LMFDB - Embedded newform 2736.2.s.z.1873.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(577,2736)]

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, 0, 2]))

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

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

chi := DirichletCharacter("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.s (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1873.3
Root			$0.403374 - 1.68443i$ of defining polynomial
Character	$\chi$	$=$	2736.1873
Dual form			2736.2.s.z.577.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.66044 - 2.87597i) q^{5} -2.32088 q^{7} +O(q^{10})$	$q+(1.66044 - 2.87597i) q^{5} -2.32088 q^{7} -1.70739 q^{11} +(-2.01414 - 3.48859i) q^{13} +(-0.193252 - 4.35461i) q^{19} +(1.17458 + 2.03443i) q^{23} +(-3.01414 - 5.22064i) q^{25} +(3.32088 + 5.75194i) q^{29} -6.70739 q^{31} +(-3.85369 + 6.67479i) q^{35} -1.00000 q^{37} +(-3.32088 + 5.75194i) q^{41} +(0.353695 - 0.612617i) q^{43} +(3.00000 + 5.19615i) q^{47} -1.61350 q^{49} +(-4.98133 - 8.62791i) q^{53} +(-2.83502 + 4.91040i) q^{55} +(-0.853695 + 1.47864i) q^{59} +(-1.69325 - 2.93280i) q^{61} -13.3774 q^{65} +(-4.18872 - 7.25507i) q^{67} +(4.70739 - 8.15344i) q^{71} +(-5.82088 + 10.0821i) q^{73} +3.96265 q^{77} +(-1.67458 + 2.90046i) q^{79} -10.0565 q^{83} +(-1.33956 - 2.32018i) q^{89} +(4.67458 + 8.09661i) q^{91} +(-12.8446 - 6.67479i) q^{95} +(-8.86330 + 15.3517i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 6 * q + 2 * q^5 + 2 * q^7	$ 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65} + 9 q^{67} + 18 q^{71} - 19 q^{73} - 24 q^{77} + 11 q^{79} - 8 q^{83} - 16 q^{89} + 7 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) $ 6 * q + 2 * q^5 + 2 * q^7 + q^13 - 4 * q^19 - 14 * q^23 - 5 * q^25 + 4 * q^29 - 30 * q^31 - 18 * q^35 - 6 * q^37 - 4 * q^41 - 3 * q^43 + 18 * q^47 - 4 * q^49 - 6 * q^53 + 12 * q^55 - 13 * q^61 - 12 * q^65 + 9 * q^67 + 18 * q^71 - 19 * q^73 - 24 * q^77 + 11 * q^79 - 8 * q^83 - 16 * q^89 + 7 * q^91 + 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$5$	1.66044	−	2.87597i	0.742572	−	1.28617i	−0.208748	−	0.977969i	$-0.566939\pi$
$5$	1.66044	−	2.87597i	0.742572	−	1.28617i	0.951320	−	0.308204i	$-0.0997278\pi$
$6$		0			0
$7$	−2.32088			−0.877212			−0.438606	−	0.898679i	$-0.644528\pi$
$7$	−2.32088			−0.877212			−0.438606	+	0.898679i	$0.644528\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.70739			−0.514797			−0.257399	−	0.966305i	$-0.582865\pi$
$11$	−1.70739			−0.514797			−0.257399	+	0.966305i	$0.582865\pi$
$12$		0			0
$13$	−2.01414	−	3.48859i	−0.558621	−	0.967560i	−0.997612	−	0.0690685i	$-0.977997\pi$
$13$	−2.01414	−	3.48859i	−0.558621	−	0.967560i	0.438991	−	0.898492i	$-0.355336\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−0.193252	−	4.35461i	−0.0443351	−	0.999017i
$19$	−0.193252	−	4.35461i	−0.0443351	−	0.999017i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.17458	+	2.03443i	0.244917	+	0.424208i	0.962108	−	0.272668i	$-0.0879061\pi$
$23$	1.17458	+	2.03443i	0.244917	+	0.424208i	−0.717191	+	0.696876i	$0.754573\pi$
$24$		0			0
$25$	−3.01414	−	5.22064i	−0.602827	−	1.04413i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.32088	+	5.75194i	0.616673	+	1.06811i	0.990089	+	0.140444i	$0.0448532\pi$
$29$	3.32088	+	5.75194i	0.616673	+	1.06811i	−0.373416	+	0.927664i	$0.621814\pi$
$30$		0			0
$31$	−6.70739			−1.20468			−0.602341	−	0.798239i	$-0.705765\pi$
$31$	−6.70739			−1.20468			−0.602341	+	0.798239i	$0.705765\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.85369	+	6.67479i	−0.651393	+	1.12825i
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.32088	+	5.75194i	−0.518635	+	0.898302i	0.481131	+	0.876649i	$0.340226\pi$
$41$	−3.32088	+	5.75194i	−0.518635	+	0.898302i	−0.999766	+	0.0216532i	$0.993107\pi$
$42$		0			0
$43$	0.353695	−	0.612617i	0.0539379	−	0.0934232i	−0.837796	−	0.545984i	$-0.816156\pi$
$43$	0.353695	−	0.612617i	0.0539379	−	0.0934232i	0.891734	+	0.452561i	$0.149489\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	−1.61350			−0.230499
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.98133	−	8.62791i	−0.684238	−	1.18513i	−0.973676	−	0.227938i	$-0.926802\pi$
$53$	−4.98133	−	8.62791i	−0.684238	−	1.18513i	0.289438	−	0.957197i	$-0.406532\pi$
$54$		0			0
$55$	−2.83502	+	4.91040i	−0.382274	+	0.662118i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.853695	+	1.47864i	−0.111142	+	0.192503i	−0.916231	−	0.400651i	$-0.868784\pi$
$59$	−0.853695	+	1.47864i	−0.111142	+	0.192503i	0.805089	+	0.593154i	$0.202117\pi$
$60$		0			0
$61$	−1.69325	−	2.93280i	−0.216799	−	0.375506i	0.737029	−	0.675861i	$-0.236228\pi$
$61$	−1.69325	−	2.93280i	−0.216799	−	0.375506i	−0.953828	+	0.300355i	$0.902895\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−13.3774			−1.65927
$66$		0			0
$67$	−4.18872	−	7.25507i	−0.511733	−	0.886348i	−0.999907	−	0.0136016i	$-0.995670\pi$
$67$	−4.18872	−	7.25507i	−0.511733	−	0.886348i	0.488174	−	0.872746i	$-0.337663\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.70739	−	8.15344i	0.558664	−	0.967635i	−0.438944	−	0.898514i	$-0.644647\pi$
$71$	4.70739	−	8.15344i	0.558664	−	0.967635i	0.997608	−	0.0691206i	$-0.0220193\pi$
$72$		0			0
$73$	−5.82088	+	10.0821i	−0.681283	+	1.18002i	0.293307	+	0.956018i	$0.405244\pi$
$73$	−5.82088	+	10.0821i	−0.681283	+	1.18002i	−0.974590	+	0.223998i	$0.928089\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.96265			0.451586
$78$		0			0
$79$	−1.67458	+	2.90046i	−0.188405	+	0.326327i	−0.944719	−	0.327882i	$-0.893665\pi$
$79$	−1.67458	+	2.90046i	−0.188405	+	0.326327i	0.756314	+	0.654209i	$0.226998\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.0565			−1.10385			−0.551925	−	0.833894i	$-0.686106\pi$
