LMFDB - Embedded newform 6160.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} +5.46410 q^{13} -0.732051 q^{15} +3.46410 q^{17} -3.26795 q^{19} -0.732051 q^{21} -2.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.26795 q^{29} -2.00000 q^{31} -0.732051 q^{33} +1.00000 q^{35} -2.73205 q^{37} +4.00000 q^{39} +8.19615 q^{41} -2.00000 q^{43} +2.46410 q^{45} +6.92820 q^{47} +1.00000 q^{49} +2.53590 q^{51} -10.7321 q^{53} +1.00000 q^{55} -2.39230 q^{57} +6.92820 q^{59} +8.92820 q^{61} +2.46410 q^{63} -5.46410 q^{65} +4.00000 q^{67} -1.60770 q^{69} -2.53590 q^{71} +6.39230 q^{73} +0.732051 q^{75} +1.00000 q^{77} +1.80385 q^{79} +4.46410 q^{81} -4.39230 q^{83} -3.46410 q^{85} -0.928203 q^{87} -3.46410 q^{89} -5.46410 q^{91} -1.46410 q^{93} +3.26795 q^{95} -16.5885 q^{97} +2.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} - 10 q^{19} + 2 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 2 q^{35} - 2 q^{37} + 8 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} + 12 q^{51} - 18 q^{53} + 2 q^{55} + 16 q^{57} + 4 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} - 24 q^{69} - 12 q^{71} - 8 q^{73} - 2 q^{75} + 2 q^{77} + 14 q^{79} + 2 q^{81} + 12 q^{83} + 12 q^{87} - 4 q^{91} + 4 q^{93} + 10 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 - 10 * q^19 + 2 * q^21 + 6 * q^23 + 2 * q^25 - 8 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 + 2 * q^35 - 2 * q^37 + 8 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 + 2 * q^49 + 12 * q^51 - 18 * q^53 + 2 * q^55 + 16 * q^57 + 4 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 - 24 * q^69 - 12 * q^71 - 8 * q^73 - 2 * q^75 + 2 * q^77 + 14 * q^79 + 2 * q^81 + 12 * q^83 + 12 * q^87 - 4 * q^91 + 4 * q^93 + 10 * q^95 - 2 * q^97 - 2 * 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−3.26795	−0.749719	−0.374859	−	0.927082i	$-0.622309\pi$
$19$	−3.26795	−0.749719	−0.374859	+	0.927082i	$0.622309\pi$
$20$	0	0
$21$	−0.732051	−0.159747
$22$	0	0
$23$	−2.19615	−0.457929	−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615	−0.457929	−0.228965	+	0.973435i	$0.573534\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−1.26795	−0.235452	−0.117726	−	0.993046i	$-0.537560\pi$
$29$	−1.26795	−0.235452	−0.117726	+	0.993046i	$0.537560\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−0.732051	−0.127434
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−2.73205	−0.449146	−0.224573	−	0.974457i	$-0.572099\pi$
$37$	−2.73205	−0.449146	−0.224573	+	0.974457i	$0.572099\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	8.19615	1.28002	0.640012	−	0.768365i	$-0.278929\pi$
$41$	8.19615	1.28002	0.640012	+	0.768365i	$0.278929\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	2.46410	0.367327
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.53590	0.355097
$52$	0	0
$53$	−10.7321	−1.47416	−0.737080	−	0.675805i	$-0.763796\pi$
$53$	−10.7321	−1.47416	−0.737080	+	0.675805i	$0.763796\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.39230	−0.316869
$58$	0	0
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	0	0
$63$	2.46410	0.310448
$64$	0	0
$65$	−5.46410	−0.677738
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−1.60770	−0.193544
$70$	0	0
$71$	−2.53590	−0.300956	−0.150478	−	0.988613i	$-0.548081\pi$
$71$	−2.53590	−0.300956	−0.150478	+	0.988613i	$0.548081\pi$
$72$	0	0
$73$	6.39230	0.748163	0.374081	−	0.927396i	$-0.377958\pi$
$73$	6.39230	0.748163	0.374081	+	0.927396i	$0.377958\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	1.80385	0.202949	0.101474	−	0.994838i	$-0.467644\pi$
$79$	1.80385	0.202949	0.101474	+	0.994838i	$0.467644\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−4.39230	−0.482118	−0.241059	−	0.970510i	$-0.577495\pi$
$83$	−4.39230	−0.482118	−0.241059	+	0.970510i	$0.577495\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	0	0
$87$	−0.928203	−0.0995138
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	−5.46410	−0.572793
$92$	0	0
$93$	−1.46410	−0.151820
$94$	0	0
$95$	3.26795	0.335285
$96$	0	0
$97$	−16.5885	−1.68430	−0.842151	−	0.539241i	$-0.818711\pi$
$97$	−16.5885	−1.68430	−0.842151	+	0.539241i	$0.818711\pi$
$98$	0	0
$99$	2.46410	0.247652
$100$	0	0
