LMFDB - Embedded newform 1849.2.a.r.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} + \cdots - 43 $ x^20 - 5x^19 - 19x^18 + 127x^17 + 95x^16 - 1293x^15 + 329x^14 + 6765x^13 - 5068x^12 - 19360x^11 + 20357x^10 + 29618x^9 - 37889x^8 - 21700x^7 + 32885x^6 + 6093x^5 - 11330x^4 - 1004x^3 + 1386x^2 + 87*x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.54977$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.54977 q^{2} +3.21447 q^{3} +0.401778 q^{4} +2.07911 q^{5} +4.98169 q^{6} -0.871568 q^{7} -2.47687 q^{8} +7.33285 q^{9} +O(q^{10})$	\(q+1.54977 q^{2} +3.21447 q^{3} +0.401778 q^{4} +2.07911 q^{5} +4.98169 q^{6} -0.871568 q^{7} -2.47687 q^{8} +7.33285 q^{9} +3.22213 q^{10} +0.118796 q^{11} +1.29151 q^{12} -1.17173 q^{13} -1.35073 q^{14} +6.68324 q^{15} -4.64213 q^{16} +6.72101 q^{17} +11.3642 q^{18} -4.91679 q^{19} +0.835340 q^{20} -2.80163 q^{21} +0.184107 q^{22} +2.58561 q^{23} -7.96184 q^{24} -0.677311 q^{25} -1.81590 q^{26} +13.9278 q^{27} -0.350177 q^{28} -0.929188 q^{29} +10.3575 q^{30} +6.92176 q^{31} -2.24048 q^{32} +0.381868 q^{33} +10.4160 q^{34} -1.81208 q^{35} +2.94618 q^{36} +0.871716 q^{37} -7.61988 q^{38} -3.76648 q^{39} -5.14968 q^{40} +1.73741 q^{41} -4.34188 q^{42} +0.0477298 q^{44} +15.2458 q^{45} +4.00709 q^{46} -9.72016 q^{47} -14.9220 q^{48} -6.24037 q^{49} -1.04967 q^{50} +21.6045 q^{51} -0.470774 q^{52} -5.24926 q^{53} +21.5849 q^{54} +0.246991 q^{55} +2.15876 q^{56} -15.8049 q^{57} -1.44003 q^{58} -3.87260 q^{59} +2.68518 q^{60} +2.03937 q^{61} +10.7271 q^{62} -6.39107 q^{63} +5.81204 q^{64} -2.43614 q^{65} +0.591807 q^{66} -1.82807 q^{67} +2.70036 q^{68} +8.31136 q^{69} -2.80831 q^{70} -12.5462 q^{71} -18.1625 q^{72} -3.30592 q^{73} +1.35096 q^{74} -2.17720 q^{75} -1.97546 q^{76} -0.103539 q^{77} -5.83717 q^{78} +6.32506 q^{79} -9.65149 q^{80} +22.7721 q^{81} +2.69258 q^{82} -2.34720 q^{83} -1.12564 q^{84} +13.9737 q^{85} -2.98685 q^{87} -0.294244 q^{88} -7.20924 q^{89} +23.6274 q^{90} +1.02124 q^{91} +1.03884 q^{92} +22.2498 q^{93} -15.0640 q^{94} -10.2225 q^{95} -7.20196 q^{96} -12.2645 q^{97} -9.67112 q^{98} +0.871116 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) $ 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9} + 21 q^{11} - 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} + 9 q^{18} - 7 q^{19} - 17 q^{20} + 8 q^{21} + 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} + 58 q^{26} - 22 q^{27} - 28 q^{28} - 12 q^{29} - 24 q^{30} + 26 q^{31} + 3 q^{32} - 17 q^{33} + 54 q^{34} + 26 q^{35} + 14 q^{36} - 19 q^{37} + 5 q^{38} - 7 q^{39} - 16 q^{40} + 27 q^{41} - 52 q^{42} + 32 q^{44} + 75 q^{45} + 59 q^{46} + 45 q^{47} - 66 q^{48} + 20 q^{49} + 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} - 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} + 30 q^{61} + 72 q^{62} - 21 q^{63} - 7 q^{64} - 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} - 35 q^{69} - 80 q^{70} - 31 q^{71} - 15 q^{72} - 26 q^{73} + 87 q^{74} + 73 q^{75} - 68 q^{76} - 21 q^{77} - 50 q^{78} + 39 q^{79} - 60 q^{80} + 16 q^{81} + 45 q^{82} + 13 q^{83} + 31 q^{84} - 22 q^{85} + 61 q^{87} + 50 q^{88} - 4 q^{89} - 13 q^{90} - 25 q^{91} + 5 q^{92} + 67 q^{93} + 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} - 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9 + 21 * q^11 - 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 + 9 * q^18 - 7 * q^19 - 17 * q^20 + 8 * q^21 + 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 + 58 * q^26 - 22 * q^27 - 28 * q^28 - 12 * q^29 - 24 * q^30 + 26 * q^31 + 3 * q^32 - 17 * q^33 + 54 * q^34 + 26 * q^35 + 14 * q^36 - 19 * q^37 + 5 * q^38 - 7 * q^39 - 16 * q^40 + 27 * q^41 - 52 * q^42 + 32 * q^44 + 75 * q^45 + 59 * q^46 + 45 * q^47 - 66 * q^48 + 20 * q^49 + 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 - 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 + 30 * q^61 + 72 * q^62 - 21 * q^63 - 7 * q^64 - 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 - 35 * q^69 - 80 * q^70 - 31 * q^71 - 15 * q^72 - 26 * q^73 + 87 * q^74 + 73 * q^75 - 68 * q^76 - 21 * q^77 - 50 * q^78 + 39 * q^79 - 60 * q^80 + 16 * q^81 + 45 * q^82 + 13 * q^83 + 31 * q^84 - 22 * q^85 + 61 * q^87 + 50 * q^88 - 4 * q^89 - 13 * q^90 - 25 * q^91 + 5 * q^92 + 67 * q^93 + 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 - 35 * q^98 + 13 * 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 $

$2$	1.54977	1.09585	0.547925	−	0.836527i	$-0.315418\pi$
$2$	1.54977	1.09585	0.547925	+	0.836527i	$0.315418\pi$
$3$	3.21447	1.85588	0.927939	−	0.372732i	$-0.121579\pi$
$3$	3.21447	1.85588	0.927939	+	0.372732i	$0.121579\pi$
$4$	0.401778	0.200889
$5$	2.07911	0.929805	0.464903	−	0.885362i	$-0.346089\pi$
$5$	2.07911	0.929805	0.464903	+	0.885362i	$0.346089\pi$
$6$	4.98169	2.03377
$7$	−0.871568	−0.329422	−0.164711	−	0.986342i	$-0.552669\pi$
$7$	−0.871568	−0.329422	−0.164711	+	0.986342i	$0.552669\pi$
$8$	−2.47687	−0.875706
$9$	7.33285	2.44428
$10$	3.22213	1.01893
$11$	0.118796	0.0358185	0.0179092	−	0.999840i	$-0.494299\pi$
$11$	0.118796	0.0358185	0.0179092	+	0.999840i	$0.494299\pi$
$12$	1.29151	0.372826
$13$	−1.17173	−0.324978	−0.162489	−	0.986710i	$-0.551952\pi$
$13$	−1.17173	−0.324978	−0.162489	+	0.986710i	$0.551952\pi$
$14$	−1.35073	−0.360997
$15$	6.68324	1.72560
$16$	−4.64213	−1.16053
$17$	6.72101	1.63008	0.815042	−	0.579402i	$-0.196714\pi$
$17$	6.72101	1.63008	0.815042	+	0.579402i	$0.196714\pi$
$18$	11.3642	2.67857
$19$	−4.91679	−1.12799	−0.563995	−	0.825778i	$-0.690736\pi$
$19$	−4.91679	−1.12799	−0.563995	+	0.825778i	$0.690736\pi$
$20$	0.835340	0.186788
$21$	−2.80163	−0.611366
$22$	0.184107	0.0392517
$23$	2.58561	0.539136	0.269568	−	0.962981i	$-0.413119\pi$
$23$	2.58561	0.539136	0.269568	+	0.962981i	$0.413119\pi$
$24$	−7.96184	−1.62520
$25$	−0.677311	−0.135462
$26$	−1.81590	−0.356127
$27$	13.9278	2.68041
$28$	−0.350177	−0.0661772
$29$	−0.929188	−0.172546	−0.0862729	−	0.996272i	$-0.527496\pi$
$29$	−0.929188	−0.172546	−0.0862729	+	0.996272i	$0.527496\pi$
$30$	10.3575	1.89101
$31$	6.92176	1.24318	0.621592	−	0.783341i	$-0.286486\pi$