$83$	−10.0565			−1.10385			−0.551925	+	0.833894i	$0.686106\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.33956	−	2.32018i	−0.141993	−	0.245939i	0.786254	−	0.617903i	$-0.212018\pi$
$89$	−1.33956	−	2.32018i	−0.141993	−	0.245939i	−0.928247	+	0.371964i	$0.878684\pi$
$90$		0			0
$91$	4.67458	+	8.09661i	0.490029	+	0.848755i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−12.8446	−	6.67479i	−1.31783	−	0.684820i
$96$		0			0
$97$	−8.86330	+	15.3517i	−0.899931	+	1.55873i	−0.0723511	+	0.997379i	$0.523050\pi$
$97$	−8.86330	+	15.3517i	−0.899931	+	1.55873i	−0.827580	+	0.561347i	$0.810283\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.02827	+	13.9054i	0.798843	+	1.38364i	0.920370	+	0.391049i	$0.127888\pi$
$101$	8.02827	+	13.9054i	0.798843	+	1.38364i	−0.121527	+	0.992588i	$0.538779\pi$
$102$		0			0
$103$	7.54787			0.743714			0.371857	−	0.928290i	$-0.378721\pi$
$103$	7.54787			0.743714			0.371857	+	0.928290i	$0.378721\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.28354			0.704126			0.352063	−	0.935976i	$-0.385480\pi$
$107$	7.28354			0.704126			0.352063	+	0.935976i	$0.385480\pi$
$108$		0			0
$109$	6.12763	−	10.6134i	0.586921	−	1.01658i	−0.407712	−	0.913110i	$-0.633673\pi$
$109$	6.12763	−	10.6134i	0.586921	−	1.01658i	0.994633	−	0.103466i	$-0.0329933\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.37743			−0.694010			−0.347005	−	0.937863i	$-0.612801\pi$
$113$	−7.37743			−0.694010			−0.347005	+	0.937863i	$0.612801\pi$
$114$		0			0
$115$	7.80128			0.727473
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−8.08482			−0.734984
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.41478			−0.305427
$126$		0			0
$127$	7.32088	+	12.6801i	0.649623	+	1.12518i	0.983213	+	0.182462i	$0.0584068\pi$
$127$	7.32088	+	12.6801i	0.649623	+	1.12518i	−0.333589	+	0.942718i	$0.608260\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.320884	+	0.555788i	−0.0280358	+	0.0485594i	−0.879703	−	0.475524i	$-0.842259\pi$
$131$	−0.320884	+	0.555788i	−0.0280358	+	0.0485594i	0.851667	+	0.524083i	$0.175592\pi$
$132$		0			0
$133$	0.448517	+	10.1066i	0.0388913	+	0.876349i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.29261	+	2.23887i	0.110435	+	0.191279i	0.915946	−	0.401302i	$-0.131442\pi$
$137$	1.29261	+	2.23887i	0.110435	+	0.191279i	−0.805511	+	0.592581i	$0.798109\pi$
$138$		0			0
$139$	−9.28807	−	16.0874i	−0.787804	−	1.36452i	−0.927309	−	0.374296i	$-0.877885\pi$
$139$	−9.28807	−	16.0874i	−0.787804	−	1.36452i	0.139505	−	0.990221i	$-0.455449\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.43892	+	5.95638i	0.287577	+	0.498097i
$144$		0			0
$145$	22.0565			1.83170
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.98133	+	3.43176i	−0.162317	+	0.281141i	−0.935699	−	0.352799i	$-0.885230\pi$
$149$	−1.98133	+	3.43176i	−0.162317	+	0.281141i	0.773382	+	0.633940i	$0.218563\pi$
$150$		0			0
$151$	8.69832			0.707859			0.353929	−	0.935272i	$-0.384845\pi$
$151$	8.69832			0.707859			0.353929	+	0.935272i	$0.384845\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−11.1372	+	19.2903i	−0.894564	+	1.54943i
$156$		0			0
$157$	2.84916	−	4.93489i	0.227388	−	0.393847i	−0.729645	−	0.683826i	$-0.760315\pi$
$157$	2.84916	−	4.93489i	0.227388	−	0.393847i	0.957033	+	0.289979i	$0.0936482\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.72606	−	4.72168i	−0.214844	−	0.372120i
$162$		0			0
$163$	−18.3774			−1.43943			−0.719716	−	0.694269i	$-0.755728\pi$
$163$	−18.3774			−1.43943			−0.719716	+	0.694269i	$0.755728\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.8163	−	18.7345i	−0.836994	−	1.44972i	−0.892396	−	0.451252i	$-0.850977\pi$
$167$	−10.8163	−	18.7345i	−0.836994	−	1.44972i	0.0554023	−	0.998464i	$-0.482356\pi$
$168$		0			0
$169$	−1.61350	+	2.79466i	−0.124115	+	0.214974i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.34916	+	4.06886i	−0.178603	+	0.309350i	−0.941402	−	0.337286i	$-0.890491\pi$
$173$	−2.34916	+	4.06886i	−0.178603	+	0.309350i	0.762799	+	0.646636i	$0.223825\pi$
$174$		0			0
$175$	6.99546	+	12.1165i	0.528807	+	0.915921i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.06562			−0.0796482			−0.0398241	−	0.999207i	$-0.512680\pi$
$179$	−1.06562			−0.0796482			−0.0398241	+	0.999207i	$0.512680\pi$
$180$		0			0
$181$	1.83502	+	3.17835i	0.136396	+	0.236245i	0.926130	−	0.377205i	$-0.123115\pi$
$181$	1.83502	+	3.17835i	0.136396	+	0.236245i	−0.789734	+	0.613450i	$0.789781\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.66044	+	2.87597i	−0.122078	+	0.211446i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.877832			−0.0635177			−0.0317588	−	0.999496i	$-0.510111\pi$
$191$	−0.877832			−0.0635177			−0.0317588	+	0.999496i	$0.510111\pi$
$192$		0			0
$193$	4.30675	−	7.45951i	0.310006	−	0.536947i	−0.668357	−	0.743841i	$-0.733002\pi$
$193$	4.30675	−	7.45951i	0.310006	−	0.536947i	0.978363	+	0.206894i	$0.0663354\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−24.7357			−1.76234			−0.881172	−	0.472797i	$-0.843244\pi$
$197$	−24.7357			−1.76234			−0.881172	+	0.472797i	$0.843244\pi$
$198$		0			0
$199$	−10.9955	−	19.0447i	−0.779448	−	1.35004i	−0.932260	−	0.361788i	$-0.882166\pi$
$199$	−10.9955	−	19.0447i	−0.779448	−	1.35004i	0.152813	−	0.988255i	$-0.451167\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−7.70739	−	13.3496i	−0.540953	−	0.936958i
$204$		0			0
$205$	11.0283	+	19.1015i	0.770248	+	1.33411i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.329957	+	7.43502i	0.0228236	+	0.514291i
$210$		0			0
$211$	−6.60896	+	11.4471i	−0.454979	+	0.788048i	−0.998687	−	0.0512272i	$-0.983687\pi$
