LMFDB - Embedded newform 8820.2.a.bl.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8820,2,Mod(1,8820)] 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("8820.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8820.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,0,0,0,0,2,0,10,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 980)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8820.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +3.82843 q^{11} +3.58579 q^{13} -6.41421 q^{17} +3.65685 q^{19} -0.585786 q^{23} +1.00000 q^{25} -6.65685 q^{29} +4.58579 q^{31} -3.41421 q^{37} -0.585786 q^{41} +11.6569 q^{43} +8.89949 q^{47} +3.75736 q^{53} +3.82843 q^{55} -3.41421 q^{59} +5.17157 q^{61} +3.58579 q^{65} -11.0711 q^{67} -6.48528 q^{71} +5.17157 q^{73} +13.1421 q^{79} +8.00000 q^{83} -6.41421 q^{85} -16.9706 q^{89} +3.65685 q^{95} +15.7279 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{11} + 10 q^{13} - 10 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{29} + 12 q^{31} - 4 q^{37} - 4 q^{41} + 12 q^{43} - 2 q^{47} + 16 q^{53} + 2 q^{55} - 4 q^{59} + 16 q^{61} + 10 q^{65}+ \cdots + 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^11 + 10 * q^13 - 10 * q^17 - 4 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^29 + 12 * q^31 - 4 * q^37 - 4 * q^41 + 12 * q^43 - 2 * q^47 + 16 * q^53 + 2 * q^55 - 4 * q^59 + 16 * q^61 + 10 * q^65 - 8 * q^67 + 4 * q^71 + 16 * q^73 - 2 * q^79 + 16 * q^83 - 10 * q^85 - 4 * q^95 + 6 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.82843	1.15431	0.577157	−	0.816633i	$-0.304162\pi$
$11$	3.82843	1.15431	0.577157	+	0.816633i	$0.304162\pi$
$12$	0	0
$13$	3.58579	0.994518	0.497259	−	0.867602i	$-0.334340\pi$
$13$	3.58579	0.994518	0.497259	+	0.867602i	$0.334340\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.41421	−1.55568	−0.777838	−	0.628465i	$-0.783683\pi$
$17$	−6.41421	−1.55568	−0.777838	+	0.628465i	$0.783683\pi$
$18$	0	0
$19$	3.65685	0.838940	0.419470	−	0.907769i	$-0.362216\pi$
$19$	3.65685	0.838940	0.419470	+	0.907769i	$0.362216\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.585786	−0.122145	−0.0610725	−	0.998133i	$-0.519452\pi$
$23$	−0.585786	−0.122145	−0.0610725	+	0.998133i	$0.519452\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.65685	−1.23615	−0.618073	−	0.786120i	$-0.712087\pi$
$29$	−6.65685	−1.23615	−0.618073	+	0.786120i	$0.712087\pi$
$30$	0	0
$31$	4.58579	0.823632	0.411816	−	0.911267i	$-0.364895\pi$
$31$	4.58579	0.823632	0.411816	+	0.911267i	$0.364895\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.41421	−0.561293	−0.280647	−	0.959811i	$-0.590549\pi$
$37$	−3.41421	−0.561293	−0.280647	+	0.959811i	$0.590549\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.585786	−0.0914845	−0.0457422	−	0.998953i	$-0.514565\pi$
$41$	−0.585786	−0.0914845	−0.0457422	+	0.998953i	$0.514565\pi$
$42$	0	0
$43$	11.6569	1.77765	0.888827	−	0.458243i	$-0.151521\pi$
$43$	11.6569	1.77765	0.888827	+	0.458243i	$0.151521\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.89949	1.29812	0.649062	−	0.760735i	$-0.275161\pi$
$47$	8.89949	1.29812	0.649062	+	0.760735i	$0.275161\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.75736	0.516113	0.258056	−	0.966130i	$-0.416918\pi$
$53$	3.75736	0.516113	0.258056	+	0.966130i	$0.416918\pi$
$54$	0	0
$55$	3.82843	0.516225
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.41421	−0.444493	−0.222246	−	0.974991i	$-0.571339\pi$
$59$	−3.41421	−0.444493	−0.222246	+	0.974991i	$0.571339\pi$
$60$	0	0
$61$	5.17157	0.662152	0.331076	−	0.943604i	$-0.392588\pi$
$61$	5.17157	0.662152	0.331076	+	0.943604i	$0.392588\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.58579	0.444762
$66$	0	0
$67$	−11.0711	−1.35255	−0.676273	−	0.736651i	$-0.736406\pi$
$67$	−11.0711	−1.35255	−0.676273	+	0.736651i	$0.736406\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.48528	−0.769661	−0.384831	−	0.922987i	$-0.625740\pi$
$71$	−6.48528	−0.769661	−0.384831	+	0.922987i	$0.625740\pi$
$72$	0	0
$73$	5.17157	0.605287	0.302643	−	0.953104i	$-0.402131\pi$
$73$	5.17157	0.605287	0.302643	+	0.953104i	$0.402131\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.1421	1.47861	0.739303	−	0.673373i	$-0.235155\pi$
