LMFDB - Embedded newform 4009.2.a.c.1.33

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			1.33
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)

$f(q)$	$=$	$q-0.477696 q^{2} -3.43272 q^{3} -1.77181 q^{4} +3.57141 q^{5} +1.63980 q^{6} -1.07831 q^{7} +1.80178 q^{8} +8.78359 q^{9} +O(q^{10})$	\(q-0.477696 q^{2} -3.43272 q^{3} -1.77181 q^{4} +3.57141 q^{5} +1.63980 q^{6} -1.07831 q^{7} +1.80178 q^{8} +8.78359 q^{9} -1.70605 q^{10} -3.09339 q^{11} +6.08212 q^{12} -0.489920 q^{13} +0.515105 q^{14} -12.2597 q^{15} +2.68291 q^{16} +2.22157 q^{17} -4.19589 q^{18} +1.00000 q^{19} -6.32785 q^{20} +3.70155 q^{21} +1.47770 q^{22} -0.992978 q^{23} -6.18500 q^{24} +7.75498 q^{25} +0.234033 q^{26} -19.8535 q^{27} +1.91056 q^{28} +1.09778 q^{29} +5.85640 q^{30} -9.13290 q^{31} -4.88517 q^{32} +10.6188 q^{33} -1.06124 q^{34} -3.85110 q^{35} -15.5628 q^{36} +0.471947 q^{37} -0.477696 q^{38} +1.68176 q^{39} +6.43489 q^{40} +2.64620 q^{41} -1.76821 q^{42} -4.29792 q^{43} +5.48089 q^{44} +31.3698 q^{45} +0.474342 q^{46} +6.91878 q^{47} -9.20969 q^{48} -5.83724 q^{49} -3.70452 q^{50} -7.62604 q^{51} +0.868044 q^{52} -10.3456 q^{53} +9.48393 q^{54} -11.0478 q^{55} -1.94288 q^{56} -3.43272 q^{57} -0.524404 q^{58} -8.50408 q^{59} +21.7218 q^{60} +9.16327 q^{61} +4.36275 q^{62} -9.47145 q^{63} -3.03220 q^{64} -1.74971 q^{65} -5.07254 q^{66} +15.6673 q^{67} -3.93620 q^{68} +3.40862 q^{69} +1.83965 q^{70} -6.16527 q^{71} +15.8261 q^{72} +1.02757 q^{73} -0.225447 q^{74} -26.6207 q^{75} -1.77181 q^{76} +3.33564 q^{77} -0.803370 q^{78} +9.16306 q^{79} +9.58178 q^{80} +41.8007 q^{81} -1.26408 q^{82} +12.7276 q^{83} -6.55842 q^{84} +7.93415 q^{85} +2.05310 q^{86} -3.76837 q^{87} -5.57360 q^{88} -2.03157 q^{89} -14.9852 q^{90} +0.528287 q^{91} +1.75936 q^{92} +31.3507 q^{93} -3.30507 q^{94} +3.57141 q^{95} +16.7694 q^{96} -11.6021 q^{97} +2.78843 q^{98} -27.1711 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.477696	−0.337782	−0.168891	−	0.985635i	$-0.554019\pi$
$2$	−0.477696	−0.337782	−0.168891	+	0.985635i	$0.554019\pi$
$3$	−3.43272	−1.98188	−0.990942	−	0.134290i	$-0.957125\pi$
$3$	−3.43272	−1.98188	−0.990942	+	0.134290i	$0.957125\pi$
$4$	−1.77181	−0.885903
$5$	3.57141	1.59718	0.798592	−	0.601873i	$-0.205579\pi$
$5$	3.57141	1.59718	0.798592	+	0.601873i	$0.205579\pi$
$6$	1.63980	0.669445
$7$	−1.07831	−0.407564	−0.203782	−	0.979016i	$-0.565323\pi$
$7$	−1.07831	−0.407564	−0.203782	+	0.979016i	$0.565323\pi$
$8$	1.80178	0.637024
$9$	8.78359	2.92786
$10$	−1.70605	−0.539500
$11$	−3.09339	−0.932692	−0.466346	−	0.884602i	$-0.654430\pi$
$11$	−3.09339	−0.932692	−0.466346	+	0.884602i	$0.654430\pi$
$12$	6.08212	1.75576
$13$	−0.489920	−0.135879	−0.0679397	−	0.997689i	$-0.521643\pi$
$13$	−0.489920	−0.135879	−0.0679397	+	0.997689i	$0.521643\pi$
$14$	0.515105	0.137668
$15$	−12.2597	−3.16543
$16$	2.68291	0.670728
$17$	2.22157	0.538810	0.269405	−	0.963027i	$-0.413173\pi$
$17$	2.22157	0.538810	0.269405	+	0.963027i	$0.413173\pi$
$18$	−4.19589	−0.988980
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−6.32785	−1.41495
$21$	3.70155	0.807744
$22$	1.47770	0.315047
$23$	−0.992978	−0.207050	−0.103525	−	0.994627i	$-0.533012\pi$
$23$	−0.992978	−0.207050	−0.103525	+	0.994627i	$0.533012\pi$
$24$	−6.18500	−1.26251
$25$	7.75498	1.55100
$26$	0.234033	0.0458976
$27$	−19.8535	−3.82080
$28$	1.91056	0.361062
$29$	1.09778	0.203852	0.101926	−	0.994792i	$-0.467499\pi$
$29$	1.09778	0.203852	0.101926	+	0.994792i	$0.467499\pi$
$30$	5.85640	1.06923
$31$	−9.13290	−1.64032	−0.820158	−	0.572136i	$-0.806115\pi$
$31$	−9.13290	−1.64032	−0.820158	+	0.572136i	$0.806115\pi$
$32$	−4.88517	−0.863584
$33$	10.6188	1.84849
$34$	−1.06124	−0.182000
$35$	−3.85110	−0.650954
$36$	−15.5628	−2.59380
$37$	0.471947	0.0775876	0.0387938	−	0.999247i	$-0.487648\pi$
$37$	0.471947	0.0775876	0.0387938	+	0.999247i	$0.487648\pi$
$38$	−0.477696	−0.0774925
$39$	1.68176	0.269297
$40$	6.43489	1.01745
$41$	2.64620	0.413267	0.206633	−	0.978418i	$-0.433749\pi$
$41$	2.64620	0.413267	0.206633	+	0.978418i	$0.433749\pi$
$42$	−1.76821	−0.272841
$43$	−4.29792	−0.655427	−0.327714	−	0.944777i	$-0.606278\pi$
$43$	−4.29792	−0.655427	−0.327714	+	0.944777i	$0.606278\pi$
$44$	5.48089	0.826275
$45$	31.3698	4.67634
$46$	0.474342	0.0699379
$47$	6.91878	1.00921	0.504604	−	0.863351i	$-0.331639\pi$
$47$	6.91878	1.00921	0.504604	+	0.863351i	$0.331639\pi$
$48$	−9.20969	−1.32930
$49$	−5.83724	−0.833892
$50$	−3.70452	−0.523899
$51$	−7.62604	−1.06786
$52$	0.868044	0.120376
$53$	−10.3456	−1.42108	−0.710538	−	0.703659i	$-0.751548\pi$
$53$	−10.3456	−1.42108	−0.710538	+	0.703659i	$0.751548\pi$
$54$	9.48393	1.29060
$55$	−11.0478	−1.48968
$56$	−1.94288	−0.259628
$57$	−3.43272	−0.454675
$58$	−0.524404	−0.0688577
$59$	−8.50408	−1.10714	−0.553569	−	0.832803i	$-0.686734\pi$
$59$	−8.50408	−1.10714	−0.553569	+	0.832803i	$0.686734\pi$
$60$	21.7218	2.80427
$61$	9.16327	1.17324	0.586618	−	0.809864i	$-0.300459\pi$
$61$	9.16327	1.17324	0.586618	+	0.809864i	$0.300459\pi$
$62$	4.36275	0.554070
$63$	−9.47145	−1.19329
$64$	−3.03220	−0.379024
$65$	−1.74971	−0.217024
$66$	−5.07254	−0.624386
$67$	15.6673	1.91407	0.957035	−	0.289972i	$-0.0936459\pi$
$67$	15.6673	1.91407	0.957035	+	0.289972i	$0.0936459\pi$
$68$	−3.93620	−0.477334
$69$	3.40862	0.410350
$70$	1.83965	0.219881
$71$	−6.16527	−0.731683	−0.365841	−	0.930677i	$-0.619219\pi$
$71$	−6.16527	−0.731683	−0.365841	+	0.930677i	$0.619219\pi$
$72$	15.8261	1.86512
$73$	1.02757	0.120268	0.0601341	−	0.998190i	$-0.480847\pi$
