LMFDB - Embedded newform 4009.2.a.c.1.22

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			1.22
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)

$f(q)$	$=$	$q-1.44569 q^{2} +0.647795 q^{3} +0.0900311 q^{4} -3.22671 q^{5} -0.936514 q^{6} +4.87053 q^{7} +2.76123 q^{8} -2.58036 q^{9} +O(q^{10})$	\(q-1.44569 q^{2} +0.647795 q^{3} +0.0900311 q^{4} -3.22671 q^{5} -0.936514 q^{6} +4.87053 q^{7} +2.76123 q^{8} -2.58036 q^{9} +4.66484 q^{10} -0.712533 q^{11} +0.0583217 q^{12} -0.152742 q^{13} -7.04130 q^{14} -2.09025 q^{15} -4.17196 q^{16} +3.90440 q^{17} +3.73041 q^{18} +1.00000 q^{19} -0.290505 q^{20} +3.15511 q^{21} +1.03010 q^{22} -1.01831 q^{23} +1.78871 q^{24} +5.41168 q^{25} +0.220818 q^{26} -3.61493 q^{27} +0.438499 q^{28} -8.18185 q^{29} +3.02186 q^{30} -4.42855 q^{31} +0.508911 q^{32} -0.461576 q^{33} -5.64456 q^{34} -15.7158 q^{35} -0.232313 q^{36} +2.64407 q^{37} -1.44569 q^{38} -0.0989456 q^{39} -8.90970 q^{40} -9.97907 q^{41} -4.56132 q^{42} +12.8265 q^{43} -0.0641501 q^{44} +8.32609 q^{45} +1.47217 q^{46} +1.55194 q^{47} -2.70257 q^{48} +16.7221 q^{49} -7.82364 q^{50} +2.52925 q^{51} -0.0137515 q^{52} -1.55328 q^{53} +5.22609 q^{54} +2.29914 q^{55} +13.4487 q^{56} +0.647795 q^{57} +11.8285 q^{58} -1.37475 q^{59} -0.188188 q^{60} +10.1835 q^{61} +6.40233 q^{62} -12.5677 q^{63} +7.60818 q^{64} +0.492855 q^{65} +0.667297 q^{66} -13.1428 q^{67} +0.351517 q^{68} -0.659658 q^{69} +22.7203 q^{70} -5.13287 q^{71} -7.12497 q^{72} -9.13435 q^{73} -3.82251 q^{74} +3.50566 q^{75} +0.0900311 q^{76} -3.47042 q^{77} +0.143045 q^{78} +8.60696 q^{79} +13.4617 q^{80} +5.39935 q^{81} +14.4267 q^{82} +1.02751 q^{83} +0.284058 q^{84} -12.5984 q^{85} -18.5431 q^{86} -5.30017 q^{87} -1.96747 q^{88} -6.86424 q^{89} -12.0370 q^{90} -0.743936 q^{91} -0.0916797 q^{92} -2.86879 q^{93} -2.24363 q^{94} -3.22671 q^{95} +0.329671 q^{96} +8.72232 q^{97} -24.1750 q^{98} +1.83859 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.44569	−1.02226	−0.511130	−	0.859503i	$-0.670773\pi$
$2$	−1.44569	−1.02226	−0.511130	+	0.859503i	$0.670773\pi$
$3$	0.647795	0.374005	0.187002	−	0.982359i	$-0.440123\pi$
$3$	0.647795	0.374005	0.187002	+	0.982359i	$0.440123\pi$
$4$	0.0900311	0.0450155
$5$	−3.22671	−1.44303	−0.721515	−	0.692399i	$-0.756554\pi$
$5$	−3.22671	−1.44303	−0.721515	+	0.692399i	$0.756554\pi$
$6$	−0.936514	−0.382330
$7$	4.87053	1.84089	0.920444	−	0.390874i	$-0.127827\pi$
$7$	4.87053	1.84089	0.920444	+	0.390874i	$0.127827\pi$
$8$	2.76123	0.976242
$9$	−2.58036	−0.860120
$10$	4.66484	1.47515
$11$	−0.712533	−0.214837	−0.107418	−	0.994214i	$-0.534258\pi$
$11$	−0.712533	−0.214837	−0.107418	+	0.994214i	$0.534258\pi$
$12$	0.0583217	0.0168360
$13$	−0.152742	−0.0423630	−0.0211815	−	0.999776i	$-0.506743\pi$
$13$	−0.152742	−0.0423630	−0.0211815	+	0.999776i	$0.506743\pi$
$14$	−7.04130	−1.88187
$15$	−2.09025	−0.539700
$16$	−4.17196	−1.04299
$17$	3.90440	0.946955	0.473478	−	0.880806i	$-0.342998\pi$
$17$	3.90440	0.946955	0.473478	+	0.880806i	$0.342998\pi$
$18$	3.73041	0.879267
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.290505	−0.0649588
$21$	3.15511	0.688501
$22$	1.03010	0.219619
$23$	−1.01831	−0.212333	−0.106166	−	0.994348i	$-0.533858\pi$
$23$	−1.01831	−0.212333	−0.106166	+	0.994348i	$0.533858\pi$
$24$	1.78871	0.365119
$25$	5.41168	1.08234
$26$	0.220818	0.0433060
$27$	−3.61493	−0.695694
$28$	0.438499	0.0828686
$29$	−8.18185	−1.51933	−0.759666	−	0.650314i	$-0.774638\pi$
$29$	−8.18185	−1.51933	−0.759666	+	0.650314i	$0.774638\pi$
$30$	3.02186	0.551714
$31$	−4.42855	−0.795391	−0.397695	−	0.917518i	$-0.630190\pi$
$31$	−4.42855	−0.795391	−0.397695	+	0.917518i	$0.630190\pi$
$32$	0.508911	0.0899637
$33$	−0.461576	−0.0803500
$34$	−5.64456	−0.968034
$35$	−15.7158	−2.65646
$36$	−0.232313	−0.0387188
$37$	2.64407	0.434682	0.217341	−	0.976096i	$-0.430262\pi$
$37$	2.64407	0.434682	0.217341	+	0.976096i	$0.430262\pi$
$38$	−1.44569	−0.234523
$39$	−0.0989456	−0.0158440
$40$	−8.90970	−1.40875
$41$	−9.97907	−1.55847	−0.779235	−	0.626732i	$-0.784392\pi$
$41$	−9.97907	−1.55847	−0.779235	+	0.626732i	$0.784392\pi$
$42$	−4.56132	−0.703827
$43$	12.8265	1.95602	0.978009	−	0.208565i	$-0.0668792\pi$
$43$	12.8265	1.95602	0.978009	+	0.208565i	$0.0668792\pi$
$44$	−0.0641501	−0.00967099
$45$	8.32609	1.24118
$46$	1.47217	0.217059
$47$	1.55194	0.226373	0.113187	−	0.993574i	$-0.463894\pi$
$47$	1.55194	0.226373	0.113187	+	0.993574i	$0.463894\pi$
$48$	−2.70257	−0.390083
$49$	16.7221	2.38887
$50$	−7.82364	−1.10643
$51$	2.52925	0.354166
$52$	−0.0137515	−0.00190700
$53$	−1.55328	−0.213360	−0.106680	−	0.994293i	$-0.534022\pi$
$53$	−1.55328	−0.213360	−0.106680	+	0.994293i	$0.534022\pi$
$54$	5.22609	0.711180
$55$	2.29914	0.310016
$56$	13.4487	1.79715
$57$	0.647795	0.0858026
$58$	11.8285	1.55315
$59$	−1.37475	−0.178977	−0.0894883	−	0.995988i	$-0.528523\pi$
$59$	−1.37475	−0.178977	−0.0894883	+	0.995988i	$0.528523\pi$
$60$	−0.188188	−0.0242949
$61$	10.1835	1.30386	0.651931	−	0.758278i	$-0.273959\pi$
$61$	10.1835	1.30386	0.651931	+	0.758278i	$0.273959\pi$
$62$	6.40233	0.813096
$63$	−12.5677	−1.58339
$64$	7.60818	0.951023
$65$	0.492855	0.0611311
$66$	0.667297	0.0821386
$67$	−13.1428	−1.60565	−0.802826	−	0.596214i	$-0.796671\pi$
$67$	−13.1428	−1.60565	−0.802826	+	0.596214i	$0.796671\pi$
$68$	0.351517	0.0426277
$69$	−0.659658	−0.0794134
$70$	22.7203	2.71559
$71$	−5.13287	−0.609160	−0.304580	−	0.952487i	$-0.598516\pi$
$71$	−5.13287	−0.609160	−0.304580	+	0.952487i	$0.598516\pi$
$72$	−7.12497	−0.839686
$73$	−9.13435	−1.06909	−0.534547	−	0.845139i	$-0.679518\pi$
