LMFDB - Embedded newform 8208.2.a.ce.1.5

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 = [6,0,0,0,-2,0,-2,0,0,0,-3] 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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 18x^{4} + 17x^{3} + 72x^{2} + 29x - 3 $ x^6 - 2x^5 - 18x^4 + 17x^3 + 72x^2 + 29*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4104)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.40017$ 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)

$f(q)$	$=$	$q+1.40017 q^{5} +4.51232 q^{7} -6.07883 q^{11} -4.91692 q^{13} -0.433490 q^{17} -1.00000 q^{19} -0.995569 q^{23} -3.03952 q^{25} +2.23826 q^{29} +1.27849 q^{31} +6.31802 q^{35} +3.80570 q^{37} +5.31802 q^{41} +1.70663 q^{43} -9.71819 q^{47} +13.3610 q^{49} -4.70755 q^{53} -8.51140 q^{55} -8.18407 q^{59} +12.2683 q^{61} -6.88453 q^{65} +12.0730 q^{67} +9.35640 q^{71} -7.79011 q^{73} -27.4296 q^{77} -10.3460 q^{79} -10.5114 q^{83} -0.606959 q^{85} -5.84500 q^{89} -22.1867 q^{91} -1.40017 q^{95} +1.47372 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 2 q^{7} - 3 q^{11} + q^{13} - 7 q^{17} - 6 q^{19} + 3 q^{23} + 10 q^{25} + 6 q^{29} - 5 q^{31} - 3 q^{35} + 11 q^{37} - 9 q^{41} - 7 q^{43} - 7 q^{47} + 20 q^{49} - 11 q^{53} - 22 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) $ 6 * q - 2 * q^5 - 2 * q^7 - 3 * q^11 + q^13 - 7 * q^17 - 6 * q^19 + 3 * q^23 + 10 * q^25 + 6 * q^29 - 5 * q^31 - 3 * q^35 + 11 * q^37 - 9 * q^41 - 7 * q^43 - 7 * q^47 + 20 * q^49 - 11 * q^53 - 22 * q^55 - 17 * q^59 + 17 * q^61 + 4 * q^65 + 4 * q^67 + 10 * q^71 + 18 * q^73 - 27 * q^77 - 27 * q^79 - 34 * q^83 + 29 * q^85 - 18 * q^89 - 17 * q^91 + 2 * q^95 + 8 * 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
$5$	1.40017	0.626175	0.313087	−	0.949724i	$-0.398637\pi$
$5$	1.40017	0.626175	0.313087	+	0.949724i	$0.398637\pi$
$6$	0	0
$7$	4.51232	1.70550	0.852749	−	0.522321i	$-0.174934\pi$
$7$	4.51232	1.70550	0.852749	+	0.522321i	$0.174934\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.07883	−1.83284	−0.916418	−	0.400221i	$-0.868933\pi$
$11$	−6.07883	−1.83284	−0.916418	+	0.400221i	$0.868933\pi$
$12$	0	0
$13$	−4.91692	−1.36371	−0.681855	−	0.731488i	$-0.738826\pi$
$13$	−4.91692	−1.36371	−0.681855	+	0.731488i	$0.738826\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.433490	−0.105137	−0.0525683	−	0.998617i	$-0.516741\pi$
$17$	−0.433490	−0.105137	−0.0525683	+	0.998617i	$0.516741\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.995569	−0.207590	−0.103795	−	0.994599i	$-0.533099\pi$
$23$	−0.995569	−0.207590	−0.103795	+	0.994599i	$0.533099\pi$
$24$	0	0
$25$	−3.03952	−0.607905
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.23826	0.415635	0.207817	−	0.978168i	$-0.433364\pi$
$29$	2.23826	0.415635	0.207817	+	0.978168i	$0.433364\pi$
$30$	0	0
$31$	1.27849	0.229624	0.114812	−	0.993387i	$-0.463373\pi$
$31$	1.27849	0.229624	0.114812	+	0.993387i	$0.463373\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.31802	1.06794
$36$	0	0
$37$	3.80570	0.625652	0.312826	−	0.949810i	$-0.398724\pi$
$37$	3.80570	0.625652	0.312826	+	0.949810i	$0.398724\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.31802	0.830535	0.415267	−	0.909699i	$-0.363688\pi$
$41$	5.31802	0.830535	0.415267	+	0.909699i	$0.363688\pi$
$42$	0	0
$43$	1.70663	0.260258	0.130129	−	0.991497i	$-0.458461\pi$
$43$	1.70663	0.260258	0.130129	+	0.991497i	$0.458461\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.71819	−1.41754	−0.708772	−	0.705438i	$-0.750750\pi$
$47$	−9.71819	−1.41754	−0.708772	+	0.705438i	$0.750750\pi$
$48$	0	0
$49$	13.3610	1.90872
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.70755	−0.646632	−0.323316	−	0.946291i	$-0.604798\pi$
$53$	−4.70755	−0.646632	−0.323316	+	0.946291i	$0.604798\pi$
$54$	0	0
$55$	−8.51140	−1.14768
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.18407	−1.06548	−0.532738	−	0.846280i	$-0.678837\pi$
$59$	−8.18407	−1.06548	−0.532738	+	0.846280i	$0.678837\pi$
$60$	0	0
$61$	12.2683	1.57079	0.785395	−	0.618995i	$-0.212460\pi$
$61$	12.2683	1.57079	0.785395	+	0.618995i	$0.212460\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.88453	−0.853921
$66$	0	0
$67$	12.0730	1.47496	0.737478	−	0.675371i	$-0.236017\pi$
$67$	12.0730	1.47496	0.737478	+	0.675371i	$0.236017\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.35640	1.11040	0.555200	−	0.831717i	$-0.312642\pi$
$71$	9.35640	1.11040	0.555200	+	0.831717i	$0.312642\pi$
$72$	0	0
$73$	−7.79011	−0.911763	−0.455882	−	0.890040i	$-0.650676\pi$
