LMFDB - Embedded newform 828.4.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [828,4,Mod(1,828)] 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("828.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(828, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 828 = 2^{2} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	828.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.8535814848$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 92)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.75153$ of defining polynomial
Character	$\chi$	$=$	828.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.78065 q^{5} +9.15411 q^{7} +44.5989 q^{11} -68.6307 q^{13} -134.544 q^{17} -148.629 q^{19} +23.0000 q^{23} -47.9002 q^{25} -98.5939 q^{29} -15.7647 q^{31} +80.3791 q^{35} +333.418 q^{37} -337.315 q^{41} +148.160 q^{43} +375.729 q^{47} -259.202 q^{49} +207.052 q^{53} +391.608 q^{55} -511.174 q^{59} -435.177 q^{61} -602.622 q^{65} -914.462 q^{67} +162.191 q^{71} +370.418 q^{73} +408.264 q^{77} -935.424 q^{79} +923.690 q^{83} -1181.39 q^{85} -61.8242 q^{89} -628.253 q^{91} -1305.06 q^{95} +103.550 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 10 q^{5} - 46 q^{7} + 64 q^{11} - 44 q^{13} + 88 q^{17} - 94 q^{19} + 69 q^{23} + 181 q^{25} - 308 q^{29} - 140 q^{31} - 192 q^{35} + 26 q^{37} - 584 q^{41} - 478 q^{43} + 28 q^{47} + 695 q^{49} - 356 q^{53}+ \cdots + 736 q^{97}+O(q^{100}) $ 3 * q + 10 * q^5 - 46 * q^7 + 64 * q^11 - 44 * q^13 + 88 * q^17 - 94 * q^19 + 69 * q^23 + 181 * q^25 - 308 * q^29 - 140 * q^31 - 192 * q^35 + 26 * q^37 - 584 * q^41 - 478 * q^43 + 28 * q^47 + 695 * q^49 - 356 * q^53 - 1004 * q^55 - 144 * q^59 - 1052 * q^61 - 150 * q^65 - 2008 * q^67 - 360 * q^71 - 252 * q^73 + 576 * q^77 - 720 * q^79 + 1404 * q^83 - 1632 * q^85 - 534 * q^89 - 1526 * q^91 + 108 * q^95 + 736 * q^97

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
$5$	8.78065	0.785365	0.392683	−	0.919674i	$-0.371547\pi$
$5$	8.78065	0.785365	0.392683	+	0.919674i	$0.371547\pi$
$6$	0	0
$7$	9.15411	0.494276	0.247138	−	0.968980i	$-0.420510\pi$
$7$	9.15411	0.494276	0.247138	+	0.968980i	$0.420510\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	44.5989	1.22246	0.611231	−	0.791452i	$-0.290675\pi$
$11$	44.5989	1.22246	0.611231	+	0.791452i	$0.290675\pi$
$12$	0	0
$13$	−68.6307	−1.46421	−0.732105	−	0.681192i	$-0.761462\pi$
$13$	−68.6307	−1.46421	−0.732105	+	0.681192i	$0.761462\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−134.544	−1.91952	−0.959758	−	0.280829i	$-0.909391\pi$
$17$	−134.544	−1.91952	−0.959758	+	0.280829i	$0.909391\pi$
$18$	0	0
$19$	−148.629	−1.79462	−0.897310	−	0.441402i	$-0.854481\pi$
$19$	−148.629	−1.79462	−0.897310	+	0.441402i	$0.854481\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−47.9002	−0.383201
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−98.5939	−0.631325	−0.315663	−	0.948871i	$-0.602227\pi$
$29$	−98.5939	−0.631325	−0.315663	+	0.948871i	$0.602227\pi$
$30$	0	0
$31$	−15.7647	−0.0913363	−0.0456682	−	0.998957i	$-0.514542\pi$
$31$	−15.7647	−0.0913363	−0.0456682	+	0.998957i	$0.514542\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	80.3791	0.388187
$36$	0	0
$37$	333.418	1.48145	0.740724	−	0.671809i	$-0.234482\pi$
$37$	333.418	1.48145	0.740724	+	0.671809i	$0.234482\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−337.315	−1.28487	−0.642436	−	0.766339i	$-0.722076\pi$
$41$	−337.315	−1.28487	−0.642436	+	0.766339i	$0.722076\pi$
$42$	0	0
$43$	148.160	0.525446	0.262723	−	0.964871i	$-0.415380\pi$
$43$	148.160	0.525446	0.262723	+	0.964871i	$0.415380\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	375.729	1.16608	0.583039	−	0.812444i	$-0.301863\pi$
$47$	375.729	1.16608	0.583039	+	0.812444i	$0.301863\pi$
$48$	0	0
$49$	−259.202	−0.755692
$50$	0	0
$51$	0	0
$52$	0	0
$53$	207.052	0.536619	0.268309	−	0.963333i	$-0.413535\pi$
$53$	207.052	0.536619	0.268309	+	0.963333i	$0.413535\pi$
$54$	0	0
$55$	391.608	0.960080
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−511.174	−1.12795	−0.563976	−	0.825791i	$-0.690729\pi$
$59$	−511.174	−1.12795	−0.563976	+	0.825791i	$0.690729\pi$
$60$	0	0
$61$	−435.177	−0.913420	−0.456710	−	0.889616i	$-0.650972\pi$
$61$	−435.177	−0.913420	−0.456710	+	0.889616i	$0.650972\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−602.622	−1.14994
$66$	0	0
$67$	−914.462	−1.66745	−0.833727	−	0.552178i	$-0.813797\pi$
$67$	−914.462	−1.66745	−0.833727	+	0.552178i	$0.813797\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	162.191	0.271106	0.135553	−	0.990770i	$-0.456719\pi$
$71$	162.191	0.271106	0.135553	+	0.990770i	$0.456719\pi$
