LMFDB - Embedded newform 8280.2.a.bq.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.655762$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +4.46569 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +4.46569 q^{7} -3.39413 q^{11} -4.70565 q^{13} +6.54830 q^{17} -6.22574 q^{19} -1.00000 q^{23} +1.00000 q^{25} -0.448521 q^{29} -5.15417 q^{31} +4.46569 q^{35} -5.98578 q^{37} -3.07157 q^{41} -4.47991 q^{43} -6.22574 q^{47} +12.9424 q^{49} +2.59166 q^{53} -3.39413 q^{55} +1.55148 q^{59} -4.91421 q^{61} -4.70565 q^{65} +6.08556 q^{67} -1.07157 q^{71} -8.70565 q^{73} -15.1571 q^{77} -13.1400 q^{79} +13.0282 q^{83} +6.54830 q^{85} -11.6199 q^{89} -21.0140 q^{91} -6.22574 q^{95} +4.30834 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} - 2 q^{13} - 2 q^{17} - 6 q^{19} - 4 q^{23} + 4 q^{25} - 4 q^{29} - 6 q^{31} - 4 q^{37} - 8 q^{41} - 20 q^{43} - 6 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 26 q^{67} - 18 q^{73} - 6 q^{77} - 18 q^{79} + 26 q^{83} - 2 q^{85} - 14 q^{89} - 38 q^{91} - 6 q^{95} - 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 2 * q^13 - 2 * q^17 - 6 * q^19 - 4 * q^23 + 4 * q^25 - 4 * q^29 - 6 * q^31 - 4 * q^37 - 8 * q^41 - 20 * q^43 - 6 * q^47 + 10 * q^49 + 4 * q^53 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 26 * q^67 - 18 * q^73 - 6 * q^77 - 18 * q^79 + 26 * q^83 - 2 * q^85 - 14 * q^89 - 38 * q^91 - 6 * q^95 - 12 * 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
$7$	4.46569	1.68787	0.843937	−	0.536443i	$-0.180232\pi$
$7$	4.46569	1.68787	0.843937	+	0.536443i	$0.180232\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.39413	−1.02337	−0.511684	−	0.859174i	$-0.670978\pi$
$11$	−3.39413	−1.02337	−0.511684	+	0.859174i	$0.670978\pi$
$12$	0	0
$13$	−4.70565	−1.30511	−0.652556	−	0.757740i	$-0.726303\pi$
$13$	−4.70565	−1.30511	−0.652556	+	0.757740i	$0.726303\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.54830	1.58819	0.794097	−	0.607790i	$-0.207944\pi$
$17$	6.54830	1.58819	0.794097	+	0.607790i	$0.207944\pi$
$18$	0	0
$19$	−6.22574	−1.42828	−0.714141	−	0.700002i	$-0.753183\pi$
$19$	−6.22574	−1.42828	−0.714141	+	0.700002i	$0.753183\pi$
$20$	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
$29$	−0.448521	−0.0832882	−0.0416441	−	0.999133i	$-0.513260\pi$
$29$	−0.448521	−0.0832882	−0.0416441	+	0.999133i	$0.513260\pi$
$30$	0	0
$31$	−5.15417	−0.925716	−0.462858	−	0.886432i	$-0.653176\pi$
$31$	−5.15417	−0.925716	−0.462858	+	0.886432i	$0.653176\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.46569	0.754840
$36$	0	0
$37$	−5.98578	−0.984057	−0.492028	−	0.870579i	$-0.663744\pi$
$37$	−5.98578	−0.984057	−0.492028	+	0.870579i	$0.663744\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.07157	−0.479698	−0.239849	−	0.970810i	$-0.577098\pi$
$41$	−3.07157	−0.479698	−0.239849	+	0.970810i	$0.577098\pi$
$42$	0	0
$43$	−4.47991	−0.683180	−0.341590	−	0.939849i	$-0.610965\pi$
$43$	−4.47991	−0.683180	−0.341590	+	0.939849i	$0.610965\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.22574	−0.908117	−0.454059	−	0.890972i	$-0.650024\pi$
$47$	−6.22574	−0.908117	−0.454059	+	0.890972i	$0.650024\pi$
$48$	0	0
$49$	12.9424	1.84892
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.59166	0.355991	0.177996	−	0.984031i	$-0.443039\pi$
$53$	2.59166	0.355991	0.177996	+	0.984031i	$0.443039\pi$
$54$	0	0
$55$	−3.39413	−0.457664
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.55148	0.201985	0.100993	−	0.994887i	$-0.467798\pi$
$59$	1.55148	0.201985	0.100993	+	0.994887i	$0.467798\pi$
$60$	0	0
$61$	−4.91421	−0.629201	−0.314600	−	0.949224i	$-0.601870\pi$
$61$	−4.91421	−0.629201	−0.314600	+	0.949224i	$0.601870\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.70565	−0.583664
$66$	0	0
$67$	6.08556	0.743469	0.371735	−	0.928339i	$-0.378763\pi$
$67$	6.08556	0.743469	0.371735	+	0.928339i	$0.378763\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.07157	−0.127172	−0.0635859	−	0.997976i	$-0.520254\pi$
$71$	−1.07157	−0.127172	−0.0635859	+	0.997976i	$0.520254\pi$
$72$	0	0
$73$	−8.70565	−1.01892	−0.509460	−	0.860495i	$-0.670155\pi$
$73$	−8.70565	−1.01892	−0.509460	+	0.860495i	$0.670155\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.1571	−1.72731
$78$	0	0
$79$	−13.1400	−1.47836	−0.739180	−	0.673508i	$-0.764787\pi$
$79$	−13.1400	−1.47836	−0.739180	+	0.673508i	$0.764787\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.0282	1.43003	0.715016	−	0.699108i	$-0.246420\pi$