$73$	−7.79011	−0.911763	−0.455882	+	0.890040i	$0.650676\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−27.4296	−3.12590
$78$	0	0
$79$	−10.3460	−1.16401	−0.582007	−	0.813184i	$-0.697732\pi$
$79$	−10.3460	−1.16401	−0.582007	+	0.813184i	$0.697732\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.5114	−1.15378	−0.576888	−	0.816823i	$-0.695733\pi$
$83$	−10.5114	−1.15378	−0.576888	+	0.816823i	$0.695733\pi$
$84$	0	0
$85$	−0.606959	−0.0658339
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.84500	−0.619569	−0.309785	−	0.950807i	$-0.600257\pi$
$89$	−5.84500	−0.619569	−0.309785	+	0.950807i	$0.600257\pi$
$90$	0	0
$91$	−22.1867	−2.32580
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.40017	−0.143654
$96$	0	0
$97$	1.47372	0.149634	0.0748169	−	0.997197i	$-0.476163\pi$
$97$	1.47372	0.149634	0.0748169	+	0.997197i	$0.476163\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.1856	−1.01351	−0.506754	−	0.862091i	$-0.669155\pi$
$101$	−10.1856	−1.01351	−0.506754	+	0.862091i	$0.669155\pi$
$102$	0	0
$103$	−15.1903	−1.49674	−0.748371	−	0.663280i	$-0.769164\pi$
$103$	−15.1903	−1.49674	−0.748371	+	0.663280i	$0.769164\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.6405	−1.22200	−0.611000	−	0.791631i	$-0.709232\pi$
$107$	−12.6405	−1.22200	−0.611000	+	0.791631i	$0.709232\pi$
$108$	0	0
$109$	−4.28913	−0.410824	−0.205412	−	0.978676i	$-0.565853\pi$
$109$	−4.28913	−0.410824	−0.205412	+	0.978676i	$0.565853\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.30757	0.781510	0.390755	−	0.920495i	$-0.372214\pi$
$113$	8.30757	0.781510	0.390755	+	0.920495i	$0.372214\pi$
$114$	0	0
$115$	−1.39397	−0.129988
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.95604	−0.179310
$120$	0	0
$121$	25.9522	2.35929
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.2567	−1.00683
$126$	0	0
$127$	−19.4653	−1.72726	−0.863631	−	0.504124i	$-0.831815\pi$
$127$	−19.4653	−1.72726	−0.863631	+	0.504124i	$0.831815\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−14.8741	−1.29955	−0.649777	−	0.760125i	$-0.725138\pi$
$131$	−14.8741	−1.29955	−0.649777	+	0.760125i	$0.725138\pi$
$132$	0	0
$133$	−4.51232	−0.391268
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.3924	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$137$	13.3924	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$138$	0	0
$139$	−18.9136	−1.60423	−0.802115	−	0.597169i	$-0.796292\pi$
$139$	−18.9136	−1.60423	−0.802115	+	0.597169i	$0.796292\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	29.8892	2.49946
$144$	0	0
$145$	3.13395	0.260260
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.60023	0.704559	0.352279	−	0.935895i	$-0.385407\pi$
$149$	8.60023	0.704559	0.352279	+	0.935895i	$0.385407\pi$
$150$	0	0
$151$	5.23494	0.426013	0.213007	−	0.977051i	$-0.431674\pi$
$151$	5.23494	0.426013	0.213007	+	0.977051i	$0.431674\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.79011	0.143785
$156$	0	0
$157$	−16.4250	−1.31086	−0.655429	−	0.755257i	$-0.727512\pi$
$157$	−16.4250	−1.31086	−0.655429	+	0.755257i	$0.727512\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.49233	−0.354045
$162$	0	0
$163$	1.27314	0.0997198	0.0498599	−	0.998756i	$-0.484123\pi$
$163$	1.27314	0.0997198	0.0498599	+	0.998756i	$0.484123\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.3520	−0.801059	−0.400530	−	0.916284i	$-0.631174\pi$
$167$	−10.3520	−0.801059	−0.400530	+	0.916284i	$0.631174\pi$
$168$	0	0
$169$	11.1761	0.859703
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−23.3181	−1.77284	−0.886422	−	0.462877i	$-0.846817\pi$
$173$	−23.3181	−1.77284	−0.886422	+	0.462877i	$0.846817\pi$
$174$	0	0
$175$	−13.7153	−1.03678
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.322187	0.0240814	0.0120407	−	0.999928i	$-0.496167\pi$
$179$	0.322187	0.0240814	0.0120407	+	0.999928i	$0.496167\pi$
$180$	0	0
$181$	−22.4319	−1.66735	−0.833675	−	0.552255i	$-0.813768\pi$
$181$	−22.4319	−1.66735	−0.833675	+	0.552255i	$0.813768\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.32862	0.391768
$186$	0	0
$187$	2.63511	0.192698
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.6440	−1.49375	−0.746873	−	0.664967i	$-0.768446\pi$
$191$	−20.6440	−1.49375	−0.746873	+	0.664967i	$0.768446\pi$
$192$	0	0
$193$	8.83614	0.636039	0.318020	−	0.948084i	$-0.396982\pi$
$193$	8.83614	0.636039	0.318020	+	0.948084i	$0.396982\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.200510	0.0142857	0.00714286	−	0.999974i	$-0.497726\pi$
$197$	0.200510	0.0142857	0.00714286	+	0.999974i	$0.497726\pi$
