LMFDB - Embedded newform 720.3.bh.f.433.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,3,Mod(433,720)] 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("720.433"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.bh (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-12,0,-20] 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:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.2
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	720.433
Dual form			720.3.bh.f.577.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.77526 - 4.67423i) q^{5} +(-2.55051 + 2.55051i) q^{7} +8.24745 q^{11} +(-12.2474 - 12.2474i) q^{13} +(12.4495 - 12.4495i) q^{17} +34.4949i q^{19} +(-17.3485 - 17.3485i) q^{23} +(-18.6969 + 16.5959i) q^{25} -9.75255i q^{29} -28.4949 q^{31} +(16.4495 + 7.39388i) q^{35} +(-7.34847 + 7.34847i) q^{37} -74.4949 q^{41} +(-34.8990 - 34.8990i) q^{43} +(-22.0454 + 22.0454i) q^{47} +35.9898i q^{49} +(64.6969 + 64.6969i) q^{53} +(-14.6413 - 38.5505i) q^{55} +15.2577i q^{59} -53.5051 q^{61} +(-35.5051 + 78.9898i) q^{65} +(-4.69694 + 4.69694i) q^{67} -117.980 q^{71} +(34.1918 + 34.1918i) q^{73} +(-21.0352 + 21.0352i) q^{77} +0.494897i q^{79} +(-18.3587 - 18.3587i) q^{83} +(-80.2929 - 36.0908i) q^{85} -136.969i q^{89} +62.4745 q^{91} +(161.237 - 61.2372i) q^{95} +(94.5959 - 94.5959i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{5} - 20 q^{7} - 16 q^{11} + 40 q^{17} - 40 q^{23} - 16 q^{25} - 16 q^{31} + 56 q^{35} - 200 q^{41} - 120 q^{43} + 200 q^{53} + 108 q^{55} - 312 q^{61} - 240 q^{65} + 40 q^{67} - 80 q^{71} - 20 q^{73}+ \cdots + 300 q^{97}+O(q^{100}) $ 4 * q - 12 * q^5 - 20 * q^7 - 16 * q^11 + 40 * q^17 - 40 * q^23 - 16 * q^25 - 16 * q^31 + 56 * q^35 - 200 * q^41 - 120 * q^43 + 200 * q^53 + 108 * q^55 - 312 * q^61 - 240 * q^65 + 40 * q^67 - 80 * q^71 - 20 * q^73 + 200 * q^77 - 240 * q^83 - 184 * q^85 - 240 * q^91 + 400 * q^95 + 300 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$	−1.77526	−	4.67423i	−0.355051	−	0.934847i
$5$	−1.77526	−	4.67423i	−0.355051	−	0.934847i
$6$		0			0
$7$	−2.55051	+	2.55051i	−0.364359	+	0.364359i	−0.865415	−	0.501056i	$-0.832945\pi$
$7$	−2.55051	+	2.55051i	−0.364359	+	0.364359i	0.501056	+	0.865415i	$0.332945\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	8.24745			0.749768			0.374884	−	0.927072i	$-0.377683\pi$
$11$	8.24745			0.749768			0.374884	+	0.927072i	$0.377683\pi$
$12$		0			0
$13$	−12.2474	−	12.2474i	−0.942111	−	0.942111i	0.0563023	−	0.998414i	$-0.482069\pi$
$13$	−12.2474	−	12.2474i	−0.942111	−	0.942111i	−0.998414	+	0.0563023i	$0.982069\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.4495	−	12.4495i	0.732323	−	0.732323i	−0.238757	−	0.971079i	$-0.576740\pi$
$17$	12.4495	−	12.4495i	0.732323	−	0.732323i	0.971079	+	0.238757i	$0.0767398\pi$
$18$		0			0
$19$			34.4949i			1.81552i	0.419489	+	0.907760i	$0.362209\pi$
$19$			34.4949i			1.81552i	−0.419489	+	0.907760i	$0.637791\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−17.3485	−	17.3485i	−0.754281	−	0.754281i	0.220994	−	0.975275i	$-0.429070\pi$
$23$	−17.3485	−	17.3485i	−0.754281	−	0.754281i	−0.975275	+	0.220994i	$0.929070\pi$
$24$		0			0
$25$	−18.6969	+	16.5959i	−0.747878	+	0.663837i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	9.75255i		−	0.336295i	−0.985762	−	0.168147i	$-0.946222\pi$
$29$		−	9.75255i		−	0.336295i	0.985762	−	0.168147i	$-0.0537785\pi$
$30$		0			0
$31$	−28.4949			−0.919190			−0.459595	−	0.888129i	$-0.652005\pi$
$31$	−28.4949			−0.919190			−0.459595	+	0.888129i	$0.652005\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	16.4495	+	7.39388i	0.469985	+	0.211254i
$36$		0			0
$37$	−7.34847	+	7.34847i	−0.198607	+	0.198607i	−0.799403	−	0.600795i	$-0.794851\pi$
$37$	−7.34847	+	7.34847i	−0.198607	+	0.198607i	0.600795	+	0.799403i	$0.294851\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−74.4949			−1.81695			−0.908474	−	0.417941i	$-0.862752\pi$
$41$	−74.4949			−1.81695			−0.908474	+	0.417941i	$0.862752\pi$
$42$		0			0
$43$	−34.8990	−	34.8990i	−0.811604	−	0.811604i	0.173270	−	0.984874i	$-0.444567\pi$
$43$	−34.8990	−	34.8990i	−0.811604	−	0.811604i	−0.984874	+	0.173270i	$0.944567\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−22.0454	+	22.0454i	−0.469051	+	0.469051i	−0.901607	−	0.432556i	$-0.857612\pi$
$47$	−22.0454	+	22.0454i	−0.469051	+	0.469051i	0.432556	+	0.901607i	$0.357612\pi$
$48$		0			0
$49$			35.9898i			0.734486i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	64.6969	+	64.6969i	1.22070	+	1.22070i	0.967385	+	0.253312i	$0.0815201\pi$
$53$	64.6969	+	64.6969i	1.22070	+	1.22070i	0.253312	+	0.967385i	$0.418480\pi$
$54$		0			0
$55$	−14.6413	−	38.5505i	−0.266206	−	0.700918i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			15.2577i			0.258604i	0.991605	+	0.129302i	$0.0412737\pi$
$59$			15.2577i			0.258604i	−0.991605	+	0.129302i	$0.958726\pi$
$60$		0			0
$61$	−53.5051			−0.877133			−0.438566	−	0.898699i	$-0.644514\pi$
$61$	−53.5051			−0.877133			−0.438566	+	0.898699i	$0.644514\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−35.5051	+	78.9898i	−0.546232	+	1.21523i
$66$		0			0
$67$	−4.69694	+	4.69694i	−0.0701036	+	0.0701036i	−0.741289	−	0.671186i	$-0.765785\pi$
$67$	−4.69694	+	4.69694i	−0.0701036	+	0.0701036i	0.671186	+	0.741289i	$0.265785\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−117.980			−1.66168			−0.830842	−	0.556508i	$-0.812141\pi$
$71$	−117.980			−1.66168			−0.830842	+	0.556508i	$0.812141\pi$
$72$		0			0
$73$	34.1918	+	34.1918i	0.468381	+	0.468381i	0.901390	−	0.433009i	$-0.142548\pi$
$73$	34.1918	+	34.1918i	0.468381	+	0.468381i	−0.433009	+	0.901390i	$0.642548\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−21.0352	+	21.0352i	−0.273184	+	0.273184i
$78$		0			0
$79$			0.494897i			0.00626452i	0.999995	+	0.00313226i	$0.000997032\pi$
