LMFDB - Embedded newform 5220.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1740)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +3.37228 q^{7} +1.37228 q^{11} -5.37228 q^{13} -7.37228 q^{17} +2.00000 q^{19} +1.00000 q^{25} -1.00000 q^{29} +4.74456 q^{31} +3.37228 q^{35} +8.00000 q^{37} -11.4891 q^{41} +8.00000 q^{43} +10.1168 q^{47} +4.37228 q^{49} +11.4891 q^{53} +1.37228 q^{55} +2.74456 q^{59} -0.744563 q^{61} -5.37228 q^{65} +0.627719 q^{67} +12.0000 q^{71} +8.00000 q^{73} +4.62772 q^{77} +14.0000 q^{79} +2.74456 q^{83} -7.37228 q^{85} +4.11684 q^{89} -18.1168 q^{91} +2.00000 q^{95} +5.25544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + q^{7} - 3 q^{11} - 5 q^{13} - 9 q^{17} + 4 q^{19} + 2 q^{25} - 2 q^{29} - 2 q^{31} + q^{35} + 16 q^{37} + 16 q^{43} + 3 q^{47} + 3 q^{49} - 3 q^{55} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 7 q^{67}+ \cdots + 22 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + q^7 - 3 * q^11 - 5 * q^13 - 9 * q^17 + 4 * q^19 + 2 * q^25 - 2 * q^29 - 2 * q^31 + q^35 + 16 * q^37 + 16 * q^43 + 3 * q^47 + 3 * q^49 - 3 * q^55 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 7 * q^67 + 24 * q^71 + 16 * q^73 + 15 * q^77 + 28 * q^79 - 6 * q^83 - 9 * q^85 - 9 * q^89 - 19 * q^91 + 4 * q^95 + 22 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.37228	1.27460	0.637301	−	0.770615i	$-0.280051\pi$
$7$	3.37228	1.27460	0.637301	+	0.770615i	$0.280051\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.37228	0.413758	0.206879	−	0.978366i	$-0.433669\pi$
$11$	1.37228	0.413758	0.206879	+	0.978366i	$0.433669\pi$
$12$	0	0
$13$	−5.37228	−1.49000	−0.745001	−	0.667063i	$-0.767551\pi$
$13$	−5.37228	−1.49000	−0.745001	+	0.667063i	$0.767551\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.37228	−1.78804	−0.894020	−	0.448026i	$-0.852127\pi$
$17$	−7.37228	−1.78804	−0.894020	+	0.448026i	$0.852127\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.74456	0.852149	0.426074	−	0.904688i	$-0.359896\pi$
$31$	4.74456	0.852149	0.426074	+	0.904688i	$0.359896\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.37228	0.570020
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.4891	−1.79430	−0.897150	−	0.441726i	$-0.854366\pi$
$41$	−11.4891	−1.79430	−0.897150	+	0.441726i	$0.854366\pi$
$42$	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
$47$	10.1168	1.47569	0.737847	−	0.674968i	$-0.235843\pi$
$47$	10.1168	1.47569	0.737847	+	0.674968i	$0.235843\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.4891	1.57815	0.789076	−	0.614295i	$-0.210560\pi$
$53$	11.4891	1.57815	0.789076	+	0.614295i	$0.210560\pi$
$54$	0	0
$55$	1.37228	0.185038
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.74456	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$59$	2.74456	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$60$	0	0
$61$	−0.744563	−0.0953315	−0.0476657	−	0.998863i	$-0.515178\pi$
$61$	−0.744563	−0.0953315	−0.0476657	+	0.998863i	$0.515178\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.37228	−0.666349
$66$	0	0
$67$	0.627719	0.0766880	0.0383440	−	0.999265i	$-0.487792\pi$
$67$	0.627719	0.0766880	0.0383440	+	0.999265i	$0.487792\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.62772	0.527377
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.74456	0.301255	0.150627	−	0.988591i	$-0.451871\pi$
$83$	2.74456	0.301255	0.150627	+	0.988591i	$0.451871\pi$
$84$	0	0
$85$	−7.37228	−0.799636
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.11684	0.436385	0.218192	−	0.975906i	$-0.429984\pi$
$89$	4.11684	0.436385	0.218192	+	0.975906i	$0.429984\pi$
$90$	0	0
$91$	−18.1168	−1.89916
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	5.25544	0.533609	0.266804	−	0.963751i	$-0.414032\pi$
$97$	5.25544	0.533609	0.266804	+	0.963751i	$0.414032\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.11684	−0.409641	−0.204821	−	0.978800i	$-0.565661\pi$
