LMFDB - Embedded newform 3420.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, 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("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.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:	$27.3088374913$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.00000 q^{7} -3.46410 q^{11} +0.732051 q^{13} -3.46410 q^{17} +1.00000 q^{19} -3.46410 q^{23} +1.00000 q^{25} +3.46410 q^{29} +5.46410 q^{31} -2.00000 q^{35} +3.26795 q^{37} +6.00000 q^{41} +8.92820 q^{43} +0.928203 q^{47} -3.00000 q^{49} +7.26795 q^{53} +3.46410 q^{55} +6.92820 q^{59} -8.39230 q^{61} -0.732051 q^{65} +3.26795 q^{67} +9.46410 q^{71} -7.46410 q^{73} -6.92820 q^{77} -10.9282 q^{79} +3.46410 q^{83} +3.46410 q^{85} +8.53590 q^{89} +1.46410 q^{91} -1.00000 q^{95} +14.5885 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^7	$ 2 q - 2 q^{5} + 4 q^{7} - 2 q^{13} + 2 q^{19} + 2 q^{25} + 4 q^{31} - 4 q^{35} + 10 q^{37} + 12 q^{41} + 4 q^{43} - 12 q^{47} - 6 q^{49} + 18 q^{53} + 4 q^{61} + 2 q^{65} + 10 q^{67} + 12 q^{71} - 8 q^{73} - 8 q^{79} + 24 q^{89} - 4 q^{91} - 2 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^7 - 2 * q^13 + 2 * q^19 + 2 * q^25 + 4 * q^31 - 4 * q^35 + 10 * q^37 + 12 * q^41 + 4 * q^43 - 12 * q^47 - 6 * q^49 + 18 * q^53 + 4 * q^61 + 2 * q^65 + 10 * q^67 + 12 * q^71 - 8 * q^73 - 8 * q^79 + 24 * q^89 - 4 * q^91 - 2 * q^95 - 2 * q^97

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	0
$3$	0	0
$4$	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
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	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	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	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
$33$	0	0
$34$	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	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	0.928203	0.135392	0.0676962	−	0.997706i	$-0.478435\pi$
$47$	0.928203	0.135392	0.0676962	+	0.997706i	$0.478435\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.26795	0.998330	0.499165	−	0.866507i	$-0.333640\pi$
$53$	7.26795	0.998330	0.499165	+	0.866507i	$0.333640\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	0	0
$57$	0	0
$58$	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
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	0	0
$63$	0	0
$64$	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
$69$	0	0
$70$	0	0
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	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	0
$76$	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
$81$	0	0
$82$	0	0
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.53590	0.904803	0.452402	−	0.891814i	$-0.350567\pi$
$89$	8.53590	0.904803	0.452402	+	0.891814i	$0.350567\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	−4.39230	−0.437051	−0.218525	−	0.975831i	$-0.570125\pi$
$101$	−4.39230	−0.437051	−0.218525	+	0.975831i	$0.570125\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	−11.6603	−1.12724	−0.563620	−	0.826034i	$-0.690592\pi$
$107$	−11.6603	−1.12724	−0.563620	+	0.826034i	$0.690592\pi$
$108$	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
$111$	0	0
$112$	0	0
$113$	18.5885	1.74865	0.874327	−	0.485336i	$-0.161303\pi$
$113$	18.5885	1.74865	0.874327	+	0.485336i	$0.161303\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	0	0
$131$	5.07180	0.443125	0.221562	−	0.975146i	$-0.428884\pi$
$131$	5.07180	0.443125	0.221562	+	0.975146i	$0.428884\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.3205	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$137$	17.3205	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$138$	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	0
