LMFDB - Embedded newform 8280.2.a.bj.1.3

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:	$3$
Coefficient field:	3.3.2597.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 8 $ x^3 - x^2 - 9*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.95759$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +3.95759 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +3.95759 q^{7} +0.957587 q^{11} -2.74732 q^{13} -5.74732 q^{17} -6.74732 q^{19} -1.00000 q^{23} +1.00000 q^{25} -5.21027 q^{29} -5.95759 q^{31} -3.95759 q^{35} +9.12544 q^{37} -0.252679 q^{41} +8.00000 q^{43} -5.49464 q^{47} +8.66249 q^{49} +7.12544 q^{53} -0.957587 q^{55} +4.78973 q^{59} +12.4522 q^{61} +2.74732 q^{65} +9.12544 q^{67} -1.66249 q^{71} +12.3357 q^{73} +3.78973 q^{77} +11.8303 q^{79} -0.704908 q^{83} +5.74732 q^{85} +15.8303 q^{89} -10.8728 q^{91} +6.74732 q^{95} -10.8728 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 2 * q^7	$ 3 q - 3 q^{5} + 2 q^{7} - 7 q^{11} - q^{13} - 10 q^{17} - 13 q^{19} - 3 q^{23} + 3 q^{25} - 13 q^{29} - 8 q^{31} - 2 q^{35} + 5 q^{37} - 8 q^{41} + 24 q^{43} - 2 q^{47} - q^{49} - q^{53} + 7 q^{55} + 17 q^{59} + 13 q^{61} + q^{65} + 5 q^{67} + 22 q^{71} + 12 q^{73} + 14 q^{77} - 4 q^{79} + 15 q^{83} + 10 q^{85} + 8 q^{89} - 3 q^{91} + 13 q^{95} - 3 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - 7 * q^11 - q^13 - 10 * q^17 - 13 * q^19 - 3 * q^23 + 3 * q^25 - 13 * q^29 - 8 * q^31 - 2 * q^35 + 5 * q^37 - 8 * q^41 + 24 * q^43 - 2 * q^47 - q^49 - q^53 + 7 * q^55 + 17 * q^59 + 13 * q^61 + q^65 + 5 * q^67 + 22 * q^71 + 12 * q^73 + 14 * q^77 - 4 * q^79 + 15 * q^83 + 10 * q^85 + 8 * q^89 - 3 * q^91 + 13 * q^95 - 3 * 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
$7$	3.95759	1.49583	0.747914	−	0.663796i	$-0.231056\pi$
$7$	3.95759	1.49583	0.747914	+	0.663796i	$0.231056\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.957587	0.288723	0.144362	−	0.989525i	$-0.453887\pi$
$11$	0.957587	0.288723	0.144362	+	0.989525i	$0.453887\pi$
$12$	0	0
$13$	−2.74732	−0.761970	−0.380985	−	0.924581i	$-0.624415\pi$
$13$	−2.74732	−0.761970	−0.380985	+	0.924581i	$0.624415\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.74732	−1.39393	−0.696965	−	0.717105i	$-0.745467\pi$
$17$	−5.74732	−1.39393	−0.696965	+	0.717105i	$0.745467\pi$
$18$	0	0
$19$	−6.74732	−1.54794	−0.773971	−	0.633221i	$-0.781732\pi$
$19$	−6.74732	−1.54794	−0.773971	+	0.633221i	$0.781732\pi$
$20$	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
$29$	−5.21027	−0.967522	−0.483761	−	0.875200i	$-0.660730\pi$
$29$	−5.21027	−0.967522	−0.483761	+	0.875200i	$0.660730\pi$
$30$	0	0
$31$	−5.95759	−1.07001	−0.535007	−	0.844848i	$-0.679691\pi$
$31$	−5.95759	−1.07001	−0.535007	+	0.844848i	$0.679691\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.95759	−0.668954
$36$	0	0
$37$	9.12544	1.50021	0.750107	−	0.661317i	$-0.230002\pi$
$37$	9.12544	1.50021	0.750107	+	0.661317i	$0.230002\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.252679	−0.0394619	−0.0197309	−	0.999805i	$-0.506281\pi$
$41$	−0.252679	−0.0394619	−0.0197309	+	0.999805i	$0.506281\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.49464	−0.801476	−0.400738	−	0.916193i	$-0.631246\pi$
$47$	−5.49464	−0.801476	−0.400738	+	0.916193i	$0.631246\pi$
$48$	0	0
$49$	8.66249	1.23750
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.12544	0.978754	0.489377	−	0.872072i	$-0.337224\pi$
$53$	7.12544	0.978754	0.489377	+	0.872072i	$0.337224\pi$
$54$	0	0
$55$	−0.957587	−0.129121
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.78973	0.623570	0.311785	−	0.950153i	$-0.399073\pi$
$59$	4.78973	0.623570	0.311785	+	0.950153i	$0.399073\pi$
$60$	0	0
$61$	12.4522	1.59434	0.797172	−	0.603752i	$-0.206328\pi$
$61$	12.4522	1.59434	0.797172	+	0.603752i	$0.206328\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.74732	0.340763
$66$	0	0
$67$	9.12544	1.11485	0.557425	−	0.830227i	$-0.311789\pi$
$67$	9.12544	1.11485	0.557425	+	0.830227i	$0.311789\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.66249	−0.197302	−0.0986509	−	0.995122i	$-0.531453\pi$
$71$	−1.66249	−0.197302	−0.0986509	+	0.995122i	$0.531453\pi$
$72$	0	0
$73$	12.3357	1.44379	0.721893	−	0.692005i	$-0.243272\pi$
$73$	12.3357	1.44379	0.721893	+	0.692005i	$0.243272\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.78973	0.431880
$78$	0	0
$79$	11.8303	1.33102	0.665509	−	0.746390i	$-0.268214\pi$
