LMFDB - Embedded newform 7920.2.a.cn.1.3

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:	$4$
Coefficient field:	4.4.48704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 4x + 6 $ x^4 - 2x^3 - 6x^2 + 4*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.26270$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} -0.704647 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -0.704647 q^{7} +1.00000 q^{11} +3.82075 q^{13} -7.15733 q^{17} -7.33659 q^{19} -5.86198 q^{23} +1.00000 q^{25} +8.97808 q^{29} +0.590706 q^{31} -0.704647 q^{35} +10.4527 q^{37} -7.92729 q^{41} -3.29535 q^{43} -3.64149 q^{47} -6.50347 q^{49} +11.8620 q^{53} +1.00000 q^{55} +10.8112 q^{59} +5.64149 q^{61} +3.82075 q^{65} +10.9128 q^{67} +11.8620 q^{71} +3.82075 q^{73} -0.704647 q^{77} +3.33659 q^{79} +4.04124 q^{83} -7.15733 q^{85} -7.05079 q^{89} -2.69228 q^{91} -7.33659 q^{95} +6.81120 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^7	$ 4 q + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 4 q^{29} - 4 q^{35} + 8 q^{37} - 4 q^{41} - 12 q^{43} + 20 q^{49} + 16 q^{53} + 4 q^{55} + 24 q^{59} + 8 q^{61} + 8 q^{65} + 16 q^{71} + 8 q^{73} - 4 q^{77} - 12 q^{79} - 8 q^{83} + 4 q^{85} - 16 q^{89} + 16 q^{91} - 4 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 4 * q^29 - 4 * q^35 + 8 * q^37 - 4 * q^41 - 12 * q^43 + 20 * q^49 + 16 * q^53 + 4 * q^55 + 24 * q^59 + 8 * q^61 + 8 * q^65 + 16 * q^71 + 8 * q^73 - 4 * q^77 - 12 * q^79 - 8 * q^83 + 4 * q^85 - 16 * q^89 + 16 * q^91 - 4 * q^95 + 8 * 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
$7$	−0.704647	−0.266332	−0.133166	−	0.991094i	$-0.542514\pi$
$7$	−0.704647	−0.266332	−0.133166	+	0.991094i	$0.542514\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.82075	1.05968	0.529842	−	0.848096i	$-0.322251\pi$
$13$	3.82075	1.05968	0.529842	+	0.848096i	$0.322251\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.15733	−1.73591	−0.867954	−	0.496644i	$-0.834565\pi$
$17$	−7.15733	−1.73591	−0.867954	+	0.496644i	$0.834565\pi$
$18$	0	0
$19$	−7.33659	−1.68313	−0.841564	−	0.540157i	$-0.818365\pi$
$19$	−7.33659	−1.68313	−0.841564	+	0.540157i	$0.818365\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.86198	−1.22231	−0.611154	−	0.791512i	$-0.709294\pi$
$23$	−5.86198	−1.22231	−0.611154	+	0.791512i	$0.709294\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.97808	1.66719	0.833594	−	0.552378i	$-0.186279\pi$
$29$	8.97808	1.66719	0.833594	+	0.552378i	$0.186279\pi$
$30$	0	0
$31$	0.590706	0.106094	0.0530470	−	0.998592i	$-0.483107\pi$
$31$	0.590706	0.106094	0.0530470	+	0.998592i	$0.483107\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.704647	−0.119107
$36$	0	0
$37$	10.4527	1.71841	0.859206	−	0.511630i	$-0.170958\pi$
$37$	10.4527	1.71841	0.859206	+	0.511630i	$0.170958\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.92729	−1.23804	−0.619018	−	0.785377i	$-0.712469\pi$
$41$	−7.92729	−1.23804	−0.619018	+	0.785377i	$0.712469\pi$
$42$	0	0
$43$	−3.29535	−0.502537	−0.251268	−	0.967917i	$-0.580848\pi$
$43$	−3.29535	−0.502537	−0.251268	+	0.967917i	$0.580848\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.64149	−0.531166	−0.265583	−	0.964088i	$-0.585564\pi$
$47$	−3.64149	−0.531166	−0.265583	+	0.964088i	$0.585564\pi$
$48$	0	0
$49$	−6.50347	−0.929068
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.8620	1.62937	0.814684	−	0.579905i	$-0.196910\pi$
$53$	11.8620	1.62937	0.814684	+	0.579905i	$0.196910\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.8112	1.40750	0.703749	−	0.710449i	$-0.251508\pi$
$59$	10.8112	1.40750	0.703749	+	0.710449i	$0.251508\pi$
$60$	0	0
$61$	5.64149	0.722319	0.361159	−	0.932504i	$-0.382381\pi$
$61$	5.64149	0.722319	0.361159	+	0.932504i	$0.382381\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.82075	0.473905
$66$	0	0
$67$	10.9128	1.33321	0.666603	−	0.745413i	$-0.267748\pi$
$67$	10.9128	1.33321	0.666603	+	0.745413i	$0.267748\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.8620	1.40776	0.703879	−	0.710320i	$-0.251450\pi$
$71$	11.8620	1.40776	0.703879	+	0.710320i	$0.251450\pi$
$72$	0	0
$73$	3.82075	0.447184	0.223592	−	0.974683i	$-0.428222\pi$
$73$	3.82075	0.447184	0.223592	+	0.974683i	$0.428222\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.704647	−0.0803020
$78$	0	0
$79$	3.33659	0.375396	0.187698	−	0.982227i	$-0.439897\pi$
$79$	3.33659	0.375396	0.187698	+	0.982227i	$0.439897\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.04124	0.443583	0.221792	−	0.975094i	$-0.428810\pi$
