LMFDB - Embedded newform 1080.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.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:	$8.62384341830$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.77200$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.77200 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.77200 q^{7} +4.77200 q^{11} +3.00000 q^{13} -6.77200 q^{17} +5.77200 q^{19} -0.772002 q^{23} +1.00000 q^{25} -4.77200 q^{29} +10.7720 q^{31} +3.77200 q^{35} +7.77200 q^{37} +7.54400 q^{41} +6.77200 q^{43} -2.77200 q^{47} +7.22800 q^{49} +7.54400 q^{53} -4.77200 q^{55} +12.0000 q^{59} -7.77200 q^{61} -3.00000 q^{65} +6.22800 q^{67} -3.54400 q^{71} -3.77200 q^{73} -18.0000 q^{77} -5.00000 q^{79} +6.00000 q^{83} +6.77200 q^{85} +8.00000 q^{89} -11.3160 q^{91} -5.77200 q^{95} +7.31601 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + q^7	$ 2 q - 2 q^{5} + q^{7} + q^{11} + 6 q^{13} - 5 q^{17} + 3 q^{19} + 7 q^{23} + 2 q^{25} - q^{29} + 13 q^{31} - q^{35} + 7 q^{37} - 2 q^{41} + 5 q^{43} + 3 q^{47} + 23 q^{49} - 2 q^{53} - q^{55} + 24 q^{59} - 7 q^{61} - 6 q^{65} + 21 q^{67} + 10 q^{71} + q^{73} - 36 q^{77} - 10 q^{79} + 12 q^{83} + 5 q^{85} + 16 q^{89} + 3 q^{91} - 3 q^{95} - 11 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + q^7 + q^11 + 6 * q^13 - 5 * q^17 + 3 * q^19 + 7 * q^23 + 2 * q^25 - q^29 + 13 * q^31 - q^35 + 7 * q^37 - 2 * q^41 + 5 * q^43 + 3 * q^47 + 23 * q^49 - 2 * q^53 - q^55 + 24 * q^59 - 7 * q^61 - 6 * q^65 + 21 * q^67 + 10 * q^71 + q^73 - 36 * q^77 - 10 * q^79 + 12 * q^83 + 5 * q^85 + 16 * q^89 + 3 * q^91 - 3 * q^95 - 11 * 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
$7$	−3.77200	−1.42568	−0.712841	−	0.701325i	$-0.752592\pi$
$7$	−3.77200	−1.42568	−0.712841	+	0.701325i	$0.752592\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.77200	1.43881	0.719406	−	0.694589i	$-0.244414\pi$
$11$	4.77200	1.43881	0.719406	+	0.694589i	$0.244414\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.77200	−1.64245	−0.821226	−	0.570603i	$-0.806709\pi$
$17$	−6.77200	−1.64245	−0.821226	+	0.570603i	$0.806709\pi$
$18$	0	0
$19$	5.77200	1.32419	0.662094	−	0.749421i	$-0.269668\pi$
$19$	5.77200	1.32419	0.662094	+	0.749421i	$0.269668\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.772002	−0.160974	−0.0804868	−	0.996756i	$-0.525647\pi$
$23$	−0.772002	−0.160974	−0.0804868	+	0.996756i	$0.525647\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.77200	−0.886139	−0.443069	−	0.896487i	$-0.646110\pi$
$29$	−4.77200	−0.886139	−0.443069	+	0.896487i	$0.646110\pi$
$30$	0	0
$31$	10.7720	1.93471	0.967354	−	0.253428i	$-0.0815580\pi$
$31$	10.7720	1.93471	0.967354	+	0.253428i	$0.0815580\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.77200	0.637585
$36$	0	0
$37$	7.77200	1.27771	0.638855	−	0.769327i	$-0.279409\pi$
$37$	7.77200	1.27771	0.638855	+	0.769327i	$0.279409\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.54400	1.17818	0.589088	−	0.808069i	$-0.299487\pi$
$41$	7.54400	1.17818	0.589088	+	0.808069i	$0.299487\pi$
$42$	0	0
$43$	6.77200	1.03272	0.516360	−	0.856371i	$-0.327287\pi$
$43$	6.77200	1.03272	0.516360	+	0.856371i	$0.327287\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.77200	−0.404338	−0.202169	−	0.979351i	$-0.564799\pi$
$47$	−2.77200	−0.404338	−0.202169	+	0.979351i	$0.564799\pi$
$48$	0	0
$49$	7.22800	1.03257
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.54400	1.03625	0.518124	−	0.855305i	$-0.326630\pi$
$53$	7.54400	1.03625	0.518124	+	0.855305i	$0.326630\pi$
$54$	0	0
$55$	−4.77200	−0.643457
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−7.77200	−0.995103	−0.497551	−	0.867434i	$-0.665767\pi$
$61$	−7.77200	−0.995103	−0.497551	+	0.867434i	$0.665767\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	6.22800	0.760871	0.380436	−	0.924807i	$-0.375774\pi$
$67$	6.22800	0.760871	0.380436	+	0.924807i	$0.375774\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.54400	−0.420596	−0.210298	−	0.977637i	$-0.567443\pi$
$71$	−3.54400	−0.420596	−0.210298	+	0.977637i	$0.567443\pi$
$72$	0	0
$73$	−3.77200	−0.441479	−0.220740	−	0.975333i	$-0.570847\pi$
$73$	−3.77200	−0.441479	−0.220740	+	0.975333i	$0.570847\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−18.0000	−2.05129
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	6.77200	0.734527