$101$	19.8564	1.97579	0.987893	−	0.155136i	$-0.0495815\pi$
$101$	19.8564	1.97579	0.987893	+	0.155136i	$0.0495815\pi$
$102$	0	0
$103$	8.39230	0.826918	0.413459	−	0.910523i	$-0.364320\pi$
$103$	8.39230	0.826918	0.413459	+	0.910523i	$0.364320\pi$
$104$	0	0
$105$	0.732051	0.0714408
$106$	0	0
$107$	19.8564	1.91959	0.959796	−	0.280700i	$-0.0905665\pi$
$107$	19.8564	1.91959	0.959796	+	0.280700i	$0.0905665\pi$
$108$	0	0
$109$	1.66025	0.159023	0.0795117	−	0.996834i	$-0.474664\pi$
$109$	1.66025	0.159023	0.0795117	+	0.996834i	$0.474664\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−19.8564	−1.86793	−0.933967	−	0.357360i	$-0.883677\pi$
$113$	−19.8564	−1.86793	−0.933967	+	0.357360i	$0.883677\pi$
$114$	0	0
$115$	2.19615	0.204792
$116$	0	0
$117$	−13.4641	−1.24476
$118$	0	0
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.9282	−1.32466	−0.662332	−	0.749211i	$-0.730433\pi$
$127$	−14.9282	−1.32466	−0.662332	+	0.749211i	$0.730433\pi$
$128$	0	0
$129$	−1.46410	−0.128907
$130$	0	0
$131$	11.6603	1.01876	0.509381	−	0.860541i	$-0.329875\pi$
$131$	11.6603	1.01876	0.509381	+	0.860541i	$0.329875\pi$
$132$	0	0
$133$	3.26795	0.283367
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	0	0
$139$	3.66025	0.310459	0.155229	−	0.987878i	$-0.450388\pi$
$139$	3.66025	0.310459	0.155229	+	0.987878i	$0.450388\pi$
$140$	0	0
$141$	5.07180	0.427122
$142$	0	0
$143$	−5.46410	−0.456931
$144$	0	0
$145$	1.26795	0.105297
$146$	0	0
$147$	0.732051	0.0603785
$148$	0	0
$149$	−10.7321	−0.879204	−0.439602	−	0.898193i	$-0.644880\pi$
$149$	−10.7321	−0.879204	−0.439602	+	0.898193i	$0.644880\pi$
$150$	0	0
$151$	13.1244	1.06804	0.534022	−	0.845470i	$-0.320680\pi$
$151$	13.1244	1.06804	0.534022	+	0.845470i	$0.320680\pi$
$152$	0	0
$153$	−8.53590	−0.690086
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	11.4641	0.914935	0.457467	−	0.889226i	$-0.348757\pi$
$157$	11.4641	0.914935	0.457467	+	0.889226i	$0.348757\pi$
$158$	0	0
$159$	−7.85641	−0.623054
$160$	0	0
$161$	2.19615	0.173081
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0.732051	0.0569901
$166$	0	0
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	8.05256	0.615795
$172$	0	0
$173$	−12.9282	−0.982913	−0.491457	−	0.870902i	$-0.663535\pi$
$173$	−12.9282	−0.982913	−0.491457	+	0.870902i	$0.663535\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	5.07180	0.381220
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	6.53590	0.483148
$184$	0	0
$185$	2.73205	0.200864
$186$	0	0
$187$	−3.46410	−0.253320
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−8.39230	−0.604091	−0.302046	−	0.953293i	$-0.597670\pi$
$193$	−8.39230	−0.604091	−0.302046	+	0.953293i	$0.597670\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	24.2487	1.72765	0.863825	−	0.503793i	$-0.168062\pi$
$197$	24.2487	1.72765	0.863825	+	0.503793i	$0.168062\pi$
$198$	0	0
$199$	10.9282	0.774680	0.387340	−	0.921937i	$-0.373394\pi$
$199$	10.9282	0.774680	0.387340	+	0.921937i	$0.373394\pi$
$200$	0	0
$201$	2.92820	0.206540
$202$	0	0
$203$	1.26795	0.0889926
$204$	0	0
$205$	−8.19615	−0.572444
$206$	0	0
$207$	5.41154	0.376128
$208$	0	0
$209$	3.26795	0.226049
$210$	0	0
$211$	−13.0718	−0.899900	−0.449950	−	0.893054i	$-0.648558\pi$
$211$	−13.0718	−0.899900	−0.449950	+	0.893054i	$0.648558\pi$
$212$	0	0
$213$	−1.85641	−0.127199
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	4.67949	0.316211
$220$	0	0
$221$	18.9282	1.27325
$222$	0	0
$223$	18.5359	1.24126	0.620628	−	0.784105i	$-0.286878\pi$
$223$	18.5359	1.24126	0.620628	+	0.784105i	$0.286878\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	6.92820	0.459841	0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820	0.459841	0.229920	+	0.973209i	$0.426153\pi$
$228$	0	0
$229$	3.60770	0.238403	0.119202	−	0.992870i	$-0.461967\pi$
$229$	3.60770	0.238403	0.119202	+	0.992870i	$0.461967\pi$
$230$	0	0
$231$	0.732051	0.0481654
$232$	0	0
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	−6.92820	−0.451946
$236$	0	0
$237$	1.32051	0.0857762
$238$	0	0
$239$	−4.73205	−0.306091	−0.153045	−	0.988219i	$-0.548908\pi$
$239$	−4.73205	−0.306091	−0.153045	+	0.988219i	$0.548908\pi$
$240$	0	0
$241$	6.73205	0.433650	0.216825	−	0.976211i	$-0.430430\pi$
