LMFDB - Embedded newform 7920.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7920,2,Mod(1,7920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, 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("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.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:	$63.2415184009$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1320)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.82843 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{11} +4.82843 q^{13} +0.828427 q^{17} +5.65685 q^{19} +5.65685 q^{23} +1.00000 q^{25} -2.00000 q^{29} -5.65685 q^{31} -2.82843 q^{35} +6.00000 q^{37} +2.00000 q^{41} -1.17157 q^{43} +5.65685 q^{47} +1.00000 q^{49} -0.343146 q^{53} +1.00000 q^{55} +9.65685 q^{59} +0.343146 q^{61} -4.82843 q^{65} -9.65685 q^{67} -13.6569 q^{71} +14.4853 q^{73} -2.82843 q^{77} -9.65685 q^{79} +10.8284 q^{83} -0.828427 q^{85} -2.00000 q^{89} +13.6569 q^{91} -5.65685 q^{95} +4.34315 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} - 2 q^{11} + 4 q^{13} - 4 q^{17} + 2 q^{25} - 4 q^{29} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 2 q^{49} - 12 q^{53} + 2 q^{55} + 8 q^{59} + 12 q^{61} - 4 q^{65} - 8 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{79} + 16 q^{83} + 4 q^{85} - 4 q^{89} + 16 q^{91} + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^11 + 4 * q^13 - 4 * q^17 + 2 * q^25 - 4 * q^29 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 2 * q^49 - 12 * q^53 + 2 * q^55 + 8 * q^59 + 12 * q^61 - 4 * q^65 - 8 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^79 + 16 * q^83 + 4 * q^85 - 4 * q^89 + 16 * q^91 + 20 * 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
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.65685	1.17954	0.589768	−	0.807573i	$-0.299219\pi$
$23$	5.65685	1.17954	0.589768	+	0.807573i	$0.299219\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	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
$43$	−1.17157	−0.178663	−0.0893316	−	0.996002i	$-0.528473\pi$
$43$	−1.17157	−0.178663	−0.0893316	+	0.996002i	$0.528473\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685	0.825137	0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685	0.825137	0.412568	+	0.910927i	$0.364632\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.343146	−0.0471347	−0.0235673	−	0.999722i	$-0.507502\pi$
$53$	−0.343146	−0.0471347	−0.0235673	+	0.999722i	$0.507502\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.65685	1.25722	0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685	1.25722	0.628608	+	0.777723i	$0.283625\pi$
$60$	0	0
$61$	0.343146	0.0439353	0.0219677	−	0.999759i	$-0.493007\pi$
$61$	0.343146	0.0439353	0.0219677	+	0.999759i	$0.493007\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.82843	−0.598893
$66$	0	0
$67$	−9.65685	−1.17977	−0.589886	−	0.807486i	$-0.700827\pi$
$67$	−9.65685	−1.17977	−0.589886	+	0.807486i	$0.700827\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.6569	−1.62077	−0.810385	−	0.585897i	$-0.800742\pi$
$71$	−13.6569	−1.62077	−0.810385	+	0.585897i	$0.800742\pi$
$72$	0	0
$73$	14.4853	1.69537	0.847687	−	0.530497i	$-0.177995\pi$
$73$	14.4853	1.69537	0.847687	+	0.530497i	$0.177995\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	−9.65685	−1.08648	−0.543240	−	0.839577i	$-0.682803\pi$
$79$	−9.65685	−1.08648	−0.543240	+	0.839577i	$0.682803\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.8284	1.18857	0.594287	−	0.804253i	$-0.297434\pi$
$83$	10.8284	1.18857	0.594287	+	0.804253i	$0.297434\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	13.6569	1.43163
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	4.34315	0.440980	0.220490	−	0.975389i	$-0.429234\pi$
$97$	4.34315	0.440980	0.220490	+	0.975389i	$0.429234\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	0	0