$142$	0	0
$143$	−2.53590	−0.212062
$144$	0	0
$145$	−3.46410	−0.287678
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.3923	−1.34291	−0.671455	−	0.741045i	$-0.734330\pi$
$149$	−16.3923	−1.34291	−0.671455	+	0.741045i	$0.734330\pi$
$150$	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
$153$	0	0
$154$	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
$159$	0	0
$160$	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
$165$	0	0
$166$	0	0
$167$	7.26795	0.562411	0.281205	−	0.959648i	$-0.409266\pi$
$167$	7.26795	0.562411	0.281205	+	0.959648i	$0.409266\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.1962	−1.07931	−0.539657	−	0.841885i	$-0.681446\pi$
$173$	−14.1962	−1.07931	−0.539657	+	0.841885i	$0.681446\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	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
$183$	0	0
$184$	0	0
$185$	−3.26795	−0.240264
$186$	0	0
$187$	12.0000	0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	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	0
$196$	0	0
$197$	12.9282	0.921096	0.460548	−	0.887635i	$-0.347653\pi$
$197$	12.9282	0.921096	0.460548	+	0.887635i	$0.347653\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	6.92820	0.486265
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	−8.92820	−0.608898
$216$	0	0
$217$	10.9282	0.741855
$218$	0	0
$219$	0	0
$220$	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
$225$	0	0
$226$	0	0
$227$	2.19615	0.145764	0.0728819	−	0.997341i	$-0.476780\pi$
$227$	2.19615	0.145764	0.0728819	+	0.997341i	$0.476780\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	−12.9282	−0.846955	−0.423477	−	0.905907i	$-0.639191\pi$
$233$	−12.9282	−0.846955	−0.423477	+	0.905907i	$0.639191\pi$
$234$	0	0
$235$	−0.928203	−0.0605493
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.8564	−0.896296	−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564	−0.896296	−0.448148	+	0.893959i	$0.647916\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	0.732051	0.0465793
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.12436	−0.569162	−0.284581	−	0.958652i	$-0.591854\pi$
$257$	−9.12436	−0.569162	−0.284581	+	0.958652i	$0.591854\pi$
$258$	0	0
$259$	6.53590	0.406121
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.4641	−0.953557	−0.476779	−	0.879023i	$-0.658196\pi$
$263$	−15.4641	−0.953557	−0.476779	+	0.879023i	$0.658196\pi$
$264$	0	0
$265$	−7.26795	−0.446467
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.3923	0.633630	0.316815	−	0.948487i	$-0.397387\pi$
$269$	10.3923	0.633630	0.316815	+	0.948487i	$0.397387\pi$
$270$	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
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	12.9282	0.771232	0.385616	−	0.922659i	$-0.373989\pi$
$281$	12.9282	0.771232	0.385616	+	0.922659i	$0.373989\pi$
$282$	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	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.6603	−0.681199	−0.340600	−	0.940208i	$-0.610630\pi$
$293$	−11.6603	−0.681199	−0.340600	+	0.940208i	$0.610630\pi$
$294$	0	0
$295$	−6.92820	−0.403376
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.53590	−0.146655
$300$	0	0
$301$	17.8564	1.02923
$302$	0	0
$303$	0	0
$304$	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
$309$	0	0
$310$	0	0
$311$	22.3923	1.26975	0.634876	−	0.772614i	$-0.281051\pi$
$311$	22.3923	1.26975	0.634876	+	0.772614i	$0.281051\pi$
$312$	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
$315$	0	0
$316$	0	0
$317$	21.1244	1.18646	0.593231	−	0.805032i	$-0.297852\pi$
$317$	21.1244	1.18646	0.593231	+	0.805032i	$0.297852\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.46410	−0.192748
$324$	0	0
$325$	0.732051	0.0406069