$198$	0	0
$199$	7.98534	0.566065	0.283033	−	0.959110i	$-0.408659\pi$
$199$	7.98534	0.566065	0.283033	+	0.959110i	$0.408659\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	10.0998	0.708864
$204$	0	0
$205$	7.44613	0.520060
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.07883	0.420482
$210$	0	0
$211$	8.78731	0.604943	0.302472	−	0.953158i	$-0.402188\pi$
$211$	8.78731	0.604943	0.302472	+	0.953158i	$0.402188\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.38957	0.162967
$216$	0	0
$217$	5.76897	0.391623
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.13143	0.143376
$222$	0	0
$223$	−10.6935	−0.716091	−0.358045	−	0.933704i	$-0.616557\pi$
$223$	−10.6935	−0.716091	−0.358045	+	0.933704i	$0.616557\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.3285	−1.08376	−0.541879	−	0.840456i	$-0.682287\pi$
$227$	−16.3285	−1.08376	−0.541879	+	0.840456i	$0.682287\pi$
$228$	0	0
$229$	19.4771	1.28708	0.643540	−	0.765413i	$-0.277465\pi$
$229$	19.4771	1.28708	0.643540	+	0.765413i	$0.277465\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.2936	0.674354	0.337177	−	0.941441i	$-0.390528\pi$
$233$	10.2936	0.674354	0.337177	+	0.941441i	$0.390528\pi$
$234$	0	0
$235$	−13.6071	−0.887630
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.16405	0.334034	0.167017	−	0.985954i	$-0.446586\pi$
$239$	5.16405	0.334034	0.167017	+	0.985954i	$0.446586\pi$
$240$	0	0
$241$	22.8636	1.47277	0.736386	−	0.676561i	$-0.236531\pi$
$241$	22.8636	1.47277	0.736386	+	0.676561i	$0.236531\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	18.7077	1.19519
$246$	0	0
$247$	4.91692	0.312856
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−14.9140	−0.941366	−0.470683	−	0.882302i	$-0.655993\pi$
$251$	−14.9140	−0.941366	−0.470683	+	0.882302i	$0.655993\pi$
$252$	0	0
$253$	6.05189	0.380479
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.9266	−1.30537	−0.652684	−	0.757630i	$-0.726357\pi$
$257$	−20.9266	−1.30537	−0.652684	+	0.757630i	$0.726357\pi$
$258$	0	0
$259$	17.1725	1.06705
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.70238	0.598275	0.299137	−	0.954210i	$-0.403301\pi$
$263$	9.70238	0.598275	0.299137	+	0.954210i	$0.403301\pi$
$264$	0	0
$265$	−6.59137	−0.404905
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.21963	−0.0743624	−0.0371812	−	0.999309i	$-0.511838\pi$
$269$	−1.21963	−0.0743624	−0.0371812	+	0.999309i	$0.511838\pi$
$270$	0	0
$271$	21.8052	1.32457	0.662287	−	0.749251i	$-0.269586\pi$
$271$	21.8052	1.32457	0.662287	+	0.749251i	$0.269586\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.4768	1.11419
$276$	0	0
$277$	−21.9216	−1.31714	−0.658571	−	0.752518i	$-0.728839\pi$
$277$	−21.9216	−1.31714	−0.658571	+	0.752518i	$0.728839\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.7242	0.818716	0.409358	−	0.912374i	$-0.365753\pi$
$281$	13.7242	0.818716	0.409358	+	0.912374i	$0.365753\pi$
$282$	0	0
$283$	−1.85479	−0.110256	−0.0551279	−	0.998479i	$-0.517557\pi$
$283$	−1.85479	−0.110256	−0.0551279	+	0.998479i	$0.517557\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.9966	1.41647
$288$	0	0
$289$	−16.8121	−0.988946
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.08440	0.180193	0.0900964	−	0.995933i	$-0.471282\pi$
$293$	3.08440	0.180193	0.0900964	+	0.995933i	$0.471282\pi$
$294$	0	0
$295$	−11.4591	−0.667174
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.89513	0.283093
$300$	0	0
$301$	7.70085	0.443870
$302$	0	0
$303$	0	0
$304$	0	0
$305$	17.1776	0.983589
$306$	0	0
$307$	8.78559	0.501420	0.250710	−	0.968062i	$-0.419336\pi$
$307$	8.78559	0.501420	0.250710	+	0.968062i	$0.419336\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−23.2902	−1.32066	−0.660332	−	0.750974i	$-0.729584\pi$
$311$	−23.2902	−1.32066	−0.660332	+	0.750974i	$0.729584\pi$
$312$	0	0
$313$	−0.255589	−0.0144467	−0.00722336	−	0.999974i	$-0.502299\pi$
$313$	−0.255589	−0.0144467	−0.00722336	+	0.999974i	$0.502299\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.0915	−0.959954	−0.479977	−	0.877281i	$-0.659355\pi$
$317$	−17.0915	−0.959954	−0.479977	+	0.877281i	$0.659355\pi$
$318$	0	0
$319$	−13.6060	−0.761790
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.433490	0.0241200
$324$	0	0
$325$	14.9451	0.829005
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−43.8516	−2.41762
$330$	0	0
$331$	−6.90362	−0.379457	−0.189729	−	0.981837i	$-0.560761\pi$
$331$	−6.90362	−0.379457	−0.189729	+	0.981837i	$0.560761\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.9043	0.923580
$336$	0	0
