LMFDB - Embedded newform 2736.2.s.j.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.577
Dual form			2736.2.s.j.1873.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{7} +O(q^{10})$	$q-1.00000 q^{7} -2.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(-2.00000 - 3.46410i) q^{17} +(4.00000 + 1.73205i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} +(-4.00000 + 6.92820i) q^{29} +3.00000 q^{31} +3.00000 q^{37} +(-6.00000 - 10.3923i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(3.00000 - 5.19615i) q^{47} -6.00000 q^{49} +(2.00000 - 3.46410i) q^{53} +(-5.00000 - 8.66025i) q^{59} +(6.50000 - 11.2583i) q^{61} +(5.50000 - 9.52628i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(5.50000 + 9.52628i) q^{73} +2.00000 q^{77} +(0.500000 + 0.866025i) q^{79} +(-3.00000 + 5.19615i) q^{89} +(2.50000 - 4.33013i) q^{91} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^7	$ 2 q - 2 q^{7} - 4 q^{11} - 5 q^{13} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 5 q^{25} - 8 q^{29} + 6 q^{31} + 6 q^{37} - 12 q^{41} - q^{43} + 6 q^{47} - 12 q^{49} + 4 q^{53} - 10 q^{59} + 13 q^{61} + 11 q^{67} - 6 q^{71} + 11 q^{73} + 4 q^{77} + q^{79} - 6 q^{89} + 5 q^{91} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 4 * q^11 - 5 * q^13 - 4 * q^17 + 8 * q^19 + 4 * q^23 + 5 * q^25 - 8 * q^29 + 6 * q^31 + 6 * q^37 - 12 * q^41 - q^43 + 6 * q^47 - 12 * q^49 + 4 * q^53 - 10 * q^59 + 13 * q^61 + 11 * q^67 - 6 * q^71 + 11 * q^73 + 4 * q^77 + q^79 - 6 * q^89 + 5 * q^91 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−1.00000			−0.377964			−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000			−0.377964			−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	0.277350	+	0.960769i	$0.410544\pi$
$13$	−2.50000	+	4.33013i	−0.693375	+	1.20096i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000	−	3.46410i	−0.485071	−	0.840168i	0.514782	−	0.857321i	$-0.327873\pi$
$17$	−2.00000	−	3.46410i	−0.485071	−	0.840168i	−0.999853	+	0.0171533i	$0.994540\pi$
$18$		0			0
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000	+	6.92820i	−0.742781	+	1.28654i	0.208443	+	0.978035i	$0.433160\pi$
$29$	−4.00000	+	6.92820i	−0.742781	+	1.28654i	−0.951224	+	0.308500i	$0.900173\pi$
$30$		0			0
$31$	3.00000			0.538816			0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000			0.538816			0.269408	+	0.963026i	$0.413172\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000			0.493197			0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000			0.493197			0.246598	+	0.969118i	$0.420687\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.00000	−	10.3923i	−0.937043	−	1.62301i	−0.770950	−	0.636895i	$-0.780218\pi$
$41$	−6.00000	−	10.3923i	−0.937043	−	1.62301i	−0.166092	−	0.986110i	$-0.553115\pi$
$42$		0			0
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	0.825380	−	0.564578i	$-0.190961\pi$
$43$	−0.500000	−	0.866025i	−0.0762493	−	0.132068i	−0.901629	+	0.432511i	$0.857628\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	−0.559908	−	0.828554i	$-0.689164\pi$
$47$	3.00000	−	5.19615i	0.437595	−	0.757937i	0.997503	+	0.0706177i	$0.0224970\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.00000	−	3.46410i	0.274721	−	0.475831i	−0.695344	−	0.718677i	$-0.744748\pi$
$53$	2.00000	−	3.46410i	0.274721	−	0.475831i	0.970065	+	0.242846i	$0.0780811\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	−0.982894	−	0.184172i	$-0.941040\pi$
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	0.331949	−	0.943297i	$-0.392294\pi$
$60$		0			0
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.50000	−	11.2583i	0.832240	−	1.44148i	0.896258	−	0.443533i	$-0.146275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00000	−	5.19615i	−0.356034	−	0.616670i	0.631260	−	0.775571i	$-0.282538\pi$
$71$	−3.00000	−	5.19615i	−0.356034	−	0.616670i	−0.987294	+	0.158901i	$0.949205\pi$
$72$		0			0
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	0.984594	+	0.174855i	$0.0559458\pi$
$73$	5.50000	+	9.52628i	0.643726	+	1.11497i	−0.340868	+	0.940111i	$0.610721\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.00000			0.227921
$78$		0			0
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	$-0.148751\pi$
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	−0.836527	+	0.547926i	$0.815418\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	−0.980071	−	0.198650i	$-0.936344\pi$
$89$	−3.00000	+	5.19615i	−0.317999	+	0.550791i	0.662071	+	0.749441i	$0.269678\pi$
$90$		0			0
