LMFDB - Embedded newform 1440.2.bi.e.847.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,2,Mod(847,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.847"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.bi (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			847.12
Character	$\chi$	$=$	1440.847
Dual form			1440.2.bi.e.1423.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.22965 - 0.169312i) q^{5} +(-0.645414 + 0.645414i) q^{7} -2.11990 q^{11} +(-1.65437 - 1.65437i) q^{13} +(4.23698 + 4.23698i) q^{17} -2.18966i q^{19} +(6.05433 + 6.05433i) q^{23} +(4.94267 - 0.755014i) q^{25} +7.93585 q^{29} -0.574128i q^{31} +(-1.32977 + 1.54832i) q^{35} +(6.90775 - 6.90775i) q^{37} -2.99799 q^{41} +(-3.18765 + 3.18765i) q^{43} +(5.04999 - 5.04999i) q^{47} +6.16688i q^{49} +(-1.19626 - 1.19626i) q^{53} +(-4.72664 + 0.358925i) q^{55} +11.5919i q^{59} +2.35096i q^{61} +(-3.96876 - 3.40855i) q^{65} +(-4.99799 - 4.99799i) q^{67} -5.01557i q^{71} +(-6.18765 + 6.18765i) q^{73} +(1.36821 - 1.36821i) q^{77} +10.1700 q^{79} +(11.7654 - 11.7654i) q^{83} +(10.1643 + 8.72960i) q^{85} +8.65485i q^{89} +2.13550 q^{91} +(-0.370736 - 4.88217i) q^{95} +(-8.06823 - 8.06823i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 8 q^{17} - 8 q^{25} - 48 q^{35} + 32 q^{43} + 8 q^{65} - 48 q^{67} - 40 q^{73} + 80 q^{83} - 64 q^{91} - 24 q^{97}+O(q^{100}) $ 24 * q - 8 * q^17 - 8 * q^25 - 48 * q^35 + 32 * q^43 + 8 * q^65 - 48 * q^67 - 40 * q^73 + 80 * q^83 - 64 * q^91 - 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$	$-1$

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
$5$	2.22965	−	0.169312i	0.997129	−	0.0757188i
$5$	2.22965	−	0.169312i	0.997129	−	0.0757188i
$6$		0			0
$7$	−0.645414	+	0.645414i	−0.243943	+	0.243943i	−0.818479	−	0.574536i	$-0.805183\pi$
$7$	−0.645414	+	0.645414i	−0.243943	+	0.243943i	0.574536	+	0.818479i	$0.305183\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.11990			−0.639174			−0.319587	−	0.947557i	$-0.603544\pi$
$11$	−2.11990			−0.639174			−0.319587	+	0.947557i	$0.603544\pi$
$12$		0			0
$13$	−1.65437	−	1.65437i	−0.458839	−	0.458839i	0.439435	−	0.898274i	$-0.355179\pi$
$13$	−1.65437	−	1.65437i	−0.458839	−	0.458839i	−0.898274	+	0.439435i	$0.855179\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.23698	+	4.23698i	1.02762	+	1.02762i	0.999608	+	0.0280106i	$0.00891721\pi$
$17$	4.23698	+	4.23698i	1.02762	+	1.02762i	0.0280106	+	0.999608i	$0.491083\pi$
$18$		0			0
$19$		−	2.18966i		−	0.502342i	−0.967943	−	0.251171i	$-0.919184\pi$
$19$		−	2.18966i		−	0.502342i	0.967943	−	0.251171i	$-0.0808157\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.05433	+	6.05433i	1.26242	+	1.26242i	0.949919	+	0.312496i	$0.101165\pi$
$23$	6.05433	+	6.05433i	1.26242	+	1.26242i	0.312496	+	0.949919i	$0.398835\pi$
$24$		0			0
$25$	4.94267	−	0.755014i	0.988533	−	0.151003i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.93585			1.47365			0.736825	−	0.676083i	$-0.236324\pi$
$29$	7.93585			1.47365			0.736825	+	0.676083i	$0.236324\pi$
$30$		0			0
$31$		−	0.574128i		−	0.103116i	−0.998670	−	0.0515582i	$-0.983581\pi$
$31$		−	0.574128i		−	0.103116i	0.998670	−	0.0515582i	$-0.0164188\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.32977	+	1.54832i	−0.224772	+	0.261714i
$36$		0			0
$37$	6.90775	−	6.90775i	1.13563	−	1.13563i	0.146402	−	0.989225i	$-0.453231\pi$
$37$	6.90775	−	6.90775i	1.13563	−	1.13563i	0.989225	−	0.146402i	$-0.0467692\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.99799			−0.468208			−0.234104	−	0.972212i	$-0.575216\pi$
$41$	−2.99799			−0.468208			−0.234104	+	0.972212i	$0.575216\pi$
$42$		0			0
$43$	−3.18765	+	3.18765i	−0.486112	+	0.486112i	−0.907077	−	0.420965i	$-0.861692\pi$
$43$	−3.18765	+	3.18765i	−0.486112	+	0.486112i	0.420965	+	0.907077i	$0.361692\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.04999	−	5.04999i	0.736617	−	0.736617i	−0.235305	−	0.971922i	$-0.575609\pi$
$47$	5.04999	−	5.04999i	0.736617	−	0.736617i	0.971922	+	0.235305i	$0.0756088\pi$
$48$		0			0
$49$			6.16688i			0.880983i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.19626	−	1.19626i	−0.164319	−	0.164319i	0.620158	−	0.784477i	$-0.287068\pi$
$53$	−1.19626	−	1.19626i	−0.164319	−	0.164319i	−0.784477	+	0.620158i	$0.787068\pi$
$54$		0			0
$55$	−4.72664	+	0.358925i	−0.637340	+	0.0483975i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			11.5919i			1.50913i	0.656225	+	0.754565i	$0.272152\pi$
$59$			11.5919i			1.50913i	−0.656225	+	0.754565i	$0.727848\pi$
$60$		0			0
$61$			2.35096i			0.301009i	0.988609	+	0.150505i	$0.0480898\pi$
$61$			2.35096i			0.301009i	−0.988609	+	0.150505i	$0.951910\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.96876	−	3.40855i	−0.492264	−	0.422779i
$66$		0			0
$67$	−4.99799	−	4.99799i	−0.610602	−	0.610602i	0.332501	−	0.943103i	$-0.392108\pi$
$67$	−4.99799	−	4.99799i	−0.610602	−	0.610602i	−0.943103	+	0.332501i	$0.892108\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.01557i		−	0.595239i	−0.954685	−	0.297619i	$-0.903807\pi$
$71$		−	5.01557i		−	0.595239i	0.954685	−	0.297619i	$-0.0961926\pi$
$72$		0			0
$73$	−6.18765	+	6.18765i	−0.724210	+	0.724210i	−0.969460	−	0.245250i	$-0.921130\pi$
$73$	−6.18765	+	6.18765i	−0.724210	+	0.724210i	0.245250	+	0.969460i	$0.421130\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.36821	−	1.36821i	0.155922	−	0.155922i
$78$		0			0
$79$	10.1700			1.14421			0.572106	−	0.820180i	$-0.306127\pi$
$79$	10.1700			1.14421			0.572106	+	0.820180i	$0.306127\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.7654	−	11.7654i	1.29142	−	1.29142i	0.357517	−	0.933906i	$-0.383623\pi$
$83$	11.7654	−	11.7654i	1.29142	−	1.29142i	0.933906	−	0.357517i	$-0.116377\pi$
$84$		0			0
