LMFDB - Embedded newform 8208.2.a.bh.1.1

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 = [3,0,0,0,-3,0,3,0,0,0,0] 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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 513)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ 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)

$f(q)$	$=$	$q-2.53209 q^{5} -0.184793 q^{7} -5.63816 q^{11} -6.63816 q^{13} +2.06418 q^{17} -1.00000 q^{19} +3.46791 q^{23} +1.41147 q^{25} -0.467911 q^{29} -5.26857 q^{31} +0.467911 q^{35} -4.41147 q^{37} -11.7246 q^{41} -5.68004 q^{43} +10.1480 q^{47} -6.96585 q^{49} -6.10607 q^{53} +14.2763 q^{55} +2.89393 q^{59} -12.6382 q^{61} +16.8084 q^{65} -0.958111 q^{67} +6.57398 q^{71} +1.08378 q^{73} +1.04189 q^{77} +5.95811 q^{79} +5.08378 q^{83} -5.22668 q^{85} -4.70233 q^{89} +1.22668 q^{91} +2.53209 q^{95} -8.86484 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 3 q^{7} - 3 q^{13} - 3 q^{17} - 3 q^{19} + 15 q^{23} - 6 q^{25} - 6 q^{29} - 6 q^{31} + 6 q^{35} - 3 q^{37} - 3 q^{41} + 3 q^{43} + 15 q^{47} - 6 q^{53} + 9 q^{55} + 21 q^{59} - 21 q^{61}+ \cdots - 3 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 3 * q^7 - 3 * q^13 - 3 * q^17 - 3 * q^19 + 15 * q^23 - 6 * q^25 - 6 * q^29 - 6 * q^31 + 6 * q^35 - 3 * q^37 - 3 * q^41 + 3 * q^43 + 15 * q^47 - 6 * q^53 + 9 * q^55 + 21 * q^59 - 21 * q^61 + 12 * q^65 - 6 * q^67 + 12 * q^71 - 3 * q^73 + 21 * q^79 + 9 * q^83 - 9 * q^85 + 12 * q^89 - 3 * q^91 + 3 * q^95 - 3 * 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
$5$	−2.53209	−1.13238	−0.566192	−	0.824273i	$-0.691584\pi$
$5$	−2.53209	−1.13238	−0.566192	+	0.824273i	$0.691584\pi$
$6$	0	0
$7$	−0.184793	−0.0698450	−0.0349225	−	0.999390i	$-0.511118\pi$
$7$	−0.184793	−0.0698450	−0.0349225	+	0.999390i	$0.511118\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.63816	−1.69997	−0.849984	−	0.526809i	$-0.823388\pi$
$11$	−5.63816	−1.69997	−0.849984	+	0.526809i	$0.823388\pi$
$12$	0	0
$13$	−6.63816	−1.84109	−0.920547	−	0.390633i	$-0.872256\pi$
$13$	−6.63816	−1.84109	−0.920547	+	0.390633i	$0.872256\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.06418	0.500637	0.250318	−	0.968164i	$-0.419465\pi$
$17$	2.06418	0.500637	0.250318	+	0.968164i	$0.419465\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.46791	0.723109	0.361555	−	0.932351i	$-0.382246\pi$
$23$	3.46791	0.723109	0.361555	+	0.932351i	$0.382246\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.467911	−0.0868889	−0.0434445	−	0.999056i	$-0.513833\pi$
$29$	−0.467911	−0.0868889	−0.0434445	+	0.999056i	$0.513833\pi$
$30$	0	0
$31$	−5.26857	−0.946263	−0.473132	−	0.880992i	$-0.656876\pi$
$31$	−5.26857	−0.946263	−0.473132	+	0.880992i	$0.656876\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.467911	0.0790914
$36$	0	0
$37$	−4.41147	−0.725242	−0.362621	−	0.931937i	$-0.618118\pi$
$37$	−4.41147	−0.725242	−0.362621	+	0.931937i	$0.618118\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.7246	−1.83108	−0.915539	−	0.402229i	$-0.868236\pi$
$41$	−11.7246	−1.83108	−0.915539	+	0.402229i	$0.868236\pi$
$42$	0	0
$43$	−5.68004	−0.866199	−0.433099	−	0.901346i	$-0.642580\pi$
$43$	−5.68004	−0.866199	−0.433099	+	0.901346i	$0.642580\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.1480	1.48023	0.740116	−	0.672479i	$-0.234771\pi$
$47$	10.1480	1.48023	0.740116	+	0.672479i	$0.234771\pi$
$48$	0	0
$49$	−6.96585	−0.995122
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.10607	−0.838733	−0.419366	−	0.907817i	$-0.637748\pi$
$53$	−6.10607	−0.838733	−0.419366	+	0.907817i	$0.637748\pi$
$54$	0	0
$55$	14.2763	1.92502
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.89393	0.376758	0.188379	−	0.982096i	$-0.439677\pi$
$59$	2.89393	0.376758	0.188379	+	0.982096i	$0.439677\pi$
$60$	0	0
$61$	−12.6382	−1.61815	−0.809075	−	0.587705i	$-0.800031\pi$
$61$	−12.6382	−1.61815	−0.809075	+	0.587705i	$0.800031\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.8084	2.08483
$66$	0	0
$67$	−0.958111	−0.117052	−0.0585259	−	0.998286i	$-0.518640\pi$
$67$	−0.958111	−0.117052	−0.0585259	+	0.998286i	$0.518640\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.57398	0.780188	0.390094	−	0.920775i	$-0.372443\pi$
$71$	6.57398	0.780188	0.390094	+	0.920775i	$0.372443\pi$
$72$	0	0
$73$	1.08378	0.126847	0.0634233	−	0.997987i	$-0.479798\pi$
$73$	1.08378	0.126847	0.0634233	+	0.997987i	$0.479798\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.04189	0.118734
$78$	0	0