$79$			0.494897i			0.00626452i	−0.999995	+	0.00313226i	$0.999003\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−18.3587	−	18.3587i	−0.221189	−	0.221189i	0.587810	−	0.808999i	$-0.299990\pi$
$83$	−18.3587	−	18.3587i	−0.221189	−	0.221189i	−0.808999	+	0.587810i	$0.799990\pi$
$84$		0			0
$85$	−80.2929	−	36.0908i	−0.944622	−	0.424598i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	136.969i		−	1.53898i	−0.638658	−	0.769491i	$-0.720510\pi$
$89$		−	136.969i		−	1.53898i	0.638658	−	0.769491i	$-0.279490\pi$
$90$		0			0
$91$	62.4745			0.686533
$92$		0			0
$93$		0			0
$94$		0			0
$95$	161.237	−	61.2372i	1.69723	−	0.644603i
$96$		0			0
$97$	94.5959	−	94.5959i	0.975216	−	0.975216i	−0.0244846	−	0.999700i	$-0.507794\pi$
$97$	94.5959	−	94.5959i	0.975216	−	0.975216i	0.999700	+	0.0244846i	$0.00779446\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−70.2474			−0.695519			−0.347760	−	0.937584i	$-0.613057\pi$
$101$	−70.2474			−0.695519			−0.347760	+	0.937584i	$0.613057\pi$
$102$		0			0
$103$	−86.8434	−	86.8434i	−0.843139	−	0.843139i	0.146126	−	0.989266i	$-0.453319\pi$
$103$	−86.8434	−	86.8434i	−0.843139	−	0.843139i	−0.989266	+	0.146126i	$0.953319\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−96.3383	+	96.3383i	−0.900358	+	0.900358i	−0.995467	−	0.0951092i	$-0.969680\pi$
$107$	−96.3383	+	96.3383i	−0.900358	+	0.900358i	0.0951092	+	0.995467i	$0.469680\pi$
$108$		0			0
$109$		−	12.5153i		−	0.114819i	−0.998351	−	0.0574097i	$-0.981716\pi$
$109$		−	12.5153i		−	0.114819i	0.998351	−	0.0574097i	$-0.0182841\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−66.9444	−	66.9444i	−0.592428	−	0.592428i	0.345858	−	0.938287i	$-0.387588\pi$
$113$	−66.9444	−	66.9444i	−0.592428	−	0.592428i	−0.938287	+	0.345858i	$0.887588\pi$
$114$		0			0
$115$	−50.2929	+	111.889i	−0.437329	+	0.972946i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			63.5051i			0.533656i
$120$		0			0
$121$	−52.9796			−0.437848
$122$		0			0
$123$		0			0
$124$		0			0
$125$	110.765	+	57.9319i	0.886120	+	0.463455i
$126$		0			0
$127$	−2.55051	+	2.55051i	−0.0200828	+	0.0200828i	−0.717077	−	0.696994i	$-0.754520\pi$
$127$	−2.55051	+	2.55051i	−0.0200828	+	0.0200828i	0.696994	+	0.717077i	$0.254520\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	44.2474			0.337767			0.168883	−	0.985636i	$-0.445984\pi$
$131$	44.2474			0.337767			0.168883	+	0.985636i	$0.445984\pi$
$132$		0			0
$133$	−87.9796	−	87.9796i	−0.661501	−	0.661501i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.3587	−	18.3587i	0.134005	−	0.134005i	−0.636923	−	0.770928i	$-0.719793\pi$
$137$	18.3587	−	18.3587i	0.134005	−	0.134005i	0.770928	+	0.636923i	$0.219793\pi$
$138$		0			0
$139$			219.980i			1.58259i	0.611437	+	0.791293i	$0.290592\pi$
$139$			219.980i			1.58259i	−0.611437	+	0.791293i	$0.709408\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−101.010	−	101.010i	−0.706365	−	0.706365i
$144$		0			0
$145$	−45.5857	+	17.3133i	−0.314384	+	0.119402i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	28.2679i		−	0.189717i	−0.995491	−	0.0948586i	$-0.969760\pi$
$149$		−	28.2679i		−	0.189717i	0.995491	−	0.0948586i	$-0.0302399\pi$
$150$		0			0
$151$	61.0102			0.404041			0.202021	−	0.979381i	$-0.435249\pi$
$151$	61.0102			0.404041			0.202021	+	0.979381i	$0.435249\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	50.5857	+	133.192i	0.326359	+	0.859302i
$156$		0			0
$157$	−22.6515	+	22.6515i	−0.144277	+	0.144277i	−0.775556	−	0.631279i	$-0.782530\pi$
$157$	−22.6515	+	22.6515i	−0.144277	+	0.144277i	0.631279	+	0.775556i	$0.282530\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	88.4949			0.549658
$162$		0			0
$163$	88.9898	+	88.9898i	0.545950	+	0.545950i	0.925267	−	0.379317i	$-0.123841\pi$
$163$	88.9898	+	88.9898i	0.545950	+	0.545950i	−0.379317	+	0.925267i	$0.623841\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	171.641	−	171.641i	1.02779	−	1.02779i	0.0281898	−	0.999603i	$-0.491026\pi$
$167$	171.641	−	171.641i	1.02779	−	1.02779i	0.999603	−	0.0281898i	$-0.00897427\pi$
$168$		0			0
$169$			131.000i			0.775148i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.6311	+	10.6311i	0.0614516	+	0.0614516i	0.737165	−	0.675713i	$-0.236164\pi$
$173$	10.6311	+	10.6311i	0.0614516	+	0.0614516i	−0.675713	+	0.737165i	$0.736164\pi$
$174$		0			0
$175$	5.35867	−	90.0148i	0.0306210	−	0.514370i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	100.247i		−	0.560042i	−0.959994	−	0.280021i	$-0.909659\pi$
$179$		−	100.247i		−	0.560042i	0.959994	−	0.280021i	$-0.0903414\pi$
$180$		0			0
$181$	259.444			1.43339			0.716696	−	0.697386i	$-0.245654\pi$
$181$	259.444			1.43339			0.716696	+	0.697386i	$0.245654\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	47.3939	+	21.3031i	0.256183	+	0.115152i
$186$		0			0
$187$	102.677	−	102.677i	0.549072	−	0.549072i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	77.4847			0.405679			0.202840	−	0.979212i	$-0.434983\pi$
$191$	77.4847			0.405679			0.202840	+	0.979212i	$0.434983\pi$
$192$		0			0
$193$	83.5857	+	83.5857i	0.433087	+	0.433087i	0.889677	−	0.456590i	$-0.150930\pi$
$193$	83.5857	+	83.5857i	0.433087	+	0.433087i	−0.456590	+	0.889677i	$0.650930\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	200.252	−	200.252i	1.01651	−	1.01651i	0.0166464	−	0.999861i	$-0.494701\pi$
$197$	200.252	−	200.252i	1.01651	−	1.01651i	0.999861	−	0.0166464i	$-0.00529894\pi$
$198$		0			0
$199$		−	162.990i		−	0.819044i	−0.912300	−	0.409522i	$-0.865695\pi$
$199$		−	162.990i		−	0.819044i	0.912300	−	0.409522i	$-0.134305\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.8740	+	24.8740i	0.122532	+	0.122532i
$204$		0			0
$205$	132.247	+	348.207i	0.645110	+	1.69857i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			284.495i			1.36122i
$210$		0			0
