LMFDB - Embedded newform 2736.2.dc.d.449.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(449,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

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

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

chi := DirichletCharacter("2736.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.dc (of order $6$, 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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 10x^{14} + 46x^{12} + 126x^{10} + 315x^{8} + 1134x^{6} + 3726x^{4} + 7290x^{2} + 6561 $ x^16 + 10x^14 + 46x^12 + 126x^10 + 315x^8 + 1134x^6 + 3726x^4 + 7290*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.6
Root			$0.243856 + 1.71480i$ of defining polynomial
Character	$\chi$	$=$	2736.449
Dual form			2736.2.dc.d.1889.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.11928 - 0.646214i) q^{5} +0.567493 q^{7} +O(q^{10})$	$q+(1.11928 - 0.646214i) q^{5} +0.567493 q^{7} +5.19868i q^{11} +(-6.03362 - 3.48351i) q^{13} +(0.318772 - 0.184043i) q^{17} +(-1.16481 + 4.20038i) q^{19} +(-6.57542 - 3.79632i) q^{23} +(-1.66481 + 2.88354i) q^{25} +(3.70169 - 6.41151i) q^{29} -6.51621i q^{31} +(0.635181 - 0.366722i) q^{35} -4.89784i q^{37} +(-5.62146 - 9.73666i) q^{41} +(2.44856 + 4.24103i) q^{43} +(-7.37592 - 4.25849i) q^{47} -6.67795 q^{49} +(3.03905 - 5.26380i) q^{53} +(3.35946 + 5.81875i) q^{55} +(-3.03905 - 5.26380i) q^{59} +(1.52983 - 2.64974i) q^{61} -9.00438 q^{65} +(-1.10959 - 0.640619i) q^{67} +(3.67423 + 6.36396i) q^{71} +(7.43630 + 12.8800i) q^{73} +2.95021i q^{77} +(-8.12651 + 4.69184i) q^{79} -13.3503i q^{83} +(0.237863 - 0.411990i) q^{85} +(0.953953 - 1.65229i) q^{89} +(-3.42404 - 1.97687i) q^{91} +(1.41060 + 5.45410i) q^{95} +(3.75279 - 2.16668i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 12 q^{19} + 4 q^{25} + 4 q^{43} - 20 q^{55} + 12 q^{61} - 36 q^{67} + 44 q^{73} + 12 q^{79} + 56 q^{85} + 60 q^{91}+O(q^{100}) $ 16 * q + 12 * q^19 + 4 * q^25 + 4 * q^43 - 20 * q^55 + 12 * q^61 - 36 * q^67 + 44 * q^73 + 12 * q^79 + 56 * q^85 + 60 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.11928	−	0.646214i	0.500555	−	0.288996i	−0.228388	−	0.973570i	$-0.573345\pi$
$5$	1.11928	−	0.646214i	0.500555	−	0.288996i	0.728943	+	0.684575i	$0.240012\pi$
$6$		0			0
$7$	0.567493			0.214492			0.107246	−	0.994233i	$-0.465797\pi$
$7$	0.567493			0.214492			0.107246	+	0.994233i	$0.465797\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.19868i			1.56746i	0.621101	+	0.783731i	$0.286686\pi$
$11$			5.19868i			1.56746i	−0.621101	+	0.783731i	$0.713314\pi$
$12$		0			0
$13$	−6.03362	−	3.48351i	−1.67343	−	0.966153i	−0.965698	−	0.259666i	$-0.916387\pi$
$13$	−6.03362	−	3.48351i	−1.67343	−	0.966153i	−0.707727	−	0.706486i	$-0.750279\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.318772	−	0.184043i	0.0773136	−	0.0446370i	−0.460845	−	0.887481i	$-0.652454\pi$
$17$	0.318772	−	0.184043i	0.0773136	−	0.0446370i	0.538158	+	0.842844i	$0.319120\pi$
$18$		0			0
$19$	−1.16481	+	4.20038i	−0.267227	+	0.963634i
$19$	−1.16481	+	4.20038i	−0.267227	+	0.963634i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.57542	−	3.79632i	−1.37107	−	0.791587i	−0.380007	−	0.924984i	$-0.624078\pi$
$23$	−6.57542	−	3.79632i	−1.37107	−	0.791587i	−0.991063	+	0.133397i	$0.957412\pi$
$24$		0			0
$25$	−1.66481	+	2.88354i	−0.332963	+	0.576709i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.70169	−	6.41151i	0.687386	−	1.19059i	−0.285295	−	0.958440i	$-0.592092\pi$
$29$	3.70169	−	6.41151i	0.687386	−	1.19059i	0.972681	−	0.232147i	$-0.0745751\pi$
$30$		0			0
$31$		−	6.51621i		−	1.17035i	−0.810908	−	0.585173i	$-0.801027\pi$
$31$		−	6.51621i		−	1.17035i	0.810908	−	0.585173i	$-0.198973\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.635181	−	0.366722i	0.107365	−	0.0619873i
$36$		0			0
$37$		−	4.89784i		−	0.805200i	−0.915376	−	0.402600i	$-0.868107\pi$
$37$		−	4.89784i		−	0.805200i	0.915376	−	0.402600i	$-0.131893\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.62146	−	9.73666i	−0.877925	−	1.52061i	−0.853614	−	0.520907i	$-0.825594\pi$
$41$	−5.62146	−	9.73666i	−0.877925	−	1.52061i	−0.0243116	−	0.999704i	$-0.507739\pi$
$42$		0			0
$43$	2.44856	+	4.24103i	0.373402	+	0.646751i	0.990086	−	0.140459i	$-0.0448578\pi$
$43$	2.44856	+	4.24103i	0.373402	+	0.646751i	−0.616684	+	0.787211i	$0.711525\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.37592	−	4.25849i	−1.07589	−	0.621165i	−0.146104	−	0.989269i	$-0.546673\pi$
$47$	−7.37592	−	4.25849i	−1.07589	−	0.621165i	−0.929784	+	0.368105i	$0.880007\pi$
$48$		0			0
$49$	−6.67795			−0.953993
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.03905	−	5.26380i	0.417446	−	0.723038i	−0.578236	−	0.815870i	$-0.696259\pi$
$53$	3.03905	−	5.26380i	0.417446	−	0.723038i	0.995682	+	0.0928318i	$0.0295919\pi$
$54$		0			0
$55$	3.35946	+	5.81875i	0.452989	+	0.784601i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.03905	−	5.26380i	−0.395651	−	0.685288i	0.597533	−	0.801844i	$-0.296148\pi$
$59$	−3.03905	−	5.26380i	−0.395651	−	0.685288i	−0.993184	+	0.116557i	$0.962814\pi$
$60$		0			0
$61$	1.52983	−	2.64974i	0.195875	−	0.339265i	−0.751312	−	0.659947i	$-0.770579\pi$
$61$	1.52983	−	2.64974i	0.195875	−	0.339265i	0.947187	+	0.320682i	$0.103912\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.00438			−1.11686
$66$		0			0
$67$	−1.10959	−	0.640619i	−0.135557	−	0.0782641i	0.430688	−	0.902501i	$-0.358271\pi$
$67$	−1.10959	−	0.640619i	−0.135557	−	0.0782641i	−0.566245	+	0.824237i	$0.691604\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	3.67423	+	6.36396i	0.436051	+	0.755263i	0.997381	−	0.0723293i	$-0.0230432\pi$
$71$	3.67423	+	6.36396i	0.436051	+	0.755263i	−0.561329	+	0.827592i	$0.689710\pi$
$72$		0			0
$73$	7.43630	+	12.8800i	0.870353	+	1.50750i	0.861632	+	0.507533i	$0.169443\pi$
