LMFDB - Embedded newform 2736.2.s.f.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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
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.f.1873.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} +5.00000 q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{5} +5.00000 q^{7} -4.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(-4.00000 - 1.73205i) q^{19} +(3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +(4.00000 - 6.92820i) q^{29} +1.00000 q^{31} +(-5.00000 - 8.66025i) q^{35} +7.00000 q^{37} +(-5.50000 - 9.52628i) q^{43} +(-5.00000 + 8.66025i) q^{47} +18.0000 q^{49} +(3.00000 - 5.19615i) q^{53} +(4.00000 + 6.92820i) q^{55} +(-4.00000 - 6.92820i) q^{59} +(0.500000 - 0.866025i) q^{61} +10.0000 q^{65} +(2.50000 - 4.33013i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(-0.500000 - 0.866025i) q^{73} -20.0000 q^{77} +(-6.50000 - 11.2583i) q^{79} -4.00000 q^{83} +(-6.00000 + 10.3923i) q^{89} +(-12.5000 + 21.6506i) q^{91} +(1.00000 + 8.66025i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 10 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 10 * q^7	$ 2 q - 2 q^{5} + 10 q^{7} - 8 q^{11} - 5 q^{13} - 8 q^{19} + 6 q^{23} + q^{25} + 8 q^{29} + 2 q^{31} - 10 q^{35} + 14 q^{37} - 11 q^{43} - 10 q^{47} + 36 q^{49} + 6 q^{53} + 8 q^{55} - 8 q^{59} + q^{61} + 20 q^{65} + 5 q^{67} - 6 q^{71} - q^{73} - 40 q^{77} - 13 q^{79} - 8 q^{83} - 12 q^{89} - 25 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 10 * q^7 - 8 * q^11 - 5 * q^13 - 8 * q^19 + 6 * q^23 + q^25 + 8 * q^29 + 2 * q^31 - 10 * q^35 + 14 * q^37 - 11 * q^43 - 10 * q^47 + 36 * q^49 + 6 * q^53 + 8 * q^55 - 8 * q^59 + q^61 + 20 * q^65 + 5 * q^67 - 6 * q^71 - q^73 - 40 * q^77 - 13 * q^79 - 8 * q^83 - 12 * q^89 - 25 * q^91 + 2 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	0.550990	−	0.834512i	$-0.314250\pi$
$5$	−1.00000	−	1.73205i	−0.447214	−	0.774597i	−0.998203	+	0.0599153i	$0.980917\pi$
$6$		0			0
$7$	5.00000			1.88982			0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000			1.88982			0.944911	+	0.327327i	$0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		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
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		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
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	−0.362892	−	0.931831i	$-0.618211\pi$
$23$	3.00000	−	5.19615i	0.625543	−	1.08347i	0.988436	−	0.151642i	$-0.0484560\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.00000	−	6.92820i	0.742781	−	1.28654i	−0.208443	−	0.978035i	$-0.566840\pi$
$29$	4.00000	−	6.92820i	0.742781	−	1.28654i	0.951224	−	0.308500i	$-0.0998271\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.00000	−	8.66025i	−0.845154	−	1.46385i
$36$		0			0
$37$	7.00000			1.15079			0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000			1.15079			0.575396	+	0.817875i	$0.304848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	−5.50000	−	9.52628i	−0.838742	−	1.45274i	−0.890947	−	0.454108i	$-0.849958\pi$
$43$	−5.50000	−	9.52628i	−0.838742	−	1.45274i	0.0522047	−	0.998636i	$-0.483375\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.00000	+	8.66025i	−0.729325	+	1.26323i	0.227844	+	0.973698i	$0.426832\pi$
$47$	−5.00000	+	8.66025i	−0.729325	+	1.26323i	−0.957169	+	0.289530i	$0.906501\pi$
$48$		0			0
$49$	18.0000			2.57143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$	4.00000	+	6.92820i	0.539360	+	0.934199i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	$-0.992317\pi$
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	0.478953	−	0.877841i	$-0.341016\pi$
$60$		0			0
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	0.500000	−	0.866025i	0.0640184	−	0.110883i	0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.0000			1.24035
$66$		0			0
$67$	2.50000	−	4.33013i	0.305424	−	0.529009i	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	2.50000	−	4.33013i	0.305424	−	0.529009i	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
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	0.835281	−	0.549823i	$-0.185305\pi$
$73$	−0.500000	−	0.866025i	−0.0585206	−	0.101361i	−0.893801	+	0.448463i	$0.851972\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−20.0000			−2.27921
$78$		0			0
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	0.350304	+	0.936636i	$0.386078\pi$
$89$	−6.00000	+	10.3923i	−0.635999	+	1.10158i	−0.986303	+	0.164946i	$0.947255\pi$
$90$		0			0
$91$	−12.5000	+	21.6506i	−1.31036	+	2.26960i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.00000	+	8.66025i	0.102598	+	0.888523i
$96$		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
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	5.00000			0.492665			0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000			0.492665			0.246332	+	0.969185i	$0.420775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	−1.00000	−	1.73205i	−0.0957826	−	0.165900i	0.814152	−	0.580651i	$-0.197202\pi$
