LMFDB - Embedded newform 1764.2.f.a.881.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,2,Mod(881,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.881"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.4
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.881
Dual form			1764.2.f.a.881.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{5} +4.24264i q^{11} +1.73205i q^{13} -4.89898 q^{17} +5.19615i q^{19} +1.00000 q^{25} +8.48528i q^{29} +1.73205i q^{31} -5.00000 q^{37} +12.2474 q^{41} -11.0000 q^{43} +2.44949 q^{47} -8.48528i q^{53} +10.3923i q^{55} +4.89898 q^{59} -3.46410i q^{61} +4.24264i q^{65} -7.00000 q^{67} +12.7279i q^{71} -15.5885i q^{73} +11.0000 q^{79} +12.2474 q^{83} -12.0000 q^{85} +14.6969 q^{89} +12.7279i q^{95} +3.46410i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{25} - 20 q^{37} - 44 q^{43} - 28 q^{67} + 44 q^{79} - 48 q^{85}+O(q^{100}) $ 4 * q + 4 * q^25 - 20 * q^37 - 44 * q^43 - 28 * q^67 + 44 * q^79 - 48 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.24264i	1.27920i	0.768706	+	0.639602i	$0.220901\pi$
$11$	4.24264i	1.27920i	−0.768706	+	0.639602i	$0.779099\pi$
$12$	0	0
$13$	1.73205i	0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$13$	1.73205i	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.89898	−1.18818	−0.594089	−	0.804400i	$-0.702487\pi$
$17$	−4.89898	−1.18818	−0.594089	+	0.804400i	$0.702487\pi$
$18$	0	0
$19$	5.19615i	1.19208i	0.802955	+	0.596040i	$0.203260\pi$
$19$	5.19615i	1.19208i	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.48528i	1.57568i	0.615882	+	0.787839i	$0.288800\pi$
$29$	8.48528i	1.57568i	−0.615882	+	0.787839i	$0.711200\pi$
$30$	0	0
$31$	1.73205i	0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	1.73205i	0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	12.2474	1.91273	0.956365	−	0.292174i	$-0.0943788\pi$
$41$	12.2474	1.91273	0.956365	+	0.292174i	$0.0943788\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.44949	0.357295	0.178647	−	0.983913i	$-0.442828\pi$
$47$	2.44949	0.357295	0.178647	+	0.983913i	$0.442828\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.48528i	− 1.16554i	−0.812636	−	0.582772i	$-0.801968\pi$
$53$	− 8.48528i	− 1.16554i	0.812636	−	0.582772i	$-0.198032\pi$
$54$	0	0
$55$	10.3923i	1.40130i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.89898	0.637793	0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898	0.637793	0.318896	+	0.947790i	$0.396688\pi$
$60$	0	0
$61$	− 3.46410i	− 0.443533i	−0.975100	−	0.221766i	$-0.928818\pi$
$61$	− 3.46410i	− 0.443533i	0.975100	−	0.221766i	$-0.0711822\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.24264i	0.526235i
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.7279i	1.51053i	0.655422	+	0.755263i	$0.272491\pi$
$71$	12.7279i	1.51053i	−0.655422	+	0.755263i	$0.727509\pi$
$72$	0	0
$73$	− 15.5885i	− 1.82449i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	− 15.5885i	− 1.82449i	0.409644	−	0.912245i	$-0.365653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.2474	1.34433	0.672166	−	0.740400i	$-0.265364\pi$
$83$	12.2474	1.34433	0.672166	+	0.740400i	$0.265364\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.6969	1.55787	0.778936	−	0.627103i	$-0.215760\pi$
$89$	14.6969	1.55787	0.778936	+	0.627103i	$0.215760\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.7279i	1.30586i
$96$	0	0
$97$	3.46410i	0.351726i	0.984415	+	0.175863i	$0.0562716\pi$
$97$	3.46410i	0.351726i	−0.984415	+	0.175863i	$0.943728\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.34847	−0.731200	−0.365600	−	0.930772i	$-0.619136\pi$
$101$	−7.34847	−0.731200	−0.365600	+	0.930772i	$0.619136\pi$
$102$	0	0
$103$	8.66025i	0.853320i	0.904412	+	0.426660i	$0.140310\pi$
$103$	8.66025i	0.853320i	−0.904412	+	0.426660i	$0.859690\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.48528i	0.820303i	0.912017	+	0.410152i	$0.134524\pi$
