LMFDB - Embedded newform 7920.2.a.ce.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.2415184009$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1320)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +5.12311 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{11} -3.12311 q^{13} -3.12311 q^{17} -4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} +5.12311 q^{35} +6.00000 q^{37} -6.00000 q^{41} -5.12311 q^{43} +4.00000 q^{47} +19.2462 q^{49} +8.24621 q^{53} +1.00000 q^{55} +4.00000 q^{59} +10.0000 q^{61} -3.12311 q^{65} +6.24621 q^{67} -6.24621 q^{71} +4.87689 q^{73} +5.12311 q^{77} -2.24621 q^{79} +11.3693 q^{83} -3.12311 q^{85} +16.2462 q^{89} -16.0000 q^{91} +12.2462 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} + 2 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{29} + 2 q^{35} + 12 q^{37} - 12 q^{41} - 2 q^{43} + 8 q^{47} + 22 q^{49} + 2 q^{55} + 8 q^{59} + 20 q^{61} + 2 q^{65} - 4 q^{67} + 4 q^{71} + 18 q^{73} + 2 q^{77} + 12 q^{79} - 2 q^{83} + 2 q^{85} + 16 q^{89} - 32 q^{91} + 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 + 2 * q^17 - 8 * q^23 + 2 * q^25 - 4 * q^29 + 2 * q^35 + 12 * q^37 - 12 * q^41 - 2 * q^43 + 8 * q^47 + 22 * q^49 + 2 * q^55 + 8 * q^59 + 20 * q^61 + 2 * q^65 - 4 * q^67 + 4 * q^71 + 18 * q^73 + 2 * q^77 + 12 * q^79 - 2 * q^83 + 2 * q^85 + 16 * q^89 - 32 * q^91 + 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
$7$	5.12311	1.93635	0.968176	−	0.250270i	$-0.0805195\pi$
$7$	5.12311	1.93635	0.968176	+	0.250270i	$0.0805195\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.12311	−0.866194	−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311	−0.866194	−0.433097	+	0.901347i	$0.642579\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.12311	0.865963
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−5.12311	−0.781266	−0.390633	−	0.920546i	$-0.627744\pi$
$43$	−5.12311	−0.781266	−0.390633	+	0.920546i	$0.627744\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.24621	1.13270	0.566352	−	0.824163i	$-0.308354\pi$
$53$	8.24621	1.13270	0.566352	+	0.824163i	$0.308354\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.12311	−0.387374
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.24621	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$71$	−6.24621	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$72$	0	0
$73$	4.87689	0.570797	0.285399	−	0.958409i	$-0.407874\pi$
$73$	4.87689	0.570797	0.285399	+	0.958409i	$0.407874\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.12311	0.583832
$78$	0	0
$79$	−2.24621	−0.252719	−0.126359	−	0.991985i	$-0.540329\pi$
$79$	−2.24621	−0.252719	−0.126359	+	0.991985i	$0.540329\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.3693	1.24794	0.623972	−	0.781446i	$-0.285518\pi$
$83$	11.3693	1.24794	0.623972	+	0.781446i	$0.285518\pi$
$84$	0	0
$85$	−3.12311	−0.338748
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.2462	1.72209	0.861047	−	0.508525i	$-0.169809\pi$
$89$	16.2462	1.72209	0.861047	+	0.508525i	$0.169809\pi$
$90$	0	0
$91$	−16.0000	−1.67726
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.2462	1.24341	0.621707	−	0.783250i	$-0.286439\pi$
$97$	12.2462	1.24341	0.621707	+	0.783250i	$0.286439\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3693	1.09911	0.549557	−	0.835456i	$-0.314797\pi$
$107$	11.3693	1.09911	0.549557	+	0.835456i	$0.314797\pi$
$108$	0	0
$109$	20.2462	1.93924	0.969618	−	0.244625i	$-0.0786650\pi$
$109$	20.2462	1.93924	0.969618	+	0.244625i	$0.0786650\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.2462	1.52831	0.764157	−	0.645030i	$-0.223155\pi$
