LMFDB - Embedded newform 380.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [380,2,Mod(1,380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("380.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	380.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.03431527681$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	380.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.46410 q^{9} +3.46410 q^{11} +0.732051 q^{13} -0.732051 q^{15} +3.46410 q^{17} +1.00000 q^{19} -1.46410 q^{21} +3.46410 q^{23} +1.00000 q^{25} +4.00000 q^{27} -3.46410 q^{29} +5.46410 q^{31} -2.53590 q^{33} +2.00000 q^{35} +3.26795 q^{37} -0.535898 q^{39} -6.00000 q^{41} +8.92820 q^{43} -2.46410 q^{45} -0.928203 q^{47} -3.00000 q^{49} -2.53590 q^{51} -7.26795 q^{53} +3.46410 q^{55} -0.732051 q^{57} -6.92820 q^{59} -8.39230 q^{61} -4.92820 q^{63} +0.732051 q^{65} +3.26795 q^{67} -2.53590 q^{69} -9.46410 q^{71} -7.46410 q^{73} -0.732051 q^{75} +6.92820 q^{77} -10.9282 q^{79} +4.46410 q^{81} -3.46410 q^{83} +3.46410 q^{85} +2.53590 q^{87} -8.53590 q^{89} +1.46410 q^{91} -4.00000 q^{93} +1.00000 q^{95} +14.5885 q^{97} -8.53590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 2 q^{19} + 4 q^{21} + 2 q^{25} + 8 q^{27} + 4 q^{31} - 12 q^{33} + 4 q^{35} + 10 q^{37} - 8 q^{39} - 12 q^{41} + 4 q^{43} + 2 q^{45} + 12 q^{47}+ \cdots - 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 2 * q^19 + 4 * q^21 + 2 * q^25 + 8 * q^27 + 4 * q^31 - 12 * q^33 + 4 * q^35 + 10 * q^37 - 8 * q^39 - 12 * q^41 + 4 * q^43 + 2 * q^45 + 12 * q^47 - 6 * q^49 - 12 * q^51 - 18 * q^53 + 2 * q^57 + 4 * q^61 + 4 * q^63 - 2 * q^65 + 10 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^89 - 4 * q^91 - 8 * q^93 + 2 * q^95 - 2 * q^97 - 24 * q^99

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	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	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
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.46410	−0.319493
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	0	0
$33$	−2.53590	−0.441443
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	3.26795	0.537248	0.268624	−	0.963245i	$-0.413431\pi$
$37$	3.26795	0.537248	0.268624	+	0.963245i	$0.413431\pi$
$38$	0	0
$39$	−0.535898	−0.0858124
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.92820	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$43$	8.92820	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$44$	0	0
$45$	−2.46410	−0.367327
$46$	0	0
$47$	−0.928203	−0.135392	−0.0676962	−	0.997706i	$-0.521565\pi$
$47$	−0.928203	−0.135392	−0.0676962	+	0.997706i	$0.521565\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−2.53590	−0.355097
$52$	0	0
$53$	−7.26795	−0.998330	−0.499165	−	0.866507i	$-0.666360\pi$
$53$	−7.26795	−0.998330	−0.499165	+	0.866507i	$0.666360\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	0	0
$57$	−0.732051	−0.0969625
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	0	0
$63$	−4.92820	−0.620895
$64$	0	0
$65$	0.732051	0.0907997
$66$	0	0
$67$	3.26795	0.399244	0.199622	−	0.979873i	$-0.436029\pi$
$67$	3.26795	0.399244	0.199622	+	0.979873i	$0.436029\pi$
$68$	0	0
$69$	−2.53590	−0.305286
$70$	0	0
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	0	0
$73$	−7.46410	−0.873607	−0.436804	−	0.899557i	$-0.643889\pi$
$73$	−7.46410	−0.873607	−0.436804	+	0.899557i	$0.643889\pi$
$74$	0	0
$75$	−0.732051	−0.0845299
$76$	0	0
$77$	6.92820	0.789542
$78$	0	0
$79$	−10.9282	−1.22952	−0.614759	−	0.788715i	$-0.710747\pi$
$79$	−10.9282	−1.22952	−0.614759	+	0.788715i	$0.710747\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	2.53590	0.271877
$88$	0	0
$89$	−8.53590	−0.904803	−0.452402	−	0.891814i	$-0.649433\pi$
$89$	−8.53590	−0.904803	−0.452402	+	0.891814i	$0.649433\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	14.5885	1.48123	0.740617	−	0.671928i	$-0.234533\pi$
$97$	14.5885	1.48123	0.740617	+	0.671928i	$0.234533\pi$
$98$	0	0
$99$	−8.53590	−0.857890
$100$	0	0
$101$	4.39230	0.437051	0.218525	−	0.975831i	$-0.429875\pi$
$101$	4.39230	0.437051	0.218525	+	0.975831i	$0.429875\pi$
$102$	0	0
$103$	−3.66025	−0.360656	−0.180328	−	0.983607i	$-0.557716\pi$
$103$	−3.66025	−0.360656	−0.180328	+	0.983607i	$0.557716\pi$
$104$	0	0
$105$	−1.46410	−0.142882
$106$	0	0
$107$	11.6603	1.12724	0.563620	−	0.826034i	$-0.309408\pi$
