LMFDB - Embedded newform 5148.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5148,2,Mod(1,5148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5148, 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("5148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5148.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:	$41.1069869606$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.815952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 9x - 2 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 9x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.63081$ of defining polynomial
Character	$\chi$	$=$	5148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47267 q^{5} -3.87430 q^{7} +O(q^{10})$	$q-1.47267 q^{5} -3.87430 q^{7} -1.00000 q^{11} -1.00000 q^{13} +0.358587 q^{17} -1.61547 q^{19} -3.63731 q^{23} -2.83125 q^{25} -4.34418 q^{29} +3.59501 q^{31} +5.70556 q^{35} +3.44115 q^{37} -7.93919 q^{41} -7.36881 q^{43} +6.30745 q^{47} +8.01022 q^{49} +6.25883 q^{53} +1.47267 q^{55} -4.46578 q^{59} -9.64810 q^{61} +1.47267 q^{65} +2.79173 q^{67} +12.3865 q^{71} -7.76497 q^{73} +3.87430 q^{77} +1.90024 q^{79} +11.2958 q^{83} -0.528079 q^{85} -0.919392 q^{89} +3.87430 q^{91} +2.37905 q^{95} -1.07651 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 2 * q^5 - 2 * q^7	$ 5 q + 2 q^{5} - 2 q^{7} - 5 q^{11} - 5 q^{13} - 2 q^{17} - 2 q^{19} + 12 q^{23} - q^{25} + 4 q^{29} - 2 q^{31} - 2 q^{35} + 6 q^{41} + 14 q^{43} + 14 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 20 q^{59} - 2 q^{65} + 10 q^{67} + 26 q^{71} + 16 q^{73} + 2 q^{77} + 2 q^{79} + 16 q^{83} - 18 q^{85} + 24 q^{89} + 2 q^{91} + 22 q^{95}+O(q^{100}) $ 5 * q + 2 * q^5 - 2 * q^7 - 5 * q^11 - 5 * q^13 - 2 * q^17 - 2 * q^19 + 12 * q^23 - q^25 + 4 * q^29 - 2 * q^31 - 2 * q^35 + 6 * q^41 + 14 * q^43 + 14 * q^47 - 7 * q^49 + 20 * q^53 - 2 * q^55 + 20 * q^59 - 2 * q^65 + 10 * q^67 + 26 * q^71 + 16 * q^73 + 2 * q^77 + 2 * q^79 + 16 * q^83 - 18 * q^85 + 24 * q^89 + 2 * q^91 + 22 * q^95

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
$5$	−1.47267	−0.658597	−0.329298	−	0.944226i	$-0.606812\pi$
$5$	−1.47267	−0.658597	−0.329298	+	0.944226i	$0.606812\pi$
$6$	0	0
$7$	−3.87430	−1.46435	−0.732174	−	0.681117i	$-0.761494\pi$
$7$	−3.87430	−1.46435	−0.732174	+	0.681117i	$0.761494\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.358587	0.0869700	0.0434850	−	0.999054i	$-0.486154\pi$
$17$	0.358587	0.0869700	0.0434850	+	0.999054i	$0.486154\pi$
$18$	0	0
$19$	−1.61547	−0.370615	−0.185307	−	0.982681i	$-0.559328\pi$
$19$	−1.61547	−0.370615	−0.185307	+	0.982681i	$0.559328\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.63731	−0.758432	−0.379216	−	0.925308i	$-0.623806\pi$
$23$	−3.63731	−0.758432	−0.379216	+	0.925308i	$0.623806\pi$
$24$	0	0
$25$	−2.83125	−0.566251
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.34418	−0.806695	−0.403347	−	0.915047i	$-0.632153\pi$
$29$	−4.34418	−0.806695	−0.403347	+	0.915047i	$0.632153\pi$
$30$	0	0
$31$	3.59501	0.645682	0.322841	−	0.946453i	$-0.395362\pi$
$31$	3.59501	0.645682	0.322841	+	0.946453i	$0.395362\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.70556	0.964415
$36$	0	0
$37$	3.44115	0.565722	0.282861	−	0.959161i	$-0.408716\pi$
$37$	3.44115	0.565722	0.282861	+	0.959161i	$0.408716\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.93919	−1.23989	−0.619947	−	0.784644i	$-0.712846\pi$
$41$	−7.93919	−1.23989	−0.619947	+	0.784644i	$0.712846\pi$
$42$	0	0
$43$	−7.36881	−1.12373	−0.561866	−	0.827228i	$-0.689916\pi$
$43$	−7.36881	−1.12373	−0.561866	+	0.827228i	$0.689916\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.30745	0.920036	0.460018	−	0.887910i	$-0.347843\pi$
$47$	6.30745	0.920036	0.460018	+	0.887910i	$0.347843\pi$
$48$	0	0
$49$	8.01022	1.14432
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.25883	0.859716	0.429858	−	0.902896i	$-0.358564\pi$
$53$	6.25883	0.859716	0.429858	+	0.902896i	$0.358564\pi$
$54$	0	0
$55$	1.47267	0.198574
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.46578	−0.581395	−0.290697	−	0.956815i	$-0.593887\pi$
$59$	−4.46578	−0.581395	−0.290697	+	0.956815i	$0.593887\pi$
$60$	0	0
$61$	−9.64810	−1.23531	−0.617657	−	0.786448i	$-0.711918\pi$
$61$	−9.64810	−1.23531	−0.617657	+	0.786448i	$0.711918\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.47267	0.182662
$66$	0	0
$67$	2.79173	0.341065	0.170532	−	0.985352i	$-0.445451\pi$
$67$	2.79173	0.341065	0.170532	+	0.985352i	$0.445451\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.3865	1.47001	0.735003	−	0.678064i	$-0.237181\pi$
