LMFDB - Embedded newform 5148.2.a.r.1.3

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:	$4$
Coefficient field:	4.4.82128.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 12x^{2} - 10x + 3 $ x^4 - 12x^2 - 10x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.18255$ 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)

$f(q)$	$=$	$q+2.18255 q^{5} +4.85250 q^{7} +O(q^{10})$	$q+2.18255 q^{5} +4.85250 q^{7} +1.00000 q^{11} +1.00000 q^{13} -3.92087 q^{17} -2.48740 q^{19} +6.10342 q^{23} -0.236477 q^{25} +6.66995 q^{29} -8.02429 q^{31} +10.5908 q^{35} -1.25092 q^{37} +2.85250 q^{41} -1.92087 q^{43} +8.95592 q^{47} +16.5467 q^{49} +7.33990 q^{53} +2.18255 q^{55} +1.13306 q^{59} +4.95592 q^{61} +2.18255 q^{65} +3.04949 q^{67} -10.2958 q^{71} +2.12230 q^{73} +4.85250 q^{77} +10.2860 q^{79} -6.20684 q^{83} -8.55749 q^{85} -4.30041 q^{89} +4.85250 q^{91} -5.42887 q^{95} -1.25092 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 + 2 * q^7	$ 4 q + 4 q^{5} + 2 q^{7} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 2 q^{19} + 8 q^{23} + 8 q^{25} + 14 q^{29} - 4 q^{31} + 18 q^{35} - 6 q^{37} - 6 q^{41} + 4 q^{43} + 2 q^{47} + 8 q^{49} + 4 q^{53} + 4 q^{55} + 12 q^{59} - 14 q^{61} + 4 q^{65} + 18 q^{71} + 10 q^{73} + 2 q^{77} + 20 q^{79} + 8 q^{83} + 18 q^{85} - 6 q^{89} + 2 q^{91} + 30 q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 2 * q^7 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 2 * q^19 + 8 * q^23 + 8 * q^25 + 14 * q^29 - 4 * q^31 + 18 * q^35 - 6 * q^37 - 6 * q^41 + 4 * q^43 + 2 * q^47 + 8 * q^49 + 4 * q^53 + 4 * q^55 + 12 * q^59 - 14 * q^61 + 4 * q^65 + 18 * q^71 + 10 * q^73 + 2 * q^77 + 20 * q^79 + 8 * q^83 + 18 * q^85 - 6 * q^89 + 2 * q^91 + 30 * q^95 - 6 * 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
$5$	2.18255	0.976066	0.488033	−	0.872825i	$-0.337715\pi$
$5$	2.18255	0.976066	0.488033	+	0.872825i	$0.337715\pi$
$6$	0	0
$7$	4.85250	1.83407	0.917036	−	0.398805i	$-0.130575\pi$
$7$	4.85250	1.83407	0.917036	+	0.398805i	$0.130575\pi$
$8$	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
$17$	−3.92087	−0.950951	−0.475475	−	0.879729i	$-0.657724\pi$
$17$	−3.92087	−0.950951	−0.475475	+	0.879729i	$0.657724\pi$
$18$	0	0
$19$	−2.48740	−0.570648	−0.285324	−	0.958431i	$-0.592101\pi$
$19$	−2.48740	−0.570648	−0.285324	+	0.958431i	$0.592101\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.10342	1.27265	0.636325	−	0.771421i	$-0.280454\pi$
$23$	6.10342	1.27265	0.636325	+	0.771421i	$0.280454\pi$
$24$	0	0
$25$	−0.236477	−0.0472953
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.66995	1.23858	0.619289	−	0.785163i	$-0.287421\pi$
$29$	6.66995	1.23858	0.619289	+	0.785163i	$0.287421\pi$
$30$	0	0
$31$	−8.02429	−1.44120	−0.720602	−	0.693348i	$-0.756135\pi$
$31$	−8.02429	−1.44120	−0.720602	+	0.693348i	$0.756135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.5908	1.79017
$36$	0	0
$37$	−1.25092	−0.205650	−0.102825	−	0.994699i	$-0.532788\pi$
$37$	−1.25092	−0.205650	−0.102825	+	0.994699i	$0.532788\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.85250	0.445485	0.222743	−	0.974877i	$-0.428499\pi$
$41$	2.85250	0.445485	0.222743	+	0.974877i	$0.428499\pi$
$42$	0	0
$43$	−1.92087	−0.292930	−0.146465	−	0.989216i	$-0.546790\pi$
$43$	−1.92087	−0.292930	−0.146465	+	0.989216i	$0.546790\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.95592	1.30635	0.653177	−	0.757205i	$-0.273436\pi$
$47$	8.95592	1.30635	0.653177	+	0.757205i	$0.273436\pi$
$48$	0	0
$49$	16.5467	2.36382
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.33990	1.00821	0.504106	−	0.863642i	$-0.331822\pi$
$53$	7.33990	1.00821	0.504106	+	0.863642i	$0.331822\pi$
$54$	0	0
$55$	2.18255	0.294295
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.13306	0.147511	0.0737557	−	0.997276i	$-0.476501\pi$
$59$	1.13306	0.147511	0.0737557	+	0.997276i	$0.476501\pi$
$60$	0	0
$61$	4.95592	0.634540	0.317270	−	0.948335i	$-0.397234\pi$
$61$	4.95592	0.634540	0.317270	+	0.948335i	$0.397234\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.18255	0.270712
$66$	0	0
$67$	3.04949	0.372555	0.186277	−	0.982497i	$-0.440358\pi$
$67$	3.04949	0.372555	0.186277	+	0.982497i	$0.440358\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.2958	−1.22189	−0.610944	−	0.791674i	$-0.709210\pi$
$71$	−10.2958	−1.22189	−0.610944	+	0.791674i	$0.709210\pi$
$72$	0	0
$73$	2.12230	0.248396	0.124198	−	0.992257i	$-0.460364\pi$
$73$	2.12230	0.248396	0.124198	+	0.992257i	$0.460364\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.85250	0.552993