$109$	−1.00000	−	1.73205i	−0.0957826	−	0.165900i	−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	−12.0000			−1.11901
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	10.0000	−	17.3205i	0.887357	−	1.53695i	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	10.0000	−	17.3205i	0.887357	−	1.53695i	0.842989	−	0.537931i	$-0.180794\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.00000	−	8.66025i	−0.436852	−	0.756650i	0.560593	−	0.828092i	$-0.310573\pi$
$131$	−5.00000	−	8.66025i	−0.436852	−	0.756650i	−0.997445	+	0.0714417i	$0.977240\pi$
$132$		0			0
$133$	−20.0000	−	8.66025i	−1.73422	−	0.750939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.00000	−	8.66025i	0.427179	−	0.739895i	−0.569442	−	0.822031i	$-0.692841\pi$
$137$	5.00000	−	8.66025i	0.427179	−	0.739895i	0.996621	+	0.0821359i	$0.0261741\pi$
$138$		0			0
$139$	−2.50000	+	4.33013i	−0.212047	+	0.367277i	−0.952355	−	0.304991i	$-0.901346\pi$
$139$	−2.50000	+	4.33013i	−0.212047	+	0.367277i	0.740308	+	0.672268i	$0.234680\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.0000	−	17.3205i	0.836242	−	1.44841i
$144$		0			0
$145$	−16.0000			−1.32873
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.00000	+	6.92820i	0.327693	+	0.567581i	0.982054	−	0.188602i	$-0.0603956\pi$
$149$	4.00000	+	6.92820i	0.327693	+	0.567581i	−0.654361	+	0.756182i	$0.727062\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.00000	−	1.73205i	−0.0803219	−	0.139122i
$156$		0			0
$157$	6.50000	+	11.2583i	0.518756	+	0.898513i	0.999762	+	0.0217953i	$0.00693820\pi$
$157$	6.50000	+	11.2583i	0.518756	+	0.898513i	−0.481006	+	0.876717i	$0.659728\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.0000	−	25.9808i	1.18217	−	2.04757i
$162$		0			0
$163$	−1.00000			−0.0783260			−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000			−0.0783260			−0.0391630	+	0.999233i	$0.512469\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		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
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$	9.00000	−	15.5885i	0.668965	−	1.15868i	−0.309229	−	0.950988i	$-0.600071\pi$
$181$	9.00000	−	15.5885i	0.668965	−	1.15868i	0.978194	−	0.207693i	$-0.0665956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.00000	−	12.1244i	−0.514650	−	0.891400i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−26.0000			−1.88129			−0.940647	−	0.339387i	$-0.889781\pi$
$191$	−26.0000			−1.88129			−0.940647	+	0.339387i	$0.889781\pi$
$192$		0			0
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	0.941865	−	0.335993i	$-0.109072\pi$
$193$	2.50000	+	4.33013i	0.179954	+	0.311689i	−0.761911	+	0.647682i	$0.775738\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	20.0000			1.42494			0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000			1.42494			0.712470	+	0.701702i	$0.247576\pi$
$198$		0			0
$199$	1.50000	−	2.59808i	0.106332	−	0.184173i	−0.807950	−	0.589252i	$-0.799423\pi$
$199$	1.50000	−	2.59808i	0.106332	−	0.184173i	0.914282	+	0.405079i	$0.132756\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	20.0000	−	34.6410i	1.40372	−	2.43132i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.0000	+	6.92820i	1.10674	+	0.479234i
$210$		0			0
$211$	−10.5000	−	18.1865i	−0.722850	−	1.25201i	−0.959853	−	0.280504i	$-0.909498\pi$
$211$	−10.5000	−	18.1865i	−0.722850	−	1.25201i	0.237003	−	0.971509i	$-0.423835\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.0000	+	19.0526i	−0.750194	+	1.29937i
$216$		0			0
$217$	5.00000			0.339422
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.500000	−	0.866025i	−0.0334825	−	0.0579934i	0.848799	−	0.528716i	$-0.177326\pi$
$223$	−0.500000	−	0.866025i	−0.0334825	−	0.0579934i	−0.882281	+	0.470723i	$0.843993\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.00000			0.132745			0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000			0.132745			0.0663723	+	0.997795i	$0.478857\pi$
$228$		0			0
$229$	−27.0000			−1.78421			−0.892105	−	0.451828i	$-0.850772\pi$
$229$	−27.0000			−1.78421			−0.892105	+	0.451828i	$0.850772\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	0.947403	−	0.320043i	$-0.103697\pi$
$233$	3.00000	+	5.19615i	0.196537	+	0.340411i	−0.750867	+	0.660454i	$0.770364\pi$
$234$		0			0
$235$	20.0000			1.30466
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.00000			0.129369			0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000			0.129369			0.0646846	+	0.997906i	$0.479396\pi$
$240$		0			0
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	−0.632709	−	0.774389i	$-0.718057\pi$
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	0.986996	+	0.160748i	$0.0513906\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−18.0000	−	31.1769i	−1.14998	−	1.99182i
$246$		0			0
