LMFDB - Embedded newform 3900.2.a.t.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3900,2,Mod(1,3900)] 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("3900.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3900.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,4,0,2,0,4,0,-2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.1416567883$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 780)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	3900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.00000 q^{7} +1.00000 q^{9} -0.449490 q^{11} -1.00000 q^{13} +2.00000 q^{17} +6.89898 q^{19} +2.00000 q^{21} -4.89898 q^{23} +1.00000 q^{27} -2.00000 q^{29} -2.89898 q^{31} -0.449490 q^{33} +10.8990 q^{37} -1.00000 q^{39} +3.55051 q^{41} +7.79796 q^{43} +5.34847 q^{47} -3.00000 q^{49} +2.00000 q^{51} -2.89898 q^{53} +6.89898 q^{57} -4.44949 q^{59} +4.00000 q^{61} +2.00000 q^{63} -7.79796 q^{67} -4.89898 q^{69} +14.2474 q^{71} -7.79796 q^{73} -0.898979 q^{77} +10.0000 q^{79} +1.00000 q^{81} -8.44949 q^{83} -2.00000 q^{87} +1.34847 q^{89} -2.00000 q^{91} -2.89898 q^{93} +6.00000 q^{97} -0.449490 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 2 q^{27} - 4 q^{29} + 4 q^{31} + 4 q^{33} + 12 q^{37} - 2 q^{39} + 12 q^{41} - 4 q^{43} - 4 q^{47} - 6 q^{49} + 4 q^{51}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 4 * q^17 + 4 * q^19 + 4 * q^21 + 2 * q^27 - 4 * q^29 + 4 * q^31 + 4 * q^33 + 12 * q^37 - 2 * q^39 + 12 * q^41 - 4 * q^43 - 4 * q^47 - 6 * q^49 + 4 * q^51 + 4 * q^53 + 4 * q^57 - 4 * q^59 + 8 * q^61 + 4 * q^63 + 4 * q^67 + 4 * q^71 + 4 * q^73 + 8 * q^77 + 20 * q^79 + 2 * q^81 - 12 * q^83 - 4 * q^87 - 12 * q^89 - 4 * q^91 + 4 * q^93 + 12 * q^97 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	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
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.449490	−0.135526	−0.0677631	−	0.997701i	$-0.521586\pi$
$11$	−0.449490	−0.135526	−0.0677631	+	0.997701i	$0.521586\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	6.89898	1.58273	0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898	1.58273	0.791367	+	0.611341i	$0.209370\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−4.89898	−1.02151	−0.510754	−	0.859727i	$-0.670634\pi$
$23$	−4.89898	−1.02151	−0.510754	+	0.859727i	$0.670634\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−2.89898	−0.520672	−0.260336	−	0.965518i	$-0.583833\pi$
$31$	−2.89898	−0.520672	−0.260336	+	0.965518i	$0.583833\pi$
$32$	0	0
$33$	−0.449490	−0.0782461
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.8990	1.79178	0.895891	−	0.444275i	$-0.146539\pi$
$37$	10.8990	1.79178	0.895891	+	0.444275i	$0.146539\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.55051	0.554497	0.277248	−	0.960798i	$-0.410578\pi$
$41$	3.55051	0.554497	0.277248	+	0.960798i	$0.410578\pi$
$42$	0	0
$43$	7.79796	1.18918	0.594589	−	0.804030i	$-0.297315\pi$
$43$	7.79796	1.18918	0.594589	+	0.804030i	$0.297315\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.34847	0.780154	0.390077	−	0.920782i	$-0.372448\pi$
$47$	5.34847	0.780154	0.390077	+	0.920782i	$0.372448\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−2.89898	−0.398205	−0.199103	−	0.979979i	$-0.563803\pi$
$53$	−2.89898	−0.398205	−0.199103	+	0.979979i	$0.563803\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.89898	0.913792
$58$	0	0
$59$	−4.44949	−0.579274	−0.289637	−	0.957137i	$-0.593535\pi$
$59$	−4.44949	−0.579274	−0.289637	+	0.957137i	$0.593535\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.79796	−0.952672	−0.476336	−	0.879263i	$-0.658035\pi$
$67$	−7.79796	−0.952672	−0.476336	+	0.879263i	$0.658035\pi$
$68$	0	0
$69$	−4.89898	−0.589768
$70$	0	0
$71$	14.2474	1.69086	0.845431	−	0.534085i	$-0.179344\pi$
$71$	14.2474	1.69086	0.845431	+	0.534085i	$0.179344\pi$
$72$	0	0
$73$	−7.79796	−0.912682	−0.456341	−	0.889805i	$-0.650840\pi$
$73$	−7.79796	−0.912682	−0.456341	+	0.889805i	$0.650840\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.898979	−0.102448
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.44949	−0.927452	−0.463726	−	0.885979i	$-0.653488\pi$