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	−11.3160	−1.18624
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.77200	−0.592195
$96$	0	0
$97$	7.31601	0.742828	0.371414	−	0.928467i	$-0.378873\pi$
$97$	7.31601	0.742828	0.371414	+	0.928467i	$0.378873\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.7720	−1.86788	−0.933942	−	0.357425i	$-0.883655\pi$
$101$	−18.7720	−1.86788	−0.933942	+	0.357425i	$0.883655\pi$
$102$	0	0
$103$	8.22800	0.810729	0.405364	−	0.914155i	$-0.367145\pi$
$103$	8.22800	0.810729	0.405364	+	0.914155i	$0.367145\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.0000	−1.35343	−0.676716	−	0.736245i	$-0.736597\pi$
$107$	−14.0000	−1.35343	−0.676716	+	0.736245i	$0.736597\pi$
$108$	0	0
$109$	−7.54400	−0.722585	−0.361292	−	0.932453i	$-0.617664\pi$
$109$	−7.54400	−0.722585	−0.361292	+	0.932453i	$0.617664\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.3160	−1.34674	−0.673368	−	0.739307i	$-0.735153\pi$
$113$	−14.3160	−1.34674	−0.673368	+	0.739307i	$0.735153\pi$
$114$	0	0
$115$	0.772002	0.0719895
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.5440	2.34161
$120$	0	0
$121$	11.7720	1.07018
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.54400	0.491951	0.245975	−	0.969276i	$-0.420892\pi$
$127$	5.54400	0.491951	0.245975	+	0.969276i	$0.420892\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.7720	1.29064	0.645318	−	0.763914i	$-0.276725\pi$
$131$	14.7720	1.29064	0.645318	+	0.763914i	$0.276725\pi$
$132$	0	0
$133$	−21.7720	−1.88787
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.0880	1.11818	0.559092	−	0.829106i	$-0.311150\pi$
$137$	13.0880	1.11818	0.559092	+	0.829106i	$0.311150\pi$
$138$	0	0
$139$	−4.22800	−0.358614	−0.179307	−	0.983793i	$-0.557386\pi$
$139$	−4.22800	−0.358614	−0.179307	+	0.983793i	$0.557386\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.3160	1.19716
$144$	0	0
$145$	4.77200	0.396293
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.22800	−0.100602	−0.0503008	−	0.998734i	$-0.516018\pi$
$149$	−1.22800	−0.100602	−0.0503008	+	0.998734i	$0.516018\pi$
$150$	0	0
$151$	10.0880	0.820950	0.410475	−	0.911872i	$-0.365363\pi$
$151$	10.0880	0.820950	0.410475	+	0.911872i	$0.365363\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.7720	−0.865228
$156$	0	0
$157$	−0.772002	−0.0616125	−0.0308062	−	0.999525i	$-0.509807\pi$
$157$	−0.772002	−0.0616125	−0.0308062	+	0.999525i	$0.509807\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.91199	0.229497
$162$	0	0
$163$	8.54400	0.669218	0.334609	−	0.942357i	$-0.391396\pi$
$163$	8.54400	0.669218	0.334609	+	0.942357i	$0.391396\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.0880	0.858016	0.429008	−	0.903301i	$-0.358863\pi$
$167$	11.0880	0.858016	0.429008	+	0.903301i	$0.358863\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.54400	−0.725617	−0.362809	−	0.931864i	$-0.618182\pi$
$173$	−9.54400	−0.725617	−0.362809	+	0.931864i	$0.618182\pi$
$174$	0	0
$175$	−3.77200	−0.285137
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.54400	−0.563865	−0.281933	−	0.959434i	$-0.590975\pi$
$179$	−7.54400	−0.563865	−0.281933	+	0.959434i	$0.590975\pi$
$180$	0	0
$181$	1.31601	0.0978179	0.0489090	−	0.998803i	$-0.484426\pi$
$181$	1.31601	0.0978179	0.0489090	+	0.998803i	$0.484426\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.77200	−0.571409
$186$	0	0
$187$	−32.3160	−2.36318
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.5440	−0.980010	−0.490005	−	0.871720i	$-0.663005\pi$
$191$	−13.5440	−0.980010	−0.490005	+	0.871720i	$0.663005\pi$
$192$	0	0
$193$	−4.22800	−0.304338	−0.152169	−	0.988354i	$-0.548626\pi$
$193$	−4.22800	−0.304338	−0.152169	+	0.988354i	$0.548626\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.0000	−1.42494	−0.712470	−	0.701702i	$-0.752424\pi$
$197$	−20.0000	−1.42494	−0.712470	+	0.701702i	$0.752424\pi$
$198$	0	0
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.0000	1.26335
$204$	0	0
$205$	−7.54400	−0.526896
$206$	0	0
$207$	0	0
$208$	0	0
$209$	27.5440	1.90526
$210$	0	0
$211$	−11.3160	−0.779026	−0.389513	−	0.921021i	$-0.627357\pi$
$211$	−11.3160	−0.779026	−0.389513	+	0.921021i	$0.627357\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.77200	−0.461847
$216$	0	0
$217$	−40.6320	−2.75828
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−20.3160	−1.36660
$222$	0	0
