LMFDB - Embedded newform 8280.2.a.bv.1.4

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 18x^{5} + 46x^{4} + 60x^{3} - 76x^{2} - 51x - 7 $ x^7 - 3x^6 - 18x^5 + 46x^4 + 60x^3 - 76x^2 - 51x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.37429$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -0.958551 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -0.958551 q^{7} -4.98608 q^{11} -5.47824 q^{13} -0.297696 q^{17} +4.44417 q^{19} -1.00000 q^{23} +1.00000 q^{25} +6.77610 q^{29} +3.44272 q^{31} +0.958551 q^{35} -11.9853 q^{37} +7.74204 q^{41} -12.9381 q^{43} +7.47247 q^{47} -6.08118 q^{49} -11.5809 q^{53} +4.98608 q^{55} -0.978689 q^{59} -4.85161 q^{61} +5.47824 q^{65} -16.0244 q^{67} -5.95940 q^{71} +8.07972 q^{73} +4.77941 q^{77} +11.2958 q^{79} -8.26985 q^{83} +0.297696 q^{85} -10.1085 q^{89} +5.25117 q^{91} -4.44417 q^{95} +13.6164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) $ 7 * q - 7 * q^5 - 6 * q^7	$ 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * 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
$7$	−0.958551	−0.362298	−0.181149	−	0.983456i	$-0.557982\pi$
$7$	−0.958551	−0.362298	−0.181149	+	0.983456i	$0.557982\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.98608	−1.50336	−0.751679	−	0.659529i	$-0.770756\pi$
$11$	−4.98608	−1.50336	−0.751679	+	0.659529i	$0.770756\pi$
$12$	0	0
$13$	−5.47824	−1.51939	−0.759695	−	0.650280i	$-0.774652\pi$
$13$	−5.47824	−1.51939	−0.759695	+	0.650280i	$0.774652\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.297696	−0.0722018	−0.0361009	−	0.999348i	$-0.511494\pi$
$17$	−0.297696	−0.0722018	−0.0361009	+	0.999348i	$0.511494\pi$
$18$	0	0
$19$	4.44417	1.01956	0.509782	−	0.860304i	$-0.329726\pi$
$19$	4.44417	1.01956	0.509782	+	0.860304i	$0.329726\pi$
$20$	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
$29$	6.77610	1.25829	0.629145	−	0.777288i	$-0.283405\pi$
$29$	6.77610	1.25829	0.629145	+	0.777288i	$0.283405\pi$
$30$	0	0
$31$	3.44272	0.618330	0.309165	−	0.951008i	$-0.399950\pi$
$31$	3.44272	0.618330	0.309165	+	0.951008i	$0.399950\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.958551	0.162025
$36$	0	0
$37$	−11.9853	−1.97037	−0.985186	−	0.171489i	$-0.945142\pi$
$37$	−11.9853	−1.97037	−0.985186	+	0.171489i	$0.945142\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.74204	1.20910	0.604551	−	0.796566i	$-0.293352\pi$
$41$	7.74204	1.20910	0.604551	+	0.796566i	$0.293352\pi$
$42$	0	0
$43$	−12.9381	−1.97304	−0.986520	−	0.163640i	$-0.947677\pi$
$43$	−12.9381	−1.97304	−0.986520	+	0.163640i	$0.947677\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.47247	1.08997	0.544986	−	0.838445i	$-0.316535\pi$
$47$	7.47247	1.08997	0.544986	+	0.838445i	$0.316535\pi$
$48$	0	0
$49$	−6.08118	−0.868740
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.5809	−1.59075	−0.795377	−	0.606115i	$-0.792727\pi$
$53$	−11.5809	−1.59075	−0.795377	+	0.606115i	$0.792727\pi$
$54$	0	0
$55$	4.98608	0.672322
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.978689	−0.127414	−0.0637072	−	0.997969i	$-0.520292\pi$
$59$	−0.978689	−0.127414	−0.0637072	+	0.997969i	$0.520292\pi$
$60$	0	0
$61$	−4.85161	−0.621185	−0.310592	−	0.950543i	$-0.600527\pi$
$61$	−4.85161	−0.621185	−0.310592	+	0.950543i	$0.600527\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.47824	0.679492
$66$	0	0
$67$	−16.0244	−1.95769	−0.978843	−	0.204613i	$-0.934407\pi$
$67$	−16.0244	−1.95769	−0.978843	+	0.204613i	$0.934407\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.95940	−0.707251	−0.353625	−	0.935387i	$-0.615051\pi$
$71$	−5.95940	−0.707251	−0.353625	+	0.935387i	$0.615051\pi$
$72$	0	0
$73$	8.07972	0.945660	0.472830	−	0.881154i	$-0.343232\pi$
$73$	8.07972	0.945660	0.472830	+	0.881154i	$0.343232\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.77941	0.544664
$78$	0	0
$79$	11.2958	1.27087	0.635437	−	0.772152i	$-0.280820\pi$
$79$	11.2958	1.27087	0.635437	+	0.772152i	$0.280820\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.26985	−0.907734	−0.453867	−	0.891069i	$-0.649956\pi$
$83$	−8.26985	−0.907734	−0.453867	+	0.891069i	$0.649956\pi$
$84$	0	0
$85$	0.297696	0.0322896
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.1085	−1.07150	−0.535752	−	0.844376i	$-0.679972\pi$
$89$	−10.1085	−1.07150	−0.535752	+	0.844376i	$0.679972\pi$
$90$	0	0
$91$	5.25117	0.550472
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.44417	−0.455963
$96$	0	0