$72$	0	0
$73$	370.418	0.593892	0.296946	−	0.954894i	$-0.404032\pi$
$73$	370.418	0.593892	0.296946	+	0.954894i	$0.404032\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	408.264	0.604233
$78$	0	0
$79$	−935.424	−1.33219	−0.666097	−	0.745865i	$-0.732036\pi$
$79$	−935.424	−1.33219	−0.666097	+	0.745865i	$0.732036\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	923.690	1.22154	0.610772	−	0.791806i	$-0.290859\pi$
$83$	923.690	1.22154	0.610772	+	0.791806i	$0.290859\pi$
$84$	0	0
$85$	−1181.39	−1.50752
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−61.8242	−0.0736332	−0.0368166	−	0.999322i	$-0.511722\pi$
$89$	−61.8242	−0.0736332	−0.0368166	+	0.999322i	$0.511722\pi$
$90$	0	0
$91$	−628.253	−0.723723
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1305.06	−1.40943
$96$	0	0
$97$	103.550	0.108391	0.0541955	−	0.998530i	$-0.482741\pi$
$97$	103.550	0.108391	0.0541955	+	0.998530i	$0.482741\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1662.94	1.63830	0.819151	−	0.573577i	$-0.194445\pi$
$101$	1662.94	1.63830	0.819151	+	0.573577i	$0.194445\pi$
$102$	0	0
$103$	−649.982	−0.621793	−0.310896	−	0.950444i	$-0.600629\pi$
$103$	−649.982	−0.621793	−0.310896	+	0.950444i	$0.600629\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−596.248	−0.538705	−0.269353	−	0.963042i	$-0.586810\pi$
$107$	−596.248	−0.538705	−0.269353	+	0.963042i	$0.586810\pi$
$108$	0	0
$109$	1273.48	1.11906	0.559529	−	0.828811i	$-0.310982\pi$
$109$	1273.48	1.11906	0.559529	+	0.828811i	$0.310982\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	581.544	0.484133	0.242067	−	0.970260i	$-0.422175\pi$
$113$	581.544	0.484133	0.242067	+	0.970260i	$0.422175\pi$
$114$	0	0
$115$	201.955	0.163760
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1231.63	−0.948770
$120$	0	0
$121$	658.066	0.494414
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1518.18	−1.08632
$126$	0	0
$127$	−1198.55	−0.837436	−0.418718	−	0.908116i	$-0.637520\pi$
$127$	−1198.55	−0.837436	−0.418718	+	0.908116i	$0.637520\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1944.94	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$131$	−1944.94	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$132$	0	0
$133$	−1360.56	−0.887037
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1141.23	−0.711691	−0.355846	−	0.934545i	$-0.615807\pi$
$137$	−1141.23	−0.711691	−0.355846	+	0.934545i	$0.615807\pi$
$138$	0	0
$139$	−1210.30	−0.738534	−0.369267	−	0.929323i	$-0.620391\pi$
$139$	−1210.30	−0.738534	−0.369267	+	0.929323i	$0.620391\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3060.86	−1.78994
$144$	0	0
$145$	−865.719	−0.495821
$146$	0	0
$147$	0	0
$148$	0	0
$149$	332.410	0.182766	0.0913829	−	0.995816i	$-0.470871\pi$
$149$	332.410	0.182766	0.0913829	+	0.995816i	$0.470871\pi$
$150$	0	0
$151$	2661.90	1.43459	0.717293	−	0.696772i	$-0.245381\pi$
$151$	2661.90	1.43459	0.717293	+	0.696772i	$0.245381\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−138.424	−0.0717324
$156$	0	0
$157$	−2310.24	−1.17438	−0.587190	−	0.809449i	$-0.699766\pi$
$157$	−2310.24	−1.17438	−0.587190	+	0.809449i	$0.699766\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	210.545	0.103064
$162$	0	0
$163$	−242.541	−0.116548	−0.0582738	−	0.998301i	$-0.518560\pi$
$163$	−242.541	−0.116548	−0.0582738	+	0.998301i	$0.518560\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1188.59	−0.550753	−0.275376	−	0.961336i	$-0.588802\pi$
$167$	−1188.59	−0.550753	−0.275376	+	0.961336i	$0.588802\pi$
$168$	0	0
$169$	2513.17	1.14391
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2409.12	1.05874	0.529371	−	0.848391i	$-0.322428\pi$
$173$	2409.12	1.05874	0.529371	+	0.848391i	$0.322428\pi$
$174$	0	0
$175$	−438.484	−0.189407
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−773.725	−0.323078	−0.161539	−	0.986866i	$-0.551646\pi$
$179$	−773.725	−0.323078	−0.161539	+	0.986866i	$0.551646\pi$
$180$	0	0
$181$	−3145.60	−1.29177	−0.645886	−	0.763434i	$-0.723512\pi$
$181$	−3145.60	−1.29177	−0.645886	+	0.763434i	$0.723512\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2927.63	1.16348
$186$	0	0
$187$	−6000.53	−2.34654
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4839.43	1.83335	0.916673	−	0.399638i	$-0.130864\pi$
$191$	4839.43	1.83335	0.916673	+	0.399638i	$0.130864\pi$
$192$	0	0
$193$	2184.38	0.814688	0.407344	−	0.913275i	$-0.366455\pi$
$193$	2184.38	0.814688	0.407344	+	0.913275i	$0.366455\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	465.129	0.168219	0.0841094	−	0.996457i	$-0.473195\pi$
$197$	465.129	0.168219	0.0841094	+	0.996457i	$0.473195\pi$
$198$	0	0
