LMFDB - Embedded newform 4009.2.a.c.1.69

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.69
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.45691 q^{2} +0.632827 q^{3} +4.03640 q^{4} -3.17038 q^{5} +1.55480 q^{6} -1.44419 q^{7} +5.00326 q^{8} -2.59953 q^{9} +O(q^{10})$	\(q+2.45691 q^{2} +0.632827 q^{3} +4.03640 q^{4} -3.17038 q^{5} +1.55480 q^{6} -1.44419 q^{7} +5.00326 q^{8} -2.59953 q^{9} -7.78933 q^{10} -1.18986 q^{11} +2.55435 q^{12} +6.81359 q^{13} -3.54825 q^{14} -2.00630 q^{15} +4.21975 q^{16} -3.93164 q^{17} -6.38681 q^{18} +1.00000 q^{19} -12.7969 q^{20} -0.913923 q^{21} -2.92339 q^{22} -4.19360 q^{23} +3.16620 q^{24} +5.05129 q^{25} +16.7404 q^{26} -3.54353 q^{27} -5.82934 q^{28} -1.23471 q^{29} -4.92930 q^{30} -8.61787 q^{31} +0.361022 q^{32} -0.752977 q^{33} -9.65968 q^{34} +4.57863 q^{35} -10.4928 q^{36} +0.114513 q^{37} +2.45691 q^{38} +4.31183 q^{39} -15.8622 q^{40} -0.501324 q^{41} -2.24543 q^{42} +6.77076 q^{43} -4.80277 q^{44} +8.24149 q^{45} -10.3033 q^{46} -6.82369 q^{47} +2.67037 q^{48} -4.91431 q^{49} +12.4106 q^{50} -2.48805 q^{51} +27.5024 q^{52} -8.73281 q^{53} -8.70614 q^{54} +3.77231 q^{55} -7.22566 q^{56} +0.632827 q^{57} -3.03357 q^{58} -3.84950 q^{59} -8.09824 q^{60} -11.7872 q^{61} -21.1733 q^{62} +3.75422 q^{63} -7.55250 q^{64} -21.6017 q^{65} -1.85000 q^{66} +0.926198 q^{67} -15.8697 q^{68} -2.65382 q^{69} +11.2493 q^{70} -3.73103 q^{71} -13.0061 q^{72} +11.2536 q^{73} +0.281347 q^{74} +3.19659 q^{75} +4.03640 q^{76} +1.71839 q^{77} +10.5938 q^{78} -8.89311 q^{79} -13.3782 q^{80} +5.55614 q^{81} -1.23171 q^{82} -5.29976 q^{83} -3.68896 q^{84} +12.4648 q^{85} +16.6351 q^{86} -0.781357 q^{87} -5.95319 q^{88} +11.9138 q^{89} +20.2486 q^{90} -9.84013 q^{91} -16.9270 q^{92} -5.45362 q^{93} -16.7652 q^{94} -3.17038 q^{95} +0.228464 q^{96} +2.47780 q^{97} -12.0740 q^{98} +3.09308 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	2.45691	1.73730	0.868649	−	0.495429i	$-0.164989\pi$
$2$	2.45691	1.73730	0.868649	+	0.495429i	$0.164989\pi$
$3$	0.632827	0.365363	0.182681	−	0.983172i	$-0.441522\pi$
$3$	0.632827	0.365363	0.182681	+	0.983172i	$0.441522\pi$
$4$	4.03640	2.01820
$5$	−3.17038	−1.41784	−0.708918	−	0.705291i	$-0.750816\pi$
$5$	−3.17038	−1.41784	−0.708918	+	0.705291i	$0.750816\pi$
$6$	1.55480	0.634744
$7$	−1.44419	−0.545853	−0.272926	−	0.962035i	$-0.587992\pi$
$7$	−1.44419	−0.545853	−0.272926	+	0.962035i	$0.587992\pi$
$8$	5.00326	1.76892
$9$	−2.59953	−0.866510
$10$	−7.78933	−2.46320
$11$	−1.18986	−0.358757	−0.179379	−	0.983780i	$-0.557409\pi$
$11$	−1.18986	−0.358757	−0.179379	+	0.983780i	$0.557409\pi$
$12$	2.55435	0.737376
$13$	6.81359	1.88975	0.944876	−	0.327430i	$-0.106182\pi$
$13$	6.81359	1.88975	0.944876	+	0.327430i	$0.106182\pi$
$14$	−3.54825	−0.948309
$15$	−2.00630	−0.518025
$16$	4.21975	1.05494
$17$	−3.93164	−0.953563	−0.476781	−	0.879022i	$-0.658197\pi$
$17$	−3.93164	−0.953563	−0.476781	+	0.879022i	$0.658197\pi$
$18$	−6.38681	−1.50539
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−12.7969	−2.86148
$21$	−0.913923	−0.199434
$22$	−2.92339	−0.623268
$23$	−4.19360	−0.874425	−0.437213	−	0.899358i	$-0.644034\pi$
$23$	−4.19360	−0.874425	−0.437213	+	0.899358i	$0.644034\pi$
$24$	3.16620	0.646298
$25$	5.05129	1.01026
$26$	16.7404	3.28306
$27$	−3.54353	−0.681953
$28$	−5.82934	−1.10164
$29$	−1.23471	−0.229279	−0.114640	−	0.993407i	$-0.536571\pi$
$29$	−1.23471	−0.229279	−0.114640	+	0.993407i	$0.536571\pi$
$30$	−4.92930	−0.899963
$31$	−8.61787	−1.54782	−0.773908	−	0.633298i	$-0.781701\pi$
$31$	−8.61787	−1.54782	−0.773908	+	0.633298i	$0.781701\pi$
$32$	0.361022	0.0638203
$33$	−0.752977	−0.131077
$34$	−9.65968	−1.65662
$35$	4.57863	0.773930
$36$	−10.4928	−1.74879
$37$	0.114513	0.0188257	0.00941287	−	0.999956i	$-0.497004\pi$
$37$	0.114513	0.0188257	0.00941287	+	0.999956i	$0.497004\pi$
$38$	2.45691	0.398563
$39$	4.31183	0.690445
$40$	−15.8622	−2.50804
$41$	−0.501324	−0.0782937	−0.0391468	−	0.999233i	$-0.512464\pi$
$41$	−0.501324	−0.0782937	−0.0391468	+	0.999233i	$0.512464\pi$
$42$	−2.24543	−0.346477
$43$	6.77076	1.03253	0.516265	−	0.856429i	$-0.327322\pi$
$43$	6.77076	1.03253	0.516265	+	0.856429i	$0.327322\pi$
$44$	−4.80277	−0.724044
$45$	8.24149	1.22857
$46$	−10.3033	−1.51914
$47$	−6.82369	−0.995337	−0.497669	−	0.867367i	$-0.665810\pi$
$47$	−6.82369	−0.995337	−0.497669	+	0.867367i	$0.665810\pi$
$48$	2.67037	0.385435
$49$	−4.91431	−0.702045
$50$	12.4106	1.75512
$51$	−2.48805	−0.348397
$52$	27.5024	3.81390
$53$	−8.73281	−1.19954	−0.599772	−	0.800171i	$-0.704742\pi$
$53$	−8.73281	−1.19954	−0.599772	+	0.800171i	$0.704742\pi$
$54$	−8.70614	−1.18476
$55$	3.77231	0.508659
$56$	−7.22566	−0.965570
$57$	0.632827	0.0838200
$58$	−3.03357	−0.398327
$59$	−3.84950	−0.501162	−0.250581	−	0.968096i	$-0.580622\pi$
$59$	−3.84950	−0.501162	−0.250581	+	0.968096i	$0.580622\pi$
$60$	−8.09824	−1.04548
$61$	−11.7872	−1.50920	−0.754598	−	0.656187i	$-0.772168\pi$
$61$	−11.7872	−1.50920	−0.754598	+	0.656187i	$0.772168\pi$
$62$	−21.1733	−2.68902
$63$	3.75422	0.472987
$64$	−7.55250	−0.944063
$65$	−21.6017	−2.67936
$66$	−1.85000	−0.227719
$67$	0.926198	0.113153	0.0565765	−	0.998398i	$-0.481982\pi$
$67$	0.926198	0.113153	0.0565765	+	0.998398i	$0.481982\pi$
$68$	−15.8697	−1.92448
$69$	−2.65382	−0.319483
$70$	11.2493	1.34455
$71$	−3.73103	−0.442792	−0.221396	−	0.975184i	$-0.571061\pi$
$71$	−3.73103	−0.442792	−0.221396	+	0.975184i	$0.571061\pi$
$72$	−13.0061	−1.53279
$73$	11.2536	1.31714	0.658569	−	0.752520i	$-0.271162\pi$
