LMFDB - Embedded newform 8049.2.a.d.1.15

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			1.15
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)

$f(q)$	$=$	$q-2.31049 q^{2} +1.00000 q^{3} +3.33834 q^{4} +3.79120 q^{5} -2.31049 q^{6} -2.69218 q^{7} -3.09222 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.31049 q^{2} +1.00000 q^{3} +3.33834 q^{4} +3.79120 q^{5} -2.31049 q^{6} -2.69218 q^{7} -3.09222 q^{8} +1.00000 q^{9} -8.75950 q^{10} +3.65347 q^{11} +3.33834 q^{12} +6.10249 q^{13} +6.22025 q^{14} +3.79120 q^{15} +0.467842 q^{16} +4.94376 q^{17} -2.31049 q^{18} -6.56560 q^{19} +12.6563 q^{20} -2.69218 q^{21} -8.44128 q^{22} +7.91766 q^{23} -3.09222 q^{24} +9.37317 q^{25} -14.0997 q^{26} +1.00000 q^{27} -8.98743 q^{28} -3.72495 q^{29} -8.75950 q^{30} +6.09886 q^{31} +5.10350 q^{32} +3.65347 q^{33} -11.4225 q^{34} -10.2066 q^{35} +3.33834 q^{36} +2.44966 q^{37} +15.1697 q^{38} +6.10249 q^{39} -11.7232 q^{40} -8.38647 q^{41} +6.22025 q^{42} -1.85999 q^{43} +12.1965 q^{44} +3.79120 q^{45} -18.2936 q^{46} +6.77790 q^{47} +0.467842 q^{48} +0.247853 q^{49} -21.6566 q^{50} +4.94376 q^{51} +20.3722 q^{52} +6.95940 q^{53} -2.31049 q^{54} +13.8510 q^{55} +8.32482 q^{56} -6.56560 q^{57} +8.60644 q^{58} +11.7875 q^{59} +12.6563 q^{60} -11.0895 q^{61} -14.0913 q^{62} -2.69218 q^{63} -12.7272 q^{64} +23.1357 q^{65} -8.44128 q^{66} -0.124387 q^{67} +16.5040 q^{68} +7.91766 q^{69} +23.5822 q^{70} +11.0241 q^{71} -3.09222 q^{72} -1.20803 q^{73} -5.65991 q^{74} +9.37317 q^{75} -21.9182 q^{76} -9.83580 q^{77} -14.0997 q^{78} -2.93387 q^{79} +1.77368 q^{80} +1.00000 q^{81} +19.3768 q^{82} +16.1917 q^{83} -8.98743 q^{84} +18.7428 q^{85} +4.29748 q^{86} -3.72495 q^{87} -11.2973 q^{88} -12.6427 q^{89} -8.75950 q^{90} -16.4290 q^{91} +26.4318 q^{92} +6.09886 q^{93} -15.6602 q^{94} -24.8915 q^{95} +5.10350 q^{96} -1.73075 q^{97} -0.572661 q^{98} +3.65347 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.31049	−1.63376	−0.816880	−	0.576808i	$-0.804298\pi$
$2$	−2.31049	−1.63376	−0.816880	+	0.576808i	$0.804298\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.33834	1.66917
$5$	3.79120	1.69547	0.847737	−	0.530416i	$-0.177964\pi$
$5$	3.79120	1.69547	0.847737	+	0.530416i	$0.177964\pi$
$6$	−2.31049	−0.943252
$7$	−2.69218	−1.01755	−0.508775	−	0.860900i	$-0.669901\pi$
$7$	−2.69218	−1.01755	−0.508775	+	0.860900i	$0.669901\pi$
$8$	−3.09222	−1.09326
$9$	1.00000	0.333333
$10$	−8.75950	−2.77000
$11$	3.65347	1.10156	0.550781	−	0.834650i	$-0.314330\pi$
$11$	3.65347	1.10156	0.550781	+	0.834650i	$0.314330\pi$
$12$	3.33834	0.963696
$13$	6.10249	1.69253	0.846263	−	0.532765i	$-0.178847\pi$
$13$	6.10249	1.69253	0.846263	+	0.532765i	$0.178847\pi$
$14$	6.22025	1.66243
$15$	3.79120	0.978883
$16$	0.467842	0.116960
$17$	4.94376	1.19904	0.599519	−	0.800361i	$-0.295359\pi$
$17$	4.94376	1.19904	0.599519	+	0.800361i	$0.295359\pi$
$18$	−2.31049	−0.544587
$19$	−6.56560	−1.50625	−0.753126	−	0.657877i	$-0.771455\pi$
$19$	−6.56560	−1.50625	−0.753126	+	0.657877i	$0.771455\pi$
$20$	12.6563	2.83004
$21$	−2.69218	−0.587483
$22$	−8.44128	−1.79969
$23$	7.91766	1.65095	0.825473	−	0.564442i	$-0.190909\pi$
$23$	7.91766	1.65095	0.825473	+	0.564442i	$0.190909\pi$
$24$	−3.09222	−0.631196
$25$	9.37317	1.87463
$26$	−14.0997	−2.76518
$27$	1.00000	0.192450
$28$	−8.98743	−1.69846
$29$	−3.72495	−0.691705	−0.345853	−	0.938289i	$-0.612410\pi$
$29$	−3.72495	−0.691705	−0.345853	+	0.938289i	$0.612410\pi$
$30$	−8.75950	−1.59926
$31$	6.09886	1.09539	0.547694	−	0.836679i	$-0.315506\pi$
$31$	6.09886	1.09539	0.547694	+	0.836679i	$0.315506\pi$
$32$	5.10350	0.902179
$33$	3.65347	0.635987
$34$	−11.4225	−1.95894
$35$	−10.2066	−1.72523
$36$	3.33834	0.556390
$37$	2.44966	0.402722	0.201361	−	0.979517i	$-0.435464\pi$
$37$	2.44966	0.402722	0.201361	+	0.979517i	$0.435464\pi$
$38$	15.1697	2.46085
$39$	6.10249	0.977180
$40$	−11.7232	−1.85360
$41$	−8.38647	−1.30975	−0.654873	−	0.755739i	$-0.727278\pi$
$41$	−8.38647	−1.30975	−0.654873	+	0.755739i	$0.727278\pi$
$42$	6.22025	0.959805
$43$	−1.85999	−0.283646	−0.141823	−	0.989892i	$-0.545296\pi$
$43$	−1.85999	−0.283646	−0.141823	+	0.989892i	$0.545296\pi$
$44$	12.1965	1.83869
$45$	3.79120	0.565158
$46$	−18.2936	−2.69725
$47$	6.77790	0.988658	0.494329	−	0.869275i	$-0.335414\pi$
$47$	6.77790	0.988658	0.494329	+	0.869275i	$0.335414\pi$
$48$	0.467842	0.0675272
$49$	0.247853	0.0354076
$50$	−21.6566	−3.06270
$51$	4.94376	0.692265
$52$	20.3722	2.82511
$53$	6.95940	0.955947	0.477973	−	0.878374i	$-0.341371\pi$
$53$	6.95940	0.955947	0.477973	+	0.878374i	$0.341371\pi$
$54$	−2.31049	−0.314417
$55$	13.8510	1.86767
$56$	8.32482	1.11245
$57$	−6.56560	−0.869634
$58$	8.60644	1.13008
$59$	11.7875	1.53460	0.767302	−	0.641286i	$-0.221599\pi$
$59$	11.7875	1.53460	0.767302	+	0.641286i	$0.221599\pi$
$60$	12.6563	1.63392
$61$	−11.0895	−1.41986	−0.709931	−	0.704271i	$-0.751274\pi$
$61$	−11.0895	−1.41986	−0.709931	+	0.704271i	$0.751274\pi$
$62$	−14.0913	−1.78960
$63$	−2.69218	−0.339183
$64$	−12.7272	−1.59090
$65$	23.1357	2.86963
$66$	−8.44128	−1.03905
$67$	−0.124387	−0.0151963	−0.00759814	−	0.999971i	$-0.502419\pi$
$67$	−0.124387	−0.0151963	−0.00759814	+	0.999971i	$0.502419\pi$
$68$	16.5040	2.00140
$69$	7.91766	0.953174
$70$	23.5822	2.81861
$71$	11.0241	1.30832	0.654160	−	0.756356i	$-0.273022\pi$
$71$	11.0241	1.30832	0.654160	+	0.756356i	$0.273022\pi$
$72$	−3.09222	−0.364421
$73$	−1.20803	−0.141389	−0.0706946	−	0.997498i	$-0.522522\pi$
$73$	−1.20803	−0.141389	−0.0706946	+	0.997498i	$0.522522\pi$