$79$	13.1421	1.47861	0.739303	+	0.673373i	$0.235155\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−6.41421	−0.695719
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.9706	−1.79888	−0.899438	−	0.437048i	$-0.856024\pi$
$89$	−16.9706	−1.79888	−0.899438	+	0.437048i	$0.856024\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.65685	0.375185
$96$	0	0
$97$	15.7279	1.59693	0.798464	−	0.602042i	$-0.205646\pi$
$97$	15.7279	1.59693	0.798464	+	0.602042i	$0.205646\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.82843	0.480446	0.240223	−	0.970718i	$-0.422779\pi$
$101$	4.82843	0.480446	0.240223	+	0.970718i	$0.422779\pi$
$102$	0	0
$103$	1.58579	0.156252	0.0781261	−	0.996943i	$-0.475106\pi$
$103$	1.58579	0.156252	0.0781261	+	0.996943i	$0.475106\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.8284	1.62687	0.813433	−	0.581659i	$-0.197596\pi$
$107$	16.8284	1.62687	0.813433	+	0.581659i	$0.197596\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.07107	0.477046	0.238523	−	0.971137i	$-0.423337\pi$
$113$	5.07107	0.477046	0.238523	+	0.971137i	$0.423337\pi$
$114$	0	0
$115$	−0.585786	−0.0546249
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	3.65685	0.332441
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−21.8995	−1.94327	−0.971633	−	0.236494i	$-0.924002\pi$
$127$	−21.8995	−1.94327	−0.971633	+	0.236494i	$0.924002\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.7574	1.02725	0.513623	−	0.858016i	$-0.328303\pi$
$131$	11.7574	1.02725	0.513623	+	0.858016i	$0.328303\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.9706	−1.10815	−0.554075	−	0.832467i	$-0.686928\pi$
$137$	−12.9706	−1.10815	−0.554075	+	0.832467i	$0.686928\pi$
$138$	0	0
$139$	−13.8995	−1.17894	−0.589470	−	0.807790i	$-0.700663\pi$
$139$	−13.8995	−1.17894	−0.589470	+	0.807790i	$0.700663\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.7279	1.14799
$144$	0	0
$145$	−6.65685	−0.552822
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.51472	−0.287937	−0.143968	−	0.989582i	$-0.545986\pi$
$149$	−3.51472	−0.287937	−0.143968	+	0.989582i	$0.545986\pi$
$150$	0	0
$151$	−11.8284	−0.962584	−0.481292	−	0.876560i	$-0.659832\pi$
$151$	−11.8284	−0.962584	−0.481292	+	0.876560i	$0.659832\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.58579	0.368339
$156$	0	0
$157$	10.4853	0.836817	0.418408	−	0.908259i	$-0.362588\pi$
$157$	10.4853	0.836817	0.418408	+	0.908259i	$0.362588\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.5563	1.68842	0.844212	−	0.536010i	$-0.180069\pi$
$163$	21.5563	1.68842	0.844212	+	0.536010i	$0.180069\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.41421	−0.186817	−0.0934087	−	0.995628i	$-0.529776\pi$
$167$	−2.41421	−0.186817	−0.0934087	+	0.995628i	$0.529776\pi$
$168$	0	0
$169$	−0.142136	−0.0109335
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.5563	−1.41081	−0.705407	−	0.708803i	$-0.749236\pi$
$173$	−18.5563	−1.41081	−0.705407	+	0.708803i	$0.749236\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	0	0
$181$	1.75736	0.130623	0.0653117	−	0.997865i	$-0.479196\pi$
$181$	1.75736	0.130623	0.0653117	+	0.997865i	$0.479196\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.41421	−0.251018
$186$	0	0
$187$	−24.5563	−1.79574
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.6569	−1.20525	−0.602624	−	0.798025i	$-0.705878\pi$
$191$	−16.6569	−1.20525	−0.602624	+	0.798025i	$0.705878\pi$
$192$	0	0
$193$	5.65685	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$193$	5.65685	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.5563	1.96331	0.981654	−	0.190669i	$-0.0610659\pi$
$197$	27.5563	1.96331	0.981654	+	0.190669i	$0.0610659\pi$
$198$	0	0
$199$	−26.7279	−1.89469	−0.947346	−	0.320212i	$-0.896246\pi$
$199$	−26.7279	−1.89469	−0.947346	+	0.320212i	$0.896246\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.585786	−0.0409131
$206$	0	0
$207$	0	0
$208$	0	0
$209$	14.0000	0.968400
$210$	0	0
$211$	−24.3137	−1.67382	−0.836912	−	0.547337i	$-0.815642\pi$
$211$	−24.3137	−1.67382	−0.836912	+	0.547337i	$0.815642\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.6569	0.794991
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−23.0000	−1.54715
$222$	0	0