$91$	2.50000	−	4.33013i	0.262071	−	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.00000	−	12.1244i	0.696526	−	1.20642i	−0.273138	−	0.961975i	$-0.588061\pi$
$101$	7.00000	−	12.1244i	0.696526	−	1.20642i	0.969664	−	0.244443i	$-0.0786053\pi$
$102$		0			0
$103$	−13.0000			−1.28093			−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000			−1.28093			−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18.0000			−1.74013			−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000			−1.74013			−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	0.909935	−	0.414751i	$-0.136131\pi$
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.00000	+	3.46410i	0.183340	+	0.317554i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−4.00000	+	6.92820i	−0.354943	+	0.614779i	−0.987108	−	0.160055i	$-0.948833\pi$
$127$	−4.00000	+	6.92820i	−0.354943	+	0.614779i	0.632166	+	0.774833i	$0.282166\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.00000	+	12.1244i	0.611593	+	1.05931i	0.990972	+	0.134069i	$0.0428042\pi$
$131$	7.00000	+	12.1244i	0.611593	+	1.05931i	−0.379379	+	0.925241i	$0.623862\pi$
$132$		0			0
$133$	−4.00000	−	1.73205i	−0.346844	−	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	−0.169226	−	0.985577i	$-0.554127\pi$
$137$	9.00000	−	15.5885i	0.768922	−	1.33181i	0.938148	−	0.346235i	$-0.112540\pi$
$138$		0			0
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	−0.740308	−	0.672268i	$-0.765320\pi$
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	0.952355	+	0.304991i	$0.0986536\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.00000	−	8.66025i	0.418121	−	0.724207i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.50000	−	2.59808i	−0.119713	−	0.207349i	0.799941	−	0.600079i	$-0.204864\pi$
$157$	−1.50000	−	2.59808i	−0.119713	−	0.207349i	−0.919654	+	0.392730i	$0.871531\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000	+	3.46410i	−0.157622	+	0.273009i
$162$		0			0
$163$	−3.00000			−0.234978			−0.117489	−	0.993074i	$-0.537485\pi$
$163$	−3.00000			−0.234978			−0.117489	+	0.993074i	$0.537485\pi$
$164$		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
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−2.50000	+	4.33013i	−0.188982	+	0.327327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	1.00000	−	1.73205i	0.0743294	−	0.128742i	−0.826465	−	0.562988i	$-0.809652\pi$
$181$	1.00000	−	1.73205i	0.0743294	−	0.128742i	0.900794	+	0.434246i	$0.142985\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.00000	+	6.92820i	0.292509	+	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	−0.973805	−	0.227387i	$-0.926982\pi$
$193$	−9.50000	−	16.4545i	−0.683825	−	1.18442i	0.289980	−	0.957033i	$-0.406351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	26.0000			1.85242			0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000			1.85242			0.926212	+	0.377004i	$0.123046\pi$
$198$		0			0
$199$	10.5000	−	18.1865i	0.744325	−	1.28921i	−0.206184	−	0.978513i	$-0.566105\pi$
$199$	10.5000	−	18.1865i	0.744325	−	1.28921i	0.950509	−	0.310696i	$-0.100562\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.00000	−	6.92820i	0.280745	−	0.486265i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000	−	3.46410i	−0.553372	−	0.239617i
$210$		0			0
$211$	−7.50000	−	12.9904i	−0.516321	−	0.894295i	−0.999820	−	0.0189499i	$-0.993968\pi$
$211$	−7.50000	−	12.9904i	−0.516321	−	0.894295i	0.483499	−	0.875345i	$-0.339366\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.00000			−0.203653
$218$		0			0
$219$		0			0
$220$		0			0
$221$	20.0000			1.34535
$222$		0			0
$223$	4.50000	+	7.79423i	0.301342	+	0.521940i	0.976440	−	0.215788i	$-0.0692320\pi$
$223$	4.50000	+	7.79423i	0.301342	+	0.521940i	−0.675098	+	0.737728i	$0.735899\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−24.0000			−1.59294			−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000			−1.59294			−0.796468	+	0.604681i	$0.793301\pi$
$228$		0			0
$229$	13.0000			0.859064			0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000			0.859064			0.429532	+	0.903052i	$0.358679\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	0.750867	−	0.660454i	$-0.229636\pi$
$233$	−3.00000	−	5.19615i	−0.196537	−	0.340411i	−0.947403	+	0.320043i	$0.896303\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	3.50000	−	6.06218i	0.225455	−	0.390499i	−0.731001	−	0.682376i	$-0.760947\pi$
$241$	3.50000	−	6.06218i	0.225455	−	0.390499i	0.956456	+	0.291877i	$0.0942799\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−17.5000	+	12.9904i	−1.11350	+	0.826558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.00000	−	1.73205i	0.0631194	−	0.109326i	−0.832739	−	0.553666i	$-0.813228\pi$
