LMFDB - Embedded newform 1764.3.bk.a.1745.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(557,1764)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1745.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.1745
Dual form			1764.3.bk.a.557.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.67423 - 2.12132i) q^{5} +O(q^{10})$	$q+(-3.67423 - 2.12132i) q^{5} +(-18.3712 + 10.6066i) q^{11} -23.0000 q^{13} +(14.6969 - 8.48528i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(14.6969 + 8.48528i) q^{23} +(-3.50000 - 6.06218i) q^{25} +33.9411i q^{29} +(-24.5000 - 42.4352i) q^{31} +(-8.50000 + 14.7224i) q^{37} +21.2132i q^{41} +47.0000 q^{43} +(33.0681 + 19.0919i) q^{47} +(73.4847 - 42.4264i) q^{53} +90.0000 q^{55} +(44.0908 - 25.4558i) q^{59} +(-20.0000 + 34.6410i) q^{61} +(84.5074 + 48.7904i) q^{65} +(-11.5000 - 19.9186i) q^{67} -63.6396i q^{71} +(8.50000 + 14.7224i) q^{73} +(39.5000 - 68.4160i) q^{79} +106.066i q^{83} -72.0000 q^{85} +(-117.576 - 67.8823i) q^{89} +(3.67423 - 2.12132i) q^{95} +40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 92 q^{13} - 2 q^{19} - 14 q^{25} - 98 q^{31} - 34 q^{37} + 188 q^{43} + 360 q^{55} - 80 q^{61} - 46 q^{67} + 34 q^{73} + 158 q^{79} - 288 q^{85} + 160 q^{97}+O(q^{100}) $ 4 * q - 92 * q^13 - 2 * q^19 - 14 * q^25 - 98 * q^31 - 34 * q^37 + 188 * q^43 + 360 * q^55 - 80 * q^61 - 46 * q^67 + 34 * q^73 + 158 * q^79 - 288 * q^85 + 160 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−3.67423	−	2.12132i	−0.734847	−	0.424264i	0.0853458	−	0.996351i	$-0.472801\pi$
$5$	−3.67423	−	2.12132i	−0.734847	−	0.424264i	−0.820193	+	0.572087i	$0.806134\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−18.3712	+	10.6066i	−1.67011	+	0.964237i	−0.702532	+	0.711652i	$0.747947\pi$
$11$	−18.3712	+	10.6066i	−1.67011	+	0.964237i	−0.967575	+	0.252584i	$0.918720\pi$
$12$		0			0
$13$	−23.0000			−1.76923			−0.884615	−	0.466321i	$-0.845579\pi$
$13$	−23.0000			−1.76923			−0.884615	+	0.466321i	$0.845579\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.6969	−	8.48528i	0.864526	−	0.499134i	−0.000999453	−	1.00000i	$-0.500318\pi$
$17$	14.6969	−	8.48528i	0.864526	−	0.499134i	0.865525	+	0.500865i	$0.166985\pi$
$18$		0			0
$19$	−0.500000	+	0.866025i	−0.0263158	+	0.0455803i	−0.878883	−	0.477037i	$-0.841711\pi$
$19$	−0.500000	+	0.866025i	−0.0263158	+	0.0455803i	0.852568	+	0.522617i	$0.175044\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	14.6969	+	8.48528i	0.638997	+	0.368925i	0.784228	−	0.620473i	$-0.213059\pi$
$23$	14.6969	+	8.48528i	0.638997	+	0.368925i	−0.145231	+	0.989398i	$0.546392\pi$
$24$		0			0
$25$	−3.50000	−	6.06218i	−0.140000	−	0.242487i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			33.9411i			1.17038i	0.810895	+	0.585192i	$0.198981\pi$
$29$			33.9411i			1.17038i	−0.810895	+	0.585192i	$0.801019\pi$
$30$		0			0
$31$	−24.5000	−	42.4352i	−0.790323	−	1.36888i	−0.925767	−	0.378094i	$-0.876580\pi$
$31$	−24.5000	−	42.4352i	−0.790323	−	1.36888i	0.135445	−	0.990785i	$-0.456754\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−8.50000	+	14.7224i	−0.229730	+	0.397904i	−0.957728	−	0.287675i	$-0.907118\pi$
$37$	−8.50000	+	14.7224i	−0.229730	+	0.397904i	0.727998	+	0.685579i	$0.240451\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			21.2132i			0.517395i	0.965958	+	0.258698i	$0.0832933\pi$
$41$			21.2132i			0.517395i	−0.965958	+	0.258698i	$0.916707\pi$
$42$		0			0
$43$	47.0000			1.09302			0.546512	−	0.837452i	$-0.315955\pi$
$43$	47.0000			1.09302			0.546512	+	0.837452i	$0.315955\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	33.0681	+	19.0919i	0.703577	+	0.406210i	0.808678	−	0.588251i	$-0.200183\pi$
$47$	33.0681	+	19.0919i	0.703577	+	0.406210i	−0.105101	+	0.994462i	$0.533517\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	73.4847	−	42.4264i	1.38650	−	0.800498i	0.393584	−	0.919289i	$-0.371235\pi$
$53$	73.4847	−	42.4264i	1.38650	−	0.800498i	0.992919	+	0.118790i	$0.0379016\pi$
$54$		0			0
$55$	90.0000			1.63636
$56$		0			0
$57$		0			0
$58$		0			0
$59$	44.0908	−	25.4558i	0.747302	−	0.431455i	−0.0774163	−	0.996999i	$-0.524667\pi$
$59$	44.0908	−	25.4558i	0.747302	−	0.431455i	0.824718	+	0.565544i	$0.191334\pi$
$60$		0			0
$61$	−20.0000	+	34.6410i	−0.327869	+	0.567886i	−0.982089	−	0.188419i	$-0.939664\pi$
$61$	−20.0000	+	34.6410i	−0.327869	+	0.567886i	0.654220	+	0.756304i	$0.272997\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	84.5074	+	48.7904i	1.30011	+	0.750621i
$66$		0			0
$67$	−11.5000	−	19.9186i	−0.171642	−	0.297292i	0.767352	−	0.641226i	$-0.221574\pi$
$67$	−11.5000	−	19.9186i	−0.171642	−	0.297292i	−0.938994	+	0.343934i	$0.888240\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	63.6396i		−	0.896333i	−0.893950	−	0.448166i	$-0.852077\pi$
$71$		−	63.6396i		−	0.896333i	0.893950	−	0.448166i	$-0.147923\pi$
$72$		0			0
$73$	8.50000	+	14.7224i	0.116438	+	0.201677i	0.918354	−	0.395760i	$-0.129519\pi$
$73$	8.50000	+	14.7224i	0.116438	+	0.201677i	−0.801915	+	0.597438i	$0.796186\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	39.5000	−	68.4160i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	39.5000	−	68.4160i	0.500000	−	0.866025i	1.00000			$0$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			106.066i			1.27790i	0.769247	+	0.638952i	$0.220632\pi$
$83$			106.066i			1.27790i	−0.769247	+	0.638952i	$0.779368\pi$
$84$		0			0
$85$	−72.0000			−0.847059