$73$	−9.13435	−1.06909	−0.534547	+	0.845139i	$0.679518\pi$
$74$	−3.82251	−0.444358
$75$	3.50566	0.404799
$76$	0.0900311	0.0103273
$77$	−3.47042	−0.395491
$78$	0.143045	0.0161967
$79$	8.60696	0.968359	0.484179	−	0.874969i	$-0.339118\pi$
$79$	8.60696	0.968359	0.484179	+	0.874969i	$0.339118\pi$
$80$	13.4617	1.50507
$81$	5.39935	0.599927
$82$	14.4267	1.59316
$83$	1.02751	0.112784	0.0563920	−	0.998409i	$-0.482040\pi$
$83$	1.02751	0.112784	0.0563920	+	0.998409i	$0.482040\pi$
$84$	0.284058	0.0309933
$85$	−12.5984	−1.36649
$86$	−18.5431	−1.99956
$87$	−5.30017	−0.568238
$88$	−1.96747	−0.209733
$89$	−6.86424	−0.727608	−0.363804	−	0.931476i	$-0.618522\pi$
$89$	−6.86424	−0.727608	−0.363804	+	0.931476i	$0.618522\pi$
$90$	−12.0370	−1.26881
$91$	−0.743936	−0.0779856
$92$	−0.0916797	−0.00955827
$93$	−2.86879	−0.297480
$94$	−2.24363	−0.231412
$95$	−3.22671	−0.331054
$96$	0.329671	0.0336469
$97$	8.72232	0.885617	0.442809	−	0.896616i	$-0.353982\pi$
$97$	8.72232	0.885617	0.442809	+	0.896616i	$0.353982\pi$
$98$	−24.1750	−2.44205
$99$	1.83859	0.184785
$100$	0.487220	0.0487220
$101$	17.6003	1.75130	0.875648	−	0.482950i	$-0.160435\pi$
$101$	17.6003	1.75130	0.875648	+	0.482950i	$0.160435\pi$
$102$	−3.65652	−0.362050
$103$	1.67271	0.164817	0.0824085	−	0.996599i	$-0.473739\pi$
$103$	1.67271	0.164817	0.0824085	+	0.996599i	$0.473739\pi$
$104$	−0.421756	−0.0413566
$105$	−10.1806	−0.993528
$106$	2.24557	0.218109
$107$	13.1985	1.27595	0.637975	−	0.770057i	$-0.279772\pi$
$107$	13.1985	1.27595	0.637975	+	0.770057i	$0.279772\pi$
$108$	−0.325456	−0.0313171
$109$	−11.6050	−1.11155	−0.555777	−	0.831331i	$-0.687579\pi$
$109$	−11.6050	−1.11155	−0.555777	+	0.831331i	$0.687579\pi$
$110$	−3.32385	−0.316917
$111$	1.71281	0.162573
$112$	−20.3197	−1.92003
$113$	0.612651	0.0576334	0.0288167	−	0.999585i	$-0.490826\pi$
$113$	0.612651	0.0576334	0.0288167	+	0.999585i	$0.490826\pi$
$114$	−0.936514	−0.0877126
$115$	3.28580	0.306402
$116$	−0.736621	−0.0683936
$117$	0.394130	0.0364373
$118$	1.98746	0.182961
$119$	19.0165	1.74324
$120$	−5.77166	−0.526878
$121$	−10.4923	−0.953845
$122$	−14.7222	−1.33289
$123$	−6.46440	−0.582875
$124$	−0.398707	−0.0358049
$125$	−1.32839	−0.118814
$126$	18.1691	1.61863
$127$	−13.5409	−1.20156	−0.600782	−	0.799413i	$-0.705144\pi$
$127$	−13.5409	−1.20156	−0.600782	+	0.799413i	$0.705144\pi$
$128$	−12.0169	−1.06216
$129$	8.30892	0.731560
$130$	−0.712518	−0.0624919
$131$	−14.7150	−1.28565	−0.642826	−	0.766012i	$-0.722238\pi$
$131$	−14.7150	−1.28565	−0.642826	+	0.766012i	$0.722238\pi$
$132$	−0.0415562	−0.00361700
$133$	4.87053	0.422329
$134$	19.0005	1.64139
$135$	11.6644	1.00391
$136$	10.7809	0.924458
$137$	0.843360	0.0720531	0.0360266	−	0.999351i	$-0.488530\pi$
$137$	0.843360	0.0720531	0.0360266	+	0.999351i	$0.488530\pi$
$138$	0.953663	0.0811812
$139$	−16.2757	−1.38049	−0.690245	−	0.723576i	$-0.742497\pi$
$139$	−16.2757	−1.38049	−0.690245	+	0.723576i	$0.742497\pi$
$140$	−1.41491	−0.119582
$141$	1.00534	0.0846647
$142$	7.42056	0.622720
$143$	0.108834	0.00910114
$144$	10.7652	0.897096
$145$	26.4005	2.19244
$146$	13.2055	1.09289
$147$	10.8325	0.893449
$148$	0.238048	0.0195674
$149$	15.7852	1.29317	0.646585	−	0.762842i	$-0.276196\pi$
$149$	15.7852	1.29317	0.646585	+	0.762842i	$0.276196\pi$
$150$	−5.06812	−0.413810
$151$	−14.9222	−1.21435	−0.607176	−	0.794568i	$-0.707698\pi$
$151$	−14.9222	−1.21435	−0.607176	+	0.794568i	$0.707698\pi$
$152$	2.76123	0.223965
$153$	−10.0748	−0.814495
$154$	5.01716	0.404294
$155$	14.2897	1.14777
$156$	−0.00890818	−0.000713226 0
$157$	−19.3550	−1.54470	−0.772348	−	0.635200i	$-0.780918\pi$
$157$	−19.3550	−1.54470	−0.772348	+	0.635200i	$0.780918\pi$
$158$	−12.4430	−0.989914
$159$	−1.00621	−0.0797977
$160$	−1.64211	−0.129820
$161$	−4.95972	−0.390881
$162$	−7.80580	−0.613282
$163$	15.0081	1.17553	0.587764	−	0.809032i	$-0.300008\pi$
$163$	15.0081	1.17553	0.587764	+	0.809032i	$0.300008\pi$
$164$	−0.898427	−0.0701554
$165$	1.48937	0.115947
$166$	−1.48547	−0.115295
$167$	−23.8284	−1.84390	−0.921949	−	0.387311i	$-0.873404\pi$
$167$	−23.8284	−1.84390	−0.921949	+	0.387311i	$0.873404\pi$
$168$	8.71199	0.672144
$169$	−12.9767	−0.998205
$170$	18.2134	1.39690
$171$	−2.58036	−0.197325
$172$	1.15478	0.0880512
$173$	4.54022	0.345187	0.172593	−	0.984993i	$-0.444785\pi$
$173$	4.54022	0.345187	0.172593	+	0.984993i	$0.444785\pi$
$174$	7.66242	0.580887
$175$	26.3578	1.99246
$176$	2.97266	0.224072
$177$	−0.890554	−0.0669381
$178$	9.92359	0.743805
$179$	22.7719	1.70205	0.851027	−	0.525122i	$-0.175980\pi$
$179$	22.7719	1.70205	0.851027	+	0.525122i	$0.175980\pi$
$180$	0.749607	0.0558724
$181$	8.23250	0.611917	0.305958	−	0.952045i	$-0.401023\pi$
$181$	8.23250	0.611917	0.305958	+	0.952045i	$0.401023\pi$
$182$	1.07550	0.0797216
$183$	6.59682	0.487651
$184$	−2.81179	−0.207288
$185$	−8.53165	−0.627259
$186$	4.14740	0.304102
$187$	−2.78201	−0.203441
$188$	0.139723	0.0101903
$189$	−17.6067	−1.28070
$190$	4.66484	0.338423
$191$	−4.23017	−0.306084	−0.153042	−	0.988220i	$-0.548907\pi$
$191$	−4.23017	−0.306084	−0.153042	+	0.988220i	$0.548907\pi$
$192$	4.92855	0.355687
$193$	4.25599	0.306353	0.153176	−	0.988199i	$-0.451050\pi$
$193$	4.25599	0.306353	0.153176	+	0.988199i	$0.451050\pi$
$194$	−12.6098	−0.905331
$195$	0.319269	0.0228633
$196$	1.50551	0.107536
$197$	−14.9495	−1.06511	−0.532553	−	0.846397i	$-0.678767\pi$
$197$	−14.9495	−1.06511	−0.532553	+	0.846397i	$0.678767\pi$
$198$	−2.65804	−0.188899
$199$	8.17675	0.579634	0.289817	−	0.957082i	$-0.406405\pi$
$199$	8.17675	0.579634	0.289817	+	0.957082i	$0.406405\pi$
$200$	14.9429	1.05662
$201$	−8.51386	−0.600521
$202$	−25.4447	−1.79028