$107$	8.48528i	0.820303i	−0.912017	+	0.410152i	$0.865476\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4.24264i	− 0.399114i	−0.979886	−	0.199557i	$-0.936050\pi$
$113$	− 4.24264i	− 0.399114i	0.979886	−	0.199557i	$-0.0639503\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.79796	−0.876356
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.34847	0.642039	0.321019	−	0.947073i	$-0.395975\pi$
$131$	7.34847	0.642039	0.321019	+	0.947073i	$0.395975\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 1.73205i	− 0.146911i	−0.997299	−	0.0734553i	$-0.976597\pi$
$139$	− 1.73205i	− 0.146911i	0.997299	−	0.0734553i	$-0.0234026\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.34847	−0.614510
$144$	0	0
$145$	20.7846i	1.72607i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 16.9706i	− 1.39028i	−0.718873	−	0.695141i	$-0.755342\pi$
$149$	− 16.9706i	− 1.39028i	0.718873	−	0.695141i	$-0.244658\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.24264i	0.340777i
$156$	0	0
$157$	10.3923i	0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$	10.3923i	0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.1464	1.32683	0.663415	−	0.748251i	$-0.269106\pi$
$167$	17.1464	1.32683	0.663415	+	0.748251i	$0.269106\pi$
$168$	0	0
$169$	10.0000	0.769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.7279i	0.951330i	0.879627	+	0.475665i	$0.157792\pi$
$179$	12.7279i	0.951330i	−0.879627	+	0.475665i	$0.842208\pi$
$180$	0	0
$181$	− 5.19615i	− 0.386227i	−0.981176	−	0.193113i	$-0.938141\pi$
$181$	− 5.19615i	− 0.386227i	0.981176	−	0.193113i	$-0.0618586\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.2474	−0.900450
$186$	0	0
$187$	− 20.7846i	− 1.51992i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.24264i	0.306987i	0.988150	+	0.153493i	$0.0490524\pi$
$191$	4.24264i	0.306987i	−0.988150	+	0.153493i	$0.950948\pi$
$192$	0	0
$193$	19.0000	1.36765	0.683825	−	0.729646i	$-0.260315\pi$
$193$	19.0000	1.36765	0.683825	+	0.729646i	$0.260315\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.48528i	0.604551i	0.953221	+	0.302276i	$0.0977463\pi$
$197$	8.48528i	0.604551i	−0.953221	+	0.302276i	$0.902254\pi$
$198$	0	0
$199$	13.8564i	0.982255i	0.871088	+	0.491127i	$0.163415\pi$
$199$	13.8564i	0.982255i	−0.871088	+	0.491127i	$0.836585\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	30.0000	2.09529
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−22.0454	−1.52491
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−26.9444	−1.83759
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 8.48528i	− 0.570782i
$222$	0	0
$223$	20.7846i	1.39184i	0.718119	+	0.695920i	$0.245003\pi$
$223$	20.7846i	1.39184i	−0.718119	+	0.695920i	$0.754997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.0454	1.46321	0.731603	−	0.681731i	$-0.238773\pi$
$227$	22.0454	1.46321	0.731603	+	0.681731i	$0.238773\pi$
$228$	0	0
$229$	1.73205i	0.114457i	0.998361	+	0.0572286i	$0.0182264\pi$
$229$	1.73205i	0.114457i	−0.998361	+	0.0572286i	$0.981774\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.24264i	0.277945i	0.990296	+	0.138972i	$0.0443799\pi$
$233$	4.24264i	0.277945i	−0.990296	+	0.138972i	$0.955620\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 4.24264i	− 0.274434i	−0.990541	−	0.137217i	$-0.956184\pi$
$239$	− 4.24264i	− 0.274434i	0.990541	−	0.137217i	$-0.0438157\pi$
$240$	0	0
$241$	− 13.8564i	− 0.892570i	−0.894891	−	0.446285i	$-0.852747\pi$
$241$	− 13.8564i	− 0.892570i	0.894891	−	0.446285i	$-0.147253\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.00000	−0.572656
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.79796	0.618442	0.309221	−	0.950990i	$-0.399932\pi$
$251$	9.79796	0.618442	0.309221	+	0.950990i	$0.399932\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.34847	−0.458385	−0.229192	−	0.973381i	$-0.573609\pi$
