LMFDB - Embedded newform 4320.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, 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("4320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.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:	$34.4953736732$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.551929$ of defining polynomial
Character	$\chi$	$=$	4320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -5.24730 q^{7} +5.14344 q^{11} +2.10386 q^{13} +5.14344 q^{17} -7.24730 q^{19} +6.24730 q^{23} +1.00000 q^{25} -4.24730 q^{29} -3.14344 q^{31} -5.24730 q^{35} -1.24730 q^{37} +2.20772 q^{41} +6.03959 q^{43} -10.2473 q^{47} +20.5342 q^{49} +4.00000 q^{53} +5.14344 q^{55} -6.49461 q^{59} -1.24730 q^{61} +2.10386 q^{65} -5.03959 q^{67} -1.79228 q^{71} -11.7419 q^{73} -26.9892 q^{77} +6.10386 q^{79} +14.7023 q^{83} +5.14344 q^{85} +8.28689 q^{89} -11.0396 q^{91} -7.24730 q^{95} +13.2473 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - q^{7} + 2 q^{11} + 5 q^{13} + 2 q^{17} - 7 q^{19} + 4 q^{23} + 3 q^{25} + 2 q^{29} + 4 q^{31} - q^{35} + 11 q^{37} + 4 q^{41} + 6 q^{43} - 16 q^{47} + 20 q^{49} + 12 q^{53} + 2 q^{55}+ \cdots + 25 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - q^7 + 2 * q^11 + 5 * q^13 + 2 * q^17 - 7 * q^19 + 4 * q^23 + 3 * q^25 + 2 * q^29 + 4 * q^31 - q^35 + 11 * q^37 + 4 * q^41 + 6 * q^43 - 16 * q^47 + 20 * q^49 + 12 * q^53 + 2 * q^55 + 10 * q^59 + 11 * q^61 + 5 * q^65 - 3 * q^67 - 8 * q^71 + 9 * q^73 - 22 * q^77 + 17 * q^79 + 12 * q^83 + 2 * q^85 - 2 * q^89 - 21 * q^91 - 7 * q^95 + 25 * 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
$7$	−5.24730	−1.98329	−0.991647	−	0.128981i	$-0.958830\pi$
$7$	−5.24730	−1.98329	−0.991647	+	0.128981i	$0.958830\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.14344	1.55081	0.775403	−	0.631466i	$-0.217547\pi$
$11$	5.14344	1.55081	0.775403	+	0.631466i	$0.217547\pi$
$12$	0	0
$13$	2.10386	0.583505	0.291753	−	0.956494i	$-0.405762\pi$
$13$	2.10386	0.583505	0.291753	+	0.956494i	$0.405762\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.14344	1.24747	0.623734	−	0.781636i	$-0.285615\pi$
$17$	5.14344	1.24747	0.623734	+	0.781636i	$0.285615\pi$
$18$	0	0
$19$	−7.24730	−1.66265	−0.831323	−	0.555790i	$-0.812416\pi$
$19$	−7.24730	−1.66265	−0.831323	+	0.555790i	$0.812416\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.24730	1.30265	0.651326	−	0.758798i	$-0.274213\pi$
$23$	6.24730	1.30265	0.651326	+	0.758798i	$0.274213\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.24730	−0.788704	−0.394352	−	0.918959i	$-0.629031\pi$
$29$	−4.24730	−0.788704	−0.394352	+	0.918959i	$0.629031\pi$
$30$	0	0
$31$	−3.14344	−0.564579	−0.282290	−	0.959329i	$-0.591094\pi$
$31$	−3.14344	−0.564579	−0.282290	+	0.959329i	$0.591094\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.24730	−0.886956
$36$	0	0
$37$	−1.24730	−0.205055	−0.102528	−	0.994730i	$-0.532693\pi$
$37$	−1.24730	−0.205055	−0.102528	+	0.994730i	$0.532693\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.20772	0.344788	0.172394	−	0.985028i	$-0.444850\pi$
$41$	2.20772	0.344788	0.172394	+	0.985028i	$0.444850\pi$
$42$	0	0
$43$	6.03959	0.921028	0.460514	−	0.887652i	$-0.347665\pi$
$43$	6.03959	0.921028	0.460514	+	0.887652i	$0.347665\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.2473	−1.49472	−0.747361	−	0.664418i	$-0.768680\pi$
$47$	−10.2473	−1.49472	−0.747361	+	0.664418i	$0.768680\pi$
$48$	0	0
$49$	20.5342	2.93346
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	5.14344	0.693542
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.49461	−0.845526	−0.422763	−	0.906240i	$-0.638940\pi$
$59$	−6.49461	−0.845526	−0.422763	+	0.906240i	$0.638940\pi$
$60$	0	0
$61$	−1.24730	−0.159701	−0.0798504	−	0.996807i	$-0.525444\pi$
$61$	−1.24730	−0.159701	−0.0798504	+	0.996807i	$0.525444\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.10386	0.260952
$66$	0	0
$67$	−5.03959	−0.615683	−0.307842	−	0.951438i	$-0.599607\pi$
$67$	−5.03959	−0.615683	−0.307842	+	0.951438i	$0.599607\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.79228	−0.212705	−0.106352	−	0.994329i	$-0.533917\pi$
$71$	−1.79228	−0.212705	−0.106352	+	0.994329i	$0.533917\pi$
$72$	0	0
$73$	−11.7419	−1.37429	−0.687143	−	0.726522i	$-0.741136\pi$
$73$	−11.7419	−1.37429	−0.687143	+	0.726522i	$0.741136\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−26.9892	−3.07571
$78$	0	0
$79$	6.10386	0.686738	0.343369	−	0.939201i	$-0.388432\pi$
