LMFDB - Embedded newform 5400.2.f.be.649.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5400,2,Mod(649,5400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("5400.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5400.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$43.1192170915$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{73})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 37x^{2} + 324 $ x^4 + 37*x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.3
Root			$3.77200i$ of defining polynomial
Character	$\chi$	$=$	5400.649
Dual form			5400.2.f.be.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.77200i q^{7} +O(q^{10})$	$q+3.77200i q^{7} +4.77200 q^{11} +3.00000i q^{13} +6.77200i q^{17} -5.77200 q^{19} -0.772002i q^{23} +4.77200 q^{29} +10.7720 q^{31} -7.77200i q^{37} +7.54400 q^{41} +6.77200i q^{43} +2.77200i q^{47} -7.22800 q^{49} +7.54400i q^{53} -12.0000 q^{59} -7.77200 q^{61} -6.22800i q^{67} -3.54400 q^{71} -3.77200i q^{73} +18.0000i q^{77} +5.00000 q^{79} +6.00000i q^{83} -8.00000 q^{89} -11.3160 q^{91} -7.31601i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 2 q^{11} - 6 q^{19} + 2 q^{29} + 26 q^{31} - 4 q^{41} - 46 q^{49} - 48 q^{59} - 14 q^{61} + 20 q^{71} + 20 q^{79} - 32 q^{89} + 6 q^{91}+O(q^{100}) $ 4 * q + 2 * q^11 - 6 * q^19 + 2 * q^29 + 26 * q^31 - 4 * q^41 - 46 * q^49 - 48 * q^59 - 14 * q^61 + 20 * q^71 + 20 * q^79 - 32 * q^89 + 6 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5400\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$	$2701$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.77200i	1.42568i	0.701325	+	0.712841i	$0.252592\pi$
$7$	3.77200i	1.42568i	−0.701325	+	0.712841i	$0.747408\pi$
$8$	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
$13$	3.00000i	0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$13$	3.00000i	0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.77200i	1.64245i	0.570603	+	0.821226i	$0.306709\pi$
$17$	6.77200i	1.64245i	−0.570603	+	0.821226i	$0.693291\pi$
$18$	0	0
$19$	−5.77200	−1.32419	−0.662094	−	0.749421i	$-0.730332\pi$
$19$	−5.77200	−1.32419	−0.662094	+	0.749421i	$0.730332\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 0.772002i	− 0.160974i	−0.996756	−	0.0804868i	$-0.974353\pi$
$23$	− 0.772002i	− 0.160974i	0.996756	−	0.0804868i	$-0.0256475\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.77200	0.886139	0.443069	−	0.896487i	$-0.353890\pi$
$29$	4.77200	0.886139	0.443069	+	0.896487i	$0.353890\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	− 7.77200i	− 1.27771i	−0.769327	−	0.638855i	$-0.779409\pi$
$37$	− 7.77200i	− 1.27771i	0.769327	−	0.638855i	$-0.220591\pi$
$38$	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
$43$	6.77200i	1.03272i	0.856371	+	0.516360i	$0.172713\pi$
$43$	6.77200i	1.03272i	−0.856371	+	0.516360i	$0.827287\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.77200i	0.404338i	0.979351	+	0.202169i	$0.0647990\pi$
$47$	2.77200i	0.404338i	−0.979351	+	0.202169i	$0.935201\pi$
$48$	0	0
$49$	−7.22800	−1.03257
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.54400i	1.03625i	0.855305	+	0.518124i	$0.173370\pi$
$53$	7.54400i	1.03625i	−0.855305	+	0.518124i	$0.826630\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	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
$65$	0	0
$66$	0	0
$67$	− 6.22800i	− 0.760871i	−0.924807	−	0.380436i	$-0.875774\pi$
$67$	− 6.22800i	− 0.760871i	0.924807	−	0.380436i	$-0.124226\pi$
$68$	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