$31$	6.92176	1.24318	0.621592	+	0.783341i	$0.286486\pi$
$32$	−2.24048	−0.396064
$33$	0.381868	0.0664747
$34$	10.4160	1.78633
$35$	−1.81208	−0.306298
$36$	2.94618	0.491030
$37$	0.871716	0.143309	0.0716546	−	0.997430i	$-0.477172\pi$
$37$	0.871716	0.143309	0.0716546	+	0.997430i	$0.477172\pi$
$38$	−7.61988	−1.23611
$39$	−3.76648	−0.603120
$40$	−5.14968	−0.814236
$41$	1.73741	0.271338	0.135669	−	0.990754i	$-0.456682\pi$
$41$	1.73741	0.271338	0.135669	+	0.990754i	$0.456682\pi$
$42$	−4.34188	−0.669966
$43$	0	0
$43$	0	0
$44$	0.0477298	0.00719554
$45$	15.2458	2.27271
$46$	4.00709	0.590813
$47$	−9.72016	−1.41783	−0.708916	−	0.705293i	$-0.750815\pi$
$47$	−9.72016	−1.41783	−0.708916	+	0.705293i	$0.750815\pi$
$48$	−14.9220	−2.15381
$49$	−6.24037	−0.891481
$50$	−1.04967	−0.148446
$51$	21.6045	3.02524
$52$	−0.470774	−0.0652846
$53$	−5.24926	−0.721041	−0.360520	−	0.932751i	$-0.617401\pi$
$53$	−5.24926	−0.721041	−0.360520	+	0.932751i	$0.617401\pi$
$54$	21.5849	2.93733
$55$	0.246991	0.0333042
$56$	2.15876	0.288477
$57$	−15.8049	−2.09341
$58$	−1.44003	−0.189085
$59$	−3.87260	−0.504169	−0.252085	−	0.967705i	$-0.581116\pi$
$59$	−3.87260	−0.504169	−0.252085	+	0.967705i	$0.581116\pi$
$60$	2.68518	0.346655
$61$	2.03937	0.261114	0.130557	−	0.991441i	$-0.458323\pi$
$61$	2.03937	0.261114	0.130557	+	0.991441i	$0.458323\pi$
$62$	10.7271	1.36234
$63$	−6.39107	−0.805200
$64$	5.81204	0.726505
$65$	−2.43614	−0.302166
$66$	0.591807	0.0728464
$67$	−1.82807	−0.223335	−0.111667	−	0.993746i	$-0.535619\pi$
$67$	−1.82807	−0.223335	−0.111667	+	0.993746i	$0.535619\pi$
$68$	2.70036	0.327466
$69$	8.31136	1.00057
$70$	−2.80831	−0.335657
$71$	−12.5462	−1.48896	−0.744481	−	0.667644i	$-0.767303\pi$
$71$	−12.5462	−1.48896	−0.744481	+	0.667644i	$0.767303\pi$
$72$	−18.1625	−2.14047
$73$	−3.30592	−0.386928	−0.193464	−	0.981107i	$-0.561972\pi$
$73$	−3.30592	−0.386928	−0.193464	+	0.981107i	$0.561972\pi$
$74$	1.35096	0.157046
$75$	−2.17720	−0.251401
$76$	−1.97546	−0.226601
$77$	−0.103539	−0.0117994
$78$	−5.83717	−0.660929
$79$	6.32506	0.711625	0.355813	−	0.934557i	$-0.384204\pi$
$79$	6.32506	0.711625	0.355813	+	0.934557i	$0.384204\pi$
$80$	−9.65149	−1.07907
$81$	22.7721	2.53023
$82$	2.69258	0.297346
$83$	−2.34720	−0.257639	−0.128820	−	0.991668i	$-0.541119\pi$
$83$	−2.34720	−0.257639	−0.128820	+	0.991668i	$0.541119\pi$
$84$	−1.12564	−0.122817
$85$	13.9737	1.51566
$86$	0	0
$87$	−2.98685	−0.320224
$88$	−0.294244	−0.0313665
$89$	−7.20924	−0.764178	−0.382089	−	0.924126i	$-0.624795\pi$
$89$	−7.20924	−0.764178	−0.382089	+	0.924126i	$0.624795\pi$
$90$	23.6274	2.49055
$91$	1.02124	0.107055
$92$	1.03884	0.108307
$93$	22.2498	2.30720
$94$	−15.0640	−1.55373
$95$	−10.2225	−1.04881
$96$	−7.20196	−0.735047
$97$	−12.2645	−1.24527	−0.622637	−	0.782511i	$-0.713939\pi$
$97$	−12.2645	−1.24527	−0.622637	+	0.782511i	$0.713939\pi$
$98$	−9.67112	−0.976931
$99$	0.871116	0.0875505
$100$	−0.272129	−0.0272129
$101$	15.0311	1.49565	0.747823	−	0.663898i	$-0.231099\pi$
$101$	15.0311	1.49565	0.747823	+	0.663898i	$0.231099\pi$
$102$	33.4820	3.31521
$103$	5.37576	0.529690	0.264845	−	0.964291i	$-0.414679\pi$
$103$	5.37576	0.529690	0.264845	+	0.964291i	$0.414679\pi$
$104$	2.90221	0.284585
$105$	−5.82490	−0.568452
$106$	−8.13512	−0.790153
$107$	4.18065	0.404159	0.202079	−	0.979369i	$-0.435230\pi$
$107$	4.18065	0.404159	0.202079	+	0.979369i	$0.435230\pi$
$108$	5.59590	0.538465
$109$	−15.0109	−1.43778	−0.718892	−	0.695122i	$-0.755350\pi$
$109$	−15.0109	−1.43778	−0.718892	+	0.695122i	$0.755350\pi$
$110$	0.382778	0.0364965
$111$	2.80211	0.265964
$112$	4.04593	0.382305
$113$	−10.5214	−0.989772	−0.494886	−	0.868958i	$-0.664790\pi$
$113$	−10.5214	−0.989772	−0.494886	+	0.868958i	$0.664790\pi$
$114$	−24.4939	−2.29407
$115$	5.37575	0.501292
$116$	−0.373328	−0.0346626
$117$	−8.59208	−0.794338
$118$	−6.00163	−0.552494
$119$	−5.85782	−0.536985
$120$	−16.5535	−1.51112
$121$	−10.9859	−0.998717
$122$	3.16054	0.286142
$123$	5.58486	0.503570
$124$	2.78101	0.249742
$125$	−11.8037	−1.05576
$126$	−9.90468	−0.882379
$127$	18.5498	1.64603	0.823016	−	0.568018i	$-0.192290\pi$
$127$	18.5498	1.64603	0.823016	+	0.568018i	$0.192290\pi$
$128$	13.4883	1.19221
$129$	0	0
$130$	−3.77545	−0.331129
$131$	−19.2658	−1.68326	−0.841628	−	0.540057i	$-0.818403\pi$
$131$	−19.2658	−1.68326	−0.841628	+	0.540057i	$0.818403\pi$
$132$	0.153426	0.0133541
$133$	4.28532	0.371584
$134$	−2.83309	−0.244741
$135$	28.9574	2.49226
$136$	−16.6471	−1.42748
$137$	−10.4252	−0.890685	−0.445343	−	0.895360i	$-0.646918\pi$
$137$	−10.4252	−0.890685	−0.445343	+	0.895360i	$0.646918\pi$
$138$	12.8807	1.09648
$139$	12.6616	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$139$	12.6616	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$140$	−0.728056	−0.0615319
$141$	−31.2452	−2.63132
$142$	−19.4437	−1.63168
$143$	−0.139197	−0.0116402
$144$	−34.0400	−2.83667
$145$	−1.93188	−0.160434
$146$	−5.12340	−0.424016
$147$	−20.0595	−1.65448
$148$	0.350237	0.0287893
$149$	5.73581	0.469896	0.234948	−	0.972008i	$-0.424508\pi$
$149$	5.73581	0.469896	0.234948	+	0.972008i	$0.424508\pi$
$150$	−3.37415	−0.275498
$151$	12.0649	0.981827	0.490913	−	0.871208i	$-0.336663\pi$
$151$	12.0649	0.981827	0.490913	+	0.871208i	$0.336663\pi$
$152$	12.1783	0.987788
$153$	49.2841	3.98439
$154$	−0.160462	−0.0129304
$155$	14.3911	1.15592
$156$	−1.51329	−0.121160
$157$	17.9121	1.42954	0.714772	−	0.699357i	$-0.246530\pi$
$157$	17.9121	1.42954	0.714772	+	0.699357i	$0.246530\pi$
$158$	9.80238	0.779835
$159$	−16.8736	−1.33816
$160$	−4.65820	−0.368263
$161$	−2.25353	−0.177603
$162$	35.2915	2.77276
$163$	14.2861	1.11898	0.559489	−	0.828838i	$-0.310998\pi$
$163$	14.2861	1.11898	0.559489	+	0.828838i	$0.310998\pi$
$164$	0.698053	0.0545088
$165$	0.793945	0.0618086
$166$	−3.63762	−0.282334
$167$	−3.46008	−0.267749	−0.133874	−	0.990998i	$-0.542742\pi$
$167$	−3.46008	−0.267749	−0.133874	+	0.990998i	$0.542742\pi$
$168$	6.93928	0.535377
$169$	−11.6271	−0.894389
$170$	21.6560	1.66094
$171$	−36.0541	−2.75712
$172$	0	0
$173$	10.9867	0.835304	0.417652	−	0.908607i	$-0.362853\pi$
$173$	10.9867	0.835304	0.417652	+	0.908607i	$0.362853\pi$
