LMFDB - Embedded newform 381.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [381,2,Mod(1,381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(381, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 381 = 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	381.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:	$3.04230031701$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
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			1.3
Root			$-1.15351$ of defining polynomial
Character	$\chi$	$=$	381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.484093 q^{2} -1.00000 q^{3} -1.76565 q^{4} +2.26452 q^{5} -0.484093 q^{6} +1.63760 q^{7} -1.82293 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.484093 q^{2} -1.00000 q^{3} -1.76565 q^{4} +2.26452 q^{5} -0.484093 q^{6} +1.63760 q^{7} -1.82293 q^{8} +1.00000 q^{9} +1.09624 q^{10} +0.792751 q^{11} +1.76565 q^{12} +0.637602 q^{13} +0.792751 q^{14} -2.26452 q^{15} +2.64884 q^{16} -0.222022 q^{17} +0.484093 q^{18} +7.75333 q^{19} -3.99836 q^{20} -1.63760 q^{21} +0.383765 q^{22} +6.98768 q^{23} +1.82293 q^{24} +0.128052 q^{25} +0.308658 q^{26} -1.00000 q^{27} -2.89144 q^{28} +3.66588 q^{29} -1.09624 q^{30} -1.51182 q^{31} +4.92814 q^{32} -0.792751 q^{33} -0.107479 q^{34} +3.70838 q^{35} -1.76565 q^{36} -2.52306 q^{37} +3.75333 q^{38} -0.637602 q^{39} -4.12805 q^{40} -3.38268 q^{41} -0.792751 q^{42} -4.81549 q^{43} -1.39972 q^{44} +2.26452 q^{45} +3.38268 q^{46} +5.25220 q^{47} -2.64884 q^{48} -4.31826 q^{49} +0.0619892 q^{50} +0.222022 q^{51} -1.12578 q^{52} -8.50194 q^{53} -0.484093 q^{54} +1.79520 q^{55} -2.98523 q^{56} -7.75333 q^{57} +1.77463 q^{58} +6.54363 q^{59} +3.99836 q^{60} -10.8248 q^{61} -0.731860 q^{62} +1.63760 q^{63} -2.91201 q^{64} +1.44386 q^{65} -0.383765 q^{66} -14.9099 q^{67} +0.392015 q^{68} -6.98768 q^{69} +1.79520 q^{70} -2.54608 q^{71} -1.82293 q^{72} -0.150431 q^{73} -1.22139 q^{74} -0.128052 q^{75} -13.6897 q^{76} +1.29821 q^{77} -0.308658 q^{78} +4.22202 q^{79} +5.99836 q^{80} +1.00000 q^{81} -1.63753 q^{82} +3.93449 q^{83} +2.89144 q^{84} -0.502774 q^{85} -2.33114 q^{86} -3.66588 q^{87} -1.44513 q^{88} +12.1762 q^{89} +1.09624 q^{90} +1.04414 q^{91} -12.3378 q^{92} +1.51182 q^{93} +2.54255 q^{94} +17.5576 q^{95} -4.92814 q^{96} -5.97045 q^{97} -2.09044 q^{98} +0.792751 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + 2 * q^2 - 5 * q^3 + 6 * q^4 + q^5 - 2 * q^6 + 6 * q^8 + 5 * q^9	\( 5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9} - 2 q^{10} + 14 q^{11} - 6 q^{12} - 5 q^{13} + 14 q^{14} - q^{15} + 8 q^{16} + 4 q^{17} + 2 q^{18} + 4 q^{19} + 6 q^{20} - 2 q^{22} + 15 q^{23} - 6 q^{24} - 6 q^{25} + 12 q^{26} - 5 q^{27} - 2 q^{28} + 9 q^{29} + 2 q^{30} + 3 q^{31} + 14 q^{32} - 14 q^{33} + 4 q^{34} + 4 q^{35} + 6 q^{36} - 5 q^{37} - 16 q^{38} + 5 q^{39} - 14 q^{40} + 4 q^{41} - 14 q^{42} + 10 q^{43} + 18 q^{44} + q^{45} - 4 q^{46} - 4 q^{47} - 8 q^{48} - 9 q^{49} - 24 q^{50} - 4 q^{51} - 8 q^{52} + 3 q^{53} - 2 q^{54} + 4 q^{55} - 10 q^{56} - 4 q^{57} + 6 q^{58} + 23 q^{59} - 6 q^{60} - 15 q^{61} - 24 q^{62} + 3 q^{65} + 2 q^{66} + 18 q^{67} - 24 q^{68} - 15 q^{69} + 4 q^{70} + 12 q^{71} + 6 q^{72} - 43 q^{73} - 36 q^{74} + 6 q^{75} - 32 q^{76} + 4 q^{77} - 12 q^{78} + 16 q^{79} + 4 q^{80} + 5 q^{81} - 44 q^{82} + 11 q^{83} + 2 q^{84} - 24 q^{85} - 28 q^{86} - 9 q^{87} - 14 q^{88} + 9 q^{89} - 2 q^{90} + 26 q^{91} + 14 q^{92} - 3 q^{93} - 14 q^{94} + 16 q^{95} - 14 q^{96} - 20 q^{97} - 10 q^{98} + 14 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 - 5 * q^3 + 6 * q^4 + q^5 - 2 * q^6 + 6 * q^8 + 5 * q^9 - 2 * q^10 + 14 * q^11 - 6 * q^12 - 5 * q^13 + 14 * q^14 - q^15 + 8 * q^16 + 4 * q^17 + 2 * q^18 + 4 * q^19 + 6 * q^20 - 2 * q^22 + 15 * q^23 - 6 * q^24 - 6 * q^25 + 12 * q^26 - 5 * q^27 - 2 * q^28 + 9 * q^29 + 2 * q^30 + 3 * q^31 + 14 * q^32 - 14 * q^33 + 4 * q^34 + 4 * q^35 + 6 * q^36 - 5 * q^37 - 16 * q^38 + 5 * q^39 - 14 * q^40 + 4 * q^41 - 14 * q^42 + 10 * q^43 + 18 * q^44 + q^45 - 4 * q^46 - 4 * q^47 - 8 * q^48 - 9 * q^49 - 24 * q^50 - 4 * q^51 - 8 * q^52 + 3 * q^53 - 2 * q^54 + 4 * q^55 - 10 * q^56 - 4 * q^57 + 6 * q^58 + 23 * q^59 - 6 * q^60 - 15 * q^61 - 24 * q^62 + 3 * q^65 + 2 * q^66 + 18 * q^67 - 24 * q^68 - 15 * q^69 + 4 * q^70 + 12 * q^71 + 6 * q^72 - 43 * q^73 - 36 * q^74 + 6 * q^75 - 32 * q^76 + 4 * q^77 - 12 * q^78 + 16 * q^79 + 4 * q^80 + 5 * q^81 - 44 * q^82 + 11 * q^83 + 2 * q^84 - 24 * q^85 - 28 * q^86 - 9 * q^87 - 14 * q^88 + 9 * q^89 - 2 * q^90 + 26 * q^91 + 14 * q^92 - 3 * q^93 - 14 * q^94 + 16 * q^95 - 14 * q^96 - 20 * q^97 - 10 * q^98 + 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.484093	0.342305	0.171153	−	0.985245i	$-0.445251\pi$
$2$	0.484093	0.342305	0.171153	+	0.985245i	$0.445251\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.76565	−0.882827
$5$	2.26452	1.01272	0.506362	−	0.862321i	$-0.330990\pi$
$5$	2.26452	1.01272	0.506362	+	0.862321i	$0.330990\pi$
$6$	−0.484093	−0.197630
$7$	1.63760	0.618955	0.309478	−	0.950907i	$-0.399846\pi$
$7$	1.63760	0.618955	0.309478	+	0.950907i	$0.399846\pi$
$8$	−1.82293	−0.644502
$9$	1.00000	0.333333
$10$	1.09624	0.346661
$11$	0.792751	0.239023	0.119512	−	0.992833i	$-0.461867\pi$
$11$	0.792751	0.239023	0.119512	+	0.992833i	$0.461867\pi$
$12$	1.76565	0.509700
$13$	0.637602	0.176839	0.0884195	−	0.996083i	$-0.471818\pi$
$13$	0.637602	0.176839	0.0884195	+	0.996083i	$0.471818\pi$
$14$	0.792751	0.211872
$15$	−2.26452	−0.584697
$16$	2.64884	0.662211
$17$	−0.222022	−0.0538483	−0.0269242	−	0.999637i	$-0.508571\pi$
$17$	−0.222022	−0.0538483	−0.0269242	+	0.999637i	$0.508571\pi$
$18$	0.484093	0.114102
$19$	7.75333	1.77874	0.889368	−	0.457192i	$-0.151145\pi$
$19$	7.75333	1.77874	0.889368	+	0.457192i	$0.151145\pi$
$20$	−3.99836	−0.894060
$21$	−1.63760	−0.357354
$22$	0.383765	0.0818190
$23$	6.98768	1.45703	0.728516	−	0.685029i	$-0.240211\pi$
$23$	6.98768	1.45703	0.728516	+	0.685029i	$0.240211\pi$
$24$	1.82293	0.372103
$25$	0.128052	0.0256105
$26$	0.308658	0.0605329
$27$	−1.00000	−0.192450
$28$	−2.89144	−0.546431
$29$	3.66588	0.680738	0.340369	−	0.940292i	$-0.389448\pi$
$29$	3.66588	0.680738	0.340369	+	0.940292i	$0.389448\pi$
$30$	−1.09624	−0.200145
$31$	−1.51182	−0.271530	−0.135765	−	0.990741i	$-0.543349\pi$
