LMFDB - Embedded newform 8820.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8820,2,Mod(1,8820)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 420)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{5} +O(q^{10})$	$q+1.00000 q^{5} -4.24264 q^{11} -5.24264 q^{13} +4.24264 q^{17} -7.00000 q^{19} -4.24264 q^{23} +1.00000 q^{25} +10.2426 q^{29} +7.48528 q^{31} -5.24264 q^{37} -4.24264 q^{41} -5.24264 q^{43} +6.00000 q^{47} +8.48528 q^{53} -4.24264 q^{55} +1.75736 q^{59} -12.4853 q^{61} -5.24264 q^{65} +3.24264 q^{67} -12.7279 q^{71} +0.757359 q^{73} +11.0000 q^{79} +1.75736 q^{83} +4.24264 q^{85} +1.75736 q^{89} -7.00000 q^{95} +16.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 2 q^{13} - 14 q^{19} + 2 q^{25} + 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{43} + 12 q^{47} + 12 q^{59} - 8 q^{61} - 2 q^{65} - 2 q^{67} + 10 q^{73} + 22 q^{79} + 12 q^{83} + 12 q^{89} - 14 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^13 - 14 * q^19 + 2 * q^25 + 12 * q^29 - 2 * q^31 - 2 * q^37 - 2 * q^43 + 12 * q^47 + 12 * q^59 - 8 * q^61 - 2 * q^65 - 2 * q^67 + 10 * q^73 + 22 * q^79 + 12 * q^83 + 12 * q^89 - 14 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	−5.24264	−1.45405	−0.727023	−	0.686613i	$-0.759097\pi$
$13$	−5.24264	−1.45405	−0.727023	+	0.686613i	$0.759097\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.24264	−0.884652	−0.442326	−	0.896854i	$-0.645847\pi$
$23$	−4.24264	−0.884652	−0.442326	+	0.896854i	$0.645847\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10.2426	1.90201	0.951005	−	0.309175i	$-0.100053\pi$
$29$	10.2426	1.90201	0.951005	+	0.309175i	$0.100053\pi$
$30$	0	0
$31$	7.48528	1.34440	0.672198	−	0.740371i	$-0.265350\pi$
$31$	7.48528	1.34440	0.672198	+	0.740371i	$0.265350\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.24264	−0.861885	−0.430942	−	0.902379i	$-0.641819\pi$
$37$	−5.24264	−0.861885	−0.430942	+	0.902379i	$0.641819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	0	0
$43$	−5.24264	−0.799495	−0.399748	−	0.916625i	$-0.630902\pi$
$43$	−5.24264	−0.799495	−0.399748	+	0.916625i	$0.630902\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.48528	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$54$	0	0
$55$	−4.24264	−0.572078
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.75736	0.228789	0.114394	−	0.993435i	$-0.463507\pi$
$59$	1.75736	0.228789	0.114394	+	0.993435i	$0.463507\pi$
$60$	0	0
$61$	−12.4853	−1.59858	−0.799288	−	0.600948i	$-0.794790\pi$
$61$	−12.4853	−1.59858	−0.799288	+	0.600948i	$0.794790\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.24264	−0.650270
$66$	0	0
$67$	3.24264	0.396152	0.198076	−	0.980187i	$-0.436531\pi$
$67$	3.24264	0.396152	0.198076	+	0.980187i	$0.436531\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.7279	−1.51053	−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279	−1.51053	−0.755263	+	0.655422i	$0.772491\pi$
$72$	0	0
$73$	0.757359	0.0886422	0.0443211	−	0.999017i	$-0.485888\pi$
$73$	0.757359	0.0886422	0.0443211	+	0.999017i	$0.485888\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.75736	0.192895	0.0964476	−	0.995338i	$-0.469252\pi$
$83$	1.75736	0.192895	0.0964476	+	0.995338i	$0.469252\pi$
$84$	0	0
$85$	4.24264	0.460179
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.75736	0.186280	0.0931399	−	0.995653i	$-0.470310\pi$
$89$	1.75736	0.186280	0.0931399	+	0.995653i	$0.470310\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.00000	−0.718185
$96$	0	0
$97$	16.4853	1.67383	0.836913	−	0.547335i	$-0.184358\pi$
$97$	16.4853	1.67383	0.836913	+	0.547335i	$0.184358\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.2426	1.61620	0.808102	−	0.589043i	$-0.200495\pi$
