LMFDB - Embedded newform 1764.3.c.e.197.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,3,Mod(197,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.197"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-23})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 17x^{2} - 16x + 18 $ x^4 - 2x^3 + 17x^2 - 16*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			197.2
Root			$0.500000 + 0.983702i$ of defining polynomial
Character	$\chi$	$=$	1764.197
Dual form			1764.3.c.e.197.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.96740i q^{5} +14.6953i q^{11} -7.21767 q^{13} +12.1123i q^{17} -18.3470 q^{19} -36.9524i q^{23} +21.1293 q^{25} -1.96740i q^{29} +1.78233 q^{31} -19.6530 q^{37} -45.7456i q^{41} +10.3470 q^{43} -38.1838i q^{47} -22.8728i q^{53} +28.9116 q^{55} -111.353i q^{59} +45.4763 q^{61} +14.2001i q^{65} +8.87068 q^{67} +79.9945i q^{71} -139.823 q^{73} -28.2177 q^{79} -21.0928i q^{83} +23.8297 q^{85} +128.751i q^{89} +36.0959i q^{95} -177.517 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 56 q^{13} + 8 q^{19} - 24 q^{25} - 20 q^{31} - 160 q^{37} - 40 q^{43} - 20 q^{55} - 8 q^{61} + 144 q^{67} - 288 q^{73} - 140 q^{79} + 448 q^{85} - 276 q^{97}+O(q^{100}) $ 4 * q - 56 * q^13 + 8 * q^19 - 24 * q^25 - 20 * q^31 - 160 * q^37 - 40 * q^43 - 20 * q^55 - 8 * q^61 + 144 * q^67 - 288 * q^73 - 140 * q^79 + 448 * q^85 - 276 * q^97

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
$5$	− 1.96740i	− 0.393481i	−0.980456	−	0.196740i	$-0.936964\pi$
$5$	− 1.96740i	− 0.393481i	0.980456	−	0.196740i	$-0.0630356\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.6953i	1.33594i	0.744189	+	0.667969i	$0.232836\pi$
$11$	14.6953i	1.33594i	−0.744189	+	0.667969i	$0.767164\pi$
$12$	0	0
$13$	−7.21767	−0.555205	−0.277603	−	0.960696i	$-0.589540\pi$
$13$	−7.21767	−0.555205	−0.277603	+	0.960696i	$0.589540\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.1123i	0.712486i	0.934393	+	0.356243i	$0.115942\pi$
$17$	12.1123i	0.712486i	−0.934393	+	0.356243i	$0.884058\pi$
$18$	0	0
$19$	−18.3470	−0.965631	−0.482816	−	0.875722i	$-0.660386\pi$
$19$	−18.3470	−0.965631	−0.482816	+	0.875722i	$0.660386\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 36.9524i	− 1.60663i	−0.595556	−	0.803314i	$-0.703068\pi$
$23$	− 36.9524i	− 1.60663i	0.595556	−	0.803314i	$-0.296932\pi$
$24$	0	0
$25$	21.1293	0.845173
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1.96740i	− 0.0678415i	−0.999425	−	0.0339208i	$-0.989201\pi$
$29$	− 1.96740i	− 0.0678415i	0.999425	−	0.0339208i	$-0.0107994\pi$
$30$	0	0
$31$	1.78233	0.0574945	0.0287473	−	0.999587i	$-0.490848\pi$
$31$	1.78233	0.0574945	0.0287473	+	0.999587i	$0.490848\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−19.6530	−0.531162	−0.265581	−	0.964088i	$-0.585564\pi$
$37$	−19.6530	−0.531162	−0.265581	+	0.964088i	$0.585564\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 45.7456i	− 1.11575i	−0.829927	−	0.557873i	$-0.811618\pi$
$41$	− 45.7456i	− 1.11575i	0.829927	−	0.557873i	$-0.188382\pi$
$42$	0	0
$43$	10.3470	0.240628	0.120314	−	0.992736i	$-0.461610\pi$
$43$	10.3470	0.240628	0.120314	+	0.992736i	$0.461610\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 38.1838i	− 0.812421i	−0.913780	−	0.406210i	$-0.866850\pi$
$47$	− 38.1838i	− 0.812421i	0.913780	−	0.406210i	$-0.133150\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 22.8728i	− 0.431562i	−0.976442	−	0.215781i	$-0.930770\pi$
$53$	− 22.8728i	− 0.431562i	0.976442	−	0.215781i	$-0.0692297\pi$
$54$	0	0
$55$	28.9116	0.525666
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 111.353i	− 1.88733i	−0.330900	−	0.943666i	$-0.607352\pi$
$59$	− 111.353i	− 1.88733i	0.330900	−	0.943666i	$-0.392648\pi$
$60$	0	0
$61$	45.4763	0.745513	0.372757	−	0.927929i	$-0.378413\pi$
$61$	45.4763	0.745513	0.372757	+	0.927929i	$0.378413\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.2001i	0.218463i
$66$	0	0
$67$	8.87068	0.132398	0.0661991	−	0.997806i	$-0.478913\pi$
$67$	8.87068	0.132398	0.0661991	+	0.997806i	$0.478913\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	79.9945i	1.12668i	0.826224	+	0.563342i	$0.190484\pi$
$71$	79.9945i	1.12668i	−0.826224	+	0.563342i	$0.809516\pi$
$72$	0	0
$73$	−139.823	−1.91539	−0.957694	−	0.287789i	$-0.907080\pi$
$73$	−139.823	−1.91539	−0.957694	+	0.287789i	$0.907080\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−28.2177	−0.357186	−0.178593	−	0.983923i	$-0.557154\pi$