$83$	13.0282	1.43003	0.715016	+	0.699108i	$0.246420\pi$
$84$	0	0
$85$	6.54830	0.710262
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.6199	−1.23170	−0.615852	−	0.787862i	$-0.711188\pi$
$89$	−11.6199	−1.23170	−0.615852	+	0.787862i	$0.711188\pi$
$90$	0	0
$91$	−21.0140	−2.20286
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.22574	−0.638747
$96$	0	0
$97$	4.30834	0.437446	0.218723	−	0.975787i	$-0.429811\pi$
$97$	4.30834	0.437446	0.218723	+	0.975787i	$0.429811\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.3397	−1.02884	−0.514421	−	0.857538i	$-0.671993\pi$
$101$	−10.3397	−1.02884	−0.514421	+	0.857538i	$0.671993\pi$
$102$	0	0
$103$	−10.8316	−1.06727	−0.533635	−	0.845715i	$-0.679174\pi$
$103$	−10.8316	−1.06727	−0.533635	+	0.845715i	$0.679174\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.76004	0.170150	0.0850750	−	0.996375i	$-0.472887\pi$
$107$	1.76004	0.170150	0.0850750	+	0.996375i	$0.472887\pi$
$108$	0	0
$109$	−1.39413	−0.133533	−0.0667665	−	0.997769i	$-0.521268\pi$
$109$	−1.39413	−0.133533	−0.0667665	+	0.997769i	$0.521268\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.17134	−0.110191	−0.0550954	−	0.998481i	$-0.517546\pi$
$113$	−1.17134	−0.110191	−0.0550954	+	0.998481i	$0.517546\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	29.2427	2.68067
$120$	0	0
$121$	0.520089	0.0472808
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.9454	1.23745	0.618726	−	0.785607i	$-0.287649\pi$
$127$	13.9454	1.23745	0.618726	+	0.785607i	$0.287649\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−13.8628	−1.21120	−0.605598	−	0.795771i	$-0.707066\pi$
$131$	−13.8628	−1.21120	−0.605598	+	0.795771i	$0.707066\pi$
$132$	0	0
$133$	−27.8022	−2.41076
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.4081	1.23097	0.615484	−	0.788149i	$-0.288961\pi$
$137$	14.4081	1.23097	0.615484	+	0.788149i	$0.288961\pi$
$138$	0	0
$139$	−10.1140	−0.857858	−0.428929	−	0.903338i	$-0.641109\pi$
$139$	−10.1140	−0.857858	−0.428929	+	0.903338i	$0.641109\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	15.9716	1.33561
$144$	0	0
$145$	−0.448521	−0.0372476
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.32870	−0.272697	−0.136349	−	0.990661i	$-0.543537\pi$
$149$	−3.32870	−0.272697	−0.136349	+	0.990661i	$0.543537\pi$
$150$	0	0
$151$	−22.3427	−1.81822	−0.909111	−	0.416554i	$-0.863238\pi$
$151$	−22.3427	−1.81822	−0.909111	+	0.416554i	$0.863238\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.15417	−0.413993
$156$	0	0
$157$	6.63090	0.529203	0.264602	−	0.964358i	$-0.414760\pi$
$157$	6.63090	0.529203	0.264602	+	0.964358i	$0.414760\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.46569	−0.351946
$162$	0	0
$163$	−22.3083	−1.74732	−0.873662	−	0.486533i	$-0.838261\pi$
$163$	−22.3083	−1.74732	−0.873662	+	0.486533i	$0.838261\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	24.0885	1.86403	0.932013	−	0.362426i	$-0.118051\pi$
$167$	24.0885	1.86403	0.932013	+	0.362426i	$0.118051\pi$
$168$	0	0
$169$	9.14314	0.703318
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.9314	1.43933	0.719663	−	0.694323i	$-0.244296\pi$
$173$	18.9314	1.43933	0.719663	+	0.694323i	$0.244296\pi$
$174$	0	0
$175$	4.46569	0.337575
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.7598	1.40217	0.701087	−	0.713075i	$-0.252698\pi$
$179$	18.7598	1.40217	0.701087	+	0.713075i	$0.252698\pi$
$180$	0	0
$181$	−21.1964	−1.57551	−0.787757	−	0.615986i	$-0.788758\pi$
$181$	−21.1964	−1.57551	−0.787757	+	0.615986i	$0.788758\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.98578	−0.440083
$186$	0	0
$187$	−22.2257	−1.62531
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.3909	−1.04129	−0.520646	−	0.853773i	$-0.674309\pi$
$191$	−14.3909	−1.04129	−0.520646	+	0.853773i	$0.674309\pi$
$192$	0	0
$193$	11.0966	0.798750	0.399375	−	0.916788i	$-0.369227\pi$
$193$	11.0966	0.798750	0.399375	+	0.916788i	$0.369227\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.1652	−1.00923	−0.504614	−	0.863345i	$-0.668365\pi$
$197$	−14.1652	−1.00923	−0.504614	+	0.863345i	$0.668365\pi$
$198$	0	0
$199$	8.66004	0.613894	0.306947	−	0.951727i	$-0.400693\pi$
$199$	8.66004	0.613894	0.306947	+	0.951727i	$0.400693\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.00296	−0.140580
$204$	0	0
$205$	−3.07157	−0.214528
$206$	0	0
$207$	0	0
$208$	0	0
$209$	21.1309	1.46166
$210$	0	0
$211$	−11.1321	−0.766366	−0.383183	−	0.923673i	$-0.625172\pi$