$203$	−39.8500	−2.79692
$204$	0.227711	0.0159430
$205$	32.1996	2.24892
$206$	−2.41823	−0.168486
$207$	2.62761	0.182632
$208$	0.637233	0.0441842
$209$	−0.712533	−0.0492869
$210$	14.7181	1.01564
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−0.139844	−0.00960452
$213$	−3.32505	−0.227829
$214$	−19.0810	−1.30435
$215$	−41.3873	−2.82259
$216$	−9.98166	−0.679166
$217$	−21.5694	−1.46423
$218$	16.7772	1.13630
$219$	−5.91719	−0.399847
$220$	0.206994	0.0139555
$221$	−0.596366	−0.0401159
$222$	−2.47621	−0.166192
$223$	7.39353	0.495107	0.247554	−	0.968874i	$-0.420373\pi$
$223$	7.39353	0.495107	0.247554	+	0.968874i	$0.420373\pi$
$224$	2.47867	0.165613
$225$	−13.9641	−0.930940
$226$	−0.885706	−0.0589163
$227$	−14.3357	−0.951491	−0.475745	−	0.879583i	$-0.657822\pi$
$227$	−14.3357	−0.951491	−0.475745	+	0.879583i	$0.657822\pi$
$228$	0.0583217	0.00386245
$229$	−25.4910	−1.68449	−0.842245	−	0.539095i	$-0.818766\pi$
$229$	−25.4910	−1.68449	−0.842245	+	0.539095i	$0.818766\pi$
$230$	−4.75026	−0.313223
$231$	−2.24812	−0.147915
$232$	−22.5920	−1.48324
$233$	−17.1535	−1.12376	−0.561881	−	0.827218i	$-0.689922\pi$
$233$	−17.1535	−1.12376	−0.561881	+	0.827218i	$0.689922\pi$
$234$	−0.569791	−0.0372484
$235$	−5.00766	−0.326663
$236$	−0.123770	−0.00805673
$237$	5.57555	0.362171
$238$	−27.4920	−1.78204
$239$	−10.6416	−0.688347	−0.344173	−	0.938906i	$-0.611841\pi$
$239$	−10.6416	−0.688347	−0.344173	+	0.938906i	$0.611841\pi$
$240$	8.72044	0.562902
$241$	3.08506	0.198726	0.0993632	−	0.995051i	$-0.468319\pi$
$241$	3.08506	0.198726	0.0993632	+	0.995051i	$0.468319\pi$
$242$	15.1687	0.975078
$243$	14.3425	0.920070
$244$	0.916830	0.0586940
$245$	−53.9574	−3.44721
$246$	9.34554	0.595850
$247$	−0.152742	−0.00971875
$248$	−12.2282	−0.776494
$249$	0.665617	0.0421818
$250$	1.92044	0.121459
$251$	5.53724	0.349507	0.174754	−	0.984612i	$-0.444087\pi$
$251$	5.53724	0.349507	0.174754	+	0.984612i	$0.444087\pi$
$252$	−1.13149	−0.0712770
$253$	0.725580	0.0456168
$254$	19.5760	1.22831
$255$	−8.16117	−0.511072
$256$	2.15643	0.134777
$257$	−20.7933	−1.29705	−0.648524	−	0.761194i	$-0.724613\pi$
$257$	−20.7933	−1.29705	−0.648524	+	0.761194i	$0.724613\pi$
$258$	−12.0122	−0.747845
$259$	12.8780	0.800201
$260$	0.0443723	0.00275185
$261$	21.1121	1.30681
$262$	21.2733	1.31427
$263$	−17.8273	−1.09928	−0.549641	−	0.835401i	$-0.685235\pi$
$263$	−17.8273	−1.09928	−0.549641	+	0.835401i	$0.685235\pi$
$264$	−1.27452	−0.0784411
$265$	5.01200	0.307885
$266$	−7.04130	−0.431730
$267$	−4.44662	−0.272129
$268$	−1.18326	−0.0722793
$269$	−11.1437	−0.679441	−0.339720	−	0.940526i	$-0.610332\pi$
$269$	−11.1437	−0.679441	−0.339720	+	0.940526i	$0.610332\pi$
$270$	−16.8631	−1.02625
$271$	3.32063	0.201714	0.100857	−	0.994901i	$-0.467842\pi$
$271$	3.32063	0.201714	0.100857	+	0.994901i	$0.467842\pi$
$272$	−16.2890	−0.987664
$273$	−0.481918	−0.0291670
$274$	−1.21924	−0.0736570
$275$	−3.85600	−0.232526
$276$	−0.0593897	−0.00357484
$277$	−25.6220	−1.53948	−0.769738	−	0.638360i	$-0.779613\pi$
$277$	−25.6220	−1.53948	−0.769738	+	0.638360i	$0.779613\pi$
$278$	23.5297	1.41122
$279$	11.4273	0.684132
$280$	−43.3950	−2.59335
$281$	−13.3412	−0.795870	−0.397935	−	0.917414i	$-0.630273\pi$
$281$	−13.3412	−0.795870	−0.397935	+	0.917414i	$0.630273\pi$
$282$	−1.45341	−0.0865493
$283$	22.8245	1.35678	0.678388	−	0.734704i	$-0.262679\pi$
$283$	22.8245	1.35678	0.678388	+	0.734704i	$0.262679\pi$
$284$	−0.462118	−0.0274217
$285$	−2.09025	−0.123816
$286$	−0.157340	−0.00930373
$287$	−48.6034	−2.86897
$288$	−1.31318	−0.0773796
$289$	−1.75569	−0.103276
$290$	−38.1670	−2.24125
$291$	5.65028	0.331225
$292$	−0.822375	−0.0481259
$293$	−28.5135	−1.66578	−0.832890	−	0.553439i	$-0.813315\pi$
$293$	−28.5135	−1.66578	−0.832890	+	0.553439i	$0.813315\pi$
$294$	−15.6605	−0.913338
$295$	4.43591	0.258269
$296$	7.30088	0.424355
$297$	2.57576	0.149461
$298$	−22.8205	−1.32196
$299$	0.155539	0.00899505
$300$	0.315619	0.0182223
$301$	62.4717	3.60081
$302$	21.5729	1.24138
$303$	11.4014	0.654993
$304$	−4.17196	−0.239278
$305$	−32.8592	−1.88151
$306$	14.5650	0.832626
$307$	9.28185	0.529743	0.264872	−	0.964284i	$-0.414670\pi$
$307$	9.28185	0.529743	0.264872	+	0.964284i	$0.414670\pi$
$308$	−0.312445	−0.0178032
$309$	1.08357	0.0616424
$310$	−20.6585	−1.17332
$311$	22.4062	1.27054	0.635268	−	0.772292i	$-0.280890\pi$
$311$	22.4062	1.27054	0.635268	+	0.772292i	$0.280890\pi$
$312$	−0.273212	−0.0154676
$313$	9.98437	0.564350	0.282175	−	0.959363i	$-0.408944\pi$
$313$	9.98437	0.564350	0.282175	+	0.959363i	$0.408944\pi$
$314$	27.9814	1.57908
$315$	40.5525	2.28487
$316$	0.774894	0.0435912
$317$	−24.6895	−1.38670	−0.693351	−	0.720600i	$-0.743867\pi$
$317$	−24.6895	−1.38670	−0.693351	+	0.720600i	$0.743867\pi$
$318$	1.45467	0.0815740
$319$	5.82984	0.326408
$320$	−24.5494	−1.37235
$321$	8.54995	0.477212
$322$	7.17024	0.399582
$323$	3.90440	0.217246
$324$	0.486109	0.0270061
$325$	−0.826592	−0.0458511
$326$	−21.6972	−1.20170
$327$	−7.51764	−0.415727
$328$	−27.5545	−1.52144
$329$	7.55876	0.416728
$330$	−2.15318	−0.118528
$331$	−3.22365	−0.177188	−0.0885938	−	0.996068i	$-0.528237\pi$
$331$	−3.22365	−0.177188	−0.0885938	+	0.996068i	$0.528237\pi$
$332$	0.0925080	0.00507703
$333$	−6.82265	−0.373879
$334$	34.4486	1.88494
$335$	42.4081	2.31700
$336$	−13.1630	−0.718099
$337$	−9.64977	−0.525656	−0.262828	−	0.964843i	$-0.584655\pi$
$337$	−9.64977	−0.525656	−0.262828	+	0.964843i	$0.584655\pi$
$338$	18.7603	1.02043
$339$	0.396873	0.0215552
$340$	−1.13425	−0.0615131
$341$	3.15549	0.170879
$342$	3.73041	0.201718
$343$	47.3518	2.55676
$344$	35.4168	1.90955
