LMFDB - Embedded newform 2592.2.i.bc.1729.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(865,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2592.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 4x^{2} + 3x + 9 $ x^4 - x^3 + 4x^2 + 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1729.1
Root			$1.15139 - 1.99426i$ of defining polynomial
Character	$\chi$	$=$	2592.1729
Dual form			2592.2.i.bc.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.80278 + 3.12250i) q^{5} +(1.80278 + 3.12250i) q^{7} +O(q^{10})$	$q+(-1.80278 + 3.12250i) q^{5} +(1.80278 + 3.12250i) q^{7} +(0.500000 + 0.866025i) q^{11} +(-2.00000 + 3.46410i) q^{13} +7.21110 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-4.00000 - 6.92820i) q^{25} +(3.60555 + 6.24500i) q^{29} +(-1.80278 + 3.12250i) q^{31} -13.0000 q^{35} +10.0000 q^{37} +(3.60555 - 6.24500i) q^{41} +(3.60555 + 6.24500i) q^{43} +(-5.00000 - 8.66025i) q^{47} +(-3.00000 + 5.19615i) q^{49} -3.60555 q^{53} -3.60555 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-7.21110 - 12.4900i) q^{65} +(3.60555 - 6.24500i) q^{67} -8.00000 q^{71} -3.00000 q^{73} +(-1.80278 + 3.12250i) q^{77} +(-7.21110 - 12.4900i) q^{79} +(4.50000 + 7.79423i) q^{83} +7.21110 q^{89} -14.4222 q^{91} +(-13.0000 + 22.5167i) q^{95} +(-3.50000 - 6.06218i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 2 q^{11} - 8 q^{13} - 12 q^{23} - 16 q^{25} - 52 q^{35} + 40 q^{37} - 20 q^{47} - 12 q^{49} - 8 q^{59} - 32 q^{71} - 12 q^{73} + 18 q^{83} - 52 q^{95} - 14 q^{97}+O(q^{100}) $ 4 * q + 2 * q^11 - 8 * q^13 - 12 * q^23 - 16 * q^25 - 52 * q^35 + 40 * q^37 - 20 * q^47 - 12 * q^49 - 8 * q^59 - 32 * q^71 - 12 * q^73 + 18 * q^83 - 52 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.80278	+	3.12250i	−0.806226	+	1.39642i	0.109235	+	0.994016i	$0.465160\pi$
$5$	−1.80278	+	3.12250i	−0.806226	+	1.39642i	−0.915460	+	0.402408i	$0.868173\pi$
$6$		0			0
$7$	1.80278	+	3.12250i	0.681385	+	1.18019i	0.974558	+	0.224134i	$0.0719554\pi$
$7$	1.80278	+	3.12250i	0.681385	+	1.18019i	−0.293173	+	0.956059i	$0.594711\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	0.931505	−	0.363727i	$-0.118496\pi$
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	−0.780750	+	0.624844i	$0.785163\pi$
$12$		0			0
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	0.443227	+	0.896410i	$0.353834\pi$
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	7.21110			1.65434			0.827170	−	0.561951i	$-0.189949\pi$
$19$	7.21110			1.65434			0.827170	+	0.561951i	$0.189949\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	0.362892	+	0.931831i	$0.381789\pi$
$23$	−3.00000	+	5.19615i	−0.625543	+	1.08347i	−0.988436	+	0.151642i	$0.951544\pi$
$24$		0			0
$25$	−4.00000	−	6.92820i	−0.800000	−	1.38564i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.60555	+	6.24500i	0.669534	+	1.15967i	0.978035	+	0.208443i	$0.0668395\pi$
$29$	3.60555	+	6.24500i	0.669534	+	1.15967i	−0.308500	+	0.951224i	$0.599827\pi$
$30$		0			0
$31$	−1.80278	+	3.12250i	−0.323788	+	0.560817i	−0.981266	−	0.192656i	$-0.938290\pi$
$31$	−1.80278	+	3.12250i	−0.323788	+	0.560817i	0.657478	+	0.753474i	$0.271623\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−13.0000			−2.19740
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.60555	−	6.24500i	0.563093	−	0.975305i	−0.434132	−	0.900849i	$-0.642945\pi$
$41$	3.60555	−	6.24500i	0.563093	−	0.975305i	0.997224	−	0.0744555i	$-0.0237219\pi$
$42$		0			0
$43$	3.60555	+	6.24500i	0.549841	+	0.952353i	0.998285	+	0.0585421i	$0.0186452\pi$
$43$	3.60555	+	6.24500i	0.549841	+	0.952353i	−0.448444	+	0.893811i	$0.648021\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.00000	−	8.66025i	−0.729325	−	1.26323i	−0.957169	−	0.289530i	$-0.906501\pi$
$47$	−5.00000	−	8.66025i	−0.729325	−	1.26323i	0.227844	−	0.973698i	$-0.426832\pi$
$48$		0			0
$49$	−3.00000	+	5.19615i	−0.428571	+	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.60555			−0.495261			−0.247630	−	0.968855i	$-0.579652\pi$
$53$	−3.60555			−0.495261			−0.247630	+	0.968855i	$0.579652\pi$
$54$		0			0
$55$	−3.60555			−0.486172
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.21110	−	12.4900i	−0.894427	−	1.54919i
$66$		0			0
$67$	3.60555	−	6.24500i	0.440488	−	0.762948i	−0.557237	−	0.830353i	$-0.688139\pi$
$67$	3.60555	−	6.24500i	0.440488	−	0.762948i	0.997726	+	0.0674052i	$0.0214720\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	−3.00000			−0.351123			−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000			−0.351123			−0.175562	+	0.984468i	$0.556174\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.80278	+	3.12250i	−0.205445	+	0.355842i
$78$		0			0
$79$	−7.21110	−	12.4900i	−0.811312	−	1.40523i	−0.911946	−	0.410311i	$-0.865420\pi$
$79$	−7.21110	−	12.4900i	−0.811312	−	1.40523i	0.100633	−	0.994924i	$-0.467913\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	0.999976	−	0.00698436i	$-0.00222321\pi$
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	−0.506036	+	0.862512i	$0.668890\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.21110			0.764375			0.382188	−	0.924085i	$-0.375171\pi$