$107$	11.6603	1.12724	0.563620	+	0.826034i	$0.309408\pi$
$108$	0	0
$109$	6.39230	0.612272	0.306136	−	0.951988i	$-0.400964\pi$
$109$	6.39230	0.612272	0.306136	+	0.951988i	$0.400964\pi$
$110$	0	0
$111$	−2.39230	−0.227068
$112$	0	0
$113$	−18.5885	−1.74865	−0.874327	−	0.485336i	$-0.838697\pi$
$113$	−18.5885	−1.74865	−0.874327	+	0.485336i	$0.838697\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	0	0
$117$	−1.80385	−0.166766
$118$	0	0
$119$	6.92820	0.635107
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.39230	0.396041
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.5885	−0.939574	−0.469787	−	0.882780i	$-0.655669\pi$
$127$	−10.5885	−0.939574	−0.469787	+	0.882780i	$0.655669\pi$
$128$	0	0
$129$	−6.53590	−0.575454
$130$	0	0
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−17.3205	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$137$	−17.3205	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$138$	0	0
$139$	4.53590	0.384730	0.192365	−	0.981323i	$-0.438384\pi$
$139$	4.53590	0.384730	0.192365	+	0.981323i	$0.438384\pi$
$140$	0	0
$141$	0.679492	0.0572235
$142$	0	0
$143$	2.53590	0.212062
$144$	0	0
$145$	−3.46410	−0.287678
$146$	0	0
$147$	2.19615	0.181136
$148$	0	0
$149$	16.3923	1.34291	0.671455	−	0.741045i	$-0.265670\pi$
$149$	16.3923	1.34291	0.671455	+	0.741045i	$0.265670\pi$
$150$	0	0
$151$	12.3923	1.00847	0.504236	−	0.863566i	$-0.331774\pi$
$151$	12.3923	1.00847	0.504236	+	0.863566i	$0.331774\pi$
$152$	0	0
$153$	−8.53590	−0.690086
$154$	0	0
$155$	5.46410	0.438887
$156$	0	0
$157$	16.5359	1.31971	0.659854	−	0.751394i	$-0.270618\pi$
$157$	16.5359	1.31971	0.659854	+	0.751394i	$0.270618\pi$
$158$	0	0
$159$	5.32051	0.421944
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	20.9282	1.63922	0.819612	−	0.572919i	$-0.194189\pi$
$163$	20.9282	1.63922	0.819612	+	0.572919i	$0.194189\pi$
$164$	0	0
$165$	−2.53590	−0.197419
$166$	0	0
$167$	−7.26795	−0.562411	−0.281205	−	0.959648i	$-0.590734\pi$
$167$	−7.26795	−0.562411	−0.281205	+	0.959648i	$0.590734\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	−2.46410	−0.188435
$172$	0	0
$173$	14.1962	1.07931	0.539657	−	0.841885i	$-0.318554\pi$
$173$	14.1962	1.07931	0.539657	+	0.841885i	$0.318554\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	5.07180	0.381220
$178$	0	0
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	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
$183$	6.14359	0.454148
$184$	0	0
$185$	3.26795	0.240264
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	8.00000	0.581914
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	−18.1962	−1.30979	−0.654894	−	0.755721i	$-0.727287\pi$
$193$	−18.1962	−1.30979	−0.654894	+	0.755721i	$0.727287\pi$
$194$	0	0
$195$	−0.535898	−0.0383765
$196$	0	0
$197$	−12.9282	−0.921096	−0.460548	−	0.887635i	$-0.652347\pi$
$197$	−12.9282	−0.921096	−0.460548	+	0.887635i	$0.652347\pi$
$198$	0	0
$199$	13.0718	0.926635	0.463318	−	0.886192i	$-0.346659\pi$
$199$	13.0718	0.926635	0.463318	+	0.886192i	$0.346659\pi$
$200$	0	0
$201$	−2.39230	−0.168740
$202$	0	0
$203$	−6.92820	−0.486265
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−8.53590	−0.593286
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	−15.3205	−1.05471	−0.527354	−	0.849646i	$-0.676816\pi$
$211$	−15.3205	−1.05471	−0.527354	+	0.849646i	$0.676816\pi$
$212$	0	0
$213$	6.92820	0.474713
$214$	0	0
$215$	8.92820	0.608898
$216$	0	0
$217$	10.9282	0.741855
$218$	0	0
$219$	5.46410	0.369230
$220$	0	0
$221$	2.53590	0.170583
$222$	0	0
$223$	7.66025	0.512969	0.256484	−	0.966548i	$-0.417436\pi$
$223$	7.66025	0.512969	0.256484	+	0.966548i	$0.417436\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	−2.19615	−0.145764	−0.0728819	−	0.997341i	$-0.523220\pi$
$227$	−2.19615	−0.145764	−0.0728819	+	0.997341i	$0.523220\pi$
$228$	0	0
$229$	10.5359	0.696232	0.348116	−	0.937452i	$-0.386822\pi$
$229$	10.5359	0.696232	0.348116	+	0.937452i	$0.386822\pi$
$230$	0	0
$231$	−5.07180	−0.333700
$232$	0	0
$233$	12.9282	0.846955	0.423477	−	0.905907i	$-0.360809\pi$
$233$	12.9282	0.846955	0.423477	+	0.905907i	$0.360809\pi$
$234$	0	0
$235$	−0.928203	−0.0605493
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	13.8564	0.896296	0.448148	−	0.893959i	$-0.352084\pi$
$239$	13.8564	0.896296	0.448148	+	0.893959i	$0.352084\pi$
