LMFDB - Embedded newform 92.3.d.a.45.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [92,3,Mod(45,92)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(92, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("92.45"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 92 = 2^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	92.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$2.50681843211$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.53792.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 2 $ x^4 + 7*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			45.1
Root			$2.58874i$ of defining polynomial
Character	$\chi$	$=$	92.45
Dual form			92.3.d.a.45.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70156 q^{3} -1.54515i q^{5} -10.3550i q^{7} -1.70156 q^{9} -16.0744i q^{11} -2.70156 q^{13} +4.17433i q^{15} +13.4453i q^{17} -11.9001i q^{19} +27.9745i q^{21} +(-5.80625 + 22.2551i) q^{23} +22.6125 q^{25} +28.9109 q^{27} +2.91093 q^{29} -40.3141 q^{31} +43.4261i q^{33} -16.0000 q^{35} -30.6037i q^{37} +7.29844 q^{39} +24.5234 q^{41} -6.64175i q^{43} +2.62918i q^{45} +58.9109 q^{47} -58.2250 q^{49} -36.3232i q^{51} +14.0681i q^{53} -24.8375 q^{55} +32.1489i q^{57} -49.2250 q^{59} -67.8492i q^{61} +17.6196i q^{63} +4.17433i q^{65} -110.353i q^{67} +(15.6859 - 60.1234i) q^{69} +86.1359 q^{71} +120.523 q^{73} -61.0891 q^{75} -166.450 q^{77} -86.8522i q^{79} -62.7906 q^{81} +105.095i q^{83} +20.7750 q^{85} -7.86407 q^{87} +7.10288i q^{89} +27.9745i q^{91} +108.911 q^{93} -18.3875 q^{95} +121.331i q^{97} +27.3517i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{9} + 2 q^{13} + 28 q^{23} - 12 q^{25} + 26 q^{27} - 78 q^{29} - 46 q^{31} - 64 q^{35} + 42 q^{39} - 94 q^{41} + 146 q^{47} - 28 q^{49} + 208 q^{55} + 8 q^{59} + 178 q^{69} + 50 q^{71}+ \cdots - 176 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^9 + 2 * q^13 + 28 * q^23 - 12 * q^25 + 26 * q^27 - 78 * q^29 - 46 * q^31 - 64 * q^35 + 42 * q^39 - 94 * q^41 + 146 * q^47 - 28 * q^49 + 208 * q^55 + 8 * q^59 + 178 * q^69 + 50 * q^71 + 290 * q^73 - 334 * q^75 - 256 * q^77 - 328 * q^81 + 288 * q^85 - 326 * q^87 + 346 * q^93 - 176 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/92\mathbb{Z}\right)^\times$.

$n$	$5$	$47$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.70156			−0.900521			−0.450260	−	0.892897i	$-0.648669\pi$
$3$	−2.70156			−0.900521			−0.450260	+	0.892897i	$0.648669\pi$
$4$		0			0
$5$		−	1.54515i		−	0.309031i	−0.987990	−	0.154515i	$-0.950618\pi$
$5$		−	1.54515i		−	0.309031i	0.987990	−	0.154515i	$-0.0493816\pi$
$6$		0			0
$7$		−	10.3550i		−	1.47928i	−0.673003	−	0.739639i	$-0.734996\pi$
$7$		−	10.3550i		−	1.47928i	0.673003	−	0.739639i	$-0.265004\pi$
$8$		0			0
$9$	−1.70156			−0.189062
$10$		0			0
$11$		−	16.0744i		−	1.46131i	−0.682746	−	0.730656i	$-0.739214\pi$
$11$		−	16.0744i		−	1.46131i	0.682746	−	0.730656i	$-0.260786\pi$
$12$		0			0
$13$	−2.70156			−0.207812			−0.103906	−	0.994587i	$-0.533134\pi$
$13$	−2.70156			−0.207812			−0.103906	+	0.994587i	$0.533134\pi$
$14$		0			0
$15$			4.17433i			0.278289i
$16$		0			0
$17$			13.4453i			0.790898i	0.918488	+	0.395449i	$0.129411\pi$
$17$			13.4453i			0.790898i	−0.918488	+	0.395449i	$0.870589\pi$
$18$		0			0
$19$		−	11.9001i		−	0.626321i	−0.949700	−	0.313161i	$-0.898612\pi$
$19$		−	11.9001i		−	0.626321i	0.949700	−	0.313161i	$-0.101388\pi$
$20$		0			0
$21$			27.9745i			1.33212i
$22$		0			0
$23$	−5.80625	+	22.2551i	−0.252446	+	0.967611i
$23$	−5.80625	+	22.2551i	−0.252446	+	0.967611i
$24$		0			0
$25$	22.6125			0.904500
$26$		0			0
$27$	28.9109			1.07078
$28$		0			0
$29$	2.91093			0.100377			0.0501885	−	0.998740i	$-0.484018\pi$
$29$	2.91093			0.100377			0.0501885	+	0.998740i	$0.484018\pi$
$30$		0			0
$31$	−40.3141			−1.30045			−0.650227	−	0.759740i	$-0.725326\pi$
$31$	−40.3141			−1.30045			−0.650227	+	0.759740i	$0.725326\pi$
$32$		0			0
$33$			43.4261i			1.31594i
$34$		0			0
$35$	−16.0000			−0.457143
$36$		0			0
$37$		−	30.6037i		−	0.827128i	−0.910475	−	0.413564i	$-0.864284\pi$
$37$		−	30.6037i		−	0.827128i	0.910475	−	0.413564i	$-0.135716\pi$
$38$		0			0
$39$	7.29844			0.187139
$40$		0			0
$41$	24.5234			0.598132			0.299066	−	0.954232i	$-0.403325\pi$
$41$	24.5234			0.598132			0.299066	+	0.954232i	$0.403325\pi$
$42$		0			0
$43$		−	6.64175i		−	0.154459i	−0.997013	−	0.0772297i	$-0.975393\pi$
$43$		−	6.64175i		−	0.154459i	0.997013	−	0.0772297i	$-0.0246075\pi$
$44$		0			0
$45$			2.62918i			0.0584261i
$46$		0			0
$47$	58.9109			1.25342			0.626712	−	0.779251i	$-0.284400\pi$
$47$	58.9109			1.25342			0.626712	+	0.779251i	$0.284400\pi$
$48$		0			0
$49$	−58.2250			−1.18827
$50$		0			0
$51$		−	36.3232i		−	0.712220i
$52$		0			0
$53$			14.0681i			0.265437i	0.991154	+	0.132718i	$0.0423706\pi$
$53$			14.0681i			0.265437i	−0.991154	+	0.132718i	$0.957629\pi$
$54$		0			0
$55$	−24.8375			−0.451591
$56$		0			0
$57$			32.1489i			0.564015i
$58$		0			0
$59$	−49.2250			−0.834322			−0.417161	−	0.908833i	$-0.636975\pi$
$59$	−49.2250			−0.834322			−0.417161	+	0.908833i	$0.636975\pi$
$60$		0			0
$61$		−	67.8492i		−	1.11228i	−0.831088	−	0.556141i	$-0.812281\pi$
$61$		−	67.8492i		−	1.11228i	0.831088	−	0.556141i	$-0.187719\pi$
$62$		0			0
$63$			17.6196i			0.279676i
$64$		0			0
$65$			4.17433i			0.0642205i
$66$		0			0
$67$		−	110.353i		−	1.64706i	−0.567273	−	0.823530i	$-0.692001\pi$
$67$		−	110.353i		−	1.64706i	0.567273	−	0.823530i	$-0.307999\pi$