$73$	11.2536	1.31714	0.658569	+	0.752520i	$0.271162\pi$
$74$	0.281347	0.0327059
$75$	3.19659	0.369111
$76$	4.03640	0.463007
$77$	1.71839	0.195829
$78$	10.5938	1.19951
$79$	−8.89311	−1.00055	−0.500277	−	0.865866i	$-0.666768\pi$
$79$	−8.89311	−1.00055	−0.500277	+	0.865866i	$0.666768\pi$
$80$	−13.3782	−1.49573
$81$	5.55614	0.617349
$82$	−1.23171	−0.136019
$83$	−5.29976	−0.581724	−0.290862	−	0.956765i	$-0.593942\pi$
$83$	−5.29976	−0.581724	−0.290862	+	0.956765i	$0.593942\pi$
$84$	−3.68896	−0.402499
$85$	12.4648	1.35200
$86$	16.6351	1.79381
$87$	−0.781357	−0.0837702
$88$	−5.95319	−0.634613
$89$	11.9138	1.26286	0.631430	−	0.775433i	$-0.282468\pi$
$89$	11.9138	1.26286	0.631430	+	0.775433i	$0.282468\pi$
$90$	20.2486	2.13439
$91$	−9.84013	−1.03153
$92$	−16.9270	−1.76477
$93$	−5.45362	−0.565514
$94$	−16.7652	−1.72920
$95$	−3.17038	−0.325274
$96$	0.228464	0.0233176
$97$	2.47780	0.251582	0.125791	−	0.992057i	$-0.459853\pi$
$97$	2.47780	0.251582	0.125791	+	0.992057i	$0.459853\pi$
$98$	−12.0740	−1.21966
$99$	3.09308	0.310867
$100$	20.3890	2.03890
$101$	5.00369	0.497886	0.248943	−	0.968518i	$-0.419917\pi$
$101$	5.00369	0.497886	0.248943	+	0.968518i	$0.419917\pi$
$102$	−6.11291	−0.605268
$103$	13.2122	1.30184	0.650921	−	0.759146i	$-0.274383\pi$
$103$	13.2122	1.30184	0.650921	+	0.759146i	$0.274383\pi$
$104$	34.0902	3.34282
$105$	2.89748	0.282765
$106$	−21.4557	−2.08396
$107$	−1.16256	−0.112389	−0.0561947	−	0.998420i	$-0.517897\pi$
$107$	−1.16256	−0.112389	−0.0561947	+	0.998420i	$0.517897\pi$
$108$	−14.3031	−1.37632
$109$	3.69295	0.353721	0.176860	−	0.984236i	$-0.443406\pi$
$109$	3.69295	0.353721	0.176860	+	0.984236i	$0.443406\pi$
$110$	9.26823	0.883691
$111$	0.0724666	0.00687823
$112$	−6.09413	−0.575841
$113$	9.83846	0.925525	0.462762	−	0.886482i	$-0.346858\pi$
$113$	9.83846	0.925525	0.462762	+	0.886482i	$0.346858\pi$
$114$	1.55480	0.145620
$115$	13.2953	1.23979
$116$	−4.98378	−0.462732
$117$	−17.7121	−1.63749
$118$	−9.45787	−0.870667
$119$	5.67804	0.520505
$120$	−10.0380	−0.916344
$121$	−9.58423	−0.871293
$122$	−28.9601	−2.62192
$123$	−0.317251	−0.0286056
$124$	−34.7852	−3.12380
$125$	−0.162606	−0.0145439
$126$	9.22377	0.821719
$127$	4.98326	0.442193	0.221097	−	0.975252i	$-0.429036\pi$
$127$	4.98326	0.442193	0.221097	+	0.975252i	$0.429036\pi$
$128$	−19.2779	−1.70394
$129$	4.28472	0.377248
$130$	−53.0733	−4.65484
$131$	2.23418	0.195201	0.0976005	−	0.995226i	$-0.468883\pi$
$131$	2.23418	0.195201	0.0976005	+	0.995226i	$0.468883\pi$
$132$	−3.03932	−0.264539
$133$	−1.44419	−0.125227
$134$	2.27558	0.196581
$135$	11.2343	0.966898
$136$	−19.6710	−1.68678
$137$	13.2363	1.13086	0.565429	−	0.824797i	$-0.308711\pi$
$137$	13.2363	1.13086	0.565429	+	0.824797i	$0.308711\pi$
$138$	−6.52020	−0.555036
$139$	−10.7539	−0.912130	−0.456065	−	0.889947i	$-0.650742\pi$
$139$	−10.7539	−0.912130	−0.456065	+	0.889947i	$0.650742\pi$
$140$	18.4812	1.56195
$141$	−4.31822	−0.363659
$142$	−9.16680	−0.769261
$143$	−8.10724	−0.677962
$144$	−10.9694	−0.914114
$145$	3.91449	0.325081
$146$	27.6492	2.28826
$147$	−3.10991	−0.256501
$148$	0.462219	0.0379942
$149$	19.4470	1.59316	0.796582	−	0.604531i	$-0.206639\pi$
$149$	19.4470	1.59316	0.796582	+	0.604531i	$0.206639\pi$
$150$	7.85374	0.641255
$151$	17.6586	1.43704	0.718518	−	0.695508i	$-0.244821\pi$
$151$	17.6586	1.43704	0.718518	+	0.695508i	$0.244821\pi$
$152$	5.00326	0.405818
$153$	10.2204	0.826272
$154$	4.22193	0.340213
$155$	27.3219	2.19455
$156$	17.4043	1.39346
$157$	−14.3949	−1.14884	−0.574420	−	0.818561i	$-0.694772\pi$
$157$	−14.3949	−1.14884	−0.574420	+	0.818561i	$0.694772\pi$
$158$	−21.8496	−1.73826
$159$	−5.52636	−0.438269
$160$	−1.14458	−0.0904867
$161$	6.05635	0.477308
$162$	13.6509	1.07252
$163$	10.8275	0.848078	0.424039	−	0.905644i	$-0.360612\pi$
$163$	10.8275	0.848078	0.424039	+	0.905644i	$0.360612\pi$
$164$	−2.02355	−0.158012
$165$	2.38722	0.185845
$166$	−13.0210	−1.01063
$167$	−23.0462	−1.78336	−0.891682	−	0.452662i	$-0.850475\pi$
$167$	−23.0462	−1.78336	−0.891682	+	0.452662i	$0.850475\pi$
$168$	−4.57260	−0.352783
$169$	33.4251	2.57116
$170$	30.6248	2.34882
$171$	−2.59953	−0.198791
$172$	27.3295	2.08386
$173$	−14.2356	−1.08231	−0.541157	−	0.840921i	$-0.682014\pi$
$173$	−14.2356	−1.08231	−0.541157	+	0.840921i	$0.682014\pi$
$174$	−1.91972	−0.145534
$175$	−7.29503	−0.551452
$176$	−5.02092	−0.378466
$177$	−2.43607	−0.183106
$178$	29.2711	2.19396
$179$	−19.3497	−1.44627	−0.723133	−	0.690709i	$-0.757299\pi$
$179$	−19.3497	−1.44627	−0.723133	+	0.690709i	$0.757299\pi$
$180$	33.2660	2.47950
$181$	13.5069	1.00396	0.501980	−	0.864879i	$-0.332605\pi$
$181$	13.5069	1.00396	0.501980	+	0.864879i	$0.332605\pi$
$182$	−24.1763	−1.79207
$183$	−7.45926	−0.551405
$184$	−20.9817	−1.54679
$185$	−0.363048	−0.0266918
$186$	−13.3991	−0.982467
$187$	4.67811	0.342098
$188$	−27.5432	−2.00879
$189$	5.11754	0.372246
$190$	−7.78933	−0.565097
$191$	−11.5285	−0.834173	−0.417086	−	0.908867i	$-0.636949\pi$
$191$	−11.5285	−0.834173	−0.417086	+	0.908867i	$0.636949\pi$
$192$	−4.77943	−0.344925
$193$	−24.5776	−1.76913	−0.884567	−	0.466412i	$-0.845546\pi$
$193$	−24.5776	−1.76913	−0.884567	+	0.466412i	$0.845546\pi$
$194$	6.08773	0.437073
$195$	−13.6701	−0.978937
$196$	−19.8361	−1.41687
$197$	−8.16372	−0.581641	−0.290820	−	0.956778i	$-0.593928\pi$
$197$	−8.16372	−0.581641	−0.290820	+	0.956778i	$0.593928\pi$
$198$	7.59943	0.540068
$199$	0.756088	0.0535976	0.0267988	−	0.999641i	$-0.491469\pi$
$199$	0.756088	0.0535976	0.0267988	+	0.999641i	$0.491469\pi$
$200$	25.2729	1.78706