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−117.576	−	67.8823i	−1.32107	−	0.762722i	−0.337173	−	0.941443i	$-0.609471\pi$
$89$	−117.576	−	67.8823i	−1.32107	−	0.762722i	−0.983900	+	0.178721i	$0.942804\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.67423	−	2.12132i	0.0386762	−	0.0223297i
$96$		0			0
$97$	40.0000			0.412371			0.206186	−	0.978513i	$-0.433895\pi$
$97$	40.0000			0.412371			0.206186	+	0.978513i	$0.433895\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.3712	−	10.6066i	0.181893	−	0.105016i	−0.406289	−	0.913745i	$-0.633177\pi$
$101$	18.3712	−	10.6066i	0.181893	−	0.105016i	0.588182	+	0.808729i	$0.299844\pi$
$102$		0			0
$103$	11.5000	−	19.9186i	0.111650	−	0.193384i	−0.804785	−	0.593566i	$-0.797720\pi$
$103$	11.5000	−	19.9186i	0.111650	−	0.193384i	0.916436	+	0.400182i	$0.131053\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.6969	−	8.48528i	−0.137355	−	0.0793017i	0.429748	−	0.902949i	$-0.358602\pi$
$107$	−14.6969	−	8.48528i	−0.137355	−	0.0793017i	−0.567103	+	0.823647i	$0.691936\pi$
$108$		0			0
$109$	−35.5000	−	61.4878i	−0.325688	−	0.564108i	0.655963	−	0.754793i	$-0.272263\pi$
$109$	−35.5000	−	61.4878i	−0.325688	−	0.564108i	−0.981651	+	0.190684i	$0.938929\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			131.522i			1.16391i	0.813221	+	0.581955i	$0.197712\pi$
$113$			131.522i			1.16391i	−0.813221	+	0.581955i	$0.802288\pi$
$114$		0			0
$115$	−36.0000	−	62.3538i	−0.313043	−	0.542207i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	164.500	−	284.922i	1.35950	−	2.35473i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			135.765i			1.08612i
$126$		0			0
$127$	17.0000			0.133858			0.0669291	−	0.997758i	$-0.478680\pi$
$127$	17.0000			0.133858			0.0669291	+	0.997758i	$0.478680\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−47.7650	−	27.5772i	−0.364619	−	0.210513i	0.306486	−	0.951875i	$-0.400847\pi$
$131$	−47.7650	−	27.5772i	−0.364619	−	0.210513i	−0.671105	+	0.741362i	$0.734180\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	58.7878	−	33.9411i	0.429108	−	0.247745i	−0.269859	−	0.962900i	$-0.586977\pi$
$137$	58.7878	−	33.9411i	0.429108	−	0.247745i	0.698966	+	0.715154i	$0.253644\pi$
$138$		0			0
$139$	31.0000			0.223022			0.111511	−	0.993763i	$-0.464431\pi$
$139$	31.0000			0.223022			0.111511	+	0.993763i	$0.464431\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	422.537	−	243.952i	2.95480	−	1.70596i
$144$		0			0
$145$	72.0000	−	124.708i	0.496552	−	0.860053i
$146$		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
$151$	20.0000	+	34.6410i	0.132450	+	0.229411i	0.924621	−	0.380889i	$-0.124382\pi$
$151$	20.0000	+	34.6410i	0.132450	+	0.229411i	−0.792170	+	0.610300i	$0.791049\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			207.889i			1.34122i
$156$		0			0
$157$	−140.000	−	242.487i	−0.891720	−	1.54450i	−0.837813	−	0.545958i	$-0.816166\pi$
$157$	−140.000	−	242.487i	−0.891720	−	1.54450i	−0.0539072	−	0.998546i	$-0.517168\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−100.000	+	173.205i	−0.613497	+	1.06261i	0.377149	+	0.926152i	$0.376904\pi$
$163$	−100.000	+	173.205i	−0.613497	+	1.06261i	−0.990646	+	0.136455i	$0.956429\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	106.066i		−	0.635126i	−0.948237	−	0.317563i	$-0.897136\pi$
$167$		−	106.066i		−	0.635126i	0.948237	−	0.317563i	$-0.102864\pi$
$168$		0			0
$169$	360.000			2.13018
$170$		0			0
$171$		0			0
$172$		0			0
$173$	220.454	+	127.279i	1.27430	+	0.735718i	0.975794	−	0.218690i	$-0.0701782\pi$
$173$	220.454	+	127.279i	1.27430	+	0.735718i	0.298507	+	0.954408i	$0.403512\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	172.689	−	99.7021i	0.964743	−	0.556995i	0.0671136	−	0.997745i	$-0.478621\pi$
$179$	172.689	−	99.7021i	0.964743	−	0.556995i	0.897630	+	0.440751i	$0.145288\pi$
$180$		0			0
$181$	271.000			1.49724			0.748619	−	0.663001i	$-0.230717\pi$
$181$	271.000			1.49724			0.748619	+	0.663001i	$0.230717\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	62.4620	−	36.0624i	0.337632	−	0.194932i
$186$		0			0
$187$	−180.000	+	311.769i	−0.962567	+	1.66721i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	209.431	+	120.915i	1.09650	+	0.633064i	0.935299	−	0.353858i	$-0.115130\pi$
$191$	209.431	+	120.915i	1.09650	+	0.633064i	0.161200	+	0.986922i	$0.448464\pi$
$192$		0			0
$193$	51.5000	+	89.2006i	0.266839	+	0.462179i	0.968044	−	0.250781i	$-0.0806873\pi$
$193$	51.5000	+	89.2006i	0.266839	+	0.462179i	−0.701205	+	0.712960i	$0.747354\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			152.735i			0.775305i	0.921806	+	0.387652i	$0.126714\pi$
$197$			152.735i			0.775305i	−0.921806	+	0.387652i	$0.873286\pi$
$198$		0			0
$199$	−80.0000	−	138.564i	−0.402010	−	0.696302i	0.591958	−	0.805969i	$-0.298355\pi$
$199$	−80.0000	−	138.564i	−0.402010	−	0.696302i	−0.993968	+	0.109667i	$0.965022\pi$
$200$		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
$209$		−	21.2132i		−	0.101499i
$210$		0			0
$211$	80.0000			0.379147			0.189573	−	0.981867i	$-0.439289\pi$
$211$	80.0000			0.379147			0.189573	+	0.981867i	$0.439289\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−172.689	−	99.7021i	−0.803205	−	0.463730i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−338.030	+	195.161i	−1.52955	+	0.883084i
$222$		0			0
$223$	−104.000			−0.466368			−0.233184	−	0.972433i	$-0.574914\pi$