$68$		0			0
$69$	15.6859	−	60.1234i	0.227332	−	0.871354i
$70$		0			0
$71$	86.1359			1.21318			0.606591	−	0.795014i	$-0.292537\pi$
$71$	86.1359			1.21318			0.606591	+	0.795014i	$0.292537\pi$
$72$		0			0
$73$	120.523			1.65101			0.825503	−	0.564398i	$-0.190892\pi$
$73$	120.523			1.65101			0.825503	+	0.564398i	$0.190892\pi$
$74$		0			0
$75$	−61.0891			−0.814521
$76$		0			0
$77$	−166.450			−2.16169
$78$		0			0
$79$		−	86.8522i		−	1.09939i	−0.835364	−	0.549697i	$-0.814743\pi$
$79$		−	86.8522i		−	1.09939i	0.835364	−	0.549697i	$-0.185257\pi$
$80$		0			0
$81$	−62.7906			−0.775193
$82$		0			0
$83$			105.095i			1.26620i	0.774070	+	0.633100i	$0.218218\pi$
$83$			105.095i			1.26620i	−0.774070	+	0.633100i	$0.781782\pi$
$84$		0			0
$85$	20.7750			0.244412
$86$		0			0
$87$	−7.86407			−0.0903916
$88$		0			0
$89$			7.10288i			0.0798077i	0.999204	+	0.0399038i	$0.0127052\pi$
$89$			7.10288i			0.0798077i	−0.999204	+	0.0399038i	$0.987295\pi$
$90$		0			0
$91$			27.9745i			0.307413i
$92$		0			0
$93$	108.911			1.17109
$94$		0			0
$95$	−18.3875			−0.193553
$96$		0			0
$97$			121.331i			1.25083i	0.780291	+	0.625417i	$0.215071\pi$
$97$			121.331i			1.25083i	−0.780291	+	0.625417i	$0.784929\pi$
$98$		0			0
$99$			27.3517i			0.276279i
$100$		0			0
$101$	74.0000			0.732673			0.366337	−	0.930482i	$-0.380612\pi$
$101$	74.0000			0.732673			0.366337	+	0.930482i	$0.380612\pi$
$102$		0			0
$103$			111.736i			1.08482i	0.840114	+	0.542410i	$0.182488\pi$
$103$			111.736i			1.08482i	−0.840114	+	0.542410i	$0.817512\pi$
$104$		0			0
$105$	43.2250			0.411667
$106$		0			0
$107$			156.869i			1.46607i	0.680191	+	0.733035i	$0.261897\pi$
$107$			156.869i			1.46607i	−0.680191	+	0.733035i	$0.738103\pi$
$108$		0			0
$109$		−	171.399i		−	1.57247i	−0.617931	−	0.786233i	$-0.712029\pi$
$109$		−	171.399i		−	1.57247i	0.617931	−	0.786233i	$-0.287971\pi$
$110$		0			0
$111$			82.6778i			0.744845i
$112$		0			0
$113$			125.343i			1.10923i	0.832106	+	0.554617i	$0.187135\pi$
$113$			125.343i			1.10923i	−0.832106	+	0.554617i	$0.812865\pi$
$114$		0			0
$115$	34.3875	+	8.97155i	0.299022	+	0.0780135i
$116$		0			0
$117$	4.59688			0.0392895
$118$		0			0
$119$	139.225			1.16996
$120$		0			0
$121$	−137.388			−1.13543
$122$		0			0
$123$	−66.2516			−0.538631
$124$		0			0
$125$		−	73.5687i		−	0.588549i
$126$		0			0
$127$	−144.314			−1.13633			−0.568166	−	0.822914i	$-0.692347\pi$
$127$	−144.314			−1.13633			−0.568166	+	0.822914i	$0.692347\pi$
$128$		0			0
$129$			17.9431i			0.139094i
$130$		0			0
$131$	94.1359			0.718595			0.359297	−	0.933223i	$-0.383016\pi$
$131$	94.1359			0.718595			0.359297	+	0.933223i	$0.383016\pi$
$132$		0			0
$133$	−123.225			−0.926504
$134$		0			0
$135$		−	44.6719i		−	0.330903i
$136$		0			0
$137$		−	195.984i		−	1.43054i	−0.698850	−	0.715268i	$-0.746304\pi$
$137$		−	195.984i		−	1.43054i	0.698850	−	0.715268i	$-0.253696\pi$
$138$		0			0
$139$	221.298			1.59208			0.796038	−	0.605247i	$-0.206926\pi$
$139$	221.298			1.59208			0.796038	+	0.605247i	$0.206926\pi$
$140$		0			0
$141$	−159.152			−1.12873
$142$		0			0
$143$			43.4261i			0.303679i
$144$		0			0
$145$		−	4.49784i		−	0.0310196i
$146$		0			0
$147$	157.298			1.07006
$148$		0			0
$149$		−	273.702i		−	1.83693i	−0.395503	−	0.918464i	$-0.629430\pi$
$149$		−	273.702i		−	1.83693i	0.395503	−	0.918464i	$-0.370570\pi$
$150$		0			0
$151$	−115.539			−0.765159			−0.382580	−	0.923923i	$-0.624964\pi$
$151$	−115.539			−0.765159			−0.382580	+	0.923923i	$0.624964\pi$
$152$		0			0
$153$		−	22.8779i		−	0.149529i
$154$		0			0
$155$			62.2915i			0.401880i
$156$		0			0
$157$		−	48.9838i		−	0.311999i	−0.987757	−	0.155999i	$-0.950140\pi$
$157$		−	48.9838i		−	0.311999i	0.987757	−	0.155999i	$-0.0498598\pi$
$158$		0			0
$159$		−	38.0060i		−	0.239031i
$160$		0			0
$161$	230.450	+	60.1234i	1.43137	+	0.373437i
$162$		0			0
$163$	12.4609			0.0764475			0.0382238	−	0.999269i	$-0.487830\pi$
$163$	12.4609			0.0764475			0.0382238	+	0.999269i	$0.487830\pi$
$164$		0			0
$165$	67.1000			0.406667
$166$		0			0
$167$	−276.450			−1.65539			−0.827695	−	0.561179i	$-0.810348\pi$
$167$	−276.450			−1.65539			−0.827695	+	0.561179i	$0.810348\pi$
$168$		0			0
$169$	−161.702			−0.956814
$170$		0			0
$171$			20.2488i			0.118414i
$172$		0			0
$173$	−106.775			−0.617197			−0.308598	−	0.951192i	$-0.599860\pi$
$173$	−106.775			−0.617197			−0.308598	+	0.951192i	$0.599860\pi$
$174$		0			0
$175$		−	234.151i		−	1.33801i
$176$		0			0
$177$	132.984			0.751324
$178$		0			0
$179$	−325.214			−1.81684			−0.908419	−	0.418061i	$-0.862710\pi$
$179$	−325.214			−1.81684			−0.908419	+	0.418061i	$0.862710\pi$
$180$		0			0
$181$			176.819i			0.976899i	0.872592	+	0.488450i	$0.162437\pi$
$181$			176.819i			0.976899i	−0.872592	+	0.488450i	$0.837563\pi$
$182$		0			0
$183$			183.299i			1.00163i
$184$		0			0
$185$	−47.2875			−0.255608
$186$		0			0
$187$	216.125			1.15575
$188$		0			0
$189$		−	299.371i		−	1.58398i
$190$		0			0
$191$		−	154.078i		−	0.806694i	−0.915047	−	0.403347i	$-0.867847\pi$
$191$		−	154.078i		−	0.806694i	0.915047	−	0.403347i	$-0.132153\pi$
$192$		0			0
$193$	296.648			1.53704			0.768519	−	0.639827i	$-0.220994\pi$
$193$	296.648			1.53704			0.768519	+	0.639827i	$0.220994\pi$
$194$		0			0
$195$		−	11.2772i		−	0.0578319i
$196$		0			0
$197$	228.586			1.16033			0.580167	−	0.814497i	$-0.302987\pi$
$197$	228.586			1.16033			0.580167	+	0.814497i	$0.302987\pi$
$198$		0			0