$83$	−8.44949	−0.927452	−0.463726	+	0.885979i	$0.653488\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	1.34847	0.142937	0.0714687	−	0.997443i	$-0.477231\pi$
$89$	1.34847	0.142937	0.0714687	+	0.997443i	$0.477231\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−2.89898	−0.300610
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	−0.449490	−0.0451754
$100$	0	0
$101$	14.8990	1.48250	0.741252	−	0.671227i	$-0.234232\pi$
$101$	14.8990	1.48250	0.741252	+	0.671227i	$0.234232\pi$
$102$	0	0
$103$	3.79796	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$103$	3.79796	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.7980	−1.33390	−0.666950	−	0.745103i	$-0.732400\pi$
$107$	−13.7980	−1.33390	−0.666950	+	0.745103i	$0.732400\pi$
$108$	0	0
$109$	10.8990	1.04393	0.521966	−	0.852966i	$-0.325199\pi$
$109$	10.8990	1.04393	0.521966	+	0.852966i	$0.325199\pi$
$110$	0	0
$111$	10.8990	1.03449
$112$	0	0
$113$	8.69694	0.818139	0.409070	−	0.912503i	$-0.365853\pi$
$113$	8.69694	0.818139	0.409070	+	0.912503i	$0.365853\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.7980	−0.981633
$122$	0	0
$123$	3.55051	0.320139
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	7.79796	0.686572
$130$	0	0
$131$	−4.89898	−0.428026	−0.214013	−	0.976831i	$-0.568653\pi$
$131$	−4.89898	−0.428026	−0.214013	+	0.976831i	$0.568653\pi$
$132$	0	0
$133$	13.7980	1.19643
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.2474	0.875499	0.437749	−	0.899097i	$-0.355776\pi$
$137$	10.2474	0.875499	0.437749	+	0.899097i	$0.355776\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	5.34847	0.450422
$142$	0	0
$143$	0.449490	0.0375882
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−6.24745	−0.511811	−0.255905	−	0.966702i	$-0.582374\pi$
$149$	−6.24745	−0.511811	−0.255905	+	0.966702i	$0.582374\pi$
$150$	0	0
$151$	18.8990	1.53798	0.768989	−	0.639263i	$-0.220760\pi$
$151$	18.8990	1.53798	0.768989	+	0.639263i	$0.220760\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.79796	−0.462728	−0.231364	−	0.972867i	$-0.574319\pi$
$157$	−5.79796	−0.462728	−0.231364	+	0.972867i	$0.574319\pi$
$158$	0	0
$159$	−2.89898	−0.229904
$160$	0	0
$161$	−9.79796	−0.772187
$162$	0	0
$163$	−6.89898	−0.540370	−0.270185	−	0.962808i	$-0.587085\pi$
$163$	−6.89898	−0.540370	−0.270185	+	0.962808i	$0.587085\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.55051	−0.274747	−0.137373	−	0.990519i	$-0.543866\pi$
$167$	−3.55051	−0.274747	−0.137373	+	0.990519i	$0.543866\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.89898	0.527578
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.44949	−0.334444
$178$	0	0
$179$	−19.5959	−1.46467	−0.732334	−	0.680946i	$-0.761569\pi$
$179$	−19.5959	−1.46467	−0.732334	+	0.680946i	$0.761569\pi$
$180$	0	0
$181$	−3.79796	−0.282300	−0.141150	−	0.989988i	$-0.545080\pi$
$181$	−3.79796	−0.282300	−0.141150	+	0.989988i	$0.545080\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.898979	−0.0657399
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	16.8990	1.22277	0.611384	−	0.791334i	$-0.290613\pi$
$191$	16.8990	1.22277	0.611384	+	0.791334i	$0.290613\pi$
$192$	0	0
$193$	1.10102	0.0792532	0.0396266	−	0.999215i	$-0.487383\pi$
$193$	1.10102	0.0792532	0.0396266	+	0.999215i	$0.487383\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.5505	−0.822940	−0.411470	−	0.911423i	$-0.634984\pi$
$197$	−11.5505	−0.822940	−0.411470	+	0.911423i	$0.634984\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	−7.79796	−0.550026
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.89898	−0.340503
$208$	0	0
$209$	−3.10102	−0.214502
$210$	0	0
$211$	11.7980	0.812205	0.406102	−	0.913828i	$-0.366888\pi$
$211$	11.7980	0.812205	0.406102	+	0.913828i	$0.366888\pi$
$212$	0	0
$213$	14.2474	0.976219
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.79796	−0.393591
$218$	0	0
$219$	−7.79796	−0.526937
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	14.8990	0.997709	0.498855	−	0.866686i	$-0.333754\pi$
$223$	14.8990	0.997709	0.498855	+	0.866686i	$0.333754\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.4495	−1.35728	−0.678640	−	0.734471i	$-0.737430\pi$
$227$	−20.4495	−1.35728	−0.678640	+	0.734471i	$0.737430\pi$