$79$	6.10386	0.686738	0.343369	+	0.939201i	$0.388432\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.7023	1.61379	0.806895	−	0.590695i	$-0.201146\pi$
$83$	14.7023	1.61379	0.806895	+	0.590695i	$0.201146\pi$
$84$	0	0
$85$	5.14344	0.557885
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.28689	0.878408	0.439204	−	0.898387i	$-0.355260\pi$
$89$	8.28689	0.878408	0.439204	+	0.898387i	$0.355260\pi$
$90$	0	0
$91$	−11.0396	−1.15726
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.24730	−0.743558
$96$	0	0
$97$	13.2473	1.34506	0.672530	−	0.740070i	$-0.265208\pi$
$97$	13.2473	1.34506	0.672530	+	0.740070i	$0.265208\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.24730	0.422622	0.211311	−	0.977419i	$-0.432227\pi$
$101$	4.24730	0.422622	0.211311	+	0.977419i	$0.432227\pi$
$102$	0	0
$103$	13.5342	1.33356	0.666782	−	0.745253i	$-0.267671\pi$
$103$	13.5342	1.33356	0.666782	+	0.745253i	$0.267671\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−2.49461	−0.238940	−0.119470	−	0.992838i	$-0.538120\pi$
$109$	−2.49461	−0.238940	−0.119470	+	0.992838i	$0.538120\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.1434	1.23643	0.618216	−	0.786009i	$-0.287856\pi$
$113$	13.1434	1.23643	0.618216	+	0.786009i	$0.287856\pi$
$114$	0	0
$115$	6.24730	0.582564
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−26.9892	−2.47410
$120$	0	0
$121$	15.4550	1.40500
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.28689	0.202929	0.101464	−	0.994839i	$-0.467647\pi$
$127$	2.28689	0.202929	0.101464	+	0.994839i	$0.467647\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.3512	1.16650	0.583248	−	0.812294i	$-0.301781\pi$
$131$	13.3512	1.16650	0.583248	+	0.812294i	$0.301781\pi$
$132$	0	0
$133$	38.0288	3.29752
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.7815	1.09200	0.545998	−	0.837786i	$-0.316150\pi$
$137$	12.7815	1.09200	0.545998	+	0.837786i	$0.316150\pi$
$138$	0	0
$139$	13.5342	1.14796	0.573978	−	0.818871i	$-0.305400\pi$
$139$	13.5342	1.14796	0.573978	+	0.818871i	$0.305400\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.8211	0.904904
$144$	0	0
$145$	−4.24730	−0.352719
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.74191	0.716165	0.358083	−	0.933690i	$-0.383431\pi$
$149$	8.74191	0.716165	0.358083	+	0.933690i	$0.383431\pi$
$150$	0	0
$151$	10.1039	0.822240	0.411120	−	0.911581i	$-0.365138\pi$
$151$	10.1039	0.822240	0.411120	+	0.911581i	$0.365138\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.14344	−0.252488
$156$	0	0
$157$	15.3512	1.22516	0.612578	−	0.790410i	$-0.290133\pi$
$157$	15.3512	1.22516	0.612578	+	0.790410i	$0.290133\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−32.7815	−2.58354
$162$	0	0
$163$	3.49461	0.273719	0.136859	−	0.990590i	$-0.456299\pi$
$163$	3.49461	0.273719	0.136859	+	0.990590i	$0.456299\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.4946	0.812097	0.406049	−	0.913852i	$-0.366906\pi$
$167$	10.4946	0.812097	0.406049	+	0.913852i	$0.366906\pi$
$168$	0	0
$169$	−8.57378	−0.659521
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.494607	−0.0376042	−0.0188021	−	0.999823i	$-0.505985\pi$
$173$	−0.494607	−0.0376042	−0.0188021	+	0.999823i	$0.505985\pi$
$174$	0	0
$175$	−5.24730	−0.396659
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.9892	−0.970859	−0.485430	−	0.874276i	$-0.661337\pi$
$179$	−12.9892	−0.970859	−0.485430	+	0.874276i	$0.661337\pi$
$180$	0	0
$181$	8.75270	0.650583	0.325291	−	0.945614i	$-0.394538\pi$
$181$	8.75270	0.650583	0.325291	+	0.945614i	$0.394538\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.24730	−0.0917036
$186$	0	0
$187$	26.4550	1.93458
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.2077	−0.883319	−0.441660	−	0.897183i	$-0.645610\pi$
$191$	−12.2077	−0.883319	−0.441660	+	0.897183i	$0.645610\pi$
$192$	0	0
$193$	−16.0288	−1.15378	−0.576889	−	0.816822i	$-0.695734\pi$
$193$	−16.0288	−1.15378	−0.576889	+	0.816822i	$0.695734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.70232	0.620015	0.310007	−	0.950734i	$-0.399668\pi$
$197$	8.70232	0.620015	0.310007	+	0.950734i	$0.399668\pi$
$198$	0	0
$199$	−9.68842	−0.686794	−0.343397	−	0.939190i	$-0.611578\pi$
$199$	−9.68842	−0.686794	−0.343397	+	0.939190i	$0.611578\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	22.2869	1.56423
$204$	0	0
$205$	2.20772	0.154194
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−37.2761	−2.57844
$210$	0	0
$211$	−15.4550	−1.06397	−0.531984	−	0.846755i	$-0.678553\pi$
