LMFDB - Embedded newform 2736.2.bm.p.559.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(559,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.bm (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.897122304.10
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.3
Root			$-2.07341 - 1.19709i$ of defining polynomial
Character	$\chi$	$=$	2736.559
Dual form			2736.2.bm.p.1855.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.876327 - 1.51784i) q^{5} -1.00000i q^{7} +O(q^{10})$	$q+(0.876327 - 1.51784i) q^{5} -1.00000i q^{7} -1.75265i q^{11} +(-1.50000 + 0.866025i) q^{13} +(-2.39417 + 4.14682i) q^{17} +(-1.73205 + 4.00000i) q^{19} +(-5.66467 + 3.27050i) q^{23} +(0.964102 + 1.66987i) q^{25} +(-7.18251 + 4.14682i) q^{29} -7.73205 q^{31} +(-1.51784 - 0.876327i) q^{35} +7.73205i q^{37} +(0.401924 + 0.232051i) q^{43} +(8.29365 - 4.78834i) q^{47} +6.00000 q^{49} +(9.81149 - 5.66467i) q^{53} +(-2.66025 - 1.53590i) q^{55} +(5.66467 - 9.81149i) q^{59} +(4.96410 + 8.59808i) q^{61} +3.03569i q^{65} +(-0.401924 - 0.696152i) q^{67} +(-7.18251 + 12.4405i) q^{71} +(-7.69615 + 13.3301i) q^{73} -1.75265 q^{77} +(-1.33013 + 2.30385i) q^{79} +1.28303i q^{83} +(4.19615 + 7.26795i) q^{85} +(4.55353 - 2.62898i) q^{89} +(0.866025 + 1.50000i) q^{91} +(4.55353 + 6.13429i) q^{95} +(5.19615 + 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 12 q^{13} - 20 q^{25} - 48 q^{31} + 24 q^{43} + 48 q^{49} + 48 q^{55} + 12 q^{61} - 24 q^{67} - 20 q^{73} + 24 q^{79} - 8 q^{85}+O(q^{100}) $ 8 * q - 12 * q^13 - 20 * q^25 - 48 * q^31 + 24 * q^43 + 48 * q^49 + 48 * q^55 + 12 * q^61 - 24 * q^67 - 20 * q^73 + 24 * q^79 - 8 * q^85

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.876327	−	1.51784i	0.391905	−	0.678800i	−0.600795	−	0.799403i	$-0.705149\pi$
$5$	0.876327	−	1.51784i	0.391905	−	0.678800i	0.992701	+	0.120603i	$0.0384827\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.75265i		−	0.528445i	−0.964462	−	0.264223i	$-0.914885\pi$
$11$		−	1.75265i		−	0.528445i	0.964462	−	0.264223i	$-0.0851153\pi$
$12$		0			0
$13$	−1.50000	+	0.866025i	−0.416025	+	0.240192i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−1.50000	+	0.866025i	−0.416025	+	0.240192i	0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.39417	+	4.14682i	−0.580672	+	1.00575i	0.414728	+	0.909945i	$0.363877\pi$
$17$	−2.39417	+	4.14682i	−0.580672	+	1.00575i	−0.995400	+	0.0958074i	$0.969457\pi$
$18$		0			0
$19$	−1.73205	+	4.00000i	−0.397360	+	0.917663i
$19$	−1.73205	+	4.00000i	−0.397360	+	0.917663i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−5.66467	+	3.27050i	−1.18116	+	0.681946i	−0.956284	−	0.292440i	$-0.905533\pi$
$23$	−5.66467	+	3.27050i	−1.18116	+	0.681946i	−0.224881	+	0.974386i	$0.572199\pi$
$24$		0			0
$25$	0.964102	+	1.66987i	0.192820	+	0.333975i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.18251	+	4.14682i	−1.33376	+	0.770046i	−0.985874	−	0.167491i	$-0.946433\pi$
$29$	−7.18251	+	4.14682i	−1.33376	+	0.770046i	−0.347885	+	0.937537i	$0.613100\pi$
$30$		0			0
$31$	−7.73205			−1.38872			−0.694359	−	0.719629i	$-0.744312\pi$
$31$	−7.73205			−1.38872			−0.694359	+	0.719629i	$0.744312\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.51784	−	0.876327i	−0.256562	−	0.148126i
$36$		0			0
$37$			7.73205i			1.27114i	0.772043	+	0.635571i	$0.219235\pi$
$37$			7.73205i			1.27114i	−0.772043	+	0.635571i	$0.780765\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	0.401924	+	0.232051i	0.0612928	+	0.0353874i	0.530333	−	0.847789i	$-0.322067\pi$
$43$	0.401924	+	0.232051i	0.0612928	+	0.0353874i	−0.469040	+	0.883177i	$0.655400\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	8.29365	−	4.78834i	1.20975	−	0.698451i	0.247048	−	0.969003i	$-0.420540\pi$
$47$	8.29365	−	4.78834i	1.20975	−	0.698451i	0.962705	+	0.270552i	$0.0872062\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.81149	−	5.66467i	1.34771	−	0.778102i	0.359787	−	0.933034i	$-0.382849\pi$
$53$	9.81149	−	5.66467i	1.34771	−	0.778102i	0.987925	+	0.154933i	$0.0495160\pi$
$54$		0			0
$55$	−2.66025	−	1.53590i	−0.358709	−	0.207100i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.66467	−	9.81149i	0.737477	−	1.27735i	−0.216151	−	0.976360i	$-0.569350\pi$
$59$	5.66467	−	9.81149i	0.737477	−	1.27735i	0.953628	−	0.300988i	$-0.0973163\pi$
$60$		0			0
$61$	4.96410	+	8.59808i	0.635588	+	1.10087i	0.986390	+	0.164421i	$0.0525756\pi$
$61$	4.96410	+	8.59808i	0.635588	+	1.10087i	−0.350802	+	0.936450i	$0.614091\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			3.03569i			0.376531i
$66$		0			0
$67$	−0.401924	−	0.696152i	−0.0491028	−	0.0850486i	0.840429	−	0.541921i	$-0.182303\pi$
$67$	−0.401924	−	0.696152i	−0.0491028	−	0.0850486i	−0.889532	+	0.456873i	$0.848970\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.18251	+	12.4405i	−0.852407	+	1.47641i	0.0266225	+	0.999646i	$0.491525\pi$
$71$	−7.18251	+	12.4405i	−0.852407	+	1.47641i	−0.879030	+	0.476767i	$0.841809\pi$
$72$		0			0
$73$	−7.69615	+	13.3301i	−0.900767	+	1.56017i	−0.0742652	+	0.997239i	$0.523661\pi$
$73$	−7.69615	+	13.3301i	−0.900767	+	1.56017i	−0.826501	+	0.562935i	$0.809672\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.75265			−0.199733
$78$		0			0
$79$	−1.33013	+	2.30385i	−0.149651	+	0.259203i	−0.931098	−	0.364768i	$-0.881148\pi$
$79$	−1.33013	+	2.30385i	−0.149651	+	0.259203i	0.781448	+	0.623971i	$0.214482\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.28303i			0.140831i	0.997518	+	0.0704155i	$0.0224325\pi$
$83$			1.28303i			0.140831i	−0.997518	+	0.0704155i	$0.977567\pi$
$84$		0			0
$85$	4.19615	+	7.26795i	0.455137	+	0.788320i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.55353	−	2.62898i	0.482673	−	0.278671i	−0.238857	−	0.971055i	$-0.576773\pi$