$247$	17.5000	−	12.9904i	1.11350	−	0.826558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.0000	+	22.5167i	−0.820553	+	1.42124i	0.0847185	+	0.996405i	$0.473001\pi$
$251$	−13.0000	+	22.5167i	−0.820553	+	1.42124i	−0.905271	+	0.424834i	$0.860332\pi$
$252$		0			0
$253$	−12.0000	+	20.7846i	−0.754434	+	1.30672i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.00000	−	8.66025i	0.311891	−	0.540212i	−0.666880	−	0.745165i	$-0.732371\pi$
$257$	5.00000	−	8.66025i	0.311891	−	0.540212i	0.978772	+	0.204953i	$0.0657041\pi$
$258$		0			0
$259$	35.0000			2.17479
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.00000	+	12.1244i	0.431638	+	0.747620i	0.997015	−	0.0772134i	$-0.0246023\pi$
$263$	7.00000	+	12.1244i	0.431638	+	0.747620i	−0.565376	+	0.824833i	$0.691269\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000	−	10.3923i	−0.365826	−	0.633630i	0.623082	−	0.782157i	$-0.285880\pi$
$269$	−6.00000	−	10.3923i	−0.365826	−	0.633630i	−0.988908	+	0.148527i	$0.952547\pi$
$270$		0			0
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	0.513905	−	0.857847i	$-0.328199\pi$
$271$	−8.00000	−	13.8564i	−0.485965	−	0.841717i	−0.999870	+	0.0161307i	$0.994865\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.00000	+	3.46410i	−0.120605	+	0.208893i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	−0.975741	−	0.218926i	$-0.929745\pi$
$281$	−5.00000	+	8.66025i	−0.298275	+	0.516627i	0.677466	+	0.735554i	$0.263078\pi$
$282$		0			0
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	0.993626	+	0.112728i	$0.0359589\pi$
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	−0.399188	+	0.916869i	$0.630708\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.0000			0.584206			0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000			0.584206			0.292103	+	0.956387i	$0.405645\pi$
$294$		0			0
$295$	−8.00000	+	13.8564i	−0.465778	+	0.806751i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	15.0000	+	25.9808i	0.867472	+	1.50251i
$300$		0			0
$301$	−27.5000	−	47.6314i	−1.58507	−	2.74543i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.00000			−0.114520
$306$		0			0
$307$	6.00000	+	10.3923i	0.342438	+	0.593120i	0.984885	−	0.173210i	$-0.0554140\pi$
$307$	6.00000	+	10.3923i	0.342438	+	0.593120i	−0.642447	+	0.766330i	$0.722081\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−5.00000	+	8.66025i	−0.282617	+	0.489506i	0.689412	+	0.724370i	$0.257869\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	−16.0000	+	27.7128i	−0.895828	+	1.55162i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.50000	+	4.33013i	0.138675	+	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−25.0000	+	43.3013i	−1.37829	+	2.38728i
$330$		0			0
$331$	−5.00000			−0.274825			−0.137412	−	0.990514i	$-0.543879\pi$
$331$	−5.00000			−0.274825			−0.137412	+	0.990514i	$0.543879\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−10.0000			−0.546358
$336$		0			0
$337$	3.50000	+	6.06218i	0.190657	+	0.330228i	0.945468	−	0.325714i	$-0.105605\pi$
$337$	3.50000	+	6.06218i	0.190657	+	0.330228i	−0.754811	+	0.655942i	$0.772271\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.00000			−0.216612
$342$		0			0
$343$	55.0000			2.96972
$344$		0			0
$345$		0			0
$346$		0			0
$347$	15.0000	+	25.9808i	0.805242	+	1.39472i	0.916127	+	0.400887i	$0.131298\pi$
$347$	15.0000	+	25.9808i	0.805242	+	1.39472i	−0.110885	+	0.993833i	$0.535369\pi$
$348$		0			0
$349$	11.0000			0.588817			0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000			0.588817			0.294408	+	0.955680i	$0.404877\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.0000			0.532246			0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000			0.532246			0.266123	+	0.963939i	$0.414257\pi$
$354$		0			0
$355$	−6.00000	+	10.3923i	−0.318447	+	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	−0.999476	−	0.0323801i	$-0.989691\pi$
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	0.471696	−	0.881761i	$-0.343642\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.00000	+	1.73205i	−0.0523424	+	0.0906597i
$366$		0			0
$367$	−7.50000	+	12.9904i	−0.391497	+	0.678092i	−0.992647	−	0.121044i	$-0.961376\pi$
$367$	−7.50000	+	12.9904i	−0.391497	+	0.678092i	0.601150	+	0.799136i	$0.294709\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	15.0000	−	25.9808i	0.778761	−	1.34885i
$372$		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
$379$	−13.0000			−0.667765			−0.333883	−	0.942615i	$-0.608359\pi$
$379$	−13.0000			−0.667765			−0.333883	+	0.942615i	$0.608359\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.0000	−	17.3205i	−0.510976	−	0.885037i	−0.999919	−	0.0127209i	$-0.995951\pi$
$383$	−10.0000	−	17.3205i	−0.510976	−	0.885037i	0.488943	−	0.872316i	$-0.337383\pi$
$384$		0			0
$385$	20.0000	+	34.6410i	1.01929	+	1.76547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.0000	+	19.0526i	−0.557722	+	0.966003i	0.439964	+	0.898015i	$0.354991\pi$
