LMFDB - Embedded newform 7920.2.a.cm.1.4

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			1.4
Root			$-1.69696$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +4.13176 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +4.13176 q^{7} -1.00000 q^{11} +2.73785 q^{13} -2.41541 q^{17} +1.15325 q^{19} -8.54717 q^{23} +1.00000 q^{25} +1.67756 q^{29} +10.2635 q^{31} -4.13176 q^{35} +5.71636 q^{37} +9.11028 q^{41} -8.13176 q^{43} +1.47569 q^{47} +10.0715 q^{49} +2.54717 q^{53} +1.00000 q^{55} -8.24066 q^{59} +3.47569 q^{61} -2.73785 q^{65} -15.3350 q^{67} +2.54717 q^{71} +2.73785 q^{73} -4.13176 q^{77} -5.15325 q^{79} +9.28502 q^{83} +2.41541 q^{85} -4.78783 q^{89} +11.3121 q^{91} -1.15325 q^{95} +4.24066 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	4.13176	1.56166	0.780830	−	0.624744i	$-0.214796\pi$
$7$	4.13176	1.56166	0.780830	+	0.624744i	$0.214796\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.73785	0.759342	0.379671	−	0.925122i	$-0.376037\pi$
$13$	2.73785	0.759342	0.379671	+	0.925122i	$0.376037\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.41541	−0.585822	−0.292911	−	0.956140i	$-0.594624\pi$
$17$	−2.41541	−0.585822	−0.292911	+	0.956140i	$0.594624\pi$
$18$	0	0
$19$	1.15325	0.264574	0.132287	−	0.991211i	$-0.457768\pi$
$19$	1.15325	0.264574	0.132287	+	0.991211i	$0.457768\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.54717	−1.78221	−0.891104	−	0.453799i	$-0.850068\pi$
$23$	−8.54717	−1.78221	−0.891104	+	0.453799i	$0.850068\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.67756	0.311515	0.155757	−	0.987795i	$-0.450218\pi$
$29$	1.67756	0.311515	0.155757	+	0.987795i	$0.450218\pi$
$30$	0	0
$31$	10.2635	1.84338	0.921692	−	0.387922i	$-0.126807\pi$
$31$	10.2635	1.84338	0.921692	+	0.387922i	$0.126807\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.13176	−0.698396
$36$	0	0
$37$	5.71636	0.939764	0.469882	−	0.882729i	$-0.344297\pi$
$37$	5.71636	0.939764	0.469882	+	0.882729i	$0.344297\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.11028	1.42279	0.711393	−	0.702794i	$-0.248065\pi$
$41$	9.11028	1.42279	0.711393	+	0.702794i	$0.248065\pi$
$42$	0	0
$43$	−8.13176	−1.24008	−0.620041	−	0.784569i	$-0.712884\pi$
$43$	−8.13176	−1.24008	−0.620041	+	0.784569i	$0.712884\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.47569	0.215252	0.107626	−	0.994191i	$-0.465675\pi$
$47$	1.47569	0.215252	0.107626	+	0.994191i	$0.465675\pi$
$48$	0	0
$49$	10.0715	1.43878
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.54717	0.349881	0.174940	−	0.984579i	$-0.444027\pi$
$53$	2.54717	0.349881	0.174940	+	0.984579i	$0.444027\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.24066	−1.07284	−0.536422	−	0.843950i	$-0.680224\pi$
$59$	−8.24066	−1.07284	−0.536422	+	0.843950i	$0.680224\pi$
$60$	0	0
$61$	3.47569	0.445017	0.222509	−	0.974931i	$-0.428575\pi$
$61$	3.47569	0.445017	0.222509	+	0.974931i	$0.428575\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.73785	−0.339588
$66$	0	0
$67$	−15.3350	−1.87347	−0.936734	−	0.350041i	$-0.886168\pi$
$67$	−15.3350	−1.87347	−0.936734	+	0.350041i	$0.886168\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.54717	0.302293	0.151147	−	0.988511i	$-0.451703\pi$
$71$	2.54717	0.302293	0.151147	+	0.988511i	$0.451703\pi$
$72$	0	0
$73$	2.73785	0.320441	0.160220	−	0.987081i	$-0.448780\pi$
$73$	2.73785	0.320441	0.160220	+	0.987081i	$0.448780\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.13176	−0.470858
$78$	0	0
$79$	−5.15325	−0.579786	−0.289893	−	0.957059i	$-0.593620\pi$
$79$	−5.15325	−0.579786	−0.289893	+	0.957059i	$0.593620\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.28502	1.01916	0.509582	−	0.860422i	$-0.329800\pi$
$83$	9.28502	1.01916	0.509582	+	0.860422i	$0.329800\pi$
$84$	0	0
$85$	2.41541	0.261988
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.78783	−0.507509	−0.253755	−	0.967269i	$-0.581666\pi$
$89$	−4.78783	−0.507509	−0.253755	+	0.967269i	$0.581666\pi$
$90$	0	0
$91$	11.3121	1.18583
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.15325	−0.118321
$96$	0	0
$97$	4.24066	0.430574	0.215287	−	0.976551i	$-0.430931\pi$
$97$	4.24066	0.430574	0.215287	+	0.976551i	$0.430931\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.9411	1.38719	0.693595	−	0.720365i	$-0.256026\pi$
$101$	13.9411	1.38719	0.693595	+	0.720365i	$0.256026\pi$