$79$	−28.2177	−0.357186	−0.178593	+	0.983923i	$0.557154\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 21.0928i	− 0.254130i	−0.991894	−	0.127065i	$-0.959444\pi$
$83$	− 21.0928i	− 0.254130i	0.991894	−	0.127065i	$-0.0405557\pi$
$84$	0	0
$85$	23.8297	0.280350
$86$	0	0
$87$	0	0
$88$	0	0
$89$	128.751i	1.44664i	0.690511	+	0.723322i	$0.257386\pi$
$89$	128.751i	1.44664i	−0.690511	+	0.723322i	$0.742614\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	36.0959i	0.379957i
$96$	0	0
$97$	−177.517	−1.83008	−0.915038	−	0.403369i	$-0.867839\pi$
$97$	−177.517	−1.83008	−0.915038	+	0.403369i	$0.867839\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 189.380i	− 1.87505i	−0.347923	−	0.937523i	$-0.613113\pi$
$101$	− 189.380i	− 1.87505i	0.347923	−	0.937523i	$-0.386887\pi$
$102$	0	0
$103$	−58.0000	−0.563107	−0.281553	−	0.959546i	$-0.590850\pi$
$103$	−58.0000	−0.563107	−0.281553	+	0.959546i	$0.590850\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 142.590i	− 1.33262i	−0.745675	−	0.666309i	$-0.767873\pi$
$107$	− 142.590i	− 1.33262i	0.745675	−	0.666309i	$-0.232127\pi$
$108$	0	0
$109$	−101.211	−0.928544	−0.464272	−	0.885693i	$-0.653684\pi$
$109$	−101.211	−0.928544	−0.464272	+	0.885693i	$0.653684\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	39.7229i	0.351530i	0.984432	+	0.175765i	$0.0562399\pi$
$113$	39.7229i	0.351530i	−0.984432	+	0.175765i	$0.943760\pi$
$114$	0	0
$115$	−72.7004	−0.632177
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−94.9526	−0.784732
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 90.7550i	− 0.726040i
$126$	0	0
$127$	152.558	1.20125	0.600623	−	0.799532i	$-0.294919\pi$
$127$	152.558	1.20125	0.600623	+	0.799532i	$0.294919\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 189.192i	− 1.44422i	−0.691781	−	0.722108i	$-0.743173\pi$
$131$	− 189.192i	− 1.44422i	0.691781	−	0.722108i	$-0.256827\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	147.810i	1.07890i	0.842017	+	0.539452i	$0.181368\pi$
$137$	147.810i	1.07890i	−0.842017	+	0.539452i	$0.818632\pi$
$138$	0	0
$139$	−133.823	−0.962758	−0.481379	−	0.876513i	$-0.659864\pi$
$139$	−133.823	−0.962758	−0.481379	+	0.876513i	$0.659864\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 106.066i	− 0.741720i
$144$	0	0
$145$	−3.87068	−0.0266943
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 169.706i	− 1.13896i	−0.822004	−	0.569482i	$-0.807144\pi$
$149$	− 169.706i	− 1.13896i	0.822004	−	0.569482i	$-0.192856\pi$
$150$	0	0
$151$	182.994	1.21188	0.605939	−	0.795511i	$-0.292798\pi$
$151$	182.994	1.21188	0.605939	+	0.795511i	$0.292798\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.50656i	− 0.0226230i
$156$	0	0
$157$	24.6940	0.157286	0.0786432	−	0.996903i	$-0.474941\pi$
$157$	24.6940	0.157286	0.0786432	+	0.996903i	$0.474941\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−257.041	−1.57694	−0.788469	−	0.615074i	$-0.789126\pi$
$163$	−257.041	−1.57694	−0.788469	+	0.615074i	$0.789126\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 130.358i	− 0.780584i	−0.920691	−	0.390292i	$-0.872374\pi$
$167$	− 130.358i	− 0.780584i	0.920691	−	0.390292i	$-0.127626\pi$
$168$	0	0
$169$	−116.905	−0.691747
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 153.592i	− 0.887812i	−0.896073	−	0.443906i	$-0.853592\pi$
$173$	− 153.592i	− 0.887812i	0.896073	−	0.443906i	$-0.146408\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	155.679i	0.869717i	0.900499	+	0.434859i	$0.143202\pi$
$179$	155.679i	0.869717i	−0.900499	+	0.434859i	$0.856798\pi$
$180$	0	0
$181$	240.599	1.32928	0.664639	−	0.747165i	$-0.268585\pi$
$181$	240.599	1.32928	0.664639	+	0.747165i	$0.268585\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	38.6654i	0.209002i
$186$	0	0
$187$	−177.994	−0.951837
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 198.106i	− 1.03720i	−0.855016	−	0.518602i	$-0.826453\pi$
$191$	− 198.106i	− 1.03720i	0.855016	−	0.518602i	$-0.173547\pi$
$192$	0	0
$193$	286.470	1.48430	0.742150	−	0.670234i	$-0.233806\pi$
$193$	286.470	1.48430	0.742150	+	0.670234i	$0.233806\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	304.480i	1.54558i	0.634661	+	0.772791i	$0.281140\pi$
$197$	304.480i	1.54558i	−0.634661	+	0.772791i	$0.718860\pi$
$198$	0	0
$199$	−264.082	−1.32704	−0.663522	−	0.748156i	$-0.730939\pi$
$199$	−264.082	−1.32704	−0.663522	+	0.748156i	$0.730939\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−90.0000	−0.439024