$101$	−4.11684	−0.409641	−0.204821	+	0.978800i	$0.565661\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.74456	−0.265327	−0.132663	−	0.991161i	$-0.542353\pi$
$107$	−2.74456	−0.265327	−0.132663	+	0.991161i	$0.542353\pi$
$108$	0	0
$109$	14.8614	1.42346	0.711732	−	0.702451i	$-0.247911\pi$
$109$	14.8614	1.42346	0.711732	+	0.702451i	$0.247911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.62772	0.435339	0.217670	−	0.976023i	$-0.430154\pi$
$113$	4.62772	0.435339	0.217670	+	0.976023i	$0.430154\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−24.8614	−2.27904
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.2337	1.44051	0.720253	−	0.693711i	$-0.244026\pi$
$127$	16.2337	1.44051	0.720253	+	0.693711i	$0.244026\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.37228	0.119897	0.0599484	−	0.998201i	$-0.480906\pi$
$131$	1.37228	0.119897	0.0599484	+	0.998201i	$0.480906\pi$
$132$	0	0
$133$	6.74456	0.584828
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.74456	−0.234484	−0.117242	−	0.993103i	$-0.537405\pi$
$137$	−2.74456	−0.234484	−0.117242	+	0.993103i	$0.537405\pi$
$138$	0	0
$139$	−2.11684	−0.179548	−0.0897742	−	0.995962i	$-0.528615\pi$
$139$	−2.11684	−0.179548	−0.0897742	+	0.995962i	$0.528615\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.37228	−0.616501
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.2337	−1.16607	−0.583035	−	0.812447i	$-0.698135\pi$
$149$	−14.2337	−1.16607	−0.583035	+	0.812447i	$0.698135\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.74456	0.381092
$156$	0	0
$157$	−18.7446	−1.49598	−0.747989	−	0.663711i	$-0.768981\pi$
$157$	−18.7446	−1.49598	−0.747989	+	0.663711i	$0.768981\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.74456	−0.528275	−0.264137	−	0.964485i	$-0.585087\pi$
$163$	−6.74456	−0.528275	−0.264137	+	0.964485i	$0.585087\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	15.8614	1.22011
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.25544	0.247506	0.123753	−	0.992313i	$-0.460507\pi$
$173$	3.25544	0.247506	0.123753	+	0.992313i	$0.460507\pi$
$174$	0	0
$175$	3.37228	0.254921
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−17.3723	−1.29127	−0.645636	−	0.763646i	$-0.723408\pi$
$181$	−17.3723	−1.29127	−0.645636	+	0.763646i	$0.723408\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	−10.1168	−0.739817
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−14.2337	−1.02991	−0.514957	−	0.857216i	$-0.672192\pi$
$191$	−14.2337	−1.02991	−0.514957	+	0.857216i	$0.672192\pi$
$192$	0	0
$193$	−12.2337	−0.880600	−0.440300	−	0.897851i	$-0.645128\pi$
$193$	−12.2337	−0.880600	−0.440300	+	0.897851i	$0.645128\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.4891	1.67353	0.836765	−	0.547561i	$-0.184444\pi$
$197$	23.4891	1.67353	0.836765	+	0.547561i	$0.184444\pi$
$198$	0	0
$199$	−11.3723	−0.806160	−0.403080	−	0.915165i	$-0.632060\pi$
$199$	−11.3723	−0.806160	−0.403080	+	0.915165i	$0.632060\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.37228	−0.236688
$204$	0	0
$205$	−11.4891	−0.802435
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.74456	0.189845
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	39.6060	2.66419
$222$	0	0
$223$	18.1168	1.21319	0.606597	−	0.795010i	$-0.292534\pi$
$223$	18.1168	1.21319	0.606597	+	0.795010i	$0.292534\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.7446	−1.77510	−0.887549	−	0.460712i	$-0.847594\pi$
$227$	−26.7446	−1.77510	−0.887549	+	0.460712i	$0.847594\pi$
$228$	0	0
$229$	−20.9783	−1.38628	−0.693141	−	0.720802i	$-0.743774\pi$
$229$	−20.9783	−1.38628	−0.693141	+	0.720802i	$0.743774\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.510875	0.0334685	0.0167343	−	0.999860i	$-0.494673\pi$
$233$	0.510875	0.0334685	0.0167343	+	0.999860i	$0.494673\pi$
$234$	0	0
$235$	10.1168	0.659950