$113$	16.2462	1.52831	0.764157	+	0.645030i	$0.223155\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16.0000	−1.46672
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.12311	−0.454602	−0.227301	−	0.973825i	$-0.572990\pi$
$127$	−5.12311	−0.454602	−0.227301	+	0.973825i	$0.572990\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.75379	0.153229	0.0766146	−	0.997061i	$-0.475589\pi$
$131$	1.75379	0.153229	0.0766146	+	0.997061i	$0.475589\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.2462	−1.72975	−0.864875	−	0.501987i	$-0.832603\pi$
$137$	−20.2462	−1.72975	−0.864875	+	0.501987i	$0.832603\pi$
$138$	0	0
$139$	2.24621	0.190521	0.0952606	−	0.995452i	$-0.469632\pi$
$139$	2.24621	0.190521	0.0952606	+	0.995452i	$0.469632\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.12311	−0.261167
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.2462	1.33094	0.665471	−	0.746424i	$-0.268231\pi$
$149$	16.2462	1.33094	0.665471	+	0.746424i	$0.268231\pi$
$150$	0	0
$151$	−10.2462	−0.833825	−0.416912	−	0.908947i	$-0.636888\pi$
$151$	−10.2462	−0.833825	−0.416912	+	0.908947i	$0.636888\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.24621	−0.338885	−0.169442	−	0.985540i	$-0.554197\pi$
$157$	−4.24621	−0.338885	−0.169442	+	0.985540i	$0.554197\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−20.4924	−1.61503
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.8769	−0.841679	−0.420840	−	0.907135i	$-0.638265\pi$
$167$	−10.8769	−0.841679	−0.420840	+	0.907135i	$0.638265\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.6155	−1.49134	−0.745670	−	0.666315i	$-0.767871\pi$
$173$	−19.6155	−1.49134	−0.745670	+	0.666315i	$0.767871\pi$
$174$	0	0
$175$	5.12311	0.387270
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−16.4924	−1.23270	−0.616351	−	0.787472i	$-0.711390\pi$
$179$	−16.4924	−1.23270	−0.616351	+	0.787472i	$0.711390\pi$
$180$	0	0
$181$	24.2462	1.80221	0.901103	−	0.433604i	$-0.142758\pi$
$181$	24.2462	1.80221	0.901103	+	0.433604i	$0.142758\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	−3.12311	−0.228384
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.2462	1.03082	0.515410	−	0.856944i	$-0.327640\pi$
$191$	14.2462	1.03082	0.515410	+	0.856944i	$0.327640\pi$
$192$	0	0
$193$	1.36932	0.0985656	0.0492828	−	0.998785i	$-0.484306\pi$
$193$	1.36932	0.0985656	0.0492828	+	0.998785i	$0.484306\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.12311	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$197$	−7.12311	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$198$	0	0
$199$	20.4924	1.45267	0.726335	−	0.687341i	$-0.241222\pi$
$199$	20.4924	1.45267	0.726335	+	0.687341i	$0.241222\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.2462	−0.719143
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.12311	−0.349393
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.75379	0.656111
$222$	0	0
$223$	−22.7386	−1.52269	−0.761346	−	0.648346i	$-0.775461\pi$
$223$	−22.7386	−1.52269	−0.761346	+	0.648346i	$0.775461\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.1231	1.66748	0.833740	−	0.552158i	$-0.186195\pi$
$227$	25.1231	1.66748	0.833740	+	0.552158i	$0.186195\pi$
$228$	0	0
$229$	18.4924	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$229$	18.4924	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.1231	1.51485	0.757423	−	0.652925i	$-0.226458\pi$
$233$	23.1231	1.51485	0.757423	+	0.652925i	$0.226458\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.2462	−1.69773	−0.848863	−	0.528613i	$-0.822712\pi$
$239$	−26.2462	−1.69773	−0.848863	+	0.528613i	$0.822712\pi$
$240$	0	0
$241$	−28.7386	−1.85122	−0.925609	−	0.378481i	$-0.876447\pi$
