0.345862
$439$	5.85612	0.279497	0.139749	−	0.990187i	$-0.455371\pi$
$439$	5.85612	0.279497	0.139749	+	0.990187i	$0.455371\pi$
$440$	−5.01456	−0.239060
$441$	0.975464	0.0464507
$442$	3.26837	0.155460
$443$	14.0734	0.668648	0.334324	−	0.942458i	$-0.391492\pi$
$443$	14.0734	0.668648	0.334324	+	0.942458i	$0.391492\pi$
$444$	15.4910	0.735171
$445$	−36.5290	−1.73164
$446$	19.4803	0.922421
$447$	−45.0962	−2.13298
$448$	−2.88832	−0.136460
$449$	26.3755	1.24474	0.622368	−	0.782725i	$-0.286171\pi$
$449$	26.3755	1.24474	0.622368	+	0.782725i	$0.286171\pi$
$450$	5.25728	0.247831
$451$	−11.7949	−0.555402
$452$	−10.6739	−0.502058
$453$	−47.0294	−2.20963
$454$	22.8084	1.07045
$455$	54.7757	2.56792
$456$	−1.33121	−0.0623396
$457$	0.588990	0.0275518	0.0137759	−	0.999905i	$-0.495615\pi$
$457$	0.588990	0.0275518	0.0137759	+	0.999905i	$0.495615\pi$
$458$	−24.9537	−1.16601
$459$	−2.64553	−0.123483
$460$	2.54096	0.118473
$461$	18.1297	0.844383	0.422191	−	0.906507i	$-0.361261\pi$
$461$	18.1297	0.844383	0.422191	+	0.906507i	$0.361261\pi$
$462$	−7.99352	−0.371892
$463$	42.1883	1.96066	0.980329	−	0.197370i	$-0.0632399\pi$
$463$	42.1883	1.96066	0.980329	+	0.197370i	$0.0632399\pi$
$464$	−0.00220431	−0.000102332 0
$465$	54.2211	2.51444
$466$	18.5593	0.859742
$467$	20.5822	0.952430	0.476215	−	0.879329i	$-0.342008\pi$
$467$	20.5822	0.952430	0.476215	+	0.879329i	$0.342008\pi$
$468$	3.93979	0.182117
$469$	3.41791	0.157824
$470$	12.3118	0.567899
$471$	−16.9703	−0.781949
$472$	−5.25720	−0.241982
$473$	−4.83975	−0.222532
$474$	−15.0580	−0.691639
$475$	−4.98905	−0.228913
$476$	−1.74114	−0.0798049
$477$	4.85065	0.222096
$478$	−26.8834	−1.22962
$479$	−36.1970	−1.65388	−0.826942	−	0.562287i	$-0.809921\pi$
$479$	−36.1970	−1.65388	−0.826942	+	0.562287i	$0.809921\pi$
$480$	−6.75242	−0.308204
$481$	43.5074	1.98377
$482$	19.9119	0.906961
$483$	4.05046	0.184302
$484$	−8.94475	−0.406579
$485$	−4.14964	−0.188425
$486$	−7.39734	−0.335550
$487$	−21.8948	−0.992147	−0.496074	−	0.868280i	$-0.665225\pi$
$487$	−21.8948	−0.992147	−0.496074	+	0.868280i	$0.665225\pi$
$488$	−9.36485	−0.423927
$489$	24.1995	1.09434
$490$	−4.69549	−0.212121
$491$	−14.5534	−0.656788	−0.328394	−	0.944541i	$-0.606507\pi$
$491$	−14.5534	−0.656788	−0.328394	+	0.944541i	$0.606507\pi$
$492$	−15.8826	−0.716044
$493$	−0.00132880	−5.98462e−5 0
$494$	−3.73878	−0.168215
$495$	−3.64387	−0.163780
$496$	−8.02988	−0.360552
$497$	−23.9029	−1.07219
$498$	8.76916	0.392955
$499$	13.6323	0.610268	0.305134	−	0.952310i	$-0.401299\pi$
$499$	13.6323	0.610268	0.305134	+	0.952310i	$0.401299\pi$
$500$	−7.81723	−0.349597
$501$	2.56444	0.114571
$502$	14.8311	0.661945
$503$	19.4529	0.867362	0.433681	−	0.901067i	$-0.357215\pi$
$503$	19.4529	0.867362	0.433681	+	0.901067i	$0.357215\pi$
$504$	−2.09882	−0.0934889
$505$	55.0730	2.45072
$506$	−1.04143	−0.0462974
$507$	31.6514	1.40569
$508$	−16.3458	−0.725228
$509$	−33.6520	−1.49160	−0.745800	−	0.666170i	$-0.767932\pi$
$509$	−33.6520	−1.49160	−0.745800	+	0.666170i	$0.767932\pi$