$89$	4.55353	−	2.62898i	0.482673	−	0.278671i	0.721530	+	0.692383i	$0.243439\pi$
$90$		0			0
$91$	0.866025	+	1.50000i	0.0907841	+	0.157243i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.55353	+	6.13429i	0.467182	+	0.629365i
$96$		0			0
$97$	5.19615	+	3.00000i	0.527589	+	0.304604i	0.740034	−	0.672569i	$-0.234809\pi$
$97$	5.19615	+	3.00000i	0.527589	+	0.304604i	−0.212445	+	0.977173i	$0.568143\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.641516	−	1.11114i	−0.0638332	−	0.110562i	0.832343	−	0.554261i	$-0.186999\pi$
$101$	−0.641516	−	1.11114i	−0.0638332	−	0.110562i	−0.896176	+	0.443699i	$0.853666\pi$
$102$		0			0
$103$	−2.66025			−0.262123			−0.131061	−	0.991374i	$-0.541838\pi$
$103$	−2.66025			−0.262123			−0.131061	+	0.991374i	$0.541838\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.22228			−0.214836			−0.107418	−	0.994214i	$-0.534258\pi$
$107$	−2.22228			−0.214836			−0.107418	+	0.994214i	$0.534258\pi$
$108$		0			0
$109$	−0.803848	−	0.464102i	−0.0769946	−	0.0444529i	0.461008	−	0.887396i	$-0.347488\pi$
$109$	−0.803848	−	0.464102i	−0.0769946	−	0.0444529i	−0.538003	+	0.842943i	$0.680821\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.3293i		−	1.06577i	−0.846186	−	0.532887i	$-0.821107\pi$
$113$		−	11.3293i		−	1.06577i	0.846186	−	0.532887i	$-0.178893\pi$
$114$		0			0
$115$			11.4641i			1.06903i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	4.14682	+	2.39417i	0.380139	+	0.219473i
$120$		0			0
$121$	7.92820			0.720746
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1427			1.08608
$126$		0			0
$127$	−6.92820	−	12.0000i	−0.614779	−	1.06483i	−0.990423	−	0.138064i	$-0.955912\pi$
$127$	−6.92820	−	12.0000i	−0.614779	−	1.06483i	0.375645	−	0.926764i	$-0.377421\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.14682	−	2.39417i	−0.362310	−	0.209180i	0.307784	−	0.951456i	$-0.400413\pi$
$131$	−4.14682	−	2.39417i	−0.362310	−	0.209180i	−0.670093	+	0.742277i	$0.733746\pi$
$132$		0			0
$133$	4.00000	+	1.73205i	0.346844	+	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.93516	−	15.4762i	−0.763383	−	1.32222i	−0.941097	−	0.338135i	$-0.890204\pi$
$137$	−8.93516	−	15.4762i	−0.763383	−	1.32222i	0.177715	−	0.984082i	$-0.443130\pi$
$138$		0			0
$139$	−8.13397	+	4.69615i	−0.689915	+	0.398322i	−0.803580	−	0.595197i	$-0.797074\pi$
$139$	−8.13397	+	4.69615i	−0.689915	+	0.398322i	0.113665	+	0.993519i	$0.463741\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.51784	+	2.62898i	0.126928	+	0.219846i
$144$		0			0
$145$			14.5359i			1.20714i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.41732	+	12.8472i	−0.607651	+	1.05248i	0.383976	+	0.923343i	$0.374555\pi$
$149$	−7.41732	+	12.8472i	−0.607651	+	1.05248i	−0.991627	+	0.129139i	$0.958779\pi$
$150$		0			0
$151$	−18.9282			−1.54036			−0.770178	−	0.637829i	$-0.779832\pi$
$151$	−18.9282			−1.54036			−0.770178	+	0.637829i	$0.779832\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−6.77581	+	11.7360i	−0.544246	+	0.942661i
$156$		0			0
$157$	−3.69615	+	6.40192i	−0.294985	+	0.510929i	−0.974981	−	0.222286i	$-0.928648\pi$
$157$	−3.69615	+	6.40192i	−0.294985	+	0.510929i	0.679996	+	0.733216i	$0.261981\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.27050	+	5.66467i	0.257751	+	0.446438i
$162$		0			0
$163$			5.00000i			0.391630i	0.980641	+	0.195815i	$0.0627352\pi$
$163$			5.00000i			0.391630i	−0.980641	+	0.195815i	$0.937265\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.9226	−	18.9186i	−0.845218	−	1.46396i	−0.885431	−	0.464770i	$-0.846137\pi$
$167$	−10.9226	−	18.9186i	−0.845218	−	1.46396i	0.0402128	−	0.999191i	$-0.487196\pi$
$168$		0			0
$169$	−5.00000	+	8.66025i	−0.384615	+	0.666173i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.92455	−	1.11114i	−0.146321	−	0.0844783i	0.425052	−	0.905169i	$-0.360256\pi$
$173$	−1.92455	−	1.11114i	−0.146321	−	0.0844783i	−0.571373	+	0.820691i	$0.693589\pi$
$174$		0			0
$175$	1.66987	−	0.964102i	0.126231	−	0.0728792i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.8453			1.63279			0.816396	−	0.577493i	$-0.195969\pi$
$179$	21.8453			1.63279			0.816396	+	0.577493i	$0.195969\pi$
$180$		0			0
$181$	−1.39230	+	0.803848i	−0.103489	+	0.0597495i	−0.550851	−	0.834603i	$-0.685697\pi$
$181$	−1.39230	+	0.803848i	−0.103489	+	0.0597495i	0.447362	+	0.894353i	$0.352364\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	11.7360	+	6.77581i	0.862851	+	0.498167i
$186$		0			0
$187$	7.26795	+	4.19615i	0.531485	+	0.306853i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	14.8346i		−	1.07340i	−0.843774	−	0.536699i	$-0.819671\pi$
$191$		−	14.8346i		−	1.07340i	0.843774	−	0.536699i	$-0.180329\pi$
$192$		0			0
$193$	8.89230	+	5.13397i	0.640082	+	0.369552i	0.784646	−	0.619944i	$-0.212845\pi$
$193$	8.89230	+	5.13397i	0.640082	+	0.369552i	−0.144564	+	0.989495i	$0.546178\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.54099			−0.466027			−0.233013	−	0.972474i	$-0.574859\pi$
$197$	−6.54099			−0.466027			−0.233013	+	0.972474i	$0.574859\pi$
$198$		0			0
$199$	−5.59808	+	3.23205i	−0.396837	+	0.229114i	−0.685118	−	0.728432i	$-0.740250\pi$
$199$	−5.59808	+	3.23205i	−0.396837	+	0.229114i	0.288281	+	0.957546i	$0.406916\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.14682	+	7.18251i	0.291050	+	0.504113i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.01062	+	3.03569i	0.484935	+	0.209983i
$210$		0			0
$211$	−1.33013	+	2.30385i	−0.0915697	+	0.158603i	−0.908172	−	0.418598i	$-0.862522\pi$
$211$	−1.33013	+	2.30385i	−0.0915697	+	0.158603i	0.816602	+	0.577201i	$0.195855\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.704433	−	0.406705i	0.0480420	−	0.0277370i