$74$	−5.65991	−0.657951
$75$	9.37317	1.08232
$76$	−21.9182	−2.51419
$77$	−9.83580	−1.12089
$78$	−14.0997	−1.59648
$79$	−2.93387	−0.330086	−0.165043	−	0.986286i	$-0.552776\pi$
$79$	−2.93387	−0.330086	−0.165043	+	0.986286i	$0.552776\pi$
$80$	1.77368	0.198303
$81$	1.00000	0.111111
$82$	19.3768	2.13981
$83$	16.1917	1.77727	0.888634	−	0.458617i	$-0.151655\pi$
$83$	16.1917	1.77727	0.888634	+	0.458617i	$0.151655\pi$
$84$	−8.98743	−0.980609
$85$	18.7428	2.03294
$86$	4.29748	0.463409
$87$	−3.72495	−0.399356
$88$	−11.2973	−1.20430
$89$	−12.6427	−1.34012	−0.670061	−	0.742306i	$-0.733732\pi$
$89$	−12.6427	−1.34012	−0.670061	+	0.742306i	$0.733732\pi$
$90$	−8.75950	−0.923333
$91$	−16.4290	−1.72223
$92$	26.4318	2.75571
$93$	6.09886	0.632422
$94$	−15.6602	−1.61523
$95$	−24.8915	−2.55381
$96$	5.10350	0.520873
$97$	−1.73075	−0.175731	−0.0878654	−	0.996132i	$-0.528005\pi$
$97$	−1.73075	−0.175731	−0.0878654	+	0.996132i	$0.528005\pi$
$98$	−0.572661	−0.0578475
$99$	3.65347	0.367187
$100$	31.2908	3.12908
$101$	−1.03519	−0.103006	−0.0515028	−	0.998673i	$-0.516401\pi$
$101$	−1.03519	−0.103006	−0.0515028	+	0.998673i	$0.516401\pi$
$102$	−11.4225	−1.13099
$103$	0.0359112	0.00353844	0.00176922	−	0.999998i	$-0.499437\pi$
$103$	0.0359112	0.00353844	0.00176922	+	0.999998i	$0.499437\pi$
$104$	−18.8702	−1.85038
$105$	−10.2066	−0.996062
$106$	−16.0796	−1.56179
$107$	−17.5199	−1.69371	−0.846857	−	0.531821i	$-0.821508\pi$
$107$	−17.5199	−1.69371	−0.846857	+	0.531821i	$0.821508\pi$
$108$	3.33834	0.321232
$109$	−5.02926	−0.481716	−0.240858	−	0.970560i	$-0.577429\pi$
$109$	−5.02926	−0.481716	−0.240858	+	0.970560i	$0.577429\pi$
$110$	−32.0026	−3.05132
$111$	2.44966	0.232512
$112$	−1.25952	−0.119013
$113$	−3.65332	−0.343675	−0.171838	−	0.985125i	$-0.554970\pi$
$113$	−3.65332	−0.343675	−0.171838	+	0.985125i	$0.554970\pi$
$114$	15.1697	1.42077
$115$	30.0174	2.79914
$116$	−12.4351	−1.15457
$117$	6.10249	0.564175
$118$	−27.2349	−2.50718
$119$	−13.3095	−1.22008
$120$	−11.7232	−1.07018
$121$	2.34782	0.213438
$122$	25.6221	2.31971
$123$	−8.38647	−0.756182
$124$	20.3601	1.82839
$125$	16.5795	1.48292
$126$	6.22025	0.554144
$127$	−9.56639	−0.848880	−0.424440	−	0.905456i	$-0.639529\pi$
$127$	−9.56639	−0.848880	−0.424440	+	0.905456i	$0.639529\pi$
$128$	19.1991	1.69698
$129$	−1.85999	−0.163763
$130$	−53.4548	−4.68829
$131$	4.53522	0.396244	0.198122	−	0.980177i	$-0.436516\pi$
$131$	4.53522	0.396244	0.198122	+	0.980177i	$0.436516\pi$
$132$	12.1965	1.06157
$133$	17.6758	1.53269
$134$	0.287394	0.0248271
$135$	3.79120	0.326294
$136$	−15.2872	−1.31086
$137$	−15.9895	−1.36608	−0.683040	−	0.730381i	$-0.739342\pi$
$137$	−15.9895	−1.36608	−0.683040	+	0.730381i	$0.739342\pi$
$138$	−18.2936	−1.55726
$139$	13.6800	1.16033	0.580163	−	0.814500i	$-0.302989\pi$
$139$	13.6800	1.16033	0.580163	+	0.814500i	$0.302989\pi$
$140$	−34.0731	−2.87970
$141$	6.77790	0.570802
$142$	−25.4710	−2.13748
$143$	22.2952	1.86442
$144$	0.467842	0.0389868
$145$	−14.1220	−1.17277
$146$	2.79114	0.230996
$147$	0.247853	0.0204426
$148$	8.17782	0.672212
$149$	−20.6206	−1.68931	−0.844654	−	0.535313i	$-0.820194\pi$
$149$	−20.6206	−1.68931	−0.844654	+	0.535313i	$0.820194\pi$
$150$	−21.6566	−1.76825
$151$	6.25555	0.509069	0.254535	−	0.967064i	$-0.418078\pi$
$151$	6.25555	0.509069	0.254535	+	0.967064i	$0.418078\pi$
$152$	20.3023	1.64673
$153$	4.94376	0.399679
$154$	22.7255	1.83127
$155$	23.1220	1.85720
$156$	20.3722	1.63108
$157$	11.6277	0.927992	0.463996	−	0.885837i	$-0.346415\pi$
$157$	11.6277	0.927992	0.463996	+	0.885837i	$0.346415\pi$
$158$	6.77866	0.539281
$159$	6.95940	0.551916
$160$	19.3484	1.52962
$161$	−21.3158	−1.67992
$162$	−2.31049	−0.181529
$163$	−17.5072	−1.37127	−0.685635	−	0.727945i	$-0.740476\pi$
$163$	−17.5072	−1.37127	−0.685635	+	0.727945i	$0.740476\pi$
$164$	−27.9969	−2.18619
$165$	13.8510	1.07830
$166$	−37.4107	−2.90363
$167$	−8.86064	−0.685657	−0.342828	−	0.939398i	$-0.611385\pi$
$167$	−8.86064	−0.685657	−0.342828	+	0.939398i	$0.611385\pi$
$168$	8.32482	0.642274
$169$	24.2404	1.86464
$170$	−43.3049	−3.32133
$171$	−6.56560	−0.502084
$172$	−6.20929	−0.473454
$173$	9.98193	0.758912	0.379456	−	0.925210i	$-0.376111\pi$
$173$	9.98193	0.758912	0.379456	+	0.925210i	$0.376111\pi$
$174$	8.60644	0.652452
$175$	−25.2343	−1.90753
$176$	1.70924	0.128839
$177$	11.7875	0.886004
$178$	29.2107	2.18944
$179$	−1.15664	−0.0864512	−0.0432256	−	0.999065i	$-0.513763\pi$
$179$	−1.15664	−0.0864512	−0.0432256	+	0.999065i	$0.513763\pi$
$180$	12.6563	0.943346
$181$	3.13086	0.232715	0.116358	−	0.993207i	$-0.462878\pi$
$181$	3.13086	0.232715	0.116358	+	0.993207i	$0.462878\pi$
$182$	37.9590	2.81371
$183$	−11.0895	−0.819758
$184$	−24.4831	−1.80492
$185$	9.28716	0.682805
$186$	−14.0913	−1.03323
$187$	18.0619	1.32081
$188$	22.6269	1.65024
$189$	−2.69218	−0.195828
$190$	57.5114	4.17231
$191$	0.249116	0.0180254	0.00901270	−	0.999959i	$-0.497131\pi$
$191$	0.249116	0.0180254	0.00901270	+	0.999959i	$0.497131\pi$
$192$	−12.7272	−0.918509
$193$	17.1017	1.23101	0.615503	−	0.788135i	$-0.288953\pi$
$193$	17.1017	1.23101	0.615503	+	0.788135i	$0.288953\pi$
$194$	3.99887	0.287102
$195$	23.1357	1.65678
$196$	0.827418	0.0591013
$197$	−27.4548	−1.95607	−0.978035	−	0.208439i	$-0.933162\pi$
$197$	−27.4548	−1.95607	−0.978035	+	0.208439i	$0.933162\pi$
$198$	−8.44128	−0.599896
$199$	11.7742	0.834651	0.417326	−	0.908757i	$-0.362967\pi$
$199$	11.7742	0.834651	0.417326	+	0.908757i	$0.362967\pi$
$200$	−28.9839	−2.04947
$201$	−0.124387	−0.00877357
$202$	2.39180	0.168286