$257$	−7.34847	−0.458385	−0.229192	+	0.973381i	$0.573609\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 25.4558i	− 1.56967i	−0.619702	−	0.784837i	$-0.712746\pi$
$263$	− 25.4558i	− 1.56967i	0.619702	−	0.784837i	$-0.287254\pi$
$264$	0	0
$265$	− 20.7846i	− 1.27679i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.2474	0.746740	0.373370	−	0.927682i	$-0.378202\pi$
$269$	12.2474	0.746740	0.373370	+	0.927682i	$0.378202\pi$
$270$	0	0
$271$	− 27.7128i	− 1.68343i	−0.539919	−	0.841717i	$-0.681545\pi$
$271$	− 27.7128i	− 1.68343i	0.539919	−	0.841717i	$-0.318455\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24264i	0.255841i
$276$	0	0
$277$	11.0000	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 16.9706i	− 1.01238i	−0.862422	−	0.506189i	$-0.831054\pi$
$281$	− 16.9706i	− 1.01238i	0.862422	−	0.506189i	$-0.168946\pi$
$282$	0	0
$283$	8.66025i	0.514799i	0.966305	+	0.257399i	$0.0828656\pi$
$283$	8.66025i	0.514799i	−0.966305	+	0.257399i	$0.917134\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.4949	−1.43101	−0.715504	−	0.698609i	$-0.753803\pi$
$293$	−24.4949	−1.43101	−0.715504	+	0.698609i	$0.753803\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.48528i	− 0.485866i
$306$	0	0
$307$	− 19.0526i	− 1.08739i	−0.839284	−	0.543693i	$-0.817025\pi$
$307$	− 19.0526i	− 1.08739i	0.839284	−	0.543693i	$-0.182975\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.1464	0.972285	0.486142	−	0.873880i	$-0.338404\pi$
$311$	17.1464	0.972285	0.486142	+	0.873880i	$0.338404\pi$
$312$	0	0
$313$	− 15.5885i	− 0.881112i	−0.897725	−	0.440556i	$-0.854781\pi$
$313$	− 15.5885i	− 0.881112i	0.897725	−	0.440556i	$-0.145219\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.48528i	− 0.476581i	−0.971194	−	0.238290i	$-0.923413\pi$
$317$	− 8.48528i	− 0.476581i	0.971194	−	0.238290i	$-0.0765870\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 25.4558i	− 1.41640i
$324$	0	0
$325$	1.73205i	0.0960769i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5.00000	−0.274825	−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000	−0.274825	−0.137412	+	0.990514i	$0.543879\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.1464	−0.936809
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.34847	−0.397942
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.48528i	0.455514i	0.973718	+	0.227757i	$0.0731391\pi$
$347$	8.48528i	0.455514i	−0.973718	+	0.227757i	$0.926861\pi$
$348$	0	0
$349$	− 24.2487i	− 1.29800i	−0.760787	−	0.649002i	$-0.775187\pi$
$349$	− 24.2487i	− 1.29800i	0.760787	−	0.649002i	$-0.224813\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−26.9444	−1.43411	−0.717053	−	0.697019i	$-0.754509\pi$
$353$	−26.9444	−1.43411	−0.717053	+	0.697019i	$0.754509\pi$
$354$	0	0
$355$	31.1769i	1.65470i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 16.9706i	− 0.895672i	−0.894116	−	0.447836i	$-0.852195\pi$
$359$	− 16.9706i	− 0.895672i	0.894116	−	0.447836i	$-0.147805\pi$
$360$	0	0
$361$	−8.00000	−0.421053
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 38.1838i	− 1.99863i
$366$	0	0
$367$	− 25.9808i	− 1.35618i	−0.734977	−	0.678092i	$-0.762807\pi$
$367$	− 25.9808i	− 1.35618i	0.734977	−	0.678092i	$-0.237193\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.6969	−0.756931
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.79796	−0.500652	−0.250326	−	0.968162i	$-0.580538\pi$
$383$	−9.79796	−0.500652	−0.250326	+	0.968162i	$0.580538\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.24264i	0.215110i	0.994199	+	0.107555i	$0.0343022\pi$
$389$	4.24264i	0.215110i	−0.994199	+	0.107555i	$0.965698\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	26.9444	1.35572
$396$	0	0
$397$	8.66025i	0.434646i	0.976100	+	0.217323i	$0.0697324\pi$
