LMFDB - Embedded newform 8049.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$0$
Dimension:	$129$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16763 q^{2} +1.00000 q^{3} +2.69860 q^{4} +2.25786 q^{5} -2.16763 q^{6} +0.531131 q^{7} -1.51431 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.16763 q^{2} +1.00000 q^{3} +2.69860 q^{4} +2.25786 q^{5} -2.16763 q^{6} +0.531131 q^{7} -1.51431 q^{8} +1.00000 q^{9} -4.89419 q^{10} -4.73685 q^{11} +2.69860 q^{12} -3.83734 q^{13} -1.15129 q^{14} +2.25786 q^{15} -2.11475 q^{16} -3.61524 q^{17} -2.16763 q^{18} -8.00178 q^{19} +6.09306 q^{20} +0.531131 q^{21} +10.2677 q^{22} +6.38319 q^{23} -1.51431 q^{24} +0.0979245 q^{25} +8.31792 q^{26} +1.00000 q^{27} +1.43331 q^{28} +2.95547 q^{29} -4.89419 q^{30} +1.64398 q^{31} +7.61260 q^{32} -4.73685 q^{33} +7.83650 q^{34} +1.19922 q^{35} +2.69860 q^{36} +11.1710 q^{37} +17.3449 q^{38} -3.83734 q^{39} -3.41909 q^{40} -2.13397 q^{41} -1.15129 q^{42} +1.89554 q^{43} -12.7829 q^{44} +2.25786 q^{45} -13.8364 q^{46} -4.06313 q^{47} -2.11475 q^{48} -6.71790 q^{49} -0.212264 q^{50} -3.61524 q^{51} -10.3555 q^{52} +6.63627 q^{53} -2.16763 q^{54} -10.6951 q^{55} -0.804297 q^{56} -8.00178 q^{57} -6.40636 q^{58} -9.87714 q^{59} +6.09306 q^{60} -10.4563 q^{61} -3.56353 q^{62} +0.531131 q^{63} -12.2718 q^{64} -8.66417 q^{65} +10.2677 q^{66} +4.82213 q^{67} -9.75611 q^{68} +6.38319 q^{69} -2.59946 q^{70} -3.53528 q^{71} -1.51431 q^{72} +14.9317 q^{73} -24.2145 q^{74} +0.0979245 q^{75} -21.5936 q^{76} -2.51589 q^{77} +8.31792 q^{78} +8.12817 q^{79} -4.77481 q^{80} +1.00000 q^{81} +4.62566 q^{82} +6.07505 q^{83} +1.43331 q^{84} -8.16271 q^{85} -4.10883 q^{86} +2.95547 q^{87} +7.17305 q^{88} -3.68571 q^{89} -4.89419 q^{90} -2.03813 q^{91} +17.2257 q^{92} +1.64398 q^{93} +8.80734 q^{94} -18.0669 q^{95} +7.61260 q^{96} +10.7150 q^{97} +14.5619 q^{98} -4.73685 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) $ 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9 + 20 * q^10 + 48 * q^11 + 158 * q^12 + 77 * q^13 + 13 * q^14 + 11 * q^15 + 212 * q^16 + 9 * q^17 + 8 * q^18 + 68 * q^19 + 19 * q^20 + 40 * q^21 + 45 * q^22 + 64 * q^23 + 18 * q^24 + 188 * q^25 + 19 * q^26 + 129 * q^27 + 69 * q^28 + 23 * q^29 + 20 * q^30 + 133 * q^31 + 24 * q^32 + 48 * q^33 + 63 * q^34 + 26 * q^35 + 158 * q^36 + 147 * q^37 + 9 * q^38 + 77 * q^39 + 58 * q^40 + 21 * q^41 + 13 * q^42 + 76 * q^43 + 110 * q^44 + 11 * q^45 + 48 * q^46 + 85 * q^47 + 212 * q^48 + 213 * q^49 + 17 * q^50 + 9 * q^51 + 139 * q^52 + 30 * q^53 + 8 * q^54 + 103 * q^55 + 19 * q^56 + 68 * q^57 + 94 * q^58 + 64 * q^59 + 19 * q^60 + 110 * q^61 - 10 * q^62 + 40 * q^63 + 288 * q^64 - 8 * q^65 + 45 * q^66 + 118 * q^67 - 15 * q^68 + 64 * q^69 + 75 * q^70 + 154 * q^71 + 18 * q^72 + 137 * q^73 + 28 * q^74 + 188 * q^75 + 156 * q^76 + 17 * q^77 + 19 * q^78 + 157 * q^79 + 2 * q^80 + 129 * q^81 + 72 * q^82 + 39 * q^83 + 69 * q^84 + 127 * q^85 + 54 * q^86 + 23 * q^87 + 97 * q^88 + 31 * q^89 + 20 * q^90 + 137 * q^91 + 82 * q^92 + 133 * q^93 + 40 * q^94 + 68 * q^95 + 24 * q^96 + 170 * q^97 - 21 * q^98 + 48 * 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 $

$2$	−2.16763	−1.53274	−0.766372	−	0.642398i	$-0.777940\pi$
$2$	−2.16763	−1.53274	−0.766372	+	0.642398i	$0.777940\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.69860	1.34930
$5$	2.25786	1.00974	0.504872	−	0.863194i	$-0.331540\pi$
$5$	2.25786	1.00974	0.504872	+	0.863194i	$0.331540\pi$
$6$	−2.16763	−0.884930
$7$	0.531131	0.200749	0.100374	−	0.994950i	$-0.467996\pi$
$7$	0.531131	0.200749	0.100374	+	0.994950i	$0.467996\pi$
$8$	−1.51431	−0.535389
$9$	1.00000	0.333333
$10$	−4.89419	−1.54768
$11$	−4.73685	−1.42821	−0.714107	−	0.700037i	$-0.753167\pi$
$11$	−4.73685	−1.42821	−0.714107	+	0.700037i	$0.753167\pi$
$12$	2.69860	0.779019
$13$	−3.83734	−1.06429	−0.532143	−	0.846654i	$-0.678613\pi$
$13$	−3.83734	−1.06429	−0.532143	+	0.846654i	$0.678613\pi$
$14$	−1.15129	−0.307696
$15$	2.25786	0.582977
$16$	−2.11475	−0.528688
$17$	−3.61524	−0.876825	−0.438413	−	0.898774i	$-0.644459\pi$
$17$	−3.61524	−0.876825	−0.438413	+	0.898774i	$0.644459\pi$
$18$	−2.16763	−0.510914
$19$	−8.00178	−1.83573	−0.917867	−	0.396887i	$-0.870090\pi$
$19$	−8.00178	−1.83573	−0.917867	+	0.396887i	$0.870090\pi$
$20$	6.09306	1.36245
$21$	0.531131	0.115902
$22$	10.2677	2.18908
$23$	6.38319	1.33099	0.665494	−	0.746404i	$-0.268221\pi$
$23$	6.38319	1.33099	0.665494	+	0.746404i	$0.268221\pi$
$24$	−1.51431	−0.309107
$25$	0.0979245	0.0195849
$26$	8.31792	1.63128
$27$	1.00000	0.192450
$28$	1.43331	0.270871
$29$	2.95547	0.548818	0.274409	−	0.961613i	$-0.411518\pi$
$29$	2.95547	0.548818	0.274409	+	0.961613i	$0.411518\pi$
$30$	−4.89419	−0.893553
$31$	1.64398	0.295267	0.147633	−	0.989042i	$-0.452834\pi$
$31$	1.64398	0.295267	0.147633	+	0.989042i	$0.452834\pi$
$32$	7.61260	1.34573
$33$	−4.73685	−0.824580
$34$	7.83650	1.34395
$35$	1.19922	0.202705
$36$	2.69860	0.449767
$37$	11.1710	1.83650	0.918248	−	0.396006i	$-0.129604\pi$
$37$	11.1710	1.83650	0.918248	+	0.396006i	$0.129604\pi$
$38$	17.3449	2.81371
$39$	−3.83734	−0.614466
$40$	−3.41909	−0.540606
$41$	−2.13397	−0.333271	−0.166635	−	0.986019i	$-0.553290\pi$
$41$	−2.13397	−0.333271	−0.166635	+	0.986019i	$0.553290\pi$
$42$	−1.15129	−0.177649
$43$	1.89554	0.289067	0.144534	−	0.989500i	$-0.453832\pi$
$43$	1.89554	0.289067	0.144534	+	0.989500i	$0.453832\pi$
$44$	−12.7829	−1.92709
$45$	2.25786	0.336582
$46$	−13.8364	−2.04006
$47$	−4.06313	−0.592668	−0.296334	−	0.955084i	$-0.595764\pi$
$47$	−4.06313	−0.592668	−0.296334	+	0.955084i	$0.595764\pi$
$48$	−2.11475	−0.305238
$49$	−6.71790	−0.959700
$50$	−0.212264	−0.0300186
$51$	−3.61524	−0.506235
$52$	−10.3555	−1.43604
$53$	6.63627	0.911561	0.455781	−	0.890092i	$-0.349360\pi$
$53$	6.63627	0.911561	0.455781	+	0.890092i	$0.349360\pi$
