LMFDB - Embedded newform 8042.2.a.c.1.12

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:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.38541 q^{3} +1.00000 q^{4} +2.54225 q^{5} +2.38541 q^{6} +3.05592 q^{7} -1.00000 q^{8} +2.69019 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.38541 q^{3} +1.00000 q^{4} +2.54225 q^{5} +2.38541 q^{6} +3.05592 q^{7} -1.00000 q^{8} +2.69019 q^{9} -2.54225 q^{10} +5.40659 q^{11} -2.38541 q^{12} +2.16059 q^{13} -3.05592 q^{14} -6.06432 q^{15} +1.00000 q^{16} +0.860456 q^{17} -2.69019 q^{18} -6.33765 q^{19} +2.54225 q^{20} -7.28963 q^{21} -5.40659 q^{22} -2.60042 q^{23} +2.38541 q^{24} +1.46305 q^{25} -2.16059 q^{26} +0.739034 q^{27} +3.05592 q^{28} -9.11723 q^{29} +6.06432 q^{30} +5.79999 q^{31} -1.00000 q^{32} -12.8969 q^{33} -0.860456 q^{34} +7.76893 q^{35} +2.69019 q^{36} -3.68703 q^{37} +6.33765 q^{38} -5.15389 q^{39} -2.54225 q^{40} -2.55894 q^{41} +7.28963 q^{42} -2.74552 q^{43} +5.40659 q^{44} +6.83913 q^{45} +2.60042 q^{46} -8.15126 q^{47} -2.38541 q^{48} +2.33866 q^{49} -1.46305 q^{50} -2.05254 q^{51} +2.16059 q^{52} +13.6247 q^{53} -0.739034 q^{54} +13.7449 q^{55} -3.05592 q^{56} +15.1179 q^{57} +9.11723 q^{58} +4.92751 q^{59} -6.06432 q^{60} +8.90175 q^{61} -5.79999 q^{62} +8.22100 q^{63} +1.00000 q^{64} +5.49276 q^{65} +12.8969 q^{66} -8.71794 q^{67} +0.860456 q^{68} +6.20308 q^{69} -7.76893 q^{70} -5.52150 q^{71} -2.69019 q^{72} -1.03624 q^{73} +3.68703 q^{74} -3.48998 q^{75} -6.33765 q^{76} +16.5221 q^{77} +5.15389 q^{78} +15.5592 q^{79} +2.54225 q^{80} -9.83346 q^{81} +2.55894 q^{82} +6.14988 q^{83} -7.28963 q^{84} +2.18750 q^{85} +2.74552 q^{86} +21.7483 q^{87} -5.40659 q^{88} +15.3032 q^{89} -6.83913 q^{90} +6.60259 q^{91} -2.60042 q^{92} -13.8354 q^{93} +8.15126 q^{94} -16.1119 q^{95} +2.38541 q^{96} -10.1728 q^{97} -2.33866 q^{98} +14.5447 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.38541	−1.37722	−0.688609	−	0.725133i	$-0.741778\pi$
$3$	−2.38541	−1.37722	−0.688609	+	0.725133i	$0.741778\pi$
$4$	1.00000	0.500000
$5$	2.54225	1.13693	0.568465	−	0.822707i	$-0.307537\pi$
$5$	2.54225	1.13693	0.568465	+	0.822707i	$0.307537\pi$
$6$	2.38541	0.973840
$7$	3.05592	1.15503	0.577515	−	0.816380i	$-0.304023\pi$
$7$	3.05592	1.15503	0.577515	+	0.816380i	$0.304023\pi$
$8$	−1.00000	−0.353553
$9$	2.69019	0.896729
$10$	−2.54225	−0.803931
$11$	5.40659	1.63015	0.815075	−	0.579356i	$-0.196696\pi$
$11$	5.40659	1.63015	0.815075	+	0.579356i	$0.196696\pi$
$12$	−2.38541	−0.688609
$13$	2.16059	0.599239	0.299620	−	0.954059i	$-0.403140\pi$
$13$	2.16059	0.599239	0.299620	+	0.954059i	$0.403140\pi$
$14$	−3.05592	−0.816730
$15$	−6.06432	−1.56580
$16$	1.00000	0.250000
$17$	0.860456	0.208691	0.104346	−	0.994541i	$-0.466725\pi$
$17$	0.860456	0.208691	0.104346	+	0.994541i	$0.466725\pi$
$18$	−2.69019	−0.634083
$19$	−6.33765	−1.45396	−0.726978	−	0.686661i	$-0.759076\pi$
$19$	−6.33765	−1.45396	−0.726978	+	0.686661i	$0.759076\pi$
$20$	2.54225	0.568465
$21$	−7.28963	−1.59073
$22$	−5.40659	−1.15269
$23$	−2.60042	−0.542226	−0.271113	−	0.962548i	$-0.587392\pi$
$23$	−2.60042	−0.542226	−0.271113	+	0.962548i	$0.587392\pi$
$24$	2.38541	0.486920
$25$	1.46305	0.292610
$26$	−2.16059	−0.423726
$27$	0.739034	0.142227
$28$	3.05592	0.577515
$29$	−9.11723	−1.69303	−0.846513	−	0.532367i	$-0.821302\pi$
$29$	−9.11723	−1.69303	−0.846513	+	0.532367i	$0.821302\pi$
$30$	6.06432	1.10719
$31$	5.79999	1.04171	0.520855	−	0.853645i	$-0.325613\pi$
$31$	5.79999	1.04171	0.520855	+	0.853645i	$0.325613\pi$
$32$	−1.00000	−0.176777
$33$	−12.8969	−2.24507
$34$	−0.860456	−0.147567
$35$	7.76893	1.31319
$36$	2.69019	0.448364
$37$	−3.68703	−0.606145	−0.303072	−	0.952968i	$-0.598012\pi$
$37$	−3.68703	−0.606145	−0.303072	+	0.952968i	$0.598012\pi$
$38$	6.33765	1.02810
$39$	−5.15389	−0.825283
$40$	−2.54225	−0.401966
$41$	−2.55894	−0.399639	−0.199820	−	0.979833i	$-0.564036\pi$
$41$	−2.55894	−0.399639	−0.199820	+	0.979833i	$0.564036\pi$
$42$	7.28963	1.12481
$43$	−2.74552	−0.418688	−0.209344	−	0.977842i	$-0.567133\pi$
$43$	−2.74552	−0.418688	−0.209344	+	0.977842i	$0.567133\pi$
$44$	5.40659	0.815075
$45$	6.83913	1.01952
$46$	2.60042	0.383411
$47$	−8.15126	−1.18898	−0.594492	−	0.804102i	$-0.702647\pi$
$47$	−8.15126	−1.18898	−0.594492	+	0.804102i	$0.702647\pi$
$48$	−2.38541	−0.344304
$49$	2.33866	0.334094
$50$	−1.46305	−0.206907
$51$	−2.05254	−0.287413
$52$	2.16059	0.299620
$53$	13.6247	1.87149	0.935746	−	0.352674i	$-0.114728\pi$
$53$	13.6247	1.87149	0.935746	+	0.352674i	$0.114728\pi$
$54$	−0.739034	−0.100570
$55$	13.7449	1.85337
$56$	−3.05592	−0.408365
$57$	15.1179	2.00241
$58$	9.11723	1.19715
$59$	4.92751	0.641507	0.320754	−	0.947163i	$-0.396064\pi$
$59$	4.92751	0.641507	0.320754	+	0.947163i	$0.396064\pi$
$60$	−6.06432	−0.782900
$61$	8.90175	1.13975	0.569876	−	0.821730i	$-0.306991\pi$
$61$	8.90175	1.13975	0.569876	+	0.821730i	$0.306991\pi$
$62$	−5.79999	−0.736600
$63$	8.22100	1.03575
$64$	1.00000	0.125000
$65$	5.49276	0.681293
$66$	12.8969	1.58750
$67$	−8.71794	−1.06507	−0.532533	−	0.846409i	$-0.678760\pi$
$67$	−8.71794	−1.06507	−0.532533	+	0.846409i	$0.678760\pi$
$68$	0.860456	0.104346
$69$	6.20308	0.746763
$70$	−7.76893	−0.928564
$71$	−5.52150	−0.655282	−0.327641	−	0.944802i	$-0.606254\pi$
$71$	−5.52150	−0.655282	−0.327641	+	0.944802i	$0.606254\pi$
$72$	−2.69019	−0.317041
$73$	−1.03624	−0.121283	−0.0606413	−	0.998160i	$-0.519315\pi$
$73$	−1.03624	−0.121283	−0.0606413	+	0.998160i	$0.519315\pi$
$74$	3.68703	0.428609
$75$	−3.48998	−0.402988