$79$	11.8303	1.33102	0.665509	+	0.746390i	$0.268214\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.704908	−0.0773737	−0.0386868	−	0.999251i	$-0.512317\pi$
$83$	−0.704908	−0.0773737	−0.0386868	+	0.999251i	$0.512317\pi$
$84$	0	0
$85$	5.74732	0.623384
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.8303	1.67801	0.839007	−	0.544121i	$-0.183137\pi$
$89$	15.8303	1.67801	0.839007	+	0.544121i	$0.183137\pi$
$90$	0	0
$91$	−10.8728	−1.13978
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.74732	0.692261
$96$	0	0
$97$	−10.8728	−1.10396	−0.551981	−	0.833857i	$-0.686128\pi$
$97$	−10.8728	−1.10396	−0.551981	+	0.833857i	$0.686128\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.21027	−0.916456	−0.458228	−	0.888835i	$-0.651516\pi$
$101$	−9.21027	−0.916456	−0.458228	+	0.888835i	$0.651516\pi$
$102$	0	0
$103$	12.4522	1.22695	0.613477	−	0.789712i	$-0.289770\pi$
$103$	12.4522	1.22695	0.613477	+	0.789712i	$0.289770\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.63080	−0.351003	−0.175501	−	0.984479i	$-0.556155\pi$
$107$	−3.63080	−0.351003	−0.175501	+	0.984479i	$0.556155\pi$
$108$	0	0
$109$	16.6625	1.59598	0.797989	−	0.602672i	$-0.205897\pi$
$109$	16.6625	1.59598	0.797989	+	0.602672i	$0.205897\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.1147	1.70409	0.852045	−	0.523469i	$-0.175362\pi$
$113$	18.1147	1.70409	0.852045	+	0.523469i	$0.175362\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−22.7455	−2.08508
$120$	0	0
$121$	−10.0830	−0.916639
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.91517	−0.524887	−0.262443	−	0.964947i	$-0.584528\pi$
$127$	−5.91517	−0.524887	−0.262443	+	0.964947i	$0.584528\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.40982	−0.297917	−0.148958	−	0.988843i	$-0.547592\pi$
$131$	−3.40982	−0.297917	−0.148958	+	0.988843i	$0.547592\pi$
$132$	0	0
$133$	−26.7031	−2.31545
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.67321	0.142952	0.0714761	−	0.997442i	$-0.477229\pi$
$137$	1.67321	0.142952	0.0714761	+	0.997442i	$0.477229\pi$
$138$	0	0
$139$	8.62008	0.731146	0.365573	−	0.930783i	$-0.380873\pi$
$139$	8.62008	0.731146	0.365573	+	0.930783i	$0.380873\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.63080	−0.219998
$144$	0	0
$145$	5.21027	0.432689
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.0830	1.89104	0.945518	−	0.325571i	$-0.105556\pi$
$149$	23.0830	1.89104	0.945518	+	0.325571i	$0.105556\pi$
$150$	0	0
$151$	10.7879	0.877910	0.438955	−	0.898509i	$-0.355349\pi$
$151$	10.7879	0.877910	0.438955	+	0.898509i	$0.355349\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.95759	0.478525
$156$	0	0
$157$	8.19955	0.654395	0.327198	−	0.944956i	$-0.393896\pi$
$157$	8.19955	0.654395	0.327198	+	0.944956i	$0.393896\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.95759	−0.311902
$162$	0	0
$163$	16.6625	1.30511	0.652554	−	0.757743i	$-0.273698\pi$
$163$	16.6625	1.30511	0.652554	+	0.757743i	$0.273698\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−5.45223	−0.419402
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.87276	−0.674584	−0.337292	−	0.941400i	$-0.609511\pi$
$173$	−8.87276	−0.674584	−0.337292	+	0.941400i	$0.609511\pi$
$174$	0	0
$175$	3.95759	0.299165
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.49464	−0.111715	−0.0558574	−	0.998439i	$-0.517789\pi$
$179$	−1.49464	−0.111715	−0.0558574	+	0.998439i	$0.517789\pi$
$180$	0	0
$181$	−12.5777	−0.934891	−0.467445	−	0.884022i	$-0.654826\pi$
$181$	−12.5777	−0.934891	−0.467445	+	0.884022i	$0.654826\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.12544	−0.670916
$186$	0	0
$187$	−5.50356	−0.402460
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.9045	−1.22316	−0.611582	−	0.791181i	$-0.709466\pi$
$191$	−16.9045	−1.22316	−0.611582	+	0.791181i	$0.709466\pi$
$192$	0	0
$193$	9.91517	0.713710	0.356855	−	0.934160i	$-0.383849\pi$
$193$	9.91517	0.713710	0.356855	+	0.934160i	$0.383849\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.53705	−0.323252	−0.161626	−	0.986852i	$-0.551674\pi$
$197$	−4.53705	−0.323252	−0.161626	+	0.986852i	$0.551674\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−20.6201	−1.44725
$204$	0	0
$205$	0.252679	0.0176479
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.46115	−0.446927
$210$	0	0
$211$	8.70491	0.599271	0.299635	−	0.954054i	$-0.403135\pi$
$211$	8.70491	0.599271	0.299635	+	0.954054i	$0.403135\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	−23.5777	−1.60056