$83$	4.04124	0.443583	0.221792	+	0.975094i	$0.428810\pi$
$84$	0	0
$85$	−7.15733	−0.776322
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.05079	−0.747382	−0.373691	−	0.927553i	$-0.621908\pi$
$89$	−7.05079	−0.747382	−0.373691	+	0.927553i	$0.621908\pi$
$90$	0	0
$91$	−2.69228	−0.282227
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−7.33659	−0.752718
$96$	0	0
$97$	6.81120	0.691572	0.345786	−	0.938313i	$-0.387612\pi$
$97$	6.81120	0.691572	0.345786	+	0.938313i	$0.387612\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	6.38737	0.635567	0.317784	−	0.948163i	$-0.397061\pi$
$101$	6.38737	0.635567	0.317784	+	0.948163i	$0.397061\pi$
$102$	0	0
$103$	11.4019	1.12346	0.561731	−	0.827320i	$-0.310135\pi$
$103$	11.4019	1.12346	0.561731	+	0.827320i	$0.310135\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.58116	−0.539551	−0.269775	−	0.962923i	$-0.586949\pi$
$107$	−5.58116	−0.539551	−0.269775	+	0.962923i	$0.586949\pi$
$108$	0	0
$109$	0.949215	0.0909183	0.0454591	−	0.998966i	$-0.485525\pi$
$109$	0.949215	0.0909183	0.0454591	+	0.998966i	$0.485525\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.27128	0.872168	0.436084	−	0.899906i	$-0.356365\pi$
$113$	9.27128	0.872168	0.436084	+	0.899906i	$0.356365\pi$
$114$	0	0
$115$	−5.86198	−0.546633
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.04339	0.462327
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.0193	0.977806	0.488903	−	0.872338i	$-0.337397\pi$
$127$	11.0193	0.977806	0.488903	+	0.872338i	$0.337397\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.64149	0.318159	0.159079	−	0.987266i	$-0.449147\pi$
$131$	3.64149	0.318159	0.159079	+	0.987266i	$0.449147\pi$
$132$	0	0
$133$	5.16970	0.448270
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	2.07271	0.175805	0.0879023	−	0.996129i	$-0.471984\pi$
$139$	2.07271	0.175805	0.0879023	+	0.996129i	$0.471984\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.82075	0.319507
$144$	0	0
$145$	8.97808	0.745589
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.4382	−1.59244	−0.796218	−	0.605010i	$-0.793169\pi$
$149$	−19.4382	−1.59244	−0.796218	+	0.605010i	$0.793169\pi$
$150$	0	0
$151$	−7.33659	−0.597043	−0.298522	−	0.954403i	$-0.596493\pi$
$151$	−7.33659	−0.597043	−0.298522	+	0.954403i	$0.596493\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.590706	0.0474467
$156$	0	0
$157$	−11.2639	−0.898956	−0.449478	−	0.893291i	$-0.648390\pi$
$157$	−11.2639	−0.898956	−0.449478	+	0.893291i	$0.648390\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.13063	0.325539
$162$	0	0
$163$	13.5035	1.05767	0.528837	−	0.848724i	$-0.322628\pi$
$163$	13.5035	1.05767	0.528837	+	0.848724i	$0.322628\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.0412	−1.24131	−0.620654	−	0.784085i	$-0.713133\pi$
$167$	−16.0412	−1.24131	−0.620654	+	0.784085i	$0.713133\pi$
$168$	0	0
$169$	1.59810	0.122931
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.84267	0.368181	0.184091	−	0.982909i	$-0.441066\pi$
$173$	4.84267	0.368181	0.184091	+	0.982909i	$0.441066\pi$
$174$	0	0
$175$	−0.704647	−0.0532663
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.05079	0.0785394	0.0392697	−	0.999229i	$-0.487497\pi$
$179$	1.05079	0.0785394	0.0392697	+	0.999229i	$0.487497\pi$
$180$	0	0
$181$	0.598098	0.0444563	0.0222281	−	0.999753i	$-0.492924\pi$
$181$	0.598098	0.0444563	0.0222281	+	0.999753i	$0.492924\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.4527	0.768497
$186$	0	0
$187$	−7.15733	−0.523396
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.7747	1.79264	0.896319	−	0.443410i	$-0.146231\pi$
$191$	24.7747	1.79264	0.896319	+	0.443410i	$0.146231\pi$
$192$	0	0
$193$	23.9032	1.72059	0.860296	−	0.509796i	$-0.170279\pi$
$193$	23.9032	1.72059	0.860296	+	0.509796i	$0.170279\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5666	1.18032	0.590162	−	0.807285i	$-0.299064\pi$
$197$	16.5666	1.18032	0.590162	+	0.807285i	$0.299064\pi$
$198$	0	0
$199$	5.05079	0.358041	0.179020	−	0.983845i	$-0.442707\pi$
$199$	5.05079	0.358041	0.179020	+	0.983845i	$0.442707\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.32638	−0.444025
$204$	0	0
$205$	−7.92729	−0.553666
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.33659	−0.507482
$210$	0	0
$211$	10.1552	0.699111	0.349556	−	0.936916i	$-0.386333\pi$
$211$	10.1552	0.699111	0.349556	+	0.936916i	$0.386333\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.29535	−0.224741
$216$	0	0
$217$	−0.416239	−0.0282562
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−27.3464	−1.83951
$222$	0	0