$211$	−11.1321	−0.766366	−0.383183	+	0.923673i	$0.625172\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.47991	−0.305527
$216$	0	0
$217$	−23.0169	−1.56249
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−30.8140	−2.07277
$222$	0	0
$223$	20.1711	1.35076	0.675379	−	0.737471i	$-0.263980\pi$
$223$	20.1711	1.35076	0.675379	+	0.737471i	$0.263980\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.523273	0.0347308	0.0173654	−	0.999849i	$-0.494472\pi$
$227$	0.523273	0.0347308	0.0173654	+	0.999849i	$0.494472\pi$
$228$	0	0
$229$	−3.33359	−0.220290	−0.110145	−	0.993916i	$-0.535132\pi$
$229$	−3.33359	−0.220290	−0.110145	+	0.993916i	$0.535132\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.5481	−1.01859	−0.509294	−	0.860593i	$-0.670093\pi$
$233$	−15.5481	−1.01859	−0.509294	+	0.860593i	$0.670093\pi$
$234$	0	0
$235$	−6.22574	−0.406122
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.5451	−1.78175	−0.890873	−	0.454253i	$-0.849906\pi$
$239$	−27.5451	−1.78175	−0.890873	+	0.454253i	$0.849906\pi$
$240$	0	0
$241$	8.74000	0.562993	0.281496	−	0.959562i	$-0.409169\pi$
$241$	8.74000	0.562993	0.281496	+	0.959562i	$0.409169\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.9424	0.826861
$246$	0	0
$247$	29.2961	1.86407
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.0312	−0.948759	−0.474379	−	0.880321i	$-0.657327\pi$
$251$	−15.0312	−0.948759	−0.474379	+	0.880321i	$0.657327\pi$
$252$	0	0
$253$	3.39413	0.213387
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.1571	0.820719	0.410359	−	0.911924i	$-0.365403\pi$
$257$	13.1571	0.820719	0.410359	+	0.911924i	$0.365403\pi$
$258$	0	0
$259$	−26.7307	−1.66096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	23.5795	1.45397	0.726986	−	0.686652i	$-0.240921\pi$
$263$	23.5795	1.45397	0.726986	+	0.686652i	$0.240921\pi$
$264$	0	0
$265$	2.59166	0.159204
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.52304	0.458688	0.229344	−	0.973345i	$-0.426342\pi$
$269$	7.52304	0.458688	0.229344	+	0.973345i	$0.426342\pi$
$270$	0	0
$271$	18.5370	1.12604	0.563022	−	0.826442i	$-0.309639\pi$
$271$	18.5370	1.12604	0.563022	+	0.826442i	$0.309639\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.39413	−0.204673
$276$	0	0
$277$	23.8628	1.43378	0.716888	−	0.697189i	$-0.245566\pi$
$277$	23.8628	1.43378	0.716888	+	0.697189i	$0.245566\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.6118	1.34891	0.674453	−	0.738318i	$-0.264380\pi$
$281$	22.6118	1.34891	0.674453	+	0.738318i	$0.264380\pi$
$282$	0	0
$283$	13.6684	0.812504	0.406252	−	0.913761i	$-0.366836\pi$
$283$	13.6684	0.812504	0.406252	+	0.913761i	$0.366836\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−13.7167	−0.809670
$288$	0	0
$289$	25.8802	1.52236
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.0431	0.878829	0.439415	−	0.898284i	$-0.355186\pi$
$293$	15.0431	0.878829	0.439415	+	0.898284i	$0.355186\pi$
$294$	0	0
$295$	1.55148	0.0903306
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.70565	0.272135
$300$	0	0
$301$	−20.0059	−1.15312
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.91421	−0.281387
$306$	0	0
$307$	−31.9674	−1.82448	−0.912239	−	0.409658i	$-0.865648\pi$
$307$	−31.9674	−1.82448	−0.912239	+	0.409658i	$0.865648\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	34.1932	1.93892	0.969459	−	0.245254i	$-0.0788715\pi$
$311$	34.1932	1.93892	0.969459	+	0.245254i	$0.0788715\pi$
$312$	0	0
$313$	−11.5713	−0.654049	−0.327024	−	0.945016i	$-0.606046\pi$
$313$	−11.5713	−0.654049	−0.327024	+	0.945016i	$0.606046\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.12915	−0.175750	−0.0878752	−	0.996131i	$-0.528008\pi$
$317$	−3.12915	−0.175750	−0.0878752	+	0.996131i	$0.528008\pi$
$318$	0	0
$319$	1.52234	0.0852344
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−40.7680	−2.26839
$324$	0	0
$325$	−4.70565	−0.261022
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−27.8022	−1.53279
$330$	0	0
$331$	−0.988966	−0.0543585	−0.0271793	−	0.999631i	$-0.508652\pi$
$331$	−0.988966	−0.0543585	−0.0271793	+	0.999631i	$0.508652\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.08556	0.332490
$336$	0	0
$337$	21.2397	1.15700	0.578501	−	0.815682i	$-0.303638\pi$
$337$	21.2397	1.15700	0.578501	+	0.815682i	$0.303638\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.4939	0.947348
$342$	0	0
$343$	26.5370	1.43287
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.78825	0.149681	0.0748406	−	0.997196i	$-0.476155\pi$