$78$	0	0
$79$	10.2860	1.15726	0.578631	−	0.815589i	$-0.303587\pi$
$79$	10.2860	1.15726	0.578631	+	0.815589i	$0.303587\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−6.20684	−0.681289	−0.340645	−	0.940192i	$-0.610645\pi$
$83$	−6.20684	−0.681289	−0.340645	+	0.940192i	$0.610645\pi$
$84$	0	0
$85$	−8.55749	−0.928191
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.30041	−0.455843	−0.227922	−	0.973680i	$-0.573193\pi$
$89$	−4.30041	−0.455843	−0.227922	+	0.973680i	$0.573193\pi$
$90$	0	0
$91$	4.85250	0.508680
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.42887	−0.556990
$96$	0	0
$97$	−1.25092	−0.127012	−0.0635059	−	0.997981i	$-0.520228\pi$
$97$	−1.25092	−0.127012	−0.0635059	+	0.997981i	$0.520228\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.1277	1.60477	0.802384	−	0.596809i	$-0.203565\pi$
$101$	16.1277	1.60477	0.802384	+	0.596809i	$0.203565\pi$
$102$	0	0
$103$	4.47295	0.440733	0.220367	−	0.975417i	$-0.429275\pi$
$103$	4.47295	0.440733	0.220367	+	0.975417i	$0.429275\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.47295	−0.239069	−0.119535	−	0.992830i	$-0.538140\pi$
$107$	−2.47295	−0.239069	−0.119535	+	0.992830i	$0.538140\pi$
$108$	0	0
$109$	−15.8273	−1.51598	−0.757990	−	0.652266i	$-0.773818\pi$
$109$	−15.8273	−1.51598	−0.757990	+	0.652266i	$0.773818\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.838053	−0.0788374	−0.0394187	−	0.999223i	$-0.512551\pi$
$113$	−0.838053	−0.0788374	−0.0394187	+	0.999223i	$0.512551\pi$
$114$	0	0
$115$	13.3210	1.24219
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.0260	−1.74411
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.4289	−1.02223
$126$	0	0
$127$	−10.3749	−0.920627	−0.460314	−	0.887756i	$-0.652263\pi$
$127$	−10.3749	−0.920627	−0.460314	+	0.887756i	$0.652263\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19.0449	1.66396	0.831980	−	0.554806i	$-0.187207\pi$
$131$	19.0449	1.66396	0.831980	+	0.554806i	$0.187207\pi$
$132$	0	0
$133$	−12.0701	−1.04661
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.9765	1.53584	0.767919	−	0.640547i	$-0.221292\pi$
$137$	17.9765	1.53584	0.767919	+	0.640547i	$0.221292\pi$
$138$	0	0
$139$	−21.2419	−1.80171	−0.900857	−	0.434117i	$-0.857061\pi$
$139$	−21.2419	−1.80171	−0.900857	+	0.434117i	$0.857061\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	14.5575	1.20893
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.9515	0.897180	0.448590	−	0.893738i	$-0.351926\pi$
$149$	10.9515	0.897180	0.448590	+	0.893738i	$0.351926\pi$
$150$	0	0
$151$	0.989242	0.0805034	0.0402517	−	0.999190i	$-0.487184\pi$
$151$	0.989242	0.0805034	0.0402517	+	0.999190i	$0.487184\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−17.5134	−1.40671
$156$	0	0
$157$	−20.6754	−1.65007	−0.825037	−	0.565079i	$-0.808846\pi$
$157$	−20.6754	−1.65007	−0.825037	+	0.565079i	$0.808846\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	29.6168	2.33413
$162$	0	0
$163$	17.7104	1.38719	0.693593	−	0.720367i	$-0.256027\pi$
$163$	17.7104	1.38719	0.693593	+	0.720367i	$0.256027\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.26980	0.253025	0.126512	−	0.991965i	$-0.459622\pi$
$167$	3.26980	0.253025	0.126512	+	0.991965i	$0.459622\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.5629	−1.41131	−0.705656	−	0.708555i	$-0.749347\pi$
$173$	−18.5629	−1.41131	−0.705656	+	0.708555i	$0.749347\pi$
$174$	0	0
$175$	−1.14750	−0.0867430
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.67536	−0.349453	−0.174726	−	0.984617i	$-0.555904\pi$
$179$	−4.67536	−0.349453	−0.174726	+	0.984617i	$0.555904\pi$
$180$	0	0
$181$	−9.26537	−0.688689	−0.344345	−	0.938843i	$-0.611899\pi$
$181$	−9.26537	−0.688689	−0.344345	+	0.938843i	$0.611899\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.73020	−0.200728
$186$	0	0
$187$	−3.92087	−0.286722
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.33990	0.531096	0.265548	−	0.964098i	$-0.414447\pi$
$191$	7.33990	0.531096	0.265548	+	0.964098i	$0.414447\pi$
$192$	0	0
$193$	−15.7895	−1.13656	−0.568278	−	0.822837i	$-0.692390\pi$
$193$	−15.7895	−1.13656	−0.568278	+	0.822837i	$0.692390\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.1042	−1.57486	−0.787431	−	0.616403i	$-0.788589\pi$
$197$	−22.1042	−1.57486	−0.787431	+	0.616403i	$0.788589\pi$
$198$	0	0
$199$	−6.97480	−0.494430	−0.247215	−	0.968961i	$-0.579515\pi$
$199$	−6.97480	−0.494430	−0.247215	+	0.968961i	$0.579515\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	32.3659	2.27164
$204$	0	0