$337$	30.3357	1.65249	0.826245	−	0.563311i	$-0.190473\pi$
$337$	30.3357	1.65249	0.826245	+	0.563311i	$0.190473\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7.77174	−0.420863
$342$	0	0
$343$	28.7031	1.54982
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.1070	−1.18677	−0.593383	−	0.804921i	$-0.702208\pi$
$347$	−22.1070	−1.18677	−0.593383	+	0.804921i	$0.702208\pi$
$348$	0	0
$349$	24.7770	1.32628	0.663140	−	0.748495i	$-0.269223\pi$
$349$	24.7770	1.32628	0.663140	+	0.748495i	$0.269223\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.6368	0.619363	0.309681	−	0.950840i	$-0.399778\pi$
$353$	11.6368	0.619363	0.309681	+	0.950840i	$0.399778\pi$
$354$	0	0
$355$	13.1006	0.695305
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.11123	0.480872	0.240436	−	0.970665i	$-0.422710\pi$
$359$	9.11123	0.480872	0.240436	+	0.970665i	$0.422710\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.9075	−0.570923
$366$	0	0
$367$	−23.8322	−1.24403	−0.622016	−	0.783005i	$-0.713686\pi$
$367$	−23.8322	−1.24403	−0.622016	+	0.783005i	$0.713686\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−21.2420	−1.10283
$372$	0	0
$373$	−2.47834	−0.128324	−0.0641618	−	0.997940i	$-0.520437\pi$
$373$	−2.47834	−0.128324	−0.0641618	+	0.997940i	$0.520437\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11.0054	−0.566805
$378$	0	0
$379$	−2.61986	−0.134573	−0.0672865	−	0.997734i	$-0.521434\pi$
$379$	−2.61986	−0.134573	−0.0672865	+	0.997734i	$0.521434\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	35.4838	1.81314	0.906569	−	0.422057i	$-0.138692\pi$
$383$	35.4838	1.81314	0.906569	+	0.422057i	$0.138692\pi$
$384$	0	0
$385$	−38.4062	−1.95736
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.22515	0.417032	0.208516	−	0.978019i	$-0.433137\pi$
$389$	8.22515	0.417032	0.208516	+	0.978019i	$0.433137\pi$
$390$	0	0
$391$	0.431569	0.0218254
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.4861	−0.728876
$396$	0	0
$397$	5.19080	0.260519	0.130259	−	0.991480i	$-0.458419\pi$
$397$	5.19080	0.260519	0.130259	+	0.991480i	$0.458419\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.19737	0.0597936	0.0298968	−	0.999553i	$-0.490482\pi$
$401$	1.19737	0.0597936	0.0298968	+	0.999553i	$0.490482\pi$
$402$	0	0
$403$	−6.28625	−0.313140
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.1342	−1.14672
$408$	0	0
$409$	−4.70663	−0.232728	−0.116364	−	0.993207i	$-0.537124\pi$
$409$	−4.70663	−0.232728	−0.116364	+	0.993207i	$0.537124\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−36.9292	−1.81717
$414$	0	0
$415$	−14.7177	−0.722466
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.4557	0.950471	0.475236	−	0.879859i	$-0.342363\pi$
$419$	19.4557	0.950471	0.475236	+	0.879859i	$0.342363\pi$
$420$	0	0
$421$	−9.23763	−0.450214	−0.225107	−	0.974334i	$-0.572273\pi$
$421$	−9.23763	−0.450214	−0.225107	+	0.974334i	$0.572273\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.31760	0.0639131
$426$	0	0
$427$	55.3583	2.67898
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.37375	0.0661713	0.0330857	−	0.999453i	$-0.489467\pi$
$431$	1.37375	0.0661713	0.0330857	+	0.999453i	$0.489467\pi$
$432$	0	0
$433$	9.50033	0.456557	0.228278	−	0.973596i	$-0.426690\pi$
$433$	9.50033	0.456557	0.228278	+	0.973596i	$0.426690\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.995569	0.0476245
$438$	0	0
$439$	21.8188	1.04135	0.520677	−	0.853754i	$-0.325680\pi$
$439$	21.8188	1.04135	0.520677	+	0.853754i	$0.325680\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.9065	−0.565695	−0.282848	−	0.959165i	$-0.591279\pi$
$443$	−11.9065	−0.565695	−0.282848	+	0.959165i	$0.591279\pi$
$444$	0	0
$445$	−8.18400	−0.387959
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.8050	0.651497	0.325748	−	0.945457i	$-0.394384\pi$
$449$	13.8050	0.651497	0.325748	+	0.945457i	$0.394384\pi$
$450$	0	0
$451$	−32.3273	−1.52223
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−31.0652	−1.45636
$456$	0	0
$457$	4.69602	0.219671	0.109835	−	0.993950i	$-0.464968\pi$
$457$	4.69602	0.219671	0.109835	+	0.993950i	$0.464968\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.7060	1.43012	0.715061	−	0.699062i	$-0.246399\pi$
$461$	30.7060	1.43012	0.715061	+	0.699062i	$0.246399\pi$
$462$	0	0
$463$	42.9295	1.99510	0.997552	−	0.0699288i	$-0.0222772\pi$
$463$	42.9295	1.99510	0.997552	+	0.0699288i	$0.0222772\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.1851	−1.67445	−0.837224	−	0.546861i	$-0.815823\pi$
$467$	−36.1851	−1.67445	−0.837224	+	0.546861i	$0.815823\pi$
$468$	0	0
$469$	54.4774	2.51553
