LMFDB - Embedded newform 6080.2.a.bu.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.21432 q^{3} -1.00000 q^{5} -1.52543 q^{7} +1.90321 q^{9} +O(q^{10})$	$q-2.21432 q^{3} -1.00000 q^{5} -1.52543 q^{7} +1.90321 q^{9} +2.90321 q^{11} +1.31111 q^{13} +2.21432 q^{15} -7.80642 q^{17} +1.00000 q^{19} +3.37778 q^{21} -4.28100 q^{23} +1.00000 q^{25} +2.42864 q^{27} +9.18421 q^{29} +7.80642 q^{31} -6.42864 q^{33} +1.52543 q^{35} -8.16839 q^{37} -2.90321 q^{39} -11.0509 q^{41} +1.09679 q^{43} -1.90321 q^{45} -4.28100 q^{47} -4.67307 q^{49} +17.2859 q^{51} -5.54617 q^{53} -2.90321 q^{55} -2.21432 q^{57} -1.86665 q^{59} -0.709636 q^{61} -2.90321 q^{63} -1.31111 q^{65} +1.45875 q^{67} +9.47949 q^{69} -2.29529 q^{71} +14.0415 q^{73} -2.21432 q^{75} -4.42864 q^{77} +6.13335 q^{79} -11.0874 q^{81} -11.7605 q^{83} +7.80642 q^{85} -20.3368 q^{87} +3.67307 q^{89} -2.00000 q^{91} -17.2859 q^{93} -1.00000 q^{95} -11.2192 q^{97} +5.52543 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 + 2 * q^7 - q^9	$ 3 q - 3 q^{5} + 2 q^{7} - q^{9} + 2 q^{11} + 4 q^{13} - 10 q^{17} + 3 q^{19} + 10 q^{21} - 6 q^{23} + 3 q^{25} - 6 q^{27} + 14 q^{29} + 10 q^{31} - 6 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{39} - 20 q^{41} + 10 q^{43} + q^{45} - 6 q^{47} - q^{49} + 12 q^{51} + 10 q^{53} - 2 q^{55} - 6 q^{59} + 18 q^{61} - 2 q^{63} - 4 q^{65} - 2 q^{67} + 2 q^{69} + 6 q^{71} + 2 q^{73} + 18 q^{79} - 13 q^{81} - 2 q^{83} + 10 q^{85} - 8 q^{87} - 2 q^{89} - 6 q^{91} - 12 q^{93} - 3 q^{95} + 6 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - q^9 + 2 * q^11 + 4 * q^13 - 10 * q^17 + 3 * q^19 + 10 * q^21 - 6 * q^23 + 3 * q^25 - 6 * q^27 + 14 * q^29 + 10 * q^31 - 6 * q^33 - 2 * q^35 + 2 * q^37 - 2 * q^39 - 20 * q^41 + 10 * q^43 + q^45 - 6 * q^47 - q^49 + 12 * q^51 + 10 * q^53 - 2 * q^55 - 6 * q^59 + 18 * q^61 - 2 * q^63 - 4 * q^65 - 2 * q^67 + 2 * q^69 + 6 * q^71 + 2 * q^73 + 18 * q^79 - 13 * q^81 - 2 * q^83 + 10 * q^85 - 8 * q^87 - 2 * q^89 - 6 * q^91 - 12 * q^93 - 3 * q^95 + 6 * q^97 + 10 * 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	0
$2$	0	0
$3$	−2.21432	−1.27844	−0.639219	−	0.769025i	$-0.720742\pi$
$3$	−2.21432	−1.27844	−0.639219	+	0.769025i	$0.720742\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.52543	−0.576557	−0.288279	−	0.957547i	$-0.593083\pi$
$7$	−1.52543	−0.576557	−0.288279	+	0.957547i	$0.593083\pi$
$8$	0	0
$9$	1.90321	0.634404
$10$	0	0
$11$	2.90321	0.875351	0.437676	−	0.899133i	$-0.355802\pi$
$11$	2.90321	0.875351	0.437676	+	0.899133i	$0.355802\pi$
$12$	0	0
$13$	1.31111	0.363636	0.181818	−	0.983332i	$-0.441802\pi$
$13$	1.31111	0.363636	0.181818	+	0.983332i	$0.441802\pi$
$14$	0	0
$15$	2.21432	0.571735
$16$	0	0
$17$	−7.80642	−1.89334	−0.946668	−	0.322211i	$-0.895574\pi$
$17$	−7.80642	−1.89334	−0.946668	+	0.322211i	$0.895574\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.37778	0.737093
$22$	0	0
$23$	−4.28100	−0.892649	−0.446325	−	0.894871i	$-0.647267\pi$
$23$	−4.28100	−0.892649	−0.446325	+	0.894871i	$0.647267\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.42864	0.467392
$28$	0	0
$29$	9.18421	1.70546	0.852732	−	0.522348i	$-0.174944\pi$
$29$	9.18421	1.70546	0.852732	+	0.522348i	$0.174944\pi$
$30$	0	0
$31$	7.80642	1.40208	0.701038	−	0.713124i	$-0.252721\pi$
$31$	7.80642	1.40208	0.701038	+	0.713124i	$0.252721\pi$
$32$	0	0
$33$	−6.42864	−1.11908
$34$	0	0
$35$	1.52543	0.257844
$36$	0	0
$37$	−8.16839	−1.34287	−0.671437	−	0.741061i	$-0.734323\pi$
$37$	−8.16839	−1.34287	−0.671437	+	0.741061i	$0.734323\pi$
$38$	0	0
$39$	−2.90321	−0.464886
$40$	0	0
$41$	−11.0509	−1.72585	−0.862927	−	0.505329i	$-0.831371\pi$
$41$	−11.0509	−1.72585	−0.862927	+	0.505329i	$0.831371\pi$
$42$	0	0
$43$	1.09679	0.167259	0.0836293	−	0.996497i	$-0.473349\pi$
$43$	1.09679	0.167259	0.0836293	+	0.996497i	$0.473349\pi$
$44$	0	0
$45$	−1.90321	−0.283714
$46$	0	0
$47$	−4.28100	−0.624447	−0.312224	−	0.950009i	$-0.601074\pi$
$47$	−4.28100	−0.624447	−0.312224	+	0.950009i	$0.601074\pi$
$48$	0	0
$49$	−4.67307	−0.667582
$50$	0	0
$51$	17.2859	2.42051
$52$	0	0
$53$	−5.54617	−0.761825	−0.380913	−	0.924611i	$-0.624390\pi$
$53$	−5.54617	−0.761825	−0.380913	+	0.924611i	$0.624390\pi$
$54$	0	0
$55$	−2.90321	−0.391469
$56$	0	0
$57$	−2.21432	−0.293294
$58$	0	0
$59$	−1.86665	−0.243017	−0.121508	−	0.992590i	$-0.538773\pi$
$59$	−1.86665	−0.243017	−0.121508	+	0.992590i	$0.538773\pi$
$60$	0	0
$61$	−0.709636	−0.0908596	−0.0454298	−	0.998968i	$-0.514466\pi$
$61$	−0.709636	−0.0908596	−0.0454298	+	0.998968i	$0.514466\pi$
$62$	0	0
$63$	−2.90321	−0.365770
$64$	0	0
$65$	−1.31111	−0.162623
$66$	0	0
$67$	1.45875	0.178215	0.0891074	−	0.996022i	$-0.471599\pi$
