LMFDB - Embedded newform 5220.2.a.s.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{5} -3.37228 q^{7} -5.37228 q^{11} +1.37228 q^{13} +0.627719 q^{17} -4.74456 q^{19} +1.00000 q^{25} -1.00000 q^{29} +4.74456 q^{31} -3.37228 q^{35} +2.00000 q^{41} -6.74456 q^{43} +4.62772 q^{47} +4.37228 q^{49} +4.74456 q^{53} -5.37228 q^{55} +12.0000 q^{59} +6.00000 q^{61} +1.37228 q^{65} -8.62772 q^{67} +2.74456 q^{71} +18.1168 q^{77} -8.74456 q^{79} +12.0000 q^{83} +0.627719 q^{85} -0.116844 q^{89} -4.62772 q^{91} -4.74456 q^{95} -9.48913 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - q^{7} - 5 q^{11} - 3 q^{13} + 7 q^{17} + 2 q^{19} + 2 q^{25} - 2 q^{29} - 2 q^{31} - q^{35} + 4 q^{41} - 2 q^{43} + 15 q^{47} + 3 q^{49} - 2 q^{53} - 5 q^{55} + 24 q^{59} + 12 q^{61} - 3 q^{65}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - q^7 - 5 * q^11 - 3 * q^13 + 7 * q^17 + 2 * q^19 + 2 * q^25 - 2 * q^29 - 2 * q^31 - q^35 + 4 * q^41 - 2 * q^43 + 15 * q^47 + 3 * q^49 - 2 * q^53 - 5 * q^55 + 24 * q^59 + 12 * q^61 - 3 * q^65 - 23 * q^67 - 6 * q^71 + 19 * q^77 - 6 * q^79 + 24 * q^83 + 7 * q^85 + 17 * q^89 - 15 * q^91 + 2 * q^95 + 4 * 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	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.37228	−1.61980	−0.809902	−	0.586565i	$-0.800480\pi$
$11$	−5.37228	−1.61980	−0.809902	+	0.586565i	$0.800480\pi$
$12$	0	0
$13$	1.37228	0.380602	0.190301	−	0.981726i	$-0.439054\pi$
$13$	1.37228	0.380602	0.190301	+	0.981726i	$0.439054\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.627719	0.152244	0.0761221	−	0.997099i	$-0.475746\pi$
$17$	0.627719	0.152244	0.0761221	+	0.997099i	$0.475746\pi$
$18$	0	0
$19$	−4.74456	−1.08848	−0.544239	−	0.838930i	$-0.683181\pi$
$19$	−4.74456	−1.08848	−0.544239	+	0.838930i	$0.683181\pi$
$20$	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
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−6.74456	−1.02854	−0.514268	−	0.857629i	$-0.671936\pi$
$43$	−6.74456	−1.02854	−0.514268	+	0.857629i	$0.671936\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.62772	0.675022	0.337511	−	0.941322i	$-0.390415\pi$
$47$	4.62772	0.675022	0.337511	+	0.941322i	$0.390415\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.74456	0.651716	0.325858	−	0.945419i	$-0.394347\pi$
$53$	4.74456	0.651716	0.325858	+	0.945419i	$0.394347\pi$
$54$	0	0
$55$	−5.37228	−0.724398
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.37228	0.170211
$66$	0	0
$67$	−8.62772	−1.05404	−0.527022	−	0.849852i	$-0.676691\pi$
$67$	−8.62772	−1.05404	−0.527022	+	0.849852i	$0.676691\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.74456	0.325720	0.162860	−	0.986649i	$-0.447928\pi$
$71$	2.74456	0.325720	0.162860	+	0.986649i	$0.447928\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.1168	2.06461
$78$	0	0
$79$	−8.74456	−0.983840	−0.491920	−	0.870640i	$-0.663705\pi$
$79$	−8.74456	−0.983840	−0.491920	+	0.870640i	$0.663705\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	0.627719	0.0680856
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.116844	−0.0123854	−0.00619272	−	0.999981i	$-0.501971\pi$
$89$	−0.116844	−0.0123854	−0.00619272	+	0.999981i	$0.501971\pi$
$90$	0	0
$91$	−4.62772	−0.485117
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.74456	−0.486782
$96$	0	0
$97$	−9.48913	−0.963475	−0.481737	−	0.876316i	$-0.659994\pi$
$97$	−9.48913	−0.963475	−0.481737	+	0.876316i	$0.659994\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.1168	1.60369	0.801843	−	0.597535i	$-0.203853\pi$
$101$	16.1168	1.60369	0.801843	+	0.597535i	$0.203853\pi$
$102$	0	0
$103$	5.48913	0.540860	0.270430	−	0.962740i	$-0.412834\pi$
$103$	5.48913	0.540860	0.270430	+	0.962740i	$0.412834\pi$
