LMFDB - Embedded newform 2736.3.o.o.721.3

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			721.3
Root			$-5.92486i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.o.721.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.55726 q^{5} +7.34590 q^{7} +O(q^{10})$	$q-3.55726 q^{5} +7.34590 q^{7} -9.93627 q^{11} -7.98112i q^{13} +29.0733 q^{17} +(-7.34590 + 17.5225i) q^{19} -10.5515 q^{23} -12.3459 q^{25} -35.4087i q^{29} +57.2482i q^{31} -26.1313 q^{35} +38.1655i q^{37} -9.95282i q^{41} +26.0377 q^{43} -19.9928 q^{47} +4.96229 q^{49} -60.8645i q^{53} +35.3459 q^{55} -79.3026i q^{59} +24.7295 q^{61} +28.3909i q^{65} +4.86063i q^{67} +62.3321i q^{71} +90.0377 q^{73} -72.9909 q^{77} +46.1466i q^{79} +65.9966 q^{83} -103.421 q^{85} +145.717i q^{89} -58.6286i q^{91} +(26.1313 - 62.3321i) q^{95} +1.74014i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.55726			−0.711452			−0.355726	−	0.934590i	$-0.615766\pi$
$5$	−3.55726			−0.711452			−0.355726	+	0.934590i	$0.615766\pi$
$6$		0			0
$7$	7.34590			1.04941			0.524707	−	0.851283i	$-0.324175\pi$
$7$	7.34590			1.04941			0.524707	+	0.851283i	$0.324175\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.93627			−0.903298			−0.451649	−	0.892196i	$-0.649164\pi$
$11$	−9.93627			−0.903298			−0.451649	+	0.892196i	$0.649164\pi$
$12$		0			0
$13$		−	7.98112i		−	0.613933i	−0.951720	−	0.306966i	$-0.900686\pi$
$13$		−	7.98112i		−	0.613933i	0.951720	−	0.306966i	$-0.0993139\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	29.0733			1.71019			0.855097	−	0.518467i	$-0.173497\pi$
$17$	29.0733			1.71019			0.855097	+	0.518467i	$0.173497\pi$
$18$		0			0
$19$	−7.34590	+	17.5225i	−0.386626	+	0.922236i
$19$	−7.34590	+	17.5225i	−0.386626	+	0.922236i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−10.5515			−0.458761			−0.229381	−	0.973337i	$-0.573670\pi$
$23$	−10.5515			−0.458761			−0.229381	+	0.973337i	$0.573670\pi$
$24$		0			0
$25$	−12.3459			−0.493836
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	35.4087i		−	1.22099i	−0.792021	−	0.610494i	$-0.790971\pi$
$29$		−	35.4087i		−	1.22099i	0.792021	−	0.610494i	$-0.209029\pi$
$30$		0			0
$31$			57.2482i			1.84672i	0.383940	+	0.923358i	$0.374567\pi$
$31$			57.2482i			1.84672i	−0.383940	+	0.923358i	$0.625433\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−26.1313			−0.746608
$36$		0			0
$37$			38.1655i			1.03150i	0.856739	+	0.515750i	$0.172487\pi$
$37$			38.1655i			1.03150i	−0.856739	+	0.515750i	$0.827513\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.95282i		−	0.242752i	−0.992607	−	0.121376i	$-0.961269\pi$
$41$		−	9.95282i		−	0.242752i	0.992607	−	0.121376i	$-0.0387306\pi$
$42$		0			0
$43$	26.0377			0.605528			0.302764	−	0.953066i	$-0.402091\pi$
$43$	26.0377			0.605528			0.302764	+	0.953066i	$0.402091\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−19.9928			−0.425379			−0.212690	−	0.977120i	$-0.568222\pi$
$47$	−19.9928			−0.425379			−0.212690	+	0.977120i	$0.568222\pi$
$48$		0			0
$49$	4.96229			0.101271
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	60.8645i		−	1.14839i	−0.818720	−	0.574194i	$-0.805316\pi$
$53$		−	60.8645i		−	1.14839i	0.818720	−	0.574194i	$-0.194684\pi$
$54$		0			0
$55$	35.3459			0.642653
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	79.3026i		−	1.34411i	−0.740500	−	0.672056i	$-0.765411\pi$
$59$		−	79.3026i		−	1.34411i	0.740500	−	0.672056i	$-0.234589\pi$
$60$		0			0
$61$	24.7295			0.405402			0.202701	−	0.979241i	$-0.435028\pi$
$61$	24.7295			0.405402			0.202701	+	0.979241i	$0.435028\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			28.3909i			0.436784i
$66$		0			0
$67$			4.86063i			0.0725468i	0.999342	+	0.0362734i	$0.0115487\pi$
$67$			4.86063i			0.0725468i	−0.999342	+	0.0362734i	$0.988451\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			62.3321i			0.877916i	0.898508	+	0.438958i	$0.144652\pi$
$71$			62.3321i			0.877916i	−0.898508	+	0.438958i	$0.855348\pi$
$72$		0			0
$73$	90.0377			1.23339			0.616697	−	0.787201i	$-0.288471\pi$
$73$	90.0377			1.23339			0.616697	+	0.787201i	$0.288471\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−72.9909			−0.947934
$78$		0			0
$79$			46.1466i			0.584134i	0.956398	+	0.292067i	$0.0943430\pi$
$79$			46.1466i			0.584134i	−0.956398	+	0.292067i	$0.905657\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	65.9966			0.795140			0.397570	−	0.917572i	$-0.369854\pi$
$83$	65.9966			0.795140			0.397570	+	0.917572i	$0.369854\pi$
$84$		0			0
$85$	−103.421			−1.21672
$86$		0			0
$87$		0			0
$88$		0			0
$89$			145.717i			1.63727i	0.574312	+	0.818637i	$0.305270\pi$
$89$			145.717i			1.63727i	−0.574312	+	0.818637i	$0.694730\pi$
$90$		0			0
$91$		−	58.6286i		−	0.644270i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	26.1313	−	62.3321i	0.275066	−	0.656127i
$96$		0			0