$85$	10.1643	+	8.72960i	1.10248	+	0.946858i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			8.65485i			0.917412i	0.888588	+	0.458706i	$0.151687\pi$
$89$			8.65485i			0.917412i	−0.888588	+	0.458706i	$0.848313\pi$
$90$		0			0
$91$	2.13550			0.223861
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.370736	−	4.88217i	−0.0380367	−	0.500900i
$96$		0			0
$97$	−8.06823	−	8.06823i	−0.819205	−	0.819205i	0.166788	−	0.985993i	$-0.446660\pi$
$97$	−8.06823	−	8.06823i	−0.819205	−	0.819205i	−0.985993	+	0.166788i	$0.946660\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.83542i		−	0.282135i	−0.990000	−	0.141067i	$-0.954947\pi$
$101$		−	2.83542i		−	0.282135i	0.990000	−	0.141067i	$-0.0450534\pi$
$102$		0			0
$103$	7.52594	+	7.52594i	0.741553	+	0.741553i	0.972877	−	0.231324i	$-0.0743057\pi$
$103$	7.52594	+	7.52594i	0.741553	+	0.741553i	−0.231324	+	0.972877i	$0.574306\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.38812	−	2.38812i	−0.230868	−	0.230868i	0.582187	−	0.813055i	$-0.302197\pi$
$107$	−2.38812	−	2.38812i	−0.230868	−	0.230868i	−0.813055	+	0.582187i	$0.802197\pi$
$108$		0			0
$109$	−3.69420			−0.353840			−0.176920	−	0.984225i	$-0.556613\pi$
$109$	−3.69420			−0.353840			−0.176920	+	0.984225i	$0.556613\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.14033	−	3.14033i	0.295418	−	0.295418i	−0.543798	−	0.839216i	$-0.683014\pi$
$113$	3.14033	−	3.14033i	0.295418	−	0.295418i	0.839216	+	0.543798i	$0.183014\pi$
$114$		0			0
$115$	14.5241	+	12.4740i	1.35438	+	1.16320i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.46921			−0.501362
$120$		0			0
$121$	−6.50602			−0.591456
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.8926	−	2.52027i	0.974262	−	0.225420i
$126$		0			0
$127$	12.1799	−	12.1799i	1.08080	−	1.08080i	0.0843601	−	0.996435i	$-0.473115\pi$
$127$	12.1799	−	12.1799i	1.08080	−	1.08080i	0.996435	−	0.0843601i	$-0.0268846\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.440307			−0.0384698			−0.0192349	−	0.999815i	$-0.506123\pi$
$131$	−0.440307			−0.0384698			−0.0192349	+	0.999815i	$0.506123\pi$
$132$		0			0
$133$	1.41324	+	1.41324i	0.122543	+	0.122543i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.57812	+	1.57812i	0.134828	+	0.134828i	0.771300	−	0.636472i	$-0.219607\pi$
$137$	1.57812	+	1.57812i	0.134828	+	0.134828i	−0.636472	+	0.771300i	$0.719607\pi$
$138$		0			0
$139$			12.6228i			1.07065i	0.844647	+	0.535324i	$0.179810\pi$
$139$			12.6228i			1.07065i	−0.844647	+	0.535324i	$0.820190\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.50709	+	3.50709i	0.293278	+	0.293278i
$144$		0			0
$145$	17.6942	−	1.34364i	1.46942	−	0.111583i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.4367			−1.51039			−0.755195	−	0.655500i	$-0.772458\pi$
$149$	−18.4367			−1.51039			−0.755195	+	0.655500i	$0.772458\pi$
$150$		0			0
$151$		−	0.00374102i		−	0.000304440i	−1.00000		0.000152220i	$-0.999952\pi$
$151$		−	0.00374102i		−	0.000304440i	1.00000		0.000152220i	$-4.84532e-5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.0972070	−	1.28010i	−0.00780785	−	0.102820i
$156$		0			0
$157$	−11.3149	+	11.3149i	−0.903028	+	0.903028i	−0.995697	−	0.0926688i	$-0.970460\pi$
$157$	−11.3149	+	11.3149i	−0.903028	+	0.903028i	0.0926688	+	0.995697i	$0.470460\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.81510			−0.615916
$162$		0			0
$163$	5.79433	−	5.79433i	0.453847	−	0.453847i	−0.442782	−	0.896629i	$-0.646009\pi$
$163$	5.79433	−	5.79433i	0.453847	−	0.453847i	0.896629	+	0.442782i	$0.146009\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.21418	−	2.21418i	0.171339	−	0.171339i	−0.616229	−	0.787567i	$-0.711340\pi$
$167$	2.21418	−	2.21418i	0.171339	−	0.171339i	0.787567	+	0.616229i	$0.211340\pi$
$168$		0			0
$169$		−	7.52614i		−	0.578934i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.02012	+	7.02012i	0.533730	+	0.533730i	0.921680	−	0.387951i	$-0.126817\pi$
$173$	7.02012	+	7.02012i	0.533730	+	0.533730i	−0.387951	+	0.921680i	$0.626817\pi$
$174$		0			0
$175$	−2.70277	+	3.67736i	−0.204310	+	0.277982i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			13.2165i			0.987851i	0.869504	+	0.493926i	$0.164438\pi$
$179$			13.2165i			0.987851i	−0.869504	+	0.493926i	$0.835562\pi$
$180$		0			0
$181$		−	7.24786i		−	0.538729i	−0.963038	−	0.269365i	$-0.913186\pi$
$181$		−	7.24786i		−	0.538729i	0.963038	−	0.269365i	$-0.0868137\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.2323	−	16.5714i	1.04638	−	1.21836i
$186$		0			0
$187$	−8.98198	−	8.98198i	−0.656827	−	0.656827i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			4.60879i			0.333480i	0.986001	+	0.166740i	$0.0533241\pi$
$191$			4.60879i			0.333480i	−0.986001	+	0.166740i	$0.946676\pi$
$192$		0			0
$193$	2.88058	−	2.88058i	0.207349	−	0.207349i	−0.595791	−	0.803140i	$-0.703161\pi$
$193$	2.88058	−	2.88058i	0.207349	−	0.207349i	0.803140	+	0.595791i	$0.203161\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.3607	+	13.3607i	−0.951911	+	0.951911i	−0.998896	−	0.0469844i	$-0.985039\pi$
$197$	−13.3607	+	13.3607i	−0.951911	+	0.951911i	0.0469844	+	0.998896i	$0.485039\pi$
$198$		0			0
$199$	−21.3015			−1.51002			−0.755010	−	0.655713i	$-0.772368\pi$
$199$	−21.3015			−1.51002			−0.755010	+	0.655713i	$0.772368\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.12191	+	5.12191i	−0.359487	+	0.359487i
$204$		0			0
$205$	−6.68447	+	0.507597i	−0.466864	+	0.0354521i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			4.64186i			0.321084i
$210$		0			0
$211$	−10.2796			−0.707677			−0.353839	−	0.935306i	$-0.615124\pi$
$211$	−10.2796			−0.707677			−0.353839	+	0.935306i	$0.615124\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.56764	+	7.64705i	−0.447909	+	0.521525i
$216$		0			0
$217$	0.370550	+	0.370550i	0.0251546	+	0.0251546i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	14.0190i		−	0.943022i
