LMFDB - Embedded newform 8208.2.a.cb.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8208,2,Mod(1,8208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8208.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			1.5
Root			$-0.757877$ of defining polynomial
Character	$\chi$	$=$	8208.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83464 q^{5} -1.39043 q^{7} +5.50239 q^{11} +3.03519 q^{13} -3.51575 q^{17} +1.00000 q^{19} +7.65070 q^{23} -1.63409 q^{25} +5.01814 q^{29} +1.12532 q^{31} -2.55094 q^{35} +2.12532 q^{37} -1.11312 q^{41} +6.57915 q^{43} +8.55248 q^{47} -5.06670 q^{49} -4.63256 q^{53} +10.0949 q^{55} -4.38559 q^{59} -12.9110 q^{61} +5.56848 q^{65} -10.9110 q^{67} +6.63409 q^{71} +5.02452 q^{73} -7.65070 q^{77} +12.8895 q^{79} +4.40263 q^{83} -6.45015 q^{85} -1.28216 q^{89} -4.22023 q^{91} +1.83464 q^{95} +7.94138 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{5} - q^{7} + 2 q^{11} + 3 q^{13} - 8 q^{17} + 5 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} + 2 q^{31} + 9 q^{35} + 7 q^{37} - 7 q^{41} + 10 q^{47} + 6 q^{49} - 22 q^{53} + 8 q^{55} + 12 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) $ 5 * q - 3 * q^5 - q^7 + 2 * q^11 + 3 * q^13 - 8 * q^17 + 5 * q^19 + 2 * q^23 + 4 * q^25 - 10 * q^29 + 2 * q^31 + 9 * q^35 + 7 * q^37 - 7 * q^41 + 10 * q^47 + 6 * q^49 - 22 * q^53 + 8 * q^55 + 12 * q^59 + 2 * q^61 - 11 * q^65 + 12 * q^67 + 21 * q^71 + 7 * q^73 - 2 * q^77 - 6 * q^79 + 11 * q^83 + 4 * q^85 - 27 * q^89 + 25 * q^91 - 3 * q^95 + 12 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.83464	0.820476	0.410238	−	0.911978i	$-0.365446\pi$
$5$	1.83464	0.820476	0.410238	+	0.911978i	$0.365446\pi$
$6$	0	0
$7$	−1.39043	−0.525534	−0.262767	−	0.964859i	$-0.584635\pi$
$7$	−1.39043	−0.525534	−0.262767	+	0.964859i	$0.584635\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.50239	1.65903	0.829516	−	0.558483i	$-0.188616\pi$
$11$	5.50239	1.65903	0.829516	+	0.558483i	$0.188616\pi$
$12$	0	0
$13$	3.03519	0.841810	0.420905	−	0.907105i	$-0.361713\pi$
$13$	3.03519	0.841810	0.420905	+	0.907105i	$0.361713\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.51575	−0.852696	−0.426348	−	0.904559i	$-0.640200\pi$
$17$	−3.51575	−0.852696	−0.426348	+	0.904559i	$0.640200\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.65070	1.59528	0.797640	−	0.603134i	$-0.206081\pi$
$23$	7.65070	1.59528	0.797640	+	0.603134i	$0.206081\pi$
$24$	0	0
$25$	−1.63409	−0.326818
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.01814	0.931845	0.465923	−	0.884825i	$-0.345722\pi$
$29$	5.01814	0.931845	0.465923	+	0.884825i	$0.345722\pi$
$30$	0	0
$31$	1.12532	0.202114	0.101057	−	0.994881i	$-0.467778\pi$
$31$	1.12532	0.202114	0.101057	+	0.994881i	$0.467778\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.55094	−0.431188
$36$	0	0
$37$	2.12532	0.349401	0.174700	−	0.984622i	$-0.444104\pi$
$37$	2.12532	0.349401	0.174700	+	0.984622i	$0.444104\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.11312	−0.173840	−0.0869200	−	0.996215i	$-0.527702\pi$
$41$	−1.11312	−0.173840	−0.0869200	+	0.996215i	$0.527702\pi$
$42$	0	0
$43$	6.57915	1.00331	0.501656	−	0.865067i	$-0.332724\pi$
$43$	6.57915	1.00331	0.501656	+	0.865067i	$0.332724\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.55248	1.24751	0.623754	−	0.781621i	$-0.285607\pi$
$47$	8.55248	1.24751	0.623754	+	0.781621i	$0.285607\pi$
$48$	0	0
$49$	−5.06670	−0.723814
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.63256	−0.636330	−0.318165	−	0.948035i	$-0.603067\pi$
$53$	−4.63256	−0.636330	−0.318165	+	0.948035i	$0.603067\pi$
$54$	0	0
$55$	10.0949	1.36120
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.38559	−0.570955	−0.285477	−	0.958385i	$-0.592152\pi$
$59$	−4.38559	−0.570955	−0.285477	+	0.958385i	$0.592152\pi$
$60$	0	0
$61$	−12.9110	−1.65308	−0.826539	−	0.562879i	$-0.809694\pi$
$61$	−12.9110	−1.65308	−0.826539	+	0.562879i	$0.809694\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.56848	0.690686
$66$	0	0
$67$	−10.9110	−1.33299	−0.666493	−	0.745511i	$-0.732205\pi$
$67$	−10.9110	−1.33299	−0.666493	+	0.745511i	$0.732205\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.63409	0.787322	0.393661	−	0.919256i	$-0.371208\pi$
$71$	6.63409	0.787322	0.393661	+	0.919256i	$0.371208\pi$
$72$	0	0
$73$	5.02452	0.588076	0.294038	−	0.955794i	$-0.405001\pi$
$73$	5.02452	0.588076	0.294038	+	0.955794i	$0.405001\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−7.65070	−0.871878
$78$	0	0