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 269.615i	− 1.29002i
$210$	0	0
$211$	−72.3470	−0.342877	−0.171438	−	0.985195i	$-0.554841\pi$
$211$	−72.3470	−0.342877	−0.171438	+	0.985195i	$0.554841\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 20.3567i	− 0.0946824i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 87.4223i	− 0.395576i
$222$	0	0
$223$	−291.347	−1.30649	−0.653244	−	0.757147i	$-0.726593\pi$
$223$	−291.347	−1.30649	−0.653244	+	0.757147i	$0.726593\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 256.218i	− 1.12871i	−0.825531	−	0.564357i	$-0.809124\pi$
$227$	− 256.218i	− 1.12871i	0.825531	−	0.564357i	$-0.190876\pi$
$228$	0	0
$229$	170.871	0.746160	0.373080	−	0.927799i	$-0.378302\pi$
$229$	170.871	0.746160	0.373080	+	0.927799i	$0.378302\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 253.876i	− 1.08960i	−0.838567	−	0.544798i	$-0.816606\pi$
$233$	− 253.876i	− 1.08960i	0.838567	−	0.544798i	$-0.183394\pi$
$234$	0	0
$235$	−75.1229	−0.319672
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 239.234i	− 1.00098i	−0.865743	−	0.500489i	$-0.833153\pi$
$239$	− 239.234i	− 1.00098i	0.865743	−	0.500489i	$-0.166847\pi$
$240$	0	0
$241$	74.3880	0.308664	0.154332	−	0.988019i	$-0.450678\pi$
$241$	74.3880	0.308664	0.154332	+	0.988019i	$0.450678\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	132.423	0.536124
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 140.368i	− 0.559237i	−0.960111	−	0.279618i	$-0.909792\pi$
$251$	− 140.368i	− 0.559237i	0.960111	−	0.279618i	$-0.0902079\pi$
$252$	0	0
$253$	543.028	2.14636
$254$	0	0
$255$	0	0
$256$	0	0
$257$	202.175i	0.786672i	0.919395	+	0.393336i	$0.128679\pi$
$257$	202.175i	0.786672i	−0.919395	+	0.393336i	$0.871321\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.02081i	− 0.00768369i	−0.999993	−	0.00384185i	$-0.998777\pi$
$263$	− 2.02081i	− 0.00768369i	0.999993	−	0.00384185i	$-0.00122290\pi$
$264$	0	0
$265$	−45.0000	−0.169811
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 247.291i	− 0.919297i	−0.888101	−	0.459648i	$-0.847975\pi$
$269$	− 247.291i	− 0.919297i	0.888101	−	0.459648i	$-0.152025\pi$
$270$	0	0
$271$	−15.4418	−0.0569806	−0.0284903	−	0.999594i	$-0.509070\pi$
$271$	−15.4418	−0.0569806	−0.0284903	+	0.999594i	$0.509070\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	310.502i	1.12910i
$276$	0	0
$277$	−356.694	−1.28770	−0.643852	−	0.765150i	$-0.722665\pi$
$277$	−356.694	−1.28770	−0.643852	+	0.765150i	$0.722665\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	228.487i	0.813121i	0.913624	+	0.406560i	$0.133272\pi$
$281$	228.487i	0.813121i	−0.913624	+	0.406560i	$0.866728\pi$
$282$	0	0
$283$	−322.170	−1.13841	−0.569205	−	0.822195i	$-0.692749\pi$
$283$	−322.170	−1.13841	−0.569205	+	0.822195i	$0.692749\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	142.293	0.492364
$290$	0	0
$291$	0	0
$292$	0	0
$293$	493.605i	1.68466i	0.538963	+	0.842329i	$0.318816\pi$
$293$	493.605i	1.68466i	−0.538963	+	0.842329i	$0.681184\pi$
$294$	0	0
$295$	−219.076	−0.742629
$296$	0	0
$297$	0	0
$298$	0	0
$299$	266.711i	0.892008i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 89.4703i	− 0.293345i
$306$	0	0
$307$	115.300	0.375569	0.187784	−	0.982210i	$-0.439869\pi$
$307$	115.300	0.375569	0.187784	+	0.982210i	$0.439869\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 231.806i	− 0.745357i	−0.927960	−	0.372679i	$-0.878439\pi$
$311$	− 231.806i	− 0.745357i	0.927960	−	0.372679i	$-0.121561\pi$
$312$	0	0
$313$	−154.823	−0.494643	−0.247322	−	0.968933i	$-0.579550\pi$
$313$	−154.823	−0.494643	−0.247322	+	0.968933i	$0.579550\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 123.224i	− 0.388719i	−0.980930	−	0.194360i	$-0.937737\pi$
$317$	− 123.224i	− 0.388719i	0.980930	−	0.194360i	$-0.0622629\pi$
$318$	0	0
$319$	28.9116	0.0906321
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 222.223i	− 0.687998i
$324$	0	0
$325$	−152.504	−0.469244
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	399.306	1.20636	0.603181	−	0.797604i	$-0.293900\pi$
$331$	399.306	1.20636	0.603181	+	0.797604i	$0.293900\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 17.4522i	− 0.0520962i
$336$	0	0
$337$	−99.1767	−0.294293	−0.147146	−	0.989115i	$-0.547009\pi$
$337$	−99.1767	−0.294293	−0.147146	+	0.989115i	$0.547009\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.1919i	0.0768092i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 176.344i	− 0.508196i	−0.967179	−	0.254098i	$-0.918221\pi$
