LMFDB - Embedded newform 6047.2.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6047,2,Mod(1,6047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6047 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6047.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:	$48.2855381023$
Analytic rank:	$1$
Dimension:	$217$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54610 q^{2} -2.31345 q^{3} +4.48264 q^{4} -3.02578 q^{5} +5.89027 q^{6} -2.15804 q^{7} -6.32106 q^{8} +2.35204 q^{9} +O(q^{10})$	\(q-2.54610 q^{2} -2.31345 q^{3} +4.48264 q^{4} -3.02578 q^{5} +5.89027 q^{6} -2.15804 q^{7} -6.32106 q^{8} +2.35204 q^{9} +7.70396 q^{10} -1.11866 q^{11} -10.3704 q^{12} -5.26486 q^{13} +5.49458 q^{14} +6.99999 q^{15} +7.12878 q^{16} -6.53216 q^{17} -5.98853 q^{18} -6.25317 q^{19} -13.5635 q^{20} +4.99250 q^{21} +2.84823 q^{22} -1.84125 q^{23} +14.6234 q^{24} +4.15537 q^{25} +13.4049 q^{26} +1.49903 q^{27} -9.67370 q^{28} +1.34606 q^{29} -17.8227 q^{30} +0.169506 q^{31} -5.50850 q^{32} +2.58797 q^{33} +16.6315 q^{34} +6.52975 q^{35} +10.5433 q^{36} -9.24432 q^{37} +15.9212 q^{38} +12.1800 q^{39} +19.1262 q^{40} +11.1562 q^{41} -12.7114 q^{42} -1.50820 q^{43} -5.01456 q^{44} -7.11676 q^{45} +4.68802 q^{46} -4.48501 q^{47} -16.4921 q^{48} -2.34288 q^{49} -10.5800 q^{50} +15.1118 q^{51} -23.6005 q^{52} +6.89328 q^{53} -3.81668 q^{54} +3.38483 q^{55} +13.6411 q^{56} +14.4664 q^{57} -3.42721 q^{58} -2.12259 q^{59} +31.3784 q^{60} +8.00933 q^{61} -0.431580 q^{62} -5.07578 q^{63} -0.232355 q^{64} +15.9303 q^{65} -6.58923 q^{66} -13.2349 q^{67} -29.2813 q^{68} +4.25964 q^{69} -16.6254 q^{70} -0.196267 q^{71} -14.8674 q^{72} -9.73800 q^{73} +23.5370 q^{74} -9.61323 q^{75} -28.0307 q^{76} +2.41411 q^{77} -31.0115 q^{78} -7.98632 q^{79} -21.5702 q^{80} -10.5240 q^{81} -28.4048 q^{82} +14.9679 q^{83} +22.3796 q^{84} +19.7649 q^{85} +3.84003 q^{86} -3.11404 q^{87} +7.07113 q^{88} -4.50608 q^{89} +18.1200 q^{90} +11.3617 q^{91} -8.25368 q^{92} -0.392143 q^{93} +11.4193 q^{94} +18.9208 q^{95} +12.7436 q^{96} +2.24387 q^{97} +5.96522 q^{98} -2.63114 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) $ 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * 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 $

$2$	−2.54610	−1.80037	−0.900183	−	0.435511i	$-0.856568\pi$
$2$	−2.54610	−1.80037	−0.900183	+	0.435511i	$0.856568\pi$
$3$	−2.31345	−1.33567	−0.667835	−	0.744310i	$-0.732779\pi$
$3$	−2.31345	−1.33567	−0.667835	+	0.744310i	$0.732779\pi$
$4$	4.48264	2.24132
$5$	−3.02578	−1.35317	−0.676586	−	0.736364i	$-0.736541\pi$
$5$	−3.02578	−1.35317	−0.676586	+	0.736364i	$0.736541\pi$
$6$	5.89027	2.40469
$7$	−2.15804	−0.815661	−0.407830	−	0.913058i	$-0.633715\pi$
$7$	−2.15804	−0.815661	−0.407830	+	0.913058i	$0.633715\pi$
$8$	−6.32106	−2.23483
$9$	2.35204	0.784013
$10$	7.70396	2.43621
$11$	−1.11866	−0.337290	−0.168645	−	0.985677i	$-0.553939\pi$
$11$	−1.11866	−0.337290	−0.168645	+	0.985677i	$0.553939\pi$
$12$	−10.3704	−2.99366
$13$	−5.26486	−1.46021	−0.730104	−	0.683336i	$-0.760529\pi$
$13$	−5.26486	−1.46021	−0.730104	+	0.683336i	$0.760529\pi$
$14$	5.49458	1.46849
$15$	6.99999	1.80739
$16$	7.12878	1.78220
$17$	−6.53216	−1.58428	−0.792140	−	0.610339i	$-0.791033\pi$
$17$	−6.53216	−1.58428	−0.792140	+	0.610339i	$0.791033\pi$
$18$	−5.98853	−1.41151
$19$	−6.25317	−1.43458	−0.717288	−	0.696777i	$-0.754617\pi$
$19$	−6.25317	−1.43458	−0.717288	+	0.696777i	$0.754617\pi$
$20$	−13.5635	−3.03289
$21$	4.99250	1.08945
$22$	2.84823	0.607245
$23$	−1.84125	−0.383928	−0.191964	−	0.981402i	$-0.561486\pi$
$23$	−1.84125	−0.383928	−0.191964	+	0.981402i	$0.561486\pi$
$24$	14.6234	2.98500
$25$	4.15537	0.831074
$26$	13.4049	2.62891
$27$	1.49903	0.288488
$28$	−9.67370	−1.82816
$29$	1.34606	0.249957	0.124979	−	0.992159i	$-0.460114\pi$
$29$	1.34606	0.249957	0.124979	+	0.992159i	$0.460114\pi$
$30$	−17.8227	−3.25397
$31$	0.169506	0.0304442	0.0152221	−	0.999884i	$-0.495154\pi$
$31$	0.169506	0.0304442	0.0152221	+	0.999884i	$0.495154\pi$
$32$	−5.50850	−0.973775
$33$	2.58797	0.450507
$34$	16.6315	2.85229
$35$	6.52975	1.10373
$36$	10.5433	1.75722
$37$	−9.24432	−1.51976	−0.759878	−	0.650065i	$-0.774741\pi$
$37$	−9.24432	−1.51976	−0.759878	+	0.650065i	$0.774741\pi$
$38$	15.9212	2.58276
$39$	12.1800	1.95036
$40$	19.1262	3.02411
$41$	11.1562	1.74230	0.871152	−	0.491014i	$-0.163374\pi$
$41$	11.1562	1.74230	0.871152	+	0.491014i	$0.163374\pi$
$42$	−12.7114	−1.96142
$43$	−1.50820	−0.229998	−0.114999	−	0.993366i	$-0.536687\pi$
$43$	−1.50820	−0.229998	−0.114999	+	0.993366i	$0.536687\pi$
$44$	−5.01456	−0.755974
$45$	−7.11676	−1.06090
$46$	4.68802	0.691211
$47$	−4.48501	−0.654206	−0.327103	−	0.944989i	$-0.606072\pi$
$47$	−4.48501	−0.654206	−0.327103	+	0.944989i	$0.606072\pi$
$48$	−16.4921	−2.38042
$49$	−2.34288	−0.334697
$50$	−10.5800	−1.49624
$51$	15.1118	2.11607
$52$	−23.6005	−3.27280
$53$	6.89328	0.946864	0.473432	−	0.880830i	$-0.343015\pi$
$53$	6.89328	0.946864	0.473432	+	0.880830i	$0.343015\pi$
$54$	−3.81668	−0.519384
$55$	3.38483	0.456411
$56$	13.6411	1.82286
$57$	14.4664	1.91612
$58$	−3.42721	−0.450015
$59$	−2.12259	−0.276338	−0.138169	−	0.990409i	$-0.544122\pi$
$59$	−2.12259	−0.276338	−0.138169	+	0.990409i	$0.544122\pi$
$60$	31.3784	4.05094
$61$	8.00933	1.02549	0.512745	−	0.858541i	$-0.328629\pi$
$61$	8.00933	1.02549	0.512745	+	0.858541i	$0.328629\pi$
$62$	−0.431580	−0.0548107
$63$	−5.07578	−0.639488
$64$	−0.232355	−0.0290444
$65$	15.9303	1.97591
$66$	−6.58923	−0.811078
$67$	−13.2349	−1.61690	−0.808450	−	0.588564i	$-0.799693\pi$
$67$	−13.2349	−1.61690	−0.808450	+	0.588564i	$0.799693\pi$
$68$	−29.2813	−3.55088
$69$	4.25964	0.512801
$70$	−16.6254	−1.98712
$71$	−0.196267	−0.0232925	−0.0116463	−	0.999932i	$-0.503707\pi$
$71$	−0.196267	−0.0232925	−0.0116463	+	0.999932i	$0.503707\pi$
$72$	−14.8674	−1.75214
$73$	−9.73800	−1.13975	−0.569874	−	0.821732i	$-0.693008\pi$