$89$	7.21110			0.764375			0.382188	+	0.924085i	$0.375171\pi$
$90$		0			0
$91$	−14.4222			−1.51186
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.0000	+	22.5167i	−1.33377	+	2.31016i
$96$		0			0
$97$	−3.50000	−	6.06218i	−0.355371	−	0.615521i	0.631810	−	0.775123i	$-0.282312\pi$
$97$	−3.50000	−	6.06218i	−0.355371	−	0.615521i	−0.987181	+	0.159602i	$0.948979\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.40833	−	9.36750i	−0.538149	−	0.932101i	−0.999004	−	0.0446254i	$-0.985791\pi$
$101$	−5.40833	−	9.36750i	−0.538149	−	0.932101i	0.460855	−	0.887475i	$-0.347543\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−17.0000			−1.64345			−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000			−1.64345			−0.821726	+	0.569883i	$0.806989\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.60555	−	6.24500i	0.339182	−	0.587480i	−0.645097	−	0.764100i	$-0.723183\pi$
$113$	3.60555	−	6.24500i	0.339182	−	0.587480i	0.984279	+	0.176620i	$0.0565165\pi$
$114$		0			0
$115$	−10.8167	−	18.7350i	−1.00866	−	1.74705i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.8167			0.967471
$126$		0			0
$127$	3.60555			0.319941			0.159970	−	0.987122i	$-0.448860\pi$
$127$	3.60555			0.319941			0.159970	+	0.987122i	$0.448860\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.500000	−	0.866025i	0.0436852	−	0.0756650i	−0.843356	−	0.537355i	$-0.819423\pi$
$131$	0.500000	−	0.866025i	0.0436852	−	0.0756650i	0.887041	+	0.461690i	$0.152757\pi$
$132$		0			0
$133$	13.0000	+	22.5167i	1.12724	+	1.95244i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.8167	+	18.7350i	0.924129	+	1.60064i	0.792956	+	0.609279i	$0.208541\pi$
$137$	10.8167	+	18.7350i	0.924129	+	1.60064i	0.131173	+	0.991359i	$0.458126\pi$
$138$		0			0
$139$	−7.21110	+	12.4900i	−0.611638	+	1.05939i	0.379327	+	0.925263i	$0.376156\pi$
$139$	−7.21110	+	12.4900i	−0.611638	+	1.05939i	−0.990964	+	0.134125i	$0.957178\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.00000			−0.334497
$144$		0			0
$145$	−26.0000			−2.15918
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.40833	−	9.36750i	0.443067	−	0.767415i	−0.554848	−	0.831952i	$-0.687224\pi$
$149$	5.40833	−	9.36750i	0.443067	−	0.767415i	0.997915	+	0.0645365i	$0.0205569\pi$
$150$		0			0
$151$	5.40833	+	9.36750i	0.440123	+	0.762316i	0.997698	−	0.0678106i	$-0.0216014\pi$
$151$	5.40833	+	9.36750i	0.440123	+	0.762316i	−0.557575	+	0.830127i	$0.688268\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.50000	−	11.2583i	−0.522093	−	0.904291i
$156$		0			0
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	−0.775113	−	0.631822i	$-0.782307\pi$
$157$	2.00000	−	3.46410i	0.159617	−	0.276465i	0.934731	+	0.355357i	$0.115641\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−21.6333			−1.70494
$162$		0			0
$163$	−14.4222			−1.12963			−0.564817	−	0.825216i	$-0.691053\pi$
$163$	−14.4222			−1.12963			−0.564817	+	0.825216i	$0.691053\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000	−	1.73205i	0.0773823	−	0.134030i	−0.824737	−	0.565516i	$-0.808677\pi$
$167$	1.00000	−	1.73205i	0.0773823	−	0.134030i	0.902120	+	0.431486i	$0.142010\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.40833	+	9.36750i	0.411187	+	0.712198i	0.995020	−	0.0996766i	$-0.0317808\pi$
$173$	5.40833	+	9.36750i	0.411187	+	0.712198i	−0.583832	+	0.811874i	$0.698447\pi$
$174$		0			0
$175$	14.4222	−	24.9800i	1.09022	−	1.88831i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.0000			−1.12115			−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000			−1.12115			−0.560576	+	0.828103i	$0.689420\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−18.0278	+	31.2250i	−1.32543	+	2.29571i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	0.563081	−	0.826402i	$-0.309616\pi$
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	−0.997225	+	0.0744412i	$0.976283\pi$
$192$		0			0
$193$	7.50000	−	12.9904i	0.539862	−	0.935068i	−0.459049	−	0.888411i	$-0.651810\pi$
$193$	7.50000	−	12.9904i	0.539862	−	0.935068i	0.998911	−	0.0466572i	$-0.0148568\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0278			−1.28442			−0.642212	−	0.766527i	$-0.721983\pi$
$197$	−18.0278			−1.28442			−0.642212	+	0.766527i	$0.721983\pi$
$198$		0			0
$199$	3.60555			0.255591			0.127795	−	0.991801i	$-0.459210\pi$
$199$	3.60555			0.255591			0.127795	+	0.991801i	$0.459210\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−13.0000	+	22.5167i	−0.912421	+	1.58036i
$204$		0			0
$205$	13.0000	+	22.5167i	0.907959	+	1.57263i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.60555	+	6.24500i	0.249401	+	0.431976i
$210$		0			0
$211$	−3.60555	+	6.24500i	−0.248216	+	0.429923i	−0.963031	−	0.269391i	$-0.913178\pi$
$211$	−3.60555	+	6.24500i	−0.248216	+	0.429923i	0.714815	+	0.699314i	$0.246511\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−26.0000			−1.77319
$216$		0			0
$217$	−13.0000			−0.882498
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	7.21110	+	12.4900i	0.482891	+	0.836392i	0.999807	−	0.0196442i	$-0.00625334\pi$
$223$	7.21110	+	12.4900i	0.482891	+	0.836392i	−0.516916	+	0.856036i	$0.672920\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	0.993508	−	0.113761i	$-0.0362899\pi$