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	−18.9282	−1.02502
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.1769	1.67366	0.836832	−	0.547459i	$-0.184405\pi$
$347$	31.1769	1.67366	0.836832	+	0.547459i	$0.184405\pi$
$348$	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
$351$	0	0
$352$	0	0
$353$	−22.3923	−1.19182	−0.595911	−	0.803050i	$-0.703209\pi$
$353$	−22.3923	−1.19182	−0.595911	+	0.803050i	$0.703209\pi$
$354$	0	0
$355$	−9.46410	−0.502302
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.4641	0.816164	0.408082	−	0.912945i	$-0.366198\pi$
$359$	15.4641	0.816164	0.408082	+	0.912945i	$0.366198\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	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
$369$	0	0
$370$	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	0
$376$	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
$381$	0	0
$382$	0	0
$383$	−23.6603	−1.20898	−0.604491	−	0.796612i	$-0.706624\pi$
$383$	−23.6603	−1.20898	−0.604491	+	0.796612i	$0.706624\pi$
$384$	0	0
$385$	6.92820	0.353094
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	2.78461	0.139057	0.0695284	−	0.997580i	$-0.477851\pi$
$401$	2.78461	0.139057	0.0695284	+	0.997580i	$0.477851\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	13.8564	0.681829
$414$	0	0
$415$	−3.46410	−0.170046
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	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
$423$	0	0
$424$	0	0
$425$	−3.46410	−0.168034
$426$	0	0
$427$	−16.7846	−0.812264
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.5359	1.27819	0.639095	−	0.769128i	$-0.279309\pi$
$431$	26.5359	1.27819	0.639095	+	0.769128i	$0.279309\pi$
$432$	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
$435$	0	0
$436$	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
$441$	0	0
$442$	0	0
$443$	−34.3923	−1.63403	−0.817014	−	0.576618i	$-0.804372\pi$
$443$	−34.3923	−1.63403	−0.817014	+	0.576618i	$0.804372\pi$
$444$	0	0
$445$	−8.53590	−0.404640
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.46410	−0.163481	−0.0817405	−	0.996654i	$-0.526048\pi$
$449$	−3.46410	−0.163481	−0.0817405	+	0.996654i	$0.526048\pi$
$450$	0	0
$451$	−20.7846	−0.978709
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	−33.7128	−1.57016	−0.785081	−	0.619393i	$-0.787379\pi$
$461$	−33.7128	−1.57016	−0.785081	+	0.619393i	$0.787379\pi$
$462$	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
$465$	0	0
$466$	0	0
$467$	−5.32051	−0.246204	−0.123102	−	0.992394i	$-0.539284\pi$
$467$	−5.32051	−0.246204	−0.123102	+	0.992394i	$0.539284\pi$
$468$	0	0
$469$	6.53590	0.301800
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−30.9282	−1.42208
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.60770	0.0734575	0.0367287	−	0.999325i	$-0.488306\pi$
$479$	1.60770	0.0734575	0.0367287	+	0.999325i	$0.488306\pi$
$480$	0	0
$481$	2.39230	0.109080
$482$	0	0
$483$	0	0
$484$	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
$489$	0	0
$490$	0	0
$491$	−5.07180	−0.228887	−0.114443	−	0.993430i	$-0.536508\pi$
$491$	−5.07180	−0.228887	−0.114443	+	0.993430i	$0.536508\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	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
$501$	0	0
$502$	0	0
$503$	−8.53590	−0.380597	−0.190298	−	0.981726i	$-0.560946\pi$
$503$	−8.53590	−0.380597	−0.190298	+	0.981726i	$0.560946\pi$
$504$	0	0
$505$	4.39230	0.195455
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−29.3205	−1.29961	−0.649804	−	0.760102i	$-0.725149\pi$
$509$	−29.3205	−1.29961	−0.649804	+	0.760102i	$0.725149\pi$
$510$	0	0
$511$	−14.9282	−0.660385
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.66025	0.161290
$516$	0	0
$517$	−3.21539	−0.141413