$216$		0			0
$217$			7.73205i			0.524886i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	8.29365i		−	0.557891i
$222$		0			0
$223$	−5.13397	+	8.89230i	−0.343796	+	0.595473i	−0.985134	−	0.171786i	$-0.945046\pi$
$223$	−5.13397	+	8.89230i	−0.343796	+	0.595473i	0.641338	+	0.767259i	$0.278380\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.6230			−1.30242			−0.651212	−	0.758896i	$-0.725739\pi$
$227$	−19.6230			−1.30242			−0.651212	+	0.758896i	$0.725739\pi$
$228$		0			0
$229$	8.32051			0.549835			0.274917	−	0.961468i	$-0.411350\pi$
$229$	8.32051			0.549835			0.274917	+	0.961468i	$0.411350\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.9709	+	20.7341i	−0.784237	+	1.35834i	0.145218	+	0.989400i	$0.453612\pi$
$233$	−11.9709	+	20.7341i	−0.784237	+	1.35834i	−0.929454	+	0.368938i	$0.879722\pi$
$234$		0			0
$235$		−	16.7846i		−	1.09491i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			13.8954i			0.898819i	0.893326	+	0.449409i	$0.148366\pi$
$239$			13.8954i			0.898819i	−0.893326	+	0.449409i	$0.851634\pi$
$240$		0			0
$241$	−0.696152	+	0.401924i	−0.0448431	+	0.0258902i	−0.522254	−	0.852790i	$-0.674909\pi$
$241$	−0.696152	+	0.401924i	−0.0448431	+	0.0258902i	0.477411	+	0.878680i	$0.341575\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.25796	−	9.10706i	0.335919	−	0.581829i
$246$		0			0
$247$	−0.866025	−	7.50000i	−0.0551039	−	0.477214i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.07137	−	3.50531i	0.383222	−	0.221253i	−0.295997	−	0.955189i	$-0.595652\pi$
$251$	6.07137	−	3.50531i	0.383222	−	0.221253i	0.679219	+	0.733936i	$0.262319\pi$
$252$		0			0
$253$	5.73205	+	9.92820i	0.360371	+	0.624181i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.704433	−	0.406705i	0.0439413	−	0.0253695i	−0.477868	−	0.878431i	$-0.658590\pi$
$257$	0.704433	−	0.406705i	0.0439413	−	0.0253695i	0.521810	+	0.853062i	$0.325257\pi$
$258$		0			0
$259$	7.73205			0.480446
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.36910	+	3.67720i	0.392736	+	0.226746i	0.683345	−	0.730096i	$-0.260525\pi$
$263$	6.36910	+	3.67720i	0.392736	+	0.226746i	−0.290609	+	0.956842i	$0.593858\pi$
$264$		0			0
$265$		−	19.8564i		−	1.21977i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−15.0695	−	8.70035i	−0.918801	−	0.530470i	−0.0355485	−	0.999368i	$-0.511318\pi$
$269$	−15.0695	−	8.70035i	−0.918801	−	0.530470i	−0.883252	+	0.468898i	$0.844651\pi$
$270$		0			0
$271$	−19.3923	−	11.1962i	−1.17800	−	0.680118i	−0.222448	−	0.974945i	$-0.571405\pi$
$271$	−19.3923	−	11.1962i	−1.17800	−	0.680118i	−0.955551	+	0.294827i	$0.904738\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.92671	−	1.68974i	0.176487	−	0.101895i
$276$		0			0
$277$	−14.5359			−0.873377			−0.436689	−	0.899613i	$-0.643849\pi$
$277$	−14.5359			−0.873377			−0.436689	+	0.899613i	$0.643849\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.47808	+	3.74012i	−0.386450	+	0.223117i	−0.680621	−	0.732636i	$-0.738290\pi$
$281$	−6.47808	+	3.74012i	−0.386450	+	0.223117i	0.294171	+	0.955753i	$0.404956\pi$
$282$		0			0
$283$	21.5885	+	12.4641i	1.28330	+	0.740914i	0.977450	−	0.211166i	$-0.0677260\pi$
$283$	21.5885	+	12.4641i	1.28330	+	0.740914i	0.305850	+	0.952080i	$0.401059\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.96410	−	5.13397i	−0.174359	−	0.301999i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			6.07137i			0.354693i	0.984148	+	0.177347i	$0.0567514\pi$
$293$			6.07137i			0.354693i	−0.984148	+	0.177347i	$0.943249\pi$
$294$		0			0
$295$	−9.92820	−	17.1962i	−0.578042	−	1.00120i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.66467	−	9.81149i	0.327596	−	0.567413i
$300$		0			0
$301$	0.232051	−	0.401924i	0.0133752	−	0.0231665i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	17.4007			0.996362
$306$		0			0
$307$	−13.7321	+	23.7846i	−0.783730	+	1.35746i	0.146026	+	0.989281i	$0.453352\pi$
$307$	−13.7321	+	23.7846i	−0.783730	+	1.35746i	−0.929755	+	0.368179i	$0.879982\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			18.6837i			1.05946i	0.848167	+	0.529729i	$0.177706\pi$
$311$			18.6837i			1.05946i	−0.848167	+	0.529729i	$0.822294\pi$
$312$		0			0
$313$	−2.19615	−	3.80385i	−0.124134	−	0.215006i	0.797260	−	0.603636i	$-0.206282\pi$
$313$	−2.19615	−	3.80385i	−0.124134	−	0.215006i	−0.921394	+	0.388630i	$0.872949\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.7360	+	6.77581i	−0.659162	+	0.380567i	−0.791957	−	0.610576i	$-0.790938\pi$
$317$	−11.7360	+	6.77581i	−0.659162	+	0.380567i	0.132796	+	0.991143i	$0.457605\pi$
$318$		0			0
$319$	7.26795	+	12.5885i	0.406927	+	0.704818i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.4405	−	16.7592i	−0.692206	−	0.932506i
$324$		0			0
$325$	−2.89230	−	1.66987i	−0.160436	−	0.0926279i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.78834	−	8.29365i	−0.263990	−	0.457244i
$330$		0			0
$331$	25.9808			1.42803			0.714016	−	0.700129i	$-0.246874\pi$
$331$	25.9808			1.42803			0.714016	+	0.700129i	$0.246874\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.40887			−0.0769746
$336$		0			0
$337$	−5.08846	−	2.93782i	−0.277186	−	0.160033i	0.354963	−	0.934880i	$-0.384494\pi$
$337$	−5.08846	−	2.93782i	−0.277186	−	0.160033i	−0.632149	+	0.774847i	$0.717827\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			13.5516i			0.733861i
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.8472	−	7.41732i	−0.689673	−	0.398183i	0.113817	−	0.993502i	$-0.463692\pi$
$347$	−12.8472	−	7.41732i	−0.689673	−	0.398183i	−0.803489	+	0.595319i	$0.797026\pi$
$348$		0			0