$97$			1.74014i			0.0179396i	0.999960	+	0.00896981i	$0.00285522\pi$
$97$			1.74014i			0.0179396i	−0.999960	+	0.00896981i	$0.997145\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−25.5161			−0.252634			−0.126317	−	0.991990i	$-0.540316\pi$
$101$	−25.5161			−0.252634			−0.126317	+	0.991990i	$0.540316\pi$
$102$		0			0
$103$			200.549i			1.94707i	0.228529	+	0.973537i	$0.426608\pi$
$103$			200.549i			1.94707i	−0.228529	+	0.973537i	$0.573392\pi$
$104$		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
$109$		−	144.681i		−	1.32735i	−0.748023	−	0.663673i	$-0.768997\pi$
$109$		−	144.681i		−	1.32735i	0.748023	−	0.663673i	$-0.231003\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			38.3438i			0.339325i	0.985502	+	0.169663i	$0.0542678\pi$
$113$			38.3438i			0.339325i	−0.985502	+	0.169663i	$0.945732\pi$
$114$		0			0
$115$	37.5344			0.326387
$116$		0			0
$117$		0			0
$118$		0			0
$119$	213.570			1.79470
$120$		0			0
$121$	−22.2705			−0.184054
$122$		0			0
$123$		0			0
$124$		0			0
$125$	132.849			1.06279
$126$		0			0
$127$			27.0639i			0.213101i	0.994307	+	0.106551i	$0.0339806\pi$
$127$			27.0639i			0.213101i	−0.994307	+	0.106551i	$0.966019\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−28.8189			−0.219992			−0.109996	−	0.993932i	$-0.535084\pi$
$131$	−28.8189			−0.219992			−0.109996	+	0.993932i	$0.535084\pi$
$132$		0			0
$133$	−53.9623	+	128.719i	−0.405732	+	0.967808i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−71.2793			−0.520287			−0.260144	−	0.965570i	$-0.583770\pi$
$137$	−71.2793			−0.520287			−0.260144	+	0.965570i	$0.583770\pi$
$138$		0			0
$139$	155.497			1.11868			0.559341	−	0.828938i	$-0.311054\pi$
$139$	155.497			1.11868			0.559341	+	0.828938i	$0.311054\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			79.3026i			0.554564i
$144$		0			0
$145$			125.958i			0.868675i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−200.571			−1.34612			−0.673058	−	0.739590i	$-0.735019\pi$
$149$	−200.571			−1.34612			−0.673058	+	0.739590i	$0.735019\pi$
$150$		0			0
$151$		−	14.2221i		−	0.0941861i	−0.998891	−	0.0470931i	$-0.985004\pi$
$151$		−	14.2221i		−	0.0941861i	0.998891	−	0.0470931i	$-0.0149957\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	203.647i		−	1.31385i
$156$		0			0
$157$	150.843			0.960781			0.480391	−	0.877055i	$-0.340495\pi$
$157$	150.843			0.960781			0.480391	+	0.877055i	$0.340495\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−77.5103			−0.481431
$162$		0			0
$163$	−158.692			−0.973569			−0.486785	−	0.873522i	$-0.661830\pi$
$163$	−158.692			−0.973569			−0.486785	+	0.873522i	$0.661830\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			56.4619i			0.338095i	0.985608	+	0.169048i	$0.0540691\pi$
$167$			56.4619i			0.338095i	−0.985608	+	0.169048i	$0.945931\pi$
$168$		0			0
$169$	105.302			0.623087
$170$		0			0
$171$		0			0
$172$		0			0
$173$			74.9000i			0.432948i	0.976288	+	0.216474i	$0.0694556\pi$
$173$			74.9000i			0.432948i	−0.976288	+	0.216474i	$0.930544\pi$
$174$		0			0
$175$	−90.6918			−0.518239
$176$		0			0
$177$		0			0
$178$		0			0
$179$			130.214i			0.727454i	0.931506	+	0.363727i	$0.118496\pi$
$179$			130.214i			0.727454i	−0.931506	+	0.363727i	$0.881504\pi$
$180$		0			0
$181$			194.308i			1.07352i	0.843734	+	0.536761i	$0.180352\pi$
$181$			194.308i			1.07352i	−0.843734	+	0.536761i	$0.819648\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	135.765i		−	0.733862i
$186$		0			0
$187$	−288.880			−1.54481
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−326.185			−1.70778			−0.853889	−	0.520456i	$-0.825762\pi$
$191$	−326.185			−1.70778			−0.853889	+	0.520456i	$0.825762\pi$
$192$		0			0
$193$			291.102i			1.50830i	0.656703	+	0.754149i	$0.271951\pi$
$193$			291.102i			1.50830i	−0.656703	+	0.754149i	$0.728049\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	289.503			1.46956			0.734778	−	0.678307i	$-0.237286\pi$
$197$	289.503			1.46956			0.734778	+	0.678307i	$0.237286\pi$
$198$		0			0
$199$	366.189			1.84014			0.920072	−	0.391750i	$-0.128130\pi$
$199$	366.189			1.84014			0.920072	+	0.391750i	$0.128130\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	260.109i		−	1.28132i
$204$		0			0
$205$			35.4048i			0.172706i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	72.9909	−	174.108i	0.349239	−	0.833054i
$210$		0			0
$211$			97.1538i			0.460445i	0.973138	+	0.230222i	$0.0739453\pi$
$211$			97.1538i			0.460445i	−0.973138	+	0.230222i	$0.926055\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−92.6229			−0.430804
$216$		0			0
$217$			420.540i			1.93797i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	232.038i		−	1.04994i
$222$		0			0
$223$		−	47.8867i		−	0.214739i	−0.994219	−	0.107369i	$-0.965757\pi$
$223$		−	47.8867i		−	0.214739i	0.994219	−	0.107369i	$-0.0342427\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			407.613i			1.79565i	0.440348	+	0.897827i	$0.354855\pi$
