LMFDB - Embedded newform 8280.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.20087896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 $ x^5 - x^4 - 21x^3 + 5x^2 + 84*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.27399$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.55148 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.55148 q^{7} +2.79304 q^{11} +3.73151 q^{13} +7.83097 q^{17} +6.27949 q^{19} +1.00000 q^{23} +1.00000 q^{25} -2.75844 q^{29} -2.48995 q^{31} +3.55148 q^{35} -1.55148 q^{37} -5.78954 q^{41} -4.54798 q^{43} +6.27949 q^{47} +5.61301 q^{49} -8.89250 q^{53} -2.79304 q^{55} -3.31342 q^{59} +11.2180 q^{61} -3.73151 q^{65} +5.52105 q^{67} -6.34452 q^{71} +15.3824 q^{73} -9.91943 q^{77} -9.62051 q^{79} -5.83097 q^{83} -7.83097 q^{85} -0.390487 q^{89} -13.2524 q^{91} -6.27949 q^{95} +0.961896 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} - 4 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 - 4 * q^7	$ 5 q - 5 q^{5} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 10 q^{17} - 4 q^{19} + 5 q^{23} + 5 q^{25} - 10 q^{29} + 6 q^{31} + 4 q^{35} + 6 q^{37} - 12 q^{41} - 2 q^{43} - 4 q^{47} + 19 q^{49} - 4 q^{55} - 6 q^{59} + 16 q^{61} - 4 q^{65} - 4 q^{67} - 8 q^{71} + 14 q^{73} - 16 q^{77} + 18 q^{79} + 20 q^{83} + 10 q^{85} - 18 q^{89} + 20 q^{91} + 4 q^{95} + 4 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 10 * q^17 - 4 * q^19 + 5 * q^23 + 5 * q^25 - 10 * q^29 + 6 * q^31 + 4 * q^35 + 6 * q^37 - 12 * q^41 - 2 * q^43 - 4 * q^47 + 19 * q^49 - 4 * q^55 - 6 * q^59 + 16 * q^61 - 4 * q^65 - 4 * q^67 - 8 * q^71 + 14 * q^73 - 16 * q^77 + 18 * q^79 + 20 * q^83 + 10 * q^85 - 18 * q^89 + 20 * q^91 + 4 * q^95 + 4 * 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
$7$	−3.55148	−1.34233	−0.671167	−	0.741306i	$-0.734207\pi$
$7$	−3.55148	−1.34233	−0.671167	+	0.741306i	$0.734207\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.79304	0.842134	0.421067	−	0.907030i	$-0.361656\pi$
$11$	2.79304	0.842134	0.421067	+	0.907030i	$0.361656\pi$
$12$	0	0
$13$	3.73151	1.03493	0.517467	−	0.855703i	$-0.326875\pi$
$13$	3.73151	1.03493	0.517467	+	0.855703i	$0.326875\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.83097	1.89929	0.949644	−	0.313330i	$-0.101445\pi$
$17$	7.83097	1.89929	0.949644	+	0.313330i	$0.101445\pi$
$18$	0	0
$19$	6.27949	1.44061	0.720307	−	0.693656i	$-0.244001\pi$
$19$	6.27949	1.44061	0.720307	+	0.693656i	$0.244001\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.75844	−0.512229	−0.256115	−	0.966646i	$-0.582442\pi$
$29$	−2.75844	−0.512229	−0.256115	+	0.966646i	$0.582442\pi$
$30$	0	0
$31$	−2.48995	−0.447208	−0.223604	−	0.974680i	$-0.571782\pi$
$31$	−2.48995	−0.447208	−0.223604	+	0.974680i	$0.571782\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.55148	0.600310
$36$	0	0
$37$	−1.55148	−0.255062	−0.127531	−	0.991835i	$-0.540705\pi$
$37$	−1.55148	−0.255062	−0.127531	+	0.991835i	$0.540705\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.78954	−0.904174	−0.452087	−	0.891974i	$-0.649320\pi$
$41$	−5.78954	−0.904174	−0.452087	+	0.891974i	$0.649320\pi$
$42$	0	0
$43$	−4.54798	−0.693560	−0.346780	−	0.937946i	$-0.612725\pi$
$43$	−4.54798	−0.693560	−0.346780	+	0.937946i	$0.612725\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.27949	0.915957	0.457979	−	0.888963i	$-0.348574\pi$
$47$	6.27949	0.915957	0.457979	+	0.888963i	$0.348574\pi$
$48$	0	0
$49$	5.61301	0.801859
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.89250	−1.22148	−0.610739	−	0.791832i	$-0.709128\pi$
$53$	−8.89250	−1.22148	−0.610739	+	0.791832i	$0.709128\pi$
$54$	0	0
$55$	−2.79304	−0.376614
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.31342	−0.431371	−0.215685	−	0.976463i	$-0.569199\pi$
$59$	−3.31342	−0.431371	−0.215685	+	0.976463i	$0.569199\pi$
$60$	0	0
$61$	11.2180	1.43631	0.718156	−	0.695882i	$-0.244986\pi$
$61$	11.2180	1.43631	0.718156	+	0.695882i	$0.244986\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.73151	−0.462837
$66$	0	0
$67$	5.52105	0.674503	0.337252	−	0.941415i	$-0.390503\pi$
$67$	5.52105	0.674503	0.337252	+	0.941415i	$0.390503\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.34452	−0.752956	−0.376478	−	0.926426i	$-0.622865\pi$
$71$	−6.34452	−0.752956	−0.376478	+	0.926426i	$0.622865\pi$
$72$	0	0
$73$	15.3824	1.80038	0.900190	−	0.435498i	$-0.143428\pi$
$73$	15.3824	1.80038	0.900190	+	0.435498i	$0.143428\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.91943	−1.13042