$97$	13.6164	1.38253	0.691267	−	0.722599i	$-0.257053\pi$
$97$	13.6164	1.38253	0.691267	+	0.722599i	$0.257053\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.62292	0.659005	0.329502	−	0.944155i	$-0.393119\pi$
$101$	6.62292	0.659005	0.329502	+	0.944155i	$0.393119\pi$
$102$	0	0
$103$	−12.1846	−1.20058	−0.600292	−	0.799781i	$-0.704949\pi$
$103$	−12.1846	−1.20058	−0.600292	+	0.799781i	$0.704949\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.23928	0.216479	0.108239	−	0.994125i	$-0.465479\pi$
$107$	2.23928	0.216479	0.108239	+	0.994125i	$0.465479\pi$
$108$	0	0
$109$	6.57188	0.629472	0.314736	−	0.949179i	$-0.398084\pi$
$109$	6.57188	0.629472	0.314736	+	0.949179i	$0.398084\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.54912	0.333873	0.166937	−	0.985968i	$-0.446612\pi$
$113$	3.54912	0.333873	0.166937	+	0.985968i	$0.446612\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.285356	0.0261586
$120$	0	0
$121$	13.8610	1.26009
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.70291	−0.151109	−0.0755544	−	0.997142i	$-0.524073\pi$
$127$	−1.70291	−0.151109	−0.0755544	+	0.997142i	$0.524073\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.6504	1.62950	0.814748	−	0.579815i	$-0.196875\pi$
$131$	18.6504	1.62950	0.814748	+	0.579815i	$0.196875\pi$
$132$	0	0
$133$	−4.25997	−0.369386
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.5472	1.07198	0.535990	−	0.844224i	$-0.319938\pi$
$137$	12.5472	1.07198	0.535990	+	0.844224i	$0.319938\pi$
$138$	0	0
$139$	15.1215	1.28259	0.641293	−	0.767296i	$-0.278398\pi$
$139$	15.1215	1.28259	0.641293	+	0.767296i	$0.278398\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	27.3149	2.28419
$144$	0	0
$145$	−6.77610	−0.562724
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.73958	−0.306358	−0.153179	−	0.988198i	$-0.548951\pi$
$149$	−3.73958	−0.306358	−0.153179	+	0.988198i	$0.548951\pi$
$150$	0	0
$151$	−14.9586	−1.21732	−0.608658	−	0.793433i	$-0.708292\pi$
$151$	−14.9586	−1.21732	−0.608658	+	0.793433i	$0.708292\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.44272	−0.276526
$156$	0	0
$157$	18.7208	1.49408	0.747041	−	0.664777i	$-0.231474\pi$
$157$	18.7208	1.49408	0.747041	+	0.664777i	$0.231474\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.958551	0.0755444
$162$	0	0
$163$	−1.19592	−0.0936718	−0.0468359	−	0.998903i	$-0.514914\pi$
$163$	−1.19592	−0.0936718	−0.0468359	+	0.998903i	$0.514914\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.5167	1.66501	0.832507	−	0.554014i	$-0.186905\pi$
$167$	21.5167	1.66501	0.832507	+	0.554014i	$0.186905\pi$
$168$	0	0
$169$	17.0111	1.30854
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.3707	1.24464	0.622321	−	0.782762i	$-0.286190\pi$
$173$	16.3707	1.24464	0.622321	+	0.782762i	$0.286190\pi$
$174$	0	0
$175$	−0.958551	−0.0724596
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.25143	0.392510	0.196255	−	0.980553i	$-0.437122\pi$
$179$	5.25143	0.392510	0.196255	+	0.980553i	$0.437122\pi$
$180$	0	0
$181$	−9.51892	−0.707535	−0.353768	−	0.935333i	$-0.615100\pi$
$181$	−9.51892	−0.707535	−0.353768	+	0.935333i	$0.615100\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	11.9853	0.881177
$186$	0	0
$187$	1.48433	0.108545
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.6093	1.56360	0.781798	−	0.623531i	$-0.214303\pi$
$191$	21.6093	1.56360	0.781798	+	0.623531i	$0.214303\pi$
$192$	0	0
$193$	15.6709	1.12802	0.564009	−	0.825769i	$-0.309258\pi$
$193$	15.6709	1.12802	0.564009	+	0.825769i	$0.309258\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.81885	0.343329	0.171665	−	0.985155i	$-0.445085\pi$
$197$	4.81885	0.343329	0.171665	+	0.985155i	$0.445085\pi$
$198$	0	0
$199$	−5.07527	−0.359776	−0.179888	−	0.983687i	$-0.557574\pi$
$199$	−5.07527	−0.359776	−0.179888	+	0.983687i	$0.557574\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.49523	−0.455876
$204$	0	0
$205$	−7.74204	−0.540727
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−22.1590	−1.53277
$210$	0	0
$211$	6.31999	0.435086	0.217543	−	0.976051i	$-0.430196\pi$
$211$	6.31999	0.435086	0.217543	+	0.976051i	$0.430196\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.9381	0.882370
$216$	0	0
$217$	−3.30002	−0.224020
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.63085	0.109703
$222$	0	0
$223$	−18.6284	−1.24745	−0.623724	−	0.781645i	$-0.714381\pi$