$201$	0.586123	0.0413419
$202$	12.2936	0.864976
$203$	1.78315	0.125153
$204$	−10.0428	−0.703135
$205$	1.58939	0.111008
$206$	32.4613	2.26169
$207$	10.9014	0.757698
$208$	28.7517	1.99357
$209$	−1.18986	−0.0823045
$210$	7.11885	0.491247
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−35.2491	−2.42092
$213$	−2.36110	−0.161780
$214$	−2.85632	−0.195254
$215$	−21.4658	−1.46396
$216$	−17.7292	−1.20632
$217$	12.4459	0.844880
$218$	9.07325	0.614518
$219$	7.12161	0.481233
$220$	15.2266	1.02658
$221$	−26.7886	−1.80200
$222$	0.178044	0.0119495
$223$	0.326734	0.0218797	0.0109399	−	0.999940i	$-0.496518\pi$
$223$	0.326734	0.0218797	0.0109399	+	0.999940i	$0.496518\pi$
$224$	−0.521385	−0.0348365
$225$	−13.1310	−0.875398
$226$	24.1722	1.60791
$227$	−10.8447	−0.719789	−0.359895	−	0.932993i	$-0.617187\pi$
$227$	−10.8447	−0.719789	−0.359895	+	0.932993i	$0.617187\pi$
$228$	2.55435	0.169166
$229$	28.8566	1.90690	0.953448	−	0.301556i	$-0.0975060\pi$
$229$	28.8566	1.90690	0.953448	+	0.301556i	$0.0975060\pi$
$230$	32.6653	2.15389
$231$	1.08744	0.0715485
$232$	−6.17756	−0.405577
$233$	2.01093	0.131740	0.0658701	−	0.997828i	$-0.479018\pi$
$233$	2.01093	0.131740	0.0658701	+	0.997828i	$0.479018\pi$
$234$	−43.5171	−2.84480
$235$	21.6337	1.41122
$236$	−15.5381	−1.01145
$237$	−5.62780	−0.365565
$238$	13.9504	0.904272
$239$	17.8083	1.15192	0.575961	−	0.817477i	$-0.304628\pi$
$239$	17.8083	1.15192	0.575961	+	0.817477i	$0.304628\pi$
$240$	−8.46609	−0.546483
$241$	15.6403	1.00748	0.503741	−	0.863855i	$-0.331957\pi$
$241$	15.6403	1.00748	0.503741	+	0.863855i	$0.331957\pi$
$242$	−23.5476	−1.51370
$243$	14.1467	0.907510
$244$	−47.5779	−3.04586
$245$	15.5802	0.995384
$246$	−0.779458	−0.0496964
$247$	6.81359	0.433539
$248$	−43.1175	−2.73796
$249$	−3.35383	−0.212540
$250$	−0.399508	−0.0252671
$251$	−4.79966	−0.302952	−0.151476	−	0.988461i	$-0.548403\pi$
$251$	−4.79966	−0.302952	−0.151476	+	0.988461i	$0.548403\pi$
$252$	15.1535	0.954583
$253$	4.98980	0.313706
$254$	12.2434	0.768221
$255$	7.88805	0.493969
$256$	−32.2589	−2.01618
$257$	−20.8039	−1.29771	−0.648856	−	0.760911i	$-0.724752\pi$
$257$	−20.8039	−1.29771	−0.648856	+	0.760911i	$0.724752\pi$
$258$	10.5272	0.655393
$259$	−0.165378	−0.0102761
$260$	−87.1930	−5.40748
$261$	3.20966	0.198673
$262$	5.48917	0.339122
$263$	3.53608	0.218044	0.109022	−	0.994039i	$-0.465228\pi$
$263$	3.53608	0.218044	0.109022	+	0.994039i	$0.465228\pi$
$264$	−3.76734	−0.231864
$265$	27.6863	1.70076
$266$	−3.54825	−0.217557
$267$	7.53937	0.461402
$268$	3.73851	0.228366
$269$	13.9109	0.848165	0.424083	−	0.905623i	$-0.360597\pi$
$269$	13.9109	0.848165	0.424083	+	0.905623i	$0.360597\pi$
$270$	27.6018	1.67979
$271$	12.3576	0.750669	0.375335	−	0.926889i	$-0.377528\pi$
$271$	12.3576	0.750669	0.375335	+	0.926889i	$0.377528\pi$
$272$	−16.5905	−1.00595
$273$	−6.22710	−0.376881
$274$	32.5205	1.96464
$275$	−6.01034	−0.362437
$276$	−10.7119	−0.644780
$277$	−21.8645	−1.31371	−0.656855	−	0.754017i	$-0.728114\pi$
$277$	−21.8645	−1.31371	−0.656855	+	0.754017i	$0.728114\pi$
$278$	−26.4212	−1.58464
$279$	22.4024	1.34120
$280$	22.9081	1.36902
$281$	6.94705	0.414426	0.207213	−	0.978296i	$-0.433561\pi$
$281$	6.94705	0.414426	0.207213	+	0.978296i	$0.433561\pi$
$282$	−10.6095	−0.631784
$283$	−1.06427	−0.0632645	−0.0316323	−	0.999500i	$-0.510071\pi$
$283$	−1.06427	−0.0632645	−0.0316323	+	0.999500i	$0.510071\pi$
$284$	−15.0599	−0.893643
$285$	−2.00630	−0.118843
$286$	−19.9188	−1.17782
$287$	0.724008	0.0427368
$288$	−0.938487	−0.0553009
$289$	−1.54220	−0.0907178
$290$	9.61754	0.564762
$291$	1.56802	0.0919188
$292$	45.4242	2.65825
$293$	14.9259	0.871982	0.435991	−	0.899951i	$-0.356398\pi$
$293$	14.9259	0.871982	0.435991	+	0.899951i	$0.356398\pi$
$294$	−7.64077	−0.445619
$295$	12.2044	0.710565
$296$	0.572936	0.0333012
$297$	4.21632	0.244656
$298$	47.7796	2.76780
$299$	−28.5735	−1.65245
$300$	12.9027	0.744940
$301$	−9.77827	−0.563610
$302$	43.3856	2.49656
$303$	3.16647	0.181909
$304$	4.21975	0.242019
$305$	37.3699	2.13979
$306$	25.1106	1.43548
$307$	−19.5712	−1.11699	−0.558494	−	0.829509i	$-0.688620\pi$
$307$	−19.5712	−1.11699	−0.558494	+	0.829509i	$0.688620\pi$
$308$	6.93611	0.395222
$309$	8.36107	0.475645
$310$	67.1274	3.81258
$311$	20.0504	1.13695	0.568477	−	0.822699i	$-0.307533\pi$
$311$	20.0504	1.13695	0.568477	+	0.822699i	$0.307533\pi$
$312$	21.5732	1.22134
$313$	23.0507	1.30290	0.651450	−	0.758691i	$-0.274161\pi$
$313$	23.0507	1.30290	0.651450	+	0.758691i	$0.274161\pi$
$314$	−35.3670	−1.99588
$315$	−11.9023	−0.670618
$316$	−35.8962	−2.01932
$317$	−2.67234	−0.150094	−0.0750468	−	0.997180i	$-0.523911\pi$
$317$	−2.67234	−0.150094	−0.0750468	+	0.997180i	$0.523911\pi$
$318$	−13.5778	−0.761403
$319$	1.46913	0.0822557
$320$	23.9443	1.33853
$321$	−0.735703	−0.0410629
$322$	14.8799	0.829225
$323$	−3.93164	−0.218762
$324$	22.4268	1.24594
$325$	34.4174	1.90914
$326$	26.6023	1.47336
$327$	2.33700	0.129236
$328$	−2.50825	−0.138495
$329$	9.85471	0.543308
$330$	5.86519	0.322868
$331$	−9.46645	−0.520323	−0.260161	−	0.965565i	$-0.583776\pi$
$331$	−9.46645	−0.520323	−0.260161	+	0.965565i	$0.583776\pi$
$332$	−21.3920	−1.17404
$333$	−0.297679	−0.0163127
$334$	−56.6223	−3.09823
$335$	−2.93640	−0.160432
$336$	−3.85653	−0.210391
$337$	−9.59519	−0.522683	−0.261342	−	0.965246i	$-0.584165\pi$
$337$	−9.59519	−0.522683	−0.261342	+	0.965246i	$0.584165\pi$
$338$	82.1224	4.46687
$339$	6.22605	0.338152
$340$	50.3129	2.72860
$341$	10.2541	0.555290
$342$	−6.38681	−0.345359
$343$	17.2065	0.929066