$227$			407.613i			1.79565i	−0.440348	+	0.897827i	$0.645145\pi$
$228$		0			0
$229$	202.654			0.884952			0.442476	−	0.896780i	$-0.354100\pi$
$229$	202.654			0.884952			0.442476	+	0.896780i	$0.354100\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	402.988			1.72956			0.864781	−	0.502149i	$-0.167457\pi$
$233$	402.988			1.72956			0.864781	+	0.502149i	$0.167457\pi$
$234$		0			0
$235$	71.1196			0.302637
$236$		0			0
$237$		0			0
$238$		0			0
$239$	137.517			0.575383			0.287691	−	0.957723i	$-0.407112\pi$
$239$	137.517			0.575383			0.287691	+	0.957723i	$0.407112\pi$
$240$		0			0
$241$			15.9622i			0.0662334i	0.999451	+	0.0331167i	$0.0105433\pi$
$241$			15.9622i			0.0662334i	−0.999451	+	0.0331167i	$0.989457\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−17.6522			−0.0720496
$246$		0			0
$247$	139.849	+	58.6286i	0.566191	+	0.237363i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−226.342			−0.901759			−0.450880	−	0.892585i	$-0.648890\pi$
$251$	−226.342			−0.901759			−0.450880	+	0.892585i	$0.648890\pi$
$252$		0			0
$253$	104.843			0.414398
$254$		0			0
$255$		0			0
$256$		0			0
$257$			428.987i			1.66921i	0.550850	+	0.834604i	$0.314304\pi$
$257$			428.987i			1.66921i	−0.550850	+	0.834604i	$0.685696\pi$
$258$		0			0
$259$			280.360i			1.08247i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−86.4844			−0.328838			−0.164419	−	0.986391i	$-0.552575\pi$
$263$	−86.4844			−0.328838			−0.164419	+	0.986391i	$0.552575\pi$
$264$		0			0
$265$			216.511i			0.817022i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			437.152i			1.62510i	0.582891	+	0.812550i	$0.301921\pi$
$269$			437.152i			1.62510i	−0.582891	+	0.812550i	$0.698079\pi$
$270$		0			0
$271$	152.377			0.562277			0.281139	−	0.959667i	$-0.409288\pi$
$271$	152.377			0.562277			0.281139	+	0.959667i	$0.409288\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	122.672			0.446081
$276$		0			0
$277$	6.50326			0.0234775			0.0117387	−	0.999931i	$-0.496263\pi$
$277$	6.50326			0.0234775			0.0117387	+	0.999931i	$0.496263\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	179.658i		−	0.639354i	−0.947527	−	0.319677i	$-0.896426\pi$
$281$		−	179.658i		−	0.639354i	0.947527	−	0.319677i	$-0.103574\pi$
$282$		0			0
$283$	250.339			0.884591			0.442296	−	0.896869i	$-0.354164\pi$
$283$	250.339			0.884591			0.442296	+	0.896869i	$0.354164\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	73.1125i		−	0.254747i
$288$		0			0
$289$	556.257			1.92477
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	188.464i		−	0.643221i	−0.946872	−	0.321610i	$-0.895776\pi$
$293$		−	188.464i		−	0.643221i	0.946872	−	0.321610i	$-0.104224\pi$
$294$		0			0
$295$			282.100i			0.956271i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			84.2129i			0.281648i
$300$		0			0
$301$	191.270			0.635450
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−87.9693			−0.288424
$306$		0			0
$307$			150.922i			0.491602i	0.969320	+	0.245801i	$0.0790509\pi$
$307$			150.922i			0.491602i	−0.969320	+	0.245801i	$0.920949\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	222.891			0.716691			0.358345	−	0.933589i	$-0.383341\pi$
$311$	222.891			0.716691			0.358345	+	0.933589i	$0.383341\pi$
$312$		0			0
$313$	−257.157			−0.821589			−0.410795	−	0.911728i	$-0.634749\pi$
$313$	−257.157			−0.821589			−0.410795	+	0.911728i	$0.634749\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			337.944i			1.06607i	0.846094	+	0.533034i	$0.178948\pi$
$317$			337.944i			1.06607i	−0.846094	+	0.533034i	$0.821052\pi$
$318$		0			0
$319$			351.830i			1.10292i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−213.570	+	509.437i	−0.661207	+	1.57720i
$324$		0			0
$325$			98.5342i			0.303182i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−146.865			−0.446399
$330$		0			0
$331$		−	23.2238i		−	0.0701625i	−0.999384	−	0.0350812i	$-0.988831\pi$
$331$		−	23.2238i		−	0.0701625i	0.999384	−	0.0350812i	$-0.0111690\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	17.2905i		−	0.0516135i
$336$		0			0
$337$			456.246i			1.35384i	0.736055	+	0.676922i	$0.236687\pi$
$337$			456.246i			1.35384i	−0.736055	+	0.676922i	$0.763313\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	568.834i		−	1.66813i
$342$		0			0
$343$	−323.497			−0.943139
$344$		0			0
$345$		0			0
$346$		0			0
$347$	533.496			1.53745			0.768727	−	0.639577i	$-0.220891\pi$
$347$	533.496			1.53745			0.768727	+	0.639577i	$0.220891\pi$
$348$		0			0
$349$	−492.641			−1.41158			−0.705789	−	0.708422i	$-0.749408\pi$
$349$	−492.641			−1.41158			−0.705789	+	0.708422i	$0.749408\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	298.329			0.845124			0.422562	−	0.906334i	$-0.361131\pi$
$353$	298.329			0.845124			0.422562	+	0.906334i	$0.361131\pi$