$205$	6.22572	0.434823
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.48740	−0.172057
$210$	0	0
$211$	2.68883	0.185107	0.0925533	−	0.995708i	$-0.470497\pi$
$211$	2.68883	0.185107	0.0925533	+	0.995708i	$0.470497\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.19239	−0.285919
$216$	0	0
$217$	−38.9378	−2.64327
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.92087	−0.263746
$222$	0	0
$223$	4.97940	0.333445	0.166723	−	0.986004i	$-0.446682\pi$
$223$	4.97940	0.333445	0.166723	+	0.986004i	$0.446682\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.71575	0.445740	0.222870	−	0.974848i	$-0.428457\pi$
$227$	6.71575	0.445740	0.222870	+	0.974848i	$0.428457\pi$
$228$	0	0
$229$	−7.63754	−0.504703	−0.252351	−	0.967636i	$-0.581204\pi$
$229$	−7.63754	−0.504703	−0.252351	+	0.967636i	$0.581204\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.4450	−1.47042	−0.735212	−	0.677838i	$-0.762917\pi$
$233$	−22.4450	−1.47042	−0.735212	+	0.677838i	$0.762917\pi$
$234$	0	0
$235$	19.5467	1.27509
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.6906	1.01494	0.507469	−	0.861670i	$-0.330581\pi$
$239$	15.6906	1.01494	0.507469	+	0.861670i	$0.330581\pi$
$240$	0	0
$241$	16.9370	1.09101	0.545505	−	0.838107i	$-0.316338\pi$
$241$	16.9370	1.09101	0.545505	+	0.838107i	$0.316338\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	36.1141	2.30724
$246$	0	0
$247$	−2.48740	−0.158269
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.34358	0.526642	0.263321	−	0.964708i	$-0.415182\pi$
$251$	8.34358	0.526642	0.263321	+	0.964708i	$0.415182\pi$
$252$	0	0
$253$	6.10342	0.383719
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.9407	−0.994355	−0.497178	−	0.867649i	$-0.665630\pi$
$257$	−15.9407	−0.994355	−0.497178	+	0.867649i	$0.665630\pi$
$258$	0	0
$259$	−6.07010	−0.377177
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.81285	−0.235110	−0.117555	−	0.993066i	$-0.537506\pi$
$263$	−3.81285	−0.235110	−0.117555	+	0.993066i	$0.537506\pi$
$264$	0	0
$265$	16.0197	0.984082
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.54674	−0.0943061	−0.0471531	−	0.998888i	$-0.515015\pi$
$269$	−1.54674	−0.0943061	−0.0471531	+	0.998888i	$0.515015\pi$
$270$	0	0
$271$	−14.3992	−0.874691	−0.437346	−	0.899294i	$-0.644081\pi$
$271$	−14.3992	−0.874691	−0.437346	+	0.899294i	$0.644081\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.236477	−0.0142601
$276$	0	0
$277$	27.8829	1.67532	0.837662	−	0.546189i	$-0.183922\pi$
$277$	27.8829	1.67532	0.837662	+	0.546189i	$0.183922\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.3255	0.675620	0.337810	−	0.941214i	$-0.390314\pi$
$281$	11.3255	0.675620	0.337810	+	0.941214i	$0.390314\pi$
$282$	0	0
$283$	18.0584	1.07346	0.536731	−	0.843753i	$-0.319659\pi$
$283$	18.0584	1.07346	0.536731	+	0.843753i	$0.319659\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.8417	0.817052
$288$	0	0
$289$	−1.62678	−0.0956929
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.4496	1.31152	0.655761	−	0.754969i	$-0.272348\pi$
$293$	22.4496	1.31152	0.655761	+	0.754969i	$0.272348\pi$
$294$	0	0
$295$	2.47295	0.143981
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.10342	0.352970
$300$	0	0
$301$	−9.32102	−0.537254
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.8165	0.619353
$306$	0	0
$307$	−26.6565	−1.52137	−0.760683	−	0.649124i	$-0.775136\pi$
$307$	−26.6565	−1.52137	−0.760683	+	0.649124i	$0.775136\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.07822	−0.174550	−0.0872748	−	0.996184i	$-0.527816\pi$
$311$	−3.07822	−0.174550	−0.0872748	+	0.996184i	$0.527816\pi$
$312$	0	0
$313$	1.85882	0.105067	0.0525334	−	0.998619i	$-0.483270\pi$
$313$	1.85882	0.105067	0.0525334	+	0.998619i	$0.483270\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.9802	1.06603	0.533017	−	0.846104i	$-0.321058\pi$
$317$	18.9802	1.06603	0.533017	+	0.846104i	$0.321058\pi$
$318$	0	0
$319$	6.66995	0.373445
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.75277	0.542658
$324$	0	0
$325$	−0.236477	−0.0131174
$326$	0	0
$327$	0	0
$328$	0	0
$329$	43.4586	2.39595
$330$	0	0
$331$	−10.2715	−0.564574	−0.282287	−	0.959330i	$-0.591093\pi$
$331$	−10.2715	−0.564574	−0.282287	+	0.959330i	$0.591093\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.65567	0.363638
$336$	0	0
$337$	−19.3110	−1.05194	−0.525969	−	0.850504i	$-0.676297\pi$
$337$	−19.3110	−1.05194	−0.525969	+	0.850504i	$0.676297\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.02429	−0.434540
$342$	0	0
$343$	46.3255	2.50134