$67$	1.45875	0.178215	0.0891074	+	0.996022i	$0.471599\pi$
$68$	0	0
$69$	9.47949	1.14120
$70$	0	0
$71$	−2.29529	−0.272400	−0.136200	−	0.990681i	$-0.543489\pi$
$71$	−2.29529	−0.272400	−0.136200	+	0.990681i	$0.543489\pi$
$72$	0	0
$73$	14.0415	1.64343	0.821716	−	0.569897i	$-0.193017\pi$
$73$	14.0415	1.64343	0.821716	+	0.569897i	$0.193017\pi$
$74$	0	0
$75$	−2.21432	−0.255688
$76$	0	0
$77$	−4.42864	−0.504690
$78$	0	0
$79$	6.13335	0.690056	0.345028	−	0.938592i	$-0.387869\pi$
$79$	6.13335	0.690056	0.345028	+	0.938592i	$0.387869\pi$
$80$	0	0
$81$	−11.0874	−1.23194
$82$	0	0
$83$	−11.7605	−1.29088	−0.645441	−	0.763810i	$-0.723326\pi$
$83$	−11.7605	−1.29088	−0.645441	+	0.763810i	$0.723326\pi$
$84$	0	0
$85$	7.80642	0.846726
$86$	0	0
$87$	−20.3368	−2.18033
$88$	0	0
$89$	3.67307	0.389345	0.194672	−	0.980868i	$-0.437636\pi$
$89$	3.67307	0.389345	0.194672	+	0.980868i	$0.437636\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−17.2859	−1.79247
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−11.2192	−1.13914	−0.569571	−	0.821942i	$-0.692891\pi$
$97$	−11.2192	−1.13914	−0.569571	+	0.821942i	$0.692891\pi$
$98$	0	0
$99$	5.52543	0.555326
$100$	0	0
$101$	0.963435	0.0958654	0.0479327	−	0.998851i	$-0.484737\pi$
$101$	0.963435	0.0958654	0.0479327	+	0.998851i	$0.484737\pi$
$102$	0	0
$103$	17.8874	1.76250	0.881248	−	0.472653i	$-0.156704\pi$
$103$	17.8874	1.76250	0.881248	+	0.472653i	$0.156704\pi$
$104$	0	0
$105$	−3.37778	−0.329638
$106$	0	0
$107$	13.5002	1.30512	0.652559	−	0.757738i	$-0.273696\pi$
$107$	13.5002	1.30512	0.652559	+	0.757738i	$0.273696\pi$
$108$	0	0
$109$	−10.5303	−1.00862	−0.504312	−	0.863521i	$-0.668254\pi$
$109$	−10.5303	−1.00862	−0.504312	+	0.863521i	$0.668254\pi$
$110$	0	0
$111$	18.0874	1.71678
$112$	0	0
$113$	−18.5970	−1.74946	−0.874731	−	0.484610i	$-0.838962\pi$
$113$	−18.5970	−1.74946	−0.874731	+	0.484610i	$0.838962\pi$
$114$	0	0
$115$	4.28100	0.399205
$116$	0	0
$117$	2.49532	0.230692
$118$	0	0
$119$	11.9081	1.09162
$120$	0	0
$121$	−2.57136	−0.233760
$122$	0	0
$123$	24.4701	2.20640
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.2050	−1.34922	−0.674611	−	0.738174i	$-0.735688\pi$
$127$	−15.2050	−1.34922	−0.674611	+	0.738174i	$0.735688\pi$
$128$	0	0
$129$	−2.42864	−0.213830
$130$	0	0
$131$	−11.0923	−0.969142	−0.484571	−	0.874752i	$-0.661024\pi$
$131$	−11.0923	−0.969142	−0.484571	+	0.874752i	$0.661024\pi$
$132$	0	0
$133$	−1.52543	−0.132271
$134$	0	0
$135$	−2.42864	−0.209024
$136$	0	0
$137$	12.1017	1.03392	0.516959	−	0.856010i	$-0.327064\pi$
$137$	12.1017	1.03392	0.516959	+	0.856010i	$0.327064\pi$
$138$	0	0
$139$	10.3412	0.877131	0.438565	−	0.898699i	$-0.355487\pi$
$139$	10.3412	0.877131	0.438565	+	0.898699i	$0.355487\pi$
$140$	0	0
$141$	9.47949	0.798317
$142$	0	0
$143$	3.80642	0.318309
$144$	0	0
$145$	−9.18421	−0.762707
$146$	0	0
$147$	10.3477	0.853462
$148$	0	0
$149$	5.65878	0.463585	0.231793	−	0.972765i	$-0.425541\pi$
$149$	5.65878	0.463585	0.231793	+	0.972765i	$0.425541\pi$
$150$	0	0
$151$	11.4795	0.934188	0.467094	−	0.884208i	$-0.345301\pi$
$151$	11.4795	0.934188	0.467094	+	0.884208i	$0.345301\pi$
$152$	0	0
$153$	−14.8573	−1.20114
$154$	0	0
$155$	−7.80642	−0.627027
$156$	0	0
$157$	0.488863	0.0390155	0.0195077	−	0.999810i	$-0.493790\pi$
$157$	0.488863	0.0390155	0.0195077	+	0.999810i	$0.493790\pi$
$158$	0	0
$159$	12.2810	0.973946
$160$	0	0
$161$	6.53035	0.514664
$162$	0	0
$163$	−21.1798	−1.65893	−0.829464	−	0.558561i	$-0.811354\pi$
$163$	−21.1798	−1.65893	−0.829464	+	0.558561i	$0.811354\pi$
$164$	0	0
$165$	6.42864	0.500469
$166$	0	0
$167$	16.4494	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$167$	16.4494	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$168$	0	0
$169$	−11.2810	−0.867769
$170$	0	0
$171$	1.90321	0.145542
$172$	0	0
$173$	19.8415	1.50852	0.754259	−	0.656577i	$-0.227996\pi$
$173$	19.8415	1.50852	0.754259	+	0.656577i	$0.227996\pi$
$174$	0	0
$175$	−1.52543	−0.115311
$176$	0	0
$177$	4.13335	0.310682
$178$	0	0
$179$	−20.7052	−1.54758	−0.773789	−	0.633443i	$-0.781641\pi$
$179$	−20.7052	−1.54758	−0.773789	+	0.633443i	$0.781641\pi$
$180$	0	0
$181$	11.5111	0.855616	0.427808	−	0.903870i	$-0.359286\pi$
$181$	11.5111	0.855616	0.427808	+	0.903870i	$0.359286\pi$
$182$	0	0
$183$	1.57136	0.116158
$184$	0	0
$185$	8.16839	0.600552
$186$	0	0
$187$	−22.6637	−1.65733
$188$	0	0
$189$	−3.70471	−0.269478
$190$	0	0
$191$	−10.0731	−0.728866	−0.364433	−	0.931230i	$-0.618737\pi$
$191$	−10.0731	−0.728866	−0.364433	+	0.931230i	$0.618737\pi$
$192$	0	0
$193$	−12.7304	−0.916353	−0.458176	−	0.888861i	$-0.651497\pi$
$193$	−12.7304	−0.916353	−0.458176	+	0.888861i	$0.651497\pi$
$194$	0	0
$195$	2.90321	0.207903
$196$	0	0
$197$	−12.2351	−0.871712	−0.435856	−	0.900016i	$-0.643554\pi$