$241$	−28.7386	−1.85122	−0.925609	+	0.378481i	$0.876447\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	19.2462	1.22960
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	30.7386	1.91001
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−17.6155	−1.08622	−0.543110	−	0.839662i	$-0.682753\pi$
$263$	−17.6155	−1.08622	−0.543110	+	0.839662i	$0.682753\pi$
$264$	0	0
$265$	8.24621	0.506561
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.24621	−0.258896	−0.129448	−	0.991586i	$-0.541321\pi$
$269$	−4.24621	−0.258896	−0.129448	+	0.991586i	$0.541321\pi$
$270$	0	0
$271$	−18.2462	−1.10838	−0.554189	−	0.832391i	$-0.686972\pi$
$271$	−18.2462	−1.10838	−0.554189	+	0.832391i	$0.686972\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	−23.6155	−1.41892	−0.709460	−	0.704746i	$-0.751061\pi$
$277$	−23.6155	−1.41892	−0.709460	+	0.704746i	$0.751061\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	10.8769	0.646564	0.323282	−	0.946303i	$-0.395214\pi$
$283$	10.8769	0.646564	0.323282	+	0.946303i	$0.395214\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−30.7386	−1.81444
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.3693	−1.01473	−0.507363	−	0.861732i	$-0.669380\pi$
$293$	−17.3693	−1.01473	−0.507363	+	0.861732i	$0.669380\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.4924	0.722455
$300$	0	0
$301$	−26.2462	−1.51281
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	−13.1231	−0.748975	−0.374488	−	0.927232i	$-0.622181\pi$
$307$	−13.1231	−0.748975	−0.374488	+	0.927232i	$0.622181\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.2462	1.26147	0.630733	−	0.776000i	$-0.282754\pi$
$311$	22.2462	1.26147	0.630733	+	0.776000i	$0.282754\pi$
$312$	0	0
$313$	−30.9848	−1.75137	−0.875683	−	0.482886i	$-0.839589\pi$
$313$	−30.9848	−1.75137	−0.875683	+	0.482886i	$0.839589\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−3.12311	−0.173239
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.4924	1.12978
$330$	0	0
$331$	−32.4924	−1.78595	−0.892973	−	0.450111i	$-0.851384\pi$
$331$	−32.4924	−1.78595	−0.892973	+	0.450111i	$0.851384\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.24621	0.341267
$336$	0	0
$337$	12.8769	0.701449	0.350725	−	0.936479i	$-0.385935\pi$
$337$	12.8769	0.701449	0.350725	+	0.936479i	$0.385935\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	62.7386	3.38757
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.36932	0.180874	0.0904372	−	0.995902i	$-0.471174\pi$
$347$	3.36932	0.180874	0.0904372	+	0.995902i	$0.471174\pi$
$348$	0	0
$349$	−2.49242	−0.133416	−0.0667082	−	0.997773i	$-0.521250\pi$
$349$	−2.49242	−0.133416	−0.0667082	+	0.997773i	$0.521250\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.49242	−0.345557	−0.172778	−	0.984961i	$-0.555274\pi$
$353$	−6.49242	−0.345557	−0.172778	+	0.984961i	$0.555274\pi$
$354$	0	0
$355$	−6.24621	−0.331514
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.75379	−0.303673	−0.151837	−	0.988406i	$-0.548519\pi$
$359$	−5.75379	−0.303673	−0.151837	+	0.988406i	$0.548519\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.87689	0.255268
$366$	0	0
$367$	−22.7386	−1.18695	−0.593474	−	0.804854i	$-0.702244\pi$
$367$	−22.7386	−1.18695	−0.593474	+	0.804854i	$0.702244\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	42.2462	2.19331
$372$	0	0
$373$	27.6155	1.42988	0.714939	−	0.699187i	$-0.246454\pi$
$373$	27.6155	1.42988	0.714939	+	0.699187i	$0.246454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.24621	0.321696
$378$	0	0
$379$	−32.4924	−1.66902	−0.834512	−	0.550990i	$-0.814250\pi$
$379$	−32.4924	−1.66902	−0.834512	+	0.550990i	$0.814250\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.24621	−0.319166	−0.159583	−	0.987184i	$-0.551015\pi$
