LMFDB - Embedded newform 1728.2.p.a.1439.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(287,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("1728.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.p (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:	$13.7981494693$
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} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 $ x^16 + 11x^14 + 85x^12 + 332x^10 + 940x^8 + 1064x^6 + 880x^4 + 128*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1439.1
Root			$-0.192865 + 0.334053i$ of defining polynomial
Character	$\chi$	$=$	1728.1439
Dual form			1728.2.p.a.287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.06470 - 3.57617i) q^{5} +(-0.287429 - 0.165947i) q^{7} +O(q^{10})$	$q+(-2.06470 - 3.57617i) q^{5} +(-0.287429 - 0.165947i) q^{7} +(3.62268 + 2.09155i) q^{11} +(4.27682 - 2.46922i) q^{13} -3.20639i q^{17} +5.42029 q^{19} +(-1.39438 - 2.41514i) q^{23} +(-6.02601 + 10.4374i) q^{25} +(1.03570 - 1.79388i) q^{29} +(-3.60828 + 2.08324i) q^{31} +1.37053i q^{35} +2.21390i q^{37} +(-1.41298 + 0.815784i) q^{41} +(2.99758 - 5.19195i) q^{43} +(3.97567 - 6.88607i) q^{47} +(-3.44492 - 5.96678i) q^{49} +2.05801 q^{53} -17.2738i q^{55} +(-5.40497 + 3.12056i) q^{59} +(-9.10709 - 5.25798i) q^{61} +(-17.6607 - 10.1964i) q^{65} +(-5.83658 - 10.1093i) q^{67} +7.66299 q^{71} -6.21742 q^{73} +(-0.694175 - 1.20235i) q^{77} +(-0.719039 - 0.415137i) q^{79} +(5.96543 + 3.44414i) q^{83} +(-11.4666 + 6.62025i) q^{85} +9.14211i q^{89} -1.63904 q^{91} +(-11.1913 - 19.3839i) q^{95} +(-0.749190 + 1.29763i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{5}+O(q^{10}) $ 16 * q - 6 * q^5	$ 16 q - 6 q^{5} + 6 q^{13} - 14 q^{25} - 18 q^{29} + 6 q^{49} + 48 q^{53} - 42 q^{61} - 54 q^{65} + 28 q^{73} - 66 q^{77} - 36 q^{85} + 8 q^{97}+O(q^{100}) $ 16 * q - 6 * q^5 + 6 * q^13 - 14 * q^25 - 18 * q^29 + 6 * q^49 + 48 * q^53 - 42 * q^61 - 54 * q^65 + 28 * q^73 - 66 * q^77 - 36 * q^85 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.06470	−	3.57617i	−0.923364	−	1.59931i	−0.794172	−	0.607693i	$-0.792095\pi$
$5$	−2.06470	−	3.57617i	−0.923364	−	1.59931i	−0.129192	−	0.991620i	$-0.541238\pi$
$6$		0			0
$7$	−0.287429	−	0.165947i	−0.108638	−	0.0627222i	0.444696	−	0.895681i	$-0.353312\pi$
$7$	−0.287429	−	0.165947i	−0.108638	−	0.0627222i	−0.553335	+	0.832959i	$0.686645\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.62268	+	2.09155i	1.09228	+	0.630627i	0.934182	−	0.356797i	$-0.116131\pi$
$11$	3.62268	+	2.09155i	1.09228	+	0.630627i	0.158096	+	0.987424i	$0.449464\pi$
$12$		0			0
$13$	4.27682	−	2.46922i	1.18618	−	0.684839i	0.228741	−	0.973487i	$-0.426539\pi$
$13$	4.27682	−	2.46922i	1.18618	−	0.684839i	0.957435	+	0.288648i	$0.0932059\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.20639i		−	0.777664i	−0.921309	−	0.388832i	$-0.872879\pi$
$17$		−	3.20639i		−	0.777664i	0.921309	−	0.388832i	$-0.127121\pi$
$18$		0			0
$19$	5.42029			1.24350			0.621750	−	0.783215i	$-0.286422\pi$
$19$	5.42029			1.24350			0.621750	+	0.783215i	$0.286422\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.39438	−	2.41514i	−0.290748	−	0.503591i	0.683239	−	0.730195i	$-0.260571\pi$
$23$	−1.39438	−	2.41514i	−0.290748	−	0.503591i	−0.973987	+	0.226604i	$0.927238\pi$
$24$		0			0
$25$	−6.02601	+	10.4374i	−1.20520	+	2.08747i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.03570	−	1.79388i	0.192324	−	0.333115i	−0.753696	−	0.657223i	$-0.771731\pi$
$29$	1.03570	−	1.79388i	0.192324	−	0.333115i	0.946020	+	0.324108i	$0.105064\pi$
$30$		0			0
$31$	−3.60828	+	2.08324i	−0.648067	+	0.374161i	−0.787715	−	0.616040i	$-0.788736\pi$
$31$	−3.60828	+	2.08324i	−0.648067	+	0.374161i	0.139649	+	0.990201i	$0.455403\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			1.37053i			0.231662i
$36$		0			0
$37$			2.21390i			0.363963i	0.983302	+	0.181982i	$0.0582511\pi$
$37$			2.21390i			0.363963i	−0.983302	+	0.181982i	$0.941749\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.41298	+	0.815784i	−0.220670	+	0.127404i	−0.606261	−	0.795266i	$-0.707331\pi$
$41$	−1.41298	+	0.815784i	−0.220670	+	0.127404i	0.385590	+	0.922670i	$0.373998\pi$
$42$		0			0
$43$	2.99758	−	5.19195i	0.457126	−	0.791766i	−0.541682	−	0.840584i	$-0.682212\pi$
$43$	2.99758	−	5.19195i	0.457126	−	0.791766i	0.998808	+	0.0488181i	$0.0155455\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.97567	−	6.88607i	0.579912	−	1.00444i	−0.415577	−	0.909558i	$-0.636420\pi$
$47$	3.97567	−	6.88607i	0.579912	−	1.00444i	0.995489	−	0.0948784i	$-0.0302462\pi$
$48$		0			0
$49$	−3.44492	−	5.96678i	−0.492132	−	0.852397i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.05801			0.282690			0.141345	−	0.989960i	$-0.454857\pi$
$53$	2.05801			0.282690			0.141345	+	0.989960i	$0.454857\pi$
$54$		0			0
$55$		−	17.2738i		−	2.32919i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.40497	+	3.12056i	−0.703667	+	0.406262i	−0.808712	−	0.588205i	$-0.799835\pi$
$59$	−5.40497	+	3.12056i	−0.703667	+	0.406262i	0.105045	+	0.994467i	$0.466501\pi$
$60$		0			0
$61$	−9.10709	−	5.25798i	−1.16604	−	0.673216i	−0.213299	−	0.976987i	$-0.568421\pi$
$61$	−9.10709	−	5.25798i	−1.16604	−	0.673216i	−0.952745	+	0.303771i	$0.901754\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−17.6607	−	10.1964i	−2.19054	−	1.26471i
$66$		0			0
$67$	−5.83658	−	10.1093i	−0.713052	−	1.23504i	−0.963706	−	0.266965i	$-0.913979\pi$
$67$	−5.83658	−	10.1093i	−0.713052	−	1.23504i	0.250655	−	0.968077i	$-0.419354\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.66299			0.909429			0.454715	−	0.890637i	$-0.349741\pi$
$71$	7.66299			0.909429			0.454715	+	0.890637i	$0.349741\pi$
$72$		0			0
$73$	−6.21742			−0.727695			−0.363847	−	0.931459i	$-0.618537\pi$
$73$	−6.21742			−0.727695			−0.363847	+	0.931459i	$0.618537\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.694175	−	1.20235i	−0.0791086	−	0.137020i
$78$		0			0