$79$	12.8895	1.45018	0.725092	−	0.688653i	$-0.241797\pi$
$79$	12.8895	1.45018	0.725092	+	0.688653i	$0.241797\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.40263	0.483252	0.241626	−	0.970369i	$-0.422319\pi$
$83$	4.40263	0.483252	0.241626	+	0.970369i	$0.422319\pi$
$84$	0	0
$85$	−6.45015	−0.699617
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.28216	−0.135909	−0.0679544	−	0.997688i	$-0.521647\pi$
$89$	−1.28216	−0.135909	−0.0679544	+	0.997688i	$0.521647\pi$
$90$	0	0
$91$	−4.22023	−0.442400
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.83464	0.188230
$96$	0	0
$97$	7.94138	0.806325	0.403162	−	0.915128i	$-0.367911\pi$
$97$	7.94138	0.806325	0.403162	+	0.915128i	$0.367911\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.803133	−0.0799147	−0.0399574	−	0.999201i	$-0.512722\pi$
$101$	−0.803133	−0.0799147	−0.0399574	+	0.999201i	$0.512722\pi$
$102$	0	0
$103$	1.90288	0.187497	0.0937484	−	0.995596i	$-0.470115\pi$
$103$	1.90288	0.187497	0.0937484	+	0.995596i	$0.470115\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.58509	−0.636605	−0.318302	−	0.947989i	$-0.603113\pi$
$107$	−6.58509	−0.636605	−0.318302	+	0.947989i	$0.603113\pi$
$108$	0	0
$109$	7.54396	0.722580	0.361290	−	0.932453i	$-0.382336\pi$
$109$	7.54396	0.722580	0.361290	+	0.932453i	$0.382336\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.86829	0.269826	0.134913	−	0.990857i	$-0.456924\pi$
$113$	2.86829	0.269826	0.134913	+	0.990857i	$0.456924\pi$
$114$	0	0
$115$	14.0363	1.30889
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.88842	0.448121
$120$	0	0
$121$	19.2763	1.75239
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1712	−1.08862
$126$	0	0
$127$	−15.8591	−1.40727	−0.703633	−	0.710563i	$-0.748440\pi$
$127$	−15.8591	−1.40727	−0.703633	+	0.710563i	$0.748440\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.76903	−0.591413	−0.295707	−	0.955279i	$-0.595555\pi$
$131$	−6.76903	−0.591413	−0.295707	+	0.955279i	$0.595555\pi$
$132$	0	0
$133$	−1.39043	−0.120566
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.18504	0.272116	0.136058	−	0.990701i	$-0.456557\pi$
$137$	3.18504	0.272116	0.136058	+	0.990701i	$0.456557\pi$
$138$	0	0
$139$	−9.93917	−0.843029	−0.421515	−	0.906822i	$-0.638501\pi$
$139$	−9.93917	−0.843029	−0.421515	+	0.906822i	$0.638501\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.7008	1.39659
$144$	0	0
$145$	9.20649	0.764557
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.52251	−0.124729	−0.0623644	−	0.998053i	$-0.519864\pi$
$149$	−1.52251	−0.124729	−0.0623644	+	0.998053i	$0.519864\pi$
$150$	0	0
$151$	−8.66560	−0.705197	−0.352598	−	0.935775i	$-0.614702\pi$
$151$	−8.66560	−0.705197	−0.352598	+	0.935775i	$0.614702\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.06456	0.165830
$156$	0	0
$157$	3.45911	0.276067	0.138034	−	0.990428i	$-0.455922\pi$
$157$	3.45911	0.276067	0.138034	+	0.990428i	$0.455922\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−10.6378	−0.838374
$162$	0	0
$163$	17.5031	1.37095	0.685475	−	0.728097i	$-0.259595\pi$
$163$	17.5031	1.37095	0.685475	+	0.728097i	$0.259595\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.8208	1.45640	0.728200	−	0.685365i	$-0.240357\pi$
$167$	18.8208	1.45640	0.728200	+	0.685365i	$0.240357\pi$
$168$	0	0
$169$	−3.78762	−0.291355
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.4693	−1.10008	−0.550042	−	0.835137i	$-0.685388\pi$
$173$	−14.4693	−1.10008	−0.550042	+	0.835137i	$0.685388\pi$
$174$	0	0
$175$	2.27209	0.171754
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.4560	1.52895	0.764475	−	0.644653i	$-0.222998\pi$
$179$	20.4560	1.52895	0.764475	+	0.644653i	$0.222998\pi$
$180$	0	0
$181$	8.58283	0.637957	0.318978	−	0.947762i	$-0.396660\pi$
$181$	8.58283	0.637957	0.318978	+	0.947762i	$0.396660\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.89920	0.286675
$186$	0	0
$187$	−19.3450	−1.41465
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.7910	−1.79381	−0.896907	−	0.442219i	$-0.854192\pi$
$191$	−24.7910	−1.79381	−0.896907	+	0.442219i	$0.854192\pi$
$192$	0	0
$193$	−8.43738	−0.607336	−0.303668	−	0.952778i	$-0.598211\pi$
$193$	−8.43738	−0.607336	−0.303668	+	0.952778i	$0.598211\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.5205	−1.74702	−0.873508	−	0.486810i	$-0.838160\pi$
$197$	−24.5205	−1.74702	−0.873508	+	0.486810i	$0.838160\pi$
$198$	0	0
$199$	13.7292	0.973239	0.486619	−	0.873614i	$-0.338230\pi$
$199$	13.7292	0.973239	0.486619	+	0.873614i	$0.338230\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.97738	−0.489716