$389$	−11.0000	+	19.0526i	−0.557722	+	0.966003i	−0.997686	+	0.0679877i	$0.978342\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.0000	+	22.5167i	−0.654101	+	1.13294i
$396$		0			0
$397$	12.5000	+	21.6506i	0.627357	+	1.08661i	0.988080	+	0.153941i	$0.0491966\pi$
$397$	12.5000	+	21.6506i	0.627357	+	1.08661i	−0.360723	+	0.932673i	$0.617470\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	−0.992932	−	0.118686i	$-0.962132\pi$
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	0.393680	−	0.919247i	$-0.371202\pi$
$402$		0			0
$403$	−2.50000	+	4.33013i	−0.124534	+	0.215699i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−28.0000			−1.38791
$408$		0			0
$409$	−11.0000	+	19.0526i	−0.543915	+	0.942088i	0.454759	+	0.890614i	$0.349725\pi$
$409$	−11.0000	+	19.0526i	−0.543915	+	0.942088i	−0.998674	+	0.0514740i	$0.983608\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−20.0000	−	34.6410i	−0.984136	−	1.70457i
$414$		0			0
$415$	4.00000	+	6.92820i	0.196352	+	0.340092i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−3.00000	−	5.19615i	−0.146211	−	0.253245i	0.783613	−	0.621249i	$-0.213375\pi$
$421$	−3.00000	−	5.19615i	−0.146211	−	0.253245i	−0.929824	+	0.368004i	$0.880041\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.50000	−	4.33013i	0.120983	−	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.00000	+	5.19615i	−0.144505	+	0.250290i	−0.929188	−	0.369607i	$-0.879492\pi$
$431$	−3.00000	+	5.19615i	−0.144505	+	0.250290i	0.784683	+	0.619897i	$0.212826\pi$
$432$		0			0
$433$	2.50000	−	4.33013i	0.120142	−	0.208093i	−0.799681	−	0.600425i	$-0.794998\pi$
$433$	2.50000	−	4.33013i	0.120142	−	0.208093i	0.919824	+	0.392332i	$0.128332\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.0000	+	15.5885i	−1.00457	+	0.745697i
$438$		0			0
$439$	6.50000	+	11.2583i	0.310228	+	0.537331i	0.978412	−	0.206666i	$-0.0662612\pi$
$439$	6.50000	+	11.2583i	0.310228	+	0.537331i	−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$	24.0000			1.13771
$446$		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
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	50.0000			2.34404
$456$		0			0
$457$	−15.0000			−0.701670			−0.350835	−	0.936437i	$-0.614102\pi$
$457$	−15.0000			−0.701670			−0.350835	+	0.936437i	$0.614102\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.00000	−	13.8564i	−0.372597	−	0.645357i	0.617367	−	0.786675i	$-0.288199\pi$
$461$	−8.00000	−	13.8564i	−0.372597	−	0.645357i	−0.989964	+	0.141318i	$0.954866\pi$
$462$		0			0
$463$	−25.0000			−1.16185			−0.580924	−	0.813958i	$-0.697309\pi$
$463$	−25.0000			−1.16185			−0.580924	+	0.813958i	$0.697309\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$	12.5000	−	21.6506i	0.577196	−	0.999733i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	22.0000	+	38.1051i	1.01156	+	1.75208i
$474$		0			0
$475$	−3.50000	+	2.59808i	−0.160591	+	0.119208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	−0.583803	−	0.811895i	$-0.698436\pi$
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	0.995023	+	0.0996406i	$0.0317693\pi$
$480$		0			0
$481$	−17.5000	+	30.3109i	−0.797931	+	1.38206i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000	+	3.46410i	−0.0908153	+	0.157297i
$486$		0			0
$487$	−20.0000			−0.906287			−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000			−0.906287			−0.453143	+	0.891438i	$0.649697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.0000	+	22.5167i	0.586682	+	1.01616i	0.994663	+	0.103173i	$0.0328994\pi$
$491$	13.0000	+	22.5167i	0.586682	+	1.01616i	−0.407982	+	0.912990i	$0.633767\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15.0000	−	25.9808i	−0.672842	−	1.16540i
$498$		0			0
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	0.804627	−	0.593780i	$-0.202365\pi$
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	−0.916542	+	0.399937i	$0.869032\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	0.314459	+	0.949271i	$0.398177\pi$
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	−0.979322	+	0.202306i	$0.935156\pi$
$510$		0			0
$511$	−2.50000	−	4.33013i	−0.110593	−	0.191554i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.00000	−	8.66025i	−0.220326	−	0.381616i
$516$		0			0
$517$	20.0000	−	34.6410i	0.879599	−	1.52351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	16.0000			0.700973			0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000			0.700973			0.350486	+	0.936568i	$0.386016\pi$
$522$		0			0
$523$	−4.50000	+	7.79423i	−0.196771	+	0.340818i	−0.947480	−	0.319816i	$-0.896379\pi$
$523$	−4.50000	+	7.79423i	−0.196771	+	0.340818i	0.750708	+	0.660634i	$0.229712\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−6.50000	−	11.2583i	−0.282609	−	0.489493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−18.0000	−	31.1769i	−0.778208	−	1.34790i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−72.0000			−3.10126