$199$	−742.721	−0.264573	−0.132287	−	0.991211i	$-0.542232\pi$
$199$	−742.721	−0.264573	−0.132287	+	0.991211i	$0.542232\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−902.540	−0.312049
$204$	0	0
$205$	−2961.85	−1.00909
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6628.68	−2.19385
$210$	0	0
$211$	4943.94	1.61306	0.806529	−	0.591195i	$-0.201344\pi$
$211$	4943.94	1.61306	0.806529	+	0.591195i	$0.201344\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1300.94	0.412667
$216$	0	0
$217$	−144.312	−0.0451453
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9233.86	2.81057
$222$	0	0
$223$	−6004.69	−1.80316	−0.901578	−	0.432617i	$-0.857590\pi$
$223$	−6004.69	−1.80316	−0.901578	+	0.432617i	$0.857590\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1222.31	−0.357391	−0.178696	−	0.983904i	$-0.557188\pi$
$227$	−1222.31	−0.357391	−0.178696	+	0.983904i	$0.557188\pi$
$228$	0	0
$229$	−9.58750	−0.00276664	−0.00138332	−	0.999999i	$-0.500440\pi$
$229$	−9.58750	−0.00276664	−0.00138332	+	0.999999i	$0.500440\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3060.45	−0.860501	−0.430250	−	0.902710i	$-0.641575\pi$
$233$	−3060.45	−0.860501	−0.430250	+	0.902710i	$0.641575\pi$
$234$	0	0
$235$	3299.14	0.915797
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−799.347	−0.216341	−0.108170	−	0.994132i	$-0.534499\pi$
$239$	−799.347	−0.216341	−0.108170	+	0.994132i	$0.534499\pi$
$240$	0	0
$241$	−125.753	−0.0336120	−0.0168060	−	0.999859i	$-0.505350\pi$
$241$	−125.753	−0.0336120	−0.0168060	+	0.999859i	$0.505350\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2275.96	−0.593494
$246$	0	0
$247$	10200.5	2.62770
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6311.94	1.58727	0.793637	−	0.608391i	$-0.208185\pi$
$251$	6311.94	1.58727	0.793637	+	0.608391i	$0.208185\pi$
$252$	0	0
$253$	1025.78	0.254901
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2510.97	−0.609456	−0.304728	−	0.952439i	$-0.598566\pi$
$257$	−2510.97	−0.609456	−0.304728	+	0.952439i	$0.598566\pi$
$258$	0	0
$259$	3052.15	0.732244
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1038.03	0.243375	0.121687	−	0.992568i	$-0.461169\pi$
$263$	1038.03	0.243375	0.121687	+	0.992568i	$0.461169\pi$
$264$	0	0
$265$	1818.05	0.421442
$266$	0	0
$267$	0	0
$268$	0	0
$269$	643.196	0.145786	0.0728929	−	0.997340i	$-0.476777\pi$
$269$	643.196	0.145786	0.0728929	+	0.997340i	$0.476777\pi$
$270$	0	0
$271$	−3765.24	−0.843992	−0.421996	−	0.906598i	$-0.638670\pi$
$271$	−3765.24	−0.843992	−0.421996	+	0.906598i	$0.638670\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2136.30	−0.468449
$276$	0	0
$277$	907.271	0.196796	0.0983982	−	0.995147i	$-0.468628\pi$
$277$	907.271	0.196796	0.0983982	+	0.995147i	$0.468628\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1902.40	−0.403871	−0.201935	−	0.979399i	$-0.564723\pi$
$281$	−1902.40	−0.403871	−0.201935	+	0.979399i	$0.564723\pi$
$282$	0	0
$283$	−884.273	−0.185741	−0.0928703	−	0.995678i	$-0.529604\pi$
$283$	−884.273	−0.185741	−0.0928703	+	0.995678i	$0.529604\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3087.82	−0.635081
$288$	0	0
$289$	13189.1	2.68454
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3099.02	−0.617908	−0.308954	−	0.951077i	$-0.599979\pi$
$293$	−3099.02	−0.617908	−0.308954	+	0.951077i	$0.599979\pi$
$294$	0	0
$295$	−4488.44	−0.885855
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1578.51	−0.305309
$300$	0	0
$301$	1356.27	0.259715
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3821.13	−0.717369
$306$	0	0
$307$	−4078.41	−0.758200	−0.379100	−	0.925356i	$-0.623766\pi$
$307$	−4078.41	−0.758200	−0.379100	+	0.925356i	$0.623766\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2215.72	0.403994	0.201997	−	0.979386i	$-0.435257\pi$
$311$	2215.72	0.403994	0.201997	+	0.979386i	$0.435257\pi$
$312$	0	0
$313$	−1103.21	−0.199223	−0.0996116	−	0.995026i	$-0.531760\pi$
$313$	−1103.21	−0.199223	−0.0996116	+	0.995026i	$0.531760\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3876.78	0.686883	0.343441	−	0.939174i	$-0.388407\pi$
$317$	3876.78	0.686883	0.343441	+	0.939174i	$0.388407\pi$
$318$	0	0
$319$	−4397.18	−0.771771
$320$	0	0
$321$	0	0
$322$	0	0
$323$	19997.1	3.44480
$324$	0	0
$325$	3287.42	0.561087
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3439.46	0.576364
$330$	0	0
$331$	1973.89	0.327779	0.163890	−	0.986479i	$-0.447596\pi$
$331$	1973.89	0.327779	0.163890	+	0.986479i	$0.447596\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8029.57	−1.30956
$336$	0	0
$337$	−10237.2	−1.65477	−0.827385	−	0.561635i	$-0.810173\pi$
