LMFDB - Embedded newform 1849.2.a.p.1.17

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			1.17
Root			$-1.90109$ 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)

$f(q)$	$=$	$q+1.90109 q^{2} -2.82058 q^{3} +1.61414 q^{4} -3.37266 q^{5} -5.36217 q^{6} -1.68862 q^{7} -0.733548 q^{8} +4.95566 q^{9} +O(q^{10})$	\(q+1.90109 q^{2} -2.82058 q^{3} +1.61414 q^{4} -3.37266 q^{5} -5.36217 q^{6} -1.68862 q^{7} -0.733548 q^{8} +4.95566 q^{9} -6.41173 q^{10} -2.68837 q^{11} -4.55282 q^{12} -1.63521 q^{13} -3.21021 q^{14} +9.51285 q^{15} -4.62283 q^{16} +0.886205 q^{17} +9.42115 q^{18} -0.0521266 q^{19} -5.44396 q^{20} +4.76287 q^{21} -5.11083 q^{22} -6.14849 q^{23} +2.06903 q^{24} +6.37484 q^{25} -3.10868 q^{26} -5.51609 q^{27} -2.72567 q^{28} -2.73270 q^{29} +18.0848 q^{30} +8.21703 q^{31} -7.32132 q^{32} +7.58275 q^{33} +1.68476 q^{34} +5.69513 q^{35} +7.99914 q^{36} -0.691432 q^{37} -0.0990973 q^{38} +4.61223 q^{39} +2.47401 q^{40} +8.95440 q^{41} +9.05465 q^{42} -4.33941 q^{44} -16.7138 q^{45} -11.6888 q^{46} +9.18009 q^{47} +13.0390 q^{48} -4.14857 q^{49} +12.1191 q^{50} -2.49961 q^{51} -2.63946 q^{52} -4.71607 q^{53} -10.4866 q^{54} +9.06695 q^{55} +1.23868 q^{56} +0.147027 q^{57} -5.19511 q^{58} +11.3107 q^{59} +15.3551 q^{60} -9.69602 q^{61} +15.6213 q^{62} -8.36821 q^{63} -4.67282 q^{64} +5.51500 q^{65} +14.4155 q^{66} -5.11532 q^{67} +1.43046 q^{68} +17.3423 q^{69} +10.8270 q^{70} +8.68000 q^{71} -3.63521 q^{72} -7.27808 q^{73} -1.31447 q^{74} -17.9807 q^{75} -0.0841398 q^{76} +4.53962 q^{77} +8.76827 q^{78} +4.37248 q^{79} +15.5912 q^{80} +0.691576 q^{81} +17.0231 q^{82} +14.6813 q^{83} +7.68796 q^{84} -2.98887 q^{85} +7.70780 q^{87} +1.97205 q^{88} +8.65498 q^{89} -31.7743 q^{90} +2.76124 q^{91} -9.92454 q^{92} -23.1768 q^{93} +17.4522 q^{94} +0.175805 q^{95} +20.6503 q^{96} -12.9655 q^{97} -7.88681 q^{98} -13.3226 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.90109	1.34427	0.672137	−	0.740427i	$-0.265377\pi$
$2$	1.90109	1.34427	0.672137	+	0.740427i	$0.265377\pi$
$3$	−2.82058	−1.62846	−0.814231	−	0.580542i	$-0.802841\pi$
$3$	−2.82058	−1.62846	−0.814231	+	0.580542i	$0.802841\pi$
$4$	1.61414	0.807072
$5$	−3.37266	−1.50830	−0.754150	−	0.656703i	$-0.771951\pi$
$5$	−3.37266	−1.50830	−0.754150	+	0.656703i	$0.771951\pi$
$6$	−5.36217	−2.18910
$7$	−1.68862	−0.638237	−0.319118	−	0.947715i	$-0.603387\pi$
$7$	−1.68862	−0.638237	−0.319118	+	0.947715i	$0.603387\pi$
$8$	−0.733548	−0.259348
$9$	4.95566	1.65189
$10$	−6.41173	−2.02757
$11$	−2.68837	−0.810573	−0.405287	−	0.914190i	$-0.632828\pi$
$11$	−2.68837	−0.810573	−0.405287	+	0.914190i	$0.632828\pi$
$12$	−4.55282	−1.31428
$13$	−1.63521	−0.453525	−0.226763	−	0.973950i	$-0.572814\pi$
$13$	−1.63521	−0.453525	−0.226763	+	0.973950i	$0.572814\pi$
$14$	−3.21021	−0.857965
$15$	9.51285	2.45621
$16$	−4.62283	−1.15571
$17$	0.886205	0.214936	0.107468	−	0.994209i	$-0.465726\pi$
$17$	0.886205	0.214936	0.107468	+	0.994209i	$0.465726\pi$
$18$	9.42115	2.22059
$19$	−0.0521266	−0.0119587	−0.00597933	−	0.999982i	$-0.501903\pi$
$19$	−0.0521266	−0.0119587	−0.00597933	+	0.999982i	$0.501903\pi$
$20$	−5.44396	−1.21731
$21$	4.76287	1.03934
$22$	−5.11083	−1.08963
$23$	−6.14849	−1.28205	−0.641024	−	0.767521i	$-0.721490\pi$
$23$	−6.14849	−1.28205	−0.641024	+	0.767521i	$0.721490\pi$
$24$	2.06903	0.422339
$25$	6.37484	1.27497
$26$	−3.10868	−0.609662
$27$	−5.51609	−1.06157
$28$	−2.72567	−0.515103
$29$	−2.73270	−0.507450	−0.253725	−	0.967276i	$-0.581656\pi$
$29$	−2.73270	−0.507450	−0.253725	+	0.967276i	$0.581656\pi$
$30$	18.0848	3.30181
$31$	8.21703	1.47582	0.737911	−	0.674898i	$-0.235812\pi$
$31$	8.21703	1.47582	0.737911	+	0.674898i	$0.235812\pi$
$32$	−7.32132	−1.29424
$33$	7.58275	1.31999
$34$	1.68476	0.288933
$35$	5.69513	0.962652
$36$	7.99914	1.33319
$37$	−0.691432	−0.113671	−0.0568354	−	0.998384i	$-0.518101\pi$
$37$	−0.691432	−0.113671	−0.0568354	+	0.998384i	$0.518101\pi$
$38$	−0.0990973	−0.0160757
$39$	4.61223	0.738548
$40$	2.47401	0.391175
$41$	8.95440	1.39844	0.699221	−	0.714905i	$-0.253530\pi$
$41$	8.95440	1.39844	0.699221	+	0.714905i	$0.253530\pi$
$42$	9.05465	1.39716
$43$	0	0
$43$	0	0
$44$	−4.33941	−0.654191
$45$	−16.7138	−2.49154
$46$	−11.6888	−1.72342
$47$	9.18009	1.33905	0.669527	−	0.742788i	$-0.266497\pi$
$47$	9.18009	1.33905	0.669527	+	0.742788i	$0.266497\pi$
$48$	13.0390	1.88202
$49$	−4.14857	−0.592654
$50$	12.1191	1.71390
$51$	−2.49961	−0.350016
$52$	−2.63946	−0.366027
$53$	−4.71607	−0.647801	−0.323901	−	0.946091i	$-0.604994\pi$
$53$	−4.71607	−0.647801	−0.323901	+	0.946091i	$0.604994\pi$
$54$	−10.4866	−1.42704
$55$	9.06695	1.22259
$56$	1.23868	0.165526
$57$	0.147027	0.0194742
$58$	−5.19511	−0.682152
$59$	11.3107	1.47252	0.736262	−	0.676697i	$-0.236589\pi$
$59$	11.3107	1.47252	0.736262	+	0.676697i	$0.236589\pi$
$60$	15.3551	1.98234
$61$	−9.69602	−1.24145	−0.620724	−	0.784029i	$-0.713161\pi$
$61$	−9.69602	−1.24145	−0.620724	+	0.784029i	$0.713161\pi$
$62$	15.6213	1.98391
$63$	−8.36821	−1.05429
$64$	−4.67282	−0.584103
$65$	5.51500	0.684052
$66$	14.4155	1.77442
$67$	−5.11532	−0.624935	−0.312468	−	0.949928i	$-0.601156\pi$
$67$	−5.11532	−0.624935	−0.312468	+	0.949928i	$0.601156\pi$
$68$	1.43046	0.173469
$69$	17.3423	2.08777
$70$	10.8270	1.29407
$71$	8.68000	1.03013	0.515063	−	0.857152i	$-0.327769\pi$
$71$	8.68000	1.03013	0.515063	+	0.857152i	$0.327769\pi$
$72$	−3.63521	−0.428414
$73$	−7.27808	−0.851834	−0.425917	−	0.904762i	$-0.640048\pi$
$73$	−7.27808	−0.851834	−0.425917	+	0.904762i	$0.640048\pi$