$344$	33.8759	1.82646
$345$	8.41361	0.452974
$346$	−34.9756	−1.88030
$347$	1.44683	0.0776699	0.0388350	−	0.999246i	$-0.487635\pi$
$347$	1.44683	0.0776699	0.0388350	+	0.999246i	$0.487635\pi$
$348$	−3.15387	−0.169065
$349$	7.24676	0.387910	0.193955	−	0.981010i	$-0.437868\pi$
$349$	7.24676	0.387910	0.193955	+	0.981010i	$0.437868\pi$
$350$	−17.9232	−0.958036
$351$	−24.1442	−1.28872
$352$	−0.429567	−0.0228960
$353$	−8.78561	−0.467611	−0.233805	−	0.972283i	$-0.575118\pi$
$353$	−8.78561	−0.467611	−0.233805	+	0.972283i	$0.575118\pi$
$354$	−5.98519	−0.318109
$355$	11.8288	0.627806
$356$	48.0889	2.54871
$357$	3.59322	0.190173
$358$	−47.5405	−2.51260
$359$	12.0442	0.635669	0.317835	−	0.948146i	$-0.397044\pi$
$359$	12.0442	0.635669	0.317835	+	0.948146i	$0.397044\pi$
$360$	41.2343	2.17324
$361$	1.00000	0.0526316
$362$	33.1852	1.74418
$363$	−6.06516	−0.318338
$364$	−39.7188	−2.08183
$365$	−35.6783	−1.86749
$366$	−18.3267	−0.957954
$367$	10.2972	0.537507	0.268753	−	0.963209i	$-0.413388\pi$
$367$	10.2972	0.537507	0.268753	+	0.963209i	$0.413388\pi$
$368$	−17.6959	−0.922464
$369$	1.30321	0.0678422
$370$	−0.891976	−0.0463716
$371$	12.6118	0.654774
$372$	−22.0130	−1.14132
$373$	−5.63516	−0.291777	−0.145889	−	0.989301i	$-0.546604\pi$
$373$	−5.63516	−0.291777	−0.145889	+	0.989301i	$0.546604\pi$
$374$	11.4937	0.594325
$375$	−0.102902	−0.00531381
$376$	−34.1407	−1.76067
$377$	−8.41280	−0.433281
$378$	12.5733	0.646703
$379$	5.10838	0.262400	0.131200	−	0.991356i	$-0.458117\pi$
$379$	5.10838	0.262400	0.131200	+	0.991356i	$0.458117\pi$
$380$	−12.7969	−0.656468
$381$	3.15354	0.161561
$382$	−28.3245	−1.44921
$383$	−3.86050	−0.197262	−0.0986311	−	0.995124i	$-0.531446\pi$
$383$	−3.86050	−0.197262	−0.0986311	+	0.995124i	$0.531446\pi$
$384$	−12.1995	−0.622556
$385$	−5.44794	−0.277653
$386$	−60.3850	−3.07351
$387$	−17.6008	−0.894698
$388$	10.0014	0.507744
$389$	−32.6721	−1.65654	−0.828269	−	0.560330i	$-0.810674\pi$
$389$	−32.6721	−1.65654	−0.828269	+	0.560330i	$0.810674\pi$
$390$	−33.5862	−1.70071
$391$	16.4877	0.833820
$392$	−24.5876	−1.24186
$393$	1.41385	0.0713192
$394$	−20.0575	−1.01048
$395$	28.1945	1.41862
$396$	12.4849	0.627392
$397$	18.3656	0.921741	0.460871	−	0.887467i	$-0.347537\pi$
$397$	18.3656	0.921741	0.460871	+	0.887467i	$0.347537\pi$
$398$	1.85764	0.0931150
$399$	−0.913923	−0.0457534
$400$	21.3152	1.06576
$401$	−26.7229	−1.33448	−0.667240	−	0.744843i	$-0.732525\pi$
$401$	−26.7229	−1.33448	−0.667240	+	0.744843i	$0.732525\pi$
$402$	1.44005	0.0718232
$403$	−58.7187	−2.92499
$404$	20.1969	1.00483
$405$	−17.6151	−0.875300
$406$	4.38105	0.217428
$407$	−0.136254	−0.00675387
$408$	−12.4484	−0.616285
$409$	7.03292	0.347756	0.173878	−	0.984767i	$-0.444370\pi$
$409$	7.03292	0.347756	0.173878	+	0.984767i	$0.444370\pi$
$410$	3.90498	0.192853
$411$	8.37632	0.413173
$412$	53.3300	2.62738
$413$	5.55941	0.273561
$414$	26.7837	1.31635
$415$	16.8022	0.824789
$416$	2.45986	0.120604
$417$	−6.80533	−0.333258
$418$	−2.92339	−0.142987
$419$	6.34369	0.309910	0.154955	−	0.987922i	$-0.450477\pi$
$419$	6.34369	0.309910	0.154955	+	0.987922i	$0.450477\pi$
$420$	11.6954	0.570677
$421$	−22.3001	−1.08684	−0.543420	−	0.839461i	$-0.682871\pi$
$421$	−22.3001	−1.08684	−0.543420	+	0.839461i	$0.682871\pi$
$422$	2.45691	0.119600
$423$	17.7384	0.862470
$424$	−43.6925	−2.12190
$425$	−19.8599	−0.963344
$426$	−5.80100	−0.281059
$427$	17.0230	0.823800
$428$	−4.69258	−0.226825
$429$	−5.13048	−0.247702
$430$	−52.7396	−2.54333
$431$	−27.0874	−1.30475	−0.652376	−	0.757895i	$-0.726228\pi$
$431$	−27.0874	−1.30475	−0.652376	+	0.757895i	$0.726228\pi$
$432$	−14.9528	−0.719418
$433$	35.7679	1.71890	0.859449	−	0.511222i	$-0.170807\pi$
$433$	35.7679	1.71890	0.859449	+	0.511222i	$0.170807\pi$
$434$	30.5783	1.46781
$435$	2.47719	0.118772
$436$	14.9062	0.713879
$437$	−4.19360	−0.200607
$438$	17.4971	0.836046
$439$	−37.4667	−1.78819	−0.894095	−	0.447878i	$-0.852180\pi$
$439$	−37.4667	−1.78819	−0.894095	+	0.447878i	$0.852180\pi$
$440$	18.8739	0.899776
$441$	12.7749	0.608329
$442$	−65.8172	−3.13060
$443$	8.87269	0.421554	0.210777	−	0.977534i	$-0.432401\pi$
$443$	8.87269	0.421554	0.210777	+	0.977534i	$0.432401\pi$
$444$	0.292505	0.0138817
$445$	−37.7712	−1.79053
$446$	0.802756	0.0380116
$447$	12.3066	0.582083
$448$	10.9073	0.515319
$449$	−26.8620	−1.26770	−0.633848	−	0.773458i	$-0.718525\pi$
$449$	−26.8620	−1.26770	−0.633848	+	0.773458i	$0.718525\pi$
$450$	−32.2616	−1.52083
$451$	0.596507	0.0280884
$452$	39.7120	1.86790
$453$	11.1748	0.525040
$454$	−26.6445	−1.25049
$455$	31.1969	1.46253
$456$	3.16620	0.148271
$457$	−38.7958	−1.81479	−0.907396	−	0.420277i	$-0.861933\pi$
$457$	−38.7958	−1.81479	−0.907396	+	0.420277i	$0.861933\pi$
$458$	70.8980	3.31285
$459$	13.9319	0.650286
$460$	53.6651	2.50215
$461$	35.6060	1.65834	0.829168	−	0.558999i	$-0.188814\pi$
$461$	35.6060	1.65834	0.829168	+	0.558999i	$0.188814\pi$
$462$	2.67175	0.124301
$463$	−21.5093	−0.999624	−0.499812	−	0.866134i	$-0.666598\pi$
$463$	−21.5093	−0.999624	−0.499812	+	0.866134i	$0.666598\pi$
$464$	−5.21016	−0.241876
$465$	17.2900	0.801806
$466$	4.94066	0.228872
$467$	−5.05633	−0.233979	−0.116990	−	0.993133i	$-0.537324\pi$
$467$	−5.05633	−0.233979	−0.116990	+	0.993133i	$0.537324\pi$
$468$	−71.4934	−3.30478
$469$	−1.33761	−0.0617649
$470$	53.1520	2.45172
$471$	−9.10949	−0.419743
$472$	−19.2600	−0.886515
$473$	−8.05627	−0.370428
$474$	−13.8270	−0.635095
$475$	5.05129	0.231769
$476$	22.9189	1.05048
$477$	22.7012	1.03942
$478$	43.7534	2.00123