$251$	1.00000	−	1.73205i	0.0631194	−	0.109326i	0.895858	+	0.444340i	$0.146562\pi$
$252$		0			0
$253$	−4.00000	+	6.92820i	−0.251478	+	0.435572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−10.0000	+	17.3205i	−0.623783	+	1.08042i	0.364992	+	0.931011i	$0.381072\pi$
$257$	−10.0000	+	17.3205i	−0.623783	+	1.08042i	−0.988775	+	0.149413i	$0.952262\pi$
$258$		0			0
$259$	−3.00000			−0.186411
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.0000	−	22.5167i	−0.801614	−	1.38844i	−0.918553	−	0.395298i	$-0.870641\pi$
$263$	−13.0000	−	22.5167i	−0.801614	−	1.38844i	0.116939	−	0.993139i	$-0.462692\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	0.977229	−	0.212187i	$-0.0680585\pi$
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	−0.672374	+	0.740212i	$0.734725\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.00000	+	8.66025i	−0.301511	+	0.522233i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.00000	+	6.92820i	−0.238620	+	0.413302i	−0.960319	−	0.278906i	$-0.910028\pi$
$281$	−4.00000	+	6.92820i	−0.238620	+	0.413302i	0.721699	+	0.692207i	$0.243362\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.00000	+	10.3923i	0.354169	+	0.613438i
$288$		0			0
$289$	0.500000	−	0.866025i	0.0294118	−	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−14.0000			−0.817889			−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000			−0.817889			−0.408944	+	0.912559i	$0.634103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.0000	+	17.3205i	0.578315	+	1.00167i
$300$		0			0
$301$	0.500000	+	0.866025i	0.0288195	+	0.0499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−6.00000	−	10.3923i	−0.342438	−	0.593120i	0.642447	−	0.766330i	$-0.277919\pi$
$307$	−6.00000	−	10.3923i	−0.342438	−	0.593120i	−0.984885	+	0.173210i	$0.944586\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		0			0
$313$	−11.0000	+	19.0526i	−0.621757	+	1.07691i	0.367402	+	0.930062i	$0.380247\pi$
$313$	−11.0000	+	19.0526i	−0.621757	+	1.07691i	−0.989158	+	0.146852i	$0.953086\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2.00000	+	3.46410i	−0.112331	+	0.194563i	−0.916710	−	0.399554i	$-0.869165\pi$
$317$	−2.00000	+	3.46410i	−0.112331	+	0.194563i	0.804379	+	0.594117i	$0.202498\pi$
$318$		0			0
$319$	8.00000	−	13.8564i	0.447914	−	0.775810i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.00000	−	17.3205i	−0.111283	−	0.963739i
$324$		0			0
$325$	12.5000	+	21.6506i	0.693375	+	1.20096i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.00000	+	5.19615i	−0.165395	+	0.286473i
$330$		0			0
$331$	25.0000			1.37412			0.687062	−	0.726599i	$-0.258900\pi$
$331$	25.0000			1.37412			0.687062	+	0.726599i	$0.258900\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−6.50000	−	11.2583i	−0.354078	−	0.613280i	0.632882	−	0.774248i	$-0.281872\pi$
$337$	−6.50000	−	11.2583i	−0.354078	−	0.613280i	−0.986960	+	0.160968i	$0.948538\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$	13.0000			0.701934
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.00000	+	13.8564i	0.429463	+	0.743851i	0.996826	−	0.0796169i	$-0.0253697\pi$
$347$	8.00000	+	13.8564i	0.429463	+	0.743851i	−0.567363	+	0.823468i	$0.692036\pi$
$348$		0			0
$349$	−21.0000			−1.12410			−0.562052	−	0.827102i	$-0.689988\pi$
$349$	−21.0000			−1.12410			−0.562052	+	0.827102i	$0.689988\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.00000			−0.212899			−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000			−0.212899			−0.106449	+	0.994318i	$0.533948\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	0.999476	+	0.0323801i	$0.0103087\pi$
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	−0.471696	+	0.881761i	$0.656358\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	−0.760100	−	0.649806i	$-0.774850\pi$
$367$	3.50000	−	6.06218i	0.182699	−	0.316443i	0.942799	+	0.333363i	$0.108183\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.00000	+	3.46410i	−0.103835	+	0.179847i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.0000	−	34.6410i	−1.03005	−	1.78410i
$378$		0			0
$379$	5.00000			0.256833			0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000			0.256833			0.128416	+	0.991720i	$0.459011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7.00000	−	12.1244i	−0.357683	−	0.619526i	0.629890	−	0.776684i	$-0.283100\pi$
$383$	−7.00000	−	12.1244i	−0.357683	−	0.619526i	−0.987573	+	0.157159i	$0.949767\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0000	+	20.7846i	−0.608424	+	1.05382i	0.383076	+	0.923717i	$0.374865\pi$