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.9018	1.71258	0.856290	−	0.516496i	$-0.172764\pi$
$347$	31.9018	1.71258	0.856290	+	0.516496i	$0.172764\pi$
$348$	0	0
$349$	−12.9748	−0.694525	−0.347262	−	0.937768i	$-0.612889\pi$
$349$	−12.9748	−0.694525	−0.347262	+	0.937768i	$0.612889\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.9954	−1.59649	−0.798247	−	0.602330i	$-0.794239\pi$
$353$	−29.9954	−1.59649	−0.798247	+	0.602330i	$0.794239\pi$
$354$	0	0
$355$	−22.4711	−1.19264
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.03596	0.213010	0.106505	−	0.994312i	$-0.466034\pi$
$359$	4.03596	0.213010	0.106505	+	0.994312i	$0.466034\pi$
$360$	0	0
$361$	−12.8128	−0.674360
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.63202	0.242451
$366$	0	0
$367$	30.1888	1.57584	0.787920	−	0.615777i	$-0.211158\pi$
$367$	30.1888	1.57584	0.787920	+	0.615777i	$0.211158\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	35.6168	1.84913
$372$	0	0
$373$	7.82022	0.404916	0.202458	−	0.979291i	$-0.435107\pi$
$373$	7.82022	0.404916	0.202458	+	0.979291i	$0.435107\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.66995	0.343520
$378$	0	0
$379$	29.2875	1.50440	0.752200	−	0.658935i	$-0.228993\pi$
$379$	29.2875	1.50440	0.752200	+	0.658935i	$0.228993\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−27.4389	−1.40206	−0.701031	−	0.713131i	$-0.747276\pi$
$383$	−27.4389	−1.40206	−0.701031	+	0.713131i	$0.747276\pi$
$384$	0	0
$385$	10.5908	0.539758
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−16.3651	−0.829743	−0.414872	−	0.909880i	$-0.636174\pi$
$389$	−16.3651	−0.829743	−0.414872	+	0.909880i	$0.636174\pi$
$390$	0	0
$391$	−23.9307	−1.21023
$392$	0	0
$393$	0	0
$394$	0	0
$395$	22.4496	1.12956
$396$	0	0
$397$	−20.1664	−1.01212	−0.506062	−	0.862497i	$-0.668899\pi$
$397$	−20.1664	−1.01212	−0.506062	+	0.862497i	$0.668899\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−21.4720	−1.07226	−0.536131	−	0.844135i	$-0.680115\pi$
$401$	−21.4720	−1.07226	−0.536131	+	0.844135i	$0.680115\pi$
$402$	0	0
$403$	−8.02429	−0.399718
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.25092	−0.0620059
$408$	0	0
$409$	12.8740	0.636579	0.318290	−	0.947994i	$-0.396892\pi$
$409$	12.8740	0.636579	0.318290	+	0.947994i	$0.396892\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.49816	0.270547
$414$	0	0
$415$	−13.5467	−0.664983
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.3892	1.48461	0.742305	−	0.670062i	$-0.233732\pi$
$419$	30.3892	1.48461	0.742305	+	0.670062i	$0.233732\pi$
$420$	0	0
$421$	−34.0305	−1.65855	−0.829273	−	0.558844i	$-0.811245\pi$
$421$	−34.0305	−1.65855	−0.829273	+	0.558844i	$0.811245\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.927194	0.0449755
$426$	0	0
$427$	24.0486	1.16379
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.73020	−0.420519	−0.210259	−	0.977646i	$-0.567431\pi$
$431$	−8.73020	−0.420519	−0.210259	+	0.977646i	$0.567431\pi$
$432$	0	0
$433$	23.1439	1.11222	0.556112	−	0.831107i	$-0.312293\pi$
$433$	23.1439	1.11222	0.556112	+	0.831107i	$0.312293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.1816	−0.726236
$438$	0	0
$439$	−26.3345	−1.25688	−0.628440	−	0.777858i	$-0.716306\pi$
$439$	−26.3345	−1.25688	−0.628440	+	0.777858i	$0.716306\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.3633	−1.44260	−0.721301	−	0.692621i	$-0.756456\pi$
$443$	−30.3633	−1.44260	−0.721301	+	0.692621i	$0.756456\pi$
$444$	0	0
$445$	−9.38587	−0.444933
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.6592	0.833389	0.416694	−	0.909047i	$-0.363189\pi$
$449$	17.6592	0.833389	0.416694	+	0.909047i	$0.363189\pi$
$450$	0	0
$451$	2.85250	0.134319
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.5908	0.496505
$456$	0	0
$457$	12.9586	0.606176	0.303088	−	0.952963i	$-0.401982\pi$
$457$	12.9586	0.606176	0.303088	+	0.952963i	$0.401982\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.52860	0.210918	0.105459	−	0.994424i	$-0.466369\pi$
$461$	4.52860	0.210918	0.105459	+	0.994424i	$0.466369\pi$
$462$	0	0
$463$	−32.0054	−1.48742	−0.743709	−	0.668504i	$-0.766935\pi$
$463$	−32.0054	−1.48742	−0.743709	+	0.668504i	$0.766935\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.0405	1.25128	0.625642	−	0.780111i	$-0.284837\pi$
$467$	27.0405	1.25128	0.625642	+	0.780111i	$0.284837\pi$
$468$	0	0
$469$	14.7977	0.683292
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.92087	−0.0883217
$474$	0	0
$475$	0.588211	0.0269890
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.4029	1.06931	0.534653	−	0.845071i	$-0.320442\pi$