$223$	−18.6284	−1.24745	−0.623724	+	0.781645i	$0.714381\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.8258	0.851282	0.425641	−	0.904892i	$-0.360049\pi$
$227$	12.8258	0.851282	0.425641	+	0.904892i	$0.360049\pi$
$228$	0	0
$229$	7.53523	0.497942	0.248971	−	0.968511i	$-0.419908\pi$
$229$	7.53523	0.497942	0.248971	+	0.968511i	$0.419908\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.6931	0.831550	0.415775	−	0.909467i	$-0.363510\pi$
$233$	12.6931	0.831550	0.415775	+	0.909467i	$0.363510\pi$
$234$	0	0
$235$	−7.47247	−0.487450
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.4418	0.998845	0.499423	−	0.866359i	$-0.333546\pi$
$239$	15.4418	0.998845	0.499423	+	0.866359i	$0.333546\pi$
$240$	0	0
$241$	16.1529	1.04050	0.520250	−	0.854014i	$-0.325839\pi$
$241$	16.1529	1.04050	0.520250	+	0.854014i	$0.325839\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.08118	0.388512
$246$	0	0
$247$	−24.3462	−1.54911
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.62326	−0.291818	−0.145909	−	0.989298i	$-0.546611\pi$
$251$	−4.62326	−0.291818	−0.145909	+	0.989298i	$0.546611\pi$
$252$	0	0
$253$	4.98608	0.313472
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.0954	0.941625	0.470813	−	0.882233i	$-0.343961\pi$
$257$	15.0954	0.941625	0.470813	+	0.882233i	$0.343961\pi$
$258$	0	0
$259$	11.4885	0.713862
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.0502	−1.54466	−0.772329	−	0.635222i	$-0.780909\pi$
$263$	−25.0502	−1.54466	−0.772329	+	0.635222i	$0.780909\pi$
$264$	0	0
$265$	11.5809	0.711407
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.17557	0.376531	0.188266	−	0.982118i	$-0.439713\pi$
$269$	6.17557	0.376531	0.188266	+	0.982118i	$0.439713\pi$
$270$	0	0
$271$	−14.6086	−0.887411	−0.443706	−	0.896173i	$-0.646336\pi$
$271$	−14.6086	−0.887411	−0.443706	+	0.896173i	$0.646336\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.98608	−0.300672
$276$	0	0
$277$	−10.7020	−0.643022	−0.321511	−	0.946906i	$-0.604191\pi$
$277$	−10.7020	−0.643022	−0.321511	+	0.946906i	$0.604191\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.4538	−1.22017	−0.610087	−	0.792335i	$-0.708865\pi$
$281$	−20.4538	−1.22017	−0.610087	+	0.792335i	$0.708865\pi$
$282$	0	0
$283$	12.5175	0.744088	0.372044	−	0.928215i	$-0.378657\pi$
$283$	12.5175	0.744088	0.372044	+	0.928215i	$0.378657\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.42113	−0.438056
$288$	0	0
$289$	−16.9114	−0.994787
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.52365	0.439536	0.219768	−	0.975552i	$-0.429470\pi$
$293$	7.52365	0.439536	0.219768	+	0.975552i	$0.429470\pi$
$294$	0	0
$295$	0.978689	0.0569815
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.47824	0.316815
$300$	0	0
$301$	12.4018	0.714829
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.85161	0.277802
$306$	0	0
$307$	7.95901	0.454245	0.227122	−	0.973866i	$-0.427068\pi$
$307$	7.95901	0.454245	0.227122	+	0.973866i	$0.427068\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.72758	−0.494895	−0.247448	−	0.968901i	$-0.579592\pi$
$311$	−8.72758	−0.494895	−0.247448	+	0.968901i	$0.579592\pi$
$312$	0	0
$313$	0.374546	0.0211706	0.0105853	−	0.999944i	$-0.496631\pi$
$313$	0.374546	0.0211706	0.0105853	+	0.999944i	$0.496631\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.722813	−0.0405972	−0.0202986	−	0.999794i	$-0.506462\pi$
$317$	−0.722813	−0.0405972	−0.0202986	+	0.999794i	$0.506462\pi$
$318$	0	0
$319$	−33.7861	−1.89166
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.32301	−0.0736143
$324$	0	0
$325$	−5.47824	−0.303878
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.16274	−0.394895
$330$	0	0
$331$	2.06596	0.113555	0.0567776	−	0.998387i	$-0.481917\pi$
$331$	2.06596	0.113555	0.0567776	+	0.998387i	$0.481917\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	16.0244	0.875504
$336$	0	0
$337$	32.9428	1.79451	0.897255	−	0.441513i	$-0.145558\pi$
$337$	32.9428	1.79451	0.897255	+	0.441513i	$0.145558\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−17.1657	−0.929572
$342$	0	0
$343$	12.5390	0.677041
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.24754	0.442751	0.221376	−	0.975189i	$-0.428945\pi$
$347$	8.24754	0.442751	0.221376	+	0.975189i	$0.428945\pi$
$348$	0	0
$349$	17.5159	0.937605	0.468803	−	0.883303i	$-0.344686\pi$
$349$	17.5159	0.937605	0.468803	+	0.883303i	$0.344686\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.4549	1.88708	0.943538	−	0.331264i	$-0.107475\pi$