$211$	−6.60896	+	11.4471i	−0.454979	+	0.788048i	0.543708	+	0.839275i	$0.317020\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.17458	−	2.03443i	−0.0801056	−	0.138747i
$216$		0			0
$217$	15.5671			1.05676
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.74020	+	4.74616i	−0.183497	+	0.317827i	−0.943069	−	0.332597i	$-0.892075\pi$
$223$	−2.74020	+	4.74616i	−0.183497	+	0.317827i	0.759572	+	0.650423i	$0.225409\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.70739			0.511557			0.255779	−	0.966735i	$-0.417668\pi$
$227$	7.70739			0.511557			0.255779	+	0.966735i	$0.417668\pi$
$228$		0			0
$229$	4.02827			0.266196			0.133098	−	0.991103i	$-0.457508\pi$
$229$	4.02827			0.266196			0.133098	+	0.991103i	$0.457508\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.0565	−	22.6146i	0.855363	−	1.48153i	−0.0209451	−	0.999781i	$-0.506668\pi$
$233$	13.0565	−	22.6146i	0.855363	−	1.48153i	0.876308	−	0.481751i	$-0.159999\pi$
$234$		0			0
$235$	19.9253			1.29978
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.70739			−0.498550			−0.249275	−	0.968433i	$-0.580192\pi$
$239$	−7.70739			−0.498550			−0.249275	+	0.968433i	$0.580192\pi$
$240$		0			0
$241$	−10.8492	−	18.7913i	−0.698856	−	1.21045i	−0.968863	−	0.247595i	$-0.920360\pi$
$241$	−10.8492	−	18.7913i	−0.698856	−	1.21045i	0.270008	−	0.962858i	$-0.412974\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.67912	+	4.64036i	−0.171162	+	0.296462i
$246$		0			0
$247$	−14.8022	+	9.44496i	−0.941842	+	0.600969i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.70739	−	8.15344i	−0.297128	−	0.514640i	0.678350	−	0.734739i	$-0.262695\pi$
$251$	−4.70739	−	8.15344i	−0.297128	−	0.514640i	−0.975478	+	0.220099i	$0.929362\pi$
$252$		0			0
$253$	−2.00546	−	3.47357i	−0.126082	−	0.218381i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.07522	−	8.79054i	−0.316584	−	0.548339i	0.663189	−	0.748452i	$-0.269202\pi$
$257$	−5.07522	−	8.79054i	−0.316584	−	0.548339i	−0.979773	+	0.200113i	$0.935869\pi$
$258$		0			0
$259$	2.32088			0.144213
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.29261	+	2.23887i	−0.0797058	+	0.138054i	−0.903123	−	0.429382i	$-0.858731\pi$
$263$	−1.29261	+	2.23887i	−0.0797058	+	0.138054i	0.823417	+	0.567436i	$0.192065\pi$
$264$		0			0
$265$	−33.0848			−2.03238
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.273937	−	0.474473i	0.0167023	−	0.0289292i	−0.857553	−	0.514395i	$-0.828017\pi$
$269$	0.273937	−	0.474473i	0.0167023	−	0.0289292i	0.874256	+	0.485466i	$0.161350\pi$
$270$		0			0
$271$	−10.4431	+	18.0879i	−0.634370	+	1.09876i	0.352278	+	0.935895i	$0.385407\pi$
$271$	−10.4431	+	18.0879i	−0.634370	+	1.09876i	−0.986648	+	0.162866i	$0.947926\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.14631	+	8.91366i	0.310334	+	0.537514i
$276$		0			0
$277$	−23.7831			−1.42899			−0.714495	−	0.699640i	$-0.753344\pi$
$277$	−23.7831			−1.42899			−0.714495	+	0.699640i	$0.753344\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.95305	−	10.3110i	−0.355129	−	0.615102i	0.632011	−	0.774960i	$-0.282230\pi$
$281$	−5.95305	−	10.3110i	−0.355129	−	0.615102i	−0.987140	+	0.159858i	$0.948897\pi$
$282$		0			0
$283$	9.02827	−	15.6374i	0.536675	−	0.929549i	−0.462405	−	0.886669i	$-0.653013\pi$
$283$	9.02827	−	15.6374i	0.536675	−	0.929549i	0.999080	−	0.0428799i	$-0.0136533\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.70739	−	13.3496i	0.454953	−	0.788001i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	27.3966			1.60053			0.800264	−	0.599648i	$-0.204693\pi$
$293$	27.3966			1.60053			0.800264	+	0.599648i	$0.204693\pi$
$294$		0			0
$295$	2.83502	+	4.91040i	0.165061	+	0.285895i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.73153	−	8.19524i	0.273631	−	0.473943i
$300$		0			0
$301$	−0.820884	+	1.42181i	−0.0473150	+	0.0819520i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.2462			−0.643955
$306$		0			0
$307$	1.80675	−	3.12938i	0.103117	−	0.178603i	−0.809851	−	0.586636i	$-0.800452\pi$
$307$	1.80675	−	3.12938i	0.103117	−	0.178603i	0.912967	+	0.408033i	$0.133785\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.4057			1.04369			0.521846	−	0.853040i	$-0.325244\pi$
$311$	18.4057			1.04369			0.521846	+	0.853040i	$0.325244\pi$
$312$		0			0
$313$	−11.5424	−	19.9920i	−0.652416	−	1.13002i	−0.982535	−	0.186078i	$-0.940422\pi$
$313$	−11.5424	−	19.9920i	−0.652416	−	1.13002i	0.330119	−	0.943939i	$-0.392911\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.6887	−	21.9775i	−0.712669	−	1.23438i	−0.963852	−	0.266440i	$-0.914153\pi$
$317$	−12.6887	−	21.9775i	−0.712669	−	1.23438i	0.251182	−	0.967940i	$-0.419181\pi$
$318$		0			0
$319$	−5.67004	−	9.82080i	−0.317461	−	0.549859i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−12.1418	+	21.0302i	−0.673504	+	1.16654i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.96265	−	12.0597i	−0.383864	−	0.664871i
$330$		0			0
$331$	21.3492			1.17346			0.586728	−	0.809784i	$-0.300416\pi$
$331$	21.3492			1.17346			0.586728	+	0.809784i	$0.300416\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−27.8205			−1.52000
$336$		0			0
$337$	2.04241	−	3.53756i	0.111257	−	0.192703i	−0.805020	−	0.593247i	$-0.797846\pi$
$337$	2.04241	−	3.53756i	0.111257	−	0.192703i	0.916277	+	0.400544i	$0.131179\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	11.4521			0.620167
$342$		0			0
$343$	19.9909			1.07941
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.85369	+	17.0671i	−0.528974	+	0.916210i	0.470455	+	0.882424i	$0.344090\pi$
$347$	−9.85369	+	17.0671i	−0.528974	+	0.916210i	−0.999429	+	0.0337860i	$0.989244\pi$
$348$		0			0
$349$	23.3118			1.24785			0.623926	−	0.781483i	$-0.285537\pi$