$228$	0	0
$229$	−24.6969	−1.63202	−0.816010	−	0.578038i	$-0.803819\pi$
$229$	−24.6969	−1.63202	−0.816010	+	0.578038i	$0.803819\pi$
$230$	0	0
$231$	−0.898979	−0.0591485
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.0000	0.649570
$238$	0	0
$239$	20.4495	1.32277	0.661384	−	0.750048i	$-0.269969\pi$
$239$	20.4495	1.32277	0.661384	+	0.750048i	$0.269969\pi$
$240$	0	0
$241$	10.8990	0.702065	0.351032	−	0.936363i	$-0.385831\pi$
$241$	10.8990	0.702065	0.351032	+	0.936363i	$0.385831\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.89898	−0.438972
$248$	0	0
$249$	−8.44949	−0.535465
$250$	0	0
$251$	19.5959	1.23688	0.618442	−	0.785831i	$-0.287764\pi$
$251$	19.5959	1.23688	0.618442	+	0.785831i	$0.287764\pi$
$252$	0	0
$253$	2.20204	0.138441
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.79796	0.236910	0.118455	−	0.992959i	$-0.462206\pi$
$257$	3.79796	0.236910	0.118455	+	0.992959i	$0.462206\pi$
$258$	0	0
$259$	21.7980	1.35446
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−17.7980	−1.09747	−0.548735	−	0.835997i	$-0.684890\pi$
$263$	−17.7980	−1.09747	−0.548735	+	0.835997i	$0.684890\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.34847	0.0825250
$268$	0	0
$269$	31.7980	1.93876	0.969378	−	0.245574i	$-0.0789764\pi$
$269$	31.7980	1.93876	0.969378	+	0.245574i	$0.0789764\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.7980	0.829039	0.414520	−	0.910040i	$-0.363950\pi$
$277$	13.7980	0.829039	0.414520	+	0.910040i	$0.363950\pi$
$278$	0	0
$279$	−2.89898	−0.173557
$280$	0	0
$281$	30.2474	1.80441	0.902206	−	0.431306i	$-0.141947\pi$
$281$	30.2474	1.80441	0.902206	+	0.431306i	$0.141947\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.10102	0.419160
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−7.55051	−0.441106	−0.220553	−	0.975375i	$-0.570786\pi$
$293$	−7.55051	−0.441106	−0.220553	+	0.975375i	$0.570786\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.449490	−0.0260820
$298$	0	0
$299$	4.89898	0.283315
$300$	0	0
$301$	15.5959	0.898934
$302$	0	0
$303$	14.8990	0.855924
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.59592	0.547668	0.273834	−	0.961777i	$-0.411708\pi$
$307$	9.59592	0.547668	0.273834	+	0.961777i	$0.411708\pi$
$308$	0	0
$309$	3.79796	0.216058
$310$	0	0
$311$	−23.5959	−1.33800	−0.669001	−	0.743262i	$-0.733278\pi$
$311$	−23.5959	−1.33800	−0.669001	+	0.743262i	$0.733278\pi$
$312$	0	0
$313$	12.2020	0.689700	0.344850	−	0.938658i	$-0.387930\pi$
$313$	12.2020	0.689700	0.344850	+	0.938658i	$0.387930\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.1464	−1.30003	−0.650016	−	0.759920i	$-0.725238\pi$
$317$	−23.1464	−1.30003	−0.650016	+	0.759920i	$0.725238\pi$
$318$	0	0
$319$	0.898979	0.0503332
$320$	0	0
$321$	−13.7980	−0.770127
$322$	0	0
$323$	13.7980	0.767739
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.8990	0.602715
$328$	0	0
$329$	10.6969	0.589741
$330$	0	0
$331$	−25.5959	−1.40688	−0.703439	−	0.710755i	$-0.748353\pi$
$331$	−25.5959	−1.40688	−0.703439	+	0.710755i	$0.748353\pi$
$332$	0	0
$333$	10.8990	0.597260
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	8.69694	0.472353
$340$	0	0
$341$	1.30306	0.0705647
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−12.2020	−0.653160	−0.326580	−	0.945170i	$-0.605896\pi$
$349$	−12.2020	−0.653160	−0.326580	+	0.945170i	$0.605896\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−36.9444	−1.96635	−0.983176	−	0.182663i	$-0.941528\pi$
$353$	−36.9444	−1.96635	−0.983176	+	0.182663i	$0.941528\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	9.34847	0.493393	0.246697	−	0.969093i	$-0.420655\pi$
$359$	9.34847	0.493393	0.246697	+	0.969093i	$0.420655\pi$
$360$	0	0
$361$	28.5959	1.50505
$362$	0	0
$363$	−10.7980	−0.566746
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.7980	0.615848	0.307924	−	0.951411i	$-0.400366\pi$
$367$	11.7980	0.615848	0.307924	+	0.951411i	$0.400366\pi$
$368$	0	0
$369$	3.55051	0.184832
$370$	0	0
$371$	−5.79796	−0.301015
$372$	0	0
$373$	23.7980	1.23221	0.616106	−	0.787663i	$-0.288709\pi$
$373$	23.7980	1.23221	0.616106	+	0.787663i	$0.288709\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−0.696938	−0.0357993	−0.0178997	−	0.999840i	$-0.505698\pi$
