LMFDB - Embedded newform 2736.3.b.e.2735.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(2735,2736)]

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([1, 0, 1, 1]))

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

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

chi := DirichletCharacter("2736.2735");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.b (of order $2$, degree $1$, 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:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2735.16
Character	$\chi$	$=$	2736.2735
Dual form			2736.3.b.e.2735.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.60057i q^{5} +2.03549i q^{7} +O(q^{10})$	$q+5.60057i q^{5} +2.03549i q^{7} -1.92287 q^{11} -5.11778i q^{13} -17.3173i q^{17} +(-18.8026 + 2.73143i) q^{19} +28.7178 q^{23} -6.36637 q^{25} -22.2559 q^{29} +18.4633 q^{31} -11.3999 q^{35} +63.4541i q^{37} -46.0569 q^{41} -16.5390i q^{43} -57.8149 q^{47} +44.8568 q^{49} -84.0450 q^{53} -10.7692i q^{55} -117.362i q^{59} -95.7069 q^{61} +28.6625 q^{65} -84.1311 q^{67} +50.6045i q^{71} +3.29031 q^{73} -3.91399i q^{77} -25.0961 q^{79} +119.982 q^{83} +96.9869 q^{85} -53.2945 q^{89} +10.4172 q^{91} +(-15.2976 - 105.305i) q^{95} -91.9959i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 96 q^{25} + 336 q^{49} - 256 q^{61} - 560 q^{73} - 752 q^{85}+O(q^{100}) $ 32 * q + 96 * q^25 + 336 * q^49 - 256 * q^61 - 560 * q^73 - 752 * 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
$5$			5.60057i			1.12011i	0.828454	+	0.560057i	$0.189221\pi$
$5$			5.60057i			1.12011i	−0.828454	+	0.560057i	$0.810779\pi$
$6$		0			0
$7$			2.03549i			0.290784i	0.989374	+	0.145392i	$0.0464444\pi$
$7$			2.03549i			0.290784i	−0.989374	+	0.145392i	$0.953556\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.92287			−0.174807			−0.0874033	−	0.996173i	$-0.527857\pi$
$11$	−1.92287			−0.174807			−0.0874033	+	0.996173i	$0.527857\pi$
$12$		0			0
$13$		−	5.11778i		−	0.393675i	−0.980436	−	0.196838i	$-0.936933\pi$
$13$		−	5.11778i		−	0.393675i	0.980436	−	0.196838i	$-0.0630672\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	17.3173i		−	1.01867i	−0.860569	−	0.509333i	$-0.829892\pi$
$17$		−	17.3173i		−	1.01867i	0.860569	−	0.509333i	$-0.170108\pi$
$18$		0			0
$19$	−18.8026	+	2.73143i	−0.989613	+	0.143760i
$19$	−18.8026	+	2.73143i	−0.989613	+	0.143760i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	28.7178			1.24860			0.624301	−	0.781184i	$-0.285384\pi$
$23$	28.7178			1.24860			0.624301	+	0.781184i	$0.285384\pi$
$24$		0			0
$25$	−6.36637			−0.254655
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−22.2559			−0.767446			−0.383723	−	0.923448i	$-0.625358\pi$
$29$	−22.2559			−0.767446			−0.383723	+	0.923448i	$0.625358\pi$
$30$		0			0
$31$	18.4633			0.595592			0.297796	−	0.954630i	$-0.403748\pi$
$31$	18.4633			0.595592			0.297796	+	0.954630i	$0.403748\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−11.3999			−0.325711
$36$		0			0
$37$			63.4541i			1.71498i	0.514504	+	0.857488i	$0.327976\pi$
$37$			63.4541i			1.71498i	−0.514504	+	0.857488i	$0.672024\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−46.0569			−1.12334			−0.561669	−	0.827362i	$-0.689841\pi$
$41$	−46.0569			−1.12334			−0.561669	+	0.827362i	$0.689841\pi$
$42$		0			0
$43$		−	16.5390i		−	0.384629i	−0.981333	−	0.192314i	$-0.938401\pi$
$43$		−	16.5390i		−	0.384629i	0.981333	−	0.192314i	$-0.0615993\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−57.8149			−1.23010			−0.615052	−	0.788486i	$-0.710865\pi$
$47$	−57.8149			−1.23010			−0.615052	+	0.788486i	$0.710865\pi$
$48$		0			0
$49$	44.8568			0.915445
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−84.0450			−1.58576			−0.792878	−	0.609381i	$-0.791418\pi$
$53$	−84.0450			−1.58576			−0.792878	+	0.609381i	$0.791418\pi$
$54$		0			0
$55$		−	10.7692i		−	0.195803i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	117.362i		−	1.98918i	−0.103880	−	0.994590i	$-0.533126\pi$
$59$		−	117.362i		−	1.98918i	0.103880	−	0.994590i	$-0.466874\pi$
$60$		0			0
$61$	−95.7069			−1.56896			−0.784482	−	0.620151i	$-0.787071\pi$
$61$	−95.7069			−1.56896			−0.784482	+	0.620151i	$0.787071\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	28.6625			0.440961
$66$		0			0
$67$	−84.1311			−1.25569			−0.627844	−	0.778339i	$-0.716062\pi$
$67$	−84.1311			−1.25569			−0.627844	+	0.778339i	$0.716062\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			50.6045i			0.712740i	0.934345	+	0.356370i	$0.115986\pi$
$71$			50.6045i			0.712740i	−0.934345	+	0.356370i	$0.884014\pi$
$72$		0			0
$73$	3.29031			0.0450727			0.0225364	−	0.999746i	$-0.492826\pi$
$73$	3.29031			0.0450727			0.0225364	+	0.999746i	$0.492826\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.91399i		−	0.0508310i
$78$		0			0
$79$	−25.0961			−0.317672			−0.158836	−	0.987305i	$-0.550774\pi$
$79$	−25.0961			−0.317672			−0.158836	+	0.987305i	$0.550774\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	119.982			1.44557			0.722785	−	0.691073i	$-0.242862\pi$