$337$	−10237.2	−1.65477	−0.827385	+	0.561635i	$0.810173\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−703.090	−0.111655
$342$	0	0
$343$	−5512.63	−0.867796
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8093.36	−1.25209	−0.626043	−	0.779788i	$-0.715327\pi$
$347$	−8093.36	−1.25209	−0.626043	+	0.779788i	$0.715327\pi$
$348$	0	0
$349$	7313.87	1.12178	0.560892	−	0.827889i	$-0.310458\pi$
$349$	7313.87	1.12178	0.560892	+	0.827889i	$0.310458\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3652.16	0.550665	0.275333	−	0.961349i	$-0.411212\pi$
$353$	3652.16	0.550665	0.275333	+	0.961349i	$0.411212\pi$
$354$	0	0
$355$	1424.14	0.212917
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4165.92	−0.612448	−0.306224	−	0.951960i	$-0.599066\pi$
$359$	−4165.92	−0.612448	−0.306224	+	0.951960i	$0.599066\pi$
$360$	0	0
$361$	15231.5	2.22066
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3252.51	0.466422
$366$	0	0
$367$	−6904.48	−0.982047	−0.491023	−	0.871146i	$-0.663377\pi$
$367$	−6904.48	−0.982047	−0.491023	+	0.871146i	$0.663377\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1895.38	0.265238
$372$	0	0
$373$	7825.16	1.08625	0.543125	−	0.839652i	$-0.317241\pi$
$373$	7825.16	1.08625	0.543125	+	0.839652i	$0.317241\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6766.57	0.924393
$378$	0	0
$379$	1265.52	0.171518	0.0857588	−	0.996316i	$-0.472669\pi$
$379$	1265.52	0.171518	0.0857588	+	0.996316i	$0.472669\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12077.3	1.61129	0.805644	−	0.592399i	$-0.201819\pi$
$383$	12077.3	1.61129	0.805644	+	0.592399i	$0.201819\pi$
$384$	0	0
$385$	3584.82	0.474544
$386$	0	0
$387$	0	0
$388$	0	0
$389$	95.9629	0.0125077	0.00625387	−	0.999980i	$-0.498009\pi$
$389$	95.9629	0.0125077	0.00625387	+	0.999980i	$0.498009\pi$
$390$	0	0
$391$	−3094.52	−0.400247
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8213.63	−1.04626
$396$	0	0
$397$	12331.2	1.55890	0.779449	−	0.626465i	$-0.215499\pi$
$397$	12331.2	1.55890	0.779449	+	0.626465i	$0.215499\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3418.48	0.425712	0.212856	−	0.977084i	$-0.431723\pi$
$401$	3418.48	0.425712	0.212856	+	0.977084i	$0.431723\pi$
$402$	0	0
$403$	1081.94	0.133736
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14870.1	1.81102
$408$	0	0
$409$	13426.3	1.62320	0.811600	−	0.584214i	$-0.198597\pi$
$409$	13426.3	1.62320	0.811600	+	0.584214i	$0.198597\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4679.35	−0.557520
$414$	0	0
$415$	8110.60	0.959359
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2513.46	0.293057	0.146528	−	0.989206i	$-0.453190\pi$
$419$	2513.46	0.293057	0.146528	+	0.989206i	$0.453190\pi$
$420$	0	0
$421$	−7188.87	−0.832218	−0.416109	−	0.909315i	$-0.636607\pi$
$421$	−7188.87	−0.832218	−0.416109	+	0.909315i	$0.636607\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6444.69	0.735561
$426$	0	0
$427$	−3983.66	−0.451482
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11651.2	−1.30214	−0.651068	−	0.759019i	$-0.725679\pi$
$431$	−11651.2	−1.30214	−0.651068	+	0.759019i	$0.725679\pi$
$432$	0	0
$433$	−2335.44	−0.259201	−0.129600	−	0.991566i	$-0.541369\pi$
$433$	−2335.44	−0.259201	−0.129600	+	0.991566i	$0.541369\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3418.46	−0.374204
$438$	0	0
$439$	17600.4	1.91348	0.956742	−	0.290939i	$-0.0939676\pi$
$439$	17600.4	1.91348	0.956742	+	0.290939i	$0.0939676\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5825.93	−0.624827	−0.312413	−	0.949946i	$-0.601137\pi$
$443$	−5825.93	−0.624827	−0.312413	+	0.949946i	$0.601137\pi$
$444$	0	0
$445$	−542.857	−0.0578290
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7964.71	−0.837145	−0.418572	−	0.908183i	$-0.637469\pi$
$449$	−7964.71	−0.837145	−0.418572	+	0.908183i	$0.637469\pi$
$450$	0	0
$451$	−15043.9	−1.57071
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5516.47	−0.568387
$456$	0	0
$457$	−8398.37	−0.859648	−0.429824	−	0.902913i	$-0.641424\pi$
$457$	−8398.37	−0.859648	−0.429824	+	0.902913i	$0.641424\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7147.43	−0.722102	−0.361051	−	0.932546i	$-0.617582\pi$
$461$	−7147.43	−0.722102	−0.361051	+	0.932546i	$0.617582\pi$
$462$	0	0
$463$	11250.0	1.12922	0.564611	−	0.825357i	$-0.309026\pi$
$463$	11250.0	1.12922	0.564611	+	0.825357i	$0.309026\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15341.7	−1.52019	−0.760097	−	0.649810i	$-0.774849\pi$
$467$	−15341.7	−1.52019	−0.760097	+	0.649810i	$0.774849\pi$
$468$	0	0
$469$	−8371.09	−0.824182
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6607.77	0.642337
$474$	0	0
$475$	7119.34	0.687701
$476$	0	0