$354$		0			0
$355$		−	221.731i		−	0.624595i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−203.980			−0.568191			−0.284095	−	0.958796i	$-0.591693\pi$
$359$	−203.980			−0.568191			−0.284095	+	0.958796i	$0.591693\pi$
$360$		0			0
$361$	−253.075	−	257.437i	−0.701040	−	0.713122i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−320.288			−0.877500
$366$		0			0
$367$	104.377			0.284406			0.142203	−	0.989837i	$-0.454581\pi$
$367$	104.377			0.284406			0.142203	+	0.989837i	$0.454581\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	447.105i		−	1.20513i
$372$		0			0
$373$			279.640i			0.749706i	0.927084	+	0.374853i	$0.122307\pi$
$373$			279.640i			0.749706i	−0.927084	+	0.374853i	$0.877693\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−282.601			−0.749605
$378$		0			0
$379$			461.466i			1.21759i	0.793328	+	0.608794i	$0.208346\pi$
$379$			461.466i			1.21759i	−0.793328	+	0.608794i	$0.791654\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	571.769i		−	1.49287i	−0.665459	−	0.746435i	$-0.731764\pi$
$383$		−	571.769i		−	1.49287i	0.665459	−	0.746435i	$-0.268236\pi$
$384$		0			0
$385$	259.648			0.674409
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−633.382			−1.62823			−0.814115	−	0.580703i	$-0.802778\pi$
$389$	−633.382			−1.62823			−0.814115	+	0.580703i	$0.802778\pi$
$390$		0			0
$391$	−306.767			−0.784571
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	164.155i		−	0.415583i
$396$		0			0
$397$	−251.736			−0.634096			−0.317048	−	0.948410i	$-0.602692\pi$
$397$	−251.736			−0.634096			−0.317048	+	0.948410i	$0.602692\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			335.009i			0.835433i	0.908577	+	0.417716i	$0.137169\pi$
$401$			335.009i			0.835433i	−0.908577	+	0.417716i	$0.862831\pi$
$402$		0			0
$403$	456.905			1.13376
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	379.223i		−	0.931751i
$408$		0			0
$409$		−	5.52139i		−	0.0134997i	−0.999977	−	0.00674986i	$-0.997851\pi$
$409$		−	5.52139i		−	0.0134997i	0.999977	−	0.00674986i	$-0.00214856\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	582.549i		−	1.41053i
$414$		0			0
$415$	−234.767			−0.565704
$416$		0			0
$417$		0			0
$418$		0			0
$419$	205.854			0.491298			0.245649	−	0.969359i	$-0.420999\pi$
$419$	205.854			0.491298			0.245649	+	0.969359i	$0.420999\pi$
$420$		0			0
$421$		−	808.436i		−	1.92027i	−0.279529	−	0.960137i	$-0.590178\pi$
$421$		−	808.436i		−	1.92027i	0.279529	−	0.960137i	$-0.409822\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−358.936			−0.844556
$426$		0			0
$427$	181.661			0.425435
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	520.537i		−	1.20774i	−0.797082	−	0.603872i	$-0.793624\pi$
$431$		−	520.537i		−	1.20774i	0.797082	−	0.603872i	$-0.206376\pi$
$432$		0			0
$433$			606.866i			1.40154i	0.713388	+	0.700769i	$0.247160\pi$
$433$			606.866i			1.40154i	−0.713388	+	0.700769i	$0.752840\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	77.5103	−	184.889i	0.177369	−	0.423086i
$438$		0			0
$439$		−	67.6302i		−	0.154055i	−0.997029	−	0.0770276i	$-0.975457\pi$
$439$		−	67.6302i		−	0.154055i	0.997029	−	0.0770276i	$-0.0245430\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−78.8888			−0.178079			−0.0890393	−	0.996028i	$-0.528380\pi$
$443$	−78.8888			−0.178079			−0.0890393	+	0.996028i	$0.528380\pi$
$444$		0			0
$445$		−	518.354i		−	1.16484i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	360.464i		−	0.802816i	−0.915899	−	0.401408i	$-0.868521\pi$
$449$		−	360.464i		−	0.802816i	0.915899	−	0.401408i	$-0.131479\pi$
$450$		0			0
$451$			98.8940i			0.219277i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			208.557i			0.458367i
$456$		0			0
$457$	186.038			0.407085			0.203542	−	0.979066i	$-0.434755\pi$
$457$	186.038			0.407085			0.203542	+	0.979066i	$0.434755\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	455.944			0.989034			0.494517	−	0.869168i	$-0.335345\pi$
$461$	455.944			0.989034			0.494517	+	0.869168i	$0.335345\pi$
$462$		0			0
$463$	−562.415			−1.21472			−0.607359	−	0.794427i	$-0.707771\pi$
$463$	−562.415			−1.21472			−0.607359	+	0.794427i	$0.707771\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	709.888			1.52010			0.760052	−	0.649862i	$-0.225174\pi$
$467$	709.888			1.52010			0.760052	+	0.649862i	$0.225174\pi$
$468$		0			0
$469$			35.7057i			0.0761316i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−258.718			−0.546972
$474$		0			0
$475$	90.6918	−	216.331i	0.190930	−	0.455434i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	606.247			1.26565			0.632825	−	0.774294i	$-0.281895\pi$
$479$	606.247			1.26565			0.632825	+	0.774294i	$0.281895\pi$
$480$		0			0
$481$	304.603			0.633271
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	6.19014i		−	0.0127632i
$486$		0			0
$487$		−	541.938i		−	1.11281i	−0.830912	−	0.556404i	$-0.812181\pi$
