LMFDB - Embedded newform 2736.3.o.o.721.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.5661086492196864.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 54x^{6} + 753x^{4} + 3152x^{2} + 144 $ x^8 + 54x^6 + 753x^4 + 3152*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 342)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.7
Root			$-3.04335i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.o.721.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.50871 q^{5} -10.3459 q^{7} +O(q^{10})$	$q+5.50871 q^{5} -10.3459 q^{7} +3.20476 q^{11} -22.0975i q^{13} +3.70710 q^{17} +(10.3459 + 15.9362i) q^{19} -44.5720 q^{23} +5.34590 q^{25} +39.6513i q^{29} -36.9681i q^{31} -56.9926 q^{35} -24.6454i q^{37} +65.1072i q^{41} -27.0377 q^{43} +67.5077 q^{47} +58.0377 q^{49} +14.1955i q^{53} +17.6541 q^{55} +70.8173i q^{59} -63.7295 q^{61} -121.729i q^{65} +78.6154i q^{67} -87.7879i q^{71} +36.9623 q^{73} -33.1561 q^{77} -2.54787i q^{79} -16.9246 q^{83} +20.4213 q^{85} +70.6573i q^{89} +228.619i q^{91} +(56.9926 + 87.7879i) q^{95} +135.133i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{7}+O(q^{10}) $ 8 * q - 12 * q^7	$ 8 q - 12 q^{7} + 12 q^{19} - 28 q^{25} - 4 q^{43} + 252 q^{49} + 212 q^{55} - 156 q^{61} + 508 q^{73} - 332 q^{85}+O(q^{100}) $ 8 * q - 12 * q^7 + 12 * q^19 - 28 * q^25 - 4 * q^43 + 252 * q^49 + 212 * q^55 - 156 * q^61 + 508 * q^73 - 332 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	5.50871			1.10174			0.550871	−	0.834590i	$-0.314296\pi$
$5$	5.50871			1.10174			0.550871	+	0.834590i	$0.314296\pi$
$6$		0			0
$7$	−10.3459			−1.47799			−0.738993	−	0.673713i	$-0.764698\pi$
$7$	−10.3459			−1.47799			−0.738993	+	0.673713i	$0.764698\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.20476			0.291342			0.145671	−	0.989333i	$-0.453466\pi$
$11$	3.20476			0.291342			0.145671	+	0.989333i	$0.453466\pi$
$12$		0			0
$13$		−	22.0975i		−	1.69981i	−0.526935	−	0.849906i	$-0.676659\pi$
$13$		−	22.0975i		−	1.69981i	0.526935	−	0.849906i	$-0.323341\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.70710			0.218064			0.109032	−	0.994038i	$-0.465225\pi$
$17$	3.70710			0.218064			0.109032	+	0.994038i	$0.465225\pi$
$18$		0			0
$19$	10.3459	+	15.9362i	0.544521	+	0.838747i
$19$	10.3459	+	15.9362i	0.544521	+	0.838747i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−44.5720			−1.93791			−0.968957	−	0.247229i	$-0.920480\pi$
$23$	−44.5720			−1.93791			−0.968957	+	0.247229i	$0.920480\pi$
$24$		0			0
$25$	5.34590			0.213836
$26$		0			0
$27$		0			0
$28$		0			0
$29$			39.6513i			1.36729i	0.729816	+	0.683643i	$0.239606\pi$
$29$			39.6513i			1.36729i	−0.729816	+	0.683643i	$0.760394\pi$
$30$		0			0
$31$		−	36.9681i		−	1.19252i	−0.802791	−	0.596260i	$-0.796653\pi$
$31$		−	36.9681i		−	1.19252i	0.802791	−	0.596260i	$-0.203347\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−56.9926			−1.62836
$36$		0			0
$37$		−	24.6454i		−	0.666092i	−0.942911	−	0.333046i	$-0.891923\pi$
$37$		−	24.6454i		−	0.666092i	0.942911	−	0.333046i	$-0.108077\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			65.1072i			1.58798i	0.607931	+	0.793990i	$0.292000\pi$
$41$			65.1072i			1.58798i	−0.607931	+	0.793990i	$0.708000\pi$
$42$		0			0
$43$	−27.0377			−0.628784			−0.314392	−	0.949293i	$-0.601801\pi$
$43$	−27.0377			−0.628784			−0.314392	+	0.949293i	$0.601801\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	67.5077			1.43633			0.718167	−	0.695871i	$-0.244981\pi$
$47$	67.5077			1.43633			0.718167	+	0.695871i	$0.244981\pi$
$48$		0			0
$49$	58.0377			1.18444
$50$		0			0
$51$		0			0
$52$		0			0
$53$			14.1955i			0.267839i	0.990992	+	0.133919i	$0.0427563\pi$
$53$			14.1955i			0.267839i	−0.990992	+	0.133919i	$0.957244\pi$
$54$		0			0
$55$	17.6541			0.320984
$56$		0			0
$57$		0			0
$58$		0			0
$59$			70.8173i			1.20029i	0.799890	+	0.600147i	$0.204891\pi$
$59$			70.8173i			1.20029i	−0.799890	+	0.600147i	$0.795109\pi$
$60$		0			0
$61$	−63.7295			−1.04475			−0.522373	−	0.852717i	$-0.674953\pi$
$61$	−63.7295			−1.04475			−0.522373	+	0.852717i	$0.674953\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	121.729i		−	1.87275i
$66$		0			0
$67$			78.6154i			1.17336i	0.809818	+	0.586682i	$0.199566\pi$
$67$			78.6154i			1.17336i	−0.809818	+	0.586682i	$0.800434\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	87.7879i		−	1.23645i	−0.786001	−	0.618225i	$-0.787852\pi$
$71$		−	87.7879i		−	1.23645i	0.786001	−	0.618225i	$-0.212148\pi$
$72$		0			0
$73$	36.9623			0.506333			0.253166	−	0.967423i	$-0.418528\pi$
$73$	36.9623			0.506333			0.253166	+	0.967423i	$0.418528\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−33.1561			−0.430599
$78$		0			0
$79$		−	2.54787i		−	0.0322515i	−0.999870	−	0.0161258i	$-0.994867\pi$
$79$		−	2.54787i		−	0.0322515i	0.999870	−	0.0161258i	$-0.00513321\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−16.9246			−0.203911			−0.101955	−	0.994789i	$-0.532510\pi$