$102$	0	0
$103$	18.5042	1.82327	0.911636	−	0.410998i	$-0.134820\pi$
$103$	18.5042	1.82327	0.911636	+	0.410998i	$0.134820\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.7663	1.33084	0.665421	−	0.746468i	$-0.268252\pi$
$107$	13.7663	1.33084	0.665421	+	0.746468i	$0.268252\pi$
$108$	0	0
$109$	12.7878	1.22485	0.612426	−	0.790528i	$-0.290194\pi$
$109$	12.7878	1.22485	0.612426	+	0.790528i	$0.290194\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.8107	1.39327	0.696637	−	0.717424i	$-0.254679\pi$
$113$	14.8107	1.39327	0.696637	+	0.717424i	$0.254679\pi$
$114$	0	0
$115$	8.54717	0.797028
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.97989	−0.914855
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.9626	−1.15024	−0.575121	−	0.818068i	$-0.695045\pi$
$127$	−12.9626	−1.15024	−0.575121	+	0.818068i	$0.695045\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.47569	−0.128932	−0.0644660	−	0.997920i	$-0.520534\pi$
$131$	−1.47569	−0.128932	−0.0644660	+	0.997920i	$0.520534\pi$
$132$	0	0
$133$	4.76497	0.413175
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	0.889725	0.0754655	0.0377327	−	0.999288i	$-0.487986\pi$
$139$	0.889725	0.0754655	0.0377327	+	0.999288i	$0.487986\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.73785	−0.228950
$144$	0	0
$145$	−1.67756	−0.139314
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.7289	−1.04279	−0.521397	−	0.853314i	$-0.674589\pi$
$149$	−12.7289	−1.04279	−0.521397	+	0.853314i	$0.674589\pi$
$150$	0	0
$151$	1.15325	0.0938504	0.0469252	−	0.998898i	$-0.485058\pi$
$151$	1.15325	0.0938504	0.0469252	+	0.998898i	$0.485058\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.2635	−0.824386
$156$	0	0
$157$	−3.95702	−0.315805	−0.157902	−	0.987455i	$-0.550473\pi$
$157$	−3.95702	−0.315805	−0.157902	+	0.987455i	$0.550473\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−35.3149	−2.78320
$162$	0	0
$163$	−3.07148	−0.240577	−0.120288	−	0.992739i	$-0.538382\pi$
$163$	−3.07148	−0.240577	−0.120288	+	0.992739i	$0.538382\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.71498	0.210092	0.105046	−	0.994467i	$-0.466501\pi$
$167$	2.71498	0.210092	0.105046	+	0.994467i	$0.466501\pi$
$168$	0	0
$169$	−5.50419	−0.423399
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.4154	−1.09598	−0.547992	−	0.836484i	$-0.684607\pi$
$173$	−14.4154	−1.09598	−0.547992	+	0.836484i	$0.684607\pi$
$174$	0	0
$175$	4.13176	0.312332
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.7878	0.806321	0.403160	−	0.915129i	$-0.367912\pi$
$179$	10.7878	0.806321	0.403160	+	0.915129i	$0.367912\pi$
$180$	0	0
$181$	−6.50419	−0.483453	−0.241726	−	0.970344i	$-0.577714\pi$
$181$	−6.50419	−0.483453	−0.241726	+	0.970344i	$0.577714\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.71636	−0.420275
$186$	0	0
$187$	2.41541	0.176632
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.8822	1.14919	0.574597	−	0.818437i	$-0.305159\pi$
$191$	15.8822	1.14919	0.574597	+	0.818437i	$0.305159\pi$
$192$	0	0
$193$	−3.83219	−0.275847	−0.137923	−	0.990443i	$-0.544043\pi$
$193$	−3.83219	−0.275847	−0.137923	+	0.990443i	$0.544043\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.67893	0.190866	0.0954331	−	0.995436i	$-0.469576\pi$
$197$	2.67893	0.190866	0.0954331	+	0.995436i	$0.469576\pi$
$198$	0	0
$199$	−6.78783	−0.481177	−0.240588	−	0.970627i	$-0.577340\pi$
$199$	−6.78783	−0.481177	−0.240588	+	0.970627i	$0.577340\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.93128	0.486480
$204$	0	0
$205$	−9.11028	−0.636289
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.15325	−0.0797722
$210$	0	0
$211$	−17.6803	−1.21716	−0.608581	−	0.793491i	$-0.708261\pi$
$211$	−17.6803	−1.21716	−0.608581	+	0.793491i	$0.708261\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.13176	0.554582
$216$	0	0
$217$	42.4065	2.87874
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.61301	−0.444839
$222$	0	0
$223$	8.83081	0.591355	0.295677	−	0.955288i	$-0.404455\pi$
$223$	8.83081	0.591355	0.295677	+	0.955288i	$0.404455\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.4972	0.696723	0.348361	−	0.937360i	$-0.386738\pi$
$227$	10.4972	0.696723	0.348361	+	0.937360i	$0.386738\pi$
$228$	0	0
$229$	4.95139	0.327197	0.163598	−	0.986527i	$-0.447690\pi$