$389$	−12.0000	+	20.7846i	−0.608424	+	1.05382i	−0.991500	+	0.130105i	$0.958469\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	0.764730	−	0.644351i	$-0.222873\pi$
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	−0.940389	+	0.340099i	$0.889539\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	0.781345	−	0.624099i	$-0.214534\pi$
$401$	−3.00000	−	5.19615i	−0.149813	−	0.259483i	−0.931158	+	0.364615i	$0.881200\pi$
$402$		0			0
$403$	−7.50000	+	12.9904i	−0.373602	+	0.647097i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6.00000			−0.297409
$408$		0			0
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	−0.639430	−	0.768849i	$-0.720830\pi$
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	0.985558	+	0.169338i	$0.0541630\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.00000	+	8.66025i	0.246034	+	0.426143i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.0000			−0.683945			−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000			−0.683945			−0.341972	+	0.939710i	$0.611095\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−20.0000			−0.970143
$426$		0			0
$427$	−6.50000	+	11.2583i	−0.314557	+	0.544829i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.00000	−	8.66025i	0.240842	−	0.417150i	−0.720113	−	0.693857i	$-0.755910\pi$
$431$	5.00000	−	8.66025i	0.240842	−	0.417150i	0.960954	+	0.276707i	$0.0892433\pi$
$432$		0			0
$433$	4.50000	−	7.79423i	0.216256	−	0.374567i	−0.737404	−	0.675452i	$-0.763949\pi$
$433$	4.50000	−	7.79423i	0.216256	−	0.374567i	0.953660	+	0.300885i	$0.0972820\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.0000	−	10.3923i	0.669711	−	0.497131i
$438$		0			0
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	0.893843	+	0.448379i	$0.147999\pi$
$439$	17.5000	+	30.3109i	0.835229	+	1.44666i	−0.0586141	+	0.998281i	$0.518668\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.0000	−	27.7128i	0.760183	−	1.31668i	−0.182573	−	0.983192i	$-0.558443\pi$
$443$	16.0000	−	27.7128i	0.760183	−	1.31668i	0.942756	−	0.333483i	$-0.108224\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$		0			0
$451$	12.0000	+	20.7846i	0.565058	+	0.978709i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.0000			−0.514558			−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000			−0.514558			−0.257279	+	0.966337i	$0.582826\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	0.968945	+	0.247276i	$0.0795353\pi$
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	−0.270326	+	0.962769i	$0.587131\pi$
$462$		0			0
$463$	−23.0000			−1.06890			−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000			−1.06890			−0.534450	+	0.845200i	$0.679481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	−5.50000	+	9.52628i	−0.253966	+	0.439883i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.00000	+	1.73205i	0.0459800	+	0.0796398i
$474$		0			0
$475$	17.5000	−	12.9904i	0.802955	−	0.596040i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.00000	+	12.1244i	−0.319838	+	0.553976i	−0.980454	−	0.196748i	$-0.936962\pi$
$479$	−7.00000	+	12.1244i	−0.319838	+	0.553976i	0.660616	+	0.750724i	$0.270295\pi$
$480$		0			0
$481$	−7.50000	+	12.9904i	−0.341971	+	0.592310i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	0.790358	−	0.612646i	$-0.209895\pi$
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	−0.925746	+	0.378147i	$0.876561\pi$
$492$		0			0
$493$	32.0000			1.44121
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.00000	+	5.19615i	0.134568	+	0.233079i
$498$		0			0
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	0.983336	+	0.181797i	$0.0581915\pi$
$499$	14.5000	+	25.1147i	0.649109	+	1.12429i	−0.334227	+	0.942493i	$0.608475\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20.0000	+	34.6410i	−0.891756	+	1.54457i	−0.0539870	+	0.998542i	$0.517193\pi$
$503$	−20.0000	+	34.6410i	−0.891756	+	1.54457i	−0.837769	+	0.546025i	$0.816140\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	−0.924821	−	0.380402i	$-0.875786\pi$
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	0.791849	+	0.610718i	$0.209119\pi$
$510$		0			0
$511$	−5.50000	−	9.52628i	−0.243306	−	0.421418i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−6.00000	+	10.3923i	−0.263880	+	0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	−0.352677	−	0.935745i	$-0.614728\pi$