$76$	−6.33765	−0.726978
$77$	16.5221	1.88287
$78$	5.15389	0.583563
$79$	15.5592	1.75055	0.875276	−	0.483624i	$-0.160680\pi$
$79$	15.5592	1.75055	0.875276	+	0.483624i	$0.160680\pi$
$80$	2.54225	0.284233
$81$	−9.83346	−1.09261
$82$	2.55894	0.282588
$83$	6.14988	0.675037	0.337519	−	0.941319i	$-0.390412\pi$
$83$	6.14988	0.675037	0.337519	+	0.941319i	$0.390412\pi$
$84$	−7.28963	−0.795364
$85$	2.18750	0.237267
$86$	2.74552	0.296057
$87$	21.7483	2.33167
$88$	−5.40659	−0.576345
$89$	15.3032	1.62213	0.811067	−	0.584953i	$-0.198887\pi$
$89$	15.3032	1.62213	0.811067	+	0.584953i	$0.198887\pi$
$90$	−6.83913	−0.720908
$91$	6.60259	0.692139
$92$	−2.60042	−0.271113
$93$	−13.8354	−1.43466
$94$	8.15126	0.840738
$95$	−16.1119	−1.65305
$96$	2.38541	0.243460
$97$	−10.1728	−1.03289	−0.516445	−	0.856320i	$-0.672745\pi$
$97$	−10.1728	−1.03289	−0.516445	+	0.856320i	$0.672745\pi$
$98$	−2.33866	−0.236240
$99$	14.5447	1.46180
$100$	1.46305	0.146305
$101$	−6.81110	−0.677730	−0.338865	−	0.940835i	$-0.610043\pi$
$101$	−6.81110	−0.677730	−0.338865	+	0.940835i	$0.610043\pi$
$102$	2.05254	0.203232
$103$	17.4455	1.71896	0.859479	−	0.511171i	$-0.170788\pi$
$103$	17.4455	1.71896	0.859479	+	0.511171i	$0.170788\pi$
$104$	−2.16059	−0.211863
$105$	−18.5321	−1.80855
$106$	−13.6247	−1.32335
$107$	10.7161	1.03597	0.517983	−	0.855391i	$-0.326683\pi$
$107$	10.7161	1.03597	0.517983	+	0.855391i	$0.326683\pi$
$108$	0.739034	0.0711136
$109$	11.1958	1.07237	0.536184	−	0.844101i	$-0.319865\pi$
$109$	11.1958	1.07237	0.536184	+	0.844101i	$0.319865\pi$
$110$	−13.7449	−1.31053
$111$	8.79509	0.834793
$112$	3.05592	0.288757
$113$	18.7388	1.76280	0.881398	−	0.472375i	$-0.156603\pi$
$113$	18.7388	1.76280	0.881398	+	0.472375i	$0.156603\pi$
$114$	−15.1179	−1.41592
$115$	−6.61093	−0.616473
$116$	−9.11723	−0.846513
$117$	5.81238	0.537355
$118$	−4.92751	−0.453614
$119$	2.62949	0.241045
$120$	6.06432	0.553594
$121$	18.2313	1.65739
$122$	−8.90175	−0.805927
$123$	6.10413	0.550390
$124$	5.79999	0.520855
$125$	−8.99182	−0.804253
$126$	−8.22100	−0.732385
$127$	−3.66483	−0.325201	−0.162600	−	0.986692i	$-0.551988\pi$
$127$	−3.66483	−0.325201	−0.162600	+	0.986692i	$0.551988\pi$
$128$	−1.00000	−0.0883883
$129$	6.54920	0.576625
$130$	−5.49276	−0.481747
$131$	−17.9734	−1.57034	−0.785172	−	0.619277i	$-0.787426\pi$
$131$	−17.9734	−1.57034	−0.785172	+	0.619277i	$0.787426\pi$
$132$	−12.8969	−1.12254
$133$	−19.3674	−1.67936
$134$	8.71794	0.753116
$135$	1.87881	0.161702
$136$	−0.860456	−0.0737835
$137$	0.651530	0.0556640	0.0278320	−	0.999613i	$-0.491140\pi$
$137$	0.651530	0.0556640	0.0278320	+	0.999613i	$0.491140\pi$
$138$	−6.20308	−0.528041
$139$	6.35989	0.539438	0.269719	−	0.962939i	$-0.413069\pi$
$139$	6.35989	0.539438	0.269719	+	0.962939i	$0.413069\pi$
$140$	7.76893	0.656594
$141$	19.4441	1.63749
$142$	5.52150	0.463354
$143$	11.6814	0.976849
$144$	2.69019	0.224182
$145$	−23.1783	−1.92485
$146$	1.03624	0.0857597
$147$	−5.57866	−0.460120
$148$	−3.68703	−0.303072
$149$	−0.224420	−0.0183852	−0.00919260	−	0.999958i	$-0.502926\pi$
$149$	−0.224420	−0.0183852	−0.00919260	+	0.999958i	$0.502926\pi$
$150$	3.48998	0.284956
$151$	2.55670	0.208061	0.104031	−	0.994574i	$-0.466826\pi$
$151$	2.55670	0.208061	0.104031	+	0.994574i	$0.466826\pi$
$152$	6.33765	0.514051
$153$	2.31479	0.187139
$154$	−16.5221	−1.33139
$155$	14.7450	1.18435
$156$	−5.15389	−0.412641
$157$	1.52156	0.121434	0.0607170	−	0.998155i	$-0.480661\pi$
$157$	1.52156	0.121434	0.0607170	+	0.998155i	$0.480661\pi$
$158$	−15.5592	−1.23783
$159$	−32.5004	−2.57745
$160$	−2.54225	−0.200983
$161$	−7.94669	−0.626287
$162$	9.83346	0.772589
$163$	12.0908	0.947028	0.473514	−	0.880786i	$-0.342985\pi$
$163$	12.0908	0.947028	0.473514	+	0.880786i	$0.342985\pi$
$164$	−2.55894	−0.199820
$165$	−32.7873	−2.55249
$166$	−6.14988	−0.477324
$167$	−0.854626	−0.0661329	−0.0330665	−	0.999453i	$-0.510527\pi$
$167$	−0.854626	−0.0661329	−0.0330665	+	0.999453i	$0.510527\pi$
$168$	7.28963	0.562407
$169$	−8.33186	−0.640912
$170$	−2.18750	−0.167773
$171$	−17.0494	−1.30380
$172$	−2.74552	−0.209344
$173$	18.2898	1.39055	0.695274	−	0.718745i	$-0.255283\pi$
$173$	18.2898	1.39055	0.695274	+	0.718745i	$0.255283\pi$
$174$	−21.7483	−1.64874
$175$	4.47097	0.337974
$176$	5.40659	0.407537
$177$	−11.7541	−0.883495
$178$	−15.3032	−1.14702
$179$	11.6079	0.867615	0.433807	−	0.901006i	$-0.357170\pi$
$179$	11.6079	0.867615	0.433807	+	0.901006i	$0.357170\pi$
$180$	6.83913	0.509759
$181$	11.7305	0.871918	0.435959	−	0.899967i	$-0.356409\pi$
$181$	11.7305	0.871918	0.435959	+	0.899967i	$0.356409\pi$
$182$	−6.60259	−0.489416
$183$	−21.2343	−1.56969
$184$	2.60042	0.191706
$185$	−9.37337	−0.689144
$186$	13.8354	1.01446
$187$	4.65214	0.340198
$188$	−8.15126	−0.594492
$189$	2.25843	0.164277
$190$	16.1119	1.16888
$191$	18.4808	1.33722	0.668611	−	0.743612i	$-0.266889\pi$
$191$	18.4808	1.33722	0.668611	+	0.743612i	$0.266889\pi$
$192$	−2.38541	−0.172152
$193$	−5.84771	−0.420928	−0.210464	−	0.977602i	$-0.567497\pi$
$193$	−5.84771	−0.420928	−0.210464	+	0.977602i	$0.567497\pi$
$194$	10.1728	0.730364
$195$	−13.1025	−0.938289
$196$	2.33866	0.167047
$197$	19.8406	1.41358	0.706790	−	0.707423i	$-0.250142\pi$
$197$	19.8406	1.41358	0.706790	+	0.707423i	$0.250142\pi$
$198$	−14.5447	−1.03365
$199$	25.3449	1.79665	0.898326	−	0.439329i	$-0.144784\pi$
$199$	25.3449	1.79665	0.898326	+	0.439329i	$0.144784\pi$
$200$	−1.46305	−0.103453
$201$	20.7959	1.46683
$202$	6.81110	0.479227
$203$	−27.8615	−1.95550
$204$	−2.05254	−0.143707
$205$	−6.50548	−0.454362