$83$	−16.9246			−0.203911			−0.101955	+	0.994789i	$0.532510\pi$
$84$		0			0
$85$	20.4213			0.240251
$86$		0			0
$87$		0			0
$88$		0			0
$89$			70.6573i			0.793903i	0.917840	+	0.396951i	$0.129932\pi$
$89$			70.6573i			0.793903i	−0.917840	+	0.396951i	$0.870068\pi$
$90$		0			0
$91$			228.619i			2.51230i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	56.9926	+	87.7879i	0.599922	+	0.924083i
$96$		0			0
$97$			135.133i			1.39313i	0.717496	+	0.696563i	$0.245288\pi$
$97$			135.133i			1.39313i	−0.717496	+	0.696563i	$0.754712\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.21581			−0.0912456			−0.0456228	−	0.998959i	$-0.514527\pi$
$101$	−9.21581			−0.0912456			−0.0456228	+	0.998959i	$0.514527\pi$
$102$		0			0
$103$			34.0036i			0.330132i	0.986283	+	0.165066i	$0.0527838\pi$
$103$			34.0036i			0.330132i	−0.986283	+	0.165066i	$0.947216\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			84.8528i			0.793017i	0.918031	+	0.396508i	$0.129778\pi$
$107$			84.8528i			0.793017i	−0.918031	+	0.396508i	$0.870222\pi$
$108$		0			0
$109$			120.679i			1.10715i	0.832800	+	0.553574i	$0.186737\pi$
$109$			120.679i			1.10715i	−0.832800	+	0.553574i	$0.813263\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	186.836i		−	1.65342i	−0.562630	−	0.826709i	$-0.690210\pi$
$113$		−	186.836i		−	1.65342i	0.562630	−	0.826709i	$-0.309790\pi$
$114$		0			0
$115$	−245.534			−2.13508
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−38.3533			−0.322296
$120$		0			0
$121$	−110.730			−0.915120
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−108.269			−0.866150
$126$		0			0
$127$			9.77484i			0.0769672i	0.999259	+	0.0384836i	$0.0122527\pi$
$127$			9.77484i			0.0769672i	−0.999259	+	0.0384836i	$0.987747\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	227.364			1.73560			0.867802	−	0.496910i	$-0.165532\pi$
$131$	227.364			1.73560			0.867802	+	0.496910i	$0.165532\pi$
$132$		0			0
$133$	−107.038	−	164.874i	−0.804795	−	1.23966i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−181.995			−1.32843			−0.664216	−	0.747541i	$-0.731235\pi$
$137$	−181.995			−1.32843			−0.664216	+	0.747541i	$0.731235\pi$
$138$		0			0
$139$	−74.4967			−0.535948			−0.267974	−	0.963426i	$-0.586354\pi$
$139$	−74.4967			−0.535948			−0.267974	+	0.963426i	$0.586354\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	70.8173i		−	0.495226i
$144$		0			0
$145$			218.428i			1.50640i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−79.2352			−0.531780			−0.265890	−	0.964003i	$-0.585666\pi$
$149$	−79.2352			−0.531780			−0.265890	+	0.964003i	$0.585666\pi$
$150$		0			0
$151$			90.9381i			0.602239i	0.953586	+	0.301119i	$0.0973603\pi$
$151$			90.9381i			0.602239i	−0.953586	+	0.301119i	$0.902640\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	203.647i		−	1.31385i
$156$		0			0
$157$	−96.8426			−0.616832			−0.308416	−	0.951252i	$-0.599799\pi$
$157$	−96.8426			−0.616832			−0.308416	+	0.951252i	$0.599799\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	461.138			2.86421
$162$		0			0
$163$	−123.308			−0.756492			−0.378246	−	0.925705i	$-0.623473\pi$
$163$	−123.308			−0.756492			−0.378246	+	0.925705i	$0.623473\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			206.582i			1.23702i	0.785778	+	0.618508i	$0.212263\pi$
$167$			206.582i			1.23702i	−0.785778	+	0.618508i	$0.787737\pi$
$168$		0			0
$169$	−319.302			−1.88936
$170$		0			0
$171$		0			0
$172$		0			0
$173$			149.960i			0.866821i	0.901197	+	0.433410i	$0.142690\pi$
$173$			149.960i			0.866821i	−0.901197	+	0.433410i	$0.857310\pi$
$174$		0			0
$175$	−55.3082			−0.316047
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19.9056i		−	0.111205i	−0.998453	−	0.0556024i	$-0.982292\pi$
$179$		−	19.9056i		−	0.111205i	0.998453	−	0.0556024i	$-0.0177079\pi$
$180$		0			0
$181$			147.039i			0.812371i	0.913791	+	0.406186i	$0.133141\pi$
$181$			147.039i			0.812371i	−0.913791	+	0.406186i	$0.866859\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	135.765i		−	0.733862i
$186$		0			0
$187$	11.8804			0.0635313
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−43.0820			−0.225560			−0.112780	−	0.993620i	$-0.535976\pi$
$191$	−43.0820			−0.225560			−0.112780	+	0.993620i	$0.535976\pi$
$192$		0			0
$193$		−	106.225i		−	0.550390i	−0.961388	−	0.275195i	$-0.911258\pi$
$193$		−	106.225i		−	0.550390i	0.961388	−	0.275195i	$-0.0887424\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−58.4826			−0.296866			−0.148433	−	0.988922i	$-0.547423\pi$
$197$	−58.4826			−0.296866			−0.148433	+	0.988922i	$0.547423\pi$
$198$		0			0
$199$	100.811			0.506590			0.253295	−	0.967389i	$-0.418486\pi$
$199$	100.811			0.506590			0.253295	+	0.967389i	$0.418486\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	410.229i		−	2.02083i
$204$		0			0
$205$			358.657i			1.74954i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	33.1561	+	51.0717i	0.158642	+	0.244362i
$210$		0			0