$223$	24.0711	1.61192	0.805959	−	0.591971i	$-0.201650\pi$
$223$	24.0711	1.61192	0.805959	+	0.591971i	$0.201650\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.72792	−0.512920	−0.256460	−	0.966555i	$-0.582556\pi$
$227$	−7.72792	−0.512920	−0.256460	+	0.966555i	$0.582556\pi$
$228$	0	0
$229$	23.8995	1.57932	0.789662	−	0.613543i	$-0.210256\pi$
$229$	23.8995	1.57932	0.789662	+	0.613543i	$0.210256\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.17157	0.600850	0.300425	−	0.953805i	$-0.402872\pi$
$233$	9.17157	0.600850	0.300425	+	0.953805i	$0.402872\pi$
$234$	0	0
$235$	8.89949	0.580539
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.1716	−0.916683	−0.458341	−	0.888776i	$-0.651556\pi$
$239$	−14.1716	−0.916683	−0.458341	+	0.888776i	$0.651556\pi$
$240$	0	0
$241$	−3.55635	−0.229085	−0.114542	−	0.993418i	$-0.536540\pi$
$241$	−3.55635	−0.229085	−0.114542	+	0.993418i	$0.536540\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.1127	0.834341
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.0711	1.70871	0.854355	−	0.519689i	$-0.173952\pi$
$251$	27.0711	1.70871	0.854355	+	0.519689i	$0.173952\pi$
$252$	0	0
$253$	−2.24264	−0.140994
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.7990	1.60930	0.804648	−	0.593752i	$-0.202354\pi$
$257$	25.7990	1.60930	0.804648	+	0.593752i	$0.202354\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	0	0
$265$	3.75736	0.230813
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.7279	1.01992	0.509960	−	0.860198i	$-0.329660\pi$
$269$	16.7279	1.01992	0.509960	+	0.860198i	$0.329660\pi$
$270$	0	0
$271$	28.9706	1.75984	0.879918	−	0.475125i	$-0.157597\pi$
$271$	28.9706	1.75984	0.879918	+	0.475125i	$0.157597\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.82843	0.230863
$276$	0	0
$277$	−7.41421	−0.445477	−0.222738	−	0.974878i	$-0.571500\pi$
$277$	−7.41421	−0.445477	−0.222738	+	0.974878i	$0.571500\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.00000	0.0596550	0.0298275	−	0.999555i	$-0.490504\pi$
$281$	1.00000	0.0596550	0.0298275	+	0.999555i	$0.490504\pi$
$282$	0	0
$283$	−18.7574	−1.11501	−0.557505	−	0.830174i	$-0.688241\pi$
$283$	−18.7574	−1.11501	−0.557505	+	0.830174i	$0.688241\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	24.1421	1.42013
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.4142	0.842087	0.421044	−	0.907040i	$-0.361664\pi$
$293$	14.4142	0.842087	0.421044	+	0.907040i	$0.361664\pi$
$294$	0	0
$295$	−3.41421	−0.198783
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.10051	−0.121475
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.17157	0.296123
$306$	0	0
$307$	3.58579	0.204652	0.102326	−	0.994751i	$-0.467372\pi$
$307$	3.58579	0.204652	0.102326	+	0.994751i	$0.467372\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.97056	−0.395264	−0.197632	−	0.980276i	$-0.563325\pi$
$311$	−6.97056	−0.395264	−0.197632	+	0.980276i	$0.563325\pi$
$312$	0	0
$313$	16.5563	0.935820	0.467910	−	0.883776i	$-0.345007\pi$
$313$	16.5563	0.935820	0.467910	+	0.883776i	$0.345007\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.9706	0.840831	0.420415	−	0.907332i	$-0.361884\pi$
$317$	14.9706	0.840831	0.420415	+	0.907332i	$0.361884\pi$
$318$	0	0
$319$	−25.4853	−1.42690
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−23.4558	−1.30512
$324$	0	0
$325$	3.58579	0.198904
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.82843	0.265394	0.132697	−	0.991157i	$-0.457636\pi$
$331$	4.82843	0.265394	0.132697	+	0.991157i	$0.457636\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.0711	−0.604877
$336$	0	0
$337$	−18.7279	−1.02017	−0.510087	−	0.860123i	$-0.670387\pi$
$337$	−18.7279	−1.02017	−0.510087	+	0.860123i	$0.670387\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.5563	0.950730
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.41421	0.398016	0.199008	−	0.979998i	$-0.436228\pi$
$347$	7.41421	0.398016	0.199008	+	0.979998i	$0.436228\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.07107	0.110232	0.0551159	−	0.998480i	$-0.482447\pi$
$353$	2.07107	0.110232	0.0551159	+	0.998480i	$0.482447\pi$
$354$	0	0
$355$	−6.48528	−0.344203
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3137	−0.597115	−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137	−0.597115	−0.298557	+	0.954392i	$0.596505\pi$
