LMFDB - Embedded newform 273.2.cd.a.44.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(44,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 4, 9]))

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

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

chi := DirichletCharacter("273.44");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.cd (of order $12$, degree $4$, 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:	$2.17991597518$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			44.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	273.44
Dual form			273.2.cd.a.242.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q+(0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 + 2.59808i) q^{9} +(-3.00000 - 1.73205i) q^{12} +(-0.866025 + 3.50000i) q^{13} +(2.00000 - 3.46410i) q^{16} +(0.330127 + 1.23205i) q^{19} +(-3.00000 - 3.46410i) q^{21} +(-4.33013 + 2.50000i) q^{25} -5.19615 q^{27} +(4.00000 - 3.46410i) q^{28} +(10.6962 + 2.86603i) q^{31} -6.00000i q^{36} +(8.96410 - 2.40192i) q^{37} +(-6.00000 + 1.73205i) q^{39} +12.1244i q^{43} +6.92820 q^{48} +(6.50000 - 2.59808i) q^{49} +(-2.00000 - 6.92820i) q^{52} +(-1.56218 + 1.56218i) q^{57} +(3.46410 - 6.00000i) q^{61} +(2.59808 - 7.50000i) q^{63} +8.00000i q^{64} +(-0.767949 - 0.205771i) q^{67} +(4.13397 - 15.4282i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(-1.80385 - 1.80385i) q^{76} +(-6.06218 + 10.5000i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(8.66025 + 3.00000i) q^{84} +(0.500000 - 9.52628i) q^{91} +(4.96410 + 18.5263i) q^{93} +(0.0717968 - 0.0717968i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} - 12 q^{12} + 8 q^{16} - 16 q^{19} - 12 q^{21} + 16 q^{28} + 22 q^{31} + 22 q^{37} - 24 q^{39} + 26 q^{49} - 8 q^{52} + 18 q^{57} - 10 q^{67} + 20 q^{73} - 30 q^{75} - 28 q^{76} - 18 q^{81} + 2 q^{91} + 6 q^{93} + 28 q^{97}+O(q^{100}) $ 4 * q - 6 * q^9 - 12 * q^12 + 8 * q^16 - 16 * q^19 - 12 * q^21 + 16 * q^28 + 22 * q^31 + 22 * q^37 - 24 * q^39 + 26 * q^49 - 8 * q^52 + 18 * q^57 - 10 * q^67 + 20 * q^73 - 30 * q^75 - 28 * q^76 - 18 * q^81 + 2 * q^91 + 6 * q^93 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$.

$n$	$92$	$106$	$157$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{1}{3}\right)$

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		−0.258819	−	0.965926i	$-0.583333\pi$
$2$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$3$	0.866025	+	1.50000i	0.500000	+	0.866025i
$3$	0.866025	+	1.50000i	0.500000	+	0.866025i
$4$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$5$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$5$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$6$		0			0
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$8$		0			0
$9$	−1.50000	+	2.59808i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$11$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$12$	−3.00000	−	1.73205i	−0.866025	−	0.500000i
$13$	−0.866025	+	3.50000i	−0.240192	+	0.970725i
$13$	−0.866025	+	3.50000i	−0.240192	+	0.970725i
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	0.330127	+	1.23205i	0.0757363	+	0.282652i	0.993399	−	0.114708i	$-0.0365932\pi$
$19$	0.330127	+	1.23205i	0.0757363	+	0.282652i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	−3.00000	−	3.46410i	−0.654654	−	0.755929i
$22$		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
$25$	−4.33013	+	2.50000i	−0.866025	+	0.500000i
$26$		0			0
$27$	−5.19615			−1.00000
$28$	4.00000	−	3.46410i	0.755929	−	0.654654i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	10.6962	+	2.86603i	1.92109	+	0.514753i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	10.6962	+	2.86603i	1.92109	+	0.514753i	0.933257	+	0.359211i	$0.116954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		−	6.00000i		−	1.00000i
$37$	8.96410	−	2.40192i	1.47369	−	0.394874i	0.569495	−	0.821995i	$-0.307139\pi$
$37$	8.96410	−	2.40192i	1.47369	−	0.394874i	0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$	−6.00000	+	1.73205i	−0.960769	+	0.277350i
$40$		0			0