$347$	2.78825	0.149681	0.0748406	+	0.997196i	$0.476155\pi$
$348$	0	0
$349$	30.3939	1.62695	0.813474	−	0.581601i	$-0.197574\pi$
$349$	30.3939	1.62695	0.813474	+	0.581601i	$0.197574\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.6993	−0.569465	−0.284733	−	0.958607i	$-0.591905\pi$
$353$	−10.6993	−0.569465	−0.284733	+	0.958607i	$0.591905\pi$
$354$	0	0
$355$	−1.07157	−0.0568729
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.5120	−0.977027	−0.488513	−	0.872556i	$-0.662461\pi$
$359$	−18.5120	−0.977027	−0.488513	+	0.872556i	$0.662461\pi$
$360$	0	0
$361$	19.7598	1.03999
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.70565	−0.455675
$366$	0	0
$367$	−24.6937	−1.28900	−0.644500	−	0.764605i	$-0.722934\pi$
$367$	−24.6937	−1.28900	−0.644500	+	0.764605i	$0.722934\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.5735	0.600869
$372$	0	0
$373$	−24.9968	−1.29429	−0.647143	−	0.762369i	$-0.724036\pi$
$373$	−24.9968	−1.29429	−0.647143	+	0.762369i	$0.724036\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.11058	0.108700
$378$	0	0
$379$	20.6167	1.05901	0.529504	−	0.848308i	$-0.322378\pi$
$379$	20.6167	1.05901	0.529504	+	0.848308i	$0.322378\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.42645	−0.226181	−0.113091	−	0.993585i	$-0.536075\pi$
$383$	−4.42645	−0.226181	−0.113091	+	0.993585i	$0.536075\pi$
$384$	0	0
$385$	−15.1571	−0.772479
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.7417	1.00094	0.500472	−	0.865753i	$-0.333160\pi$
$389$	19.7417	1.00094	0.500472	+	0.865753i	$0.333160\pi$
$390$	0	0
$391$	−6.54830	−0.331162
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.1400	−0.661143
$396$	0	0
$397$	−5.68529	−0.285337	−0.142668	−	0.989771i	$-0.545568\pi$
$397$	−5.68529	−0.285337	−0.142668	+	0.989771i	$0.545568\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.1341	−0.605949	−0.302975	−	0.952999i	$-0.597980\pi$
$401$	−12.1341	−0.605949	−0.302975	+	0.952999i	$0.597980\pi$
$402$	0	0
$403$	24.2537	1.20816
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.3165	1.00705
$408$	0	0
$409$	25.0454	1.23841	0.619207	−	0.785228i	$-0.287454\pi$
$409$	25.0454	1.23841	0.619207	+	0.785228i	$0.287454\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.92843	0.340926
$414$	0	0
$415$	13.0282	0.639530
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−32.3314	−1.57949	−0.789747	−	0.613433i	$-0.789788\pi$
$419$	−32.3314	−1.57949	−0.789747	+	0.613433i	$0.789788\pi$
$420$	0	0
$421$	24.8858	1.21286	0.606429	−	0.795137i	$-0.292601\pi$
$421$	24.8858	1.21286	0.606429	+	0.795137i	$0.292601\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.54830	0.317639
$426$	0	0
$427$	−21.9454	−1.06201
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.0343	−1.35037	−0.675183	−	0.737650i	$-0.735936\pi$
$431$	−28.0343	−1.35037	−0.675183	+	0.737650i	$0.735936\pi$
$432$	0	0
$433$	−18.7370	−0.900445	−0.450222	−	0.892916i	$-0.648655\pi$
$433$	−18.7370	−0.900445	−0.450222	+	0.892916i	$0.648655\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.22574	0.297817
$438$	0	0
$439$	−1.01811	−0.0485918	−0.0242959	−	0.999705i	$-0.507734\pi$
$439$	−1.01811	−0.0485918	−0.0242959	+	0.999705i	$0.507734\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.4311	−0.828177	−0.414089	−	0.910237i	$-0.635900\pi$
$443$	−17.4311	−0.828177	−0.414089	+	0.910237i	$0.635900\pi$
$444$	0	0
$445$	−11.6199	−0.550834
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.6138	1.58633	0.793167	−	0.609005i	$-0.208431\pi$
$449$	33.6138	1.58633	0.793167	+	0.609005i	$0.208431\pi$
$450$	0	0
$451$	10.4253	0.490908
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−21.0140	−0.985151
$456$	0	0
$457$	−5.01695	−0.234683	−0.117341	−	0.993092i	$-0.537437\pi$
$457$	−5.01695	−0.234683	−0.117341	+	0.993092i	$0.537437\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.49165	0.255772	0.127886	−	0.991789i	$-0.459181\pi$
$461$	5.49165	0.255772	0.127886	+	0.991789i	$0.459181\pi$
$462$	0	0
$463$	−42.3684	−1.96903	−0.984514	−	0.175308i	$-0.943908\pi$
$463$	−42.3684	−1.96903	−0.984514	+	0.175308i	$0.943908\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.95959	0.275777	0.137889	−	0.990448i	$-0.455968\pi$
$467$	5.95959	0.275777	0.137889	+	0.990448i	$0.455968\pi$
$468$	0	0
$469$	27.1762	1.25488
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.2054	0.699144
$474$	0	0
$475$	−6.22574	−0.285656
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.2135	−1.51757	−0.758783	−	0.651344i	$-0.774206\pi$