$78$	0	0
$79$	−9.62051	−1.08239	−0.541196	−	0.840897i	$-0.682028\pi$
$79$	−9.62051	−1.08239	−0.541196	+	0.840897i	$0.682028\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.83097	−0.640032	−0.320016	−	0.947412i	$-0.603688\pi$
$83$	−5.83097	−0.640032	−0.320016	+	0.947412i	$0.603688\pi$
$84$	0	0
$85$	−7.83097	−0.849388
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.390487	−0.0413915	−0.0206958	−	0.999786i	$-0.506588\pi$
$89$	−0.390487	−0.0413915	−0.0206958	+	0.999786i	$0.506588\pi$
$90$	0	0
$91$	−13.2524	−1.38923
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.27949	−0.644262
$96$	0	0
$97$	0.961896	0.0976657	0.0488329	−	0.998807i	$-0.484450\pi$
$97$	0.961896	0.0976657	0.0488329	+	0.998807i	$0.484450\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.9035	1.68196	0.840980	−	0.541066i	$-0.181979\pi$
$101$	16.9035	1.68196	0.840980	+	0.541066i	$0.181979\pi$
$102$	0	0
$103$	−6.04143	−0.595280	−0.297640	−	0.954678i	$-0.596199\pi$
$103$	−6.04143	−0.595280	−0.297640	+	0.954678i	$0.596199\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.3141	−1.19045	−0.595224	−	0.803560i	$-0.702937\pi$
$107$	−12.3141	−1.19045	−0.595224	+	0.803560i	$0.702937\pi$
$108$	0	0
$109$	14.8579	1.42313	0.711564	−	0.702621i	$-0.247987\pi$
$109$	14.8579	1.42313	0.711564	+	0.702621i	$0.247987\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.21113	−0.302077	−0.151039	−	0.988528i	$-0.548262\pi$
$113$	−3.21113	−0.302077	−0.151039	+	0.988528i	$0.548262\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−27.8115	−2.54948
$120$	0	0
$121$	−3.19892	−0.290811
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.145425	0.0129043	0.00645217	−	0.999979i	$-0.497946\pi$
$127$	0.145425	0.0129043	0.00645217	+	0.999979i	$0.497946\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−22.3015	−1.93378
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.16450	0.526668	0.263334	−	0.964705i	$-0.415178\pi$
$137$	6.16450	0.526668	0.263334	+	0.964705i	$0.415178\pi$
$138$	0	0
$139$	14.0690	1.19332	0.596660	−	0.802494i	$-0.296494\pi$
$139$	14.0690	1.19332	0.596660	+	0.802494i	$0.296494\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.4223	0.871553
$144$	0	0
$145$	2.75844	0.229076
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.7964	−1.13024	−0.565121	−	0.825008i	$-0.691171\pi$
$149$	−13.7964	−1.13024	−0.565121	+	0.825008i	$0.691171\pi$
$150$	0	0
$151$	16.1989	1.31825	0.659125	−	0.752034i	$-0.270927\pi$
$151$	16.1989	1.31825	0.659125	+	0.752034i	$0.270927\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.48995	0.199997
$156$	0	0
$157$	−14.1105	−1.12614	−0.563068	−	0.826410i	$-0.690379\pi$
$157$	−14.1105	−1.12614	−0.563068	+	0.826410i	$0.690379\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.55148	−0.279896
$162$	0	0
$163$	21.2480	1.66427	0.832137	−	0.554571i	$-0.187118\pi$
$163$	21.2480	1.66427	0.832137	+	0.554571i	$0.187118\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.3895	−1.50040	−0.750200	−	0.661211i	$-0.770043\pi$
$167$	−19.3895	−1.50040	−0.750200	+	0.661211i	$0.770043\pi$
$168$	0	0
$169$	0.924149	0.0710884
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.1030	0.996200	0.498100	−	0.867120i	$-0.334031\pi$
$173$	13.1030	0.996200	0.498100	+	0.867120i	$0.334031\pi$
$174$	0	0
$175$	−3.55148	−0.268467
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.0759	1.05208	0.526039	−	0.850460i	$-0.323676\pi$
$179$	14.0759	1.05208	0.526039	+	0.850460i	$0.323676\pi$
$180$	0	0
$181$	21.1374	1.57113	0.785565	−	0.618779i	$-0.212373\pi$
$181$	21.1374	1.57113	0.785565	+	0.618779i	$0.212373\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.55148	0.114067
$186$	0	0
$187$	21.8722	1.59945
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.81647	−0.203793	−0.101896	−	0.994795i	$-0.532491\pi$
$191$	−2.81647	−0.203793	−0.101896	+	0.994795i	$0.532491\pi$
$192$	0	0
$193$	15.4560	1.11255	0.556274	−	0.830999i	$-0.312230\pi$
$193$	15.4560	1.11255	0.556274	+	0.830999i	$0.312230\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.7311	−1.83327	−0.916634	−	0.399728i	$-0.869105\pi$
$197$	−25.7311	−1.83327	−0.916634	+	0.399728i	$0.869105\pi$
$198$	0	0
$199$	5.95857	0.422392	0.211196	−	0.977444i	$-0.432264\pi$
$199$	5.95857	0.422392	0.211196	+	0.977444i	$0.432264\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.79654	0.687583
$204$	0	0
$205$	5.78954	0.404359
$206$	0	0
$207$	0	0
$208$	0	0
$209$	17.5389	1.21319
$210$	0	0