$73$	−9.73800	−1.13975	−0.569874	+	0.821732i	$0.693008\pi$
$74$	23.5370	2.73612
$75$	−9.61323	−1.11004
$76$	−28.0307	−3.21534
$77$	2.41411	0.275114
$78$	−31.0115	−3.51136
$79$	−7.98632	−0.898531	−0.449266	−	0.893398i	$-0.648314\pi$
$79$	−7.98632	−0.898531	−0.449266	+	0.893398i	$0.648314\pi$
$80$	−21.5702	−2.41162
$81$	−10.5240	−1.16934
$82$	−28.4048	−3.13679
$83$	14.9679	1.64295	0.821473	−	0.570248i	$-0.193153\pi$
$83$	14.9679	1.64295	0.821473	+	0.570248i	$0.193153\pi$
$84$	22.3796	2.44181
$85$	19.7649	2.14380
$86$	3.84003	0.414081
$87$	−3.11404	−0.333860
$88$	7.07113	0.753785
$89$	−4.50608	−0.477643	−0.238822	−	0.971063i	$-0.576761\pi$
$89$	−4.50608	−0.477643	−0.238822	+	0.971063i	$0.576761\pi$
$90$	18.1200	1.91002
$91$	11.3617	1.19103
$92$	−8.25368	−0.860506
$93$	−0.392143	−0.0406633
$94$	11.4193	1.17781
$95$	18.9208	1.94123
$96$	12.7436	1.30064
$97$	2.24387	0.227830	0.113915	−	0.993490i	$-0.463661\pi$
$97$	2.24387	0.227830	0.113915	+	0.993490i	$0.463661\pi$
$98$	5.96522	0.602578
$99$	−2.63114	−0.264439
$100$	18.6270	1.86270
$101$	1.54808	0.154039	0.0770196	−	0.997030i	$-0.475460\pi$
$101$	1.54808	0.154039	0.0770196	+	0.997030i	$0.475460\pi$
$102$	−38.4762	−3.80971
$103$	14.8966	1.46781	0.733905	−	0.679252i	$-0.237696\pi$
$103$	14.8966	1.46781	0.733905	+	0.679252i	$0.237696\pi$
$104$	33.2795	3.26332
$105$	−15.1062	−1.47422
$106$	−17.5510	−1.70470
$107$	−4.51998	−0.436963	−0.218482	−	0.975841i	$-0.570110\pi$
$107$	−4.51998	−0.436963	−0.218482	+	0.975841i	$0.570110\pi$
$108$	6.71960	0.646594
$109$	9.46619	0.906696	0.453348	−	0.891334i	$-0.350229\pi$
$109$	9.46619	0.906696	0.453348	+	0.891334i	$0.350229\pi$
$110$	−8.61813	−0.821707
$111$	21.3862	2.02989
$112$	−15.3842	−1.45367
$113$	−3.25496	−0.306200	−0.153100	−	0.988211i	$-0.548926\pi$
$113$	−3.25496	−0.306200	−0.153100	+	0.988211i	$0.548926\pi$
$114$	−36.8329	−3.44972
$115$	5.57124	0.519521
$116$	6.03391	0.560234
$117$	−12.3831	−1.14482
$118$	5.40434	0.497510
$119$	14.0966	1.29224
$120$	−44.2474	−4.03921
$121$	−9.74859	−0.886236
$122$	−20.3926	−1.84626
$123$	−25.8092	−2.32714
$124$	0.759834	0.0682351
$125$	2.55566	0.228586
$126$	12.9235	1.15131
$127$	−4.37884	−0.388559	−0.194279	−	0.980946i	$-0.562237\pi$
$127$	−4.37884	−0.388559	−0.194279	+	0.980946i	$0.562237\pi$
$128$	11.6086	1.02607
$129$	3.48914	0.307202
$130$	−40.5602	−3.55737
$131$	2.76271	0.241379	0.120689	−	0.992690i	$-0.461490\pi$
$131$	2.76271	0.241379	0.120689	+	0.992690i	$0.461490\pi$
$132$	11.6009	1.00973
$133$	13.4946	1.17013
$134$	33.6974	2.91101
$135$	−4.53573	−0.390374
$136$	41.2901	3.54060
$137$	−11.2697	−0.962839	−0.481419	−	0.876490i	$-0.659879\pi$
$137$	−11.2697	−0.962839	−0.481419	+	0.876490i	$0.659879\pi$
$138$	−10.8455	−0.923230
$139$	17.9048	1.51866	0.759331	−	0.650705i	$-0.225527\pi$
$139$	17.9048	1.51866	0.759331	+	0.650705i	$0.225527\pi$
$140$	29.2705	2.47381
$141$	10.3758	0.873803
$142$	0.499715	0.0419351
$143$	5.88960	0.492513
$144$	16.7672	1.39726
$145$	−4.07289	−0.338235
$146$	24.7940	2.05196
$147$	5.42013	0.447045
$148$	−41.4390	−3.40626
$149$	0.878309	0.0719539	0.0359769	−	0.999353i	$-0.488546\pi$
$149$	0.878309	0.0719539	0.0359769	+	0.999353i	$0.488546\pi$
$150$	24.4763	1.99848
$151$	−21.3497	−1.73741	−0.868707	−	0.495326i	$-0.835049\pi$
$151$	−21.3497	−1.73741	−0.868707	+	0.495326i	$0.835049\pi$
$152$	39.5267	3.20604
$153$	−15.3639	−1.24210
$154$	−6.14658	−0.495306
$155$	−0.512888	−0.0411962
$156$	54.5984	4.37137
$157$	11.1660	0.891143	0.445572	−	0.895246i	$-0.353000\pi$
$157$	11.1660	0.891143	0.445572	+	0.895246i	$0.353000\pi$
$158$	20.3340	1.61769
$159$	−15.9472	−1.26470
$160$	16.6675	1.31768
$161$	3.97349	0.313155
$162$	26.7953	2.10524
$163$	12.3511	0.967411	0.483705	−	0.875231i	$-0.339291\pi$
$163$	12.3511	0.967411	0.483705	+	0.875231i	$0.339291\pi$
$164$	50.0092	3.90506
$165$	−7.83063	−0.609614
$166$	−38.1099	−2.95790
$167$	19.1792	1.48413	0.742067	−	0.670326i	$-0.233846\pi$
$167$	19.1792	1.48413	0.742067	+	0.670326i	$0.233846\pi$
$168$	−31.5579	−2.43474
$169$	14.7187	1.13221
$170$	−50.3235	−3.85963
$171$	−14.7077	−1.12473
$172$	−6.76072	−0.515500
$173$	11.1174	0.845239	0.422619	−	0.906307i	$-0.361111\pi$
$173$	11.1174	0.845239	0.422619	+	0.906307i	$0.361111\pi$
$174$	7.92867	0.601071
$175$	−8.96744	−0.677875
$176$	−7.97471	−0.601116
$177$	4.91051	0.369097
$178$	11.4729	0.859933
$179$	17.3640	1.29784	0.648921	−	0.760855i	$-0.275220\pi$
$179$	17.3640	1.29784	0.648921	+	0.760855i	$0.275220\pi$
$180$	−31.9019	−2.37783
$181$	−20.7141	−1.53967	−0.769833	−	0.638245i	$-0.779661\pi$
$181$	−20.7141	−1.53967	−0.769833	+	0.638245i	$0.779661\pi$
$182$	−28.9282	−2.14430
$183$	−18.5292	−1.36972
$184$	11.6387	0.858014
$185$	27.9713	2.05649
$186$	0.998437	0.0732089
$187$	7.30728	0.534361
$188$	−20.1047	−1.46629
$189$	−3.23495	−0.235308
$190$	−48.1742	−3.49492
$191$	−0.136049	−0.00984415	−0.00492208	−	0.999988i	$-0.501567\pi$
$191$	−0.136049	−0.00984415	−0.00492208	+	0.999988i	$0.501567\pi$
$192$	0.537541	0.0387937
$193$	−4.33713	−0.312193	−0.156097	−	0.987742i	$-0.549891\pi$
$193$	−4.33713	−0.312193	−0.156097	+	0.987742i	$0.549891\pi$
$194$	−5.71312	−0.410178
$195$	−36.8540	−2.63917
$196$	−10.5023	−0.750164
$197$	−17.8907	−1.27466	−0.637331	−	0.770590i	$-0.719962\pi$
$197$	−17.8907	−1.27466	−0.637331	+	0.770590i	$0.719962\pi$
$198$	6.69915	0.476088
$199$	−12.0588	−0.854825	−0.427413	−	0.904057i	$-0.640575\pi$
$199$	−12.0588	−0.854825	−0.427413	+	0.904057i	$0.640575\pi$
$200$	−26.2663	−1.85731
$201$	30.6182	2.15964
$202$	−3.94156	−0.277327
$203$	−2.90485	−0.203880
$204$	67.7408	4.74280
$205$	−33.7562	−2.35764
$206$	−37.9284	−2.64260
$207$	−4.33070	−0.301004
$208$	−37.5320	−2.60238
$209$	6.99519	0.483868
$210$	38.4620	2.65413
$211$	18.4728	1.27172	0.635859	−	0.771806i	$-0.280646\pi$
$211$	18.4728	1.27172	0.635859	+	0.771806i	$0.280646\pi$
$212$	30.9001	2.12223