$479$	−33.2135	−1.51757	−0.758783	+	0.651344i	$0.774206\pi$
$480$	0	0
$481$	28.1670	1.28430
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.30834	0.195632
$486$	0	0
$487$	−28.7336	−1.30204	−0.651022	−	0.759058i	$-0.725660\pi$
$487$	−28.7336	−1.30204	−0.651022	+	0.759058i	$0.725660\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19.4427	−0.877436	−0.438718	−	0.898625i	$-0.644567\pi$
$491$	−19.4427	−0.877436	−0.438718	+	0.898625i	$0.644567\pi$
$492$	0	0
$493$	−2.93705	−0.132278
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.78530	−0.214650
$498$	0	0
$499$	−5.21695	−0.233543	−0.116771	−	0.993159i	$-0.537254\pi$
$499$	−5.21695	−0.233543	−0.116771	+	0.993159i	$0.537254\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.06574	0.270458	0.135229	−	0.990814i	$-0.456823\pi$
$503$	6.06574	0.270458	0.135229	+	0.990814i	$0.456823\pi$
$504$	0	0
$505$	−10.3397	−0.460112
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12.5079	−0.554402	−0.277201	−	0.960812i	$-0.589407\pi$
$509$	−12.5079	−0.554402	−0.277201	+	0.960812i	$0.589407\pi$
$510$	0	0
$511$	−38.8768	−1.71981
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.8316	−0.477298
$516$	0	0
$517$	21.1309	0.929338
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.1539	1.05820	0.529102	−	0.848558i	$-0.322529\pi$
$521$	24.1539	1.05820	0.529102	+	0.848558i	$0.322529\pi$
$522$	0	0
$523$	−28.1427	−1.23059	−0.615297	−	0.788296i	$-0.710964\pi$
$523$	−28.1427	−1.23059	−0.615297	+	0.788296i	$0.710964\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.7510	−1.47022
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.4537	0.626060
$534$	0	0
$535$	1.76004	0.0760934
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−43.9282	−1.89212
$540$	0	0
$541$	−17.0117	−0.731392	−0.365696	−	0.930734i	$-0.619169\pi$
$541$	−17.0117	−0.731392	−0.365696	+	0.930734i	$0.619169\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.39413	−0.0597178
$546$	0	0
$547$	−19.0402	−0.814099	−0.407050	−	0.913406i	$-0.633442\pi$
$547$	−19.0402	−0.814099	−0.407050	+	0.913406i	$0.633442\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.79237	0.118959
$552$	0	0
$553$	−58.6790	−2.49529
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−23.6882	−1.00370	−0.501852	−	0.864954i	$-0.667348\pi$
$557$	−23.6882	−1.00370	−0.501852	+	0.864954i	$0.667348\pi$
$558$	0	0
$559$	21.0809	0.891627
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.8630	0.542111	0.271055	−	0.962564i	$-0.412627\pi$
$563$	12.8630	0.542111	0.271055	+	0.962564i	$0.412627\pi$
$564$	0	0
$565$	−1.17134	−0.0492788
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.4709	0.438963	0.219481	−	0.975617i	$-0.429563\pi$
$569$	10.4709	0.438963	0.219481	+	0.975617i	$0.429563\pi$
$570$	0	0
$571$	−11.0361	−0.461845	−0.230922	−	0.972972i	$-0.574174\pi$
$571$	−11.0361	−0.461845	−0.230922	+	0.972972i	$0.574174\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−45.9535	−1.91307	−0.956535	−	0.291616i	$-0.905807\pi$
$577$	−45.9535	−1.91307	−0.956535	+	0.291616i	$0.905807\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	58.1800	2.41371
$582$	0	0
$583$	−8.79641	−0.364310
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.2578	1.00123	0.500614	−	0.865671i	$-0.333108\pi$
$587$	24.2578	1.00123	0.500614	+	0.865671i	$0.333108\pi$
$588$	0	0
$589$	32.0885	1.32218
$590$	0	0
$591$	0	0
$592$	0	0
$593$	27.5282	1.13045	0.565225	−	0.824937i	$-0.308789\pi$
$593$	27.5282	1.13045	0.565225	+	0.824937i	$0.308789\pi$
$594$	0	0
$595$	29.2427	1.19883
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.2741	−1.27782	−0.638912	−	0.769280i	$-0.720615\pi$
$599$	−31.2741	−1.27782	−0.638912	+	0.769280i	$0.720615\pi$
$600$	0	0
$601$	43.3253	1.76728	0.883638	−	0.468171i	$-0.155087\pi$
$601$	43.3253	1.76728	0.883638	+	0.468171i	$0.155087\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.520089	0.0211446
$606$	0	0
$607$	29.7802	1.20874	0.604370	−	0.796704i	$-0.293425\pi$
$607$	29.7802	1.20874	0.604370	+	0.796704i	$0.293425\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	29.2961	1.18520
$612$	0	0
$613$	13.6763	0.552380	0.276190	−	0.961103i	$-0.410928\pi$
$613$	13.6763	0.552380	0.276190	+	0.961103i	$0.410928\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.9130	−0.600377	−0.300188	−	0.953880i	$-0.597050\pi$
$617$	−14.9130	−0.600377	−0.300188	+	0.953880i	$0.597050\pi$
$618$	0	0
$619$	−41.3021	−1.66007	−0.830035	−	0.557712i	$-0.811680\pi$