$397$	8.66025i	0.434646i	−0.976100	+	0.217323i	$0.930268\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.48528i	0.423735i	0.977298	+	0.211867i	$0.0679545\pi$
$401$	8.48528i	0.423735i	−0.977298	+	0.211867i	$0.932046\pi$
$402$	0	0
$403$	−3.00000	−0.149441
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 21.2132i	− 1.05150i
$408$	0	0
$409$	8.66025i	0.428222i	0.976809	+	0.214111i	$0.0686854\pi$
$409$	8.66025i	0.428222i	−0.976809	+	0.214111i	$0.931315\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	30.0000	1.47264
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.9444	−1.31632	−0.658160	−	0.752878i	$-0.728665\pi$
$419$	−26.9444	−1.31632	−0.658160	+	0.752878i	$0.728665\pi$
$420$	0	0
$421$	35.0000	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.89898	−0.237635
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.2132i	1.02180i	0.859639	+	0.510902i	$0.170689\pi$
$431$	21.2132i	1.02180i	−0.859639	+	0.510902i	$0.829311\pi$
$432$	0	0
$433$	− 12.1244i	− 0.582659i	−0.956623	−	0.291330i	$-0.905902\pi$
$433$	− 12.1244i	− 0.582659i	0.956623	−	0.291330i	$-0.0940977\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 41.5692i	− 1.98399i	−0.126275	−	0.991995i	$-0.540302\pi$
$439$	− 41.5692i	− 1.98399i	0.126275	−	0.991995i	$-0.459698\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	36.0000	1.70656
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 12.7279i	− 0.600668i	−0.953834	−	0.300334i	$-0.902902\pi$
$449$	− 12.7279i	− 0.600668i	0.953834	−	0.300334i	$-0.0970981\pi$
$450$	0	0
$451$	51.9615i	2.44677i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.89898	0.228168	0.114084	−	0.993471i	$-0.463607\pi$
$461$	4.89898	0.228168	0.114084	+	0.993471i	$0.463607\pi$
$462$	0	0
$463$	29.0000	1.34774	0.673872	−	0.738848i	$-0.264630\pi$
$463$	29.0000	1.34774	0.673872	+	0.738848i	$0.264630\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.34847	0.340047	0.170023	−	0.985440i	$-0.445616\pi$
$467$	7.34847	0.340047	0.170023	+	0.985440i	$0.445616\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 46.6690i	− 2.14585i
$474$	0	0
$475$	5.19615i	0.238416i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.89898	0.223840	0.111920	−	0.993717i	$-0.464300\pi$
$479$	4.89898	0.223840	0.111920	+	0.993717i	$0.464300\pi$
$480$	0	0
$481$	− 8.66025i	− 0.394874i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.48528i	0.385297i
$486$	0	0
$487$	7.00000	0.317200	0.158600	−	0.987343i	$-0.449302\pi$
$487$	7.00000	0.317200	0.158600	+	0.987343i	$0.449302\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	33.9411i	1.53174i	0.642995	+	0.765871i	$0.277692\pi$
$491$	33.9411i	1.53174i	−0.642995	+	0.765871i	$0.722308\pi$
$492$	0	0
$493$	− 41.5692i	− 1.87218i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−35.0000	−1.56682	−0.783408	−	0.621508i	$-0.786520\pi$
$499$	−35.0000	−1.56682	−0.783408	+	0.621508i	$0.786520\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.44949	−0.109217	−0.0546087	−	0.998508i	$-0.517391\pi$
$503$	−2.44949	−0.109217	−0.0546087	+	0.998508i	$0.517391\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17.1464	−0.760002	−0.380001	−	0.924986i	$-0.624076\pi$
$509$	−17.1464	−0.760002	−0.380001	+	0.924986i	$0.624076\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.2132i	0.934765i
$516$	0	0
$517$	10.3923i	0.457053i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.89898	0.214628	0.107314	−	0.994225i	$-0.465775\pi$
$521$	4.89898	0.214628	0.107314	+	0.994225i	$0.465775\pi$
$522$	0	0
$523$	− 8.66025i	− 0.378686i	−0.981911	−	0.189343i	$-0.939364\pi$
$523$	− 8.66025i	− 0.378686i	0.981911	−	0.189343i	$-0.0606359\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.48528i	− 0.369625i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	21.2132i	0.918846i
$534$	0	0
$535$	20.7846i	0.898597i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.2474	−0.524623