$104$	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
$113$	10.1168	0.951713	0.475856	−	0.879523i	$-0.342138\pi$
$113$	10.1168	0.951713	0.475856	+	0.879523i	$0.342138\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.11684	−0.194051
$120$	0	0
$121$	17.8614	1.62376
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.25544	−0.466345	−0.233172	−	0.972435i	$-0.574911\pi$
$127$	−5.25544	−0.466345	−0.233172	+	0.972435i	$0.574911\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.6277	0.928548	0.464274	−	0.885692i	$-0.346315\pi$
$131$	10.6277	0.928548	0.464274	+	0.885692i	$0.346315\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.25544	0.449002	0.224501	−	0.974474i	$-0.427925\pi$
$137$	5.25544	0.449002	0.224501	+	0.974474i	$0.427925\pi$
$138$	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
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	9.48913	0.772214	0.386107	−	0.922454i	$-0.373820\pi$
$151$	9.48913	0.772214	0.386107	+	0.922454i	$0.373820\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.74456	0.381092
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−5.48913	−0.429941	−0.214971	−	0.976620i	$-0.568966\pi$
$163$	−5.48913	−0.429941	−0.214971	+	0.976620i	$0.568966\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−11.1168	−0.855142
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.2337	−0.778053	−0.389026	−	0.921227i	$-0.627189\pi$
$173$	−10.2337	−0.778053	−0.389026	+	0.921227i	$0.627189\pi$
$174$	0	0
$175$	−3.37228	−0.254921
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.74456	0.205138	0.102569	−	0.994726i	$-0.467294\pi$
$179$	2.74456	0.205138	0.102569	+	0.994726i	$0.467294\pi$
$180$	0	0
$181$	14.6277	1.08727	0.543635	−	0.839322i	$-0.317048\pi$
$181$	14.6277	1.08727	0.543635	+	0.839322i	$0.317048\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.37228	−0.246606
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.25544	−0.235555	−0.117778	−	0.993040i	$-0.537577\pi$
$191$	−3.25544	−0.235555	−0.117778	+	0.993040i	$0.537577\pi$
$192$	0	0
$193$	6.74456	0.485484	0.242742	−	0.970091i	$-0.421953\pi$
$193$	6.74456	0.485484	0.242742	+	0.970091i	$0.421953\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.48913	0.533578	0.266789	−	0.963755i	$-0.414037\pi$
$197$	7.48913	0.533578	0.266789	+	0.963755i	$0.414037\pi$
$198$	0	0
$199$	−24.8614	−1.76238	−0.881189	−	0.472764i	$-0.843256\pi$
$199$	−24.8614	−1.76238	−0.881189	+	0.472764i	$0.843256\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.37228	0.236688
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25.4891	1.76312
$210$	0	0
$211$	−7.48913	−0.515573	−0.257786	−	0.966202i	$-0.582993\pi$
$211$	−7.48913	−0.515573	−0.257786	+	0.966202i	$0.582993\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.74456	−0.459975
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.861407	0.0579445
$222$	0	0
$223$	24.8614	1.66484	0.832421	−	0.554143i	$-0.186954\pi$
$223$	24.8614	1.66484	0.832421	+	0.554143i	$0.186954\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.48913	0.629815	0.314908	−	0.949122i	$-0.398026\pi$
$227$	9.48913	0.629815	0.314908	+	0.949122i	$0.398026\pi$
$228$	0	0
$229$	3.48913	0.230568	0.115284	−	0.993333i	$-0.463222\pi$
$229$	3.48913	0.230568	0.115284	+	0.993333i	$0.463222\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	4.62772	0.301879
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.74456	0.436269	0.218135	−	0.975919i	$-0.430003\pi$
$239$	6.74456	0.436269	0.218135	+	0.975919i	$0.430003\pi$
$240$	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
$247$	−6.51087	−0.414277