$73$	− 3.77200i	− 0.441479i	−0.975333	−	0.220740i	$-0.929153\pi$
$73$	− 3.77200i	− 0.441479i	0.975333	−	0.220740i	$-0.0708471\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.0000i	2.05129i
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	0	0
$91$	−11.3160	−1.18624
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.31601i	− 0.742828i	−0.928467	−	0.371414i	$-0.878873\pi$
$97$	− 7.31601i	− 0.742828i	0.928467	−	0.371414i	$-0.121127\pi$
$98$	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
$103$	8.22800i	0.810729i	0.914155	+	0.405364i	$0.132855\pi$
$103$	8.22800i	0.810729i	−0.914155	+	0.405364i	$0.867145\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.0000i	1.35343i	0.736245	+	0.676716i	$0.236597\pi$
$107$	14.0000i	1.35343i	−0.736245	+	0.676716i	$0.763403\pi$
$108$	0	0
$109$	7.54400	0.722585	0.361292	−	0.932453i	$-0.382336\pi$
$109$	7.54400	0.722585	0.361292	+	0.932453i	$0.382336\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 14.3160i	− 1.34674i	−0.739307	−	0.673368i	$-0.764847\pi$
$113$	− 14.3160i	− 1.34674i	0.739307	−	0.673368i	$-0.235153\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	0	0
$127$	− 5.54400i	− 0.491951i	−0.969276	−	0.245975i	$-0.920892\pi$
$127$	− 5.54400i	− 0.491951i	0.969276	−	0.245975i	$-0.0791082\pi$
$128$	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
$133$	− 21.7720i	− 1.88787i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.0880i	− 1.11818i	−0.829106	−	0.559092i	$-0.811150\pi$
$137$	− 13.0880i	− 1.11818i	0.829106	−	0.559092i	$-0.188850\pi$
$138$	0	0
$139$	4.22800	0.358614	0.179307	−	0.983793i	$-0.442614\pi$
$139$	4.22800	0.358614	0.179307	+	0.983793i	$0.442614\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.3160i	1.19716i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.22800	0.100602	0.0503008	−	0.998734i	$-0.483982\pi$
$149$	1.22800	0.100602	0.0503008	+	0.998734i	$0.483982\pi$
$150$	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
$155$	0	0
$156$	0	0
$157$	0.772002i	0.0616125i	0.999525	+	0.0308062i	$0.00980748\pi$
$157$	0.772002i	0.0616125i	−0.999525	+	0.0308062i	$0.990193\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.91199	0.229497
$162$	0	0
$163$	8.54400i	0.669218i	0.942357	+	0.334609i	$0.108604\pi$
$163$	8.54400i	0.669218i	−0.942357	+	0.334609i	$0.891396\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 11.0880i	− 0.858016i	−0.903301	−	0.429008i	$-0.858863\pi$
$167$	− 11.0880i	− 0.858016i	0.903301	−	0.429008i	$-0.141137\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 9.54400i	− 0.725617i	−0.931864	−	0.362809i	$-0.881818\pi$
$173$	− 9.54400i	− 0.725617i	0.931864	−	0.362809i	$-0.118182\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.54400	0.563865	0.281933	−	0.959434i	$-0.409025\pi$
$179$	7.54400	0.563865	0.281933	+	0.959434i	$0.409025\pi$
$180$	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
$185$	0	0
$186$	0	0
$187$	32.3160i	2.36318i
$188$	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
$193$	− 4.22800i	− 0.304338i	−0.988354	−	0.152169i	$-0.951374\pi$
$193$	− 4.22800i	− 0.304338i	0.988354	−	0.152169i	$-0.0486258\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0000i	1.42494i	0.701702	+	0.712470i	$0.252424\pi$
$197$	20.0000i	1.42494i	−0.701702	+	0.712470i	$0.747576\pi$
$198$	0	0
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.0000i	1.26335i
$204$	0	0
$205$	0	0
$206$	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
$215$	0	0
$216$	0	0
$217$	40.6320i	2.75828i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−20.3160	−1.36660