$73$	7.43630	+	12.8800i	0.870353	+	1.50750i	0.00872085	+	0.999962i	$0.497224\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			2.95021i			0.336208i
$78$		0			0
$79$	−8.12651	+	4.69184i	−0.914304	+	0.527874i	−0.881814	−	0.471598i	$-0.843677\pi$
$79$	−8.12651	+	4.69184i	−0.914304	+	0.527874i	−0.0324907	+	0.999472i	$0.510344\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	13.3503i		−	1.46539i	−0.680559	−	0.732693i	$-0.738263\pi$
$83$		−	13.3503i		−	1.46539i	0.680559	−	0.732693i	$-0.261737\pi$
$84$		0			0
$85$	0.237863	−	0.411990i	0.0257998	−	0.0446866i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.953953	−	1.65229i	0.101119	−	0.175143i	−0.811027	−	0.585009i	$-0.801091\pi$
$89$	0.953953	−	1.65229i	0.101119	−	0.175143i	0.912146	+	0.409866i	$0.134424\pi$
$90$		0			0
$91$	−3.42404	−	1.97687i	−0.358936	−	0.207232i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.41060	+	5.45410i	0.144724	+	0.559579i
$96$		0			0
$97$	3.75279	−	2.16668i	0.381038	−	0.219993i	−0.297232	−	0.954805i	$-0.596063\pi$
$97$	3.75279	−	2.16668i	0.381038	−	0.219993i	0.678270	+	0.734813i	$0.262730\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.21759	+	1.85768i	0.320162	+	0.184846i	0.651465	−	0.758679i	$-0.274155\pi$
$101$	3.21759	+	1.85768i	0.320162	+	0.184846i	−0.331303	+	0.943525i	$0.607488\pi$
$102$		0			0
$103$		−	4.14872i		−	0.408785i	−0.978889	−	0.204393i	$-0.934478\pi$
$103$		−	4.14872i		−	0.408785i	0.978889	−	0.204393i	$-0.0655219\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.98601			0.772037			0.386019	−	0.922491i	$-0.373850\pi$
$107$	7.98601			0.772037			0.386019	+	0.922491i	$0.373850\pi$
$108$		0			0
$109$	−10.7920	+	6.23075i	−1.03368	+	0.596797i	−0.918038	−	0.396493i	$-0.870227\pi$
$109$	−10.7920	+	6.23075i	−1.03368	+	0.596797i	−0.115645	+	0.993291i	$0.536894\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.47237			0.420726			0.210363	−	0.977623i	$-0.432535\pi$
$113$	4.47237			0.420726			0.210363	+	0.977623i	$0.432535\pi$
$114$		0			0
$115$	−9.81294			−0.915061
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.180901	−	0.104443i	0.0165832	−	0.00957429i
$120$		0			0
$121$	−16.0263			−1.45693
$122$		0			0
$123$		0			0
$124$		0			0
$125$			10.7654i			0.962891i
$126$		0			0
$127$	−11.8089	−	6.81787i	−1.04787	−	0.604988i	−0.125818	−	0.992053i	$-0.540155\pi$
$127$	−11.8089	−	6.81787i	−1.04787	−	0.604988i	−0.922052	+	0.387065i	$0.873489\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.37592	−	4.25849i	0.644437	−	0.372066i	−0.141885	−	0.989883i	$-0.545316\pi$
$131$	7.37592	−	4.25849i	0.644437	−	0.372066i	0.786322	+	0.617817i	$0.211983\pi$
$132$		0			0
$133$	−0.661024	+	2.38369i	−0.0573180	+	0.206692i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.72914	−	2.15302i	−0.318602	−	0.183945i	0.332167	−	0.943220i	$-0.392220\pi$
$137$	−3.72914	−	2.15302i	−0.318602	−	0.183945i	−0.650769	+	0.759276i	$0.725553\pi$
$138$		0			0
$139$	3.05523	−	5.29181i	0.259141	−	0.448846i	−0.706871	−	0.707343i	$-0.749894\pi$
$139$	3.05523	−	5.29181i	0.259141	−	0.448846i	0.966012	+	0.258497i	$0.0832272\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.1097	−	31.3669i	1.51441	−	2.62303i
$144$		0			0
$145$		−	9.56832i		−	0.794606i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.5249	+	8.96330i	−1.27185	+	0.734302i	−0.975336	−	0.220727i	$-0.929157\pi$
$149$	−15.5249	+	8.96330i	−1.27185	+	0.734302i	−0.296513	+	0.955029i	$0.595824\pi$
$150$		0			0
$151$			8.59574i			0.699512i	0.936841	+	0.349756i	$0.113735\pi$
$151$			8.59574i			0.699512i	−0.936841	+	0.349756i	$0.886265\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.21087	−	7.29344i	−0.338225	−	0.585823i
$156$		0			0
$157$	0.829630	+	1.43696i	0.0662117	+	0.114682i	0.897231	−	0.441562i	$-0.145575\pi$
$157$	0.829630	+	1.43696i	0.0662117	+	0.114682i	−0.831019	+	0.556244i	$0.812242\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.73150	−	2.15438i	−0.294083	−	0.169789i
$162$		0			0
$163$	−15.1777			−1.18881			−0.594405	−	0.804166i	$-0.702612\pi$
$163$	−15.1777			−1.18881			−0.594405	+	0.804166i	$0.702612\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.22919	+	10.7893i	−0.482029	+	0.834900i	−0.999787	−	0.0206277i	$-0.993434\pi$
$167$	−6.22919	+	10.7893i	−0.482029	+	0.834900i	0.517758	+	0.855527i	$0.326767\pi$
$168$		0			0
$169$	17.7697	+	30.7781i	1.36690	+	2.36754i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.23855	+	3.87728i	0.170194	+	0.294784i	0.938488	−	0.345313i	$-0.112227\pi$
$173$	2.23855	+	3.87728i	0.170194	+	0.294784i	−0.768294	+	0.640098i	$0.778894\pi$
$174$		0			0
$175$	−0.944770	+	1.63639i	−0.0714179	+	0.123699i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−19.9167			−1.48864			−0.744321	−	0.667822i	$-0.767227\pi$
$179$	−19.9167			−1.48864			−0.744321	+	0.667822i	$0.767227\pi$
$180$		0			0
$181$	13.5336	+	7.81364i	1.00595	+	0.580783i	0.910002	−	0.414603i	$-0.136080\pi$
$181$	13.5336	+	7.81364i	1.00595	+	0.580783i	0.0959439	+	0.995387i	$0.469413\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.16505	−	5.48203i	−0.232699	−	0.403047i
$186$		0			0
$187$	0.956782	+	1.65720i	0.0699668	+	0.121186i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.8749i		−	1.00395i	−0.864882	−	0.501975i	$-0.832607\pi$
$191$		−	13.8749i		−	1.00395i	0.864882	−	0.501975i	$-0.167393\pi$
$192$		0			0
$193$	−7.73610	+	4.46644i	−0.556857	+	0.321501i	−0.751883	−	0.659297i	$-0.770854\pi$
$193$	−7.73610	+	4.46644i	−0.556857	+	0.321501i	0.195026	+	0.980798i	$0.437521\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	27.7967i		−	1.98043i	−0.139546	−	0.990216i	$-0.544564\pi$
$197$		−	27.7967i		−	1.98043i	0.139546	−	0.990216i	$-0.455436\pi$
$198$		0			0
$199$	−11.5795	+	20.0563i	−0.820849	+	1.42175i	0.0842013	+	0.996449i	$0.473166\pi$
