LMFDB - Embedded newform 540.4.i.c.361.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [540,4,Mod(181,540)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(540, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("540.181");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 540 = 2^{2} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	540.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$31.8610314031$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 7 x^{13} + 180 x^{12} - 989 x^{11} + 11627 x^{10} - 49236 x^{9} + 328637 x^{8} - 1029725 x^{7} + \cdots + 1484973 $ x^14 - 7x^13 + 180x^12 - 989x^11 + 11627x^10 - 49236x^9 + 328637x^8 - 1029725x^7 + 3973395x^6 - 8512360x^5 + 18130585x^4 - 23182065x^3 + 18357462x^2 - 8027505*x + 1484973
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{17} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.5
Root			$0.500000 + 7.22388i$ of defining polynomial
Character	$\chi$	$=$	540.361
Dual form			540.4.i.c.181.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{5} +(6.86720 + 11.8943i) q^{7} +(10.8215 + 18.7434i) q^{11} +(39.8133 - 68.9586i) q^{13} +64.3628 q^{17} -144.503 q^{19} +(-13.4725 + 23.3351i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(148.503 + 257.215i) q^{29} +(48.3254 - 83.7020i) q^{31} +68.6720 q^{35} +401.580 q^{37} +(-5.84448 + 10.1229i) q^{41} +(-146.136 - 253.114i) q^{43} +(87.6812 + 151.868i) q^{47} +(77.1832 - 133.685i) q^{49} -155.406 q^{53} +108.215 q^{55} +(297.582 - 515.427i) q^{59} +(31.0535 + 53.7862i) q^{61} +(-199.066 - 344.793i) q^{65} +(80.3318 - 139.139i) q^{67} +959.723 q^{71} +763.424 q^{73} +(-148.627 + 257.429i) q^{77} +(-31.9825 - 55.3952i) q^{79} +(458.663 + 794.427i) q^{83} +(160.907 - 278.699i) q^{85} +1318.88 q^{89} +1093.62 q^{91} +(-361.256 + 625.714i) q^{95} +(64.0845 + 110.998i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 35 q^{5} - 8 q^{7} + 27 q^{11} - 32 q^{13} - 246 q^{17} - 134 q^{19} + 42 q^{23} - 175 q^{25} + 324 q^{29} - 98 q^{31} - 80 q^{35} + 712 q^{37} + 339 q^{41} - 119 q^{43} - 96 q^{47} - 813 q^{49}+ \cdots - 1991 q^{97}+O(q^{100}) $ 14 * q + 35 * q^5 - 8 * q^7 + 27 * q^11 - 32 * q^13 - 246 * q^17 - 134 * q^19 + 42 * q^23 - 175 * q^25 + 324 * q^29 - 98 * q^31 - 80 * q^35 + 712 * q^37 + 339 * q^41 - 119 * q^43 - 96 * q^47 - 813 * q^49 - 1716 * q^53 + 270 * q^55 + 549 * q^59 - 260 * q^61 + 160 * q^65 - 881 * q^67 - 684 * q^71 + 1474 * q^73 - 456 * q^77 - 1886 * q^79 - 132 * q^83 - 615 * q^85 - 792 * q^89 + 2528 * q^91 - 335 * q^95 - 1991 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$461$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	6.86720	+	11.8943i	0.370794	+	0.642234i	0.989688	−	0.143241i	$-0.0457523\pi$
$7$	6.86720	+	11.8943i	0.370794	+	0.642234i	−0.618894	+	0.785475i	$0.712419\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.8215	+	18.7434i	0.296619	+	0.513759i	0.975360	−	0.220619i	$-0.0708076\pi$
$11$	10.8215	+	18.7434i	0.296619	+	0.513759i	−0.678741	+	0.734377i	$0.737474\pi$
$12$		0			0
$13$	39.8133	−	68.9586i	0.849401	−	1.47121i	−0.0323418	−	0.999477i	$-0.510297\pi$
$13$	39.8133	−	68.9586i	0.849401	−	1.47121i	0.881743	−	0.471730i	$-0.156370\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	64.3628			0.918251			0.459125	−	0.888371i	$-0.348163\pi$
$17$	64.3628			0.918251			0.459125	+	0.888371i	$0.348163\pi$
$18$		0			0
$19$	−144.503			−1.74480			−0.872399	−	0.488795i	$-0.837437\pi$
$19$	−144.503			−1.74480			−0.872399	+	0.488795i	$0.837437\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−13.4725	+	23.3351i	−0.122140	+	0.211552i	−0.920611	−	0.390480i	$-0.872309\pi$
$23$	−13.4725	+	23.3351i	−0.122140	+	0.211552i	0.798471	+	0.602033i	$0.205642\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	148.503	+	257.215i	0.950908	+	1.64702i	0.743465	+	0.668775i	$0.233181\pi$
$29$	148.503	+	257.215i	0.950908	+	1.64702i	0.207443	+	0.978247i	$0.433486\pi$
$30$		0			0
$31$	48.3254	−	83.7020i	0.279984	−	0.484946i	−0.691397	−	0.722475i	$-0.743004\pi$
$31$	48.3254	−	83.7020i	0.279984	−	0.484946i	0.971380	+	0.237529i	$0.0763376\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	68.6720			0.331648
$36$		0			0
$37$	401.580			1.78431			0.892153	−	0.451734i	$-0.149194\pi$
$37$	401.580			1.78431			0.892153	+	0.451734i	$0.149194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.84448	+	10.1229i	−0.0222623	+	0.0385595i	−0.876942	−	0.480596i	$-0.840420\pi$
$41$	−5.84448	+	10.1229i	−0.0222623	+	0.0385595i	0.854680	+	0.519156i	$0.173754\pi$
$42$		0			0
$43$	−146.136	−	253.114i	−0.518267	−	0.897665i	−0.999775	−	0.0212231i	$-0.993244\pi$
$43$	−146.136	−	253.114i	−0.518267	−	0.897665i	0.481508	−	0.876442i	$-0.340089\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	87.6812	+	151.868i	0.272119	+	0.471325i	0.969404	−	0.245469i	$-0.0789421\pi$
$47$	87.6812	+	151.868i	0.272119	+	0.471325i	−0.697285	+	0.716794i	$0.745609\pi$
$48$		0			0
$49$	77.1832	−	133.685i	0.225024	−	0.389753i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−155.406			−0.402767			−0.201383	−	0.979513i	$-0.564544\pi$
$53$	−155.406			−0.402767			−0.201383	+	0.979513i	$0.564544\pi$
$54$		0			0
$55$	108.215			0.265304
$56$		0			0
$57$		0			0
$58$		0			0
$59$	297.582	−	515.427i	0.656641	−	1.13734i	−0.324838	−	0.945770i	$-0.605310\pi$
$59$	297.582	−	515.427i	0.656641	−	1.13734i	0.981480	−	0.191567i	$-0.0613568\pi$
$60$		0			0
$61$	31.0535	+	53.7862i	0.0651802	+	0.112895i	0.896774	−	0.442489i	$-0.145904\pi$
$61$	31.0535	+	53.7862i	0.0651802	+	0.112895i	−0.831594	+	0.555384i	$0.812571\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−199.066	−	344.793i	−0.379864	−	0.657944i
$66$		0			0
$67$	80.3318	−	139.139i	0.146479	−	0.253709i	−0.783445	−	0.621461i	$-0.786539\pi$