$223$	−104.000			−0.466368			−0.233184	+	0.972433i	$0.574914\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	128.598	−	74.2462i	0.566512	−	0.327076i	−0.189243	−	0.981930i	$-0.560603\pi$
$227$	128.598	−	74.2462i	0.566512	−	0.327076i	0.755755	+	0.654854i	$0.227270\pi$
$228$		0			0
$229$	−15.5000	+	26.8468i	−0.0676856	+	0.117235i	−0.897882	−	0.440236i	$-0.854895\pi$
$229$	−15.5000	+	26.8468i	−0.0676856	+	0.117235i	0.830197	+	0.557471i	$0.188228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	150.644	+	86.9741i	0.646539	+	0.373280i	0.787129	−	0.616788i	$-0.211567\pi$
$233$	150.644	+	86.9741i	0.646539	+	0.373280i	−0.140590	+	0.990068i	$0.544900\pi$
$234$		0			0
$235$	−81.0000	−	140.296i	−0.344681	−	0.597005i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	326.683i		−	1.36688i	−0.730009	−	0.683438i	$-0.760484\pi$
$239$		−	326.683i		−	1.36688i	0.730009	−	0.683438i	$-0.239516\pi$
$240$		0			0
$241$	−65.0000	−	112.583i	−0.269710	−	0.467151i	0.699077	−	0.715046i	$-0.253594\pi$
$241$	−65.0000	−	112.583i	−0.269710	−	0.467151i	−0.968787	+	0.247896i	$0.920261\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	11.5000	−	19.9186i	0.0465587	−	0.0806420i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			458.205i			1.82552i	0.408498	+	0.912759i	$0.366053\pi$
$251$			458.205i			1.82552i	−0.408498	+	0.912759i	$0.633947\pi$
$252$		0			0
$253$	−360.000			−1.42292
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−91.8559	−	53.0330i	−0.357416	−	0.206354i	0.310531	−	0.950563i	$-0.399493\pi$
$257$	−91.8559	−	53.0330i	−0.357416	−	0.206354i	−0.667947	+	0.744209i	$0.732827\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	146.969	−	84.8528i	0.558819	−	0.322634i	−0.193852	−	0.981031i	$-0.562098\pi$
$263$	146.969	−	84.8528i	0.558819	−	0.322634i	0.752671	+	0.658396i	$0.228765\pi$
$264$		0			0
$265$	−360.000			−1.35849
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−194.734	+	112.430i	−0.723920	+	0.417955i	−0.816194	−	0.577778i	$-0.803920\pi$
$269$	−194.734	+	112.430i	−0.723920	+	0.417955i	0.0922739	+	0.995734i	$0.470586\pi$
$270$		0			0
$271$	100.000	−	173.205i	0.369004	−	0.639133i	−0.620406	−	0.784281i	$-0.713032\pi$
$271$	100.000	−	173.205i	0.369004	−	0.639133i	0.989410	+	0.145147i	$0.0463656\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	128.598	+	74.2462i	0.467630	+	0.269986i
$276$		0			0
$277$	−68.5000	−	118.645i	−0.247292	−	0.428323i	0.715481	−	0.698632i	$-0.246207\pi$
$277$	−68.5000	−	118.645i	−0.247292	−	0.428323i	−0.962774	+	0.270309i	$0.912874\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			84.8528i			0.301967i	0.988536	+	0.150984i	$0.0482441\pi$
$281$			84.8528i			0.301967i	−0.988536	+	0.150984i	$0.951756\pi$
$282$		0			0
$283$	128.500	+	222.569i	0.454064	+	0.786461i	0.998634	−	0.0522540i	$-0.0166405\pi$
$283$	128.500	+	222.569i	0.454064	+	0.786461i	−0.544570	+	0.838715i	$0.683307\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.00173010	+	0.00299663i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			339.411i			1.15840i	0.815185	+	0.579200i	$0.196635\pi$
$293$			339.411i			1.15840i	−0.815185	+	0.579200i	$0.803365\pi$
$294$		0			0
$295$	−216.000			−0.732203
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−338.030	−	195.161i	−1.13053	−	0.652714i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	146.969	−	84.8528i	0.481867	−	0.278206i
$306$		0			0
$307$	127.000			0.413681			0.206840	−	0.978375i	$-0.433682\pi$
$307$	127.000			0.413681			0.206840	+	0.978375i	$0.433682\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−275.568	+	159.099i	−0.886069	+	0.511572i	−0.872655	−	0.488338i	$-0.837603\pi$
$311$	−275.568	+	159.099i	−0.886069	+	0.511572i	−0.0134146	+	0.999910i	$0.504270\pi$
$312$		0			0
$313$	68.5000	−	118.645i	0.218850	−	0.379059i	−0.735607	−	0.677409i	$-0.763103\pi$
$313$	68.5000	−	118.645i	0.218850	−	0.379059i	0.954457	+	0.298350i	$0.0964362\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−352.727	−	203.647i	−1.11270	−	0.642419i	−0.173174	−	0.984891i	$-0.555402\pi$
$317$	−352.727	−	203.647i	−1.11270	−	0.642419i	−0.939528	+	0.342472i	$0.888736\pi$
$318$		0			0
$319$	−360.000	−	623.538i	−1.12853	−	1.95467i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			16.9706i			0.0525404i
$324$		0			0
$325$	80.5000	+	139.430i	0.247692	+	0.429016i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	120.500	−	208.712i	0.364048	−	0.630550i	−0.624575	−	0.780965i	$-0.714728\pi$
$331$	120.500	−	208.712i	0.364048	−	0.630550i	0.988623	+	0.150415i	$0.0480610\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			97.5807i			0.291286i
$336$		0			0
$337$	527.000			1.56380			0.781899	−	0.623405i	$-0.214251\pi$
$337$	527.000			1.56380			0.781899	+	0.623405i	$0.214251\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	900.187	+	519.723i	2.63985	+	1.52412i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	220.454	−	127.279i	0.635314	−	0.366799i	−0.147493	−	0.989063i	$-0.547120\pi$
$347$	220.454	−	127.279i	0.635314	−	0.366799i	0.782807	+	0.622264i	$0.213787\pi$
$348$		0			0
$349$	400.000			1.14613			0.573066	−	0.819509i	$-0.305754\pi$
$349$	400.000			1.14613			0.573066	+	0.819509i	$0.305754\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	349.052	−	201.525i	0.988817	−	0.570894i	0.0838963	−	0.996474i	$-0.473264\pi$
$353$	349.052	−	201.525i	0.988817	−	0.570894i	0.904920	+	0.425581i	$0.139930\pi$
$354$		0			0