$197$	−12.2351	−0.871712	−0.435856	+	0.900016i	$0.643554\pi$
$198$	0	0
$199$	5.24443	0.371768	0.185884	−	0.982572i	$-0.440485\pi$
$199$	5.24443	0.371768	0.185884	+	0.982572i	$0.440485\pi$
$200$	0	0
$201$	−3.23014	−0.227837
$202$	0	0
$203$	−14.0098	−0.983298
$204$	0	0
$205$	11.0509	0.771825
$206$	0	0
$207$	−8.14764	−0.566300
$208$	0	0
$209$	2.90321	0.200819
$210$	0	0
$211$	16.7239	1.15132	0.575661	−	0.817688i	$-0.304745\pi$
$211$	16.7239	1.15132	0.575661	+	0.817688i	$0.304745\pi$
$212$	0	0
$213$	5.08250	0.348247
$214$	0	0
$215$	−1.09679	−0.0748003
$216$	0	0
$217$	−11.9081	−0.808377
$218$	0	0
$219$	−31.0923	−2.10103
$220$	0	0
$221$	−10.2351	−0.688485
$222$	0	0
$223$	27.5002	1.84155	0.920776	−	0.390092i	$-0.127557\pi$
$223$	27.5002	1.84155	0.920776	+	0.390092i	$0.127557\pi$
$224$	0	0
$225$	1.90321	0.126881
$226$	0	0
$227$	19.3067	1.28143	0.640714	−	0.767780i	$-0.278638\pi$
$227$	19.3067	1.28143	0.640714	+	0.767780i	$0.278638\pi$
$228$	0	0
$229$	19.2400	1.27141	0.635707	−	0.771930i	$-0.280709\pi$
$229$	19.2400	1.27141	0.635707	+	0.771930i	$0.280709\pi$
$230$	0	0
$231$	9.80642	0.645215
$232$	0	0
$233$	−14.9906	−0.982069	−0.491034	−	0.871140i	$-0.663381\pi$
$233$	−14.9906	−0.982069	−0.491034	+	0.871140i	$0.663381\pi$
$234$	0	0
$235$	4.28100	0.279261
$236$	0	0
$237$	−13.5812	−0.882194
$238$	0	0
$239$	11.7748	0.761647	0.380824	−	0.924648i	$-0.375641\pi$
$239$	11.7748	0.761647	0.380824	+	0.924648i	$0.375641\pi$
$240$	0	0
$241$	−10.4889	−0.675647	−0.337824	−	0.941209i	$-0.609691\pi$
$241$	−10.4889	−0.675647	−0.337824	+	0.941209i	$0.609691\pi$
$242$	0	0
$243$	17.2652	1.10756
$244$	0	0
$245$	4.67307	0.298552
$246$	0	0
$247$	1.31111	0.0834238
$248$	0	0
$249$	26.0415	1.65031
$250$	0	0
$251$	−11.0509	−0.697524	−0.348762	−	0.937211i	$-0.613398\pi$
$251$	−11.0509	−0.697524	−0.348762	+	0.937211i	$0.613398\pi$
$252$	0	0
$253$	−12.4286	−0.781382
$254$	0	0
$255$	−17.2859	−1.08249
$256$	0	0
$257$	17.6795	1.10282	0.551409	−	0.834235i	$-0.314090\pi$
$257$	17.6795	1.10282	0.551409	+	0.834235i	$0.314090\pi$
$258$	0	0
$259$	12.4603	0.774244
$260$	0	0
$261$	17.4795	1.08195
$262$	0	0
$263$	21.1383	1.30344	0.651721	−	0.758459i	$-0.274047\pi$
$263$	21.1383	1.30344	0.651721	+	0.758459i	$0.274047\pi$
$264$	0	0
$265$	5.54617	0.340699
$266$	0	0
$267$	−8.13335	−0.497753
$268$	0	0
$269$	17.5812	1.07194	0.535972	−	0.844235i	$-0.319945\pi$
$269$	17.5812	1.07194	0.535972	+	0.844235i	$0.319945\pi$
$270$	0	0
$271$	−21.0049	−1.27596	−0.637979	−	0.770054i	$-0.720229\pi$
$271$	−21.0049	−1.27596	−0.637979	+	0.770054i	$0.720229\pi$
$272$	0	0
$273$	4.42864	0.268033
$274$	0	0
$275$	2.90321	0.175070
$276$	0	0
$277$	6.85728	0.412014	0.206007	−	0.978551i	$-0.433953\pi$
$277$	6.85728	0.412014	0.206007	+	0.978551i	$0.433953\pi$
$278$	0	0
$279$	14.8573	0.889482
$280$	0	0
$281$	17.0321	1.01605	0.508026	−	0.861342i	$-0.330376\pi$
$281$	17.0321	1.01605	0.508026	+	0.861342i	$0.330376\pi$
$282$	0	0
$283$	27.7605	1.65019	0.825095	−	0.564994i	$-0.191122\pi$
$283$	27.7605	1.65019	0.825095	+	0.564994i	$0.191122\pi$
$284$	0	0
$285$	2.21432	0.131165
$286$	0	0
$287$	16.8573	0.995054
$288$	0	0
$289$	43.9403	2.58472
$290$	0	0
$291$	24.8430	1.45632
$292$	0	0
$293$	32.3432	1.88951	0.944756	−	0.327775i	$-0.106299\pi$
$293$	32.3432	1.88951	0.944756	+	0.327775i	$0.106299\pi$
$294$	0	0
$295$	1.86665	0.108680
$296$	0	0
$297$	7.05086	0.409132
$298$	0	0
$299$	−5.61285	−0.324599
$300$	0	0
$301$	−1.67307	−0.0964342
$302$	0	0
$303$	−2.13335	−0.122558
$304$	0	0
$305$	0.709636	0.0406336
$306$	0	0
$307$	26.1541	1.49269	0.746347	−	0.665558i	$-0.231806\pi$
$307$	26.1541	1.49269	0.746347	+	0.665558i	$0.231806\pi$
$308$	0	0
$309$	−39.6084	−2.25324
$310$	0	0
$311$	−14.1289	−0.801177	−0.400588	−	0.916258i	$-0.631194\pi$
$311$	−14.1289	−0.801177	−0.400588	+	0.916258i	$0.631194\pi$
$312$	0	0
$313$	−6.99063	−0.395134	−0.197567	−	0.980289i	$-0.563304\pi$
$313$	−6.99063	−0.395134	−0.197567	+	0.980289i	$0.563304\pi$
$314$	0	0
$315$	2.90321	0.163577
$316$	0	0
$317$	8.62867	0.484634	0.242317	−	0.970197i	$-0.422093\pi$
$317$	8.62867	0.484634	0.242317	+	0.970197i	$0.422093\pi$
$318$	0	0
$319$	26.6637	1.49288
$320$	0	0
$321$	−29.8938	−1.66851
$322$	0	0
$323$	−7.80642	−0.434361
$324$	0	0
$325$	1.31111	0.0727272
$326$	0	0
$327$	23.3176	1.28946
$328$	0	0
$329$	6.53035	0.360030
$330$	0	0
$331$	−12.4286	−0.683140	−0.341570	−	0.939856i	$-0.610959\pi$
$331$	−12.4286	−0.683140	−0.341570	+	0.939856i	$0.610959\pi$
$332$	0	0
$333$	−15.5462	−0.851925
$334$	0	0
$335$	−1.45875	−0.0797001
$336$	0	0
$337$	−5.05731	−0.275489	−0.137745	−	0.990468i	$-0.543985\pi$
$337$	−5.05731	−0.275489	−0.137745	+	0.990468i	$0.543985\pi$
$338$	0	0