$383$	−6.24621	−0.319166	−0.159583	+	0.987184i	$0.551015\pi$
$384$	0	0
$385$	5.12311	0.261098
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.7386	−1.25430	−0.627149	−	0.778899i	$-0.715778\pi$
$389$	−24.7386	−1.25430	−0.627149	+	0.778899i	$0.715778\pi$
$390$	0	0
$391$	12.4924	0.631769
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.24621	−0.113019
$396$	0	0
$397$	20.7386	1.04084	0.520421	−	0.853910i	$-0.325775\pi$
$397$	20.7386	1.04084	0.520421	+	0.853910i	$0.325775\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.24621	0.411796	0.205898	−	0.978573i	$-0.433988\pi$
$401$	8.24621	0.411796	0.205898	+	0.978573i	$0.433988\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	34.9848	1.72989	0.864945	−	0.501867i	$-0.167353\pi$
$409$	34.9848	1.72989	0.864945	+	0.501867i	$0.167353\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	20.4924	1.00837
$414$	0	0
$415$	11.3693	0.558098
$416$	0	0
$417$	0	0
$418$	0	0
$419$	28.9848	1.41600	0.708001	−	0.706211i	$-0.249597\pi$
$419$	28.9848	1.41600	0.708001	+	0.706211i	$0.249597\pi$
$420$	0	0
$421$	−15.7538	−0.767793	−0.383896	−	0.923376i	$-0.625418\pi$
$421$	−15.7538	−0.767793	−0.383896	+	0.923376i	$0.625418\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.12311	−0.151493
$426$	0	0
$427$	51.2311	2.47924
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	−28.7386	−1.38109	−0.690545	−	0.723289i	$-0.742629\pi$
$433$	−28.7386	−1.38109	−0.690545	+	0.723289i	$0.742629\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−28.4924	−1.35987	−0.679935	−	0.733273i	$-0.737992\pi$
$439$	−28.4924	−1.35987	−0.679935	+	0.733273i	$0.737992\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	16.2462	0.770144
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.2462	0.766706	0.383353	−	0.923602i	$-0.374769\pi$
$449$	16.2462	0.766706	0.383353	+	0.923602i	$0.374769\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−16.0000	−0.750092
$456$	0	0
$457$	27.6155	1.29180	0.645900	−	0.763422i	$-0.276482\pi$
$457$	27.6155	1.29180	0.645900	+	0.763422i	$0.276482\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.24621	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$461$	8.24621	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$462$	0	0
$463$	−10.2462	−0.476182	−0.238091	−	0.971243i	$-0.576522\pi$
$463$	−10.2462	−0.476182	−0.238091	+	0.971243i	$0.576522\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.24621	−0.289040	−0.144520	−	0.989502i	$-0.546164\pi$
$467$	−6.24621	−0.289040	−0.144520	+	0.989502i	$0.546164\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.12311	−0.235561
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.24621	−0.102632	−0.0513160	−	0.998682i	$-0.516342\pi$
$479$	−2.24621	−0.102632	−0.0513160	+	0.998682i	$0.516342\pi$
$480$	0	0
$481$	−18.7386	−0.854408
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.2462	0.556072
$486$	0	0
$487$	−26.2462	−1.18933	−0.594665	−	0.803974i	$-0.702715\pi$
$487$	−26.2462	−1.18933	−0.594665	+	0.803974i	$0.702715\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	6.24621	0.281315
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−32.0000	−1.43540
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0.630683	0.0281208	0.0140604	−	0.999901i	$-0.495524\pi$
$503$	0.630683	0.0281208	0.0140604	+	0.999901i	$0.495524\pi$
$504$	0	0
$505$	0.246211	0.0109563
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.7538	1.23017	0.615083	−	0.788463i	$-0.289123\pi$
$509$	27.7538	1.23017	0.615083	+	0.788463i	$0.289123\pi$
$510$	0	0
$511$	24.9848	1.10526
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	4.00000	0.175920