$479$	26.0544	1.19046	0.595229	−	0.803556i	$-0.297061\pi$
$479$	26.0544	1.19046	0.595229	+	0.803556i	$0.297061\pi$
$480$	−0.724319	−0.0330605
$481$	0.780242	0.0355760
$482$	38.4268	1.75029
$483$	3.83263	0.174391
$484$	−38.6858	−1.75845
$485$	−7.85555	−0.356702
$486$	34.7571	1.57661
$487$	−5.48045	−0.248343	−0.124172	−	0.992261i	$-0.539627\pi$
$487$	−5.48045	−0.248343	−0.124172	+	0.992261i	$0.539627\pi$
$488$	−58.9745	−2.66965
$489$	6.85196	0.309856
$490$	38.2792	1.72928
$491$	−24.1965	−1.09197	−0.545986	−	0.837794i	$-0.683845\pi$
$491$	−24.1965	−1.09197	−0.545986	+	0.837794i	$0.683845\pi$
$492$	−1.28055	−0.0577319
$493$	4.85443	0.218632
$494$	16.7404	0.753186
$495$	−9.80624	−0.440758
$496$	−36.3653	−1.63285
$497$	5.38832	0.241699
$498$	−8.24006	−0.369246
$499$	−37.1213	−1.66178	−0.830889	−	0.556438i	$-0.812168\pi$
$499$	−37.1213	−1.66178	−0.830889	+	0.556438i	$0.812168\pi$
$500$	−0.656344	−0.0293526
$501$	−14.5842	−0.651575
$502$	−11.7923	−0.526317
$503$	−31.1154	−1.38737	−0.693683	−	0.720281i	$-0.744013\pi$
$503$	−31.1154	−1.38737	−0.693683	+	0.720281i	$0.744013\pi$
$504$	18.7833	0.836676
$505$	−15.8636	−0.705920
$506$	12.2595	0.545001
$507$	21.1523	0.939406
$508$	20.1145	0.892435
$509$	−31.1211	−1.37942	−0.689709	−	0.724086i	$-0.742262\pi$
$509$	−31.1211	−1.37942	−0.689709	+	0.724086i	$0.742262\pi$
$510$	19.3802	0.858171
$511$	−16.2524	−0.718964
$512$	−40.7016	−1.79877
$513$	−3.54353	−0.156451
$514$	−51.1133	−2.25451
$515$	−41.8878	−1.84580
$516$	17.2949	0.761363
$517$	8.11925	0.357084
$518$	−0.406319	−0.0178526
$519$	−9.00869	−0.395438
$520$	−108.079	−4.73957
$521$	4.77132	0.209035	0.104518	−	0.994523i	$-0.466670\pi$
$521$	4.77132	0.209035	0.104518	+	0.994523i	$0.466670\pi$
$522$	7.88584	0.345154
$523$	11.9811	0.523896	0.261948	−	0.965082i	$-0.415635\pi$
$523$	11.9811	0.523896	0.261948	+	0.965082i	$0.415635\pi$
$524$	9.01804	0.393955
$525$	−4.61649	−0.201480
$526$	8.68782	0.378807
$527$	33.8824	1.47594
$528$	−3.17738	−0.138278
$529$	−5.41375	−0.235380
$530$	68.0227	2.95472
$531$	10.0069	0.434262
$532$	−5.82934	−0.252734
$533$	−3.41582	−0.147956
$534$	18.5236	0.801592
$535$	3.68577	0.159350
$536$	4.63401	0.200159
$537$	−12.2450	−0.528412
$538$	34.1779	1.47352
$539$	5.84736	0.251864
$540$	45.3463	1.95140
$541$	−26.6248	−1.14469	−0.572346	−	0.820013i	$-0.693966\pi$
$541$	−26.6248	−1.14469	−0.572346	+	0.820013i	$0.693966\pi$
$542$	30.3614	1.30414
$543$	8.54753	0.366810
$544$	−1.41941	−0.0608566
$545$	−11.7080	−0.501518
$546$	−15.2994	−0.654755
$547$	18.2119	0.778684	0.389342	−	0.921093i	$-0.372702\pi$
$547$	18.2119	0.778684	0.389342	+	0.921093i	$0.372702\pi$
$548$	53.4273	2.28230
$549$	30.6412	1.30773
$550$	−14.7669	−0.629661
$551$	−1.23471	−0.0526003
$552$	−13.2778	−0.565139
$553$	12.8434	0.546155
$554$	−53.7191	−2.28231
$555$	−0.229747	−0.00975220
$556$	−43.4069	−1.84086
$557$	−18.9583	−0.803289	−0.401644	−	0.915796i	$-0.631561\pi$
$557$	−18.9583	−0.803289	−0.401644	+	0.915796i	$0.631561\pi$
$558$	55.0407	2.33006
$559$	46.1332	1.95123
$560$	19.3207	0.816447
$561$	2.96044	0.124990
$562$	17.0683	0.719981
$563$	25.8985	1.09149	0.545746	−	0.837951i	$-0.316246\pi$
$563$	25.8985	1.09149	0.545746	+	0.837951i	$0.316246\pi$
$564$	−17.4301	−0.733938
$565$	−31.1916	−1.31224
$566$	−2.61483	−0.109909
$567$	−8.02414	−0.336982
$568$	−18.6673	−0.783263
$569$	2.75849	0.115642	0.0578210	−	0.998327i	$-0.481585\pi$
$569$	2.75849	0.115642	0.0578210	+	0.998327i	$0.481585\pi$
$570$	−4.92930	−0.206466
$571$	18.6025	0.778491	0.389246	−	0.921134i	$-0.372736\pi$
$571$	18.6025	0.778491	0.389246	+	0.921134i	$0.372736\pi$
$572$	−32.7241	−1.36826
$573$	−7.29555	−0.304776
$574$	1.77882	0.0742466
$575$	−21.1831	−0.883395
$576$	19.6330	0.818040
$577$	7.02896	0.292619	0.146310	−	0.989239i	$-0.453260\pi$
$577$	7.02896	0.292619	0.146310	+	0.989239i	$0.453260\pi$
$578$	−3.78905	−0.157604
$579$	−15.5534	−0.646376
$580$	15.8005	0.656078
$581$	7.65387	0.317536
$582$	3.85248	0.159690
$583$	10.3908	0.430345
$584$	56.3049	2.32991
$585$	56.1542	2.32169
$586$	36.6717	1.51489
$587$	−27.3028	−1.12691	−0.563453	−	0.826148i	$-0.690528\pi$
$587$	−27.3028	−1.12691	−0.563453	+	0.826148i	$0.690528\pi$
$588$	−12.5529	−0.517671
$589$	−8.61787	−0.355093
$590$	29.9850	1.23446
$591$	−5.16622	−0.212510
$592$	0.483214	0.0198600
$593$	−22.7869	−0.935748	−0.467874	−	0.883795i	$-0.654980\pi$
$593$	−22.7869	−0.935748	−0.467874	+	0.883795i	$0.654980\pi$
$594$	10.3591	0.425040
$595$	−18.0015	−0.737991
$596$	78.4961	3.21533
$597$	0.478473	0.0195826
$598$	−70.2024	−2.87079
$599$	1.49797	0.0612052	0.0306026	−	0.999532i	$-0.490257\pi$
$599$	1.49797	0.0612052	0.0306026	+	0.999532i	$0.490257\pi$
$600$	15.9934	0.652927
$601$	−27.3661	−1.11629	−0.558143	−	0.829745i	$-0.688486\pi$
$601$	−27.3661	−1.11629	−0.558143	+	0.829745i	$0.688486\pi$
$602$	−24.0243	−0.979158
$603$	−2.40768	−0.0980483
$604$	71.2772	2.90023
$605$	30.3856	1.23535
$606$	7.77973	0.316030
$607$	34.2937	1.39194	0.695968	−	0.718072i	$-0.254975\pi$
$607$	34.2937	1.39194	0.695968	+	0.718072i	$0.254975\pi$
$608$	0.361022	0.0146414
$609$	1.12843	0.0457262
$610$	91.8144	3.71746
$611$	−46.4939	−1.88094
$612$	41.2537	1.66758
$613$	−26.8359	−1.08389	−0.541945	−	0.840414i	$-0.682312\pi$
$613$	−26.8359	−1.08389	−0.541945	+	0.840414i	$0.682312\pi$
$614$	−48.0847	−1.94054
$615$	1.00581	0.0405580
$616$	8.59755	0.346405
$617$	−17.4588	−0.702865	−0.351432	−	0.936213i	$-0.614305\pi$
$617$	−17.4588	−0.702865	−0.351432	+	0.936213i	$0.614305\pi$
$618$	20.5424	0.826336