$222$	0	0
$223$	− 19.0880i	− 1.27823i	−0.769112	−	0.639114i	$-0.779301\pi$
$223$	− 19.0880i	− 1.27823i	0.769112	−	0.639114i	$-0.220699\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.5440i	1.03169i	0.856681	+	0.515846i	$0.172522\pi$
$227$	15.5440i	1.03169i	−0.856681	+	0.515846i	$0.827478\pi$
$228$	0	0
$229$	−27.5440	−1.82016	−0.910080	−	0.414434i	$-0.863980\pi$
$229$	−27.5440	−1.82016	−0.910080	+	0.414434i	$0.863980\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	29.5440i	1.93549i	0.251928	+	0.967746i	$0.418935\pi$
$233$	29.5440i	1.93549i	−0.251928	+	0.967746i	$0.581065\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	− 17.3160i	− 1.10179i
$248$	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
$253$	− 3.68399i	− 0.231611i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.7720i	0.671939i	0.941873	+	0.335970i	$0.109064\pi$
$257$	10.7720i	0.671939i	−0.941873	+	0.335970i	$0.890936\pi$
$258$	0	0
$259$	29.3160	1.82161
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.3160	−1.23869	−0.619344	−	0.785119i	$-0.712602\pi$
$269$	−20.3160	−1.23869	−0.619344	+	0.785119i	$0.712602\pi$
$270$	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
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	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
$283$	23.0880i	1.37244i	0.727394	+	0.686220i	$0.240731\pi$
$283$	23.0880i	1.37244i	−0.727394	+	0.686220i	$0.759269\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	28.4560i	1.67970i
$288$	0	0
$289$	−28.8600	−1.69765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.08801i	0.297245i	0.988894	+	0.148622i	$0.0474839\pi$
$293$	5.08801i	0.297245i	−0.988894	+	0.148622i	$0.952516\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	0	0
$307$	− 2.77200i	− 0.158207i	−0.996866	−	0.0791033i	$-0.974794\pi$
$307$	− 2.77200i	− 0.158207i	0.996866	−	0.0791033i	$-0.0252057\pi$
$308$	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
$313$	− 2.68399i	− 0.151708i	−0.997119	−	0.0758542i	$-0.975832\pi$
$313$	− 2.68399i	− 0.151708i	0.997119	−	0.0758542i	$-0.0241683\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.0880i	1.29675i	0.761320	+	0.648376i	$0.224551\pi$
$317$	23.0880i	1.29675i	−0.761320	+	0.648376i	$0.775449\pi$
$318$	0	0
$319$	22.7720	1.27499
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 39.0880i	− 2.17491i
$324$	0	0
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	18.2280i	0.992942i	0.868053	+	0.496471i	$0.165371\pi$
$337$	18.2280i	0.992942i	−0.868053	+	0.496471i	$0.834629\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	51.4040	2.78368
$342$	0	0
$343$	− 0.860009i	− 0.0464361i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.6320i	1.21495i	0.794339	+	0.607475i	$0.207818\pi$
$347$	22.6320i	1.21495i	−0.794339	+	0.607475i	$0.792182\pi$
$348$	0	0
$349$	−34.4040	−1.84160	−0.920802	−	0.390030i	$-0.872465\pi$
$349$	−34.4040	−1.84160	−0.920802	+	0.390030i	$0.872465\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.7720i	1.21203i	0.795453	+	0.606016i	$0.207233\pi$
$353$	22.7720i	1.21203i	−0.795453	+	0.606016i	$0.792767\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.5440	1.03149	0.515747	−	0.856741i	$-0.327515\pi$
$359$	19.5440	1.03149	0.515747	+	0.856741i	$0.327515\pi$
$360$	0	0
$361$	14.3160	0.753474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 15.3160i	− 0.799489i	−0.916627	−	0.399744i	$-0.869099\pi$
$367$	− 15.3160i	− 0.799489i	0.916627	−	0.399744i	$-0.130901\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−28.4560	−1.47736