$355$	−135.000	+	233.827i	−0.380282	+	0.658667i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	29.3939	+	16.9706i	0.0818771	+	0.0472718i	0.540380	−	0.841421i	$-0.318281\pi$
$359$	29.3939	+	16.9706i	0.0818771	+	0.0472718i	−0.458502	+	0.888693i	$0.651614\pi$
$360$		0			0
$361$	180.000	+	311.769i	0.498615	+	0.863626i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	72.1249i		−	0.197602i
$366$		0			0
$367$	−111.500	−	193.124i	−0.303815	−	0.526223i	0.673182	−	0.739477i	$-0.264927\pi$
$367$	−111.500	−	193.124i	−0.303815	−	0.526223i	−0.976997	+	0.213254i	$0.931594\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−308.500	+	534.338i	−0.827078	+	1.43254i	0.0732432	+	0.997314i	$0.476665\pi$
$373$	−308.500	+	534.338i	−0.827078	+	1.43254i	−0.900321	+	0.435227i	$0.856668\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	780.646i		−	2.07068i
$378$		0			0
$379$	−361.000			−0.952507			−0.476253	−	0.879308i	$-0.658005\pi$
$379$	−361.000			−0.952507			−0.476253	+	0.879308i	$0.658005\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−367.423	−	212.132i	−0.959330	−	0.553870i	−0.0633634	−	0.997991i	$-0.520183\pi$
$383$	−367.423	−	212.132i	−0.959330	−	0.553870i	−0.895967	+	0.444121i	$0.853516\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	91.8559	−	53.0330i	0.236133	−	0.136332i	−0.377265	−	0.926105i	$-0.623135\pi$
$389$	91.8559	−	53.0330i	0.236133	−	0.136332i	0.613398	+	0.789774i	$0.289802\pi$
$390$		0			0
$391$	288.000			0.736573
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−290.265	+	167.584i	−0.734847	+	0.424264i
$396$		0			0
$397$	188.500	−	326.492i	0.474811	−	0.822397i	−0.524773	−	0.851242i	$-0.675850\pi$
$397$	188.500	−	326.492i	0.474811	−	0.822397i	0.999584	+	0.0288454i	$0.00918305\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−73.4847	−	42.4264i	−0.183254	−	0.105802i	0.405567	−	0.914065i	$-0.367074\pi$
$401$	−73.4847	−	42.4264i	−0.183254	−	0.105802i	−0.588820	+	0.808264i	$0.700407\pi$
$402$		0			0
$403$	563.500	+	976.011i	1.39826	+	2.42186i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	360.624i		−	0.886055i
$408$		0			0
$409$	344.500	+	596.692i	0.842298	+	1.45890i	0.887947	+	0.459946i	$0.152131\pi$
$409$	344.500	+	596.692i	0.842298	+	1.45890i	−0.0456487	+	0.998958i	$0.514535\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	225.000	−	389.711i	0.542169	−	0.939064i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	691.550i		−	1.65048i	−0.564783	−	0.825239i	$-0.691040\pi$
$419$		−	691.550i		−	1.65048i	0.564783	−	0.825239i	$-0.308960\pi$
$420$		0			0
$421$	281.000			0.667458			0.333729	−	0.942669i	$-0.391693\pi$
$421$	281.000			0.667458			0.333729	+	0.942669i	$0.391693\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−102.879	−	59.3970i	−0.242067	−	0.139758i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−378.446	+	218.496i	−0.878065	+	0.506951i	−0.870020	−	0.493016i	$-0.835894\pi$
$431$	−378.446	+	218.496i	−0.878065	+	0.506951i	−0.00804532	+	0.999968i	$0.502561\pi$
$432$		0			0
$433$	−593.000			−1.36952			−0.684758	−	0.728771i	$-0.740092\pi$
$433$	−593.000			−1.36952			−0.684758	+	0.728771i	$0.740092\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−14.6969	+	8.48528i	−0.0336314	+	0.0194171i
$438$		0			0
$439$	−236.000	+	408.764i	−0.537585	+	0.931125i	0.461448	+	0.887167i	$0.347330\pi$
$439$	−236.000	+	408.764i	−0.537585	+	0.931125i	−0.999033	+	0.0439580i	$0.986003\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−661.362	−	381.838i	−1.49292	−	0.861936i	−0.492950	−	0.870058i	$-0.664081\pi$
$443$	−661.362	−	381.838i	−1.49292	−	0.861936i	−0.999967	+	0.00812162i	$0.997415\pi$
$444$		0			0
$445$	288.000	+	498.831i	0.647191	+	1.12097i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			649.124i			1.44571i	0.690999	+	0.722855i	$0.257171\pi$
$449$			649.124i			1.44571i	−0.690999	+	0.722855i	$0.742829\pi$
$450$		0			0
$451$	−225.000	−	389.711i	−0.498891	−	0.864105i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−176.500	+	305.707i	−0.386214	+	0.668943i	−0.991937	−	0.126733i	$-0.959551\pi$
$457$	−176.500	+	305.707i	−0.386214	+	0.668943i	0.605722	+	0.795676i	$0.292884\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	543.058i		−	1.17800i	−0.808133	−	0.589000i	$-0.799522\pi$
$461$		−	543.058i		−	1.17800i	0.808133	−	0.589000i	$-0.200478\pi$
$462$		0			0
$463$	−337.000			−0.727862			−0.363931	−	0.931426i	$-0.618566\pi$
$463$	−337.000			−0.727862			−0.363931	+	0.931426i	$0.618566\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	3.67423	+	2.12132i	0.00786774	+	0.00454244i	0.503929	−	0.863745i	$-0.331887\pi$
$467$	3.67423	+	2.12132i	0.00786774	+	0.00454244i	−0.496061	+	0.868288i	$0.665221\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−863.445	+	498.510i	−1.82547	+	1.05393i
$474$		0			0
$475$	7.00000			0.0147368
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−367.423	+	212.132i	−0.767064	+	0.442864i	−0.831826	−	0.555036i	$-0.812704\pi$
$479$	−367.423	+	212.132i	−0.767064	+	0.442864i	0.0647625	+	0.997901i	$0.479371\pi$
$480$		0			0
$481$	195.500	−	338.616i	0.406445	−	0.703983i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−146.969	−	84.8528i	−0.303030	−	0.174954i
$486$		0			0
$487$	−143.500	−	248.549i	−0.294661	−	0.510368i	0.680245	−	0.732985i	$-0.261873\pi$
$487$	−143.500	−	248.549i	−0.294661	−	0.510368i	−0.974906	+	0.222617i	$0.928540\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	339.411i		−	0.691265i	−0.938370	−	0.345633i	$-0.887664\pi$