$174$	−4.62892	−0.350918
$175$	0.590323	0.0446242
$176$	−0.551469	−0.0415685
$177$	−12.4484	−0.935677
$178$	−11.1726	−0.837425
$179$	10.2806	0.768412	0.384206	−	0.923247i	$-0.374475\pi$
$179$	10.2806	0.768412	0.384206	+	0.923247i	$0.374475\pi$
$180$	6.12542	0.456562
$181$	−13.1597	−0.978154	−0.489077	−	0.872241i	$-0.662666\pi$
$181$	−13.1597	−0.978154	−0.489077	+	0.872241i	$0.662666\pi$
$182$	1.58268	0.117316
$183$	6.55549	0.484596
$184$	−6.40421	−0.472125
$185$	1.81239	0.133250
$186$	34.4820	2.52834
$187$	0.798432	0.0583872
$188$	−3.90535	−0.284827
$189$	−12.1390	−0.882986
$190$	−15.8426	−1.14934
$191$	−6.04540	−0.437430	−0.218715	−	0.975789i	$-0.570186\pi$
$191$	−6.04540	−0.437430	−0.218715	+	0.975789i	$0.570186\pi$
$192$	18.6827	1.34830
$193$	−23.6408	−1.70170	−0.850852	−	0.525406i	$-0.823913\pi$
$193$	−23.6408	−1.70170	−0.850852	+	0.525406i	$0.823913\pi$
$194$	−19.0072	−1.36463
$195$	−7.83092	−0.560784
$196$	−2.50724	−0.179089
$197$	2.96314	0.211115	0.105558	−	0.994413i	$-0.466337\pi$
$197$	2.96314	0.211115	0.105558	+	0.994413i	$0.466337\pi$
$198$	1.35003	0.0959423
$199$	21.4455	1.52023	0.760117	−	0.649786i	$-0.225142\pi$
$199$	21.4455	1.52023	0.760117	+	0.649786i	$0.225142\pi$
$200$	1.67761	0.118625
$201$	−5.87629	−0.414482
$202$	23.2946	1.63900
$203$	0.809850	0.0568404
$204$	8.68022	0.607737
$205$	3.61226	0.252291
$206$	8.33118	0.580461
$207$	18.9599	1.31780
$208$	5.43930	0.377148
$209$	−0.584098	−0.0404029
$210$	−9.02723	−0.622938
$211$	1.41265	0.0972506	0.0486253	−	0.998817i	$-0.484516\pi$
$211$	1.41265	0.0972506	0.0486253	+	0.998817i	$0.484516\pi$
$212$	−2.10904	−0.144849
$213$	−40.3295	−2.76333
$214$	6.47903	0.442898
$215$	0	0
$216$	−34.4974	−2.34725
$217$	−6.03278	−0.409532
$218$	−23.2634	−1.57560
$219$	−10.6268	−0.718092
$220$	0.0992355	0.00669046
$221$	−7.87518	−0.529742
$222$	4.34262	0.291457
$223$	−18.4579	−1.23603	−0.618016	−	0.786166i	$-0.712063\pi$
$223$	−18.4579	−1.23603	−0.618016	+	0.786166i	$0.712063\pi$
$224$	1.95273	0.130472
$225$	−4.96662	−0.331108
$226$	−16.3058	−1.08464
$227$	17.4397	1.15752	0.578758	−	0.815499i	$-0.303538\pi$
$227$	17.4397	1.15752	0.578758	+	0.815499i	$0.303538\pi$
$228$	−6.35007	−0.420543
$229$	16.5342	1.09261	0.546306	−	0.837586i	$-0.316034\pi$
$229$	16.5342	1.09261	0.546306	+	0.837586i	$0.316034\pi$
$230$	8.33117	0.549341
$231$	−0.332824	−0.0218982
$232$	2.30148	0.151100
$233$	10.5158	0.688914	0.344457	−	0.938802i	$-0.388063\pi$
$233$	10.5158	0.688914	0.344457	+	0.938802i	$0.388063\pi$
$234$	−13.3157	−0.870476
$235$	−20.2093	−1.31831
$236$	−1.55593	−0.101282
$237$	20.3318	1.32069
$238$	−9.07825	−0.588456
$239$	−2.10552	−0.136195	−0.0680973	−	0.997679i	$-0.521693\pi$
$239$	−2.10552	−0.136195	−0.0680973	+	0.997679i	$0.521693\pi$
$240$	−31.0245	−2.00262
$241$	−24.9373	−1.60635	−0.803176	−	0.595742i	$-0.796858\pi$
$241$	−24.9373	−1.60635	−0.803176	+	0.595742i	$0.796858\pi$
$242$	−17.0256	−1.09444
$243$	31.4169	2.01539
$244$	0.819373	0.0524550
$245$	−12.9744	−0.828904
$246$	8.65523	0.551837
$247$	5.76113	0.366572
$248$	−17.1443	−1.08866
$249$	−7.54502	−0.478147
$250$	−18.2931	−1.15695
$251$	26.0926	1.64695	0.823474	−	0.567354i	$-0.192033\pi$
$251$	26.0926	1.64695	0.823474	+	0.567354i	$0.192033\pi$
$252$	−2.56779	−0.161756
$253$	0.307161	0.0193110
$254$	28.7479	1.80381
$255$	44.9181	2.81288
$256$	9.27959	0.579975
$257$	−0.883164	−0.0550903	−0.0275451	−	0.999621i	$-0.508769\pi$
$257$	−0.883164	−0.0550903	−0.0275451	+	0.999621i	$0.508769\pi$
$258$	0	0
$259$	−0.759760	−0.0472092
$260$	−0.978789	−0.0607019
$261$	−6.81359	−0.421751
$262$	−29.8574	−1.84460
$263$	−2.92298	−0.180239	−0.0901193	−	0.995931i	$-0.528725\pi$
$263$	−2.92298	−0.180239	−0.0901193	+	0.995931i	$0.528725\pi$
$264$	−0.945839	−0.0582123
$265$	−10.9138	−0.670427
$266$	6.64125	0.407201
$267$	−23.1739	−1.41822
$268$	−0.734479	−0.0448655
$269$	−5.32307	−0.324553	−0.162277	−	0.986745i	$-0.551884\pi$
$269$	−5.32307	−0.324553	−0.162277	+	0.986745i	$0.551884\pi$
$270$	44.8773	2.73115
$271$	17.9472	1.09022	0.545108	−	0.838366i	$-0.316489\pi$
$271$	17.9472	1.09022	0.545108	+	0.838366i	$0.316489\pi$
$272$	−31.1998	−1.89177
$273$	3.28274	0.198681
$274$	−16.1566	−0.976058
$275$	−0.0804622	−0.00485205
$276$	3.33933	0.201004
$277$	24.6071	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$277$	24.6071	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$278$	19.6226	1.17688
$279$	50.7562	3.03869
$280$	4.48830	0.268227
$281$	21.3772	1.27526	0.637629	−	0.770344i	$-0.279915\pi$
$281$	21.3772	1.27526	0.637629	+	0.770344i	$0.279915\pi$
$282$	−48.4228	−2.88354
$283$	−9.40482	−0.559059	−0.279529	−	0.960137i	$-0.590178\pi$
$283$	−9.40482	−0.559059	−0.279529	+	0.960137i	$0.590178\pi$
$284$	−5.04080	−0.299116
$285$	−32.8601	−1.94646
$286$	−0.215723	−0.0127559
$287$	−1.51427	−0.0893845
$288$	−16.4291	−0.968093
$289$	28.1720	1.65717
$290$	−2.99397	−0.175812
$291$	−39.4240	−2.31108
$292$	−1.32825	−0.0777297
$293$	−6.07616	−0.354973	−0.177486	−	0.984123i	$-0.556797\pi$
$293$	−6.07616	−0.354973	−0.177486	+	0.984123i	$0.556797\pi$
$294$	−31.0876	−1.81306
$295$	−8.05155	−0.468779
$296$	−2.15913	−0.125497
$297$	1.65458	0.0960083
$298$	8.88916	0.514935
$299$	−3.02962	−0.175207
$300$	−0.874752	−0.0505038
$301$	0	0
$302$	18.6978	1.07594
$303$	48.3169	2.77574
$304$	22.8244	1.30907
$305$	4.24006	0.242785
$306$	76.3789	4.36629
$307$	19.0136	1.08516	0.542582	−	0.840003i	$-0.317447\pi$
$307$	19.0136	1.08516	0.542582	+	0.840003i	$0.317447\pi$
$308$	−0.0415998	−0.00237037
$309$	17.2803	0.983039
$310$	22.3028	1.26671
$311$	8.92295	0.505974	0.252987	−	0.967470i	$-0.418587\pi$
$311$	8.92295	0.505974	0.252987	+	0.967470i	$0.418587\pi$
$312$	9.32909	0.528156
$313$	25.4924	1.44092	0.720458	−	0.693499i	$-0.243932\pi$
$313$	25.4924	1.44092	0.720458	+	0.693499i	$0.243932\pi$
$314$	27.7597	1.56657
$315$	−13.2877	−0.748679
$316$	2.54127	0.142958
$317$	0.227021	0.0127508	0.00637539	−	0.999980i	$-0.497971\pi$
$317$	0.227021	0.0127508	0.00637539	+	0.999980i	$0.497971\pi$
$318$	−26.1501	−1.46643
$319$	−0.110384	−0.00618033