$213$	0.454052	0.0311111
$214$	11.5083	0.786694
$215$	4.56349	0.311227
$216$	−9.47543	−0.644722
$217$	−0.365800	−0.0248321
$218$	−24.1019	−1.63239
$219$	22.5284	1.52233
$220$	15.1730	1.02296
$221$	34.3909	2.31338
$222$	−54.4516	−3.65455
$223$	10.1377	0.678870	0.339435	−	0.940630i	$-0.389764\pi$
$223$	10.1377	0.678870	0.339435	+	0.940630i	$0.389764\pi$
$224$	11.8875	0.794270
$225$	9.77359	0.651573
$226$	8.28745	0.551273
$227$	−18.9122	−1.25525	−0.627623	−	0.778517i	$-0.715972\pi$
$227$	−18.9122	−1.25525	−0.627623	+	0.778517i	$0.715972\pi$
$228$	64.8476	4.29464
$229$	12.0998	0.799575	0.399787	−	0.916608i	$-0.369084\pi$
$229$	12.0998	0.799575	0.399787	+	0.916608i	$0.369084\pi$
$230$	−14.1849	−0.935328
$231$	−5.58493	−0.367461
$232$	−8.50853	−0.558612
$233$	−0.123894	−0.00811658	−0.00405829	−	0.999992i	$-0.501292\pi$
$233$	−0.123894	−0.00811658	−0.00405829	+	0.999992i	$0.501292\pi$
$234$	31.5288	2.06110
$235$	13.5707	0.885254
$236$	−9.51483	−0.619363
$237$	18.4759	1.20014
$238$	−35.8915	−2.32650
$239$	−4.64682	−0.300578	−0.150289	−	0.988642i	$-0.548020\pi$
$239$	−4.64682	−0.300578	−0.150289	+	0.988642i	$0.548020\pi$
$240$	49.9014	3.22112
$241$	8.69431	0.560049	0.280025	−	0.959993i	$-0.409657\pi$
$241$	8.69431	0.560049	0.280025	+	0.959993i	$0.409657\pi$
$242$	24.8209	1.59555
$243$	19.8497	1.27336
$244$	35.9029	2.29845
$245$	7.08906	0.452903
$246$	65.7130	4.18971
$247$	32.9221	2.09478
$248$	−1.07146	−0.0680376
$249$	−34.6276	−2.19443
$250$	−6.50698	−0.411538
$251$	14.6695	0.925929	0.462965	−	0.886377i	$-0.346786\pi$
$251$	14.6695	0.925929	0.462965	+	0.886377i	$0.346786\pi$
$252$	−22.7529	−1.43330
$253$	2.05974	0.129495
$254$	11.1490	0.699548
$255$	−45.7250	−2.86341
$256$	−29.0920	−1.81825
$257$	14.2408	0.888319	0.444159	−	0.895948i	$-0.353502\pi$
$257$	14.2408	0.888319	0.444159	+	0.895948i	$0.353502\pi$
$258$	−8.88371	−0.553076
$259$	19.9496	1.23961
$260$	71.4099	4.42865
$261$	3.16599	0.195970
$262$	−7.03413	−0.434570
$263$	23.6715	1.45965	0.729823	−	0.683636i	$-0.239602\pi$
$263$	23.6715	1.45965	0.729823	+	0.683636i	$0.239602\pi$
$264$	−16.3587	−1.00681
$265$	−20.8576	−1.28127
$266$	−34.3586	−2.10666
$267$	10.4246	0.637973
$268$	−59.3273	−3.62399
$269$	17.6373	1.07536	0.537681	−	0.843148i	$-0.319300\pi$
$269$	17.6373	1.07536	0.537681	+	0.843148i	$0.319300\pi$
$270$	11.5484	0.702816
$271$	−21.3323	−1.29584	−0.647922	−	0.761706i	$-0.724362\pi$
$271$	−21.3323	−1.29584	−0.647922	+	0.761706i	$0.724362\pi$
$272$	−46.5663	−2.82350
$273$	−26.2848	−1.59083
$274$	28.6939	1.73346
$275$	−4.64846	−0.280313
$276$	19.0945	1.14935
$277$	12.9240	0.776526	0.388263	−	0.921549i	$-0.373075\pi$
$277$	12.9240	0.776526	0.388263	+	0.921549i	$0.373075\pi$
$278$	−45.5873	−2.73415
$279$	0.398684	0.0238686
$280$	−41.2749	−2.46665
$281$	3.65752	0.218189	0.109095	−	0.994031i	$-0.465205\pi$
$281$	3.65752	0.218189	0.109095	+	0.994031i	$0.465205\pi$
$282$	−26.4180	−1.57317
$283$	0.0449931	0.00267456	0.00133728	−	0.999999i	$-0.499574\pi$
$283$	0.0449931	0.00267456	0.00133728	+	0.999999i	$0.499574\pi$
$284$	−0.879792	−0.0522061
$285$	−43.7722	−2.59284
$286$	−14.9955	−0.886704
$287$	−24.0754	−1.42113
$288$	−12.9562	−0.763452
$289$	25.6691	1.50994
$290$	10.3700	0.608947
$291$	−5.19107	−0.304306
$292$	−43.6520	−2.55454
$293$	0.922965	0.0539202	0.0269601	−	0.999637i	$-0.491417\pi$
$293$	0.922965	0.0539202	0.0269601	+	0.999637i	$0.491417\pi$
$294$	−13.8002	−0.804845
$295$	6.42251	0.373933
$296$	58.4339	3.39640
$297$	−1.67691	−0.0973039
$298$	−2.23627	−0.129543
$299$	9.69394	0.560615
$300$	−43.0927	−2.48796
$301$	3.25475	0.187601
$302$	54.3585	3.12798
$303$	−3.58139	−0.205745
$304$	−44.5775	−2.55670
$305$	−24.2345	−1.38766
$306$	39.1180	2.23623
$307$	−7.64583	−0.436371	−0.218185	−	0.975907i	$-0.570014\pi$
$307$	−7.64583	−0.436371	−0.218185	+	0.975907i	$0.570014\pi$
$308$	10.8216	0.616618
$309$	−34.4626	−1.96051
$310$	1.30587	0.0741682
$311$	−27.1648	−1.54037	−0.770187	−	0.637819i	$-0.779837\pi$
$311$	−27.1648	−1.54037	−0.770187	+	0.637819i	$0.779837\pi$
$312$	−76.9903	−4.35872
$313$	22.7039	1.28330	0.641651	−	0.766997i	$-0.278250\pi$
$313$	22.7039	1.28330	0.641651	+	0.766997i	$0.278250\pi$
$314$	−28.4298	−1.60438
$315$	15.3582	0.865338
$316$	−35.7998	−2.01390
$317$	−19.8243	−1.11345	−0.556723	−	0.830698i	$-0.687941\pi$
$317$	−19.8243	−1.11345	−0.556723	+	0.830698i	$0.687941\pi$
$318$	40.6033	2.27692
$319$	−1.50579	−0.0843080
$320$	0.703056	0.0393020
$321$	10.4567	0.583638
$322$	−10.1169	−0.563794
$323$	40.8467	2.27277
$324$	−47.1754	−2.62086
$325$	−21.8774	−1.21354
$326$	−31.4471	−1.74169
$327$	−21.8995	−1.21105
$328$	−70.5189	−3.89375
$329$	9.67882	0.533610
$330$	19.9376	1.09753
$331$	5.94560	0.326800	0.163400	−	0.986560i	$-0.447754\pi$
$331$	5.94560	0.326800	0.163400	+	0.986560i	$0.447754\pi$
$332$	67.0959	3.68237
$333$	−21.7430	−1.19151
$334$	−48.8323	−2.67198
$335$	40.0460	2.18794
$336$	35.5905	1.94162
$337$	11.7125	0.638020	0.319010	−	0.947751i	$-0.396650\pi$
$337$	11.7125	0.638020	0.319010	+	0.947751i	$0.396650\pi$
$338$	−37.4754	−2.03839
$339$	7.53017	0.408983
$340$	88.5989	4.80495
$341$	−0.189620	−0.0102685
$342$	37.4473	2.02492
$343$	20.1623	1.08866
$344$	9.53342	0.514007
$345$	−12.8888	−0.693908
$346$	−28.3060	−1.52174
$347$	−18.1724	−0.975548	−0.487774	−	0.872970i	$-0.662191\pi$
$347$	−18.1724	−0.975548	−0.487774	+	0.872970i	$0.662191\pi$
$348$	−13.9591	−0.748288
$349$	−0.0279519	−0.00149623	−0.000748116	−	1.00000i	$-0.500238\pi$
$349$	−0.0279519	−0.00149623	−0.000748116		1.00000i	$0.500238\pi$
$350$	22.8320	1.22042
$351$	−7.89216	−0.421252
$352$	6.16216	0.328444
$353$	3.52057	0.187381	0.0936905	−	0.995601i	$-0.470134\pi$
$353$	3.52057	0.187381	0.0936905	+	0.995601i	$0.470134\pi$
$354$	−12.5027	−0.664509
$355$	0.593860	0.0315188
$356$	−20.1991	−1.07055
$357$	−32.6118	−1.72600
$358$	−44.2104	−2.33659
$359$	−33.3893	−1.76222	−0.881109	−	0.472914i	$-0.843202\pi$