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.7897	1.06213
$222$	0	0
$223$	−5.40982	−0.362268	−0.181134	−	0.983458i	$-0.557977\pi$
$223$	−5.40982	−0.362268	−0.181134	+	0.983458i	$0.557977\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.3250	1.14990	0.574950	−	0.818189i	$-0.305022\pi$
$227$	17.3250	1.14990	0.574950	+	0.818189i	$0.305022\pi$
$228$	0	0
$229$	−11.4098	−0.753982	−0.376991	−	0.926217i	$-0.623041\pi$
$229$	−11.4098	−0.753982	−0.376991	+	0.926217i	$0.623041\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.08483	−0.136581	−0.0682907	−	0.997665i	$-0.521755\pi$
$233$	−2.08483	−0.136581	−0.0682907	+	0.997665i	$0.521755\pi$
$234$	0	0
$235$	5.49464	0.358431
$236$	0	0
$237$	0	0
$238$	0	0
$239$	18.5353	1.19895	0.599473	−	0.800395i	$-0.295377\pi$
$239$	18.5353	1.19895	0.599473	+	0.800395i	$0.295377\pi$
$240$	0	0
$241$	10.2509	0.660317	0.330159	−	0.943925i	$-0.392898\pi$
$241$	10.2509	0.660317	0.330159	+	0.943925i	$0.392898\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−8.66249	−0.553426
$246$	0	0
$247$	18.5371	1.17948
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.2527	1.08898	0.544490	−	0.838768i	$-0.316723\pi$
$251$	17.2527	1.08898	0.544490	+	0.838768i	$0.316723\pi$
$252$	0	0
$253$	−0.957587	−0.0602030
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.9045	−1.80301	−0.901505	−	0.432768i	$-0.857537\pi$
$257$	−28.9045	−1.80301	−0.901505	+	0.432768i	$0.857537\pi$
$258$	0	0
$259$	36.1147	2.24406
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.24196	−0.323233	−0.161617	−	0.986854i	$-0.551671\pi$
$263$	−5.24196	−0.323233	−0.161617	+	0.986854i	$0.551671\pi$
$264$	0	0
$265$	−7.12544	−0.437712
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.37992	0.328019	0.164010	−	0.986459i	$-0.447557\pi$
$269$	5.37992	0.328019	0.164010	+	0.986459i	$0.447557\pi$
$270$	0	0
$271$	−11.5371	−0.700826	−0.350413	−	0.936595i	$-0.613959\pi$
$271$	−11.5371	−0.700826	−0.350413	+	0.936595i	$0.613959\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.957587	0.0577447
$276$	0	0
$277$	5.83035	0.350312	0.175156	−	0.984541i	$-0.443957\pi$
$277$	5.83035	0.350312	0.175156	+	0.984541i	$0.443957\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.0741	−0.660626	−0.330313	−	0.943871i	$-0.607154\pi$
$281$	−11.0741	−0.660626	−0.330313	+	0.943871i	$0.607154\pi$
$282$	0	0
$283$	−3.86384	−0.229682	−0.114841	−	0.993384i	$-0.536636\pi$
$283$	−3.86384	−0.229682	−0.114841	+	0.993384i	$0.536636\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	16.0317	0.943041
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.54597	0.0903167	0.0451583	−	0.998980i	$-0.485621\pi$
$293$	1.54597	0.0903167	0.0451583	+	0.998980i	$0.485621\pi$
$294$	0	0
$295$	−4.78973	−0.278869
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.74732	0.158882
$300$	0	0
$301$	31.6607	1.82489
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.4522	−0.713013
$306$	0	0
$307$	−6.45223	−0.368248	−0.184124	−	0.982903i	$-0.558945\pi$
$307$	−6.45223	−0.368248	−0.184124	+	0.982903i	$0.558945\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.8196	−1.29398	−0.646991	−	0.762497i	$-0.723973\pi$
$311$	−22.8196	−1.29398	−0.646991	+	0.762497i	$0.723973\pi$
$312$	0	0
$313$	−16.8620	−0.953099	−0.476550	−	0.879148i	$-0.658113\pi$
$313$	−16.8620	−0.953099	−0.476550	+	0.879148i	$0.658113\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.33751	0.0751218	0.0375609	−	0.999294i	$-0.488041\pi$
$317$	1.33751	0.0751218	0.0375609	+	0.999294i	$0.488041\pi$
$318$	0	0
$319$	−4.98928	−0.279346
$320$	0	0
$321$	0	0
$322$	0	0
$323$	38.7790	2.15772
$324$	0	0
$325$	−2.74732	−0.152394
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−21.7455	−1.19887
$330$	0	0
$331$	−32.2808	−1.77431	−0.887156	−	0.461470i	$-0.847322\pi$
$331$	−32.2808	−1.77431	−0.887156	+	0.461470i	$0.847322\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.12544	−0.498576
$336$	0	0
$337$	−4.83215	−0.263224	−0.131612	−	0.991301i	$-0.542015\pi$
$337$	−4.83215	−0.263224	−0.131612	+	0.991301i	$0.542015\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.70491	−0.308938
$342$	0	0
$343$	6.57947	0.355258
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.6183	1.85841	0.929203	−	0.369569i	$-0.120495\pi$
$347$	34.6183	1.85841	0.929203	+	0.369569i	$0.120495\pi$
$348$	0	0
$349$	26.9558	1.44291	0.721455	−	0.692461i	$-0.243474\pi$