$619$	−41.3021	−1.66007	−0.830035	+	0.557712i	$0.811680\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−51.8908	−2.07896
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−39.1967	−1.56287
$630$	0	0
$631$	44.2104	1.75999	0.879993	−	0.474986i	$-0.157547\pi$
$631$	44.2104	1.75999	0.879993	+	0.474986i	$0.157547\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.9454	0.553405
$636$	0	0
$637$	−60.9025	−2.41304
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.2750	−1.27479	−0.637393	−	0.770539i	$-0.719987\pi$
$641$	−32.2750	−1.27479	−0.637393	+	0.770539i	$0.719987\pi$
$642$	0	0
$643$	−23.2689	−0.917635	−0.458817	−	0.888531i	$-0.651727\pi$
$643$	−23.2689	−0.917635	−0.458817	+	0.888531i	$0.651727\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.49390	0.0587313	0.0293656	−	0.999569i	$-0.490651\pi$
$647$	1.49390	0.0587313	0.0293656	+	0.999569i	$0.490651\pi$
$648$	0	0
$649$	−5.26592	−0.206705
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.30663	−0.364197	−0.182098	−	0.983280i	$-0.558289\pi$
$653$	−9.30663	−0.364197	−0.182098	+	0.983280i	$0.558289\pi$
$654$	0	0
$655$	−13.8628	−0.541663
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−34.8903	−1.35913	−0.679566	−	0.733614i	$-0.737832\pi$
$659$	−34.8903	−1.35913	−0.679566	+	0.733614i	$0.737832\pi$
$660$	0	0
$661$	2.14578	0.0834613	0.0417307	−	0.999129i	$-0.486713\pi$
$661$	2.14578	0.0834613	0.0417307	+	0.999129i	$0.486713\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−27.8022	−1.07812
$666$	0	0
$667$	0.448521	0.0173668
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.6795	0.643903
$672$	0	0
$673$	27.0483	1.04264	0.521318	−	0.853362i	$-0.325440\pi$
$673$	27.0483	1.04264	0.521318	+	0.853362i	$0.325440\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−45.9442	−1.76578	−0.882890	−	0.469580i	$-0.844405\pi$
$677$	−45.9442	−1.76578	−0.882890	+	0.469580i	$0.844405\pi$
$678$	0	0
$679$	19.2397	0.738353
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.9571	0.954958	0.477479	−	0.878643i	$-0.341551\pi$
$683$	24.9571	0.954958	0.477479	+	0.878643i	$0.341551\pi$
$684$	0	0
$685$	14.4081	0.550506
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12.1954	−0.464609
$690$	0	0
$691$	−18.8167	−0.715820	−0.357910	−	0.933756i	$-0.616511\pi$
$691$	−18.8167	−0.715820	−0.357910	+	0.933756i	$0.616511\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.1140	−0.383646
$696$	0	0
$697$	−20.1135	−0.761855
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−42.0601	−1.58859	−0.794294	−	0.607534i	$-0.792159\pi$
$701$	−42.0601	−1.58859	−0.794294	+	0.607534i	$0.792159\pi$
$702$	0	0
$703$	37.2659	1.40551
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−46.1741	−1.73655
$708$	0	0
$709$	−29.3133	−1.10088	−0.550442	−	0.834873i	$-0.685541\pi$
$709$	−29.3133	−1.10088	−0.550442	+	0.834873i	$0.685541\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.15417	0.193025
$714$	0	0
$715$	15.9716	0.597303
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.95641	0.334018	0.167009	−	0.985955i	$-0.446589\pi$
$719$	8.95641	0.334018	0.167009	+	0.985955i	$0.446589\pi$
$720$	0	0
$721$	−48.3707	−1.80142
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.448521	−0.0166576
$726$	0	0
$727$	−16.4313	−0.609405	−0.304702	−	0.952448i	$-0.598557\pi$
$727$	−16.4313	−0.609405	−0.304702	+	0.952448i	$0.598557\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−29.3358	−1.08502
$732$	0	0
$733$	8.41519	0.310822	0.155411	−	0.987850i	$-0.450330\pi$
$733$	8.41519	0.310822	0.155411	+	0.987850i	$0.450330\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.6551	−0.760842
$738$	0	0
$739$	10.3375	0.380270	0.190135	−	0.981758i	$-0.439107\pi$
$739$	10.3375	0.380270	0.190135	+	0.981758i	$0.439107\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.6388	1.30746	0.653731	−	0.756727i	$-0.273203\pi$
$743$	35.6388	1.30746	0.653731	+	0.756727i	$0.273203\pi$
$744$	0	0
$745$	−3.32870	−0.121954
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.85982	0.287192
$750$	0	0
$751$	30.4402	1.11078	0.555390	−	0.831590i	$-0.312569\pi$
$751$	30.4402	1.11078	0.555390	+	0.831590i	$0.312569\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−22.3427	−0.813134
$756$	0	0
$757$	−10.4941	−0.381416	−0.190708	−	0.981647i	$-0.561078\pi$
$757$	−10.4941	−0.381416	−0.190708	+	0.981647i	$0.561078\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.99704	0.144893	0.0724464	−	0.997372i	$-0.476919\pi$