$71$	12.3865	1.47001	0.735003	+	0.678064i	$0.237181\pi$
$72$	0	0
$73$	−7.76497	−0.908821	−0.454410	−	0.890792i	$-0.650150\pi$
$73$	−7.76497	−0.908821	−0.454410	+	0.890792i	$0.650150\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.87430	0.441518
$78$	0	0
$79$	1.90024	0.213794	0.106897	−	0.994270i	$-0.465908\pi$
$79$	1.90024	0.213794	0.106897	+	0.994270i	$0.465908\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.2958	1.23988	0.619939	−	0.784650i	$-0.287157\pi$
$83$	11.2958	1.23988	0.619939	+	0.784650i	$0.287157\pi$
$84$	0	0
$85$	−0.528079	−0.0572782
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.919392	−0.0974554	−0.0487277	−	0.998812i	$-0.515517\pi$
$89$	−0.919392	−0.0974554	−0.0487277	+	0.998812i	$0.515517\pi$
$90$	0	0
$91$	3.87430	0.406137
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.37905	0.244086
$96$	0	0
$97$	−1.07651	−0.109303	−0.0546513	−	0.998506i	$-0.517405\pi$
$97$	−1.07651	−0.109303	−0.0546513	+	0.998506i	$0.517405\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.58675	−0.257391	−0.128695	−	0.991684i	$-0.541079\pi$
$101$	−2.58675	−0.257391	−0.128695	+	0.991684i	$0.541079\pi$
$102$	0	0
$103$	−6.20695	−0.611589	−0.305794	−	0.952098i	$-0.598922\pi$
$103$	−6.20695	−0.611589	−0.305794	+	0.952098i	$0.598922\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.38396	0.907181	0.453591	−	0.891210i	$-0.350143\pi$
$107$	9.38396	0.907181	0.453591	+	0.891210i	$0.350143\pi$
$108$	0	0
$109$	1.49313	0.143016	0.0715081	−	0.997440i	$-0.477219\pi$
$109$	1.49313	0.143016	0.0715081	+	0.997440i	$0.477219\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.105819	−0.00995466	−0.00497733	−	0.999988i	$-0.501584\pi$
$113$	−0.105819	−0.00995466	−0.00497733	+	0.999988i	$0.501584\pi$
$114$	0	0
$115$	5.35655	0.499501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.38927	−0.127354
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.5328	1.03153
$126$	0	0
$127$	−3.56212	−0.316087	−0.158044	−	0.987432i	$-0.550519\pi$
$127$	−3.56212	−0.316087	−0.158044	+	0.987432i	$0.550519\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.4630	1.35101	0.675504	−	0.737356i	$-0.263926\pi$
$131$	15.4630	1.35101	0.675504	+	0.737356i	$0.263926\pi$
$132$	0	0
$133$	6.25883	0.542709
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.28126	0.536644	0.268322	−	0.963329i	$-0.413531\pi$
$137$	6.28126	0.536644	0.268322	+	0.963329i	$0.413531\pi$
$138$	0	0
$139$	−2.66604	−0.226130	−0.113065	−	0.993588i	$-0.536067\pi$
$139$	−2.66604	−0.226130	−0.113065	+	0.993588i	$0.536067\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	6.39753	0.531286
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.55058	0.127029	0.0635144	−	0.997981i	$-0.479769\pi$
$149$	1.55058	0.127029	0.0635144	+	0.997981i	$0.479769\pi$
$150$	0	0
$151$	−6.36686	−0.518128	−0.259064	−	0.965860i	$-0.583414\pi$
$151$	−6.36686	−0.518128	−0.259064	+	0.965860i	$0.583414\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.29425	−0.425244
$156$	0	0
$157$	−7.68362	−0.613220	−0.306610	−	0.951835i	$-0.599195\pi$
$157$	−7.68362	−0.613220	−0.306610	+	0.951835i	$0.599195\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	14.0920	1.11061
$162$	0	0
$163$	2.71244	0.212455	0.106227	−	0.994342i	$-0.466123\pi$
$163$	2.71244	0.212455	0.106227	+	0.994342i	$0.466123\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.0409	1.16390	0.581950	−	0.813225i	$-0.302290\pi$
$167$	15.0409	1.16390	0.581950	+	0.813225i	$0.302290\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−0.644199	−0.0489775	−0.0244888	−	0.999700i	$-0.507796\pi$
$173$	−0.644199	−0.0489775	−0.0244888	+	0.999700i	$0.507796\pi$
$174$	0	0
$175$	10.9691	0.829188
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.1515	1.65568	0.827839	−	0.560966i	$-0.189570\pi$
$179$	22.1515	1.65568	0.827839	+	0.560966i	$0.189570\pi$
$180$	0	0
$181$	16.3312	1.21389	0.606946	−	0.794743i	$-0.292394\pi$
$181$	16.3312	1.21389	0.606946	+	0.794743i	$0.292394\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.06767	−0.372583
$186$	0	0
$187$	−0.358587	−0.0262225
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.25326	0.307755	0.153878	−	0.988090i	$-0.450824\pi$
$191$	4.25326	0.307755	0.153878	+	0.988090i	$0.450824\pi$
$192$	0	0
$193$	−5.50408	−0.396192	−0.198096	−	0.980183i	$-0.563476\pi$
$193$	−5.50408	−0.396192	−0.198096	+	0.980183i	$0.563476\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.88174	−0.134068	−0.0670341	−	0.997751i	$-0.521354\pi$
$197$	−1.88174	−0.134068	−0.0670341	+	0.997751i	$0.521354\pi$