$101$	16.2426	1.61620	0.808102	+	0.589043i	$0.200495\pi$
$102$	0	0
$103$	−8.75736	−0.862888	−0.431444	−	0.902140i	$-0.641996\pi$
$103$	−8.75736	−0.862888	−0.431444	+	0.902140i	$0.641996\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.7279	1.23045	0.615227	−	0.788350i	$-0.289064\pi$
$107$	12.7279	1.23045	0.615227	+	0.788350i	$0.289064\pi$
$108$	0	0
$109$	7.48528	0.716960	0.358480	−	0.933537i	$-0.383295\pi$
$109$	7.48528	0.716960	0.358480	+	0.933537i	$0.383295\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−4.24264	−0.395628
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.2426	−0.997623	−0.498812	−	0.866710i	$-0.666230\pi$
$127$	−11.2426	−0.997623	−0.498812	+	0.866710i	$0.666230\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.48528	0.217140	0.108570	−	0.994089i	$-0.465373\pi$
$131$	2.48528	0.217140	0.108570	+	0.994089i	$0.465373\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.24264	0.362473	0.181237	−	0.983440i	$-0.441990\pi$
$137$	4.24264	0.362473	0.181237	+	0.983440i	$0.441990\pi$
$138$	0	0
$139$	1.48528	0.125980	0.0629900	−	0.998014i	$-0.479936\pi$
$139$	1.48528	0.125980	0.0629900	+	0.998014i	$0.479936\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.2426	1.86002
$144$	0	0
$145$	10.2426	0.850605
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	5.51472	0.448781	0.224391	−	0.974499i	$-0.427961\pi$
$151$	5.51472	0.448781	0.224391	+	0.974499i	$0.427961\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.48528	0.601232
$156$	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
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.72792	0.520622	0.260311	−	0.965525i	$-0.416175\pi$
$167$	6.72792	0.520622	0.260311	+	0.965525i	$0.416175\pi$
$168$	0	0
$169$	14.4853	1.11425
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.51472	0.267219	0.133610	−	0.991034i	$-0.457343\pi$
$173$	3.51472	0.267219	0.133610	+	0.991034i	$0.457343\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.24264	−0.385447
$186$	0	0
$187$	−18.0000	−1.31629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	15.2426	1.09719	0.548595	−	0.836088i	$-0.315163\pi$
$193$	15.2426	1.09719	0.548595	+	0.836088i	$0.315163\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.75736	0.552689	0.276344	−	0.961059i	$-0.410877\pi$
$197$	7.75736	0.552689	0.276344	+	0.961059i	$0.410877\pi$
$198$	0	0
$199$	−6.48528	−0.459729	−0.229865	−	0.973223i	$-0.573828\pi$
$199$	−6.48528	−0.459729	−0.229865	+	0.973223i	$0.573828\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.24264	−0.296319
$206$	0	0
$207$	0	0
$208$	0	0
$209$	29.6985	2.05429
$210$	0	0
$211$	5.51472	0.379649	0.189824	−	0.981818i	$-0.439208\pi$
$211$	5.51472	0.379649	0.189824	+	0.981818i	$0.439208\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.24264	−0.357545
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−22.2426	−1.49620
$222$	0	0
$223$	−24.4853	−1.63966	−0.819828	−	0.572610i	$-0.805931\pi$
$223$	−24.4853	−1.63966	−0.819828	+	0.572610i	$0.805931\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−27.2132	−1.80620	−0.903102	−	0.429426i	$-0.858716\pi$
$227$	−27.2132	−1.80620	−0.903102	+	0.429426i	$0.858716\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.48528	0.162816	0.0814081	−	0.996681i	$-0.474058\pi$
$233$	2.48528	0.162816	0.0814081	+	0.996681i	$0.474058\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.9706	1.48584	0.742921	−	0.669379i	$-0.233440\pi$
$239$	22.9706	1.48584	0.742921	+	0.669379i	$0.233440\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	36.6985	2.33507
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.72792	−0.424663	−0.212331	−	0.977198i	$-0.568106\pi$