$206$	−17.4455	−1.21549
$207$	−6.99562	−0.486229
$208$	2.16059	0.149810
$209$	−34.2651	−2.37016
$210$	18.5321	1.27884
$211$	−9.99447	−0.688048	−0.344024	−	0.938961i	$-0.611790\pi$
$211$	−9.99447	−0.688048	−0.344024	+	0.938961i	$0.611790\pi$
$212$	13.6247	0.935746
$213$	13.1711	0.902466
$214$	−10.7161	−0.732539
$215$	−6.97982	−0.476020
$216$	−0.739034	−0.0502849
$217$	17.7243	1.20321
$218$	−11.1958	−0.758279
$219$	2.47185	0.167032
$220$	13.7449	0.926683
$221$	1.85909	0.125056
$222$	−8.79509	−0.590288
$223$	15.3316	1.02668	0.513341	−	0.858185i	$-0.328408\pi$
$223$	15.3316	1.02668	0.513341	+	0.858185i	$0.328408\pi$
$224$	−3.05592	−0.204182
$225$	3.93588	0.262392
$226$	−18.7388	−1.24648
$227$	11.8746	0.788148	0.394074	−	0.919079i	$-0.371065\pi$
$227$	11.8746	0.788148	0.394074	+	0.919079i	$0.371065\pi$
$228$	15.1179	1.00121
$229$	−14.3238	−0.946540	−0.473270	−	0.880917i	$-0.656927\pi$
$229$	−14.3238	−0.946540	−0.473270	+	0.880917i	$0.656927\pi$
$230$	6.61093	0.435912
$231$	−39.4121	−2.59312
$232$	9.11723	0.598575
$233$	20.7093	1.35671	0.678356	−	0.734733i	$-0.262693\pi$
$233$	20.7093	1.35671	0.678356	+	0.734733i	$0.262693\pi$
$234$	−5.81238	−0.379967
$235$	−20.7226	−1.35179
$236$	4.92751	0.320754
$237$	−37.1152	−2.41089
$238$	−2.62949	−0.170444
$239$	−20.9530	−1.35534	−0.677668	−	0.735368i	$-0.737009\pi$
$239$	−20.9530	−1.35534	−0.677668	+	0.735368i	$0.737009\pi$
$240$	−6.06432	−0.391450
$241$	−21.7198	−1.39910	−0.699548	−	0.714586i	$-0.746615\pi$
$241$	−21.7198	−1.39910	−0.699548	+	0.714586i	$0.746615\pi$
$242$	−18.2313	−1.17195
$243$	21.2397	1.36253
$244$	8.90175	0.569876
$245$	5.94546	0.379842
$246$	−6.10413	−0.389185
$247$	−13.6930	−0.871267
$248$	−5.79999	−0.368300
$249$	−14.6700	−0.929674
$250$	8.99182	0.568693
$251$	−15.1485	−0.956163	−0.478082	−	0.878315i	$-0.658668\pi$
$251$	−15.1485	−0.956163	−0.478082	+	0.878315i	$0.658668\pi$
$252$	8.22100	0.517874
$253$	−14.0594	−0.883909
$254$	3.66483	0.229952
$255$	−5.21808	−0.326769
$256$	1.00000	0.0625000
$257$	−27.7721	−1.73237	−0.866187	−	0.499720i	$-0.833436\pi$
$257$	−27.7721	−1.73237	−0.866187	+	0.499720i	$0.833436\pi$
$258$	−6.54920	−0.407736
$259$	−11.2673	−0.700115
$260$	5.49276	0.340647
$261$	−24.5270	−1.51819
$262$	17.9734	1.11040
$263$	−0.192744	−0.0118851	−0.00594254	−	0.999982i	$-0.501892\pi$
$263$	−0.192744	−0.0118851	−0.00594254	+	0.999982i	$0.501892\pi$
$264$	12.8969	0.793752
$265$	34.6374	2.12776
$266$	19.3674	1.18749
$267$	−36.5044	−2.23403
$268$	−8.71794	−0.532533
$269$	11.7799	0.718233	0.359116	−	0.933293i	$-0.383078\pi$
$269$	11.7799	0.718233	0.359116	+	0.933293i	$0.383078\pi$
$270$	−1.87881	−0.114341
$271$	24.8245	1.50798	0.753991	−	0.656885i	$-0.228126\pi$
$271$	24.8245	1.50798	0.753991	+	0.656885i	$0.228126\pi$
$272$	0.860456	0.0521728
$273$	−15.7499	−0.953227
$274$	−0.651530	−0.0393604
$275$	7.91013	0.476999
$276$	6.20308	0.373381
$277$	−28.3862	−1.70556	−0.852780	−	0.522270i	$-0.825085\pi$
$277$	−28.3862	−1.70556	−0.852780	+	0.522270i	$0.825085\pi$
$278$	−6.35989	−0.381441
$279$	15.6031	0.934130
$280$	−7.76893	−0.464282
$281$	7.51813	0.448494	0.224247	−	0.974532i	$-0.428008\pi$
$281$	7.51813	0.448494	0.224247	+	0.974532i	$0.428008\pi$
$282$	−19.4441	−1.15788
$283$	28.0175	1.66547	0.832735	−	0.553672i	$-0.186774\pi$
$283$	28.0175	1.66547	0.832735	+	0.553672i	$0.186774\pi$
$284$	−5.52150	−0.327641
$285$	38.4335	2.27660
$286$	−11.6814	−0.690737
$287$	−7.81992	−0.461595
$288$	−2.69019	−0.158521
$289$	−16.2596	−0.956448
$290$	23.1783	1.36108
$291$	24.2663	1.42252
$292$	−1.03624	−0.0606413
$293$	−15.7191	−0.918318	−0.459159	−	0.888354i	$-0.651849\pi$
$293$	−15.7191	−0.918318	−0.459159	+	0.888354i	$0.651849\pi$
$294$	5.57866	0.325354
$295$	12.5270	0.729349
$296$	3.68703	0.214304
$297$	3.99566	0.231852
$298$	0.224420	0.0130003
$299$	−5.61844	−0.324923
$300$	−3.48998	−0.201494
$301$	−8.39011	−0.483598
$302$	−2.55670	−0.147121
$303$	16.2473	0.933381
$304$	−6.33765	−0.363489
$305$	22.6305	1.29582
$306$	−2.31479	−0.132328
$307$	24.0414	1.37212	0.686059	−	0.727546i	$-0.259339\pi$
$307$	24.0414	1.37212	0.686059	+	0.727546i	$0.259339\pi$
$308$	16.5221	0.941436
$309$	−41.6147	−2.36738
$310$	−14.7450	−0.837462
$311$	−12.2328	−0.693661	−0.346831	−	0.937928i	$-0.612742\pi$
$311$	−12.2328	−0.693661	−0.346831	+	0.937928i	$0.612742\pi$
$312$	5.15389	0.291782
$313$	−14.1524	−0.799941	−0.399970	−	0.916528i	$-0.630980\pi$
$313$	−14.1524	−0.799941	−0.399970	+	0.916528i	$0.630980\pi$
$314$	−1.52156	−0.0858668
$315$	20.8999	1.17757
$316$	15.5592	0.875276
$317$	17.2514	0.968936	0.484468	−	0.874809i	$-0.339013\pi$
$317$	17.2514	0.968936	0.484468	+	0.874809i	$0.339013\pi$
$318$	32.5004	1.82253
$319$	−49.2931	−2.75989
$320$	2.54225	0.142116
$321$	−25.5623	−1.42675
$322$	7.94669	0.442852
$323$	−5.45327	−0.303428
$324$	−9.83346	−0.546303
$325$	3.16105	0.175344
$326$	−12.0908	−0.669650
$327$	−26.7067	−1.47688
$328$	2.55894	0.141294
$329$	−24.9096	−1.37331
$330$	32.7873	1.80488
$331$	3.91181	0.215013	0.107506	−	0.994204i	$-0.465713\pi$
$331$	3.91181	0.215013	0.107506	+	0.994204i	$0.465713\pi$
$332$	6.14988	0.337519
$333$	−9.91880	−0.543547
$334$	0.854626	0.0467630
$335$	−22.1632	−1.21091
$336$	−7.28963	−0.397682
$337$	5.89251	0.320986	0.160493	−	0.987037i	$-0.448692\pi$
$337$	5.89251	0.320986	0.160493	+	0.987037i	$0.448692\pi$
$338$	8.33186	0.453193
$339$	−44.6997	−2.42775
$340$	2.18750	0.118634
$341$	31.3582	1.69814
$342$	17.0494	0.921928
$343$	−14.2447	−0.769141
$344$	2.74552	0.148029