$540$		0			0
$541$	−0.500000	+	0.866025i	−0.0214967	+	0.0372333i	−0.876574	−	0.481268i	$-0.840176\pi$
$541$	−0.500000	+	0.866025i	−0.0214967	+	0.0372333i	0.855077	+	0.518501i	$0.173510\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000	+	3.46410i	−0.0856706	+	0.148386i
$546$		0			0
$547$	8.50000	−	14.7224i	0.363434	−	0.629486i	−0.625090	−	0.780553i	$-0.714938\pi$
$547$	8.50000	−	14.7224i	0.363434	−	0.629486i	0.988524	+	0.151067i	$0.0482710\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.0000	+	20.7846i	−1.19284	+	0.885454i
$552$		0			0
$553$	−32.5000	−	56.2917i	−1.38204	−	2.39376i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.0000	−	36.3731i	0.889799	−	1.54118i	0.0496855	−	0.998765i	$-0.484178\pi$
$557$	21.0000	−	36.3731i	0.889799	−	1.54118i	0.840113	−	0.542411i	$-0.182489\pi$
$558$		0			0
$559$	55.0000			2.32625
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.0000			0.590030			0.295015	−	0.955493i	$-0.404675\pi$
$563$	14.0000			0.590030			0.295015	+	0.955493i	$0.404675\pi$
$564$		0			0
$565$	−4.00000	−	6.92820i	−0.168281	−	0.291472i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.0000			−0.503066			−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000			−0.503066			−0.251533	+	0.967849i	$0.580935\pi$
$570$		0			0
$571$	−11.0000			−0.460336			−0.230168	−	0.973151i	$-0.573928\pi$
$571$	−11.0000			−0.460336			−0.230168	+	0.973151i	$0.573928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.00000	−	5.19615i	−0.125109	−	0.216695i
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20.0000			−0.829740
$582$		0			0
$583$	−12.0000	+	20.7846i	−0.496989	+	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.0000	−	17.3205i	−0.412744	−	0.714894i	0.582445	−	0.812870i	$-0.302096\pi$
$587$	−10.0000	−	17.3205i	−0.412744	−	0.714894i	−0.995189	+	0.0979766i	$0.968763\pi$
$588$		0			0
$589$	−4.00000	−	1.73205i	−0.164817	−	0.0713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.0000	−	17.3205i	0.408589	−	0.707697i	−0.586143	−	0.810208i	$-0.699354\pi$
$599$	10.0000	−	17.3205i	0.408589	−	0.707697i	0.994732	+	0.102511i	$0.0326876\pi$
$600$		0			0
$601$	31.0000			1.26452			0.632258	−	0.774758i	$-0.282128\pi$
$601$	31.0000			1.26452			0.632258	+	0.774758i	$0.282128\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.00000	−	8.66025i	−0.203279	−	0.352089i
$606$		0			0
$607$	5.00000			0.202944			0.101472	−	0.994838i	$-0.467645\pi$
$607$	5.00000			0.202944			0.101472	+	0.994838i	$0.467645\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−25.0000	−	43.3013i	−1.01139	−	1.75178i
$612$		0			0
$613$	23.0000	+	39.8372i	0.928961	+	1.60901i	0.785063	+	0.619416i	$0.212630\pi$
$613$	23.0000	+	39.8372i	0.928961	+	1.60901i	0.143898	+	0.989593i	$0.454036\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.0000	−	41.5692i	0.966204	−	1.67351i	0.259858	−	0.965647i	$-0.416324\pi$
$617$	24.0000	−	41.5692i	0.966204	−	1.67351i	0.706346	−	0.707867i	$-0.250342\pi$
$618$		0			0
$619$	45.0000			1.80870			0.904351	−	0.426789i	$-0.140355\pi$
$619$	45.0000			1.80870			0.904351	+	0.426789i	$0.140355\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−30.0000	+	51.9615i	−1.20192	+	2.08179i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−14.5000	+	25.1147i	−0.577236	+	0.999802i	0.418559	+	0.908190i	$0.362535\pi$
$631$	−14.5000	+	25.1147i	−0.577236	+	0.999802i	−0.995795	+	0.0916122i	$0.970798\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−40.0000			−1.58735
$636$		0			0
$637$	−45.0000	+	77.9423i	−1.78296	+	3.08819i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.0000	+	39.8372i	0.908445	+	1.57347i	0.816224	+	0.577735i	$0.196063\pi$
$641$	23.0000	+	39.8372i	0.908445	+	1.57347i	0.0922210	+	0.995739i	$0.470603\pi$
$642$		0			0
$643$	−2.50000	−	4.33013i	−0.0985904	−	0.170764i	0.812511	−	0.582946i	$-0.198100\pi$
$643$	−2.50000	−	4.33013i	−0.0985904	−	0.170764i	−0.911101	+	0.412182i	$0.864767\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	40.0000			1.57256			0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000			1.57256			0.786281	+	0.617869i	$0.212004\pi$
$648$		0			0
$649$	16.0000	+	27.7128i	0.628055	+	1.08782i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−34.0000			−1.33052			−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000			−1.33052			−0.665261	+	0.746611i	$0.731680\pi$
$654$		0			0
$655$	−10.0000	+	17.3205i	−0.390732	+	0.676768i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−22.0000	+	38.1051i	−0.856998	+	1.48436i	0.0177803	+	0.999842i	$0.494340\pi$
$659$	−22.0000	+	38.1051i	−0.856998	+	1.48436i	−0.874779	+	0.484523i	$0.838993\pi$
$660$		0			0
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	−0.697174	−	0.716902i	$-0.745559\pi$
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	0.969442	+	0.245319i	$0.0788928\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.00000	+	43.3013i	0.193892	+	1.67915i