$73$	1.02757	0.120268	0.0601341	+	0.998190i	$0.480847\pi$
$74$	−0.225447	−0.0262077
$75$	−26.6207	−3.07389
$76$	−1.77181	−0.203240
$77$	3.33564	0.380131
$78$	−0.803370	−0.0909638
$79$	9.16306	1.03092	0.515462	−	0.856912i	$-0.327620\pi$
$79$	9.16306	1.03092	0.515462	+	0.856912i	$0.327620\pi$
$80$	9.58178	1.07128
$81$	41.8007	4.64453
$82$	−1.26408	−0.139594
$83$	12.7276	1.39703	0.698516	−	0.715595i	$-0.253844\pi$
$83$	12.7276	1.39703	0.698516	+	0.715595i	$0.253844\pi$
$84$	−6.55842	−0.715583
$85$	7.93415	0.860579
$86$	2.05310	0.221392
$87$	−3.76837	−0.404012
$88$	−5.57360	−0.594148
$89$	−2.03157	−0.215346	−0.107673	−	0.994186i	$-0.534340\pi$
$89$	−2.03157	−0.215346	−0.107673	+	0.994186i	$0.534340\pi$
$90$	−14.9852	−1.57958
$91$	0.528287	0.0553795
$92$	1.75936	0.183426
$93$	31.3507	3.25092
$94$	−3.30507	−0.340892
$95$	3.57141	0.366419
$96$	16.7694	1.71152
$97$	−11.6021	−1.17801	−0.589006	−	0.808129i	$-0.700480\pi$
$97$	−11.6021	−1.17801	−0.589006	+	0.808129i	$0.700480\pi$
$98$	2.78843	0.281674
$99$	−27.1711	−2.73080
$100$	−13.7403	−1.37403
$101$	4.88219	0.485796	0.242898	−	0.970052i	$-0.421902\pi$
$101$	4.88219	0.485796	0.242898	+	0.970052i	$0.421902\pi$
$102$	3.64293	0.360704
$103$	5.84554	0.575979	0.287989	−	0.957634i	$-0.407013\pi$
$103$	5.84554	0.575979	0.287989	+	0.957634i	$0.407013\pi$
$104$	−0.882727	−0.0865585
$105$	13.2197	1.29012
$106$	4.94205	0.480014
$107$	12.6103	1.21908	0.609540	−	0.792755i	$-0.291354\pi$
$107$	12.6103	1.21908	0.609540	+	0.792755i	$0.291354\pi$
$108$	35.1765	3.38486
$109$	6.73618	0.645209	0.322605	−	0.946534i	$-0.395442\pi$
$109$	6.73618	0.645209	0.322605	+	0.946534i	$0.395442\pi$
$110$	5.27747	0.503187
$111$	−1.62006	−0.153770
$112$	−2.89301	−0.273364
$113$	−18.3419	−1.72546	−0.862731	−	0.505663i	$-0.831248\pi$
$113$	−18.3419	−1.72546	−0.862731	+	0.505663i	$0.831248\pi$
$114$	1.63980	0.153581
$115$	−3.54633	−0.330697
$116$	−1.94505	−0.180593
$117$	−4.30326	−0.397836
$118$	4.06237	0.373971
$119$	−2.39555	−0.219599
$120$	−22.0892	−2.01646
$121$	−1.43094	−0.130086
$122$	−4.37726	−0.396298
$123$	−9.08367	−0.819047
$124$	16.1817	1.45316
$125$	9.83917	0.880042
$126$	4.52448	0.403072
$127$	13.8823	1.23185	0.615927	−	0.787803i	$-0.288782\pi$
$127$	13.8823	1.23185	0.615927	+	0.787803i	$0.288782\pi$
$128$	11.2188	0.991612
$129$	14.7536	1.29898
$130$	0.835828	0.0733070
$131$	6.53457	0.570928	0.285464	−	0.958389i	$-0.407852\pi$
$131$	6.53457	0.570928	0.285464	+	0.958389i	$0.407852\pi$
$132$	−18.8144	−1.63758
$133$	−1.07831	−0.0935015
$134$	−7.48423	−0.646539
$135$	−70.9049	−6.10253
$136$	4.00278	0.343235
$137$	11.1895	0.955982	0.477991	−	0.878365i	$-0.341365\pi$
$137$	11.1895	0.955982	0.477991	+	0.878365i	$0.341365\pi$
$138$	−1.62828	−0.138609
$139$	−20.9956	−1.78082	−0.890410	−	0.455159i	$-0.849582\pi$
$139$	−20.9956	−1.78082	−0.890410	+	0.455159i	$0.849582\pi$
$140$	6.82340	0.576682
$141$	−23.7503	−2.00013
$142$	2.94512	0.247149
$143$	1.51551	0.126734
$144$	23.5656	1.96380
$145$	3.92062	0.325590
$146$	−0.490867	−0.0406244
$147$	20.0376	1.65268
$148$	−0.836199	−0.0687351
$149$	−5.81807	−0.476635	−0.238317	−	0.971187i	$-0.576596\pi$
$149$	−5.81807	−0.476635	−0.238317	+	0.971187i	$0.576596\pi$
$150$	12.7166	1.03831
$151$	−19.1238	−1.55627	−0.778137	−	0.628095i	$-0.783835\pi$
$151$	−19.1238	−1.55627	−0.778137	+	0.628095i	$0.783835\pi$
$152$	1.80178	0.146143
$153$	19.5134	1.57756
$154$	−1.59342	−0.128402
$155$	−32.6173	−2.61989
$156$	−2.97975	−0.238571
$157$	−7.71003	−0.615327	−0.307664	−	0.951495i	$-0.599547\pi$
$157$	−7.71003	−0.615327	−0.307664	+	0.951495i	$0.599547\pi$
$158$	−4.37716	−0.348228
$159$	35.5135	2.81641
$160$	−17.4470	−1.37930
$161$	1.07074	0.0843861
$162$	−19.9680	−1.56884
$163$	−2.85120	−0.223323	−0.111662	−	0.993746i	$-0.535617\pi$
$163$	−2.85120	−0.223323	−0.111662	+	0.993746i	$0.535617\pi$
$164$	−4.68855	−0.366114
$165$	37.9239	2.95237
$166$	−6.07990	−0.471892
$167$	21.1936	1.64001	0.820004	−	0.572358i	$-0.193971\pi$
$167$	21.1936	1.64001	0.820004	+	0.572358i	$0.193971\pi$
$168$	6.66936	0.514552
$169$	−12.7600	−0.981537
$170$	−3.79011	−0.290688
$171$	8.78359	0.671698
$172$	7.61509	0.580645
$173$	6.03775	0.459041	0.229521	−	0.973304i	$-0.426284\pi$
$173$	6.03775	0.459041	0.229521	+	0.973304i	$0.426284\pi$
$174$	1.80014	0.136468
$175$	−8.36229	−0.632130
$176$	−8.29929	−0.625582
$177$	29.1922	2.19422
$178$	0.970473	0.0727400
$179$	−8.00934	−0.598646	−0.299323	−	0.954152i	$-0.596761\pi$
$179$	−8.00934	−0.598646	−0.299323	+	0.954152i	$0.596761\pi$
$180$	−55.5813	−4.14278
$181$	−7.76436	−0.577121	−0.288560	−	0.957462i	$-0.593177\pi$
$181$	−7.76436	−0.577121	−0.288560	+	0.957462i	$0.593177\pi$
$182$	−0.252360	−0.0187062
$183$	−31.4550	−2.32522
$184$	−1.78913	−0.131896
$185$	1.68552	0.123922
$186$	−14.9761	−1.09810
$187$	−6.87219	−0.502544
$188$	−12.2587	−0.894060
$189$	21.4082	1.55722
$190$	−1.70605	−0.123770
$191$	16.7595	1.21267	0.606336	−	0.795209i	$-0.292639\pi$
$191$	16.7595	1.21267	0.606336	+	0.795209i	$0.292639\pi$
$192$	10.4087	0.751182
$193$	−19.3664	−1.39402	−0.697011	−	0.717061i	$-0.745487\pi$
$193$	−19.3664	−1.39402	−0.697011	+	0.717061i	$0.745487\pi$
$194$	5.54226	0.397911
$195$	6.00626	0.430117
$196$	10.3425	0.738748
$197$	−2.97153	−0.211712	−0.105856	−	0.994381i	$-0.533758\pi$
$197$	−2.97153	−0.211712	−0.105856	+	0.994381i	$0.533758\pi$
$198$	12.9795	0.922414
$199$	−20.6291	−1.46236	−0.731178	−	0.682187i	$-0.761029\pi$
$199$	−20.6291	−1.46236	−0.731178	+	0.682187i	$0.761029\pi$
$200$	13.9727	0.988023
$201$	−53.7817	−3.79347