$619$	−17.3420	−0.697036	−0.348518	−	0.937302i	$-0.613315\pi$
$619$	−17.3420	−0.697036	−0.348518	+	0.937302i	$0.613315\pi$
$620$	110.282	4.42904
$621$	14.8602	0.596317
$622$	49.2620	1.97523
$623$	−17.2058	−0.689336
$624$	18.1948	0.728376
$625$	−24.7409	−0.989637
$626$	56.6334	2.26353
$627$	−0.752977	−0.0300710
$628$	−58.1037	−2.31859
$629$	−0.450222	−0.0179515
$630$	−29.2428	−1.16506
$631$	−10.9897	−0.437493	−0.218746	−	0.975782i	$-0.570197\pi$
$631$	−10.9897	−0.437493	−0.218746	+	0.975782i	$0.570197\pi$
$632$	−44.4946	−1.76990
$633$	0.632827	0.0251526
$634$	−6.56570	−0.260757
$635$	−15.7988	−0.626957
$636$	−22.3066	−0.884515
$637$	−33.4841	−1.32669
$638$	3.60953	0.142903
$639$	9.69892	0.383683
$640$	61.1181	2.41590
$641$	32.0694	1.26666	0.633332	−	0.773880i	$-0.281687\pi$
$641$	32.0694	1.26666	0.633332	+	0.773880i	$0.281687\pi$
$642$	−1.80755	−0.0713385
$643$	5.11784	0.201828	0.100914	−	0.994895i	$-0.467823\pi$
$643$	5.11784	0.201828	0.100914	+	0.994895i	$0.467823\pi$
$644$	24.4459	0.963303
$645$	−13.5842	−0.534876
$646$	−9.65968	−0.380055
$647$	−24.2597	−0.953745	−0.476873	−	0.878972i	$-0.658230\pi$
$647$	−24.2597	−0.953745	−0.476873	+	0.878972i	$0.658230\pi$
$648$	27.7988	1.09204
$649$	4.58037	0.179795
$650$	84.5605	3.31674
$651$	7.87607	0.308688
$652$	43.7043	1.71159
$653$	−2.23568	−0.0874888	−0.0437444	−	0.999043i	$-0.513929\pi$
$653$	−2.23568	−0.0874888	−0.0437444	+	0.999043i	$0.513929\pi$
$654$	5.74180	0.224522
$655$	−7.08319	−0.276763
$656$	−2.11546	−0.0825949
$657$	−29.2542	−1.14131
$658$	24.2121	0.943887
$659$	−15.7496	−0.613517	−0.306759	−	0.951787i	$-0.599244\pi$
$659$	−15.7496	−0.613517	−0.306759	+	0.951787i	$0.599244\pi$
$660$	9.63579	0.375073
$661$	−0.968862	−0.0376844	−0.0188422	−	0.999822i	$-0.505998\pi$
$661$	−0.968862	−0.0376844	−0.0188422	+	0.999822i	$0.505998\pi$
$662$	−23.2582	−0.903956
$663$	−16.9526	−0.658383
$664$	−26.5161	−1.02902
$665$	4.57863	0.177552
$666$	−0.731370	−0.0283400
$667$	5.17787	0.200488
$668$	−93.0236	−3.59919
$669$	0.206766	0.00799404
$670$	−7.21446	−0.278719
$671$	14.0252	0.541435
$672$	−0.329946	−0.0127280
$673$	36.7201	1.41545	0.707727	−	0.706486i	$-0.249721\pi$
$673$	36.7201	1.41545	0.707727	+	0.706486i	$0.249721\pi$
$674$	−23.5745	−0.908057
$675$	−17.8994	−0.688949
$676$	134.917	5.18912
$677$	−26.5654	−1.02099	−0.510495	−	0.859880i	$-0.670538\pi$
$677$	−26.5654	−1.02099	−0.510495	+	0.859880i	$0.670538\pi$
$678$	15.2968	0.587471
$679$	−3.57841	−0.137327
$680$	62.3646	2.39157
$681$	−6.86283	−0.262984
$682$	25.1934	0.964704
$683$	34.6181	1.32462	0.662312	−	0.749228i	$-0.269575\pi$
$683$	34.6181	1.32462	0.662312	+	0.749228i	$0.269575\pi$
$684$	−10.4928	−0.401200
$685$	−41.9642	−1.60337
$686$	42.2749	1.61406
$687$	18.2612	0.696709
$688$	28.5709	1.08926
$689$	−59.5018	−2.26684
$690$	20.6715	0.786950
$691$	14.9796	0.569851	0.284925	−	0.958550i	$-0.408031\pi$
$691$	14.9796	0.569851	0.284925	+	0.958550i	$0.408031\pi$
$692$	−57.4607	−2.18433
$693$	−4.46700	−0.169687
$694$	3.55473	0.134936
$695$	34.0938	1.29325
$696$	−3.90933	−0.148183
$697$	1.97103	0.0746579
$698$	17.8046	0.673915
$699$	1.27257	0.0481330
$700$	−29.4457	−1.11294
$701$	−26.5428	−1.00251	−0.501255	−	0.865300i	$-0.667128\pi$
$701$	−26.5428	−1.00251	−0.501255	+	0.865300i	$0.667128\pi$
$702$	−59.3201	−2.23889
$703$	0.114513	0.00431892
$704$	8.98644	0.338689
$705$	13.6904	0.515609
$706$	−21.5854	−0.812379
$707$	−7.22629	−0.271773
$708$	−9.83295	−0.369545
$709$	−14.4190	−0.541515	−0.270758	−	0.962648i	$-0.587274\pi$
$709$	−14.4190	−0.541515	−0.270758	+	0.962648i	$0.587274\pi$
$710$	29.0622	1.09068
$711$	23.1179	0.866989
$712$	59.6078	2.23390
$713$	36.1399	1.35345
$714$	8.82821	0.330388
$715$	25.7030	0.961238
$716$	−78.1033	−2.91886
$717$	11.2696	0.420870
$718$	29.5915	1.10435
$719$	−38.1466	−1.42263	−0.711314	−	0.702874i	$-0.751900\pi$
$719$	−38.1466	−1.42263	−0.711314	+	0.702874i	$0.751900\pi$
$720$	34.7770	1.29606
$721$	−19.0810	−0.710614
$722$	2.45691	0.0914367
$723$	9.89762	0.368096
$724$	54.5193	2.02619
$725$	−6.23687	−0.231631
$726$	−14.9015	−0.553048
$727$	53.1435	1.97098	0.985492	−	0.169721i	$-0.0542866\pi$
$727$	53.1435	1.97098	0.985492	+	0.169721i	$0.0542866\pi$
$728$	−49.2327	−1.82469
$729$	−7.71603	−0.285779
$730$	−87.6583	−3.24438
$731$	−26.6202	−0.984583
$732$	−30.1086	−1.11285
$733$	−48.5293	−1.79247	−0.896236	−	0.443578i	$-0.853709\pi$
$733$	−48.5293	−1.79247	−0.896236	+	0.443578i	$0.853709\pi$
$734$	25.2992	0.933809
$735$	9.85959	0.363676
$736$	−1.51398	−0.0558061
$737$	−1.10205	−0.0405945
$738$	3.20186	0.117862
$739$	33.7901	1.24299	0.621495	−	0.783418i	$-0.286525\pi$
$739$	33.7901	1.24299	0.621495	+	0.783418i	$0.286525\pi$
$740$	−1.46541	−0.0538695
$741$	4.31183	0.158399
$742$	30.9862	1.13754
$743$	7.36374	0.270149	0.135075	−	0.990835i	$-0.456873\pi$
$743$	7.36374	0.270149	0.135075	+	0.990835i	$0.456873\pi$
$744$	−27.2859	−1.00035
$745$	−61.6544	−2.25884
$746$	−13.8451	−0.506904
$747$	13.7769	0.504070
$748$	18.8828	0.690422
$749$	1.67897	0.0613481
$750$	−0.252820	−0.00923167
$751$	9.72556	0.354891	0.177445	−	0.984131i	$-0.443217\pi$
$751$	9.72556	0.354891	0.177445	+	0.984131i	$0.443217\pi$
$752$	−28.7943	−1.05002
$753$	−3.03735	−0.110687
$754$	−20.6695	−0.752738
$755$	−55.9844	−2.03748
$756$	20.6565	0.751268
$757$	−50.6279	−1.84010	−0.920051	−	0.391799i	$-0.871853\pi$
$757$	−50.6279	−1.84010	−0.920051	+	0.391799i	$0.871853\pi$
$758$	12.5508	0.455866
$759$	3.15768	0.114617
$760$	−15.8622	−0.575383