$223$	−10.3147	−0.690721	−0.345361	−	0.938470i	$-0.612243\pi$
$223$	−10.3147	−0.690721	−0.345361	+	0.938470i	$0.612243\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.00955	−0.597985	−0.298992	−	0.954255i	$-0.596651\pi$
$227$	−9.00955	−0.597985	−0.298992	+	0.954255i	$0.596651\pi$
$228$	0	0
$229$	9.28298	0.613437	0.306718	−	0.951800i	$-0.400769\pi$
$229$	9.28298	0.613437	0.306718	+	0.951800i	$0.400769\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.51584	0.230331	0.115165	−	0.993346i	$-0.463260\pi$
$233$	3.51584	0.230331	0.115165	+	0.993346i	$0.463260\pi$
$234$	0	0
$235$	−3.64149	−0.237545
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.3655	−0.993909	−0.496954	−	0.867777i	$-0.665548\pi$
$239$	−15.3655	−0.993909	−0.496954	+	0.867777i	$0.665548\pi$
$240$	0	0
$241$	−0.590706	−0.0380507	−0.0190254	−	0.999819i	$-0.506056\pi$
$241$	−0.590706	−0.0380507	−0.0190254	+	0.999819i	$0.506056\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.50347	−0.415492
$246$	0	0
$247$	−28.0312	−1.78359
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−23.5859	−1.48873	−0.744366	−	0.667772i	$-0.767248\pi$
$251$	−23.5859	−1.48873	−0.744366	+	0.667772i	$0.767248\pi$
$252$	0	0
$253$	−5.86198	−0.368540
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.3147	−1.26719	−0.633597	−	0.773663i	$-0.718422\pi$
$257$	−20.3147	−1.26719	−0.633597	+	0.773663i	$0.718422\pi$
$258$	0	0
$259$	−7.36545	−0.457667
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.8958	−1.22683	−0.613415	−	0.789761i	$-0.710205\pi$
$263$	−19.8958	−1.22683	−0.613415	+	0.789761i	$0.710205\pi$
$264$	0	0
$265$	11.8620	0.728676
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.2830	−1.17570	−0.587852	−	0.808968i	$-0.700026\pi$
$269$	−19.2830	−1.17570	−0.587852	+	0.808968i	$0.700026\pi$
$270$	0	0
$271$	−9.43816	−0.573327	−0.286664	−	0.958031i	$-0.592546\pi$
$271$	−9.43816	−0.573327	−0.286664	+	0.958031i	$0.592546\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	22.5764	1.35648	0.678242	−	0.734839i	$-0.262742\pi$
$277$	22.5764	1.35648	0.678242	+	0.734839i	$0.262742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.65126	0.456436	0.228218	−	0.973610i	$-0.426710\pi$
$281$	7.65126	0.456436	0.228218	+	0.973610i	$0.426710\pi$
$282$	0	0
$283$	30.8887	1.83614	0.918071	−	0.396416i	$-0.129746\pi$
$283$	30.8887	1.83614	0.918071	+	0.396416i	$0.129746\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.58594	0.329728
$288$	0	0
$289$	34.2274	2.01338
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.3344	0.603744	0.301872	−	0.953348i	$-0.402388\pi$
$293$	10.3344	0.603744	0.301872	+	0.953348i	$0.402388\pi$
$294$	0	0
$295$	10.8112	0.629452
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−22.3971	−1.29526
$300$	0	0
$301$	2.32206	0.133841
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.64149	0.323031
$306$	0	0
$307$	13.3969	0.764603	0.382301	−	0.924038i	$-0.375132\pi$
$307$	13.3969	0.764603	0.382301	+	0.924038i	$0.375132\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.37022	−0.361222	−0.180611	−	0.983555i	$-0.557807\pi$
$311$	−6.37022	−0.361222	−0.180611	+	0.983555i	$0.557807\pi$
$312$	0	0
$313$	−1.27128	−0.0718567	−0.0359284	−	0.999354i	$-0.511439\pi$
$313$	−1.27128	−0.0718567	−0.0359284	+	0.999354i	$0.511439\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.55426	−0.255793	−0.127896	−	0.991788i	$-0.540822\pi$
$317$	−4.55426	−0.255793	−0.127896	+	0.991788i	$0.540822\pi$
$318$	0	0
$319$	8.97808	0.502676
$320$	0	0
$321$	0	0
$322$	0	0
$323$	52.5104	2.92176
$324$	0	0
$325$	3.82075	0.211937
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.56597	0.141466
$330$	0	0
$331$	30.5469	1.67901	0.839504	−	0.543354i	$-0.182846\pi$
$331$	30.5469	1.67901	0.839504	+	0.543354i	$0.182846\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.9128	0.596228
$336$	0	0
$337$	−22.2179	−1.21029	−0.605143	−	0.796117i	$-0.706884\pi$
$337$	−22.2179	−1.21029	−0.605143	+	0.796117i	$0.706884\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.590706	0.0319885
$342$	0	0
$343$	9.51518	0.513771
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.186646	−0.0100197	−0.00500984	−	0.999987i	$-0.501595\pi$
$347$	−0.186646	−0.0100197	−0.00500984	+	0.999987i	$0.501595\pi$
$348$	0	0
$349$	8.44530	0.452066	0.226033	−	0.974120i	$-0.427424\pi$
$349$	8.44530	0.452066	0.226033	+	0.974120i	$0.427424\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−17.0434	−0.907128	−0.453564	−	0.891224i	$-0.649848\pi$
$353$	−17.0434	−0.907128	−0.453564	+	0.891224i	$0.649848\pi$
