LMFDB - Embedded newform 8280.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.20087896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 $ x^5 - x^4 - 21x^3 + 5x^2 + 84*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.0475116$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -5.02263 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -5.02263 q^{7} +3.71008 q^{11} -6.49408 q^{13} -5.37642 q^{17} -8.39905 q^{19} +1.00000 q^{23} +1.00000 q^{25} -3.31255 q^{29} +7.18153 q^{31} +5.02263 q^{35} -3.02263 q^{37} -0.782477 q^{41} -0.0950231 q^{43} -8.39905 q^{47} +18.2268 q^{49} -6.82773 q^{53} -3.71008 q^{55} -11.2628 q^{59} -14.6032 q^{61} +6.49408 q^{65} -9.71160 q^{67} -8.73271 q^{71} +3.64620 q^{73} -18.6344 q^{77} +8.59395 q^{79} +7.37642 q^{83} +5.37642 q^{85} +6.29918 q^{89} +32.6173 q^{91} +8.39905 q^{95} -5.32514 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 4 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 - 4 * q^7	$ 5 q - 5 q^{5} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 10 q^{17} - 4 q^{19} + 5 q^{23} + 5 q^{25} - 10 q^{29} + 6 q^{31} + 4 q^{35} + 6 q^{37} - 12 q^{41} - 2 q^{43} - 4 q^{47} + 19 q^{49} - 4 q^{55} - 6 q^{59} + 16 q^{61} - 4 q^{65} - 4 q^{67} - 8 q^{71} + 14 q^{73} - 16 q^{77} + 18 q^{79} + 20 q^{83} + 10 q^{85} - 18 q^{89} + 20 q^{91} + 4 q^{95} + 4 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 10 * q^17 - 4 * q^19 + 5 * q^23 + 5 * q^25 - 10 * q^29 + 6 * q^31 + 4 * q^35 + 6 * q^37 - 12 * q^41 - 2 * q^43 - 4 * q^47 + 19 * q^49 - 4 * q^55 - 6 * q^59 + 16 * q^61 - 4 * q^65 - 4 * q^67 - 8 * q^71 + 14 * q^73 - 16 * q^77 + 18 * q^79 + 20 * q^83 + 10 * q^85 - 18 * q^89 + 20 * q^91 + 4 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−5.02263	−1.89837	−0.949187	−	0.314712i	$-0.898092\pi$
$7$	−5.02263	−1.89837	−0.949187	+	0.314712i	$0.898092\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.71008	1.11863	0.559316	−	0.828955i	$-0.311064\pi$
$11$	3.71008	1.11863	0.559316	+	0.828955i	$0.311064\pi$
$12$	0	0
$13$	−6.49408	−1.80113	−0.900566	−	0.434719i	$-0.856848\pi$
$13$	−6.49408	−1.80113	−0.900566	+	0.434719i	$0.856848\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.37642	−1.30397	−0.651987	−	0.758230i	$-0.726064\pi$
$17$	−5.37642	−1.30397	−0.651987	+	0.758230i	$0.726064\pi$
$18$	0	0
$19$	−8.39905	−1.92687	−0.963437	−	0.267934i	$-0.913659\pi$
$19$	−8.39905	−1.92687	−0.963437	+	0.267934i	$0.913659\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.31255	−0.615124	−0.307562	−	0.951528i	$-0.599513\pi$
$29$	−3.31255	−0.615124	−0.307562	+	0.951528i	$0.599513\pi$
$30$	0	0
$31$	7.18153	1.28984	0.644920	−	0.764250i	$-0.276891\pi$
$31$	7.18153	1.28984	0.644920	+	0.764250i	$0.276891\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.02263	0.848979
$36$	0	0
$37$	−3.02263	−0.496917	−0.248458	−	0.968643i	$-0.579924\pi$
$37$	−3.02263	−0.496917	−0.248458	+	0.968643i	$0.579924\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.782477	−0.122202	−0.0611012	−	0.998132i	$-0.519461\pi$
$41$	−0.782477	−0.122202	−0.0611012	+	0.998132i	$0.519461\pi$
$42$	0	0
$43$	−0.0950231	−0.0144909	−0.00724544	−	0.999974i	$-0.502306\pi$
$43$	−0.0950231	−0.0144909	−0.00724544	+	0.999974i	$0.502306\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.39905	−1.22513	−0.612564	−	0.790421i	$-0.709862\pi$
$47$	−8.39905	−1.22513	−0.612564	+	0.790421i	$0.709862\pi$
$48$	0	0
$49$	18.2268	2.60383
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.82773	−0.937861	−0.468930	−	0.883235i	$-0.655361\pi$
$53$	−6.82773	−0.937861	−0.468930	+	0.883235i	$0.655361\pi$
$54$	0	0
$55$	−3.71008	−0.500267
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.2628	−1.46629	−0.733144	−	0.680073i	$-0.761948\pi$
$59$	−11.2628	−1.46629	−0.733144	+	0.680073i	$0.761948\pi$
$60$	0	0
$61$	−14.6032	−1.86975	−0.934875	−	0.354978i	$-0.884488\pi$
$61$	−14.6032	−1.86975	−0.934875	+	0.354978i	$0.884488\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.49408	0.805491
$66$	0	0
$67$	−9.71160	−1.18646	−0.593230	−	0.805033i	$-0.702148\pi$
$67$	−9.71160	−1.18646	−0.593230	+	0.805033i	$0.702148\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.73271	−1.03638	−0.518191	−	0.855265i	$-0.673394\pi$
$71$	−8.73271	−1.03638	−0.518191	+	0.855265i	$0.673394\pi$
$72$	0	0
$73$	3.64620	0.426756	0.213378	−	0.976970i	$-0.431553\pi$
$73$	3.64620	0.426756	0.213378	+	0.976970i	$0.431553\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−18.6344	−2.12358