$339$	41.1798	2.23658
$340$	0	0
$341$	22.6637	1.22731
$342$	0	0
$343$	17.8064	0.961457
$344$	0	0
$345$	−9.47949	−0.510359
$346$	0	0
$347$	−3.59856	−0.193181	−0.0965903	−	0.995324i	$-0.530794\pi$
$347$	−3.59856	−0.193181	−0.0965903	+	0.995324i	$0.530794\pi$
$348$	0	0
$349$	−0.917502	−0.0491128	−0.0245564	−	0.999698i	$-0.507817\pi$
$349$	−0.917502	−0.0491128	−0.0245564	+	0.999698i	$0.507817\pi$
$350$	0	0
$351$	3.18421	0.169960
$352$	0	0
$353$	20.9304	1.11401	0.557007	−	0.830508i	$-0.311950\pi$
$353$	20.9304	1.11401	0.557007	+	0.830508i	$0.311950\pi$
$354$	0	0
$355$	2.29529	0.121821
$356$	0	0
$357$	−26.3684	−1.39556
$358$	0	0
$359$	4.41435	0.232980	0.116490	−	0.993192i	$-0.462836\pi$
$359$	4.41435	0.232980	0.116490	+	0.993192i	$0.462836\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	5.69381	0.298848
$364$	0	0
$365$	−14.0415	−0.734965
$366$	0	0
$367$	17.8336	0.930907	0.465454	−	0.885072i	$-0.345891\pi$
$367$	17.8336	0.930907	0.465454	+	0.885072i	$0.345891\pi$
$368$	0	0
$369$	−21.0321	−1.09489
$370$	0	0
$371$	8.46028	0.439236
$372$	0	0
$373$	−3.73975	−0.193637	−0.0968184	−	0.995302i	$-0.530867\pi$
$373$	−3.73975	−0.193637	−0.0968184	+	0.995302i	$0.530867\pi$
$374$	0	0
$375$	2.21432	0.114347
$376$	0	0
$377$	12.0415	0.620168
$378$	0	0
$379$	29.3274	1.50645	0.753224	−	0.657764i	$-0.228498\pi$
$379$	29.3274	1.50645	0.753224	+	0.657764i	$0.228498\pi$
$380$	0	0
$381$	33.6686	1.72490
$382$	0	0
$383$	−26.7447	−1.36659	−0.683294	−	0.730143i	$-0.739453\pi$
$383$	−26.7447	−1.36659	−0.683294	+	0.730143i	$0.739453\pi$
$384$	0	0
$385$	4.42864	0.225704
$386$	0	0
$387$	2.08742	0.106110
$388$	0	0
$389$	17.3876	0.881588	0.440794	−	0.897608i	$-0.354697\pi$
$389$	17.3876	0.881588	0.440794	+	0.897608i	$0.354697\pi$
$390$	0	0
$391$	33.4193	1.69009
$392$	0	0
$393$	24.5620	1.23899
$394$	0	0
$395$	−6.13335	−0.308602
$396$	0	0
$397$	−0.152089	−0.00763313	−0.00381656	−	0.999993i	$-0.501215\pi$
$397$	−0.152089	−0.00763313	−0.00381656	+	0.999993i	$0.501215\pi$
$398$	0	0
$399$	3.37778	0.169101
$400$	0	0
$401$	−34.3783	−1.71677	−0.858384	−	0.513007i	$-0.828531\pi$
$401$	−34.3783	−1.71677	−0.858384	+	0.513007i	$0.828531\pi$
$402$	0	0
$403$	10.2351	0.509845
$404$	0	0
$405$	11.0874	0.550938
$406$	0	0
$407$	−23.7146	−1.17549
$408$	0	0
$409$	16.3368	0.807801	0.403901	−	0.914803i	$-0.367654\pi$
$409$	16.3368	0.807801	0.403901	+	0.914803i	$0.367654\pi$
$410$	0	0
$411$	−26.7971	−1.32180
$412$	0	0
$413$	2.84743	0.140113
$414$	0	0
$415$	11.7605	0.577300
$416$	0	0
$417$	−22.8988	−1.12136
$418$	0	0
$419$	0.161933	0.00791096	0.00395548	−	0.999992i	$-0.498741\pi$
$419$	0.161933	0.00791096	0.00395548	+	0.999992i	$0.498741\pi$
$420$	0	0
$421$	14.9590	0.729057	0.364528	−	0.931192i	$-0.381230\pi$
$421$	14.9590	0.729057	0.364528	+	0.931192i	$0.381230\pi$
$422$	0	0
$423$	−8.14764	−0.396152
$424$	0	0
$425$	−7.80642	−0.378667
$426$	0	0
$427$	1.08250	0.0523857
$428$	0	0
$429$	−8.42864	−0.406939
$430$	0	0
$431$	−10.3269	−0.497431	−0.248715	−	0.968577i	$-0.580008\pi$
$431$	−10.3269	−0.497431	−0.248715	+	0.968577i	$0.580008\pi$
$432$	0	0
$433$	12.7491	0.612683	0.306342	−	0.951922i	$-0.400895\pi$
$433$	12.7491	0.612683	0.306342	+	0.951922i	$0.400895\pi$
$434$	0	0
$435$	20.3368	0.975074
$436$	0	0
$437$	−4.28100	−0.204788
$438$	0	0
$439$	13.5714	0.647726	0.323863	−	0.946104i	$-0.395018\pi$
$439$	13.5714	0.647726	0.323863	+	0.946104i	$0.395018\pi$
$440$	0	0
$441$	−8.89384	−0.423516
$442$	0	0
$443$	0.280996	0.0133505	0.00667527	−	0.999978i	$-0.497875\pi$
$443$	0.280996	0.0133505	0.00667527	+	0.999978i	$0.497875\pi$
$444$	0	0
$445$	−3.67307	−0.174120
$446$	0	0
$447$	−12.5303	−0.592665
$448$	0	0
$449$	−12.4286	−0.586544	−0.293272	−	0.956029i	$-0.594744\pi$
$449$	−12.4286	−0.586544	−0.293272	+	0.956029i	$0.594744\pi$
$450$	0	0
$451$	−32.0830	−1.51073
$452$	0	0
$453$	−25.4193	−1.19430
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	7.24443	0.338880	0.169440	−	0.985540i	$-0.445804\pi$
$457$	7.24443	0.338880	0.169440	+	0.985540i	$0.445804\pi$
$458$	0	0
$459$	−18.9590	−0.884930
$460$	0	0
$461$	−7.32741	−0.341271	−0.170636	−	0.985334i	$-0.554582\pi$
$461$	−7.32741	−0.341271	−0.170636	+	0.985334i	$0.554582\pi$
$462$	0	0
$463$	−22.8528	−1.06206	−0.531031	−	0.847353i	$-0.678195\pi$
$463$	−22.8528	−1.06206	−0.531031	+	0.847353i	$0.678195\pi$
$464$	0	0
$465$	17.2859	0.801615
$466$	0	0
$467$	10.6494	0.492796	0.246398	−	0.969169i	$-0.420753\pi$
$467$	10.6494	0.492796	0.246398	+	0.969169i	$0.420753\pi$
$468$	0	0
$469$	−2.22522	−0.102751
$470$	0	0
$471$	−1.08250	−0.0498789
$472$	0	0
$473$	3.18421	0.146410
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−10.5555	−0.483305
$478$	0	0