$103$	3.31371	0.326509	0.163255	−	0.986584i	$-0.447801\pi$
$103$	3.31371	0.326509	0.163255	+	0.986584i	$0.447801\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.48528	−0.820303	−0.410152	−	0.912017i	$-0.634524\pi$
$107$	−8.48528	−0.820303	−0.410152	+	0.912017i	$0.634524\pi$
$108$	0	0
$109$	−5.31371	−0.508961	−0.254480	−	0.967078i	$-0.581904\pi$
$109$	−5.31371	−0.508961	−0.254480	+	0.967078i	$0.581904\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.3137	−1.25245	−0.626224	−	0.779643i	$-0.715401\pi$
$113$	−13.3137	−1.25245	−0.626224	+	0.779643i	$0.715401\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.34315	−0.554203	−0.277102	−	0.960841i	$-0.589374\pi$
$131$	−6.34315	−0.554203	−0.277102	+	0.960841i	$0.589374\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	0	0
$139$	−11.3137	−0.959616	−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137	−0.959616	−0.479808	+	0.877373i	$0.659294\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.82843	−0.403773
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.343146	0.0281116	0.0140558	−	0.999901i	$-0.495526\pi$
$149$	0.343146	0.0281116	0.0140558	+	0.999901i	$0.495526\pi$
$150$	0	0
$151$	−12.9706	−1.05553	−0.527765	−	0.849391i	$-0.676970\pi$
$151$	−12.9706	−1.05553	−0.527765	+	0.849391i	$0.676970\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.65685	0.454369
$156$	0	0
$157$	22.9706	1.83325	0.916625	−	0.399748i	$-0.130902\pi$
$157$	22.9706	1.83325	0.916625	+	0.399748i	$0.130902\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.1421	−1.40388	−0.701940	−	0.712236i	$-0.747683\pi$
$167$	−18.1421	−1.40388	−0.701940	+	0.712236i	$0.747683\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.51472	0.115162	0.0575810	−	0.998341i	$-0.481661\pi$
$173$	1.51472	0.115162	0.0575810	+	0.998341i	$0.481661\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	9.31371	0.692283	0.346141	−	0.938182i	$-0.387492\pi$
$181$	9.31371	0.692283	0.346141	+	0.938182i	$0.387492\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−0.828427	−0.0605806
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	3.17157	0.228295	0.114147	−	0.993464i	$-0.463586\pi$
$193$	3.17157	0.228295	0.114147	+	0.993464i	$0.463586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.8284	1.48396	0.741982	−	0.670420i	$-0.233886\pi$
$197$	20.8284	1.48396	0.741982	+	0.670420i	$0.233886\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.65685	−0.397033
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	16.9706	1.16830	0.584151	−	0.811645i	$-0.301428\pi$
$211$	16.9706	1.16830	0.584151	+	0.811645i	$0.301428\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.17157	0.0799006
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	24.9706	1.67215	0.836076	−	0.548613i	$-0.184844\pi$
$223$	24.9706	1.67215	0.836076	+	0.548613i	$0.184844\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.48528	0.563188	0.281594	−	0.959534i	$-0.409137\pi$
$227$	8.48528	0.563188	0.281594	+	0.959534i	$0.409137\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.4853	0.948962	0.474481	−	0.880266i	$-0.342636\pi$
$233$	14.4853	0.948962	0.474481	+	0.880266i	$0.342636\pi$
$234$	0	0
$235$	−5.65685	−0.369012
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.65685	0.365911	0.182956	−	0.983121i	$-0.441433\pi$
$239$	5.65685	0.365911	0.182956	+	0.983121i	$0.441433\pi$
$240$	0	0
$241$	−3.65685	−0.235559	−0.117779	−	0.993040i	$-0.537578\pi$
$241$	−3.65685	−0.235559	−0.117779	+	0.993040i	$0.537578\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	27.3137	1.73793
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.9706	0.818695	0.409347	−	0.912379i	$-0.365756\pi$
$251$	12.9706	0.818695	0.409347	+	0.912379i	$0.365756\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	30.9706	1.93189	0.965945	−	0.258746i	$-0.0833094\pi$
