LMFDB - Embedded newform 1764.3.bk.f.1745.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,3,Mod(557,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.557"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.bk (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,112] 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.92844527616.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} - 2x^{5} + 239x^{4} - 200x^{3} - 50x^{2} - 288x + 324 $ x^8 - 2x^7 - 13x^6 - 2x^5 + 239x^4 - 200x^3 - 50x^2 - 288*x + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1745.2
Root			$1.10191 + 0.0588384i$ of defining polynomial
Character	$\chi$	$=$	1764.1745
Dual form			1764.3.bk.f.557.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70382 - 0.983702i) q^{5} +(12.7265 - 7.34766i) q^{11} +7.21767 q^{13} +(-10.4895 + 6.05613i) q^{17} +(-9.17349 + 15.8890i) q^{19} +(32.0018 + 18.4762i) q^{23} +(-10.5647 - 18.2985i) q^{25} -1.96740i q^{29} +(0.891165 + 1.54354i) q^{31} +(9.82651 - 17.0200i) q^{37} +45.7456i q^{41} +10.3470 q^{43} +(-33.0681 - 19.0919i) q^{47} +(-19.8084 + 11.4364i) q^{53} -28.9116 q^{55} +(96.4342 - 55.6763i) q^{59} +(22.7382 - 39.3836i) q^{61} +(-12.2976 - 7.10004i) q^{65} +(-4.43534 - 7.68223i) q^{67} +79.9945i q^{71} +(-69.9116 - 121.091i) q^{73} +(14.1088 - 24.4372i) q^{79} +21.0928i q^{83} +23.8297 q^{85} +(111.502 + 64.3757i) q^{89} +(31.2600 - 18.0480i) q^{95} +177.517 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 112 q^{13} + 8 q^{19} + 24 q^{25} - 20 q^{31} + 160 q^{37} - 80 q^{43} + 40 q^{55} - 8 q^{61} - 144 q^{67} - 288 q^{73} + 140 q^{79} + 896 q^{85} + 552 q^{97}+O(q^{100}) $ 8 * q + 112 * q^13 + 8 * q^19 + 24 * q^25 - 20 * q^31 + 160 * q^37 - 80 * q^43 + 40 * q^55 - 8 * q^61 - 144 * q^67 - 288 * q^73 + 140 * q^79 + 896 * q^85 + 552 * q^97

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.70382	−	0.983702i	−0.340764	−	0.196740i	0.319846	−	0.947470i	$-0.396369\pi$
$5$	−1.70382	−	0.983702i	−0.340764	−	0.196740i	−0.660610	+	0.750729i	$0.729702\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.7265	−	7.34766i	1.15696	−	0.667969i	0.206384	−	0.978471i	$-0.433830\pi$
$11$	12.7265	−	7.34766i	1.15696	−	0.667969i	0.950573	+	0.310502i	$0.100497\pi$
$12$		0			0
$13$	7.21767			0.555205			0.277603	−	0.960696i	$-0.410460\pi$
$13$	7.21767			0.555205			0.277603	+	0.960696i	$0.410460\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−10.4895	+	6.05613i	−0.617031	+	0.356243i	−0.775712	−	0.631087i	$-0.782609\pi$
$17$	−10.4895	+	6.05613i	−0.617031	+	0.356243i	0.158681	+	0.987330i	$0.449276\pi$
$18$		0			0
$19$	−9.17349	+	15.8890i	−0.482816	+	0.836261i	−0.999805	−	0.0197305i	$-0.993719\pi$
$19$	−9.17349	+	15.8890i	−0.482816	+	0.836261i	0.516990	+	0.855992i	$0.327053\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	32.0018	+	18.4762i	1.39138	+	0.803314i	0.993468	−	0.114110i	$-0.0364015\pi$
$23$	32.0018	+	18.4762i	1.39138	+	0.803314i	0.397912	+	0.917423i	$0.369735\pi$
$24$		0			0
$25$	−10.5647	−	18.2985i	−0.422586	−	0.731941i
$26$		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
$31$	0.891165	+	1.54354i	0.0287473	+	0.0497917i	0.880041	−	0.474898i	$-0.157515\pi$
$31$	0.891165	+	1.54354i	0.0287473	+	0.0497917i	−0.851294	+	0.524689i	$0.824182\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	9.82651	−	17.0200i	0.265581	−	0.460000i	−0.702135	−	0.712044i	$-0.747769\pi$
$37$	9.82651	−	17.0200i	0.265581	−	0.460000i	0.967716	+	0.252044i	$0.0811028\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			45.7456i			1.11575i	0.829927	+	0.557873i	$0.188382\pi$
$41$			45.7456i			1.11575i	−0.829927	+	0.557873i	$0.811618\pi$
$42$		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
$47$	−33.0681	−	19.0919i	−0.703577	−	0.406210i	0.105101	−	0.994462i	$-0.466483\pi$
$47$	−33.0681	−	19.0919i	−0.703577	−	0.406210i	−0.808678	+	0.588251i	$0.799817\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−19.8084	+	11.4364i	−0.373743	+	0.215781i	−0.675093	−	0.737733i	$-0.735896\pi$
$53$	−19.8084	+	11.4364i	−0.373743	+	0.215781i	0.301349	+	0.953514i	$0.402563\pi$
$54$		0			0
$55$	−28.9116			−0.525666
$56$		0			0
$57$		0			0
$58$		0			0
$59$	96.4342	−	55.6763i	1.63448	−	0.943666i	0.651789	−	0.758401i	$-0.274019\pi$
$59$	96.4342	−	55.6763i	1.63448	−	0.943666i	0.982689	−	0.185265i	$-0.0593144\pi$
$60$		0			0
$61$	22.7382	−	39.3836i	0.372757	−	0.645633i	−0.617232	−	0.786781i	$-0.711746\pi$
$61$	22.7382	−	39.3836i	0.372757	−	0.645633i	0.989989	+	0.141148i	$0.0450793\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.2976	−	7.10004i	−0.189194	−	0.109231i
$66$		0			0
$67$	−4.43534	−	7.68223i	−0.0661991	−	0.114660i	0.831026	−	0.556233i	$-0.187754\pi$
$67$	−4.43534	−	7.68223i	−0.0661991	−	0.114660i	−0.897225	+	0.441573i	$0.854421\pi$
$68$		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