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−44.0908	−1.87833
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.2132i	− 0.898832i	−0.893323	−	0.449416i	$-0.851632\pi$
$557$	− 21.2132i	− 0.898832i	0.893323	−	0.449416i	$-0.148368\pi$
$558$	0	0
$559$	− 19.0526i	− 0.805837i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.34847	−0.309701	−0.154851	−	0.987938i	$-0.549490\pi$
$563$	−7.34847	−0.309701	−0.154851	+	0.987938i	$0.549490\pi$
$564$	0	0
$565$	− 10.3923i	− 0.437208i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 21.2132i	− 0.889304i	−0.895703	−	0.444652i	$-0.853327\pi$
$569$	− 21.2132i	− 0.889304i	0.895703	−	0.444652i	$-0.146673\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.8372i	1.65844i	0.558920	+	0.829222i	$0.311216\pi$
$577$	39.8372i	1.65844i	−0.558920	+	0.829222i	$0.688784\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.6969	0.606608	0.303304	−	0.952894i	$-0.401910\pi$
$587$	14.6969	0.606608	0.303304	+	0.952894i	$0.401910\pi$
$588$	0	0
$589$	−9.00000	−0.370839
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.44949	−0.100588	−0.0502942	−	0.998734i	$-0.516016\pi$
$593$	−2.44949	−0.100588	−0.0502942	+	0.998734i	$0.516016\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16.9706i	0.693398i	0.937976	+	0.346699i	$0.112698\pi$
$599$	16.9706i	0.693398i	−0.937976	+	0.346699i	$0.887302\pi$
$600$	0	0
$601$	43.3013i	1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$601$	43.3013i	1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.1464	−0.697101
$606$	0	0
$607$	25.9808i	1.05453i	0.849702	+	0.527263i	$0.176782\pi$
$607$	25.9808i	1.05453i	−0.849702	+	0.527263i	$0.823218\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.24264i	0.171639i
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.2132i	0.854011i	0.904249	+	0.427006i	$0.140432\pi$
$617$	21.2132i	0.854011i	−0.904249	+	0.427006i	$0.859568\pi$
$618$	0	0
$619$	− 8.66025i	− 0.348085i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	− 8.66025i	− 0.348085i	0.984738	−	0.174042i	$-0.0556830\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.4949	0.976676
$630$	0	0
$631$	−22.0000	−0.875806	−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000	−0.875806	−0.437903	+	0.899022i	$0.644279\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.2474	0.486025
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.48528i	0.335148i	0.985859	+	0.167574i	$0.0535934\pi$
$641$	8.48528i	0.335148i	−0.985859	+	0.167574i	$0.946407\pi$
$642$	0	0
$643$	− 32.9090i	− 1.29780i	−0.760872	−	0.648901i	$-0.775229\pi$
$643$	− 32.9090i	− 1.29780i	0.760872	−	0.648901i	$-0.224771\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.0454	−0.866694	−0.433347	−	0.901227i	$-0.642668\pi$
$647$	−22.0454	−0.866694	−0.433347	+	0.901227i	$0.642668\pi$
$648$	0	0
$649$	20.7846i	0.815867i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.2132i	0.830137i	0.909790	+	0.415068i	$0.136242\pi$
$653$	21.2132i	0.830137i	−0.909790	+	0.415068i	$0.863758\pi$
$654$	0	0
$655$	18.0000	0.703318
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 16.9706i	− 0.661079i	−0.943792	−	0.330540i	$-0.892769\pi$
$659$	− 16.9706i	− 0.661079i	0.943792	−	0.330540i	$-0.107231\pi$
$660$	0	0
$661$	− 5.19615i	− 0.202107i	−0.994881	−	0.101053i	$-0.967779\pi$
$661$	− 5.19615i	− 0.202107i	0.994881	−	0.101053i	$-0.0322213\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	14.6969	0.567369
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.3939	1.12970	0.564849	−	0.825194i	$-0.308934\pi$
$677$	29.3939	1.12970	0.564849	+	0.825194i	$0.308934\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 8.48528i	− 0.324680i	−0.986735	−	0.162340i	$-0.948096\pi$
$683$	− 8.48528i	− 0.324680i	0.986735	−	0.162340i	$-0.0519042\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.6969	0.559909