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.246211	0.0107867	0.00539336	−	0.999985i	$-0.498283\pi$
$521$	0.246211	0.0107867	0.00539336	+	0.999985i	$0.498283\pi$
$522$	0	0
$523$	21.1231	0.923649	0.461824	−	0.886971i	$-0.347195\pi$
$523$	21.1231	0.923649	0.461824	+	0.886971i	$0.347195\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	18.7386	0.811660
$534$	0	0
$535$	11.3693	0.491538
$536$	0	0
$537$	0	0
$538$	0	0
$539$	19.2462	0.828993
$540$	0	0
$541$	−11.7538	−0.505335	−0.252667	−	0.967553i	$-0.581308\pi$
$541$	−11.7538	−0.505335	−0.252667	+	0.967553i	$0.581308\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	20.2462	0.867252
$546$	0	0
$547$	0.630683	0.0269661	0.0134830	−	0.999909i	$-0.495708\pi$
$547$	0.630683	0.0269661	0.0134830	+	0.999909i	$0.495708\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−11.5076	−0.489352
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.6307	−0.450437	−0.225218	−	0.974308i	$-0.572310\pi$
$557$	−10.6307	−0.450437	−0.225218	+	0.974308i	$0.572310\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.3693	−0.816319	−0.408160	−	0.912911i	$-0.633829\pi$
$563$	−19.3693	−0.816319	−0.408160	+	0.912911i	$0.633829\pi$
$564$	0	0
$565$	16.2462	0.683483
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.7386	0.701720	0.350860	−	0.936428i	$-0.385889\pi$
$569$	16.7386	0.701720	0.350860	+	0.936428i	$0.385889\pi$
$570$	0	0
$571$	−46.7386	−1.95595	−0.977975	−	0.208720i	$-0.933070\pi$
$571$	−46.7386	−1.95595	−0.977975	+	0.208720i	$0.933070\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	23.7538	0.988883	0.494442	−	0.869211i	$-0.335373\pi$
$577$	23.7538	0.988883	0.494442	+	0.869211i	$0.335373\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	58.2462	2.41646
$582$	0	0
$583$	8.24621	0.341523
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.7386	−0.443231	−0.221615	−	0.975134i	$-0.571133\pi$
$587$	−10.7386	−0.443231	−0.221615	+	0.975134i	$0.571133\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−43.1231	−1.77085	−0.885427	−	0.464779i	$-0.846134\pi$
$593$	−43.1231	−1.77085	−0.885427	+	0.464779i	$0.846134\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.7386	0.438769	0.219384	−	0.975639i	$-0.429595\pi$
$599$	10.7386	0.438769	0.219384	+	0.975639i	$0.429595\pi$
$600$	0	0
$601$	24.7386	1.00911	0.504555	−	0.863380i	$-0.331657\pi$
$601$	24.7386	1.00911	0.504555	+	0.863380i	$0.331657\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	10.8769	0.441480	0.220740	−	0.975333i	$-0.429153\pi$
$607$	10.8769	0.441480	0.220740	+	0.975333i	$0.429153\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.4924	−0.505389
$612$	0	0
$613$	15.1231	0.610816	0.305408	−	0.952222i	$-0.401207\pi$
$613$	15.1231	0.610816	0.305408	+	0.952222i	$0.401207\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.2462	0.976116	0.488058	−	0.872811i	$-0.337706\pi$
$617$	24.2462	0.976116	0.488058	+	0.872811i	$0.337706\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	83.2311	3.33458
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.7386	−0.747158
$630$	0	0
$631$	3.50758	0.139634	0.0698172	−	0.997560i	$-0.477758\pi$
$631$	3.50758	0.139634	0.0698172	+	0.997560i	$0.477758\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.12311	−0.203304
$636$	0	0
$637$	−60.1080	−2.38156
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20.2462	−0.799677	−0.399839	−	0.916586i	$-0.630934\pi$
$641$	−20.2462	−0.799677	−0.399839	+	0.916586i	$0.630934\pi$
$642$	0	0
$643$	−32.4924	−1.28138	−0.640688	−	0.767801i	$-0.721351\pi$