$31$	−1.51182	−0.271530	−0.135765	+	0.990741i	$0.543349\pi$
$32$	4.92814	0.871180
$33$	−0.792751	−0.138000
$34$	−0.107479	−0.0184326
$35$	3.70838	0.626831
$36$	−1.76565	−0.294276
$37$	−2.52306	−0.414788	−0.207394	−	0.978257i	$-0.566498\pi$
$37$	−2.52306	−0.414788	−0.207394	+	0.978257i	$0.566498\pi$
$38$	3.75333	0.608871
$39$	−0.637602	−0.102098
$40$	−4.12805	−0.652702
$41$	−3.38268	−0.528286	−0.264143	−	0.964483i	$-0.585089\pi$
$41$	−3.38268	−0.528286	−0.264143	+	0.964483i	$0.585089\pi$
$42$	−0.792751	−0.122324
$43$	−4.81549	−0.734355	−0.367177	−	0.930151i	$-0.619676\pi$
$43$	−4.81549	−0.734355	−0.367177	+	0.930151i	$0.619676\pi$
$44$	−1.39972	−0.211016
$45$	2.26452	0.337575
$46$	3.38268	0.498749
$47$	5.25220	0.766112	0.383056	−	0.923725i	$-0.374872\pi$
$47$	5.25220	0.766112	0.383056	+	0.923725i	$0.374872\pi$
$48$	−2.64884	−0.382328
$49$	−4.31826	−0.616894
$50$	0.0619892	0.00876660
$51$	0.222022	0.0310893
$52$	−1.12578	−0.156118
$53$	−8.50194	−1.16783	−0.583916	−	0.811814i	$-0.698480\pi$
$53$	−8.50194	−1.16783	−0.583916	+	0.811814i	$0.698480\pi$
$54$	−0.484093	−0.0658767
$55$	1.79520	0.242065
$56$	−2.98523	−0.398918
$57$	−7.75333	−1.02695
$58$	1.77463	0.233020
$59$	6.54363	0.851908	0.425954	−	0.904745i	$-0.359938\pi$
$59$	6.54363	0.851908	0.425954	+	0.904745i	$0.359938\pi$
$60$	3.99836	0.516186
$61$	−10.8248	−1.38598	−0.692988	−	0.720949i	$-0.743706\pi$
$61$	−10.8248	−1.38598	−0.692988	+	0.720949i	$0.743706\pi$
$62$	−0.731860	−0.0929463
$63$	1.63760	0.206318
$64$	−2.91201	−0.364001
$65$	1.44386	0.179089
$66$	−0.383765	−0.0472382
$67$	−14.9099	−1.82154	−0.910768	−	0.412918i	$-0.864510\pi$
$67$	−14.9099	−1.82154	−0.910768	+	0.412918i	$0.864510\pi$
$68$	0.392015	0.0475387
$69$	−6.98768	−0.841217
$70$	1.79520	0.214568
$71$	−2.54608	−0.302164	−0.151082	−	0.988521i	$-0.548276\pi$
$71$	−2.54608	−0.302164	−0.151082	+	0.988521i	$0.548276\pi$
$72$	−1.82293	−0.214834
$73$	−0.150431	−0.0176067	−0.00880334	−	0.999961i	$-0.502802\pi$
$73$	−0.150431	−0.0176067	−0.00880334	+	0.999961i	$0.502802\pi$
$74$	−1.22139	−0.141984
$75$	−0.128052	−0.0147862
$76$	−13.6897	−1.57032
$77$	1.29821	0.147945
$78$	−0.308658	−0.0349487
$79$	4.22202	0.475015	0.237507	−	0.971386i	$-0.423670\pi$
$79$	4.22202	0.475015	0.237507	+	0.971386i	$0.423670\pi$
$80$	5.99836	0.670637
$81$	1.00000	0.111111
$82$	−1.63753	−0.180835
$83$	3.93449	0.431867	0.215933	−	0.976408i	$-0.430721\pi$
$83$	3.93449	0.431867	0.215933	+	0.976408i	$0.430721\pi$
$84$	2.89144	0.315482
$85$	−0.502774	−0.0545335
$86$	−2.33114	−0.251373
$87$	−3.66588	−0.393024
$88$	−1.44513	−0.154051
$89$	12.1762	1.29068	0.645340	−	0.763896i	$-0.276716\pi$
$89$	12.1762	1.29068	0.645340	+	0.763896i	$0.276716\pi$
$90$	1.09624	0.115554
$91$	1.04414	0.109455
$92$	−12.3378	−1.28631
$93$	1.51182	0.156768
$94$	2.54255	0.262244
$95$	17.5576	1.80137
$96$	−4.92814	−0.502976
$97$	−5.97045	−0.606208	−0.303104	−	0.952958i	$-0.598023\pi$
$97$	−5.97045	−0.606208	−0.303104	+	0.952958i	$0.598023\pi$
$98$	−2.09044	−0.211166
$99$	0.792751	0.0796745
$100$	−0.226096	−0.0226096
$101$	−8.45536	−0.841339	−0.420670	−	0.907214i	$-0.638205\pi$
$101$	−8.45536	−0.841339	−0.420670	+	0.907214i	$0.638205\pi$
$102$	0.107479	0.0106420
$103$	4.75560	0.468583	0.234292	−	0.972166i	$-0.424723\pi$
$103$	4.75560	0.468583	0.234292	+	0.972166i	$0.424723\pi$
$104$	−1.16230	−0.113973
$105$	−3.70838	−0.361901
$106$	−4.11573	−0.399755
$107$	17.2736	1.66990	0.834949	−	0.550327i	$-0.185497\pi$
$107$	17.2736	1.66990	0.834949	+	0.550327i	$0.185497\pi$
$108$	1.76565	0.169900
$109$	10.6920	1.02410	0.512052	−	0.858954i	$-0.328885\pi$
$109$	10.6920	1.02410	0.512052	+	0.858954i	$0.328885\pi$
$110$	0.869044	0.0828601
$111$	2.52306	0.239478
$112$	4.33775	0.409879
$113$	1.86409	0.175359	0.0876794	−	0.996149i	$-0.472055\pi$
$113$	1.86409	0.175359	0.0876794	+	0.996149i	$0.472055\pi$
$114$	−3.75333	−0.351532
$115$	15.8237	1.47557
$116$	−6.47269	−0.600974
$117$	0.637602	0.0589463
$118$	3.16772	0.291613
$119$	−0.363584	−0.0333297
$120$	4.12805	0.376838
$121$	−10.3715	−0.942868
$122$	−5.24022	−0.474427
$123$	3.38268	0.305006
$124$	2.66935	0.239714
$125$	−11.0326	−0.986788
$126$	0.792751	0.0706239
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.2660	−0.995779
$129$	4.81549	0.423980
$130$	0.698963	0.0613031
$131$	−8.92760	−0.780008	−0.390004	−	0.920813i	$-0.627526\pi$
$131$	−8.92760	−0.780008	−0.390004	+	0.920813i	$0.627526\pi$
$132$	1.39972	0.121830
$133$	12.6969	1.10096
$134$	−7.21778	−0.623521
$135$	−2.26452	−0.194899
$136$	0.404730	0.0347053
$137$	−5.15767	−0.440649	−0.220325	−	0.975427i	$-0.570712\pi$
$137$	−5.15767	−0.440649	−0.220325	+	0.975427i	$0.570712\pi$
$138$	−3.38268	−0.287953
$139$	−0.614037	−0.0520819	−0.0260410	−	0.999661i	$-0.508290\pi$
$139$	−0.614037	−0.0520819	−0.0260410	+	0.999661i	$0.508290\pi$
$140$	−6.54772	−0.553384
$141$	−5.25220	−0.442315
$142$	−1.23254	−0.103432
$143$	0.505460	0.0422687
$144$	2.64884	0.220737
$145$	8.30147	0.689400
$146$	−0.0728228	−0.00602686
$147$	4.31826	0.356164
$148$	4.45485	0.366186
$149$	6.88697	0.564203	0.282101	−	0.959385i	$-0.408969\pi$
$149$	6.88697	0.564203	0.282101	+	0.959385i	$0.408969\pi$
$150$	−0.0619892	−0.00506140
$151$	−9.88589	−0.804502	−0.402251	−	0.915529i	$-0.631772\pi$
$151$	−9.88589	−0.804502	−0.402251	+	0.915529i	$0.631772\pi$
$152$	−14.1337	−1.14640
$153$	−0.222022	−0.0179494
$154$	0.628454	0.0506423
$155$	−3.42354	−0.274985
$156$	1.12578	0.0901349
$157$	5.96382	0.475965	0.237982	−	0.971269i	$-0.423514\pi$
$157$	5.96382	0.475965	0.237982	+	0.971269i	$0.423514\pi$
$158$	2.04385	0.162600
$159$	8.50194	0.674248
$160$	11.1599	0.882265
$161$	11.4430	0.901837
$162$	0.484093	0.0380339
$163$	−17.3210	−1.35668	−0.678341	−	0.734747i	$-0.737301\pi$
$163$	−17.3210	−1.35668	−0.678341	+	0.734747i	$0.737301\pi$
$164$	5.97265	0.466386
$165$	−1.79520	−0.139756
$166$	1.90466	0.147830
$167$	−20.4843	−1.58512	−0.792561	−	0.609793i	$-0.791253\pi$
$167$	−20.4843	−1.58512	−0.792561	+	0.609793i	$0.791253\pi$
$168$	2.98523	0.230315
$169$	−12.5935	−0.968728
$170$	−0.243389	−0.0186671
$171$	7.75333	0.592912
$172$	8.50248	0.648308
$173$	14.0587	1.06886	0.534432	−	0.845212i	$-0.320526\pi$
$173$	14.0587	1.06886	0.534432	+	0.845212i	$0.320526\pi$