$510$	−4.07050	−0.180245
$511$	−10.8299	−0.479088
$512$	1.00000	0.0441942
$513$	3.02629	0.133614
$514$	−25.0530	−1.10504
$515$	−36.3167	−1.60030
$516$	−6.51703	−0.286896
$517$	−5.04607	−0.221926
$518$	−23.1774	−1.01836
$519$	42.7551	1.87674
$520$	−18.9645	−0.831650
$521$	25.6094	1.12197	0.560984	−	0.827827i	$-0.310423\pi$
$521$	25.6094	1.12197	0.560984	+	0.827827i	$0.310423\pi$
$522$	−0.00160178	−7.01079e−5 0
$523$	−5.89387	−0.257721	−0.128860	−	0.991663i	$-0.541132\pi$
$523$	−5.89387	−0.257721	−0.128860	+	0.991663i	$0.541132\pi$
$524$	16.8693	0.736939
$525$	−40.3400	−1.76058
$526$	−17.8987	−0.780422
$527$	−4.84057	−0.210859
$528$	2.76753	0.120441
$529$	−22.4723	−0.977056
$530$	−23.3491	−1.01422
$531$	−3.82019	−0.165782
$532$	1.99174	0.0863528
$533$	−44.6073	−1.93215
$534$	20.1603	0.872422
$535$	53.8650	2.32879
$536$	−1.18335	−0.0511131
$537$	−2.50955	−0.108295
$538$	−10.6461	−0.458987
$539$	1.92448	0.0828934
$540$	15.3506	0.660583
$541$	−34.9308	−1.50179	−0.750895	−	0.660421i	$-0.770378\pi$
$541$	−34.9308	−1.50179	−0.750895	+	0.660421i	$0.770378\pi$
$542$	−18.7684	−0.806171
$543$	−24.4154	−1.04776
$544$	0.602820	0.0258457
$545$	38.3396	1.64229
$546$	−30.2307	−1.29375
$547$	−12.2841	−0.525229	−0.262614	−	0.964901i	$-0.584585\pi$
$547$	−12.2841	−0.525229	−0.262614	+	0.964901i	$0.584585\pi$
$548$	−18.9766	−0.810641
$549$	−6.80504	−0.290432
$550$	10.3720	0.442265
$551$	0.00152005	6.47564e−5 0
$552$	−1.40236	−0.0596883
$553$	22.5296	0.958058
$554$	−17.4142	−0.739857
$555$	−54.1851	−2.30003
$556$	−13.4422	−0.570076
$557$	−29.0946	−1.23278	−0.616389	−	0.787442i	$-0.711405\pi$
$557$	−29.0946	−1.23278	−0.616389	+	0.787442i	$0.711405\pi$
$558$	−5.83498	−0.247014
$559$	−18.3034	−0.774153
$560$	10.1029	0.426925
$561$	1.66832	0.0704367
$562$	−8.58728	−0.362233
$563$	34.6694	1.46114	0.730570	−	0.682838i	$-0.239254\pi$
$563$	34.6694	1.46114	0.730570	+	0.682838i	$0.239254\pi$
$564$	−6.79486	−0.286115
$565$	37.3356	1.57072
$566$	13.6714	0.574651
$567$	30.7662	1.29206
$568$	8.27572	0.347241
$569$	45.1611	1.89325	0.946626	−	0.322334i	$-0.104467\pi$
$569$	45.1611	1.89325	0.946626	+	0.322334i	$0.104467\pi$
$570$	4.65636	0.195033
$571$	−16.5720	−0.693516	−0.346758	−	0.937955i	$-0.612718\pi$
$571$	−16.5720	−0.693516	−0.346758	+	0.937955i	$0.612718\pi$
$572$	7.77276	0.324996
$573$	51.8849	2.16752
$574$	23.7634	0.991864
$575$	−5.25569	−0.219177
$576$	0.726658	0.0302774
$577$	43.6437	1.81691	0.908456	−	0.417980i	$-0.137262\pi$
$577$	43.6437	1.81691	0.908456	+	0.417980i	$0.137262\pi$
$578$	−16.6366	−0.691992
$579$	9.41431	0.391245
$580$	0.00771031	0.000320153 0
$581$	−13.1203	−0.544322
$582$	2.29018	0.0949312
$583$	9.56979	0.396340
$584$	3.74956	0.155158
$585$	−13.7807	−0.569763
$586$	−24.0618	−0.993984
$587$	28.0452	1.15755	0.578775	−	0.815488i	$-0.303531\pi$
$587$	28.0452	1.15755	0.578775	+	0.815488i	$0.303531\pi$
$588$	2.59144	0.106869
$589$	5.53727	0.228159
$590$	18.3888	0.757056