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.3743	−0.477011
$474$	0	0
$475$	3.03952	0.139463
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.100541	−0.00459382	−0.00229691	−	0.999997i	$-0.500731\pi$
$479$	−0.100541	−0.00459382	−0.00229691	+	0.999997i	$0.500731\pi$
$480$	0	0
$481$	−18.7123	−0.853208
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.06346	0.0936969
$486$	0	0
$487$	27.4446	1.24363	0.621816	−	0.783163i	$-0.286395\pi$
$487$	27.4446	1.24363	0.621816	+	0.783163i	$0.286395\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.0350	1.22007	0.610037	−	0.792373i	$-0.291155\pi$
$491$	27.0350	1.22007	0.610037	+	0.792373i	$0.291155\pi$
$492$	0	0
$493$	−0.970262	−0.0436984
$494$	0	0
$495$	0	0
$496$	0	0
$497$	42.2191	1.89379
$498$	0	0
$499$	27.3864	1.22598	0.612991	−	0.790090i	$-0.289966\pi$
$499$	27.3864	1.22598	0.612991	+	0.790090i	$0.289966\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−41.6863	−1.85870	−0.929351	−	0.369198i	$-0.879632\pi$
$503$	−41.6863	−1.85870	−0.929351	+	0.369198i	$0.879632\pi$
$504$	0	0
$505$	−14.2616	−0.634633
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.53948	0.289857	0.144929	−	0.989442i	$-0.453705\pi$
$509$	6.53948	0.289857	0.144929	+	0.989442i	$0.453705\pi$
$510$	0	0
$511$	−35.1515	−1.55501
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−21.2690	−0.937223
$516$	0	0
$517$	59.0752	2.59813
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−45.2302	−1.98157	−0.990785	−	0.135443i	$-0.956754\pi$
$521$	−45.2302	−1.98157	−0.990785	+	0.135443i	$0.956754\pi$
$522$	0	0
$523$	33.4715	1.46361	0.731803	−	0.681517i	$-0.238679\pi$
$523$	33.4715	1.46361	0.731803	+	0.681517i	$0.238679\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.554213	−0.0241419
$528$	0	0
$529$	−22.0088	−0.956906
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−26.1483	−1.13261
$534$	0	0
$535$	−17.6988	−0.765186
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−81.2196	−3.49837
$540$	0	0
$541$	18.8996	0.812557	0.406279	−	0.913749i	$-0.366826\pi$
$541$	18.8996	0.812557	0.406279	+	0.913749i	$0.366826\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.00551	−0.257248
$546$	0	0
$547$	−40.7476	−1.74224	−0.871122	−	0.491067i	$-0.836607\pi$
$547$	−40.7476	−1.74224	−0.871122	+	0.491067i	$0.836607\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.23826	−0.0953531
$552$	0	0
$553$	−46.6844	−1.98522
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.9595	−1.65077	−0.825383	−	0.564573i	$-0.809041\pi$
$557$	−38.9595	−1.65077	−0.825383	+	0.564573i	$0.809041\pi$
$558$	0	0
$559$	−8.39135	−0.354916
$560$	0	0
$561$	0	0
$562$	0	0
$563$	41.0964	1.73201	0.866004	−	0.500038i	$-0.166680\pi$
$563$	41.0964	1.73201	0.866004	+	0.500038i	$0.166680\pi$
$564$	0	0
$565$	11.6320	0.489362
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.23428	0.303277	0.151638	−	0.988436i	$-0.451545\pi$
$569$	7.23428	0.303277	0.151638	+	0.988436i	$0.451545\pi$
$570$	0	0
$571$	−32.6729	−1.36732	−0.683659	−	0.729802i	$-0.739612\pi$
$571$	−32.6729	−1.36732	−0.683659	+	0.729802i	$0.739612\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.02606	0.126195
$576$	0	0
$577$	17.0315	0.709030	0.354515	−	0.935050i	$-0.384646\pi$
$577$	17.0315	0.709030	0.354515	+	0.935050i	$0.384646\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−47.4308	−1.96776
$582$	0	0
$583$	28.6164	1.18517
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.5192	−1.17711	−0.588557	−	0.808456i	$-0.700304\pi$
$587$	−28.5192	−1.17711	−0.588557	+	0.808456i	$0.700304\pi$
$588$	0	0
$589$	−1.27849	−0.0526794
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5.49950	−0.225837	−0.112919	−	0.993604i	$-0.536020\pi$
$593$	−5.49950	−0.225837	−0.112919	+	0.993604i	$0.536020\pi$
$594$	0	0
$595$	−2.73879	−0.112280
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.3464	−0.831331	−0.415665	−	0.909518i	$-0.636451\pi$
$599$	−20.3464	−0.831331	−0.415665	+	0.909518i	$0.636451\pi$
$600$	0	0
$601$	−16.5825	−0.676415	−0.338207	−	0.941072i	$-0.609821\pi$
$601$	−16.5825	−0.676415	−0.338207	+	0.941072i	$0.609821\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	36.3375	1.47733
$606$	0	0
$607$	−36.9107	−1.49816	−0.749079	−	0.662480i	$-0.769504\pi$
$607$	−36.9107	−1.49816	−0.749079	+	0.662480i	$0.769504\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	47.7836	1.93312
$612$	0	0
$613$	−36.0386	−1.45558	−0.727792	−	0.685798i	$-0.759453\pi$
