LMFDB - Embedded newform 2720.2.l.b.2481.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2720,2,Mod(2481,2720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2720.2481"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2720 = 2^{5} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2720.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [36,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$21.7193093498$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 680)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2481.5
Character	$\chi$	$=$	2720.2481
Dual form			2720.2.l.b.2481.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54711 q^{3} +1.00000 q^{5} +0.654431i q^{7} +3.48778 q^{9} -2.31821 q^{11} +0.495577i q^{13} -2.54711 q^{15} +(2.19658 - 3.48928i) q^{17} +2.57745i q^{19} -1.66691i q^{21} +0.188252i q^{23} +1.00000 q^{25} -1.24244 q^{27} -6.85794 q^{29} -3.96966i q^{31} +5.90475 q^{33} +0.654431i q^{35} -3.06408 q^{37} -1.26229i q^{39} +3.21537i q^{41} -3.55727i q^{43} +3.48778 q^{45} +3.43065 q^{47} +6.57172 q^{49} +(-5.59493 + 8.88758i) q^{51} +8.47932i q^{53} -2.31821 q^{55} -6.56507i q^{57} -0.432821i q^{59} -3.55435 q^{61} +2.28252i q^{63} +0.495577i q^{65} -8.66134i q^{67} -0.479499i q^{69} +9.72787i q^{71} +3.67345i q^{73} -2.54711 q^{75} -1.51711i q^{77} -4.08394i q^{79} -7.29871 q^{81} -15.3164i q^{83} +(2.19658 - 3.48928i) q^{85} +17.4679 q^{87} +8.98386 q^{89} -0.324321 q^{91} +10.1112i q^{93} +2.57745i q^{95} -11.6566i q^{97} -8.08543 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 4 q^{3} + 36 q^{5} + 36 q^{9} + 8 q^{11} + 4 q^{15} + 36 q^{25} + 16 q^{27} - 8 q^{33} + 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} + 4 q^{75} + 44 q^{81} - 24 q^{87} + 56 q^{91} + 40 q^{99}+O(q^{100}) $ 36 * q + 4 * q^3 + 36 * q^5 + 36 * q^9 + 8 * q^11 + 4 * q^15 + 36 * q^25 + 16 * q^27 - 8 * q^33 + 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 + 4 * q^75 + 44 * q^81 - 24 * q^87 + 56 * q^91 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2720\mathbb{Z}\right)^\times$.

$n$	$511$	$1601$	$1701$	$2177$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−2.54711			−1.47058			−0.735288	−	0.677755i	$-0.762953\pi$
$3$	−2.54711			−1.47058			−0.735288	+	0.677755i	$0.762953\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			0.654431i			0.247352i	0.992323	+	0.123676i	$0.0394683\pi$
$7$			0.654431i			0.247352i	−0.992323	+	0.123676i	$0.960532\pi$
$8$		0			0
$9$	3.48778			1.16259
$10$		0			0
$11$	−2.31821			−0.698967			−0.349484	−	0.936942i	$-0.613643\pi$
$11$	−2.31821			−0.698967			−0.349484	+	0.936942i	$0.613643\pi$
$12$		0			0
$13$			0.495577i			0.137448i	0.997636	+	0.0687241i	$0.0218928\pi$
$13$			0.495577i			0.137448i	−0.997636	+	0.0687241i	$0.978107\pi$
$14$		0			0
$15$	−2.54711			−0.657662
$16$		0			0
$17$	2.19658	−	3.48928i	0.532748	−	0.846274i
$17$	2.19658	−	3.48928i	0.532748	−	0.846274i
$18$		0			0
$19$			2.57745i			0.591308i	0.955295	+	0.295654i	$0.0955376\pi$
$19$			2.57745i			0.591308i	−0.955295	+	0.295654i	$0.904462\pi$
$20$		0			0
$21$		−	1.66691i		−	0.363750i
$22$		0			0
$23$			0.188252i			0.0392532i	0.999807	+	0.0196266i	$0.00624775\pi$
$23$			0.188252i			0.0392532i	−0.999807	+	0.0196266i	$0.993752\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.24244			−0.239108
$28$		0			0
$29$	−6.85794			−1.27349			−0.636743	−	0.771076i	$-0.719719\pi$
$29$	−6.85794			−1.27349			−0.636743	+	0.771076i	$0.719719\pi$
$30$		0			0
$31$		−	3.96966i		−	0.712972i	−0.934301	−	0.356486i	$-0.883975\pi$
$31$		−	3.96966i		−	0.712972i	0.934301	−	0.356486i	$-0.116025\pi$
$32$		0			0
$33$	5.90475			1.02789
$34$		0			0
$35$			0.654431i			0.110619i
$36$		0			0
$37$	−3.06408			−0.503732			−0.251866	−	0.967762i	$-0.581044\pi$
$37$	−3.06408			−0.503732			−0.251866	+	0.967762i	$0.581044\pi$
$38$		0			0
$39$		−	1.26229i		−	0.202128i
$40$		0			0
$41$			3.21537i			0.502156i	0.967967	+	0.251078i	$0.0807851\pi$
$41$			3.21537i			0.502156i	−0.967967	+	0.251078i	$0.919215\pi$
$42$		0			0
$43$		−	3.55727i		−	0.542479i	−0.962512	−	0.271239i	$-0.912566\pi$
$43$		−	3.55727i		−	0.542479i	0.962512	−	0.271239i	$-0.0874335\pi$
$44$		0			0
$45$	3.48778			0.519928
$46$		0			0
$47$	3.43065			0.500411			0.250206	−	0.968193i	$-0.419502\pi$
$47$	3.43065			0.500411			0.250206	+	0.968193i	$0.419502\pi$
$48$		0			0
$49$	6.57172			0.938817
$50$		0			0
$51$	−5.59493	+	8.88758i	−0.783447	+	1.24451i
$52$		0			0
$53$			8.47932i			1.16472i	0.812930	+	0.582362i	$0.197871\pi$
$53$			8.47932i			1.16472i	−0.812930	+	0.582362i	$0.802129\pi$
$54$		0			0
$55$	−2.31821			−0.312588
$56$		0			0
$57$		−	6.56507i		−	0.869564i
$58$		0			0
$59$		−	0.432821i		−	0.0563485i	−0.999603	−	0.0281743i	$-0.991031\pi$
$59$		−	0.432821i		−	0.0563485i	0.999603	−	0.0281743i	$-0.00896933\pi$
$60$		0			0
$61$	−3.55435			−0.455089			−0.227544	−	0.973768i	$-0.573070\pi$
$61$	−3.55435			−0.455089			−0.227544	+	0.973768i	$0.573070\pi$
$62$		0			0
$63$			2.28252i			0.287570i
$64$		0			0
$65$			0.495577i			0.0614687i
$66$		0			0
$67$		−	8.66134i		−	1.05815i	−0.848575	−	0.529076i	$-0.822539\pi$
$67$		−	8.66134i		−	1.05815i	0.848575	−	0.529076i	$-0.177461\pi$
$68$		0			0
$69$		−	0.479499i		−	0.0577249i
$70$		0			0
$71$			9.72787i			1.15449i	0.816573	+	0.577243i	$0.195871\pi$
$71$			9.72787i			1.15449i	−0.816573	+	0.577243i	$0.804129\pi$