$54$	−2.16763	−0.294977
$55$	−10.6951	−1.44213
$56$	−0.804297	−0.107479
$57$	−8.00178	−1.05986
$58$	−6.40636	−0.841196
$59$	−9.87714	−1.28589	−0.642947	−	0.765910i	$-0.722289\pi$
$59$	−9.87714	−1.28589	−0.642947	+	0.765910i	$0.722289\pi$
$60$	6.09306	0.786611
$61$	−10.4563	−1.33879	−0.669394	−	0.742908i	$-0.733446\pi$
$61$	−10.4563	−1.33879	−0.669394	+	0.742908i	$0.733446\pi$
$62$	−3.56353	−0.452568
$63$	0.531131	0.0669163
$64$	−12.2718	−1.53397
$65$	−8.66417	−1.07466
$66$	10.2677	1.26387
$67$	4.82213	0.589116	0.294558	−	0.955634i	$-0.404828\pi$
$67$	4.82213	0.589116	0.294558	+	0.955634i	$0.404828\pi$
$68$	−9.75611	−1.18310
$69$	6.38319	0.768446
$70$	−2.59946	−0.310695
$71$	−3.53528	−0.419560	−0.209780	−	0.977749i	$-0.567275\pi$
$71$	−3.53528	−0.419560	−0.209780	+	0.977749i	$0.567275\pi$
$72$	−1.51431	−0.178463
$73$	14.9317	1.74763	0.873814	−	0.486261i	$-0.161639\pi$
$73$	14.9317	1.74763	0.873814	+	0.486261i	$0.161639\pi$
$74$	−24.2145	−2.81488
$75$	0.0979245	0.0113073
$76$	−21.5936	−2.47696
$77$	−2.51589	−0.286712
$78$	8.31792	0.941819
$79$	8.12817	0.914490	0.457245	−	0.889341i	$-0.348836\pi$
$79$	8.12817	0.914490	0.457245	+	0.889341i	$0.348836\pi$
$80$	−4.77481	−0.533840
$81$	1.00000	0.111111
$82$	4.62566	0.510818
$83$	6.07505	0.666823	0.333412	−	0.942781i	$-0.391800\pi$
$83$	6.07505	0.666823	0.333412	+	0.942781i	$0.391800\pi$
$84$	1.43331	0.156387
$85$	−8.16271	−0.885370
$86$	−4.10883	−0.443066
$87$	2.95547	0.316860
$88$	7.17305	0.764650
$89$	−3.68571	−0.390684	−0.195342	−	0.980735i	$-0.562582\pi$
$89$	−3.68571	−0.390684	−0.195342	+	0.980735i	$0.562582\pi$
$90$	−4.89419	−0.515893
$91$	−2.03813	−0.213654
$92$	17.2257	1.79590
$93$	1.64398	0.170472
$94$	8.80734	0.908407
$95$	−18.0669	−1.85362
$96$	7.61260	0.776958
$97$	10.7150	1.08794	0.543972	−	0.839104i	$-0.316920\pi$
$97$	10.7150	1.08794	0.543972	+	0.839104i	$0.316920\pi$
$98$	14.5619	1.47097
$99$	−4.73685	−0.476071
$100$	0.264259	0.0264259
$101$	−11.9847	−1.19252	−0.596259	−	0.802792i	$-0.703347\pi$
$101$	−11.9847	−1.19252	−0.596259	+	0.802792i	$0.703347\pi$
$102$	7.83650	0.775929
$103$	16.1419	1.59051	0.795256	−	0.606274i	$-0.207336\pi$
$103$	16.1419	1.59051	0.795256	+	0.606274i	$0.207336\pi$
$104$	5.81092	0.569807
$105$	1.19922	0.117032
$106$	−14.3849	−1.39719
$107$	−0.486805	−0.0470612	−0.0235306	−	0.999723i	$-0.507491\pi$
$107$	−0.486805	−0.0470612	−0.0235306	+	0.999723i	$0.507491\pi$
$108$	2.69860	0.259673
$109$	13.9354	1.33477	0.667387	−	0.744711i	$-0.267413\pi$
$109$	13.9354	1.33477	0.667387	+	0.744711i	$0.267413\pi$
$110$	23.1831	2.21042
$111$	11.1710	1.06030
$112$	−1.12321	−0.106133
$113$	6.89368	0.648503	0.324251	−	0.945971i	$-0.394888\pi$
$113$	6.89368	0.648503	0.324251	+	0.945971i	$0.394888\pi$
$114$	17.3449	1.62450
$115$	14.4123	1.34396
$116$	7.97565	0.740520
$117$	−3.83734	−0.354762
$118$	21.4100	1.97095
$119$	−1.92017	−0.176022
$120$	−3.41909	−0.312119
$121$	11.4377	1.03979
$122$	22.6653	2.05202
$123$	−2.13397	−0.192414
$124$	4.43644	0.398404
$125$	−11.0682	−0.989969
$126$	−1.15129	−0.102565
$127$	18.3412	1.62752	0.813759	−	0.581202i	$-0.197417\pi$
$127$	18.3412	1.62752	0.813759	+	0.581202i	$0.197417\pi$
$128$	11.3754	1.00545
$129$	1.89554	0.166893
$130$	18.7807	1.64717
$131$	−0.657604	−0.0574551	−0.0287275	−	0.999587i	$-0.509146\pi$
$131$	−0.657604	−0.0574551	−0.0287275	+	0.999587i	$0.509146\pi$
$132$	−12.7829	−1.11261
$133$	−4.25000	−0.368521
$134$	−10.4526	−0.902964
$135$	2.25786	0.194326
$136$	5.47459	0.469443
$137$	17.9522	1.53376	0.766879	−	0.641791i	$-0.221809\pi$
$137$	17.9522	1.53376	0.766879	+	0.641791i	$0.221809\pi$
$138$	−13.8364	−1.17783
$139$	13.7599	1.16710	0.583548	−	0.812078i	$-0.301664\pi$
$139$	13.7599	1.16710	0.583548	+	0.812078i	$0.301664\pi$
$140$	3.23622	0.273510
$141$	−4.06313	−0.342177
$142$	7.66316	0.643078
$143$	18.1769	1.52003
$144$	−2.11475	−0.176229
$145$	6.67304	0.554166
$146$	−32.3664	−2.67866
$147$	−6.71790	−0.554083
$148$	30.1460	2.47799
$149$	−0.289473	−0.0237146	−0.0118573	−	0.999930i	$-0.503774\pi$
$149$	−0.289473	−0.0237146	−0.0118573	+	0.999930i	$0.503774\pi$
$150$	−0.212264	−0.0173313
$151$	14.6661	1.19351	0.596757	−	0.802422i	$-0.296456\pi$
$151$	14.6661	1.19351	0.596757	+	0.802422i	$0.296456\pi$
$152$	12.1172	0.982832
$153$	−3.61524	−0.292275
$154$	5.45351	0.439456
$155$	3.71187	0.298144
$156$	−10.3555	−0.829100
$157$	−16.0576	−1.28154	−0.640769	−	0.767734i	$-0.721384\pi$
$157$	−16.0576	−1.28154	−0.640769	+	0.767734i	$0.721384\pi$
$158$	−17.6188	−1.40168
$159$	6.63627	0.526290
$160$	17.1882	1.35885
$161$	3.39031	0.267194
$162$	−2.16763	−0.170305
$163$	0.440922	0.0345357	0.0172678	−	0.999851i	$-0.494503\pi$
$163$	0.440922	0.0345357	0.0172678	+	0.999851i	$0.494503\pi$
$164$	−5.75875	−0.449682
$165$	−10.6951	−0.832615
$166$	−13.1684	−1.02207
$167$	−5.14807	−0.398370	−0.199185	−	0.979962i	$-0.563829\pi$
$167$	−5.14807	−0.398370	−0.199185	+	0.979962i	$0.563829\pi$
$168$	−0.804297	−0.0620528
$169$	1.72518	0.132706
$170$	17.6937	1.35704
$171$	−8.00178	−0.611912
$172$	5.11531	0.390039
$173$	−3.07074	−0.233464	−0.116732	−	0.993163i	$-0.537242\pi$
$173$	−3.07074	−0.233464	−0.116732	+	0.993163i	$0.537242\pi$
$174$	−6.40636	−0.485665
$175$	0.0520108	0.00393164
$176$	10.0173	0.755079
$177$	−9.87714	−0.742412
$178$	7.98923	0.598818
$179$	6.50294	0.486053	0.243026	−	0.970020i	$-0.421860\pi$
$179$	6.50294	0.486053	0.243026	+	0.970020i	$0.421860\pi$
$180$	6.09306	0.454150
$181$	2.75193	0.204550	0.102275	−	0.994756i	$-0.467388\pi$
$181$	2.75193	0.204550	0.102275	+	0.994756i	$0.467388\pi$
$182$	4.41791	0.327477
$183$	−10.4563	−0.772950
$184$	−9.66612	−0.712596
$185$	25.2225	1.85439
$186$	−3.56353	−0.261290
$187$	17.1249	1.25229
$188$	−10.9648	−0.799687
$189$	0.531131	0.0386341
$190$	39.1623	2.84113
$191$	24.2754	1.75651	0.878253	−	0.478196i	$-0.158709\pi$