$477$	0	0
$478$	0	0
$479$	299.049	0.0285259	0.0142630	−	0.999898i	$-0.495460\pi$
$479$	299.049	0.0285259	0.0142630	+	0.999898i	$0.495460\pi$
$480$	0	0
$481$	−22882.7	−2.16915
$482$	0	0
$483$	0	0
$484$	0	0
$485$	909.238	0.0851265
$486$	0	0
$487$	4670.42	0.434573	0.217286	−	0.976108i	$-0.430279\pi$
$487$	4670.42	0.434573	0.217286	+	0.976108i	$0.430279\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11483.1	1.05545	0.527726	−	0.849414i	$-0.323045\pi$
$491$	11483.1	1.05545	0.527726	+	0.849414i	$0.323045\pi$
$492$	0	0
$493$	13265.2	1.21184
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1484.72	0.134001
$498$	0	0
$499$	−1200.58	−0.107706	−0.0538531	−	0.998549i	$-0.517150\pi$
$499$	−1200.58	−0.107706	−0.0538531	+	0.998549i	$0.517150\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2684.90	−0.238000	−0.119000	−	0.992894i	$-0.537969\pi$
$503$	−2684.90	−0.238000	−0.119000	+	0.992894i	$0.537969\pi$
$504$	0	0
$505$	14601.7	1.28667
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19320.1	1.68241	0.841206	−	0.540715i	$-0.181846\pi$
$509$	19320.1	1.68241	0.841206	+	0.540715i	$0.181846\pi$
$510$	0	0
$511$	3390.85	0.293547
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5707.27	−0.488335
$516$	0	0
$517$	16757.1	1.42549
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6212.53	−0.522411	−0.261205	−	0.965283i	$-0.584120\pi$
$521$	−6212.53	−0.522411	−0.261205	+	0.965283i	$0.584120\pi$
$522$	0	0
$523$	−10728.4	−0.896983	−0.448492	−	0.893787i	$-0.648039\pi$
$523$	−10728.4	−0.896983	−0.448492	+	0.893787i	$0.648039\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2121.05	0.175322
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	23150.2	1.88132
$534$	0	0
$535$	−5235.44	−0.423080
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−11560.1	−0.923804
$540$	0	0
$541$	−9857.53	−0.783379	−0.391690	−	0.920097i	$-0.628109\pi$
$541$	−9857.53	−0.783379	−0.391690	+	0.920097i	$0.628109\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11182.0	0.878869
$546$	0	0
$547$	−13709.3	−1.07160	−0.535802	−	0.844343i	$-0.679991\pi$
$547$	−13709.3	−1.07160	−0.535802	+	0.844343i	$0.679991\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14653.9	1.13299
$552$	0	0
$553$	−8562.97	−0.658471
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2471.87	−0.188037	−0.0940185	−	0.995570i	$-0.529971\pi$
$557$	−2471.87	−0.188037	−0.0940185	+	0.995570i	$0.529971\pi$
$558$	0	0
$559$	−10168.3	−0.769363
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4911.12	0.367636	0.183818	−	0.982960i	$-0.441154\pi$
$563$	4911.12	0.367636	0.183818	+	0.982960i	$0.441154\pi$
$564$	0	0
$565$	5106.34	0.380221
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6667.28	0.491225	0.245612	−	0.969368i	$-0.421011\pi$
$569$	6667.28	0.491225	0.245612	+	0.969368i	$0.421011\pi$
$570$	0	0
$571$	14658.5	1.07433	0.537163	−	0.843478i	$-0.319496\pi$
$571$	14658.5	1.07433	0.537163	+	0.843478i	$0.319496\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1101.70	−0.0799030
$576$	0	0
$577$	4593.54	0.331424	0.165712	−	0.986174i	$-0.447008\pi$
$577$	4593.54	0.331424	0.165712	+	0.986174i	$0.447008\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8455.57	0.603780
$582$	0	0
$583$	9234.31	0.655996
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9227.49	−0.648823	−0.324412	−	0.945916i	$-0.605166\pi$
$587$	−9227.49	−0.648823	−0.324412	+	0.945916i	$0.605166\pi$
$588$	0	0
$589$	2343.09	0.163914
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16921.2	1.17179	0.585895	−	0.810387i	$-0.300743\pi$
$593$	16921.2	1.17179	0.585895	+	0.810387i	$0.300743\pi$
$594$	0	0
$595$	−10814.5	−0.745131
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6264.86	0.427337	0.213669	−	0.976906i	$-0.431459\pi$
$599$	6264.86	0.427337	0.213669	+	0.976906i	$0.431459\pi$
$600$	0	0
$601$	−16317.7	−1.10751	−0.553756	−	0.832679i	$-0.686806\pi$
$601$	−16317.7	−1.10751	−0.553756	+	0.832679i	$0.686806\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5778.24	0.388296
$606$	0	0
$607$	−27547.7	−1.84205	−0.921027	−	0.389499i	$-0.872648\pi$
$607$	−27547.7	−1.84205	−0.921027	+	0.389499i	$0.872648\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−25786.5	−1.70738
$612$	0	0
$613$	−5537.21	−0.364838	−0.182419	−	0.983221i	$-0.558393\pi$
$613$	−5537.21	−0.364838	−0.182419	+	0.983221i	$0.558393\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17182.6	1.12115	0.560573	−	0.828105i	$-0.310581\pi$
$617$	17182.6	1.12115	0.560573	+	0.828105i	$0.310581\pi$
$618$	0	0
$619$	14375.4	0.933434	0.466717	−	0.884407i	$-0.345437\pi$