$174$	−1.77463	−0.134534
$175$	0.209699	0.0158517
$176$	2.09987	0.158284
$177$	−6.54363	−0.491850
$178$	5.89443	0.441806
$179$	9.72542	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$179$	9.72542	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$180$	−3.99836	−0.298020
$181$	12.0731	0.897384	0.448692	−	0.893687i	$-0.351890\pi$
$181$	12.0731	0.897384	0.448692	+	0.893687i	$0.351890\pi$
$182$	0.505460	0.0374672
$183$	10.8248	0.800194
$184$	−12.7380	−0.939059
$185$	−5.71352	−0.420066
$186$	0.731860	0.0536626
$187$	−0.176008	−0.0128710
$188$	−9.27356	−0.676344
$189$	−1.63760	−0.119118
$190$	8.49949	0.616618
$191$	−2.78025	−0.201172	−0.100586	−	0.994928i	$-0.532072\pi$
$191$	−2.78025	−0.201172	−0.100586	+	0.994928i	$0.532072\pi$
$192$	2.91201	0.210156
$193$	−17.8448	−1.28450	−0.642249	−	0.766496i	$-0.721998\pi$
$193$	−17.8448	−1.28450	−0.642249	+	0.766496i	$0.721998\pi$
$194$	−2.89025	−0.207508
$195$	−1.44386	−0.103397
$196$	7.62455	0.544611
$197$	2.24904	0.160238	0.0801188	−	0.996785i	$-0.474470\pi$
$197$	2.24904	0.160238	0.0801188	+	0.996785i	$0.474470\pi$
$198$	0.383765	0.0272730
$199$	−10.6598	−0.755653	−0.377826	−	0.925876i	$-0.623328\pi$
$199$	−10.6598	−0.755653	−0.377826	+	0.925876i	$0.623328\pi$
$200$	−0.233430	−0.0165060
$201$	14.9099	1.05166
$202$	−4.09318	−0.287995
$203$	6.00326	0.421346
$204$	−0.392015	−0.0274465
$205$	−7.66015	−0.535008
$206$	2.30215	0.160398
$207$	6.98768	0.485677
$208$	1.68891	0.117105
$209$	6.14646	0.425160
$210$	−1.79520	−0.123881
$211$	15.3797	1.05878	0.529391	−	0.848378i	$-0.322421\pi$
$211$	15.3797	1.05878	0.529391	+	0.848378i	$0.322421\pi$
$212$	15.0115	1.03099
$213$	2.54608	0.174455
$214$	8.36201	0.571615
$215$	−10.9048	−0.743699
$216$	1.82293	0.124034
$217$	−2.47576	−0.168065
$218$	5.17590	0.350556
$219$	0.150431	0.0101652
$220$	−3.16970	−0.213701
$221$	−0.141562	−0.00952248
$222$	1.22139	0.0819746
$223$	26.0744	1.74607	0.873034	−	0.487659i	$-0.162149\pi$
$223$	26.0744	1.74607	0.873034	+	0.487659i	$0.162149\pi$
$224$	8.07033	0.539221
$225$	0.128052	0.00853682
$226$	0.902392	0.0600263
$227$	13.0423	0.865648	0.432824	−	0.901479i	$-0.357517\pi$
$227$	13.0423	0.865648	0.432824	+	0.901479i	$0.357517\pi$
$228$	13.6897	0.906623
$229$	−10.9338	−0.722524	−0.361262	−	0.932464i	$-0.617654\pi$
$229$	−10.9338	−0.722524	−0.361262	+	0.932464i	$0.617654\pi$
$230$	7.66015	0.505096
$231$	−1.29821	−0.0854160
$232$	−6.68264	−0.438737
$233$	−14.8371	−0.972009	−0.486005	−	0.873956i	$-0.661546\pi$
$233$	−14.8371	−0.972009	−0.486005	+	0.873956i	$0.661546\pi$
$234$	0.308658	0.0201776
$235$	11.8937	0.775860
$236$	−11.5538	−0.752088
$237$	−4.22202	−0.274250
$238$	−0.176008	−0.0114089
$239$	16.5235	1.06882	0.534409	−	0.845226i	$-0.320534\pi$
$239$	16.5235	1.06882	0.534409	+	0.845226i	$0.320534\pi$
$240$	−5.99836	−0.387192
$241$	−0.204439	−0.0131691	−0.00658453	−	0.999978i	$-0.502096\pi$
$241$	−0.204439	−0.0131691	−0.00658453	+	0.999978i	$0.502096\pi$
$242$	−5.02079	−0.322749
$243$	−1.00000	−0.0641500
$244$	19.1129	1.22358
$245$	−9.77879	−0.624744
$246$	1.63753	0.104405
$247$	4.94354	0.314550
$248$	2.75593	0.175002
$249$	−3.93449	−0.249338
$250$	−5.34081	−0.337783
$251$	−7.20418	−0.454724	−0.227362	−	0.973810i	$-0.573010\pi$
$251$	−7.20418	−0.454724	−0.227362	+	0.973810i	$0.573010\pi$
$252$	−2.89144	−0.182144
$253$	5.53949	0.348265
$254$	0.484093	0.0303747
$255$	0.502774	0.0314849
$256$	0.370256	0.0231410
$257$	−29.0206	−1.81026	−0.905129	−	0.425137i	$-0.860226\pi$
$257$	−29.0206	−1.81026	−0.905129	+	0.425137i	$0.860226\pi$
$258$	2.33114	0.145131
$259$	−4.13177	−0.256735
$260$	−2.54936	−0.158105
$261$	3.66588	0.226913
$262$	−4.32178	−0.267001
$263$	−13.5877	−0.837851	−0.418926	−	0.908021i	$-0.637593\pi$
$263$	−13.5877	−0.837851	−0.418926	+	0.908021i	$0.637593\pi$
$264$	1.44513	0.0889414
$265$	−19.2528	−1.18269
$266$	6.14646	0.376864
$267$	−12.1762	−0.745174
$268$	26.3258	1.60810
$269$	24.4150	1.48861	0.744304	−	0.667840i	$-0.232781\pi$
$269$	24.4150	1.48861	0.744304	+	0.667840i	$0.232781\pi$
$270$	−1.09624	−0.0667149
$271$	6.42127	0.390065	0.195032	−	0.980797i	$-0.437519\pi$
$271$	6.42127	0.390065	0.195032	+	0.980797i	$0.437519\pi$
$272$	−0.588102	−0.0356589
$273$	−1.04414	−0.0631941
$274$	−2.49679	−0.150837
$275$	0.101514	0.00612150
$276$	12.3378	0.742650
$277$	21.9567	1.31925	0.659624	−	0.751596i	$-0.270716\pi$
$277$	21.9567	1.31925	0.659624	+	0.751596i	$0.270716\pi$
$278$	−0.297251	−0.0178279
$279$	−1.51182	−0.0905101
$280$	−6.76011	−0.403994
$281$	22.1860	1.32350	0.661752	−	0.749723i	$-0.269813\pi$
$281$	22.1860	1.32350	0.661752	+	0.749723i	$0.269813\pi$
$282$	−2.54255	−0.151407
$283$	−25.7123	−1.52844	−0.764218	−	0.644958i	$-0.776875\pi$
$283$	−25.7123	−1.52844	−0.764218	+	0.644958i	$0.776875\pi$
$284$	4.49550	0.266759
$285$	−17.5576	−1.04002
$286$	0.244689	0.0144688
$287$	−5.53949	−0.326986
$288$	4.92814	0.290393
$289$	−16.9507	−0.997100
$290$	4.01868	0.235985
$291$	5.97045	0.349994
$292$	0.265610	0.0155436
$293$	−17.8700	−1.04398	−0.521989	−	0.852952i	$-0.674810\pi$
$293$	−17.8700	−1.04398	−0.521989	+	0.852952i	$0.674810\pi$
$294$	2.09044	0.121917
$295$	14.8182	0.862748
$296$	4.59935	0.267332
$297$	−0.792751	−0.0460001
$298$	3.33393	0.193130
$299$	4.45536	0.257660
$300$	0.226096	0.0130537
$301$	−7.88585	−0.454533
$302$	−4.78569	−0.275385
$303$	8.45536	0.485748
$304$	20.5374	1.17790
$305$	−24.5130	−1.40361
$306$	−0.107479	−0.00614419
$307$	−21.9118	−1.25057	−0.625286	−	0.780396i	$-0.715018\pi$
$307$	−21.9118	−1.25057	−0.625286	+	0.780396i	$0.715018\pi$
$308$	−2.29219	−0.130610
$309$	−4.75560	−0.270537
$310$	−1.65731	−0.0941290
$311$	−12.3845	−0.702260	−0.351130	−	0.936327i	$-0.614202\pi$
$311$	−12.3845	−0.702260	−0.351130	+	0.936327i	$0.614202\pi$
$312$	1.16230	0.0658023
$313$	−4.25938	−0.240755	−0.120377	−	0.992728i	$-0.538410\pi$
$313$	−4.25938	−0.240755	−0.120377	+	0.992728i	$0.538410\pi$
$314$	2.88704	0.162925
$315$	3.70838	0.208944
$316$	−7.45463	−0.419356
$317$	−5.97298	−0.335476	−0.167738	−	0.985832i	$-0.553646\pi$
$317$	−5.97298	−0.335476	−0.167738	+	0.985832i	$0.553646\pi$
$318$	4.11573	0.230799
$319$	2.90613	0.162712
$320$	−6.59431	−0.368633
$321$	−17.2736	−0.964116
$322$	5.53949	0.308704
$323$	−1.72141	−0.0957819