$72$		0			0
$73$			3.67345i			0.429945i	0.976620	+	0.214973i	$0.0689662\pi$
$73$			3.67345i			0.429945i	−0.976620	+	0.214973i	$0.931034\pi$
$74$		0			0
$75$	−2.54711			−0.294115
$76$		0			0
$77$		−	1.51711i		−	0.172891i
$78$		0			0
$79$		−	4.08394i		−	0.459480i	−0.973252	−	0.229740i	$-0.926212\pi$
$79$		−	4.08394i		−	0.459480i	0.973252	−	0.229740i	$-0.0737875\pi$
$80$		0			0
$81$	−7.29871			−0.810968
$82$		0			0
$83$		−	15.3164i		−	1.68119i	−0.541665	−	0.840595i	$-0.682206\pi$
$83$		−	15.3164i		−	1.68119i	0.541665	−	0.840595i	$-0.317794\pi$
$84$		0			0
$85$	2.19658	−	3.48928i	0.238252	−	0.378465i
$86$		0			0
$87$	17.4679			1.87276
$88$		0			0
$89$	8.98386			0.952287			0.476144	−	0.879368i	$-0.342034\pi$
$89$	8.98386			0.952287			0.476144	+	0.879368i	$0.342034\pi$
$90$		0			0
$91$	−0.324321			−0.0339981
$92$		0			0
$93$			10.1112i			1.04848i
$94$		0			0
$95$			2.57745i			0.264441i
$96$		0			0
$97$		−	11.6566i		−	1.18355i	−0.806102	−	0.591776i	$-0.798427\pi$
$97$		−	11.6566i		−	1.18355i	0.806102	−	0.591776i	$-0.201573\pi$
$98$		0			0
$99$	−8.08543			−0.812616
$100$		0			0
$101$			7.75678i			0.771828i	0.922535	+	0.385914i	$0.126114\pi$
$101$			7.75678i			0.771828i	−0.922535	+	0.385914i	$0.873886\pi$
$102$		0			0
$103$	−5.75307			−0.566866			−0.283433	−	0.958992i	$-0.591473\pi$
$103$	−5.75307			−0.566866			−0.283433	+	0.958992i	$0.591473\pi$
$104$		0			0
$105$		−	1.66691i		−	0.162674i
$106$		0			0
$107$	11.3343			1.09572			0.547862	−	0.836569i	$-0.315442\pi$
$107$	11.3343			1.09572			0.547862	+	0.836569i	$0.315442\pi$
$108$		0			0
$109$	−2.73908			−0.262357			−0.131178	−	0.991359i	$-0.541876\pi$
$109$	−2.73908			−0.262357			−0.131178	+	0.991359i	$0.541876\pi$
$110$		0			0
$111$	7.80456			0.740776
$112$		0			0
$113$		−	13.9883i		−	1.31591i	−0.753059	−	0.657953i	$-0.771423\pi$
$113$		−	13.9883i		−	1.31591i	0.753059	−	0.657953i	$-0.228577\pi$
$114$		0			0
$115$			0.188252i			0.0175546i
$116$		0			0
$117$			1.72846i			0.159797i
$118$		0			0
$119$	2.28349	+	1.43751i	0.209327	+	0.131776i
$120$		0			0
$121$	−5.62589			−0.511445
$122$		0			0
$123$		−	8.18990i		−	0.738459i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	16.7241			1.48403			0.742013	−	0.670385i	$-0.233871\pi$
$127$	16.7241			1.48403			0.742013	+	0.670385i	$0.233871\pi$
$128$		0			0
$129$			9.06077i			0.797757i
$130$		0			0
$131$	1.25462			0.109617			0.0548084	−	0.998497i	$-0.482545\pi$
$131$	1.25462			0.109617			0.0548084	+	0.998497i	$0.482545\pi$
$132$		0			0
$133$	−1.68677			−0.146261
$134$		0			0
$135$	−1.24244			−0.106932
$136$		0			0
$137$	15.8578			1.35483			0.677413	−	0.735603i	$-0.263101\pi$
$137$	15.8578			1.35483			0.677413	+	0.735603i	$0.263101\pi$
$138$		0			0
$139$	−5.49102			−0.465742			−0.232871	−	0.972508i	$-0.574812\pi$
$139$	−5.49102			−0.465742			−0.232871	+	0.972508i	$0.574812\pi$
$140$		0			0
$141$	−8.73824			−0.735893
$142$		0			0
$143$		−	1.14885i		−	0.0960718i
$144$		0			0
$145$	−6.85794			−0.569521
$146$		0			0
$147$	−16.7389			−1.38060
$148$		0			0
$149$		−	3.99288i		−	0.327109i	−0.986534	−	0.163555i	$-0.947704\pi$
$149$		−	3.99288i		−	0.327109i	0.986534	−	0.163555i	$-0.0522960\pi$
$150$		0			0
$151$	6.42385			0.522765			0.261383	−	0.965235i	$-0.415822\pi$
$151$	6.42385			0.522765			0.261383	+	0.965235i	$0.415822\pi$
$152$		0			0
$153$	7.66119	−	12.1698i	0.619370	−	0.983874i
$154$		0			0
$155$		−	3.96966i		−	0.318851i
$156$		0			0
$157$		−	22.9114i		−	1.82853i	−0.405116	−	0.914265i	$-0.632769\pi$
$157$		−	22.9114i		−	1.82853i	0.405116	−	0.914265i	$-0.367231\pi$
$158$		0			0
$159$		−	21.5978i		−	1.71282i
$160$		0			0
$161$	−0.123198			−0.00970936
$162$		0			0
$163$	16.1246			1.26297			0.631487	−	0.775387i	$-0.282445\pi$
$163$	16.1246			1.26297			0.631487	+	0.775387i	$0.282445\pi$
$164$		0			0
$165$	5.90475			0.459684
$166$		0			0
$167$		−	13.2324i		−	1.02395i	−0.859000	−	0.511976i	$-0.828914\pi$
$167$		−	13.2324i		−	1.02395i	0.859000	−	0.511976i	$-0.171086\pi$
$168$		0			0
$169$	12.7544			0.981108
$170$		0			0
$171$			8.98960i			0.687452i
$172$		0			0
$173$	21.3027			1.61961			0.809806	−	0.586697i	$-0.199572\pi$
$173$	21.3027			1.61961			0.809806	+	0.586697i	$0.199572\pi$
$174$		0			0
$175$			0.654431i			0.0494704i
$176$		0			0
$177$			1.10244i			0.0828648i
$178$		0			0
$179$		−	11.4748i		−	0.857667i	−0.903384	−	0.428833i	$-0.858925\pi$
$179$		−	11.4748i		−	0.857667i	0.903384	−	0.428833i	$-0.141075\pi$
$180$		0			0
$181$	8.44241			0.627519			0.313760	−	0.949502i	$-0.398411\pi$
$181$	8.44241			0.627519			0.313760	+	0.949502i	$0.398411\pi$
$182$		0			0
$183$	9.05334			0.669242
$184$		0			0
$185$	−3.06408			−0.225276
$186$		0			0
$187$	−5.09213	+	8.08888i	−0.372374	+	0.591518i
$188$		0			0
$189$		−	0.813094i		−	0.0591439i
$190$		0			0
$191$	−26.7666			−1.93676			−0.968382	−	0.249473i	$-0.919743\pi$
$191$	−26.7666			−1.93676			−0.968382	+	0.249473i	$0.919743\pi$
$192$		0			0
$193$		−	19.5588i		−	1.40788i	−0.710261	−	0.703938i	$-0.751423\pi$
$193$		−	19.5588i		−	1.40788i	0.710261	−	0.703938i	$-0.248577\pi$
$194$		0			0
$195$		−	1.26229i		−	0.0903944i
$196$		0			0
$197$	8.04276			0.573023			0.286512	−	0.958077i	$-0.407504\pi$
$197$	8.04276			0.573023			0.286512	+	0.958077i	$0.407504\pi$
$198$		0			0
$199$		−	23.3064i		−	1.65215i	−0.563561	−	0.826075i	$-0.690569\pi$