$761$	3.99704	0.144893	0.0724464	+	0.997372i	$0.476919\pi$
$762$	0	0
$763$	−6.22574	−0.225387
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.30072	−0.263614
$768$	0	0
$769$	−15.6307	−0.563656	−0.281828	−	0.959465i	$-0.590941\pi$
$769$	−15.6307	−0.563656	−0.281828	+	0.959465i	$0.590941\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.6912	−1.21179	−0.605894	−	0.795545i	$-0.707185\pi$
$773$	−33.6912	−1.21179	−0.605894	+	0.795545i	$0.707185\pi$
$774$	0	0
$775$	−5.15417	−0.185143
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19.1228	0.685145
$780$	0	0
$781$	3.63704	0.130143
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.63090	0.236667
$786$	0	0
$787$	1.57721	0.0562215	0.0281108	−	0.999605i	$-0.491051\pi$
$787$	1.57721	0.0562215	0.0281108	+	0.999605i	$0.491051\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.23086	−0.185988
$792$	0	0
$793$	23.1246	0.821178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.5677	0.516015	0.258007	−	0.966143i	$-0.416934\pi$
$797$	14.5677	0.516015	0.258007	+	0.966143i	$0.416934\pi$
$798$	0	0
$799$	−40.7680	−1.44227
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29.5481	1.04273
$804$	0	0
$805$	−4.46569	−0.157395
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−29.1280	−1.02409	−0.512043	−	0.858960i	$-0.671111\pi$
$809$	−29.1280	−1.02409	−0.512043	+	0.858960i	$0.671111\pi$
$810$	0	0
$811$	−28.0351	−0.984444	−0.492222	−	0.870470i	$-0.663815\pi$
$811$	−28.0351	−0.984444	−0.492222	+	0.870470i	$0.663815\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−22.3083	−0.781427
$816$	0	0
$817$	27.8908	0.975774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.5015	−1.48331	−0.741657	−	0.670780i	$-0.765960\pi$
$821$	−42.5015	−1.48331	−0.741657	+	0.670780i	$0.765960\pi$
$822$	0	0
$823$	−14.5662	−0.507745	−0.253873	−	0.967238i	$-0.581704\pi$
$823$	−14.5662	−0.507745	−0.253873	+	0.967238i	$0.581704\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.7880	−0.409910	−0.204955	−	0.978771i	$-0.565705\pi$
$827$	−11.7880	−0.409910	−0.204955	+	0.978771i	$0.565705\pi$
$828$	0	0
$829$	−2.34385	−0.0814053	−0.0407027	−	0.999171i	$-0.512960\pi$
$829$	−2.34385	−0.0814053	−0.0407027	+	0.999171i	$0.512960\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	84.7508	2.93644
$834$	0	0
$835$	24.0885	0.833617
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.97157	−0.275209	−0.137604	−	0.990487i	$-0.543940\pi$
$839$	−7.97157	−0.275209	−0.137604	+	0.990487i	$0.543940\pi$
$840$	0	0
$841$	−28.7988	−0.993063
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.14314	0.314533
$846$	0	0
$847$	2.32256	0.0798040
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.98578	0.205190
$852$	0	0
$853$	49.4736	1.69394	0.846972	−	0.531637i	$-0.178423\pi$
$853$	49.4736	1.69394	0.846972	+	0.531637i	$0.178423\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.7359	−1.25487	−0.627437	−	0.778667i	$-0.715896\pi$
$857$	−36.7359	−1.25487	−0.627437	+	0.778667i	$0.715896\pi$
$858$	0	0
$859$	13.8277	0.471796	0.235898	−	0.971778i	$-0.424197\pi$
$859$	13.8277	0.471796	0.235898	+	0.971778i	$0.424197\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.43337	−0.116873	−0.0584366	−	0.998291i	$-0.518612\pi$
$863$	−3.43337	−0.116873	−0.0584366	+	0.998291i	$0.518612\pi$
$864$	0	0
$865$	18.9314	0.643686
$866$	0	0
$867$	0	0
$868$	0	0
$869$	44.5987	1.51291
$870$	0	0
$871$	−28.6365	−0.970311
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.46569	0.150968
$876$	0	0
$877$	27.9838	0.944947	0.472474	−	0.881345i	$-0.343361\pi$
$877$	27.9838	0.944947	0.472474	+	0.881345i	$0.343361\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.77108	0.0933600	0.0466800	−	0.998910i	$-0.485136\pi$
$881$	2.77108	0.0933600	0.0466800	+	0.998910i	$0.485136\pi$
$882$	0	0
$883$	42.9052	1.44387	0.721937	−	0.691958i	$-0.243252\pi$
$883$	42.9052	1.44387	0.721937	+	0.691958i	$0.243252\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−54.1991	−1.81983	−0.909914	−	0.414798i	$-0.863852\pi$
$887$	−54.1991	−1.81983	−0.909914	+	0.414798i	$0.863852\pi$
$888$	0	0
$889$	62.2758	2.08866
$890$	0	0
$891$	0	0
$892$	0	0
$893$	38.7598	1.29705
$894$	0	0
$895$	18.7598	0.627072
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.31175	0.0771012
$900$	0	0
$901$	16.9709	0.565384
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.1964	−0.704591
$906$	0	0
$907$	−26.4003	−0.876606	−0.438303	−	0.898827i	$-0.644420\pi$