$79$	−0.719039	−	0.415137i	−0.0808982	−	0.0467066i	0.459005	−	0.888434i	$-0.348206\pi$
$79$	−0.719039	−	0.415137i	−0.0808982	−	0.0467066i	−0.539903	+	0.841727i	$0.681539\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.96543	+	3.44414i	0.654791	+	0.378044i	0.790289	−	0.612734i	$-0.209930\pi$
$83$	5.96543	+	3.44414i	0.654791	+	0.378044i	−0.135498	+	0.990778i	$0.543263\pi$
$84$		0			0
$85$	−11.4666	+	6.62025i	−1.24373	+	0.718067i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.14211i			0.969061i	0.874774	+	0.484531i	$0.161010\pi$
$89$			9.14211i			0.969061i	−0.874774	+	0.484531i	$0.838990\pi$
$90$		0			0
$91$	−1.63904			−0.171818
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.1913	−	19.3839i	−1.14820	−	1.98875i
$96$		0			0
$97$	−0.749190	+	1.29763i	−0.0760687	+	0.131755i	−0.901551	−	0.432674i	$-0.857570\pi$
$97$	−0.749190	+	1.29763i	−0.0760687	+	0.131755i	0.825482	+	0.564429i	$0.190904\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.82358	+	8.35469i	−0.479965	+	0.831323i	−0.999736	−	0.0229825i	$-0.992684\pi$
$101$	−4.82358	+	8.35469i	−0.479965	+	0.831323i	0.519771	+	0.854305i	$0.326017\pi$
$102$		0			0
$103$	−4.32639	+	2.49784i	−0.426292	+	0.246120i	−0.697766	−	0.716326i	$-0.745822\pi$
$103$	−4.32639	+	2.49784i	−0.426292	+	0.246120i	0.271474	+	0.962446i	$0.412489\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.1294i		−	1.26927i	−0.772813	−	0.634634i	$-0.781151\pi$
$107$		−	13.1294i		−	1.26927i	0.772813	−	0.634634i	$-0.218849\pi$
$108$		0			0
$109$		−	10.7401i		−	1.02872i	−0.857576	−	0.514358i	$-0.828030\pi$
$109$		−	10.7401i		−	1.02872i	0.857576	−	0.514358i	$-0.171970\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.02870	+	2.32597i	−0.378989	+	0.218809i	−0.677378	−	0.735635i	$-0.736884\pi$
$113$	−4.02870	+	2.32597i	−0.378989	+	0.218809i	0.298390	+	0.954444i	$0.403551\pi$
$114$		0			0
$115$	−5.75797	+	9.97309i	−0.536933	+	0.929995i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.532092	+	0.921611i	−0.0487768	+	0.0844839i
$120$		0			0
$121$	3.24919	+	5.62776i	0.295381	+	0.511615i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	29.1207			2.60463
$126$		0			0
$127$		−	0.663789i		−	0.0589018i	−0.999566	−	0.0294509i	$-0.990624\pi$
$127$		−	0.663789i		−	0.0589018i	0.999566	−	0.0294509i	$-0.00937587\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	4.77467	−	2.75666i	0.417165	−	0.240850i	−0.276699	−	0.960957i	$-0.589240\pi$
$131$	4.77467	−	2.75666i	0.417165	−	0.240850i	0.693864	+	0.720106i	$0.255907\pi$
$132$		0			0
$133$	−1.55795	−	0.899484i	−0.135092	−	0.0779951i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.41730	−	1.97298i	−0.291959	−	0.168563i	0.346866	−	0.937915i	$-0.387246\pi$
$137$	−3.41730	−	1.97298i	−0.291959	−	0.168563i	−0.638825	+	0.769352i	$0.720579\pi$
$138$		0			0
$139$	9.09321	+	15.7499i	0.771276	+	1.33589i	0.936864	+	0.349695i	$0.113715\pi$
$139$	9.09321	+	15.7499i	0.771276	+	1.33589i	−0.165588	+	0.986195i	$0.552952\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.6580			1.72751
$144$		0			0
$145$	−8.55364			−0.710341
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.33052	−	12.6968i	−0.600539	−	1.04016i	−0.992739	−	0.120285i	$-0.961619\pi$
$149$	−7.33052	−	12.6968i	−0.600539	−	1.04016i	0.392200	−	0.919880i	$-0.371714\pi$
$150$		0			0
$151$	−18.4236	−	10.6369i	−1.49929	−	0.865617i	−0.499293	−	0.866433i	$-0.666407\pi$
$151$	−18.4236	−	10.6369i	−1.49929	−	0.865617i	−1.00000		0.000816542i	$0.999740\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.9001	+	8.60256i	1.19680	+	0.690974i
$156$		0			0
$157$	9.74343	−	5.62537i	0.777611	−	0.448954i	−0.0579722	−	0.998318i	$-0.518463\pi$
$157$	9.74343	−	5.62537i	0.777611	−	0.448954i	0.835583	+	0.549365i	$0.185130\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.925575i			0.0729455i
$162$		0			0
$163$	13.3410			1.04495			0.522473	−	0.852656i	$-0.325009\pi$
$163$	13.3410			1.04495			0.522473	+	0.852656i	$0.325009\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.3782	+	21.4397i	0.957855	+	1.65905i	0.727694	+	0.685901i	$0.240592\pi$
$167$	12.3782	+	21.4397i	0.957855	+	1.65905i	0.230161	+	0.973153i	$0.426075\pi$
$168$		0			0
$169$	5.69411	−	9.86249i	0.438009	−	0.758653i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.13610	−	7.16393i	0.314462	−	0.544664i	−0.664861	−	0.746967i	$-0.731509\pi$
$173$	4.13610	−	7.16393i	0.314462	−	0.544664i	0.979323	+	0.202303i	$0.0648426\pi$
$174$		0			0
$175$	3.46410	−	2.00000i	0.261861	−	0.151186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.3748i		−	0.924939i	−0.886635	−	0.462470i	$-0.846963\pi$
$179$		−	12.3748i		−	0.924939i	0.886635	−	0.462470i	$-0.153037\pi$
$180$		0			0
$181$		−	2.41487i		−	0.179496i	−0.995965	−	0.0897478i	$-0.971394\pi$
$181$		−	2.41487i		−	0.179496i	0.995965	−	0.0897478i	$-0.0286061\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.91730	−	4.57105i	0.582091	−	0.336070i
$186$		0			0
$187$	6.70634	−	11.6157i	0.490416	−	0.849426i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.21563	+	12.4978i	−0.522105	+	0.904312i	0.477564	+	0.878597i	$0.341520\pi$
$191$	−7.21563	+	12.4978i	−0.522105	+	0.904312i	−0.999669	+	0.0257154i	$0.991814\pi$
$192$		0			0
$193$	8.63904	+	14.9632i	0.621851	+	1.07708i	0.989141	+	0.146971i	$0.0469526\pi$
$193$	8.63904	+	14.9632i	0.621851	+	1.07708i	−0.367289	+	0.930107i	$0.619714\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.1427			−1.07887			−0.539435	−	0.842027i	$-0.681362\pi$
$197$	−15.1427			−1.07887			−0.539435	+	0.842027i	$0.681362\pi$
$198$		0			0
$199$			21.6057i			1.53158i	0.643088	+	0.765792i	$0.277653\pi$
$199$			21.6057i			1.53158i	−0.643088	+	0.765792i	$0.722347\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.595379	+	0.343742i	−0.0417874	+	0.0241260i
$204$		0			0
$205$	5.83477	+	3.36871i	0.407518	+	0.235281i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	19.6360	+	11.3368i	1.35825	+	0.784185i
$210$		0			0