$240$	0	0
$241$	15.8564	1.02140	0.510700	−	0.859759i	$-0.329386\pi$
$241$	15.8564	1.02140	0.510700	+	0.859759i	$0.329386\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	0.732051	0.0465793
$248$	0	0
$249$	2.53590	0.160706
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	−2.53590	−0.158804
$256$	0	0
$257$	9.12436	0.569162	0.284581	−	0.958652i	$-0.408146\pi$
$257$	9.12436	0.569162	0.284581	+	0.958652i	$0.408146\pi$
$258$	0	0
$259$	6.53590	0.406121
$260$	0	0
$261$	8.53590	0.528359
$262$	0	0
$263$	15.4641	0.953557	0.476779	−	0.879023i	$-0.341804\pi$
$263$	15.4641	0.953557	0.476779	+	0.879023i	$0.341804\pi$
$264$	0	0
$265$	−7.26795	−0.446467
$266$	0	0
$267$	6.24871	0.382415
$268$	0	0
$269$	−10.3923	−0.633630	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923	−0.633630	−0.316815	+	0.948487i	$0.602613\pi$
$270$	0	0
$271$	18.3923	1.11725	0.558626	−	0.829419i	$-0.311329\pi$
$271$	18.3923	1.11725	0.558626	+	0.829419i	$0.311329\pi$
$272$	0	0
$273$	−1.07180	−0.0648681
$274$	0	0
$275$	3.46410	0.208893
$276$	0	0
$277$	−2.39230	−0.143740	−0.0718698	−	0.997414i	$-0.522897\pi$
$277$	−2.39230	−0.143740	−0.0718698	+	0.997414i	$0.522897\pi$
$278$	0	0
$279$	−13.4641	−0.806075
$280$	0	0
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	0	0
$283$	−2.39230	−0.142208	−0.0711039	−	0.997469i	$-0.522652\pi$
$283$	−2.39230	−0.142208	−0.0711039	+	0.997469i	$0.522652\pi$
$284$	0	0
$285$	−0.732051	−0.0433629
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−10.6795	−0.626043
$292$	0	0
$293$	11.6603	0.681199	0.340600	−	0.940208i	$-0.389370\pi$
$293$	11.6603	0.681199	0.340600	+	0.940208i	$0.389370\pi$
$294$	0	0
$295$	−6.92820	−0.403376
$296$	0	0
$297$	13.8564	0.804030
$298$	0	0
$299$	2.53590	0.146655
$300$	0	0
$301$	17.8564	1.02923
$302$	0	0
$303$	−3.21539	−0.184719
$304$	0	0
$305$	−8.39230	−0.480542
$306$	0	0
$307$	−18.1962	−1.03851	−0.519255	−	0.854620i	$-0.673790\pi$
$307$	−18.1962	−1.03851	−0.519255	+	0.854620i	$0.673790\pi$
$308$	0	0
$309$	2.67949	0.152431
$310$	0	0
$311$	−22.3923	−1.26975	−0.634876	−	0.772614i	$-0.718949\pi$
$311$	−22.3923	−1.26975	−0.634876	+	0.772614i	$0.718949\pi$
$312$	0	0
$313$	−30.7846	−1.74005	−0.870025	−	0.493008i	$-0.835897\pi$
$313$	−30.7846	−1.74005	−0.870025	+	0.493008i	$0.835897\pi$
$314$	0	0
$315$	−4.92820	−0.277673
$316$	0	0
$317$	−21.1244	−1.18646	−0.593231	−	0.805032i	$-0.702148\pi$
$317$	−21.1244	−1.18646	−0.593231	+	0.805032i	$0.702148\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	−8.53590	−0.476427
$322$	0	0
$323$	3.46410	0.192748
$324$	0	0
$325$	0.732051	0.0406069
$326$	0	0
$327$	−4.67949	−0.258776
$328$	0	0
$329$	−1.85641	−0.102347
$330$	0	0
$331$	26.2487	1.44276	0.721380	−	0.692540i	$-0.243508\pi$
$331$	26.2487	1.44276	0.721380	+	0.692540i	$0.243508\pi$
$332$	0	0
$333$	−8.05256	−0.441278
$334$	0	0
$335$	3.26795	0.178547
$336$	0	0
$337$	−20.7321	−1.12935	−0.564673	−	0.825314i	$-0.690998\pi$
$337$	−20.7321	−1.12935	−0.564673	+	0.825314i	$0.690998\pi$
$338$	0	0
$339$	13.6077	0.739069
$340$	0	0
$341$	18.9282	1.02502
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−2.53590	−0.136528
$346$	0	0
$347$	−31.1769	−1.67366	−0.836832	−	0.547459i	$-0.815595\pi$
$347$	−31.1769	−1.67366	−0.836832	+	0.547459i	$0.815595\pi$
$348$	0	0
$349$	7.07180	0.378545	0.189272	−	0.981925i	$-0.439387\pi$
$349$	7.07180	0.378545	0.189272	+	0.981925i	$0.439387\pi$
$350$	0	0
$351$	2.92820	0.156296
$352$	0	0
$353$	22.3923	1.19182	0.595911	−	0.803050i	$-0.296791\pi$
$353$	22.3923	1.19182	0.595911	+	0.803050i	$0.296791\pi$
$354$	0	0
$355$	−9.46410	−0.502302
$356$	0	0
$357$	−5.07180	−0.268428
$358$	0	0
$359$	−15.4641	−0.816164	−0.408082	−	0.912945i	$-0.633802\pi$
$359$	−15.4641	−0.816164	−0.408082	+	0.912945i	$0.633802\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.732051	−0.0384227
$364$	0	0
$365$	−7.46410	−0.390689
$366$	0	0
$367$	−23.8564	−1.24529	−0.622647	−	0.782503i	$-0.713943\pi$
$367$	−23.8564	−1.24529	−0.622647	+	0.782503i	$0.713943\pi$
$368$	0	0
$369$	14.7846	0.769656
$370$	0	0
$371$	−14.5359	−0.754666
$372$	0	0
$373$	14.5885	0.755362	0.377681	−	0.925936i	$-0.376722\pi$
$373$	14.5885	0.755362	0.377681	+	0.925936i	$0.376722\pi$
$374$	0	0
$375$	−0.732051	−0.0378029
$376$	0	0
$377$	−2.53590	−0.130605
$378$	0	0