$211$	−207.980			−0.985685			−0.492843	−	0.870118i	$-0.664042\pi$
$211$	−207.980			−0.985685			−0.492843	+	0.870118i	$0.664042\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−101.171	+	225.081i	−0.470565	+	1.04689i
$216$		0			0
$217$	72.6765	−	72.6765i	0.334915	−	0.334915i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−304.949			−1.37986
$222$		0			0
$223$	−29.7219	−	29.7219i	−0.133282	−	0.133282i	0.637318	−	0.770601i	$-0.280044\pi$
$223$	−29.7219	−	29.7219i	−0.133282	−	0.133282i	−0.770601	+	0.637318i	$0.780044\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	197.980	−	197.980i	0.872157	−	0.872157i	−0.120550	−	0.992707i	$-0.538466\pi$
$227$	197.980	−	197.980i	0.872157	−	0.872157i	0.992707	+	0.120550i	$0.0384659\pi$
$228$		0			0
$229$		−	8.96938i		−	0.0391676i	−0.999808	−	0.0195838i	$-0.993766\pi$
$229$		−	8.96938i		−	0.0391676i	0.999808	−	0.0195838i	$-0.00623412\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−28.7628	−	28.7628i	−0.123445	−	0.123445i	0.642685	−	0.766130i	$-0.277820\pi$
$233$	−28.7628	−	28.7628i	−0.123445	−	0.123445i	−0.766130	+	0.642685i	$0.777820\pi$
$234$		0			0
$235$	142.182	+	63.9092i	0.605028	+	0.271954i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	242.969i		−	1.01661i	−0.861178	−	0.508304i	$-0.830273\pi$
$239$		−	242.969i		−	1.01661i	0.861178	−	0.508304i	$-0.169727\pi$
$240$		0			0
$241$	−32.0000			−0.132780			−0.0663900	−	0.997794i	$-0.521148\pi$
$241$	−32.0000			−0.132780			−0.0663900	+	0.997794i	$0.521148\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	168.225	−	63.8911i	0.686632	−	0.260780i
$246$		0			0
$247$	422.474	−	422.474i	1.71042	−	1.71042i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−250.763			−0.999055			−0.499527	−	0.866298i	$-0.666493\pi$
$251$	−250.763			−0.999055			−0.499527	+	0.866298i	$0.666493\pi$
$252$		0			0
$253$	−143.081	−	143.081i	−0.565536	−	0.565536i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	73.8638	−	73.8638i	0.287408	−	0.287408i	−0.548647	−	0.836054i	$-0.684857\pi$
$257$	73.8638	−	73.8638i	0.287408	−	0.287408i	0.836054	+	0.548647i	$0.184857\pi$
$258$		0			0
$259$		−	37.4847i		−	0.144729i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	49.1668	+	49.1668i	0.186946	+	0.186946i	0.794374	−	0.607428i	$-0.207799\pi$
$263$	49.1668	+	49.1668i	0.186946	+	0.186946i	−0.607428	+	0.794374i	$0.707799\pi$
$264$		0			0
$265$	187.555	−	417.262i	0.707755	−	1.57457i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			278.742i			1.03622i	0.855315	+	0.518108i	$0.173364\pi$
$269$			278.742i			1.03622i	−0.855315	+	0.518108i	$0.826636\pi$
$270$		0			0
$271$	66.0000			0.243542			0.121771	−	0.992558i	$-0.461143\pi$
$271$	66.0000			0.243542			0.121771	+	0.992558i	$0.461143\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−154.202	+	136.874i	−0.560735	+	0.497724i
$276$		0			0
$277$	211.843	−	211.843i	0.764777	−	0.764777i	−0.212404	−	0.977182i	$-0.568129\pi$
$277$	211.843	−	211.843i	0.764777	−	0.764777i	0.977182	+	0.212404i	$0.0681294\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	312.434			1.11186			0.555932	−	0.831228i	$-0.312362\pi$
$281$	312.434			1.11186			0.555932	+	0.831228i	$0.312362\pi$
$282$		0			0
$283$	204.747	+	204.747i	0.723487	+	0.723487i	0.969314	−	0.245826i	$-0.0790593\pi$
$283$	204.747	+	204.747i	0.723487	+	0.723487i	−0.245826	+	0.969314i	$0.579059\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	190.000	−	190.000i	0.662021	−	0.662021i
$288$		0			0
$289$		−	20.9796i		−	0.0725937i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−325.151	−	325.151i	−1.10973	−	1.10973i	−0.993185	−	0.116545i	$-0.962818\pi$
$293$	−325.151	−	325.151i	−1.10973	−	1.10973i	−0.116545	−	0.993185i	$-0.537182\pi$
$294$		0			0
$295$	71.3179	−	27.0862i	0.241755	−	0.0918177i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			424.949i			1.42123i
$300$		0			0
$301$	178.020			0.591430
$302$		0			0
$303$		0			0
$304$		0			0
$305$	94.9852	+	250.095i	0.311427	+	0.819985i
$306$		0			0
$307$	−179.394	+	179.394i	−0.584345	+	0.584345i	−0.936094	−	0.351749i	$-0.885587\pi$
$307$	−179.394	+	179.394i	−0.584345	+	0.584345i	0.351749	+	0.936094i	$0.385587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−213.980			−0.688037			−0.344019	−	0.938963i	$-0.611788\pi$
$311$	−213.980			−0.688037			−0.344019	+	0.938963i	$0.611788\pi$
$312$		0			0
$313$	−186.414	−	186.414i	−0.595573	−	0.595573i	0.343558	−	0.939131i	$-0.388368\pi$
$313$	−186.414	−	186.414i	−0.595573	−	0.595573i	−0.939131	+	0.343558i	$0.888368\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−57.5505	+	57.5505i	−0.181547	+	0.181547i	−0.792030	−	0.610482i	$-0.790976\pi$
$317$	−57.5505	+	57.5505i	−0.181547	+	0.181547i	0.610482	+	0.792030i	$0.290976\pi$
$318$		0			0
$319$		−	80.4337i		−	0.252143i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	429.444	+	429.444i	1.32955	+	1.32955i
$324$		0			0
$325$	432.247	+	25.7321i	1.32999	+	0.0791758i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	112.454i		−	0.341806i
$330$		0			0
$331$	214.413			0.647774			0.323887	−	0.946096i	$-0.395010\pi$
$331$	214.413			0.647774			0.323887	+	0.946096i	$0.395010\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	30.2929	+	13.6163i	0.0904264	+	0.0406458i
$336$		0			0
$337$	−94.5959	+	94.5959i	−0.280700	+	0.280700i	−0.833388	−	0.552688i	$-0.813602\pi$
$337$	−94.5959	+	94.5959i	−0.280700	+	0.280700i	0.552688	+	0.833388i	$0.313602\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−235.010			−0.689179
$342$		0			0
$343$	−216.767	−	216.767i	−0.631975	−	0.631975i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	226.919	−	226.919i	0.653946	−	0.653946i	−0.299995	−	0.953941i	$-0.596985\pi$
$347$	226.919	−	226.919i	0.653946	−	0.653946i	0.953941	+	0.299995i	$0.0969849\pi$