$619$	14375.4	0.933434	0.466717	+	0.884407i	$0.345437\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−565.946	−0.0363951
$624$	0	0
$625$	−7343.05	−0.469955
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−44859.5	−2.84366
$630$	0	0
$631$	9337.75	0.589112	0.294556	−	0.955634i	$-0.404828\pi$
$631$	9337.75	0.589112	0.294556	+	0.955634i	$0.404828\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10524.1	−0.657693
$636$	0	0
$637$	17789.2	1.10649
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27089.9	−1.66924	−0.834622	−	0.550823i	$-0.814314\pi$
$641$	−27089.9	−1.66924	−0.834622	+	0.550823i	$0.814314\pi$
$642$	0	0
$643$	−30809.4	−1.88959	−0.944794	−	0.327663i	$-0.893739\pi$
$643$	−30809.4	−1.88959	−0.944794	+	0.327663i	$0.893739\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13252.1	−0.805246	−0.402623	−	0.915366i	$-0.631902\pi$
$647$	−13252.1	−0.805246	−0.402623	+	0.915366i	$0.631902\pi$
$648$	0	0
$649$	−22797.8	−1.37888
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19773.8	−1.18501	−0.592503	−	0.805568i	$-0.701860\pi$
$653$	−19773.8	−1.18501	−0.592503	+	0.805568i	$0.701860\pi$
$654$	0	0
$655$	−17077.9	−1.01876
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−11796.1	−0.697286	−0.348643	−	0.937256i	$-0.613358\pi$
$659$	−11796.1	−0.697286	−0.348643	+	0.937256i	$0.613358\pi$
$660$	0	0
$661$	1378.16	0.0810955	0.0405477	−	0.999178i	$-0.487090\pi$
$661$	1378.16	0.0810955	0.0405477	+	0.999178i	$0.487090\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−11946.6	−0.696648
$666$	0	0
$667$	−2267.66	−0.131640
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19408.4	−1.11662
$672$	0	0
$673$	7023.66	0.402291	0.201146	−	0.979561i	$-0.435534\pi$
$673$	7023.66	0.402291	0.201146	+	0.979561i	$0.435534\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8986.94	0.510186	0.255093	−	0.966916i	$-0.417894\pi$
$677$	8986.94	0.510186	0.255093	+	0.966916i	$0.417894\pi$
$678$	0	0
$679$	947.910	0.0535750
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5617.70	−0.314722	−0.157361	−	0.987541i	$-0.550299\pi$
$683$	−5617.70	−0.314722	−0.157361	+	0.987541i	$0.550299\pi$
$684$	0	0
$685$	−10020.7	−0.558938
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14210.1	−0.785723
$690$	0	0
$691$	17838.7	0.982080	0.491040	−	0.871137i	$-0.336617\pi$
$691$	17838.7	0.982080	0.491040	+	0.871137i	$0.336617\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10627.2	−0.580019
$696$	0	0
$697$	45383.8	2.46633
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23519.6	1.26722	0.633611	−	0.773652i	$-0.281572\pi$
$701$	23519.6	1.26722	0.633611	+	0.773652i	$0.281572\pi$
$702$	0	0
$703$	−49555.5	−2.65864
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15222.7	0.809773
$708$	0	0
$709$	11200.6	0.593297	0.296648	−	0.954987i	$-0.404131\pi$
$709$	11200.6	0.593297	0.296648	+	0.954987i	$0.404131\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−362.588	−0.0190449
$714$	0	0
$715$	−26876.3	−1.40576
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26086.3	1.35307	0.676533	−	0.736412i	$-0.263482\pi$
$719$	26086.3	1.35307	0.676533	+	0.736412i	$0.263482\pi$
$720$	0	0
$721$	−5950.01	−0.307337
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4722.67	0.241925
$726$	0	0
$727$	−3684.94	−0.187987	−0.0939937	−	0.995573i	$-0.529963\pi$
$727$	−3684.94	−0.187987	−0.0939937	+	0.995573i	$0.529963\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−19934.0	−1.00860
$732$	0	0
$733$	6626.75	0.333922	0.166961	−	0.985964i	$-0.446605\pi$
$733$	6626.75	0.333922	0.166961	+	0.985964i	$0.446605\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40784.1	−2.03840
$738$	0	0
$739$	−16061.3	−0.799494	−0.399747	−	0.916626i	$-0.630902\pi$
$739$	−16061.3	−0.799494	−0.399747	+	0.916626i	$0.630902\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10457.5	−0.516351	−0.258176	−	0.966098i	$-0.583121\pi$
$743$	−10457.5	−0.516351	−0.258176	+	0.966098i	$0.583121\pi$
$744$	0	0
$745$	2918.78	0.143538
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5458.12	−0.266269
$750$	0	0
$751$	14075.4	0.683912	0.341956	−	0.939716i	$-0.388911\pi$
$751$	14075.4	0.683912	0.341956	+	0.939716i	$0.388911\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23373.2	1.12667
$756$	0	0
$757$	15015.4	0.720932	0.360466	−	0.932772i	$-0.382618\pi$
$757$	15015.4	0.720932	0.360466	+	0.932772i	$0.382618\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	26658.3	1.26986	0.634930	−	0.772570i	$-0.281029\pi$
$761$	26658.3	1.26986	0.634930	+	0.772570i	$0.281029\pi$
$762$	0	0
$763$	11657.6	0.553123
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35082.2	1.65156