$73$	−69.9116	−	121.091i	−0.957694	−	1.65877i	−0.728080	−	0.685493i	$-0.759587\pi$
$73$	−69.9116	−	121.091i	−0.957694	−	1.65877i	−0.229614	−	0.973282i	$-0.573746\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	14.1088	−	24.4372i	0.178593	−	0.309332i	−0.762806	−	0.646627i	$-0.776179\pi$
$79$	14.1088	−	24.4372i	0.178593	−	0.309332i	0.941399	+	0.337296i	$0.109512\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			21.0928i			0.254130i	0.991894	+	0.127065i	$0.0405557\pi$
$83$			21.0928i			0.254130i	−0.991894	+	0.127065i	$0.959444\pi$
$84$		0			0
$85$	23.8297			0.280350
$86$		0			0
$87$		0			0
$88$		0			0
$89$	111.502	+	64.3757i	1.25283	+	0.723322i	0.971671	−	0.236339i	$-0.0759474\pi$
$89$	111.502	+	64.3757i	1.25283	+	0.723322i	0.281160	+	0.959661i	$0.409281\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	31.2600	−	18.0480i	0.329053	−	0.189979i
$96$		0			0
$97$	177.517			1.83008			0.915038	−	0.403369i	$-0.132161\pi$
$97$	177.517			1.83008			0.915038	+	0.403369i	$0.132161\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	164.008	−	94.6898i	1.62384	−	0.937523i	0.637957	−	0.770072i	$-0.279780\pi$
$101$	164.008	−	94.6898i	1.62384	−	0.937523i	0.985880	−	0.167451i	$-0.0535537\pi$
$102$		0			0
$103$	−29.0000	+	50.2295i	−0.281553	+	0.487665i	−0.971768	−	0.235940i	$-0.924183\pi$
$103$	−29.0000	+	50.2295i	−0.281553	+	0.487665i	0.690214	+	0.723605i	$0.257516\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	123.487	+	71.2951i	1.15408	+	0.666309i	0.949879	−	0.312619i	$-0.101206\pi$
$107$	123.487	+	71.2951i	1.15408	+	0.666309i	0.204203	+	0.978929i	$0.434540\pi$
$108$		0			0
$109$	50.6056	+	87.6515i	0.464272	+	0.804142i	0.999168	−	0.0407752i	$-0.0129827\pi$
$109$	50.6056	+	87.6515i	0.464272	+	0.804142i	−0.534897	+	0.844918i	$0.679649\pi$
$110$		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
$115$	−36.3502	−	62.9604i	−0.316089	−	0.547482i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	47.4763	−	82.2314i	0.392366	−	0.679598i
$122$		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
$131$	−163.845	−	94.5961i	−1.25073	−	0.722108i	−0.279473	−	0.960153i	$-0.590160\pi$
$131$	−163.845	−	94.5961i	−1.25073	−	0.722108i	−0.971254	+	0.238046i	$0.923493\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	128.007	−	73.9049i	0.934358	−	0.539452i	0.0461706	−	0.998934i	$-0.485298\pi$
$137$	128.007	−	73.9049i	0.934358	−	0.539452i	0.888187	+	0.459482i	$0.151965\pi$
$138$		0			0
$139$	133.823			0.962758			0.481379	−	0.876513i	$-0.340136\pi$
$139$	133.823			0.962758			0.481379	+	0.876513i	$0.340136\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	91.8559	−	53.0330i	0.642349	−	0.370860i
$144$		0			0
$145$	−1.93534	+	3.35211i	−0.0133472	+	0.0231180i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	146.969	+	84.8528i	0.986372	+	0.569482i	0.904188	−	0.427135i	$-0.140477\pi$
$149$	146.969	+	84.8528i	0.986372	+	0.569482i	0.0821839	+	0.996617i	$0.473811\pi$
$150$		0			0
$151$	−91.4968	−	158.477i	−0.605939	−	1.04952i	−0.991902	−	0.127003i	$-0.959464\pi$
$151$	−91.4968	−	158.477i	−0.605939	−	1.04952i	0.385963	−	0.922514i	$-0.373869\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3.50656i		−	0.0226230i
$156$		0			0
$157$	12.3470	+	21.3856i	0.0786432	+	0.136214i	0.902665	−	0.430344i	$-0.141608\pi$
$157$	12.3470	+	21.3856i	0.0786432	+	0.136214i	−0.824022	+	0.566558i	$0.808275\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	128.520	−	222.604i	0.788469	−	1.36567i	−0.138435	−	0.990371i	$-0.544207\pi$
$163$	128.520	−	222.604i	0.788469	−	1.36567i	0.926904	−	0.375297i	$-0.122459\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			130.358i			0.780584i	0.920691	+	0.390292i	$0.127626\pi$
$167$			130.358i			0.780584i	−0.920691	+	0.390292i	$0.872374\pi$
$168$		0			0
$169$	−116.905			−0.691747
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−133.014	−	76.7958i	−0.768868	−	0.443906i	0.0636026	−	0.997975i	$-0.479741\pi$
$173$	−133.014	−	76.7958i	−0.768868	−	0.443906i	−0.832471	+	0.554069i	$0.813074\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	134.822	−	77.8397i	0.753197	−	0.434859i	−0.0736508	−	0.997284i	$-0.523465\pi$
$179$	134.822	−	77.8397i	0.753197	−	0.434859i	0.826848	+	0.562426i	$0.190132\pi$
$180$		0			0
$181$	−240.599			−1.32928			−0.664639	−	0.747165i	$-0.731415\pi$
$181$	−240.599			−1.32928			−0.664639	+	0.747165i	$0.731415\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−33.4852	+	19.3327i	−0.181001	+	0.104501i
$186$		0			0
$187$	−88.9968	+	154.147i	−0.475919	+	0.824315i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	171.565	+	99.0529i	0.898244	+	0.518602i	0.876630	−	0.481165i	$-0.159786\pi$
$191$	171.565	+	99.0529i	0.898244	+	0.518602i	0.0216141	+	0.999766i	$0.493119\pi$
$192$		0			0
$193$	−143.235	−	248.090i	−0.742150	−	1.28544i	−0.951515	−	0.307604i	$-0.900473\pi$