$211$			73.5196i			0.348434i	0.984707	+	0.174217i	$0.0557395\pi$
$211$			73.5196i			0.348434i	−0.984707	+	0.174217i	$0.944261\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−148.943			−0.692758
$216$		0			0
$217$			382.469i			1.76253i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	81.9177i		−	0.370668i
$222$		0			0
$223$		−	132.585i		−	0.594553i	−0.954791	−	0.297276i	$-0.903922\pi$
$223$		−	132.585i		−	0.594553i	0.954791	−	0.297276i	$-0.0960783\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	42.7464i		−	0.188310i	−0.995558	−	0.0941550i	$-0.969985\pi$
$227$		−	42.7464i		−	0.188310i	0.995558	−	0.0941550i	$-0.0300149\pi$
$228$		0			0
$229$	220.346			0.962209			0.481105	−	0.876663i	$-0.340236\pi$
$229$	220.346			0.962209			0.481105	+	0.876663i	$0.340236\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	301.801			1.29528			0.647641	−	0.761946i	$-0.275756\pi$
$233$	301.801			1.29528			0.647641	+	0.761946i	$0.275756\pi$
$234$		0			0
$235$	371.880			1.58247
$236$		0			0
$237$		0			0
$238$		0			0
$239$	42.8743			0.179390			0.0896951	−	0.995969i	$-0.471411\pi$
$239$	42.8743			0.179390			0.0896951	+	0.995969i	$0.471411\pi$
$240$		0			0
$241$			44.1951i			0.183382i	0.995788	+	0.0916911i	$0.0292272\pi$
$241$			44.1951i			0.183382i	−0.995788	+	0.0916911i	$0.970773\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	319.713			1.30495
$246$		0			0
$247$	352.151	−	228.619i	1.42571	−	0.925583i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−319.522			−1.27300			−0.636499	−	0.771278i	$-0.719618\pi$
$251$	−319.522			−1.27300			−0.636499	+	0.771278i	$0.719618\pi$
$252$		0			0
$253$	−142.843			−0.564595
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	246.553i		−	0.959351i	−0.877446	−	0.479675i	$-0.840754\pi$
$257$		−	246.553i		−	0.959351i	0.877446	−	0.479675i	$-0.159246\pi$
$258$		0			0
$259$			254.979i			0.984475i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−24.4427			−0.0929379			−0.0464689	−	0.998920i	$-0.514797\pi$
$263$	−24.4427			−0.0929379			−0.0464689	+	0.998920i	$0.514797\pi$
$264$		0			0
$265$			78.1987i			0.295090i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			211.972i			0.788000i	0.919110	+	0.394000i	$0.128909\pi$
$269$			211.972i			0.788000i	−0.919110	+	0.394000i	$0.871091\pi$
$270$		0			0
$271$	−378.377			−1.39623			−0.698113	−	0.715988i	$-0.745977\pi$
$271$	−378.377			−1.39623			−0.698113	+	0.715988i	$0.745977\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	17.1323			0.0622994
$276$		0			0
$277$	236.497			0.853779			0.426889	−	0.904304i	$-0.359609\pi$
$277$	236.497			0.853779			0.426889	+	0.904304i	$0.359609\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	104.598i		−	0.372237i	−0.982527	−	0.186118i	$-0.940409\pi$
$281$		−	104.598i		−	0.372237i	0.982527	−	0.186118i	$-0.0595908\pi$
$282$		0			0
$283$	−227.339			−0.803319			−0.401660	−	0.915789i	$-0.631567\pi$
$283$	−227.339			−0.803319			−0.401660	+	0.915789i	$0.631567\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	673.592i		−	2.34701i
$288$		0			0
$289$	−275.257			−0.952448
$290$		0			0
$291$		0			0
$292$		0			0
$293$			336.956i			1.15002i	0.818146	+	0.575010i	$0.195002\pi$
$293$			336.956i			1.15002i	−0.818146	+	0.575010i	$0.804998\pi$
$294$		0			0
$295$			390.112i			1.32241i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			984.933i			3.29409i
$300$		0			0
$301$	279.730			0.929334
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−351.068			−1.15104
$306$		0			0
$307$		−	233.715i		−	0.761286i	−0.924722	−	0.380643i	$-0.875703\pi$
$307$		−	233.715i		−	0.761286i	0.924722	−	0.380643i	$-0.124297\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−89.3348			−0.287250			−0.143625	−	0.989632i	$-0.545876\pi$
$311$	−89.3348			−0.287250			−0.143625	+	0.989632i	$0.545876\pi$
$312$		0			0
$313$	−504.843			−1.61292			−0.806458	−	0.591292i	$-0.798618\pi$
$313$	−504.843			−1.61292			−0.806458	+	0.591292i	$0.798618\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			413.004i			1.30285i	0.758713	+	0.651425i	$0.225829\pi$
$317$			413.004i			1.30285i	−0.758713	+	0.651425i	$0.774171\pi$
$318$		0			0
$319$			127.073i			0.398348i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	38.3533	+	59.0770i	0.118741	+	0.182901i
$324$		0			0
$325$		−	118.131i		−	0.363481i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−698.428			−2.12288
$330$		0			0
$331$			587.276i			1.77425i	0.461533	+	0.887123i	$0.347300\pi$
$331$			587.276i			1.77425i	−0.461533	+	0.887123i	$0.652700\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			433.069i			1.29274i
$336$		0			0
$337$		−	430.878i		−	1.27857i	−0.768970	−	0.639285i	$-0.779230\pi$
$337$		−	430.878i		−	1.27857i	0.768970	−	0.639285i	$-0.220770\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	118.474i		−	0.347431i
$342$		0			0
$343$	−93.5033			−0.272604
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−58.6734			−0.169088			−0.0845438	−	0.996420i	$-0.526943\pi$
