LMFDB - Embedded newform 8047.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59413 q^{2} -3.39586 q^{3} +4.72950 q^{4} +2.81515 q^{5} +8.80929 q^{6} -0.0603743 q^{7} -7.08068 q^{8} +8.53185 q^{9} +O(q^{10})$	\(q-2.59413 q^{2} -3.39586 q^{3} +4.72950 q^{4} +2.81515 q^{5} +8.80929 q^{6} -0.0603743 q^{7} -7.08068 q^{8} +8.53185 q^{9} -7.30287 q^{10} -4.65929 q^{11} -16.0607 q^{12} +1.00000 q^{13} +0.156619 q^{14} -9.55986 q^{15} +8.90918 q^{16} +7.88612 q^{17} -22.1327 q^{18} +1.10991 q^{19} +13.3143 q^{20} +0.205023 q^{21} +12.0868 q^{22} +5.56200 q^{23} +24.0450 q^{24} +2.92510 q^{25} -2.59413 q^{26} -18.7854 q^{27} -0.285540 q^{28} +1.21533 q^{29} +24.7995 q^{30} +8.34952 q^{31} -8.95020 q^{32} +15.8223 q^{33} -20.4576 q^{34} -0.169963 q^{35} +40.3514 q^{36} +0.350660 q^{37} -2.87924 q^{38} -3.39586 q^{39} -19.9332 q^{40} -3.04744 q^{41} -0.531855 q^{42} +4.39418 q^{43} -22.0361 q^{44} +24.0185 q^{45} -14.4285 q^{46} +11.9885 q^{47} -30.2543 q^{48} -6.99635 q^{49} -7.58807 q^{50} -26.7801 q^{51} +4.72950 q^{52} +4.36119 q^{53} +48.7317 q^{54} -13.1166 q^{55} +0.427491 q^{56} -3.76909 q^{57} -3.15273 q^{58} -3.90890 q^{59} -45.2134 q^{60} +5.18236 q^{61} -21.6597 q^{62} -0.515104 q^{63} +5.39960 q^{64} +2.81515 q^{65} -41.0450 q^{66} +1.14247 q^{67} +37.2974 q^{68} -18.8878 q^{69} +0.440906 q^{70} -11.4914 q^{71} -60.4113 q^{72} +9.71788 q^{73} -0.909656 q^{74} -9.93321 q^{75} +5.24931 q^{76} +0.281301 q^{77} +8.80929 q^{78} -2.34698 q^{79} +25.0807 q^{80} +38.1969 q^{81} +7.90544 q^{82} -10.8299 q^{83} +0.969654 q^{84} +22.2006 q^{85} -11.3991 q^{86} -4.12710 q^{87} +32.9909 q^{88} +14.3548 q^{89} -62.3070 q^{90} -0.0603743 q^{91} +26.3055 q^{92} -28.3538 q^{93} -31.0996 q^{94} +3.12456 q^{95} +30.3936 q^{96} +0.0921691 q^{97} +18.1494 q^{98} -39.7523 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * 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 $

$2$	−2.59413	−1.83433	−0.917163	−	0.398513i	$-0.869526\pi$
$2$	−2.59413	−1.83433	−0.917163	+	0.398513i	$0.869526\pi$
$3$	−3.39586	−1.96060	−0.980300	−	0.197516i	$-0.936713\pi$
$3$	−3.39586	−1.96060	−0.980300	+	0.197516i	$0.936713\pi$
$4$	4.72950	2.36475
$5$	2.81515	1.25898	0.629488	−	0.777010i	$-0.283265\pi$
$5$	2.81515	1.25898	0.629488	+	0.777010i	$0.283265\pi$
$6$	8.80929	3.59638
$7$	−0.0603743	−0.0228193	−0.0114097	−	0.999935i	$-0.503632\pi$
$7$	−0.0603743	−0.0228193	−0.0114097	+	0.999935i	$0.503632\pi$
$8$	−7.08068	−2.50340
$9$	8.53185	2.84395
$10$	−7.30287	−2.30937
$11$	−4.65929	−1.40483	−0.702414	−	0.711769i	$-0.747894\pi$
$11$	−4.65929	−1.40483	−0.702414	+	0.711769i	$0.747894\pi$
$12$	−16.0607	−4.63633
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.156619	0.0418581
$15$	−9.55986	−2.46835
$16$	8.90918	2.22729
$17$	7.88612	1.91267	0.956333	−	0.292281i	$-0.0944142\pi$
$17$	7.88612	1.91267	0.956333	+	0.292281i	$0.0944142\pi$
$18$	−22.1327	−5.21673
$19$	1.10991	0.254630	0.127315	−	0.991862i	$-0.459364\pi$
$19$	1.10991	0.254630	0.127315	+	0.991862i	$0.459364\pi$
$20$	13.3143	2.97716
$21$	0.205023	0.0447396
$22$	12.0868	2.57691
$23$	5.56200	1.15976	0.579879	−	0.814703i	$-0.303100\pi$
$23$	5.56200	1.15976	0.579879	+	0.814703i	$0.303100\pi$
$24$	24.0450	4.90816
$25$	2.92510	0.585019
$26$	−2.59413	−0.508750
$27$	−18.7854	−3.61525
$28$	−0.285540	−0.0539620
$29$	1.21533	0.225682	0.112841	−	0.993613i	$-0.464005\pi$
$29$	1.21533	0.225682	0.112841	+	0.993613i	$0.464005\pi$
$30$	24.7995	4.52775
$31$	8.34952	1.49962	0.749809	−	0.661655i	$-0.230146\pi$
$31$	8.34952	1.49962	0.749809	+	0.661655i	$0.230146\pi$
$32$	−8.95020	−1.58219
$33$	15.8223	2.75430
$34$	−20.4576	−3.50845
$35$	−0.169963	−0.0287290
$36$	40.3514	6.72523
$37$	0.350660	0.0576481	0.0288240	−	0.999585i	$-0.490824\pi$
$37$	0.350660	0.0576481	0.0288240	+	0.999585i	$0.490824\pi$
$38$	−2.87924	−0.467075
$39$	−3.39586	−0.543772
$40$	−19.9332	−3.15171
$41$	−3.04744	−0.475929	−0.237965	−	0.971274i	$-0.576480\pi$
$41$	−3.04744	−0.475929	−0.237965	+	0.971274i	$0.576480\pi$
$42$	−0.531855	−0.0820670
$43$	4.39418	0.670106	0.335053	−	0.942199i	$-0.391246\pi$
$43$	4.39418	0.670106	0.335053	+	0.942199i	$0.391246\pi$
$44$	−22.0361	−3.32207
$45$	24.0185	3.58046
$46$	−14.4285	−2.12737
$47$	11.9885	1.74870	0.874348	−	0.485299i	$-0.161289\pi$
$47$	11.9885	1.74870	0.874348	+	0.485299i	$0.161289\pi$
$48$	−30.2543	−4.36683
$49$	−6.99635	−0.999479
$50$	−7.58807	−1.07312
$51$	−26.7801	−3.74997
$52$	4.72950	0.655864
$53$	4.36119	0.599056	0.299528	−	0.954088i	$-0.403171\pi$
$53$	4.36119	0.599056	0.299528	+	0.954088i	$0.403171\pi$
$54$	48.7317	6.63154
$55$	−13.1166	−1.76864
$56$	0.427491	0.0571259
$57$	−3.76909	−0.499228
$58$	−3.15273	−0.413974
$59$	−3.90890	−0.508895	−0.254448	−	0.967087i	$-0.581894\pi$
$59$	−3.90890	−0.508895	−0.254448	+	0.967087i	$0.581894\pi$
$60$	−45.2134	−5.83702
$61$	5.18236	0.663534	0.331767	−	0.943361i	$-0.392355\pi$
$61$	5.18236	0.663534	0.331767	+	0.943361i	$0.392355\pi$
$62$	−21.6597	−2.75079
$63$	−0.515104	−0.0648971
$64$	5.39960	0.674951
$65$	2.81515	0.349177
$66$	−41.0450	−5.05229
$67$	1.14247	0.139575	0.0697877	−	0.997562i	$-0.477768\pi$
$67$	1.14247	0.139575	0.0697877	+	0.997562i	$0.477768\pi$
$68$	37.2974	4.52298
$69$	−18.8878	−2.27382
$70$	0.440906	0.0526983
$71$	−11.4914	−1.36378	−0.681891	−	0.731454i	$-0.738842\pi$
$71$	−11.4914	−1.36378	−0.681891	+	0.731454i	$0.738842\pi$
$72$	−60.4113	−7.11954
$73$	9.71788	1.13739	0.568696	−	0.822548i	$-0.307448\pi$
$73$	9.71788	1.13739	0.568696	+	0.822548i	$0.307448\pi$