$523$	14.5000	−	25.1147i	0.634041	−	1.09819i	0.986718	−	0.162446i	$-0.0519382\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000	−	10.3923i	−0.261364	−	0.452696i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	60.0000			2.59889
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.0000			0.516877
$540$		0			0
$541$	19.5000	−	33.7750i	0.838370	−	1.45210i	−0.0528859	−	0.998601i	$-0.516842\pi$
$541$	19.5000	−	33.7750i	0.838370	−	1.45210i	0.891256	−	0.453500i	$-0.149825\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−14.5000	+	25.1147i	−0.619975	+	1.07383i	0.369514	+	0.929225i	$0.379524\pi$
$547$	−14.5000	+	25.1147i	−0.619975	+	1.07383i	−0.989490	+	0.144604i	$0.953809\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.0000	+	20.7846i	−1.19284	+	0.885454i
$552$		0			0
$553$	−0.500000	−	0.866025i	−0.0212622	−	0.0368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.0000	+	32.9090i	−0.805056	+	1.39440i	0.111198	+	0.993798i	$0.464531\pi$
$557$	−19.0000	+	32.9090i	−0.805056	+	1.39440i	−0.916253	+	0.400599i	$0.868802\pi$
$558$		0			0
$559$	5.00000			0.211477
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.0000			0.758610			0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000			0.758610			0.379305	+	0.925272i	$0.376163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$		0			0
$571$	−33.0000			−1.38101			−0.690504	−	0.723329i	$-0.742611\pi$
$571$	−33.0000			−1.38101			−0.690504	+	0.723329i	$0.742611\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.0000	−	17.3205i	−0.417029	−	0.722315i
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4.00000	+	6.92820i	−0.165663	+	0.286937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.0000	+	22.5167i	0.536567	+	0.929362i	0.999086	+	0.0427523i	$0.0136126\pi$
$587$	13.0000	+	22.5167i	0.536567	+	0.929362i	−0.462518	+	0.886610i	$0.653054\pi$
$588$		0			0
$589$	12.0000	+	5.19615i	0.494451	+	0.214104i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.00000	+	5.19615i	−0.123195	+	0.213380i	−0.921026	−	0.389501i	$-0.872647\pi$
$593$	−3.00000	+	5.19615i	−0.123195	+	0.213380i	0.797831	+	0.602881i	$0.205981\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.00000	+	15.5885i	−0.367730	+	0.636927i	−0.989210	−	0.146503i	$-0.953198\pi$
$599$	−9.00000	+	15.5885i	−0.367730	+	0.636927i	0.621480	+	0.783430i	$0.286532\pi$
$600$		0			0
$601$	−21.0000			−0.856608			−0.428304	−	0.903635i	$-0.640889\pi$
$601$	−21.0000			−0.856608			−0.428304	+	0.903635i	$0.640889\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	43.0000			1.74532			0.872658	−	0.488332i	$-0.162394\pi$
$607$	43.0000			1.74532			0.872658	+	0.488332i	$0.162394\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.0000	+	25.9808i	0.606835	+	1.05107i
$612$		0			0
$613$	15.0000	+	25.9808i	0.605844	+	1.04935i	0.991917	+	0.126885i	$0.0404979\pi$
$613$	15.0000	+	25.9808i	0.605844	+	1.04935i	−0.386073	+	0.922468i	$0.626169\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	−	5.19615i	0.120775	−	0.209189i	−0.799298	−	0.600935i	$-0.794795\pi$
$617$	3.00000	−	5.19615i	0.120775	−	0.209189i	0.920074	+	0.391745i	$0.128129\pi$
$618$		0			0
$619$	−25.0000			−1.00483			−0.502417	−	0.864625i	$-0.667556\pi$
$619$	−25.0000			−1.00483			−0.502417	+	0.864625i	$0.667556\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.00000	−	5.19615i	0.120192	−	0.208179i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.00000	−	10.3923i	−0.239236	−	0.414368i
$630$		0			0
$631$	−7.50000	+	12.9904i	−0.298570	+	0.517139i	−0.975809	−	0.218624i	$-0.929843\pi$
$631$	−7.50000	+	12.9904i	−0.298570	+	0.517139i	0.677239	+	0.735763i	$0.263176\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.0000	−	25.9808i	0.594322	−	1.02940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	−0.993899	−	0.110291i	$-0.964822\pi$
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	0.401435	−	0.915888i	$-0.368512\pi$
$642$		0			0
$643$	−9.50000	−	16.4545i	−0.374643	−	0.648901i	0.615630	−	0.788035i	$-0.288902\pi$
$643$	−9.50000	−	16.4545i	−0.374643	−	0.648901i	−0.990274	+	0.139134i	$0.955568\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$	10.0000	+	17.3205i	0.392534	+	0.679889i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−17.0000	+	29.4449i	−0.662226	+	1.14701i	0.317803	+	0.948157i	$0.397055\pi$
$659$	−17.0000	+	29.4449i	−0.662226	+	1.14701i	−0.980029	+	0.198852i	$0.936279\pi$
$660$		0			0
$661$	−21.0000	+	36.3731i	−0.816805	+	1.41475i	0.0912190	+	0.995831i	$0.470924\pi$