$79$	5.95811	0.670340	0.335170	−	0.942158i	$-0.391206\pi$
$79$	5.95811	0.670340	0.335170	+	0.942158i	$0.391206\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.08378	0.558017	0.279009	−	0.960289i	$-0.409994\pi$
$83$	5.08378	0.558017	0.279009	+	0.960289i	$0.409994\pi$
$84$	0	0
$85$	−5.22668	−0.566913
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.70233	−0.498446	−0.249223	−	0.968446i	$-0.580175\pi$
$89$	−4.70233	−0.498446	−0.249223	+	0.968446i	$0.580175\pi$
$90$	0	0
$91$	1.22668	0.128591
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.53209	0.259787
$96$	0	0
$97$	−8.86484	−0.900088	−0.450044	−	0.893006i	$-0.648592\pi$
$97$	−8.86484	−0.900088	−0.450044	+	0.893006i	$0.648592\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.5740	1.54967	0.774834	−	0.632164i	$-0.217833\pi$
$101$	15.5740	1.54967	0.774834	+	0.632164i	$0.217833\pi$
$102$	0	0
$103$	−10.2267	−1.00766	−0.503832	−	0.863801i	$-0.668077\pi$
$103$	−10.2267	−1.00766	−0.503832	+	0.863801i	$0.668077\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.9564	−1.73591	−0.867953	−	0.496646i	$-0.834565\pi$
$107$	−17.9564	−1.73591	−0.867953	+	0.496646i	$0.834565\pi$
$108$	0	0
$109$	−10.3354	−0.989955	−0.494978	−	0.868906i	$-0.664824\pi$
$109$	−10.3354	−0.989955	−0.494978	+	0.868906i	$0.664824\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.2344	−0.962773	−0.481387	−	0.876508i	$-0.659867\pi$
$113$	−10.2344	−0.962773	−0.481387	+	0.876508i	$0.659867\pi$
$114$	0	0
$115$	−8.78106	−0.818838
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.381445	−0.0349670
$120$	0	0
$121$	20.7888	1.88989
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.08647	0.812718
$126$	0	0
$127$	−10.5621	−0.937236	−0.468618	−	0.883401i	$-0.655248\pi$
$127$	−10.5621	−0.937236	−0.468618	+	0.883401i	$0.655248\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	7.25402	0.633787	0.316893	−	0.948461i	$-0.397360\pi$
$131$	7.25402	0.633787	0.316893	+	0.948461i	$0.397360\pi$
$132$	0	0
$133$	0.184793	0.0160235
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.49020	0.127316	0.0636582	−	0.997972i	$-0.479723\pi$
$137$	1.49020	0.127316	0.0636582	+	0.997972i	$0.479723\pi$
$138$	0	0
$139$	−12.1088	−1.02705	−0.513526	−	0.858074i	$-0.671661\pi$
$139$	−12.1088	−1.02705	−0.513526	+	0.858074i	$0.671661\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	37.4270	3.12980
$144$	0	0
$145$	1.18479	0.0983917
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.61856	0.460290	0.230145	−	0.973156i	$-0.426080\pi$
$149$	5.61856	0.460290	0.230145	+	0.973156i	$0.426080\pi$
$150$	0	0
$151$	16.5107	1.34362	0.671812	−	0.740721i	$-0.265516\pi$
$151$	16.5107	1.34362	0.671812	+	0.740721i	$0.265516\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.3405	1.07153
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.640844	−0.0505056
$162$	0	0
$163$	25.0496	1.96204	0.981019	−	0.193911i	$-0.0621172\pi$
$163$	25.0496	1.96204	0.981019	+	0.193911i	$0.0621172\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.0223	1.00769	0.503847	−	0.863793i	$-0.331917\pi$
$167$	13.0223	1.00769	0.503847	+	0.863793i	$0.331917\pi$
$168$	0	0
$169$	31.0651	2.38962
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.9777	−0.834620	−0.417310	−	0.908764i	$-0.637027\pi$
$173$	−10.9777	−0.834620	−0.417310	+	0.908764i	$0.637027\pi$
$174$	0	0
$175$	−0.260830	−0.0197169
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.63816	−0.421416	−0.210708	−	0.977549i	$-0.567577\pi$
$179$	−5.63816	−0.421416	−0.210708	+	0.977549i	$0.567577\pi$
$180$	0	0
$181$	22.2003	1.65013	0.825067	−	0.565035i	$-0.191137\pi$
$181$	22.2003	1.65013	0.825067	+	0.565035i	$0.191137\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.1702	0.821253
$186$	0	0
$187$	−11.6382	−0.851066
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.44562	0.176959	0.0884795	−	0.996078i	$-0.471799\pi$
$191$	2.44562	0.176959	0.0884795	+	0.996078i	$0.471799\pi$
$192$	0	0
$193$	−8.47060	−0.609727	−0.304864	−	0.952396i	$-0.598611\pi$
$193$	−8.47060	−0.609727	−0.304864	+	0.952396i	$0.598611\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.9564	1.06560	0.532798	−	0.846242i	$-0.321141\pi$
$197$	14.9564	1.06560	0.532798	+	0.846242i	$0.321141\pi$
$198$	0	0
$199$	6.51249	0.461658	0.230829	−	0.972994i	$-0.425856\pi$
$199$	6.51249	0.461658	0.230829	+	0.972994i	$0.425856\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.0864665	0.00606876
$204$	0	0
$205$	29.6878	2.07348
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.63816	0.389999