$591$	−21.5679	−0.887186
$592$	8.02454	0.329807
$593$	1.73507	0.0712507	0.0356253	−	0.999365i	$-0.488658\pi$
$593$	1.73507	0.0712507	0.0356253	+	0.999365i	$0.488658\pi$
$594$	−6.29155	−0.258145
$595$	6.09022	0.249675
$596$	−23.3604	−0.956880
$597$	−30.8293	−1.26176
$598$	−3.93860	−0.161061
$599$	21.9382	0.896372	0.448186	−	0.893940i	$-0.352070\pi$
$599$	21.9382	0.896372	0.448186	+	0.893940i	$0.352070\pi$
$600$	13.9666	0.570184
$601$	13.5118	0.551157	0.275579	−	0.961279i	$-0.411131\pi$
$601$	13.5118	0.551157	0.275579	+	0.961279i	$0.411131\pi$
$602$	9.75070	0.397409
$603$	−0.859893	−0.0350176
$604$	−24.3618	−0.991268
$605$	31.2873	1.27201
$606$	−30.3948	−1.23470
$607$	19.1460	0.777113	0.388556	−	0.921425i	$-0.372974\pi$
$607$	19.1460	0.777113	0.388556	+	0.921425i	$0.372974\pi$
$608$	−0.689583	−0.0279663
$609$	0.0122907	0.000498045 0
$610$	32.7567	1.32628
$611$	−19.0837	−0.772045
$612$	0.438044	0.0177069
$613$	−21.0614	−0.850661	−0.425330	−	0.905038i	$-0.639842\pi$
$613$	−21.0614	−0.850661	−0.425330	+	0.905038i	$0.639842\pi$
$614$	−28.3216	−1.14297
$615$	55.5549	2.24019
$616$	−4.14074	−0.166835
$617$	9.21630	0.371034	0.185517	−	0.982641i	$-0.440604\pi$
$617$	9.21630	0.371034	0.185517	+	0.982641i	$0.440604\pi$
$618$	20.0432	0.806254
$619$	−11.0095	−0.442511	−0.221255	−	0.975216i	$-0.571015\pi$
$619$	−11.0095	−0.442511	−0.221255	+	0.975216i	$0.571015\pi$
$620$	28.0872	1.12801
$621$	3.18804	0.127931
$622$	3.99269	0.160092
$623$	−30.1636	−1.20848
$624$	10.4665	0.418996
$625$	−8.83095	−0.353238
$626$	16.9204	0.676275
$627$	−1.90844	−0.0762159
$628$	−8.79082	−0.350792
$629$	4.83735	0.192878
$630$	7.34134	0.292486
$631$	−6.82725	−0.271789	−0.135894	−	0.990723i	$-0.543391\pi$
$631$	−6.82725	−0.271789	−0.135894	+	0.990723i	$0.543391\pi$
$632$	−7.80026	−0.310278
$633$	42.1609	1.67574
$634$	−0.439446	−0.0174526
$635$	57.1750	2.26892
$636$	12.8863	0.510976
$637$	7.27820	0.288373
$638$	−0.00316013	−0.000125111 0
$639$	6.01361	0.237895
$640$	−3.49784	−0.138264
$641$	18.3383	0.724319	0.362159	−	0.932116i	$-0.382040\pi$
$641$	18.3383	0.724319	0.362159	+	0.932116i	$0.382040\pi$
$642$	−29.7281	−1.17327
$643$	27.8691	1.09905	0.549525	−	0.835477i	$-0.314809\pi$
$643$	27.8691	1.09905	0.549525	+	0.835477i	$0.314809\pi$
$644$	2.09819	0.0826801
$645$	22.7955	0.897573
$646$	−0.415695	−0.0163553
$647$	−24.5281	−0.964300	−0.482150	−	0.876089i	$-0.660144\pi$
$647$	−24.5281	−0.964300	−0.482150	+	0.876089i	$0.660144\pi$
$648$	−10.6519	−0.418448
$649$	−7.53681	−0.295845
$650$	39.2260	1.53857
$651$	44.7728	1.75478
$652$	12.5357	0.490934
$653$	−3.20657	−0.125483	−0.0627415	−	0.998030i	$-0.519984\pi$
$653$	−3.20657	−0.125483	−0.0627415	+	0.998030i	$0.519984\pi$
$654$	−21.1596	−0.827405
$655$	−59.0061	−2.30556
$656$	−8.22740	−0.321226
$657$	2.72465	0.106299
$658$	10.1664	0.396327
$659$	−13.5766	−0.528868	−0.264434	−	0.964404i	$-0.585185\pi$
$659$	−13.5766	−0.528868	−0.264434	+	0.964404i	$0.585185\pi$
$660$	−9.68038	−0.376808