$353$	35.4549	1.88708	0.943538	+	0.331264i	$0.107475\pi$
$354$	0	0
$355$	5.95940	0.316292
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.9148	−1.47329	−0.736645	−	0.676280i	$-0.763591\pi$
$359$	−27.9148	−1.47329	−0.736645	+	0.676280i	$0.763591\pi$
$360$	0	0
$361$	0.750687	0.0395099
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.07972	−0.422912
$366$	0	0
$367$	−18.7065	−0.976471	−0.488235	−	0.872712i	$-0.662359\pi$
$367$	−18.7065	−0.976471	−0.488235	+	0.872712i	$0.662359\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.1008	0.576327
$372$	0	0
$373$	18.9310	0.980210	0.490105	−	0.871663i	$-0.336958\pi$
$373$	18.9310	0.980210	0.490105	+	0.871663i	$0.336958\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−37.1211	−1.91183
$378$	0	0
$379$	−4.30636	−0.221203	−0.110601	−	0.993865i	$-0.535278\pi$
$379$	−4.30636	−0.221203	−0.110601	+	0.993865i	$0.535278\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−19.1934	−0.980735	−0.490368	−	0.871516i	$-0.663138\pi$
$383$	−19.1934	−0.980735	−0.490368	+	0.871516i	$0.663138\pi$
$384$	0	0
$385$	−4.77941	−0.243581
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.6953	0.795785	0.397893	−	0.917432i	$-0.369742\pi$
$389$	15.6953	0.795785	0.397893	+	0.917432i	$0.369742\pi$
$390$	0	0
$391$	0.297696	0.0150551
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.2958	−0.568353
$396$	0	0
$397$	−38.3922	−1.92685	−0.963424	−	0.267983i	$-0.913643\pi$
$397$	−38.3922	−1.92685	−0.963424	+	0.267983i	$0.913643\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.2085	−0.559725	−0.279863	−	0.960040i	$-0.590289\pi$
$401$	−11.2085	−0.559725	−0.279863	+	0.960040i	$0.590289\pi$
$402$	0	0
$403$	−18.8600	−0.939485
$404$	0	0
$405$	0	0
$406$	0	0
$407$	59.7596	2.96218
$408$	0	0
$409$	−11.0399	−0.545889	−0.272945	−	0.962030i	$-0.587998\pi$
$409$	−11.0399	−0.545889	−0.272945	+	0.962030i	$0.587998\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.938123	0.0461620
$414$	0	0
$415$	8.26985	0.405951
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.9715	1.80617	0.903087	−	0.429458i	$-0.141295\pi$
$419$	36.9715	1.80617	0.903087	+	0.429458i	$0.141295\pi$
$420$	0	0
$421$	−25.6177	−1.24853	−0.624265	−	0.781213i	$-0.714601\pi$
$421$	−25.6177	−1.24853	−0.624265	+	0.781213i	$0.714601\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.297696	−0.0144404
$426$	0	0
$427$	4.65051	0.225054
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.3458	0.883685	0.441843	−	0.897093i	$-0.354325\pi$
$431$	18.3458	0.883685	0.441843	+	0.897093i	$0.354325\pi$
$432$	0	0
$433$	22.6925	1.09053	0.545267	−	0.838262i	$-0.316428\pi$
$433$	22.6925	1.09053	0.545267	+	0.838262i	$0.316428\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.44417	−0.212594
$438$	0	0
$439$	8.71754	0.416066	0.208033	−	0.978122i	$-0.433294\pi$
$439$	8.71754	0.416066	0.208033	+	0.978122i	$0.433294\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−19.9261	−0.946717	−0.473359	−	0.880870i	$-0.656958\pi$
$443$	−19.9261	−0.946717	−0.473359	+	0.880870i	$0.656958\pi$
$444$	0	0
$445$	10.1085	0.479191
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.4647	−0.682629	−0.341315	−	0.939949i	$-0.610872\pi$
$449$	−14.4647	−0.682629	−0.341315	+	0.939949i	$0.610872\pi$
$450$	0	0
$451$	−38.6024	−1.81772
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.25117	−0.246178
$456$	0	0
$457$	−20.5286	−0.960286	−0.480143	−	0.877190i	$-0.659415\pi$
$457$	−20.5286	−0.960286	−0.480143	+	0.877190i	$0.659415\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.0734	1.63353	0.816766	−	0.576969i	$-0.195765\pi$
$461$	35.0734	1.63353	0.816766	+	0.576969i	$0.195765\pi$
$462$	0	0
$463$	4.22790	0.196487	0.0982435	−	0.995162i	$-0.468678\pi$
$463$	4.22790	0.196487	0.0982435	+	0.995162i	$0.468678\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.51556	−0.301504	−0.150752	−	0.988572i	$-0.548170\pi$
$467$	−6.51556	−0.301504	−0.150752	+	0.988572i	$0.548170\pi$
$468$	0	0
$469$	15.3602	0.709266
$470$	0	0
$471$	0	0
$472$	0	0
$473$	64.5103	2.96619
$474$	0	0
$475$	4.44417	0.203913
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.8414	1.27210	0.636052	−	0.771646i	$-0.280566\pi$
$479$	27.8414	1.27210	0.636052	+	0.771646i	$0.280566\pi$
$480$	0	0
$481$	65.6583	2.99376
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.6164	−0.618288
$486$	0	0
$487$	−25.1562	−1.13994	−0.569969	−	0.821666i	$-0.693045\pi$