$211$	3.70129	+	6.41082i	0.254807	+	0.441339i	0.964843	−	0.262826i	$-0.0846546\pi$
$211$	3.70129	+	6.41082i	0.254807	+	0.441339i	−0.710036	+	0.704166i	$0.751321\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−24.7564			−1.68837
$216$		0			0
$217$	1.38283			0.0938729
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.91730	−	13.7132i	−0.532575	−	0.922447i
$222$		0			0
$223$	−2.60181	−	1.50216i	−0.174230	−	0.100592i	0.410349	−	0.911929i	$-0.365407\pi$
$223$	−2.60181	−	1.50216i	−0.174230	−	0.100592i	−0.584579	+	0.811337i	$0.698740\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.28027	+	4.20326i	0.483208	+	0.278980i	0.721753	−	0.692151i	$-0.243337\pi$
$227$	7.28027	+	4.20326i	0.483208	+	0.278980i	−0.238544	+	0.971132i	$0.576670\pi$
$228$		0			0
$229$	−2.74775	+	1.58641i	−0.181576	+	0.104833i	−0.588033	−	0.808837i	$-0.700098\pi$
$229$	−2.74775	+	1.58641i	−0.181576	+	0.104833i	0.406457	+	0.913670i	$0.366764\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	10.0341i		−	0.657357i	−0.944442	−	0.328678i	$-0.893397\pi$
$233$		−	10.0341i		−	0.657357i	0.944442	−	0.328678i	$-0.106603\pi$
$234$		0			0
$235$	−32.8344			−2.14188
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.95896	−	8.58918i	−0.320769	−	0.555588i	0.659878	−	0.751373i	$-0.270608\pi$
$239$	−4.95896	−	8.58918i	−0.320769	−	0.555588i	−0.980647	+	0.195785i	$0.937275\pi$
$240$		0			0
$241$	0.334590	−	0.579528i	0.0215529	−	0.0373307i	−0.855048	−	0.518549i	$-0.826472\pi$
$241$	0.334590	−	0.579528i	0.0215529	−	0.0373307i	0.876601	+	0.481219i	$0.159806\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.2255	+	24.6393i	−0.908833	+	1.57415i
$246$		0			0
$247$	23.1816	−	13.3839i	1.47501	−	0.851598i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			3.38823i			0.213863i	0.994266	+	0.106931i	$0.0341025\pi$
$251$			3.38823i			0.213863i	−0.994266	+	0.106931i	$0.965897\pi$
$252$		0			0
$253$		−	11.6657i		−	0.733415i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	19.5531	−	11.2890i	1.21969	−	0.704187i	0.254837	−	0.966984i	$-0.417978\pi$
$257$	19.5531	−	11.2890i	1.21969	−	0.704187i	0.964851	+	0.262797i	$0.0846448\pi$
$258$		0			0
$259$	0.367391	−	0.636340i	0.0228286	−	0.0395403i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.70876	−	6.42377i	0.228692	−	0.396107i	−0.728729	−	0.684803i	$-0.759888\pi$
$263$	3.70876	−	6.42377i	0.228692	−	0.396107i	0.957421	+	0.288696i	$0.0932217\pi$
$264$		0			0
$265$	−4.24919	−	7.35981i	−0.261026	−	0.452110i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	19.9774			1.21804			0.609021	−	0.793154i	$-0.291562\pi$
$269$	19.9774			1.21804			0.609021	+	0.793154i	$0.291562\pi$
$270$		0			0
$271$			26.1040i			1.58571i	0.609412	+	0.792853i	$0.291405\pi$
$271$			26.1040i			1.58571i	−0.609412	+	0.792853i	$0.708595\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−43.6606	+	25.2074i	−2.63283	+	1.52007i
$276$		0			0
$277$	21.2472	+	12.2671i	1.27662	+	0.737057i	0.976225	−	0.216757i	$-0.0695481\pi$
$277$	21.2472	+	12.2671i	1.27662	+	0.737057i	0.300395	+	0.953815i	$0.402881\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−23.8348	−	13.7610i	−1.42186	−	0.820913i	−0.425405	−	0.905003i	$-0.639868\pi$
$281$	−23.8348	−	13.7610i	−1.42186	−	0.820913i	−0.996458	+	0.0840901i	$0.973202\pi$
$282$		0			0
$283$	−0.480920	−	0.832977i	−0.0285877	−	0.0495154i	0.851378	−	0.524553i	$-0.175768\pi$
$283$	−0.480920	−	0.832977i	−0.0285877	−	0.0495154i	−0.879965	+	0.475038i	$0.842434\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.541509			0.0319643
$288$		0			0
$289$	6.71904			0.395238
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.36384	−	2.36224i	−0.0796763	−	0.138003i	0.823434	−	0.567412i	$-0.192055\pi$
$293$	−1.36384	−	2.36224i	−0.0796763	−	0.138003i	−0.903110	+	0.429409i	$0.858722\pi$
$294$		0			0
$295$	22.3193	+	12.8861i	1.29948	+	0.750256i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11.9270	−	6.88607i	−0.689757	−	0.398232i
$300$		0			0
$301$	−1.72318	+	0.994880i	−0.0993226	+	0.0573439i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			43.4247i			2.48649i
$306$		0			0
$307$	−5.29186			−0.302022			−0.151011	−	0.988532i	$-0.548253\pi$
$307$	−5.29186			−0.302022			−0.151011	+	0.988532i	$0.548253\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.1708	−	21.0804i	−0.690140	−	1.19536i	−0.971792	−	0.235841i	$-0.924216\pi$
$311$	−12.1708	−	21.0804i	−0.690140	−	1.19536i	0.281652	−	0.959517i	$-0.409118\pi$
$312$		0			0
$313$	15.2158	−	26.3545i	0.860048	−	1.48965i	−0.0118333	−	0.999930i	$-0.503767\pi$
$313$	15.2158	−	26.3545i	0.860048	−	1.48965i	0.871881	−	0.489717i	$-0.162900\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.8885	+	25.7876i	−0.836220	+	1.44837i	0.0568140	+	0.998385i	$0.481906\pi$
$317$	−14.8885	+	25.7876i	−0.836220	+	1.44837i	−0.893034	+	0.449990i	$0.851428\pi$
$318$		0			0
$319$	7.50399	−	4.33243i	0.420143	−	0.242570i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	17.3796i		−	0.967026i
$324$		0			0
$325$			59.5182i			3.30148i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.28545	+	1.31950i	−0.126001	+	0.0727467i
$330$		0			0
$331$	9.53922	−	16.5224i	0.524323	−	0.908154i	−0.475276	−	0.879837i	$-0.657652\pi$
$331$	9.53922	−	16.5224i	0.524323	−	0.908154i	0.999599	−	0.0283170i	$-0.00901480\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−24.1016	+	41.7452i	−1.31681	+	2.28079i
$336$		0			0
$337$	−1.33027	−	2.30410i	−0.0724647	−	0.125512i	0.827516	−	0.561442i	$-0.189753\pi$
$337$	−1.33027	−	2.30410i	−0.0724647	−	0.125512i	−0.899981	+	0.435929i	$0.856420\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−17.4289			−0.943825
$342$		0			0
$343$			4.60997i			0.248915i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.1413	+	11.0512i	−1.02756	+	0.593261i	−0.916284	−	0.400528i	$-0.868827\pi$
$347$	−19.1413	+	11.0512i	−1.02756	+	0.593261i	−0.111275	+	0.993790i	$0.535493\pi$
$348$		0			0