$661$	1.39103	0.0541048	0.0270524	−	0.999634i	$-0.491388\pi$
$661$	1.39103	0.0541048	0.0270524	+	0.999634i	$0.491388\pi$
$662$	−14.3632	−0.558243
$663$	6.30943	0.245038
$664$	4.54253	0.176285
$665$	−6.96678	−0.270160
$666$	5.83110	0.225950
$667$	0.00160129	6.20023e−5 0
$668$	1.32841	0.0513978
$669$	37.6059	1.45393
$670$	4.13918	0.159911
$671$	−13.4256	−0.518289
$672$	−5.57577	−0.215090
$673$	31.3682	1.20915	0.604577	−	0.796547i	$-0.293342\pi$
$673$	31.3682	1.20915	0.604577	+	0.796547i	$0.293342\pi$
$674$	2.24411	0.0864398
$675$	−31.7509	−1.22209
$676$	16.3958	0.630608
$677$	38.2267	1.46917	0.734586	−	0.678516i	$-0.237376\pi$
$677$	38.2267	1.46917	0.734586	+	0.678516i	$0.237376\pi$
$678$	−20.6055	−0.791349
$679$	−3.42654	−0.131499
$680$	−2.10857	−0.0808599
$681$	44.0305	1.68725
$682$	−11.5118	−0.440808
$683$	−11.9873	−0.458682	−0.229341	−	0.973346i	$-0.573657\pi$
$683$	−11.9873	−0.458682	−0.229341	+	0.973346i	$0.573657\pi$
$684$	−0.501091	−0.0191597
$685$	66.3772	2.53614
$686$	16.3410	0.623901
$687$	−48.1720	−1.83788
$688$	−3.37590	−0.128705
$689$	36.1920	1.37880
$690$	4.90522	0.186738
$691$	−5.96902	−0.227072	−0.113536	−	0.993534i	$-0.536218\pi$
$691$	−5.96902	−0.227072	−0.113536	+	0.993534i	$0.536218\pi$
$692$	22.1477	0.841928
$693$	−3.00890	−0.114299
$694$	21.8444	0.829201
$695$	47.0186	1.78352
$696$	−0.00425532	−0.000161297 0
$697$	−4.95965	−0.187860
$698$	33.9036	1.28327
$699$	35.8278	1.35513
$700$	−20.8966	−0.789819
$701$	13.2811	0.501620	0.250810	−	0.968036i	$-0.419303\pi$
$701$	13.2811	0.501620	0.250810	+	0.968036i	$0.419303\pi$
$702$	−23.7940	−0.898046
$703$	−5.53359	−0.208703
$704$	1.43362	0.0540314
$705$	23.7673	0.895129
$706$	28.4989	1.07257
$707$	45.4762	1.71031
$708$	−10.1488	−0.381415
$709$	31.0177	1.16489	0.582447	−	0.812869i	$-0.302095\pi$
$709$	31.0177	1.16489	0.582447	+	0.812869i	$0.302095\pi$
$710$	−28.9471	−1.08637
$711$	−5.66812	−0.212571
$712$	10.4433	0.391379
$713$	5.83321	0.218455
$714$	−3.36119	−0.125789
$715$	−27.1879	−1.01677
$716$	−1.29998	−0.0485825
$717$	−51.8973	−1.93814
$718$	−11.5177	−0.429837
$719$	−37.2582	−1.38950	−0.694749	−	0.719253i	$-0.744484\pi$
$719$	−37.2582	−1.38950	−0.694749	+	0.719253i	$0.744484\pi$
$720$	−2.54173	−0.0947248
$721$	−29.9883	−1.11682
$722$	−18.5245	−0.689410
$723$	38.4390	1.42956
$724$	−12.6475	−0.470040
$725$	−0.0159479	−0.000592290 0
$726$	−17.2674	−0.640855
$727$	51.8502	1.92302	0.961509	−	0.274772i	$-0.0886024\pi$
$727$	51.8502	1.92302	0.961509	+	0.274772i	$0.0886024\pi$
$728$	−15.6599	−0.580393
$729$	17.6756	0.654651
$730$	−13.1153	−0.485421
$731$	−2.03506	−0.0752696
$732$	−18.0784	−0.668198
$733$	9.78692	0.361488	0.180744	−	0.983530i	$-0.442149\pi$
$733$	9.78692	0.361488	0.180744	+	0.983530i	$0.442149\pi$
$734$	17.6374	0.651009
$735$	−9.06444	−0.334347
$736$	−0.726438	−0.0267769
$737$	−1.69647	−0.0624905
$738$	−5.97851	−0.220072
$739$	34.4199	1.26616	0.633078	−	0.774088i	$-0.281791\pi$
$739$	34.4199	1.26616	0.633078	+	0.774088i	$0.281791\pi$