$487$	−25.1562	−1.13994	−0.569969	+	0.821666i	$0.693045\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−9.44599	−0.426292	−0.213146	−	0.977020i	$-0.568371\pi$
$491$	−9.44599	−0.426292	−0.213146	+	0.977020i	$0.568371\pi$
$492$	0	0
$493$	−2.01721	−0.0908508
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.71239	0.256236
$498$	0	0
$499$	−10.1761	−0.455543	−0.227771	−	0.973715i	$-0.573144\pi$
$499$	−10.1761	−0.455543	−0.227771	+	0.973715i	$0.573144\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	27.6355	1.23220	0.616102	−	0.787666i	$-0.288711\pi$
$503$	27.6355	1.23220	0.616102	+	0.787666i	$0.288711\pi$
$504$	0	0
$505$	−6.62292	−0.294716
$506$	0	0
$507$	0	0
$508$	0	0
$509$	32.9102	1.45872	0.729360	−	0.684130i	$-0.239818\pi$
$509$	32.9102	1.45872	0.729360	+	0.684130i	$0.239818\pi$
$510$	0	0
$511$	−7.74482	−0.342611
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.1846	0.536917
$516$	0	0
$517$	−37.2583	−1.63862
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.59651	−0.332809	−0.166405	−	0.986058i	$-0.553216\pi$
$521$	−7.59651	−0.332809	−0.166405	+	0.986058i	$0.553216\pi$
$522$	0	0
$523$	−9.23305	−0.403733	−0.201866	−	0.979413i	$-0.564701\pi$
$523$	−9.23305	−0.403733	−0.201866	+	0.979413i	$0.564701\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.02488	−0.0446446
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−42.4127	−1.83710
$534$	0	0
$535$	−2.23928	−0.0968123
$536$	0	0
$537$	0	0
$538$	0	0
$539$	30.3212	1.30603
$540$	0	0
$541$	23.8783	1.02661	0.513305	−	0.858206i	$-0.328421\pi$
$541$	23.8783	1.02661	0.513305	+	0.858206i	$0.328421\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.57188	−0.281509
$546$	0	0
$547$	−1.82276	−0.0779354	−0.0389677	−	0.999240i	$-0.512407\pi$
$547$	−1.82276	−0.0779354	−0.0389677	+	0.999240i	$0.512407\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	30.1142	1.28291
$552$	0	0
$553$	−10.8276	−0.460436
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−47.0518	−1.99365	−0.996824	−	0.0796321i	$-0.974625\pi$
$557$	−47.0518	−1.99365	−0.996824	+	0.0796321i	$0.974625\pi$
$558$	0	0
$559$	70.8779	2.99782
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.80746	0.160465	0.0802326	−	0.996776i	$-0.474434\pi$
$563$	3.80746	0.160465	0.0802326	+	0.996776i	$0.474434\pi$
$564$	0	0
$565$	−3.54912	−0.149313
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.5314	1.44763	0.723816	−	0.689993i	$-0.242387\pi$
$569$	34.5314	1.44763	0.723816	+	0.689993i	$0.242387\pi$
$570$	0	0
$571$	−34.9028	−1.46064	−0.730319	−	0.683106i	$-0.760629\pi$
$571$	−34.9028	−1.46064	−0.730319	+	0.683106i	$0.760629\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−32.1089	−1.33671	−0.668357	−	0.743841i	$-0.733002\pi$
$577$	−32.1089	−1.33671	−0.668357	+	0.743841i	$0.733002\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.92707	0.328870
$582$	0	0
$583$	57.7431	2.39147
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.3358	−0.426606	−0.213303	−	0.976986i	$-0.568422\pi$
$587$	−10.3358	−0.426606	−0.213303	+	0.976986i	$0.568422\pi$
$588$	0	0
$589$	15.3000	0.630427
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−0.735145	−0.0301888	−0.0150944	−	0.999886i	$-0.504805\pi$
$593$	−0.735145	−0.0301888	−0.0150944	+	0.999886i	$0.504805\pi$
$594$	0	0
$595$	−0.285356	−0.0116985
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.5251	−1.86011	−0.930053	−	0.367425i	$-0.880240\pi$
$599$	−45.5251	−1.86011	−0.930053	+	0.367425i	$0.880240\pi$
$600$	0	0
$601$	−16.9508	−0.691436	−0.345718	−	0.938338i	$-0.612365\pi$
$601$	−16.9508	−0.691436	−0.345718	+	0.938338i	$0.612365\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13.8610	−0.563528
$606$	0	0
$607$	−20.1782	−0.819010	−0.409505	−	0.912308i	$-0.634299\pi$
$607$	−20.1782	−0.819010	−0.409505	+	0.912308i	$0.634299\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−40.9360	−1.65609
$612$	0	0
$613$	23.0592	0.931354	0.465677	−	0.884955i	$-0.345811\pi$
$613$	23.0592	0.931354	0.465677	+	0.884955i	$0.345811\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.55021	0.303960	0.151980	−	0.988384i	$-0.451435\pi$
$617$	7.55021	0.303960	0.151980	+	0.988384i	$0.451435\pi$
$618$	0	0
$619$	25.4433	1.02265	0.511327	−	0.859386i	$-0.329154\pi$
$619$	25.4433	1.02265	0.511327	+	0.859386i	$0.329154\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.68955	0.388204
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.56797	0.142264
$630$	0	0