$349$	17.9088	+	10.3397i	0.958638	+	0.553470i	0.895754	−	0.444551i	$-0.146637\pi$
$349$	17.9088	+	10.3397i	0.958638	+	0.553470i	0.0628846	+	0.998021i	$0.479970\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.7969	+	6.81094i	0.627885	+	0.362510i	0.779933	−	0.625864i	$-0.215253\pi$
$353$	11.7969	+	6.81094i	0.627885	+	0.362510i	−0.152047	+	0.988373i	$0.548587\pi$
$354$		0			0
$355$	−15.8218	−	27.4042i	−0.839734	−	1.45446i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	17.3818			0.917377			0.458688	−	0.888597i	$-0.348319\pi$
$359$	17.3818			0.917377			0.458688	+	0.888597i	$0.348319\pi$
$360$		0			0
$361$	10.3796			0.546294
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.8371	+	22.2346i	0.671927	+	1.16381i
$366$		0			0
$367$	16.3976	+	9.46715i	0.855947	+	0.494181i	0.862653	−	0.505796i	$-0.168801\pi$
$367$	16.3976	+	9.46715i	0.855947	+	0.494181i	−0.00670584	+	0.999978i	$0.502135\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.591533	−	0.341522i	−0.0307109	−	0.0177309i
$372$		0			0
$373$	−24.2230	+	13.9852i	−1.25422	+	0.724124i	−0.971945	−	0.235210i	$-0.924422\pi$
$373$	−24.2230	+	13.9852i	−1.25422	+	0.724124i	−0.282275	+	0.959334i	$0.591089\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	10.2295i		−	0.526844i
$378$		0			0
$379$	11.7997			0.606111			0.303056	−	0.952973i	$-0.401993\pi$
$379$	11.7997			0.606111			0.303056	+	0.952973i	$0.401993\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.87782	+	17.1089i	0.504733	+	0.874223i	0.999985	+	0.00547380i	$0.00174237\pi$
$383$	9.87782	+	17.1089i	0.504733	+	0.874223i	−0.495252	+	0.868749i	$0.664924\pi$
$384$		0			0
$385$	−2.86653	+	4.96498i	−0.146092	+	0.253039i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.76557	+	4.79011i	−0.140220	+	0.242868i	−0.927579	−	0.373626i	$-0.878114\pi$
$389$	−2.76557	+	4.79011i	−0.140220	+	0.242868i	0.787359	+	0.616494i	$0.211448\pi$
$390$		0			0
$391$	−7.74388	+	4.47093i	−0.391625	+	0.226105i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			3.42854i			0.172509i
$396$		0			0
$397$		−	31.5247i		−	1.58218i	−0.611700	−	0.791090i	$-0.709514\pi$
$397$		−	31.5247i		−	1.58218i	0.611700	−	0.791090i	$-0.290486\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	17.9709	−	10.3755i	0.897425	−	0.518129i	0.0210614	−	0.999778i	$-0.493295\pi$
$401$	17.9709	−	10.3755i	0.897425	−	0.518129i	0.876364	+	0.481649i	$0.159962\pi$
$402$		0			0
$403$	−10.2880	+	17.8193i	−0.512481	+	0.887642i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.63049	+	8.02025i	−0.229525	+	0.397549i
$408$		0			0
$409$	−5.99838	−	10.3895i	−0.296601	−	0.513728i	0.678755	−	0.734365i	$-0.262520\pi$
$409$	−5.99838	−	10.3895i	−0.296601	−	0.513728i	−0.975356	+	0.220637i	$0.929186\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.07139			0.101927
$414$		0			0
$415$		−	28.4446i		−	1.39629i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.8617	−	12.0445i	1.01916	−	0.588412i	0.105299	−	0.994441i	$-0.466420\pi$
$419$	20.8617	−	12.0445i	1.01916	−	0.588412i	0.913860	+	0.406029i	$0.133087\pi$
$420$		0			0
$421$	−26.3884	−	15.2354i	−1.28609	−	0.742526i	−0.308137	−	0.951342i	$-0.599706\pi$
$421$	−26.3884	−	15.2354i	−1.28609	−	0.742526i	−0.977955	+	0.208816i	$0.933039\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	33.4662	+	19.3217i	1.62335	+	0.937242i
$426$		0			0
$427$	1.74510	+	3.02260i	0.0844511	+	0.146274i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.902398			0.0434670			0.0217335	−	0.999764i	$-0.493081\pi$
$431$	0.902398			0.0434670			0.0217335	+	0.999764i	$0.493081\pi$
$432$		0			0
$433$	0.559027			0.0268651			0.0134326	−	0.999910i	$-0.495724\pi$
$433$	0.559027			0.0268651			0.0134326	+	0.999910i	$0.495724\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.55795	−	13.0908i	−0.361546	−	0.626216i
$438$		0			0
$439$	20.1482	+	11.6326i	0.961621	+	0.555192i	0.896671	−	0.442697i	$-0.145978\pi$
$439$	20.1482	+	11.6326i	0.961621	+	0.555192i	0.0649491	+	0.997889i	$0.479311\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.0835	−	12.7499i	−1.04922	−	0.605766i	−0.126788	−	0.991930i	$-0.540467\pi$
$443$	−22.0835	−	12.7499i	−1.04922	−	0.605766i	−0.922430	+	0.386164i	$0.873800\pi$
$444$		0			0
$445$	32.6937	−	18.8757i	1.54983	−	0.894796i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	16.6758i		−	0.786981i	−0.919329	−	0.393490i	$-0.871268\pi$
$449$		−	16.6758i		−	0.786981i	0.919329	−	0.393490i	$-0.128732\pi$
$450$		0			0
$451$	−6.82502			−0.321378
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.38414	+	5.86150i	0.158651	+	0.274791i
$456$		0			0
$457$	3.30283	−	5.72066i	0.154500	−	0.267601i	−0.778377	−	0.627797i	$-0.783957\pi$
$457$	3.30283	−	5.72066i	0.154500	−	0.267601i	0.932877	+	0.360196i	$0.117290\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.40191	−	16.2846i	0.437891	−	0.758449i	−0.559636	−	0.828739i	$-0.689059\pi$
$461$	9.40191	−	16.2846i	0.437891	−	0.758449i	0.997527	+	0.0702897i	$0.0223924\pi$
$462$		0			0
$463$	−21.5994	+	12.4704i	−1.00381	+	0.579548i	−0.909372	−	0.415983i	$-0.863438\pi$
$463$	−21.5994	+	12.4704i	−1.00381	+	0.579548i	−0.0944346	+	0.995531i	$0.530104\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			10.2003i			0.472015i	0.971751	+	0.236007i	$0.0758389\pi$
$467$			10.2003i			0.472015i	−0.971751	+	0.236007i	$0.924161\pi$
$468$		0			0
$469$			3.87426i			0.178897i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.7185	−	12.5392i	0.998618	−	0.576552i
$474$		0			0
$475$	−32.6627	+	56.5735i	−1.49867	+	2.59577i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.43334	+	9.41082i	−0.248256	+	0.429991i	−0.963042	−	0.269352i	$-0.913191\pi$
$479$	−5.43334	+	9.41082i	−0.248256	+	0.429991i	0.714786	+	0.699343i	$0.246524\pi$
$480$		0			0
$481$	5.46661	+	9.46845i	0.249256	+	0.431724i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.18742			0.280956
$486$		0			0
$487$			21.1073i			0.956462i	0.878234	+	0.478231i	$0.158722\pi$