$199$		−	141.118i		−	0.709138i	−0.935030	−	0.354569i	$-0.884628\pi$
$199$		−	141.118i		−	0.709138i	0.935030	−	0.354569i	$-0.115372\pi$
$200$		0			0
$201$			298.126i			1.48321i
$202$		0			0
$203$		−	30.1426i		−	0.148486i
$204$		0			0
$205$		−	37.8925i		−	0.184841i
$206$		0			0
$207$	9.87969	−	37.8684i	0.0477280	−	0.182939i
$208$		0			0
$209$	−191.287			−0.915251
$210$		0			0
$211$	30.7750			0.145853			0.0729266	−	0.997337i	$-0.476766\pi$
$211$	30.7750			0.145853			0.0729266	+	0.997337i	$0.476766\pi$
$212$		0			0
$213$	−232.702			−1.09250
$214$		0			0
$215$	−10.2625			−0.0477327
$216$		0			0
$217$			417.450i			1.92373i
$218$		0			0
$219$	−325.602			−1.48677
$220$		0			0
$221$		−	36.3232i		−	0.164358i
$222$		0			0
$223$	161.287			0.723262			0.361631	−	0.932321i	$-0.382220\pi$
$223$	161.287			0.723262			0.361631	+	0.932321i	$0.382220\pi$
$224$		0			0
$225$	−38.4766			−0.171007
$226$		0			0
$227$		−	100.160i		−	0.441233i	−0.975361	−	0.220616i	$-0.929193\pi$
$227$		−	100.160i		−	0.441233i	0.975361	−	0.220616i	$-0.0708069\pi$
$228$		0			0
$229$			411.892i			1.79866i	0.437274	+	0.899328i	$0.355944\pi$
$229$			411.892i			1.79866i	−0.437274	+	0.899328i	$0.644056\pi$
$230$		0			0
$231$	449.675			1.94664
$232$		0			0
$233$	−116.252			−0.498934			−0.249467	−	0.968383i	$-0.580255\pi$
$233$	−116.252			−0.498934			−0.249467	+	0.968383i	$0.580255\pi$
$234$		0			0
$235$		−	91.0265i		−	0.387347i
$236$		0			0
$237$			234.637i			0.990028i
$238$		0			0
$239$	279.686			1.17023			0.585117	−	0.810949i	$-0.301049\pi$
$239$	279.686			1.17023			0.585117	+	0.810949i	$0.301049\pi$
$240$		0			0
$241$			313.577i			1.30115i	0.759442	+	0.650575i	$0.225472\pi$
$241$			313.577i			1.30115i	−0.759442	+	0.650575i	$0.774528\pi$
$242$		0			0
$243$	−90.5656			−0.372698
$244$		0			0
$245$			89.9666i			0.367211i
$246$		0			0
$247$			32.1489i			0.130157i
$248$		0			0
$249$		−	283.920i		−	1.14024i
$250$		0			0
$251$			311.271i			1.24013i	0.784552	+	0.620063i	$0.212893\pi$
$251$			311.271i			1.24013i	−0.784552	+	0.620063i	$0.787107\pi$
$252$		0			0
$253$	357.737	+	93.3322i	1.41398	+	0.368902i
$254$		0			0
$255$	−56.1250			−0.220098
$256$		0			0
$257$	194.073			0.755150			0.377575	−	0.925979i	$-0.376758\pi$
$257$	194.073			0.755150			0.377575	+	0.925979i	$0.376758\pi$
$258$		0			0
$259$	−316.900			−1.22355
$260$		0			0
$261$	−4.95314			−0.0189775
$262$		0			0
$263$		−	46.6782i		−	0.177483i	−0.996055	−	0.0887417i	$-0.971715\pi$
$263$		−	46.6782i		−	0.177483i	0.996055	−	0.0887417i	$-0.0282846\pi$
$264$		0			0
$265$	21.7375			0.0820282
$266$		0			0
$267$		−	19.1889i		−	0.0718685i
$268$		0			0
$269$	−31.6015			−0.117478			−0.0587389	−	0.998273i	$-0.518708\pi$
$269$	−31.6015			−0.117478			−0.0587389	+	0.998273i	$0.518708\pi$
$270$		0			0
$271$	−378.900			−1.39815			−0.699077	−	0.715046i	$-0.746406\pi$
$271$	−378.900			−1.39815			−0.699077	+	0.715046i	$0.746406\pi$
$272$		0			0
$273$		−	75.5750i		−	0.276831i
$274$		0			0
$275$		−	363.483i		−	1.32176i
$276$		0			0
$277$	−132.377			−0.477894			−0.238947	−	0.971033i	$-0.576802\pi$
$277$	−132.377			−0.477894			−0.238947	+	0.971033i	$0.576802\pi$
$278$		0			0
$279$	68.5969			0.245867
$280$		0			0
$281$			138.627i			0.493334i	0.969100	+	0.246667i	$0.0793354\pi$
$281$			138.627i			0.493334i	−0.969100	+	0.246667i	$0.920665\pi$
$282$		0			0
$283$			95.5002i			0.337457i	0.985663	+	0.168728i	$0.0539660\pi$
$283$			95.5002i			0.337457i	−0.985663	+	0.168728i	$0.946034\pi$
$284$		0			0
$285$	49.6750			0.174298
$286$		0			0
$287$		−	253.939i		−	0.884805i
$288$		0			0
$289$	108.225			0.374481
$290$		0			0
$291$		−	327.783i		−	1.12640i
$292$		0			0
$293$		−	251.471i		−	0.858264i	−0.903242	−	0.429132i	$-0.858819\pi$
$293$		−	251.471i		−	0.858264i	0.903242	−	0.429132i	$-0.141181\pi$
$294$		0			0
$295$			76.0602i			0.257831i
$296$		0			0
$297$		−	464.727i		−	1.56474i
$298$		0			0
$299$	15.6859	−	60.1234i	0.0524613	−	0.201082i
$300$		0			0
$301$	−68.7750			−0.228488
$302$		0			0
$303$	−199.916			−0.659787
$304$		0			0
$305$	−104.837			−0.343729
$306$		0			0
$307$	16.4500			0.0535830			0.0267915	−	0.999641i	$-0.491471\pi$
$307$	16.4500			0.0535830			0.0267915	+	0.999641i	$0.491471\pi$
$308$		0			0
$309$		−	301.863i		−	0.976902i
$310$		0			0
$311$	87.6859			0.281948			0.140974	−	0.990013i	$-0.454977\pi$
$311$	87.6859			0.281948			0.140974	+	0.990013i	$0.454977\pi$
$312$		0			0
$313$			210.513i			0.672565i	0.941761	+	0.336282i	$0.109170\pi$
$313$			210.513i			0.672565i	−0.941761	+	0.336282i	$0.890830\pi$
$314$		0			0
$315$	27.2250			0.0864286
$316$		0			0
$317$	−82.9000			−0.261514			−0.130757	−	0.991414i	$-0.541741\pi$
$317$	−82.9000			−0.261514			−0.130757	+	0.991414i	$0.541741\pi$
$318$		0			0
$319$		−	46.7916i		−	0.146682i
$320$		0			0
$321$		−	423.792i		−	1.32023i
$322$		0			0
$323$	160.000			0.495356
$324$		0			0
$325$	−61.0891			−0.187966
$326$		0			0
$327$			463.044i			1.41604i
$328$		0			0
$329$		−	610.020i		−	1.85416i
$330$		0			0
$331$	−31.6015			−0.0954729			−0.0477365	−	0.998860i	$-0.515201\pi$
$331$	−31.6015			−0.0954729			−0.0477365	+	0.998860i	$0.515201\pi$
$332$		0			0
$333$			52.0741i			0.156379i
$334$		0			0
$335$	−170.512			−0.508992
$336$		0			0
$337$			395.656i			1.17405i	0.809567	+	0.587027i	$0.199702\pi$
$337$			395.656i			1.17405i	−0.809567	+	0.587027i	$0.800298\pi$
$338$		0			0
$339$		−	338.623i		−	0.998888i
$340$		0			0
$341$			648.026i			1.90037i
$342$		0			0