$211$	−15.4550	−1.06397	−0.531984	+	0.846755i	$0.678553\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.03959	0.411896
$216$	0	0
$217$	16.4946	1.11973
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.8211	0.727905
$222$	0	0
$223$	18.2077	1.21928	0.609639	−	0.792679i	$-0.291314\pi$
$223$	18.2077	1.21928	0.609639	+	0.792679i	$0.291314\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.7815	1.77755	0.888775	−	0.458343i	$-0.151557\pi$
$227$	26.7815	1.77755	0.888775	+	0.458343i	$0.151557\pi$
$228$	0	0
$229$	16.2869	1.07627	0.538134	−	0.842859i	$-0.319129\pi$
$229$	16.2869	1.07627	0.538134	+	0.842859i	$0.319129\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.71311	0.112230	0.0561148	−	0.998424i	$-0.482129\pi$
$233$	1.71311	0.112230	0.0561148	+	0.998424i	$0.482129\pi$
$234$	0	0
$235$	−10.2473	−0.668460
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−28.4946	−1.84316	−0.921581	−	0.388185i	$-0.873102\pi$
$239$	−28.4946	−1.84316	−0.921581	+	0.388185i	$0.873102\pi$
$240$	0	0
$241$	25.5738	1.64735	0.823676	−	0.567061i	$-0.191920\pi$
$241$	25.5738	1.64735	0.823676	+	0.567061i	$0.191920\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.5342	1.31188
$246$	0	0
$247$	−15.2473	−0.970163
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.6381	1.74450	0.872249	−	0.489062i	$-0.162661\pi$
$251$	27.6381	1.74450	0.872249	+	0.489062i	$0.162661\pi$
$252$	0	0
$253$	32.1327	2.02016
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.3404	−1.64307	−0.821534	−	0.570160i	$-0.806881\pi$
$257$	−26.3404	−1.64307	−0.821534	+	0.570160i	$0.806881\pi$
$258$	0	0
$259$	6.54498	0.406685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.415435	0.0256168	0.0128084	−	0.999918i	$-0.495923\pi$
$263$	0.415435	0.0256168	0.0128084	+	0.999918i	$0.495923\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−22.8211	−1.39143	−0.695713	−	0.718320i	$-0.744911\pi$
$269$	−22.8211	−1.39143	−0.695713	+	0.718320i	$0.744911\pi$
$270$	0	0
$271$	−9.45502	−0.574352	−0.287176	−	0.957878i	$-0.592716\pi$
$271$	−9.45502	−0.574352	−0.287176	+	0.957878i	$0.592716\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.14344	0.310161
$276$	0	0
$277$	−21.1969	−1.27360	−0.636800	−	0.771029i	$-0.719742\pi$
$277$	−21.1969	−1.27360	−0.636800	+	0.771029i	$0.719742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−32.4946	−1.93847	−0.969233	−	0.246146i	$-0.920836\pi$
$281$	−32.4946	−1.93847	−0.969233	+	0.246146i	$0.920836\pi$
$282$	0	0
$283$	−20.9892	−1.24768	−0.623840	−	0.781552i	$-0.714428\pi$
$283$	−20.9892	−1.24768	−0.623840	+	0.781552i	$0.714428\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.5846	−0.683815
$288$	0	0
$289$	9.45502	0.556178
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.1969	1.70570	0.852851	−	0.522154i	$-0.174871\pi$
$293$	29.1969	1.70570	0.852851	+	0.522154i	$0.174871\pi$
$294$	0	0
$295$	−6.49461	−0.378131
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.1434	0.760105
$300$	0	0
$301$	−31.6915	−1.82667
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.24730	−0.0714204
$306$	0	0
$307$	−1.96041	−0.111887	−0.0559434	−	0.998434i	$-0.517817\pi$
$307$	−1.96041	−0.111887	−0.0559434	+	0.998434i	$0.517817\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.28689	0.243087	0.121544	−	0.992586i	$-0.461216\pi$
$311$	4.28689	0.243087	0.121544	+	0.992586i	$0.461216\pi$
$312$	0	0
$313$	6.83187	0.386160	0.193080	−	0.981183i	$-0.438152\pi$
$313$	6.83187	0.386160	0.193080	+	0.981183i	$0.438152\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	25.4046	1.42687	0.713434	−	0.700723i	$-0.247139\pi$
$317$	25.4046	1.42687	0.713434	+	0.700723i	$0.247139\pi$
$318$	0	0
$319$	−21.8458	−1.22313
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−37.2761	−2.07410
$324$	0	0
$325$	2.10386	0.116701
$326$	0	0
$327$	0	0
$328$	0	0
$329$	53.7707	2.96448
$330$	0	0
$331$	6.75270	0.371162	0.185581	−	0.982629i	$-0.440583\pi$
$331$	6.75270	0.371162	0.185581	+	0.982629i	$0.440583\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.03959	−0.275342
$336$	0	0
$337$	2.54498	0.138634	0.0693169	−	0.997595i	$-0.477918\pi$
$337$	2.54498	0.138634	0.0693169	+	0.997595i	$0.477918\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.1681	−0.875553
$342$	0	0
$343$	−71.0180	−3.83461
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.78150	−0.149319	−0.0746593	−	0.997209i	$-0.523787\pi$
$347$	−2.78150	−0.149319	−0.0746593	+	0.997209i	$0.523787\pi$