$487$		−	541.938i		−	1.11281i	0.830912	−	0.556404i	$-0.187819\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−833.403			−1.69736			−0.848679	−	0.528909i	$-0.822601\pi$
$491$	−833.403			−1.69736			−0.848679	+	0.528909i	$0.822601\pi$
$492$		0			0
$493$		−	1029.45i		−	2.08813i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			457.885i			0.921298i
$498$		0			0
$499$	563.497			1.12925			0.564626	−	0.825347i	$-0.309020\pi$
$499$	563.497			1.12925			0.564626	+	0.825347i	$0.309020\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−924.716			−1.83840			−0.919201	−	0.393788i	$-0.871164\pi$
$503$	−924.716			−1.83840			−0.919201	+	0.393788i	$0.871164\pi$
$504$		0			0
$505$	90.7672			0.179737
$506$		0			0
$507$		0			0
$508$		0			0
$509$			352.619i			0.692768i	0.938093	+	0.346384i	$0.112591\pi$
$509$			352.619i			0.692768i	−0.938093	+	0.346384i	$0.887409\pi$
$510$		0			0
$511$	661.408			1.29434
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	713.404i		−	1.38525i
$516$		0			0
$517$	198.654			0.384244
$518$		0			0
$519$		0			0
$520$		0			0
$521$			479.578i			0.920496i	0.887790	+	0.460248i	$0.152239\pi$
$521$			479.578i			0.920496i	−0.887790	+	0.460248i	$0.847761\pi$
$522$		0			0
$523$			330.648i			0.632213i	0.948724	+	0.316107i	$0.102376\pi$
$523$			330.648i			0.632213i	−0.948724	+	0.316107i	$0.897624\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1664.40i			3.15824i
$528$		0			0
$529$	−417.666			−0.789538
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−79.4347			−0.149033
$534$		0			0
$535$		−	301.843i		−	0.564193i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−49.3067			−0.0914781
$540$		0			0
$541$	−526.566			−0.973319			−0.486660	−	0.873592i	$-0.661785\pi$
$541$	−526.566			−0.973319			−0.486660	+	0.873592i	$0.661785\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			514.667i			0.944343i
$546$		0			0
$547$			676.237i			1.23626i	0.786074	+	0.618132i	$0.212110\pi$
$547$			676.237i			1.23626i	−0.786074	+	0.618132i	$0.787890\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	620.448	+	260.109i	1.12604	+	0.472066i
$552$		0			0
$553$			338.988i			0.612999i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	496.976			0.892236			0.446118	−	0.894974i	$-0.352806\pi$
$557$	496.976			0.892236			0.446118	+	0.894974i	$0.352806\pi$
$558$		0			0
$559$		−	207.810i		−	0.371753i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			985.573i			1.75057i	0.483605	+	0.875286i	$0.339327\pi$
$563$			985.573i			1.75057i	−0.483605	+	0.875286i	$0.660673\pi$
$564$		0			0
$565$		−	136.399i		−	0.241414i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	855.866i		−	1.50416i	−0.659073	−	0.752079i	$-0.729051\pi$
$569$		−	855.866i		−	1.50416i	0.659073	−	0.752079i	$-0.270949\pi$
$570$		0			0
$571$	−316.075			−0.553547			−0.276774	−	0.960935i	$-0.589265\pi$
$571$	−316.075			−0.553547			−0.276774	+	0.960935i	$0.589265\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	130.268			0.226553
$576$		0			0
$577$	576.730			0.999531			0.499766	−	0.866161i	$-0.333419\pi$
$577$	576.730			0.999531			0.499766	+	0.866161i	$0.333419\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	484.805			0.834432
$582$		0			0
$583$			604.766i			1.03734i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−856.379			−1.45891			−0.729454	−	0.684030i	$-0.760226\pi$
$587$	−856.379			−1.45891			−0.729454	+	0.684030i	$0.760226\pi$
$588$		0			0
$589$	−1003.13	−	420.540i	−1.70311	−	0.713989i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	707.548			1.19317			0.596583	−	0.802551i	$-0.296525\pi$
$593$	707.548			1.19317			0.596583	+	0.802551i	$0.296525\pi$
$594$		0			0
$595$	−759.723			−1.27685
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	582.549i		−	0.972536i	−0.873810	−	0.486268i	$-0.838358\pi$
$599$		−	582.549i		−	0.972536i	0.873810	−	0.486268i	$-0.161642\pi$
$600$		0			0
$601$		−	802.495i		−	1.33527i	−0.744490	−	0.667634i	$-0.767307\pi$
$601$		−	802.495i		−	1.33527i	0.744490	−	0.667634i	$-0.232693\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	79.2219			0.130945
$606$		0			0
$607$		−	785.572i		−	1.29419i	−0.762411	−	0.647094i	$-0.775984\pi$
$607$		−	785.572i		−	1.29419i	0.762411	−	0.647094i	$-0.224016\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			159.565i			0.261154i
$612$		0			0
$613$	57.0312			0.0930362			0.0465181	−	0.998917i	$-0.485187\pi$
$613$	57.0312			0.0930362			0.0465181	+	0.998917i	$0.485187\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	744.272			1.20628			0.603138	−	0.797637i	$-0.293917\pi$
$617$	744.272			1.20628			0.603138	+	0.797637i	$0.293917\pi$
$618$		0			0
$619$	−79.2952			−0.128102			−0.0640510	−	0.997947i	$-0.520402\pi$
$619$	−79.2952			−0.128102			−0.0640510	+	0.997947i	$0.520402\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1070.43i			1.71818i