$740$	−28.0685	−1.03182
$741$	−7.21754	−0.265143
$742$	−19.2804	−0.707804
$743$	−31.6046	−1.15946	−0.579730	−	0.814809i	$-0.696842\pi$
$743$	−31.6046	−1.15946	−0.579730	+	0.814809i	$0.696842\pi$
$744$	−15.5013	−0.568306
$745$	81.7110	2.99366
$746$	−32.9489	−1.20635
$747$	3.30087	0.120772
$748$	0.864213	0.0315987
$749$	44.4787	1.62522
$750$	−15.0908	−0.551039
$751$	−8.18616	−0.298717	−0.149359	−	0.988783i	$-0.547721\pi$
$751$	−8.18616	−0.298717	−0.149359	+	0.988783i	$0.547721\pi$
$752$	−3.51982	−0.128355
$753$	28.6308	1.04336
$754$	−0.0119513	−0.000435240 0
$755$	85.2137	3.10124
$756$	12.6756	0.461009
$757$	−8.21538	−0.298593	−0.149296	−	0.988792i	$-0.547701\pi$
$757$	−8.21538	−0.298593	−0.149296	+	0.988792i	$0.547701\pi$
$758$	8.57770	0.311556
$759$	−2.01044	−0.0729744
$760$	2.41205	0.0874943
$761$	4.31259	0.156331	0.0781656	−	0.996940i	$-0.475094\pi$
$761$	4.31259	0.156331	0.0781656	+	0.996940i	$0.475094\pi$
$762$	−31.5549	−1.14311
$763$	31.6587	1.14612
$764$	26.8770	0.972376
$765$	−1.53221	−0.0553971
$766$	25.6327	0.926147
$767$	−28.5034	−1.02920
$768$	1.93046	0.0696593
$769$	−32.4464	−1.17005	−0.585023	−	0.811017i	$-0.698914\pi$
$769$	−32.4464	−1.17005	−0.585023	+	0.811017i	$0.698914\pi$
$770$	14.4837	0.521955
$771$	−48.3637	−1.74178
$772$	4.87673	0.175517
$773$	3.71137	0.133489	0.0667443	−	0.997770i	$-0.478739\pi$
$773$	3.71137	0.133489	0.0667443	+	0.997770i	$0.478739\pi$
$774$	−2.45313	−0.0881759
$775$	−58.0952	−2.08684
$776$	1.18634	0.0425873
$777$	−44.7430	−1.60515
$778$	21.1460	0.758120
$779$	5.67348	0.203273
$780$	−36.6102	−1.31086
$781$	11.8642	0.424534
$782$	−0.437912	−0.0156597
$783$	0.00967379	0.000345713 0
$784$	1.34240	0.0479428
$785$	30.7489	1.09747
$786$	32.5654	1.16157
$787$	19.3891	0.691146	0.345573	−	0.938392i	$-0.387685\pi$
$787$	19.3891	0.691146	0.345573	+	0.938392i	$0.387685\pi$
$788$	−11.1725	−0.398002
$789$	−34.5527	−1.23011
$790$	27.2840	0.970722
$791$	30.8297	1.09618
$792$	1.04175	0.0370169
$793$	−50.7742	−1.80305
$794$	−26.1637	−0.928515
$795$	−45.0743	−1.59862
$796$	−15.9700	−0.566040
$797$	−9.53053	−0.337589	−0.168794	−	0.985651i	$-0.553987\pi$
$797$	−9.53053	−0.337589	−0.168794	+	0.985651i	$0.553987\pi$
$798$	3.84496	0.136110
$799$	−2.12182	−0.0750646
$800$	7.23488	0.255791
$801$	7.58871	0.268134
$802$	−34.9966	−1.23577
$803$	5.37542	0.189695
$804$	−2.28441	−0.0805650
$805$	−7.33912	−0.258670
$806$	−43.5363	−1.53350
$807$	−20.5519	−0.723460
$808$	−15.7449	−0.553903
$809$	12.9291	0.454561	0.227281	−	0.973829i	$-0.427017\pi$
$809$	12.9291	0.454561	0.227281	+	0.973829i	$0.427017\pi$
$810$	37.2588	1.30914
$811$	−5.89341	−0.206946	−0.103473	−	0.994632i	$-0.532995\pi$
$811$	−5.89341	−0.206946	−0.103473	+	0.994632i	$0.532995\pi$
$812$	0.00636675	0.000223429 0
$813$	−36.2315	−1.27069
$814$	11.5041	0.403219
$815$	−43.8477	−1.53592
$816$	1.16372	0.0407383
$817$	2.32797	0.0814453
$818$	−22.3525	−0.781538
$819$	−11.3794	−0.397627
$820$	28.7781	1.00498