$372$	0	0
$373$	− 14.0880i	− 0.729449i	−0.931115	−	0.364725i	$-0.881163\pi$
$373$	− 14.0880i	− 0.729449i	0.931115	−	0.364725i	$-0.118837\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.3160i	0.737312i
$378$	0	0
$379$	25.3160	1.30040	0.650198	−	0.759765i	$-0.274686\pi$
$379$	25.3160	1.30040	0.650198	+	0.759765i	$0.274686\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 5.22800i	− 0.267138i	−0.991040	−	0.133569i	$-0.957356\pi$
$383$	− 5.22800i	− 0.267138i	0.991040	−	0.133569i	$-0.0426438\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.68399	−0.186786	−0.0933930	−	0.995629i	$-0.529771\pi$
$389$	−3.68399	−0.186786	−0.0933930	+	0.995629i	$0.529771\pi$
$390$	0	0
$391$	5.22800	0.264391
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.68399i	− 0.285272i	−0.989775	−	0.142636i	$-0.954442\pi$
$397$	− 5.68399i	− 0.285272i	0.989775	−	0.142636i	$-0.0455577\pi$
$398$	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
$403$	32.3160i	1.60977i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 37.0880i	− 1.83838i
$408$	0	0
$409$	−3.45600	−0.170888	−0.0854440	−	0.996343i	$-0.527231\pi$
$409$	−3.45600	−0.170888	−0.0854440	+	0.996343i	$0.527231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 45.2640i	− 2.22730i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.77200	−0.330834	−0.165417	−	0.986224i	$-0.552897\pi$
$419$	−6.77200	−0.330834	−0.165417	+	0.986224i	$0.552897\pi$
$420$	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
$425$	0	0
$426$	0	0
$427$	− 29.3160i	− 1.41870i
$428$	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
$433$	− 33.0880i	− 1.59011i	−0.606539	−	0.795054i	$-0.707442\pi$
$433$	− 33.0880i	− 1.59011i	0.606539	−	0.795054i	$-0.292558\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.45600i	0.213159i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.08801i	0.336761i	0.985722	+	0.168381i	$0.0538538\pi$
$443$	7.08801i	0.336761i	−0.985722	+	0.168381i	$0.946146\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.4560	−0.493449	−0.246724	−	0.969086i	$-0.579354\pi$
$449$	−10.4560	−0.493449	−0.246724	+	0.969086i	$0.579354\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 0.455996i	− 0.0213306i	−0.999943	−	0.0106653i	$-0.996605\pi$
$457$	− 0.455996i	− 0.0213306i	0.999943	−	0.0106653i	$-0.00339494\pi$
$458$	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
$463$	− 34.8600i	− 1.62008i	−0.586373	−	0.810041i	$-0.699445\pi$
$463$	− 34.8600i	− 1.62008i	0.586373	−	0.810041i	$-0.300555\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.0000i	1.48078i	0.672176	+	0.740392i	$0.265360\pi$
$467$	32.0000i	1.48078i	−0.672176	+	0.740392i	$0.734640\pi$
$468$	0	0
$469$	23.4920	1.08476
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.3160i	1.48589i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−27.5440	−1.25852	−0.629259	−	0.777196i	$-0.716641\pi$
$479$	−27.5440	−1.25852	−0.629259	+	0.777196i	$0.716641\pi$
$480$	0	0
$481$	23.3160	1.06312
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.77200i	− 0.0802971i	−0.999194	−	0.0401485i	$-0.987217\pi$
$487$	− 1.77200i	− 0.0802971i	0.999194	−	0.0401485i	$-0.0127831\pi$
$488$	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
$493$	32.3160i	1.45544i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13.3680i	− 0.599636i
$498$	0	0
$499$	35.0880	1.57075	0.785377	−	0.619017i	$-0.212469\pi$
$499$	35.0880	1.57075	0.785377	+	0.619017i	$0.212469\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 23.8600i	− 1.06387i	−0.846787	−	0.531933i	$-0.821466\pi$