$624$		0			0
$625$	−163.931			−0.262290
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1109.60i			1.76406i
$630$		0			0
$631$	616.553			0.977104			0.488552	−	0.872535i	$-0.337525\pi$
$631$	616.553			0.977104			0.488552	+	0.872535i	$0.337525\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	96.2732i		−	0.151611i
$636$		0			0
$637$		−	39.6047i		−	0.0621737i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1161.34i		−	1.81176i	−0.423538	−	0.905878i	$-0.639212\pi$
$641$		−	1161.34i		−	1.81176i	0.423538	−	0.905878i	$-0.360788\pi$
$642$		0			0
$643$	−262.867			−0.408814			−0.204407	−	0.978886i	$-0.565527\pi$
$643$	−262.867			−0.408814			−0.204407	+	0.978886i	$0.565527\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−202.750			−0.313369			−0.156685	−	0.987649i	$-0.550081\pi$
$647$	−202.750			−0.313369			−0.156685	+	0.987649i	$0.550081\pi$
$648$		0			0
$649$			787.972i			1.21413i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	258.505			0.395873			0.197936	−	0.980215i	$-0.436576\pi$
$653$	258.505			0.395873			0.197936	+	0.980215i	$0.436576\pi$
$654$		0			0
$655$	102.516			0.156513
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	758.445i		−	1.15090i	−0.817836	−	0.575452i	$-0.804826\pi$
$659$		−	758.445i		−	1.15090i	0.817836	−	0.575452i	$-0.195174\pi$
$660$		0			0
$661$		−	589.884i		−	0.892411i	−0.894931	−	0.446205i	$-0.852775\pi$
$661$		−	589.884i		−	0.892411i	0.894931	−	0.446205i	$-0.147225\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	191.958	−	457.885i	0.288658	−	0.688549i
$666$		0			0
$667$			373.615i			0.560142i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−245.719			−0.366199
$672$		0			0
$673$			576.983i			0.857330i	0.903464	+	0.428665i	$0.141016\pi$
$673$			576.983i			0.857330i	−0.903464	+	0.428665i	$0.858984\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			375.460i			0.554594i	0.960784	+	0.277297i	$0.0894385\pi$
$677$			375.460i			0.554594i	−0.960784	+	0.277297i	$0.910561\pi$
$678$		0			0
$679$			12.7829i			0.0188261i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			10.7804i			0.0157839i	0.999969	+	0.00789196i	$0.00251212\pi$
$683$			10.7804i			0.0157839i	−0.999969	+	0.00789196i	$0.997488\pi$
$684$		0			0
$685$	253.559			0.370159
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−485.767			−0.705032
$690$		0			0
$691$	−750.264			−1.08577			−0.542883	−	0.839809i	$-0.682667\pi$
$691$	−750.264			−1.08577			−0.542883	+	0.839809i	$0.682667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−553.142			−0.795888
$696$		0			0
$697$		−	289.362i		−	0.415153i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−938.154			−1.33831			−0.669154	−	0.743123i	$-0.733344\pi$
$701$	−938.154			−1.33831			−0.669154	+	0.743123i	$0.733344\pi$
$702$		0			0
$703$	−668.754	−	280.360i	−0.951286	−	0.398805i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−187.438			−0.265118
$708$		0			0
$709$	579.623			0.817522			0.408761	−	0.912642i	$-0.365961\pi$
$709$	579.623			0.817522			0.408761	+	0.912642i	$0.365961\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	604.055i		−	0.847202i
$714$		0			0
$715$		−	282.100i		−	0.394545i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1057.32			1.47055			0.735275	−	0.677769i	$-0.237053\pi$
$719$	1057.32			1.47055			0.735275	+	0.677769i	$0.237053\pi$
$720$		0			0
$721$			1473.21i			2.04329i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			437.152i			0.602968i
$726$		0			0
$727$	486.189			0.668760			0.334380	−	0.942438i	$-0.391473\pi$
$727$	486.189			0.668760			0.334380	+	0.942438i	$0.391473\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	757.002			1.03557
$732$		0			0
$733$	−892.679			−1.21784			−0.608921	−	0.793231i	$-0.708398\pi$
$733$	−892.679			−1.21784			−0.608921	+	0.793231i	$0.708398\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	48.2966i		−	0.0655313i
$738$		0			0
$739$	395.497			0.535178			0.267589	−	0.963533i	$-0.413773\pi$
$739$	395.497			0.535178			0.267589	+	0.963533i	$0.413773\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	491.826i		−	0.661947i	−0.943640	−	0.330973i	$-0.892623\pi$
$743$		−	491.826i		−	0.661947i	0.943640	−	0.330973i	$-0.107377\pi$
$744$		0			0
$745$	713.484			0.957696
$746$		0			0
$747$		0			0
$748$		0			0
$749$			623.321i			0.832204i
$750$		0			0
$751$		−	970.819i		−	1.29270i	−0.763041	−	0.646351i	$-0.776294\pi$
$751$		−	970.819i		−	1.29270i	0.763041	−	0.646351i	$-0.223706\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			50.5917i			0.0670089i
$756$		0			0
$757$	−902.251			−1.19188			−0.595939	−	0.803030i	$-0.703220\pi$
$757$	−902.251			−1.19188			−0.595939	+	0.803030i	$0.703220\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	105.621			0.138793			0.0693965	−	0.997589i	$-0.477893\pi$