$360$	0	0
$361$	−5.62742	−0.296180
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.17157	0.270692
$366$	0	0
$367$	19.7279	1.02979	0.514895	−	0.857254i	$-0.327831\pi$
$367$	19.7279	1.02979	0.514895	+	0.857254i	$0.327831\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.4853	0.853576	0.426788	−	0.904352i	$-0.359645\pi$
$373$	16.4853	0.853576	0.426788	+	0.904352i	$0.359645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.8701	−1.22937
$378$	0	0
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.51472	0.179594	0.0897969	−	0.995960i	$-0.471378\pi$
$383$	3.51472	0.179594	0.0897969	+	0.995960i	$0.471378\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.1421	−0.666333	−0.333166	−	0.942868i	$-0.608117\pi$
$389$	−13.1421	−0.666333	−0.333166	+	0.942868i	$0.608117\pi$
$390$	0	0
$391$	3.75736	0.190018
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.1421	0.661253
$396$	0	0
$397$	14.4142	0.723429	0.361714	−	0.932289i	$-0.382192\pi$
$397$	14.4142	0.723429	0.361714	+	0.932289i	$0.382192\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.514719	−0.0257038	−0.0128519	−	0.999917i	$-0.504091\pi$
$401$	−0.514719	−0.0257038	−0.0128519	+	0.999917i	$0.504091\pi$
$402$	0	0
$403$	16.4437	0.819117
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.0711	−0.647909
$408$	0	0
$409$	32.8284	1.62326	0.811631	−	0.584171i	$-0.198580\pi$
$409$	32.8284	1.62326	0.811631	+	0.584171i	$0.198580\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.55635	0.466858	0.233429	−	0.972374i	$-0.425005\pi$
$419$	9.55635	0.466858	0.233429	+	0.972374i	$0.425005\pi$
$420$	0	0
$421$	20.3137	0.990030	0.495015	−	0.868885i	$-0.335163\pi$
$421$	20.3137	0.990030	0.495015	+	0.868885i	$0.335163\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.41421	−0.311135
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.82843	0.473419	0.236709	−	0.971581i	$-0.423931\pi$
$431$	9.82843	0.473419	0.236709	+	0.971581i	$0.423931\pi$
$432$	0	0
$433$	−22.2843	−1.07091	−0.535457	−	0.844563i	$-0.679861\pi$
$433$	−22.2843	−1.07091	−0.535457	+	0.844563i	$0.679861\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.14214	−0.102472
$438$	0	0
$439$	24.2426	1.15704	0.578519	−	0.815669i	$-0.303631\pi$
$439$	24.2426	1.15704	0.578519	+	0.815669i	$0.303631\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.4558	1.68456	0.842279	−	0.539042i	$-0.181214\pi$
$443$	35.4558	1.68456	0.842279	+	0.539042i	$0.181214\pi$
$444$	0	0
$445$	−16.9706	−0.804482
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−23.8284	−1.12453	−0.562267	−	0.826956i	$-0.690070\pi$
$449$	−23.8284	−1.12453	−0.562267	+	0.826956i	$0.690070\pi$
$450$	0	0
$451$	−2.24264	−0.105602
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.0711	−0.704995	−0.352497	−	0.935813i	$-0.614667\pi$
$457$	−15.0711	−0.704995	−0.352497	+	0.935813i	$0.614667\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.3431	−1.04062	−0.520312	−	0.853976i	$-0.674184\pi$
$461$	−22.3431	−1.04062	−0.520312	+	0.853976i	$0.674184\pi$
$462$	0	0
$463$	−13.4558	−0.625346	−0.312673	−	0.949861i	$-0.601224\pi$
$463$	−13.4558	−0.625346	−0.312673	+	0.949861i	$0.601224\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.899495	−0.0416237	−0.0208118	−	0.999783i	$-0.506625\pi$
$467$	−0.899495	−0.0416237	−0.0208118	+	0.999783i	$0.506625\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	44.6274	2.05197
$474$	0	0
$475$	3.65685	0.167788
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.41421	−0.430146	−0.215073	−	0.976598i	$-0.568999\pi$
$479$	−9.41421	−0.430146	−0.215073	+	0.976598i	$0.568999\pi$
$480$	0	0
$481$	−12.2426	−0.558216
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.7279	0.714168
$486$	0	0
$487$	−7.41421	−0.335970	−0.167985	−	0.985790i	$-0.553726\pi$
$487$	−7.41421	−0.335970	−0.167985	+	0.985790i	$0.553726\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.2843	−0.780028	−0.390014	−	0.920809i	$-0.627530\pi$
$491$	−17.2843	−0.780028	−0.390014	+	0.920809i	$0.627530\pi$
$492$	0	0
$493$	42.6985	1.92304
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.6274	−0.610047	−0.305023	−	0.952345i	$-0.598664\pi$