$345$	2.12853	0.114596
$346$	−6.56377	−0.352870
$347$	−0.601478	−0.0322890	−0.0161445	−	0.999870i	$-0.505139\pi$
$347$	−0.601478	−0.0322890	−0.0161445	+	0.999870i	$0.505139\pi$
$348$	−0.477180	−0.0255795
$349$	2.78016	0.148818	0.0744092	−	0.997228i	$-0.476293\pi$
$349$	2.78016	0.148818	0.0744092	+	0.997228i	$0.476293\pi$
$350$	−38.1053	−2.03681
$351$	0.552152	0.0294717
$352$	−0.362616	−0.0193275
$353$	−31.6428	−1.68417	−0.842087	−	0.539341i	$-0.818673\pi$
$353$	−31.6428	−1.68417	−0.842087	+	0.539341i	$0.818673\pi$
$354$	1.28747	0.0684282
$355$	16.5623	0.879036
$356$	−0.617995	−0.0327537
$357$	12.3188	0.651980
$358$	−32.9212	−1.73994
$359$	1.52022	0.0802342	0.0401171	−	0.999195i	$-0.487227\pi$
$359$	1.52022	0.0802342	0.0401171	+	0.999195i	$0.487227\pi$
$360$	22.9902	1.21169
$361$	1.00000	0.0526316
$362$	−11.9017	−0.625538
$363$	−6.79686	−0.356743
$364$	−0.0669773	−0.00351057
$365$	29.4739	1.54274
$366$	−9.53698	−0.498506
$367$	−29.8988	−1.56070	−0.780352	−	0.625340i	$-0.784960\pi$
$367$	−29.8988	−1.56070	−0.780352	+	0.625340i	$0.784960\pi$
$368$	4.24835	0.221461
$369$	25.7496	1.34047
$370$	12.3342	0.641222
$371$	−7.56532	−0.392772
$372$	−0.258281	−0.0133912
$373$	−5.62580	−0.291293	−0.145646	−	0.989337i	$-0.546526\pi$
$373$	−5.62580	−0.291293	−0.145646	+	0.989337i	$0.546526\pi$
$374$	4.02194	0.207969
$375$	−0.860522	−0.0444372
$376$	4.28525	0.220995
$377$	1.24971	0.0643635
$378$	25.4538	1.30920
$379$	17.4450	0.896089	0.448044	−	0.894011i	$-0.352121\pi$
$379$	17.4450	0.896089	0.448044	+	0.894011i	$0.352121\pi$
$380$	−0.290505	−0.0149026
$381$	−8.77176	−0.449391
$382$	6.11553	0.312898
$383$	−7.95713	−0.406590	−0.203295	−	0.979118i	$-0.565165\pi$
$383$	−7.95713	−0.406590	−0.203295	+	0.979118i	$0.565165\pi$
$384$	−7.78451	−0.397252
$385$	11.1980	0.570705
$386$	−6.15285	−0.313172
$387$	−33.0969	−1.68241
$388$	0.785280	0.0398665
$389$	−2.82590	−0.143279	−0.0716394	−	0.997431i	$-0.522823\pi$
$389$	−2.82590	−0.143279	−0.0716394	+	0.997431i	$0.522823\pi$
$390$	−0.461566	−0.0233723
$391$	−3.97589	−0.201069
$392$	46.1736	2.33212
$393$	−9.53228	−0.480840
$394$	21.6124	1.08882
$395$	−27.7722	−1.39737
$396$	0.165530	0.00831822
$397$	17.9702	0.901898	0.450949	−	0.892550i	$-0.351086\pi$
$397$	17.9702	0.901898	0.450949	+	0.892550i	$0.351086\pi$
$398$	−11.8211	−0.592537
$399$	3.15511	0.157953
$400$	−22.5773	−1.12887
$401$	2.08713	0.104226	0.0521131	−	0.998641i	$-0.483404\pi$
$401$	2.08713	0.104226	0.0521131	+	0.998641i	$0.483404\pi$
$402$	12.3084	0.613889
$403$	0.676426	0.0336952
$404$	1.58457	0.0788356
$405$	−17.4221	−0.865713
$406$	57.6109	2.85918
$407$	−1.88398	−0.0933857
$408$	6.98384	0.345752
$409$	−30.2661	−1.49656	−0.748282	−	0.663381i	$-0.769121\pi$
$409$	−30.2661	−1.49656	−0.748282	+	0.663381i	$0.769121\pi$
$410$	−46.5508	−2.29898
$411$	0.546325	0.0269482
$412$	0.150596	0.00741933
$413$	−6.69574	−0.329476
$414$	−3.79872	−0.186697
$415$	−3.31549	−0.162751
$416$	−0.0777322	−0.00381113
$417$	−10.5433	−0.516310
$418$	1.03010	0.0503841
$419$	−10.2927	−0.502829	−0.251415	−	0.967879i	$-0.580896\pi$
$419$	−10.2927	−0.502829	−0.251415	+	0.967879i	$0.580896\pi$
$420$	−0.916574	−0.0447242
$421$	−35.1936	−1.71523	−0.857616	−	0.514291i	$-0.828055\pi$
$421$	−35.1936	−1.71523	−0.857616	+	0.514291i	$0.828055\pi$
$422$	−1.44569	−0.0703753
$423$	−4.00456	−0.194708
$424$	−4.28898	−0.208291
$425$	21.1294	1.02492
$426$	4.80701	0.232900
$427$	49.5990	2.40026
$428$	1.18828	0.0574376
$429$	0.0705020	0.00340387
$430$	59.8334	2.88542
$431$	18.2254	0.877886	0.438943	−	0.898515i	$-0.355353\pi$
$431$	18.2254	0.877886	0.438943	+	0.898515i	$0.355353\pi$
$432$	15.0813	0.725601
$433$	−27.2121	−1.30773	−0.653866	−	0.756611i	$-0.726854\pi$
$433$	−27.2121	−1.30773	−0.653866	+	0.756611i	$0.726854\pi$
$434$	31.1827	1.49682
$435$	17.1021	0.819984
$436$	−1.04481	−0.0500372
$437$	−1.01831	−0.0487124
$438$	8.55444	0.408747
$439$	25.8540	1.23395	0.616973	−	0.786985i	$-0.288359\pi$
$439$	25.8540	1.23395	0.616973	+	0.786985i	$0.288359\pi$
$440$	6.34845	0.302651
$441$	−43.1490	−2.05472
$442$	0.862162	0.0410089
$443$	0.180972	0.00859825	0.00429912	−	0.999991i	$-0.498632\pi$
$443$	0.180972	0.00859825	0.00429912	+	0.999991i	$0.498632\pi$
$444$	0.154207	0.00731832
$445$	22.1489	1.04996
$446$	−10.6888	−0.506128
$447$	10.2256	0.483652
$448$	37.0559	1.75073
$449$	−26.1152	−1.23245	−0.616226	−	0.787569i	$-0.711339\pi$
$449$	−26.1152	−1.23245	−0.616226	+	0.787569i	$0.711339\pi$
$450$	20.1878	0.951663
$451$	7.11042	0.334817
$452$	0.0551577	0.00259440
$453$	−9.66653	−0.454173
$454$	20.7250	0.972671
$455$	2.40047	0.112536
$456$	1.78871	0.0837641
$457$	33.8526	1.58356	0.791779	−	0.610808i	$-0.209155\pi$
$457$	33.8526	1.58356	0.791779	+	0.610808i	$0.209155\pi$
$458$	36.8521	1.72199
$459$	−14.1141	−0.658791
$460$	0.295824	0.0137929
$461$	13.7269	0.639324	0.319662	−	0.947532i	$-0.396431\pi$
$461$	13.7269	0.639324	0.319662	+	0.947532i	$0.396431\pi$
$462$	3.25009	0.151208
$463$	−0.673663	−0.0313078	−0.0156539	−	0.999877i	$-0.504983\pi$
$463$	−0.673663	−0.0313078	−0.0156539	+	0.999877i	$0.504983\pi$
$464$	34.1343	1.58465
$465$	9.25678	0.429273
$466$	24.7987	1.14878
$467$	31.9637	1.47910	0.739551	−	0.673101i	$-0.235038\pi$
$467$	31.9637	1.47910	0.739551	+	0.673101i	$0.235038\pi$
$468$	0.0354839	0.00164025
$469$	−64.0125	−2.95582
$470$	7.23954	0.333935
$471$	−12.5381	−0.577724
$472$	−3.79599	−0.174725
$473$	−9.13928	−0.420224
$474$	−8.06054	−0.370233
$475$	5.41168	0.248305
$476$	1.71208	0.0784729
$477$	4.00803	0.183515
$478$	15.3845	0.703669
$479$	−5.14748	−0.235194	−0.117597	−	0.993061i	$-0.537519\pi$