$199$	−11.5795	+	20.0563i	−0.820849	+	1.42175i	−0.905051	+	0.425304i	$0.860167\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.10068	−	3.63848i	0.147439	−	0.255371i
$204$		0			0
$205$	−12.5839	−	7.26534i	−0.878900	−	0.507433i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.8364	−	6.05550i	−1.51046	−	0.418868i
$210$		0			0
$211$	17.1938	−	9.92682i	1.18367	−	0.683390i	0.226807	−	0.973940i	$-0.427171\pi$
$211$	17.1938	−	9.92682i	1.18367	−	0.683390i	0.956860	+	0.290549i	$0.0938380\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	5.48123	+	3.16459i	0.373817	+	0.215823i
$216$		0			0
$217$		−	3.69790i		−	0.251030i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.56447			−0.172505
$222$		0			0
$223$	−5.45210	+	3.14777i	−0.365100	+	0.210790i	−0.671315	−	0.741172i	$-0.734270\pi$
$223$	−5.45210	+	3.14777i	−0.365100	+	0.210790i	0.306216	+	0.951962i	$0.400937\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.27036			−0.0843168			−0.0421584	−	0.999111i	$-0.513423\pi$
$227$	−1.27036			−0.0843168			−0.0421584	+	0.999111i	$0.513423\pi$
$228$		0			0
$229$	−2.33721			−0.154447			−0.0772237	−	0.997014i	$-0.524606\pi$
$229$	−2.33721			−0.154447			−0.0772237	+	0.997014i	$0.524606\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	14.2677	−	8.23749i	0.934711	−	0.539656i	0.0464127	−	0.998922i	$-0.485221\pi$
$233$	14.2677	−	8.23749i	0.934711	−	0.539656i	0.888298	+	0.459267i	$0.151888\pi$
$234$		0			0
$235$	−11.0076			−0.718055
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7.22455i		−	0.467317i	−0.972319	−	0.233659i	$-0.924930\pi$
$239$		−	7.22455i		−	0.467317i	0.972319	−	0.233659i	$-0.0750698\pi$
$240$		0			0
$241$	−5.53918	−	3.19804i	−0.356810	−	0.206004i	0.310871	−	0.950452i	$-0.399379\pi$
$241$	−5.53918	−	3.19804i	−0.356810	−	0.206004i	−0.667680	+	0.744448i	$0.732713\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−7.47447	+	4.31539i	−0.477526	+	0.275700i
$246$		0			0
$247$	21.6601	−	21.2859i	1.37820	−	1.35439i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.24080	−	5.33518i	−0.583274	−	0.336753i	0.179159	−	0.983820i	$-0.442662\pi$
$251$	−9.24080	−	5.33518i	−0.583274	−	0.336753i	−0.762433	+	0.647067i	$0.775996\pi$
$252$		0			0
$253$	19.7358	−	34.1835i	1.24078	−	2.14910i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.61211	+	16.6487i	−0.599587	+	1.03851i	0.393295	+	0.919412i	$0.371335\pi$
$257$	−9.61211	+	16.6487i	−0.599587	+	1.03851i	−0.992882	+	0.119103i	$0.961998\pi$
$258$		0			0
$259$		−	2.77949i		−	0.172709i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.62360	+	4.40149i	−0.470091	+	0.271407i	−0.716278	−	0.697815i	$-0.754156\pi$
$263$	−7.62360	+	4.40149i	−0.470091	+	0.271407i	0.246187	+	0.969222i	$0.420822\pi$
$264$		0			0
$265$		−	7.85552i		−	0.482561i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.57542	+	11.3890i	0.400910	+	0.694397i	0.993836	−	0.110860i	$-0.0353606\pi$
$269$	6.57542	+	11.3890i	0.400910	+	0.694397i	−0.592926	+	0.805257i	$0.702027\pi$
$270$		0			0
$271$	1.00000	+	1.73205i	0.0607457	+	0.105215i	0.894799	−	0.446469i	$-0.147319\pi$
$271$	1.00000	+	1.73205i	0.0607457	+	0.105215i	−0.834053	+	0.551684i	$0.813985\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.9906	−	8.65484i	−0.903969	−	0.521907i
$276$		0			0
$277$	−17.3372			−1.04169			−0.520846	−	0.853651i	$-0.674383\pi$
$277$	−17.3372			−1.04169			−0.520846	+	0.853651i	$0.674383\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.86674	−	11.8935i	0.409635	−	0.709509i	−0.585214	−	0.810879i	$-0.698989\pi$
$281$	6.86674	−	11.8935i	0.409635	−	0.709509i	0.994849	+	0.101370i	$0.0323226\pi$
$282$		0			0
$283$	−4.38929	−	7.60247i	−0.260916	−	0.451920i	0.705569	−	0.708641i	$-0.250691\pi$
$283$	−4.38929	−	7.60247i	−0.260916	−	0.451920i	−0.966486	+	0.256721i	$0.917358\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.19014	−	5.52548i	−0.188308	−	0.326159i
$288$		0			0
$289$	−8.43226	+	14.6051i	−0.496015	+	0.859123i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.3394			1.07140			0.535700	−	0.844409i	$-0.320048\pi$
$293$	18.3394			1.07140			0.535700	+	0.844409i	$0.320048\pi$
$294$		0			0
$295$	−6.80308	−	3.92776i	−0.396090	−	0.228683i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	26.4490	+	45.8111i	1.52959	+	2.64932i
$300$		0			0
$301$	1.38954	+	2.40675i	0.0800918	+	0.138723i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.95439i		−	0.226428i
$306$		0			0
$307$	19.0784	−	11.0149i	1.08886	−	0.628654i	0.155588	−	0.987822i	$-0.450273\pi$
$307$	19.0784	−	11.0149i	1.08886	−	0.628654i	0.933273	+	0.359168i	$0.116940\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			19.2852i			1.09356i	0.837276	+	0.546780i	$0.184147\pi$
$311$			19.2852i			1.09356i	−0.837276	+	0.546780i	$0.815853\pi$
$312$		0			0
$313$	6.53766	−	11.3236i	0.369531	−	0.640046i	−0.619962	−	0.784632i	$-0.712852\pi$
$313$	6.53766	−	11.3236i	0.369531	−	0.640046i	0.989492	+	0.144586i	$0.0461852\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0434	−	20.8598i	0.676427	−	1.17161i	−0.299623	−	0.954058i	$-0.596861\pi$
$317$	12.0434	−	20.8598i	0.676427	−	1.17161i	0.976050	−	0.217548i	$-0.0698057\pi$
$318$		0			0
$319$	33.3314	+	19.2439i	1.86620	+	1.07745i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.401741	+	1.55334i	0.0223535	+	0.0864302i
$324$		0			0
$325$	20.0897	−	11.5988i	1.11438	−	0.643386i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.18578	−	2.41666i	−0.230770	−	0.133235i
$330$		0			0
$331$			8.91609i			0.490072i	0.969514	+	0.245036i	$0.0787999\pi$
$331$			8.91609i			0.490072i	−0.969514	+	0.245036i	$0.921200\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.65591			−0.0904720
$336$		0			0
$337$	−13.4778	+	7.78138i	−0.734180	+	0.423879i	−0.819949	−	0.572436i	$-0.805998\pi$