$199$		−	23.3064i		−	1.65215i	0.563561	−	0.826075i	$-0.309431\pi$
$200$		0			0
$201$			22.0614i			1.55609i
$202$		0			0
$203$		−	4.48805i		−	0.314999i
$204$		0			0
$205$			3.21537i			0.224571i
$206$		0			0
$207$			0.656582i			0.0456356i
$208$		0			0
$209$		−	5.97509i		−	0.413305i
$210$		0			0
$211$	21.8256			1.50254			0.751268	−	0.659998i	$-0.229443\pi$
$211$	21.8256			1.50254			0.751268	+	0.659998i	$0.229443\pi$
$212$		0			0
$213$		−	24.7780i		−	1.69776i
$214$		0			0
$215$		−	3.55727i		−	0.242604i
$216$		0			0
$217$	2.59787			0.176355
$218$		0			0
$219$		−	9.35670i		−	0.632267i
$220$		0			0
$221$	1.72920	+	1.08857i	0.116319	+	0.0732253i
$222$		0			0
$223$	14.9113			0.998536			0.499268	−	0.866448i	$-0.333602\pi$
$223$	14.9113			0.998536			0.499268	+	0.866448i	$0.333602\pi$
$224$		0			0
$225$	3.48778			0.232519
$226$		0			0
$227$	6.36332			0.422348			0.211174	−	0.977448i	$-0.432271\pi$
$227$	6.36332			0.422348			0.211174	+	0.977448i	$0.432271\pi$
$228$		0			0
$229$		−	6.95553i		−	0.459634i	−0.973234	−	0.229817i	$-0.926187\pi$
$229$		−	6.95553i		−	0.459634i	0.973234	−	0.229817i	$-0.0738128\pi$
$230$		0			0
$231$			3.86425i			0.254249i
$232$		0			0
$233$		−	4.82255i		−	0.315936i	−0.987444	−	0.157968i	$-0.949506\pi$
$233$		−	4.82255i		−	0.315936i	0.987444	−	0.157968i	$-0.0504942\pi$
$234$		0			0
$235$	3.43065			0.223791
$236$		0			0
$237$			10.4023i			0.675700i
$238$		0			0
$239$	−2.02631			−0.131071			−0.0655355	−	0.997850i	$-0.520876\pi$
$239$	−2.02631			−0.131071			−0.0655355	+	0.997850i	$0.520876\pi$
$240$		0			0
$241$		−	5.62463i		−	0.362315i	−0.983454	−	0.181157i	$-0.942016\pi$
$241$		−	5.62463i		−	0.362315i	0.983454	−	0.181157i	$-0.0579843\pi$
$242$		0			0
$243$	22.3180			1.43170
$244$		0			0
$245$	6.57172			0.419852
$246$		0			0
$247$	−1.27733			−0.0812743
$248$		0			0
$249$			39.0125i			2.47232i
$250$		0			0
$251$		−	21.2370i		−	1.34047i	−0.742151	−	0.670233i	$-0.766194\pi$
$251$		−	21.2370i		−	1.34047i	0.742151	−	0.670233i	$-0.233806\pi$
$252$		0			0
$253$		−	0.436408i		−	0.0274367i
$254$		0			0
$255$	−5.59493	+	8.88758i	−0.350368	+	0.556562i
$256$		0			0
$257$	−3.56840			−0.222591			−0.111295	−	0.993787i	$-0.535500\pi$
$257$	−3.56840			−0.222591			−0.111295	+	0.993787i	$0.535500\pi$
$258$		0			0
$259$		−	2.00523i		−	0.124599i
$260$		0			0
$261$	−23.9190			−1.48055
$262$		0			0
$263$	−1.31194			−0.0808976			−0.0404488	−	0.999182i	$-0.512879\pi$
$263$	−1.31194			−0.0808976			−0.0404488	+	0.999182i	$0.512879\pi$
$264$		0			0
$265$			8.47932i			0.520880i
$266$		0			0
$267$	−22.8829			−1.40041
$268$		0			0
$269$	22.5763			1.37650			0.688249	−	0.725474i	$-0.258380\pi$
$269$	22.5763			1.37650			0.688249	+	0.725474i	$0.258380\pi$
$270$		0			0
$271$	−7.24361			−0.440018			−0.220009	−	0.975498i	$-0.570609\pi$
$271$	−7.24361			−0.440018			−0.220009	+	0.975498i	$0.570609\pi$
$272$		0			0
$273$	0.826082			0.0499967
$274$		0			0
$275$	−2.31821			−0.139793
$276$		0			0
$277$	−15.4507			−0.928342			−0.464171	−	0.885746i	$-0.653648\pi$
$277$	−15.4507			−0.928342			−0.464171	+	0.885746i	$0.653648\pi$
$278$		0			0
$279$		−	13.8453i		−	0.828897i
$280$		0			0
$281$	−19.5260			−1.16482			−0.582412	−	0.812894i	$-0.697891\pi$
$281$	−19.5260			−1.16482			−0.582412	+	0.812894i	$0.697891\pi$
$282$		0			0
$283$	21.5407			1.28046			0.640230	−	0.768183i	$-0.278839\pi$
$283$	21.5407			1.28046			0.640230	+	0.768183i	$0.278839\pi$
$284$		0			0
$285$		−	6.56507i		−	0.388881i
$286$		0			0
$287$	−2.10424			−0.124209
$288$		0			0
$289$	−7.35010	−	15.3289i	−0.432359	−	0.901702i
$290$		0			0
$291$			29.6908i			1.74050i
$292$		0			0
$293$		−	21.2734i		−	1.24280i	−0.783492	−	0.621401i	$-0.786564\pi$
$293$		−	21.2734i		−	1.24280i	0.783492	−	0.621401i	$-0.213436\pi$
$294$		0			0
$295$		−	0.432821i		−	0.0251998i
$296$		0			0
$297$	2.88025			0.167129
$298$		0			0
$299$	−0.0932932			−0.00539529
$300$		0			0
$301$	2.32799			0.134183
$302$		0			0
$303$		−	19.7574i		−	1.13503i
$304$		0			0
$305$	−3.55435			−0.203522
$306$		0			0
$307$			24.5694i			1.40225i	0.713037	+	0.701126i	$0.247319\pi$
$307$			24.5694i			1.40225i	−0.713037	+	0.701126i	$0.752681\pi$
$308$		0			0
$309$	14.6537			0.833620
$310$		0			0
$311$			31.0746i			1.76208i	0.473044	+	0.881039i	$0.343155\pi$
$311$			31.0746i			1.76208i	−0.473044	+	0.881039i	$0.656845\pi$
$312$		0			0
$313$			15.7952i			0.892798i	0.894834	+	0.446399i	$0.147294\pi$
$313$			15.7952i			0.892798i	−0.894834	+	0.446399i	$0.852706\pi$
$314$		0			0
$315$			2.28252i			0.128605i
$316$		0			0
$317$	−3.40672			−0.191341			−0.0956704	−	0.995413i	$-0.530499\pi$
$317$	−3.40672			−0.191341			−0.0956704	+	0.995413i	$0.530499\pi$
$318$		0			0
$319$	15.8982			0.890126
$320$		0			0
$321$	−28.8696			−1.61135
$322$		0			0
$323$	8.99345	+	5.66158i	0.500409	+	0.315018i
$324$		0			0
$325$			0.495577i			0.0274896i
$326$		0			0
$327$	6.97676			0.385816
$328$		0			0
$329$			2.24512i			0.123778i
$330$		0			0
$331$			17.2530i			0.948310i	0.880441	+	0.474155i	$0.157246\pi$
$331$			17.2530i			0.948310i	−0.880441	+	0.474155i	$0.842754\pi$
$332$		0			0
$333$	−10.6869			−0.585636
$334$		0			0
$335$		−	8.66134i		−	0.473220i
$336$		0			0
$337$		−	13.8486i		−	0.754383i	−0.926135	−	0.377192i	$-0.876890\pi$
$337$		−	13.8486i		−	0.754383i	0.926135	−	0.377192i	$-0.123110\pi$
$338$		0			0
$339$			35.6297i			1.93514i