$821$	−38.5990	−1.34711	−0.673557	−	0.739135i	$-0.735235\pi$
$821$	−38.5990	−1.34711	−0.673557	+	0.739135i	$0.735235\pi$
$822$	−36.6335	−1.27774
$823$	42.5732	1.48401	0.742005	−	0.670394i	$-0.233875\pi$
$823$	42.5732	1.48401	0.742005	+	0.670394i	$0.233875\pi$
$824$	10.3826	0.361695
$825$	20.0227	0.697103
$826$	15.1845	0.528336
$827$	18.0877	0.628971	0.314486	−	0.949262i	$-0.398168\pi$
$827$	18.0877	0.628971	0.314486	+	0.949262i	$0.398168\pi$
$828$	−0.527872	−0.0183448
$829$	50.0747	1.73917	0.869584	−	0.493786i	$-0.164387\pi$
$829$	50.0747	1.73917	0.869584	+	0.493786i	$0.164387\pi$
$830$	−15.8891	−0.551517
$831$	−33.6172	−1.16617
$832$	5.42179	0.187967
$833$	0.809225	0.0280380
$834$	−25.9495	−0.898559
$835$	−4.64657	−0.160801
$836$	−0.988597	−0.0341914
$837$	35.2398	1.21807
$838$	−4.20461	−0.145246
$839$	12.9308	0.446420	0.223210	−	0.974770i	$-0.428346\pi$
$839$	12.9308	0.446420	0.223210	+	0.974770i	$0.428346\pi$
$840$	19.5032	0.672923
$841$	−29.0000	−1.00000
$842$	5.01054	0.172675
$843$	−16.5774	−0.570955
$844$	21.8399	0.751759
$845$	−57.3499	−1.97290
$846$	−2.55771	−0.0879357
$847$	25.8353	0.887711
$848$	6.67528	0.229230
$849$	26.3920	0.905771
$850$	4.36133	0.149592
$851$	−5.82933	−0.199827
$852$	15.9759	0.547325
$853$	−25.0199	−0.856664	−0.428332	−	0.903621i	$-0.640899\pi$
$853$	−25.0199	−0.856664	−0.428332	+	0.903621i	$0.640899\pi$
$854$	27.0487	0.925587
$855$	1.75274	0.0599423
$856$	−15.3995	−0.526344
$857$	7.77043	0.265433	0.132716	−	0.991154i	$-0.457630\pi$
$857$	7.77043	0.265433	0.132716	+	0.991154i	$0.457630\pi$
$858$	15.0050	0.512261
$859$	−9.12848	−0.311460	−0.155730	−	0.987800i	$-0.549773\pi$
$859$	−9.12848	−0.311460	−0.155730	+	0.987800i	$0.549773\pi$
$860$	11.8084	0.402662
$861$	45.8741	1.56339
$862$	31.0104	1.05622
$863$	−45.5787	−1.55152	−0.775758	−	0.631030i	$-0.782632\pi$
$863$	−45.5787	−1.55152	−0.775758	+	0.631030i	$0.782632\pi$
$864$	−4.38859	−0.149303
$865$	−77.4690	−2.63403
$866$	21.6898	0.737050
$867$	−32.1162	−1.09072
$868$	23.1929	0.787217
$869$	−11.1826	−0.379343
$870$	0.0148844	0.000504629 0
$871$	−6.41590	−0.217394
$872$	−10.9609	−0.371184
$873$	0.862066	0.0291765
$874$	0.500940	0.0169445
$875$	22.5787	0.763299
$876$	7.23835	0.244561
$877$	13.8677	0.468280	0.234140	−	0.972203i	$-0.424773\pi$
$877$	13.8677	0.468280	0.234140	+	0.972203i	$0.424773\pi$
$878$	5.85612	0.197635
$879$	−46.4502	−1.56673
$880$	−5.01456	−0.169041
$881$	52.0020	1.75199	0.875996	−	0.482319i	$-0.160205\pi$
$881$	52.0020	1.75199	0.875996	+	0.482319i	$0.160205\pi$
$882$	0.975464	0.0328456
$883$	−16.9717	−0.571142	−0.285571	−	0.958358i	$-0.592183\pi$
$883$	−16.9717	−0.571142	−0.285571	+	0.958358i	$0.592183\pi$
$884$	3.26837	0.109927
$885$	35.4988	1.19328
$886$	14.0734	0.472806
$887$	18.0116	0.604771	0.302385	−	0.953186i	$-0.402217\pi$
$887$	18.0116	0.604771	0.302385	+	0.953186i	$0.402217\pi$
$888$	15.4910	0.519844
$889$	47.2120	1.58344
$890$	−36.5290	−1.22445
$891$	−15.2708	−0.511591
$892$	19.4803	0.652250