$359$	−33.3893	−1.76222	−0.881109	+	0.472914i	$0.843202\pi$
$360$	44.9854	2.37094
$361$	20.1022	1.05801
$362$	52.7402	2.77196
$363$	22.5529	1.18372
$364$	50.9306	2.66949
$365$	29.4651	1.54227
$366$	47.1772	2.46599
$367$	−9.05997	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$367$	−9.05997	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$368$	−13.1259	−0.684235
$369$	26.2398	1.36599
$370$	−71.2178	−3.70244
$371$	−14.8759	−0.772320
$372$	−1.75784	−0.0911396
$373$	−22.6728	−1.17396	−0.586978	−	0.809603i	$-0.699682\pi$
$373$	−22.6728	−1.17396	−0.586978	+	0.809603i	$0.699682\pi$
$374$	−18.6051	−0.962046
$375$	−5.91239	−0.305315
$376$	28.3500	1.46204
$377$	−7.08682	−0.364990
$378$	8.23652	0.423641
$379$	6.89468	0.354156	0.177078	−	0.984197i	$-0.443335\pi$
$379$	6.89468	0.354156	0.177078	+	0.984197i	$0.443335\pi$
$380$	84.8149	4.35091
$381$	10.1302	0.518986
$382$	0.346395	0.0177231
$383$	28.3483	1.44853	0.724265	−	0.689521i	$-0.242179\pi$
$383$	28.3483	1.44853	0.724265	+	0.689521i	$0.242179\pi$
$384$	−26.8559	−1.37048
$385$	−7.30459	−0.372276
$386$	11.0428	0.562062
$387$	−3.54734	−0.180322
$388$	10.0584	0.510640
$389$	3.61541	0.183309	0.0916543	−	0.995791i	$-0.470785\pi$
$389$	3.61541	0.183309	0.0916543	+	0.995791i	$0.470785\pi$
$390$	93.8340	4.75147
$391$	12.0274	0.608250
$392$	14.8095	0.747992
$393$	−6.39137	−0.322402
$394$	45.5517	2.29486
$395$	24.1649	1.21587
$396$	−11.7944	−0.592693
$397$	24.5672	1.23299	0.616497	−	0.787357i	$-0.288551\pi$
$397$	24.5672	1.23299	0.616497	+	0.787357i	$0.288551\pi$
$398$	30.7029	1.53900
$399$	−31.2190	−1.56290
$400$	29.6227	1.48114
$401$	30.6801	1.53209	0.766045	−	0.642787i	$-0.222222\pi$
$401$	30.6801	1.53209	0.766045	+	0.642787i	$0.222222\pi$
$402$	−77.9572	−3.88815
$403$	−0.892425	−0.0444548
$404$	6.93946	0.345251
$405$	31.8435	1.58231
$406$	7.39604	0.367059
$407$	10.3413	0.512598
$408$	−95.5226	−4.72907
$409$	0.984912	0.0487007	0.0243504	−	0.999703i	$-0.492248\pi$
$409$	0.984912	0.0487007	0.0243504	+	0.999703i	$0.492248\pi$
$410$	85.9468	4.24461
$411$	26.0719	1.28603
$412$	66.7763	3.28983
$413$	4.58064	0.225398
$414$	11.0264	0.541918
$415$	−45.2898	−2.22319
$416$	29.0015	1.42191
$417$	−41.4217	−2.02843
$418$	−17.8105	−0.871139
$419$	−2.30116	−0.112419	−0.0562096	−	0.998419i	$-0.517902\pi$
$419$	−2.30116	−0.112419	−0.0562096	+	0.998419i	$0.517902\pi$
$420$	−67.7158	−3.30419
$421$	29.9506	1.45970	0.729851	−	0.683606i	$-0.239589\pi$
$421$	29.9506	1.45970	0.729851	+	0.683606i	$0.239589\pi$
$422$	−47.0335	−2.28956
$423$	−10.5489	−0.512906
$424$	−43.5728	−2.11608
$425$	−27.1435	−1.31665
$426$	−1.15606	−0.0560115
$427$	−17.2844	−0.836452
$428$	−20.2615	−0.979374
$429$	−13.6253	−0.657835
$430$	−11.6191	−0.560323
$431$	−20.8300	−1.00335	−0.501673	−	0.865057i	$-0.667282\pi$
$431$	−20.8300	−1.00335	−0.501673	+	0.865057i	$0.667282\pi$
$432$	10.6862	0.514142
$433$	−33.8117	−1.62488	−0.812442	−	0.583042i	$-0.801862\pi$
$433$	−33.8117	−1.62488	−0.812442	+	0.583042i	$0.801862\pi$
$434$	0.931364	0.0447069
$435$	9.42242	0.451770
$436$	42.4335	2.03220
$437$	11.5137	0.550774
$438$	−57.3595	−2.74074
$439$	−12.3996	−0.591798	−0.295899	−	0.955219i	$-0.595619\pi$
$439$	−12.3996	−0.591798	−0.295899	+	0.955219i	$0.595619\pi$
$440$	−21.3957	−1.02000
$441$	−5.51055	−0.262407
$442$	−87.5627	−4.16493
$443$	−26.9209	−1.27905	−0.639526	−	0.768770i	$-0.720869\pi$
$443$	−26.9209	−1.27905	−0.639526	+	0.768770i	$0.720869\pi$
$444$	95.8668	4.54964
$445$	13.6344	0.646333
$446$	−25.8116	−1.22222
$447$	−2.03192	−0.0961066
$448$	0.501430	0.0236903
$449$	29.9104	1.41156	0.705779	−	0.708432i	$-0.250597\pi$
$449$	29.9104	1.41156	0.705779	+	0.708432i	$0.250597\pi$
$450$	−24.8846	−1.17307
$451$	−12.4800	−0.587661
$452$	−14.5908	−0.686293
$453$	49.3914	2.32061
$454$	48.1524	2.25990
$455$	−34.3782	−1.61168
$456$	−91.4429	−4.28220
$457$	4.79167	0.224145	0.112072	−	0.993700i	$-0.464251\pi$
$457$	4.79167	0.224145	0.112072	+	0.993700i	$0.464251\pi$
$458$	−30.8072	−1.43953
$459$	−9.79188	−0.457046
$460$	24.9739	1.16441
$461$	6.91366	0.322001	0.161001	−	0.986954i	$-0.448528\pi$
$461$	6.91366	0.322001	0.161001	+	0.986954i	$0.448528\pi$
$462$	14.2198	0.661565
$463$	−29.1807	−1.35614	−0.678071	−	0.734997i	$-0.737184\pi$
$463$	−29.1807	−1.35614	−0.678071	+	0.734997i	$0.737184\pi$
$464$	9.59578	0.445473
$465$	1.18654	0.0550245
$466$	0.315447	0.0146128
$467$	−12.0740	−0.558719	−0.279360	−	0.960187i	$-0.590122\pi$
$467$	−12.0740	−0.558719	−0.279360	+	0.960187i	$0.590122\pi$
$468$	−55.5092	−2.56591
$469$	28.5614	1.31884
$470$	−34.5523	−1.59378
$471$	−25.8319	−1.19027
$472$	13.4170	0.617570
$473$	1.68717	0.0775760
$474$	−47.0416	−2.16069
$475$	−25.9843	−1.19224
$476$	63.1901	2.89631
$477$	16.2132	0.742354
$478$	11.8313	0.541150
$479$	−25.2183	−1.15225	−0.576127	−	0.817360i	$-0.695437\pi$
$479$	−25.2183	−1.15225	−0.576127	+	0.817360i	$0.695437\pi$
$480$	−38.5595	−1.75999
$481$	48.6700	2.21916
$482$	−22.1366	−1.00829
$483$	−9.19246	−0.418272
$484$	−43.6994	−1.98634
$485$	−6.78946	−0.308293
$486$	−50.5394	−2.29251
$487$	9.99182	0.452772	0.226386	−	0.974038i	$-0.427309\pi$
$487$	9.99182	0.452772	0.226386	+	0.974038i	$0.427309\pi$
$488$	−50.6274	−2.29180
$489$	−28.5736	−1.29214
$490$	−18.0495	−0.815392
$491$	7.59133	0.342592	0.171296	−	0.985220i	$-0.445205\pi$
$491$	7.59133	0.342592	0.171296	+	0.985220i	$0.445205\pi$
$492$	−115.694	−5.21587
$493$	−8.79268	−0.396003
$494$	−83.8230	−3.77137
$495$	7.96125	0.357832
$496$	1.20837	0.0542575
$497$	0.423550	0.0189988
$498$	88.1653	3.95078
$499$	8.02626	0.359305	0.179652	−	0.983730i	$-0.442503\pi$
$499$	8.02626	0.359305	0.179652	+	0.983730i	$0.442503\pi$
$500$	11.4561	0.512333
$501$	−44.3701	−1.98231
$502$	−37.3500	−1.66701
$503$	28.8795	1.28767	0.643836	−	0.765163i	$-0.277342\pi$
$503$	28.8795	1.28767	0.643836	+	0.765163i	$0.277342\pi$
$504$	32.0843	1.42915