$74$	−1.31447	−0.152805
$75$	−17.9807	−2.07623
$76$	−0.0841398	−0.00965150
$77$	4.53962	0.517338
$78$	8.76827	0.992811
$79$	4.37248	0.491942	0.245971	−	0.969277i	$-0.420893\pi$
$79$	4.37248	0.491942	0.245971	+	0.969277i	$0.420893\pi$
$80$	15.5912	1.74315
$81$	0.691576	0.0768418
$82$	17.0231	1.87989
$83$	14.6813	1.61148	0.805740	−	0.592269i	$-0.201768\pi$
$83$	14.6813	1.61148	0.805740	+	0.592269i	$0.201768\pi$
$84$	7.68796	0.838825
$85$	−2.98887	−0.324188
$86$	0	0
$87$	7.70780	0.826363
$88$	1.97205	0.210221
$89$	8.65498	0.917426	0.458713	−	0.888584i	$-0.348311\pi$
$89$	8.65498	0.917426	0.458713	+	0.888584i	$0.348311\pi$
$90$	−31.7743	−3.34931
$91$	2.76124	0.289457
$92$	−9.92454	−1.03470
$93$	−23.1768	−2.40332
$94$	17.4522	1.80006
$95$	0.175805	0.0180372
$96$	20.6503	2.10762
$97$	−12.9655	−1.31645	−0.658224	−	0.752822i	$-0.728692\pi$
$97$	−12.9655	−1.31645	−0.658224	+	0.752822i	$0.728692\pi$
$98$	−7.88681	−0.796689
$99$	−13.3226	−1.33897
$100$	10.2899	1.02899
$101$	−14.5701	−1.44978	−0.724892	−	0.688863i	$-0.758110\pi$
$101$	−14.5701	−1.44978	−0.724892	+	0.688863i	$0.758110\pi$
$102$	−4.75199	−0.470517
$103$	3.37055	0.332110	0.166055	−	0.986116i	$-0.446897\pi$
$103$	3.37055	0.332110	0.166055	+	0.986116i	$0.446897\pi$
$104$	1.19950	0.117621
$105$	−16.0636	−1.56764
$106$	−8.96566	−0.870822
$107$	−7.76033	−0.750219	−0.375110	−	0.926980i	$-0.622395\pi$
$107$	−7.76033	−0.750219	−0.375110	+	0.926980i	$0.622395\pi$
$108$	−8.90375	−0.856764
$109$	8.28134	0.793209	0.396604	−	0.917990i	$-0.370188\pi$
$109$	8.28134	0.793209	0.396604	+	0.917990i	$0.370188\pi$
$110$	17.2371	1.64349
$111$	1.95024	0.185108
$112$	7.80618	0.737615
$113$	2.07646	0.195337	0.0976683	−	0.995219i	$-0.468862\pi$
$113$	2.07646	0.195337	0.0976683	+	0.995219i	$0.468862\pi$
$114$	0.279512	0.0261787
$115$	20.7368	1.93371
$116$	−4.41097	−0.409548
$117$	−8.10353	−0.749172
$118$	21.5026	1.97947
$119$	−1.49646	−0.137180
$120$	−6.97813	−0.637014
$121$	−3.77268	−0.342971
$122$	−18.4330	−1.66885
$123$	−25.2566	−2.27731
$124$	13.2635	1.19109
$125$	−4.63685	−0.414733
$126$	−15.9087	−1.41726
$127$	−11.2912	−1.00193	−0.500964	−	0.865468i	$-0.667021\pi$
$127$	−11.2912	−1.00193	−0.500964	+	0.865468i	$0.667021\pi$
$128$	5.75917	0.509044
$129$	0	0
$130$	10.4845	0.919553
$131$	−11.7393	−1.02567	−0.512834	−	0.858488i	$-0.671404\pi$
$131$	−11.7393	−1.02567	−0.512834	+	0.858488i	$0.671404\pi$
$132$	12.2396	1.06532
$133$	0.0880218	0.00763246
$134$	−9.72468	−0.840084
$135$	18.6039	1.60117
$136$	−0.650074	−0.0557434
$137$	−2.98373	−0.254917	−0.127459	−	0.991844i	$-0.540682\pi$
$137$	−2.98373	−0.254917	−0.127459	+	0.991844i	$0.540682\pi$
$138$	32.9693	2.80653
$139$	−9.87911	−0.837935	−0.418968	−	0.908001i	$-0.637608\pi$
$139$	−9.87911	−0.837935	−0.418968	+	0.908001i	$0.637608\pi$
$140$	9.19275	0.776930
$141$	−25.8932	−2.18060
$142$	16.5015	1.38477
$143$	4.39604	0.367615
$144$	−22.9092	−1.90910
$145$	9.21647	0.765386
$146$	−13.8363	−1.14510
$147$	11.7014	0.965113
$148$	−1.11607	−0.0917404
$149$	23.8152	1.95102	0.975510	−	0.219955i	$-0.0705909\pi$
$149$	23.8152	1.95102	0.975510	+	0.219955i	$0.0705909\pi$
$150$	−34.1830	−2.79103
$151$	11.9409	0.971734	0.485867	−	0.874033i	$-0.338504\pi$
$151$	11.9409	0.971734	0.485867	+	0.874033i	$0.338504\pi$
$152$	0.0382374	0.00310146
$153$	4.39173	0.355050
$154$	8.63023	0.695444
$155$	−27.7132	−2.22598
$156$	7.44480	0.596061
$157$	4.48669	0.358077	0.179039	−	0.983842i	$-0.442701\pi$
$157$	4.48669	0.358077	0.179039	+	0.983842i	$0.442701\pi$
$158$	8.31248	0.661305
$159$	13.3020	1.05492
$160$	24.6923	1.95210
$161$	10.3824	0.818251
$162$	1.31475	0.103296
$163$	2.85903	0.223937	0.111968	−	0.993712i	$-0.464284\pi$
$163$	2.85903	0.223937	0.111968	+	0.993712i	$0.464284\pi$
$164$	14.4537	1.12864
$165$	−25.5740	−1.99094
$166$	27.9104	2.16627
$167$	−1.92534	−0.148988	−0.0744938	−	0.997221i	$-0.523734\pi$
$167$	−1.92534	−0.148988	−0.0744938	+	0.997221i	$0.523734\pi$
$168$	−3.49380	−0.269552
$169$	−10.3261	−0.794315
$170$	−5.68211	−0.435798
$171$	−0.258322	−0.0197543
$172$	0	0
$173$	2.98860	0.227219	0.113610	−	0.993525i	$-0.463759\pi$
$173$	2.98860	0.227219	0.113610	+	0.993525i	$0.463759\pi$
$174$	14.6532	1.11086
$175$	−10.7647	−0.813731
$176$	12.4279	0.936785
$177$	−31.9026	−2.39795
$178$	16.4539	1.23327
$179$	4.40660	0.329364	0.164682	−	0.986347i	$-0.447340\pi$
$179$	4.40660	0.329364	0.164682	+	0.986347i	$0.447340\pi$
$180$	−26.9784	−2.01085
$181$	−25.4863	−1.89438	−0.947191	−	0.320670i	$-0.896092\pi$
$181$	−25.4863	−1.89438	−0.947191	+	0.320670i	$0.896092\pi$
$182$	5.24936	0.389109
$183$	27.3484	2.02165
$184$	4.51021	0.332497
$185$	2.33197	0.171449
$186$	−44.0611	−3.23072
$187$	−2.38245	−0.174222
$188$	14.8180	1.08071
$189$	9.31455	0.677534
$190$	0.334222	0.0242470
$191$	−1.62590	−0.117646	−0.0588231	−	0.998268i	$-0.518735\pi$
$191$	−1.62590	−0.117646	−0.0588231	+	0.998268i	$0.518735\pi$
$192$	13.1801	0.951189
$193$	−6.53242	−0.470214	−0.235107	−	0.971969i	$-0.575544\pi$
$193$	−6.53242	−0.470214	−0.235107	+	0.971969i	$0.575544\pi$
$194$	−24.6486	−1.76967
$195$	−15.5555	−1.11395
$196$	−6.69639	−0.478314
$197$	11.6943	0.833183	0.416592	−	0.909094i	$-0.363225\pi$
$197$	11.6943	0.833183	0.416592	+	0.909094i	$0.363225\pi$
$198$	−25.3275	−1.79995
$199$	7.38810	0.523729	0.261864	−	0.965105i	$-0.415663\pi$
$199$	7.38810	0.523729	0.261864	+	0.965105i	$0.415663\pi$
$200$	−4.67625	−0.330661
$201$	14.4281	1.01768
$202$	−27.6992	−1.94891
$203$	4.61448	0.323873