$83$	119.982			1.44557			0.722785	+	0.691073i	$0.242862\pi$
$84$		0			0
$85$	96.9869			1.14102
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−53.2945			−0.598815			−0.299407	−	0.954125i	$-0.596789\pi$
$89$	−53.2945			−0.598815			−0.299407	+	0.954125i	$0.596789\pi$
$90$		0			0
$91$	10.4172			0.114475
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.2976	−	105.305i	−0.161027	−	1.10848i
$96$		0			0
$97$		−	91.9959i		−	0.948412i	−0.880414	−	0.474206i	$-0.842735\pi$
$97$		−	91.9959i		−	0.948412i	0.880414	−	0.474206i	$-0.157265\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	17.8113i		−	0.176349i	−0.996105	−	0.0881746i	$-0.971897\pi$
$101$		−	17.8113i		−	0.176349i	0.996105	−	0.0881746i	$-0.0281033\pi$
$102$		0			0
$103$	165.918			1.61086			0.805428	−	0.592694i	$-0.201936\pi$
$103$	165.918			1.61086			0.805428	+	0.592694i	$0.201936\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.4956i		−	0.126127i	−0.998010	−	0.0630636i	$-0.979913\pi$
$107$		−	13.4956i		−	0.126127i	0.998010	−	0.0630636i	$-0.0200871\pi$
$108$		0			0
$109$		−	165.686i		−	1.52005i	−0.649893	−	0.760026i	$-0.725186\pi$
$109$		−	165.686i		−	1.52005i	0.649893	−	0.760026i	$-0.274814\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−121.607			−1.07617			−0.538086	−	0.842890i	$-0.680852\pi$
$113$	−121.607			−1.07617			−0.538086	+	0.842890i	$0.680852\pi$
$114$		0			0
$115$			160.836i			1.39858i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	35.2493			0.296212
$120$		0			0
$121$	−117.303			−0.969443
$122$		0			0
$123$		0			0
$124$		0			0
$125$			104.359i			0.834871i
$126$		0			0
$127$	102.455			0.806732			0.403366	−	0.915039i	$-0.367840\pi$
$127$	102.455			0.806732			0.403366	+	0.915039i	$0.367840\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−72.6073			−0.554254			−0.277127	−	0.960833i	$-0.589382\pi$
$131$	−72.6073			−0.554254			−0.277127	+	0.960833i	$0.589382\pi$
$132$		0			0
$133$	−5.55980	−	38.2726i	−0.0418030	−	0.287764i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	76.2677i		−	0.556698i	−0.960480	−	0.278349i	$-0.910213\pi$
$137$		−	76.2677i		−	0.556698i	0.960480	−	0.278349i	$-0.0897872\pi$
$138$		0			0
$139$		−	124.076i		−	0.892635i	−0.894875	−	0.446318i	$-0.852735\pi$
$139$		−	124.076i		−	0.892635i	0.894875	−	0.446318i	$-0.147265\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			9.84085i			0.0688171i
$144$		0			0
$145$		−	124.646i		−	0.859627i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	129.953i		−	0.872169i	−0.899906	−	0.436085i	$-0.856365\pi$
$149$		−	129.953i		−	0.872169i	0.899906	−	0.436085i	$-0.143635\pi$
$150$		0			0
$151$	249.910			1.65503			0.827515	−	0.561443i	$-0.189754\pi$
$151$	249.910			1.65503			0.827515	+	0.561443i	$0.189754\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			103.405i			0.667131i
$156$		0			0
$157$	190.291			1.21204			0.606021	−	0.795449i	$-0.292765\pi$
$157$	190.291			1.21204			0.606021	+	0.795449i	$0.292765\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			58.4548i			0.363074i
$162$		0			0
$163$		−	82.9371i		−	0.508816i	−0.967097	−	0.254408i	$-0.918119\pi$
$163$		−	82.9371i		−	0.508816i	0.967097	−	0.254408i	$-0.0818807\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	168.465i		−	1.00877i	−0.863478	−	0.504386i	$-0.831719\pi$
$167$		−	168.465i		−	1.00877i	0.863478	−	0.504386i	$-0.168281\pi$
$168$		0			0
$169$	142.808			0.845020
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−49.1469			−0.284086			−0.142043	−	0.989860i	$-0.545367\pi$
$173$	−49.1469			−0.284086			−0.142043	+	0.989860i	$0.545367\pi$
$174$		0			0
$175$		−	12.9587i		−	0.0740496i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	180.041i		−	1.00582i	−0.864339	−	0.502909i	$-0.832263\pi$
$179$		−	180.041i		−	1.00582i	0.864339	−	0.502909i	$-0.167737\pi$
$180$		0			0
$181$		−	175.436i		−	0.969262i	−0.874719	−	0.484631i	$-0.838954\pi$
$181$		−	175.436i		−	0.969262i	0.874719	−	0.484631i	$-0.161046\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−355.379			−1.92097
$186$		0			0
$187$			33.2990i			0.178070i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−107.606			−0.563382			−0.281691	−	0.959505i	$-0.590895\pi$
$191$	−107.606			−0.563382			−0.281691	+	0.959505i	$0.590895\pi$
$192$		0			0
$193$		−	142.932i		−	0.740581i	−0.928916	−	0.370290i	$-0.879258\pi$
$193$		−	142.932i		−	0.740581i	0.928916	−	0.370290i	$-0.120742\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	250.495i		−	1.27155i	−0.771875	−	0.635775i	$-0.780681\pi$
$197$		−	250.495i		−	1.27155i	0.771875	−	0.635775i	$-0.219319\pi$
$198$		0			0
$199$			297.036i			1.49264i	0.665586	+	0.746321i	$0.268182\pi$
$199$			297.036i			1.49264i	−0.665586	+	0.746321i	$0.731818\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	45.3017i		−	0.223161i
$204$		0			0
$205$		−	257.945i		−	1.25827i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	36.1551	−	5.25220i	0.172991	−	0.0251301i