$613$	−36.0386	−1.45558	−0.727792	+	0.685798i	$0.759453\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.1665	−1.01317	−0.506583	−	0.862191i	$-0.669091\pi$
$617$	−25.1665	−1.01317	−0.506583	+	0.862191i	$0.669091\pi$
$618$	0	0
$619$	−41.9575	−1.68642	−0.843208	−	0.537588i	$-0.819336\pi$
$619$	−41.9575	−1.68642	−0.843208	+	0.537588i	$0.819336\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−26.3745	−1.05667
$624$	0	0
$625$	−0.563670	−0.0225468
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.64973	−0.0657790
$630$	0	0
$631$	−24.4114	−0.971802	−0.485901	−	0.874014i	$-0.661509\pi$
$631$	−24.4114	−0.971802	−0.485901	+	0.874014i	$0.661509\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−27.2547	−1.08157
$636$	0	0
$637$	−65.6953	−2.60294
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28.4323	−1.12301	−0.561504	−	0.827474i	$-0.689777\pi$
$641$	−28.4323	−1.12301	−0.561504	+	0.827474i	$0.689777\pi$
$642$	0	0
$643$	38.8708	1.53292	0.766458	−	0.642295i	$-0.222018\pi$
$643$	38.8708	1.53292	0.766458	+	0.642295i	$0.222018\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11.0604	0.434829	0.217414	−	0.976079i	$-0.430238\pi$
$647$	11.0604	0.434829	0.217414	+	0.976079i	$0.430238\pi$
$648$	0	0
$649$	49.7496	1.95284
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.2832	−0.480678	−0.240339	−	0.970689i	$-0.577259\pi$
$653$	−12.2832	−0.480678	−0.240339	+	0.970689i	$0.577259\pi$
$654$	0	0
$655$	−20.8262	−0.813748
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.4723	0.563762	0.281881	−	0.959449i	$-0.409042\pi$
$659$	14.4723	0.563762	0.281881	+	0.959449i	$0.409042\pi$
$660$	0	0
$661$	−22.9169	−0.891365	−0.445683	−	0.895191i	$-0.647039\pi$
$661$	−22.9169	−0.891365	−0.445683	+	0.895191i	$0.647039\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.31802	−0.245002
$666$	0	0
$667$	−2.22834	−0.0862817
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−74.5767	−2.87900
$672$	0	0
$673$	5.74536	0.221467	0.110734	−	0.993850i	$-0.464680\pi$
$673$	5.74536	0.221467	0.110734	+	0.993850i	$0.464680\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.7599	1.64340	0.821698	−	0.569923i	$-0.193027\pi$
$677$	42.7599	1.64340	0.821698	+	0.569923i	$0.193027\pi$
$678$	0	0
$679$	6.64991	0.255200
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.8394	0.873925	0.436963	−	0.899480i	$-0.356054\pi$
$683$	22.8394	0.873925	0.436963	+	0.899480i	$0.356054\pi$
$684$	0	0
$685$	18.7517	0.716464
$686$	0	0
$687$	0	0
$688$	0	0
$689$	23.1467	0.881818
$690$	0	0
$691$	10.9691	0.417282	0.208641	−	0.977992i	$-0.433096\pi$
$691$	10.9691	0.417282	0.208641	+	0.977992i	$0.433096\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−26.4823	−1.00453
$696$	0	0
$697$	−2.30530	−0.0873196
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.8911	−0.486892	−0.243446	−	0.969914i	$-0.578278\pi$
$701$	−12.8911	−0.486892	−0.243446	+	0.969914i	$0.578278\pi$
$702$	0	0
$703$	−3.80570	−0.143535
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−45.9608	−1.72854
$708$	0	0
$709$	2.67218	0.100356	0.0501780	−	0.998740i	$-0.484021\pi$
$709$	2.67218	0.100356	0.0501780	+	0.998740i	$0.484021\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.27283	−0.0476678
$714$	0	0
$715$	41.8499	1.56510
$716$	0	0
$717$	0	0
$718$	0	0
$719$	28.4430	1.06074	0.530372	−	0.847765i	$-0.322052\pi$
$719$	28.4430	1.06074	0.530372	+	0.847765i	$0.322052\pi$
$720$	0	0
$721$	−68.5434	−2.55269
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.80325	−0.252666
$726$	0	0
$727$	6.22594	0.230907	0.115454	−	0.993313i	$-0.463168\pi$
$727$	6.22594	0.230907	0.115454	+	0.993313i	$0.463168\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.739805	−0.0273627
$732$	0	0
$733$	−30.7155	−1.13450	−0.567251	−	0.823545i	$-0.691993\pi$
$733$	−30.7155	−1.13450	−0.567251	+	0.823545i	$0.691993\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−73.3899	−2.70335
$738$	0	0
$739$	−35.1503	−1.29303	−0.646513	−	0.762903i	$-0.723773\pi$
$739$	−35.1503	−1.29303	−0.646513	+	0.762903i	$0.723773\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.3170	1.55246	0.776231	−	0.630449i	$-0.217129\pi$
$743$	42.3170	1.55246	0.776231	+	0.630449i	$0.217129\pi$
$744$	0	0
$745$	12.0418	0.441177
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−57.0378	−2.08412
$750$	0	0
$751$	−4.04726	−0.147686	−0.0738432	−	0.997270i	$-0.523526\pi$
$751$	−4.04726	−0.147686	−0.0738432	+	0.997270i	$0.523526\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.32981	0.266759
$756$	0	0
$757$	8.60653	0.312810	0.156405	−	0.987693i	$-0.450010\pi$