$379$	1.07180	0.0550545	0.0275273	−	0.999621i	$-0.491237\pi$
$379$	1.07180	0.0550545	0.0275273	+	0.999621i	$0.491237\pi$
$380$	0	0
$381$	7.75129	0.397111
$382$	0	0
$383$	23.6603	1.20898	0.604491	−	0.796612i	$-0.293376\pi$
$383$	23.6603	1.20898	0.604491	+	0.796612i	$0.293376\pi$
$384$	0	0
$385$	6.92820	0.353094
$386$	0	0
$387$	−22.0000	−1.11832
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	3.71281	0.187287
$394$	0	0
$395$	−10.9282	−0.549858
$396$	0	0
$397$	−12.5359	−0.629159	−0.314579	−	0.949231i	$-0.601863\pi$
$397$	−12.5359	−0.629159	−0.314579	+	0.949231i	$0.601863\pi$
$398$	0	0
$399$	−1.46410	−0.0732968
$400$	0	0
$401$	−2.78461	−0.139057	−0.0695284	−	0.997580i	$-0.522149\pi$
$401$	−2.78461	−0.139057	−0.0695284	+	0.997580i	$0.522149\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	4.46410	0.221823
$406$	0	0
$407$	11.3205	0.561137
$408$	0	0
$409$	6.39230	0.316079	0.158040	−	0.987433i	$-0.449483\pi$
$409$	6.39230	0.316079	0.158040	+	0.987433i	$0.449483\pi$
$410$	0	0
$411$	12.6795	0.625433
$412$	0	0
$413$	−13.8564	−0.681829
$414$	0	0
$415$	−3.46410	−0.170046
$416$	0	0
$417$	−3.32051	−0.162606
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	2.28719	0.111207
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	−16.7846	−0.812264
$428$	0	0
$429$	−1.85641	−0.0896281
$430$	0	0
$431$	−26.5359	−1.27819	−0.639095	−	0.769128i	$-0.720691\pi$
$431$	−26.5359	−1.27819	−0.639095	+	0.769128i	$0.720691\pi$
$432$	0	0
$433$	−39.6603	−1.90595	−0.952975	−	0.303049i	$-0.901996\pi$
$433$	−39.6603	−1.90595	−0.952975	+	0.303049i	$0.901996\pi$
$434$	0	0
$435$	2.53590	0.121587
$436$	0	0
$437$	3.46410	0.165710
$438$	0	0
$439$	23.7128	1.13175	0.565875	−	0.824491i	$-0.308538\pi$
$439$	23.7128	1.13175	0.565875	+	0.824491i	$0.308538\pi$
$440$	0	0
$441$	7.39230	0.352015
$442$	0	0
$443$	34.3923	1.63403	0.817014	−	0.576618i	$-0.195628\pi$
$443$	34.3923	1.63403	0.817014	+	0.576618i	$0.195628\pi$
$444$	0	0
$445$	−8.53590	−0.404640
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	3.46410	0.163481	0.0817405	−	0.996654i	$-0.473952\pi$
$449$	3.46410	0.163481	0.0817405	+	0.996654i	$0.473952\pi$
$450$	0	0
$451$	−20.7846	−0.978709
$452$	0	0
$453$	−9.07180	−0.426230
$454$	0	0
$455$	1.46410	0.0686381
$456$	0	0
$457$	−6.78461	−0.317371	−0.158685	−	0.987329i	$-0.550726\pi$
$457$	−6.78461	−0.317371	−0.158685	+	0.987329i	$0.550726\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	33.7128	1.57016	0.785081	−	0.619393i	$-0.212621\pi$
$461$	33.7128	1.57016	0.785081	+	0.619393i	$0.212621\pi$
$462$	0	0
$463$	11.4641	0.532782	0.266391	−	0.963865i	$-0.414169\pi$
$463$	11.4641	0.532782	0.266391	+	0.963865i	$0.414169\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	5.32051	0.246204	0.123102	−	0.992394i	$-0.460716\pi$
$467$	5.32051	0.246204	0.123102	+	0.992394i	$0.460716\pi$
$468$	0	0
$469$	6.53590	0.301800
$470$	0	0
$471$	−12.1051	−0.557774
$472$	0	0
$473$	30.9282	1.42208
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	17.9090	0.819995
$478$	0	0
$479$	−1.60770	−0.0734575	−0.0367287	−	0.999325i	$-0.511694\pi$
$479$	−1.60770	−0.0734575	−0.0367287	+	0.999325i	$0.511694\pi$
$480$	0	0
$481$	2.39230	0.109080
$482$	0	0
$483$	−5.07180	−0.230775
$484$	0	0
$485$	14.5885	0.662428
$486$	0	0
$487$	−1.80385	−0.0817401	−0.0408701	−	0.999164i	$-0.513013\pi$
$487$	−1.80385	−0.0817401	−0.0408701	+	0.999164i	$0.513013\pi$
$488$	0	0
$489$	−15.3205	−0.692817
$490$	0	0
$491$	5.07180	0.228887	0.114443	−	0.993430i	$-0.463492\pi$
$491$	5.07180	0.228887	0.114443	+	0.993430i	$0.463492\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	−8.53590	−0.383660
$496$	0	0
$497$	−18.9282	−0.849046
$498$	0	0
$499$	16.5359	0.740248	0.370124	−	0.928982i	$-0.379315\pi$
$499$	16.5359	0.740248	0.370124	+	0.928982i	$0.379315\pi$
$500$	0	0
$501$	5.32051	0.237703
$502$	0	0
$503$	8.53590	0.380597	0.190298	−	0.981726i	$-0.439054\pi$
$503$	8.53590	0.380597	0.190298	+	0.981726i	$0.439054\pi$
$504$	0	0
$505$	4.39230	0.195455
$506$	0	0
$507$	9.12436	0.405227
$508$	0	0
$509$	29.3205	1.29961	0.649804	−	0.760102i	$-0.274851\pi$
$509$	29.3205	1.29961	0.649804	+	0.760102i	$0.274851\pi$
$510$	0	0
$511$	−14.9282	−0.660385
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	−3.66025	−0.161290
$516$	0	0
$517$	−3.21539	−0.141413
$518$	0	0
$519$	−10.3923	−0.456172