$479$	23.4029	1.06931	0.534653	+	0.845071i	$0.320442\pi$
$480$	0	0
$481$	−1.25092	−0.0570371
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.73020	−0.123972
$486$	0	0
$487$	42.1618	1.91054	0.955268	−	0.295742i	$-0.0955670\pi$
$487$	42.1618	1.91054	0.955268	+	0.295742i	$0.0955670\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.0475	1.22064	0.610319	−	0.792156i	$-0.291041\pi$
$491$	27.0475	1.22064	0.610319	+	0.792156i	$0.291041\pi$
$492$	0	0
$493$	−26.1520	−1.17783
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−49.9604	−2.24103
$498$	0	0
$499$	4.97940	0.222908	0.111454	−	0.993770i	$-0.464449\pi$
$499$	4.97940	0.222908	0.111454	+	0.993770i	$0.464449\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.2769	−0.458226	−0.229113	−	0.973400i	$-0.573583\pi$
$503$	−10.2769	−0.458226	−0.229113	+	0.973400i	$0.573583\pi$
$504$	0	0
$505$	35.1995	1.56636
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.0718	−1.37723	−0.688617	−	0.725125i	$-0.741782\pi$
$509$	−31.0718	−1.37723	−0.688617	+	0.725125i	$0.741782\pi$
$510$	0	0
$511$	10.2985	0.455577
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.76244	0.430185
$516$	0	0
$517$	8.95592	0.393881
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.0935	0.661257	0.330628	−	0.943761i	$-0.392739\pi$
$521$	15.0935	0.661257	0.330628	+	0.943761i	$0.392739\pi$
$522$	0	0
$523$	−30.2390	−1.32226	−0.661130	−	0.750272i	$-0.729923\pi$
$523$	−30.2390	−1.32226	−0.661130	+	0.750272i	$0.729923\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.4622	1.37051
$528$	0	0
$529$	14.2517	0.619640
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.85250	0.123555
$534$	0	0
$535$	−5.39734	−0.233348
$536$	0	0
$537$	0	0
$538$	0	0
$539$	16.5467	0.712718
$540$	0	0
$541$	−3.70500	−0.159290	−0.0796451	−	0.996823i	$-0.525379\pi$
$541$	−3.70500	−0.159290	−0.0796451	+	0.996823i	$0.525379\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−34.5439	−1.47970
$546$	0	0
$547$	−12.1377	−0.518971	−0.259486	−	0.965747i	$-0.583553\pi$
$547$	−12.1377	−0.518971	−0.259486	+	0.965747i	$0.583553\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.5908	−0.706793
$552$	0	0
$553$	49.9126	2.12250
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.5809	−0.829668	−0.414834	−	0.909897i	$-0.636160\pi$
$557$	−19.5809	−0.829668	−0.414834	+	0.909897i	$0.636160\pi$
$558$	0	0
$559$	−1.92087	−0.0812441
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.3848	−0.859116	−0.429558	−	0.903039i	$-0.641331\pi$
$563$	−20.3848	−0.859116	−0.429558	+	0.903039i	$0.641331\pi$
$564$	0	0
$565$	−1.82909	−0.0769505
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.0881	1.76443	0.882213	−	0.470851i	$-0.156053\pi$
$569$	42.0881	1.76443	0.882213	+	0.470851i	$0.156053\pi$
$570$	0	0
$571$	24.4910	1.02492	0.512458	−	0.858712i	$-0.328735\pi$
$571$	24.4910	1.02492	0.512458	+	0.858712i	$0.328735\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.44332	−0.0601904
$576$	0	0
$577$	3.89215	0.162032	0.0810161	−	0.996713i	$-0.474183\pi$
$577$	3.89215	0.162032	0.0810161	+	0.996713i	$0.474183\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.1187	−1.24953
$582$	0	0
$583$	7.33990	0.303987
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.6872	0.688753	0.344377	−	0.938832i	$-0.388090\pi$
$587$	16.6872	0.688753	0.344377	+	0.938832i	$0.388090\pi$
$588$	0	0
$589$	19.9596	0.822421
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−37.1672	−1.52627	−0.763137	−	0.646237i	$-0.776342\pi$
$593$	−37.1672	−1.52627	−0.763137	+	0.646237i	$0.776342\pi$
$594$	0	0
$595$	−41.5252	−1.70237
$596$	0	0
$597$	0	0
$598$	0	0
$599$	39.8938	1.63002	0.815008	−	0.579450i	$-0.196733\pi$
$599$	39.8938	1.63002	0.815008	+	0.579450i	$0.196733\pi$
$600$	0	0
$601$	−29.2517	−1.19320	−0.596602	−	0.802538i	$-0.703483\pi$
$601$	−29.2517	−1.19320	−0.596602	+	0.802538i	$0.703483\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.18255	0.0887333
$606$	0	0
$607$	−2.15924	−0.0876407	−0.0438204	−	0.999039i	$-0.513953\pi$
$607$	−2.15924	−0.0876407	−0.0438204	+	0.999039i	$0.513953\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.95592	0.362318
$612$	0	0
$613$	−8.52860	−0.344467	−0.172234	−	0.985056i	$-0.555098\pi$
$613$	−8.52860	−0.344467	−0.172234	+	0.985056i	$0.555098\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.972023	−0.0391322	−0.0195661	−	0.999809i	$-0.506228\pi$
$617$	−0.972023	−0.0391322	−0.0195661	+	0.999809i	$0.506228\pi$
$618$	0	0
$619$	−2.63662	−0.105975	−0.0529874	−	0.998595i	$-0.516874\pi$
