LMFDB - Embedded newform 1368.2.s.c.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(505,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1368.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.s (of order $3$, 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:	$10.9235349965$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1368.577
Dual form			1368.2.s.c.505.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} +3.00000 q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{5} +3.00000 q^{7} -6.00000 q^{11} +(0.500000 - 0.866025i) q^{13} +(-1.00000 - 1.73205i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(0.500000 - 0.866025i) q^{25} +(-1.00000 + 1.73205i) q^{29} -1.00000 q^{31} +(-3.00000 - 5.19615i) q^{35} -7.00000 q^{37} +(0.500000 + 0.866025i) q^{43} +2.00000 q^{49} +(-2.00000 + 3.46410i) q^{53} +(6.00000 + 10.3923i) q^{55} +(-4.00000 - 6.92820i) q^{59} +(5.50000 - 9.52628i) q^{61} -2.00000 q^{65} +(-7.50000 + 12.9904i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(-4.50000 - 7.79423i) q^{73} -18.0000 q^{77} +(6.50000 + 11.2583i) q^{79} -14.0000 q^{83} +(-2.00000 + 3.46410i) q^{85} +(6.00000 - 10.3923i) q^{89} +(1.50000 - 2.59808i) q^{91} +(1.00000 + 8.66025i) q^{95} +(-5.00000 - 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 6 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 6 * q^7	$ 2 q - 2 q^{5} + 6 q^{7} - 12 q^{11} + q^{13} - 2 q^{17} - 8 q^{19} + q^{25} - 2 q^{29} - 2 q^{31} - 6 q^{35} - 14 q^{37} + q^{43} + 4 q^{49} - 4 q^{53} + 12 q^{55} - 8 q^{59} + 11 q^{61} - 4 q^{65} - 15 q^{67} - 6 q^{71} - 9 q^{73} - 36 q^{77} + 13 q^{79} - 28 q^{83} - 4 q^{85} + 12 q^{89} + 3 q^{91} + 2 q^{95} - 10 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^7 - 12 * q^11 + q^13 - 2 * q^17 - 8 * q^19 + q^25 - 2 * q^29 - 2 * q^31 - 6 * q^35 - 14 * q^37 + q^43 + 4 * q^49 - 4 * q^53 + 12 * q^55 - 8 * q^59 + 11 * q^61 - 4 * q^65 - 15 * q^67 - 6 * q^71 - 9 * q^73 - 36 * q^77 + 13 * q^79 - 28 * q^83 - 4 * q^85 + 12 * q^89 + 3 * q^91 + 2 * q^95 - 10 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	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
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.00000			−1.80907			−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000			−1.80907			−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	$-0.244646\pi$
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	−0.961436	+	0.275029i	$0.911312\pi$
$18$		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
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.00000	+	1.73205i	−0.185695	+	0.321634i	−0.943811	−	0.330487i	$-0.892787\pi$
$29$	−1.00000	+	1.73205i	−0.185695	+	0.321634i	0.758115	+	0.652121i	$0.226120\pi$
$30$		0			0
$31$	−1.00000			−0.179605			−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000			−0.179605			−0.0898027	+	0.995960i	$0.528624\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000	−	5.19615i	−0.507093	−	0.878310i
$36$		0			0
$37$	−7.00000			−1.15079			−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000			−1.15079			−0.575396	+	0.817875i	$0.695152\pi$
$38$		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
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	0.695344	+	0.718677i	$0.255252\pi$
$54$		0			0
$55$	6.00000	+	10.3923i	0.809040	+	1.40130i
$56$		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
$61$	5.50000	−	9.52628i	0.704203	−	1.21972i	−0.262776	−	0.964857i	$-0.584638\pi$
$61$	5.50000	−	9.52628i	0.704203	−	1.21972i	0.966978	−	0.254858i	$-0.0820288\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000			−0.248069
$66$		0			0
$67$	−7.50000	+	12.9904i	−0.916271	+	1.58703i	−0.111241	+	0.993793i	$0.535483\pi$
$67$	−7.50000	+	12.9904i	−0.916271	+	1.58703i	−0.805030	+	0.593234i	$0.797851\pi$
$68$		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
$73$	−4.50000	−	7.79423i	−0.526685	−	0.912245i	−0.999517	−	0.0310925i	$-0.990101\pi$
$73$	−4.50000	−	7.79423i	−0.526685	−	0.912245i	0.472831	−	0.881153i	$-0.343232\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−18.0000			−2.05129
$78$		0			0
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	0.956325	+	0.292306i	$0.0944227\pi$
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−14.0000			−1.53670			−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000			−1.53670			−0.768350	+	0.640030i	$0.778922\pi$
$84$		0			0
$85$	−2.00000	+	3.46410i	−0.216930	+	0.375735i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	−0.350304	−	0.936636i	$-0.613922\pi$
$89$	6.00000	−	10.3923i	0.635999	−	1.10158i	0.986303	−	0.164946i	$-0.0527450\pi$
$90$		0			0