$210$	0	0
$211$	−16.1506	−1.11186	−0.555928	−	0.831230i	$-0.687637\pi$
$211$	−16.1506	−1.11186	−0.555928	+	0.831230i	$0.687637\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.3824	0.980870
$216$	0	0
$217$	0.973593	0.0660918
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.7023	−0.921719
$222$	0	0
$223$	13.0669	0.875022	0.437511	−	0.899213i	$-0.355860\pi$
$223$	13.0669	0.875022	0.437511	+	0.899213i	$0.355860\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.76558	0.117186	0.0585928	−	0.998282i	$-0.481339\pi$
$227$	1.76558	0.117186	0.0585928	+	0.998282i	$0.481339\pi$
$228$	0	0
$229$	−12.1334	−0.801798	−0.400899	−	0.916122i	$-0.631302\pi$
$229$	−12.1334	−0.801798	−0.400899	+	0.916122i	$0.631302\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.4260	0.748544	0.374272	−	0.927319i	$-0.377893\pi$
$233$	11.4260	0.748544	0.374272	+	0.927319i	$0.377893\pi$
$234$	0	0
$235$	−25.6955	−1.67619
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.4270	−0.674464	−0.337232	−	0.941422i	$-0.609491\pi$
$239$	−10.4270	−0.674464	−0.337232	+	0.941422i	$0.609491\pi$
$240$	0	0
$241$	9.40373	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$241$	9.40373	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	17.6382	1.12686
$246$	0	0
$247$	6.63816	0.422376
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.53478	0.601830	0.300915	−	0.953651i	$-0.402708\pi$
$251$	9.53478	0.601830	0.300915	+	0.953651i	$0.402708\pi$
$252$	0	0
$253$	−19.5526	−1.22926
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.3628	0.708791	0.354395	−	0.935096i	$-0.384687\pi$
$257$	11.3628	0.708791	0.354395	+	0.935096i	$0.384687\pi$
$258$	0	0
$259$	0.815207	0.0506545
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.0205	−1.23452	−0.617260	−	0.786760i	$-0.711757\pi$
$263$	−20.0205	−1.23452	−0.617260	+	0.786760i	$0.711757\pi$
$264$	0	0
$265$	15.4611	0.949768
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.45100	0.271382	0.135691	−	0.990751i	$-0.456675\pi$
$269$	4.45100	0.271382	0.135691	+	0.990751i	$0.456675\pi$
$270$	0	0
$271$	26.9905	1.63956	0.819778	−	0.572681i	$-0.194097\pi$
$271$	26.9905	1.63956	0.819778	+	0.572681i	$0.194097\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.95811	−0.479892
$276$	0	0
$277$	17.3182	1.04055	0.520275	−	0.853999i	$-0.325829\pi$
$277$	17.3182	1.04055	0.520275	+	0.853999i	$0.325829\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.06418	−0.123139	−0.0615693	−	0.998103i	$-0.519611\pi$
$281$	−2.06418	−0.123139	−0.0615693	+	0.998103i	$0.519611\pi$
$282$	0	0
$283$	−1.17705	−0.0699685	−0.0349842	−	0.999388i	$-0.511138\pi$
$283$	−1.17705	−0.0699685	−0.0349842	+	0.999388i	$0.511138\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.16662	0.127892
$288$	0	0
$289$	−12.7392	−0.749363
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.5107	−1.43193	−0.715966	−	0.698135i	$-0.754014\pi$
$293$	−24.5107	−1.43193	−0.715966	+	0.698135i	$0.754014\pi$
$294$	0	0
$295$	−7.32770	−0.426635
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−23.0205	−1.33131
$300$	0	0
$301$	1.04963	0.0604997
$302$	0	0
$303$	0	0
$304$	0	0
$305$	32.0009	1.83237
$306$	0	0
$307$	−19.6209	−1.11983	−0.559913	−	0.828552i	$-0.689165\pi$
$307$	−19.6209	−1.11983	−0.559913	+	0.828552i	$0.689165\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.40373	0.419827	0.209914	−	0.977720i	$-0.432682\pi$
$311$	7.40373	0.419827	0.209914	+	0.977720i	$0.432682\pi$
$312$	0	0
$313$	−24.3756	−1.37779	−0.688894	−	0.724862i	$-0.741904\pi$
$313$	−24.3756	−1.37779	−0.688894	+	0.724862i	$0.741904\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.23173	−0.350009	−0.175005	−	0.984568i	$-0.555994\pi$
$317$	−6.23173	−0.350009	−0.175005	+	0.984568i	$0.555994\pi$
$318$	0	0
$319$	2.63816	0.147708
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.06418	−0.114854
$324$	0	0
$325$	−9.36959	−0.519731
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.87527	−0.103387
$330$	0	0
$331$	21.5185	1.18276	0.591381	−	0.806392i	$-0.298583\pi$
$331$	21.5185	1.18276	0.591381	+	0.806392i	$0.298583\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.42602	0.132548
$336$	0	0
$337$	17.3851	0.947025	0.473512	−	0.880787i	$-0.342986\pi$
$337$	17.3851	0.947025	0.473512	+	0.880787i	$0.342986\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	29.7050	1.60862
$342$	0	0
$343$	2.58079	0.139349
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.00093	0.268464	0.134232	−	0.990950i	$-0.457143\pi$