$204$	−4.03473	−0.282488
$205$	−30.2002	−2.10927
$206$	6.40772	0.446447
$207$	−30.4698	−2.11780
$208$	7.55929	0.524142
$209$	0.140135	0.00969337
$210$	−30.5383	−2.10734
$211$	−14.1914	−0.976975	−0.488488	−	0.872571i	$-0.662451\pi$
$211$	−14.1914	−0.976975	−0.488488	+	0.872571i	$0.662451\pi$
$212$	−7.61241	−0.522822
$213$	−24.4826	−1.67752
$214$	−14.7531	−1.00850
$215$	0	0
$216$	4.04632	0.275317
$217$	−13.8754	−0.941924
$218$	15.7436	1.06629
$219$	20.5284	1.38718
$220$	14.6354	0.986715
$221$	−1.44913	−0.0974791
$222$	3.70758	0.248836
$223$	3.46847	0.232266	0.116133	−	0.993234i	$-0.462950\pi$
$223$	3.46847	0.232266	0.116133	+	0.993234i	$0.462950\pi$
$224$	12.3629	0.826031
$225$	31.5915	2.10610
$226$	3.94753	0.262586
$227$	−17.4816	−1.16029	−0.580147	−	0.814512i	$-0.697005\pi$
$227$	−17.4816	−1.16029	−0.580147	+	0.814512i	$0.697005\pi$
$228$	0.237323	0.0157171
$229$	13.9382	0.921063	0.460531	−	0.887643i	$-0.347659\pi$
$229$	13.9382	0.921063	0.460531	+	0.887643i	$0.347659\pi$
$230$	39.4224	2.59944
$231$	−12.8044	−0.842465
$232$	2.00457	0.131606
$233$	−23.2023	−1.52004	−0.760018	−	0.649902i	$-0.774810\pi$
$233$	−23.2023	−1.52004	−0.760018	+	0.649902i	$0.774810\pi$
$234$	−15.4055	−1.00709
$235$	−30.9613	−2.01969
$236$	18.2570	1.18843
$237$	−12.3329	−0.801109
$238$	−2.84491	−0.184408
$239$	11.5715	0.748497	0.374248	−	0.927328i	$-0.377901\pi$
$239$	11.5715	0.748497	0.374248	+	0.927328i	$0.377901\pi$
$240$	−43.9763	−2.83866
$241$	29.1016	1.87460	0.937298	−	0.348528i	$-0.113318\pi$
$241$	29.1016	1.87460	0.937298	+	0.348528i	$0.113318\pi$
$242$	−7.17221	−0.461047
$243$	14.5976	0.936438
$244$	−15.6508	−1.00194
$245$	13.9917	0.893899
$246$	−48.0150	−3.06133
$247$	0.0852378	0.00542355
$248$	−6.02759	−0.382752
$249$	−41.4097	−2.62423
$250$	−8.81507	−0.557514
$251$	13.1355	0.829104	0.414552	−	0.910026i	$-0.363938\pi$
$251$	13.1355	0.829104	0.414552	+	0.910026i	$0.363938\pi$
$252$	−13.5075	−0.850891
$253$	16.5294	1.03919
$254$	−21.4655	−1.34687
$255$	8.43034	0.527928
$256$	20.2944	1.26840
$257$	11.2265	0.700290	0.350145	−	0.936695i	$-0.386132\pi$
$257$	11.2265	0.700290	0.350145	+	0.936695i	$0.386132\pi$
$258$	0	0
$259$	1.16756	0.0725489
$260$	8.90200	0.552079
$261$	−13.5423	−0.838250
$262$	−22.3175	−1.37878
$263$	11.3232	0.698217	0.349109	−	0.937082i	$-0.386484\pi$
$263$	11.3232	0.698217	0.349109	+	0.937082i	$0.386484\pi$
$264$	−5.56231	−0.342337
$265$	15.9057	0.977078
$266$	0.167337	0.0102601
$267$	−24.4120	−1.49399
$268$	−8.25685	−0.504368
$269$	16.6704	1.01641	0.508205	−	0.861236i	$-0.330309\pi$
$269$	16.6704	1.01641	0.508205	+	0.861236i	$0.330309\pi$
$270$	35.3677	2.15241
$271$	−12.4955	−0.759046	−0.379523	−	0.925182i	$-0.623912\pi$
$271$	−12.4955	−0.759046	−0.379523	+	0.925182i	$0.623912\pi$
$272$	−4.09677	−0.248403
$273$	−7.78829	−0.471369
$274$	−5.67234	−0.342679
$275$	−17.1379	−1.03345
$276$	27.9929	1.68498
$277$	17.0842	1.02649	0.513246	−	0.858242i	$-0.328443\pi$
$277$	17.0842	1.02649	0.513246	+	0.858242i	$0.328443\pi$
$278$	−18.7811	−1.12641
$279$	40.7208	2.43789
$280$	−4.17765	−0.249662
$281$	10.8965	0.650029	0.325015	−	0.945709i	$-0.394631\pi$
$281$	10.8965	0.650029	0.325015	+	0.945709i	$0.394631\pi$
$282$	−49.2252	−2.93132
$283$	−12.9997	−0.772753	−0.386377	−	0.922341i	$-0.626274\pi$
$283$	−12.9997	−0.772753	−0.386377	+	0.922341i	$0.626274\pi$
$284$	14.0108	0.831385
$285$	−0.495872	−0.0293729
$286$	8.35727	0.494176
$287$	−15.1206	−0.892538
$288$	−36.2819	−2.13793
$289$	−16.2146	−0.953802
$290$	17.5213	1.02889
$291$	36.5702	2.14379
$292$	−11.7479	−0.687491
$293$	14.9815	0.875231	0.437616	−	0.899162i	$-0.355823\pi$
$293$	14.9815	0.875231	0.437616	+	0.899162i	$0.355823\pi$
$294$	22.2454	1.29738
$295$	−38.1470	−2.22101
$296$	0.507199	0.0294803
$297$	14.8293	0.860481
$298$	45.2749	2.62271
$299$	10.0541	0.581441
$300$	−29.0235	−1.67567
$301$	0	0
$302$	22.7007	1.30628
$303$	41.0962	2.36092
$304$	0.240972	0.0138207
$305$	32.7014	1.87247
$306$	8.34908	0.477285
$307$	−0.0407881	−0.00232790	−0.00116395	−	0.999999i	$-0.500370\pi$
$307$	−0.0407881	−0.00232790	−0.00116395	+	0.999999i	$0.500370\pi$
$308$	7.32760	0.417529
$309$	−9.50690	−0.540829
$310$	−52.6854	−2.99233
$311$	22.5862	1.28075	0.640374	−	0.768063i	$-0.278779\pi$
$311$	22.5862	1.28075	0.640374	+	0.768063i	$0.278779\pi$
$312$	−3.38329	−0.191541
$313$	23.6277	1.33552	0.667759	−	0.744377i	$-0.267254\pi$
$313$	23.6277	1.33552	0.667759	+	0.744377i	$0.267254\pi$
$314$	8.52961	0.481354
$315$	28.2231	1.59019
$316$	7.05781	0.397033
$317$	−22.3450	−1.25502	−0.627510	−	0.778609i	$-0.715926\pi$
$317$	−22.3450	−1.25502	−0.627510	+	0.778609i	$0.715926\pi$
$318$	25.2884	1.41810
$319$	7.34651	0.411325
$320$	15.7598	0.881002
$321$	21.8886	1.22170
$322$	19.7379	1.09995
$323$	−0.0461949	−0.00257035
$324$	1.11630	0.0620168
$325$	−10.4242	−0.578230
$326$	5.43528	0.301032
$327$	−23.3582	−1.29171
$328$	−6.56849	−0.362684
$329$	−15.5017	−0.854634
$330$	−48.6185	−2.67636
$331$	−10.1515	−0.557978	−0.278989	−	0.960294i	$-0.589999\pi$
$331$	−10.1515	−0.557978	−0.278989	+	0.960294i	$0.589999\pi$
$332$	23.6977	1.30058
$333$	−3.42650	−0.187771
$334$	−3.66025	−0.200280
$335$	17.2522	0.942590
$336$	−22.0179	−1.20118
$337$	1.25513	0.0683710	0.0341855	−	0.999416i	$-0.489116\pi$
$337$	1.25513	0.0683710	0.0341855	+	0.999416i	$0.489116\pi$
$338$	−19.6308	−1.06778
$339$	−5.85680	−0.318098
$340$	−4.82446	−0.261643
$341$	−22.0904	−1.19626
$342$	−0.491093	−0.0265552
$343$	18.8257	1.01649
$344$	0	0
$345$	−58.4896	−3.14898