$223$	−19.0880	−1.27823	−0.639114	−	0.769112i	$-0.720699\pi$
$223$	−19.0880	−1.27823	−0.639114	+	0.769112i	$0.720699\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.5440	−1.03169	−0.515846	−	0.856681i	$-0.672522\pi$
$227$	−15.5440	−1.03169	−0.515846	+	0.856681i	$0.672522\pi$
$228$	0	0
$229$	27.5440	1.82016	0.910080	−	0.414434i	$-0.136020\pi$
$229$	27.5440	1.82016	0.910080	+	0.414434i	$0.136020\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.5440	1.93549	0.967746	−	0.251928i	$-0.0810646\pi$
$233$	29.5440	1.93549	0.967746	+	0.251928i	$0.0810646\pi$
$234$	0	0
$235$	2.77200	0.180825
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−26.0880	−1.68048	−0.840238	−	0.542218i	$-0.817585\pi$
$241$	−26.0880	−1.68048	−0.840238	+	0.542218i	$0.817585\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−7.22800	−0.461780
$246$	0	0
$247$	17.3160	1.10179
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.77200	0.553684	0.276842	−	0.960915i	$-0.410712\pi$
$251$	8.77200	0.553684	0.276842	+	0.960915i	$0.410712\pi$
$252$	0	0
$253$	−3.68399	−0.231611
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.7720	−0.671939	−0.335970	−	0.941873i	$-0.609064\pi$
$257$	−10.7720	−0.671939	−0.335970	+	0.941873i	$0.609064\pi$
$258$	0	0
$259$	−29.3160	−1.82161
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	−7.54400	−0.463424
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.3160	1.23869	0.619344	−	0.785119i	$-0.287398\pi$
$269$	20.3160	1.23869	0.619344	+	0.785119i	$0.287398\pi$
$270$	0	0
$271$	−2.22800	−0.135341	−0.0676706	−	0.997708i	$-0.521557\pi$
$271$	−2.22800	−0.135341	−0.0676706	+	0.997708i	$0.521557\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.77200	0.287763
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.54400	0.330728	0.165364	−	0.986233i	$-0.447120\pi$
$281$	5.54400	0.330728	0.165364	+	0.986233i	$0.447120\pi$
$282$	0	0
$283$	23.0880	1.37244	0.686220	−	0.727394i	$-0.259269\pi$
$283$	23.0880	1.37244	0.686220	+	0.727394i	$0.259269\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.4560	−1.67970
$288$	0	0
$289$	28.8600	1.69765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.08801	0.297245	0.148622	−	0.988894i	$-0.452516\pi$
$293$	5.08801	0.297245	0.148622	+	0.988894i	$0.452516\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.31601	−0.133938
$300$	0	0
$301$	−25.5440	−1.47233
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.77200	0.445024
$306$	0	0
$307$	2.77200	0.158207	0.0791033	−	0.996866i	$-0.474794\pi$
$307$	2.77200	0.158207	0.0791033	+	0.996866i	$0.474794\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	−2.68399	−0.151708	−0.0758542	−	0.997119i	$-0.524168\pi$
$313$	−2.68399	−0.151708	−0.0758542	+	0.997119i	$0.524168\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.0880	−1.29675	−0.648376	−	0.761320i	$-0.724551\pi$
$317$	−23.0880	−1.29675	−0.648376	+	0.761320i	$0.724551\pi$
$318$	0	0
$319$	−22.7720	−1.27499
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−39.0880	−2.17491
$324$	0	0
$325$	3.00000	0.166410
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.4560	0.576458
$330$	0	0
$331$	11.7720	0.647048	0.323524	−	0.946220i	$-0.395132\pi$
$331$	11.7720	0.647048	0.323524	+	0.946220i	$0.395132\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.22800	−0.340272
$336$	0	0
$337$	−18.2280	−0.992942	−0.496471	−	0.868053i	$-0.665371\pi$
$337$	−18.2280	−0.992942	−0.496471	+	0.868053i	$0.665371\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	51.4040	2.78368
$342$	0	0
$343$	−0.860009	−0.0464361
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.6320	−1.21495	−0.607475	−	0.794339i	$-0.707818\pi$
$347$	−22.6320	−1.21495	−0.607475	+	0.794339i	$0.707818\pi$
$348$	0	0
$349$	34.4040	1.84160	0.920802	−	0.390030i	$-0.127535\pi$
$349$	34.4040	1.84160	0.920802	+	0.390030i	$0.127535\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.7720	1.21203	0.606016	−	0.795453i	$-0.292767\pi$
$353$	22.7720	1.21203	0.606016	+	0.795453i	$0.292767\pi$
$354$	0	0
$355$	3.54400	0.188096
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.5440	−1.03149	−0.515747	−	0.856741i	$-0.672485\pi$
$359$	−19.5440	−1.03149	−0.515747	+	0.856741i	$0.672485\pi$
$360$	0	0
$361$	14.3160	0.753474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.77200	0.197436
$366$	0	0