$320$	12.0839	0.675508
$321$	13.4386	0.750069
$322$	−3.49245	−0.194627
$323$	−33.0458	−1.83872
$324$	9.14933	0.508296
$325$	0.793623	0.0440223
$326$	22.1402	1.22623
$327$	−48.2522	−2.66835
$328$	−4.30334	−0.237612
$329$	8.47178	0.467065
$330$	1.23043	0.0677330
$331$	−9.38415	−0.515799	−0.257900	−	0.966172i	$-0.583030\pi$
$331$	−9.38415	−0.515799	−0.257900	+	0.966172i	$0.583030\pi$
$332$	−0.943055	−0.0517569
$333$	6.39216	0.350288
$334$	−5.36232	−0.293413
$335$	−3.80076	−0.207658
$336$	13.0055	0.709511
$337$	−6.36872	−0.346926	−0.173463	−	0.984840i	$-0.555496\pi$
$337$	−6.36872	−0.346926	−0.173463	+	0.984840i	$0.555496\pi$
$338$	−18.0192	−0.980117
$339$	−33.8208	−1.83690
$340$	5.61433	0.304480
$341$	0.822280	0.0445290
$342$	−55.8754	−3.02140
$343$	11.5399	0.623095
$344$	0	0
$345$	17.2802	0.930336
$346$	17.0268	0.915369
$347$	13.6088	0.730556	0.365278	−	0.930898i	$-0.380974\pi$
$347$	13.6088	0.730556	0.365278	+	0.930898i	$0.380974\pi$
$348$	−1.20005	−0.0643295
$349$	−21.5883	−1.15560	−0.577799	−	0.816179i	$-0.696088\pi$
$349$	−21.5883	−1.15560	−0.577799	+	0.816179i	$0.696088\pi$
$350$	0.914863	0.0489015
$351$	−16.3196	−0.871075
$352$	−0.266161	−0.0141864
$353$	2.59435	0.138083	0.0690416	−	0.997614i	$-0.478006\pi$
$353$	2.59435	0.138083	0.0690416	+	0.997614i	$0.478006\pi$
$354$	−19.2921	−1.02536
$355$	−26.0849	−1.38444
$356$	−2.89652	−0.153515
$357$	−18.8298	−0.996579
$358$	15.9326	0.842065
$359$	21.7304	1.14689	0.573445	−	0.819244i	$-0.305607\pi$
$359$	21.7304	1.14689	0.573445	+	0.819244i	$0.305607\pi$
$360$	−37.7618	−1.99022
$361$	5.17485	0.272361
$362$	−20.3945	−1.07191
$363$	−35.3139	−1.85350
$364$	0.410311	0.0215062
$365$	−6.87336	−0.359768
$366$	10.1595	0.531044
$367$	−7.28324	−0.380182	−0.190091	−	0.981766i	$-0.560878\pi$
$367$	−7.28324	−0.380182	−0.190091	+	0.981766i	$0.560878\pi$
$368$	−12.0027	−0.625685
$369$	12.7402	0.663226
$370$	2.80879	0.146022
$371$	4.57508	0.237526
$372$	8.93949	0.463491
$373$	−15.2019	−0.787124	−0.393562	−	0.919298i	$-0.628757\pi$
$373$	−15.2019	−0.787124	−0.393562	+	0.919298i	$0.628757\pi$
$374$	1.23738	0.0639836
$375$	−37.9428	−1.95936
$376$	24.0756	1.24160
$377$	1.08875	0.0560736
$378$	−18.8127	−0.967621
$379$	20.5398	1.05506	0.527529	−	0.849537i	$-0.323118\pi$
$379$	20.5398	1.05506	0.527529	+	0.849537i	$0.323118\pi$
$380$	−4.10719	−0.210695
$381$	59.6280	3.05483
$382$	−9.36896	−0.479358
$383$	8.07630	0.412680	0.206340	−	0.978480i	$-0.433845\pi$
$383$	8.07630	0.412680	0.206340	+	0.978480i	$0.433845\pi$
$384$	43.3577	2.21259
$385$	−0.215269	−0.0109711
$386$	−36.6378	−1.86481
$387$	0	0
$388$	−4.92762	−0.250162
$389$	−33.5847	−1.70281	−0.851406	−	0.524507i	$-0.824249\pi$
$389$	−33.5847	−1.70281	−0.851406	+	0.524507i	$0.824249\pi$
$390$	−12.1361	−0.614535
$391$	17.3779	0.878837
$392$	15.4566	0.780676
$393$	−61.9293	−3.12392
$394$	4.59218	0.231351
$395$	13.1505	0.661673
$396$	0.349996	0.0175879
$397$	23.7641	1.19269	0.596343	−	0.802730i	$-0.296620\pi$
$397$	23.7641	1.19269	0.596343	+	0.802730i	$0.296620\pi$
$398$	33.2356	1.66595
$399$	13.7750	0.689615
$400$	3.14417	0.157208
$401$	5.89139	0.294202	0.147101	−	0.989121i	$-0.453006\pi$
$401$	5.89139	0.294202	0.147101	+	0.989121i	$0.453006\pi$
$402$	−9.10688	−0.454210
$403$	−8.11040	−0.404008
$404$	6.03915	0.300459
$405$	47.3456	2.35262
$406$	1.25508	0.0622886
$407$	0.103557	0.00513312
$408$	−53.5116	−2.64922
$409$	−15.8785	−0.785144	−0.392572	−	0.919721i	$-0.628415\pi$
$409$	−15.8785	−0.785144	−0.392572	+	0.919721i	$0.628415\pi$
$410$	5.59816	0.276474
$411$	−33.5116	−1.65300
$412$	2.15987	0.106409
$413$	3.37523	0.166084
$414$	29.3834	1.44411
$415$	−4.88009	−0.239554
$416$	2.62522	0.128712
$417$	40.7005	1.99311
$418$	−0.905215	−0.0442755
$419$	−35.1526	−1.71731	−0.858657	−	0.512550i	$-0.828701\pi$
$419$	−35.1526	−1.71731	−0.858657	+	0.512550i	$0.828701\pi$
$420$	−2.34032	−0.114196
$421$	5.89826	0.287464	0.143732	−	0.989617i	$-0.454090\pi$
$421$	5.89826	0.287464	0.143732	+	0.989617i	$0.454090\pi$
$422$	2.18927	0.106572
$423$	−71.2765	−3.46558
$424$	13.0017	0.631420
$425$	−4.55222	−0.220815
$426$	−62.5013	−3.02820
$427$	−1.77745	−0.0860166
$428$	1.67969	0.0811911
$429$	−0.447445	−0.0216028
$430$	0	0
$431$	17.6191	0.848684	0.424342	−	0.905502i	$-0.360505\pi$
$431$	17.6191	0.848684	0.424342	+	0.905502i	$0.360505\pi$
$432$	−64.6548	−3.11070
$433$	12.1201	0.582456	0.291228	−	0.956654i	$-0.405936\pi$
$433$	12.1201	0.582456	0.291228	+	0.956654i	$0.405936\pi$
$434$	−9.34941	−0.448786
$435$	−6.20998	−0.297746
$436$	−6.03105	−0.288835
$437$	−12.7129	−0.608140
$438$	−16.4690	−0.786922
$439$	25.2649	1.20583	0.602915	−	0.797806i	$-0.294006\pi$
$439$	25.2649	1.20583	0.602915	+	0.797806i	$0.294006\pi$
$440$	−0.611764	−0.0291647
$441$	−45.7597	−2.17903
$442$	−12.2047	−0.580518
$443$	−19.5814	−0.930339	−0.465170	−	0.885222i	$-0.654007\pi$
$443$	−19.5814	−0.930339	−0.465170	+	0.885222i	$0.654007\pi$
$444$	1.12583	0.0534294
$445$	−14.9888	−0.710537
$446$	−28.6054	−1.35451
$447$	18.4376	0.872069
$448$	−5.06559	−0.239327
$449$	1.44077	0.0679943	0.0339971	−	0.999422i	$-0.489176\pi$
$449$	1.44077	0.0679943	0.0339971	+	0.999422i	$0.489176\pi$
$450$	−7.69711	−0.362845
$451$	0.206398	0.00971891
$452$	−4.22728	−0.198834
$453$	38.7823	1.82215
$454$	27.0275	1.26846
$455$	2.12326	0.0995401
$456$	39.1467	1.83321
$457$	27.8686	1.30364	0.651820	−	0.758374i	$-0.274006\pi$
$457$	27.8686	1.30364	0.651820	+	0.758374i	$0.274006\pi$
$458$	25.6242	1.19734
$459$	93.6090	4.36930
$460$	2.15986	0.100704
$461$	−41.5360	−1.93452	−0.967262	−	0.253780i	$-0.918326\pi$
$461$	−41.5360	−1.93452	−0.967262	+	0.253780i	$0.918326\pi$
$462$	−0.515800	−0.0239972
$463$	27.7716	1.29066	0.645328	−	0.763906i	$-0.276721\pi$
$463$	27.7716	1.29066	0.645328	+	0.763906i	$0.276721\pi$
$464$	4.31341	0.200245
$465$	46.2597	2.14524
$466$	16.2971	0.754947
$467$	−33.0499	−1.52937	−0.764684	−	0.644405i	$-0.777105\pi$
$467$	−33.0499	−1.52937	−0.764684	+	0.644405i	$0.777105\pi$
$468$	−3.45211	−0.159574
$469$	1.59329	0.0735712
$470$	−31.3197	−1.44467
$471$	57.5781	2.65306
$472$	9.59193	0.441504