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$			12.1244i			1.84895i	0.381246	+	0.924473i	$0.375495\pi$
$43$			12.1244i			1.84895i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$47$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$48$	6.92820			1.00000
$49$	6.50000	−	2.59808i	0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$	−2.00000	−	6.92820i	−0.277350	−	0.960769i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.56218	+	1.56218i	−0.206916	+	0.206916i
$58$		0			0
$59$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$59$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$60$		0			0
$61$	3.46410	−	6.00000i	0.443533	−	0.768221i	−0.554416	−	0.832240i	$-0.687058\pi$
$61$	3.46410	−	6.00000i	0.443533	−	0.768221i	0.997949	+	0.0640184i	$0.0203916\pi$
$62$		0			0
$63$	2.59808	−	7.50000i	0.327327	−	0.944911i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−0.767949	−	0.205771i	−0.0938199	−	0.0251390i	0.211604	−	0.977356i	$-0.432131\pi$
$67$	−0.767949	−	0.205771i	−0.0938199	−	0.0251390i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$71$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$72$		0			0
$73$	4.13397	−	15.4282i	0.483845	−	1.80573i	−0.101361	−	0.994850i	$-0.532320\pi$
$73$	4.13397	−	15.4282i	0.483845	−	1.80573i	0.585206	−	0.810885i	$-0.301014\pi$
$74$		0			0
$75$	−7.50000	−	4.33013i	−0.866025	−	0.500000i
$76$	−1.80385	−	1.80385i	−0.206916	−	0.206916i
$77$		0			0
$78$		0			0
$79$	−6.06218	+	10.5000i	−0.682048	+	1.18134i	0.292306	+	0.956325i	$0.405577\pi$
$79$	−6.06218	+	10.5000i	−0.682048	+	1.18134i	−0.974355	+	0.225018i	$0.927756\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$	8.66025	+	3.00000i	0.944911	+	0.327327i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$89$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$90$		0			0
$91$	0.500000	−	9.52628i	0.0524142	−	0.998625i
$92$		0			0
$93$	4.96410	+	18.5263i	0.514753	+	1.92109i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.0717968	−	0.0717968i	0.00728986	−	0.00728986i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	0.0717968	−	0.0717968i	0.00728986	−	0.00728986i	0.710742	+	0.703452i	$0.248359\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−16.5000	−	9.52628i	−1.62579	−	0.938652i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−16.5000	−	9.52628i	−1.62579	−	0.938652i	−0.640464	−	0.767988i	$-0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	9.00000	−	5.19615i	0.866025	−	0.500000i
$109$	19.8923	+	5.33013i	1.90534	+	0.510534i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$	19.8923	+	5.33013i	1.90534	+	0.510534i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	11.3660	+	11.3660i	1.07882	+	1.07882i
$112$	−3.46410	+	10.0000i	−0.327327	+	0.944911i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.79423	−	7.50000i	−0.720577	−	0.693375i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.52628	−	5.50000i	−0.866025	−	0.500000i
$122$		0			0
$123$		0			0
$124$	−21.3923	+	5.73205i	−1.92109	+	0.514753i
$125$		0			0
$126$		0			0
$127$			19.0000i			1.68598i	0.537931	+	0.842989i	$0.319206\pi$
$127$			19.0000i			1.68598i	−0.537931	+	0.842989i	$0.680794\pi$
$128$		0			0
$129$	−18.1865	+	10.5000i	−1.60123	+	0.924473i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	−1.47372	−	3.03590i	−0.127788	−	0.263246i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$137$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$138$		0			0
$139$	−7.00000			−0.593732			−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000			−0.593732			−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.00000	+	10.3923i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	9.52628	+	7.50000i	0.785714	+	0.618590i
$148$	−13.1244	+	13.1244i	−1.07882	+	1.07882i
$149$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$149$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$150$		0			0
$151$	−5.16987	+	19.2942i	−0.420718	+	1.57014i	0.352381	+	0.935857i	$0.385372\pi$
$151$	−5.16987	+	19.2942i	−0.420718	+	1.57014i	−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	8.66025	−	9.00000i	0.693375	−	0.720577i