$367$	15.3160	0.799489	0.399744	−	0.916627i	$-0.369099\pi$
$367$	15.3160	0.799489	0.399744	+	0.916627i	$0.369099\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−28.4560	−1.47736
$372$	0	0
$373$	−14.0880	−0.729449	−0.364725	−	0.931115i	$-0.618837\pi$
$373$	−14.0880	−0.729449	−0.364725	+	0.931115i	$0.618837\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.3160	−0.737312
$378$	0	0
$379$	−25.3160	−1.30040	−0.650198	−	0.759765i	$-0.725314\pi$
$379$	−25.3160	−1.30040	−0.650198	+	0.759765i	$0.725314\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.22800	−0.267138	−0.133569	−	0.991040i	$-0.542644\pi$
$383$	−5.22800	−0.267138	−0.133569	+	0.991040i	$0.542644\pi$
$384$	0	0
$385$	18.0000	0.917365
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.68399	0.186786	0.0933930	−	0.995629i	$-0.470229\pi$
$389$	3.68399	0.186786	0.0933930	+	0.995629i	$0.470229\pi$
$390$	0	0
$391$	5.22800	0.264391
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.00000	0.251577
$396$	0	0
$397$	5.68399	0.285272	0.142636	−	0.989775i	$-0.454442\pi$
$397$	5.68399	0.285272	0.142636	+	0.989775i	$0.454442\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.5440	0.776231	0.388115	−	0.921611i	$-0.373126\pi$
$401$	15.5440	0.776231	0.388115	+	0.921611i	$0.373126\pi$
$402$	0	0
$403$	32.3160	1.60977
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.0880	1.83838
$408$	0	0
$409$	3.45600	0.170888	0.0854440	−	0.996343i	$-0.472769\pi$
$409$	3.45600	0.170888	0.0854440	+	0.996343i	$0.472769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−45.2640	−2.22730
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.77200	0.330834	0.165417	−	0.986224i	$-0.447103\pi$
$419$	6.77200	0.330834	0.165417	+	0.986224i	$0.447103\pi$
$420$	0	0
$421$	−13.7720	−0.671206	−0.335603	−	0.942003i	$-0.608940\pi$
$421$	−13.7720	−0.671206	−0.335603	+	0.942003i	$0.608940\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.77200	−0.328490
$426$	0	0
$427$	29.3160	1.41870
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.6320	1.28282	0.641409	−	0.767199i	$-0.278350\pi$
$431$	26.6320	1.28282	0.641409	+	0.767199i	$0.278350\pi$
$432$	0	0
$433$	−33.0880	−1.59011	−0.795054	−	0.606539i	$-0.792558\pi$
$433$	−33.0880	−1.59011	−0.795054	+	0.606539i	$0.792558\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.45600	−0.213159
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.08801	0.336761	0.168381	−	0.985722i	$-0.446146\pi$
$443$	7.08801	0.336761	0.168381	+	0.985722i	$0.446146\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.4560	0.493449	0.246724	−	0.969086i	$-0.420646\pi$
$449$	10.4560	0.493449	0.246724	+	0.969086i	$0.420646\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11.3160	0.530503
$456$	0	0
$457$	0.455996	0.0213306	0.0106653	−	0.999943i	$-0.496605\pi$
$457$	0.455996	0.0213306	0.0106653	+	0.999943i	$0.496605\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18.4560	−0.859581	−0.429791	−	0.902929i	$-0.641413\pi$
$461$	−18.4560	−0.859581	−0.429791	+	0.902929i	$0.641413\pi$
$462$	0	0
$463$	−34.8600	−1.62008	−0.810041	−	0.586373i	$-0.800555\pi$
$463$	−34.8600	−1.62008	−0.810041	+	0.586373i	$0.800555\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	0	0
$469$	−23.4920	−1.08476
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.3160	1.48589
$474$	0	0
$475$	5.77200	0.264838
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.5440	1.25852	0.629259	−	0.777196i	$-0.283359\pi$
$479$	27.5440	1.25852	0.629259	+	0.777196i	$0.283359\pi$
$480$	0	0
$481$	23.3160	1.06312
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.31601	−0.332203
$486$	0	0
$487$	1.77200	0.0802971	0.0401485	−	0.999194i	$-0.487217\pi$
$487$	1.77200	0.0802971	0.0401485	+	0.999194i	$0.487217\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.4560	0.742649	0.371324	−	0.928503i	$-0.378904\pi$
$491$	16.4560	0.742649	0.371324	+	0.928503i	$0.378904\pi$
$492$	0	0
$493$	32.3160	1.45544
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.3680	0.599636
$498$	0	0
$499$	−35.0880	−1.57075	−0.785377	−	0.619017i	$-0.787531\pi$
$499$	−35.0880	−1.57075	−0.785377	+	0.619017i	$0.787531\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−23.8600	−1.06387	−0.531933	−	0.846787i	$-0.678534\pi$
$503$	−23.8600	−1.06387	−0.531933	+	0.846787i	$0.678534\pi$
$504$	0	0
$505$	18.7720	0.835343
