LMFDB - Embedded newform 8208.2.a.be.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,-3,0,0,0,2] 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:	$1$
Dimension:	$3$
Coefficient field:	3.3.993.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 3 $ x^3 - x^2 - 6*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2052)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.77339$ 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)

$f(q)$	$=$	$q+0.773387 q^{5} -1.91829 q^{7} +6.46507 q^{11} +0.0817097 q^{13} -3.00000 q^{17} +1.00000 q^{19} -0.773387 q^{23} -4.40187 q^{25} -3.77339 q^{29} +0.918290 q^{31} -1.48358 q^{35} -9.23845 q^{37} -2.93681 q^{41} -6.32016 q^{43} +4.54677 q^{47} -3.32016 q^{49} -3.30832 q^{53} +5.00000 q^{55} -3.40187 q^{59} +0.0817097 q^{61} +0.0631932 q^{65} -6.40187 q^{67} +4.79190 q^{71} +11.4769 q^{73} -12.4019 q^{77} -9.23845 q^{79} +3.00000 q^{83} -2.32016 q^{85} -7.08171 q^{89} -0.156743 q^{91} +0.773387 q^{95} +1.40187 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{5} - 3 q^{7} + 2 q^{11} + 3 q^{13} - 9 q^{17} + 3 q^{19} + 5 q^{23} + 6 q^{25} - 4 q^{29} + 12 q^{35} - 3 q^{37} - 7 q^{41} + 3 q^{43} - q^{47} + 12 q^{49} - 20 q^{53} + 15 q^{55} + 9 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) $ 3 * q - 5 * q^5 - 3 * q^7 + 2 * q^11 + 3 * q^13 - 9 * q^17 + 3 * q^19 + 5 * q^23 + 6 * q^25 - 4 * q^29 + 12 * q^35 - 3 * q^37 - 7 * q^41 + 3 * q^43 - q^47 + 12 * q^49 - 20 * q^53 + 15 * q^55 + 9 * q^59 + 3 * q^61 + 2 * q^65 + 8 * q^71 - 15 * q^73 - 18 * q^77 - 3 * q^79 + 9 * q^83 + 15 * q^85 - 24 * q^89 + 27 * q^91 - 5 * q^95 - 15 * 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
$5$	0.773387	0.345869	0.172935	−	0.984933i	$-0.444675\pi$
$5$	0.773387	0.345869	0.172935	+	0.984933i	$0.444675\pi$
$6$	0	0
$7$	−1.91829	−0.725046	−0.362523	−	0.931975i	$-0.618085\pi$
$7$	−1.91829	−0.725046	−0.362523	+	0.931975i	$0.618085\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.46507	1.94929	0.974645	−	0.223756i	$-0.0718318\pi$
$11$	6.46507	1.94929	0.974645	+	0.223756i	$0.0718318\pi$
$12$	0	0
$13$	0.0817097	0.0226622	0.0113311	−	0.999936i	$-0.496393\pi$
$13$	0.0817097	0.0226622	0.0113311	+	0.999936i	$0.496393\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.773387	−0.161262	−0.0806312	−	0.996744i	$-0.525694\pi$
$23$	−0.773387	−0.161262	−0.0806312	+	0.996744i	$0.525694\pi$
$24$	0	0
$25$	−4.40187	−0.880374
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.77339	−0.700700	−0.350350	−	0.936619i	$-0.613937\pi$
$29$	−3.77339	−0.700700	−0.350350	+	0.936619i	$0.613937\pi$
$30$	0	0
$31$	0.918290	0.164930	0.0824649	−	0.996594i	$-0.473721\pi$
$31$	0.918290	0.164930	0.0824649	+	0.996594i	$0.473721\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.48358	−0.250771
$36$	0	0
$37$	−9.23845	−1.51879	−0.759396	−	0.650629i	$-0.774506\pi$
$37$	−9.23845	−1.51879	−0.759396	+	0.650629i	$0.774506\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.93681	−0.458652	−0.229326	−	0.973350i	$-0.573652\pi$
$41$	−2.93681	−0.458652	−0.229326	+	0.973350i	$0.573652\pi$
$42$	0	0
$43$	−6.32016	−0.963816	−0.481908	−	0.876222i	$-0.660056\pi$
$43$	−6.32016	−0.963816	−0.481908	+	0.876222i	$0.660056\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.54677	0.663215	0.331608	−	0.943417i	$-0.392409\pi$
$47$	4.54677	0.663215	0.331608	+	0.943417i	$0.392409\pi$
$48$	0	0
$49$	−3.32016	−0.474309
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.30832	−0.454433	−0.227217	−	0.973844i	$-0.572963\pi$
$53$	−3.30832	−0.454433	−0.227217	+	0.973844i	$0.572963\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.40187	−0.442886	−0.221443	−	0.975173i	$-0.571077\pi$
$59$	−3.40187	−0.442886	−0.221443	+	0.975173i	$0.571077\pi$
$60$	0	0
$61$	0.0817097	0.0104619	0.00523093	−	0.999986i	$-0.498335\pi$
$61$	0.0817097	0.0104619	0.00523093	+	0.999986i	$0.498335\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.0631932	0.00783816
$66$	0	0
$67$	−6.40187	−0.782113	−0.391057	−	0.920367i	$-0.627890\pi$
$67$	−6.40187	−0.782113	−0.391057	+	0.920367i	$0.627890\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.79190	0.568694	0.284347	−	0.958721i	$-0.408223\pi$
$71$	4.79190	0.568694	0.284347	+	0.958721i	$0.408223\pi$
$72$	0	0
$73$	11.4769	1.34327	0.671635	−	0.740882i	$-0.265592\pi$
$73$	11.4769	1.34327	0.671635	+	0.740882i	$0.265592\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.4019	−1.41332
$78$	0	0