$761$	105.621			0.138793			0.0693965	+	0.997589i	$0.477893\pi$
$762$		0			0
$763$		−	1062.81i		−	1.39294i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−632.924			−0.825194
$768$		0			0
$769$	1137.94			1.47976			0.739880	−	0.672738i	$-0.234882\pi$
$769$	1137.94			1.47976			0.739880	+	0.672738i	$0.234882\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			609.473i			0.788451i	0.919014	+	0.394226i	$0.128987\pi$
$773$			609.473i			0.788451i	−0.919014	+	0.394226i	$0.871013\pi$
$774$		0			0
$775$		−	706.781i		−	0.911975i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	174.398	+	73.1125i	0.223875	+	0.0938543i
$780$		0			0
$781$		−	619.348i		−	0.793020i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−536.586			−0.683550
$786$		0			0
$787$		−	408.418i		−	0.518955i	−0.965749	−	0.259478i	$-0.916450\pi$
$787$		−	408.418i		−	0.518955i	0.965749	−	0.259478i	$-0.0835503\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			281.669i			0.356093i
$792$		0			0
$793$		−	197.369i		−	0.248889i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	238.415i		−	0.299141i	−0.988751	−	0.149571i	$-0.952211\pi$
$797$		−	238.415i		−	0.299141i	0.988751	−	0.149571i	$-0.0477891\pi$
$798$		0			0
$799$	−581.257			−0.727481
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−894.639			−1.11412
$804$		0			0
$805$	275.724			0.342515
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1142.74			−1.41253			−0.706267	−	0.707945i	$-0.749622\pi$
$809$	−1142.74			−1.41253			−0.706267	+	0.707945i	$0.749622\pi$
$810$		0			0
$811$		−	1173.11i		−	1.44649i	−0.690589	−	0.723247i	$-0.742649\pi$
$811$		−	1173.11i		−	1.44649i	0.690589	−	0.723247i	$-0.257351\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	564.508			0.692648
$816$		0			0
$817$	−191.270	+	456.246i	−0.234113	+	0.558440i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1018.05			1.24001			0.620004	−	0.784598i	$-0.287131\pi$
$821$	1018.05			1.24001			0.620004	+	0.784598i	$0.287131\pi$
$822$		0			0
$823$	−504.956			−0.613555			−0.306778	−	0.951781i	$-0.599251\pi$
$823$	−504.956			−0.613555			−0.306778	+	0.951781i	$0.599251\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			152.415i			0.184299i	0.995745	+	0.0921494i	$0.0293737\pi$
$827$			152.415i			0.184299i	−0.995745	+	0.0921494i	$0.970626\pi$
$828$		0			0
$829$			123.498i			0.148972i	0.997222	+	0.0744862i	$0.0237317\pi$
$829$			123.498i			0.148972i	−0.997222	+	0.0744862i	$0.976268\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	144.270			0.173194
$834$		0			0
$835$		−	200.850i		−	0.240538i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1091.99i		−	1.30153i	−0.759278	−	0.650766i	$-0.774448\pi$
$839$		−	1091.99i		−	1.30153i	0.759278	−	0.650766i	$-0.225552\pi$
$840$		0			0
$841$	−412.774			−0.490813
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−374.585			−0.443296
$846$		0			0
$847$	−163.597			−0.193149
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	402.703i		−	0.473212i
$852$		0			0
$853$	−458.151			−0.537105			−0.268553	−	0.963265i	$-0.586545\pi$
$853$	−458.151			−0.537105			−0.268553	+	0.963265i	$0.586545\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			244.926i			0.285794i	0.989738	+	0.142897i	$0.0456418\pi$
$857$			244.926i			0.285794i	−0.989738	+	0.142897i	$0.954358\pi$
$858$		0			0
$859$	−1037.11			−1.20734			−0.603671	−	0.797234i	$-0.706296\pi$
$859$	−1037.11			−1.20734			−0.603671	+	0.797234i	$0.706296\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	317.210i		−	0.367567i	−0.982967	−	0.183784i	$-0.941165\pi$
$863$		−	317.210i		−	0.367567i	0.982967	−	0.183784i	$-0.0588346\pi$
$864$		0			0
$865$		−	266.439i		−	0.308022i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	458.525i		−	0.527647i
$870$		0			0
$871$	38.7933			0.0445388
$872$		0			0
$873$		0			0
$874$		0			0
$875$	975.896			1.11531
$876$		0			0
$877$		−	1385.12i		−	1.57938i	−0.613505	−	0.789691i	$-0.710241\pi$
$877$		−	1385.12i		−	1.57938i	0.613505	−	0.789691i	$-0.289759\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−660.073			−0.749231			−0.374616	−	0.927180i	$-0.622225\pi$
$881$	−660.073			−0.749231			−0.374616	+	0.927180i	$0.622225\pi$
$882$		0			0
$883$	779.497			0.882782			0.441391	−	0.897315i	$-0.354485\pi$
$883$	779.497			0.882782			0.441391	+	0.897315i	$0.354485\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	769.226i		−	0.867222i	−0.901100	−	0.433611i	$-0.857239\pi$
$887$		−	769.226i		−	0.867222i	0.901100	−	0.433611i	$-0.142761\pi$
$888$		0			0
$889$			198.808i			0.223632i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	146.865	−	350.324i	0.164463	−	0.392300i
$894$		0			0
$895$		−	463.206i		−	0.517549i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2027.08			2.25482
$900$		0			0