$67$	80.3318	−	139.139i	0.146479	−	0.253709i	0.929924	+	0.367752i	$0.119873\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	959.723			1.60420			0.802100	−	0.597190i	$-0.203716\pi$
$71$	959.723			1.60420			0.802100	+	0.597190i	$0.203716\pi$
$72$		0			0
$73$	763.424			1.22400			0.612000	−	0.790858i	$-0.290365\pi$
$73$	763.424			1.22400			0.612000	+	0.790858i	$0.290365\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−148.627	+	257.429i	−0.219969	+	0.380997i
$78$		0			0
$79$	−31.9825	−	55.3952i	−0.0455482	−	0.0788918i	0.842352	−	0.538927i	$-0.181170\pi$
$79$	−31.9825	−	55.3952i	−0.0455482	−	0.0788918i	−0.887901	+	0.460035i	$0.847837\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	458.663	+	794.427i	0.606564	+	1.05060i	0.991802	+	0.127782i	$0.0407858\pi$
$83$	458.663	+	794.427i	0.606564	+	1.05060i	−0.385239	+	0.922817i	$0.625881\pi$
$84$		0			0
$85$	160.907	−	278.699i	0.205327	−	0.355637i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1318.88			1.57080			0.785399	−	0.618990i	$-0.212458\pi$
$89$	1318.88			1.57080			0.785399	+	0.618990i	$0.212458\pi$
$90$		0			0
$91$	1093.62			1.25981
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−361.256	+	625.714i	−0.390149	+	0.675757i
$96$		0			0
$97$	64.0845	+	110.998i	0.0670804	+	0.116187i	0.897615	−	0.440780i	$-0.145298\pi$
$97$	64.0845	+	110.998i	0.0670804	+	0.116187i	−0.830535	+	0.556967i	$0.811965\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−230.204	−	398.725i	−0.226793	−	0.392818i	0.730063	−	0.683380i	$-0.239491\pi$
$101$	−230.204	−	398.725i	−0.226793	−	0.392818i	−0.956856	+	0.290563i	$0.906158\pi$
$102$		0			0
$103$	−358.731	+	621.340i	−0.343173	+	0.594393i	−0.985020	−	0.172440i	$-0.944835\pi$
$103$	−358.731	+	621.340i	−0.343173	+	0.594393i	0.641847	+	0.766832i	$0.278168\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1361.76			−1.23034			−0.615171	−	0.788393i	$-0.710913\pi$
$107$	−1361.76			−1.23034			−0.615171	+	0.788393i	$0.710913\pi$
$108$		0			0
$109$	225.862			0.198474			0.0992369	−	0.995064i	$-0.468360\pi$
$109$	225.862			0.198474			0.0992369	+	0.995064i	$0.468360\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	984.234	−	1704.74i	0.819371	−	1.41919i	−0.0867746	−	0.996228i	$-0.527656\pi$
$113$	984.234	−	1704.74i	0.819371	−	1.41919i	0.906146	−	0.422965i	$-0.139011\pi$
$114$		0			0
$115$	67.3627	+	116.676i	0.0546226	+	0.0946091i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	441.992	+	765.552i	0.340482	+	0.589732i
$120$		0			0
$121$	431.290	−	747.017i	0.324035	−	0.561245i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	751.204			0.524871			0.262435	−	0.964950i	$-0.415474\pi$
$127$	751.204			0.524871			0.262435	+	0.964950i	$0.415474\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−39.7899	+	68.9181i	−0.0265378	+	0.0459649i	−0.878989	−	0.476841i	$-0.841782\pi$
$131$	−39.7899	+	68.9181i	−0.0265378	+	0.0459649i	0.852451	+	0.522806i	$0.175115\pi$
$132$		0			0
$133$	−992.328	−	1718.76i	−0.646960	−	1.12057i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	401.792	+	695.924i	0.250565	+	0.433991i	0.963682	−	0.267054i	$-0.0860503\pi$
$137$	401.792	+	695.924i	0.250565	+	0.433991i	−0.713116	+	0.701046i	$0.752717\pi$
$138$		0			0
$139$	−880.833	+	1525.65i	−0.537491	+	0.930962i	0.461547	+	0.887116i	$0.347294\pi$
$139$	−880.833	+	1525.65i	−0.537491	+	0.930962i	−0.999038	+	0.0438460i	$0.986039\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1723.36			1.00779
$144$		0			0
$145$	1485.03			0.850518
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1204.96	+	2087.04i	−0.662509	+	1.14750i	0.317446	+	0.948276i	$0.397175\pi$
$149$	−1204.96	+	2087.04i	−0.662509	+	1.14750i	−0.979954	+	0.199222i	$0.936158\pi$
$150$		0			0
$151$	−923.839	−	1600.14i	−0.497887	−	0.862366i	0.502110	−	0.864804i	$-0.332557\pi$
$151$	−923.839	−	1600.14i	−0.497887	−	0.862366i	−0.999997	+	0.00243799i	$0.999224\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−241.627	−	418.510i	−0.125212	−	0.216874i
$156$		0			0
$157$	43.0700	−	74.5995i	0.0218940	−	0.0379216i	−0.854871	−	0.518841i	$-0.826364\pi$
$157$	43.0700	−	74.5995i	0.0218940	−	0.0379216i	0.876765	+	0.480919i	$0.159697\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−370.074			−0.181155
$162$		0			0
$163$	−1127.15			−0.541625			−0.270812	−	0.962632i	$-0.587292\pi$
$163$	−1127.15			−0.541625			−0.270812	+	0.962632i	$0.587292\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−484.322	+	838.871i	−0.224419	+	0.388705i	−0.956145	−	0.292894i	$-0.905382\pi$
$167$	−484.322	+	838.871i	−0.224419	+	0.388705i	0.731726	+	0.681599i	$0.238715\pi$
$168$		0			0
$169$	−2071.70	−	3588.28i	−0.942966	−	1.63326i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	876.037	+	1517.34i	0.384993	+	0.666828i	0.991768	−	0.128046i	$-0.0408704\pi$
$173$	876.037	+	1517.34i	0.384993	+	0.666828i	−0.606775	+	0.794874i	$0.707537\pi$
$174$		0			0
$175$	171.680	−	297.358i	0.0741588	−	0.128447i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	880.297			0.367578			0.183789	−	0.982966i	$-0.441164\pi$
$179$	880.297			0.367578			0.183789	+	0.982966i	$0.441164\pi$
$180$		0			0
$181$	335.846			0.137919			0.0689593	−	0.997619i	$-0.478032\pi$
$181$	335.846			0.137919			0.0689593	+	0.997619i	$0.478032\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1003.95	−	1738.89i	0.398983	−	0.691059i
$186$		0			0
$187$	696.502	+	1206.38i	0.272370	+	0.471759i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1988.65	−	3444.43i	−0.753368	−	1.30487i	−0.946181	−	0.323637i	$-0.895094\pi$
$191$	−1988.65	−	3444.43i	−0.753368	−	1.30487i	0.192813	−	0.981235i	$-0.438239\pi$
$192$		0			0
$193$	−2079.55	+	3601.89i	−0.775593	+	1.34337i	0.158867	+	0.987300i	$0.449216\pi$
$193$	−2079.55	+	3601.89i	−0.775593	+	1.34337i	−0.934460	+	0.356067i	$0.884118\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1581.17			−0.571846			−0.285923	−	0.958253i	$-0.592300\pi$