$643$	−32.4924	−1.28138	−0.640688	+	0.767801i	$0.721351\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.7386	1.05120	0.525602	−	0.850731i	$-0.323840\pi$
$647$	26.7386	1.05120	0.525602	+	0.850731i	$0.323840\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.7386	−0.655033	−0.327517	−	0.944845i	$-0.606212\pi$
$653$	−16.7386	−0.655033	−0.327517	+	0.944845i	$0.606212\pi$
$654$	0	0
$655$	1.75379	0.0685262
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	−14.6307	−0.563971	−0.281986	−	0.959419i	$-0.590993\pi$
$673$	−14.6307	−0.563971	−0.281986	+	0.959419i	$0.590993\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.61553	0.292689	0.146344	−	0.989234i	$-0.453249\pi$
$677$	7.61553	0.292689	0.146344	+	0.989234i	$0.453249\pi$
$678$	0	0
$679$	62.7386	2.40769
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.2462	1.46345	0.731725	−	0.681600i	$-0.238715\pi$
$683$	38.2462	1.46345	0.731725	+	0.681600i	$0.238715\pi$
$684$	0	0
$685$	−20.2462	−0.773568
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−25.7538	−0.981141
$690$	0	0
$691$	8.49242	0.323067	0.161533	−	0.986867i	$-0.448356\pi$
$691$	8.49242	0.323067	0.161533	+	0.986867i	$0.448356\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.24621	0.0852036
$696$	0	0
$697$	18.7386	0.709776
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.26137	0.0474386
$708$	0	0
$709$	3.75379	0.140976	0.0704882	−	0.997513i	$-0.477544\pi$
$709$	3.75379	0.140976	0.0704882	+	0.997513i	$0.477544\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−3.12311	−0.116798
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.24621	0.232944	0.116472	−	0.993194i	$-0.462841\pi$
$719$	6.24621	0.232944	0.116472	+	0.993194i	$0.462841\pi$
$720$	0	0
$721$	40.9848	1.52636
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−24.9848	−0.926637	−0.463318	−	0.886192i	$-0.653341\pi$
$727$	−24.9848	−0.926637	−0.463318	+	0.886192i	$0.653341\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	23.1231	0.854071	0.427036	−	0.904235i	$-0.359558\pi$
$733$	23.1231	0.854071	0.427036	+	0.904235i	$0.359558\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.24621	0.230082
$738$	0	0
$739$	−26.2462	−0.965482	−0.482741	−	0.875763i	$-0.660359\pi$
$739$	−26.2462	−0.965482	−0.482741	+	0.875763i	$0.660359\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.3693	0.857337	0.428669	−	0.903462i	$-0.358983\pi$
$743$	23.3693	0.857337	0.428669	+	0.903462i	$0.358983\pi$
$744$	0	0
$745$	16.2462	0.595215
$746$	0	0
$747$	0	0
$748$	0	0
$749$	58.2462	2.12827
$750$	0	0
$751$	24.9848	0.911710	0.455855	−	0.890054i	$-0.349334\pi$
$751$	24.9848	0.911710	0.455855	+	0.890054i	$0.349334\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.2462	−0.372898
$756$	0	0
$757$	−4.24621	−0.154331	−0.0771656	−	0.997018i	$-0.524587\pi$
$757$	−4.24621	−0.154331	−0.0771656	+	0.997018i	$0.524587\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.24621	−0.298925	−0.149462	−	0.988767i	$-0.547754\pi$
$761$	−8.24621	−0.298925	−0.149462	+	0.988767i	$0.547754\pi$
$762$	0	0
$763$	103.723	3.75504
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.4924	−0.451075
$768$	0	0
$769$	−48.2462	−1.73980	−0.869901	−	0.493226i	$-0.835818\pi$
$769$	−48.2462	−1.73980	−0.869901	+	0.493226i	$0.835818\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.7538	0.710494	0.355247	−	0.934772i	$-0.384397\pi$
$773$	19.7538	0.710494	0.355247	+	0.934772i	$0.384397\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−6.24621	−0.223507
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.24621	−0.151554
$786$	0	0
$787$	−24.6307	−0.877989	−0.438995	−	0.898490i	$-0.644665\pi$
$787$	−24.6307	−0.877989	−0.438995	+	0.898490i	$0.644665\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	83.2311	2.95936