$479$	−5.14748	−0.235194	−0.117597	+	0.993061i	$0.537519\pi$
$480$	−1.06375	−0.0485534
$481$	−0.403860	−0.0184144
$482$	−4.46006	−0.203150
$483$	−3.21288	−0.146191
$484$	−0.944633	−0.0429379
$485$	−28.1444	−1.27797
$486$	−20.7348	−0.940551
$487$	−33.9757	−1.53959	−0.769793	−	0.638294i	$-0.779640\pi$
$487$	−33.9757	−1.53959	−0.769793	+	0.638294i	$0.779640\pi$
$488$	28.1189	1.27288
$489$	9.72220	0.439653
$490$	78.0059	3.52395
$491$	8.88196	0.400837	0.200419	−	0.979710i	$-0.435770\pi$
$491$	8.88196	0.400837	0.200419	+	0.979710i	$0.435770\pi$
$492$	−0.581997	−0.0262384
$493$	−31.9452	−1.43874
$494$	0.220818	0.00993509
$495$	−5.93261	−0.266651
$496$	18.4757	0.829584
$497$	−24.9998	−1.12140
$498$	−0.962279	−0.0431207
$499$	−11.2571	−0.503938	−0.251969	−	0.967735i	$-0.581078\pi$
$499$	−11.2571	−0.503938	−0.251969	+	0.967735i	$0.581078\pi$
$500$	−0.119596	−0.00534850
$501$	−15.4359	−0.689627
$502$	−8.00515	−0.357287
$503$	−9.63152	−0.429448	−0.214724	−	0.976675i	$-0.568885\pi$
$503$	−9.63152	−0.429448	−0.214724	+	0.976675i	$0.568885\pi$
$504$	−34.7024	−1.54577
$505$	−56.7912	−2.52717
$506$	−1.04897	−0.0466323
$507$	−8.40623	−0.373334
$508$	−1.21911	−0.0540890
$509$	9.87436	0.437673	0.218837	−	0.975761i	$-0.429774\pi$
$509$	9.87436	0.437673	0.218837	+	0.975761i	$0.429774\pi$
$510$	11.7986	0.522449
$511$	−44.4891	−1.96808
$512$	20.9163	0.924379
$513$	−3.61493	−0.159603
$514$	30.0607	1.32592
$515$	−5.39736	−0.237836
$516$	0.748062	0.0329316
$517$	−1.10581	−0.0486333
$518$	−18.6177	−0.818014
$519$	2.94113	0.129101
$520$	1.36089	0.0596788
$521$	−8.30675	−0.363925	−0.181963	−	0.983305i	$-0.558245\pi$
$521$	−8.30675	−0.363925	−0.181963	+	0.983305i	$0.558245\pi$
$522$	−30.5217	−1.33590
$523$	8.90799	0.389519	0.194760	−	0.980851i	$-0.437607\pi$
$523$	8.90799	0.389519	0.194760	+	0.980851i	$0.437607\pi$
$524$	−1.32480	−0.0578743
$525$	17.0745	0.745190
$526$	25.7729	1.12375
$527$	−17.2908	−0.753199
$528$	1.92567	0.0838042
$529$	−21.9630	−0.954915
$530$	−7.24582	−0.314739
$531$	3.54734	0.153941
$532$	0.438499	0.0190114
$533$	1.52422	0.0660215
$534$	6.42846	0.278187
$535$	−42.5879	−1.84123
$536$	−36.2904	−1.56750
$537$	14.7516	0.636577
$538$	16.1103	0.694565
$539$	−11.9150	−0.513217
$540$	1.05015	0.0451915
$541$	37.7153	1.62151	0.810753	−	0.585388i	$-0.199058\pi$
$541$	37.7153	1.62151	0.810753	+	0.585388i	$0.199058\pi$
$542$	−4.80062	−0.206204
$543$	5.33298	0.228860
$544$	1.98699	0.0851916
$545$	37.4459	1.60401
$546$	0.696706	0.0298163
$547$	−27.7499	−1.18650	−0.593250	−	0.805018i	$-0.702155\pi$
$547$	−27.7499	−1.18650	−0.593250	+	0.805018i	$0.702155\pi$
$548$	0.0759286	0.00324351
$549$	−26.2771	−1.12148
$550$	5.57460	0.237702
$551$	−8.18185	−0.348559
$552$	−1.82147	−0.0775268
$553$	41.9205	1.78264
$554$	37.0415	1.57374
$555$	−5.52676	−0.234598
$556$	−1.46532	−0.0621435
$557$	−36.3777	−1.54137	−0.770686	−	0.637215i	$-0.780086\pi$
$557$	−36.3777	−1.54137	−0.770686	+	0.637215i	$0.780086\pi$
$558$	−16.5203	−0.699361
$559$	−1.95914	−0.0828628
$560$	65.5657	2.77066
$561$	−1.80217	−0.0760878
$562$	19.2873	0.813586
$563$	12.4688	0.525496	0.262748	−	0.964864i	$-0.415371\pi$
$563$	12.4688	0.525496	0.262748	+	0.964864i	$0.415371\pi$
$564$	0.0905116	0.00381123
$565$	−1.97685	−0.0831667
$566$	−32.9972	−1.38698
$567$	26.2977	1.10440
$568$	−14.1730	−0.594688
$569$	16.6375	0.697480	0.348740	−	0.937220i	$-0.386610\pi$
$569$	16.6375	0.697480	0.348740	+	0.937220i	$0.386610\pi$
$570$	3.02186	0.126572
$571$	−3.26902	−0.136804	−0.0684020	−	0.997658i	$-0.521790\pi$
$571$	−3.26902	−0.136804	−0.0684020	+	0.997658i	$0.521790\pi$
$572$	0.00979842	0.000409693 0
$573$	−2.74028	−0.114477
$574$	70.2657	2.93283
$575$	−5.51078	−0.229815
$576$	−19.6319	−0.817994
$577$	22.0607	0.918398	0.459199	−	0.888333i	$-0.348136\pi$
$577$	22.0607	0.918398	0.459199	+	0.888333i	$0.348136\pi$
$578$	2.53819	0.105575
$579$	2.75701	0.114577
$580$	2.37687	0.0986940
$581$	5.00453	0.207623
$582$	−8.16857	−0.338598
$583$	1.10677	0.0458376
$584$	−25.2220	−1.04370
$585$	−1.27174	−0.0525801
$586$	41.2219	1.70286
$587$	−29.9258	−1.23517	−0.617584	−	0.786505i	$-0.711889\pi$
$587$	−29.9258	−1.23517	−0.617584	+	0.786505i	$0.711889\pi$
$588$	0.975262	0.0402191
$589$	−4.42855	−0.182475
$590$	−6.41297	−0.264018
$591$	−9.68420	−0.398355
$592$	−11.0309	−0.453369
$593$	35.0850	1.44077	0.720384	−	0.693576i	$-0.243966\pi$
$593$	35.0850	1.44077	0.720384	+	0.693576i	$0.243966\pi$
$594$	−3.72376	−0.152788
$595$	−61.3608	−2.51555
$596$	1.42116	0.0582128
$597$	5.29686	0.216786
$598$	−0.224862	−0.00919528
$599$	3.98537	0.162838	0.0814189	−	0.996680i	$-0.474055\pi$
$599$	3.98537	0.162838	0.0814189	+	0.996680i	$0.474055\pi$
$600$	9.67995	0.395182
$601$	−38.2191	−1.55899	−0.779496	−	0.626408i	$-0.784525\pi$
$601$	−38.2191	−1.55899	−0.779496	+	0.626408i	$0.784525\pi$
$602$	−90.3150	−3.68096
$603$	33.9132	1.38105
$604$	−1.34346	−0.0546647
$605$	33.8556	1.37643
$606$	−16.4829	−0.669574
$607$	36.8645	1.49628	0.748141	−	0.663539i	$-0.230947\pi$
$607$	36.8645	1.49628	0.748141	+	0.663539i	$0.230947\pi$
$608$	0.508911	0.0206391
$609$	−25.8146	−1.04606
$610$	47.5043	1.92339
$611$	−0.237046	−0.00958986
$612$	−0.907041	−0.0366650
$613$	−32.0654	−1.29511	−0.647555	−	0.762018i	$-0.724209\pi$
$613$	−32.0654	−1.29511	−0.647555	+	0.762018i	$0.724209\pi$
$614$	−13.4187	−0.541535
$615$	20.8588	0.841107
$616$	−9.58262	−0.386095
$617$	−37.3775	−1.50476	−0.752381	−	0.658728i	$-0.771095\pi$
$617$	−37.3775	−1.50476	−0.752381	+	0.658728i	$0.771095\pi$
$618$	−1.56652	−0.0630146
$619$	31.2617	1.25651	0.628257	−	0.778006i	$-0.283769\pi$
