LMFDB - Embedded newform 8208.2.a.cb.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] 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("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8208, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8208 = 2^{4} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8208.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-3,0,-1,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$65.5412099791$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.4873296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 8x^{2} + 15x + 3 $ x^5 - x^4 - 10x^3 + 8x^2 + 15*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.04032$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17927 q^{5} -4.44641 q^{7} +0.698144 q^{11} -3.60931 q^{13} +2.08064 q^{17} +1.00000 q^{19} +3.10424 q^{23} -0.250766 q^{25} -5.38249 q^{29} -7.52705 q^{31} +9.68995 q^{35} -6.52705 q^{37} -8.01541 q^{41} +0.559190 q^{43} -10.9259 q^{47} +12.7706 q^{49} -10.4867 q^{53} -1.52145 q^{55} +11.8692 q^{59} -0.762065 q^{61} +7.86568 q^{65} +1.23794 q^{67} +5.25077 q^{71} +6.69718 q^{73} -3.10424 q^{77} -15.4070 q^{79} -8.09605 q^{83} -4.53428 q^{85} -16.7466 q^{89} +16.0485 q^{91} -2.17927 q^{95} -1.24354 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} - q^{7} + 2 q^{11} + 3 q^{13} - 8 q^{17} + 5 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} + 2 q^{31} + 9 q^{35} + 7 q^{37} - 7 q^{41} + 10 q^{47} + 6 q^{49} - 22 q^{53} + 8 q^{55} + 12 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) $ 5 * q - 3 * q^5 - q^7 + 2 * q^11 + 3 * q^13 - 8 * q^17 + 5 * q^19 + 2 * q^23 + 4 * q^25 - 10 * q^29 + 2 * q^31 + 9 * q^35 + 7 * q^37 - 7 * q^41 + 10 * q^47 + 6 * q^49 - 22 * q^53 + 8 * q^55 + 12 * q^59 + 2 * q^61 - 11 * q^65 + 12 * q^67 + 21 * q^71 + 7 * q^73 - 2 * q^77 - 6 * q^79 + 11 * q^83 + 4 * q^85 - 27 * q^89 + 25 * q^91 - 3 * q^95 + 12 * 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
$5$	−2.17927	−0.974601	−0.487300	−	0.873234i	$-0.662018\pi$
$5$	−2.17927	−0.974601	−0.487300	+	0.873234i	$0.662018\pi$
$6$	0	0
$7$	−4.44641	−1.68059	−0.840293	−	0.542132i	$-0.817617\pi$
$7$	−4.44641	−1.68059	−0.840293	+	0.542132i	$0.817617\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.698144	0.210498	0.105249	−	0.994446i	$-0.466436\pi$
$11$	0.698144	0.210498	0.105249	+	0.994446i	$0.466436\pi$
$12$	0	0
$13$	−3.60931	−1.00104	−0.500522	−	0.865724i	$-0.666858\pi$
$13$	−3.60931	−1.00104	−0.500522	+	0.865724i	$0.666858\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.08064	0.504629	0.252314	−	0.967645i	$-0.418808\pi$
$17$	2.08064	0.504629	0.252314	+	0.967645i	$0.418808\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.10424	0.647279	0.323639	−	0.946181i	$-0.395094\pi$
$23$	3.10424	0.647279	0.323639	+	0.946181i	$0.395094\pi$
$24$	0	0
$25$	−0.250766	−0.0501532
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.38249	−0.999504	−0.499752	−	0.866169i	$-0.666575\pi$
$29$	−5.38249	−0.999504	−0.499752	+	0.866169i	$0.666575\pi$
$30$	0	0
$31$	−7.52705	−1.35190	−0.675949	−	0.736948i	$-0.736266\pi$
$31$	−7.52705	−1.35190	−0.675949	+	0.736948i	$0.736266\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.68995	1.63790
$36$	0	0
$37$	−6.52705	−1.07304	−0.536520	−	0.843887i	$-0.680261\pi$
$37$	−6.52705	−1.07304	−0.536520	+	0.843887i	$0.680261\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.01541	−1.25180	−0.625898	−	0.779905i	$-0.715267\pi$
$41$	−8.01541	−1.25180	−0.625898	+	0.779905i	$0.715267\pi$
$42$	0	0
$43$	0.559190	0.0852756	0.0426378	−	0.999091i	$-0.486424\pi$
$43$	0.559190	0.0852756	0.0426378	+	0.999091i	$0.486424\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.9259	−1.59371	−0.796854	−	0.604171i	$-0.793504\pi$
$47$	−10.9259	−1.59371	−0.796854	+	0.604171i	$0.793504\pi$
$48$	0	0
$49$	12.7706	1.82437
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.4867	−1.44046	−0.720232	−	0.693734i	$-0.755965\pi$
$53$	−10.4867	−1.44046	−0.720232	+	0.693734i	$0.755965\pi$
$54$	0	0
$55$	−1.52145	−0.205152
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.8692	1.54524	0.772621	−	0.634868i	$-0.218945\pi$
$59$	11.8692	1.54524	0.772621	+	0.634868i	$0.218945\pi$
$60$	0	0
$61$	−0.762065	−0.0975724	−0.0487862	−	0.998809i	$-0.515535\pi$
$61$	−0.762065	−0.0975724	−0.0487862	+	0.998809i	$0.515535\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.86568	0.975618
$66$	0	0
$67$	1.23794	0.151238	0.0756189	−	0.997137i	$-0.475907\pi$