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.6060	0.858801	0.429401	−	0.903114i	$-0.358725\pi$
$251$	13.6060	0.858801	0.429401	+	0.903114i	$0.358725\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.4891	1.46521	0.732606	−	0.680653i	$-0.238304\pi$
$257$	23.4891	1.46521	0.732606	+	0.680653i	$0.238304\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.48913	−0.585125	−0.292562	−	0.956246i	$-0.594508\pi$
$263$	−9.48913	−0.585125	−0.292562	+	0.956246i	$0.594508\pi$
$264$	0	0
$265$	4.74456	0.291456
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.3723	−0.815322	−0.407661	−	0.913133i	$-0.633656\pi$
$269$	−13.3723	−0.815322	−0.407661	+	0.913133i	$0.633656\pi$
$270$	0	0
$271$	−20.9783	−1.27434	−0.637169	−	0.770724i	$-0.719895\pi$
$271$	−20.9783	−1.27434	−0.637169	+	0.770724i	$0.719895\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.37228	−0.323961
$276$	0	0
$277$	20.1168	1.20870	0.604352	−	0.796717i	$-0.293432\pi$
$277$	20.1168	1.20870	0.604352	+	0.796717i	$0.293432\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−30.2337	−1.80359	−0.901795	−	0.432163i	$-0.857750\pi$
$281$	−30.2337	−1.80359	−0.901795	+	0.432163i	$0.857750\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.74456	−0.398119
$288$	0	0
$289$	−16.6060	−0.976822
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.3723	0.898058	0.449029	−	0.893517i	$-0.351770\pi$
$293$	15.3723	0.898058	0.449029	+	0.893517i	$0.351770\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	22.7446	1.31097
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	22.9783	1.31144	0.655719	−	0.755005i	$-0.272366\pi$
$307$	22.9783	1.31144	0.655719	+	0.755005i	$0.272366\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.3723	−0.985092	−0.492546	−	0.870286i	$-0.663934\pi$
$311$	−17.3723	−0.985092	−0.492546	+	0.870286i	$0.663934\pi$
$312$	0	0
$313$	25.3723	1.43413	0.717063	−	0.697008i	$-0.245486\pi$
$313$	25.3723	1.43413	0.717063	+	0.697008i	$0.245486\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.37228	0.189406	0.0947031	−	0.995506i	$-0.469810\pi$
$317$	3.37228	0.189406	0.0947031	+	0.995506i	$0.469810\pi$
$318$	0	0
$319$	5.37228	0.300790
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.97825	−0.165714
$324$	0	0
$325$	1.37228	0.0761205
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−15.6060	−0.860385
$330$	0	0
$331$	15.4891	0.851359	0.425680	−	0.904874i	$-0.360035\pi$
$331$	15.4891	0.851359	0.425680	+	0.904874i	$0.360035\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.62772	−0.471383
$336$	0	0
$337$	−24.2337	−1.32009	−0.660047	−	0.751225i	$-0.729463\pi$
$337$	−24.2337	−1.32009	−0.660047	+	0.751225i	$0.729463\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−25.4891	−1.38031
$342$	0	0
$343$	8.86141	0.478471
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.9783	−0.804075	−0.402037	−	0.915623i	$-0.631698\pi$
$347$	−14.9783	−0.804075	−0.402037	+	0.915623i	$0.631698\pi$
$348$	0	0
$349$	−12.9783	−0.694710	−0.347355	−	0.937734i	$-0.612920\pi$
$349$	−12.9783	−0.694710	−0.347355	+	0.937734i	$0.612920\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.4891	0.611504	0.305752	−	0.952111i	$-0.401092\pi$
$353$	11.4891	0.611504	0.305752	+	0.952111i	$0.401092\pi$
$354$	0	0
$355$	2.74456	0.145666
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.25544	−0.171815	−0.0859077	−	0.996303i	$-0.527379\pi$
$359$	−3.25544	−0.171815	−0.0859077	+	0.996303i	$0.527379\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.9783	0.573060	0.286530	−	0.958071i	$-0.407498\pi$
$367$	10.9783	0.573060	0.286530	+	0.958071i	$0.407498\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−16.0000	−0.830679
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.37228	−0.0706761