$487$			21.1073i			0.956462i	−0.878234	+	0.478231i	$0.841278\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.5189	−	10.6919i	0.835746	−	0.482518i	−0.0200698	−	0.999799i	$-0.506389\pi$
$491$	18.5189	−	10.6919i	0.835746	−	0.482518i	0.855816	+	0.517280i	$0.173056\pi$
$492$		0			0
$493$	−5.75189	−	3.32085i	−0.259052	−	0.149564i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.20257	−	1.27165i	−0.0987986	−	0.0570414i
$498$		0			0
$499$	12.9981	+	22.5134i	0.581876	+	1.00784i	0.995257	+	0.0972805i	$0.0310144\pi$
$499$	12.9981	+	22.5134i	0.581876	+	1.00784i	−0.413381	+	0.910558i	$0.635652\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.4476			−1.26842			−0.634208	−	0.773163i	$-0.718674\pi$
$503$	−28.4476			−1.26842			−0.634208	+	0.773163i	$0.718674\pi$
$504$		0			0
$505$	39.8371			1.77273
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.928302	+	1.60787i	0.0411462	+	0.0712674i	0.885865	−	0.463943i	$-0.153566\pi$
$509$	0.928302	+	1.60787i	0.0411462	+	0.0712674i	−0.844719	+	0.535210i	$0.820232\pi$
$510$		0			0
$511$	1.78707	+	1.03177i	0.0790553	+	0.0456426i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.8654	+	10.3146i	0.787245	+	0.454516i
$516$		0			0
$517$	28.8052	−	16.6307i	1.26685	−	0.731416i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			40.2415i			1.76301i	0.472173	+	0.881506i	$0.343470\pi$
$521$			40.2415i			1.76301i	−0.472173	+	0.881506i	$0.656530\pi$
$522$		0			0
$523$	21.0040			0.918440			0.459220	−	0.888323i	$-0.348129\pi$
$523$	21.0040			0.918440			0.459220	+	0.888323i	$0.348129\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.67969	+	11.5696i	0.290972	+	0.503978i
$528$		0			0
$529$	7.61141	−	13.1833i	0.330931	−	0.573189i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.02870	+	6.97792i	−0.174503	+	0.302247i
$534$		0			0
$535$	−46.9530	+	27.1083i	−2.02996	+	1.17200i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	28.8210i		−	1.24141i
$540$		0			0
$541$			5.33020i			0.229163i	0.993414	+	0.114582i	$0.0365528\pi$
$541$			5.33020i			0.229163i	−0.993414	+	0.114582i	$0.963447\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−38.4085	+	22.1751i	−1.64524	+	0.949879i
$546$		0			0
$547$	−1.09162	+	1.89075i	−0.0466745	+	0.0808426i	−0.888419	−	0.459034i	$-0.848196\pi$
$547$	−1.09162	+	1.89075i	−0.0466745	+	0.0808426i	0.841744	+	0.539876i	$0.181529\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.61378	−	9.72336i	0.239155	−	0.414229i
$552$		0			0
$553$	0.137782	+	0.238645i	0.00585908	+	0.0101482i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14.6744			0.621775			0.310887	−	0.950447i	$-0.399374\pi$
$557$	14.6744			0.621775			0.310887	+	0.950447i	$0.399374\pi$
$558$		0			0
$559$		−	29.6067i		−	1.25223i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.62742	−	2.67164i	0.195023	−	0.112596i	−0.399309	−	0.916816i	$-0.630750\pi$
$563$	4.62742	−	2.67164i	0.195023	−	0.112596i	0.594332	+	0.804220i	$0.297417\pi$
$564$		0			0
$565$	16.6362	+	9.60489i	0.699889	+	0.404081i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.77232	−	1.02325i	−0.0742997	−	0.0428969i	0.462390	−	0.886677i	$-0.346992\pi$
$569$	−1.77232	−	1.02325i	−0.0742997	−	0.0428969i	−0.536690	+	0.843780i	$0.680325\pi$
$570$		0			0
$571$	4.29802	+	7.44439i	0.179866	+	0.311538i	0.941835	−	0.336077i	$-0.109100\pi$
$571$	4.29802	+	7.44439i	0.179866	+	0.311538i	−0.761968	+	0.647614i	$0.775767\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	33.6102			1.40164
$576$		0			0
$577$	26.6609			1.10991			0.554954	−	0.831881i	$-0.312736\pi$
$577$	26.6609			1.10991			0.554954	+	0.831881i	$0.312736\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.14309	−	1.97990i	−0.0474235	−	0.0821399i
$582$		0			0
$583$	7.45552	+	4.30445i	0.308776	+	0.178272i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	39.9759	+	23.0801i	1.64998	+	0.952618i	0.977076	+	0.212892i	$0.0682883\pi$
$587$	39.9759	+	23.0801i	1.64998	+	0.952618i	0.672908	+	0.739726i	$0.265045\pi$
$588$		0			0
$589$	−19.5580	+	11.2918i	−0.805871	+	0.465270i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	15.7409i		−	0.646402i	−0.946330	−	0.323201i	$-0.895241\pi$
$593$		−	15.7409i		−	0.646402i	0.946330	−	0.323201i	$-0.104759\pi$
$594$		0			0
$595$	4.39445			0.180155
$596$		0			0
$597$		0			0
$598$		0			0
$599$	21.5203	+	37.2743i	0.879297	+	1.52299i	0.852114	+	0.523356i	$0.175320\pi$
$599$	21.5203	+	37.2743i	0.879297	+	1.52299i	0.0271823	+	0.999630i	$0.491347\pi$
$600$		0			0
$601$	−19.6867	+	34.0984i	−0.803039	+	1.39090i	0.114569	+	0.993415i	$0.463451\pi$
$601$	−19.6867	+	34.0984i	−0.803039	+	1.39090i	−0.917607	+	0.397488i	$0.869882\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	13.4172	−	23.2393i	0.545488	−	0.944813i
$606$		0			0
$607$	25.7834	−	14.8861i	1.04652	−	0.604207i	0.124845	−	0.992176i	$-0.460157\pi$
$607$	25.7834	−	14.8861i	1.04652	−	0.604207i	0.921672	+	0.387969i	$0.126823\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	39.2673i		−	1.58858i
$612$		0			0
$613$		−	0.615900i		−	0.0248760i	−0.999923	−	0.0124380i	$-0.996041\pi$
$613$		−	0.615900i		−	0.0248760i	0.999923	−	0.0124380i	$-0.00395924\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.03339	+	1.17398i	−0.0818610	+	0.0472625i	−0.540372	−	0.841426i	$-0.681716\pi$
$617$	−2.03339	+	1.17398i	−0.0818610	+	0.0472625i	0.458511	+	0.888689i	$0.348383\pi$
$618$		0			0
$619$	9.58718	−	16.6055i	0.385341	−	0.667431i	−0.606475	−	0.795102i	$-0.707417\pi$
$619$	9.58718	−	16.6055i	0.385341	−	0.667431i	0.991816	+	0.127672i	$0.0407504\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.51711	−	2.62771i	0.0607817	−	0.105277i
$624$		0			0
$625$	−29.9955	−	51.9537i	−1.19982	−	2.07815i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.09864			0.283041
$630$		0			0
$631$		−	3.77969i		−	0.150467i	−0.997166	−	0.0752336i	$-0.976030\pi$
$631$		−	3.77969i		−	0.150467i	0.997166	−	0.0752336i	$-0.0239702\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.37383	+	1.37053i	−0.0942024	+	0.0543878i