$203$	10.0282	0.703845
$204$	16.5040	1.15551
$205$	−31.7947	−2.22064
$206$	−0.0829723	−0.00578095
$207$	7.91766	0.550315
$208$	2.85500	0.197959
$209$	−23.9872	−1.65923
$210$	23.5822	1.62733
$211$	7.21090	0.496419	0.248210	−	0.968706i	$-0.420158\pi$
$211$	7.21090	0.496419	0.248210	+	0.968706i	$0.420158\pi$
$212$	23.2329	1.59564
$213$	11.0241	0.755359
$214$	40.4795	2.76712
$215$	−7.05159	−0.480915
$216$	−3.09222	−0.210399
$217$	−16.4192	−1.11461
$218$	11.6200	0.787008
$219$	−1.20803	−0.0816311
$220$	46.2394	3.11746
$221$	30.1692	2.02940
$222$	−5.65991	−0.379868
$223$	−18.8270	−1.26075	−0.630375	−	0.776291i	$-0.717099\pi$
$223$	−18.8270	−1.26075	−0.630375	+	0.776291i	$0.717099\pi$
$224$	−13.7395	−0.918012
$225$	9.37317	0.624878
$226$	8.44094	0.561483
$227$	8.71210	0.578243	0.289121	−	0.957292i	$-0.406637\pi$
$227$	8.71210	0.578243	0.289121	+	0.957292i	$0.406637\pi$
$228$	−21.9182	−1.45157
$229$	−9.92460	−0.655836	−0.327918	−	0.944706i	$-0.606347\pi$
$229$	−9.92460	−0.655836	−0.327918	+	0.944706i	$0.606347\pi$
$230$	−69.3547	−4.57312
$231$	−9.83580	−0.647148
$232$	11.5184	0.756217
$233$	22.9477	1.50335	0.751677	−	0.659532i	$-0.229245\pi$
$233$	22.9477	1.50335	0.751677	+	0.659532i	$0.229245\pi$
$234$	−14.0997	−0.921727
$235$	25.6963	1.67625
$236$	39.3508	2.56152
$237$	−2.93387	−0.190575
$238$	30.7514	1.99332
$239$	−8.60455	−0.556582	−0.278291	−	0.960497i	$-0.589768\pi$
$239$	−8.60455	−0.556582	−0.278291	+	0.960497i	$0.589768\pi$
$240$	1.77368	0.114491
$241$	12.1191	0.780663	0.390331	−	0.920674i	$-0.372360\pi$
$241$	12.1191	0.780663	0.390331	+	0.920674i	$0.372360\pi$
$242$	−5.42461	−0.348707
$243$	1.00000	0.0641500
$244$	−37.0205	−2.36999
$245$	0.939659	0.0600326
$246$	19.3768	1.23542
$247$	−40.0665	−2.54937
$248$	−18.8590	−1.19755
$249$	16.1917	1.02611
$250$	−38.3068	−2.42273
$251$	−2.03951	−0.128733	−0.0643663	−	0.997926i	$-0.520503\pi$
$251$	−2.03951	−0.128733	−0.0643663	+	0.997926i	$0.520503\pi$
$252$	−8.98743	−0.566155
$253$	28.9269	1.81862
$254$	22.1030	1.38687
$255$	18.7428	1.17372
$256$	−18.9048	−1.18155
$257$	−8.80741	−0.549391	−0.274696	−	0.961531i	$-0.588577\pi$
$257$	−8.80741	−0.549391	−0.274696	+	0.961531i	$0.588577\pi$
$258$	4.29748	0.267550
$259$	−6.59495	−0.409790
$260$	77.2350	4.78991
$261$	−3.72495	−0.230568
$262$	−10.4786	−0.647368
$263$	−25.8380	−1.59324	−0.796618	−	0.604483i	$-0.793380\pi$
$263$	−25.8380	−1.59324	−0.796618	+	0.604483i	$0.793380\pi$
$264$	−11.2973	−0.695302
$265$	26.3844	1.62078
$266$	−40.8397	−2.50404
$267$	−12.6427	−0.773720
$268$	−0.415246	−0.0253652
$269$	−16.8777	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$269$	−16.8777	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$270$	−8.75950	−0.533086
$271$	11.8235	0.718227	0.359114	−	0.933294i	$-0.383079\pi$
$271$	11.8235	0.718227	0.359114	+	0.933294i	$0.383079\pi$
$272$	2.31290	0.140240
$273$	−16.4290	−0.994330
$274$	36.9436	2.23185
$275$	34.2446	2.06502
$276$	26.4318	1.59101
$277$	−0.656720	−0.0394585	−0.0197292	−	0.999805i	$-0.506280\pi$
$277$	−0.656720	−0.0394585	−0.0197292	+	0.999805i	$0.506280\pi$
$278$	−31.6075	−1.89569
$279$	6.09886	0.365129
$280$	31.5610	1.88613
$281$	17.4366	1.04018	0.520091	−	0.854111i	$-0.325898\pi$
$281$	17.4366	1.04018	0.520091	+	0.854111i	$0.325898\pi$
$282$	−15.6602	−0.932554
$283$	31.7051	1.88467	0.942337	−	0.334666i	$-0.108624\pi$
$283$	31.7051	1.88467	0.942337	+	0.334666i	$0.108624\pi$
$284$	36.8022	2.18381
$285$	−24.8915	−1.47444
$286$	−51.5128	−3.04602
$287$	22.5779	1.33273
$288$	5.10350	0.300726
$289$	7.44074	0.437691
$290$	32.6287	1.91602
$291$	−1.73075	−0.101458
$292$	−4.03282	−0.236003
$293$	−7.92224	−0.462822	−0.231411	−	0.972856i	$-0.574334\pi$
$293$	−7.92224	−0.462822	−0.231411	+	0.972856i	$0.574334\pi$
$294$	−0.572661	−0.0333982
$295$	44.6888	2.60188
$296$	−7.57490	−0.440282
$297$	3.65347	0.211996
$298$	47.6437	2.75992
$299$	48.3174	2.79427
$300$	31.2908	1.80658
$301$	5.00744	0.288624
$302$	−14.4533	−0.831697
$303$	−1.03519	−0.0594703
$304$	−3.07166	−0.176172
$305$	−42.0424	−2.40734
$306$	−11.4225	−0.652980
$307$	20.6559	1.17889	0.589447	−	0.807807i	$-0.299346\pi$
$307$	20.6559	1.17889	0.589447	+	0.807807i	$0.299346\pi$
$308$	−32.8353	−1.87096
$309$	0.0359112	0.00204292
$310$	−53.4230	−3.03422
$311$	12.5969	0.714305	0.357152	−	0.934046i	$-0.383748\pi$
$311$	12.5969	0.714305	0.357152	+	0.934046i	$0.383748\pi$
$312$	−18.8702	−1.06832
$313$	4.32504	0.244466	0.122233	−	0.992501i	$-0.460995\pi$
$313$	4.32504	0.244466	0.122233	+	0.992501i	$0.460995\pi$
$314$	−26.8657	−1.51612
$315$	−10.2066	−0.575077
$316$	−9.79426	−0.550970
$317$	−16.6217	−0.933568	−0.466784	−	0.884371i	$-0.654588\pi$
$317$	−16.6217	−0.933568	−0.466784	+	0.884371i	$0.654588\pi$
$318$	−16.0796	−0.901699
$319$	−13.6090	−0.761956
$320$	−48.2514	−2.69734
$321$	−17.5199	−0.977866
$322$	49.2498	2.74458
$323$	−32.4587	−1.80605
$324$	3.33834	0.185463
$325$	57.1997	3.17287
$326$	40.4502	2.24033
$327$	−5.02926	−0.278119
$328$	25.9328	1.43190
$329$	−18.2474	−1.00601
$330$	−32.0026	−1.76168
$331$	−20.6548	−1.13529	−0.567646	−	0.823273i	$-0.692146\pi$
$331$	−20.6548	−1.13529	−0.567646	+	0.823273i	$0.692146\pi$
$332$	54.0534	2.96656
$333$	2.44966	0.134241
$334$	20.4724	1.12020
$335$	−0.471575	−0.0257649
$336$	−1.25952	−0.0687122
$337$	−22.3721	−1.21869	−0.609343	−	0.792907i	$-0.708567\pi$
$337$	−22.3721	−1.21869	−0.609343	+	0.792907i	$0.708567\pi$
$338$	−56.0070	−3.04638
$339$	−3.65332	−0.198421
$340$	62.5697	3.39332
$341$	22.2820	1.20664
$342$	15.1697	0.820284