$346$	5.68160	0.305445
$347$	15.8168	0.849088	0.424544	−	0.905407i	$-0.360434\pi$
$347$	15.8168	0.849088	0.424544	+	0.905407i	$0.360434\pi$
$348$	12.4415	0.666934
$349$	16.1432	0.864125	0.432062	−	0.901844i	$-0.357786\pi$
$349$	16.1432	0.864125	0.432062	+	0.901844i	$0.357786\pi$
$350$	−20.4646	−1.09388
$351$	9.01995	0.481449
$352$	19.6824	1.04907
$353$	10.6460	0.566630	0.283315	−	0.959027i	$-0.408566\pi$
$353$	10.6460	0.566630	0.283315	+	0.959027i	$0.408566\pi$
$354$	−60.6497	−3.22350
$355$	−29.2747	−1.55374
$356$	13.9704	0.740429
$357$	4.22088	0.223393
$358$	8.37734	0.442756
$359$	9.38632	0.495391	0.247696	−	0.968838i	$-0.420327\pi$
$359$	9.38632	0.495391	0.247696	+	0.968838i	$0.420327\pi$
$360$	12.2603	0.646177
$361$	−18.9973	−0.999857
$362$	−48.4518	−2.54657
$363$	10.6411	0.558515
$364$	4.45704	0.233612
$365$	24.5465	1.28482
$366$	51.9917	2.71765
$367$	31.7133	1.65542	0.827710	−	0.561156i	$-0.189643\pi$
$367$	31.7133	1.65542	0.827710	+	0.561156i	$0.189643\pi$
$368$	28.4234	1.48167
$369$	44.3750	2.31007
$370$	4.43328	0.230475
$371$	7.96362	0.413451
$372$	−37.4106	−1.93965
$373$	2.36635	0.122525	0.0612626	−	0.998122i	$-0.480487\pi$
$373$	2.36635	0.122525	0.0612626	+	0.998122i	$0.480487\pi$
$374$	−4.52924	−0.234202
$375$	13.0786	0.675376
$376$	−6.73404	−0.347282
$377$	4.46854	0.230141
$378$	17.7078	0.910791
$379$	−15.7225	−0.807611	−0.403805	−	0.914845i	$-0.632313\pi$
$379$	−15.7225	−0.807611	−0.403805	+	0.914845i	$0.632313\pi$
$380$	0.283775	0.0145573
$381$	31.8476	1.63160
$382$	−3.09099	−0.158149
$383$	7.97855	0.407685	0.203842	−	0.979004i	$-0.434657\pi$
$383$	7.97855	0.407685	0.203842	+	0.979004i	$0.434657\pi$
$384$	−16.2442	−0.828958
$385$	−15.3106	−0.780300
$386$	−12.4187	−0.632097
$387$	0	0
$388$	−20.9282	−1.06247
$389$	−13.8198	−0.700689	−0.350345	−	0.936621i	$-0.613936\pi$
$389$	−13.8198	−0.700689	−0.350345	+	0.936621i	$0.613936\pi$
$390$	−29.5724	−1.49746
$391$	−5.44882	−0.275559
$392$	3.04318	0.153704
$393$	33.1116	1.67026
$394$	22.2319	1.12003
$395$	−14.7469	−0.741996
$396$	−21.5046	−1.08065
$397$	−28.1747	−1.41405	−0.707023	−	0.707190i	$-0.749962\pi$
$397$	−28.1747	−1.41405	−0.707023	+	0.707190i	$0.749962\pi$
$398$	14.0454	0.704035
$399$	−0.248272	−0.0124292
$400$	−29.4698	−1.47349
$401$	−9.23181	−0.461015	−0.230507	−	0.973071i	$-0.574039\pi$
$401$	−9.23181	−0.461015	−0.230507	+	0.973071i	$0.574039\pi$
$402$	27.4292	1.36804
$403$	−13.4366	−0.669323
$404$	−23.5183	−1.17008
$405$	−2.33245	−0.115900
$406$	8.77255	0.435374
$407$	1.85882	0.0921384
$408$	1.83359	0.0907760
$409$	15.3188	0.757467	0.378733	−	0.925506i	$-0.376360\pi$
$409$	15.3188	0.757467	0.378733	+	0.925506i	$0.376360\pi$
$410$	−57.4132	−2.83544
$411$	8.41584	0.415123
$412$	5.44055	0.268037
$413$	−19.0994	−0.939819
$414$	−57.9258	−2.84690
$415$	−49.5150	−2.43059
$416$	11.9719	0.586970
$417$	27.8648	1.36454
$418$	0.266410	0.0130305
$419$	−0.596119	−0.0291223	−0.0145612	−	0.999894i	$-0.504635\pi$
$419$	−0.596119	−0.0291223	−0.0145612	+	0.999894i	$0.504635\pi$
$420$	−25.9289	−1.26520
$421$	30.8627	1.50415	0.752077	−	0.659075i	$-0.229052\pi$
$421$	30.8627	1.50415	0.752077	+	0.659075i	$0.229052\pi$
$422$	−26.9791	−1.31332
$423$	45.4934	2.21197
$424$	3.45946	0.168006
$425$	5.64941	0.274037
$426$	−46.5436	−2.25505
$427$	16.3729	0.792338
$428$	−12.5263	−0.605481
$429$	−12.3994	−0.598647
$430$	0	0
$431$	27.7925	1.33872	0.669359	−	0.742939i	$-0.266569\pi$
$431$	27.7925	1.33872	0.669359	+	0.742939i	$0.266569\pi$
$432$	25.4999	1.22687
$433$	23.4693	1.12786	0.563932	−	0.825821i	$-0.309288\pi$
$433$	23.4693	1.12786	0.563932	+	0.825821i	$0.309288\pi$
$434$	−26.3784	−1.26620
$435$	−25.9958	−1.24640
$436$	13.3673	0.640176
$437$	0.320500	0.0153316
$438$	39.0263	1.86475
$439$	27.1891	1.29767	0.648834	−	0.760930i	$-0.275257\pi$
$439$	27.1891	1.29767	0.648834	+	0.760930i	$0.275257\pi$
$440$	−6.65104	−0.317076
$441$	−20.5589	−0.978996
$442$	−2.75493	−0.131039
$443$	−12.5086	−0.594302	−0.297151	−	0.954831i	$-0.596036\pi$
$443$	−12.5086	−0.594302	−0.297151	+	0.954831i	$0.596036\pi$
$444$	3.14796	0.149396
$445$	−29.1903	−1.38375
$446$	6.59386	0.312229
$447$	−67.1727	−3.17716
$448$	7.89061	0.372796
$449$	−41.1059	−1.93991	−0.969954	−	0.243286i	$-0.921774\pi$
$449$	−41.1059	−1.93991	−0.969954	+	0.243286i	$0.921774\pi$
$450$	60.0583	2.83118
$451$	−24.0727	−1.13354
$452$	3.35170	0.157651
$453$	−33.6802	−1.58243
$454$	−33.2341	−1.55975
$455$	−9.31272	−0.436587
$456$	−0.107851	−0.00505061
$457$	−37.2883	−1.74427	−0.872137	−	0.489262i	$-0.837266\pi$
$457$	−37.2883	−1.74427	−0.872137	+	0.489262i	$0.837266\pi$
$458$	26.4978	1.23816
$459$	−4.88839	−0.228170
$460$	33.4721	1.56064
$461$	−6.56035	−0.305546	−0.152773	−	0.988261i	$-0.548820\pi$
$461$	−6.56035	−0.305546	−0.152773	+	0.988261i	$0.548820\pi$
$462$	−24.3422	−1.13250
$463$	17.3213	0.804988	0.402494	−	0.915423i	$-0.368144\pi$
$463$	17.3213	0.804988	0.402494	+	0.915423i	$0.368144\pi$
$464$	12.6328	0.586463
$465$	78.1674	3.62493
$466$	−44.1097	−2.04334
$467$	17.2171	0.796712	0.398356	−	0.917231i	$-0.369581\pi$
$467$	17.2171	0.796712	0.398356	+	0.917231i	$0.369581\pi$
$468$	−13.0803	−0.604635
$469$	8.63781	0.398857
$470$	−58.8603	−2.71502
$471$	−12.6551	−0.583115
$472$	−8.29692	−0.381897
$473$	0	0
$474$	−23.4460	−1.07691
$475$	−0.332298	−0.0152469
$476$	−2.41550	−0.110714
$477$	−23.3712	−1.07009
$478$	21.9984	1.00618
$479$	−37.9019	−1.73178	−0.865890	−	0.500234i	$-0.833247\pi$
$479$	−37.9019	−1.73178	−0.865890	+	0.500234i	$0.833247\pi$