$251$	−6.72792	−0.424663	−0.212331	+	0.977198i	$0.568106\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.75736	−0.109621	−0.0548105	−	0.998497i	$-0.517455\pi$
$257$	−1.75736	−0.109621	−0.0548105	+	0.998497i	$0.517455\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	8.48528	0.521247
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.4853	−0.883183	−0.441592	−	0.897216i	$-0.645586\pi$
$269$	−14.4853	−0.883183	−0.441592	+	0.897216i	$0.645586\pi$
$270$	0	0
$271$	27.4558	1.66782	0.833912	−	0.551898i	$-0.186096\pi$
$271$	27.4558	1.66782	0.833912	+	0.551898i	$0.186096\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	−7.72792	−0.464326	−0.232163	−	0.972677i	$-0.574580\pi$
$277$	−7.72792	−0.464326	−0.232163	+	0.972677i	$0.574580\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.97056	0.296519	0.148259	−	0.988948i	$-0.452633\pi$
$281$	4.97056	0.296519	0.148259	+	0.988948i	$0.452633\pi$
$282$	0	0
$283$	−1.72792	−0.102714	−0.0513572	−	0.998680i	$-0.516355\pi$
$283$	−1.72792	−0.102714	−0.0513572	+	0.998680i	$0.516355\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−28.9706	−1.69248	−0.846239	−	0.532803i	$-0.821139\pi$
$293$	−28.9706	−1.69248	−0.846239	+	0.532803i	$0.821139\pi$
$294$	0	0
$295$	1.75736	0.102317
$296$	0	0
$297$	0	0
$298$	0	0
$299$	22.2426	1.28633
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.4853	−0.714905
$306$	0	0
$307$	−5.24264	−0.299213	−0.149607	−	0.988746i	$-0.547801\pi$
$307$	−5.24264	−0.299213	−0.149607	+	0.988746i	$0.547801\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.2132	1.20289	0.601445	−	0.798914i	$-0.294592\pi$
$311$	21.2132	1.20289	0.601445	+	0.798914i	$0.294592\pi$
$312$	0	0
$313$	−1.72792	−0.0976679	−0.0488340	−	0.998807i	$-0.515551\pi$
$313$	−1.72792	−0.0976679	−0.0488340	+	0.998807i	$0.515551\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.727922	−0.0408842	−0.0204421	−	0.999791i	$-0.506507\pi$
$317$	−0.727922	−0.0408842	−0.0204421	+	0.999791i	$0.506507\pi$
$318$	0	0
$319$	−43.4558	−2.43306
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−29.6985	−1.65247
$324$	0	0
$325$	−5.24264	−0.290809
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.24264	0.177164
$336$	0	0
$337$	11.7279	0.638861	0.319430	−	0.947610i	$-0.396508\pi$
$337$	11.7279	0.638861	0.319430	+	0.947610i	$0.396508\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−31.7574	−1.71976
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.75736	0.0935348	0.0467674	−	0.998906i	$-0.485108\pi$
$353$	1.75736	0.0935348	0.0467674	+	0.998906i	$0.485108\pi$
$354$	0	0
$355$	−12.7279	−0.675528
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.2426	−0.540586	−0.270293	−	0.962778i	$-0.587121\pi$
$359$	−10.2426	−0.540586	−0.270293	+	0.962778i	$0.587121\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.757359	0.0396420
$366$	0	0
$367$	10.2721	0.536198	0.268099	−	0.963391i	$-0.413605\pi$
$367$	10.2721	0.536198	0.268099	+	0.963391i	$0.413605\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−7.72792	−0.400137	−0.200068	−	0.979782i	$-0.564116\pi$
$373$	−7.72792	−0.400137	−0.200068	+	0.979782i	$0.564116\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−53.6985	−2.76561
$378$	0	0
$379$	30.4558	1.56441	0.782206	−	0.623020i	$-0.214095\pi$
$379$	30.4558	1.56441	0.782206	+	0.623020i	$0.214095\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.4558	1.30073	0.650366	−	0.759621i	$-0.274615\pi$
$383$	25.4558	1.30073	0.650366	+	0.759621i	$0.274615\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−21.2132	−1.07555	−0.537776	−	0.843088i	$-0.680735\pi$
$389$	−21.2132	−1.07555	−0.537776	+	0.843088i	$0.680735\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.0000	0.553470
$396$	0	0
$397$	−17.2426	−0.865383	−0.432692	−	0.901542i	$-0.642436\pi$