$661$	−21.0000	+	36.3731i	−0.816805	+	1.41475i	−0.908024	+	0.418917i	$0.862410\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.0000	+	27.7128i	0.619522	+	1.07304i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−13.0000	+	22.5167i	−0.501859	+	0.869246i
$672$		0			0
$673$	−33.0000			−1.27206			−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000			−1.27206			−0.636028	+	0.771666i	$0.719424\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−34.0000			−1.30673			−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000			−1.30673			−0.653363	+	0.757045i	$0.726642\pi$
$678$		0			0
$679$	1.00000	+	1.73205i	0.0383765	+	0.0664700i
$680$		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
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.0000	+	17.3205i	0.380970	+	0.659859i
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−24.0000	+	41.5692i	−0.909065	+	1.57455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.0000	+	17.3205i	0.377695	+	0.654187i	0.990726	−	0.135872i	$-0.0433835\pi$
$701$	10.0000	+	17.3205i	0.377695	+	0.654187i	−0.613032	+	0.790058i	$0.710050\pi$
$702$		0			0
$703$	12.0000	+	5.19615i	0.452589	+	0.195977i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.00000	+	12.1244i	−0.263262	+	0.455983i
$708$		0			0
$709$	19.5000	−	33.7750i	0.732338	−	1.26845i	−0.223544	−	0.974694i	$-0.571763\pi$
$709$	19.5000	−	33.7750i	0.732338	−	1.26845i	0.955882	−	0.293752i	$-0.0949041\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.00000	−	10.3923i	0.224702	−	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	13.0000			0.484145
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.0000	+	34.6410i	0.742781	+	1.28654i
$726$		0			0
$727$	−23.5000	−	40.7032i	−0.871567	−	1.50960i	−0.860376	−	0.509661i	$-0.829771\pi$
$727$	−23.5000	−	40.7032i	−0.871567	−	1.50960i	−0.0111912	−	0.999937i	$-0.503562\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.00000	+	3.46410i	−0.0739727	+	0.128124i
$732$		0			0
$733$	−50.0000			−1.84679			−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000			−1.84679			−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−11.0000	+	19.0526i	−0.405190	+	0.701810i
$738$		0			0
$739$	9.50000	+	16.4545i	0.349463	+	0.605288i	0.986154	−	0.165831i	$-0.0530307\pi$
$739$	9.50000	+	16.4545i	0.349463	+	0.605288i	−0.636691	+	0.771119i	$0.719697\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25.0000	+	43.3013i	0.917161	+	1.58857i	0.803706	+	0.595026i	$0.202858\pi$
$743$	25.0000	+	43.3013i	0.917161	+	1.58857i	0.113455	+	0.993543i	$0.463808\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000			0.657706
$750$		0			0
$751$	22.5000	−	38.9711i	0.821037	−	1.42208i	−0.0838743	−	0.996476i	$-0.526729\pi$
$751$	22.5000	−	38.9711i	0.821037	−	1.42208i	0.904911	−	0.425601i	$-0.139937\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−6.50000	−	11.2583i	−0.236247	−	0.409191i	0.723388	−	0.690442i	$-0.242584\pi$
$757$	−6.50000	−	11.2583i	−0.236247	−	0.409191i	−0.959634	+	0.281251i	$0.909251\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−6.00000			−0.217500			−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000			−0.217500			−0.108750	+	0.994069i	$0.534685\pi$
$762$		0			0
$763$	−1.00000	−	1.73205i	−0.0362024	−	0.0627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	50.0000			1.80540
$768$		0			0
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	−0.311577	−	0.950221i	$-0.600857\pi$
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	0.978704	−	0.205277i	$-0.0658095\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.00000	−	12.1244i	0.251773	−	0.436083i	−0.712241	−	0.701935i	$-0.752320\pi$
$773$	7.00000	−	12.1244i	0.251773	−	0.436083i	0.964014	+	0.265852i	$0.0856532\pi$
$774$		0			0
$775$	7.50000	−	12.9904i	0.269408	−	0.466628i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.00000	−	51.9615i	−0.214972	−	1.86171i
$780$		0			0
$781$	6.00000	+	10.3923i	0.214697	+	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−25.0000			−0.891154			−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000			−0.891154			−0.445577	+	0.895244i	$0.647001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2.00000			0.0711118
$792$		0			0
$793$	32.5000	+	56.2917i	1.15411	+	1.99898i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.0000			0.495905			0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000			0.495905			0.247953	+	0.968772i	$0.420242\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.0000	−	19.0526i	−0.388182	−	0.672350i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.0000			0.562530			0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000			0.562530			0.281265	+	0.959630i	$0.409246\pi$