$78$	0	0
$79$	8.59395	0.966895	0.483447	−	0.875373i	$-0.339384\pi$
$79$	8.59395	0.966895	0.483447	+	0.875373i	$0.339384\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.37642	0.809668	0.404834	−	0.914390i	$-0.367329\pi$
$83$	7.37642	0.809668	0.404834	+	0.914390i	$0.367329\pi$
$84$	0	0
$85$	5.37642	0.583155
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.29918	0.667712	0.333856	−	0.942624i	$-0.391650\pi$
$89$	6.29918	0.667712	0.333856	+	0.942624i	$0.391650\pi$
$90$	0	0
$91$	32.6173	3.41922
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.39905	0.861725
$96$	0	0
$97$	−5.32514	−0.540686	−0.270343	−	0.962764i	$-0.587137\pi$
$97$	−5.32514	−0.540686	−0.270343	+	0.962764i	$0.587137\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0654	−1.00154	−0.500772	−	0.865579i	$-0.666951\pi$
$101$	−10.0654	−1.00154	−0.500772	+	0.865579i	$0.666951\pi$
$102$	0	0
$103$	2.15890	0.212723	0.106361	−	0.994328i	$-0.466080\pi$
$103$	2.15890	0.212723	0.106361	+	0.994328i	$0.466080\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.00152	0.193494	0.0967470	−	0.995309i	$-0.469156\pi$
$107$	2.00152	0.193494	0.0967470	+	0.995309i	$0.469156\pi$
$108$	0	0
$109$	12.4302	1.19060	0.595298	−	0.803505i	$-0.297034\pi$
$109$	12.4302	1.19060	0.595298	+	0.803505i	$0.297034\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.0468	1.32141	0.660705	−	0.750646i	$-0.270258\pi$
$113$	14.0468	1.32141	0.660705	+	0.750646i	$0.270258\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	27.0038	2.47543
$120$	0	0
$121$	2.76470	0.251336
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−11.9142	−1.05722	−0.528609	−	0.848866i	$-0.677286\pi$
$127$	−11.9142	−1.05722	−0.528609	+	0.848866i	$0.677286\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	42.1853	3.65793
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.2494	1.73002	0.865012	−	0.501751i	$-0.167311\pi$
$137$	20.2494	1.73002	0.865012	+	0.501751i	$0.167311\pi$
$138$	0	0
$139$	−5.61658	−0.476392	−0.238196	−	0.971217i	$-0.576556\pi$
$139$	−5.61658	−0.476392	−0.238196	+	0.971217i	$0.576556\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.0935	−2.01480
$144$	0	0
$145$	3.31255	0.275092
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.226041	−0.0185180	−0.00925898	−	0.999957i	$-0.502947\pi$
$149$	−0.226041	−0.0185180	−0.00925898	+	0.999957i	$0.502947\pi$
$150$	0	0
$151$	10.2353	0.832937	0.416468	−	0.909150i	$-0.363268\pi$
$151$	10.2353	0.832937	0.416468	+	0.909150i	$0.363268\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.18153	−0.576834
$156$	0	0
$157$	13.7755	1.09940	0.549701	−	0.835361i	$-0.314742\pi$
$157$	13.7755	1.09940	0.549701	+	0.835361i	$0.314742\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.02263	−0.395838
$162$	0	0
$163$	−3.33269	−0.261036	−0.130518	−	0.991446i	$-0.541664\pi$
$163$	−3.33269	−0.261036	−0.130518	+	0.991446i	$0.541664\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.5014	−1.50906	−0.754532	−	0.656263i	$-0.772136\pi$
$167$	−19.5014	−1.50906	−0.754532	+	0.656263i	$0.772136\pi$
$168$	0	0
$169$	29.1730	2.24408
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.0453	1.21990	0.609949	−	0.792441i	$-0.291190\pi$
$173$	16.0453	1.21990	0.609949	+	0.792441i	$0.291190\pi$
$174$	0	0
$175$	−5.02263	−0.379675
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−14.1730	−1.05934	−0.529670	−	0.848204i	$-0.677684\pi$
$179$	−14.1730	−1.05934	−0.529670	+	0.848204i	$0.677684\pi$
$180$	0	0
$181$	4.03114	0.299633	0.149816	−	0.988714i	$-0.452132\pi$
$181$	4.03114	0.299633	0.149816	+	0.988714i	$0.452132\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.02263	0.222228
$186$	0	0
$187$	−19.9470	−1.45867
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.58910	−0.621485	−0.310743	−	0.950494i	$-0.600578\pi$
$191$	−8.58910	−0.621485	−0.310743	+	0.950494i	$0.600578\pi$
$192$	0	0
$193$	−16.8434	−1.21241	−0.606206	−	0.795308i	$-0.707309\pi$
$193$	−16.8434	−1.21241	−0.606206	+	0.795308i	$0.707309\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.0422192	−0.00300800	−0.00150400	−	0.999999i	$-0.500479\pi$
$197$	−0.0422192	−0.00300800	−0.00150400	+	0.999999i	$0.500479\pi$
$198$	0	0
$199$	14.1589	1.00370	0.501849	−	0.864955i	$-0.332653\pi$
$199$	14.1589	1.00370	0.501849	+	0.864955i	$0.332653\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.6377	1.16774
$204$	0	0
$205$	0.782477	0.0546506
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−31.1612	−2.15546