$211$	−6.06903	−0.417809	−0.208905	−	0.977936i	$-0.566990\pi$
$211$	−6.06903	−0.417809	−0.208905	+	0.977936i	$0.566990\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.54798	0.310170
$216$	0	0
$217$	8.84300	0.600302
$218$	0	0
$219$	0	0
$220$	0	0
$221$	29.2213	1.96564
$222$	0	0
$223$	−1.57509	−0.105476	−0.0527379	−	0.998608i	$-0.516795\pi$
$223$	−1.57509	−0.105476	−0.0527379	+	0.998608i	$0.516795\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	21.7034	1.44050	0.720251	−	0.693713i	$-0.244026\pi$
$227$	21.7034	1.44050	0.720251	+	0.693713i	$0.244026\pi$
$228$	0	0
$229$	23.2136	1.53400	0.766999	−	0.641649i	$-0.221749\pi$
$229$	23.2136	1.53400	0.766999	+	0.641649i	$0.221749\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.1989	1.19225	0.596125	−	0.802891i	$-0.296706\pi$
$233$	18.1989	1.19225	0.596125	+	0.802891i	$0.296706\pi$
$234$	0	0
$235$	−6.27949	−0.409629
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.9236	−1.41812	−0.709060	−	0.705148i	$-0.750880\pi$
$239$	−21.9236	−1.41812	−0.709060	+	0.705148i	$0.750880\pi$
$240$	0	0
$241$	17.7964	1.14636	0.573182	−	0.819428i	$-0.305709\pi$
$241$	17.7964	1.14636	0.573182	+	0.819428i	$0.305709\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.61301	−0.358602
$246$	0	0
$247$	23.4320	1.49094
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.2785	1.46932	0.734661	−	0.678434i	$-0.237341\pi$
$251$	23.2785	1.46932	0.734661	+	0.678434i	$0.237341\pi$
$252$	0	0
$253$	2.79304	0.175597
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.5884	−1.59616	−0.798079	−	0.602552i	$-0.794150\pi$
$257$	−25.5884	−1.59616	−0.798079	+	0.602552i	$0.794150\pi$
$258$	0	0
$259$	5.51005	0.342378
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.57051	0.528481	0.264240	−	0.964457i	$-0.414879\pi$
$263$	8.57051	0.528481	0.264240	+	0.964457i	$0.414879\pi$
$264$	0	0
$265$	8.89250	0.546262
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.4405	0.819481	0.409740	−	0.912202i	$-0.365619\pi$
$269$	13.4405	0.819481	0.409740	+	0.912202i	$0.365619\pi$
$270$	0	0
$271$	3.38699	0.205745	0.102872	−	0.994695i	$-0.467197\pi$
$271$	3.38699	0.205745	0.102872	+	0.994695i	$0.467197\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.79304	0.168427
$276$	0	0
$277$	−18.7469	−1.12639	−0.563196	−	0.826323i	$-0.690428\pi$
$277$	−18.7469	−1.12639	−0.563196	+	0.826323i	$0.690428\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.21396	0.311039	0.155519	−	0.987833i	$-0.450295\pi$
$281$	5.21396	0.311039	0.155519	+	0.987833i	$0.450295\pi$
$282$	0	0
$283$	−18.9771	−1.12807	−0.564035	−	0.825751i	$-0.690752\pi$
$283$	−18.9771	−1.12807	−0.564035	+	0.825751i	$0.690752\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.5614	1.21370
$288$	0	0
$289$	44.3240	2.60730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	27.3976	1.60059	0.800293	−	0.599609i	$-0.204677\pi$
$293$	27.3976	1.60059	0.800293	+	0.599609i	$0.204677\pi$
$294$	0	0
$295$	3.31342	0.192915
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.73151	0.215799
$300$	0	0
$301$	16.1521	0.930989
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−11.2180	−0.642338
$306$	0	0
$307$	1.18353	0.0675475	0.0337738	−	0.999430i	$-0.489247\pi$
$307$	1.18353	0.0675475	0.0337738	+	0.999430i	$0.489247\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.5770	−1.05340	−0.526702	−	0.850050i	$-0.676572\pi$
$311$	−18.5770	−1.05340	−0.526702	+	0.850050i	$0.676572\pi$
$312$	0	0
$313$	−8.50094	−0.480502	−0.240251	−	0.970711i	$-0.577230\pi$
$313$	−8.50094	−0.480502	−0.240251	+	0.970711i	$0.577230\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.5884	−1.21252	−0.606262	−	0.795265i	$-0.707332\pi$
$317$	−21.5884	−1.21252	−0.606262	+	0.795265i	$0.707332\pi$
$318$	0	0
$319$	−7.70443	−0.431366
$320$	0	0
$321$	0	0
$322$	0	0
$323$	49.1745	2.73614
$324$	0	0
$325$	3.73151	0.206987
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−22.3015	−1.22952
$330$	0	0
$331$	24.0288	1.32074	0.660372	−	0.750939i	$-0.270399\pi$
$331$	24.0288	1.32074	0.660372	+	0.750939i	$0.270399\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.52105	−0.301647
$336$	0	0
$337$	−19.7960	−1.07836	−0.539178	−	0.842192i	$-0.681265\pi$
$337$	−19.7960	−1.07836	−0.539178	+	0.842192i	$0.681265\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.95452	−0.376609
$342$	0	0
$343$	4.92585	0.265971
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.52388	−0.189172	−0.0945859	−	0.995517i	$-0.530153\pi$