$204$	0	0
$205$	−2.04217	−0.142632
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.50239	0.380608
$210$	0	0
$211$	23.6986	1.63148	0.815739	−	0.578421i	$-0.196331\pi$
$211$	23.6986	1.63148	0.815739	+	0.578421i	$0.196331\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.0704	0.823193
$216$	0	0
$217$	−1.56468	−0.106218
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.6710	−0.717808
$222$	0	0
$223$	−22.4756	−1.50508	−0.752540	−	0.658546i	$-0.771172\pi$
$223$	−22.4756	−1.50508	−0.752540	+	0.658546i	$0.771172\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.08485	0.470238	0.235119	−	0.971967i	$-0.424452\pi$
$227$	7.08485	0.470238	0.235119	+	0.971967i	$0.424452\pi$
$228$	0	0
$229$	13.4411	0.888210	0.444105	−	0.895975i	$-0.353522\pi$
$229$	13.4411	0.888210	0.444105	+	0.895975i	$0.353522\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.56755	0.102693	0.0513467	−	0.998681i	$-0.483649\pi$
$233$	1.56755	0.102693	0.0513467	+	0.998681i	$0.483649\pi$
$234$	0	0
$235$	15.6907	1.02355
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.4940	−1.06691	−0.533453	−	0.845830i	$-0.679106\pi$
$239$	−16.4940	−1.06691	−0.533453	+	0.845830i	$0.679106\pi$
$240$	0	0
$241$	30.5887	1.97039	0.985196	−	0.171434i	$-0.0548400\pi$
$241$	30.5887	1.97039	0.985196	+	0.171434i	$0.0548400\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−9.29557	−0.593872
$246$	0	0
$247$	3.03519	0.193125
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.94697	−0.186011	−0.0930055	−	0.995666i	$-0.529647\pi$
$251$	−2.94697	−0.186011	−0.0930055	+	0.995666i	$0.529647\pi$
$252$	0	0
$253$	42.0971	2.64662
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.5371	0.844423	0.422212	−	0.906497i	$-0.361254\pi$
$257$	13.5371	0.844423	0.422212	+	0.906497i	$0.361254\pi$
$258$	0	0
$259$	−2.95512	−0.183622
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.13539	−0.0700110	−0.0350055	−	0.999387i	$-0.511145\pi$
$263$	−1.13539	−0.0700110	−0.0350055	+	0.999387i	$0.511145\pi$
$264$	0	0
$265$	−8.49908	−0.522094
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.42679	−0.574761	−0.287381	−	0.957816i	$-0.592784\pi$
$269$	−9.42679	−0.574761	−0.287381	+	0.957816i	$0.592784\pi$
$270$	0	0
$271$	−0.275168	−0.0167152	−0.00835762	−	0.999965i	$-0.502660\pi$
$271$	−0.275168	−0.0167152	−0.00835762	+	0.999965i	$0.502660\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.99141	−0.542202
$276$	0	0
$277$	12.9355	0.777218	0.388609	−	0.921403i	$-0.372956\pi$
$277$	12.9355	0.777218	0.388609	+	0.921403i	$0.372956\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.1384	−0.903079	−0.451539	−	0.892251i	$-0.649125\pi$
$281$	−15.1384	−0.903079	−0.451539	+	0.892251i	$0.649125\pi$
$282$	0	0
$283$	25.7129	1.52847	0.764237	−	0.644935i	$-0.223116\pi$
$283$	25.7129	1.52847	0.764237	+	0.644935i	$0.223116\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.54772	0.0913588
$288$	0	0
$289$	−4.63947	−0.272910
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.12495	0.416244	0.208122	−	0.978103i	$-0.433265\pi$
$293$	7.12495	0.416244	0.208122	+	0.978103i	$0.433265\pi$
$294$	0	0
$295$	−8.04598	−0.468455
$296$	0	0
$297$	0	0
$298$	0	0
$299$	23.2213	1.34292
$300$	0	0
$301$	−9.14786	−0.527274
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−23.6870	−1.35631
$306$	0	0
$307$	−0.252343	−0.0144020	−0.00720099	−	0.999974i	$-0.502292\pi$
$307$	−0.252343	−0.0144020	−0.00720099	+	0.999974i	$0.502292\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.2492	0.694586	0.347293	−	0.937757i	$-0.387101\pi$
$311$	12.2492	0.694586	0.347293	+	0.937757i	$0.387101\pi$
$312$	0	0
$313$	−7.83348	−0.442774	−0.221387	−	0.975186i	$-0.571058\pi$
$313$	−7.83348	−0.442774	−0.221387	+	0.975186i	$0.571058\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.9788	−0.841294	−0.420647	−	0.907224i	$-0.638197\pi$
$317$	−14.9788	−0.841294	−0.420647	+	0.907224i	$0.638197\pi$
$318$	0	0
$319$	27.6117	1.54596
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.51575	−0.195622
$324$	0	0
$325$	−4.95978	−0.275119
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.8916	−0.655608
$330$	0	0
$331$	30.3056	1.66574	0.832872	−	0.553466i	$-0.186695\pi$
$331$	30.3056	1.66574	0.832872	+	0.553466i	$0.186695\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−20.0177	−1.09368
$336$	0	0
$337$	31.4934	1.71556	0.857778	−	0.514021i	$-0.171845\pi$
$337$	31.4934	1.71556	0.857778	+	0.514021i	$0.171845\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.19196	0.335313