$210$	0	0
$211$	13.6166	0.937404	0.468702	−	0.883356i	$-0.344722\pi$
$211$	13.6166	0.937404	0.468702	+	0.883356i	$0.344722\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.0950231	0.00648052
$216$	0	0
$217$	−36.0701	−2.44860
$218$	0	0
$219$	0	0
$220$	0	0
$221$	34.9149	2.34863
$222$	0	0
$223$	−28.3133	−1.89600	−0.947999	−	0.318273i	$-0.896897\pi$
$223$	−28.3133	−1.89600	−0.947999	+	0.318273i	$0.896897\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.9118	−0.856983	−0.428492	−	0.903546i	$-0.640955\pi$
$227$	−12.9118	−0.856983	−0.428492	+	0.903546i	$0.640955\pi$
$228$	0	0
$229$	18.6814	1.23450	0.617252	−	0.786766i	$-0.288246\pi$
$229$	18.6814	1.23450	0.617252	+	0.786766i	$0.288246\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.2353	0.801561	0.400781	−	0.916174i	$-0.368739\pi$
$233$	12.2353	0.801561	0.400781	+	0.916174i	$0.368739\pi$
$234$	0	0
$235$	8.39905	0.547894
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.2977	−0.924839	−0.462419	−	0.886661i	$-0.653019\pi$
$239$	−14.2977	−0.924839	−0.462419	+	0.886661i	$0.653019\pi$
$240$	0	0
$241$	4.22604	0.272223	0.136112	−	0.990694i	$-0.456539\pi$
$241$	4.22604	0.272223	0.136112	+	0.990694i	$0.456539\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−18.2268	−1.16447
$246$	0	0
$247$	54.5441	3.47056
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.4015	0.972136	0.486068	−	0.873921i	$-0.338431\pi$
$251$	15.4015	0.972136	0.486068	+	0.873921i	$0.338431\pi$
$252$	0	0
$253$	3.71008	0.233251
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.7367	−1.23114	−0.615571	−	0.788081i	$-0.711075\pi$
$257$	−19.7367	−1.23114	−0.615571	+	0.788081i	$0.711075\pi$
$258$	0	0
$259$	15.1815	0.943334
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.81588	−0.605273	−0.302637	−	0.953106i	$-0.597867\pi$
$263$	−9.81588	−0.605273	−0.302637	+	0.953106i	$0.597867\pi$
$264$	0	0
$265$	6.82773	0.419424
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.92275	0.422088	0.211044	−	0.977477i	$-0.432314\pi$
$269$	6.92275	0.422088	0.211044	+	0.977477i	$0.432314\pi$
$270$	0	0
$271$	−9.22678	−0.560487	−0.280244	−	0.959929i	$-0.590415\pi$
$271$	−9.22678	−0.560487	−0.280244	+	0.959929i	$0.590415\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.71008	0.223726
$276$	0	0
$277$	−8.33032	−0.500521	−0.250260	−	0.968179i	$-0.580516\pi$
$277$	−8.33032	−0.500521	−0.250260	+	0.968179i	$0.580516\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.1451	0.963138	0.481569	−	0.876408i	$-0.340067\pi$
$281$	16.1451	0.963138	0.481569	+	0.876408i	$0.340067\pi$
$282$	0	0
$283$	28.5550	1.69742	0.848708	−	0.528862i	$-0.177381\pi$
$283$	28.5550	1.69742	0.848708	+	0.528862i	$0.177381\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.93009	0.231986
$288$	0	0
$289$	11.9059	0.700350
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−25.5836	−1.49461	−0.747305	−	0.664481i	$-0.768653\pi$
$293$	−25.5836	−1.49461	−0.747305	+	0.664481i	$0.768653\pi$
$294$	0	0
$295$	11.2628	0.655744
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.49408	−0.375562
$300$	0	0
$301$	0.477266	0.0275091
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.6032	0.836177
$306$	0	0
$307$	−4.58910	−0.261914	−0.130957	−	0.991388i	$-0.541805\pi$
$307$	−4.58910	−0.261914	−0.130957	+	0.991388i	$0.541805\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.8360	1.35162	0.675808	−	0.737077i	$-0.263795\pi$
$311$	23.8360	1.35162	0.675808	+	0.737077i	$0.263795\pi$
$312$	0	0
$313$	26.0747	1.47383	0.736913	−	0.675987i	$-0.236283\pi$
$313$	26.0747	1.47383	0.736913	+	0.675987i	$0.236283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.7367	−0.883862	−0.441931	−	0.897049i	$-0.645706\pi$
$317$	−15.7367	−0.883862	−0.441931	+	0.897049i	$0.645706\pi$
$318$	0	0
$319$	−12.2898	−0.688098
$320$	0	0
$321$	0	0
$322$	0	0
$323$	45.1569	2.51260
$324$	0	0
$325$	−6.49408	−0.360226
$326$	0	0
$327$	0	0
$328$	0	0
$329$	42.1853	2.32575
$330$	0	0
$331$	−34.3427	−1.88764	−0.943822	−	0.330453i	$-0.892799\pi$
$331$	−34.3427	−1.88764	−0.943822	+	0.330453i	$0.892799\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.71160	0.530601
$336$	0	0
$337$	9.23767	0.503208	0.251604	−	0.967830i	$-0.419042\pi$
$337$	9.23767	0.503208	0.251604	+	0.967830i	$0.419042\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	26.6441	1.44286
$342$	0	0
$343$	−56.3879	−3.04466
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.4803	−0.884709	−0.442354	−	0.896840i	$-0.645857\pi$