$347$	− 176.344i	− 0.508196i	0.967179	−	0.254098i	$-0.0817785\pi$
$348$	0	0
$349$	−132.517	−0.379706	−0.189853	−	0.981813i	$-0.560801\pi$
$349$	−132.517	−0.379706	−0.189853	+	0.981813i	$0.560801\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	551.543i	1.56245i	0.624252	+	0.781223i	$0.285404\pi$
$353$	551.543i	1.56245i	−0.624252	+	0.781223i	$0.714596\pi$
$354$	0	0
$355$	157.382	0.443328
$356$	0	0
$357$	0	0
$358$	0	0
$359$	596.780i	1.66234i	0.556019	+	0.831170i	$0.312328\pi$
$359$	596.780i	1.66234i	−0.556019	+	0.831170i	$0.687672\pi$
$360$	0	0
$361$	−24.3880	−0.0675567
$362$	0	0
$363$	0	0
$364$	0	0
$365$	275.089i	0.753668i
$366$	0	0
$367$	214.735	0.585109	0.292554	−	0.956249i	$-0.405495\pi$
$367$	214.735	0.585109	0.292554	+	0.956249i	$0.405495\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−627.375	−1.68197	−0.840985	−	0.541058i	$-0.818024\pi$
$373$	−627.375	−1.68197	−0.840985	+	0.541058i	$0.818024\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.2001i	0.0376660i
$378$	0	0
$379$	−471.375	−1.24373	−0.621867	−	0.783123i	$-0.713626\pi$
$379$	−471.375	−1.24373	−0.621867	+	0.783123i	$0.713626\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 555.826i	− 1.45124i	−0.688095	−	0.725621i	$-0.741553\pi$
$383$	− 555.826i	− 1.45124i	0.688095	−	0.725621i	$-0.258447\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 296.744i	− 0.762838i	−0.924402	−	0.381419i	$-0.875436\pi$
$389$	− 296.744i	− 0.762838i	0.924402	−	0.381419i	$-0.124564\pi$
$390$	0	0
$391$	447.577	1.14470
$392$	0	0
$393$	0	0
$394$	0	0
$395$	55.5156i	0.140546i
$396$	0	0
$397$	252.353	0.635651	0.317825	−	0.948149i	$-0.397047\pi$
$397$	252.353	0.635651	0.317825	+	0.948149i	$0.397047\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 258.600i	− 0.644888i	−0.946589	−	0.322444i	$-0.895496\pi$
$401$	− 258.600i	− 0.644888i	0.946589	−	0.322444i	$-0.104504\pi$
$402$	0	0
$403$	−12.8643	−0.0319213
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 288.807i	− 0.709600i
$408$	0	0
$409$	738.798	1.80635	0.903176	−	0.429271i	$-0.141230\pi$
$409$	738.798	1.80635	0.903176	+	0.429271i	$0.141230\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−41.4980	−0.0999953
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 410.238i	− 0.979088i	−0.871979	−	0.489544i	$-0.837163\pi$
$419$	− 410.238i	− 0.979088i	0.871979	−	0.489544i	$-0.162837\pi$
$420$	0	0
$421$	543.722	1.29150	0.645751	−	0.763548i	$-0.276545\pi$
$421$	543.722	1.29150	0.645751	+	0.763548i	$0.276545\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	255.924i	0.602174i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	596.539i	1.38408i	0.721858	+	0.692041i	$0.243288\pi$
$431$	596.539i	1.38408i	−0.721858	+	0.692041i	$0.756712\pi$
$432$	0	0
$433$	137.918	0.318517	0.159259	−	0.987237i	$-0.449090\pi$
$433$	137.918	0.318517	0.159259	+	0.987237i	$0.449090\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	677.966i	1.55141i
$438$	0	0
$439$	−78.0538	−0.177799	−0.0888995	−	0.996041i	$-0.528335\pi$
$439$	−78.0538	−0.177799	−0.0888995	+	0.996041i	$0.528335\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	196.955i	0.444594i	0.974979	+	0.222297i	$0.0713554\pi$
$443$	196.955i	0.444594i	−0.974979	+	0.222297i	$0.928645\pi$
$444$	0	0
$445$	253.306	0.569227
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 484.317i	− 1.07866i	−0.842096	−	0.539328i	$-0.818678\pi$
$449$	− 484.317i	− 1.07866i	0.842096	−	0.539328i	$-0.181322\pi$
$450$	0	0
$451$	672.246	1.49057
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	65.0948	0.142439	0.0712197	−	0.997461i	$-0.477311\pi$
$457$	65.0948	0.142439	0.0712197	+	0.997461i	$0.477311\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	674.620i	1.46338i	0.681636	+	0.731692i	$0.261269\pi$
$461$	674.620i	1.46338i	−0.681636	+	0.731692i	$0.738731\pi$
$462$	0	0
$463$	−412.965	−0.891934	−0.445967	−	0.895049i	$-0.647140\pi$
$463$	−412.965	−0.891934	−0.445967	+	0.895049i	$0.647140\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 616.588i	− 1.32032i	−0.751126	−	0.660159i	$-0.770489\pi$
$467$	− 616.588i	− 1.32032i	0.751126	−	0.660159i	$-0.229511\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	152.052i	0.321464i
$474$	0	0
$475$	−387.659	−0.816125
$476$	0	0
$477$	0	0
$478$	0	0
$479$	455.916i	0.951809i	0.879497	+	0.475904i	$0.157879\pi$
$479$	455.916i	0.951809i	−0.879497	+	0.475904i	$0.842121\pi$