$67$	1.23794	0.151238	0.0756189	+	0.997137i	$0.475907\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.25077	0.623151	0.311576	−	0.950221i	$-0.399143\pi$
$71$	5.25077	0.623151	0.311576	+	0.950221i	$0.399143\pi$
$72$	0	0
$73$	6.69718	0.783846	0.391923	−	0.919998i	$-0.371810\pi$
$73$	6.69718	0.783846	0.391923	+	0.919998i	$0.371810\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.10424	−0.353761
$78$	0	0
$79$	−15.4070	−1.73343	−0.866714	−	0.498805i	$-0.833772\pi$
$79$	−15.4070	−1.73343	−0.866714	+	0.498805i	$0.833772\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.09605	−0.888656	−0.444328	−	0.895864i	$-0.646558\pi$
$83$	−8.09605	−0.888656	−0.444328	+	0.895864i	$0.646558\pi$
$84$	0	0
$85$	−4.53428	−0.491812
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.7466	−1.77514	−0.887570	−	0.460672i	$-0.847608\pi$
$89$	−16.7466	−1.77514	−0.887570	+	0.460672i	$0.847608\pi$
$90$	0	0
$91$	16.0485	1.68234
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.17927	−0.223589
$96$	0	0
$97$	−1.24354	−0.126262	−0.0631311	−	0.998005i	$-0.520109\pi$
$97$	−1.24354	−0.126262	−0.0631311	+	0.998005i	$0.520109\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.98200	−0.794239	−0.397119	−	0.917767i	$-0.629990\pi$
$101$	−7.98200	−0.794239	−0.397119	+	0.917767i	$0.629990\pi$
$102$	0	0
$103$	16.7762	1.65301	0.826504	−	0.562931i	$-0.190326\pi$
$103$	16.7762	1.65301	0.826504	+	0.562931i	$0.190326\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.71915	−0.552892	−0.276446	−	0.961030i	$-0.589157\pi$
$107$	−5.71915	−0.552892	−0.276446	+	0.961030i	$0.589157\pi$
$108$	0	0
$109$	8.16850	0.782401	0.391200	−	0.920306i	$-0.372060\pi$
$109$	8.16850	0.782401	0.391200	+	0.920306i	$0.372060\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.552622	−0.0519863	−0.0259931	−	0.999662i	$-0.508275\pi$
$113$	−0.552622	−0.0519863	−0.0259931	+	0.999662i	$0.508275\pi$
$114$	0	0
$115$	−6.76499	−0.630838
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.25137	−0.848072
$120$	0	0
$121$	−10.5126	−0.955690
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.4429	1.02348
$126$	0	0
$127$	15.4014	1.36666	0.683329	−	0.730111i	$-0.260532\pi$
$127$	15.4014	1.36666	0.683329	+	0.730111i	$0.260532\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.43564	−0.562285	−0.281142	−	0.959666i	$-0.590713\pi$
$131$	−6.43564	−0.562285	−0.281142	+	0.959666i	$0.590713\pi$
$132$	0	0
$133$	−4.44641	−0.385553
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.4392	−0.891880	−0.445940	−	0.895063i	$-0.647130\pi$
$137$	−10.4392	−0.891880	−0.445940	+	0.895063i	$0.647130\pi$
$138$	0	0
$139$	−4.01120	−0.340226	−0.170113	−	0.985425i	$-0.554413\pi$
$139$	−4.01120	−0.340226	−0.170113	+	0.985425i	$0.554413\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.51982	−0.210718
$144$	0	0
$145$	11.7299	0.974117
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.0006	1.14697	0.573487	−	0.819215i	$-0.305590\pi$
$149$	14.0006	1.14697	0.573487	+	0.819215i	$0.305590\pi$
$150$	0	0
$151$	3.91051	0.318233	0.159116	−	0.987260i	$-0.449136\pi$
$151$	3.91051	0.318233	0.159116	+	0.987260i	$0.449136\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.4035	1.31756
$156$	0	0
$157$	−11.6404	−0.929008	−0.464504	−	0.885571i	$-0.653767\pi$
$157$	−11.6404	−0.929008	−0.464504	+	0.885571i	$0.653767\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−13.8027	−1.08781
$162$	0	0
$163$	20.5421	1.60898	0.804491	−	0.593964i	$-0.202438\pi$
$163$	20.5421	1.60898	0.804491	+	0.593964i	$0.202438\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.67980	0.671663	0.335832	−	0.941922i	$-0.390983\pi$
$167$	8.67980	0.671663	0.335832	+	0.941922i	$0.390983\pi$
$168$	0	0
$169$	0.0271439	0.00208799
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.8666	1.20632	0.603159	−	0.797621i	$-0.293908\pi$
$173$	15.8666	1.20632	0.603159	+	0.797621i	$0.293908\pi$
$174$	0	0
$175$	1.11501	0.0842868
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.08785	−0.679258	−0.339629	−	0.940559i	$-0.610302\pi$
$179$	−9.08785	−0.679258	−0.339629	+	0.940559i	$0.610302\pi$
$180$	0	0
$181$	7.11115	0.528568	0.264284	−	0.964445i	$-0.414864\pi$
$181$	7.11115	0.528568	0.264284	+	0.964445i	$0.414864\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.2242	1.04579
$186$	0	0
$187$	1.45259	0.106224
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.2527	0.958933	0.479467	−	0.877560i	$-0.340830\pi$
$191$	13.2527	0.958933	0.479467	+	0.877560i	$0.340830\pi$
$192$	0	0
$193$	17.6251	1.26868	0.634340	−	0.773054i	$-0.281272\pi$
$193$	17.6251	1.26868	0.634340	+	0.773054i	$0.281272\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.31565	−0.663713	−0.331856	−	0.943330i	$-0.607675\pi$