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	34.9783	1.78731	0.893653	−	0.448760i	$-0.148134\pi$
$383$	34.9783	1.78731	0.893653	+	0.448760i	$0.148134\pi$
$384$	0	0
$385$	18.1168	0.923320
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.6060	−1.70389	−0.851945	−	0.523631i	$-0.824577\pi$
$389$	−33.6060	−1.70389	−0.851945	+	0.523631i	$0.824577\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.74456	−0.439987
$396$	0	0
$397$	−20.9783	−1.05287	−0.526434	−	0.850216i	$-0.676471\pi$
$397$	−20.9783	−1.05287	−0.526434	+	0.850216i	$0.676471\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.7228	−1.18466	−0.592330	−	0.805695i	$-0.701792\pi$
$401$	−23.7228	−1.18466	−0.592330	+	0.805695i	$0.701792\pi$
$402$	0	0
$403$	6.51087	0.324330
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	3.25544	0.160971	0.0804855	−	0.996756i	$-0.474353\pi$
$409$	3.25544	0.160971	0.0804855	+	0.996756i	$0.474353\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−40.4674	−1.99127
$414$	0	0
$415$	12.0000	0.589057
$416$	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
$421$	1.76631	0.0860848	0.0430424	−	0.999073i	$-0.486295\pi$
$421$	1.76631	0.0860848	0.0430424	+	0.999073i	$0.486295\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.627719	0.0304488
$426$	0	0
$427$	−20.2337	−0.979177
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.48913	−0.0717286	−0.0358643	−	0.999357i	$-0.511418\pi$
$431$	−1.48913	−0.0717286	−0.0358643	+	0.999357i	$0.511418\pi$
$432$	0	0
$433$	−9.48913	−0.456018	−0.228009	−	0.973659i	$-0.573222\pi$
$433$	−9.48913	−0.456018	−0.228009	+	0.973659i	$0.573222\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	22.1168	1.05558	0.527790	−	0.849375i	$-0.323021\pi$
$439$	22.1168	1.05558	0.527790	+	0.849375i	$0.323021\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.3505	0.871860	0.435930	−	0.899981i	$-0.356420\pi$
$443$	18.3505	0.871860	0.435930	+	0.899981i	$0.356420\pi$
$444$	0	0
$445$	−0.116844	−0.00553894
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.1168	1.51569	0.757844	−	0.652436i	$-0.226253\pi$
$449$	32.1168	1.51569	0.757844	+	0.652436i	$0.226253\pi$
$450$	0	0
$451$	−10.7446	−0.505942
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.62772	−0.216951
$456$	0	0
$457$	37.8397	1.77006	0.885032	−	0.465530i	$-0.154136\pi$
$457$	37.8397	1.77006	0.885032	+	0.465530i	$0.154136\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.4891	−1.09400	−0.546999	−	0.837133i	$-0.684230\pi$
$461$	−23.4891	−1.09400	−0.546999	+	0.837133i	$0.684230\pi$
$462$	0	0
$463$	19.3723	0.900306	0.450153	−	0.892951i	$-0.351369\pi$
$463$	19.3723	0.900306	0.450153	+	0.892951i	$0.351369\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	29.0951	1.34349
$470$	0	0
$471$	0	0
$472$	0	0
$473$	36.2337	1.66603
$474$	0	0
$475$	−4.74456	−0.217695
$476$	0	0
$477$	0	0
$478$	0	0
$479$	39.7228	1.81498	0.907491	−	0.420072i	$-0.137995\pi$
$479$	39.7228	1.81498	0.907491	+	0.420072i	$0.137995\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.48913	−0.430879
$486$	0	0
$487$	−42.9783	−1.94753	−0.973765	−	0.227555i	$-0.926927\pi$
$487$	−42.9783	−1.94753	−0.973765	+	0.227555i	$0.926927\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.7228	−1.61215	−0.806074	−	0.591815i	$-0.798412\pi$
$491$	−35.7228	−1.61215	−0.806074	+	0.591815i	$0.798412\pi$
$492$	0	0
$493$	−0.627719	−0.0282710
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.25544	−0.415163
$498$	0	0
$499$	34.3505	1.53774	0.768870	−	0.639405i	$-0.220819\pi$
$499$	34.3505	1.53774	0.768870	+	0.639405i	$0.220819\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22.1168	0.986141	0.493071	−	0.869989i	$-0.335874\pi$