$349$	26.9558	1.44291	0.721455	+	0.692461i	$0.243474\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	1.66249	0.0882361
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.420532	−0.0221949	−0.0110974	−	0.999938i	$-0.503532\pi$
$359$	−0.420532	−0.0221949	−0.0110974	+	0.999938i	$0.503532\pi$
$360$	0	0
$361$	26.5263	1.39612
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.3357	−0.645680
$366$	0	0
$367$	11.7156	0.611551	0.305775	−	0.952104i	$-0.401084\pi$
$367$	11.7156	0.611551	0.305775	+	0.952104i	$0.401084\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.1995	1.46405
$372$	0	0
$373$	−28.1661	−1.45838	−0.729192	−	0.684310i	$-0.760104\pi$
$373$	−28.1661	−1.45838	−0.729192	+	0.684310i	$0.760104\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.3143	0.737223
$378$	0	0
$379$	27.3781	1.40632	0.703160	−	0.711032i	$-0.251772\pi$
$379$	27.3781	1.40632	0.703160	+	0.711032i	$0.251772\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.7754	1.62365	0.811824	−	0.583902i	$-0.198475\pi$
$383$	31.7754	1.62365	0.811824	+	0.583902i	$0.198475\pi$
$384$	0	0
$385$	−3.78973	−0.193143
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.7031	0.948285	0.474143	−	0.880448i	$-0.342758\pi$
$389$	18.7031	0.948285	0.474143	+	0.880448i	$0.342758\pi$
$390$	0	0
$391$	5.74732	0.290655
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.8303	−0.595249
$396$	0	0
$397$	−39.6924	−1.99210	−0.996052	−	0.0887714i	$-0.971706\pi$
$397$	−39.6924	−1.99210	−0.996052	+	0.0887714i	$0.971706\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.3250	0.565543	0.282771	−	0.959187i	$-0.408746\pi$
$401$	11.3250	0.565543	0.282771	+	0.959187i	$0.408746\pi$
$402$	0	0
$403$	16.3674	0.815318
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.73840	0.433147
$408$	0	0
$409$	9.45223	0.467383	0.233691	−	0.972311i	$-0.424919\pi$
$409$	9.45223	0.467383	0.233691	+	0.972311i	$0.424919\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	18.9558	0.932753
$414$	0	0
$415$	0.704908	0.0346026
$416$	0	0
$417$	0	0
$418$	0	0
$419$	33.2402	1.62389	0.811944	−	0.583735i	$-0.198409\pi$
$419$	33.2402	1.62389	0.811944	+	0.583735i	$0.198409\pi$
$420$	0	0
$421$	−38.8285	−1.89239	−0.946194	−	0.323600i	$-0.895107\pi$
$421$	−38.8285	−1.89239	−0.946194	+	0.323600i	$0.895107\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.74732	−0.278786
$426$	0	0
$427$	49.2808	2.38486
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.4839	0.601329	0.300665	−	0.953730i	$-0.402791\pi$
$431$	12.4839	0.601329	0.300665	+	0.953730i	$0.402791\pi$
$432$	0	0
$433$	−5.78793	−0.278150	−0.139075	−	0.990282i	$-0.544413\pi$
$433$	−5.78793	−0.278150	−0.139075	+	0.990282i	$0.544413\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.74732	0.322768
$438$	0	0
$439$	−11.9241	−0.569106	−0.284553	−	0.958660i	$-0.591845\pi$
$439$	−11.9241	−0.569106	−0.284553	+	0.958660i	$0.591845\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−39.3973	−1.87182	−0.935911	−	0.352236i	$-0.885421\pi$
$443$	−39.3973	−1.87182	−0.935911	+	0.352236i	$0.885421\pi$
$444$	0	0
$445$	−15.8303	−0.750430
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.3674	−1.00839	−0.504195	−	0.863590i	$-0.668211\pi$
$449$	−21.3674	−1.00839	−0.504195	+	0.863590i	$0.668211\pi$
$450$	0	0
$451$	−0.241962	−0.0113936
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.8728	0.509723
$456$	0	0
$457$	−20.3656	−0.952663	−0.476331	−	0.879266i	$-0.658034\pi$
$457$	−20.3656	−0.952663	−0.476331	+	0.879266i	$0.658034\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.4946	1.28055	0.640277	−	0.768144i	$-0.278820\pi$
$461$	27.4946	1.28055	0.640277	+	0.768144i	$0.278820\pi$
$462$	0	0
$463$	−16.8196	−0.781675	−0.390837	−	0.920460i	$-0.627814\pi$
$463$	−16.8196	−0.781675	−0.390837	+	0.920460i	$0.627814\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.4504	1.13143	0.565715	−	0.824601i	$-0.308600\pi$
$467$	24.4504	1.13143	0.565715	+	0.824601i	$0.308600\pi$
$468$	0	0
$469$	36.1147	1.66762
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.66070	0.352239
$474$	0	0
$475$	−6.74732	−0.309588
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.5688	−0.482899	−0.241449	−	0.970413i	$-0.577623\pi$
$479$	−10.5688	−0.482899	−0.241449	+	0.970413i	$0.577623\pi$
$480$	0	0
$481$	−25.0705	−1.14312
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.8728	0.493707
$486$	0	0