$229$	4.95139	0.327197	0.163598	+	0.986527i	$0.447690\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.89110	0.254914	0.127457	−	0.991844i	$-0.459318\pi$
$233$	3.89110	0.254914	0.127457	+	0.991844i	$0.459318\pi$
$234$	0	0
$235$	−1.47569	−0.0962637
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.6186	−1.01029	−0.505143	−	0.863036i	$-0.668560\pi$
$239$	−15.6186	−1.01029	−0.505143	+	0.863036i	$0.668560\pi$
$240$	0	0
$241$	−10.2635	−0.661132	−0.330566	−	0.943783i	$-0.607240\pi$
$241$	−10.2635	−0.661132	−0.330566	+	0.943783i	$0.607240\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.0715	−0.643443
$246$	0	0
$247$	3.15743	0.200902
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.6415	−1.23976	−0.619881	−	0.784696i	$-0.712819\pi$
$251$	−19.6415	−1.23976	−0.619881	+	0.784696i	$0.712819\pi$
$252$	0	0
$253$	8.54717	0.537356
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.16919	0.0729320	0.0364660	−	0.999335i	$-0.488390\pi$
$257$	1.16919	0.0729320	0.0364660	+	0.999335i	$0.488390\pi$
$258$	0	0
$259$	23.6186	1.46759
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.93553	0.550989	0.275494	−	0.961303i	$-0.411158\pi$
$263$	8.93553	0.550989	0.275494	+	0.961303i	$0.411158\pi$
$264$	0	0
$265$	−2.54717	−0.156471
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.9514	0.911602	0.455801	−	0.890082i	$-0.349353\pi$
$269$	14.9514	0.911602	0.455801	+	0.890082i	$0.349353\pi$
$270$	0	0
$271$	22.7289	1.38068	0.690342	−	0.723483i	$-0.257460\pi$
$271$	22.7289	1.38068	0.690342	+	0.723483i	$0.257460\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−22.1387	−1.33019	−0.665093	−	0.746761i	$-0.731608\pi$
$277$	−22.1387	−1.33019	−0.665093	+	0.746761i	$0.731608\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.9841	1.19215	0.596075	−	0.802929i	$-0.296726\pi$
$281$	19.9841	1.19215	0.596075	+	0.802929i	$0.296726\pi$
$282$	0	0
$283$	−24.2775	−1.44315	−0.721573	−	0.692339i	$-0.756580\pi$
$283$	−24.2775	−1.44315	−0.721573	+	0.692339i	$0.756580\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	37.6415	2.22191
$288$	0	0
$289$	−11.1658	−0.656813
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.4182	0.959159	0.479579	−	0.877498i	$-0.340789\pi$
$293$	16.4182	0.959159	0.479579	+	0.877498i	$0.340789\pi$
$294$	0	0
$295$	8.24066	0.479790
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−23.4008	−1.35331
$300$	0	0
$301$	−33.5985	−1.93659
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.47569	−0.199018
$306$	0	0
$307$	−5.44390	−0.310700	−0.155350	−	0.987859i	$-0.549651\pi$
$307$	−5.44390	−0.310700	−0.155350	+	0.987859i	$0.549651\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.2864	1.60397	0.801987	−	0.597341i	$-0.203776\pi$
$311$	28.2864	1.60397	0.801987	+	0.597341i	$0.203776\pi$
$312$	0	0
$313$	22.8107	1.28934	0.644668	−	0.764462i	$-0.276995\pi$
$313$	22.8107	1.28934	0.644668	+	0.764462i	$0.276995\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.8593	−1.34007	−0.670036	−	0.742328i	$-0.733721\pi$
$317$	−23.8593	−1.34007	−0.670036	+	0.742328i	$0.733721\pi$
$318$	0	0
$319$	−1.67756	−0.0939252
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.78557	−0.154993
$324$	0	0
$325$	2.73785	0.151868
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.09722	0.336151
$330$	0	0
$331$	18.9084	1.03930	0.519650	−	0.854379i	$-0.326062\pi$
$331$	18.9084	1.03930	0.519650	+	0.854379i	$0.326062\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	15.3350	0.837841
$336$	0	0
$337$	24.6630	1.34348	0.671740	−	0.740787i	$-0.265547\pi$
$337$	24.6630	1.34348	0.671740	+	0.740787i	$0.265547\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−10.2635	−0.555801
$342$	0	0
$343$	12.6906	0.685229
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.5056	−0.832383	−0.416191	−	0.909277i	$-0.636635\pi$
$347$	−15.5056	−0.832383	−0.416191	+	0.909277i	$0.636635\pi$
$348$	0	0
$349$	20.4841	1.09649	0.548244	−	0.836319i	$-0.315297\pi$
$349$	20.4841	1.09649	0.548244	+	0.836319i	$0.315297\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.9799	1.16987	0.584936	−	0.811080i	$-0.301120\pi$
$353$	21.9799	1.16987	0.584936	+	0.811080i	$0.301120\pi$
$354$	0	0
$355$	−2.54717	−0.135190
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.6962	−1.03953	−0.519764	−	0.854310i	$-0.673980\pi$
$359$	−19.6962	−1.03953	−0.519764	+	0.854310i	$0.673980\pi$
$360$	0	0