$690$	0	0
$691$	19.0526i	0.724793i	0.932024	+	0.362397i	$0.118041\pi$
$691$	19.0526i	0.724793i	−0.932024	+	0.362397i	$0.881959\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 4.24264i	− 0.160933i
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 33.9411i	− 1.28194i	−0.767567	−	0.640969i	$-0.778533\pi$
$701$	− 33.9411i	− 1.28194i	0.767567	−	0.640969i	$-0.221467\pi$
$702$	0	0
$703$	− 25.9808i	− 0.979883i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−18.0000	−0.673162
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26.9444	1.00486	0.502428	−	0.864619i	$-0.332440\pi$
$719$	26.9444	1.00486	0.502428	+	0.864619i	$0.332440\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.48528i	0.315135i
$726$	0	0
$727$	39.8372i	1.47748i	0.673991	+	0.738739i	$0.264579\pi$
$727$	39.8372i	1.47748i	−0.673991	+	0.738739i	$0.735421\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	53.8888	1.99315
$732$	0	0
$733$	8.66025i	0.319874i	0.987127	+	0.159937i	$0.0511291\pi$
$733$	8.66025i	0.319874i	−0.987127	+	0.159937i	$0.948871\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 29.6985i	− 1.09396i
$738$	0	0
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.7279i	0.466942i	0.972364	+	0.233471i	$0.0750084\pi$
$743$	12.7279i	0.466942i	−0.972364	+	0.233471i	$0.924992\pi$
$744$	0	0
$745$	− 41.5692i	− 1.52298i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.89898	−0.178292
$756$	0	0
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.2929	1.24312	0.621558	−	0.783369i	$-0.286500\pi$
$761$	34.2929	1.24312	0.621558	+	0.783369i	$0.286500\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.48528i	0.306386i
$768$	0	0
$769$	− 15.5885i	− 0.562134i	−0.959688	−	0.281067i	$-0.909312\pi$
$769$	− 15.5885i	− 0.562134i	0.959688	−	0.281067i	$-0.0906883\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−41.6413	−1.49773	−0.748867	−	0.662720i	$-0.769402\pi$
$773$	−41.6413	−1.49773	−0.748867	+	0.662720i	$0.769402\pi$
$774$	0	0
$775$	1.73205i	0.0622171i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	63.6396i	2.28013i
$780$	0	0
$781$	−54.0000	−1.93227
$782$	0	0
$783$	0	0
$784$	0	0
$785$	25.4558i	0.908558i
$786$	0	0
$787$	− 17.3205i	− 0.617409i	−0.951158	−	0.308705i	$-0.900105\pi$
$787$	− 17.3205i	− 0.617409i	0.951158	−	0.308705i	$-0.0998955\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	0	0
$802$	0	0
$803$	66.1362	2.33390
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.2132i	0.745817i	0.927868	+	0.372908i	$0.121639\pi$
$809$	21.2132i	0.745817i	−0.927868	+	0.372908i	$0.878361\pi$
$810$	0	0
$811$	31.1769i	1.09477i	0.836881	+	0.547385i	$0.184377\pi$
$811$	31.1769i	1.09477i	−0.836881	+	0.547385i	$0.815623\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−53.8888	−1.88764
$816$	0	0
$817$	− 57.1577i	− 1.99969i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.2132i	0.740346i	0.928963	+	0.370173i	$0.120702\pi$
$821$	21.2132i	0.740346i	−0.928963	+	0.370173i	$0.879298\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.6985i	1.03272i	0.856372	+	0.516359i	$0.172713\pi$
$827$	29.6985i	1.03272i	−0.856372	+	0.516359i	$0.827287\pi$
$828$	0	0
$829$	− 36.3731i	− 1.26329i	−0.775258	−	0.631644i	$-0.782380\pi$
$829$	− 36.3731i	− 1.26329i	0.775258	−	0.631644i	$-0.217620\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	42.0000	1.45347
$836$	0	0
$837$	0	0
$838$	0	0
$839$	53.8888	1.86045	0.930224	−	0.366993i	$-0.119613\pi$
$839$	53.8888	1.86045	0.930224	+	0.366993i	$0.119613\pi$
$840$	0	0
$841$	−43.0000	−1.48276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	24.4949	0.842650
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 25.9808i	− 0.889564i	−0.895639	−	0.444782i	$-0.853281\pi$