$347$	−16.4803	−0.884709	−0.442354	+	0.896840i	$0.645857\pi$
$348$	0	0
$349$	3.48882	0.186752	0.0933761	−	0.995631i	$-0.470234\pi$
$349$	3.48882	0.186752	0.0933761	+	0.995631i	$0.470234\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.7503	−1.58345	−0.791723	−	0.610880i	$-0.790816\pi$
$353$	−29.7503	−1.58345	−0.791723	+	0.610880i	$0.790816\pi$
$354$	0	0
$355$	8.73271	0.463484
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.5014	−0.607021	−0.303511	−	0.952828i	$-0.598159\pi$
$359$	−11.5014	−0.607021	−0.303511	+	0.952828i	$0.598159\pi$
$360$	0	0
$361$	51.5441	2.71285
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.64620	−0.190851
$366$	0	0
$367$	2.58758	0.135071	0.0675353	−	0.997717i	$-0.478486\pi$
$367$	2.58758	0.135071	0.0675353	+	0.997717i	$0.478486\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	34.2931	1.78041
$372$	0	0
$373$	−12.6392	−0.654433	−0.327217	−	0.944949i	$-0.606111\pi$
$373$	−12.6392	−0.654433	−0.327217	+	0.944949i	$0.606111\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21.5119	1.10792
$378$	0	0
$379$	13.8858	0.713265	0.356633	−	0.934245i	$-0.383925\pi$
$379$	13.8858	0.713265	0.356633	+	0.934245i	$0.383925\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.20264	−0.316940	−0.158470	−	0.987364i	$-0.550656\pi$
$383$	−6.20264	−0.316940	−0.158470	+	0.987364i	$0.550656\pi$
$384$	0	0
$385$	18.6344	0.949695
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−0.147827	−0.00749513	−0.00374756	−	0.999993i	$-0.501193\pi$
$389$	−0.147827	−0.00749513	−0.00374756	+	0.999993i	$0.501193\pi$
$390$	0	0
$391$	−5.37642	−0.271897
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.59395	−0.432408
$396$	0	0
$397$	29.0259	1.45677	0.728383	−	0.685170i	$-0.240272\pi$
$397$	29.0259	1.45677	0.728383	+	0.685170i	$0.240272\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.2145	−0.859652	−0.429826	−	0.902912i	$-0.641425\pi$
$401$	−17.2145	−0.859652	−0.429826	+	0.902912i	$0.641425\pi$
$402$	0	0
$403$	−46.6374	−2.32317
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.2142	−0.555867
$408$	0	0
$409$	−23.8348	−1.17856	−0.589279	−	0.807930i	$-0.700588\pi$
$409$	−23.8348	−1.17856	−0.589279	+	0.807930i	$0.700588\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	56.5687	2.78357
$414$	0	0
$415$	−7.37642	−0.362094
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.99064	−0.390368	−0.195184	−	0.980767i	$-0.562530\pi$
$419$	−7.99064	−0.390368	−0.195184	+	0.980767i	$0.562530\pi$
$420$	0	0
$421$	−11.5326	−0.562062	−0.281031	−	0.959699i	$-0.590676\pi$
$421$	−11.5326	−0.562062	−0.281031	+	0.959699i	$0.590676\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.37642	−0.260795
$426$	0	0
$427$	73.3465	3.54948
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.3661	1.27001	0.635005	−	0.772508i	$-0.280998\pi$
$431$	26.3661	1.27001	0.635005	+	0.772508i	$0.280998\pi$
$432$	0	0
$433$	2.24618	0.107945	0.0539723	−	0.998542i	$-0.482812\pi$
$433$	2.24618	0.107945	0.0539723	+	0.998542i	$0.482812\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−8.39905	−0.401781
$438$	0	0
$439$	−15.3512	−0.732673	−0.366337	−	0.930482i	$-0.619388\pi$
$439$	−15.3512	−0.732673	−0.366337	+	0.930482i	$0.619388\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.0948	1.23980	0.619901	−	0.784680i	$-0.287173\pi$
$443$	26.0948	1.23980	0.619901	+	0.784680i	$0.287173\pi$
$444$	0	0
$445$	−6.29918	−0.298610
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.67109	−0.267635	−0.133818	−	0.991006i	$-0.542724\pi$
$449$	−5.67109	−0.267635	−0.133818	+	0.991006i	$0.542724\pi$
$450$	0	0
$451$	−2.90305	−0.136699
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−32.6173	−1.52912
$456$	0	0
$457$	30.2024	1.41281	0.706405	−	0.707808i	$-0.250316\pi$
$457$	30.2024	1.41281	0.706405	+	0.707808i	$0.250316\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.0526	1.25996	0.629982	−	0.776609i	$-0.283062\pi$
$461$	27.0526	1.25996	0.629982	+	0.776609i	$0.283062\pi$
$462$	0	0
$463$	10.6671	0.495742	0.247871	−	0.968793i	$-0.420269\pi$
$463$	10.6671	0.495742	0.247871	+	0.968793i	$0.420269\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.9696	0.970357	0.485179	−	0.874415i	$-0.338755\pi$
$467$	20.9696	0.970357	0.485179	+	0.874415i	$0.338755\pi$
$468$	0	0
$469$	48.7777	2.25235
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.352543	−0.0162100
$474$	0	0
$475$	−8.39905	−0.385375
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.7218	0.855422	0.427711	−	0.903916i	$-0.359320\pi$