$361$	−17.6700	−0.930000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.73785	−0.143305
$366$	0	0
$367$	28.2635	1.47534	0.737672	−	0.675159i	$-0.235925\pi$
$367$	28.2635	1.47534	0.737672	+	0.675159i	$0.235925\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.5243	0.546395
$372$	0	0
$373$	−9.78921	−0.506866	−0.253433	−	0.967353i	$-0.581560\pi$
$373$	−9.78921	−0.506866	−0.253433	+	0.967353i	$0.581560\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.59290	0.236546
$378$	0	0
$379$	−5.66162	−0.290818	−0.145409	−	0.989372i	$-0.546450\pi$
$379$	−5.66162	−0.290818	−0.145409	+	0.989372i	$0.546450\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−26.5042	−1.35430	−0.677150	−	0.735845i	$-0.736785\pi$
$383$	−26.5042	−1.35430	−0.677150	+	0.735845i	$0.736785\pi$
$384$	0	0
$385$	4.13176	0.210574
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.3551	0.778535	0.389268	−	0.921125i	$-0.372728\pi$
$389$	15.3551	0.778535	0.389268	+	0.921125i	$0.372728\pi$
$390$	0	0
$391$	20.6449	1.04406
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.15325	0.259288
$396$	0	0
$397$	16.7677	0.841548	0.420774	−	0.907166i	$-0.361759\pi$
$397$	16.7677	0.841548	0.420774	+	0.907166i	$0.361759\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.0457	1.30066	0.650331	−	0.759651i	$-0.274630\pi$
$401$	26.0457	1.30066	0.650331	+	0.759651i	$0.274630\pi$
$402$	0	0
$403$	28.1000	1.39976
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.71636	−0.283349
$408$	0	0
$409$	22.4495	1.11005	0.555027	−	0.831832i	$-0.312708\pi$
$409$	22.4495	1.11005	0.555027	+	0.831832i	$0.312708\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−34.0485	−1.67542
$414$	0	0
$415$	−9.28502	−0.455784
$416$	0	0
$417$	0	0
$418$	0	0
$419$	27.8822	1.36213	0.681067	−	0.732221i	$-0.261516\pi$
$419$	27.8822	1.36213	0.681067	+	0.732221i	$0.261516\pi$
$420$	0	0
$421$	15.5985	0.760226	0.380113	−	0.924940i	$-0.375885\pi$
$421$	15.5985	0.760226	0.380113	+	0.924940i	$0.375885\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.41541	−0.117164
$426$	0	0
$427$	14.3608	0.694965
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.95139	0.142163	0.0710817	−	0.997470i	$-0.477355\pi$
$431$	2.95139	0.142163	0.0710817	+	0.997470i	$0.477355\pi$
$432$	0	0
$433$	24.2864	1.16713	0.583565	−	0.812067i	$-0.301657\pi$
$433$	24.2864	1.16713	0.583565	+	0.812067i	$0.301657\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.85705	−0.471527
$438$	0	0
$439$	19.1103	0.912084	0.456042	−	0.889958i	$-0.349267\pi$
$439$	19.1103	0.912084	0.456042	+	0.889958i	$0.349267\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.97714	−0.0939365	−0.0469683	−	0.998896i	$-0.514956\pi$
$443$	−1.97714	−0.0939365	−0.0469683	+	0.998896i	$0.514956\pi$
$444$	0	0
$445$	4.78783	0.226965
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.1719	−0.716008	−0.358004	−	0.933720i	$-0.616543\pi$
$449$	−15.1719	−0.716008	−0.358004	+	0.933720i	$0.616543\pi$
$450$	0	0
$451$	−9.11028	−0.428986
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11.3121	−0.530321
$456$	0	0
$457$	−6.17056	−0.288647	−0.144323	−	0.989531i	$-0.546101\pi$
$457$	−6.17056	−0.288647	−0.144323	+	0.989531i	$0.546101\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.1990	1.08048	0.540242	−	0.841510i	$-0.318333\pi$
$461$	23.1990	1.08048	0.540242	+	0.841510i	$0.318333\pi$
$462$	0	0
$463$	1.39809	0.0649750	0.0324875	−	0.999472i	$-0.489657\pi$
$463$	1.39809	0.0649750	0.0324875	+	0.999472i	$0.489657\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.6901	−1.04997	−0.524987	−	0.851110i	$-0.675930\pi$
$467$	−22.6901	−1.04997	−0.524987	+	0.851110i	$0.675930\pi$
$468$	0	0
$469$	−63.3606	−2.92572
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.13176	0.373899
$474$	0	0
$475$	1.15325	0.0529149
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.7906	−1.13271	−0.566355	−	0.824161i	$-0.691647\pi$
$479$	−24.7906	−1.13271	−0.566355	+	0.824161i	$0.691647\pi$
$480$	0	0
$481$	15.6505	0.713602
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.24066	−0.192559
$486$	0	0
$487$	33.7191	1.52796	0.763979	−	0.645241i	$-0.223243\pi$
$487$	33.7191	1.52796	0.763979	+	0.645241i	$0.223243\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.90566	0.311648	0.155824	−	0.987785i	$-0.450197\pi$
$491$	6.90566	0.311648	0.155824	+	0.987785i	$0.450197\pi$
$492$	0	0
$493$	−4.05198	−0.182492