$479$	14.9032	0.680945	0.340473	−	0.940254i	$-0.389413\pi$
$479$	14.9032	0.680945	0.340473	+	0.940254i	$0.389413\pi$
$480$	0	0
$481$	−10.7096	−0.488317
$482$	0	0
$483$	−14.4603	−0.657966
$484$	0	0
$485$	11.2192	0.509440
$486$	0	0
$487$	−34.7576	−1.57502	−0.787508	−	0.616304i	$-0.788629\pi$
$487$	−34.7576	−1.57502	−0.787508	+	0.616304i	$0.788629\pi$
$488$	0	0
$489$	46.8988	2.12084
$490$	0	0
$491$	31.1338	1.40505	0.702525	−	0.711659i	$-0.252056\pi$
$491$	31.1338	1.40505	0.702525	+	0.711659i	$0.252056\pi$
$492$	0	0
$493$	−71.6958	−3.22902
$494$	0	0
$495$	−5.52543	−0.248350
$496$	0	0
$497$	3.50129	0.157054
$498$	0	0
$499$	20.8015	0.931203	0.465602	−	0.884994i	$-0.345838\pi$
$499$	20.8015	0.931203	0.465602	+	0.884994i	$0.345838\pi$
$500$	0	0
$501$	−36.4242	−1.62731
$502$	0	0
$503$	11.5986	0.517154	0.258577	−	0.965991i	$-0.416746\pi$
$503$	11.5986	0.517154	0.258577	+	0.965991i	$0.416746\pi$
$504$	0	0
$505$	−0.963435	−0.0428723
$506$	0	0
$507$	24.9797	1.10939
$508$	0	0
$509$	12.5018	0.554131	0.277066	−	0.960851i	$-0.410638\pi$
$509$	12.5018	0.554131	0.277066	+	0.960851i	$0.410638\pi$
$510$	0	0
$511$	−21.4193	−0.947533
$512$	0	0
$513$	2.42864	0.107227
$514$	0	0
$515$	−17.8874	−0.788213
$516$	0	0
$517$	−12.4286	−0.546611
$518$	0	0
$519$	−43.9353	−1.92855
$520$	0	0
$521$	−15.8064	−0.692492	−0.346246	−	0.938144i	$-0.612544\pi$
$521$	−15.8064	−0.692492	−0.346246	+	0.938144i	$0.612544\pi$
$522$	0	0
$523$	−2.90967	−0.127231	−0.0636154	−	0.997974i	$-0.520263\pi$
$523$	−2.90967	−0.127231	−0.0636154	+	0.997974i	$0.520263\pi$
$524$	0	0
$525$	3.37778	0.147419
$526$	0	0
$527$	−60.9403	−2.65460
$528$	0	0
$529$	−4.67307	−0.203177
$530$	0	0
$531$	−3.55262	−0.154171
$532$	0	0
$533$	−14.4889	−0.627582
$534$	0	0
$535$	−13.5002	−0.583666
$536$	0	0
$537$	45.8479	1.97848
$538$	0	0
$539$	−13.5669	−0.584368
$540$	0	0
$541$	−4.18913	−0.180105	−0.0900524	−	0.995937i	$-0.528703\pi$
$541$	−4.18913	−0.180105	−0.0900524	+	0.995937i	$0.528703\pi$
$542$	0	0
$543$	−25.4893	−1.09385
$544$	0	0
$545$	10.5303	0.451071
$546$	0	0
$547$	−18.2143	−0.778788	−0.389394	−	0.921071i	$-0.627316\pi$
$547$	−18.2143	−0.778788	−0.389394	+	0.921071i	$0.627316\pi$
$548$	0	0
$549$	−1.35059	−0.0576417
$550$	0	0
$551$	9.18421	0.391260
$552$	0	0
$553$	−9.35599	−0.397857
$554$	0	0
$555$	−18.0874	−0.767768
$556$	0	0
$557$	40.6548	1.72260	0.861300	−	0.508097i	$-0.169651\pi$
$557$	40.6548	1.72260	0.861300	+	0.508097i	$0.169651\pi$
$558$	0	0
$559$	1.43801	0.0608212
$560$	0	0
$561$	50.1847	2.11880
$562$	0	0
$563$	21.4400	0.903589	0.451794	−	0.892122i	$-0.350784\pi$
$563$	21.4400	0.903589	0.451794	+	0.892122i	$0.350784\pi$
$564$	0	0
$565$	18.5970	0.782383
$566$	0	0
$567$	16.9131	0.710282
$568$	0	0
$569$	25.1941	1.05619	0.528095	−	0.849185i	$-0.322907\pi$
$569$	25.1941	1.05619	0.528095	+	0.849185i	$0.322907\pi$
$570$	0	0
$571$	5.27163	0.220611	0.110305	−	0.993898i	$-0.464817\pi$
$571$	5.27163	0.220611	0.110305	+	0.993898i	$0.464817\pi$
$572$	0	0
$573$	22.3051	0.931810
$574$	0	0
$575$	−4.28100	−0.178530
$576$	0	0
$577$	23.3145	0.970595	0.485298	−	0.874349i	$-0.338711\pi$
$577$	23.3145	0.970595	0.485298	+	0.874349i	$0.338711\pi$
$578$	0	0
$579$	28.1891	1.17150
$580$	0	0
$581$	17.9398	0.744267
$582$	0	0
$583$	−16.1017	−0.666865
$584$	0	0
$585$	−2.49532	−0.103169
$586$	0	0
$587$	−28.9131	−1.19337	−0.596685	−	0.802476i	$-0.703516\pi$
$587$	−28.9131	−1.19337	−0.596685	+	0.802476i	$0.703516\pi$
$588$	0	0
$589$	7.80642	0.321658
$590$	0	0
$591$	27.0923	1.11443
$592$	0	0
$593$	−17.6128	−0.723273	−0.361636	−	0.932319i	$-0.617782\pi$
$593$	−17.6128	−0.723273	−0.361636	+	0.932319i	$0.617782\pi$
$594$	0	0
$595$	−11.9081	−0.488186
$596$	0	0
$597$	−11.6128	−0.475282
$598$	0	0
$599$	30.4286	1.24328	0.621640	−	0.783303i	$-0.286467\pi$
$599$	30.4286	1.24328	0.621640	+	0.783303i	$0.286467\pi$
$600$	0	0
$601$	−17.4479	−0.711713	−0.355856	−	0.934541i	$-0.615811\pi$
$601$	−17.4479	−0.711713	−0.355856	+	0.934541i	$0.615811\pi$
$602$	0	0
$603$	2.77631	0.113060
$604$	0	0
$605$	2.57136	0.104541
$606$	0	0
$607$	24.4592	0.992769	0.496385	−	0.868103i	$-0.334661\pi$
$607$	24.4592	0.992769	0.496385	+	0.868103i	$0.334661\pi$
$608$	0	0
$609$	31.0223	1.25709
$610$	0	0
$611$	−5.61285	−0.227072
$612$	0	0
$613$	12.1432	0.490459	0.245230	−	0.969465i	$-0.421137\pi$
$613$	12.1432	0.490459	0.245230	+	0.969465i	$0.421137\pi$
$614$	0	0
$615$	−24.4701	−0.986731
$616$	0	0
$617$	25.3876	1.02207	0.511034	−	0.859561i	$-0.329263\pi$
$617$	25.3876	1.02207	0.511034	+	0.859561i	$0.329263\pi$
$618$	0	0
$619$	20.4429	0.821671	0.410835	−	0.911710i	$-0.365237\pi$
$619$	20.4429	0.821671	0.410835	+	0.911710i	$0.365237\pi$
$620$	0	0
$621$	−10.3970	−0.417217
$622$	0	0
$623$	−5.60300	−0.224480