$79$	−9.23845	−1.03941	−0.519704	−	0.854347i	$-0.673958\pi$
$79$	−9.23845	−1.03941	−0.519704	+	0.854347i	$0.673958\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	−2.32016	−0.251657
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.08171	−0.750660	−0.375330	−	0.926891i	$-0.622471\pi$
$89$	−7.08171	−0.750660	−0.375330	+	0.926891i	$0.622471\pi$
$90$	0	0
$91$	−0.156743	−0.0164311
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.773387	0.0793479
$96$	0	0
$97$	1.40187	0.142339	0.0711693	−	0.997464i	$-0.477327\pi$
$97$	1.40187	0.142339	0.0711693	+	0.997464i	$0.477327\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.01184	0.498697	0.249348	−	0.968414i	$-0.419784\pi$
$101$	5.01184	0.498697	0.249348	+	0.968414i	$0.419784\pi$
$102$	0	0
$103$	−9.15674	−0.902241	−0.451120	−	0.892463i	$-0.648975\pi$
$103$	−9.15674	−0.902241	−0.451120	+	0.892463i	$0.648975\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.60997	−0.155641	−0.0778207	−	0.996967i	$-0.524796\pi$
$107$	−1.60997	−0.155641	−0.0778207	+	0.996967i	$0.524796\pi$
$108$	0	0
$109$	−14.3202	−1.37162	−0.685811	−	0.727779i	$-0.740553\pi$
$109$	−14.3202	−1.37162	−0.685811	+	0.727779i	$0.740553\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.98148	−0.468619	−0.234309	−	0.972162i	$-0.575283\pi$
$113$	−4.98148	−0.468619	−0.234309	+	0.972162i	$0.575283\pi$
$114$	0	0
$115$	−0.598128	−0.0557757
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.75487	0.527548
$120$	0	0
$121$	30.7971	2.79973
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.27129	−0.650364
$126$	0	0
$127$	12.8366	1.13906	0.569531	−	0.821970i	$-0.307125\pi$
$127$	12.8366	1.13906	0.569531	+	0.821970i	$0.307125\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.0750	1.75396	0.876982	−	0.480523i	$-0.159553\pi$
$131$	20.0750	1.75396	0.876982	+	0.480523i	$0.159553\pi$
$132$	0	0
$133$	−1.91829	−0.166337
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.30165	0.623822	0.311911	−	0.950111i	$-0.399031\pi$
$137$	7.30165	0.623822	0.311911	+	0.950111i	$0.399031\pi$
$138$	0	0
$139$	4.67316	0.396372	0.198186	−	0.980164i	$-0.436495\pi$
$139$	4.67316	0.396372	0.198186	+	0.980164i	$0.436495\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.528258	0.0441752
$144$	0	0
$145$	−2.91829	−0.242351
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−20.2318	−1.65745	−0.828726	−	0.559655i	$-0.810934\pi$
$149$	−20.2318	−1.65745	−0.828726	+	0.559655i	$0.810934\pi$
$150$	0	0
$151$	19.7154	1.60441	0.802207	−	0.597047i	$-0.203659\pi$
$151$	19.7154	1.60441	0.802207	+	0.597047i	$0.203659\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.710194	0.0570442
$156$	0	0
$157$	8.67316	0.692194	0.346097	−	0.938199i	$-0.387507\pi$
$157$	8.67316	0.692194	0.346097	+	0.938199i	$0.387507\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.48358	0.116923
$162$	0	0
$163$	9.32016	0.730011	0.365006	−	0.931005i	$-0.381067\pi$
$163$	9.32016	0.730011	0.365006	+	0.931005i	$0.381067\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.7482	−1.45078	−0.725389	−	0.688339i	$-0.758340\pi$
$167$	−18.7482	−1.45078	−0.725389	+	0.688339i	$0.758340\pi$
$168$	0	0
$169$	−12.9933	−0.999486
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.9116	−1.58988	−0.794940	−	0.606688i	$-0.792498\pi$
$173$	−20.9116	−1.58988	−0.794940	+	0.606688i	$0.792498\pi$
$174$	0	0
$175$	8.44407	0.638312
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.9183	−1.26453	−0.632266	−	0.774751i	$-0.717875\pi$
$179$	−16.9183	−1.26453	−0.632266	+	0.774751i	$0.717875\pi$
$180$	0	0
$181$	−12.8855	−0.957768	−0.478884	−	0.877878i	$-0.658959\pi$
$181$	−12.8855	−0.957768	−0.478884	+	0.877878i	$0.658959\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.14490	−0.525304
$186$	0	0
$187$	−19.3952	−1.41832
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.1054	−0.803558	−0.401779	−	0.915737i	$-0.631608\pi$
$191$	−11.1054	−0.803558	−0.401779	+	0.915737i	$0.631608\pi$
$192$	0	0
$193$	17.7549	1.27802	0.639012	−	0.769197i	$-0.279343\pi$
$193$	17.7549	1.27802	0.639012	+	0.769197i	$0.279343\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.8669	−1.13047	−0.565236	−	0.824929i	$-0.691215\pi$
$197$	−15.8669	−1.13047	−0.565236	+	0.824929i	$0.691215\pi$
$198$	0	0
$199$	14.8299	1.05126	0.525632	−	0.850712i	$-0.323829\pi$
$199$	14.8299	1.05126	0.525632	+	0.850712i	$0.323829\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.23845	0.508040
$204$	0	0
$205$	−2.27129	−0.158634