$379$	−0.696938	−0.0357993	−0.0178997	+	0.999840i	$0.505698\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	−27.5505	−1.40777	−0.703883	−	0.710316i	$-0.748552\pi$
$383$	−27.5505	−1.40777	−0.703883	+	0.710316i	$0.748552\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.79796	0.396393
$388$	0	0
$389$	−6.89898	−0.349792	−0.174896	−	0.984587i	$-0.555959\pi$
$389$	−6.89898	−0.349792	−0.174896	+	0.984587i	$0.555959\pi$
$390$	0	0
$391$	−9.79796	−0.495504
$392$	0	0
$393$	−4.89898	−0.247121
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.10102	−0.0552586	−0.0276293	−	0.999618i	$-0.508796\pi$
$397$	−1.10102	−0.0552586	−0.0276293	+	0.999618i	$0.508796\pi$
$398$	0	0
$399$	13.7980	0.690762
$400$	0	0
$401$	−35.1464	−1.75513	−0.877564	−	0.479459i	$-0.840833\pi$
$401$	−35.1464	−1.75513	−0.877564	+	0.479459i	$0.840833\pi$
$402$	0	0
$403$	2.89898	0.144408
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.89898	−0.242833
$408$	0	0
$409$	−21.1010	−1.04338	−0.521689	−	0.853136i	$-0.674698\pi$
$409$	−21.1010	−1.04338	−0.521689	+	0.853136i	$0.674698\pi$
$410$	0	0
$411$	10.2474	0.505469
$412$	0	0
$413$	−8.89898	−0.437890
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	18.6969	0.913405	0.456703	−	0.889619i	$-0.349030\pi$
$419$	18.6969	0.913405	0.456703	+	0.889619i	$0.349030\pi$
$420$	0	0
$421$	0.696938	0.0339667	0.0169834	−	0.999856i	$-0.494594\pi$
$421$	0.696938	0.0339667	0.0169834	+	0.999856i	$0.494594\pi$
$422$	0	0
$423$	5.34847	0.260051
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	0.449490	0.0217016
$430$	0	0
$431$	−19.1464	−0.922251	−0.461125	−	0.887335i	$-0.652554\pi$
$431$	−19.1464	−0.922251	−0.461125	+	0.887335i	$0.652554\pi$
$432$	0	0
$433$	−37.5959	−1.80674	−0.903372	−	0.428857i	$-0.858916\pi$
$433$	−37.5959	−1.80674	−0.903372	+	0.428857i	$0.858916\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−33.7980	−1.61678
$438$	0	0
$439$	−35.5959	−1.69890	−0.849450	−	0.527669i	$-0.823066\pi$
$439$	−35.5959	−1.69890	−0.849450	+	0.527669i	$0.823066\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	11.5959	0.550939	0.275469	−	0.961310i	$-0.411167\pi$
$443$	11.5959	0.550939	0.275469	+	0.961310i	$0.411167\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.24745	−0.295494
$448$	0	0
$449$	−35.1464	−1.65866	−0.829331	−	0.558757i	$-0.811278\pi$
$449$	−35.1464	−1.65866	−0.829331	+	0.558757i	$0.811278\pi$
$450$	0	0
$451$	−1.59592	−0.0751488
$452$	0	0
$453$	18.8990	0.887952
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.3939	0.907207	0.453604	−	0.891204i	$-0.350138\pi$
$457$	19.3939	0.907207	0.453604	+	0.891204i	$0.350138\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	19.1464	0.891738	0.445869	−	0.895098i	$-0.352895\pi$
$461$	19.1464	0.891738	0.445869	+	0.895098i	$0.352895\pi$
$462$	0	0
$463$	−25.5959	−1.18954	−0.594772	−	0.803895i	$-0.702758\pi$
$463$	−25.5959	−1.18954	−0.594772	+	0.803895i	$0.702758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.89898	0.411796	0.205898	−	0.978573i	$-0.433989\pi$
$467$	8.89898	0.411796	0.205898	+	0.978573i	$0.433989\pi$
$468$	0	0
$469$	−15.5959	−0.720153
$470$	0	0
$471$	−5.79796	−0.267156
$472$	0	0
$473$	−3.50510	−0.161165
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.89898	−0.132735
$478$	0	0
$479$	0.853572	0.0390007	0.0195003	−	0.999810i	$-0.493792\pi$
$479$	0.853572	0.0390007	0.0195003	+	0.999810i	$0.493792\pi$
$480$	0	0
$481$	−10.8990	−0.496951
$482$	0	0
$483$	−9.79796	−0.445823
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.10102	0.231149	0.115575	−	0.993299i	$-0.463129\pi$
$487$	5.10102	0.231149	0.115575	+	0.993299i	$0.463129\pi$
$488$	0	0
$489$	−6.89898	−0.311983
$490$	0	0
$491$	37.3939	1.68756	0.843781	−	0.536688i	$-0.180325\pi$
$491$	37.3939	1.68756	0.843781	+	0.536688i	$0.180325\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28.4949	1.27817
$498$	0	0
$499$	23.3939	1.04725	0.523627	−	0.851948i	$-0.324579\pi$
$499$	23.3939	1.04725	0.523627	+	0.851948i	$0.324579\pi$
$500$	0	0
$501$	−3.55051	−0.158625
$502$	0	0
$503$	−19.5959	−0.873739	−0.436869	−	0.899525i	$-0.643913\pi$