$792$	0	0
$793$	−31.2311	−1.10905
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.50758	0.336776	0.168388	−	0.985721i	$-0.446144\pi$
$797$	9.50758	0.336776	0.168388	+	0.985721i	$0.446144\pi$
$798$	0	0
$799$	−12.4924	−0.441950
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.87689	0.172102
$804$	0	0
$805$	−20.4924	−0.722263
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	−35.2311	−1.23713	−0.618565	−	0.785734i	$-0.712286\pi$
$811$	−35.2311	−1.23713	−0.618565	+	0.785734i	$0.712286\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.9848	1.08138	0.540689	−	0.841222i	$-0.318163\pi$
$821$	30.9848	1.08138	0.540689	+	0.841222i	$0.318163\pi$
$822$	0	0
$823$	−24.9848	−0.870917	−0.435458	−	0.900209i	$-0.643414\pi$
$823$	−24.9848	−0.870917	−0.435458	+	0.900209i	$0.643414\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.6307	−0.439212	−0.219606	−	0.975589i	$-0.570477\pi$
$827$	−12.6307	−0.439212	−0.219606	+	0.975589i	$0.570477\pi$
$828$	0	0
$829$	−48.7386	−1.69276	−0.846381	−	0.532577i	$-0.821224\pi$
$829$	−48.7386	−1.69276	−0.846381	+	0.532577i	$0.821224\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−60.1080	−2.08262
$834$	0	0
$835$	−10.8769	−0.376410
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.7386	−0.923120	−0.461560	−	0.887109i	$-0.652710\pi$
$839$	−26.7386	−0.923120	−0.461560	+	0.887109i	$0.652710\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3.24621	−0.111673
$846$	0	0
$847$	5.12311	0.176032
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	26.6307	0.911817	0.455909	−	0.890027i	$-0.349314\pi$
$853$	26.6307	0.911817	0.455909	+	0.890027i	$0.349314\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.6155	−1.62652	−0.813258	−	0.581904i	$-0.802308\pi$
$857$	−47.6155	−1.62652	−0.813258	+	0.581904i	$0.802308\pi$
$858$	0	0
$859$	8.49242	0.289758	0.144879	−	0.989449i	$-0.453721\pi$
$859$	8.49242	0.289758	0.144879	+	0.989449i	$0.453721\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.7386	1.45484	0.727420	−	0.686192i	$-0.240719\pi$
$863$	42.7386	1.45484	0.727420	+	0.686192i	$0.240719\pi$
$864$	0	0
$865$	−19.6155	−0.666948
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.24621	−0.0761975
$870$	0	0
$871$	−19.5076	−0.660989
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.12311	0.173193
$876$	0	0
$877$	−37.3693	−1.26187	−0.630936	−	0.775835i	$-0.717329\pi$
$877$	−37.3693	−1.26187	−0.630936	+	0.775835i	$0.717329\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.3693	−1.32189	−0.660946	−	0.750433i	$-0.729845\pi$
$887$	−39.3693	−1.32189	−0.660946	+	0.750433i	$0.729845\pi$
$888$	0	0
$889$	−26.2462	−0.880270
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−16.4924	−0.551281
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−25.7538	−0.857983
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.2462	0.805971
$906$	0	0
$907$	1.75379	0.0582336	0.0291168	−	0.999576i	$-0.490731\pi$
$907$	1.75379	0.0582336	0.0291168	+	0.999576i	$0.490731\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.75379	0.0581056	0.0290528	−	0.999578i	$-0.490751\pi$
$911$	1.75379	0.0581056	0.0290528	+	0.999578i	$0.490751\pi$
$912$	0	0
$913$	11.3693	0.376269
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.98485	0.296706
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	19.5076	0.642100
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.9848	−0.885344	−0.442672	−	0.896684i	$-0.645969\pi$
$929$	−26.9848	−0.885344	−0.442672	+	0.896684i	$0.645969\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.12311	−0.102136
$936$	0	0
$937$	2.63068	0.0859407	0.0429703	−	0.999076i	$-0.486318\pi$