$666$		0			0
$667$	−24.0000	−	41.5692i	−0.929284	−	1.60957i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.00000	+	3.46410i	−0.0772091	+	0.133730i
$672$		0			0
$673$	35.0000			1.34915			0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000			1.34915			0.674575	+	0.738206i	$0.264327\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	30.0000			1.15299			0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000			1.15299			0.576497	+	0.817099i	$0.304419\pi$
$678$		0			0
$679$	−5.00000	−	8.66025i	−0.191882	−	0.332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.0000			0.382639			0.191320	−	0.981528i	$-0.438723\pi$
$683$	10.0000			0.382639			0.191320	+	0.981528i	$0.438723\pi$
$684$		0			0
$685$	−20.0000			−0.764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.0000	+	25.9808i	0.571454	+	0.989788i
$690$		0			0
$691$	44.0000			1.67384			0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000			1.67384			0.836919	+	0.547326i	$0.184354\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.0000			0.379322
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.00000	−	1.73205i	−0.0377695	−	0.0654187i	0.846523	−	0.532353i	$-0.178692\pi$
$701$	−1.00000	−	1.73205i	−0.0377695	−	0.0654187i	−0.884292	+	0.466934i	$0.845359\pi$
$702$		0			0
$703$	−28.0000	−	12.1244i	−1.05604	−	0.457279i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15.0000	+	25.9808i	−0.564133	+	0.977107i
$708$		0			0
$709$	13.5000	−	23.3827i	0.507003	−	0.878155i	−0.492964	−	0.870050i	$-0.664087\pi$
$709$	13.5000	−	23.3827i	0.507003	−	0.878155i	0.999967	−	0.00810550i	$-0.00258009\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.00000	−	5.19615i	0.112351	−	0.194597i
$714$		0			0
$715$	−40.0000			−1.49592
$716$		0			0
$717$		0			0
$718$		0			0
$719$	21.0000	+	36.3731i	0.783168	+	1.35649i	0.930087	+	0.367338i	$0.119731\pi$
$719$	21.0000	+	36.3731i	0.783168	+	1.35649i	−0.146920	+	0.989148i	$0.546936\pi$
$720$		0			0
$721$	25.0000			0.931049
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.00000	−	6.92820i	−0.148556	−	0.257307i
$726$		0			0
$727$	−18.5000	−	32.0429i	−0.686127	−	1.18841i	−0.973081	−	0.230463i	$-0.925976\pi$
$727$	−18.5000	−	32.0429i	−0.686127	−	1.18841i	0.286954	−	0.957944i	$-0.407357\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−26.0000			−0.960332			−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000			−0.960332			−0.480166	+	0.877178i	$0.659424\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.0000	+	17.3205i	−0.368355	+	0.638009i
$738$		0			0
$739$	−1.50000	−	2.59808i	−0.0551784	−	0.0955718i	0.837117	−	0.547024i	$-0.184239\pi$
$739$	−1.50000	−	2.59808i	−0.0551784	−	0.0955718i	−0.892295	+	0.451452i	$0.850906\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.0000	−	20.7846i	−0.440237	−	0.762513i	0.557470	−	0.830197i	$-0.311772\pi$
$743$	−12.0000	−	20.7846i	−0.440237	−	0.762513i	−0.997707	+	0.0676840i	$0.978439\pi$
$744$		0			0
$745$	8.00000	−	13.8564i	0.293097	−	0.507659i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	90.0000			3.28853
$750$		0			0
$751$	−14.5000	+	25.1147i	−0.529113	+	0.916450i	0.470311	+	0.882501i	$0.344142\pi$
$751$	−14.5000	+	25.1147i	−0.529113	+	0.916450i	−0.999424	+	0.0339490i	$0.989192\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000	+	27.7128i	0.582300	+	1.00857i
$756$		0			0
$757$	7.50000	+	12.9904i	0.272592	+	0.472143i	0.969525	−	0.244993i	$-0.0787857\pi$
$757$	7.50000	+	12.9904i	0.272592	+	0.472143i	−0.696933	+	0.717137i	$0.745452\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		0			0
$763$	−5.00000	−	8.66025i	−0.181012	−	0.313522i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	40.0000			1.44432
$768$		0			0
$769$	−21.5000	+	37.2391i	−0.775310	+	1.34288i	0.159310	+	0.987229i	$0.449073\pi$
$769$	−21.5000	+	37.2391i	−0.775310	+	1.34288i	−0.934620	+	0.355647i	$0.884260\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15.0000	+	25.9808i	−0.539513	+	0.934463i	0.459418	+	0.888220i	$0.348058\pi$
$773$	−15.0000	+	25.9808i	−0.539513	+	0.934463i	−0.998930	+	0.0462427i	$0.985275\pi$
$774$		0			0
$775$	0.500000	−	0.866025i	0.0179605	−	0.0311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	12.0000	+	20.7846i	0.429394	+	0.743732i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.0000	−	22.5167i	0.463990	−	0.803654i
$786$		0			0
$787$	5.00000			0.178231			0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000			0.178231			0.0891154	+	0.996021i	$0.471596\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	20.0000			0.711118
$792$		0			0
$793$	2.50000	+	4.33013i	0.0887776	+	0.153767i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.00000	+	3.46410i	0.0705785	+	0.122245i
$804$		0			0
$805$	−60.0000			−2.11472