$193$	−143.235	−	248.090i	−0.742150	−	1.28544i	0.209365	−	0.977838i	$-0.432860\pi$
$194$		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
$199$	−132.041	−	228.702i	−0.663522	−	1.14925i	−0.979684	−	0.200549i	$-0.935727\pi$
$199$	−132.041	−	228.702i	−0.663522	−	1.14925i	0.316161	−	0.948705i	$-0.397606\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	45.0000	−	77.9423i	0.219512	−	0.380206i
$206$		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
$215$	−17.6294	−	10.1784i	−0.0819974	−	0.0473412i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−75.7099	+	43.7111i	−0.342579	+	0.197788i
$222$		0			0
$223$	291.347			1.30649			0.653244	−	0.757147i	$-0.273407\pi$
$223$	291.347			1.30649			0.653244	+	0.757147i	$0.273407\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	221.891	−	128.109i	0.977495	−	0.564357i	0.0759819	−	0.997109i	$-0.475791\pi$
$227$	221.891	−	128.109i	0.977495	−	0.564357i	0.901513	+	0.432752i	$0.142458\pi$
$228$		0			0
$229$	85.4353	−	147.978i	0.373080	−	0.646194i	−0.616958	−	0.786996i	$-0.711635\pi$
$229$	85.4353	−	147.978i	0.373080	−	0.646194i	0.990038	+	0.140803i	$0.0449683\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	219.863	+	126.938i	0.943617	+	0.544798i	0.891092	−	0.453822i	$-0.149940\pi$
$233$	219.863	+	126.938i	0.943617	+	0.544798i	0.0525250	+	0.998620i	$0.483273\pi$
$234$		0			0
$235$	37.5615	+	65.0583i	0.159836	+	0.276844i
$236$		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
$241$	37.1940	+	64.4219i	0.154332	+	0.267311i	0.932816	−	0.360354i	$-0.117344\pi$
$241$	37.1940	+	64.4219i	0.154332	+	0.267311i	−0.778484	+	0.627665i	$0.784011\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−66.2113	+	114.681i	−0.268062	+	0.464297i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			140.368i			0.559237i	0.960111	+	0.279618i	$0.0902079\pi$
$251$			140.368i			0.559237i	−0.960111	+	0.279618i	$0.909792\pi$
$252$		0			0
$253$	543.028			2.14636
$254$		0			0
$255$		0			0
$256$		0			0
$257$	175.088	+	101.087i	0.681278	+	0.393336i	0.800336	−	0.599551i	$-0.204654\pi$
$257$	175.088	+	101.087i	0.681278	+	0.393336i	−0.119059	+	0.992887i	$0.537988\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.75007	+	1.01041i	−0.00665427	+	0.00384185i	−0.503323	−	0.864098i	$-0.667890\pi$
$263$	−1.75007	+	1.01041i	−0.00665427	+	0.00384185i	0.496669	+	0.867940i	$0.334556\pi$
$264$		0			0
$265$	45.0000			0.169811
$266$		0			0
$267$		0			0
$268$		0			0
$269$	214.160	−	123.645i	0.796135	−	0.459648i	−0.0459832	−	0.998942i	$-0.514642\pi$
$269$	214.160	−	123.645i	0.796135	−	0.459648i	0.842118	+	0.539294i	$0.181309\pi$
$270$		0			0
$271$	−7.72088	+	13.3729i	−0.0284903	+	0.0493467i	−0.879919	−	0.475124i	$-0.842403\pi$
$271$	−7.72088	+	13.3729i	−0.0284903	+	0.0493467i	0.851429	+	0.524470i	$0.175737\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−268.903	−	155.251i	−0.977829	−	0.564550i
$276$		0			0
$277$	178.347	+	308.906i	0.643852	+	1.11518i	0.984565	+	0.175017i	$0.0559981\pi$
$277$	178.347	+	308.906i	0.643852	+	1.11518i	−0.340713	+	0.940167i	$0.610669\pi$
$278$		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
$283$	−161.085	−	279.008i	−0.569205	−	0.985893i	−0.996645	−	0.0818487i	$-0.973918\pi$
$283$	−161.085	−	279.008i	−0.569205	−	0.985893i	0.427439	−	0.904044i	$-0.359416\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−71.1466	+	123.230i	−0.246182	+	0.426400i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	493.605i		−	1.68466i	−0.538963	−	0.842329i	$-0.681184\pi$
$293$		−	493.605i		−	1.68466i	0.538963	−	0.842329i	$-0.318816\pi$
$294$		0			0
$295$	−219.076			−0.742629
$296$		0			0
$297$		0			0
$298$		0			0
$299$	230.978	+	133.355i	0.772502	+	0.446004i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−77.4835	+	44.7351i	−0.254044	+	0.146673i
$306$		0			0
$307$	−115.300			−0.375569			−0.187784	−	0.982210i	$-0.560131\pi$
$307$	−115.300			−0.375569			−0.187784	+	0.982210i	$0.560131\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	200.750	−	115.903i	0.645498	−	0.372679i	−0.141231	−	0.989977i	$-0.545106\pi$
$311$	200.750	−	115.903i	0.645498	−	0.372679i	0.786729	+	0.617298i	$0.211773\pi$
$312$		0			0
$313$	−77.4116	+	134.081i	−0.247322	+	0.428374i	−0.962782	−	0.270280i	$-0.912884\pi$
$313$	−77.4116	+	134.081i	−0.247322	+	0.428374i	0.715460	+	0.698653i	$0.246217\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	106.715	+	61.6120i	0.336641	+	0.194360i	0.658786	−	0.752331i	$-0.271070\pi$
$317$	106.715	+	61.6120i	0.336641	+	0.194360i	−0.322145	+	0.946690i	$0.604404\pi$
$318$		0			0
$319$	−14.4558	−	25.0382i	−0.0453161	−	0.0784897i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	222.223i		−	0.687998i
$324$		0			0
$325$	−76.2522	−	132.073i	−0.234622	−	0.406378i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−199.653	+	345.809i	−0.603181	+	1.04474i	0.389155	+	0.921172i	$0.372767\pi$