$345$	15.7698	0.849017
$346$	−18.2898	−0.983266
$347$	−4.25322	−0.228325	−0.114162	−	0.993462i	$-0.536418\pi$
$347$	−4.25322	−0.228325	−0.114162	+	0.993462i	$0.536418\pi$
$348$	21.7483	1.16583
$349$	−29.7292	−1.59137	−0.795683	−	0.605714i	$-0.792888\pi$
$349$	−29.7292	−1.59137	−0.795683	+	0.605714i	$0.792888\pi$
$350$	−4.47097	−0.238984
$351$	1.59675	0.0852282
$352$	−5.40659	−0.288172
$353$	3.01361	0.160398	0.0801991	−	0.996779i	$-0.474444\pi$
$353$	3.01361	0.160398	0.0801991	+	0.996779i	$0.474444\pi$
$354$	11.7541	0.624725
$355$	−14.0371	−0.745010
$356$	15.3032	0.811067
$357$	−6.27241	−0.331971
$358$	−11.6079	−0.613496
$359$	17.5768	0.927669	0.463834	−	0.885922i	$-0.346473\pi$
$359$	17.5768	0.927669	0.463834	+	0.885922i	$0.346473\pi$
$360$	−6.83913	−0.360454
$361$	21.1658	1.11399
$362$	−11.7305	−0.616539
$363$	−43.4890	−2.28258
$364$	6.60259	0.346070
$365$	−2.63438	−0.137890
$366$	21.2343	1.10994
$367$	25.6358	1.33818	0.669089	−	0.743182i	$-0.266684\pi$
$367$	25.6358	1.33818	0.669089	+	0.743182i	$0.266684\pi$
$368$	−2.60042	−0.135556
$369$	−6.88403	−0.358368
$370$	9.37337	0.487298
$371$	41.6359	2.16163
$372$	−13.8354	−0.717330
$373$	10.4814	0.542705	0.271353	−	0.962480i	$-0.412529\pi$
$373$	10.4814	0.542705	0.271353	+	0.962480i	$0.412529\pi$
$374$	−4.65214	−0.240556
$375$	21.4492	1.10763
$376$	8.15126	0.420369
$377$	−19.6986	−1.01453
$378$	−2.25843	−0.116161
$379$	−12.4358	−0.638786	−0.319393	−	0.947622i	$-0.603479\pi$
$379$	−12.4358	−0.638786	−0.319393	+	0.947622i	$0.603479\pi$
$380$	−16.1119	−0.826523
$381$	8.74211	0.447872
$382$	−18.4808	−0.945559
$383$	18.2512	0.932593	0.466297	−	0.884628i	$-0.345588\pi$
$383$	18.2512	0.932593	0.466297	+	0.884628i	$0.345588\pi$
$384$	2.38541	0.121730
$385$	42.0034	2.14069
$386$	5.84771	0.297641
$387$	−7.38597	−0.375450
$388$	−10.1728	−0.516445
$389$	14.5636	0.738406	0.369203	−	0.929349i	$-0.379631\pi$
$389$	14.5636	0.738406	0.369203	+	0.929349i	$0.379631\pi$
$390$	13.1025	0.663471
$391$	−2.23755	−0.113158
$392$	−2.33866	−0.118120
$393$	42.8740	2.16271
$394$	−19.8406	−0.999553
$395$	39.5555	1.99026
$396$	14.5447	0.730901
$397$	−3.75094	−0.188254	−0.0941272	−	0.995560i	$-0.530006\pi$
$397$	−3.75094	−0.188254	−0.0941272	+	0.995560i	$0.530006\pi$
$398$	−25.3449	−1.27043
$399$	46.1991	2.31285
$400$	1.46305	0.0731526
$401$	−24.8048	−1.23869	−0.619347	−	0.785117i	$-0.712603\pi$
$401$	−24.8048	−1.23869	−0.619347	+	0.785117i	$0.712603\pi$
$402$	−20.7959	−1.03720
$403$	12.5314	0.624233
$404$	−6.81110	−0.338865
$405$	−24.9991	−1.24222
$406$	27.8615	1.38274
$407$	−19.9343	−0.988106
$408$	2.05254	0.101616
$409$	25.0789	1.24007	0.620036	−	0.784574i	$-0.287118\pi$
$409$	25.0789	1.24007	0.620036	+	0.784574i	$0.287118\pi$
$410$	6.50548	0.321283
$411$	−1.55417	−0.0766614
$412$	17.4455	0.859479
$413$	15.0581	0.740960
$414$	6.99562	0.343816
$415$	15.6346	0.767471
$416$	−2.16059	−0.105932
$417$	−15.1709	−0.742924
$418$	34.2651	1.67596
$419$	17.8485	0.871955	0.435978	−	0.899957i	$-0.356403\pi$
$419$	17.8485	0.871955	0.435978	+	0.899957i	$0.356403\pi$
$420$	−18.5321	−0.904273
$421$	18.9872	0.925381	0.462690	−	0.886520i	$-0.346884\pi$
$421$	18.9872	0.925381	0.462690	+	0.886520i	$0.346884\pi$
$422$	9.99447	0.486523
$423$	−21.9284	−1.06620
$424$	−13.6247	−0.661673
$425$	1.25889	0.0610652
$426$	−13.1711	−0.638140
$427$	27.2031	1.31645
$428$	10.7161	0.517983
$429$	−27.8650	−1.34533
$430$	6.97982	0.336597
$431$	26.1878	1.26142	0.630710	−	0.776018i	$-0.282764\pi$
$431$	26.1878	1.26142	0.630710	+	0.776018i	$0.282764\pi$
$432$	0.739034	0.0355568
$433$	−20.0820	−0.965077	−0.482539	−	0.875875i	$-0.660285\pi$
$433$	−20.0820	−0.965077	−0.482539	+	0.875875i	$0.660285\pi$
$434$	−17.7243	−0.850795
$435$	55.2898	2.65094
$436$	11.1958	0.536184
$437$	16.4806	0.788372
$438$	−2.47185	−0.118110
$439$	39.1621	1.86911	0.934553	−	0.355824i	$-0.115800\pi$
$439$	39.1621	1.86911	0.934553	+	0.355824i	$0.115800\pi$
$440$	−13.7449	−0.655264
$441$	6.29143	0.299592
$442$	−1.85909	−0.0884280
$443$	−20.4838	−0.973217	−0.486608	−	0.873620i	$-0.661766\pi$
$443$	−20.4838	−0.973217	−0.486608	+	0.873620i	$0.661766\pi$
$444$	8.79509	0.417397
$445$	38.9046	1.84425
$446$	−15.3316	−0.725973
$447$	0.535334	0.0253204
$448$	3.05592	0.144379
$449$	−8.81757	−0.416127	−0.208063	−	0.978115i	$-0.566716\pi$
$449$	−8.81757	−0.416127	−0.208063	+	0.978115i	$0.566716\pi$
$450$	−3.93588	−0.185539
$451$	−13.8352	−0.651472
$452$	18.7388	0.881398
$453$	−6.09877	−0.286545
$454$	−11.8746	−0.557305
$455$	16.7855	0.786914
$456$	−15.1179	−0.707960
$457$	13.0062	0.608403	0.304201	−	0.952608i	$-0.401610\pi$
$457$	13.0062	0.608403	0.304201	+	0.952608i	$0.401610\pi$
$458$	14.3238	0.669305
$459$	0.635907	0.0296816
$460$	−6.61093	−0.308236
$461$	7.97413	0.371392	0.185696	−	0.982607i	$-0.440546\pi$
$461$	7.97413	0.371392	0.185696	+	0.982607i	$0.440546\pi$
$462$	39.4121	1.83362
$463$	16.7681	0.779278	0.389639	−	0.920968i	$-0.372600\pi$
$463$	16.7681	0.779278	0.389639	+	0.920968i	$0.372600\pi$
$464$	−9.11723	−0.423257
$465$	−35.1730	−1.63111
$466$	−20.7093	−0.959340
$467$	−29.8711	−1.38227	−0.691134	−	0.722726i	$-0.742889\pi$
$467$	−29.8711	−1.38227	−0.691134	+	0.722726i	$0.742889\pi$
$468$	5.81238	0.268677
$469$	−26.6414	−1.23018
$470$	20.7226	0.955861
$471$	−3.62955	−0.167241
$472$	−4.92751	−0.226807
$473$	−14.8439	−0.682525
$474$	37.1152	1.70476
$475$	−9.27230	−0.425442
$476$	2.62949	0.120522
$477$	36.6529	1.67822
$478$	20.9530	0.958368
$479$	−32.0686	−1.46525	−0.732625	−	0.680632i	$-0.761705\pi$