$499$	−13.6274	−0.610047	−0.305023	+	0.952345i	$0.598664\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−39.0416	−1.74078	−0.870390	−	0.492363i	$-0.836133\pi$
$503$	−39.0416	−1.74078	−0.870390	+	0.492363i	$0.836133\pi$
$504$	0	0
$505$	4.82843	0.214862
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.0711	1.55450	0.777249	−	0.629193i	$-0.216615\pi$
$509$	35.0711	1.55450	0.777249	+	0.629193i	$0.216615\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.58579	0.0698781
$516$	0	0
$517$	34.0711	1.49844
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.68629	−0.292932	−0.146466	−	0.989216i	$-0.546790\pi$
$521$	−6.68629	−0.292932	−0.146466	+	0.989216i	$0.546790\pi$
$522$	0	0
$523$	1.85786	0.0812387	0.0406194	−	0.999175i	$-0.487067\pi$
$523$	1.85786	0.0812387	0.0406194	+	0.999175i	$0.487067\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.4142	−1.28130
$528$	0	0
$529$	−22.6569	−0.985081
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.10051	−0.0909830
$534$	0	0
$535$	16.8284	0.727556
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	15.9706	0.686628	0.343314	−	0.939221i	$-0.388450\pi$
$541$	15.9706	0.686628	0.343314	+	0.939221i	$0.388450\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.00000	0.385518
$546$	0	0
$547$	7.51472	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24.3431	−1.03705
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.79899	0.330454	0.165227	−	0.986256i	$-0.447164\pi$
$557$	7.79899	0.330454	0.165227	+	0.986256i	$0.447164\pi$
$558$	0	0
$559$	41.7990	1.76791
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20.6274	0.869342	0.434671	−	0.900589i	$-0.356865\pi$
$563$	20.6274	0.869342	0.434671	+	0.900589i	$0.356865\pi$
$564$	0	0
$565$	5.07107	0.213341
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.7990	1.33308	0.666542	−	0.745468i	$-0.267774\pi$
$569$	31.7990	1.33308	0.666542	+	0.745468i	$0.267774\pi$
$570$	0	0
$571$	4.82843	0.202063	0.101032	−	0.994883i	$-0.467786\pi$
$571$	4.82843	0.202063	0.101032	+	0.994883i	$0.467786\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.585786	−0.0244290
$576$	0	0
$577$	−14.0711	−0.585786	−0.292893	−	0.956145i	$-0.594618\pi$
$577$	−14.0711	−0.585786	−0.292893	+	0.956145i	$0.594618\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	14.3848	0.595757
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.8284	1.27243	0.636213	−	0.771514i	$-0.280500\pi$
$587$	30.8284	1.27243	0.636213	+	0.771514i	$0.280500\pi$
$588$	0	0
$589$	16.7696	0.690977
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−17.7279	−0.727999	−0.363999	−	0.931399i	$-0.618589\pi$
$593$	−17.7279	−0.727999	−0.363999	+	0.931399i	$0.618589\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.7990	0.522953	0.261476	−	0.965210i	$-0.415791\pi$
$599$	12.7990	0.522953	0.261476	+	0.965210i	$0.415791\pi$
$600$	0	0
$601$	39.6569	1.61764	0.808818	−	0.588058i	$-0.200108\pi$
$601$	39.6569	1.61764	0.808818	+	0.588058i	$0.200108\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.65685	0.148672
$606$	0	0
$607$	13.9289	0.565358	0.282679	−	0.959215i	$-0.408777\pi$
$607$	13.9289	0.565358	0.282679	+	0.959215i	$0.408777\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.9117	1.29101
$612$	0	0
$613$	−41.3137	−1.66864	−0.834322	−	0.551277i	$-0.814141\pi$
$613$	−41.3137	−1.66864	−0.834322	+	0.551277i	$0.814141\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.8701	−1.64537	−0.822683	−	0.568500i	$-0.807524\pi$
$617$	−40.8701	−1.64537	−0.822683	+	0.568500i	$0.807524\pi$
$618$	0	0
$619$	10.8701	0.436905	0.218452	−	0.975848i	$-0.429899\pi$
$619$	10.8701	0.436905	0.218452	+	0.975848i	$0.429899\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	21.8995	0.873190
$630$	0	0
$631$	−39.4853	−1.57188	−0.785942	−	0.618300i	$-0.787822\pi$
$631$	−39.4853	−1.57188	−0.785942	+	0.618300i	$0.787822\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21.8995	−0.869055
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.3137	−1.55280	−0.776399	−	0.630242i	$-0.782956\pi$
$641$	−39.3137	−1.55280	−0.776399	+	0.630242i	$0.782956\pi$
$642$	0	0
$643$	4.21320	0.166153	0.0830763	−	0.996543i	$-0.473525\pi$