$197$	−9.31565	−0.663713	−0.331856	+	0.943330i	$0.607675\pi$
$198$	0	0
$199$	−2.20917	−0.156604	−0.0783018	−	0.996930i	$-0.524950\pi$
$199$	−2.20917	−0.156604	−0.0783018	+	0.996930i	$0.524950\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	23.9328	1.67975
$204$	0	0
$205$	17.4678	1.22000
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.698144	0.0482917
$210$	0	0
$211$	7.73492	0.532494	0.266247	−	0.963905i	$-0.414216\pi$
$211$	7.73492	0.532494	0.266247	+	0.963905i	$0.414216\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.21863	−0.0831097
$216$	0	0
$217$	33.4684	2.27198
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.50967	−0.505155
$222$	0	0
$223$	24.7063	1.65446	0.827228	−	0.561866i	$-0.189916\pi$
$223$	24.7063	1.65446	0.827228	+	0.561866i	$0.189916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.8089	1.51388	0.756941	−	0.653483i	$-0.226693\pi$
$227$	22.8089	1.51388	0.756941	+	0.653483i	$0.226693\pi$
$228$	0	0
$229$	−8.07310	−0.533486	−0.266743	−	0.963768i	$-0.585947\pi$
$229$	−8.07310	−0.533486	−0.266743	+	0.963768i	$0.585947\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.8365	−0.906458	−0.453229	−	0.891394i	$-0.649728\pi$
$233$	−13.8365	−0.906458	−0.453229	+	0.891394i	$0.649728\pi$
$234$	0	0
$235$	23.8106	1.55323
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.6126	−1.33332	−0.666660	−	0.745362i	$-0.732277\pi$
$239$	−20.6126	−1.33332	−0.666660	+	0.745362i	$0.732277\pi$
$240$	0	0
$241$	5.35177	0.344738	0.172369	−	0.985032i	$-0.444858\pi$
$241$	5.35177	0.344738	0.172369	+	0.985032i	$0.444858\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−27.8306	−1.77803
$246$	0	0
$247$	−3.60931	−0.229655
$248$	0	0
$249$	0	0
$250$	0	0
$251$	30.9272	1.95211	0.976053	−	0.217532i	$-0.0698005\pi$
$251$	30.9272	1.95211	0.976053	+	0.217532i	$0.0698005\pi$
$252$	0	0
$253$	2.16721	0.136251
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.83088	−0.301342	−0.150671	−	0.988584i	$-0.548143\pi$
$257$	−4.83088	−0.301342	−0.150671	+	0.988584i	$0.548143\pi$
$258$	0	0
$259$	29.0220	1.80334
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.10458	−0.561413	−0.280706	−	0.959794i	$-0.590569\pi$
$263$	−9.10458	−0.561413	−0.280706	+	0.959794i	$0.590569\pi$
$264$	0	0
$265$	22.8535	1.40388
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.6008	−1.25605	−0.628026	−	0.778193i	$-0.716137\pi$
$269$	−20.6008	−1.25605	−0.628026	+	0.778193i	$0.716137\pi$
$270$	0	0
$271$	15.3569	0.932867	0.466433	−	0.884556i	$-0.345539\pi$
$271$	15.3569	0.932867	0.466433	+	0.884556i	$0.345539\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.175071	−0.0105572
$276$	0	0
$277$	2.45924	0.147762	0.0738808	−	0.997267i	$-0.476462\pi$
$277$	2.45924	0.147762	0.0738808	+	0.997267i	$0.476462\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−31.5566	−1.88251	−0.941253	−	0.337702i	$-0.890350\pi$
$281$	−31.5566	−1.88251	−0.941253	+	0.337702i	$0.890350\pi$
$282$	0	0
$283$	5.98039	0.355497	0.177749	−	0.984076i	$-0.443119\pi$
$283$	5.98039	0.355497	0.177749	+	0.984076i	$0.443119\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	35.6398	2.10375
$288$	0	0
$289$	−12.6709	−0.745350
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.4726	1.19602	0.598011	−	0.801488i	$-0.295958\pi$
$293$	20.4726	1.19602	0.598011	+	0.801488i	$0.295958\pi$
$294$	0	0
$295$	−25.8663	−1.50599
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.2042	−0.647954
$300$	0	0
$301$	−2.48639	−0.143313
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.66075	0.0950941
$306$	0	0
$307$	12.1859	0.695485	0.347742	−	0.937590i	$-0.386948\pi$
$307$	12.1859	0.695485	0.347742	+	0.937590i	$0.386948\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.04461	0.342758	0.171379	−	0.985205i	$-0.445178\pi$
$311$	6.04461	0.342758	0.171379	+	0.985205i	$0.445178\pi$
$312$	0	0
$313$	10.9429	0.618532	0.309266	−	0.950976i	$-0.399917\pi$
$313$	10.9429	0.618532	0.309266	+	0.950976i	$0.399917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.244848	0.0137520	0.00687601	−	0.999976i	$-0.497811\pi$
$317$	0.244848	0.0137520	0.00687601	+	0.999976i	$0.497811\pi$
$318$	0	0
$319$	−3.75776	−0.210394
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.08064	0.115770
$324$	0	0
$325$	0.905093	0.0502055
$326$	0	0
$327$	0	0
$328$	0	0
$329$	48.5811	2.67836
$330$	0	0
$331$	−7.41913	−0.407792	−0.203896	−	0.978993i	$-0.565360\pi$
$331$	−7.41913	−0.407792	−0.203896	+	0.978993i	$0.565360\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.69780	−0.147397
