LMFDB - Embedded newform 8208.2.a.bp.1.3

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 = [3,0,0,0,5,0,-1,0,0,0,-8] 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:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*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			1.3
Root			$-1.69963$ 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)

$f(q)$	$=$	$q+3.69963 q^{5} +1.58836 q^{7} -4.58836 q^{11} +1.58836 q^{13} -5.57598 q^{17} +1.00000 q^{19} -9.09888 q^{23} +8.68725 q^{25} -7.65383 q^{29} +2.58836 q^{31} +5.87636 q^{35} +0.287992 q^{37} +3.69963 q^{41} -9.32141 q^{43} -9.00000 q^{47} -4.47710 q^{49} +8.24219 q^{53} -16.9752 q^{55} -7.46472 q^{59} +0.189108 q^{61} +5.87636 q^{65} -8.08650 q^{67} +6.21015 q^{71} -1.22253 q^{73} -7.28799 q^{77} +5.90978 q^{79} +2.84431 q^{83} -20.6291 q^{85} +1.03342 q^{89} +2.52290 q^{91} +3.69963 q^{95} -15.4647 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{5} - q^{7} - 8 q^{11} - q^{13} + 7 q^{17} + 3 q^{19} - 9 q^{23} + 2 q^{25} - 6 q^{29} + 2 q^{31} - 11 q^{37} + 5 q^{41} - 9 q^{43} - 27 q^{47} - 8 q^{49} + 2 q^{53} - 15 q^{55} + q^{59} + 7 q^{61}+ \cdots - 23 q^{97}+O(q^{100}) $ 3 * q + 5 * q^5 - q^7 - 8 * q^11 - q^13 + 7 * q^17 + 3 * q^19 - 9 * q^23 + 2 * q^25 - 6 * q^29 + 2 * q^31 - 11 * q^37 + 5 * q^41 - 9 * q^43 - 27 * q^47 - 8 * q^49 + 2 * q^53 - 15 * q^55 + q^59 + 7 * q^61 + 12 * q^67 - 3 * q^73 - 10 * q^77 - 7 * q^79 - 5 * q^83 - 9 * q^85 - 4 * q^89 + 13 * q^91 + 5 * q^95 - 23 * 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
$5$	3.69963	1.65452	0.827262	−	0.561816i	$-0.189897\pi$
$5$	3.69963	1.65452	0.827262	+	0.561816i	$0.189897\pi$
$6$	0	0
$7$	1.58836	0.600345	0.300173	−	0.953885i	$-0.402956\pi$
$7$	1.58836	0.600345	0.300173	+	0.953885i	$0.402956\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.58836	−1.38344	−0.691722	−	0.722164i	$-0.743148\pi$
$11$	−4.58836	−1.38344	−0.691722	+	0.722164i	$0.743148\pi$
$12$	0	0
$13$	1.58836	0.440533	0.220266	−	0.975440i	$-0.429307\pi$
$13$	1.58836	0.440533	0.220266	+	0.975440i	$0.429307\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.57598	−1.35237	−0.676187	−	0.736730i	$-0.736369\pi$
$17$	−5.57598	−1.35237	−0.676187	+	0.736730i	$0.736369\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.09888	−1.89725	−0.948624	−	0.316405i	$-0.897524\pi$
$23$	−9.09888	−1.89725	−0.948624	+	0.316405i	$0.897524\pi$
$24$	0	0
$25$	8.68725	1.73745
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.65383	−1.42128	−0.710640	−	0.703556i	$-0.751595\pi$
$29$	−7.65383	−1.42128	−0.710640	+	0.703556i	$0.751595\pi$
$30$	0	0
$31$	2.58836	0.464884	0.232442	−	0.972610i	$-0.425328\pi$
$31$	2.58836	0.464884	0.232442	+	0.972610i	$0.425328\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.87636	0.993285
$36$	0	0
$37$	0.287992	0.0473456	0.0236728	−	0.999720i	$-0.492464\pi$
$37$	0.287992	0.0473456	0.0236728	+	0.999720i	$0.492464\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.69963	0.577785	0.288892	−	0.957362i	$-0.406713\pi$
$41$	3.69963	0.577785	0.288892	+	0.957362i	$0.406713\pi$
$42$	0	0
$43$	−9.32141	−1.42150	−0.710751	−	0.703444i	$-0.751645\pi$
$43$	−9.32141	−1.42150	−0.710751	+	0.703444i	$0.751645\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	−4.47710	−0.639586
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.24219	1.13215	0.566076	−	0.824353i	$-0.308461\pi$
$53$	8.24219	1.13215	0.566076	+	0.824353i	$0.308461\pi$
$54$	0	0
$55$	−16.9752	−2.28894
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.46472	−0.971824	−0.485912	−	0.874008i	$-0.661512\pi$
$59$	−7.46472	−0.971824	−0.485912	+	0.874008i	$0.661512\pi$
$60$	0	0
$61$	0.189108	0.0242128	0.0121064	−	0.999927i	$-0.496146\pi$
$61$	0.189108	0.0242128	0.0121064	+	0.999927i	$0.496146\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.87636	0.728872
$66$	0	0
$67$	−8.08650	−0.987924	−0.493962	−	0.869484i	$-0.664452\pi$
$67$	−8.08650	−0.987924	−0.493962	+	0.869484i	$0.664452\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.21015	0.737009	0.368505	−	0.929626i	$-0.379870\pi$
$71$	6.21015	0.737009	0.368505	+	0.929626i	$0.379870\pi$
$72$	0	0
$73$	−1.22253	−0.143086	−0.0715431	−	0.997438i	$-0.522792\pi$
$73$	−1.22253	−0.143086	−0.0715431	+	0.997438i	$0.522792\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.28799	−0.830544
$78$	0	0
$79$	5.90978	0.664902	0.332451	−	0.943121i	$-0.392124\pi$
$79$	5.90978	0.664902	0.332451	+	0.943121i	$0.392124\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.84431	0.312204	0.156102	−	0.987741i	$-0.450107\pi$