$343$			95.5244i			0.278497i
$344$		0			0
$345$	−92.9000	−	24.2372i	−0.269275	−	0.0702528i
$346$		0			0
$347$	299.675			0.863617			0.431808	−	0.901965i	$-0.357876\pi$
$347$	299.675			0.863617			0.431808	+	0.901965i	$0.357876\pi$
$348$		0			0
$349$	372.586			1.06758			0.533791	−	0.845617i	$-0.320767\pi$
$349$	372.586			1.06758			0.533791	+	0.845617i	$0.320767\pi$
$350$		0			0
$351$	−78.1047			−0.222520
$352$		0			0
$353$	−468.502			−1.32720			−0.663600	−	0.748088i	$-0.730972\pi$
$353$	−468.502			−1.32720			−0.663600	+	0.748088i	$0.730972\pi$
$354$		0			0
$355$		−	133.093i		−	0.374911i
$356$		0			0
$357$	−376.125			−1.05357
$358$		0			0
$359$			566.594i			1.57826i	0.614229	+	0.789128i	$0.289467\pi$
$359$			566.594i			1.57826i	−0.614229	+	0.789128i	$0.710533\pi$
$360$		0			0
$361$	219.388			0.607722
$362$		0			0
$363$	371.161			1.02248
$364$		0			0
$365$		−	186.227i		−	0.510212i
$366$		0			0
$367$		−	198.637i		−	0.541245i	−0.962686	−	0.270622i	$-0.912771\pi$
$367$		−	198.637i		−	0.541245i	0.962686	−	0.270622i	$-0.0872295\pi$
$368$		0			0
$369$	−41.7281			−0.113084
$370$		0			0
$371$	145.675			0.392655
$372$		0			0
$373$		−	68.0109i		−	0.182335i	−0.995836	−	0.0911675i	$-0.970940\pi$
$373$		−	68.0109i		−	0.182335i	0.995836	−	0.0911675i	$-0.0290599\pi$
$374$		0			0
$375$			198.750i			0.530001i
$376$		0			0
$377$	−7.86407			−0.0208596
$378$		0			0
$379$			185.604i			0.489722i	0.969558	+	0.244861i	$0.0787423\pi$
$379$			185.604i			0.489722i	−0.969558	+	0.244861i	$0.921258\pi$
$380$		0			0
$381$	389.873			1.02329
$382$		0			0
$383$			175.872i			0.459197i	0.973285	+	0.229598i	$0.0737413\pi$
$383$			175.872i			0.459197i	−0.973285	+	0.229598i	$0.926259\pi$
$384$		0			0
$385$			257.191i			0.668029i
$386$		0			0
$387$			11.3014i			0.0292025i
$388$		0			0
$389$			47.7863i			0.122844i	0.998112	+	0.0614220i	$0.0195635\pi$
$389$			47.7863i			0.122844i	−0.998112	+	0.0614220i	$0.980436\pi$
$390$		0			0
$391$	−299.225	−	78.0665i	−0.765281	−	0.199659i
$392$		0			0
$393$	−254.314			−0.647110
$394$		0			0
$395$	−134.200			−0.339747
$396$		0			0
$397$	407.936			1.02755			0.513773	−	0.857926i	$-0.328247\pi$
$397$	407.936			1.02755			0.513773	+	0.857926i	$0.328247\pi$
$398$		0			0
$399$	332.900			0.834336
$400$		0			0
$401$		−	655.129i		−	1.63374i	−0.576824	−	0.816869i	$-0.695708\pi$
$401$		−	655.129i		−	1.63374i	0.576824	−	0.816869i	$-0.304292\pi$
$402$		0			0
$403$	108.911			0.270250
$404$		0			0
$405$			97.0212i			0.239559i
$406$		0			0
$407$	−491.938			−1.20869
$408$		0			0
$409$	224.523			0.548957			0.274479	−	0.961593i	$-0.411495\pi$
$409$	224.523			0.548957			0.274479	+	0.961593i	$0.411495\pi$
$410$		0			0
$411$			529.462i			1.28823i
$412$		0			0
$413$			509.722i			1.23419i
$414$		0			0
$415$	162.388			0.391295
$416$		0			0
$417$	−597.851			−1.43370
$418$		0			0
$419$			186.850i			0.445943i	0.974825	+	0.222972i	$0.0715757\pi$
$419$			186.850i			0.445943i	−0.974825	+	0.222972i	$0.928424\pi$
$420$		0			0
$421$			342.174i			0.812766i	0.913703	+	0.406383i	$0.133210\pi$
$421$			342.174i			0.812766i	−0.913703	+	0.406383i	$0.866790\pi$
$422$		0			0
$423$	−100.241			−0.236975
$424$		0			0
$425$			304.031i			0.715367i
$426$		0			0
$427$	−702.575			−1.64537
$428$		0			0
$429$		−	117.318i		−	0.273469i
$430$		0			0
$431$		−	170.339i		−	0.395218i	−0.980281	−	0.197609i	$-0.936682\pi$
$431$		−	170.339i		−	0.395218i	0.980281	−	0.197609i	$-0.0633176\pi$
$432$		0			0
$433$		−	417.450i		−	0.964088i	−0.876147	−	0.482044i	$-0.839895\pi$
$433$		−	417.450i		−	0.964088i	0.876147	−	0.482044i	$-0.160105\pi$
$434$		0			0
$435$			12.1512i			0.0279338i
$436$		0			0
$437$	264.837	+	69.0950i	0.606035	+	0.158112i
$438$		0			0
$439$	603.161			1.37394			0.686971	−	0.726684i	$-0.258940\pi$
$439$	603.161			1.37394			0.686971	+	0.726684i	$0.258940\pi$
$440$		0			0
$441$	99.0734			0.224656
$442$		0			0
$443$	699.873			1.57985			0.789925	−	0.613204i	$-0.210120\pi$
$443$	699.873			1.57985			0.789925	+	0.613204i	$0.210120\pi$
$444$		0			0
$445$	10.9751			0.0246630
$446$		0			0
$447$			739.424i			1.65419i
$448$		0			0
$449$	681.412			1.51762			0.758811	−	0.651311i	$-0.225781\pi$
$449$	681.412			1.51762			0.758811	+	0.651311i	$0.225781\pi$
$450$		0			0
$451$		−	394.200i		−	0.874058i
$452$		0			0
$453$	312.136			0.689042
$454$		0			0
$455$	43.2250			0.0950000
$456$		0			0
$457$		−	589.035i		−	1.28892i	−0.764640	−	0.644458i	$-0.777083\pi$
$457$		−	589.035i		−	1.28892i	0.764640	−	0.644458i	$-0.222917\pi$
$458$		0			0
$459$			388.715i			0.846874i
$460$		0			0
$461$	−712.314			−1.54515			−0.772575	−	0.634924i	$-0.781032\pi$
$461$	−712.314			−1.54515			−0.772575	+	0.634924i	$0.781032\pi$
$462$		0			0
$463$	326.775			0.705778			0.352889	−	0.935665i	$-0.385199\pi$
$463$	326.775			0.705778			0.352889	+	0.935665i	$0.385199\pi$
$464$		0			0
$465$		−	168.284i		−	0.361902i
$466$		0			0
$467$			309.265i			0.662238i	0.943589	+	0.331119i	$0.107426\pi$
$467$			309.265i			0.662238i	−0.943589	+	0.331119i	$0.892574\pi$
$468$		0			0
$469$	−1142.70			−2.43646
$470$		0			0
$471$			132.333i			0.280961i
$472$		0			0
$473$	−106.762			−0.225713
$474$		0			0
$475$		−	269.091i		−	0.566508i
$476$		0			0
$477$		−	23.9378i		−	0.0501841i
$478$		0			0
$479$		−	623.352i		−	1.30136i	−0.759352	−	0.650680i	$-0.774484\pi$
$479$		−	623.352i		−	1.30136i	0.759352	−	0.650680i	$-0.225516\pi$
$480$		0			0
$481$			82.6778i			0.171887i
$482$		0			0