$810$		0			0
$811$	−22.0000	+	38.1051i	−0.772524	+	1.33805i	0.163651	+	0.986518i	$0.447673\pi$
$811$	−22.0000	+	38.1051i	−0.772524	+	1.33805i	−0.936175	+	0.351533i	$0.885660\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.500000	−	4.33013i	−0.0174928	−	0.151492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	−0.577016	−	0.816733i	$-0.695783\pi$
$821$	12.0000	−	20.7846i	0.418803	−	0.725388i	0.995819	+	0.0913446i	$0.0291165\pi$
$822$		0			0
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	−0.829060	−	0.559159i	$-0.811124\pi$
$823$	2.00000	−	3.46410i	0.0697156	−	0.120751i	0.898776	+	0.438408i	$0.144457\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.0000	−	31.1769i	0.625921	−	1.08413i	−0.362441	−	0.932007i	$-0.618056\pi$
$827$	18.0000	−	31.1769i	0.625921	−	1.08413i	0.988362	−	0.152121i	$-0.0486102\pi$
$828$		0			0
$829$	25.0000			0.868286			0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000			0.868286			0.434143	+	0.900844i	$0.357051\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.0000	+	20.7846i	0.415775	+	0.720144i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	0.950813	−	0.309764i	$-0.100250\pi$
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	−0.743670	+	0.668546i	$0.766917\pi$
$840$		0			0
$841$	−17.5000	−	30.3109i	−0.603448	−	1.04520i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	7.00000			0.240523
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	−	10.3923i	0.205677	−	0.356244i
$852$		0			0
$853$	2.50000	+	4.33013i	0.0855984	+	0.148261i	0.905646	−	0.424034i	$-0.139386\pi$
$853$	2.50000	+	4.33013i	0.0855984	+	0.148261i	−0.820048	+	0.572295i	$0.806053\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.00000	+	8.66025i	0.170797	+	0.295829i	0.938699	−	0.344739i	$-0.112033\pi$
$857$	5.00000	+	8.66025i	0.170797	+	0.295829i	−0.767902	+	0.640567i	$0.778699\pi$
$858$		0			0
$859$	7.50000	−	12.9904i	0.255897	−	0.443226i	−0.709242	−	0.704965i	$-0.750963\pi$
$859$	7.50000	−	12.9904i	0.255897	−	0.443226i	0.965139	+	0.261739i	$0.0842960\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.0000			0.340404			0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000			0.340404			0.170202	+	0.985409i	$0.445558\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.00000	−	1.73205i	−0.0339227	−	0.0587558i
$870$		0			0
$871$	27.5000	+	47.6314i	0.931802	+	1.61393i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.50000	−	12.9904i	−0.253257	−	0.438654i	0.711164	−	0.703027i	$-0.248168\pi$
$877$	−7.50000	−	12.9904i	−0.253257	−	0.438654i	−0.964421	+	0.264373i	$0.914835\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	28.0000			0.943344			0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000			0.943344			0.471672	+	0.881774i	$0.343651\pi$
$882$		0			0
$883$	−0.500000	+	0.866025i	−0.0168263	+	0.0291441i	−0.874316	−	0.485357i	$-0.838690\pi$
$883$	−0.500000	+	0.866025i	−0.0168263	+	0.0291441i	0.857490	+	0.514501i	$0.172023\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.00000	−	5.19615i	0.100730	−	0.174470i	−0.811256	−	0.584692i	$-0.801215\pi$
$887$	3.00000	−	5.19615i	0.100730	−	0.174470i	0.911986	+	0.410222i	$0.134549\pi$
$888$		0			0
$889$	4.00000	−	6.92820i	0.134156	−	0.232364i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	21.0000	−	15.5885i	0.702738	−	0.521648i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0000	+	20.7846i	−0.400222	+	0.693206i
$900$		0			0
$901$	−16.0000			−0.533037
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	−0.956671	−	0.291172i	$-0.905955\pi$
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	0.226173	−	0.974087i	$-0.427379\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−7.00000	−	12.1244i	−0.231160	−	0.400381i
$918$		0			0
$919$	25.0000			0.824674			0.412337	−	0.911031i	$-0.364713\pi$
$919$	25.0000			0.824674			0.412337	+	0.911031i	$0.364713\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	30.0000			0.987462
$924$		0			0
$925$	7.50000	−	12.9904i	0.246598	−	0.427121i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.0000	+	24.2487i	0.459325	+	0.795574i	0.998925	−	0.0463469i	$-0.0147580\pi$
$929$	14.0000	+	24.2487i	0.459325	+	0.795574i	−0.539600	+	0.841921i	$0.681425\pi$
$930$		0			0
$931$	−24.0000	−	10.3923i	−0.786568	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.50000	−	7.79423i	0.147009	−	0.254626i	−0.783112	−	0.621881i	$-0.786369\pi$