$348$	0	0
$349$	−21.8211	−1.16806	−0.584028	−	0.811734i	$-0.698524\pi$
$349$	−21.8211	−1.16806	−0.584028	+	0.811734i	$0.698524\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.9357	1.11430	0.557148	−	0.830413i	$-0.311896\pi$
$353$	20.9357	1.11430	0.557148	+	0.830413i	$0.311896\pi$
$354$	0	0
$355$	−1.79228	−0.0951245
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.286889	−0.0151414	−0.00757071	−	0.999971i	$-0.502410\pi$
$359$	−0.286889	−0.0151414	−0.00757071	+	0.999971i	$0.502410\pi$
$360$	0	0
$361$	33.5234	1.76439
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.7419	−0.614600
$366$	0	0
$367$	−7.45502	−0.389149	−0.194574	−	0.980888i	$-0.562333\pi$
$367$	−7.45502	−0.389149	−0.194574	+	0.980888i	$0.562333\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.9892	−1.08971
$372$	0	0
$373$	−1.68842	−0.0874233	−0.0437116	−	0.999044i	$-0.513918\pi$
$373$	−1.68842	−0.0874233	−0.0437116	+	0.999044i	$0.513918\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.93573	−0.460213
$378$	0	0
$379$	−30.0288	−1.54248	−0.771238	−	0.636547i	$-0.780362\pi$
$379$	−30.0288	−1.54248	−0.771238	+	0.636547i	$0.780362\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.8211	−1.67708	−0.838539	−	0.544841i	$-0.816590\pi$
$383$	−32.8211	−1.67708	−0.838539	+	0.544841i	$0.816590\pi$
$384$	0	0
$385$	−26.9892	−1.37550
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13.8319	0.701304	0.350652	−	0.936506i	$-0.385960\pi$
$389$	13.8319	0.701304	0.350652	+	0.936506i	$0.385960\pi$
$390$	0	0
$391$	32.1327	1.62502
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.10386	0.307118
$396$	0	0
$397$	21.6381	1.08598	0.542991	−	0.839738i	$-0.317292\pi$
$397$	21.6381	1.08598	0.542991	+	0.839738i	$0.317292\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.41544	−0.420247	−0.210123	−	0.977675i	$-0.567387\pi$
$401$	−8.41544	−0.420247	−0.210123	+	0.977675i	$0.567387\pi$
$402$	0	0
$403$	−6.61336	−0.329435
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.41544	−0.318001
$408$	0	0
$409$	−15.9100	−0.786701	−0.393350	−	0.919389i	$-0.628684\pi$
$409$	−15.9100	−0.786701	−0.393350	+	0.919389i	$0.628684\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	34.0792	1.67693
$414$	0	0
$415$	14.7023	0.721709
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.154231	−0.00753468	−0.00376734	−	0.999993i	$-0.501199\pi$
$419$	−0.154231	−0.00753468	−0.00376734	+	0.999993i	$0.501199\pi$
$420$	0	0
$421$	−2.83187	−0.138017	−0.0690084	−	0.997616i	$-0.521984\pi$
$421$	−2.83187	−0.138017	−0.0690084	+	0.997616i	$0.521984\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.14344	0.249494
$426$	0	0
$427$	6.54498	0.316734
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.0792	−1.25619	−0.628095	−	0.778137i	$-0.716165\pi$
$431$	−26.0792	−1.25619	−0.628095	+	0.778137i	$0.716165\pi$
$432$	0	0
$433$	28.7815	1.38315	0.691575	−	0.722304i	$-0.256917\pi$
$433$	28.7815	1.38315	0.691575	+	0.722304i	$0.256917\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−45.2761	−2.16585
$438$	0	0
$439$	36.5738	1.74557	0.872786	−	0.488104i	$-0.162311\pi$
$439$	36.5738	1.74557	0.872786	+	0.488104i	$0.162311\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	25.7923	1.22543	0.612714	−	0.790305i	$-0.290078\pi$
$443$	25.7923	1.22543	0.612714	+	0.790305i	$0.290078\pi$
$444$	0	0
$445$	8.28689	0.392836
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.207718	0.00980281	0.00490140	−	0.999988i	$-0.498440\pi$
$449$	0.207718	0.00980281	0.00490140	+	0.999988i	$0.498440\pi$
$450$	0	0
$451$	11.3553	0.534699
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11.0396	−0.517544
$456$	0	0
$457$	14.9100	0.697462	0.348731	−	0.937223i	$-0.386613\pi$
$457$	14.9100	0.697462	0.348731	+	0.937223i	$0.386613\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.9892	−1.44331	−0.721656	−	0.692252i	$-0.756619\pi$
$461$	−30.9892	−1.44331	−0.721656	+	0.692252i	$0.756619\pi$
$462$	0	0
$463$	−10.6242	−0.493746	−0.246873	−	0.969048i	$-0.579403\pi$
$463$	−10.6242	−0.493746	−0.246873	+	0.969048i	$0.579403\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.28689	0.383471	0.191736	−	0.981447i	$-0.438588\pi$
$467$	8.28689	0.383471	0.191736	+	0.981447i	$0.438588\pi$
$468$	0	0
$469$	26.4442	1.22108
$470$	0	0
$471$	0	0
$472$	0	0
$473$	31.0643	1.42834
$474$	0	0
$475$	−7.24730	−0.332529
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.7707	1.26888	0.634438	−	0.772974i	$-0.281232\pi$