$347$	−58.6734			−0.169088			−0.0845438	+	0.996420i	$0.526943\pi$
$348$		0			0
$349$	409.641			1.17376			0.586878	−	0.809675i	$-0.300357\pi$
$349$	409.641			1.17376			0.586878	+	0.809675i	$0.300357\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−218.339			−0.618524			−0.309262	−	0.950977i	$-0.600082\pi$
$353$	−218.339			−0.618524			−0.309262	+	0.950977i	$0.600082\pi$
$354$		0			0
$355$		−	483.598i		−	1.36225i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	571.711			1.59251			0.796254	−	0.604962i	$-0.206812\pi$
$359$	571.711			1.59251			0.796254	+	0.604962i	$0.206812\pi$
$360$		0			0
$361$	−146.925	+	329.749i	−0.406993	+	0.913431i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	203.615			0.557848
$366$		0			0
$367$	−426.377			−1.16179			−0.580895	−	0.813978i	$-0.697297\pi$
$367$	−426.377			−1.16179			−0.580895	+	0.813978i	$0.697297\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	146.865i		−	0.395862i
$372$		0			0
$373$		−	398.589i		−	1.06860i	−0.845294	−	0.534302i	$-0.820575\pi$
$373$		−	398.589i		−	1.06860i	0.845294	−	0.534302i	$-0.179425\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	876.197			2.32413
$378$		0			0
$379$		−	25.4787i		−	0.0672261i	−0.999435	−	0.0336131i	$-0.989299\pi$
$379$		−	25.4787i		−	0.0672261i	0.999435	−	0.0336131i	$-0.0107014\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			28.7109i			0.0749632i	0.999297	+	0.0374816i	$0.0119336\pi$
$383$			28.7109i			0.0749632i	−0.999297	+	0.0374816i	$0.988066\pi$
$384$		0			0
$385$	−182.648			−0.474409
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−724.689			−1.86295			−0.931477	−	0.363800i	$-0.881479\pi$
$389$	−724.689			−1.86295			−0.931477	+	0.363800i	$0.881479\pi$
$390$		0			0
$391$	−165.233			−0.422590
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	14.0355i		−	0.0355329i
$396$		0			0
$397$	−623.264			−1.56993			−0.784967	−	0.619537i	$-0.787320\pi$
$397$	−623.264			−1.56993			−0.784967	+	0.619537i	$0.787320\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			560.189i			1.39698i	0.715620	+	0.698489i	$0.246144\pi$
$401$			560.189i			1.39698i	−0.715620	+	0.698489i	$0.753856\pi$
$402$		0			0
$403$	−816.905			−2.02706
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	78.9826i		−	0.194061i
$408$		0			0
$409$			766.604i			1.87434i	0.348877	+	0.937169i	$0.386563\pi$
$409$			766.604i			1.87434i	−0.348877	+	0.937169i	$0.613437\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	732.669i		−	1.77402i
$414$		0			0
$415$	−93.2328			−0.224657
$416$		0			0
$417$		0			0
$418$		0			0
$419$	278.155			0.663854			0.331927	−	0.943305i	$-0.392301\pi$
$419$	278.155			0.663854			0.331927	+	0.943305i	$0.392301\pi$
$420$		0			0
$421$		−	22.9789i		−	0.0545816i	−0.999628	−	0.0272908i	$-0.991312\pi$
$421$		−	22.9789i		−	0.0545816i	0.999628	−	0.0272908i	$-0.00868801\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	19.8178			0.0466301
$426$		0			0
$427$	659.339			1.54412
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	370.417i		−	0.859437i	−0.902963	−	0.429718i	$-0.858613\pi$
$431$		−	370.417i		−	0.859437i	0.902963	−	0.429718i	$-0.141387\pi$
$432$		0			0
$433$			507.410i			1.17185i	0.810366	+	0.585924i	$0.199268\pi$
$433$			507.410i			1.17185i	−0.810366	+	0.585924i	$0.800732\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−461.138	−	710.309i	−1.05524	−	1.62542i
$438$		0			0
$439$			724.957i			1.65138i	0.564123	+	0.825691i	$0.309214\pi$
$439$			724.957i			1.65138i	−0.564123	+	0.825691i	$0.690786\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−279.853			−0.631722			−0.315861	−	0.948806i	$-0.602293\pi$
$443$	−279.853			−0.631722			−0.315861	+	0.948806i	$0.602293\pi$
$444$		0			0
$445$			389.231i			0.874676i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	585.644i		−	1.30433i	−0.758077	−	0.652165i	$-0.773861\pi$
$449$		−	585.644i		−	1.30433i	0.758077	−	0.652165i	$-0.226139\pi$
$450$		0			0
$451$			208.653i			0.462645i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			1259.40i			2.76790i
$456$		0			0
$457$	132.962			0.290946			0.145473	−	0.989362i	$-0.453530\pi$
$457$	132.962			0.290946			0.145473	+	0.989362i	$0.453530\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−657.338			−1.42590			−0.712948	−	0.701217i	$-0.752641\pi$
$461$	−657.338			−1.42590			−0.712948	+	0.701217i	$0.752641\pi$
$462$		0			0
$463$	21.4148			0.0462523			0.0231261	−	0.999733i	$-0.492638\pi$
$463$	21.4148			0.0462523			0.0231261	+	0.999733i	$0.492638\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−307.466			−0.658386			−0.329193	−	0.944263i	$-0.606777\pi$
$467$	−307.466			−0.658386			−0.329193	+	0.944263i	$0.606777\pi$
$468$		0			0
$469$		−	813.347i		−	1.73422i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−86.6494			−0.183191
$474$		0			0
$475$	55.3082	+	85.1933i	0.116438	+	0.179354i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	96.6791			0.201835			0.100918	−	0.994895i	$-0.467822\pi$