$257$	30.9706	1.93189	0.965945	+	0.258746i	$0.0833094\pi$
$258$	0	0
$259$	16.9706	1.05450
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.17157	−0.565543	−0.282772	−	0.959187i	$-0.591254\pi$
$263$	−9.17157	−0.565543	−0.282772	+	0.959187i	$0.591254\pi$
$264$	0	0
$265$	0.343146	0.0210793
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.68629	0.407670	0.203835	−	0.979005i	$-0.434659\pi$
$269$	6.68629	0.407670	0.203835	+	0.979005i	$0.434659\pi$
$270$	0	0
$271$	25.6569	1.55854	0.779271	−	0.626687i	$-0.215589\pi$
$271$	25.6569	1.55854	0.779271	+	0.626687i	$0.215589\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	12.8284	0.770785	0.385393	−	0.922753i	$-0.374066\pi$
$277$	12.8284	0.770785	0.385393	+	0.922753i	$0.374066\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−20.6274	−1.23053	−0.615264	−	0.788321i	$-0.710951\pi$
$281$	−20.6274	−1.23053	−0.615264	+	0.788321i	$0.710951\pi$
$282$	0	0
$283$	−28.4853	−1.69327	−0.846637	−	0.532171i	$-0.821376\pi$
$283$	−28.4853	−1.69327	−0.846637	+	0.532171i	$0.821376\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.4853	1.54729	0.773643	−	0.633621i	$-0.218432\pi$
$293$	26.4853	1.54729	0.773643	+	0.633621i	$0.218432\pi$
$294$	0	0
$295$	−9.65685	−0.562244
$296$	0	0
$297$	0	0
$298$	0	0
$299$	27.3137	1.57959
$300$	0	0
$301$	−3.31371	−0.190999
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.343146	−0.0196485
$306$	0	0
$307$	3.51472	0.200596	0.100298	−	0.994957i	$-0.468020\pi$
$307$	3.51472	0.200596	0.100298	+	0.994957i	$0.468020\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.6569	−1.22805	−0.614024	−	0.789288i	$-0.710450\pi$
$311$	−21.6569	−1.22805	−0.614024	+	0.789288i	$0.710450\pi$
$312$	0	0
$313$	−17.3137	−0.978629	−0.489314	−	0.872107i	$-0.662753\pi$
$313$	−17.3137	−0.978629	−0.489314	+	0.872107i	$0.662753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.6274	−1.15855	−0.579276	−	0.815132i	$-0.696664\pi$
$317$	−20.6274	−1.15855	−0.579276	+	0.815132i	$0.696664\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.68629	0.260752
$324$	0	0
$325$	4.82843	0.267833
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	28.9706	1.59237	0.796183	−	0.605056i	$-0.206849\pi$
$331$	28.9706	1.59237	0.796183	+	0.605056i	$0.206849\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.65685	0.527610
$336$	0	0
$337$	−32.1421	−1.75089	−0.875447	−	0.483314i	$-0.839433\pi$
$337$	−32.1421	−1.75089	−0.875447	+	0.483314i	$0.839433\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.65685	0.306336
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.51472	0.403411	0.201706	−	0.979446i	$-0.435352\pi$
$347$	7.51472	0.403411	0.201706	+	0.979446i	$0.435352\pi$
$348$	0	0
$349$	−26.9706	−1.44370	−0.721851	−	0.692049i	$-0.756708\pi$
$349$	−26.9706	−1.44370	−0.721851	+	0.692049i	$0.756708\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.02944	−0.0547914	−0.0273957	−	0.999625i	$-0.508721\pi$
$353$	−1.02944	−0.0547914	−0.0273957	+	0.999625i	$0.508721\pi$
$354$	0	0
$355$	13.6569	0.724831
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.2843	−1.49279	−0.746393	−	0.665505i	$-0.768216\pi$
$359$	−28.2843	−1.49279	−0.746393	+	0.665505i	$0.768216\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.4853	−0.758194
$366$	0	0
$367$	−24.9706	−1.30345	−0.651726	−	0.758454i	$-0.725955\pi$
$367$	−24.9706	−1.30345	−0.651726	+	0.758454i	$0.725955\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.970563	−0.0503891
$372$	0	0
$373$	18.4853	0.957132	0.478566	−	0.878052i	$-0.341157\pi$
$373$	18.4853	0.957132	0.478566	+	0.878052i	$0.341157\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.65685	−0.497353
$378$	0	0
$379$	0.686292	0.0352524	0.0176262	−	0.999845i	$-0.494389\pi$
$379$	0.686292	0.0352524	0.0176262	+	0.999845i	$0.494389\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	2.82843	0.144150