$619$	31.2617	1.25651	0.628257	+	0.778006i	$0.283769\pi$
$620$	1.28651	0.0516676
$621$	3.68113	0.147719
$622$	−32.3924	−1.29882
$623$	−33.4325	−1.33945
$624$	0.412797	0.0165251
$625$	−22.7721	−0.910884
$626$	−14.4343	−0.576912
$627$	−0.461576	−0.0184336
$628$	−1.74255	−0.0695353
$629$	10.3235	0.411624
$630$	−58.6265	−2.33574
$631$	20.2028	0.804262	0.402131	−	0.915582i	$-0.368270\pi$
$631$	20.2028	0.804262	0.402131	+	0.915582i	$0.368270\pi$
$632$	23.7658	0.945353
$633$	0.647795	0.0257476
$634$	35.6935	1.41757
$635$	43.6927	1.73389
$636$	−0.0905902	−0.00359214
$637$	−2.55417	−0.101200
$638$	−8.42816	−0.333674
$639$	13.2447	0.523951
$640$	38.7752	1.53272
$641$	−35.6934	−1.40980	−0.704901	−	0.709305i	$-0.749009\pi$
$641$	−35.6934	−1.40980	−0.704901	+	0.709305i	$0.749009\pi$
$642$	−12.3606	−0.487834
$643$	−29.4314	−1.16066	−0.580330	−	0.814381i	$-0.697077\pi$
$643$	−29.4314	−1.16066	−0.580330	+	0.814381i	$0.697077\pi$
$644$	−0.446529	−0.0175957
$645$	−26.8105	−1.05566
$646$	−5.64456	−0.222082
$647$	−33.7389	−1.32641	−0.663207	−	0.748436i	$-0.730805\pi$
$647$	−33.7389	−1.32641	−0.663207	+	0.748436i	$0.730805\pi$
$648$	14.9088	0.585675
$649$	0.979551	0.0384507
$650$	1.19500	0.0468717
$651$	−13.9726	−0.547628
$652$	1.35120	0.0529170
$653$	8.15003	0.318935	0.159468	−	0.987203i	$-0.449022\pi$
$653$	8.15003	0.318935	0.159468	+	0.987203i	$0.449022\pi$
$654$	10.8682	0.424981
$655$	47.4810	1.85523
$656$	41.6323	1.62547
$657$	23.5699	0.919550
$658$	−10.9277	−0.426004
$659$	−34.8127	−1.35611	−0.678054	−	0.735012i	$-0.737177\pi$
$659$	−34.8127	−1.35611	−0.678054	+	0.735012i	$0.737177\pi$
$660$	0.134090	0.00521944
$661$	9.58489	0.372809	0.186404	−	0.982473i	$-0.440317\pi$
$661$	9.58489	0.372809	0.186404	+	0.982473i	$0.440317\pi$
$662$	4.66040	0.181132
$663$	−0.386323	−0.0150035
$664$	2.83720	0.110105
$665$	−15.7158	−0.609433
$666$	9.86346	0.382201
$667$	8.33167	0.322604
$668$	−2.14530	−0.0830041
$669$	4.78950	0.185173
$670$	−61.3092	−2.36858
$671$	−7.25607	−0.280117
$672$	1.60567	0.0619401
$673$	13.1214	0.505792	0.252896	−	0.967494i	$-0.418617\pi$
$673$	13.1214	0.505792	0.252896	+	0.967494i	$0.418617\pi$
$674$	13.9506	0.537358
$675$	−19.5629	−0.752975
$676$	−1.16830	−0.0449348
$677$	9.19334	0.353329	0.176664	−	0.984271i	$-0.443469\pi$
$677$	9.19334	0.353329	0.176664	+	0.984271i	$0.443469\pi$
$678$	−0.573756	−0.0220350
$679$	42.4823	1.63032
$680$	−34.7870	−1.33402
$681$	−9.28657	−0.355862
$682$	−4.56187	−0.174683
$683$	25.3910	0.971561	0.485781	−	0.874081i	$-0.338535\pi$
$683$	25.3910	0.971561	0.485781	+	0.874081i	$0.338535\pi$
$684$	−0.232313	−0.00888270
$685$	−2.72128	−0.103975
$686$	−68.4562	−2.61367
$687$	−16.5129	−0.630008
$688$	−53.5114	−2.04010
$689$	0.237252	0.00903858
$690$	−3.07720	−0.117147
$691$	−3.19741	−0.121635	−0.0608177	−	0.998149i	$-0.519371\pi$
$691$	−3.19741	−0.121635	−0.0608177	+	0.998149i	$0.519371\pi$
$692$	0.408761	0.0155388
$693$	8.95492	0.340169
$694$	0.869554	0.0330078
$695$	52.5171	1.99209
$696$	−14.6350	−0.554738
$697$	−38.9623	−1.47580
$698$	−4.01926	−0.152131
$699$	−11.1120	−0.420293
$700$	2.37302	0.0896917
$701$	−15.2308	−0.575261	−0.287630	−	0.957741i	$-0.592867\pi$
$701$	−15.2308	−0.575261	−0.287630	+	0.957741i	$0.592867\pi$
$702$	−0.798243	−0.0301278
$703$	2.64407	0.0997229
$704$	−5.42108	−0.204315
$705$	−3.24394	−0.122174
$706$	45.7458	1.72166
$707$	85.7229	3.22394
$708$	−0.0801775	−0.00301326
$709$	−1.50549	−0.0565398	−0.0282699	−	0.999600i	$-0.509000\pi$
$709$	−1.50549	−0.0565398	−0.0282699	+	0.999600i	$0.509000\pi$
$710$	−23.9440	−0.898604
$711$	−22.2091	−0.832905
$712$	−18.9537	−0.710322
$713$	4.50964	0.168887
$714$	−17.8092	−0.666493
$715$	−0.351175	−0.0131332
$716$	2.05018	0.0766189
$717$	−6.89357	−0.257445
$718$	−2.19777	−0.0820202
$719$	−47.2135	−1.76077	−0.880383	−	0.474264i	$-0.842714\pi$
$719$	−47.2135	−1.76077	−0.880383	+	0.474264i	$0.842714\pi$
$720$	−34.7361	−1.29454
$721$	8.14699	0.303410
$722$	−1.44569	−0.0538032
$723$	1.99849	0.0743247
$724$	0.741181	0.0275458
$725$	−44.2776	−1.64443
$726$	9.82618	0.364684
$727$	−12.4743	−0.462648	−0.231324	−	0.972877i	$-0.574306\pi$
$727$	−12.4743	−0.462648	−0.231324	+	0.972877i	$0.574306\pi$
$728$	−2.05418	−0.0761329
$729$	−6.90705	−0.255817
$730$	−42.6103	−1.57708
$731$	50.0796	1.85226
$732$	0.593918	0.0219519
$733$	21.4730	0.793125	0.396562	−	0.918008i	$-0.370203\pi$
$733$	21.4730	0.793125	0.396562	+	0.918008i	$0.370203\pi$
$734$	43.2245	1.59545
$735$	−34.9534	−1.28927
$736$	−0.518230	−0.0191022
$737$	9.36469	0.344953
$738$	−37.2261	−1.37031
$739$	8.35561	0.307366	0.153683	−	0.988120i	$-0.450887\pi$
$739$	8.35561	0.307366	0.153683	+	0.988120i	$0.450887\pi$
$740$	−0.768114	−0.0282364
$741$	−0.0989456	−0.00363486
$742$	10.9371	0.401515
$743$	43.0491	1.57932	0.789659	−	0.613546i	$-0.210257\pi$
$743$	43.0491	1.57932	0.789659	+	0.613546i	$0.210257\pi$
$744$	−7.92140	−0.290413
$745$	−50.9342	−1.86608
$746$	8.13318	0.297777
$747$	−2.65135	−0.0970078
$748$	−0.250467	−0.00915800
$749$	64.2839	2.34888
$750$	1.24405	0.0454263
$751$	25.0511	0.914128	0.457064	−	0.889434i	$-0.348901\pi$
$751$	25.0511	0.914128	0.457064	+	0.889434i	$0.348901\pi$
$752$	−6.47461	−0.236105
$753$	3.58700	0.130717
$754$	−1.80670	−0.0657962
$755$	48.1497	1.75235
$756$	−1.58515	−0.0576512
$757$	37.6430	1.36816	0.684079	−	0.729408i	$-0.260204\pi$
$757$	37.6430	1.36816	0.684079	+	0.729408i	$0.260204\pi$
$758$	−25.2201	−0.916036
$759$	0.470028	0.0170609
$760$	−8.90970	−0.323189
$761$	−37.4264	−1.35671	−0.678353	−	0.734736i	$-0.737306\pi$
$761$	−37.4264	−1.35671	−0.678353	+	0.734736i	$0.737306\pi$