$349$	23.3118			1.24785			0.623926	+	0.781483i	$0.285537\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−25.1896			−1.34071			−0.670355	−	0.742041i	$-0.733858\pi$
$353$	−25.1896			−1.34071			−0.670355	+	0.742041i	$0.733858\pi$
$354$		0			0
$355$	−15.6327	−	27.0766i	−0.829697	−	1.43708i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.61350	−	2.79466i	0.0851570	−	0.147496i	−0.820301	−	0.571932i	$-0.806194\pi$
$359$	1.61350	−	2.79466i	0.0851570	−	0.147496i	0.905458	+	0.424436i	$0.139527\pi$
$360$		0			0
$361$	−18.9253	+	1.68308i	−0.996069	+	0.0885831i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	19.3305	+	33.4814i	1.01180	+	1.75250i
$366$		0			0
$367$	−14.7311	−	25.5151i	−0.768959	−	1.33188i	−0.938128	−	0.346288i	$-0.887442\pi$
$367$	−14.7311	−	25.5151i	−0.768959	−	1.33188i	0.169170	−	0.985587i	$-0.445891\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	11.5611	+	20.0244i	0.600222	+	1.03961i
$372$		0			0
$373$	19.6700			1.01848			0.509238	−	0.860626i	$-0.329927\pi$
$373$	19.6700			1.01848			0.509238	+	0.860626i	$0.329927\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	13.3774	−	23.1704i	0.688973	−	1.19334i
$378$		0			0
$379$	0.763937			0.0392408			0.0196204	−	0.999808i	$-0.493754\pi$
$379$	0.763937			0.0392408			0.0196204	+	0.999808i	$0.493754\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.91932	+	3.32435i	−0.0980724	+	0.169866i	−0.910887	−	0.412656i	$-0.864601\pi$
$383$	−1.91932	+	3.32435i	−0.0980724	+	0.169866i	0.812814	+	0.582523i	$0.197934\pi$
$384$		0			0
$385$	6.57976	−	11.3965i	0.335335	−	0.580818i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.1035	+	27.8921i	0.816480	+	1.41418i	0.908261	+	0.418405i	$0.137411\pi$
$389$	16.1035	+	27.8921i	0.816480	+	1.41418i	−0.0917810	+	0.995779i	$0.529256\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.56108	+	9.63208i	0.279809	+	0.484643i
$396$		0			0
$397$	13.2977	−	23.0322i	0.667391	−	1.15596i	−0.311240	−	0.950331i	$-0.600744\pi$
$397$	13.2977	−	23.0322i	0.667391	−	1.15596i	0.978631	−	0.205624i	$-0.0659224\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−4.66044	+	8.07212i	−0.232731	+	0.403103i	−0.958611	−	0.284719i	$-0.908100\pi$
$401$	−4.66044	+	8.07212i	−0.232731	+	0.403103i	0.725880	+	0.687822i	$0.241433\pi$
$402$		0			0
$403$	13.5096	+	23.3993i	0.672961	+	1.16560i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.70739			0.0846321
$408$		0			0
$409$	−10.9006	−	18.8805i	−0.539002	−	0.933579i	−0.998958	−	0.0456372i	$-0.985468\pi$
$409$	−10.9006	−	18.8805i	−0.539002	−	0.933579i	0.459956	−	0.887942i	$-0.347865\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.98133	−	3.43176i	0.0974947	−	0.168866i
$414$		0			0
$415$	−16.6983	+	28.9223i	−0.819688	+	1.41974i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.93438			0.241060			0.120530	−	0.992710i	$-0.461541\pi$
$419$	4.93438			0.241060			0.120530	+	0.992710i	$0.461541\pi$
$420$		0			0
$421$	−2.12763	+	3.68517i	−0.103694	+	0.179604i	−0.913204	−	0.407503i	$-0.866400\pi$
$421$	−2.12763	+	3.68517i	−0.103694	+	0.179604i	0.809510	+	0.587107i	$0.199733\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	3.92984	+	6.80669i	0.190178	+	0.329399i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.414779	+	0.718418i	0.0199792	+	0.0346050i	0.875842	−	0.482598i	$-0.160307\pi$
$431$	0.414779	+	0.718418i	0.0199792	+	0.0346050i	−0.855863	+	0.517203i	$0.826973\pi$
$432$		0			0
$433$	−8.24113	−	14.2741i	−0.396043	−	0.685967i	0.597190	−	0.802099i	$-0.296284\pi$
$433$	−8.24113	−	14.2741i	−0.396043	−	0.685967i	−0.993234	+	0.116132i	$0.962950\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.63217	−	5.50800i	0.412933	−	0.263483i
$438$		0			0
$439$	14.2170	−	24.6245i	0.678540	−	1.17527i	−0.296881	−	0.954915i	$-0.595946\pi$
$439$	14.2170	−	24.6245i	0.678540	−	1.17527i	0.975421	−	0.220351i	$-0.0707203\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.70739	+	13.3496i	0.366189	+	0.634258i	0.988966	−	0.148141i	$-0.0473290\pi$
$443$	7.70739	+	13.3496i	0.366189	+	0.634258i	−0.622777	+	0.782399i	$0.713996\pi$
$444$		0			0
$445$	−8.89703			−0.421760
$446$		0			0
$447$		0			0
$448$		0			0
$449$	23.1523			1.09262			0.546312	−	0.837582i	$-0.316031\pi$
$449$	23.1523			1.09262			0.546312	+	0.837582i	$0.316031\pi$
$450$		0			0
$451$	5.67004	−	9.82080i	0.266992	−	0.462444i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	31.0475			1.45553
$456$		0			0
$457$	4.68819			0.219304			0.109652	−	0.993970i	$-0.465026\pi$
$457$	4.68819			0.219304			0.109652	+	0.993970i	$0.465026\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.27394	−	10.8668i	0.292206	−	0.506116i	−0.682125	−	0.731236i	$-0.738944\pi$
$461$	6.27394	−	10.8668i	0.292206	−	0.506116i	0.974331	+	0.225119i	$0.0722772\pi$
$462$		0			0
$463$	−22.8880			−1.06369			−0.531847	−	0.846841i	$-0.678502\pi$
$463$	−22.8880			−1.06369			−0.531847	+	0.846841i	$0.678502\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.8122			−0.546604			−0.273302	−	0.961928i	$-0.588116\pi$
$467$	−11.8122			−0.546604			−0.273302	+	0.961928i	$0.588116\pi$
$468$		0			0
$469$	9.72153	+	16.8382i	0.448898	+	0.777515i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.603895	+	1.04598i	−0.0277671	+	0.0480940i
$474$		0			0
$475$	−22.1514	+	14.1343i	−1.01637	+	0.648526i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.0192	+	29.4781i	0.777627	+	1.34689i	0.933306	+	0.359082i	$0.116910\pi$
$479$	17.0192	+	29.4781i	0.777627	+	1.34689i	−0.155679	+	0.987808i	$0.549756\pi$
$480$		0			0
$481$	2.01414	+	3.48859i	0.0918367	+	0.159066i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.4340	+	50.9811i	1.33653	+	2.31493i