$197$	−1581.17			−0.571846			−0.285923	+	0.958253i	$0.592300\pi$
$198$		0			0
$199$	−2139.37			−0.762090			−0.381045	−	0.924557i	$-0.624436\pi$
$199$	−2139.37			−0.762090			−0.381045	+	0.924557i	$0.624436\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2039.60	+	3532.69i	−0.705182	+	1.22141i
$204$		0			0
$205$	29.2224	+	50.6147i	0.00995601	+	0.0172443i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1563.73	−	2708.47i	−0.517539	−	0.896405i
$210$		0			0
$211$	88.1421	−	152.667i	0.0287581	−	0.0498105i	−0.851288	−	0.524698i	$-0.824178\pi$
$211$	88.1421	−	152.667i	0.0287581	−	0.0498105i	0.880046	+	0.474888i	$0.157511\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1461.36			−0.463552
$216$		0			0
$217$	1327.44			0.415265
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2562.49	−	4438.37i	0.779964	−	1.35094i
$222$		0			0
$223$	1811.18	+	3137.06i	0.543883	+	0.942033i	0.998676	+	0.0514358i	$0.0163798\pi$
$223$	1811.18	+	3137.06i	0.543883	+	0.942033i	−0.454793	+	0.890597i	$0.650287\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	388.172	+	672.333i	0.113497	+	0.196583i	0.917178	−	0.398478i	$-0.130461\pi$
$227$	388.172	+	672.333i	0.113497	+	0.196583i	−0.803681	+	0.595061i	$0.797128\pi$
$228$		0			0
$229$	−2543.04	+	4404.68i	−0.733838	+	1.27105i	0.221393	+	0.975185i	$0.428940\pi$
$229$	−2543.04	+	4404.68i	−0.733838	+	1.27105i	−0.955231	+	0.295861i	$0.904394\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3896.93			−1.09569			−0.547846	−	0.836579i	$-0.684552\pi$
$233$	−3896.93			−1.09569			−0.547846	+	0.836579i	$0.684552\pi$
$234$		0			0
$235$	876.812			0.243391
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−617.365	+	1069.31i	−0.167088	+	0.289405i	−0.937395	−	0.348268i	$-0.886770\pi$
$239$	−617.365	+	1069.31i	−0.167088	+	0.289405i	0.770307	+	0.637673i	$0.220103\pi$
$240$		0			0
$241$	−959.223	−	1661.42i	−0.256386	−	0.444073i	0.708885	−	0.705324i	$-0.249198\pi$
$241$	−959.223	−	1661.42i	−0.256386	−	0.444073i	−0.965271	+	0.261251i	$0.915865\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−385.916	−	668.426i	−0.100634	−	0.174303i
$246$		0			0
$247$	−5753.12	+	9964.70i	−1.48203	+	2.56696i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7109.99			−1.78796			−0.893981	−	0.448105i	$-0.852099\pi$
$251$	−7109.99			−1.78796			−0.893981	+	0.448105i	$0.852099\pi$
$252$		0			0
$253$	−583.172			−0.144916
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2091.14	+	3621.96i	−0.507555	+	0.879111i	0.492407	+	0.870365i	$0.336117\pi$
$257$	−2091.14	+	3621.96i	−0.507555	+	0.879111i	−0.999962	+	0.00874607i	$0.997216\pi$
$258$		0			0
$259$	2757.73	+	4776.52i	0.661610	+	1.14594i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2108.30	−	3651.69i	−0.494310	−	0.856170i	0.505668	−	0.862728i	$-0.331246\pi$
$263$	−2108.30	−	3651.69i	−0.494310	−	0.856170i	−0.999978	+	0.00655782i	$0.997913\pi$
$264$		0			0
$265$	−388.515	+	672.927i	−0.0900614	+	0.155991i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3271.14			0.741431			0.370715	−	0.928747i	$-0.379113\pi$
$269$	3271.14			0.741431			0.370715	+	0.928747i	$0.379113\pi$
$270$		0			0
$271$	6407.18			1.43619			0.718096	−	0.695944i	$-0.245014\pi$
$271$	6407.18			1.43619			0.718096	+	0.695944i	$0.245014\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	270.538	−	468.585i	0.0593237	−	0.102752i
$276$		0			0
$277$	−3163.13	−	5478.70i	−0.686116	−	1.18839i	−0.973085	−	0.230448i	$-0.925981\pi$
$277$	−3163.13	−	5478.70i	−0.686116	−	1.18839i	0.286969	−	0.957940i	$-0.407352\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1991.29	−	3449.01i	−0.422741	−	0.732208i	0.573466	−	0.819230i	$-0.305599\pi$
$281$	−1991.29	−	3449.01i	−0.422741	−	0.732208i	−0.996206	+	0.0870212i	$0.972265\pi$
$282$		0			0
$283$	1839.84	−	3186.70i	0.386456	−	0.669362i	−0.605514	−	0.795835i	$-0.707032\pi$
$283$	1839.84	−	3186.70i	0.386456	−	0.669362i	0.991970	+	0.126473i	$0.0403657\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−160.541			−0.0330189
$288$		0			0
$289$	−770.434			−0.156815
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4473.14	−	7747.71i	0.891890	−	1.54480i	0.0542823	−	0.998526i	$-0.482713\pi$
$293$	4473.14	−	7747.71i	0.891890	−	1.54480i	0.837607	−	0.546273i	$-0.183954\pi$
$294$		0			0
$295$	−1487.91	−	2577.13i	−0.293659	−	0.508632i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1072.77	+	1858.09i	0.207492	+	0.359386i
$300$		0			0
$301$	2007.09	−	3476.37i	0.384341	−	0.665697i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	310.535			0.0582989
$306$		0			0
$307$	−9746.88			−1.81200			−0.906000	−	0.423278i	$-0.860879\pi$
$307$	−9746.88			−1.81200			−0.906000	+	0.423278i	$0.860879\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2639.51	−	4571.77i	0.481264	−	0.833573i	−0.518505	−	0.855074i	$-0.673511\pi$
$311$	2639.51	−	4571.77i	0.481264	−	0.833573i	0.999769	+	0.0215015i	$0.00684466\pi$
$312$		0			0
$313$	−2267.20	−	3926.90i	−0.409424	−	0.709142i	0.585402	−	0.810743i	$-0.300937\pi$
$313$	−2267.20	−	3926.90i	−0.409424	−	0.709142i	−0.994825	+	0.101601i	$0.967603\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1117.38	+	1935.37i	0.197977	+	0.342905i	0.947872	−	0.318651i	$-0.103230\pi$
$317$	1117.38	+	1935.37i	0.197977	+	0.342905i	−0.749896	+	0.661556i	$0.769896\pi$
$318$		0			0
$319$	−3214.05	+	5566.91i	−0.564114	+	0.977075i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−9300.58			−1.60216
$324$		0			0
$325$	−1990.66			−0.339761
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1204.25	+	2085.82i	−0.201800	+	0.349529i
$330$		0			0
$331$	920.763	+	1594.81i	0.152899	+	0.264829i	0.932292	−	0.361706i	$-0.117806\pi$
$331$	920.763	+	1594.81i	0.152899	+	0.264829i	−0.779393	+	0.626536i	$0.784472\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−401.659	−	695.694i	−0.0655074	−	0.113462i