$191$	24.2754	1.75651	0.878253	+	0.478196i	$0.158709\pi$
$192$	−12.2718	−0.885639
$193$	−19.6083	−1.41144	−0.705720	−	0.708491i	$-0.749376\pi$
$193$	−19.6083	−1.41144	−0.705720	+	0.708491i	$0.749376\pi$
$194$	−23.2261	−1.66754
$195$	−8.66417	−0.620454
$196$	−18.1289	−1.29492
$197$	−3.01867	−0.215071	−0.107536	−	0.994201i	$-0.534296\pi$
$197$	−3.01867	−0.215071	−0.107536	+	0.994201i	$0.534296\pi$
$198$	10.2677	0.729695
$199$	−0.174871	−0.0123963	−0.00619815	−	0.999981i	$-0.501973\pi$
$199$	−0.174871	−0.0123963	−0.00619815	+	0.999981i	$0.501973\pi$
$200$	−0.148288	−0.0104855
$201$	4.82213	0.340127
$202$	25.9782	1.82782
$203$	1.56974	0.110174
$204$	−9.75611	−0.683064
$205$	−4.81821	−0.336518
$206$	−34.9897	−2.43785
$207$	6.38319	0.443662
$208$	8.11502	0.562675
$209$	37.9032	2.62182
$210$	−2.59946	−0.179380
$211$	−7.47636	−0.514694	−0.257347	−	0.966319i	$-0.582848\pi$
$211$	−7.47636	−0.514694	−0.257347	+	0.966319i	$0.582848\pi$
$212$	17.9086	1.22997
$213$	−3.53528	−0.242233
$214$	1.05521	0.0721328
$215$	4.27987	0.291884
$216$	−1.51431	−0.103036
$217$	0.873167	0.0592745
$218$	−30.2068	−2.04586
$219$	14.9317	1.00899
$220$	−28.8619	−1.94587
$221$	13.8729	0.933194
$222$	−24.2145	−1.62517
$223$	11.6175	0.777965	0.388983	−	0.921245i	$-0.372827\pi$
$223$	11.6175	0.777965	0.388983	+	0.921245i	$0.372827\pi$
$224$	4.04329	0.270154
$225$	0.0979245	0.00652830
$226$	−14.9429	−0.993988
$227$	16.4496	1.09180	0.545900	−	0.837851i	$-0.316188\pi$
$227$	16.4496	1.09180	0.545900	+	0.837851i	$0.316188\pi$
$228$	−21.5936	−1.43007
$229$	5.75053	0.380006	0.190003	−	0.981784i	$-0.439150\pi$
$229$	5.75053	0.380006	0.190003	+	0.981784i	$0.439150\pi$
$230$	−31.2406	−2.05994
$231$	−2.51589	−0.165533
$232$	−4.47550	−0.293831
$233$	4.48676	0.293937	0.146969	−	0.989141i	$-0.453048\pi$
$233$	4.48676	0.293937	0.146969	+	0.989141i	$0.453048\pi$
$234$	8.31792	0.543759
$235$	−9.17396	−0.598443
$236$	−26.6545	−1.73506
$237$	8.12817	0.527981
$238$	4.16221	0.269796
$239$	−21.9022	−1.41673	−0.708366	−	0.705845i	$-0.750568\pi$
$239$	−21.9022	−1.41673	−0.708366	+	0.705845i	$0.750568\pi$
$240$	−4.77481	−0.308212
$241$	7.72184	0.497408	0.248704	−	0.968580i	$-0.419995\pi$
$241$	7.72184	0.497408	0.248704	+	0.968580i	$0.419995\pi$
$242$	−24.7927	−1.59374
$243$	1.00000	0.0641500
$244$	−28.2173	−1.80643
$245$	−15.1681	−0.969052
$246$	4.62566	0.294921
$247$	30.7056	1.95375
$248$	−2.48949	−0.158083
$249$	6.07505	0.384991
$250$	23.9917	1.51737
$251$	13.5804	0.857190	0.428595	−	0.903497i	$-0.359009\pi$
$251$	13.5804	0.857190	0.428595	+	0.903497i	$0.359009\pi$
$252$	1.43331	0.0902902
$253$	−30.2362	−1.90093
$254$	−39.7569	−2.49457
$255$	−8.16271	−0.511169
$256$	−0.114092	−0.00713076
$257$	−5.75969	−0.359280	−0.179640	−	0.983732i	$-0.557493\pi$
$257$	−5.75969	−0.359280	−0.179640	+	0.983732i	$0.557493\pi$
$258$	−4.10883	−0.255804
$259$	5.93325	0.368674
$260$	−23.3812	−1.45004
$261$	2.95547	0.182939
$262$	1.42544	0.0880639
$263$	18.1254	1.11766	0.558830	−	0.829282i	$-0.311251\pi$
$263$	18.1254	1.11766	0.558830	+	0.829282i	$0.311251\pi$
$264$	7.17305	0.441471
$265$	14.9837	0.920444
$266$	9.21240	0.564849
$267$	−3.68571	−0.225561
$268$	13.0130	0.794896
$269$	−19.0092	−1.15901	−0.579504	−	0.814969i	$-0.696754\pi$
$269$	−19.0092	−1.15901	−0.579504	+	0.814969i	$0.696754\pi$
$270$	−4.89419	−0.297851
$271$	−1.87949	−0.114171	−0.0570856	−	0.998369i	$-0.518181\pi$
$271$	−1.87949	−0.114171	−0.0570856	+	0.998369i	$0.518181\pi$
$272$	7.64534	0.463567
$273$	−2.03813	−0.123353
$274$	−38.9136	−2.35086
$275$	−0.463854	−0.0279714
$276$	17.2257	1.03686
$277$	17.3329	1.04143	0.520716	−	0.853730i	$-0.325665\pi$
$277$	17.3329	1.04143	0.520716	+	0.853730i	$0.325665\pi$
$278$	−29.8262	−1.78886
$279$	1.64398	0.0984223
$280$	−1.81599	−0.108526
$281$	10.8614	0.647939	0.323969	−	0.946068i	$-0.394982\pi$
$281$	10.8614	0.647939	0.323969	+	0.946068i	$0.394982\pi$
$282$	8.80734	0.524469
$283$	−23.5067	−1.39733	−0.698663	−	0.715451i	$-0.746221\pi$
$283$	−23.5067	−1.39733	−0.698663	+	0.715451i	$0.746221\pi$
$284$	−9.54031	−0.566113
$285$	−18.0669	−1.07019
$286$	−39.4007	−2.32981
$287$	−1.13342	−0.0669037
$288$	7.61260	0.448577
$289$	−3.93001	−0.231177
$290$	−14.4647	−0.849394
$291$	10.7150	0.628124
$292$	40.2948	2.35808
$293$	9.96542	0.582186	0.291093	−	0.956695i	$-0.405981\pi$
$293$	9.96542	0.582186	0.291093	+	0.956695i	$0.405981\pi$
$294$	14.5619	0.849267
$295$	−22.3012	−1.29843
$296$	−16.9163	−0.983240
$297$	−4.73685	−0.274860
$298$	0.627470	0.0363483
$299$	−24.4945	−1.41655
$300$	0.264259	0.0152570
$301$	1.00678	0.0580299
$302$	−31.7907	−1.82935
$303$	−11.9847	−0.688500
$304$	16.9218	0.970530
$305$	−23.6088	−1.35183
$306$	7.83650	0.447983
$307$	4.64324	0.265004	0.132502	−	0.991183i	$-0.457699\pi$
$307$	4.64324	0.265004	0.132502	+	0.991183i	$0.457699\pi$
$308$	−6.78938	−0.386861
$309$	16.1419	0.918283
$310$	−8.04594	−0.456978
$311$	20.7187	1.17485	0.587426	−	0.809278i	$-0.300141\pi$
$311$	20.7187	1.17485	0.587426	+	0.809278i	$0.300141\pi$
$312$	5.81092	0.328978
$313$	18.9171	1.06926	0.534630	−	0.845087i	$-0.320451\pi$
$313$	18.9171	1.06926	0.534630	+	0.845087i	$0.320451\pi$
$314$	34.8069	1.96427
$315$	1.19922	0.0675684
$316$	21.9347	1.23392
$317$	11.0816	0.622406	0.311203	−	0.950343i	$-0.399268\pi$
$317$	11.0816	0.622406	0.311203	+	0.950343i	$0.399268\pi$
$318$	−14.3849	−0.806668
$319$	−13.9996	−0.783829
$320$	−27.7079	−1.54892
$321$	−0.486805	−0.0271708
$322$	−7.34893	−0.409540
$323$	28.9284	1.60962
$324$	2.69860	0.149922
$325$	−0.375770	−0.0208440
$326$	−0.955754	−0.0529343
$327$	13.9354	0.770632
$328$	3.23149	0.178429
$329$	−2.15805	−0.118977
$330$	23.1831	1.27619
$331$	14.8964	0.818780	0.409390	−	0.912359i	$-0.365742\pi$
$331$	14.8964	0.818780	0.409390	+	0.912359i	$0.365742\pi$
$332$	16.3941	0.899745
$333$	11.1710	0.612165