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	−0.595274	+	0.803523i	$0.702957\pi$
$228$		0			0
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	−0.652183	−	0.758062i	$-0.726147\pi$
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	0.982592	+	0.185776i	$0.0594799\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.6333			1.41725			0.708623	−	0.705588i	$-0.249317\pi$
$233$	21.6333			1.41725			0.708623	+	0.705588i	$0.249317\pi$
$234$		0			0
$235$	36.0555			2.35200
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	−0.752536	−	0.658551i	$-0.771170\pi$
$239$	3.00000	−	5.19615i	0.194054	−	0.336111i	0.946590	+	0.322440i	$0.104503\pi$
$240$		0			0
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	−0.965387	−	0.260822i	$-0.916006\pi$
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	0.256814	−	0.966461i	$-0.417327\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.8167	−	18.7350i	−0.691051	−	1.19693i
$246$		0			0
$247$	−14.4222	+	24.9800i	−0.917663	+	1.58944i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$	−6.00000			−0.377217
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.21110	+	12.4900i	−0.449816	+	0.779105i	−0.998374	−	0.0570083i	$-0.981844\pi$
$257$	−7.21110	+	12.4900i	−0.449816	+	0.779105i	0.548557	+	0.836113i	$0.315177\pi$
$258$		0			0
$259$	18.0278	+	31.2250i	1.12019	+	1.94023i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	0.997906	+	0.0646755i	$0.0206012\pi$
$263$	9.00000	+	15.5885i	0.554964	+	0.961225i	−0.442943	+	0.896550i	$0.646065\pi$
$264$		0			0
$265$	6.50000	−	11.2583i	0.399292	−	0.691594i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	21.6333			1.31901			0.659503	−	0.751702i	$-0.270767\pi$
$269$	21.6333			1.31901			0.659503	+	0.751702i	$0.270767\pi$
$270$		0			0
$271$	10.8167			0.657065			0.328532	−	0.944493i	$-0.393446\pi$
$271$	10.8167			0.657065			0.328532	+	0.944493i	$0.393446\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.00000	−	6.92820i	0.241209	−	0.417786i
$276$		0			0
$277$	4.00000	+	6.92820i	0.240337	+	0.416275i	0.960810	−	0.277207i	$-0.0894088\pi$
$277$	4.00000	+	6.92820i	0.240337	+	0.416275i	−0.720473	+	0.693482i	$0.756075\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$	3.60555	−	6.24500i	0.214328	−	0.371227i	−0.738737	−	0.673994i	$-0.764577\pi$
$283$	3.60555	−	6.24500i	0.214328	−	0.371227i	0.953064	+	0.302768i	$0.0979106\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	26.0000			1.53473
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.8167	−	18.7350i	0.631916	−	1.09451i	−0.355244	−	0.934774i	$-0.615602\pi$
$293$	10.8167	−	18.7350i	0.631916	−	1.09451i	0.987160	−	0.159736i	$-0.0510644\pi$
$294$		0			0
$295$	−7.21110	−	12.4900i	−0.419847	−	0.727196i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.0000	−	20.7846i	−0.693978	−	1.20201i
$300$		0			0
$301$	−13.0000	+	22.5167i	−0.749308	+	1.29784i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	−0.489585	−	0.871956i	$-0.662852\pi$
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	0.999928	−	0.0119847i	$-0.00381495\pi$
$312$		0			0
$313$	−0.500000	−	0.866025i	−0.0282617	−	0.0489506i	0.851549	−	0.524276i	$-0.175664\pi$
$313$	−0.500000	−	0.866025i	−0.0282617	−	0.0489506i	−0.879810	+	0.475325i	$0.842331\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.40833	−	9.36750i	−0.303762	−	0.526131i	0.673223	−	0.739440i	$-0.264909\pi$
$317$	−5.40833	−	9.36750i	−0.303762	−	0.526131i	−0.976985	+	0.213308i	$0.931576\pi$
$318$		0			0
$319$	−3.60555	+	6.24500i	−0.201872	+	0.349653i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	32.0000			1.77504
$326$		0			0
$327$		0			0
$328$		0			0
$329$	18.0278	−	31.2250i	0.993902	−	1.72149i
$330$		0			0
$331$	3.60555	+	6.24500i	0.198179	+	0.343256i	0.947938	−	0.318455i	$-0.103164\pi$
$331$	3.60555	+	6.24500i	0.198179	+	0.343256i	−0.749759	+	0.661711i	$0.769831\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	13.0000	+	22.5167i	0.710266	+	1.23022i
$336$		0			0
$337$	−13.0000	+	22.5167i	−0.708155	+	1.22656i	0.257386	+	0.966309i	$0.417139\pi$
$337$	−13.0000	+	22.5167i	−0.708155	+	1.22656i	−0.965541	+	0.260252i	$0.916194\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−3.60555			−0.195252
$342$		0			0
$343$	3.60555			0.194681
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.500000	+	0.866025i	−0.0268414	+	0.0464907i	−0.879134	−	0.476575i	$-0.841878\pi$
$347$	−0.500000	+	0.866025i	−0.0268414	+	0.0464907i	0.852293	+	0.523065i	$0.175212\pi$
$348$		0			0
$349$	1.00000	+	1.73205i	0.0535288	+	0.0927146i	0.891548	−	0.452926i	$-0.149620\pi$
$349$	1.00000	+	1.73205i	0.0535288	+	0.0927146i	−0.838019	+	0.545640i	$0.816286\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.60555	−	6.24500i	−0.191904	−	0.332388i	0.753977	−	0.656901i	$-0.228133\pi$
$353$	−3.60555	−	6.24500i	−0.191904	−	0.332388i	−0.945881	+	0.324513i	$0.894800\pi$
$354$		0			0
$355$	14.4222	−	24.9800i	0.765451	−	1.32580i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000			−0.527780			−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000			−0.527780			−0.263890	+	0.964553i	$0.585006\pi$