$473$	0	0
$474$	31.5095	1.44728
$475$	3.33020	0.152800
$476$	−2.35354	−0.107874
$477$	−38.4920	−1.76243
$478$	−3.26306	−0.149249
$479$	−3.48410	−0.159192	−0.0795962	−	0.996827i	$-0.525363\pi$
$479$	−3.48410	−0.159192	−0.0795962	+	0.996827i	$0.525363\pi$
$480$	−14.9737	−0.683451
$481$	−1.02141	−0.0465724
$482$	−38.6470	−1.76032
$483$	−7.24392	−0.329610
$484$	−4.41389	−0.200631
$485$	−25.4993	−1.15786
$486$	48.6888	2.20857
$487$	31.0843	1.40856	0.704282	−	0.709920i	$-0.251269\pi$
$487$	31.0843	1.40856	0.704282	+	0.709920i	$0.251269\pi$
$488$	−5.05125	−0.228659
$489$	45.9225	2.07669
$490$	−20.1073	−0.908355
$491$	5.96490	0.269192	0.134596	−	0.990901i	$-0.457026\pi$
$491$	5.96490	0.269192	0.134596	+	0.990901i	$0.457026\pi$
$492$	2.24387	0.101162
$493$	−6.24508	−0.281264
$494$	8.92841	0.401708
$495$	1.81114	0.0814049
$496$	−32.1317	−1.44276
$497$	10.9349	0.490496
$498$	−11.6930	−0.523977
$499$	5.77735	0.258630	0.129315	−	0.991604i	$-0.458722\pi$
$499$	5.77735	0.258630	0.129315	+	0.991604i	$0.458722\pi$
$500$	−4.74249	−0.212090
$501$	−11.1223	−0.496909
$502$	40.4374	1.80481
$503$	−9.74952	−0.434710	−0.217355	−	0.976093i	$-0.569743\pi$
$503$	−9.74952	−0.434710	−0.217355	+	0.976093i	$0.569743\pi$
$504$	15.8299	0.705118
$505$	31.2512	1.39066
$506$	0.476028	0.0211620
$507$	−37.3749	−1.65988
$508$	7.45292	0.330670
$509$	−8.21780	−0.364248	−0.182124	−	0.983276i	$-0.558297\pi$
$509$	−8.21780	−0.364248	−0.182124	+	0.983276i	$0.558297\pi$
$510$	69.6126	3.08250
$511$	2.88133	0.127463
$512$	−12.5953	−0.556640
$513$	−68.4802	−3.02348
$514$	−1.36870	−0.0603707
$515$	11.1768	0.492508
$516$	0	0
$517$	−1.15472	−0.0507846
$518$	−1.17745	−0.0517342
$519$	35.3165	1.55022
$520$	6.03401	0.264609
$521$	36.6124	1.60402	0.802010	−	0.597311i	$-0.203764\pi$
$521$	36.6124	1.60402	0.802010	+	0.597311i	$0.203764\pi$
$522$	−10.5595	−0.462176
$523$	−16.0999	−0.704002	−0.352001	−	0.936000i	$-0.614499\pi$
$523$	−16.0999	−0.704002	−0.352001	+	0.936000i	$0.614499\pi$
$524$	−7.74056	−0.338148
$525$	1.89758	0.0828171
$526$	−4.52994	−0.197515
$527$	46.5212	2.02649
$528$	−1.77268	−0.0771461
$529$	−16.3146	−0.709332
$530$	−16.9138	−0.734688
$531$	−28.3972	−1.23233
$532$	1.72175	0.0746472
$533$	−2.03577	−0.0881788
$534$	−35.9142	−1.55416
$535$	8.69202	0.375789
$536$	4.52790	0.195575
$537$	33.0469	1.42608
$538$	−8.24952	−0.355662
$539$	−0.741334	−0.0319315
$540$	11.6345	0.500668
$541$	35.3518	1.51989	0.759946	−	0.649986i	$-0.225225\pi$
$541$	35.3518	1.51989	0.759946	+	0.649986i	$0.225225\pi$
$542$	27.8140	1.19471
$543$	−42.3016	−1.81533
$544$	−15.0583	−0.645618
$545$	−31.2093	−1.33686
$546$	5.08749	0.217724
$547$	8.12884	0.347564	0.173782	−	0.984784i	$-0.444401\pi$
$547$	8.12884	0.347564	0.173782	+	0.984784i	$0.444401\pi$
$548$	−4.18862	−0.178929
$549$	14.9544	0.638236
$550$	−0.124698	−0.00531713
$551$	4.56862	0.194630
$552$	−20.5862	−0.876206
$553$	−5.51272	−0.234425
$554$	38.1353	1.62021
$555$	5.82589	0.247295
$556$	5.08716	0.215744
$557$	−25.5405	−1.08219	−0.541093	−	0.840963i	$-0.681989\pi$
$557$	−25.5405	−1.08219	−0.541093	+	0.840963i	$0.681989\pi$
$558$	78.6603	3.32995
$559$	0	0
$560$	8.41193	0.355469
$561$	2.56654	0.108359
$562$	33.1297	1.39749
$563$	33.8216	1.42541	0.712706	−	0.701463i	$-0.247469\pi$
$563$	33.8216	1.42541	0.712706	+	0.701463i	$0.247469\pi$
$564$	−12.5537	−0.528604
$565$	−21.8752	−0.920295
$566$	−14.5753	−0.612645
$567$	−19.8474	−0.833514
$568$	31.0754	1.30389
$569$	−6.18173	−0.259151	−0.129576	−	0.991570i	$-0.541362\pi$
$569$	−6.18173	−0.259151	−0.129576	+	0.991570i	$0.541362\pi$
$570$	−50.9255	−2.13303
$571$	3.78544	0.158416	0.0792079	−	0.996858i	$-0.474761\pi$
$571$	3.78544	0.158416	0.0792079	+	0.996858i	$0.474761\pi$
$572$	−0.0559263	−0.00233839
$573$	−19.4328	−0.811816
$574$	−2.34677	−0.0979521
$575$	−1.75126	−0.0730326
$576$	42.6188	1.77578
$577$	−9.43111	−0.392622	−0.196311	−	0.980542i	$-0.562896\pi$
$577$	−9.43111	−0.392622	−0.196311	+	0.980542i	$0.562896\pi$
$578$	43.6600	1.81602
$579$	−75.9928	−3.15815
$580$	−0.776188	−0.0322295
$581$	2.04575	0.0848719
$582$	−61.0980	−2.53259
$583$	−0.623593	−0.0258266
$584$	8.18833	0.338836
$585$	−17.8639	−0.738580
$586$	−9.41663	−0.388997
$587$	0.925629	0.0382048	0.0191024	−	0.999818i	$-0.493919\pi$
$587$	0.925629	0.0382048	0.0191024	+	0.999818i	$0.493919\pi$
$588$	−8.05947	−0.332367
$589$	−34.0328	−1.40230
$590$	−12.4780	−0.513712
$591$	9.52495	0.391804
$592$	−4.04662	−0.166315
$593$	−18.9168	−0.776820	−0.388410	−	0.921487i	$-0.626976\pi$
$593$	−18.9168	−0.776820	−0.388410	+	0.921487i	$0.626976\pi$
$594$	2.56421	0.105211
$595$	−12.1790	−0.499292
$596$	2.30452	0.0943969
$597$	68.9361	2.82137
$598$	−4.69520	−0.192001
$599$	25.8820	1.05751	0.528754	−	0.848775i	$-0.322659\pi$
$599$	25.8820	1.05751	0.528754	+	0.848775i	$0.322659\pi$
$600$	5.39265	0.220154
$601$	−26.9085	−1.09762	−0.548809	−	0.835948i	$-0.684919\pi$
$601$	−26.9085	−1.09762	−0.548809	+	0.835948i	$0.684919\pi$
$602$	0	0
$603$	−13.4050	−0.545893
$604$	4.84741	0.197238
$605$	−22.8408	−0.928612
$606$	74.8800	3.04179
$607$	−38.4078	−1.55893	−0.779463	−	0.626448i	$-0.784508\pi$
$607$	−38.4078	−1.55893	−0.779463	+	0.626448i	$0.784508\pi$
$608$	11.0160	0.446757
$609$	2.60324	0.105489
$610$	6.57111	0.266056
$611$	11.3894	0.460764
$612$	19.8013	0.800420
$613$	−19.8510	−0.801775	−0.400887	−	0.916127i	$-0.631298\pi$
$613$	−19.8510	−0.801775	−0.400887	+	0.916127i	$0.631298\pi$
$614$	29.4667	1.18918
$615$	11.6115	0.468222
$616$	0.256453	0.0103328
$617$	22.7681	0.916611	0.458306	−	0.888795i	$-0.348457\pi$
$617$	22.7681	0.916611	0.458306	+	0.888795i	$0.348457\pi$
$618$	26.7804	1.07726
$619$	−13.0377	−0.524031	−0.262015	−	0.965064i	$-0.584387\pi$
$619$	−13.0377	−0.524031	−0.262015	+	0.965064i	$0.584387\pi$
$620$	5.78202	0.232212
$621$	36.0119	1.44511
$622$	13.8285	0.554472
$623$	6.28334	0.251737
$624$	17.4845	0.699940
$625$	−21.1547	−0.846188
$626$	39.5073	1.57903
$627$	−1.87757	−0.0749828
$628$	7.19671	0.287180
$629$	5.85881	0.233606
$630$	−20.5929	−0.820440
$631$	−36.4675	−1.45175	−0.725874	−	0.687827i	$-0.758565\pi$