$334$	11.1591	0.610598
$335$	10.8877	0.594857
$336$	−1.12321	−0.0612761
$337$	−22.6930	−1.23617	−0.618084	−	0.786112i	$-0.712091\pi$
$337$	−22.6930	−1.23617	−0.618084	+	0.786112i	$0.712091\pi$
$338$	−3.73955	−0.203405
$339$	6.89368	0.374413
$340$	−22.0279	−1.19463
$341$	−7.78727	−0.421704
$342$	17.3449	0.937903
$343$	−7.28601	−0.393407
$344$	−2.87044	−0.154764
$345$	14.4123	0.775934
$346$	6.65621	0.357840
$347$	−33.3068	−1.78800	−0.894002	−	0.448063i	$-0.852114\pi$
$347$	−33.3068	−1.78800	−0.894002	+	0.448063i	$0.852114\pi$
$348$	7.97565	0.427540
$349$	−23.2317	−1.24356	−0.621782	−	0.783191i	$-0.713591\pi$
$349$	−23.2317	−1.24356	−0.621782	+	0.783191i	$0.713591\pi$
$350$	−0.112740	−0.00602620
$351$	−3.83734	−0.204822
$352$	−36.0598	−1.92199
$353$	−15.8573	−0.844001	−0.422000	−	0.906596i	$-0.638672\pi$
$353$	−15.8573	−0.844001	−0.422000	+	0.906596i	$0.638672\pi$
$354$	21.4100	1.13793
$355$	−7.98216	−0.423649
$356$	−9.94625	−0.527150
$357$	−1.92017	−0.101626
$358$	−14.0960	−0.744994
$359$	1.72106	0.0908342	0.0454171	−	0.998968i	$-0.485538\pi$
$359$	1.72106	0.0908342	0.0454171	+	0.998968i	$0.485538\pi$
$360$	−3.41909	−0.180202
$361$	45.0285	2.36992
$362$	−5.96516	−0.313522
$363$	11.4377	0.600326
$364$	−5.50011	−0.288284
$365$	33.7138	1.76466
$366$	22.6653	1.18473
$367$	−5.77145	−0.301267	−0.150634	−	0.988590i	$-0.548131\pi$
$367$	−5.77145	−0.301267	−0.150634	+	0.988590i	$0.548131\pi$
$368$	−13.4989	−0.703676
$369$	−2.13397	−0.111090
$370$	−54.6729	−2.84231
$371$	3.52473	0.182995
$372$	4.43644	0.230019
$373$	29.9848	1.55255	0.776276	−	0.630393i	$-0.217106\pi$
$373$	29.9848	1.55255	0.776276	+	0.630393i	$0.217106\pi$
$374$	−37.1203	−1.91945
$375$	−11.0682	−0.571559
$376$	6.15283	0.317308
$377$	−11.3412	−0.584099
$378$	−1.15129	−0.0592162
$379$	34.1275	1.75301	0.876507	−	0.481389i	$-0.159868\pi$
$379$	34.1275	1.75301	0.876507	+	0.481389i	$0.159868\pi$
$380$	−48.7553	−2.50110
$381$	18.3412	0.939648
$382$	−52.6200	−2.69227
$383$	11.4158	0.583319	0.291659	−	0.956522i	$-0.405793\pi$
$383$	11.4158	0.583319	0.291659	+	0.956522i	$0.405793\pi$
$384$	11.3754	0.580499
$385$	−5.68052	−0.289506
$386$	42.5036	2.16337
$387$	1.89554	0.0963558
$388$	28.9155	1.46796
$389$	−15.3670	−0.779137	−0.389569	−	0.920997i	$-0.627376\pi$
$389$	−15.3670	−0.779137	−0.389569	+	0.920997i	$0.627376\pi$
$390$	18.7807	0.950997
$391$	−23.0768	−1.16704
$392$	10.1730	0.513813
$393$	−0.657604	−0.0331717
$394$	6.54335	0.329649
$395$	18.3523	0.923402
$396$	−12.7829	−0.642364
$397$	37.4007	1.87709	0.938544	−	0.345159i	$-0.112175\pi$
$397$	37.4007	1.87709	0.938544	+	0.345159i	$0.112175\pi$
$398$	0.379055	0.0190003
$399$	−4.25000	−0.212766
$400$	−0.207086	−0.0103543
$401$	−17.1098	−0.854424	−0.427212	−	0.904152i	$-0.640504\pi$
$401$	−17.1098	−0.854424	−0.427212	+	0.904152i	$0.640504\pi$
$402$	−10.4526	−0.521327
$403$	−6.30850	−0.314249
$404$	−32.3418	−1.60907
$405$	2.25786	0.112194
$406$	−3.40262	−0.168869
$407$	−52.9152	−2.62291
$408$	5.47459	0.271033
$409$	−3.82427	−0.189098	−0.0945489	−	0.995520i	$-0.530141\pi$
$409$	−3.82427	−0.189098	−0.0945489	+	0.995520i	$0.530141\pi$
$410$	10.4441	0.515796
$411$	17.9522	0.885516
$412$	43.5607	2.14608
$413$	−5.24606	−0.258142
$414$	−13.8364	−0.680020
$415$	13.7166	0.673321
$416$	−29.2122	−1.43224
$417$	13.7599	0.673824
$418$	−82.1600	−4.01858
$419$	−16.9997	−0.830491	−0.415246	−	0.909709i	$-0.636304\pi$
$419$	−16.9997	−0.830491	−0.415246	+	0.909709i	$0.636304\pi$
$420$	3.23622	0.157911
$421$	26.4399	1.28860	0.644301	−	0.764772i	$-0.277148\pi$
$421$	26.4399	1.28860	0.644301	+	0.764772i	$0.277148\pi$
$422$	16.2060	0.788894
$423$	−4.06313	−0.197556
$424$	−10.0494	−0.488040
$425$	−0.354021	−0.0171725
$426$	7.66316	0.371281
$427$	−5.55365	−0.268760
$428$	−1.31369	−0.0634998
$429$	18.1769	0.877589
$430$	−9.27715	−0.447384
$431$	34.0709	1.64114	0.820568	−	0.571549i	$-0.193657\pi$
$431$	34.0709	1.64114	0.820568	+	0.571549i	$0.193657\pi$
$432$	−2.11475	−0.101746
$433$	16.9297	0.813592	0.406796	−	0.913519i	$-0.366646\pi$
$433$	16.9297	0.813592	0.406796	+	0.913519i	$0.366646\pi$
$434$	−1.89270	−0.0908525
$435$	6.67304	0.319948
$436$	37.6062	1.80101
$437$	−51.0769	−2.44334
$438$	−32.3664	−1.54653
$439$	37.2154	1.77619	0.888096	−	0.459657i	$-0.152028\pi$
$439$	37.2154	1.77619	0.888096	+	0.459657i	$0.152028\pi$
$440$	16.1957	0.772101
$441$	−6.71790	−0.319900
$442$	−30.0713	−1.43035
$443$	−20.2444	−0.961843	−0.480921	−	0.876764i	$-0.659698\pi$
$443$	−20.2444	−0.961843	−0.480921	+	0.876764i	$0.659698\pi$
$444$	30.1460	1.43067
$445$	−8.32180	−0.394491
$446$	−25.1824	−1.19242
$447$	−0.289473	−0.0136916
$448$	−6.51793	−0.307943
$449$	−13.6299	−0.643236	−0.321618	−	0.946870i	$-0.604227\pi$
$449$	−13.6299	−0.643236	−0.321618	+	0.946870i	$0.604227\pi$
$450$	−0.212264	−0.0100062
$451$	10.1083	0.475982
$452$	18.6033	0.875026
$453$	14.6661	0.689075
$454$	−35.6566	−1.67345
$455$	−4.60181	−0.215736
$456$	12.1172	0.567438
$457$	3.51381	0.164369	0.0821847	−	0.996617i	$-0.473810\pi$
$457$	3.51381	0.164369	0.0821847	+	0.996617i	$0.473810\pi$
$458$	−12.4650	−0.582451
$459$	−3.61524	−0.168745
$460$	38.8932	1.81340
$461$	−5.62026	−0.261762	−0.130881	−	0.991398i	$-0.541781\pi$
$461$	−5.62026	−0.261762	−0.130881	+	0.991398i	$0.541781\pi$
$462$	5.45351	0.253720
$463$	11.9385	0.554830	0.277415	−	0.960750i	$-0.410522\pi$
$463$	11.9385	0.554830	0.277415	+	0.960750i	$0.410522\pi$
$464$	−6.25009	−0.290153
$465$	3.71187	0.172134
$466$	−9.72561	−0.450530
$467$	−2.39730	−0.110934	−0.0554670	−	0.998461i	$-0.517665\pi$
$467$	−2.39730	−0.110934	−0.0554670	+	0.998461i	$0.517665\pi$
$468$	−10.3555	−0.478681
$469$	2.56118	0.118264
$470$	19.8857	0.917260
$471$	−16.0576	−0.739896
$472$	14.9570	0.688454
$473$	−8.97890	−0.412850
$474$	−17.6188	−0.809260
$475$	−0.783570	−0.0359527
$476$	−5.18177	−0.237506