$337$	−13.4778	+	7.78138i	−0.734180	+	0.423879i	0.0857694	+	0.996315i	$0.472665\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	33.8757			1.83447
$342$		0			0
$343$	−7.76214			−0.419116
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.33896	+	4.23715i	−0.393976	+	0.227462i	−0.683881	−	0.729593i	$-0.739709\pi$
$347$	−7.33896	+	4.23715i	−0.393976	+	0.227462i	0.289905	+	0.957055i	$0.406376\pi$
$348$		0			0
$349$	−3.72700			−0.199502			−0.0997510	−	0.995012i	$-0.531805\pi$
$349$	−3.72700			−0.199502			−0.0997510	+	0.995012i	$0.531805\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4.08071i		−	0.217194i	−0.994086	−	0.108597i	$-0.965364\pi$
$353$		−	4.08071i		−	0.217194i	0.994086	−	0.108597i	$-0.0346359\pi$
$354$		0			0
$355$	8.22496	+	4.74868i	0.436536	+	0.252034i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.5822	+	8.99636i	−0.822395	+	0.474810i	−0.851242	−	0.524774i	$-0.824150\pi$
$359$	−15.5822	+	8.99636i	−0.822395	+	0.474810i	0.0288467	+	0.999584i	$0.490817\pi$
$360$		0			0
$361$	−16.2864	−	9.78534i	−0.857180	−	0.515018i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.6465	+	9.61088i	0.871319	+	0.503056i
$366$		0			0
$367$	4.68238	−	8.11012i	0.244418	−	0.423345i	−0.717550	−	0.696507i	$-0.754736\pi$
$367$	4.68238	−	8.11012i	0.244418	−	0.423345i	0.961968	+	0.273162i	$0.0880697\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.72464	−	2.98717i	0.0895389	−	0.155086i
$372$		0			0
$373$			10.5873i			0.548189i	0.961703	+	0.274094i	$0.0883781\pi$
$373$			10.5873i			0.548189i	−0.961703	+	0.274094i	$0.911622\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−44.6691	+	25.7897i	−2.30058	+	1.32824i
$378$		0			0
$379$			15.3186i			0.786865i	0.919354	+	0.393432i	$0.128712\pi$
$379$			15.3186i			0.786865i	−0.919354	+	0.393432i	$0.871288\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.3940	+	19.7350i	0.582207	+	1.00841i	0.995217	+	0.0976857i	$0.0311440\pi$
$383$	11.3940	+	19.7350i	0.582207	+	1.00841i	−0.413010	+	0.910726i	$0.635523\pi$
$384$		0			0
$385$	1.90647	+	3.30210i	0.0971626	+	0.168291i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	28.9670	+	16.7241i	1.46869	+	0.847947i	0.999384	−	0.0350897i	$-0.0111717\pi$
$389$	28.9670	+	16.7241i	1.46869	+	0.847947i	0.469303	+	0.883037i	$0.344505\pi$
$390$		0			0
$391$	−2.79475			−0.141336
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.06387	+	10.5029i	−0.305106	+	0.528460i
$396$		0			0
$397$	−11.8033	−	20.4440i	−0.592393	−	1.02605i	−0.993909	−	0.110202i	$-0.964850\pi$
$397$	−11.8033	−	20.4440i	−0.592393	−	1.02605i	0.401517	−	0.915852i	$-0.368483\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.52315	−	6.10227i	−0.175938	−	0.304733i	0.764548	−	0.644567i	$-0.222962\pi$
$401$	−3.52315	−	6.10227i	−0.175938	−	0.304733i	−0.940485	+	0.339834i	$0.889629\pi$
$402$		0			0
$403$	−22.6993	+	39.3164i	−1.13073	+	1.95849i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	25.4623			1.26212
$408$		0			0
$409$	−10.8592	−	6.26957i	−0.536953	−	0.310010i	0.206890	−	0.978364i	$-0.433666\pi$
$409$	−10.8592	−	6.26957i	−0.536953	−	0.310010i	−0.743843	+	0.668354i	$0.766999\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.72464	−	2.98717i	−0.0848640	−	0.146989i
$414$		0			0
$415$	−8.62715	−	14.9427i	−0.423490	−	0.733507i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			6.58347i			0.321624i	0.986985	+	0.160812i	$0.0514112\pi$
$419$			6.58347i			0.321624i	−0.986985	+	0.160812i	$0.948589\pi$
$420$		0			0
$421$	1.01114	−	0.583781i	0.0492798	−	0.0284517i	−0.475158	−	0.879901i	$-0.657609\pi$
$421$	1.01114	−	0.583781i	0.0492798	−	0.0284517i	0.524437	+	0.851449i	$0.324276\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			1.22559i			0.0594499i
$426$		0			0
$427$	0.868167	−	1.50371i	0.0420136	−	0.0727696i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.8530	+	20.5300i	−0.570940	+	0.988897i	0.425530	+	0.904944i	$0.360088\pi$
$431$	−11.8530	+	20.5300i	−0.570940	+	0.988897i	−0.996470	+	0.0839526i	$0.973246\pi$
$432$		0			0
$433$	26.9098	+	15.5364i	1.29320	+	0.746630i	0.979220	−	0.202800i	$-0.0650041\pi$
$433$	26.9098	+	15.5364i	1.29320	+	0.746630i	0.313980	+	0.949430i	$0.398337\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	23.6051	−	23.1973i	1.12919	−	1.10968i
$438$		0			0
$439$	−11.2663	+	6.50459i	−0.537711	+	0.310447i	−0.744151	−	0.668012i	$-0.767146\pi$
$439$	−11.2663	+	6.50459i	−0.537711	+	0.310447i	0.206440	+	0.978459i	$0.433812\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.1466	−	10.4770i	−0.862172	−	0.497775i	0.00256685	−	0.999997i	$-0.499183\pi$
$443$	−18.1466	−	10.4770i	−0.862172	−	0.497775i	−0.864739	+	0.502221i	$0.832516\pi$
$444$		0			0
$445$		−	2.46583i		−	0.116892i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−26.3017			−1.24125			−0.620626	−	0.784107i	$-0.713122\pi$
$449$	−26.3017			−1.24125			−0.620626	+	0.784107i	$0.713122\pi$
$450$		0			0
$451$	50.6178	−	29.2242i	2.38350	−	1.37611i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.10992			−0.239557
$456$		0			0
$457$	14.5839			0.682208			0.341104	−	0.940026i	$-0.389199\pi$
$457$	14.5839			0.682208			0.341104	+	0.940026i	$0.389199\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.0887	−	11.0209i	0.889050	−	0.513293i	0.0154182	−	0.999881i	$-0.495092\pi$
$461$	19.0887	−	11.0209i	0.889050	−	0.513293i	0.873632	+	0.486588i	$0.161759\pi$
$462$		0			0
$463$	1.59279			0.0740234			0.0370117	−	0.999315i	$-0.488216\pi$
$463$	1.59279			0.0740234			0.0370117	+	0.999315i	$0.488216\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			27.9641i			1.29402i	0.762480	+	0.647011i	$0.223981\pi$
$467$			27.9641i			1.29402i	−0.762480	+	0.647011i	$0.776019\pi$
$468$		0			0
$469$	−0.629681	−	0.363547i	−0.0290760	−	0.0167870i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−22.0478	+	12.7293i	−1.01376	+	0.585293i
$474$		0			0