$343$	18.1780	0.981521
$344$	5.75150	0.310100
$345$	30.0174	1.61608
$346$	−23.0631	−1.23988
$347$	−22.5262	−1.20927	−0.604635	−	0.796502i	$-0.706681\pi$
$347$	−22.5262	−1.20927	−0.604635	+	0.796502i	$0.706681\pi$
$348$	−12.4351	−0.666594
$349$	10.3554	0.554310	0.277155	−	0.960825i	$-0.410609\pi$
$349$	10.3554	0.554310	0.277155	+	0.960825i	$0.410609\pi$
$350$	58.3035	3.11645
$351$	6.10249	0.325727
$352$	18.6455	0.993806
$353$	1.39428	0.0742098	0.0371049	−	0.999311i	$-0.488186\pi$
$353$	1.39428	0.0742098	0.0371049	+	0.999311i	$0.488186\pi$
$354$	−27.2349	−1.44752
$355$	41.7945	2.21822
$356$	−42.2056	−2.23689
$357$	−13.3095	−0.704414
$358$	2.67239	0.141240
$359$	−27.8828	−1.47160	−0.735799	−	0.677200i	$-0.763193\pi$
$359$	−27.8828	−1.47160	−0.735799	+	0.677200i	$0.763193\pi$
$360$	−11.7232	−0.617867
$361$	24.1071	1.26879
$362$	−7.23382	−0.380201
$363$	2.34782	0.123229
$364$	−54.8457	−2.87470
$365$	−4.57988	−0.239722
$366$	25.6221	1.33929
$367$	14.5964	0.761924	0.380962	−	0.924591i	$-0.375593\pi$
$367$	14.5964	0.761924	0.380962	+	0.924591i	$0.375593\pi$
$368$	3.70421	0.193095
$369$	−8.38647	−0.436582
$370$	−21.4578	−1.11554
$371$	−18.7360	−0.972724
$372$	20.3601	1.05562
$373$	−30.7963	−1.59457	−0.797285	−	0.603603i	$-0.793731\pi$
$373$	−30.7963	−1.59457	−0.797285	+	0.603603i	$0.793731\pi$
$374$	−41.7316	−2.15789
$375$	16.5795	0.856164
$376$	−20.9587	−1.08086
$377$	−22.7315	−1.17073
$378$	6.22025	0.319935
$379$	−33.5665	−1.72419	−0.862097	−	0.506743i	$-0.830849\pi$
$379$	−33.5665	−1.72419	−0.862097	+	0.506743i	$0.830849\pi$
$380$	−83.0962	−4.26275
$381$	−9.56639	−0.490101
$382$	−0.575579	−0.0294492
$383$	31.6631	1.61791	0.808953	−	0.587873i	$-0.200034\pi$
$383$	31.6631	1.61791	0.808953	+	0.587873i	$0.200034\pi$
$384$	19.1991	0.979750
$385$	−37.2895	−1.90045
$386$	−39.5132	−2.01117
$387$	−1.85999	−0.0945487
$388$	−5.77783	−0.293325
$389$	−2.77471	−0.140684	−0.0703418	−	0.997523i	$-0.522409\pi$
$389$	−2.77471	−0.140684	−0.0703418	+	0.997523i	$0.522409\pi$
$390$	−53.4548	−2.70679
$391$	39.1430	1.97955
$392$	−0.766415	−0.0387098
$393$	4.53522	0.228772
$394$	63.4338	3.19575
$395$	−11.1229	−0.559653
$396$	12.1965	0.612898
$397$	7.62610	0.382743	0.191372	−	0.981518i	$-0.438706\pi$
$397$	7.62610	0.382743	0.191372	+	0.981518i	$0.438706\pi$
$398$	−27.2041	−1.36362
$399$	17.6758	0.884896
$400$	4.38516	0.219258
$401$	−8.52869	−0.425903	−0.212951	−	0.977063i	$-0.568308\pi$
$401$	−8.52869	−0.425903	−0.212951	+	0.977063i	$0.568308\pi$
$402$	0.287394	0.0143339
$403$	37.2182	1.85397
$404$	−3.45583	−0.171934
$405$	3.79120	0.188386
$406$	−23.1701	−1.14991
$407$	8.94977	0.443623
$408$	−15.2872	−0.756828
$409$	−7.19718	−0.355878	−0.177939	−	0.984042i	$-0.556943\pi$
$409$	−7.19718	−0.355878	−0.177939	+	0.984042i	$0.556943\pi$
$410$	73.4613	3.62799
$411$	−15.9895	−0.788706
$412$	0.119884	0.00590625
$413$	−31.7342	−1.56154
$414$	−18.2936	−0.899083
$415$	61.3859	3.01331
$416$	31.1440	1.52696
$417$	13.6800	0.669915
$418$	55.4220	2.71078
$419$	6.24769	0.305219	0.152610	−	0.988287i	$-0.451232\pi$
$419$	6.24769	0.305219	0.152610	+	0.988287i	$0.451232\pi$
$420$	−34.0731	−1.66260
$421$	−35.7879	−1.74419	−0.872097	−	0.489334i	$-0.837240\pi$
$421$	−35.7879	−1.74419	−0.872097	+	0.489334i	$0.837240\pi$
$422$	−16.6607	−0.811029
$423$	6.77790	0.329553
$424$	−21.5200	−1.04510
$425$	46.3387	2.24776
$426$	−25.4710	−1.23408
$427$	29.8549	1.44478
$428$	−58.4874	−2.82710
$429$	22.2952	1.07642
$430$	16.2926	0.785699
$431$	3.60942	0.173860	0.0869299	−	0.996214i	$-0.472294\pi$
$431$	3.60942	0.173860	0.0869299	+	0.996214i	$0.472294\pi$
$432$	0.467842	0.0225091
$433$	−20.0005	−0.961162	−0.480581	−	0.876950i	$-0.659574\pi$
$433$	−20.0005	−0.961162	−0.480581	+	0.876950i	$0.659574\pi$
$434$	37.9364	1.82101
$435$	−14.1220	−0.677099
$436$	−16.7894	−0.804067
$437$	−51.9841	−2.48674
$438$	2.79114	0.133366
$439$	−35.6925	−1.70351	−0.851755	−	0.523940i	$-0.824462\pi$
$439$	−35.6925	−1.70351	−0.851755	+	0.523940i	$0.824462\pi$
$440$	−42.8304	−2.04186
$441$	0.247853	0.0118025
$442$	−69.7055	−3.31555
$443$	−22.5323	−1.07054	−0.535270	−	0.844681i	$-0.679790\pi$
$443$	−22.5323	−1.07054	−0.535270	+	0.844681i	$0.679790\pi$
$444$	8.17782	0.388102
$445$	−47.9309	−2.27214
$446$	43.4996	2.05976
$447$	−20.6206	−0.975322
$448$	34.2641	1.61882
$449$	9.05975	0.427556	0.213778	−	0.976882i	$-0.431423\pi$
$449$	9.05975	0.427556	0.213778	+	0.976882i	$0.431423\pi$
$450$	−21.6566	−1.02090
$451$	−30.6397	−1.44277
$452$	−12.1960	−0.573653
$453$	6.25555	0.293911
$454$	−20.1292	−0.944709
$455$	−62.2856	−2.92000
$456$	20.3023	0.950740
$457$	−29.7019	−1.38940	−0.694699	−	0.719301i	$-0.744462\pi$
$457$	−29.7019	−1.38940	−0.694699	+	0.719301i	$0.744462\pi$
$458$	22.9306	1.07148
$459$	4.94376	0.230755
$460$	100.208	4.67224
$461$	7.97068	0.371231	0.185616	−	0.982622i	$-0.440572\pi$
$461$	7.97068	0.371231	0.185616	+	0.982622i	$0.440572\pi$
$462$	22.7255	1.05728
$463$	−39.9371	−1.85603	−0.928017	−	0.372538i	$-0.878488\pi$
$463$	−39.9371	−1.85603	−0.928017	+	0.372538i	$0.878488\pi$
$464$	−1.74269	−0.0809022
$465$	23.1220	1.07226
$466$	−53.0203	−2.45612
$467$	−15.0693	−0.697323	−0.348661	−	0.937249i	$-0.613364\pi$
$467$	−15.0693	−0.697323	−0.348661	+	0.937249i	$0.613364\pi$
$468$	20.3722	0.941705
$469$	0.334872	0.0154630
$470$	−59.3710	−2.73858
$471$	11.6277	0.535777
$472$	−36.4496	−1.67773
$473$	−6.79542	−0.312454
$474$	6.77866	0.311354
$475$	−61.5404	−2.82367
$476$	−44.4317	−2.03652
$477$	6.95940	0.318649
$478$	19.8807	0.909322