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.46507	0.447198
$210$	0	0
$211$	10.2385	0.704844	0.352422	−	0.935841i	$-0.385358\pi$
$211$	10.2385	0.704844	0.352422	+	0.935841i	$0.385358\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.88793	−0.333354
$216$	0	0
$217$	−1.76155	−0.119582
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.245129	−0.0164892
$222$	0	0
$223$	−11.4019	−0.763526	−0.381763	−	0.924260i	$-0.624683\pi$
$223$	−11.4019	−0.763526	−0.381763	+	0.924260i	$0.624683\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.30832	−0.219581	−0.109791	−	0.993955i	$-0.535018\pi$
$227$	−3.30832	−0.219581	−0.109791	+	0.993955i	$0.535018\pi$
$228$	0	0
$229$	11.6798	0.771825	0.385913	−	0.922535i	$-0.373887\pi$
$229$	11.6798	0.771825	0.385913	+	0.922535i	$0.373887\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.8855	−1.30274	−0.651370	−	0.758761i	$-0.725805\pi$
$233$	−19.8855	−1.30274	−0.651370	+	0.758761i	$0.725805\pi$
$234$	0	0
$235$	3.51642	0.229386
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.40187	−0.608157	−0.304078	−	0.952647i	$-0.598348\pi$
$239$	−9.40187	−0.608157	−0.304078	+	0.952647i	$0.598348\pi$
$240$	0	0
$241$	8.82990	0.568784	0.284392	−	0.958708i	$-0.408208\pi$
$241$	8.82990	0.568784	0.284392	+	0.958708i	$0.408208\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.56777	−0.164049
$246$	0	0
$247$	0.0817097	0.00519906
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.20810	−0.265613	−0.132806	−	0.991142i	$-0.542399\pi$
$251$	−4.20810	−0.265613	−0.132806	+	0.991142i	$0.542399\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.0052	1.87167	0.935835	−	0.352438i	$-0.114647\pi$
$257$	30.0052	1.87167	0.935835	+	0.352438i	$0.114647\pi$
$258$	0	0
$259$	17.7220	1.10119
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−20.3202	−1.25299	−0.626497	−	0.779423i	$-0.715512\pi$
$263$	−20.3202	−1.25299	−0.626497	+	0.779423i	$0.715512\pi$
$264$	0	0
$265$	−2.55861	−0.157174
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.5586	1.31445	0.657226	−	0.753694i	$-0.271730\pi$
$269$	21.5586	1.31445	0.657226	+	0.753694i	$0.271730\pi$
$270$	0	0
$271$	−12.4769	−0.757918	−0.378959	−	0.925413i	$-0.623718\pi$
$271$	−12.4769	−0.757918	−0.378959	+	0.925413i	$0.623718\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−28.4584	−1.71611
$276$	0	0
$277$	−1.07503	−0.0645925	−0.0322962	−	0.999478i	$-0.510282\pi$
$277$	−1.07503	−0.0645925	−0.0322962	+	0.999478i	$0.510282\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.8973	−1.48525	−0.742624	−	0.669709i	$-0.766419\pi$
$281$	−24.8973	−1.48525	−0.742624	+	0.669709i	$0.766419\pi$
$282$	0	0
$283$	−28.1501	−1.67335	−0.836674	−	0.547701i	$-0.815503\pi$
$283$	−28.1501	−1.67335	−0.836674	+	0.547701i	$0.815503\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.63365	0.332544
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.07503	−0.296487	−0.148243	−	0.988951i	$-0.547362\pi$
$293$	−5.07503	−0.296487	−0.148243	+	0.988951i	$0.547362\pi$
$294$	0	0
$295$	−2.63096	−0.153181
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.0631932	−0.00365456
$300$	0	0
$301$	12.1239	0.698810
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.0631932	0.00361843
$306$	0	0
$307$	15.0000	0.856095	0.428048	−	0.903756i	$-0.359202\pi$
$307$	15.0000	0.856095	0.428048	+	0.903756i	$0.359202\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.0304	−1.53275	−0.766375	−	0.642393i	$-0.777942\pi$
$311$	−27.0304	−1.53275	−0.766375	+	0.642393i	$0.777942\pi$
$312$	0	0
$313$	1.47691	0.0834796	0.0417398	−	0.999129i	$-0.486710\pi$
$313$	1.47691	0.0834796	0.0417398	+	0.999129i	$0.486710\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.4769	0.644607	0.322304	−	0.946636i	$-0.395543\pi$
$317$	11.4769	0.644607	0.322304	+	0.946636i	$0.395543\pi$
$318$	0	0
$319$	−24.3952	−1.36587
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−0.359676	−0.0199512
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.72203	−0.480861
$330$	0	0
$331$	−20.1239	−1.10611	−0.553055	−	0.833145i	$-0.686538\pi$
$331$	−20.1239	−1.10611	−0.553055	+	0.833145i	$0.686538\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.95113	−0.270509
$336$	0	0
$337$	−9.15007	−0.498436	−0.249218	−	0.968447i	$-0.580174\pi$
$337$	−9.15007	−0.498436	−0.249218	+	0.968447i	$0.580174\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.93681	0.321496
$342$	0	0
$343$	19.7971	1.06894