$241$	6.73205	0.433650	0.216825	+	0.976211i	$0.430430\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−17.8564	−1.13618
$248$	0	0
$249$	−3.21539	−0.203767
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	2.19615	0.138071
$254$	0	0
$255$	−2.53590	−0.158804
$256$	0	0
$257$	−6.33975	−0.395462	−0.197731	−	0.980256i	$-0.563357\pi$
$257$	−6.33975	−0.395462	−0.197731	+	0.980256i	$0.563357\pi$
$258$	0	0
$259$	2.73205	0.169761
$260$	0	0
$261$	3.12436	0.193393
$262$	0	0
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	10.7321	0.659265
$266$	0	0
$267$	−2.53590	−0.155194
$268$	0	0
$269$	7.60770	0.463849	0.231925	−	0.972734i	$-0.425498\pi$
$269$	7.60770	0.463849	0.231925	+	0.972734i	$0.425498\pi$
$270$	0	0
$271$	20.3923	1.23874	0.619372	−	0.785098i	$-0.287387\pi$
$271$	20.3923	1.23874	0.619372	+	0.785098i	$0.287387\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	20.9282	1.25745	0.628727	−	0.777626i	$-0.283576\pi$
$277$	20.9282	1.25745	0.628727	+	0.777626i	$0.283576\pi$
$278$	0	0
$279$	4.92820	0.295044
$280$	0	0
$281$	1.60770	0.0959071	0.0479535	−	0.998850i	$-0.484730\pi$
$281$	1.60770	0.0959071	0.0479535	+	0.998850i	$0.484730\pi$
$282$	0	0
$283$	−23.7128	−1.40958	−0.704790	−	0.709416i	$-0.748959\pi$
$283$	−23.7128	−1.40958	−0.704790	+	0.709416i	$0.748959\pi$
$284$	0	0
$285$	2.39230	0.141708
$286$	0	0
$287$	−8.19615	−0.483804
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−12.1436	−0.711870
$292$	0	0
$293$	14.5359	0.849196	0.424598	−	0.905382i	$-0.360415\pi$
$293$	14.5359	0.849196	0.424598	+	0.905382i	$0.360415\pi$
$294$	0	0
$295$	−6.92820	−0.403376
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	14.5359	0.835066
$304$	0	0
$305$	−8.92820	−0.511227
$306$	0	0
$307$	−3.60770	−0.205902	−0.102951	−	0.994686i	$-0.532828\pi$
$307$	−3.60770	−0.205902	−0.102951	+	0.994686i	$0.532828\pi$
$308$	0	0
$309$	6.14359	0.349497
$310$	0	0
$311$	−11.0718	−0.627824	−0.313912	−	0.949452i	$-0.601640\pi$
$311$	−11.0718	−0.627824	−0.313912	+	0.949452i	$0.601640\pi$
$312$	0	0
$313$	11.8038	0.667193	0.333596	−	0.942716i	$-0.391738\pi$
$313$	11.8038	0.667193	0.333596	+	0.942716i	$0.391738\pi$
$314$	0	0
$315$	−2.46410	−0.138836
$316$	0	0
$317$	−0.588457	−0.0330511	−0.0165255	−	0.999863i	$-0.505260\pi$
$317$	−0.588457	−0.0330511	−0.0165255	+	0.999863i	$0.505260\pi$
$318$	0	0
$319$	1.26795	0.0709915
$320$	0	0
$321$	14.5359	0.811315
$322$	0	0
$323$	−11.3205	−0.629890
$324$	0	0
$325$	5.46410	0.303094
$326$	0	0
$327$	1.21539	0.0672112
$328$	0	0
$329$	−6.92820	−0.381964
$330$	0	0
$331$	−22.7846	−1.25236	−0.626178	−	0.779680i	$-0.715382\pi$
$331$	−22.7846	−1.25236	−0.626178	+	0.779680i	$0.715382\pi$
$332$	0	0
$333$	6.73205	0.368914
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−18.7846	−1.02326	−0.511631	−	0.859205i	$-0.670959\pi$
$337$	−18.7846	−1.02326	−0.511631	+	0.859205i	$0.670959\pi$
$338$	0	0
$339$	−14.5359	−0.789482
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.60770	0.0865554
$346$	0	0
$347$	36.9282	1.98241	0.991205	−	0.132336i	$-0.0422478\pi$
$347$	36.9282	1.98241	0.991205	+	0.132336i	$0.0422478\pi$
$348$	0	0
$349$	−26.3923	−1.41275	−0.706374	−	0.707839i	$-0.749670\pi$
$349$	−26.3923	−1.41275	−0.706374	+	0.707839i	$0.749670\pi$
$350$	0	0
$351$	−21.8564	−1.16661
$352$	0	0
$353$	3.80385	0.202458	0.101229	−	0.994863i	$-0.467722\pi$
$353$	3.80385	0.202458	0.101229	+	0.994863i	$0.467722\pi$
$354$	0	0
$355$	2.53590	0.134592
$356$	0	0
$357$	−2.53590	−0.134214
$358$	0	0
$359$	−4.05256	−0.213886	−0.106943	−	0.994265i	$-0.534106\pi$
$359$	−4.05256	−0.213886	−0.106943	+	0.994265i	$0.534106\pi$
$360$	0	0
$361$	−8.32051	−0.437921
$362$	0	0
$363$	0.732051	0.0384227
$364$	0	0
$365$	−6.39230	−0.334589
$366$	0	0
$367$	8.39230	0.438075	0.219037	−	0.975716i	$-0.429708\pi$
$367$	8.39230	0.438075	0.219037	+	0.975716i	$0.429708\pi$
$368$	0	0
$369$	−20.1962	−1.05137
$370$	0	0
$371$	10.7321	0.557180
$372$	0	0
$373$	−2.39230	−0.123869	−0.0619344	−	0.998080i	$-0.519727\pi$
$373$	−2.39230	−0.123869	−0.0619344	+	0.998080i	$0.519727\pi$
$374$	0	0
$375$	−0.732051	−0.0378029
$376$	0	0
$377$	−6.92820	−0.356821
$378$	0	0
$379$	−33.8564	−1.73909	−0.869543	−	0.493857i	$-0.835587\pi$
$379$	−33.8564	−1.73909	−0.869543	+	0.493857i	$0.835587\pi$
$380$	0	0