$480$	0	0
$481$	141.849	0.294904
$482$	0	0
$483$	0	0
$484$	0	0
$485$	349.248i	0.720100i
$486$	0	0
$487$	462.287	0.949254	0.474627	−	0.880187i	$-0.342583\pi$
$487$	462.287	0.949254	0.474627	+	0.880187i	$0.342583\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 59.9592i	− 0.122117i	−0.998134	−	0.0610583i	$-0.980552\pi$
$491$	− 59.9592i	− 0.122117i	0.998134	−	0.0610583i	$-0.0194475\pi$
$492$	0	0
$493$	23.8297	0.0483361
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	559.035	1.12031	0.560155	−	0.828388i	$-0.310742\pi$
$499$	559.035	1.12031	0.560155	+	0.828388i	$0.310742\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	726.214i	1.44377i	0.692015	+	0.721883i	$0.256723\pi$
$503$	726.214i	1.44377i	−0.692015	+	0.721883i	$0.743277\pi$
$504$	0	0
$505$	−372.586	−0.737795
$506$	0	0
$507$	0	0
$508$	0	0
$509$	896.107i	1.76052i	0.474487	+	0.880262i	$0.342633\pi$
$509$	896.107i	1.76052i	−0.474487	+	0.880262i	$0.657367\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	114.109i	0.221572i
$516$	0	0
$517$	561.123	1.08534
$518$	0	0
$519$	0	0
$520$	0	0
$521$	755.739i	1.45055i	0.688457	+	0.725277i	$0.258288\pi$
$521$	755.739i	1.45055i	−0.688457	+	0.725277i	$0.741712\pi$
$522$	0	0
$523$	−776.675	−1.48504	−0.742519	−	0.669825i	$-0.766369\pi$
$523$	−776.675	−1.48504	−0.742519	+	0.669825i	$0.766369\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.5880i	0.0409640i
$528$	0	0
$529$	−836.483	−1.58125
$530$	0	0
$531$	0	0
$532$	0	0
$533$	330.176i	0.619468i
$534$	0	0
$535$	−280.533	−0.524360
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	566.416	1.04698	0.523490	−	0.852032i	$-0.324630\pi$
$541$	566.416	1.04698	0.523490	+	0.852032i	$0.324630\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	199.123i	0.365364i
$546$	0	0
$547$	189.476	0.346392	0.173196	−	0.984887i	$-0.444591\pi$
$547$	189.476	0.346392	0.173196	+	0.984887i	$0.444591\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	36.0959i	0.0655099i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 689.998i	− 1.23877i	−0.785085	−	0.619387i	$-0.787381\pi$
$557$	− 689.998i	− 1.23877i	0.785085	−	0.619387i	$-0.212619\pi$
$558$	0	0
$559$	−74.6812	−0.133598
$560$	0	0
$561$	0	0
$562$	0	0
$563$	181.457i	0.322303i	0.986930	+	0.161152i	$0.0515208\pi$
$563$	181.457i	0.322303i	−0.986930	+	0.161152i	$0.948479\pi$
$564$	0	0
$565$	78.1511	0.138320
$566$	0	0
$567$	0	0
$568$	0	0
$569$	936.124i	1.64521i	0.568614	+	0.822605i	$0.307480\pi$
$569$	936.124i	1.64521i	−0.568614	+	0.822605i	$0.692520\pi$
$570$	0	0
$571$	−690.933	−1.21004	−0.605020	−	0.796210i	$-0.706835\pi$
$571$	−690.933	−1.21004	−0.605020	+	0.796210i	$0.706835\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 780.780i	− 1.35788i
$576$	0	0
$577$	742.035	1.28602	0.643011	−	0.765857i	$-0.277685\pi$
$577$	742.035	1.28602	0.643011	+	0.765857i	$0.277685\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	336.123	0.576540
$584$	0	0
$585$	0	0
$586$	0	0
$587$	237.160i	0.404020i	0.979383	+	0.202010i	$0.0647473\pi$
$587$	237.160i	0.404020i	−0.979383	+	0.202010i	$0.935253\pi$
$588$	0	0
$589$	−32.7004	−0.0555185
$590$	0	0
$591$	0	0
$592$	0	0
$593$	283.132i	0.477458i	0.971086	+	0.238729i	$0.0767307\pi$
$593$	283.132i	0.477458i	−0.971086	+	0.238729i	$0.923269\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 617.926i	− 1.03160i	−0.856710	−	0.515798i	$-0.827495\pi$
$599$	− 617.926i	− 1.03160i	0.856710	−	0.515798i	$-0.172505\pi$
$600$	0	0
$601$	−81.6120	−0.135794	−0.0678969	−	0.997692i	$-0.521629\pi$
$601$	−81.6120	−0.135794	−0.0678969	+	0.997692i	$0.521629\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	186.810i	0.308777i
$606$	0	0
$607$	419.416	0.690966	0.345483	−	0.938425i	$-0.387715\pi$
$607$	419.416	0.690966	0.345483	+	0.938425i	$0.387715\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	275.598i	0.451060i
$612$	0	0
$613$	−60.4418	−0.0985999	−0.0493000	−	0.998784i	$-0.515699\pi$
$613$	−60.4418	−0.0985999	−0.0493000	+	0.998784i	$0.515699\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 254.598i	− 0.412639i	−0.978485	−	0.206319i	$-0.933851\pi$
$617$	− 254.598i	− 0.412639i	0.978485	−	0.206319i	$-0.0661486\pi$
$618$	0	0
$619$	−497.192	−0.803218	−0.401609	−	0.915811i	$-0.631549\pi$
$619$	−497.192	−0.803218	−0.401609	+	0.915811i	$0.631549\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	349.681	0.559490
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 238.042i	− 0.378446i