$336$	0	0
$337$	−4.08916	−0.222751	−0.111375	−	0.993778i	$-0.535526\pi$
$337$	−4.08916	−0.222751	−0.111375	+	0.993778i	$0.535526\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.25497	−0.284573
$342$	0	0
$343$	−25.6584	−1.38542
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.7071	−0.896883	−0.448441	−	0.893812i	$-0.648021\pi$
$347$	−16.7071	−0.896883	−0.448441	+	0.893812i	$0.648021\pi$
$348$	0	0
$349$	−2.10778	−0.112827	−0.0564135	−	0.998407i	$-0.517967\pi$
$349$	−2.10778	−0.112827	−0.0564135	+	0.998407i	$0.517967\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.92951	−0.155922	−0.0779611	−	0.996956i	$-0.524841\pi$
$353$	−2.92951	−0.155922	−0.0779611	+	0.996956i	$0.524841\pi$
$354$	0	0
$355$	−11.4429	−0.607324
$356$	0	0
$357$	0	0
$358$	0	0
$359$	34.4953	1.82059	0.910297	−	0.413955i	$-0.135853\pi$
$359$	34.4953	1.82059	0.910297	+	0.413955i	$0.135853\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.5950	−0.763936
$366$	0	0
$367$	−4.94719	−0.258241	−0.129121	−	0.991629i	$-0.541215\pi$
$367$	−4.94719	−0.258241	−0.129121	+	0.991629i	$0.541215\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	46.6283	2.42082
$372$	0	0
$373$	13.1681	0.681816	0.340908	−	0.940097i	$-0.389266\pi$
$373$	13.1681	0.681816	0.340908	+	0.940097i	$0.389266\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.4271	1.00055
$378$	0	0
$379$	−1.55389	−0.0798179	−0.0399090	−	0.999203i	$-0.512707\pi$
$379$	−1.55389	−0.0798179	−0.0399090	+	0.999203i	$0.512707\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.5415	0.896328	0.448164	−	0.893951i	$-0.352078\pi$
$383$	17.5415	0.896328	0.448164	+	0.893951i	$0.352078\pi$
$384$	0	0
$385$	6.76499	0.344776
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.53459	0.128509	0.0642545	−	0.997934i	$-0.479533\pi$
$389$	2.53459	0.128509	0.0642545	+	0.997934i	$0.479533\pi$
$390$	0	0
$391$	6.45880	0.326635
$392$	0	0
$393$	0	0
$394$	0	0
$395$	33.5762	1.68940
$396$	0	0
$397$	30.7430	1.54295	0.771473	−	0.636262i	$-0.219520\pi$
$397$	30.7430	1.54295	0.771473	+	0.636262i	$0.219520\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.8265	0.890212	0.445106	−	0.895478i	$-0.353166\pi$
$401$	17.8265	0.890212	0.445106	+	0.895478i	$0.353166\pi$
$402$	0	0
$403$	27.1675	1.35331
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.55682	−0.225873
$408$	0	0
$409$	−31.8944	−1.57708	−0.788539	−	0.614985i	$-0.789162\pi$
$409$	−31.8944	−1.57708	−0.788539	+	0.614985i	$0.789162\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−52.7755	−2.59691
$414$	0	0
$415$	17.6435	0.866085
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.73689	−0.377972	−0.188986	−	0.981980i	$-0.560520\pi$
$419$	−7.73689	−0.377972	−0.188986	+	0.981980i	$0.560520\pi$
$420$	0	0
$421$	−25.0326	−1.22001	−0.610006	−	0.792397i	$-0.708833\pi$
$421$	−25.0326	−1.22001	−0.610006	+	0.792397i	$0.708833\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.521753	−0.0253088
$426$	0	0
$427$	3.38845	0.163979
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.7224	−0.757320	−0.378660	−	0.925536i	$-0.623615\pi$
$431$	−15.7224	−0.757320	−0.378660	+	0.925536i	$0.623615\pi$
$432$	0	0
$433$	15.1478	0.727955	0.363977	−	0.931408i	$-0.381419\pi$
$433$	15.1478	0.727955	0.363977	+	0.931408i	$0.381419\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.10424	0.148496
$438$	0	0
$439$	−32.0476	−1.52955	−0.764775	−	0.644298i	$-0.777150\pi$
$439$	−32.0476	−1.52955	−0.764775	+	0.644298i	$0.777150\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.3626	−1.20501	−0.602506	−	0.798114i	$-0.705831\pi$
$443$	−25.3626	−1.20501	−0.602506	+	0.798114i	$0.705831\pi$
$444$	0	0
$445$	36.4955	1.73005
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.5037	1.53394	0.766972	−	0.641680i	$-0.221762\pi$
$449$	32.5037	1.53394	0.766972	+	0.641680i	$0.221762\pi$
$450$	0	0
$451$	−5.59591	−0.263501
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−34.9741	−1.63961
$456$	0	0
$457$	−14.4992	−0.678245	−0.339122	−	0.940742i	$-0.610130\pi$
$457$	−14.4992	−0.678245	−0.339122	+	0.940742i	$0.610130\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.0547	0.980616	0.490308	−	0.871549i	$-0.336884\pi$
$461$	21.0547	0.980616	0.490308	+	0.871549i	$0.336884\pi$
$462$	0	0
$463$	31.6539	1.47108	0.735540	−	0.677481i	$-0.236928\pi$
$463$	31.6539	1.47108	0.735540	+	0.677481i	$0.236928\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.0803	−0.929204	−0.464602	−	0.885520i	$-0.653803\pi$
$467$	−20.0803	−0.929204	−0.464602	+	0.885520i	$0.653803\pi$
$468$	0	0