$360$		0			0
$361$	33.0000			1.73684
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5.40833	−	9.36750i	0.283085	−	0.490317i
$366$		0			0
$367$	5.40833	+	9.36750i	0.282312	+	0.488979i	0.971954	−	0.235172i	$-0.0755652\pi$
$367$	5.40833	+	9.36750i	0.282312	+	0.488979i	−0.689642	+	0.724151i	$0.742232\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.50000	−	11.2583i	−0.337463	−	0.584503i
$372$		0			0
$373$	8.00000	−	13.8564i	0.414224	−	0.717458i	−0.581122	−	0.813816i	$-0.697386\pi$
$373$	8.00000	−	13.8564i	0.414224	−	0.717458i	0.995347	+	0.0963587i	$0.0307196\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−28.8444			−1.48556
$378$		0			0
$379$	−14.4222			−0.740819			−0.370409	−	0.928869i	$-0.620783\pi$
$379$	−14.4222			−0.740819			−0.370409	+	0.928869i	$0.620783\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.00000	−	6.92820i	0.204390	−	0.354015i	−0.745548	−	0.666452i	$-0.767812\pi$
$383$	4.00000	−	6.92820i	0.204390	−	0.354015i	0.949938	+	0.312437i	$0.101145\pi$
$384$		0			0
$385$	−6.50000	−	11.2583i	−0.331271	−	0.573778i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.40833	+	9.36750i	0.274213	+	0.474951i	0.969936	−	0.243359i	$-0.0782493\pi$
$389$	5.40833	+	9.36750i	0.274213	+	0.474951i	−0.695723	+	0.718310i	$0.744916\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	52.0000			2.61640
$396$		0			0
$397$	4.00000			0.200754			0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000			0.200754			0.100377	+	0.994949i	$0.467995\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.21110	−	12.4900i	0.360105	−	0.623721i	−0.627873	−	0.778316i	$-0.716074\pi$
$401$	7.21110	−	12.4900i	0.360105	−	0.623721i	0.987978	+	0.154596i	$0.0494074\pi$
$402$		0			0
$403$	−7.21110	−	12.4900i	−0.359211	−	0.622171i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.00000	+	8.66025i	0.247841	+	0.429273i
$408$		0			0
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−3.50000	+	6.06218i	−0.173064	+	0.299755i	0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14.4222			−0.709670
$414$		0			0
$415$	−32.4500			−1.59291
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.00000	−	3.46410i	0.0977064	−	0.169232i	−0.813029	−	0.582224i	$-0.802183\pi$
$419$	2.00000	−	3.46410i	0.0977064	−	0.169232i	0.910735	+	0.412991i	$0.135516\pi$
$420$		0			0
$421$	−16.0000	−	27.7128i	−0.779792	−	1.35064i	−0.932061	−	0.362301i	$-0.881991\pi$
$421$	−16.0000	−	27.7128i	−0.779792	−	1.35064i	0.152269	−	0.988339i	$-0.451342\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.0000			−0.867029			−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000			−0.867029			−0.433515	+	0.901146i	$0.642727\pi$
$432$		0			0
$433$	−35.0000			−1.68199			−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−35.0000			−1.68199			−0.840996	+	0.541041i	$0.818030\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.6333	+	37.4700i	−1.03486	+	1.79243i
$438$		0			0
$439$	−9.01388	−	15.6125i	−0.430209	−	0.745144i	0.566682	−	0.823937i	$-0.308227\pi$
$439$	−9.01388	−	15.6125i	−0.430209	−	0.745144i	−0.996891	+	0.0787928i	$0.974893\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		0			0
$445$	−13.0000	+	22.5167i	−0.616259	+	1.06739i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	7.21110			0.340313			0.170156	−	0.985417i	$-0.445573\pi$
$449$	7.21110			0.340313			0.170156	+	0.985417i	$0.445573\pi$
$450$		0			0
$451$	7.21110			0.339558
$452$		0			0
$453$		0			0
$454$		0			0
$455$	26.0000	−	45.0333i	1.21890	−	2.11119i
$456$		0			0
$457$	−11.5000	−	19.9186i	−0.537947	−	0.931752i	−0.999014	−	0.0443868i	$-0.985867\pi$
$457$	−11.5000	−	19.9186i	−0.537947	−	0.931752i	0.461067	−	0.887365i	$-0.347467\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.40833	+	9.36750i	0.251891	+	0.436288i	0.964046	−	0.265734i	$-0.0856143\pi$
$461$	5.40833	+	9.36750i	0.251891	+	0.436288i	−0.712156	+	0.702022i	$0.752281\pi$
$462$		0			0
$463$	−12.6194	+	21.8575i	−0.586475	+	1.01580i	0.408215	+	0.912886i	$0.366151\pi$
$463$	−12.6194	+	21.8575i	−0.586475	+	1.01580i	−0.994690	+	0.102918i	$0.967182\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	23.0000			1.06431			0.532157	−	0.846646i	$-0.321382\pi$
$467$	23.0000			1.06431			0.532157	+	0.846646i	$0.321382\pi$
$468$		0			0
$469$	26.0000			1.20057
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.60555	+	6.24500i	−0.165783	+	0.287145i
$474$		0			0
$475$	−28.8444	−	49.9600i	−1.32347	−	2.29232i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.0000	−	36.3731i	−0.959514	−	1.66193i	−0.723681	−	0.690134i	$-0.757551\pi$
$479$	−21.0000	−	36.3731i	−0.959514	−	1.66193i	−0.235833	−	0.971794i	$-0.575782\pi$
$480$		0			0
$481$	−20.0000	+	34.6410i	−0.911922	+	1.57949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	25.2389			1.14604
$486$		0			0
$487$	−14.4222			−0.653532			−0.326766	−	0.945105i	$-0.605959\pi$
$487$	−14.4222			−0.653532			−0.326766	+	0.945105i	$0.605959\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.50000	−	6.06218i	0.157953	−	0.273582i	−0.776178	−	0.630514i	$-0.782844\pi$