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.7846	−1.17346	−0.586728	−	0.809784i	$-0.699584\pi$
$521$	−26.7846	−1.17346	−0.586728	+	0.809784i	$0.699584\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	−18.9282	−0.824525
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.39230	0.190252
$534$	0	0
$535$	11.6603	0.504117
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	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
$549$	0	0
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	−21.8564	−0.929429
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.3205	1.24235	0.621175	−	0.783672i	$-0.286656\pi$
$557$	29.3205	1.24235	0.621175	+	0.783672i	$0.286656\pi$
$558$	0	0
$559$	6.53590	0.276439
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.8038	−1.42466	−0.712331	−	0.701844i	$-0.752361\pi$
$563$	−33.8038	−1.42466	−0.712331	+	0.701844i	$0.752361\pi$
$564$	0	0
$565$	−18.5885	−0.782022
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.32051	−0.223047	−0.111524	−	0.993762i	$-0.535573\pi$
$569$	−5.32051	−0.223047	−0.111524	+	0.993762i	$0.535573\pi$
$570$	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
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	6.92820	0.287430
$582$	0	0
$583$	−25.1769	−1.04272
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.1769	0.791516	0.395758	−	0.918355i	$-0.370482\pi$
$587$	19.1769	0.791516	0.395758	+	0.918355i	$0.370482\pi$
$588$	0	0
$589$	5.46410	0.225144
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.14359	−0.170157	−0.0850785	−	0.996374i	$-0.527114\pi$
$593$	−4.14359	−0.170157	−0.0850785	+	0.996374i	$0.527114\pi$
$594$	0	0
$595$	6.92820	0.284029
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	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
$603$	0	0
$604$	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
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	−22.3923	−0.901480	−0.450740	−	0.892655i	$-0.648840\pi$
$617$	−22.3923	−0.901480	−0.450740	+	0.892655i	$0.648840\pi$
$618$	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
$621$	0	0
$622$	0	0
$623$	17.0718	0.683967
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	10.5885	0.420190
$636$	0	0
$637$	−2.19615	−0.0870147
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.0718	0.437310	0.218655	−	0.975802i	$-0.429833\pi$
$641$	11.0718	0.437310	0.218655	+	0.975802i	$0.429833\pi$
$642$	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
$645$	0	0
$646$	0	0
$647$	−8.53590	−0.335581	−0.167790	−	0.985823i	$-0.553663\pi$
$647$	−8.53590	−0.335581	−0.167790	+	0.985823i	$0.553663\pi$
$648$	0	0
$649$	−24.0000	−0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.5359	−0.803632	−0.401816	−	0.915720i	$-0.631621\pi$
$653$	−20.5359	−0.803632	−0.401816	+	0.915720i	$0.631621\pi$
$654$	0	0
$655$	−5.07180	−0.198171
$656$	0	0
$657$	0	0
$658$	0	0
$659$	32.7846	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$659$	32.7846	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	34.9808	1.34442	0.672210	−	0.740361i	$-0.265345\pi$
$677$	34.9808	1.34442	0.672210	+	0.740361i	$0.265345\pi$
$678$	0	0
$679$	29.1769	1.11971
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.339746	0.0130000	0.00650001	−	0.999979i	$-0.497931\pi$
$683$	0.339746	0.0130000	0.00650001	+	0.999979i	$0.497931\pi$
$684$	0	0
$685$	−17.3205	−0.661783
$686$	0	0
$687$	0	0
$688$	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
$693$	0	0
$694$	0	0
$695$	−4.53590	−0.172056
$696$	0	0
$697$	−20.7846	−0.787273
$698$	0	0
$699$	0	0
$700$	0	0
$701$	33.4641	1.26392	0.631961	−	0.775000i	$-0.282250\pi$
$701$	33.4641	1.26392	0.631961	+	0.775000i	$0.282250\pi$
$702$	0	0
$703$	3.26795	0.123253