$479$	27.7707	1.26888	0.634438	+	0.772974i	$0.281232\pi$
$480$	0	0
$481$	−2.62415	−0.119651
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.2473	0.601529
$486$	0	0
$487$	−14.5450	−0.659096	−0.329548	−	0.944139i	$-0.606896\pi$
$487$	−14.5450	−0.659096	−0.329548	+	0.944139i	$0.606896\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.0684	−0.680027	−0.340013	−	0.940421i	$-0.610432\pi$
$491$	−15.0684	−0.680027	−0.340013	+	0.940421i	$0.610432\pi$
$492$	0	0
$493$	−21.8458	−0.983884
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.40465	0.421856
$498$	0	0
$499$	29.4046	1.31633	0.658166	−	0.752872i	$-0.271332\pi$
$499$	29.4046	1.31633	0.658166	+	0.752872i	$0.271332\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−31.1573	−1.38924	−0.694619	−	0.719378i	$-0.744427\pi$
$503$	−31.1573	−1.38924	−0.694619	+	0.719378i	$0.744427\pi$
$504$	0	0
$505$	4.24730	0.189003
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.33726	−0.413867	−0.206933	−	0.978355i	$-0.566348\pi$
$509$	−9.33726	−0.413867	−0.206933	+	0.978355i	$0.566348\pi$
$510$	0	0
$511$	61.6134	2.72561
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.5342	0.596388
$516$	0	0
$517$	−52.7064	−2.31803
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.2761	0.932123	0.466061	−	0.884752i	$-0.345673\pi$
$521$	21.2761	0.932123	0.466061	+	0.884752i	$0.345673\pi$
$522$	0	0
$523$	26.4838	1.15806	0.579028	−	0.815307i	$-0.303432\pi$
$523$	26.4838	1.15806	0.579028	+	0.815307i	$0.303432\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.1681	−0.704295
$528$	0	0
$529$	16.0288	0.696904
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.64473	0.201185
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	105.616	4.54922
$540$	0	0
$541$	−21.5342	−0.925827	−0.462914	−	0.886403i	$-0.653196\pi$
$541$	−21.5342	−0.925827	−0.462914	+	0.886403i	$0.653196\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.49461	−0.106857
$546$	0	0
$547$	35.0792	1.49988	0.749939	−	0.661508i	$-0.230083\pi$
$547$	35.0792	1.49988	0.749939	+	0.661508i	$0.230083\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	30.7815	1.31134
$552$	0	0
$553$	−32.0288	−1.36200
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.87145	0.164039	0.0820194	−	0.996631i	$-0.473863\pi$
$557$	3.87145	0.164039	0.0820194	+	0.996631i	$0.473863\pi$
$558$	0	0
$559$	12.7064	0.537425
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.8607	1.80636	0.903181	−	0.429260i	$-0.141226\pi$
$563$	42.8607	1.80636	0.903181	+	0.429260i	$0.141226\pi$
$564$	0	0
$565$	13.1434	0.552949
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.4946	−1.27840	−0.639200	−	0.769040i	$-0.720734\pi$
$569$	−30.4946	−1.27840	−0.639200	+	0.769040i	$0.720734\pi$
$570$	0	0
$571$	12.4442	0.520775	0.260388	−	0.965504i	$-0.416150\pi$
$571$	12.4442	0.520775	0.260388	+	0.965504i	$0.416150\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.24730	0.260531
$576$	0	0
$577$	−31.5342	−1.31279	−0.656393	−	0.754419i	$-0.727919\pi$
$577$	−31.5342	−1.31279	−0.656393	+	0.754419i	$0.727919\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−77.1476	−3.20062
$582$	0	0
$583$	20.5738	0.852079
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.3553	−1.21162	−0.605811	−	0.795609i	$-0.707151\pi$
$587$	−29.3553	−1.21162	−0.605811	+	0.795609i	$0.707151\pi$
$588$	0	0
$589$	22.7815	0.938695
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.6381	1.29922	0.649610	−	0.760268i	$-0.274932\pi$
$593$	31.6381	1.29922	0.649610	+	0.760268i	$0.274932\pi$
$594$	0	0
$595$	−26.9892	−1.10645
$596$	0	0
$597$	0	0
$598$	0	0
$599$	40.2653	1.64520	0.822598	−	0.568623i	$-0.192524\pi$
$599$	40.2653	1.64520	0.822598	+	0.568623i	$0.192524\pi$
$600$	0	0
$601$	−0.455021	−0.0185607	−0.00928035	−	0.999957i	$-0.502954\pi$
$601$	−0.455021	−0.0185607	−0.00928035	+	0.999957i	$0.502954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.4550	0.628336
$606$	0	0
$607$	5.94963	0.241488	0.120744	−	0.992684i	$-0.461472\pi$
$607$	5.94963	0.241488	0.120744	+	0.992684i	$0.461472\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.5589	−0.872179
$612$	0	0
$613$	−32.3907	−1.30825	−0.654125	−	0.756386i	$-0.726963\pi$
$613$	−32.3907	−1.30825	−0.654125	+	0.756386i	$0.726963\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.1327	−0.568959	−0.284480	−	0.958682i	$-0.591821\pi$
$617$	−14.1327	−0.568959	−0.284480	+	0.958682i	$0.591821\pi$
$618$	0	0
$619$	25.6627	1.03147	0.515736	−	0.856747i	$-0.327518\pi$