$631$	−36.4675	−1.45175	−0.725874	+	0.687827i	$0.758565\pi$
$632$	−15.6664	−0.623175
$633$	4.54092	0.180485
$634$	0.351830	0.0139730
$635$	38.5671	1.53049
$636$	−6.77945	−0.268822
$637$	7.31200	0.289712
$638$	−0.171070	−0.00677272
$639$	−91.9995	−3.63944
$640$	28.0436	1.10852
$641$	29.0371	1.14690	0.573449	−	0.819241i	$-0.305605\pi$
$641$	29.0371	1.14690	0.573449	+	0.819241i	$0.305605\pi$
$642$	20.8267	0.821964
$643$	−37.8479	−1.49257	−0.746287	−	0.665624i	$-0.768165\pi$
$643$	−37.8479	−1.49257	−0.746287	+	0.665624i	$0.768165\pi$
$644$	−0.905420	−0.0356785
$645$	0	0
$646$	−51.2133	−2.01496
$647$	−22.5991	−0.888461	−0.444230	−	0.895913i	$-0.646523\pi$
$647$	−22.5991	−0.888461	−0.444230	+	0.895913i	$0.646523\pi$
$648$	−56.4036	−2.21574
$649$	−0.460051	−0.0180586
$650$	1.22993	0.0482418
$651$	−19.3922	−0.760041
$652$	5.73986	0.224790
$653$	−5.01382	−0.196206	−0.0981030	−	0.995176i	$-0.531277\pi$
$653$	−5.01382	−0.196206	−0.0981030	+	0.995176i	$0.531277\pi$
$654$	−74.7796	−2.92411
$655$	−40.0556	−1.56510
$656$	−8.06528	−0.314896
$657$	−24.2418	−0.945762
$658$	13.1293	0.511833
$659$	−29.6913	−1.15661	−0.578304	−	0.815821i	$-0.696285\pi$
$659$	−29.6913	−1.15661	−0.578304	+	0.815821i	$0.696285\pi$
$660$	0.318990	0.0124167
$661$	25.1099	0.976660	0.488330	−	0.872659i	$-0.337606\pi$
$661$	25.1099	0.976660	0.488330	+	0.872659i	$0.337606\pi$
$662$	−14.5432	−0.565239
$663$	−25.3146	−0.983136
$664$	5.81372	0.225616
$665$	8.90964	0.345501
$666$	9.90636	0.383864
$667$	−2.40251	−0.0930257
$668$	−1.39018	−0.0537879
$669$	−59.3324	−2.29392
$670$	−5.89029	−0.227562
$671$	0.242269	0.00935271
$672$	6.27700	0.242140
$673$	−14.6041	−0.562948	−0.281474	−	0.959569i	$-0.590823\pi$
$673$	−14.6041	−0.562948	−0.281474	+	0.959569i	$0.590823\pi$
$674$	−9.87003	−0.380179
$675$	−9.43347	−0.363095
$676$	−4.67150	−0.179673
$677$	31.6070	1.21476	0.607379	−	0.794412i	$-0.292221\pi$
$677$	31.6070	1.21476	0.607379	+	0.794412i	$0.292221\pi$
$678$	−52.4144	−2.01296
$679$	10.6894	0.410220
$680$	−34.6111	−1.32727
$681$	56.0596	2.14821
$682$	1.27434	0.0487971
$683$	−17.3172	−0.662626	−0.331313	−	0.943521i	$-0.607492\pi$
$683$	−17.3172	−0.662626	−0.331313	+	0.943521i	$0.607492\pi$
$684$	−14.4857	−0.553876
$685$	−21.6751	−0.828164
$686$	17.8841	0.682819
$687$	53.1488	2.02775
$688$	0	0
$689$	6.15068	0.234322
$690$	26.7803	1.01951
$691$	−12.5940	−0.479099	−0.239549	−	0.970884i	$-0.577000\pi$
$691$	−12.5940	−0.479099	−0.239549	+	0.970884i	$0.577000\pi$
$692$	4.41422	0.167804
$693$	−0.759237	−0.0288410
$694$	21.0904	0.800581
$695$	26.3249	0.998559
$696$	7.39805	0.280422
$697$	11.6771	0.442303
$698$	−33.4569	−1.26636
$699$	33.8028	1.27854
$700$	0.237179	0.00896452
$701$	44.2374	1.67082	0.835412	−	0.549625i	$-0.185229\pi$
$701$	44.2374	1.67082	0.835412	+	0.549625i	$0.185229\pi$
$702$	−25.2916	−0.954568
$703$	−4.28605	−0.161651
$704$	0.690450	0.0260223
$705$	−64.9622	−2.44662
$706$	4.02063	0.151318
$707$	−13.1006	−0.492698
$708$	−5.00148	−0.187967
$709$	−3.57636	−0.134313	−0.0671565	−	0.997742i	$-0.521393\pi$
$709$	−3.57636	−0.134313	−0.0671565	+	0.997742i	$0.521393\pi$
$710$	−40.4256	−1.51714
$711$	46.3807	1.73941
$712$	17.8564	0.669196
$713$	17.8969	0.670245
$714$	−29.1818	−1.09210
$715$	−0.289405	−0.0108231
$716$	4.13054	0.154366
$717$	−6.76814	−0.252761
$718$	33.6771	1.25682
$719$	−42.4273	−1.58227	−0.791136	−	0.611641i	$-0.790510\pi$
$719$	−42.4273	−1.58227	−0.791136	+	0.611641i	$0.790510\pi$
$720$	−70.7729	−2.63755
$721$	−4.68534	−0.174491
$722$	8.01981	0.298467
$723$	−80.1603	−2.98119
$724$	−5.28729	−0.196501
$725$	0.629350	0.0233735
$726$	−54.7283	−2.03116
$727$	−22.2087	−0.823673	−0.411837	−	0.911258i	$-0.635113\pi$
$727$	−22.2087	−0.823673	−0.411837	+	0.911258i	$0.635113\pi$
$728$	−2.52948	−0.0937486
$729$	32.6724	1.21009
$730$	−10.6521	−0.394252
$731$	0	0
$732$	2.63385	0.0973500
$733$	−2.51655	−0.0929509	−0.0464754	−	0.998919i	$-0.514799\pi$
$733$	−2.51655	−0.0929509	−0.0464754	+	0.998919i	$0.514799\pi$
$734$	−11.2873	−0.416623
$735$	−41.7059	−1.53834
$736$	−5.79299	−0.213533
$737$	−0.217168	−0.00799950
$738$	19.7443	0.726797
$739$	36.3733	1.33801	0.669006	−	0.743257i	$-0.266720\pi$
$739$	36.3733	1.33801	0.669006	+	0.743257i	$0.266720\pi$
$740$	0.728180	0.0267684
$741$	18.5190	0.680313
$742$	7.09031	0.260294
$743$	7.15992	0.262672	0.131336	−	0.991338i	$-0.458073\pi$
$743$	7.15992	0.262672	0.131336	+	0.991338i	$0.458073\pi$
$744$	−55.1099	−2.02043
$745$	11.9254	0.436911
$746$	−23.5594	−0.862570
$747$	−17.2117	−0.629742
$748$	0.320793	0.0117293
$749$	−3.64372	−0.133139
$750$	−58.8025	−2.14717
$751$	−33.6779	−1.22892	−0.614461	−	0.788947i	$-0.710627\pi$
$751$	−33.6779	−1.22892	−0.614461	+	0.788947i	$0.710627\pi$
$752$	45.1223	1.64544
$753$	83.8739	3.05653
$754$	1.68731	0.0614483
$755$	25.0842	0.912908
$756$	−4.87720	−0.177382
$757$	−11.1908	−0.406738	−0.203369	−	0.979102i	$-0.565189\pi$
$757$	−11.1908	−0.406738	−0.203369	+	0.979102i	$0.565189\pi$
$758$	31.8319	1.15619
$759$	0.987361	0.0358389
$760$	25.3199	0.918450
$761$	18.5968	0.674134	0.337067	−	0.941481i	$-0.390565\pi$
$761$	18.5968	0.674134	0.337067	+	0.941481i	$0.390565\pi$
$762$	92.4095	3.34764
$763$	13.0830	0.473637
$764$	−2.42891	−0.0878749
$765$	102.467	3.70470
$766$	12.5164	0.452235
$767$	4.53762	0.163844
$768$	29.8290	1.07636
$769$	−16.4974	−0.594912	−0.297456	−	0.954736i	$-0.596138\pi$
$769$	−16.4974	−0.594912	−0.297456	+	0.954736i	$0.596138\pi$
$770$	−0.333617	−0.0120227
$771$	−2.83891	−0.102241
$772$	−9.49837	−0.341854
$773$	−24.9319	−0.896740	−0.448370	−	0.893848i	$-0.647995\pi$
$773$	−24.9319	−0.896740	−0.448370	+	0.893848i	$0.647995\pi$
$774$	0	0
$775$	−4.68818	−0.168405
$776$	30.3777	1.09049
$777$	−2.44223	−0.0876145
$778$	−52.0485	−1.86603
$779$	−8.54248	−0.306066
$780$	−3.14629	−0.112655
$781$	−1.49045	−0.0533324
$782$	26.9317	0.963075
$783$	−12.9416	−0.462494
$784$	28.9686	1.03459
$785$	37.2413	1.32920
$786$	−95.9760	−3.42335
$787$	−18.6314	−0.664137	−0.332068	−	0.943255i	$-0.607746\pi$
$787$	−18.6314	−0.664137	−0.332068	+	0.943255i	$0.607746\pi$
$788$	1.19053	0.0424108