$340$		0			0
$341$			9.20251i			0.498344i
$342$		0			0
$343$			8.88176i			0.479570i
$344$		0			0
$345$		−	0.479499i		−	0.0258154i
$346$		0			0
$347$	−6.09457			−0.327174			−0.163587	−	0.986529i	$-0.552306\pi$
$347$	−6.09457			−0.327174			−0.163587	+	0.986529i	$0.552306\pi$
$348$		0			0
$349$		−	34.5283i		−	1.84826i	−0.382080	−	0.924129i	$-0.624792\pi$
$349$		−	34.5283i		−	1.84826i	0.382080	−	0.924129i	$-0.375208\pi$
$350$		0			0
$351$		−	0.615726i		−	0.0328650i
$352$		0			0
$353$	−33.5676			−1.78662			−0.893312	−	0.449437i	$-0.851625\pi$
$353$	−33.5676			−1.78662			−0.893312	+	0.449437i	$0.851625\pi$
$354$		0			0
$355$			9.72787i			0.516302i
$356$		0			0
$357$	−5.81631	−	3.66150i	−0.307832	−	0.193787i
$358$		0			0
$359$	−9.02404			−0.476271			−0.238135	−	0.971232i	$-0.576536\pi$
$359$	−9.02404			−0.476271			−0.238135	+	0.971232i	$0.576536\pi$
$360$		0			0
$361$	12.3567			0.650354
$362$		0			0
$363$	14.3298			0.752118
$364$		0			0
$365$			3.67345i			0.192277i
$366$		0			0
$367$			35.2634i			1.84073i	0.391055	+	0.920367i	$0.372110\pi$
$367$			35.2634i			1.84073i	−0.391055	+	0.920367i	$0.627890\pi$
$368$		0			0
$369$			11.2145i			0.583804i
$370$		0			0
$371$	−5.54913			−0.288097
$372$		0			0
$373$		−	28.7637i		−	1.48933i	−0.667439	−	0.744664i	$-0.732610\pi$
$373$		−	28.7637i		−	1.48933i	0.667439	−	0.744664i	$-0.267390\pi$
$374$		0			0
$375$	−2.54711			−0.131532
$376$		0			0
$377$		−	3.39863i		−	0.175038i
$378$		0			0
$379$	−2.79911			−0.143780			−0.0718902	−	0.997413i	$-0.522903\pi$
$379$	−2.79911			−0.143780			−0.0718902	+	0.997413i	$0.522903\pi$
$380$		0			0
$381$	−42.5982			−2.18237
$382$		0			0
$383$	−4.68600			−0.239444			−0.119722	−	0.992807i	$-0.538200\pi$
$383$	−4.68600			−0.239444			−0.119722	+	0.992807i	$0.538200\pi$
$384$		0			0
$385$		−	1.51711i		−	0.0773191i
$386$		0			0
$387$		−	12.4070i		−	0.630683i
$388$		0			0
$389$			35.9648i			1.82349i	0.410758	+	0.911744i	$0.365264\pi$
$389$			35.9648i			1.82349i	−0.410758	+	0.911744i	$0.634736\pi$
$390$		0			0
$391$	0.656863	+	0.413510i	0.0332190	+	0.0209121i
$392$		0			0
$393$	−3.19566			−0.161200
$394$		0			0
$395$		−	4.08394i		−	0.205486i
$396$		0			0
$397$	−21.8718			−1.09772			−0.548858	−	0.835916i	$-0.684937\pi$
$397$	−21.8718			−1.09772			−0.548858	+	0.835916i	$0.684937\pi$
$398$		0			0
$399$	4.29638			0.215088
$400$		0			0
$401$			3.78040i			0.188784i	0.995535	+	0.0943922i	$0.0300908\pi$
$401$			3.78040i			0.188784i	−0.995535	+	0.0943922i	$0.969909\pi$
$402$		0			0
$403$	1.96727			0.0979967
$404$		0			0
$405$	−7.29871			−0.362676
$406$		0			0
$407$	7.10319			0.352092
$408$		0			0
$409$	23.7023			1.17200			0.586001	−	0.810311i	$-0.300702\pi$
$409$	23.7023			1.17200			0.586001	+	0.810311i	$0.300702\pi$
$410$		0			0
$411$	−40.3917			−1.99238
$412$		0			0
$413$	0.283252			0.0139379
$414$		0			0
$415$		−	15.3164i		−	0.751851i
$416$		0			0
$417$	13.9862			0.684909
$418$		0			0
$419$	3.14894			0.153836			0.0769180	−	0.997037i	$-0.475492\pi$
$419$	3.14894			0.153836			0.0769180	+	0.997037i	$0.475492\pi$
$420$		0			0
$421$			13.3377i			0.650042i	0.945707	+	0.325021i	$0.105371\pi$
$421$			13.3377i			0.650042i	−0.945707	+	0.325021i	$0.894629\pi$
$422$		0			0
$423$	11.9654			0.581775
$424$		0			0
$425$	2.19658	−	3.48928i	0.106550	−	0.169255i
$426$		0			0
$427$		−	2.32608i		−	0.112567i
$428$		0			0
$429$			2.92626i			0.141281i
$430$		0			0
$431$		−	7.13978i		−	0.343911i	−0.985105	−	0.171955i	$-0.944991\pi$
$431$		−	7.13978i		−	0.343911i	0.985105	−	0.171955i	$-0.0550085\pi$
$432$		0			0
$433$	−12.0593			−0.579534			−0.289767	−	0.957097i	$-0.593578\pi$
$433$	−12.0593			−0.579534			−0.289767	+	0.957097i	$0.593578\pi$
$434$		0			0
$435$	17.4679			0.837523
$436$		0			0
$437$	−0.485211			−0.0232108
$438$		0			0
$439$			16.9899i			0.810885i	0.914121	+	0.405443i	$0.132883\pi$
$439$			16.9899i			0.810885i	−0.914121	+	0.405443i	$0.867117\pi$
$440$		0			0
$441$	22.9207			1.09146
$442$		0			0
$443$		−	6.08646i		−	0.289176i	−0.989492	−	0.144588i	$-0.953814\pi$
$443$		−	6.08646i		−	0.289176i	0.989492	−	0.144588i	$-0.0461857\pi$
$444$		0			0
$445$	8.98386			0.425876
$446$		0			0
$447$			10.1703i			0.481039i
$448$		0			0
$449$		−	15.7621i		−	0.743859i	−0.928261	−	0.371929i	$-0.878696\pi$
$449$		−	15.7621i		−	0.743859i	0.928261	−	0.371929i	$-0.121304\pi$
$450$		0			0
$451$		−	7.45391i		−	0.350991i
$452$		0			0
$453$	−16.3623			−0.768766
$454$		0			0
$455$	−0.324321			−0.0152044
$456$		0			0
$457$	7.72090			0.361169			0.180584	−	0.983560i	$-0.442201\pi$
$457$	7.72090			0.361169			0.180584	+	0.983560i	$0.442201\pi$
$458$		0			0
$459$	−2.72912	+	4.33523i	−0.127385	+	0.202351i
$460$		0			0
$461$			34.2083i			1.59324i	0.604481	+	0.796619i	$0.293380\pi$
$461$			34.2083i			1.59324i	−0.604481	+	0.796619i	$0.706620\pi$
$462$		0			0
$463$	−2.77955			−0.129177			−0.0645883	−	0.997912i	$-0.520573\pi$
$463$	−2.77955			−0.129177			−0.0645883	+	0.997912i	$0.520573\pi$
$464$		0			0
$465$			10.1112i			0.468894i
$466$		0			0
$467$			25.2523i			1.16854i	0.811560	+	0.584269i	$0.198619\pi$
$467$			25.2523i			1.16854i	−0.811560	+	0.584269i	$0.801381\pi$
$468$		0			0
$469$	5.66825			0.261736
$470$		0			0
$471$			58.3580i			2.68899i
$472$		0			0
$473$			8.24651i			0.379175i
$474$		0			0
$475$			2.57745i			0.118262i
$476$		0			0
$477$			29.5740i			1.35410i
$478$		0			0