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.42864	−0.256735
$628$	0	0
$629$	63.7659	2.54251
$630$	0	0
$631$	−5.65878	−0.225273	−0.112636	−	0.993636i	$-0.535929\pi$
$631$	−5.65878	−0.225273	−0.112636	+	0.993636i	$0.535929\pi$
$632$	0	0
$633$	−37.0321	−1.47189
$634$	0	0
$635$	15.2050	0.603390
$636$	0	0
$637$	−6.12690	−0.242757
$638$	0	0
$639$	−4.36842	−0.172812
$640$	0	0
$641$	8.20342	0.324016	0.162008	−	0.986789i	$-0.448203\pi$
$641$	8.20342	0.324016	0.162008	+	0.986789i	$0.448203\pi$
$642$	0	0
$643$	10.0874	0.397809	0.198905	−	0.980019i	$-0.436262\pi$
$643$	10.0874	0.397809	0.198905	+	0.980019i	$0.436262\pi$
$644$	0	0
$645$	2.42864	0.0956276
$646$	0	0
$647$	31.3733	1.23341	0.616707	−	0.787193i	$-0.288466\pi$
$647$	31.3733	1.23341	0.616707	+	0.787193i	$0.288466\pi$
$648$	0	0
$649$	−5.41927	−0.212725
$650$	0	0
$651$	26.3684	1.03346
$652$	0	0
$653$	−28.4099	−1.11177	−0.555883	−	0.831261i	$-0.687620\pi$
$653$	−28.4099	−1.11177	−0.555883	+	0.831261i	$0.687620\pi$
$654$	0	0
$655$	11.0923	0.433414
$656$	0	0
$657$	26.7239	1.04260
$658$	0	0
$659$	−9.74620	−0.379658	−0.189829	−	0.981817i	$-0.560793\pi$
$659$	−9.74620	−0.379658	−0.189829	+	0.981817i	$0.560793\pi$
$660$	0	0
$661$	−15.6128	−0.607269	−0.303635	−	0.952789i	$-0.598200\pi$
$661$	−15.6128	−0.607269	−0.303635	+	0.952789i	$0.598200\pi$
$662$	0	0
$663$	22.6637	0.880185
$664$	0	0
$665$	1.52543	0.0591535
$666$	0	0
$667$	−39.3176	−1.52238
$668$	0	0
$669$	−60.8943	−2.35431
$670$	0	0
$671$	−2.06022	−0.0795340
$672$	0	0
$673$	22.0350	0.849388	0.424694	−	0.905337i	$-0.360382\pi$
$673$	22.0350	0.849388	0.424694	+	0.905337i	$0.360382\pi$
$674$	0	0
$675$	2.42864	0.0934784
$676$	0	0
$677$	−30.4953	−1.17203	−0.586015	−	0.810300i	$-0.699304\pi$
$677$	−30.4953	−1.17203	−0.586015	+	0.810300i	$0.699304\pi$
$678$	0	0
$679$	17.1141	0.656780
$680$	0	0
$681$	−42.7511	−1.63823
$682$	0	0
$683$	−11.2968	−0.432261	−0.216130	−	0.976365i	$-0.569344\pi$
$683$	−11.2968	−0.432261	−0.216130	+	0.976365i	$0.569344\pi$
$684$	0	0
$685$	−12.1017	−0.462383
$686$	0	0
$687$	−42.6035	−1.62542
$688$	0	0
$689$	−7.27163	−0.277027
$690$	0	0
$691$	18.5161	0.704384	0.352192	−	0.935928i	$-0.385436\pi$
$691$	18.5161	0.704384	0.352192	+	0.935928i	$0.385436\pi$
$692$	0	0
$693$	−8.42864	−0.320178
$694$	0	0
$695$	−10.3412	−0.392265
$696$	0	0
$697$	86.2677	3.26762
$698$	0	0
$699$	33.1941	1.25551
$700$	0	0
$701$	27.0781	1.02272	0.511362	−	0.859365i	$-0.329141\pi$
$701$	27.0781	1.02272	0.511362	+	0.859365i	$0.329141\pi$
$702$	0	0
$703$	−8.16839	−0.308077
$704$	0	0
$705$	−9.47949	−0.357018
$706$	0	0
$707$	−1.46965	−0.0552719
$708$	0	0
$709$	7.01921	0.263612	0.131806	−	0.991276i	$-0.457922\pi$
$709$	7.01921	0.263612	0.131806	+	0.991276i	$0.457922\pi$
$710$	0	0
$711$	11.6731	0.437774
$712$	0	0
$713$	−33.4193	−1.25156
$714$	0	0
$715$	−3.80642	−0.142352
$716$	0	0
$717$	−26.0731	−0.973719
$718$	0	0
$719$	−36.0272	−1.34359	−0.671794	−	0.740738i	$-0.734476\pi$
$719$	−36.0272	−1.34359	−0.671794	+	0.740738i	$0.734476\pi$
$720$	0	0
$721$	−27.2859	−1.01618
$722$	0	0
$723$	23.2257	0.863773
$724$	0	0
$725$	9.18421	0.341093
$726$	0	0
$727$	−15.9442	−0.591338	−0.295669	−	0.955290i	$-0.595543\pi$
$727$	−15.9442	−0.591338	−0.295669	+	0.955290i	$0.595543\pi$
$728$	0	0
$729$	−4.96836	−0.184013
$730$	0	0
$731$	−8.56199	−0.316677
$732$	0	0
$733$	10.9621	0.404893	0.202446	−	0.979293i	$-0.435111\pi$
$733$	10.9621	0.404893	0.202446	+	0.979293i	$0.435111\pi$
$734$	0	0
$735$	−10.3477	−0.381680
$736$	0	0
$737$	4.23506	0.156001
$738$	0	0
$739$	−13.7146	−0.504498	−0.252249	−	0.967662i	$-0.581170\pi$
$739$	−13.7146	−0.504498	−0.252249	+	0.967662i	$0.581170\pi$
$740$	0	0
$741$	−2.90321	−0.106652
$742$	0	0
$743$	33.6523	1.23458	0.617292	−	0.786734i	$-0.288230\pi$
$743$	33.6523	1.23458	0.617292	+	0.786734i	$0.288230\pi$
$744$	0	0
$745$	−5.65878	−0.207322
$746$	0	0
$747$	−22.3827	−0.818940
$748$	0	0
$749$	−20.5936	−0.752475
$750$	0	0
$751$	−36.7150	−1.33975	−0.669875	−	0.742474i	$-0.733652\pi$
$751$	−36.7150	−1.33975	−0.669875	+	0.742474i	$0.733652\pi$
$752$	0	0
$753$	24.4701	0.891741
$754$	0	0
$755$	−11.4795	−0.417782
$756$	0	0
$757$	−40.6222	−1.47644	−0.738220	−	0.674560i	$-0.764333\pi$
$757$	−40.6222	−1.47644	−0.738220	+	0.674560i	$0.764333\pi$
$758$	0	0
$759$	27.5210	0.998948
$760$	0	0
$761$	−35.1481	−1.27412	−0.637059	−	0.770815i	$-0.719849\pi$
$761$	−35.1481	−1.27412	−0.637059	+	0.770815i	$0.719849\pi$
$762$	0	0
$763$	16.0633	0.581530
$764$	0	0
$765$	14.8573	0.537166
$766$	0	0
$767$	−2.44738	−0.0883696
$768$	0	0
$769$	41.1383	1.48348	0.741742	−	0.670685i	$-0.234000\pi$
$769$	41.1383	1.48348	0.741742	+	0.670685i	$0.234000\pi$
$770$	0	0
$771$	−39.1481	−1.40989