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	4.68629	0.236996
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.65685	0.485889
$396$	0	0
$397$	0.343146	0.0172220	0.00861100	−	0.999963i	$-0.497259\pi$
$397$	0.343146	0.0172220	0.00861100	+	0.999963i	$0.497259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	−27.3137	−1.36059
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−2.68629	−0.132829	−0.0664143	−	0.997792i	$-0.521156\pi$
$409$	−2.68629	−0.132829	−0.0664143	+	0.997792i	$0.521156\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	27.3137	1.34402
$414$	0	0
$415$	−10.8284	−0.531547
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.68629	0.424353	0.212177	−	0.977231i	$-0.431945\pi$
$419$	8.68629	0.424353	0.212177	+	0.977231i	$0.431945\pi$
$420$	0	0
$421$	36.6274	1.78511	0.892556	−	0.450937i	$-0.148910\pi$
$421$	36.6274	1.78511	0.892556	+	0.450937i	$0.148910\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.828427	0.0401846
$426$	0	0
$427$	0.970563	0.0469688
$428$	0	0
$429$	0	0
$430$	0	0
$431$	35.3137	1.70100	0.850501	−	0.525974i	$-0.176299\pi$
$431$	35.3137	1.70100	0.850501	+	0.525974i	$0.176299\pi$
$432$	0	0
$433$	26.9706	1.29612	0.648061	−	0.761588i	$-0.275580\pi$
$433$	26.9706	1.29612	0.648061	+	0.761588i	$0.275580\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	32.0000	1.53077
$438$	0	0
$439$	−18.6274	−0.889038	−0.444519	−	0.895769i	$-0.646625\pi$
$439$	−18.6274	−0.889038	−0.444519	+	0.895769i	$0.646625\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.9411	1.42255	0.711273	−	0.702916i	$-0.248119\pi$
$443$	29.9411	1.42255	0.711273	+	0.702916i	$0.248119\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−37.3137	−1.76094	−0.880471	−	0.474099i	$-0.842774\pi$
$449$	−37.3137	−1.76094	−0.880471	+	0.474099i	$0.842774\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13.6569	−0.640243
$456$	0	0
$457$	−3.85786	−0.180463	−0.0902316	−	0.995921i	$-0.528761\pi$
$457$	−3.85786	−0.180463	−0.0902316	+	0.995921i	$0.528761\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.34315	0.388579	0.194290	−	0.980944i	$-0.437760\pi$
$461$	8.34315	0.388579	0.194290	+	0.980944i	$0.437760\pi$
$462$	0	0
$463$	−13.6569	−0.634688	−0.317344	−	0.948311i	$-0.602791\pi$
$463$	−13.6569	−0.634688	−0.317344	+	0.948311i	$0.602791\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.9706	1.34060	0.670299	−	0.742091i	$-0.266166\pi$
$467$	28.9706	1.34060	0.670299	+	0.742091i	$0.266166\pi$
$468$	0	0
$469$	−27.3137	−1.26123
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.17157	0.0538690
$474$	0	0
$475$	5.65685	0.259554
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.3431	1.20365	0.601825	−	0.798628i	$-0.294441\pi$
$479$	26.3431	1.20365	0.601825	+	0.798628i	$0.294441\pi$
$480$	0	0
$481$	28.9706	1.32094
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.34315	−0.197212
$486$	0	0
$487$	36.2843	1.64420	0.822099	−	0.569345i	$-0.192803\pi$
$487$	36.2843	1.64420	0.822099	+	0.569345i	$0.192803\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−1.65685	−0.0746210
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−38.6274	−1.73268
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.48528	0.199989	0.0999944	−	0.994988i	$-0.468118\pi$
$503$	4.48528	0.199989	0.0999944	+	0.994988i	$0.468118\pi$
$504$	0	0
$505$	−3.65685	−0.162728
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.62742	0.382404	0.191202	−	0.981551i	$-0.438762\pi$
$509$	8.62742	0.382404	0.191202	+	0.981551i	$0.438762\pi$
$510$	0	0
$511$	40.9706	1.81243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.31371	−0.146019
$516$	0	0
$517$	−5.65685	−0.248788
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.6274	1.25419	0.627095	−	0.778943i	$-0.284244\pi$
$521$	28.6274	1.25419	0.627095	+	0.778943i	$0.284244\pi$
$522$	0	0
$523$	−37.4558	−1.63783	−0.818915	−	0.573915i	$-0.805424\pi$