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.7720	−0.743406	−0.371703	−	0.928352i	$-0.621226\pi$
$509$	−16.7720	−0.743406	−0.371703	+	0.928352i	$0.621226\pi$
$510$	0	0
$511$	14.2280	0.629410
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.22800	−0.362569
$516$	0	0
$517$	−13.2280	−0.581767
$518$	0	0
$519$	0	0
$520$	0	0
$521$	43.5440	1.90770	0.953849	−	0.300288i	$-0.0970826\pi$
$521$	43.5440	1.90770	0.953849	+	0.300288i	$0.0970826\pi$
$522$	0	0
$523$	18.5440	0.810873	0.405436	−	0.914123i	$-0.367120\pi$
$523$	18.5440	0.810873	0.405436	+	0.914123i	$0.367120\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−72.9480	−3.17767
$528$	0	0
$529$	−22.4040	−0.974088
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22.6320	0.980301
$534$	0	0
$535$	14.0000	0.605273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	34.4920	1.48568
$540$	0	0
$541$	−8.22800	−0.353749	−0.176875	−	0.984233i	$-0.556599\pi$
$541$	−8.22800	−0.353749	−0.176875	+	0.984233i	$0.556599\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.54400	0.323150
$546$	0	0
$547$	16.5440	0.707371	0.353685	−	0.935364i	$-0.384928\pi$
$547$	16.5440	0.707371	0.353685	+	0.935364i	$0.384928\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.5440	−1.17341
$552$	0	0
$553$	18.8600	0.802009
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.1760	1.27860	0.639299	−	0.768958i	$-0.279224\pi$
$557$	30.1760	1.27860	0.639299	+	0.768958i	$0.279224\pi$
$558$	0	0
$559$	20.3160	0.859275
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.54400	0.402232	0.201116	−	0.979567i	$-0.435543\pi$
$563$	9.54400	0.402232	0.201116	+	0.979567i	$0.435543\pi$
$564$	0	0
$565$	14.3160	0.602279
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.0880	−0.632522	−0.316261	−	0.948672i	$-0.602428\pi$
$569$	−15.0880	−0.632522	−0.316261	+	0.948672i	$0.602428\pi$
$570$	0	0
$571$	−27.7720	−1.16222	−0.581111	−	0.813824i	$-0.697382\pi$
$571$	−27.7720	−1.16222	−0.581111	+	0.813824i	$0.697382\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.772002	−0.0321947
$576$	0	0
$577$	40.4040	1.68204	0.841021	−	0.541003i	$-0.181955\pi$
$577$	40.4040	1.68204	0.841021	+	0.541003i	$0.181955\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22.6320	−0.938934
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.0880	1.53079	0.765393	−	0.643563i	$-0.222545\pi$
$587$	37.0880	1.53079	0.765393	+	0.643563i	$0.222545\pi$
$588$	0	0
$589$	62.1760	2.56192
$590$	0	0
$591$	0	0
$592$	0	0
$593$	27.8600	1.14407	0.572037	−	0.820228i	$-0.306153\pi$
$593$	27.8600	1.14407	0.572037	+	0.820228i	$0.306153\pi$
$594$	0	0
$595$	−25.5440	−1.04720
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−29.0880	−1.18850	−0.594252	−	0.804279i	$-0.702552\pi$
$599$	−29.0880	−1.18850	−0.594252	+	0.804279i	$0.702552\pi$
$600$	0	0
$601$	46.3160	1.88927	0.944635	−	0.328124i	$-0.106416\pi$
$601$	46.3160	1.88927	0.944635	+	0.328124i	$0.106416\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.7720	−0.478600
$606$	0	0
$607$	−2.86001	−0.116084	−0.0580421	−	0.998314i	$-0.518486\pi$
$607$	−2.86001	−0.116084	−0.0580421	+	0.998314i	$0.518486\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.31601	−0.336430
$612$	0	0
$613$	14.5440	0.587427	0.293713	−	0.955894i	$-0.405109\pi$
$613$	14.5440	0.587427	0.293713	+	0.955894i	$0.405109\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.3160	0.737375	0.368687	−	0.929553i	$-0.379807\pi$
$617$	18.3160	0.737375	0.368687	+	0.929553i	$0.379807\pi$
$618$	0	0
$619$	31.3160	1.25870	0.629348	−	0.777123i	$-0.283322\pi$
$619$	31.3160	1.25870	0.629348	+	0.777123i	$0.283322\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−30.1760	−1.20898
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−52.6320	−2.09858
$630$	0	0
$631$	5.77200	0.229780	0.114890	−	0.993378i	$-0.463348\pi$
$631$	5.77200	0.229780	0.114890	+	0.993378i	$0.463348\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.54400	−0.220007
$636$	0	0
$637$	21.6840	0.859151
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−40.6320	−1.60487	−0.802434	−	0.596741i	$-0.796462\pi$
$641$	−40.6320	−1.60487	−0.802434	+	0.596741i	$0.796462\pi$
$642$	0	0
$643$	−1.22800	−0.0484275	−0.0242138	−	0.999707i	$-0.507708\pi$
$643$	−1.22800	−0.0484275	−0.0242138	+	0.999707i	$0.507708\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.6320	−0.889756	−0.444878	−	0.895591i	$-0.646753\pi$