$619$	−2.63662	−0.105975	−0.0529874	+	0.998595i	$0.516874\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−20.8678	−0.836049
$624$	0	0
$625$	−23.7617	−0.950468
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.90470	0.195563
$630$	0	0
$631$	−6.91011	−0.275087	−0.137544	−	0.990496i	$-0.543921\pi$
$631$	−6.91011	−0.275087	−0.137544	+	0.990496i	$0.543921\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22.6438	−0.898593
$636$	0	0
$637$	16.5467	0.655606
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.9118	−1.18145	−0.590723	−	0.806874i	$-0.701157\pi$
$641$	−29.9118	−1.18145	−0.590723	+	0.806874i	$0.701157\pi$
$642$	0	0
$643$	5.72929	0.225941	0.112970	−	0.993598i	$-0.463963\pi$
$643$	5.72929	0.225941	0.112970	+	0.993598i	$0.463963\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.9567	−1.21703	−0.608517	−	0.793541i	$-0.708235\pi$
$647$	−30.9567	−1.21703	−0.608517	+	0.793541i	$0.708235\pi$
$648$	0	0
$649$	1.13306	0.0444764
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−37.3433	−1.46136	−0.730679	−	0.682721i	$-0.760796\pi$
$653$	−37.3433	−1.46136	−0.730679	+	0.682721i	$0.760796\pi$
$654$	0	0
$655$	41.5664	1.62413
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.56930	−0.333813	−0.166906	−	0.985973i	$-0.553378\pi$
$659$	−8.56930	−0.333813	−0.166906	+	0.985973i	$0.553378\pi$
$660$	0	0
$661$	11.3885	0.442960	0.221480	−	0.975165i	$-0.428911\pi$
$661$	11.3885	0.442960	0.221480	+	0.975165i	$0.428911\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−26.3436	−1.02156
$666$	0	0
$667$	40.7095	1.57628
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.95592	0.191321
$672$	0	0
$673$	6.77059	0.260987	0.130494	−	0.991449i	$-0.458344\pi$
$673$	6.77059	0.260987	0.130494	+	0.991449i	$0.458344\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.2796	−0.894709	−0.447355	−	0.894357i	$-0.647634\pi$
$677$	−23.2796	−0.894709	−0.447355	+	0.894357i	$0.647634\pi$
$678$	0	0
$679$	−6.07010	−0.232949
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.1160	1.53500	0.767499	−	0.641050i	$-0.221501\pi$
$683$	40.1160	1.53500	0.767499	+	0.641050i	$0.221501\pi$
$684$	0	0
$685$	39.2346	1.49908
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.33990	0.279628
$690$	0	0
$691$	15.7805	0.600319	0.300159	−	0.953889i	$-0.402960\pi$
$691$	15.7805	0.600319	0.300159	+	0.953889i	$0.402960\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−46.3615	−1.75859
$696$	0	0
$697$	−11.1843	−0.423635
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0.00720737	0.000272218 0	0.000136109	−	1.00000i	$-0.499957\pi$
$701$	0.00720737	0.000272218 0	0.000136109		1.00000i	$0.499957\pi$
$702$	0	0
$703$	3.11154	0.117354
$704$	0	0
$705$	0	0
$706$	0	0
$707$	78.2597	2.94326
$708$	0	0
$709$	41.6268	1.56333	0.781664	−	0.623699i	$-0.214371\pi$
$709$	41.6268	1.56333	0.781664	+	0.623699i	$0.214371\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−48.9756	−1.83415
$714$	0	0
$715$	2.18255	0.0816227
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30.2878	−1.12954	−0.564771	−	0.825247i	$-0.691036\pi$
$719$	−30.2878	−1.12954	−0.564771	+	0.825247i	$0.691036\pi$
$720$	0	0
$721$	21.7050	0.808336
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.57729	−0.0585790
$726$	0	0
$727$	−37.2732	−1.38239	−0.691194	−	0.722669i	$-0.742915\pi$
$727$	−37.2732	−1.38239	−0.691194	+	0.722669i	$0.742915\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.53148	0.278562
$732$	0	0
$733$	28.4189	1.04968	0.524838	−	0.851202i	$-0.324126\pi$
$733$	28.4189	1.04968	0.524838	+	0.851202i	$0.324126\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.04949	0.112330
$738$	0	0
$739$	14.3795	0.528960	0.264480	−	0.964391i	$-0.414800\pi$
$739$	14.3795	0.528960	0.264480	+	0.964391i	$0.414800\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.4352	−0.749695	−0.374847	−	0.927087i	$-0.622305\pi$
$743$	−20.4352	−0.749695	−0.374847	+	0.927087i	$0.622305\pi$
$744$	0	0
$745$	23.9022	0.875707
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−41.6277	−1.51901	−0.759507	−	0.650499i	$-0.774560\pi$
$751$	−41.6277	−1.51901	−0.759507	+	0.650499i	$0.774560\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.15907	0.0785766
$756$	0	0
$757$	−34.6969	−1.26108	−0.630540	−	0.776157i	$-0.717166\pi$
$757$	−34.6969	−1.26108	−0.630540	+	0.776157i	$0.717166\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−49.8255	−1.80617	−0.903086	−	0.429460i	$-0.858704\pi$
$761$	−49.8255	−1.80617	−0.903086	+	0.429460i	$0.858704\pi$