$772$	0	0
$773$	−45.0484	−1.62028	−0.810139	−	0.586237i	$-0.800609\pi$
$773$	−45.0484	−1.62028	−0.810139	+	0.586237i	$0.800609\pi$
$774$	0	0
$775$	7.80642	0.280415
$776$	0	0
$777$	−27.5910	−0.989823
$778$	0	0
$779$	−11.0509	−0.395938
$780$	0	0
$781$	−6.66370	−0.238446
$782$	0	0
$783$	22.3051	0.797120
$784$	0	0
$785$	−0.488863	−0.0174483
$786$	0	0
$787$	−25.4400	−0.906839	−0.453419	−	0.891297i	$-0.649796\pi$
$787$	−25.4400	−0.906839	−0.453419	+	0.891297i	$0.649796\pi$
$788$	0	0
$789$	−46.8069	−1.66637
$790$	0	0
$791$	28.3684	1.00866
$792$	0	0
$793$	−0.930409	−0.0330398
$794$	0	0
$795$	−12.2810	−0.435562
$796$	0	0
$797$	−20.5684	−0.728572	−0.364286	−	0.931287i	$-0.618687\pi$
$797$	−20.5684	−0.728572	−0.364286	+	0.931287i	$0.618687\pi$
$798$	0	0
$799$	33.4193	1.18229
$800$	0	0
$801$	6.99063	0.247002
$802$	0	0
$803$	40.7654	1.43858
$804$	0	0
$805$	−6.53035	−0.230165
$806$	0	0
$807$	−38.9304	−1.37042
$808$	0	0
$809$	53.2039	1.87055	0.935275	−	0.353923i	$-0.115152\pi$
$809$	53.2039	1.87055	0.935275	+	0.353923i	$0.115152\pi$
$810$	0	0
$811$	−24.7467	−0.868973	−0.434487	−	0.900678i	$-0.643070\pi$
$811$	−24.7467	−0.868973	−0.434487	+	0.900678i	$0.643070\pi$
$812$	0	0
$813$	46.5116	1.63123
$814$	0	0
$815$	21.1798	0.741895
$816$	0	0
$817$	1.09679	0.0383718
$818$	0	0
$819$	−3.80642	−0.133007
$820$	0	0
$821$	6.61237	0.230773	0.115387	−	0.993321i	$-0.463189\pi$
$821$	6.61237	0.230773	0.115387	+	0.993321i	$0.463189\pi$
$822$	0	0
$823$	37.7418	1.31559	0.657797	−	0.753195i	$-0.271488\pi$
$823$	37.7418	1.31559	0.657797	+	0.753195i	$0.271488\pi$
$824$	0	0
$825$	−6.42864	−0.223816
$826$	0	0
$827$	−25.1318	−0.873919	−0.436960	−	0.899481i	$-0.643945\pi$
$827$	−25.1318	−0.873919	−0.436960	+	0.899481i	$0.643945\pi$
$828$	0	0
$829$	12.5116	0.434546	0.217273	−	0.976111i	$-0.430284\pi$
$829$	12.5116	0.434546	0.217273	+	0.976111i	$0.430284\pi$
$830$	0	0
$831$	−15.1842	−0.526734
$832$	0	0
$833$	36.4800	1.26396
$834$	0	0
$835$	−16.4494	−0.569254
$836$	0	0
$837$	18.9590	0.655319
$838$	0	0
$839$	16.3269	0.563668	0.281834	−	0.959463i	$-0.409057\pi$
$839$	16.3269	0.563668	0.281834	+	0.959463i	$0.409057\pi$
$840$	0	0
$841$	55.3497	1.90861
$842$	0	0
$843$	−37.7146	−1.29896
$844$	0	0
$845$	11.2810	0.388078
$846$	0	0
$847$	3.92242	0.134776
$848$	0	0
$849$	−61.4706	−2.10967
$850$	0	0
$851$	34.9688	1.19872
$852$	0	0
$853$	−14.8287	−0.507725	−0.253862	−	0.967240i	$-0.581701\pi$
$853$	−14.8287	−0.507725	−0.253862	+	0.967240i	$0.581701\pi$
$854$	0	0
$855$	−1.90321	−0.0650885
$856$	0	0
$857$	−10.0796	−0.344312	−0.172156	−	0.985070i	$-0.555073\pi$
$857$	−10.0796	−0.344312	−0.172156	+	0.985070i	$0.555073\pi$
$858$	0	0
$859$	36.7783	1.25486	0.627430	−	0.778673i	$-0.284107\pi$
$859$	36.7783	1.25486	0.627430	+	0.778673i	$0.284107\pi$
$860$	0	0
$861$	−37.3274	−1.27211
$862$	0	0
$863$	−17.7857	−0.605432	−0.302716	−	0.953081i	$-0.597893\pi$
$863$	−17.7857	−0.605432	−0.302716	+	0.953081i	$0.597893\pi$
$864$	0	0
$865$	−19.8415	−0.674630
$866$	0	0
$867$	−97.2978	−3.30441
$868$	0	0
$869$	17.8064	0.604042
$870$	0	0
$871$	1.91258	0.0648053
$872$	0	0
$873$	−21.3526	−0.722676
$874$	0	0
$875$	1.52543	0.0515689
$876$	0	0
$877$	26.8321	0.906055	0.453028	−	0.891496i	$-0.350344\pi$
$877$	26.8321	0.906055	0.453028	+	0.891496i	$0.350344\pi$
$878$	0	0
$879$	−71.6182	−2.41562
$880$	0	0
$881$	−27.8524	−0.938370	−0.469185	−	0.883100i	$-0.655452\pi$
$881$	−27.8524	−0.938370	−0.469185	+	0.883100i	$0.655452\pi$
$882$	0	0
$883$	−13.8751	−0.466935	−0.233467	−	0.972365i	$-0.575007\pi$
$883$	−13.8751	−0.466935	−0.233467	+	0.972365i	$0.575007\pi$
$884$	0	0
$885$	−4.13335	−0.138941
$886$	0	0
$887$	−26.1827	−0.879128	−0.439564	−	0.898211i	$-0.644867\pi$
$887$	−26.1827	−0.879128	−0.439564	+	0.898211i	$0.644867\pi$
$888$	0	0
$889$	23.1941	0.777904
$890$	0	0
$891$	−32.1891	−1.07838
$892$	0	0
$893$	−4.28100	−0.143258
$894$	0	0
$895$	20.7052	0.692098
$896$	0	0
$897$	12.4286	0.414980
$898$	0	0
$899$	71.6958	2.39119
$900$	0	0
$901$	43.2958	1.44239
$902$	0	0
$903$	3.70471	0.123285
$904$	0	0
$905$	−11.5111	−0.382643
$906$	0	0
$907$	−1.84590	−0.0612922	−0.0306461	−	0.999530i	$-0.509756\pi$
$907$	−1.84590	−0.0612922	−0.0306461	+	0.999530i	$0.509756\pi$
$908$	0	0
$909$	1.83362	0.0608174
$910$	0	0
$911$	47.1467	1.56204	0.781021	−	0.624505i	$-0.214699\pi$
$911$	47.1467	1.56204	0.781021	+	0.624505i	$0.214699\pi$
$912$	0	0
$913$	−34.1432	−1.12997
$914$	0	0
$915$	−1.57136	−0.0519476
$916$	0	0
$917$	16.9206	0.558766
$918$	0	0
$919$	15.4509	0.509679	0.254839	−	0.966983i	$-0.417977\pi$
$919$	15.4509	0.509679	0.254839	+	0.966983i	$0.417977\pi$
$920$	0	0
$921$	−57.9135	−1.90832
$922$	0	0
$923$	−3.00937	−0.0990546
$924$	0	0
$925$	−8.16839	−0.268575