$83$	2.84431	0.312204	0.156102	+	0.987741i	$0.450107\pi$
$84$	0	0
$85$	−20.6291	−2.23754
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.03342	0.109542	0.0547712	−	0.998499i	$-0.482557\pi$
$89$	1.03342	0.109542	0.0547712	+	0.998499i	$0.482557\pi$
$90$	0	0
$91$	2.52290	0.264472
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.69963	0.379574
$96$	0	0
$97$	−15.4647	−1.57020	−0.785102	−	0.619366i	$-0.787390\pi$
$97$	−15.4647	−1.57020	−0.785102	+	0.619366i	$0.787390\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.36584	0.732928	0.366464	−	0.930432i	$-0.380568\pi$
$101$	7.36584	0.732928	0.366464	+	0.930432i	$0.380568\pi$
$102$	0	0
$103$	5.74543	0.566114	0.283057	−	0.959103i	$-0.408651\pi$
$103$	5.74543	0.566114	0.283057	+	0.959103i	$0.408651\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.699628	−0.0676356	−0.0338178	−	0.999428i	$-0.510767\pi$
$107$	−0.699628	−0.0676356	−0.0338178	+	0.999428i	$0.510767\pi$
$108$	0	0
$109$	−8.49814	−0.813974	−0.406987	−	0.913434i	$-0.633421\pi$
$109$	−8.49814	−0.813974	−0.406987	+	0.913434i	$0.633421\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.4647	−0.984438	−0.492219	−	0.870471i	$-0.663814\pi$
$113$	−10.4647	−0.984438	−0.492219	+	0.870471i	$0.663814\pi$
$114$	0	0
$115$	−33.6625	−3.13904
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.85669	−0.811892
$120$	0	0
$121$	10.0531	0.913917
$122$	0	0
$123$	0	0
$124$	0	0
$125$	13.6414	1.22013
$126$	0	0
$127$	10.7527	0.954149	0.477074	−	0.878863i	$-0.341697\pi$
$127$	10.7527	0.954149	0.477074	+	0.878863i	$0.341697\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.3090	−1.42493	−0.712463	−	0.701709i	$-0.752421\pi$
$131$	−16.3090	−1.42493	−0.712463	+	0.701709i	$0.752421\pi$
$132$	0	0
$133$	1.58836	0.137729
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.2101	−1.29949	−0.649745	−	0.760152i	$-0.725124\pi$
$137$	−15.2101	−1.29949	−0.649745	+	0.760152i	$0.725124\pi$
$138$	0	0
$139$	−2.62178	−0.222377	−0.111188	−	0.993799i	$-0.535466\pi$
$139$	−2.62178	−0.222377	−0.111188	+	0.993799i	$0.535466\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.28799	−0.609453
$144$	0	0
$145$	−28.3163	−2.35154
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.7651	−1.37345	−0.686725	−	0.726917i	$-0.740952\pi$
$149$	−16.7651	−1.37345	−0.686725	+	0.726917i	$0.740952\pi$
$150$	0	0
$151$	15.8182	1.28726	0.643632	−	0.765335i	$-0.277427\pi$
$151$	15.8182	1.28726	0.643632	+	0.765335i	$0.277427\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.57598	0.769162
$156$	0	0
$157$	14.9084	1.18982	0.594910	−	0.803792i	$-0.297188\pi$
$157$	14.9084	1.18982	0.594910	+	0.803792i	$0.297188\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−14.4523	−1.13900
$162$	0	0
$163$	−11.3004	−0.885113	−0.442557	−	0.896741i	$-0.645928\pi$
$163$	−11.3004	−0.885113	−0.442557	+	0.896741i	$0.645928\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.4647	−1.50623	−0.753113	−	0.657892i	$-0.771448\pi$
$167$	−19.4647	−1.50623	−0.753113	+	0.657892i	$0.771448\pi$
$168$	0	0
$169$	−10.4771	−0.805931
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−15.9098	−1.20960	−0.604799	−	0.796378i	$-0.706747\pi$
$173$	−15.9098	−1.20960	−0.604799	+	0.796378i	$0.706747\pi$
$174$	0	0
$175$	13.7985	1.04307
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.7861	0.955680	0.477840	−	0.878447i	$-0.341420\pi$
$179$	12.7861	0.955680	0.477840	+	0.878447i	$0.341420\pi$
$180$	0	0
$181$	19.7861	1.47069	0.735346	−	0.677692i	$-0.237020\pi$
$181$	19.7861	1.47069	0.735346	+	0.677692i	$0.237020\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.06546	0.0783345
$186$	0	0
$187$	25.5846	1.87093
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.43268	0.610167	0.305084	−	0.952326i	$-0.401316\pi$
$191$	8.43268	0.610167	0.305084	+	0.952326i	$0.401316\pi$
$192$	0	0
$193$	−6.76509	−0.486962	−0.243481	−	0.969906i	$-0.578289\pi$
$193$	−6.76509	−0.486962	−0.243481	+	0.969906i	$0.578289\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.2509	1.44281	0.721407	−	0.692512i	$-0.243496\pi$
$197$	20.2509	1.44281	0.721407	+	0.692512i	$0.243496\pi$
$198$	0	0
$199$	−0.0320432	−0.00227148	−0.00113574	−	0.999999i	$-0.500362\pi$
$199$	−0.0320432	−0.00227148	−0.00113574	+	0.999999i	$0.500362\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.1571	−0.853259
$204$	0	0
$205$	13.6872	0.955959
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.58836	−0.317384
$210$	0	0