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.2337	1.30881	0.654404	−	0.756145i	$-0.272919\pi$
$239$	20.2337	1.30881	0.654404	+	0.756145i	$0.272919\pi$
$240$	0	0
$241$	−20.1168	−1.29584	−0.647920	−	0.761708i	$-0.724361\pi$
$241$	−20.1168	−1.29584	−0.647920	+	0.761708i	$0.724361\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.37228	0.279335
$246$	0	0
$247$	−10.7446	−0.683660
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.86141	0.433088	0.216544	−	0.976273i	$-0.430521\pi$
$251$	6.86141	0.433088	0.216544	+	0.976273i	$0.430521\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.25544	0.203069	0.101534	−	0.994832i	$-0.467625\pi$
$257$	3.25544	0.203069	0.101534	+	0.994832i	$0.467625\pi$
$258$	0	0
$259$	26.9783	1.67635
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.4891	−1.07843	−0.539213	−	0.842170i	$-0.681278\pi$
$263$	−17.4891	−1.07843	−0.539213	+	0.842170i	$0.681278\pi$
$264$	0	0
$265$	11.4891	0.705771
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.86141	0.418347	0.209174	−	0.977879i	$-0.432923\pi$
$269$	6.86141	0.418347	0.209174	+	0.977879i	$0.432923\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.37228	0.0827517
$276$	0	0
$277$	−2.62772	−0.157884	−0.0789422	−	0.996879i	$-0.525154\pi$
$277$	−2.62772	−0.157884	−0.0789422	+	0.996879i	$0.525154\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−9.48913	−0.564070	−0.282035	−	0.959404i	$-0.591009\pi$
$283$	−9.48913	−0.564070	−0.282035	+	0.959404i	$0.591009\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−38.7446	−2.28702
$288$	0	0
$289$	37.3505	2.19709
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.37228	0.430693	0.215347	−	0.976538i	$-0.430912\pi$
$293$	7.37228	0.430693	0.215347	+	0.976538i	$0.430912\pi$
$294$	0	0
$295$	2.74456	0.159795
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	26.9783	1.55500
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.744563	−0.0426335
$306$	0	0
$307$	30.9783	1.76802	0.884011	−	0.467466i	$-0.154833\pi$
$307$	30.9783	1.76802	0.884011	+	0.467466i	$0.154833\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.8614	1.06953	0.534766	−	0.845000i	$-0.320400\pi$
$311$	18.8614	1.06953	0.534766	+	0.845000i	$0.320400\pi$
$312$	0	0
$313$	29.6060	1.67343	0.836714	−	0.547640i	$-0.184474\pi$
$313$	29.6060	1.67343	0.836714	+	0.547640i	$0.184474\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.8614	1.39636	0.698178	−	0.715924i	$-0.253994\pi$
$317$	24.8614	1.39636	0.698178	+	0.715924i	$0.253994\pi$
$318$	0	0
$319$	−1.37228	−0.0768330
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14.7446	−0.820409
$324$	0	0
$325$	−5.37228	−0.298001
$326$	0	0
$327$	0	0
$328$	0	0
$329$	34.1168	1.88092
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.627719	0.0342959
$336$	0	0
$337$	−18.7446	−1.02108	−0.510541	−	0.859854i	$-0.670555\pi$
$337$	−18.7446	−1.02108	−0.510541	+	0.859854i	$0.670555\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.51087	0.352584
$342$	0	0
$343$	−8.86141	−0.478471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.4891	−0.938865	−0.469433	−	0.882968i	$-0.655542\pi$
$347$	−17.4891	−0.938865	−0.469433	+	0.882968i	$0.655542\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.2337	1.38456	0.692281	−	0.721628i	$-0.256606\pi$
$359$	26.2337	1.38456	0.692281	+	0.721628i	$0.256606\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	−32.4674	−1.69478	−0.847392	−	0.530968i	$-0.821828\pi$
$367$	−32.4674	−1.69478	−0.847392	+	0.530968i	$0.821828\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	38.7446	2.01152
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.37228	0.276687
$378$	0	0
$379$	8.51087	0.437174	0.218587	−	0.975817i	$-0.429855\pi$
$379$	8.51087	0.437174	0.218587	+	0.975817i	$0.429855\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−25.7228	−1.31437	−0.657187	−	0.753727i	$-0.728254\pi$