$348$		0			0
$349$		−	182.454i		−	0.522791i	−0.965232	−	0.261396i	$-0.915817\pi$
$349$		−	182.454i		−	0.522791i	0.965232	−	0.261396i	$-0.0841827\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−263.914	−	263.914i	−0.747631	−	0.747631i	0.226403	−	0.974034i	$-0.427303\pi$
$353$	−263.914	−	263.914i	−0.747631	−	0.747631i	−0.974034	+	0.226403i	$0.927303\pi$
$354$		0			0
$355$	209.444	+	551.464i	0.589983	+	1.55342i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			285.485i			0.795222i	0.917554	+	0.397611i	$0.130161\pi$
$359$			285.485i			0.795222i	−0.917554	+	0.397611i	$0.869839\pi$
$360$		0			0
$361$	−828.898			−2.29612
$362$		0			0
$363$		0			0
$364$		0			0
$365$	99.1214	−	220.520i	0.271566	−	0.604164i
$366$		0			0
$367$	−418.964	+	418.964i	−1.14159	+	1.14159i	−0.153431	+	0.988159i	$0.549032\pi$
$367$	−418.964	+	418.964i	−1.14159	+	1.14159i	−0.988159	+	0.153431i	$0.950968\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−330.020			−0.889543
$372$		0			0
$373$	283.106	+	283.106i	0.758996	+	0.758996i	0.976140	−	0.217143i	$-0.0696739\pi$
$373$	283.106	+	283.106i	0.758996	+	0.758996i	−0.217143	+	0.976140i	$0.569674\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−119.444	+	119.444i	−0.316827	+	0.316827i
$378$		0			0
$379$		−	194.000i		−	0.511873i	−0.966694	−	0.255937i	$-0.917616\pi$
$379$		−	194.000i		−	0.511873i	0.966694	−	0.255937i	$-0.0823839\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−531.489	−	531.489i	−1.38770	−	1.38770i	−0.830129	−	0.557572i	$-0.811733\pi$
$383$	−531.489	−	531.489i	−1.38770	−	1.38770i	−0.557572	−	0.830129i	$-0.688267\pi$
$384$		0			0
$385$	135.666	+	60.9806i	0.352380	+	0.158391i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			479.196i			1.23187i	0.787798	+	0.615934i	$0.211221\pi$
$389$			479.196i			1.23187i	−0.787798	+	0.615934i	$0.788779\pi$
$390$		0			0
$391$	−431.959			−1.10475
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.31327	−	0.878569i	0.00585637	−	0.00222423i
$396$		0			0
$397$	−389.267	+	389.267i	−0.980521	+	0.980521i	−0.999814	−	0.0192929i	$-0.993858\pi$
$397$	−389.267	+	389.267i	−0.980521	+	0.980521i	0.0192929	+	0.999814i	$0.493858\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	603.898			1.50598			0.752990	−	0.658032i	$-0.228611\pi$
$401$	603.898			1.50598			0.752990	+	0.658032i	$0.228611\pi$
$402$		0			0
$403$	348.990	+	348.990i	0.865980	+	0.865980i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−60.6061	+	60.6061i	−0.148909	+	0.148909i
$408$		0			0
$409$			183.959i			0.449778i	0.974384	+	0.224889i	$0.0722019\pi$
$409$			183.959i			0.449778i	−0.974384	+	0.224889i	$0.927798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−38.9148	−	38.9148i	−0.0942247	−	0.0942247i
$414$		0			0
$415$	−53.2214	+	118.404i	−0.128244	+	0.285311i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	197.814i		−	0.472109i	−0.971740	−	0.236055i	$-0.924146\pi$
$419$		−	197.814i		−	0.472109i	0.971740	−	0.236055i	$-0.0758544\pi$
$420$		0			0
$421$	−114.041			−0.270881			−0.135440	−	0.990785i	$-0.543245\pi$
$421$	−114.041			−0.270881			−0.135440	+	0.990785i	$0.543245\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−26.1566	+	439.378i	−0.0615450	+	1.03383i
$426$		0			0
$427$	136.465	−	136.465i	0.319591	−	0.319591i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	348.454			0.808478			0.404239	−	0.914653i	$-0.367536\pi$
$431$	348.454			0.808478			0.404239	+	0.914653i	$0.367536\pi$
$432$		0			0
$433$	60.1010	+	60.1010i	0.138801	+	0.138801i	0.773094	−	0.634292i	$-0.218708\pi$
$433$	60.1010	+	60.1010i	0.138801	+	0.138801i	−0.634292	+	0.773094i	$0.718708\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	598.434	−	598.434i	1.36941	−	1.36941i
$438$		0			0
$439$		−	556.929i		−	1.26863i	−0.773075	−	0.634315i	$-0.781282\pi$
$439$		−	556.929i		−	1.26863i	0.773075	−	0.634315i	$-0.218718\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	95.5551	+	95.5551i	0.215700	+	0.215700i	0.806684	−	0.590984i	$-0.201260\pi$
$443$	95.5551	+	95.5551i	0.215700	+	0.215700i	−0.590984	+	0.806684i	$0.701260\pi$
$444$		0			0
$445$	−640.227	+	243.156i	−1.43871	+	0.546417i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			213.423i			0.475331i	0.971347	+	0.237665i	$0.0763821\pi$
$449$			213.423i			0.475331i	−0.971347	+	0.237665i	$0.923618\pi$
$450$		0			0
$451$	−614.393			−1.36229
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−110.908	−	292.020i	−0.243754	−	0.641803i
$456$		0			0
$457$	−38.9388	+	38.9388i	−0.0852052	+	0.0852052i	−0.748425	−	0.663220i	$-0.769190\pi$
$457$	−38.9388	+	38.9388i	−0.0852052	+	0.0852052i	0.663220	+	0.748425i	$0.269190\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−509.712			−1.10567			−0.552833	−	0.833292i	$-0.686453\pi$
$461$	−509.712			−1.10567			−0.552833	+	0.833292i	$0.686453\pi$
$462$		0			0
$463$	−492.955	−	492.955i	−1.06470	−	1.06470i	−0.997757	−	0.0669397i	$-0.978676\pi$
$463$	−492.955	−	492.955i	−1.06470	−	1.06470i	−0.0669397	−	0.997757i	$-0.521324\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	647.044	−	647.044i	1.38553	−	1.38553i	0.551085	−	0.834449i	$-0.314214\pi$
$467$	647.044	−	647.044i	1.38553	−	1.38553i	0.834449	−	0.551085i	$-0.185786\pi$
$468$		0			0
$469$		−	23.9592i		−	0.0510857i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−287.828	−	287.828i	−0.608515	−	0.608515i
$474$		0			0
$475$	−572.474	−	644.949i	−1.20521	−	1.35779i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			198.969i			0.415385i	0.978194	+	0.207692i	$0.0665953\pi$
$479$			198.969i			0.415385i	−0.978194	+	0.207692i	$0.933405\pi$
$480$		0			0
$481$	180.000			0.374220
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−610.095	−	274.232i	−1.25793	−	0.565426i
$486$		0			0
$487$	34.6209	−	34.6209i	0.0710902	−	0.0710902i	−0.670668	−	0.741758i	$-0.733992\pi$