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	−18.9282	−0.708867
$714$	0	0
$715$	2.53590	0.0948372
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.3205	0.645946	0.322973	−	0.946408i	$-0.395318\pi$
$719$	17.3205	0.645946	0.322973	+	0.946408i	$0.395318\pi$
$720$	0	0
$721$	−7.32051	−0.272630
$722$	0	0
$723$	0	0
$724$	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
$729$	0	0
$730$	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
$735$	0	0
$736$	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	0
$742$	0	0
$743$	−45.1244	−1.65545	−0.827726	−	0.561132i	$-0.810366\pi$
$743$	−45.1244	−1.65545	−0.827726	+	0.561132i	$0.810366\pi$
$744$	0	0
$745$	16.3923	0.600568
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	−40.3923	−1.46422	−0.732110	−	0.681186i	$-0.761464\pi$
$761$	−40.3923	−1.46422	−0.732110	+	0.681186i	$0.761464\pi$
$762$	0	0
$763$	12.7846	0.462834
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	16.0526	0.577370	0.288685	−	0.957424i	$-0.406782\pi$
$773$	16.0526	0.577370	0.288685	+	0.957424i	$0.406782\pi$
$774$	0	0
$775$	5.46410	0.196276
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	−32.7846	−1.17313
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	37.1769	1.32186
$792$	0	0
$793$	−6.14359	−0.218165
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.6603	−0.413027	−0.206514	−	0.978444i	$-0.566212\pi$
$797$	−11.6603	−0.413027	−0.206514	+	0.978444i	$0.566212\pi$
$798$	0	0
$799$	−3.21539	−0.113752
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25.8564	0.912453
$804$	0	0
$805$	6.92820	0.244187
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.7128	1.60718	0.803588	−	0.595185i	$-0.202921\pi$
$809$	45.7128	1.60718	0.803588	+	0.595185i	$0.202921\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	−20.9282	−0.733083
$816$	0	0
$817$	8.92820	0.312358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.8564	1.53060	0.765300	−	0.643674i	$-0.222591\pi$
$821$	43.8564	1.53060	0.765300	+	0.643674i	$0.222591\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	12.3397	0.429095	0.214548	−	0.976714i	$-0.431172\pi$
$827$	12.3397	0.429095	0.214548	+	0.976714i	$0.431172\pi$
$828$	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
$831$	0	0
$832$	0	0
$833$	10.3923	0.360072
$834$	0	0
$835$	−7.26795	−0.251518
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.07180	−0.175098	−0.0875489	−	0.996160i	$-0.527903\pi$
$839$	−5.07180	−0.175098	−0.0875489	+	0.996160i	$0.527903\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.4641	0.428778
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	−14.1962	−0.484931	−0.242466	−	0.970160i	$-0.577956\pi$
$857$	−14.1962	−0.484931	−0.242466	+	0.970160i	$0.577956\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	40.7321	1.38654	0.693268	−	0.720680i	$-0.256170\pi$
$863$	40.7321	1.38654	0.693268	+	0.720680i	$0.256170\pi$
$864$	0	0
$865$	14.1962	0.482684
$866$	0	0
$867$	0	0
$868$	0	0
$869$	37.8564	1.28419
$870$	0	0
$871$	2.39230	0.0810602
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	35.3205	1.18998	0.594989	−	0.803734i	$-0.297156\pi$
$881$	35.3205	1.18998	0.594989	+	0.803734i	$0.297156\pi$
$882$	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
$885$	0	0
$886$	0	0
$887$	−54.5885	−1.83290	−0.916451	−	0.400148i	$-0.868959\pi$
$887$	−54.5885	−1.83290	−0.916451	+	0.400148i	$0.868959\pi$
$888$	0	0
$889$	−21.1769	−0.710251
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.928203	0.0310611
$894$	0	0
$895$	−20.7846	−0.694753
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.9282	0.631291
$900$	0	0
$901$	−25.1769	−0.838765
$902$	0	0
$903$	0	0
$904$	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
$909$	0	0
$910$	0	0