$487$	−13.0741	−0.592444	−0.296222	−	0.955119i	$-0.595727\pi$
$487$	−13.0741	−0.592444	−0.296222	+	0.955119i	$0.595727\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.22098	−0.100232	−0.0501158	−	0.998743i	$-0.515959\pi$
$491$	−2.22098	−0.100232	−0.0501158	+	0.998743i	$0.515959\pi$
$492$	0	0
$493$	29.9451	1.34866
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.57947	−0.295129
$498$	0	0
$499$	2.87456	0.128683	0.0643415	−	0.997928i	$-0.479505\pi$
$499$	2.87456	0.128683	0.0643415	+	0.997928i	$0.479505\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.36740	−0.417672	−0.208836	−	0.977951i	$-0.566967\pi$
$503$	−9.36740	−0.417672	−0.208836	+	0.977951i	$0.566967\pi$
$504$	0	0
$505$	9.21027	0.409851
$506$	0	0
$507$	0	0
$508$	0	0
$509$	40.5866	1.79897	0.899484	−	0.436953i	$-0.143942\pi$
$509$	40.5866	1.79897	0.899484	+	0.436953i	$0.143942\pi$
$510$	0	0
$511$	48.8196	2.15965
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.4522	−0.548711
$516$	0	0
$517$	−5.26160	−0.231405
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−34.6714	−1.51898	−0.759491	−	0.650518i	$-0.774552\pi$
$521$	−34.6714	−1.51898	−0.759491	+	0.650518i	$0.774552\pi$
$522$	0	0
$523$	35.4098	1.54836	0.774182	−	0.632964i	$-0.218162\pi$
$523$	35.4098	1.54836	0.774182	+	0.632964i	$0.218162\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	34.2402	1.49152
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.694191	0.0300687
$534$	0	0
$535$	3.63080	0.156973
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8.29509	0.357295
$540$	0	0
$541$	37.0705	1.59379	0.796893	−	0.604121i	$-0.206475\pi$
$541$	37.0705	1.59379	0.796893	+	0.604121i	$0.206475\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.6625	−0.713743
$546$	0	0
$547$	21.4187	0.915799	0.457899	−	0.889004i	$-0.348602\pi$
$547$	21.4187	0.915799	0.457899	+	0.889004i	$0.348602\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	35.1553	1.49767
$552$	0	0
$553$	46.8196	1.99097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−20.9558	−0.887925	−0.443963	−	0.896045i	$-0.646428\pi$
$557$	−20.9558	−0.887925	−0.443963	+	0.896045i	$0.646428\pi$
$558$	0	0
$559$	−21.9786	−0.929594
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.70491	−0.282578	−0.141289	−	0.989968i	$-0.545125\pi$
$563$	−6.70491	−0.282578	−0.141289	+	0.989968i	$0.545125\pi$
$564$	0	0
$565$	−18.1147	−0.762092
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	13.2933	0.556307	0.278154	−	0.960537i	$-0.410278\pi$
$571$	13.2933	0.556307	0.278154	+	0.960537i	$0.410278\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	23.2402	0.967501	0.483750	−	0.875206i	$-0.339274\pi$
$577$	23.2402	0.967501	0.483750	+	0.875206i	$0.339274\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.78973	−0.115738
$582$	0	0
$583$	6.82323	0.282589
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.6112	−0.726891	−0.363445	−	0.931616i	$-0.618400\pi$
$587$	−17.6112	−0.726891	−0.363445	+	0.931616i	$0.618400\pi$
$588$	0	0
$589$	40.1978	1.65632
$590$	0	0
$591$	0	0
$592$	0	0
$593$	41.5759	1.70732	0.853658	−	0.520834i	$-0.174379\pi$
$593$	41.5759	1.70732	0.853658	+	0.520834i	$0.174379\pi$
$594$	0	0
$595$	22.7455	0.932475
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17.7138	−0.723767	−0.361884	−	0.932223i	$-0.617866\pi$
$599$	−17.7138	−0.723767	−0.361884	+	0.932223i	$0.617866\pi$
$600$	0	0
$601$	−46.7772	−1.90808	−0.954041	−	0.299675i	$-0.903122\pi$
$601$	−46.7772	−1.90808	−0.954041	+	0.299675i	$0.903122\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.0830	0.409933
$606$	0	0
$607$	42.9893	1.74488	0.872441	−	0.488720i	$-0.162536\pi$
$607$	42.9893	1.74488	0.872441	+	0.488720i	$0.162536\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15.0955	0.610700
$612$	0	0
$613$	−32.6500	−1.31872	−0.659360	−	0.751827i	$-0.729173\pi$
$613$	−32.6500	−1.31872	−0.659360	+	0.751827i	$0.729173\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.3232	0.455854	0.227927	−	0.973678i	$-0.426805\pi$
$617$	11.3232	0.455854	0.227927	+	0.973678i	$0.426805\pi$
$618$	0	0
$619$	−10.6625	−0.428562	−0.214281	−	0.976772i	$-0.568741\pi$
$619$	−10.6625	−0.428562	−0.214281	+	0.976772i	$0.568741\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	62.6500	2.51002
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−52.4468	−2.09119
$630$	0	0
$631$	5.24376	0.208751	0.104375	−	0.994538i	$-0.466716\pi$