$503$	−19.5959	−0.873739	−0.436869	+	0.899525i	$0.643913\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	9.34847	0.414364	0.207182	−	0.978302i	$-0.433571\pi$
$509$	9.34847	0.414364	0.207182	+	0.978302i	$0.433571\pi$
$510$	0	0
$511$	−15.5959	−0.689923
$512$	0	0
$513$	6.89898	0.304597
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.40408	−0.105731
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	0.202041	0.00883464	0.00441732	−	0.999990i	$-0.498594\pi$
$523$	0.202041	0.00883464	0.00441732	+	0.999990i	$0.498594\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.79796	−0.252563
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.44949	−0.193091
$532$	0	0
$533$	−3.55051	−0.153790
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−19.5959	−0.845626
$538$	0	0
$539$	1.34847	0.0580827
$540$	0	0
$541$	29.1010	1.25115	0.625575	−	0.780164i	$-0.284864\pi$
$541$	29.1010	1.25115	0.625575	+	0.780164i	$0.284864\pi$
$542$	0	0
$543$	−3.79796	−0.162986
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.40408	0.102791	0.0513956	−	0.998678i	$-0.483633\pi$
$547$	2.40408	0.102791	0.0513956	+	0.998678i	$0.483633\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	−13.7980	−0.587813
$552$	0	0
$553$	20.0000	0.850487
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.9444	−1.22641	−0.613207	−	0.789923i	$-0.710121\pi$
$557$	−28.9444	−1.22641	−0.613207	+	0.789923i	$0.710121\pi$
$558$	0	0
$559$	−7.79796	−0.329819
$560$	0	0
$561$	−0.898979	−0.0379549
$562$	0	0
$563$	−3.10102	−0.130692	−0.0653462	−	0.997863i	$-0.520815\pi$
$563$	−3.10102	−0.130692	−0.0653462	+	0.997863i	$0.520815\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−13.1010	−0.549223	−0.274612	−	0.961555i	$-0.588549\pi$
$569$	−13.1010	−0.549223	−0.274612	+	0.961555i	$0.588549\pi$
$570$	0	0
$571$	1.59592	0.0667871	0.0333935	−	0.999442i	$-0.489369\pi$
$571$	1.59592	0.0667871	0.0333935	+	0.999442i	$0.489369\pi$
$572$	0	0
$573$	16.8990	0.705965
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.6969	−1.19467	−0.597335	−	0.801992i	$-0.703774\pi$
$577$	−28.6969	−1.19467	−0.597335	+	0.801992i	$0.703774\pi$
$578$	0	0
$579$	1.10102	0.0457569
$580$	0	0
$581$	−16.8990	−0.701088
$582$	0	0
$583$	1.30306	0.0539673
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.2474	0.753153	0.376576	−	0.926386i	$-0.377101\pi$
$587$	18.2474	0.753153	0.376576	+	0.926386i	$0.377101\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	−11.5505	−0.475125
$592$	0	0
$593$	−21.3485	−0.876677	−0.438338	−	0.898810i	$-0.644433\pi$
$593$	−21.3485	−0.876677	−0.438338	+	0.898810i	$0.644433\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.00000	0.245564
$598$	0	0
$599$	22.6969	0.927372	0.463686	−	0.886000i	$-0.346527\pi$
$599$	22.6969	0.927372	0.463686	+	0.886000i	$0.346527\pi$
$600$	0	0
$601$	−6.20204	−0.252987	−0.126493	−	0.991967i	$-0.540372\pi$
$601$	−6.20204	−0.252987	−0.126493	+	0.991967i	$0.540372\pi$
$602$	0	0
$603$	−7.79796	−0.317557
$604$	0	0
$605$	0	0
$606$	0	0
$607$	25.5959	1.03891	0.519453	−	0.854499i	$-0.326136\pi$
$607$	25.5959	1.03891	0.519453	+	0.854499i	$0.326136\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	−5.34847	−0.216376
$612$	0	0
$613$	46.4949	1.87791	0.938956	−	0.344038i	$-0.111795\pi$
$613$	46.4949	1.87791	0.938956	+	0.344038i	$0.111795\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.4495	−0.662232	−0.331116	−	0.943590i	$-0.607425\pi$
$617$	−16.4495	−0.662232	−0.331116	+	0.943590i	$0.607425\pi$
$618$	0	0
$619$	−1.59592	−0.0641454	−0.0320727	−	0.999486i	$-0.510211\pi$
$619$	−1.59592	−0.0641454	−0.0320727	+	0.999486i	$0.510211\pi$
$620$	0	0
$621$	−4.89898	−0.196589
$622$	0	0
$623$	2.69694	0.108051
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.10102	−0.123843
$628$	0	0
$629$	21.7980	0.869142
$630$	0	0
$631$	20.2929	0.807846	0.403923	−	0.914793i	$-0.367646\pi$
$631$	20.2929	0.807846	0.403923	+	0.914793i	$0.367646\pi$
$632$	0	0
$633$	11.7980	0.468927
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	14.2474	0.563620
$640$	0	0
$641$	−33.1010	−1.30741	−0.653706	−	0.756749i	$-0.726787\pi$
$641$	−33.1010	−1.30741	−0.653706	+	0.756749i	$0.726787\pi$
$642$	0	0