$647$	−22.6320	−0.889756	−0.444878	+	0.895591i	$0.646753\pi$
$648$	0	0
$649$	57.2640	2.24781
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.08801	0.277375	0.138688	−	0.990336i	$-0.455712\pi$
$653$	7.08801	0.277375	0.138688	+	0.990336i	$0.455712\pi$
$654$	0	0
$655$	−14.7720	−0.577190
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	0	0
$661$	−15.7720	−0.613460	−0.306730	−	0.951797i	$-0.599235\pi$
$661$	−15.7720	−0.613460	−0.306730	+	0.951797i	$0.599235\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	21.7720	0.844282
$666$	0	0
$667$	3.68399	0.142645
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−37.0880	−1.43177
$672$	0	0
$673$	−46.4040	−1.78874	−0.894372	−	0.447325i	$-0.852377\pi$
$673$	−46.4040	−1.78874	−0.894372	+	0.447325i	$0.852377\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	43.0880	1.65601	0.828003	−	0.560723i	$-0.189477\pi$
$677$	43.0880	1.65601	0.828003	+	0.560723i	$0.189477\pi$
$678$	0	0
$679$	−27.5960	−1.05904
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.5440	1.05394	0.526971	−	0.849883i	$-0.323328\pi$
$683$	27.5440	1.05394	0.526971	+	0.849883i	$0.323328\pi$
$684$	0	0
$685$	−13.0880	−0.500067
$686$	0	0
$687$	0	0
$688$	0	0
$689$	22.6320	0.862211
$690$	0	0
$691$	−12.6320	−0.480544	−0.240272	−	0.970706i	$-0.577237\pi$
$691$	−12.6320	−0.480544	−0.240272	+	0.970706i	$0.577237\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.22800	0.160377
$696$	0	0
$697$	−51.0880	−1.93510
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11.2280	0.424076	0.212038	−	0.977261i	$-0.431990\pi$
$701$	11.2280	0.424076	0.212038	+	0.977261i	$0.431990\pi$
$702$	0	0
$703$	44.8600	1.69193
$704$	0	0
$705$	0	0
$706$	0	0
$707$	70.8080	2.66301
$708$	0	0
$709$	7.13999	0.268148	0.134074	−	0.990971i	$-0.457194\pi$
$709$	7.13999	0.268148	0.134074	+	0.990971i	$0.457194\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.31601	−0.311437
$714$	0	0
$715$	−14.3160	−0.535388
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	0	0
$721$	−31.0360	−1.15584
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.77200	−0.177228
$726$	0	0
$727$	−35.0880	−1.30134	−0.650671	−	0.759360i	$-0.725512\pi$
$727$	−35.0880	−1.30134	−0.650671	+	0.759360i	$0.725512\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−45.8600	−1.69619
$732$	0	0
$733$	−41.0880	−1.51762	−0.758810	−	0.651312i	$-0.774219\pi$
$733$	−41.0880	−1.51762	−0.758810	+	0.651312i	$0.774219\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	29.7200	1.09475
$738$	0	0
$739$	−0.911993	−0.0335482	−0.0167741	−	0.999859i	$-0.505340\pi$
$739$	−0.911993	−0.0335482	−0.0167741	+	0.999859i	$0.505340\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.4040	0.565118	0.282559	−	0.959250i	$-0.408817\pi$
$743$	15.4040	0.565118	0.282559	+	0.959250i	$0.408817\pi$
$744$	0	0
$745$	1.22800	0.0449904
$746$	0	0
$747$	0	0
$748$	0	0
$749$	52.8080	1.92956
$750$	0	0
$751$	29.1760	1.06465	0.532324	−	0.846541i	$-0.321319\pi$
$751$	29.1760	1.06465	0.532324	+	0.846541i	$0.321319\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.0880	−0.367140
$756$	0	0
$757$	38.7200	1.40730	0.703652	−	0.710545i	$-0.251552\pi$
$757$	38.7200	1.40730	0.703652	+	0.710545i	$0.251552\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.08801	−0.184440	−0.0922201	−	0.995739i	$-0.529396\pi$
$761$	−5.08801	−0.184440	−0.0922201	+	0.995739i	$0.529396\pi$
$762$	0	0
$763$	28.4560	1.03018
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.0000	1.29988
$768$	0	0
$769$	−54.2640	−1.95681	−0.978405	−	0.206695i	$-0.933729\pi$
$769$	−54.2640	−1.95681	−0.978405	+	0.206695i	$0.933729\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−45.0880	−1.62170	−0.810851	−	0.585252i	$-0.800996\pi$
$773$	−45.0880	−1.62170	−0.810851	+	0.585252i	$0.800996\pi$
$774$	0	0
$775$	10.7720	0.386942
$776$	0	0
$777$	0	0
$778$	0	0
$779$	43.5440	1.56013
$780$	0	0
$781$	−16.9120	−0.605159
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.772002	0.0275539
$786$	0	0
$787$	−29.1760	−1.04001	−0.520006	−	0.854162i	$-0.674070\pi$
$787$	−29.1760	−1.04001	−0.520006	+	0.854162i	$0.674070\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	54.0000	1.92002
$792$	0	0
$793$	−23.3160	−0.827976
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22.1760	−0.785515	−0.392757	−	0.919642i	$-0.628479\pi$