$331$	−199.653	+	345.809i	−0.603181	+	1.04474i	−0.992336	+	0.123568i	$0.960566\pi$
$332$		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
$341$	22.6829	+	13.0960i	0.0665187	+	0.0384046i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−152.718	+	88.1720i	−0.440110	+	0.254098i	−0.703644	−	0.710552i	$-0.748445\pi$
$347$	−152.718	+	88.1720i	−0.440110	+	0.254098i	0.263534	+	0.964650i	$0.415112\pi$
$348$		0			0
$349$	132.517			0.379706			0.189853	−	0.981813i	$-0.439199\pi$
$349$	132.517			0.379706			0.189853	+	0.981813i	$0.439199\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−477.650	+	275.772i	−1.35312	+	0.781223i	−0.988685	−	0.150007i	$-0.952070\pi$
$353$	−477.650	+	275.772i	−1.35312	+	0.781223i	−0.364433	+	0.931230i	$0.618737\pi$
$354$		0			0
$355$	78.6908	−	136.296i	0.221664	−	0.383934i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−516.827	−	298.390i	−1.43963	−	0.831170i	−0.441805	−	0.897111i	$-0.645662\pi$
$359$	−516.827	−	298.390i	−1.43963	−	0.831170i	−0.997824	+	0.0659412i	$0.978995\pi$
$360$		0			0
$361$	12.1940	+	21.1206i	0.0337783	+	0.0585058i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			275.089i			0.753668i
$366$		0			0
$367$	107.367	+	185.966i	0.292554	+	0.506719i	0.974413	−	0.224765i	$-0.0721614\pi$
$367$	107.367	+	185.966i	0.292554	+	0.506719i	−0.681859	+	0.731484i	$0.738828\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	313.688	−	543.323i	0.840985	−	1.45663i	−0.0480770	−	0.998844i	$-0.515309\pi$
$373$	313.688	−	543.323i	0.840985	−	1.45663i	0.889062	−	0.457786i	$-0.151357\pi$
$374$		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
$383$	−481.359	−	277.913i	−1.25681	−	0.725621i	−0.284359	−	0.958718i	$-0.591781\pi$
$383$	−481.359	−	277.913i	−1.25681	−	0.725621i	−0.972454	+	0.233097i	$0.925114\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−256.988	+	148.372i	−0.660637	+	0.381419i	−0.792520	−	0.609846i	$-0.791231\pi$
$389$	−256.988	+	148.372i	−0.660637	+	0.381419i	0.131883	+	0.991265i	$0.457898\pi$
$390$		0			0
$391$	−447.577			−1.14470
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−48.0779	+	27.7578i	−0.121716	+	0.0702729i
$396$		0			0
$397$	126.177	−	218.544i	0.317825	−	0.550490i	−0.662209	−	0.749320i	$-0.730381\pi$
$397$	126.177	−	218.544i	0.317825	−	0.550490i	0.980034	+	0.198830i	$0.0637141\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	223.954	+	129.300i	0.558489	+	0.322444i	0.752539	−	0.658548i	$-0.228829\pi$
$401$	223.954	+	129.300i	0.558489	+	0.322444i	−0.194050	+	0.980992i	$0.562162\pi$
$402$		0			0
$403$	6.43213	+	11.1408i	0.0159606	+	0.0276446i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	288.807i		−	0.709600i
$408$		0			0
$409$	369.399	+	639.818i	0.903176	+	1.56435i	0.823348	+	0.567537i	$0.192104\pi$
$409$	369.399	+	639.818i	0.903176	+	1.56435i	0.0798280	+	0.996809i	$0.474563\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	20.7490	−	35.9384i	0.0499976	−	0.0865985i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			410.238i			0.979088i	0.871979	+	0.489544i	$0.162837\pi$
$419$			410.238i			0.979088i	−0.871979	+	0.489544i	$0.837163\pi$
$420$		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
$425$	221.636	+	127.962i	0.521498	+	0.301087i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	516.618	−	298.270i	1.19865	−	0.692041i	0.238396	−	0.971168i	$-0.423378\pi$
$431$	516.618	−	298.270i	1.19865	−	0.692041i	0.960254	+	0.279127i	$0.0900451\pi$
$432$		0			0
$433$	−137.918			−0.318517			−0.159259	−	0.987237i	$-0.550910\pi$
$433$	−137.918			−0.318517			−0.159259	+	0.987237i	$0.550910\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−587.136	+	338.983i	−1.34356	+	0.775705i
$438$		0			0
$439$	−39.0269	+	67.5966i	−0.0888995	+	0.153979i	−0.907046	−	0.421031i	$-0.861668\pi$
$439$	−39.0269	+	67.5966i	−0.0888995	+	0.153979i	0.818147	+	0.575010i	$0.195002\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−170.568	−	98.4775i	−0.385030	−	0.222297i	0.294975	−	0.955505i	$-0.404689\pi$
$443$	−170.568	−	98.4775i	−0.385030	−	0.222297i	−0.680004	+	0.733208i	$0.738022\pi$
$444$		0			0
$445$	−126.653	−	219.369i	−0.284614	−	0.492965i
$446$		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
$451$	336.123	+	582.182i	0.745284	+	1.29087i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−32.5474	+	56.3737i	−0.0712197	+	0.123356i	−0.899436	−	0.437052i	$-0.856022\pi$
$457$	−32.5474	+	56.3737i	−0.0712197	+	0.123356i	0.828216	+	0.560408i	$0.189356\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	674.620i		−	1.46338i	−0.681636	−	0.731692i	$-0.738731\pi$
$461$		−	674.620i		−	1.46338i	0.681636	−	0.731692i	$-0.261269\pi$
$462$		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
$467$	−533.981	−	308.294i	−1.14343	−	0.660159i	−0.196151	−	0.980574i	$-0.562844\pi$