$74$	−0.909656	−0.105745
$75$	−9.93321	−1.14699
$76$	5.24931	0.602137
$77$	0.281301	0.0320572
$78$	8.80929	0.997456
$79$	−2.34698	−0.264055	−0.132028	−	0.991246i	$-0.542149\pi$
$79$	−2.34698	−0.264055	−0.132028	+	0.991246i	$0.542149\pi$
$80$	25.0807	2.80411
$81$	38.1969	4.24410
$82$	7.90544	0.873010
$83$	−10.8299	−1.18874	−0.594369	−	0.804192i	$-0.702598\pi$
$83$	−10.8299	−1.18874	−0.594369	+	0.804192i	$0.702598\pi$
$84$	0.969654	0.105798
$85$	22.2006	2.40800
$86$	−11.3991	−1.22919
$87$	−4.12710	−0.442472
$88$	32.9909	3.51684
$89$	14.3548	1.52160	0.760801	−	0.648985i	$-0.224806\pi$
$89$	14.3548	1.52160	0.760801	+	0.648985i	$0.224806\pi$
$90$	−62.3070	−6.56773
$91$	−0.0603743	−0.00632895
$92$	26.3055	2.74254
$93$	−28.3538	−2.94015
$94$	−31.0996	−3.20768
$95$	3.12456	0.320573
$96$	30.3936	3.10203
$97$	0.0921691	0.00935836	0.00467918	−	0.999989i	$-0.498511\pi$
$97$	0.0921691	0.00935836	0.00467918	+	0.999989i	$0.498511\pi$
$98$	18.1494	1.83337
$99$	−39.7523	−3.99526
$100$	13.8342	1.38342
$101$	10.1950	1.01444	0.507218	−	0.861818i	$-0.330674\pi$
$101$	10.1950	1.01444	0.507218	+	0.861818i	$0.330674\pi$
$102$	69.4711	6.87867
$103$	3.26638	0.321846	0.160923	−	0.986967i	$-0.448553\pi$
$103$	3.26638	0.321846	0.160923	+	0.986967i	$0.448553\pi$
$104$	−7.08068	−0.694317
$105$	0.577170	0.0563260
$106$	−11.3135	−1.09886
$107$	18.7756	1.81511	0.907553	−	0.419938i	$-0.137948\pi$
$107$	18.7756	1.81511	0.907553	+	0.419938i	$0.137948\pi$
$108$	−88.8454	−8.54916
$109$	8.34440	0.799249	0.399624	−	0.916679i	$-0.369141\pi$
$109$	8.34440	0.799249	0.399624	+	0.916679i	$0.369141\pi$
$110$	34.0262	3.24427
$111$	−1.19079	−0.113025
$112$	−0.537885	−0.0508254
$113$	−0.511824	−0.0481483	−0.0240742	−	0.999710i	$-0.507664\pi$
$113$	−0.511824	−0.0481483	−0.0240742	+	0.999710i	$0.507664\pi$
$114$	9.77749	0.915746
$115$	15.6579	1.46011
$116$	5.74792	0.533681
$117$	8.53185	0.788770
$118$	10.1402	0.933480
$119$	−0.476119	−0.0436457
$120$	67.6903	6.17925
$121$	10.7089	0.973540
$122$	−13.4437	−1.21714
$123$	10.3487	0.933107
$124$	39.4890	3.54622
$125$	−5.84118	−0.522451
$126$	1.33625	0.119042
$127$	10.6235	0.942685	0.471343	−	0.881950i	$-0.343770\pi$
$127$	10.6235	0.942685	0.471343	+	0.881950i	$0.343770\pi$
$128$	3.89313	0.344107
$129$	−14.9220	−1.31381
$130$	−7.30287	−0.640504
$131$	14.9467	1.30590	0.652948	−	0.757403i	$-0.273532\pi$
$131$	14.9467	1.30590	0.652948	+	0.757403i	$0.273532\pi$
$132$	74.8315	6.51324
$133$	−0.0670099	−0.00581049
$134$	−2.96372	−0.256027
$135$	−52.8837	−4.55151
$136$	−55.8391	−4.78816
$137$	6.55484	0.560018	0.280009	−	0.959997i	$-0.409663\pi$
$137$	6.55484	0.560018	0.280009	+	0.959997i	$0.409663\pi$
$138$	48.9973	4.17093
$139$	−15.6068	−1.32376	−0.661878	−	0.749612i	$-0.730240\pi$
$139$	−15.6068	−1.32376	−0.661878	+	0.749612i	$0.730240\pi$
$140$	−0.803840	−0.0679369
$141$	−40.7111	−3.42849
$142$	29.8102	2.50162
$143$	−4.65929	−0.389629
$144$	76.0118	6.33431
$145$	3.42135	0.284128
$146$	−25.2094	−2.08635
$147$	23.7586	1.95958
$148$	1.65845	0.136323
$149$	14.6421	1.19953	0.599765	−	0.800176i	$-0.295261\pi$
$149$	14.6421	1.19953	0.599765	+	0.800176i	$0.295261\pi$
$150$	25.7680	2.10395
$151$	−20.0070	−1.62814	−0.814072	−	0.580764i	$-0.802754\pi$
$151$	−20.0070	−1.62814	−0.814072	+	0.580764i	$0.802754\pi$
$152$	−7.85889	−0.637440
$153$	67.2832	5.43952
$154$	−0.729731	−0.0588034
$155$	23.5052	1.88798
$156$	−16.0607	−1.28589
$157$	−2.17014	−0.173196	−0.0865978	−	0.996243i	$-0.527600\pi$
$157$	−2.17014	−0.173196	−0.0865978	+	0.996243i	$0.527600\pi$
$158$	6.08836	0.484364
$159$	−14.8100	−1.17451
$160$	−25.1962	−1.99193
$161$	−0.335802	−0.0264649
$162$	−99.0877	−7.78506
$163$	22.2244	1.74075	0.870376	−	0.492388i	$-0.163876\pi$
$163$	22.2244	1.74075	0.870376	+	0.492388i	$0.163876\pi$
$164$	−14.4128	−1.12545
$165$	44.5421	3.46760
$166$	28.0942	2.18053
$167$	7.70594	0.596303	0.298152	−	0.954519i	$-0.403630\pi$
$167$	7.70594	0.596303	0.298152	+	0.954519i	$0.403630\pi$
$168$	−1.45170	−0.112001
$169$	1.00000	0.0769231
$170$	−57.5913	−4.41705
$171$	9.46956	0.724155
$172$	20.7823	1.58463
$173$	−3.49250	−0.265530	−0.132765	−	0.991148i	$-0.542386\pi$
$173$	−3.49250	−0.265530	−0.132765	+	0.991148i	$0.542386\pi$
$174$	10.7062	0.811637
$175$	−0.176601	−0.0133497
$176$	−41.5104	−3.12896
$177$	13.2741	0.997740
$178$	−37.2381	−2.79111
$179$	−2.84490	−0.212638	−0.106319	−	0.994332i	$-0.533906\pi$
$179$	−2.84490	−0.212638	−0.106319	+	0.994332i	$0.533906\pi$
$180$	113.595	8.46690
$181$	−22.8425	−1.69787	−0.848935	−	0.528497i	$-0.822756\pi$
$181$	−22.8425	−1.69787	−0.848935	+	0.528497i	$0.822756\pi$
$182$	0.156619	0.0116093
$183$	−17.5986	−1.30092
$184$	−39.3827	−2.90333
$185$	0.987161	0.0725775
$186$	73.5533	5.39319
$187$	−36.7437	−2.68696
$188$	56.6994	4.13523
$189$	1.13415	0.0824975
$190$	−8.10551	−0.588035
$191$	−10.8008	−0.781520	−0.390760	−	0.920493i	$-0.627788\pi$
$191$	−10.8008	−0.781520	−0.390760	+	0.920493i	$0.627788\pi$
$192$	−18.3363	−1.32331
$193$	−18.4220	−1.32605	−0.663024	−	0.748598i	$-0.730727\pi$
$193$	−18.4220	−1.32605	−0.663024	+	0.748598i	$0.730727\pi$
$194$	−0.239098	−0.0171663
$195$	−9.55986	−0.684596
$196$	−33.0893	−2.36352
$197$	9.77984	0.696785	0.348392	−	0.937349i	$-0.386728\pi$
$197$	9.77984	0.696785	0.348392	+	0.937349i	$0.386728\pi$
$198$	103.123	7.32861
$199$	25.9705	1.84100	0.920500	−	0.390744i	$-0.127782\pi$
$199$	25.9705	1.84100	0.920500	+	0.390744i	$0.127782\pi$
$200$	−20.7117	−1.46453
$201$	−3.87968	−0.273652
$202$	−26.4470	−1.86081