$210$		0			0
$211$	138.786			0.657755			0.328877	−	0.944373i	$-0.393330\pi$
$211$	138.786			0.657755			0.328877	+	0.944373i	$0.393330\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	92.6280			0.430828
$216$		0			0
$217$			37.5820i			0.173189i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−88.6263			−0.401024
$222$		0			0
$223$	−12.6702			−0.0568169			−0.0284084	−	0.999596i	$-0.509044\pi$
$223$	−12.6702			−0.0568169			−0.0284084	+	0.999596i	$0.509044\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	94.4859i		−	0.416237i	−0.978104	−	0.208119i	$-0.933266\pi$
$227$		−	94.4859i		−	0.416237i	0.978104	−	0.208119i	$-0.0667340\pi$
$228$		0			0
$229$	−278.839			−1.21764			−0.608820	−	0.793309i	$-0.708357\pi$
$229$	−278.839			−1.21764			−0.608820	+	0.793309i	$0.708357\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			228.437i			0.980416i	0.871605	+	0.490208i	$0.163079\pi$
$233$			228.437i			0.980416i	−0.871605	+	0.490208i	$0.836921\pi$
$234$		0			0
$235$		−	323.796i		−	1.37786i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	373.960			1.56469			0.782344	−	0.622847i	$-0.214024\pi$
$239$	373.960			1.56469			0.782344	+	0.622847i	$0.214024\pi$
$240$		0			0
$241$		−	176.349i		−	0.731737i	−0.930667	−	0.365869i	$-0.880772\pi$
$241$		−	176.349i		−	0.731737i	0.930667	−	0.365869i	$-0.119228\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			251.224i			1.02540i
$246$		0			0
$247$	13.9789	+	96.2278i	0.0565946	+	0.389586i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−135.240			−0.538804			−0.269402	−	0.963028i	$-0.586826\pi$
$251$	−135.240			−0.538804			−0.269402	+	0.963028i	$0.586826\pi$
$252$		0			0
$253$	−55.2207			−0.218264
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−309.723			−1.20515			−0.602574	−	0.798063i	$-0.705858\pi$
$257$	−309.723			−1.20515			−0.602574	+	0.798063i	$0.705858\pi$
$258$		0			0
$259$	−129.160			−0.498688
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−141.881			−0.539472			−0.269736	−	0.962934i	$-0.586936\pi$
$263$	−141.881			−0.539472			−0.269736	+	0.962934i	$0.586936\pi$
$264$		0			0
$265$		−	470.700i		−	1.77623i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−105.620			−0.392640			−0.196320	−	0.980540i	$-0.562899\pi$
$269$	−105.620			−0.392640			−0.196320	+	0.980540i	$0.562899\pi$
$270$		0			0
$271$		−	302.476i		−	1.11615i	−0.829792	−	0.558074i	$-0.811541\pi$
$271$		−	302.476i		−	1.11615i	0.829792	−	0.558074i	$-0.188459\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.2417			0.0445154
$276$		0			0
$277$	282.695			1.02056			0.510281	−	0.860008i	$-0.329542\pi$
$277$	282.695			1.02056			0.510281	+	0.860008i	$0.329542\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	40.2186			0.143127			0.0715634	−	0.997436i	$-0.477201\pi$
$281$	40.2186			0.143127			0.0715634	+	0.997436i	$0.477201\pi$
$282$		0			0
$283$			500.954i			1.77016i	0.465443	+	0.885078i	$0.345895\pi$
$283$			500.954i			1.77016i	−0.465443	+	0.885078i	$0.654105\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	93.7483i		−	0.326649i
$288$		0			0
$289$	−10.8901			−0.0376819
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−434.359			−1.48245			−0.741226	−	0.671255i	$-0.765755\pi$
$293$	−434.359			−1.48245			−0.741226	+	0.671255i	$0.765755\pi$
$294$		0			0
$295$	657.292			2.22811
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	146.972i		−	0.491544i
$300$		0			0
$301$	33.6650			0.111844
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	536.013i		−	1.75742i
$306$		0			0
$307$	7.42383			0.0241819			0.0120909	−	0.999927i	$-0.496151\pi$
$307$	7.42383			0.0241819			0.0120909	+	0.999927i	$0.496151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−113.742			−0.365730			−0.182865	−	0.983138i	$-0.558537\pi$
$311$	−113.742			−0.365730			−0.182865	+	0.983138i	$0.558537\pi$
$312$		0			0
$313$	−109.444			−0.349662			−0.174831	−	0.984598i	$-0.555938\pi$
$313$	−109.444			−0.349662			−0.174831	+	0.984598i	$0.555938\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−527.650			−1.66451			−0.832255	−	0.554392i	$-0.812951\pi$
$317$	−527.650			−1.66451			−0.832255	+	0.554392i	$0.812951\pi$
$318$		0			0
$319$	42.7954			0.134155
$320$		0			0
$321$		0			0
$322$		0			0
$323$	47.3011	+	325.612i	0.146443	+	1.00809i
$324$		0			0
$325$			32.5817i			0.100251i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	117.682i		−	0.357695i
$330$		0			0
$331$	−159.854			−0.482942			−0.241471	−	0.970408i	$-0.577630\pi$
$331$	−159.854			−0.482942			−0.241471	+	0.970408i	$0.577630\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	471.182i		−	1.40651i
$336$		0			0
$337$			314.698i			0.933823i	0.884304	+	0.466911i	$0.154633\pi$
$337$			314.698i			0.933823i	−0.884304	+	0.466911i	$0.845367\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−35.5027			−0.104113
$342$		0			0