$491$		−	339.411i		−	0.691265i	0.938370	−	0.345633i	$-0.112336\pi$
$492$		0			0
$493$	288.000	+	498.831i	0.584178	+	1.01183i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	60.5000	−	104.789i	0.121242	−	0.209998i	−0.799015	−	0.601311i	$-0.794645\pi$
$499$	60.5000	−	104.789i	0.121242	−	0.209998i	0.920258	+	0.391312i	$0.127979\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	292.742i		−	0.581992i	−0.956724	−	0.290996i	$-0.906013\pi$
$503$		−	292.742i		−	0.581992i	0.956724	−	0.290996i	$-0.0939867\pi$
$504$		0			0
$505$	−90.0000			−0.178218
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−25.7196	−	14.8492i	−0.0505297	−	0.0291734i	0.474522	−	0.880243i	$-0.342621\pi$
$509$	−25.7196	−	14.8492i	−0.0505297	−	0.0291734i	−0.525052	+	0.851070i	$0.675954\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−84.5074	+	48.7904i	−0.164092	+	0.0947386i
$516$		0			0
$517$	−810.000			−1.56673
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−734.847	+	424.264i	−1.41045	+	0.814326i	−0.995431	−	0.0954841i	$-0.969560\pi$
$521$	−734.847	+	424.264i	−1.41045	+	0.814326i	−0.415024	+	0.909811i	$0.636227\pi$
$522$		0			0
$523$	−63.5000	+	109.985i	−0.121415	+	0.210297i	−0.920326	−	0.391153i	$-0.872076\pi$
$523$	−63.5000	+	109.985i	−0.121415	+	0.210297i	0.798911	+	0.601449i	$0.205410\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−720.150	−	415.779i	−1.36651	−	0.788954i
$528$		0			0
$529$	−120.500	−	208.712i	−0.227788	−	0.394541i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	487.904i		−	0.915392i
$534$		0			0
$535$	36.0000	+	62.3538i	0.0672897	+	0.116549i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.500000	−	0.866025i	0.000924214	−	0.00160079i	−0.865563	−	0.500800i	$-0.833039\pi$
$541$	0.500000	−	0.866025i	0.000924214	−	0.00160079i	0.866487	+	0.499199i	$0.166372\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			301.227i			0.552711i
$546$		0			0
$547$	−544.000			−0.994516			−0.497258	−	0.867603i	$-0.665660\pi$
$547$	−544.000			−0.994516			−0.497258	+	0.867603i	$0.665660\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−29.3939	−	16.9706i	−0.0533464	−	0.0307996i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−224.128	+	129.401i	−0.402385	+	0.232317i	−0.687512	−	0.726173i	$-0.741297\pi$
$557$	−224.128	+	129.401i	−0.402385	+	0.232317i	0.285128	+	0.958490i	$0.407964\pi$
$558$		0			0
$559$	−1081.00			−1.93381
$560$		0			0
$561$		0			0
$562$		0			0
$563$	473.976	−	273.650i	0.841876	−	0.486057i	−0.0160254	−	0.999872i	$-0.505101\pi$
$563$	473.976	−	273.650i	0.841876	−	0.486057i	0.857902	+	0.513814i	$0.171768\pi$
$564$		0			0
$565$	279.000	−	483.242i	0.493805	−	0.855296i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	606.249	+	350.018i	1.06546	+	0.615146i	0.926938	−	0.375213i	$-0.122431\pi$
$569$	606.249	+	350.018i	1.06546	+	0.615146i	0.138525	+	0.990359i	$0.455764\pi$
$570$		0			0
$571$	−344.500	−	596.692i	−0.603327	−	1.04499i	−0.992313	−	0.123750i	$-0.960508\pi$
$571$	−344.500	−	596.692i	−0.603327	−	1.04499i	0.388986	−	0.921244i	$-0.372825\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	118.794i		−	0.206598i
$576$		0			0
$577$	476.500	+	825.322i	0.825823	+	1.43037i	0.901289	+	0.433219i	$0.142622\pi$
$577$	476.500	+	825.322i	0.825823	+	1.43037i	−0.0754653	+	0.997148i	$0.524044\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−900.000	+	1558.85i	−1.54374	+	2.67383i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			780.646i			1.32989i	0.746892	+	0.664945i	$0.231545\pi$
$587$			780.646i			1.32989i	−0.746892	+	0.664945i	$0.768455\pi$
$588$		0			0
$589$	49.0000			0.0831919
$590$		0			0
$591$		0			0
$592$		0			0
$593$	753.218	+	434.871i	1.27018	+	0.733340i	0.975022	−	0.222108i	$-0.0712937\pi$
$593$	753.218	+	434.871i	1.27018	+	0.733340i	0.295160	+	0.955448i	$0.404627\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	881.816	−	509.117i	1.47215	−	0.849945i	0.472638	−	0.881257i	$-0.343302\pi$
$599$	881.816	−	509.117i	1.47215	−	0.849945i	0.999510	+	0.0313121i	$0.00996858\pi$
$600$		0			0
$601$	961.000			1.59900			0.799501	−	0.600665i	$-0.205097\pi$
$601$	961.000			1.59900			0.799501	+	0.600665i	$0.205097\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1208.82	+	697.914i	−1.99805	+	1.15358i
$606$		0			0
$607$	371.500	−	643.457i	0.612026	−	1.06006i	−0.378872	−	0.925449i	$-0.623688\pi$
$607$	371.500	−	643.457i	0.612026	−	1.06006i	0.990898	−	0.134612i	$-0.0429787\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−760.567	−	439.113i	−1.24479	−	0.718680i
$612$		0			0
$613$	−280.000	−	484.974i	−0.456770	−	0.791149i	0.542018	−	0.840367i	$-0.317660\pi$
$613$	−280.000	−	484.974i	−0.456770	−	0.791149i	−0.998788	+	0.0492180i	$0.984327\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	46.6690i		−	0.0756387i	−0.999285	−	0.0378193i	$-0.987959\pi$
$617$		−	46.6690i		−	0.0756387i	0.999285	−	0.0378193i	$-0.0120411\pi$
$618$		0			0
$619$	104.500	+	180.999i	0.168821	+	0.292406i	0.938005	−	0.346620i	$-0.112671\pi$
$619$	104.500	+	180.999i	0.168821	+	0.292406i	−0.769185	+	0.639026i	$0.779337\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	200.500	−	347.276i	0.320800	−	0.555642i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			288.500i			0.458664i
$630$		0			0
$631$	560.000			0.887480			0.443740	−	0.896156i	$-0.353651\pi$