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	48.4949	1.90653	0.953266	−	0.302134i	$-0.0976989\pi$
$647$	48.4949	1.90653	0.953266	+	0.302134i	$0.0976989\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	−5.79796	−0.227240
$652$	0	0
$653$	2.49490	0.0976329	0.0488164	−	0.998808i	$-0.484455\pi$
$653$	2.49490	0.0976329	0.0488164	+	0.998808i	$0.484455\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−7.79796	−0.304227
$658$	0	0
$659$	−10.6969	−0.416694	−0.208347	−	0.978055i	$-0.566808\pi$
$659$	−10.6969	−0.416694	−0.208347	+	0.978055i	$0.566808\pi$
$660$	0	0
$661$	−15.7980	−0.614469	−0.307235	−	0.951634i	$-0.599404\pi$
$661$	−15.7980	−0.614469	−0.307235	+	0.951634i	$0.599404\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.79796	0.379378
$668$	0	0
$669$	14.8990	0.576028
$670$	0	0
$671$	−1.79796	−0.0694094
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.3031	−0.741877	−0.370938	−	0.928657i	$-0.620964\pi$
$677$	−19.3031	−0.741877	−0.370938	+	0.928657i	$0.620964\pi$
$678$	0	0
$679$	12.0000	0.460518
$680$	0	0
$681$	−20.4495	−0.783626
$682$	0	0
$683$	0.853572	0.0326610	0.0163305	−	0.999867i	$-0.494802\pi$
$683$	0.853572	0.0326610	0.0163305	+	0.999867i	$0.494802\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−24.6969	−0.942247
$688$	0	0
$689$	2.89898	0.110442
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	0	0
$693$	−0.898979	−0.0341494
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.10102	0.268970
$698$	0	0
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−34.8990	−1.31812	−0.659058	−	0.752092i	$-0.729045\pi$
$701$	−34.8990	−1.31812	−0.659058	+	0.752092i	$0.729045\pi$
$702$	0	0
$703$	75.1918	2.83591
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.7980	1.12067
$708$	0	0
$709$	−14.4949	−0.544367	−0.272184	−	0.962245i	$-0.587746\pi$
$709$	−14.4949	−0.544367	−0.272184	+	0.962245i	$0.587746\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	14.2020	0.531871
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.4495	0.763700
$718$	0	0
$719$	−9.79796	−0.365402	−0.182701	−	0.983169i	$-0.558484\pi$
$719$	−9.79796	−0.365402	−0.182701	+	0.983169i	$0.558484\pi$
$720$	0	0
$721$	7.59592	0.282887
$722$	0	0
$723$	10.8990	0.405337
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−25.3939	−0.941807	−0.470903	−	0.882185i	$-0.656072\pi$
$727$	−25.3939	−0.941807	−0.470903	+	0.882185i	$0.656072\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.5959	0.576836
$732$	0	0
$733$	11.7980	0.435768	0.217884	−	0.975975i	$-0.430085\pi$
$733$	11.7980	0.435768	0.217884	+	0.975975i	$0.430085\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.50510	0.129112
$738$	0	0
$739$	15.3939	0.566273	0.283136	−	0.959080i	$-0.408625\pi$
$739$	15.3939	0.566273	0.283136	+	0.959080i	$0.408625\pi$
$740$	0	0
$741$	−6.89898	−0.253440
$742$	0	0
$743$	−6.65153	−0.244021	−0.122010	−	0.992529i	$-0.538934\pi$
$743$	−6.65153	−0.244021	−0.122010	+	0.992529i	$0.538934\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.44949	−0.309151
$748$	0	0
$749$	−27.5959	−1.00833
$750$	0	0
$751$	−4.40408	−0.160707	−0.0803536	−	0.996766i	$-0.525605\pi$
$751$	−4.40408	−0.160707	−0.0803536	+	0.996766i	$0.525605\pi$
$752$	0	0
$753$	19.5959	0.714115
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−53.1918	−1.93329	−0.966645	−	0.256120i	$-0.917556\pi$
$757$	−53.1918	−1.93329	−0.966645	+	0.256120i	$0.917556\pi$
$758$	0	0
$759$	2.20204	0.0799290
$760$	0	0
$761$	−43.5505	−1.57870	−0.789352	−	0.613940i	$-0.789584\pi$
$761$	−43.5505	−1.57870	−0.789352	+	0.613940i	$0.789584\pi$
$762$	0	0
$763$	21.7980	0.789139
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.44949	0.160662
$768$	0	0
$769$	−39.3939	−1.42058	−0.710290	−	0.703909i	$-0.751436\pi$
$769$	−39.3939	−1.42058	−0.710290	+	0.703909i	$0.751436\pi$
$770$	0	0
$771$	3.79796	0.136780
$772$	0	0
$773$	−26.2474	−0.944055	−0.472028	−	0.881584i	$-0.656478\pi$
$773$	−26.2474	−0.944055	−0.472028	+	0.881584i	$0.656478\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	21.7980	0.781997
$778$	0	0
$779$	24.4949	0.877621
$780$	0	0
$781$	−6.40408	−0.229156
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.2020	−0.577540	−0.288770	−	0.957398i	$-0.593246\pi$