$520$	0	0
$521$	26.7846	1.17346	0.586728	−	0.809784i	$-0.300416\pi$
$521$	26.7846	1.17346	0.586728	+	0.809784i	$0.300416\pi$
$522$	0	0
$523$	35.3731	1.54676	0.773378	−	0.633945i	$-0.218565\pi$
$523$	35.3731	1.54676	0.773378	+	0.633945i	$0.218565\pi$
$524$	0	0
$525$	−1.46410	−0.0638986
$526$	0	0
$527$	18.9282	0.824525
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	17.0718	0.740853
$532$	0	0
$533$	−4.39230	−0.190252
$534$	0	0
$535$	11.6603	0.504117
$536$	0	0
$537$	15.2154	0.656593
$538$	0	0
$539$	−10.3923	−0.447628
$540$	0	0
$541$	33.1769	1.42639	0.713193	−	0.700967i	$-0.247248\pi$
$541$	33.1769	1.42639	0.713193	+	0.700967i	$0.247248\pi$
$542$	0	0
$543$	−10.2487	−0.439814
$544$	0	0
$545$	6.39230	0.273816
$546$	0	0
$547$	3.26795	0.139727	0.0698637	−	0.997557i	$-0.477744\pi$
$547$	3.26795	0.139727	0.0698637	+	0.997557i	$0.477744\pi$
$548$	0	0
$549$	20.6795	0.882579
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	−21.8564	−0.929429
$554$	0	0
$555$	−2.39230	−0.101548
$556$	0	0
$557$	−29.3205	−1.24235	−0.621175	−	0.783672i	$-0.713344\pi$
$557$	−29.3205	−1.24235	−0.621175	+	0.783672i	$0.713344\pi$
$558$	0	0
$559$	6.53590	0.276439
$560$	0	0
$561$	−8.78461	−0.370887
$562$	0	0
$563$	33.8038	1.42466	0.712331	−	0.701844i	$-0.247639\pi$
$563$	33.8038	1.42466	0.712331	+	0.701844i	$0.247639\pi$
$564$	0	0
$565$	−18.5885	−0.782022
$566$	0	0
$567$	8.92820	0.374949
$568$	0	0
$569$	5.32051	0.223047	0.111524	−	0.993762i	$-0.464427\pi$
$569$	5.32051	0.223047	0.111524	+	0.993762i	$0.464427\pi$
$570$	0	0
$571$	−2.39230	−0.100115	−0.0500574	−	0.998746i	$-0.515940\pi$
$571$	−2.39230	−0.100115	−0.0500574	+	0.998746i	$0.515940\pi$
$572$	0	0
$573$	5.07180	0.211877
$574$	0	0
$575$	3.46410	0.144463
$576$	0	0
$577$	34.7846	1.44810	0.724051	−	0.689746i	$-0.242278\pi$
$577$	34.7846	1.44810	0.724051	+	0.689746i	$0.242278\pi$
$578$	0	0
$579$	13.3205	0.553581
$580$	0	0
$581$	−6.92820	−0.287430
$582$	0	0
$583$	−25.1769	−1.04272
$584$	0	0
$585$	−1.80385	−0.0745799
$586$	0	0
$587$	−19.1769	−0.791516	−0.395758	−	0.918355i	$-0.629518\pi$
$587$	−19.1769	−0.791516	−0.395758	+	0.918355i	$0.629518\pi$
$588$	0	0
$589$	5.46410	0.225144
$590$	0	0
$591$	9.46410	0.389301
$592$	0	0
$593$	4.14359	0.170157	0.0850785	−	0.996374i	$-0.472886\pi$
$593$	4.14359	0.170157	0.0850785	+	0.996374i	$0.472886\pi$
$594$	0	0
$595$	6.92820	0.284029
$596$	0	0
$597$	−9.56922	−0.391642
$598$	0	0
$599$	20.7846	0.849236	0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846	0.849236	0.424618	+	0.905373i	$0.360408\pi$
$600$	0	0
$601$	−44.6410	−1.82095	−0.910473	−	0.413570i	$-0.864282\pi$
$601$	−44.6410	−1.82095	−0.910473	+	0.413570i	$0.864282\pi$
$602$	0	0
$603$	−8.05256	−0.327926
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	25.9090	1.05161	0.525806	−	0.850604i	$-0.323764\pi$
$607$	25.9090	1.05161	0.525806	+	0.850604i	$0.323764\pi$
$608$	0	0
$609$	5.07180	0.205520
$610$	0	0
$611$	−0.679492	−0.0274893
$612$	0	0
$613$	−14.3923	−0.581300	−0.290650	−	0.956829i	$-0.593871\pi$
$613$	−14.3923	−0.581300	−0.290650	+	0.956829i	$0.593871\pi$
$614$	0	0
$615$	4.39230	0.177115
$616$	0	0
$617$	22.3923	0.901480	0.450740	−	0.892655i	$-0.351160\pi$
$617$	22.3923	0.901480	0.450740	+	0.892655i	$0.351160\pi$
$618$	0	0
$619$	18.3923	0.739249	0.369625	−	0.929181i	$-0.379486\pi$
$619$	18.3923	0.739249	0.369625	+	0.929181i	$0.379486\pi$
$620$	0	0
$621$	13.8564	0.556038
$622$	0	0
$623$	−17.0718	−0.683967
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−2.53590	−0.101274
$628$	0	0
$629$	11.3205	0.451378
$630$	0	0
$631$	4.53590	0.180571	0.0902856	−	0.995916i	$-0.471222\pi$
$631$	4.53590	0.180571	0.0902856	+	0.995916i	$0.471222\pi$
$632$	0	0
$633$	11.2154	0.445772
$634$	0	0
$635$	−10.5885	−0.420190
$636$	0	0
$637$	−2.19615	−0.0870147
$638$	0	0
$639$	23.3205	0.922545
$640$	0	0
$641$	−11.0718	−0.437310	−0.218655	−	0.975802i	$-0.570167\pi$
$641$	−11.0718	−0.437310	−0.218655	+	0.975802i	$0.570167\pi$
$642$	0	0
$643$	6.39230	0.252088	0.126044	−	0.992025i	$-0.459772\pi$
$643$	6.39230	0.252088	0.126044	+	0.992025i	$0.459772\pi$
$644$	0	0
$645$	−6.53590	−0.257351
$646$	0	0
$647$	8.53590	0.335581	0.167790	−	0.985823i	$-0.446337\pi$
$647$	8.53590	0.335581	0.167790	+	0.985823i	$0.446337\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	20.5359	0.803632	0.401816	−	0.915720i	$-0.368379\pi$