$202$	−2.33220	−0.164093
$203$	−1.18375	−0.0830828
$204$	13.5119	0.946020
$205$	9.45066	0.660063
$206$	−2.79239	−0.194555
$207$	−8.72191	−0.606215
$208$	−1.31441	−0.0911381
$209$	−3.09339	−0.213974
$210$	−6.31502	−0.435778
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	18.3304	1.25894
$213$	21.1637	1.45011
$214$	−6.02387	−0.411784
$215$	−15.3497	−1.04684
$216$	−35.7715	−2.43395
$217$	9.84811	0.668533
$218$	−3.21785	−0.217940
$219$	−3.52737	−0.238358
$220$	19.5745	1.31971
$221$	−1.08839	−0.0732132
$222$	0.773898	0.0519406
$223$	1.25783	0.0842306	0.0421153	−	0.999113i	$-0.486590\pi$
$223$	1.25783	0.0842306	0.0421153	+	0.999113i	$0.486590\pi$
$224$	5.26774	0.351965
$225$	68.1166	4.54111
$226$	8.76186	0.582830
$227$	−4.95345	−0.328772	−0.164386	−	0.986396i	$-0.552564\pi$
$227$	−4.95345	−0.328772	−0.164386	+	0.986396i	$0.552564\pi$
$228$	6.08212	0.402798
$229$	−7.53952	−0.498226	−0.249113	−	0.968474i	$-0.580139\pi$
$229$	−7.53952	−0.498226	−0.249113	+	0.968474i	$0.580139\pi$
$230$	1.69407	0.111704
$231$	−11.4503	−0.753376
$232$	1.97795	0.129859
$233$	−1.83507	−0.120220	−0.0601098	−	0.998192i	$-0.519145\pi$
$233$	−1.83507	−0.120220	−0.0601098	+	0.998192i	$0.519145\pi$
$234$	2.05565	0.134382
$235$	24.7098	1.61189
$236$	15.0676	0.980817
$237$	−31.4542	−2.04317
$238$	1.14434	0.0741768
$239$	1.10734	0.0716280	0.0358140	−	0.999358i	$-0.488598\pi$
$239$	1.10734	0.0716280	0.0358140	+	0.999358i	$0.488598\pi$
$240$	−32.8916	−2.12314
$241$	−19.8041	−1.27569	−0.637847	−	0.770163i	$-0.720175\pi$
$241$	−19.8041	−1.27569	−0.637847	+	0.770163i	$0.720175\pi$
$242$	0.683556	0.0439407
$243$	−83.9299	−5.38411
$244$	−16.2355	−1.03937
$245$	−20.8472	−1.33188
$246$	4.33923	0.276659
$247$	−0.489920	−0.0311729
$248$	−16.4554	−1.04492
$249$	−43.6902	−2.76875
$250$	−4.70013	−0.297263
$251$	−25.2512	−1.59384	−0.796922	−	0.604083i	$-0.793540\pi$
$251$	−25.2512	−1.59384	−0.796922	+	0.604083i	$0.793540\pi$
$252$	16.7816	1.05714
$253$	3.07167	0.193114
$254$	−6.63151	−0.416098
$255$	−27.2357	−1.70557
$256$	0.705210	0.0440756
$257$	−16.9086	−1.05473	−0.527364	−	0.849640i	$-0.676819\pi$
$257$	−16.9086	−1.05473	−0.527364	+	0.849640i	$0.676819\pi$
$258$	−7.04773	−0.438772
$259$	−0.508906	−0.0316219
$260$	3.10014	0.192263
$261$	9.64244	0.596852
$262$	−3.12154	−0.192849
$263$	−21.7230	−1.33949	−0.669747	−	0.742589i	$-0.733598\pi$
$263$	−21.7230	−1.33949	−0.669747	+	0.742589i	$0.733598\pi$
$264$	19.1326	1.17753
$265$	−36.9483	−2.26972
$266$	0.515105	0.0315831
$267$	6.97382	0.426791
$268$	−27.7595	−1.69568
$269$	1.66752	0.101671	0.0508353	−	0.998707i	$-0.483812\pi$
$269$	1.66752	0.101671	0.0508353	+	0.998707i	$0.483812\pi$
$270$	33.8710	2.06132
$271$	30.6961	1.86465	0.932327	−	0.361616i	$-0.117775\pi$
$271$	30.6961	1.86465	0.932327	+	0.361616i	$0.117775\pi$
$272$	5.96028	0.361395
$273$	−1.81346	−0.109756
$274$	−5.34517	−0.322914
$275$	−23.9892	−1.44660
$276$	−6.03941	−0.363530
$277$	−11.6796	−0.701758	−0.350879	−	0.936421i	$-0.614117\pi$
$277$	−11.6796	−0.701758	−0.350879	+	0.936421i	$0.614117\pi$
$278$	10.0295	0.601529
$279$	−80.2197	−4.80263
$280$	−6.93882	−0.414674
$281$	−2.56241	−0.152860	−0.0764302	−	0.997075i	$-0.524352\pi$
$281$	−2.56241	−0.152860	−0.0764302	+	0.997075i	$0.524352\pi$
$282$	11.3454	0.675609
$283$	1.25417	0.0745527	0.0372763	−	0.999305i	$-0.488132\pi$
$283$	1.25417	0.0745527	0.0372763	+	0.999305i	$0.488132\pi$
$284$	10.9237	0.648200
$285$	−12.2597	−0.726200
$286$	−0.723955	−0.0428084
$287$	−2.85343	−0.168432
$288$	−42.9093	−2.52846
$289$	−12.0646	−0.709683
$290$	−1.87286	−0.109978
$291$	39.8267	2.33468
$292$	−1.82066	−0.106546
$293$	29.0523	1.69725	0.848627	−	0.528992i	$-0.177430\pi$
$293$	29.0523	1.69725	0.848627	+	0.528992i	$0.177430\pi$
$294$	−9.57190	−0.558245
$295$	−30.3716	−1.76830
$296$	0.850343	0.0494252
$297$	61.4145	3.56363
$298$	2.77927	0.160999
$299$	0.486480	0.0281339
$300$	47.1667	2.72317
$301$	4.63450	0.267128
$302$	9.13537	0.525681
$303$	−16.7592	−0.962791
$304$	2.68291	0.153875
$305$	32.7258	1.87387
$306$	−9.32147	−0.532873
$307$	−8.96527	−0.511675	−0.255838	−	0.966720i	$-0.582351\pi$
$307$	−8.96527	−0.511675	−0.255838	+	0.966720i	$0.582351\pi$
$308$	−5.91010	−0.336759
$309$	−20.0661	−1.14152
$310$	15.5812	0.884951
$311$	1.73655	0.0984705	0.0492353	−	0.998787i	$-0.484322\pi$
$311$	1.73655	0.0984705	0.0492353	+	0.998787i	$0.484322\pi$
$312$	3.03016	0.171549
$313$	−16.4056	−0.927301	−0.463650	−	0.886018i	$-0.653461\pi$
$313$	−16.4056	−0.927301	−0.463650	+	0.886018i	$0.653461\pi$
$314$	3.68305	0.207847
$315$	−33.8265	−1.90590
$316$	−16.2352	−0.913299
$317$	5.79497	0.325478	0.162739	−	0.986669i	$-0.447967\pi$
$317$	5.79497	0.325478	0.162739	+	0.986669i	$0.447967\pi$
$318$	−16.9647	−0.951332
$319$	−3.39585	−0.190131
$320$	−10.8292	−0.605372
$321$	−43.2876	−2.41608
$322$	−0.511488	−0.0285041
$323$	2.22157	0.123612
$324$	−74.0628	−4.11460
$325$	−3.79932	−0.210748
$326$	1.36201	0.0754346
$327$	−23.1235	−1.27873
$328$	4.76786	0.263261
$329$	−7.46060	−0.411316
$330$	−18.1161	−0.997259
$331$	5.89923	0.324251	0.162125	−	0.986770i	$-0.448165\pi$
$331$	5.89923	0.324251	0.162125	+	0.986770i	$0.448165\pi$
$332$	−22.5508	−1.23763
$333$	4.14539	0.227166
$334$	−10.1241	−0.553965
$335$	55.9545	3.05712
$336$	9.93092	0.541776
$337$	12.7958	0.697032	0.348516	−	0.937303i	$-0.386686\pi$
$337$	12.7958	0.697032	0.348516	+	0.937303i	$0.386686\pi$
$338$	6.09539	0.331546
$339$	62.9627	3.41967
$340$	−14.0578	−0.762390
$341$	28.2516	1.52991
$342$	−4.19589	−0.226888
$343$	13.8426	0.747428