$475$	−10.1728	−	10.3517i	−0.466759	−	0.474966i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.9582	−	9.79083i	−0.774841	−	0.447354i	0.0597581	−	0.998213i	$-0.480967\pi$
$479$	−16.9582	−	9.79083i	−0.774841	−	0.447354i	−0.834599	+	0.550859i	$0.814300\pi$
$480$		0			0
$481$	−17.0617	+	29.5517i	−0.777946	+	1.34744i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.80027	−	4.85021i	0.127154	−	0.220237i
$486$		0			0
$487$		−	6.40254i		−	0.290127i	−0.989422	−	0.145063i	$-0.953661\pi$
$487$		−	6.40254i		−	0.290127i	0.989422	−	0.145063i	$-0.0463386\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	27.4305	−	15.8370i	1.23792	−	0.714713i	0.269251	−	0.963070i	$-0.413224\pi$
$491$	27.4305	−	15.8370i	1.23792	−	0.714713i	0.968668	+	0.248357i	$0.0798907\pi$
$492$		0			0
$493$		−	2.72508i		−	0.122731i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.08510	+	3.61150i	0.0935296	+	0.161998i
$498$		0			0
$499$	15.6526	+	27.1110i	0.700704	+	1.21366i	0.968219	+	0.250102i	$0.0804642\pi$
$499$	15.6526	+	27.1110i	0.700704	+	1.21366i	−0.267515	+	0.963554i	$0.586202\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.6512	−	7.30415i	−0.564087	−	0.325676i	0.190697	−	0.981649i	$-0.438925\pi$
$503$	−12.6512	−	7.30415i	−0.564087	−	0.325676i	−0.754784	+	0.655973i	$0.772259\pi$
$504$		0			0
$505$	4.80183			0.213679
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.73838	−	11.6712i	0.298673	−	0.517317i	−0.677159	−	0.735836i	$-0.736789\pi$
$509$	6.73838	−	11.6712i	0.298673	−	0.517317i	0.975833	+	0.218519i	$0.0701226\pi$
$510$		0			0
$511$	4.22004	+	7.30933i	0.186684	+	0.323346i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.68096	−	4.64355i	−0.118137	−	0.204619i
$516$		0			0
$517$	22.1385	−	38.3451i	0.973651	−	1.68641i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−33.9855			−1.48893			−0.744466	−	0.667660i	$-0.767296\pi$
$521$	−33.9855			−1.48893			−0.744466	+	0.667660i	$0.767296\pi$
$522$		0			0
$523$	−24.3345	−	14.0496i	−1.06408	−	0.614344i	−0.137519	−	0.990499i	$-0.543913\pi$
$523$	−24.3345	−	14.0496i	−1.06408	−	0.614344i	−0.926557	+	0.376155i	$0.877246\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.19927	−	2.07719i	−0.0522408	−	0.0904837i
$528$		0			0
$529$	17.3241	+	30.0062i	0.753221	+	1.30462i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			78.3298i			3.39284i
$534$		0			0
$535$	8.93855	−	5.16067i	0.386447	−	0.223115i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	34.7165i		−	1.49535i
$540$		0			0
$541$	7.28083	−	12.6108i	0.313027	−	0.542179i	−0.665989	−	0.745962i	$-0.731990\pi$
$541$	7.28083	−	12.6108i	0.313027	−	0.542179i	0.979016	+	0.203782i	$0.0653235\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.05279	+	13.9478i	−0.344944	+	0.597460i
$546$		0			0
$547$	25.2049	+	14.5520i	1.07768	+	0.622200i	0.930270	−	0.366875i	$-0.119572\pi$
$547$	25.2049	+	14.5520i	1.07768	+	0.622200i	0.147412	+	0.989075i	$0.452906\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.6190	+	23.0167i	0.963602	+	0.980545i
$552$		0			0
$553$	−4.61174	+	2.66259i	−0.196111	+	0.113225i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.81683	+	2.20365i	0.161724	+	0.0933717i	0.578678	−	0.815556i	$-0.303569\pi$
$557$	3.81683	+	2.20365i	0.161724	+	0.0933717i	−0.416953	+	0.908928i	$0.636902\pi$
$558$		0			0
$559$		−	34.1184i		−	1.44305i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	25.6092			1.07930			0.539650	−	0.841889i	$-0.318557\pi$
$563$	25.6092			1.07930			0.539650	+	0.841889i	$0.318557\pi$
$564$		0			0
$565$	5.00582	−	2.89011i	0.210596	−	0.121588i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	25.5543			1.07129			0.535646	−	0.844442i	$-0.320068\pi$
$569$	25.5543			1.07129			0.535646	+	0.844442i	$0.320068\pi$
$570$		0			0
$571$	−4.73586			−0.198190			−0.0990949	−	0.995078i	$-0.531595\pi$
$571$	−4.73586			−0.198190			−0.0990949	+	0.995078i	$0.531595\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	21.8937	−	12.6403i	0.913031	−	0.527139i
$576$		0			0
$577$	3.68553			0.153431			0.0767154	−	0.997053i	$-0.475557\pi$
$577$	3.68553			0.153431			0.0767154	+	0.997053i	$0.475557\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	7.57620i		−	0.314314i
$582$		0			0
$583$	27.3648	+	15.7991i	1.13333	+	0.654331i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	36.2610	−	20.9353i	1.49665	−	0.864093i	0.496660	−	0.867945i	$-0.334560\pi$
$587$	36.2610	−	20.9353i	1.49665	−	0.864093i	0.999993	+	0.00385243i	$0.00122627\pi$
$588$		0			0
$589$	27.3706	+	7.59018i	1.12779	+	0.312748i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	39.3479	+	22.7175i	1.61582	+	0.932896i	0.987984	+	0.154554i	$0.0493940\pi$
$593$	39.3479	+	22.7175i	1.61582	+	0.932896i	0.627840	+	0.778343i	$0.283939\pi$
$594$		0			0
$595$	0.134985	−	0.233801i	0.00553386	−	0.00958492i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.87586	+	8.44524i	−0.199222	+	0.345063i	−0.948277	−	0.317446i	$-0.897175\pi$
$599$	−4.87586	+	8.44524i	−0.199222	+	0.345063i	0.749054	+	0.662509i	$0.230508\pi$
$600$		0			0
$601$			23.4702i			0.957371i	0.877986	+	0.478686i	$0.158887\pi$
$601$			23.4702i			0.957371i	−0.877986	+	0.478686i	$0.841113\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.9378	+	10.3564i	−0.729276	+	0.421048i
$606$		0			0
$607$		−	36.5195i		−	1.48228i	−0.671351	−	0.741139i	$-0.734286\pi$
$607$		−	36.5195i		−	1.48228i	0.671351	−	0.741139i	$-0.265714\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.6690	+	51.3882i	1.20028	+	2.07894i
$612$		0			0
$613$	−6.29576	−	10.9046i	−0.254283	−	0.440432i	0.710417	−	0.703781i	$-0.248506\pi$
$613$	−6.29576	−	10.9046i	−0.254283	−	0.440432i	−0.964701	+	0.263349i	$0.915173\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.9375	+	14.9750i	1.04420	+	0.602872i	0.921021	−	0.389512i	$-0.127356\pi$