$762$	12.6813	0.459394
$763$	−56.5224	−2.04625
$764$	−0.380846	−0.0137785
$765$	32.5083	1.17534
$766$	11.5036	0.415641
$767$	0.209981	0.00758199
$768$	1.39693	0.0504073
$769$	1.11817	0.0403223	0.0201611	−	0.999797i	$-0.493582\pi$
$769$	1.11817	0.0403223	0.0201611	+	0.999797i	$0.493582\pi$
$770$	−16.1889	−0.583409
$771$	−13.4698	−0.485102
$772$	0.383171	0.0137906
$773$	−33.5139	−1.20541	−0.602706	−	0.797963i	$-0.705911\pi$
$773$	−33.5139	−1.20541	−0.602706	+	0.797963i	$0.705911\pi$
$774$	47.8480	1.71986
$775$	−23.9659	−0.860881
$776$	24.0843	0.864577
$777$	8.34232	0.299279
$778$	4.08539	0.146468
$779$	−9.97907	−0.357537
$780$	0.0287442	0.00102921
$781$	3.65734	0.130870
$782$	5.74792	0.205545
$783$	29.5768	1.05699
$784$	−69.7639	−2.49157
$785$	62.4530	2.22904
$786$	13.7808	0.491544
$787$	48.8325	1.74069	0.870345	−	0.492443i	$-0.163896\pi$
$787$	48.8325	1.74069	0.870345	+	0.492443i	$0.163896\pi$
$788$	−1.34592	−0.0479463
$789$	−11.5485	−0.411137
$790$	40.1501	1.42848
$791$	2.98394	0.106097
$792$	5.07678	0.180395
$793$	−1.55545	−0.0552355
$794$	−25.9794	−0.921975
$795$	3.24675	0.115150
$796$	0.736162	0.0260926
$797$	1.82567	0.0646687	0.0323343	−	0.999477i	$-0.489706\pi$
$797$	1.82567	0.0646687	0.0323343	+	0.999477i	$0.489706\pi$
$798$	−4.56132	−0.161469
$799$	6.05937	0.214365
$800$	2.75407	0.0973710
$801$	17.7122	0.625830
$802$	−3.01735	−0.106546
$803$	6.50852	0.229681
$804$	−0.766512	−0.0270328
$805$	16.0036	0.564053
$806$	−0.977905	−0.0344452
$807$	−7.21881	−0.254114
$808$	48.5985	1.70969
$809$	−0.941738	−0.0331097	−0.0165549	−	0.999863i	$-0.505270\pi$
$809$	−0.941738	−0.0331097	−0.0165549	+	0.999863i	$0.505270\pi$
$810$	25.1871	0.884984
$811$	−17.6324	−0.619158	−0.309579	−	0.950874i	$-0.600188\pi$
$811$	−17.6324	−0.619158	−0.309579	+	0.950874i	$0.600188\pi$
$812$	−3.58774	−0.125905
$813$	2.15109	0.0754421
$814$	2.72367	0.0954644
$815$	−48.4270	−1.69632
$816$	−10.5519	−0.369391
$817$	12.8265	0.448741
$818$	43.7556	1.52988
$819$	1.91962	0.0670770
$820$	2.89897	0.101236
$821$	50.5858	1.76546	0.882728	−	0.469885i	$-0.155705\pi$
$821$	50.5858	1.76546	0.882728	+	0.469885i	$0.155705\pi$
$822$	−0.789818	−0.0275481
$823$	2.80992	0.0979475	0.0489738	−	0.998800i	$-0.484405\pi$
$823$	2.80992	0.0979475	0.0489738	+	0.998800i	$0.484405\pi$
$824$	4.61874	0.160901
$825$	−2.49790	−0.0869657
$826$	9.68000	0.336810
$827$	33.8878	1.17840	0.589198	−	0.807989i	$-0.299444\pi$
$827$	33.8878	1.17840	0.589198	+	0.807989i	$0.299444\pi$
$828$	0.236567	0.00822126
$829$	35.1180	1.21970	0.609849	−	0.792518i	$-0.291230\pi$
$829$	35.1180	1.21970	0.609849	+	0.792518i	$0.291230\pi$
$830$	4.79318	0.166374
$831$	−16.5978	−0.575771
$832$	−1.16209	−0.0402882
$833$	65.2897	2.26215
$834$	15.2424	0.527803
$835$	76.8875	2.66080
$836$	−0.0641501	−0.00221868
$837$	16.0089	0.553349
$838$	14.8800	0.514022
$839$	−2.62489	−0.0906214	−0.0453107	−	0.998973i	$-0.514428\pi$
$839$	−2.62489	−0.0906214	−0.0453107	+	0.998973i	$0.514428\pi$
$840$	−28.1111	−0.969925
$841$	37.9427	1.30837
$842$	50.8792	1.75341
$843$	−8.64237	−0.297659
$844$	0.0900311	0.00309900
$845$	41.8720	1.44044
$846$	5.78936	0.199042
$847$	−51.1031	−1.75592
$848$	6.48023	0.222532
$849$	14.7856	0.507441
$850$	−30.5466	−1.04774
$851$	−2.69248	−0.0922972
$852$	−0.299358	−0.0102558
$853$	−54.2566	−1.85771	−0.928855	−	0.370443i	$-0.879206\pi$
$853$	−54.2566	−1.85771	−0.928855	+	0.370443i	$0.879206\pi$
$854$	−71.7050	−2.45369
$855$	8.32609	0.284746
$856$	36.4442	1.24564
$857$	−22.9258	−0.783132	−0.391566	−	0.920150i	$-0.628066\pi$
$857$	−22.9258	−0.783132	−0.391566	+	0.920150i	$0.628066\pi$
$858$	−0.101924	−0.00347964
$859$	−41.4411	−1.41395	−0.706975	−	0.707238i	$-0.749941\pi$
$859$	−41.4411	−1.41395	−0.706975	+	0.707238i	$0.749941\pi$
$860$	−3.72615	−0.127061
$861$	−31.4851	−1.07301
$862$	−26.3483	−0.897427
$863$	0.807340	0.0274822	0.0137411	−	0.999906i	$-0.495626\pi$
$863$	0.807340	0.0274822	0.0137411	+	0.999906i	$0.495626\pi$
$864$	−1.83968	−0.0625872
$865$	−14.6500	−0.498115
$866$	39.3404	1.33684
$867$	−1.13733	−0.0386257
$868$	−1.94192	−0.0659129
$869$	−6.13274	−0.208039
$870$	−24.7244	−0.838237
$871$	2.00746	0.0680203
$872$	−32.0440	−1.08515
$873$	−22.5067	−0.761737
$874$	1.47217	0.0497968
$875$	−6.46995	−0.218724
$876$	−0.532731	−0.0179993
$877$	55.2669	1.86623	0.933116	−	0.359575i	$-0.117078\pi$
$877$	55.2669	1.86623	0.933116	+	0.359575i	$0.117078\pi$
$878$	−37.3770	−1.26141
$879$	−18.4709	−0.623010
$880$	−9.59191	−0.323343
$881$	43.1514	1.45381	0.726903	−	0.686740i	$-0.240959\pi$
$881$	43.1514	1.45381	0.726903	+	0.686740i	$0.240959\pi$
$882$	62.3803	2.10045
$883$	37.8187	1.27270	0.636351	−	0.771400i	$-0.280443\pi$
$883$	37.8187	1.27270	0.636351	+	0.771400i	$0.280443\pi$
$884$	−0.0536915	−0.00180584
$885$	2.87356	0.0965937
$886$	−0.261630	−0.00878965
$887$	−0.230257	−0.00773129	−0.00386564	−	0.999993i	$-0.501230\pi$
$887$	−0.230257	−0.00773129	−0.00386564	+	0.999993i	$0.501230\pi$
$888$	4.72948	0.158711
$889$	−65.9516	−2.21194
$890$	−32.0206	−1.07333
$891$	−3.84721	−0.128886
$892$	0.665648	0.0222875
$893$	1.55194	0.0519336
$894$	−14.7830	−0.494418
$895$	−73.4785	−2.45612
$896$	−58.5288	−1.95531
$897$	0.100757	0.00336419
$898$	37.7546	1.25989
$899$	36.2337	1.20846
$900$	−1.25720	−0.0419068
$901$	−6.06464	−0.202042
$902$	−10.2795	−0.342270
$903$	40.4689	1.34672
$904$	1.69167	0.0562642
$905$	−26.5639	−0.883015
$906$	13.9749	0.464283
$907$	−2.22442	−0.0738606	−0.0369303	−	0.999318i	$-0.511758\pi$
$907$	−2.22442	−0.0738606	−0.0369303	+	0.999318i	$0.511758\pi$
$908$	−1.29065	−0.0428319