$222$		0			0
$223$	−8.17006	−	8.17006i	−0.547108	−	0.547108i	0.378495	−	0.925603i	$-0.376442\pi$
$223$	−8.17006	−	8.17006i	−0.547108	−	0.547108i	−0.925603	+	0.378495i	$0.876442\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.26621	−	5.26621i	−0.349531	−	0.349531i	0.510404	−	0.859935i	$-0.329496\pi$
$227$	−5.26621	−	5.26621i	−0.349531	−	0.349531i	−0.859935	+	0.510404i	$0.829496\pi$
$228$		0			0
$229$	3.12556			0.206543			0.103271	−	0.994653i	$-0.467069\pi$
$229$	3.12556			0.206543			0.103271	+	0.994653i	$0.467069\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.41787	−	2.41787i	0.158400	−	0.158400i	−0.623457	−	0.781857i	$-0.714273\pi$
$233$	2.41787	−	2.41787i	0.158400	−	0.158400i	0.781857	+	0.623457i	$0.214273\pi$
$234$		0			0
$235$	10.4047	−	12.1147i	0.678726	−	0.790278i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.68630			0.432501			0.216251	−	0.976338i	$-0.430617\pi$
$239$	6.68630			0.432501			0.216251	+	0.976338i	$0.430617\pi$
$240$		0			0
$241$	−15.8501			−1.02100			−0.510499	−	0.859878i	$-0.670539\pi$
$241$	−15.8501			−1.02100			−0.510499	+	0.859878i	$0.670539\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.04413	+	13.7500i	0.0667070	+	0.878454i
$246$		0			0
$247$	−3.62250	+	3.62250i	−0.230494	+	0.230494i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.25540			0.268599			0.134299	−	0.990941i	$-0.457122\pi$
$251$	4.25540			0.268599			0.134299	+	0.990941i	$0.457122\pi$
$252$		0			0
$253$	−12.8346	−	12.8346i	−0.806904	−	0.806904i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.15838	−	3.15838i	−0.197015	−	0.197015i	0.601704	−	0.798719i	$-0.294489\pi$
$257$	−3.15838	−	3.15838i	−0.197015	−	0.197015i	−0.798719	+	0.601704i	$0.794489\pi$
$258$		0			0
$259$			8.91671i			0.554058i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.25036	−	9.25036i	−0.570402	−	0.570402i	0.361839	−	0.932241i	$-0.382149\pi$
$263$	−9.25036	−	9.25036i	−0.570402	−	0.570402i	−0.932241	+	0.361839i	$0.882149\pi$
$264$		0			0
$265$	−2.86978	−	2.46470i	−0.176289	−	0.151405i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.69801			0.164501			0.0822504	−	0.996612i	$-0.473789\pi$
$269$	2.69801			0.164501			0.0822504	+	0.996612i	$0.473789\pi$
$270$		0			0
$271$		−	23.9762i		−	1.45645i	−0.685336	−	0.728227i	$-0.740345\pi$
$271$		−	23.9762i		−	1.45645i	0.685336	−	0.728227i	$-0.259655\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.4780	+	1.60056i	−0.631845	+	0.0965171i
$276$		0			0
$277$	−0.392928	+	0.392928i	−0.0236087	+	0.0236087i	−0.718813	−	0.695204i	$-0.755314\pi$
$277$	−0.392928	+	0.392928i	−0.0236087	+	0.0236087i	0.695204	+	0.718813i	$0.255314\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.8077			−1.47990			−0.739952	−	0.672659i	$-0.765152\pi$
$281$	−24.8077			−1.47990			−0.739952	+	0.672659i	$0.765152\pi$
$282$		0			0
$283$	9.23780	−	9.23780i	0.549130	−	0.549130i	−0.377059	−	0.926189i	$-0.623065\pi$
$283$	9.23780	−	9.23780i	0.549130	−	0.549130i	0.926189	+	0.377059i	$0.123065\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.93495	−	1.93495i	0.114216	−	0.114216i
$288$		0			0
$289$			18.9040i			1.11200i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.1431	−	11.1431i	−0.650987	−	0.650987i	0.302244	−	0.953231i	$-0.402264\pi$
$293$	−11.1431	−	11.1431i	−0.650987	−	0.650987i	−0.953231	+	0.302244i	$0.902264\pi$
$294$		0			0
$295$	1.96264	+	25.8458i	0.114270	+	1.50480i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	20.0322i		−	1.15849i
$300$		0			0
$301$		−	4.11471i		−	0.237168i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.398046	+	5.24181i	0.0227921	+	0.300145i
$306$		0			0
$307$	−12.0381	−	12.0381i	−0.687053	−	0.687053i	0.274527	−	0.961580i	$-0.411479\pi$
$307$	−12.0381	−	12.0381i	−0.687053	−	0.687053i	−0.961580	+	0.274527i	$0.911479\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			30.9541i			1.75525i	0.479350	+	0.877624i	$0.340872\pi$
$311$			30.9541i			1.75525i	−0.479350	+	0.877624i	$0.659128\pi$
$312$		0			0
$313$	8.62544	−	8.62544i	0.487539	−	0.487539i	−0.419990	−	0.907529i	$-0.637966\pi$
$313$	8.62544	−	8.62544i	0.487539	−	0.487539i	0.907529	+	0.419990i	$0.137966\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.34682	+	2.34682i	−0.131811	+	0.131811i	−0.769934	−	0.638123i	$-0.779711\pi$
$317$	−2.34682	+	2.34682i	−0.131811	+	0.131811i	0.638123	+	0.769934i	$0.279711\pi$
$318$		0			0
$319$	−16.8232			−0.941920
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.27754	−	9.27754i	0.516216	−	0.516216i
$324$		0			0
$325$	−9.42605	−	6.92791i	−0.522863	−	0.384291i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			6.51867i			0.359386i
$330$		0			0
$331$	−12.1856			−0.669784			−0.334892	−	0.942257i	$-0.608700\pi$
$331$	−12.1856			−0.669784			−0.334892	+	0.942257i	$0.608700\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.9900	−	10.2975i	−0.655083	−	0.562615i
$336$		0			0
$337$	−8.63207	−	8.63207i	−0.470219	−	0.470219i	0.431767	−	0.901985i	$-0.357890\pi$
$337$	−8.63207	−	8.63207i	−0.470219	−	0.470219i	−0.901985	+	0.431767i	$0.857890\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.21710i			0.0659094i
$342$		0			0
$343$	−8.49809	−	8.49809i	−0.458854	−	0.458854i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.30238	−	7.30238i	−0.392013	−	0.392013i	0.483392	−	0.875404i	$-0.339405\pi$
$347$	−7.30238	−	7.30238i	−0.392013	−	0.392013i	−0.875404	+	0.483392i	$0.839405\pi$
$348$		0			0
$349$	−14.2003			−0.760124			−0.380062	−	0.924961i	$-0.624097\pi$
$349$	−14.2003			−0.760124			−0.380062	+	0.924961i	$0.624097\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.21893	+	4.21893i	−0.224551	+	0.224551i	−0.810412	−	0.585861i	$-0.800757\pi$
$353$	−4.21893	+	4.21893i	−0.224551	+	0.224551i	0.585861	+	0.810412i	$0.300757\pi$