$768$	0	0
$769$	−31732.7	−1.48805	−0.744025	−	0.668151i	$-0.767086\pi$
$769$	−31732.7	−1.48805	−0.744025	+	0.668151i	$0.767086\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13523.8	−0.629261	−0.314631	−	0.949214i	$-0.601881\pi$
$773$	−13523.8	−0.629261	−0.314631	+	0.949214i	$0.601881\pi$
$774$	0	0
$775$	755.133	0.0350002
$776$	0	0
$777$	0	0
$778$	0	0
$779$	50134.7	2.30586
$780$	0	0
$781$	7233.55	0.331417
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−20285.4	−0.922317
$786$	0	0
$787$	−19946.5	−0.903452	−0.451726	−	0.892157i	$-0.649191\pi$
$787$	−19946.5	−0.903452	−0.451726	+	0.892157i	$0.649191\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5323.52	0.239295
$792$	0	0
$793$	29866.5	1.33744
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1025.98	−0.0455988	−0.0227994	−	0.999740i	$-0.507258\pi$
$797$	−1025.98	−0.0455988	−0.0227994	+	0.999740i	$0.507258\pi$
$798$	0	0
$799$	−50552.1	−2.23831
$800$	0	0
$801$	0	0
$802$	0	0
$803$	16520.2	0.726011
$804$	0	0
$805$	1848.72	0.0809426
$806$	0	0
$807$	0	0
$808$	0	0
$809$	33646.9	1.46225	0.731126	−	0.682243i	$-0.238995\pi$
$809$	33646.9	1.46225	0.731126	+	0.682243i	$0.238995\pi$
$810$	0	0
$811$	−11264.7	−0.487740	−0.243870	−	0.969808i	$-0.578417\pi$
$811$	−11264.7	−0.487740	−0.243870	+	0.969808i	$0.578417\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2129.66	−0.0915324
$816$	0	0
$817$	−22020.8	−0.942975
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13577.3	−0.577163	−0.288582	−	0.957455i	$-0.593184\pi$
$821$	−13577.3	−0.577163	−0.288582	+	0.957455i	$0.593184\pi$
$822$	0	0
$823$	17638.0	0.747051	0.373525	−	0.927620i	$-0.378149\pi$
$823$	17638.0	0.747051	0.373525	+	0.927620i	$0.378149\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35211.1	−1.48054	−0.740272	−	0.672307i	$-0.765303\pi$
$827$	−35211.1	−1.48054	−0.740272	+	0.672307i	$0.765303\pi$
$828$	0	0
$829$	−7480.76	−0.313411	−0.156705	−	0.987645i	$-0.550087\pi$
$829$	−7480.76	−0.313411	−0.156705	+	0.987645i	$0.550087\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	34874.2	1.45056
$834$	0	0
$835$	−10436.6	−0.432542
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1772.03	−0.0729167	−0.0364584	−	0.999335i	$-0.511608\pi$
$839$	−1772.03	−0.0729167	−0.0364584	+	0.999335i	$0.511608\pi$
$840$	0	0
$841$	−14668.2	−0.601428
$842$	0	0
$843$	0	0
$844$	0	0
$845$	22067.3	0.898388
$846$	0	0
$847$	6024.01	0.244377
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7668.62	0.308903
$852$	0	0
$853$	−29288.4	−1.17564	−0.587818	−	0.808993i	$-0.700013\pi$
$853$	−29288.4	−1.17564	−0.587818	+	0.808993i	$0.700013\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19474.1	0.776224	0.388112	−	0.921612i	$-0.373127\pi$
$857$	19474.1	0.776224	0.388112	+	0.921612i	$0.373127\pi$
$858$	0	0
$859$	−12695.9	−0.504282	−0.252141	−	0.967691i	$-0.581135\pi$
$859$	−12695.9	−0.504282	−0.252141	+	0.967691i	$0.581135\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42212.9	−1.66506	−0.832529	−	0.553982i	$-0.813108\pi$
$863$	−42212.9	−1.66506	−0.832529	+	0.553982i	$0.813108\pi$
$864$	0	0
$865$	21153.7	0.831499
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−41718.9	−1.62856
$870$	0	0
$871$	62760.2	2.44150
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−13897.6	−0.536941
$876$	0	0
$877$	−17818.7	−0.686083	−0.343041	−	0.939320i	$-0.611457\pi$
$877$	−17818.7	−0.686083	−0.343041	+	0.939320i	$0.611457\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20699.6	0.791584	0.395792	−	0.918340i	$-0.370470\pi$
$881$	20699.6	0.791584	0.395792	+	0.918340i	$0.370470\pi$
$882$	0	0
$883$	31592.7	1.20405	0.602027	−	0.798476i	$-0.294360\pi$
$883$	31592.7	1.20405	0.602027	+	0.798476i	$0.294360\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22226.5	0.841366	0.420683	−	0.907208i	$-0.361790\pi$
$887$	22226.5	0.841366	0.420683	+	0.907208i	$0.361790\pi$
$888$	0	0
$889$	−10971.7	−0.413924
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−55844.1	−2.09267
$894$	0	0
$895$	−6793.81	−0.253734
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1554.31	0.0576629
$900$	0	0
$901$	−27857.7	−1.03005
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−27620.4	−1.01451
$906$	0	0
$907$	8906.15	0.326046	0.163023	−	0.986622i	$-0.447876\pi$
$907$	8906.15	0.326046	0.163023	+	0.986622i	$0.447876\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23772.4	−0.864561	−0.432280	−	0.901739i	$-0.642291\pi$
$911$	−23772.4	−0.864561	−0.432280	+	0.901739i	$0.642291\pi$
$912$	0	0
$913$	41195.6	1.49329
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17804.2	−0.641165
$918$	0	0