$893$	2.42721	0.0812235
$894$	−45.0962	−1.50824
$895$	4.54712	0.151993
$896$	−2.88832	−0.0964920
$897$	−7.60328	−0.253866
$898$	26.3755	0.880161
$899$	0.0177003	0.000590339 0
$900$	5.25728	0.175243
$901$	4.02399	0.134059
$902$	−11.7949	−0.392728
$903$	18.8233	0.626400
$904$	−10.6739	−0.355009
$905$	44.2388	1.47055
$906$	−47.0294	−1.56245
$907$	33.7041	1.11913	0.559563	−	0.828788i	$-0.310969\pi$
$907$	33.7041	1.11913	0.559563	+	0.828788i	$0.310969\pi$
$908$	22.8084	0.756922
$909$	−11.4411	−0.379478
$910$	54.7757	1.81580
$911$	48.0622	1.59237	0.796186	−	0.605052i	$-0.206848\pi$
$911$	48.0622	1.59237	0.796186	+	0.605052i	$0.206848\pi$
$912$	−1.33121	−0.0440808
$913$	6.51225	0.215524
$914$	0.588990	0.0194821
$915$	63.2354	2.09050
$916$	−24.9537	−0.824493
$917$	−48.7239	−1.60901
$918$	−2.64553	−0.0873154
$919$	15.8068	0.521418	0.260709	−	0.965417i	$-0.416044\pi$
$919$	15.8068	0.521418	0.260709	+	0.965417i	$0.416044\pi$
$920$	2.54096	0.0837731
$921$	−54.6736	−1.80155
$922$	18.1297	0.597069
$923$	44.8692	1.47689
$924$	−7.99352	−0.262967
$925$	58.0565	1.90889
$926$	42.1883	1.38639
$927$	7.54460	0.247797
$928$	−0.00220431	−7.23599e−5 0
$929$	−59.1690	−1.94127	−0.970636	−	0.240555i	$-0.922671\pi$
$929$	−59.1690	−1.94127	−0.970636	+	0.240555i	$0.922671\pi$
$930$	54.2211	1.77798
$931$	−0.925695	−0.0303384
$932$	18.5593	0.607929
$933$	7.70772	0.252339
$934$	20.5822	0.673469
$935$	−3.02288	−0.0988586
$936$	3.93979	0.128776
$937$	−29.7217	−0.970965	−0.485483	−	0.874246i	$-0.661356\pi$
$937$	−29.7217	−0.970965	−0.485483	+	0.874246i	$0.661356\pi$
$938$	3.41791	0.111599
$939$	32.6641	1.06595
$940$	12.3118	0.401566
$941$	8.58232	0.279776	0.139888	−	0.990167i	$-0.455326\pi$
$941$	8.58232	0.279776	0.139888	+	0.990167i	$0.455326\pi$
$942$	−16.9703	−0.552922
$943$	5.97670	0.194628
$944$	−5.25720	−0.171107
$945$	−44.3374	−1.44229
$946$	−4.83975	−0.157354
$947$	−23.4060	−0.760592	−0.380296	−	0.924865i	$-0.624178\pi$
$947$	−23.4060	−0.760592	−0.380296	+	0.924865i	$0.624178\pi$
$948$	−15.0580	−0.489063
$949$	20.3293	0.659917
$950$	−4.98905	−0.161866
$951$	−0.848331	−0.0275090
$952$	−1.74114	−0.0564306
$953$	4.94572	0.160208	0.0801038	−	0.996787i	$-0.474475\pi$
$953$	4.94572	0.160208	0.0801038	+	0.996787i	$0.474475\pi$
$954$	4.85065	0.157045
$955$	−94.0114	−3.04214
$956$	−26.8834	−0.869472
$957$	−0.00610049	−0.000197201 0
$958$	−36.1970	−1.16947
$959$	54.8106	1.76993
$960$	−6.75242	−0.217933
$961$	33.4790	1.07997
$962$	43.5074	1.40273
$963$	−11.1902	−0.360598
$964$	19.9119	0.641319
$965$	−17.0580	−0.549117
$966$	4.05046	0.130321
$967$	36.0190	1.15829	0.579147	−	0.815223i	$-0.303386\pi$
$967$	36.0190	1.15829	0.579147	+	0.815223i	$0.303386\pi$
$968$	−8.94475	−0.287495
$969$	−0.802480	−0.0257794
$970$	−4.14964	−0.133237
$971$	26.7199	0.857484	0.428742	−	0.903427i	$-0.358957\pi$
$971$	26.7199	0.857484	0.428742	+	0.903427i	$0.358957\pi$
$972$	−7.39734	−0.237270
$973$	38.8253	1.24468
$974$	−21.8948	−0.701554
$975$	75.7240	2.42511