$762$	0	0
$763$	−76.8019	−2.78042
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.13306	0.0409123
$768$	0	0
$769$	−52.5324	−1.89437	−0.947183	−	0.320695i	$-0.896084\pi$
$769$	−52.5324	−1.89437	−0.947183	+	0.320695i	$0.896084\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.0684	−0.470037	−0.235018	−	0.971991i	$-0.575515\pi$
$773$	−13.0684	−0.470037	−0.235018	+	0.971991i	$0.575515\pi$
$774$	0	0
$775$	1.89756	0.0681622
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.09530	−0.254215
$780$	0	0
$781$	−10.2958	−0.368413
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−45.1250	−1.61058
$786$	0	0
$787$	−2.06302	−0.0735389	−0.0367694	−	0.999324i	$-0.511707\pi$
$787$	−2.06302	−0.0735389	−0.0367694	+	0.999324i	$0.511707\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.06665	−0.144593
$792$	0	0
$793$	4.95592	0.175990
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.52705	0.124934	0.0624672	−	0.998047i	$-0.480103\pi$
$797$	3.52705	0.124934	0.0624672	+	0.998047i	$0.480103\pi$
$798$	0	0
$799$	−35.1150	−1.24228
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.12230	0.0748943
$804$	0	0
$805$	64.6402	2.27827
$806$	0	0
$807$	0	0
$808$	0	0
$809$	23.1348	0.813378	0.406689	−	0.913567i	$-0.366683\pi$
$809$	23.1348	0.813378	0.406689	+	0.913567i	$0.366683\pi$
$810$	0	0
$811$	−9.54149	−0.335047	−0.167524	−	0.985868i	$-0.553577\pi$
$811$	−9.54149	−0.335047	−0.167524	+	0.985868i	$0.553577\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	38.6538	1.35399
$816$	0	0
$817$	4.77797	0.167160
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.52342	0.122968	0.0614841	−	0.998108i	$-0.480417\pi$
$821$	3.52342	0.122968	0.0614841	+	0.998108i	$0.480417\pi$
$822$	0	0
$823$	−3.62603	−0.126396	−0.0631978	−	0.998001i	$-0.520130\pi$
$823$	−3.62603	−0.126396	−0.0631978	+	0.998001i	$0.520130\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.6276	0.856385	0.428193	−	0.903688i	$-0.359150\pi$
$827$	24.6276	0.856385	0.428193	+	0.903688i	$0.359150\pi$
$828$	0	0
$829$	15.6761	0.544454	0.272227	−	0.962233i	$-0.412240\pi$
$829$	15.6761	0.544454	0.272227	+	0.962233i	$0.412240\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−64.8776	−2.24788
$834$	0	0
$835$	7.13650	0.246969
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.0234	−1.41628	−0.708142	−	0.706070i	$-0.750466\pi$
$839$	−41.0234	−1.41628	−0.708142	+	0.706070i	$0.750466\pi$
$840$	0	0
$841$	15.4882	0.534076
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.18255	0.0750820
$846$	0	0
$847$	4.85250	0.166734
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7.63490	−0.261721
$852$	0	0
$853$	−5.22317	−0.178838	−0.0894190	−	0.995994i	$-0.528501\pi$
$853$	−5.22317	−0.178838	−0.0894190	+	0.995994i	$0.528501\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.6392	−1.38821	−0.694105	−	0.719874i	$-0.744200\pi$
$857$	−40.6392	−1.38821	−0.694105	+	0.719874i	$0.744200\pi$
$858$	0	0
$859$	−0.652730	−0.0222708	−0.0111354	−	0.999938i	$-0.503545\pi$
$859$	−0.652730	−0.0222708	−0.0111354	+	0.999938i	$0.503545\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.1224	−0.650933	−0.325466	−	0.945554i	$-0.605521\pi$
$863$	−19.1224	−0.650933	−0.325466	+	0.945554i	$0.605521\pi$
$864$	0	0
$865$	−40.5145	−1.37753
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10.2860	0.348928
$870$	0	0
$871$	3.04949	0.103328
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−55.4586	−1.87484
$876$	0	0
$877$	−34.5701	−1.16735	−0.583675	−	0.811987i	$-0.698386\pi$
$877$	−34.5701	−1.16735	−0.583675	+	0.811987i	$0.698386\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29.8741	−1.00648	−0.503242	−	0.864146i	$-0.667859\pi$
$881$	−29.8741	−1.00648	−0.503242	+	0.864146i	$0.667859\pi$
$882$	0	0
$883$	8.50184	0.286110	0.143055	−	0.989715i	$-0.454307\pi$
$883$	8.50184	0.286110	0.143055	+	0.989715i	$0.454307\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.1195	1.75000	0.875000	−	0.484122i	$-0.160861\pi$
$887$	52.1195	1.75000	0.875000	+	0.484122i	$0.160861\pi$
$888$	0	0
$889$	−50.3444	−1.68850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−22.2769	−0.745469
$894$	0	0
$895$	−10.2042	−0.341089
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−53.5216	−1.78505
$900$	0	0
$901$	−28.7788	−0.958760
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20.2221	−0.672206
$906$	0	0
$907$	−18.6150	−0.618100	−0.309050	−	0.951046i	$-0.600011\pi$
$907$	−18.6150	−0.618100	−0.309050	+	0.951046i	$0.600011\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−39.5178	−1.30928	−0.654642	−	0.755939i	$-0.727181\pi$