$344$	−7.74390	−0.417523
$345$	12.1736	0.655404
$346$	−2.88421	−0.155056
$347$	1.49160	0.0800732	0.0400366	−	0.999198i	$-0.487253\pi$
$347$	1.49160	0.0800732	0.0400366	+	0.999198i	$0.487253\pi$
$348$	6.67682	0.357915
$349$	33.3071	1.78289	0.891443	−	0.453132i	$-0.149693\pi$
$349$	33.3071	1.78289	0.891443	+	0.453132i	$0.149693\pi$
$350$	3.99463	0.213522
$351$	9.72662	0.519168
$352$	15.1117	0.805458
$353$	1.20283	0.0640200	0.0320100	−	0.999488i	$-0.489809\pi$
$353$	1.20283	0.0640200	0.0320100	+	0.999488i	$0.489809\pi$
$354$	−13.9450	−0.741168
$355$	−22.0187	−1.16863
$356$	3.59955	0.190776
$357$	8.22325	0.435221
$358$	3.82603	0.202212
$359$	8.55566	0.451550	0.225775	−	0.974179i	$-0.427509\pi$
$359$	8.55566	0.451550	0.225775	+	0.974179i	$0.427509\pi$
$360$	56.5214	2.97894
$361$	1.00000	0.0526316
$362$	3.70901	0.194941
$363$	4.91204	0.257815
$364$	−0.936022	−0.0490609
$365$	3.66988	0.192090
$366$	15.0259	0.785417
$367$	−24.5273	−1.28032	−0.640158	−	0.768243i	$-0.721131\pi$
$367$	−24.5273	−1.28032	−0.640158	+	0.768243i	$0.721131\pi$
$368$	−2.66407	−0.138874
$369$	23.2431	1.20999
$370$	−0.805165	−0.0418585
$371$	11.1558	0.579179
$372$	−55.5474	−2.88000
$373$	−25.6165	−1.32637	−0.663187	−	0.748454i	$-0.730797\pi$
$373$	−25.6165	−1.32637	−0.663187	+	0.748454i	$0.730797\pi$
$374$	3.28282	0.169750
$375$	−33.7752	−1.74414
$376$	12.4661	0.642890
$377$	−0.537824	−0.0276993
$378$	−10.2266	−0.526001
$379$	14.8882	0.764755	0.382378	−	0.924006i	$-0.375105\pi$
$379$	14.8882	0.764755	0.382378	+	0.924006i	$0.375105\pi$
$380$	−6.32785	−0.324612
$381$	−47.6541	−2.44139
$382$	−8.00593	−0.409619
$383$	10.2512	0.523810	0.261905	−	0.965094i	$-0.415649\pi$
$383$	10.2512	0.523810	0.261905	+	0.965094i	$0.415649\pi$
$384$	−38.5111	−1.96526
$385$	11.9129	0.607139
$386$	9.25124	0.470876
$387$	−37.7512	−1.91900
$388$	20.5566	1.04360
$389$	−18.2445	−0.925034	−0.462517	−	0.886610i	$-0.653054\pi$
$389$	−18.2445	−0.925034	−0.462517	+	0.886610i	$0.653054\pi$
$390$	−2.86917	−0.145286
$391$	−2.20597	−0.111561
$392$	−10.5174	−0.531210
$393$	−22.4314	−1.13151
$394$	1.41949	0.0715127
$395$	32.7250	1.64658
$396$	48.1419	2.41922
$397$	8.46729	0.424961	0.212481	−	0.977165i	$-0.431846\pi$
$397$	8.46729	0.424961	0.212481	+	0.977165i	$0.431846\pi$
$398$	9.85443	0.493958
$399$	3.70155	0.185309
$400$	20.8059	1.04030
$401$	−33.2589	−1.66087	−0.830435	−	0.557116i	$-0.811908\pi$
$401$	−33.2589	−1.66087	−0.830435	+	0.557116i	$0.811908\pi$
$402$	25.6913	1.28136
$403$	4.47439	0.222885
$404$	−8.65029	−0.430368
$405$	149.288	7.41816
$406$	0.565471	0.0280639
$407$	−1.45992	−0.0723653
$408$	−13.7404	−0.680253
$409$	−25.5313	−1.26244	−0.631220	−	0.775604i	$-0.717446\pi$
$409$	−25.5313	−1.26244	−0.631220	+	0.775604i	$0.717446\pi$
$410$	−4.51454	−0.222957
$411$	−38.4104	−1.89465
$412$	−10.3572	−0.510261
$413$	9.17006	0.451229
$414$	4.16642	0.204769
$415$	45.4554	2.23132
$416$	2.39334	0.117343
$417$	72.0720	3.52938
$418$	1.47770	0.0722767
$419$	15.6418	0.764154	0.382077	−	0.924130i	$-0.375209\pi$
$419$	15.6418	0.764154	0.382077	+	0.924130i	$0.375209\pi$
$420$	−23.4228	−1.14292
$421$	14.0798	0.686208	0.343104	−	0.939298i	$-0.388522\pi$
$421$	14.0798	0.686208	0.343104	+	0.939298i	$0.388522\pi$
$422$	−0.477696	−0.0232539
$423$	60.7717	2.95482
$424$	−18.6404	−0.905260
$425$	17.2282	0.835693
$426$	−10.1098	−0.489821
$427$	−9.88086	−0.478169
$428$	−22.3430	−1.07999
$429$	−5.20234	−0.251171
$430$	7.33247	0.353603
$431$	−26.1901	−1.26153	−0.630767	−	0.775973i	$-0.717260\pi$
$431$	−26.1901	−1.26153	−0.630767	+	0.775973i	$0.717260\pi$
$432$	−53.2651	−2.56272
$433$	−24.2435	−1.16507	−0.582534	−	0.812806i	$-0.697939\pi$
$433$	−24.2435	−1.16507	−0.582534	+	0.812806i	$0.697939\pi$
$434$	−4.70440	−0.225819
$435$	−13.4584	−0.645281
$436$	−11.9352	−0.571593
$437$	−0.992978	−0.0475006
$438$	1.68501	0.0805129
$439$	−23.8448	−1.13805	−0.569025	−	0.822320i	$-0.692679\pi$
$439$	−23.8448	−1.13805	−0.569025	+	0.822320i	$0.692679\pi$
$440$	−19.9056	−0.948963
$441$	−51.2720	−2.44152
$442$	0.519921	0.0247301
$443$	−9.94655	−0.472575	−0.236287	−	0.971683i	$-0.575931\pi$
$443$	−9.94655	−0.472575	−0.236287	+	0.971683i	$0.575931\pi$
$444$	2.87044	0.136225
$445$	−7.25557	−0.343947
$446$	−0.600861	−0.0284516
$447$	19.9718	0.944635
$448$	3.26965	0.154477
$449$	−14.2822	−0.674020	−0.337010	−	0.941501i	$-0.609416\pi$
$449$	−14.2822	−0.674020	−0.337010	+	0.941501i	$0.609416\pi$
$450$	−32.5390	−1.53390
$451$	−8.18572	−0.385450
$452$	32.4983	1.52859
$453$	65.6468	3.08435
$454$	2.36624	0.111053
$455$	1.88673	0.0884512
$456$	−6.18500	−0.289639
$457$	−14.9950	−0.701437	−0.350718	−	0.936481i	$-0.614063\pi$
$457$	−14.9950	−0.701437	−0.350718	+	0.936481i	$0.614063\pi$
$458$	3.60160	0.168292
$459$	−44.1059	−2.05869
$460$	6.28342	0.292966
$461$	−7.19583	−0.335143	−0.167572	−	0.985860i	$-0.553593\pi$
$461$	−7.19583	−0.335143	−0.167572	+	0.985860i	$0.553593\pi$
$462$	5.46977	0.254477
$463$	26.9376	1.25189	0.625947	−	0.779865i	$-0.284712\pi$
$463$	26.9376	1.25189	0.625947	+	0.779865i	$0.284712\pi$
$464$	2.94524	0.136729
$465$	111.966	5.19231
$466$	0.876607	0.0406080
$467$	−28.3575	−1.31223	−0.656115	−	0.754661i	$-0.727801\pi$
$467$	−28.3575	−1.31223	−0.656115	+	0.754661i	$0.727801\pi$
$468$	7.62454	0.352445
$469$	−16.8943	−0.780105
$470$	−11.8038	−0.544468
$471$	26.4664	1.21951
$472$	−15.3225	−0.705274
$473$	13.2951	0.611312
$474$	15.0256	0.690147
$475$	7.75498	0.355823
$476$	4.24445	0.194544
$477$	−90.8714	−4.16072
$478$	−0.528973	−0.0241947