$336$		0			0
$337$	−3517.09	+	6091.78i	−0.568511	+	0.984690i	0.428203	+	0.903683i	$0.359147\pi$
$337$	−3517.09	+	6091.78i	−0.568511	+	0.984690i	−0.996714	+	0.0810072i	$0.974186\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2091.81			0.332193
$342$		0			0
$343$	6831.03			1.07534
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5887.57	+	10197.6i	−0.910839	+	1.57762i	−0.0979579	+	0.995191i	$0.531231\pi$
$347$	−5887.57	+	10197.6i	−0.910839	+	1.57762i	−0.812881	+	0.582429i	$0.802102\pi$
$348$		0			0
$349$	−3949.29	−	6840.37i	−0.605733	−	1.04916i	−0.991935	−	0.126746i	$-0.959547\pi$
$349$	−3949.29	−	6840.37i	−0.605733	−	1.04916i	0.386202	−	0.922414i	$-0.373787\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4483.96	+	7766.44i	0.676082	+	1.17101i	0.976151	+	0.217091i	$0.0696568\pi$
$353$	4483.96	+	7766.44i	0.676082	+	1.17101i	−0.300070	+	0.953917i	$0.597010\pi$
$354$		0			0
$355$	2399.31	−	4155.72i	0.358710	−	0.621304i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8351.48			−1.22778			−0.613892	−	0.789390i	$-0.710397\pi$
$359$	−8351.48			−1.22778			−0.613892	+	0.789390i	$0.710397\pi$
$360$		0			0
$361$	14022.0			2.04432
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1908.56	−	3305.72i	0.273695	−	0.474053i
$366$		0			0
$367$	−55.2588	−	95.7111i	−0.00785964	−	0.0136133i	0.862069	−	0.506791i	$-0.169168\pi$
$367$	−55.2588	−	95.7111i	−0.00785964	−	0.0136133i	−0.869928	+	0.493178i	$0.835835\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1067.20	−	1848.45i	−0.149343	−	0.258670i
$372$		0			0
$373$	−2447.97	+	4240.01i	−0.339816	+	0.588578i	−0.984398	−	0.175957i	$-0.943698\pi$
$373$	−2447.97	+	4240.01i	−0.339816	+	0.588578i	0.644582	+	0.764535i	$0.277031\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	23649.6			3.23081
$378$		0			0
$379$	10326.8			1.39962			0.699808	−	0.714331i	$-0.253269\pi$
$379$	10326.8			1.39962			0.699808	+	0.714331i	$0.253269\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4166.26	+	7216.17i	−0.555838	+	0.962739i	0.442000	+	0.897015i	$0.354269\pi$
$383$	−4166.26	+	7216.17i	−0.555838	+	0.962739i	−0.997838	+	0.0657244i	$0.979064\pi$
$384$		0			0
$385$	743.134	+	1287.15i	0.0983730	+	0.170387i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1035.75	+	1793.97i	0.134999	+	0.233825i	0.925597	−	0.378510i	$-0.123564\pi$
$389$	1035.75	+	1793.97i	0.134999	+	0.233825i	−0.790598	+	0.612335i	$0.790230\pi$
$390$		0			0
$391$	−867.129	+	1501.91i	−0.112155	+	0.194258i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−319.825			−0.0407396
$396$		0			0
$397$	−7891.46			−0.997635			−0.498818	−	0.866707i	$-0.666232\pi$
$397$	−7891.46			−0.997635			−0.498818	+	0.866707i	$0.666232\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	866.818	−	1501.37i	0.107947	−	0.186970i	−0.806991	−	0.590563i	$-0.798906\pi$
$401$	866.818	−	1501.37i	0.107947	−	0.186970i	0.914939	+	0.403593i	$0.132239\pi$
$402$		0			0
$403$	−3847.98	−	6664.90i	−0.475637	−	0.823827i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4345.70	+	7526.97i	0.529258	+	0.916702i
$408$		0			0
$409$	4706.30	−	8151.55i	0.568977	−	0.985497i	−0.427690	−	0.903925i	$-0.640673\pi$
$409$	4706.30	−	8151.55i	0.568977	−	0.985497i	0.996667	−	0.0815720i	$-0.0259941\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8174.21			0.973914
$414$		0			0
$415$	4586.63			0.542527
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1934.35	+	3350.40i	−0.225535	+	0.390639i	−0.956480	−	0.291798i	$-0.905746\pi$
$419$	−1934.35	+	3350.40i	−0.225535	+	0.390639i	0.730945	+	0.682437i	$0.239080\pi$
$420$		0			0
$421$	−6462.48	−	11193.4i	−0.748129	−	1.29580i	−0.948719	−	0.316122i	$-0.897619\pi$
$421$	−6462.48	−	11193.4i	−0.748129	−	1.29580i	0.200590	−	0.979675i	$-0.435714\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−804.535	−	1393.49i	−0.0918251	−	0.159046i
$426$		0			0
$427$	−426.501	+	738.721i	−0.0483368	+	0.0837219i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13183.6			1.47339			0.736697	−	0.676223i	$-0.236384\pi$
$431$	13183.6			1.47339			0.736697	+	0.676223i	$0.236384\pi$
$432$		0			0
$433$	−7584.46			−0.841769			−0.420884	−	0.907114i	$-0.638280\pi$
$433$	−7584.46			−0.841769			−0.420884	+	0.907114i	$0.638280\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1946.81	−	3371.98i	0.213109	−	0.369116i
$438$		0			0
$439$	3773.23	+	6535.43i	0.410220	+	0.710521i	0.994914	−	0.100733i	$-0.0321187\pi$
$439$	3773.23	+	6535.43i	0.410220	+	0.710521i	−0.584694	+	0.811254i	$0.698785\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1022.52	+	1771.06i	0.109665	+	0.189945i	0.915634	−	0.402012i	$-0.131689\pi$
$443$	1022.52	+	1771.06i	0.109665	+	0.189945i	−0.805970	+	0.591957i	$0.798356\pi$
$444$		0			0
$445$	3297.20	−	5710.92i	0.351241	−	0.608367i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10723.2			−1.12708			−0.563541	−	0.826088i	$-0.690561\pi$
$449$	−10723.2			−1.12708			−0.563541	+	0.826088i	$0.690561\pi$
$450$		0			0
$451$	−252.984			−0.0264137
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2734.06	−	4735.53i	0.281702	−	0.487923i
$456$		0			0
$457$	2330.29	+	4036.18i	0.238526	+	0.413139i	0.960292	−	0.278998i	$-0.0900024\pi$
$457$	2330.29	+	4036.18i	0.238526	+	0.413139i	−0.721766	+	0.692138i	$0.756669\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8320.18	−	14411.0i	−0.840584	−	1.45593i	−0.889401	−	0.457127i	$-0.848878\pi$
$461$	−8320.18	−	14411.0i	−0.840584	−	1.45593i	0.0488170	−	0.998808i	$-0.484455\pi$
$462$		0			0
$463$	1734.35	−	3003.97i	0.174086	−	0.301526i	−0.765759	−	0.643128i	$-0.777636\pi$
$463$	1734.35	−	3003.97i	0.174086	−	0.301526i	0.939845	+	0.341602i	$0.110970\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9920.55			0.983016			0.491508	−	0.870873i	$-0.336446\pi$