$383$	−25.7228	−1.31437	−0.657187	+	0.753727i	$0.728254\pi$
$384$	0	0
$385$	4.62772	0.235850
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.13859	0.260537	0.130269	−	0.991479i	$-0.458416\pi$
$389$	5.13859	0.260537	0.130269	+	0.991479i	$0.458416\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.0000	0.704416
$396$	0	0
$397$	19.4891	0.978131	0.489066	−	0.872247i	$-0.337338\pi$
$397$	19.4891	0.978131	0.489066	+	0.872247i	$0.337338\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−25.4891	−1.26970
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.9783	0.544172
$408$	0	0
$409$	−32.9783	−1.63067	−0.815335	−	0.578990i	$-0.803447\pi$
$409$	−32.9783	−1.63067	−0.815335	+	0.578990i	$0.803447\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.25544	0.455430
$414$	0	0
$415$	2.74456	0.134725
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.25544	0.452158	0.226079	−	0.974109i	$-0.427409\pi$
$419$	9.25544	0.452158	0.226079	+	0.974109i	$0.427409\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.37228	−0.357608
$426$	0	0
$427$	−2.51087	−0.121510
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−13.2554	−0.637016	−0.318508	−	0.947920i	$-0.603182\pi$
$433$	−13.2554	−0.637016	−0.318508	+	0.947920i	$0.603182\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−23.3723	−1.11550	−0.557749	−	0.830010i	$-0.688335\pi$
$439$	−23.3723	−1.11550	−0.557749	+	0.830010i	$0.688335\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.8614	0.611064	0.305532	−	0.952182i	$-0.401166\pi$
$443$	12.8614	0.611064	0.305532	+	0.952182i	$0.401166\pi$
$444$	0	0
$445$	4.11684	0.195157
$446$	0	0
$447$	0	0
$448$	0	0
$449$	6.86141	0.323810	0.161905	−	0.986806i	$-0.448236\pi$
$449$	6.86141	0.323810	0.161905	+	0.986806i	$0.448236\pi$
$450$	0	0
$451$	−15.7663	−0.742407
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.1168	−0.849331
$456$	0	0
$457$	36.1168	1.68947	0.844737	−	0.535181i	$-0.179757\pi$
$457$	36.1168	1.68947	0.844737	+	0.535181i	$0.179757\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	42.1168	1.95734	0.978668	−	0.205449i	$-0.0658654\pi$
$463$	42.1168	1.95734	0.978668	+	0.205449i	$0.0658654\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	2.11684	0.0977468
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.9783	0.504780
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−32.7446	−1.49614	−0.748069	−	0.663621i	$-0.769019\pi$
$479$	−32.7446	−1.49614	−0.748069	+	0.663621i	$0.769019\pi$
$480$	0	0
$481$	−42.9783	−1.95964
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.25544	0.238637
$486$	0	0
$487$	−26.9783	−1.22250	−0.611251	−	0.791437i	$-0.709333\pi$
$487$	−26.9783	−1.22250	−0.611251	+	0.791437i	$0.709333\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−8.74456	−0.394637	−0.197318	−	0.980339i	$-0.563223\pi$
$491$	−8.74456	−0.394637	−0.197318	+	0.980339i	$0.563223\pi$
$492$	0	0
$493$	7.37228	0.332031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40.4674	1.81521
$498$	0	0
$499$	−19.6060	−0.877684	−0.438842	−	0.898564i	$-0.644611\pi$
$499$	−19.6060	−0.877684	−0.438842	+	0.898564i	$0.644611\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.88316	−0.0839658	−0.0419829	−	0.999118i	$-0.513367\pi$
$503$	−1.88316	−0.0839658	−0.0419829	+	0.999118i	$0.513367\pi$
$504$	0	0
$505$	−4.11684	−0.183197
$506$	0	0
$507$	0	0
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	26.9783	1.19345
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	13.8832	0.610581
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.74456	0.383106	0.191553	−	0.981482i	$-0.438648\pi$
$521$	8.74456	0.383106	0.191553	+	0.981482i	$0.438648\pi$
$522$	0	0
$523$	39.3723	1.72163	0.860815	−	0.508918i	$-0.169954\pi$
$523$	39.3723	1.72163	0.860815	+	0.508918i	$0.169954\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−34.9783	−1.52368