$479$	17.8675	0.816388	0.408194	−	0.912895i	$-0.366159\pi$
$479$	17.8675	0.816388	0.408194	+	0.912895i	$0.366159\pi$
$480$	59.8906	2.73362
$481$	−0.231216	−0.0105426
$482$	9.46033	0.430906
$483$	−3.67555	−0.167244
$484$	2.53536	0.115243
$485$	−41.4358	−1.88150
$486$	40.0930	1.81866
$487$	−39.0180	−1.76807	−0.884037	−	0.467416i	$-0.845185\pi$
$487$	−39.0180	−1.76807	−0.884037	+	0.467416i	$0.845185\pi$
$488$	16.5102	0.747380
$489$	9.78738	0.442600
$490$	9.95863	0.449885
$491$	9.50040	0.428747	0.214373	−	0.976752i	$-0.431229\pi$
$491$	9.50040	0.428747	0.214373	+	0.976752i	$0.431229\pi$
$492$	16.0945	0.725596
$493$	2.43879	0.109838
$494$	0.234033	0.0105296
$495$	−97.0391	−4.36158
$496$	−24.5028	−1.10021
$497$	6.64808	0.298207
$498$	20.8706	0.935236
$499$	14.0268	0.627927	0.313964	−	0.949435i	$-0.398343\pi$
$499$	14.0268	0.627927	0.313964	+	0.949435i	$0.398343\pi$
$500$	−17.4331	−0.779632
$501$	−72.7517	−3.25030
$502$	12.0624	0.538372
$503$	17.1227	0.763463	0.381731	−	0.924273i	$-0.375328\pi$
$503$	17.1227	0.763463	0.381731	+	0.924273i	$0.375328\pi$
$504$	−17.0654	−0.760155
$505$	17.4363	0.775905
$506$	−1.46732	−0.0652305
$507$	43.8015	1.94529
$508$	−24.5967	−1.09130
$509$	19.9917	0.886116	0.443058	−	0.896493i	$-0.353894\pi$
$509$	19.9917	0.886116	0.443058	+	0.896493i	$0.353894\pi$
$510$	13.0104	0.576110
$511$	−1.10804	−0.0490169
$512$	−22.7745	−1.00650
$513$	−19.8535	−0.876552
$514$	8.07715	0.356268
$515$	20.8768	0.919944
$516$	−26.1405	−1.15077
$517$	−21.4025	−0.941280
$518$	0.243102	0.0106813
$519$	−20.7259	−0.909767
$520$	−3.15258	−0.138250
$521$	−35.7326	−1.56547	−0.782737	−	0.622353i	$-0.786177\pi$
$521$	−35.7326	−1.56547	−0.782737	+	0.622353i	$0.786177\pi$
$522$	−4.60615	−0.201606
$523$	−22.3925	−0.979155	−0.489577	−	0.871960i	$-0.662849\pi$
$523$	−22.3925	−0.979155	−0.489577	+	0.871960i	$0.662849\pi$
$524$	−11.5780	−0.505787
$525$	28.7054	1.25281
$526$	10.3770	0.452457
$527$	−20.2894	−0.883820
$528$	28.4892	1.23983
$529$	−22.0140	−0.957130
$530$	17.6501	0.766671
$531$	−74.6964	−3.24155
$532$	1.91056	0.0828333
$533$	−1.29643	−0.0561544
$534$	−3.33136	−0.144162
$535$	45.0365	1.94710
$536$	28.2291	1.21931
$537$	27.4938	1.18645
$538$	−0.796569	−0.0343425
$539$	18.0569	0.777764
$540$	125.630	5.40625
$541$	12.2865	0.528240	0.264120	−	0.964490i	$-0.414919\pi$
$541$	12.2865	0.528240	0.264120	+	0.964490i	$0.414919\pi$
$542$	−14.6634	−0.629847
$543$	26.6529	1.14379
$544$	−10.8528	−0.465308
$545$	24.0577	1.03052
$546$	0.866284	0.0370735
$547$	12.7837	0.546591	0.273295	−	0.961930i	$-0.411886\pi$
$547$	12.7837	0.546591	0.273295	+	0.961930i	$0.411886\pi$
$548$	−19.8256	−0.846907
$549$	80.4864	3.43508
$550$	11.4595	0.488636
$551$	1.09778	0.0467669
$552$	6.14157	0.261403
$553$	−9.88063	−0.420167
$554$	5.57929	0.237041
$555$	−5.78591	−0.245598
$556$	37.2001	1.57763
$557$	17.5203	0.742358	0.371179	−	0.928561i	$-0.378954\pi$
$557$	17.5203	0.742358	0.371179	+	0.928561i	$0.378954\pi$
$558$	38.3206	1.62224
$559$	2.10564	0.0890590
$560$	−10.3321	−0.436613
$561$	23.5903	0.995984
$562$	1.22405	0.0516335
$563$	28.8949	1.21778	0.608888	−	0.793256i	$-0.291616\pi$
$563$	28.8949	1.21778	0.608888	+	0.793256i	$0.291616\pi$
$564$	42.0809	1.77192
$565$	−65.5065	−2.75588
$566$	−0.599112	−0.0251826
$567$	−45.0742	−1.89294
$568$	−11.1084	−0.466100
$569$	45.6849	1.91521	0.957606	−	0.288082i	$-0.0930175\pi$
$569$	45.6849	1.91521	0.957606	+	0.288082i	$0.0930175\pi$
$570$	5.85640	0.245297
$571$	−16.6310	−0.695985	−0.347993	−	0.937497i	$-0.613137\pi$
$571$	−16.6310	−0.695985	−0.347993	+	0.937497i	$0.613137\pi$
$572$	−2.68520	−0.112274
$573$	−57.5306	−2.40338
$574$	1.36307	0.0568935
$575$	−7.70053	−0.321134
$576$	−26.6336	−1.10973
$577$	14.0984	0.586926	0.293463	−	0.955970i	$-0.405192\pi$
$577$	14.0984	0.586926	0.293463	+	0.955970i	$0.405192\pi$
$578$	5.76322	0.239718
$579$	66.4794	2.76279
$580$	−6.94657	−0.288441
$581$	−13.7243	−0.569379
$582$	−19.0251	−0.788614
$583$	32.0029	1.32543
$584$	1.85146	0.0766138
$585$	−15.3687	−0.635418
$586$	−13.8782	−0.573302
$587$	8.73909	0.360701	0.180350	−	0.983602i	$-0.442277\pi$
$587$	8.73909	0.360701	0.180350	+	0.983602i	$0.442277\pi$
$588$	−35.5028	−1.46411
$589$	−9.13290	−0.376314
$590$	14.5084	0.597301
$591$	10.2004	0.419590
$592$	1.26619	0.0520402
$593$	−23.4448	−0.962763	−0.481382	−	0.876511i	$-0.659865\pi$
$593$	−23.4448	−0.962763	−0.481382	+	0.876511i	$0.659865\pi$
$594$	−29.3375	−1.20373
$595$	−8.55549	−0.350741
$596$	10.3085	0.422252
$597$	70.8139	2.89822
$598$	−0.232390	−0.00950312
$599$	−9.28051	−0.379191	−0.189596	−	0.981862i	$-0.560718\pi$
$599$	−9.28051	−0.379191	−0.189596	+	0.981862i	$0.560718\pi$
$600$	−47.9646	−1.95815
$601$	−35.3191	−1.44070	−0.720349	−	0.693612i	$-0.756018\pi$
$601$	−35.3191	−1.44070	−0.720349	+	0.693612i	$0.756018\pi$
$602$	−2.21388	−0.0902311
$603$	137.616	5.60414
$604$	33.8837	1.37871
$605$	−5.11049	−0.207771
$606$	8.00580	0.325213
$607$	−25.4694	−1.03377	−0.516885	−	0.856055i	$-0.672909\pi$
$607$	−25.4694	−1.03377	−0.516885	+	0.856055i	$0.672909\pi$
$608$	−4.88517	−0.198120
$609$	4.06348	0.164660
$610$	−15.6330	−0.632961
$611$	−3.38965	−0.137131
$612$	−34.5739	−1.39757
$613$	−9.83666	−0.397299	−0.198649	−	0.980071i	$-0.563656\pi$
$613$	−9.83666	−0.397299	−0.198649	+	0.980071i	$0.563656\pi$
$614$	4.28268	0.172835
$615$	−32.4415	−1.30817
$616$	6.01008	0.242153
$617$	32.4759	1.30743	0.653715	−	0.756741i	$-0.273209\pi$
$617$	32.4759	1.30743	0.653715	+	0.756741i	$0.273209\pi$
$618$	9.58551	0.385586
$619$	13.3942	0.538358	0.269179	−	0.963090i	$-0.413248\pi$