$343$			191.045i			0.556981i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	92.4423			0.266404			0.133202	−	0.991089i	$-0.457474\pi$
$347$	92.4423			0.266404			0.133202	+	0.991089i	$0.457474\pi$
$348$		0			0
$349$	100.178			0.287043			0.143522	−	0.989647i	$-0.454157\pi$
$349$	100.178			0.287043			0.143522	+	0.989647i	$0.454157\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			30.9310i			0.0876232i	0.999040	+	0.0438116i	$0.0139501\pi$
$353$			30.9310i			0.0876232i	−0.999040	+	0.0438116i	$0.986050\pi$
$354$		0			0
$355$	−283.414			−0.798350
$356$		0			0
$357$		0			0
$358$		0			0
$359$	105.868			0.294897			0.147448	−	0.989070i	$-0.452894\pi$
$359$	105.868			0.294897			0.147448	+	0.989070i	$0.452894\pi$
$360$		0			0
$361$	346.079	−	102.716i	0.958666	−	0.284532i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			18.4276i			0.0504866i
$366$		0			0
$367$		−	395.018i		−	1.07634i	−0.842835	−	0.538172i	$-0.819115\pi$
$367$		−	395.018i		−	1.07634i	0.842835	−	0.538172i	$-0.180885\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	171.073i		−	0.461113i
$372$		0			0
$373$			286.369i			0.767745i	0.923386	+	0.383873i	$0.125410\pi$
$373$			286.369i			0.767745i	−0.923386	+	0.383873i	$0.874590\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			113.901i			0.302125i
$378$		0			0
$379$	−480.082			−1.26671			−0.633354	−	0.773862i	$-0.718322\pi$
$379$	−480.082			−1.26671			−0.633354	+	0.773862i	$0.718322\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	557.892i		−	1.45664i	−0.685239	−	0.728318i	$-0.740302\pi$
$383$		−	557.892i		−	1.45664i	0.685239	−	0.728318i	$-0.259698\pi$
$384$		0			0
$385$	21.9206			0.0569365
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	114.774i		−	0.295048i	−0.989058	−	0.147524i	$-0.952870\pi$
$389$		−	114.774i		−	0.295048i	0.989058	−	0.147524i	$-0.0471304\pi$
$390$		0			0
$391$		−	497.316i		−	1.27191i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	140.552i		−	0.355829i
$396$		0			0
$397$	−397.320			−1.00081			−0.500403	−	0.865793i	$-0.666815\pi$
$397$	−397.320			−1.00081			−0.500403	+	0.865793i	$0.666815\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−624.793			−1.55809			−0.779044	−	0.626969i	$-0.784295\pi$
$401$	−624.793			−1.55809			−0.779044	+	0.626969i	$0.784295\pi$
$402$		0			0
$403$		−	94.4914i		−	0.234470i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	122.014i		−	0.299789i
$408$		0			0
$409$			411.111i			1.00516i	0.864530	+	0.502581i	$0.167616\pi$
$409$			411.111i			1.00516i	−0.864530	+	0.502581i	$0.832384\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	238.888			0.578422
$414$		0			0
$415$			671.969i			1.61920i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−528.326			−1.26092			−0.630460	−	0.776221i	$-0.717134\pi$
$419$	−528.326			−1.26092			−0.630460	+	0.776221i	$0.717134\pi$
$420$		0			0
$421$		−	115.321i		−	0.273922i	−0.990576	−	0.136961i	$-0.956267\pi$
$421$		−	115.321i		−	0.273922i	0.990576	−	0.136961i	$-0.0437335\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			110.249i			0.259408i
$426$		0			0
$427$		−	194.810i		−	0.456230i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	5.49770i		−	0.0127557i	−0.999980	−	0.00637784i	$-0.997970\pi$
$431$		−	5.49770i		−	0.0127557i	0.999980	−	0.00637784i	$-0.00203014\pi$
$432$		0			0
$433$			243.056i			0.561330i	0.959806	+	0.280665i	$0.0905550\pi$
$433$			243.056i			0.561330i	−0.959806	+	0.280665i	$0.909445\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−539.971	+	78.4408i	−1.23563	+	0.179498i
$438$		0			0
$439$	−76.2189			−0.173619			−0.0868097	−	0.996225i	$-0.527667\pi$
$439$	−76.2189			−0.173619			−0.0868097	+	0.996225i	$0.527667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	464.886			1.04940			0.524702	−	0.851286i	$-0.324177\pi$
$443$	464.886			1.04940			0.524702	+	0.851286i	$0.324177\pi$
$444$		0			0
$445$		−	298.480i		−	0.670741i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−336.034			−0.748406			−0.374203	−	0.927347i	$-0.622084\pi$
$449$	−336.034			−0.748406			−0.374203	+	0.927347i	$0.622084\pi$
$450$		0			0
$451$	88.5615			0.196367
$452$		0			0
$453$		0			0
$454$		0			0
$455$			58.3422i			0.128225i
$456$		0			0
$457$	−671.906			−1.47025			−0.735127	−	0.677929i	$-0.762878\pi$
$457$	−671.906			−1.47025			−0.735127	+	0.677929i	$0.762878\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	444.022i		−	0.963172i	−0.876399	−	0.481586i	$-0.840061\pi$
$461$		−	444.022i		−	0.963172i	0.876399	−	0.481586i	$-0.159939\pi$
$462$		0			0
$463$			697.591i			1.50668i	0.657634	+	0.753338i	$0.271557\pi$
$463$			697.591i			1.50668i	−0.657634	+	0.753338i	$0.728443\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	415.168			0.889010			0.444505	−	0.895776i	$-0.353380\pi$
$467$	415.168			0.889010			0.444505	+	0.895776i	$0.353380\pi$
$468$		0			0
$469$		−	171.248i		−	0.365134i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			31.8025i			0.0672356i
$474$		0			0