$397$	−17.2426	−0.865383	−0.432692	+	0.901542i	$0.642436\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.4853	1.02299	0.511493	−	0.859287i	$-0.329093\pi$
$401$	20.4853	1.02299	0.511493	+	0.859287i	$0.329093\pi$
$402$	0	0
$403$	−39.2426	−1.95482
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.2426	1.10253
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.75736	0.0862654
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.48528	−0.121414	−0.0607070	−	0.998156i	$-0.519336\pi$
$419$	−2.48528	−0.121414	−0.0607070	+	0.998156i	$0.519336\pi$
$420$	0	0
$421$	14.5147	0.707404	0.353702	−	0.935358i	$-0.384923\pi$
$421$	14.5147	0.707404	0.353702	+	0.935358i	$0.384923\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.24264	0.205798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.4558	1.51517	0.757587	−	0.652734i	$-0.226378\pi$
$431$	31.4558	1.51517	0.757587	+	0.652734i	$0.226378\pi$
$432$	0	0
$433$	24.7574	1.18976	0.594881	−	0.803814i	$-0.297199\pi$
$433$	24.7574	1.18976	0.594881	+	0.803814i	$0.297199\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	29.6985	1.42067
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.51472	−0.166989	−0.0834947	−	0.996508i	$-0.526608\pi$
$443$	−3.51472	−0.166989	−0.0834947	+	0.996508i	$0.526608\pi$
$444$	0	0
$445$	1.75736	0.0833068
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.2132	−0.477754	−0.238877	−	0.971050i	$-0.576779\pi$
$457$	−10.2132	−0.477754	−0.238877	+	0.971050i	$0.576779\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.72792	0.313351	0.156675	−	0.987650i	$-0.449922\pi$
$461$	6.72792	0.313351	0.156675	+	0.987650i	$0.449922\pi$
$462$	0	0
$463$	35.7279	1.66042	0.830209	−	0.557453i	$-0.188221\pi$
$463$	35.7279	1.66042	0.830209	+	0.557453i	$0.188221\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.9706	−1.06295	−0.531475	−	0.847074i	$-0.678362\pi$
$467$	−22.9706	−1.06295	−0.531475	+	0.847074i	$0.678362\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22.2426	1.02272
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	27.4853	1.25322
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.4853	0.748558
$486$	0	0
$487$	−12.2721	−0.556101	−0.278050	−	0.960566i	$-0.589688\pi$
$487$	−12.2721	−0.556101	−0.278050	+	0.960566i	$0.589688\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.4853	−1.46604	−0.733020	−	0.680207i	$-0.761890\pi$
$491$	−32.4853	−1.46604	−0.733020	+	0.680207i	$0.761890\pi$
$492$	0	0
$493$	43.4558	1.95715
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2.51472	0.112574	0.0562871	−	0.998415i	$-0.482074\pi$
$499$	2.51472	0.112574	0.0562871	+	0.998415i	$0.482074\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.51472	0.424240	0.212120	−	0.977244i	$-0.431963\pi$
$503$	9.51472	0.424240	0.212120	+	0.977244i	$0.431963\pi$
$504$	0	0
$505$	16.2426	0.722788
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.4558	−1.39426	−0.697128	−	0.716947i	$-0.745539\pi$
$509$	−31.4558	−1.39426	−0.697128	+	0.716947i	$0.745539\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.75736	−0.385895
$516$	0	0
$517$	−25.4558	−1.11955
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−37.4558	−1.64097	−0.820485	−	0.571668i	$-0.806297\pi$
$521$	−37.4558	−1.64097	−0.820485	+	0.571668i	$0.806297\pi$
$522$	0	0
$523$	3.24264	0.141791	0.0708954	−	0.997484i	$-0.477414\pi$
$523$	3.24264	0.141791	0.0708954	+	0.997484i	$0.477414\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.7574	1.38337
$528$	0	0
$529$	−5.00000	−0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22.2426	0.963436
$534$	0	0
$535$	12.7279	0.550276
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	26.9411	1.15829	0.579145	−	0.815225i	$-0.303387\pi$