$354$		0			0
$355$	−0.849198	−	11.1830i	−0.0450707	−	0.593530i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−34.5078			−1.82125			−0.910626	−	0.413232i	$-0.864400\pi$
$359$	−34.5078			−1.82125			−0.910626	+	0.413232i	$0.864400\pi$
$360$		0			0
$361$	14.2054			0.747652
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.7486	+	14.8439i	−0.667295	+	0.776967i
$366$		0			0
$367$	3.79229	−	3.79229i	0.197956	−	0.197956i	−0.601167	−	0.799123i	$-0.705297\pi$
$367$	3.79229	−	3.79229i	0.197956	−	0.197956i	0.799123	+	0.601167i	$0.205297\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.54417			0.0801691
$372$		0			0
$373$	15.0702	+	15.0702i	0.780305	+	0.780305i	0.979882	−	0.199577i	$-0.0639568\pi$
$373$	15.0702	+	15.0702i	0.780305	+	0.780305i	−0.199577	+	0.979882i	$0.563957\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13.1288	−	13.1288i	−0.676168	−	0.676168i
$378$		0			0
$379$			25.4415i			1.30684i	0.756995	+	0.653421i	$0.226667\pi$
$379$			25.4415i			1.30684i	−0.756995	+	0.653421i	$0.773333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.40762	−	5.40762i	−0.276317	−	0.276317i	0.555320	−	0.831637i	$-0.312596\pi$
$383$	−5.40762	−	5.40762i	−0.276317	−	0.276317i	−0.831637	+	0.555320i	$0.812596\pi$
$384$		0			0
$385$	2.81898	−	3.28229i	0.143669	−	0.167281i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.35502			−0.220809			−0.110404	−	0.993887i	$-0.535215\pi$
$389$	−4.35502			−0.220809			−0.110404	+	0.993887i	$0.535215\pi$
$390$		0			0
$391$			51.3041i			2.59456i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	22.6755	−	1.72190i	1.14093	−	0.0866383i
$396$		0			0
$397$	12.1053	−	12.1053i	0.607550	−	0.607550i	−0.334755	−	0.942305i	$-0.608654\pi$
$397$	12.1053	−	12.1053i	0.607550	−	0.607550i	0.942305	+	0.334755i	$0.108654\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	26.5884			1.32776			0.663880	−	0.747839i	$-0.268909\pi$
$401$	26.5884			1.32776			0.663880	+	0.747839i	$0.268909\pi$
$402$		0			0
$403$	−0.949819	+	0.949819i	−0.0473138	+	0.0473138i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14.6438	+	14.6438i	−0.725864	+	0.725864i
$408$		0			0
$409$			9.14423i			0.452153i	0.974110	+	0.226077i	$0.0725900\pi$
$409$			9.14423i			0.452153i	−0.974110	+	0.226077i	$0.927410\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−7.48154	−	7.48154i	−0.368143	−	0.368143i
$414$		0			0
$415$	24.2407	−	28.2248i	1.18993	−	1.38550i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			4.32353i			0.211218i	0.994408	+	0.105609i	$0.0336793\pi$
$419$			4.32353i			0.211218i	−0.994408	+	0.105609i	$0.966321\pi$
$420$		0			0
$421$		−	8.49742i		−	0.414139i	−0.978326	−	0.207069i	$-0.933607\pi$
$421$		−	8.49742i		−	0.414139i	0.978326	−	0.207069i	$-0.0663926\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	24.1409	+	17.7430i	1.17101	+	0.860662i
$426$		0			0
$427$	−1.51734	−	1.51734i	−0.0734293	−	0.0734293i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	25.8697i		−	1.24610i	−0.782182	−	0.623050i	$-0.785893\pi$
$431$		−	25.8697i		−	1.24610i	0.782182	−	0.623050i	$-0.214107\pi$
$432$		0			0
$433$	18.1990	−	18.1990i	0.874590	−	0.874590i	−0.118378	−	0.992969i	$-0.537770\pi$
$433$	18.1990	−	18.1990i	0.874590	−	0.874590i	0.992969	+	0.118378i	$0.0377696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13.2569	−	13.2569i	0.634164	−	0.634164i
$438$		0			0
$439$	−16.7069			−0.797377			−0.398688	−	0.917086i	$-0.630535\pi$
$439$	−16.7069			−0.797377			−0.398688	+	0.917086i	$0.630535\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.25017	+	4.25017i	−0.201932	+	0.201932i	−0.800827	−	0.598896i	$-0.795606\pi$
$443$	−4.25017	+	4.25017i	−0.201932	+	0.201932i	0.598896	+	0.800827i	$0.295606\pi$
$444$		0			0
$445$	1.46537	+	19.2973i	0.0694653	+	0.914778i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	36.7452i		−	1.73412i	−0.498208	−	0.867058i	$-0.666008\pi$
$449$		−	36.7452i		−	1.73412i	0.498208	−	0.867058i	$-0.333992\pi$
$450$		0			0
$451$	6.35545			0.299267
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.76142	−	0.361567i	0.223219	−	0.0169505i
$456$		0			0
$457$	19.5101	+	19.5101i	0.912645	+	0.912645i	0.996480	−	0.0838344i	$-0.0267167\pi$
$457$	19.5101	+	19.5101i	0.912645	+	0.912645i	−0.0838344	+	0.996480i	$0.526717\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	5.42713i		−	0.252767i	−0.991981	−	0.126383i	$-0.959663\pi$
$461$		−	5.42713i		−	0.252767i	0.991981	−	0.126383i	$-0.0403370\pi$
$462$		0			0
$463$	24.1397	+	24.1397i	1.12187	+	1.12187i	0.991461	+	0.130407i	$0.0416285\pi$
$463$	24.1397	+	24.1397i	1.12187	+	1.12187i	0.130407	+	0.991461i	$0.458372\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.6661	−	19.6661i	−0.910039	−	0.910039i	0.0862361	−	0.996275i	$-0.472516\pi$
$467$	−19.6661	−	19.6661i	−0.910039	−	0.910039i	−0.996275	+	0.0862361i	$0.972516\pi$
$468$		0			0
$469$	6.45155			0.297905
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.75751	−	6.75751i	0.310711	−	0.310711i
$474$		0			0
$475$	−1.65322	−	10.8228i	−0.0758551	−	0.496582i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.12247			−0.188360			−0.0941802	−	0.995555i	$-0.530023\pi$
$479$	−4.12247			−0.188360			−0.0941802	+	0.995555i	$0.530023\pi$
$480$		0			0
$481$	−22.8559			−1.04214
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−19.3554	−	16.6233i	−0.878882	−	0.754824i
$486$		0			0
$487$	−17.0261	+	17.0261i	−0.771525	+	0.771525i	−0.978373	−	0.206848i	$-0.933679\pi$
$487$	−17.0261	+	17.0261i	−0.771525	+	0.771525i	0.206848	+	0.978373i	$0.433679\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.33601			−0.285940			−0.142970	−	0.989727i	$-0.545665\pi$
$491$	−6.33601			−0.285940			−0.142970	+	0.989727i	$0.545665\pi$