$354$	0	0
$355$	11.8620	0.629569
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.4961	1.02896	0.514482	−	0.857501i	$-0.327984\pi$
$359$	19.4961	1.02896	0.514482	+	0.857501i	$0.327984\pi$
$360$	0	0
$361$	34.8255	1.83292
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.82075	0.199987
$366$	0	0
$367$	18.5907	0.970427	0.485213	−	0.874396i	$-0.338742\pi$
$367$	18.5907	0.970427	0.485213	+	0.874396i	$0.338742\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.35851	−0.433952
$372$	0	0
$373$	10.6393	0.550884	0.275442	−	0.961318i	$-0.411176\pi$
$373$	10.6393	0.550884	0.275442	+	0.961318i	$0.411176\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	34.3030	1.76669
$378$	0	0
$379$	32.6293	1.67606	0.838028	−	0.545627i	$-0.183708\pi$
$379$	32.6293	1.67606	0.838028	+	0.545627i	$0.183708\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.4019	0.991391	0.495695	−	0.868496i	$-0.334913\pi$
$383$	19.4019	0.991391	0.495695	+	0.868496i	$0.334913\pi$
$384$	0	0
$385$	−0.704647	−0.0359121
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.95616	0.301989	0.150995	−	0.988535i	$-0.451752\pi$
$389$	5.95616	0.301989	0.150995	+	0.988535i	$0.451752\pi$
$390$	0	0
$391$	41.9562	2.12181
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.33659	0.167882
$396$	0	0
$397$	−0.00739165	−0.000370976 0	−0.000185488	−	1.00000i	$-0.500059\pi$
$397$	−0.00739165	−0.000370976 0	−0.000185488		1.00000i	$0.500059\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.55902	−0.0778538	−0.0389269	−	0.999242i	$-0.512394\pi$
$401$	−1.55902	−0.0778538	−0.0389269	+	0.999242i	$0.512394\pi$
$402$	0	0
$403$	2.25694	0.112426
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.4527	0.518120
$408$	0	0
$409$	−27.6801	−1.36869	−0.684347	−	0.729156i	$-0.739913\pi$
$409$	−27.6801	−1.36869	−0.684347	+	0.729156i	$0.739913\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.61808	−0.374861
$414$	0	0
$415$	4.04124	0.198376
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.7747	0.624087	0.312044	−	0.950068i	$-0.398986\pi$
$419$	12.7747	0.624087	0.312044	+	0.950068i	$0.398986\pi$
$420$	0	0
$421$	−20.3221	−0.990437	−0.495218	−	0.868769i	$-0.664912\pi$
$421$	−20.3221	−0.990437	−0.495218	+	0.868769i	$0.664912\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.15733	−0.347182
$426$	0	0
$427$	−3.97526	−0.192376
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.28298	−0.350809	−0.175404	−	0.984496i	$-0.556123\pi$
$431$	−7.28298	−0.350809	−0.175404	+	0.984496i	$0.556123\pi$
$432$	0	0
$433$	2.37022	0.113905	0.0569527	−	0.998377i	$-0.481862\pi$
$433$	2.37022	0.113905	0.0569527	+	0.998377i	$0.481862\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	43.0069	2.05730
$438$	0	0
$439$	17.9273	0.855623	0.427812	−	0.903868i	$-0.359285\pi$
$439$	17.9273	0.855623	0.427812	+	0.903868i	$0.359285\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.2205	0.675636	0.337818	−	0.941211i	$-0.390311\pi$
$443$	14.2205	0.675636	0.337818	+	0.941211i	$0.390311\pi$
$444$	0	0
$445$	−7.05079	−0.334239
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.1376	0.808772	0.404386	−	0.914588i	$-0.367485\pi$
$449$	17.1376	0.808772	0.404386	+	0.914588i	$0.367485\pi$
$450$	0	0
$451$	−7.92729	−0.373282
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.69228	−0.126216
$456$	0	0
$457$	−16.7261	−0.782415	−0.391207	−	0.920303i	$-0.627943\pi$
$457$	−16.7261	−0.782415	−0.391207	+	0.920303i	$0.627943\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.77757	0.455387	0.227693	−	0.973733i	$-0.426882\pi$
$461$	9.77757	0.455387	0.227693	+	0.973733i	$0.426882\pi$
$462$	0	0
$463$	−27.2200	−1.26502	−0.632511	−	0.774551i	$-0.717976\pi$
$463$	−27.2200	−1.26502	−0.632511	+	0.774551i	$0.717976\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.8689	−1.15080	−0.575398	−	0.817873i	$-0.695153\pi$
$467$	−24.8689	−1.15080	−0.575398	+	0.817873i	$0.695153\pi$
$468$	0	0
$469$	−7.68965	−0.355075
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.29535	−0.151520
$474$	0	0
$475$	−7.33659	−0.336626
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.22788	−0.193177	−0.0965884	−	0.995324i	$-0.530793\pi$
$479$	−4.22788	−0.193177	−0.0965884	+	0.995324i	$0.530793\pi$
$480$	0	0
$481$	39.9371	1.82097
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.81120	0.309280
$486$	0	0
$487$	21.2756	0.964089	0.482045	−	0.876147i	$-0.339894\pi$
$487$	21.2756	0.964089	0.482045	+	0.876147i	$0.339894\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.7240	−1.61220	−0.806100	−	0.591779i	$-0.798426\pi$