$541$	26.9411	1.15829	0.579145	+	0.815225i	$0.303387\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.48528	0.320634
$546$	0	0
$547$	−17.4558	−0.746358	−0.373179	−	0.927759i	$-0.621732\pi$
$547$	−17.4558	−0.746358	−0.373179	+	0.927759i	$0.621732\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−71.6985	−3.05446
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	27.4853	1.16250
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.00000	0.252870	0.126435	−	0.991975i	$-0.459647\pi$
$563$	6.00000	0.252870	0.126435	+	0.991975i	$0.459647\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.6985	−0.741959	−0.370980	−	0.928641i	$-0.620978\pi$
$569$	−17.6985	−0.741959	−0.370980	+	0.928641i	$0.620978\pi$
$570$	0	0
$571$	38.9411	1.62964	0.814818	−	0.579717i	$-0.196837\pi$
$571$	38.9411	1.62964	0.814818	+	0.579717i	$0.196837\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.24264	−0.176930
$576$	0	0
$577$	21.2426	0.884343	0.442171	−	0.896931i	$-0.354208\pi$
$577$	21.2426	0.884343	0.442171	+	0.896931i	$0.354208\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.78680	−0.115023	−0.0575117	−	0.998345i	$-0.518317\pi$
$587$	−2.78680	−0.115023	−0.0575117	+	0.998345i	$0.518317\pi$
$588$	0	0
$589$	−52.3970	−2.15898
$590$	0	0
$591$	0	0
$592$	0	0
$593$	39.2132	1.61029	0.805147	−	0.593076i	$-0.202087\pi$
$593$	39.2132	1.61029	0.805147	+	0.593076i	$0.202087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15.5147	0.633914	0.316957	−	0.948440i	$-0.397339\pi$
$599$	15.5147	0.633914	0.316957	+	0.948440i	$0.397339\pi$
$600$	0	0
$601$	13.4853	0.550076	0.275038	−	0.961433i	$-0.411310\pi$
$601$	13.4853	0.550076	0.275038	+	0.961433i	$0.411310\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−20.7574	−0.842515	−0.421258	−	0.906941i	$-0.638411\pi$
$607$	−20.7574	−0.842515	−0.421258	+	0.906941i	$0.638411\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.4558	−1.27257
$612$	0	0
$613$	45.4558	1.83594	0.917972	−	0.396645i	$-0.129826\pi$
$613$	45.4558	1.83594	0.917972	+	0.396645i	$0.129826\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.51472	0.383048	0.191524	−	0.981488i	$-0.438657\pi$
$617$	9.51472	0.383048	0.191524	+	0.981488i	$0.438657\pi$
$618$	0	0
$619$	21.9706	0.883071	0.441536	−	0.897244i	$-0.354434\pi$
$619$	21.9706	0.883071	0.441536	+	0.897244i	$0.354434\pi$
$620$	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
$629$	−22.2426	−0.886872
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.2426	−0.446151
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25.7574	−1.01735	−0.508677	−	0.860957i	$-0.669865\pi$
$641$	−25.7574	−1.01735	−0.508677	+	0.860957i	$0.669865\pi$
$642$	0	0
$643$	5.72792	0.225887	0.112944	−	0.993601i	$-0.463972\pi$
$643$	5.72792	0.225887	0.112944	+	0.993601i	$0.463972\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.75736	−0.0690889	−0.0345445	−	0.999403i	$-0.510998\pi$
$647$	−1.75736	−0.0690889	−0.0345445	+	0.999403i	$0.510998\pi$
$648$	0	0
$649$	−7.45584	−0.292667
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.75736	−0.0687708	−0.0343854	−	0.999409i	$-0.510947\pi$
$653$	−1.75736	−0.0687708	−0.0343854	+	0.999409i	$0.510947\pi$
$654$	0	0
$655$	2.48528	0.0971080
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.02944	−0.273828	−0.136914	−	0.990583i	$-0.543718\pi$
$659$	−7.02944	−0.273828	−0.136914	+	0.990583i	$0.543718\pi$
$660$	0	0
$661$	−35.9706	−1.39909	−0.699546	−	0.714587i	$-0.746615\pi$
$661$	−35.9706	−1.39909	−0.699546	+	0.714587i	$0.746615\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−43.4558	−1.68262
$668$	0	0
$669$	0	0
$670$	0	0
$671$	52.9706	2.04491
$672$	0	0
$673$	4.27208	0.164677	0.0823383	−	0.996604i	$-0.473761\pi$