$211$	5.35483	0.368642	0.184321	−	0.982866i	$-0.440991\pi$
$211$	5.35483	0.368642	0.184321	+	0.982866i	$0.440991\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−34.4858	−2.35191
$216$	0	0
$217$	4.11126	0.279091
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.85669	−0.595766
$222$	0	0
$223$	7.84294	0.525202	0.262601	−	0.964905i	$-0.415420\pi$
$223$	7.84294	0.525202	0.262601	+	0.964905i	$0.415420\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−8.55632	−0.567903	−0.283951	−	0.958839i	$-0.591645\pi$
$227$	−8.55632	−0.567903	−0.283951	+	0.958839i	$0.591645\pi$
$228$	0	0
$229$	−5.03204	−0.332527	−0.166263	−	0.986081i	$-0.553170\pi$
$229$	−5.03204	−0.332527	−0.166263	+	0.986081i	$0.553170\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.2559	0.999450	0.499725	−	0.866184i	$-0.333434\pi$
$233$	15.2559	0.999450	0.499725	+	0.866184i	$0.333434\pi$
$234$	0	0
$235$	−33.2967	−2.17203
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.6167	1.59232	0.796161	−	0.605085i	$-0.206861\pi$
$239$	24.6167	1.59232	0.796161	+	0.605085i	$0.206861\pi$
$240$	0	0
$241$	−11.5229	−0.742255	−0.371128	−	0.928582i	$-0.621029\pi$
$241$	−11.5229	−0.742255	−0.371128	+	0.928582i	$0.621029\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−16.5636	−1.05821
$246$	0	0
$247$	1.58836	0.101065
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.6094	1.67957	0.839785	−	0.542919i	$-0.182681\pi$
$251$	26.6094	1.67957	0.839785	+	0.542919i	$0.182681\pi$
$252$	0	0
$253$	41.7490	2.62474
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.6167	1.66030	0.830152	−	0.557538i	$-0.188254\pi$
$257$	26.6167	1.66030	0.830152	+	0.557538i	$0.188254\pi$
$258$	0	0
$259$	0.457436	0.0284237
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.45234	0.336206	0.168103	−	0.985769i	$-0.446236\pi$
$263$	5.45234	0.336206	0.168103	+	0.985769i	$0.446236\pi$
$264$	0	0
$265$	30.4930	1.87317
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.2312	1.17255	0.586273	−	0.810113i	$-0.300595\pi$
$269$	19.2312	1.17255	0.586273	+	0.810113i	$0.300595\pi$
$270$	0	0
$271$	24.1730	1.46841	0.734203	−	0.678930i	$-0.237556\pi$
$271$	24.1730	1.46841	0.734203	+	0.678930i	$0.237556\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−39.8603	−2.40366
$276$	0	0
$277$	−0.00137742	−8.27610e−5 0	−4.13805e−5	−	1.00000i	$-0.500013\pi$
$277$	−0.00137742	−8.27610e−5 0	−4.13805e−5		1.00000i	$0.500013\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.24729	−0.432337	−0.216168	−	0.976356i	$-0.569356\pi$
$281$	−7.24729	−0.432337	−0.216168	+	0.976356i	$0.569356\pi$
$282$	0	0
$283$	−5.37450	−0.319481	−0.159740	−	0.987159i	$-0.551066\pi$
$283$	−5.37450	−0.319481	−0.159740	+	0.987159i	$0.551066\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.87636	0.346870
$288$	0	0
$289$	14.0916	0.828918
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.13602	−0.416891	−0.208445	−	0.978034i	$-0.566840\pi$
$293$	−7.13602	−0.416891	−0.208445	+	0.978034i	$0.566840\pi$
$294$	0	0
$295$	−27.6167	−1.60791
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.4523	−0.835800
$300$	0	0
$301$	−14.8058	−0.853392
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.699628	0.0400606
$306$	0	0
$307$	10.7280	0.612277	0.306138	−	0.951987i	$-0.400963\pi$
$307$	10.7280	0.612277	0.306138	+	0.951987i	$0.400963\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.83056	0.500735	0.250367	−	0.968151i	$-0.419449\pi$
$311$	8.83056	0.500735	0.250367	+	0.968151i	$0.419449\pi$
$312$	0	0
$313$	−12.8196	−0.724604	−0.362302	−	0.932061i	$-0.618009\pi$
$313$	−12.8196	−0.724604	−0.362302	+	0.932061i	$0.618009\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.5512	−0.761113	−0.380556	−	0.924758i	$-0.624267\pi$
$317$	−13.5512	−0.761113	−0.380556	+	0.924758i	$0.624267\pi$
$318$	0	0
$319$	35.1185	1.96626
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.57598	−0.310256
$324$	0	0
$325$	13.7985	0.765404
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−14.2953	−0.788124
$330$	0	0
$331$	−2.94692	−0.161977	−0.0809886	−	0.996715i	$-0.525808\pi$
$331$	−2.94692	−0.161977	−0.0809886	+	0.996715i	$0.525808\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−29.9171	−1.63454
$336$	0	0
$337$	−29.0173	−1.58067	−0.790337	−	0.612672i	$-0.790095\pi$
$337$	−29.0173	−1.58067	−0.790337	+	0.612672i	$0.790095\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.8764	−0.643141
$342$	0	0
$343$	−18.2298	−0.984317
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.119925	0.00643793	0.00321897	−	0.999995i	$-0.498975\pi$