$643$	4.21320	0.166153	0.0830763	+	0.996543i	$0.473525\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.1127	−1.38042	−0.690211	−	0.723608i	$-0.742482\pi$
$647$	−35.1127	−1.38042	−0.690211	+	0.723608i	$0.742482\pi$
$648$	0	0
$649$	−13.0711	−0.513084
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.6863	1.04432	0.522158	−	0.852849i	$-0.325127\pi$
$653$	26.6863	1.04432	0.522158	+	0.852849i	$0.325127\pi$
$654$	0	0
$655$	11.7574	0.459398
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.6569	0.804677	0.402338	−	0.915491i	$-0.368198\pi$
$659$	20.6569	0.804677	0.402338	+	0.915491i	$0.368198\pi$
$660$	0	0
$661$	−34.1421	−1.32798	−0.663988	−	0.747744i	$-0.731137\pi$
$661$	−34.1421	−1.32798	−0.663988	+	0.747744i	$0.731137\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.89949	0.150989
$668$	0	0
$669$	0	0
$670$	0	0
$671$	19.7990	0.764332
$672$	0	0
$673$	−27.5147	−1.06061	−0.530307	−	0.847806i	$-0.677923\pi$
$673$	−27.5147	−1.06061	−0.530307	+	0.847806i	$0.677923\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.2721	0.394788	0.197394	−	0.980324i	$-0.436752\pi$
$677$	10.2721	0.394788	0.197394	+	0.980324i	$0.436752\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−35.1127	−1.34355	−0.671775	−	0.740755i	$-0.734468\pi$
$683$	−35.1127	−1.34355	−0.671775	+	0.740755i	$0.734468\pi$
$684$	0	0
$685$	−12.9706	−0.495580
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.4731	0.513284
$690$	0	0
$691$	3.51472	0.133706	0.0668531	−	0.997763i	$-0.478704\pi$
$691$	3.51472	0.133706	0.0668531	+	0.997763i	$0.478704\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.8995	−0.527238
$696$	0	0
$697$	3.75736	0.142320
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.514719	−0.0194407	−0.00972033	−	0.999953i	$-0.503094\pi$
$701$	−0.514719	−0.0194407	−0.00972033	+	0.999953i	$0.503094\pi$
$702$	0	0
$703$	−12.4853	−0.470891
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.62742	0.136231	0.0681153	−	0.997677i	$-0.478301\pi$
$709$	3.62742	0.136231	0.0681153	+	0.997677i	$0.478301\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.68629	−0.100602
$714$	0	0
$715$	13.7279	0.513395
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−15.7574	−0.587650	−0.293825	−	0.955859i	$-0.594928\pi$
$719$	−15.7574	−0.587650	−0.293825	+	0.955859i	$0.594928\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.65685	−0.247229
$726$	0	0
$727$	−18.0000	−0.667583	−0.333792	−	0.942647i	$-0.608328\pi$
$727$	−18.0000	−0.667583	−0.333792	+	0.942647i	$0.608328\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−74.7696	−2.76545
$732$	0	0
$733$	−30.6985	−1.13387	−0.566937	−	0.823761i	$-0.691872\pi$
$733$	−30.6985	−1.13387	−0.566937	+	0.823761i	$0.691872\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−42.3848	−1.56126
$738$	0	0
$739$	20.6569	0.759875	0.379937	−	0.925012i	$-0.375946\pi$
$739$	20.6569	0.759875	0.379937	+	0.925012i	$0.375946\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.92893	0.327571	0.163785	−	0.986496i	$-0.447630\pi$
$743$	8.92893	0.327571	0.163785	+	0.986496i	$0.447630\pi$
$744$	0	0
$745$	−3.51472	−0.128769
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.14214	0.0416771	0.0208386	−	0.999783i	$-0.493366\pi$
$751$	1.14214	0.0416771	0.0208386	+	0.999783i	$0.493366\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.8284	−0.430481
$756$	0	0
$757$	37.1716	1.35102	0.675512	−	0.737349i	$-0.263923\pi$
$757$	37.1716	1.35102	0.675512	+	0.737349i	$0.263923\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.8701	0.684039	0.342020	−	0.939693i	$-0.388889\pi$
$761$	18.8701	0.684039	0.342020	+	0.939693i	$0.388889\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.2426	−0.442056
$768$	0	0
$769$	28.1421	1.01483	0.507416	−	0.861701i	$-0.330601\pi$
$769$	28.1421	1.01483	0.507416	+	0.861701i	$0.330601\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	43.7279	1.57278	0.786392	−	0.617728i	$-0.211947\pi$
$773$	43.7279	1.57278	0.786392	+	0.617728i	$0.211947\pi$
$774$	0	0
$775$	4.58579	0.164726
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.14214	−0.0767500
$780$	0	0
$781$	−24.8284	−0.888431
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.4853	0.374236
$786$	0	0
$787$	−43.7279	−1.55873	−0.779366	−	0.626569i	$-0.784459\pi$