$492$		0			0
$493$	33.6240	+	33.6240i	1.51435	+	1.51435i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.23712	+	3.23712i	0.145205	+	0.145205i
$498$		0			0
$499$			17.1014i			0.765564i	0.923839	+	0.382782i	$0.125034\pi$
$499$			17.1014i			0.765564i	−0.923839	+	0.382782i	$0.874966\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.10917	+	7.10917i	0.316982	+	0.316982i	0.847607	−	0.530625i	$-0.178043\pi$
$503$	7.10917	+	7.10917i	0.316982	+	0.316982i	−0.530625	+	0.847607i	$0.678043\pi$
$504$		0			0
$505$	−0.480071	−	6.32198i	−0.0213629	−	0.281325i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.3100			−0.722927			−0.361464	−	0.932386i	$-0.617723\pi$
$509$	−16.3100			−0.722927			−0.361464	+	0.932386i	$0.617723\pi$
$510$		0			0
$511$		−	7.98719i		−	0.353333i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.0544	+	15.5060i	0.795574	+	0.683275i
$516$		0			0
$517$	−10.7055	+	10.7055i	−0.470827	+	0.470827i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.97735			0.261872			0.130936	−	0.991391i	$-0.458202\pi$
$521$	5.97735			0.261872			0.130936	+	0.991391i	$0.458202\pi$
$522$		0			0
$523$	15.8757	−	15.8757i	0.694195	−	0.694195i	−0.268957	−	0.963152i	$-0.586679\pi$
$523$	15.8757	−	15.8757i	0.694195	−	0.694195i	0.963152	+	0.268957i	$0.0866791\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.43257	−	2.43257i	0.105964	−	0.105964i
$528$		0			0
$529$			50.3098i			2.18738i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.95978	+	4.95978i	0.214832	+	0.214832i
$534$		0			0
$535$	−5.72901	−	4.92033i	−0.247687	−	0.212724i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	13.0732i		−	0.563102i
$540$		0			0
$541$		−	32.5138i		−	1.39788i	−0.715182	−	0.698938i	$-0.753656\pi$
$541$		−	32.5138i		−	1.39788i	0.715182	−	0.698938i	$-0.246344\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.23677	+	0.625474i	−0.352825	+	0.0267924i
$546$		0			0
$547$	−1.85008	−	1.85008i	−0.0791037	−	0.0791037i	0.666448	−	0.745552i	$-0.267814\pi$
$547$	−1.85008	−	1.85008i	−0.0791037	−	0.0791037i	−0.745552	+	0.666448i	$0.767814\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	17.3768i		−	0.740277i
$552$		0			0
$553$	−6.56384	+	6.56384i	−0.279123	+	0.279123i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.55944	−	4.55944i	0.193190	−	0.193190i	−0.603883	−	0.797073i	$-0.706381\pi$
$557$	4.55944	−	4.55944i	0.193190	−	0.193190i	0.797073	+	0.603883i	$0.206381\pi$
$558$		0			0
$559$	10.5471			0.446094
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.3506	+	28.3506i	−1.19483	+	1.19483i	−0.219142	+	0.975693i	$0.570326\pi$
$563$	−28.3506	+	28.3506i	−1.19483	+	1.19483i	−0.975693	+	0.219142i	$0.929674\pi$
$564$		0			0
$565$	6.47014	−	7.53354i	0.272201	−	0.316938i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			23.9629i			1.00458i	0.864700	+	0.502289i	$0.167509\pi$
$569$			23.9629i			1.00458i	−0.864700	+	0.502289i	$0.832491\pi$
$570$		0			0
$571$	−30.4111			−1.27266			−0.636332	−	0.771415i	$-0.719549\pi$
$571$	−30.4111			−1.27266			−0.636332	+	0.771415i	$0.719549\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	34.4956	+	25.3534i	1.43857	+	1.05731i
$576$		0			0
$577$	−21.4532	−	21.4532i	−0.893108	−	0.893108i	0.101707	−	0.994814i	$-0.467570\pi$
$577$	−21.4532	−	21.4532i	−0.893108	−	0.893108i	−0.994814	+	0.101707i	$0.967570\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			15.1871i			0.630069i
$582$		0			0
$583$	2.53595	+	2.53595i	0.105028	+	0.105028i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.9607	+	20.9607i	0.865142	+	0.865142i	0.991930	−	0.126788i	$-0.0404668\pi$
$587$	20.9607	+	20.9607i	0.865142	+	0.865142i	−0.126788	+	0.991930i	$0.540467\pi$
$588$		0			0
$589$	−1.25715			−0.0517998
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.7528	+	16.7528i	−0.687953	+	0.687953i	−0.961779	−	0.273826i	$-0.911711\pi$
$593$	−16.7528	+	16.7528i	−0.687953	+	0.687953i	0.273826	+	0.961779i	$0.411711\pi$
$594$		0			0
$595$	−12.1944	+	0.926004i	−0.499922	+	0.0379625i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	28.7818			1.17599			0.587997	−	0.808863i	$-0.299917\pi$
$599$	28.7818			1.17599			0.587997	+	0.808863i	$0.299917\pi$
$600$		0			0
$601$	−23.8948			−0.974691			−0.487346	−	0.873209i	$-0.662035\pi$
$601$	−23.8948			−0.974691			−0.487346	+	0.873209i	$0.662035\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.5061	+	1.10155i	−0.589758	+	0.0447843i
$606$		0			0
$607$	24.0023	−	24.0023i	0.974224	−	0.974224i	−0.0254522	−	0.999676i	$-0.508103\pi$
$607$	24.0023	−	24.0023i	0.974224	−	0.974224i	0.999676	+	0.0254522i	$0.00810257\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.7091			−0.675977
$612$		0			0
$613$	−33.6909	−	33.6909i	−1.36076	−	1.36076i	−0.872946	−	0.487816i	$-0.837794\pi$
$613$	−33.6909	−	33.6909i	−1.36076	−	1.36076i	−0.487816	−	0.872946i	$-0.662206\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.5125	−	22.5125i	−0.906319	−	0.906319i	0.0896535	−	0.995973i	$-0.471424\pi$
$617$	−22.5125	−	22.5125i	−0.906319	−	0.906319i	−0.995973	+	0.0896535i	$0.971424\pi$
$618$		0			0
$619$		−	21.0797i		−	0.847265i	−0.905834	−	0.423633i	$-0.860755\pi$
$619$		−	21.0797i		−	0.847265i	0.905834	−	0.423633i	$-0.139245\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5.58596	−	5.58596i	−0.223797	−	0.223797i
$624$		0			0
$625$	23.8599	−	7.46356i	0.954396	−	0.298543i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	58.5360			2.33398
$630$		0			0
$631$		−	42.6546i		−	1.69805i	−0.528351	−	0.849026i	$-0.677190\pi$
$631$		−	42.6546i		−	1.69805i	0.528351	−	0.849026i	$-0.322810\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	25.0948	−	29.2192i	0.995856	−	1.15953i
$636$		0			0
$637$	10.2023	−	10.2023i	0.404229	−	0.404229i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.687931			−0.0271716			−0.0135858	−	0.999908i	$-0.504325\pi$