$469$	−5.50437	−0.254168
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.390395	0.0179504
$474$	0	0
$475$	−0.250766	−0.0115059
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.3940	−0.886136	−0.443068	−	0.896488i	$-0.646110\pi$
$479$	−19.3940	−0.886136	−0.443068	+	0.896488i	$0.646110\pi$
$480$	0	0
$481$	23.5582	1.07416
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.71001	0.123055
$486$	0	0
$487$	−1.67643	−0.0759664	−0.0379832	−	0.999278i	$-0.512093\pi$
$487$	−1.67643	−0.0759664	−0.0379832	+	0.999278i	$0.512093\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.6681	−1.06812	−0.534062	−	0.845445i	$-0.679335\pi$
$491$	−23.6681	−1.06812	−0.534062	+	0.845445i	$0.679335\pi$
$492$	0	0
$493$	−11.1990	−0.504378
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23.3471	−1.04726
$498$	0	0
$499$	−2.64766	−0.118526	−0.0592628	−	0.998242i	$-0.518875\pi$
$499$	−2.64766	−0.118526	−0.0592628	+	0.998242i	$0.518875\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.38056	−0.150732	−0.0753659	−	0.997156i	$-0.524012\pi$
$503$	−3.38056	−0.150732	−0.0753659	+	0.997156i	$0.524012\pi$
$504$	0	0
$505$	17.3950	0.774066
$506$	0	0
$507$	0	0
$508$	0	0
$509$	42.8179	1.89787	0.948934	−	0.315474i	$-0.102163\pi$
$509$	42.8179	1.89787	0.948934	+	0.315474i	$0.102163\pi$
$510$	0	0
$511$	−29.7784	−1.31732
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−36.5599	−1.61102
$516$	0	0
$517$	−7.62787	−0.335473
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.3936	−1.15632	−0.578162	−	0.815922i	$-0.696230\pi$
$521$	−26.3936	−1.15632	−0.578162	+	0.815922i	$0.696230\pi$
$522$	0	0
$523$	−20.1256	−0.880030	−0.440015	−	0.897990i	$-0.645027\pi$
$523$	−20.1256	−0.880030	−0.440015	+	0.897990i	$0.645027\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.6611	−0.682207
$528$	0	0
$529$	−13.3637	−0.581031
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.9301	1.25310
$534$	0	0
$535$	12.4636	0.538849
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.91572	0.384027
$540$	0	0
$541$	25.5598	1.09890	0.549451	−	0.835526i	$-0.314837\pi$
$541$	25.5598	1.09890	0.549451	+	0.835526i	$0.314837\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.8014	−0.762528
$546$	0	0
$547$	22.8128	0.975405	0.487702	−	0.873010i	$-0.337835\pi$
$547$	22.8128	0.975405	0.487702	+	0.873010i	$0.337835\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.38249	−0.229302
$552$	0	0
$553$	68.5061	2.91318
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.46839	0.104589	0.0522945	−	0.998632i	$-0.483347\pi$
$557$	2.46839	0.104589	0.0522945	+	0.998632i	$0.483347\pi$
$558$	0	0
$559$	−2.01829	−0.0853646
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13.7211	−0.578275	−0.289137	−	0.957288i	$-0.593368\pi$
$563$	−13.7211	−0.578275	−0.289137	+	0.957288i	$0.593368\pi$
$564$	0	0
$565$	1.20431	0.0506659
$566$	0	0
$567$	0	0
$568$	0	0
$569$	40.3694	1.69237	0.846187	−	0.532886i	$-0.178892\pi$
$569$	40.3694	1.69237	0.846187	+	0.532886i	$0.178892\pi$
$570$	0	0
$571$	−26.4839	−1.10832	−0.554158	−	0.832411i	$-0.686960\pi$
$571$	−26.4839	−1.10832	−0.554158	+	0.832411i	$0.686960\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.778438	−0.0324631
$576$	0	0
$577$	20.9369	0.871613	0.435806	−	0.900040i	$-0.356463\pi$
$577$	20.9369	0.871613	0.435806	+	0.900040i	$0.356463\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	35.9984	1.49346
$582$	0	0
$583$	−7.32125	−0.303215
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.6683	−0.522876	−0.261438	−	0.965220i	$-0.584197\pi$
$587$	−12.6683	−0.522876	−0.261438	+	0.965220i	$0.584197\pi$
$588$	0	0
$589$	−7.52705	−0.310147
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.96020	0.326886	0.163443	−	0.986553i	$-0.447740\pi$
$593$	7.96020	0.326886	0.163443	+	0.986553i	$0.447740\pi$
$594$	0	0
$595$	20.1613	0.826532
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−28.3614	−1.15881	−0.579407	−	0.815038i	$-0.696716\pi$
$599$	−28.3614	−1.15881	−0.579407	+	0.815038i	$0.696716\pi$
$600$	0	0
$601$	−7.60140	−0.310067	−0.155034	−	0.987909i	$-0.549549\pi$
$601$	−7.60140	−0.310067	−0.155034	+	0.987909i	$0.549549\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.9098	0.931417
$606$	0	0
$607$	12.4610	0.505778	0.252889	−	0.967495i	$-0.418619\pi$
$607$	12.4610	0.505778	0.252889	+	0.967495i	$0.418619\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	39.4351	1.59537
$612$	0	0
$613$	−32.1606	−1.29895	−0.649476	−	0.760382i	$-0.725012\pi$
$613$	−32.1606	−1.29895	−0.649476	+	0.760382i	$0.725012\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.63957	−0.347816	−0.173908	−	0.984762i	$-0.555640\pi$