$911$	−39.5178	−1.30928	−0.654642	+	0.755939i	$0.727181\pi$
$912$	0	0
$913$	−6.20684	−0.205416
$914$	0	0
$915$	0	0
$916$	0	0
$917$	92.4153	3.05182
$918$	0	0
$919$	31.0621	1.02464	0.512322	−	0.858793i	$-0.328785\pi$
$919$	31.0621	1.02464	0.512322	+	0.858793i	$0.328785\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.2958	−0.338891
$924$	0	0
$925$	0.295814	0.00972629
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−21.6295	−0.709641	−0.354820	−	0.934935i	$-0.615458\pi$
$929$	−21.6295	−0.709641	−0.354820	+	0.934935i	$0.615458\pi$
$930$	0	0
$931$	−41.1583	−1.34891
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.55749	−0.279860
$936$	0	0
$937$	22.9147	0.748591	0.374296	−	0.927309i	$-0.377885\pi$
$937$	22.9147	0.748591	0.374296	+	0.927309i	$0.377885\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	28.9423	0.943491	0.471746	−	0.881735i	$-0.343624\pi$
$941$	28.9423	0.943491	0.471746	+	0.881735i	$0.343624\pi$
$942$	0	0
$943$	17.4100	0.566947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.7024	−0.900206	−0.450103	−	0.892977i	$-0.648613\pi$
$947$	−27.7024	−0.900206	−0.450103	+	0.892977i	$0.648613\pi$
$948$	0	0
$949$	2.12230	0.0688927
$950$	0	0
$951$	0	0
$952$	0	0
$953$	49.1248	1.59131	0.795655	−	0.605750i	$-0.207127\pi$
$953$	49.1248	1.59131	0.795655	+	0.605750i	$0.207127\pi$
$954$	0	0
$955$	16.0197	0.518385
$956$	0	0
$957$	0	0
$958$	0	0
$959$	87.2310	2.81684
$960$	0	0
$961$	33.3892	1.07707
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−34.4614	−1.10935
$966$	0	0
$967$	−49.6436	−1.59643	−0.798215	−	0.602372i	$-0.794222\pi$
$967$	−49.6436	−1.59643	−0.798215	+	0.602372i	$0.794222\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	52.4720	1.68391	0.841953	−	0.539551i	$-0.181406\pi$
$971$	52.4720	1.68391	0.841953	+	0.539551i	$0.181406\pi$
$972$	0	0
$973$	−103.076	−3.30447
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.1565	−0.612872	−0.306436	−	0.951891i	$-0.599137\pi$
$977$	−19.1565	−0.612872	−0.306436	+	0.951891i	$0.599137\pi$
$978$	0	0
$979$	−4.30041	−0.137442
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25.1636	0.802593	0.401297	−	0.915948i	$-0.368560\pi$
$983$	25.1636	0.802593	0.401297	+	0.915948i	$0.368560\pi$
$984$	0	0
$985$	−48.2436	−1.53717
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.7239	−0.372798
$990$	0	0
$991$	−31.5683	−1.00280	−0.501399	−	0.865216i	$-0.667181\pi$
$991$	−31.5683	−1.00280	−0.501399	+	0.865216i	$0.667181\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.2228	−0.482596
$996$	0	0
$997$	−28.1187	−0.890527	−0.445264	−	0.895399i	$-0.646890\pi$
$997$	−28.1187	−0.890527	−0.445264	+	0.895399i	$0.646890\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5148.2.a.r.1.3	yes	4
3.2	odd	2		5148.2.a.p.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5148.2.a.p.1.2	✓	4	3.2	odd	2
5148.2.a.r.1.3	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5148 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5148.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18255 q^{5} +4.85250 q^{7} +O(q^{10})\)	\(q+2.18255 q^{5} +4.85250 q^{7} +1.00000 q^{11} +1.00000 q^{13} -3.92087 q^{17} -2.48740 q^{19} +6.10342 q^{23} -0.236477 q^{25} +6.66995 q^{29} -8.02429 q^{31} +10.5908 q^{35} -1.25092 q^{37} +2.85250 q^{41} -1.92087 q^{43} +8.95592 q^{47} +16.5467 q^{49} +7.33990 q^{53} +2.18255 q^{55} +1.13306 q^{59} +4.95592 q^{61} +2.18255 q^{65} +3.04949 q^{67} -10.2958 q^{71} +2.12230 q^{73} +4.85250 q^{77} +10.2860 q^{79} -6.20684 q^{83} -8.55749 q^{85} -4.30041 q^{89} +4.85250 q^{91} -5.42887 q^{95} -1.25092 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 + 2 * q^7	\( 4 q + 4 q^{5} + 2 q^{7} + 4 q^{11} + 4 q^{13} - 4 q^{17} - 2 q^{19} + 8 q^{23} + 8 q^{25} + 14 q^{29} - 4 q^{31} + 18 q^{35} - 6 q^{37} - 6 q^{41} + 4 q^{43} + 2 q^{47} + 8 q^{49} + 4 q^{53} + 4 q^{55} + 12 q^{59} - 14 q^{61} + 4 q^{65} + 18 q^{71} + 10 q^{73} + 2 q^{77} + 20 q^{79} + 8 q^{83} + 18 q^{85} - 6 q^{89} + 2 q^{91} + 30 q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 2 * q^7 + 4 * q^11 + 4 * q^13 - 4 * q^17 - 2 * q^19 + 8 * q^23 + 8 * q^25 + 14 * q^29 - 4 * q^31 + 18 * q^35 - 6 * q^37 - 6 * q^41 + 4 * q^43 + 2 * q^47 + 8 * q^49 + 4 * q^53 + 4 * q^55 + 12 * q^59 - 14 * q^61 + 4 * q^65 + 18 * q^71 + 10 * q^73 + 2 * q^77 + 20 * q^79 + 8 * q^83 + 18 * q^85 - 6 * q^89 + 2 * q^91 + 30 * q^95 - 6 * q^97

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