$347$	0.119925	0.00643793	0.00321897	+	0.999995i	$0.498975\pi$
$348$	0	0
$349$	−11.0902	−0.593646	−0.296823	−	0.954933i	$-0.595927\pi$
$349$	−11.0902	−0.593646	−0.296823	+	0.954933i	$0.595927\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.6291	−0.672177	−0.336089	−	0.941830i	$-0.609104\pi$
$353$	−12.6291	−0.672177	−0.336089	+	0.941830i	$0.609104\pi$
$354$	0	0
$355$	22.9752	1.21940
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.8726	0.890504	0.445252	−	0.895405i	$-0.353114\pi$
$359$	16.8726	0.890504	0.445252	+	0.895405i	$0.353114\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.52290	−0.236739
$366$	0	0
$367$	29.8319	1.55721	0.778607	−	0.627512i	$-0.215927\pi$
$367$	29.8319	1.55721	0.778607	+	0.627512i	$0.215927\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.0916	0.679682
$372$	0	0
$373$	10.6625	0.552083	0.276041	−	0.961146i	$-0.410977\pi$
$373$	10.6625	0.552083	0.276041	+	0.961146i	$0.410977\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.1571	−0.626121
$378$	0	0
$379$	4.25457	0.218543	0.109271	−	0.994012i	$-0.465148\pi$
$379$	4.25457	0.218543	0.109271	+	0.994012i	$0.465148\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.8233	−0.553043	−0.276522	−	0.961008i	$-0.589182\pi$
$383$	−10.8233	−0.553043	−0.276522	+	0.961008i	$0.589182\pi$
$384$	0	0
$385$	−26.9629	−1.37415
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.2595	1.68632	0.843162	−	0.537659i	$-0.180691\pi$
$389$	33.2595	1.68632	0.843162	+	0.537659i	$0.180691\pi$
$390$	0	0
$391$	50.7352	2.56579
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.8640	1.10010
$396$	0	0
$397$	−31.1024	−1.56099	−0.780494	−	0.625164i	$-0.785032\pi$
$397$	−31.1024	−1.56099	−0.780494	+	0.625164i	$0.785032\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−25.0334	−1.25011	−0.625055	−	0.780581i	$-0.714923\pi$
$401$	−25.0334	−1.25011	−0.625055	+	0.780581i	$0.714923\pi$
$402$	0	0
$403$	4.11126	0.204797
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.32141	−0.0655000
$408$	0	0
$409$	38.7417	1.91565	0.957827	−	0.287345i	$-0.0927726\pi$
$409$	38.7417	1.91565	0.957827	+	0.287345i	$0.0927726\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11.8567	−0.583430
$414$	0	0
$415$	10.5229	0.516549
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.41892	−0.215878	−0.107939	−	0.994157i	$-0.534425\pi$
$419$	−4.41892	−0.215878	−0.107939	+	0.994157i	$0.534425\pi$
$420$	0	0
$421$	−13.4574	−0.655875	−0.327938	−	0.944699i	$-0.606354\pi$
$421$	−13.4574	−0.655875	−0.327938	+	0.944699i	$0.606354\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−48.4400	−2.34968
$426$	0	0
$427$	0.300372	0.0145360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.6945	−1.67118	−0.835588	−	0.549356i	$-0.814873\pi$
$431$	−34.6945	−1.67118	−0.835588	+	0.549356i	$0.814873\pi$
$432$	0	0
$433$	23.7848	1.14302	0.571511	−	0.820594i	$-0.306357\pi$
$433$	23.7848	1.14302	0.571511	+	0.820594i	$0.306357\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.09888	−0.435259
$438$	0	0
$439$	−1.95420	−0.0932689	−0.0466344	−	0.998912i	$-0.514850\pi$
$439$	−1.95420	−0.0932689	−0.0466344	+	0.998912i	$0.514850\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.9281	0.946811	0.473405	−	0.880845i	$-0.343025\pi$
$443$	19.9281	0.946811	0.473405	+	0.880845i	$0.343025\pi$
$444$	0	0
$445$	3.82327	0.181240
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−40.2522	−1.89962	−0.949810	−	0.312827i	$-0.898724\pi$
$449$	−40.2522	−1.89962	−0.949810	+	0.312827i	$0.898724\pi$
$450$	0	0
$451$	−16.9752	−0.799333
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9.33379	0.437575
$456$	0	0
$457$	24.0159	1.12342	0.561709	−	0.827335i	$-0.310144\pi$
$457$	24.0159	1.12342	0.561709	+	0.827335i	$0.310144\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.9876	0.744618	0.372309	−	0.928109i	$-0.378566\pi$
$461$	15.9876	0.744618	0.372309	+	0.928109i	$0.378566\pi$
$462$	0	0
$463$	−13.0617	−0.607031	−0.303515	−	0.952827i	$-0.598160\pi$
$463$	−13.0617	−0.607031	−0.303515	+	0.952827i	$0.598160\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.40063	−0.0648136	−0.0324068	−	0.999475i	$-0.510317\pi$
$467$	−1.40063	−0.0648136	−0.0324068	+	0.999475i	$0.510317\pi$
$468$	0	0
$469$	−12.8443	−0.593095
$470$	0	0
$471$	0	0
$472$	0	0
$473$	42.7700	1.96657
$474$	0	0
$475$	8.68725	0.398598
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.3287	−0.746077	−0.373039	−	0.927816i	$-0.621684\pi$