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.70352	−0.413546	−0.206773	−	0.978389i	$-0.566296\pi$
$347$	−7.70352	−0.413546	−0.206773	+	0.978389i	$0.566296\pi$
$348$	0	0
$349$	0.0750332	0.00401644	0.00200822	−	0.999998i	$-0.499361\pi$
$349$	0.0750332	0.00401644	0.00200822	+	0.999998i	$0.499361\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.8299	−0.736092	−0.368046	−	0.929808i	$-0.619973\pi$
$353$	−13.8299	−0.736092	−0.368046	+	0.929808i	$0.619973\pi$
$354$	0	0
$355$	3.70600	0.196694
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.82990	0.254913	0.127456	−	0.991844i	$-0.459319\pi$
$359$	4.82990	0.254913	0.127456	+	0.991844i	$0.459319\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.87609	0.464596
$366$	0	0
$367$	−12.7549	−0.665799	−0.332899	−	0.942962i	$-0.608027\pi$
$367$	−12.7549	−0.665799	−0.332899	+	0.942962i	$0.608027\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.34632	0.329485
$372$	0	0
$373$	16.8788	0.873950	0.436975	−	0.899474i	$-0.356050\pi$
$373$	16.8788	0.873950	0.436975	+	0.899474i	$0.356050\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.308322	−0.0158794
$378$	0	0
$379$	7.48358	0.384406	0.192203	−	0.981355i	$-0.438437\pi$
$379$	7.48358	0.384406	0.192203	+	0.981355i	$0.438437\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.3505	−1.03986	−0.519931	−	0.854208i	$-0.674043\pi$
$383$	−20.3505	−1.03986	−0.519931	+	0.854208i	$0.674043\pi$
$384$	0	0
$385$	−9.59145	−0.488826
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.5519	−0.687111	−0.343555	−	0.939132i	$-0.611631\pi$
$389$	−13.5519	−0.687111	−0.343555	+	0.939132i	$0.611631\pi$
$390$	0	0
$391$	2.32016	0.117336
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.14490	−0.359499
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.9420	1.04579	0.522896	−	0.852396i	$-0.324852\pi$
$401$	20.9420	1.04579	0.522896	+	0.852396i	$0.324852\pi$
$402$	0	0
$403$	0.0750332	0.00373767
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−59.7272	−2.96057
$408$	0	0
$409$	−10.6665	−0.527424	−0.263712	−	0.964601i	$-0.584947\pi$
$409$	−10.6665	−0.527424	−0.263712	+	0.964601i	$0.584947\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.52578	0.321113
$414$	0	0
$415$	2.32016	0.113892
$416$	0	0
$417$	0	0
$418$	0	0
$419$	35.3885	1.72884	0.864421	−	0.502768i	$-0.167685\pi$
$419$	35.3885	1.72884	0.864421	+	0.502768i	$0.167685\pi$
$420$	0	0
$421$	−10.5586	−0.514595	−0.257298	−	0.966332i	$-0.582832\pi$
$421$	−10.5586	−0.514595	−0.257298	+	0.966332i	$0.582832\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.2056	0.640566
$426$	0	0
$427$	−0.156743	−0.00758532
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.4085	−0.549530	−0.274765	−	0.961511i	$-0.588600\pi$
$431$	−11.4085	−0.549530	−0.274765	+	0.961511i	$0.588600\pi$
$432$	0	0
$433$	−29.4836	−1.41689	−0.708445	−	0.705766i	$-0.750603\pi$
$433$	−29.4836	−1.41689	−0.708445	+	0.705766i	$0.750603\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.773387	−0.0369961
$438$	0	0
$439$	−2.83658	−0.135383	−0.0676913	−	0.997706i	$-0.521563\pi$
$439$	−2.83658	−0.135383	−0.0676913	+	0.997706i	$0.521563\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.2950	−1.10678	−0.553389	−	0.832923i	$-0.686666\pi$
$443$	−23.2950	−1.10678	−0.553389	+	0.832923i	$0.686666\pi$
$444$	0	0
$445$	−5.47691	−0.259630
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.1054	0.524096	0.262048	−	0.965055i	$-0.415602\pi$
$449$	11.1054	0.524096	0.262048	+	0.965055i	$0.415602\pi$
$450$	0	0
$451$	−18.9866	−0.894046
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.121223	−0.00568302
$456$	0	0
$457$	−1.89213	−0.0885101	−0.0442551	−	0.999020i	$-0.514091\pi$
$457$	−1.89213	−0.0885101	−0.0442551	+	0.999020i	$0.514091\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.9553	−0.882837	−0.441419	−	0.897301i	$-0.645525\pi$
$461$	−18.9553	−0.882837	−0.441419	+	0.897301i	$0.645525\pi$
$462$	0	0
$463$	28.0422	1.30323	0.651616	−	0.758549i	$-0.274092\pi$
$463$	28.0422	1.30323	0.651616	+	0.758549i	$0.274092\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.1819	0.563713	0.281856	−	0.959457i	$-0.409050\pi$
$467$	12.1819	0.563713	0.281856	+	0.959457i	$0.409050\pi$
$468$	0	0
$469$	12.2806	0.567068
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−40.8603	−1.87876
$474$	0	0
$475$	−4.40187	−0.201972
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.0699	−0.779942	−0.389971	−	0.920827i	$-0.627515\pi$
$479$	−17.0699	−0.779942	−0.389971	+	0.920827i	$0.627515\pi$