$349$	23.0000			1.23116			0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000			1.23116			0.615581	+	0.788074i	$0.288921\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−31.4219			−1.67242			−0.836211	−	0.548408i	$-0.815234\pi$
$353$	−31.4219			−1.67242			−0.836211	+	0.548408i	$0.815234\pi$
$354$		0			0
$355$	12.5885	+	21.8038i	0.668126	+	1.15723i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.6944	−	14.8346i	−1.35610	−	0.782943i	−0.367001	−	0.930221i	$-0.619615\pi$
$359$	−25.6944	−	14.8346i	−1.35610	−	0.782943i	−0.989095	+	0.147278i	$0.952949\pi$
$360$		0			0
$361$	−13.0000	−	13.8564i	−0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.4887	+	23.3631i	0.706031	+	1.22288i
$366$		0			0
$367$	−3.99038	+	2.30385i	−0.208296	+	0.120260i	−0.600519	−	0.799610i	$-0.705039\pi$
$367$	−3.99038	+	2.30385i	−0.208296	+	0.120260i	0.392223	+	0.919870i	$0.371706\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5.66467	−	9.81149i	−0.294095	−	0.509387i
$372$		0			0
$373$		−	27.4641i		−	1.42204i	−0.703173	−	0.711019i	$-0.748234\pi$
$373$		−	27.4641i		−	1.42204i	0.703173	−	0.711019i	$-0.251766\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.18251	−	12.4405i	0.369918	−	0.640717i
$378$		0			0
$379$	9.58846			0.492526			0.246263	−	0.969203i	$-0.420797\pi$
$379$	9.58846			0.492526			0.246263	+	0.969203i	$0.420797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.58922	−	13.1449i	0.387791	−	0.671673i	−0.604361	−	0.796710i	$-0.706572\pi$
$383$	7.58922	−	13.1449i	0.387791	−	0.671673i	0.992152	+	0.125037i	$0.0399050\pi$
$384$		0			0
$385$	−1.53590	+	2.66025i	−0.0782766	+	0.135579i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.2811	+	17.8074i	0.521273	+	0.902872i	0.999694	+	0.0247409i	$0.00787607\pi$
$389$	10.2811	+	17.8074i	0.521273	+	0.902872i	−0.478421	+	0.878131i	$0.658791\pi$
$390$		0			0
$391$		−	31.3205i		−	1.58395i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.33125	+	4.03785i	0.117298	+	0.203166i
$396$		0			0
$397$	−7.89230	+	13.6699i	−0.396103	+	0.686071i	−0.993241	−	0.116067i	$-0.962971\pi$
$397$	−7.89230	+	13.6699i	−0.396103	+	0.686071i	0.597138	+	0.802139i	$0.296304\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0695	+	8.70035i	0.752533	+	0.434475i	0.826608	−	0.562778i	$-0.190267\pi$
$401$	15.0695	+	8.70035i	0.752533	+	0.434475i	−0.0740756	+	0.997253i	$0.523601\pi$
$402$		0			0
$403$	11.5981	−	6.69615i	0.577741	−	0.333559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.5516			0.671728
$408$		0			0
$409$	19.9808	−	11.5359i	0.987985	−	0.570413i	0.0833137	−	0.996523i	$-0.473450\pi$
$409$	19.9808	−	11.5359i	0.987985	−	0.570413i	0.904671	+	0.426110i	$0.140116\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.81149	−	5.66467i	−0.482792	−	0.278740i
$414$		0			0
$415$	1.94744	+	1.12436i	0.0955961	+	0.0551924i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			31.4219i			1.53506i	0.641012	+	0.767531i	$0.278515\pi$
$419$			31.4219i			1.53506i	−0.641012	+	0.767531i	$0.721485\pi$
$420$		0			0
$421$	−9.00000	−	5.19615i	−0.438633	−	0.253245i	0.264385	−	0.964417i	$-0.414831\pi$
$421$	−9.00000	−	5.19615i	−0.438633	−	0.253245i	−0.703018	+	0.711172i	$0.748165\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.23289			−0.447861
$426$		0			0
$427$	8.59808	−	4.96410i	0.416090	−	0.240230i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.29365	+	14.3650i	0.399491	+	0.691939i	0.993663	−	0.112399i	$-0.0358535\pi$
$431$	8.29365	+	14.3650i	0.399491	+	0.691939i	−0.594172	+	0.804338i	$0.702520\pi$
$432$		0			0
$433$	−19.5000	+	11.2583i	−0.937110	+	0.541041i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−19.5000	+	11.2583i	−0.937110	+	0.541041i	−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.27050	−	28.3233i	−0.156449	−	1.35489i
$438$		0			0
$439$	13.7942	−	23.8923i	0.658363	−	1.14032i	−0.322677	−	0.946509i	$-0.604583\pi$
$439$	13.7942	−	23.8923i	0.658363	−	1.14032i	0.981039	−	0.193808i	$-0.0620840\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.14682	−	2.39417i	0.197022	−	0.113750i	−0.398244	−	0.917280i	$-0.630380\pi$
$443$	4.14682	−	2.39417i	0.197022	−	0.113750i	0.595265	+	0.803529i	$0.297047\pi$
$444$		0			0
$445$		−	9.21539i		−	0.436851i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	37.0237i		−	1.74726i	−0.486594	−	0.873628i	$-0.661761\pi$
$449$		−	37.0237i		−	1.74726i	0.486594	−	0.873628i	$-0.338239\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.03569			0.142315
$456$		0			0
$457$	−3.39230			−0.158685			−0.0793427	−	0.996847i	$-0.525282\pi$
$457$	−3.39230			−0.158685			−0.0793427	+	0.996847i	$0.525282\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	19.3882	−	33.5813i	0.902997	−	1.56404i	0.0794139	−	0.996842i	$-0.474695\pi$
$461$	19.3882	−	33.5813i	0.902997	−	1.56404i	0.823583	−	0.567195i	$-0.191972\pi$
$462$		0			0
$463$		−	14.3205i		−	0.665530i	−0.943010	−	0.332765i	$-0.892018\pi$
$463$		−	14.3205i		−	0.665530i	0.943010	−	0.332765i	$-0.107982\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			32.2354i			1.49167i	0.666128	+	0.745837i	$0.267950\pi$
$467$			32.2354i			1.49167i	−0.666128	+	0.745837i	$0.732050\pi$
$468$		0			0
$469$	−0.696152	+	0.401924i	−0.0321453	+	0.0185591i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.406705	−	0.704433i	0.0187003	−	0.0323899i
$474$		0			0
$475$	−8.34936	+	0.964102i	−0.383095	+	0.0442360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.6627	+	8.46554i	−0.669958	+	0.386801i	−0.796061	−	0.605217i	$-0.793086\pi$
$479$	−14.6627	+	8.46554i	−0.669958	+	0.386801i	0.126103	+	0.992017i	$0.459753\pi$
$480$		0			0
$481$	−6.69615	−	11.5981i	−0.305318	−	0.528827i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.10706	−	5.25796i	0.413530	−	0.238752i