$324$	−1.76565	−0.0980919
$325$	0.0816464	0.00452893
$326$	−8.38495	−0.464399
$327$	−10.6920	−0.591267
$328$	6.16638	0.340481
$329$	8.60101	0.474189
$330$	−0.869044	−0.0478393
$331$	21.3536	1.17370	0.586849	−	0.809696i	$-0.300368\pi$
$331$	21.3536	1.17370	0.586849	+	0.809696i	$0.300368\pi$
$332$	−6.94696	−0.381264
$333$	−2.52306	−0.138263
$334$	−9.91629	−0.542595
$335$	−33.7638	−1.84471
$336$	−4.33775	−0.236644
$337$	5.73734	0.312533	0.156266	−	0.987715i	$-0.450054\pi$
$337$	5.73734	0.312533	0.156266	+	0.987715i	$0.450054\pi$
$338$	−6.09640	−0.331601
$339$	−1.86409	−0.101243
$340$	0.887725	0.0481436
$341$	−1.19849	−0.0649021
$342$	3.75333	0.202957
$343$	−18.5348	−1.00079
$344$	8.77827	0.473293
$345$	−15.8237	−0.851921
$346$	6.80572	0.365878
$347$	−24.6516	−1.32337	−0.661684	−	0.749783i	$-0.730158\pi$
$347$	−24.6516	−1.32337	−0.661684	+	0.749783i	$0.730158\pi$
$348$	6.47269	0.346972
$349$	3.51701	0.188261	0.0941305	−	0.995560i	$-0.469993\pi$
$349$	3.51701	0.188261	0.0941305	+	0.995560i	$0.469993\pi$
$350$	0.101514	0.00542613
$351$	−0.637602	−0.0340327
$352$	3.90679	0.208232
$353$	8.95060	0.476392	0.238196	−	0.971217i	$-0.423444\pi$
$353$	8.95060	0.476392	0.238196	+	0.971217i	$0.423444\pi$
$354$	−3.16772	−0.168363
$355$	−5.76565	−0.306009
$356$	−21.4990	−1.13945
$357$	0.363584	0.0192429
$358$	4.70801	0.248826
$359$	13.4086	0.707678	0.353839	−	0.935306i	$-0.384876\pi$
$359$	13.4086	0.707678	0.353839	+	0.935306i	$0.384876\pi$
$360$	−4.12805	−0.217567
$361$	41.1141	2.16390
$362$	5.84448	0.307179
$363$	10.3715	0.544365
$364$	−1.84359	−0.0966302
$365$	−0.340655	−0.0178307
$366$	5.24022	0.273910
$367$	−12.6515	−0.660402	−0.330201	−	0.943911i	$-0.607116\pi$
$367$	−12.6515	−0.660402	−0.330201	+	0.943911i	$0.607116\pi$
$368$	18.5093	0.964862
$369$	−3.38268	−0.176095
$370$	−2.76587	−0.143791
$371$	−13.9228	−0.722836
$372$	−2.66935	−0.138399
$373$	−13.9114	−0.720303	−0.360151	−	0.932894i	$-0.617275\pi$
$373$	−13.9114	−0.720303	−0.360151	+	0.932894i	$0.617275\pi$
$374$	−0.0852044	−0.00440581
$375$	11.0326	0.569722
$376$	−9.57436	−0.493760
$377$	2.33738	0.120381
$378$	−0.792751	−0.0407747
$379$	22.5759	1.15965	0.579823	−	0.814743i	$-0.303122\pi$
$379$	22.5759	1.15965	0.579823	+	0.814743i	$0.303122\pi$
$380$	−31.0006	−1.59030
$381$	−1.00000	−0.0512316
$382$	−1.34590	−0.0688621
$383$	7.83325	0.400260	0.200130	−	0.979769i	$-0.435863\pi$
$383$	7.83325	0.400260	0.200130	+	0.979769i	$0.435863\pi$
$384$	11.2660	0.574914
$385$	2.93982	0.149827
$386$	−8.63854	−0.439690
$387$	−4.81549	−0.244785
$388$	10.5418	0.535177
$389$	3.67320	0.186239	0.0931194	−	0.995655i	$-0.470316\pi$
$389$	3.67320	0.186239	0.0931194	+	0.995655i	$0.470316\pi$
$390$	−0.698963	−0.0353934
$391$	−1.55142	−0.0784587
$392$	7.87187	0.397589
$393$	8.92760	0.450338
$394$	1.08874	0.0548502
$395$	9.56086	0.481059
$396$	−1.39972	−0.0703388
$397$	−21.4528	−1.07668	−0.538341	−	0.842727i	$-0.680949\pi$
$397$	−21.4528	−1.07668	−0.538341	+	0.842727i	$0.680949\pi$
$398$	−5.16033	−0.258664
$399$	−12.6969	−0.635639
$400$	0.339191	0.0169595
$401$	−37.0201	−1.84870	−0.924348	−	0.381551i	$-0.875390\pi$
$401$	−37.0201	−1.84870	−0.924348	+	0.381551i	$0.875390\pi$
$402$	7.21778	0.359990
$403$	−0.963938	−0.0480172
$404$	14.9292	0.742757
$405$	2.26452	0.112525
$406$	2.90613	0.144229
$407$	−2.00016	−0.0991441
$408$	−0.404730	−0.0200371
$409$	−29.3888	−1.45318	−0.726592	−	0.687069i	$-0.758897\pi$
$409$	−29.3888	−1.45318	−0.726592	+	0.687069i	$0.758897\pi$
$410$	−3.70822	−0.183136
$411$	5.15767	0.254409
$412$	−8.39674	−0.413678
$413$	10.7159	0.527293
$414$	3.38268	0.166250
$415$	8.90974	0.437362
$416$	3.14219	0.154059
$417$	0.614037	0.0300695
$418$	2.97546	0.145534
$419$	−38.2693	−1.86958	−0.934788	−	0.355206i	$-0.884411\pi$
$419$	−38.2693	−1.86958	−0.934788	+	0.355206i	$0.884411\pi$
$420$	6.54772	0.319496
$421$	22.6777	1.10524	0.552621	−	0.833433i	$-0.313628\pi$
$421$	22.6777	1.10524	0.552621	+	0.833433i	$0.313628\pi$
$422$	7.44520	0.362426
$423$	5.25220	0.255371
$424$	15.4984	0.752669
$425$	−0.0284305	−0.00137908
$426$	1.23254	0.0597167
$427$	−17.7267	−0.857857
$428$	−30.4991	−1.47423
$429$	−0.505460	−0.0244038
$430$	−5.27892	−0.254572
$431$	25.9447	1.24971	0.624855	−	0.780741i	$-0.285158\pi$
$431$	25.9447	1.24971	0.624855	+	0.780741i	$0.285158\pi$
$432$	−2.64884	−0.127443
$433$	−22.9121	−1.10108	−0.550542	−	0.834808i	$-0.685579\pi$
$433$	−22.9121	−1.10108	−0.550542	+	0.834808i	$0.685579\pi$
$434$	−1.19849	−0.0575296
$435$	−8.30147	−0.398025
$436$	−18.8783	−0.904108
$437$	54.1778	2.59167
$438$	0.0728228	0.00347961
$439$	−19.0507	−0.909242	−0.454621	−	0.890685i	$-0.650225\pi$
$439$	−19.0507	−0.909242	−0.454621	+	0.890685i	$0.650225\pi$
$440$	−3.27252	−0.156011
$441$	−4.31826	−0.205631
$442$	−0.0685290	−0.00325959
$443$	16.1706	0.768287	0.384143	−	0.923273i	$-0.374497\pi$
$443$	16.1706	0.768287	0.384143	+	0.923273i	$0.374497\pi$
$444$	−4.45485	−0.211418
$445$	27.5734	1.30710
$446$	12.6224	0.597688
$447$	−6.88697	−0.325743
$448$	−4.76872	−0.225301
$449$	38.1729	1.80149	0.900745	−	0.434349i	$-0.143022\pi$
$449$	38.1729	1.80149	0.900745	+	0.434349i	$0.143022\pi$
$450$	0.0619892	0.00292220
$451$	−2.68163	−0.126273
$452$	−3.29134	−0.154812
$453$	9.88589	0.464480
$454$	6.31368	0.296316
$455$	2.36447	0.110848
$456$	14.1337	0.661873
$457$	−15.3347	−0.717325	−0.358662	−	0.933467i	$-0.616767\pi$
$457$	−15.3347	−0.717325	−0.358662	+	0.933467i	$0.616767\pi$
$458$	−5.29296	−0.247324
$459$	0.222022	0.0103631
$460$	−27.9392	−1.30267
$461$	−28.5406	−1.32927	−0.664635	−	0.747168i	$-0.731413\pi$
$461$	−28.5406	−1.32927	−0.664635	+	0.747168i	$0.731413\pi$
$462$	−0.628454	−0.0292383
$463$	−12.6429	−0.587567	−0.293783	−	0.955872i	$-0.594914\pi$
$463$	−12.6429	−0.587567	−0.293783	+	0.955872i	$0.594914\pi$
$464$	9.71035	0.450792
$465$	3.42354	0.158763
$466$	−7.18252	−0.332724
$467$	−2.53092	−0.117117	−0.0585584	−	0.998284i	$-0.518650\pi$
$467$	−2.53092	−0.117117	−0.0585584	+	0.998284i	$0.518650\pi$
$468$	−1.12578	−0.0520394
$469$	−24.4165	−1.12745
$470$	5.75766	0.265581
$471$	−5.96382	−0.274798
$472$	−11.9286	−0.549056
$473$	−3.81748	−0.175528
$474$	−2.04385	−0.0938771
$475$	0.992832	0.0455543
$476$	0.641964	0.0294244
$477$	−8.50194	−0.389277