$479$	27.5955	1.26087	0.630436	−	0.776241i	$-0.282876\pi$
$479$	27.5955	1.26087	0.630436	+	0.776241i	$0.282876\pi$
$480$	19.3484	0.883127
$481$	14.9490	0.681618
$482$	−28.0011	−1.27542
$483$	−21.3158	−0.969902
$484$	7.83783	0.356265
$485$	−6.56160	−0.297947
$486$	−2.31049	−0.104806
$487$	37.5697	1.70245	0.851224	−	0.524803i	$-0.175861\pi$
$487$	37.5697	1.70245	0.851224	+	0.524803i	$0.175861\pi$
$488$	34.2911	1.55228
$489$	−17.5072	−0.791704
$490$	−2.17107	−0.0980789
$491$	−17.8199	−0.804201	−0.402100	−	0.915596i	$-0.631720\pi$
$491$	−17.8199	−0.804201	−0.402100	+	0.915596i	$0.631720\pi$
$492$	−27.9969	−1.26220
$493$	−18.4152	−0.829381
$494$	92.5730	4.16506
$495$	13.8510	0.622557
$496$	2.85330	0.128117
$497$	−29.6789	−1.33128
$498$	−37.4107	−1.67641
$499$	−6.24847	−0.279720	−0.139860	−	0.990171i	$-0.544665\pi$
$499$	−6.24847	−0.279720	−0.139860	+	0.990171i	$0.544665\pi$
$500$	55.3482	2.47525
$501$	−8.86064	−0.395864
$502$	4.71226	0.210318
$503$	−5.72708	−0.255358	−0.127679	−	0.991816i	$-0.540753\pi$
$503$	−5.72708	−0.255358	−0.127679	+	0.991816i	$0.540753\pi$
$504$	8.32482	0.370817
$505$	−3.92462	−0.174643
$506$	−66.8352	−2.97119
$507$	24.2404	1.07655
$508$	−31.9359	−1.41693
$509$	−35.7323	−1.58380	−0.791902	−	0.610648i	$-0.790909\pi$
$509$	−35.7323	−1.58380	−0.791902	+	0.610648i	$0.790909\pi$
$510$	−43.3049	−1.91757
$511$	3.25224	0.143871
$512$	5.28097	0.233388
$513$	−6.56560	−0.289878
$514$	20.3494	0.897573
$515$	0.136146	0.00599933
$516$	−6.20929	−0.273349
$517$	24.7628	1.08907
$518$	15.2375	0.669498
$519$	9.98193	0.438158
$520$	−71.5407	−3.13727
$521$	23.4573	1.02768	0.513841	−	0.857886i	$-0.328222\pi$
$521$	23.4573	1.02768	0.513841	+	0.857886i	$0.328222\pi$
$522$	8.60644	0.376694
$523$	7.27524	0.318124	0.159062	−	0.987269i	$-0.449153\pi$
$523$	7.27524	0.318124	0.159062	+	0.987269i	$0.449153\pi$
$524$	15.1401	0.661399
$525$	−25.2343	−1.10131
$526$	59.6982	2.60297
$527$	30.1513	1.31341
$528$	1.70924	0.0743853
$529$	39.6893	1.72562
$530$	−60.9609	−2.64797
$531$	11.7875	0.511535
$532$	59.0078	2.55831
$533$	−51.1783	−2.21678
$534$	29.2107	1.26407
$535$	−66.4214	−2.87165
$536$	0.384631	0.0166135
$537$	−1.15664	−0.0499126
$538$	38.9957	1.68122
$539$	0.905523	0.0390036
$540$	12.6563	0.544641
$541$	−8.07212	−0.347047	−0.173524	−	0.984830i	$-0.555515\pi$
$541$	−8.07212	−0.347047	−0.173524	+	0.984830i	$0.555515\pi$
$542$	−27.3180	−1.17341
$543$	3.13086	0.134358
$544$	25.2304	1.08175
$545$	−19.0669	−0.816737
$546$	37.9590	1.62450
$547$	−22.9487	−0.981217	−0.490609	−	0.871380i	$-0.663225\pi$
$547$	−22.9487	−0.981217	−0.490609	+	0.871380i	$0.663225\pi$
$548$	−53.3786	−2.28022
$549$	−11.0895	−0.473287
$550$	−79.1216	−3.37375
$551$	24.4565	1.04188
$552$	−24.4831	−1.04207
$553$	7.89852	0.335879
$554$	1.51734	0.0644656
$555$	9.28716	0.394218
$556$	45.6687	1.93678
$557$	−1.60182	−0.0678711	−0.0339355	−	0.999424i	$-0.510804\pi$
$557$	−1.60182	−0.0678711	−0.0339355	+	0.999424i	$0.510804\pi$
$558$	−14.0913	−0.596533
$559$	−11.3506	−0.480078
$560$	−4.77507	−0.201784
$561$	18.0619	0.762572
$562$	−40.2871	−1.69941
$563$	−12.4038	−0.522756	−0.261378	−	0.965237i	$-0.584177\pi$
$563$	−12.4038	−0.522756	−0.261378	+	0.965237i	$0.584177\pi$
$564$	22.6269	0.952766
$565$	−13.8504	−0.582693
$566$	−73.2542	−3.07910
$567$	−2.69218	−0.113061
$568$	−34.0889	−1.43034
$569$	−3.28241	−0.137606	−0.0688029	−	0.997630i	$-0.521918\pi$
$569$	−3.28241	−0.137606	−0.0688029	+	0.997630i	$0.521918\pi$
$570$	57.5114	2.40889
$571$	31.3906	1.31365	0.656827	−	0.754041i	$-0.271898\pi$
$571$	31.3906	1.31365	0.656827	+	0.754041i	$0.271898\pi$
$572$	74.4291	3.11204
$573$	0.249116	0.0104070
$574$	−52.1659	−2.17736
$575$	74.2135	3.09492
$576$	−12.7272	−0.530301
$577$	−8.24257	−0.343143	−0.171571	−	0.985172i	$-0.554884\pi$
$577$	−8.24257	−0.343143	−0.171571	+	0.985172i	$0.554884\pi$
$578$	−17.1917	−0.715081
$579$	17.1017	0.710721
$580$	−47.1441	−1.95755
$581$	−43.5910	−1.80846
$582$	3.99887	0.165758
$583$	25.4259	1.05303
$584$	3.73549	0.154576
$585$	23.1357	0.956545
$586$	18.3042	0.756141
$587$	24.0579	0.992976	0.496488	−	0.868043i	$-0.334623\pi$
$587$	24.0579	0.992976	0.496488	+	0.868043i	$0.334623\pi$
$588$	0.827418	0.0341221
$589$	−40.0426	−1.64993
$590$	−103.253	−4.25085
$591$	−27.4548	−1.12934
$592$	1.14606	0.0471026
$593$	20.0588	0.823718	0.411859	−	0.911248i	$-0.364880\pi$
$593$	20.0588	0.823718	0.411859	+	0.911248i	$0.364880\pi$
$594$	−8.44128	−0.346350
$595$	−50.4589	−2.06861
$596$	−68.8387	−2.81974
$597$	11.7742	0.481886
$598$	−111.637	−4.56516
$599$	−13.2564	−0.541643	−0.270822	−	0.962630i	$-0.587295\pi$
$599$	−13.2564	−0.541643	−0.270822	+	0.962630i	$0.587295\pi$
$600$	−28.9839	−1.18326
$601$	44.2255	1.80400	0.901998	−	0.431741i	$-0.142101\pi$
$601$	44.2255	1.80400	0.901998	+	0.431741i	$0.142101\pi$
$602$	−11.5696	−0.471542
$603$	−0.124387	−0.00506543
$604$	20.8832	0.849723
$605$	8.90105	0.361879
$606$	2.39180	0.0971601
$607$	−29.0524	−1.17920	−0.589600	−	0.807696i	$-0.700714\pi$
$607$	−29.0524	−1.17920	−0.589600	+	0.807696i	$0.700714\pi$
$608$	−33.5075	−1.35891
$609$	10.0282	0.406365
$610$	97.1383	3.93302
$611$	41.3621	1.67333
$612$	16.5040	0.667133
$613$	−12.0547	−0.486885	−0.243443	−	0.969915i	$-0.578277\pi$
$613$	−12.0547	−0.486885	−0.243443	+	0.969915i	$0.578277\pi$
$614$	−47.7252	−1.92603
$615$	−31.7947	−1.28209
$616$	30.4145	1.22543
$617$	31.6796	1.27537	0.637687	−	0.770295i	$-0.279891\pi$
$617$	31.6796	1.27537	0.637687	+	0.770295i	$0.279891\pi$
$618$	−0.0829723	−0.00333763