$486$		0			0
$487$	7.41478			0.335996			0.167998	−	0.985787i	$-0.446270\pi$
$487$	7.41478			0.335996			0.167998	+	0.985787i	$0.446270\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.93438	+	3.35044i	−0.0872973	+	0.151203i	−0.906368	−	0.422489i	$-0.861156\pi$
$491$	−1.93438	+	3.35044i	−0.0872973	+	0.151203i	0.819071	+	0.573693i	$0.194490\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10.9253	+	18.9232i	−0.490067	+	0.848821i
$498$		0			0
$499$	−17.7931	+	30.8186i	−0.796530	+	1.37963i	0.125333	+	0.992115i	$0.460000\pi$
$499$	−17.7931	+	30.8186i	−0.796530	+	1.37963i	−0.921863	+	0.387516i	$0.873333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.29261	−	7.43502i	−0.191398	−	0.331511i	0.754316	−	0.656512i	$-0.227969\pi$
$503$	−4.29261	−	7.43502i	−0.191398	−	0.331511i	−0.945714	+	0.325001i	$0.894635\pi$
$504$		0			0
$505$	53.3219			2.37280
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.97173	−	6.87923i	−0.176044	−	0.304917i	0.764478	−	0.644649i	$-0.222997\pi$
$509$	−3.97173	−	6.87923i	−0.176044	−	0.304917i	−0.940522	+	0.339733i	$0.889663\pi$
$510$		0			0
$511$	13.5096	−	23.3993i	0.597630	−	1.03512i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.5328	−	21.7075i	0.552262	−	0.956545i
$516$		0			0
$517$	−5.12217	−	8.87186i	−0.225273	−	0.390184i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.0757489			0.00331862			0.00165931	−	0.999999i	$-0.499472\pi$
$521$	0.0757489			0.00331862			0.00165931	+	0.999999i	$0.499472\pi$
$522$		0			0
$523$	10.1514	+	17.5827i	0.443888	+	0.768837i	0.997974	−	0.0636224i	$-0.0202653\pi$
$523$	10.1514	+	17.5827i	0.443888	+	0.768837i	−0.554086	+	0.832460i	$0.686932\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	8.74073	−	15.1394i	0.380032	−	0.658234i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	26.7549			1.15888
$534$		0			0
$535$	12.0939	−	20.9472i	0.522865	−	0.905628i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.75486			0.118660
$540$		0			0
$541$	16.5475	+	28.6611i	0.711432	+	1.23224i	0.964320	+	0.264740i	$0.0852862\pi$
$541$	16.5475	+	28.6611i	0.711432	+	1.23224i	−0.252888	+	0.967495i	$0.581381\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−20.3492	−	35.2458i	−0.871662	−	1.50976i
$546$		0			0
$547$	−17.9581	−	31.1044i	−0.767834	−	1.32993i	−0.938735	−	0.344639i	$-0.888001\pi$
$547$	−17.9581	−	31.1044i	−0.767834	−	1.32993i	0.170902	−	0.985288i	$-0.445332\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	24.4057	−	15.5727i	1.03972	−	0.663421i
$552$		0			0
$553$	3.88650	−	6.73162i	0.165271	−	0.286258i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.962653	−	1.66736i	−0.0407889	−	0.0706485i	0.844910	−	0.534908i	$-0.179654\pi$
$557$	−0.962653	−	1.66736i	−0.0407889	−	0.0706485i	−0.885699	+	0.464260i	$0.846320\pi$
$558$		0			0
$559$	−2.84956			−0.120523
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.8861			0.458795			0.229397	−	0.973333i	$-0.426324\pi$
$563$	10.8861			0.458795			0.229397	+	0.973333i	$0.426324\pi$
$564$		0			0
$565$	−12.2498	+	21.2173i	−0.515353	+	0.892618i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	36.4358			1.52747			0.763735	−	0.645530i	$-0.223364\pi$
$569$	36.4358			1.52747			0.763735	+	0.645530i	$0.223364\pi$
$570$		0			0
$571$	7.54787			0.315869			0.157934	−	0.987450i	$-0.449517\pi$
$571$	7.54787			0.315869			0.157934	+	0.987450i	$0.449517\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.08068	−	12.2641i	0.295285	−	0.511449i
$576$		0			0
$577$	−15.4823			−0.644535			−0.322267	−	0.946649i	$-0.604445\pi$
$577$	−15.4823			−0.644535			−0.322267	+	0.946649i	$0.604445\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	23.3401			0.968310
$582$		0			0
$583$	8.50506	+	14.7312i	0.352244	+	0.610104i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.8729	−	25.7606i	0.613870	−	1.06325i	−0.376712	−	0.926331i	$-0.622945\pi$
$587$	14.8729	−	25.7606i	0.613870	−	1.06325i	0.990582	−	0.136924i	$-0.0437215\pi$
$588$		0			0
$589$	1.29622	+	29.2081i	0.0534098	+	1.20350i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.273937	−	0.474473i	−0.0112493	−	0.0194843i	0.860346	−	0.509711i	$-0.170247\pi$
$593$	−0.273937	−	0.474473i	−0.0112493	−	0.0194843i	−0.871595	+	0.490226i	$0.836914\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.8163	+	39.5191i	0.932251	+	1.61471i	0.779465	+	0.626446i	$0.215491\pi$
$599$	22.8163	+	39.5191i	0.932251	+	1.61471i	0.152786	+	0.988259i	$0.451175\pi$
$600$		0			0
$601$	−6.85783			−0.279737			−0.139868	−	0.990170i	$-0.544668\pi$
$601$	−6.85783			−0.279737			−0.139868	+	0.990170i	$0.544668\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−13.4244	+	23.2517i	−0.545779	+	0.945316i
$606$		0			0
$607$	6.76394			0.274540			0.137270	−	0.990534i	$-0.456167\pi$
$607$	6.76394			0.274540			0.137270	+	0.990534i	$0.456167\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	12.0848	−	20.9315i	0.488900	−	0.846799i
$612$		0			0
$613$	−2.22153	+	3.84780i	−0.0897266	+	0.155411i	−0.907396	−	0.420278i	$-0.861933\pi$
$613$	−2.22153	+	3.84780i	−0.0897266	+	0.155411i	0.817669	+	0.575689i	$0.195266\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16.0005	−	27.7137i	−0.644157	−	1.11571i	−0.984496	−	0.175410i	$-0.943875\pi$
$617$	−16.0005	−	27.7137i	−0.644157	−	1.11571i	0.340339	−	0.940303i	$-0.389458\pi$
$618$		0			0
$619$	14.3774			0.577878			0.288939	−	0.957348i	$-0.406698\pi$
$619$	14.3774			0.577878			0.288939	+	0.957348i	$0.406698\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.10896	+	5.38487i	0.124558	+	0.215740i
$624$		0			0
$625$	9.40064	−	16.2824i	0.376026	−	0.651296i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	5.54695	+	9.60759i	0.220820	+	0.382472i	0.955057	−	0.296421i	$-0.0957932\pi$