$342$	0	0
$343$	16.7779	0.905923
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.0011	−1.44950	−0.724748	−	0.689014i	$-0.758044\pi$
$347$	−27.0011	−1.44950	−0.724748	+	0.689014i	$0.758044\pi$
$348$	0	0
$349$	7.30337	0.390941	0.195470	−	0.980710i	$-0.437377\pi$
$349$	7.30337	0.390941	0.195470	+	0.980710i	$0.437377\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.8560	0.737481	0.368741	−	0.929532i	$-0.379789\pi$
$353$	13.8560	0.737481	0.368741	+	0.929532i	$0.379789\pi$
$354$	0	0
$355$	12.1712	0.645979
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.4880	1.08131	0.540657	−	0.841243i	$-0.318176\pi$
$359$	20.4880	1.08131	0.540657	+	0.841243i	$0.318176\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.21820	0.482503
$366$	0	0
$367$	18.3025	0.955382	0.477691	−	0.878528i	$-0.341474\pi$
$367$	18.3025	0.955382	0.477691	+	0.878528i	$0.341474\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.44125	0.334413
$372$	0	0
$373$	−31.2895	−1.62011	−0.810054	−	0.586355i	$-0.800562\pi$
$373$	−31.2895	−1.62011	−0.810054	+	0.586355i	$0.800562\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.2310	0.784437
$378$	0	0
$379$	16.2628	0.835363	0.417681	−	0.908594i	$-0.362843\pi$
$379$	16.2628	0.835363	0.417681	+	0.908594i	$0.362843\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.49007	−0.229432	−0.114716	−	0.993398i	$-0.536596\pi$
$383$	−4.49007	−0.229432	−0.114716	+	0.993398i	$0.536596\pi$
$384$	0	0
$385$	−14.0363	−0.715355
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.0935	0.917376	0.458688	−	0.888597i	$-0.348320\pi$
$389$	18.0935	0.917376	0.458688	+	0.888597i	$0.348320\pi$
$390$	0	0
$391$	−26.8980	−1.36029
$392$	0	0
$393$	0	0
$394$	0	0
$395$	23.6476	1.18984
$396$	0	0
$397$	−27.1125	−1.36074	−0.680369	−	0.732869i	$-0.738181\pi$
$397$	−27.1125	−1.36074	−0.680369	+	0.732869i	$0.738181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.5517	−1.72543	−0.862716	−	0.505689i	$-0.831238\pi$
$401$	−34.5517	−1.72543	−0.862716	+	0.505689i	$0.831238\pi$
$402$	0	0
$403$	3.41557	0.170141
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.6943	0.579667
$408$	0	0
$409$	35.6446	1.76251	0.881256	−	0.472639i	$-0.156698\pi$
$409$	35.6446	1.76251	0.881256	+	0.472639i	$0.156698\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.09786	0.300056
$414$	0	0
$415$	8.07726	0.396497
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−18.7892	−0.917910	−0.458955	−	0.888459i	$-0.651776\pi$
$419$	−18.7892	−0.917910	−0.458955	+	0.888459i	$0.651776\pi$
$420$	0	0
$421$	−14.5066	−0.707006	−0.353503	−	0.935433i	$-0.615010\pi$
$421$	−14.5066	−0.707006	−0.353503	+	0.935433i	$0.615010\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.74507	0.278677
$426$	0	0
$427$	17.9518	0.868749
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.0123	0.530444	0.265222	−	0.964187i	$-0.414555\pi$
$431$	11.0123	0.530444	0.265222	+	0.964187i	$0.414555\pi$
$432$	0	0
$433$	15.4541	0.742675	0.371337	−	0.928498i	$-0.378899\pi$
$433$	15.4541	0.742675	0.371337	+	0.928498i	$0.378899\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.65070	0.365982
$438$	0	0
$439$	−27.1350	−1.29508	−0.647541	−	0.762030i	$-0.724203\pi$
$439$	−27.1350	−1.29508	−0.647541	+	0.762030i	$0.724203\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.17073	0.245669	0.122834	−	0.992427i	$-0.460802\pi$
$443$	5.17073	0.245669	0.122834	+	0.992427i	$0.460802\pi$
$444$	0	0
$445$	−2.35230	−0.111510
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.8147	−1.12388	−0.561941	−	0.827177i	$-0.689945\pi$
$449$	−23.8147	−1.12388	−0.561941	+	0.827177i	$0.689945\pi$
$450$	0	0
$451$	−6.12481	−0.288406
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.74260	−0.362979
$456$	0	0
$457$	−34.6929	−1.62287	−0.811433	−	0.584445i	$-0.801312\pi$
$457$	−34.6929	−1.62287	−0.811433	+	0.584445i	$0.801312\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.7732	−0.548330	−0.274165	−	0.961683i	$-0.588402\pi$
$461$	−11.7732	−0.548330	−0.274165	+	0.961683i	$0.588402\pi$
$462$	0	0
$463$	14.5634	0.676820	0.338410	−	0.940999i	$-0.390111\pi$
$463$	14.5634	0.676820	0.338410	+	0.940999i	$0.390111\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	42.0318	1.94500	0.972500	−	0.232903i	$-0.0748224\pi$
$467$	42.0318	1.94500	0.972500	+	0.232903i	$0.0748224\pi$
$468$	0	0
$469$	15.1709	0.700529
$470$	0	0
$471$	0	0
$472$	0	0
$473$	36.2010	1.66453
$474$	0	0
$475$	−1.63409	−0.0749773