$198$	0	0
$199$	12.5716	0.891177	0.445588	−	0.895238i	$-0.352994\pi$
$199$	12.5716	0.891177	0.445588	+	0.895238i	$0.352994\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.8307	1.18128
$204$	0	0
$205$	11.6918	0.816589
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.61547	0.111745
$210$	0	0
$211$	6.36602	0.438255	0.219128	−	0.975696i	$-0.429679\pi$
$211$	6.36602	0.438255	0.219128	+	0.975696i	$0.429679\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.8518	0.740086
$216$	0	0
$217$	−13.9281	−0.945504
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−0.358587	−0.0241212
$222$	0	0
$223$	−6.64410	−0.444922	−0.222461	−	0.974942i	$-0.571409\pi$
$223$	−6.64410	−0.444922	−0.222461	+	0.974942i	$0.571409\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.5943	0.902283	0.451141	−	0.892452i	$-0.351017\pi$
$227$	13.5943	0.902283	0.451141	+	0.892452i	$0.351017\pi$
$228$	0	0
$229$	2.63788	0.174316	0.0871581	−	0.996194i	$-0.472221\pi$
$229$	2.63788	0.174316	0.0871581	+	0.996194i	$0.472221\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.00390	−0.458841	−0.229421	−	0.973327i	$-0.573683\pi$
$233$	−7.00390	−0.458841	−0.229421	+	0.973327i	$0.573683\pi$
$234$	0	0
$235$	−9.28877	−0.605933
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.7992	1.66881	0.834405	−	0.551151i	$-0.185811\pi$
$239$	25.7992	1.66881	0.834405	+	0.551151i	$0.185811\pi$
$240$	0	0
$241$	18.3423	1.18153	0.590766	−	0.806843i	$-0.298826\pi$
$241$	18.3423	1.18153	0.590766	+	0.806843i	$0.298826\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−11.7964	−0.753643
$246$	0	0
$247$	1.61547	0.102790
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.6467	−1.24009	−0.620045	−	0.784567i	$-0.712885\pi$
$251$	−19.6467	−1.24009	−0.620045	+	0.784567i	$0.712885\pi$
$252$	0	0
$253$	3.63731	0.228676
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.560963	0.0349919	0.0174960	−	0.999847i	$-0.494431\pi$
$257$	0.560963	0.0349919	0.0174960	+	0.999847i	$0.494431\pi$
$258$	0	0
$259$	−13.3321	−0.828415
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.7183	1.21588	0.607941	−	0.793982i	$-0.291996\pi$
$263$	19.7183	1.21588	0.607941	+	0.793982i	$0.291996\pi$
$264$	0	0
$265$	−9.21717	−0.566206
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.46021	−0.393886	−0.196943	−	0.980415i	$-0.563101\pi$
$269$	−6.46021	−0.393886	−0.196943	+	0.980415i	$0.563101\pi$
$270$	0	0
$271$	−6.19802	−0.376503	−0.188251	−	0.982121i	$-0.560282\pi$
$271$	−6.19802	−0.376503	−0.188251	+	0.982121i	$0.560282\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.83125	0.170731
$276$	0	0
$277$	12.5272	0.752684	0.376342	−	0.926481i	$-0.377182\pi$
$277$	12.5272	0.752684	0.376342	+	0.926481i	$0.377182\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.2555	−0.731100	−0.365550	−	0.930792i	$-0.619119\pi$
$281$	−12.2555	−0.731100	−0.365550	+	0.930792i	$0.619119\pi$
$282$	0	0
$283$	−20.4527	−1.21579	−0.607895	−	0.794018i	$-0.707986\pi$
$283$	−20.4527	−1.21579	−0.607895	+	0.794018i	$0.707986\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.7588	1.81564
$288$	0	0
$289$	−16.8714	−0.992436
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.1591	1.06087	0.530434	−	0.847726i	$-0.322029\pi$
$293$	18.1591	1.06087	0.530434	+	0.847726i	$0.322029\pi$
$294$	0	0
$295$	6.57660	0.382905
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.63731	0.210351
$300$	0	0
$301$	28.5490	1.64554
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.2084	0.813573
$306$	0	0
$307$	−3.64263	−0.207896	−0.103948	−	0.994583i	$-0.533148\pi$
$307$	−3.64263	−0.207896	−0.103948	+	0.994583i	$0.533148\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.7439	−1.23298	−0.616490	−	0.787362i	$-0.711446\pi$
$311$	−21.7439	−1.23298	−0.616490	+	0.787362i	$0.711446\pi$
$312$	0	0
$313$	25.0408	1.41539	0.707696	−	0.706517i	$-0.249735\pi$
$313$	25.0408	1.41539	0.707696	+	0.706517i	$0.249735\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.86751	−0.273387	−0.136693	−	0.990613i	$-0.543647\pi$
$317$	−4.86751	−0.273387	−0.136693	+	0.990613i	$0.543647\pi$
$318$	0	0
$319$	4.34418	0.243228
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.579287	−0.0322324
$324$	0	0
$325$	2.83125	0.157050
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−24.4370	−1.34725
$330$	0	0
$331$	−10.8638	−0.597129	−0.298564	−	0.954390i	$-0.596508\pi$
$331$	−10.8638	−0.597129	−0.298564	+	0.954390i	$0.596508\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.11129	−0.224624
$336$	0	0
$337$	7.46856	0.406839	0.203419	−	0.979092i	$-0.434795\pi$