$203$	−0.0733749	−0.00514991
$204$	−126.657	−8.86774
$205$	−8.57900	−0.599183
$206$	−8.47341	−0.590371
$207$	47.4542	3.29829
$208$	8.90918	0.617740
$209$	−5.17137	−0.357711
$210$	−1.49725	−0.103320
$211$	20.5503	1.41474	0.707370	−	0.706844i	$-0.249882\pi$
$211$	20.5503	1.41474	0.707370	+	0.706844i	$0.249882\pi$
$212$	20.6263	1.41662
$213$	39.0232	2.67383
$214$	−48.7063	−3.32950
$215$	12.3703	0.843647
$216$	133.013	9.05040
$217$	−0.504096	−0.0342203
$218$	−21.6464	−1.46608
$219$	−33.0006	−2.22997
$220$	−62.0350	−4.18240
$221$	7.88612	0.530478
$222$	3.08906	0.207324
$223$	−26.6354	−1.78363	−0.891817	−	0.452395i	$-0.850570\pi$
$223$	−26.6354	−1.78363	−0.891817	+	0.452395i	$0.850570\pi$
$224$	0.540362	0.0361044
$225$	24.9565	1.66376
$226$	1.32774	0.0883197
$227$	−8.66265	−0.574960	−0.287480	−	0.957787i	$-0.592818\pi$
$227$	−8.66265	−0.574960	−0.287480	+	0.957787i	$0.592818\pi$
$228$	−17.8259	−1.18055
$229$	−10.4936	−0.693439	−0.346720	−	0.937969i	$-0.612704\pi$
$229$	−10.4936	−0.693439	−0.346720	+	0.937969i	$0.612704\pi$
$230$	−40.6186	−2.67831
$231$	−0.955258	−0.0628514
$232$	−8.60538	−0.564971
$233$	15.6904	1.02791	0.513957	−	0.857816i	$-0.328179\pi$
$233$	15.6904	1.02791	0.513957	+	0.857816i	$0.328179\pi$
$234$	−22.1327	−1.44686
$235$	33.7494	2.20157
$236$	−18.4871	−1.20341
$237$	7.97000	0.517707
$238$	1.23511	0.0800605
$239$	−16.0203	−1.03627	−0.518133	−	0.855300i	$-0.673373\pi$
$239$	−16.0203	−1.03627	−0.518133	+	0.855300i	$0.673373\pi$
$240$	−85.1705	−5.49773
$241$	6.23970	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$241$	6.23970	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$242$	−27.7804	−1.78579
$243$	−73.3551	−4.70573
$244$	24.5100	1.56909
$245$	−19.6958	−1.25832
$246$	−26.8457	−1.71162
$247$	1.10991	0.0706217
$248$	−59.1202	−3.75414
$249$	36.7769	2.33064
$250$	15.1528	0.958345
$251$	11.5960	0.731930	0.365965	−	0.930629i	$-0.380739\pi$
$251$	11.5960	0.731930	0.365965	+	0.930629i	$0.380739\pi$
$252$	−2.43619	−0.153465
$253$	−25.9150	−1.62926
$254$	−27.5588	−1.72919
$255$	−75.3902	−4.72112
$256$	−20.8985	−1.30616
$257$	19.5067	1.21680	0.608398	−	0.793632i	$-0.291812\pi$
$257$	19.5067	1.21680	0.608398	+	0.793632i	$0.291812\pi$
$258$	38.7096	2.40995
$259$	−0.0211708	−0.00131549
$260$	13.3143	0.825716
$261$	10.3690	0.641828
$262$	−38.7735	−2.39544
$263$	−5.56656	−0.343249	−0.171625	−	0.985162i	$-0.554902\pi$
$263$	−5.56656	−0.343249	−0.171625	+	0.985162i	$0.554902\pi$
$264$	−112.032	−6.89512
$265$	12.2774	0.754196
$266$	0.173832	0.0106583
$267$	−48.7467	−2.98325
$268$	5.40333	0.330061
$269$	−5.43645	−0.331466	−0.165733	−	0.986171i	$-0.552999\pi$
$269$	−5.43645	−0.331466	−0.165733	+	0.986171i	$0.552999\pi$
$270$	137.187	8.34895
$271$	−12.8608	−0.781236	−0.390618	−	0.920553i	$-0.627739\pi$
$271$	−12.8608	−0.781236	−0.390618	+	0.920553i	$0.627739\pi$
$272$	70.2588	4.26007
$273$	0.205023	0.0124085
$274$	−17.0041	−1.02725
$275$	−13.6289	−0.821851
$276$	−89.3297	−5.37702
$277$	20.3833	1.22471	0.612356	−	0.790582i	$-0.290222\pi$
$277$	20.3833	1.22471	0.612356	+	0.790582i	$0.290222\pi$
$278$	40.4862	2.42820
$279$	71.2368	4.26484
$280$	1.20345	0.0719200
$281$	−6.84459	−0.408314	−0.204157	−	0.978938i	$-0.565445\pi$
$281$	−6.84459	−0.408314	−0.204157	+	0.978938i	$0.565445\pi$
$282$	105.610	6.28897
$283$	−11.9877	−0.712596	−0.356298	−	0.934372i	$-0.615961\pi$
$283$	−11.9877	−0.712596	−0.356298	+	0.934372i	$0.615961\pi$
$284$	−54.3487	−3.22500
$285$	−10.6106	−0.628515
$286$	12.0868	0.714707
$287$	0.183987	0.0108604
$288$	−76.3617	−4.49966
$289$	45.1909	2.65829
$290$	−8.87543	−0.521183
$291$	−0.312993	−0.0183480
$292$	45.9607	2.68965
$293$	6.83182	0.399119	0.199560	−	0.979886i	$-0.436049\pi$
$293$	6.83182	0.399119	0.199560	+	0.979886i	$0.436049\pi$
$294$	−61.6329	−3.59450
$295$	−11.0042	−0.640687
$296$	−2.48291	−0.144316
$297$	87.5264	5.07880
$298$	−37.9835	−2.20033
$299$	5.56200	0.321659
$300$	−46.9791	−2.71234
$301$	−0.265295	−0.0152914
$302$	51.9006	2.98655
$303$	−34.6206	−1.98890
$304$	9.88836	0.567136
$305$	14.5892	0.835372
$306$	−174.541	−9.97786
$307$	2.02636	0.115651	0.0578253	−	0.998327i	$-0.481583\pi$
$307$	2.02636	0.115651	0.0578253	+	0.998327i	$0.481583\pi$
$308$	1.33041	0.0758074
$309$	−11.0922	−0.631011
$310$	−60.9754	−3.46317
$311$	4.08332	0.231544	0.115772	−	0.993276i	$-0.463066\pi$
$311$	4.08332	0.231544	0.115772	+	0.993276i	$0.463066\pi$
$312$	24.0450	1.36128
$313$	4.51668	0.255298	0.127649	−	0.991819i	$-0.459257\pi$
$313$	4.51668	0.255298	0.127649	+	0.991819i	$0.459257\pi$
$314$	5.62961	0.317697
$315$	−1.45010	−0.0817038
$316$	−11.1000	−0.624425
$317$	−19.8298	−1.11375	−0.556875	−	0.830596i	$-0.688000\pi$
$317$	−19.8298	−1.11375	−0.556875	+	0.830596i	$0.688000\pi$
$318$	38.4190	2.15443
$319$	−5.66259	−0.317044
$320$	15.2007	0.849746
$321$	−63.7593	−3.55870
$322$	0.871113	0.0485452
$323$	8.75286	0.487022
$324$	180.652	10.0362
$325$	2.92510	0.162255
$326$	−57.6530	−3.19311
$327$	−28.3364	−1.56701
$328$	21.5779	1.19144
$329$	−0.723795	−0.0399041
$330$	−115.548	−6.36071
$331$	−25.4310	−1.39781	−0.698907	−	0.715212i	$-0.746330\pi$
$331$	−25.4310	−1.39781	−0.698907	+	0.715212i	$0.746330\pi$
$332$	−51.2201	−2.81107
$333$	2.99178	0.163948
$334$	−19.9902	−1.09381
$335$	3.21624	0.175722
$336$	1.82658	0.0996482
$337$	−5.85874	−0.319146	−0.159573	−	0.987186i	$-0.551012\pi$
$337$	−5.85874	−0.319146	−0.159573	+	0.987186i	$0.551012\pi$
$338$	−2.59413	−0.141102
$339$	1.73808	0.0943995
$340$	104.998	5.69432