$381$	−10.9282	−0.559869
$382$	0	0
$383$	26.5359	1.35592	0.677961	−	0.735098i	$-0.262864\pi$
$383$	26.5359	1.35592	0.677961	+	0.735098i	$0.262864\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	4.92820	0.250515
$388$	0	0
$389$	−22.3923	−1.13533	−0.567667	−	0.823258i	$-0.692154\pi$
$389$	−22.3923	−1.13533	−0.567667	+	0.823258i	$0.692154\pi$
$390$	0	0
$391$	−7.60770	−0.384738
$392$	0	0
$393$	8.53590	0.430579
$394$	0	0
$395$	−1.80385	−0.0907614
$396$	0	0
$397$	9.60770	0.482196	0.241098	−	0.970501i	$-0.422492\pi$
$397$	9.60770	0.482196	0.241098	+	0.970501i	$0.422492\pi$
$398$	0	0
$399$	2.39230	0.119765
$400$	0	0
$401$	9.46410	0.472615	0.236307	−	0.971678i	$-0.424063\pi$
$401$	9.46410	0.472615	0.236307	+	0.971678i	$0.424063\pi$
$402$	0	0
$403$	−10.9282	−0.544373
$404$	0	0
$405$	−4.46410	−0.221823
$406$	0	0
$407$	2.73205	0.135423
$408$	0	0
$409$	35.1244	1.73679	0.868394	−	0.495875i	$-0.165153\pi$
$409$	35.1244	1.73679	0.868394	+	0.495875i	$0.165153\pi$
$410$	0	0
$411$	9.46410	0.466830
$412$	0	0
$413$	−6.92820	−0.340915
$414$	0	0
$415$	4.39230	0.215610
$416$	0	0
$417$	2.67949	0.131215
$418$	0	0
$419$	−17.0718	−0.834012	−0.417006	−	0.908904i	$-0.636921\pi$
$419$	−17.0718	−0.834012	−0.417006	+	0.908904i	$0.636921\pi$
$420$	0	0
$421$	−8.14359	−0.396894	−0.198447	−	0.980112i	$-0.563590\pi$
$421$	−8.14359	−0.396894	−0.198447	+	0.980112i	$0.563590\pi$
$422$	0	0
$423$	−17.0718	−0.830059
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	−8.92820	−0.432066
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	21.1244	1.01752	0.508762	−	0.860907i	$-0.330103\pi$
$431$	21.1244	1.01752	0.508762	+	0.860907i	$0.330103\pi$
$432$	0	0
$433$	−7.80385	−0.375029	−0.187514	−	0.982262i	$-0.560043\pi$
$433$	−7.80385	−0.375029	−0.187514	+	0.982262i	$0.560043\pi$
$434$	0	0
$435$	0.928203	0.0445039
$436$	0	0
$437$	7.17691	0.343318
$438$	0	0
$439$	22.2487	1.06187	0.530937	−	0.847412i	$-0.321840\pi$
$439$	22.2487	1.06187	0.530937	+	0.847412i	$0.321840\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	3.46410	0.164214
$446$	0	0
$447$	−7.85641	−0.371595
$448$	0	0
$449$	19.6077	0.925344	0.462672	−	0.886529i	$-0.346891\pi$
$449$	19.6077	0.925344	0.462672	+	0.886529i	$0.346891\pi$
$450$	0	0
$451$	−8.19615	−0.385942
$452$	0	0
$453$	9.60770	0.451409
$454$	0	0
$455$	5.46410	0.256161
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	−13.8564	−0.646762
$460$	0	0
$461$	1.60770	0.0748778	0.0374389	−	0.999299i	$-0.488080\pi$
$461$	1.60770	0.0748778	0.0374389	+	0.999299i	$0.488080\pi$
$462$	0	0
$463$	17.5167	0.814068	0.407034	−	0.913413i	$-0.366563\pi$
$463$	17.5167	0.814068	0.407034	+	0.913413i	$0.366563\pi$
$464$	0	0
$465$	1.46410	0.0678961
$466$	0	0
$467$	−4.05256	−0.187530	−0.0937650	−	0.995594i	$-0.529890\pi$
$467$	−4.05256	−0.187530	−0.0937650	+	0.995594i	$0.529890\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	8.39230	0.386697
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	−3.26795	−0.149944
$476$	0	0
$477$	26.4449	1.21083
$478$	0	0
$479$	−8.78461	−0.401379	−0.200690	−	0.979655i	$-0.564318\pi$
$479$	−8.78461	−0.401379	−0.200690	+	0.979655i	$0.564318\pi$
$480$	0	0
$481$	−14.9282	−0.680667
$482$	0	0
$483$	1.60770	0.0731527
$484$	0	0
$485$	16.5885	0.753243
$486$	0	0
$487$	6.19615	0.280774	0.140387	−	0.990097i	$-0.455165\pi$
$487$	6.19615	0.280774	0.140387	+	0.990097i	$0.455165\pi$
$488$	0	0
$489$	2.92820	0.132418
$490$	0	0
$491$	−27.7128	−1.25066	−0.625331	−	0.780360i	$-0.715036\pi$
$491$	−27.7128	−1.25066	−0.625331	+	0.780360i	$0.715036\pi$
$492$	0	0
$493$	−4.39230	−0.197819
$494$	0	0
$495$	−2.46410	−0.110753
$496$	0	0
$497$	2.53590	0.113751
$498$	0	0
$499$	−39.8564	−1.78422	−0.892109	−	0.451820i	$-0.850775\pi$
$499$	−39.8564	−1.78422	−0.892109	+	0.451820i	$0.850775\pi$
$500$	0	0
$501$	10.1436	0.453182
$502$	0	0
$503$	32.7846	1.46179	0.730897	−	0.682488i	$-0.239102\pi$
$503$	32.7846	1.46179	0.730897	+	0.682488i	$0.239102\pi$
$504$	0	0
$505$	−19.8564	−0.883598
$506$	0	0
$507$	12.3397	0.548027
$508$	0	0
$509$	24.9282	1.10492	0.552462	−	0.833538i	$-0.313689\pi$
$509$	24.9282	1.10492	0.552462	+	0.833538i	$0.313689\pi$
$510$	0	0
$511$	−6.39230	−0.282779
$512$	0	0
$513$	13.0718	0.577134
$514$	0	0