$653$	20.5359	0.803632	0.401816	+	0.915720i	$0.368379\pi$
$654$	0	0
$655$	−5.07180	−0.198171
$656$	0	0
$657$	18.3923	0.717552
$658$	0	0
$659$	−32.7846	−1.27711	−0.638554	−	0.769577i	$-0.720467\pi$
$659$	−32.7846	−1.27711	−0.638554	+	0.769577i	$0.720467\pi$
$660$	0	0
$661$	46.7846	1.81971	0.909855	−	0.414926i	$-0.136193\pi$
$661$	46.7846	1.81971	0.909855	+	0.414926i	$0.136193\pi$
$662$	0	0
$663$	−1.85641	−0.0720969
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	−5.60770	−0.216806
$670$	0	0
$671$	−29.0718	−1.12230
$672$	0	0
$673$	−47.2679	−1.82205	−0.911023	−	0.412356i	$-0.864706\pi$
$673$	−47.2679	−1.82205	−0.911023	+	0.412356i	$0.864706\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−34.9808	−1.34442	−0.672210	−	0.740361i	$-0.734655\pi$
$677$	−34.9808	−1.34442	−0.672210	+	0.740361i	$0.734655\pi$
$678$	0	0
$679$	29.1769	1.11971
$680$	0	0
$681$	1.60770	0.0616070
$682$	0	0
$683$	−0.339746	−0.0130000	−0.00650001	−	0.999979i	$-0.502069\pi$
$683$	−0.339746	−0.0130000	−0.00650001	+	0.999979i	$0.502069\pi$
$684$	0	0
$685$	−17.3205	−0.661783
$686$	0	0
$687$	−7.71281	−0.294262
$688$	0	0
$689$	−5.32051	−0.202695
$690$	0	0
$691$	−11.1769	−0.425190	−0.212595	−	0.977140i	$-0.568191\pi$
$691$	−11.1769	−0.425190	−0.212595	+	0.977140i	$0.568191\pi$
$692$	0	0
$693$	−17.0718	−0.648504
$694$	0	0
$695$	4.53590	0.172056
$696$	0	0
$697$	−20.7846	−0.787273
$698$	0	0
$699$	−9.46410	−0.357965
$700$	0	0
$701$	−33.4641	−1.26392	−0.631961	−	0.775000i	$-0.717750\pi$
$701$	−33.4641	−1.26392	−0.631961	+	0.775000i	$0.717750\pi$
$702$	0	0
$703$	3.26795	0.123253
$704$	0	0
$705$	0.679492	0.0255911
$706$	0	0
$707$	8.78461	0.330379
$708$	0	0
$709$	10.7846	0.405025	0.202512	−	0.979280i	$-0.435089\pi$
$709$	10.7846	0.405025	0.202512	+	0.979280i	$0.435089\pi$
$710$	0	0
$711$	26.9282	1.00989
$712$	0	0
$713$	18.9282	0.708867
$714$	0	0
$715$	2.53590	0.0948372
$716$	0	0
$717$	−10.1436	−0.378819
$718$	0	0
$719$	−17.3205	−0.645946	−0.322973	−	0.946408i	$-0.604682\pi$
$719$	−17.3205	−0.645946	−0.322973	+	0.946408i	$0.604682\pi$
$720$	0	0
$721$	−7.32051	−0.272630
$722$	0	0
$723$	−11.6077	−0.431695
$724$	0	0
$725$	−3.46410	−0.128654
$726$	0	0
$727$	27.8564	1.03314	0.516568	−	0.856246i	$-0.327209\pi$
$727$	27.8564	1.03314	0.516568	+	0.856246i	$0.327209\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	30.9282	1.14392
$732$	0	0
$733$	−30.7846	−1.13706	−0.568528	−	0.822664i	$-0.692487\pi$
$733$	−30.7846	−1.13706	−0.568528	+	0.822664i	$0.692487\pi$
$734$	0	0
$735$	2.19615	0.0810063
$736$	0	0
$737$	11.3205	0.416996
$738$	0	0
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	−0.535898	−0.0196867
$742$	0	0
$743$	45.1244	1.65545	0.827726	−	0.561132i	$-0.189634\pi$
$743$	45.1244	1.65545	0.827726	+	0.561132i	$0.189634\pi$
$744$	0	0
$745$	16.3923	0.600568
$746$	0	0
$747$	8.53590	0.312312
$748$	0	0
$749$	23.3205	0.852113
$750$	0	0
$751$	−18.5359	−0.676385	−0.338192	−	0.941077i	$-0.609815\pi$
$751$	−18.5359	−0.676385	−0.338192	+	0.941077i	$0.609815\pi$
$752$	0	0
$753$	17.5692	0.640258
$754$	0	0
$755$	12.3923	0.451002
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	−8.78461	−0.318861
$760$	0	0
$761$	40.3923	1.46422	0.732110	−	0.681186i	$-0.238536\pi$
$761$	40.3923	1.46422	0.732110	+	0.681186i	$0.238536\pi$
$762$	0	0
$763$	12.7846	0.462834
$764$	0	0
$765$	−8.53590	−0.308616
$766$	0	0
$767$	−5.07180	−0.183132
$768$	0	0
$769$	−37.4641	−1.35099	−0.675495	−	0.737365i	$-0.736070\pi$
$769$	−37.4641	−1.35099	−0.675495	+	0.737365i	$0.736070\pi$
$770$	0	0
$771$	−6.67949	−0.240556
$772$	0	0
$773$	−16.0526	−0.577370	−0.288685	−	0.957424i	$-0.593218\pi$
$773$	−16.0526	−0.577370	−0.288685	+	0.957424i	$0.593218\pi$
$774$	0	0
$775$	5.46410	0.196276
$776$	0	0
$777$	−4.78461	−0.171647
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	−32.7846	−1.17313
$782$	0	0
$783$	−13.8564	−0.495188
$784$	0	0
$785$	16.5359	0.590192
$786$	0	0
$787$	−1.80385	−0.0643002	−0.0321501	−	0.999483i	$-0.510235\pi$
$787$	−1.80385	−0.0643002	−0.0321501	+	0.999483i	$0.510235\pi$
$788$	0	0
$789$	−11.3205	−0.403021
$790$	0	0
$791$	−37.1769	−1.32186
$792$	0	0
$793$	−6.14359	−0.218165
$794$	0	0
$795$	5.32051	0.188699
$796$	0	0
$797$	11.6603	0.413027	0.206514	−	0.978444i	$-0.433788\pi$