$630$	0	0
$631$	−599.183	−0.949577	−0.474789	−	0.880100i	$-0.657475\pi$
$631$	−599.183	−0.949577	−0.474789	+	0.880100i	$0.657475\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 300.144i	− 0.472667i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 771.103i	− 1.20297i	−0.798884	−	0.601484i	$-0.794576\pi$
$641$	− 771.103i	− 1.20297i	0.798884	−	0.601484i	$-0.205424\pi$
$642$	0	0
$643$	1169.35	1.81858	0.909292	−	0.416159i	$-0.136624\pi$
$643$	1169.35	1.81858	0.909292	+	0.416159i	$0.136624\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 145.896i	− 0.225496i	−0.993624	−	0.112748i	$-0.964035\pi$
$647$	− 145.896i	− 0.225496i	0.993624	−	0.112748i	$-0.0359652\pi$
$648$	0	0
$649$	1636.36	2.52136
$650$	0	0
$651$	0	0
$652$	0	0
$653$	388.115i	0.594356i	0.954822	+	0.297178i	$0.0960455\pi$
$653$	388.115i	0.594356i	−0.954822	+	0.297178i	$0.903954\pi$
$654$	0	0
$655$	−372.218	−0.568271
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 271.395i	− 0.411829i	−0.978570	−	0.205914i	$-0.933983\pi$
$659$	− 271.395i	− 0.411829i	0.978570	−	0.205914i	$-0.0660168\pi$
$660$	0	0
$661$	526.442	0.796432	0.398216	−	0.917292i	$-0.369629\pi$
$661$	526.442	0.796432	0.398216	+	0.917292i	$0.369629\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−72.7004	−0.108996
$668$	0	0
$669$	0	0
$670$	0	0
$671$	668.289i	0.995960i
$672$	0	0
$673$	714.022	1.06095	0.530477	−	0.847699i	$-0.322013\pi$
$673$	714.022	1.06095	0.530477	+	0.847699i	$0.322013\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	934.840i	1.38086i	0.723401	+	0.690428i	$0.242578\pi$
$677$	934.840i	1.38086i	−0.723401	+	0.690428i	$0.757422\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 394.552i	− 0.577675i	−0.957378	−	0.288837i	$-0.906731\pi$
$683$	− 394.552i	− 0.577675i	0.957378	−	0.288837i	$-0.0932687\pi$
$684$	0	0
$685$	290.802	0.424528
$686$	0	0
$687$	0	0
$688$	0	0
$689$	165.088i	0.239605i
$690$	0	0
$691$	−1087.88	−1.57436	−0.787178	−	0.616726i	$-0.788459\pi$
$691$	−1087.88	−1.57436	−0.787178	+	0.616726i	$0.788459\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	263.285i	0.378827i
$696$	0	0
$697$	554.082	0.794953
$698$	0	0
$699$	0	0
$700$	0	0
$701$	670.592i	0.956621i	0.878191	+	0.478311i	$0.158751\pi$
$701$	670.592i	0.956621i	−0.878191	+	0.478311i	$0.841249\pi$
$702$	0	0
$703$	360.574	0.512907
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1297.96	1.83069	0.915347	−	0.402667i	$-0.131917\pi$
$709$	1297.96	1.83069	0.915347	+	0.402667i	$0.131917\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 65.8614i	− 0.0923723i
$714$	0	0
$715$	−208.675	−0.291853
$716$	0	0
$717$	0	0
$718$	0	0
$719$	39.8569i	0.0554338i	0.999616	+	0.0277169i	$0.00882370\pi$
$719$	39.8569i	0.0554338i	−0.999616	+	0.0277169i	$0.991176\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 41.5699i	− 0.0573378i
$726$	0	0
$727$	282.627	0.388758	0.194379	−	0.980926i	$-0.437731\pi$
$727$	282.627	0.388758	0.194379	+	0.980926i	$0.437731\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	125.325i	0.171444i
$732$	0	0
$733$	−62.2394	−0.0849105	−0.0424553	−	0.999098i	$-0.513518\pi$
$733$	−62.2394	−0.0849105	−0.0424553	+	0.999098i	$0.513518\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	130.358i	0.176876i
$738$	0	0
$739$	1252.23	1.69450	0.847248	−	0.531197i	$-0.178258\pi$
$739$	1252.23	1.69450	0.847248	+	0.531197i	$0.178258\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 230.602i	− 0.310366i	−0.987886	−	0.155183i	$-0.950403\pi$
$743$	− 230.602i	− 0.310366i	0.987886	−	0.155183i	$-0.0495967\pi$
$744$	0	0
$745$	−333.880	−0.448161
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−711.688	−0.947653	−0.473827	−	0.880618i	$-0.657128\pi$
$751$	−711.688	−0.947653	−0.473827	+	0.880618i	$0.657128\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 360.022i	− 0.476851i
$756$	0	0
$757$	−1073.10	−1.41757	−0.708783	−	0.705427i	$-0.750755\pi$
$757$	−1073.10	−1.41757	−0.708783	+	0.705427i	$0.750755\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	436.751i	0.573917i	0.957943	+	0.286959i	$0.0926443\pi$
$761$	436.751i	0.573917i	−0.957943	+	0.286959i	$0.907356\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	803.706i	1.04786i
$768$	0	0
$769$	284.237	0.369619	0.184809	−	0.982774i	$-0.440833\pi$
$769$	284.237	0.369619	0.184809	+	0.982774i	$0.440833\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	411.951i	0.532925i	0.963845	+	0.266462i	$0.0858548\pi$