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.40572	−0.109920	−0.0549600	−	0.998489i	$-0.517503\pi$
$479$	−2.40572	−0.109920	−0.0549600	+	0.998489i	$0.517503\pi$
$480$	0	0
$481$	6.45076	0.294129
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.5696	0.661570
$486$	0	0
$487$	−0.934621	−0.0423517	−0.0211759	−	0.999776i	$-0.506741\pi$
$487$	−0.934621	−0.0423517	−0.0211759	+	0.999776i	$0.506741\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−30.7216	−1.38644	−0.693222	−	0.720724i	$-0.743810\pi$
$491$	−30.7216	−1.38644	−0.693222	+	0.720724i	$0.743810\pi$
$492$	0	0
$493$	−17.6425	−0.794580
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.22426	−0.413764
$498$	0	0
$499$	1.88364	0.0843235	0.0421618	−	0.999111i	$-0.486576\pi$
$499$	1.88364	0.0843235	0.0421618	+	0.999111i	$0.486576\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.9739	0.890591	0.445295	−	0.895384i	$-0.353099\pi$
$503$	19.9739	0.890591	0.445295	+	0.895384i	$0.353099\pi$
$504$	0	0
$505$	−1.47346	−0.0655682
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.5836	1.44424	0.722121	−	0.691766i	$-0.243167\pi$
$509$	32.5836	1.44424	0.722121	+	0.691766i	$0.243167\pi$
$510$	0	0
$511$	−6.98626	−0.309054
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.49111	0.153837
$516$	0	0
$517$	47.0591	2.06965
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.4277	1.50831	0.754153	−	0.656699i	$-0.228048\pi$
$521$	34.4277	1.50831	0.754153	+	0.656699i	$0.228048\pi$
$522$	0	0
$523$	26.2976	1.14991	0.574957	−	0.818184i	$-0.305019\pi$
$523$	26.2976	1.14991	0.574957	+	0.818184i	$0.305019\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.95635	−0.172341
$528$	0	0
$529$	35.5331	1.54492
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.37853	−0.146340
$534$	0	0
$535$	−12.0813	−0.522319
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−27.8789	−1.20083
$540$	0	0
$541$	16.0566	0.690329	0.345164	−	0.938542i	$-0.387823\pi$
$541$	16.0566	0.690329	0.345164	+	0.938542i	$0.387823\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.8405	0.592860
$546$	0	0
$547$	6.77411	0.289640	0.144820	−	0.989458i	$-0.453740\pi$
$547$	6.77411	0.289640	0.144820	+	0.989458i	$0.453740\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.01814	0.213780
$552$	0	0
$553$	−17.9220	−0.762120
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.95100	0.0826664	0.0413332	−	0.999145i	$-0.486839\pi$
$557$	1.95100	0.0826664	0.0413332	+	0.999145i	$0.486839\pi$
$558$	0	0
$559$	19.9690	0.844598
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−27.5408	−1.16071	−0.580353	−	0.814365i	$-0.697086\pi$
$563$	−27.5408	−1.16071	−0.580353	+	0.814365i	$0.697086\pi$
$564$	0	0
$565$	5.26229	0.221386
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.59564	−0.0668925	−0.0334463	−	0.999441i	$-0.510648\pi$
$569$	−1.59564	−0.0668925	−0.0334463	+	0.999441i	$0.510648\pi$
$570$	0	0
$571$	−40.8972	−1.71149	−0.855747	−	0.517394i	$-0.826902\pi$
$571$	−40.8972	−1.71149	−0.855747	+	0.517394i	$0.826902\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.5019	−0.521367
$576$	0	0
$577$	10.3939	0.432703	0.216352	−	0.976316i	$-0.430584\pi$
$577$	10.3939	0.432703	0.216352	+	0.976316i	$0.430584\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.12157	−0.253965
$582$	0	0
$583$	−25.4901	−1.05569
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.23833	−0.133660	−0.0668301	−	0.997764i	$-0.521289\pi$
$587$	−3.23833	−0.133660	−0.0668301	+	0.997764i	$0.521289\pi$
$588$	0	0
$589$	1.12532	0.0463681
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.5977	−1.09224	−0.546118	−	0.837708i	$-0.683895\pi$
$593$	−26.5977	−1.09224	−0.546118	+	0.837708i	$0.683895\pi$
$594$	0	0
$595$	8.96849	0.367672
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.6241	1.53728	0.768639	−	0.639682i	$-0.220934\pi$
$599$	37.6241	1.53728	0.768639	+	0.639682i	$0.220934\pi$
$600$	0	0
$601$	−15.4200	−0.628993	−0.314497	−	0.949259i	$-0.601836\pi$
$601$	−15.4200	−0.628993	−0.314497	+	0.949259i	$0.601836\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	35.3650	1.43779
$606$	0	0
$607$	−28.8146	−1.16955	−0.584774	−	0.811196i	$-0.698817\pi$
$607$	−28.8146	−1.16955	−0.584774	+	0.811196i	$0.698817\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	25.9584	1.05016
$612$	0	0
$613$	23.0323	0.930264	0.465132	−	0.885241i	$-0.346007\pi$
$613$	23.0323	0.930264	0.465132	+	0.885241i	$0.346007\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.4118	−1.74769	−0.873846	−	0.486202i	$-0.838382\pi$