$757$	8.60653	0.312810	0.156405	+	0.987693i	$0.450010\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−0.197419	−0.00715642	−0.00357821	−	0.999994i	$-0.501139\pi$
$761$	−0.197419	−0.00715642	−0.00357821	+	0.999994i	$0.501139\pi$
$762$	0	0
$763$	−19.3539	−0.700659
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.2405	1.45300
$768$	0	0
$769$	16.2221	0.584984	0.292492	−	0.956268i	$-0.405516\pi$
$769$	16.2221	0.584984	0.292492	+	0.956268i	$0.405516\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.6353	1.17381	0.586906	−	0.809655i	$-0.300346\pi$
$773$	32.6353	1.17381	0.586906	+	0.809655i	$0.300346\pi$
$774$	0	0
$775$	−3.88601	−0.139590
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.31802	−0.190538
$780$	0	0
$781$	−56.8760	−2.03518
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.9978	−0.820826
$786$	0	0
$787$	49.9844	1.78175	0.890875	−	0.454248i	$-0.150092\pi$
$787$	49.9844	1.78175	0.890875	+	0.454248i	$0.150092\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	37.4864	1.33286
$792$	0	0
$793$	−60.3221	−2.14210
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.4853	1.46949	0.734743	−	0.678346i	$-0.237303\pi$
$797$	41.4853	1.46949	0.734743	+	0.678346i	$0.237303\pi$
$798$	0	0
$799$	4.21273	0.149036
$800$	0	0
$801$	0	0
$802$	0	0
$803$	47.3548	1.67111
$804$	0	0
$805$	−6.29002	−0.221694
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.2625	−0.782708	−0.391354	−	0.920240i	$-0.627993\pi$
$809$	−22.2625	−0.782708	−0.391354	+	0.920240i	$0.627993\pi$
$810$	0	0
$811$	−19.6724	−0.690790	−0.345395	−	0.938457i	$-0.612255\pi$
$811$	−19.6724	−0.690790	−0.345395	+	0.938457i	$0.612255\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.78261	0.0624420
$816$	0	0
$817$	−1.70663	−0.0597073
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.2205	0.705701	0.352850	−	0.935680i	$-0.385213\pi$
$821$	20.2205	0.705701	0.352850	+	0.935680i	$0.385213\pi$
$822$	0	0
$823$	53.8546	1.87726	0.938628	−	0.344932i	$-0.112098\pi$
$823$	53.8546	1.87726	0.938628	+	0.344932i	$0.112098\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.7204	1.20735	0.603674	−	0.797232i	$-0.293703\pi$
$827$	34.7204	1.20735	0.603674	+	0.797232i	$0.293703\pi$
$828$	0	0
$829$	12.0078	0.417048	0.208524	−	0.978017i	$-0.433134\pi$
$829$	12.0078	0.417048	0.208524	+	0.978017i	$0.433134\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.79187	−0.200677
$834$	0	0
$835$	−14.4945	−0.501603
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21.2306	−0.732961	−0.366481	−	0.930426i	$-0.619437\pi$
$839$	−21.2306	−0.732961	−0.366481	+	0.930426i	$0.619437\pi$
$840$	0	0
$841$	−23.9902	−0.827248
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.6485	0.538324
$846$	0	0
$847$	117.105	4.02376
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.78883	−0.129879
$852$	0	0
$853$	−11.8153	−0.404550	−0.202275	−	0.979329i	$-0.564833\pi$
$853$	−11.8153	−0.404550	−0.202275	+	0.979329i	$0.564833\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.4050	−0.970296	−0.485148	−	0.874432i	$-0.661234\pi$
$857$	−28.4050	−0.970296	−0.485148	+	0.874432i	$0.661234\pi$
$858$	0	0
$859$	10.7265	0.365982	0.182991	−	0.983115i	$-0.441422\pi$
$859$	10.7265	0.365982	0.182991	+	0.983115i	$0.441422\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.0745	1.05779	0.528894	−	0.848688i	$-0.322607\pi$
$863$	31.0745	1.05779	0.528894	+	0.848688i	$0.322607\pi$
$864$	0	0
$865$	−32.6493	−1.11011
$866$	0	0
$867$	0	0
$868$	0	0
$869$	62.8915	2.13345
$870$	0	0
$871$	−59.3622	−2.01141
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−50.7939	−1.71715
$876$	0	0
$877$	18.0091	0.608125	0.304063	−	0.952652i	$-0.401657\pi$
$877$	18.0091	0.608125	0.304063	+	0.952652i	$0.401657\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.67751	0.123898	0.0619492	−	0.998079i	$-0.480268\pi$
$881$	3.67751	0.123898	0.0619492	+	0.998079i	$0.480268\pi$
$882$	0	0
$883$	47.3039	1.59190	0.795951	−	0.605361i	$-0.206971\pi$
$883$	47.3039	1.59190	0.795951	+	0.605361i	$0.206971\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44.9316	1.50865	0.754327	−	0.656498i	$-0.227963\pi$
$887$	44.9316	1.50865	0.754327	+	0.656498i	$0.227963\pi$
$888$	0	0
$889$	−87.8335	−2.94584
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.71819	0.325207
$894$	0	0
$895$	0.451116	0.0150791
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.86160	0.0954397
$900$	0	0
$901$	2.04067	0.0679847
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.4085	−1.04405
$906$	0	0
$907$	51.1566	1.69863	0.849314	−	0.527889i	$-0.177016\pi$