$479$	−32.0686	−1.46525	−0.732625	+	0.680632i	$0.761705\pi$
$480$	6.06432	0.276797
$481$	−7.96616	−0.363226
$482$	21.7198	0.989310
$483$	18.9561	0.862533
$484$	18.2313	0.828693
$485$	−25.8618	−1.17432
$486$	−21.2397	−0.963454
$487$	8.43016	0.382007	0.191003	−	0.981589i	$-0.438826\pi$
$487$	8.43016	0.382007	0.191003	+	0.981589i	$0.438826\pi$
$488$	−8.90175	−0.402963
$489$	−28.8416	−1.30426
$490$	−5.94546	−0.268589
$491$	41.9399	1.89272	0.946360	−	0.323115i	$-0.104730\pi$
$491$	41.9399	1.89272	0.946360	+	0.323115i	$0.104730\pi$
$492$	6.10413	0.275195
$493$	−7.84498	−0.353320
$494$	13.6930	0.616079
$495$	36.9764	1.66197
$496$	5.79999	0.260427
$497$	−16.8733	−0.756870
$498$	14.6700	0.657378
$499$	28.2315	1.26382	0.631908	−	0.775043i	$-0.282272\pi$
$499$	28.2315	1.26382	0.631908	+	0.775043i	$0.282272\pi$
$500$	−8.99182	−0.402126
$501$	2.03863	0.0910794
$502$	15.1485	0.676110
$503$	−1.57312	−0.0701420	−0.0350710	−	0.999385i	$-0.511166\pi$
$503$	−1.57312	−0.0701420	−0.0350710	+	0.999385i	$0.511166\pi$
$504$	−8.22100	−0.366192
$505$	−17.3155	−0.770531
$506$	14.0594	0.625018
$507$	19.8749	0.882676
$508$	−3.66483	−0.162600
$509$	−29.4176	−1.30391	−0.651956	−	0.758257i	$-0.726051\pi$
$509$	−29.4176	−1.30391	−0.651956	+	0.758257i	$0.726051\pi$
$510$	5.21808	0.231060
$511$	−3.16666	−0.140085
$512$	−1.00000	−0.0441942
$513$	−4.68374	−0.206792
$514$	27.7721	1.22497
$515$	44.3509	1.95434
$516$	6.54920	0.288313
$517$	−44.0705	−1.93822
$518$	11.2673	0.495056
$519$	−43.6287	−1.91509
$520$	−5.49276	−0.240874
$521$	−11.6727	−0.511388	−0.255694	−	0.966758i	$-0.582304\pi$
$521$	−11.6727	−0.511388	−0.255694	+	0.966758i	$0.582304\pi$
$522$	24.5270	1.07352
$523$	25.8144	1.12878	0.564392	−	0.825507i	$-0.309111\pi$
$523$	25.8144	1.12878	0.564392	+	0.825507i	$0.309111\pi$
$524$	−17.9734	−0.785172
$525$	−10.6651	−0.465463
$526$	0.192744	0.00840402
$527$	4.99064	0.217396
$528$	−12.8969	−0.561268
$529$	−16.2378	−0.705991
$530$	−34.6374	−1.50455
$531$	13.2559	0.575258
$532$	−19.3674	−0.839681
$533$	−5.52882	−0.239480
$534$	36.5044	1.57970
$535$	27.2431	1.17782
$536$	8.71794	0.376558
$537$	−27.6896	−1.19489
$538$	−11.7799	−0.507867
$539$	12.6442	0.544623
$540$	1.87881	0.0808512
$541$	−6.36999	−0.273867	−0.136934	−	0.990580i	$-0.543725\pi$
$541$	−6.36999	−0.273867	−0.136934	+	0.990580i	$0.543725\pi$
$542$	−24.8245	−1.06630
$543$	−27.9820	−1.20082
$544$	−0.860456	−0.0368918
$545$	28.4627	1.21921
$546$	15.7499	0.674033
$547$	−15.3332	−0.655602	−0.327801	−	0.944747i	$-0.606308\pi$
$547$	−15.3332	−0.655602	−0.327801	+	0.944747i	$0.606308\pi$
$548$	0.651530	0.0278320
$549$	23.9474	1.02205
$550$	−7.91013	−0.337289
$551$	57.7818	2.46159
$552$	−6.20308	−0.264021
$553$	47.5478	2.02194
$554$	28.3862	1.20601
$555$	22.3593	0.949101
$556$	6.35989	0.269719
$557$	−44.6158	−1.89043	−0.945215	−	0.326448i	$-0.894148\pi$
$557$	−44.6158	−1.89043	−0.945215	+	0.326448i	$0.894148\pi$
$558$	−15.6031	−0.660530
$559$	−5.93195	−0.250895
$560$	7.76893	0.328297
$561$	−11.0973	−0.468527
$562$	−7.51813	−0.317133
$563$	−6.50862	−0.274306	−0.137153	−	0.990550i	$-0.543795\pi$
$563$	−6.50862	−0.274306	−0.137153	+	0.990550i	$0.543795\pi$
$564$	19.4441	0.818744
$565$	47.6387	2.00418
$566$	−28.0175	−1.17767
$567$	−30.0503	−1.26199
$568$	5.52150	0.231677
$569$	−33.2160	−1.39249	−0.696243	−	0.717806i	$-0.745147\pi$
$569$	−33.2160	−1.39249	−0.696243	+	0.717806i	$0.745147\pi$
$570$	−38.4335	−1.60980
$571$	−9.69938	−0.405906	−0.202953	−	0.979188i	$-0.565054\pi$
$571$	−9.69938	−0.405906	−0.202953	+	0.979188i	$0.565054\pi$
$572$	11.6814	0.488425
$573$	−44.0843	−1.84165
$574$	7.81992	0.326397
$575$	−3.80455	−0.158661
$576$	2.69019	0.112091
$577$	36.1366	1.50439	0.752194	−	0.658942i	$-0.228996\pi$
$577$	36.1366	1.50439	0.752194	+	0.658942i	$0.228996\pi$
$578$	16.2596	0.676311
$579$	13.9492	0.579709
$580$	−23.1783	−0.962427
$581$	18.7936	0.779688
$582$	−24.2663	−1.00587
$583$	73.6631	3.05081
$584$	1.03624	0.0428799
$585$	14.7765	0.610935
$586$	15.7191	0.649349
$587$	17.2526	0.712093	0.356046	−	0.934468i	$-0.384125\pi$
$587$	17.2526	0.712093	0.356046	+	0.934468i	$0.384125\pi$
$588$	−5.57866	−0.230060
$589$	−36.7583	−1.51460
$590$	−12.5270	−0.515727
$591$	−47.3279	−1.94681
$592$	−3.68703	−0.151536
$593$	9.74335	0.400112	0.200056	−	0.979785i	$-0.435888\pi$
$593$	9.74335	0.400112	0.200056	+	0.979785i	$0.435888\pi$
$594$	−3.99566	−0.163944
$595$	6.68482	0.274051
$596$	−0.224420	−0.00919260
$597$	−60.4580	−2.47438
$598$	5.61844	0.229755
$599$	−1.69261	−0.0691582	−0.0345791	−	0.999402i	$-0.511009\pi$
$599$	−1.69261	−0.0691582	−0.0345791	+	0.999402i	$0.511009\pi$
$600$	3.48998	0.142478
$601$	37.1925	1.51712	0.758558	−	0.651606i	$-0.225904\pi$
$601$	37.1925	1.51712	0.758558	+	0.651606i	$0.225904\pi$
$602$	8.39011	0.341955
$603$	−23.4529	−0.955075
$604$	2.55670	0.104031
$605$	46.3485	1.88433
$606$	−16.2473	−0.660000
$607$	6.99304	0.283839	0.141919	−	0.989878i	$-0.454673\pi$
$607$	6.99304	0.283839	0.141919	+	0.989878i	$0.454673\pi$
$608$	6.33765	0.257025
$609$	66.4612	2.69314
$610$	−22.6305	−0.916283
$611$	−17.6115	−0.712486
$612$	2.31479	0.0935697
$613$	−4.78897	−0.193425	−0.0967123	−	0.995312i	$-0.530833\pi$
$613$	−4.78897	−0.193425	−0.0967123	+	0.995312i	$0.530833\pi$
$614$	−24.0414	−0.970233
$615$	15.5182	0.625756
$616$	−16.5221	−0.665696
$617$	−37.1562	−1.49585	−0.747926	−	0.663782i	$-0.768950\pi$
$617$	−37.1562	−1.49585	−0.747926	+	0.663782i	$0.768950\pi$
$618$	41.6147	1.67399