$503$	22.1168	0.986141	0.493071	+	0.869989i	$0.335874\pi$
$504$	0	0
$505$	16.1168	0.717190
$506$	0	0
$507$	0	0
$508$	0	0
$509$	34.2337	1.51738	0.758691	−	0.651451i	$-0.225839\pi$
$509$	34.2337	1.51738	0.758691	+	0.651451i	$0.225839\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.48913	0.241880
$516$	0	0
$517$	−24.8614	−1.09340
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.9783	0.568587	0.284294	−	0.958737i	$-0.408241\pi$
$521$	12.9783	0.568587	0.284294	+	0.958737i	$0.408241\pi$
$522$	0	0
$523$	−12.8614	−0.562390	−0.281195	−	0.959651i	$-0.590731\pi$
$523$	−12.8614	−0.562390	−0.281195	+	0.959651i	$0.590731\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.97825	0.129735
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.74456	0.118880
$534$	0	0
$535$	−2.74456	−0.118658
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−23.4891	−1.01175
$540$	0	0
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	14.8614	0.636593
$546$	0	0
$547$	3.13859	0.134197	0.0670983	−	0.997746i	$-0.478626\pi$
$547$	3.13859	0.134197	0.0670983	+	0.997746i	$0.478626\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.74456	0.202125
$552$	0	0
$553$	29.4891	1.25401
$554$	0	0
$555$	0	0
$556$	0	0
$557$	23.2554	0.985365	0.492682	−	0.870209i	$-0.336016\pi$
$557$	23.2554	0.985365	0.492682	+	0.870209i	$0.336016\pi$
$558$	0	0
$559$	−9.25544	−0.391463
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.3505	1.27912	0.639561	−	0.768740i	$-0.279116\pi$
$563$	30.3505	1.27912	0.639561	+	0.768740i	$0.279116\pi$
$564$	0	0
$565$	10.1168	0.425619
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−22.6277	−0.948603	−0.474302	−	0.880362i	$-0.657299\pi$
$569$	−22.6277	−0.948603	−0.474302	+	0.880362i	$0.657299\pi$
$570$	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
$577$	−2.51087	−0.104529	−0.0522645	−	0.998633i	$-0.516644\pi$
$577$	−2.51087	−0.104529	−0.0522645	+	0.998633i	$0.516644\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−40.4674	−1.67887
$582$	0	0
$583$	−25.4891	−1.05565
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.9783	−0.948414	−0.474207	−	0.880413i	$-0.657265\pi$
$587$	−22.9783	−0.948414	−0.474207	+	0.880413i	$0.657265\pi$
$588$	0	0
$589$	−22.5109	−0.927544
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.7446	1.18040	0.590199	−	0.807258i	$-0.299049\pi$
$593$	28.7446	1.18040	0.590199	+	0.807258i	$0.299049\pi$
$594$	0	0
$595$	−2.11684	−0.0867821
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.60597	0.392489	0.196245	−	0.980555i	$-0.437125\pi$
$599$	9.60597	0.392489	0.196245	+	0.980555i	$0.437125\pi$
$600$	0	0
$601$	3.48913	0.142324	0.0711622	−	0.997465i	$-0.477329\pi$
$601$	3.48913	0.142324	0.0711622	+	0.997465i	$0.477329\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	17.8614	0.726169
$606$	0	0
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.35053	0.256915
$612$	0	0
$613$	−21.6060	−0.872657	−0.436328	−	0.899787i	$-0.643721\pi$
$613$	−21.6060	−0.872657	−0.436328	+	0.899787i	$0.643721\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.7446	−1.07670	−0.538348	−	0.842723i	$-0.680951\pi$
$617$	−26.7446	−1.07670	−0.538348	+	0.842723i	$0.680951\pi$
$618$	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
$623$	0.394031	0.0157865
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−48.8614	−1.94514	−0.972571	−	0.232606i	$-0.925275\pi$
$631$	−48.8614	−1.94514	−0.972571	+	0.232606i	$0.925275\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.25544	−0.208556
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	0	0
$640$	0	0
$641$	32.3505	1.27777	0.638885	−	0.769303i	$-0.279396\pi$
$641$	32.3505	1.27777	0.638885	+	0.769303i	$0.279396\pi$