$631$	5.54695	+	9.60759i	0.220820	+	0.382472i	−0.734237	+	0.678893i	$0.762460\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	48.6236			1.92957
$636$		0			0
$637$	3.24980	+	5.62882i	0.128762	+	0.223022i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21.7357	−	37.6473i	0.858507	−	1.48698i	−0.0148458	−	0.999890i	$-0.504726\pi$
$641$	21.7357	−	37.6473i	0.858507	−	1.48698i	0.873353	−	0.487088i	$-0.161941\pi$
$642$		0			0
$643$	−11.8113	+	20.4577i	−0.465792	+	0.806775i	−0.999237	−	0.0390599i	$-0.987564\pi$
$643$	−11.8113	+	20.4577i	−0.465792	+	0.806775i	0.533445	+	0.845835i	$0.320897\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−19.9253			−0.783345			−0.391672	−	0.920105i	$-0.628103\pi$
$647$	−19.9253			−0.783345			−0.391672	+	0.920105i	$0.628103\pi$
$648$		0			0
$649$	1.45759	−	2.52462i	0.0572154	−	0.0990999i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.11310			0.0826918			0.0413459	−	0.999145i	$-0.486835\pi$
$653$	2.11310			0.0826918			0.0413459	+	0.999145i	$0.486835\pi$
$654$		0			0
$655$	1.06562	+	1.84571i	0.0416372	+	0.0721178i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−21.7507	−	37.6734i	−0.847288	−	1.46755i	−0.883619	−	0.468206i	$-0.844900\pi$
$659$	−21.7507	−	37.6734i	−0.847288	−	1.46755i	0.0363312	−	0.999340i	$-0.488433\pi$
$660$		0			0
$661$	−11.0565	−	19.1505i	−0.430050	−	0.744868i	0.566827	−	0.823837i	$-0.308171\pi$
$661$	−11.0565	−	19.1505i	−0.430050	−	0.744868i	−0.996877	+	0.0789685i	$0.974837\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	29.8109	+	15.4914i	1.15602	+	0.600732i
$666$		0			0
$667$	−7.80128	+	13.5122i	−0.302067	+	0.523195i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.89104	+	5.00743i	0.111607	+	0.193310i
$672$		0			0
$673$	12.5953			0.485515			0.242758	−	0.970087i	$-0.421948\pi$
$673$	12.5953			0.485515			0.242758	+	0.970087i	$0.421948\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	32.9427			1.26609			0.633045	−	0.774115i	$-0.281805\pi$
$677$	32.9427			1.26609			0.633045	+	0.774115i	$0.281805\pi$
$678$		0			0
$679$	20.5707	−	35.6295i	0.789430	−	1.36733i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18.3876			−0.703580			−0.351790	−	0.936079i	$-0.614427\pi$
$683$	−18.3876			−0.703580			−0.351790	+	0.936079i	$0.614427\pi$
$684$		0			0
$685$	8.58522			0.328024
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−20.0661	+	34.7556i	−0.764459	+	1.32408i
$690$		0			0
$691$	16.6599			0.633773			0.316887	−	0.948463i	$-0.397363\pi$
$691$	16.6599			0.633773			0.316887	+	0.948463i	$0.397363\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−61.6892			−2.34001
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−24.3588	+	42.1906i	−0.920018	+	1.59352i	−0.120633	+	0.992697i	$0.538493\pi$
$701$	−24.3588	+	42.1906i	−0.920018	+	1.59352i	−0.799384	+	0.600820i	$0.794841\pi$
$702$		0			0
$703$	0.193252	+	4.35461i	0.00728865	+	0.164237i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18.6327	−	32.2728i	−0.700755	−	1.21374i
$708$		0			0
$709$	−14.7498	−	25.5474i	−0.553940	−	0.959453i	−0.997985	−	0.0634483i	$-0.979790\pi$
$709$	−14.7498	−	25.5474i	−0.553940	−	0.959453i	0.444045	−	0.896005i	$-0.353543\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.87836	−	13.6457i	−0.295047	−	0.511036i
$714$		0			0
$715$	22.8405			0.854186
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−11.3250	+	19.6155i	−0.422352	+	0.731535i	−0.996169	−	0.0874484i	$-0.972129\pi$
$719$	−11.3250	+	19.6155i	−0.422352	+	0.731535i	0.573817	+	0.818984i	$0.305462\pi$
$720$		0			0
$721$	−17.5177			−0.652395
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.0192	−	34.6743i	0.743494	−	1.28777i
$726$		0			0
$727$	26.3638	−	45.6635i	0.977780	−	1.69356i	0.307342	−	0.951599i	$-0.400560\pi$
$727$	26.3638	−	45.6635i	0.977780	−	1.69356i	0.670438	−	0.741966i	$-0.266106\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	5.72659			0.211516			0.105758	−	0.994392i	$-0.466273\pi$
$733$	5.72659			0.211516			0.105758	+	0.994392i	$0.466273\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.15177	+	12.3872i	0.263439	+	0.456289i
$738$		0			0
$739$	16.7083	−	28.9397i	0.614625	−	1.06456i	−0.375825	−	0.926691i	$-0.622641\pi$
$739$	16.7083	−	28.9397i	0.614625	−	1.06456i	0.990450	−	0.137872i	$-0.0440261\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17.9759	−	31.1351i	0.659470	−	1.14224i	−0.321282	−	0.946983i	$-0.604114\pi$
$743$	17.9759	−	31.1351i	0.659470	−	1.14224i	0.980753	−	0.195253i	$-0.0625528\pi$
$744$		0			0
$745$	6.57976	+	11.3965i	0.241064	+	0.417534i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−16.9043			−0.617668
$750$		0			0
$751$	−11.5752	−	20.0489i	−0.422386	−	0.731594i	0.573787	−	0.819005i	$-0.305474\pi$
$751$	−11.5752	−	20.0489i	−0.422386	−	0.731594i	−0.996172	+	0.0874112i	$0.972141\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.4431	−	25.0161i	0.525637	−	0.910429i
$756$		0			0
$757$	−6.49454	+	11.2489i	−0.236048	+	0.408847i	−0.959577	−	0.281447i	$-0.909186\pi$
$757$	−6.49454	+	11.2489i	−0.236048	+	0.408847i	0.723529	+	0.690294i	$0.242519\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.3593			−0.484274			−0.242137	−	0.970242i	$-0.577848\pi$
$761$	−13.3593			−0.484274			−0.242137	+	0.970242i	$0.577848\pi$
$762$		0			0
$763$	−14.2215	+	24.6324i	−0.514854	+	0.891753i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.87783			0.248344
$768$		0			0
$769$	−5.81181	−	10.0664i	−0.209579	−	0.363002i	0.742003	−	0.670397i	$-0.233876\pi$