$479$	−16.3287	−0.746077	−0.373039	+	0.927816i	$0.621684\pi$
$480$	0	0
$481$	0.457436	0.0208573
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−57.2137	−2.59794
$486$	0	0
$487$	−35.3832	−1.60336	−0.801682	−	0.597751i	$-0.796061\pi$
$487$	−35.3832	−1.60336	−0.801682	+	0.597751i	$0.796061\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.64654	0.345084	0.172542	−	0.985002i	$-0.444802\pi$
$491$	7.64654	0.345084	0.172542	+	0.985002i	$0.444802\pi$
$492$	0	0
$493$	42.6776	1.92210
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.86398	0.442460
$498$	0	0
$499$	12.1447	0.543671	0.271835	−	0.962344i	$-0.412369\pi$
$499$	12.1447	0.543671	0.271835	+	0.962344i	$0.412369\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.2261	1.21395	0.606976	−	0.794720i	$-0.292382\pi$
$503$	27.2261	1.21395	0.606976	+	0.794720i	$0.292382\pi$
$504$	0	0
$505$	27.2509	1.21265
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.5141	−1.70711	−0.853553	−	0.521005i	$-0.825557\pi$
$509$	−38.5141	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$510$	0	0
$511$	−1.94182	−0.0859011
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.2559	0.936649
$516$	0	0
$517$	41.2953	1.81616
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.3535	−0.979323	−0.489661	−	0.871913i	$-0.662880\pi$
$521$	−22.3535	−0.979323	−0.489661	+	0.871913i	$0.662880\pi$
$522$	0	0
$523$	6.12227	0.267708	0.133854	−	0.991001i	$-0.457265\pi$
$523$	6.12227	0.267708	0.133854	+	0.991001i	$0.457265\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.4327	−0.628697
$528$	0	0
$529$	59.7897	2.59955
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.87636	0.254533
$534$	0	0
$535$	−2.58836	−0.111905
$536$	0	0
$537$	0	0
$538$	0	0
$539$	20.5426	0.884831
$540$	0	0
$541$	−23.0073	−0.989160	−0.494580	−	0.869132i	$-0.664678\pi$
$541$	−23.0073	−0.989160	−0.494580	+	0.869132i	$0.664678\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−31.4400	−1.34674
$546$	0	0
$547$	13.1716	0.563178	0.281589	−	0.959535i	$-0.409138\pi$
$547$	13.1716	0.563178	0.281589	+	0.959535i	$0.409138\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.65383	−0.326064
$552$	0	0
$553$	9.38688	0.399171
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.18773	−0.0503257	−0.0251629	−	0.999683i	$-0.508010\pi$
$557$	−1.18773	−0.0503257	−0.0251629	+	0.999683i	$0.508010\pi$
$558$	0	0
$559$	−14.8058	−0.626218
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.2929	1.82458	0.912290	−	0.409545i	$-0.134313\pi$
$563$	43.2929	1.82458	0.912290	+	0.409545i	$0.134313\pi$
$564$	0	0
$565$	−38.7156	−1.62878
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.54256	0.106590	0.0532949	−	0.998579i	$-0.483028\pi$
$569$	2.54256	0.106590	0.0532949	+	0.998579i	$0.483028\pi$
$570$	0	0
$571$	−28.3621	−1.18692	−0.593459	−	0.804864i	$-0.702238\pi$
$571$	−28.3621	−1.18692	−0.593459	+	0.804864i	$0.702238\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−79.0443	−3.29637
$576$	0	0
$577$	−1.67859	−0.0698805	−0.0349403	−	0.999389i	$-0.511124\pi$
$577$	−1.67859	−0.0698805	−0.0349403	+	0.999389i	$0.511124\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.51780	0.187430
$582$	0	0
$583$	−37.8182	−1.56627
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.6662	−0.935535	−0.467767	−	0.883852i	$-0.654941\pi$
$587$	−22.6662	−0.935535	−0.467767	+	0.883852i	$0.654941\pi$
$588$	0	0
$589$	2.58836	0.106652
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.77747	0.278317	0.139159	−	0.990270i	$-0.455560\pi$
$593$	6.77747	0.278317	0.139159	+	0.990270i	$0.455560\pi$
$594$	0	0
$595$	−32.7665	−1.34329
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.30766	−0.176006	−0.0880030	−	0.996120i	$-0.528049\pi$
$599$	−4.30766	−0.176006	−0.0880030	+	0.996120i	$0.528049\pi$
$600$	0	0
$601$	−15.5215	−0.633136	−0.316568	−	0.948570i	$-0.602531\pi$
$601$	−15.5215	−0.633136	−0.316568	+	0.948570i	$0.602531\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	37.1927	1.51210
$606$	0	0
$607$	−24.7600	−1.00498	−0.502489	−	0.864584i	$-0.667582\pi$
$607$	−24.7600	−1.00498	−0.502489	+	0.864584i	$0.667582\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−14.2953	−0.578325
$612$	0	0
$613$	−21.7775	−0.879584	−0.439792	−	0.898100i	$-0.644948\pi$
$613$	−21.7775	−0.879584	−0.439792	+	0.898100i	$0.644948\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.452340	−0.0182105	−0.00910527	−	0.999959i	$-0.502898\pi$
$617$	−0.452340	−0.0182105	−0.00910527	+	0.999959i	$0.502898\pi$
$618$	0	0