$617$	25.9375	+	14.9750i	1.04420	+	0.602872i	0.123183	+	0.992384i	$0.460690\pi$
$618$		0			0
$619$	7.39512			0.297235			0.148618	−	0.988895i	$-0.452518\pi$
$619$	7.39512			0.297235			0.148618	+	0.988895i	$0.452518\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.541361	−	0.937665i	0.0216892	−	0.0375668i
$624$		0			0
$625$	−1.36729	−	2.36822i	−0.0546917	−	0.0947288i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.901414	−	1.56130i	−0.0359417	−	0.0622529i
$630$		0			0
$631$	−8.08910	+	14.0107i	−0.322022	+	0.557759i	−0.980905	−	0.194487i	$-0.937696\pi$
$631$	−8.08910	+	14.0107i	−0.322022	+	0.557759i	0.658883	+	0.752245i	$0.271029\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17.6232			−0.699356
$636$		0			0
$637$	40.2922	+	23.2627i	1.59644	+	0.921703i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.0432	−	26.0555i	−0.594169	−	1.02913i	−0.993664	−	0.112396i	$-0.964148\pi$
$641$	−15.0432	−	26.0555i	−0.594169	−	1.02913i	0.399494	−	0.916736i	$-0.369186\pi$
$642$		0			0
$643$	−17.1523	−	29.7087i	−0.676421	−	1.17160i	−0.976051	−	0.217540i	$-0.930197\pi$
$643$	−17.1523	−	29.7087i	−0.676421	−	1.17160i	0.299631	−	0.954055i	$-0.403137\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2.64279i		−	0.103899i	−0.998650	−	0.0519494i	$-0.983457\pi$
$647$		−	2.64279i		−	0.103899i	0.998650	−	0.0519494i	$-0.0165435\pi$
$648$		0			0
$649$	27.3648	−	15.7991i	1.07416	−	0.620168i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			5.11932i			0.200334i	0.994971	+	0.100167i	$0.0319378\pi$
$653$			5.11932i			0.200334i	−0.994971	+	0.100167i	$0.968062\pi$
$654$		0			0
$655$	5.50379	−	9.53285i	0.215051	−	0.372479i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.54797	+	11.3414i	−0.255073	+	0.441799i	−0.964915	−	0.262562i	$-0.915433\pi$
$659$	−6.54797	+	11.3414i	−0.255073	+	0.441799i	0.709843	+	0.704360i	$0.248766\pi$
$660$		0			0
$661$	−3.92165	−	2.26416i	−0.152534	−	0.0880658i	0.421790	−	0.906694i	$-0.361402\pi$
$661$	−3.92165	−	2.26416i	−0.152534	−	0.0880658i	−0.574325	+	0.818628i	$0.694735\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.800503	+	3.09516i	0.0310422	+	0.120025i
$666$		0			0
$667$	−48.6803	+	28.1056i	−1.88491	+	1.08825i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.7752	+	7.95310i	0.531784	+	0.307026i
$672$		0			0
$673$		−	19.7011i		−	0.759423i	−0.925105	−	0.379712i	$-0.876023\pi$
$673$		−	19.7011i		−	0.759423i	0.925105	−	0.379712i	$-0.123977\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.1374			0.581777			0.290889	−	0.956757i	$-0.406049\pi$
$677$	15.1374			0.581777			0.290889	+	0.956757i	$0.406049\pi$
$678$		0			0
$679$	2.12968	−	1.22957i	0.0817297	−	0.0471866i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.68384			−0.294014			−0.147007	−	0.989135i	$-0.546964\pi$
$683$	−7.68384			−0.294014			−0.147007	+	0.989135i	$0.546964\pi$
$684$		0			0
$685$	−5.56524			−0.212637
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−36.6730	+	21.1732i	−1.39713	+	0.806633i
$690$		0			0
$691$	−4.59504			−0.174804			−0.0874019	−	0.996173i	$-0.527856\pi$
$691$	−4.59504			−0.174804			−0.0874019	+	0.996173i	$0.527856\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	7.89733i		−	0.299563i
$696$		0			0
$697$	−3.58393	−	2.06919i	−0.135751	−	0.0783760i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−35.0953	+	20.2623i	−1.32553	+	0.765296i	−0.984605	−	0.174793i	$-0.944074\pi$
$701$	−35.0953	+	20.2623i	−1.32553	+	0.765296i	−0.340927	+	0.940090i	$0.610741\pi$
$702$		0			0
$703$	20.5728	+	5.70508i	0.775918	+	0.215171i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.82596	+	1.05422i	0.0686723	+	0.0396480i
$708$		0			0
$709$	−5.07280	+	8.78634i	−0.190513	+	0.329978i	−0.945420	−	0.325853i	$-0.894348\pi$
$709$	−5.07280	+	8.78634i	−0.190513	+	0.329978i	0.754907	+	0.655831i	$0.227682\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−24.7376	+	42.8468i	−0.926431	+	1.60463i
$714$		0			0
$715$		−	46.8109i		−	1.75063i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	17.9443	−	10.3602i	0.669211	−	0.386369i	−0.126567	−	0.991958i	$-0.540396\pi$
$719$	17.9443	−	10.3602i	0.669211	−	0.386369i	0.795778	+	0.605589i	$0.207062\pi$
$720$		0			0
$721$		−	2.35437i		−	0.0876811i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.3252	+	21.3479i	0.457748	+	0.792843i
$726$		0			0
$727$	−7.64321	−	13.2384i	−0.283471	−	0.490986i	0.688766	−	0.724983i	$-0.258153\pi$
$727$	−7.64321	−	13.2384i	−0.283471	−	0.490986i	−0.972237	+	0.233997i	$0.924819\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.56107	+	0.901282i	0.0577381	+	0.0333351i
$732$		0			0
$733$	24.6289			0.909689			0.454845	−	0.890571i	$-0.349695\pi$
$733$	24.6289			0.909689			0.454845	+	0.890571i	$0.349695\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.33038	−	5.76838i	0.122676	−	0.212481i
$738$		0			0
$739$	−0.955908	−	1.65568i	−0.0351636	−	0.0609052i	0.847908	−	0.530143i	$-0.177862\pi$
$739$	−0.955908	−	1.65568i	−0.0351636	−	0.0609052i	−0.883072	+	0.469238i	$0.844529\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.9492	−	38.0172i	−0.805239	−	1.39471i	−0.916130	−	0.400882i	$-0.868704\pi$
$743$	−21.9492	−	38.0172i	−0.805239	−	1.39471i	0.110891	−	0.993833i	$-0.464630\pi$
$744$		0			0
$745$	−11.5844	+	20.0648i	−0.424420	+	0.735118i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.53200			0.165596
$750$		0			0
$751$	−18.4632	−	10.6598i	−0.673733	−	0.388980i	0.123757	−	0.992313i	$-0.460506\pi$
$751$	−18.4632	−	10.6598i	−0.673733	−	0.388980i	−0.797490	+	0.603333i	$0.793839\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5.55469	+	9.62100i	0.202156	+	0.350144i
$756$		0			0
$757$	−4.43453	−	7.68084i	−0.161176	−	0.279165i	0.774115	−	0.633045i	$-0.218195\pi$
$757$	−4.43453	−	7.68084i	−0.161176	−	0.279165i	−0.935291	+	0.353880i	$0.884862\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6.98326i		−	0.253143i	−0.991957	−	0.126571i	$-0.959603\pi$