$477$	6.63627	0.303854
$478$	47.4757	2.17149
$479$	−7.62228	−0.348271	−0.174136	−	0.984722i	$-0.555713\pi$
$479$	−7.62228	−0.348271	−0.174136	+	0.984722i	$0.555713\pi$
$480$	17.1882	0.784530
$481$	−42.8668	−1.95456
$482$	−16.7381	−0.762398
$483$	3.39031	0.154265
$484$	30.8659	1.40300
$485$	24.1930	1.09855
$486$	−2.16763	−0.0983255
$487$	−38.5894	−1.74865	−0.874326	−	0.485339i	$-0.838696\pi$
$487$	−38.5894	−1.74865	−0.874326	+	0.485339i	$0.838696\pi$
$488$	15.8340	0.716772
$489$	0.440922	0.0199392
$490$	32.8787	1.48531
$491$	−26.5870	−1.19985	−0.599926	−	0.800055i	$-0.704803\pi$
$491$	−26.5870	−1.19985	−0.599926	+	0.800055i	$0.704803\pi$
$492$	−5.75875	−0.259624
$493$	−10.6848	−0.481217
$494$	−66.5582	−2.99459
$495$	−10.6951	−0.480711
$496$	−3.47660	−0.156104
$497$	−1.87770	−0.0842262
$498$	−13.1684	−0.590092
$499$	−1.02526	−0.0458970	−0.0229485	−	0.999737i	$-0.507305\pi$
$499$	−1.02526	−0.0458970	−0.0229485	+	0.999737i	$0.507305\pi$
$500$	−29.8686	−1.33577
$501$	−5.14807	−0.229999
$502$	−29.4373	−1.31385
$503$	−31.3546	−1.39803	−0.699016	−	0.715106i	$-0.746378\pi$
$503$	−31.3546	−1.39803	−0.699016	+	0.715106i	$0.746378\pi$
$504$	−0.804297	−0.0358262
$505$	−27.0596	−1.20414
$506$	65.5408	2.91364
$507$	1.72518	0.0766181
$508$	49.4956	2.19601
$509$	−35.4348	−1.57062	−0.785310	−	0.619102i	$-0.787497\pi$
$509$	−35.4348	−1.57062	−0.785310	+	0.619102i	$0.787497\pi$
$510$	17.6937	0.783490
$511$	7.93071	0.350834
$512$	−22.5035	−0.994525
$513$	−8.00178	−0.353287
$514$	12.4849	0.550684
$515$	36.4462	1.60601
$516$	5.11531	0.225189
$517$	19.2464	0.846456
$518$	−12.8611	−0.565083
$519$	−3.07074	−0.134790
$520$	13.1202	0.575360
$521$	18.0832	0.792237	0.396119	−	0.918199i	$-0.370357\pi$
$521$	18.0832	0.792237	0.396119	+	0.918199i	$0.370357\pi$
$522$	−6.40636	−0.280399
$523$	38.7765	1.69558	0.847790	−	0.530332i	$-0.177933\pi$
$523$	38.7765	1.69558	0.847790	+	0.530332i	$0.177933\pi$
$524$	−1.77461	−0.0775242
$525$	0.0520108	0.00226994
$526$	−39.2891	−1.71308
$527$	−5.94338	−0.258898
$528$	10.0173	0.435945
$529$	17.7451	0.771527
$530$	−32.4792	−1.41080
$531$	−9.87714	−0.428632
$532$	−11.4691	−0.497246
$533$	8.18878	0.354696
$534$	7.98923	0.345728
$535$	−1.09914	−0.0475198
$536$	−7.30219	−0.315406
$537$	6.50294	0.280623
$538$	41.2047	1.77646
$539$	31.8217	1.37066
$540$	6.09306	0.262204
$541$	−31.9870	−1.37523	−0.687613	−	0.726077i	$-0.741342\pi$
$541$	−31.9870	−1.37523	−0.687613	+	0.726077i	$0.741342\pi$
$542$	4.07404	0.174995
$543$	2.75193	0.118097
$544$	−27.5214	−1.17997
$545$	31.4643	1.34778
$546$	4.41791	0.189069
$547$	23.0664	0.986249	0.493124	−	0.869959i	$-0.335855\pi$
$547$	23.0664	0.986249	0.493124	+	0.869959i	$0.335855\pi$
$548$	48.4458	2.06950
$549$	−10.4563	−0.446263
$550$	1.00546	0.0428730
$551$	−23.6491	−1.00748
$552$	−9.66612	−0.411417
$553$	4.31712	0.183583
$554$	−37.5712	−1.59625
$555$	25.2225	1.07063
$556$	37.1324	1.57476
$557$	−19.3551	−0.820102	−0.410051	−	0.912063i	$-0.634489\pi$
$557$	−19.3551	−0.820102	−0.410051	+	0.912063i	$0.634489\pi$
$558$	−3.56353	−0.150856
$559$	−7.27384	−0.307651
$560$	−2.53605	−0.107168
$561$	17.1249	0.723012
$562$	−23.5435	−0.993124
$563$	2.63613	0.111100	0.0555498	−	0.998456i	$-0.482309\pi$
$563$	2.63613	0.111100	0.0555498	+	0.998456i	$0.482309\pi$
$564$	−10.9648	−0.461700
$565$	15.5650	0.654823
$566$	50.9536	2.14174
$567$	0.531131	0.0223054
$568$	5.35350	0.224628
$569$	4.23186	0.177409	0.0887043	−	0.996058i	$-0.471727\pi$
$569$	4.23186	0.177409	0.0887043	+	0.996058i	$0.471727\pi$
$570$	39.1623	1.64033
$571$	−14.1532	−0.592294	−0.296147	−	0.955142i	$-0.595702\pi$
$571$	−14.1532	−0.592294	−0.296147	+	0.955142i	$0.595702\pi$
$572$	49.0522	2.05098
$573$	24.2754	1.01412
$574$	2.45683	0.102546
$575$	0.625071	0.0260673
$576$	−12.2718	−0.511324
$577$	34.6321	1.44175	0.720877	−	0.693063i	$-0.243739\pi$
$577$	34.6321	1.44175	0.720877	+	0.693063i	$0.243739\pi$
$578$	8.51879	0.354335
$579$	−19.6083	−0.814895
$580$	18.0079	0.747737
$581$	3.22665	0.133864
$582$	−23.2261	−0.962753
$583$	−31.4350	−1.30190
$584$	−22.6113	−0.935661
$585$	−8.66417	−0.358219
$586$	−21.6013	−0.892341
$587$	−29.1177	−1.20182	−0.600909	−	0.799317i	$-0.705195\pi$
$587$	−29.1177	−1.20182	−0.600909	+	0.799317i	$0.705195\pi$
$588$	−18.1289	−0.747625
$589$	−13.1547	−0.542032
$590$	48.3406	1.99015
$591$	−3.01867	−0.124172
$592$	−23.6238	−0.970933
$593$	23.6028	0.969252	0.484626	−	0.874722i	$-0.338956\pi$
$593$	23.6028	0.969252	0.484626	+	0.874722i	$0.338956\pi$
$594$	10.2677	0.421290
$595$	−4.33547	−0.177737
$596$	−0.781173	−0.0319981
$597$	−0.174871	−0.00715700
$598$	53.0949	2.17121
$599$	7.73075	0.315870	0.157935	−	0.987450i	$-0.449516\pi$
$599$	7.73075	0.315870	0.157935	+	0.987450i	$0.449516\pi$
$600$	−0.148288	−0.00605383
$601$	−13.3134	−0.543067	−0.271533	−	0.962429i	$-0.587531\pi$
$601$	−13.3134	−0.543067	−0.271533	+	0.962429i	$0.587531\pi$
$602$	−2.18233	−0.0889450
$603$	4.82213	0.196372
$604$	39.5781	1.61041
$605$	25.8248	1.04993
$606$	25.9782	1.05529
$607$	−35.6968	−1.44889	−0.724445	−	0.689333i	$-0.757904\pi$
$607$	−35.6968	−1.44889	−0.724445	+	0.689333i	$0.757904\pi$
$608$	−60.9144	−2.47040
$609$	1.56974	0.0636093
$610$	51.1750	2.07201
$611$	15.5916	0.630768
$612$	−9.75611	−0.394367
$613$	−25.6418	−1.03566	−0.517830	−	0.855483i	$-0.673260\pi$
$613$	−25.6418	−1.03566	−0.517830	+	0.855483i	$0.673260\pi$
$614$	−10.0648	−0.406182
$615$	−4.81821	−0.194289
$616$	3.80983	0.153503
$617$	12.3278	0.496300	0.248150	−	0.968722i	$-0.420177\pi$
$617$	12.3278	0.496300	0.248150	+	0.968722i	$0.420177\pi$
$618$	−34.9897	−1.40749
$619$	21.6759	0.871228	0.435614	−	0.900133i	$-0.356531\pi$
$619$	21.6759	0.871228	0.435614	+	0.900133i	$0.356531\pi$
$620$	10.0168	0.402286
$621$	6.38319	0.256149
$622$	−44.9105	−1.80075
$623$	−1.95759	−0.0784293
$624$	8.11502	0.324861