$486$		0			0
$487$	−16.3923			−0.742806			−0.371403	−	0.928472i	$-0.621123\pi$
$487$	−16.3923			−0.742806			−0.371403	+	0.928472i	$0.621123\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	35.0991	+	20.2645i	1.58400	+	0.914524i	0.994267	+	0.106927i	$0.0341013\pi$
$491$	35.0991	+	20.2645i	1.58400	+	0.914524i	0.589735	+	0.807597i	$0.299232\pi$
$492$		0			0
$493$		−	39.7128i		−	1.78858i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.4405	+	7.18251i	0.558031	+	0.322180i
$498$		0			0
$499$	30.3109	+	17.5000i	1.35690	+	0.783408i	0.989205	−	0.146538i	$-0.0468131\pi$
$499$	30.3109	+	17.5000i	1.35690	+	0.783408i	0.367697	+	0.929946i	$0.380146\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14.6627	−	8.46554i	0.653780	−	0.377460i	−0.136123	−	0.990692i	$-0.543464\pi$
$503$	14.6627	−	8.46554i	0.653780	−	0.377460i	0.789903	+	0.613232i	$0.210131\pi$
$504$		0			0
$505$	−2.24871			−0.100066
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.25796	−	3.03569i	0.233055	−	0.134554i	−0.378926	−	0.925427i	$-0.623706\pi$
$509$	5.25796	−	3.03569i	0.233055	−	0.134554i	0.611981	+	0.790873i	$0.290373\pi$
$510$		0			0
$511$	13.3301	+	7.69615i	0.589690	+	0.340458i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.33125	+	4.03785i	−0.102727	+	0.177929i
$516$		0			0
$517$	−8.39230	−	14.5359i	−0.369093	−	0.639288i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15.1784i			0.664979i	0.943107	+	0.332490i	$0.107889\pi$
$521$			15.1784i			0.664979i	−0.943107	+	0.332490i	$0.892111\pi$
$522$		0			0
$523$	−0.866025	−	1.50000i	−0.0378686	−	0.0655904i	0.846470	−	0.532437i	$-0.178724\pi$
$523$	−0.866025	−	1.50000i	−0.0378686	−	0.0655904i	−0.884339	+	0.466846i	$0.845390\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.5118	−	32.0635i	0.806389	−	1.39671i
$528$		0			0
$529$	9.89230	−	17.1340i	0.430100	−	0.744955i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.94744	+	3.37307i	−0.0841952	+	0.145830i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	10.5159i		−	0.452953i
$540$		0			0
$541$	−12.8923	−	22.3301i	−0.554283	−	0.960047i	−0.997959	−	0.0638596i	$-0.979659\pi$
$541$	−12.8923	−	22.3301i	−0.554283	−	0.960047i	0.443675	−	0.896188i	$-0.353674\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.40887	+	0.813410i	−0.0603492	+	0.0348426i
$546$		0			0
$547$	13.7942	+	23.8923i	0.589799	+	1.02156i	0.994258	+	0.107005i	$0.0341262\pi$
$547$	13.7942	+	23.8923i	0.589799	+	1.02156i	−0.404460	+	0.914556i	$0.632540\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.14682	−	35.9126i	−0.176661	−	1.52993i
$552$		0			0
$553$	2.30385	+	1.33013i	0.0979696	+	0.0565628i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.9458	+	27.6189i	0.675644	+	1.17025i	0.976280	+	0.216511i	$0.0694679\pi$
$557$	15.9458	+	27.6189i	0.675644	+	1.17025i	−0.300636	+	0.953739i	$0.597199\pi$
$558$		0			0
$559$	−0.803848			−0.0339991
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−30.9523			−1.30448			−0.652242	−	0.758011i	$-0.726172\pi$
$563$	−30.9523			−1.30448			−0.652242	+	0.758011i	$0.726172\pi$
$564$		0			0
$565$	−17.1962	−	9.92820i	−0.723448	−	0.417683i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			16.5873i			0.695376i	0.937610	+	0.347688i	$0.113033\pi$
$569$			16.5873i			0.695376i	−0.937610	+	0.347688i	$0.886967\pi$
$570$		0			0
$571$			23.7846i			0.995355i	0.867362	+	0.497677i	$0.165814\pi$
$571$			23.7846i			0.995355i	−0.867362	+	0.497677i	$0.834186\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.9226	−	6.30618i	−0.455505	−	0.262986i
$576$		0			0
$577$	−11.3205			−0.471279			−0.235639	−	0.971841i	$-0.575718\pi$
$577$	−11.3205			−0.471279			−0.235639	+	0.971841i	$0.575718\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.28303			0.0532291
$582$		0			0
$583$	−9.92820	−	17.1962i	−0.411184	−	0.712192i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	32.1724	+	18.5748i	1.32790	+	0.766663i	0.984974	−	0.172701i	$-0.0552493\pi$
$587$	32.1724	+	18.5748i	1.32790	+	0.766663i	0.342924	+	0.939363i	$0.388583\pi$
$588$		0			0
$589$	13.3923	−	30.9282i	0.551820	−	1.27437i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.27050	+	5.66467i	0.134303	+	0.232620i	0.925331	−	0.379160i	$-0.123787\pi$
$593$	3.27050	+	5.66467i	0.134303	+	0.232620i	−0.791028	+	0.611780i	$0.790454\pi$
$594$		0			0
$595$	7.26795	−	4.19615i	0.297957	−	0.172025i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	4.55353	+	7.88694i	0.186052	+	0.322252i	0.943931	−	0.330144i	$-0.107097\pi$
$599$	4.55353	+	7.88694i	0.186052	+	0.322252i	−0.757878	+	0.652396i	$0.773764\pi$
$600$		0			0
$601$		−	16.2679i		−	0.663583i	−0.943353	−	0.331792i	$-0.892347\pi$
$601$		−	16.2679i		−	0.663583i	0.943353	−	0.331792i	$-0.107653\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.94770	−	12.0338i	0.282464	−	0.489242i
$606$		0			0
$607$	−38.6603			−1.56917			−0.784586	−	0.620020i	$-0.787124\pi$
$607$	−38.6603			−1.56917			−0.784586	+	0.620020i	$0.787124\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.29365	+	14.3650i	−0.335525	+	0.581147i
$612$		0			0
$613$	−3.46410	+	6.00000i	−0.139914	+	0.242338i	−0.927464	−	0.373913i	$-0.878016\pi$
$613$	−3.46410	+	6.00000i	−0.139914	+	0.242338i	0.787550	+	0.616251i	$0.211349\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.876327	+	1.51784i	0.0352796	+	0.0611061i	0.883126	−	0.469136i	$-0.155434\pi$
$617$	0.876327	+	1.51784i	0.0352796	+	0.0611061i	−0.847846	+	0.530242i	$0.822101\pi$
$618$		0			0
$619$		−	15.3923i		−	0.618669i	−0.950953	−	0.309334i	$-0.899894\pi$
$619$		−	15.3923i		−	0.618669i	0.950953	−	0.309334i	$-0.100106\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.62898	−	4.55353i	−0.105328	−	0.182433i