$619$	−25.1323	−1.01015	−0.505076	−	0.863075i	$-0.668536\pi$
$619$	−25.1323	−1.01015	−0.505076	+	0.863075i	$0.668536\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.64145	0.0657632
$624$	0	0
$625$	7.03204	0.281282
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.60584	−0.0640290
$630$	0	0
$631$	6.26461	0.249390	0.124695	−	0.992195i	$-0.460205\pi$
$631$	6.26461	0.249390	0.124695	+	0.992195i	$0.460205\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	39.7810	1.57866
$636$	0	0
$637$	−7.11126	−0.281759
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19.4858	0.769641	0.384821	−	0.922991i	$-0.374263\pi$
$641$	19.4858	0.769641	0.384821	+	0.922991i	$0.374263\pi$
$642$	0	0
$643$	26.8887	1.06039	0.530194	−	0.847876i	$-0.322119\pi$
$643$	26.8887	1.06039	0.530194	+	0.847876i	$0.322119\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−43.7280	−1.71912	−0.859562	−	0.511032i	$-0.829263\pi$
$647$	−43.7280	−1.71912	−0.859562	+	0.511032i	$0.829263\pi$
$648$	0	0
$649$	34.2509	1.34446
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.84294	0.189519	0.0947594	−	0.995500i	$-0.469792\pi$
$653$	4.84294	0.189519	0.0947594	+	0.995500i	$0.469792\pi$
$654$	0	0
$655$	−60.3374	−2.35758
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−26.5846	−1.03559	−0.517795	−	0.855504i	$-0.673247\pi$
$659$	−26.5846	−1.03559	−0.517795	+	0.855504i	$0.673247\pi$
$660$	0	0
$661$	7.97524	0.310201	0.155100	−	0.987899i	$-0.450430\pi$
$661$	7.97524	0.310201	0.155100	+	0.987899i	$0.450430\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.87636	0.227875
$666$	0	0
$667$	69.6413	2.69652
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.867695	−0.0334970
$672$	0	0
$673$	−32.7476	−1.26233	−0.631164	−	0.775649i	$-0.717422\pi$
$673$	−32.7476	−1.26233	−0.631164	+	0.775649i	$0.717422\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.8282	1.41542	0.707712	−	0.706501i	$-0.249728\pi$
$677$	36.8282	1.41542	0.707712	+	0.706501i	$0.249728\pi$
$678$	0	0
$679$	−24.5636	−0.942665
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−41.6784	−1.59478	−0.797390	−	0.603464i	$-0.793787\pi$
$683$	−41.6784	−1.59478	−0.797390	+	0.603464i	$0.793787\pi$
$684$	0	0
$685$	−56.2719	−2.15004
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.0916	0.498750
$690$	0	0
$691$	−18.3077	−0.696456	−0.348228	−	0.937410i	$-0.613217\pi$
$691$	−18.3077	−0.696456	−0.348228	+	0.937410i	$0.613217\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.69963	−0.367928
$696$	0	0
$697$	−20.6291	−0.781382
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−14.8589	−0.561212	−0.280606	−	0.959823i	$-0.590535\pi$
$701$	−14.8589	−0.561212	−0.280606	+	0.959823i	$0.590535\pi$
$702$	0	0
$703$	0.287992	0.0108618
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.6996	0.440010
$708$	0	0
$709$	−1.97168	−0.0740478	−0.0370239	−	0.999314i	$-0.511788\pi$
$709$	−1.97168	−0.0740478	−0.0370239	+	0.999314i	$0.511788\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−23.5512	−0.882000
$714$	0	0
$715$	−26.9629	−1.00835
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.49814	0.130459	0.0652293	−	0.997870i	$-0.479222\pi$
$719$	3.49814	0.130459	0.0652293	+	0.997870i	$0.479222\pi$
$720$	0	0
$721$	9.12583	0.339864
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−66.4907	−2.46940
$726$	0	0
$727$	8.69453	0.322462	0.161231	−	0.986917i	$-0.448454\pi$
$727$	8.69453	0.322462	0.161231	+	0.986917i	$0.448454\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	51.9761	1.92240
$732$	0	0
$733$	15.0124	0.554495	0.277247	−	0.960799i	$-0.410578\pi$
$733$	15.0124	0.554495	0.277247	+	0.960799i	$0.410578\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	37.1038	1.36674
$738$	0	0
$739$	−8.81955	−0.324433	−0.162216	−	0.986755i	$-0.551864\pi$
$739$	−8.81955	−0.324433	−0.162216	+	0.986755i	$0.551864\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.60940	0.315848	0.157924	−	0.987451i	$-0.449520\pi$
$743$	8.60940	0.315848	0.157924	+	0.987451i	$0.449520\pi$
$744$	0	0
$745$	−62.0246	−2.27241
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.11126	−0.0406047
$750$	0	0
$751$	3.39788	0.123990	0.0619952	−	0.998076i	$-0.480254\pi$
$751$	3.39788	0.123990	0.0619952	+	0.998076i	$0.480254\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	58.5214	2.12981
$756$	0	0
$757$	−39.5585	−1.43778	−0.718889	−	0.695125i	$-0.755349\pi$
$757$	−39.5585	−1.43778	−0.718889	+	0.695125i	$0.755349\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.8355	−0.791536	−0.395768	−	0.918350i	$-0.629522\pi$