$617$	−8.63957	−0.347816	−0.173908	+	0.984762i	$0.555640\pi$
$618$	0	0
$619$	38.3717	1.54229	0.771144	−	0.636661i	$-0.219685\pi$
$619$	38.3717	1.54229	0.771144	+	0.636661i	$0.219685\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	74.4625	2.98328
$624$	0	0
$625$	−23.6833	−0.947331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.5804	−0.541487
$630$	0	0
$631$	−9.78436	−0.389509	−0.194755	−	0.980852i	$-0.562391\pi$
$631$	−9.78436	−0.389509	−0.194755	+	0.980852i	$0.562391\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−33.5640	−1.33195
$636$	0	0
$637$	−46.0931	−1.82627
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−15.4965	−0.612074	−0.306037	−	0.952020i	$-0.599003\pi$
$641$	−15.4965	−0.612074	−0.306037	+	0.952020i	$0.599003\pi$
$642$	0	0
$643$	−0.868603	−0.0342544	−0.0171272	−	0.999853i	$-0.505452\pi$
$643$	−0.868603	−0.0342544	−0.0171272	+	0.999853i	$0.505452\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.1489	−0.752819	−0.376409	−	0.926453i	$-0.622841\pi$
$647$	−19.1489	−0.752819	−0.376409	+	0.926453i	$0.622841\pi$
$648$	0	0
$649$	8.28643	0.325271
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.8026	−1.47933	−0.739665	−	0.672975i	$-0.765016\pi$
$653$	−37.8026	−1.47933	−0.739665	+	0.672975i	$0.765016\pi$
$654$	0	0
$655$	14.0250	0.548003
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.67264	0.0651568	0.0325784	−	0.999469i	$-0.489628\pi$
$659$	1.67264	0.0651568	0.0325784	+	0.999469i	$0.489628\pi$
$660$	0	0
$661$	−6.08318	−0.236608	−0.118304	−	0.992977i	$-0.537746\pi$
$661$	−6.08318	−0.236608	−0.118304	+	0.992977i	$0.537746\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.68995	0.375760
$666$	0	0
$667$	−16.7085	−0.646957
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.532031	−0.0205388
$672$	0	0
$673$	−12.5462	−0.483619	−0.241810	−	0.970324i	$-0.577741\pi$
$673$	−12.5462	−0.483619	−0.241810	+	0.970324i	$0.577741\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.9176	1.38043	0.690214	−	0.723606i	$-0.257517\pi$
$677$	35.9176	1.38043	0.690214	+	0.723606i	$0.257517\pi$
$678$	0	0
$679$	5.52928	0.212194
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.21569	0.276101	0.138050	−	0.990425i	$-0.455916\pi$
$683$	7.21569	0.276101	0.138050	+	0.990425i	$0.455916\pi$
$684$	0	0
$685$	22.7498	0.869227
$686$	0	0
$687$	0	0
$688$	0	0
$689$	37.8499	1.44197
$690$	0	0
$691$	38.6934	1.47197	0.735983	−	0.677000i	$-0.236720\pi$
$691$	38.6934	1.47197	0.735983	+	0.677000i	$0.236720\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.74151	0.331585
$696$	0	0
$697$	−16.6772	−0.631692
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11.1955	0.422847	0.211424	−	0.977395i	$-0.432190\pi$
$701$	11.1955	0.422847	0.211424	+	0.977395i	$0.432190\pi$
$702$	0	0
$703$	−6.52705	−0.246172
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35.4913	1.33479
$708$	0	0
$709$	27.5312	1.03396	0.516978	−	0.855999i	$-0.327057\pi$
$709$	27.5312	1.03396	0.516978	+	0.855999i	$0.327057\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−23.3658	−0.875055
$714$	0	0
$715$	5.49138	0.205366
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.8647	−1.56129	−0.780645	−	0.624975i	$-0.785109\pi$
$719$	−41.8647	−1.56129	−0.780645	+	0.624975i	$0.785109\pi$
$720$	0	0
$721$	−74.5939	−2.77802
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.34975	0.0501283
$726$	0	0
$727$	−39.7391	−1.47384	−0.736920	−	0.675980i	$-0.763721\pi$
$727$	−39.7391	−1.47384	−0.736920	+	0.675980i	$0.763721\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.16347	0.0430325
$732$	0	0
$733$	−28.1825	−1.04094	−0.520472	−	0.853879i	$-0.674244\pi$
$733$	−28.1825	−1.04094	−0.520472	+	0.853879i	$0.674244\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.864258	0.0318353
$738$	0	0
$739$	3.80866	0.140104	0.0700519	−	0.997543i	$-0.477684\pi$
$739$	3.80866	0.140104	0.0700519	+	0.997543i	$0.477684\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.75597	−0.321225	−0.160613	−	0.987017i	$-0.551347\pi$
$743$	−8.75597	−0.321225	−0.160613	+	0.987017i	$0.551347\pi$
$744$	0	0
$745$	−30.5112	−1.11784
$746$	0	0
$747$	0	0
$748$	0	0
$749$	25.4297	0.929182
$750$	0	0
$751$	−21.7050	−0.792027	−0.396014	−	0.918245i	$-0.629607\pi$
$751$	−21.7050	−0.792027	−0.396014	+	0.918245i	$0.629607\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.52207	−0.310150
$756$	0	0
$757$	12.5910	0.457629	0.228814	−	0.973470i	$-0.426515\pi$