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	61.7228	2.67351
$534$	0	0
$535$	−2.74456	−0.118658
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−16.5109	−0.709858	−0.354929	−	0.934893i	$-0.615495\pi$
$541$	−16.5109	−0.709858	−0.354929	+	0.934893i	$0.615495\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14.8614	0.636593
$546$	0	0
$547$	9.88316	0.422573	0.211287	−	0.977424i	$-0.432235\pi$
$547$	9.88316	0.422573	0.211287	+	0.977424i	$0.432235\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	47.2119	2.00766
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.510875	0.0216465	0.0108232	−	0.999941i	$-0.496555\pi$
$557$	0.510875	0.0216465	0.0108232	+	0.999941i	$0.496555\pi$
$558$	0	0
$559$	−42.9783	−1.81779
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.62772	−0.195035	−0.0975175	−	0.995234i	$-0.531090\pi$
$563$	−4.62772	−0.195035	−0.0975175	+	0.995234i	$0.531090\pi$
$564$	0	0
$565$	4.62772	0.194690
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.8614	−1.79684	−0.898422	−	0.439134i	$-0.855286\pi$
$569$	−42.8614	−1.79684	−0.898422	+	0.439134i	$0.855286\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−40.0000	−1.66522	−0.832611	−	0.553858i	$-0.813155\pi$
$577$	−40.0000	−1.66522	−0.832611	+	0.553858i	$0.813155\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.25544	0.383980
$582$	0	0
$583$	15.7663	0.652974
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.2337	−1.33043	−0.665213	−	0.746653i	$-0.731659\pi$
$587$	−32.2337	−1.33043	−0.665213	+	0.746653i	$0.731659\pi$
$588$	0	0
$589$	9.48913	0.390993
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−34.4674	−1.41541	−0.707703	−	0.706510i	$-0.750268\pi$
$593$	−34.4674	−1.41541	−0.707703	+	0.706510i	$0.750268\pi$
$594$	0	0
$595$	−24.8614	−1.01922
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.6060	−0.882796	−0.441398	−	0.897311i	$-0.645517\pi$
$599$	−21.6060	−0.882796	−0.441398	+	0.897311i	$0.645517\pi$
$600$	0	0
$601$	−39.4891	−1.61080	−0.805398	−	0.592735i	$-0.798048\pi$
$601$	−39.4891	−1.61080	−0.805398	+	0.592735i	$0.798048\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.11684	−0.370652
$606$	0	0
$607$	−36.2337	−1.47068	−0.735340	−	0.677698i	$-0.762978\pi$
$607$	−36.2337	−1.47068	−0.735340	+	0.677698i	$0.762978\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−54.3505	−2.19879
$612$	0	0
$613$	12.1168	0.489395	0.244697	−	0.969600i	$-0.421311\pi$
$613$	12.1168	0.489395	0.244697	+	0.969600i	$0.421311\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.2119	−1.73965	−0.869824	−	0.493363i	$-0.835767\pi$
$617$	−43.2119	−1.73965	−0.869824	+	0.493363i	$0.835767\pi$
$618$	0	0
$619$	42.4674	1.70691	0.853454	−	0.521168i	$-0.174504\pi$
$619$	42.4674	1.70691	0.853454	+	0.521168i	$0.174504\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	13.8832	0.556217
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−58.9783	−2.35162
$630$	0	0
$631$	−16.8614	−0.671242	−0.335621	−	0.941997i	$-0.608946\pi$
$631$	−16.8614	−0.671242	−0.335621	+	0.941997i	$0.608946\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.2337	0.644214
$636$	0	0
$637$	−23.4891	−0.930673
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.60597	0.379413	0.189706	−	0.981841i	$-0.439246\pi$
$641$	9.60597	0.379413	0.189706	+	0.981841i	$0.439246\pi$
$642$	0	0
$643$	−34.3505	−1.35465	−0.677326	−	0.735683i	$-0.736861\pi$
$643$	−34.3505	−1.35465	−0.677326	+	0.735683i	$0.736861\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.48913	0.215800	0.107900	−	0.994162i	$-0.465587\pi$
$647$	5.48913	0.215800	0.107900	+	0.994162i	$0.465587\pi$
$648$	0	0
$649$	3.76631	0.147841
$650$	0	0
$651$	0	0
$652$	0	0
$653$	48.8614	1.91209	0.956047	−	0.293212i	$-0.0947243\pi$
$653$	48.8614	1.91209	0.956047	+	0.293212i	$0.0947243\pi$
$654$	0	0
$655$	1.37228	0.0536195