$789$	−9.39584	−0.334501
$790$	20.3802	0.725095
$791$	9.17013	0.326052
$792$	−2.15764	−0.0766685
$793$	−2.38958	−0.0848563
$794$	36.8288	1.30701
$795$	−35.0820	−1.24423
$796$	8.61635	0.305398
$797$	54.0840	1.91575	0.957876	−	0.287181i	$-0.0927181\pi$
$797$	54.0840	1.91575	0.957876	+	0.287181i	$0.0927181\pi$
$798$	21.3481	0.755715
$799$	−65.3293	−2.31119
$800$	1.51750	0.0536518
$801$	−52.8643	−1.86787
$802$	9.13028	0.322401
$803$	−0.392731	−0.0138592
$804$	−2.36097	−0.0832648
$805$	−4.68533	−0.165136
$806$	−12.5692	−0.442732
$807$	−17.1109	−0.602332
$808$	−37.2300	−1.30975
$809$	27.7176	0.974499	0.487249	−	0.873263i	$-0.338000\pi$
$809$	27.7176	0.974499	0.487249	+	0.873263i	$0.338000\pi$
$810$	73.3747	2.57813
$811$	−1.13784	−0.0399551	−0.0199776	−	0.999800i	$-0.506359\pi$
$811$	−1.13784	−0.0399551	−0.0199776	+	0.999800i	$0.506359\pi$
$812$	0.325380	0.0114186
$813$	57.6909	2.02331
$814$	0.160489	0.00562513
$815$	29.7024	1.04043
$816$	−100.291	−3.51089
$817$	0	0
$818$	−24.6080	−0.860400
$819$	7.48858	0.261672
$820$	1.45133	0.0506826
$821$	1.25444	0.0437803	0.0218902	−	0.999760i	$-0.493032\pi$
$821$	1.25444	0.0437803	0.0218902	+	0.999760i	$0.493032\pi$
$822$	−51.9351	−1.81144
$823$	41.3981	1.44305	0.721523	−	0.692391i	$-0.243443\pi$
$823$	41.3981	1.44305	0.721523	+	0.692391i	$0.243443\pi$
$824$	−13.3151	−0.463853
$825$	−0.258644	−0.00900482
$826$	5.23082	0.182004
$827$	25.2468	0.877918	0.438959	−	0.898507i	$-0.355347\pi$
$827$	25.2468	0.877918	0.438959	+	0.898507i	$0.355347\pi$
$828$	7.61766	0.264732
$829$	6.34472	0.220361	0.110181	−	0.993912i	$-0.464857\pi$
$829$	6.34472	0.220361	0.110181	+	0.993912i	$0.464857\pi$
$830$	−7.56300	−0.262516
$831$	79.0989	2.74391
$832$	−6.81011	−0.236098
$833$	−41.9416	−1.45319
$834$	63.0762	2.18415
$835$	−7.19388	−0.248954
$836$	−0.234678	−0.00811650
$837$	96.4050	3.33224
$838$	−54.4783	−1.88192
$839$	3.18926	0.110106	0.0550528	−	0.998483i	$-0.482467\pi$
$839$	3.18926	0.110106	0.0550528	+	0.998483i	$0.482467\pi$
$840$	14.4275	0.497797
$841$	−28.1366	−0.970228
$842$	9.14094	0.315018
$843$	68.7165	2.36672
$844$	0.567571	0.0195366
$845$	−24.1739	−0.831608
$846$	−110.462	−3.79776
$847$	9.57495	0.328999
$848$	24.3677	0.836791
$849$	−30.2316	−1.03754
$850$	−7.05488	−0.241980
$851$	2.25391	0.0772632
$852$	−16.2035	−0.555123
$853$	5.14257	0.176078	0.0880391	−	0.996117i	$-0.471940\pi$
$853$	5.14257	0.176078	0.0880391	+	0.996117i	$0.471940\pi$
$854$	−2.75463	−0.0942614
$855$	−74.9603	−2.56359
$856$	−10.3549	−0.353924
$857$	26.9541	0.920736	0.460368	−	0.887728i	$-0.347717\pi$
$857$	26.9541	0.920736	0.460368	+	0.887728i	$0.347717\pi$
$858$	−0.693435	−0.0236735
$859$	−31.7257	−1.08247	−0.541234	−	0.840872i	$-0.682043\pi$
$859$	−31.7257	−1.08247	−0.541234	+	0.840872i	$0.682043\pi$
$860$	0	0
$861$	−4.86758	−0.165887
$862$	27.3056	0.930031
$863$	−34.4558	−1.17289	−0.586445	−	0.809989i	$-0.699473\pi$
$863$	−34.4558	−1.17289	−0.586445	+	0.809989i	$0.699473\pi$
$864$	−31.2050	−1.06162
$865$	22.8426	0.776670
$866$	18.7834	0.638285
$867$	90.5581	3.07551
$868$	−2.42384	−0.0822705
$869$	0.751395	0.0254893
$870$	−9.62403	−0.326285
$871$	2.14200	0.0725788
$872$	37.1801	1.25908
$873$	−89.9339	−3.04380
$874$	−19.7020	−0.666431
$875$	10.2878	0.347790
$876$	−4.26961	−0.144257
$877$	33.8882	1.14432	0.572161	−	0.820141i	$-0.306105\pi$
$877$	33.8882	1.14432	0.572161	+	0.820141i	$0.306105\pi$
$878$	39.1548	1.32141
$879$	−19.5317	−0.658786
$880$	−1.14656	−0.0386506
$881$	−26.2950	−0.885901	−0.442950	−	0.896546i	$-0.646068\pi$
$881$	−26.2950	−0.885901	−0.442950	+	0.896546i	$0.646068\pi$
$882$	−70.9168	−2.38789
$883$	−5.79598	−0.195050	−0.0975252	−	0.995233i	$-0.531093\pi$
$883$	−5.79598	−0.195050	−0.0975252	+	0.995233i	$0.531093\pi$
$884$	−3.16407	−0.106419
$885$	−25.8815	−0.869997
$886$	−30.3466	−1.01951
$887$	11.4913	0.385840	0.192920	−	0.981214i	$-0.438204\pi$
$887$	11.4913	0.385840	0.192920	+	0.981214i	$0.438204\pi$
$888$	−6.94046	−0.232907
$889$	−16.1674	−0.542239
$890$	−23.2291	−0.778642
$891$	2.70525	0.0906291
$892$	−7.41598	−0.248305
$893$	47.7920	1.59930
$894$	28.5740	0.955657
$895$	21.3746	0.714473
$896$	−11.7559	−0.392738
$897$	−9.73863	−0.325164
$898$	2.23286	0.0745116
$899$	−6.43161	−0.214506
$900$	−1.99548	−0.0665160
$901$	−35.2803	−1.17536
$902$	0.319869	0.0106505
$903$	0	0
$904$	26.0602	0.866750
$905$	−27.3605	−0.909493
$906$	60.1035	1.99681
$907$	47.6023	1.58061	0.790304	−	0.612716i	$-0.209923\pi$
$907$	47.6023	1.58061	0.790304	+	0.612716i	$0.209923\pi$
$908$	7.00691	0.232532
$909$	110.220	3.65578
$910$	3.29056	0.109081
$911$	−30.6746	−1.01629	−0.508147	−	0.861271i	$-0.669669\pi$
$911$	−30.6746	−1.01629	−0.508147	+	0.861271i	$0.669669\pi$
$912$	73.3684	2.42947
$913$	−0.278839	−0.00922824
$914$	43.1899	1.42859
$915$	13.6296	0.450579
$916$	6.64309	0.219494
$917$	16.7914	0.554501
$918$	145.072	4.78810
$919$	26.9074	0.887594	0.443797	−	0.896127i	$-0.353631\pi$
$919$	26.9074	0.887594	0.443797	+	0.896127i	$0.353631\pi$
$920$	−13.3150	−0.438984
$921$	61.1188	2.01393
$922$	−64.3711	−2.11995
$923$	14.7007	0.483880
$924$	−0.133721	−0.00439911
$925$	−0.590423	−0.0194130
$926$	43.0395	1.41437
$927$	39.4197	1.29471
$928$	2.08183	0.0683393
$929$	−23.8762	−0.783354	−0.391677	−	0.920103i	$-0.628105\pi$
$929$	−23.8762	−0.783354	−0.391677	+	0.920103i	$0.628105\pi$
$930$	71.6918	2.35087
$931$	30.6826	1.00558
$932$	4.22503	0.138395
$933$	28.6826	0.939026
$934$	−51.2197	−1.67596
$935$	1.66003	0.0542887
$936$	21.2815	0.695607
$937$	−15.6416	−0.510990	−0.255495	−	0.966810i	$-0.582238\pi$
$937$	−15.6416	−0.510990	−0.255495	+	0.966810i	$0.582238\pi$
$938$	2.46923	0.0806231
$939$	81.9447	2.67416
$940$	−8.11964	−0.264834
$941$	52.3791	1.70751	0.853755	−	0.520675i	$-0.174320\pi$
$941$	52.3791	1.70751	0.853755	+	0.520675i	$0.174320\pi$
$942$	89.2327	2.90736
$943$	4.49225	0.146288
$944$	17.9771	0.585105
$945$	−25.2384	−0.821005
$946$	0	0
$947$	2.92049	0.0949031	0.0474516	−	0.998874i	$-0.484890\pi$
$947$	2.92049	0.0949031	0.0474516	+	0.998874i	$0.484890\pi$
$948$	8.16886	0.265312
$949$	3.87363	0.125743
$950$	5.16103	0.167446