$909$	−45.4151	−1.50633
$910$	−3.47034	−0.115041
$911$	18.1195	0.600325	0.300162	−	0.953888i	$-0.402959\pi$
$911$	18.1195	0.600325	0.300162	+	0.953888i	$0.402959\pi$
$912$	−2.70257	−0.0894912
$913$	−0.732136	−0.0242302
$914$	−48.9405	−1.61881
$915$	−21.2860	−0.703695
$916$	−2.29498	−0.0758282
$917$	−71.6697	−2.36674
$918$	20.4047	0.673456
$919$	43.0293	1.41941	0.709703	−	0.704501i	$-0.248829\pi$
$919$	43.0293	1.41941	0.709703	+	0.704501i	$0.248829\pi$
$920$	9.07285	0.299123
$921$	6.01274	0.198127
$922$	−19.8448	−0.653555
$923$	0.784006	0.0258059
$924$	−0.202401	−0.00665849
$925$	14.3089	0.470472
$926$	0.973910	0.0320047
$927$	−4.31620	−0.141763
$928$	−4.16384	−0.136685
$929$	−36.7462	−1.20560	−0.602802	−	0.797891i	$-0.705949\pi$
$929$	−36.7462	−1.20560	−0.602802	+	0.797891i	$0.705949\pi$
$930$	−13.3825	−0.438828
$931$	16.7221	0.548045
$932$	−1.54435	−0.0505868
$933$	14.5146	0.475187
$934$	−46.2097	−1.51203
$935$	8.97675	0.293571
$936$	1.08828	0.0355716
$937$	43.5115	1.42146	0.710730	−	0.703465i	$-0.248365\pi$
$937$	43.5115	1.42146	0.710730	+	0.703465i	$0.248365\pi$
$938$	92.5426	3.02162
$939$	6.46783	0.211070
$940$	−0.450845	−0.0147049
$941$	18.5622	0.605110	0.302555	−	0.953132i	$-0.402160\pi$
$941$	18.5622	0.605110	0.302555	+	0.953132i	$0.402160\pi$
$942$	18.1262	0.590584
$943$	10.1618	0.330914
$944$	5.73538	0.186671
$945$	56.8116	1.84808
$946$	13.2126	0.429579
$947$	40.9035	1.32919	0.664593	−	0.747206i	$-0.268605\pi$
$947$	40.9035	1.32919	0.664593	+	0.747206i	$0.268605\pi$
$948$	0.501973	0.0163033
$949$	1.39520	0.0452901
$950$	−7.82364	−0.253832
$951$	−15.9938	−0.518634
$952$	52.5089	1.70182
$953$	56.0977	1.81718	0.908591	−	0.417687i	$-0.137159\pi$
$953$	56.0977	1.81718	0.908591	+	0.417687i	$0.137159\pi$
$954$	−5.79439	−0.187600
$955$	13.6495	0.441689
$956$	−0.958073	−0.0309863
$957$	3.77654	0.122078
$958$	7.44168	0.240430
$959$	4.10761	0.132642
$960$	−15.9030	−0.513267
$961$	−11.3880	−0.367354
$962$	0.583859	0.0188244
$963$	−34.0570	−1.09747
$964$	0.277752	0.00894578
$965$	−13.7328	−0.442076
$966$	4.64485	0.149446
$967$	8.64354	0.277957	0.138979	−	0.990295i	$-0.455618\pi$
$967$	8.64354	0.277957	0.138979	+	0.990295i	$0.455618\pi$
$968$	−28.9717	−0.931184
$969$	2.52925	0.0812512
$970$	40.6882	1.30642
$971$	7.81243	0.250713	0.125356	−	0.992112i	$-0.459993\pi$
$971$	7.81243	0.250713	0.125356	+	0.992112i	$0.459993\pi$
$972$	1.29127	0.0414174
$973$	−79.2715	−2.54133
$974$	49.1185	1.57386
$975$	−0.535463	−0.0171485
$976$	−42.4851	−1.35991
$977$	−44.8908	−1.43618	−0.718092	−	0.695949i	$-0.754984\pi$
$977$	−44.8908	−1.43618	−0.718092	+	0.695949i	$0.754984\pi$
$978$	−14.0553	−0.449440
$979$	4.89100	0.156317
$980$	−4.85785	−0.155178
$981$	29.9450	0.956070
$982$	−12.8406	−0.409760
$983$	−5.97242	−0.190491	−0.0952453	−	0.995454i	$-0.530364\pi$
$983$	−5.97242	−0.190491	−0.0952453	+	0.995454i	$0.530364\pi$
$984$	−17.8497	−0.569028
$985$	48.2377	1.53698
$986$	46.1830	1.47077
$987$	4.89653	0.155858
$988$	−0.0137515	−0.000437495 0
$989$	−13.0613	−0.415326
$990$	8.57674	0.272587
$991$	4.43513	0.140887	0.0704433	−	0.997516i	$-0.477559\pi$
$991$	4.43513	0.140887	0.0704433	+	0.997516i	$0.477559\pi$
$992$	−2.25374	−0.0715563
$993$	−2.08826	−0.0662690
$994$	36.1421	1.14636
$995$	−26.3840	−0.836430
$996$	0.0599263	0.00189884
$997$	−18.2769	−0.578836	−0.289418	−	0.957203i	$-0.593462\pi$
$997$	−18.2769	−0.578836	−0.289418	+	0.957203i	$0.593462\pi$
$998$	16.2744	0.515156
$999$	−9.55813	−0.302406

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.22	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.22	✓	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)

\(f(q)\)	\(=\)	\(q-1.44569 q^{2} +0.647795 q^{3} +0.0900311 q^{4} -3.22671 q^{5} -0.936514 q^{6} +4.87053 q^{7} +2.76123 q^{8} -2.58036 q^{9} +O(q^{10})\)	\(q-1.44569 q^{2} +0.647795 q^{3} +0.0900311 q^{4} -3.22671 q^{5} -0.936514 q^{6} +4.87053 q^{7} +2.76123 q^{8} -2.58036 q^{9} +4.66484 q^{10} -0.712533 q^{11} +0.0583217 q^{12} -0.152742 q^{13} -7.04130 q^{14} -2.09025 q^{15} -4.17196 q^{16} +3.90440 q^{17} +3.73041 q^{18} +1.00000 q^{19} -0.290505 q^{20} +3.15511 q^{21} +1.03010 q^{22} -1.01831 q^{23} +1.78871 q^{24} +5.41168 q^{25} +0.220818 q^{26} -3.61493 q^{27} +0.438499 q^{28} -8.18185 q^{29} +3.02186 q^{30} -4.42855 q^{31} +0.508911 q^{32} -0.461576 q^{33} -5.64456 q^{34} -15.7158 q^{35} -0.232313 q^{36} +2.64407 q^{37} -1.44569 q^{38} -0.0989456 q^{39} -8.90970 q^{40} -9.97907 q^{41} -4.56132 q^{42} +12.8265 q^{43} -0.0641501 q^{44} +8.32609 q^{45} +1.47217 q^{46} +1.55194 q^{47} -2.70257 q^{48} +16.7221 q^{49} -7.82364 q^{50} +2.52925 q^{51} -0.0137515 q^{52} -1.55328 q^{53} +5.22609 q^{54} +2.29914 q^{55} +13.4487 q^{56} +0.647795 q^{57} +11.8285 q^{58} -1.37475 q^{59} -0.188188 q^{60} +10.1835 q^{61} +6.40233 q^{62} -12.5677 q^{63} +7.60818 q^{64} +0.492855 q^{65} +0.667297 q^{66} -13.1428 q^{67} +0.351517 q^{68} -0.659658 q^{69} +22.7203 q^{70} -5.13287 q^{71} -7.12497 q^{72} -9.13435 q^{73} -3.82251 q^{74} +3.50566 q^{75} +0.0900311 q^{76} -3.47042 q^{77} +0.143045 q^{78} +8.60696 q^{79} +13.4617 q^{80} +5.39935 q^{81} +14.4267 q^{82} +1.02751 q^{83} +0.284058 q^{84} -12.5984 q^{85} -18.5431 q^{86} -5.30017 q^{87} -1.96747 q^{88} -6.86424 q^{89} -12.0370 q^{90} -0.743936 q^{91} -0.0916797 q^{92} -2.86879 q^{93} -2.24363 q^{94} -3.22671 q^{95} +0.329671 q^{96} +8.72232 q^{97} -24.1750 q^{98} +1.83859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 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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