$337$	7.46856	0.406839	0.203419	+	0.979092i	$0.434795\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.59501	−0.194681
$342$	0	0
$343$	−3.91390	−0.211331
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−31.8039	−1.70732	−0.853661	−	0.520829i	$-0.825623\pi$
$347$	−31.8039	−1.70732	−0.853661	+	0.520829i	$0.825623\pi$
$348$	0	0
$349$	−26.0688	−1.39543	−0.697716	−	0.716375i	$-0.745800\pi$
$349$	−26.0688	−1.39543	−0.697716	+	0.716375i	$0.745800\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.04111	−0.0554124	−0.0277062	−	0.999616i	$-0.508820\pi$
$353$	−1.04111	−0.0554124	−0.0277062	+	0.999616i	$0.508820\pi$
$354$	0	0
$355$	−18.2412	−0.968141
$356$	0	0
$357$	0	0
$358$	0	0
$359$	27.9301	1.47409	0.737047	−	0.675841i	$-0.236219\pi$
$359$	27.9301	1.47409	0.737047	+	0.675841i	$0.236219\pi$
$360$	0	0
$361$	−16.3902	−0.862645
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.4352	0.598546
$366$	0	0
$367$	0.127912	0.00667693	0.00333846	−	0.999994i	$-0.498937\pi$
$367$	0.127912	0.00667693	0.00333846	+	0.999994i	$0.498937\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−24.2486	−1.25892
$372$	0	0
$373$	17.5491	0.908658	0.454329	−	0.890834i	$-0.349879\pi$
$373$	17.5491	0.908658	0.454329	+	0.890834i	$0.349879\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.34418	0.223737
$378$	0	0
$379$	7.93633	0.407662	0.203831	−	0.979006i	$-0.434661\pi$
$379$	7.93633	0.407662	0.203831	+	0.979006i	$0.434661\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.88396	0.249559	0.124779	−	0.992185i	$-0.460178\pi$
$383$	4.88396	0.249559	0.124779	+	0.992185i	$0.460178\pi$
$384$	0	0
$385$	−5.70556	−0.290782
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−23.5314	−1.19309	−0.596545	−	0.802579i	$-0.703460\pi$
$389$	−23.5314	−1.19309	−0.596545	+	0.802579i	$0.703460\pi$
$390$	0	0
$391$	−1.30429	−0.0659608
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.79842	−0.140804
$396$	0	0
$397$	26.7374	1.34191	0.670955	−	0.741498i	$-0.265884\pi$
$397$	26.7374	1.34191	0.670955	+	0.741498i	$0.265884\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11.3306	−0.565823	−0.282912	−	0.959146i	$-0.591300\pi$
$401$	−11.3306	−0.565823	−0.282912	+	0.959146i	$0.591300\pi$
$402$	0	0
$403$	−3.59501	−0.179080
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.44115	−0.170572
$408$	0	0
$409$	−15.1722	−0.750215	−0.375108	−	0.926981i	$-0.622394\pi$
$409$	−15.1722	−0.750215	−0.375108	+	0.926981i	$0.622394\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	17.3018	0.851365
$414$	0	0
$415$	−16.6350	−0.816580
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.3469	−0.603185	−0.301593	−	0.953437i	$-0.597518\pi$
$419$	−12.3469	−0.603185	−0.301593	+	0.953437i	$0.597518\pi$
$420$	0	0
$421$	−27.3375	−1.33235	−0.666174	−	0.745797i	$-0.732069\pi$
$421$	−27.3375	−1.33235	−0.666174	+	0.745797i	$0.732069\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1.01525	−0.0492468
$426$	0	0
$427$	37.3797	1.80893
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.8646	−0.523332	−0.261666	−	0.965159i	$-0.584272\pi$
$431$	−10.8646	−0.523332	−0.261666	+	0.965159i	$0.584272\pi$
$432$	0	0
$433$	13.5697	0.652118	0.326059	−	0.945349i	$-0.394279\pi$
$433$	13.5697	0.652118	0.326059	+	0.945349i	$0.394279\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.87598	0.281086
$438$	0	0
$439$	0.0316692	0.00151149	0.000755743	−	1.00000i	$-0.499759\pi$
$439$	0.0316692	0.00151149	0.000755743		1.00000i	$0.499759\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.47120	0.354967	0.177484	−	0.984124i	$-0.443204\pi$
$443$	7.47120	0.354967	0.177484	+	0.984124i	$0.443204\pi$
$444$	0	0
$445$	1.35396	0.0641838
$446$	0	0
$447$	0	0
$448$	0	0
$449$	35.3196	1.66684	0.833418	−	0.552644i	$-0.186381\pi$
$449$	35.3196	1.66684	0.833418	+	0.552644i	$0.186381\pi$
$450$	0	0
$451$	7.93919	0.373842
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.70556	−0.267481
$456$	0	0
$457$	−16.7897	−0.785389	−0.392694	−	0.919669i	$-0.628457\pi$
$457$	−16.7897	−0.785389	−0.392694	+	0.919669i	$0.628457\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.80827	0.270518	0.135259	−	0.990810i	$-0.456813\pi$
$461$	5.80827	0.270518	0.135259	+	0.990810i	$0.456813\pi$
$462$	0	0
$463$	7.15664	0.332597	0.166299	−	0.986075i	$-0.446818\pi$
$463$	7.15664	0.332597	0.166299	+	0.986075i	$0.446818\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.424222	0.0196307	0.00981534	−	0.999952i	$-0.496876\pi$
$467$	0.424222	0.0196307	0.00981534	+	0.999952i	$0.496876\pi$
$468$	0	0
$469$	−10.8160	−0.499438