$761$		−	6.98326i		−	0.253143i	0.991957	−	0.126571i	$-0.0403973\pi$
$762$		0			0
$763$	−6.12436	+	3.53590i	−0.221717	+	0.128008i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			42.3463i			1.52904i
$768$		0			0
$769$	23.7022	−	41.0535i	0.854724	−	1.48043i	−0.0221765	−	0.999754i	$-0.507060\pi$
$769$	23.7022	−	41.0535i	0.854724	−	1.48043i	0.876901	−	0.480672i	$-0.159607\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−21.0383	+	36.4394i	−0.756695	+	1.31063i	0.187832	+	0.982201i	$0.439854\pi$
$773$	−21.0383	+	36.4394i	−0.756695	+	1.31063i	−0.944527	+	0.328433i	$0.893479\pi$
$774$		0			0
$775$	18.7898	+	10.8483i	0.674949	+	0.389682i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	47.4457	−	12.2709i	1.69992	−	0.439650i
$780$		0			0
$781$	−33.0842	+	19.1012i	−1.18385	+	0.683494i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.85717	+	1.07224i	0.0662852	+	0.0382698i
$786$		0			0
$787$			36.3759i			1.29666i	0.761360	+	0.648330i	$0.224532\pi$
$787$			36.3759i			1.29666i	−0.761360	+	0.648330i	$0.775468\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.53804			0.0902423
$792$		0			0
$793$	−18.4608	+	10.6584i	−0.655563	+	0.378490i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36.4903			1.29255			0.646277	−	0.763103i	$-0.276325\pi$
$797$	36.4903			1.29255			0.646277	+	0.763103i	$0.276325\pi$
$798$		0			0
$799$	−3.13499			−0.110908
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−66.9592	+	38.6589i	−2.36294	+	1.36424i
$804$		0			0
$805$	−5.56877			−0.196273
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	16.2246i		−	0.570426i	−0.958464	−	0.285213i	$-0.907936\pi$
$809$		−	16.2246i		−	0.570426i	0.958464	−	0.285213i	$-0.0920644\pi$
$810$		0			0
$811$	16.8200	+	9.71103i	0.590630	+	0.341001i	0.765347	−	0.643618i	$-0.222568\pi$
$811$	16.8200	+	9.71103i	0.590630	+	0.341001i	−0.174716	+	0.984619i	$0.555901\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−16.9880	+	9.80804i	−0.595065	+	0.343561i
$816$		0			0
$817$	−20.6661	+	5.34487i	−0.723014	+	0.186993i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.0185	−	15.0218i	−0.908052	−	0.524264i	−0.0282484	−	0.999601i	$-0.508993\pi$
$821$	−26.0185	−	15.0218i	−0.908052	−	0.524264i	−0.879804	+	0.475337i	$0.842326\pi$
$822$		0			0
$823$	−12.4360	+	21.5399i	−0.433493	+	0.750833i	−0.997171	−	0.0751621i	$-0.976053\pi$
$823$	−12.4360	+	21.5399i	−0.433493	+	0.750833i	0.563678	+	0.825995i	$0.309386\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.9634	−	20.7213i	0.416010	−	0.720550i	−0.579524	−	0.814955i	$-0.696762\pi$
$827$	11.9634	−	20.7213i	0.416010	−	0.720550i	0.995534	+	0.0944052i	$0.0300949\pi$
$828$		0			0
$829$			9.30883i			0.323309i	0.986847	+	0.161655i	$0.0516830\pi$
$829$			9.30883i			0.323309i	−0.986847	+	0.161655i	$0.948317\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.12875	+	1.22903i	−0.0737567	+	0.0425834i
$834$		0			0
$835$			16.1016i			0.557218i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−13.9788	−	24.2120i	−0.482601	−	0.835890i	0.517199	−	0.855865i	$-0.326975\pi$
$839$	−13.9788	−	24.2120i	−0.482601	−	0.835890i	−0.999800	+	0.0199750i	$0.993641\pi$
$840$		0			0
$841$	−12.9050	−	22.3520i	−0.444998	−	0.770760i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	39.7784	+	22.9661i	1.36842	+	0.790057i
$846$		0			0
$847$	−9.09479			−0.312501
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−18.5938	+	32.2053i	−0.637386	+	1.10398i
$852$		0			0
$853$	13.9441	+	24.1519i	0.477438	+	0.826947i	0.999666	−	0.0258593i	$-0.00823219\pi$
$853$	13.9441	+	24.1519i	0.477438	+	0.826947i	−0.522228	+	0.852806i	$0.674899\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17.5128	−	30.3330i	−0.598225	−	1.03616i	−0.993083	−	0.117415i	$-0.962539\pi$
$857$	−17.5128	−	30.3330i	−0.598225	−	1.03616i	0.394858	−	0.918742i	$-0.370794\pi$
$858$		0			0
$859$	24.0062	−	41.5800i	0.819081	−	1.41869i	−0.0872788	−	0.996184i	$-0.527817\pi$
$859$	24.0062	−	41.5800i	0.819081	−	1.41869i	0.906360	−	0.422506i	$-0.138850\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	4.42220			0.150533			0.0752667	−	0.997163i	$-0.476019\pi$
$863$	4.42220			0.150533			0.0752667	+	0.997163i	$0.476019\pi$
$864$		0			0
$865$	5.01111	+	2.89317i	0.170383	+	0.0983706i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.3914	−	42.2471i	−0.827422	−	1.43314i
$870$		0			0
$871$	4.46321	+	7.73051i	0.151230	+	0.261938i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			6.10931i			0.206532i
$876$		0			0
$877$	−26.4386	+	15.2643i	−0.892768	+	0.515440i	−0.874847	−	0.484399i	$-0.839038\pi$
$877$	−26.4386	+	15.2643i	−0.892768	+	0.515440i	−0.0179213	+	0.999839i	$0.505705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	43.4912i		−	1.46526i	−0.680629	−	0.732629i	$-0.738293\pi$
$881$		−	43.4912i		−	1.46526i	0.680629	−	0.732629i	$-0.261707\pi$
$882$		0			0
$883$	19.0588	−	33.0108i	0.641379	−	1.11090i	−0.343747	−	0.939062i	$-0.611696\pi$
$883$	19.0588	−	33.0108i	0.641379	−	1.11090i	0.985125	−	0.171838i	$-0.0549706\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.77728	+	10.0065i	−0.193982	+	0.335987i	−0.946566	−	0.322509i	$-0.895474\pi$
$887$	−5.77728	+	10.0065i	−0.193982	+	0.335987i	0.752584	+	0.658496i	$0.228807\pi$
$888$		0			0
$889$	−6.70146	−	3.86909i	−0.224760	−	0.129765i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26.4789	−	26.0213i	0.886081	−	0.870771i
$894$		0			0
$895$	−22.2922	+	12.8704i	−0.745147	+	0.430211i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−41.7788	−	24.1210i	−1.39340	−	0.804480i
$900$		0			0
$901$		−	2.23727i		−	0.0745343i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.1971			0.671375
$906$		0			0
$907$	31.0561	−	17.9303i	1.03120	−	0.595365i	0.113873	−	0.993495i	$-0.463674\pi$