$491$	3.50000	−	6.06218i	0.157953	−	0.273582i	0.934130	+	0.356932i	$0.116177\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−14.4222	−	24.9800i	−0.646924	−	1.12051i
$498$		0			0
$499$	−10.8167	+	18.7350i	−0.484220	+	0.838694i	−0.999836	−	0.0181264i	$-0.994230\pi$
$499$	−10.8167	+	18.7350i	−0.484220	+	0.838694i	0.515616	+	0.856820i	$0.327563\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	42.0000			1.87269			0.936344	−	0.351085i	$-0.114187\pi$
$503$	42.0000			1.87269			0.936344	+	0.351085i	$0.114187\pi$
$504$		0			0
$505$	39.0000			1.73548
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.40833	+	9.36750i	−0.239720	+	0.415207i	−0.960634	−	0.277818i	$-0.910389\pi$
$509$	−5.40833	+	9.36750i	−0.239720	+	0.415207i	0.720914	+	0.693025i	$0.243722\pi$
$510$		0			0
$511$	−5.40833	−	9.36750i	−0.239250	−	0.414394i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	5.00000	−	8.66025i	0.219900	−	0.380878i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	43.2666			1.89554			0.947772	−	0.318947i	$-0.103329\pi$
$521$	43.2666			1.89554			0.947772	+	0.318947i	$0.103329\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14.4222	+	24.9800i	0.624695	+	1.08200i
$534$		0			0
$535$	30.6472	−	53.0825i	1.32499	−	2.29496i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$	4.00000			0.171973			0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000			0.171973			0.0859867	+	0.996296i	$0.472596\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.60555	+	6.24500i	−0.154445	+	0.267506i
$546$		0			0
$547$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$547$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	26.0000	+	45.0333i	1.10764	+	1.91848i
$552$		0			0
$553$	26.0000	−	45.0333i	1.10563	−	1.91501i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−39.6611			−1.68049			−0.840247	−	0.542204i	$-0.817590\pi$
$557$	−39.6611			−1.68049			−0.840247	+	0.542204i	$0.817590\pi$
$558$		0			0
$559$	−28.8444			−1.21999
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	−0.0824933	−	0.996592i	$-0.526288\pi$
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	0.904320	−	0.426855i	$-0.140378\pi$
$564$		0			0
$565$	13.0000	+	22.5167i	0.546914	+	0.947283i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7.21110	−	12.4900i	−0.302305	−	0.523608i	0.674353	−	0.738410i	$-0.264423\pi$
$569$	−7.21110	−	12.4900i	−0.302305	−	0.523608i	−0.976658	+	0.214802i	$0.931089\pi$
$570$		0			0
$571$	7.21110	−	12.4900i	0.301775	−	0.522690i	−0.674763	−	0.738035i	$-0.735754\pi$
$571$	7.21110	−	12.4900i	0.301775	−	0.522690i	0.976538	+	0.215345i	$0.0690874\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	48.0000			2.00174
$576$		0			0
$577$	6.00000			0.249783			0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000			0.249783			0.124892	+	0.992170i	$0.460142\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−16.2250	+	28.1025i	−0.673126	+	1.16589i
$582$		0			0
$583$	−1.80278	−	3.12250i	−0.0746633	−	0.129321i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.50000	−	2.59808i	−0.0619116	−	0.107234i	0.833408	−	0.552658i	$-0.186386\pi$
$587$	−1.50000	−	2.59808i	−0.0619116	−	0.107234i	−0.895320	+	0.445424i	$0.853053\pi$
$588$		0			0
$589$	−13.0000	+	22.5167i	−0.535656	+	0.927783i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.6333			0.888373			0.444187	−	0.895934i	$-0.353493\pi$
$593$	21.6333			0.888373			0.444187	+	0.895934i	$0.353493\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.00000	−	12.1244i	0.286012	−	0.495388i	−0.686842	−	0.726807i	$-0.741004\pi$
$599$	7.00000	−	12.1244i	0.286012	−	0.495388i	0.972854	+	0.231419i	$0.0743369\pi$
$600$		0			0
$601$	8.50000	+	14.7224i	0.346722	+	0.600541i	0.985665	−	0.168714i	$-0.0539613\pi$
$601$	8.50000	+	14.7224i	0.346722	+	0.600541i	−0.638943	+	0.769254i	$0.720628\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	18.0278	+	31.2250i	0.732933	+	1.26948i
$606$		0			0
$607$	−7.21110	+	12.4900i	−0.292690	+	0.506953i	−0.974445	−	0.224627i	$-0.927884\pi$
$607$	−7.21110	+	12.4900i	−0.292690	+	0.506953i	0.681755	+	0.731580i	$0.261217\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	40.0000			1.61823
$612$		0			0
$613$	−18.0000			−0.727013			−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000			−0.727013			−0.363507	+	0.931592i	$0.618421\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.8167	−	18.7350i	0.435462	−	0.754242i	−0.561871	−	0.827225i	$-0.689918\pi$
$617$	10.8167	−	18.7350i	0.435462	−	0.754242i	0.997333	+	0.0729823i	$0.0232517\pi$
$618$		0			0
$619$	21.6333	+	37.4700i	0.869516	+	1.50605i	0.862492	+	0.506071i	$0.168902\pi$
$619$	21.6333	+	37.4700i	0.869516	+	1.50605i	0.00702402	+	0.999975i	$0.497764\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	13.0000	+	22.5167i	0.520834	+	0.902111i
$624$		0			0
$625$	0.500000	−	0.866025i	0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	25.2389			1.00474			0.502372	−	0.864652i	$-0.332461\pi$
$631$	25.2389			1.00474			0.502372	+	0.864652i	$0.332461\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.50000	+	11.2583i	−0.257945	+	0.446773i