$505$	−4.68414	−0.208442
$506$	−5.24432	−0.233138
$507$	−34.0510	−1.51226
$508$	−19.6287	−0.870885
$509$	13.5685	0.601412	0.300706	−	0.953717i	$-0.402778\pi$
$509$	13.5685	0.601412	0.300706	+	0.953717i	$0.402778\pi$
$510$	116.421	5.15519
$511$	21.0150	0.929647
$512$	50.8540	2.24745
$513$	−9.37367	−0.413858
$514$	−36.2586	−1.59930
$515$	−45.0740	−1.98620
$516$	15.6406	0.688537
$517$	5.01722	0.220657
$518$	−50.7937	−2.23174
$519$	−25.7195	−1.12896
$520$	−100.696	−4.41583
$521$	−8.64002	−0.378526	−0.189263	−	0.981926i	$-0.560610\pi$
$521$	−8.64002	−0.378526	−0.189263	+	0.981926i	$0.560610\pi$
$522$	−8.06093	−0.352817
$523$	38.3582	1.67729	0.838645	−	0.544679i	$-0.183349\pi$
$523$	38.3582	1.67729	0.838645	+	0.544679i	$0.183349\pi$
$524$	12.3842	0.541007
$525$	20.7457	0.905416
$526$	−60.2701	−2.62790
$527$	−1.10724	−0.0482321
$528$	18.4491	0.802892
$529$	−19.6098	−0.852599
$530$	53.1055	2.30676
$531$	−4.99242	−0.216653
$532$	60.4913	2.62263
$533$	−58.7357	−2.54413
$534$	−26.5420	−1.14859
$535$	13.6765	0.591286
$536$	83.6586	3.61350
$537$	−40.1706	−1.73349
$538$	−44.9063	−1.93605
$539$	2.62090	0.112890
$540$	−20.3321	−0.874952
$541$	−12.0929	−0.519916	−0.259958	−	0.965620i	$-0.583709\pi$
$541$	−12.0929	−0.519916	−0.259958	+	0.965620i	$0.583709\pi$
$542$	54.3142	2.33300
$543$	47.9210	2.05649
$544$	35.9824	1.54273
$545$	−28.6426	−1.22692
$546$	66.9238	2.86408
$547$	6.79584	0.290569	0.145284	−	0.989390i	$-0.453590\pi$
$547$	6.79584	0.290569	0.145284	+	0.989390i	$0.453590\pi$
$548$	−50.5182	−2.15803
$549$	18.8382	0.803997
$550$	11.8355	0.504666
$551$	−8.41715	−0.358583
$552$	−26.9255	−1.14602
$553$	17.2348	0.732897
$554$	−32.9058	−1.39803
$555$	−64.7102	−2.74679
$556$	80.2606	3.40381
$557$	−5.81733	−0.246488	−0.123244	−	0.992376i	$-0.539330\pi$
$557$	−5.81733	−0.246488	−0.123244	+	0.992376i	$0.539330\pi$
$558$	−1.01509	−0.0429722
$559$	7.94045	0.335845
$560$	46.5492	1.96706
$561$	−16.9050	−0.713730
$562$	−9.31243	−0.392821
$563$	−27.3558	−1.15291	−0.576456	−	0.817128i	$-0.695565\pi$
$563$	−27.3558	−1.15291	−0.576456	+	0.817128i	$0.695565\pi$
$564$	46.5112	1.95847
$565$	9.84879	0.414342
$566$	−0.114557	−0.00481519
$567$	22.7112	0.953782
$568$	1.24061	0.0520549
$569$	−13.7344	−0.575775	−0.287887	−	0.957664i	$-0.592953\pi$
$569$	−13.7344	−0.575775	−0.287887	+	0.957664i	$0.592953\pi$
$570$	111.448	4.66806
$571$	0.218398	0.00913966	0.00456983	−	0.999990i	$-0.498545\pi$
$571$	0.218398	0.00913966	0.00456983	+	0.999990i	$0.498545\pi$
$572$	26.4010	1.10388
$573$	0.314742	0.0131485
$574$	61.2986	2.55855
$575$	−7.65109	−0.319073
$576$	−0.546507	−0.0227711
$577$	−37.2260	−1.54974	−0.774870	−	0.632121i	$-0.782185\pi$
$577$	−37.2260	−1.54974	−0.774870	+	0.632121i	$0.782185\pi$
$578$	−65.3561	−2.71845
$579$	10.0337	0.416987
$580$	−18.2573	−0.758093
$581$	−32.3014	−1.34009
$582$	13.2170	0.547862
$583$	−7.71125	−0.319367
$584$	61.5545	2.54714
$585$	37.4687	1.54914
$586$	−2.34996	−0.0970761
$587$	27.7337	1.14469	0.572346	−	0.820012i	$-0.306033\pi$
$587$	27.7337	1.14469	0.572346	+	0.820012i	$0.306033\pi$
$588$	24.2965	1.00197
$589$	−1.05995	−0.0436745
$590$	−16.3524	−0.673217
$591$	41.3893	1.70253
$592$	−65.9007	−2.70850
$593$	−27.9161	−1.14638	−0.573189	−	0.819423i	$-0.694294\pi$
$593$	−27.9161	−1.14638	−0.573189	+	0.819423i	$0.694294\pi$
$594$	4.26957	0.175183
$595$	−42.6533	−1.74862
$596$	3.93714	0.161272
$597$	27.8974	1.14176
$598$	−24.6818	−1.00931
$599$	−4.71962	−0.192838	−0.0964192	−	0.995341i	$-0.530739\pi$
$599$	−4.71962	−0.192838	−0.0964192	+	0.995341i	$0.530739\pi$
$600$	60.7658	2.48075
$601$	3.85608	0.157293	0.0786463	−	0.996903i	$-0.474940\pi$
$601$	3.85608	0.157293	0.0786463	+	0.996903i	$0.474940\pi$
$602$	−8.28692	−0.337750
$603$	−31.1290	−1.26767
$604$	−95.7030	−3.89410
$605$	29.4971	1.19923
$606$	9.11859	0.370417
$607$	35.6084	1.44530	0.722650	−	0.691214i	$-0.242924\pi$
$607$	35.6084	1.44530	0.722650	+	0.691214i	$0.242924\pi$
$608$	34.4456	1.39695
$609$	6.72021	0.272317
$610$	61.7036	2.49830
$611$	23.6129	0.955278
$612$	−68.8707	−2.78393
$613$	−3.38207	−0.136601	−0.0683003	−	0.997665i	$-0.521758\pi$
$613$	−3.38207	−0.136601	−0.0683003	+	0.997665i	$0.521758\pi$
$614$	19.4671	0.785627
$615$	78.0932	3.14902
$616$	−15.2598	−0.614833
$617$	−43.2857	−1.74262	−0.871308	−	0.490737i	$-0.836728\pi$
$617$	−43.2857	−1.74262	−0.871308	+	0.490737i	$0.836728\pi$
$618$	87.7453	3.52963
$619$	−23.6178	−0.949280	−0.474640	−	0.880180i	$-0.657422\pi$
$619$	−23.6178	−0.949280	−0.474640	+	0.880180i	$0.657422\pi$
$620$	−2.29909	−0.0923338
$621$	−2.76009	−0.110759
$622$	69.1643	2.77324
$623$	9.72427	0.389595
$624$	86.8284	3.47592
$625$	−28.5097	−1.14039
$626$	−57.8065	−2.31041
$627$	−16.1830	−0.646287
$628$	50.0531	1.99734
$629$	60.3853	2.40772
$630$	−39.1036	−1.55792
$631$	2.28587	0.0909990	0.0454995	−	0.998964i	$-0.485512\pi$
$631$	2.28587	0.0909990	0.0454995	+	0.998964i	$0.485512\pi$
$632$	50.4820	2.00807
$633$	−42.7358	−1.69859
$634$	50.4748	2.00461
$635$	13.2494	0.525787
$636$	−71.4857	−2.83459
$637$	12.3349	0.488728
$638$	3.83389	0.151785
$639$	−0.461626	−0.0182617
$640$	−35.1251	−1.38844
$641$	−39.0478	−1.54229	−0.771147	−	0.636657i	$-0.780317\pi$
$641$	−39.0478	−1.54229	−0.771147	+	0.636657i	$0.780317\pi$
$642$	−26.6239	−1.05076
$643$	−3.64472	−0.143734	−0.0718669	−	0.997414i	$-0.522896\pi$
$643$	−3.64472	−0.143734	−0.0718669	+	0.997414i	$0.522896\pi$
$644$	17.8117	0.701881
$645$	−10.5574	−0.415697
$646$	−104.000	−4.09182
$647$	2.37465	0.0933571	0.0466785	−	0.998910i	$-0.485136\pi$
$647$	2.37465	0.0933571	0.0466785	+	0.998910i	$0.485136\pi$
$648$	66.5230	2.61327
$649$	2.37447	0.0932060
$650$	55.7022	2.18482
$651$	0.846259	0.0331675
$652$	55.3654	2.16828
$653$	−41.9990	−1.64355	−0.821774	−	0.569814i	$-0.807015\pi$
$653$	−41.9990	−1.64355	−0.821774	+	0.569814i	$0.807015\pi$
$654$	55.7584	2.18033
$655$	−8.35935	−0.326627