$797$	−22.1760	−0.785515	−0.392757	+	0.919642i	$0.628479\pi$
$798$	0	0
$799$	18.7720	0.664106
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	−2.91199	−0.102634
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	−14.4560	−0.507619	−0.253809	−	0.967254i	$-0.581684\pi$
$811$	−14.4560	−0.507619	−0.253809	+	0.967254i	$0.581684\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.54400	−0.299283
$816$	0	0
$817$	39.0880	1.36752
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.45600	−0.225316	−0.112658	−	0.993634i	$-0.535936\pi$
$821$	−6.45600	−0.225316	−0.112658	+	0.993634i	$0.535936\pi$
$822$	0	0
$823$	20.4040	0.711239	0.355620	−	0.934631i	$-0.384270\pi$
$823$	20.4040	0.711239	0.355620	+	0.934631i	$0.384270\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.5440	0.470971	0.235486	−	0.971878i	$-0.424332\pi$
$827$	13.5440	0.470971	0.235486	+	0.971878i	$0.424332\pi$
$828$	0	0
$829$	30.2280	1.04986	0.524931	−	0.851145i	$-0.324091\pi$
$829$	30.2280	1.04986	0.524931	+	0.851145i	$0.324091\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−48.9480	−1.69595
$834$	0	0
$835$	−11.0880	−0.383716
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.36799	0.116276	0.0581379	−	0.998309i	$-0.481484\pi$
$839$	3.36799	0.116276	0.0581379	+	0.998309i	$0.481484\pi$
$840$	0	0
$841$	−6.22800	−0.214759
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.00000	0.137604
$846$	0	0
$847$	−44.4040	−1.52574
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	−32.5440	−1.11429	−0.557143	−	0.830417i	$-0.688102\pi$
$853$	−32.5440	−1.11429	−0.557143	+	0.830417i	$0.688102\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.91199	0.236109	0.118055	−	0.993007i	$-0.462334\pi$
$857$	6.91199	0.236109	0.118055	+	0.993007i	$0.462334\pi$
$858$	0	0
$859$	−16.2280	−0.553692	−0.276846	−	0.960914i	$-0.589289\pi$
$859$	−16.2280	−0.553692	−0.276846	+	0.960914i	$0.589289\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.316006	0.0107570	0.00537848	−	0.999986i	$-0.498288\pi$
$863$	0.316006	0.0107570	0.00537848	+	0.999986i	$0.498288\pi$
$864$	0	0
$865$	9.54400	0.324506
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.8600	−0.809395
$870$	0	0
$871$	18.6840	0.633083
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.77200	0.127517
$876$	0	0
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41.0880	−1.38429	−0.692145	−	0.721758i	$-0.743334\pi$
$881$	−41.0880	−1.38429	−0.692145	+	0.721758i	$0.743334\pi$
$882$	0	0
$883$	−13.3160	−0.448119	−0.224060	−	0.974575i	$-0.571931\pi$
$883$	−13.3160	−0.448119	−0.224060	+	0.974575i	$0.571931\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.3160	1.08507	0.542533	−	0.840035i	$-0.317465\pi$
$887$	32.3160	1.08507	0.542533	+	0.840035i	$0.317465\pi$
$888$	0	0
$889$	−20.9120	−0.701366
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	7.54400	0.252168
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−51.4040	−1.71442
$900$	0	0
$901$	−51.0880	−1.70199
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.31601	−0.0437455
$906$	0	0
$907$	−35.0000	−1.16216	−0.581078	−	0.813848i	$-0.697369\pi$
$907$	−35.0000	−1.16216	−0.581078	+	0.813848i	$0.697369\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30.1760	0.999776	0.499888	−	0.866090i	$-0.333375\pi$
$911$	30.1760	0.999776	0.499888	+	0.866090i	$0.333375\pi$
$912$	0	0
$913$	28.6320	0.947581
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−55.7200	−1.84004
$918$	0	0
$919$	−37.8600	−1.24889	−0.624443	−	0.781070i	$-0.714674\pi$
$919$	−37.8600	−1.24889	−0.624443	+	0.781070i	$0.714674\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.6320	−0.349957
$924$	0	0
$925$	7.77200	0.255542
$926$	0	0
$927$	0	0
$928$	0	0
$929$	34.6320	1.13624	0.568120	−	0.822946i	$-0.307671\pi$
$929$	34.6320	1.13624	0.568120	+	0.822946i	$0.307671\pi$
$930$	0	0
$931$	41.7200	1.36732
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32.3160	1.05685
$936$	0	0
$937$	−8.22800	−0.268797	−0.134398	−	0.990927i	$-0.542910\pi$
$937$	−8.22800	−0.268797	−0.134398	+	0.990927i	$0.542910\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	37.4040	1.21934	0.609668	−	0.792657i	$-0.291303\pi$
$941$	37.4040	1.21934	0.609668	+	0.792657i	$0.291303\pi$
$942$	0	0
$943$	−5.82399	−0.189655
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.5440	0.830069	0.415034	−	0.909806i	$-0.363770\pi$