$91$	1.50000	−	2.59808i	0.157243	−	0.272352i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.00000	+	8.66025i	0.102598	+	0.888523i
$96$		0			0
$97$	−5.00000	−	8.66025i	−0.507673	−	0.879316i	−0.999961	−	0.00888289i	$-0.997172\pi$
$97$	−5.00000	−	8.66025i	−0.507673	−	0.879316i	0.492287	−	0.870433i	$-0.336161\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	−0.677284	−	0.735721i	$-0.736843\pi$
$101$	3.00000	−	5.19615i	0.298511	−	0.517036i	0.975796	+	0.218685i	$0.0701767\pi$
$102$		0			0
$103$	−5.00000			−0.492665			−0.246332	−	0.969185i	$-0.579225\pi$
$103$	−5.00000			−0.492665			−0.246332	+	0.969185i	$0.579225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.0000			0.966736			0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000			0.966736			0.483368	+	0.875417i	$0.339413\pi$
$108$		0			0
$109$	−9.00000	−	15.5885i	−0.862044	−	1.49310i	−0.869953	−	0.493135i	$-0.835851\pi$
$109$	−9.00000	−	15.5885i	−0.862044	−	1.49310i	0.00790932	−	0.999969i	$-0.497482\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−16.0000			−1.50515			−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000			−1.50515			−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.00000	−	5.19615i	−0.275010	−	0.476331i
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$127$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.00000	+	15.5885i	0.786334	+	1.36197i	0.928199	+	0.372084i	$0.121357\pi$
$131$	9.00000	+	15.5885i	0.786334	+	1.36197i	−0.141865	+	0.989886i	$0.545310\pi$
$132$		0			0
$133$	−12.0000	−	5.19615i	−1.04053	−	0.450564i
$134$		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	−0.952355	−	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
$143$	−3.00000	+	5.19615i	−0.250873	+	0.434524i
$144$		0			0
$145$	4.00000			0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	0.822153	−	0.569267i	$-0.192773\pi$
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	−0.904076	+	0.427372i	$0.859440\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.00000	+	1.73205i	0.0803219	+	0.139122i
$156$		0			0
$157$	−2.50000	−	4.33013i	−0.199522	−	0.345582i	0.748852	−	0.662738i	$-0.230606\pi$
$157$	−2.50000	−	4.33013i	−0.199522	−	0.345582i	−0.948373	+	0.317156i	$0.897272\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	19.0000			1.48819			0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000			1.48819			0.744097	+	0.668071i	$0.232880\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.00000	−	6.92820i	0.309529	−	0.536120i	−0.668730	−	0.743505i	$-0.733162\pi$
$167$	4.00000	−	6.92820i	0.309529	−	0.536120i	0.978259	+	0.207385i	$0.0664952\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.00000	+	12.1244i	0.532200	+	0.921798i	0.999293	+	0.0375896i	$0.0119679\pi$
$173$	7.00000	+	12.1244i	0.532200	+	0.921798i	−0.467093	+	0.884208i	$0.654699\pi$
$174$		0			0
$175$	1.50000	−	2.59808i	0.113389	−	0.196396i
$176$		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
$181$	5.00000	−	8.66025i	0.371647	−	0.643712i	−0.618172	−	0.786043i	$-0.712126\pi$
$181$	5.00000	−	8.66025i	0.371647	−	0.643712i	0.989819	+	0.142331i	$0.0454598\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.00000	+	12.1244i	0.514650	+	0.891400i
$186$		0			0
$187$	6.00000	+	10.3923i	0.438763	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$		0			0
$193$	4.50000	+	7.79423i	0.323917	+	0.561041i	0.981293	−	0.192522i	$-0.0616668\pi$
$193$	4.50000	+	7.79423i	0.323917	+	0.561041i	−0.657376	+	0.753563i	$0.728333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−1.50000	+	2.59808i	−0.106332	+	0.184173i	−0.914282	−	0.405079i	$-0.867244\pi$
$199$	−1.50000	+	2.59808i	−0.106332	+	0.184173i	0.807950	+	0.589252i	$0.200577\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.00000	+	5.19615i	−0.210559	+	0.364698i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	24.0000	+	10.3923i	1.66011	+	0.718851i
$210$		0			0
$211$	−4.50000	−	7.79423i	−0.309793	−	0.536577i	0.668524	−	0.743690i	$-0.266926\pi$
$211$	−4.50000	−	7.79423i	−0.309793	−	0.536577i	−0.978317	+	0.207114i	$0.933593\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.00000	−	1.73205i	0.0681994	−	0.118125i
$216$		0			0
$217$	−3.00000			−0.203653
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000			−0.134535
$222$		0			0
$223$	2.50000	+	4.33013i	0.167412	+	0.289967i	0.937509	−	0.347960i	$-0.113126\pi$
$223$	2.50000	+	4.33013i	0.167412	+	0.289967i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	22.0000			1.46019			0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000			1.46019			0.730096	+	0.683345i	$0.239475\pi$
$228$		0			0