$625$	−25.4800	−1.01920
$626$	−41.0052	−1.63890
$627$	37.9032	1.51371
$628$	−43.3332	−1.72918
$629$	−40.3858	−1.61029
$630$	−2.59946	−0.103565
$631$	20.2850	0.807534	0.403767	−	0.914862i	$-0.367701\pi$
$631$	20.2850	0.807534	0.403767	+	0.914862i	$0.367701\pi$
$632$	−12.3086	−0.489608
$633$	−7.47636	−0.297159
$634$	−24.0208	−0.953989
$635$	41.4118	1.64338
$636$	17.9086	0.710124
$637$	25.7789	1.02140
$638$	30.3460	1.20141
$639$	−3.53528	−0.139853
$640$	25.6841	1.01525
$641$	17.1707	0.678202	0.339101	−	0.940750i	$-0.389877\pi$
$641$	17.1707	0.678202	0.339101	+	0.940750i	$0.389877\pi$
$642$	1.05521	0.0416459
$643$	45.4817	1.79362	0.896812	−	0.442411i	$-0.145877\pi$
$643$	45.4817	1.79362	0.896812	+	0.442411i	$0.145877\pi$
$644$	9.14910	0.360525
$645$	4.27987	0.168520
$646$	−62.7059	−2.46713
$647$	0.232303	0.00913278	0.00456639	−	0.999990i	$-0.498546\pi$
$647$	0.232303	0.00913278	0.00456639	+	0.999990i	$0.498546\pi$
$648$	−1.51431	−0.0594877
$649$	46.7865	1.83653
$650$	0.814528	0.0319484
$651$	0.873167	0.0342221
$652$	1.18987	0.0465990
$653$	−22.2338	−0.870077	−0.435038	−	0.900412i	$-0.643265\pi$
$653$	−22.2338	−0.870077	−0.435038	+	0.900412i	$0.643265\pi$
$654$	−30.2068	−1.18118
$655$	−1.48478	−0.0580150
$656$	4.51282	0.176196
$657$	14.9317	0.582543
$658$	4.67785	0.182362
$659$	33.0428	1.28716	0.643582	−	0.765377i	$-0.277447\pi$
$659$	33.0428	1.28716	0.643582	+	0.765377i	$0.277447\pi$
$660$	−28.8619	−1.12345
$661$	−20.6059	−0.801477	−0.400738	−	0.916193i	$-0.631246\pi$
$661$	−20.6059	−0.801477	−0.400738	+	0.916193i	$0.631246\pi$
$662$	−32.2898	−1.25498
$663$	13.8729	0.538780
$664$	−9.19950	−0.357010
$665$	−9.59589	−0.372113
$666$	−24.2145	−0.938292
$667$	18.8654	0.730469
$668$	−13.8926	−0.537521
$669$	11.6175	0.449158
$670$	−23.6004	−0.911764
$671$	49.5298	1.91208
$672$	4.04329	0.155973
$673$	14.9342	0.575671	0.287835	−	0.957680i	$-0.407064\pi$
$673$	14.9342	0.575671	0.287835	+	0.957680i	$0.407064\pi$
$674$	49.1900	1.89473
$675$	0.0979245	0.00376912
$676$	4.65558	0.179061
$677$	35.6081	1.36853	0.684265	−	0.729234i	$-0.260123\pi$
$677$	35.6081	1.36853	0.684265	+	0.729234i	$0.260123\pi$
$678$	−14.9429	−0.573879
$679$	5.69107	0.218403
$680$	12.3609	0.474017
$681$	16.4496	0.630351
$682$	16.8799	0.646364
$683$	29.8998	1.14408	0.572042	−	0.820224i	$-0.306151\pi$
$683$	29.8998	1.14408	0.572042	+	0.820224i	$0.306151\pi$
$684$	−21.5936	−0.825653
$685$	40.5335	1.54870
$686$	15.7933	0.602992
$687$	5.75053	0.219396
$688$	−4.00860	−0.152826
$689$	−25.4656	−0.970163
$690$	−31.2406	−1.18931
$691$	12.4852	0.474961	0.237481	−	0.971392i	$-0.423678\pi$
$691$	12.4852	0.474961	0.237481	+	0.971392i	$0.423678\pi$
$692$	−8.28669	−0.315013
$693$	−2.51589	−0.0955707
$694$	72.1967	2.74055
$695$	31.0678	1.17847
$696$	−4.47550	−0.169643
$697$	7.71483	0.292220
$698$	50.3576	1.90606
$699$	4.48676	0.169705
$700$	0.140356	0.00530497
$701$	−35.1200	−1.32646	−0.663232	−	0.748414i	$-0.730816\pi$
$701$	−35.1200	−1.32646	−0.663232	+	0.748414i	$0.730816\pi$
$702$	8.31792	0.313940
$703$	−89.3877	−3.37132
$704$	58.1296	2.19084
$705$	−9.17396	−0.345511
$706$	34.3728	1.29364
$707$	−6.36542	−0.239396
$708$	−26.6545	−1.00174
$709$	16.1300	0.605774	0.302887	−	0.953026i	$-0.402049\pi$
$709$	16.1300	0.605774	0.302887	+	0.953026i	$0.402049\pi$
$710$	17.3023	0.649345
$711$	8.12817	0.304830
$712$	5.58129	0.209168
$713$	10.4938	0.392996
$714$	4.16221	0.155767
$715$	41.0409	1.53484
$716$	17.5489	0.655832
$717$	−21.9022	−0.817951
$718$	−3.73062	−0.139226
$719$	7.93689	0.295996	0.147998	−	0.988988i	$-0.452717\pi$
$719$	7.93689	0.295996	0.147998	+	0.988988i	$0.452717\pi$
$720$	−4.77481	−0.177947
$721$	8.57349	0.319293
$722$	−97.6050	−3.63248
$723$	7.72184	0.287178
$724$	7.42638	0.275999
$725$	0.289413	0.0107485
$726$	−24.7927	−0.920145
$727$	−18.1680	−0.673815	−0.336907	−	0.941538i	$-0.609381\pi$
$727$	−18.1680	−0.673815	−0.336907	+	0.941538i	$0.609381\pi$
$728$	3.08636	0.114388
$729$	1.00000	0.0370370
$730$	−73.0788	−2.70477
$731$	−6.85285	−0.253462
$732$	−28.2173	−1.04294
$733$	−0.723096	−0.0267082	−0.0133541	−	0.999911i	$-0.504251\pi$
$733$	−0.723096	−0.0267082	−0.0133541	+	0.999911i	$0.504251\pi$
$734$	12.5103	0.461765
$735$	−15.1681	−0.559483
$736$	48.5927	1.79115
$737$	−22.8417	−0.841384
$738$	4.62566	0.170273
$739$	−27.8362	−1.02397	−0.511985	−	0.858994i	$-0.671090\pi$
$739$	−27.8362	−1.02397	−0.511985	+	0.858994i	$0.671090\pi$
$740$	68.0654	2.50213
$741$	30.7056	1.12800
$742$	−7.64029	−0.280484
$743$	−23.1863	−0.850623	−0.425311	−	0.905047i	$-0.639835\pi$
$743$	−23.1863	−0.850623	−0.425311	+	0.905047i	$0.639835\pi$
$744$	−2.48949	−0.0912690
$745$	−0.653590	−0.0239457
$746$	−64.9958	−2.37966
$747$	6.07505	0.222274
$748$	46.2132	1.68972
$749$	−0.258557	−0.00944748
$750$	23.9917	0.876053
$751$	−34.7295	−1.26730	−0.633649	−	0.773621i	$-0.718444\pi$
$751$	−34.7295	−1.26730	−0.633649	+	0.773621i	$0.718444\pi$
$752$	8.59249	0.313336
$753$	13.5804	0.494899
$754$	24.5834	0.895274
$755$	33.1141	1.20514
$756$	1.43331	0.0521291
$757$	−4.96079	−0.180303	−0.0901516	−	0.995928i	$-0.528735\pi$
$757$	−4.96079	−0.180303	−0.0901516	+	0.995928i	$0.528735\pi$
$758$	−73.9757	−2.68692
$759$	−30.2362	−1.09750
$760$	27.3588	0.992410
$761$	14.6042	0.529401	0.264700	−	0.964331i	$-0.414727\pi$
$761$	14.6042	0.529401	0.264700	+	0.964331i	$0.414727\pi$
$762$	−39.7569	−1.44024
$763$	7.40155	0.267954
$764$	65.5097	2.37006
$765$	−8.16271	−0.295123
$766$	−24.7451	−0.894078
$767$	37.9020	1.36856
$768$	−0.114092	−0.00411695
$769$	8.22073	0.296447	0.148224	−	0.988954i	$-0.452644\pi$
$769$	8.22073	0.296447	0.148224	+	0.988954i	$0.452644\pi$
$770$	12.3132	0.443739
$771$	−5.75969	−0.207430
$772$	−52.9151	−1.90446
$773$	−17.9167	−0.644419	−0.322210	−	0.946668i	$-0.604426\pi$
$773$	−17.9167	−0.644419	−0.322210	+	0.946668i	$0.604426\pi$