$467$	9920.55			0.983016			0.491508	+	0.870873i	$0.336446\pi$
$468$		0			0
$469$	2206.62			0.217254
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3162.82	−	5478.16i	0.307455	−	0.532528i
$474$		0			0
$475$	1806.28	+	3128.57i	0.174480	+	0.302208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9132.33	+	15817.7i	0.871120	+	1.50882i	0.860839	+	0.508877i	$0.169939\pi$
$479$	9132.33	+	15817.7i	0.871120	+	1.50882i	0.0102812	+	0.999947i	$0.496727\pi$
$480$		0			0
$481$	15988.2	−	27692.4i	1.51559	−	2.62508i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	640.845			0.0599985
$486$		0			0
$487$	−3244.55			−0.301899			−0.150949	−	0.988542i	$-0.548233\pi$
$487$	−3244.55			−0.301899			−0.150949	+	0.988542i	$0.548233\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−295.547	+	511.902i	−0.0271646	+	0.0470505i	−0.879288	−	0.476290i	$-0.841981\pi$
$491$	−295.547	+	511.902i	−0.0271646	+	0.0470505i	0.852124	+	0.523341i	$0.175315\pi$
$492$		0			0
$493$	9558.08	+	16555.1i	0.873173	+	1.51238i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6590.61	+	11415.3i	0.594827	+	1.03027i
$498$		0			0
$499$	4822.49	−	8352.79i	0.432634	−	0.749343i	−0.564466	−	0.825457i	$-0.690918\pi$
$499$	4822.49	−	8352.79i	0.432634	−	0.749343i	0.997099	+	0.0761132i	$0.0242510\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−723.099			−0.0640982			−0.0320491	−	0.999486i	$-0.510203\pi$
$503$	−723.099			−0.0640982			−0.0320491	+	0.999486i	$0.510203\pi$
$504$		0			0
$505$	−2302.04			−0.202850
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1342.27	+	2324.88i	−0.116886	+	0.202453i	−0.918532	−	0.395346i	$-0.870625\pi$
$509$	−1342.27	+	2324.88i	−0.116886	+	0.202453i	0.801646	+	0.597799i	$0.203958\pi$
$510$		0			0
$511$	5242.58	+	9080.42i	0.453852	+	0.786094i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1793.65	+	3106.70i	0.153472	+	0.265820i
$516$		0			0
$517$	−1897.68	+	3286.88i	−0.161431	+	0.279607i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4203.80			0.353497			0.176748	−	0.984256i	$-0.443442\pi$
$521$	4203.80			0.353497			0.176748	+	0.984256i	$0.443442\pi$
$522$		0			0
$523$	10571.8			0.883883			0.441942	−	0.897044i	$-0.354290\pi$
$523$	10571.8			0.883883			0.441942	+	0.897044i	$0.354290\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3110.35	−	5387.29i	0.257095	−	0.445302i
$528$		0			0
$529$	5720.48	+	9908.17i	0.470164	+	0.814347i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	465.376	+	806.055i	0.0378193	+	0.0655049i
$534$		0			0
$535$	−3404.41	+	5896.61i	−0.275113	+	0.476510i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3340.95			0.266985
$540$		0			0
$541$	15310.3			1.21671			0.608356	−	0.793664i	$-0.291829\pi$
$541$	15310.3			1.21671			0.608356	+	0.793664i	$0.291829\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	564.654	−	978.010i	0.0443801	−	0.0768685i
$546$		0			0
$547$	447.172	+	774.525i	0.0349538	+	0.0605417i	0.882973	−	0.469424i	$-0.155538\pi$
$547$	447.172	+	774.525i	0.0349538	+	0.0605417i	−0.848019	+	0.529965i	$0.822205\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−21459.1	−	37168.2i	−1.65914	−	2.87372i
$552$		0			0
$553$	439.260	−	760.820i	0.0337780	−	0.0585052i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25371.2			−1.93000			−0.965001	−	0.262245i	$-0.915537\pi$
$557$	−25371.2			−1.93000			−0.965001	+	0.262245i	$0.915537\pi$
$558$		0			0
$559$	−23272.6			−1.76087
$560$		0			0
$561$		0			0
$562$		0			0
$563$	424.449	−	735.167i	0.0317733	−	0.0550330i	−0.849701	−	0.527264i	$-0.823218\pi$
$563$	424.449	−	735.167i	0.0317733	−	0.0550330i	0.881475	+	0.472231i	$0.156551\pi$
$564$		0			0
$565$	−4921.17	−	8523.72i	−0.366434	−	0.634682i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4318.33	−	7479.57i	−0.318161	−	0.551072i	0.661943	−	0.749554i	$-0.269732\pi$
$569$	−4318.33	−	7479.57i	−0.318161	−	0.551072i	−0.980104	+	0.198482i	$0.936399\pi$
$570$		0			0
$571$	3615.99	−	6263.08i	0.265017	−	0.459022i	−0.702551	−	0.711633i	$-0.747956\pi$
$571$	3615.99	−	6263.08i	0.265017	−	0.459022i	0.967568	+	0.252611i	$0.0812892\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	673.627			0.0488559
$576$		0			0
$577$	−5882.23			−0.424403			−0.212201	−	0.977226i	$-0.568063\pi$
$577$	−5882.23			−0.424403			−0.212201	+	0.977226i	$0.568063\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6299.46	+	10911.0i	−0.449820	+	0.779111i
$582$		0			0
$583$	−1681.72	−	2912.83i	−0.119468	−	0.206925i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9666.38	+	16742.7i	0.679684	+	1.17725i	0.975076	+	0.221871i	$0.0712163\pi$
$587$	9666.38	+	16742.7i	0.679684	+	1.17725i	−0.295392	+	0.955376i	$0.595450\pi$
$588$		0			0
$589$	−6983.14	+	12095.1i	−0.488515	+	0.846132i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4318.26			0.299039			0.149519	−	0.988759i	$-0.452227\pi$
$593$	4318.26			0.299039			0.149519	+	0.988759i	$0.452227\pi$
$594$		0			0
$595$	4419.92			0.304536
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7690.82	−	13320.9i	0.524605	−	0.908643i	−0.474984	−	0.879994i	$-0.657546\pi$
$599$	7690.82	−	13320.9i	0.524605	−	0.908643i	0.999590	−	0.0286487i	$-0.00912042\pi$
$600$		0			0
$601$	−2086.27	−	3613.53i	−0.141599	−	0.245256i	0.786500	−	0.617590i	$-0.211891\pi$
$601$	−2086.27	−	3613.53i	−0.141599	−	0.245256i	−0.928099	+	0.372334i	$0.878558\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2156.45	−	3735.08i	−0.144913	−	0.250996i
$606$		0			0
$607$	−346.410	+	600.000i	−0.0231637	+	0.0401207i	−0.877375	−	0.479806i	$-0.840707\pi$
$607$	−346.410	+	600.000i	−0.0231637	+	0.0401207i	0.854211	+	0.519926i	$0.174041\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13963.5			0.924555
$612$		0			0
$613$	11649.8			0.767584			0.383792	−	0.923419i	$-0.374618\pi$