$347$	5.00093	0.268464	0.134232	+	0.990950i	$0.457143\pi$
$348$	0	0
$349$	−5.57903	−0.298639	−0.149319	−	0.988789i	$-0.547708\pi$
$349$	−5.57903	−0.298639	−0.149319	+	0.988789i	$0.547708\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28.0651	−1.49376	−0.746878	−	0.664962i	$-0.768448\pi$
$353$	−28.0651	−1.49376	−0.746878	+	0.664962i	$0.768448\pi$
$354$	0	0
$355$	−16.6459	−0.883472
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.63547	−0.244651	−0.122325	−	0.992490i	$-0.539035\pi$
$359$	−4.63547	−0.244651	−0.122325	+	0.992490i	$0.539035\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.74422	−0.143639
$366$	0	0
$367$	−16.8307	−0.878555	−0.439277	−	0.898351i	$-0.644765\pi$
$367$	−16.8307	−0.878555	−0.439277	+	0.898351i	$0.644765\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.12836	0.0585813
$372$	0	0
$373$	−19.5107	−1.01023	−0.505114	−	0.863053i	$-0.668549\pi$
$373$	−19.5107	−1.01023	−0.505114	+	0.863053i	$0.668549\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.10607	0.159971
$378$	0	0
$379$	−23.9718	−1.23135	−0.615675	−	0.788000i	$-0.711117\pi$
$379$	−23.9718	−1.23135	−0.615675	+	0.788000i	$0.711117\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.31551	−0.271610	−0.135805	−	0.990736i	$-0.543362\pi$
$383$	−5.31551	−0.271610	−0.135805	+	0.990736i	$0.543362\pi$
$384$	0	0
$385$	−2.63816	−0.134453
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.49113	−0.481220	−0.240610	−	0.970622i	$-0.577347\pi$
$389$	−9.49113	−0.481220	−0.240610	+	0.970622i	$0.577347\pi$
$390$	0	0
$391$	7.15839	0.362015
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.0865	−0.759083
$396$	0	0
$397$	32.3851	1.62536	0.812680	−	0.582710i	$-0.198008\pi$
$397$	32.3851	1.62536	0.812680	+	0.582710i	$0.198008\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.2550	1.36105	0.680524	−	0.732726i	$-0.261752\pi$
$401$	27.2550	1.36105	0.680524	+	0.732726i	$0.261752\pi$
$402$	0	0
$403$	34.9736	1.74216
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.8726	1.23289
$408$	0	0
$409$	−3.03415	−0.150029	−0.0750145	−	0.997182i	$-0.523900\pi$
$409$	−3.03415	−0.150029	−0.0750145	+	0.997182i	$0.523900\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.534777	−0.0263147
$414$	0	0
$415$	−12.8726	−0.631890
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.2344	0.646544	0.323272	−	0.946306i	$-0.395217\pi$
$419$	13.2344	0.646544	0.323272	+	0.946306i	$0.395217\pi$
$420$	0	0
$421$	−2.18479	−0.106480	−0.0532401	−	0.998582i	$-0.516955\pi$
$421$	−2.18479	−0.106480	−0.0532401	+	0.998582i	$0.516955\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.91353	0.141327
$426$	0	0
$427$	2.33544	0.113020
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.06242	0.292017	0.146008	−	0.989283i	$-0.453357\pi$
$431$	6.06242	0.292017	0.146008	+	0.989283i	$0.453357\pi$
$432$	0	0
$433$	−17.4037	−0.836370	−0.418185	−	0.908362i	$-0.637334\pi$
$433$	−17.4037	−0.836370	−0.418185	+	0.908362i	$0.637334\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.46791	−0.165893
$438$	0	0
$439$	10.9162	0.521003	0.260501	−	0.965473i	$-0.416112\pi$
$439$	10.9162	0.521003	0.260501	+	0.965473i	$0.416112\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.89393	−0.280029	−0.140015	−	0.990149i	$-0.544715\pi$
$443$	−5.89393	−0.280029	−0.140015	+	0.990149i	$0.544715\pi$
$444$	0	0
$445$	11.9067	0.564433
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6.99825	−0.330268	−0.165134	−	0.986271i	$-0.552806\pi$
$449$	−6.99825	−0.330268	−0.165134	+	0.986271i	$0.552806\pi$
$450$	0	0
$451$	66.1052	3.11277
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.10607	−0.145615
$456$	0	0
$457$	−24.1566	−1.13000	−0.565000	−	0.825091i	$-0.691124\pi$
$457$	−24.1566	−1.13000	−0.565000	+	0.825091i	$0.691124\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.1497	0.565868	0.282934	−	0.959139i	$-0.408692\pi$
$461$	12.1497	0.565868	0.282934	+	0.959139i	$0.408692\pi$
$462$	0	0
$463$	−36.1411	−1.67962	−0.839811	−	0.542879i	$-0.817334\pi$
$463$	−36.1411	−1.67962	−0.839811	+	0.542879i	$0.817334\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.4020	1.08291	0.541457	−	0.840728i	$-0.317873\pi$
$467$	23.4020	1.08291	0.541457	+	0.840728i	$0.317873\pi$
$468$	0	0
$469$	0.177052	0.00817549
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.0250	1.47251
$474$	0	0
$475$	−1.41147	−0.0647629
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.40467	−0.155563	−0.0777816	−	0.996970i	$-0.524784\pi$