$631$	560.000			0.887480			0.443740	+	0.896156i	$0.353651\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−62.4620	−	36.0624i	−0.0983653	−	0.0567913i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	102.879	−	59.3970i	0.160497	−	0.0926630i	−0.417600	−	0.908631i	$-0.637129\pi$
$641$	102.879	−	59.3970i	0.160497	−	0.0926630i	0.578097	+	0.815968i	$0.303795\pi$
$642$		0			0
$643$	−113.000			−0.175739			−0.0878694	−	0.996132i	$-0.528006\pi$
$643$	−113.000			−0.175739			−0.0878694	+	0.996132i	$0.528006\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−459.279	+	265.165i	−0.709860	+	0.409838i	−0.811009	−	0.585033i	$-0.801081\pi$
$647$	−459.279	+	265.165i	−0.709860	+	0.409838i	0.101149	+	0.994871i	$0.467748\pi$
$648$		0			0
$649$	−540.000	+	935.307i	−0.832049	+	1.44115i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.3712	−	10.6066i	−0.0281335	−	0.0162429i	0.485867	−	0.874033i	$-0.338504\pi$
$653$	−18.3712	−	10.6066i	−0.0281335	−	0.0162429i	−0.514001	+	0.857790i	$0.671837\pi$
$654$		0			0
$655$	117.000	+	202.650i	0.178626	+	0.309389i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	933.381i		−	1.41636i	−0.706032	−	0.708180i	$-0.749517\pi$
$659$		−	933.381i		−	1.41636i	0.706032	−	0.708180i	$-0.250483\pi$
$660$		0			0
$661$	155.500	+	269.334i	0.235250	+	0.407464i	0.959345	−	0.282235i	$-0.0910759\pi$
$661$	155.500	+	269.334i	0.235250	+	0.407464i	−0.724096	+	0.689700i	$0.757743\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−288.000	+	498.831i	−0.431784	+	0.747872i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	848.528i		−	1.26457i
$672$		0			0
$673$	−223.000			−0.331352			−0.165676	−	0.986180i	$-0.552981\pi$
$673$	−223.000			−0.331352			−0.165676	+	0.986180i	$0.552981\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	352.727	+	203.647i	0.521014	+	0.300808i	0.737349	−	0.675511i	$-0.236077\pi$
$677$	352.727	+	203.647i	0.521014	+	0.300808i	−0.216335	+	0.976319i	$0.569410\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−279.242	+	161.220i	−0.408846	+	0.236047i	−0.690294	−	0.723529i	$-0.742519\pi$
$683$	−279.242	+	161.220i	−0.408846	+	0.236047i	0.281448	+	0.959577i	$0.409185\pi$
$684$		0			0
$685$	−288.000			−0.420438
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1690.15	+	975.807i	−2.45304	+	1.41627i
$690$		0			0
$691$	−399.500	+	691.954i	−0.578148	+	1.00138i	0.417544	+	0.908657i	$0.362891\pi$
$691$	−399.500	+	691.954i	−0.578148	+	1.00138i	−0.995692	+	0.0927244i	$0.970442\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−113.901	−	65.7609i	−0.163887	−	0.0946200i
$696$		0			0
$697$	180.000	+	311.769i	0.258250	+	0.447301i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	458.205i		−	0.653645i	−0.945086	−	0.326823i	$-0.894022\pi$
$701$		−	458.205i		−	0.653645i	0.945086	−	0.326823i	$-0.105978\pi$
$702$		0			0
$703$	−8.50000	−	14.7224i	−0.0120910	−	0.0209423i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	20.0000	−	34.6410i	0.0282087	−	0.0488590i	−0.851576	−	0.524231i	$-0.824353\pi$
$709$	20.0000	−	34.6410i	0.0282087	−	0.0488590i	0.879785	+	0.475372i	$0.157686\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	831.558i		−	1.16628i
$714$		0			0
$715$	−2070.00			−2.89510
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−723.824	−	417.900i	−1.00671	−	0.581224i	−0.0964830	−	0.995335i	$-0.530759\pi$
$719$	−723.824	−	417.900i	−1.00671	−	0.581224i	−0.910227	+	0.414111i	$0.864093\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	205.757	−	118.794i	0.283803	−	0.163854i
$726$		0			0
$727$	943.000			1.29711			0.648556	−	0.761167i	$-0.275373\pi$
$727$	943.000			1.29711			0.648556	+	0.761167i	$0.275373\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	690.756	−	398.808i	0.944947	−	0.545565i
$732$		0			0
$733$	263.500	−	456.395i	0.359482	−	0.622640i	−0.628393	−	0.777896i	$-0.716287\pi$
$733$	263.500	−	456.395i	0.359482	−	0.622640i	0.987874	+	0.155256i	$0.0496202\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	422.537	+	243.952i	0.573320	+	0.331007i
$738$		0			0
$739$	495.500	+	858.231i	0.670501	+	1.16134i	0.977762	+	0.209716i	$0.0672540\pi$
$739$	495.500	+	858.231i	0.670501	+	1.16134i	−0.307262	+	0.951625i	$0.599413\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	530.330i		−	0.713769i	−0.934149	−	0.356884i	$-0.883839\pi$
$743$		−	530.330i		−	0.713769i	0.934149	−	0.356884i	$-0.116161\pi$
$744$		0			0
$745$	−360.000	−	623.538i	−0.483221	−	0.836964i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−380.500	+	659.045i	−0.506658	+	0.877557i	0.493313	+	0.869852i	$0.335786\pi$
$751$	−380.500	+	659.045i	−0.506658	+	0.877557i	−0.999970	+	0.00770489i	$0.997547\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	169.706i		−	0.224776i
$756$		0			0
$757$	1190.00			1.57199			0.785997	−	0.618230i	$-0.212150\pi$
$757$	1190.00			1.57199			0.785997	+	0.618230i	$0.212150\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	690.756	+	398.808i	0.907695	+	0.524058i	0.879689	−	0.475550i	$-0.157751\pi$
$761$	690.756	+	398.808i	0.907695	+	0.524058i	0.0280064	+	0.999608i	$0.491084\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1014.09	+	585.484i	−1.32215	+	0.763343i
$768$		0			0
$769$	−599.000			−0.778934			−0.389467	−	0.921040i	$-0.627341\pi$
$769$	−599.000			−0.778934			−0.389467	+	0.921040i	$0.627341\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	312.310	−	180.312i	0.404023	−	0.233263i	−0.284195	−	0.958766i	$-0.591726\pi$