$619$	25.6627	1.03147	0.515736	+	0.856747i	$0.327518\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−43.4838	−1.74214
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.41544	−0.255800
$630$	0	0
$631$	−21.6134	−0.860414	−0.430207	−	0.902730i	$-0.641560\pi$
$631$	−21.6134	−0.860414	−0.430207	+	0.902730i	$0.641560\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.28689	0.0907524
$636$	0	0
$637$	43.2010	1.71169
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.27610	0.129398	0.0646991	−	0.997905i	$-0.479391\pi$
$641$	3.27610	0.129398	0.0646991	+	0.997905i	$0.479391\pi$
$642$	0	0
$643$	−14.5342	−0.573173	−0.286586	−	0.958054i	$-0.592521\pi$
$643$	−14.5342	−0.573173	−0.286586	+	0.958054i	$0.592521\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.4046	−0.841504	−0.420752	−	0.907176i	$-0.638234\pi$
$647$	−21.4046	−0.841504	−0.420752	+	0.907176i	$0.638234\pi$
$648$	0	0
$649$	−33.4046	−1.31125
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.37685	0.210412	0.105206	−	0.994450i	$-0.466450\pi$
$653$	5.37685	0.210412	0.105206	+	0.994450i	$0.466450\pi$
$654$	0	0
$655$	13.3512	0.521673
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.415435	0.0161831	0.00809153	−	0.999967i	$-0.497424\pi$
$659$	0.415435	0.0161831	0.00809153	+	0.999967i	$0.497424\pi$
$660$	0	0
$661$	−0.673525	−0.0261971	−0.0130986	−	0.999914i	$-0.504170\pi$
$661$	−0.673525	−0.0261971	−0.0130986	+	0.999914i	$0.504170\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	38.0288	1.47469
$666$	0	0
$667$	−26.5342	−1.02741
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.41544	−0.247665
$672$	0	0
$673$	−47.6134	−1.83536	−0.917680	−	0.397320i	$-0.869940\pi$
$673$	−47.6134	−1.83536	−0.917680	+	0.397320i	$0.869940\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.6529	1.17809	0.589044	−	0.808101i	$-0.299504\pi$
$677$	30.6529	1.17809	0.589044	+	0.808101i	$0.299504\pi$
$678$	0	0
$679$	−69.5126	−2.66765
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.5846	−0.443271	−0.221636	−	0.975130i	$-0.571140\pi$
$683$	−11.5846	−0.443271	−0.221636	+	0.975130i	$0.571140\pi$
$684$	0	0
$685$	12.7815	0.488356
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.41544	0.320603
$690$	0	0
$691$	−26.2869	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−26.2869	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.5342	0.513381
$696$	0	0
$697$	11.3553	0.430112
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.66274	0.100570	0.0502851	−	0.998735i	$-0.483987\pi$
$701$	2.66274	0.100570	0.0502851	+	0.998735i	$0.483987\pi$
$702$	0	0
$703$	9.03959	0.340934
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−22.2869	−0.838185
$708$	0	0
$709$	−0.881243	−0.0330958	−0.0165479	−	0.999863i	$-0.505268\pi$
$709$	−0.881243	−0.0330958	−0.0165479	+	0.999863i	$0.505268\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−19.6381	−0.735451
$714$	0	0
$715$	10.8211	0.404685
$716$	0	0
$717$	0	0
$718$	0	0
$719$	33.0684	1.23324	0.616621	−	0.787260i	$-0.288501\pi$
$719$	33.0684	1.23324	0.616621	+	0.787260i	$0.288501\pi$
$720$	0	0
$721$	−71.0180	−2.64485
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.24730	−0.157741
$726$	0	0
$727$	37.4046	1.38726	0.693631	−	0.720331i	$-0.256010\pi$
$727$	37.4046	1.38726	0.693631	+	0.720331i	$0.256010\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31.0643	1.14895
$732$	0	0
$733$	−4.20772	−0.155416	−0.0777078	−	0.996976i	$-0.524760\pi$
$733$	−4.20772	−0.155416	−0.0777078	+	0.996976i	$0.524760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.9208	−0.954806
$738$	0	0
$739$	−2.62315	−0.0964942	−0.0482471	−	0.998835i	$-0.515364\pi$
$739$	−2.62315	−0.0964942	−0.0482471	+	0.998835i	$0.515364\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.8103	−0.873515	−0.436757	−	0.899579i	$-0.643873\pi$
$743$	−23.8103	−0.873515	−0.436757	+	0.899579i	$0.643873\pi$
$744$	0	0
$745$	8.74191	0.320279
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−31.4838	−1.15039
$750$	0	0
$751$	21.6884	0.791422	0.395711	−	0.918375i	$-0.370498\pi$
$751$	21.6884	0.791422	0.395711	+	0.918375i	$0.370498\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	10.1039	0.367717
$756$	0	0
$757$	−41.5877	−1.51153	−0.755765	−	0.654843i	$-0.772735\pi$
$757$	−41.5877	−1.51153	−0.755765	+	0.654843i	$0.772735\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.7923	−0.427470	−0.213735	−	0.976892i	$-0.568563\pi$
$761$	−11.7923	−0.427470	−0.213735	+	0.976892i	$0.568563\pi$