$919$	−52391.7	−1.88057	−0.940285	−	0.340389i	$-0.889441\pi$
$919$	−52391.7	−1.88057	−0.940285	+	0.340389i	$0.889441\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−11131.3	−0.396956
$924$	0	0
$925$	−15970.8	−0.567693
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34847.2	1.23068	0.615338	−	0.788263i	$-0.289019\pi$
$929$	34847.2	1.23068	0.615338	+	0.788263i	$0.289019\pi$
$930$	0	0
$931$	38524.9	1.35618
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−52688.6	−1.84289
$936$	0	0
$937$	40160.9	1.40021	0.700106	−	0.714039i	$-0.253136\pi$
$937$	40160.9	1.40021	0.700106	+	0.714039i	$0.253136\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−48726.3	−1.68803	−0.844014	−	0.536322i	$-0.819813\pi$
$941$	−48726.3	−1.68803	−0.844014	+	0.536322i	$0.819813\pi$
$942$	0	0
$943$	−7758.25	−0.267914
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47185.8	1.61915	0.809574	−	0.587018i	$-0.199698\pi$
$947$	47185.8	1.61915	0.809574	+	0.587018i	$0.199698\pi$
$948$	0	0
$949$	−25422.0	−0.869583
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15122.7	0.514030	0.257015	−	0.966407i	$-0.417261\pi$
$953$	15122.7	0.514030	0.257015	+	0.966407i	$0.417261\pi$
$954$	0	0
$955$	42493.4	1.43985
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10446.9	−0.351772
$960$	0	0
$961$	−29542.5	−0.991658
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19180.2	0.639828
$966$	0	0
$967$	−21384.5	−0.711148	−0.355574	−	0.934648i	$-0.615715\pi$
$967$	−21384.5	−0.711148	−0.355574	+	0.934648i	$0.615715\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14788.3	0.488752	0.244376	−	0.969681i	$-0.421417\pi$
$971$	14788.3	0.488752	0.244376	+	0.969681i	$0.421417\pi$
$972$	0	0
$973$	−11079.2	−0.365039
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46409.7	1.51973	0.759867	−	0.650079i	$-0.225264\pi$
$977$	46409.7	1.51973	0.759867	+	0.650079i	$0.225264\pi$
$978$	0	0
$979$	−2757.29	−0.0900138
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15076.8	0.489190	0.244595	−	0.969625i	$-0.421345\pi$
$983$	15076.8	0.489190	0.244595	+	0.969625i	$0.421345\pi$
$984$	0	0
$985$	4084.14	0.132113
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3407.68	0.109563
$990$	0	0
$991$	30525.4	0.978476	0.489238	−	0.872150i	$-0.337275\pi$
$991$	30525.4	0.978476	0.489238	+	0.872150i	$0.337275\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6521.58	−0.207787
$996$	0	0
$997$	6084.40	0.193275	0.0966374	−	0.995320i	$-0.469191\pi$
$997$	6084.40	0.193275	0.0966374	+	0.995320i	$0.469191\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	828.4.a.f.1.2		3
3.2	odd	2		92.4.a.a.1.2	✓	3
12.11	even	2		368.4.a.k.1.2		3
15.2	even	4		2300.4.c.b.1749.4		6
15.8	even	4		2300.4.c.b.1749.3		6
15.14	odd	2		2300.4.a.b.1.2		3
24.5	odd	2		1472.4.a.w.1.2		3
24.11	even	2		1472.4.a.p.1.2		3
69.68	even	2		2116.4.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.4.a.a.1.2	✓	3	3.2	odd	2
368.4.a.k.1.2		3	12.11	even	2
828.4.a.f.1.2		3	1.1	even	1	trivial
1472.4.a.p.1.2		3	24.11	even	2
1472.4.a.w.1.2		3	24.5	odd	2
2116.4.a.a.1.2		3	69.68	even	2
2300.4.a.b.1.2		3	15.14	odd	2
2300.4.c.b.1749.3		6	15.8	even	4
2300.4.c.b.1749.4		6	15.2	even	4

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:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	828.a (trivial)

\(f(q)\)	\(=\)	\(q+8.78065 q^{5} +9.15411 q^{7} +44.5989 q^{11} -68.6307 q^{13} -134.544 q^{17} -148.629 q^{19} +23.0000 q^{23} -47.9002 q^{25} -98.5939 q^{29} -15.7647 q^{31} +80.3791 q^{35} +333.418 q^{37} -337.315 q^{41} +148.160 q^{43} +375.729 q^{47} -259.202 q^{49} +207.052 q^{53} +391.608 q^{55} -511.174 q^{59} -435.177 q^{61} -602.622 q^{65} -914.462 q^{67} +162.191 q^{71} +370.418 q^{73} +408.264 q^{77} -935.424 q^{79} +923.690 q^{83} -1181.39 q^{85} -61.8242 q^{89} -628.253 q^{91} -1305.06 q^{95} +103.550 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 10 q^{5} - 46 q^{7} + 64 q^{11} - 44 q^{13} + 88 q^{17} - 94 q^{19} + 69 q^{23} + 181 q^{25} - 308 q^{29} - 140 q^{31} - 192 q^{35} + 26 q^{37} - 584 q^{41} - 478 q^{43} + 28 q^{47} + 695 q^{49} - 356 q^{53}+ \cdots + 736 q^{97}+O(q^{100}) \) 3 * q + 10 * q^5 - 46 * q^7 + 64 * q^11 - 44 * q^13 + 88 * q^17 - 94 * q^19 + 69 * q^23 + 181 * q^25 - 308 * q^29 - 140 * q^31 - 192 * q^35 + 26 * q^37 - 584 * q^41 - 478 * q^43 + 28 * q^47 + 695 * q^49 - 356 * q^53 - 1004 * q^55 - 144 * q^59 - 1052 * q^61 - 150 * q^65 - 2008 * q^67 - 360 * q^71 - 252 * q^73 + 576 * q^77 - 720 * q^79 + 1404 * q^83 - 1632 * q^85 - 534 * q^89 - 1526 * q^91 + 108 * q^95 + 736 * q^97

Introduction

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