$641$	−0.687931			−0.0271716			−0.0135858	+	0.999908i	$0.504325\pi$
$642$		0			0
$643$	−9.24755	+	9.24755i	−0.364688	+	0.364688i	−0.865535	−	0.500848i	$-0.833022\pi$
$643$	−9.24755	+	9.24755i	−0.364688	+	0.364688i	0.500848	+	0.865535i	$0.333022\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	17.0187	−	17.0187i	0.669073	−	0.669073i	−0.288429	−	0.957501i	$-0.593133\pi$
$647$	17.0187	−	17.0187i	0.669073	−	0.669073i	0.957501	+	0.288429i	$0.0931328\pi$
$648$		0			0
$649$		−	24.5736i		−	0.964598i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.9095	+	11.9095i	0.466056	+	0.466056i	0.900634	−	0.434578i	$-0.143102\pi$
$653$	11.9095	+	11.9095i	0.466056	+	0.466056i	−0.434578	+	0.900634i	$0.643102\pi$
$654$		0			0
$655$	−0.981730	+	0.0745494i	−0.0383594	+	0.00291289i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			27.4244i			1.06830i	0.845389	+	0.534151i	$0.179369\pi$
$659$			27.4244i			1.06830i	−0.845389	+	0.534151i	$0.820631\pi$
$660$		0			0
$661$			39.1425i			1.52247i	0.648478	+	0.761233i	$0.275406\pi$
$661$			39.1425i			1.52247i	−0.648478	+	0.761233i	$0.724594\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.39030	+	2.91174i	0.131470	+	0.112912i
$666$		0			0
$667$	48.0463	+	48.0463i	1.86036	+	1.86036i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	4.98380i		−	0.192397i
$672$		0			0
$673$	23.5686	−	23.5686i	0.908503	−	0.908503i	−0.0876488	−	0.996151i	$-0.527935\pi$
$673$	23.5686	−	23.5686i	0.908503	−	0.908503i	0.996151	+	0.0876488i	$0.0279353\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	21.9738	−	21.9738i	0.844522	−	0.844522i	−0.144921	−	0.989443i	$-0.546293\pi$
$677$	21.9738	−	21.9738i	0.844522	−	0.844522i	0.989443	+	0.144921i	$0.0462928\pi$
$678$		0			0
$679$	10.4147			0.399679
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.94901	+	6.94901i	−0.265896	+	0.265896i	−0.827444	−	0.561548i	$-0.810206\pi$
$683$	−6.94901	+	6.94901i	−0.265896	+	0.265896i	0.561548	+	0.827444i	$0.310206\pi$
$684$		0			0
$685$	3.78584	+	3.25145i	0.144650	+	0.124232i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			3.95810i			0.150792i
$690$		0			0
$691$	16.5423			0.629299			0.314650	−	0.949208i	$-0.398113\pi$
$691$	16.5423			0.629299			0.314650	+	0.949208i	$0.398113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.13719	+	28.1443i	0.0810681	+	1.06757i
$696$		0			0
$697$	−12.7024	−	12.7024i	−0.481139	−	0.481139i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			35.1981i			1.32941i	0.747105	+	0.664707i	$0.231443\pi$
$701$			35.1981i			1.32941i	−0.747105	+	0.664707i	$0.768557\pi$
$702$		0			0
$703$	−15.1256	−	15.1256i	−0.570473	−	0.570473i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.83002	+	1.83002i	0.0688249	+	0.0688249i
$708$		0			0
$709$	−45.4352			−1.70636			−0.853178	−	0.521620i	$-0.825328\pi$
$709$	−45.4352			−1.70636			−0.853178	+	0.521620i	$0.825328\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.47596	−	3.47596i	0.130176	−	0.130176i
$714$		0			0
$715$	8.41338	+	7.22579i	0.314643	+	0.270229i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.1926			0.603880			0.301940	−	0.953327i	$-0.402366\pi$
$719$	16.1926			0.603880			0.301940	+	0.953327i	$0.402366\pi$
$720$		0			0
$721$	−9.71469			−0.361794
$722$		0			0
$723$		0			0
$724$		0			0
$725$	39.2243	−	5.99168i	1.45675	−	0.222525i
$726$		0			0
$727$	−10.5017	+	10.5017i	−0.389488	+	0.389488i	−0.874505	−	0.485017i	$-0.838813\pi$
$727$	−10.5017	+	10.5017i	−0.389488	+	0.389488i	0.485017	+	0.874505i	$0.338813\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−27.0120			−0.999076
$732$		0			0
$733$	10.0752	+	10.0752i	0.372134	+	0.372134i	0.868254	−	0.496120i	$-0.165242\pi$
$733$	10.0752	+	10.0752i	0.372134	+	0.372134i	−0.496120	+	0.868254i	$0.665242\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	10.5953	+	10.5953i	0.390281	+	0.390281i
$738$		0			0
$739$			26.4753i			0.973909i	0.873427	+	0.486955i	$0.161892\pi$
$739$			26.4753i			0.973909i	−0.873427	+	0.486955i	$0.838108\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.15658	−	6.15658i	−0.225863	−	0.225863i	0.585099	−	0.810962i	$-0.301056\pi$
$743$	−6.15658	−	6.15658i	−0.225863	−	0.225863i	−0.810962	+	0.585099i	$0.801056\pi$
$744$		0			0
$745$	−41.1073	+	3.12155i	−1.50605	+	0.114365i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.08265			0.112638
$750$		0			0
$751$		−	13.9490i		−	0.509008i	−0.967072	−	0.254504i	$-0.918088\pi$
$751$		−	13.9490i		−	0.509008i	0.967072	−	0.254504i	$-0.0819122\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.000633401	−	0.00834117i	−2.30518e−5	−	0.000303566i
$756$		0			0
$757$	7.61475	−	7.61475i	0.276763	−	0.276763i	−0.555053	−	0.831815i	$-0.687302\pi$
$757$	7.61475	−	7.61475i	0.276763	−	0.276763i	0.831815	+	0.555053i	$0.187302\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.57254			0.0932543			0.0466272	−	0.998912i	$-0.485153\pi$
$761$	2.57254			0.0932543			0.0466272	+	0.998912i	$0.485153\pi$
$762$		0			0
$763$	2.38429	−	2.38429i	0.0863171	−	0.0863171i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	19.1772	−	19.1772i	0.692448	−	0.692448i
$768$		0			0
$769$			5.28253i			0.190493i	0.995454	+	0.0952465i	$0.0303639\pi$
$769$			5.28253i			0.190493i	−0.995454	+	0.0952465i	$0.969636\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.4982	+	19.4982i	0.701302	+	0.701302i	0.964690	−	0.263388i	$-0.0848398\pi$
$773$	19.4982	+	19.4982i	0.701302	+	0.701302i	−0.263388	+	0.964690i	$0.584840\pi$
$774$		0			0
$775$	−0.433475	−	2.83773i	−0.0155709	−	0.101934i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			6.56458i			0.235201i
$780$		0			0