$656$	79.5300	3.10513
$657$	−22.9042	−0.893576
$658$	−24.6433	−0.960694
$659$	41.3762	1.61179	0.805894	−	0.592059i	$-0.201685\pi$
$659$	41.3762	1.61179	0.805894	+	0.592059i	$0.201685\pi$
$660$	−35.1019	−1.36634
$661$	−47.6937	−1.85507	−0.927535	−	0.373735i	$-0.878077\pi$
$661$	−47.6937	−1.85507	−0.927535	+	0.373735i	$0.878077\pi$
$662$	−15.1381	−0.588360
$663$	−79.5615	−3.08991
$664$	−94.6133	−3.67171
$665$	−40.8317	−1.58338
$666$	55.3599	2.14515
$667$	−2.47844	−0.0959656
$668$	85.9736	3.32642
$669$	−23.4530	−0.906746
$670$	−101.961	−3.93910
$671$	−8.95974	−0.345887
$672$	−27.5012	−1.06088
$673$	−48.6598	−1.87570	−0.937850	−	0.347042i	$-0.887186\pi$
$673$	−48.6598	−1.87570	−0.937850	+	0.347042i	$0.887186\pi$
$674$	−29.8212	−1.14867
$675$	6.22901	0.239755
$676$	65.9787	2.53764
$677$	−15.9423	−0.612711	−0.306355	−	0.951917i	$-0.599110\pi$
$677$	−15.9423	−0.612711	−0.306355	+	0.951917i	$0.599110\pi$
$678$	−19.1726	−0.736319
$679$	−4.84235	−0.185832
$680$	−124.935	−4.79104
$681$	43.7524	1.67659
$682$	0.482792	0.0184871
$683$	3.83176	0.146618	0.0733092	−	0.997309i	$-0.476644\pi$
$683$	3.83176	0.146618	0.0733092	+	0.997309i	$0.476644\pi$
$684$	−65.9293	−2.52087
$685$	34.0998	1.30289
$686$	−51.3352	−1.95999
$687$	−27.9922	−1.06797
$688$	−10.7516	−0.409902
$689$	−36.2921	−1.38262
$690$	32.8161	1.24929
$691$	−11.6515	−0.443245	−0.221622	−	0.975133i	$-0.571135\pi$
$691$	−11.6515	−0.443245	−0.221622	+	0.975133i	$0.571135\pi$
$692$	49.8352	1.89445
$693$	5.67809	0.215693
$694$	46.2689	1.75634
$695$	−54.1759	−2.05501
$696$	19.6840	0.746122
$697$	−72.8739	−2.76030
$698$	0.0711685	0.00269377
$699$	0.286623	0.0108411
$700$	−40.1978	−1.51933
$701$	−31.6364	−1.19489	−0.597445	−	0.801910i	$-0.703817\pi$
$701$	−31.6364	−1.19489	−0.597445	+	0.801910i	$0.703817\pi$
$702$	20.0943	0.758409
$703$	57.8063	2.18021
$704$	0.259927	0.00979636
$705$	−31.3951	−1.18241
$706$	−8.96374	−0.337355
$707$	−3.34080	−0.125644
$708$	22.0121	0.827264
$709$	−32.7582	−1.23026	−0.615130	−	0.788426i	$-0.710897\pi$
$709$	−32.7582	−1.23026	−0.615130	+	0.788426i	$0.710897\pi$
$710$	−1.51203	−0.0567454
$711$	−18.7841	−0.704460
$712$	28.4832	1.06745
$713$	−0.312104	−0.0116884
$714$	83.0330	3.10743
$715$	−17.8207	−0.666455
$716$	77.8364	2.90888
$717$	10.7502	0.401472
$718$	85.0125	3.17264
$719$	−26.0367	−0.971006	−0.485503	−	0.874235i	$-0.661363\pi$
$719$	−26.0367	−0.971006	−0.485503	+	0.874235i	$0.661363\pi$
$720$	−50.7338	−1.89074
$721$	−32.1475	−1.19723
$722$	−51.1822	−1.90480
$723$	−20.1138	−0.748041
$724$	−92.8539	−3.45089
$725$	5.59338	0.207733
$726$	−57.4219	−2.13113
$727$	−20.2033	−0.749298	−0.374649	−	0.927167i	$-0.622237\pi$
$727$	−20.2033	−0.749298	−0.374649	+	0.927167i	$0.622237\pi$
$728$	−71.8183	−2.66176
$729$	−14.3492	−0.531450
$730$	−75.0212	−2.77666
$731$	9.85179	0.364382
$732$	−83.0596	−3.06997
$733$	−26.7064	−0.986423	−0.493212	−	0.869909i	$-0.664177\pi$
$733$	−26.7064	−0.986423	−0.493212	+	0.869909i	$0.664177\pi$
$734$	23.0676	0.851442
$735$	−16.4002	−0.604929
$736$	10.1426	0.373859
$737$	14.8054	0.545364
$738$	−66.8092	−2.45928
$739$	−6.02930	−0.221791	−0.110896	−	0.993832i	$-0.535372\pi$
$739$	−6.02930	−0.221791	−0.110896	+	0.993832i	$0.535372\pi$
$740$	125.385	4.60926
$741$	−76.1634	−2.79793
$742$	37.8757	1.39046
$743$	2.64757	0.0971300	0.0485650	−	0.998820i	$-0.484535\pi$
$743$	2.64757	0.0971300	0.0485650	+	0.998820i	$0.484535\pi$
$744$	2.47876	0.0908757
$745$	−2.65757	−0.0973660
$746$	57.7274	2.11355
$747$	35.2052	1.28809
$748$	32.7559	1.19767
$749$	9.75428	0.356414
$750$	15.0536	0.549678
$751$	45.9015	1.67497	0.837485	−	0.546460i	$-0.184025\pi$
$751$	45.9015	1.67497	0.837485	+	0.546460i	$0.184025\pi$
$752$	−31.9727	−1.16592
$753$	−33.9371	−1.23674
$754$	18.0438	0.657116
$755$	64.5996	2.35102
$756$	−14.5011	−0.527401
$757$	23.7709	0.863966	0.431983	−	0.901882i	$-0.357814\pi$
$757$	23.7709	0.863966	0.431983	+	0.901882i	$0.357814\pi$
$758$	−17.5546	−0.637611
$759$	−4.76511	−0.172962
$760$	−119.599	−4.33832
$761$	−14.3488	−0.520144	−0.260072	−	0.965589i	$-0.583746\pi$
$761$	−14.3488	−0.520144	−0.260072	+	0.965589i	$0.583746\pi$
$762$	−25.7925	−0.934365
$763$	−20.4284	−0.739556
$764$	−0.609858	−0.0220639
$765$	46.4878	1.68077
$766$	−72.1777	−2.60789
$767$	11.1752	0.403512
$768$	67.3028	2.42858
$769$	−15.8726	−0.572382	−0.286191	−	0.958173i	$-0.592389\pi$
$769$	−15.8726	−0.572382	−0.286191	+	0.958173i	$0.592389\pi$
$770$	18.5982	0.670234
$771$	−32.9454	−1.18650
$772$	−19.4418	−0.699725
$773$	42.7407	1.53728	0.768639	−	0.639683i	$-0.220934\pi$
$773$	42.7407	1.53728	0.768639	+	0.639683i	$0.220934\pi$
$774$	9.03190	0.324645
$775$	0.704360	0.0253014
$776$	−14.1836	−0.509162
$777$	−46.1523	−1.65570
$778$	−9.20521	−0.330023
$779$	−69.7616	−2.49947
$780$	−165.203	−5.91522
$781$	0.219556	0.00785633
$782$	−30.6229	−1.09507
$783$	2.01778	0.0721096
$784$	−16.7019	−0.596497
$785$	−33.7859	−1.20587
$786$	16.2731	0.580442
$787$	17.0715	0.608533	0.304266	−	0.952587i	$-0.401589\pi$
$787$	17.0715	0.608533	0.304266	+	0.952587i	$0.401589\pi$
$788$	−80.1978	−2.85693
$789$	−54.7627	−1.94961
$790$	−61.5263	−2.18901
$791$	7.02431	0.249756
$792$	16.6316	0.590977
$793$	−42.1680	−1.49743
$794$	−62.5507	−2.21984
$795$	48.2529	1.71135
$796$	−54.0552	−1.91594
$797$	34.1268	1.20883	0.604417	−	0.796668i	$-0.293406\pi$
$797$	34.1268	1.20883	0.604417	+	0.796668i	$0.293406\pi$
$798$	79.4867	2.81380
$799$	29.2968	1.03645
$800$	−22.8899	−0.809279
$801$	−10.5985	−0.374478
$802$	−78.1146	−2.75832
$803$	10.8935	0.384425
$804$	137.251	4.84046
$805$	−12.0229	−0.423753
$806$	2.27220	0.0800350
$807$	−40.8029	−1.43633
$808$	−9.78547	−0.344252
$809$	43.8724	1.54247	0.771236	−	0.636549i	$-0.219639\pi$
$809$	43.8724	1.54247	0.771236	+	0.636549i	$0.219639\pi$
$810$	−81.0767	−2.84875
$811$	−0.487573	−0.0171210	−0.00856051	−	0.999963i	$-0.502725\pi$