$911$	−21.4641	−0.711137	−0.355569	−	0.934650i	$-0.615713\pi$
$911$	−21.4641	−0.711137	−0.355569	+	0.934650i	$0.615713\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	0	0
$916$	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
$921$	0	0
$922$	0	0
$923$	6.92820	0.228045
$924$	0	0
$925$	3.26795	0.107450
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.5692	−1.16699	−0.583494	−	0.812117i	$-0.698315\pi$
$929$	−35.5692	−1.16699	−0.583494	+	0.812117i	$0.698315\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	16.6410	0.542482	0.271241	−	0.962512i	$-0.412566\pi$
$941$	16.6410	0.542482	0.271241	+	0.962512i	$0.412566\pi$
$942$	0	0
$943$	−20.7846	−0.676840
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.3205	−1.34274	−0.671368	−	0.741124i	$-0.734293\pi$
$947$	−41.3205	−1.34274	−0.671368	+	0.741124i	$0.734293\pi$
$948$	0	0
$949$	−5.46410	−0.177372
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.4115	−0.952733	−0.476367	−	0.879247i	$-0.658047\pi$
$953$	−29.4115	−0.952733	−0.476367	+	0.879247i	$0.658047\pi$
$954$	0	0
$955$	−6.92820	−0.224191
$956$	0	0
$957$	0	0
$958$	0	0
$959$	34.6410	1.11862
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	0	0
$971$	21.4641	0.688816	0.344408	−	0.938820i	$-0.388080\pi$
$971$	21.4641	0.688816	0.344408	+	0.938820i	$0.388080\pi$
$972$	0	0
$973$	9.07180	0.290828
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−21.8038	−0.697567	−0.348783	−	0.937203i	$-0.613405\pi$
$977$	−21.8038	−0.697567	−0.348783	+	0.937203i	$0.613405\pi$
$978$	0	0
$979$	−29.5692	−0.945036
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−56.4449	−1.80031	−0.900156	−	0.435568i	$-0.856548\pi$
$983$	−56.4449	−1.80031	−0.900156	+	0.435568i	$0.856548\pi$
$984$	0	0
$985$	−12.9282	−0.411927
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.a.h.1.1		2
3.2	odd	2		380.2.a.d.1.1	✓	2
12.11	even	2		1520.2.a.l.1.2		2
15.2	even	4		1900.2.c.e.1749.3		4
15.8	even	4		1900.2.c.e.1749.2		4
15.14	odd	2		1900.2.a.d.1.2		2
24.5	odd	2		6080.2.a.z.1.2		2
24.11	even	2		6080.2.a.bj.1.1		2
57.56	even	2		7220.2.a.h.1.2		2
60.59	even	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	3.2	odd	2
1520.2.a.l.1.2		2	12.11	even	2
1900.2.a.d.1.2		2	15.14	odd	2
1900.2.c.e.1749.2		4	15.8	even	4
1900.2.c.e.1749.3		4	15.2	even	4
3420.2.a.h.1.1		2	1.1	even	1	trivial
6080.2.a.z.1.2		2	24.5	odd	2
6080.2.a.bj.1.1		2	24.11	even	2
7220.2.a.h.1.2		2	57.56	even	2
7600.2.a.bf.1.1		2	60.59	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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.00000 q^{7} -3.46410 q^{11} +0.732051 q^{13} -3.46410 q^{17} +1.00000 q^{19} -3.46410 q^{23} +1.00000 q^{25} +3.46410 q^{29} +5.46410 q^{31} -2.00000 q^{35} +3.26795 q^{37} +6.00000 q^{41} +8.92820 q^{43} +0.928203 q^{47} -3.00000 q^{49} +7.26795 q^{53} +3.46410 q^{55} +6.92820 q^{59} -8.39230 q^{61} -0.732051 q^{65} +3.26795 q^{67} +9.46410 q^{71} -7.46410 q^{73} -6.92820 q^{77} -10.9282 q^{79} +3.46410 q^{83} +3.46410 q^{85} +8.53590 q^{89} +1.46410 q^{91} -1.00000 q^{95} +14.5885 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^7	\( 2 q - 2 q^{5} + 4 q^{7} - 2 q^{13} + 2 q^{19} + 2 q^{25} + 4 q^{31} - 4 q^{35} + 10 q^{37} + 12 q^{41} + 4 q^{43} - 12 q^{47} - 6 q^{49} + 18 q^{53} + 4 q^{61} + 2 q^{65} + 10 q^{67} + 12 q^{71} - 8 q^{73} - 8 q^{79} + 24 q^{89} - 4 q^{91} - 2 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^7 - 2 * q^13 + 2 * q^19 + 2 * q^25 + 4 * q^31 - 4 * q^35 + 10 * q^37 + 12 * q^41 + 4 * q^43 - 12 * q^47 - 6 * q^49 + 18 * q^53 + 4 * q^61 + 2 * q^65 + 10 * q^67 + 12 * q^71 - 8 * q^73 - 8 * q^79 + 24 * q^89 - 4 * q^91 - 2 * q^95 - 2 * q^97

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