$642$	0	0
$643$	28.8614	1.13818	0.569091	−	0.822274i	$-0.307295\pi$
$643$	28.8614	1.13818	0.569091	+	0.822274i	$0.307295\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.2337	1.73901	0.869503	−	0.493928i	$-0.164439\pi$
$647$	44.2337	1.73901	0.869503	+	0.493928i	$0.164439\pi$
$648$	0	0
$649$	−64.4674	−2.53057
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.8832	0.543290	0.271645	−	0.962398i	$-0.412432\pi$
$653$	13.8832	0.543290	0.271645	+	0.962398i	$0.412432\pi$
$654$	0	0
$655$	10.6277	0.415259
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.37228	0.0534565	0.0267282	−	0.999643i	$-0.491491\pi$
$659$	1.37228	0.0534565	0.0267282	+	0.999643i	$0.491491\pi$
$660$	0	0
$661$	5.60597	0.218047	0.109023	−	0.994039i	$-0.465228\pi$
$661$	5.60597	0.218047	0.109023	+	0.994039i	$0.465228\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−32.2337	−1.24437
$672$	0	0
$673$	30.8614	1.18962	0.594810	−	0.803866i	$-0.297227\pi$
$673$	30.8614	1.18962	0.594810	+	0.803866i	$0.297227\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.3505	−1.62766	−0.813832	−	0.581101i	$-0.802622\pi$
$677$	−42.3505	−1.62766	−0.813832	+	0.581101i	$0.802622\pi$
$678$	0	0
$679$	32.0000	1.22805
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.48913	0.0569798	0.0284899	−	0.999594i	$-0.490930\pi$
$683$	1.48913	0.0569798	0.0284899	+	0.999594i	$0.490930\pi$
$684$	0	0
$685$	5.25544	0.200800
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.51087	0.248045
$690$	0	0
$691$	11.6060	0.441512	0.220756	−	0.975329i	$-0.429148\pi$
$691$	11.6060	0.441512	0.220756	+	0.975329i	$0.429148\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.11684	−0.0802965
$696$	0	0
$697$	1.25544	0.0475531
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−16.5109	−0.623607	−0.311804	−	0.950147i	$-0.600933\pi$
$701$	−16.5109	−0.623607	−0.311804	+	0.950147i	$0.600933\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−54.3505	−2.04406
$708$	0	0
$709$	−16.5109	−0.620079	−0.310039	−	0.950724i	$-0.600342\pi$
$709$	−16.5109	−0.620079	−0.310039	+	0.950724i	$0.600342\pi$
$710$	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
$719$	1.25544	0.0468199	0.0234100	−	0.999726i	$-0.492548\pi$
$719$	1.25544	0.0468199	0.0234100	+	0.999726i	$0.492548\pi$
$720$	0	0
$721$	−18.5109	−0.689381
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	13.4891	0.500284	0.250142	−	0.968209i	$-0.419523\pi$
$727$	13.4891	0.500284	0.250142	+	0.968209i	$0.419523\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.23369	−0.156589
$732$	0	0
$733$	13.2554	0.489601	0.244800	−	0.969574i	$-0.421278\pi$
$733$	13.2554	0.489601	0.244800	+	0.969574i	$0.421278\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.3505	1.70734
$738$	0	0
$739$	4.97825	0.183128	0.0915640	−	0.995799i	$-0.470813\pi$
$739$	4.97825	0.183128	0.0915640	+	0.995799i	$0.470813\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.8614	−1.35231	−0.676157	−	0.736758i	$-0.736356\pi$
$743$	−36.8614	−1.35231	−0.676157	+	0.736758i	$0.736356\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.25544	0.338186
$750$	0	0
$751$	−23.2554	−0.848603	−0.424301	−	0.905521i	$-0.639480\pi$
$751$	−23.2554	−0.848603	−0.424301	+	0.905521i	$0.639480\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.48913	0.345345
$756$	0	0
$757$	−13.4891	−0.490271	−0.245135	−	0.969489i	$-0.578832\pi$
$757$	−13.4891	−0.490271	−0.245135	+	0.969489i	$0.578832\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.72281	0.279952	0.139976	−	0.990155i	$-0.455297\pi$
$761$	7.72281	0.279952	0.139976	+	0.990155i	$0.455297\pi$
$762$	0	0
$763$	−50.1168	−1.81435
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.4674	0.594602