$617$	−43.4118	−1.74769	−0.873846	+	0.486202i	$0.838382\pi$
$618$	0	0
$619$	−19.8058	−0.796060	−0.398030	−	0.917372i	$-0.630306\pi$
$619$	−19.8058	−0.796060	−0.398030	+	0.917372i	$0.630306\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.78276	0.0714246
$624$	0	0
$625$	−14.1593	−0.566371
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−7.47211	−0.297932
$630$	0	0
$631$	−38.1149	−1.51733	−0.758664	−	0.651482i	$-0.774148\pi$
$631$	−38.1149	−1.51733	−0.758664	+	0.651482i	$0.774148\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.0957	−1.15463
$636$	0	0
$637$	−15.3784	−0.609314
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.8744	0.666500	0.333250	−	0.942838i	$-0.391855\pi$
$641$	16.8744	0.666500	0.333250	+	0.942838i	$0.391855\pi$
$642$	0	0
$643$	34.3788	1.35577	0.677883	−	0.735170i	$-0.262898\pi$
$643$	34.3788	1.35577	0.677883	+	0.735170i	$0.262898\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.8998	−1.17548	−0.587742	−	0.809048i	$-0.699983\pi$
$647$	−29.8998	−1.17548	−0.587742	+	0.809048i	$0.699983\pi$
$648$	0	0
$649$	−24.1312	−0.947232
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.8855	0.973844	0.486922	−	0.873445i	$-0.338120\pi$
$653$	24.8855	0.973844	0.486922	+	0.873445i	$0.338120\pi$
$654$	0	0
$655$	−12.4187	−0.485241
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−42.7784	−1.66641	−0.833205	−	0.552964i	$-0.813497\pi$
$659$	−42.7784	−1.66641	−0.833205	+	0.552964i	$0.813497\pi$
$660$	0	0
$661$	2.08254	0.0810014	0.0405007	−	0.999180i	$-0.487105\pi$
$661$	2.08254	0.0810014	0.0405007	+	0.999180i	$0.487105\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.55094	−0.0989214
$666$	0	0
$667$	38.3923	1.48655
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−71.0411	−2.74251
$672$	0	0
$673$	4.64130	0.178909	0.0894546	−	0.995991i	$-0.471488\pi$
$673$	4.64130	0.178909	0.0894546	+	0.995991i	$0.471488\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.90229	0.342143	0.171071	−	0.985259i	$-0.445277\pi$
$677$	8.90229	0.342143	0.171071	+	0.985259i	$0.445277\pi$
$678$	0	0
$679$	−11.0419	−0.423751
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.63882	−0.292291	−0.146146	−	0.989263i	$-0.546687\pi$
$683$	−7.63882	−0.292291	−0.146146	+	0.989263i	$0.546687\pi$
$684$	0	0
$685$	5.84340	0.223265
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.0607	−0.535669
$690$	0	0
$691$	−12.5126	−0.476001	−0.238001	−	0.971265i	$-0.576492\pi$
$691$	−12.5126	−0.476001	−0.238001	+	0.971265i	$0.576492\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.2348	−0.691685
$696$	0	0
$697$	3.91345	0.148233
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.5966	1.19339	0.596694	−	0.802469i	$-0.296481\pi$
$701$	31.5966	1.19339	0.596694	+	0.802469i	$0.296481\pi$
$702$	0	0
$703$	2.12532	0.0801580
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.11670	0.0419979
$708$	0	0
$709$	−45.1176	−1.69443	−0.847213	−	0.531253i	$-0.821721\pi$
$709$	−45.1176	−1.69443	−0.847213	+	0.531253i	$0.821721\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.60949	0.322428
$714$	0	0
$715$	30.6400	1.14587
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.42506	0.0531457	0.0265728	−	0.999647i	$-0.491541\pi$
$719$	1.42506	0.0531457	0.0265728	+	0.999647i	$0.491541\pi$
$720$	0	0
$721$	−2.64583	−0.0985360
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.20010	−0.304544
$726$	0	0
$727$	8.29368	0.307596	0.153798	−	0.988102i	$-0.450850\pi$
$727$	8.29368	0.307596	0.153798	+	0.988102i	$0.450850\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23.1307	−0.855519
$732$	0	0
$733$	37.9477	1.40163	0.700816	−	0.713343i	$-0.252820\pi$
$733$	37.9477	1.40163	0.700816	+	0.713343i	$0.252820\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−60.0363	−2.21147
$738$	0	0
$739$	51.7664	1.90426	0.952128	−	0.305699i	$-0.0988900\pi$
$739$	51.7664	1.90426	0.952128	+	0.305699i	$0.0988900\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−29.1383	−1.06898	−0.534491	−	0.845174i	$-0.679497\pi$
$743$	−29.1383	−1.06898	−0.534491	+	0.845174i	$0.679497\pi$
$744$	0	0
$745$	−2.79326	−0.102337
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.15612	0.334557
$750$	0	0
$751$	−15.0775	−0.550185	−0.275093	−	0.961418i	$-0.588709\pi$
$751$	−15.0775	−0.550185	−0.275093	+	0.961418i	$0.588709\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.8983	−0.578597
$756$	0	0
$757$	27.9338	1.01527	0.507635	−	0.861572i	$-0.330520\pi$