$624$		0			0
$625$	5.82051	−	10.0814i	0.232820	−	0.403257i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−32.0635	−	18.5118i	−1.27845	−	0.738116i
$630$		0			0
$631$	37.9186	−	21.8923i	1.50952	−	0.871519i	0.509577	−	0.860425i	$-0.329802\pi$
$631$	37.9186	−	21.8923i	1.50952	−	0.871519i	0.999938	−	0.0110936i	$-0.00353126\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.2855			−0.963740
$636$		0			0
$637$	−9.00000	+	5.19615i	−0.356593	+	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−19.6230	−	11.3293i	−0.775061	−	0.447482i	0.0596159	−	0.998221i	$-0.481012\pi$
$641$	−19.6230	−	11.3293i	−0.775061	−	0.447482i	−0.834677	+	0.550740i	$0.814346\pi$
$642$		0			0
$643$	36.7750	+	21.2321i	1.45026	+	0.837310i	0.998496	−	0.0548251i	$-0.0174601\pi$
$643$	36.7750	+	21.2321i	1.45026	+	0.837310i	0.451768	+	0.892135i	$0.350793\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			7.35440i			0.289131i	0.989495	+	0.144566i	$0.0461785\pi$
$647$			7.35440i			0.289131i	−0.989495	+	0.144566i	$0.953821\pi$
$648$		0			0
$649$	−17.1962	−	9.92820i	−0.675008	−	0.389716i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.35440			−0.287800			−0.143900	−	0.989592i	$-0.545964\pi$
$653$	−7.35440			−0.287800			−0.143900	+	0.989592i	$0.545964\pi$
$654$		0			0
$655$	−7.26795	+	4.19615i	−0.283982	+	0.163957i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.8472	−	22.2520i	−0.500455	−	0.866813i	−1.00000		0.000525469i	$-0.999833\pi$
$659$	−12.8472	−	22.2520i	−0.500455	−	0.866813i	0.499545	−	0.866288i	$-0.333501\pi$
$660$		0			0
$661$	20.7846	−	12.0000i	0.808428	−	0.466746i	−0.0379819	−	0.999278i	$-0.512093\pi$
$661$	20.7846	−	12.0000i	0.808428	−	0.466746i	0.846410	+	0.532533i	$0.178760\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.13429	−	4.55353i	0.237878	−	0.176578i
$666$		0			0
$667$	27.1244	−	46.9808i	1.05026	−	1.81910i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	15.0695	−	8.70035i	0.581750	−	0.335873i
$672$		0			0
$673$		−	48.3731i		−	1.86464i	−0.361628	−	0.932322i	$-0.617779\pi$
$673$		−	48.3731i		−	1.86464i	0.361628	−	0.932322i	$-0.382221\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			33.1746i			1.27500i	0.770449	+	0.637502i	$0.220032\pi$
$677$			33.1746i			1.27500i	−0.770449	+	0.637502i	$0.779968\pi$
$678$		0			0
$679$	3.00000	−	5.19615i	0.115129	−	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21.2498			0.813101			0.406551	−	0.913628i	$-0.366732\pi$
$683$	21.2498			0.813101			0.406551	+	0.913628i	$0.366732\pi$
$684$		0			0
$685$	−31.3205			−1.19670
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−9.81149	+	16.9940i	−0.373788	+	0.647420i
$690$		0			0
$691$		−	12.2487i		−	0.465963i	−0.972481	−	0.232981i	$-0.925152\pi$
$691$		−	12.2487i		−	0.465963i	0.972481	−	0.232981i	$-0.0748482\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			16.4615i			0.624419i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.34595	−	2.33125i	0.0508358	−	0.0880502i	−0.839488	−	0.543379i	$-0.817145\pi$
$701$	1.34595	−	2.33125i	0.0508358	−	0.0880502i	0.890324	+	0.455328i	$0.150478\pi$
$702$		0			0
$703$	−30.9282	−	13.3923i	−1.16648	−	0.505100i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.11114	+	0.641516i	−0.0417887	+	0.0241267i
$708$		0			0
$709$	−12.0885	−	20.9378i	−0.453992	−	0.786336i	0.544638	−	0.838671i	$-0.316667\pi$
$709$	−12.0885	−	20.9378i	−0.453992	−	0.786336i	−0.998630	+	0.0523348i	$0.983334\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	43.7995	−	25.2877i	1.64030	−	0.947030i
$714$		0			0
$715$	5.32051			0.198976
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−23.0654	−	13.3168i	−0.860193	−	0.496633i	0.00388370	−	0.999992i	$-0.498764\pi$
$719$	−23.0654	−	13.3168i	−0.860193	−	0.496633i	−0.864077	+	0.503360i	$0.832097\pi$
$720$		0			0
$721$			2.66025i			0.0990730i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13.8493	−	7.99592i	−0.514352	−	0.296961i
$726$		0			0
$727$	13.2058	+	7.62436i	0.489775	+	0.282772i	0.724481	−	0.689295i	$-0.242079\pi$
$727$	13.2058	+	7.62436i	0.489775	+	0.282772i	−0.234706	+	0.972066i	$0.575413\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.92455	+	1.11114i	−0.0711820	+	0.0410969i
$732$		0			0
$733$	36.3923			1.34418			0.672090	−	0.740469i	$-0.265397\pi$
$733$	36.3923			1.34418			0.672090	+	0.740469i	$0.265397\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1.22011	+	0.704433i	−0.0449435	+	0.0259481i
$738$		0			0
$739$	31.3301	+	18.0885i	1.15250	+	0.665395i	0.949495	−	0.313784i	$-0.101597\pi$
$739$	31.3301	+	18.0885i	1.15250	+	0.665395i	0.203003	+	0.979178i	$0.434930\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.51784	−	2.62898i	0.0556843	−	0.0964480i	−0.836840	−	0.547448i	$-0.815599\pi$
$743$	1.51784	−	2.62898i	0.0556843	−	0.0964480i	0.892524	+	0.451000i	$0.148933\pi$
$744$		0			0
$745$	13.0000	+	22.5167i	0.476283	+	0.824947i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			2.22228i			0.0812002i
$750$		0			0
$751$	6.06218	+	10.5000i	0.221212	+	0.383150i	0.955176	−	0.296038i	$-0.0956654\pi$
$751$	6.06218	+	10.5000i	0.221212	+	0.383150i	−0.733964	+	0.679188i	$0.762332\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.5873	+	28.7300i	−0.603674	+	1.04559i
$756$		0			0
$757$	−25.6244	+	44.3827i	−0.931333	+	1.61312i	−0.150287	+	0.988642i	$0.548020\pi$
$757$	−25.6244	+	44.3827i	−0.931333	+	1.61312i	−0.781046	+	0.624474i	$0.785313\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−47.4138			−1.71875			−0.859374	−	0.511347i	$-0.829147\pi$
$761$	−47.4138			−1.71875			−0.859374	+	0.511347i	$0.829147\pi$
$762$		0			0