$229$	−17.0000			−1.12339			−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000			−1.12339			−0.561696	+	0.827344i	$0.689851\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.00000	+	8.66025i	0.327561	+	0.567352i	0.982027	−	0.188739i	$-0.0604400\pi$
$233$	5.00000	+	8.66025i	0.327561	+	0.567352i	−0.654466	+	0.756091i	$0.727107\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−18.0000			−1.16432			−0.582162	−	0.813073i	$-0.697793\pi$
$239$	−18.0000			−1.16432			−0.582162	+	0.813073i	$0.697793\pi$
$240$		0			0
$241$	−4.50000	+	7.79423i	−0.289870	+	0.502070i	−0.973779	−	0.227498i	$-0.926946\pi$
$241$	−4.50000	+	7.79423i	−0.289870	+	0.502070i	0.683908	+	0.729568i	$0.260279\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.00000	−	3.46410i	−0.127775	−	0.221313i
$246$		0			0
$247$	−3.50000	+	2.59808i	−0.222700	+	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.00000	−	13.8564i	0.504956	−	0.874609i	−0.495028	−	0.868877i	$-0.664842\pi$
$251$	8.00000	−	13.8564i	0.504956	−	0.874609i	0.999984	−	0.00573163i	$-0.00182444\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	−0.990217	−	0.139533i	$-0.955440\pi$
$257$	−6.00000	+	10.3923i	−0.374270	+	0.648254i	0.615948	+	0.787787i	$0.288773\pi$
$258$		0			0
$259$	−21.0000			−1.30488
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.00000	−	8.66025i	−0.308313	−	0.534014i	0.669680	−	0.742650i	$-0.266431\pi$
$263$	−5.00000	−	8.66025i	−0.308313	−	0.534014i	−0.977993	+	0.208635i	$0.933098\pi$
$264$		0			0
$265$	8.00000			0.491436
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.0000	+	27.7128i	0.975537	+	1.68968i	0.678151	+	0.734923i	$0.262782\pi$
$269$	16.0000	+	27.7128i	0.975537	+	1.68968i	0.297386	+	0.954757i	$0.403885\pi$
$270$		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
$275$	−3.00000	+	5.19615i	−0.180907	+	0.313340i
$276$		0			0
$277$	−30.0000			−1.80253			−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000			−1.80253			−0.901263	+	0.433273i	$0.857359\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	−0.677466	−	0.735554i	$-0.736922\pi$
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	0.975741	+	0.218926i	$0.0702554\pi$
$282$		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
$287$		0			0
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	20.0000			1.16841			0.584206	−	0.811605i	$-0.301406\pi$
$293$	20.0000			1.16841			0.584206	+	0.811605i	$0.301406\pi$
$294$		0			0
$295$	−8.00000	+	13.8564i	−0.465778	+	0.806751i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.50000	+	2.59808i	0.0864586	+	0.149751i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−22.0000			−1.25972
$306$		0			0
$307$	10.0000	+	17.3205i	0.570730	+	0.988534i	0.996491	+	0.0836980i	$0.0266731\pi$
$307$	10.0000	+	17.3205i	0.570730	+	0.988534i	−0.425761	+	0.904836i	$0.639994\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	−0.689412	−	0.724370i	$-0.742131\pi$
$313$	5.00000	−	8.66025i	0.282617	−	0.489506i	0.972028	+	0.234863i	$0.0754642\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	−0.0453045	−	0.998973i	$-0.514426\pi$
$317$	15.0000	−	25.9808i	0.842484	−	1.45922i	0.887788	−	0.460252i	$-0.152241\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.00000	+	8.66025i	0.0556415	+	0.481869i
$324$		0			0
$325$	−0.500000	−	0.866025i	−0.0277350	−	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.00000			0.384755			0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000			0.384755			0.192377	+	0.981321i	$0.438380\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	30.0000			1.63908
$336$		0			0
$337$	7.50000	+	12.9904i	0.408551	+	0.707631i	0.994728	−	0.102552i	$-0.0327009\pi$
$337$	7.50000	+	12.9904i	0.408551	+	0.707631i	−0.586177	+	0.810183i	$0.699368\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$	−15.0000			−0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9.00000	+	15.5885i	0.483145	+	0.836832i	0.999813	−	0.0193540i	$-0.00616095\pi$
$347$	9.00000	+	15.5885i	0.483145	+	0.836832i	−0.516667	+	0.856186i	$0.672828\pi$
$348$		0			0
$349$	5.00000			0.267644			0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000			0.267644			0.133822	+	0.991005i	$0.457275\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.0000			0.745145			0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000			0.745145			0.372572	+	0.928003i	$0.378476\pi$
$354$		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	$0.0103087\pi$
$359$	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
$365$	−9.00000	+	15.5885i	−0.471082	+	0.815937i
$366$		0			0
$367$	17.5000	−	30.3109i	0.913493	−	1.58222i	0.104399	−	0.994535i	$-0.466708\pi$