$907$	31.0561	−	17.9303i	1.03120	−	0.595365i	0.917329	+	0.398131i	$0.130341\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	42.0217			1.39224			0.696121	−	0.717925i	$-0.254908\pi$
$911$	42.0217			1.39224			0.696121	+	0.717925i	$0.254908\pi$
$912$		0			0
$913$	69.4040			2.29694
$914$		0			0
$915$		0			0
$916$		0			0
$917$	4.18578	−	2.41666i	0.138227	−	0.0798052i
$918$		0			0
$919$	8.57457			0.282849			0.141425	−	0.989949i	$-0.454832\pi$
$919$	8.57457			0.282849			0.141425	+	0.989949i	$0.454832\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	51.1970i		−	1.68517i
$924$		0			0
$925$	14.1231	+	8.15400i	0.464366	+	0.268102i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	6.20113	−	3.58022i	0.203452	−	0.117463i	−0.394812	−	0.918762i	$-0.629190\pi$
$929$	6.20113	−	3.58022i	0.203452	−	0.117463i	0.598265	+	0.801298i	$0.295857\pi$
$930$		0			0
$931$	7.77858	−	28.0499i	0.254933	−	0.919300i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.14181	+	1.23657i	0.0700445	+	0.0404402i
$936$		0			0
$937$	−2.73610	+	4.73906i	−0.0893844	+	0.154818i	−0.907251	−	0.420589i	$-0.861823\pi$
$937$	−2.73610	+	4.73906i	−0.0893844	+	0.154818i	0.817867	+	0.575408i	$0.195157\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.9764	+	24.2079i	−0.455618	+	0.789154i	−0.998724	−	0.0505105i	$-0.983915\pi$
$941$	−13.9764	+	24.2079i	−0.455618	+	0.789154i	0.543105	+	0.839665i	$0.317249\pi$
$942$		0			0
$943$			85.3635i			2.77982i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.45438	+	2.57174i	−0.144748	+	0.0835702i	−0.570625	−	0.821211i	$-0.693299\pi$
$947$	−4.45438	+	2.57174i	−0.144748	+	0.0835702i	0.425877	+	0.904781i	$0.359966\pi$
$948$		0			0
$949$		−	103.618i		−	3.36357i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.85765	+	13.6099i	0.254534	+	0.440866i	0.964769	−	0.263099i	$-0.0847445\pi$
$953$	7.85765	+	13.6099i	0.254534	+	0.440866i	−0.710235	+	0.703965i	$0.751411\pi$
$954$		0			0
$955$	−8.96613	−	15.5298i	−0.290137	−	0.502532i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.11626	−	1.22182i	−0.0683375	−	0.0394547i
$960$		0			0
$961$	−11.4610			−0.369711
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.77255	+	9.99835i	−0.185825	+	0.321858i
$966$		0			0
$967$	6.83027	+	11.8304i	0.219647	+	0.380439i	0.954700	−	0.297570i	$-0.0961762\pi$
$967$	6.83027	+	11.8304i	0.219647	+	0.380439i	−0.735053	+	0.678009i	$0.762843\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.857770	+	1.48570i	0.0275271	+	0.0476784i	0.879461	−	0.475971i	$-0.157903\pi$
$971$	0.857770	+	1.48570i	0.0275271	+	0.0476784i	−0.851934	+	0.523650i	$0.824570\pi$
$972$		0			0
$973$	1.73382	−	3.00307i	0.0555837	−	0.0962738i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	31.0857			0.994519			0.497259	−	0.867602i	$-0.334340\pi$
$977$	31.0857			0.994519			0.497259	+	0.867602i	$0.334340\pi$
$978$		0			0
$979$	8.58975	+	4.95930i	0.274530	+	0.158500i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.745601	+	1.29142i	0.0237810	+	0.0411898i	0.877671	−	0.479264i	$-0.159096\pi$
$983$	0.745601	+	1.29142i	0.0237810	+	0.0411898i	−0.853890	+	0.520454i	$0.825763\pi$
$984$		0			0
$985$	−17.9626	−	31.1121i	−0.572336	−	0.991315i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	37.1821i		−	1.18232i
$990$		0			0
$991$	37.1101	−	21.4255i	1.17884	−	0.680604i	0.223095	−	0.974797i	$-0.428384\pi$
$991$	37.1101	−	21.4255i	1.17884	−	0.680604i	0.955746	+	0.294193i	$0.0950508\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			29.9314i			0.948888i
$996$		0			0
$997$	9.82963	−	17.0254i	0.311308	−	0.539200i	−0.667338	−	0.744755i	$-0.732566\pi$
$997$	9.82963	−	17.0254i	0.311308	−	0.539200i	0.978646	+	0.205554i	$0.0658998\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.dc.d.449.6		16
3.2	odd	2	inner	2736.2.dc.d.449.3		16
4.3	odd	2		684.2.bk.a.449.6	yes	16
12.11	even	2		684.2.bk.a.449.3	✓	16
19.8	odd	6	inner	2736.2.dc.d.1889.3		16
57.8	even	6	inner	2736.2.dc.d.1889.6		16
76.27	even	6		684.2.bk.a.521.3	yes	16
228.179	odd	6		684.2.bk.a.521.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.bk.a.449.3	✓	16	12.11	even	2
684.2.bk.a.449.6	yes	16	4.3	odd	2
684.2.bk.a.521.3	yes	16	76.27	even	6
684.2.bk.a.521.6	yes	16	228.179	odd	6
2736.2.dc.d.449.3		16	3.2	odd	2	inner
2736.2.dc.d.449.6		16	1.1	even	1	trivial
2736.2.dc.d.1889.3		16	19.8	odd	6	inner
2736.2.dc.d.1889.6		16	57.8	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.dc (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.11928 - 0.646214i) q^{5} +0.567493 q^{7} +O(q^{10})\)	\(q+(1.11928 - 0.646214i) q^{5} +0.567493 q^{7} +5.19868i q^{11} +(-6.03362 - 3.48351i) q^{13} +(0.318772 - 0.184043i) q^{17} +(-1.16481 + 4.20038i) q^{19} +(-6.57542 - 3.79632i) q^{23} +(-1.66481 + 2.88354i) q^{25} +(3.70169 - 6.41151i) q^{29} -6.51621i q^{31} +(0.635181 - 0.366722i) q^{35} -4.89784i q^{37} +(-5.62146 - 9.73666i) q^{41} +(2.44856 + 4.24103i) q^{43} +(-7.37592 - 4.25849i) q^{47} -6.67795 q^{49} +(3.03905 - 5.26380i) q^{53} +(3.35946 + 5.81875i) q^{55} +(-3.03905 - 5.26380i) q^{59} +(1.52983 - 2.64974i) q^{61} -9.00438 q^{65} +(-1.10959 - 0.640619i) q^{67} +(3.67423 + 6.36396i) q^{71} +(7.43630 + 12.8800i) q^{73} +2.95021i q^{77} +(-8.12651 + 4.69184i) q^{79} -13.3503i q^{83} +(0.237863 - 0.411990i) q^{85} +(0.953953 - 1.65229i) q^{89} +(-3.42404 - 1.97687i) q^{91} +(1.41060 + 5.45410i) q^{95} +(3.75279 - 2.16668i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 12 q^{19} + 4 q^{25} + 4 q^{43} - 20 q^{55} + 12 q^{61} - 36 q^{67} + 44 q^{73} + 12 q^{79} + 56 q^{85} + 60 q^{91}+O(q^{100}) \) 16 * q + 12 * q^19 + 4 * q^25 + 4 * q^43 - 20 * q^55 + 12 * q^61 - 36 * q^67 + 44 * q^73 + 12 * q^79 + 56 * q^85 + 60 * q^91

Introduction

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