$487$	34.6209	−	34.6209i	0.0710902	−	0.0710902i	0.741758	+	0.670668i	$0.233992\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−155.217			−0.316124			−0.158062	−	0.987429i	$-0.550525\pi$
$491$	−155.217			−0.316124			−0.158062	+	0.987429i	$0.550525\pi$
$492$		0			0
$493$	−121.414	−	121.414i	−0.246276	−	0.246276i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	300.908	−	300.908i	0.605449	−	0.605449i
$498$		0			0
$499$			547.444i			1.09708i	0.836124	+	0.548541i	$0.184817\pi$
$499$			547.444i			1.09708i	−0.836124	+	0.548541i	$0.815183\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−526.590	−	526.590i	−1.04690	−	1.04690i	−0.998845	−	0.0480545i	$-0.984698\pi$
$503$	−526.590	−	526.590i	−1.04690	−	1.04690i	−0.0480545	−	0.998845i	$-0.515302\pi$
$504$		0			0
$505$	124.707	+	328.353i	0.246945	+	0.650204i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	115.773i		−	0.227452i	−0.993512	−	0.113726i	$-0.963721\pi$
$509$		−	115.773i		−	0.227452i	0.993512	−	0.113726i	$-0.0362786\pi$
$510$		0			0
$511$	−174.413			−0.341318
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−251.757	+	560.095i	−0.488849	+	1.08756i
$516$		0			0
$517$	−181.818	+	181.818i	−0.351680	+	0.351680i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−791.444			−1.51909			−0.759543	−	0.650457i	$-0.774577\pi$
$521$	−791.444			−1.51909			−0.759543	+	0.650457i	$0.774577\pi$
$522$		0			0
$523$	−93.6867	−	93.6867i	−0.179133	−	0.179133i	0.611845	−	0.790978i	$-0.290428\pi$
$523$	−93.6867	−	93.6867i	−0.179133	−	0.179133i	−0.790978	+	0.611845i	$0.790428\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−354.747	+	354.747i	−0.673144	+	0.673144i
$528$		0			0
$529$			72.9388i			0.137880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	912.372	+	912.372i	1.71177	+	1.71177i
$534$		0			0
$535$	621.333	+	279.283i	1.16137	+	0.522024i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			296.824i			0.550694i
$540$		0			0
$541$	−359.526			−0.664557			−0.332279	−	0.943181i	$-0.607817\pi$
$541$	−359.526			−0.664557			−0.332279	+	0.943181i	$0.607817\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−58.4995	+	22.2179i	−0.107339	+	0.0407667i
$546$		0			0
$547$	−164.647	+	164.647i	−0.301000	+	0.301000i	−0.841405	−	0.540405i	$-0.818271\pi$
$547$	−164.647	+	164.647i	−0.301000	+	0.301000i	0.540405	+	0.841405i	$0.318271\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	336.413			0.610550
$552$		0			0
$553$	−1.26224	−	1.26224i	−0.00228253	−	0.00228253i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−139.040	+	139.040i	−0.249623	+	0.249623i	−0.820816	−	0.571193i	$-0.806481\pi$
$557$	−139.040	+	139.040i	−0.249623	+	0.249623i	0.571193	+	0.820816i	$0.306481\pi$
$558$		0			0
$559$			854.847i			1.52924i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−787.423	−	787.423i	−1.39862	−	1.39862i	−0.804024	−	0.594596i	$-0.797312\pi$
$563$	−787.423	−	787.423i	−1.39862	−	1.39862i	−0.594596	−	0.804024i	$-0.702688\pi$
$564$		0			0
$565$	−194.070	+	431.757i	−0.343487	+	0.764172i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			4.92856i			0.00866180i	0.999991	+	0.00433090i	$0.00137857\pi$
$569$			4.92856i			0.00866180i	−0.999991	+	0.00433090i	$0.998621\pi$
$570$		0			0
$571$	−205.505			−0.359904			−0.179952	−	0.983675i	$-0.557594\pi$
$571$	−205.505			−0.359904			−0.179952	+	0.983675i	$0.557594\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	612.277	+	36.4495i	1.06483	+	0.0633904i
$576$		0			0
$577$	689.999	−	689.999i	1.19584	−	1.19584i	0.220438	−	0.975401i	$-0.429251\pi$
$577$	689.999	−	689.999i	1.19584	−	1.19584i	0.975401	−	0.220438i	$-0.0707486\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	93.6480			0.161184
$582$		0			0
$583$	533.585	+	533.585i	0.915240	+	0.915240i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−672.347	+	672.347i	−1.14540	+	1.14540i	−0.157949	+	0.987447i	$0.550488\pi$
$587$	−672.347	+	672.347i	−1.14540	+	1.14540i	−0.987447	+	0.157949i	$0.949512\pi$
$588$		0			0
$589$		−	982.929i		−	1.66881i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−344.570	−	344.570i	−0.581062	−	0.581062i	0.354133	−	0.935195i	$-0.384776\pi$
$593$	−344.570	−	344.570i	−0.581062	−	0.581062i	−0.935195	+	0.354133i	$0.884776\pi$
$594$		0			0
$595$	296.838	−	112.738i	0.498887	−	0.189475i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			739.464i			1.23450i	0.786768	+	0.617249i	$0.211753\pi$
$599$			739.464i			1.23450i	−0.786768	+	0.617249i	$0.788247\pi$
$600$		0			0
$601$	−642.908			−1.06973			−0.534865	−	0.844937i	$-0.679638\pi$
$601$	−642.908			−1.06973			−0.534865	+	0.844937i	$0.679638\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	94.0523	+	247.639i	0.155458	+	0.409321i
$606$		0			0
$607$	362.803	−	362.803i	0.597698	−	0.597698i	−0.342002	−	0.939699i	$-0.611105\pi$
$607$	362.803	−	362.803i	0.597698	−	0.597698i	0.939699	+	0.342002i	$0.111105\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	540.000			0.883797
$612$		0			0
$613$	−721.943	−	721.943i	−1.17772	−	1.17772i	−0.980323	−	0.197398i	$-0.936751\pi$
$613$	−721.943	−	721.943i	−1.17772	−	1.17772i	−0.197398	−	0.980323i	$-0.563249\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−72.9036	+	72.9036i	−0.118158	+	0.118158i	−0.763713	−	0.645555i	$-0.776626\pi$
$617$	−72.9036	+	72.9036i	−0.118158	+	0.118158i	0.645555	+	0.763713i	$0.276626\pi$
$618$		0			0
$619$		−	228.061i		−	0.368435i	−0.982886	−	0.184217i	$-0.941025\pi$
$619$		−	228.061i		−	0.368435i	0.982886	−	0.184217i	$-0.0589751\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	349.342	+	349.342i	0.560741	+	0.560741i
$624$		0			0
$625$	74.1510	−	620.586i	0.118642	−	0.992937i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			182.969i			0.290889i
$630$		0			0