$467$	−533.981	−	308.294i	−1.14343	−	0.660159i	−0.947277	+	0.320415i	$0.896178\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	131.681	−	76.0262i	0.278396	−	0.160732i
$474$		0			0
$475$	387.659			0.816125
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−394.835	+	227.958i	−0.824290	+	0.475904i	−0.851894	−	0.523715i	$-0.824546\pi$
$479$	−394.835	+	227.958i	−0.824290	+	0.475904i	0.0276033	+	0.999619i	$0.491212\pi$
$480$		0			0
$481$	70.9245	−	122.845i	0.147452	−	0.255395i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−302.458	−	174.624i	−0.623624	−	0.360050i
$486$		0			0
$487$	−231.143	−	400.352i	−0.474627	−	0.822078i	0.524951	−	0.851133i	$-0.324084\pi$
$487$	−231.143	−	400.352i	−0.474627	−	0.822078i	−0.999578	+	0.0290544i	$0.990750\pi$
$488$		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
$493$	11.9149	+	20.6371i	0.0241681	+	0.0418603i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−279.517	+	484.138i	−0.560155	+	0.970217i	0.437328	+	0.899302i	$0.355925\pi$
$499$	−279.517	+	484.138i	−0.560155	+	0.970217i	−0.997482	+	0.0709144i	$0.977408\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	726.214i		−	1.44377i	−0.692015	−	0.721883i	$-0.743277\pi$
$503$		−	726.214i		−	1.44377i	0.692015	−	0.721883i	$-0.256723\pi$
$504$		0			0
$505$	−372.586			−0.737795
$506$		0			0
$507$		0			0
$508$		0			0
$509$	776.051	+	448.054i	1.52466	+	0.880262i	0.999573	+	0.0292131i	$0.00930016\pi$
$509$	776.051	+	448.054i	1.52466	+	0.880262i	0.525086	+	0.851049i	$0.324033\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	98.8217	−	57.0547i	0.191887	−	0.110786i
$516$		0			0
$517$	−561.123			−1.08534
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−654.489	+	377.869i	−1.25622	+	0.725277i	−0.972337	−	0.233583i	$-0.924955\pi$
$521$	−654.489	+	377.869i	−1.25622	+	0.725277i	−0.283880	+	0.958860i	$0.591622\pi$
$522$		0			0
$523$	−388.337	+	672.620i	−0.742519	+	1.28608i	0.208826	+	0.977953i	$0.433036\pi$
$523$	−388.337	+	672.620i	−0.742519	+	1.28608i	−0.951345	+	0.308128i	$0.900298\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.6958	−	10.7940i	−0.0354759	−	0.0204820i
$528$		0			0
$529$	418.241	+	724.415i	0.790626	+	1.36941i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			330.176i			0.619468i
$534$		0			0
$535$	−140.266	−	242.948i	−0.262180	−	0.454109i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−283.208	+	490.531i	−0.523490	+	0.906711i	0.476136	+	0.879372i	$0.342037\pi$
$541$	−283.208	+	490.531i	−0.523490	+	0.906711i	−0.999626	+	0.0273397i	$0.991296\pi$
$542$		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
$551$	31.2600	+	18.0480i	0.0567332	+	0.0327549i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−597.556	+	344.999i	−1.07281	+	0.619387i	−0.928948	−	0.370210i	$-0.879286\pi$
$557$	−597.556	+	344.999i	−1.07281	+	0.619387i	−0.143863	+	0.989598i	$0.545952\pi$
$558$		0			0
$559$	74.6812			0.133598
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−157.146	+	90.7283i	−0.279123	+	0.161152i	−0.633026	−	0.774130i	$-0.718187\pi$
$563$	−157.146	+	90.7283i	−0.279123	+	0.161152i	0.353903	+	0.935282i	$0.384854\pi$
$564$		0			0
$565$	39.0755	−	67.6808i	0.0691602	−	0.119789i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−810.707	−	468.062i	−1.42479	−	0.822605i	−0.428090	−	0.903736i	$-0.640813\pi$
$569$	−810.707	−	468.062i	−1.42479	−	0.822605i	−0.996703	+	0.0811314i	$0.974147\pi$
$570$		0			0
$571$	345.467	+	598.366i	0.605020	+	1.04793i	0.992048	+	0.125858i	$0.0401684\pi$
$571$	345.467	+	598.366i	0.605020	+	1.04793i	−0.387028	+	0.922068i	$0.626498\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	780.780i		−	1.35788i
$576$		0			0
$577$	371.017	+	642.621i	0.643011	+	1.11373i	0.984757	+	0.173935i	$0.0556483\pi$
$577$	371.017	+	642.621i	0.643011	+	1.11373i	−0.341746	+	0.939792i	$0.611018\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−168.061	+	291.091i	−0.288270	+	0.499298i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	237.160i		−	0.404020i	−0.979383	−	0.202010i	$-0.935253\pi$
$587$		−	237.160i		−	0.404020i	0.979383	−	0.202010i	$-0.0647473\pi$
$588$		0			0
$589$	−32.7004			−0.0555185
$590$		0			0
$591$		0			0
$592$		0			0
$593$	245.200	+	141.566i	0.413490	+	0.238729i	0.692288	−	0.721621i	$-0.256603\pi$
$593$	245.200	+	141.566i	0.413490	+	0.238729i	−0.278798	+	0.960350i	$0.589936\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−535.140	+	308.963i	−0.893389	+	0.515798i	−0.875049	−	0.484034i	$-0.839171\pi$
$599$	−535.140	+	308.963i	−0.893389	+	0.515798i	−0.0183392	+	0.999832i	$0.505838\pi$
$600$		0			0
$601$	81.6120			0.135794			0.0678969	−	0.997692i	$-0.478371\pi$
$601$	81.6120			0.135794			0.0678969	+	0.997692i	$0.478371\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−161.782	+	93.4051i	−0.267409	+	0.154389i
$606$		0			0
$607$	209.708	−	363.225i	0.345483	−	0.598394i	−0.639959	−	0.768409i	$-0.721049\pi$