$480$	0	0
$481$	−0.754871	−0.0344192
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.08419	0.0492305
$486$	0	0
$487$	5.59145	0.253373	0.126686	−	0.991943i	$-0.459566\pi$
$487$	5.59145	0.253373	0.126686	+	0.991943i	$0.459566\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.8603	0.896281	0.448141	−	0.893963i	$-0.352086\pi$
$491$	19.8603	0.896281	0.448141	+	0.893963i	$0.352086\pi$
$492$	0	0
$493$	11.3202	0.509834
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.19226	−0.412329
$498$	0	0
$499$	12.4836	0.558842	0.279421	−	0.960169i	$-0.409858\pi$
$499$	12.4836	0.558842	0.279421	+	0.960169i	$0.409858\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.5164	−0.468904	−0.234452	−	0.972128i	$-0.575330\pi$
$503$	−10.5164	−0.468904	−0.234452	+	0.972128i	$0.575330\pi$
$504$	0	0
$505$	3.87609	0.172484
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.5258	1.08709	0.543543	−	0.839381i	$-0.317082\pi$
$509$	24.5258	1.08709	0.543543	+	0.839381i	$0.317082\pi$
$510$	0	0
$511$	−22.0160	−0.973932
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7.08171	−0.312057
$516$	0	0
$517$	29.3952	1.29280
$518$	0	0
$519$	0	0
$520$	0	0
$521$	34.7667	1.52316	0.761579	−	0.648073i	$-0.224425\pi$
$521$	34.7667	1.52316	0.761579	+	0.648073i	$0.224425\pi$
$522$	0	0
$523$	−33.3624	−1.45883	−0.729417	−	0.684069i	$-0.760209\pi$
$523$	−33.3624	−1.45883	−0.729417	+	0.684069i	$0.760209\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.75487	−0.120004
$528$	0	0
$529$	−22.4019	−0.973994
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.239966	−0.0103941
$534$	0	0
$535$	−1.24513	−0.0538316
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−21.4651	−0.924566
$540$	0	0
$541$	20.4702	0.880084	0.440042	−	0.897977i	$-0.354964\pi$
$541$	20.4702	0.880084	0.440042	+	0.897977i	$0.354964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11.0750	−0.474402
$546$	0	0
$547$	−17.7615	−0.759429	−0.379714	−	0.925104i	$-0.623978\pi$
$547$	−17.7615	−0.759429	−0.379714	+	0.925104i	$0.623978\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.77339	−0.160752
$552$	0	0
$553$	17.7220	0.753618
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.39003	0.0588975	0.0294488	−	0.999566i	$-0.490625\pi$
$557$	1.39003	0.0588975	0.0294488	+	0.999566i	$0.490625\pi$
$558$	0	0
$559$	−0.516418	−0.0218422
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.4836	0.821135	0.410568	−	0.911830i	$-0.365331\pi$
$563$	19.4836	0.821135	0.410568	+	0.911830i	$0.365331\pi$
$564$	0	0
$565$	−3.85262	−0.162081
$566$	0	0
$567$	0	0
$568$	0	0
$569$	37.3016	1.56377	0.781883	−	0.623425i	$-0.214259\pi$
$569$	37.3016	1.56377	0.781883	+	0.623425i	$0.214259\pi$
$570$	0	0
$571$	44.7087	1.87100	0.935500	−	0.353327i	$-0.114950\pi$
$571$	44.7087	1.87100	0.935500	+	0.353327i	$0.114950\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.40435	0.141971
$576$	0	0
$577$	−10.9933	−0.457658	−0.228829	−	0.973467i	$-0.573490\pi$
$577$	−10.9933	−0.457658	−0.228829	+	0.973467i	$0.573490\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.75487	−0.238752
$582$	0	0
$583$	−21.3885	−0.885822
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.5519	−0.559348	−0.279674	−	0.960095i	$-0.590226\pi$
$587$	−13.5519	−0.559348	−0.279674	+	0.960095i	$0.590226\pi$
$588$	0	0
$589$	0.918290	0.0378375
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.5139	0.554951	0.277475	−	0.960733i	$-0.410502\pi$
$593$	13.5139	0.554951	0.277475	+	0.960733i	$0.410502\pi$
$594$	0	0
$595$	4.45074	0.182463
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−43.6403	−1.78310	−0.891548	−	0.452927i	$-0.850380\pi$
$599$	−43.6403	−1.78310	−0.891548	+	0.452927i	$0.850380\pi$
$600$	0	0
$601$	19.0355	0.776475	0.388238	−	0.921559i	$-0.373084\pi$
$601$	19.0355	0.776475	0.388238	+	0.921559i	$0.373084\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	23.8181	0.968342
$606$	0	0
$607$	−43.9800	−1.78509	−0.892546	−	0.450956i	$-0.851083\pi$
$607$	−43.9800	−1.78509	−0.892546	+	0.450956i	$0.851083\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.371516	0.0150299
$612$	0	0
$613$	−6.15007	−0.248399	−0.124199	−	0.992257i	$-0.539636\pi$
$613$	−6.15007	−0.248399	−0.124199	+	0.992257i	$0.539636\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.4374	0.983812	0.491906	−	0.870648i	$-0.336300\pi$
$617$	24.4374	0.983812	0.491906	+	0.870648i	$0.336300\pi$
$618$	0	0
$619$	−5.27129	−0.211871	−0.105936	−	0.994373i	$-0.533784\pi$