$619$	13.3942	0.538358	0.269179	+	0.963090i	$0.413248\pi$
$620$	57.7916	2.32097
$621$	19.7141	0.791098
$622$	−0.829541	−0.0332616
$623$	2.19067	0.0877672
$624$	4.51201	0.180625
$625$	−3.63517	−0.145407
$626$	7.83690	0.313226
$627$	10.6188	0.424072
$628$	13.6607	0.545120
$629$	1.04846	0.0418050
$630$	16.1588	0.643781
$631$	−28.9428	−1.15220	−0.576098	−	0.817381i	$-0.695425\pi$
$631$	−28.9428	−1.15220	−0.576098	+	0.817381i	$0.695425\pi$
$632$	16.5098	0.656724
$633$	−3.43272	−0.136439
$634$	−2.76824	−0.109941
$635$	49.5794	1.96750
$636$	−62.9231	−2.49506
$637$	2.85978	0.113309
$638$	1.62219	0.0642230
$639$	−54.1532	−2.14227
$640$	40.0670	1.58379
$641$	12.2804	0.485048	0.242524	−	0.970145i	$-0.422025\pi$
$641$	12.2804	0.485048	0.242524	+	0.970145i	$0.422025\pi$
$642$	20.6783	0.816107
$643$	26.2972	1.03706	0.518531	−	0.855059i	$-0.326479\pi$
$643$	26.2972	1.03706	0.518531	+	0.855059i	$0.326479\pi$
$644$	−1.89714	−0.0747579
$645$	52.6911	2.07471
$646$	−1.06124	−0.0417538
$647$	−25.4235	−0.999502	−0.499751	−	0.866169i	$-0.666575\pi$
$647$	−25.4235	−0.999502	−0.499751	+	0.866169i	$0.666575\pi$
$648$	75.3156	2.95868
$649$	26.3064	1.03262
$650$	1.81492	0.0711871
$651$	−33.8058	−1.32496
$652$	5.05177	0.197843
$653$	−31.7080	−1.24083	−0.620416	−	0.784273i	$-0.713036\pi$
$653$	−31.7080	−1.24083	−0.620416	+	0.784273i	$0.713036\pi$
$654$	11.0460	0.431932
$655$	23.3376	0.911877
$656$	7.09951	0.277189
$657$	9.02577	0.352129
$658$	3.56390	0.138935
$659$	−42.9593	−1.67346	−0.836728	−	0.547619i	$-0.815534\pi$
$659$	−42.9593	−1.67346	−0.836728	+	0.547619i	$0.815534\pi$
$660$	−67.1939	−2.61552
$661$	−41.8791	−1.62891	−0.814455	−	0.580227i	$-0.802964\pi$
$661$	−41.8791	−1.62891	−0.814455	+	0.580227i	$0.802964\pi$
$662$	−2.81804	−0.109526
$663$	3.73615	0.145100
$664$	22.9322	0.889943
$665$	−3.85110	−0.149339
$666$	−1.98024	−0.0767326
$667$	−1.09007	−0.0422077
$668$	−37.5509	−1.45289
$669$	−4.31778	−0.166935
$670$	−26.7293	−1.03264
$671$	−28.3456	−1.09427
$672$	−18.0827	−0.697555
$673$	−9.86159	−0.380136	−0.190068	−	0.981771i	$-0.560871\pi$
$673$	−9.86159	−0.380136	−0.190068	+	0.981771i	$0.560871\pi$
$674$	−6.11251	−0.235445
$675$	−153.963	−5.92605
$676$	22.6082	0.869547
$677$	−18.6115	−0.715298	−0.357649	−	0.933856i	$-0.616422\pi$
$677$	−18.6115	−0.715298	−0.357649	+	0.933856i	$0.616422\pi$
$678$	−30.0770	−1.15510
$679$	12.5106	0.480114
$680$	14.2956	0.548210
$681$	17.0038	0.651587
$682$	−13.4957	−0.516776
$683$	18.6655	0.714217	0.357109	−	0.934063i	$-0.383763\pi$
$683$	18.6655	0.714217	0.357109	+	0.934063i	$0.383763\pi$
$684$	−15.5628	−0.595060
$685$	39.9622	1.52688
$686$	−6.61253	−0.252468
$687$	25.8811	0.987425
$688$	−11.5309	−0.439613
$689$	5.06851	0.193095
$690$	−5.81527	−0.221384
$691$	0.691609	0.0263101	0.0131550	−	0.999913i	$-0.495813\pi$
$691$	0.691609	0.0263101	0.0131550	+	0.999913i	$0.495813\pi$
$692$	−10.6977	−0.406666
$693$	29.2989	1.11297
$694$	−0.712530	−0.0270473
$695$	−74.9838	−2.84430
$696$	−6.78976	−0.257365
$697$	5.87872	0.222672
$698$	−15.9107	−0.602227
$699$	6.29930	0.238261
$700$	14.8164	0.560006
$701$	−35.6770	−1.34750	−0.673750	−	0.738959i	$-0.735318\pi$
$701$	−35.6770	−1.34750	−0.673750	+	0.738959i	$0.735318\pi$
$702$	−4.64637	−0.175366
$703$	0.471947	0.0177998
$704$	9.37976	0.353513
$705$	−84.8219	−3.19458
$706$	−0.574586	−0.0216248
$707$	−5.26452	−0.197993
$708$	−51.7229	−1.94387
$709$	28.8059	1.08183	0.540915	−	0.841077i	$-0.318078\pi$
$709$	28.8059	1.08183	0.540915	+	0.841077i	$0.318078\pi$
$710$	10.5183	0.394743
$711$	80.4846	3.01841
$712$	−3.66044	−0.137181
$713$	9.06877	0.339628
$714$	−3.92822	−0.147010
$715$	5.41252	0.202417
$716$	14.1910	0.530342
$717$	−3.80120	−0.141958
$718$	−4.08701	−0.152526
$719$	−39.9630	−1.49037	−0.745184	−	0.666859i	$-0.767638\pi$
$719$	−39.9630	−1.49037	−0.745184	+	0.666859i	$0.767638\pi$
$720$	84.1625	3.13655
$721$	−6.30332	−0.234748
$722$	−0.477696	−0.0177780
$723$	67.9820	2.52828
$724$	13.7570	0.511273
$725$	8.51325	0.316174
$726$	−2.34646	−0.0870853
$727$	21.8746	0.811285	0.405643	−	0.914032i	$-0.367048\pi$
$727$	21.8746	0.811285	0.405643	+	0.914032i	$0.367048\pi$
$728$	0.951855	0.0352781
$729$	162.706	6.02615
$730$	−1.75309	−0.0648847
$731$	−9.54814	−0.353151
$732$	55.7321	2.05992
$733$	29.0044	1.07130	0.535651	−	0.844440i	$-0.320066\pi$
$733$	29.0044	1.07130	0.535651	+	0.844440i	$0.320066\pi$
$734$	11.7166	0.432468
$735$	71.5627	2.63963
$736$	4.85087	0.178805
$737$	−48.4652	−1.78524
$738$	−11.1031	−0.408713
$739$	−42.9048	−1.57828	−0.789139	−	0.614215i	$-0.789473\pi$
$739$	−42.9048	−1.57828	−0.789139	+	0.614215i	$0.789473\pi$
$740$	−2.98641	−0.109783
$741$	1.68176	0.0617810
$742$	−5.32907	−0.195636
$743$	17.4071	0.638605	0.319302	−	0.947653i	$-0.396551\pi$
$743$	17.4071	0.638605	0.319302	+	0.947653i	$0.396551\pi$
$744$	56.4870	2.07091
$745$	−20.7787	−0.761274
$746$	12.2369	0.448025
$747$	111.794	4.09032
$748$	12.1762	0.445205
$749$	−13.5978	−0.496853
$750$	16.1343	0.589140
$751$	−4.00910	−0.146294	−0.0731471	−	0.997321i	$-0.523304\pi$
$751$	−4.00910	−0.146294	−0.0731471	+	0.997321i	$0.523304\pi$
$752$	18.5625	0.676904
$753$	86.6805	3.15881
$754$	0.256916	0.00935634
$755$	−68.2990	−2.48565
$756$	−37.9313	−1.37955
$757$	35.1343	1.27698	0.638489	−	0.769631i	$-0.279560\pi$
$757$	35.1343	1.27698	0.638489	+	0.769631i	$0.279560\pi$
$758$	−7.11203	−0.258321
$759$	−10.5442	−0.382730
$760$	6.43489	0.233418
$761$	19.1294	0.693440	0.346720	−	0.937969i	$-0.387295\pi$