$503$	− 23.8600i	− 1.06387i	0.846787	−	0.531933i	$-0.178534\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.7720	0.743406	0.371703	−	0.928352i	$-0.378774\pi$
$509$	16.7720	0.743406	0.371703	+	0.928352i	$0.378774\pi$
$510$	0	0
$511$	14.2280	0.629410
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	13.2280i	0.581767i
$518$	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
$523$	18.5440i	0.810873i	0.914123	+	0.405436i	$0.132880\pi$
$523$	18.5440i	0.810873i	−0.914123	+	0.405436i	$0.867120\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	72.9480i	3.17767i
$528$	0	0
$529$	22.4040	0.974088
$530$	0	0
$531$	0	0
$532$	0	0
$533$	22.6320i	0.980301i
$534$	0	0
$535$	0	0
$536$	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
$545$	0	0
$546$	0	0
$547$	− 16.5440i	− 0.707371i	−0.935364	−	0.353685i	$-0.884928\pi$
$547$	− 16.5440i	− 0.707371i	0.935364	−	0.353685i	$-0.115072\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−27.5440	−1.17341
$552$	0	0
$553$	18.8600i	0.802009i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 30.1760i	− 1.27860i	−0.768958	−	0.639299i	$-0.779224\pi$
$557$	− 30.1760i	− 1.27860i	0.768958	−	0.639299i	$-0.220776\pi$
$558$	0	0
$559$	−20.3160	−0.859275
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.54400i	0.402232i	0.979567	+	0.201116i	$0.0644568\pi$
$563$	9.54400i	0.402232i	−0.979567	+	0.201116i	$0.935543\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.0880	0.632522	0.316261	−	0.948672i	$-0.397572\pi$
$569$	15.0880	0.632522	0.316261	+	0.948672i	$0.397572\pi$
$570$	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	0
$576$	0	0
$577$	− 40.4040i	− 1.68204i	−0.541003	−	0.841021i	$-0.681955\pi$
$577$	− 40.4040i	− 1.68204i	0.541003	−	0.841021i	$-0.318045\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−22.6320	−0.938934
$582$	0	0
$583$	36.0000i	1.49097i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.0880i	− 1.53079i	−0.643563	−	0.765393i	$-0.722545\pi$
$587$	− 37.0880i	− 1.53079i	0.643563	−	0.765393i	$-0.277455\pi$
$588$	0	0
$589$	−62.1760	−2.56192
$590$	0	0
$591$	0	0
$592$	0	0
$593$	27.8600i	1.14407i	0.820228	+	0.572037i	$0.193847\pi$
$593$	27.8600i	1.14407i	−0.820228	+	0.572037i	$0.806153\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.0880	1.18850	0.594252	−	0.804279i	$-0.297448\pi$
$599$	29.0880	1.18850	0.594252	+	0.804279i	$0.297448\pi$
$600$	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
$605$	0	0
$606$	0	0
$607$	2.86001i	0.116084i	0.998314	+	0.0580421i	$0.0184858\pi$
$607$	2.86001i	0.116084i	−0.998314	+	0.0580421i	$0.981514\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.31601	−0.336430
$612$	0	0
$613$	14.5440i	0.587427i	0.955894	+	0.293713i	$0.0948911\pi$
$613$	14.5440i	0.587427i	−0.955894	+	0.293713i	$0.905109\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 18.3160i	− 0.737375i	−0.929553	−	0.368687i	$-0.879807\pi$
$617$	− 18.3160i	− 0.737375i	0.929553	−	0.368687i	$-0.120193\pi$
$618$	0	0
$619$	−31.3160	−1.25870	−0.629348	−	0.777123i	$-0.716678\pi$
$619$	−31.3160	−1.25870	−0.629348	+	0.777123i	$0.716678\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 30.1760i	− 1.20898i
$624$	0	0
$625$	0	0
$626$	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
$635$	0	0
$636$	0	0
$637$	− 21.6840i	− 0.859151i
$638$	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
$643$	− 1.22800i	− 0.0484275i	−0.999707	−	0.0242138i	$-0.992292\pi$