$483$	−622.575	−	162.427i	−1.28898	−	0.336288i
$484$		0			0
$485$	187.475			0.386546
$486$		0			0
$487$	−124.964			−0.256600			−0.128300	−	0.991735i	$-0.540952\pi$
$487$	−124.964			−0.256600			−0.128300	+	0.991735i	$0.540952\pi$
$488$		0			0
$489$	−33.6640			−0.0688426
$490$		0			0
$491$	−632.564			−1.28832			−0.644159	−	0.764892i	$-0.722792\pi$
$491$	−632.564			−1.28832			−0.644159	+	0.764892i	$0.722792\pi$
$492$		0			0
$493$			39.1383i			0.0793880i
$494$		0			0
$495$	42.2625			0.0853789
$496$		0			0
$497$		−	891.933i		−	1.79463i
$498$		0			0
$499$	−401.277			−0.804161			−0.402081	−	0.915604i	$-0.631713\pi$
$499$	−401.277			−0.804161			−0.402081	+	0.915604i	$0.631713\pi$
$500$		0			0
$501$	746.847			1.49071
$502$		0			0
$503$			527.018i			1.04775i	0.851795	+	0.523875i	$0.175514\pi$
$503$			527.018i			1.04775i	−0.851795	+	0.523875i	$0.824486\pi$
$504$		0			0
$505$		−	114.341i		−	0.226419i
$506$		0			0
$507$	436.847			0.861631
$508$		0			0
$509$	−327.477			−0.643372			−0.321686	−	0.946846i	$-0.604250\pi$
$509$	−327.477			−0.643372			−0.321686	+	0.946846i	$0.604250\pi$
$510$		0			0
$511$		−	1248.01i		−	2.44230i
$512$		0			0
$513$		−	344.043i		−	0.670649i
$514$		0			0
$515$	172.650			0.335243
$516$		0			0
$517$		−	946.960i		−	1.83164i
$518$		0			0
$519$	288.459			0.555798
$520$		0			0
$521$		−	403.730i		−	0.774913i	−0.921888	−	0.387456i	$-0.873354\pi$
$521$		−	403.730i		−	0.774913i	0.921888	−	0.387456i	$-0.126646\pi$
$522$		0			0
$523$			336.964i			0.644291i	0.946690	+	0.322146i	$0.104404\pi$
$523$			336.964i			0.644291i	−0.946690	+	0.322146i	$0.895596\pi$
$524$		0			0
$525$			632.574i			1.20490i
$526$		0			0
$527$		−	542.033i		−	1.02853i
$528$		0			0
$529$	−461.575	−	258.437i	−0.872542	−	0.488538i
$530$		0			0
$531$	83.7594			0.157739
$532$		0			0
$533$	−66.2516			−0.124299
$534$		0			0
$535$	242.388			0.453061
$536$		0			0
$537$	878.586			1.63610
$538$		0			0
$539$			935.934i			1.73643i
$540$		0			0
$541$	−85.2140			−0.157512			−0.0787560	−	0.996894i	$-0.525095\pi$
$541$	−85.2140			−0.157512			−0.0787560	+	0.996894i	$0.525095\pi$
$542$		0			0
$543$		−	477.687i		−	0.879718i
$544$		0			0
$545$	−264.837			−0.485940
$546$		0			0
$547$	194.073			0.354796			0.177398	−	0.984139i	$-0.443232\pi$
$547$	194.073			0.354796			0.177398	+	0.984139i	$0.443232\pi$
$548$		0			0
$549$			115.450i			0.210291i
$550$		0			0
$551$		−	34.6404i		−	0.0628683i
$552$		0			0
$553$	−899.350			−1.62631
$554$		0			0
$555$	127.750			0.230180
$556$		0			0
$557$			340.168i			0.610715i	0.952238	+	0.305357i	$0.0987759\pi$
$557$			340.168i			0.610715i	−0.952238	+	0.305357i	$0.901224\pi$
$558$		0			0
$559$			17.9431i			0.0320986i
$560$		0			0
$561$	−583.875			−1.04078
$562$		0			0
$563$		−	379.695i		−	0.674414i	−0.941430	−	0.337207i	$-0.890518\pi$
$563$		−	379.695i		−	0.674414i	0.941430	−	0.337207i	$-0.109482\pi$
$564$		0			0
$565$	193.675			0.342788
$566$		0			0
$567$			650.194i			1.14673i
$568$		0			0
$569$			731.512i			1.28561i	0.766030	+	0.642805i	$0.222230\pi$
$569$			731.512i			1.28561i	−0.766030	+	0.642805i	$0.777770\pi$
$570$		0			0
$571$		−	13.5829i		−	0.0237879i	−0.999929	−	0.0118939i	$-0.996214\pi$
$571$		−	13.5829i		−	0.0237879i	0.999929	−	0.0118939i	$-0.00378605\pi$
$572$		0			0
$573$			416.253i			0.726444i
$574$		0			0
$575$	−131.294	+	503.242i	−0.228337	+	0.875204i
$576$		0			0
$577$	−329.152			−0.570453			−0.285227	−	0.958460i	$-0.592069\pi$
$577$	−329.152			−0.570453			−0.285227	+	0.958460i	$0.592069\pi$
$578$		0			0
$579$	−801.414			−1.38413
$580$		0			0
$581$	1088.25			1.87306
$582$		0			0
$583$	226.138			0.387886
$584$		0			0
$585$		−	7.10288i		−	0.0121417i
$586$		0			0
$587$	924.711			1.57532			0.787658	−	0.616112i	$-0.211293\pi$
$587$	924.711			1.57532			0.787658	+	0.616112i	$0.211293\pi$
$588$		0			0
$589$			479.742i			0.814502i
$590$		0			0
$591$	−617.539			−1.04491
$592$		0			0
$593$	820.637			1.38387			0.691937	−	0.721958i	$-0.256758\pi$
$593$	820.637			1.38387			0.691937	+	0.721958i	$0.256758\pi$
$594$		0			0
$595$		−	215.124i		−	0.361553i
$596$		0			0
$597$			381.240i			0.638594i
$598$		0			0
$599$	−472.512			−0.788836			−0.394418	−	0.918931i	$-0.629054\pi$
$599$	−472.512			−0.788836			−0.394418	+	0.918931i	$0.629054\pi$
$600$		0			0
$601$	530.323			0.882402			0.441201	−	0.897408i	$-0.354553\pi$
$601$	530.323			0.882402			0.441201	+	0.897408i	$0.354553\pi$
$602$		0			0
$603$			187.773i			0.311397i
$604$		0			0
$605$			212.285i			0.350884i
$606$		0			0
$607$	−214.837			−0.353933			−0.176967	−	0.984217i	$-0.556628\pi$
$607$	−214.837			−0.353933			−0.176967	+	0.984217i	$0.556628\pi$
$608$		0			0
$609$			81.4321i			0.133714i
$610$		0			0
$611$	−159.152			−0.260477
$612$		0			0
$613$		−	9.57030i		−	0.0156122i	−0.999970	−	0.00780612i	$-0.997515\pi$
$613$		−	9.57030i		−	0.0156122i	0.999970	−	0.00780612i	$-0.00248479\pi$
$614$		0			0
$615$			102.369i			0.166454i
$616$		0			0
$617$		−	184.496i		−	0.299022i	−0.988760	−	0.149511i	$-0.952230\pi$
$617$		−	184.496i		−	0.299022i	0.988760	−	0.149511i	$-0.0477699\pi$
$618$		0			0
$619$			708.335i			1.14432i	0.820141	+	0.572161i	$0.193895\pi$
$619$			708.335i			1.14432i	−0.820141	+	0.572161i	$0.806105\pi$
$620$		0			0
$621$	−167.864	+	643.414i	−0.270313	+	1.03609i
$622$		0			0
$623$	73.5500			0.118058
$624$		0			0
$625$	451.637			0.722620
$626$		0			0
$627$	516.775			0.824203
$628$		0			0
$629$	411.475			0.654173
$630$		0			0