$347$	−3.52388	−0.189172	−0.0945859	+	0.995517i	$0.530153\pi$
$348$	0	0
$349$	9.10979	0.487636	0.243818	−	0.969821i	$-0.421600\pi$
$349$	9.10979	0.487636	0.243818	+	0.969821i	$0.421600\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.7224	1.31584	0.657920	−	0.753088i	$-0.271436\pi$
$353$	24.7224	1.31584	0.657920	+	0.753088i	$0.271436\pi$
$354$	0	0
$355$	6.34452	0.336732
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3895	−0.601112	−0.300556	−	0.953764i	$-0.597172\pi$
$359$	−11.3895	−0.601112	−0.300556	+	0.953764i	$0.597172\pi$
$360$	0	0
$361$	20.4320	1.07537
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−15.3824	−0.805154
$366$	0	0
$367$	11.1306	0.581011	0.290505	−	0.956873i	$-0.406177\pi$
$367$	11.1306	0.581011	0.290505	+	0.956873i	$0.406177\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.5815	1.63963
$372$	0	0
$373$	8.51755	0.441022	0.220511	−	0.975385i	$-0.429228\pi$
$373$	8.51755	0.441022	0.220511	+	0.975385i	$0.429228\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.2931	−0.530123
$378$	0	0
$379$	−21.1320	−1.08548	−0.542738	−	0.839902i	$-0.682612\pi$
$379$	−21.1320	−1.08548	−0.542738	+	0.839902i	$0.682612\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.37562	−0.479072	−0.239536	−	0.970887i	$-0.576995\pi$
$383$	−9.37562	−0.479072	−0.239536	+	0.970887i	$0.576995\pi$
$384$	0	0
$385$	9.91943	0.505541
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.6352	0.843437	0.421719	−	0.906727i	$-0.361427\pi$
$389$	16.6352	0.843437	0.421719	+	0.906727i	$0.361427\pi$
$390$	0	0
$391$	7.83097	0.396029
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.62051	0.484060
$396$	0	0
$397$	23.9261	1.20081	0.600407	−	0.799694i	$-0.295005\pi$
$397$	23.9261	1.20081	0.600407	+	0.799694i	$0.295005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12.4457	0.621508	0.310754	−	0.950490i	$-0.399418\pi$
$401$	12.4457	0.621508	0.310754	+	0.950490i	$0.399418\pi$
$402$	0	0
$403$	−9.29125	−0.462830
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.33335	−0.214796
$408$	0	0
$409$	27.0419	1.33714	0.668568	−	0.743651i	$-0.266907\pi$
$409$	27.0419	1.33714	0.668568	+	0.743651i	$0.266907\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.7676	0.579043
$414$	0	0
$415$	5.83097	0.286231
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.0949	−0.493169	−0.246585	−	0.969121i	$-0.579308\pi$
$419$	−10.0949	−0.493169	−0.246585	+	0.969121i	$0.579308\pi$
$420$	0	0
$421$	−28.5268	−1.39031	−0.695156	−	0.718858i	$-0.744665\pi$
$421$	−28.5268	−1.39031	−0.695156	+	0.718858i	$0.744665\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.83097	0.379858
$426$	0	0
$427$	−39.8403	−1.92801
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.6081	−1.04082	−0.520412	−	0.853915i	$-0.674222\pi$
$431$	−21.6081	−1.04082	−0.520412	+	0.853915i	$0.674222\pi$
$432$	0	0
$433$	−8.21009	−0.394552	−0.197276	−	0.980348i	$-0.563209\pi$
$433$	−8.21009	−0.394552	−0.197276	+	0.980348i	$0.563209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.27949	0.300389
$438$	0	0
$439$	24.4429	1.16660	0.583298	−	0.812258i	$-0.301762\pi$
$439$	24.4429	1.16660	0.583298	+	0.812258i	$0.301762\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−32.5074	−1.54447	−0.772237	−	0.635335i	$-0.780862\pi$
$443$	−32.5074	−1.54447	−0.772237	+	0.635335i	$0.780862\pi$
$444$	0	0
$445$	0.390487	0.0185109
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.5635	1.15922	0.579612	−	0.814893i	$-0.303204\pi$
$449$	24.5635	1.15922	0.579612	+	0.814893i	$0.303204\pi$
$450$	0	0
$451$	−16.1704	−0.761436
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.2524	0.621281
$456$	0	0
$457$	−29.6798	−1.38836	−0.694180	−	0.719801i	$-0.744233\pi$
$457$	−29.6798	−1.38836	−0.694180	+	0.719801i	$0.744233\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	34.2239	1.59397	0.796983	−	0.604001i	$-0.206428\pi$
$461$	34.2239	1.59397	0.796983	+	0.604001i	$0.206428\pi$
$462$	0	0
$463$	−27.8074	−1.29232	−0.646159	−	0.763203i	$-0.723626\pi$
$463$	−27.8074	−1.29232	−0.646159	+	0.763203i	$0.723626\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.3207	−1.03288	−0.516440	−	0.856323i	$-0.672743\pi$
$467$	−22.3207	−1.03288	−0.516440	+	0.856323i	$0.672743\pi$
$468$	0	0
$469$	−19.6079	−0.905408
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12.7027	−0.584070
$474$	0	0
$475$	6.27949	0.288123
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.7535	1.49655	0.748274	−	0.663390i	$-0.230883\pi$
$479$	32.7535	1.49655	0.748274	+	0.663390i	$0.230883\pi$