$787$	−16.2020	−0.577540	−0.288770	+	0.957398i	$0.593246\pi$
$788$	0	0
$789$	−17.7980	−0.633624
$790$	0	0
$791$	17.3939	0.618455
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.3939	−0.970341	−0.485170	−	0.874420i	$-0.661242\pi$
$797$	−27.3939	−0.970341	−0.485170	+	0.874420i	$0.661242\pi$
$798$	0	0
$799$	10.6969	0.378430
$800$	0	0
$801$	1.34847	0.0476458
$802$	0	0
$803$	3.50510	0.123692
$804$	0	0
$805$	0	0
$806$	0	0
$807$	31.7980	1.11934
$808$	0	0
$809$	−53.5959	−1.88433	−0.942166	−	0.335146i	$-0.891214\pi$
$809$	−53.5959	−1.88433	−0.942166	+	0.335146i	$0.891214\pi$
$810$	0	0
$811$	−16.6969	−0.586309	−0.293154	−	0.956065i	$-0.594705\pi$
$811$	−16.6969	−0.586309	−0.293154	+	0.956065i	$0.594705\pi$
$812$	0	0
$813$	10.0000	0.350715
$814$	0	0
$815$	0	0
$816$	0	0
$817$	53.7980	1.88215
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	16.0454	0.559989	0.279994	−	0.960002i	$-0.409667\pi$
$821$	16.0454	0.559989	0.279994	+	0.960002i	$0.409667\pi$
$822$	0	0
$823$	−7.39388	−0.257734	−0.128867	−	0.991662i	$-0.541134\pi$
$823$	−7.39388	−0.257734	−0.128867	+	0.991662i	$0.541134\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.4495	−1.54566	−0.772830	−	0.634613i	$-0.781160\pi$
$827$	−44.4495	−1.54566	−0.772830	+	0.634613i	$0.781160\pi$
$828$	0	0
$829$	−17.3939	−0.604114	−0.302057	−	0.953290i	$-0.597673\pi$
$829$	−17.3939	−0.604114	−0.302057	+	0.953290i	$0.597673\pi$
$830$	0	0
$831$	13.7980	0.478646
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.89898	−0.100203
$838$	0	0
$839$	−45.3485	−1.56560	−0.782802	−	0.622271i	$-0.786210\pi$
$839$	−45.3485	−1.56560	−0.782802	+	0.622271i	$0.786210\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	30.2474	1.04178
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.5959	−0.742045
$848$	0	0
$849$	16.0000	0.549119
$850$	0	0
$851$	−53.3939	−1.83032
$852$	0	0
$853$	−44.6969	−1.53039	−0.765197	−	0.643796i	$-0.777358\pi$
$853$	−44.6969	−1.53039	−0.765197	+	0.643796i	$0.777358\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−57.5959	−1.96744	−0.983720	−	0.179711i	$-0.942484\pi$
$857$	−57.5959	−1.96744	−0.983720	+	0.179711i	$0.942484\pi$
$858$	0	0
$859$	47.7980	1.63085	0.815423	−	0.578866i	$-0.196505\pi$
$859$	47.7980	1.63085	0.815423	+	0.578866i	$0.196505\pi$
$860$	0	0
$861$	7.10102	0.242002
$862$	0	0
$863$	−0.853572	−0.0290559	−0.0145280	−	0.999894i	$-0.504625\pi$
$863$	−0.853572	−0.0290559	−0.0145280	+	0.999894i	$0.504625\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	−4.49490	−0.152479
$870$	0	0
$871$	7.79796	0.264224
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.0908	1.28624	0.643118	−	0.765767i	$-0.277641\pi$
$877$	38.0908	1.28624	0.643118	+	0.765767i	$0.277641\pi$
$878$	0	0
$879$	−7.55051	−0.254672
$880$	0	0
$881$	−32.2929	−1.08797	−0.543987	−	0.839094i	$-0.683086\pi$
$881$	−32.2929	−1.08797	−0.543987	+	0.839094i	$0.683086\pi$
$882$	0	0
$883$	5.39388	0.181518	0.0907592	−	0.995873i	$-0.471071\pi$
$883$	5.39388	0.181518	0.0907592	+	0.995873i	$0.471071\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	−0.449490	−0.0150585
$892$	0	0
$893$	36.8990	1.23478
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.89898	0.163572
$898$	0	0
$899$	5.79796	0.193373
$900$	0	0
$901$	−5.79796	−0.193158
$902$	0	0
$903$	15.5959	0.519000
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−31.1918	−1.03571	−0.517854	−	0.855469i	$-0.673269\pi$
$907$	−31.1918	−1.03571	−0.517854	+	0.855469i	$0.673269\pi$
$908$	0	0
$909$	14.8990	0.494168
$910$	0	0
$911$	−15.1010	−0.500319	−0.250160	−	0.968205i	$-0.580483\pi$
$911$	−15.1010	−0.500319	−0.250160	+	0.968205i	$0.580483\pi$
$912$	0	0
$913$	3.79796	0.125694
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.79796	−0.323557
$918$	0	0
$919$	13.5959	0.448488	0.224244	−	0.974533i	$-0.428009\pi$
$919$	13.5959	0.448488	0.224244	+	0.974533i	$0.428009\pi$
$920$	0	0
$921$	9.59592	0.316196
$922$	0	0
$923$	−14.2474	−0.468960
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.79796	0.124741
$928$	0	0
$929$	−21.3485	−0.700421	−0.350210	−	0.936671i	$-0.613890\pi$
$929$	−21.3485	−0.700421	−0.350210	+	0.936671i	$0.613890\pi$