$619$	−5.27129	−0.211871	−0.105936	+	0.994373i	$0.533784\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.5848	0.544263
$624$	0	0
$625$	16.3858	0.655433
$626$	0	0
$627$	0	0
$628$	0	0
$629$	27.7154	1.10508
$630$	0	0
$631$	41.4307	1.64933	0.824665	−	0.565621i	$-0.191363\pi$
$631$	41.4307	1.64933	0.824665	+	0.565621i	$0.191363\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9.92765	0.393967
$636$	0	0
$637$	−0.271289	−0.0107489
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.6825	−1.09339	−0.546697	−	0.837331i	$-0.684115\pi$
$641$	−27.6825	−1.09339	−0.546697	+	0.837331i	$0.684115\pi$
$642$	0	0
$643$	43.3885	1.71108	0.855538	−	0.517740i	$-0.173226\pi$
$643$	43.3885	1.71108	0.855538	+	0.517740i	$0.173226\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.6640	1.52004	0.760019	−	0.649900i	$-0.225189\pi$
$647$	38.6640	1.52004	0.760019	+	0.649900i	$0.225189\pi$
$648$	0	0
$649$	−21.9933	−0.863313
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19.2056	−0.751574	−0.375787	−	0.926706i	$-0.622628\pi$
$653$	−19.2056	−0.751574	−0.375787	+	0.926706i	$0.622628\pi$
$654$	0	0
$655$	15.5258	0.606642
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.50210	−0.214331	−0.107166	−	0.994241i	$-0.534178\pi$
$659$	−5.50210	−0.214331	−0.107166	+	0.994241i	$0.534178\pi$
$660$	0	0
$661$	25.8366	1.00493	0.502463	−	0.864599i	$-0.332427\pi$
$661$	25.8366	1.00493	0.502463	+	0.864599i	$0.332427\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.48358	−0.0575308
$666$	0	0
$667$	2.91829	0.112997
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.528258	0.0203932
$672$	0	0
$673$	−42.3557	−1.63269	−0.816346	−	0.577563i	$-0.804004\pi$
$673$	−42.3557	−1.63269	−0.816346	+	0.577563i	$0.804004\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.8156	−1.49180	−0.745902	−	0.666056i	$-0.767981\pi$
$677$	−38.8156	−1.49180	−0.745902	+	0.666056i	$0.767981\pi$
$678$	0	0
$679$	−2.68920	−0.103202
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.6640	−0.790686	−0.395343	−	0.918534i	$-0.629374\pi$
$683$	−20.6640	−0.790686	−0.395343	+	0.918534i	$0.629374\pi$
$684$	0	0
$685$	5.64700	0.215761
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.270322	−0.0102984
$690$	0	0
$691$	38.8232	1.47691	0.738453	−	0.674305i	$-0.235557\pi$
$691$	38.8232	1.47691	0.738453	+	0.674305i	$0.235557\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.61416	0.137093
$696$	0	0
$697$	8.81042	0.333718
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.0370	−0.416863	−0.208431	−	0.978037i	$-0.566836\pi$
$701$	−11.0370	−0.416863	−0.208431	+	0.978037i	$0.566836\pi$
$702$	0	0
$703$	−9.23845	−0.348435
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.61416	−0.361578
$708$	0	0
$709$	20.6336	0.774913	0.387456	−	0.921888i	$-0.373354\pi$
$709$	20.6336	0.774913	0.387456	+	0.921888i	$0.373354\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.710194	−0.0265970
$714$	0	0
$715$	0.408548	0.0152788
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−37.1804	−1.38660	−0.693298	−	0.720651i	$-0.743843\pi$
$719$	−37.1804	−1.38660	−0.693298	+	0.720651i	$0.743843\pi$
$720$	0	0
$721$	17.5653	0.654166
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16.6100	0.616879
$726$	0	0
$727$	20.9049	0.775321	0.387661	−	0.921802i	$-0.373283\pi$
$727$	20.9049	0.775321	0.387661	+	0.921802i	$0.373283\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	18.9605	0.701279
$732$	0	0
$733$	−10.0817	−0.372376	−0.186188	−	0.982514i	$-0.559613\pi$
$733$	−10.0817	−0.372376	−0.186188	+	0.982514i	$0.559613\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−41.3885	−1.52457
$738$	0	0
$739$	−23.3135	−0.857600	−0.428800	−	0.903399i	$-0.641064\pi$
$739$	−23.3135	−0.857600	−0.428800	+	0.903399i	$0.641064\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.1947	−1.32786	−0.663928	−	0.747796i	$-0.731112\pi$
$743$	−36.1947	−1.32786	−0.663928	+	0.747796i	$0.731112\pi$
$744$	0	0
$745$	−15.6470	−0.573262
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.08839	0.112847
$750$	0	0
$751$	45.8788	1.67414	0.837070	−	0.547096i	$-0.184267\pi$
$751$	45.8788	1.67414	0.837070	+	0.547096i	$0.184267\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.2476	0.554917
$756$	0	0
$757$	−32.4702	−1.18015	−0.590075	−	0.807348i	$-0.700902\pi$
$757$	−32.4702	−1.18015	−0.590075	+	0.807348i	$0.700902\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.5981	−0.746682	−0.373341	−	0.927694i	$-0.621788\pi$