$479$	18.7218	0.855422	0.427711	+	0.903916i	$0.359320\pi$
$480$	0	0
$481$	19.6292	0.895013
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.32514	0.241802
$486$	0	0
$487$	−10.5363	−0.477445	−0.238723	−	0.971088i	$-0.576729\pi$
$487$	−10.5363	−0.477445	−0.238723	+	0.971088i	$0.576729\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.77063	−0.350683	−0.175342	−	0.984508i	$-0.556103\pi$
$491$	−7.77063	−0.350683	−0.175342	+	0.984508i	$0.556103\pi$
$492$	0	0
$493$	17.8097	0.802107
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.8611	1.96744
$498$	0	0
$499$	20.8370	0.932792	0.466396	−	0.884576i	$-0.345552\pi$
$499$	20.8370	0.932792	0.466396	+	0.884576i	$0.345552\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.6377	0.741838	0.370919	−	0.928665i	$-0.379043\pi$
$503$	16.6377	0.741838	0.370919	+	0.928665i	$0.379043\pi$
$504$	0	0
$505$	10.0654	0.447904
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−25.9235	−1.14904	−0.574519	−	0.818491i	$-0.694811\pi$
$509$	−25.9235	−1.14904	−0.574519	+	0.818491i	$0.694811\pi$
$510$	0	0
$511$	−18.3135	−0.810142
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.15890	−0.0951326
$516$	0	0
$517$	−31.1612	−1.37047
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.91501	0.390574	0.195287	−	0.980746i	$-0.437436\pi$
$521$	8.91501	0.390574	0.195287	+	0.980746i	$0.437436\pi$
$522$	0	0
$523$	−14.6481	−0.640518	−0.320259	−	0.947330i	$-0.603770\pi$
$523$	−14.6481	−0.640518	−0.320259	+	0.947330i	$0.603770\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−38.6110	−1.68192
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.08146	0.220103
$534$	0	0
$535$	−2.00152	−0.0865331
$536$	0	0
$537$	0	0
$538$	0	0
$539$	67.6228	2.91272
$540$	0	0
$541$	−19.0571	−0.819329	−0.409664	−	0.912236i	$-0.634354\pi$
$541$	−19.0571	−0.819329	−0.409664	+	0.912236i	$0.634354\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.4302	−0.532451
$546$	0	0
$547$	2.84032	0.121443	0.0607217	−	0.998155i	$-0.480660\pi$
$547$	2.84032	0.121443	0.0607217	+	0.998155i	$0.480660\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	27.8222	1.18527
$552$	0	0
$553$	−43.1642	−1.83553
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−35.1365	−1.48878	−0.744391	−	0.667744i	$-0.767260\pi$
$557$	−35.1365	−1.48878	−0.744391	+	0.667744i	$0.767260\pi$
$558$	0	0
$559$	0.617087	0.0261000
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.96960	−0.209444	−0.104722	−	0.994502i	$-0.533395\pi$
$563$	−4.96960	−0.209444	−0.104722	+	0.994502i	$0.533395\pi$
$564$	0	0
$565$	−14.0468	−0.590952
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.6769	1.45373	0.726866	−	0.686780i	$-0.240976\pi$
$569$	34.6769	1.45373	0.726866	+	0.686780i	$0.240976\pi$
$570$	0	0
$571$	−11.2514	−0.470858	−0.235429	−	0.971892i	$-0.575650\pi$
$571$	−11.2514	−0.470858	−0.235429	+	0.971892i	$0.575650\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−14.2872	−0.594785	−0.297392	−	0.954755i	$-0.596117\pi$
$577$	−14.2872	−0.594785	−0.297392	+	0.954755i	$0.596117\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−37.0490	−1.53705
$582$	0	0
$583$	−25.3314	−1.04912
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.8856	−1.19224	−0.596118	−	0.802897i	$-0.703291\pi$
$587$	−28.8856	−1.19224	−0.596118	+	0.802897i	$0.703291\pi$
$588$	0	0
$589$	−60.3180	−2.48536
$590$	0	0
$591$	0	0
$592$	0	0
$593$	37.6225	1.54497	0.772486	−	0.635032i	$-0.219013\pi$
$593$	37.6225	1.54497	0.772486	+	0.635032i	$0.219013\pi$
$594$	0	0
$595$	−27.0038	−1.10705
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.7483	−1.09291	−0.546454	−	0.837489i	$-0.684023\pi$
$599$	−26.7483	−1.09291	−0.546454	+	0.837489i	$0.684023\pi$
$600$	0	0
$601$	21.4621	0.875457	0.437728	−	0.899107i	$-0.355783\pi$
$601$	21.4621	0.875457	0.437728	+	0.899107i	$0.355783\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.76470	−0.112401
$606$	0	0
$607$	25.1644	1.02139	0.510696	−	0.859761i	$-0.329388\pi$
$607$	25.1644	1.02139	0.510696	+	0.859761i	$0.329388\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	54.5441	2.20662
$612$	0	0
$613$	−9.62432	−0.388723	−0.194361	−	0.980930i	$-0.562263\pi$
$613$	−9.62432	−0.388723	−0.194361	+	0.980930i	$0.562263\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.2368	1.21729	0.608644	−	0.793443i	$-0.291714\pi$
$617$	30.2368	1.21729	0.608644	+	0.793443i	$0.291714\pi$
$618$	0	0
$619$	−22.6503	−0.910391	−0.455196	−	0.890391i	$-0.650431\pi$