$926$	0	0
$927$	34.0435	1.11814
$928$	0	0
$929$	17.8163	0.584533	0.292266	−	0.956337i	$-0.405591\pi$
$929$	17.8163	0.584533	0.292266	+	0.956337i	$0.405591\pi$
$930$	0	0
$931$	−4.67307	−0.153154
$932$	0	0
$933$	31.2859	1.02425
$934$	0	0
$935$	22.6637	0.741182
$936$	0	0
$937$	30.6953	1.00277	0.501387	−	0.865223i	$-0.332823\pi$
$937$	30.6953	1.00277	0.501387	+	0.865223i	$0.332823\pi$
$938$	0	0
$939$	15.4795	0.505154
$940$	0	0
$941$	11.0696	0.360858	0.180429	−	0.983588i	$-0.442251\pi$
$941$	11.0696	0.360858	0.180429	+	0.983588i	$0.442251\pi$
$942$	0	0
$943$	47.3087	1.54058
$944$	0	0
$945$	3.70471	0.120514
$946$	0	0
$947$	−11.3033	−0.367307	−0.183653	−	0.982991i	$-0.558792\pi$
$947$	−11.3033	−0.367307	−0.183653	+	0.982991i	$0.558792\pi$
$948$	0	0
$949$	18.4099	0.597611
$950$	0	0
$951$	−19.1066	−0.619575
$952$	0	0
$953$	−20.8637	−0.675843	−0.337921	−	0.941174i	$-0.609724\pi$
$953$	−20.8637	−0.675843	−0.337921	+	0.941174i	$0.609724\pi$
$954$	0	0
$955$	10.0731	0.325959
$956$	0	0
$957$	−59.0420	−1.90856
$958$	0	0
$959$	−18.4603	−0.596114
$960$	0	0
$961$	29.9403	0.965815
$962$	0	0
$963$	25.6938	0.827972
$964$	0	0
$965$	12.7304	0.409805
$966$	0	0
$967$	−56.7007	−1.82337	−0.911687	−	0.410886i	$-0.865219\pi$
$967$	−56.7007	−1.82337	−0.911687	+	0.410886i	$0.865219\pi$
$968$	0	0
$969$	17.2859	0.555304
$970$	0	0
$971$	−24.8760	−0.798309	−0.399155	−	0.916884i	$-0.630696\pi$
$971$	−24.8760	−0.798309	−0.399155	+	0.916884i	$0.630696\pi$
$972$	0	0
$973$	−15.7748	−0.505716
$974$	0	0
$975$	−2.90321	−0.0929772
$976$	0	0
$977$	−21.9847	−0.703351	−0.351676	−	0.936122i	$-0.614388\pi$
$977$	−21.9847	−0.703351	−0.351676	+	0.936122i	$0.614388\pi$
$978$	0	0
$979$	10.6637	0.340813
$980$	0	0
$981$	−20.0415	−0.639875
$982$	0	0
$983$	29.7540	0.949006	0.474503	−	0.880254i	$-0.342628\pi$
$983$	29.7540	0.949006	0.474503	+	0.880254i	$0.342628\pi$
$984$	0	0
$985$	12.2351	0.389842
$986$	0	0
$987$	−14.4603	−0.460276
$988$	0	0
$989$	−4.69535	−0.149303
$990$	0	0
$991$	−38.9719	−1.23798	−0.618992	−	0.785398i	$-0.712458\pi$
$991$	−38.9719	−1.23798	−0.618992	+	0.785398i	$0.712458\pi$
$992$	0	0
$993$	27.5210	0.873352
$994$	0	0
$995$	−5.24443	−0.166260
$996$	0	0
$997$	18.8287	0.596311	0.298155	−	0.954517i	$-0.403629\pi$
$997$	18.8287	0.596311	0.298155	+	0.954517i	$0.403629\pi$
$998$	0	0
$999$	−19.8381	−0.627649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bu.1.1		3
4.3	odd	2		6080.2.a.bt.1.3		3
8.3	odd	2		3040.2.a.l.1.1	✓	3
8.5	even	2		3040.2.a.m.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.l.1.1	✓	3	8.3	odd	2
3040.2.a.m.1.3	yes	3	8.5	even	2
6080.2.a.bt.1.3		3	4.3	odd	2
6080.2.a.bu.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q-2.21432 q^{3} -1.00000 q^{5} -1.52543 q^{7} +1.90321 q^{9} +O(q^{10})\)	\(q-2.21432 q^{3} -1.00000 q^{5} -1.52543 q^{7} +1.90321 q^{9} +2.90321 q^{11} +1.31111 q^{13} +2.21432 q^{15} -7.80642 q^{17} +1.00000 q^{19} +3.37778 q^{21} -4.28100 q^{23} +1.00000 q^{25} +2.42864 q^{27} +9.18421 q^{29} +7.80642 q^{31} -6.42864 q^{33} +1.52543 q^{35} -8.16839 q^{37} -2.90321 q^{39} -11.0509 q^{41} +1.09679 q^{43} -1.90321 q^{45} -4.28100 q^{47} -4.67307 q^{49} +17.2859 q^{51} -5.54617 q^{53} -2.90321 q^{55} -2.21432 q^{57} -1.86665 q^{59} -0.709636 q^{61} -2.90321 q^{63} -1.31111 q^{65} +1.45875 q^{67} +9.47949 q^{69} -2.29529 q^{71} +14.0415 q^{73} -2.21432 q^{75} -4.42864 q^{77} +6.13335 q^{79} -11.0874 q^{81} -11.7605 q^{83} +7.80642 q^{85} -20.3368 q^{87} +3.67307 q^{89} -2.00000 q^{91} -17.2859 q^{93} -1.00000 q^{95} -11.2192 q^{97} +5.52543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 + 2 * q^7 - q^9	\( 3 q - 3 q^{5} + 2 q^{7} - q^{9} + 2 q^{11} + 4 q^{13} - 10 q^{17} + 3 q^{19} + 10 q^{21} - 6 q^{23} + 3 q^{25} - 6 q^{27} + 14 q^{29} + 10 q^{31} - 6 q^{33} - 2 q^{35} + 2 q^{37} - 2 q^{39} - 20 q^{41} + 10 q^{43} + q^{45} - 6 q^{47} - q^{49} + 12 q^{51} + 10 q^{53} - 2 q^{55} - 6 q^{59} + 18 q^{61} - 2 q^{63} - 4 q^{65} - 2 q^{67} + 2 q^{69} + 6 q^{71} + 2 q^{73} + 18 q^{79} - 13 q^{81} - 2 q^{83} + 10 q^{85} - 8 q^{87} - 2 q^{89} - 6 q^{91} - 12 q^{93} - 3 q^{95} + 6 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - q^9 + 2 * q^11 + 4 * q^13 - 10 * q^17 + 3 * q^19 + 10 * q^21 - 6 * q^23 + 3 * q^25 - 6 * q^27 + 14 * q^29 + 10 * q^31 - 6 * q^33 - 2 * q^35 + 2 * q^37 - 2 * q^39 - 20 * q^41 + 10 * q^43 + q^45 - 6 * q^47 - q^49 + 12 * q^51 + 10 * q^53 - 2 * q^55 - 6 * q^59 + 18 * q^61 - 2 * q^63 - 4 * q^65 - 2 * q^67 + 2 * q^69 + 6 * q^71 + 2 * q^73 + 18 * q^79 - 13 * q^81 - 2 * q^83 + 10 * q^85 - 8 * q^87 - 2 * q^89 - 6 * q^91 - 12 * q^93 - 3 * q^95 + 6 * q^97 + 10 * q^99

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