$480$	0	0
$481$	−5.78936	−0.263972
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.961896	−0.0436774
$486$	0	0
$487$	−25.9996	−1.17816	−0.589078	−	0.808076i	$-0.700509\pi$
$487$	−25.9996	−1.17816	−0.589078	+	0.808076i	$0.700509\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.67348	0.346299	0.173150	−	0.984896i	$-0.444606\pi$
$491$	7.67348	0.346299	0.173150	+	0.984896i	$0.444606\pi$
$492$	0	0
$493$	−21.6012	−0.972871
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.5324	1.01072
$498$	0	0
$499$	15.2951	0.684701	0.342350	−	0.939572i	$-0.388777\pi$
$499$	15.2951	0.684701	0.342350	+	0.939572i	$0.388777\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9.79654	0.436806	0.218403	−	0.975859i	$-0.429915\pi$
$503$	9.79654	0.436806	0.218403	+	0.975859i	$0.429915\pi$
$504$	0	0
$505$	−16.9035	−0.752196
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.25713	−0.277342	−0.138671	−	0.990338i	$-0.544283\pi$
$509$	−6.25713	−0.277342	−0.138671	+	0.990338i	$0.544283\pi$
$510$	0	0
$511$	−54.6305	−2.41671
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.04143	0.266217
$516$	0	0
$517$	17.5389	0.771358
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.72384	0.382198	0.191099	−	0.981571i	$-0.438795\pi$
$521$	8.72384	0.382198	0.191099	+	0.981571i	$0.438795\pi$
$522$	0	0
$523$	−8.66404	−0.378852	−0.189426	−	0.981895i	$-0.560663\pi$
$523$	−8.66404	−0.378852	−0.189426	+	0.981895i	$0.560663\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.4987	−0.849376
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−21.6037	−0.935761
$534$	0	0
$535$	12.3141	0.532384
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15.6774	0.675273
$540$	0	0
$541$	−36.5660	−1.57209	−0.786047	−	0.618167i	$-0.787876\pi$
$541$	−36.5660	−1.57209	−0.786047	+	0.618167i	$0.787876\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.8579	−0.636442
$546$	0	0
$547$	−0.827834	−0.0353956	−0.0176978	−	0.999843i	$-0.505634\pi$
$547$	−0.827834	−0.0353956	−0.0176978	+	0.999843i	$0.505634\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−17.3216	−0.737924
$552$	0	0
$553$	34.1670	1.45293
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−0.125533	−0.00531902	−0.00265951	−	0.999996i	$-0.500847\pi$
$557$	−0.125533	−0.00531902	−0.00265951	+	0.999996i	$0.500847\pi$
$558$	0	0
$559$	−16.9708	−0.717789
$560$	0	0
$561$	0	0
$562$	0	0
$563$	38.3207	1.61503	0.807513	−	0.589849i	$-0.200813\pi$
$563$	38.3207	1.61503	0.807513	+	0.589849i	$0.200813\pi$
$564$	0	0
$565$	3.21113	0.135093
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.8715	1.21036	0.605179	−	0.796090i	$-0.293102\pi$
$569$	28.8715	1.21036	0.605179	+	0.796090i	$0.293102\pi$
$570$	0	0
$571$	15.0514	0.629881	0.314940	−	0.949111i	$-0.398015\pi$
$571$	15.0514	0.629881	0.314940	+	0.949111i	$0.398015\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−21.0561	−0.876577	−0.438288	−	0.898834i	$-0.644415\pi$
$577$	−21.0561	−0.876577	−0.438288	+	0.898834i	$0.644415\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	20.7086	0.859136
$582$	0	0
$583$	−24.8371	−1.02865
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.2751	−0.919393	−0.459696	−	0.888076i	$-0.652042\pi$
$587$	−22.2751	−0.919393	−0.459696	+	0.888076i	$0.652042\pi$
$588$	0	0
$589$	−15.6356	−0.644253
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.45642	0.347263	0.173632	−	0.984811i	$-0.444450\pi$
$593$	8.45642	0.347263	0.173632	+	0.984811i	$0.444450\pi$
$594$	0	0
$595$	27.8115	1.14016
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.0040	0.408752	0.204376	−	0.978892i	$-0.434483\pi$
$599$	10.0040	0.408752	0.204376	+	0.978892i	$0.434483\pi$
$600$	0	0
$601$	14.8119	0.604191	0.302096	−	0.953278i	$-0.402314\pi$
$601$	14.8119	0.604191	0.302096	+	0.953278i	$0.402314\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.19892	0.130055
$606$	0	0
$607$	10.8883	0.441944	0.220972	−	0.975280i	$-0.429077\pi$
$607$	10.8883	0.441944	0.220972	+	0.975280i	$0.429077\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23.4320	0.947955
$612$	0	0
$613$	3.35238	0.135401	0.0677007	−	0.997706i	$-0.478434\pi$
$613$	3.35238	0.135401	0.0677007	+	0.997706i	$0.478434\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.8848	0.881050	0.440525	−	0.897740i	$-0.354792\pi$
$617$	21.8848	0.881050	0.440525	+	0.897740i	$0.354792\pi$
$618$	0	0
$619$	−10.0762	−0.404997	−0.202498	−	0.979283i	$-0.564906\pi$