$769$	−5.81181	−	10.0664i	−0.209579	−	0.363002i	−0.951582	+	0.307395i	$0.900543\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.87783	−	6.71660i	−0.139476	−	0.241579i	0.787822	−	0.615902i	$-0.211208\pi$
$773$	−3.87783	−	6.71660i	−0.139476	−	0.241579i	−0.927298	+	0.374323i	$0.877875\pi$
$774$		0			0
$775$	20.2170	+	35.0169i	0.726216	+	1.25784i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	25.6892	+	13.3496i	0.920413	+	0.478299i
$780$		0			0
$781$	−8.03735	+	13.9211i	−0.287599	+	0.498136i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.46173	−	16.3882i	−0.337703	−	0.584920i
$786$		0			0
$787$	18.8880			0.673283			0.336642	−	0.941633i	$-0.390709\pi$
$787$	18.8880			0.673283			0.336642	+	0.941633i	$0.390709\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	17.1222			0.608794
$792$		0			0
$793$	−6.82088	+	11.8141i	−0.242217	+	0.419532i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−34.5105			−1.22243			−0.611213	−	0.791466i	$-0.709318\pi$
$797$	−34.5105			−1.22243			−0.611213	+	0.791466i	$0.709318\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.93852	−	17.2140i	0.350723	−	0.607469i
$804$		0			0
$805$	−18.1059			−0.638148
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−40.6044			−1.42758			−0.713788	−	0.700362i	$-0.753022\pi$
$809$	−40.6044			−1.42758			−0.713788	+	0.700362i	$0.753022\pi$
$810$		0			0
$811$	−8.54787	−	14.8054i	−0.300156	−	0.519886i	0.676015	−	0.736888i	$-0.263706\pi$
$811$	−8.54787	−	14.8054i	−0.300156	−	0.519886i	−0.976171	+	0.217002i	$0.930372\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−30.5147	+	52.8529i	−1.06888	+	1.85136i
$816$		0			0
$817$	−2.73606	−	1.42181i	−0.0957227	−	0.0497430i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.0475	−	32.9912i	−0.664761	−	1.15140i	−0.979350	−	0.202172i	$-0.935200\pi$
$821$	−19.0475	−	32.9912i	−0.664761	−	1.15140i	0.314588	−	0.949228i	$-0.398133\pi$
$822$		0			0
$823$	−6.60442	−	11.4392i	−0.230216	−	0.398745i	0.727656	−	0.685942i	$-0.240610\pi$
$823$	−6.60442	−	11.4392i	−0.230216	−	0.398745i	−0.957871	+	0.287197i	$0.907277\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.9344	+	29.3312i	0.588866	+	1.01995i	0.994381	+	0.105858i	$0.0337588\pi$
$827$	16.9344	+	29.3312i	0.588866	+	1.01995i	−0.405515	+	0.914088i	$0.632908\pi$
$828$		0			0
$829$	16.8397			0.584866			0.292433	−	0.956286i	$-0.405535\pi$
$829$	16.8397			0.584866			0.292433	+	0.956286i	$0.405535\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−71.8397			−2.48611
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.26847	−	2.19706i	0.0437926	−	0.0758509i	−0.843298	−	0.537446i	$-0.819389\pi$
$839$	1.26847	−	2.19706i	0.0437926	−	0.0758509i	0.887091	+	0.461595i	$0.152723\pi$
$840$		0			0
$841$	−7.55655	+	13.0883i	−0.260571	+	0.451322i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.35823	+	9.28073i	0.184329	+	0.319267i
$846$		0			0
$847$	18.7639			0.644737
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.17458	−	2.03443i	−0.0402641	−	0.0697394i
$852$		0			0
$853$	0.312212	−	0.540766i	0.0106899	−	0.0185155i	−0.860631	−	0.509229i	$-0.829931\pi$
$853$	0.312212	−	0.540766i	0.0106899	−	0.0185155i	0.871321	+	0.490714i	$0.163264\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	−0.912705	−	0.408619i	$-0.866010\pi$
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	0.810227	+	0.586116i	$0.199344\pi$
$858$		0			0
$859$	−13.6035	−	23.5619i	−0.464145	−	0.803923i	0.535017	−	0.844841i	$-0.320305\pi$
$859$	−13.6035	−	23.5619i	−0.464145	−	0.803923i	−0.999163	+	0.0409180i	$0.986972\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26.1131			−0.888900			−0.444450	−	0.895804i	$-0.646601\pi$
$863$	−26.1131			−0.888900			−0.444450	+	0.895804i	$0.646601\pi$
$864$		0			0
$865$	7.80128	+	13.5122i	0.265252	+	0.459429i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.85916	−	4.95221i	0.0969903	−	0.167992i
$870$		0			0
$871$	−16.8733	+	29.2254i	−0.571730	+	0.990265i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.92531			0.267924
$876$		0			0
$877$	13.3296	−	23.0875i	0.450107	−	0.779608i	−0.548285	−	0.836292i	$-0.684719\pi$
$877$	13.3296	−	23.0875i	0.450107	−	0.779608i	0.998392	+	0.0566830i	$0.0180524\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−6.73566			−0.226930			−0.113465	−	0.993542i	$-0.536195\pi$
$881$	−6.73566			−0.226930			−0.113465	+	0.993542i	$0.536195\pi$
$882$		0			0
$883$	16.0803	+	27.8519i	0.541145	+	0.937290i	0.998839	+	0.0481803i	$0.0153422\pi$
$883$	16.0803	+	27.8519i	0.541145	+	0.937290i	−0.457694	+	0.889110i	$0.651324\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.91932	−	3.32435i	−0.0644443	−	0.111621i	0.832003	−	0.554771i	$-0.187194\pi$
$887$	−1.91932	−	3.32435i	−0.0644443	−	0.111621i	−0.896447	+	0.443150i	$0.853861\pi$
$888$		0			0
$889$	−16.9909	−	29.4291i	−0.569857	−	0.987022i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	22.0475	−	14.0680i	0.737791	−	0.470768i
$894$		0			0
$895$	−1.76940	+	3.06469i	−0.0591446	+	0.102441i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−22.2745	−	38.5805i	−0.742895	−	1.28673i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.1878			0.405136
$906$		0			0
$907$	−19.9819	+	34.6096i	−0.663487	+	1.14919i	0.316207	+	0.948690i	$0.397591\pi$
$907$	−19.9819	+	34.6096i	−0.663487	+	1.14919i	−0.979693	+	0.200502i	$0.935743\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	29.3219			0.971479			0.485740	−	0.874104i	$-0.338550\pi$
$911$	29.3219			0.971479			0.485740	+	0.874104i	$0.338550\pi$
$912$		0			0
$913$	17.1704			0.568259
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.744736	−	1.28992i	0.0245933	−	0.0425969i
$918$		0			0
$919$	30.6218			1.01012			0.505059	−	0.863085i	$-0.331471\pi$