$491$	−35.7240	−1.61220	−0.806100	+	0.591779i	$0.798426\pi$
$492$	0	0
$493$	−64.2591	−2.89409
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.35851	−0.374930
$498$	0	0
$499$	9.76780	0.437267	0.218633	−	0.975807i	$-0.429840\pi$
$499$	9.76780	0.437267	0.218633	+	0.975807i	$0.429840\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.9106	1.06612	0.533061	−	0.846077i	$-0.321042\pi$
$503$	23.9106	1.06612	0.533061	+	0.846077i	$0.321042\pi$
$504$	0	0
$505$	6.38737	0.284234
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.9054	−0.749318	−0.374659	−	0.927163i	$-0.622240\pi$
$509$	−16.9054	−0.749318	−0.374659	+	0.927163i	$0.622240\pi$
$510$	0	0
$511$	−2.69228	−0.119099
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.4019	0.502428
$516$	0	0
$517$	−3.64149	−0.160153
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.8038	−0.911431	−0.455716	−	0.890125i	$-0.650617\pi$
$521$	−20.8038	−0.911431	−0.455716	+	0.890125i	$0.650617\pi$
$522$	0	0
$523$	−31.9247	−1.39597	−0.697985	−	0.716113i	$-0.745920\pi$
$523$	−31.9247	−1.39597	−0.697985	+	0.716113i	$0.745920\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.22788	−0.184169
$528$	0	0
$529$	11.3628	0.494036
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−30.2882	−1.31193
$534$	0	0
$535$	−5.58116	−0.241294
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.50347	−0.280124
$540$	0	0
$541$	−35.8255	−1.54026	−0.770130	−	0.637887i	$-0.779809\pi$
$541$	−35.8255	−1.54026	−0.770130	+	0.637887i	$0.779809\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.949215	0.0406599
$546$	0	0
$547$	20.8930	0.893320	0.446660	−	0.894704i	$-0.352613\pi$
$547$	20.8930	0.893320	0.446660	+	0.894704i	$0.352613\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−65.8685	−2.80609
$552$	0	0
$553$	−2.35112	−0.0999797
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.2589	0.900769	0.450384	−	0.892835i	$-0.351287\pi$
$557$	21.2589	0.900769	0.450384	+	0.892835i	$0.351287\pi$
$558$	0	0
$559$	−12.5907	−0.532530
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.2543	1.19078	0.595389	−	0.803438i	$-0.296998\pi$
$563$	28.2543	1.19078	0.595389	+	0.803438i	$0.296998\pi$
$564$	0	0
$565$	9.27128	0.390045
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.75804	0.0737007	0.0368504	−	0.999321i	$-0.488268\pi$
$569$	1.75804	0.0737007	0.0368504	+	0.999321i	$0.488268\pi$
$570$	0	0
$571$	12.7459	0.533399	0.266699	−	0.963780i	$-0.414067\pi$
$571$	12.7459	0.533399	0.266699	+	0.963780i	$0.414067\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.86198	−0.244462
$576$	0	0
$577$	12.9245	0.538053	0.269026	−	0.963133i	$-0.413298\pi$
$577$	12.9245	0.538053	0.269026	+	0.963133i	$0.413298\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.84764	−0.118140
$582$	0	0
$583$	11.8620	0.491273
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.8472	−1.06683	−0.533414	−	0.845854i	$-0.679091\pi$
$587$	−25.8472	−1.06683	−0.533414	+	0.845854i	$0.679091\pi$
$588$	0	0
$589$	−4.33377	−0.178570
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−40.4789	−1.66227	−0.831136	−	0.556070i	$-0.812309\pi$
$593$	−40.4789	−1.66227	−0.831136	+	0.556070i	$0.812309\pi$
$594$	0	0
$595$	5.04339	0.206759
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.2830	1.27819	0.639094	−	0.769129i	$-0.279309\pi$
$599$	31.2830	1.27819	0.639094	+	0.769129i	$0.279309\pi$
$600$	0	0
$601$	12.9245	0.527200	0.263600	−	0.964632i	$-0.415090\pi$
$601$	12.9245	0.527200	0.263600	+	0.964632i	$0.415090\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	11.0193	0.447260	0.223630	−	0.974674i	$-0.428209\pi$
$607$	11.0193	0.447260	0.223630	+	0.974674i	$0.428209\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.9132	−0.562868
$612$	0	0
$613$	−18.0529	−0.729152	−0.364576	−	0.931174i	$-0.618786\pi$
$613$	−18.0529	−0.729152	−0.364576	+	0.931174i	$0.618786\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.0820	1.25132	0.625658	−	0.780098i	$-0.284831\pi$
$617$	31.0820	1.25132	0.625658	+	0.780098i	$0.284831\pi$
$618$	0	0
$619$	30.5469	1.22778	0.613891	−	0.789391i	$-0.289603\pi$
$619$	30.5469	1.22778	0.613891	+	0.789391i	$0.289603\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.96831	0.199051
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−74.8134	−2.98300
$630$	0	0
$631$	−4.33377	−0.172525	−0.0862623	−	0.996272i	$-0.527492\pi$
$631$	−4.33377	−0.172525	−0.0862623	+	0.996272i	$0.527492\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.0193	0.437288
$636$	0	0
$637$	−24.8481	−0.984518