$619$	−10.0762	−0.404997	−0.202498	+	0.979283i	$0.564906\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.38681	0.0555613
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12.1496	−0.484436
$630$	0	0
$631$	6.50485	0.258954	0.129477	−	0.991582i	$-0.458670\pi$
$631$	6.50485	0.258954	0.129477	+	0.991582i	$0.458670\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.145425	−0.00577100
$636$	0	0
$637$	20.9450	0.829871
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.60723	0.0634816	0.0317408	−	0.999496i	$-0.489895\pi$
$641$	1.60723	0.0634816	0.0317408	+	0.999496i	$0.489895\pi$
$642$	0	0
$643$	−7.73605	−0.305080	−0.152540	−	0.988297i	$-0.548745\pi$
$643$	−7.73605	−0.305080	−0.152540	+	0.988297i	$0.548745\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.89232	0.349593	0.174797	−	0.984605i	$-0.444073\pi$
$647$	8.89232	0.349593	0.174797	+	0.984605i	$0.444073\pi$
$648$	0	0
$649$	−9.25452	−0.363272
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.2055	−1.22117	−0.610583	−	0.791952i	$-0.709065\pi$
$653$	−31.2055	−1.22117	−0.610583	+	0.791952i	$0.709065\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.7051	0.845509	0.422755	−	0.906244i	$-0.361063\pi$
$659$	21.7051	0.845509	0.422755	+	0.906244i	$0.361063\pi$
$660$	0	0
$661$	30.0564	1.16906	0.584529	−	0.811372i	$-0.301279\pi$
$661$	30.0564	1.16906	0.584529	+	0.811372i	$0.301279\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	22.3015	0.864814
$666$	0	0
$667$	−2.75844	−0.106807
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31.3322	1.20957
$672$	0	0
$673$	16.2795	0.627528	0.313764	−	0.949501i	$-0.398410\pi$
$673$	16.2795	0.627528	0.313764	+	0.949501i	$0.398410\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.70723	0.180914	0.0904568	−	0.995900i	$-0.471167\pi$
$677$	4.70723	0.180914	0.0904568	+	0.995900i	$0.471167\pi$
$678$	0	0
$679$	−3.41616	−0.131100
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−9.82536	−0.375957	−0.187978	−	0.982173i	$-0.560193\pi$
$683$	−9.82536	−0.375957	−0.187978	+	0.982173i	$0.560193\pi$
$684$	0	0
$685$	−6.16450	−0.235533
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−33.1824	−1.26415
$690$	0	0
$691$	−43.1718	−1.64233	−0.821167	−	0.570689i	$-0.806676\pi$
$691$	−43.1718	−1.64233	−0.821167	+	0.570689i	$0.806676\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.0690	−0.533669
$696$	0	0
$697$	−45.3377	−1.71729
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.44539	0.0545915	0.0272957	−	0.999627i	$-0.491310\pi$
$701$	1.44539	0.0545915	0.0272957	+	0.999627i	$0.491310\pi$
$702$	0	0
$703$	−9.74250	−0.367445
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−60.0324	−2.25775
$708$	0	0
$709$	−28.9697	−1.08798	−0.543991	−	0.839091i	$-0.683087\pi$
$709$	−28.9697	−1.08798	−0.543991	+	0.839091i	$0.683087\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.48995	−0.0932492
$714$	0	0
$715$	−10.4223	−0.389770
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−42.4742	−1.58402	−0.792011	−	0.610507i	$-0.790966\pi$
$719$	−42.4742	−1.58402	−0.792011	+	0.610507i	$0.790966\pi$
$720$	0	0
$721$	21.4560	0.799064
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.75844	−0.102446
$726$	0	0
$727$	−3.10857	−0.115291	−0.0576453	−	0.998337i	$-0.518359\pi$
$727$	−3.10857	−0.115291	−0.0576453	+	0.998337i	$0.518359\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−35.6151	−1.31727
$732$	0	0
$733$	6.20453	0.229170	0.114585	−	0.993413i	$-0.463446\pi$
$733$	6.20453	0.229170	0.114585	+	0.993413i	$0.463446\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.4205	0.568022
$738$	0	0
$739$	−21.5010	−0.790926	−0.395463	−	0.918482i	$-0.629416\pi$
$739$	−21.5010	−0.790926	−0.395463	+	0.918482i	$0.629416\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.4469	0.933558	0.466779	−	0.884374i	$-0.345414\pi$
$743$	25.4469	0.933558	0.466779	+	0.884374i	$0.345414\pi$
$744$	0	0
$745$	13.7964	0.505460
$746$	0	0
$747$	0	0
$748$	0	0
$749$	43.7332	1.59798
$750$	0	0
$751$	49.9066	1.82112	0.910560	−	0.413378i	$-0.135651\pi$
$751$	49.9066	1.82112	0.910560	+	0.413378i	$0.135651\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.1989	−0.589539
$756$	0	0
$757$	31.8486	1.15756	0.578778	−	0.815485i	$-0.303530\pi$
$757$	31.8486	1.15756	0.578778	+	0.815485i	$0.303530\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.63541	−0.276783	−0.138392	−	0.990378i	$-0.544193\pi$
$761$	−7.63541	−0.276783	−0.138392	+	0.990378i	$0.544193\pi$
$762$	0	0
$763$	−52.7675	−1.91031