$636$		0			0
$637$	−12.0000	−	20.7846i	−0.475457	−	0.823516i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.0278	+	31.2250i	0.712054	+	1.23331i	0.964085	+	0.265594i	$0.0855680\pi$
$641$	18.0278	+	31.2250i	0.712054	+	1.23331i	−0.252031	+	0.967719i	$0.581099\pi$
$642$		0			0
$643$	−21.6333	+	37.4700i	−0.853134	+	1.47767i	0.0252307	+	0.999682i	$0.491968\pi$
$643$	−21.6333	+	37.4700i	−0.853134	+	1.47767i	−0.878365	+	0.477990i	$0.841365\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.01388	+	15.6125i	−0.352740	+	0.610964i	−0.986729	−	0.162379i	$-0.948083\pi$
$653$	−9.01388	+	15.6125i	−0.352740	+	0.610964i	0.633988	+	0.773343i	$0.281417\pi$
$654$		0			0
$655$	1.80278	+	3.12250i	0.0704403	+	0.122006i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.5000	+	23.3827i	0.525885	+	0.910860i	0.999545	+	0.0301523i	$0.00959924\pi$
$659$	13.5000	+	23.3827i	0.525885	+	0.910860i	−0.473660	+	0.880708i	$0.657067\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−93.7443			−3.63525
$666$		0			0
$667$	−43.2666			−1.67529
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	9.50000	+	16.4545i	0.366198	+	0.634274i	0.988968	−	0.148132i	$-0.0473259\pi$
$673$	9.50000	+	16.4545i	0.366198	+	0.634274i	−0.622770	+	0.782405i	$0.713993\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.60555	−	6.24500i	−0.138573	−	0.240015i	0.788384	−	0.615184i	$-0.210918\pi$
$677$	−3.60555	−	6.24500i	−0.138573	−	0.240015i	−0.926957	+	0.375169i	$0.877585\pi$
$678$		0			0
$679$	12.6194	−	21.8575i	0.484289	−	0.838814i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$	−78.0000			−2.98023
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.21110	−	12.4900i	0.274721	−	0.475831i
$690$		0			0
$691$	7.21110	+	12.4900i	0.274323	+	0.475142i	0.969964	−	0.243248i	$-0.0782128\pi$
$691$	7.21110	+	12.4900i	0.274323	+	0.475142i	−0.695641	+	0.718390i	$0.744880\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−26.0000	−	45.0333i	−0.986236	−	1.70821i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.8167			0.408539			0.204270	−	0.978915i	$-0.434518\pi$
$701$	10.8167			0.408539			0.204270	+	0.978915i	$0.434518\pi$
$702$		0			0
$703$	72.1110			2.71972
$704$		0			0
$705$		0			0
$706$		0			0
$707$	19.5000	−	33.7750i	0.733373	−	1.27024i
$708$		0			0
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	0.826023	−	0.563636i	$-0.190598\pi$
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	−0.901135	+	0.433539i	$0.857265\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10.8167	−	18.7350i	−0.405087	−	0.701631i
$714$		0			0
$715$	7.21110	−	12.4900i	0.269680	−	0.467099i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.00000			0.149175			0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000			0.149175			0.0745874	+	0.997214i	$0.476236\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	28.8444	−	49.9600i	1.07125	−	1.85547i
$726$		0			0
$727$	16.2250	+	28.1025i	0.601751	+	1.04226i	0.992556	+	0.121790i	$0.0388635\pi$
$727$	16.2250	+	28.1025i	0.601751	+	1.04226i	−0.390805	+	0.920474i	$0.627803\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	23.0000	−	39.8372i	0.849524	−	1.47142i	−0.0321090	−	0.999484i	$-0.510222\pi$
$733$	23.0000	−	39.8372i	0.849524	−	1.47142i	0.881633	−	0.471935i	$-0.156444\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7.21110			0.265624
$738$		0			0
$739$	28.8444			1.06106			0.530529	−	0.847667i	$-0.321993\pi$
$739$	28.8444			1.06106			0.530529	+	0.847667i	$0.321993\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.0000	+	31.1769i	−0.660356	+	1.14377i	0.320166	+	0.947361i	$0.396261\pi$
$743$	−18.0000	+	31.1769i	−0.660356	+	1.14377i	−0.980522	+	0.196409i	$0.937072\pi$
$744$		0			0
$745$	19.5000	+	33.7750i	0.714425	+	1.23742i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30.6472	−	53.0825i	−1.11982	−	1.93959i
$750$		0			0
$751$	−23.4361	+	40.5925i	−0.855195	+	1.48124i	0.0212693	+	0.999774i	$0.493229\pi$
$751$	−23.4361	+	40.5925i	−0.855195	+	1.48124i	−0.876464	+	0.481467i	$0.840104\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−39.0000			−1.41936
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	3.60555	+	6.24500i	0.130530	+	0.226084i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.00000	−	13.8564i	−0.288863	−	0.500326i
$768$		0			0
$769$	5.50000	−	9.52628i	0.198335	−	0.343526i	−0.749654	−	0.661830i	$-0.769780\pi$
$769$	5.50000	−	9.52628i	0.198335	−	0.343526i	0.947989	+	0.318304i	$0.103113\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21.6333			0.778096			0.389048	−	0.921217i	$-0.372804\pi$
$773$	21.6333			0.778096			0.389048	+	0.921217i	$0.372804\pi$
$774$		0			0
$775$	28.8444			1.03612
$776$		0			0
$777$		0			0
$778$		0			0
$779$	26.0000	−	45.0333i	0.931547	−	1.61349i
$780$		0			0
$781$	−4.00000	−	6.92820i	−0.143131	−	0.247911i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.21110	+	12.4900i	0.257375	+	0.445787i
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	26.0000			0.924454
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.40833	−	9.36750i	0.191573	−	0.331814i	−0.754199	−	0.656646i	$-0.771975\pi$