$607$	209.708	−	363.225i	0.345483	−	0.598394i	0.985441	+	0.170016i	$0.0543818\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−238.675	−	137.799i	−0.390630	−	0.225530i
$612$		0			0
$613$	30.2209	+	52.3441i	0.0493000	+	0.0853900i	0.889622	−	0.456697i	$-0.150968\pi$
$613$	30.2209	+	52.3441i	0.0493000	+	0.0853900i	−0.840322	+	0.542087i	$0.817634\pi$
$614$		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
$619$	−248.596	−	430.581i	−0.401609	−	0.695607i	0.592311	−	0.805709i	$-0.298216\pi$
$619$	−248.596	−	430.581i	−0.401609	−	0.695607i	−0.993920	+	0.110102i	$0.964882\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−174.841	+	302.833i	−0.279745	+	0.484532i
$626$		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
$635$	−259.932	−	150.072i	−0.409342	−	0.236334i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−667.795	+	385.552i	−1.04180	+	0.601484i	−0.920343	−	0.391112i	$-0.872090\pi$
$641$	−667.795	+	385.552i	−1.04180	+	0.601484i	−0.121459	+	0.992596i	$0.538757\pi$
$642$		0			0
$643$	−1169.35			−1.81858			−0.909292	−	0.416159i	$-0.863376\pi$
$643$	−1169.35			−1.81858			−0.909292	+	0.416159i	$0.863376\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	126.349	−	72.9479i	0.195285	−	0.112748i	−0.399169	−	0.916877i	$-0.630701\pi$
$647$	126.349	−	72.9479i	0.195285	−	0.112748i	0.594454	+	0.804129i	$0.297368\pi$
$648$		0			0
$649$	818.181	−	1417.13i	1.26068	−	2.18356i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−336.117	−	194.057i	−0.514728	−	0.297178i	0.220047	−	0.975489i	$-0.429379\pi$
$653$	−336.117	−	194.057i	−0.514728	−	0.297178i	−0.734775	+	0.678311i	$0.762712\pi$
$654$		0			0
$655$	186.109	+	322.350i	0.284136	+	0.492137i
$656$		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
$661$	263.221	+	455.912i	0.398216	+	0.689731i	0.993506	−	0.113781i	$-0.0362961\pi$
$661$	263.221	+	455.912i	0.398216	+	0.689731i	−0.595290	+	0.803511i	$0.702963\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	36.3502	−	62.9604i	0.0544980	−	0.0943934i
$668$		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
$677$	809.595	+	467.420i	1.19586	+	0.690428i	0.959629	−	0.281270i	$-0.0907556\pi$
$677$	809.595	+	467.420i	1.19586	+	0.690428i	0.236228	+	0.971698i	$0.424089\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−341.692	+	197.276i	−0.500281	+	0.288837i	−0.728830	−	0.684695i	$-0.759935\pi$
$683$	−341.692	+	197.276i	−0.500281	+	0.288837i	0.228548	+	0.973533i	$0.426602\pi$
$684$		0			0
$685$	−290.802			−0.424528
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−142.971	+	82.5441i	−0.207504	+	0.119803i
$690$		0			0
$691$	−543.940	+	942.131i	−0.787178	+	1.36343i	0.140512	+	0.990079i	$0.455125\pi$
$691$	−543.940	+	942.131i	−0.787178	+	1.36343i	−0.927689	+	0.373353i	$0.878208\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−228.011	−	131.642i	−0.328074	−	0.189413i
$696$		0			0
$697$	−277.041	−	479.849i	−0.397476	−	0.688449i
$698$		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
$703$	180.287	+	312.266i	0.256453	+	0.444190i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−648.981	+	1124.07i	−0.915347	+	1.58543i	−0.108954	+	0.994047i	$0.534750\pi$
$709$	−648.981	+	1124.07i	−0.915347	+	1.58543i	−0.806393	+	0.591380i	$0.798583\pi$
$710$		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
$719$	34.5171	+	19.9285i	0.0480071	+	0.0277169i	0.523811	−	0.851834i	$-0.324510\pi$
$719$	34.5171	+	19.9285i	0.0480071	+	0.0277169i	−0.475804	+	0.879551i	$0.657843\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−36.0006	+	20.7850i	−0.0496560	+	0.0286689i
$726$		0			0
$727$	−282.627			−0.388758			−0.194379	−	0.980926i	$-0.562269\pi$
$727$	−282.627			−0.388758			−0.194379	+	0.980926i	$0.562269\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−108.535	+	62.6627i	−0.148475	+	0.0857219i
$732$		0			0
$733$	−31.1197	+	53.9009i	−0.0424553	+	0.0735347i	−0.886472	−	0.462782i	$-0.846851\pi$
$733$	−31.1197	+	53.9009i	−0.0424553	+	0.0735347i	0.844017	+	0.536317i	$0.180185\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−112.893	−	65.1788i	−0.153179	−	0.0884380i
$738$		0			0
$739$	−626.116	−	1084.47i	−0.847248	−	1.46748i	−0.883654	−	0.468140i	$-0.844924\pi$
$739$	−626.116	−	1084.47i	−0.847248	−	1.46748i	0.0364060	−	0.999337i	$-0.488409\pi$
$740$		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
$745$	−166.940	−	289.148i	−0.224080	−	0.388118i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	355.844	−	616.340i	0.473827	−	0.820692i	−0.525724	−	0.850655i	$-0.676206\pi$
$751$	355.844	−	616.340i	0.473827	−	0.820692i	0.999551	+	0.0299632i	$0.00953901\pi$
$752$		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
$761$	378.238	+	218.376i	0.497027	+	0.286959i	0.727485	−	0.686124i	$-0.240689\pi$
$761$	378.238	+	218.376i	0.497027	+	0.286959i	−0.230458	+	0.973082i	$0.574022\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	696.030	−	401.853i	0.907471	−	0.523928i