$631$	5.24376	0.208751	0.104375	+	0.994538i	$0.466716\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.91517	0.234737
$636$	0	0
$637$	−23.7987	−0.942937
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.4205	−1.20154	−0.600769	−	0.799422i	$-0.705139\pi$
$641$	−30.4205	−1.20154	−0.600769	+	0.799422i	$0.705139\pi$
$642$	0	0
$643$	−9.37992	−0.369908	−0.184954	−	0.982747i	$-0.559214\pi$
$643$	−9.37992	−0.369908	−0.184954	+	0.982747i	$0.559214\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.6714	0.734049	0.367024	−	0.930211i	$-0.380377\pi$
$647$	18.6714	0.734049	0.367024	+	0.930211i	$0.380377\pi$
$648$	0	0
$649$	4.58659	0.180039
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.0723	−0.472426	−0.236213	−	0.971701i	$-0.575906\pi$
$653$	−12.0723	−0.472426	−0.236213	+	0.971701i	$0.575906\pi$
$654$	0	0
$655$	3.40982	0.133233
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21.2402	−0.827399	−0.413700	−	0.910413i	$-0.635764\pi$
$659$	−21.2402	−0.827399	−0.413700	+	0.910413i	$0.635764\pi$
$660$	0	0
$661$	−26.9362	−1.04769	−0.523847	−	0.851812i	$-0.675504\pi$
$661$	−26.9362	−1.04769	−0.523847	+	0.851812i	$0.675504\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	26.7031	1.03550
$666$	0	0
$667$	5.21027	0.201742
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.9241	0.460324
$672$	0	0
$673$	−11.4098	−0.439816	−0.219908	−	0.975521i	$-0.570576\pi$
$673$	−11.4098	−0.439816	−0.219908	+	0.975521i	$0.570576\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	11.8638	0.455965	0.227982	−	0.973665i	$-0.426787\pi$
$677$	11.8638	0.455965	0.227982	+	0.973665i	$0.426787\pi$
$678$	0	0
$679$	−43.0299	−1.65134
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.0723	−0.385406	−0.192703	−	0.981257i	$-0.561725\pi$
$683$	−10.0723	−0.385406	−0.192703	+	0.981257i	$0.561725\pi$
$684$	0	0
$685$	−1.67321	−0.0639301
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−19.5759	−0.745781
$690$	0	0
$691$	−45.9964	−1.74979	−0.874893	−	0.484317i	$-0.839068\pi$
$691$	−45.9964	−1.74979	−0.874893	+	0.484317i	$0.839068\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.62008	−0.326978
$696$	0	0
$697$	1.45223	0.0550071
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.2312	0.801893	0.400947	−	0.916101i	$-0.368681\pi$
$701$	21.2312	0.801893	0.400947	+	0.916101i	$0.368681\pi$
$702$	0	0
$703$	−61.5723	−2.32224
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−36.4504	−1.37086
$708$	0	0
$709$	31.3781	1.17843	0.589215	−	0.807976i	$-0.299437\pi$
$709$	31.3781	1.17843	0.589215	+	0.807976i	$0.299437\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.95759	0.223113
$714$	0	0
$715$	2.63080	0.0983863
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.4629	0.763139	0.381570	−	0.924340i	$-0.375384\pi$
$719$	20.4629	0.763139	0.381570	+	0.924340i	$0.375384\pi$
$720$	0	0
$721$	49.2808	1.83531
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.21027	−0.193504
$726$	0	0
$727$	14.9875	0.555855	0.277928	−	0.960602i	$-0.410352\pi$
$727$	14.9875	0.555855	0.277928	+	0.960602i	$0.410352\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−45.9786	−1.70058
$732$	0	0
$733$	−22.2844	−0.823092	−0.411546	−	0.911389i	$-0.635011\pi$
$733$	−22.2844	−0.823092	−0.411546	+	0.911389i	$0.635011\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.73840	0.321883
$738$	0	0
$739$	30.9344	1.13794	0.568969	−	0.822359i	$-0.307342\pi$
$739$	30.9344	1.13794	0.568969	+	0.822359i	$0.307342\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.8285	1.64460	0.822300	−	0.569054i	$-0.192691\pi$
$743$	44.8285	1.64460	0.822300	+	0.569054i	$0.192691\pi$
$744$	0	0
$745$	−23.0830	−0.845697
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14.3692	−0.525039
$750$	0	0
$751$	32.1661	1.17376	0.586878	−	0.809675i	$-0.300357\pi$
$751$	32.1661	1.17376	0.586878	+	0.809675i	$0.300357\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.7879	−0.392613
$756$	0	0
$757$	40.6835	1.47867	0.739333	−	0.673340i	$-0.235141\pi$
$757$	40.6835	1.47867	0.739333	+	0.673340i	$0.235141\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.4415	−1.03100	−0.515502	−	0.856888i	$-0.672395\pi$
$761$	−28.4415	−1.03100	−0.515502	+	0.856888i	$0.672395\pi$
$762$	0	0
$763$	65.9433	2.38731
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.1589	−0.475142
$768$	0	0
$769$	−40.4205	−1.45760	−0.728801	−	0.684726i	$-0.759922\pi$