$797$	5.40833	−	9.36750i	0.191573	−	0.331814i	0.945772	+	0.324832i	$0.105308\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.50000	−	2.59808i	−0.0529339	−	0.0916841i
$804$		0			0
$805$	39.0000	−	67.5500i	1.37457	−	2.38082i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	50.4777			1.77470			0.887351	−	0.461095i	$-0.152543\pi$
$809$	50.4777			1.77470			0.887351	+	0.461095i	$0.152543\pi$
$810$		0			0
$811$	7.21110			0.253216			0.126608	−	0.991953i	$-0.459591\pi$
$811$	7.21110			0.253216			0.126608	+	0.991953i	$0.459591\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	26.0000	−	45.0333i	0.910740	−	1.57745i
$816$		0			0
$817$	26.0000	+	45.0333i	0.909625	+	1.57552i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.60555	+	6.24500i	0.125835	+	0.217952i	0.922059	−	0.387050i	$-0.126506\pi$
$821$	3.60555	+	6.24500i	0.125835	+	0.217952i	−0.796224	+	0.605002i	$0.793172\pi$
$822$		0			0
$823$	−9.01388	+	15.6125i	−0.314204	+	0.544217i	−0.979268	−	0.202569i	$-0.935071\pi$
$823$	−9.01388	+	15.6125i	−0.314204	+	0.544217i	0.665064	+	0.746786i	$0.268404\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−16.0000			−0.556375			−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000			−0.556375			−0.278187	+	0.960527i	$0.589734\pi$
$828$		0			0
$829$	−46.0000			−1.59765			−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000			−1.59765			−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	3.60555	+	6.24500i	0.124775	+	0.216117i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	−0.971682	−	0.236293i	$-0.924067\pi$
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	0.281205	−	0.959648i	$-0.409266\pi$
$840$		0			0
$841$	−11.5000	+	19.9186i	−0.396552	+	0.686848i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.8167			0.372104
$846$		0			0
$847$	36.0555			1.23888
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−30.0000	+	51.9615i	−1.02839	+	1.78122i
$852$		0			0
$853$	21.0000	+	36.3731i	0.719026	+	1.24539i	0.961386	+	0.275204i	$0.0887453\pi$
$853$	21.0000	+	36.3731i	0.719026	+	1.24539i	−0.242360	+	0.970186i	$0.577921\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.4222	−	24.9800i	−0.492653	−	0.853300i	0.507311	−	0.861763i	$-0.330640\pi$
$857$	−14.4222	−	24.9800i	−0.492653	−	0.853300i	−0.999964	+	0.00846275i	$0.997306\pi$
$858$		0			0
$859$	14.4222	−	24.9800i	0.492079	−	0.852306i	−0.507879	−	0.861428i	$-0.669570\pi$
$859$	14.4222	−	24.9800i	0.492079	−	0.852306i	0.999958	+	0.00912203i	$0.00290367\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	28.0000			0.953131			0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000			0.953131			0.476566	+	0.879139i	$0.341881\pi$
$864$		0			0
$865$	−39.0000			−1.32604
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.21110	−	12.4900i	0.244620	−	0.423694i
$870$		0			0
$871$	14.4222	+	24.9800i	0.488678	+	0.846415i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	19.5000	+	33.7750i	0.659220	+	1.14180i
$876$		0			0
$877$	−17.0000	+	29.4449i	−0.574049	+	0.994282i	0.422095	+	0.906552i	$0.361295\pi$
$877$	−17.0000	+	29.4449i	−0.574049	+	0.994282i	−0.996144	+	0.0877308i	$0.972038\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	43.2666			1.45769			0.728845	−	0.684679i	$-0.240058\pi$
$881$	43.2666			1.45769			0.728845	+	0.684679i	$0.240058\pi$
$882$		0			0
$883$	−50.4777			−1.69871			−0.849355	−	0.527822i	$-0.823009\pi$
$883$	−50.4777			−1.69871			−0.849355	+	0.527822i	$0.823009\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	0.387768	+	0.921757i	$0.373246\pi$
$887$	−18.0000	+	31.1769i	−0.604381	+	1.04682i	−0.992149	+	0.125061i	$0.960087\pi$
$888$		0			0
$889$	6.50000	+	11.2583i	0.218003	+	0.377592i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−36.0555	−	62.4500i	−1.20655	−	2.08981i
$894$		0			0
$895$	27.0416	−	46.8375i	0.903902	−	1.56560i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−26.0000			−0.867149
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.4222	+	24.9800i	−0.479410	+	0.830363i
$906$		0			0
$907$	−3.60555	−	6.24500i	−0.119720	−	0.207362i	0.799936	−	0.600085i	$-0.204866\pi$
$907$	−3.60555	−	6.24500i	−0.119720	−	0.207362i	−0.919657	+	0.392723i	$0.871533\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.0000	+	25.9808i	0.496972	+	0.860781i	0.999994	−	0.00349271i	$-0.00111177\pi$
$911$	15.0000	+	25.9808i	0.496972	+	0.860781i	−0.503022	+	0.864274i	$0.667778\pi$
$912$		0			0
$913$	−4.50000	+	7.79423i	−0.148928	+	0.257951i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.60555			0.119066
$918$		0			0
$919$	−54.0833			−1.78404			−0.892021	−	0.451994i	$-0.850713\pi$
$919$	−54.0833			−1.78404			−0.892021	+	0.451994i	$0.850713\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16.0000	−	27.7128i	0.526646	−	0.912178i
$924$		0			0
$925$	−40.0000	−	69.2820i	−1.31519	−	2.27798i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21.6333	+	37.4700i	0.709766	+	1.22935i	0.964944	+	0.262456i	$0.0845326\pi$
$929$	21.6333	+	37.4700i	0.709766	+	1.22935i	−0.255178	+	0.966894i	$0.582134\pi$