$761$	19.1294	0.693440	0.346720	+	0.937969i	$0.387295\pi$
$762$	22.7642	0.824658
$763$	−7.26370	−0.262964
$764$	−29.6945	−1.07431
$765$	69.6903	2.51966
$766$	−4.89694	−0.176934
$767$	4.16632	0.150437
$768$	−2.42079	−0.0873527
$769$	25.0850	0.904588	0.452294	−	0.891869i	$-0.350606\pi$
$769$	25.0850	0.904588	0.452294	+	0.891869i	$0.350606\pi$
$770$	−5.69076	−0.205081
$771$	58.0424	2.09035
$772$	34.3135	1.23497
$773$	−35.5065	−1.27708	−0.638539	−	0.769589i	$-0.720461\pi$
$773$	−35.5065	−1.27708	−0.638539	+	0.769589i	$0.720461\pi$
$774$	18.0336	0.648204
$775$	−70.8255	−2.54413
$776$	−20.9043	−0.750422
$777$	1.74693	0.0626709
$778$	8.71534	0.312460
$779$	2.64620	0.0948099
$780$	−10.6419	−0.381042
$781$	19.0716	0.682435
$782$	1.05378	0.0376832
$783$	−21.7947	−0.778879
$784$	−15.6608	−0.559314
$785$	−27.5357	−0.982791
$786$	10.7154	0.382205
$787$	33.5834	1.19712	0.598559	−	0.801079i	$-0.295740\pi$
$787$	33.5834	1.19712	0.598559	+	0.801079i	$0.295740\pi$
$788$	5.26497	0.187557
$789$	74.5689	2.65472
$790$	−15.6326	−0.556184
$791$	19.7783	0.703235
$792$	−48.9562	−1.73958
$793$	−4.48927	−0.159419
$794$	−4.04479	−0.143544
$795$	126.833	4.49832
$796$	36.5507	1.29551
$797$	12.1566	0.430610	0.215305	−	0.976547i	$-0.430925\pi$
$797$	12.1566	0.430610	0.215305	+	0.976547i	$0.430925\pi$
$798$	−1.76821	−0.0625941
$799$	15.3706	0.543771
$800$	−37.8844	−1.33942
$801$	−17.8445	−0.630504
$802$	15.8876	0.561012
$803$	−3.17868	−0.112173
$804$	95.2907	3.36064
$805$	3.82405	0.134780
$806$	−2.13740	−0.0752867
$807$	−5.72414	−0.201499
$808$	8.79661	0.309464
$809$	1.78653	0.0628111	0.0314055	−	0.999507i	$-0.490002\pi$
$809$	1.78653	0.0628111	0.0314055	+	0.999507i	$0.490002\pi$
$810$	−71.3141	−2.50572
$811$	21.5202	0.755677	0.377838	−	0.925872i	$-0.376668\pi$
$811$	21.5202	0.755677	0.377838	+	0.925872i	$0.376668\pi$
$812$	2.09737	0.0736033
$813$	−105.371	−3.69553
$814$	0.697396	0.0244437
$815$	−10.1828	−0.356688
$816$	−20.4600	−0.716243
$817$	−4.29792	−0.150365
$818$	12.1962	0.426430
$819$	4.64026	0.162144
$820$	−16.7447	−0.584752
$821$	−33.8888	−1.18273	−0.591363	−	0.806405i	$-0.701410\pi$
$821$	−33.8888	−1.18273	−0.591363	+	0.806405i	$0.701410\pi$
$822$	18.3485	0.639977
$823$	26.5944	0.927021	0.463511	−	0.886091i	$-0.346590\pi$
$823$	26.5944	0.927021	0.463511	+	0.886091i	$0.346590\pi$
$824$	10.5324	0.366912
$825$	82.3482	2.86700
$826$	−4.38050	−0.152417
$827$	−22.6980	−0.789286	−0.394643	−	0.918834i	$-0.629132\pi$
$827$	−22.6980	−0.789286	−0.394643	+	0.918834i	$0.629132\pi$
$828$	15.4535	0.537048
$829$	−32.1149	−1.11540	−0.557698	−	0.830044i	$-0.688315\pi$
$829$	−32.1149	−1.11540	−0.557698	+	0.830044i	$0.688315\pi$
$830$	−21.7138	−0.753699
$831$	40.0928	1.39080
$832$	1.48553	0.0515016
$833$	−12.9679	−0.449310
$834$	−34.4285	−1.19216
$835$	75.6909	2.61939
$836$	5.48089	0.189560
$837$	181.320	6.26733
$838$	−7.47205	−0.258118
$839$	−2.01092	−0.0694247	−0.0347123	−	0.999397i	$-0.511052\pi$
$839$	−2.01092	−0.0694247	−0.0347123	+	0.999397i	$0.511052\pi$
$840$	23.8190	0.821835
$841$	−27.7949	−0.958444
$842$	−6.72587	−0.231789
$843$	8.79603	0.302951
$844$	−1.77181	−0.0609881
$845$	−45.5711	−1.56769
$846$	−29.0304	−0.998086
$847$	1.54300	0.0530182
$848$	−27.7563	−0.953155
$849$	−4.30522	−0.147755
$850$	−8.22987	−0.282282
$851$	−0.468633	−0.0160645
$852$	−37.4979	−1.28466
$853$	27.8124	0.952277	0.476139	−	0.879370i	$-0.342036\pi$
$853$	27.8124	0.952277	0.476139	+	0.879370i	$0.342036\pi$
$854$	4.72005	0.161517
$855$	31.3698	1.07283
$856$	22.7209	0.776584
$857$	−21.4248	−0.731857	−0.365928	−	0.930643i	$-0.619248\pi$
$857$	−21.4248	−0.731857	−0.365928	+	0.930643i	$0.619248\pi$
$858$	2.48514	0.0848412
$859$	5.72305	0.195268	0.0976339	−	0.995222i	$-0.468873\pi$
$859$	5.72305	0.195268	0.0976339	+	0.995222i	$0.468873\pi$
$860$	27.1966	0.927397
$861$	9.79502	0.333814
$862$	12.5109	0.426123
$863$	16.2488	0.553114	0.276557	−	0.960997i	$-0.410806\pi$
$863$	16.2488	0.553114	0.276557	+	0.960997i	$0.410806\pi$
$864$	96.9876	3.29959
$865$	21.5633	0.733173
$866$	11.5810	0.393539
$867$	41.4145	1.40651
$868$	−17.4489	−0.592256
$869$	−28.3449	−0.961535
$870$	6.42902	0.217964
$871$	−7.67575	−0.260083
$872$	12.1371	0.411014
$873$	−101.908	−3.44906
$874$	0.474342	0.0160448
$875$	−10.6097	−0.358673
$876$	6.24982	0.211162
$877$	15.2654	0.515477	0.257739	−	0.966215i	$-0.417023\pi$
$877$	15.2654	0.515477	0.257739	+	0.966215i	$0.417023\pi$
$878$	11.3906	0.384413
$879$	−99.7285	−3.36376
$880$	−29.6402	−0.999170
$881$	−30.1780	−1.01672	−0.508361	−	0.861144i	$-0.669749\pi$
$881$	−30.1780	−1.01672	−0.508361	+	0.861144i	$0.669749\pi$
$882$	24.4924	0.824703
$883$	−29.3636	−0.988165	−0.494083	−	0.869415i	$-0.664496\pi$
$883$	−29.3636	−0.988165	−0.494083	+	0.869415i	$0.664496\pi$
$884$	1.92842	0.0648598
$885$	104.257	3.50457
$886$	4.75143	0.159627
$887$	18.9917	0.637679	0.318839	−	0.947809i	$-0.396707\pi$
$887$	18.9917	0.637679	0.318839	+	0.947809i	$0.396707\pi$
$888$	−2.91899	−0.0979550
$889$	−14.9694	−0.502059
$890$	3.46596	0.116179
$891$	−129.306	−4.33191
$892$	−2.22863	−0.0746201
$893$	6.91878	0.231528
$894$	−9.54047	−0.319081
$895$	−28.6046	−0.956148
$896$	−12.0974	−0.404145
$897$	−1.66995	−0.0557580
$898$	6.82256	0.227672
$899$	−10.0259	−0.334382
$900$	−120.689	−4.02298
$901$	−22.9835	−0.765690
$902$	3.91029	0.130198
$903$	−15.9090	−0.529417
$904$	−33.0480	−1.09916
$905$	−27.7297	−0.921768
$906$	−31.3592	−1.04184
$907$	35.2955	1.17197	0.585985	−	0.810322i	$-0.300708\pi$
$907$	35.2955	1.17197	0.585985	+	0.810322i	$0.300708\pi$
$908$	8.77655	0.291260