$613$	11649.8			0.767584			0.383792	+	0.923419i	$0.374618\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−10893.6	+	18868.2i	−0.710793	+	1.23113i	0.253767	+	0.967265i	$0.418330\pi$
$617$	−10893.6	+	18868.2i	−0.710793	+	1.23113i	−0.964560	+	0.263864i	$0.915003\pi$
$618$		0			0
$619$	2485.00	+	4304.15i	0.161358	+	0.279480i	0.935356	−	0.353708i	$-0.115079\pi$
$619$	2485.00	+	4304.15i	0.161358	+	0.279480i	−0.773998	+	0.633188i	$0.781746\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	9057.01	+	15687.2i	0.582442	+	1.00882i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	25846.8			1.63844
$630$		0			0
$631$	−16907.2			−1.06666			−0.533331	−	0.845906i	$-0.679060\pi$
$631$	−16907.2			−1.06666			−0.533331	+	0.845906i	$0.679060\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1878.01	−	3252.81i	0.117365	−	0.203282i
$636$		0			0
$637$	−6145.83	−	10644.9i	−0.382271	−	0.662113i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10734.9	+	18593.5i	0.661474	+	1.14571i	0.980229	+	0.197869i	$0.0634020\pi$
$641$	10734.9	+	18593.5i	0.661474	+	1.14571i	−0.318755	+	0.947837i	$0.603265\pi$
$642$		0			0
$643$	8856.22	−	15339.4i	0.543165	−	0.940790i	−0.455555	−	0.890208i	$-0.650559\pi$
$643$	8856.22	−	15339.4i	0.543165	−	0.940790i	0.998720	−	0.0505819i	$-0.0161076\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−12632.9			−0.767619			−0.383809	−	0.923412i	$-0.625388\pi$
$647$	−12632.9			−0.767619			−0.383809	+	0.923412i	$0.625388\pi$
$648$		0			0
$649$	12881.1			0.779088
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8085.08	−	14003.8i	0.484523	−	0.839219i	−0.515319	−	0.856999i	$-0.672326\pi$
$653$	8085.08	−	14003.8i	0.484523	−	0.839219i	0.999842	+	0.0177796i	$0.00565971\pi$
$654$		0			0
$655$	198.949	+	344.590i	0.0118681	+	0.0205561i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3876.38	+	6714.09i	0.229139	+	0.396880i	0.957553	−	0.288257i	$-0.0930758\pi$
$659$	3876.38	+	6714.09i	0.229139	+	0.396880i	−0.728414	+	0.685137i	$0.759742\pi$
$660$		0			0
$661$	−5846.77	+	10126.9i	−0.344044	+	0.595901i	−0.985180	−	0.171526i	$-0.945130\pi$
$661$	−5846.77	+	10126.9i	−0.344044	+	0.595901i	0.641136	+	0.767427i	$0.278464\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9923.28			−0.578659
$666$		0			0
$667$	−8002.85			−0.464575
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−672.091	+	1164.10i	−0.0386673	+	0.0669738i
$672$		0			0
$673$	2059.46	+	3567.08i	0.117959	+	0.204310i	0.918959	−	0.394354i	$-0.129032\pi$
$673$	2059.46	+	3567.08i	0.117959	+	0.204310i	−0.801000	+	0.598664i	$0.795698\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6801.97	−	11781.4i	−0.386146	−	0.668825i	0.605781	−	0.795631i	$-0.292861\pi$
$677$	−6801.97	−	11781.4i	−0.386146	−	0.668825i	−0.991928	+	0.126806i	$0.959527\pi$
$678$		0			0
$679$	−880.162	+	1524.49i	−0.0497460	+	0.0861626i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−28073.3			−1.57276			−0.786381	−	0.617742i	$-0.788048\pi$
$683$	−28073.3			−1.57276			−0.786381	+	0.617742i	$0.788048\pi$
$684$		0			0
$685$	4017.92			0.224112
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−6187.22	+	10716.6i	−0.342111	+	0.592553i
$690$		0			0
$691$	7958.75	+	13785.0i	0.438155	+	0.758907i	0.997547	−	0.0699960i	$-0.0222987\pi$
$691$	7958.75	+	13785.0i	0.438155	+	0.758907i	−0.559392	+	0.828903i	$0.688965\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4404.16	+	7628.23i	0.240373	+	0.416339i
$696$		0			0
$697$	−376.167	+	651.541i	−0.0204424	+	0.0354073i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−17451.6			−0.940280			−0.470140	−	0.882592i	$-0.655797\pi$
$701$	−17451.6			−0.940280			−0.470140	+	0.882592i	$0.655797\pi$
$702$		0			0
$703$	−58029.3			−3.11325
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3161.71	−	5476.24i	0.168187	−	0.291309i
$708$		0			0
$709$	−15089.1	−	26135.1i	−0.799273	−	1.38438i	−0.920090	−	0.391706i	$-0.871885\pi$
$709$	−15089.1	−	26135.1i	−0.799273	−	1.38438i	0.120818	−	0.992675i	$-0.461448\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1302.13	+	2255.36i	0.0683943	+	0.118462i
$714$		0			0
$715$	4308.40	−	7462.36i	0.225349	−	0.390317i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7009.26			0.363562			0.181781	−	0.983339i	$-0.441814\pi$
$719$	7009.26			0.363562			0.181781	+	0.983339i	$0.441814\pi$
$720$		0			0
$721$	−9853.90			−0.508985
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3712.58	−	6430.38i	0.190182	−	0.329404i
$726$		0			0
$727$	15583.0	+	26990.6i	0.794968	+	1.37693i	0.922859	+	0.385137i	$0.125846\pi$
$727$	15583.0	+	26990.6i	0.794968	+	1.37693i	−0.127891	+	0.991788i	$0.540821\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−9405.70	−	16291.1i	−0.475899	−	0.824282i
$732$		0			0
$733$	−16038.1	+	27778.7i	−0.808158	+	1.39977i	0.105981	+	0.994368i	$0.466202\pi$
$733$	−16038.1	+	27778.7i	−0.808158	+	1.39977i	−0.914139	+	0.405402i	$0.867132\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3477.24			0.173794
$738$		0			0
$739$	−35798.6			−1.78197			−0.890984	−	0.454035i	$-0.849984\pi$
$739$	−35798.6			−1.78197			−0.890984	+	0.454035i	$0.849984\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−521.962	+	904.064i	−0.0257724	+	0.0446392i	−0.878624	−	0.477514i	$-0.841538\pi$
$743$	−521.962	+	904.064i	−0.0257724	+	0.0446392i	0.852852	+	0.522153i	$0.174871\pi$
$744$		0			0
$745$	6024.78	+	10435.2i	0.296283	+	0.513177i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9351.50	−	16197.3i	−0.456204	−	0.790168i
$750$		0			0
$751$	20122.4	−	34853.0i	0.977731	−	1.69348i	0.307118	−	0.951672i	$-0.400635\pi$
$751$	20122.4	−	34853.0i	0.977731	−	1.69348i	0.670613	−	0.741808i	$-0.266031\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9238.39			−0.445324
$756$		0			0
$757$	1971.31			0.0946480			0.0473240	−	0.998880i	$-0.484931\pi$