$515$	−8.39230	−0.369809
$516$	0	0
$517$	−6.92820	−0.304702
$518$	0	0
$519$	−9.46410	−0.415428
$520$	0	0
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	0	0
$523$	−33.1769	−1.45073	−0.725363	−	0.688367i	$-0.758328\pi$
$523$	−33.1769	−1.45073	−0.725363	+	0.688367i	$0.758328\pi$
$524$	0	0
$525$	−0.732051	−0.0319493
$526$	0	0
$527$	−6.92820	−0.301797
$528$	0	0
$529$	−18.1769	−0.790301
$530$	0	0
$531$	−17.0718	−0.740853
$532$	0	0
$533$	44.7846	1.93984
$534$	0	0
$535$	−19.8564	−0.858467
$536$	0	0
$537$	4.39230	0.189542
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−17.2679	−0.742407	−0.371204	−	0.928552i	$-0.621055\pi$
$541$	−17.2679	−0.742407	−0.371204	+	0.928552i	$0.621055\pi$
$542$	0	0
$543$	10.2487	0.439814
$544$	0	0
$545$	−1.66025	−0.0711175
$546$	0	0
$547$	12.7846	0.546630	0.273315	−	0.961925i	$-0.411880\pi$
$547$	12.7846	0.546630	0.273315	+	0.961925i	$0.411880\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	0	0
$551$	4.14359	0.176523
$552$	0	0
$553$	−1.80385	−0.0767074
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	−46.3923	−1.96571	−0.982853	−	0.184393i	$-0.940968\pi$
$557$	−46.3923	−1.96571	−0.982853	+	0.184393i	$0.940968\pi$
$558$	0	0
$559$	−10.9282	−0.462214
$560$	0	0
$561$	−2.53590	−0.107066
$562$	0	0
$563$	−18.9282	−0.797729	−0.398864	−	0.917010i	$-0.630596\pi$
$563$	−18.9282	−0.797729	−0.398864	+	0.917010i	$0.630596\pi$
$564$	0	0
$565$	19.8564	0.835365
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−24.3923	−1.02079	−0.510393	−	0.859941i	$-0.670500\pi$
$571$	−24.3923	−1.02079	−0.510393	+	0.859941i	$0.670500\pi$
$572$	0	0
$573$	8.78461	0.366982
$574$	0	0
$575$	−2.19615	−0.0915859
$576$	0	0
$577$	−3.41154	−0.142024	−0.0710122	−	0.997475i	$-0.522623\pi$
$577$	−3.41154	−0.142024	−0.0710122	+	0.997475i	$0.522623\pi$
$578$	0	0
$579$	−6.14359	−0.255319
$580$	0	0
$581$	4.39230	0.182224
$582$	0	0
$583$	10.7321	0.444476
$584$	0	0
$585$	13.4641	0.556672
$586$	0	0
$587$	42.5885	1.75781	0.878907	−	0.476993i	$-0.158273\pi$
$587$	42.5885	1.75781	0.878907	+	0.476993i	$0.158273\pi$
$588$	0	0
$589$	6.53590	0.269307
$590$	0	0
$591$	17.7513	0.730190
$592$	0	0
$593$	−48.2487	−1.98134	−0.990669	−	0.136293i	$-0.956481\pi$
$593$	−48.2487	−1.98134	−0.990669	+	0.136293i	$0.956481\pi$
$594$	0	0
$595$	3.46410	0.142014
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	25.1769	1.02870	0.514350	−	0.857580i	$-0.328033\pi$
$599$	25.1769	1.02870	0.514350	+	0.857580i	$0.328033\pi$
$600$	0	0
$601$	39.5167	1.61192	0.805959	−	0.591971i	$-0.201650\pi$
$601$	39.5167	1.61192	0.805959	+	0.591971i	$0.201650\pi$
$602$	0	0
$603$	−9.85641	−0.401384
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−7.07180	−0.287035	−0.143518	−	0.989648i	$-0.545841\pi$
$607$	−7.07180	−0.287035	−0.143518	+	0.989648i	$0.545841\pi$
$608$	0	0
$609$	0.928203	0.0376127
$610$	0	0
$611$	37.8564	1.53151
$612$	0	0
$613$	27.1769	1.09767	0.548833	−	0.835932i	$-0.315072\pi$
$613$	27.1769	1.09767	0.548833	+	0.835932i	$0.315072\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−17.3205	−0.697297	−0.348649	−	0.937253i	$-0.613359\pi$
$617$	−17.3205	−0.697297	−0.348649	+	0.937253i	$0.613359\pi$
$618$	0	0
$619$	12.7846	0.513857	0.256928	−	0.966430i	$-0.417290\pi$
$619$	12.7846	0.513857	0.256928	+	0.966430i	$0.417290\pi$
$620$	0	0
$621$	8.78461	0.352514
$622$	0	0
$623$	3.46410	0.138786
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.39230	0.0955395
$628$	0	0
$629$	−9.46410	−0.377358
$630$	0	0
$631$	33.0718	1.31657	0.658284	−	0.752770i	$-0.271283\pi$
$631$	33.0718	1.31657	0.658284	+	0.752770i	$0.271283\pi$
$632$	0	0
$633$	−9.56922	−0.380342
$634$	0	0
$635$	14.9282	0.592408
$636$	0	0
$637$	5.46410	0.216496
$638$	0	0
$639$	6.24871	0.247195
$640$	0	0
$641$	47.5692	1.87887	0.939436	−	0.342725i	$-0.111350\pi$
$641$	47.5692	1.87887	0.939436	+	0.342725i	$0.111350\pi$
$642$	0	0
$643$	−36.7321	−1.44857	−0.724285	−	0.689500i	$-0.757830\pi$
$643$	−36.7321	−1.44857	−0.724285	+	0.689500i	$0.757830\pi$
$644$	0	0
$645$	1.46410	0.0576489
$646$	0	0
$647$	33.4641	1.31561	0.657805	−	0.753188i	$-0.271485\pi$
$647$	33.4641	1.31561	0.657805	+	0.753188i	$0.271485\pi$
$648$	0	0
$649$	−6.92820	−0.271956
$650$	0	0