$478$	7.99892	0.365862
$479$	−10.7367	−0.490573	−0.245286	−	0.969451i	$-0.578882\pi$
$479$	−10.7367	−0.490573	−0.245286	+	0.969451i	$0.578882\pi$
$480$	−11.1599	−0.509376
$481$	−1.60871	−0.0733507
$482$	−0.0989674	−0.00450784
$483$	−11.4430	−0.520676
$484$	18.3126	0.832389
$485$	−13.5202	−0.613921
$486$	−0.484093	−0.0219589
$487$	−30.8119	−1.39622	−0.698110	−	0.715991i	$-0.745975\pi$
$487$	−30.8119	−1.39622	−0.698110	+	0.715991i	$0.745975\pi$
$488$	19.7328	0.893264
$489$	17.3210	0.783281
$490$	−4.73384	−0.213853
$491$	−1.03671	−0.0467863	−0.0233931	−	0.999726i	$-0.507447\pi$
$491$	−1.03671	−0.0467863	−0.0233931	+	0.999726i	$0.507447\pi$
$492$	−5.97265	−0.269268
$493$	−0.813908	−0.0366566
$494$	2.39313	0.107672
$495$	1.79520	0.0806883
$496$	−4.00457	−0.179810
$497$	−4.16947	−0.187026
$498$	−1.90466	−0.0853499
$499$	42.2215	1.89010	0.945048	−	0.326932i	$-0.106015\pi$
$499$	42.2215	1.89010	0.945048	+	0.326932i	$0.106015\pi$
$500$	19.4798	0.871163
$501$	20.4843	0.915170
$502$	−3.48749	−0.155654
$503$	−32.3148	−1.44085	−0.720423	−	0.693535i	$-0.756052\pi$
$503$	−32.3148	−1.44085	−0.720423	+	0.693535i	$0.756052\pi$
$504$	−2.98523	−0.132973
$505$	−19.1473	−0.852045
$506$	2.68163	0.119213
$507$	12.5935	0.559295
$508$	−1.76565	−0.0783382
$509$	1.47752	0.0654899	0.0327449	−	0.999464i	$-0.489575\pi$
$509$	1.47752	0.0654899	0.0327449	+	0.999464i	$0.489575\pi$
$510$	0.243389	0.0107775
$511$	−0.246347	−0.0108977
$512$	22.7112	1.00370
$513$	−7.75333	−0.342318
$514$	−14.0487	−0.619661
$515$	10.7691	0.474545
$516$	−8.50248	−0.374301
$517$	4.16368	0.183119
$518$	−2.00016	−0.0878819
$519$	−14.0587	−0.617109
$520$	−2.63205	−0.115423
$521$	26.7752	1.17304	0.586521	−	0.809934i	$-0.300497\pi$
$521$	26.7752	1.17304	0.586521	+	0.809934i	$0.300497\pi$
$522$	1.77463	0.0776734
$523$	27.8665	1.21852	0.609258	−	0.792972i	$-0.291467\pi$
$523$	27.8665	1.21852	0.609258	+	0.792972i	$0.291467\pi$
$524$	15.7630	0.688612
$525$	−0.209699	−0.00915201
$526$	−6.57769	−0.286801
$527$	0.335657	0.0146215
$528$	−2.09987	−0.0913853
$529$	25.8276	1.12294
$530$	−9.32015	−0.404842
$531$	6.54363	0.283969
$532$	−22.4183	−0.971956
$533$	−2.15681	−0.0934216
$534$	−5.89443	−0.255077
$535$	39.1163	1.69115
$536$	27.1797	1.17398
$537$	−9.72542	−0.419683
$538$	11.8191	0.509559
$539$	−3.42331	−0.147452
$540$	3.99836	0.172062
$541$	26.6789	1.14702	0.573508	−	0.819200i	$-0.305582\pi$
$541$	26.6789	1.14702	0.573508	+	0.819200i	$0.305582\pi$
$542$	3.10849	0.133521
$543$	−12.0731	−0.518105
$544$	−1.09416	−0.0469116
$545$	24.2122	1.03714
$546$	−0.505460	−0.0216317
$547$	39.9946	1.71005	0.855023	−	0.518589i	$-0.173543\pi$
$547$	39.9946	1.71005	0.855023	+	0.518589i	$0.173543\pi$
$548$	9.10666	0.389017
$549$	−10.8248	−0.461992
$550$	0.0491420	0.00209542
$551$	28.4228	1.21085
$552$	12.7380	0.542166
$553$	6.91399	0.294013
$554$	10.6291	0.451585
$555$	5.71352	0.242525
$556$	1.08418	0.0459793
$557$	28.1551	1.19297	0.596485	−	0.802624i	$-0.296563\pi$
$557$	28.1551	1.19297	0.596485	+	0.802624i	$0.296563\pi$
$558$	−0.731860	−0.0309821
$559$	−3.07036	−0.129863
$560$	9.82293	0.415094
$561$	0.176008	0.00743108
$562$	10.7401	0.453042
$563$	14.6527	0.617536	0.308768	−	0.951137i	$-0.400083\pi$
$563$	14.6527	0.617536	0.308768	+	0.951137i	$0.400083\pi$
$564$	9.27356	0.390487
$565$	4.22127	0.177590
$566$	−12.4471	−0.523192
$567$	1.63760	0.0687728
$568$	4.64132	0.194745
$569$	34.8435	1.46071	0.730357	−	0.683065i	$-0.239354\pi$
$569$	34.8435	1.46071	0.730357	+	0.683065i	$0.239354\pi$
$570$	−8.49949	−0.356005
$571$	−0.228468	−0.00956110	−0.00478055	−	0.999989i	$-0.501522\pi$
$571$	−0.228468	−0.00956110	−0.00478055	+	0.999989i	$0.501522\pi$
$572$	−0.892467	−0.0373159
$573$	2.78025	0.116146
$574$	−2.68163	−0.111929
$575$	0.894788	0.0373153
$576$	−2.91201	−0.121334
$577$	−18.0338	−0.750757	−0.375378	−	0.926872i	$-0.622487\pi$
$577$	−18.0338	−0.750757	−0.375378	+	0.926872i	$0.622487\pi$
$578$	−8.20571	−0.341313
$579$	17.8448	0.741605
$580$	−14.6575	−0.608621
$581$	6.44314	0.267306
$582$	2.89025	0.119805
$583$	−6.73993	−0.279139
$584$	0.274225	0.0113475
$585$	1.44386	0.0596964
$586$	−8.65075	−0.357359
$587$	14.7821	0.610122	0.305061	−	0.952333i	$-0.401323\pi$
$587$	14.7821	0.610122	0.305061	+	0.952333i	$0.401323\pi$
$588$	−7.62455	−0.314431
$589$	−11.7216	−0.482981
$590$	7.17338	0.295323
$591$	−2.24904	−0.0925132
$592$	−6.68319	−0.274677
$593$	32.0102	1.31450	0.657251	−	0.753672i	$-0.271719\pi$
$593$	32.0102	1.31450	0.657251	+	0.753672i	$0.271719\pi$
$594$	−0.383765	−0.0157461
$595$	−0.823344	−0.0337538
$596$	−12.1600	−0.498094
$597$	10.6598	0.436276
$598$	2.15681	0.0881983
$599$	21.3788	0.873513	0.436757	−	0.899580i	$-0.356127\pi$
$599$	21.3788	0.873513	0.436757	+	0.899580i	$0.356127\pi$
$600$	0.233430	0.00952974
$601$	−8.02879	−0.327501	−0.163751	−	0.986502i	$-0.552359\pi$
$601$	−8.02879	−0.327501	−0.163751	+	0.986502i	$0.552359\pi$
$602$	−3.81748	−0.155589
$603$	−14.9099	−0.607179
$604$	17.4551	0.710237
$605$	−23.4866	−0.954865
$606$	4.09318	0.166274
$607$	−27.9309	−1.13368	−0.566840	−	0.823828i	$-0.691834\pi$
$607$	−27.9309	−1.13368	−0.566840	+	0.823828i	$0.691834\pi$
$608$	38.2095	1.54960
$609$	−6.00326	−0.243264
$610$	−11.8666	−0.480464
$611$	3.34881	0.135478
$612$	0.392015	0.0158462
$613$	−10.9321	−0.441544	−0.220772	−	0.975325i	$-0.570858\pi$
$613$	−10.9321	−0.441544	−0.220772	+	0.975325i	$0.570858\pi$
$614$	−10.6073	−0.428077
$615$	7.66015	0.308887
$616$	−2.36654	−0.0953507
$617$	−6.32757	−0.254739	−0.127369	−	0.991855i	$-0.540653\pi$
$617$	−6.32757	−0.254739	−0.127369	+	0.991855i	$0.540653\pi$
$618$	−2.30215	−0.0926061
$619$	39.3623	1.58210	0.791051	−	0.611750i	$-0.209534\pi$
$619$	39.3623	1.58210	0.791051	+	0.611750i	$0.209534\pi$
$620$	6.04479	0.242765
$621$	−6.98768	−0.280406
$622$	−5.99524	−0.240387
$623$	19.9398	0.798873
$624$	−1.68891	−0.0676104
$625$	−25.6239	−1.02495
$626$	−2.06194	−0.0824116
$627$	−6.14646	−0.245466
$628$	−10.5301	−0.420195
$629$	0.560175	0.0223357
$630$	1.79520	0.0715225
$631$	39.6978	1.58034	0.790172	−	0.612886i	$-0.209991\pi$
$631$	39.6978	1.58034	0.790172	+	0.612886i	$0.209991\pi$
$632$	−7.69643	−0.306148
$633$	−15.3797	−0.611288
$634$	−2.89148	−0.114835
$635$	2.26452	0.0898647
$636$	−15.0115	−0.595244