$631$		−	1159.48i		−	1.83753i	−0.394809	−	0.918763i	$-0.629189\pi$
$631$		−	1159.48i		−	1.83753i	0.394809	−	0.918763i	$-0.370811\pi$
$632$		0			0
$633$	−83.1406			−0.131344
$634$		0			0
$635$			222.988i			0.351161i
$636$		0			0
$637$	157.298			0.246936
$638$		0			0
$639$	−146.566			−0.229367
$640$		0			0
$641$			760.733i			1.18679i	0.804911	+	0.593395i	$0.202213\pi$
$641$			760.733i			1.18679i	−0.804911	+	0.593395i	$0.797787\pi$
$642$		0			0
$643$		−	529.534i		−	0.823537i	−0.911289	−	0.411768i	$-0.864911\pi$
$643$		−	529.534i		−	0.823537i	0.911289	−	0.411768i	$-0.135089\pi$
$644$		0			0
$645$	27.7249			0.0429843
$646$		0			0
$647$	−91.4141			−0.141289			−0.0706446	−	0.997502i	$-0.522506\pi$
$647$	−91.4141			−0.141289			−0.0706446	+	0.997502i	$0.522506\pi$
$648$		0			0
$649$			791.264i			1.21920i
$650$		0			0
$651$		−	1127.77i		−	1.73236i
$652$		0			0
$653$	−711.727			−1.08993			−0.544967	−	0.838458i	$-0.683458\pi$
$653$	−711.727			−1.08993			−0.544967	+	0.838458i	$0.683458\pi$
$654$		0			0
$655$		−	145.455i		−	0.222068i
$656$		0			0
$657$	−205.078			−0.312143
$658$		0			0
$659$			342.546i			0.519797i	0.965636	+	0.259899i	$0.0836891\pi$
$659$			342.546i			0.519797i	−0.965636	+	0.259899i	$0.916311\pi$
$660$		0			0
$661$		−	621.111i		−	0.939654i	−0.882759	−	0.469827i	$-0.844316\pi$
$661$		−	621.111i		−	0.939654i	0.882759	−	0.469827i	$-0.155684\pi$
$662$		0			0
$663$			98.1294i			0.148008i
$664$		0			0
$665$			190.402i			0.286318i
$666$		0			0
$667$	−16.9016	+	64.7830i	−0.0253397	+	0.0971260i
$668$		0			0
$669$	−435.728			−0.651313
$670$		0			0
$671$	−1090.64			−1.62539
$672$		0			0
$673$	−34.9515			−0.0519339			−0.0259669	−	0.999663i	$-0.508266\pi$
$673$	−34.9515			−0.0519339			−0.0259669	+	0.999663i	$0.508266\pi$
$674$		0			0
$675$	653.748			0.968516
$676$		0			0
$677$			313.876i			0.463628i	0.972760	+	0.231814i	$0.0744661\pi$
$677$			313.876i			0.463628i	−0.972760	+	0.231814i	$0.925534\pi$
$678$		0			0
$679$	1256.37			1.85033
$680$		0			0
$681$			270.588i			0.397339i
$682$		0			0
$683$	−469.802			−0.687850			−0.343925	−	0.938997i	$-0.611757\pi$
$683$	−469.802			−0.687850			−0.343925	+	0.938997i	$0.611757\pi$
$684$		0			0
$685$	−302.825			−0.442080
$686$		0			0
$687$		−	1112.75i		−	1.61973i
$688$		0			0
$689$		−	38.0060i		−	0.0551611i
$690$		0			0
$691$	−483.150			−0.699204			−0.349602	−	0.936898i	$-0.613683\pi$
$691$	−483.150			−0.699204			−0.349602	+	0.936898i	$0.613683\pi$
$692$		0			0
$693$	283.225			0.408694
$694$		0			0
$695$		−	341.940i		−	0.492000i
$696$		0			0
$697$			329.724i			0.473062i
$698$		0			0
$699$	314.061			0.449300
$700$		0			0
$701$			1139.30i			1.62524i	0.582792	+	0.812622i	$0.301960\pi$
$701$			1139.30i			1.62524i	−0.582792	+	0.812622i	$0.698040\pi$
$702$		0			0
$703$	−364.187			−0.518048
$704$		0			0
$705$			245.914i			0.348814i
$706$		0			0
$707$		−	766.266i		−	1.08383i
$708$		0			0
$709$		−	167.224i		−	0.235859i	−0.993022	−	0.117930i	$-0.962374\pi$
$709$		−	167.224i		−	0.235859i	0.993022	−	0.117930i	$-0.0376258\pi$
$710$		0			0
$711$			147.784i			0.207854i
$712$		0			0
$713$	234.073	−	897.192i	0.328294	−	1.25833i
$714$		0			0
$715$	67.1000			0.0938462
$716$		0			0
$717$	−755.589			−1.05382
$718$		0			0
$719$	402.713			0.560101			0.280050	−	0.959985i	$-0.409649\pi$
$719$	402.713			0.560101			0.280050	+	0.959985i	$0.409649\pi$
$720$		0			0
$721$	1157.02			1.60475
$722$		0			0
$723$		−	847.148i		−	1.17171i
$724$		0			0
$725$	65.8235			0.0907910
$726$		0			0
$727$			838.104i			1.15283i	0.817159	+	0.576413i	$0.195548\pi$
$727$			838.104i			1.15283i	−0.817159	+	0.576413i	$0.804452\pi$
$728$		0			0
$729$	809.784			1.11082
$730$		0			0
$731$	89.3001			0.122162
$732$		0			0
$733$		−	303.246i		−	0.413706i	−0.978372	−	0.206853i	$-0.933678\pi$
$733$		−	303.246i		−	0.413706i	0.978372	−	0.206853i	$-0.0663221\pi$
$734$		0			0
$735$		−	243.050i		−	0.330681i
$736$		0			0
$737$	−1773.86			−2.40687
$738$		0			0
$739$	796.461			1.07776			0.538878	−	0.842384i	$-0.318849\pi$
$739$	796.461			1.07776			0.538878	+	0.842384i	$0.318849\pi$
$740$		0			0
$741$		−	86.8522i		−	0.117209i
$742$		0			0
$743$			166.811i			0.224511i	0.993679	+	0.112255i	$0.0358075\pi$
$743$			166.811i			0.224511i	−0.993679	+	0.112255i	$0.964193\pi$
$744$		0			0
$745$	−422.913			−0.567668
$746$		0			0
$747$		−	178.825i		−	0.239391i
$748$		0			0
$749$	1624.37			2.16872
$750$		0			0
$751$		−	283.111i		−	0.376979i	−0.982075	−	0.188489i	$-0.939641\pi$
$751$		−	283.111i		−	0.376979i	0.982075	−	0.188489i	$-0.0603591\pi$
$752$		0			0
$753$		−	840.919i		−	1.11676i
$754$		0			0
$755$			178.526i			0.236458i
$756$		0			0
$757$		−	648.535i		−	0.856718i	−0.903609	−	0.428359i	$-0.859092\pi$
$757$		−	648.535i		−	0.856718i	0.903609	−	0.428359i	$-0.140908\pi$
$758$		0			0
$759$	−966.450	−	252.143i	−1.27332	−	0.332204i
$760$		0			0
$761$	−1193.53			−1.56837			−0.784183	−	0.620530i	$-0.786918\pi$
$761$	−1193.53			−1.56837			−0.784183	+	0.620530i	$0.786918\pi$
$762$		0			0
$763$	−1774.82			−2.32611
$764$		0			0
$765$	−35.3500			−0.0462091
$766$		0			0
$767$	132.984			0.173383
$768$		0			0
$769$			293.514i			0.381683i	0.981621	+	0.190841i	$0.0611216\pi$
$769$			293.514i			0.381683i	−0.981621	+	0.190841i	$0.938878\pi$
$770$		0			0
$771$	−524.301			−0.680028
$772$		0			0
$773$			306.498i			0.396505i	0.980151	+	0.198252i	$0.0635266\pi$
$773$			306.498i			0.396505i	−0.980151	+	0.198252i	$0.936473\pi$