$757$	1971.31			0.0946480			0.0473240	+	0.998880i	$0.484931\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7438.23	+	12883.4i	−0.354317	+	0.613696i	−0.987001	−	0.160715i	$-0.948620\pi$
$761$	−7438.23	+	12883.4i	−0.354317	+	0.613696i	0.632683	+	0.774410i	$0.281953\pi$
$762$		0			0
$763$	1551.04	+	2686.48i	0.0735928	+	0.127467i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−23695.4	−	41041.7i	−1.11550	−	1.93211i
$768$		0			0
$769$	−1142.37	+	1978.64i	−0.0535695	+	0.0927851i	−0.891567	−	0.452890i	$-0.850393\pi$
$769$	−1142.37	+	1978.64i	−0.0535695	+	0.0927851i	0.837997	+	0.545675i	$0.183727\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15649.7			−0.728177			−0.364089	−	0.931364i	$-0.618620\pi$
$773$	−15649.7			−0.728177			−0.364089	+	0.931364i	$0.618620\pi$
$774$		0			0
$775$	−2416.27			−0.111993
$776$		0			0
$777$		0			0
$778$		0			0
$779$	844.543	−	1462.79i	0.0388432	−	0.0672785i
$780$		0			0
$781$	10385.6	+	17988.5i	0.475836	+	0.824171i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−215.350	−	372.997i	−0.00979131	−	0.0169590i
$786$		0			0
$787$	15210.3	−	26345.1i	0.688932	−	1.19327i	−0.283251	−	0.959046i	$-0.591413\pi$
$787$	15210.3	−	26345.1i	0.688932	−	1.19327i	0.972184	−	0.234220i	$-0.0752536\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	27035.7			1.21527
$792$		0			0
$793$	4945.37			0.221457
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9307.98	+	16121.9i	−0.413683	+	0.716521i	−0.995289	−	0.0969501i	$-0.969091\pi$
$797$	−9307.98	+	16121.9i	−0.413683	+	0.716521i	0.581606	+	0.813471i	$0.302425\pi$
$798$		0			0
$799$	5643.40	+	9774.66i	0.249874	+	0.432794i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8261.39	+	14309.2i	0.363061	+	0.628841i
$804$		0			0
$805$	−925.185	+	1602.47i	−0.0405075	+	0.0701610i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−42989.2			−1.86826			−0.934129	−	0.356935i	$-0.883822\pi$
$809$	−42989.2			−1.86826			−0.934129	+	0.356935i	$0.883822\pi$
$810$		0			0
$811$	−28303.4			−1.22548			−0.612741	−	0.790284i	$-0.709933\pi$
$811$	−28303.4			−1.22548			−0.612741	+	0.790284i	$0.709933\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2817.86	+	4880.68i	−0.121111	+	0.209770i
$816$		0			0
$817$	21117.0	+	36575.7i	0.904271	+	1.56624i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5297.09	+	9174.83i	0.225176	+	0.390017i	0.956372	−	0.292151i	$-0.0943708\pi$
$821$	5297.09	+	9174.83i	0.225176	+	0.390017i	−0.731196	+	0.682167i	$0.761038\pi$
$822$		0			0
$823$	12299.2	−	21302.8i	0.520925	−	0.902269i	−0.478779	−	0.877936i	$-0.658920\pi$
$823$	12299.2	−	21302.8i	0.520925	−	0.902269i	0.999704	−	0.0243334i	$-0.00774633\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	15104.9			0.635126			0.317563	−	0.948237i	$-0.397135\pi$
$827$	15104.9			0.635126			0.317563	+	0.948237i	$0.397135\pi$
$828$		0			0
$829$	−33824.6			−1.41710			−0.708551	−	0.705660i	$-0.750651\pi$
$829$	−33824.6			−1.41710			−0.708551	+	0.705660i	$0.750651\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4967.72	−	8604.35i	0.206628	−	0.357891i
$834$		0			0
$835$	2421.61	+	4194.36i	0.100363	+	0.173834i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2683.31	+	4647.63i	0.110415	+	0.191244i	0.915938	−	0.401321i	$-0.131449\pi$
$839$	2683.31	+	4647.63i	0.110415	+	0.191244i	−0.805523	+	0.592565i	$0.798115\pi$
$840$		0			0
$841$	−31911.9	+	55273.0i	−1.30845	+	2.26631i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−20717.0			−0.843414
$846$		0			0
$847$	11847.0			0.480600
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5410.30	+	9370.91i	−0.217935	+	0.377474i
$852$		0			0
$853$	−8191.63	−	14188.3i	−0.328812	−	0.569518i	0.653465	−	0.756957i	$-0.273315\pi$
$853$	−8191.63	−	14188.3i	−0.328812	−	0.569518i	−0.982276	+	0.187439i	$0.939981\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14059.2	−	24351.2i	−0.560389	−	0.970622i	−0.997462	−	0.0711960i	$-0.977318\pi$
$857$	−14059.2	−	24351.2i	−0.560389	−	0.970622i	0.437074	−	0.899426i	$-0.356015\pi$
$858$		0			0
$859$	−16111.5	+	27905.9i	−0.639949	+	1.10842i	0.345494	+	0.938421i	$0.387711\pi$
$859$	−16111.5	+	27905.9i	−0.639949	+	1.10842i	−0.985443	+	0.170003i	$0.945622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−37447.4			−1.47709			−0.738543	−	0.674206i	$-0.764486\pi$
$863$	−37447.4			−1.47709			−0.738543	+	0.674206i	$0.764486\pi$
$864$		0			0
$865$	8760.37			0.344349
$866$		0			0
$867$		0			0
$868$		0			0
$869$	692.196	−	1198.92i	0.0270209	−	0.0468016i
$870$		0			0
$871$	−6396.55	−	11079.1i	−0.248839	−	0.431002i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−858.400	−	1486.79i	−0.0331648	−	0.0574431i
$876$		0			0
$877$	3916.87	−	6784.21i	0.150813	−	0.261216i	−0.780713	−	0.624889i	$-0.785144\pi$
$877$	3916.87	−	6784.21i	0.150813	−	0.261216i	0.931527	+	0.363673i	$0.118477\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3195.87			0.122215			0.0611076	−	0.998131i	$-0.480537\pi$
$881$	3195.87			0.122215			0.0611076	+	0.998131i	$0.480537\pi$
$882$		0			0
$883$	6033.49			0.229947			0.114973	−	0.993369i	$-0.463322\pi$
$883$	6033.49			0.229947			0.114973	+	0.993369i	$0.463322\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1739.61	+	3013.09i	−0.0658515	+	0.114058i	−0.897071	−	0.441886i	$-0.854310\pi$
$887$	−1739.61	+	3013.09i	−0.0658515	+	0.114058i	0.831220	+	0.555944i	$0.187643\pi$
$888$		0			0
$889$	5158.67	+	8935.08i	0.194619	+	0.337090i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12670.2	−	21945.3i	−0.474793	−	0.822366i
$894$		0			0
$895$	2200.74	−	3811.80i	0.0821930	−	0.142362i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	28705.9			1.06496
$900$		0			0
$901$	−10002.3			−0.369841
$902$		0			0
$903$		0			0
$904$		0			0
$905$	839.616	−	1454.26i	0.0308395	−	0.0534156i
$906$		0			0
$907$	−2191.65	−	3796.06i	−0.0802345	−	0.138970i	0.823116	−	0.567873i	$-0.192234\pi$