$757$	27.9338	1.01527	0.507635	+	0.861572i	$0.330520\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.395369	0.0143321	0.00716606	−	0.999974i	$-0.497719\pi$
$761$	0.395369	0.0143321	0.00716606	+	0.999974i	$0.497719\pi$
$762$	0	0
$763$	−10.4894	−0.379741
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.3111	−0.480635
$768$	0	0
$769$	37.6078	1.35617	0.678087	−	0.734982i	$-0.262809\pi$
$769$	37.6078	1.35617	0.678087	+	0.734982i	$0.262809\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.0830937	0.00298867	0.00149434	−	0.999999i	$-0.499524\pi$
$773$	0.0830937	0.00298867	0.00149434	+	0.999999i	$0.499524\pi$
$774$	0	0
$775$	−1.83888	−0.0660545
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.11312	−0.0398816
$780$	0	0
$781$	36.5033	1.30619
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.34623	0.226507
$786$	0	0
$787$	8.06032	0.287319	0.143660	−	0.989627i	$-0.454113\pi$
$787$	8.06032	0.287319	0.143660	+	0.989627i	$0.454113\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.98817	−0.141803
$792$	0	0
$793$	−39.1872	−1.39158
$794$	0	0
$795$	0	0
$796$	0	0
$797$	38.2685	1.35554	0.677770	−	0.735274i	$-0.262947\pi$
$797$	38.2685	1.35554	0.677770	+	0.735274i	$0.262947\pi$
$798$	0	0
$799$	−30.0684	−1.06374
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27.6469	0.975637
$804$	0	0
$805$	−19.5165	−0.687866
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.55617	−0.160186	−0.0800932	−	0.996787i	$-0.525522\pi$
$809$	−4.55617	−0.160186	−0.0800932	+	0.996787i	$0.525522\pi$
$810$	0	0
$811$	15.3819	0.540130	0.270065	−	0.962842i	$-0.412955\pi$
$811$	15.3819	0.540130	0.270065	+	0.962842i	$0.412955\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	32.1119	1.12483
$816$	0	0
$817$	6.57915	0.230175
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−33.6469	−1.17428	−0.587142	−	0.809484i	$-0.699747\pi$
$821$	−33.6469	−1.17428	−0.587142	+	0.809484i	$0.699747\pi$
$822$	0	0
$823$	8.74381	0.304790	0.152395	−	0.988320i	$-0.451301\pi$
$823$	8.74381	0.304790	0.152395	+	0.988320i	$0.451301\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.05746	0.245412	0.122706	−	0.992443i	$-0.460843\pi$
$827$	7.05746	0.245412	0.122706	+	0.992443i	$0.460843\pi$
$828$	0	0
$829$	−3.22500	−0.112009	−0.0560044	−	0.998431i	$-0.517836\pi$
$829$	−3.22500	−0.112009	−0.0560044	+	0.998431i	$0.517836\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	17.8133	0.617193
$834$	0	0
$835$	34.5295	1.19494
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.2090	−0.904837	−0.452418	−	0.891806i	$-0.649439\pi$
$839$	−26.2090	−0.904837	−0.452418	+	0.891806i	$0.649439\pi$
$840$	0	0
$841$	−3.81827	−0.131664
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.94892	−0.239050
$846$	0	0
$847$	−26.8023	−0.920939
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16.2602	0.557392
$852$	0	0
$853$	3.21506	0.110082	0.0550408	−	0.998484i	$-0.482471\pi$
$853$	3.21506	0.110082	0.0550408	+	0.998484i	$0.482471\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.8135	0.574337	0.287168	−	0.957880i	$-0.407286\pi$
$857$	16.8135	0.574337	0.287168	+	0.957880i	$0.407286\pi$
$858$	0	0
$859$	−8.42341	−0.287403	−0.143702	−	0.989621i	$-0.545901\pi$
$859$	−8.42341	−0.287403	−0.143702	+	0.989621i	$0.545901\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.2762	1.13274	0.566368	−	0.824152i	$-0.308348\pi$
$863$	33.2762	1.13274	0.566368	+	0.824152i	$0.308348\pi$
$864$	0	0
$865$	−26.5460	−0.902592
$866$	0	0
$867$	0	0
$868$	0	0
$869$	70.9231	2.40590
$870$	0	0
$871$	−33.1168	−1.12212
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16.9232	0.572109
$876$	0	0
$877$	−28.4894	−0.962018	−0.481009	−	0.876716i	$-0.659730\pi$
$877$	−28.4894	−0.962018	−0.481009	+	0.876716i	$0.659730\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−38.3765	−1.29294	−0.646469	−	0.762940i	$-0.723755\pi$
$881$	−38.3765	−1.29294	−0.646469	+	0.762940i	$0.723755\pi$
$882$	0	0
$883$	−28.5747	−0.961616	−0.480808	−	0.876826i	$-0.659657\pi$
$883$	−28.5747	−0.961616	−0.480808	+	0.876826i	$0.659657\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.5713	1.26152	0.630761	−	0.775977i	$-0.282743\pi$
$887$	37.5713	1.26152	0.630761	+	0.775977i	$0.282743\pi$
$888$	0	0
$889$	22.0510	0.739567
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.55248	0.286198
$894$	0	0
$895$	37.5294	1.25447
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.64702	0.188339
$900$	0	0
$901$	16.2869	0.542596
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.7464	0.523429
$906$	0	0