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.3641	−0.446440
$768$	0	0
$769$	−3.36880	−0.121482	−0.0607410	−	0.998154i	$-0.519346\pi$
$769$	−3.36880	−0.121482	−0.0607410	+	0.998154i	$0.519346\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.77800	−0.207820	−0.103910	−	0.994587i	$-0.533135\pi$
$773$	−5.77800	−0.207820	−0.103910	+	0.994587i	$0.533135\pi$
$774$	0	0
$775$	−2.48995	−0.0894415
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.3553	−1.30257
$780$	0	0
$781$	−17.7205	−0.634090
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.1105	0.503624
$786$	0	0
$787$	6.46988	0.230626	0.115313	−	0.993329i	$-0.463213\pi$
$787$	6.46988	0.230626	0.115313	+	0.993329i	$0.463213\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.4043	0.405489
$792$	0	0
$793$	41.8599	1.48649
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.9636	0.777990	0.388995	−	0.921240i	$-0.372822\pi$
$797$	21.9636	0.777990	0.388995	+	0.921240i	$0.372822\pi$
$798$	0	0
$799$	49.1745	1.73967
$800$	0	0
$801$	0	0
$802$	0	0
$803$	42.9638	1.51616
$804$	0	0
$805$	3.55148	0.125173
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.7222	1.50203	0.751017	−	0.660283i	$-0.229564\pi$
$809$	42.7222	1.50203	0.751017	+	0.660283i	$0.229564\pi$
$810$	0	0
$811$	−30.2347	−1.06169	−0.530843	−	0.847470i	$-0.678124\pi$
$811$	−30.2347	−1.06169	−0.530843	+	0.847470i	$0.678124\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21.2480	−0.744286
$816$	0	0
$817$	−28.5590	−0.999152
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.3260	0.639581	0.319791	−	0.947488i	$-0.396387\pi$
$821$	18.3260	0.639581	0.319791	+	0.947488i	$0.396387\pi$
$822$	0	0
$823$	−14.5081	−0.505721	−0.252861	−	0.967503i	$-0.581371\pi$
$823$	−14.5081	−0.505721	−0.252861	+	0.967503i	$0.581371\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0452	0.975228	0.487614	−	0.873059i	$-0.337867\pi$
$827$	28.0452	0.975228	0.487614	+	0.873059i	$0.337867\pi$
$828$	0	0
$829$	−26.2281	−0.910939	−0.455470	−	0.890251i	$-0.650529\pi$
$829$	−26.2281	−0.910939	−0.455470	+	0.890251i	$0.650529\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	43.9553	1.52296
$834$	0	0
$835$	19.3895	0.671000
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.2129	−0.421637	−0.210819	−	0.977525i	$-0.567613\pi$
$839$	−12.2129	−0.421637	−0.210819	+	0.977525i	$0.567613\pi$
$840$	0	0
$841$	−21.3910	−0.737621
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.924149	−0.0317917
$846$	0	0
$847$	11.3609	0.390365
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.55148	−0.0531841
$852$	0	0
$853$	54.2928	1.85895	0.929474	−	0.368886i	$-0.120261\pi$
$853$	54.2928	1.85895	0.929474	+	0.368886i	$0.120261\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.0491	−0.514067	−0.257034	−	0.966402i	$-0.582745\pi$
$857$	−15.0491	−0.514067	−0.257034	+	0.966402i	$0.582745\pi$
$858$	0	0
$859$	34.2351	1.16809	0.584043	−	0.811723i	$-0.301470\pi$
$859$	34.2351	1.16809	0.584043	+	0.811723i	$0.301470\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.6488	−0.498652	−0.249326	−	0.968420i	$-0.580209\pi$
$863$	−14.6488	−0.498652	−0.249326	+	0.968420i	$0.580209\pi$
$864$	0	0
$865$	−13.1030	−0.445514
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−26.8705	−0.911518
$870$	0	0
$871$	20.6018	0.698066
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.55148	0.120062
$876$	0	0
$877$	−54.2319	−1.83128	−0.915641	−	0.401998i	$-0.868316\pi$
$877$	−54.2319	−1.83128	−0.915641	+	0.401998i	$0.868316\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−44.3162	−1.49305	−0.746525	−	0.665357i	$-0.768279\pi$
$881$	−44.3162	−1.49305	−0.746525	+	0.665357i	$0.768279\pi$
$882$	0	0
$883$	38.2704	1.28790	0.643951	−	0.765067i	$-0.277294\pi$
$883$	38.2704	1.28790	0.643951	+	0.765067i	$0.277294\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	5.80699	0.194980	0.0974898	−	0.995237i	$-0.468919\pi$
$887$	5.80699	0.194980	0.0974898	+	0.995237i	$0.468919\pi$
$888$	0	0
$889$	−0.516473	−0.0173219
$890$	0	0
$891$	0	0
$892$	0	0
$893$	39.4320	1.31954
$894$	0	0
$895$	−14.0759	−0.470504
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.86837	0.229073
$900$	0	0
$901$	−69.6369	−2.31994
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.1374	−0.702630
$906$	0	0
$907$	−12.3419	−0.409805	−0.204903	−	0.978782i	$-0.565688\pi$
$907$	−12.3419	−0.409805	−0.204903	+	0.978782i	$0.565688\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27.2190	−0.901807	−0.450903	−	0.892573i	$-0.648898\pi$