$367$	17.5000	−	30.3109i	0.913493	−	1.58222i	0.809093	−	0.587680i	$-0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.00000	+	10.3923i	−0.311504	+	0.539542i
$372$		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
$377$	1.00000	+	1.73205i	0.0515026	+	0.0892052i
$378$		0			0
$379$	27.0000			1.38690			0.693448	−	0.720506i	$-0.256091\pi$
$379$	27.0000			1.38690			0.693448	+	0.720506i	$0.256091\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.00000	+	12.1244i	0.357683	+	0.619526i	0.987573	−	0.157159i	$-0.0502334\pi$
$383$	7.00000	+	12.1244i	0.357683	+	0.619526i	−0.629890	+	0.776684i	$0.716900\pi$
$384$		0			0
$385$	18.0000	+	31.1769i	0.917365	+	1.58892i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.0000	−	27.7128i	0.811232	−	1.40510i	−0.100770	−	0.994910i	$-0.532131\pi$
$389$	16.0000	−	27.7128i	0.811232	−	1.40510i	0.912002	−	0.410186i	$-0.134536\pi$
$390$		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	$-0.950803\pi$
$397$	−12.5000	−	21.6506i	−0.627357	−	1.08661i	0.360723	−	0.932673i	$-0.382530\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	0.992932	+	0.118686i	$0.0378683\pi$
$401$	12.0000	+	20.7846i	0.599251	+	1.03793i	−0.393680	+	0.919247i	$0.628798\pi$
$402$		0			0
$403$	−0.500000	+	0.866025i	−0.0249068	+	0.0431398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	42.0000			2.08186
$408$		0			0
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.173064	−	0.984911i	$-0.444633\pi$
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.766426	−	0.642333i	$-0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12.0000	−	20.7846i	−0.590481	−	1.02274i
$414$		0			0
$415$	14.0000	+	24.2487i	0.687233	+	1.19032i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$	−13.0000	−	22.5167i	−0.633581	−	1.09739i	−0.986814	−	0.161859i	$-0.948251\pi$
$421$	−13.0000	−	22.5167i	−0.633581	−	1.09739i	0.353233	−	0.935536i	$-0.385082\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$	16.5000	−	28.5788i	0.798491	−	1.38303i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−18.0000	+	31.1769i	−0.867029	+	1.50174i	−0.00201168	+	0.999998i	$0.500640\pi$
$431$	−18.0000	+	31.1769i	−0.867029	+	1.50174i	−0.865018	+	0.501741i	$0.832693\pi$
$432$		0			0
$433$	−9.50000	+	16.4545i	−0.456541	+	0.790752i	−0.998775	−	0.0494752i	$-0.984245\pi$
$433$	−9.50000	+	16.4545i	−0.456541	+	0.790752i	0.542234	+	0.840227i	$0.317578\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.00000	+	8.66025i	−0.237557	+	0.411461i	−0.960013	−	0.279956i	$-0.909680\pi$
$443$	−5.00000	+	8.66025i	−0.237557	+	0.411461i	0.722456	+	0.691417i	$0.243013\pi$
$444$		0			0
$445$	−24.0000			−1.13771
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.00000			−0.281284
$456$		0			0
$457$	25.0000			1.16945			0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000			1.16945			0.584725	+	0.811231i	$0.301202\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.0000	+	31.1769i	0.838344	+	1.45205i	0.891279	+	0.453456i	$0.149809\pi$
$461$	18.0000	+	31.1769i	0.838344	+	1.45205i	−0.0529352	+	0.998598i	$0.516858\pi$
$462$		0			0
$463$	−31.0000			−1.44069			−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000			−1.44069			−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.00000			0.370196			0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000			0.370196			0.185098	+	0.982720i	$0.440740\pi$
$468$		0			0
$469$	−22.5000	+	38.9711i	−1.03895	+	1.79952i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.00000	−	5.19615i	−0.137940	−	0.238919i
$474$		0			0
$475$	−3.50000	+	2.59808i	−0.160591	+	0.119208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.0000	−	19.0526i	0.502603	−	0.870534i	−0.497393	−	0.867526i	$-0.665709\pi$
$479$	11.0000	−	19.0526i	0.502603	−	0.870534i	0.999995	−	0.00300810i	$-0.000957509\pi$
$480$		0			0
$481$	−3.50000	+	6.06218i	−0.159586	+	0.276412i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.0000	+	17.3205i	−0.454077	+	0.786484i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.0000	−	29.4449i	−0.767199	−	1.32883i	−0.939076	−	0.343710i	$-0.888316\pi$
$491$	−17.0000	−	29.4449i	−0.767199	−	1.32883i	0.171877	−	0.985118i	$-0.445017\pi$
$492$		0			0
$493$	4.00000			0.180151
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.00000	−	15.5885i	−0.403705	−	0.699238i
$498$		0			0
$499$	−12.5000	−	21.6506i	−0.559577	−	0.969216i	−0.997532	−	0.0702185i	$-0.977630\pi$
$499$	−12.5000	−	21.6506i	−0.559577	−	0.969216i	0.437955	−	0.898997i	$-0.355703\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	15.0000	−	25.9808i	0.668817	−	1.15842i	−0.309418	−	0.950926i	$-0.600134\pi$