$930$		0			0
$931$	−21.6333	+	37.4700i	−0.709003	+	1.22803i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	35.0000			1.14340			0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000			1.14340			0.571700	+	0.820463i	$0.306284\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.6194	−	21.8575i	0.411382	−	0.712534i	−0.583659	−	0.811999i	$-0.698380\pi$
$941$	12.6194	−	21.8575i	0.411382	−	0.712534i	0.995041	+	0.0994646i	$0.0317130\pi$
$942$		0			0
$943$	21.6333	+	37.4700i	0.704477	+	1.22019i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.5000	+	23.3827i	0.438691	+	0.759835i	0.997589	−	0.0694014i	$-0.0221089\pi$
$947$	13.5000	+	23.3827i	0.438691	+	0.759835i	−0.558898	+	0.829237i	$0.688776\pi$
$948$		0			0
$949$	6.00000	−	10.3923i	0.194768	−	0.337348i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−43.2666			−1.40154			−0.700772	−	0.713386i	$-0.747161\pi$
$953$	−43.2666			−1.40154			−0.700772	+	0.713386i	$0.747161\pi$
$954$		0			0
$955$	43.2666			1.40007
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−39.0000	+	67.5500i	−1.25938	+	2.18130i
$960$		0			0
$961$	9.00000	+	15.5885i	0.290323	+	0.502853i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	27.0416	+	46.8375i	0.870501	+	1.50775i
$966$		0			0
$967$	−5.40833	+	9.36750i	−0.173920	+	0.301238i	−0.939787	−	0.341761i	$-0.888977\pi$
$967$	−5.40833	+	9.36750i	−0.173920	+	0.301238i	0.765867	+	0.642999i	$0.222310\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−25.0000			−0.802288			−0.401144	−	0.916015i	$-0.631387\pi$
$971$	−25.0000			−0.802288			−0.401144	+	0.916015i	$0.631387\pi$
$972$		0			0
$973$	−52.0000			−1.66704
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−18.0278	+	31.2250i	−0.576759	+	0.998976i	0.419089	+	0.907945i	$0.362350\pi$
$977$	−18.0278	+	31.2250i	−0.576759	+	0.998976i	−0.995848	+	0.0910308i	$0.970984\pi$
$978$		0			0
$979$	3.60555	+	6.24500i	0.115234	+	0.199591i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	0.909894	−	0.414840i	$-0.136162\pi$
$983$	3.00000	+	5.19615i	0.0956851	+	0.165732i	−0.814209	+	0.580572i	$0.802829\pi$
$984$		0			0
$985$	32.5000	−	56.2917i	1.03554	−	1.79360i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−43.2666			−1.37580
$990$		0			0
$991$	−61.2944			−1.94708			−0.973540	−	0.228517i	$-0.926612\pi$
$991$	−61.2944			−1.94708			−0.973540	+	0.228517i	$0.926612\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.50000	+	11.2583i	−0.206064	+	0.356913i
$996$		0			0
$997$	−18.0000	−	31.1769i	−0.570066	−	0.987383i	−0.996559	−	0.0828918i	$-0.973584\pi$
$997$	−18.0000	−	31.1769i	−0.570066	−	0.987383i	0.426493	−	0.904491i	$-0.359749\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.i.bc.1729.1		4
3.2	odd	2		2592.2.i.bb.1729.2		4
4.3	odd	2		2592.2.i.bb.1729.1		4
9.2	odd	6		2592.2.i.bb.865.2		4
9.4	even	3		864.2.a.m.1.2	yes	2
9.5	odd	6		864.2.a.n.1.1	yes	2
9.7	even	3	inner	2592.2.i.bc.865.1		4
12.11	even	2	inner	2592.2.i.bc.1729.2		4
36.7	odd	6		2592.2.i.bb.865.1		4
36.11	even	6	inner	2592.2.i.bc.865.2		4
36.23	even	6		864.2.a.m.1.1	✓	2
36.31	odd	6		864.2.a.n.1.2	yes	2
72.5	odd	6		1728.2.a.bc.1.2		2
72.13	even	6		1728.2.a.bd.1.1		2
72.59	even	6		1728.2.a.bd.1.2		2
72.67	odd	6		1728.2.a.bc.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.a.m.1.1	✓	2	36.23	even	6
864.2.a.m.1.2	yes	2	9.4	even	3
864.2.a.n.1.1	yes	2	9.5	odd	6
864.2.a.n.1.2	yes	2	36.31	odd	6
1728.2.a.bc.1.1		2	72.67	odd	6
1728.2.a.bc.1.2		2	72.5	odd	6
1728.2.a.bd.1.1		2	72.13	even	6
1728.2.a.bd.1.2		2	72.59	even	6
2592.2.i.bb.865.1		4	36.7	odd	6
2592.2.i.bb.865.2		4	9.2	odd	6
2592.2.i.bb.1729.1		4	4.3	odd	2
2592.2.i.bb.1729.2		4	3.2	odd	2
2592.2.i.bc.865.1		4	9.7	even	3	inner
2592.2.i.bc.865.2		4	36.11	even	6	inner
2592.2.i.bc.1729.1		4	1.1	even	1	trivial
2592.2.i.bc.1729.2		4	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.80278 + 3.12250i) q^{5} +(1.80278 + 3.12250i) q^{7} +O(q^{10})\)	\(q+(-1.80278 + 3.12250i) q^{5} +(1.80278 + 3.12250i) q^{7} +(0.500000 + 0.866025i) q^{11} +(-2.00000 + 3.46410i) q^{13} +7.21110 q^{19} +(-3.00000 + 5.19615i) q^{23} +(-4.00000 - 6.92820i) q^{25} +(3.60555 + 6.24500i) q^{29} +(-1.80278 + 3.12250i) q^{31} -13.0000 q^{35} +10.0000 q^{37} +(3.60555 - 6.24500i) q^{41} +(3.60555 + 6.24500i) q^{43} +(-5.00000 - 8.66025i) q^{47} +(-3.00000 + 5.19615i) q^{49} -3.60555 q^{53} -3.60555 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-7.21110 - 12.4900i) q^{65} +(3.60555 - 6.24500i) q^{67} -8.00000 q^{71} -3.00000 q^{73} +(-1.80278 + 3.12250i) q^{77} +(-7.21110 - 12.4900i) q^{79} +(4.50000 + 7.79423i) q^{83} +7.21110 q^{89} -14.4222 q^{91} +(-13.0000 + 22.5167i) q^{95} +(-3.50000 - 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 2 q^{11} - 8 q^{13} - 12 q^{23} - 16 q^{25} - 52 q^{35} + 40 q^{37} - 20 q^{47} - 12 q^{49} - 8 q^{59} - 32 q^{71} - 12 q^{73} + 18 q^{83} - 52 q^{95} - 14 q^{97}+O(q^{100}) \) 4 * q + 2 * q^11 - 8 * q^13 - 12 * q^23 - 16 * q^25 - 52 * q^35 + 40 * q^37 - 20 * q^47 - 12 * q^49 - 8 * q^59 - 32 * q^71 - 12 * q^73 + 18 * q^83 - 52 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more