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.85459	−0.389233	−0.194616	−	0.980879i	$-0.562346\pi$
$641$	−9.85459	−0.389233	−0.194616	+	0.980879i	$0.562346\pi$
$642$	0	0
$643$	40.0312	1.57868	0.789339	−	0.613958i	$-0.210423\pi$
$643$	40.0312	1.57868	0.789339	+	0.613958i	$0.210423\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−31.1259	−1.22368	−0.611842	−	0.790980i	$-0.709571\pi$
$647$	−31.1259	−1.22368	−0.611842	+	0.790980i	$0.709571\pi$
$648$	0	0
$649$	10.8112	0.424377
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.2131	0.712734	0.356367	−	0.934346i	$-0.384015\pi$
$653$	18.2131	0.712734	0.356367	+	0.934346i	$0.384015\pi$
$654$	0	0
$655$	3.64149	0.142285
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.1524	−0.590252	−0.295126	−	0.955458i	$-0.595362\pi$
$659$	−15.1524	−0.590252	−0.295126	+	0.955458i	$0.595362\pi$
$660$	0	0
$661$	−26.6293	−1.03576	−0.517881	−	0.855453i	$-0.673279\pi$
$661$	−26.6293	−1.03576	−0.517881	+	0.855453i	$0.673279\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.16970	0.200473
$666$	0	0
$667$	−52.6293	−2.03782
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.64149	0.217787
$672$	0	0
$673$	24.3924	0.940256	0.470128	−	0.882598i	$-0.344208\pi$
$673$	24.3924	0.940256	0.470128	+	0.882598i	$0.344208\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.20549	0.123197	0.0615985	−	0.998101i	$-0.480380\pi$
$677$	3.20549	0.123197	0.0615985	+	0.998101i	$0.480380\pi$
$678$	0	0
$679$	−4.79949	−0.184187
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.36545	−0.128776	−0.0643878	−	0.997925i	$-0.520509\pi$
$683$	−3.36545	−0.128776	−0.0643878	+	0.997925i	$0.520509\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.3216	1.72662
$690$	0	0
$691$	−47.8889	−1.82178	−0.910890	−	0.412649i	$-0.864603\pi$
$691$	−47.8889	−1.82178	−0.910890	+	0.412649i	$0.864603\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.07271	0.0786222
$696$	0	0
$697$	56.7383	2.14912
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.3175	0.880689	0.440345	−	0.897829i	$-0.354856\pi$
$701$	23.3175	0.880689	0.440345	+	0.897829i	$0.354856\pi$
$702$	0	0
$703$	−76.6871	−2.89231
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.50084	−0.169272
$708$	0	0
$709$	17.2274	0.646990	0.323495	−	0.946230i	$-0.395142\pi$
$709$	17.2274	0.646990	0.323495	+	0.946230i	$0.395142\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.46271	−0.129679
$714$	0	0
$715$	3.82075	0.142888
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.586390	0.0218687	0.0109343	−	0.999940i	$-0.496519\pi$
$719$	0.586390	0.0218687	0.0109343	+	0.999940i	$0.496519\pi$
$720$	0	0
$721$	−8.03432	−0.299214
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.97808	0.333438
$726$	0	0
$727$	32.1649	1.19293	0.596466	−	0.802638i	$-0.296571\pi$
$727$	32.1649	1.19293	0.596466	+	0.802638i	$0.296571\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	23.5859	0.872358
$732$	0	0
$733$	31.3993	1.15976	0.579880	−	0.814702i	$-0.303100\pi$
$733$	31.3993	1.15976	0.579880	+	0.814702i	$0.303100\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.9128	0.401977
$738$	0	0
$739$	−38.0922	−1.40125	−0.700623	−	0.713532i	$-0.747094\pi$
$739$	−38.0922	−1.40125	−0.700623	+	0.713532i	$0.747094\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.23743	−0.0453969	−0.0226985	−	0.999742i	$-0.507226\pi$
$743$	−1.23743	−0.0453969	−0.0226985	+	0.999742i	$0.507226\pi$
$744$	0	0
$745$	−19.4382	−0.712159
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.93274	0.143699
$750$	0	0
$751$	−35.8546	−1.30835	−0.654176	−	0.756342i	$-0.726985\pi$
$751$	−35.8546	−1.30835	−0.654176	+	0.756342i	$0.726985\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.33659	−0.267006
$756$	0	0
$757$	14.5235	0.527864	0.263932	−	0.964541i	$-0.414981\pi$
$757$	14.5235	0.527864	0.263932	+	0.964541i	$0.414981\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.8448	−0.864374	−0.432187	−	0.901784i	$-0.642258\pi$
$761$	−23.8448	−0.864374	−0.432187	+	0.901784i	$0.642258\pi$
$762$	0	0
$763$	−0.668861	−0.0242144
$764$	0	0
$765$	0	0
$766$	0	0
$767$	41.3068	1.49150
$768$	0	0
$769$	32.4453	1.17001	0.585004	−	0.811031i	$-0.301093\pi$
$769$	32.4453	1.17001	0.585004	+	0.811031i	$0.301093\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	51.3216	1.84591	0.922955	−	0.384908i	$-0.125767\pi$
$773$	51.3216	1.84591	0.922955	+	0.384908i	$0.125767\pi$
$774$	0	0
$775$	0.590706	0.0212188
$776$	0	0
$777$	0	0
$778$	0	0
$779$	58.1593	2.08377
$780$	0	0
$781$	11.8620	0.424455