$637$	−2.75333	−0.109091
$638$	1.40684	0.0556973
$639$	−2.54608	−0.100721
$640$	−25.5120	−1.00845
$641$	−14.4905	−0.572339	−0.286169	−	0.958179i	$-0.592382\pi$
$641$	−14.4905	−0.572339	−0.286169	+	0.958179i	$0.592382\pi$
$642$	−8.36201	−0.330022
$643$	12.5017	0.493019	0.246509	−	0.969140i	$-0.420716\pi$
$643$	12.5017	0.493019	0.246509	+	0.969140i	$0.420716\pi$
$644$	−20.2044	−0.796166
$645$	10.9048	0.429375
$646$	−0.833323	−0.0327867
$647$	22.6426	0.890172	0.445086	−	0.895488i	$-0.353173\pi$
$647$	22.6426	0.890172	0.445086	+	0.895488i	$0.353173\pi$
$648$	−1.82293	−0.0716113
$649$	5.18747	0.203626
$650$	0.0395244	0.00155028
$651$	2.47576	0.0970325
$652$	30.5828	1.19772
$653$	−24.0103	−0.939595	−0.469797	−	0.882774i	$-0.655673\pi$
$653$	−24.0103	−0.939595	−0.469797	+	0.882774i	$0.655673\pi$
$654$	−5.17590	−0.202394
$655$	−20.2167	−0.789933
$656$	−8.96020	−0.349837
$657$	−0.150431	−0.00586889
$658$	4.16368	0.162317
$659$	−27.5076	−1.07155	−0.535773	−	0.844362i	$-0.679980\pi$
$659$	−27.5076	−1.07155	−0.535773	+	0.844362i	$0.679980\pi$
$660$	3.16970	0.123381
$661$	−14.9028	−0.579651	−0.289826	−	0.957079i	$-0.593597\pi$
$661$	−14.9028	−0.579651	−0.289826	+	0.957079i	$0.593597\pi$
$662$	10.3371	0.401763
$663$	0.141562	0.00549781
$664$	−7.17229	−0.278339
$665$	28.7523	1.11497
$666$	−1.22139	−0.0473281
$667$	25.6160	0.991856
$668$	36.1682	1.39939
$669$	−26.0744	−1.00809
$670$	−16.3448	−0.631455
$671$	−8.58139	−0.331281
$672$	−8.07033	−0.311320
$673$	−25.8180	−0.995210	−0.497605	−	0.867404i	$-0.665787\pi$
$673$	−25.8180	−0.995210	−0.497605	+	0.867404i	$0.665787\pi$
$674$	2.77740	0.106982
$675$	−0.128052	−0.00492874
$676$	22.2357	0.855219
$677$	−31.1586	−1.19752	−0.598762	−	0.800927i	$-0.704340\pi$
$677$	−31.1586	−1.19752	−0.598762	+	0.800927i	$0.704340\pi$
$678$	−0.902392	−0.0346562
$679$	−9.77723	−0.375215
$680$	0.916520	0.0351469
$681$	−13.0423	−0.499782
$682$	−0.580183	−0.0222163
$683$	−21.1324	−0.808609	−0.404304	−	0.914624i	$-0.632486\pi$
$683$	−21.1324	−0.808609	−0.404304	+	0.914624i	$0.632486\pi$
$684$	−13.6897	−0.523439
$685$	−11.6796	−0.446256
$686$	−8.97256	−0.342574
$687$	10.9338	0.417149
$688$	−12.7555	−0.486298
$689$	−5.42086	−0.206518
$690$	−7.66015	−0.291617
$691$	25.8346	0.982794	0.491397	−	0.870936i	$-0.336486\pi$
$691$	25.8346	0.982794	0.491397	+	0.870936i	$0.336486\pi$
$692$	−24.8228	−0.943622
$693$	1.29821	0.0493149
$694$	−11.9337	−0.452996
$695$	−1.39050	−0.0527446
$696$	6.68264	0.253305
$697$	0.751031	0.0284473
$698$	1.70256	0.0644427
$699$	14.8371	0.561190
$700$	−0.370256	−0.0139943
$701$	−5.04292	−0.190468	−0.0952342	−	0.995455i	$-0.530360\pi$
$701$	−5.04292	−0.190468	−0.0952342	+	0.995455i	$0.530360\pi$
$702$	−0.308658	−0.0116496
$703$	−19.5621	−0.737799
$704$	−2.30850	−0.0870049
$705$	−11.8937	−0.447943
$706$	4.33292	0.163072
$707$	−13.8465	−0.520751
$708$	11.5538	0.434218
$709$	−5.28671	−0.198546	−0.0992732	−	0.995060i	$-0.531652\pi$
$709$	−5.28671	−0.198546	−0.0992732	+	0.995060i	$0.531652\pi$
$710$	−2.79111	−0.104749
$711$	4.22202	0.158338
$712$	−22.1964	−0.831845
$713$	−10.5641	−0.395628
$714$	0.176008	0.00658695
$715$	1.14462	0.0428065
$716$	−17.1717	−0.641738
$717$	−16.5235	−0.617082
$718$	6.49100	0.242242
$719$	−44.0765	−1.64377	−0.821887	−	0.569650i	$-0.807079\pi$
$719$	−44.0765	−1.64377	−0.821887	+	0.569650i	$0.807079\pi$
$720$	5.99836	0.223546
$721$	7.78778	0.290032
$722$	19.9031	0.740715
$723$	0.204439	0.00760316
$724$	−21.3169	−0.792235
$725$	0.469425	0.0174340
$726$	5.02079	0.186339
$727$	3.68927	0.136828	0.0684138	−	0.997657i	$-0.478206\pi$
$727$	3.68927	0.136828	0.0684138	+	0.997657i	$0.478206\pi$
$728$	−1.90339	−0.0705442
$729$	1.00000	0.0370370
$730$	−0.164909	−0.00610354
$731$	1.06915	0.0395438
$732$	−19.1129	−0.706433
$733$	−10.9178	−0.403258	−0.201629	−	0.979462i	$-0.564624\pi$
$733$	−10.9178	−0.403258	−0.201629	+	0.979462i	$0.564624\pi$
$734$	−6.12449	−0.226059
$735$	9.77879	0.360696
$736$	34.4362	1.26934
$737$	−11.8199	−0.435390
$738$	−1.63753	−0.0602784
$739$	13.5495	0.498428	0.249214	−	0.968448i	$-0.419828\pi$
$739$	13.5495	0.498428	0.249214	+	0.968448i	$0.419828\pi$
$740$	10.0881	0.370846
$741$	−4.94354	−0.181605
$742$	−6.73993	−0.247430
$743$	25.0140	0.917674	0.458837	−	0.888521i	$-0.348266\pi$
$743$	25.0140	0.917674	0.458837	+	0.888521i	$0.348266\pi$
$744$	−2.75593	−0.101037
$745$	15.5957	0.571382
$746$	−6.73439	−0.246563
$747$	3.93449	0.143956
$748$	0.310770	0.0113629
$749$	28.2872	1.03359
$750$	5.34081	0.195019
$751$	−18.7752	−0.685116	−0.342558	−	0.939497i	$-0.611293\pi$
$751$	−18.7752	−0.685116	−0.342558	+	0.939497i	$0.611293\pi$
$752$	13.9122	0.507327
$753$	7.20418	0.262535
$754$	1.13151	0.0412070
$755$	−22.3868	−0.814739
$756$	2.89144	0.105161
$757$	15.1998	0.552446	0.276223	−	0.961094i	$-0.410917\pi$
$757$	15.1998	0.552446	0.276223	+	0.961094i	$0.410917\pi$
$758$	10.9288	0.396953
$759$	−5.53949	−0.201071
$760$	−32.0062	−1.16099
$761$	27.5896	1.00012	0.500061	−	0.865990i	$-0.333311\pi$
$761$	27.5896	1.00012	0.500061	+	0.865990i	$0.333311\pi$
$762$	−0.484093	−0.0175368
$763$	17.5092	0.633875
$764$	4.90895	0.177600
$765$	−0.502774	−0.0181778
$766$	3.79202	0.137011
$767$	4.17223	0.150651
$768$	−0.370256	−0.0133604
$769$	17.2440	0.621834	0.310917	−	0.950437i	$-0.399364\pi$
$769$	17.2440	0.621834	0.310917	+	0.950437i	$0.399364\pi$
$770$	1.42315	0.0512867
$771$	29.0206	1.04515
$772$	31.5078	1.13399
$773$	−2.64575	−0.0951609	−0.0475805	−	0.998867i	$-0.515151\pi$
$773$	−2.64575	−0.0951609	−0.0475805	+	0.998867i	$0.515151\pi$
$774$	−2.33114	−0.0837912
$775$	−0.193592	−0.00695402
$776$	10.8837	0.390702
$777$	4.13177	0.148226
$778$	1.77817	0.0637505
$779$	−26.2271	−0.939682
$780$	2.54936	0.0912818
$781$	−2.01841	−0.0722244
$782$	−0.751031	−0.0268568
$783$	−3.66588	−0.131008
$784$	−11.4384	−0.408514
$785$	13.5052	0.482021
$786$	4.32178	0.154153
$787$	−50.1003	−1.78588	−0.892942	−	0.450172i	$-0.851363\pi$
$787$	−50.1003	−1.78588	−0.892942	+	0.450172i	$0.851363\pi$
$788$	−3.97103	−0.141462
$789$	13.5877	0.483734
$790$	4.62834	0.164669
$791$	3.05264	0.108539
$792$	−1.44513	−0.0513503
$793$	−6.90192	−0.245095
$794$	−10.3851	−0.368554
$795$	19.2528	0.682827
$796$	18.8215	0.667111
$797$	28.0183	0.992459	0.496229	−	0.868191i	$-0.334717\pi$