$773$	312.310	−	180.312i	0.404023	−	0.233263i	0.688218	+	0.725504i	$0.258393\pi$
$774$		0			0
$775$	−171.500	+	297.047i	−0.221290	+	0.383286i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18.3712	−	10.6066i	−0.0235830	−	0.0136157i
$780$		0			0
$781$	675.000	+	1169.13i	0.864277	+	1.49697i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1187.94i			1.51330i
$786$		0			0
$787$	85.0000	+	147.224i	0.108005	+	0.187070i	0.914962	−	0.403540i	$-0.132220\pi$
$787$	85.0000	+	147.224i	0.108005	+	0.187070i	−0.806957	+	0.590610i	$0.798887\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	460.000	−	796.743i	0.580076	−	1.00472i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	322.441i		−	0.404568i	−0.979327	−	0.202284i	$-0.935164\pi$
$797$		−	322.441i		−	0.404568i	0.979327	−	0.202284i	$-0.0648364\pi$
$798$		0			0
$799$	648.000			0.811014
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−312.310	−	180.312i	−0.388929	−	0.224548i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−268.219	+	154.856i	−0.331544	+	0.191417i	−0.656526	−	0.754303i	$-0.727975\pi$
$809$	−268.219	+	154.856i	−0.331544	+	0.191417i	0.324982	+	0.945720i	$0.394642\pi$
$810$		0			0
$811$	−1190.00			−1.46732			−0.733662	−	0.679514i	$-0.762190\pi$
$811$	−1190.00			−1.46732			−0.733662	+	0.679514i	$0.762190\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	734.847	−	424.264i	0.901653	−	0.520569i
$816$		0			0
$817$	−23.5000	+	40.7032i	−0.0287638	+	0.0498203i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−496.022	−	286.378i	−0.604168	−	0.348816i	0.166512	−	0.986039i	$-0.446750\pi$
$821$	−496.022	−	286.378i	−0.604168	−	0.348816i	−0.770679	+	0.637223i	$0.780083\pi$
$822$		0			0
$823$	−580.000	−	1004.59i	−0.704739	−	1.22064i	−0.966786	−	0.255588i	$-0.917731\pi$
$823$	−580.000	−	1004.59i	−0.704739	−	1.22064i	0.262047	−	0.965055i	$-0.415602\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	343.654i		−	0.415543i	−0.978177	−	0.207771i	$-0.933379\pi$
$827$		−	343.654i		−	0.415543i	0.978177	−	0.207771i	$-0.0666210\pi$
$828$		0			0
$829$	539.500	+	934.441i	0.650784	+	1.12719i	0.982933	+	0.183964i	$0.0588930\pi$
$829$	539.500	+	934.441i	0.650784	+	1.12719i	−0.332149	+	0.943227i	$0.607774\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−225.000	+	389.711i	−0.269461	+	0.466720i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	33.9411i		−	0.0404543i	−0.999795	−	0.0202271i	$-0.993561\pi$
$839$		−	33.9411i		−	0.0404543i	0.999795	−	0.0202271i	$-0.00643893\pi$
$840$		0			0
$841$	−311.000			−0.369798
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1322.72	−	763.675i	−1.56535	−	0.903758i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−249.848	+	144.250i	−0.293593	+	0.169506i
$852$		0			0
$853$	1033.00			1.21102			0.605510	−	0.795838i	$-0.292969\pi$
$853$	1033.00			1.21102			0.605510	+	0.795838i	$0.292969\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	205.757	−	118.794i	0.240090	−	0.138616i	−0.375128	−	0.926973i	$-0.622401\pi$
$857$	205.757	−	118.794i	0.240090	−	0.138616i	0.615218	+	0.788357i	$0.289068\pi$
$858$		0			0
$859$	445.000	−	770.763i	0.518044	−	0.897279i	−0.481736	−	0.876316i	$-0.659994\pi$
$859$	445.000	−	770.763i	0.518044	−	0.897279i	0.999780	−	0.0209626i	$-0.00667308\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−400.492	−	231.224i	−0.464069	−	0.267930i	0.249685	−	0.968327i	$-0.419673\pi$
$863$	−400.492	−	231.224i	−0.464069	−	0.267930i	−0.713754	+	0.700397i	$0.753006\pi$
$864$		0			0
$865$	−540.000	−	935.307i	−0.624277	−	1.08128i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1675.84i			1.92847i
$870$		0			0
$871$	264.500	+	458.127i	0.303674	+	0.525979i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	752.000	−	1302.50i	0.857469	−	1.48518i	−0.0168671	−	0.999858i	$-0.505369\pi$
$877$	752.000	−	1302.50i	0.857469	−	1.48518i	0.874336	−	0.485322i	$-0.161297\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	254.558i		−	0.288943i	−0.989509	−	0.144471i	$-0.953852\pi$
$881$		−	254.558i		−	0.288943i	0.989509	−	0.144471i	$-0.0461481\pi$
$882$		0			0
$883$	1313.00			1.48698			0.743488	−	0.668749i	$-0.233170\pi$
$883$	1313.00			1.48698			0.743488	+	0.668749i	$0.233170\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1010.41	+	583.363i	1.13914	+	0.657681i	0.946217	−	0.323533i	$-0.104871\pi$
$887$	1010.41	+	583.363i	1.13914	+	0.657681i	0.192920	+	0.981214i	$0.438204\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.0681	+	19.0919i	−0.0370304	+	0.0213795i
$894$		0			0
$895$	−846.000			−0.945251
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1440.30	−	831.558i	1.60211	−	0.924981i
$900$		0			0
$901$	720.000	−	1247.08i	0.799112	−	1.38410i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−995.718	−	574.878i	−1.10024	−	0.635224i
$906$		0			0
$907$	408.500	+	707.543i	0.450386	+	0.780091i	0.998410	−	0.0563714i	$-0.0179531\pi$
$907$	408.500	+	707.543i	0.450386	+	0.780091i	−0.548024	+	0.836463i	$0.684620\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			899.440i			0.987310i	0.869658	+	0.493655i	$0.164340\pi$
$911$			899.440i			0.987310i	−0.869658	+	0.493655i	$0.835660\pi$
$912$		0			0
$913$	−1125.00	−	1948.56i	−1.23220	−	2.13424i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	864.500	−	1497.36i	0.940696	−	1.62933i	0.176549	−	0.984292i	$-0.443506\pi$