$523$	−37.4558	−1.63783	−0.818915	+	0.573915i	$0.805424\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.68629	−0.204138
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.65685	0.418285
$534$	0	0
$535$	8.48528	0.366851
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.31371	0.227614
$546$	0	0
$547$	−0.201010	−0.00859457	−0.00429729	−	0.999991i	$-0.501368\pi$
$547$	−0.201010	−0.00859457	−0.00429729	+	0.999991i	$0.501368\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	−27.3137	−1.16150
$554$	0	0
$555$	0	0
$556$	0	0
$557$	25.5147	1.08109	0.540547	−	0.841314i	$-0.318217\pi$
$557$	25.5147	1.08109	0.540547	+	0.841314i	$0.318217\pi$
$558$	0	0
$559$	−5.65685	−0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17.4558	−0.735676	−0.367838	−	0.929890i	$-0.619902\pi$
$563$	−17.4558	−0.735676	−0.367838	+	0.929890i	$0.619902\pi$
$564$	0	0
$565$	13.3137	0.560112
$566$	0	0
$567$	0	0
$568$	0	0
$569$	30.2843	1.26958	0.634791	−	0.772684i	$-0.281086\pi$
$569$	30.2843	1.26958	0.634791	+	0.772684i	$0.281086\pi$
$570$	0	0
$571$	−3.31371	−0.138674	−0.0693372	−	0.997593i	$-0.522088\pi$
$571$	−3.31371	−0.138674	−0.0693372	+	0.997593i	$0.522088\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.65685	0.235907
$576$	0	0
$577$	−5.02944	−0.209378	−0.104689	−	0.994505i	$-0.533385\pi$
$577$	−5.02944	−0.209378	−0.104689	+	0.994505i	$0.533385\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	30.6274	1.27064
$582$	0	0
$583$	0.343146	0.0142116
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.3431	1.25240	0.626198	−	0.779664i	$-0.284610\pi$
$587$	30.3431	1.25240	0.626198	+	0.779664i	$0.284610\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−41.1127	−1.68830	−0.844148	−	0.536110i	$-0.819893\pi$
$593$	−41.1127	−1.68830	−0.844148	+	0.536110i	$0.819893\pi$
$594$	0	0
$595$	−2.34315	−0.0960596
$596$	0	0
$597$	0	0
$598$	0	0
$599$	38.6274	1.57827	0.789137	−	0.614218i	$-0.210528\pi$
$599$	38.6274	1.57827	0.789137	+	0.614218i	$0.210528\pi$
$600$	0	0
$601$	−35.6569	−1.45447	−0.727237	−	0.686387i	$-0.759196\pi$
$601$	−35.6569	−1.45447	−0.727237	+	0.686387i	$0.759196\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	19.7990	0.803616	0.401808	−	0.915724i	$-0.368382\pi$
$607$	19.7990	0.803616	0.401808	+	0.915724i	$0.368382\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.3137	1.10499
$612$	0	0
$613$	25.1127	1.01429	0.507146	−	0.861860i	$-0.330700\pi$
$613$	25.1127	1.01429	0.507146	+	0.861860i	$0.330700\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.627417	−0.0252589	−0.0126294	−	0.999920i	$-0.504020\pi$
$617$	−0.627417	−0.0252589	−0.0126294	+	0.999920i	$0.504020\pi$
$618$	0	0
$619$	6.34315	0.254953	0.127476	−	0.991842i	$-0.459312\pi$
$619$	6.34315	0.254953	0.127476	+	0.991842i	$0.459312\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.65685	−0.226637
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.97056	0.198189
$630$	0	0
$631$	35.3137	1.40582	0.702908	−	0.711281i	$-0.251884\pi$
$631$	35.3137	1.40582	0.702908	+	0.711281i	$0.251884\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.82843	0.112243
$636$	0	0
$637$	4.82843	0.191309
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	2.62742	0.103615	0.0518076	−	0.998657i	$-0.483502\pi$
$643$	2.62742	0.103615	0.0518076	+	0.998657i	$0.483502\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	−9.65685	−0.379065
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.0294	0.666413	0.333207	−	0.942854i	$-0.391869\pi$
$653$	17.0294	0.666413	0.333207	+	0.942854i	$0.391869\pi$
$654$	0	0
$655$	6.34315	0.247847
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.3137	−0.596537	−0.298269	−	0.954482i	$-0.596409\pi$
$659$	−15.3137	−0.596537	−0.298269	+	0.954482i	$0.596409\pi$
$660$	0	0
$661$	49.3137	1.91808	0.959040	−	0.283269i	$-0.0914189\pi$