$797$	11.6603	0.413027	0.206514	+	0.978444i	$0.433788\pi$
$798$	0	0
$799$	−3.21539	−0.113752
$800$	0	0
$801$	21.0333	0.743176
$802$	0	0
$803$	−25.8564	−0.912453
$804$	0	0
$805$	6.92820	0.244187
$806$	0	0
$807$	7.60770	0.267804
$808$	0	0
$809$	−45.7128	−1.60718	−0.803588	−	0.595185i	$-0.797079\pi$
$809$	−45.7128	−1.60718	−0.803588	+	0.595185i	$0.797079\pi$
$810$	0	0
$811$	36.3923	1.27791	0.638953	−	0.769246i	$-0.279368\pi$
$811$	36.3923	1.27791	0.638953	+	0.769246i	$0.279368\pi$
$812$	0	0
$813$	−13.4641	−0.472207
$814$	0	0
$815$	20.9282	0.733083
$816$	0	0
$817$	8.92820	0.312358
$818$	0	0
$819$	−3.60770	−0.126063
$820$	0	0
$821$	−43.8564	−1.53060	−0.765300	−	0.643674i	$-0.777409\pi$
$821$	−43.8564	−1.53060	−0.765300	+	0.643674i	$0.777409\pi$
$822$	0	0
$823$	43.5692	1.51873	0.759364	−	0.650666i	$-0.225510\pi$
$823$	43.5692	1.51873	0.759364	+	0.650666i	$0.225510\pi$
$824$	0	0
$825$	−2.53590	−0.0882886
$826$	0	0
$827$	−12.3397	−0.429095	−0.214548	−	0.976714i	$-0.568828\pi$
$827$	−12.3397	−0.429095	−0.214548	+	0.976714i	$0.568828\pi$
$828$	0	0
$829$	−40.2487	−1.39790	−0.698948	−	0.715173i	$-0.746348\pi$
$829$	−40.2487	−1.39790	−0.698948	+	0.715173i	$0.746348\pi$
$830$	0	0
$831$	1.75129	0.0607515
$832$	0	0
$833$	−10.3923	−0.360072
$834$	0	0
$835$	−7.26795	−0.251518
$836$	0	0
$837$	21.8564	0.755468
$838$	0	0
$839$	5.07180	0.175098	0.0875489	−	0.996160i	$-0.472097\pi$
$839$	5.07180	0.175098	0.0875489	+	0.996160i	$0.472097\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	9.46410	0.325961
$844$	0	0
$845$	−12.4641	−0.428778
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	1.75129	0.0601041
$850$	0	0
$851$	11.3205	0.388062
$852$	0	0
$853$	24.6410	0.843692	0.421846	−	0.906667i	$-0.361382\pi$
$853$	24.6410	0.843692	0.421846	+	0.906667i	$0.361382\pi$
$854$	0	0
$855$	−2.46410	−0.0842705
$856$	0	0
$857$	14.1962	0.484931	0.242466	−	0.970160i	$-0.422044\pi$
$857$	14.1962	0.484931	0.242466	+	0.970160i	$0.422044\pi$
$858$	0	0
$859$	−7.71281	−0.263158	−0.131579	−	0.991306i	$-0.542005\pi$
$859$	−7.71281	−0.263158	−0.131579	+	0.991306i	$0.542005\pi$
$860$	0	0
$861$	8.78461	0.299379
$862$	0	0
$863$	−40.7321	−1.38654	−0.693268	−	0.720680i	$-0.743830\pi$
$863$	−40.7321	−1.38654	−0.693268	+	0.720680i	$0.743830\pi$
$864$	0	0
$865$	14.1962	0.482684
$866$	0	0
$867$	3.66025	0.124309
$868$	0	0
$869$	−37.8564	−1.28419
$870$	0	0
$871$	2.39230	0.0810602
$872$	0	0
$873$	−35.9474	−1.21664
$874$	0	0
$875$	2.00000	0.0676123
$876$	0	0
$877$	−50.9808	−1.72150	−0.860749	−	0.509030i	$-0.830004\pi$
$877$	−50.9808	−1.72150	−0.860749	+	0.509030i	$0.830004\pi$
$878$	0	0
$879$	−8.53590	−0.287909
$880$	0	0
$881$	−35.3205	−1.18998	−0.594989	−	0.803734i	$-0.702844\pi$
$881$	−35.3205	−1.18998	−0.594989	+	0.803734i	$0.702844\pi$
$882$	0	0
$883$	−22.0000	−0.740359	−0.370179	−	0.928960i	$-0.620704\pi$
$883$	−22.0000	−0.740359	−0.370179	+	0.928960i	$0.620704\pi$
$884$	0	0
$885$	5.07180	0.170487
$886$	0	0
$887$	54.5885	1.83290	0.916451	−	0.400148i	$-0.131041\pi$
$887$	54.5885	1.83290	0.916451	+	0.400148i	$0.131041\pi$
$888$	0	0
$889$	−21.1769	−0.710251
$890$	0	0
$891$	15.4641	0.518067
$892$	0	0
$893$	−0.928203	−0.0310611
$894$	0	0
$895$	−20.7846	−0.694753
$896$	0	0
$897$	−1.85641	−0.0619836
$898$	0	0
$899$	−18.9282	−0.631291
$900$	0	0
$901$	−25.1769	−0.838765
$902$	0	0
$903$	−13.0718	−0.435002
$904$	0	0
$905$	14.0000	0.465376
$906$	0	0
$907$	38.5885	1.28131	0.640654	−	0.767829i	$-0.278663\pi$
$907$	38.5885	1.28131	0.640654	+	0.767829i	$0.278663\pi$
$908$	0	0
$909$	−10.8231	−0.358979
$910$	0	0
$911$	21.4641	0.711137	0.355569	−	0.934650i	$-0.384287\pi$
$911$	21.4641	0.711137	0.355569	+	0.934650i	$0.384287\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	6.14359	0.203101
$916$	0	0
$917$	−10.1436	−0.334971
$918$	0	0
$919$	−52.4974	−1.73173	−0.865865	−	0.500278i	$-0.833231\pi$
$919$	−52.4974	−1.73173	−0.865865	+	0.500278i	$0.833231\pi$
$920$	0	0
$921$	13.3205	0.438926
$922$	0	0
$923$	−6.92820	−0.228045
$924$	0	0
$925$	3.26795	0.107450
$926$	0	0
$927$	9.01924	0.296231
$928$	0	0
$929$	35.5692	1.16699	0.583494	−	0.812117i	$-0.301685\pi$
$929$	35.5692	1.16699	0.583494	+	0.812117i	$0.301685\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	16.3923	0.536660
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	−11.1769	−0.365134	−0.182567	−	0.983193i	$-0.558441\pi$