$619$	−22.6503	−0.910391	−0.455196	+	0.890391i	$0.650431\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−31.6384	−1.26757
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.2509	0.647967
$630$	0	0
$631$	39.3188	1.56526	0.782629	−	0.622489i	$-0.213878\pi$
$631$	39.3188	1.56526	0.782629	+	0.622489i	$0.213878\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.9142	0.472802
$636$	0	0
$637$	−118.366	−4.68984
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.9826	0.631276	0.315638	−	0.948880i	$-0.397781\pi$
$641$	15.9826	0.631276	0.315638	+	0.948880i	$0.397781\pi$
$642$	0	0
$643$	3.36706	0.132784	0.0663919	−	0.997794i	$-0.478851\pi$
$643$	3.36706	0.132784	0.0663919	+	0.997794i	$0.478851\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.5839	−0.534039	−0.267019	−	0.963691i	$-0.586039\pi$
$647$	−13.5839	−0.534039	−0.267019	+	0.963691i	$0.586039\pi$
$648$	0	0
$649$	−41.7858	−1.64024
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.3754	0.953881	0.476941	−	0.878936i	$-0.341746\pi$
$653$	24.3754	0.953881	0.476941	+	0.878936i	$0.341746\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.9766	−1.63518	−0.817589	−	0.575803i	$-0.804690\pi$
$659$	−41.9766	−1.63518	−0.817589	+	0.575803i	$0.804690\pi$
$660$	0	0
$661$	−39.8004	−1.54805	−0.774027	−	0.633152i	$-0.781761\pi$
$661$	−39.8004	−1.54805	−0.774027	+	0.633152i	$0.781761\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−42.1853	−1.63588
$666$	0	0
$667$	−3.31255	−0.128262
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−54.1791	−2.09156
$672$	0	0
$673$	1.60095	0.0617120	0.0308560	−	0.999524i	$-0.490177\pi$
$673$	1.60095	0.0617120	0.0308560	+	0.999524i	$0.490177\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.52044	−0.327467	−0.163734	−	0.986505i	$-0.552354\pi$
$677$	−8.52044	−0.327467	−0.163734	+	0.986505i	$0.552354\pi$
$678$	0	0
$679$	26.7462	1.02642
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.7050	1.59580	0.797899	−	0.602791i	$-0.205945\pi$
$683$	41.7050	1.59580	0.797899	+	0.602791i	$0.205945\pi$
$684$	0	0
$685$	−20.2494	−0.773690
$686$	0	0
$687$	0	0
$688$	0	0
$689$	44.3398	1.68921
$690$	0	0
$691$	−6.01703	−0.228899	−0.114449	−	0.993429i	$-0.536510\pi$
$691$	−6.01703	−0.228899	−0.114449	+	0.993429i	$0.536510\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.61658	0.213049
$696$	0	0
$697$	4.20693	0.159349
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.51347	0.359319	0.179659	−	0.983729i	$-0.442500\pi$
$701$	9.51347	0.359319	0.179659	+	0.983729i	$0.442500\pi$
$702$	0	0
$703$	25.3872	0.957496
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50.5547	1.90131
$708$	0	0
$709$	27.8187	1.04475	0.522376	−	0.852715i	$-0.325046\pi$
$709$	27.8187	1.04475	0.522376	+	0.852715i	$0.325046\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.18153	0.268950
$714$	0	0
$715$	24.0935	0.901047
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−38.1729	−1.42361	−0.711805	−	0.702377i	$-0.752122\pi$
$719$	−38.1729	−1.42361	−0.711805	+	0.702377i	$0.752122\pi$
$720$	0	0
$721$	−10.8434	−0.403828
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.31255	−0.123025
$726$	0	0
$727$	−44.3738	−1.64573	−0.822867	−	0.568234i	$-0.807627\pi$
$727$	−44.3738	−1.64573	−0.822867	+	0.568234i	$0.807627\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.510885	0.0188957
$732$	0	0
$733$	38.5639	1.42439	0.712195	−	0.701982i	$-0.247701\pi$
$733$	38.5639	1.42439	0.712195	+	0.701982i	$0.247701\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.0308	−1.32721
$738$	0	0
$739$	−32.9275	−1.21126	−0.605629	−	0.795747i	$-0.707078\pi$
$739$	−32.9275	−1.21126	−0.605629	+	0.795747i	$0.707078\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.09739	−0.187005	−0.0935025	−	0.995619i	$-0.529806\pi$
$743$	−5.09739	−0.187005	−0.0935025	+	0.995619i	$0.529806\pi$
$744$	0	0
$745$	0.226041	0.00828149
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−10.0529	−0.367324
$750$	0	0
$751$	−11.9611	−0.436466	−0.218233	−	0.975897i	$-0.570029\pi$
$751$	−11.9611	−0.436466	−0.218233	+	0.975897i	$0.570029\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.2353	−0.372501
$756$	0	0
$757$	−9.87805	−0.359024	−0.179512	−	0.983756i	$-0.557452\pi$
$757$	−9.87805	−0.359024	−0.179512	+	0.983756i	$0.557452\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.9064	−1.15661	−0.578303	−	0.815822i	$-0.696285\pi$
$761$	−31.9064	−1.15661	−0.578303	+	0.815822i	$0.696285\pi$
$762$	0	0
$763$	−62.4322	−2.26020