$919$	864.500	−	1497.36i	0.940696	−	1.62933i	0.764147	−	0.645042i	$-0.223160\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			1463.71i			1.58582i
$924$		0			0
$925$	119.000			0.128649
$926$		0			0
$927$		0			0
$928$		0			0
$929$	826.703	+	477.297i	0.889885	+	0.513775i	0.873905	−	0.486097i	$-0.161580\pi$
$929$	826.703	+	477.297i	0.889885	+	0.513775i	0.0159798	+	0.999872i	$0.494913\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1322.72	−	763.675i	1.41468	−	0.816765i
$936$		0			0
$937$	−377.000			−0.402348			−0.201174	−	0.979556i	$-0.564476\pi$
$937$	−377.000			−0.402348			−0.201174	+	0.979556i	$0.564476\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	837.725	−	483.661i	0.890250	−	0.513986i	0.0162259	−	0.999868i	$-0.494835\pi$
$941$	837.725	−	483.661i	0.890250	−	0.513986i	0.874024	+	0.485882i	$0.161502\pi$
$942$		0			0
$943$	−180.000	+	311.769i	−0.190880	+	0.330614i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1157.38	+	668.216i	1.22216	+	0.705613i	0.965378	−	0.260856i	$-0.0840049\pi$
$947$	1157.38	+	668.216i	1.22216	+	0.705613i	0.256781	+	0.966470i	$0.417338\pi$
$948$		0			0
$949$	−195.500	−	338.616i	−0.206006	−	0.356813i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	763.675i		−	0.801338i	−0.916223	−	0.400669i	$-0.868778\pi$
$953$		−	763.675i		−	0.801338i	0.916223	−	0.400669i	$-0.131222\pi$
$954$		0			0
$955$	−513.000	−	888.542i	−0.537173	−	0.930411i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−720.000	+	1247.08i	−0.749220	+	1.29769i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	436.992i		−	0.452841i
$966$		0			0
$967$	−1057.00			−1.09307			−0.546536	−	0.837436i	$-0.684054\pi$
$967$	−1057.00			−1.09307			−0.546536	+	0.837436i	$0.684054\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−955.301	−	551.543i	−0.983832	−	0.568016i	−0.0804072	−	0.996762i	$-0.525622\pi$
$971$	−955.301	−	551.543i	−0.983832	−	0.568016i	−0.903425	+	0.428746i	$0.858955\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	275.568	−	159.099i	0.282055	−	0.162844i	−0.352298	−	0.935888i	$-0.614600\pi$
$977$	275.568	−	159.099i	0.282055	−	0.162844i	0.634353	+	0.773043i	$0.281266\pi$
$978$		0			0
$979$	2880.00			2.94178
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−455.605	+	263.044i	−0.463484	+	0.267593i	−0.713508	−	0.700647i	$-0.752895\pi$
$983$	−455.605	+	263.044i	−0.463484	+	0.267593i	0.250024	+	0.968240i	$0.419562\pi$
$984$		0			0
$985$	324.000	−	561.184i	0.328934	−	0.569730i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	690.756	+	398.808i	0.698439	+	0.403244i
$990$		0			0
$991$	−215.500	−	373.257i	−0.217457	−	0.376647i	0.736573	−	0.676358i	$-0.236443\pi$
$991$	−215.500	−	373.257i	−0.217457	−	0.376647i	−0.954030	+	0.299712i	$0.903110\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			678.823i			0.682234i
$996$		0			0
$997$	68.5000	+	118.645i	0.0687061	+	0.119002i	0.898332	−	0.439317i	$-0.144780\pi$
$997$	68.5000	+	118.645i	0.0687061	+	0.119002i	−0.829626	+	0.558320i	$0.811446\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.bk.a.1745.1		4
3.2	odd	2	inner	1764.3.bk.a.1745.2		4
7.2	even	3		1764.3.c.a.197.2		2
7.3	odd	6		252.3.bk.a.53.1	✓	4
7.4	even	3	inner	1764.3.bk.a.557.2		4
7.5	odd	6		1764.3.c.d.197.1		2
7.6	odd	2		252.3.bk.a.233.2	yes	4
21.2	odd	6		1764.3.c.a.197.1		2
21.5	even	6		1764.3.c.d.197.2		2
21.11	odd	6	inner	1764.3.bk.a.557.1		4
21.17	even	6		252.3.bk.a.53.2	yes	4
21.20	even	2		252.3.bk.a.233.1	yes	4
28.3	even	6		1008.3.dc.c.305.1		4
28.27	even	2		1008.3.dc.c.737.2		4
84.59	odd	6		1008.3.dc.c.305.2		4
84.83	odd	2		1008.3.dc.c.737.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.bk.a.53.1	✓	4	7.3	odd	6
252.3.bk.a.53.2	yes	4	21.17	even	6
252.3.bk.a.233.1	yes	4	21.20	even	2
252.3.bk.a.233.2	yes	4	7.6	odd	2
1008.3.dc.c.305.1		4	28.3	even	6
1008.3.dc.c.305.2		4	84.59	odd	6
1008.3.dc.c.737.1		4	84.83	odd	2
1008.3.dc.c.737.2		4	28.27	even	2
1764.3.c.a.197.1		2	21.2	odd	6
1764.3.c.a.197.2		2	7.2	even	3
1764.3.c.d.197.1		2	7.5	odd	6
1764.3.c.d.197.2		2	21.5	even	6
1764.3.bk.a.557.1		4	21.11	odd	6	inner
1764.3.bk.a.557.2		4	7.4	even	3	inner
1764.3.bk.a.1745.1		4	1.1	even	1	trivial
1764.3.bk.a.1745.2		4	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q+(-3.67423 - 2.12132i) q^{5} +O(q^{10})\)	\(q+(-3.67423 - 2.12132i) q^{5} +(-18.3712 + 10.6066i) q^{11} -23.0000 q^{13} +(14.6969 - 8.48528i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(14.6969 + 8.48528i) q^{23} +(-3.50000 - 6.06218i) q^{25} +33.9411i q^{29} +(-24.5000 - 42.4352i) q^{31} +(-8.50000 + 14.7224i) q^{37} +21.2132i q^{41} +47.0000 q^{43} +(33.0681 + 19.0919i) q^{47} +(73.4847 - 42.4264i) q^{53} +90.0000 q^{55} +(44.0908 - 25.4558i) q^{59} +(-20.0000 + 34.6410i) q^{61} +(84.5074 + 48.7904i) q^{65} +(-11.5000 - 19.9186i) q^{67} -63.6396i q^{71} +(8.50000 + 14.7224i) q^{73} +(39.5000 - 68.4160i) q^{79} +106.066i q^{83} -72.0000 q^{85} +(-117.576 - 67.8823i) q^{89} +(3.67423 - 2.12132i) q^{95} +40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 92 q^{13} - 2 q^{19} - 14 q^{25} - 98 q^{31} - 34 q^{37} + 188 q^{43} + 360 q^{55} - 80 q^{61} - 46 q^{67} + 34 q^{73} + 158 q^{79} - 288 q^{85} + 160 q^{97}+O(q^{100}) \) 4 * q - 92 * q^13 - 2 * q^19 - 14 * q^25 - 98 * q^31 - 34 * q^37 + 188 * q^43 + 360 * q^55 - 80 * q^61 - 46 * q^67 + 34 * q^73 + 158 * q^79 - 288 * q^85 + 160 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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