$762$	0	0
$763$	13.0900	0.473888
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.6637	−0.493369
$768$	0	0
$769$	−12.5846	−0.453811	−0.226906	−	0.973917i	$-0.572861\pi$
$769$	−12.5846	−0.453811	−0.226906	+	0.973917i	$0.572861\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.71311	0.205486	0.102743	−	0.994708i	$-0.467238\pi$
$773$	5.71311	0.205486	0.102743	+	0.994708i	$0.467238\pi$
$774$	0	0
$775$	−3.14344	−0.112916
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−9.21850	−0.329864
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.3512	0.547906
$786$	0	0
$787$	21.1583	0.754213	0.377107	−	0.926170i	$-0.376919\pi$
$787$	21.1583	0.754213	0.377107	+	0.926170i	$0.376919\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−68.9676	−2.45221
$792$	0	0
$793$	−2.62415	−0.0931863
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.21850	−0.326536	−0.163268	−	0.986582i	$-0.552204\pi$
$797$	−9.21850	−0.326536	−0.163268	+	0.986582i	$0.552204\pi$
$798$	0	0
$799$	−52.7064	−1.86462
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−60.3939	−2.13125
$804$	0	0
$805$	−32.7815	−1.15540
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−33.4046	−1.17444	−0.587222	−	0.809426i	$-0.699779\pi$
$809$	−33.4046	−1.17444	−0.587222	+	0.809426i	$0.699779\pi$
$810$	0	0
$811$	−34.2869	−1.20398	−0.601988	−	0.798505i	$-0.705624\pi$
$811$	−34.2869	−1.20398	−0.601988	+	0.798505i	$0.705624\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.49461	0.122411
$816$	0	0
$817$	−43.7707	−1.53134
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.5738	−0.508628	−0.254314	−	0.967122i	$-0.581850\pi$
$821$	−14.5738	−0.508628	−0.254314	+	0.967122i	$0.581850\pi$
$822$	0	0
$823$	28.1080	0.979782	0.489891	−	0.871784i	$-0.337037\pi$
$823$	28.1080	0.979782	0.489891	+	0.871784i	$0.337037\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.9892	1.21669	0.608347	−	0.793671i	$-0.291833\pi$
$827$	34.9892	1.21669	0.608347	+	0.793671i	$0.291833\pi$
$828$	0	0
$829$	−50.8597	−1.76643	−0.883215	−	0.468969i	$-0.844626\pi$
$829$	−50.8597	−1.76643	−0.883215	+	0.468969i	$0.844626\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	105.616	3.65939
$834$	0	0
$835$	10.4946	0.363181
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19.1178	0.660018	0.330009	−	0.943978i	$-0.392948\pi$
$839$	19.1178	0.660018	0.330009	+	0.943978i	$0.392948\pi$
$840$	0	0
$841$	−10.9604	−0.377945
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.57378	−0.294947
$846$	0	0
$847$	−81.0972	−2.78653
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7.79228	−0.267116
$852$	0	0
$853$	−3.97531	−0.136112	−0.0680561	−	0.997681i	$-0.521680\pi$
$853$	−3.97531	−0.136112	−0.0680561	+	0.997681i	$0.521680\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.7023	−1.59532	−0.797661	−	0.603107i	$-0.793929\pi$
$857$	−46.7023	−1.59532	−0.797661	+	0.603107i	$0.793929\pi$
$858$	0	0
$859$	23.7419	0.810064	0.405032	−	0.914303i	$-0.367260\pi$
$859$	23.7419	0.810064	0.405032	+	0.914303i	$0.367260\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.84265	−0.164846	−0.0824229	−	0.996597i	$-0.526266\pi$
$863$	−4.84265	−0.164846	−0.0824229	+	0.996597i	$0.526266\pi$
$864$	0	0
$865$	−0.494607	−0.0168171
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31.3949	1.06500
$870$	0	0
$871$	−10.6026	−0.359255
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.24730	−0.177391
$876$	0	0
$877$	36.2900	1.22543	0.612713	−	0.790305i	$-0.290078\pi$
$877$	36.2900	1.22543	0.612713	+	0.790305i	$0.290078\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.01079	−0.0340543	−0.0170271	−	0.999855i	$-0.505420\pi$
$881$	−1.01079	−0.0340543	−0.0170271	+	0.999855i	$0.505420\pi$
$882$	0	0
$883$	6.44423	0.216866	0.108433	−	0.994104i	$-0.465417\pi$
$883$	6.44423	0.216866	0.108433	+	0.994104i	$0.465417\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.6627	0.962401	0.481200	−	0.876611i	$-0.340201\pi$
$887$	28.6627	0.962401	0.481200	+	0.876611i	$0.340201\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	74.2653	2.48519
$894$	0	0
$895$	−12.9892	−0.434182
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.3512	0.445286
$900$	0	0
$901$	20.5738	0.685412
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.75270	0.290949
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11.5846	−0.383814	−0.191907	−	0.981413i	$-0.561467\pi$
$911$	−11.5846	−0.383814	−0.191907	+	0.981413i	$0.561467\pi$
$912$	0	0
$913$	75.6206	2.50268