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.00000			0.0703163			0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000			0.0703163			0.0351581	+	0.999382i	$0.488807\pi$
$810$		0			0
$811$	−8.00000	+	13.8564i	−0.280918	+	0.486564i	−0.971611	−	0.236584i	$-0.923972\pi$
$811$	−8.00000	+	13.8564i	−0.280918	+	0.486564i	0.690693	+	0.723148i	$0.257306\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.00000	+	1.73205i	0.0350285	+	0.0606711i
$816$		0			0
$817$	5.50000	+	47.6314i	0.192421	+	1.66641i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.0000	+	20.7846i	−0.418803	+	0.725388i	−0.995819	−	0.0913446i	$-0.970884\pi$
$821$	−12.0000	+	20.7846i	−0.418803	+	0.725388i	0.577016	+	0.816733i	$0.304217\pi$
$822$		0			0
$823$	−20.0000	+	34.6410i	−0.697156	+	1.20751i	0.272292	+	0.962215i	$0.412218\pi$
$823$	−20.0000	+	34.6410i	−0.697156	+	1.20751i	−0.969448	+	0.245295i	$0.921115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10.0000	+	17.3205i	−0.347734	+	0.602293i	−0.985847	−	0.167650i	$-0.946382\pi$
$827$	−10.0000	+	17.3205i	−0.347734	+	0.602293i	0.638112	+	0.769943i	$0.279715\pi$
$828$		0			0
$829$	45.0000			1.56291			0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000			1.56291			0.781457	+	0.623959i	$0.214477\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.0000	+	25.9808i	0.517858	+	0.896956i	0.999785	+	0.0207443i	$0.00660359\pi$
$839$	15.0000	+	25.9808i	0.517858	+	0.896956i	−0.481927	+	0.876211i	$0.660063\pi$
$840$		0			0
$841$	−17.5000	−	30.3109i	−0.603448	−	1.04520i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0000	+	20.7846i	−0.412813	+	0.715012i
$846$		0			0
$847$	25.0000			0.859010
$848$		0			0
$849$		0			0
$850$		0			0
$851$	21.0000	−	36.3731i	0.719871	−	1.24685i
$852$		0			0
$853$	0.500000	+	0.866025i	0.0171197	+	0.0296521i	0.874458	−	0.485101i	$-0.161217\pi$
$853$	0.500000	+	0.866025i	0.0171197	+	0.0296521i	−0.857339	+	0.514753i	$0.827884\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.0000	+	36.3731i	0.717346	+	1.24248i	0.962048	+	0.272882i	$0.0879768\pi$
$857$	21.0000	+	36.3731i	0.717346	+	1.24248i	−0.244701	+	0.969599i	$0.578690\pi$
$858$		0			0
$859$	8.50000	−	14.7224i	0.290016	−	0.502323i	−0.683797	−	0.729672i	$-0.739673\pi$
$859$	8.50000	−	14.7224i	0.290016	−	0.502323i	0.973813	+	0.227349i	$0.0730059\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34.0000			1.15737			0.578687	−	0.815550i	$-0.303565\pi$
$863$	34.0000			1.15737			0.578687	+	0.815550i	$0.303565\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	26.0000	+	45.0333i	0.881990	+	1.52765i
$870$		0			0
$871$	12.5000	+	21.6506i	0.423546	+	0.733604i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−60.0000			−2.02837
$876$		0			0
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	0.954652	−	0.297724i	$-0.0962275\pi$
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	−0.735163	+	0.677891i	$0.762894\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.0000			−0.471672			−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000			−0.471672			−0.235836	+	0.971793i	$0.575783\pi$
$882$		0			0
$883$	−17.5000	+	30.3109i	−0.588922	+	1.02004i	0.405452	+	0.914116i	$0.367114\pi$
$883$	−17.5000	+	30.3109i	−0.588922	+	1.02004i	−0.994374	+	0.105926i	$0.966219\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.0000	−	17.3205i	0.335767	−	0.581566i	−0.647865	−	0.761755i	$-0.724338\pi$
$887$	10.0000	−	17.3205i	0.335767	−	0.581566i	0.983632	+	0.180190i	$0.0576711\pi$
$888$		0			0
$889$	50.0000	−	86.6025i	1.67695	−	2.90456i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	35.0000	−	25.9808i	1.17123	−	0.869413i
$894$		0			0
$895$	−18.0000	−	31.1769i	−0.601674	−	1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.00000	−	6.92820i	0.133407	−	0.231069i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−36.0000			−1.19668
$906$		0			0
$907$	−10.0000	−	17.3205i	−0.332045	−	0.575118i	0.650868	−	0.759191i	$-0.274405\pi$
$907$	−10.0000	−	17.3205i	−0.332045	−	0.575118i	−0.982913	+	0.184073i	$0.941072\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$		0			0
$913$	16.0000			0.529523
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−25.0000	−	43.3013i	−0.825573	−	1.42993i
$918$		0			0
$919$	55.0000			1.81428			0.907141	−	0.420826i	$-0.138260\pi$
$919$	55.0000			1.81428			0.907141	+	0.420826i	$0.138260\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	30.0000			0.987462
$924$		0			0
$925$	3.50000	−	6.06218i	0.115079	−	0.199323i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	0.679770	−	0.733426i	$-0.262080\pi$
$929$	−9.00000	−	15.5885i	−0.295280	−	0.511441i	−0.975050	+	0.221985i	$0.928746\pi$
$930$		0			0
$931$	−72.0000	−	31.1769i	−2.35970	−	1.02178i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−11.5000	+	19.9186i	−0.375689	+	0.650712i	−0.990430	−	0.138017i	$-0.955927\pi$