$768$		0			0
$769$	−284.237			−0.369619			−0.184809	−	0.982774i	$-0.559167\pi$
$769$	−284.237			−0.369619			−0.184809	+	0.982774i	$0.559167\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−356.760	+	205.975i	−0.461526	+	0.266462i	−0.712686	−	0.701483i	$-0.752521\pi$
$773$	−356.760	+	205.975i	−0.461526	+	0.266462i	0.251159	+	0.967946i	$0.419188\pi$
$774$		0			0
$775$	18.8297	−	32.6140i	0.0242964	−	0.0420826i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−726.849	−	419.647i	−0.933054	−	0.538699i
$780$		0			0
$781$	587.773	+	1018.05i	0.752590	+	1.30352i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	48.5830i		−	0.0618892i
$786$		0			0
$787$	−24.6088	−	42.6238i	−0.0312692	−	0.0541598i	0.849967	−	0.526835i	$-0.176622\pi$
$787$	−24.6088	−	42.6238i	−0.0312692	−	0.0541598i	−0.881237	+	0.472676i	$0.843288\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	164.116	−	284.258i	0.206956	−	0.358459i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			22.4979i			0.0282283i	0.999900	+	0.0141141i	$0.00449282\pi$
$797$			22.4979i			0.0282283i	−0.999900	+	0.0141141i	$0.995507\pi$
$798$		0			0
$799$	462.492			0.578838
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1779.46	−	1027.37i	−2.21602	−	1.27942i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	315.868	−	182.367i	0.390443	−	0.225422i	−0.291909	−	0.956446i	$-0.594291\pi$
$809$	315.868	−	182.367i	0.390443	−	0.225422i	0.682352	+	0.731024i	$0.260957\pi$
$810$		0			0
$811$	1084.15			1.33681			0.668404	−	0.743799i	$-0.266978\pi$
$811$	1084.15			1.33681			0.668404	+	0.743799i	$0.266978\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−437.952	+	252.852i	−0.537365	+	0.310248i
$816$		0			0
$817$	−94.9181	+	164.403i	−0.116179	+	0.201228i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	816.851	+	471.609i	0.994946	+	0.574432i	0.906749	−	0.421671i	$-0.138556\pi$
$821$	816.851	+	471.609i	0.994946	+	0.574432i	0.0881970	+	0.996103i	$0.471889\pi$
$822$		0			0
$823$	−357.006	−	618.353i	−0.433787	−	0.751341i	0.563409	−	0.826178i	$-0.309490\pi$
$823$	−357.006	−	618.353i	−0.433787	−	0.751341i	−0.997196	+	0.0748376i	$0.976156\pi$
$824$		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
$829$	475.284	+	823.215i	0.573322	+	0.993022i	0.996222	+	0.0868458i	$0.0276788\pi$
$829$	475.284	+	823.215i	0.573322	+	0.993022i	−0.422900	+	0.906176i	$0.638988\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	128.233	−	222.106i	0.153572	−	0.265995i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	338.622i		−	0.403602i	−0.979427	−	0.201801i	$-0.935321\pi$
$839$		−	338.622i		−	0.403602i	0.979427	−	0.201801i	$-0.0646794\pi$
$840$		0			0
$841$	837.129			0.995398
$842$		0			0
$843$		0			0
$844$		0			0
$845$	199.186	+	115.000i	0.235723	+	0.136095i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	628.931	−	363.113i	0.739049	−	0.426690i
$852$		0			0
$853$	1048.53			1.22923			0.614613	−	0.788829i	$-0.289312\pi$
$853$	1048.53			1.22923			0.614613	+	0.788829i	$0.289312\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−122.525	+	70.7396i	−0.142969	+	0.0825433i	−0.569778	−	0.821798i	$-0.692971\pi$
$857$	−122.525	+	70.7396i	−0.142969	+	0.0825433i	0.426809	+	0.904342i	$0.359638\pi$
$858$		0			0
$859$	−80.6626	+	139.712i	−0.0939029	+	0.162645i	−0.909150	−	0.416468i	$-0.863268\pi$
$859$	−80.6626	+	139.712i	−0.0939029	+	0.162645i	0.815247	+	0.579113i	$0.196601\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1110.01	+	640.866i	1.28623	+	0.742603i	0.977979	−	0.208704i	$-0.0669245\pi$
$863$	1110.01	+	640.866i	1.28623	+	0.742603i	0.308246	+	0.951307i	$0.400258\pi$
$864$		0			0
$865$	151.088	+	261.693i	0.174669	+	0.302535i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	414.668i		−	0.477178i
$870$		0			0
$871$	−32.0128	−	55.4478i	−0.0367541	−	0.0636600i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	509.974	−	883.301i	0.581499	−	1.00719i	−0.413803	−	0.910366i	$-0.635800\pi$
$877$	509.974	−	883.301i	0.581499	−	1.00719i	0.995302	−	0.0968190i	$-0.0308668\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	461.497i		−	0.523833i	−0.965090	−	0.261917i	$-0.915645\pi$
$881$		−	461.497i		−	0.523833i	0.965090	−	0.261917i	$-0.0843546\pi$
$882$		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
$887$	−57.3623	−	33.1181i	−0.0646700	−	0.0373372i	0.467316	−	0.884090i	$-0.345221\pi$
$887$	−57.3623	−	33.1181i	−0.0646700	−	0.0373372i	−0.531986	+	0.846753i	$0.678554\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	606.700	−	350.279i	0.679396	−	0.392249i
$894$		0			0
$895$	−306.284			−0.342217
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3.03677	−	1.75328i	0.00337795	−	0.00195026i
$900$		0			0
$901$	138.520	−	239.925i	0.153741	−	0.266287i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	409.938	+	236.678i	0.452970	+	0.261523i
$906$		0			0