$761$	−21.8355	−0.791536	−0.395768	+	0.918350i	$0.629522\pi$
$762$	0	0
$763$	−13.4981	−0.488666
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−11.8567	−0.428120
$768$	0	0
$769$	−6.19049	−0.223235	−0.111617	−	0.993751i	$-0.535603\pi$
$769$	−6.19049	−0.223235	−0.111617	+	0.993751i	$0.535603\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40.9184	1.47173	0.735867	−	0.677126i	$-0.236775\pi$
$773$	40.9184	1.47173	0.735867	+	0.677126i	$0.236775\pi$
$774$	0	0
$775$	22.4858	0.807712
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.69963	0.132553
$780$	0	0
$781$	−28.4944	−1.01961
$782$	0	0
$783$	0	0
$784$	0	0
$785$	55.1555	1.96859
$786$	0	0
$787$	28.9171	1.03078	0.515391	−	0.856955i	$-0.327647\pi$
$787$	28.9171	1.03078	0.515391	+	0.856955i	$0.327647\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.6218	−0.591003
$792$	0	0
$793$	0.300372	0.0106665
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.4596	−0.441343	−0.220671	−	0.975348i	$-0.570825\pi$
$797$	−12.4596	−0.441343	−0.220671	+	0.975348i	$0.570825\pi$
$798$	0	0
$799$	50.1839	1.77538
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.60940	0.197952
$804$	0	0
$805$	−53.4683	−1.88451
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.07922	−0.319208	−0.159604	−	0.987181i	$-0.551022\pi$
$809$	−9.07922	−0.319208	−0.159604	+	0.987181i	$0.551022\pi$
$810$	0	0
$811$	−25.5622	−0.897611	−0.448806	−	0.893629i	$-0.648150\pi$
$811$	−25.5622	−0.897611	−0.448806	+	0.893629i	$0.648150\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−41.8072	−1.46444
$816$	0	0
$817$	−9.32141	−0.326115
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.10026	0.317601	0.158801	−	0.987311i	$-0.449237\pi$
$821$	9.10026	0.317601	0.158801	+	0.987311i	$0.449237\pi$
$822$	0	0
$823$	29.0865	1.01389	0.506946	−	0.861978i	$-0.330774\pi$
$823$	29.0865	1.01389	0.506946	+	0.861978i	$0.330774\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.94182	−0.137071	−0.0685353	−	0.997649i	$-0.521833\pi$
$827$	−3.94182	−0.137071	−0.0685353	+	0.997649i	$0.521833\pi$
$828$	0	0
$829$	56.1184	1.94907	0.974536	−	0.224230i	$-0.0719868\pi$
$829$	56.1184	1.94907	0.974536	+	0.224230i	$0.0719868\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.9642	0.864960
$834$	0	0
$835$	−72.0122	−2.49209
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30.1868	−1.04216	−0.521081	−	0.853507i	$-0.674471\pi$
$839$	−30.1868	−1.04216	−0.521081	+	0.853507i	$0.674471\pi$
$840$	0	0
$841$	29.5811	1.02004
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−38.7614	−1.33343
$846$	0	0
$847$	15.9680	0.548665
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.62041	−0.0898264
$852$	0	0
$853$	19.3200	0.661505	0.330753	−	0.943717i	$-0.392697\pi$
$853$	19.3200	0.661505	0.330753	+	0.943717i	$0.392697\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−49.4472	−1.68909	−0.844543	−	0.535488i	$-0.820128\pi$
$857$	−49.4472	−1.68909	−0.844543	+	0.535488i	$0.820128\pi$
$858$	0	0
$859$	35.1075	1.19785	0.598927	−	0.800804i	$-0.295594\pi$
$859$	35.1075	1.19785	0.598927	+	0.800804i	$0.295594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−43.8158	−1.49151	−0.745754	−	0.666221i	$-0.767911\pi$
$863$	−43.8158	−1.49151	−0.745754	+	0.666221i	$0.767911\pi$
$864$	0	0
$865$	−58.8603	−2.00131
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−27.1162	−0.919854
$870$	0	0
$871$	−12.8443	−0.435213
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21.6676	0.732498
$876$	0	0
$877$	−9.82327	−0.331708	−0.165854	−	0.986150i	$-0.553038\pi$
$877$	−9.82327	−0.331708	−0.165854	+	0.986150i	$0.553038\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.6648	0.494071	0.247035	−	0.969006i	$-0.420544\pi$
$881$	14.6648	0.494071	0.247035	+	0.969006i	$0.420544\pi$
$882$	0	0
$883$	14.9432	0.502879	0.251439	−	0.967873i	$-0.419096\pi$
$883$	14.9432	0.502879	0.251439	+	0.967873i	$0.419096\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.3004	−0.681620	−0.340810	−	0.940132i	$-0.610701\pi$
$887$	−20.3004	−0.681620	−0.340810	+	0.940132i	$0.610701\pi$
$888$	0	0
$889$	17.0792	0.572819
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.00000	−0.301174
$894$	0	0
$895$	47.3039	1.58120
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.8109	−0.660730
$900$	0	0
$901$	−45.9583	−1.53109
$902$	0	0
$903$	0	0
$904$	0	0
$905$	73.2013	2.43329
$906$	0	0
$907$	−54.9120	−1.82332	−0.911661	−	0.410943i	$-0.865199\pi$