$673$	4.27208	0.164677	0.0823383	+	0.996604i	$0.473761\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.7279	0.489174	0.244587	−	0.969627i	$-0.421348\pi$
$677$	12.7279	0.489174	0.244587	+	0.969627i	$0.421348\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.21320	0.352533	0.176267	−	0.984342i	$-0.443598\pi$
$683$	9.21320	0.352533	0.176267	+	0.984342i	$0.443598\pi$
$684$	0	0
$685$	4.24264	0.162103
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−44.4853	−1.69475
$690$	0	0
$691$	−40.9411	−1.55747	−0.778737	−	0.627351i	$-0.784139\pi$
$691$	−40.9411	−1.55747	−0.778737	+	0.627351i	$0.784139\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.48528	0.0563399
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−51.2132	−1.93430	−0.967148	−	0.254214i	$-0.918183\pi$
$701$	−51.2132	−1.93430	−0.967148	+	0.254214i	$0.918183\pi$
$702$	0	0
$703$	36.6985	1.38411
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−19.5147	−0.732891	−0.366445	−	0.930440i	$-0.619425\pi$
$709$	−19.5147	−0.732891	−0.366445	+	0.930440i	$0.619425\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31.7574	−1.18932
$714$	0	0
$715$	22.2426	0.831828
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9.51472	−0.354839	−0.177420	−	0.984135i	$-0.556775\pi$
$719$	−9.51472	−0.354839	−0.177420	+	0.984135i	$0.556775\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.2426	0.380402
$726$	0	0
$727$	9.24264	0.342791	0.171395	−	0.985202i	$-0.445172\pi$
$727$	9.24264	0.342791	0.171395	+	0.985202i	$0.445172\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−22.2426	−0.822674
$732$	0	0
$733$	44.2132	1.63305	0.816526	−	0.577309i	$-0.195897\pi$
$733$	44.2132	1.63305	0.816526	+	0.577309i	$0.195897\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.7574	−0.506759
$738$	0	0
$739$	1.48528	0.0546370	0.0273185	−	0.999627i	$-0.491303\pi$
$739$	1.48528	0.0546370	0.0273185	+	0.999627i	$0.491303\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.2132	0.998356	0.499178	−	0.866500i	$-0.333635\pi$
$743$	27.2132	0.998356	0.499178	+	0.866500i	$0.333635\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−22.9411	−0.837134	−0.418567	−	0.908186i	$-0.637467\pi$
$751$	−22.9411	−0.837134	−0.418567	+	0.908186i	$0.637467\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.51472	0.200701
$756$	0	0
$757$	42.9706	1.56179	0.780896	−	0.624661i	$-0.214763\pi$
$757$	42.9706	1.56179	0.780896	+	0.624661i	$0.214763\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.7574	−1.36870	−0.684352	−	0.729152i	$-0.739915\pi$
$761$	−37.7574	−1.36870	−0.684352	+	0.729152i	$0.739915\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.21320	−0.332669
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	47.6985	1.71560	0.857798	−	0.513988i	$-0.171832\pi$
$773$	47.6985	1.71560	0.857798	+	0.513988i	$0.171832\pi$
$774$	0	0
$775$	7.48528	0.268879
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.6985	1.06406
$780$	0	0
$781$	54.0000	1.93227
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.4853	0.374236
$786$	0	0
$787$	11.5147	0.410455	0.205228	−	0.978714i	$-0.434207\pi$
$787$	11.5147	0.410455	0.205228	+	0.978714i	$0.434207\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	65.4558	2.32441
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.2426	1.42547	0.712734	−	0.701435i	$-0.247457\pi$
$797$	40.2426	1.42547	0.712734	+	0.701435i	$0.247457\pi$
$798$	0	0
$799$	25.4558	0.900563
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.21320	−0.113391
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.9411	1.40426	0.702128	−	0.712051i	$-0.252234\pi$
$809$	39.9411	1.40426	0.702128	+	0.712051i	$0.252234\pi$
$810$	0	0