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.36881	0.338818
$474$	0	0
$475$	4.57381	0.209861
$476$	0	0
$477$	0	0
$478$	0	0
$479$	35.2157	1.60905	0.804523	−	0.593922i	$-0.202421\pi$
$479$	35.2157	1.60905	0.804523	+	0.593922i	$0.202421\pi$
$480$	0	0
$481$	−3.44115	−0.156903
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.58533	0.0719863
$486$	0	0
$487$	−5.24388	−0.237623	−0.118811	−	0.992917i	$-0.537908\pi$
$487$	−5.24388	−0.237623	−0.118811	+	0.992917i	$0.537908\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−23.1111	−1.04299	−0.521495	−	0.853255i	$-0.674625\pi$
$491$	−23.1111	−1.04299	−0.521495	+	0.853255i	$0.674625\pi$
$492$	0	0
$493$	−1.55777	−0.0701583
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−47.9890	−2.15260
$498$	0	0
$499$	35.2763	1.57918	0.789592	−	0.613632i	$-0.210292\pi$
$499$	35.2763	1.57918	0.789592	+	0.613632i	$0.210292\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.12032	0.183716	0.0918579	−	0.995772i	$-0.470719\pi$
$503$	4.12032	0.183716	0.0918579	+	0.995772i	$0.470719\pi$
$504$	0	0
$505$	3.80941	0.169517
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.8017	0.877695	0.438848	−	0.898562i	$-0.355387\pi$
$509$	19.8017	0.877695	0.438848	+	0.898562i	$0.355387\pi$
$510$	0	0
$511$	30.0838	1.33083
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.14077	0.402790
$516$	0	0
$517$	−6.30745	−0.277401
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.0260	1.05260	0.526299	−	0.850299i	$-0.323579\pi$
$521$	24.0260	1.05260	0.526299	+	0.850299i	$0.323579\pi$
$522$	0	0
$523$	−24.7130	−1.08062	−0.540312	−	0.841465i	$-0.681694\pi$
$523$	−24.7130	−1.08062	−0.540312	+	0.841465i	$0.681694\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.28912	0.0561550
$528$	0	0
$529$	−9.76997	−0.424781
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.93919	0.343885
$534$	0	0
$535$	−13.8194	−0.597466
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8.01022	−0.345025
$540$	0	0
$541$	1.07660	0.0462866	0.0231433	−	0.999732i	$-0.492633\pi$
$541$	1.07660	0.0462866	0.0231433	+	0.999732i	$0.492633\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.19889	−0.0941899
$546$	0	0
$547$	25.9622	1.11007	0.555033	−	0.831829i	$-0.312706\pi$
$547$	25.9622	1.11007	0.555033	+	0.831829i	$0.312706\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.01791	0.298973
$552$	0	0
$553$	−7.36212	−0.313069
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.2787	0.859238	0.429619	−	0.903010i	$-0.358648\pi$
$557$	20.2787	0.859238	0.429619	+	0.903010i	$0.358648\pi$
$558$	0	0
$559$	7.36881	0.311667
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.3183	−0.477009	−0.238505	−	0.971141i	$-0.576657\pi$
$563$	−11.3183	−0.477009	−0.238505	+	0.971141i	$0.576657\pi$
$564$	0	0
$565$	0.155837	0.00655610
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.4664	−1.65452	−0.827259	−	0.561820i	$-0.810101\pi$
$569$	−39.4664	−1.65452	−0.827259	+	0.561820i	$0.810101\pi$
$570$	0	0
$571$	−22.9002	−0.958345	−0.479173	−	0.877721i	$-0.659063\pi$
$571$	−22.9002	−0.958345	−0.479173	+	0.877721i	$0.659063\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.2981	0.429462
$576$	0	0
$577$	−8.49293	−0.353565	−0.176783	−	0.984250i	$-0.556569\pi$
$577$	−8.49293	−0.353565	−0.176783	+	0.984250i	$0.556569\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−43.7635	−1.81561
$582$	0	0
$583$	−6.25883	−0.259214
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−26.4232	−1.09060	−0.545301	−	0.838240i	$-0.683585\pi$
$587$	−26.4232	−1.09060	−0.545301	+	0.838240i	$0.683585\pi$
$588$	0	0
$589$	−5.80764	−0.239299
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.1371	0.580539	0.290270	−	0.956945i	$-0.406255\pi$
$593$	14.1371	0.580539	0.290270	+	0.956945i	$0.406255\pi$
$594$	0	0
$595$	2.04594	0.0838752
$596$	0	0
$597$	0	0
$598$	0	0
$599$	42.6374	1.74212	0.871058	−	0.491179i	$-0.163434\pi$
$599$	42.6374	1.74212	0.871058	+	0.491179i	$0.163434\pi$
$600$	0	0
$601$	−40.6487	−1.65810	−0.829048	−	0.559177i	$-0.811117\pi$
$601$	−40.6487	−1.65810	−0.829048	+	0.559177i	$0.811117\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.47267	−0.0598724
$606$	0	0
$607$	−32.6346	−1.32460	−0.662299	−	0.749240i	$-0.730419\pi$
$607$	−32.6346	−1.32460	−0.662299	+	0.749240i	$0.730419\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.30745	−0.255172
$612$	0	0
$613$	−24.7363	−0.999089	−0.499545	−	0.866288i	$-0.666499\pi$
$613$	−24.7363	−0.999089	−0.499545	+	0.866288i	$0.666499\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.131133	0.00527921	0.00263960	−	0.999997i	$-0.499160\pi$