$976$	−9.36485	−0.299762
$977$	40.1760	1.28534	0.642672	−	0.766142i	$-0.277826\pi$
$977$	40.1760	1.28534	0.642672	+	0.766142i	$0.277826\pi$
$978$	24.1995	0.773815
$979$	14.9717	0.478497
$980$	−4.69549	−0.149992
$981$	−7.96485	−0.254298
$982$	−14.5534	−0.464419
$983$	15.7049	0.500908	0.250454	−	0.968128i	$-0.419420\pi$
$983$	15.7049	0.500908	0.250454	+	0.968128i	$0.419420\pi$
$984$	−15.8826	−0.506320
$985$	39.0794	1.24517
$986$	−0.00132880	−4.23177e−5 0
$987$	19.6257	0.624694
$988$	−3.73878	−0.118946
$989$	2.45239	0.0779813
$990$	−3.64387	−0.115810
$991$	−13.2973	−0.422402	−0.211201	−	0.977443i	$-0.567737\pi$
$991$	−13.2973	−0.422402	−0.211201	+	0.977443i	$0.567737\pi$
$992$	−8.02988	−0.254949
$993$	−27.7276	−0.879908
$994$	−23.9029	−0.758155
$995$	55.8603	1.77089
$996$	8.76916	0.277861
$997$	−50.0079	−1.58377	−0.791883	−	0.610673i	$-0.790899\pi$
$997$	−50.0079	−1.58377	−0.791883	+	0.610673i	$0.790899\pi$
$998$	13.6323	0.431524
$999$	−35.2164	−1.11420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.57	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.57	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.93046 q^{3} +1.00000 q^{4} -3.49784 q^{5} +1.93046 q^{6} -2.88832 q^{7} +1.00000 q^{8} +0.726658 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.93046 q^{3} +1.00000 q^{4} -3.49784 q^{5} +1.93046 q^{6} -2.88832 q^{7} +1.00000 q^{8} +0.726658 q^{9} -3.49784 q^{10} +1.43362 q^{11} +1.93046 q^{12} +5.42179 q^{13} -2.88832 q^{14} -6.75242 q^{15} +1.00000 q^{16} +0.602820 q^{17} +0.726658 q^{18} -0.689583 q^{19} -3.49784 q^{20} -5.57577 q^{21} +1.43362 q^{22} -0.726438 q^{23} +1.93046 q^{24} +7.23488 q^{25} +5.42179 q^{26} -4.38859 q^{27} -2.88832 q^{28} -0.00220431 q^{29} -6.75242 q^{30} -8.02988 q^{31} +1.00000 q^{32} +2.76753 q^{33} +0.602820 q^{34} +10.1029 q^{35} +0.726658 q^{36} +8.02454 q^{37} -0.689583 q^{38} +10.4665 q^{39} -3.49784 q^{40} -8.22740 q^{41} -5.57577 q^{42} -3.37590 q^{43} +1.43362 q^{44} -2.54173 q^{45} -0.726438 q^{46} -3.51982 q^{47} +1.93046 q^{48} +1.34240 q^{49} +7.23488 q^{50} +1.16372 q^{51} +5.42179 q^{52} +6.67528 q^{53} -4.38859 q^{54} -5.01456 q^{55} -2.88832 q^{56} -1.33121 q^{57} -0.00220431 q^{58} -5.25720 q^{59} -6.75242 q^{60} -9.36485 q^{61} -8.02988 q^{62} -2.09882 q^{63} +1.00000 q^{64} -18.9645 q^{65} +2.76753 q^{66} -1.18335 q^{67} +0.602820 q^{68} -1.40236 q^{69} +10.1029 q^{70} +8.27572 q^{71} +0.726658 q^{72} +3.74956 q^{73} +8.02454 q^{74} +13.9666 q^{75} -0.689583 q^{76} -4.14074 q^{77} +10.4665 q^{78} -7.80026 q^{79} -3.49784 q^{80} -10.6519 q^{81} -8.22740 q^{82} +4.54253 q^{83} -5.57577 q^{84} -2.10857 q^{85} -3.37590 q^{86} -0.00425532 q^{87} +1.43362 q^{88} +10.4433 q^{89} -2.54173 q^{90} -15.6599 q^{91} -0.726438 q^{92} -15.5013 q^{93} -3.51982 q^{94} +2.41205 q^{95} +1.93046 q^{96} +1.18634 q^{97} +1.34240 q^{98} +1.04175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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