$769$	−40.4205	−1.45760	−0.728801	+	0.684726i	$0.759922\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.5688	−0.883677	−0.441838	−	0.897095i	$-0.645673\pi$
$773$	−24.5688	−0.883677	−0.441838	+	0.897095i	$0.645673\pi$
$774$	0	0
$775$	−5.95759	−0.214003
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.70491	0.0610847
$780$	0	0
$781$	−1.59198	−0.0569656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.19955	−0.292654
$786$	0	0
$787$	−11.7790	−0.419877	−0.209938	−	0.977715i	$-0.567326\pi$
$787$	−11.7790	−0.419877	−0.209938	+	0.977715i	$0.567326\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	71.6906	2.54902
$792$	0	0
$793$	−34.2103	−1.21484
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.8602	0.490955	0.245478	−	0.969402i	$-0.421055\pi$
$797$	13.8602	0.490955	0.245478	+	0.969402i	$0.421055\pi$
$798$	0	0
$799$	31.5795	1.11720
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.8125	0.416854
$804$	0	0
$805$	3.95759	0.139487
$806$	0	0
$807$	0	0
$808$	0	0
$809$	52.0830	1.83114	0.915571	−	0.402157i	$-0.131739\pi$
$809$	52.0830	1.83114	0.915571	+	0.402157i	$0.131739\pi$
$810$	0	0
$811$	−40.7013	−1.42922	−0.714608	−	0.699525i	$-0.753395\pi$
$811$	−40.7013	−1.42922	−0.714608	+	0.699525i	$0.753395\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.6625	−0.583662
$816$	0	0
$817$	−53.9786	−1.88847
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	−42.9009	−1.49543	−0.747715	−	0.664020i	$-0.768849\pi$
$823$	−42.9009	−1.49543	−0.747715	+	0.664020i	$0.768849\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39.4397	−1.37145	−0.685727	−	0.727859i	$-0.740515\pi$
$827$	−39.4397	−1.37145	−0.685727	+	0.727859i	$0.740515\pi$
$828$	0	0
$829$	29.5460	1.02617	0.513087	−	0.858337i	$-0.328502\pi$
$829$	29.5460	1.02617	0.513087	+	0.858337i	$0.328502\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−49.7861	−1.72499
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.09554	0.0378223	0.0189112	−	0.999821i	$-0.493980\pi$
$839$	1.09554	0.0378223	0.0189112	+	0.999821i	$0.493980\pi$
$840$	0	0
$841$	−1.85313	−0.0639009
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.45223	0.187562
$846$	0	0
$847$	−39.9045	−1.37113
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.12544	−0.312816
$852$	0	0
$853$	−10.0937	−0.345603	−0.172802	−	0.984957i	$-0.555282\pi$
$853$	−10.0937	−0.345603	−0.172802	+	0.984957i	$0.555282\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.8303	−1.42890	−0.714449	−	0.699688i	$-0.753322\pi$
$857$	−41.8303	−1.42890	−0.714449	+	0.699688i	$0.753322\pi$
$858$	0	0
$859$	−3.35848	−0.114590	−0.0572950	−	0.998357i	$-0.518248\pi$
$859$	−3.35848	−0.114590	−0.0572950	+	0.998357i	$0.518248\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.83395	0.130509	0.0652545	−	0.997869i	$-0.479214\pi$
$863$	3.83395	0.130509	0.0652545	+	0.997869i	$0.479214\pi$
$864$	0	0
$865$	8.87276	0.301683
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.3286	0.384296
$870$	0	0
$871$	−25.0705	−0.849482
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.95759	−0.133791
$876$	0	0
$877$	15.5263	0.524287	0.262144	−	0.965029i	$-0.415571\pi$
$877$	15.5263	0.524287	0.262144	+	0.965029i	$0.415571\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.9893	−0.909292	−0.454646	−	0.890672i	$-0.650234\pi$
$881$	−26.9893	−0.909292	−0.454646	+	0.890672i	$0.650234\pi$
$882$	0	0
$883$	21.5884	0.726507	0.363254	−	0.931690i	$-0.381666\pi$
$883$	21.5884	0.726507	0.363254	+	0.931690i	$0.381666\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−49.4098	−1.65902	−0.829510	−	0.558492i	$-0.811380\pi$
$887$	−49.4098	−1.65902	−0.829510	+	0.558492i	$0.811380\pi$
$888$	0	0
$889$	−23.4098	−0.785140
$890$	0	0
$891$	0	0
$892$	0	0
$893$	37.0741	1.24064
$894$	0	0
$895$	1.49464	0.0499604
$896$	0	0
$897$	0	0
$898$	0	0
$899$	31.0406	1.03526
$900$	0	0
$901$	−40.9522	−1.36432
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.5777	0.418096
$906$	0	0
$907$	−9.54957	−0.317088	−0.158544	−	0.987352i	$-0.550680\pi$
$907$	−9.54957	−0.317088	−0.158544	+	0.987352i	$0.550680\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47.3036	1.56724	0.783618	−	0.621243i	$-0.213372\pi$
$911$	47.3036	1.56724	0.783618	+	0.621243i	$0.213372\pi$
$912$	0	0
$913$	−0.675011	−0.0223396
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13.4946	−0.445632
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.56741	0.150338
$924$	0	0
$925$	9.12544	0.300043