$768$	0	0
$769$	−23.4891	−0.847040	−0.423520	−	0.905887i	$-0.639206\pi$
$769$	−23.4891	−0.847040	−0.423520	+	0.905887i	$0.639206\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17.2554	−0.620635	−0.310317	−	0.950633i	$-0.600435\pi$
$773$	−17.2554	−0.620635	−0.310317	+	0.950633i	$0.600435\pi$
$774$	0	0
$775$	4.74456	0.170430
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.48913	−0.339983
$780$	0	0
$781$	−14.7446	−0.527602
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−34.1168	−1.21306
$792$	0	0
$793$	8.23369	0.292387
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.7446	0.805654	0.402827	−	0.915276i	$-0.368028\pi$
$797$	22.7446	0.805654	0.402827	+	0.915276i	$0.368028\pi$
$798$	0	0
$799$	2.90491	0.102768
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.6277	0.795548	0.397774	−	0.917483i	$-0.369783\pi$
$809$	22.6277	0.795548	0.397774	+	0.917483i	$0.369783\pi$
$810$	0	0
$811$	−44.8614	−1.57530	−0.787649	−	0.616125i	$-0.788702\pi$
$811$	−44.8614	−1.57530	−0.787649	+	0.616125i	$0.788702\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.48913	−0.192276
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.4891	0.400973	0.200487	−	0.979696i	$-0.435748\pi$
$821$	11.4891	0.400973	0.200487	+	0.979696i	$0.435748\pi$
$822$	0	0
$823$	−51.2119	−1.78514	−0.892568	−	0.450913i	$-0.851099\pi$
$823$	−51.2119	−1.78514	−0.892568	+	0.450913i	$0.851099\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.9783	−0.659938	−0.329969	−	0.943992i	$-0.607038\pi$
$827$	−18.9783	−0.659938	−0.329969	+	0.943992i	$0.607038\pi$
$828$	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
$833$	2.74456	0.0950935
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.8614	−1.20355	−0.601775	−	0.798666i	$-0.705539\pi$
$839$	−34.8614	−1.20355	−0.601775	+	0.798666i	$0.705539\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.1168	−0.382431
$846$	0	0
$847$	−60.2337	−2.06965
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−16.2337	−0.555831	−0.277915	−	0.960606i	$-0.589644\pi$
$853$	−16.2337	−0.555831	−0.277915	+	0.960606i	$0.589644\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.4674	−1.31402	−0.657010	−	0.753882i	$-0.728179\pi$
$857$	−38.4674	−1.31402	−0.657010	+	0.753882i	$0.728179\pi$
$858$	0	0
$859$	46.2337	1.57747	0.788737	−	0.614731i	$-0.210736\pi$
$859$	46.2337	1.57747	0.788737	+	0.614731i	$0.210736\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	55.2119	1.87944	0.939718	−	0.341950i	$-0.111087\pi$
$863$	55.2119	1.87944	0.939718	+	0.341950i	$0.111087\pi$
$864$	0	0
$865$	−10.2337	−0.347956
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46.9783	1.59363
$870$	0	0
$871$	−11.8397	−0.401171
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.37228	−0.114004
$876$	0	0
$877$	26.0000	0.877958	0.438979	−	0.898497i	$-0.355340\pi$
$877$	26.0000	0.877958	0.438979	+	0.898497i	$0.355340\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	13.3723	0.450524	0.225262	−	0.974298i	$-0.427676\pi$
$881$	13.3723	0.450524	0.225262	+	0.974298i	$0.427676\pi$
$882$	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
$887$	−22.1168	−0.742611	−0.371305	−	0.928511i	$-0.621090\pi$
$887$	−22.1168	−0.742611	−0.371305	+	0.928511i	$0.621090\pi$
$888$	0	0
$889$	17.7228	0.594404
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.9565	−0.734746
$894$	0	0
$895$	2.74456	0.0917406
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.74456	−0.158240
$900$	0	0
$901$	2.97825	0.0992199
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.6277	0.486242
$906$	0	0
$907$	56.4674	1.87497	0.937484	−	0.348028i	$-0.113149\pi$