$937$	−11.1769	−0.365134	−0.182567	+	0.983193i	$0.558441\pi$
$938$	0	0
$939$	22.5359	0.735431
$940$	0	0
$941$	−16.6410	−0.542482	−0.271241	−	0.962512i	$-0.587434\pi$
$941$	−16.6410	−0.542482	−0.271241	+	0.962512i	$0.587434\pi$
$942$	0	0
$943$	−20.7846	−0.676840
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	41.3205	1.34274	0.671368	−	0.741124i	$-0.265707\pi$
$947$	41.3205	1.34274	0.671368	+	0.741124i	$0.265707\pi$
$948$	0	0
$949$	−5.46410	−0.177372
$950$	0	0
$951$	15.4641	0.501458
$952$	0	0
$953$	29.4115	0.952733	0.476367	−	0.879247i	$-0.341953\pi$
$953$	29.4115	0.952733	0.476367	+	0.879247i	$0.341953\pi$
$954$	0	0
$955$	−6.92820	−0.224191
$956$	0	0
$957$	8.78461	0.283966
$958$	0	0
$959$	−34.6410	−1.11862
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	−28.7321	−0.925877
$964$	0	0
$965$	−18.1962	−0.585755
$966$	0	0
$967$	−10.6795	−0.343429	−0.171715	−	0.985147i	$-0.554931\pi$
$967$	−10.6795	−0.343429	−0.171715	+	0.985147i	$0.554931\pi$
$968$	0	0
$969$	−2.53590	−0.0814648
$970$	0	0
$971$	−21.4641	−0.688816	−0.344408	−	0.938820i	$-0.611920\pi$
$971$	−21.4641	−0.688816	−0.344408	+	0.938820i	$0.611920\pi$
$972$	0	0
$973$	9.07180	0.290828
$974$	0	0
$975$	−0.535898	−0.0171625
$976$	0	0
$977$	21.8038	0.697567	0.348783	−	0.937203i	$-0.386595\pi$
$977$	21.8038	0.697567	0.348783	+	0.937203i	$0.386595\pi$
$978$	0	0
$979$	−29.5692	−0.945036
$980$	0	0
$981$	−15.7513	−0.502900
$982$	0	0
$983$	56.4449	1.80031	0.900156	−	0.435568i	$-0.143452\pi$
$983$	56.4449	1.80031	0.900156	+	0.435568i	$0.143452\pi$
$984$	0	0
$985$	−12.9282	−0.411927
$986$	0	0
$987$	1.35898	0.0432569
$988$	0	0
$989$	30.9282	0.983460
$990$	0	0
$991$	12.3923	0.393655	0.196827	−	0.980438i	$-0.436936\pi$
$991$	12.3923	0.393655	0.196827	+	0.980438i	$0.436936\pi$
$992$	0	0
$993$	−19.2154	−0.609782
$994$	0	0
$995$	13.0718	0.414404
$996$	0	0
$997$	9.60770	0.304279	0.152139	−	0.988359i	$-0.451384\pi$
$997$	9.60770	0.304279	0.152139	+	0.988359i	$0.451384\pi$
$998$	0	0
$999$	13.0718	0.413573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.2.a.d.1.1	✓	2
3.2	odd	2		3420.2.a.h.1.1		2
4.3	odd	2		1520.2.a.l.1.2		2
5.2	odd	4		1900.2.c.e.1749.3		4
5.3	odd	4		1900.2.c.e.1749.2		4
5.4	even	2		1900.2.a.d.1.2		2
8.3	odd	2		6080.2.a.bj.1.1		2
8.5	even	2		6080.2.a.z.1.2		2
19.18	odd	2		7220.2.a.h.1.2		2
20.19	odd	2		7600.2.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.d.1.1	✓	2	1.1	even	1	trivial
1520.2.a.l.1.2		2	4.3	odd	2
1900.2.a.d.1.2		2	5.4	even	2
1900.2.c.e.1749.2		4	5.3	odd	4
1900.2.c.e.1749.3		4	5.2	odd	4
3420.2.a.h.1.1		2	3.2	odd	2
6080.2.a.z.1.2		2	8.5	even	2
6080.2.a.bj.1.1		2	8.3	odd	2
7220.2.a.h.1.2		2	19.18	odd	2
7600.2.a.bf.1.1		2	20.19	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +1.00000 q^{5} +2.00000 q^{7} -2.46410 q^{9} +3.46410 q^{11} +0.732051 q^{13} -0.732051 q^{15} +3.46410 q^{17} +1.00000 q^{19} -1.46410 q^{21} +3.46410 q^{23} +1.00000 q^{25} +4.00000 q^{27} -3.46410 q^{29} +5.46410 q^{31} -2.53590 q^{33} +2.00000 q^{35} +3.26795 q^{37} -0.535898 q^{39} -6.00000 q^{41} +8.92820 q^{43} -2.46410 q^{45} -0.928203 q^{47} -3.00000 q^{49} -2.53590 q^{51} -7.26795 q^{53} +3.46410 q^{55} -0.732051 q^{57} -6.92820 q^{59} -8.39230 q^{61} -4.92820 q^{63} +0.732051 q^{65} +3.26795 q^{67} -2.53590 q^{69} -9.46410 q^{71} -7.46410 q^{73} -0.732051 q^{75} +6.92820 q^{77} -10.9282 q^{79} +4.46410 q^{81} -3.46410 q^{83} +3.46410 q^{85} +2.53590 q^{87} -8.53590 q^{89} +1.46410 q^{91} -4.00000 q^{93} +1.00000 q^{95} +14.5885 q^{97} -8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{15} + 2 q^{19} + 4 q^{21} + 2 q^{25} + 8 q^{27} + 4 q^{31} - 12 q^{33} + 4 q^{35} + 10 q^{37} - 8 q^{39} - 12 q^{41} + 4 q^{43} + 2 q^{45} + 12 q^{47}+ \cdots - 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^13 + 2 * q^15 + 2 * q^19 + 4 * q^21 + 2 * q^25 + 8 * q^27 + 4 * q^31 - 12 * q^33 + 4 * q^35 + 10 * q^37 - 8 * q^39 - 12 * q^41 + 4 * q^43 + 2 * q^45 + 12 * q^47 - 6 * q^49 - 12 * q^51 - 18 * q^53 + 2 * q^57 + 4 * q^61 + 4 * q^63 - 2 * q^65 + 10 * q^67 - 12 * q^69 - 12 * q^71 - 8 * q^73 + 2 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^87 - 24 * q^89 - 4 * q^91 - 8 * q^93 + 2 * q^95 - 2 * q^97 - 24 * q^99

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