$503$	15.0000	−	25.9808i	0.668817	−	1.15842i	0.978235	−	0.207499i	$-0.0665323\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.0000	−	17.3205i	0.443242	−	0.767718i	−0.554686	−	0.832060i	$-0.687161\pi$
$509$	10.0000	−	17.3205i	0.443242	−	0.767718i	0.997928	+	0.0643419i	$0.0204948\pi$
$510$		0			0
$511$	−13.5000	−	23.3827i	−0.597205	−	1.03439i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5.00000	+	8.66025i	0.220326	+	0.381616i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14.0000			−0.613351			−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000			−0.613351			−0.306676	+	0.951814i	$0.599217\pi$
$522$		0			0
$523$	15.5000	−	26.8468i	0.677768	−	1.17393i	−0.297884	−	0.954602i	$-0.596281\pi$
$523$	15.5000	−	26.8468i	0.677768	−	1.17393i	0.975652	−	0.219326i	$-0.0703858\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.00000	+	1.73205i	0.0435607	+	0.0754493i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−10.0000	−	17.3205i	−0.432338	−	0.748831i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−15.5000	+	26.8468i	−0.666397	+	1.15423i	0.312507	+	0.949915i	$0.398831\pi$
$541$	−15.5000	+	26.8468i	−0.666397	+	1.15423i	−0.978905	+	0.204318i	$0.934502\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−18.0000	+	31.1769i	−0.771035	+	1.33547i
$546$		0			0
$547$	2.50000	−	4.33013i	0.106892	−	0.185143i	−0.807617	−	0.589707i	$-0.799243\pi$
$547$	2.50000	−	4.33013i	0.106892	−	0.185143i	0.914510	+	0.404564i	$0.132577\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	7.00000	−	5.19615i	0.298210	−	0.221364i
$552$		0			0
$553$	19.5000	+	33.7750i	0.829224	+	1.43626i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.00000	−	8.66025i	0.211857	−	0.366947i	−0.740439	−	0.672124i	$-0.765382\pi$
$557$	5.00000	−	8.66025i	0.211857	−	0.366947i	0.952296	+	0.305177i	$0.0987156\pi$
$558$		0			0
$559$	1.00000			0.0422955
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−46.0000			−1.93867			−0.969334	−	0.245745i	$-0.920967\pi$
$563$	−46.0000			−1.93867			−0.969334	+	0.245745i	$0.920967\pi$
$564$		0			0
$565$	16.0000	+	27.7128i	0.673125	+	1.16589i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.0000			−1.17382			−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000			−1.17382			−0.586911	+	0.809652i	$0.699656\pi$
$570$		0			0
$571$	−19.0000			−0.795125			−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000			−0.795125			−0.397563	+	0.917575i	$0.630144\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−42.0000			−1.74245
$582$		0			0
$583$	12.0000	−	20.7846i	0.496989	−	0.860811i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14.0000	−	24.2487i	−0.577842	−	1.00085i	−0.995726	−	0.0923513i	$-0.970562\pi$
$587$	−14.0000	−	24.2487i	−0.577842	−	1.00085i	0.417885	−	0.908500i	$-0.362772\pi$
$588$		0			0
$589$	4.00000	+	1.73205i	0.164817	+	0.0713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.0000	−	31.1769i	0.739171	−	1.28028i	−0.213697	−	0.976900i	$-0.568551\pi$
$593$	18.0000	−	31.1769i	0.739171	−	1.28028i	0.952869	−	0.303383i	$-0.0981160\pi$
$594$		0			0
$595$	−6.00000	+	10.3923i	−0.245976	+	0.426043i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.00000	+	12.1244i	−0.286012	+	0.495388i	−0.972854	−	0.231419i	$-0.925663\pi$
$599$	−7.00000	+	12.1244i	−0.286012	+	0.495388i	0.686842	+	0.726807i	$0.258996\pi$
$600$		0			0
$601$	−1.00000			−0.0407909			−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000			−0.0407909			−0.0203954	+	0.999792i	$0.506493\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−25.0000	−	43.3013i	−1.01639	−	1.76045i
$606$		0			0
$607$	27.0000			1.09590			0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000			1.09590			0.547948	+	0.836512i	$0.315409\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7.00000	−	12.1244i	−0.282727	−	0.489698i	0.689328	−	0.724449i	$-0.257906\pi$
$613$	−7.00000	−	12.1244i	−0.282727	−	0.489698i	−0.972056	+	0.234751i	$0.924572\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−15.0000	+	25.9808i	−0.603877	+	1.04595i	0.388351	+	0.921512i	$0.373045\pi$
$617$	−15.0000	+	25.9808i	−0.603877	+	1.04595i	−0.992228	+	0.124434i	$0.960288\pi$
$618$		0			0
$619$	1.00000			0.0401934			0.0200967	−	0.999798i	$-0.493603\pi$
$619$	1.00000			0.0401934			0.0200967	+	0.999798i	$0.493603\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.0000	−	31.1769i	0.721155	−	1.24908i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.00000	+	12.1244i	0.279108	+	0.483430i
$630$		0			0
$631$	−15.5000	+	26.8468i	−0.617045	+	1.06875i	0.372977	+	0.927841i	$0.378337\pi$