$853$	− 25.9808i	− 0.889564i	0.895639	−	0.444782i	$-0.146719\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.1918	1.33877	0.669384	−	0.742917i	$-0.266558\pi$
$857$	39.1918	1.33877	0.669384	+	0.742917i	$0.266558\pi$
$858$	0	0
$859$	45.0333i	1.53652i	0.640140	+	0.768259i	$0.278876\pi$
$859$	45.0333i	1.53652i	−0.640140	+	0.768259i	$0.721124\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 38.1838i	− 1.29979i	−0.760024	−	0.649895i	$-0.774813\pi$
$863$	− 38.1838i	− 1.29979i	0.760024	−	0.649895i	$-0.225187\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.6690i	1.58314i
$870$	0	0
$871$	− 12.1244i	− 0.410818i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.0000	−0.945493	−0.472746	−	0.881199i	$-0.656737\pi$
$877$	−28.0000	−0.945493	−0.472746	+	0.881199i	$0.656737\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.3939	0.990305	0.495152	−	0.868806i	$-0.335112\pi$
$881$	29.3939	0.990305	0.495152	+	0.868806i	$0.335112\pi$
$882$	0	0
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−56.3383	−1.89165	−0.945827	−	0.324671i	$-0.894746\pi$
$887$	−56.3383	−1.89165	−0.945827	+	0.324671i	$0.894746\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.7279i	0.425924i
$894$	0	0
$895$	31.1769i	1.04213i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14.6969	−0.490170
$900$	0	0
$901$	41.5692i	1.38487i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 12.7279i	− 0.423090i
$906$	0	0
$907$	7.00000	0.232431	0.116216	−	0.993224i	$-0.462924\pi$
$907$	7.00000	0.232431	0.116216	+	0.993224i	$0.462924\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 8.48528i	− 0.281130i	−0.990071	−	0.140565i	$-0.955108\pi$
$911$	− 8.48528i	− 0.281130i	0.990071	−	0.140565i	$-0.0448919\pi$
$912$	0	0
$913$	51.9615i	1.71968i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−41.0000	−1.35247	−0.676233	−	0.736688i	$-0.736389\pi$
$919$	−41.0000	−1.35247	−0.676233	+	0.736688i	$0.736389\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−22.0454	−0.725633
$924$	0	0
$925$	−5.00000	−0.164399
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.9444	−0.884017	−0.442008	−	0.897011i	$-0.645734\pi$
$929$	−26.9444	−0.884017	−0.442008	+	0.897011i	$0.645734\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 50.9117i	− 1.66499i
$936$	0	0
$937$	19.0526i	0.622420i	0.950341	+	0.311210i	$0.100734\pi$
$937$	19.0526i	0.622420i	−0.950341	+	0.311210i	$0.899266\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.5959	0.638809	0.319404	−	0.947619i	$-0.396517\pi$
$941$	19.5959	0.638809	0.319404	+	0.947619i	$0.396517\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.1543i	1.79227i	0.443777	+	0.896137i	$0.353638\pi$
$947$	55.1543i	1.79227i	−0.443777	+	0.896137i	$0.646362\pi$
$948$	0	0
$949$	27.0000	0.876457
$950$	0	0
$951$	0	0
$952$	0	0
$953$	33.9411i	1.09946i	0.835342	+	0.549730i	$0.185270\pi$
$953$	33.9411i	1.09946i	−0.835342	+	0.549730i	$0.814730\pi$
$954$	0	0
$955$	10.3923i	0.336287i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	28.0000	0.903226
$962$	0	0
$963$	0	0
$964$	0	0
$965$	46.5403	1.49819
$966$	0	0
$967$	−43.0000	−1.38279	−0.691393	−	0.722478i	$-0.743003\pi$
$967$	−43.0000	−1.38279	−0.691393	+	0.722478i	$0.743003\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9.79796	−0.314431	−0.157216	−	0.987564i	$-0.550252\pi$
$971$	−9.79796	−0.314431	−0.157216	+	0.987564i	$0.550252\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	38.1838i	1.22161i	0.791782	+	0.610803i	$0.209153\pi$
$977$	38.1838i	1.22161i	−0.791782	+	0.610803i	$0.790847\pi$
$978$	0	0
$979$	62.3538i	1.99284i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−53.8888	−1.71878	−0.859392	−	0.511316i	$-0.829158\pi$
$983$	−53.8888	−1.71878	−0.859392	+	0.511316i	$0.829158\pi$
$984$	0	0
$985$	20.7846i	0.662253i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	13.0000	0.412959	0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000	0.412959	0.206479	+	0.978451i	$0.433799\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	33.9411i	1.07601i