$341$	−38.9028	−2.10670
$342$	−24.5653	−1.32834
$343$	0.845020	0.0456268
$344$	−31.1137	−1.67754
$345$	−53.1720	−2.86268
$346$	9.05999	0.487068
$347$	−10.5077	−0.564085	−0.282042	−	0.959402i	$-0.591012\pi$
$347$	−10.5077	−0.564085	−0.282042	+	0.959402i	$0.591012\pi$
$348$	−19.5191	−1.04633
$349$	−28.4512	−1.52296	−0.761479	−	0.648189i	$-0.775526\pi$
$349$	−28.4512	−1.52296	−0.761479	+	0.648189i	$0.775526\pi$
$350$	0.458124	0.0244878
$351$	−18.7854	−1.00269
$352$	41.7015	2.22270
$353$	−8.88003	−0.472636	−0.236318	−	0.971676i	$-0.575941\pi$
$353$	−8.88003	−0.472636	−0.236318	+	0.971676i	$0.575941\pi$
$354$	−34.4346	−1.83018
$355$	−32.3501	−1.71697
$356$	67.8909	3.59821
$357$	1.61683	0.0855718
$358$	7.38004	0.390047
$359$	32.9993	1.74164	0.870818	−	0.491606i	$-0.163590\pi$
$359$	32.9993	1.74164	0.870818	+	0.491606i	$0.163590\pi$
$360$	−170.067	−8.96332
$361$	−17.7681	−0.935163
$362$	59.2564	3.11445
$363$	−36.3661	−1.90872
$364$	−0.285540	−0.0149664
$365$	27.3573	1.43195
$366$	45.6529	2.38632
$367$	−21.3781	−1.11593	−0.557963	−	0.829866i	$-0.688417\pi$
$367$	−21.3781	−1.11593	−0.557963	+	0.829866i	$0.688417\pi$
$368$	49.5529	2.58312
$369$	−26.0003	−1.35352
$370$	−2.56082	−0.133131
$371$	−0.263304	−0.0136701
$372$	−134.099	−6.95272
$373$	−21.3047	−1.10312	−0.551558	−	0.834136i	$-0.685967\pi$
$373$	−21.3047	−1.10312	−0.551558	+	0.834136i	$0.685967\pi$
$374$	95.3178	4.92877
$375$	19.8358	1.02432
$376$	−84.8864	−4.37768
$377$	1.21533	0.0625929
$378$	−2.94214	−0.151327
$379$	−24.5892	−1.26306	−0.631531	−	0.775351i	$-0.717573\pi$
$379$	−24.5892	−1.26306	−0.631531	+	0.775351i	$0.717573\pi$
$380$	14.7776	0.758075
$381$	−36.0760	−1.84823
$382$	28.0187	1.43356
$383$	13.7295	0.701545	0.350772	−	0.936461i	$-0.385919\pi$
$383$	13.7295	0.701545	0.350772	+	0.936461i	$0.385919\pi$
$384$	−13.2205	−0.674656
$385$	0.791906	0.0403593
$386$	47.7891	2.43240
$387$	37.4905	1.90575
$388$	0.435914	0.0221302
$389$	−30.6149	−1.55224	−0.776118	−	0.630588i	$-0.782814\pi$
$389$	−30.6149	−1.55224	−0.776118	+	0.630588i	$0.782814\pi$
$390$	24.7995	1.25577
$391$	43.8626	2.21823
$392$	49.5389	2.50209
$393$	−50.7567	−2.56034
$394$	−25.3702	−1.27813
$395$	−6.60710	−0.332439
$396$	−188.009	−9.44779
$397$	−35.7309	−1.79328	−0.896642	−	0.442757i	$-0.854000\pi$
$397$	−35.7309	−1.79328	−0.896642	+	0.442757i	$0.854000\pi$
$398$	−67.3708	−3.37699
$399$	0.227556	0.0113920
$400$	26.0602	1.30301
$401$	10.6255	0.530611	0.265305	−	0.964164i	$-0.414527\pi$
$401$	10.6255	0.530611	0.265305	+	0.964164i	$0.414527\pi$
$402$	10.0644	0.501966
$403$	8.34952	0.415919
$404$	48.2171	2.39889
$405$	107.530	5.34322
$406$	0.190344	0.00944661
$407$	−1.63382	−0.0809856
$408$	189.621	9.38766
$409$	5.32384	0.263247	0.131623	−	0.991300i	$-0.457981\pi$
$409$	5.32384	0.263247	0.131623	+	0.991300i	$0.457981\pi$
$410$	22.2550	1.09910
$411$	−22.2593	−1.09797
$412$	15.4484	0.761086
$413$	0.235997	0.0116127
$414$	−123.102	−6.05014
$415$	−30.4879	−1.49659
$416$	−8.95020	−0.438820
$417$	52.9986	2.59535
$418$	13.4152	0.656159
$419$	−6.26991	−0.306305	−0.153153	−	0.988203i	$-0.548943\pi$
$419$	−6.26991	−0.306305	−0.153153	+	0.988203i	$0.548943\pi$
$420$	2.72973	0.133197
$421$	21.0879	1.02776	0.513881	−	0.857861i	$-0.328207\pi$
$421$	21.0879	1.02776	0.513881	+	0.857861i	$0.328207\pi$
$422$	−53.3100	−2.59509
$423$	102.284	4.97320
$424$	−30.8802	−1.49967
$425$	23.0677	1.11895
$426$	−101.231	−4.90467
$427$	−0.312881	−0.0151414
$428$	88.7992	4.29227
$429$	15.8223	0.763906
$430$	−32.0901	−1.54752
$431$	−1.28482	−0.0618874	−0.0309437	−	0.999521i	$-0.509851\pi$
$431$	−1.28482	−0.0618874	−0.0309437	+	0.999521i	$0.509851\pi$
$432$	−167.362	−8.05222
$433$	4.87274	0.234169	0.117085	−	0.993122i	$-0.462645\pi$
$433$	4.87274	0.234169	0.117085	+	0.993122i	$0.462645\pi$
$434$	1.30769	0.0627711
$435$	−11.6184	−0.557061
$436$	39.4648	1.89002
$437$	6.17330	0.295309
$438$	85.6077	4.09049
$439$	−17.4435	−0.832531	−0.416265	−	0.909243i	$-0.636661\pi$
$439$	−17.4435	−0.832531	−0.416265	+	0.909243i	$0.636661\pi$
$440$	92.8745	4.42762
$441$	−59.6918	−2.84247
$442$	−20.4576	−0.973069
$443$	−0.909203	−0.0431975	−0.0215988	−	0.999767i	$-0.506876\pi$
$443$	−0.909203	−0.0431975	−0.0215988	+	0.999767i	$0.506876\pi$
$444$	−5.63184	−0.267275
$445$	40.4109	1.91566
$446$	69.0955	3.27177
$447$	−49.7226	−2.35180
$448$	−0.325997	−0.0154019
$449$	11.6052	0.547682	0.273841	−	0.961775i	$-0.411706\pi$
$449$	11.6052	0.547682	0.273841	+	0.961775i	$0.411706\pi$
$450$	−64.7403	−3.05189
$451$	14.1989	0.668599
$452$	−2.42067	−0.113859
$453$	67.9408	3.19214
$454$	22.4720	1.05466
$455$	−0.169963	−0.00796799
$456$	26.6877	1.24976
$457$	−19.4370	−0.909222	−0.454611	−	0.890690i	$-0.650222\pi$
$457$	−19.4370	−0.909222	−0.454611	+	0.890690i	$0.650222\pi$
$458$	27.2218	1.27199
$459$	−148.144	−6.91476
$460$	74.0540	3.45279
$461$	22.7813	1.06103	0.530517	−	0.847675i	$-0.321998\pi$
$461$	22.7813	1.06103	0.530517	+	0.847675i	$0.321998\pi$
$462$	2.47806	0.115290
$463$	17.2725	0.802721	0.401360	−	0.915920i	$-0.368537\pi$
$463$	17.2725	0.802721	0.401360	+	0.915920i	$0.368537\pi$
$464$	10.8276	0.502660
$465$	−79.8202	−3.70158
$466$	−40.7030	−1.88553
$467$	−3.32740	−0.153974	−0.0769868	−	0.997032i	$-0.524530\pi$
$467$	−3.32740	−0.153974	−0.0769868	+	0.997032i	$0.524530\pi$
$468$	40.3514	1.86524
$469$	−0.0689761	−0.00318502
$470$	−87.5502	−4.03839
$471$	7.36947	0.339567
$472$	27.6776	1.27397
$473$	−20.4737	−0.941383
$474$	−20.6752	−0.949643
$475$	3.24658	0.148963
$476$	−2.25180	−0.103211