$479$	−3.40467	−0.155563	−0.0777816	+	0.996970i	$0.524784\pi$
$480$	0	0
$481$	29.2841	1.33524
$482$	0	0
$483$	0	0
$484$	0	0
$485$	22.4466	1.01925
$486$	0	0
$487$	−25.1070	−1.13771	−0.568853	−	0.822439i	$-0.692613\pi$
$487$	−25.1070	−1.13771	−0.568853	+	0.822439i	$0.692613\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.70409	−0.167163	−0.0835816	−	0.996501i	$-0.526636\pi$
$491$	−3.70409	−0.167163	−0.0835816	+	0.996501i	$0.526636\pi$
$492$	0	0
$493$	−0.965852	−0.0434998
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.21482	−0.0544922
$498$	0	0
$499$	8.78880	0.393441	0.196720	−	0.980460i	$-0.436971\pi$
$499$	8.78880	0.393441	0.196720	+	0.980460i	$0.436971\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.49289	0.244916	0.122458	−	0.992474i	$-0.460922\pi$
$503$	5.49289	0.244916	0.122458	+	0.992474i	$0.460922\pi$
$504$	0	0
$505$	−39.4347	−1.75482
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.0178	1.24187	0.620935	−	0.783862i	$-0.286753\pi$
$509$	28.0178	1.24187	0.620935	+	0.783862i	$0.286753\pi$
$510$	0	0
$511$	−0.200274	−0.00885960
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25.8949	1.14106
$516$	0	0
$517$	−57.2158	−2.51635
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−30.3851	−1.33119	−0.665597	−	0.746311i	$-0.731823\pi$
$521$	−30.3851	−1.33119	−0.665597	+	0.746311i	$0.731823\pi$
$522$	0	0
$523$	−2.16931	−0.0948573	−0.0474287	−	0.998875i	$-0.515103\pi$
$523$	−2.16931	−0.0948573	−0.0474287	+	0.998875i	$0.515103\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−10.8753	−0.473734
$528$	0	0
$529$	−10.9736	−0.477113
$530$	0	0
$531$	0	0
$532$	0	0
$533$	77.8299	3.37119
$534$	0	0
$535$	45.4671	1.96571
$536$	0	0
$537$	0	0
$538$	0	0
$539$	39.2746	1.69167
$540$	0	0
$541$	17.3414	0.745566	0.372783	−	0.927919i	$-0.378404\pi$
$541$	17.3414	0.745566	0.372783	+	0.927919i	$0.378404\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	26.1702	1.12101
$546$	0	0
$547$	−11.9831	−0.512360	−0.256180	−	0.966629i	$-0.582464\pi$
$547$	−11.9831	−0.512360	−0.256180	+	0.966629i	$0.582464\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.467911	0.0199337
$552$	0	0
$553$	−1.10101	−0.0468199
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.4233	0.822993	0.411497	−	0.911411i	$-0.365006\pi$
$557$	19.4233	0.822993	0.411497	+	0.911411i	$0.365006\pi$
$558$	0	0
$559$	37.7050	1.59475
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−23.9564	−1.00964	−0.504820	−	0.863225i	$-0.668441\pi$
$563$	−23.9564	−1.00964	−0.504820	+	0.863225i	$0.668441\pi$
$564$	0	0
$565$	25.9145	1.09023
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.0232	−1.13287	−0.566436	−	0.824106i	$-0.691678\pi$
$569$	−27.0232	−1.13287	−0.566436	+	0.824106i	$0.691678\pi$
$570$	0	0
$571$	13.6477	0.571136	0.285568	−	0.958358i	$-0.407818\pi$
$571$	13.6477	0.571136	0.285568	+	0.958358i	$0.407818\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.89487	0.204130
$576$	0	0
$577$	−30.4252	−1.26662	−0.633309	−	0.773899i	$-0.718304\pi$
$577$	−30.4252	−1.26662	−0.633309	+	0.773899i	$0.718304\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.939444	−0.0389747
$582$	0	0
$583$	34.4270	1.42582
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.3378	0.509236	0.254618	−	0.967042i	$-0.418050\pi$
$587$	12.3378	0.509236	0.254618	+	0.967042i	$0.418050\pi$
$588$	0	0
$589$	5.26857	0.217088
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.51155	−0.144202	−0.0721011	−	0.997397i	$-0.522970\pi$
$593$	−3.51155	−0.144202	−0.0721011	+	0.997397i	$0.522970\pi$
$594$	0	0
$595$	0.965852	0.0395961
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−34.7060	−1.41805	−0.709023	−	0.705185i	$-0.750864\pi$
$599$	−34.7060	−1.41805	−0.709023	+	0.705185i	$0.750864\pi$
$600$	0	0
$601$	38.6536	1.57671	0.788357	−	0.615218i	$-0.210932\pi$
$601$	38.6536	1.57671	0.788357	+	0.615218i	$0.210932\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−52.6391	−2.14008
$606$	0	0
$607$	−26.3354	−1.06892	−0.534461	−	0.845193i	$-0.679485\pi$
$607$	−26.3354	−1.06892	−0.534461	+	0.845193i	$0.679485\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−67.3637	−2.72524
$612$	0	0
$613$	3.51216	0.141855	0.0709275	−	0.997481i	$-0.477404\pi$
$613$	3.51216	0.141855	0.0709275	+	0.997481i	$0.477404\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.6658	0.952750	0.476375	−	0.879242i	$-0.341950\pi$
$617$	23.6658	0.952750	0.476375	+	0.879242i	$0.341950\pi$
$618$	0	0
$619$	4.94263	0.198661	0.0993305	−	0.995054i	$-0.468330\pi$