$661$	49.3137	1.91808	0.959040	+	0.283269i	$0.0914189\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	−11.3137	−0.438069
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.343146	−0.0132470
$672$	0	0
$673$	−35.4558	−1.36672	−0.683361	−	0.730080i	$-0.739483\pi$
$673$	−35.4558	−1.36672	−0.683361	+	0.730080i	$0.739483\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.4853	0.710447	0.355224	−	0.934781i	$-0.384405\pi$
$677$	18.4853	0.710447	0.355224	+	0.934781i	$0.384405\pi$
$678$	0	0
$679$	12.2843	0.471427
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.3431	0.854937	0.427468	−	0.904030i	$-0.359406\pi$
$683$	22.3431	0.854937	0.427468	+	0.904030i	$0.359406\pi$
$684$	0	0
$685$	−17.3137	−0.661523
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.65685	−0.0631211
$690$	0	0
$691$	−7.31371	−0.278227	−0.139113	−	0.990276i	$-0.544425\pi$
$691$	−7.31371	−0.278227	−0.139113	+	0.990276i	$0.544425\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.3137	0.429153
$696$	0	0
$697$	1.65685	0.0627578
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.31371	−0.200696	−0.100348	−	0.994952i	$-0.531996\pi$
$701$	−5.31371	−0.200696	−0.100348	+	0.994952i	$0.531996\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.3431	0.388994
$708$	0	0
$709$	−22.6863	−0.852002	−0.426001	−	0.904723i	$-0.640078\pi$
$709$	−22.6863	−0.852002	−0.426001	+	0.904723i	$0.640078\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	4.82843	0.180573
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.6863	0.771468	0.385734	−	0.922610i	$-0.373948\pi$
$719$	20.6863	0.771468	0.385734	+	0.922610i	$0.373948\pi$
$720$	0	0
$721$	9.37258	0.349053
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−33.9411	−1.25881	−0.629403	−	0.777079i	$-0.716701\pi$
$727$	−33.9411	−1.25881	−0.629403	+	0.777079i	$0.716701\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.970563	−0.0358976
$732$	0	0
$733$	−13.5147	−0.499178	−0.249589	−	0.968352i	$-0.580295\pi$
$733$	−13.5147	−0.499178	−0.249589	+	0.968352i	$0.580295\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.65685	0.355715
$738$	0	0
$739$	−22.6274	−0.832363	−0.416181	−	0.909282i	$-0.636632\pi$
$739$	−22.6274	−0.832363	−0.416181	+	0.909282i	$0.636632\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.1716	−1.21695	−0.608473	−	0.793574i	$-0.708218\pi$
$743$	−33.1716	−1.21695	−0.608473	+	0.793574i	$0.708218\pi$
$744$	0	0
$745$	−0.343146	−0.0125719
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−17.9411	−0.654681	−0.327340	−	0.944906i	$-0.606152\pi$
$751$	−17.9411	−0.654681	−0.327340	+	0.944906i	$0.606152\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.9706	0.472047
$756$	0	0
$757$	−41.5980	−1.51190	−0.755952	−	0.654627i	$-0.772826\pi$
$757$	−41.5980	−1.51190	−0.755952	+	0.654627i	$0.772826\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.6569	−0.712560	−0.356280	−	0.934379i	$-0.615955\pi$
$761$	−19.6569	−0.712560	−0.356280	+	0.934379i	$0.615955\pi$
$762$	0	0
$763$	−15.0294	−0.544102
$764$	0	0
$765$	0	0
$766$	0	0
$767$	46.6274	1.68362
$768$	0	0
$769$	49.5980	1.78855	0.894274	−	0.447519i	$-0.147692\pi$
$769$	49.5980	1.78855	0.894274	+	0.447519i	$0.147692\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.9706	0.394584	0.197292	−	0.980345i	$-0.436785\pi$
$773$	10.9706	0.394584	0.197292	+	0.980345i	$0.436785\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.3137	0.405356
$780$	0	0
$781$	13.6569	0.488681
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22.9706	−0.819855
$786$	0	0
$787$	−43.1127	−1.53680	−0.768401	−	0.639969i	$-0.778947\pi$
$787$	−43.1127	−1.53680	−0.768401	+	0.639969i	$0.778947\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−37.6569	−1.33892
$792$	0	0
$793$	1.65685	0.0588366
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.6274	0.872348	0.436174	−	0.899862i	$-0.356333\pi$