$781$			10.6325i			0.380461i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−23.3125	+	27.1440i	−0.832060	+	0.968812i
$786$		0			0
$787$	3.02573	+	3.02573i	0.107856	+	0.107856i	0.758975	−	0.651120i	$-0.225700\pi$
$787$	3.02573	+	3.02573i	0.107856	+	0.107856i	−0.651120	+	0.758975i	$0.725700\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			4.05363i			0.144130i
$792$		0			0
$793$	3.88935	−	3.88935i	0.138115	−	0.138115i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.35305	−	5.35305i	0.189615	−	0.189615i	−0.605915	−	0.795529i	$-0.707193\pi$
$797$	5.35305	−	5.35305i	0.189615	−	0.189615i	0.795529	+	0.605915i	$0.207193\pi$
$798$		0			0
$799$	42.7934			1.51392
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13.1172	−	13.1172i	0.462897	−	0.462897i
$804$		0			0
$805$	−17.4249	+	1.32319i	−0.614148	+	0.0466364i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			25.2432i			0.887504i	0.896150	+	0.443752i	$0.146353\pi$
$809$			25.2432i			0.887504i	−0.896150	+	0.443752i	$0.853647\pi$
$810$		0			0
$811$	1.16655			0.0409632			0.0204816	−	0.999790i	$-0.493480\pi$
$811$	1.16655			0.0409632			0.0204816	+	0.999790i	$0.493480\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	11.9383	−	13.9004i	0.418179	−	0.486908i
$816$		0			0
$817$	6.97987	+	6.97987i	0.244195	+	0.244195i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			13.4087i			0.467968i	0.972240	+	0.233984i	$0.0751763\pi$
$821$			13.4087i			0.467968i	−0.972240	+	0.233984i	$0.924824\pi$
$822$		0			0
$823$	1.36211	+	1.36211i	0.0474803	+	0.0474803i	0.730448	−	0.682968i	$-0.239311\pi$
$823$	1.36211	+	1.36211i	0.0474803	+	0.0474803i	−0.682968	+	0.730448i	$0.739311\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.7402	+	25.7402i	0.895073	+	0.895073i	0.994995	−	0.0999219i	$-0.0318593\pi$
$827$	25.7402	+	25.7402i	0.895073	+	0.895073i	−0.0999219	+	0.994995i	$0.531859\pi$
$828$		0			0
$829$	11.4288			0.396940			0.198470	−	0.980107i	$-0.436403\pi$
$829$	11.4288			0.396940			0.198470	+	0.980107i	$0.436403\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26.1289	+	26.1289i	−0.905314	+	0.905314i
$834$		0			0
$835$	4.56196	−	5.31174i	0.157873	−	0.183820i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	47.7970			1.65014			0.825069	−	0.565032i	$-0.191136\pi$
$839$	47.7970			1.65014			0.825069	+	0.565032i	$0.191136\pi$
$840$		0			0
$841$	33.9777			1.17165
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.27427	−	16.7807i	−0.0438362	−	0.577272i
$846$		0			0
$847$	4.19907	−	4.19907i	0.144282	−	0.144282i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	83.6436			2.86727
$852$		0			0
$853$	22.3165	+	22.3165i	0.764103	+	0.764103i	0.977061	−	0.212959i	$-0.0683099\pi$
$853$	22.3165	+	22.3165i	0.764103	+	0.764103i	−0.212959	+	0.977061i	$0.568310\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.47209	−	8.47209i	−0.289401	−	0.289401i	0.547442	−	0.836843i	$-0.315602\pi$
$857$	−8.47209	−	8.47209i	−0.289401	−	0.289401i	−0.836843	+	0.547442i	$0.815602\pi$
$858$		0			0
$859$		−	37.2555i		−	1.27114i	−0.772042	−	0.635571i	$-0.780765\pi$
$859$		−	37.2555i		−	1.27114i	0.772042	−	0.635571i	$-0.219235\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.5386	+	10.5386i	0.358738	+	0.358738i	0.863348	−	0.504610i	$-0.168364\pi$
$863$	10.5386	+	10.5386i	0.358738	+	0.358738i	−0.504610	+	0.863348i	$0.668364\pi$
$864$		0			0
$865$	16.8410	+	14.4638i	0.572611	+	0.491784i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−21.5593			−0.731351
$870$		0			0
$871$			16.5370i			0.560336i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.40360	+	8.65684i	−0.182675	+	0.292654i
$876$		0			0
$877$	16.9202	−	16.9202i	0.571353	−	0.571353i	−0.361153	−	0.932506i	$-0.617617\pi$
$877$	16.9202	−	16.9202i	0.571353	−	0.571353i	0.932506	+	0.361153i	$0.117617\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	34.6268			1.16661			0.583304	−	0.812254i	$-0.301760\pi$
$881$	34.6268			1.16661			0.583304	+	0.812254i	$0.301760\pi$
$882$		0			0
$883$	23.6485	−	23.6485i	0.795835	−	0.795835i	−0.186601	−	0.982436i	$-0.559747\pi$
$883$	23.6485	−	23.6485i	0.795835	−	0.795835i	0.982436	+	0.186601i	$0.0597472\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.5275	+	28.5275i	−0.957859	+	0.957859i	−0.999147	−	0.0412884i	$-0.986854\pi$
$887$	−28.5275	+	28.5275i	−0.957859	+	0.957859i	0.0412884	+	0.999147i	$0.486854\pi$
$888$		0			0
$889$			15.7222i			0.527306i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−11.0578	−	11.0578i	−0.370034	−	0.370034i
$894$		0			0
$895$	2.23772	+	29.4683i	0.0747989	+	0.985015i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	4.55620i		−	0.151958i
$900$		0			0
$901$		−	10.1371i		−	0.337714i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.22715	−	16.1602i	−0.0407919	−	0.537183i
$906$		0			0
$907$	−15.0125	−	15.0125i	−0.498481	−	0.498481i	0.412484	−	0.910965i	$-0.364661\pi$
$907$	−15.0125	−	15.0125i	−0.498481	−	0.498481i	−0.910965	+	0.412484i	$0.864661\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	17.5986i		−	0.583068i	−0.956560	−	0.291534i	$-0.905834\pi$
$911$		−	17.5986i		−	0.583068i	0.956560	−	0.291534i	$-0.0941657\pi$
$912$		0			0
$913$	−24.9416	+	24.9416i	−0.825445	+	0.825445i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.284180	−	0.284180i	0.00938446	−	0.00938446i
$918$		0			0
$919$	5.93406			0.195747			0.0978733	−	0.995199i	$-0.468796\pi$
$919$	5.93406			0.195747			0.0978733	+	0.995199i	$0.468796\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.29759	+	8.29759i	−0.273119	+	0.273119i
$924$		0			0
$925$	28.9273	−	39.3581i	0.951122	−	1.29409i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	43.1598i		−	1.41603i	−0.706198	−	0.708014i	$-0.749591\pi$
$929$		−	43.1598i		−	1.41603i	0.706198	−	0.708014i	$-0.250409\pi$
$930$		0			0
$931$	13.5034			0.442555