$631$	33.1181	1.31841	0.659205	−	0.751963i	$-0.270893\pi$
$631$	33.1181	1.31841	0.659205	+	0.751963i	$0.270893\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.70291	0.0675779
$636$	0	0
$637$	33.3141	1.31995
$638$	0	0
$639$	0	0
$640$	0	0
$641$	45.0812	1.78060	0.890300	−	0.455374i	$-0.150494\pi$
$641$	45.0812	1.78060	0.890300	+	0.455374i	$0.150494\pi$
$642$	0	0
$643$	30.2437	1.19269	0.596347	−	0.802727i	$-0.296618\pi$
$643$	30.2437	1.19269	0.596347	+	0.802727i	$0.296618\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.4145	−0.802577	−0.401288	−	0.915952i	$-0.631437\pi$
$647$	−20.4145	−0.802577	−0.401288	+	0.915952i	$0.631437\pi$
$648$	0	0
$649$	4.87982	0.191550
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.5785	1.62709	0.813546	−	0.581500i	$-0.197534\pi$
$653$	41.5785	1.62709	0.813546	+	0.581500i	$0.197534\pi$
$654$	0	0
$655$	−18.6504	−0.728733
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.7738	−1.43251	−0.716253	−	0.697841i	$-0.754144\pi$
$659$	−36.7738	−1.43251	−0.716253	+	0.697841i	$0.754144\pi$
$660$	0	0
$661$	−20.9428	−0.814580	−0.407290	−	0.913299i	$-0.633526\pi$
$661$	−20.9428	−0.814580	−0.407290	+	0.913299i	$0.633526\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.25997	0.165194
$666$	0	0
$667$	−6.77610	−0.262372
$668$	0	0
$669$	0	0
$670$	0	0
$671$	24.1905	0.933864
$672$	0	0
$673$	42.7366	1.64737	0.823687	−	0.567044i	$-0.191913\pi$
$673$	42.7366	1.64737	0.823687	+	0.567044i	$0.191913\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.8417	−1.30064	−0.650320	−	0.759660i	$-0.725365\pi$
$677$	−33.8417	−1.30064	−0.650320	+	0.759660i	$0.725365\pi$
$678$	0	0
$679$	−13.0520	−0.500889
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.58452	0.366742	0.183371	−	0.983044i	$-0.441299\pi$
$683$	9.58452	0.366742	0.183371	+	0.983044i	$0.441299\pi$
$684$	0	0
$685$	−12.5472	−0.479404
$686$	0	0
$687$	0	0
$688$	0	0
$689$	63.4427	2.41697
$690$	0	0
$691$	−7.47145	−0.284228	−0.142114	−	0.989850i	$-0.545390\pi$
$691$	−7.47145	−0.284228	−0.142114	+	0.989850i	$0.545390\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.1215	−0.573590
$696$	0	0
$697$	−2.30477	−0.0872994
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.1846	−0.422436	−0.211218	−	0.977439i	$-0.567743\pi$
$701$	−11.1846	−0.422436	−0.211218	+	0.977439i	$0.567743\pi$
$702$	0	0
$703$	−53.2648	−2.00892
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.34840	−0.238756
$708$	0	0
$709$	−52.0310	−1.95406	−0.977032	−	0.213093i	$-0.931646\pi$
$709$	−52.0310	−1.95406	−0.977032	+	0.213093i	$0.931646\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.44272	−0.128931
$714$	0	0
$715$	−27.3149	−1.02152
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.8868	0.965414	0.482707	−	0.875782i	$-0.339654\pi$
$719$	25.8868	0.965414	0.482707	+	0.875782i	$0.339654\pi$
$720$	0	0
$721$	11.6795	0.434969
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.77610	0.251658
$726$	0	0
$727$	−35.7977	−1.32766	−0.663831	−	0.747883i	$-0.731071\pi$
$727$	−35.7977	−1.32766	−0.663831	+	0.747883i	$0.731071\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.85161	0.142457
$732$	0	0
$733$	−2.62765	−0.0970546	−0.0485273	−	0.998822i	$-0.515453\pi$
$733$	−2.62765	−0.0970546	−0.0485273	+	0.998822i	$0.515453\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	79.8986	2.94310
$738$	0	0
$739$	−48.6458	−1.78946	−0.894732	−	0.446604i	$-0.852633\pi$
$739$	−48.6458	−1.78946	−0.894732	+	0.446604i	$0.852633\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−52.5199	−1.92677	−0.963385	−	0.268122i	$-0.913597\pi$
$743$	−52.5199	−1.92677	−0.963385	+	0.268122i	$0.913597\pi$
$744$	0	0
$745$	3.73958	0.137008
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.14646	−0.0784299
$750$	0	0
$751$	41.3738	1.50975	0.754875	−	0.655868i	$-0.227697\pi$
$751$	41.3738	1.50975	0.754875	+	0.655868i	$0.227697\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.9586	0.544400
$756$	0	0
$757$	41.4634	1.50701	0.753506	−	0.657441i	$-0.228361\pi$
$757$	41.4634	1.50701	0.753506	+	0.657441i	$0.228361\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	24.3719	0.883480	0.441740	−	0.897143i	$-0.354361\pi$
$761$	24.3719	0.883480	0.441740	+	0.897143i	$0.354361\pi$
$762$	0	0
$763$	−6.29948	−0.228057
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.36149	0.193592
$768$	0	0
$769$	9.21844	0.332425	0.166213	−	0.986090i	$-0.446846\pi$