$619$	35.6402	1.43250	0.716250	−	0.697844i	$-0.245857\pi$
$619$	35.6402	1.43250	0.716250	+	0.697844i	$0.245857\pi$
$620$	77.1890	3.09999
$621$	7.91766	0.317725
$622$	−29.1049	−1.16700
$623$	34.0364	1.36364
$624$	2.85500	0.114291
$625$	15.9905	0.639618
$626$	−9.99294	−0.399398
$627$	−23.9872	−0.957956
$628$	38.8173	1.54898
$629$	12.1105	0.482879
$630$	23.5822	0.939537
$631$	27.6767	1.10179	0.550896	−	0.834574i	$-0.314286\pi$
$631$	27.6767	1.10179	0.550896	+	0.834574i	$0.314286\pi$
$632$	9.07217	0.360871
$633$	7.21090	0.286608
$634$	38.4042	1.52523
$635$	−36.2681	−1.43925
$636$	23.2329	0.921243
$637$	1.51252	0.0599282
$638$	31.4433	1.24485
$639$	11.0241	0.436107
$640$	72.7875	2.87718
$641$	13.4439	0.531002	0.265501	−	0.964111i	$-0.414463\pi$
$641$	13.4439	0.531002	0.265501	+	0.964111i	$0.414463\pi$
$642$	40.4795	1.59760
$643$	43.5972	1.71931	0.859654	−	0.510877i	$-0.170679\pi$
$643$	43.5972	1.71931	0.859654	+	0.510877i	$0.170679\pi$
$644$	−71.1594	−2.80407
$645$	−7.05159	−0.277656
$646$	74.9954	2.95065
$647$	33.1500	1.30326	0.651630	−	0.758537i	$-0.274085\pi$
$647$	33.1500	1.30326	0.651630	+	0.758537i	$0.274085\pi$
$648$	−3.09222	−0.121474
$649$	43.0653	1.69046
$650$	−132.159	−5.18370
$651$	−16.4192	−0.643521
$652$	−58.4451	−2.28889
$653$	−39.1603	−1.53246	−0.766230	−	0.642566i	$-0.777870\pi$
$653$	−39.1603	−1.53246	−0.766230	+	0.642566i	$0.777870\pi$
$654$	11.6200	0.454380
$655$	17.1939	0.671822
$656$	−3.92354	−0.153188
$657$	−1.20803	−0.0471297
$658$	42.1602	1.64358
$659$	48.0240	1.87075	0.935376	−	0.353656i	$-0.115062\pi$
$659$	48.0240	1.87075	0.935376	+	0.353656i	$0.115062\pi$
$660$	46.2394	1.79987
$661$	−6.14055	−0.238840	−0.119420	−	0.992844i	$-0.538103\pi$
$661$	−6.14055	−0.238840	−0.119420	+	0.992844i	$0.538103\pi$
$662$	47.7227	1.85479
$663$	30.1692	1.17168
$664$	−50.0682	−1.94302
$665$	67.0124	2.59863
$666$	−5.65991	−0.219317
$667$	−29.4929	−1.14197
$668$	−29.5798	−1.14448
$669$	−18.8270	−0.727895
$670$	1.08957	0.0420937
$671$	−40.5150	−1.56407
$672$	−13.7395	−0.530015
$673$	32.6779	1.25964	0.629820	−	0.776741i	$-0.283129\pi$
$673$	32.6779	1.25964	0.629820	+	0.776741i	$0.283129\pi$
$674$	51.6904	1.99104
$675$	9.37317	0.360773
$676$	80.9226	3.11241
$677$	3.53727	0.135948	0.0679742	−	0.997687i	$-0.478346\pi$
$677$	3.53727	0.135948	0.0679742	+	0.997687i	$0.478346\pi$
$678$	8.44094	0.324172
$679$	4.65949	0.178815
$680$	−57.9567	−2.22254
$681$	8.71210	0.333849
$682$	−51.4822	−1.97135
$683$	40.5635	1.55212	0.776059	−	0.630660i	$-0.217216\pi$
$683$	40.5635	1.55212	0.776059	+	0.630660i	$0.217216\pi$
$684$	−21.9182	−0.838063
$685$	−60.6195	−2.31615
$686$	−42.0000	−1.60357
$687$	−9.92460	−0.378647
$688$	−0.870182	−0.0331754
$689$	42.4697	1.61797
$690$	−69.3547	−2.64029
$691$	−25.9880	−0.988629	−0.494314	−	0.869283i	$-0.664581\pi$
$691$	−25.9880	−0.988629	−0.494314	+	0.869283i	$0.664581\pi$
$692$	33.3231	1.26675
$693$	−9.83580	−0.373631
$694$	52.0465	1.97566
$695$	51.8637	1.96730
$696$	11.5184	0.436602
$697$	−41.4607	−1.57043
$698$	−23.9259	−0.905609
$699$	22.9477	0.867962
$700$	−84.2407	−3.18400
$701$	32.0276	1.20967	0.604833	−	0.796352i	$-0.293240\pi$
$701$	32.0276	1.20967	0.604833	+	0.796352i	$0.293240\pi$
$702$	−14.0997	−0.532159
$703$	−16.0835	−0.606601
$704$	−46.4985	−1.75248
$705$	25.6963	0.967781
$706$	−3.22145	−0.121241
$707$	2.78693	0.104813
$708$	39.3508	1.47889
$709$	0.174941	0.00657006	0.00328503	−	0.999995i	$-0.498954\pi$
$709$	0.174941	0.00657006	0.00328503	+	0.999995i	$0.498954\pi$
$710$	−96.5657	−3.62405
$711$	−2.93387	−0.110029
$712$	39.0939	1.46511
$713$	48.2887	1.80842
$714$	30.7514	1.15084
$715$	84.5256	3.16108
$716$	−3.86125	−0.144302
$717$	−8.60455	−0.321343
$718$	64.4228	2.40424
$719$	6.26233	0.233546	0.116773	−	0.993159i	$-0.462745\pi$
$719$	6.26233	0.233546	0.116773	+	0.993159i	$0.462745\pi$
$720$	1.77368	0.0661012
$721$	−0.0966795	−0.00360053
$722$	−55.6990	−2.07290
$723$	12.1191	0.450716
$724$	10.4519	0.388442
$725$	−34.9146	−1.29669
$726$	−5.42461	−0.201326
$727$	−33.5795	−1.24540	−0.622698	−	0.782462i	$-0.713963\pi$
$727$	−33.5795	−1.24540	−0.622698	+	0.782462i	$0.713963\pi$
$728$	50.8021	1.88285
$729$	1.00000	0.0370370
$730$	10.5817	0.391648
$731$	−9.19535	−0.340102
$732$	−37.0205	−1.36832
$733$	9.84452	0.363616	0.181808	−	0.983334i	$-0.441805\pi$
$733$	9.84452	0.363616	0.181808	+	0.983334i	$0.441805\pi$
$734$	−33.7247	−1.24480
$735$	0.939659	0.0346599
$736$	40.4077	1.48945
$737$	−0.454443	−0.0167396
$738$	19.3768	0.713270
$739$	−26.3502	−0.969307	−0.484653	−	0.874706i	$-0.661054\pi$
$739$	−26.3502	−0.969307	−0.484653	+	0.874706i	$0.661054\pi$
$740$	31.0037	1.13972
$741$	−40.0665	−1.47188
$742$	43.2892	1.58920
$743$	35.2627	1.29366	0.646831	−	0.762633i	$-0.276094\pi$
$743$	35.2627	1.29366	0.646831	+	0.762633i	$0.276094\pi$
$744$	−18.8590	−0.691405
$745$	−78.1768	−2.86418
$746$	71.1543	2.60514
$747$	16.1917	0.592423
$748$	60.2966	2.20466
$749$	47.1668	1.72344
$750$	−38.3068	−1.39877
$751$	−12.0301	−0.438986	−0.219493	−	0.975614i	$-0.570440\pi$
$751$	−12.0301	−0.438986	−0.219493	+	0.975614i	$0.570440\pi$
$752$	3.17099	0.115634
$753$	−2.03951	−0.0743238
$754$	52.5207	1.91269
$755$	23.7160	0.863114
$756$	−8.98743	−0.326870
$757$	−16.8769	−0.613400	−0.306700	−	0.951806i	$-0.599225\pi$
$757$	−16.8769	−0.613400	−0.306700	+	0.951806i	$0.599225\pi$
$758$	77.5548	2.81692
$759$	28.9269	1.04998
$760$	76.9699	2.79199
$761$	−16.2389	−0.588661	−0.294331	−	0.955704i	$-0.595097\pi$
$761$	−16.2389	−0.588661	−0.294331	+	0.955704i	$0.595097\pi$