$811$	−55.9411	−1.96436	−0.982179	−	0.187946i	$-0.939817\pi$
$811$	−55.9411	−1.96436	−0.982179	+	0.187946i	$0.939817\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	36.6985	1.28392
$818$	0	0
$819$	0	0
$820$	0	0
$821$	53.6985	1.87409	0.937045	−	0.349209i	$-0.113550\pi$
$821$	53.6985	1.87409	0.937045	+	0.349209i	$0.113550\pi$
$822$	0	0
$823$	−5.02944	−0.175315	−0.0876576	−	0.996151i	$-0.527938\pi$
$823$	−5.02944	−0.175315	−0.0876576	+	0.996151i	$0.527938\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.45584	0.259265	0.129633	−	0.991562i	$-0.458620\pi$
$827$	7.45584	0.259265	0.129633	+	0.991562i	$0.458620\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6.72792	0.232829
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.2426	−1.18219	−0.591094	−	0.806603i	$-0.701304\pi$
$839$	−34.2426	−1.18219	−0.591094	+	0.806603i	$0.701304\pi$
$840$	0	0
$841$	75.9117	2.61764
$842$	0	0
$843$	0	0
$844$	0	0
$845$	14.4853	0.498309
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22.2426	0.762468
$852$	0	0
$853$	−0.272078	−0.00931577	−0.00465789	−	0.999989i	$-0.501483\pi$
$853$	−0.272078	−0.00931577	−0.00465789	+	0.999989i	$0.501483\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.24264	0.144926	0.0724629	−	0.997371i	$-0.476914\pi$
$857$	4.24264	0.144926	0.0724629	+	0.997371i	$0.476914\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.4853	1.31005	0.655027	−	0.755605i	$-0.272657\pi$
$863$	38.4853	1.31005	0.655027	+	0.755605i	$0.272657\pi$
$864$	0	0
$865$	3.51472	0.119504
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−46.6690	−1.58314
$870$	0	0
$871$	−17.0000	−0.576023
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.02944	−0.236828	−0.118414	−	0.992964i	$-0.537781\pi$
$881$	−7.02944	−0.236828	−0.118414	+	0.992964i	$0.537781\pi$
$882$	0	0
$883$	−19.7279	−0.663897	−0.331949	−	0.943297i	$-0.607706\pi$
$883$	−19.7279	−0.663897	−0.331949	+	0.943297i	$0.607706\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.2132	0.913730	0.456865	−	0.889536i	$-0.348972\pi$
$887$	27.2132	0.913730	0.456865	+	0.889536i	$0.348972\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−42.0000	−1.40548
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	0	0
$898$	0	0
$899$	76.6690	2.55706
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.0000	−0.432135
$906$	0	0
$907$	−55.7279	−1.85042	−0.925208	−	0.379461i	$-0.876109\pi$
$907$	−55.7279	−1.85042	−0.925208	+	0.379461i	$0.876109\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.78680	−0.291120	−0.145560	−	0.989349i	$-0.546498\pi$
$911$	−8.78680	−0.291120	−0.145560	+	0.989349i	$0.546498\pi$
$912$	0	0
$913$	−7.45584	−0.246752
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0.0294373	0.000971045 0	0.000485523	−	1.00000i	$-0.499845\pi$
$919$	0.0294373	0.000971045 0	0.000485523		1.00000i	$0.499845\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	66.7279	2.19638
$924$	0	0
$925$	−5.24264	−0.172377
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−11.2721	−0.369825	−0.184912	−	0.982755i	$-0.559200\pi$
$929$	−11.2721	−0.369825	−0.184912	+	0.982755i	$0.559200\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.0000	−0.588663
$936$	0	0
$937$	40.6985	1.32956	0.664781	−	0.747039i	$-0.268525\pi$
$937$	40.6985	1.32956	0.664781	+	0.747039i	$0.268525\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−11.6985	−0.381360	−0.190680	−	0.981652i	$-0.561069\pi$
$941$	−11.6985	−0.381360	−0.190680	+	0.981652i	$0.561069\pi$
$942$	0	0
$943$	18.0000	0.586161
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.2132	−1.27426	−0.637129	−	0.770757i	$-0.719878\pi$
$947$	−39.2132	−1.27426	−0.637129	+	0.770757i	$0.719878\pi$
$948$	0	0
$949$	−3.97056	−0.128890
$950$	0	0
$951$	0	0