$769$	9.21844	0.332425	0.166213	+	0.986090i	$0.446846\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.7489	0.926122	0.463061	−	0.886326i	$-0.346751\pi$
$773$	25.7489	0.926122	0.463061	+	0.886326i	$0.346751\pi$
$774$	0	0
$775$	3.44272	0.123666
$776$	0	0
$777$	0	0
$778$	0	0
$779$	34.4070	1.23276
$780$	0	0
$781$	29.7140	1.06325
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.7208	−0.668174
$786$	0	0
$787$	41.8564	1.49202	0.746009	−	0.665936i	$-0.231967\pi$
$787$	41.8564	1.49202	0.746009	+	0.665936i	$0.231967\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.40201	−0.120962
$792$	0	0
$793$	26.5783	0.943822
$794$	0	0
$795$	0	0
$796$	0	0
$797$	28.0511	0.993623	0.496811	−	0.867859i	$-0.334504\pi$
$797$	28.0511	0.993623	0.496811	+	0.867859i	$0.334504\pi$
$798$	0	0
$799$	−2.22452	−0.0786979
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−40.2861	−1.42167
$804$	0	0
$805$	−0.958551	−0.0337845
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−32.4300	−1.14018	−0.570089	−	0.821583i	$-0.693091\pi$
$809$	−32.4300	−1.14018	−0.570089	+	0.821583i	$0.693091\pi$
$810$	0	0
$811$	5.06160	0.177737	0.0888684	−	0.996043i	$-0.471675\pi$
$811$	5.06160	0.177737	0.0888684	+	0.996043i	$0.471675\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.19592	0.0418913
$816$	0	0
$817$	−57.4991	−2.01164
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.50814	0.296936	0.148468	−	0.988917i	$-0.452566\pi$
$821$	8.50814	0.296936	0.148468	+	0.988917i	$0.452566\pi$
$822$	0	0
$823$	−2.79005	−0.0972550	−0.0486275	−	0.998817i	$-0.515485\pi$
$823$	−2.79005	−0.0972550	−0.0486275	+	0.998817i	$0.515485\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.68263	−0.336698	−0.168349	−	0.985727i	$-0.553844\pi$
$827$	−9.68263	−0.336698	−0.168349	+	0.985727i	$0.553844\pi$
$828$	0	0
$829$	39.9395	1.38716	0.693579	−	0.720381i	$-0.256033\pi$
$829$	39.9395	1.38716	0.693579	+	0.720381i	$0.256033\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.81034	0.0627246
$834$	0	0
$835$	−21.5167	−0.744617
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.2897	−0.804050	−0.402025	−	0.915629i	$-0.631694\pi$
$839$	−23.2897	−0.804050	−0.402025	+	0.915629i	$0.631694\pi$
$840$	0	0
$841$	16.9155	0.583293
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17.0111	−0.585199
$846$	0	0
$847$	−13.2864	−0.456527
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.9853	0.410851
$852$	0	0
$853$	16.8748	0.577783	0.288892	−	0.957362i	$-0.406713\pi$
$853$	16.8748	0.577783	0.288892	+	0.957362i	$0.406713\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−48.2872	−1.64946	−0.824730	−	0.565526i	$-0.808673\pi$
$857$	−48.2872	−1.64946	−0.824730	+	0.565526i	$0.808673\pi$
$858$	0	0
$859$	4.73596	0.161589	0.0807945	−	0.996731i	$-0.474254\pi$
$859$	4.73596	0.161589	0.0807945	+	0.996731i	$0.474254\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.6114	0.565459	0.282730	−	0.959200i	$-0.408760\pi$
$863$	16.6114	0.565459	0.282730	+	0.959200i	$0.408760\pi$
$864$	0	0
$865$	−16.3707	−0.556621
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−56.3216	−1.91058
$870$	0	0
$871$	87.7852	2.97449
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.958551	0.0324049
$876$	0	0
$877$	−45.5301	−1.53744	−0.768721	−	0.639584i	$-0.779107\pi$
$877$	−45.5301	−1.53744	−0.768721	+	0.639584i	$0.779107\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44.4138	1.49634	0.748170	−	0.663507i	$-0.230933\pi$
$881$	44.4138	1.49634	0.748170	+	0.663507i	$0.230933\pi$
$882$	0	0
$883$	1.47401	0.0496045	0.0248022	−	0.999692i	$-0.492104\pi$
$883$	1.47401	0.0496045	0.0248022	+	0.999692i	$0.492104\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.4805	−0.821974	−0.410987	−	0.911641i	$-0.634816\pi$
$887$	−24.4805	−0.821974	−0.410987	+	0.911641i	$0.634816\pi$
$888$	0	0
$889$	1.63232	0.0547464
$890$	0	0
$891$	0	0
$892$	0	0
$893$	33.2090	1.11130
$894$	0	0
$895$	−5.25143	−0.175536
$896$	0	0
$897$	0	0
$898$	0	0
$899$	23.3282	0.778039
$900$	0	0
$901$	3.44757	0.114855
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.51892	0.316419
$906$	0	0
$907$	7.79935	0.258973	0.129487	−	0.991581i	$-0.458667\pi$
$907$	7.79935	0.258973	0.129487	+	0.991581i	$0.458667\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.8374	−1.61806	−0.809028	−	0.587770i	$-0.800006\pi$
$911$	−48.8374	−1.61806	−0.809028	+	0.587770i	$0.800006\pi$
$912$	0	0
$913$	41.2341	1.36465