$617$	0.131133	0.00527921	0.00263960	+	0.999997i	$0.499160\pi$
$618$	0	0
$619$	−21.9776	−0.883355	−0.441677	−	0.897174i	$-0.645616\pi$
$619$	−21.9776	−0.883355	−0.441677	+	0.897174i	$0.645616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.56200	0.142709
$624$	0	0
$625$	−2.82774	−0.113110
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.23395	0.0492009
$630$	0	0
$631$	−12.1059	−0.481927	−0.240963	−	0.970534i	$-0.577463\pi$
$631$	−12.1059	−0.481927	−0.240963	+	0.970534i	$0.577463\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.24582	0.208174
$636$	0	0
$637$	−8.01022	−0.317377
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−39.6952	−1.56787	−0.783934	−	0.620845i	$-0.786790\pi$
$641$	−39.6952	−1.56787	−0.783934	+	0.620845i	$0.786790\pi$
$642$	0	0
$643$	37.7326	1.48803	0.744014	−	0.668164i	$-0.232920\pi$
$643$	37.7326	1.48803	0.744014	+	0.668164i	$0.232920\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−13.5335	−0.532055	−0.266027	−	0.963965i	$-0.585711\pi$
$647$	−13.5335	−0.532055	−0.266027	+	0.963965i	$0.585711\pi$
$648$	0	0
$649$	4.46578	0.175297
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.12921	−0.357253	−0.178627	−	0.983917i	$-0.557165\pi$
$653$	−9.12921	−0.357253	−0.178627	+	0.983917i	$0.557165\pi$
$654$	0	0
$655$	−22.7718	−0.889769
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.5551	0.839669	0.419834	−	0.907601i	$-0.362088\pi$
$659$	21.5551	0.839669	0.419834	+	0.907601i	$0.362088\pi$
$660$	0	0
$661$	11.4701	0.446133	0.223067	−	0.974803i	$-0.428393\pi$
$661$	11.4701	0.446133	0.223067	+	0.974803i	$0.428393\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.21717	−0.357427
$666$	0	0
$667$	15.8011	0.611823
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.64810	0.372461
$672$	0	0
$673$	−7.42666	−0.286277	−0.143138	−	0.989703i	$-0.545719\pi$
$673$	−7.42666	−0.286277	−0.143138	+	0.989703i	$0.545719\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.4725	0.940553	0.470277	−	0.882519i	$-0.344154\pi$
$677$	24.4725	0.940553	0.470277	+	0.882519i	$0.344154\pi$
$678$	0	0
$679$	4.17071	0.160057
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.7755	0.718423	0.359212	−	0.933256i	$-0.383046\pi$
$683$	18.7755	0.718423	0.359212	+	0.933256i	$0.383046\pi$
$684$	0	0
$685$	−9.25020	−0.353432
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.25883	−0.238442
$690$	0	0
$691$	35.1286	1.33635	0.668177	−	0.744002i	$-0.267075\pi$
$691$	35.1286	1.33635	0.668177	+	0.744002i	$0.267075\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.92618	0.148929
$696$	0	0
$697$	−2.84689	−0.107834
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6.54581	−0.247232	−0.123616	−	0.992330i	$-0.539449\pi$
$701$	−6.54581	−0.247232	−0.123616	+	0.992330i	$0.539449\pi$
$702$	0	0
$703$	−5.55909	−0.209665
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.0218	0.376910
$708$	0	0
$709$	−1.59417	−0.0598701	−0.0299351	−	0.999552i	$-0.509530\pi$
$709$	−1.59417	−0.0598701	−0.0299351	+	0.999552i	$0.509530\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−13.0762	−0.489706
$714$	0	0
$715$	−1.47267	−0.0550746
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.1874	0.640981	0.320491	−	0.947252i	$-0.396152\pi$
$719$	17.1874	0.640981	0.320491	+	0.947252i	$0.396152\pi$
$720$	0	0
$721$	24.0476	0.895579
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.2995	0.456791
$726$	0	0
$727$	11.1746	0.414444	0.207222	−	0.978294i	$-0.433558\pi$
$727$	11.1746	0.414444	0.207222	+	0.978294i	$0.433558\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.64236	−0.0977311
$732$	0	0
$733$	41.1965	1.52163	0.760814	−	0.648970i	$-0.224800\pi$
$733$	41.1965	1.52163	0.760814	+	0.648970i	$0.224800\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.79173	−0.102835
$738$	0	0
$739$	−3.11177	−0.114468	−0.0572341	−	0.998361i	$-0.518228\pi$
$739$	−3.11177	−0.114468	−0.0572341	+	0.998361i	$0.518228\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.27261	0.156747	0.0783734	−	0.996924i	$-0.475027\pi$
$743$	4.27261	0.156747	0.0783734	+	0.996924i	$0.475027\pi$
$744$	0	0
$745$	−2.28349	−0.0836608
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−36.3563	−1.32843
$750$	0	0
$751$	−15.9379	−0.581581	−0.290790	−	0.956787i	$-0.593918\pi$
$751$	−15.9379	−0.581581	−0.290790	+	0.956787i	$0.593918\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9.37627	0.341237
$756$	0	0
$757$	8.64398	0.314171	0.157085	−	0.987585i	$-0.449790\pi$