$907$	56.4674	1.87497	0.937484	+	0.348028i	$0.113149\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.1168	0.533975	0.266987	−	0.963700i	$-0.413972\pi$
$911$	16.1168	0.533975	0.266987	+	0.963700i	$0.413972\pi$
$912$	0	0
$913$	−64.4674	−2.13356
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−35.8397	−1.18353
$918$	0	0
$919$	29.0951	0.959759	0.479879	−	0.877335i	$-0.340680\pi$
$919$	29.0951	0.959759	0.479879	+	0.877335i	$0.340680\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.76631	0.123970
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.9783	0.425803	0.212901	−	0.977074i	$-0.431709\pi$
$929$	12.9783	0.425803	0.212901	+	0.977074i	$0.431709\pi$
$930$	0	0
$931$	−20.7446	−0.679876
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.37228	−0.110285
$936$	0	0
$937$	−6.39403	−0.208884	−0.104442	−	0.994531i	$-0.533306\pi$
$937$	−6.39403	−0.208884	−0.104442	+	0.994531i	$0.533306\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.3940	−0.662717	−0.331358	−	0.943505i	$-0.607507\pi$
$947$	−20.3940	−0.662717	−0.331358	+	0.943505i	$0.607507\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−23.4891	−0.760887	−0.380444	−	0.924804i	$-0.624229\pi$
$953$	−23.4891	−0.760887	−0.380444	+	0.924804i	$0.624229\pi$
$954$	0	0
$955$	−3.25544	−0.105343
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−17.7228	−0.572299
$960$	0	0
$961$	−8.48913	−0.273843
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.74456	0.217115
$966$	0	0
$967$	−44.4674	−1.42997	−0.714987	−	0.699138i	$-0.753568\pi$
$967$	−44.4674	−1.42997	−0.714987	+	0.699138i	$0.753568\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.7446	0.537359	0.268679	−	0.963230i	$-0.413413\pi$
$971$	16.7446	0.537359	0.268679	+	0.963230i	$0.413413\pi$
$972$	0	0
$973$	7.13859	0.228853
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.7228	0.630989	0.315494	−	0.948927i	$-0.397830\pi$
$977$	19.7228	0.630989	0.315494	+	0.948927i	$0.397830\pi$
$978$	0	0
$979$	0.627719	0.0200620
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.4891	−1.32330	−0.661649	−	0.749814i	$-0.730143\pi$
$983$	−41.4891	−1.32330	−0.661649	+	0.749814i	$0.730143\pi$
$984$	0	0
$985$	7.48913	0.238623
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	20.8614	0.662684	0.331342	−	0.943511i	$-0.392499\pi$
$991$	20.8614	0.662684	0.331342	+	0.943511i	$0.392499\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24.8614	−0.788160
$996$	0	0
$997$	−53.9565	−1.70882	−0.854410	−	0.519600i	$-0.826081\pi$
$997$	−53.9565	−1.70882	−0.854410	+	0.519600i	$0.826081\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.37228 q^{7} -5.37228 q^{11} +1.37228 q^{13} +0.627719 q^{17} -4.74456 q^{19} +1.00000 q^{25} -1.00000 q^{29} +4.74456 q^{31} -3.37228 q^{35} +2.00000 q^{41} -6.74456 q^{43} +4.62772 q^{47} +4.37228 q^{49} +4.74456 q^{53} -5.37228 q^{55} +12.0000 q^{59} +6.00000 q^{61} +1.37228 q^{65} -8.62772 q^{67} +2.74456 q^{71} +18.1168 q^{77} -8.74456 q^{79} +12.0000 q^{83} +0.627719 q^{85} -0.116844 q^{89} -4.62772 q^{91} -4.74456 q^{95} -9.48913 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - q^{7} - 5 q^{11} - 3 q^{13} + 7 q^{17} + 2 q^{19} + 2 q^{25} - 2 q^{29} - 2 q^{31} - q^{35} + 4 q^{41} - 2 q^{43} + 15 q^{47} + 3 q^{49} - 2 q^{53} - 5 q^{55} + 24 q^{59} + 12 q^{61} - 3 q^{65}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - q^7 - 5 * q^11 - 3 * q^13 + 7 * q^17 + 2 * q^19 + 2 * q^25 - 2 * q^29 - 2 * q^31 - q^35 + 4 * q^41 - 2 * q^43 + 15 * q^47 + 3 * q^49 - 2 * q^53 - 5 * q^55 + 24 * q^59 + 12 * q^61 - 3 * q^65 - 23 * q^67 - 6 * q^71 + 19 * q^77 - 6 * q^79 + 24 * q^83 + 7 * q^85 + 17 * q^89 - 15 * q^91 + 2 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more