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.5243	0.472080
$498$	0	0
$499$	2.26078	0.101206	0.0506031	−	0.998719i	$-0.483886\pi$
$499$	2.26078	0.101206	0.0506031	+	0.998719i	$0.483886\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.5999	0.918504	0.459252	−	0.888306i	$-0.348117\pi$
$503$	20.5999	0.918504	0.459252	+	0.888306i	$0.348117\pi$
$504$	0	0
$505$	−13.9411	−0.620370
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.43272	0.329449	0.164725	−	0.986340i	$-0.447326\pi$
$509$	7.43272	0.329449	0.164725	+	0.986340i	$0.447326\pi$
$510$	0	0
$511$	11.3121	0.500420
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−18.5042	−0.815392
$516$	0	0
$517$	−1.47569	−0.0649010
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.0084	1.53375	0.766873	−	0.641799i	$-0.221812\pi$
$521$	35.0084	1.53375	0.766873	+	0.641799i	$0.221812\pi$
$522$	0	0
$523$	1.52986	0.0668960	0.0334480	−	0.999440i	$-0.489351\pi$
$523$	1.52986	0.0668960	0.0334480	+	0.999440i	$0.489351\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.7906	−1.07989
$528$	0	0
$529$	50.0541	2.17627
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.9425	1.08038
$534$	0	0
$535$	−13.7663	−0.595171
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.0715	−0.433809
$540$	0	0
$541$	16.6700	0.716700	0.358350	−	0.933587i	$-0.383340\pi$
$541$	16.6700	0.716700	0.358350	+	0.933587i	$0.383340\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.7878	−0.547771
$546$	0	0
$547$	2.25234	0.0963032	0.0481516	−	0.998840i	$-0.484667\pi$
$547$	2.25234	0.0963032	0.0481516	+	0.998840i	$0.484667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.93465	0.0824188
$552$	0	0
$553$	−21.2920	−0.905429
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.9911	0.508078	0.254039	−	0.967194i	$-0.418241\pi$
$557$	11.9911	0.508078	0.254039	+	0.967194i	$0.418241\pi$
$558$	0	0
$559$	−22.2635	−0.941647
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.4598	−0.820134	−0.410067	−	0.912055i	$-0.634495\pi$
$563$	−19.4598	−0.820134	−0.410067	+	0.912055i	$0.634495\pi$
$564$	0	0
$565$	−14.8107	−0.623091
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.7205	−0.826728	−0.413364	−	0.910566i	$-0.635646\pi$
$569$	−19.7205	−0.826728	−0.413364	+	0.910566i	$0.635646\pi$
$570$	0	0
$571$	−5.41678	−0.226685	−0.113343	−	0.993556i	$-0.536156\pi$
$571$	−5.41678	−0.226685	−0.113343	+	0.993556i	$0.536156\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.54717	−0.356442
$576$	0	0
$577$	6.42708	0.267563	0.133781	−	0.991011i	$-0.457288\pi$
$577$	6.42708	0.267563	0.133781	+	0.991011i	$0.457288\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	38.3635	1.59159
$582$	0	0
$583$	−2.54717	−0.105493
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.9883	1.85686	0.928432	−	0.371502i	$-0.121157\pi$
$587$	44.9883	1.85686	0.928432	+	0.371502i	$0.121157\pi$
$588$	0	0
$589$	11.8364	0.487712
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.3892	−0.878348	−0.439174	−	0.898402i	$-0.644729\pi$
$593$	−21.3892	−0.878348	−0.439174	+	0.898402i	$0.644729\pi$
$594$	0	0
$595$	9.97989	0.409135
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.9514	−1.10120	−0.550602	−	0.834768i	$-0.685602\pi$
$599$	−26.9514	−1.10120	−0.550602	+	0.834768i	$0.685602\pi$
$600$	0	0
$601$	6.42708	0.262166	0.131083	−	0.991371i	$-0.458155\pi$
$601$	6.42708	0.262166	0.131083	+	0.991371i	$0.458155\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−12.9626	−0.526135	−0.263067	−	0.964777i	$-0.584734\pi$
$607$	−12.9626	−0.526135	−0.263067	+	0.964777i	$0.584734\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.04023	0.163450
$612$	0	0
$613$	−24.4771	−0.988620	−0.494310	−	0.869286i	$-0.664579\pi$
$613$	−24.4771	−0.988620	−0.494310	+	0.869286i	$0.664579\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.9453	0.480898	0.240449	−	0.970662i	$-0.422705\pi$
$617$	11.9453	0.480898	0.240449	+	0.970662i	$0.422705\pi$
$618$	0	0
$619$	18.9084	0.759993	0.379997	−	0.924988i	$-0.375925\pi$
$619$	18.9084	0.759993	0.379997	+	0.924988i	$0.375925\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−19.7822	−0.792557
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.8073	−0.550534
$630$	0	0
$631$	11.8364	0.471201	0.235601	−	0.971850i	$-0.424294\pi$