$631$	−15.5000	+	26.8468i	−0.617045	+	1.06875i	−0.990022	+	0.140913i	$0.954996\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.00000	−	1.73205i	0.0396214	−	0.0686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000	−	20.7846i	−0.473972	−	0.820943i	0.525584	−	0.850741i	$-0.323847\pi$
$641$	−12.0000	−	20.7846i	−0.473972	−	0.820943i	−0.999556	+	0.0297987i	$0.990513\pi$
$642$		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
$647$	−38.0000			−1.49393			−0.746967	−	0.664861i	$-0.768491\pi$
$647$	−38.0000			−1.49393			−0.746967	+	0.664861i	$0.768491\pi$
$648$		0			0
$649$	24.0000	+	41.5692i	0.942082	+	1.63173i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$	18.0000	−	31.1769i	0.703318	−	1.21818i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.00000	−	5.19615i	0.116863	−	0.202413i	−0.801660	−	0.597781i	$-0.796049\pi$
$659$	3.00000	−	5.19615i	0.116863	−	0.202413i	0.918523	+	0.395367i	$0.129383\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.00000	+	25.9808i	0.116335	+	1.00749i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−33.0000	+	57.1577i	−1.27395	+	2.20655i
$672$		0			0
$673$	15.0000			0.578208			0.289104	−	0.957298i	$-0.406643\pi$
$673$	15.0000			0.578208			0.289104	+	0.957298i	$0.406643\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.00000			−0.307465			−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000			−0.307465			−0.153732	+	0.988113i	$0.549129\pi$
$678$		0			0
$679$	−15.0000	−	25.9808i	−0.575647	−	0.997050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.0000			−0.382639			−0.191320	−	0.981528i	$-0.561277\pi$
$683$	−10.0000			−0.382639			−0.191320	+	0.981528i	$0.561277\pi$
$684$		0			0
$685$	−36.0000			−1.37549
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.00000	+	3.46410i	0.0761939	+	0.131972i
$690$		0			0
$691$	−36.0000			−1.36950			−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000			−1.36950			−0.684752	+	0.728776i	$0.740090\pi$
$692$		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
$701$	−15.0000	−	25.9808i	−0.566542	−	0.981280i	−0.996904	−	0.0786236i	$-0.974947\pi$
$701$	−15.0000	−	25.9808i	−0.566542	−	0.981280i	0.430362	−	0.902656i	$-0.358386\pi$
$702$		0			0
$703$	28.0000	+	12.1244i	1.05604	+	0.457279i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.00000	−	15.5885i	0.338480	−	0.586264i
$708$		0			0
$709$	16.5000	−	28.5788i	0.619671	−	1.07330i	−0.369875	−	0.929081i	$-0.620600\pi$
$709$	16.5000	−	28.5788i	0.619671	−	1.07330i	0.989546	−	0.144219i	$-0.0460671\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.00000	−	6.92820i	−0.149175	−	0.258378i	0.781748	−	0.623595i	$-0.214328\pi$
$719$	−4.00000	−	6.92820i	−0.149175	−	0.258378i	−0.930923	+	0.365216i	$0.880995\pi$
$720$		0			0
$721$	−15.0000			−0.558629
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.00000	+	1.73205i	0.0371391	+	0.0643268i
$726$		0			0
$727$	8.50000	+	14.7224i	0.315248	+	0.546025i	0.979490	−	0.201492i	$-0.0645791\pi$
$727$	8.50000	+	14.7224i	0.315248	+	0.546025i	−0.664243	+	0.747517i	$0.731246\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.00000	−	1.73205i	0.0369863	−	0.0640622i
$732$		0			0
$733$	−14.0000			−0.517102			−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000			−0.517102			−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	45.0000	−	77.9423i	1.65760	−	2.87104i
$738$		0			0
$739$	−3.50000	−	6.06218i	−0.128750	−	0.223001i	0.794443	−	0.607339i	$-0.207763\pi$
$739$	−3.50000	−	6.06218i	−0.128750	−	0.223001i	−0.923192	+	0.384338i	$0.874430\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	0.805735	−	0.592277i	$-0.201771\pi$
$743$	−3.00000	−	5.19615i	−0.110059	−	0.190628i	−0.915794	+	0.401648i	$0.868437\pi$
$744$		0			0
$745$	−2.00000	+	3.46410i	−0.0732743	+	0.126915i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	30.0000			1.09618
$750$		0			0
$751$	−5.50000	+	9.52628i	−0.200698	+	0.347619i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−5.50000	+	9.52628i	−0.200698	+	0.347619i	0.748056	+	0.663636i	$0.230988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.00000	+	6.92820i	0.145575	+	0.252143i
$756$		0			0
$757$	−23.5000	−	40.7032i	−0.854122	−	1.47938i	−0.877457	−	0.479655i	$-0.840762\pi$
$757$	−23.5000	−	40.7032i	−0.854122	−	1.47938i	0.0233351	−	0.999728i	$-0.492572\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−40.0000			−1.45000			−0.724999	−	0.688749i	$-0.758160\pi$
$761$	−40.0000			−1.45000			−0.724999	+	0.688749i	$0.758160\pi$
$762$		0			0
$763$	−27.0000	−	46.7654i	−0.977466	−	1.69302i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.00000			−0.288863
$768$		0			0
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	−0.724701	−	0.689063i	$-0.758022\pi$