$919$	30.6218			1.01012			0.505059	+	0.863085i	$0.331471\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−37.9253			−1.24833
$924$		0			0
$925$	3.01414	+	5.22064i	0.0991042	+	0.171654i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.96213	+	5.13055i	−0.0971842	+	0.168328i	−0.910518	−	0.413469i	$-0.864317\pi$
$929$	−2.96213	+	5.13055i	−0.0971842	+	0.168328i	0.813334	+	0.581797i	$0.197650\pi$
$930$		0			0
$931$	0.311812	+	7.02615i	0.0102192	+	0.230273i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.50546	+	4.33959i	0.0818499	+	0.141768i	0.904045	−	0.427438i	$-0.140584\pi$
$937$	2.50546	+	4.33959i	0.0818499	+	0.141768i	−0.822195	+	0.569206i	$0.807251\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4.19872	+	7.27239i	0.136874	+	0.237073i	0.926312	−	0.376758i	$-0.122961\pi$
$941$	4.19872	+	7.27239i	0.136874	+	0.237073i	−0.789438	+	0.613831i	$0.789628\pi$
$942$		0			0
$943$	−15.6026			−0.508089
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.73566	−	16.8627i	0.316367	−	0.547963i	−0.663360	−	0.748300i	$-0.730870\pi$
$947$	9.73566	−	16.8627i	0.316367	−	0.547963i	0.979727	+	0.200337i	$0.0642037\pi$
$948$		0			0
$949$	46.8962			1.52232
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.7543	−	18.6271i	0.348367	−	0.603390i	−0.637592	−	0.770374i	$-0.720070\pi$
$953$	10.7543	−	18.6271i	0.348367	−	0.603390i	0.985960	+	0.166984i	$0.0534029\pi$
$954$		0			0
$955$	−1.45759	+	2.52462i	−0.0471665	+	0.0816947i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	13.9891			0.451260
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.3022	−	24.7722i	−0.460404	−	0.797444i
$966$		0			0
$967$	−5.41024	+	9.37081i	−0.173982	+	0.301345i	−0.939808	−	0.341702i	$-0.888997\pi$
$967$	−5.41024	+	9.37081i	−0.173982	+	0.301345i	0.765827	+	0.643047i	$0.222330\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.4996	+	26.8461i	−0.497406	+	0.861532i	−0.999996	−	0.00299289i	$-0.999047\pi$
$971$	−15.4996	+	26.8461i	−0.497406	+	0.861532i	0.502590	+	0.864525i	$0.332381\pi$
$972$		0			0
$973$	21.5565	+	37.3370i	0.691071	+	1.19697i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−56.0950			−1.79464			−0.897318	−	0.441384i	$-0.854488\pi$
$977$	−56.0950			−1.79464			−0.897318	+	0.441384i	$0.854488\pi$
$978$		0			0
$979$	2.28715	+	3.96145i	0.0730975	+	0.126609i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.7225	−	28.9641i	0.533363	−	0.923813i	−0.465877	−	0.884849i	$-0.654261\pi$
$983$	16.7225	−	28.9641i	0.533363	−	0.923813i	0.999241	−	0.0389632i	$-0.0124055\pi$
$984$		0			0
$985$	−41.0721	+	71.1390i	−1.30867	+	2.26668i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.66177			0.0528412
$990$		0			0
$991$	4.22245	−	7.31350i	0.134131	−	0.232321i	−0.791134	−	0.611642i	$-0.790509\pi$
$991$	4.22245	−	7.31350i	0.134131	−	0.232321i	0.925265	+	0.379321i	$0.123842\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−73.0293			−2.31519
$996$		0			0
$997$	−25.2074	−	43.6605i	−0.798326	−	1.38274i	−0.920706	−	0.390258i	$-0.872386\pi$
$997$	−25.2074	−	43.6605i	−0.798326	−	1.38274i	0.122380	−	0.992483i	$-0.460947\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.z.1873.3		6
3.2	odd	2		912.2.q.l.49.1		6
4.3	odd	2		171.2.f.b.163.1		6
12.11	even	2		57.2.e.b.49.3	yes	6
19.7	even	3	inner	2736.2.s.z.577.3		6
57.26	odd	6		912.2.q.l.577.1		6
76.7	odd	6		171.2.f.b.64.1		6
76.11	odd	6		3249.2.a.y.1.3		3
76.27	even	6		3249.2.a.t.1.1		3
228.11	even	6		1083.2.a.l.1.1		3
228.83	even	6		57.2.e.b.7.3	✓	6
228.179	odd	6		1083.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.b.7.3	✓	6	228.83	even	6
57.2.e.b.49.3	yes	6	12.11	even	2
171.2.f.b.64.1		6	76.7	odd	6
171.2.f.b.163.1		6	4.3	odd	2
912.2.q.l.49.1		6	3.2	odd	2
912.2.q.l.577.1		6	57.26	odd	6
1083.2.a.l.1.1		3	228.11	even	6
1083.2.a.o.1.3		3	228.179	odd	6
2736.2.s.z.577.3		6	19.7	even	3	inner
2736.2.s.z.1873.3		6	1.1	even	1	trivial
3249.2.a.t.1.1		3	76.27	even	6
3249.2.a.y.1.3		3	76.11	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.66044 - 2.87597i) q^{5} -2.32088 q^{7} +O(q^{10})\)	\(q+(1.66044 - 2.87597i) q^{5} -2.32088 q^{7} -1.70739 q^{11} +(-2.01414 - 3.48859i) q^{13} +(-0.193252 - 4.35461i) q^{19} +(1.17458 + 2.03443i) q^{23} +(-3.01414 - 5.22064i) q^{25} +(3.32088 + 5.75194i) q^{29} -6.70739 q^{31} +(-3.85369 + 6.67479i) q^{35} -1.00000 q^{37} +(-3.32088 + 5.75194i) q^{41} +(0.353695 - 0.612617i) q^{43} +(3.00000 + 5.19615i) q^{47} -1.61350 q^{49} +(-4.98133 - 8.62791i) q^{53} +(-2.83502 + 4.91040i) q^{55} +(-0.853695 + 1.47864i) q^{59} +(-1.69325 - 2.93280i) q^{61} -13.3774 q^{65} +(-4.18872 - 7.25507i) q^{67} +(4.70739 - 8.15344i) q^{71} +(-5.82088 + 10.0821i) q^{73} +3.96265 q^{77} +(-1.67458 + 2.90046i) q^{79} -10.0565 q^{83} +(-1.33956 - 2.32018i) q^{89} +(4.67458 + 8.09661i) q^{91} +(-12.8446 - 6.67479i) q^{95} +(-8.86330 + 15.3517i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 6 * q + 2 * q^5 + 2 * q^7	\( 6 q + 2 q^{5} + 2 q^{7} + q^{13} - 4 q^{19} - 14 q^{23} - 5 q^{25} + 4 q^{29} - 30 q^{31} - 18 q^{35} - 6 q^{37} - 4 q^{41} - 3 q^{43} + 18 q^{47} - 4 q^{49} - 6 q^{53} + 12 q^{55} - 13 q^{61} - 12 q^{65} + 9 q^{67} + 18 q^{71} - 19 q^{73} - 24 q^{77} + 11 q^{79} - 8 q^{83} - 16 q^{89} + 7 q^{91} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 6 * q + 2 * q^5 + 2 * q^7 + q^13 - 4 * q^19 - 14 * q^23 - 5 * q^25 + 4 * q^29 - 30 * q^31 - 18 * q^35 - 6 * q^37 - 4 * q^41 - 3 * q^43 + 18 * q^47 - 4 * q^49 - 6 * q^53 + 12 * q^55 - 13 * q^61 - 12 * q^65 + 9 * q^67 + 18 * q^71 - 19 * q^73 - 24 * q^77 + 11 * q^79 - 8 * q^83 - 16 * q^89 + 7 * q^91 + 2 * q^95 + 2 * q^97

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