$631$	11.8364	0.471201	0.235601	+	0.971850i	$0.424294\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12.9626	0.514404
$636$	0	0
$637$	27.5742	1.09253
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.2206	0.482683	0.241341	−	0.970440i	$-0.422413\pi$
$641$	12.2206	0.482683	0.241341	+	0.970440i	$0.422413\pi$
$642$	0	0
$643$	8.84257	0.348717	0.174358	−	0.984682i	$-0.444215\pi$
$643$	8.84257	0.348717	0.174358	+	0.984682i	$0.444215\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.40985	0.369939	0.184970	−	0.982744i	$-0.440781\pi$
$647$	9.40985	0.369939	0.184970	+	0.982744i	$0.440781\pi$
$648$	0	0
$649$	8.24066	0.323474
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.7449	−0.890075	−0.445038	−	0.895512i	$-0.646810\pi$
$653$	−22.7449	−0.890075	−0.445038	+	0.895512i	$0.646810\pi$
$654$	0	0
$655$	1.47569	0.0576602
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.3635	−0.793249	−0.396625	−	0.917981i	$-0.629819\pi$
$659$	−20.3635	−0.793249	−0.396625	+	0.917981i	$0.629819\pi$
$660$	0	0
$661$	11.6616	0.453585	0.226792	−	0.973943i	$-0.427176\pi$
$661$	11.6616	0.453585	0.226792	+	0.973943i	$0.427176\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.76497	−0.184778
$666$	0	0
$667$	−14.3384	−0.555184
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.47569	−0.134178
$672$	0	0
$673$	30.0070	1.15669	0.578343	−	0.815794i	$-0.303700\pi$
$673$	30.0070	1.15669	0.578343	+	0.815794i	$0.303700\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−51.4695	−1.97813	−0.989067	−	0.147466i	$-0.952888\pi$
$677$	−51.4695	−1.97813	−0.989067	+	0.147466i	$0.952888\pi$
$678$	0	0
$679$	17.5214	0.672411
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−27.6186	−1.05680	−0.528399	−	0.848996i	$-0.677207\pi$
$683$	−27.6186	−1.05680	−0.528399	+	0.848996i	$0.677207\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.97376	0.265679
$690$	0	0
$691$	34.2344	1.30234	0.651169	−	0.758933i	$-0.274279\pi$
$691$	34.2344	1.30234	0.651169	+	0.758933i	$0.274279\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.889725	−0.0337492
$696$	0	0
$697$	−22.0050	−0.833499
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.8524	−0.447658	−0.223829	−	0.974628i	$-0.571856\pi$
$701$	−11.8524	−0.447658	−0.223829	+	0.974628i	$0.571856\pi$
$702$	0	0
$703$	6.59241	0.248637
$704$	0	0
$705$	0	0
$706$	0	0
$707$	57.6013	2.16632
$708$	0	0
$709$	−28.1658	−1.05779	−0.528895	−	0.848687i	$-0.677393\pi$
$709$	−28.1658	−1.05779	−0.528895	+	0.848687i	$0.677393\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−87.7241	−3.28529
$714$	0	0
$715$	2.73785	0.102390
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26.2663	0.979567	0.489783	−	0.871844i	$-0.337076\pi$
$719$	26.2663	0.979567	0.489783	+	0.871844i	$0.337076\pi$
$720$	0	0
$721$	76.4550	2.84733
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.67756	0.0623030
$726$	0	0
$727$	−21.1401	−0.784042	−0.392021	−	0.919956i	$-0.628224\pi$
$727$	−21.1401	−0.784042	−0.392021	+	0.919956i	$0.628224\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.6415	0.726467
$732$	0	0
$733$	3.86406	0.142722	0.0713611	−	0.997451i	$-0.477266\pi$
$733$	3.86406	0.142722	0.0713611	+	0.997451i	$0.477266\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.3350	0.564872
$738$	0	0
$739$	14.0298	0.516094	0.258047	−	0.966132i	$-0.416921\pi$
$739$	14.0298	0.516094	0.258047	+	0.966132i	$0.416921\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.2934	−0.964611	−0.482306	−	0.876003i	$-0.660201\pi$
$743$	−26.2934	−0.964611	−0.482306	+	0.876003i	$0.660201\pi$
$744$	0	0
$745$	12.7289	0.466352
$746$	0	0
$747$	0	0
$748$	0	0
$749$	56.8793	2.07832
$750$	0	0
$751$	−38.2206	−1.39469	−0.697344	−	0.716737i	$-0.745635\pi$
$751$	−38.2206	−1.39469	−0.697344	+	0.716737i	$0.745635\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.15325	−0.0419712
$756$	0	0
$757$	−36.6158	−1.33082	−0.665411	−	0.746477i	$-0.731744\pi$
$757$	−36.6158	−1.33082	−0.665411	+	0.746477i	$0.731744\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	51.6803	1.87341	0.936705	−	0.350120i	$-0.113859\pi$
$761$	51.6803	1.87341	0.936705	+	0.350120i	$0.113859\pi$
$762$	0	0
$763$	52.8363	1.91280
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.5617	−0.814655
$768$	0	0
$769$	44.4841	1.60414	0.802068	−	0.597232i	$-0.203733\pi$