$636$		0			0
$637$	−29.4666	−	17.0126i	−1.16751	−	0.674062i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.1609	−	7.02110i	−0.480327	−	0.277317i	0.240226	−	0.970717i	$-0.422779\pi$
$641$	−12.1609	−	7.02110i	−0.480327	−	0.277317i	−0.720553	+	0.693400i	$0.756112\pi$
$642$		0			0
$643$	4.39850	+	7.61843i	0.173460	+	0.300441i	0.939627	−	0.342200i	$-0.111172\pi$
$643$	4.39850	+	7.61843i	0.173460	+	0.300441i	−0.766167	+	0.642641i	$0.777839\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−29.5118			−1.16023			−0.580114	−	0.814535i	$-0.696992\pi$
$647$	−29.5118			−1.16023			−0.580114	+	0.814535i	$0.696992\pi$
$648$		0			0
$649$	−26.1073			−1.02480
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.00669049	−	0.0115883i	−0.000261819	−	0.000453484i	0.865894	−	0.500227i	$-0.166750\pi$
$653$	−0.00669049	−	0.0115883i	−0.000261819	−	0.000453484i	−0.866156	+	0.499773i	$0.833417\pi$
$654$		0			0
$655$	−19.7166	−	11.3834i	−0.770390	−	0.444785i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−34.3244	−	19.8172i	−1.33709	−	0.771970i	−0.350715	−	0.936482i	$-0.614062\pi$
$659$	−34.3244	−	19.8172i	−1.33709	−	0.771970i	−0.986375	+	0.164513i	$0.947395\pi$
$660$		0			0
$661$	−11.8008	+	6.81322i	−0.458999	+	0.265003i	−0.711623	−	0.702561i	$-0.752040\pi$
$661$	−11.8008	+	6.81322i	−0.458999	+	0.265003i	0.252624	+	0.967565i	$0.418706\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			7.42867i			0.288071i
$666$		0			0
$667$	−5.77662			−0.223672
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−21.9947	−	38.0959i	−0.849096	−	1.47068i
$672$		0			0
$673$	−3.22156	+	5.57991i	−0.124182	+	0.215090i	−0.921413	−	0.388585i	$-0.872964\pi$
$673$	−3.22156	+	5.57991i	−0.124182	+	0.215090i	0.797231	+	0.603675i	$0.206297\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.67204	+	13.2884i	−0.294860	+	0.510713i	−0.974952	−	0.222414i	$-0.928606\pi$
$677$	−7.67204	+	13.2884i	−0.294860	+	0.510713i	0.680092	+	0.733127i	$0.261940\pi$
$678$		0			0
$679$	0.430678	−	0.248652i	0.0165279	−	0.00954239i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			17.2454i			0.659878i	0.944002	+	0.329939i	$0.107028\pi$
$683$			17.2454i			0.659878i	−0.944002	+	0.329939i	$0.892972\pi$
$684$		0			0
$685$			16.2945i			0.622579i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.80175	−	5.08169i	0.335320	−	0.193597i
$690$		0			0
$691$	9.97083	−	17.2700i	0.379308	−	0.656981i	−0.611654	−	0.791126i	$-0.709495\pi$
$691$	9.97083	−	17.2700i	0.379308	−	0.656981i	0.990962	+	0.134145i	$0.0428287\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	37.5496	−	65.0378i	1.42434	−	2.46702i
$696$		0			0
$697$	2.61572	+	4.53057i	0.0990776	+	0.171607i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.45975			−0.130673			−0.0653364	−	0.997863i	$-0.520812\pi$
$701$	−3.45975			−0.130673			−0.0653364	+	0.997863i	$0.520812\pi$
$702$		0			0
$703$			12.0000i			0.452589i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.77288	−	1.60092i	0.104285	−	0.0602089i
$708$		0			0
$709$	−34.2429	−	19.7701i	−1.28602	−	0.742483i	−0.308077	−	0.951361i	$-0.599685\pi$
$709$	−34.2429	−	19.7701i	−1.28602	−	0.742483i	−0.977942	+	0.208878i	$0.933019\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	10.0626	+	5.80966i	0.376849	+	0.217574i
$714$		0			0
$715$	−42.6527	−	73.8767i	−1.59512	−	2.76283i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.62877			0.172624			0.0863120	−	0.996268i	$-0.472492\pi$
$719$	4.62877			0.172624			0.0863120	+	0.996268i	$0.472492\pi$
$720$		0			0
$721$	1.65804			0.0617487
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.4822	+	21.6199i	0.463579	+	0.802942i
$726$		0			0
$727$	−5.33286	−	3.07893i	−0.197785	−	0.114191i	0.397837	−	0.917456i	$-0.369761\pi$
$727$	−5.33286	−	3.07893i	−0.197785	−	0.114191i	−0.595622	+	0.803265i	$0.703094\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−16.6474	−	9.61141i	−0.615728	−	0.355491i
$732$		0			0
$733$	−18.9417	+	10.9360i	−0.699627	+	0.403930i	−0.807208	−	0.590266i	$-0.799023\pi$
$733$	−18.9417	+	10.9360i	−0.699627	+	0.403930i	0.107581	+	0.994196i	$0.465689\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	48.8301i		−	1.79868i
$738$		0			0
$739$	−24.5677			−0.903738			−0.451869	−	0.892084i	$-0.649243\pi$
$739$	−24.5677			−0.903738			−0.451869	+	0.892084i	$0.649243\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.52447	−	6.10457i	−0.129300	−	0.223955i	0.794105	−	0.607780i	$-0.207940\pi$
$743$	−3.52447	−	6.10457i	−0.129300	−	0.223955i	−0.923406	+	0.383825i	$0.874606\pi$
$744$		0			0
$745$	−30.2707	+	52.4304i	−1.10903	+	1.92090i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.17879	+	3.77378i	−0.0796113	+	0.137891i
$750$		0			0
$751$	20.9770	−	12.1110i	0.765460	−	0.441938i	−0.0657928	−	0.997833i	$-0.520958\pi$
$751$	20.9770	−	12.1110i	0.765460	−	0.441938i	0.831253	+	0.555895i	$0.187624\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			87.8480i			3.19712i
$756$		0			0
$757$		−	29.8780i		−	1.08593i	−0.839754	−	0.542967i	$-0.817301\pi$
$757$		−	29.8780i		−	1.08593i	0.839754	−	0.542967i	$-0.182699\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−36.2631	+	20.9365i	−1.31454	+	0.758949i	−0.982844	−	0.184438i	$-0.940953\pi$
$761$	−36.2631	+	20.9365i	−1.31454	+	0.758949i	−0.331694	+	0.943387i	$0.607620\pi$
$762$		0			0
$763$	−1.78229	+	3.08702i	−0.0645233	+	0.111758i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.4107	+	26.6921i	−0.556449	+	0.963797i
$768$		0			0
$769$	6.30858	+	10.9268i	0.227493	+	0.394030i	0.957065	−	0.289875i	$-0.0936137\pi$
$769$	6.30858	+	10.9268i	0.227493	+	0.394030i	−0.729571	+	0.683905i	$0.760280\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−28.6390			−1.03007			−0.515037	−	0.857168i	$-0.672222\pi$
$773$	−28.6390			−1.03007			−0.515037	+	0.857168i	$0.672222\pi$
$774$		0			0