$477$	37.2090	1.70368
$478$	41.5587	1.90085
$479$	−34.2732	−1.56598	−0.782991	−	0.622033i	$-0.786307\pi$
$479$	−34.2732	−1.56598	−0.782991	+	0.622033i	$0.786307\pi$
$480$	85.5627	3.90538
$481$	0.350660	0.0159887
$482$	−16.1866	−0.737278
$483$	1.14034	0.0518871
$484$	50.6480	2.30218
$485$	0.259470	0.0117819
$486$	190.293	8.63185
$487$	−8.28800	−0.375565	−0.187783	−	0.982211i	$-0.560130\pi$
$487$	−8.28800	−0.375565	−0.187783	+	0.982211i	$0.560130\pi$
$488$	−36.6946	−1.66109
$489$	−75.4710	−3.41292
$490$	51.0935	2.30817
$491$	−23.0921	−1.04213	−0.521066	−	0.853517i	$-0.674465\pi$
$491$	−23.0921	−1.04213	−0.521066	+	0.853517i	$0.674465\pi$
$492$	48.9440	2.20657
$493$	9.58427	0.431654
$494$	−2.87924	−0.129543
$495$	−111.909	−5.02993
$496$	74.3873	3.34009
$497$	0.693787	0.0311206
$498$	−95.4039	−4.27515
$499$	37.0140	1.65697	0.828486	−	0.560010i	$-0.189203\pi$
$499$	37.0140	1.65697	0.828486	+	0.560010i	$0.189203\pi$
$500$	−27.6259	−1.23547
$501$	−26.1683	−1.16911
$502$	−30.0814	−1.34260
$503$	38.5478	1.71876	0.859380	−	0.511338i	$-0.170850\pi$
$503$	38.5478	1.71876	0.859380	+	0.511338i	$0.170850\pi$
$504$	3.64729	0.162463
$505$	28.7004	1.27715
$506$	67.2267	2.98859
$507$	−3.39586	−0.150815
$508$	50.2440	2.22922
$509$	5.62784	0.249449	0.124725	−	0.992191i	$-0.460195\pi$
$509$	5.62784	0.249449	0.124725	+	0.992191i	$0.460195\pi$
$510$	195.572	8.66007
$511$	−0.586710	−0.0259545
$512$	46.4271	2.05181
$513$	−20.8500	−0.920551
$514$	−50.6030	−2.23200
$515$	9.19537	0.405196
$516$	−70.5736	−3.10683
$517$	−55.8576	−2.45662
$518$	0.0549198	0.00241304
$519$	11.8600	0.520598
$520$	−19.9332	−0.874128
$521$	33.0478	1.44785	0.723924	−	0.689879i	$-0.242336\pi$
$521$	33.0478	1.44785	0.723924	+	0.689879i	$0.242336\pi$
$522$	−26.8986	−1.17732
$523$	38.7982	1.69653	0.848263	−	0.529575i	$-0.177648\pi$
$523$	38.7982	1.69653	0.848263	+	0.529575i	$0.177648\pi$
$524$	70.6902	3.08812
$525$	0.599710	0.0261735
$526$	14.4404	0.629631
$527$	65.8453	2.86827
$528$	140.963	6.13465
$529$	7.93586	0.345037
$530$	−31.8492	−1.38344
$531$	−33.3501	−1.44727
$532$	−0.316923	−0.0137404
$533$	−3.04744	−0.131999
$534$	126.455	5.47226
$535$	52.8562	2.28517
$536$	−8.08949	−0.349413
$537$	9.66088	0.416898
$538$	14.1028	0.608017
$539$	32.5980	1.40410
$540$	−250.114	−10.7632
$541$	−17.4737	−0.751251	−0.375626	−	0.926771i	$-0.622572\pi$
$541$	−17.4737	−0.751251	−0.375626	+	0.926771i	$0.622572\pi$
$542$	33.3625	1.43304
$543$	77.5699	3.32884
$544$	−70.5823	−3.02619
$545$	23.4908	1.00623
$546$	−0.531855	−0.0227613
$547$	−30.4782	−1.30315	−0.651577	−	0.758582i	$-0.725892\pi$
$547$	−30.4782	−1.30315	−0.651577	+	0.758582i	$0.725892\pi$
$548$	31.0011	1.32430
$549$	44.2151	1.88706
$550$	35.3550	1.50754
$551$	1.34891	0.0574654
$552$	133.738	5.69227
$553$	0.141697	0.00602557
$554$	−52.8768	−2.24652
$555$	−3.35226	−0.142295
$556$	−73.8126	−3.13035
$557$	−38.7688	−1.64269	−0.821344	−	0.570434i	$-0.806775\pi$
$557$	−38.7688	−1.64269	−0.821344	+	0.570434i	$0.806775\pi$
$558$	−184.797	−7.82310
$559$	4.39418	0.185854
$560$	−1.51423	−0.0639879
$561$	124.776	5.26806
$562$	17.7558	0.748981
$563$	−18.2161	−0.767715	−0.383858	−	0.923392i	$-0.625405\pi$
$563$	−18.2161	−0.767715	−0.383858	+	0.923392i	$0.625405\pi$
$564$	−192.543	−8.10753
$565$	−1.44086	−0.0606175
$566$	31.0977	1.30713
$567$	−2.30611	−0.0968476
$568$	81.3671	3.41409
$569$	13.5336	0.567357	0.283678	−	0.958920i	$-0.408445\pi$
$569$	13.5336	0.567357	0.283678	+	0.958920i	$0.408445\pi$
$570$	27.5252	1.15290
$571$	−26.3624	−1.10323	−0.551616	−	0.834098i	$-0.685989\pi$
$571$	−26.3624	−1.10323	−0.551616	+	0.834098i	$0.685989\pi$
$572$	−22.0361	−0.921376
$573$	36.6780	1.53225
$574$	−0.477285	−0.0199215
$575$	16.2694	0.678480
$576$	46.0686	1.91953
$577$	1.33427	0.0555463	0.0277732	−	0.999614i	$-0.491158\pi$
$577$	1.33427	0.0555463	0.0277732	+	0.999614i	$0.491158\pi$
$578$	−117.231	−4.87616
$579$	62.5586	2.59985
$580$	16.1813	0.671891
$581$	0.653849	0.0271262
$582$	0.811944	0.0336562
$583$	−20.3200	−0.841570
$584$	−68.8092	−2.84734
$585$	24.0185	0.993042
$586$	−17.7226	−0.732114
$587$	−14.6656	−0.605315	−0.302657	−	0.953099i	$-0.597874\pi$
$587$	−14.6656	−0.605315	−0.302657	+	0.953099i	$0.597874\pi$
$588$	112.366	4.63391
$589$	9.26719	0.381848
$590$	28.5462	1.17523
$591$	−33.2109	−1.36612
$592$	3.12409	0.128399
$593$	16.8854	0.693400	0.346700	−	0.937976i	$-0.387302\pi$
$593$	16.8854	0.693400	0.346700	+	0.937976i	$0.387302\pi$
$594$	−227.055	−9.31617
$595$	−1.34035	−0.0549489
$596$	69.2499	2.83659
$597$	−88.1921	−3.60946
$598$	−14.4285	−0.590027
$599$	−12.9685	−0.529877	−0.264938	−	0.964265i	$-0.585352\pi$
$599$	−12.9685	−0.529877	−0.264938	+	0.964265i	$0.585352\pi$
$600$	70.3338	2.87137
$601$	18.1208	0.739163	0.369582	−	0.929198i	$-0.379501\pi$
$601$	18.1208	0.739163	0.369582	+	0.929198i	$0.379501\pi$
$602$	0.688210	0.0280493
$603$	9.74742	0.396946
$604$	−94.6230	−3.85015
$605$	30.1473	1.22566
$606$	89.8103	3.64830
$607$	−24.4641	−0.992966	−0.496483	−	0.868047i	$-0.665375\pi$
$607$	−24.4641	−0.992966	−0.496483	+	0.868047i	$0.665375\pi$
$608$	−9.93389	−0.402872
$609$	0.249171	0.0100969
$610$	−37.8461	−1.53235
$611$	11.9885	0.485001
$612$	318.216	12.8631
$613$	44.7147	1.80601	0.903004	−	0.429631i	$-0.141357\pi$
$613$	44.7147	1.80601	0.903004	+	0.429631i	$0.141357\pi$
$614$	−5.25665	−0.212141
$615$	29.1331	1.17476
$616$	−1.99180	−0.0802520
$617$	−3.84541	−0.154810	−0.0774052	−	0.997000i	$-0.524664\pi$
$617$	−3.84541	−0.154810	−0.0774052	+	0.997000i	$0.524664\pi$