$996$	0	0
$997$	50.2295i	1.59078i	0.606096	+	0.795392i	$0.292735\pi$
$997$	50.2295i	1.59078i	−0.606096	+	0.795392i	$0.707265\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.f.a.881.4		4
3.2	odd	2	inner	1764.2.f.a.881.1		4
4.3	odd	2		7056.2.k.a.881.3		4
7.2	even	3		252.2.t.a.17.1	✓	4
7.3	odd	6		252.2.t.a.89.2	yes	4
7.4	even	3		1764.2.t.a.1097.1		4
7.5	odd	6		1764.2.t.a.521.2		4
7.6	odd	2	inner	1764.2.f.a.881.2		4
12.11	even	2		7056.2.k.a.881.2		4
21.2	odd	6		252.2.t.a.17.2	yes	4
21.5	even	6		1764.2.t.a.521.1		4
21.11	odd	6		1764.2.t.a.1097.2		4
21.17	even	6		252.2.t.a.89.1	yes	4
21.20	even	2	inner	1764.2.f.a.881.3		4
28.3	even	6		1008.2.bt.a.593.2		4
28.23	odd	6		1008.2.bt.a.17.1		4
28.27	even	2		7056.2.k.a.881.1		4
35.2	odd	12		6300.2.dd.a.4049.3		8
35.3	even	12		6300.2.dd.a.1349.4		8
35.9	even	6		6300.2.ch.a.4301.1		4
35.17	even	12		6300.2.dd.a.1349.2		8
35.23	odd	12		6300.2.dd.a.4049.1		8
35.24	odd	6		6300.2.ch.a.1601.2		4
63.2	odd	6		2268.2.bm.g.1025.1		4
63.16	even	3		2268.2.bm.g.1025.2		4
63.23	odd	6		2268.2.w.h.269.2		4
63.31	odd	6		2268.2.bm.g.593.1		4
63.38	even	6		2268.2.w.h.1349.1		4
63.52	odd	6		2268.2.w.h.1349.2		4
63.58	even	3		2268.2.w.h.269.1		4
63.59	even	6		2268.2.bm.g.593.2		4
84.23	even	6		1008.2.bt.a.17.2		4
84.59	odd	6		1008.2.bt.a.593.1		4
84.83	odd	2		7056.2.k.a.881.4		4
105.2	even	12		6300.2.dd.a.4049.4		8
105.17	odd	12		6300.2.dd.a.1349.1		8
105.23	even	12		6300.2.dd.a.4049.2		8
105.38	odd	12		6300.2.dd.a.1349.3		8
105.44	odd	6		6300.2.ch.a.4301.2		4
105.59	even	6		6300.2.ch.a.1601.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.t.a.17.1	✓	4	7.2	even	3
252.2.t.a.17.2	yes	4	21.2	odd	6
252.2.t.a.89.1	yes	4	21.17	even	6
252.2.t.a.89.2	yes	4	7.3	odd	6
1008.2.bt.a.17.1		4	28.23	odd	6
1008.2.bt.a.17.2		4	84.23	even	6
1008.2.bt.a.593.1		4	84.59	odd	6
1008.2.bt.a.593.2		4	28.3	even	6
1764.2.f.a.881.1		4	3.2	odd	2	inner
1764.2.f.a.881.2		4	7.6	odd	2	inner
1764.2.f.a.881.3		4	21.20	even	2	inner
1764.2.f.a.881.4		4	1.1	even	1	trivial
1764.2.t.a.521.1		4	21.5	even	6
1764.2.t.a.521.2		4	7.5	odd	6
1764.2.t.a.1097.1		4	7.4	even	3
1764.2.t.a.1097.2		4	21.11	odd	6
2268.2.w.h.269.1		4	63.58	even	3
2268.2.w.h.269.2		4	63.23	odd	6
2268.2.w.h.1349.1		4	63.38	even	6
2268.2.w.h.1349.2		4	63.52	odd	6
2268.2.bm.g.593.1		4	63.31	odd	6
2268.2.bm.g.593.2		4	63.59	even	6
2268.2.bm.g.1025.1		4	63.2	odd	6
2268.2.bm.g.1025.2		4	63.16	even	3
6300.2.ch.a.1601.1		4	105.59	even	6
6300.2.ch.a.1601.2		4	35.24	odd	6
6300.2.ch.a.4301.1		4	35.9	even	6
6300.2.ch.a.4301.2		4	105.44	odd	6
6300.2.dd.a.1349.1		8	105.17	odd	12
6300.2.dd.a.1349.2		8	35.17	even	12
6300.2.dd.a.1349.3		8	105.38	odd	12
6300.2.dd.a.1349.4		8	35.3	even	12
6300.2.dd.a.4049.1		8	35.23	odd	12
6300.2.dd.a.4049.2		8	105.23	even	12
6300.2.dd.a.4049.3		8	35.2	odd	12
6300.2.dd.a.4049.4		8	105.2	even	12
7056.2.k.a.881.1		4	28.27	even	2
7056.2.k.a.881.2		4	12.11	even	2
7056.2.k.a.881.3		4	4.3	odd	2
7056.2.k.a.881.4		4	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.44949 q^{5} +4.24264i q^{11} +1.73205i q^{13} -4.89898 q^{17} +5.19615i q^{19} +1.00000 q^{25} +8.48528i q^{29} +1.73205i q^{31} -5.00000 q^{37} +12.2474 q^{41} -11.0000 q^{43} +2.44949 q^{47} -8.48528i q^{53} +10.3923i q^{55} +4.89898 q^{59} -3.46410i q^{61} +4.24264i q^{65} -7.00000 q^{67} +12.7279i q^{71} -15.5885i q^{73} +11.0000 q^{79} +12.2474 q^{83} -12.0000 q^{85} +14.6969 q^{89} +12.7279i q^{95} +3.46410i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{25} - 20 q^{37} - 44 q^{43} - 28 q^{67} + 44 q^{79} - 48 q^{85}+O(q^{100}) \) 4 * q + 4 * q^25 - 20 * q^37 - 44 * q^43 - 28 * q^67 + 44 * q^79 - 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more