$631$	86.4337			0.136979			0.0684894	−	0.997652i	$-0.478182\pi$
$631$	86.4337			0.136979			0.0684894	+	0.997652i	$0.478182\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	16.4495	+	7.39388i	0.0259047	+	0.0116439i
$636$		0			0
$637$	440.783	−	440.783i	0.691967	−	0.691967i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−340.061			−0.530517			−0.265258	−	0.964177i	$-0.585457\pi$
$641$	−340.061			−0.530517			−0.265258	+	0.964177i	$0.585457\pi$
$642$		0			0
$643$	−769.040	−	769.040i	−1.19602	−	1.19602i	−0.975349	−	0.220670i	$-0.929176\pi$
$643$	−769.040	−	769.040i	−1.19602	−	1.19602i	−0.220670	−	0.975349i	$-0.570824\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−602.854	+	602.854i	−0.931767	+	0.931767i	−0.997816	−	0.0660489i	$-0.978961\pi$
$647$	−602.854	+	602.854i	−0.931767	+	0.931767i	0.0660489	+	0.997816i	$0.478961\pi$
$648$		0			0
$649$			125.837i			0.193893i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	75.8842	+	75.8842i	0.116209	+	0.116209i	0.762820	−	0.646611i	$-0.223814\pi$
$653$	75.8842	+	75.8842i	0.116209	+	0.116209i	−0.646611	+	0.762820i	$0.723814\pi$
$654$		0			0
$655$	−78.5505	−	206.823i	−0.119924	−	0.315760i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			61.1964i			0.0928626i	0.998921	+	0.0464313i	$0.0147849\pi$
$659$			61.1964i			0.0928626i	−0.998921	+	0.0464313i	$0.985215\pi$
$660$		0			0
$661$	1158.45			1.75258			0.876289	−	0.481786i	$-0.160012\pi$
$661$	1158.45			1.75258			0.876289	+	0.481786i	$0.160012\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−255.051	+	567.423i	−0.383535	+	0.853268i
$666$		0			0
$667$	−169.192	+	169.192i	−0.253661	+	0.253661i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−441.281			−0.657646
$672$		0			0
$673$	711.413	+	711.413i	1.05708	+	1.05708i	0.998269	+	0.0588084i	$0.0187301\pi$
$673$	711.413	+	711.413i	1.05708	+	1.05708i	0.0588084	+	0.998269i	$0.481270\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−351.943	+	351.943i	−0.519857	+	0.519857i	−0.917528	−	0.397671i	$-0.869819\pi$
$677$	−351.943	+	351.943i	−0.519857	+	0.519857i	0.397671	+	0.917528i	$0.369819\pi$
$678$		0			0
$679$			482.536i			0.710656i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	277.905	+	277.905i	0.406888	+	0.406888i	0.880652	−	0.473764i	$-0.157105\pi$
$683$	277.905	+	277.905i	0.406888	+	0.406888i	−0.473764	+	0.880652i	$0.657105\pi$
$684$		0			0
$685$	−118.404	−	53.2214i	−0.172853	−	0.0776955i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1584.74i		−	2.30007i
$690$		0			0
$691$	1097.51			1.58829			0.794143	−	0.607731i	$-0.207920\pi$
$691$	1097.51			1.58829			0.794143	+	0.607731i	$0.207920\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1028.24	−	390.520i	1.47948	−	0.561899i
$696$		0			0
$697$	−927.423	+	927.423i	−1.33059	+	1.33059i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	354.783			0.506110			0.253055	−	0.967452i	$-0.418565\pi$
$701$	354.783			0.506110			0.253055	+	0.967452i	$0.418565\pi$
$702$		0			0
$703$	−253.485	−	253.485i	−0.360576	−	0.360576i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	179.167	−	179.167i	0.253418	−	0.253418i
$708$		0			0
$709$			368.061i			0.519127i	0.965726	+	0.259564i	$0.0835787\pi$
$709$			368.061i			0.519127i	−0.965726	+	0.259564i	$0.916421\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	494.343	+	494.343i	0.693328	+	0.693328i
$714$		0			0
$715$	−292.827	+	651.464i	−0.409548	+	0.911139i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	382.969i		−	0.532642i	−0.963884	−	0.266321i	$-0.914192\pi$
$719$		−	382.969i		−	0.532642i	0.963884	−	0.266321i	$-0.0858081\pi$
$720$		0			0
$721$	442.990			0.614410
$722$		0			0
$723$		0			0
$724$		0			0
$725$	161.853	+	182.343i	0.223245	+	0.251507i
$726$		0			0
$727$	−837.297	+	837.297i	−1.15172	+	1.15172i	−0.165507	+	0.986209i	$0.552926\pi$
$727$	−837.297	+	837.297i	−1.15172	+	1.15172i	−0.986209	+	0.165507i	$0.947074\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−868.949			−1.18871
$732$		0			0
$733$	−29.8230	−	29.8230i	−0.0406862	−	0.0406862i	0.686471	−	0.727157i	$-0.259159\pi$
$733$	−29.8230	−	29.8230i	−0.0406862	−	0.0406862i	−0.727157	+	0.686471i	$0.759159\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−38.7378	+	38.7378i	−0.0525614	+	0.0525614i
$738$		0			0
$739$		−	1183.28i		−	1.60119i	−0.599205	−	0.800596i	$-0.704516\pi$
$739$		−	1183.28i		−	1.60119i	0.599205	−	0.800596i	$-0.295484\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	65.7821	+	65.7821i	0.0885358	+	0.0885358i	0.749988	−	0.661452i	$-0.230059\pi$
$743$	65.7821	+	65.7821i	0.0885358	+	0.0885358i	−0.661452	+	0.749988i	$0.730059\pi$
$744$		0			0
$745$	−132.131	+	50.1827i	−0.177357	+	0.0673593i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	491.423i		−	0.656106i
$750$		0			0
$751$	850.270			1.13218			0.566092	−	0.824342i	$-0.308455\pi$
$751$	850.270			1.13218			0.566092	+	0.824342i	$0.308455\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−108.309	−	285.176i	−0.143455	−	0.377717i
$756$		0			0
$757$	145.832	−	145.832i	0.192645	−	0.192645i	−0.604193	−	0.796838i	$-0.706504\pi$
$757$	145.832	−	145.832i	0.192645	−	0.192645i	0.796838	+	0.604193i	$0.206504\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−157.505			−0.206971			−0.103486	−	0.994631i	$-0.533000\pi$
$761$	−157.505			−0.206971			−0.103486	+	0.994631i	$0.533000\pi$
$762$		0			0
$763$	31.9204	+	31.9204i	0.0418354	+	0.0418354i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	186.867	−	186.867i	0.243634	−	0.243634i
$768$		0			0
$769$			559.031i			0.726958i	0.931602	+	0.363479i	$0.118411\pi$
$769$			559.031i			0.726958i	−0.931602	+	0.363479i	$0.881589\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	348.965	+	348.965i	0.451442	+	0.451442i	0.895833	−	0.444391i	$-0.146580\pi$