$763$	−0.464102	+	0.803848i	−0.0168016	+	0.0291012i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			19.6230i			0.708545i
$768$		0			0
$769$	−13.9641	−	24.1865i	−0.503558	−	0.872189i	−0.999992	−	0.00411375i	$-0.998691\pi$
$769$	−13.9641	−	24.1865i	−0.503558	−	0.872189i	0.496433	−	0.868075i	$-0.334643\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.84910	+	2.22228i	−0.138442	+	0.0799297i	−0.567621	−	0.823290i	$-0.692136\pi$
$773$	−3.84910	+	2.22228i	−0.138442	+	0.0799297i	0.429179	+	0.903219i	$0.358803\pi$
$774$		0			0
$775$	−7.45448	−	12.9115i	−0.267773	−	0.463796i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	21.8038	+	12.5885i	0.780203	+	0.450450i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	6.47808	+	11.2204i	0.231212	+	0.400472i
$786$		0			0
$787$	14.4115			0.513716			0.256858	−	0.966449i	$-0.417313\pi$
$787$	14.4115			0.513716			0.256858	+	0.966449i	$0.417313\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−11.3293			−0.402825
$792$		0			0
$793$	−14.8923	−	8.59808i	−0.528841	−	0.305327i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	24.8809i		−	0.881328i	−0.897672	−	0.440664i	$-0.854743\pi$
$797$		−	24.8809i		−	0.881328i	0.897672	−	0.440664i	$-0.145257\pi$
$798$		0			0
$799$			45.8564i			1.62228i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.3631	+	13.4887i	0.824466	+	0.476006i
$804$		0			0
$805$	11.4641			0.404056
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.19721			0.217882			0.108941	−	0.994048i	$-0.465254\pi$
$809$	6.19721			0.217882			0.108941	+	0.994048i	$0.465254\pi$
$810$		0			0
$811$	5.07180	+	8.78461i	0.178095	+	0.308469i	0.941228	−	0.337772i	$-0.109673\pi$
$811$	5.07180	+	8.78461i	0.178095	+	0.308469i	−0.763133	+	0.646241i	$0.776340\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7.58922	+	4.38164i	0.265839	+	0.153482i
$816$		0			0
$817$	−1.62436	+	1.20577i	−0.0568290	+	0.0421846i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.6878	−	18.5118i	−0.373007	−	0.646068i	0.617019	−	0.786948i	$-0.288340\pi$
$821$	−10.6878	−	18.5118i	−0.373007	−	0.646068i	−0.990027	+	0.140880i	$0.955007\pi$
$822$		0			0
$823$	18.7128	−	10.8038i	0.652288	−	0.376598i	−0.137044	−	0.990565i	$-0.543760\pi$
$823$	18.7128	−	10.8038i	0.652288	−	0.376598i	0.789332	+	0.613966i	$0.210427\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.5832	−	42.5794i	−0.854842	−	1.48063i	−0.876791	−	0.480871i	$-0.840320\pi$
$827$	−24.5832	−	42.5794i	−0.854842	−	1.48063i	0.0219491	−	0.999759i	$-0.493013\pi$
$828$		0			0
$829$			7.73205i			0.268545i	0.990944	+	0.134273i	$0.0428698\pi$
$829$			7.73205i			0.268545i	−0.990944	+	0.134273i	$0.957130\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.3650	+	24.8809i	−0.497718	+	0.862074i
$834$		0			0
$835$	−38.2872			−1.32498
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.1408	+	36.6170i	−0.729862	+	1.26416i	0.227079	+	0.973876i	$0.427083\pi$
$839$	−21.1408	+	36.6170i	−0.729862	+	1.26416i	−0.956941	+	0.290282i	$0.906251\pi$
$840$		0			0
$841$	19.8923	−	34.4545i	0.685942	−	1.18809i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.76327	+	15.1784i	0.301466	+	0.522154i
$846$		0			0
$847$		−	7.92820i		−	0.272416i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−25.2877	−	43.7995i	−0.866850	−	1.50143i
$852$		0			0
$853$	3.89230	−	6.74167i	0.133270	−	0.230830i	−0.791665	−	0.610955i	$-0.790786\pi$
$853$	3.89230	−	6.74167i	0.133270	−	0.230830i	0.924935	+	0.380125i	$0.124119\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	33.9880	+	19.6230i	1.16101	+	0.670308i	0.951545	−	0.307508i	$-0.0994952\pi$
$857$	33.9880	+	19.6230i	1.16101	+	0.670308i	0.209463	+	0.977817i	$0.432829\pi$
$858$		0			0
$859$	8.13397	−	4.69615i	0.277528	−	0.160231i	−0.354776	−	0.934951i	$-0.615443\pi$
$859$	8.13397	−	4.69615i	0.277528	−	0.160231i	0.632304	+	0.774721i	$0.282110\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.5159			0.357966			0.178983	−	0.983852i	$-0.442719\pi$
$863$	10.5159			0.357966			0.178983	+	0.983852i	$0.442719\pi$
$864$		0			0
$865$	−3.37307	+	1.94744i	−0.114688	+	0.0662150i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.03785	+	2.33125i	0.136975	+	0.0790823i
$870$		0			0
$871$	1.20577	+	0.696152i	0.0408560	+	0.0235882i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	12.1427i		−	0.410500i
$876$		0			0
$877$	−41.8923	−	24.1865i	−1.41460	−	0.816721i	−0.418785	−	0.908085i	$-0.637544\pi$
$877$	−41.8923	−	24.1865i	−1.41460	−	0.816721i	−0.995818	+	0.0913642i	$0.970877\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.2686			−0.413339			−0.206669	−	0.978411i	$-0.566262\pi$
$881$	−12.2686			−0.413339			−0.206669	+	0.978411i	$0.566262\pi$
$882$		0			0
$883$	−40.3301	+	23.2846i	−1.35722	+	0.783590i	−0.989248	−	0.146249i	$-0.953280\pi$
$883$	−40.3301	+	23.2846i	−1.35722	+	0.783590i	−0.367969	+	0.929838i	$0.619947\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.66467	−	9.81149i	−0.190201	−	0.329438i	0.755116	−	0.655591i	$-0.227581\pi$
$887$	−5.66467	−	9.81149i	−0.190201	−	0.329438i	−0.945317	+	0.326154i	$0.894247\pi$
$888$		0			0
$889$	−12.0000	+	6.92820i	−0.402467	+	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.78834	+	41.4682i	0.160236	+	1.38768i
$894$		0			0
$895$	19.1436	−	33.1577i	0.639900	−	1.10834i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	55.5355	−	32.0635i	1.85221	−	1.06938i
$900$		0			0
$901$			54.2487i			1.80729i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			2.81773i			0.0936646i
$906$		0			0
$907$	6.58846	−	11.4115i	0.218766	−	0.378914i	−0.735665	−	0.677346i	$-0.763130\pi$