$773$	411.951i	0.532925i	−0.963845	+	0.266462i	$0.914145\pi$
$774$	0	0
$775$	37.6594	0.0485928
$776$	0	0
$777$	0	0
$778$	0	0
$779$	839.293i	1.07740i
$780$	0	0
$781$	−1175.55	−1.50518
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 48.5830i	− 0.0618892i
$786$	0	0
$787$	−49.2177	−0.0625383	−0.0312692	−	0.999511i	$-0.509955\pi$
$787$	−49.2177	−0.0625383	−0.0312692	+	0.999511i	$0.509955\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−328.233	−0.413913
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 22.4979i	− 0.0282283i	−0.999900	−	0.0141141i	$-0.995507\pi$
$797$	− 22.4979i	− 0.0282283i	0.999900	−	0.0141141i	$-0.00449282\pi$
$798$	0	0
$799$	462.492	0.578838
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2054.75i	− 2.55884i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	364.733i	0.450844i	0.974261	+	0.225422i	$0.0723761\pi$
$809$	364.733i	0.450844i	−0.974261	+	0.225422i	$0.927624\pi$
$810$	0	0
$811$	−1084.15	−1.33681	−0.668404	−	0.743799i	$-0.733022\pi$
$811$	−1084.15	−1.33681	−0.668404	+	0.743799i	$0.733022\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	505.704i	0.620495i
$816$	0	0
$817$	−189.836	−0.232358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 943.218i	− 1.14886i	−0.818552	−	0.574432i	$-0.805223\pi$
$821$	− 943.218i	− 1.14886i	0.818552	−	0.574432i	$-0.194777\pi$
$822$	0	0
$823$	714.013	0.867573	0.433787	−	0.901016i	$-0.357177\pi$
$823$	714.013	0.867573	0.433787	+	0.901016i	$0.357177\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	186.864i	0.225954i	0.993598	+	0.112977i	$0.0360386\pi$
$827$	186.864i	0.225954i	−0.993598	+	0.112977i	$0.963961\pi$
$828$	0	0
$829$	950.567	1.14664	0.573322	−	0.819330i	$-0.305655\pi$
$829$	950.567	1.14664	0.573322	+	0.819330i	$0.305655\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−256.466	−0.307145
$836$	0	0
$837$	0	0
$838$	0	0
$839$	338.622i	0.403602i	0.979427	+	0.201801i	$0.0646794\pi$
$839$	338.622i	0.403602i	−0.979427	+	0.201801i	$0.935321\pi$
$840$	0	0
$841$	837.129	0.995398
$842$	0	0
$843$	0	0
$844$	0	0
$845$	230.000i	0.272189i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	726.227i	0.853380i
$852$	0	0
$853$	−1048.53	−1.22923	−0.614613	−	0.788829i	$-0.710688\pi$
$853$	−1048.53	−1.22923	−0.614613	+	0.788829i	$0.710688\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	141.479i	0.165087i	0.996587	+	0.0825433i	$0.0263043\pi$
$857$	141.479i	0.165087i	−0.996587	+	0.0825433i	$0.973696\pi$
$858$	0	0
$859$	−161.325	−0.187806	−0.0939029	−	0.995581i	$-0.529934\pi$
$859$	−161.325	−0.187806	−0.0939029	+	0.995581i	$0.529934\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1281.73i	− 1.48521i	−0.669732	−	0.742603i	$-0.733591\pi$
$863$	− 1281.73i	− 1.48521i	0.669732	−	0.742603i	$-0.266409\pi$
$864$	0	0
$865$	−302.177	−0.349337
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 414.668i	− 0.477178i
$870$	0	0
$871$	−64.0256	−0.0735082
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1019.95	−1.16300	−0.581499	−	0.813547i	$-0.697533\pi$
$877$	−1019.95	−1.16300	−0.581499	+	0.813547i	$0.697533\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	461.497i	0.523833i	0.965090	+	0.261917i	$0.0843546\pi$
$881$	461.497i	0.523833i	−0.965090	+	0.261917i	$0.915645\pi$
$882$	0	0
$883$	−449.836	−0.509441	−0.254720	−	0.967015i	$-0.581983\pi$
$883$	−449.836	−0.509441	−0.254720	+	0.967015i	$0.581983\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 66.2363i	− 0.0746745i	−0.999303	−	0.0373372i	$-0.988112\pi$
$887$	− 66.2363i	− 0.0746745i	0.999303	−	0.0373372i	$-0.0118876\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	700.557i	0.784499i
$894$	0	0
$895$	306.284	0.342217
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.50656i	− 0.00390052i
$900$	0	0
$901$	277.041	0.307482
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 473.356i	− 0.523045i
$906$	0	0
$907$	−1672.32	−1.84379	−0.921897	−	0.387435i	$-0.873361\pi$
$907$	−1672.32	−1.84379	−0.921897	+	0.387435i	$0.873361\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 780.445i	− 0.856690i	−0.903615	−	0.428345i	$-0.859097\pi$
$911$	− 780.445i	− 0.856690i	0.903615	−	0.428345i	$-0.140903\pi$
$912$	0	0
$913$	309.965	0.339502
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−134.884	−0.146772	−0.0733860	−	0.997304i	$-0.523381\pi$
$919$	−134.884	−0.146772	−0.0733860	+	0.997304i	$0.523381\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 577.374i	− 0.625541i