$907$	836.161	+	1448.27i	0.921897	+	1.59677i	0.796477	+	0.604669i	$0.206695\pi$
$907$	836.161	+	1448.27i	0.921897	+	1.59677i	0.125420	+	0.992104i	$0.459972\pi$
$908$		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
$913$	154.983	+	268.438i	0.169751	+	0.294017i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	67.4418	−	116.813i	0.0733860	−	0.127108i	−0.826997	−	0.562206i	$-0.809953\pi$
$919$	67.4418	−	116.813i	0.0733860	−	0.127108i	0.900383	+	0.435098i	$0.143286\pi$
$920$		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
$929$	−80.3580	−	46.3947i	−0.0864995	−	0.0499405i	0.456126	−	0.889915i	$-0.349237\pi$
$929$	−80.3580	−	46.3947i	−0.0864995	−	0.0499405i	−0.542626	+	0.839975i	$0.682570\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	303.269	−	175.093i	0.324352	−	0.187265i
$936$		0			0
$937$	−80.1421			−0.0855306			−0.0427653	−	0.999085i	$-0.513617\pi$
$937$	−80.1421			−0.0855306			−0.0427653	+	0.999085i	$0.513617\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−368.919	+	212.995i	−0.392050	+	0.226350i	−0.683048	−	0.730374i	$-0.739346\pi$
$941$	−368.919	+	212.995i	−0.392050	+	0.226350i	0.290998	+	0.956724i	$0.406013\pi$
$942$		0			0
$943$	−845.205	+	1463.94i	−0.896294	+	1.55243i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.0761	−	15.6324i	−0.0285915	−	0.0165073i	0.485636	−	0.874161i	$-0.338588\pi$
$947$	−27.0761	−	15.6324i	−0.0285915	−	0.0165073i	−0.514228	+	0.857654i	$0.671921\pi$
$948$		0			0
$949$	−504.599	−	873.991i	−0.531717	−	0.920960i
$950$		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
$955$	−194.877	−	337.537i	−0.204060	−	0.353442i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	478.912	−	829.499i	0.498347	−	0.863163i
$962$		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
$971$	491.211	+	283.601i	0.505882	+	0.292071i	0.731139	−	0.682228i	$-0.238989\pi$
$971$	491.211	+	283.601i	0.505882	+	0.292071i	−0.225257	+	0.974299i	$0.572322\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−173.338	+	100.077i	−0.177419	+	0.102433i	−0.586079	−	0.810254i	$-0.699329\pi$
$977$	−173.338	+	100.077i	−0.177419	+	0.102433i	0.408661	+	0.912686i	$0.365996\pi$
$978$		0			0
$979$	1892.04			1.93263
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1620.49	−	935.589i	1.64851	−	0.951769i	0.670849	−	0.741594i	$-0.265930\pi$
$983$	1620.49	−	935.589i	1.64851	−	0.951769i	0.977664	−	0.210175i	$-0.0674035\pi$
$984$		0			0
$985$	299.517	−	518.779i	0.304078	−	0.526679i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	331.122	+	191.173i	0.334805	+	0.193300i
$990$		0			0
$991$	−808.307	−	1400.03i	−0.815648	−	1.41274i	−0.908862	−	0.417098i	$-0.863047\pi$
$991$	−808.307	−	1400.03i	−0.815648	−	1.41274i	0.0932135	−	0.995646i	$-0.470286\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			519.556i			0.522167i
$996$		0			0
$997$	−600.694	−	1040.43i	−0.602501	−	1.04356i	−0.992441	−	0.122723i	$-0.960837\pi$
$997$	−600.694	−	1040.43i	−0.602501	−	1.04356i	0.389940	−	0.920840i	$-0.372496\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.bk.b.53.2	✓	8	7.3	odd	6
252.3.bk.b.53.3	yes	8	21.17	even	6
252.3.bk.b.233.2	yes	8	21.20	even	2
252.3.bk.b.233.3	yes	8	7.6	odd	2
1008.3.dc.d.305.2		8	28.3	even	6
1008.3.dc.d.305.3		8	84.59	odd	6
1008.3.dc.d.737.2		8	84.83	odd	2
1008.3.dc.d.737.3		8	28.27	even	2
1764.3.c.e.197.2		4	7.5	odd	6
1764.3.c.e.197.3		4	21.5	even	6
1764.3.c.g.197.2		4	21.2	odd	6
1764.3.c.g.197.3		4	7.2	even	3
1764.3.bk.f.557.2		8	21.11	odd	6	inner
1764.3.bk.f.557.3		8	7.4	even	3	inner
1764.3.bk.f.1745.2		8	1.1	even	1	trivial
1764.3.bk.f.1745.3		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.bk (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.70382 - 0.983702i) q^{5} +(12.7265 - 7.34766i) q^{11} +7.21767 q^{13} +(-10.4895 + 6.05613i) q^{17} +(-9.17349 + 15.8890i) q^{19} +(32.0018 + 18.4762i) q^{23} +(-10.5647 - 18.2985i) q^{25} -1.96740i q^{29} +(0.891165 + 1.54354i) q^{31} +(9.82651 - 17.0200i) q^{37} +45.7456i q^{41} +10.3470 q^{43} +(-33.0681 - 19.0919i) q^{47} +(-19.8084 + 11.4364i) q^{53} -28.9116 q^{55} +(96.4342 - 55.6763i) q^{59} +(22.7382 - 39.3836i) q^{61} +(-12.2976 - 7.10004i) q^{65} +(-4.43534 - 7.68223i) q^{67} +79.9945i q^{71} +(-69.9116 - 121.091i) q^{73} +(14.1088 - 24.4372i) q^{79} +21.0928i q^{83} +23.8297 q^{85} +(111.502 + 64.3757i) q^{89} +(31.2600 - 18.0480i) q^{95} +177.517 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 112 q^{13} + 8 q^{19} + 24 q^{25} - 20 q^{31} + 160 q^{37} - 80 q^{43} + 40 q^{55} - 8 q^{61} - 144 q^{67} - 288 q^{73} + 140 q^{79} + 896 q^{85} + 552 q^{97}+O(q^{100}) \) 8 * q + 112 * q^13 + 8 * q^19 + 24 * q^25 - 20 * q^31 + 160 * q^37 - 80 * q^43 + 40 * q^55 - 8 * q^61 - 144 * q^67 - 288 * q^73 + 140 * q^79 + 896 * q^85 + 552 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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