$937$	4.50000	−	7.79423i	0.147009	−	0.254626i	0.930121	+	0.367254i	$0.119702\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$	−48.0000			−1.56310
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−25.0000	−	43.3013i	−0.812391	−	1.40710i	−0.911186	−	0.411994i	$-0.864832\pi$
$947$	−25.0000	−	43.3013i	−0.812391	−	1.40710i	0.0987955	−	0.995108i	$-0.468501\pi$
$948$		0			0
$949$	−55.0000			−1.78538
$950$		0			0
$951$		0			0
$952$		0			0
$953$	17.0000	+	29.4449i	0.550684	+	0.953813i	0.998225	+	0.0595495i	$0.0189664\pi$
$953$	17.0000	+	29.4449i	0.550684	+	0.953813i	−0.447541	+	0.894263i	$0.647700\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.00000	+	15.5885i	−0.290625	+	0.503378i
$960$		0			0
$961$	−22.0000			−0.709677
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−14.5000	−	25.1147i	−0.466289	−	0.807635i	0.532970	−	0.846134i	$-0.321076\pi$
$967$	−14.5000	−	25.1147i	−0.466289	−	0.807635i	−0.999259	+	0.0384986i	$0.987742\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21.0000	+	36.3731i	0.673922	+	1.16727i	0.976783	+	0.214232i	$0.0687250\pi$
$971$	21.0000	+	36.3731i	0.673922	+	1.16727i	−0.302861	+	0.953035i	$0.597942\pi$
$972$		0			0
$973$	−2.50000	+	4.33013i	−0.0801463	+	0.138817i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−54.0000			−1.72761			−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000			−1.72761			−0.863807	+	0.503824i	$0.831926\pi$
$978$		0			0
$979$	6.00000	−	10.3923i	0.191761	−	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.00000	−	8.66025i	−0.159475	−	0.276219i	0.775204	−	0.631711i	$-0.217647\pi$
$983$	−5.00000	−	8.66025i	−0.159475	−	0.276219i	−0.934680	+	0.355491i	$0.884314\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.00000			−0.127193
$990$		0			0
$991$	23.5000	+	40.7032i	0.746502	+	1.29298i	0.949490	+	0.313798i	$0.101602\pi$
$991$	23.5000	+	40.7032i	0.746502	+	1.29298i	−0.202988	+	0.979181i	$0.565065\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.50000	+	6.06218i	−0.110846	+	0.191991i	−0.916112	−	0.400923i	$-0.868689\pi$
$997$	−3.50000	+	6.06218i	−0.110846	+	0.191991i	0.805266	+	0.592914i	$0.202023\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.j.577.1		2
3.2	odd	2		912.2.q.a.577.1		2
4.3	odd	2		171.2.f.a.64.1		2
12.11	even	2		57.2.e.a.7.1	✓	2
19.11	even	3	inner	2736.2.s.j.1873.1		2
57.11	odd	6		912.2.q.a.49.1		2
76.7	odd	6		3249.2.a.c.1.1		1
76.11	odd	6		171.2.f.a.163.1		2
76.31	even	6		3249.2.a.f.1.1		1
228.11	even	6		57.2.e.a.49.1	yes	2
228.83	even	6		1083.2.a.c.1.1		1
228.107	odd	6		1083.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.a.7.1	✓	2	12.11	even	2
57.2.e.a.49.1	yes	2	228.11	even	6
171.2.f.a.64.1		2	4.3	odd	2
171.2.f.a.163.1		2	76.11	odd	6
912.2.q.a.49.1		2	57.11	odd	6
912.2.q.a.577.1		2	3.2	odd	2
1083.2.a.b.1.1		1	228.107	odd	6
1083.2.a.c.1.1		1	228.83	even	6
2736.2.s.j.577.1		2	1.1	even	1	trivial
2736.2.s.j.1873.1		2	19.11	even	3	inner
3249.2.a.c.1.1		1	76.7	odd	6
3249.2.a.f.1.1		1	76.31	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{7} +O(q^{10})\)	\(q-1.00000 q^{7} -2.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(-2.00000 - 3.46410i) q^{17} +(4.00000 + 1.73205i) q^{19} +(2.00000 - 3.46410i) q^{23} +(2.50000 - 4.33013i) q^{25} +(-4.00000 + 6.92820i) q^{29} +3.00000 q^{31} +3.00000 q^{37} +(-6.00000 - 10.3923i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(3.00000 - 5.19615i) q^{47} -6.00000 q^{49} +(2.00000 - 3.46410i) q^{53} +(-5.00000 - 8.66025i) q^{59} +(6.50000 - 11.2583i) q^{61} +(5.50000 - 9.52628i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(5.50000 + 9.52628i) q^{73} +2.00000 q^{77} +(0.500000 + 0.866025i) q^{79} +(-3.00000 + 5.19615i) q^{89} +(2.50000 - 4.33013i) q^{91} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^7	\( 2 q - 2 q^{7} - 4 q^{11} - 5 q^{13} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 5 q^{25} - 8 q^{29} + 6 q^{31} + 6 q^{37} - 12 q^{41} - q^{43} + 6 q^{47} - 12 q^{49} + 4 q^{53} - 10 q^{59} + 13 q^{61} + 11 q^{67} - 6 q^{71} + 11 q^{73} + 4 q^{77} + q^{79} - 6 q^{89} + 5 q^{91} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 4 * q^11 - 5 * q^13 - 4 * q^17 + 8 * q^19 + 4 * q^23 + 5 * q^25 - 8 * q^29 + 6 * q^31 + 6 * q^37 - 12 * q^41 - q^43 + 6 * q^47 - 12 * q^49 + 4 * q^53 - 10 * q^59 + 13 * q^61 + 11 * q^67 - 6 * q^71 + 11 * q^73 + 4 * q^77 + q^79 - 6 * q^89 + 5 * q^91 - 2 * q^97

Introduction

L-functions

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Groups

Database

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