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−22.0263	+	5.90192i	−1.72523	+	0.462274i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−22.0263	+	5.90192i	−1.72523	+	0.462274i	−0.746156	+	0.665771i	$0.768103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		0			0
$169$	−11.5000	−	6.06218i	−0.884615	−	0.466321i
$170$		0			0
$171$	−3.69615	−	0.990381i	−0.282652	−	0.0757363i
$172$	−12.1244	−	21.0000i	−0.924473	−	1.60123i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	10.0000	−	8.66025i	0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$		−	25.9808i		−	1.93113i	−0.260153	−	0.965567i	$-0.583773\pi$
$181$		−	25.9808i		−	1.93113i	0.260153	−	0.965567i	$-0.416227\pi$
$182$		0			0
$183$	12.0000			0.887066
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.5000	−	2.59808i	0.981981	−	0.188982i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−12.0000	+	6.92820i	−0.866025	+	0.500000i
$193$	2.35641	−	8.79423i	0.169618	−	0.633022i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	2.35641	−	8.79423i	0.169618	−	0.633022i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$		0			0
$195$		0			0
$196$	−8.66025	+	11.0000i	−0.618590	+	0.785714i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$	24.2487	−	14.0000i	1.71895	−	0.992434i	0.798082	−	0.602549i	$-0.205848\pi$
$199$	24.2487	−	14.0000i	1.71895	−	0.992434i	0.920864	−	0.389885i	$-0.127485\pi$
$200$		0			0
$201$	−0.356406	−	1.33013i	−0.0251390	−	0.0938199i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	10.3923	+	10.0000i	0.720577	+	0.693375i
$209$		0			0
$210$		0			0
$211$	24.2487			1.66935			0.834675	−	0.550743i	$-0.185655\pi$
$211$	24.2487			1.66935			0.834675	+	0.550743i	$0.185655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−29.2224	−	2.09808i	−1.98375	−	0.142427i
$218$		0			0
$219$	26.7224	−	7.16025i	1.80573	−	0.483845i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.80385	−	8.80385i	0.589549	−	0.589549i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	8.80385	−	8.80385i	0.589549	−	0.589549i	0.937509	+	0.347960i	$0.113126\pi$
$224$		0			0
$225$		−	15.0000i		−	1.00000i
$226$		0			0
$227$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$227$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$228$	1.14359	−	4.26795i	0.0757363	−	0.282652i
$229$	15.3301	−	4.10770i	1.01304	−	0.271444i	0.286143	−	0.958187i	$-0.407627\pi$
$229$	15.3301	−	4.10770i	1.01304	−	0.271444i	0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−21.0000			−1.36410
$238$		0			0
$239$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$239$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−2.50962	+	9.36603i	−0.161659	+	0.603319i	0.836784	+	0.547533i	$0.184433\pi$
$241$	−2.50962	+	9.36603i	−0.161659	+	0.603319i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$	7.79423	−	13.5000i	0.500000	−	0.866025i
$244$			13.8564i			0.887066i
$245$		0			0
$246$		0			0
$247$	−4.59808	+	0.0884573i	−0.292569	+	0.00562840i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	3.00000	+	15.5885i	0.188982	+	0.981981i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−22.0885	+	10.7224i	−1.37251	+	0.666259i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.53590	−	0.411543i	0.0938199	−	0.0251390i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	30.9545	−	8.29423i	1.88035	−	0.503839i	0.880812	−	0.473466i	$-0.156997\pi$
$271$	30.9545	−	8.29423i	1.88035	−	0.503839i	0.999539	−	0.0303728i	$-0.00966946\pi$
$272$		0			0
$273$	14.7224	−	7.50000i	0.891042	−	0.453921i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	10.5000	−	6.06218i	0.630884	−	0.364241i	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	10.5000	−	6.06218i	0.630884	−	0.364241i	0.781094	+	0.624413i	$0.214662\pi$
$278$		0			0
$279$	−23.4904	+	23.4904i	−1.40633	+	1.40633i
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$		0			0
$283$	−6.06218	+	3.50000i	−0.360359	+	0.208053i	−0.669238	−	0.743048i	$-0.733379\pi$
$283$	−6.06218	+	3.50000i	−0.360359	+	0.208053i	0.308879	+	0.951101i	$0.400046\pi$
$284$		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