$757$	8.64398	0.314171	0.157085	+	0.987585i	$0.449790\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.0798	−1.23539	−0.617695	−	0.786418i	$-0.711933\pi$
$761$	−34.0798	−1.23539	−0.617695	+	0.786418i	$0.711933\pi$
$762$	0	0
$763$	−5.78485	−0.209426
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.46578	0.161250
$768$	0	0
$769$	48.7430	1.75772	0.878859	−	0.477081i	$-0.158305\pi$
$769$	48.7430	1.75772	0.878859	+	0.477081i	$0.158305\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	55.1652	1.98415	0.992077	−	0.125635i	$-0.0400967\pi$
$773$	55.1652	1.98415	0.992077	+	0.125635i	$0.0400967\pi$
$774$	0	0
$775$	−10.1784	−0.365618
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.8255	0.459523
$780$	0	0
$781$	−12.3865	−0.443223
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.3154	0.403864
$786$	0	0
$787$	−24.1185	−0.859731	−0.429865	−	0.902893i	$-0.641439\pi$
$787$	−24.1185	−0.859731	−0.429865	+	0.902893i	$0.641439\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.409977	0.0145771
$792$	0	0
$793$	9.64810	0.342614
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.1069	1.52692	0.763462	−	0.645852i	$-0.223498\pi$
$797$	43.1069	1.52692	0.763462	+	0.645852i	$0.223498\pi$
$798$	0	0
$799$	2.26177	0.0800156
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.76497	0.274020
$804$	0	0
$805$	−20.7529	−0.731443
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28.5330	−1.00317	−0.501584	−	0.865109i	$-0.667249\pi$
$809$	−28.5330	−1.00317	−0.501584	+	0.865109i	$0.667249\pi$
$810$	0	0
$811$	42.0062	1.47504	0.737519	−	0.675326i	$-0.235997\pi$
$811$	42.0062	1.47504	0.737519	+	0.675326i	$0.235997\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.99453	−0.139922
$816$	0	0
$817$	11.9041	0.416472
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.00522	−0.209584	−0.104792	−	0.994494i	$-0.533418\pi$
$821$	−6.00522	−0.209584	−0.104792	+	0.994494i	$0.533418\pi$
$822$	0	0
$823$	18.2735	0.636975	0.318487	−	0.947927i	$-0.396825\pi$
$823$	18.2735	0.636975	0.318487	+	0.947927i	$0.396825\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.4278	0.501705	0.250852	−	0.968025i	$-0.419289\pi$
$827$	14.4278	0.501705	0.250852	+	0.968025i	$0.419289\pi$
$828$	0	0
$829$	12.3470	0.428829	0.214414	−	0.976743i	$-0.431216\pi$
$829$	12.3470	0.428829	0.214414	+	0.976743i	$0.431216\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.87236	0.0995213
$834$	0	0
$835$	−22.1502	−0.766540
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−38.6121	−1.33304	−0.666518	−	0.745489i	$-0.732216\pi$
$839$	−38.6121	−1.33304	−0.666518	+	0.745489i	$0.732216\pi$
$840$	0	0
$841$	−10.1281	−0.349244
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.47267	−0.0506613
$846$	0	0
$847$	−3.87430	−0.133123
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.5165	−0.429062
$852$	0	0
$853$	16.4552	0.563415	0.281707	−	0.959500i	$-0.409099\pi$
$853$	16.4552	0.563415	0.281707	+	0.959500i	$0.409099\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.3498	−0.763453	−0.381727	−	0.924275i	$-0.624670\pi$
$857$	−22.3498	−0.763453	−0.381727	+	0.924275i	$0.624670\pi$
$858$	0	0
$859$	7.20787	0.245929	0.122965	−	0.992411i	$-0.460760\pi$
$859$	7.20787	0.245929	0.122965	+	0.992411i	$0.460760\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.223096	−0.00759427	−0.00379714	−	0.999993i	$-0.501209\pi$
$863$	−0.223096	−0.00759427	−0.00379714	+	0.999993i	$0.501209\pi$
$864$	0	0
$865$	0.948690	0.0322564
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.90024	−0.0644613
$870$	0	0
$871$	−2.79173	−0.0945943
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−44.6817	−1.51052
$876$	0	0
$877$	−18.6907	−0.631141	−0.315571	−	0.948902i	$-0.602196\pi$
$877$	−18.6907	−0.631141	−0.315571	+	0.948902i	$0.602196\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.6184	−1.26740	−0.633699	−	0.773580i	$-0.718464\pi$
$881$	−37.6184	−1.26740	−0.633699	+	0.773580i	$0.718464\pi$
$882$	0	0
$883$	1.28355	0.0431948	0.0215974	−	0.999767i	$-0.493125\pi$
$883$	1.28355	0.0431948	0.0215974	+	0.999767i	$0.493125\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.54431	−0.186160	−0.0930798	−	0.995659i	$-0.529671\pi$
$887$	−5.54431	−0.186160	−0.0930798	+	0.995659i	$0.529671\pi$
$888$	0	0
$889$	13.8007	0.462862
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.1895	−0.340979
$894$	0	0
$895$	−32.6217	−1.09042
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.6174	−0.520868
$900$	0	0
$901$	2.24433	0.0747696
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0505	−0.799465
$906$	0	0
$907$	30.6975	1.01929	0.509647	−	0.860384i	$-0.329776\pi$