$757$	12.5910	0.457629	0.228814	+	0.973470i	$0.426515\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−44.1239	−1.59949	−0.799745	−	0.600340i	$-0.795032\pi$
$761$	−44.1239	−1.59949	−0.799745	+	0.600340i	$0.795032\pi$
$762$	0	0
$763$	−36.3205	−1.31489
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−42.8398	−1.54685
$768$	0	0
$769$	4.62876	0.166918	0.0834588	−	0.996511i	$-0.473403\pi$
$769$	4.62876	0.166918	0.0834588	+	0.996511i	$0.473403\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36.8840	−1.32663	−0.663314	−	0.748342i	$-0.730850\pi$
$773$	−36.8840	−1.32663	−0.663314	+	0.748342i	$0.730850\pi$
$774$	0	0
$775$	1.88753	0.0678020
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.01541	−0.287182
$780$	0	0
$781$	3.66579	0.131172
$782$	0	0
$783$	0	0
$784$	0	0
$785$	25.3677	0.905412
$786$	0	0
$787$	22.1118	0.788199	0.394100	−	0.919068i	$-0.371057\pi$
$787$	22.1118	0.788199	0.394100	+	0.919068i	$0.371057\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.45718	0.0873674
$792$	0	0
$793$	2.75053	0.0976742
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.4175	−0.546114	−0.273057	−	0.961998i	$-0.588035\pi$
$797$	−15.4175	−0.546114	−0.273057	+	0.961998i	$0.588035\pi$
$798$	0	0
$799$	−22.7329	−0.804231
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.67560	0.164998
$804$	0	0
$805$	30.0799	1.06018
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−33.5881	−1.18089	−0.590447	−	0.807077i	$-0.701048\pi$
$809$	−33.5881	−1.18089	−0.590447	+	0.807077i	$0.701048\pi$
$810$	0	0
$811$	−9.28393	−0.326003	−0.163001	−	0.986626i	$-0.552118\pi$
$811$	−9.28393	−0.326003	−0.163001	+	0.986626i	$0.552118\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−44.7669	−1.56812
$816$	0	0
$817$	0.559190	0.0195636
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.2864	1.16170	0.580852	−	0.814009i	$-0.302719\pi$
$821$	33.2864	1.16170	0.580852	+	0.814009i	$0.302719\pi$
$822$	0	0
$823$	54.6005	1.90325	0.951627	−	0.307256i	$-0.0994109\pi$
$823$	54.6005	1.90325	0.951627	+	0.307256i	$0.0994109\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.47986	0.0514598	0.0257299	−	0.999669i	$-0.491809\pi$
$827$	1.47986	0.0514598	0.0257299	+	0.999669i	$0.491809\pi$
$828$	0	0
$829$	26.6522	0.925669	0.462835	−	0.886445i	$-0.346832\pi$
$829$	26.6522	0.925669	0.462835	+	0.886445i	$0.346832\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	26.5710	0.920629
$834$	0	0
$835$	−18.9157	−0.654604
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.3810	−0.772676	−0.386338	−	0.922357i	$-0.626260\pi$
$839$	−22.3810	−0.772676	−0.386338	+	0.922357i	$0.626260\pi$
$840$	0	0
$841$	−0.0287695	−0.000992052 0
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.0591540	−0.00203496
$846$	0	0
$847$	46.7433	1.60612
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20.2615	−0.694556
$852$	0	0
$853$	33.4603	1.14566	0.572829	−	0.819675i	$-0.305846\pi$
$853$	33.4603	1.14566	0.572829	+	0.819675i	$0.305846\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.3861	−1.58452	−0.792260	−	0.610183i	$-0.791096\pi$
$857$	−46.3861	−1.58452	−0.792260	+	0.610183i	$0.791096\pi$
$858$	0	0
$859$	−8.09184	−0.276090	−0.138045	−	0.990426i	$-0.544082\pi$
$859$	−8.09184	−0.276090	−0.138045	+	0.990426i	$0.544082\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.5692	−0.734224	−0.367112	−	0.930177i	$-0.619653\pi$
$863$	−21.5692	−0.734224	−0.367112	+	0.930177i	$0.619653\pi$
$864$	0	0
$865$	−34.5777	−1.17568
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−10.7563	−0.364884
$870$	0	0
$871$	−4.46810	−0.151396
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−50.8797	−1.72005
$876$	0	0
$877$	41.4847	1.40084	0.700420	−	0.713731i	$-0.252996\pi$
$877$	41.4847	1.40084	0.700420	+	0.713731i	$0.252996\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−6.38614	−0.215155	−0.107577	−	0.994197i	$-0.534309\pi$
$881$	−6.38614	−0.215155	−0.107577	+	0.994197i	$0.534309\pi$
$882$	0	0
$883$	58.8554	1.98064	0.990320	−	0.138801i	$-0.0443248\pi$
$883$	58.8554	1.98064	0.990320	+	0.138801i	$0.0443248\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.4410	1.15641	0.578207	−	0.815890i	$-0.303752\pi$
$887$	34.4410	1.15641	0.578207	+	0.815890i	$0.303752\pi$
$888$	0	0
$889$	−68.4812	−2.29678
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.9259	−0.365622
$894$	0	0
$895$	19.8049	0.662005
$896$	0	0
$897$	0	0
$898$	0	0
$899$	40.5143	1.35123
$900$	0	0
$901$	−21.8191	−0.726899
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−15.4971	−0.515143
$906$	0	0
$907$	39.2022	1.30169	0.650843	−	0.759212i	$-0.274416\pi$