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.1168	1.56273	0.781365	−	0.624074i	$-0.214524\pi$
$659$	40.1168	1.56273	0.781365	+	0.624074i	$0.214524\pi$
$660$	0	0
$661$	24.1168	0.938037	0.469018	−	0.883188i	$-0.344608\pi$
$661$	24.1168	0.938037	0.469018	+	0.883188i	$0.344608\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.74456	0.261543
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.02175	−0.0394442
$672$	0	0
$673$	−16.3505	−0.630267	−0.315133	−	0.949047i	$-0.602049\pi$
$673$	−16.3505	−0.630267	−0.315133	+	0.949047i	$0.602049\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.60597	0.138589	0.0692943	−	0.997596i	$-0.477925\pi$
$677$	3.60597	0.138589	0.0692943	+	0.997596i	$0.477925\pi$
$678$	0	0
$679$	17.7228	0.680139
$680$	0	0
$681$	0	0
$682$	0	0
$683$	46.9783	1.79757	0.898786	−	0.438387i	$-0.144450\pi$
$683$	46.9783	1.79757	0.898786	+	0.438387i	$0.144450\pi$
$684$	0	0
$685$	−2.74456	−0.104864
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−61.7228	−2.35145
$690$	0	0
$691$	30.1168	1.14570	0.572849	−	0.819661i	$-0.305838\pi$
$691$	30.1168	1.14570	0.572849	+	0.819661i	$0.305838\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.11684	−0.0802965
$696$	0	0
$697$	84.7011	3.20828
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−2.23369	−0.0843652	−0.0421826	−	0.999110i	$-0.513431\pi$
$701$	−2.23369	−0.0843652	−0.0421826	+	0.999110i	$0.513431\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.8832	−0.522130
$708$	0	0
$709$	−38.4674	−1.44467	−0.722336	−	0.691542i	$-0.756932\pi$
$709$	−38.4674	−1.44467	−0.722336	+	0.691542i	$0.756932\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−7.37228	−0.275708
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.9783	−1.30447	−0.652234	−	0.758017i	$-0.726168\pi$
$719$	−34.9783	−1.30447	−0.652234	+	0.758017i	$0.726168\pi$
$720$	0	0
$721$	26.9783	1.00472
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	37.4891	1.39039	0.695197	−	0.718819i	$-0.255317\pi$
$727$	37.4891	1.39039	0.695197	+	0.718819i	$0.255317\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−58.9783	−2.18139
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.861407	0.0317303
$738$	0	0
$739$	−1.76631	−0.0649748	−0.0324874	−	0.999472i	$-0.510343\pi$
$739$	−1.76631	−0.0649748	−0.0324874	+	0.999472i	$0.510343\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.88316	−0.0690863	−0.0345431	−	0.999403i	$-0.510998\pi$
$743$	−1.88316	−0.0690863	−0.0345431	+	0.999403i	$0.510998\pi$
$744$	0	0
$745$	−14.2337	−0.521482
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.25544	−0.338186
$750$	0	0
$751$	22.2337	0.811319	0.405659	−	0.914024i	$-0.367042\pi$
$751$	22.2337	0.811319	0.405659	+	0.914024i	$0.367042\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	48.4674	1.76158	0.880788	−	0.473510i	$-0.157013\pi$
$757$	48.4674	1.76158	0.880788	+	0.473510i	$0.157013\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.4891	1.28648	0.643240	−	0.765665i	$-0.277590\pi$
$761$	35.4891	1.28648	0.643240	+	0.765665i	$0.277590\pi$
$762$	0	0
$763$	50.1168	1.81435
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.7446	−0.532395
$768$	0	0
$769$	−11.7228	−0.422735	−0.211368	−	0.977407i	$-0.567792\pi$
$769$	−11.7228	−0.422735	−0.211368	+	0.977407i	$0.567792\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−49.7228	−1.78841	−0.894203	−	0.447662i	$-0.852257\pi$
$773$	−49.7228	−1.78841	−0.894203	+	0.447662i	$0.852257\pi$
$774$	0	0
$775$	4.74456	0.170430
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−22.9783	−0.823281
$780$	0	0
$781$	16.4674	0.589249
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.7446	−0.669022
$786$	0	0
$787$	41.9565	1.49559	0.747794	−	0.663931i	$-0.231113\pi$
$787$	41.9565	1.49559	0.747794	+	0.663931i	$0.231113\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15.6060	0.554884