$775$		−	50.2145i		−	1.80376i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.65876	+	4.42179i	−0.274404	+	0.158427i
$780$		0			0
$781$	27.7605	+	16.0275i	0.993349	+	0.573511i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−40.2346	−	23.2295i	−1.43603	−	0.829095i
$786$		0			0
$787$	19.7520	+	34.2115i	0.704083	+	1.21951i	0.967021	+	0.254695i	$0.0819751\pi$
$787$	19.7520	+	34.2115i	0.704083	+	1.21951i	−0.262938	+	0.964813i	$0.584692\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.54396			0.0548968
$792$		0			0
$793$	−51.9325			−1.84418
$794$		0			0
$795$		0			0
$796$		0			0
$797$	13.6253	+	23.5998i	0.482634	+	0.835947i	0.999801	−	0.0199377i	$-0.00634678\pi$
$797$	13.6253	+	23.5998i	0.482634	+	0.835947i	−0.517167	+	0.855884i	$0.673013\pi$
$798$		0			0
$799$	−22.0794	−	12.7476i	−0.781114	−	0.450977i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22.5237	−	13.0041i	−0.794845	−	0.458904i
$804$		0			0
$805$	3.31002	−	1.91104i	0.116663	−	0.0673552i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	32.7182i		−	1.15031i	−0.818045	−	0.575155i	$-0.804942\pi$
$809$		−	32.7182i		−	1.15031i	0.818045	−	0.575155i	$-0.195058\pi$
$810$		0			0
$811$	23.3901			0.821336			0.410668	−	0.911785i	$-0.365296\pi$
$811$	23.3901			0.821336			0.410668	+	0.911785i	$0.365296\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−27.5452	−	47.7097i	−0.964866	−	1.67120i
$816$		0			0
$817$	16.2477	−	28.1419i	0.568437	−	0.984561i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.4953	−	26.8387i	0.540790	−	0.936676i	−0.458069	−	0.888917i	$-0.651459\pi$
$821$	15.4953	−	26.8387i	0.540790	−	0.936676i	0.998859	−	0.0477594i	$-0.0152081\pi$
$822$		0			0
$823$	41.9067	−	24.1948i	1.46077	−	0.843379i	0.461728	−	0.887022i	$-0.347230\pi$
$823$	41.9067	−	24.1948i	1.46077	−	0.843379i	0.999047	+	0.0436430i	$0.0138964\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			53.6957i			1.86718i	0.358340	+	0.933591i	$0.383343\pi$
$827$			53.6957i			1.86718i	−0.358340	+	0.933591i	$0.616657\pi$
$828$		0			0
$829$			14.1371i			0.491001i	0.969396	+	0.245500i	$0.0789523\pi$
$829$			14.1371i			0.491001i	−0.969396	+	0.245500i	$0.921048\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−19.1318	+	11.0458i	−0.662879	+	0.382713i
$834$		0			0
$835$	51.1147	−	88.5333i	1.76890	−	3.06382i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.1957	−	36.7120i	0.731757	−	1.26744i	−0.224375	−	0.974503i	$-0.572034\pi$
$839$	21.1957	−	36.7120i	0.731757	−	1.26744i	0.956132	−	0.292937i	$-0.0946326\pi$
$840$		0			0
$841$	12.3547	+	21.3989i	0.426023	+	0.737893i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−47.0266			−1.61777
$846$		0			0
$847$		−	2.15678i		−	0.0741078i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.34688	−	3.08702i	0.183289	−	0.105822i
$852$		0			0
$853$	20.7766	+	11.9954i	0.711378	+	0.410714i	0.811571	−	0.584254i	$-0.198613\pi$
$853$	20.7766	+	11.9954i	0.711378	+	0.410714i	−0.100193	+	0.994968i	$0.531946\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.4336	−	8.33324i	−0.493042	−	0.284658i	0.232793	−	0.972526i	$-0.425213\pi$
$857$	−14.4336	−	8.33324i	−0.493042	−	0.284658i	−0.725836	+	0.687868i	$0.758547\pi$
$858$		0			0
$859$	9.78253	+	16.9438i	0.333776	+	0.578116i	0.983249	−	0.182268i	$-0.0583439\pi$
$859$	9.78253	+	16.9438i	0.333776	+	0.578116i	−0.649473	+	0.760384i	$0.725011\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.5794			1.04094			0.520468	−	0.853881i	$-0.325758\pi$
$863$	30.5794			1.04094			0.520468	+	0.853881i	$0.325758\pi$
$864$		0			0
$865$	−34.1593			−1.16145
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.73656	−	3.00782i	−0.0589089	−	0.102033i
$870$		0			0
$871$	−49.9240	−	28.8236i	−1.69161	−	0.976651i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.37013	−	4.83250i	−0.282962	−	0.163368i
$876$		0			0
$877$	33.9915	−	19.6250i	1.14781	−	0.662690i	0.199460	−	0.979906i	$-0.436081\pi$
$877$	33.9915	−	19.6250i	1.14781	−	0.662690i	0.948353	+	0.317216i	$0.102748\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			20.2971i			0.683828i	0.939731	+	0.341914i	$0.111075\pi$
$881$			20.2971i			0.683828i	−0.939731	+	0.341914i	$0.888925\pi$
$882$		0			0
$883$	−13.2658			−0.446431			−0.223216	−	0.974769i	$-0.571655\pi$
$883$	−13.2658			−0.446431			−0.223216	+	0.974769i	$0.571655\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.2973	+	21.2996i	0.412904	+	0.715170i	0.995206	−	0.0978022i	$-0.0311812\pi$
$887$	12.2973	+	21.2996i	0.412904	+	0.715170i	−0.582302	+	0.812972i	$0.697848\pi$
$888$		0			0
$889$	−0.110154	+	0.190792i	−0.00369445	+	0.00639897i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	21.5493	−	37.3245i	0.721120	−	1.24902i
$894$		0			0
$895$	−44.2546	+	25.5504i	−1.47927	+	0.854056i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			8.63044i			0.287841i
$900$		0			0
$901$		−	6.59880i		−	0.219838i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8.63598	+	4.98599i	−0.287070	+	0.165740i
$906$		0			0
$907$	−4.21712	+	7.30427i	−0.140027	+	0.242534i	−0.927507	−	0.373807i	$-0.878052\pi$
$907$	−4.21712	+	7.30427i	−0.140027	+	0.242534i	0.787479	+	0.616341i	$0.211386\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	7.35597	−	12.7409i	0.243714	−	0.422125i	−0.718055	−	0.695986i	$-0.754967\pi$
$911$	7.35597	−	12.7409i	0.243714	−	0.422125i	0.961769	+	0.273861i	$0.0883008\pi$
$912$		0			0
$913$	14.4072	+	24.9540i	0.476809	+	0.825858i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.82984			−0.0604267
$918$		0			0
$919$			38.4521i			1.26842i	0.773162	+	0.634209i	$0.218674\pi$
$919$			38.4521i			1.26842i	−0.773162	+	0.634209i	$0.781326\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	32.7732	−	18.9216i	1.07874	−	0.622812i
$924$		0			0
$925$	−23.1073	−	13.3410i	−0.759762	−	0.438649i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.88859	+	2.24508i	0.127581	+	0.0736587i	0.562432	−	0.826844i	$-0.309866\pi$
$929$	3.88859	+	2.24508i	0.127581	+	0.0736587i	−0.434851	+	0.900502i	$0.643199\pi$