$907$	30.6975	1.01929	0.509647	+	0.860384i	$0.329776\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.7108	−0.421129	−0.210564	−	0.977580i	$-0.567530\pi$
$911$	−12.7108	−0.421129	−0.210564	+	0.977580i	$0.567530\pi$
$912$	0	0
$913$	−11.2958	−0.373838
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−59.9083	−1.97835
$918$	0	0
$919$	−41.6384	−1.37352	−0.686761	−	0.726883i	$-0.740968\pi$
$919$	−41.6384	−1.37352	−0.686761	+	0.726883i	$0.740968\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−12.3865	−0.407706
$924$	0	0
$925$	−9.74278	−0.320341
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−0.294049	−0.00964744	−0.00482372	−	0.999988i	$-0.501535\pi$
$929$	−0.294049	−0.00964744	−0.00482372	+	0.999988i	$0.501535\pi$
$930$	0	0
$931$	−12.9403	−0.424101
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.528079	0.0172700
$936$	0	0
$937$	−19.9679	−0.652323	−0.326161	−	0.945314i	$-0.605755\pi$
$937$	−19.9679	−0.652323	−0.326161	+	0.945314i	$0.605755\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.2835	0.889417	0.444708	−	0.895675i	$-0.353307\pi$
$941$	27.2835	0.889417	0.444708	+	0.895675i	$0.353307\pi$
$942$	0	0
$943$	28.8773	0.940374
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.9164	−1.00465	−0.502324	−	0.864680i	$-0.667521\pi$
$947$	−30.9164	−1.00465	−0.502324	+	0.864680i	$0.667521\pi$
$948$	0	0
$949$	7.76497	0.252062
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33.1519	−1.07389	−0.536947	−	0.843616i	$-0.680423\pi$
$953$	−33.1519	−1.07389	−0.536947	+	0.843616i	$0.680423\pi$
$954$	0	0
$955$	−6.26363	−0.202686
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24.3355	−0.785834
$960$	0	0
$961$	−18.0759	−0.583094
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.10568	0.260931
$966$	0	0
$967$	5.22167	0.167918	0.0839588	−	0.996469i	$-0.473244\pi$
$967$	5.22167	0.167918	0.0839588	+	0.996469i	$0.473244\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.8532	−0.733394	−0.366697	−	0.930340i	$-0.619511\pi$
$971$	−22.8532	−0.733394	−0.366697	+	0.930340i	$0.619511\pi$
$972$	0	0
$973$	10.3290	0.331134
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.9044	0.348861	0.174431	−	0.984669i	$-0.444191\pi$
$977$	10.9044	0.348861	0.174431	+	0.984669i	$0.444191\pi$
$978$	0	0
$979$	0.919392	0.0293839
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.3042	1.28550	0.642752	−	0.766074i	$-0.277792\pi$
$983$	40.3042	1.28550	0.642752	+	0.766074i	$0.277792\pi$
$984$	0	0
$985$	2.77117	0.0882969
$986$	0	0
$987$	0	0
$988$	0	0
$989$	26.8026	0.852275
$990$	0	0
$991$	54.7525	1.73927	0.869635	−	0.493695i	$-0.164354\pi$
$991$	54.7525	1.73927	0.869635	+	0.493695i	$0.164354\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−18.5138	−0.586926
$996$	0	0
$997$	10.3700	0.328422	0.164211	−	0.986425i	$-0.447492\pi$
$997$	10.3700	0.328422	0.164211	+	0.986425i	$0.447492\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.t.1.2	yes	5
3.2	odd	2		5148.2.a.s.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5148.2.a.s.1.4	✓	5	3.2	odd	2
5148.2.a.t.1.2	yes	5	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:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q-1.47267 q^{5} -3.87430 q^{7} +O(q^{10})\)	\(q-1.47267 q^{5} -3.87430 q^{7} -1.00000 q^{11} -1.00000 q^{13} +0.358587 q^{17} -1.61547 q^{19} -3.63731 q^{23} -2.83125 q^{25} -4.34418 q^{29} +3.59501 q^{31} +5.70556 q^{35} +3.44115 q^{37} -7.93919 q^{41} -7.36881 q^{43} +6.30745 q^{47} +8.01022 q^{49} +6.25883 q^{53} +1.47267 q^{55} -4.46578 q^{59} -9.64810 q^{61} +1.47267 q^{65} +2.79173 q^{67} +12.3865 q^{71} -7.76497 q^{73} +3.87430 q^{77} +1.90024 q^{79} +11.2958 q^{83} -0.528079 q^{85} -0.919392 q^{89} +3.87430 q^{91} +2.37905 q^{95} -1.07651 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 2 * q^5 - 2 * q^7	\( 5 q + 2 q^{5} - 2 q^{7} - 5 q^{11} - 5 q^{13} - 2 q^{17} - 2 q^{19} + 12 q^{23} - q^{25} + 4 q^{29} - 2 q^{31} - 2 q^{35} + 6 q^{41} + 14 q^{43} + 14 q^{47} - 7 q^{49} + 20 q^{53} - 2 q^{55} + 20 q^{59} - 2 q^{65} + 10 q^{67} + 26 q^{71} + 16 q^{73} + 2 q^{77} + 2 q^{79} + 16 q^{83} - 18 q^{85} + 24 q^{89} + 2 q^{91} + 22 q^{95}+O(q^{100}) \) 5 * q + 2 * q^5 - 2 * q^7 - 5 * q^11 - 5 * q^13 - 2 * q^17 - 2 * q^19 + 12 * q^23 - q^25 + 4 * q^29 - 2 * q^31 - 2 * q^35 + 6 * q^41 + 14 * q^43 + 14 * q^47 - 7 * q^49 + 20 * q^53 - 2 * q^55 + 20 * q^59 - 2 * q^65 + 10 * q^67 + 26 * q^71 + 16 * q^73 + 2 * q^77 + 2 * q^79 + 16 * q^83 - 18 * q^85 + 24 * q^89 + 2 * q^91 + 22 * q^95

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