$619$	4.94263	0.198661	0.0993305	+	0.995054i	$0.468330\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.868956	0.0348140
$624$	0	0
$625$	−30.0651	−1.20260
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.10607	−0.363083
$630$	0	0
$631$	0.468845	0.0186644	0.00933221	−	0.999956i	$-0.497029\pi$
$631$	0.468845	0.0186644	0.00933221	+	0.999956i	$0.497029\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	26.7442	1.06131
$636$	0	0
$637$	46.2404	1.83211
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.5303	−0.731904	−0.365952	−	0.930634i	$-0.619257\pi$
$641$	−18.5303	−0.731904	−0.365952	+	0.930634i	$0.619257\pi$
$642$	0	0
$643$	−4.68180	−0.184632	−0.0923161	−	0.995730i	$-0.529427\pi$
$643$	−4.68180	−0.184632	−0.0923161	+	0.995730i	$0.529427\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.4688	1.74825	0.874125	−	0.485700i	$-0.161435\pi$
$647$	44.4688	1.74825	0.874125	+	0.485700i	$0.161435\pi$
$648$	0	0
$649$	−16.3164	−0.640477
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.5107	0.606982	0.303491	−	0.952834i	$-0.401848\pi$
$653$	15.5107	0.606982	0.303491	+	0.952834i	$0.401848\pi$
$654$	0	0
$655$	−18.3678	−0.717691
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−27.8253	−1.08392	−0.541960	−	0.840404i	$-0.682318\pi$
$659$	−27.8253	−1.08392	−0.541960	+	0.840404i	$0.682318\pi$
$660$	0	0
$661$	8.92221	0.347034	0.173517	−	0.984831i	$-0.444487\pi$
$661$	8.92221	0.347034	0.173517	+	0.984831i	$0.444487\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.467911	−0.0181448
$666$	0	0
$667$	−1.62267	−0.0628302
$668$	0	0
$669$	0	0
$670$	0	0
$671$	71.2559	2.75080
$672$	0	0
$673$	24.8648	0.958469	0.479235	−	0.877687i	$-0.340914\pi$
$673$	24.8648	0.958469	0.479235	+	0.877687i	$0.340914\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.2995	1.12607	0.563036	−	0.826432i	$-0.309633\pi$
$677$	29.2995	1.12607	0.563036	+	0.826432i	$0.309633\pi$
$678$	0	0
$679$	1.63816	0.0628666
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.4688	1.01280	0.506401	−	0.862298i	$-0.330976\pi$
$683$	26.4688	1.01280	0.506401	+	0.862298i	$0.330976\pi$
$684$	0	0
$685$	−3.77332	−0.144171
$686$	0	0
$687$	0	0
$688$	0	0
$689$	40.5330	1.54419
$690$	0	0
$691$	3.80747	0.144843	0.0724214	−	0.997374i	$-0.476927\pi$
$691$	3.80747	0.144843	0.0724214	+	0.997374i	$0.476927\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	30.6604	1.16302
$696$	0	0
$697$	−24.2017	−0.916705
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−38.6222	−1.45874	−0.729370	−	0.684120i	$-0.760187\pi$
$701$	−38.6222	−1.45874	−0.729370	+	0.684120i	$0.760187\pi$
$702$	0	0
$703$	4.41147	0.166382
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.87795	−0.108237
$708$	0	0
$709$	16.2327	0.609631	0.304815	−	0.952411i	$-0.401405\pi$
$709$	16.2327	0.609631	0.304815	+	0.952411i	$0.401405\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−18.2709	−0.684252
$714$	0	0
$715$	−94.7684	−3.54414
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.06324	0.114240	0.0571199	−	0.998367i	$-0.481808\pi$
$719$	3.06324	0.114240	0.0571199	+	0.998367i	$0.481808\pi$
$720$	0	0
$721$	1.88981	0.0703804
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.660444	−0.0245283
$726$	0	0
$727$	26.9145	0.998202	0.499101	−	0.866544i	$-0.333664\pi$
$727$	26.9145	0.998202	0.499101	+	0.866544i	$0.333664\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.7246	−0.433651
$732$	0	0
$733$	12.8557	0.474835	0.237417	−	0.971408i	$-0.423699\pi$
$733$	12.8557	0.474835	0.237417	+	0.971408i	$0.423699\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.40198	0.198984
$738$	0	0
$739$	20.0215	0.736502	0.368251	−	0.929726i	$-0.379957\pi$
$739$	20.0215	0.736502	0.368251	+	0.929726i	$0.379957\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.78611	−0.359018	−0.179509	−	0.983756i	$-0.557451\pi$
$743$	−9.78611	−0.359018	−0.179509	+	0.983756i	$0.557451\pi$
$744$	0	0
$745$	−14.2267	−0.521225
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.31820	0.121244
$750$	0	0
$751$	5.23442	0.191007	0.0955034	−	0.995429i	$-0.469554\pi$
$751$	5.23442	0.191007	0.0955034	+	0.995429i	$0.469554\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−41.8066	−1.52150
$756$	0	0
$757$	−4.73885	−0.172236	−0.0861181	−	0.996285i	$-0.527446\pi$
$757$	−4.73885	−0.172236	−0.0861181	+	0.996285i	$0.527446\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.23442	−0.0447478	−0.0223739	−	0.999750i	$-0.507122\pi$
$761$	−1.23442	−0.0447478	−0.0223739	+	0.999750i	$0.507122\pi$