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.2554	−1.17797	−0.588984	−	0.808145i	$-0.700472\pi$
$797$	−33.2554	−1.17797	−0.588984	+	0.808145i	$0.700472\pi$
$798$	0	0
$799$	−74.5842	−2.63860
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.9783	0.387414
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.13859	−0.180663	−0.0903317	−	0.995912i	$-0.528793\pi$
$809$	−5.13859	−0.180663	−0.0903317	+	0.995912i	$0.528793\pi$
$810$	0	0
$811$	0.627719	0.0220422	0.0110211	−	0.999939i	$-0.496492\pi$
$811$	0.627719	0.0220422	0.0110211	+	0.999939i	$0.496492\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.74456	−0.236252
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.02175	−0.245061	−0.122530	−	0.992465i	$-0.539101\pi$
$821$	−7.02175	−0.245061	−0.122530	+	0.992465i	$0.539101\pi$
$822$	0	0
$823$	1.48913	0.0519076	0.0259538	−	0.999663i	$-0.491738\pi$
$823$	1.48913	0.0519076	0.0259538	+	0.999663i	$0.491738\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−40.4674	−1.40719	−0.703594	−	0.710602i	$-0.748423\pi$
$827$	−40.4674	−1.40719	−0.703594	+	0.710602i	$0.748423\pi$
$828$	0	0
$829$	−18.2337	−0.633282	−0.316641	−	0.948545i	$-0.602555\pi$
$829$	−18.2337	−0.633282	−0.316641	+	0.948545i	$0.602555\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−32.2337	−1.11683
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−17.1386	−0.591690	−0.295845	−	0.955236i	$-0.595601\pi$
$839$	−17.1386	−0.591690	−0.295845	+	0.955236i	$0.595601\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.8614	0.545649
$846$	0	0
$847$	−30.7446	−1.05640
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−14.9783	−0.512846	−0.256423	−	0.966565i	$-0.582544\pi$
$853$	−14.9783	−0.512846	−0.256423	+	0.966565i	$0.582544\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.4674	−0.767471	−0.383735	−	0.923443i	$-0.625363\pi$
$857$	−22.4674	−0.767471	−0.383735	+	0.923443i	$0.625363\pi$
$858$	0	0
$859$	18.4674	0.630099	0.315049	−	0.949075i	$-0.397979\pi$
$859$	18.4674	0.630099	0.315049	+	0.949075i	$0.397979\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.9783	1.19067	0.595337	−	0.803476i	$-0.297019\pi$
$863$	34.9783	1.19067	0.595337	+	0.803476i	$0.297019\pi$
$864$	0	0
$865$	3.25544	0.110688
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.2119	0.651720
$870$	0	0
$871$	−3.37228	−0.114265
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.37228	0.114004
$876$	0	0
$877$	−32.9783	−1.11360	−0.556798	−	0.830648i	$-0.687970\pi$
$877$	−32.9783	−1.11360	−0.556798	+	0.830648i	$0.687970\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−54.8614	−1.84833	−0.924164	−	0.381996i	$-0.875237\pi$
$881$	−54.8614	−1.84833	−0.924164	+	0.381996i	$0.875237\pi$
$882$	0	0
$883$	41.9565	1.41195	0.705974	−	0.708237i	$-0.250509\pi$
$883$	41.9565	1.41195	0.705974	+	0.708237i	$0.250509\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.3505	1.42199	0.710996	−	0.703196i	$-0.248244\pi$
$887$	42.3505	1.42199	0.710996	+	0.703196i	$0.248244\pi$
$888$	0	0
$889$	54.7446	1.83607
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.2337	0.677095
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.74456	−0.158240
$900$	0	0
$901$	−84.7011	−2.82180
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.3723	−0.577474
$906$	0	0
$907$	−52.7011	−1.74991	−0.874955	−	0.484204i	$-0.839109\pi$
$907$	−52.7011	−1.74991	−0.874955	+	0.484204i	$0.839109\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6.86141	0.227329	0.113664	−	0.993519i	$-0.463741\pi$
$911$	6.86141	0.227329	0.113664	+	0.993519i	$0.463741\pi$
$912$	0	0
$913$	3.76631	0.124647
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.62772	0.152821
$918$	0	0
$919$	−29.8832	−0.985754	−0.492877	−	0.870099i	$-0.664055\pi$
$919$	−29.8832	−0.985754	−0.492877	+	0.870099i	$0.664055\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−64.4674	−2.12197