$479$			16.9072i			0.772511i	0.922392	+	0.386255i	$0.126232\pi$
$479$			16.9072i			0.772511i	−0.922392	+	0.386255i	$0.873768\pi$
$480$		0			0
$481$		−	1.51849i		−	0.0692370i
$482$		0			0
$483$	0.313799			0.0142784
$484$		0			0
$485$		−	11.6566i		−	0.529301i
$486$		0			0
$487$		−	29.1269i		−	1.31987i	−0.751324	−	0.659933i	$-0.770585\pi$
$487$		−	29.1269i		−	1.31987i	0.751324	−	0.659933i	$-0.229415\pi$
$488$		0			0
$489$	−41.0711			−1.85730
$490$		0			0
$491$		−	23.7259i		−	1.07074i	−0.844619	−	0.535368i	$-0.820173\pi$
$491$		−	23.7259i		−	1.07074i	0.844619	−	0.535368i	$-0.179827\pi$
$492$		0			0
$493$	−15.0640	+	23.9292i	−0.678448	+	1.07772i
$494$		0			0
$495$	−8.08543			−0.363413
$496$		0			0
$497$	−6.36622			−0.285564
$498$		0			0
$499$	3.53161			0.158097			0.0790483	−	0.996871i	$-0.474812\pi$
$499$	3.53161			0.158097			0.0790483	+	0.996871i	$0.474812\pi$
$500$		0			0
$501$			33.7043i			1.50580i
$502$		0			0
$503$			27.6635i			1.23346i	0.787177	+	0.616728i	$0.211542\pi$
$503$			27.6635i			1.23346i	−0.787177	+	0.616728i	$0.788458\pi$
$504$		0			0
$505$			7.75678i			0.345172i
$506$		0			0
$507$	−32.4869			−1.44279
$508$		0			0
$509$		−	10.4113i		−	0.461472i	−0.973016	−	0.230736i	$-0.925887\pi$
$509$		−	10.4113i		−	0.461472i	0.973016	−	0.230736i	$-0.0741134\pi$
$510$		0			0
$511$	−2.40402			−0.106348
$512$		0			0
$513$		−	3.20234i		−	0.141387i
$514$		0			0
$515$	−5.75307			−0.253510
$516$		0			0
$517$	−7.95297			−0.349771
$518$		0			0
$519$	−54.2603			−2.38176
$520$		0			0
$521$			13.3749i			0.585965i	0.956118	+	0.292983i	$0.0946478\pi$
$521$			13.3749i			0.585965i	−0.956118	+	0.292983i	$0.905352\pi$
$522$		0			0
$523$		−	31.3021i		−	1.36875i	−0.729132	−	0.684373i	$-0.760076\pi$
$523$		−	31.3021i		−	1.36875i	0.729132	−	0.684373i	$-0.239924\pi$
$524$		0			0
$525$		−	1.66691i		−	0.0727499i
$526$		0			0
$527$	−13.8512	−	8.71966i	−0.603369	−	0.379834i
$528$		0			0
$529$	22.9646			0.998459
$530$		0			0
$531$		−	1.50959i		−	0.0655105i
$532$		0			0
$533$	−1.59346			−0.0690204
$534$		0			0
$535$	11.3343			0.490023
$536$		0			0
$537$			29.2276i			1.26126i
$538$		0			0
$539$	−15.2346			−0.656203
$540$		0			0
$541$	−12.3847			−0.532461			−0.266231	−	0.963909i	$-0.585778\pi$
$541$	−12.3847			−0.532461			−0.266231	+	0.963909i	$0.585778\pi$
$542$		0			0
$543$	−21.5038			−0.922815
$544$		0			0
$545$	−2.73908			−0.117329
$546$		0			0
$547$	7.06558			0.302102			0.151051	−	0.988526i	$-0.451734\pi$
$547$	7.06558			0.302102			0.151051	+	0.988526i	$0.451734\pi$
$548$		0			0
$549$	−12.3968			−0.529084
$550$		0			0
$551$		−	17.6760i		−	0.753023i
$552$		0			0
$553$	2.67266			0.113653
$554$		0			0
$555$	7.80456			0.331285
$556$		0			0
$557$		−	7.14741i		−	0.302845i	−0.988469	−	0.151423i	$-0.951615\pi$
$557$		−	7.14741i		−	0.302845i	0.988469	−	0.151423i	$-0.0483855\pi$
$558$		0			0
$559$	1.76290			0.0745627
$560$		0			0
$561$	12.9702	−	20.6033i	0.547604	−	0.869872i
$562$		0			0
$563$			40.7336i			1.71672i	0.513049	+	0.858359i	$0.328516\pi$
$563$			40.7336i			1.71672i	−0.513049	+	0.858359i	$0.671484\pi$
$564$		0			0
$565$		−	13.9883i		−	0.588491i
$566$		0			0
$567$		−	4.77651i		−	0.200594i
$568$		0			0
$569$	13.5629			0.568587			0.284293	−	0.958737i	$-0.408241\pi$
$569$	13.5629			0.568587			0.284293	+	0.958737i	$0.408241\pi$
$570$		0			0
$571$	−12.9863			−0.543460			−0.271730	−	0.962374i	$-0.587596\pi$
$571$	−12.9863			−0.543460			−0.271730	+	0.962374i	$0.587596\pi$
$572$		0			0
$573$	68.1776			2.84816
$574$		0			0
$575$			0.188252i			0.00785065i
$576$		0			0
$577$	−41.0167			−1.70755			−0.853773	−	0.520645i	$-0.825692\pi$
$577$	−41.0167			−1.70755			−0.853773	+	0.520645i	$0.825692\pi$
$578$		0			0
$579$			49.8186i			2.07039i
$580$		0			0
$581$	10.0235			0.415845
$582$		0			0
$583$		−	19.6569i		−	0.814104i
$584$		0			0
$585$			1.72846i			0.0714632i
$586$		0			0
$587$		−	34.8343i		−	1.43777i	−0.695131	−	0.718884i	$-0.744653\pi$
$587$		−	34.8343i		−	1.43777i	0.695131	−	0.718884i	$-0.255347\pi$
$588$		0			0
$589$	10.2316			0.421586
$590$		0			0
$591$	−20.4858			−0.842674
$592$		0			0
$593$	−17.2670			−0.709072			−0.354536	−	0.935042i	$-0.615361\pi$
$593$	−17.2670			−0.709072			−0.354536	+	0.935042i	$0.615361\pi$
$594$		0			0
$595$	2.28349	+	1.43751i	0.0936140	+	0.0589321i
$596$		0			0
$597$			59.3641i			2.42961i
$598$		0			0
$599$	−14.3306			−0.585531			−0.292766	−	0.956184i	$-0.594576\pi$
$599$	−14.3306			−0.585531			−0.292766	+	0.956184i	$0.594576\pi$
$600$		0			0
$601$			1.65416i			0.0674746i	0.999431	+	0.0337373i	$0.0107409\pi$
$601$			1.65416i			0.0674746i	−0.999431	+	0.0337373i	$0.989259\pi$
$602$		0			0
$603$		−	30.2089i		−	1.23020i
$604$		0			0
$605$	−5.62589			−0.228725
$606$		0			0
$607$			15.0062i			0.609082i	0.952499	+	0.304541i	$0.0985031\pi$
$607$			15.0062i			0.609082i	−0.952499	+	0.304541i	$0.901497\pi$
$608$		0			0
$609$			11.4316i			0.463230i
$610$		0			0
$611$			1.70015i			0.0687806i
$612$		0			0
$613$		−	17.7388i		−	0.716462i	−0.933633	−	0.358231i	$-0.883380\pi$
$613$		−	17.7388i		−	0.716462i	0.933633	−	0.358231i	$-0.116620\pi$
$614$		0			0
$615$		−	8.18990i		−	0.330249i
$616$		0			0
$617$			43.1029i			1.73526i	0.497211	+	0.867629i	$0.334357\pi$
$617$			43.1029i			1.73526i	−0.497211	+	0.867629i	$0.665643\pi$
$618$		0			0
$619$	−27.0444			−1.08700			−0.543502	−	0.839408i	$-0.682902\pi$