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	0.959097	+	0.283078i	$0.0913554\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.0000	+	34.6410i	−0.719350	+	1.24595i	0.241908	+	0.970299i	$0.422227\pi$
$773$	−20.0000	+	34.6410i	−0.719350	+	1.24595i	−0.961258	+	0.275651i	$0.911106\pi$
$774$		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
$781$	18.0000	+	31.1769i	0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5.00000	+	8.66025i	−0.178458	+	0.309098i
$786$		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
$791$	−48.0000			−1.70668
$792$		0			0
$793$	−5.50000	−	9.52628i	−0.195311	−	0.338288i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	42.0000			1.48772			0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000			1.48772			0.743858	+	0.668338i	$0.232994\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	27.0000	+	46.7654i	0.952809	+	1.65031i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	18.0000	−	31.1769i	0.632065	−	1.09477i	−0.355063	−	0.934842i	$-0.615541\pi$
$811$	18.0000	−	31.1769i	0.632065	−	1.09477i	0.987129	−	0.159927i	$-0.0511260\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.0000	−	32.9090i	−0.665541	−	1.15275i
$816$		0			0
$817$	−0.500000	−	4.33013i	−0.0174928	−	0.151492i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−23.0000	+	39.8372i	−0.802706	+	1.39033i	0.115124	+	0.993351i	$0.463274\pi$
$821$	−23.0000	+	39.8372i	−0.802706	+	1.39033i	−0.917829	+	0.396976i	$0.870060\pi$
$822$		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
$827$	5.00000	−	8.66025i	0.173867	−	0.301147i	−0.765902	−	0.642958i	$-0.777707\pi$
$827$	5.00000	−	8.66025i	0.173867	−	0.301147i	0.939769	+	0.341811i	$0.111040\pi$
$828$		0			0
$829$	15.0000			0.520972			0.260486	−	0.965478i	$-0.416117\pi$
$829$	15.0000			0.520972			0.260486	+	0.965478i	$0.416117\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.00000	−	3.46410i	−0.0692959	−	0.120024i
$834$		0			0
$835$	−16.0000			−0.553703
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.0000	−	25.9808i	−0.517858	−	0.896956i	−0.999785	−	0.0207443i	$-0.993396\pi$
$839$	−15.0000	−	25.9808i	−0.517858	−	0.896956i	0.481927	−	0.876211i	$-0.339937\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.0000	−	20.7846i	0.412813	−	0.715012i
$846$		0			0
$847$	75.0000			2.57703
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−4.50000	−	7.79423i	−0.154077	−	0.266869i	0.778646	−	0.627464i	$-0.215907\pi$
$853$	−4.50000	−	7.79423i	−0.154077	−	0.266869i	−0.932723	+	0.360595i	$0.882574\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.00000	+	6.92820i	0.136637	+	0.236663i	0.926222	−	0.376979i	$-0.123037\pi$
$857$	4.00000	+	6.92820i	0.136637	+	0.236663i	−0.789584	+	0.613642i	$0.789704\pi$
$858$		0			0
$859$	6.50000	−	11.2583i	0.221777	−	0.384129i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	6.50000	−	11.2583i	0.221777	−	0.384129i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	36.0000			1.22545			0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000			1.22545			0.612727	+	0.790295i	$0.290072\pi$
$864$		0			0
$865$	14.0000	−	24.2487i	0.476014	−	0.824481i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−39.0000	−	67.5500i	−1.32298	−	2.29148i
$870$		0			0
$871$	7.50000	+	12.9904i	0.254128	+	0.440162i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−36.0000			−1.21702
$876$		0			0
$877$	−12.5000	−	21.6506i	−0.422095	−	0.731090i	0.574049	−	0.818821i	$-0.305372\pi$
$877$	−12.5000	−	21.6506i	−0.422095	−	0.731090i	−0.996144	+	0.0877308i	$0.972038\pi$
$878$		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	−0.983632	−	0.180190i	$-0.942329\pi$
$887$	−10.0000	+	17.3205i	−0.335767	+	0.581566i	0.647865	+	0.761755i	$0.275662\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−12.0000	−	20.7846i	−0.401116	−	0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.00000	−	1.73205i	0.0333519	−	0.0577671i
$900$		0			0
$901$	8.00000			0.266519
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−20.0000			−0.664822
$906$		0			0
$907$	10.0000	+	17.3205i	0.332045	+	0.575118i	0.982913	−	0.184073i	$-0.0589282\pi$
$907$	10.0000	+	17.3205i	0.332045	+	0.575118i	−0.650868	+	0.759191i	$0.725595\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−56.0000			−1.85536			−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000			−1.85536			−0.927681	+	0.373373i	$0.878201\pi$
$912$		0			0
$913$	84.0000			2.77999
$914$		0			0
$915$		0			0
$916$		0			0
$917$	27.0000	+	46.7654i	0.891619	+	1.54433i
$918$		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
$923$	−6.00000			−0.197492
$924$		0			0