$811$	−0.487573	−0.0171210	−0.00856051	+	0.999963i	$0.502725\pi$
$812$	−13.0214	−0.456961
$813$	49.3511	1.73082
$814$	−26.3300	−0.922864
$815$	−37.3717	−1.30907
$816$	107.729	3.77126
$817$	9.43103	0.329950
$818$	−2.50769	−0.0876792
$819$	26.7233	0.933786
$820$	−151.317	−5.28422
$821$	−3.39195	−0.118380	−0.0591900	−	0.998247i	$-0.518852\pi$
$821$	−3.39195	−0.118380	−0.0591900	+	0.998247i	$0.518852\pi$
$822$	−66.3818	−2.31533
$823$	2.28824	0.0797629	0.0398814	−	0.999204i	$-0.487302\pi$
$823$	2.28824	0.0797629	0.0398814	+	0.999204i	$0.487302\pi$
$824$	−94.1625	−3.28031
$825$	10.7540	0.374405
$826$	−11.6628	−0.405800
$827$	14.9974	0.521511	0.260756	−	0.965405i	$-0.416028\pi$
$827$	14.9974	0.521511	0.260756	+	0.965405i	$0.416028\pi$
$828$	−19.4130	−0.674647
$829$	−46.8989	−1.62887	−0.814433	−	0.580257i	$-0.802952\pi$
$829$	−46.8989	−1.62887	−0.814433	+	0.580257i	$0.802952\pi$
$830$	115.312	4.00255
$831$	−29.8989	−1.03718
$832$	1.22332	0.0424108
$833$	15.3041	0.530255
$834$	105.464	3.65192
$835$	−58.0322	−2.00829
$836$	31.3569	1.08450
$837$	0.254094	0.00878277
$838$	5.85900	0.202396
$839$	34.0396	1.17518	0.587588	−	0.809160i	$-0.300078\pi$
$839$	34.0396	1.17518	0.587588	+	0.809160i	$0.300078\pi$
$840$	95.4874	3.29463
$841$	−27.1881	−0.937521
$842$	−76.2573	−2.62800
$843$	−8.46148	−0.291429
$844$	82.8067	2.85033
$845$	−44.5357	−1.53207
$846$	26.8586	0.923419
$847$	21.0378	0.722868
$848$	49.1407	1.68750
$849$	−0.104089	−0.00357233
$850$	69.1102	2.37046
$851$	17.0211	0.583477
$852$	2.03535	0.0697300
$853$	−5.66104	−0.193830	−0.0969151	−	0.995293i	$-0.530898\pi$
$853$	−5.66104	−0.193830	−0.0969151	+	0.995293i	$0.530898\pi$
$854$	44.0079	1.50592
$855$	44.5023	1.52195
$856$	28.5711	0.976539
$857$	10.3392	0.353181	0.176590	−	0.984284i	$-0.443493\pi$
$857$	10.3392	0.353181	0.176590	+	0.984284i	$0.443493\pi$
$858$	34.6914	1.18434
$859$	51.8642	1.76958	0.884791	−	0.465988i	$-0.154301\pi$
$859$	51.8642	1.76958	0.884791	+	0.465988i	$0.154301\pi$
$860$	20.4565	0.697560
$861$	55.6973	1.89816
$862$	53.0354	1.80639
$863$	15.6714	0.533460	0.266730	−	0.963771i	$-0.414057\pi$
$863$	15.6714	0.533460	0.266730	+	0.963771i	$0.414057\pi$
$864$	−8.25739	−0.280922
$865$	−33.6388	−1.14375
$866$	86.0880	2.92539
$867$	−59.3840	−2.01679
$868$	−1.63975	−0.0556567
$869$	8.93400	0.303065
$870$	−23.9904	−0.813352
$871$	69.6799	2.36101
$872$	−59.8363	−2.02631
$873$	5.27766	0.178622
$874$	−29.3150	−0.991595
$875$	−5.51521	−0.186448
$876$	100.987	3.41202
$877$	41.7811	1.41085	0.705424	−	0.708786i	$-0.250757\pi$
$877$	41.7811	1.41085	0.705424	+	0.708786i	$0.250757\pi$
$878$	31.5705	1.06545
$879$	−2.13523	−0.0720195
$880$	24.1297	0.813413
$881$	50.2244	1.69210	0.846052	−	0.533100i	$-0.178973\pi$
$881$	50.2244	1.69210	0.846052	+	0.533100i	$0.178973\pi$
$882$	14.0304	0.472429
$883$	33.1248	1.11474	0.557369	−	0.830265i	$-0.311811\pi$
$883$	33.1248	1.11474	0.557369	+	0.830265i	$0.311811\pi$
$884$	154.162	5.18503
$885$	−14.8581	−0.499451
$886$	68.5434	2.30276
$887$	−32.0100	−1.07479	−0.537396	−	0.843330i	$-0.680592\pi$
$887$	−32.0100	−1.07479	−0.537396	+	0.843330i	$0.680592\pi$
$888$	−135.184	−4.53647
$889$	9.44968	0.316932
$890$	−34.7146	−1.16364
$891$	11.7728	0.394405
$892$	45.4436	1.52157
$893$	28.0456	0.938509
$894$	5.17348	0.173027
$895$	−52.5396	−1.75620
$896$	−25.0518	−0.836921
$897$	−22.4264	−0.748796
$898$	−76.1549	−2.54132
$899$	0.228165	0.00760974
$900$	43.8115	1.46038
$901$	−45.0280	−1.50010
$902$	31.7754	1.05800
$903$	−7.52969	−0.250572
$904$	20.5748	0.684306
$905$	62.6764	2.08343
$906$	−125.756	−4.17795
$907$	−51.8138	−1.72045	−0.860225	−	0.509915i	$-0.829677\pi$
$907$	−51.8138	−1.72045	−0.860225	+	0.509915i	$0.829677\pi$
$908$	−84.7766	−2.81341
$909$	3.64113	0.120769
$910$	87.5304	2.90161
$911$	−1.01322	−0.0335694	−0.0167847	−	0.999859i	$-0.505343\pi$
$911$	−1.01322	−0.0335694	−0.0167847	+	0.999859i	$0.505343\pi$
$912$	103.128	3.41490
$913$	−16.7441	−0.554148
$914$	−12.2001	−0.403543
$915$	56.0653	1.85346
$916$	54.2389	1.79210
$917$	−5.96202	−0.196883
$918$	24.9311	0.822850
$919$	44.4151	1.46512	0.732559	−	0.680704i	$-0.238326\pi$
$919$	44.4151	1.46512	0.732559	+	0.680704i	$0.238326\pi$
$920$	−35.2161	−1.16104
$921$	17.6882	0.582847
$922$	−17.6029	−0.579721
$923$	1.03332	0.0340120
$924$	−25.0352	−0.823598
$925$	−38.4136	−1.26303
$926$	74.2970	2.44155
$927$	35.0375	1.15078
$928$	−7.41478	−0.243402
$929$	−3.62621	−0.118972	−0.0594860	−	0.998229i	$-0.518946\pi$
$929$	−3.62621	−0.118972	−0.0594860	+	0.998229i	$0.518946\pi$
$930$	−3.02105	−0.0990643
$931$	14.6504	0.480149
$932$	−0.555373	−0.0181918
$933$	62.8443	2.05743
$934$	30.7417	1.00590
$935$	−22.1103	−0.723083
$936$	78.2746	2.55848
$937$	13.5578	0.442914	0.221457	−	0.975170i	$-0.428919\pi$
$937$	13.5578	0.442914	0.221457	+	0.975170i	$0.428919\pi$
$938$	−72.7202	−2.37440
$939$	−52.5243	−1.71407
$940$	60.8325	1.98414
$941$	22.5394	0.734764	0.367382	−	0.930070i	$-0.380254\pi$
$941$	22.5394	0.734764	0.367382	+	0.930070i	$0.380254\pi$
$942$	65.7708	2.14293
$943$	−20.5414	−0.668919
$944$	−15.1315	−0.492489
$945$	9.78827	0.318412
$946$	−4.29570	−0.139665
$947$	29.0633	0.944429	0.472214	−	0.881484i	$-0.343455\pi$
$947$	29.0633	0.944429	0.472214	+	0.881484i	$0.343455\pi$
$948$	82.8210	2.68990
$949$	51.2692	1.66427
$950$	66.1586	2.14647
$951$	45.8625	1.48719
$952$	−89.1056	−2.88793
$953$	−0.340675	−0.0110356	−0.00551778	−	0.999985i	$-0.501756\pi$
$953$	−0.340675	−0.0110356	−0.00551778	+	0.999985i	$0.501756\pi$
$954$	−41.2806	−1.33651
$955$	0.411655	0.0133208
$956$	−20.8300	−0.673690
$957$	3.48356	0.112608
$958$	64.2084	2.07448
$959$	24.3205	0.785350
$960$	−1.62648	−0.0524945
$961$	−30.9713	−0.999073
$962$	−123.919	−3.99530
$963$	−10.6312	−0.342585
$964$	38.9734	1.25525
$965$	13.1232	0.422451
$966$	23.4050	0.753042
$967$	0.227348	0.00731102	0.00365551	−	0.999993i	$-0.498836\pi$
$967$	0.227348	0.00731102	0.00365551	+	0.999993i	$0.498836\pi$