$937$	−11.5000	+	19.9186i	−0.375689	+	0.650712i	0.614741	+	0.788729i	$0.289260\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.0000	+	48.4974i	−0.912774	+	1.58097i	−0.102646	+	0.994718i	$0.532731\pi$
$941$	−28.0000	+	48.4974i	−0.912774	+	1.58097i	−0.810128	+	0.586253i	$0.800603\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.0000	+	25.9808i	0.487435	+	0.844261i	0.999896	−	0.0144491i	$-0.00459946\pi$
$947$	15.0000	+	25.9808i	0.487435	+	0.844261i	−0.512461	+	0.858710i	$0.671266\pi$
$948$		0			0
$949$	5.00000			0.162307
$950$		0			0
$951$		0			0
$952$		0			0
$953$	30.0000	+	51.9615i	0.971795	+	1.68320i	0.690129	+	0.723686i	$0.257554\pi$
$953$	30.0000	+	51.9615i	0.971795	+	1.68320i	0.281666	+	0.959512i	$0.409113\pi$
$954$		0			0
$955$	26.0000	+	45.0333i	0.841340	+	1.45724i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	25.0000	−	43.3013i	0.807292	−	1.39827i
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.00000	−	8.66025i	0.160956	−	0.278783i
$966$		0			0
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	0.619721	−	0.784822i	$-0.287246\pi$
$967$	−11.5000	−	19.9186i	−0.369815	−	0.640538i	−0.989536	+	0.144283i	$0.953912\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−11.0000	−	19.0526i	−0.353007	−	0.611426i	0.633768	−	0.773523i	$-0.281507\pi$
$971$	−11.0000	−	19.0526i	−0.353007	−	0.611426i	−0.986775	+	0.162098i	$0.948174\pi$
$972$		0			0
$973$	−12.5000	+	21.6506i	−0.400732	+	0.694087i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−50.0000			−1.59964			−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000			−1.59964			−0.799821	+	0.600239i	$0.795072\pi$
$978$		0			0
$979$	24.0000	−	41.5692i	0.767043	−	1.32856i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−20.0000	−	34.6410i	−0.637901	−	1.10488i	−0.985893	−	0.167379i	$-0.946470\pi$
$983$	−20.0000	−	34.6410i	−0.637901	−	1.10488i	0.347992	−	0.937498i	$-0.386864\pi$
$984$		0			0
$985$	−20.0000	−	34.6410i	−0.637253	−	1.10375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−66.0000			−2.09868
$990$		0			0
$991$	12.5000	+	21.6506i	0.397076	+	0.687755i	0.993364	−	0.115015i	$-0.0366917\pi$
$991$	12.5000	+	21.6506i	0.397076	+	0.687755i	−0.596288	+	0.802771i	$0.703358\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.00000			−0.190213
$996$		0			0
$997$	12.5000	−	21.6506i	0.395879	−	0.685682i	−0.597334	−	0.801993i	$-0.703773\pi$
$997$	12.5000	−	21.6506i	0.395879	−	0.685682i	0.993213	+	0.116310i	$0.0371066\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.f.577.1		2
3.2	odd	2		912.2.q.c.577.1		2
4.3	odd	2		1368.2.s.b.577.1		2
12.11	even	2		456.2.q.c.121.1	yes	2
19.11	even	3	inner	2736.2.s.f.1873.1		2
57.11	odd	6		912.2.q.c.49.1		2
76.11	odd	6		1368.2.s.b.505.1		2
228.11	even	6		456.2.q.c.49.1	✓	2
228.83	even	6		8664.2.a.b.1.1		1
228.107	odd	6		8664.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.q.c.49.1	✓	2	228.11	even	6
456.2.q.c.121.1	yes	2	12.11	even	2
912.2.q.c.49.1		2	57.11	odd	6
912.2.q.c.577.1		2	3.2	odd	2
1368.2.s.b.505.1		2	76.11	odd	6
1368.2.s.b.577.1		2	4.3	odd	2
2736.2.s.f.577.1		2	1.1	even	1	trivial
2736.2.s.f.1873.1		2	19.11	even	3	inner
8664.2.a.b.1.1		1	228.83	even	6
8664.2.a.h.1.1		1	228.107	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} +5.00000 q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{5} +5.00000 q^{7} -4.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(-4.00000 - 1.73205i) q^{19} +(3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +(4.00000 - 6.92820i) q^{29} +1.00000 q^{31} +(-5.00000 - 8.66025i) q^{35} +7.00000 q^{37} +(-5.50000 - 9.52628i) q^{43} +(-5.00000 + 8.66025i) q^{47} +18.0000 q^{49} +(3.00000 - 5.19615i) q^{53} +(4.00000 + 6.92820i) q^{55} +(-4.00000 - 6.92820i) q^{59} +(0.500000 - 0.866025i) q^{61} +10.0000 q^{65} +(2.50000 - 4.33013i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(-0.500000 - 0.866025i) q^{73} -20.0000 q^{77} +(-6.50000 - 11.2583i) q^{79} -4.00000 q^{83} +(-6.00000 + 10.3923i) q^{89} +(-12.5000 + 21.6506i) q^{91} +(1.00000 + 8.66025i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 10 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 10 * q^7	\( 2 q - 2 q^{5} + 10 q^{7} - 8 q^{11} - 5 q^{13} - 8 q^{19} + 6 q^{23} + q^{25} + 8 q^{29} + 2 q^{31} - 10 q^{35} + 14 q^{37} - 11 q^{43} - 10 q^{47} + 36 q^{49} + 6 q^{53} + 8 q^{55} - 8 q^{59} + q^{61} + 20 q^{65} + 5 q^{67} - 6 q^{71} - q^{73} - 40 q^{77} - 13 q^{79} - 8 q^{83} - 12 q^{89} - 25 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 10 * q^7 - 8 * q^11 - 5 * q^13 - 8 * q^19 + 6 * q^23 + q^25 + 8 * q^29 + 2 * q^31 - 10 * q^35 + 14 * q^37 - 11 * q^43 - 10 * q^47 + 36 * q^49 + 6 * q^53 + 8 * q^55 - 8 * q^59 + q^61 + 20 * q^65 + 5 * q^67 - 6 * q^71 - q^73 - 40 * q^77 - 13 * q^79 - 8 * q^83 - 12 * q^89 - 25 * q^91 + 2 * q^95 - 2 * q^97

Introduction

L-functions

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Database

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