$651$	1.46410	0.0573827
$652$	0	0
$653$	−5.66025	−0.221503	−0.110751	−	0.993848i	$-0.535326\pi$
$653$	−5.66025	−0.221503	−0.110751	+	0.993848i	$0.535326\pi$
$654$	0	0
$655$	−11.6603	−0.455604
$656$	0	0
$657$	−15.7513	−0.614516
$658$	0	0
$659$	18.2487	0.710869	0.355434	−	0.934701i	$-0.384333\pi$
$659$	18.2487	0.710869	0.355434	+	0.934701i	$0.384333\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	0	0
$663$	13.8564	0.538138
$664$	0	0
$665$	−3.26795	−0.126726
$666$	0	0
$667$	2.78461	0.107821
$668$	0	0
$669$	13.5692	0.524616
$670$	0	0
$671$	−8.92820	−0.344669
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−31.8564	−1.22434	−0.612171	−	0.790726i	$-0.709703\pi$
$677$	−31.8564	−1.22434	−0.612171	+	0.790726i	$0.709703\pi$
$678$	0	0
$679$	16.5885	0.636607
$680$	0	0
$681$	5.07180	0.194352
$682$	0	0
$683$	42.2487	1.61660	0.808301	−	0.588769i	$-0.200387\pi$
$683$	42.2487	1.61660	0.808301	+	0.588769i	$0.200387\pi$
$684$	0	0
$685$	−12.9282	−0.493961
$686$	0	0
$687$	2.64102	0.100761
$688$	0	0
$689$	−58.6410	−2.23404
$690$	0	0
$691$	6.53590	0.248637	0.124319	−	0.992242i	$-0.460325\pi$
$691$	6.53590	0.248637	0.124319	+	0.992242i	$0.460325\pi$
$692$	0	0
$693$	−2.46410	−0.0936035
$694$	0	0
$695$	−3.66025	−0.138841
$696$	0	0
$697$	28.3923	1.07544
$698$	0	0
$699$	14.5359	0.549798
$700$	0	0
$701$	−23.9090	−0.903029	−0.451515	−	0.892264i	$-0.649116\pi$
$701$	−23.9090	−0.903029	−0.451515	+	0.892264i	$0.649116\pi$
$702$	0	0
$703$	8.92820	0.336734
$704$	0	0
$705$	−5.07180	−0.191015
$706$	0	0
$707$	−19.8564	−0.746777
$708$	0	0
$709$	−9.32051	−0.350039	−0.175020	−	0.984565i	$-0.555999\pi$
$709$	−9.32051	−0.350039	−0.175020	+	0.984565i	$0.555999\pi$
$710$	0	0
$711$	−4.44486	−0.166695
$712$	0	0
$713$	4.39230	0.164493
$714$	0	0
$715$	5.46410	0.204346
$716$	0	0
$717$	−3.46410	−0.129369
$718$	0	0
$719$	−27.7128	−1.03351	−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$720$	0	0
$721$	−8.39230	−0.312546
$722$	0	0
$723$	4.92820	0.183282
$724$	0	0
$725$	−1.26795	−0.0470905
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−6.92820	−0.256249
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	−0.732051	−0.0270021
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	43.7128	1.60800	0.804001	−	0.594628i	$-0.202701\pi$
$739$	43.7128	1.60800	0.804001	+	0.594628i	$0.202701\pi$
$740$	0	0
$741$	−13.0718	−0.480204
$742$	0	0
$743$	−8.78461	−0.322276	−0.161138	−	0.986932i	$-0.551516\pi$
$743$	−8.78461	−0.322276	−0.161138	+	0.986932i	$0.551516\pi$
$744$	0	0
$745$	10.7321	0.393192
$746$	0	0
$747$	10.8231	0.395996
$748$	0	0
$749$	−19.8564	−0.725537
$750$	0	0
$751$	32.3923	1.18201	0.591006	−	0.806667i	$-0.298731\pi$
$751$	32.3923	1.18201	0.591006	+	0.806667i	$0.298731\pi$
$752$	0	0
$753$	−8.78461	−0.320129
$754$	0	0
$755$	−13.1244	−0.477644
$756$	0	0
$757$	29.3731	1.06758	0.533791	−	0.845616i	$-0.320767\pi$
$757$	29.3731	1.06758	0.533791	+	0.845616i	$0.320767\pi$
$758$	0	0
$759$	1.60770	0.0583556
$760$	0	0
$761$	−29.6603	−1.07518	−0.537592	−	0.843205i	$-0.680666\pi$
$761$	−29.6603	−1.07518	−0.537592	+	0.843205i	$0.680666\pi$
$762$	0	0
$763$	−1.66025	−0.0601052
$764$	0	0
$765$	8.53590	0.308616
$766$	0	0
$767$	37.8564	1.36692
$768$	0	0
$769$	−7.80385	−0.281414	−0.140707	−	0.990051i	$-0.544938\pi$
$769$	−7.80385	−0.281414	−0.140707	+	0.990051i	$0.544938\pi$
$770$	0	0
$771$	−4.64102	−0.167142
$772$	0	0
$773$	12.9282	0.464995	0.232498	−	0.972597i	$-0.425310\pi$
$773$	12.9282	0.464995	0.232498	+	0.972597i	$0.425310\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−26.7846	−0.959658
$780$	0	0
$781$	2.53590	0.0907416
$782$	0	0
$783$	5.07180	0.181251
$784$	0	0
$785$	−11.4641	−0.409171
$786$	0	0
$787$	−45.8564	−1.63460	−0.817302	−	0.576209i	$-0.804531\pi$
$787$	−45.8564	−1.63460	−0.817302	+	0.576209i	$0.804531\pi$
$788$	0	0
$789$	17.5692	0.625481
$790$	0	0
$791$	19.8564	0.706013
$792$	0	0
$793$	48.7846	1.73239
$794$	0	0
$795$	7.85641	0.278638
$796$	0	0
$797$	−46.3923	−1.64330	−0.821650	−	0.569993i	$-0.806946\pi$
$797$	−46.3923	−1.64330	−0.821650	+	0.569993i	$0.806946\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	8.53590	0.301601
$802$	0	0
$803$	−6.39230	−0.225580
$804$	0	0
$805$	−2.19615	−0.0774042
$806$	0	0