$774$		0			0
$775$	−911.602			−1.17626
$776$		0			0
$777$	856.125			1.10183
$778$		0			0
$779$		−	291.831i		−	0.374623i
$780$		0			0
$781$		−	1384.59i		−	1.77284i
$782$		0			0
$783$	84.1578			0.107481
$784$		0			0
$785$	−75.6876			−0.0964173
$786$		0			0
$787$			1244.58i			1.58142i	0.612192	+	0.790709i	$0.290288\pi$
$787$			1244.58i			1.58142i	−0.612192	+	0.790709i	$0.709712\pi$
$788$		0			0
$789$			126.104i			0.159828i
$790$		0			0
$791$	1297.92			1.64087
$792$		0			0
$793$			183.299i			0.231146i
$794$		0			0
$795$	−58.7251			−0.0738681
$796$		0			0
$797$		−	1442.73i		−	1.81020i	−0.425201	−	0.905099i	$-0.639797\pi$
$797$		−	1442.73i		−	1.81020i	0.425201	−	0.905099i	$-0.360203\pi$
$798$		0			0
$799$			792.073i			0.991330i
$800$		0			0
$801$		−	12.0860i		−	0.0150886i
$802$		0			0
$803$		−	1937.35i		−	2.41264i
$804$		0			0
$805$	92.9000	−	356.081i	0.115404	−	0.442336i
$806$		0			0
$807$	85.3735			0.105791
$808$		0			0
$809$	615.275			0.760538			0.380269	−	0.924876i	$-0.375831\pi$
$809$	615.275			0.760538			0.380269	+	0.924876i	$0.375831\pi$
$810$		0			0
$811$	828.711			1.02184			0.510919	−	0.859629i	$-0.329305\pi$
$811$	828.711			1.02184			0.510919	+	0.859629i	$0.329305\pi$
$812$		0			0
$813$	1023.62			1.25907
$814$		0			0
$815$		−	19.2541i		−	0.0236247i
$816$		0			0
$817$	−79.0375			−0.0967412
$818$		0			0
$819$		−	47.6004i		−	0.0581202i
$820$		0			0
$821$	−141.875			−0.172808			−0.0864038	−	0.996260i	$-0.527538\pi$
$821$	−141.875			−0.172808			−0.0864038	+	0.996260i	$0.527538\pi$
$822$		0			0
$823$	−670.764			−0.815023			−0.407512	−	0.913200i	$-0.633603\pi$
$823$	−670.764			−0.815023			−0.407512	+	0.913200i	$0.633603\pi$
$824$		0			0
$825$			981.972i			1.19027i
$826$		0			0
$827$			1558.09i			1.88402i	0.335579	+	0.942012i	$0.391068\pi$
$827$			1558.09i			1.88402i	−0.335579	+	0.942012i	$0.608932\pi$
$828$		0			0
$829$	−29.8750			−0.0360374			−0.0180187	−	0.999838i	$-0.505736\pi$
$829$	−29.8750			−0.0360374			−0.0180187	+	0.999838i	$0.505736\pi$
$830$		0			0
$831$	357.623			0.430353
$832$		0			0
$833$		−	782.850i		−	0.939796i
$834$		0			0
$835$			427.158i			0.511566i
$836$		0			0
$837$	−1165.52			−1.39249
$838$		0			0
$839$			1039.67i			1.23918i	0.784927	+	0.619588i	$0.212700\pi$
$839$			1039.67i			1.23918i	−0.784927	+	0.619588i	$0.787300\pi$
$840$		0			0
$841$	−832.526			−0.989924
$842$		0			0
$843$		−	374.509i		−	0.444258i
$844$		0			0
$845$			249.854i			0.295685i
$846$		0			0
$847$			1422.64i			1.67962i
$848$		0			0
$849$		−	258.000i		−	0.303887i
$850$		0			0
$851$	681.087	+	177.693i	0.800338	+	0.208805i
$852$		0			0
$853$	−729.350			−0.855041			−0.427520	−	0.904006i	$-0.640613\pi$
$853$	−729.350			−0.855041			−0.427520	+	0.904006i	$0.640613\pi$
$854$		0			0
$855$	31.2875			0.0365935
$856$		0			0
$857$	−211.076			−0.246297			−0.123148	−	0.992388i	$-0.539299\pi$
$857$	−211.076			−0.246297			−0.123148	+	0.992388i	$0.539299\pi$
$858$		0			0
$859$	722.573			0.841180			0.420590	−	0.907251i	$-0.361823\pi$
$859$	722.573			0.841180			0.420590	+	0.907251i	$0.361823\pi$
$860$		0			0
$861$			686.032i			0.796785i
$862$		0			0
$863$	1521.49			1.76302			0.881510	−	0.472166i	$-0.156528\pi$
$863$	1521.49			1.76302			0.881510	+	0.472166i	$0.156528\pi$
$864$		0			0
$865$			164.984i			0.190733i
$866$		0			0
$867$	−292.377			−0.337228
$868$		0			0
$869$	−1396.10			−1.60656
$870$		0			0
$871$			298.126i			0.342280i
$872$		0			0
$873$		−	206.452i		−	0.236486i
$874$		0			0
$875$	−761.800			−0.870629
$876$		0			0
$877$	954.500			1.08837			0.544185	−	0.838965i	$-0.316839\pi$
$877$	954.500			1.08837			0.544185	+	0.838965i	$0.316839\pi$
$878$		0			0
$879$			679.366i			0.772885i
$880$		0			0
$881$			1319.79i			1.49806i	0.662539	+	0.749028i	$0.269479\pi$
$881$			1319.79i			1.49806i	−0.662539	+	0.749028i	$0.730521\pi$
$882$		0			0
$883$	634.500			0.718573			0.359286	−	0.933227i	$-0.383020\pi$
$883$	634.500			0.718573			0.359286	+	0.933227i	$0.383020\pi$
$884$		0			0
$885$		−	205.481i		−	0.232182i
$886$		0			0
$887$	847.936			0.955959			0.477980	−	0.878371i	$-0.341369\pi$
$887$	847.936			0.955959			0.477980	+	0.878371i	$0.341369\pi$
$888$		0			0
$889$			1494.36i			1.68095i
$890$		0			0
$891$			1009.32i			1.13280i
$892$		0			0
$893$		−	701.046i		−	0.785046i
$894$		0			0
$895$			502.506i			0.561459i
$896$		0			0
$897$	−42.3765	+	162.427i	−0.0472425	+	0.181078i
$898$		0			0
$899$	−117.352			−0.130536
$900$		0			0
$901$	−189.150			−0.209933
$902$		0			0
$903$	185.800			0.205759
$904$		0			0
$905$	273.212			0.301892
$906$		0			0
$907$			50.8283i			0.0560401i	0.999607	+	0.0280200i	$0.00892022\pi$
$907$			50.8283i			0.0560401i	−0.999607	+	0.0280200i	$0.991080\pi$
$908$		0			0
$909$	−125.916			−0.138521
$910$		0			0
$911$		−	1166.58i		−	1.28055i	−0.768145	−	0.640276i	$-0.778820\pi$
$911$		−	1166.58i		−	1.28055i	0.768145	−	0.640276i	$-0.221180\pi$
$912$		0			0
$913$	1689.34			1.85031
$914$		0			0
$915$	283.225			0.309536
$916$		0			0
$917$		−	974.773i		−	1.06300i
$918$		0			0
$919$			1245.15i			1.35490i	0.735570	+	0.677449i	$0.236914\pi$
$919$			1245.15i			1.35490i	−0.735570	+	0.677449i	$0.763086\pi$
$920$		0			0
$921$	−44.4407			−0.0482526
$922$		0			0
$923$	−232.702			−0.252114
$924$		0			0
$925$		−	692.026i		−	0.748137i
$926$		0			0
$927$		−	190.126i		−	0.205099i
$928$		0			0
$929$	227.748			0.245154			0.122577	−	0.992459i	$-0.460884\pi$