$475$	119.705	−	17.3893i	0.252010	−	0.0366091i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	151.389			0.316051			0.158026	−	0.987435i	$-0.449487\pi$
$479$	151.389			0.316051			0.158026	+	0.987435i	$0.449487\pi$
$480$		0			0
$481$	324.744			0.675144
$482$		0			0
$483$		0			0
$484$		0			0
$485$	515.230			1.06233
$486$		0			0
$487$	675.755			1.38759			0.693794	−	0.720174i	$-0.255938\pi$
$487$	675.755			1.38759			0.693794	+	0.720174i	$0.255938\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	66.7682			0.135984			0.0679921	−	0.997686i	$-0.478341\pi$
$491$	66.7682			0.135984			0.0679921	+	0.997686i	$0.478341\pi$
$492$		0			0
$493$			385.414i			0.781772i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−103.005			−0.207254
$498$		0			0
$499$		−	665.660i		−	1.33399i	−0.745063	−	0.666994i	$-0.767581\pi$
$499$		−	665.660i		−	1.33399i	0.745063	−	0.666994i	$-0.232419\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−462.048			−0.918585			−0.459293	−	0.888285i	$-0.651897\pi$
$503$	−462.048			−0.918585			−0.459293	+	0.888285i	$0.651897\pi$
$504$		0			0
$505$	99.7532			0.197531
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.4701			0.0205699			0.0102850	−	0.999947i	$-0.496726\pi$
$509$	10.4701			0.0205699			0.0102850	+	0.999947i	$0.496726\pi$
$510$		0			0
$511$			6.69739i			0.0131064i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			929.236i			1.80434i
$516$		0			0
$517$	111.171			0.215030
$518$		0			0
$519$		0			0
$520$		0			0
$521$	674.371			1.29438			0.647189	−	0.762330i	$-0.275945\pi$
$521$	674.371			1.29438			0.647189	+	0.762330i	$0.275945\pi$
$522$		0			0
$523$	−371.563			−0.710445			−0.355223	−	0.934782i	$-0.615595\pi$
$523$	−371.563			−0.710445			−0.355223	+	0.934782i	$0.615595\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	319.736i		−	0.606710i
$528$		0			0
$529$	295.714			0.559005
$530$		0			0
$531$		0			0
$532$		0			0
$533$			235.709i			0.442231i
$534$		0			0
$535$	75.5831			0.141277
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−86.2539			−0.160026
$540$		0			0
$541$	279.907			0.517389			0.258694	−	0.965959i	$-0.416708\pi$
$541$	279.907			0.517389			0.258694	+	0.965959i	$0.416708\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	927.934			1.70263
$546$		0			0
$547$	498.484			0.911305			0.455652	−	0.890158i	$-0.349406\pi$
$547$	498.484			0.911305			0.455652	+	0.890158i	$0.349406\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	418.470	−	60.7906i	0.759474	−	0.110328i
$552$		0			0
$553$		−	51.0828i		−	0.0923740i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			946.606i			1.69947i	0.527208	+	0.849736i	$0.323239\pi$
$557$			946.606i			1.69947i	−0.527208	+	0.849736i	$0.676761\pi$
$558$		0			0
$559$	−84.6431			−0.151419
$560$		0			0
$561$		0			0
$562$		0			0
$563$			900.136i			1.59882i	0.600786	+	0.799410i	$0.294855\pi$
$563$			900.136i			1.59882i	−0.600786	+	0.799410i	$0.705145\pi$
$564$		0			0
$565$		−	681.070i		−	1.20543i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−187.541			−0.329597			−0.164799	−	0.986327i	$-0.552697\pi$
$569$	−187.541			−0.329597			−0.164799	+	0.986327i	$0.552697\pi$
$570$		0			0
$571$			885.114i			1.55011i	0.631893	+	0.775056i	$0.282278\pi$
$571$			885.114i			1.55011i	−0.631893	+	0.775056i	$0.717722\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−182.828			−0.317962
$576$		0			0
$577$	56.1526			0.0973183			0.0486591	−	0.998815i	$-0.484505\pi$
$577$	56.1526			0.0973183			0.0486591	+	0.998815i	$0.484505\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			244.223i			0.420349i
$582$		0			0
$583$	161.608			0.277201
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1093.11			−1.86220			−0.931098	−	0.364769i	$-0.881148\pi$
$587$	−1093.11			−1.86220			−0.931098	+	0.364769i	$0.881148\pi$
$588$		0			0
$589$	−347.160	+	50.4314i	−0.589405	+	0.0856220i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			604.504i			1.01940i	0.860352	+	0.509700i	$0.170243\pi$
$593$			604.504i			1.01940i	−0.860352	+	0.509700i	$0.829757\pi$
$594$		0			0
$595$			197.416i			0.331791i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	922.740i		−	1.54047i	−0.637761	−	0.770234i	$-0.720139\pi$
$599$		−	922.740i		−	1.54047i	0.637761	−	0.770234i	$-0.279861\pi$
$600$		0			0
$601$		−	780.834i		−	1.29922i	−0.760266	−	0.649612i	$-0.774931\pi$
$601$		−	780.834i		−	1.29922i	0.760266	−	0.649612i	$-0.225069\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	656.961i		−	1.08589i
$606$		0			0
$607$	−426.722			−0.703002			−0.351501	−	0.936187i	$-0.614329\pi$
$607$	−426.722			−0.703002			−0.351501	+	0.936187i	$0.614329\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			295.884i			0.484262i
$612$		0			0
$613$	−226.053			−0.368765			−0.184383	−	0.982855i	$-0.559029\pi$
$613$	−226.053			−0.368765			−0.184383	+	0.982855i	$0.559029\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	873.370i		−	1.41551i	−0.706457	−	0.707755i	$-0.749708\pi$