$764$	0	0
$765$	0	0
$766$	0	0
$767$	73.1413	2.64098
$768$	0	0
$769$	−8.75907	−0.315860	−0.157930	−	0.987450i	$-0.550482\pi$
$769$	−8.75907	−0.315860	−0.157930	+	0.987450i	$0.550482\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.1997	0.366859	0.183430	−	0.983033i	$-0.441280\pi$
$773$	10.1997	0.366859	0.183430	+	0.983033i	$0.441280\pi$
$774$	0	0
$775$	7.18153	0.257968
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.57206	0.235469
$780$	0	0
$781$	−32.3991	−1.15933
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.7755	−0.491668
$786$	0	0
$787$	23.4576	0.836172	0.418086	−	0.908407i	$-0.362701\pi$
$787$	23.4576	0.836172	0.418086	+	0.908407i	$0.362701\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−70.5517	−2.50853
$792$	0	0
$793$	94.8343	3.36767
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−48.5265	−1.71890	−0.859449	−	0.511222i	$-0.829193\pi$
$797$	−48.5265	−1.71890	−0.859449	+	0.511222i	$0.829193\pi$
$798$	0	0
$799$	45.1569	1.59754
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.5277	0.477382
$804$	0	0
$805$	5.02263	0.177024
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32.1619	−1.13075	−0.565376	−	0.824833i	$-0.691269\pi$
$809$	−32.1619	−1.13075	−0.565376	+	0.824833i	$0.691269\pi$
$810$	0	0
$811$	22.2522	0.781380	0.390690	−	0.920522i	$-0.372236\pi$
$811$	22.2522	0.781380	0.390690	+	0.920522i	$0.372236\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.33269	0.116739
$816$	0	0
$817$	0.798104	0.0279221
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.10472	−0.0734553	−0.0367277	−	0.999325i	$-0.511693\pi$
$821$	−2.10472	−0.0734553	−0.0367277	+	0.999325i	$0.511693\pi$
$822$	0	0
$823$	−12.1922	−0.424993	−0.212497	−	0.977162i	$-0.568159\pi$
$823$	−12.1922	−0.424993	−0.212497	+	0.977162i	$0.568159\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.9593	−0.415866	−0.207933	−	0.978143i	$-0.566674\pi$
$827$	−11.9593	−0.415866	−0.207933	+	0.978143i	$0.566674\pi$
$828$	0	0
$829$	−2.71590	−0.0943271	−0.0471635	−	0.998887i	$-0.515018\pi$
$829$	−2.71590	−0.0943271	−0.0471635	+	0.998887i	$0.515018\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−97.9949	−3.39532
$834$	0	0
$835$	19.5014	0.674874
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.9457	−1.03384	−0.516920	−	0.856033i	$-0.672922\pi$
$839$	−29.9457	−1.03384	−0.516920	+	0.856033i	$0.672922\pi$
$840$	0	0
$841$	−18.0270	−0.621622
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−29.1730	−1.00358
$846$	0	0
$847$	−13.8861	−0.477130
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.02263	−0.103614
$852$	0	0
$853$	7.02437	0.240510	0.120255	−	0.992743i	$-0.461629\pi$
$853$	7.02437	0.240510	0.120255	+	0.992743i	$0.461629\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.56799	0.121880	0.0609401	−	0.998141i	$-0.480590\pi$
$857$	3.56799	0.121880	0.0609401	+	0.998141i	$0.480590\pi$
$858$	0	0
$859$	22.5711	0.770116	0.385058	−	0.922892i	$-0.374181\pi$
$859$	22.5711	0.770116	0.385058	+	0.922892i	$0.374181\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.2607	0.655642	0.327821	−	0.944740i	$-0.393686\pi$
$863$	19.2607	0.655642	0.327821	+	0.944740i	$0.393686\pi$
$864$	0	0
$865$	−16.0453	−0.545555
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31.8842	1.08160
$870$	0	0
$871$	63.0678	2.13697
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.02263	0.169796
$876$	0	0
$877$	26.4441	0.892953	0.446477	−	0.894795i	$-0.352679\pi$
$877$	26.4441	0.892953	0.446477	+	0.894795i	$0.352679\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27.2630	−0.918513	−0.459256	−	0.888304i	$-0.651884\pi$
$881$	−27.2630	−0.918513	−0.459256	+	0.888304i	$0.651884\pi$
$882$	0	0
$883$	25.3469	0.852992	0.426496	−	0.904490i	$-0.359748\pi$
$883$	25.3469	0.852992	0.426496	+	0.904490i	$0.359748\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−48.1308	−1.61607	−0.808037	−	0.589132i	$-0.799470\pi$
$887$	−48.1308	−1.61607	−0.808037	+	0.589132i	$0.799470\pi$
$888$	0	0
$889$	59.8408	2.00699
$890$	0	0
$891$	0	0
$892$	0	0
$893$	70.5441	2.36067
$894$	0	0
$895$	14.1730	0.473752
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−23.7891	−0.793412
$900$	0	0
$901$	36.7088	1.22295
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.03114	−0.134000
$906$	0	0
$907$	18.4071	0.611199	0.305599	−	0.952160i	$-0.401143\pi$
$907$	18.4071	0.611199	0.305599	+	0.952160i	$0.401143\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40.5984	−1.34508	−0.672542	−	0.740059i	$-0.734797\pi$