$951$	0.729754	0.0236639
$952$	14.5091	0.470241
$953$	−4.41502	−0.143016	−0.0715082	−	0.997440i	$-0.522781\pi$
$953$	−4.41502	−0.143016	−0.0715082	+	0.997440i	$0.522781\pi$
$954$	−59.6536	−1.93136
$955$	−12.5690	−0.406724
$956$	−0.845951	−0.0273600
$957$	−0.354827	−0.0114699
$958$	−5.39954	−0.174451
$959$	9.08627	0.293411
$960$	38.8433	1.25366
$961$	16.9107	0.545507
$962$	−1.58295	−0.0510364
$963$	30.6561	0.987878
$964$	−10.0193	−0.322699
$965$	−49.1518	−1.58225
$966$	−11.2264	−0.361203
$967$	10.7653	0.346190	0.173095	−	0.984905i	$-0.444623\pi$
$967$	10.7653	0.346190	0.173095	+	0.984905i	$0.444623\pi$
$968$	27.2106	0.874583
$969$	−106.225	−3.41244
$970$	−39.5179	−1.26884
$971$	−3.67115	−0.117813	−0.0589064	−	0.998264i	$-0.518761\pi$
$971$	−3.67115	−0.117813	−0.0589064	+	0.998264i	$0.518761\pi$
$972$	12.6226	0.404870
$973$	−11.0355	−0.353781
$974$	48.1734	1.54358
$975$	2.55108	0.0816999
$976$	−9.46700	−0.303031
$977$	47.0516	1.50531	0.752657	−	0.658413i	$-0.228772\pi$
$977$	47.0516	1.50531	0.752657	+	0.658413i	$0.228772\pi$
$978$	71.1691	2.27574
$979$	−0.856432	−0.0273717
$980$	−5.21283	−0.166518
$981$	−110.073	−3.51435
$982$	9.24420	0.294994
$983$	32.3172	1.03076	0.515379	−	0.856962i	$-0.327651\pi$
$983$	32.3172	1.03076	0.515379	+	0.856962i	$0.327651\pi$
$984$	−13.8330	−0.440979
$985$	6.16069	0.196296
$986$	−9.67842	−0.308224
$987$	27.2323	0.866815
$988$	2.31470	0.0736403
$989$	0	0
$990$	2.80685	0.0892076
$991$	46.2290	1.46851	0.734256	−	0.678873i	$-0.237531\pi$
$991$	46.2290	1.46851	0.734256	+	0.678873i	$0.237531\pi$
$992$	−15.5080	−0.492381
$993$	−30.1651	−0.957261
$994$	16.9465	0.537511
$995$	44.5876	1.41352
$996$	−3.03143	−0.0960544
$997$	−30.7575	−0.974100	−0.487050	−	0.873374i	$-0.661927\pi$
$997$	−30.7575	−0.974100	−0.487050	+	0.873374i	$0.661927\pi$
$998$	8.95354	0.283420
$999$	12.1411	0.384128

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.r.1.14	yes	20
43.42	odd	2		1849.2.a.p.1.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.7	✓	20	43.42	odd	2
1849.2.a.r.1.14	yes	20	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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+1.54977 q^{2} +3.21447 q^{3} +0.401778 q^{4} +2.07911 q^{5} +4.98169 q^{6} -0.871568 q^{7} -2.47687 q^{8} +7.33285 q^{9} +O(q^{10})\)	\(q+1.54977 q^{2} +3.21447 q^{3} +0.401778 q^{4} +2.07911 q^{5} +4.98169 q^{6} -0.871568 q^{7} -2.47687 q^{8} +7.33285 q^{9} +3.22213 q^{10} +0.118796 q^{11} +1.29151 q^{12} -1.17173 q^{13} -1.35073 q^{14} +6.68324 q^{15} -4.64213 q^{16} +6.72101 q^{17} +11.3642 q^{18} -4.91679 q^{19} +0.835340 q^{20} -2.80163 q^{21} +0.184107 q^{22} +2.58561 q^{23} -7.96184 q^{24} -0.677311 q^{25} -1.81590 q^{26} +13.9278 q^{27} -0.350177 q^{28} -0.929188 q^{29} +10.3575 q^{30} +6.92176 q^{31} -2.24048 q^{32} +0.381868 q^{33} +10.4160 q^{34} -1.81208 q^{35} +2.94618 q^{36} +0.871716 q^{37} -7.61988 q^{38} -3.76648 q^{39} -5.14968 q^{40} +1.73741 q^{41} -4.34188 q^{42} +0.0477298 q^{44} +15.2458 q^{45} +4.00709 q^{46} -9.72016 q^{47} -14.9220 q^{48} -6.24037 q^{49} -1.04967 q^{50} +21.6045 q^{51} -0.470774 q^{52} -5.24926 q^{53} +21.5849 q^{54} +0.246991 q^{55} +2.15876 q^{56} -15.8049 q^{57} -1.44003 q^{58} -3.87260 q^{59} +2.68518 q^{60} +2.03937 q^{61} +10.7271 q^{62} -6.39107 q^{63} +5.81204 q^{64} -2.43614 q^{65} +0.591807 q^{66} -1.82807 q^{67} +2.70036 q^{68} +8.31136 q^{69} -2.80831 q^{70} -12.5462 q^{71} -18.1625 q^{72} -3.30592 q^{73} +1.35096 q^{74} -2.17720 q^{75} -1.97546 q^{76} -0.103539 q^{77} -5.83717 q^{78} +6.32506 q^{79} -9.65149 q^{80} +22.7721 q^{81} +2.69258 q^{82} -2.34720 q^{83} -1.12564 q^{84} +13.9737 q^{85} -2.98685 q^{87} -0.294244 q^{88} -7.20924 q^{89} +23.6274 q^{90} +1.02124 q^{91} +1.03884 q^{92} +22.2498 q^{93} -15.0640 q^{94} -10.2225 q^{95} -7.20196 q^{96} -12.2645 q^{97} -9.67112 q^{98} +0.871116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9} + 21 q^{11} - 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} + 9 q^{18} - 7 q^{19} - 17 q^{20} + 8 q^{21} + 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} + 58 q^{26} - 22 q^{27} - 28 q^{28} - 12 q^{29} - 24 q^{30} + 26 q^{31} + 3 q^{32} - 17 q^{33} + 54 q^{34} + 26 q^{35} + 14 q^{36} - 19 q^{37} + 5 q^{38} - 7 q^{39} - 16 q^{40} + 27 q^{41} - 52 q^{42} + 32 q^{44} + 75 q^{45} + 59 q^{46} + 45 q^{47} - 66 q^{48} + 20 q^{49} + 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} - 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} + 30 q^{61} + 72 q^{62} - 21 q^{63} - 7 q^{64} - 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} - 35 q^{69} - 80 q^{70} - 31 q^{71} - 15 q^{72} - 26 q^{73} + 87 q^{74} + 73 q^{75} - 68 q^{76} - 21 q^{77} - 50 q^{78} + 39 q^{79} - 60 q^{80} + 16 q^{81} + 45 q^{82} + 13 q^{83} + 31 q^{84} - 22 q^{85} + 61 q^{87} + 50 q^{88} - 4 q^{89} - 13 q^{90} - 25 q^{91} + 5 q^{92} + 67 q^{93} + 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} - 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9 + 21 * q^11 - 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 + 9 * q^18 - 7 * q^19 - 17 * q^20 + 8 * q^21 + 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 + 58 * q^26 - 22 * q^27 - 28 * q^28 - 12 * q^29 - 24 * q^30 + 26 * q^31 + 3 * q^32 - 17 * q^33 + 54 * q^34 + 26 * q^35 + 14 * q^36 - 19 * q^37 + 5 * q^38 - 7 * q^39 - 16 * q^40 + 27 * q^41 - 52 * q^42 + 32 * q^44 + 75 * q^45 + 59 * q^46 + 45 * q^47 - 66 * q^48 + 20 * q^49 + 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 - 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 + 30 * q^61 + 72 * q^62 - 21 * q^63 - 7 * q^64 - 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 - 35 * q^69 - 80 * q^70 - 31 * q^71 - 15 * q^72 - 26 * q^73 + 87 * q^74 + 73 * q^75 - 68 * q^76 - 21 * q^77 - 50 * q^78 + 39 * q^79 - 60 * q^80 + 16 * q^81 + 45 * q^82 + 13 * q^83 + 31 * q^84 - 22 * q^85 + 61 * q^87 + 50 * q^88 - 4 * q^89 - 13 * q^90 - 25 * q^91 + 5 * q^92 + 67 * q^93 + 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 - 35 * q^98 + 13 * q^99

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