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	8.74456	0.286591
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.1168	−0.330856
$936$	0	0
$937$	10.3940	0.339558	0.169779	−	0.985482i	$-0.445695\pi$
$937$	10.3940	0.339558	0.169779	+	0.985482i	$0.445695\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.25544	0.106124	0.0530621	−	0.998591i	$-0.483102\pi$
$941$	3.25544	0.106124	0.0530621	+	0.998591i	$0.483102\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.1168	0.718701	0.359350	−	0.933203i	$-0.382998\pi$
$947$	22.1168	0.718701	0.359350	+	0.933203i	$0.382998\pi$
$948$	0	0
$949$	−42.9783	−1.39513
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15.2554	0.494172	0.247086	−	0.968994i	$-0.420527\pi$
$953$	15.2554	0.494172	0.247086	+	0.968994i	$0.420527\pi$
$954$	0	0
$955$	−14.2337	−0.460591
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.25544	−0.298874
$960$	0	0
$961$	−8.48913	−0.273843
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.2337	−0.393816
$966$	0	0
$967$	−7.76631	−0.249748	−0.124874	−	0.992173i	$-0.539853\pi$
$967$	−7.76631	−0.249748	−0.124874	+	0.992173i	$0.539853\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−15.2554	−0.489570	−0.244785	−	0.969577i	$-0.578717\pi$
$971$	−15.2554	−0.489570	−0.244785	+	0.969577i	$0.578717\pi$
$972$	0	0
$973$	−7.13859	−0.228853
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.76631	−0.312452	−0.156226	−	0.987721i	$-0.549933\pi$
$977$	−9.76631	−0.312452	−0.156226	+	0.987721i	$0.549933\pi$
$978$	0	0
$979$	5.64947	0.180558
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−46.9783	−1.49837	−0.749187	−	0.662359i	$-0.769555\pi$
$983$	−46.9783	−1.49837	−0.749187	+	0.662359i	$0.769555\pi$
$984$	0	0
$985$	23.4891	0.748426
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−38.1168	−1.21082	−0.605411	−	0.795913i	$-0.706991\pi$
$991$	−38.1168	−1.21082	−0.605411	+	0.795913i	$0.706991\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.3723	−0.360526
$996$	0	0
$997$	48.4674	1.53498	0.767489	−	0.641063i	$-0.221506\pi$
$997$	48.4674	1.53498	0.767489	+	0.641063i	$0.221506\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.t.1.2		2
3.2	odd	2		1740.2.a.l.1.2	✓	2
12.11	even	2		6960.2.a.bs.1.1		2
15.2	even	4		8700.2.g.s.349.2		4
15.8	even	4		8700.2.g.s.349.3		4
15.14	odd	2		8700.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1740.2.a.l.1.2	✓	2	3.2	odd	2
5220.2.a.t.1.2		2	1.1	even	1	trivial
6960.2.a.bs.1.1		2	12.11	even	2
8700.2.a.w.1.1		2	15.14	odd	2
8700.2.g.s.349.2		4	15.2	even	4
8700.2.g.s.349.3		4	15.8	even	4

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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +3.37228 q^{7} +1.37228 q^{11} -5.37228 q^{13} -7.37228 q^{17} +2.00000 q^{19} +1.00000 q^{25} -1.00000 q^{29} +4.74456 q^{31} +3.37228 q^{35} +8.00000 q^{37} -11.4891 q^{41} +8.00000 q^{43} +10.1168 q^{47} +4.37228 q^{49} +11.4891 q^{53} +1.37228 q^{55} +2.74456 q^{59} -0.744563 q^{61} -5.37228 q^{65} +0.627719 q^{67} +12.0000 q^{71} +8.00000 q^{73} +4.62772 q^{77} +14.0000 q^{79} +2.74456 q^{83} -7.37228 q^{85} +4.11684 q^{89} -18.1168 q^{91} +2.00000 q^{95} +5.25544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + q^{7} - 3 q^{11} - 5 q^{13} - 9 q^{17} + 4 q^{19} + 2 q^{25} - 2 q^{29} - 2 q^{31} + q^{35} + 16 q^{37} + 16 q^{43} + 3 q^{47} + 3 q^{49} - 3 q^{55} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 7 q^{67}+ \cdots + 22 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + q^7 - 3 * q^11 - 5 * q^13 - 9 * q^17 + 4 * q^19 + 2 * q^25 - 2 * q^29 - 2 * q^31 + q^35 + 16 * q^37 + 16 * q^43 + 3 * q^47 + 3 * q^49 - 3 * q^55 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 7 * q^67 + 24 * q^71 + 16 * q^73 + 15 * q^77 + 28 * q^79 - 6 * q^83 - 9 * q^85 - 9 * q^89 - 19 * q^91 + 4 * q^95 + 22 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more