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−11.2639	−0.402025
$786$	0	0
$787$	−17.1209	−0.610294	−0.305147	−	0.952305i	$-0.598706\pi$
$787$	−17.1209	−0.610294	−0.305147	+	0.952305i	$0.598706\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.53298	−0.232286
$792$	0	0
$793$	21.5547	0.765430
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.2565	−0.611256	−0.305628	−	0.952151i	$-0.598866\pi$
$797$	−17.2565	−0.611256	−0.305628	+	0.952151i	$0.598866\pi$
$798$	0	0
$799$	26.0634	0.922056
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.82075	0.134831
$804$	0	0
$805$	4.13063	0.145585
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−14.9342	−0.525060	−0.262530	−	0.964924i	$-0.584557\pi$
$809$	−14.9342	−0.525060	−0.262530	+	0.964924i	$0.584557\pi$
$810$	0	0
$811$	−21.9273	−0.769971	−0.384986	−	0.922923i	$-0.625794\pi$
$811$	−21.9273	−0.769971	−0.384986	+	0.922923i	$0.625794\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.5035	0.473006
$816$	0	0
$817$	24.1767	0.845834
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34.8036	−1.21465	−0.607327	−	0.794452i	$-0.707758\pi$
$821$	−34.8036	−1.21465	−0.607327	+	0.794452i	$0.707758\pi$
$822$	0	0
$823$	27.8429	0.970542	0.485271	−	0.874364i	$-0.338721\pi$
$823$	27.8429	0.970542	0.485271	+	0.874364i	$0.338721\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.5590	1.20174	0.600868	−	0.799348i	$-0.294822\pi$
$827$	34.5590	1.20174	0.600868	+	0.799348i	$0.294822\pi$
$828$	0	0
$829$	−21.0243	−0.730204	−0.365102	−	0.930968i	$-0.618966\pi$
$829$	−21.0243	−0.730204	−0.365102	+	0.930968i	$0.618966\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	46.5475	1.61278
$834$	0	0
$835$	−16.0412	−0.555130
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.3390	−1.28908	−0.644542	−	0.764569i	$-0.722952\pi$
$839$	−37.3390	−1.28908	−0.644542	+	0.764569i	$0.722952\pi$
$840$	0	0
$841$	51.6059	1.77951
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.59810	0.0549762
$846$	0	0
$847$	−0.704647	−0.0242120
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−61.2735	−2.10043
$852$	0	0
$853$	−22.9831	−0.786925	−0.393462	−	0.919341i	$-0.628723\pi$
$853$	−22.9831	−0.786925	−0.393462	+	0.919341i	$0.628723\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.0584	−1.57332	−0.786662	−	0.617383i	$-0.788193\pi$
$857$	−46.0584	−1.57332	−0.786662	+	0.617383i	$0.788193\pi$
$858$	0	0
$859$	−29.3655	−1.00194	−0.500968	−	0.865466i	$-0.667023\pi$
$859$	−29.3655	−1.00194	−0.500968	+	0.865466i	$0.667023\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.7795	1.14987	0.574934	−	0.818200i	$-0.305028\pi$
$863$	33.7795	1.14987	0.574934	+	0.818200i	$0.305028\pi$
$864$	0	0
$865$	4.84267	0.164656
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.33659	0.113186
$870$	0	0
$871$	41.6949	1.41278
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.704647	−0.0238214
$876$	0	0
$877$	43.9609	1.48446	0.742228	−	0.670148i	$-0.233769\pi$
$877$	43.9609	1.48446	0.742228	+	0.670148i	$0.233769\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.86937	−0.265126	−0.132563	−	0.991175i	$-0.542321\pi$
$881$	−7.86937	−0.265126	−0.132563	+	0.991175i	$0.542321\pi$
$882$	0	0
$883$	8.62934	0.290400	0.145200	−	0.989402i	$-0.453617\pi$
$883$	8.62934	0.290400	0.145200	+	0.989402i	$0.453617\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.5013	−0.486906	−0.243453	−	0.969913i	$-0.578280\pi$
$887$	−14.5013	−0.486906	−0.243453	+	0.969913i	$0.578280\pi$
$888$	0	0
$889$	−7.76473	−0.260421
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.7161	0.894021
$894$	0	0
$895$	1.05079	0.0351239
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.30341	0.176879
$900$	0	0
$901$	−84.9002	−2.82843
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.598098	0.0198814
$906$	0	0
$907$	13.5977	0.451503	0.225751	−	0.974185i	$-0.427516\pi$
$907$	13.5977	0.451503	0.225751	+	0.974185i	$0.427516\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.0126	0.828704	0.414352	−	0.910117i	$-0.364008\pi$
$911$	25.0126	0.828704	0.414352	+	0.910117i	$0.364008\pi$
$912$	0	0
$913$	4.04124	0.133745
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.56597	−0.0847357
$918$	0	0
$919$	−0.580940	−0.0191634	−0.00958172	−	0.999954i	$-0.503050\pi$
$919$	−0.580940	−0.0191634	−0.00958172	+	0.999954i	$0.503050\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	45.3216	1.49178
$924$	0	0
$925$	10.4527	0.343682
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.5786	0.904823	0.452411	−	0.891809i	$-0.350564\pi$