$619$	−27.0444			−1.08700			−0.543502	+	0.839408i	$0.682902\pi$
$620$		0			0
$621$		−	0.233892i		−	0.00938577i
$622$		0			0
$623$			5.87932i			0.235550i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$			15.2192i			0.607797i
$628$		0			0
$629$	−6.73049	+	10.6914i	−0.268362	+	0.426295i
$630$		0			0
$631$	20.9950			0.835796			0.417898	−	0.908494i	$-0.362767\pi$
$631$	20.9950			0.835796			0.417898	+	0.908494i	$0.362767\pi$
$632$		0			0
$633$	−55.5922			−2.20959
$634$		0			0
$635$	16.7241			0.663677
$636$		0			0
$637$			3.25679i			0.129039i
$638$		0			0
$639$			33.9287i			1.34220i
$640$		0			0
$641$		−	11.0768i		−	0.437509i	−0.975780	−	0.218755i	$-0.929801\pi$
$641$		−	11.0768i		−	0.437509i	0.975780	−	0.218755i	$-0.0701994\pi$
$642$		0			0
$643$	24.9194			0.982725			0.491363	−	0.870955i	$-0.336499\pi$
$643$	24.9194			0.982725			0.491363	+	0.870955i	$0.336499\pi$
$644$		0			0
$645$			9.06077i			0.356768i
$646$		0			0
$647$	−38.2146			−1.50237			−0.751185	−	0.660092i	$-0.770517\pi$
$647$	−38.2146			−1.50237			−0.751185	+	0.660092i	$0.770517\pi$
$648$		0			0
$649$			1.00337i			0.0393858i
$650$		0			0
$651$	−6.61707			−0.259343
$652$		0			0
$653$	46.6416			1.82523			0.912613	−	0.408825i	$-0.134061\pi$
$653$	46.6416			1.82523			0.912613	+	0.408825i	$0.134061\pi$
$654$		0			0
$655$	1.25462			0.0490221
$656$		0			0
$657$			12.8122i			0.499852i
$658$		0			0
$659$		−	29.2111i		−	1.13790i	−0.822371	−	0.568952i	$-0.807349\pi$
$659$		−	29.2111i		−	1.13790i	0.822371	−	0.568952i	$-0.192651\pi$
$660$		0			0
$661$			6.88515i			0.267801i	0.990995	+	0.133901i	$0.0427503\pi$
$661$			6.88515i			0.267801i	−0.990995	+	0.133901i	$0.957250\pi$
$662$		0			0
$663$	−4.40448	−	2.77272i	−0.171056	−	0.107683i
$664$		0			0
$665$	−1.68677			−0.0654100
$666$		0			0
$667$		−	1.29102i		−	0.0499885i
$668$		0			0
$669$	−37.9808			−1.46842
$670$		0			0
$671$	8.23975			0.318092
$672$		0			0
$673$		−	39.7685i		−	1.53296i	−0.642267	−	0.766481i	$-0.722006\pi$
$673$		−	39.7685i		−	1.53296i	0.642267	−	0.766481i	$-0.277994\pi$
$674$		0			0
$675$	−1.24244			−0.0478217
$676$		0			0
$677$	38.9268			1.49608			0.748039	−	0.663655i	$-0.230996\pi$
$677$	38.9268			1.49608			0.748039	+	0.663655i	$0.230996\pi$
$678$		0			0
$679$	7.62847			0.292754
$680$		0			0
$681$	−16.2081			−0.621096
$682$		0			0
$683$	−8.48371			−0.324620			−0.162310	−	0.986740i	$-0.551894\pi$
$683$	−8.48371			−0.324620			−0.162310	+	0.986740i	$0.551894\pi$
$684$		0			0
$685$	15.8578			0.605897
$686$		0			0
$687$			17.7165i			0.675927i
$688$		0			0
$689$	−4.20215			−0.160089
$690$		0			0
$691$	−46.3733			−1.76412			−0.882061	−	0.471136i	$-0.843844\pi$
$691$	−46.3733			−1.76412			−0.882061	+	0.471136i	$0.843844\pi$
$692$		0			0
$693$		−	5.29136i		−	0.201002i
$694$		0			0
$695$	−5.49102			−0.208286
$696$		0			0
$697$	11.2193	+	7.06280i	0.424961	+	0.267523i
$698$		0			0
$699$			12.2836i			0.464608i
$700$		0			0
$701$		−	10.6927i		−	0.403858i	−0.979400	−	0.201929i	$-0.935279\pi$
$701$		−	10.6927i		−	0.403858i	0.979400	−	0.201929i	$-0.0647210\pi$
$702$		0			0
$703$		−	7.89752i		−	0.297861i
$704$		0			0
$705$	−8.73824			−0.329101
$706$		0			0
$707$	−5.07628			−0.190913
$708$		0			0
$709$	30.9514			1.16240			0.581202	−	0.813759i	$-0.302582\pi$
$709$	30.9514			1.16240			0.581202	+	0.813759i	$0.302582\pi$
$710$		0			0
$711$		−	14.2439i		−	0.534189i
$712$		0			0
$713$	0.747296			0.0279865
$714$		0			0
$715$		−	1.14885i		−	0.0429646i
$716$		0			0
$717$	5.16123			0.192750
$718$		0			0
$719$			4.18798i			0.156185i	0.996946	+	0.0780927i	$0.0248830\pi$
$719$			4.18798i			0.156185i	−0.996946	+	0.0780927i	$0.975117\pi$
$720$		0			0
$721$		−	3.76499i		−	0.140215i
$722$		0			0
$723$			14.3266i			0.532811i
$724$		0			0
$725$	−6.85794			−0.254697
$726$		0			0
$727$	−26.6170			−0.987169			−0.493584	−	0.869698i	$-0.664314\pi$
$727$	−26.6170			−0.987169			−0.493584	+	0.869698i	$0.664314\pi$
$728$		0			0
$729$	−34.9503			−1.29445
$730$		0			0
$731$	−12.4123	−	7.81382i	−0.459086	−	0.289005i
$732$		0			0
$733$			24.4994i			0.904905i	0.891788	+	0.452453i	$0.149451\pi$
$733$			24.4994i			0.904905i	−0.891788	+	0.452453i	$0.850549\pi$
$734$		0			0
$735$	−16.7389			−0.617424
$736$		0			0
$737$			20.0788i			0.739613i
$738$		0			0
$739$			8.64956i			0.318179i	0.987264	+	0.159090i	$0.0508559\pi$
$739$			8.64956i			0.318179i	−0.987264	+	0.159090i	$0.949144\pi$
$740$		0			0
$741$	3.25349			0.119520
$742$		0			0
$743$			29.2437i			1.07285i	0.843949	+	0.536424i	$0.180225\pi$
$743$			29.2437i			1.07285i	−0.843949	+	0.536424i	$0.819775\pi$
$744$		0			0
$745$		−	3.99288i		−	0.146288i
$746$		0			0
$747$		−	53.4202i		−	1.95454i
$748$		0			0
$749$			7.41749i			0.271029i
$750$		0			0
$751$		−	11.4840i		−	0.419057i	−0.977802	−	0.209529i	$-0.932807\pi$
$751$		−	11.4840i		−	0.419057i	0.977802	−	0.209529i	$-0.0671930\pi$
$752$		0			0
$753$			54.0930i			1.97126i
$754$		0			0
$755$	6.42385			0.233788
$756$		0			0
$757$		−	1.40306i		−	0.0509950i	−0.999675	−	0.0254975i	$-0.991883\pi$
$757$		−	1.40306i		−	0.0509950i	0.999675	−	0.0254975i	$-0.00811699\pi$
$758$		0			0
$759$			1.11158i			0.0403478i
$760$		0			0
$761$	29.1306			1.05598			0.527991	−	0.849250i	$-0.322945\pi$
$761$	29.1306			1.05598			0.527991	+	0.849250i	$0.322945\pi$
$762$		0			0
$763$		−	1.79254i		−	0.0648944i
$764$		0			0