$773$	348.965	+	348.965i	0.451442	+	0.451442i	−0.444391	+	0.895833i	$0.646580\pi$
$774$		0			0
$775$	532.767	−	472.899i	0.687442	−	0.610192i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	2569.69i		−	3.29871i
$780$		0			0
$781$	−973.031			−1.24588
$782$		0			0
$783$		0			0
$784$		0			0
$785$	146.091	+	65.6663i	0.186103	+	0.0836514i
$786$		0			0
$787$	821.414	−	821.414i	1.04373	−	1.04373i	0.0447293	−	0.998999i	$-0.485757\pi$
$787$	821.414	−	821.414i	1.04373	−	1.04373i	0.998999	−	0.0447293i	$-0.0142425\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	341.485			0.431713
$792$		0			0
$793$	655.301	+	655.301i	0.826357	+	0.826357i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	649.065	−	649.065i	0.814385	−	0.814385i	−0.170903	−	0.985288i	$-0.554668\pi$
$797$	649.065	−	649.065i	0.814385	−	0.814385i	0.985288	+	0.170903i	$0.0546685\pi$
$798$		0			0
$799$			548.908i			0.686994i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	281.995	+	281.995i	0.351177	+	0.351177i
$804$		0			0
$805$	−157.101	−	413.646i	−0.195157	−	0.513846i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	909.918i		−	1.12474i	−0.826884	−	0.562372i	$-0.809889\pi$
$809$		−	909.918i		−	1.12474i	0.826884	−	0.562372i	$-0.190111\pi$
$810$		0			0
$811$	652.929			0.805091			0.402545	−	0.915400i	$-0.368126\pi$
$811$	652.929			0.805091			0.402545	+	0.915400i	$0.368126\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	257.980	−	573.939i	0.316539	−	0.704219i
$816$		0			0
$817$	1203.84	−	1203.84i	1.47348	−	1.47348i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	808.064			0.984243			0.492122	−	0.870526i	$-0.336222\pi$
$821$	808.064			0.984243			0.492122	+	0.870526i	$0.336222\pi$
$822$		0			0
$823$	−333.863	−	333.863i	−0.405666	−	0.405666i	0.474558	−	0.880224i	$-0.342608\pi$
$823$	−333.863	−	333.863i	−0.405666	−	0.405666i	−0.880224	+	0.474558i	$0.842608\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	481.162	−	481.162i	0.581817	−	0.581817i	−0.353586	−	0.935402i	$-0.615038\pi$
$827$	481.162	−	481.162i	0.581817	−	0.581817i	0.935402	+	0.353586i	$0.115038\pi$
$828$		0			0
$829$			1296.35i			1.56375i	0.623433	+	0.781877i	$0.285738\pi$
$829$			1296.35i			1.56375i	−0.623433	+	0.781877i	$0.714262\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	448.055	+	448.055i	0.537881	+	0.537881i
$834$		0			0
$835$	−1107.00	−	497.585i	−1.32575	−	0.595910i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			978.515i			1.16629i	0.812369	+	0.583144i	$0.198178\pi$
$839$			978.515i			1.16629i	−0.812369	+	0.583144i	$0.801822\pi$
$840$		0			0
$841$	745.888			0.886906
$842$		0			0
$843$		0			0
$844$		0			0
$845$	612.325	−	232.558i	0.724645	−	0.275217i
$846$		0			0
$847$	135.125	−	135.125i	0.159534	−	0.159534i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	254.969			0.299611
$852$		0			0
$853$	300.127	+	300.127i	0.351849	+	0.351849i	0.860797	−	0.508948i	$-0.169966\pi$
$853$	300.127	+	300.127i	0.351849	+	0.351849i	−0.508948	+	0.860797i	$0.669966\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−879.823	+	879.823i	−1.02663	+	1.02663i	−0.0269957	+	0.999636i	$0.508594\pi$
$857$	−879.823	+	879.823i	−1.02663	+	1.02663i	−0.999636	+	0.0269957i	$0.991406\pi$
$858$		0			0
$859$		−	128.888i		−	0.150044i	−0.997182	−	0.0750220i	$-0.976097\pi$
$859$		−	128.888i		−	0.150044i	0.997182	−	0.0750220i	$-0.0239027\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	204.418	+	204.418i	0.236869	+	0.236869i	0.815552	−	0.578683i	$-0.196433\pi$
$863$	204.418	+	204.418i	0.236869	+	0.236869i	−0.578683	+	0.815552i	$0.696433\pi$
$864$		0			0
$865$	30.8194	−	68.5653i	0.0356294	−	0.0792662i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			4.08164i			0.00469694i
$870$		0			0
$871$	115.051			0.132091
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−430.263	+	134.752i	−0.491729	+	0.154002i
$876$		0			0
$877$	−570.025	+	570.025i	−0.649971	+	0.649971i	−0.952986	−	0.303014i	$-0.902007\pi$
$877$	−570.025	+	570.025i	−0.649971	+	0.649971i	0.303014	+	0.952986i	$0.402007\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1235.86			1.40279			0.701395	−	0.712773i	$-0.252561\pi$
$881$	1235.86			1.40279			0.701395	+	0.712773i	$0.252561\pi$
$882$		0			0
$883$	713.131	+	713.131i	0.807622	+	0.807622i	0.984274	−	0.176651i	$-0.0565264\pi$
$883$	713.131	+	713.131i	0.807622	+	0.807622i	−0.176651	+	0.984274i	$0.556526\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	343.510	−	343.510i	0.387271	−	0.387271i	−0.486442	−	0.873713i	$-0.661705\pi$
$887$	343.510	−	343.510i	0.387271	−	0.387271i	0.873713	+	0.486442i	$0.161705\pi$
$888$		0			0
$889$		−	13.0102i		−	0.0146347i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−760.454	−	760.454i	−0.851572	−	0.851572i
$894$		0			0
$895$	−468.580	+	177.965i	−0.523553	+	0.198843i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			277.898i			0.309119i
$900$		0			0
$901$	1610.89			1.78789
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−460.579	−	1212.70i	−0.508927	−	1.34000i
$906$		0			0
$907$	918.938	−	918.938i	1.01316	−	1.01316i	0.0132496	−	0.999912i	$-0.495782\pi$
$907$	918.938	−	918.938i	1.01316	−	1.01316i	0.999912	−	0.0132496i	$-0.00421759\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	788.454			0.865482			0.432741	−	0.901518i	$-0.357546\pi$
$911$	788.454			0.865482			0.432741	+	0.901518i	$0.357546\pi$
$912$		0			0
$913$	−151.412	−	151.412i	−0.165840	−	0.165840i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−112.854	+	112.854i	−0.123068	+	0.123068i
$918$		0			0
$919$			1644.33i			1.78926i	0.446806	+	0.894631i	$0.352561\pi$
$919$			1644.33i			1.78926i	−0.446806	+	0.894631i	$0.647439\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1444.95	+	1444.95i	1.56549	+	1.56549i
$924$		0			0