$907$	−54.9120	−1.82332	−0.911661	+	0.410943i	$0.865199\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.7751	0.986494	0.493247	−	0.869889i	$-0.335810\pi$
$911$	29.7751	0.986494	0.493247	+	0.869889i	$0.335810\pi$
$912$	0	0
$913$	−13.0507	−0.431917
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25.9047	−0.855448
$918$	0	0
$919$	−19.5563	−0.645103	−0.322552	−	0.946552i	$-0.604541\pi$
$919$	−19.5563	−0.645103	−0.322552	+	0.946552i	$0.604541\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.86398	0.324677
$924$	0	0
$925$	2.50186	0.0822606
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23.0641	0.756708	0.378354	−	0.925661i	$-0.376490\pi$
$929$	23.0641	0.756708	0.378354	+	0.925661i	$0.376490\pi$
$930$	0	0
$931$	−4.47710	−0.146731
$932$	0	0
$933$	0	0
$934$	0	0
$935$	94.6537	3.09551
$936$	0	0
$937$	−4.31632	−0.141008	−0.0705040	−	0.997511i	$-0.522461\pi$
$937$	−4.31632	−0.141008	−0.0705040	+	0.997511i	$0.522461\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.9322	0.714969	0.357485	−	0.933919i	$-0.383634\pi$
$941$	21.9322	0.714969	0.357485	+	0.933919i	$0.383634\pi$
$942$	0	0
$943$	−33.6625	−1.09620
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.5288	−0.699592	−0.349796	−	0.936826i	$-0.613749\pi$
$947$	−21.5288	−0.699592	−0.349796	+	0.936826i	$0.613749\pi$
$948$	0	0
$949$	−1.94182	−0.0630341
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26.4313	−0.856194	−0.428097	−	0.903733i	$-0.640816\pi$
$953$	−26.4313	−0.856194	−0.428097	+	0.903733i	$0.640816\pi$
$954$	0	0
$955$	31.1978	1.00954
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.1593	−0.780143
$960$	0	0
$961$	−24.3004	−0.783883
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−25.0283	−0.805690
$966$	0	0
$967$	22.2953	0.716968	0.358484	−	0.933536i	$-0.383294\pi$
$967$	22.2953	0.716968	0.358484	+	0.933536i	$0.383294\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−30.6167	−0.982536	−0.491268	−	0.871008i	$-0.663466\pi$
$971$	−30.6167	−0.982536	−0.491268	+	0.871008i	$0.663466\pi$
$972$	0	0
$973$	−4.16435	−0.133503
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.1593	0.868901	0.434451	−	0.900696i	$-0.356943\pi$
$977$	27.1593	0.868901	0.434451	+	0.900696i	$0.356943\pi$
$978$	0	0
$979$	−4.74171	−0.151546
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36.3128	−1.15820	−0.579098	−	0.815258i	$-0.696595\pi$
$983$	−36.3128	−1.15820	−0.579098	+	0.815258i	$0.696595\pi$
$984$	0	0
$985$	74.9206	2.38717
$986$	0	0
$987$	0	0
$988$	0	0
$989$	84.8145	2.69694
$990$	0	0
$991$	−25.1671	−0.799459	−0.399730	−	0.916633i	$-0.630896\pi$
$991$	−25.1671	−0.799459	−0.399730	+	0.916633i	$0.630896\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.118548	−0.00375822
$996$	0	0
$997$	−31.3360	−0.992420	−0.496210	−	0.868202i	$-0.665275\pi$
$997$	−31.3360	−0.992420	−0.496210	+	0.868202i	$0.665275\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.bp.1.3		3
3.2	odd	2		8208.2.a.bf.1.1		3
4.3	odd	2		513.2.a.f.1.1	yes	3
12.11	even	2		513.2.a.e.1.3	✓	3
76.75	even	2		9747.2.a.x.1.3		3
228.227	odd	2		9747.2.a.ba.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.e.1.3	✓	3	12.11	even	2
513.2.a.f.1.1	yes	3	4.3	odd	2
8208.2.a.bf.1.1		3	3.2	odd	2
8208.2.a.bp.1.3		3	1.1	even	1	trivial
9747.2.a.x.1.3		3	76.75	even	2
9747.2.a.ba.1.1		3	228.227	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)

\(f(q)\)	\(=\)	\(q+3.69963 q^{5} +1.58836 q^{7} -4.58836 q^{11} +1.58836 q^{13} -5.57598 q^{17} +1.00000 q^{19} -9.09888 q^{23} +8.68725 q^{25} -7.65383 q^{29} +2.58836 q^{31} +5.87636 q^{35} +0.287992 q^{37} +3.69963 q^{41} -9.32141 q^{43} -9.00000 q^{47} -4.47710 q^{49} +8.24219 q^{53} -16.9752 q^{55} -7.46472 q^{59} +0.189108 q^{61} +5.87636 q^{65} -8.08650 q^{67} +6.21015 q^{71} -1.22253 q^{73} -7.28799 q^{77} +5.90978 q^{79} +2.84431 q^{83} -20.6291 q^{85} +1.03342 q^{89} +2.52290 q^{91} +3.69963 q^{95} -15.4647 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{5} - q^{7} - 8 q^{11} - q^{13} + 7 q^{17} + 3 q^{19} - 9 q^{23} + 2 q^{25} - 6 q^{29} + 2 q^{31} - 11 q^{37} + 5 q^{41} - 9 q^{43} - 27 q^{47} - 8 q^{49} + 2 q^{53} - 15 q^{55} + q^{59} + 7 q^{61}+ \cdots - 23 q^{97}+O(q^{100}) \) 3 * q + 5 * q^5 - q^7 - 8 * q^11 - q^13 + 7 * q^17 + 3 * q^19 - 9 * q^23 + 2 * q^25 - 6 * q^29 + 2 * q^31 - 11 * q^37 + 5 * q^41 - 9 * q^43 - 27 * q^47 - 8 * q^49 + 2 * q^53 - 15 * q^55 + q^59 + 7 * q^61 + 12 * q^67 - 3 * q^73 - 10 * q^77 - 7 * q^79 - 5 * q^83 - 9 * q^85 - 4 * q^89 + 13 * q^91 + 5 * q^95 - 23 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more