$762$	22.1030	0.800707
$763$	13.5397	0.490170
$764$	0.831634	0.0300875
$765$	18.7428	0.677646
$766$	−73.1570	−2.64327
$767$	71.9332	2.59736
$768$	−18.9048	−0.682167
$769$	−20.3130	−0.732507	−0.366253	−	0.930515i	$-0.619360\pi$
$769$	−20.3130	−0.732507	−0.366253	+	0.930515i	$0.619360\pi$
$770$	86.1568	3.10487
$771$	−8.80741	−0.317191
$772$	57.0912	2.05476
$773$	−44.1694	−1.58866	−0.794331	−	0.607485i	$-0.792178\pi$
$773$	−44.1694	−1.58866	−0.794331	+	0.607485i	$0.792178\pi$
$774$	4.29748	0.154470
$775$	57.1656	2.05345
$776$	5.35185	0.192120
$777$	−6.59495	−0.236592
$778$	6.41094	0.229843
$779$	55.0621	1.97281
$780$	77.2350	2.76546
$781$	40.2762	1.44120
$782$	−90.4393	−3.23410
$783$	−3.72495	−0.133119
$784$	0.115956	0.00414128
$785$	44.0829	1.57339
$786$	−10.4786	−0.373758
$787$	28.7306	1.02413	0.512067	−	0.858945i	$-0.328880\pi$
$787$	28.7306	1.02413	0.512067	+	0.858945i	$0.328880\pi$
$788$	−91.6534	−3.26502
$789$	−25.8380	−0.919856
$790$	25.6992	0.914338
$791$	9.83540	0.349707
$792$	−11.2973	−0.401433
$793$	−67.6734	−2.40315
$794$	−17.6200	−0.625310
$795$	26.3844	0.935760
$796$	39.3063	1.39318
$797$	−2.42114	−0.0857613	−0.0428806	−	0.999080i	$-0.513654\pi$
$797$	−2.42114	−0.0857613	−0.0428806	+	0.999080i	$0.513654\pi$
$798$	−40.8397	−1.44571
$799$	33.5083	1.18544
$800$	47.8359	1.69126
$801$	−12.6427	−0.446707
$802$	19.7054	0.695822
$803$	−4.41350	−0.155749
$804$	−0.415246	−0.0146446
$805$	−80.8123	−2.84826
$806$	−85.9921	−3.02894
$807$	−16.8777	−0.594123
$808$	3.20104	0.112612
$809$	−17.9945	−0.632651	−0.316326	−	0.948651i	$-0.602449\pi$
$809$	−17.9945	−0.632651	−0.316326	+	0.948651i	$0.602449\pi$
$810$	−8.75950	−0.307778
$811$	36.8163	1.29280	0.646398	−	0.763001i	$-0.276275\pi$
$811$	36.8163	1.29280	0.646398	+	0.763001i	$0.276275\pi$
$812$	33.4777	1.17484
$813$	11.8235	0.414669
$814$	−20.6783	−0.724774
$815$	−66.3733	−2.32495
$816$	2.31290	0.0809676
$817$	12.2120	0.427242
$818$	16.6290	0.581419
$819$	−16.4290	−0.574076
$820$	−106.142	−3.70663
$821$	−29.7628	−1.03873	−0.519364	−	0.854553i	$-0.673831\pi$
$821$	−29.7628	−1.03873	−0.519364	+	0.854553i	$0.673831\pi$
$822$	36.9436	1.28856
$823$	−10.4946	−0.365819	−0.182910	−	0.983130i	$-0.558552\pi$
$823$	−10.4946	−0.365819	−0.182910	+	0.983130i	$0.558552\pi$
$824$	−0.111045	−0.00386845
$825$	34.2446	1.19224
$826$	73.3213	2.55118
$827$	49.5733	1.72383	0.861916	−	0.507051i	$-0.169265\pi$
$827$	49.5733	1.72383	0.861916	+	0.507051i	$0.169265\pi$
$828$	26.4318	0.918570
$829$	−22.9050	−0.795522	−0.397761	−	0.917489i	$-0.630213\pi$
$829$	−22.9050	−0.795522	−0.397761	+	0.917489i	$0.630213\pi$
$830$	−141.831	−4.92303
$831$	−0.656720	−0.0227814
$832$	−77.6678	−2.69265
$833$	1.22532	0.0424550
$834$	−31.6075	−1.09448
$835$	−33.5924	−1.16251
$836$	−80.0774	−2.76954
$837$	6.09886	0.210807
$838$	−14.4352	−0.498655
$839$	31.0896	1.07333	0.536667	−	0.843794i	$-0.319683\pi$
$839$	31.0896	1.07333	0.536667	+	0.843794i	$0.319683\pi$
$840$	31.5610	1.08896
$841$	−15.1248	−0.521544
$842$	82.6873	2.84959
$843$	17.4366	0.600550
$844$	24.0725	0.828608
$845$	91.9000	3.16146
$846$	−15.6602	−0.538410
$847$	−6.32076	−0.217184
$848$	3.25590	0.111808
$849$	31.7051	1.08812
$850$	−107.065	−3.67229
$851$	19.3956	0.664873
$852$	36.8022	1.26082
$853$	48.2634	1.65251	0.826253	−	0.563299i	$-0.190468\pi$
$853$	48.2634	1.65251	0.826253	+	0.563299i	$0.190468\pi$
$854$	−68.9793	−2.36042
$855$	−24.8915	−0.851270
$856$	54.1754	1.85168
$857$	5.34429	0.182558	0.0912788	−	0.995825i	$-0.470905\pi$
$857$	5.34429	0.182558	0.0912788	+	0.995825i	$0.470905\pi$
$858$	−51.5128	−1.75862
$859$	6.93458	0.236605	0.118302	−	0.992978i	$-0.462255\pi$
$859$	6.93458	0.236605	0.118302	+	0.992978i	$0.462255\pi$
$860$	−23.5406	−0.802729
$861$	22.5779	0.769453
$862$	−8.33952	−0.284045
$863$	−13.9341	−0.474321	−0.237160	−	0.971470i	$-0.576217\pi$
$863$	−13.9341	−0.474321	−0.237160	+	0.971470i	$0.576217\pi$
$864$	5.10350	0.173624
$865$	37.8434	1.28672
$866$	46.2108	1.57031
$867$	7.44074	0.252701
$868$	−54.8130	−1.86048
$869$	−10.7188	−0.363610
$870$	32.6287	1.10622
$871$	−0.759070	−0.0257201
$872$	15.5516	0.526643
$873$	−1.73075	−0.0585769
$874$	120.109	4.06273
$875$	−44.6352	−1.50894
$876$	−4.03282	−0.136256
$877$	17.1964	0.580680	0.290340	−	0.956923i	$-0.406232\pi$
$877$	17.1964	0.580680	0.290340	+	0.956923i	$0.406232\pi$
$878$	82.4670	2.78313
$879$	−7.92224	−0.267211
$880$	6.48008	0.218444
$881$	13.3824	0.450864	0.225432	−	0.974259i	$-0.427621\pi$
$881$	13.3824	0.450864	0.225432	+	0.974259i	$0.427621\pi$
$882$	−0.572661	−0.0192825
$883$	−43.0600	−1.44908	−0.724542	−	0.689231i	$-0.757949\pi$
$883$	−43.0600	−1.44908	−0.724542	+	0.689231i	$0.757949\pi$
$884$	100.715	3.38742
$885$	44.6888	1.50220
$886$	52.0605	1.74901
$887$	−24.4570	−0.821186	−0.410593	−	0.911819i	$-0.634678\pi$
$887$	−24.4570	−0.821186	−0.410593	+	0.911819i	$0.634678\pi$
$888$	−7.57490	−0.254197
$889$	25.7545	0.863778
$890$	110.744	3.71214
$891$	3.65347	0.122396
$892$	−62.8510	−2.10441
$893$	−44.5010	−1.48917
$894$	47.6437	1.59344
$895$	−4.38504	−0.146576
$896$	−51.6875	−1.72676
$897$	48.3174	1.61327
$898$	−20.9324	−0.698524
$899$	−22.7179	−0.757685
$900$	31.2908	1.04303
$901$	34.4056	1.14622
$902$	70.7925	2.35713
$903$	5.00744	0.166637
$904$	11.2969	0.375728
$905$	11.8697	0.394563
$906$	−14.4533	−0.480180
$907$	−53.8753	−1.78890	−0.894450	−	0.447167i	$-0.852433\pi$
$907$	−53.8753	−1.78890	−0.894450	+	0.447167i	$0.852433\pi$
$908$	29.0840	0.965186
$909$	−1.03519	−0.0343352