$617$		−	873.370i		−	1.41551i	0.706457	−	0.707755i	$-0.250292\pi$
$618$		0			0
$619$		−	453.397i		−	0.732468i	−0.930523	−	0.366234i	$-0.880647\pi$
$619$		−	453.397i		−	0.732468i	0.930523	−	0.366234i	$-0.119353\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	108.480i		−	0.174126i
$624$		0			0
$625$	−743.629			−1.18981
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1098.86			1.74699
$630$		0			0
$631$		−	482.242i		−	0.764251i	−0.924110	−	0.382125i	$-0.875192\pi$
$631$		−	482.242i		−	0.764251i	0.924110	−	0.382125i	$-0.124808\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			573.806i			0.903631i
$636$		0			0
$637$		−	229.567i		−	0.360388i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	644.325			1.00519			0.502593	−	0.864523i	$-0.332379\pi$
$641$	644.325			1.00519			0.502593	+	0.864523i	$0.332379\pi$
$642$		0			0
$643$		−	712.415i		−	1.10795i	−0.832532	−	0.553977i	$-0.813109\pi$
$643$		−	712.415i		−	1.10795i	0.832532	−	0.553977i	$-0.186891\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	395.657			0.611526			0.305763	−	0.952108i	$-0.401088\pi$
$647$	395.657			0.611526			0.305763	+	0.952108i	$0.401088\pi$
$648$		0			0
$649$			225.672i			0.347722i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	84.4300i		−	0.129296i	−0.997908	−	0.0646478i	$-0.979408\pi$
$653$		−	84.4300i		−	0.129296i	0.997908	−	0.0646478i	$-0.0205924\pi$
$654$		0			0
$655$		−	406.642i		−	0.620828i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			999.513i			1.51671i	0.651841	+	0.758356i	$0.273997\pi$
$659$			999.513i			1.51671i	−0.651841	+	0.758356i	$0.726003\pi$
$660$		0			0
$661$			735.344i			1.11247i	0.831025	+	0.556236i	$0.187755\pi$
$661$			735.344i			1.11247i	−0.831025	+	0.556236i	$0.812245\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	214.348	−	31.1380i	0.322328	−	0.0468241i
$666$		0			0
$667$	−639.142			−0.958234
$668$		0			0
$669$		0			0
$670$		0			0
$671$	184.032			0.274266
$672$		0			0
$673$		−	330.032i		−	0.490389i	−0.969474	−	0.245194i	$-0.921148\pi$
$673$		−	330.032i		−	0.490389i	0.969474	−	0.245194i	$-0.0788518\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	527.650			0.779394			0.389697	−	0.920943i	$-0.372580\pi$
$677$	527.650			0.779394			0.389697	+	0.920943i	$0.372580\pi$
$678$		0			0
$679$	187.257			0.275783
$680$		0			0
$681$		0			0
$682$		0			0
$683$			268.574i			0.393227i	0.980481	+	0.196614i	$0.0629945\pi$
$683$			268.574i			0.393227i	−0.980481	+	0.196614i	$0.937006\pi$
$684$		0			0
$685$	427.142			0.623565
$686$		0			0
$687$		0			0
$688$		0			0
$689$			430.124i			0.624273i
$690$		0			0
$691$			1084.24i			1.56908i	0.620076	+	0.784542i	$0.287102\pi$
$691$			1084.24i			1.56908i	−0.620076	+	0.784542i	$0.712898\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	694.898			0.999853
$696$		0			0
$697$			797.582i			1.14431i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	963.700i		−	1.37475i	−0.726302	−	0.687375i	$-0.758763\pi$
$701$		−	963.700i		−	1.37475i	0.726302	−	0.687375i	$-0.241237\pi$
$702$		0			0
$703$	−173.320	−	1193.10i	−0.246544	−	1.69716i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	36.2546			0.0512795
$708$		0			0
$709$	1026.74			1.44815			0.724076	−	0.689720i	$-0.242266\pi$
$709$	1026.74			1.44815			0.724076	+	0.689720i	$0.242266\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	530.227			0.743657
$714$		0			0
$715$	−55.1143			−0.0770830
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−876.207			−1.21865			−0.609323	−	0.792922i	$-0.708559\pi$
$719$	−876.207			−1.21865			−0.609323	+	0.792922i	$0.708559\pi$
$720$		0			0
$721$			337.725i			0.468411i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	141.690			0.195434
$726$		0			0
$727$			477.628i			0.656985i	0.944507	+	0.328492i	$0.106541\pi$
$727$			477.628i			0.656985i	−0.944507	+	0.328492i	$0.893459\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−286.412			−0.391808
$732$		0			0
$733$	524.677			0.715793			0.357897	−	0.933761i	$-0.383494\pi$
$733$	524.677			0.715793			0.357897	+	0.933761i	$0.383494\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	161.773			0.219503
$738$		0			0
$739$			1277.94i			1.72928i	0.502388	+	0.864642i	$0.332455\pi$
$739$			1277.94i			1.72928i	−0.502388	+	0.864642i	$0.667545\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	804.008i		−	1.08211i	−0.840987	−	0.541055i	$-0.818025\pi$
$743$		−	804.008i		−	1.08211i	0.840987	−	0.541055i	$-0.181975\pi$
$744$		0			0
$745$	727.812			0.976929
$746$		0			0
$747$		0			0
$748$		0			0
$749$	27.4702			0.0366758
$750$		0			0
$751$	−1147.57			−1.52805			−0.764027	−	0.645184i	$-0.776781\pi$
$751$	−1147.57			−1.52805			−0.764027	+	0.645184i	$0.776781\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1399.64i			1.85382i
$756$		0			0
$757$	−975.992			−1.28929			−0.644644	−	0.764483i	$-0.722995\pi$
$757$	−975.992			−1.28929			−0.644644	+	0.764483i	$0.722995\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			685.045i			0.900190i	0.892981	+	0.450095i	$0.148610\pi$