$911$	−40.5984	−1.34508	−0.672542	+	0.740059i	$0.734797\pi$
$912$	0	0
$913$	27.3671	0.905720
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	25.5485	0.842767	0.421383	−	0.906883i	$-0.361545\pi$
$919$	25.5485	0.842767	0.421383	+	0.906883i	$0.361545\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	56.7109	1.86666
$924$	0	0
$925$	−3.02263	−0.0993834
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18.7588	−0.615456	−0.307728	−	0.951474i	$-0.599569\pi$
$929$	−18.7588	−0.615456	−0.307728	+	0.951474i	$0.599569\pi$
$930$	0	0
$931$	−153.088	−5.01725
$932$	0	0
$933$	0	0
$934$	0	0
$935$	19.9470	0.652336
$936$	0	0
$937$	10.4697	0.342031	0.171015	−	0.985268i	$-0.445295\pi$
$937$	10.4697	0.342031	0.171015	+	0.985268i	$0.445295\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15.1569	0.494100	0.247050	−	0.969003i	$-0.420539\pi$
$941$	15.1569	0.494100	0.247050	+	0.969003i	$0.420539\pi$
$942$	0	0
$943$	−0.782477	−0.0254810
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.2362	0.365127	0.182563	−	0.983194i	$-0.441560\pi$
$947$	11.2362	0.365127	0.182563	+	0.983194i	$0.441560\pi$
$948$	0	0
$949$	−23.6787	−0.768643
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.5317	0.470727	0.235363	−	0.971907i	$-0.424372\pi$
$953$	14.5317	0.470727	0.235363	+	0.971907i	$0.424372\pi$
$954$	0	0
$955$	8.58910	0.277937
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−101.705	−3.28423
$960$	0	0
$961$	20.5744	0.663689
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.8434	0.542207
$966$	0	0
$967$	−23.8670	−0.767512	−0.383756	−	0.923434i	$-0.625370\pi$
$967$	−23.8670	−0.767512	−0.383756	+	0.923434i	$0.625370\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16.8670	−0.541286	−0.270643	−	0.962680i	$-0.587236\pi$
$971$	−16.8670	−0.541286	−0.270643	+	0.962680i	$0.587236\pi$
$972$	0	0
$973$	28.2100	0.904370
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28.3073	0.905629	0.452815	−	0.891605i	$-0.350420\pi$
$977$	28.3073	0.905629	0.452815	+	0.891605i	$0.350420\pi$
$978$	0	0
$979$	23.3705	0.746923
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−7.45949	−0.237921	−0.118960	−	0.992899i	$-0.537956\pi$
$983$	−7.45949	−0.237921	−0.118960	+	0.992899i	$0.537956\pi$
$984$	0	0
$985$	0.0422192	0.00134522
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.0950231	−0.00302156
$990$	0	0
$991$	32.6469	1.03706	0.518532	−	0.855058i	$-0.326479\pi$
$991$	32.6469	1.03706	0.518532	+	0.855058i	$0.326479\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.1589	−0.448867
$996$	0	0
$997$	28.7800	0.911473	0.455737	−	0.890115i	$-0.349376\pi$
$997$	28.7800	0.911473	0.455737	+	0.890115i	$0.349376\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.br.1.1		5
3.2	odd	2		2760.2.a.w.1.1	✓	5
12.11	even	2		5520.2.a.cc.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.w.1.1	✓	5	3.2	odd	2
5520.2.a.cc.1.5		5	12.11	even	2
8280.2.a.br.1.1		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -5.02263 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -5.02263 q^{7} +3.71008 q^{11} -6.49408 q^{13} -5.37642 q^{17} -8.39905 q^{19} +1.00000 q^{23} +1.00000 q^{25} -3.31255 q^{29} +7.18153 q^{31} +5.02263 q^{35} -3.02263 q^{37} -0.782477 q^{41} -0.0950231 q^{43} -8.39905 q^{47} +18.2268 q^{49} -6.82773 q^{53} -3.71008 q^{55} -11.2628 q^{59} -14.6032 q^{61} +6.49408 q^{65} -9.71160 q^{67} -8.73271 q^{71} +3.64620 q^{73} -18.6344 q^{77} +8.59395 q^{79} +7.37642 q^{83} +5.37642 q^{85} +6.29918 q^{89} +32.6173 q^{91} +8.39905 q^{95} -5.32514 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 4 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 - 4 * q^7	\( 5 q - 5 q^{5} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 10 q^{17} - 4 q^{19} + 5 q^{23} + 5 q^{25} - 10 q^{29} + 6 q^{31} + 4 q^{35} + 6 q^{37} - 12 q^{41} - 2 q^{43} - 4 q^{47} + 19 q^{49} - 4 q^{55} - 6 q^{59} + 16 q^{61} - 4 q^{65} - 4 q^{67} - 8 q^{71} + 14 q^{73} - 16 q^{77} + 18 q^{79} + 20 q^{83} + 10 q^{85} - 18 q^{89} + 20 q^{91} + 4 q^{95} + 4 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 10 * q^17 - 4 * q^19 + 5 * q^23 + 5 * q^25 - 10 * q^29 + 6 * q^31 + 4 * q^35 + 6 * q^37 - 12 * q^41 - 2 * q^43 - 4 * q^47 + 19 * q^49 - 4 * q^55 - 6 * q^59 + 16 * q^61 - 4 * q^65 - 4 * q^67 - 8 * q^71 + 14 * q^73 - 16 * q^77 + 18 * q^79 + 20 * q^83 + 10 * q^85 - 18 * q^89 + 20 * q^91 + 4 * q^95 + 4 * q^97

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