$807$	5.56922	0.196046
$808$	0	0
$809$	31.1769	1.09612	0.548061	−	0.836438i	$-0.315366\pi$
$809$	31.1769	1.09612	0.548061	+	0.836438i	$0.315366\pi$
$810$	0	0
$811$	18.8756	0.662814	0.331407	−	0.943488i	$-0.392477\pi$
$811$	18.8756	0.662814	0.331407	+	0.943488i	$0.392477\pi$
$812$	0	0
$813$	14.9282	0.523555
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	6.53590	0.228662
$818$	0	0
$819$	13.4641	0.470474
$820$	0	0
$821$	−47.9090	−1.67203	−0.836017	−	0.548703i	$-0.815122\pi$
$821$	−47.9090	−1.67203	−0.836017	+	0.548703i	$0.815122\pi$
$822$	0	0
$823$	25.1244	0.875780	0.437890	−	0.899029i	$-0.355726\pi$
$823$	25.1244	0.875780	0.437890	+	0.899029i	$0.355726\pi$
$824$	0	0
$825$	−0.732051	−0.0254867
$826$	0	0
$827$	−1.85641	−0.0645536	−0.0322768	−	0.999479i	$-0.510276\pi$
$827$	−1.85641	−0.0645536	−0.0322768	+	0.999479i	$0.510276\pi$
$828$	0	0
$829$	−46.2487	−1.60628	−0.803142	−	0.595788i	$-0.796840\pi$
$829$	−46.2487	−1.60628	−0.803142	+	0.595788i	$0.796840\pi$
$830$	0	0
$831$	15.3205	0.531463
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	−13.8564	−0.479521
$836$	0	0
$837$	8.00000	0.276520
$838$	0	0
$839$	4.14359	0.143053	0.0715264	−	0.997439i	$-0.477213\pi$
$839$	4.14359	0.143053	0.0715264	+	0.997439i	$0.477213\pi$
$840$	0	0
$841$	−27.3923	−0.944562
$842$	0	0
$843$	1.17691	0.0405351
$844$	0	0
$845$	−16.8564	−0.579878
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−17.3590	−0.595759
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	−13.2154	−0.452486	−0.226243	−	0.974071i	$-0.572644\pi$
$853$	−13.2154	−0.452486	−0.226243	+	0.974071i	$0.572644\pi$
$854$	0	0
$855$	−8.05256	−0.275392
$856$	0	0
$857$	−20.5359	−0.701493	−0.350746	−	0.936470i	$-0.614072\pi$
$857$	−20.5359	−0.701493	−0.350746	+	0.936470i	$0.614072\pi$
$858$	0	0
$859$	−28.7846	−0.982118	−0.491059	−	0.871126i	$-0.663390\pi$
$859$	−28.7846	−0.982118	−0.491059	+	0.871126i	$0.663390\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	−37.5167	−1.27708	−0.638541	−	0.769588i	$-0.720462\pi$
$863$	−37.5167	−1.27708	−0.638541	+	0.769588i	$0.720462\pi$
$864$	0	0
$865$	12.9282	0.439572
$866$	0	0
$867$	−3.66025	−0.124309
$868$	0	0
$869$	−1.80385	−0.0611913
$870$	0	0
$871$	21.8564	0.740576
$872$	0	0
$873$	40.8756	1.38343
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	13.3205	0.449802	0.224901	−	0.974382i	$-0.427794\pi$
$877$	13.3205	0.449802	0.224901	+	0.974382i	$0.427794\pi$
$878$	0	0
$879$	10.6410	0.358913
$880$	0	0
$881$	24.9282	0.839853	0.419926	−	0.907558i	$-0.362056\pi$
$881$	24.9282	0.839853	0.419926	+	0.907558i	$0.362056\pi$
$882$	0	0
$883$	56.3923	1.89775	0.948876	−	0.315649i	$-0.102222\pi$
$883$	56.3923	1.89775	0.948876	+	0.315649i	$0.102222\pi$
$884$	0	0
$885$	−5.07180	−0.170487
$886$	0	0
$887$	13.8564	0.465253	0.232626	−	0.972566i	$-0.425268\pi$
$887$	13.8564	0.465253	0.232626	+	0.972566i	$0.425268\pi$
$888$	0	0
$889$	14.9282	0.500676
$890$	0	0
$891$	−4.46410	−0.149553
$892$	0	0
$893$	−22.6410	−0.757653
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	−8.78461	−0.293310
$898$	0	0
$899$	2.53590	0.0845769
$900$	0	0
$901$	−37.1769	−1.23854
$902$	0	0
$903$	1.46410	0.0487223
$904$	0	0
$905$	−14.0000	−0.465376
$906$	0	0
$907$	−59.0333	−1.96017	−0.980085	−	0.198580i	$-0.936367\pi$
$907$	−59.0333	−1.96017	−0.980085	+	0.198580i	$0.936367\pi$
$908$	0	0
$909$	−48.9282	−1.62285
$910$	0	0
$911$	−2.53590	−0.0840181	−0.0420090	−	0.999117i	$-0.513376\pi$
$911$	−2.53590	−0.0840181	−0.0420090	+	0.999117i	$0.513376\pi$
$912$	0	0
$913$	4.39230	0.145364
$914$	0	0
$915$	−6.53590	−0.216070
$916$	0	0
$917$	−11.6603	−0.385056
$918$	0	0
$919$	−47.3731	−1.56269	−0.781347	−	0.624097i	$-0.785467\pi$
$919$	−47.3731	−1.56269	−0.781347	+	0.624097i	$0.785467\pi$
$920$	0	0
$921$	−2.64102	−0.0870244
$922$	0	0
$923$	−13.8564	−0.456089
$924$	0	0
$925$	−2.73205	−0.0898293
$926$	0	0
$927$	−20.6795	−0.679204
$928$	0	0
$929$	−46.3923	−1.52208	−0.761041	−	0.648704i	$-0.775311\pi$
$929$	−46.3923	−1.52208	−0.761041	+	0.648704i	$0.775311\pi$
$930$	0	0
$931$	−3.26795	−0.107103
$932$	0	0
$933$	−8.10512	−0.265350
$934$	0	0
$935$	3.46410	0.113288
$936$	0	0
$937$	−42.7846	−1.39771	−0.698856	−	0.715262i	$-0.746307\pi$