$774$	−4.10883	−0.147689
$775$	0.160986	0.00578277
$776$	−16.2258	−0.582473
$777$	5.93325	0.212854
$778$	33.3099	1.19422
$779$	17.0756	0.611796
$780$	−23.3812	−0.837180
$781$	16.7461	0.599222
$782$	50.0219	1.78878
$783$	2.95547	0.105620
$784$	14.2067	0.507381
$785$	−36.2558	−1.29403
$786$	1.42544	0.0508437
$787$	−49.1830	−1.75318	−0.876592	−	0.481234i	$-0.840189\pi$
$787$	−49.1830	−1.75318	−0.876592	+	0.481234i	$0.840189\pi$
$788$	−8.14620	−0.290196
$789$	18.1254	0.645281
$790$	−39.7808	−1.41534
$791$	3.66145	0.130186
$792$	7.17305	0.254883
$793$	40.1243	1.42485
$794$	−81.0708	−2.87709
$795$	14.9837	0.531419
$796$	−0.471908	−0.0167263
$797$	47.7215	1.69038	0.845191	−	0.534464i	$-0.179487\pi$
$797$	47.7215	1.69038	0.845191	+	0.534464i	$0.179487\pi$
$798$	9.21240	0.326116
$799$	14.6892	0.519666
$800$	0.745461	0.0263560
$801$	−3.68571	−0.130228
$802$	37.0877	1.30961
$803$	−70.7294	−2.49599
$804$	13.0130	0.458933
$805$	7.65485	0.269798
$806$	13.6745	0.481662
$807$	−19.0092	−0.669154
$808$	18.1485	0.638461
$809$	−18.8308	−0.662057	−0.331029	−	0.943621i	$-0.607396\pi$
$809$	−18.8308	−0.662057	−0.331029	+	0.943621i	$0.607396\pi$
$810$	−4.89419	−0.171964
$811$	−14.8785	−0.522453	−0.261227	−	0.965278i	$-0.584127\pi$
$811$	−14.8785	−0.522453	−0.261227	+	0.965278i	$0.584127\pi$
$812$	4.23612	0.148659
$813$	−1.87949	−0.0659167
$814$	114.700	4.02025
$815$	0.995540	0.0348722
$816$	7.64534	0.267640
$817$	−15.1677	−0.530651
$818$	8.28958	0.289838
$819$	−2.03813	−0.0712181
$820$	−13.0024	−0.454065
$821$	−4.19602	−0.146442	−0.0732211	−	0.997316i	$-0.523328\pi$
$821$	−4.19602	−0.146442	−0.0732211	+	0.997316i	$0.523328\pi$
$822$	−38.9136	−1.35727
$823$	19.2581	0.671294	0.335647	−	0.941988i	$-0.391045\pi$
$823$	19.2581	0.671294	0.335647	+	0.941988i	$0.391045\pi$
$824$	−24.4439	−0.851543
$825$	−0.463854	−0.0161493
$826$	11.3715	0.395665
$827$	13.9802	0.486141	0.243070	−	0.970009i	$-0.421845\pi$
$827$	13.9802	0.486141	0.243070	+	0.970009i	$0.421845\pi$
$828$	17.2257	0.598634
$829$	−54.6432	−1.89784	−0.948919	−	0.315520i	$-0.897821\pi$
$829$	−54.6432	−1.89784	−0.948919	+	0.315520i	$0.897821\pi$
$830$	−29.7325	−1.03203
$831$	17.3329	0.601271
$832$	47.0910	1.63259
$833$	24.2868	0.841489
$834$	−29.8262	−1.03280
$835$	−11.6236	−0.402252
$836$	102.286	3.53763
$837$	1.64398	0.0568241
$838$	36.8491	1.27293
$839$	−19.4882	−0.672806	−0.336403	−	0.941718i	$-0.609210\pi$
$839$	−19.4882	−0.672806	−0.336403	+	0.941718i	$0.609210\pi$
$840$	−1.81599	−0.0626575
$841$	−20.2652	−0.698799
$842$	−57.3119	−1.97510
$843$	10.8614	0.374088
$844$	−20.1757	−0.694477
$845$	3.89522	0.134000
$846$	8.80734	0.302802
$847$	6.07494	0.208738
$848$	−14.0340	−0.481931
$849$	−23.5067	−0.806746
$850$	0.767385	0.0263211
$851$	71.3064	2.44435
$852$	−9.54031	−0.326846
$853$	13.6076	0.465914	0.232957	−	0.972487i	$-0.425160\pi$
$853$	13.6076	0.465914	0.232957	+	0.972487i	$0.425160\pi$
$854$	12.0382	0.411940
$855$	−18.0669	−0.617875
$856$	0.737173	0.0251961
$857$	−26.2720	−0.897434	−0.448717	−	0.893674i	$-0.648119\pi$
$857$	−26.2720	−0.897434	−0.448717	+	0.893674i	$0.648119\pi$
$858$	−39.4007	−1.34512
$859$	8.78330	0.299682	0.149841	−	0.988710i	$-0.452124\pi$
$859$	8.78330	0.299682	0.149841	+	0.988710i	$0.452124\pi$
$860$	11.5497	0.393840
$861$	−1.13342	−0.0386268
$862$	−73.8529	−2.51544
$863$	−8.65779	−0.294715	−0.147357	−	0.989083i	$-0.547077\pi$
$863$	−8.65779	−0.294715	−0.147357	+	0.989083i	$0.547077\pi$
$864$	7.61260	0.258986
$865$	−6.93329	−0.235739
$866$	−36.6974	−1.24703
$867$	−3.93001	−0.133470
$868$	2.35633	0.0799791
$869$	−38.5019	−1.30609
$870$	−14.4647	−0.490398
$871$	−18.5041	−0.626989
$872$	−21.1026	−0.714623
$873$	10.7150	0.362648
$874$	110.716	3.74501
$875$	−5.87866	−0.198735
$876$	40.2948	1.36144
$877$	−56.1032	−1.89447	−0.947236	−	0.320537i	$-0.896137\pi$
$877$	−56.1032	−1.89447	−0.947236	+	0.320537i	$0.896137\pi$
$878$	−80.6690	−2.72245
$879$	9.96542	0.336125
$880$	22.6175	0.762437
$881$	−32.8754	−1.10760	−0.553800	−	0.832649i	$-0.686823\pi$
$881$	−32.8754	−1.10760	−0.553800	+	0.832649i	$0.686823\pi$
$882$	14.5619	0.490324
$883$	28.2499	0.950685	0.475343	−	0.879801i	$-0.342324\pi$
$883$	28.2499	0.950685	0.475343	+	0.879801i	$0.342324\pi$
$884$	37.4375	1.25916
$885$	−22.3012	−0.749646
$886$	43.8824	1.47426
$887$	−23.1616	−0.777691	−0.388845	−	0.921303i	$-0.627126\pi$
$887$	−23.1616	−0.777691	−0.388845	+	0.921303i	$0.627126\pi$
$888$	−16.9163	−0.567674
$889$	9.74159	0.326722
$890$	18.0385	0.604654
$891$	−4.73685	−0.158690
$892$	31.3510	1.04971
$893$	32.5122	1.08798
$894$	0.627470	0.0209857
$895$	14.6827	0.490789
$896$	6.04184	0.201844
$897$	−24.4945	−0.817847
$898$	29.5446	0.985915
$899$	4.85873	0.162048
$900$	0.264259	0.00880864
$901$	−23.9917	−0.799280
$902$	−21.9110	−0.729558
$903$	1.00678	0.0335036
$904$	−10.4392	−0.347201
$905$	6.21348	0.206543
$906$	−31.7907	−1.05618
$907$	17.0480	0.566071	0.283036	−	0.959109i	$-0.408659\pi$
$907$	17.0480	0.566071	0.283036	+	0.959109i	$0.408659\pi$
$908$	44.3910	1.47317
$909$	−11.9847	−0.397506
$910$	9.97501	0.330668
$911$	−27.9078	−0.924626	−0.462313	−	0.886717i	$-0.652980\pi$
$911$	−27.9078	−0.924626	−0.462313	+	0.886717i	$0.652980\pi$
$912$	16.9218	0.560336
$913$	−28.7766	−0.952366
$914$	−7.61664	−0.251936
$915$	−23.6088	−0.780482
$916$	15.5184	0.512742
$917$	−0.349274	−0.0115340
$918$	7.83650	0.258643
$919$	−2.24010	−0.0738942	−0.0369471	−	0.999317i	$-0.511763\pi$
$919$	−2.24010	−0.0738942	−0.0369471	+	0.999317i	$0.511763\pi$
$920$	−21.8247	−0.719540
$921$	4.64324	0.153000
$922$	12.1826	0.401213
$923$	13.5661	0.446532
$924$	−6.78938	−0.223354
$925$	1.09391	0.0359676
$926$	−25.8783	−0.850413
$927$	16.1419	0.530171
$928$	22.4989	0.738561
$929$	27.4588	0.900893	0.450446	−	0.892804i	$-0.351265\pi$
$929$	27.4588	0.900893	0.450446	+	0.892804i	$0.351265\pi$