$618$	28.7745	1.15748
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	111.168	4.46460
$621$	−104.484	−4.19281
$622$	−10.5927	−0.424727
$623$	−0.866659	−0.0347220
$624$	−30.2543	−1.21114
$625$	−31.0693	−1.24277
$626$	−11.7168	−0.468299
$627$	17.5613	0.701329
$628$	−10.2637	−0.409564
$629$	2.76534	0.110261
$630$	3.76174	0.149871
$631$	27.7312	1.10396	0.551982	−	0.833856i	$-0.313872\pi$
$631$	27.7312	1.10396	0.551982	+	0.833856i	$0.313872\pi$
$632$	16.6182	0.661036
$633$	−69.7858	−2.77374
$634$	51.4409	2.04298
$635$	29.9069	1.18682
$636$	−70.0438	−2.77742
$637$	−6.99635	−0.277206
$638$	14.6895	0.581562
$639$	−98.0431	−3.87853
$640$	10.9598	0.433222
$641$	19.3541	0.764443	0.382221	−	0.924071i	$-0.375159\pi$
$641$	19.3541	0.764443	0.382221	+	0.924071i	$0.375159\pi$
$642$	165.400	6.52781
$643$	24.6191	0.970883	0.485442	−	0.874269i	$-0.338659\pi$
$643$	24.6191	0.970883	0.485442	+	0.874269i	$0.338659\pi$
$644$	−1.58818	−0.0625829
$645$	−42.0077	−1.65405
$646$	−22.7060	−0.893357
$647$	−0.557741	−0.0219271	−0.0109635	−	0.999940i	$-0.503490\pi$
$647$	−0.557741	−0.0219271	−0.0109635	+	0.999940i	$0.503490\pi$
$648$	−270.460	−10.6247
$649$	18.2127	0.714910
$650$	−7.58807	−0.297629
$651$	1.71184	0.0670922
$652$	105.110	4.11644
$653$	39.5847	1.54907	0.774534	−	0.632532i	$-0.217984\pi$
$653$	39.5847	1.54907	0.774534	+	0.632532i	$0.217984\pi$
$654$	73.5082	2.87440
$655$	42.0772	1.64409
$656$	−27.1501	−1.06004
$657$	82.9115	3.23469
$658$	1.87762	0.0731971
$659$	−6.89660	−0.268653	−0.134327	−	0.990937i	$-0.542887\pi$
$659$	−6.89660	−0.268653	−0.134327	+	0.990937i	$0.542887\pi$
$660$	210.662	8.20001
$661$	−15.0505	−0.585398	−0.292699	−	0.956205i	$-0.594553\pi$
$661$	−15.0505	−0.585398	−0.292699	+	0.956205i	$0.594553\pi$
$662$	65.9713	2.56405
$663$	−26.7801	−1.04005
$664$	76.6831	2.97588
$665$	−0.188643	−0.00731527
$666$	−7.76105	−0.300735
$667$	6.75969	0.261736
$668$	36.4452	1.41011
$669$	90.4499	3.49699
$670$	−8.34334	−0.322331
$671$	−24.1461	−0.932150
$672$	−1.83499	−0.0707864
$673$	−7.77169	−0.299577	−0.149788	−	0.988718i	$-0.547859\pi$
$673$	−7.77169	−0.299577	−0.149788	+	0.988718i	$0.547859\pi$
$674$	15.1983	0.585418
$675$	−54.9490	−2.11499
$676$	4.72950	0.181904
$677$	−7.81150	−0.300220	−0.150110	−	0.988669i	$-0.547963\pi$
$677$	−7.81150	−0.300220	−0.150110	+	0.988669i	$0.547963\pi$
$678$	−4.50880	−0.173160
$679$	−0.00556464	−0.000213551 0
$680$	−157.196	−6.02818
$681$	29.4171	1.12727
$682$	100.919	3.86438
$683$	14.5561	0.556975	0.278487	−	0.960440i	$-0.410167\pi$
$683$	14.5561	0.556975	0.278487	+	0.960440i	$0.410167\pi$
$684$	44.7863	1.71245
$685$	18.4529	0.705048
$686$	−2.19209	−0.0836944
$687$	35.6349	1.35956
$688$	39.1485	1.49252
$689$	4.36119	0.166148
$690$	137.935	5.25109
$691$	−11.8433	−0.450540	−0.225270	−	0.974296i	$-0.572326\pi$
$691$	−11.8433	−0.450540	−0.225270	+	0.974296i	$0.572326\pi$
$692$	−16.5178	−0.627912
$693$	2.40002	0.0911692
$694$	27.2584	1.03471
$695$	−43.9357	−1.66658
$696$	29.2227	1.10768
$697$	−24.0324	−0.910294
$698$	73.8061	2.79360
$699$	−53.2824	−2.01533
$700$	−0.835233	−0.0315688
$701$	−30.7467	−1.16129	−0.580644	−	0.814158i	$-0.697199\pi$
$701$	−30.7467	−1.16129	−0.580644	+	0.814158i	$0.697199\pi$
$702$	48.7317	1.83926
$703$	0.389200	0.0146789
$704$	−25.1583	−0.948189
$705$	−114.608	−4.31639
$706$	23.0359	0.866969
$707$	−0.615513	−0.0231488
$708$	62.7797	2.35941
$709$	40.1533	1.50799	0.753994	−	0.656881i	$-0.228125\pi$
$709$	40.1533	1.50799	0.753994	+	0.656881i	$0.228125\pi$
$710$	83.9204	3.14948
$711$	−20.0240	−0.750960
$712$	−101.641	−3.80917
$713$	46.4400	1.73919
$714$	−4.19427	−0.156967
$715$	−13.1166	−0.490533
$716$	−13.4550	−0.502836
$717$	54.4027	2.03170
$718$	−85.6044	−3.19473
$719$	−9.82105	−0.366263	−0.183132	−	0.983088i	$-0.558623\pi$
$719$	−9.82105	−0.366263	−0.183132	+	0.983088i	$0.558623\pi$
$720$	213.985	7.97475
$721$	−0.197205	−0.00734431
$722$	46.0927	1.71539
$723$	−21.1891	−0.788032
$724$	−108.034	−4.01504
$725$	3.55497	0.132028
$726$	94.3382	3.50122
$727$	4.45121	0.165086	0.0825432	−	0.996587i	$-0.473696\pi$
$727$	4.45121	0.165086	0.0825432	+	0.996587i	$0.473696\pi$
$728$	0.427491	0.0158439
$729$	134.513	4.98196
$730$	−70.9685	−2.62666
$731$	34.6530	1.28169
$732$	−83.2324	−3.07636
$733$	−52.3042	−1.93190	−0.965949	−	0.258733i	$-0.916695\pi$
$733$	−52.3042	−1.93190	−0.965949	+	0.258733i	$0.916695\pi$
$734$	55.4575	2.04697
$735$	66.8842	2.46706
$736$	−49.7810	−1.83495
$737$	−5.32311	−0.196079
$738$	67.4480	2.48280
$739$	−45.4679	−1.67257	−0.836283	−	0.548299i	$-0.815276\pi$
$739$	−45.4679	−1.67257	−0.836283	+	0.548299i	$0.815276\pi$
$740$	4.66878	0.171628
$741$	−3.76909	−0.138461
$742$	0.683044	0.0250753
$743$	−22.0593	−0.809277	−0.404638	−	0.914477i	$-0.632603\pi$
$743$	−22.0593	−0.809277	−0.404638	+	0.914477i	$0.632603\pi$
$744$	200.764	7.36036
$745$	41.2198	1.51018
$746$	55.2672	2.02348
$747$	−92.3992	−3.38071
$748$	−173.779	−6.35400
$749$	−1.13356	−0.0414195
$750$	−51.4566	−1.87893
$751$	−21.8002	−0.795501	−0.397750	−	0.917494i	$-0.630209\pi$
$751$	−21.8002	−0.795501	−0.397750	+	0.917494i	$0.630209\pi$
$752$	106.807	3.89486
$753$	−39.3782	−1.43502
$754$	−3.15273	−0.114816
$755$	−56.3227	−2.04979
$756$	5.36398	0.195086
$757$	−4.87472	−0.177175	−0.0885873	−	0.996068i	$-0.528235\pi$
$757$	−4.87472	−0.177175	−0.0885873	+	0.996068i	$0.528235\pi$
$758$	63.7875	2.31687
$759$	88.0035	3.19432
$760$	−22.1240	−0.802522
$761$	−46.1033	−1.67124	−0.835622	−	0.549304i	$-0.814893\pi$