$761$			685.045i			0.900190i	−0.892981	+	0.450095i	$0.851390\pi$
$762$		0			0
$763$	337.251			0.442007
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−600.631			−0.783091
$768$		0			0
$769$	−557.401			−0.724839			−0.362420	−	0.932015i	$-0.618049\pi$
$769$	−557.401			−0.724839			−0.362420	+	0.932015i	$0.618049\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−215.572			−0.278877			−0.139438	−	0.990231i	$-0.544530\pi$
$773$	−215.572			−0.278877			−0.139438	+	0.990231i	$0.544530\pi$
$774$		0			0
$775$	−117.545			−0.151670
$776$		0			0
$777$		0			0
$778$		0			0
$779$	865.991	−	125.801i	1.11167	−	0.161491i
$780$		0			0
$781$		−	97.3061i		−	0.124592i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1065.74i			1.35762i
$786$		0			0
$787$	50.9293			0.0647132			0.0323566	−	0.999476i	$-0.489699\pi$
$787$	50.9293			0.0647132			0.0323566	+	0.999476i	$0.489699\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	247.530i		−	0.312934i
$792$		0			0
$793$			489.807i			0.617663i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	955.749			1.19918			0.599592	−	0.800306i	$-0.295330\pi$
$797$	955.749			1.19918			0.599592	+	0.800306i	$0.295330\pi$
$798$		0			0
$799$			1001.20i			1.25307i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.32685			−0.00787902
$804$		0			0
$805$	−327.380			−0.406684
$806$		0			0
$807$		0			0
$808$		0			0
$809$			14.2361i			0.0175972i	0.999961	+	0.00879860i	$0.00280072\pi$
$809$			14.2361i			0.0175972i	−0.999961	+	0.00879860i	$0.997199\pi$
$810$		0			0
$811$	−393.368			−0.485041			−0.242521	−	0.970146i	$-0.577974\pi$
$811$	−393.368			−0.485041			−0.242521	+	0.970146i	$0.577974\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	464.495			0.569932
$816$		0			0
$817$	45.1752	+	310.977i	0.0552940	+	0.380633i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			561.359i			0.683751i	0.939745	+	0.341875i	$0.111062\pi$
$821$			561.359i			0.683751i	−0.939745	+	0.341875i	$0.888938\pi$
$822$		0			0
$823$		−	549.746i		−	0.667978i	−0.942577	−	0.333989i	$-0.891605\pi$
$823$		−	549.746i		−	0.667978i	0.942577	−	0.333989i	$-0.108395\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1419.13i			1.71600i	0.513653	+	0.857998i	$0.328292\pi$
$827$			1419.13i			1.71600i	−0.513653	+	0.857998i	$0.671708\pi$
$828$		0			0
$829$			1301.72i			1.57023i	0.619349	+	0.785116i	$0.287397\pi$
$829$			1301.72i			1.57023i	−0.619349	+	0.785116i	$0.712603\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	776.800i		−	0.932533i
$834$		0			0
$835$	943.500			1.12994
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1247.20i			1.48653i	0.668996	+	0.743266i	$0.266724\pi$
$839$			1247.20i			1.48653i	−0.668996	+	0.743266i	$0.733276\pi$
$840$		0			0
$841$	−345.673			−0.411026
$842$		0			0
$843$		0			0
$844$		0			0
$845$			799.808i			0.946518i
$846$		0			0
$847$		−	238.768i		−	0.281899i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1822.26i			2.14132i
$852$		0			0
$853$	−1275.12			−1.49486			−0.747430	−	0.664340i	$-0.768713\pi$
$853$	−1275.12			−1.49486			−0.747430	+	0.664340i	$0.768713\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	546.505			0.637696			0.318848	−	0.947806i	$-0.396704\pi$
$857$	546.505			0.637696			0.318848	+	0.947806i	$0.396704\pi$
$858$		0			0
$859$		−	1577.83i		−	1.83683i	−0.395621	−	0.918414i	$-0.629471\pi$
$859$		−	1577.83i		−	1.83683i	0.395621	−	0.918414i	$-0.370529\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	737.992i		−	0.855147i	−0.903981	−	0.427573i	$-0.859369\pi$
$863$		−	737.992i		−	0.855147i	0.903981	−	0.427573i	$-0.140631\pi$
$864$		0			0
$865$		−	275.250i		−	0.318209i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	48.2566			0.0555312
$870$		0			0
$871$			430.564i			0.494333i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−212.421			−0.242767
$876$		0			0
$877$			975.085i			1.11184i	0.831235	+	0.555921i	$0.187634\pi$
$877$			975.085i			1.11184i	−0.831235	+	0.555921i	$0.812366\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	883.124i		−	1.00241i	−0.865328	−	0.501206i	$-0.832890\pi$
$881$		−	883.124i		−	1.00241i	0.865328	−	0.501206i	$-0.167110\pi$
$882$		0			0
$883$		−	952.901i		−	1.07916i	−0.841933	−	0.539581i	$-0.818583\pi$
$883$		−	952.901i		−	1.07916i	0.841933	−	0.539581i	$-0.181417\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1335.98i		−	1.50618i	−0.657917	−	0.753090i	$-0.728562\pi$
$887$		−	1335.98i		−	1.50618i	0.657917	−	0.753090i	$-0.271438\pi$
$888$		0			0
$889$			208.546i			0.234585i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1087.07	−	157.917i	1.21733	−	0.176839i
$894$		0			0
$895$	1008.33			1.12663
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−410.919			−0.457085
$900$		0			0
$901$			1455.44i			1.61536i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	982.544			1.08568
$906$		0			0
$907$	−1446.42			−1.59473			−0.797365	−	0.603497i	$-0.793773\pi$