$901$		−	1769.53i		−	1.96397i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	691.203i		−	0.763760i
$906$		0			0
$907$		−	21.1238i		−	0.0232898i	−0.999932	−	0.0116449i	$-0.996293\pi$
$907$		−	21.1238i		−	0.0232898i	0.999932	−	0.0116449i	$-0.00370677\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1421.58i			1.56046i	0.625494	+	0.780229i	$0.284897\pi$
$911$			1421.58i			1.56046i	−0.625494	+	0.780229i	$0.715103\pi$
$912$		0			0
$913$	−655.761			−0.718248
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−211.701			−0.230862
$918$		0			0
$919$	−1331.45			−1.44880			−0.724399	−	0.689381i	$-0.757883\pi$
$919$	−1331.45			−1.44880			−0.724399	+	0.689381i	$0.757883\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	497.480			0.538981
$924$		0			0
$925$		−	471.187i		−	0.509392i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1497.58			−1.61204			−0.806019	−	0.591890i	$-0.798382\pi$
$929$	−1497.58			−1.61204			−0.806019	+	0.591890i	$0.798382\pi$
$930$		0			0
$931$	−36.4525	+	86.9517i	−0.0391541	+	0.0933960i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1027.62			1.09906
$936$		0			0
$937$	1412.57			1.50754			0.753770	−	0.657138i	$-0.228233\pi$
$937$	1412.57			1.50754			0.753770	+	0.657138i	$0.228233\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1352.55i			1.43735i	0.695346	+	0.718675i	$0.255251\pi$
$941$			1352.55i			1.43735i	−0.695346	+	0.718675i	$0.744749\pi$
$942$		0			0
$943$			105.017i			0.111365i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1123.47			−1.18635			−0.593173	−	0.805075i	$-0.702125\pi$
$947$	−1123.47			−1.18635			−0.593173	+	0.805075i	$0.702125\pi$
$948$		0			0
$949$		−	718.602i		−	0.757220i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1241.60i			1.30283i	0.758721	+	0.651416i	$0.225825\pi$
$953$			1241.60i			1.30283i	−0.758721	+	0.651416i	$0.774175\pi$
$954$		0			0
$955$	1160.33			1.21500
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−523.611			−0.545997
$960$		0			0
$961$	−2316.36			−2.41036
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1035.52i		−	1.07308i
$966$		0			0
$967$	206.364			0.213406			0.106703	−	0.994291i	$-0.465971\pi$
$967$	206.364			0.213406			0.106703	+	0.994291i	$0.465971\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	129.894i		−	0.133774i	−0.997761	−	0.0668869i	$-0.978693\pi$
$971$		−	129.894i		−	0.133774i	0.997761	−	0.0668869i	$-0.0213067\pi$
$972$		0			0
$973$	1142.26			1.17396
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1892.67i		−	1.93723i	−0.248573	−	0.968613i	$-0.579962\pi$
$977$		−	1892.67i		−	1.93723i	0.248573	−	0.968613i	$-0.420038\pi$
$978$		0			0
$979$		−	1447.89i		−	1.47894i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	748.625i		−	0.761571i	−0.924663	−	0.380786i	$-0.875654\pi$
$983$		−	748.625i		−	0.761571i	0.924663	−	0.380786i	$-0.124346\pi$
$984$		0			0
$985$	−1029.84			−1.04552
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−274.737			−0.277793
$990$		0			0
$991$			937.213i			0.945724i	0.881136	+	0.472862i	$0.156779\pi$
$991$			937.213i			0.945724i	−0.881136	+	0.472862i	$0.843221\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1302.63			−1.30917
$996$		0			0
$997$	−1209.38			−1.21302			−0.606511	−	0.795075i	$-0.707431\pi$
$997$	−1209.38			−1.21302			−0.606511	+	0.795075i	$0.707431\pi$
$998$		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.721.3		8
3.2	odd	2	inner	2736.3.o.o.721.5		8
4.3	odd	2		342.3.d.c.37.6	yes	8
12.11	even	2		342.3.d.c.37.3	yes	8
19.18	odd	2	inner	2736.3.o.o.721.4		8
57.56	even	2	inner	2736.3.o.o.721.6		8
76.75	even	2		342.3.d.c.37.2	✓	8
228.227	odd	2		342.3.d.c.37.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
342.3.d.c.37.2	✓	8	76.75	even	2
342.3.d.c.37.3	yes	8	12.11	even	2
342.3.d.c.37.6	yes	8	4.3	odd	2
342.3.d.c.37.7	yes	8	228.227	odd	2
2736.3.o.o.721.3		8	1.1	even	1	trivial
2736.3.o.o.721.4		8	19.18	odd	2	inner
2736.3.o.o.721.5		8	3.2	odd	2	inner
2736.3.o.o.721.6		8	57.56	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.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.55726 q^{5} +7.34590 q^{7} +O(q^{10})\)	\(q-3.55726 q^{5} +7.34590 q^{7} -9.93627 q^{11} -7.98112i q^{13} +29.0733 q^{17} +(-7.34590 + 17.5225i) q^{19} -10.5515 q^{23} -12.3459 q^{25} -35.4087i q^{29} +57.2482i q^{31} -26.1313 q^{35} +38.1655i q^{37} -9.95282i q^{41} +26.0377 q^{43} -19.9928 q^{47} +4.96229 q^{49} -60.8645i q^{53} +35.3459 q^{55} -79.3026i q^{59} +24.7295 q^{61} +28.3909i q^{65} +4.86063i q^{67} +62.3321i q^{71} +90.0377 q^{73} -72.9909 q^{77} +46.1466i q^{79} +65.9966 q^{83} -103.421 q^{85} +145.717i q^{89} -58.6286i q^{91} +(26.1313 - 62.3321i) q^{95} +1.74014i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{7}+O(q^{10}) \) 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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