$761$	−20.5981	−0.746682	−0.373341	+	0.927694i	$0.621788\pi$
$762$	0	0
$763$	27.4702	0.994489
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.277966	−0.0100368
$768$	0	0
$769$	29.7971	1.07451	0.537255	−	0.843420i	$-0.319461\pi$
$769$	29.7971	1.07451	0.537255	+	0.843420i	$0.319461\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−29.2873	−1.05339	−0.526696	−	0.850054i	$-0.676569\pi$
$773$	−29.2873	−1.05339	−0.526696	+	0.850054i	$0.676569\pi$
$774$	0	0
$775$	−4.04220	−0.145200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.93681	−0.105222
$780$	0	0
$781$	30.9800	1.10855
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.70771	0.239409
$786$	0	0
$787$	27.0817	0.965359	0.482679	−	0.875797i	$-0.339664\pi$
$787$	27.0817	0.965359	0.482679	+	0.875797i	$0.339664\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.55593	0.339770
$792$	0	0
$793$	0.00667647	0.000237088 0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.3135	1.07376	0.536879	−	0.843659i	$-0.319603\pi$
$797$	30.3135	1.07376	0.536879	+	0.843659i	$0.319603\pi$
$798$	0	0
$799$	−13.6403	−0.482560
$800$	0	0
$801$	0	0
$802$	0	0
$803$	74.1989	2.61842
$804$	0	0
$805$	1.14738	0.0404399
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.91829	0.0674435	0.0337217	−	0.999431i	$-0.489264\pi$
$809$	1.91829	0.0674435	0.0337217	+	0.999431i	$0.489264\pi$
$810$	0	0
$811$	10.1373	0.355967	0.177984	−	0.984033i	$-0.443043\pi$
$811$	10.1373	0.355967	0.177984	+	0.984033i	$0.443043\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.20810	0.252489
$816$	0	0
$817$	−6.32016	−0.221114
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.6892	1.55966	0.779832	−	0.625989i	$-0.215305\pi$
$821$	44.6892	1.55966	0.779832	+	0.625989i	$0.215305\pi$
$822$	0	0
$823$	19.4019	0.676307	0.338153	−	0.941091i	$-0.390198\pi$
$823$	19.4019	0.676307	0.338153	+	0.941091i	$0.390198\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.46507	0.120492	0.0602461	−	0.998184i	$-0.480811\pi$
$827$	3.46507	0.120492	0.0602461	+	0.998184i	$0.480811\pi$
$828$	0	0
$829$	46.5519	1.61682	0.808408	−	0.588623i	$-0.200330\pi$
$829$	46.5519	1.61682	0.808408	+	0.588623i	$0.200330\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9.96049	0.345110
$834$	0	0
$835$	−14.4996	−0.501780
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.343842	0.0118708	0.00593538	−	0.999982i	$-0.498111\pi$
$839$	0.343842	0.0118708	0.00593538	+	0.999982i	$0.498111\pi$
$840$	0	0
$841$	−14.7615	−0.509019
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.0489	−0.345692
$846$	0	0
$847$	−59.0777	−2.02993
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.14490	0.244924
$852$	0	0
$853$	−39.0684	−1.33768	−0.668838	−	0.743409i	$-0.733208\pi$
$853$	−39.0684	−1.33768	−0.668838	+	0.743409i	$0.733208\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−45.0355	−1.53838	−0.769192	−	0.639018i	$-0.779341\pi$
$857$	−45.0355	−1.53838	−0.769192	+	0.639018i	$0.779341\pi$
$858$	0	0
$859$	−43.1923	−1.47370	−0.736850	−	0.676056i	$-0.763688\pi$
$859$	−43.1923	−1.47370	−0.736850	+	0.676056i	$0.763688\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.11723	0.140152	0.0700761	−	0.997542i	$-0.477676\pi$
$863$	4.11723	0.140152	0.0700761	+	0.997542i	$0.477676\pi$
$864$	0	0
$865$	−16.1728	−0.549891
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−59.7272	−2.02611
$870$	0	0
$871$	−0.523095	−0.0177244
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.9484	0.471543
$876$	0	0
$877$	6.19626	0.209233	0.104616	−	0.994513i	$-0.466639\pi$
$877$	6.19626	0.209233	0.104616	+	0.994513i	$0.466639\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−32.3001	−1.08822	−0.544110	−	0.839014i	$-0.683132\pi$
$881$	−32.3001	−1.08822	−0.544110	+	0.839014i	$0.683132\pi$
$882$	0	0
$883$	−41.4507	−1.39493	−0.697464	−	0.716620i	$-0.745688\pi$
$883$	−41.4507	−1.39493	−0.697464	+	0.716620i	$0.745688\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.60997	0.154788	0.0773938	−	0.997001i	$-0.475340\pi$
$887$	4.60997	0.154788	0.0773938	+	0.997001i	$0.475340\pi$
$888$	0	0
$889$	−24.6243	−0.825872
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.54677	0.152152
$894$	0	0
$895$	−13.0844	−0.437363
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.46507	−0.115566
$900$	0	0
$901$	9.92497	0.330649
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.96545	−0.331263
$906$	0	0
$907$	−34.1106	−1.13262	−0.566311	−	0.824192i	$-0.691630\pi$