$907$	−1446.42			−1.59473			−0.797365	+	0.603497i	$0.793773\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	550.253i		−	0.604010i	−0.953306	−	0.302005i	$-0.902344\pi$
$911$		−	550.253i		−	0.604010i	0.953306	−	0.302005i	$-0.0976559\pi$
$912$		0			0
$913$	−230.711			−0.252695
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	147.791i		−	0.161168i
$918$		0			0
$919$			670.051i			0.729109i	0.931182	+	0.364555i	$0.118779\pi$
$919$			670.051i			0.729109i	−0.931182	+	0.364555i	$0.881221\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	258.983			0.280588
$924$		0			0
$925$		−	403.973i		−	0.436727i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			354.307i			0.381385i	0.981650	+	0.190692i	$0.0610733\pi$
$929$			354.307i			0.381385i	−0.981650	+	0.190692i	$0.938927\pi$
$930$		0			0
$931$	−843.426	+	122.523i	−0.905935	+	0.131604i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−186.494			−0.199458
$936$		0			0
$937$	607.866			0.648737			0.324368	−	0.945931i	$-0.394848\pi$
$937$	607.866			0.648737			0.324368	+	0.945931i	$0.394848\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−716.670			−0.761604			−0.380802	−	0.924657i	$-0.624352\pi$
$941$	−716.670			−0.761604			−0.380802	+	0.924657i	$0.624352\pi$
$942$		0			0
$943$	−1322.65			−1.40260
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−244.834			−0.258536			−0.129268	−	0.991610i	$-0.541263\pi$
$947$	−244.834			−0.258536			−0.129268	+	0.991610i	$0.541263\pi$
$948$		0			0
$949$		−	16.8391i		−	0.0177440i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	655.921			0.688270			0.344135	−	0.938920i	$-0.388172\pi$
$953$	655.921			0.688270			0.344135	+	0.938920i	$0.388172\pi$
$954$		0			0
$955$		−	602.655i		−	0.631052i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	155.242			0.161879
$960$		0			0
$961$	−620.105			−0.645270
$962$		0			0
$963$		0			0
$964$		0			0
$965$	800.501			0.829535
$966$		0			0
$967$			992.013i			1.02587i	0.858428	+	0.512933i	$0.171441\pi$
$967$			992.013i			1.02587i	−0.858428	+	0.512933i	$0.828559\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	331.137i		−	0.341027i	−0.985355	−	0.170514i	$-0.945457\pi$
$971$		−	331.137i		−	0.341027i	0.985355	−	0.170514i	$-0.0545427\pi$
$972$		0			0
$973$	252.556			0.259564
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22.0340			−0.0225528			−0.0112764	−	0.999936i	$-0.503589\pi$
$977$	−22.0340			−0.0225528			−0.0112764	+	0.999936i	$0.503589\pi$
$978$		0			0
$979$	102.479			0.104677
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	534.361i		−	0.543602i	−0.962354	−	0.271801i	$-0.912381\pi$
$983$		−	534.361i		−	0.543602i	0.962354	−	0.271801i	$-0.0876192\pi$
$984$		0			0
$985$	1402.92			1.42428
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	474.965i		−	0.480248i
$990$		0			0
$991$	−225.440			−0.227488			−0.113744	−	0.993510i	$-0.536284\pi$
$991$	−225.440			−0.227488			−0.113744	+	0.993510i	$0.536284\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1663.57			−1.67193
$996$		0			0
$997$	694.382			0.696472			0.348236	−	0.937407i	$-0.386781\pi$
$997$	694.382			0.696472			0.348236	+	0.937407i	$0.386781\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.b.e.2735.16	yes	32
3.2	odd	2	inner	2736.3.b.e.2735.19	yes	32
4.3	odd	2	inner	2736.3.b.e.2735.18	yes	32
12.11	even	2	inner	2736.3.b.e.2735.11	✓	32
19.18	odd	2	inner	2736.3.b.e.2735.12	yes	32
57.56	even	2	inner	2736.3.b.e.2735.17	yes	32
76.75	even	2	inner	2736.3.b.e.2735.20	yes	32
228.227	odd	2	inner	2736.3.b.e.2735.15	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.3.b.e.2735.11	✓	32	12.11	even	2	inner
2736.3.b.e.2735.12	yes	32	19.18	odd	2	inner
2736.3.b.e.2735.15	yes	32	228.227	odd	2	inner
2736.3.b.e.2735.16	yes	32	1.1	even	1	trivial
2736.3.b.e.2735.17	yes	32	57.56	even	2	inner
2736.3.b.e.2735.18	yes	32	4.3	odd	2	inner
2736.3.b.e.2735.19	yes	32	3.2	odd	2	inner
2736.3.b.e.2735.20	yes	32	76.75	even	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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.60057i q^{5} +2.03549i q^{7} +O(q^{10})\)	\(q+5.60057i q^{5} +2.03549i q^{7} -1.92287 q^{11} -5.11778i q^{13} -17.3173i q^{17} +(-18.8026 + 2.73143i) q^{19} +28.7178 q^{23} -6.36637 q^{25} -22.2559 q^{29} +18.4633 q^{31} -11.3999 q^{35} +63.4541i q^{37} -46.0569 q^{41} -16.5390i q^{43} -57.8149 q^{47} +44.8568 q^{49} -84.0450 q^{53} -10.7692i q^{55} -117.362i q^{59} -95.7069 q^{61} +28.6625 q^{65} -84.1311 q^{67} +50.6045i q^{71} +3.29031 q^{73} -3.91399i q^{77} -25.0961 q^{79} +119.982 q^{83} +96.9869 q^{85} -53.2945 q^{89} +10.4172 q^{91} +(-15.2976 - 105.305i) q^{95} -91.9959i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 96 q^{25} + 336 q^{49} - 256 q^{61} - 560 q^{73} - 752 q^{85}+O(q^{100}) \) 32 * q + 96 * q^25 + 336 * q^49 - 256 * q^61 - 560 * q^73 - 752 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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