$937$	2.63068	0.0859407	0.0429703	+	0.999076i	$0.486318\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−53.2311	−1.73528	−0.867641	−	0.497190i	$-0.834365\pi$
$941$	−53.2311	−1.73528	−0.867641	+	0.497190i	$0.834365\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.2462	−1.50280	−0.751400	−	0.659847i	$-0.770621\pi$
$947$	−46.2462	−1.50280	−0.751400	+	0.659847i	$0.770621\pi$
$948$	0	0
$949$	−15.2311	−0.494421
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−12.3845	−0.401172	−0.200586	−	0.979676i	$-0.564285\pi$
$953$	−12.3845	−0.401172	−0.200586	+	0.979676i	$0.564285\pi$
$954$	0	0
$955$	14.2462	0.460997
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−103.723	−3.34941
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.36932	0.0440799
$966$	0	0
$967$	43.8617	1.41050	0.705249	−	0.708959i	$-0.250835\pi$
$967$	43.8617	1.41050	0.705249	+	0.708959i	$0.250835\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32.4924	−1.04273	−0.521366	−	0.853333i	$-0.674577\pi$
$971$	−32.4924	−1.04273	−0.521366	+	0.853333i	$0.674577\pi$
$972$	0	0
$973$	11.5076	0.368916
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−7.75379	−0.248066	−0.124033	−	0.992278i	$-0.539583\pi$
$977$	−7.75379	−0.248066	−0.124033	+	0.992278i	$0.539583\pi$
$978$	0	0
$979$	16.2462	0.519231
$980$	0	0
$981$	0	0
$982$	0	0
$983$	54.2462	1.73019	0.865093	−	0.501612i	$-0.167259\pi$
$983$	54.2462	1.73019	0.865093	+	0.501612i	$0.167259\pi$
$984$	0	0
$985$	−7.12311	−0.226961
$986$	0	0
$987$	0	0
$988$	0	0
$989$	20.4924	0.651621
$990$	0	0
$991$	45.4773	1.44463	0.722317	−	0.691563i	$-0.243077\pi$
$991$	45.4773	1.44463	0.722317	+	0.691563i	$0.243077\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	20.4924	0.649653
$996$	0	0
$997$	27.6155	0.874593	0.437296	−	0.899318i	$-0.355936\pi$
$997$	27.6155	0.874593	0.437296	+	0.899318i	$0.355936\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.ce.1.2		2
3.2	odd	2		2640.2.a.ba.1.2		2
4.3	odd	2		3960.2.a.bc.1.1		2
12.11	even	2		1320.2.a.o.1.1	✓	2
60.23	odd	4		6600.2.d.bc.1849.2		4
60.47	odd	4		6600.2.d.bc.1849.3		4
60.59	even	2		6600.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.o.1.1	✓	2	12.11	even	2
2640.2.a.ba.1.2		2	3.2	odd	2
3960.2.a.bc.1.1		2	4.3	odd	2
6600.2.a.bl.1.2		2	60.59	even	2
6600.2.d.bc.1849.2		4	60.23	odd	4
6600.2.d.bc.1849.3		4	60.47	odd	4
7920.2.a.ce.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +5.12311 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +5.12311 q^{7} +1.00000 q^{11} -3.12311 q^{13} -3.12311 q^{17} -4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} +5.12311 q^{35} +6.00000 q^{37} -6.00000 q^{41} -5.12311 q^{43} +4.00000 q^{47} +19.2462 q^{49} +8.24621 q^{53} +1.00000 q^{55} +4.00000 q^{59} +10.0000 q^{61} -3.12311 q^{65} +6.24621 q^{67} -6.24621 q^{71} +4.87689 q^{73} +5.12311 q^{77} -2.24621 q^{79} +11.3693 q^{83} -3.12311 q^{85} +16.2462 q^{89} -16.0000 q^{91} +12.2462 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} + 2 q^{17} - 8 q^{23} + 2 q^{25} - 4 q^{29} + 2 q^{35} + 12 q^{37} - 12 q^{41} - 2 q^{43} + 8 q^{47} + 22 q^{49} + 2 q^{55} + 8 q^{59} + 20 q^{61} + 2 q^{65} - 4 q^{67} + 4 q^{71} + 18 q^{73} + 2 q^{77} + 12 q^{79} - 2 q^{83} + 2 q^{85} + 16 q^{89} - 32 q^{91} + 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 + 2 * q^17 - 8 * q^23 + 2 * q^25 - 4 * q^29 + 2 * q^35 + 12 * q^37 - 12 * q^41 - 2 * q^43 + 8 * q^47 + 22 * q^49 + 2 * q^55 + 8 * q^59 + 20 * q^61 + 2 * q^65 - 4 * q^67 + 4 * q^71 + 18 * q^73 + 2 * q^77 + 12 * q^79 - 2 * q^83 + 2 * q^85 + 16 * q^89 - 32 * q^91 + 8 * q^97

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