$797$	24.6274	0.872348	0.436174	+	0.899862i	$0.356333\pi$
$798$	0	0
$799$	4.68629	0.165789
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−14.4853	−0.511174
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.0000	−0.773479	−0.386739	−	0.922189i	$-0.626399\pi$
$809$	−22.0000	−0.773479	−0.386739	+	0.922189i	$0.626399\pi$
$810$	0	0
$811$	−20.6863	−0.726394	−0.363197	−	0.931712i	$-0.618315\pi$
$811$	−20.6863	−0.726394	−0.363197	+	0.931712i	$0.618315\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20.0000	0.700569
$816$	0	0
$817$	−6.62742	−0.231864
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−28.6863	−0.999941	−0.499971	−	0.866042i	$-0.666656\pi$
$823$	−28.6863	−0.999941	−0.499971	+	0.866042i	$0.666656\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.1421	1.04814	0.524072	−	0.851674i	$-0.324412\pi$
$827$	30.1421	1.04814	0.524072	+	0.851674i	$0.324412\pi$
$828$	0	0
$829$	−27.9411	−0.970435	−0.485218	−	0.874393i	$-0.661260\pi$
$829$	−27.9411	−0.970435	−0.485218	+	0.874393i	$0.661260\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.828427	0.0287033
$834$	0	0
$835$	18.1421	0.627834
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.6569	0.747678	0.373839	−	0.927494i	$-0.378041\pi$
$839$	21.6569	0.747678	0.373839	+	0.927494i	$0.378041\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.3137	−0.354802
$846$	0	0
$847$	2.82843	0.0971859
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33.9411	1.16349
$852$	0	0
$853$	−47.4558	−1.62486	−0.812429	−	0.583061i	$-0.801855\pi$
$853$	−47.4558	−1.62486	−0.812429	+	0.583061i	$0.801855\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42.7696	1.46098	0.730490	−	0.682923i	$-0.239292\pi$
$857$	42.7696	1.46098	0.730490	+	0.682923i	$0.239292\pi$
$858$	0	0
$859$	44.9706	1.53438	0.767188	−	0.641422i	$-0.221655\pi$
$859$	44.9706	1.53438	0.767188	+	0.641422i	$0.221655\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.3137	0.385123	0.192562	−	0.981285i	$-0.438320\pi$
$863$	11.3137	0.385123	0.192562	+	0.981285i	$0.438320\pi$
$864$	0	0
$865$	−1.51472	−0.0515020
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.65685	0.327586
$870$	0	0
$871$	−46.6274	−1.57991
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.82843	−0.0956183
$876$	0	0
$877$	−21.5147	−0.726500	−0.363250	−	0.931692i	$-0.618333\pi$
$877$	−21.5147	−0.726500	−0.363250	+	0.931692i	$0.618333\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.3137	−1.25713	−0.628565	−	0.777757i	$-0.716358\pi$
$881$	−37.3137	−1.25713	−0.628565	+	0.777757i	$0.716358\pi$
$882$	0	0
$883$	13.3726	0.450023	0.225012	−	0.974356i	$-0.427758\pi$
$883$	13.3726	0.450023	0.225012	+	0.974356i	$0.427758\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.79899	0.261864	0.130932	−	0.991391i	$-0.458203\pi$
$887$	7.79899	0.261864	0.130932	+	0.991391i	$0.458203\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	−4.00000	−0.133705
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.3137	0.377333
$900$	0	0
$901$	−0.284271	−0.00947045
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.31371	−0.309598
$906$	0	0
$907$	1.65685	0.0550149	0.0275075	−	0.999622i	$-0.491243\pi$
$907$	1.65685	0.0550149	0.0275075	+	0.999622i	$0.491243\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−49.9411	−1.65462	−0.827312	−	0.561743i	$-0.810131\pi$
$911$	−49.9411	−1.65462	−0.827312	+	0.561743i	$0.810131\pi$
$912$	0	0
$913$	−10.8284	−0.358369
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.9411	−0.592468
$918$	0	0
$919$	47.3137	1.56074	0.780368	−	0.625321i	$-0.215032\pi$
$919$	47.3137	1.56074	0.780368	+	0.625321i	$0.215032\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−65.9411	−2.17048
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−53.3137	−1.74917	−0.874583	−	0.484876i	$-0.838865\pi$
$929$	−53.3137	−1.74917	−0.874583	+	0.484876i	$0.838865\pi$