$291$	0.169873	+	0.0455173i	0.00995813	+	0.00266827i
$292$	8.26795	+	30.8564i	0.483845	+	1.80573i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	17.3205			1.00000
$301$	−6.06218	−	31.5000i	−0.349418	−	1.81563i
$302$		0			0
$303$		0			0
$304$	4.92820	+	1.32051i	0.282652	+	0.0757363i
$305$		0			0
$306$		0			0
$307$	16.6340	+	16.6340i	0.949351	+	0.949351i	0.998778	−	0.0494267i	$-0.0157394\pi$
$307$	16.6340	+	16.6340i	0.949351	+	0.949351i	−0.0494267	+	0.998778i	$0.515739\pi$
$308$		0			0
$309$		−	33.0000i		−	1.87730i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−2.59808	+	4.50000i	−0.146852	+	0.254355i	−0.930062	−	0.367402i	$-0.880247\pi$
$313$	−2.59808	+	4.50000i	−0.146852	+	0.254355i	0.783210	+	0.621757i	$0.213581\pi$
$314$		0			0
$315$		0			0
$316$		−	24.2487i		−	1.36410i
$317$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$317$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	15.5885	+	9.00000i	0.866025	+	0.500000i
$325$	−5.00000	−	17.3205i	−0.277350	−	0.960769i
$326$		0			0
$327$	9.23205	+	34.4545i	0.510534	+	1.90534i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.47372	−	24.1603i	−0.355828	−	1.32797i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−6.47372	−	24.1603i	−0.355828	−	1.32797i	0.523612	−	0.851957i	$-0.324584\pi$
$332$		0			0
$333$	−7.20577	+	26.8923i	−0.394874	+	1.47369i
$334$		0			0
$335$		0			0
$336$	−18.0000	+	3.46410i	−0.981981	+	0.188982i
$337$			5.00000i			0.272367i	0.990684	+	0.136184i	$0.0434837\pi$
$337$			5.00000i			0.272367i	−0.990684	+	0.136184i	$0.956516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−15.5885	+	10.0000i	−0.841698	+	0.539949i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−24.3205	−	24.3205i	−1.30185	−	1.30185i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	−24.3205	−	24.3205i	−1.30185	−	1.30185i	−0.374701	−	0.927146i	$-0.622255\pi$
$350$		0			0
$351$	4.50000	−	18.1865i	0.240192	−	0.970725i
$352$		0			0
$353$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$353$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$359$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$360$		0			0
$361$	15.0455	−	8.68653i	0.791869	−	0.457186i
$362$		0			0
$363$		−	19.0526i		−	1.00000i
$364$	8.66025	+	17.0000i	0.453921	+	0.891042i
$365$		0			0
$366$		0			0
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.809093	+	0.587680i	$0.199959\pi$
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.104399	+	0.994535i	$0.466708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−27.1244	−	27.1244i	−1.40633	−	1.40633i
$373$	−18.1865	+	31.5000i	−0.941663	+	1.63101i	−0.179364	+	0.983783i	$0.557404\pi$
$373$	−18.1865	+	31.5000i	−0.941663	+	1.63101i	−0.762299	+	0.647225i	$0.775929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−12.4378	+	12.4378i	−0.638888	+	0.638888i	−0.950281	−	0.311393i	$-0.899204\pi$
$379$	−12.4378	+	12.4378i	−0.638888	+	0.638888i	0.311393	+	0.950281i	$0.399204\pi$
$380$		0			0
$381$	−28.5000	+	16.4545i	−1.46010	+	0.842989i
$382$		0			0
$383$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$383$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−31.5000	−	18.1865i	−1.60123	−	0.924473i
$388$	−0.0525589	+	0.196152i	−0.00266827	+	0.00995813i
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	36.9186	−	9.89230i	1.85289	−	0.496481i	0.853206	−	0.521575i	$-0.174655\pi$
$397$	36.9186	−	9.89230i	1.85289	−	0.496481i	0.999685	+	0.0250943i	$0.00798860\pi$
$398$		0			0
$399$	3.27757	−	4.83975i	0.164084	−	0.242290i
$400$			20.0000i			1.00000i
$401$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$401$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$402$		0			0
$403$	−19.2942	+	34.9545i	−0.961114	+	1.74121i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−31.9904	−	8.57180i	−1.58182	−	0.423848i	−0.642333	−	0.766426i	$-0.722033\pi$
$409$	−31.9904	−	8.57180i	−1.58182	−	0.423848i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$		0			0
$412$	38.1051			1.87730