$769$	44.4841	1.60414	0.802068	+	0.597232i	$0.203733\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.973762	0.0350238	0.0175119	−	0.999847i	$-0.494426\pi$
$773$	0.973762	0.0350238	0.0175119	+	0.999847i	$0.494426\pi$
$774$	0	0
$775$	10.2635	0.368677
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.5065	0.376433
$780$	0	0
$781$	−2.54717	−0.0911449
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.95702	0.141232
$786$	0	0
$787$	30.5382	1.08857	0.544285	−	0.838900i	$-0.316801\pi$
$787$	30.5382	1.08857	0.544285	+	0.838900i	$0.316801\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	61.1943	2.17582
$792$	0	0
$793$	9.51592	0.337920
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.7247	0.946639	0.473319	−	0.880891i	$-0.343056\pi$
$797$	26.7247	0.946639	0.473319	+	0.880891i	$0.343056\pi$
$798$	0	0
$799$	−3.56440	−0.126099
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.73785	−0.0966165
$804$	0	0
$805$	35.3149	1.24469
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17.0327	−0.598837	−0.299418	−	0.954122i	$-0.596793\pi$
$809$	−17.0327	−0.598837	−0.299418	+	0.954122i	$0.596793\pi$
$810$	0	0
$811$	−23.1103	−0.811512	−0.405756	−	0.913982i	$-0.632992\pi$
$811$	−23.1103	−0.811512	−0.405756	+	0.913982i	$0.632992\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.07148	0.107589
$816$	0	0
$817$	−9.37798	−0.328094
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.3476	−0.989337	−0.494668	−	0.869082i	$-0.664710\pi$
$821$	−28.3476	−0.989337	−0.494668	+	0.869082i	$0.664710\pi$
$822$	0	0
$823$	10.4585	0.364559	0.182280	−	0.983247i	$-0.441652\pi$
$823$	10.4585	0.364559	0.182280	+	0.983247i	$0.441652\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.2672	1.57409	0.787047	−	0.616893i	$-0.211609\pi$
$827$	45.2672	1.57409	0.787047	+	0.616893i	$0.211609\pi$
$828$	0	0
$829$	−22.9855	−0.798320	−0.399160	−	0.916881i	$-0.630698\pi$
$829$	−22.9855	−0.798320	−0.399160	+	0.916881i	$0.630698\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−24.3267	−0.842870
$834$	0	0
$835$	−2.71498	−0.0939559
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.1547	0.695818	0.347909	−	0.937528i	$-0.386892\pi$
$839$	20.1547	0.695818	0.347909	+	0.937528i	$0.386892\pi$
$840$	0	0
$841$	−26.1858	−0.902959
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.50419	0.189350
$846$	0	0
$847$	4.13176	0.141969
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−48.8587	−1.67485
$852$	0	0
$853$	−38.2705	−1.31036	−0.655179	−	0.755474i	$-0.727407\pi$
$853$	−38.2705	−1.31036	−0.655179	+	0.755474i	$0.727407\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.51250	−0.324941	−0.162470	−	0.986713i	$-0.551946\pi$
$857$	−9.51250	−0.324941	−0.162470	+	0.986713i	$0.551946\pi$
$858$	0	0
$859$	1.61865	0.0552275	0.0276137	−	0.999619i	$-0.491209\pi$
$859$	1.61865	0.0552275	0.0276137	+	0.999619i	$0.491209\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46.0229	−1.56664	−0.783318	−	0.621621i	$-0.786474\pi$
$863$	−46.0229	−1.56664	−0.783318	+	0.621621i	$0.786474\pi$
$864$	0	0
$865$	14.4154	0.490138
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.15325	0.174812
$870$	0	0
$871$	−41.9849	−1.42260
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.13176	−0.139679
$876$	0	0
$877$	−28.7630	−0.971257	−0.485628	−	0.874165i	$-0.661409\pi$
$877$	−28.7630	−0.971257	−0.485628	+	0.874165i	$0.661409\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.3149	−0.785499	−0.392749	−	0.919646i	$-0.628476\pi$
$881$	−23.3149	−0.785499	−0.392749	+	0.919646i	$0.628476\pi$
$882$	0	0
$883$	−29.6616	−0.998193	−0.499097	−	0.866546i	$-0.666335\pi$
$883$	−29.6616	−0.998193	−0.499097	+	0.866546i	$0.666335\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.3364	−0.682829	−0.341414	−	0.939913i	$-0.610906\pi$
$887$	−20.3364	−0.682829	−0.341414	+	0.939913i	$0.610906\pi$
$888$	0	0
$889$	−53.5583	−1.79629
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.70185	0.0569502
$894$	0	0
$895$	−10.7878	−0.360598
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.2177	0.574241
$900$	0	0
$901$	−6.15245	−0.204968
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.50419	0.216207
$906$	0	0
$907$	−9.87942	−0.328041	−0.164020	−	0.986457i	$-0.552446\pi$
$907$	−9.87942	−0.328041	−0.164020	+	0.986457i	$0.552446\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.22343	−0.239323	−0.119662	−	0.992815i	$-0.538181\pi$