$643$	− 1.22800i	− 0.0484275i	0.999707	−	0.0242138i	$-0.00770823\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22.6320i	0.889756i	0.895591	+	0.444878i	$0.146753\pi$
$647$	22.6320i	0.889756i	−0.895591	+	0.444878i	$0.853247\pi$
$648$	0	0
$649$	−57.2640	−2.24781
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.08801i	0.277375i	0.990336	+	0.138688i	$0.0442884\pi$
$653$	7.08801i	0.277375i	−0.990336	+	0.138688i	$0.955712\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	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
$665$	0	0
$666$	0	0
$667$	− 3.68399i	− 0.142645i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−37.0880	−1.43177
$672$	0	0
$673$	− 46.4040i	− 1.78874i	−0.447325	−	0.894372i	$-0.647623\pi$
$673$	− 46.4040i	− 1.78874i	0.447325	−	0.894372i	$-0.352377\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 43.0880i	− 1.65601i	−0.560723	−	0.828003i	$-0.689477\pi$
$677$	− 43.0880i	− 1.65601i	0.560723	−	0.828003i	$-0.310523\pi$
$678$	0	0
$679$	27.5960	1.05904
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.5440i	1.05394i	0.849883	+	0.526971i	$0.176672\pi$
$683$	27.5440i	1.05394i	−0.849883	+	0.526971i	$0.823328\pi$
$684$	0	0
$685$	0	0
$686$	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
$695$	0	0
$696$	0	0
$697$	51.0880i	1.93510i
$698$	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
$703$	44.8600i	1.69193i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 70.8080i	− 2.66301i
$708$	0	0
$709$	−7.13999	−0.268148	−0.134074	−	0.990971i	$-0.542806\pi$
$709$	−7.13999	−0.268148	−0.134074	+	0.990971i	$0.542806\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 8.31601i	− 0.311437i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	−31.0360	−1.15584
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.0880i	1.30134i	0.759360	+	0.650671i	$0.225512\pi$
$727$	35.0880i	1.30134i	−0.759360	+	0.650671i	$0.774488\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−45.8600	−1.69619
$732$	0	0
$733$	− 41.0880i	− 1.51762i	−0.651312	−	0.758810i	$-0.725781\pi$
$733$	− 41.0880i	− 1.51762i	0.651312	−	0.758810i	$-0.274219\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 29.7200i	− 1.09475i
$738$	0	0
$739$	0.911993	0.0335482	0.0167741	−	0.999859i	$-0.494660\pi$
$739$	0.911993	0.0335482	0.0167741	+	0.999859i	$0.494660\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.4040i	0.565118i	0.959250	+	0.282559i	$0.0911834\pi$
$743$	15.4040i	0.565118i	−0.959250	+	0.282559i	$0.908817\pi$
$744$	0	0
$745$	0	0
$746$	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
$755$	0	0
$756$	0	0
$757$	− 38.7200i	− 1.40730i	−0.710545	−	0.703652i	$-0.751552\pi$
$757$	− 38.7200i	− 1.40730i	0.710545	−	0.703652i	$-0.248448\pi$
$758$	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
$763$	28.4560i	1.03018i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 36.0000i	− 1.29988i
$768$	0	0
$769$	54.2640	1.95681	0.978405	−	0.206695i	$-0.0662709\pi$
$769$	54.2640	1.95681	0.978405	+	0.206695i	$0.0662709\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 45.0880i	− 1.62170i	−0.585252	−	0.810851i	$-0.699004\pi$
$773$	− 45.0880i	− 1.62170i	0.585252	−	0.810851i	$-0.300996\pi$
$774$	0	0
$775$	0	0
$776$	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	0
$786$	0	0
$787$	29.1760i	1.04001i	0.854162	+	0.520006i	$0.174070\pi$
$787$	29.1760i	1.04001i	−0.854162	+	0.520006i	$0.825930\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	54.0000	1.92002
$792$	0	0
$793$	− 23.3160i	− 0.827976i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.1760i	0.785515i	0.919642	+	0.392757i	$0.128479\pi$