$480$	−69.6466	−3.17892
$481$	1.13064	0.0515525
$482$	55.3247	2.51997
$483$	−29.2845	−1.33249
$484$	−6.08965	−0.276802
$485$	43.7283	1.98560
$486$	27.7514	1.25883
$487$	−3.26385	−0.147899	−0.0739497	−	0.997262i	$-0.523560\pi$
$487$	−3.26385	−0.147899	−0.0739497	+	0.997262i	$0.523560\pi$
$488$	7.11250	0.321968
$489$	−8.06413	−0.364673
$490$	26.5995	1.20164
$491$	34.1726	1.54219	0.771095	−	0.636721i	$-0.219710\pi$
$491$	34.1726	1.54219	0.771095	+	0.636721i	$0.219710\pi$
$492$	−40.7678	−1.83795
$493$	−2.42173	−0.109069
$494$	0.162045	0.00729074
$495$	44.9327	2.01957
$496$	−37.9859	−1.70562
$497$	−14.6572	−0.657464
$498$	−78.7236	−3.52769
$499$	36.7104	1.64338	0.821692	−	0.569932i	$-0.193030\pi$
$499$	36.7104	1.64338	0.821692	+	0.569932i	$0.193030\pi$
$500$	−7.48454	−0.334719
$501$	5.43058	0.242621
$502$	24.9717	1.11454
$503$	6.83246	0.304644	0.152322	−	0.988331i	$-0.451325\pi$
$503$	6.83246	0.304644	0.152322	+	0.988331i	$0.451325\pi$
$504$	6.13848	0.273430
$505$	49.1402	2.18671
$506$	31.4239	1.39696
$507$	29.1256	1.29351
$508$	−18.2256	−0.808628
$509$	−21.5010	−0.953014	−0.476507	−	0.879171i	$-0.658097\pi$
$509$	−21.5010	−0.953014	−0.476507	+	0.879171i	$0.658097\pi$
$510$	16.0268	0.709680
$511$	12.2899	0.543672
$512$	27.0630	1.19603
$513$	0.287535	0.0126950
$514$	21.3426	0.941382
$515$	−11.3677	−0.500922
$516$	0	0
$517$	−24.6795	−1.08540
$518$	2.21964	0.0975255
$519$	−8.42959	−0.370018
$520$	−4.04552	−0.177408
$521$	−3.09271	−0.135494	−0.0677471	−	0.997703i	$-0.521581\pi$
$521$	−3.09271	−0.135494	−0.0677471	+	0.997703i	$0.521581\pi$
$522$	−25.7452	−1.12684
$523$	−16.5023	−0.721594	−0.360797	−	0.932644i	$-0.617495\pi$
$523$	−16.5023	−0.721594	−0.360797	+	0.932644i	$0.617495\pi$
$524$	−18.9489	−0.827787
$525$	30.3625	1.32513
$526$	21.5264	0.938595
$527$	7.28198	0.317208
$528$	−35.0537	−1.52552
$529$	14.8039	0.643648
$530$	30.2381	1.31346
$531$	56.0518	2.43244
$532$	0.142080	0.00615994
$533$	−14.6423	−0.634229
$534$	−46.4095	−2.00834
$535$	26.1730	1.13156
$536$	3.75233	0.162076
$537$	−12.4291	−0.536357
$538$	31.6919	1.36633
$539$	11.1529	0.480389
$540$	30.0293	1.29226
$541$	−9.90010	−0.425639	−0.212819	−	0.977092i	$-0.568265\pi$
$541$	−9.90010	−0.425639	−0.212819	+	0.977092i	$0.568265\pi$
$542$	−23.7550	−1.02037
$543$	71.8861	3.08493
$544$	−6.48819	−0.278179
$545$	−27.9302	−1.19640
$546$	−14.8062	−0.633649
$547$	0.825233	0.0352844	0.0176422	−	0.999844i	$-0.494384\pi$
$547$	0.825233	0.0352844	0.0176422	+	0.999844i	$0.494384\pi$
$548$	−4.81617	−0.205736
$549$	−48.0501	−2.05073
$550$	−32.5807	−1.38925
$551$	0.142446	0.00606842
$552$	−12.7214	−0.541459
$553$	−7.38344	−0.313976
$554$	32.4786	1.37989
$555$	−6.57749	−0.279199
$556$	−15.9463	−0.676274
$557$	1.58630	0.0672137	0.0336069	−	0.999435i	$-0.489301\pi$
$557$	1.58630	0.0672137	0.0336069	+	0.999435i	$0.489301\pi$
$558$	77.4139	3.27719
$559$	0	0
$560$	−26.3276	−1.11254
$561$	6.71987	0.283713
$562$	20.7152	0.873817
$563$	−4.01671	−0.169284	−0.0846420	−	0.996411i	$-0.526975\pi$
$563$	−4.01671	−0.169284	−0.0846420	+	0.996411i	$0.526975\pi$
$564$	−41.7953	−1.75990
$565$	−7.00318	−0.294626
$566$	−24.7136	−1.03879
$567$	−1.16781	−0.0490433
$568$	−6.36720	−0.267162
$569$	14.8231	0.621415	0.310708	−	0.950506i	$-0.399434\pi$
$569$	14.8231	0.621415	0.310708	+	0.950506i	$0.399434\pi$
$570$	−0.942698	−0.0394853
$571$	6.04978	0.253176	0.126588	−	0.991955i	$-0.459597\pi$
$571$	6.04978	0.253176	0.126588	+	0.991955i	$0.459597\pi$
$572$	7.09584	0.296692
$573$	4.58598	0.191582
$574$	−28.7455	−1.19982
$575$	−39.1956	−1.63457
$576$	−23.1569	−0.964872
$577$	4.28375	0.178335	0.0891674	−	0.996017i	$-0.471579\pi$
$577$	4.28375	0.178335	0.0891674	+	0.996017i	$0.471579\pi$
$578$	−30.8255	−1.28217
$579$	18.4252	0.765726
$580$	14.8767	0.617722
$581$	−24.7911	−1.02851
$582$	69.5233	2.88183
$583$	12.6785	0.525090
$584$	5.33882	0.220922
$585$	27.3305	1.12998
$586$	28.4813	1.17655
$587$	47.3730	1.95529	0.977647	−	0.210251i	$-0.0674283\pi$
$587$	47.3730	1.95529	0.977647	+	0.210251i	$0.0674283\pi$
$588$	18.8877	0.778916
$589$	−0.428326	−0.0176489
$590$	−72.5209	−2.98564
$591$	−32.9846	−1.35681
$592$	3.19637	0.131370
$593$	−15.1212	−0.620955	−0.310478	−	0.950581i	$-0.600489\pi$
$593$	−15.1212	−0.620955	−0.310478	+	0.950581i	$0.600489\pi$
$594$	28.1918	1.15672
$595$	5.04705	0.206909
$596$	38.4412	1.57461
$597$	−20.8387	−0.852872
$598$	19.1137	0.781616
$599$	35.9933	1.47065	0.735323	−	0.677717i	$-0.237030\pi$
$599$	35.9933	1.47065	0.735323	+	0.677717i	$0.237030\pi$
$600$	13.1897	0.538468
$601$	−36.4362	−1.48626	−0.743132	−	0.669144i	$-0.766661\pi$
$601$	−36.4362	−1.48626	−0.743132	+	0.669144i	$0.766661\pi$
$602$	0	0
$603$	−25.3498	−1.03232
$604$	19.2743	0.784259
$605$	12.7240	0.517303
$606$	78.1276	3.17372
$607$	−13.6794	−0.555228	−0.277614	−	0.960693i	$-0.589544\pi$
$607$	−13.6794	−0.555228	−0.277614	+	0.960693i	$0.589544\pi$
$608$	0.381635	0.0154774
$609$	−13.0155	−0.527415
$610$	62.1682	2.51712
$611$	−15.0114	−0.607295
$612$	7.08888	0.286551
$613$	41.7078	1.68456	0.842282	−	0.539037i	$-0.181212\pi$
$613$	41.7078	1.68456	0.842282	+	0.539037i	$0.181212\pi$
$614$	−0.0775419	−0.00312933
$615$	85.1819	3.43487
$616$	−3.33003	−0.134171
$617$	−11.6939	−0.470777	−0.235389	−	0.971901i	$-0.575636\pi$
$617$	−11.6939	−0.470777	−0.235389	+	0.971901i	$0.575636\pi$
$618$	−18.0735	−0.727022
$619$	43.8396	1.76206	0.881031	−	0.473059i	$-0.156850\pi$
$619$	43.8396	1.76206	0.881031	+	0.473059i	$0.156850\pi$