$619$	0.270313	0.0108648	0.00543239	−	0.999985i	$-0.498271\pi$
$619$	0.270313	0.0108648	0.00543239	+	0.999985i	$0.498271\pi$
$620$	14.7450	0.592175
$621$	−1.92180	−0.0771193
$622$	12.2328	0.490493
$623$	46.7653	1.87361
$624$	−5.15389	−0.206321
$625$	−30.1747	−1.20699
$626$	14.1524	0.565643
$627$	81.7363	3.26423
$628$	1.52156	0.0607170
$629$	−3.17253	−0.126497
$630$	−20.8999	−0.832670
$631$	−38.4507	−1.53070	−0.765348	−	0.643616i	$-0.777433\pi$
$631$	−38.4507	−1.53070	−0.765348	+	0.643616i	$0.777433\pi$
$632$	−15.5592	−0.618914
$633$	23.8409	0.947592
$634$	−17.2514	−0.685141
$635$	−9.31691	−0.369730
$636$	−32.5004	−1.28873
$637$	5.05288	0.200202
$638$	49.2931	1.95153
$639$	−14.8539	−0.587610
$640$	−2.54225	−0.100491
$641$	0.745939	0.0294628	0.0147314	−	0.999891i	$-0.495311\pi$
$641$	0.745939	0.0294628	0.0147314	+	0.999891i	$0.495311\pi$
$642$	25.5623	1.00887
$643$	0.745265	0.0293904	0.0146952	−	0.999892i	$-0.495322\pi$
$643$	0.745265	0.0293904	0.0146952	+	0.999892i	$0.495322\pi$
$644$	−7.94669	−0.313143
$645$	16.6497	0.655583
$646$	5.45327	0.214556
$647$	35.5757	1.39863	0.699313	−	0.714815i	$-0.253489\pi$
$647$	35.5757	1.39863	0.699313	+	0.714815i	$0.253489\pi$
$648$	9.83346	0.386295
$649$	26.6410	1.04575
$650$	−3.16105	−0.123987
$651$	−42.2798	−1.65708
$652$	12.0908	0.473514
$653$	−29.0244	−1.13581	−0.567907	−	0.823093i	$-0.692246\pi$
$653$	−29.0244	−1.13581	−0.567907	+	0.823093i	$0.692246\pi$
$654$	26.7067	1.04431
$655$	−45.6930	−1.78537
$656$	−2.55894	−0.0999098
$657$	−2.78767	−0.108758
$658$	24.9096	0.971078
$659$	6.70174	0.261063	0.130531	−	0.991444i	$-0.458332\pi$
$659$	6.70174	0.261063	0.130531	+	0.991444i	$0.458332\pi$
$660$	−32.7873	−1.27624
$661$	−32.0963	−1.24840	−0.624201	−	0.781264i	$-0.714575\pi$
$661$	−32.0963	−1.24840	−0.624201	+	0.781264i	$0.714575\pi$
$662$	−3.91181	−0.152037
$663$	−4.43470	−0.172229
$664$	−6.14988	−0.238662
$665$	−49.2367	−1.90932
$666$	9.91880	0.384346
$667$	23.7087	0.918003
$668$	−0.854626	−0.0330665
$669$	−36.5722	−1.41396
$670$	22.1632	0.856240
$671$	48.1282	1.85797
$672$	7.28963	0.281204
$673$	18.8382	0.726159	0.363080	−	0.931758i	$-0.381725\pi$
$673$	18.8382	0.726159	0.363080	+	0.931758i	$0.381725\pi$
$674$	−5.89251	−0.226971
$675$	1.08125	0.0416172
$676$	−8.33186	−0.320456
$677$	35.2051	1.35304	0.676521	−	0.736423i	$-0.263487\pi$
$677$	35.2051	1.35304	0.676521	+	0.736423i	$0.263487\pi$
$678$	44.6997	1.71668
$679$	−31.0873	−1.19302
$680$	−2.18750	−0.0838867
$681$	−28.3259	−1.08545
$682$	−31.3582	−1.20077
$683$	29.5822	1.13193	0.565966	−	0.824429i	$-0.308503\pi$
$683$	29.5822	1.13193	0.565966	+	0.824429i	$0.308503\pi$
$684$	−17.0494	−0.651902
$685$	1.65636	0.0632861
$686$	14.2447	0.543865
$687$	34.1680	1.30359
$688$	−2.74552	−0.104672
$689$	29.4373	1.12147
$690$	−15.7698	−0.600346
$691$	11.6370	0.442692	0.221346	−	0.975195i	$-0.428955\pi$
$691$	11.6370	0.442692	0.221346	+	0.975195i	$0.428955\pi$
$692$	18.2898	0.695274
$693$	44.4476	1.68842
$694$	4.25322	0.161450
$695$	16.1684	0.613304
$696$	−21.7483	−0.824369
$697$	−2.20186	−0.0834013
$698$	29.7292	1.12527
$699$	−49.4002	−1.86849
$700$	4.47097	0.168987
$701$	−51.3777	−1.94051	−0.970255	−	0.242086i	$-0.922168\pi$
$701$	−51.3777	−1.94051	−0.970255	+	0.242086i	$0.922168\pi$
$702$	−1.59675	−0.0602654
$703$	23.3671	0.881307
$704$	5.40659	0.203769
$705$	49.4318	1.86171
$706$	−3.01361	−0.113419
$707$	−20.8142	−0.782798
$708$	−11.7541	−0.441747
$709$	17.6913	0.664409	0.332205	−	0.943207i	$-0.392208\pi$
$709$	17.6913	0.664409	0.332205	+	0.943207i	$0.392208\pi$
$710$	14.0371	0.526801
$711$	41.8573	1.56977
$712$	−15.3032	−0.573511
$713$	−15.0824	−0.564842
$714$	6.27241	0.234739
$715$	29.6971	1.11061
$716$	11.6079	0.433807
$717$	49.9815	1.86659
$718$	−17.5768	−0.655961
$719$	−32.1842	−1.20027	−0.600134	−	0.799899i	$-0.704886\pi$
$719$	−32.1842	−1.20027	−0.600134	+	0.799899i	$0.704886\pi$
$720$	6.83913	0.254879
$721$	53.3122	1.98545
$722$	−21.1658	−0.787708
$723$	51.8106	1.92686
$724$	11.7305	0.435959
$725$	−13.3390	−0.495397
$726$	43.4890	1.61403
$727$	−22.1568	−0.821750	−0.410875	−	0.911692i	$-0.634777\pi$
$727$	−22.1568	−0.821750	−0.410875	+	0.911692i	$0.634777\pi$
$728$	−6.60259	−0.244708
$729$	−21.1651	−0.783894
$730$	2.63438	0.0975028
$731$	−2.36240	−0.0873766
$732$	−21.2343	−0.784844
$733$	25.1291	0.928163	0.464081	−	0.885793i	$-0.346385\pi$
$733$	25.1291	0.928163	0.464081	+	0.885793i	$0.346385\pi$
$734$	−25.6358	−0.946235
$735$	−14.1824	−0.523125
$736$	2.60042	0.0958529
$737$	−47.1344	−1.73622
$738$	6.88403	0.253404
$739$	32.5495	1.19735	0.598676	−	0.800991i	$-0.295694\pi$
$739$	32.5495	1.19735	0.598676	+	0.800991i	$0.295694\pi$
$740$	−9.37337	−0.344572
$741$	32.6635	1.19992
$742$	−41.6359	−1.52850
$743$	−7.62050	−0.279569	−0.139785	−	0.990182i	$-0.544641\pi$
$743$	−7.62050	−0.279569	−0.139785	+	0.990182i	$0.544641\pi$
$744$	13.8354	0.507229
$745$	−0.570532	−0.0209027
$746$	−10.4814	−0.383750
$747$	16.5443	0.605325
$748$	4.65214	0.170099
$749$	32.7476	1.19657
$750$	−21.4492	−0.783213
$751$	−38.4999	−1.40488	−0.702440	−	0.711742i	$-0.747906\pi$
$751$	−38.4999	−1.40488	−0.702440	+	0.711742i	$0.747906\pi$
$752$	−8.15126	−0.297246
$753$	36.1353	1.31685
$754$	19.6986	0.717380
$755$	6.49977	0.236551
$756$	2.25843	0.0821384
$757$	−0.692942	−0.0251854	−0.0125927	−	0.999921i	$-0.504008\pi$
$757$	−0.692942	−0.0251854	−0.0125927	+	0.999921i	$0.504008\pi$
$758$	12.4358	0.451690
$759$	33.5375	1.21733
$760$	16.1119	0.584440
$761$	54.8971	1.99002	0.995010	−	0.0997772i	$-0.0318130\pi$