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−6.06218	−	10.5000i	−0.296866	−	0.514187i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	27.6865	−	27.6865i	1.34936	−	1.34936i	0.463002	−	0.886357i	$-0.346772\pi$
$421$	27.6865	−	27.6865i	1.34936	−	1.34936i	0.886357	−	0.463002i	$-0.153228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6.00000	+	17.3205i	−0.290360	+	0.838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$431$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$432$	−10.3923	+	18.0000i	−0.500000	+	0.866025i
$433$			35.0000i			1.68199i	0.541041	+	0.840996i	$0.318030\pi$
$433$			35.0000i			1.68199i	−0.541041	+	0.840996i	$0.681970\pi$
$434$		0			0
$435$		0			0
$436$	−39.7846	+	10.6603i	−1.90534	+	0.510534i
$437$		0			0
$438$		0			0
$439$	27.0000	+	15.5885i	1.28864	+	0.743996i	0.978412	−	0.206666i	$-0.0662612\pi$
$439$	27.0000	+	15.5885i	1.28864	+	0.743996i	0.310228	+	0.950662i	$0.399595\pi$
$440$		0			0
$441$	−3.00000	+	20.7846i	−0.142857	+	0.989743i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−31.0526	−	8.32051i	−1.47369	−	0.394874i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−4.00000	−	20.7846i	−0.188982	−	0.981981i
$449$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$449$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−33.4186	+	8.95448i	−1.57014	+	0.420718i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	9.72243	+	36.2846i	0.454796	+	1.69732i	0.688686	+	0.725059i	$0.258188\pi$
$457$	9.72243	+	36.2846i	0.454796	+	1.69732i	−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$	−6.68653	−	6.68653i	−0.310750	−	0.310750i	0.534450	−	0.845200i	$-0.320519\pi$
$463$	−6.68653	−	6.68653i	−0.310750	−	0.310750i	−0.845200	+	0.534450i	$0.820519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$	21.0000	+	5.19615i	0.970725	+	0.240192i
$469$	2.09808	+	0.150635i	0.0968802	+	0.00695568i
$470$		0			0
$471$	−12.1244	+	21.0000i	−0.558661	+	0.967629i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.50962	−	4.50962i	−0.206916	−	0.206916i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$479$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$480$		0			0
$481$	0.643594	+	33.4545i	0.0293453	+	1.52539i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	41.9186	+	11.2321i	1.89951	+	0.508973i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$	41.9186	+	11.2321i	1.89951	+	0.508973i	0.902597	+	0.430486i	$0.141658\pi$
$488$		0			0
$489$	−27.9282	−	27.9282i	−1.26296	−	1.26296i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	31.3205	−	31.3205i	1.40633	−	1.40633i
$497$		0			0
$498$		0			0
$499$	−10.0885	−	37.6506i	−0.451621	−	1.68547i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	−10.0885	−	37.6506i	−0.451621	−	1.68547i	0.246214	−	0.969216i	$-0.420813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.866025	−	22.5000i	−0.0384615	−	0.999260i
$508$	−19.0000	−	32.9090i	−0.842989	−	1.46010i
$509$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$509$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$510$		0			0
$511$	−3.02628	+	42.1506i	−0.133875	+	1.86463i
$512$		0			0
$513$	−1.71539	−	6.40192i	−0.0757363	−	0.282652i
$514$		0			0
$515$		0			0
$516$	21.0000	−	36.3731i	0.924473	−	1.60123i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	17.5000	−	30.3109i	0.765222	−	1.32540i	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	17.5000	−	30.3109i	0.765222	−	1.32540i	0.940129	−	0.340818i	$-0.110704\pi$
$524$		0			0
$525$	21.6506	+	7.50000i	0.944911	+	0.327327i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	5.58846	+	3.78461i	0.242290	+	0.164084i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	44.6506	−	11.9641i	1.91968	−	0.514377i	0.930834	−	0.365444i	$-0.119083\pi$
$541$	44.6506	−	11.9641i	1.91968	−	0.514377i	0.988847	−	0.148933i	$-0.0475840\pi$
$542$		0			0
$543$	38.9711	−	22.5000i	1.67241	−	0.965567i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	10.3923	+	18.0000i	0.443533	+	0.768221i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	10.5000	−	30.3109i	0.446505	−	1.28895i
$554$		0			0
$555$		0			0
$556$	12.1244	−	7.00000i	0.514187	−	0.296866i