$787$	−43.7279	−1.55873	−0.779366	+	0.626569i	$0.784459\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	18.5442	0.658522
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.3848	1.25339	0.626697	−	0.779263i	$-0.284407\pi$
$797$	35.3848	1.25339	0.626697	+	0.779263i	$0.284407\pi$
$798$	0	0
$799$	−57.0833	−2.01946
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.7990	0.698691
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.02944	−0.211984	−0.105992	−	0.994367i	$-0.533802\pi$
$809$	−6.02944	−0.211984	−0.105992	+	0.994367i	$0.533802\pi$
$810$	0	0
$811$	−19.5563	−0.686716	−0.343358	−	0.939205i	$-0.611564\pi$
$811$	−19.5563	−0.686716	−0.343358	+	0.939205i	$0.611564\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	21.5563	0.755086
$816$	0	0
$817$	42.6274	1.49134
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−47.8284	−1.66922	−0.834612	−	0.550839i	$-0.814308\pi$
$821$	−47.8284	−1.66922	−0.834612	+	0.550839i	$0.814308\pi$
$822$	0	0
$823$	36.5269	1.27325	0.636624	−	0.771174i	$-0.280330\pi$
$823$	36.5269	1.27325	0.636624	+	0.771174i	$0.280330\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.0416	0.349182	0.174591	−	0.984641i	$-0.444140\pi$
$827$	10.0416	0.349182	0.174591	+	0.984641i	$0.444140\pi$
$828$	0	0
$829$	22.7279	0.789373	0.394687	−	0.918816i	$-0.370853\pi$
$829$	22.7279	0.789373	0.394687	+	0.918816i	$0.370853\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.41421	−0.0835473
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.3848	−0.772808	−0.386404	−	0.922330i	$-0.626283\pi$
$839$	−22.3848	−0.772808	−0.386404	+	0.922330i	$0.626283\pi$
$840$	0	0
$841$	15.3137	0.528059
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.142136	−0.00488961
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	−30.2843	−1.03691	−0.518457	−	0.855104i	$-0.673493\pi$
$853$	−30.2843	−1.03691	−0.518457	+	0.855104i	$0.673493\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20.8284	0.711486	0.355743	−	0.934584i	$-0.384228\pi$
$857$	20.8284	0.711486	0.355743	+	0.934584i	$0.384228\pi$
$858$	0	0
$859$	−45.4558	−1.55093	−0.775467	−	0.631388i	$-0.782485\pi$
$859$	−45.4558	−1.55093	−0.775467	+	0.631388i	$0.782485\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.7574	0.672548	0.336274	−	0.941764i	$-0.390833\pi$
$863$	19.7574	0.672548	0.336274	+	0.941764i	$0.390833\pi$
$864$	0	0
$865$	−18.5563	−0.630935
$866$	0	0
$867$	0	0
$868$	0	0
$869$	50.3137	1.70678
$870$	0	0
$871$	−39.6985	−1.34513
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.6863	0.360850	0.180425	−	0.983589i	$-0.442253\pi$
$877$	10.6863	0.360850	0.180425	+	0.983589i	$0.442253\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	56.2843	1.89627	0.948133	−	0.317875i	$-0.102969\pi$
$881$	56.2843	1.89627	0.948133	+	0.317875i	$0.102969\pi$
$882$	0	0
$883$	−7.11270	−0.239361	−0.119681	−	0.992812i	$-0.538187\pi$
$883$	−7.11270	−0.239361	−0.119681	+	0.992812i	$0.538187\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−57.5980	−1.93395	−0.966975	−	0.254870i	$-0.917967\pi$
$887$	−57.5980	−1.93395	−0.966975	+	0.254870i	$0.917967\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.5442	1.08905
$894$	0	0
$895$	−6.48528	−0.216779
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−30.5269	−1.01813
$900$	0	0
$901$	−24.1005	−0.802904
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.75736	0.0584166
$906$	0	0
$907$	22.5269	0.747994	0.373997	−	0.927430i	$-0.377987\pi$
$907$	22.5269	0.747994	0.373997	+	0.927430i	$0.377987\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	55.5980	1.84204	0.921022	−	0.389511i	$-0.127356\pi$
$911$	55.5980	1.84204	0.921022	+	0.389511i	$0.127356\pi$
$912$	0	0
$913$	30.6274	1.01362
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.5147	−0.610744	−0.305372	−	0.952233i	$-0.598781\pi$
$919$	−18.5147	−0.610744	−0.305372	+	0.952233i	$0.598781\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−23.2548	−0.765442
$924$	0	0
$925$	−3.41421	−0.112259
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.2721	0.697914	0.348957	−	0.937139i	$-0.386536\pi$
$929$	21.2721	0.697914	0.348957	+	0.937139i	$0.386536\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.5563	−0.803078
$936$	0	0