$797$	22.1760i	0.785515i	−0.919642	+	0.392757i	$0.871521\pi$
$798$	0	0
$799$	−18.7720	−0.664106
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 18.0000i	− 0.635206i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	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
$815$	0	0
$816$	0	0
$817$	− 39.0880i	− 1.36752i
$818$	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
$823$	20.4040i	0.711239i	0.934631	+	0.355620i	$0.115730\pi$
$823$	20.4040i	0.711239i	−0.934631	+	0.355620i	$0.884270\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.5440i	− 0.470971i	−0.971878	−	0.235486i	$-0.924332\pi$
$827$	− 13.5440i	− 0.470971i	0.971878	−	0.235486i	$-0.0756680\pi$
$828$	0	0
$829$	−30.2280	−1.04986	−0.524931	−	0.851145i	$-0.675909\pi$
$829$	−30.2280	−1.04986	−0.524931	+	0.851145i	$0.675909\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 48.9480i	− 1.69595i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.36799	−0.116276	−0.0581379	−	0.998309i	$-0.518516\pi$
$839$	−3.36799	−0.116276	−0.0581379	+	0.998309i	$0.518516\pi$
$840$	0	0
$841$	−6.22800	−0.214759
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	44.4040i	1.52574i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	− 32.5440i	− 1.11429i	−0.830417	−	0.557143i	$-0.811898\pi$
$853$	− 32.5440i	− 1.11429i	0.830417	−	0.557143i	$-0.188102\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.91199i	− 0.236109i	−0.993007	−	0.118055i	$-0.962334\pi$
$857$	− 6.91199i	− 0.236109i	0.993007	−	0.118055i	$-0.0376658\pi$
$858$	0	0
$859$	16.2280	0.553692	0.276846	−	0.960914i	$-0.410711\pi$
$859$	16.2280	0.553692	0.276846	+	0.960914i	$0.410711\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.316006i	0.0107570i	0.999986	+	0.00537848i	$0.00171203\pi$
$863$	0.316006i	0.0107570i	−0.999986	+	0.00537848i	$0.998288\pi$
$864$	0	0
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	23.0000i	0.776655i	0.921521	+	0.388327i	$0.126947\pi$
$877$	23.0000i	0.776655i	−0.921521	+	0.388327i	$0.873053\pi$
$878$	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
$883$	− 13.3160i	− 0.448119i	−0.974575	−	0.224060i	$-0.928069\pi$
$883$	− 13.3160i	− 0.448119i	0.974575	−	0.224060i	$-0.0719310\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 32.3160i	− 1.08507i	−0.840035	−	0.542533i	$-0.817465\pi$
$887$	− 32.3160i	− 1.08507i	0.840035	−	0.542533i	$-0.182535\pi$
$888$	0	0
$889$	20.9120	0.701366
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 16.0000i	− 0.535420i
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	35.0000i	1.16216i	0.813848	+	0.581078i	$0.197369\pi$
$907$	35.0000i	1.16216i	−0.813848	+	0.581078i	$0.802631\pi$
$908$	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
$913$	28.6320i	0.947581i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	55.7200i	1.84004i
$918$	0	0
$919$	37.8600	1.24889	0.624443	−	0.781070i	$-0.285326\pi$
$919$	37.8600	1.24889	0.624443	+	0.781070i	$0.285326\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 10.6320i	− 0.349957i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.6320	−1.13624	−0.568120	−	0.822946i	$-0.692329\pi$
$929$	−34.6320	−1.13624	−0.568120	+	0.822946i	$0.692329\pi$
$930$	0	0
$931$	41.7200	1.36732
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.22800i	0.268797i	0.990927	+	0.134398i	$0.0429102\pi$
$937$	8.22800i	0.268797i	−0.990927	+	0.134398i	$0.957090\pi$
$938$	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
$943$	− 5.82399i	− 0.189655i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 25.5440i	− 0.830069i	−0.909806	−	0.415034i	$-0.863770\pi$