$765$	7.66119	−	12.1698i	0.276991	−	0.440002i
$766$		0			0
$767$	0.214496			0.00774500
$768$		0			0
$769$	16.7407			0.603685			0.301842	−	0.953358i	$-0.402398\pi$
$769$	16.7407			0.603685			0.301842	+	0.953358i	$0.402398\pi$
$770$		0			0
$771$	9.08912			0.327337
$772$		0			0
$773$			15.9297i			0.572951i	0.958088	+	0.286475i	$0.0924836\pi$
$773$			15.9297i			0.572951i	−0.958088	+	0.286475i	$0.907516\pi$
$774$		0			0
$775$		−	3.96966i		−	0.142594i
$776$		0			0
$777$			5.10755i			0.183232i
$778$		0			0
$779$	−8.28746			−0.296929
$780$		0			0
$781$		−	22.5513i		−	0.806948i
$782$		0			0
$783$	8.52059			0.304501
$784$		0			0
$785$		−	22.9114i		−	0.817744i
$786$		0			0
$787$	42.3403			1.50927			0.754634	−	0.656146i	$-0.227814\pi$
$787$	42.3403			1.50927			0.754634	+	0.656146i	$0.227814\pi$
$788$		0			0
$789$	3.34166			0.118966
$790$		0			0
$791$	9.15436			0.325492
$792$		0			0
$793$		−	1.76146i		−	0.0625511i
$794$		0			0
$795$		−	21.5978i		−	0.765994i
$796$		0			0
$797$		−	28.8219i		−	1.02092i	−0.859900	−	0.510462i	$-0.829474\pi$
$797$		−	28.8219i		−	1.02092i	0.859900	−	0.510462i	$-0.170526\pi$
$798$		0			0
$799$	7.53568	−	11.9705i	0.266593	−	0.423485i
$800$		0			0
$801$	31.3338			1.10712
$802$		0			0
$803$		−	8.51585i		−	0.300518i
$804$		0			0
$805$	−0.123198			−0.00434216
$806$		0			0
$807$	−57.5043			−2.02425
$808$		0			0
$809$		−	17.6944i		−	0.622103i	−0.950393	−	0.311052i	$-0.899319\pi$
$809$		−	17.6944i		−	0.622103i	0.950393	−	0.311052i	$-0.100681\pi$
$810$		0			0
$811$	−17.0795			−0.599744			−0.299872	−	0.953979i	$-0.596944\pi$
$811$	−17.0795			−0.599744			−0.299872	+	0.953979i	$0.596944\pi$
$812$		0			0
$813$	18.4503			0.647080
$814$		0			0
$815$	16.1246			0.564819
$816$		0			0
$817$	9.16870			0.320772
$818$		0			0
$819$	−1.13116			−0.0395260
$820$		0			0
$821$	15.3541			0.535861			0.267930	−	0.963438i	$-0.413660\pi$
$821$	15.3541			0.535861			0.267930	+	0.963438i	$0.413660\pi$
$822$		0			0
$823$		−	33.9882i		−	1.18475i	−0.805661	−	0.592377i	$-0.798189\pi$
$823$		−	33.9882i		−	1.18475i	0.805661	−	0.592377i	$-0.201811\pi$
$824$		0			0
$825$	5.90475			0.205577
$826$		0			0
$827$	35.0401			1.21846			0.609232	−	0.792992i	$-0.291478\pi$
$827$	35.0401			1.21846			0.609232	+	0.792992i	$0.291478\pi$
$828$		0			0
$829$		−	24.1097i		−	0.837366i	−0.908132	−	0.418683i	$-0.862492\pi$
$829$		−	24.1097i		−	0.837366i	0.908132	−	0.418683i	$-0.137508\pi$
$830$		0			0
$831$	39.3546			1.36520
$832$		0			0
$833$	14.4353	−	22.9305i	0.500153	−	0.794496i
$834$		0			0
$835$		−	13.2324i		−	0.457925i
$836$		0			0
$837$			4.93208i			0.170477i
$838$		0			0
$839$		−	23.0802i		−	0.796818i	−0.917208	−	0.398409i	$-0.869562\pi$
$839$		−	23.0802i		−	0.796818i	0.917208	−	0.398409i	$-0.130438\pi$
$840$		0			0
$841$	18.0313			0.621768
$842$		0			0
$843$	49.7350			1.71296
$844$		0			0
$845$	12.7544			0.438765
$846$		0			0
$847$		−	3.68176i		−	0.126507i
$848$		0			0
$849$	−54.8665			−1.88301
$850$		0			0
$851$		−	0.576819i		−	0.0197731i
$852$		0			0
$853$	17.7132			0.606490			0.303245	−	0.952913i	$-0.401930\pi$
$853$	17.7132			0.606490			0.303245	+	0.952913i	$0.401930\pi$
$854$		0			0
$855$			8.98960i			0.307438i
$856$		0			0
$857$		−	21.2303i		−	0.725214i	−0.931942	−	0.362607i	$-0.881887\pi$
$857$		−	21.2303i		−	0.725214i	0.931942	−	0.362607i	$-0.118113\pi$
$858$		0			0
$859$		−	55.6788i		−	1.89974i	−0.312652	−	0.949868i	$-0.601218\pi$
$859$		−	55.6788i		−	1.89974i	0.312652	−	0.949868i	$-0.398782\pi$
$860$		0			0
$861$	5.35973			0.182659
$862$		0			0
$863$	−30.0014			−1.02126			−0.510630	−	0.859801i	$-0.670588\pi$
$863$	−30.0014			−1.02126			−0.510630	+	0.859801i	$0.670588\pi$
$864$		0			0
$865$	21.3027			0.724313
$866$		0			0
$867$	18.7215	+	39.0445i	0.635816	+	1.32602i
$868$		0			0
$869$			9.46745i			0.321161i
$870$		0			0
$871$	4.29236			0.145441
$872$		0			0
$873$		−	40.6559i		−	1.37599i
$874$		0			0
$875$			0.654431i			0.0221238i
$876$		0			0
$877$	24.9771			0.843417			0.421708	−	0.906732i	$-0.361431\pi$
$877$	24.9771			0.843417			0.421708	+	0.906732i	$0.361431\pi$
$878$		0			0
$879$			54.1856i			1.82764i
$880$		0			0
$881$			51.0071i			1.71847i	0.511580	+	0.859236i	$0.329061\pi$
$881$			51.0071i			1.71847i	−0.511580	+	0.859236i	$0.670939\pi$
$882$		0			0
$883$		−	32.9436i		−	1.10864i	−0.832304	−	0.554320i	$-0.812978\pi$
$883$		−	32.9436i		−	1.10864i	0.832304	−	0.554320i	$-0.187022\pi$
$884$		0			0
$885$			1.10244i			0.0370583i
$886$		0			0
$887$		−	36.3257i		−	1.21970i	−0.792518	−	0.609848i	$-0.791230\pi$
$887$		−	36.3257i		−	1.21970i	0.792518	−	0.609848i	$-0.208770\pi$
$888$		0			0
$889$			10.9448i			0.367077i
$890$		0			0
$891$	16.9200			0.566840
$892$		0			0
$893$			8.84233i			0.295897i
$894$		0			0
$895$		−	11.4748i		−	0.383560i
$896$		0			0
$897$	0.237628			0.00793418
$898$		0			0
$899$			27.2237i			0.907960i
$900$		0			0
$901$	29.5867	+	18.6255i	0.985675	+	0.620505i
$902$		0			0
$903$	−5.92965			−0.197326
$904$		0			0
$905$	8.44241			0.280635
$906$		0			0
$907$	5.38304			0.178741			0.0893704	−	0.995998i	$-0.471515\pi$
$907$	5.38304			0.178741			0.0893704	+	0.995998i	$0.471515\pi$
$908$		0			0
$909$			27.0540i			0.897323i
$910$		0			0
$911$		−	22.7399i		−	0.753405i	−0.926334	−	0.376703i	$-0.877058\pi$
$911$		−	22.7399i		−	0.753405i	0.926334	−	0.376703i	$-0.122942\pi$
$912$		0			0
$913$			35.5066i			1.17510i
$914$		0			0