$947$	25.5440	0.830069	0.415034	+	0.909806i	$0.363770\pi$
$948$	0	0
$949$	−11.3160	−0.367333
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6.13999	−0.198894	−0.0994469	−	0.995043i	$-0.531707\pi$
$953$	−6.13999	−0.198894	−0.0994469	+	0.995043i	$0.531707\pi$
$954$	0	0
$955$	13.5440	0.438274
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−49.3680	−1.59418
$960$	0	0
$961$	85.0360	2.74310
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.22800	0.136104
$966$	0	0
$967$	3.77200	0.121299	0.0606497	−	0.998159i	$-0.480683\pi$
$967$	3.77200	0.121299	0.0606497	+	0.998159i	$0.480683\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.2280	0.937971	0.468986	−	0.883206i	$-0.344620\pi$
$971$	29.2280	0.937971	0.468986	+	0.883206i	$0.344620\pi$
$972$	0	0
$973$	15.9480	0.511270
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.77200	0.280641	0.140321	−	0.990106i	$-0.455187\pi$
$977$	8.77200	0.280641	0.140321	+	0.990106i	$0.455187\pi$
$978$	0	0
$979$	38.1760	1.22011
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.4040	1.44816	0.724082	−	0.689714i	$-0.242264\pi$
$983$	45.4040	1.44816	0.724082	+	0.689714i	$0.242264\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.22800	−0.166241
$990$	0	0
$991$	−26.0880	−0.828713	−0.414356	−	0.910115i	$-0.635993\pi$
$991$	−26.0880	−0.828713	−0.414356	+	0.910115i	$0.635993\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.0000	0.475532
$996$	0	0
$997$	38.3160	1.21348	0.606740	−	0.794900i	$-0.292477\pi$
$997$	38.3160	1.21348	0.606740	+	0.794900i	$0.292477\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.a.m.1.1	✓	2
3.2	odd	2		1080.2.a.n.1.1	yes	2
4.3	odd	2		2160.2.a.z.1.2		2
5.2	odd	4		5400.2.f.be.649.2		4
5.3	odd	4		5400.2.f.be.649.3		4
5.4	even	2		5400.2.a.cb.1.2		2
8.3	odd	2		8640.2.a.db.1.2		2
8.5	even	2		8640.2.a.de.1.1		2
9.2	odd	6		3240.2.q.z.1081.2		4
9.4	even	3		3240.2.q.bc.2161.2		4
9.5	odd	6		3240.2.q.z.2161.2		4
9.7	even	3		3240.2.q.bc.1081.2		4
12.11	even	2		2160.2.a.bb.1.2		2
15.2	even	4		5400.2.f.bd.649.2		4
15.8	even	4		5400.2.f.bd.649.3		4
15.14	odd	2		5400.2.a.ca.1.2		2
24.5	odd	2		8640.2.a.cq.1.1		2
24.11	even	2		8640.2.a.cn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.m.1.1	✓	2	1.1	even	1	trivial
1080.2.a.n.1.1	yes	2	3.2	odd	2
2160.2.a.z.1.2		2	4.3	odd	2
2160.2.a.bb.1.2		2	12.11	even	2
3240.2.q.z.1081.2		4	9.2	odd	6
3240.2.q.z.2161.2		4	9.5	odd	6
3240.2.q.bc.1081.2		4	9.7	even	3
3240.2.q.bc.2161.2		4	9.4	even	3
5400.2.a.ca.1.2		2	15.14	odd	2
5400.2.a.cb.1.2		2	5.4	even	2
5400.2.f.bd.649.2		4	15.2	even	4
5400.2.f.bd.649.3		4	15.8	even	4
5400.2.f.be.649.2		4	5.2	odd	4
5400.2.f.be.649.3		4	5.3	odd	4
8640.2.a.cn.1.2		2	24.11	even	2
8640.2.a.cq.1.1		2	24.5	odd	2
8640.2.a.db.1.2		2	8.3	odd	2
8640.2.a.de.1.1		2	8.5	even	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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.77200 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.77200 q^{7} +4.77200 q^{11} +3.00000 q^{13} -6.77200 q^{17} +5.77200 q^{19} -0.772002 q^{23} +1.00000 q^{25} -4.77200 q^{29} +10.7720 q^{31} +3.77200 q^{35} +7.77200 q^{37} +7.54400 q^{41} +6.77200 q^{43} -2.77200 q^{47} +7.22800 q^{49} +7.54400 q^{53} -4.77200 q^{55} +12.0000 q^{59} -7.77200 q^{61} -3.00000 q^{65} +6.22800 q^{67} -3.54400 q^{71} -3.77200 q^{73} -18.0000 q^{77} -5.00000 q^{79} +6.00000 q^{83} +6.77200 q^{85} +8.00000 q^{89} -11.3160 q^{91} -5.77200 q^{95} +7.31601 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + q^7	\( 2 q - 2 q^{5} + q^{7} + q^{11} + 6 q^{13} - 5 q^{17} + 3 q^{19} + 7 q^{23} + 2 q^{25} - q^{29} + 13 q^{31} - q^{35} + 7 q^{37} - 2 q^{41} + 5 q^{43} + 3 q^{47} + 23 q^{49} - 2 q^{53} - q^{55} + 24 q^{59} - 7 q^{61} - 6 q^{65} + 21 q^{67} + 10 q^{71} + q^{73} - 36 q^{77} - 10 q^{79} + 12 q^{83} + 5 q^{85} + 16 q^{89} + 3 q^{91} - 3 q^{95} - 11 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + q^7 + q^11 + 6 * q^13 - 5 * q^17 + 3 * q^19 + 7 * q^23 + 2 * q^25 - q^29 + 13 * q^31 - q^35 + 7 * q^37 - 2 * q^41 + 5 * q^43 + 3 * q^47 + 23 * q^49 - 2 * q^53 - q^55 + 24 * q^59 - 7 * q^61 - 6 * q^65 + 21 * q^67 + 10 * q^71 + q^73 - 36 * q^77 - 10 * q^79 + 12 * q^83 + 5 * q^85 + 16 * q^89 + 3 * q^91 - 3 * q^95 - 11 * q^97

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