$761$	−46.1033	−1.67124	−0.835622	+	0.549304i	$0.814893\pi$
$762$	93.5857	3.39025
$763$	−0.503787	−0.0182383
$764$	−51.0825	−1.84810
$765$	189.413	6.84823
$766$	−35.6161	−1.28686
$767$	−3.90890	−0.141142
$768$	70.9683	2.56085
$769$	−15.6734	−0.565197	−0.282599	−	0.959238i	$-0.591196\pi$
$769$	−15.6734	−0.565197	−0.282599	+	0.959238i	$0.591196\pi$
$770$	−2.05431	−0.0740320
$771$	−66.2421	−2.38565
$772$	−87.1271	−3.13577
$773$	30.5303	1.09810	0.549049	−	0.835790i	$-0.314990\pi$
$773$	30.5303	1.09810	0.549049	+	0.835790i	$0.314990\pi$
$774$	−97.2550	−3.49576
$775$	24.4231	0.877305
$776$	−0.652620	−0.0234277
$777$	0.0718931	0.00257915
$778$	79.4189	2.84731
$779$	−3.38237	−0.121186
$780$	−45.2134	−1.61890
$781$	53.5418	1.91588
$782$	−113.785	−4.06895
$783$	−22.8305	−0.815895
$784$	−62.3318	−2.22613
$785$	−6.10927	−0.218049
$786$	131.669	4.69649
$787$	−32.3135	−1.15185	−0.575926	−	0.817502i	$-0.695358\pi$
$787$	−32.3135	−1.15185	−0.575926	+	0.817502i	$0.695358\pi$
$788$	46.2538	1.64772
$789$	18.9033	0.672974
$790$	17.1397	0.609802
$791$	0.0309010	0.00109871
$792$	281.473	10.0017
$793$	5.18236	0.184031
$794$	92.6906	3.28947
$795$	−41.6924	−1.47868
$796$	122.827	4.35350
$797$	44.7721	1.58591	0.792955	−	0.609281i	$-0.208542\pi$
$797$	44.7721	1.58591	0.792955	+	0.609281i	$0.208542\pi$
$798$	−0.590309	−0.0208967
$799$	94.5424	3.34467
$800$	−26.1802	−0.925609
$801$	122.473	4.32736
$802$	−27.5638	−0.973313
$803$	−45.2784	−1.59784
$804$	−18.3490	−0.647118
$805$	−0.945334	−0.0333187
$806$	−21.6597	−0.762931
$807$	18.4614	0.649872
$808$	−72.1872	−2.53954
$809$	23.3146	0.819699	0.409849	−	0.912153i	$-0.365581\pi$
$809$	23.3146	0.819699	0.409849	+	0.912153i	$0.365581\pi$
$810$	−278.947	−9.80120
$811$	−34.2956	−1.20428	−0.602141	−	0.798390i	$-0.705685\pi$
$811$	−34.2956	−1.20428	−0.602141	+	0.798390i	$0.705685\pi$
$812$	−0.347027	−0.0121783
$813$	43.6733	1.53169
$814$	4.23835	0.148554
$815$	62.5652	2.19156
$816$	−238.589	−8.35229
$817$	4.87713	0.170629
$818$	−13.8107	−0.482880
$819$	−0.515104	−0.0179992
$820$	−40.5744	−1.41692
$821$	−38.9950	−1.36094	−0.680468	−	0.732778i	$-0.738224\pi$
$821$	−38.9950	−1.36094	−0.680468	+	0.732778i	$0.738224\pi$
$822$	57.7435	2.01403
$823$	0.373978	0.0130361	0.00651803	−	0.999979i	$-0.497925\pi$
$823$	0.373978	0.0130361	0.00651803	+	0.999979i	$0.497925\pi$
$824$	−23.1282	−0.805708
$825$	46.2817	1.61132
$826$	−0.612207	−0.0213014
$827$	−18.9622	−0.659380	−0.329690	−	0.944089i	$-0.606944\pi$
$827$	−18.9622	−0.659380	−0.329690	+	0.944089i	$0.606944\pi$
$828$	224.434	7.79964
$829$	30.1782	1.04813	0.524066	−	0.851678i	$-0.324415\pi$
$829$	30.1782	1.04813	0.524066	+	0.851678i	$0.324415\pi$
$830$	79.0895	2.74524
$831$	−69.2186	−2.40117
$832$	5.39960	0.187198
$833$	−55.1741	−1.91167
$834$	−137.485	−4.76072
$835$	21.6934	0.750731
$836$	−24.4580	−0.845898
$837$	−156.849	−5.42149
$838$	16.2649	0.561863
$839$	14.8431	0.512440	0.256220	−	0.966618i	$-0.417523\pi$
$839$	14.8431	0.512440	0.256220	+	0.966618i	$0.417523\pi$
$840$	−4.08675	−0.141006
$841$	−27.5230	−0.949068
$842$	−54.7048	−1.88525
$843$	23.2433	0.800541
$844$	97.1925	3.34551
$845$	2.81515	0.0968443
$846$	−265.337	−9.12248
$847$	−0.646545	−0.0222155
$848$	38.8546	1.33427
$849$	40.7086	1.39712
$850$	−59.8404	−2.05251
$851$	1.95037	0.0668578
$852$	184.560	6.32294
$853$	−50.2042	−1.71896	−0.859480	−	0.511170i	$-0.829212\pi$
$853$	−50.2042	−1.71896	−0.859480	+	0.511170i	$0.829212\pi$
$854$	0.811655	0.0277743
$855$	26.6583	0.911694
$856$	−132.944	−4.54393
$857$	−42.7693	−1.46097	−0.730486	−	0.682927i	$-0.760706\pi$
$857$	−42.7693	−1.46097	−0.730486	+	0.682927i	$0.760706\pi$
$858$	−41.0450	−1.40125
$859$	3.60547	0.123017	0.0615086	−	0.998107i	$-0.480409\pi$
$859$	3.60547	0.123017	0.0615086	+	0.998107i	$0.480409\pi$
$860$	58.5053	1.99501
$861$	−0.624793	−0.0212929
$862$	3.33298	0.113522
$863$	27.2897	0.928951	0.464475	−	0.885586i	$-0.346243\pi$
$863$	27.2897	0.928951	0.464475	+	0.885586i	$0.346243\pi$
$864$	168.133	5.71999
$865$	−9.83193	−0.334296
$866$	−12.6405	−0.429542
$867$	−153.462	−5.21184
$868$	−2.38412	−0.0809224
$869$	10.9352	0.370952
$870$	30.1397	1.02183
$871$	1.14247	0.0387113
$872$	−59.0840	−2.00084
$873$	0.786373	0.0266147
$874$	−16.0143	−0.541693
$875$	0.352657	0.0119220
$876$	−156.076	−5.27332
$877$	−26.0101	−0.878297	−0.439149	−	0.898414i	$-0.644720\pi$
$877$	−26.0101	−0.878297	−0.439149	+	0.898414i	$0.644720\pi$
$878$	45.2506	1.52713
$879$	−23.1999	−0.782512
$880$	−116.858	−3.93929
$881$	−48.0713	−1.61956	−0.809782	−	0.586731i	$-0.800414\pi$
$881$	−48.0713	−1.61956	−0.809782	+	0.586731i	$0.800414\pi$
$882$	154.848	5.21401
$883$	18.5975	0.625857	0.312929	−	0.949777i	$-0.398690\pi$
$883$	18.5975	0.625857	0.312929	+	0.949777i	$0.398690\pi$
$884$	37.2974	1.25445
$885$	37.3685	1.25613
$886$	2.35859	0.0792383
$887$	−45.9277	−1.54210	−0.771050	−	0.636775i	$-0.780268\pi$
$887$	−45.9277	−1.54210	−0.771050	+	0.636775i	$0.780268\pi$
$888$	8.43160	0.282946
$889$	−0.641388	−0.0215115
$890$	−104.831	−3.51394
$891$	−177.970	−5.96223
$892$	−125.972	−4.21785
$893$	13.3061	0.445271
$894$	128.987	4.31396
$895$	−8.00884	−0.267706
$896$	−0.235045	−0.00785230
$897$	−18.8878	−0.630644
$898$	−30.1053	−1.00463
$899$	10.1474	0.338436
$900$	118.032	3.93439
$901$	34.3929	1.14579
$902$	−36.8337	−1.22643
$903$	0.900905	0.0299802
$904$	3.62406	0.120534
$905$	−64.3052	−2.13758
$906$	−176.247	−5.85542
$907$	−2.56258	−0.0850891	−0.0425446	−	0.999095i	$-0.513546\pi$