$930$	−8.04594	−0.263837
$931$	53.7552	1.76175
$932$	12.1080	0.396610
$933$	20.7187	0.678301
$934$	5.19646	0.170033
$935$	38.6655	1.26450
$936$	5.81092	0.189936
$937$	28.0914	0.917706	0.458853	−	0.888512i	$-0.348261\pi$
$937$	28.0914	0.917706	0.458853	+	0.888512i	$0.348261\pi$
$938$	−5.55169	−0.181269
$939$	18.9171	0.617337
$940$	−24.7569	−0.807480
$941$	−21.8288	−0.711598	−0.355799	−	0.934563i	$-0.615791\pi$
$941$	−21.8288	−0.711598	−0.355799	+	0.934563i	$0.615791\pi$
$942$	34.8069	1.13407
$943$	−13.6216	−0.443579
$944$	20.8877	0.679837
$945$	1.19922	0.0390106
$946$	19.4629	0.632793
$947$	41.3420	1.34343	0.671717	−	0.740808i	$-0.265557\pi$
$947$	41.3420	1.34343	0.671717	+	0.740808i	$0.265557\pi$
$948$	21.9347	0.712406
$949$	−57.2982	−1.85998
$950$	1.69849	0.0551062
$951$	11.0816	0.359346
$952$	2.90773	0.0942400
$953$	4.61466	0.149484	0.0747418	−	0.997203i	$-0.476187\pi$
$953$	4.61466	0.149484	0.0747418	+	0.997203i	$0.476187\pi$
$954$	−14.3849	−0.465730
$955$	54.8104	1.77362
$956$	−59.1052	−1.91160
$957$	−13.9996	−0.452544
$958$	16.5223	0.533810
$959$	9.53497	0.307900
$960$	−27.7079	−0.894270
$961$	−28.2973	−0.912817
$962$	92.9192	2.99584
$963$	−0.486805	−0.0156871
$964$	20.8382	0.671153
$965$	−44.2729	−1.42519
$966$	−7.34893	−0.236448
$967$	−13.1897	−0.424152	−0.212076	−	0.977253i	$-0.568022\pi$
$967$	−13.1897	−0.424152	−0.212076	+	0.977253i	$0.568022\pi$
$968$	−17.3203	−0.556695
$969$	28.9284	0.929314
$970$	−52.4413	−1.68379
$971$	−17.2381	−0.553197	−0.276598	−	0.960986i	$-0.589207\pi$
$971$	−17.2381	−0.553197	−0.276598	+	0.960986i	$0.589207\pi$
$972$	2.69860	0.0865577
$973$	7.30830	0.234293
$974$	83.6473	2.68023
$975$	−0.375770	−0.0120343
$976$	22.1124	0.707801
$977$	52.1429	1.66820	0.834099	−	0.551615i	$-0.185988\pi$
$977$	52.1429	1.66820	0.834099	+	0.551615i	$0.185988\pi$
$978$	−0.955754	−0.0305617
$979$	17.4586	0.557980
$980$	−40.9326	−1.30754
$981$	13.9354	0.444924
$982$	57.6306	1.83907
$983$	−9.42369	−0.300569	−0.150284	−	0.988643i	$-0.548019\pi$
$983$	−9.42369	−0.300569	−0.150284	+	0.988643i	$0.548019\pi$
$984$	3.23149	0.103016
$985$	−6.81573	−0.217167
$986$	23.1606	0.737582
$987$	−2.15805	−0.0686916
$988$	82.8621	2.63619
$989$	12.0996	0.384745
$990$	23.1831	0.736806
$991$	−38.0084	−1.20738	−0.603688	−	0.797221i	$-0.706303\pi$
$991$	−38.0084	−1.20738	−0.603688	+	0.797221i	$0.706303\pi$
$992$	12.5149	0.397350
$993$	14.8964	0.472723
$994$	4.07014	0.129097
$995$	−0.394834	−0.0125171
$996$	16.3941	0.519468
$997$	22.5462	0.714046	0.357023	−	0.934096i	$-0.383792\pi$
$997$	22.5462	0.714046	0.357023	+	0.934096i	$0.383792\pi$
$998$	2.22238	0.0703483
$999$	11.1710	0.353434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.d.1.19	✓	129

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.d.1.19	✓	129	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16763 q^{2} +1.00000 q^{3} +2.69860 q^{4} +2.25786 q^{5} -2.16763 q^{6} +0.531131 q^{7} -1.51431 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.16763 q^{2} +1.00000 q^{3} +2.69860 q^{4} +2.25786 q^{5} -2.16763 q^{6} +0.531131 q^{7} -1.51431 q^{8} +1.00000 q^{9} -4.89419 q^{10} -4.73685 q^{11} +2.69860 q^{12} -3.83734 q^{13} -1.15129 q^{14} +2.25786 q^{15} -2.11475 q^{16} -3.61524 q^{17} -2.16763 q^{18} -8.00178 q^{19} +6.09306 q^{20} +0.531131 q^{21} +10.2677 q^{22} +6.38319 q^{23} -1.51431 q^{24} +0.0979245 q^{25} +8.31792 q^{26} +1.00000 q^{27} +1.43331 q^{28} +2.95547 q^{29} -4.89419 q^{30} +1.64398 q^{31} +7.61260 q^{32} -4.73685 q^{33} +7.83650 q^{34} +1.19922 q^{35} +2.69860 q^{36} +11.1710 q^{37} +17.3449 q^{38} -3.83734 q^{39} -3.41909 q^{40} -2.13397 q^{41} -1.15129 q^{42} +1.89554 q^{43} -12.7829 q^{44} +2.25786 q^{45} -13.8364 q^{46} -4.06313 q^{47} -2.11475 q^{48} -6.71790 q^{49} -0.212264 q^{50} -3.61524 q^{51} -10.3555 q^{52} +6.63627 q^{53} -2.16763 q^{54} -10.6951 q^{55} -0.804297 q^{56} -8.00178 q^{57} -6.40636 q^{58} -9.87714 q^{59} +6.09306 q^{60} -10.4563 q^{61} -3.56353 q^{62} +0.531131 q^{63} -12.2718 q^{64} -8.66417 q^{65} +10.2677 q^{66} +4.82213 q^{67} -9.75611 q^{68} +6.38319 q^{69} -2.59946 q^{70} -3.53528 q^{71} -1.51431 q^{72} +14.9317 q^{73} -24.2145 q^{74} +0.0979245 q^{75} -21.5936 q^{76} -2.51589 q^{77} +8.31792 q^{78} +8.12817 q^{79} -4.77481 q^{80} +1.00000 q^{81} +4.62566 q^{82} +6.07505 q^{83} +1.43331 q^{84} -8.16271 q^{85} -4.10883 q^{86} +2.95547 q^{87} +7.17305 q^{88} -3.68571 q^{89} -4.89419 q^{90} -2.03813 q^{91} +17.2257 q^{92} +1.64398 q^{93} +8.80734 q^{94} -18.0669 q^{95} +7.61260 q^{96} +10.7150 q^{97} +14.5619 q^{98} -4.73685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9 + 20 * q^10 + 48 * q^11 + 158 * q^12 + 77 * q^13 + 13 * q^14 + 11 * q^15 + 212 * q^16 + 9 * q^17 + 8 * q^18 + 68 * q^19 + 19 * q^20 + 40 * q^21 + 45 * q^22 + 64 * q^23 + 18 * q^24 + 188 * q^25 + 19 * q^26 + 129 * q^27 + 69 * q^28 + 23 * q^29 + 20 * q^30 + 133 * q^31 + 24 * q^32 + 48 * q^33 + 63 * q^34 + 26 * q^35 + 158 * q^36 + 147 * q^37 + 9 * q^38 + 77 * q^39 + 58 * q^40 + 21 * q^41 + 13 * q^42 + 76 * q^43 + 110 * q^44 + 11 * q^45 + 48 * q^46 + 85 * q^47 + 212 * q^48 + 213 * q^49 + 17 * q^50 + 9 * q^51 + 139 * q^52 + 30 * q^53 + 8 * q^54 + 103 * q^55 + 19 * q^56 + 68 * q^57 + 94 * q^58 + 64 * q^59 + 19 * q^60 + 110 * q^61 - 10 * q^62 + 40 * q^63 + 288 * q^64 - 8 * q^65 + 45 * q^66 + 118 * q^67 - 15 * q^68 + 64 * q^69 + 75 * q^70 + 154 * q^71 + 18 * q^72 + 137 * q^73 + 28 * q^74 + 188 * q^75 + 156 * q^76 + 17 * q^77 + 19 * q^78 + 157 * q^79 + 2 * q^80 + 129 * q^81 + 72 * q^82 + 39 * q^83 + 69 * q^84 + 127 * q^85 + 54 * q^86 + 23 * q^87 + 97 * q^88 + 31 * q^89 + 20 * q^90 + 137 * q^91 + 82 * q^92 + 133 * q^93 + 40 * q^94 + 68 * q^95 + 24 * q^96 + 170 * q^97 - 21 * q^98 + 48 * q^99

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