$907$	−26.4003	−0.876606	−0.438303	+	0.898827i	$0.644420\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38.5662	1.27775	0.638877	−	0.769309i	$-0.279399\pi$
$911$	38.5662	1.27775	0.638877	+	0.769309i	$0.279399\pi$
$912$	0	0
$913$	−44.2194	−1.46345
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−61.9069	−2.04435
$918$	0	0
$919$	−7.56345	−0.249495	−0.124748	−	0.992189i	$-0.539812\pi$
$919$	−7.56345	−0.249495	−0.124748	+	0.992189i	$0.539812\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.04242	0.165973
$924$	0	0
$925$	−5.98578	−0.196811
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.5329	0.936135	0.468067	−	0.883693i	$-0.344950\pi$
$929$	28.5329	0.936135	0.468067	+	0.883693i	$0.344950\pi$
$930$	0	0
$931$	−80.5761	−2.64078
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−22.2257	−0.726859
$936$	0	0
$937$	−37.9535	−1.23989	−0.619944	−	0.784646i	$-0.712845\pi$
$937$	−37.9535	−1.23989	−0.619944	+	0.784646i	$0.712845\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.79633	0.188955	0.0944775	−	0.995527i	$-0.469882\pi$
$941$	5.79633	0.188955	0.0944775	+	0.995527i	$0.469882\pi$
$942$	0	0
$943$	3.07157	0.100024
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.1588	1.11001	0.555007	−	0.831846i	$-0.312716\pi$
$947$	34.1588	1.11001	0.555007	+	0.831846i	$0.312716\pi$
$948$	0	0
$949$	40.9657	1.32980
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.3110	0.722724	0.361362	−	0.932426i	$-0.382312\pi$
$953$	22.3110	0.722724	0.361362	+	0.932426i	$0.382312\pi$
$954$	0	0
$955$	−14.3909	−0.465680
$956$	0	0
$957$	0	0
$958$	0	0
$959$	64.3422	2.07772
$960$	0	0
$961$	−4.43453	−0.143049
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.0966	0.357212
$966$	0	0
$967$	21.5282	0.692302	0.346151	−	0.938179i	$-0.387489\pi$
$967$	21.5282	0.692302	0.346151	+	0.938179i	$0.387489\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.0939	0.869486	0.434743	−	0.900555i	$-0.356839\pi$
$971$	27.0939	0.869486	0.434743	+	0.900555i	$0.356839\pi$
$972$	0	0
$973$	−45.1660	−1.44796
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.9880	0.895416	0.447708	−	0.894180i	$-0.352240\pi$
$977$	27.9880	0.895416	0.447708	+	0.894180i	$0.352240\pi$
$978$	0	0
$979$	39.4393	1.26048
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.2711	−0.678443	−0.339222	−	0.940706i	$-0.610164\pi$
$983$	−21.2711	−0.678443	−0.339222	+	0.940706i	$0.610164\pi$
$984$	0	0
$985$	−14.1652	−0.451341
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.47991	0.142453
$990$	0	0
$991$	51.7023	1.64238	0.821189	−	0.570656i	$-0.193311\pi$
$991$	51.7023	1.64238	0.821189	+	0.570656i	$0.193311\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.66004	0.274542
$996$	0	0
$997$	27.8687	0.882610	0.441305	−	0.897357i	$-0.354516\pi$
$997$	27.8687	0.882610	0.441305	+	0.897357i	$0.354516\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bq.1.4		4
3.2	odd	2		2760.2.a.v.1.4	✓	4
12.11	even	2		5520.2.a.cb.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.v.1.4	✓	4	3.2	odd	2
5520.2.a.cb.1.1		4	12.11	even	2
8280.2.a.bq.1.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.46569 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +4.46569 q^{7} -3.39413 q^{11} -4.70565 q^{13} +6.54830 q^{17} -6.22574 q^{19} -1.00000 q^{23} +1.00000 q^{25} -0.448521 q^{29} -5.15417 q^{31} +4.46569 q^{35} -5.98578 q^{37} -3.07157 q^{41} -4.47991 q^{43} -6.22574 q^{47} +12.9424 q^{49} +2.59166 q^{53} -3.39413 q^{55} +1.55148 q^{59} -4.91421 q^{61} -4.70565 q^{65} +6.08556 q^{67} -1.07157 q^{71} -8.70565 q^{73} -15.1571 q^{77} -13.1400 q^{79} +13.0282 q^{83} +6.54830 q^{85} -11.6199 q^{89} -21.0140 q^{91} -6.22574 q^{95} +4.30834 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} - 2 q^{13} - 2 q^{17} - 6 q^{19} - 4 q^{23} + 4 q^{25} - 4 q^{29} - 6 q^{31} - 4 q^{37} - 8 q^{41} - 20 q^{43} - 6 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 26 q^{67} - 18 q^{73} - 6 q^{77} - 18 q^{79} + 26 q^{83} - 2 q^{85} - 14 q^{89} - 38 q^{91} - 6 q^{95} - 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 2 * q^13 - 2 * q^17 - 6 * q^19 - 4 * q^23 + 4 * q^25 - 4 * q^29 - 6 * q^31 - 4 * q^37 - 8 * q^41 - 20 * q^43 - 6 * q^47 + 10 * q^49 + 4 * q^53 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 26 * q^67 - 18 * q^73 - 6 * q^77 - 18 * q^79 + 26 * q^83 - 2 * q^85 - 14 * q^89 - 38 * q^91 - 6 * q^95 - 12 * q^97

Introduction

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