$620$	−44.7332	−1.79653
$621$	33.9156	1.36099
$622$	42.9385	1.72168
$623$	−14.6149	−0.585535
$624$	−21.3216	−0.853545
$625$	−16.2357	−0.649426
$626$	44.9184	1.79530
$627$	−0.395263	−0.0157853
$628$	7.24217	0.288994
$629$	−0.612751	−0.0244320
$630$	53.6547	2.13765
$631$	8.09337	0.322192	0.161096	−	0.986939i	$-0.448497\pi$
$631$	8.09337	0.322192	0.161096	+	0.986939i	$0.448497\pi$
$632$	−3.20742	−0.127585
$633$	40.0279	1.59097
$634$	−42.4798	−1.68709
$635$	38.0812	1.51121
$636$	21.4714	0.851396
$637$	6.78378	0.268783
$638$	13.9664	0.552934
$639$	43.0151	1.70165
$640$	−19.4237	−0.767790
$641$	25.9425	1.02467	0.512333	−	0.858787i	$-0.328781\pi$
$641$	25.9425	1.02467	0.512333	+	0.858787i	$0.328781\pi$
$642$	41.6122	1.64230
$643$	31.3696	1.23710	0.618549	−	0.785747i	$-0.287721\pi$
$643$	31.3696	1.23710	0.618549	+	0.785747i	$0.287721\pi$
$644$	16.7587	0.660387
$645$	0	0
$646$	−0.0878206	−0.00345525
$647$	−12.1068	−0.475969	−0.237985	−	0.971269i	$-0.576487\pi$
$647$	−12.1068	−0.475969	−0.237985	+	0.971269i	$0.576487\pi$
$648$	−0.507304	−0.0199288
$649$	−30.4072	−1.19359
$650$	−19.8173	−0.777299
$651$	39.1367	1.53389
$652$	4.61489	0.180733
$653$	−26.0448	−1.01921	−0.509605	−	0.860408i	$-0.670209\pi$
$653$	−26.0448	−1.01921	−0.509605	+	0.860408i	$0.670209\pi$
$654$	−44.4060	−1.73641
$655$	39.5927	1.54701
$656$	−41.3947	−1.61619
$657$	−36.0677	−1.40713
$658$	−29.4700	−1.14886
$659$	23.7088	0.923563	0.461781	−	0.886994i	$-0.347210\pi$
$659$	23.7088	0.923563	0.461781	+	0.886994i	$0.347210\pi$
$660$	−41.2802	−1.60683
$661$	22.0660	0.858266	0.429133	−	0.903241i	$-0.358819\pi$
$661$	22.0660	0.858266	0.429133	+	0.903241i	$0.358819\pi$
$662$	−19.2990	−0.750076
$663$	4.08738	0.158741
$664$	−10.7694	−0.417935
$665$	−0.296868	−0.0115120
$666$	−6.51409	−0.252416
$667$	16.8020	0.650575
$668$	−3.10778	−0.120244
$669$	−9.78308	−0.378236
$670$	32.7980	1.26710
$671$	26.0665	1.00628
$672$	−34.8705	−1.34516
$673$	−32.0895	−1.23696	−0.618479	−	0.785802i	$-0.712251\pi$
$673$	−32.0895	−1.23696	−0.618479	+	0.785802i	$0.712251\pi$
$674$	2.38611	0.0919094
$675$	−35.1641	−1.35347
$676$	−16.6678	−0.641069
$677$	−4.14125	−0.159161	−0.0795806	−	0.996828i	$-0.525358\pi$
$677$	−4.14125	−0.159161	−0.0795806	+	0.996828i	$0.525358\pi$
$678$	−11.1343	−0.427611
$679$	21.8938	0.840206
$680$	2.19248	0.0840778
$681$	49.3082	1.88950
$682$	−41.9958	−1.60810
$683$	11.8890	0.454918	0.227459	−	0.973788i	$-0.426958\pi$
$683$	11.8890	0.454918	0.227459	+	0.973788i	$0.426958\pi$
$684$	−0.416968	−0.0159432
$685$	10.0631	0.384492
$686$	35.7893	1.36644
$687$	−39.3138	−1.49992
$688$	0	0
$689$	7.71175	0.293794
$690$	−111.194	−4.23309
$691$	−26.2811	−0.999781	−0.499891	−	0.866089i	$-0.666626\pi$
$691$	−26.2811	−0.999781	−0.499891	+	0.866089i	$0.666626\pi$
$692$	4.82403	0.183382
$693$	22.4968	0.854583
$694$	30.0691	1.14141
$695$	33.3189	1.26386
$696$	−5.65404	−0.214316
$697$	7.93544	0.300576
$698$	30.6896	1.16162
$699$	65.4440	2.47532
$700$	−17.3757	−0.656739
$701$	2.70781	0.102273	0.0511364	−	0.998692i	$-0.483716\pi$
$701$	2.70781	0.102273	0.0511364	+	0.998692i	$0.483716\pi$
$702$	17.1477	0.647200
$703$	0.0360420	0.00135935
$704$	12.5623	0.473458
$705$	87.3289	3.28899
$706$	20.2390	0.761706
$707$	24.6034	0.925306
$708$	−51.4954	−1.93532
$709$	39.0604	1.46694	0.733472	−	0.679719i	$-0.237898\pi$
$709$	39.0604	1.46694	0.733472	+	0.679719i	$0.237898\pi$
$710$	−55.6538	−2.08865
$711$	21.6685	0.812633
$712$	−6.34885	−0.237933
$713$	−50.5223	−1.89208
$714$	8.02428	0.300301
$715$	−14.8263	−0.554474
$716$	7.11288	0.265821
$717$	−32.6383	−1.21890
$718$	17.8442	0.665941
$719$	20.6795	0.771217	0.385608	−	0.922663i	$-0.373992\pi$
$719$	20.6795	0.771217	0.385608	+	0.922663i	$0.373992\pi$
$720$	77.2648	2.87949
$721$	−5.69157	−0.211965
$722$	−36.1155	−1.34408
$723$	−82.0832	−3.05271
$724$	−41.1385	−1.52890
$725$	−17.4205	−0.646982
$726$	20.2298	0.750797
$727$	−3.22067	−0.119448	−0.0597240	−	0.998215i	$-0.519022\pi$
$727$	−3.22067	−0.119448	−0.0597240	+	0.998215i	$0.519022\pi$
$728$	−2.02550	−0.0750701
$729$	−43.2484	−1.60179
$730$	46.6651	1.72715
$731$	0	0
$732$	44.1442	1.63162
$733$	−21.2006	−0.783063	−0.391531	−	0.920165i	$-0.628055\pi$
$733$	−21.2006	−0.783063	−0.391531	+	0.920165i	$0.628055\pi$
$734$	60.2898	2.22534
$735$	−39.4648	−1.45568
$736$	45.0150	1.65928
$737$	13.7519	0.506556
$738$	84.3608	3.10536
$739$	−47.6889	−1.75427	−0.877133	−	0.480248i	$-0.840547\pi$
$739$	−47.6889	−1.75427	−0.877133	+	0.480248i	$0.840547\pi$
$740$	3.76413	0.138372
$741$	−0.240420	−0.00883205
$742$	15.1396	0.555791
$743$	20.2305	0.742184	0.371092	−	0.928596i	$-0.378983\pi$
$743$	20.2305	0.742184	0.371092	+	0.928596i	$0.378983\pi$
$744$	17.0013	0.623297
$745$	−80.3207	−2.94272
$746$	4.49865	0.164707
$747$	72.7554	2.66198
$748$	−3.84561	−0.140609
$749$	13.1042	0.478818
$750$	24.8636	0.907890
$751$	1.07558	0.0392483	0.0196242	−	0.999807i	$-0.493753\pi$
$751$	1.07558	0.0392483	0.0196242	+	0.999807i	$0.493753\pi$
$752$	−42.4380	−1.54755
$753$	−37.0496	−1.35016
$754$	8.49509	0.309373
$755$	−40.2725	−1.46567
$756$	15.0350	0.546819
$757$	8.60082	0.312602	0.156301	−	0.987709i	$-0.450043\pi$
$757$	8.60082	0.312602	0.156301	+	0.987709i	$0.450043\pi$
$758$	−29.8899	−1.08565
$759$	−46.6224	−1.69229
$760$	−0.128962	−0.00467793
$761$	−1.67778	−0.0608195	−0.0304097	−	0.999538i	$-0.509681\pi$
$761$	−1.67778	−0.0608195	−0.0304097	+	0.999538i	$0.509681\pi$
$762$	60.5451	2.19332
$763$	−13.9840	−0.506255
$764$	−2.62444	−0.0949489