$907$	−34.1106	−1.13262	−0.566311	+	0.824192i	$0.691630\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	17.9116	0.593438	0.296719	−	0.954965i	$-0.404107\pi$
$911$	17.9116	0.593438	0.296719	+	0.954965i	$0.404107\pi$
$912$	0	0
$913$	19.3952	0.641887
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−38.5097	−1.27170
$918$	0	0
$919$	12.1079	0.399402	0.199701	−	0.979857i	$-0.436003\pi$
$919$	12.1079	0.399402	0.199701	+	0.979857i	$0.436003\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.391545	0.0128879
$924$	0	0
$925$	40.6665	1.33711
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.6774	−1.10492	−0.552459	−	0.833540i	$-0.686310\pi$
$929$	−33.6774	−1.10492	−0.552459	+	0.833540i	$0.686310\pi$
$930$	0	0
$931$	−3.32016	−0.108814
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15.0000	−0.490552
$936$	0	0
$937$	28.6892	0.937235	0.468618	−	0.883401i	$-0.344752\pi$
$937$	28.6892	0.937235	0.468618	+	0.883401i	$0.344752\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	40.3700	1.31602	0.658012	−	0.753007i	$-0.271397\pi$
$941$	40.3700	1.31602	0.658012	+	0.753007i	$0.271397\pi$
$942$	0	0
$943$	2.27129	0.0739634
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.36904	0.206966	0.103483	−	0.994631i	$-0.467001\pi$
$947$	6.36904	0.206966	0.103483	+	0.994631i	$0.467001\pi$
$948$	0	0
$949$	0.937774	0.0304414
$950$	0	0
$951$	0	0
$952$	0	0
$953$	16.0540	0.520041	0.260021	−	0.965603i	$-0.416271\pi$
$953$	16.0540	0.520041	0.260021	+	0.965603i	$0.416271\pi$
$954$	0	0
$955$	−8.58877	−0.277926
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14.0067	−0.452299
$960$	0	0
$961$	−30.1567	−0.972798
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.7314	0.442029
$966$	0	0
$967$	6.73539	0.216595	0.108298	−	0.994119i	$-0.465460\pi$
$967$	6.73539	0.216595	0.108298	+	0.994119i	$0.465460\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.4584	0.913273	0.456637	−	0.889653i	$-0.349054\pi$
$971$	28.4584	0.913273	0.456637	+	0.889653i	$0.349054\pi$
$972$	0	0
$973$	−8.96448	−0.287388
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.4271	0.397577	0.198789	−	0.980042i	$-0.436299\pi$
$977$	12.4271	0.397577	0.198789	+	0.980042i	$0.436299\pi$
$978$	0	0
$979$	−45.7837	−1.46325
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.2056	1.28236	0.641180	−	0.767391i	$-0.278445\pi$
$983$	40.2056	1.28236	0.641180	+	0.767391i	$0.278445\pi$
$984$	0	0
$985$	−12.2713	−0.390996
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.88793	0.155427
$990$	0	0
$991$	13.4085	0.425937	0.212968	−	0.977059i	$-0.431687\pi$
$991$	13.4085	0.425937	0.212968	+	0.977059i	$0.431687\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.4693	0.363600
$996$	0	0
$997$	13.6142	0.431165	0.215582	−	0.976486i	$-0.430835\pi$
$997$	13.6142	0.431165	0.215582	+	0.976486i	$0.430835\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8208.2.a.be.1.3		3
3.2	odd	2		8208.2.a.bo.1.1		3
4.3	odd	2		2052.2.a.d.1.3	✓	3
12.11	even	2		2052.2.a.g.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2052.2.a.d.1.3	✓	3	4.3	odd	2
2052.2.a.g.1.1	yes	3	12.11	even	2
8208.2.a.be.1.3		3	1.1	even	1	trivial
8208.2.a.bo.1.1		3	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)

\(f(q)\)	\(=\)	\(q+0.773387 q^{5} -1.91829 q^{7} +6.46507 q^{11} +0.0817097 q^{13} -3.00000 q^{17} +1.00000 q^{19} -0.773387 q^{23} -4.40187 q^{25} -3.77339 q^{29} +0.918290 q^{31} -1.48358 q^{35} -9.23845 q^{37} -2.93681 q^{41} -6.32016 q^{43} +4.54677 q^{47} -3.32016 q^{49} -3.30832 q^{53} +5.00000 q^{55} -3.40187 q^{59} +0.0817097 q^{61} +0.0631932 q^{65} -6.40187 q^{67} +4.79190 q^{71} +11.4769 q^{73} -12.4019 q^{77} -9.23845 q^{79} +3.00000 q^{83} -2.32016 q^{85} -7.08171 q^{89} -0.156743 q^{91} +0.773387 q^{95} +1.40187 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{5} - 3 q^{7} + 2 q^{11} + 3 q^{13} - 9 q^{17} + 3 q^{19} + 5 q^{23} + 6 q^{25} - 4 q^{29} + 12 q^{35} - 3 q^{37} - 7 q^{41} + 3 q^{43} - q^{47} + 12 q^{49} - 20 q^{53} + 15 q^{55} + 9 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) 3 * q - 5 * q^5 - 3 * q^7 + 2 * q^11 + 3 * q^13 - 9 * q^17 + 3 * q^19 + 5 * q^23 + 6 * q^25 - 4 * q^29 + 12 * q^35 - 3 * q^37 - 7 * q^41 + 3 * q^43 - q^47 + 12 * q^49 - 20 * q^53 + 15 * q^55 + 9 * q^59 + 3 * q^61 + 2 * q^65 + 8 * q^71 - 15 * q^73 - 18 * q^77 - 3 * q^79 + 9 * q^83 + 15 * q^85 - 24 * q^89 + 27 * q^91 - 5 * q^95 - 15 * q^97

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