$930$	0	0
$931$	−20.6969	−0.678315
$932$	0	0
$933$	−23.5959	−0.772496
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.3939	1.02559	0.512797	−	0.858510i	$-0.328609\pi$
$937$	31.3939	1.02559	0.512797	+	0.858510i	$0.328609\pi$
$938$	0	0
$939$	12.2020	0.398199
$940$	0	0
$941$	−4.44949	−0.145049	−0.0725246	−	0.997367i	$-0.523106\pi$
$941$	−4.44949	−0.145049	−0.0725246	+	0.997367i	$0.523106\pi$
$942$	0	0
$943$	−17.3939	−0.566423
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.2474	0.592962	0.296481	−	0.955039i	$-0.404187\pi$
$947$	18.2474	0.592962	0.296481	+	0.955039i	$0.404187\pi$
$948$	0	0
$949$	7.79796	0.253132
$950$	0	0
$951$	−23.1464	−0.750574
$952$	0	0
$953$	35.3939	1.14652	0.573260	−	0.819373i	$-0.305678\pi$
$953$	35.3939	1.14652	0.573260	+	0.819373i	$0.305678\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.898979	0.0290599
$958$	0	0
$959$	20.4949	0.661815
$960$	0	0
$961$	−22.5959	−0.728901
$962$	0	0
$963$	−13.7980	−0.444633
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.6969	0.408306	0.204153	−	0.978939i	$-0.434556\pi$
$967$	12.6969	0.408306	0.204153	+	0.978939i	$0.434556\pi$
$968$	0	0
$969$	13.7980	0.443254
$970$	0	0
$971$	27.1010	0.869713	0.434857	−	0.900500i	$-0.356799\pi$
$971$	27.1010	0.869713	0.434857	+	0.900500i	$0.356799\pi$
$972$	0	0
$973$	32.0000	1.02587
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.5505	−0.881419	−0.440709	−	0.897650i	$-0.645273\pi$
$977$	−27.5505	−0.881419	−0.440709	+	0.897650i	$0.645273\pi$
$978$	0	0
$979$	−0.606123	−0.0193718
$980$	0	0
$981$	10.8990	0.347978
$982$	0	0
$983$	−48.4495	−1.54530	−0.772649	−	0.634833i	$-0.781069\pi$
$983$	−48.4495	−1.54530	−0.772649	+	0.634833i	$0.781069\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10.6969	0.340487
$988$	0	0
$989$	−38.2020	−1.21475
$990$	0	0
$991$	0.404082	0.0128361	0.00641804	−	0.999979i	$-0.497957\pi$
$991$	0.404082	0.0128361	0.00641804	+	0.999979i	$0.497957\pi$
$992$	0	0
$993$	−25.5959	−0.812262
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.40408	0.266160	0.133080	−	0.991105i	$-0.457513\pi$
$997$	8.40408	0.266160	0.133080	+	0.991105i	$0.457513\pi$
$998$	0	0
$999$	10.8990	0.344828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3900.2.a.t.1.1		2
5.2	odd	4		780.2.h.c.469.2	✓	4
5.3	odd	4		780.2.h.c.469.4	yes	4
5.4	even	2		3900.2.a.o.1.1		2
15.2	even	4		2340.2.h.d.469.1		4
15.8	even	4		2340.2.h.d.469.2		4
20.3	even	4		3120.2.l.j.1249.2		4
20.7	even	4		3120.2.l.j.1249.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
780.2.h.c.469.2	✓	4	5.2	odd	4
780.2.h.c.469.4	yes	4	5.3	odd	4
2340.2.h.d.469.1		4	15.2	even	4
2340.2.h.d.469.2		4	15.8	even	4
3120.2.l.j.1249.2		4	20.3	even	4
3120.2.l.j.1249.4		4	20.7	even	4
3900.2.a.o.1.1		2	5.4	even	2
3900.2.a.t.1.1		2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3900.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.00000 q^{7} +1.00000 q^{9} -0.449490 q^{11} -1.00000 q^{13} +2.00000 q^{17} +6.89898 q^{19} +2.00000 q^{21} -4.89898 q^{23} +1.00000 q^{27} -2.00000 q^{29} -2.89898 q^{31} -0.449490 q^{33} +10.8990 q^{37} -1.00000 q^{39} +3.55051 q^{41} +7.79796 q^{43} +5.34847 q^{47} -3.00000 q^{49} +2.00000 q^{51} -2.89898 q^{53} +6.89898 q^{57} -4.44949 q^{59} +4.00000 q^{61} +2.00000 q^{63} -7.79796 q^{67} -4.89898 q^{69} +14.2474 q^{71} -7.79796 q^{73} -0.898979 q^{77} +10.0000 q^{79} +1.00000 q^{81} -8.44949 q^{83} -2.00000 q^{87} +1.34847 q^{89} -2.00000 q^{91} -2.89898 q^{93} +6.00000 q^{97} -0.449490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 2 q^{27} - 4 q^{29} + 4 q^{31} + 4 q^{33} + 12 q^{37} - 2 q^{39} + 12 q^{41} - 4 q^{43} - 4 q^{47} - 6 q^{49} + 4 q^{51}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 4 * q^17 + 4 * q^19 + 4 * q^21 + 2 * q^27 - 4 * q^29 + 4 * q^31 + 4 * q^33 + 12 * q^37 - 2 * q^39 + 12 * q^41 - 4 * q^43 - 4 * q^47 - 6 * q^49 + 4 * q^51 + 4 * q^53 + 4 * q^57 - 4 * q^59 + 8 * q^61 + 4 * q^63 + 4 * q^67 + 4 * q^71 + 4 * q^73 + 8 * q^77 + 20 * q^79 + 2 * q^81 - 12 * q^83 - 4 * q^87 - 12 * q^89 - 4 * q^91 + 4 * q^93 + 12 * q^97 + 4 * q^99

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