$907$	1.33106	0.0441973	0.0220986	−	0.999756i	$-0.492965\pi$
$907$	1.33106	0.0441973	0.0220986	+	0.999756i	$0.492965\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−26.9449	−0.892724	−0.446362	−	0.894853i	$-0.647281\pi$
$911$	−26.9449	−0.892724	−0.446362	+	0.894853i	$0.647281\pi$
$912$	0	0
$913$	24.2250	0.801730
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.41188	0.310808
$918$	0	0
$919$	12.5931	0.415409	0.207704	−	0.978192i	$-0.433401\pi$
$919$	12.5931	0.415409	0.207704	+	0.978192i	$0.433401\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	20.1357	0.662776
$924$	0	0
$925$	−3.47297	−0.114191
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43.7114	1.43412	0.717062	−	0.697009i	$-0.245486\pi$
$929$	43.7114	1.43412	0.717062	+	0.697009i	$0.245486\pi$
$930$	0	0
$931$	−5.06670	−0.166054
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−35.4912	−1.16069
$936$	0	0
$937$	12.3081	0.402090	0.201045	−	0.979582i	$-0.435566\pi$
$937$	12.3081	0.402090	0.201045	+	0.979582i	$0.435566\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.5969	−0.606241	−0.303120	−	0.952952i	$-0.598028\pi$
$941$	−18.5969	−0.606241	−0.303120	+	0.952952i	$0.598028\pi$
$942$	0	0
$943$	−8.51613	−0.277324
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.1387	−1.36932	−0.684661	−	0.728861i	$-0.740050\pi$
$947$	−42.1387	−1.36932	−0.684661	+	0.728861i	$0.740050\pi$
$948$	0	0
$949$	15.2504	0.495049
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.1155	0.878356	0.439178	−	0.898400i	$-0.355270\pi$
$953$	27.1155	0.878356	0.439178	+	0.898400i	$0.355270\pi$
$954$	0	0
$955$	−45.4826	−1.47178
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−4.42858	−0.143006
$960$	0	0
$961$	−29.7337	−0.959150
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.4796	−0.498305
$966$	0	0
$967$	14.1823	0.456073	0.228036	−	0.973653i	$-0.426769\pi$
$967$	14.1823	0.456073	0.228036	+	0.973653i	$0.426769\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24.7572	0.794496	0.397248	−	0.917711i	$-0.369965\pi$
$971$	24.7572	0.794496	0.397248	+	0.917711i	$0.369965\pi$
$972$	0	0
$973$	13.8197	0.443040
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.3066	1.06557	0.532786	−	0.846250i	$-0.321145\pi$
$977$	33.3066	1.06557	0.532786	+	0.846250i	$0.321145\pi$
$978$	0	0
$979$	−7.05494	−0.225477
$980$	0	0
$981$	0	0
$982$	0	0
$983$	41.0388	1.30893	0.654467	−	0.756091i	$-0.272893\pi$
$983$	41.0388	1.30893	0.654467	+	0.756091i	$0.272893\pi$
$984$	0	0
$985$	−44.9864	−1.43338
$986$	0	0
$987$	0	0
$988$	0	0
$989$	50.3351	1.60056
$990$	0	0
$991$	−9.76520	−0.310202	−0.155101	−	0.987899i	$-0.549570\pi$
$991$	−9.76520	−0.310202	−0.155101	+	0.987899i	$0.549570\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.1882	0.798520
$996$	0	0
$997$	−31.6286	−1.00169	−0.500843	−	0.865538i	$-0.666977\pi$
$997$	−31.6286	−1.00169	−0.500843	+	0.865538i	$0.666977\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.cb.1.5		5
3.2	odd	2		8208.2.a.cc.1.1		5
4.3	odd	2		4104.2.a.o.1.5	✓	5
12.11	even	2		4104.2.a.r.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4104.2.a.o.1.5	✓	5	4.3	odd	2
4104.2.a.r.1.1	yes	5	12.11	even	2
8208.2.a.cb.1.5		5	1.1	even	1	trivial
8208.2.a.cc.1.1		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.83464 q^{5} -1.39043 q^{7} +5.50239 q^{11} +3.03519 q^{13} -3.51575 q^{17} +1.00000 q^{19} +7.65070 q^{23} -1.63409 q^{25} +5.01814 q^{29} +1.12532 q^{31} -2.55094 q^{35} +2.12532 q^{37} -1.11312 q^{41} +6.57915 q^{43} +8.55248 q^{47} -5.06670 q^{49} -4.63256 q^{53} +10.0949 q^{55} -4.38559 q^{59} -12.9110 q^{61} +5.56848 q^{65} -10.9110 q^{67} +6.63409 q^{71} +5.02452 q^{73} -7.65070 q^{77} +12.8895 q^{79} +4.40263 q^{83} -6.45015 q^{85} -1.28216 q^{89} -4.22023 q^{91} +1.83464 q^{95} +7.94138 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{5} - q^{7} + 2 q^{11} + 3 q^{13} - 8 q^{17} + 5 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} + 2 q^{31} + 9 q^{35} + 7 q^{37} - 7 q^{41} + 10 q^{47} + 6 q^{49} - 22 q^{53} + 8 q^{55} + 12 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) 5 * q - 3 * q^5 - q^7 + 2 * q^11 + 3 * q^13 - 8 * q^17 + 5 * q^19 + 2 * q^23 + 4 * q^25 - 10 * q^29 + 2 * q^31 + 9 * q^35 + 7 * q^37 - 7 * q^41 + 10 * q^47 + 6 * q^49 - 22 * q^53 + 8 * q^55 + 12 * q^59 + 2 * q^61 - 11 * q^65 + 12 * q^67 + 21 * q^71 + 7 * q^73 - 2 * q^77 - 6 * q^79 + 11 * q^83 + 4 * q^85 - 27 * q^89 + 25 * q^91 - 3 * q^95 + 12 * q^97

Introduction

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