$911$	−7.22343	−0.239323	−0.119662	+	0.992815i	$0.538181\pi$
$912$	0	0
$913$	−9.28502	−0.307289
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.09722	−0.201348
$918$	0	0
$919$	−35.7233	−1.17840	−0.589201	−	0.807986i	$-0.700557\pi$
$919$	−35.7233	−1.17840	−0.589201	+	0.807986i	$0.700557\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.97376	0.229544
$924$	0	0
$925$	5.71636	0.187953
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.12621	−0.0369498	−0.0184749	−	0.999829i	$-0.505881\pi$
$929$	−1.12621	−0.0369498	−0.0184749	+	0.999829i	$0.505881\pi$
$930$	0	0
$931$	11.6150	0.380665
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.41541	−0.0789922
$936$	0	0
$937$	−0.127587	−0.00416807	−0.00208404	−	0.999998i	$-0.500663\pi$
$937$	−0.127587	−0.00416807	−0.00208404	+	0.999998i	$0.500663\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.41403	0.0460961	0.0230480	−	0.999734i	$-0.492663\pi$
$941$	1.41403	0.0460961	0.0230480	+	0.999734i	$0.492663\pi$
$942$	0	0
$943$	−77.8671	−2.53570
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.1943	1.20865	0.604326	−	0.796737i	$-0.293442\pi$
$947$	37.1943	1.20865	0.604326	+	0.796737i	$0.293442\pi$
$948$	0	0
$949$	7.49581	0.243324
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−16.5041	−0.534621	−0.267310	−	0.963610i	$-0.586135\pi$
$953$	−16.5041	−0.534621	−0.267310	+	0.963610i	$0.586135\pi$
$954$	0	0
$955$	−15.8822	−0.513935
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.7906	0.800530
$960$	0	0
$961$	74.3400	2.39807
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.83219	0.123362
$966$	0	0
$967$	3.60471	0.115920	0.0579598	−	0.998319i	$-0.481540\pi$
$967$	3.60471	0.115920	0.0579598	+	0.998319i	$0.481540\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.6214	−0.950596	−0.475298	−	0.879825i	$-0.657660\pi$
$971$	−29.6214	−0.950596	−0.475298	+	0.879825i	$0.657660\pi$
$972$	0	0
$973$	3.67613	0.117851
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−52.7157	−1.68653	−0.843263	−	0.537501i	$-0.819368\pi$
$977$	−52.7157	−1.68653	−0.843263	+	0.537501i	$0.819368\pi$
$978$	0	0
$979$	4.78783	0.153020
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−29.2602	−0.933254	−0.466627	−	0.884454i	$-0.654531\pi$
$983$	−29.2602	−0.933254	−0.466627	+	0.884454i	$0.654531\pi$
$984$	0	0
$985$	−2.67893	−0.0853579
$986$	0	0
$987$	0	0
$988$	0	0
$989$	69.5036	2.21008
$990$	0	0
$991$	−38.1776	−1.21275	−0.606375	−	0.795179i	$-0.707377\pi$
$991$	−38.1776	−1.21275	−0.606375	+	0.795179i	$0.707377\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.78783	0.215189
$996$	0	0
$997$	−40.1500	−1.27156	−0.635781	−	0.771870i	$-0.719322\pi$
$997$	−40.1500	−1.27156	−0.635781	+	0.771870i	$0.719322\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.cm.1.4		4
3.2	odd	2		7920.2.a.cn.1.4		4
4.3	odd	2		495.2.a.f.1.1	✓	4
12.11	even	2		495.2.a.g.1.4	yes	4
20.3	even	4		2475.2.c.t.199.8		8
20.7	even	4		2475.2.c.t.199.1		8
20.19	odd	2		2475.2.a.bj.1.4		4
44.43	even	2		5445.2.a.bs.1.4		4
60.23	odd	4		2475.2.c.s.199.1		8
60.47	odd	4		2475.2.c.s.199.8		8
60.59	even	2		2475.2.a.bf.1.1		4
132.131	odd	2		5445.2.a.bh.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.a.f.1.1	✓	4	4.3	odd	2
495.2.a.g.1.4	yes	4	12.11	even	2
2475.2.a.bf.1.1		4	60.59	even	2
2475.2.a.bj.1.4		4	20.19	odd	2
2475.2.c.s.199.1		8	60.23	odd	4
2475.2.c.s.199.8		8	60.47	odd	4
2475.2.c.t.199.1		8	20.7	even	4
2475.2.c.t.199.8		8	20.3	even	4
5445.2.a.bh.1.1		4	132.131	odd	2
5445.2.a.bs.1.4		4	44.43	even	2
7920.2.a.cm.1.4		4	1.1	even	1	trivial
7920.2.a.cn.1.4		4	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.13176 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +4.13176 q^{7} -1.00000 q^{11} +2.73785 q^{13} -2.41541 q^{17} +1.15325 q^{19} -8.54717 q^{23} +1.00000 q^{25} +1.67756 q^{29} +10.2635 q^{31} -4.13176 q^{35} +5.71636 q^{37} +9.11028 q^{41} -8.13176 q^{43} +1.47569 q^{47} +10.0715 q^{49} +2.54717 q^{53} +1.00000 q^{55} -8.24066 q^{59} +3.47569 q^{61} -2.73785 q^{65} -15.3350 q^{67} +2.54717 q^{71} +2.73785 q^{73} -4.13176 q^{77} -5.15325 q^{79} +9.28502 q^{83} +2.41541 q^{85} -4.78783 q^{89} +11.3121 q^{91} -1.15325 q^{95} +4.24066 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) 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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