$479$	96.6791			0.201835			0.100918	+	0.994895i	$0.467822\pi$
$480$		0			0
$481$	−544.603			−1.13223
$482$		0			0
$483$		0			0
$484$		0			0
$485$			744.410i			1.53487i
$486$		0			0
$487$			649.722i			1.33413i	0.744998	+	0.667066i	$0.232450\pi$
$487$			649.722i			1.33413i	−0.744998	+	0.667066i	$0.767550\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	718.021			1.46236			0.731182	−	0.682182i	$-0.238969\pi$
$491$	718.021			1.46236			0.731182	+	0.682182i	$0.238969\pi$
$492$		0			0
$493$			146.991i			0.298157i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			908.245i			1.82745i
$498$		0			0
$499$	333.503			0.668343			0.334172	−	0.942512i	$-0.391543\pi$
$499$	333.503			0.668343			0.334172	+	0.942512i	$0.391543\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−456.269			−0.907096			−0.453548	−	0.891232i	$-0.649842\pi$
$503$	−456.269			−0.907096			−0.453548	+	0.891232i	$0.649842\pi$
$504$		0			0
$505$	−50.7672			−0.100529
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	322.921i		−	0.634422i	−0.948355	−	0.317211i	$-0.897254\pi$
$509$		−	322.921i		−	0.634422i	0.948355	−	0.317211i	$-0.102746\pi$
$510$		0			0
$511$	−382.408			−0.748353
$512$		0			0
$513$		0			0
$514$		0			0
$515$			187.316i			0.363721i
$516$		0			0
$517$	216.346			0.418464
$518$		0			0
$519$		0			0
$520$		0			0
$521$			254.398i			0.488289i	0.969739	+	0.244144i	$0.0785071\pi$
$521$			254.398i			0.488289i	−0.969739	+	0.244144i	$0.921493\pi$
$522$		0			0
$523$		−	322.522i		−	0.616676i	−0.951277	−	0.308338i	$-0.900227\pi$
$523$		−	322.522i		−	0.616676i	0.951277	−	0.308338i	$-0.0997728\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	137.044i		−	0.260046i
$528$		0			0
$529$	1457.67			2.75551
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1438.71			2.69927
$534$		0			0
$535$			467.430i			0.873700i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	185.997			0.345078
$540$		0			0
$541$	269.566			0.498273			0.249136	−	0.968468i	$-0.419853\pi$
$541$	269.566			0.498273			0.249136	+	0.968468i	$0.419853\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			664.787i			1.21979i
$546$		0			0
$547$		−	82.4131i		−	0.150664i	−0.997159	−	0.0753319i	$-0.975998\pi$
$547$		−	82.4131i		−	0.150664i	0.997159	−	0.0753319i	$-0.0240016\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−631.891	+	410.229i	−1.14681	+	0.744516i
$552$		0			0
$553$			26.3600i			0.0476673i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	838.467			1.50533			0.752663	−	0.658406i	$-0.228769\pi$
$557$	838.467			1.50533			0.752663	+	0.658406i	$0.228769\pi$
$558$		0			0
$559$			597.467i			1.06881i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	815.867i		−	1.44914i	−0.689200	−	0.724571i	$-0.742038\pi$
$563$		−	815.867i		−	1.44914i	0.689200	−	0.724571i	$-0.257962\pi$
$564$		0			0
$565$		−	1029.23i		−	1.82164i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	480.566i		−	0.844580i	−0.906461	−	0.422290i	$-0.861226\pi$
$569$		−	480.566i		−	0.844580i	0.906461	−	0.422290i	$-0.138774\pi$
$570$		0			0
$571$	−209.925			−0.367644			−0.183822	−	0.982960i	$-0.558847\pi$
$571$	−209.925			−0.367644			−0.183822	+	0.982960i	$0.558847\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−238.278			−0.414396
$576$		0			0
$577$	488.270			0.846223			0.423111	−	0.906078i	$-0.360938\pi$
$577$	488.270			0.846223			0.423111	+	0.906078i	$0.360938\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	175.100			0.301378
$582$		0			0
$583$			45.4931i			0.0780327i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−220.989			−0.376471			−0.188236	−	0.982124i	$-0.560277\pi$
$587$	−220.989			−0.376471			−0.188236	+	0.982124i	$0.560277\pi$
$588$		0			0
$589$	589.131	−	382.469i	1.00022	−	0.649353i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−559.672			−0.943797			−0.471898	−	0.881653i	$-0.656431\pi$
$593$	−559.672			−0.943797			−0.471898	+	0.881653i	$0.656431\pi$
$594$		0			0
$595$	−211.277			−0.355087
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	732.669i		−	1.22315i	−0.791185	−	0.611577i	$-0.790535\pi$
$599$		−	732.669i		−	1.22315i	0.791185	−	0.611577i	$-0.209465\pi$
$600$		0			0
$601$			1035.99i			1.72378i	0.507099	+	0.861888i	$0.330718\pi$
$601$			1035.99i			1.72378i	−0.507099	+	0.861888i	$0.669282\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−609.977			−1.00823
$606$		0			0
$607$		−	937.039i		−	1.54372i	−0.635792	−	0.771860i	$-0.719326\pi$
$607$		−	937.039i		−	1.54372i	0.635792	−	0.771860i	$-0.280674\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1491.75i		−	2.44150i
$612$		0			0
$613$	−456.031			−0.743933			−0.371967	−	0.928246i	$-0.621316\pi$
$613$	−456.031			−0.743933			−0.371967	+	0.928246i	$0.621316\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	601.696			0.975196			0.487598	−	0.873068i	$-0.337873\pi$
$617$	601.696			0.975196			0.487598	+	0.873068i	$0.337873\pi$