$929$	227.748			0.245154			0.122577	+	0.992459i	$0.460884\pi$
$930$		0			0
$931$			692.884i			0.744236i
$932$		0			0
$933$	−236.889			−0.253900
$934$		0			0
$935$		−	333.947i		−	0.357162i
$936$		0			0
$937$		−	1020.58i		−	1.08920i	−0.838697	−	0.544598i	$-0.816682\pi$
$937$		−	1020.58i		−	1.08920i	0.838697	−	0.544598i	$-0.183318\pi$
$938$		0			0
$939$		−	568.713i		−	0.605659i
$940$		0			0
$941$			687.463i			0.730567i	0.930896	+	0.365283i	$0.119028\pi$
$941$			687.463i			0.730567i	−0.930896	+	0.365283i	$0.880972\pi$
$942$		0			0
$943$	−142.389	+	545.770i	−0.150996	+	0.578760i
$944$		0			0
$945$	−462.575			−0.489497
$946$		0			0
$947$	−1524.63			−1.60995			−0.804977	−	0.593306i	$-0.797822\pi$
$947$	−1524.63			−1.60995			−0.804977	+	0.593306i	$0.797822\pi$
$948$		0			0
$949$	−325.602			−0.343100
$950$		0			0
$951$	223.959			0.235499
$952$		0			0
$953$			428.679i			0.449821i	0.974379	+	0.224910i	$0.0722089\pi$
$953$			428.679i			0.449821i	−0.974379	+	0.224910i	$0.927791\pi$
$954$		0			0
$955$	−238.075			−0.249293
$956$		0			0
$957$			126.411i			0.132090i
$958$		0			0
$959$	−2029.40			−2.11616
$960$		0			0
$961$	664.223			0.691179
$962$		0			0
$963$		−	266.923i		−	0.277179i
$964$		0			0
$965$		−	458.368i		−	0.474992i
$966$		0			0
$967$	345.361			0.357147			0.178573	−	0.983927i	$-0.442852\pi$
$967$	345.361			0.357147			0.178573	+	0.983927i	$0.442852\pi$
$968$		0			0
$969$	−432.250			−0.446078
$970$		0			0
$971$		−	1775.98i		−	1.82902i	−0.404563	−	0.914510i	$-0.632576\pi$
$971$		−	1775.98i		−	1.82902i	0.404563	−	0.914510i	$-0.367424\pi$
$972$		0			0
$973$		−	2291.53i		−	2.35512i
$974$		0			0
$975$	165.036			0.169268
$976$		0			0
$977$		−	1117.25i		−	1.14355i	−0.820410	−	0.571776i	$-0.806255\pi$
$977$		−	1117.25i		−	1.14355i	0.820410	−	0.571776i	$-0.193745\pi$
$978$		0			0
$979$	114.175			0.116624
$980$		0			0
$981$			291.646i			0.297294i
$982$		0			0
$983$			783.335i			0.796882i	0.917194	+	0.398441i	$0.130449\pi$
$983$			783.335i			0.796882i	−0.917194	+	0.398441i	$0.869551\pi$
$984$		0			0
$985$		−	353.201i		−	0.358579i
$986$		0			0
$987$			1648.01i			1.66971i
$988$		0			0
$989$	147.813	+	38.5637i	0.149457	+	0.0389926i
$990$		0			0
$991$	575.612			0.580840			0.290420	−	0.956899i	$-0.406205\pi$
$991$	575.612			0.580840			0.290420	+	0.956899i	$0.406205\pi$
$992$		0			0
$993$	85.3735			0.0859753
$994$		0			0
$995$	−218.050			−0.219146
$996$		0			0
$997$	673.875			0.675903			0.337951	−	0.941164i	$-0.390266\pi$
$997$	673.875			0.675903			0.337951	+	0.941164i	$0.390266\pi$
$998$		0			0
$999$		−	884.782i		−	0.885668i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	92.3.d.a.45.1	✓	4
3.2	odd	2		828.3.b.a.505.3		4
4.3	odd	2		368.3.f.b.321.3		4
5.2	odd	4		2300.3.d.a.1149.6		8
5.3	odd	4		2300.3.d.a.1149.3		8
5.4	even	2		2300.3.f.a.1701.4		4
8.3	odd	2		1472.3.f.e.321.2		4
8.5	even	2		1472.3.f.d.321.4		4
23.22	odd	2	inner	92.3.d.a.45.2	yes	4
69.68	even	2		828.3.b.a.505.2		4
92.91	even	2		368.3.f.b.321.4		4
115.22	even	4		2300.3.d.a.1149.5		8
115.68	even	4		2300.3.d.a.1149.4		8
115.114	odd	2		2300.3.f.a.1701.3		4
184.45	odd	2		1472.3.f.d.321.3		4
184.91	even	2		1472.3.f.e.321.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.3.d.a.45.1	✓	4	1.1	even	1	trivial
92.3.d.a.45.2	yes	4	23.22	odd	2	inner
368.3.f.b.321.3		4	4.3	odd	2
368.3.f.b.321.4		4	92.91	even	2
828.3.b.a.505.2		4	69.68	even	2
828.3.b.a.505.3		4	3.2	odd	2
1472.3.f.d.321.3		4	184.45	odd	2
1472.3.f.d.321.4		4	8.5	even	2
1472.3.f.e.321.1		4	184.91	even	2
1472.3.f.e.321.2		4	8.3	odd	2
2300.3.d.a.1149.3		8	5.3	odd	4
2300.3.d.a.1149.4		8	115.68	even	4
2300.3.d.a.1149.5		8	115.22	even	4
2300.3.d.a.1149.6		8	5.2	odd	4
2300.3.f.a.1701.3		4	115.114	odd	2
2300.3.f.a.1701.4		4	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 92 = 2^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	92.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.70156 q^{3} -1.54515i q^{5} -10.3550i q^{7} -1.70156 q^{9} -16.0744i q^{11} -2.70156 q^{13} +4.17433i q^{15} +13.4453i q^{17} -11.9001i q^{19} +27.9745i q^{21} +(-5.80625 + 22.2551i) q^{23} +22.6125 q^{25} +28.9109 q^{27} +2.91093 q^{29} -40.3141 q^{31} +43.4261i q^{33} -16.0000 q^{35} -30.6037i q^{37} +7.29844 q^{39} +24.5234 q^{41} -6.64175i q^{43} +2.62918i q^{45} +58.9109 q^{47} -58.2250 q^{49} -36.3232i q^{51} +14.0681i q^{53} -24.8375 q^{55} +32.1489i q^{57} -49.2250 q^{59} -67.8492i q^{61} +17.6196i q^{63} +4.17433i q^{65} -110.353i q^{67} +(15.6859 - 60.1234i) q^{69} +86.1359 q^{71} +120.523 q^{73} -61.0891 q^{75} -166.450 q^{77} -86.8522i q^{79} -62.7906 q^{81} +105.095i q^{83} +20.7750 q^{85} -7.86407 q^{87} +7.10288i q^{89} +27.9745i q^{91} +108.911 q^{93} -18.3875 q^{95} +121.331i q^{97} +27.3517i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{9} + 2 q^{13} + 28 q^{23} - 12 q^{25} + 26 q^{27} - 78 q^{29} - 46 q^{31} - 64 q^{35} + 42 q^{39} - 94 q^{41} + 146 q^{47} - 28 q^{49} + 208 q^{55} + 8 q^{59} + 178 q^{69} + 50 q^{71}+ \cdots - 176 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^9 + 2 * q^13 + 28 * q^23 - 12 * q^25 + 26 * q^27 - 78 * q^29 - 46 * q^31 - 64 * q^35 + 42 * q^39 - 94 * q^41 + 146 * q^47 - 28 * q^49 + 208 * q^55 + 8 * q^59 + 178 * q^69 + 50 * q^71 + 290 * q^73 - 334 * q^75 - 256 * q^77 - 328 * q^81 + 288 * q^85 - 326 * q^87 + 346 * q^93 - 176 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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