$761$	−5.38196	−0.195096	−0.0975479	−	0.995231i	$-0.531100\pi$
$761$	−5.38196	−0.195096	−0.0975479	+	0.995231i	$0.531100\pi$
$762$	7.74797	0.280679
$763$	−5.33333	−0.193079
$764$	−46.5337	−1.68353
$765$	−32.4026	−1.17152
$766$	−9.48489	−0.342703
$767$	−26.2289	−0.947071
$768$	−20.4143	−0.736639
$769$	16.0795	0.579843	0.289921	−	0.957051i	$-0.406371\pi$
$769$	16.0795	0.579843	0.289921	+	0.957051i	$0.406371\pi$
$770$	−13.3851	−0.482366
$771$	−13.1653	−0.474136
$772$	−99.2051	−3.57047
$773$	17.5305	0.630530	0.315265	−	0.949004i	$-0.397907\pi$
$773$	17.5305	0.630530	0.315265	+	0.949004i	$0.397907\pi$
$774$	−43.2435	−1.55436
$775$	−43.5314	−1.56369
$776$	12.3971	0.445029
$777$	−0.104656	−0.00375450
$778$	−80.2723	−2.87790
$779$	−0.501324	−0.0179618
$780$	−55.1781	−1.97569
$781$	4.43941	0.158855
$782$	40.5088	1.44859
$783$	4.37523	0.156358
$784$	−20.7372	−0.740613
$785$	45.6373	1.62887
$786$	3.47370	0.123903
$787$	38.8017	1.38313	0.691566	−	0.722313i	$-0.256921\pi$
$787$	38.8017	1.38313	0.691566	+	0.722313i	$0.256921\pi$
$788$	−32.9521	−1.17387
$789$	2.23772	0.0796651
$790$	69.2714	2.46457
$791$	−14.2086	−0.505200
$792$	15.4755	0.549898
$793$	−80.3132	−2.85201
$794$	45.1225	1.60134
$795$	17.5206	0.621393
$796$	3.05188	0.108171
$797$	15.0947	0.534682	0.267341	−	0.963602i	$-0.413855\pi$
$797$	15.0947	0.534682	0.267341	+	0.963602i	$0.413855\pi$
$798$	−2.24543	−0.0794872
$799$	26.8283	0.949117
$800$	1.82363	0.0644749
$801$	−30.9703	−1.09428
$802$	−65.6559	−2.31839
$803$	−13.3903	−0.472533
$804$	2.36583	0.0834364
$805$	−19.2009	−0.676744
$806$	−144.266	−5.08157
$807$	8.80322	0.309888
$808$	25.0348	0.880720
$809$	26.3398	0.926059	0.463029	−	0.886343i	$-0.346762\pi$
$809$	26.3398	0.926059	0.463029	+	0.886343i	$0.346762\pi$
$810$	−43.2786	−1.52066
$811$	4.11555	0.144516	0.0722582	−	0.997386i	$-0.476979\pi$
$811$	4.11555	0.144516	0.0722582	+	0.997386i	$0.476979\pi$
$812$	7.19753	0.252584
$813$	7.82021	0.274267
$814$	−0.334764	−0.0117335
$815$	−34.3274	−1.20243
$816$	−10.4989	−0.367537
$817$	6.77076	0.236879
$818$	17.2793	0.604155
$819$	25.5797	0.893828
$820$	6.41540	0.224036
$821$	4.00759	0.139866	0.0699329	−	0.997552i	$-0.477721\pi$
$821$	4.00759	0.139866	0.0699329	+	0.997552i	$0.477721\pi$
$822$	20.5799	0.717805
$823$	−34.7942	−1.21285	−0.606426	−	0.795140i	$-0.707397\pi$
$823$	−34.7942	−1.21285	−0.606426	+	0.795140i	$0.707397\pi$
$824$	66.1043	2.30285
$825$	−3.80351	−0.132421
$826$	13.6590	0.475256
$827$	−35.2036	−1.22415	−0.612074	−	0.790801i	$-0.709665\pi$
$827$	−35.2036	−1.22415	−0.612074	+	0.790801i	$0.709665\pi$
$828$	44.0024	1.52919
$829$	−29.1008	−1.01071	−0.505356	−	0.862911i	$-0.668639\pi$
$829$	−29.1008	−1.01071	−0.505356	+	0.862911i	$0.668639\pi$
$830$	41.2816	1.43290
$831$	−13.8365	−0.479981
$832$	−51.4597	−1.78404
$833$	19.3213	0.669444
$834$	−16.7201	−0.578969
$835$	73.0650	2.52852
$836$	−4.80277	−0.166107
$837$	30.5377	1.05554
$838$	15.5859	0.538405
$839$	31.4456	1.08562	0.542812	−	0.839854i	$-0.317360\pi$
$839$	31.4456	1.08562	0.542812	+	0.839854i	$0.317360\pi$
$840$	14.4969	0.500189
$841$	−27.4755	−0.947431
$842$	−54.7893	−1.88816
$843$	4.39628	0.151416
$844$	4.03640	0.138939
$845$	−105.970	−3.64548
$846$	43.5816	1.49837
$847$	13.8415	0.475598
$848$	−36.8503	−1.26544
$849$	−0.673502	−0.0231145
$850$	−48.7939	−1.67362
$851$	−0.480219	−0.0164617
$852$	−9.53034	−0.326504
$853$	−22.1091	−0.757003	−0.378501	−	0.925601i	$-0.623560\pi$
$853$	−22.1091	−0.757003	−0.378501	+	0.925601i	$0.623560\pi$
$854$	41.8239	1.43118
$855$	8.24149	0.281853
$856$	−5.81662	−0.198808
$857$	40.3409	1.37802	0.689010	−	0.724752i	$-0.258046\pi$
$857$	40.3409	1.37802	0.689010	+	0.724752i	$0.258046\pi$
$858$	−12.6051	−0.430332
$859$	−6.60859	−0.225482	−0.112741	−	0.993624i	$-0.535963\pi$
$859$	−6.60859	−0.225482	−0.112741	+	0.993624i	$0.535963\pi$
$860$	−86.6448	−2.95456
$861$	0.458172	0.0156144
$862$	−66.5512	−2.26674
$863$	−21.3077	−0.725323	−0.362661	−	0.931921i	$-0.618132\pi$
$863$	−21.3077	−0.725323	−0.362661	+	0.931921i	$0.618132\pi$
$864$	−1.27929	−0.0435225
$865$	45.1323	1.53454
$866$	87.8786	2.98624
$867$	−0.975948	−0.0331449
$868$	50.2365	1.70514
$869$	10.5816	0.358956
$870$	6.08624	0.206343
$871$	6.31074	0.213831
$872$	18.4768	0.625703
$873$	−6.44111	−0.217999
$874$	−10.3033	−0.348514
$875$	0.234834	0.00793885
$876$	28.7457	0.971226
$877$	−40.6528	−1.37275	−0.686373	−	0.727249i	$-0.740798\pi$
$877$	−40.6528	−1.37275	−0.686373	+	0.727249i	$0.740798\pi$
$878$	−92.0523	−3.10662
$879$	9.44553	0.318590
$880$	15.9182	0.536603
$881$	35.0399	1.18052	0.590262	−	0.807211i	$-0.299024\pi$
$881$	35.0399	1.18052	0.590262	+	0.807211i	$0.299024\pi$
$882$	31.3868	1.05685
$883$	1.02759	0.0345810	0.0172905	−	0.999851i	$-0.494496\pi$
$883$	1.02759	0.0345810	0.0172905	+	0.999851i	$0.494496\pi$
$884$	−108.130	−3.63679
$885$	7.72325	0.259614
$886$	21.7994	0.732365
$887$	−24.4818	−0.822020	−0.411010	−	0.911631i	$-0.634824\pi$
$887$	−24.4818	−0.822020	−0.411010	+	0.911631i	$0.634824\pi$
$888$	0.362569	0.0121670
$889$	−7.19678	−0.241372
$890$	−92.8004	−3.11068
$891$	−6.61105	−0.221479
$892$	1.31883	0.0441577
$893$	−6.82369	−0.228346
$894$	30.2362	1.01125
$895$	61.3459	2.05057
$896$	27.8409	0.930099
$897$	−18.0821	−0.603743
$898$	−65.9975	−2.20236
$899$	10.6406	0.354882
$900$	−53.0019	−1.76673
$901$	34.3343	1.14384
$902$	1.46556	0.0487979
$903$	−6.18795	−0.205922
$904$	49.2244	1.63718
$905$	−42.8219	−1.42345
$906$	27.4556	0.912150
$907$	17.0832	0.567238	0.283619	−	0.958937i	$-0.408465\pi$