$952$	0	0
$953$	44.4853	1.44102	0.720510	−	0.693445i	$-0.243908\pi$
$953$	44.4853	1.44102	0.720510	+	0.693445i	$0.243908\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	25.0294	0.807401
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.2426	0.490678
$966$	0	0
$967$	−26.7574	−0.860459	−0.430229	−	0.902720i	$-0.641567\pi$
$967$	−26.7574	−0.860459	−0.430229	+	0.902720i	$0.641567\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.0294	−0.610684	−0.305342	−	0.952243i	$-0.598771\pi$
$971$	−19.0294	−0.610684	−0.305342	+	0.952243i	$0.598771\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.2132	1.63846	0.819228	−	0.573468i	$-0.194402\pi$
$977$	51.2132	1.63846	0.819228	+	0.573468i	$0.194402\pi$
$978$	0	0
$979$	−7.45584	−0.238290
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.6985	−0.564494	−0.282247	−	0.959342i	$-0.591080\pi$
$983$	−17.6985	−0.564494	−0.282247	+	0.959342i	$0.591080\pi$
$984$	0	0
$985$	7.75736	0.247170
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.2426	0.707275
$990$	0	0
$991$	−52.9411	−1.68173	−0.840865	−	0.541245i	$-0.817953\pi$
$991$	−52.9411	−1.68173	−0.840865	+	0.541245i	$0.817953\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.48528	−0.205597
$996$	0	0
$997$	−0.272078	−0.00861679	−0.00430840	−	0.999991i	$-0.501371\pi$
$997$	−0.272078	−0.00861679	−0.00430840	+	0.999991i	$0.501371\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.bk.1.1		2
3.2	odd	2		2940.2.a.r.1.2		2
7.2	even	3		1260.2.s.e.361.1		4
7.4	even	3		1260.2.s.e.541.1		4
7.6	odd	2		8820.2.a.bf.1.1		2
21.2	odd	6		420.2.q.d.361.1	yes	4
21.5	even	6		2940.2.q.q.361.1		4
21.11	odd	6		420.2.q.d.121.1	✓	4
21.17	even	6		2940.2.q.q.961.1		4
21.20	even	2		2940.2.a.p.1.2		2
84.11	even	6		1680.2.bg.t.961.2		4
84.23	even	6		1680.2.bg.t.1201.2		4
105.2	even	12		2100.2.bc.f.949.3		8
105.23	even	12		2100.2.bc.f.949.2		8
105.32	even	12		2100.2.bc.f.1549.2		8
105.44	odd	6		2100.2.q.k.1201.2		4
105.53	even	12		2100.2.bc.f.1549.3		8
105.74	odd	6		2100.2.q.k.1801.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.q.d.121.1	✓	4	21.11	odd	6
420.2.q.d.361.1	yes	4	21.2	odd	6
1260.2.s.e.361.1		4	7.2	even	3
1260.2.s.e.541.1		4	7.4	even	3
1680.2.bg.t.961.2		4	84.11	even	6
1680.2.bg.t.1201.2		4	84.23	even	6
2100.2.q.k.1201.2		4	105.44	odd	6
2100.2.q.k.1801.2		4	105.74	odd	6
2100.2.bc.f.949.2		8	105.23	even	12
2100.2.bc.f.949.3		8	105.2	even	12
2100.2.bc.f.1549.2		8	105.32	even	12
2100.2.bc.f.1549.3		8	105.53	even	12
2940.2.a.p.1.2		2	21.20	even	2
2940.2.a.r.1.2		2	3.2	odd	2
2940.2.q.q.361.1		4	21.5	even	6
2940.2.q.q.961.1		4	21.17	even	6
8820.2.a.bf.1.1		2	7.6	odd	2
8820.2.a.bk.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +O(q^{10})\)	\(q+1.00000 q^{5} -4.24264 q^{11} -5.24264 q^{13} +4.24264 q^{17} -7.00000 q^{19} -4.24264 q^{23} +1.00000 q^{25} +10.2426 q^{29} +7.48528 q^{31} -5.24264 q^{37} -4.24264 q^{41} -5.24264 q^{43} +6.00000 q^{47} +8.48528 q^{53} -4.24264 q^{55} +1.75736 q^{59} -12.4853 q^{61} -5.24264 q^{65} +3.24264 q^{67} -12.7279 q^{71} +0.757359 q^{73} +11.0000 q^{79} +1.75736 q^{83} +4.24264 q^{85} +1.75736 q^{89} -7.00000 q^{95} +16.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 2 q^{13} - 14 q^{19} + 2 q^{25} + 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{43} + 12 q^{47} + 12 q^{59} - 8 q^{61} - 2 q^{65} - 2 q^{67} + 10 q^{73} + 22 q^{79} + 12 q^{83} + 12 q^{89} - 14 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^13 - 14 * q^19 + 2 * q^25 + 12 * q^29 - 2 * q^31 - 2 * q^37 - 2 * q^43 + 12 * q^47 + 12 * q^59 - 8 * q^61 - 2 * q^65 - 2 * q^67 + 10 * q^73 + 22 * q^79 + 12 * q^83 + 12 * q^89 - 14 * q^95 + 16 * q^97

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