$618$		0			0
$619$	805.295			1.30096			0.650481	−	0.759523i	$-0.274567\pi$
$619$	805.295			1.30096			0.650481	+	0.759523i	$0.274567\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	731.014i		−	1.17338i
$624$		0			0
$625$	−730.069			−1.16811
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	91.3629i		−	0.145251i
$630$		0			0
$631$	−1099.55			−1.74256			−0.871278	−	0.490790i	$-0.836708\pi$
$631$	−1099.55			−1.74256			−0.871278	+	0.490790i	$0.836708\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			53.8468i			0.0847981i
$636$		0			0
$637$		−	1282.49i		−	2.01333i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	786.036i		−	1.22627i	−0.789980	−	0.613133i	$-0.789909\pi$
$641$		−	786.036i		−	1.22627i	0.789980	−	0.613133i	$-0.210091\pi$
$642$		0			0
$643$	957.867			1.48968			0.744842	−	0.667240i	$-0.232525\pi$
$643$	957.867			1.48968			0.744842	+	0.667240i	$0.232525\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	667.264			1.03132			0.515660	−	0.856793i	$-0.327547\pi$
$647$	667.264			1.03132			0.515660	+	0.856793i	$0.327547\pi$
$648$		0			0
$649$			226.953i			0.349696i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	915.381			1.40181			0.700904	−	0.713255i	$-0.252780\pi$
$653$	915.381			1.40181			0.700904	+	0.713255i	$0.252780\pi$
$654$		0			0
$655$	1252.48			1.91219
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	157.965i		−	0.239705i	−0.992792	−	0.119852i	$-0.961758\pi$
$659$		−	157.965i		−	0.239705i	0.992792	−	0.119852i	$-0.0382421\pi$
$660$		0			0
$661$		−	981.650i		−	1.48510i	−0.669791	−	0.742549i	$-0.733616\pi$
$661$		−	981.650i		−	1.48510i	0.669791	−	0.742549i	$-0.266384\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−589.640	−	908.245i	−0.886676	−	1.36578i
$666$		0			0
$667$		−	1767.34i		−	2.64968i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−204.238			−0.304378
$672$		0			0
$673$		−	617.850i		−	0.918054i	−0.888422	−	0.459027i	$-0.848198\pi$
$673$		−	617.850i		−	0.918054i	0.888422	−	0.459027i	$-0.151802\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	600.320i		−	0.886735i	−0.896340	−	0.443368i	$-0.853784\pi$
$677$		−	600.320i		−	0.886735i	0.896340	−	0.443368i	$-0.146216\pi$
$678$		0			0
$679$		−	1398.07i		−	2.05902i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			761.380i			1.11476i	0.830258	+	0.557379i	$0.188193\pi$
$683$			761.380i			1.11476i	−0.830258	+	0.557379i	$0.811807\pi$
$684$		0			0
$685$	−1002.56			−1.46359
$686$		0			0
$687$		0			0
$688$		0			0
$689$	313.685			0.455276
$690$		0			0
$691$	−378.736			−0.548098			−0.274049	−	0.961716i	$-0.588363\pi$
$691$	−378.736			−0.548098			−0.274049	+	0.961716i	$0.588363\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−410.381			−0.590476
$696$		0			0
$697$			241.358i			0.346282i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	965.515			1.37734			0.688669	−	0.725075i	$-0.258195\pi$
$701$	965.515			1.37734			0.688669	+	0.725075i	$0.258195\pi$
$702$		0			0
$703$	392.754	−	254.979i	0.558683	−	0.362701i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	95.3458			0.134860
$708$		0			0
$709$	1110.38			1.56612			0.783059	−	0.621948i	$-0.213658\pi$
$709$	1110.38			1.56612			0.783059	+	0.621948i	$0.213658\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1647.74i			2.31100i
$714$		0			0
$715$		−	390.112i		−	0.545612i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	470.197			0.653960			0.326980	−	0.945031i	$-0.393969\pi$
$719$	470.197			0.653960			0.326980	+	0.945031i	$0.393969\pi$
$720$		0			0
$721$		−	351.798i		−	0.487931i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			211.972i			0.292375i
$726$		0			0
$727$	220.811			0.303730			0.151865	−	0.988401i	$-0.451472\pi$
$727$	220.811			0.303730			0.151865	+	0.988401i	$0.451472\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−100.231			−0.137115
$732$		0			0
$733$	62.6788			0.0855099			0.0427550	−	0.999086i	$-0.486387\pi$
$733$	62.6788			0.0855099			0.0427550	+	0.999086i	$0.486387\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			251.943i			0.341850i
$738$		0			0
$739$	165.503			0.223956			0.111978	−	0.993711i	$-0.464281\pi$
$739$	165.503			0.223956			0.111978	+	0.993711i	$0.464281\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	942.186i		−	1.26808i	−0.773299	−	0.634042i	$-0.781395\pi$
$743$		−	942.186i		−	1.26808i	0.773299	−	0.634042i	$-0.218605\pi$
$744$		0			0
$745$	−436.484			−0.585884
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	877.879i		−	1.17207i
$750$		0			0
$751$		−	81.6279i		−	0.108692i	−0.998522	−	0.0543461i	$-0.982693\pi$
$751$		−	81.6279i		−	0.108692i	0.998522	−	0.0543461i	$-0.0173074\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			500.952i			0.663512i
$756$		0			0
$757$	389.251			0.514202			0.257101	−	0.966385i	$-0.417233\pi$
$757$	389.251			0.514202			0.257101	+	0.966385i	$0.417233\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31.3545			0.0412017			0.0206009	−	0.999788i	$-0.493442\pi$