$797$	28.0183	0.992459	0.496229	+	0.868191i	$0.334717\pi$
$798$	−6.14646	−0.217582
$799$	−1.16610	−0.0412538
$800$	0.631060	0.0223113
$801$	12.1762	0.430226
$802$	−17.9212	−0.632818
$803$	−0.119255	−0.00420841
$804$	−26.3258	−0.928438
$805$	25.9130	0.913313
$806$	−0.466635	−0.0164365
$807$	−24.4150	−0.859449
$808$	15.4135	0.542245
$809$	19.8293	0.697159	0.348580	−	0.937279i	$-0.386664\pi$
$809$	19.8293	0.697159	0.348580	+	0.937279i	$0.386664\pi$
$810$	1.09624	0.0385179
$811$	54.8034	1.92441	0.962205	−	0.272327i	$-0.0877934\pi$
$811$	54.8034	1.92441	0.962205	+	0.272327i	$0.0877934\pi$
$812$	−10.5997	−0.371976
$813$	−6.42127	−0.225204
$814$	−0.968262	−0.0339376
$815$	−39.2237	−1.37395
$816$	0.588102	0.0205877
$817$	−37.3361	−1.30622
$818$	−14.2269	−0.497433
$819$	1.04414	0.0364851
$820$	13.5252	0.472320
$821$	−2.88058	−0.100533	−0.0502665	−	0.998736i	$-0.516007\pi$
$821$	−2.88058	−0.100533	−0.0502665	+	0.998736i	$0.516007\pi$
$822$	2.49679	0.0870855
$823$	−21.0629	−0.734208	−0.367104	−	0.930180i	$-0.619651\pi$
$823$	−21.0629	−0.734208	−0.367104	+	0.930180i	$0.619651\pi$
$824$	−8.66910	−0.302003
$825$	−0.101514	−0.00353425
$826$	5.18747	0.180495
$827$	23.4195	0.814374	0.407187	−	0.913345i	$-0.366510\pi$
$827$	23.4195	0.814374	0.407187	+	0.913345i	$0.366510\pi$
$828$	−12.3378	−0.428769
$829$	−20.8181	−0.723043	−0.361521	−	0.932364i	$-0.617743\pi$
$829$	−20.8181	−0.723043	−0.361521	+	0.932364i	$0.617743\pi$
$830$	4.31314	0.149711
$831$	−21.9567	−0.761668
$832$	−1.85670	−0.0643696
$833$	0.958750	0.0332187
$834$	0.297251	0.0102930
$835$	−46.3871	−1.60529
$836$	−10.8525	−0.375342
$837$	1.51182	0.0522561
$838$	−18.5259	−0.639966
$839$	6.26825	0.216404	0.108202	−	0.994129i	$-0.465491\pi$
$839$	6.26825	0.216404	0.108202	+	0.994129i	$0.465491\pi$
$840$	6.76011	0.233246
$841$	−15.5613	−0.536596
$842$	10.9781	0.378330
$843$	−22.1860	−0.764125
$844$	−27.1552	−0.934721
$845$	−28.5182	−0.981054
$846$	2.54255	0.0874147
$847$	−16.9845	−0.583593
$848$	−22.5203	−0.773351
$849$	25.7123	0.882443
$850$	−0.0137630	−0.000472067 0
$851$	−17.6303	−0.604360
$852$	−4.49550	−0.154013
$853$	−3.62445	−0.124099	−0.0620494	−	0.998073i	$-0.519764\pi$
$853$	−3.62445	−0.124099	−0.0620494	+	0.998073i	$0.519764\pi$
$854$	−8.58139	−0.293649
$855$	17.5576	0.600456
$856$	−31.4884	−1.07625
$857$	49.6882	1.69732	0.848658	−	0.528942i	$-0.177411\pi$
$857$	49.6882	1.69732	0.848658	+	0.528942i	$0.177411\pi$
$858$	−0.244689	−0.00835356
$859$	34.9498	1.19247	0.596235	−	0.802810i	$-0.296663\pi$
$859$	34.9498	1.19247	0.596235	+	0.802810i	$0.296663\pi$
$860$	19.2540	0.656558
$861$	5.53949	0.188785
$862$	12.5596	0.427783
$863$	−40.2792	−1.37112	−0.685559	−	0.728017i	$-0.740442\pi$
$863$	−40.2792	−1.37112	−0.685559	+	0.728017i	$0.740442\pi$
$864$	−4.92814	−0.167659
$865$	31.8362	1.08246
$866$	−11.0916	−0.376906
$867$	16.9507	0.575676
$868$	4.37133	0.148373
$869$	3.34701	0.113540
$870$	−4.01868	−0.136246
$871$	−9.50659	−0.322119
$872$	−19.4907	−0.660037
$873$	−5.97045	−0.202069
$874$	26.2271	0.887144
$875$	−18.0670	−0.610778
$876$	−0.265610	−0.00897413
$877$	34.2437	1.15633	0.578163	−	0.815921i	$-0.303770\pi$
$877$	34.2437	1.15633	0.578163	+	0.815921i	$0.303770\pi$
$878$	−9.22232	−0.311238
$879$	17.8700	0.602741
$880$	4.75521	0.160298
$881$	−42.6928	−1.43836	−0.719178	−	0.694826i	$-0.755481\pi$
$881$	−42.6928	−1.43836	−0.719178	+	0.694826i	$0.755481\pi$
$882$	−2.09044	−0.0703887
$883$	23.7904	0.800611	0.400305	−	0.916382i	$-0.368904\pi$
$883$	23.7904	0.800611	0.400305	+	0.916382i	$0.368904\pi$
$884$	0.249949	0.00840670
$885$	−14.8182	−0.498108
$886$	7.82805	0.262989
$887$	−50.6589	−1.70096	−0.850479	−	0.526008i	$-0.823688\pi$
$887$	−50.6589	−1.70096	−0.850479	+	0.526008i	$0.823688\pi$
$888$	−4.59935	−0.154344
$889$	1.63760	0.0549234
$890$	13.3481	0.447428
$891$	0.792751	0.0265582
$892$	−46.0383	−1.54148
$893$	40.7220	1.36271
$894$	−3.33393	−0.111503
$895$	22.0234	0.736162
$896$	−18.4492	−0.616343
$897$	−4.45536	−0.148760
$898$	18.4792	0.616659
$899$	−5.54215	−0.184841
$900$	−0.226096	−0.00753654
$901$	1.88762	0.0628858
$902$	−1.29816	−0.0432238
$903$	7.88585	0.262425
$904$	−3.39810	−0.113019
$905$	27.3397	0.908802
$906$	4.78569	0.158994
$907$	22.1483	0.735423	0.367712	−	0.929940i	$-0.380141\pi$
$907$	22.1483	0.735423	0.367712	+	0.929940i	$0.380141\pi$
$908$	−23.0282	−0.764217
$909$	−8.45536	−0.280446
$910$	1.14462	0.0379439
$911$	−49.4587	−1.63864	−0.819320	−	0.573337i	$-0.805649\pi$
$911$	−49.4587	−1.63864	−0.819320	+	0.573337i	$0.805649\pi$
$912$	−20.5374	−0.680060
$913$	3.11908	0.103226
$914$	−7.42339	−0.245544
$915$	24.5130	0.810375
$916$	19.3053	0.637864
$917$	−14.6198	−0.482790
$918$	0.107479	0.00354735
$919$	−0.995072	−0.0328244	−0.0164122	−	0.999865i	$-0.505224\pi$
$919$	−0.995072	−0.0328244	−0.0164122	+	0.999865i	$0.505224\pi$
$920$	−28.8455	−0.951008
$921$	21.9118	0.722018
$922$	−13.8163	−0.455016
$923$	−1.62339	−0.0534344
$924$	2.29219	0.0754076
$925$	−0.323084	−0.0106229
$926$	−6.12035	−0.201127
$927$	4.75560	0.156194
$928$	18.0660	0.593045
$929$	13.2889	0.435996	0.217998	−	0.975949i	$-0.430047\pi$
$929$	13.2889	0.435996	0.217998	+	0.975949i	$0.430047\pi$
$930$	1.65731	0.0543454
$931$	−33.4809	−1.09729
$932$	26.1972	0.858116
$933$	12.3845	0.405450
$934$	−1.22520	−0.0400897
$935$	−0.398575	−0.0130348
$936$	−1.16230	−0.0379910
$937$	12.1210	0.395977	0.197988	−	0.980204i	$-0.436559\pi$
$937$	12.1210	0.395977	0.197988	+	0.980204i	$0.436559\pi$
$938$	−11.8199	−0.385932
$939$	4.25938	0.139000
$940$	−21.0002	−0.684950
$941$	−22.3197	−0.727603	−0.363801	−	0.931477i	$-0.618521\pi$
$941$	−22.3197	−0.727603	−0.363801	+	0.931477i	$0.618521\pi$
$942$	−2.88704	−0.0940650
$943$	−23.6371	−0.769730
$944$	17.3331	0.564143
$945$	−3.70838	−0.120634
$946$	−1.84802	−0.0600842
$947$	41.1503	1.33721	0.668603	−	0.743619i	$-0.266892\pi$
$947$	41.1503	1.33721	0.668603	+	0.743619i	$0.266892\pi$
$948$	7.45463	0.242115
$949$	−0.0959154	−0.00311355
$950$	0.480623	0.0155935
$951$	5.97298	0.193687
$952$	0.662787	0.0214810
$953$	34.6749	1.12323	0.561615	−	0.827399i	$-0.310180\pi$
$953$	34.6749	1.12323	0.561615	+	0.827399i	$0.310180\pi$
$954$	−4.11573	−0.133252
$955$	−6.29592	−0.203731
$956$	−29.1748	−0.943581