$762$	0	0
$763$	1.90991	0.0691434
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19.2104	−0.693647
$768$	0	0
$769$	−32.9813	−1.18934	−0.594669	−	0.803971i	$-0.702717\pi$
$769$	−32.9813	−1.18934	−0.594669	+	0.803971i	$0.702717\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.3610	0.876205	0.438103	−	0.898925i	$-0.355651\pi$
$773$	24.3610	0.876205	0.438103	+	0.898925i	$0.355651\pi$
$774$	0	0
$775$	−7.43645	−0.267125
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.7246	0.420078
$780$	0	0
$781$	−37.0651	−1.32629
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.6604	−0.451871
$786$	0	0
$787$	3.79023	0.135107	0.0675536	−	0.997716i	$-0.478481\pi$
$787$	3.79023	0.135107	0.0675536	+	0.997716i	$0.478481\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.89124	0.0672449
$792$	0	0
$793$	83.8940	2.97916
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.36009	0.0481768	0.0240884	−	0.999710i	$-0.492332\pi$
$797$	1.36009	0.0481768	0.0240884	+	0.999710i	$0.492332\pi$
$798$	0	0
$799$	20.9472	0.741058
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.11051	−0.215635
$804$	0	0
$805$	1.62267	0.0571917
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.9982	1.82816	0.914080	−	0.405533i	$-0.132914\pi$
$809$	51.9982	1.82816	0.914080	+	0.405533i	$0.132914\pi$
$810$	0	0
$811$	−48.6611	−1.70872	−0.854360	−	0.519681i	$-0.826051\pi$
$811$	−48.6611	−1.70872	−0.854360	+	0.519681i	$0.826051\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−63.4279	−2.22178
$816$	0	0
$817$	5.68004	0.198720
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.50980	−0.0526924	−0.0263462	−	0.999653i	$-0.508387\pi$
$821$	−1.50980	−0.0526924	−0.0263462	+	0.999653i	$0.508387\pi$
$822$	0	0
$823$	−15.1352	−0.527579	−0.263789	−	0.964580i	$-0.584972\pi$
$823$	−15.1352	−0.527579	−0.263789	+	0.964580i	$0.584972\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.3672	0.395277	0.197639	−	0.980275i	$-0.436673\pi$
$827$	11.3672	0.395277	0.197639	+	0.980275i	$0.436673\pi$
$828$	0	0
$829$	37.7811	1.31219	0.656095	−	0.754678i	$-0.272207\pi$
$829$	37.7811	1.31219	0.656095	+	0.754678i	$0.272207\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.3788	−0.498194
$834$	0	0
$835$	−32.9736	−1.14110
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19.5571	0.675185	0.337592	−	0.941292i	$-0.390387\pi$
$839$	19.5571	0.675185	0.337592	+	0.941292i	$0.390387\pi$
$840$	0	0
$841$	−28.7811	−0.992450
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−78.6596	−2.70597
$846$	0	0
$847$	−3.84161	−0.131999
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.2986	−0.524429
$852$	0	0
$853$	−19.1693	−0.656345	−0.328172	−	0.944618i	$-0.606433\pi$
$853$	−19.1693	−0.656345	−0.328172	+	0.944618i	$0.606433\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.4225	0.970895	0.485447	−	0.874266i	$-0.338657\pi$
$857$	28.4225	0.970895	0.485447	+	0.874266i	$0.338657\pi$
$858$	0	0
$859$	−13.3432	−0.455263	−0.227632	−	0.973747i	$-0.573098\pi$
$859$	−13.3432	−0.455263	−0.227632	+	0.973747i	$0.573098\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.33862	0.113648	0.0568240	−	0.998384i	$-0.481903\pi$
$863$	3.33862	0.113648	0.0568240	+	0.998384i	$0.481903\pi$
$864$	0	0
$865$	27.7965	0.945111
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−33.5928	−1.13956
$870$	0	0
$871$	6.36009	0.215503
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.67911	−0.0567643
$876$	0	0
$877$	17.3851	0.587052	0.293526	−	0.955951i	$-0.405171\pi$
$877$	17.3851	0.587052	0.293526	+	0.955951i	$0.405171\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34.5526	−1.16411	−0.582054	−	0.813150i	$-0.697751\pi$
$881$	−34.5526	−1.16411	−0.582054	+	0.813150i	$0.697751\pi$
$882$	0	0
$883$	36.9564	1.24368	0.621840	−	0.783144i	$-0.286385\pi$
$883$	36.9564	1.24368	0.621840	+	0.783144i	$0.286385\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.2131	1.38380	0.691900	−	0.721994i	$-0.256774\pi$
$887$	41.2131	1.38380	0.691900	+	0.721994i	$0.256774\pi$
$888$	0	0
$889$	1.95180	0.0654613
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.1480	−0.339588
$894$	0	0
$895$	14.2763	0.477204
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.46522	0.0822198
$900$	0	0
$901$	−12.6040	−0.419900
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−56.2131	−1.86859
$906$	0	0
$907$	33.8043	1.12245	0.561226	−	0.827662i	$-0.310330\pi$
$907$	33.8043	1.12245	0.561226	+	0.827662i	$0.310330\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.3432	1.17097	0.585486	−	0.810683i	$-0.300904\pi$