$924$	0	0
$925$	−415.255	−0.448924
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 92.7894i	− 0.0998810i	−0.998752	−	0.0499405i	$-0.984097\pi$
$929$	− 92.7894i	− 0.0998810i	0.998752	−	0.0499405i	$-0.0159032\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	350.185i	0.374530i
$936$	0	0
$937$	80.1421	0.0855306	0.0427653	−	0.999085i	$-0.486383\pi$
$937$	80.1421	0.0855306	0.0427653	+	0.999085i	$0.486383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	425.991i	0.452700i	0.974046	+	0.226350i	$0.0726793\pi$
$941$	425.991i	0.452700i	−0.974046	+	0.226350i	$0.927321\pi$
$942$	0	0
$943$	−1690.41	−1.79259
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.2648i	0.0330146i	0.999864	+	0.0165073i	$0.00525468\pi$
$947$	31.2648i	0.0330146i	−0.999864	+	0.0165073i	$0.994745\pi$
$948$	0	0
$949$	1009.20	1.06343
$950$	0	0
$951$	0	0
$952$	0	0
$953$	972.394i	1.02035i	0.860070	+	0.510175i	$0.170419\pi$
$953$	972.394i	1.02035i	−0.860070	+	0.510175i	$0.829581\pi$
$954$	0	0
$955$	−389.754	−0.408120
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−957.823	−0.996694
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 563.602i	− 0.584044i
$966$	0	0
$967$	348.533	0.360427	0.180213	−	0.983628i	$-0.442321\pi$
$967$	348.533	0.360427	0.180213	+	0.983628i	$0.442321\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	567.202i	0.584142i	0.956397	+	0.292071i	$0.0943444\pi$
$971$	567.202i	0.584142i	−0.956397	+	0.292071i	$0.905656\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 200.154i	− 0.204866i	−0.994740	−	0.102433i	$-0.967337\pi$
$977$	− 200.154i	− 0.204866i	0.994740	−	0.102433i	$-0.0326627\pi$
$978$	0	0
$979$	−1892.04	−1.93263
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1871.18i	− 1.90354i	−0.306815	−	0.951769i	$-0.599263\pi$
$983$	− 1871.18i	− 1.90354i	0.306815	−	0.951769i	$-0.400737\pi$
$984$	0	0
$985$	599.035	0.608157
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 382.347i	− 0.386599i
$990$	0	0
$991$	1616.61	1.63130	0.815648	−	0.578548i	$-0.196381\pi$
$991$	1616.61	1.63130	0.815648	+	0.578548i	$0.196381\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	519.556i	0.522167i
$996$	0	0
$997$	−1201.39	−1.20500	−0.602501	−	0.798118i	$-0.705829\pi$
$997$	−1201.39	−1.20500	−0.602501	+	0.798118i	$0.705829\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.c.e.197.2		4
3.2	odd	2	inner	1764.3.c.e.197.3		4
7.2	even	3		252.3.bk.b.53.2	✓	8
7.3	odd	6		1764.3.bk.f.1745.2		8
7.4	even	3		252.3.bk.b.233.3	yes	8
7.5	odd	6		1764.3.bk.f.557.3		8
7.6	odd	2		1764.3.c.g.197.3		4
21.2	odd	6		252.3.bk.b.53.3	yes	8
21.5	even	6		1764.3.bk.f.557.2		8
21.11	odd	6		252.3.bk.b.233.2	yes	8
21.17	even	6		1764.3.bk.f.1745.3		8
21.20	even	2		1764.3.c.g.197.2		4
28.11	odd	6		1008.3.dc.d.737.3		8
28.23	odd	6		1008.3.dc.d.305.2		8
84.11	even	6		1008.3.dc.d.737.2		8
84.23	even	6		1008.3.dc.d.305.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.bk.b.53.2	✓	8	7.2	even	3
252.3.bk.b.53.3	yes	8	21.2	odd	6
252.3.bk.b.233.2	yes	8	21.11	odd	6
252.3.bk.b.233.3	yes	8	7.4	even	3
1008.3.dc.d.305.2		8	28.23	odd	6
1008.3.dc.d.305.3		8	84.23	even	6
1008.3.dc.d.737.2		8	84.11	even	6
1008.3.dc.d.737.3		8	28.11	odd	6
1764.3.c.e.197.2		4	1.1	even	1	trivial
1764.3.c.e.197.3		4	3.2	odd	2	inner
1764.3.c.g.197.2		4	21.20	even	2
1764.3.c.g.197.3		4	7.6	odd	2
1764.3.bk.f.557.2		8	21.5	even	6
1764.3.bk.f.557.3		8	7.5	odd	6
1764.3.bk.f.1745.2		8	7.3	odd	6
1764.3.bk.f.1745.3		8	21.17	even	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.96740i q^{5} +14.6953i q^{11} -7.21767 q^{13} +12.1123i q^{17} -18.3470 q^{19} -36.9524i q^{23} +21.1293 q^{25} -1.96740i q^{29} +1.78233 q^{31} -19.6530 q^{37} -45.7456i q^{41} +10.3470 q^{43} -38.1838i q^{47} -22.8728i q^{53} +28.9116 q^{55} -111.353i q^{59} +45.4763 q^{61} +14.2001i q^{65} +8.87068 q^{67} +79.9945i q^{71} -139.823 q^{73} -28.2177 q^{79} -21.0928i q^{83} +23.8297 q^{85} +128.751i q^{89} +36.0959i q^{95} -177.517 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 56 q^{13} + 8 q^{19} - 24 q^{25} - 20 q^{31} - 160 q^{37} - 40 q^{43} - 20 q^{55} - 8 q^{61} + 144 q^{67} - 288 q^{73} - 140 q^{79} + 448 q^{85} - 276 q^{97}+O(q^{100}) \) 4 * q - 56 * q^13 + 8 * q^19 - 24 * q^25 - 20 * q^31 - 160 * q^37 - 40 * q^43 - 20 * q^55 - 8 * q^61 + 144 * q^67 - 288 * q^73 - 140 * q^79 + 448 * q^85 - 276 * q^97

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