$937$	−7.72792	−0.252460	−0.126230	−	0.992001i	$-0.540288\pi$
$937$	−7.72792	−0.252460	−0.126230	+	0.992001i	$0.540288\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	4.00000	0.130396	0.0651981	−	0.997872i	$-0.479232\pi$
$941$	4.00000	0.130396	0.0651981	+	0.997872i	$0.479232\pi$
$942$	0	0
$943$	0.343146	0.0111744
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0416	1.04121	0.520607	−	0.853797i	$-0.325706\pi$
$947$	32.0416	1.04121	0.520607	+	0.853797i	$0.325706\pi$
$948$	0	0
$949$	18.5442	0.601969
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.8701	1.25912	0.629562	−	0.776950i	$-0.283234\pi$
$953$	38.8701	1.25912	0.629562	+	0.776950i	$0.283234\pi$
$954$	0	0
$955$	−16.6569	−0.539003
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−9.97056	−0.321631
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5.65685	0.182101
$966$	0	0
$967$	−7.45584	−0.239764	−0.119882	−	0.992788i	$-0.538252\pi$
$967$	−7.45584	−0.239764	−0.119882	+	0.992788i	$0.538252\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.5269	0.722923	0.361462	−	0.932387i	$-0.382278\pi$
$971$	22.5269	0.722923	0.361462	+	0.932387i	$0.382278\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.14214	0.260490	0.130245	−	0.991482i	$-0.458424\pi$
$977$	8.14214	0.260490	0.130245	+	0.991482i	$0.458424\pi$
$978$	0	0
$979$	−64.9706	−2.07647
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.0122	1.33998	0.669990	−	0.742370i	$-0.266298\pi$
$983$	42.0122	1.33998	0.669990	+	0.742370i	$0.266298\pi$
$984$	0	0
$985$	27.5563	0.878018
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.82843	−0.217131
$990$	0	0
$991$	25.3137	0.804116	0.402058	−	0.915614i	$-0.368295\pi$
$991$	25.3137	0.804116	0.402058	+	0.915614i	$0.368295\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−26.7279	−0.847332
$996$	0	0
$997$	−50.8406	−1.61014	−0.805069	−	0.593181i	$-0.797872\pi$
$997$	−50.8406	−1.61014	−0.805069	+	0.593181i	$0.797872\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.bl.1.2		2
3.2	odd	2		980.2.a.k.1.1	yes	2
7.6	odd	2		8820.2.a.bg.1.2		2
12.11	even	2		3920.2.a.bo.1.2		2
15.2	even	4		4900.2.e.r.2549.3		4
15.8	even	4		4900.2.e.r.2549.2		4
15.14	odd	2		4900.2.a.x.1.2		2
21.2	odd	6		980.2.i.k.361.2		4
21.5	even	6		980.2.i.l.361.1		4
21.11	odd	6		980.2.i.k.961.2		4
21.17	even	6		980.2.i.l.961.1		4
21.20	even	2		980.2.a.j.1.2	✓	2
84.83	odd	2		3920.2.a.bx.1.1		2
105.62	odd	4		4900.2.e.q.2549.2		4
105.83	odd	4		4900.2.e.q.2549.3		4
105.104	even	2		4900.2.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.a.j.1.2	✓	2	21.20	even	2
980.2.a.k.1.1	yes	2	3.2	odd	2
980.2.i.k.361.2		4	21.2	odd	6
980.2.i.k.961.2		4	21.11	odd	6
980.2.i.l.361.1		4	21.5	even	6
980.2.i.l.961.1		4	21.17	even	6
3920.2.a.bo.1.2		2	12.11	even	2
3920.2.a.bx.1.1		2	84.83	odd	2
4900.2.a.x.1.2		2	15.14	odd	2
4900.2.a.z.1.1		2	105.104	even	2
4900.2.e.q.2549.2		4	105.62	odd	4
4900.2.e.q.2549.3		4	105.83	odd	4
4900.2.e.r.2549.2		4	15.8	even	4
4900.2.e.r.2549.3		4	15.2	even	4
8820.2.a.bg.1.2		2	7.6	odd	2
8820.2.a.bl.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8820.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +3.82843 q^{11} +3.58579 q^{13} -6.41421 q^{17} +3.65685 q^{19} -0.585786 q^{23} +1.00000 q^{25} -6.65685 q^{29} +4.58579 q^{31} -3.41421 q^{37} -0.585786 q^{41} +11.6569 q^{43} +8.89949 q^{47} +3.75736 q^{53} +3.82843 q^{55} -3.41421 q^{59} +5.17157 q^{61} +3.58579 q^{65} -11.0711 q^{67} -6.48528 q^{71} +5.17157 q^{73} +13.1421 q^{79} +8.00000 q^{83} -6.41421 q^{85} -16.9706 q^{89} +3.65685 q^{95} +15.7279 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{11} + 10 q^{13} - 10 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{29} + 12 q^{31} - 4 q^{37} - 4 q^{41} + 12 q^{43} - 2 q^{47} + 16 q^{53} + 2 q^{55} - 4 q^{59} + 16 q^{61} + 10 q^{65}+ \cdots + 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^11 + 10 * q^13 - 10 * q^17 - 4 * q^19 - 4 * q^23 + 2 * q^25 - 2 * q^29 + 12 * q^31 - 4 * q^37 - 4 * q^41 + 12 * q^43 - 2 * q^47 + 16 * q^53 + 2 * q^55 - 4 * q^59 + 16 * q^61 + 10 * q^65 - 8 * q^67 + 4 * q^71 + 16 * q^73 - 2 * q^79 + 16 * q^83 - 10 * q^85 - 4 * q^95 + 6 * q^97

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