$761$	54.8971	1.99002	0.995010	+	0.0997772i	$0.0318130\pi$
$762$	−8.74211	−0.316693
$763$	34.2136	1.23862
$764$	18.4808	0.668611
$765$	5.88477	0.212764
$766$	−18.2512	−0.659443
$767$	10.6463	0.384416
$768$	−2.38541	−0.0860761
$769$	−30.4304	−1.09735	−0.548674	−	0.836036i	$-0.684867\pi$
$769$	−30.4304	−1.09735	−0.548674	+	0.836036i	$0.684867\pi$
$770$	−42.0034	−1.51370
$771$	66.2478	2.38586
$772$	−5.84771	−0.210464
$773$	40.5910	1.45996	0.729978	−	0.683471i	$-0.239530\pi$
$773$	40.5910	1.45996	0.729978	+	0.683471i	$0.239530\pi$
$774$	7.38597	0.265483
$775$	8.48569	0.304815
$776$	10.1728	0.365182
$777$	26.8771	0.964211
$778$	−14.5636	−0.522132
$779$	16.2177	0.581058
$780$	−13.1025	−0.469145
$781$	−29.8525	−1.06821
$782$	2.23755	0.0800146
$783$	−6.73795	−0.240795
$784$	2.33866	0.0835236
$785$	3.86820	0.138062
$786$	−42.8740	−1.52926
$787$	37.4883	1.33631	0.668157	−	0.744020i	$-0.267083\pi$
$787$	37.4883	1.33631	0.668157	+	0.744020i	$0.267083\pi$
$788$	19.8406	0.706790
$789$	0.459773	0.0163683
$790$	−39.5555	−1.40732
$791$	57.2642	2.03608
$792$	−14.5447	−0.516825
$793$	19.2330	0.682985
$794$	3.75094	0.133116
$795$	−82.6243	−2.93038
$796$	25.3449	0.898326
$797$	50.1042	1.77478	0.887390	−	0.461019i	$-0.152516\pi$
$797$	50.1042	1.77478	0.887390	+	0.461019i	$0.152516\pi$
$798$	−46.1991	−1.63543
$799$	−7.01380	−0.248130
$800$	−1.46305	−0.0517267
$801$	41.1684	1.45461
$802$	24.8048	0.875889
$803$	−5.60252	−0.197709
$804$	20.7959	0.733414
$805$	−20.2025	−0.712045
$806$	−12.5314	−0.441399
$807$	−28.0999	−0.989163
$808$	6.81110	0.239614
$809$	−55.8553	−1.96377	−0.981883	−	0.189486i	$-0.939318\pi$
$809$	−55.8553	−1.96377	−0.981883	+	0.189486i	$0.939318\pi$
$810$	24.9991	0.878380
$811$	−8.41451	−0.295473	−0.147737	−	0.989027i	$-0.547199\pi$
$811$	−8.41451	−0.295473	−0.147737	+	0.989027i	$0.547199\pi$
$812$	−27.8615	−0.977748
$813$	−59.2167	−2.07682
$814$	19.9343	0.698697
$815$	30.7380	1.07671
$816$	−2.05254	−0.0718533
$817$	17.4002	0.608754
$818$	−25.0789	−0.876863
$819$	17.7622	0.620661
$820$	−6.50548	−0.227181
$821$	−20.8210	−0.726658	−0.363329	−	0.931661i	$-0.618360\pi$
$821$	−20.8210	−0.726658	−0.363329	+	0.931661i	$0.618360\pi$
$822$	1.55417	0.0542078
$823$	31.1649	1.08634	0.543171	−	0.839622i	$-0.317224\pi$
$823$	31.1649	1.08634	0.543171	+	0.839622i	$0.317224\pi$
$824$	−17.4455	−0.607744
$825$	−18.8689	−0.656931
$826$	−15.0581	−0.523938
$827$	−19.4331	−0.675754	−0.337877	−	0.941190i	$-0.609709\pi$
$827$	−19.4331	−0.675754	−0.337877	+	0.941190i	$0.609709\pi$
$828$	−6.99562	−0.243115
$829$	−16.6418	−0.577992	−0.288996	−	0.957330i	$-0.593321\pi$
$829$	−16.6418	−0.577992	−0.288996	+	0.957330i	$0.593321\pi$
$830$	−15.6346	−0.542684
$831$	67.7127	2.34893
$832$	2.16059	0.0749049
$833$	2.01231	0.0697225
$834$	15.1709	0.525327
$835$	−2.17267	−0.0751885
$836$	−34.2651	−1.18508
$837$	4.28639	0.148159
$838$	−17.8485	−0.616566
$839$	3.75459	0.129623	0.0648115	−	0.997898i	$-0.479355\pi$
$839$	3.75459	0.129623	0.0648115	+	0.997898i	$0.479355\pi$
$840$	18.5321	0.639418
$841$	54.1239	1.86634
$842$	−18.9872	−0.654343
$843$	−17.9338	−0.617674
$844$	−9.99447	−0.344024
$845$	−21.1817	−0.728673
$846$	21.9284	0.753914
$847$	55.7133	1.91433
$848$	13.6247	0.467873
$849$	−66.8334	−2.29371
$850$	−1.25889	−0.0431796
$851$	9.58785	0.328667
$852$	13.1711	0.451233
$853$	−34.0340	−1.16530	−0.582652	−	0.812722i	$-0.697985\pi$
$853$	−34.0340	−1.16530	−0.582652	+	0.812722i	$0.697985\pi$
$854$	−27.2031	−0.930870
$855$	−43.3440	−1.48233
$856$	−10.7161	−0.366269
$857$	−39.1928	−1.33880	−0.669400	−	0.742902i	$-0.733449\pi$
$857$	−39.1928	−1.33880	−0.669400	+	0.742902i	$0.733449\pi$
$858$	27.8650	0.951295
$859$	−30.6985	−1.04742	−0.523710	−	0.851896i	$-0.675453\pi$
$859$	−30.6985	−1.04742	−0.523710	+	0.851896i	$0.675453\pi$
$860$	−6.97982	−0.238010
$861$	18.6537	0.635717
$862$	−26.1878	−0.891959
$863$	−28.1492	−0.958210	−0.479105	−	0.877758i	$-0.659039\pi$
$863$	−28.1492	−0.958210	−0.479105	+	0.877758i	$0.659039\pi$
$864$	−0.739034	−0.0251425
$865$	46.4973	1.58096
$866$	20.0820	0.682413
$867$	38.7859	1.31724
$868$	17.7243	0.601603
$869$	84.1225	2.85366
$870$	−55.2898	−1.87450
$871$	−18.8359	−0.638230
$872$	−11.1958	−0.379139
$873$	−27.3667	−0.926223
$874$	−16.4806	−0.557463
$875$	−27.4783	−0.928936
$876$	2.47185	0.0835162
$877$	43.4027	1.46561	0.732803	−	0.680441i	$-0.238212\pi$
$877$	43.4027	1.46561	0.732803	+	0.680441i	$0.238212\pi$
$878$	−39.1621	−1.32166
$879$	37.4964	1.26472
$880$	13.7449	0.463341
$881$	−19.9136	−0.670906	−0.335453	−	0.942057i	$-0.608889\pi$
$881$	−19.9136	−0.670906	−0.335453	+	0.942057i	$0.608889\pi$
$882$	−6.29143	−0.211843
$883$	27.1883	0.914961	0.457480	−	0.889220i	$-0.348752\pi$
$883$	27.1883	0.914961	0.457480	+	0.889220i	$0.348752\pi$
$884$	1.85909	0.0625280
$885$	−29.8820	−1.00447
$886$	20.4838	0.688168
$887$	−31.7353	−1.06557	−0.532783	−	0.846252i	$-0.678854\pi$
$887$	−31.7353	−1.06557	−0.532783	+	0.846252i	$0.678854\pi$
$888$	−8.79509	−0.295144
$889$	−11.1994	−0.375617
$890$	−38.9046	−1.30408
$891$	−53.1655	−1.78111
$892$	15.3316	0.513341
$893$	51.6598	1.72873
$894$	−0.535334	−0.0179042
$895$	29.5102	0.986417
$896$	−3.05592	−0.102091
$897$	13.4023	0.447490
$898$	8.81757	0.294246
$899$	−52.8798	−1.76364
$900$	3.93588	0.131196
$901$	11.7234	0.390564
$902$	13.8352	0.460660
$903$	20.0139	0.666019
$904$	−18.7388	−0.623242
$905$	29.8218	0.991310
$906$	6.09877	0.202618
$907$	−11.3881	−0.378137	−0.189069	−	0.981964i	$-0.560547\pi$