$925$	−3.50000	+	6.06218i	−0.115079	+	0.199323i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21.0000	+	36.3731i	0.688988	+	1.19336i	0.972166	+	0.234294i	$0.0752779\pi$
$929$	21.0000	+	36.3731i	0.688988	+	1.19336i	−0.283178	+	0.959067i	$0.591389\pi$
$930$		0			0
$931$	−8.00000	−	3.46410i	−0.262189	−	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	−	20.7846i	0.392442	−	0.679729i
$936$		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
$947$	−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
$949$	−9.00000			−0.292152
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.00000	−	8.66025i	−0.161966	−	0.280533i	0.773608	−	0.633665i	$-0.218450\pi$
$953$	−5.00000	−	8.66025i	−0.161966	−	0.280533i	−0.935574	+	0.353132i	$0.885117\pi$
$954$		0			0
$955$	−6.00000	−	10.3923i	−0.194155	−	0.336287i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	27.0000	−	46.7654i	0.871875	−	1.51013i
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.00000	−	15.5885i	0.289720	−	0.501810i
$966$		0			0
$967$	7.50000	+	12.9904i	0.241184	+	0.417742i	0.961052	−	0.276368i	$-0.0891310\pi$
$967$	7.50000	+	12.9904i	0.241184	+	0.417742i	−0.719868	+	0.694111i	$0.755798\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	14.0000	+	24.2487i	0.449281	+	0.778178i	0.998339	−	0.0576061i	$-0.0183467\pi$
$971$	14.0000	+	24.2487i	0.449281	+	0.778178i	−0.549058	+	0.835784i	$0.685013\pi$
$972$		0			0
$973$	−7.50000	+	12.9904i	−0.240439	+	0.416452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.0000			0.895799			0.447900	−	0.894084i	$-0.352172\pi$
$977$	28.0000			0.895799			0.447900	+	0.894084i	$0.352172\pi$
$978$		0			0
$979$	−36.0000	+	62.3538i	−1.15056	+	1.99284i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.00000	−	12.1244i	−0.223265	−	0.386707i	0.732532	−	0.680732i	$-0.238338\pi$
$983$	−7.00000	−	12.1244i	−0.223265	−	0.386707i	−0.955798	+	0.294025i	$0.905005\pi$
$984$		0			0
$985$	−12.0000	−	20.7846i	−0.382352	−	0.662253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−2.50000	−	4.33013i	−0.0794151	−	0.137551i	0.823583	−	0.567196i	$-0.191972\pi$
$991$	−2.50000	−	4.33013i	−0.0794151	−	0.137551i	−0.902998	+	0.429645i	$0.858639\pi$
$992$		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	−0.993213	−	0.116310i	$-0.962893\pi$
$997$	−12.5000	+	21.6506i	−0.395879	+	0.685682i	0.597334	+	0.801993i	$0.296227\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.s.c.577.1	yes	2
3.2	odd	2		1368.2.s.f.577.1	yes	2
4.3	odd	2		2736.2.s.e.577.1		2
12.11	even	2		2736.2.s.n.577.1		2
19.11	even	3	inner	1368.2.s.c.505.1	✓	2
57.11	odd	6		1368.2.s.f.505.1	yes	2
76.11	odd	6		2736.2.s.e.1873.1		2
228.11	even	6		2736.2.s.n.1873.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.s.c.505.1	✓	2	19.11	even	3	inner
1368.2.s.c.577.1	yes	2	1.1	even	1	trivial
1368.2.s.f.505.1	yes	2	57.11	odd	6
1368.2.s.f.577.1	yes	2	3.2	odd	2
2736.2.s.e.577.1		2	4.3	odd	2
2736.2.s.e.1873.1		2	76.11	odd	6
2736.2.s.n.577.1		2	12.11	even	2
2736.2.s.n.1873.1		2	228.11	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} +3.00000 q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{5} +3.00000 q^{7} -6.00000 q^{11} +(0.500000 - 0.866025i) q^{13} +(-1.00000 - 1.73205i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(0.500000 - 0.866025i) q^{25} +(-1.00000 + 1.73205i) q^{29} -1.00000 q^{31} +(-3.00000 - 5.19615i) q^{35} -7.00000 q^{37} +(0.500000 + 0.866025i) q^{43} +2.00000 q^{49} +(-2.00000 + 3.46410i) q^{53} +(6.00000 + 10.3923i) q^{55} +(-4.00000 - 6.92820i) q^{59} +(5.50000 - 9.52628i) q^{61} -2.00000 q^{65} +(-7.50000 + 12.9904i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(-4.50000 - 7.79423i) q^{73} -18.0000 q^{77} +(6.50000 + 11.2583i) q^{79} -14.0000 q^{83} +(-2.00000 + 3.46410i) q^{85} +(6.00000 - 10.3923i) q^{89} +(1.50000 - 2.59808i) q^{91} +(1.00000 + 8.66025i) q^{95} +(-5.00000 - 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 6 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 6 * q^7	\( 2 q - 2 q^{5} + 6 q^{7} - 12 q^{11} + q^{13} - 2 q^{17} - 8 q^{19} + q^{25} - 2 q^{29} - 2 q^{31} - 6 q^{35} - 14 q^{37} + q^{43} + 4 q^{49} - 4 q^{53} + 12 q^{55} - 8 q^{59} + 11 q^{61} - 4 q^{65} - 15 q^{67} - 6 q^{71} - 9 q^{73} - 36 q^{77} + 13 q^{79} - 28 q^{83} - 4 q^{85} + 12 q^{89} + 3 q^{91} + 2 q^{95} - 10 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^7 - 12 * q^11 + q^13 - 2 * q^17 - 8 * q^19 + q^25 - 2 * q^29 - 2 * q^31 - 6 * q^35 - 14 * q^37 + q^43 + 4 * q^49 - 4 * q^53 + 12 * q^55 - 8 * q^59 + 11 * q^61 - 4 * q^65 - 15 * q^67 - 6 * q^71 - 9 * q^73 - 36 * q^77 + 13 * q^79 - 28 * q^83 - 4 * q^85 + 12 * q^89 + 3 * q^91 + 2 * q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more