$911$	−27.2190	−0.901807	−0.450903	+	0.892573i	$0.648898\pi$
$912$	0	0
$913$	−16.2861	−0.538992
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.6836	1.27605	0.638027	−	0.770014i	$-0.279751\pi$
$919$	38.6836	1.27605	0.638027	+	0.770014i	$0.279751\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−23.6746	−0.779260
$924$	0	0
$925$	−1.55148	−0.0510124
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.1365	0.562230	0.281115	−	0.959674i	$-0.409296\pi$
$929$	17.1365	0.562230	0.281115	+	0.959674i	$0.409296\pi$
$930$	0	0
$931$	35.2468	1.15517
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−21.8722	−0.715298
$936$	0	0
$937$	44.4382	1.45173	0.725866	−	0.687836i	$-0.241439\pi$
$937$	44.4382	1.45173	0.725866	+	0.687836i	$0.241439\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	19.1745	0.625069	0.312535	−	0.949906i	$-0.398822\pi$
$941$	19.1745	0.625069	0.312535	+	0.949906i	$0.398822\pi$
$942$	0	0
$943$	−5.78954	−0.188533
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−56.7662	−1.84465	−0.922327	−	0.386410i	$-0.873715\pi$
$947$	−56.7662	−1.84465	−0.922327	+	0.386410i	$0.873715\pi$
$948$	0	0
$949$	57.3997	1.86327
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37.8953	−1.22755	−0.613774	−	0.789482i	$-0.710349\pi$
$953$	−37.8953	−1.22755	−0.613774	+	0.789482i	$0.710349\pi$
$954$	0	0
$955$	2.81647	0.0911389
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−21.8931	−0.706965
$960$	0	0
$961$	−24.8002	−0.800005
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.4560	−0.497547
$966$	0	0
$967$	58.9924	1.89707	0.948534	−	0.316674i	$-0.102566\pi$
$967$	58.9924	1.89707	0.948534	+	0.316674i	$0.102566\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19.9675	−0.640787	−0.320394	−	0.947284i	$-0.603815\pi$
$971$	−19.9675	−0.640787	−0.320394	+	0.947284i	$0.603815\pi$
$972$	0	0
$973$	−49.9659	−1.60183
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.54712	0.177468	0.0887341	−	0.996055i	$-0.471718\pi$
$977$	5.54712	0.177468	0.0887341	+	0.996055i	$0.471718\pi$
$978$	0	0
$979$	−1.09065	−0.0348572
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.1636	−0.387959	−0.193979	−	0.981006i	$-0.562139\pi$
$983$	−12.1636	−0.387959	−0.193979	+	0.981006i	$0.562139\pi$
$984$	0	0
$985$	25.7311	0.819862
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.54798	−0.144617
$990$	0	0
$991$	18.1991	0.578113	0.289057	−	0.957312i	$-0.406658\pi$
$991$	18.1991	0.578113	0.289057	+	0.957312i	$0.406658\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.95857	−0.188899
$996$	0	0
$997$	49.4097	1.56482	0.782411	−	0.622762i	$-0.213990\pi$
$997$	49.4097	1.56482	0.782411	+	0.622762i	$0.213990\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.br.1.2		5
3.2	odd	2		2760.2.a.w.1.2	✓	5
12.11	even	2		5520.2.a.cc.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.w.1.2	✓	5	3.2	odd	2
5520.2.a.cc.1.4		5	12.11	even	2
8280.2.a.br.1.2		5	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.55148 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.55148 q^{7} +2.79304 q^{11} +3.73151 q^{13} +7.83097 q^{17} +6.27949 q^{19} +1.00000 q^{23} +1.00000 q^{25} -2.75844 q^{29} -2.48995 q^{31} +3.55148 q^{35} -1.55148 q^{37} -5.78954 q^{41} -4.54798 q^{43} +6.27949 q^{47} +5.61301 q^{49} -8.89250 q^{53} -2.79304 q^{55} -3.31342 q^{59} +11.2180 q^{61} -3.73151 q^{65} +5.52105 q^{67} -6.34452 q^{71} +15.3824 q^{73} -9.91943 q^{77} -9.62051 q^{79} -5.83097 q^{83} -7.83097 q^{85} -0.390487 q^{89} -13.2524 q^{91} -6.27949 q^{95} +0.961896 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} - 4 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 - 4 * q^7	\( 5 q - 5 q^{5} - 4 q^{7} + 4 q^{11} + 4 q^{13} - 10 q^{17} - 4 q^{19} + 5 q^{23} + 5 q^{25} - 10 q^{29} + 6 q^{31} + 4 q^{35} + 6 q^{37} - 12 q^{41} - 2 q^{43} - 4 q^{47} + 19 q^{49} - 4 q^{55} - 6 q^{59} + 16 q^{61} - 4 q^{65} - 4 q^{67} - 8 q^{71} + 14 q^{73} - 16 q^{77} + 18 q^{79} + 20 q^{83} + 10 q^{85} - 18 q^{89} + 20 q^{91} + 4 q^{95} + 4 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 - 4 * q^7 + 4 * q^11 + 4 * q^13 - 10 * q^17 - 4 * q^19 + 5 * q^23 + 5 * q^25 - 10 * q^29 + 6 * q^31 + 4 * q^35 + 6 * q^37 - 12 * q^41 - 2 * q^43 - 4 * q^47 + 19 * q^49 - 4 * q^55 - 6 * q^59 + 16 * q^61 - 4 * q^65 - 4 * q^67 - 8 * q^71 + 14 * q^73 - 16 * q^77 + 18 * q^79 + 20 * q^83 + 10 * q^85 - 18 * q^89 + 20 * q^91 + 4 * q^95 + 4 * q^97

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