$761$	31.3545			0.0412017			0.0206009	+	0.999788i	$0.493442\pi$
$762$		0			0
$763$		−	1248.54i		−	1.63635i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1564.89			2.04027
$768$		0			0
$769$	−648.936			−0.843870			−0.421935	−	0.906626i	$-0.638649\pi$
$769$	−648.936			−0.843870			−0.421935	+	0.906626i	$0.638649\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			684.533i			0.885553i	0.896632	+	0.442777i	$0.146007\pi$
$773$			684.533i			0.885553i	−0.896632	+	0.442777i	$0.853993\pi$
$774$		0			0
$775$		−	197.628i		−	0.255004i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1037.56	+	673.592i	−1.33191	+	0.864688i
$780$		0			0
$781$		−	281.339i		−	0.360229i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−533.478			−0.679590
$786$		0			0
$787$		−	935.324i		−	1.18847i	−0.804292	−	0.594234i	$-0.797455\pi$
$787$		−	935.324i		−	1.18847i	0.804292	−	0.594234i	$-0.202545\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1932.99i			2.44373i
$792$		0			0
$793$			1408.27i			1.77587i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1064.08i		−	1.33510i	−0.744565	−	0.667550i	$-0.767343\pi$
$797$		−	1064.08i		−	1.33510i	0.744565	−	0.667550i	$-0.232657\pi$
$798$		0			0
$799$	250.257			0.313213
$800$		0			0
$801$		0			0
$802$		0			0
$803$	118.455			0.147516
$804$		0			0
$805$	2540.28			3.15562
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1397.79			−1.72780			−0.863900	−	0.503663i	$-0.831985\pi$
$809$	−1397.79			−1.72780			−0.863900	+	0.503663i	$0.831985\pi$
$810$		0			0
$811$		−	250.765i		−	0.309204i	−0.987977	−	0.154602i	$-0.950590\pi$
$811$		−	250.765i		−	0.309204i	0.987977	−	0.154602i	$-0.0494096\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−679.269			−0.833459
$816$		0			0
$817$	−279.730	−	430.878i	−0.342386	−	0.527391i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−114.644			−0.139640			−0.0698199	−	0.997560i	$-0.522242\pi$
$821$	−114.644			−0.139640			−0.0698199	+	0.997560i	$0.522242\pi$
$822$		0			0
$823$	−98.0442			−0.119130			−0.0595651	−	0.998224i	$-0.518971\pi$
$823$	−98.0442			−0.119130			−0.0595651	+	0.998224i	$0.518971\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			602.775i			0.728869i	0.931229	+	0.364435i	$0.118738\pi$
$827$			602.775i			0.728869i	−0.931229	+	0.364435i	$0.881262\pi$
$828$		0			0
$829$		−	570.274i		−	0.687906i	−0.938987	−	0.343953i	$-0.888234\pi$
$829$		−	570.274i		−	0.687906i	0.938987	−	0.343953i	$-0.111766\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	215.151			0.258285
$834$		0			0
$835$			1138.00i			1.36287i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	791.746i		−	0.943679i	−0.881685	−	0.471839i	$-0.843590\pi$
$839$		−	791.746i		−	0.943679i	0.881685	−	0.471839i	$-0.156410\pi$
$840$		0			0
$841$	−731.226			−0.869472
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1758.94			−2.08159
$846$		0			0
$847$	1145.60			1.35253
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1098.50i			1.29083i
$852$		0			0
$853$	−245.849			−0.288217			−0.144109	−	0.989562i	$-0.546031\pi$
$853$	−245.849			−0.288217			−0.144109	+	0.989562i	$0.546031\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	130.374i		−	0.152129i	−0.997103	−	0.0760643i	$-0.975765\pi$
$857$		−	130.374i		−	0.152129i	0.997103	−	0.0760643i	$-0.0242354\pi$
$858$		0			0
$859$	−417.893			−0.486488			−0.243244	−	0.969965i	$-0.578212\pi$
$859$	−417.893			−0.486488			−0.243244	+	0.969965i	$0.578212\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			283.269i			0.328238i	0.986441	+	0.164119i	$0.0524781\pi$
$863$			283.269i			0.328238i	−0.986441	+	0.164119i	$0.947522\pi$
$864$		0			0
$865$			826.086i			0.955013i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	8.16531i		−	0.00939621i
$870$		0			0
$871$	1737.21			1.99450
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1120.14			1.28016
$876$		0			0
$877$		−	577.132i		−	0.658075i	−0.944317	−	0.329038i	$-0.893276\pi$
$877$		−	577.132i		−	0.658075i	0.944317	−	0.329038i	$-0.106724\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	583.612			0.662442			0.331221	−	0.943553i	$-0.392539\pi$
$881$	583.612			0.662442			0.331221	+	0.943553i	$0.392539\pi$
$882$		0			0
$883$	549.503			0.622314			0.311157	−	0.950359i	$-0.399283\pi$
$883$	549.503			0.622314			0.311157	+	0.950359i	$0.399283\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	919.345i		−	1.03647i	−0.855239	−	0.518233i	$-0.826590\pi$
$887$		−	919.345i		−	1.03647i	0.855239	−	0.518233i	$-0.173410\pi$
$888$		0			0
$889$		−	101.130i		−	0.113757i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	698.428	+	1075.82i	0.782114	+	1.20472i
$894$		0			0
$895$		−	109.654i		−	0.122519i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1465.83			1.63052
$900$		0			0
$901$			52.6239i			0.0584062i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			809.997i			0.895024i