$907$	39.2022	1.30169	0.650843	+	0.759212i	$0.274416\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.1097	−0.964446	−0.482223	−	0.876049i	$-0.660170\pi$
$911$	−29.1097	−0.964446	−0.482223	+	0.876049i	$0.660170\pi$
$912$	0	0
$913$	−5.65221	−0.187061
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.6155	0.944968
$918$	0	0
$919$	−19.1577	−0.631955	−0.315977	−	0.948767i	$-0.602332\pi$
$919$	−19.1577	−0.631955	−0.315977	+	0.948767i	$0.602332\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−18.9517	−0.623802
$924$	0	0
$925$	1.63676	0.0538164
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.7457	−0.418172	−0.209086	−	0.977897i	$-0.567049\pi$
$929$	−12.7457	−0.418172	−0.209086	+	0.977897i	$0.567049\pi$
$930$	0	0
$931$	12.7706	0.418539
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.16558	−0.103526
$936$	0	0
$937$	−6.71149	−0.219255	−0.109627	−	0.993973i	$-0.534966\pi$
$937$	−6.71149	−0.219255	−0.109627	+	0.993973i	$0.534966\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20.3136	−0.662204	−0.331102	−	0.943595i	$-0.607420\pi$
$941$	−20.3136	−0.662204	−0.331102	+	0.943595i	$0.607420\pi$
$942$	0	0
$943$	−24.8817	−0.810261
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.63757	−0.248188	−0.124094	−	0.992270i	$-0.539602\pi$
$947$	−7.63757	−0.248188	−0.124094	+	0.992270i	$0.539602\pi$
$948$	0	0
$949$	−24.1722	−0.784663
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.6049	0.991391	0.495695	−	0.868496i	$-0.334913\pi$
$953$	30.6049	0.991391	0.495695	+	0.868496i	$0.334913\pi$
$954$	0	0
$955$	−28.8813	−0.934577
$956$	0	0
$957$	0	0
$958$	0	0
$959$	46.4169	1.49888
$960$	0	0
$961$	25.6565	0.827629
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−38.4098	−1.23646
$966$	0	0
$967$	−50.9289	−1.63776	−0.818882	−	0.573962i	$-0.805406\pi$
$967$	−50.9289	−1.63776	−0.818882	+	0.573962i	$0.805406\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.9785	0.576956	0.288478	−	0.957486i	$-0.406851\pi$
$971$	17.9785	0.576956	0.288478	+	0.957486i	$0.406851\pi$
$972$	0	0
$973$	17.8355	0.571779
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.5245	0.688629	0.344314	−	0.938854i	$-0.388111\pi$
$977$	21.5245	0.688629	0.344314	+	0.938854i	$0.388111\pi$
$978$	0	0
$979$	−11.6916	−0.373664
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−15.0333	−0.479488	−0.239744	−	0.970836i	$-0.577063\pi$
$983$	−15.0333	−0.479488	−0.239744	+	0.970836i	$0.577063\pi$
$984$	0	0
$985$	20.3014	0.646855
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.73586	0.0551971
$990$	0	0
$991$	−36.7415	−1.16713	−0.583566	−	0.812065i	$-0.698343\pi$
$991$	−36.7415	−1.16713	−0.583566	+	0.812065i	$0.698343\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.81438	0.152626
$996$	0	0
$997$	37.0081	1.17206	0.586029	−	0.810290i	$-0.300690\pi$
$997$	37.0081	1.17206	0.586029	+	0.810290i	$0.300690\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.cb.1.2		5
3.2	odd	2		8208.2.a.cc.1.4		5
4.3	odd	2		4104.2.a.o.1.2	✓	5
12.11	even	2		4104.2.a.r.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.o.1.2	✓	5	4.3	odd	2
4104.2.a.r.1.4	yes	5	12.11	even	2
8208.2.a.cb.1.2		5	1.1	even	1	trivial
8208.2.a.cc.1.4		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8208 = 2^{4} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8208.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17927 q^{5} -4.44641 q^{7} +0.698144 q^{11} -3.60931 q^{13} +2.08064 q^{17} +1.00000 q^{19} +3.10424 q^{23} -0.250766 q^{25} -5.38249 q^{29} -7.52705 q^{31} +9.68995 q^{35} -6.52705 q^{37} -8.01541 q^{41} +0.559190 q^{43} -10.9259 q^{47} +12.7706 q^{49} -10.4867 q^{53} -1.52145 q^{55} +11.8692 q^{59} -0.762065 q^{61} +7.86568 q^{65} +1.23794 q^{67} +5.25077 q^{71} +6.69718 q^{73} -3.10424 q^{77} -15.4070 q^{79} -8.09605 q^{83} -4.53428 q^{85} -16.7466 q^{89} +16.0485 q^{91} -2.17927 q^{95} -1.24354 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} - q^{7} + 2 q^{11} + 3 q^{13} - 8 q^{17} + 5 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} + 2 q^{31} + 9 q^{35} + 7 q^{37} - 7 q^{41} + 10 q^{47} + 6 q^{49} - 22 q^{53} + 8 q^{55} + 12 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) 5 * q - 3 * q^5 - q^7 + 2 * q^11 + 3 * q^13 - 8 * q^17 + 5 * q^19 + 2 * q^23 + 4 * q^25 - 10 * q^29 + 2 * q^31 + 9 * q^35 + 7 * q^37 - 7 * q^41 + 10 * q^47 + 6 * q^49 - 22 * q^53 + 8 * q^55 + 12 * q^59 + 2 * q^61 - 11 * q^65 + 12 * q^67 + 21 * q^71 + 7 * q^73 - 2 * q^77 - 6 * q^79 + 11 * q^83 + 4 * q^85 - 27 * q^89 + 25 * q^91 - 3 * q^95 + 12 * q^97

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