$907$	17.0832	0.567238	0.283619	+	0.958937i	$0.408465\pi$
$908$	−43.7737	−1.45268
$909$	−13.0072	−0.431423
$910$	76.6480	2.54086
$911$	−15.2290	−0.504560	−0.252280	−	0.967654i	$-0.581180\pi$
$911$	−15.2290	−0.504560	−0.252280	+	0.967654i	$0.581180\pi$
$912$	2.67037	0.0884249
$913$	6.30599	0.208698
$914$	−95.3178	−3.15283
$915$	23.6487	0.781801
$916$	116.477	3.84850
$917$	−3.22658	−0.106551
$918$	34.2294	1.12974
$919$	33.6535	1.11013	0.555064	−	0.831808i	$-0.312694\pi$
$919$	33.6535	1.11013	0.555064	+	0.831808i	$0.312694\pi$
$920$	66.5198	2.19309
$921$	−12.3852	−0.408106
$922$	87.4807	2.88102
$923$	−25.4217	−0.836766
$924$	4.38936	0.144399
$925$	0.578436	0.0190189
$926$	−52.8465	−1.73664
$927$	−34.3456	−1.12806
$928$	−0.445757	−0.0146327
$929$	15.6003	0.511831	0.255915	−	0.966699i	$-0.417623\pi$
$929$	15.6003	0.511831	0.255915	+	0.966699i	$0.417623\pi$
$930$	42.4801	1.39298
$931$	−4.91431	−0.161060
$932$	8.11691	0.265878
$933$	12.6884	0.415401
$934$	−12.4229	−0.406491
$935$	−14.8314	−0.485038
$936$	−88.6185	−2.89659
$937$	54.2708	1.77295	0.886474	−	0.462778i	$-0.153147\pi$
$937$	54.2708	1.77295	0.886474	+	0.462778i	$0.153147\pi$
$938$	−3.28638	−0.107304
$939$	14.5871	0.476032
$940$	87.3222	2.84814
$941$	19.6155	0.639447	0.319724	−	0.947511i	$-0.396410\pi$
$941$	19.6155	0.639447	0.319724	+	0.947511i	$0.396410\pi$
$942$	−22.3812	−0.729219
$943$	2.10235	0.0684620
$944$	−16.2439	−0.528694
$945$	−16.2245	−0.527784
$946$	−19.7935	−0.643543
$947$	−20.4357	−0.664071	−0.332036	−	0.943267i	$-0.607735\pi$
$947$	−20.4357	−0.664071	−0.332036	+	0.943267i	$0.607735\pi$
$948$	−22.7161	−0.737784
$949$	76.6777	2.48906
$950$	12.4106	0.402652
$951$	−1.69113	−0.0548387
$952$	28.4087	0.920732
$953$	13.7569	0.445631	0.222816	−	0.974861i	$-0.428475\pi$
$953$	13.7569	0.445631	0.222816	+	0.974861i	$0.428475\pi$
$954$	55.7748	1.80578
$955$	36.5497	1.18272
$956$	71.8815	2.32481
$957$	0.929707	0.0300532
$958$	64.0134	2.06818
$959$	−19.1158	−0.617282
$960$	15.1526	0.489048
$961$	43.2677	1.39573
$962$	1.91698	0.0618060
$963$	3.02212	0.0973865
$964$	63.1306	2.03330
$965$	77.9203	2.50834
$966$	9.41641	0.302968
$967$	−38.2945	−1.23147	−0.615733	−	0.787955i	$-0.711140\pi$
$967$	−38.2945	−1.23147	−0.615733	+	0.787955i	$0.711140\pi$
$968$	−47.9524	−1.54125
$969$	−2.48805	−0.0799276
$970$	−19.3004	−0.619698
$971$	−42.7344	−1.37141	−0.685706	−	0.727878i	$-0.740506\pi$
$971$	−42.7344	−1.37141	−0.685706	+	0.727878i	$0.740506\pi$
$972$	57.1017	1.83154
$973$	15.5306	0.497889
$974$	−13.4650	−0.431446
$975$	21.7803	0.697527
$976$	−49.7391	−1.59211
$977$	31.6879	1.01379	0.506893	−	0.862009i	$-0.330794\pi$
$977$	31.6879	1.01379	0.506893	+	0.862009i	$0.330794\pi$
$978$	16.8346	0.538312
$979$	−14.1758	−0.453060
$980$	62.8881	2.00889
$981$	−9.59994	−0.306502
$982$	−59.4486	−1.89708
$983$	42.8559	1.36689	0.683445	−	0.730002i	$-0.260481\pi$
$983$	42.8559	1.36689	0.683445	+	0.730002i	$0.260481\pi$
$984$	−1.58729	−0.0506010
$985$	25.8821	0.824671
$986$	11.9269	0.379829
$987$	6.23633	0.198505
$988$	27.5024	0.874969
$989$	−28.3938	−0.902871
$990$	−24.0930	−0.765727
$991$	2.70146	0.0858147	0.0429074	−	0.999079i	$-0.486338\pi$
$991$	2.70146	0.0858147	0.0429074	+	0.999079i	$0.486338\pi$
$992$	−3.11124	−0.0987820
$993$	−5.99062	−0.190107
$994$	13.2386	0.419903
$995$	−2.39708	−0.0759926
$996$	−13.5374	−0.428950
$997$	22.8906	0.724953	0.362477	−	0.931993i	$-0.381931\pi$
$997$	22.8906	0.724953	0.362477	+	0.931993i	$0.381931\pi$
$998$	−91.2037	−2.88700
$999$	−0.405779	−0.0128383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.69	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.69	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45691 q^{2} +0.632827 q^{3} +4.03640 q^{4} -3.17038 q^{5} +1.55480 q^{6} -1.44419 q^{7} +5.00326 q^{8} -2.59953 q^{9} +O(q^{10})\)	\(q+2.45691 q^{2} +0.632827 q^{3} +4.03640 q^{4} -3.17038 q^{5} +1.55480 q^{6} -1.44419 q^{7} +5.00326 q^{8} -2.59953 q^{9} -7.78933 q^{10} -1.18986 q^{11} +2.55435 q^{12} +6.81359 q^{13} -3.54825 q^{14} -2.00630 q^{15} +4.21975 q^{16} -3.93164 q^{17} -6.38681 q^{18} +1.00000 q^{19} -12.7969 q^{20} -0.913923 q^{21} -2.92339 q^{22} -4.19360 q^{23} +3.16620 q^{24} +5.05129 q^{25} +16.7404 q^{26} -3.54353 q^{27} -5.82934 q^{28} -1.23471 q^{29} -4.92930 q^{30} -8.61787 q^{31} +0.361022 q^{32} -0.752977 q^{33} -9.65968 q^{34} +4.57863 q^{35} -10.4928 q^{36} +0.114513 q^{37} +2.45691 q^{38} +4.31183 q^{39} -15.8622 q^{40} -0.501324 q^{41} -2.24543 q^{42} +6.77076 q^{43} -4.80277 q^{44} +8.24149 q^{45} -10.3033 q^{46} -6.82369 q^{47} +2.67037 q^{48} -4.91431 q^{49} +12.4106 q^{50} -2.48805 q^{51} +27.5024 q^{52} -8.73281 q^{53} -8.70614 q^{54} +3.77231 q^{55} -7.22566 q^{56} +0.632827 q^{57} -3.03357 q^{58} -3.84950 q^{59} -8.09824 q^{60} -11.7872 q^{61} -21.1733 q^{62} +3.75422 q^{63} -7.55250 q^{64} -21.6017 q^{65} -1.85000 q^{66} +0.926198 q^{67} -15.8697 q^{68} -2.65382 q^{69} +11.2493 q^{70} -3.73103 q^{71} -13.0061 q^{72} +11.2536 q^{73} +0.281347 q^{74} +3.19659 q^{75} +4.03640 q^{76} +1.71839 q^{77} +10.5938 q^{78} -8.89311 q^{79} -13.3782 q^{80} +5.55614 q^{81} -1.23171 q^{82} -5.29976 q^{83} -3.68896 q^{84} +12.4648 q^{85} +16.6351 q^{86} -0.781357 q^{87} -5.95319 q^{88} +11.9138 q^{89} +20.2486 q^{90} -9.84013 q^{91} -16.9270 q^{92} -5.45362 q^{93} -16.7652 q^{94} -3.17038 q^{95} +0.228464 q^{96} +2.47780 q^{97} -12.0740 q^{98} +3.09308 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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