$957$	−2.90613	−0.0939420
$958$	−5.19756	−0.167926
$959$	−8.44621	−0.272742
$960$	6.59431	0.212830
$961$	−28.7144	−0.926271
$962$	−0.778763	−0.0251083
$963$	17.2736	0.556633
$964$	0.360968	0.0116260
$965$	−40.4099	−1.30084
$966$	−5.53949	−0.178230
$967$	24.0075	0.772030	0.386015	−	0.922493i	$-0.373851\pi$
$967$	24.0075	0.772030	0.386015	+	0.922493i	$0.373851\pi$
$968$	18.9066	0.607680
$969$	1.72141	0.0552997
$970$	−6.54504	−0.210148
$971$	−7.13584	−0.229000	−0.114500	−	0.993423i	$-0.536527\pi$
$971$	−7.13584	−0.229000	−0.114500	+	0.993423i	$0.536527\pi$
$972$	1.76565	0.0566334
$973$	−1.00555	−0.0322364
$974$	−14.9158	−0.477933
$975$	−0.0816464	−0.00261478
$976$	−28.6732	−0.917808
$977$	−19.7149	−0.630734	−0.315367	−	0.948970i	$-0.602128\pi$
$977$	−19.7149	−0.630734	−0.315367	+	0.948970i	$0.602128\pi$
$978$	8.38495	0.268121
$979$	9.65273	0.308503
$980$	17.2660	0.551541
$981$	10.6920	0.341368
$982$	−0.501866	−0.0160152
$983$	25.9585	0.827947	0.413973	−	0.910289i	$-0.364141\pi$
$983$	25.9585	0.827947	0.413973	+	0.910289i	$0.364141\pi$
$984$	−6.16638	−0.196577
$985$	5.09300	0.162276
$986$	−0.394007	−0.0125477
$987$	−8.60101	−0.273773
$988$	−8.72858	−0.277693
$989$	−33.6491	−1.06998
$990$	0.869044	0.0276200
$991$	57.9578	1.84109	0.920545	−	0.390638i	$-0.127745\pi$
$991$	57.9578	1.84109	0.920545	+	0.390638i	$0.127745\pi$
$992$	−7.45044	−0.236552
$993$	−21.3536	−0.677635
$994$	−2.01841	−0.0640201
$995$	−24.1393	−0.765268
$996$	6.94696	0.220123
$997$	−42.6660	−1.35125	−0.675623	−	0.737247i	$-0.736125\pi$
$997$	−42.6660	−1.35125	−0.675623	+	0.737247i	$0.736125\pi$
$998$	20.4391	0.646990
$999$	2.52306	0.0798261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	381.2.a.d.1.3	✓	5
3.2	odd	2		1143.2.a.g.1.3		5
4.3	odd	2		6096.2.a.bf.1.5		5
5.4	even	2		9525.2.a.j.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.d.1.3	✓	5	1.1	even	1	trivial
1143.2.a.g.1.3		5	3.2	odd	2
6096.2.a.bf.1.5		5	4.3	odd	2
9525.2.a.j.1.3		5	5.4	even	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:	\( N \)	\(=\)	\( 381 = 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	381.a (trivial)

\(f(q)\)	\(=\)	\(q+0.484093 q^{2} -1.00000 q^{3} -1.76565 q^{4} +2.26452 q^{5} -0.484093 q^{6} +1.63760 q^{7} -1.82293 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.484093 q^{2} -1.00000 q^{3} -1.76565 q^{4} +2.26452 q^{5} -0.484093 q^{6} +1.63760 q^{7} -1.82293 q^{8} +1.00000 q^{9} +1.09624 q^{10} +0.792751 q^{11} +1.76565 q^{12} +0.637602 q^{13} +0.792751 q^{14} -2.26452 q^{15} +2.64884 q^{16} -0.222022 q^{17} +0.484093 q^{18} +7.75333 q^{19} -3.99836 q^{20} -1.63760 q^{21} +0.383765 q^{22} +6.98768 q^{23} +1.82293 q^{24} +0.128052 q^{25} +0.308658 q^{26} -1.00000 q^{27} -2.89144 q^{28} +3.66588 q^{29} -1.09624 q^{30} -1.51182 q^{31} +4.92814 q^{32} -0.792751 q^{33} -0.107479 q^{34} +3.70838 q^{35} -1.76565 q^{36} -2.52306 q^{37} +3.75333 q^{38} -0.637602 q^{39} -4.12805 q^{40} -3.38268 q^{41} -0.792751 q^{42} -4.81549 q^{43} -1.39972 q^{44} +2.26452 q^{45} +3.38268 q^{46} +5.25220 q^{47} -2.64884 q^{48} -4.31826 q^{49} +0.0619892 q^{50} +0.222022 q^{51} -1.12578 q^{52} -8.50194 q^{53} -0.484093 q^{54} +1.79520 q^{55} -2.98523 q^{56} -7.75333 q^{57} +1.77463 q^{58} +6.54363 q^{59} +3.99836 q^{60} -10.8248 q^{61} -0.731860 q^{62} +1.63760 q^{63} -2.91201 q^{64} +1.44386 q^{65} -0.383765 q^{66} -14.9099 q^{67} +0.392015 q^{68} -6.98768 q^{69} +1.79520 q^{70} -2.54608 q^{71} -1.82293 q^{72} -0.150431 q^{73} -1.22139 q^{74} -0.128052 q^{75} -13.6897 q^{76} +1.29821 q^{77} -0.308658 q^{78} +4.22202 q^{79} +5.99836 q^{80} +1.00000 q^{81} -1.63753 q^{82} +3.93449 q^{83} +2.89144 q^{84} -0.502774 q^{85} -2.33114 q^{86} -3.66588 q^{87} -1.44513 q^{88} +12.1762 q^{89} +1.09624 q^{90} +1.04414 q^{91} -12.3378 q^{92} +1.51182 q^{93} +2.54255 q^{94} +17.5576 q^{95} -4.92814 q^{96} -5.97045 q^{97} -2.09044 q^{98} +0.792751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + 2 * q^2 - 5 * q^3 + 6 * q^4 + q^5 - 2 * q^6 + 6 * q^8 + 5 * q^9	\( 5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9} - 2 q^{10} + 14 q^{11} - 6 q^{12} - 5 q^{13} + 14 q^{14} - q^{15} + 8 q^{16} + 4 q^{17} + 2 q^{18} + 4 q^{19} + 6 q^{20} - 2 q^{22} + 15 q^{23} - 6 q^{24} - 6 q^{25} + 12 q^{26} - 5 q^{27} - 2 q^{28} + 9 q^{29} + 2 q^{30} + 3 q^{31} + 14 q^{32} - 14 q^{33} + 4 q^{34} + 4 q^{35} + 6 q^{36} - 5 q^{37} - 16 q^{38} + 5 q^{39} - 14 q^{40} + 4 q^{41} - 14 q^{42} + 10 q^{43} + 18 q^{44} + q^{45} - 4 q^{46} - 4 q^{47} - 8 q^{48} - 9 q^{49} - 24 q^{50} - 4 q^{51} - 8 q^{52} + 3 q^{53} - 2 q^{54} + 4 q^{55} - 10 q^{56} - 4 q^{57} + 6 q^{58} + 23 q^{59} - 6 q^{60} - 15 q^{61} - 24 q^{62} + 3 q^{65} + 2 q^{66} + 18 q^{67} - 24 q^{68} - 15 q^{69} + 4 q^{70} + 12 q^{71} + 6 q^{72} - 43 q^{73} - 36 q^{74} + 6 q^{75} - 32 q^{76} + 4 q^{77} - 12 q^{78} + 16 q^{79} + 4 q^{80} + 5 q^{81} - 44 q^{82} + 11 q^{83} + 2 q^{84} - 24 q^{85} - 28 q^{86} - 9 q^{87} - 14 q^{88} + 9 q^{89} - 2 q^{90} + 26 q^{91} + 14 q^{92} - 3 q^{93} - 14 q^{94} + 16 q^{95} - 14 q^{96} - 20 q^{97} - 10 q^{98} + 14 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 - 5 * q^3 + 6 * q^4 + q^5 - 2 * q^6 + 6 * q^8 + 5 * q^9 - 2 * q^10 + 14 * q^11 - 6 * q^12 - 5 * q^13 + 14 * q^14 - q^15 + 8 * q^16 + 4 * q^17 + 2 * q^18 + 4 * q^19 + 6 * q^20 - 2 * q^22 + 15 * q^23 - 6 * q^24 - 6 * q^25 + 12 * q^26 - 5 * q^27 - 2 * q^28 + 9 * q^29 + 2 * q^30 + 3 * q^31 + 14 * q^32 - 14 * q^33 + 4 * q^34 + 4 * q^35 + 6 * q^36 - 5 * q^37 - 16 * q^38 + 5 * q^39 - 14 * q^40 + 4 * q^41 - 14 * q^42 + 10 * q^43 + 18 * q^44 + q^45 - 4 * q^46 - 4 * q^47 - 8 * q^48 - 9 * q^49 - 24 * q^50 - 4 * q^51 - 8 * q^52 + 3 * q^53 - 2 * q^54 + 4 * q^55 - 10 * q^56 - 4 * q^57 + 6 * q^58 + 23 * q^59 - 6 * q^60 - 15 * q^61 - 24 * q^62 + 3 * q^65 + 2 * q^66 + 18 * q^67 - 24 * q^68 - 15 * q^69 + 4 * q^70 + 12 * q^71 + 6 * q^72 - 43 * q^73 - 36 * q^74 + 6 * q^75 - 32 * q^76 + 4 * q^77 - 12 * q^78 + 16 * q^79 + 4 * q^80 + 5 * q^81 - 44 * q^82 + 11 * q^83 + 2 * q^84 - 24 * q^85 - 28 * q^86 - 9 * q^87 - 14 * q^88 + 9 * q^89 - 2 * q^90 + 26 * q^91 + 14 * q^92 - 3 * q^93 - 14 * q^94 + 16 * q^95 - 14 * q^96 - 20 * q^97 - 10 * q^98 + 14 * q^99

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