$911$	35.3432	1.17097	0.585486	+	0.810683i	$0.300904\pi$
$912$	0	0
$913$	−28.6631	−0.948611
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.34049	−0.0442669
$918$	0	0
$919$	4.41147	0.145521	0.0727606	−	0.997349i	$-0.476819\pi$
$919$	4.41147	0.145521	0.0727606	+	0.997349i	$0.476819\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−43.6391	−1.43640
$924$	0	0
$925$	−6.22668	−0.204732
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.491955	−0.0161405	−0.00807025	−	0.999967i	$-0.502569\pi$
$929$	−0.491955	−0.0161405	−0.00807025	+	0.999967i	$0.502569\pi$
$930$	0	0
$931$	6.96585	0.228297
$932$	0	0
$933$	0	0
$934$	0	0
$935$	29.4688	0.963734
$936$	0	0
$937$	6.52940	0.213306	0.106653	−	0.994296i	$-0.465987\pi$
$937$	6.52940	0.213306	0.106653	+	0.994296i	$0.465987\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.12836	0.0367833	0.0183917	−	0.999831i	$-0.494145\pi$
$941$	1.12836	0.0367833	0.0183917	+	0.999831i	$0.494145\pi$
$942$	0	0
$943$	−40.6599	−1.32407
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.5803	1.02622	0.513111	−	0.858322i	$-0.328493\pi$
$947$	31.5803	1.02622	0.513111	+	0.858322i	$0.328493\pi$
$948$	0	0
$949$	−7.19429	−0.233536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−34.1435	−1.10602	−0.553008	−	0.833176i	$-0.686520\pi$
$953$	−34.1435	−1.10602	−0.553008	+	0.833176i	$0.686520\pi$
$954$	0	0
$955$	−6.19253	−0.200386
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.275378	−0.00889241
$960$	0	0
$961$	−3.24216	−0.104586
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21.4483	0.690446
$966$	0	0
$967$	−56.7184	−1.82394	−0.911971	−	0.410255i	$-0.865440\pi$
$967$	−56.7184	−1.82394	−0.911971	+	0.410255i	$0.865440\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−61.6587	−1.97872	−0.989361	−	0.145483i	$-0.953526\pi$
$971$	−61.6587	−1.97872	−0.989361	+	0.145483i	$0.953526\pi$
$972$	0	0
$973$	2.23761	0.0717344
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.76382	0.0884225	0.0442113	−	0.999022i	$-0.485923\pi$
$977$	2.76382	0.0884225	0.0442113	+	0.999022i	$0.485923\pi$
$978$	0	0
$979$	26.5125	0.847343
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13.0615	0.416597	0.208298	−	0.978065i	$-0.433207\pi$
$983$	13.0615	0.416597	0.208298	+	0.978065i	$0.433207\pi$
$984$	0	0
$985$	−37.8708	−1.20666
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.6979	−0.626356
$990$	0	0
$991$	−33.4706	−1.06323	−0.531614	−	0.846987i	$-0.678414\pi$
$991$	−33.4706	−1.06323	−0.531614	+	0.846987i	$0.678414\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.4902	−0.522774
$996$	0	0
$997$	−15.4252	−0.488521	−0.244261	−	0.969710i	$-0.578545\pi$
$997$	−15.4252	−0.488521	−0.244261	+	0.969710i	$0.578545\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.bh.1.1		3
3.2	odd	2		8208.2.a.bn.1.3		3
4.3	odd	2		513.2.a.d.1.1	✓	3
12.11	even	2		513.2.a.g.1.3	yes	3
76.75	even	2		9747.2.a.bc.1.3		3
228.227	odd	2		9747.2.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.d.1.1	✓	3	4.3	odd	2
513.2.a.g.1.3	yes	3	12.11	even	2
8208.2.a.bh.1.1		3	1.1	even	1	trivial
8208.2.a.bn.1.3		3	3.2	odd	2
9747.2.a.w.1.1		3	228.227	odd	2
9747.2.a.bc.1.3		3	76.75	even	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)

\(f(q)\)	\(=\)	\(q-2.53209 q^{5} -0.184793 q^{7} -5.63816 q^{11} -6.63816 q^{13} +2.06418 q^{17} -1.00000 q^{19} +3.46791 q^{23} +1.41147 q^{25} -0.467911 q^{29} -5.26857 q^{31} +0.467911 q^{35} -4.41147 q^{37} -11.7246 q^{41} -5.68004 q^{43} +10.1480 q^{47} -6.96585 q^{49} -6.10607 q^{53} +14.2763 q^{55} +2.89393 q^{59} -12.6382 q^{61} +16.8084 q^{65} -0.958111 q^{67} +6.57398 q^{71} +1.08378 q^{73} +1.04189 q^{77} +5.95811 q^{79} +5.08378 q^{83} -5.22668 q^{85} -4.70233 q^{89} +1.22668 q^{91} +2.53209 q^{95} -8.86484 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 3 q^{7} - 3 q^{13} - 3 q^{17} - 3 q^{19} + 15 q^{23} - 6 q^{25} - 6 q^{29} - 6 q^{31} + 6 q^{35} - 3 q^{37} - 3 q^{41} + 3 q^{43} + 15 q^{47} - 6 q^{53} + 9 q^{55} + 21 q^{59} - 21 q^{61}+ \cdots - 3 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 3 * q^7 - 3 * q^13 - 3 * q^17 - 3 * q^19 + 15 * q^23 - 6 * q^25 - 6 * q^29 - 6 * q^31 + 6 * q^35 - 3 * q^37 - 3 * q^41 + 3 * q^43 + 15 * q^47 - 6 * q^53 + 9 * q^55 + 21 * q^59 - 21 * q^61 + 12 * q^65 - 6 * q^67 + 12 * q^71 - 3 * q^73 + 21 * q^79 + 9 * q^83 - 9 * q^85 + 12 * q^89 - 3 * q^91 + 3 * q^95 - 3 * q^97

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