$915$	9.05334			0.299294
$916$		0			0
$917$			0.821063i			0.0271139i
$918$		0			0
$919$	20.0866			0.662595			0.331298	−	0.943526i	$-0.392514\pi$
$919$	20.0866			0.662595			0.331298	+	0.943526i	$0.392514\pi$
$920$		0			0
$921$		−	62.5811i		−	2.06212i
$922$		0			0
$923$	−4.82090			−0.158682
$924$		0			0
$925$	−3.06408			−0.100746
$926$		0			0
$927$	−20.0655			−0.659036
$928$		0			0
$929$			32.7749i			1.07531i	0.843165	+	0.537654i	$0.180689\pi$
$929$			32.7749i			1.07531i	−0.843165	+	0.537654i	$0.819311\pi$
$930$		0			0
$931$			16.9383i			0.555130i
$932$		0			0
$933$		−	79.1504i		−	2.59127i
$934$		0			0
$935$	−5.09213	+	8.08888i	−0.166531	+	0.264535i
$936$		0			0
$937$	−16.6829			−0.545008			−0.272504	−	0.962155i	$-0.587852\pi$
$937$	−16.6829			−0.545008			−0.272504	+	0.962155i	$0.587852\pi$
$938$		0			0
$939$		−	40.2322i		−	1.31293i
$940$		0			0
$941$	−40.8872			−1.33288			−0.666442	−	0.745557i	$-0.732184\pi$
$941$	−40.8872			−1.33288			−0.666442	+	0.745557i	$0.732184\pi$
$942$		0			0
$943$	−0.605299			−0.0197112
$944$		0			0
$945$		−	0.813094i		−	0.0264499i
$946$		0			0
$947$	38.9364			1.26526			0.632632	−	0.774452i	$-0.281975\pi$
$947$	38.9364			1.26526			0.632632	+	0.774452i	$0.281975\pi$
$948$		0			0
$949$	−1.82048			−0.0590952
$950$		0			0
$951$	8.67731			0.281381
$952$		0			0
$953$	−9.09398			−0.294583			−0.147292	−	0.989093i	$-0.547056\pi$
$953$	−9.09398			−0.294583			−0.147292	+	0.989093i	$0.547056\pi$
$954$		0			0
$955$	−26.7666			−0.866147
$956$		0			0
$957$	−40.4944			−1.30900
$958$		0			0
$959$			10.3779i			0.335119i
$960$		0			0
$961$	15.2418			0.491671
$962$		0			0
$963$	39.5314			1.27388
$964$		0			0
$965$		−	19.5588i		−	0.629622i
$966$		0			0
$967$	12.8681			0.413809			0.206905	−	0.978361i	$-0.433661\pi$
$967$	12.8681			0.413809			0.206905	+	0.978361i	$0.433661\pi$
$968$		0			0
$969$	−22.9073	−	14.4207i	−0.735889	−	0.463259i
$970$		0			0
$971$			24.8605i			0.797812i	0.916992	+	0.398906i	$0.130610\pi$
$971$			24.8605i			0.797812i	−0.916992	+	0.398906i	$0.869390\pi$
$972$		0			0
$973$		−	3.59349i		−	0.115202i
$974$		0			0
$975$		−	1.26229i		−	0.0404256i
$976$		0			0
$977$	−4.01155			−0.128341			−0.0641704	−	0.997939i	$-0.520440\pi$
$977$	−4.01155			−0.128341			−0.0641704	+	0.997939i	$0.520440\pi$
$978$		0			0
$979$	−20.8265			−0.665618
$980$		0			0
$981$	−9.55334			−0.305015
$982$		0			0
$983$			47.3479i			1.51016i	0.655631	+	0.755081i	$0.272403\pi$
$983$			47.3479i			1.51016i	−0.655631	+	0.755081i	$0.727597\pi$
$984$		0			0
$985$	8.04276			0.256264
$986$		0			0
$987$		−	5.71858i		−	0.182024i
$988$		0			0
$989$	0.669663			0.0212940
$990$		0			0
$991$			4.27944i			0.135941i	0.997687	+	0.0679704i	$0.0216524\pi$
$991$			4.27944i			0.135941i	−0.997687	+	0.0679704i	$0.978348\pi$
$992$		0			0
$993$		−	43.9453i		−	1.39456i
$994$		0			0
$995$		−	23.3064i		−	0.738864i
$996$		0			0
$997$	−49.6717			−1.57312			−0.786559	−	0.617515i	$-0.788139\pi$
$997$	−49.6717			−1.57312			−0.786559	+	0.617515i	$0.788139\pi$
$998$		0			0
$999$	3.80695			0.120446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2720.2.l.b.2481.5		36
4.3	odd	2		680.2.l.a.101.21	✓	36
8.3	odd	2		680.2.l.b.101.22	yes	36
8.5	even	2		2720.2.l.a.2481.31		36
17.16	even	2		2720.2.l.a.2481.32		36
68.67	odd	2		680.2.l.b.101.21	yes	36
136.67	odd	2		680.2.l.a.101.22	yes	36
136.101	even	2	inner	2720.2.l.b.2481.6		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.l.a.101.21	✓	36	4.3	odd	2
680.2.l.a.101.22	yes	36	136.67	odd	2
680.2.l.b.101.21	yes	36	68.67	odd	2
680.2.l.b.101.22	yes	36	8.3	odd	2
2720.2.l.a.2481.31		36	8.5	even	2
2720.2.l.a.2481.32		36	17.16	even	2
2720.2.l.b.2481.5		36	1.1	even	1	trivial
2720.2.l.b.2481.6		36	136.101	even	2	inner

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:	\( N \)	\(=\)	\( 2720 = 2^{5} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2720.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.54711 q^{3} +1.00000 q^{5} +0.654431i q^{7} +3.48778 q^{9} -2.31821 q^{11} +0.495577i q^{13} -2.54711 q^{15} +(2.19658 - 3.48928i) q^{17} +2.57745i q^{19} -1.66691i q^{21} +0.188252i q^{23} +1.00000 q^{25} -1.24244 q^{27} -6.85794 q^{29} -3.96966i q^{31} +5.90475 q^{33} +0.654431i q^{35} -3.06408 q^{37} -1.26229i q^{39} +3.21537i q^{41} -3.55727i q^{43} +3.48778 q^{45} +3.43065 q^{47} +6.57172 q^{49} +(-5.59493 + 8.88758i) q^{51} +8.47932i q^{53} -2.31821 q^{55} -6.56507i q^{57} -0.432821i q^{59} -3.55435 q^{61} +2.28252i q^{63} +0.495577i q^{65} -8.66134i q^{67} -0.479499i q^{69} +9.72787i q^{71} +3.67345i q^{73} -2.54711 q^{75} -1.51711i q^{77} -4.08394i q^{79} -7.29871 q^{81} -15.3164i q^{83} +(2.19658 - 3.48928i) q^{85} +17.4679 q^{87} +8.98386 q^{89} -0.324321 q^{91} +10.1112i q^{93} +2.57745i q^{95} -11.6566i q^{97} -8.08543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 4 q^{3} + 36 q^{5} + 36 q^{9} + 8 q^{11} + 4 q^{15} + 36 q^{25} + 16 q^{27} - 8 q^{33} + 36 q^{45} - 20 q^{47} - 36 q^{49} + 8 q^{55} + 4 q^{75} + 44 q^{81} - 24 q^{87} + 56 q^{91} + 40 q^{99}+O(q^{100}) \) 36 * q + 4 * q^3 + 36 * q^5 + 36 * q^9 + 8 * q^11 + 4 * q^15 + 36 * q^25 + 16 * q^27 - 8 * q^33 + 36 * q^45 - 20 * q^47 - 36 * q^49 + 8 * q^55 + 4 * q^75 + 44 * q^81 - 24 * q^87 + 56 * q^91 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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