$909$	42.8831	1.42234
$910$	−0.901283	−0.0298772
$911$	−52.3122	−1.73318	−0.866590	−	0.499021i	$-0.833693\pi$
$911$	−52.3122	−1.73318	−0.866590	+	0.499021i	$0.833693\pi$
$912$	−9.20969	−0.304963
$913$	−39.3713	−1.30300
$914$	7.16306	0.236933
$915$	−112.339	−3.71380
$916$	13.3586	0.441380
$917$	−7.04631	−0.232689
$918$	21.0692	0.695388
$919$	−0.975751	−0.0321871	−0.0160935	−	0.999870i	$-0.505123\pi$
$919$	−0.975751	−0.0321871	−0.0160935	+	0.999870i	$0.505123\pi$
$920$	−6.38970	−0.210662
$921$	30.7753	1.01408
$922$	3.43742	0.113205
$923$	3.02049	0.0994206
$924$	20.2878	0.667418
$925$	3.65994	0.120338
$926$	−12.8680	−0.422868
$927$	51.3449	1.68639
$928$	−5.36283	−0.176044
$929$	−12.9374	−0.424463	−0.212232	−	0.977219i	$-0.568073\pi$
$929$	−12.9374	−0.424463	−0.212232	+	0.977219i	$0.568073\pi$
$930$	−53.4859	−1.75387
$931$	−5.83724	−0.191308
$932$	3.25139	0.106503
$933$	−5.96108	−0.195157
$934$	13.5463	0.443248
$935$	−24.5434	−0.802655
$936$	−7.75351	−0.253432
$937$	−25.8486	−0.844435	−0.422218	−	0.906494i	$-0.638748\pi$
$937$	−25.8486	−0.844435	−0.422218	+	0.906494i	$0.638748\pi$
$938$	8.07033	0.263506
$939$	56.3160	1.83780
$940$	−43.7810	−1.42798
$941$	−54.2313	−1.76789	−0.883945	−	0.467591i	$-0.845122\pi$
$941$	−54.2313	−1.76789	−0.883945	+	0.467591i	$0.845122\pi$
$942$	−12.6429	−0.411928
$943$	−2.62762	−0.0855669
$944$	−22.8157	−0.742588
$945$	76.4576	2.48717
$946$	−6.35104	−0.206490
$947$	−20.7873	−0.675495	−0.337747	−	0.941237i	$-0.609665\pi$
$947$	−20.7873	−0.675495	−0.337747	+	0.941237i	$0.609665\pi$
$948$	55.7308	1.81005
$949$	−0.503428	−0.0163420
$950$	−3.70452	−0.120191
$951$	−19.8925	−0.645060
$952$	−4.31624	−0.139890
$953$	−11.3636	−0.368104	−0.184052	−	0.982917i	$-0.558921\pi$
$953$	−11.3636	−0.368104	−0.184052	+	0.982917i	$0.558921\pi$
$954$	43.4089	1.40542
$955$	59.8549	1.93686
$956$	−1.96200	−0.0634555
$957$	11.6570	0.376818
$958$	−8.53524	−0.275761
$959$	−12.0657	−0.389623
$960$	37.1737	1.19978
$961$	52.4098	1.69064
$962$	0.110451	0.00356109
$963$	110.763	3.56930
$964$	35.0890	1.13014
$965$	−69.1653	−2.22651
$966$	1.75580	0.0564919
$967$	5.75606	0.185102	0.0925512	−	0.995708i	$-0.470498\pi$
$967$	5.75606	0.185102	0.0925512	+	0.995708i	$0.470498\pi$
$968$	−2.57824	−0.0828678
$969$	−7.62604	−0.244984
$970$	19.7937	0.635537
$971$	−60.8600	−1.95309	−0.976545	−	0.215314i	$-0.930923\pi$
$971$	−60.8600	−1.95309	−0.976545	+	0.215314i	$0.930923\pi$
$972$	148.708	4.76980
$973$	22.6398	0.725797
$974$	18.6388	0.597224
$975$	13.0420	0.417679
$976$	24.5842	0.786922
$977$	6.53642	0.209119	0.104559	−	0.994519i	$-0.466657\pi$
$977$	6.53642	0.209119	0.104559	+	0.994519i	$0.466657\pi$
$978$	−4.67539	−0.149503
$979$	6.28443	0.200851
$980$	36.9372	1.17992
$981$	59.1679	1.88909
$982$	−4.53830	−0.144823
$983$	−10.3917	−0.331443	−0.165721	−	0.986173i	$-0.552995\pi$
$983$	−10.3917	−0.331443	−0.165721	+	0.986173i	$0.552995\pi$
$984$	−16.3667	−0.521753
$985$	−10.6125	−0.338144
$986$	−1.16500	−0.0371012
$987$	25.6102	0.815181
$988$	0.868044	0.0276161
$989$	4.26774	0.135706
$990$	46.3552	1.47326
$991$	−60.9929	−1.93750	−0.968752	−	0.248030i	$-0.920217\pi$
$991$	−60.9929	−1.93750	−0.968752	+	0.248030i	$0.920217\pi$
$992$	44.6158	1.41655
$993$	−20.2504	−0.642628
$994$	−3.17576	−0.100729
$995$	−73.6749	−2.33565
$996$	77.4106	2.45285
$997$	−34.9737	−1.10763	−0.553814	−	0.832640i	$-0.686828\pi$
$997$	−34.9737	−1.10763	−0.553814	+	0.832640i	$0.686828\pi$
$998$	−6.70056	−0.212103
$999$	−9.36979	−0.296447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.33	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.33	✓	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)

\(f(q)\)	\(=\)	\(q-0.477696 q^{2} -3.43272 q^{3} -1.77181 q^{4} +3.57141 q^{5} +1.63980 q^{6} -1.07831 q^{7} +1.80178 q^{8} +8.78359 q^{9} +O(q^{10})\)	\(q-0.477696 q^{2} -3.43272 q^{3} -1.77181 q^{4} +3.57141 q^{5} +1.63980 q^{6} -1.07831 q^{7} +1.80178 q^{8} +8.78359 q^{9} -1.70605 q^{10} -3.09339 q^{11} +6.08212 q^{12} -0.489920 q^{13} +0.515105 q^{14} -12.2597 q^{15} +2.68291 q^{16} +2.22157 q^{17} -4.19589 q^{18} +1.00000 q^{19} -6.32785 q^{20} +3.70155 q^{21} +1.47770 q^{22} -0.992978 q^{23} -6.18500 q^{24} +7.75498 q^{25} +0.234033 q^{26} -19.8535 q^{27} +1.91056 q^{28} +1.09778 q^{29} +5.85640 q^{30} -9.13290 q^{31} -4.88517 q^{32} +10.6188 q^{33} -1.06124 q^{34} -3.85110 q^{35} -15.5628 q^{36} +0.471947 q^{37} -0.477696 q^{38} +1.68176 q^{39} +6.43489 q^{40} +2.64620 q^{41} -1.76821 q^{42} -4.29792 q^{43} +5.48089 q^{44} +31.3698 q^{45} +0.474342 q^{46} +6.91878 q^{47} -9.20969 q^{48} -5.83724 q^{49} -3.70452 q^{50} -7.62604 q^{51} +0.868044 q^{52} -10.3456 q^{53} +9.48393 q^{54} -11.0478 q^{55} -1.94288 q^{56} -3.43272 q^{57} -0.524404 q^{58} -8.50408 q^{59} +21.7218 q^{60} +9.16327 q^{61} +4.36275 q^{62} -9.47145 q^{63} -3.03220 q^{64} -1.74971 q^{65} -5.07254 q^{66} +15.6673 q^{67} -3.93620 q^{68} +3.40862 q^{69} +1.83965 q^{70} -6.16527 q^{71} +15.8261 q^{72} +1.02757 q^{73} -0.225447 q^{74} -26.6207 q^{75} -1.77181 q^{76} +3.33564 q^{77} -0.803370 q^{78} +9.16306 q^{79} +9.58178 q^{80} +41.8007 q^{81} -1.26408 q^{82} +12.7276 q^{83} -6.55842 q^{84} +7.93415 q^{85} +2.05310 q^{86} -3.76837 q^{87} -5.57360 q^{88} -2.03157 q^{89} -14.9852 q^{90} +0.528287 q^{91} +1.75936 q^{92} +31.3507 q^{93} -3.30507 q^{94} +3.57141 q^{95} +16.7694 q^{96} -11.6021 q^{97} +2.78843 q^{98} -27.1711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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