$765$	−14.8118	−0.535522
$766$	15.1679	0.548040
$767$	−18.4953	−0.667826
$768$	−57.2418	−2.06554
$769$	12.8315	0.462717	0.231358	−	0.972869i	$-0.425683\pi$
$769$	12.8315	0.462717	0.231358	+	0.972869i	$0.425683\pi$
$770$	−29.1068	−1.04894
$771$	−31.6652	−1.14040
$772$	−10.5443	−0.379497
$773$	−1.53985	−0.0553844	−0.0276922	−	0.999616i	$-0.508816\pi$
$773$	−1.53985	−0.0553844	−0.0276922	+	0.999616i	$0.508816\pi$
$774$	0	0
$775$	52.3822	1.88162
$776$	9.51083	0.341419
$777$	−3.29320	−0.118143
$778$	−26.2726	−0.941918
$779$	−0.466762	−0.0167235
$780$	−25.1088	−0.899039
$781$	−23.3350	−0.834993
$782$	−10.3587	−0.370426
$783$	15.0738	0.538694
$784$	19.1781	0.684934
$785$	−15.1321	−0.540087
$786$	62.9482	2.24529
$787$	−39.3350	−1.40214	−0.701070	−	0.713092i	$-0.747294\pi$
$787$	−39.3350	−1.40214	−0.701070	+	0.713092i	$0.747294\pi$
$788$	18.8762	0.672438
$789$	−31.9379	−1.13702
$790$	−28.0352	−0.997446
$791$	−3.50634	−0.124671
$792$	9.77279	0.347261
$793$	15.8550	0.563028
$794$	−53.5626	−1.90087
$795$	−44.8632	−1.59113
$796$	11.9255	0.422687
$797$	−9.05554	−0.320764	−0.160382	−	0.987055i	$-0.551273\pi$
$797$	−9.05554	−0.320764	−0.160382	+	0.987055i	$0.551273\pi$
$798$	−0.471988	−0.0167082
$799$	8.13545	0.287811
$800$	−46.6722	−1.65011
$801$	42.8911	1.51548
$802$	−17.5505	−0.619730
$803$	19.5661	0.690474
$804$	23.2891	0.821343
$805$	−35.0164	−1.23417
$806$	−25.5441	−0.899753
$807$	−47.0201	−1.65519
$808$	10.6879	0.375999
$809$	37.3018	1.31146	0.655730	−	0.754995i	$-0.272361\pi$
$809$	37.3018	1.31146	0.655730	+	0.754995i	$0.272361\pi$
$810$	−4.43420	−0.155802
$811$	9.71236	0.341047	0.170524	−	0.985354i	$-0.445454\pi$
$811$	9.71236	0.341047	0.170524	+	0.985354i	$0.445454\pi$
$812$	7.44844	0.261389
$813$	35.2445	1.23608
$814$	3.53379	0.123859
$815$	−9.64255	−0.337764
$816$	11.5553	0.404515
$817$	0	0
$818$	29.1225	1.01824
$819$	13.6838	0.478149
$820$	−48.7474	−1.70233
$821$	3.15605	0.110147	0.0550734	−	0.998482i	$-0.482461\pi$
$821$	3.15605	0.110147	0.0550734	+	0.998482i	$0.482461\pi$
$822$	15.9993	0.558039
$823$	13.5953	0.473903	0.236952	−	0.971521i	$-0.423852\pi$
$823$	13.5953	0.473903	0.236952	+	0.971521i	$0.423852\pi$
$824$	−2.47246	−0.0861323
$825$	48.3388	1.68294
$826$	−36.3096	−1.26337
$827$	−1.89128	−0.0657661	−0.0328831	−	0.999459i	$-0.510469\pi$
$827$	−1.89128	−0.0657661	−0.0328831	+	0.999459i	$0.510469\pi$
$828$	−49.1826	−1.70921
$829$	−10.0682	−0.349682	−0.174841	−	0.984597i	$-0.555941\pi$
$829$	−10.0682	−0.349682	−0.174841	+	0.984597i	$0.555941\pi$
$830$	−94.1324	−3.26738
$831$	−48.1874	−1.67160
$832$	7.64104	0.264905
$833$	−3.67649	−0.127383
$834$	52.9735	1.83432
$835$	6.49353	0.224718
$836$	0.226199	0.00782324
$837$	−45.3259	−1.56669
$838$	−1.13328	−0.0391483
$839$	−36.5404	−1.26151	−0.630757	−	0.775980i	$-0.717256\pi$
$839$	−36.5404	−1.26151	−0.630757	+	0.775980i	$0.717256\pi$
$840$	11.7834	0.406566
$841$	−21.5323	−0.742495
$842$	58.6727	2.02199
$843$	−30.7343	−1.05855
$844$	−22.9069	−0.788489
$845$	34.8264	1.19806
$846$	86.4871	2.97349
$847$	6.37061	0.218897
$848$	21.8016	0.748669
$849$	36.6667	1.25840
$850$	10.7400	0.368380
$851$	4.25126	0.145731
$852$	−39.5184	−1.35388
$853$	1.47142	0.0503805	0.0251903	−	0.999683i	$-0.491981\pi$
$853$	1.47142	0.0503805	0.0251903	+	0.999683i	$0.491981\pi$
$854$	31.1263	1.06512
$855$	0.871231	0.0297955
$856$	5.69258	0.194568
$857$	12.1732	0.415829	0.207915	−	0.978147i	$-0.433332\pi$
$857$	12.1732	0.415829	0.207915	+	0.978147i	$0.433332\pi$
$858$	−23.5723	−0.804746
$859$	11.0240	0.376135	0.188068	−	0.982156i	$-0.439778\pi$
$859$	11.0240	0.376135	0.188068	+	0.982156i	$0.439778\pi$
$860$	0	0
$861$	42.6487	1.45346
$862$	52.8360	1.79960
$863$	−26.8685	−0.914613	−0.457307	−	0.889309i	$-0.651186\pi$
$863$	−26.8685	−0.914613	−0.457307	+	0.889309i	$0.651186\pi$
$864$	40.3850	1.37393
$865$	−10.0795	−0.342715
$866$	44.6173	1.51616
$867$	45.7347	1.55323
$868$	−22.3969	−0.760200
$869$	−11.7548	−0.398755
$870$	−49.4203	−1.67551
$871$	8.36461	0.283424
$872$	−6.07476	−0.205717
$873$	−64.2527	−2.17462
$874$	0.609299	0.0206098
$875$	7.82986	0.264698
$876$	33.1357	1.11955
$877$	−1.81534	−0.0612995	−0.0306498	−	0.999530i	$-0.509758\pi$
$877$	−1.81534	−0.0612995	−0.0306498	+	0.999530i	$0.509758\pi$
$878$	51.6890	1.74442
$879$	−42.2566	−1.42528
$880$	−41.9149	−1.41295
$881$	17.6708	0.595345	0.297673	−	0.954668i	$-0.403790\pi$
$881$	17.6708	0.595345	0.297673	+	0.954668i	$0.403790\pi$
$882$	−39.0844	−1.31604
$883$	3.60305	0.121252	0.0606262	−	0.998161i	$-0.480690\pi$
$883$	3.60305	0.121252	0.0606262	+	0.998161i	$0.480690\pi$
$884$	−2.33910	−0.0786726
$885$	107.597	3.61682
$886$	−23.7800	−0.798904
$887$	0.358075	0.0120230	0.00601149	−	0.999982i	$-0.498086\pi$
$887$	0.358075	0.0120230	0.00601149	+	0.999982i	$0.498086\pi$
$888$	−1.43059	−0.0480076
$889$	19.0664	0.639468
$890$	−55.4934	−1.86014
$891$	−1.85921	−0.0622859
$892$	5.59860	0.187455
$893$	−0.478527	−0.0160133
$894$	−127.701	−4.27097
$895$	−14.8619	−0.496780
$896$	−9.72503	−0.324890
$897$	−28.3583	−0.946855
$898$	−78.1461	−2.60777
$899$	−22.4547	−0.748906
$900$	50.9932	1.69977
$901$	−4.17940	−0.139236
$902$	−45.7644	−1.52379
$903$	0	0
$904$	−1.52318	−0.0506602
$905$	85.9566	2.85730
$906$	−64.0290	−2.12722
$907$	−13.3010	−0.441654	−0.220827	−	0.975313i	$-0.570876\pi$
$907$	−13.3010	−0.441654	−0.220827	+	0.975313i	$0.570876\pi$
$908$	−28.2178	−0.936441
$909$	−72.2047	−2.39488
$910$	−17.7043	−0.586893
$911$	−28.4342	−0.942066	−0.471033	−	0.882116i	$-0.656119\pi$