$907$	−2191.65	−	3796.06i	−0.0802345	−	0.138970i	−0.903351	+	0.428903i	$0.858900\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2342.68	−	4057.64i	−0.0851992	−	0.147569i	0.820277	−	0.571967i	$-0.193819\pi$
$911$	−2342.68	−	4057.64i	−0.0851992	−	0.147569i	−0.905476	+	0.424397i	$0.860486\pi$
$912$		0			0
$913$	−9926.84	+	17193.8i	−0.359836	+	0.623255i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1092.98			−0.0393603
$918$		0			0
$919$	36353.0			1.30487			0.652434	−	0.757846i	$-0.273748\pi$
$919$	36353.0			1.30487			0.652434	+	0.757846i	$0.273748\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	38209.7	−	66181.2i	1.36261	−	2.36011i
$924$		0			0
$925$	−5019.75	−	8694.46i	−0.178431	−	0.309051i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4271.06	+	7397.69i	0.150838	+	0.261260i	0.931536	−	0.363649i	$-0.118469\pi$
$929$	4271.06	+	7397.69i	0.150838	+	0.261260i	−0.780698	+	0.624909i	$0.785136\pi$
$930$		0			0
$931$	−11153.2	+	19317.8i	−0.392621	+	0.680040i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6965.02			0.243615
$936$		0			0
$937$	33521.5			1.16873			0.584364	−	0.811491i	$-0.301344\pi$
$937$	33521.5			1.16873			0.584364	+	0.811491i	$0.301344\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12037.0	−	20848.7i	0.416997	−	0.722260i	−0.578639	−	0.815584i	$-0.696416\pi$
$941$	12037.0	−	20848.7i	0.416997	−	0.722260i	0.995636	+	0.0933239i	$0.0297492\pi$
$942$		0			0
$943$	−157.480	−	272.763i	−0.00543823	−	0.00941930i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20540.1	−	35576.6i	−0.704820	−	1.22078i	−0.966756	−	0.255699i	$-0.917694\pi$
$947$	−20540.1	−	35576.6i	−0.704820	−	1.22078i	0.261936	−	0.965085i	$-0.415639\pi$
$948$		0			0
$949$	30394.4	−	52644.7i	1.03967	−	1.80076i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	28047.0			0.953337			0.476668	−	0.879083i	$-0.341844\pi$
$953$	28047.0			0.953337			0.476668	+	0.879083i	$0.341844\pi$
$954$		0			0
$955$	−19886.5			−0.673833
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5518.37	+	9558.10i	−0.185816	+	0.321843i
$960$		0			0
$961$	10224.8	+	17709.9i	0.343218	+	0.594472i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10397.8	+	18009.5i	0.346856	+	0.600772i
$966$		0			0
$967$	2656.52	−	4601.22i	0.0883432	−	0.153015i	−0.818468	−	0.574552i	$-0.805176\pi$
$967$	2656.52	−	4601.22i	0.0883432	−	0.153015i	0.906811	+	0.421538i	$0.138509\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−17611.7			−0.582065			−0.291032	−	0.956713i	$-0.593999\pi$
$971$	−17611.7			−0.582065			−0.291032	+	0.956713i	$0.593999\pi$
$972$		0			0
$973$	−24195.4			−0.797193
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21636.6	+	37475.6i	−0.708511	+	1.22718i	0.256899	+	0.966438i	$0.417299\pi$
$977$	−21636.6	+	37475.6i	−0.708511	+	1.22718i	−0.965409	+	0.260738i	$0.916034\pi$
$978$		0			0
$979$	14272.3	+	24720.3i	0.465928	+	0.807011i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23359.8	+	40460.4i	0.757948	+	1.31280i	0.943895	+	0.330245i	$0.107131\pi$
$983$	23359.8	+	40460.4i	0.757948	+	1.31280i	−0.185947	+	0.982560i	$0.559535\pi$
$984$		0			0
$985$	−3952.93	+	6846.67i	−0.127869	+	0.221475i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7875.27			0.253204
$990$		0			0
$991$	−4961.78			−0.159047			−0.0795237	−	0.996833i	$-0.525340\pi$
$991$	−4961.78			−0.159047			−0.0795237	+	0.996833i	$0.525340\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5348.42	+	9263.74i	−0.170408	+	0.295156i
$996$		0			0
$997$	−14670.1	−	25409.3i	−0.466004	−	0.807142i	0.533243	−	0.845962i	$-0.320973\pi$
$997$	−14670.1	−	25409.3i	−0.466004	−	0.807142i	−0.999246	+	0.0388205i	$0.987640\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.4.i.c.361.5		14
3.2	odd	2		180.4.i.c.121.1	yes	14
9.2	odd	6		180.4.i.c.61.1	✓	14
9.4	even	3		1620.4.a.k.1.3		7
9.5	odd	6		1620.4.a.l.1.3		7
9.7	even	3	inner	540.4.i.c.181.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.c.61.1	✓	14	9.2	odd	6
180.4.i.c.121.1	yes	14	3.2	odd	2
540.4.i.c.181.5		14	9.7	even	3	inner
540.4.i.c.361.5		14	1.1	even	1	trivial
1620.4.a.k.1.3		7	9.4	even	3
1620.4.a.l.1.3		7	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	540.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 - 4.33013i) q^{5} +(6.86720 + 11.8943i) q^{7} +(10.8215 + 18.7434i) q^{11} +(39.8133 - 68.9586i) q^{13} +64.3628 q^{17} -144.503 q^{19} +(-13.4725 + 23.3351i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(148.503 + 257.215i) q^{29} +(48.3254 - 83.7020i) q^{31} +68.6720 q^{35} +401.580 q^{37} +(-5.84448 + 10.1229i) q^{41} +(-146.136 - 253.114i) q^{43} +(87.6812 + 151.868i) q^{47} +(77.1832 - 133.685i) q^{49} -155.406 q^{53} +108.215 q^{55} +(297.582 - 515.427i) q^{59} +(31.0535 + 53.7862i) q^{61} +(-199.066 - 344.793i) q^{65} +(80.3318 - 139.139i) q^{67} +959.723 q^{71} +763.424 q^{73} +(-148.627 + 257.429i) q^{77} +(-31.9825 - 55.3952i) q^{79} +(458.663 + 794.427i) q^{83} +(160.907 - 278.699i) q^{85} +1318.88 q^{89} +1093.62 q^{91} +(-361.256 + 625.714i) q^{95} +(64.0845 + 110.998i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 35 q^{5} - 8 q^{7} + 27 q^{11} - 32 q^{13} - 246 q^{17} - 134 q^{19} + 42 q^{23} - 175 q^{25} + 324 q^{29} - 98 q^{31} - 80 q^{35} + 712 q^{37} + 339 q^{41} - 119 q^{43} - 96 q^{47} - 813 q^{49}+ \cdots - 1991 q^{97}+O(q^{100}) \) 14 * q + 35 * q^5 - 8 * q^7 + 27 * q^11 - 32 * q^13 - 246 * q^17 - 134 * q^19 + 42 * q^23 - 175 * q^25 + 324 * q^29 - 98 * q^31 - 80 * q^35 + 712 * q^37 + 339 * q^41 - 119 * q^43 - 96 * q^47 - 813 * q^49 - 1716 * q^53 + 270 * q^55 + 549 * q^59 - 260 * q^61 + 160 * q^65 - 881 * q^67 - 684 * q^71 + 1474 * q^73 - 456 * q^77 - 1886 * q^79 - 132 * q^83 - 615 * q^85 - 792 * q^89 + 2528 * q^91 - 335 * q^95 - 1991 * q^97

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