$929$	27.5786	0.904823	0.452411	+	0.891809i	$0.350564\pi$
$930$	0	0
$931$	47.7133	1.56374
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.15733	−0.234070
$936$	0	0
$937$	−17.9900	−0.587708	−0.293854	−	0.955850i	$-0.594938\pi$
$937$	−17.9900	−0.587708	−0.293854	+	0.955850i	$0.594938\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−0.431214	−0.0140572	−0.00702858	−	0.999975i	$-0.502237\pi$
$941$	−0.431214	−0.0140572	−0.00702858	+	0.999975i	$0.502237\pi$
$942$	0	0
$943$	46.4697	1.51326
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17.4670	0.567602	0.283801	−	0.958883i	$-0.408404\pi$
$947$	17.4670	0.567602	0.283801	+	0.958883i	$0.408404\pi$
$948$	0	0
$949$	14.5981	0.473874
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−24.8622	−0.805366	−0.402683	−	0.915340i	$-0.631922\pi$
$953$	−24.8622	−0.805366	−0.402683	+	0.915340i	$0.631922\pi$
$954$	0	0
$955$	24.7747	0.801692
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.22788	0.136525
$960$	0	0
$961$	−30.6511	−0.988744
$962$	0	0
$963$	0	0
$964$	0	0
$965$	23.9032	0.769472
$966$	0	0
$967$	18.1139	0.582505	0.291253	−	0.956646i	$-0.405928\pi$
$967$	18.1139	0.582505	0.291253	+	0.956646i	$0.405928\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.5426	−0.595059	−0.297529	−	0.954713i	$-0.596163\pi$
$971$	−18.5426	−0.595059	−0.297529	+	0.954713i	$0.596163\pi$
$972$	0	0
$973$	−1.46053	−0.0468223
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.2665	−0.776355	−0.388177	−	0.921585i	$-0.626895\pi$
$977$	−24.2665	−0.776355	−0.388177	+	0.921585i	$0.626895\pi$
$978$	0	0
$979$	−7.05079	−0.225344
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−44.9514	−1.43373	−0.716863	−	0.697214i	$-0.754423\pi$
$983$	−44.9514	−1.43373	−0.716863	+	0.697214i	$0.754423\pi$
$984$	0	0
$985$	16.5666	0.527857
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.3173	0.614254
$990$	0	0
$991$	−43.1185	−1.36970	−0.684852	−	0.728682i	$-0.740133\pi$
$991$	−43.1185	−1.36970	−0.684852	+	0.728682i	$0.740133\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.05079	0.160121
$996$	0	0
$997$	−1.38541	−0.0438763	−0.0219381	−	0.999759i	$-0.506984\pi$
$997$	−1.38541	−0.0438763	−0.0219381	+	0.999759i	$0.506984\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.cn.1.3		4
3.2	odd	2		7920.2.a.cm.1.3		4
4.3	odd	2		495.2.a.g.1.2	yes	4
12.11	even	2		495.2.a.f.1.3	✓	4
20.3	even	4		2475.2.c.s.199.5		8
20.7	even	4		2475.2.c.s.199.4		8
20.19	odd	2		2475.2.a.bf.1.3		4
44.43	even	2		5445.2.a.bh.1.3		4
60.23	odd	4		2475.2.c.t.199.4		8
60.47	odd	4		2475.2.c.t.199.5		8
60.59	even	2		2475.2.a.bj.1.2		4
132.131	odd	2		5445.2.a.bs.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.a.f.1.3	✓	4	12.11	even	2
495.2.a.g.1.2	yes	4	4.3	odd	2
2475.2.a.bf.1.3		4	20.19	odd	2
2475.2.a.bj.1.2		4	60.59	even	2
2475.2.c.s.199.4		8	20.7	even	4
2475.2.c.s.199.5		8	20.3	even	4
2475.2.c.t.199.4		8	60.23	odd	4
2475.2.c.t.199.5		8	60.47	odd	4
5445.2.a.bh.1.3		4	44.43	even	2
5445.2.a.bs.1.2		4	132.131	odd	2
7920.2.a.cm.1.3		4	3.2	odd	2
7920.2.a.cn.1.3		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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.704647 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -0.704647 q^{7} +1.00000 q^{11} +3.82075 q^{13} -7.15733 q^{17} -7.33659 q^{19} -5.86198 q^{23} +1.00000 q^{25} +8.97808 q^{29} +0.590706 q^{31} -0.704647 q^{35} +10.4527 q^{37} -7.92729 q^{41} -3.29535 q^{43} -3.64149 q^{47} -6.50347 q^{49} +11.8620 q^{53} +1.00000 q^{55} +10.8112 q^{59} +5.64149 q^{61} +3.82075 q^{65} +10.9128 q^{67} +11.8620 q^{71} +3.82075 q^{73} -0.704647 q^{77} +3.33659 q^{79} +4.04124 q^{83} -7.15733 q^{85} -7.05079 q^{89} -2.69228 q^{91} -7.33659 q^{95} +6.81120 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^7	\( 4 q + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 4 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 4 q^{29} - 4 q^{35} + 8 q^{37} - 4 q^{41} - 12 q^{43} + 20 q^{49} + 16 q^{53} + 4 q^{55} + 24 q^{59} + 8 q^{61} + 8 q^{65} + 16 q^{71} + 8 q^{73} - 4 q^{77} - 12 q^{79} - 8 q^{83} + 4 q^{85} - 16 q^{89} + 16 q^{91} - 4 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 4 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 4 * q^29 - 4 * q^35 + 8 * q^37 - 4 * q^41 - 12 * q^43 + 20 * q^49 + 16 * q^53 + 4 * q^55 + 24 * q^59 + 8 * q^61 + 8 * q^65 + 16 * q^71 + 8 * q^73 - 4 * q^77 - 12 * q^79 - 8 * q^83 + 4 * q^85 - 16 * q^89 + 16 * q^91 - 4 * q^95 + 8 * q^97

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