$907$	−2.56258	−0.0850891	−0.0425446	+	0.999095i	$0.513546\pi$
$908$	−40.9700	−1.35964
$909$	86.9818	2.88501
$910$	0.440906	0.0146159
$911$	55.1602	1.82754	0.913770	−	0.406231i	$-0.133157\pi$
$911$	55.1602	1.82754	0.913770	+	0.406231i	$0.133157\pi$
$912$	−33.5795	−1.11193
$913$	50.4597	1.66997
$914$	50.4220	1.66781
$915$	−49.5427	−1.63783
$916$	−49.6297	−1.63981
$917$	−0.902394	−0.0297997
$918$	384.304	12.6839
$919$	10.7243	0.353762	0.176881	−	0.984232i	$-0.443399\pi$
$919$	10.7243	0.353762	0.176881	+	0.984232i	$0.443399\pi$
$920$	−110.868	−3.65522
$921$	−6.88124	−0.226745
$922$	−59.0977	−1.94628
$923$	−11.4914	−0.378245
$924$	−4.51790	−0.148628
$925$	1.02571	0.0337252
$926$	−44.8071	−1.47245
$927$	27.8683	0.915314
$928$	−10.8775	−0.357071
$929$	−16.6205	−0.545300	−0.272650	−	0.962113i	$-0.587900\pi$
$929$	−16.6205	−0.545300	−0.272650	+	0.962113i	$0.587900\pi$
$930$	207.064	6.78989
$931$	−7.76530	−0.254498
$932$	74.2079	2.43076
$933$	−13.8664	−0.453965
$934$	8.63169	0.282438
$935$	−103.439	−3.38282
$936$	−60.4113	−1.97460
$937$	−33.2595	−1.08654	−0.543270	−	0.839558i	$-0.682814\pi$
$937$	−33.2595	−1.08654	−0.543270	+	0.839558i	$0.682814\pi$
$938$	0.178933	0.00584236
$939$	−15.3380	−0.500537
$940$	159.618	5.20615
$941$	−28.6578	−0.934216	−0.467108	−	0.884200i	$-0.654704\pi$
$941$	−28.6578	−0.934216	−0.467108	+	0.884200i	$0.654704\pi$
$942$	−19.1174	−0.622877
$943$	−16.9498	−0.551963
$944$	−34.8251	−1.13346
$945$	3.19282	0.103862
$946$	53.1115	1.72680
$947$	−16.5954	−0.539279	−0.269639	−	0.962961i	$-0.586905\pi$
$947$	−16.5954	−0.539279	−0.269639	+	0.962961i	$0.586905\pi$
$948$	37.6941	1.22425
$949$	9.71788	0.315456
$950$	−8.42205	−0.273248
$951$	67.3390	2.18362
$952$	3.37124	0.109263
$953$	53.9133	1.74642	0.873212	−	0.487341i	$-0.162033\pi$
$953$	53.9133	1.74642	0.873212	+	0.487341i	$0.162033\pi$
$954$	−96.5250	−3.12511
$955$	−30.4060	−0.983914
$956$	−75.7680	−2.45051
$957$	19.2293	0.621596
$958$	88.9090	2.87252
$959$	−0.395744	−0.0127792
$960$	−51.6195	−1.66601
$961$	38.7144	1.24885
$962$	−0.909656	−0.0293285
$963$	160.191	5.16207
$964$	29.5106	0.950474
$965$	−51.8609	−1.66946
$966$	−2.95818	−0.0951778
$967$	20.1730	0.648719	0.324359	−	0.945934i	$-0.394851\pi$
$967$	20.1730	0.648719	0.324359	+	0.945934i	$0.394851\pi$
$968$	−75.8266	−2.43716
$969$	−29.7235	−0.954855
$970$	−0.673099	−0.0216119
$971$	16.6425	0.534083	0.267042	−	0.963685i	$-0.413954\pi$
$971$	16.6425	0.534083	0.267042	+	0.963685i	$0.413954\pi$
$972$	−346.933	−11.1279
$973$	0.942252	0.0302072
$974$	21.5001	0.688909
$975$	−9.93321	−0.318117
$976$	46.1706	1.47788
$977$	35.9000	1.14854	0.574272	−	0.818664i	$-0.305285\pi$
$977$	35.9000	1.14854	0.574272	+	0.818664i	$0.305285\pi$
$978$	195.781	6.26040
$979$	−66.8830	−2.13759
$980$	−93.1514	−2.97561
$981$	71.1932	2.27302
$982$	59.9039	1.91161
$983$	10.4600	0.333621	0.166811	−	0.985989i	$-0.446653\pi$
$983$	10.4600	0.333621	0.166811	+	0.985989i	$0.446653\pi$
$984$	−73.2755	−2.33594
$985$	27.5318	0.877235
$986$	−24.8628	−0.791793
$987$	2.45790	0.0782359
$988$	5.24931	0.167003
$989$	24.4404	0.777160
$990$	290.306	9.22654
$991$	51.1475	1.62475	0.812377	−	0.583132i	$-0.198173\pi$
$991$	51.1475	1.62475	0.812377	+	0.583132i	$0.198173\pi$
$992$	−74.7298	−2.37267
$993$	86.3601	2.74055
$994$	−1.79977	−0.0570853
$995$	73.1109	2.31777
$996$	173.936	5.51138
$997$	8.48569	0.268744	0.134372	−	0.990931i	$-0.457098\pi$
$997$	8.48569	0.268744	0.134372	+	0.990931i	$0.457098\pi$
$998$	−96.0190	−3.03943
$999$	−6.58727	−0.208412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.6	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.6	✓	168	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59413 q^{2} -3.39586 q^{3} +4.72950 q^{4} +2.81515 q^{5} +8.80929 q^{6} -0.0603743 q^{7} -7.08068 q^{8} +8.53185 q^{9} +O(q^{10})\)	\(q-2.59413 q^{2} -3.39586 q^{3} +4.72950 q^{4} +2.81515 q^{5} +8.80929 q^{6} -0.0603743 q^{7} -7.08068 q^{8} +8.53185 q^{9} -7.30287 q^{10} -4.65929 q^{11} -16.0607 q^{12} +1.00000 q^{13} +0.156619 q^{14} -9.55986 q^{15} +8.90918 q^{16} +7.88612 q^{17} -22.1327 q^{18} +1.10991 q^{19} +13.3143 q^{20} +0.205023 q^{21} +12.0868 q^{22} +5.56200 q^{23} +24.0450 q^{24} +2.92510 q^{25} -2.59413 q^{26} -18.7854 q^{27} -0.285540 q^{28} +1.21533 q^{29} +24.7995 q^{30} +8.34952 q^{31} -8.95020 q^{32} +15.8223 q^{33} -20.4576 q^{34} -0.169963 q^{35} +40.3514 q^{36} +0.350660 q^{37} -2.87924 q^{38} -3.39586 q^{39} -19.9332 q^{40} -3.04744 q^{41} -0.531855 q^{42} +4.39418 q^{43} -22.0361 q^{44} +24.0185 q^{45} -14.4285 q^{46} +11.9885 q^{47} -30.2543 q^{48} -6.99635 q^{49} -7.58807 q^{50} -26.7801 q^{51} +4.72950 q^{52} +4.36119 q^{53} +48.7317 q^{54} -13.1166 q^{55} +0.427491 q^{56} -3.76909 q^{57} -3.15273 q^{58} -3.90890 q^{59} -45.2134 q^{60} +5.18236 q^{61} -21.6597 q^{62} -0.515104 q^{63} +5.39960 q^{64} +2.81515 q^{65} -41.0450 q^{66} +1.14247 q^{67} +37.2974 q^{68} -18.8878 q^{69} +0.440906 q^{70} -11.4914 q^{71} -60.4113 q^{72} +9.71788 q^{73} -0.909656 q^{74} -9.93321 q^{75} +5.24931 q^{76} +0.281301 q^{77} +8.80929 q^{78} -2.34698 q^{79} +25.0807 q^{80} +38.1969 q^{81} +7.90544 q^{82} -10.8299 q^{83} +0.969654 q^{84} +22.2006 q^{85} -11.3991 q^{86} -4.12710 q^{87} +32.9909 q^{88} +14.3548 q^{89} -62.3070 q^{90} -0.0603743 q^{91} +26.3055 q^{92} -28.3538 q^{93} -31.0996 q^{94} +3.12456 q^{95} +30.3936 q^{96} +0.0921691 q^{97} +18.1494 q^{98} -39.7523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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