$907$	51.1566	1.69863	0.849314	+	0.527889i	$0.177016\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.5098	1.70660	0.853298	−	0.521424i	$-0.174599\pi$
$911$	51.5098	1.70660	0.853298	+	0.521424i	$0.174599\pi$
$912$	0	0
$913$	63.8970	2.11468
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−67.1166	−2.21639
$918$	0	0
$919$	−14.1191	−0.465745	−0.232872	−	0.972507i	$-0.574812\pi$
$919$	−14.1191	−0.465745	−0.232872	+	0.972507i	$0.574812\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−46.0047	−1.51426
$924$	0	0
$925$	−11.5675	−0.380337
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−39.8715	−1.30814	−0.654071	−	0.756433i	$-0.726940\pi$
$929$	−39.8715	−1.30814	−0.654071	+	0.756433i	$0.726940\pi$
$930$	0	0
$931$	−13.3610	−0.437891
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.68960	0.120663
$936$	0	0
$937$	−11.8984	−0.388703	−0.194351	−	0.980932i	$-0.562260\pi$
$937$	−11.8984	−0.388703	−0.194351	+	0.980932i	$0.562260\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	45.0217	1.46767	0.733833	−	0.679330i	$-0.237730\pi$
$941$	45.0217	1.46767	0.733833	+	0.679330i	$0.237730\pi$
$942$	0	0
$943$	−5.29445	−0.172411
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−44.3289	−1.44050	−0.720249	−	0.693716i	$-0.755972\pi$
$947$	−44.3289	−1.44050	−0.720249	+	0.693716i	$0.755972\pi$
$948$	0	0
$949$	38.3034	1.24338
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31.5516	1.02206	0.511028	−	0.859564i	$-0.329265\pi$
$953$	31.5516	1.02206	0.511028	+	0.859564i	$0.329265\pi$
$954$	0	0
$955$	−28.9051	−0.935346
$956$	0	0
$957$	0	0
$958$	0	0
$959$	60.4309	1.95142
$960$	0	0
$961$	−29.3655	−0.947273
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.3721	0.398272
$966$	0	0
$967$	−17.5370	−0.563951	−0.281975	−	0.959422i	$-0.590990\pi$
$967$	−17.5370	−0.563951	−0.281975	+	0.959422i	$0.590990\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.48813	−0.144031	−0.0720155	−	0.997404i	$-0.522943\pi$
$971$	−4.48813	−0.144031	−0.0720155	+	0.997404i	$0.522943\pi$
$972$	0	0
$973$	−85.3443	−2.73601
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.3731	0.747771	0.373885	−	0.927475i	$-0.378025\pi$
$977$	23.3731	0.747771	0.373885	+	0.927475i	$0.378025\pi$
$978$	0	0
$979$	35.5308	1.13557
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.70211	0.0542889	0.0271444	−	0.999632i	$-0.491359\pi$
$983$	1.70211	0.0542889	0.0271444	+	0.999632i	$0.491359\pi$
$984$	0	0
$985$	0.280747	0.00894536
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.69906	−0.0540271
$990$	0	0
$991$	58.4569	1.85694	0.928472	−	0.371402i	$-0.121123\pi$
$991$	58.4569	1.85694	0.928472	+	0.371402i	$0.121123\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.1808	0.354456
$996$	0	0
$997$	−44.1576	−1.39848	−0.699242	−	0.714885i	$-0.746479\pi$
$997$	−44.1576	−1.39848	−0.699242	+	0.714885i	$0.746479\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.ce.1.5		6
3.2	odd	2		8208.2.a.cf.1.2		6
4.3	odd	2		4104.2.a.s.1.5	✓	6
12.11	even	2		4104.2.a.t.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.s.1.5	✓	6	4.3	odd	2
4104.2.a.t.1.2	yes	6	12.11	even	2
8208.2.a.ce.1.5		6	1.1	even	1	trivial
8208.2.a.cf.1.2		6	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)

\(f(q)\)	\(=\)	\(q+1.40017 q^{5} +4.51232 q^{7} -6.07883 q^{11} -4.91692 q^{13} -0.433490 q^{17} -1.00000 q^{19} -0.995569 q^{23} -3.03952 q^{25} +2.23826 q^{29} +1.27849 q^{31} +6.31802 q^{35} +3.80570 q^{37} +5.31802 q^{41} +1.70663 q^{43} -9.71819 q^{47} +13.3610 q^{49} -4.70755 q^{53} -8.51140 q^{55} -8.18407 q^{59} +12.2683 q^{61} -6.88453 q^{65} +12.0730 q^{67} +9.35640 q^{71} -7.79011 q^{73} -27.4296 q^{77} -10.3460 q^{79} -10.5114 q^{83} -0.606959 q^{85} -5.84500 q^{89} -22.1867 q^{91} -1.40017 q^{95} +1.47372 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 2 q^{7} - 3 q^{11} + q^{13} - 7 q^{17} - 6 q^{19} + 3 q^{23} + 10 q^{25} + 6 q^{29} - 5 q^{31} - 3 q^{35} + 11 q^{37} - 9 q^{41} - 7 q^{43} - 7 q^{47} + 20 q^{49} - 11 q^{53} - 22 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) \) 6 * q - 2 * q^5 - 2 * q^7 - 3 * q^11 + q^13 - 7 * q^17 - 6 * q^19 + 3 * q^23 + 10 * q^25 + 6 * q^29 - 5 * q^31 - 3 * q^35 + 11 * q^37 - 9 * q^41 - 7 * q^43 - 7 * q^47 + 20 * q^49 - 11 * q^53 - 22 * q^55 - 17 * q^59 + 17 * q^61 + 4 * q^65 + 4 * q^67 + 10 * q^71 + 18 * q^73 - 27 * q^77 - 27 * q^79 - 34 * q^83 + 29 * q^85 - 18 * q^89 - 17 * q^91 + 2 * q^95 + 8 * q^97

Introduction

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