$930$	0	0
$931$	5.65685	0.185396
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.828427	0.0270925
$936$	0	0
$937$	−25.1127	−0.820396	−0.410198	−	0.911996i	$-0.634540\pi$
$937$	−25.1127	−0.820396	−0.410198	+	0.911996i	$0.634540\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.6569	0.901588	0.450794	−	0.892628i	$-0.351141\pi$
$941$	27.6569	0.901588	0.450794	+	0.892628i	$0.351141\pi$
$942$	0	0
$943$	11.3137	0.368425
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.5980	−0.636849	−0.318424	−	0.947948i	$-0.603154\pi$
$947$	−19.5980	−0.636849	−0.318424	+	0.947948i	$0.603154\pi$
$948$	0	0
$949$	69.9411	2.27039
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.4853	1.24666	0.623330	−	0.781959i	$-0.285779\pi$
$953$	38.4853	1.24666	0.623330	+	0.781959i	$0.285779\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	0	0
$958$	0	0
$959$	48.9706	1.58134
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.17157	−0.102097
$966$	0	0
$967$	27.7990	0.893955	0.446978	−	0.894545i	$-0.352500\pi$
$967$	27.7990	0.893955	0.446978	+	0.894545i	$0.352500\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.9411	−0.960856	−0.480428	−	0.877034i	$-0.659519\pi$
$971$	−29.9411	−0.960856	−0.480428	+	0.877034i	$0.659519\pi$
$972$	0	0
$973$	−32.0000	−1.02587
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	2.00000	0.0639203
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−12.6863	−0.404630	−0.202315	−	0.979321i	$-0.564846\pi$
$983$	−12.6863	−0.404630	−0.202315	+	0.979321i	$0.564846\pi$
$984$	0	0
$985$	−20.8284	−0.663649
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.62742	−0.210740
$990$	0	0
$991$	48.9706	1.55560	0.777801	−	0.628511i	$-0.216335\pi$
$991$	48.9706	1.55560	0.777801	+	0.628511i	$0.216335\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−4.14214	−0.131183	−0.0655914	−	0.997847i	$-0.520893\pi$
$997$	−4.14214	−0.131183	−0.0655914	+	0.997847i	$0.520893\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.bs.1.2		2
3.2	odd	2		2640.2.a.bd.1.2		2
4.3	odd	2		3960.2.a.z.1.1		2
12.11	even	2		1320.2.a.p.1.1	✓	2
60.23	odd	4		6600.2.d.z.1849.2		4
60.47	odd	4		6600.2.d.z.1849.3		4
60.59	even	2		6600.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.p.1.1	✓	2	12.11	even	2
2640.2.a.bd.1.2		2	3.2	odd	2
3960.2.a.z.1.1		2	4.3	odd	2
6600.2.a.bk.1.2		2	60.59	even	2
6600.2.d.z.1849.2		4	60.23	odd	4
6600.2.d.z.1849.3		4	60.47	odd	4
7920.2.a.bs.1.2		2	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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.82843 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.82843 q^{7} -1.00000 q^{11} +4.82843 q^{13} +0.828427 q^{17} +5.65685 q^{19} +5.65685 q^{23} +1.00000 q^{25} -2.00000 q^{29} -5.65685 q^{31} -2.82843 q^{35} +6.00000 q^{37} +2.00000 q^{41} -1.17157 q^{43} +5.65685 q^{47} +1.00000 q^{49} -0.343146 q^{53} +1.00000 q^{55} +9.65685 q^{59} +0.343146 q^{61} -4.82843 q^{65} -9.65685 q^{67} -13.6569 q^{71} +14.4853 q^{73} -2.82843 q^{77} -9.65685 q^{79} +10.8284 q^{83} -0.828427 q^{85} -2.00000 q^{89} +13.6569 q^{91} -5.65685 q^{95} +4.34315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 2 q^{11} + 4 q^{13} - 4 q^{17} + 2 q^{25} - 4 q^{29} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 2 q^{49} - 12 q^{53} + 2 q^{55} + 8 q^{59} + 12 q^{61} - 4 q^{65} - 8 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{79} + 16 q^{83} + 4 q^{85} - 4 q^{89} + 16 q^{91} + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^11 + 4 * q^13 - 4 * q^17 + 2 * q^25 - 4 * q^29 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 2 * q^49 - 12 * q^53 + 2 * q^55 + 8 * q^59 + 12 * q^61 - 4 * q^65 - 8 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^79 + 16 * q^83 + 4 * q^85 - 4 * q^89 + 16 * q^91 + 20 * q^97

Introduction

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