$907$	−11.3881	−0.378137	−0.189069	+	0.981964i	$0.560547\pi$
$908$	11.8746	0.394074
$909$	−18.3231	−0.607740
$910$	−16.7855	−0.556432
$911$	7.02167	0.232638	0.116319	−	0.993212i	$-0.462890\pi$
$911$	7.02167	0.232638	0.116319	+	0.993212i	$0.462890\pi$
$912$	15.1179	0.500603
$913$	33.2499	1.10041
$914$	−13.0062	−0.430206
$915$	−53.9831	−1.78463
$916$	−14.3238	−0.473270
$917$	−54.9254	−1.81380
$918$	−0.635907	−0.0209881
$919$	17.8583	0.589092	0.294546	−	0.955637i	$-0.404832\pi$
$919$	17.8583	0.589092	0.294546	+	0.955637i	$0.404832\pi$
$920$	6.61093	0.217956
$921$	−57.3487	−1.88970
$922$	−7.97413	−0.262614
$923$	−11.9297	−0.392671
$924$	−39.4121	−1.29656
$925$	−5.39432	−0.177364
$926$	−16.7681	−0.551033
$927$	46.9317	1.54144
$928$	9.11723	0.299288
$929$	18.6170	0.610804	0.305402	−	0.952224i	$-0.401209\pi$
$929$	18.6170	0.610804	0.305402	+	0.952224i	$0.401209\pi$
$930$	35.1730	1.15337
$931$	−14.8216	−0.485758
$932$	20.7093	0.678356
$933$	29.1804	0.955323
$934$	29.8711	0.977412
$935$	11.8269	0.386781
$936$	−5.81238	−0.189984
$937$	−1.49264	−0.0487624	−0.0243812	−	0.999703i	$-0.507762\pi$
$937$	−1.49264	−0.0487624	−0.0243812	+	0.999703i	$0.507762\pi$
$938$	26.6414	0.869871
$939$	33.7593	1.10169
$940$	−20.7226	−0.675896
$941$	−33.8954	−1.10496	−0.552480	−	0.833526i	$-0.686318\pi$
$941$	−33.8954	−1.10496	−0.552480	+	0.833526i	$0.686318\pi$
$942$	3.62955	0.118257
$943$	6.65433	0.216695
$944$	4.92751	0.160377
$945$	5.74151	0.186771
$946$	14.8439	0.482618
$947$	−46.8040	−1.52093	−0.760463	−	0.649382i	$-0.775028\pi$
$947$	−46.8040	−1.52093	−0.760463	+	0.649382i	$0.775028\pi$
$948$	−37.1152	−1.20545
$949$	−2.23888	−0.0726773
$950$	9.27230	0.300833
$951$	−41.1517	−1.33444
$952$	−2.62949	−0.0852222
$953$	51.2378	1.65975	0.829877	−	0.557946i	$-0.188410\pi$
$953$	51.2378	1.65975	0.829877	+	0.557946i	$0.188410\pi$
$954$	−36.6529	−1.18668
$955$	46.9828	1.52033
$956$	−20.9530	−0.677668
$957$	117.584	3.80096
$958$	32.0686	1.03609
$959$	1.99103	0.0642936
$960$	−6.06432	−0.195725
$961$	2.63990	0.0851581
$962$	7.96616	0.256839
$963$	28.8283	0.928980
$964$	−21.7198	−0.699548
$965$	−14.8664	−0.478565
$966$	−18.9561	−0.609903
$967$	40.2837	1.29544	0.647718	−	0.761880i	$-0.275724\pi$
$967$	40.2837	1.29544	0.647718	+	0.761880i	$0.275724\pi$
$968$	−18.2313	−0.585975
$969$	13.0083	0.417886
$970$	25.8618	0.830373
$971$	50.6728	1.62617	0.813084	−	0.582146i	$-0.197787\pi$
$971$	50.6728	1.62617	0.813084	+	0.582146i	$0.197787\pi$
$972$	21.2397	0.681265
$973$	19.4353	0.623068
$974$	−8.43016	−0.270120
$975$	−7.54041	−0.241486
$976$	8.90175	0.284938
$977$	4.90163	0.156817	0.0784085	−	0.996921i	$-0.475016\pi$
$977$	4.90163	0.156817	0.0784085	+	0.996921i	$0.475016\pi$
$978$	28.8416	0.922254
$979$	82.7381	2.64432
$980$	5.94546	0.189921
$981$	30.1189	0.961623
$982$	−41.9399	−1.33835
$983$	−7.97058	−0.254222	−0.127111	−	0.991888i	$-0.540570\pi$
$983$	−7.97058	−0.254222	−0.127111	+	0.991888i	$0.540570\pi$
$984$	−6.10413	−0.194592
$985$	50.4397	1.60714
$986$	7.84498	0.249835
$987$	59.4197	1.89135
$988$	−13.6930	−0.435634
$989$	7.13952	0.227024
$990$	−36.9764	−1.17519
$991$	−32.1858	−1.02242	−0.511208	−	0.859457i	$-0.670802\pi$
$991$	−32.1858	−1.02242	−0.511208	+	0.859457i	$0.670802\pi$
$992$	−5.79999	−0.184150
$993$	−9.33128	−0.296119
$994$	16.8733	0.535188
$995$	64.4332	2.04267
$996$	−14.6700	−0.464837
$997$	20.8543	0.660462	0.330231	−	0.943900i	$-0.392873\pi$
$997$	20.8543	0.660462	0.330231	+	0.943900i	$0.392873\pi$
$998$	−28.2315	−0.893653
$999$	−2.72484	−0.0862103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.12	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.12	✓	86	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.38541 q^{3} +1.00000 q^{4} +2.54225 q^{5} +2.38541 q^{6} +3.05592 q^{7} -1.00000 q^{8} +2.69019 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.38541 q^{3} +1.00000 q^{4} +2.54225 q^{5} +2.38541 q^{6} +3.05592 q^{7} -1.00000 q^{8} +2.69019 q^{9} -2.54225 q^{10} +5.40659 q^{11} -2.38541 q^{12} +2.16059 q^{13} -3.05592 q^{14} -6.06432 q^{15} +1.00000 q^{16} +0.860456 q^{17} -2.69019 q^{18} -6.33765 q^{19} +2.54225 q^{20} -7.28963 q^{21} -5.40659 q^{22} -2.60042 q^{23} +2.38541 q^{24} +1.46305 q^{25} -2.16059 q^{26} +0.739034 q^{27} +3.05592 q^{28} -9.11723 q^{29} +6.06432 q^{30} +5.79999 q^{31} -1.00000 q^{32} -12.8969 q^{33} -0.860456 q^{34} +7.76893 q^{35} +2.69019 q^{36} -3.68703 q^{37} +6.33765 q^{38} -5.15389 q^{39} -2.54225 q^{40} -2.55894 q^{41} +7.28963 q^{42} -2.74552 q^{43} +5.40659 q^{44} +6.83913 q^{45} +2.60042 q^{46} -8.15126 q^{47} -2.38541 q^{48} +2.33866 q^{49} -1.46305 q^{50} -2.05254 q^{51} +2.16059 q^{52} +13.6247 q^{53} -0.739034 q^{54} +13.7449 q^{55} -3.05592 q^{56} +15.1179 q^{57} +9.11723 q^{58} +4.92751 q^{59} -6.06432 q^{60} +8.90175 q^{61} -5.79999 q^{62} +8.22100 q^{63} +1.00000 q^{64} +5.49276 q^{65} +12.8969 q^{66} -8.71794 q^{67} +0.860456 q^{68} +6.20308 q^{69} -7.76893 q^{70} -5.52150 q^{71} -2.69019 q^{72} -1.03624 q^{73} +3.68703 q^{74} -3.48998 q^{75} -6.33765 q^{76} +16.5221 q^{77} +5.15389 q^{78} +15.5592 q^{79} +2.54225 q^{80} -9.83346 q^{81} +2.55894 q^{82} +6.14988 q^{83} -7.28963 q^{84} +2.18750 q^{85} +2.74552 q^{86} +21.7483 q^{87} -5.40659 q^{88} +15.3032 q^{89} -6.83913 q^{90} +6.60259 q^{91} -2.60042 q^{92} -13.8354 q^{93} +8.15126 q^{94} -16.1119 q^{95} +2.38541 q^{96} -10.1728 q^{97} -2.33866 q^{98} +14.5447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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