$968$	61.6214	1.98059
$969$	−94.4967	−3.03567
$970$	17.2867	0.555041
$971$	53.3032	1.71058	0.855291	−	0.518148i	$-0.173378\pi$
$971$	53.3032	1.71058	0.855291	+	0.518148i	$0.173378\pi$
$972$	88.9791	2.85401
$973$	−38.6391	−1.23871
$974$	−25.4402	−0.815156
$975$	50.6123	1.62089
$976$	57.0968	1.82762
$977$	−55.6819	−1.78142	−0.890712	−	0.454569i	$-0.849793\pi$
$977$	−55.6819	−1.78142	−0.890712	+	0.454569i	$0.849793\pi$
$978$	72.7512	2.32633
$979$	5.04078	0.161104
$980$	31.7777	1.01510
$981$	22.2648	0.710861
$982$	−19.3283	−0.616791
$983$	−29.7370	−0.948462	−0.474231	−	0.880400i	$-0.657274\pi$
$983$	−29.7370	−0.948462	−0.474231	+	0.880400i	$0.657274\pi$
$984$	163.142	5.20077
$985$	54.1335	1.72484
$986$	22.3871	0.712950
$987$	−22.3914	−0.712727
$988$	147.578	4.69507
$989$	2.77698	0.0883028
$990$	−20.2702	−0.644228
$991$	37.0786	1.17784	0.588920	−	0.808191i	$-0.299553\pi$
$991$	37.0786	1.17784	0.588920	+	0.808191i	$0.299553\pi$
$992$	−0.933724	−0.0296458
$993$	−13.7548	−0.436497
$994$	−1.07840	−0.0342048
$995$	36.4873	1.15673
$996$	−155.223	−4.91842
$997$	22.3345	0.707341	0.353671	−	0.935370i	$-0.384933\pi$
$997$	22.3345	0.707341	0.353671	+	0.935370i	$0.384933\pi$
$998$	−20.4357	−0.646880
$999$	−13.8575	−0.438431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6047.2.a.a.1.13	✓	217

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6047.2.a.a.1.13	✓	217	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:	\( N \)	\(=\)	\( 6047 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54610 q^{2} -2.31345 q^{3} +4.48264 q^{4} -3.02578 q^{5} +5.89027 q^{6} -2.15804 q^{7} -6.32106 q^{8} +2.35204 q^{9} +O(q^{10})\)	\(q-2.54610 q^{2} -2.31345 q^{3} +4.48264 q^{4} -3.02578 q^{5} +5.89027 q^{6} -2.15804 q^{7} -6.32106 q^{8} +2.35204 q^{9} +7.70396 q^{10} -1.11866 q^{11} -10.3704 q^{12} -5.26486 q^{13} +5.49458 q^{14} +6.99999 q^{15} +7.12878 q^{16} -6.53216 q^{17} -5.98853 q^{18} -6.25317 q^{19} -13.5635 q^{20} +4.99250 q^{21} +2.84823 q^{22} -1.84125 q^{23} +14.6234 q^{24} +4.15537 q^{25} +13.4049 q^{26} +1.49903 q^{27} -9.67370 q^{28} +1.34606 q^{29} -17.8227 q^{30} +0.169506 q^{31} -5.50850 q^{32} +2.58797 q^{33} +16.6315 q^{34} +6.52975 q^{35} +10.5433 q^{36} -9.24432 q^{37} +15.9212 q^{38} +12.1800 q^{39} +19.1262 q^{40} +11.1562 q^{41} -12.7114 q^{42} -1.50820 q^{43} -5.01456 q^{44} -7.11676 q^{45} +4.68802 q^{46} -4.48501 q^{47} -16.4921 q^{48} -2.34288 q^{49} -10.5800 q^{50} +15.1118 q^{51} -23.6005 q^{52} +6.89328 q^{53} -3.81668 q^{54} +3.38483 q^{55} +13.6411 q^{56} +14.4664 q^{57} -3.42721 q^{58} -2.12259 q^{59} +31.3784 q^{60} +8.00933 q^{61} -0.431580 q^{62} -5.07578 q^{63} -0.232355 q^{64} +15.9303 q^{65} -6.58923 q^{66} -13.2349 q^{67} -29.2813 q^{68} +4.25964 q^{69} -16.6254 q^{70} -0.196267 q^{71} -14.8674 q^{72} -9.73800 q^{73} +23.5370 q^{74} -9.61323 q^{75} -28.0307 q^{76} +2.41411 q^{77} -31.0115 q^{78} -7.98632 q^{79} -21.5702 q^{80} -10.5240 q^{81} -28.4048 q^{82} +14.9679 q^{83} +22.3796 q^{84} +19.7649 q^{85} +3.84003 q^{86} -3.11404 q^{87} +7.07113 q^{88} -4.50608 q^{89} +18.1200 q^{90} +11.3617 q^{91} -8.25368 q^{92} -0.392143 q^{93} +11.4193 q^{94} +18.9208 q^{95} +12.7436 q^{96} +2.24387 q^{97} +5.96522 q^{98} -2.63114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9}+O(q^{10}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9	\( 217 q - 20 q^{2} - 27 q^{3} + 184 q^{4} - 19 q^{5} - 17 q^{6} - 48 q^{7} - 57 q^{8} + 152 q^{9} - 46 q^{10} - 32 q^{11} - 72 q^{12} - 80 q^{13} - 22 q^{14} - 43 q^{15} + 122 q^{16} - 61 q^{17} - 88 q^{18} - 43 q^{19} - 41 q^{20} - 61 q^{21} - 93 q^{22} - 60 q^{23} - 41 q^{24} + 26 q^{25} - 9 q^{26} - 93 q^{27} - 126 q^{28} - 47 q^{29} - 36 q^{30} - 100 q^{31} - 114 q^{32} - 133 q^{33} - 75 q^{34} - 37 q^{35} + 75 q^{36} - 264 q^{37} - 35 q^{38} - 47 q^{39} - 118 q^{40} - 72 q^{41} - 64 q^{42} - 107 q^{43} - 59 q^{44} - 69 q^{45} - 111 q^{46} - 54 q^{47} - 135 q^{48} + 33 q^{49} - 42 q^{50} - 26 q^{51} - 173 q^{52} - 103 q^{53} - 28 q^{54} - 78 q^{55} - 44 q^{56} - 205 q^{57} - 189 q^{58} - 38 q^{59} - 105 q^{60} - 108 q^{61} - 14 q^{62} - 116 q^{63} + 39 q^{64} - 146 q^{65} + 5 q^{66} - 206 q^{67} - 62 q^{68} - 55 q^{69} - 125 q^{70} - 78 q^{71} - 225 q^{72} - 326 q^{73} + 3 q^{74} - 95 q^{75} - 84 q^{76} - 79 q^{77} - 86 q^{78} - 117 q^{79} - 39 q^{80} + q^{81} - 96 q^{82} - 23 q^{83} - 57 q^{84} - 224 q^{85} - 7 q^{86} - 45 q^{87} - 250 q^{88} - 104 q^{89} - 36 q^{90} - 96 q^{91} - 137 q^{92} - 155 q^{93} - 48 q^{94} - 38 q^{95} - 33 q^{96} - 447 q^{97} - 46 q^{98} - 94 q^{99}+O(q^{100}) \) 217 * q - 20 * q^2 - 27 * q^3 + 184 * q^4 - 19 * q^5 - 17 * q^6 - 48 * q^7 - 57 * q^8 + 152 * q^9 - 46 * q^10 - 32 * q^11 - 72 * q^12 - 80 * q^13 - 22 * q^14 - 43 * q^15 + 122 * q^16 - 61 * q^17 - 88 * q^18 - 43 * q^19 - 41 * q^20 - 61 * q^21 - 93 * q^22 - 60 * q^23 - 41 * q^24 + 26 * q^25 - 9 * q^26 - 93 * q^27 - 126 * q^28 - 47 * q^29 - 36 * q^30 - 100 * q^31 - 114 * q^32 - 133 * q^33 - 75 * q^34 - 37 * q^35 + 75 * q^36 - 264 * q^37 - 35 * q^38 - 47 * q^39 - 118 * q^40 - 72 * q^41 - 64 * q^42 - 107 * q^43 - 59 * q^44 - 69 * q^45 - 111 * q^46 - 54 * q^47 - 135 * q^48 + 33 * q^49 - 42 * q^50 - 26 * q^51 - 173 * q^52 - 103 * q^53 - 28 * q^54 - 78 * q^55 - 44 * q^56 - 205 * q^57 - 189 * q^58 - 38 * q^59 - 105 * q^60 - 108 * q^61 - 14 * q^62 - 116 * q^63 + 39 * q^64 - 146 * q^65 + 5 * q^66 - 206 * q^67 - 62 * q^68 - 55 * q^69 - 125 * q^70 - 78 * q^71 - 225 * q^72 - 326 * q^73 + 3 * q^74 - 95 * q^75 - 84 * q^76 - 79 * q^77 - 86 * q^78 - 117 * q^79 - 39 * q^80 + q^81 - 96 * q^82 - 23 * q^83 - 57 * q^84 - 224 * q^85 - 7 * q^86 - 45 * q^87 - 250 * q^88 - 104 * q^89 - 36 * q^90 - 96 * q^91 - 137 * q^92 - 155 * q^93 - 48 * q^94 - 38 * q^95 - 33 * q^96 - 447 * q^97 - 46 * q^98 - 94 * q^99

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