$910$	143.910	4.77057
$911$	15.7452	0.521663	0.260832	−	0.965384i	$-0.416003\pi$
$911$	15.7452	0.521663	0.260832	+	0.965384i	$0.416003\pi$
$912$	−3.07166	−0.101713
$913$	59.1558	1.95777
$914$	68.6259	2.26994
$915$	−42.0424	−1.38988
$916$	−33.1317	−1.09470
$917$	−12.2096	−0.403198
$918$	−11.4225	−0.376998
$919$	−42.8804	−1.41449	−0.707247	−	0.706967i	$-0.750063\pi$
$919$	−42.8804	−1.41449	−0.707247	+	0.706967i	$0.750063\pi$
$920$	−92.8203	−3.06020
$921$	20.6559	0.680635
$922$	−18.4161	−0.606503
$923$	67.2745	2.21437
$924$	−32.8353	−1.08020
$925$	22.9611	0.754957
$926$	92.2740	3.03231
$927$	0.0359112	0.00117948
$928$	−19.0103	−0.624042
$929$	12.7853	0.419474	0.209737	−	0.977758i	$-0.432739\pi$
$929$	12.7853	0.419474	0.209737	+	0.977758i	$0.432739\pi$
$930$	−53.4230	−1.75181
$931$	−1.62730	−0.0533327
$932$	76.6073	2.50935
$933$	12.5969	0.412404
$934$	34.8173	1.13926
$935$	68.4760	2.23941
$936$	−18.8702	−0.616793
$937$	−6.25233	−0.204255	−0.102127	−	0.994771i	$-0.532565\pi$
$937$	−6.25233	−0.204255	−0.102127	+	0.994771i	$0.532565\pi$
$938$	−0.773718	−0.0252628
$939$	4.32504	0.141142
$940$	85.7832	2.79794
$941$	42.5872	1.38830	0.694151	−	0.719829i	$-0.255780\pi$
$941$	42.5872	1.38830	0.694151	+	0.719829i	$0.255780\pi$
$942$	−26.8657	−0.875330
$943$	−66.4012	−2.16232
$944$	5.51470	0.179488
$945$	−10.2066	−0.332021
$946$	15.7007	0.510474
$947$	34.8773	1.13336	0.566680	−	0.823938i	$-0.308228\pi$
$947$	34.8773	1.13336	0.566680	+	0.823938i	$0.308228\pi$
$948$	−9.79426	−0.318103
$949$	−7.37199	−0.239305
$950$	142.188	4.61320
$951$	−16.6217	−0.538996
$952$	41.1559	1.33387
$953$	−10.8262	−0.350694	−0.175347	−	0.984507i	$-0.556105\pi$
$953$	−10.8262	−0.350694	−0.175347	+	0.984507i	$0.556105\pi$
$954$	−16.0796	−0.520596
$955$	0.944448	0.0305616
$956$	−28.7249	−0.929031
$957$	−13.6090	−0.439916
$958$	−63.7590	−2.05996
$959$	43.0468	1.39005
$960$	−48.2514	−1.55731
$961$	6.19606	0.199873
$962$	−34.5396	−1.11360
$963$	−17.5199	−0.564571
$964$	40.4578	1.30306
$965$	64.8358	2.08714
$966$	49.2498	1.58459
$967$	−1.47586	−0.0474606	−0.0237303	−	0.999718i	$-0.507554\pi$
$967$	−1.47586	−0.0474606	−0.0237303	+	0.999718i	$0.507554\pi$
$968$	−7.25998	−0.233344
$969$	−32.4587	−1.04272
$970$	15.1605	0.486774
$971$	5.15042	0.165285	0.0826425	−	0.996579i	$-0.473664\pi$
$971$	5.15042	0.165285	0.0826425	+	0.996579i	$0.473664\pi$
$972$	3.33834	0.107077
$973$	−36.8292	−1.18069
$974$	−86.8043	−2.78139
$975$	57.1997	1.83186
$976$	−5.18812	−0.166068
$977$	−48.0896	−1.53852	−0.769262	−	0.638934i	$-0.779376\pi$
$977$	−48.0896	−1.53852	−0.769262	+	0.638934i	$0.779376\pi$
$978$	40.4502	1.29345
$979$	−46.1896	−1.47623
$980$	3.13690	0.100205
$981$	−5.02926	−0.160572
$982$	41.1726	1.31387
$983$	0.490687	0.0156505	0.00782524	−	0.999969i	$-0.497509\pi$
$983$	0.490687	0.0156505	0.00782524	+	0.999969i	$0.497509\pi$
$984$	25.9328	0.826707
$985$	−104.086	−3.31647
$986$	42.5481	1.35501
$987$	−18.2474	−0.580820
$988$	−133.756	−4.25533
$989$	−14.7268	−0.468284
$990$	−32.0026	−1.01711
$991$	48.2840	1.53379	0.766896	−	0.641772i	$-0.221800\pi$
$991$	48.2840	1.53379	0.766896	+	0.641772i	$0.221800\pi$
$992$	31.1255	0.988235
$993$	−20.6548	−0.655461
$994$	68.5727	2.17499
$995$	44.6383	1.41513
$996$	54.0534	1.71275
$997$	47.1618	1.49363	0.746814	−	0.665033i	$-0.231582\pi$
$997$	47.1618	1.49363	0.746814	+	0.665033i	$0.231582\pi$
$998$	14.4370	0.456995
$999$	2.44966	0.0775039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.d.1.15	✓	129

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.d.1.15	✓	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)

\(f(q)\)	\(=\)	\(q-2.31049 q^{2} +1.00000 q^{3} +3.33834 q^{4} +3.79120 q^{5} -2.31049 q^{6} -2.69218 q^{7} -3.09222 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.31049 q^{2} +1.00000 q^{3} +3.33834 q^{4} +3.79120 q^{5} -2.31049 q^{6} -2.69218 q^{7} -3.09222 q^{8} +1.00000 q^{9} -8.75950 q^{10} +3.65347 q^{11} +3.33834 q^{12} +6.10249 q^{13} +6.22025 q^{14} +3.79120 q^{15} +0.467842 q^{16} +4.94376 q^{17} -2.31049 q^{18} -6.56560 q^{19} +12.6563 q^{20} -2.69218 q^{21} -8.44128 q^{22} +7.91766 q^{23} -3.09222 q^{24} +9.37317 q^{25} -14.0997 q^{26} +1.00000 q^{27} -8.98743 q^{28} -3.72495 q^{29} -8.75950 q^{30} +6.09886 q^{31} +5.10350 q^{32} +3.65347 q^{33} -11.4225 q^{34} -10.2066 q^{35} +3.33834 q^{36} +2.44966 q^{37} +15.1697 q^{38} +6.10249 q^{39} -11.7232 q^{40} -8.38647 q^{41} +6.22025 q^{42} -1.85999 q^{43} +12.1965 q^{44} +3.79120 q^{45} -18.2936 q^{46} +6.77790 q^{47} +0.467842 q^{48} +0.247853 q^{49} -21.6566 q^{50} +4.94376 q^{51} +20.3722 q^{52} +6.95940 q^{53} -2.31049 q^{54} +13.8510 q^{55} +8.32482 q^{56} -6.56560 q^{57} +8.60644 q^{58} +11.7875 q^{59} +12.6563 q^{60} -11.0895 q^{61} -14.0913 q^{62} -2.69218 q^{63} -12.7272 q^{64} +23.1357 q^{65} -8.44128 q^{66} -0.124387 q^{67} +16.5040 q^{68} +7.91766 q^{69} +23.5822 q^{70} +11.0241 q^{71} -3.09222 q^{72} -1.20803 q^{73} -5.65991 q^{74} +9.37317 q^{75} -21.9182 q^{76} -9.83580 q^{77} -14.0997 q^{78} -2.93387 q^{79} +1.77368 q^{80} +1.00000 q^{81} +19.3768 q^{82} +16.1917 q^{83} -8.98743 q^{84} +18.7428 q^{85} +4.29748 q^{86} -3.72495 q^{87} -11.2973 q^{88} -12.6427 q^{89} -8.75950 q^{90} -16.4290 q^{91} +26.4318 q^{92} +6.09886 q^{93} -15.6602 q^{94} -24.8915 q^{95} +5.10350 q^{96} -1.73075 q^{97} -0.572661 q^{98} +3.65347 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) \) 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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