$907$	6.58846	−	11.4115i	0.218766	−	0.378914i	0.954431	+	0.298432i	$0.0964635\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	10.5159			0.348408			0.174204	−	0.984710i	$-0.444265\pi$
$911$	10.5159			0.348408			0.174204	+	0.984710i	$0.444265\pi$
$912$		0			0
$913$	2.24871			0.0744215
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.39417	+	4.14682i	−0.0790625	+	0.136940i
$918$		0			0
$919$		−	54.1769i		−	1.78713i	−0.448932	−	0.893566i	$-0.648196\pi$
$919$		−	54.1769i		−	1.78713i	0.448932	−	0.893566i	$-0.351804\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	24.8809i		−	0.818966i
$924$		0			0
$925$	−12.9115	+	7.45448i	−0.424529	+	0.245102i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.34595	+	2.33125i	−0.0441592	+	0.0764859i	−0.887260	−	0.461269i	$-0.847394\pi$
$929$	−1.34595	+	2.33125i	−0.0441592	+	0.0764859i	0.843101	+	0.537755i	$0.180728\pi$
$930$		0			0
$931$	−10.3923	+	24.0000i	−0.340594	+	0.786568i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.7382	−	7.35440i	0.416584	−	0.240515i
$936$		0			0
$937$	−0.303848	−	0.526279i	−0.00992627	−	0.0171928i	0.861020	−	0.508572i	$-0.169826\pi$
$937$	−0.303848	−	0.526279i	−0.00992627	−	0.0171928i	−0.870946	+	0.491379i	$0.836493\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−24.8809	+	14.3650i	−0.811096	+	0.468286i	−0.847336	−	0.531057i	$-0.821795\pi$
$941$	−24.8809	+	14.3650i	−0.811096	+	0.468286i	0.0362406	+	0.999343i	$0.488462\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.99592	−	4.61645i	−0.259832	−	0.150014i	0.364426	−	0.931232i	$-0.381265\pi$
$947$	−7.99592	−	4.61645i	−0.259832	−	0.150014i	−0.624258	+	0.781218i	$0.714599\pi$
$948$		0			0
$949$		−	26.6603i		−	0.865428i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.22011	−	0.704433i	−0.0395234	−	0.0228188i	0.480108	−	0.877209i	$-0.340597\pi$
$953$	−1.22011	−	0.704433i	−0.0395234	−	0.0228188i	−0.519632	+	0.854390i	$0.673931\pi$
$954$		0			0
$955$	−22.5167	−	13.0000i	−0.728622	−	0.420670i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.4762	+	8.93516i	−0.499751	+	0.288532i
$960$		0			0
$961$	28.7846			0.928536
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15.5851	−	8.99808i	0.501703	−	0.289659i
$966$		0			0
$967$	7.08142	+	4.08846i	0.227723	+	0.131476i	0.609521	−	0.792770i	$-0.291362\pi$
$967$	7.08142	+	4.08846i	0.227723	+	0.131476i	−0.381798	+	0.924246i	$0.624695\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.25796	+	9.10706i	−0.168736	+	0.292259i	−0.937976	−	0.346701i	$-0.887302\pi$
$971$	−5.25796	+	9.10706i	−0.168736	+	0.292259i	0.769240	+	0.638960i	$0.220635\pi$
$972$		0			0
$973$	4.69615	+	8.13397i	0.150552	+	0.260763i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	43.6905i		−	1.39778i	−0.715227	−	0.698892i	$-0.753677\pi$
$977$		−	43.6905i		−	1.39778i	0.715227	−	0.698892i	$-0.246323\pi$
$978$		0			0
$979$	−4.60770	−	7.98076i	−0.147263	−	0.255066i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.55353	+	7.88694i	−0.145235	+	0.251554i	−0.929461	−	0.368921i	$-0.879727\pi$
$983$	−4.55353	+	7.88694i	−0.145235	+	0.251554i	0.784226	+	0.620476i	$0.213060\pi$
$984$		0			0
$985$	−5.73205	+	9.92820i	−0.182638	+	0.316339i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.03569			−0.0965292
$990$		0			0
$991$	−18.9904	+	32.8923i	−0.603249	+	1.04486i	0.389076	+	0.921206i	$0.372794\pi$
$991$	−18.9904	+	32.8923i	−0.603249	+	1.04486i	−0.992326	+	0.123653i	$0.960539\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			11.3293i			0.359164i
$996$		0			0
$997$	4.16025	+	7.20577i	0.131757	+	0.228209i	0.924354	−	0.381537i	$-0.124605\pi$
$997$	4.16025	+	7.20577i	0.131757	+	0.228209i	−0.792597	+	0.609746i	$0.791272\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.bm.p.559.3	yes	8
3.2	odd	2	inner	2736.2.bm.p.559.2	✓	8
4.3	odd	2		2736.2.bm.q.559.3	yes	8
12.11	even	2		2736.2.bm.q.559.2	yes	8
19.12	odd	6		2736.2.bm.q.1855.3	yes	8
57.50	even	6		2736.2.bm.q.1855.2	yes	8
76.31	even	6	inner	2736.2.bm.p.1855.3	yes	8
228.107	odd	6	inner	2736.2.bm.p.1855.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.bm.p.559.2	✓	8	3.2	odd	2	inner
2736.2.bm.p.559.3	yes	8	1.1	even	1	trivial
2736.2.bm.p.1855.2	yes	8	228.107	odd	6	inner
2736.2.bm.p.1855.3	yes	8	76.31	even	6	inner
2736.2.bm.q.559.2	yes	8	12.11	even	2
2736.2.bm.q.559.3	yes	8	4.3	odd	2
2736.2.bm.q.1855.2	yes	8	57.50	even	6
2736.2.bm.q.1855.3	yes	8	19.12	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.876327 - 1.51784i) q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+(0.876327 - 1.51784i) q^{5} -1.00000i q^{7} -1.75265i q^{11} +(-1.50000 + 0.866025i) q^{13} +(-2.39417 + 4.14682i) q^{17} +(-1.73205 + 4.00000i) q^{19} +(-5.66467 + 3.27050i) q^{23} +(0.964102 + 1.66987i) q^{25} +(-7.18251 + 4.14682i) q^{29} -7.73205 q^{31} +(-1.51784 - 0.876327i) q^{35} +7.73205i q^{37} +(0.401924 + 0.232051i) q^{43} +(8.29365 - 4.78834i) q^{47} +6.00000 q^{49} +(9.81149 - 5.66467i) q^{53} +(-2.66025 - 1.53590i) q^{55} +(5.66467 - 9.81149i) q^{59} +(4.96410 + 8.59808i) q^{61} +3.03569i q^{65} +(-0.401924 - 0.696152i) q^{67} +(-7.18251 + 12.4405i) q^{71} +(-7.69615 + 13.3301i) q^{73} -1.75265 q^{77} +(-1.33013 + 2.30385i) q^{79} +1.28303i q^{83} +(4.19615 + 7.26795i) q^{85} +(4.55353 - 2.62898i) q^{89} +(0.866025 + 1.50000i) q^{91} +(4.55353 + 6.13429i) q^{95} +(5.19615 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 12 q^{13} - 20 q^{25} - 48 q^{31} + 24 q^{43} + 48 q^{49} + 48 q^{55} + 12 q^{61} - 24 q^{67} - 20 q^{73} + 24 q^{79} - 8 q^{85}+O(q^{100}) \) 8 * q - 12 * q^13 - 20 * q^25 - 48 * q^31 + 24 * q^43 + 48 * q^49 + 48 * q^55 + 12 * q^61 - 24 * q^67 - 20 * q^73 + 24 * q^79 - 8 * q^85

Introduction

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