$906$		0			0
$907$			1049.19i			1.15677i	0.815763	+	0.578386i	$0.196317\pi$
$907$			1049.19i			1.15677i	−0.815763	+	0.578386i	$0.803683\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	980.342i		−	1.07612i	−0.842908	−	0.538058i	$-0.819158\pi$
$911$		−	980.342i		−	1.07612i	0.842908	−	0.538058i	$-0.180842\pi$
$912$		0			0
$913$	−54.2393			−0.0594078
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2352.29			−2.56520
$918$		0			0
$919$	−234.554			−0.255227			−0.127614	−	0.991824i	$-0.540732\pi$
$919$	−234.554			−0.255227			−0.127614	+	0.991824i	$0.540732\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1939.90			−2.10173
$924$		0			0
$925$		−	131.752i		−	0.142435i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−214.804			−0.231220			−0.115610	−	0.993295i	$-0.536882\pi$
$929$	−214.804			−0.231220			−0.115610	+	0.993295i	$0.536882\pi$
$930$		0			0
$931$	600.453	+	924.900i	0.644954	+	0.993448i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	65.4454			0.0699951
$936$		0			0
$937$	616.434			0.657881			0.328940	−	0.944351i	$-0.393308\pi$
$937$	616.434			0.657881			0.328940	+	0.944351i	$0.393308\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1424.67i		−	1.51400i	−0.653416	−	0.756999i	$-0.726665\pi$
$941$		−	1424.67i		−	1.51400i	0.653416	−	0.756999i	$-0.273335\pi$
$942$		0			0
$943$		−	2901.96i		−	3.07737i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1098.71			−1.16020			−0.580100	−	0.814545i	$-0.696987\pi$
$947$	−1098.71			−1.16020			−0.580100	+	0.814545i	$0.696987\pi$
$948$		0			0
$949$		−	816.776i		−	0.860670i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	634.901i		−	0.666213i	−0.942889	−	0.333106i	$-0.891903\pi$
$953$		−	634.901i		−	0.666213i	0.942889	−	0.333106i	$-0.108097\pi$
$954$		0			0
$955$	−237.326			−0.248509
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1882.90			1.96340
$960$		0			0
$961$	−405.642			−0.422105
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	585.164i		−	0.606388i
$966$		0			0
$967$	−1244.36			−1.28683			−0.643415	−	0.765518i	$-0.722483\pi$
$967$	−1244.36			−1.28683			−0.643415	+	0.765518i	$0.722483\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	430.134i		−	0.442981i	−0.975163	−	0.221490i	$-0.928908\pi$
$971$		−	430.134i		−	0.442981i	0.975163	−	0.221490i	$-0.0710921\pi$
$972$		0			0
$973$	770.736			0.792123
$974$		0			0
$975$		0			0
$976$		0			0
$977$			734.429i			0.751719i	0.926677	+	0.375859i	$0.122652\pi$
$977$			734.429i			0.751719i	−0.926677	+	0.375859i	$0.877348\pi$
$978$		0			0
$979$			226.440i			0.231297i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1953.53i			1.98732i	0.112431	+	0.993659i	$0.464136\pi$
$983$			1953.53i			1.98732i	−0.112431	+	0.993659i	$0.535864\pi$
$984$		0			0
$985$	−322.164			−0.327070
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1205.13			1.21853
$990$		0			0
$991$			1356.89i			1.36921i	0.728912	+	0.684607i	$0.240026\pi$
$991$			1356.89i			1.36921i	−0.728912	+	0.684607i	$0.759974\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	555.341			0.558132
$996$		0			0
$997$	1674.38			1.67942			0.839710	−	0.543035i	$-0.182725\pi$
$997$	1674.38			1.67942			0.839710	+	0.543035i	$0.182725\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.o.721.7		8
3.2	odd	2	inner	2736.3.o.o.721.1		8
4.3	odd	2		342.3.d.c.37.8	yes	8
12.11	even	2		342.3.d.c.37.1	✓	8
19.18	odd	2	inner	2736.3.o.o.721.8		8
57.56	even	2	inner	2736.3.o.o.721.2		8
76.75	even	2		342.3.d.c.37.4	yes	8
228.227	odd	2		342.3.d.c.37.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.d.c.37.1	✓	8	12.11	even	2
342.3.d.c.37.4	yes	8	76.75	even	2
342.3.d.c.37.5	yes	8	228.227	odd	2
342.3.d.c.37.8	yes	8	4.3	odd	2
2736.3.o.o.721.1		8	3.2	odd	2	inner
2736.3.o.o.721.2		8	57.56	even	2	inner
2736.3.o.o.721.7		8	1.1	even	1	trivial
2736.3.o.o.721.8		8	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.50871 q^{5} -10.3459 q^{7} +O(q^{10})\)	\(q+5.50871 q^{5} -10.3459 q^{7} +3.20476 q^{11} -22.0975i q^{13} +3.70710 q^{17} +(10.3459 + 15.9362i) q^{19} -44.5720 q^{23} +5.34590 q^{25} +39.6513i q^{29} -36.9681i q^{31} -56.9926 q^{35} -24.6454i q^{37} +65.1072i q^{41} -27.0377 q^{43} +67.5077 q^{47} +58.0377 q^{49} +14.1955i q^{53} +17.6541 q^{55} +70.8173i q^{59} -63.7295 q^{61} -121.729i q^{65} +78.6154i q^{67} -87.7879i q^{71} +36.9623 q^{73} -33.1561 q^{77} -2.54787i q^{79} -16.9246 q^{83} +20.4213 q^{85} +70.6573i q^{89} +228.619i q^{91} +(56.9926 + 87.7879i) q^{95} +135.133i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{7}+O(q^{10}) \) 8 * q - 12 * q^7	\( 8 q - 12 q^{7} + 12 q^{19} - 28 q^{25} - 4 q^{43} + 252 q^{49} + 212 q^{55} - 156 q^{61} + 508 q^{73} - 332 q^{85}+O(q^{100}) \) 8 * q - 12 * q^7 + 12 * q^19 - 28 * q^25 - 4 * q^43 + 252 * q^49 + 212 * q^55 - 156 * q^61 + 508 * q^73 - 332 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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