$947$	− 25.5440i	− 0.830069i	0.909806	−	0.415034i	$-0.136230\pi$
$948$	0	0
$949$	11.3160	0.367333
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 6.13999i	− 0.198894i	−0.995043	−	0.0994469i	$-0.968293\pi$
$953$	− 6.13999i	− 0.198894i	0.995043	−	0.0994469i	$-0.0317073\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	− 3.77200i	− 0.121299i	−0.998159	−	0.0606497i	$-0.980683\pi$
$967$	− 3.77200i	− 0.121299i	0.998159	−	0.0606497i	$-0.0193173\pi$
$968$	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
$973$	15.9480i	0.511270i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 8.77200i	− 0.280641i	−0.990106	−	0.140321i	$-0.955187\pi$
$977$	− 8.77200i	− 0.280641i	0.990106	−	0.140321i	$-0.0448133\pi$
$978$	0	0
$979$	−38.1760	−1.22011
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.4040i	1.44816i	0.689714	+	0.724082i	$0.257736\pi$
$983$	45.4040i	1.44816i	−0.689714	+	0.724082i	$0.742264\pi$
$984$	0	0
$985$	0	0
$986$	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
$995$	0	0
$996$	0	0
$997$	− 38.3160i	− 1.21348i	−0.794900	−	0.606740i	$-0.792477\pi$
$997$	− 38.3160i	− 1.21348i	0.794900	−	0.606740i	$-0.207523\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5400.2.f.be.649.3		4
3.2	odd	2		5400.2.f.bd.649.3		4
5.2	odd	4		1080.2.a.m.1.1	✓	2
5.3	odd	4		5400.2.a.cb.1.2		2
5.4	even	2	inner	5400.2.f.be.649.2		4
15.2	even	4		1080.2.a.n.1.1	yes	2
15.8	even	4		5400.2.a.ca.1.2		2
15.14	odd	2		5400.2.f.bd.649.2		4
20.7	even	4		2160.2.a.z.1.2		2
40.27	even	4		8640.2.a.db.1.2		2
40.37	odd	4		8640.2.a.de.1.1		2
45.2	even	12		3240.2.q.z.1081.2		4
45.7	odd	12		3240.2.q.bc.1081.2		4
45.22	odd	12		3240.2.q.bc.2161.2		4
45.32	even	12		3240.2.q.z.2161.2		4
60.47	odd	4		2160.2.a.bb.1.2		2
120.77	even	4		8640.2.a.cq.1.1		2
120.107	odd	4		8640.2.a.cn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.m.1.1	✓	2	5.2	odd	4
1080.2.a.n.1.1	yes	2	15.2	even	4
2160.2.a.z.1.2		2	20.7	even	4
2160.2.a.bb.1.2		2	60.47	odd	4
3240.2.q.z.1081.2		4	45.2	even	12
3240.2.q.z.2161.2		4	45.32	even	12
3240.2.q.bc.1081.2		4	45.7	odd	12
3240.2.q.bc.2161.2		4	45.22	odd	12
5400.2.a.ca.1.2		2	15.8	even	4
5400.2.a.cb.1.2		2	5.3	odd	4
5400.2.f.bd.649.2		4	15.14	odd	2
5400.2.f.bd.649.3		4	3.2	odd	2
5400.2.f.be.649.2		4	5.4	even	2	inner
5400.2.f.be.649.3		4	1.1	even	1	trivial
8640.2.a.cn.1.2		2	120.107	odd	4
8640.2.a.cq.1.1		2	120.77	even	4
8640.2.a.db.1.2		2	40.27	even	4
8640.2.a.de.1.1		2	40.37	odd	4

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:	\( N \)	\(=\)	\( 5400 = 2^{3} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5400.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.77200i q^{7} +O(q^{10})\)	\(q+3.77200i q^{7} +4.77200 q^{11} +3.00000i q^{13} +6.77200i q^{17} -5.77200 q^{19} -0.772002i q^{23} +4.77200 q^{29} +10.7720 q^{31} -7.77200i q^{37} +7.54400 q^{41} +6.77200i q^{43} +2.77200i q^{47} -7.22800 q^{49} +7.54400i q^{53} -12.0000 q^{59} -7.77200 q^{61} -6.22800i q^{67} -3.54400 q^{71} -3.77200i q^{73} +18.0000i q^{77} +5.00000 q^{79} +6.00000i q^{83} -8.00000 q^{89} -11.3160 q^{91} -7.31601i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 2 q^{11} - 6 q^{19} + 2 q^{29} + 26 q^{31} - 4 q^{41} - 46 q^{49} - 48 q^{59} - 14 q^{61} + 20 q^{71} + 20 q^{79} - 32 q^{89} + 6 q^{91}+O(q^{100}) \) 4 * q + 2 * q^11 - 6 * q^19 + 2 * q^29 + 26 * q^31 - 4 * q^41 - 46 * q^49 - 48 * q^59 - 14 * q^61 + 20 * q^71 + 20 * q^79 - 32 * q^89 + 6 * q^91

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