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.6942	0.646145	0.323073	−	0.946374i	$-0.395284\pi$
$929$	19.6942	0.646145	0.323073	+	0.946374i	$0.395284\pi$
$930$	0	0
$931$	−58.4486	−1.91558
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.50356	0.179986
$936$	0	0
$937$	−31.8214	−1.03956	−0.519780	−	0.854300i	$-0.673986\pi$
$937$	−31.8214	−1.03956	−0.519780	+	0.854300i	$0.673986\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.36740	0.0771751	0.0385876	−	0.999255i	$-0.487714\pi$
$941$	2.36740	0.0771751	0.0385876	+	0.999255i	$0.487714\pi$
$942$	0	0
$943$	0.252679	0.00822837
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.90266	0.321793	0.160897	−	0.986971i	$-0.448561\pi$
$947$	9.90266	0.321793	0.160897	+	0.986971i	$0.448561\pi$
$948$	0	0
$949$	−33.8901	−1.10012
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31.5442	1.02182	0.510908	−	0.859635i	$-0.329309\pi$
$953$	31.5442	1.02182	0.510908	+	0.859635i	$0.329309\pi$
$954$	0	0
$955$	16.9045	0.547015
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.62188	0.213832
$960$	0	0
$961$	4.49284	0.144930
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.91517	−0.319181
$966$	0	0
$967$	−15.5973	−0.501575	−0.250788	−	0.968042i	$-0.580690\pi$
$967$	−15.5973	−0.501575	−0.250788	+	0.968042i	$0.580690\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	47.1022	1.51158	0.755791	−	0.654813i	$-0.227253\pi$
$971$	47.1022	1.51158	0.755791	+	0.654813i	$0.227253\pi$
$972$	0	0
$973$	34.1147	1.09367
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.9469	−0.926092	−0.463046	−	0.886334i	$-0.653244\pi$
$977$	−28.9469	−0.926092	−0.463046	+	0.886334i	$0.653244\pi$
$978$	0	0
$979$	15.1589	0.484482
$980$	0	0
$981$	0	0
$982$	0	0
$983$	36.2312	1.15560	0.577799	−	0.816179i	$-0.303912\pi$
$983$	36.2312	1.15560	0.577799	+	0.816179i	$0.303912\pi$
$984$	0	0
$985$	4.53705	0.144563
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−35.1750	−1.11737	−0.558685	−	0.829380i	$-0.688694\pi$
$991$	−35.1750	−1.11737	−0.558685	+	0.829380i	$0.688694\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.0000	0.443830
$996$	0	0
$997$	19.2402	0.609342	0.304671	−	0.952458i	$-0.401453\pi$
$997$	19.2402	0.609342	0.304671	+	0.952458i	$0.401453\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bj.1.3		3
3.2	odd	2		920.2.a.h.1.1	✓	3
12.11	even	2		1840.2.a.s.1.3		3
15.2	even	4		4600.2.e.p.4049.5		6
15.8	even	4		4600.2.e.p.4049.2		6
15.14	odd	2		4600.2.a.x.1.3		3
24.5	odd	2		7360.2.a.by.1.3		3
24.11	even	2		7360.2.a.cc.1.1		3
60.59	even	2		9200.2.a.ce.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.h.1.1	✓	3	3.2	odd	2
1840.2.a.s.1.3		3	12.11	even	2
4600.2.a.x.1.3		3	15.14	odd	2
4600.2.e.p.4049.2		6	15.8	even	4
4600.2.e.p.4049.5		6	15.2	even	4
7360.2.a.by.1.3		3	24.5	odd	2
7360.2.a.cc.1.1		3	24.11	even	2
8280.2.a.bj.1.3		3	1.1	even	1	trivial
9200.2.a.ce.1.1		3	60.59	even	2

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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +3.95759 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +3.95759 q^{7} +0.957587 q^{11} -2.74732 q^{13} -5.74732 q^{17} -6.74732 q^{19} -1.00000 q^{23} +1.00000 q^{25} -5.21027 q^{29} -5.95759 q^{31} -3.95759 q^{35} +9.12544 q^{37} -0.252679 q^{41} +8.00000 q^{43} -5.49464 q^{47} +8.66249 q^{49} +7.12544 q^{53} -0.957587 q^{55} +4.78973 q^{59} +12.4522 q^{61} +2.74732 q^{65} +9.12544 q^{67} -1.66249 q^{71} +12.3357 q^{73} +3.78973 q^{77} +11.8303 q^{79} -0.704908 q^{83} +5.74732 q^{85} +15.8303 q^{89} -10.8728 q^{91} +6.74732 q^{95} -10.8728 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 2 * q^7	\( 3 q - 3 q^{5} + 2 q^{7} - 7 q^{11} - q^{13} - 10 q^{17} - 13 q^{19} - 3 q^{23} + 3 q^{25} - 13 q^{29} - 8 q^{31} - 2 q^{35} + 5 q^{37} - 8 q^{41} + 24 q^{43} - 2 q^{47} - q^{49} - q^{53} + 7 q^{55} + 17 q^{59} + 13 q^{61} + q^{65} + 5 q^{67} + 22 q^{71} + 12 q^{73} + 14 q^{77} - 4 q^{79} + 15 q^{83} + 10 q^{85} + 8 q^{89} - 3 q^{91} + 13 q^{95} - 3 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - 7 * q^11 - q^13 - 10 * q^17 - 13 * q^19 - 3 * q^23 + 3 * q^25 - 13 * q^29 - 8 * q^31 - 2 * q^35 + 5 * q^37 - 8 * q^41 + 24 * q^43 - 2 * q^47 - q^49 - q^53 + 7 * q^55 + 17 * q^59 + 13 * q^61 + q^65 + 5 * q^67 + 22 * q^71 + 12 * q^73 + 14 * q^77 - 4 * q^79 + 15 * q^83 + 10 * q^85 + 8 * q^89 - 3 * q^91 + 13 * q^95 - 3 * q^97

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