$911$	−28.4342	−0.942066	−0.471033	+	0.882116i	$0.656119\pi$
$912$	−0.679681	−0.0225065
$913$	−39.4687	−1.30622
$914$	−70.8884	−2.34478
$915$	−92.2367	−3.04925
$916$	22.4983	0.743364
$917$	19.8232	0.654619
$918$	−9.29326	−0.306723
$919$	23.2436	0.766736	0.383368	−	0.923596i	$-0.374764\pi$
$919$	23.2436	0.766736	0.383368	+	0.923596i	$0.374764\pi$
$920$	−15.2114	−0.501505
$921$	0.115046	0.00379090
$922$	−12.4718	−0.410738
$923$	−14.1936	−0.467188
$924$	−20.6681	−0.679929
$925$	−4.40777	−0.144926
$926$	32.9293	1.08212
$927$	16.7033	0.548608
$928$	20.0070	0.656761
$929$	−20.6404	−0.677190	−0.338595	−	0.940932i	$-0.609952\pi$
$929$	−20.6404	−0.677190	−0.338595	+	0.940932i	$0.609952\pi$
$930$	148.603	4.87289
$931$	0.216251	0.00708734
$932$	−37.4519	−1.22678
$933$	−63.7063	−2.08565
$934$	32.7312	1.07100
$935$	8.03518	0.262778
$936$	5.94433	0.194297
$937$	0.590111	0.0192781	0.00963904	−	0.999954i	$-0.496932\pi$
$937$	0.590111	0.0192781	0.00963904	+	0.999954i	$0.496932\pi$
$938$	16.4213	0.536173
$939$	−66.6439	−2.17484
$940$	−49.9760	−1.63004
$941$	−8.30853	−0.270850	−0.135425	−	0.990788i	$-0.543240\pi$
$941$	−8.30853	−0.270850	−0.135425	+	0.990788i	$0.543240\pi$
$942$	−24.0584	−0.783866
$943$	−55.0560	−1.79287
$944$	−52.2873	−1.70181
$945$	−31.4148	−1.02192
$946$	0	0
$947$	−18.7590	−0.609586	−0.304793	−	0.952419i	$-0.598587\pi$
$947$	−18.7590	−0.609586	−0.304793	+	0.952419i	$0.598587\pi$
$948$	−19.9071	−0.646552
$949$	11.9012	0.386328
$950$	−0.631729	−0.0204960
$951$	63.0258	2.04375
$952$	1.09773	0.0355775
$953$	15.8962	0.514929	0.257464	−	0.966288i	$-0.417113\pi$
$953$	15.8962	0.514929	0.257464	+	0.966288i	$0.417113\pi$
$954$	−44.4308	−1.43850
$955$	5.48362	0.177446
$956$	18.6780	0.604091
$957$	−20.7214	−0.669827
$958$	−72.0548	−2.32799
$959$	5.03838	0.162698
$960$	−44.4519	−1.43468
$961$	36.5196	1.17805
$962$	2.14944	0.0693007
$963$	−38.4575	−1.23928
$964$	46.9741	1.51293
$965$	22.0316	0.709224
$966$	−55.6724	−1.79123
$967$	−55.5030	−1.78486	−0.892428	−	0.451191i	$-0.850999\pi$
$967$	−55.5030	−1.78486	−0.892428	+	0.451191i	$0.850999\pi$
$968$	2.76744	0.0889490
$969$	0.130296	0.00418572
$970$	83.1314	2.66919
$971$	−38.9315	−1.24937	−0.624685	−	0.780877i	$-0.714773\pi$
$971$	−38.9315	−1.24937	−0.624685	+	0.780877i	$0.714773\pi$
$972$	23.5626	0.755772
$973$	16.6820	0.534801
$974$	−6.20488	−0.198817
$975$	29.4022	0.941625
$976$	44.8230	1.43475
$977$	−44.8825	−1.43592	−0.717959	−	0.696085i	$-0.754923\pi$
$977$	−44.8825	−1.43592	−0.717959	+	0.696085i	$0.754923\pi$
$978$	−15.3306	−0.490220
$979$	−23.2678	−0.743641
$980$	22.5847	0.721441
$981$	41.0395	1.31029
$982$	64.9652	2.07312
$983$	−7.80558	−0.248959	−0.124480	−	0.992222i	$-0.539726\pi$
$983$	−7.80558	−0.248959	−0.124480	+	0.992222i	$0.539726\pi$
$984$	18.5269	0.590617
$985$	−39.4408	−1.25669
$986$	−4.60394	−0.146619
$987$	43.7236	1.39174
$988$	0.137586	0.00437720
$989$	0	0
$990$	85.4211	2.71486
$991$	−23.9015	−0.759256	−0.379628	−	0.925139i	$-0.623948\pi$
$991$	−23.9015	−0.759256	−0.379628	+	0.925139i	$0.623948\pi$
$992$	−60.1595	−1.91007
$993$	28.6332	0.908646
$994$	−27.8646	−0.883812
$995$	−24.9176	−0.789940
$996$	−66.8412	−2.11794
$997$	14.5624	0.461195	0.230597	−	0.973049i	$-0.425932\pi$
$997$	14.5624	0.461195	0.230597	+	0.973049i	$0.425932\pi$
$998$	69.7898	2.20916
$999$	3.81400	0.120670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.p.1.17	✓	20
43.42	odd	2		1849.2.a.r.1.4	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.17	✓	20	1.1	even	1	trivial
1849.2.a.r.1.4	yes	20	43.42	odd	2

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)

\(f(q)\)	\(=\)	\(q+1.90109 q^{2} -2.82058 q^{3} +1.61414 q^{4} -3.37266 q^{5} -5.36217 q^{6} -1.68862 q^{7} -0.733548 q^{8} +4.95566 q^{9} +O(q^{10})\)	\(q+1.90109 q^{2} -2.82058 q^{3} +1.61414 q^{4} -3.37266 q^{5} -5.36217 q^{6} -1.68862 q^{7} -0.733548 q^{8} +4.95566 q^{9} -6.41173 q^{10} -2.68837 q^{11} -4.55282 q^{12} -1.63521 q^{13} -3.21021 q^{14} +9.51285 q^{15} -4.62283 q^{16} +0.886205 q^{17} +9.42115 q^{18} -0.0521266 q^{19} -5.44396 q^{20} +4.76287 q^{21} -5.11083 q^{22} -6.14849 q^{23} +2.06903 q^{24} +6.37484 q^{25} -3.10868 q^{26} -5.51609 q^{27} -2.72567 q^{28} -2.73270 q^{29} +18.0848 q^{30} +8.21703 q^{31} -7.32132 q^{32} +7.58275 q^{33} +1.68476 q^{34} +5.69513 q^{35} +7.99914 q^{36} -0.691432 q^{37} -0.0990973 q^{38} +4.61223 q^{39} +2.47401 q^{40} +8.95440 q^{41} +9.05465 q^{42} -4.33941 q^{44} -16.7138 q^{45} -11.6888 q^{46} +9.18009 q^{47} +13.0390 q^{48} -4.14857 q^{49} +12.1191 q^{50} -2.49961 q^{51} -2.63946 q^{52} -4.71607 q^{53} -10.4866 q^{54} +9.06695 q^{55} +1.23868 q^{56} +0.147027 q^{57} -5.19511 q^{58} +11.3107 q^{59} +15.3551 q^{60} -9.69602 q^{61} +15.6213 q^{62} -8.36821 q^{63} -4.67282 q^{64} +5.51500 q^{65} +14.4155 q^{66} -5.11532 q^{67} +1.43046 q^{68} +17.3423 q^{69} +10.8270 q^{70} +8.68000 q^{71} -3.63521 q^{72} -7.27808 q^{73} -1.31447 q^{74} -17.9807 q^{75} -0.0841398 q^{76} +4.53962 q^{77} +8.76827 q^{78} +4.37248 q^{79} +15.5912 q^{80} +0.691576 q^{81} +17.0231 q^{82} +14.6813 q^{83} +7.68796 q^{84} -2.98887 q^{85} +7.70780 q^{87} +1.97205 q^{88} +8.65498 q^{89} -31.7743 q^{90} +2.76124 q^{91} -9.92454 q^{92} -23.1768 q^{93} +17.4522 q^{94} +0.175805 q^{95} +20.6503 q^{96} -12.9655 q^{97} -7.88681 q^{98} -13.3226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) \) 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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