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.8774	−0.590363
$918$	0	0
$919$	−27.3637	−0.902646	−0.451323	−	0.892361i	$-0.649048\pi$
$919$	−27.3637	−0.902646	−0.451323	+	0.892361i	$0.649048\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	32.6470	1.07459
$924$	0	0
$925$	−11.9853	−0.394074
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−43.5821	−1.42988	−0.714941	−	0.699184i	$-0.753547\pi$
$929$	−43.5821	−1.42988	−0.714941	+	0.699184i	$0.753547\pi$
$930$	0	0
$931$	−27.0258	−0.885736
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.48433	−0.0485429
$936$	0	0
$937$	60.4097	1.97350	0.986749	−	0.162256i	$-0.0518770\pi$
$937$	60.4097	1.97350	0.986749	+	0.162256i	$0.0518770\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−27.0838	−0.882906	−0.441453	−	0.897284i	$-0.645537\pi$
$941$	−27.0838	−0.882906	−0.441453	+	0.897284i	$0.645537\pi$
$942$	0	0
$943$	−7.74204	−0.252115
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.1034	−0.718265	−0.359132	−	0.933287i	$-0.616927\pi$
$947$	−22.1034	−0.718265	−0.359132	+	0.933287i	$0.616927\pi$
$948$	0	0
$949$	−44.2626	−1.43683
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−54.2350	−1.75684	−0.878422	−	0.477885i	$-0.841404\pi$
$953$	−54.2350	−1.75684	−0.878422	+	0.477885i	$0.841404\pi$
$954$	0	0
$955$	−21.6093	−0.699262
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0271	−0.388376
$960$	0	0
$961$	−19.1477	−0.617668
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.6709	−0.504465
$966$	0	0
$967$	−30.8968	−0.993575	−0.496787	−	0.867872i	$-0.665487\pi$
$967$	−30.8968	−0.993575	−0.496787	+	0.867872i	$0.665487\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−59.2384	−1.90105	−0.950526	−	0.310645i	$-0.899455\pi$
$971$	−59.2384	−1.90105	−0.950526	+	0.310645i	$0.899455\pi$
$972$	0	0
$973$	−14.4947	−0.464678
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.0200	1.02441	0.512205	−	0.858863i	$-0.328829\pi$
$977$	32.0200	1.02441	0.512205	+	0.858863i	$0.328829\pi$
$978$	0	0
$979$	50.4020	1.61085
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−31.7189	−1.01168	−0.505838	−	0.862628i	$-0.668817\pi$
$983$	−31.7189	−1.01168	−0.505838	+	0.862628i	$0.668817\pi$
$984$	0	0
$985$	−4.81885	−0.153541
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.9381	0.411407
$990$	0	0
$991$	−40.8991	−1.29920	−0.649601	−	0.760275i	$-0.725064\pi$
$991$	−40.8991	−1.29920	−0.649601	+	0.760275i	$0.725064\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.07527	0.160897
$996$	0	0
$997$	−11.7200	−0.371177	−0.185589	−	0.982628i	$-0.559419\pi$
$997$	−11.7200	−0.371177	−0.185589	+	0.982628i	$0.559419\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bv.1.4	✓	7
3.2	odd	2		8280.2.a.bw.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bv.1.4	✓	7	1.1	even	1	trivial
8280.2.a.bw.1.4	yes	7	3.2	odd	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -0.958551 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -0.958551 q^{7} -4.98608 q^{11} -5.47824 q^{13} -0.297696 q^{17} +4.44417 q^{19} -1.00000 q^{23} +1.00000 q^{25} +6.77610 q^{29} +3.44272 q^{31} +0.958551 q^{35} -11.9853 q^{37} +7.74204 q^{41} -12.9381 q^{43} +7.47247 q^{47} -6.08118 q^{49} -11.5809 q^{53} +4.98608 q^{55} -0.978689 q^{59} -4.85161 q^{61} +5.47824 q^{65} -16.0244 q^{67} -5.95940 q^{71} +8.07972 q^{73} +4.77941 q^{77} +11.2958 q^{79} -8.26985 q^{83} +0.297696 q^{85} -10.1085 q^{89} +5.25117 q^{91} -4.44417 q^{95} +13.6164 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{5} - 6 q^{7}+O(q^{10}) \) 7 * q - 7 * q^5 - 6 * q^7	\( 7 q - 7 q^{5} - 6 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} - 8 q^{19} - 7 q^{23} + 7 q^{25} - 2 q^{29} - 8 q^{31} + 6 q^{35} - 16 q^{37} - 2 q^{41} - 10 q^{43} + 8 q^{47} + 19 q^{49} + 10 q^{53} - 2 q^{55} + 24 q^{59} + 8 q^{61} + 6 q^{65} - 20 q^{67} + 8 q^{71} + 2 q^{73} + 12 q^{77} - 2 q^{79} + 22 q^{83} - 4 q^{85} + 16 q^{89} - 20 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 7 * q - 7 * q^5 - 6 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 - 8 * q^19 - 7 * q^23 + 7 * q^25 - 2 * q^29 - 8 * q^31 + 6 * q^35 - 16 * q^37 - 2 * q^41 - 10 * q^43 + 8 * q^47 + 19 * q^49 + 10 * q^53 - 2 * q^55 + 24 * q^59 + 8 * q^61 + 6 * q^65 - 20 * q^67 + 8 * q^71 + 2 * q^73 + 12 * q^77 - 2 * q^79 + 22 * q^83 - 4 * q^85 + 16 * q^89 - 20 * q^91 + 8 * q^95 + 4 * q^97

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