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−21.5474	−	18.5059i	−0.704676	−	0.605208i
$936$		0			0
$937$	12.4605	+	12.4605i	0.407068	+	0.407068i	0.880715	−	0.473647i	$-0.157063\pi$
$937$	12.4605	+	12.4605i	0.407068	+	0.407068i	−0.473647	+	0.880715i	$0.657063\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	18.6088i		−	0.606630i	−0.952890	−	0.303315i	$-0.901907\pi$
$941$		−	18.6088i		−	0.606630i	0.952890	−	0.303315i	$-0.0980934\pi$
$942$		0			0
$943$	−18.1508	−	18.1508i	−0.591073	−	0.591073i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.1917	−	14.1917i	−0.461168	−	0.461168i	0.437870	−	0.899038i	$-0.355733\pi$
$947$	−14.1917	−	14.1917i	−0.461168	−	0.461168i	−0.899038	+	0.437870i	$0.855733\pi$
$948$		0			0
$949$	20.4733			0.664591
$950$		0			0
$951$		0			0
$952$		0			0
$953$	25.0919	−	25.0919i	0.812806	−	0.812806i	−0.172248	−	0.985054i	$-0.555103\pi$
$953$	25.0919	−	25.0919i	0.812806	−	0.812806i	0.985054	+	0.172248i	$0.0551029\pi$
$954$		0			0
$955$	0.780324	+	10.2760i	0.0252507	+	0.332523i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.03708			−0.0657807
$960$		0			0
$961$	30.6704			0.989367
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.93496	−	6.91039i	0.191053	−	0.222453i
$966$		0			0
$967$	−41.2729	+	41.2729i	−1.32725	+	1.32725i	−0.419482	+	0.907764i	$0.637788\pi$
$967$	−41.2729	+	41.2729i	−1.32725	+	1.32725i	−0.907764	+	0.419482i	$0.862212\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−47.1256			−1.51233			−0.756166	−	0.654380i	$-0.772930\pi$
$971$	−47.1256			−1.51233			−0.756166	+	0.654380i	$0.772930\pi$
$972$		0			0
$973$	−8.14690	−	8.14690i	−0.261178	−	0.261178i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.70084	−	2.70084i	−0.0864074	−	0.0864074i	0.662582	−	0.748989i	$-0.269461\pi$
$977$	−2.70084	−	2.70084i	−0.0864074	−	0.0864074i	−0.748989	+	0.662582i	$0.769461\pi$
$978$		0			0
$979$		−	18.3474i		−	0.586386i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	17.7831	+	17.7831i	0.567192	+	0.567192i	0.931341	−	0.364149i	$-0.118640\pi$
$983$	17.7831	+	17.7831i	0.567192	+	0.567192i	−0.364149	+	0.931341i	$0.618640\pi$
$984$		0			0
$985$	−27.5276	+	32.0518i	−0.877101	+	1.02126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−38.5982			−1.22735
$990$		0			0
$991$		−	20.7223i		−	0.658267i	−0.944283	−	0.329134i	$-0.893243\pi$
$991$		−	20.7223i		−	0.658267i	0.944283	−	0.329134i	$-0.106757\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−47.4948	+	3.60660i	−1.50569	+	0.114337i
$996$		0			0
$997$	−12.9685	+	12.9685i	−0.410715	+	0.410715i	−0.881988	−	0.471273i	$-0.843795\pi$
$997$	−12.9685	+	12.9685i	−0.410715	+	0.410715i	0.471273	+	0.881988i	$0.343795\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.bi.e.847.12		24
3.2	odd	2		480.2.bh.a.367.1		24
4.3	odd	2		360.2.w.e.307.1		24
5.3	odd	4	inner	1440.2.bi.e.1423.1		24
8.3	odd	2	inner	1440.2.bi.e.847.1		24
8.5	even	2		360.2.w.e.307.6		24
12.11	even	2		120.2.v.a.67.12	yes	24
15.2	even	4		2400.2.bh.b.943.11		24
15.8	even	4		480.2.bh.a.463.6		24
15.14	odd	2		2400.2.bh.b.1807.12		24
20.3	even	4		360.2.w.e.163.6		24
24.5	odd	2		120.2.v.a.67.7	yes	24
24.11	even	2		480.2.bh.a.367.6		24
40.3	even	4	inner	1440.2.bi.e.1423.12		24
40.13	odd	4		360.2.w.e.163.1		24
60.23	odd	4		120.2.v.a.43.7	✓	24
60.47	odd	4		600.2.v.b.43.6		24
60.59	even	2		600.2.v.b.307.1		24
120.29	odd	2		600.2.v.b.307.6		24
120.53	even	4		120.2.v.a.43.12	yes	24
120.59	even	2		2400.2.bh.b.1807.11		24
120.77	even	4		600.2.v.b.43.1		24
120.83	odd	4		480.2.bh.a.463.1		24
120.107	odd	4		2400.2.bh.b.943.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.v.a.43.7	✓	24	60.23	odd	4
120.2.v.a.43.12	yes	24	120.53	even	4
120.2.v.a.67.7	yes	24	24.5	odd	2
120.2.v.a.67.12	yes	24	12.11	even	2
360.2.w.e.163.1		24	40.13	odd	4
360.2.w.e.163.6		24	20.3	even	4
360.2.w.e.307.1		24	4.3	odd	2
360.2.w.e.307.6		24	8.5	even	2
480.2.bh.a.367.1		24	3.2	odd	2
480.2.bh.a.367.6		24	24.11	even	2
480.2.bh.a.463.1		24	120.83	odd	4
480.2.bh.a.463.6		24	15.8	even	4
600.2.v.b.43.1		24	120.77	even	4
600.2.v.b.43.6		24	60.47	odd	4
600.2.v.b.307.1		24	60.59	even	2
600.2.v.b.307.6		24	120.29	odd	2
1440.2.bi.e.847.1		24	8.3	odd	2	inner
1440.2.bi.e.847.12		24	1.1	even	1	trivial
1440.2.bi.e.1423.1		24	5.3	odd	4	inner
1440.2.bi.e.1423.12		24	40.3	even	4	inner
2400.2.bh.b.943.11		24	15.2	even	4
2400.2.bh.b.943.12		24	120.107	odd	4
2400.2.bh.b.1807.11		24	120.59	even	2
2400.2.bh.b.1807.12		24	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.bi (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.22965 - 0.169312i) q^{5} +(-0.645414 + 0.645414i) q^{7} -2.11990 q^{11} +(-1.65437 - 1.65437i) q^{13} +(4.23698 + 4.23698i) q^{17} -2.18966i q^{19} +(6.05433 + 6.05433i) q^{23} +(4.94267 - 0.755014i) q^{25} +7.93585 q^{29} -0.574128i q^{31} +(-1.32977 + 1.54832i) q^{35} +(6.90775 - 6.90775i) q^{37} -2.99799 q^{41} +(-3.18765 + 3.18765i) q^{43} +(5.04999 - 5.04999i) q^{47} +6.16688i q^{49} +(-1.19626 - 1.19626i) q^{53} +(-4.72664 + 0.358925i) q^{55} +11.5919i q^{59} +2.35096i q^{61} +(-3.96876 - 3.40855i) q^{65} +(-4.99799 - 4.99799i) q^{67} -5.01557i q^{71} +(-6.18765 + 6.18765i) q^{73} +(1.36821 - 1.36821i) q^{77} +10.1700 q^{79} +(11.7654 - 11.7654i) q^{83} +(10.1643 + 8.72960i) q^{85} +8.65485i q^{89} +2.13550 q^{91} +(-0.370736 - 4.88217i) q^{95} +(-8.06823 - 8.06823i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 8 q^{17} - 8 q^{25} - 48 q^{35} + 32 q^{43} + 8 q^{65} - 48 q^{67} - 40 q^{73} + 80 q^{83} - 64 q^{91} - 24 q^{97}+O(q^{100}) \) 24 * q - 8 * q^17 - 8 * q^25 - 48 * q^35 + 32 * q^43 + 8 * q^65 - 48 * q^67 - 40 * q^73 + 80 * q^83 - 64 * q^91 - 24 * q^97

Introduction

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