$937$	−42.7846	−1.39771	−0.698856	+	0.715262i	$0.746307\pi$
$938$	0	0
$939$	8.64102	0.281989
$940$	0	0
$941$	26.7846	0.873153	0.436577	−	0.899667i	$-0.356191\pi$
$941$	26.7846	0.873153	0.436577	+	0.899667i	$0.356191\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−12.6795	−0.412028	−0.206014	−	0.978549i	$-0.566049\pi$
$947$	−12.6795	−0.412028	−0.206014	+	0.978549i	$0.566049\pi$
$948$	0	0
$949$	34.9282	1.13382
$950$	0	0
$951$	−0.430781	−0.0139690
$952$	0	0
$953$	33.4641	1.08401	0.542004	−	0.840376i	$-0.317666\pi$
$953$	33.4641	1.08401	0.542004	+	0.840376i	$0.317666\pi$
$954$	0	0
$955$	−12.0000	−0.388311
$956$	0	0
$957$	0.928203	0.0300045
$958$	0	0
$959$	−12.9282	−0.417473
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−48.9282	−1.57669
$964$	0	0
$965$	8.39230	0.270158
$966$	0	0
$967$	−13.0718	−0.420361	−0.210180	−	0.977663i	$-0.567405\pi$
$967$	−13.0718	−0.420361	−0.210180	+	0.977663i	$0.567405\pi$
$968$	0	0
$969$	−8.28719	−0.266223
$970$	0	0
$971$	−29.0718	−0.932958	−0.466479	−	0.884532i	$-0.654478\pi$
$971$	−29.0718	−0.932958	−0.466479	+	0.884532i	$0.654478\pi$
$972$	0	0
$973$	−3.66025	−0.117342
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	21.7128	0.694654	0.347327	−	0.937744i	$-0.387089\pi$
$977$	21.7128	0.694654	0.347327	+	0.937744i	$0.387089\pi$
$978$	0	0
$979$	3.46410	0.110713
$980$	0	0
$981$	−4.09103	−0.130617
$982$	0	0
$983$	−25.1769	−0.803019	−0.401509	−	0.915855i	$-0.631514\pi$
$983$	−25.1769	−0.803019	−0.401509	+	0.915855i	$0.631514\pi$
$984$	0	0
$985$	−24.2487	−0.772628
$986$	0	0
$987$	−5.07180	−0.161437
$988$	0	0
$989$	4.39230	0.139667
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	−16.6795	−0.529308
$994$	0	0
$995$	−10.9282	−0.346447
$996$	0	0
$997$	−37.7128	−1.19438	−0.597188	−	0.802101i	$-0.703716\pi$
$997$	−37.7128	−1.19438	−0.597188	+	0.802101i	$0.703716\pi$
$998$	0	0
$999$	10.9282	0.345753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.t.1.2		2
4.3	odd	2		770.2.a.j.1.1	✓	2
12.11	even	2		6930.2.a.bv.1.2		2
20.3	even	4		3850.2.c.x.1849.1		4
20.7	even	4		3850.2.c.x.1849.4		4
20.19	odd	2		3850.2.a.bd.1.2		2
28.27	even	2		5390.2.a.bs.1.2		2
44.43	even	2		8470.2.a.br.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.j.1.1	✓	2	4.3	odd	2
3850.2.a.bd.1.2		2	20.19	odd	2
3850.2.c.x.1849.1		4	20.3	even	4
3850.2.c.x.1849.4		4	20.7	even	4
5390.2.a.bs.1.2		2	28.27	even	2
6160.2.a.t.1.2		2	1.1	even	1	trivial
6930.2.a.bv.1.2		2	12.11	even	2
8470.2.a.br.1.1		2	44.43	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} +5.46410 q^{13} -0.732051 q^{15} +3.46410 q^{17} -3.26795 q^{19} -0.732051 q^{21} -2.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.26795 q^{29} -2.00000 q^{31} -0.732051 q^{33} +1.00000 q^{35} -2.73205 q^{37} +4.00000 q^{39} +8.19615 q^{41} -2.00000 q^{43} +2.46410 q^{45} +6.92820 q^{47} +1.00000 q^{49} +2.53590 q^{51} -10.7321 q^{53} +1.00000 q^{55} -2.39230 q^{57} +6.92820 q^{59} +8.92820 q^{61} +2.46410 q^{63} -5.46410 q^{65} +4.00000 q^{67} -1.60770 q^{69} -2.53590 q^{71} +6.39230 q^{73} +0.732051 q^{75} +1.00000 q^{77} +1.80385 q^{79} +4.46410 q^{81} -4.39230 q^{83} -3.46410 q^{85} -0.928203 q^{87} -3.46410 q^{89} -5.46410 q^{91} -1.46410 q^{93} +3.26795 q^{95} -16.5885 q^{97} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} - 10 q^{19} + 2 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 2 q^{35} - 2 q^{37} + 8 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} + 12 q^{51} - 18 q^{53} + 2 q^{55} + 16 q^{57} + 4 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} - 24 q^{69} - 12 q^{71} - 8 q^{73} - 2 q^{75} + 2 q^{77} + 14 q^{79} + 2 q^{81} + 12 q^{83} + 12 q^{87} - 4 q^{91} + 4 q^{93} + 10 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 - 10 * q^19 + 2 * q^21 + 6 * q^23 + 2 * q^25 - 8 * q^27 - 6 * q^29 - 4 * q^31 + 2 * q^33 + 2 * q^35 - 2 * q^37 + 8 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 + 2 * q^49 + 12 * q^51 - 18 * q^53 + 2 * q^55 + 16 * q^57 + 4 * q^61 - 2 * q^63 - 4 * q^65 + 8 * q^67 - 24 * q^69 - 12 * q^71 - 8 * q^73 - 2 * q^75 + 2 * q^77 + 14 * q^79 + 2 * q^81 + 12 * q^83 + 12 * q^87 - 4 * q^91 + 4 * q^93 + 10 * q^95 - 2 * q^97 - 2 * q^99

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