$930$		0			0
$931$	−18.6725	−	32.3417i	−0.611966	−	1.05996i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−55.3864			−1.81133
$936$		0			0
$937$	−22.9267			−0.748984			−0.374492	−	0.927230i	$-0.622183\pi$
$937$	−22.9267			−0.748984			−0.374492	+	0.927230i	$0.622183\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−27.6316	−	47.8593i	−0.900764	−	1.56017i	−0.826504	−	0.562931i	$-0.809674\pi$
$941$	−27.6316	−	47.8593i	−0.900764	−	1.56017i	−0.0742601	−	0.997239i	$-0.523659\pi$
$942$		0			0
$943$	3.94046	+	2.27503i	0.128319	+	0.0740850i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.4482	+	11.8058i	0.664479	+	0.383637i	0.793981	−	0.607942i	$-0.208005\pi$
$947$	20.4482	+	11.8058i	0.664479	+	0.383637i	−0.129503	+	0.991579i	$0.541338\pi$
$948$		0			0
$949$	−26.5908	+	15.3522i	−0.863174	+	0.498354i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	0.892008i		−	0.0288950i	−0.999896	−	0.0144475i	$-0.995401\pi$
$953$		−	0.892008i		−	0.0288950i	0.999896	−	0.0144475i	$-0.00459894\pi$
$954$		0			0
$955$	59.5926			1.92837
$956$		0			0
$957$		0			0
$958$		0			0
$959$	0.654820	+	1.13418i	0.0211453	+	0.0366247i
$960$		0			0
$961$	−6.82020	+	11.8129i	−0.220006	+	0.381062i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	35.6741	−	61.7894i	1.14839	−	1.98907i
$966$		0			0
$967$	31.3105	−	18.0771i	1.00688	−	0.581321i	0.0966015	−	0.995323i	$-0.469203\pi$
$967$	31.3105	−	18.0771i	1.00688	−	0.581321i	0.910276	+	0.414002i	$0.135869\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	42.0395i		−	1.34911i	−0.738223	−	0.674556i	$-0.764335\pi$
$971$		−	42.0395i		−	1.34911i	0.738223	−	0.674556i	$-0.235665\pi$
$972$		0			0
$973$		−	6.03598i		−	0.193505i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	13.2808	−	7.66765i	0.424889	−	0.245310i	−0.272278	−	0.962219i	$-0.587777\pi$
$977$	13.2808	−	7.66765i	0.424889	−	0.245310i	0.697167	+	0.716909i	$0.254444\pi$
$978$		0			0
$979$	−19.1212	+	33.1189i	−0.611116	+	1.05848i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.90976	−	5.03986i	0.0928071	−	0.160747i	−0.815884	−	0.578215i	$-0.803749\pi$
$983$	2.90976	−	5.03986i	0.0928071	−	0.160747i	0.908691	+	0.417469i	$0.137083\pi$
$984$		0			0
$985$	31.2651	+	54.1528i	0.996190	+	1.72545i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.7190			−0.531635
$990$		0			0
$991$			42.7700i			1.35863i	0.733845	+	0.679316i	$0.237724\pi$
$991$			42.7700i			1.35863i	−0.733845	+	0.679316i	$0.762276\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	77.2655	−	44.6093i	2.44948	−	1.41421i
$996$		0			0
$997$	45.0247	+	25.9950i	1.42595	+	0.823271i	0.996798	−	0.0799618i	$-0.0254798\pi$
$997$	45.0247	+	25.9950i	1.42595	+	0.823271i	0.429150	+	0.903233i	$0.358813\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.p.a.1439.1		16
3.2	odd	2		576.2.p.c.479.7	yes	16
4.3	odd	2	inner	1728.2.p.a.1439.2		16
8.3	odd	2		1728.2.p.c.1439.8		16
8.5	even	2		1728.2.p.c.1439.7		16
9.2	odd	6		5184.2.f.a.2591.4		16
9.4	even	3		576.2.p.a.95.7	yes	16
9.5	odd	6		1728.2.p.c.287.8		16
9.7	even	3		5184.2.f.f.2591.16		16
12.11	even	2		576.2.p.c.479.2	yes	16
24.5	odd	2		576.2.p.a.479.2	yes	16
24.11	even	2		576.2.p.a.479.7	yes	16
36.7	odd	6		5184.2.f.f.2591.14		16
36.11	even	6		5184.2.f.a.2591.2		16
36.23	even	6		1728.2.p.c.287.7		16
36.31	odd	6		576.2.p.a.95.2	✓	16
72.5	odd	6	inner	1728.2.p.a.287.2		16
72.11	even	6		5184.2.f.f.2591.13		16
72.13	even	6		576.2.p.c.95.2	yes	16
72.29	odd	6		5184.2.f.f.2591.15		16
72.43	odd	6		5184.2.f.a.2591.1		16
72.59	even	6	inner	1728.2.p.a.287.1		16
72.61	even	6		5184.2.f.a.2591.3		16
72.67	odd	6		576.2.p.c.95.7	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.p.a.95.2	✓	16	36.31	odd	6
576.2.p.a.95.7	yes	16	9.4	even	3
576.2.p.a.479.2	yes	16	24.5	odd	2
576.2.p.a.479.7	yes	16	24.11	even	2
576.2.p.c.95.2	yes	16	72.13	even	6
576.2.p.c.95.7	yes	16	72.67	odd	6
576.2.p.c.479.2	yes	16	12.11	even	2
576.2.p.c.479.7	yes	16	3.2	odd	2
1728.2.p.a.287.1		16	72.59	even	6	inner
1728.2.p.a.287.2		16	72.5	odd	6	inner
1728.2.p.a.1439.1		16	1.1	even	1	trivial
1728.2.p.a.1439.2		16	4.3	odd	2	inner
1728.2.p.c.287.7		16	36.23	even	6
1728.2.p.c.287.8		16	9.5	odd	6
1728.2.p.c.1439.7		16	8.5	even	2
1728.2.p.c.1439.8		16	8.3	odd	2
5184.2.f.a.2591.1		16	72.43	odd	6
5184.2.f.a.2591.2		16	36.11	even	6
5184.2.f.a.2591.3		16	72.61	even	6
5184.2.f.a.2591.4		16	9.2	odd	6
5184.2.f.f.2591.13		16	72.11	even	6
5184.2.f.f.2591.14		16	36.7	odd	6
5184.2.f.f.2591.15		16	72.29	odd	6
5184.2.f.f.2591.16		16	9.7	even	3

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.06470 - 3.57617i) q^{5} +(-0.287429 - 0.165947i) q^{7} +O(q^{10})\)	\(q+(-2.06470 - 3.57617i) q^{5} +(-0.287429 - 0.165947i) q^{7} +(3.62268 + 2.09155i) q^{11} +(4.27682 - 2.46922i) q^{13} -3.20639i q^{17} +5.42029 q^{19} +(-1.39438 - 2.41514i) q^{23} +(-6.02601 + 10.4374i) q^{25} +(1.03570 - 1.79388i) q^{29} +(-3.60828 + 2.08324i) q^{31} +1.37053i q^{35} +2.21390i q^{37} +(-1.41298 + 0.815784i) q^{41} +(2.99758 - 5.19195i) q^{43} +(3.97567 - 6.88607i) q^{47} +(-3.44492 - 5.96678i) q^{49} +2.05801 q^{53} -17.2738i q^{55} +(-5.40497 + 3.12056i) q^{59} +(-9.10709 - 5.25798i) q^{61} +(-17.6607 - 10.1964i) q^{65} +(-5.83658 - 10.1093i) q^{67} +7.66299 q^{71} -6.21742 q^{73} +(-0.694175 - 1.20235i) q^{77} +(-0.719039 - 0.415137i) q^{79} +(5.96543 + 3.44414i) q^{83} +(-11.4666 + 6.62025i) q^{85} +9.14211i q^{89} -1.63904 q^{91} +(-11.1913 - 19.3839i) q^{95} +(-0.749190 + 1.29763i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{5}+O(q^{10}) \) 16 * q - 6 * q^5	\( 16 q - 6 q^{5} + 6 q^{13} - 14 q^{25} - 18 q^{29} + 6 q^{49} + 48 q^{53} - 42 q^{61} - 54 q^{65} + 28 q^{73} - 66 q^{77} - 36 q^{85} + 8 q^{97}+O(q^{100}) \) 16 * q - 6 * q^5 + 6 * q^13 - 14 * q^25 - 18 * q^29 + 6 * q^49 + 48 * q^53 - 42 * q^61 - 54 * q^65 + 28 * q^73 - 66 * q^77 - 36 * q^85 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more