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−70.0576	−2.31351
$918$	0	0
$919$	56.7064	1.87057	0.935286	−	0.353892i	$-0.115142\pi$
$919$	56.7064	1.87057	0.935286	+	0.353892i	$0.115142\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.77071	−0.124114
$924$	0	0
$925$	−1.24730	−0.0410111
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.79228	−0.0588029	−0.0294014	−	0.999568i	$-0.509360\pi$
$929$	−1.79228	−0.0588029	−0.0294014	+	0.999568i	$0.509360\pi$
$930$	0	0
$931$	−148.818	−4.87730
$932$	0	0
$933$	0	0
$934$	0	0
$935$	26.4550	0.865172
$936$	0	0
$937$	9.53419	0.311468	0.155734	−	0.987799i	$-0.450226\pi$
$937$	9.53419	0.311468	0.155734	+	0.987799i	$0.450226\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−54.3265	−1.77099	−0.885496	−	0.464647i	$-0.846181\pi$
$941$	−54.3265	−1.77099	−0.885496	+	0.464647i	$0.846181\pi$
$942$	0	0
$943$	13.7923	0.449139
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−16.6232	−0.540180	−0.270090	−	0.962835i	$-0.587053\pi$
$947$	−16.6232	−0.540180	−0.270090	+	0.962835i	$0.587053\pi$
$948$	0	0
$949$	−24.7033	−0.801904
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.1219	−0.619418	−0.309709	−	0.950831i	$-0.600232\pi$
$953$	−19.1219	−0.619418	−0.309709	+	0.950831i	$0.600232\pi$
$954$	0	0
$955$	−12.2077	−0.395032
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−67.0684	−2.16575
$960$	0	0
$961$	−21.1188	−0.681250
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.0288	−0.515985
$966$	0	0
$967$	−8.96041	−0.288148	−0.144074	−	0.989567i	$-0.546020\pi$
$967$	−8.96041	−0.288148	−0.144074	+	0.989567i	$0.546020\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.71722	0.183474	0.0917372	−	0.995783i	$-0.470758\pi$
$971$	5.71722	0.183474	0.0917372	+	0.995783i	$0.470758\pi$
$972$	0	0
$973$	−71.0180	−2.27673
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.06427	0.0980348	0.0490174	−	0.998798i	$-0.484391\pi$
$977$	3.06427	0.0980348	0.0490174	+	0.998798i	$0.484391\pi$
$978$	0	0
$979$	42.6232	1.36224
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.4056	1.35253	0.676265	−	0.736658i	$-0.263597\pi$
$983$	42.4056	1.35253	0.676265	+	0.736658i	$0.263597\pi$
$984$	0	0
$985$	8.70232	0.277279
$986$	0	0
$987$	0	0
$988$	0	0
$989$	37.7311	1.19978
$990$	0	0
$991$	−11.0931	−0.352383	−0.176192	−	0.984356i	$-0.556378\pi$
$991$	−11.0931	−0.352383	−0.176192	+	0.984356i	$0.556378\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−9.68842	−0.307144
$996$	0	0
$997$	−54.2118	−1.71691	−0.858453	−	0.512892i	$-0.828574\pi$
$997$	−54.2118	−1.71691	−0.858453	+	0.512892i	$0.828574\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.a.bi.1.1	yes	3
3.2	odd	2		4320.2.a.bg.1.1	✓	3
4.3	odd	2		4320.2.a.bj.1.3	yes	3
8.3	odd	2		8640.2.a.dl.1.3		3
8.5	even	2		8640.2.a.dk.1.1		3
12.11	even	2		4320.2.a.bh.1.3	yes	3
24.5	odd	2		8640.2.a.dm.1.1		3
24.11	even	2		8640.2.a.dn.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.a.bg.1.1	✓	3	3.2	odd	2
4320.2.a.bh.1.3	yes	3	12.11	even	2
4320.2.a.bi.1.1	yes	3	1.1	even	1	trivial
4320.2.a.bj.1.3	yes	3	4.3	odd	2
8640.2.a.dk.1.1		3	8.5	even	2
8640.2.a.dl.1.3		3	8.3	odd	2
8640.2.a.dm.1.1		3	24.5	odd	2
8640.2.a.dn.1.3		3	24.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -5.24730 q^{7} +5.14344 q^{11} +2.10386 q^{13} +5.14344 q^{17} -7.24730 q^{19} +6.24730 q^{23} +1.00000 q^{25} -4.24730 q^{29} -3.14344 q^{31} -5.24730 q^{35} -1.24730 q^{37} +2.20772 q^{41} +6.03959 q^{43} -10.2473 q^{47} +20.5342 q^{49} +4.00000 q^{53} +5.14344 q^{55} -6.49461 q^{59} -1.24730 q^{61} +2.10386 q^{65} -5.03959 q^{67} -1.79228 q^{71} -11.7419 q^{73} -26.9892 q^{77} +6.10386 q^{79} +14.7023 q^{83} +5.14344 q^{85} +8.28689 q^{89} -11.0396 q^{91} -7.24730 q^{95} +13.2473 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - q^{7} + 2 q^{11} + 5 q^{13} + 2 q^{17} - 7 q^{19} + 4 q^{23} + 3 q^{25} + 2 q^{29} + 4 q^{31} - q^{35} + 11 q^{37} + 4 q^{41} + 6 q^{43} - 16 q^{47} + 20 q^{49} + 12 q^{53} + 2 q^{55}+ \cdots + 25 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - q^7 + 2 * q^11 + 5 * q^13 + 2 * q^17 - 7 * q^19 + 4 * q^23 + 3 * q^25 + 2 * q^29 + 4 * q^31 - q^35 + 11 * q^37 + 4 * q^41 + 6 * q^43 - 16 * q^47 + 20 * q^49 + 12 * q^53 + 2 * q^55 + 10 * q^59 + 11 * q^61 + 5 * q^65 - 3 * q^67 - 8 * q^71 + 9 * q^73 - 22 * q^77 + 17 * q^79 + 12 * q^83 + 2 * q^85 - 2 * q^89 - 21 * q^91 - 7 * q^95 + 25 * q^97

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