$925$	15.4393	−	259.348i	0.0166911	−	0.280377i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1021.51i			1.09957i	0.835305	+	0.549787i	$0.185291\pi$
$929$			1021.51i			1.09957i	−0.835305	+	0.549787i	$0.814709\pi$
$930$		0			0
$931$	−1241.46			−1.33347
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−662.211	−	297.657i	−0.708247	−	0.318350i
$936$		0			0
$937$	−370.555	+	370.555i	−0.395470	+	0.395470i	−0.876632	−	0.481162i	$-0.840215\pi$
$937$	−370.555	+	370.555i	−0.395470	+	0.395470i	0.481162	+	0.876632i	$0.340215\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1524.06			−1.61962			−0.809811	−	0.586691i	$-0.800430\pi$
$941$	−1524.06			−1.61962			−0.809811	+	0.586691i	$0.800430\pi$
$942$		0			0
$943$	1292.37	+	1292.37i	1.37049	+	1.37049i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−644.545	+	644.545i	−0.680618	+	0.680618i	−0.960139	−	0.279522i	$-0.909824\pi$
$947$	−644.545	+	644.545i	−0.680618	+	0.680618i	0.279522	+	0.960139i	$0.409824\pi$
$948$		0			0
$949$		−	837.526i		−	0.882535i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	461.641	+	461.641i	0.484409	+	0.484409i	0.906536	−	0.422128i	$-0.138717\pi$
$953$	461.641	+	461.641i	0.484409	+	0.484409i	−0.422128	+	0.906536i	$0.638717\pi$
$954$		0			0
$955$	−137.555	−	362.182i	−0.144037	−	0.379248i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			93.6480i			0.0976517i
$960$		0			0
$961$	−149.041			−0.155089
$962$		0			0
$963$		0			0
$964$		0			0
$965$	242.313	−	539.085i	0.251102	−	0.558638i
$966$		0			0
$967$	−462.196	+	462.196i	−0.477969	+	0.477969i	−0.904482	−	0.426512i	$-0.859742\pi$
$967$	−462.196	+	462.196i	−0.477969	+	0.477969i	0.426512	+	0.904482i	$0.359742\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−198.390			−0.204315			−0.102158	−	0.994768i	$-0.532575\pi$
$971$	−198.390			−0.204315			−0.102158	+	0.994768i	$0.532575\pi$
$972$		0			0
$973$	−561.060	−	561.060i	−0.576629	−	0.576629i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−543.206	+	543.206i	−0.555993	+	0.555993i	−0.928164	−	0.372171i	$-0.878614\pi$
$977$	−543.206	+	543.206i	−0.555993	+	0.555993i	0.372171	+	0.928164i	$0.378614\pi$
$978$		0			0
$979$		−	1129.65i		−	1.15388i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−321.387	−	321.387i	−0.326945	−	0.326945i	0.524478	−	0.851424i	$-0.324260\pi$
$983$	−321.387	−	321.387i	−0.326945	−	0.326945i	−0.851424	+	0.524478i	$0.824260\pi$
$984$		0			0
$985$	−1291.52	−	580.527i	−1.31119	−	0.589367i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1210.89i			1.22436i
$990$		0			0
$991$	1543.82			1.55784			0.778918	−	0.627125i	$-0.215769\pi$
$991$	1543.82			1.55784			0.778918	+	0.627125i	$0.215769\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−761.853	+	289.348i	−0.765681	+	0.290802i
$996$		0			0
$997$	800.429	−	800.429i	0.802838	−	0.802838i	−0.180701	−	0.983538i	$-0.557836\pi$
$997$	800.429	−	800.429i	0.802838	−	0.802838i	0.983538	+	0.180701i	$0.0578365\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.bh.f.433.2		4
3.2	odd	2		240.3.bg.d.193.1		4
4.3	odd	2		180.3.l.b.73.2		4
5.2	odd	4	inner	720.3.bh.f.577.2		4
12.11	even	2		60.3.k.a.13.2	✓	4
15.2	even	4		240.3.bg.d.97.1		4
15.8	even	4		1200.3.bg.o.1057.2		4
15.14	odd	2		1200.3.bg.o.193.2		4
20.3	even	4		900.3.l.b.757.2		4
20.7	even	4		180.3.l.b.37.2		4
20.19	odd	2		900.3.l.b.793.2		4
24.5	odd	2		960.3.bg.a.193.2		4
24.11	even	2		960.3.bg.b.193.1		4
60.23	odd	4		300.3.k.a.157.1		4
60.47	odd	4		60.3.k.a.37.2	yes	4
60.59	even	2		300.3.k.a.193.1		4
120.77	even	4		960.3.bg.a.577.2		4
120.107	odd	4		960.3.bg.b.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.k.a.13.2	✓	4	12.11	even	2
60.3.k.a.37.2	yes	4	60.47	odd	4
180.3.l.b.37.2		4	20.7	even	4
180.3.l.b.73.2		4	4.3	odd	2
240.3.bg.d.97.1		4	15.2	even	4
240.3.bg.d.193.1		4	3.2	odd	2
300.3.k.a.157.1		4	60.23	odd	4
300.3.k.a.193.1		4	60.59	even	2
720.3.bh.f.433.2		4	1.1	even	1	trivial
720.3.bh.f.577.2		4	5.2	odd	4	inner
900.3.l.b.757.2		4	20.3	even	4
900.3.l.b.793.2		4	20.19	odd	2
960.3.bg.a.193.2		4	24.5	odd	2
960.3.bg.a.577.2		4	120.77	even	4
960.3.bg.b.193.1		4	24.11	even	2
960.3.bg.b.577.1		4	120.107	odd	4
1200.3.bg.o.193.2		4	15.14	odd	2
1200.3.bg.o.1057.2		4	15.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.bh (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.77526 - 4.67423i) q^{5} +(-2.55051 + 2.55051i) q^{7} +8.24745 q^{11} +(-12.2474 - 12.2474i) q^{13} +(12.4495 - 12.4495i) q^{17} +34.4949i q^{19} +(-17.3485 - 17.3485i) q^{23} +(-18.6969 + 16.5959i) q^{25} -9.75255i q^{29} -28.4949 q^{31} +(16.4495 + 7.39388i) q^{35} +(-7.34847 + 7.34847i) q^{37} -74.4949 q^{41} +(-34.8990 - 34.8990i) q^{43} +(-22.0454 + 22.0454i) q^{47} +35.9898i q^{49} +(64.6969 + 64.6969i) q^{53} +(-14.6413 - 38.5505i) q^{55} +15.2577i q^{59} -53.5051 q^{61} +(-35.5051 + 78.9898i) q^{65} +(-4.69694 + 4.69694i) q^{67} -117.980 q^{71} +(34.1918 + 34.1918i) q^{73} +(-21.0352 + 21.0352i) q^{77} +0.494897i q^{79} +(-18.3587 - 18.3587i) q^{83} +(-80.2929 - 36.0908i) q^{85} -136.969i q^{89} +62.4745 q^{91} +(161.237 - 61.2372i) q^{95} +(94.5959 - 94.5959i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{5} - 20 q^{7} - 16 q^{11} + 40 q^{17} - 40 q^{23} - 16 q^{25} - 16 q^{31} + 56 q^{35} - 200 q^{41} - 120 q^{43} + 200 q^{53} + 108 q^{55} - 312 q^{61} - 240 q^{65} + 40 q^{67} - 80 q^{71} - 20 q^{73}+ \cdots + 300 q^{97}+O(q^{100}) \) 4 * q - 12 * q^5 - 20 * q^7 - 16 * q^11 + 40 * q^17 - 40 * q^23 - 16 * q^25 - 16 * q^31 + 56 * q^35 - 200 * q^41 - 120 * q^43 + 200 * q^53 + 108 * q^55 - 312 * q^61 - 240 * q^65 + 40 * q^67 - 80 * q^71 - 20 * q^73 + 200 * q^77 - 240 * q^83 - 184 * q^85 - 240 * q^91 + 400 * q^95 + 300 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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