$557$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$557$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$558$		0			0
$559$	−42.4352	−	10.5000i	−1.79482	−	0.444103i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	15.5885	+	18.0000i	0.654654	+	0.755929i
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	26.8468	−	15.5000i	1.12350	−	0.648655i	0.181210	−	0.983444i	$-0.441999\pi$
$571$	26.8468	−	15.5000i	1.12350	−	0.648655i	0.942293	+	0.334790i	$0.108665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−20.7846	−	12.0000i	−0.866025	−	0.500000i
$577$	−46.3827	−	12.4282i	−1.93094	−	0.517393i	−0.973434	−	0.228968i	$-0.926465\pi$
$577$	−46.3827	−	12.4282i	−1.93094	−	0.517393i	−0.957503	−	0.288425i	$-0.906868\pi$
$578$		0			0
$579$	15.2321	−	4.08142i	0.633022	−	0.169618i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−24.0000	−	3.46410i	−0.989743	−	0.142857i
$589$			14.1244i			0.581984i
$590$		0			0
$591$		0			0
$592$	9.60770	−	35.8564i	0.394874	−	1.47369i
$593$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$593$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	42.0000	+	24.2487i	1.71895	+	0.992434i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	1.73205			0.0706518			0.0353259	−	0.999376i	$-0.488753\pi$
$601$	1.73205			0.0706518			0.0353259	+	0.999376i	$0.488753\pi$
$602$		0			0
$603$	1.68653	−	1.68653i	0.0686810	−	0.0686810i
$604$	−10.3397	−	38.5885i	−0.420718	−	1.57014i
$605$		0			0
$606$		0			0
$607$	−24.5000	+	42.4352i	−0.994424	+	1.72239i	−0.405887	+	0.913923i	$0.633038\pi$
$607$	−24.5000	+	42.4352i	−0.994424	+	1.72239i	−0.588537	+	0.808470i	$0.700296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−26.2942	−	7.04552i	−1.06201	−	0.284566i	−0.314806	−	0.949156i	$-0.601939\pi$
$613$	−26.2942	−	7.04552i	−1.06201	−	0.284566i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$	10.5526	−	39.3827i	0.424143	−	1.58292i	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	10.5526	−	39.3827i	0.424143	−	1.58292i	0.765787	−	0.643094i	$-0.222350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−6.00000	+	24.2487i	−0.240192	+	0.970725i
$625$	12.5000	−	21.6506i	0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−24.2487	−	14.0000i	−0.967629	−	0.558661i
$629$		0			0
$630$		0			0
$631$	−9.87564	−	9.87564i	−0.393143	−	0.393143i	0.482663	−	0.875806i	$-0.339670\pi$
$631$	−9.87564	−	9.87564i	−0.393143	−	0.393143i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	21.0000	+	36.3731i	0.834675	+	1.44570i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.46410	+	25.0000i	0.137253	+	0.990536i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	−21.6147	+	21.6147i	−0.852402	+	0.852402i	−0.990429	−	0.138027i	$-0.955924\pi$
$643$	−21.6147	+	21.6147i	−0.852402	+	0.852402i	0.138027	+	0.990429i	$0.455924\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−22.1603	−	45.6506i	−0.868529	−	1.78919i
$652$	32.2487	−	32.2487i	1.26296	−	1.26296i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	33.8827	+	33.8827i	1.32189	+	1.32189i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−11.8205	+	44.1147i	−0.459764	+	1.71586i	0.213925	+	0.976850i	$0.431375\pi$
$661$	−11.8205	+	44.1147i	−0.459764	+	1.71586i	−0.673690	+	0.739014i	$0.735292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	20.8301	+	5.58142i	0.805339	+	0.215790i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			36.3731i			1.40208i	0.713123	+	0.701039i	$0.247280\pi$
$673$			36.3731i			1.40208i	−0.713123	+	0.701039i	$0.752720\pi$
$674$		0			0
$675$	22.5000	−	12.9904i	0.866025	−	0.500000i
$676$	25.9808	−	1.00000i	0.999260	−	0.0384615i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−0.150635	+	0.222432i	−0.00578084	+	0.00853615i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$683$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$684$	7.39230	−	1.98076i	0.282652	−	0.0757363i
$685$		0			0
$686$		0			0
$687$	19.4378	+	19.4378i	0.741599	+	0.741599i
$688$	42.0000	+	24.2487i	1.60123	+	0.924473i
$689$		0			0
$690$		0			0
$691$	−12.4545	−	46.4808i	−0.473791	−	1.76821i	−0.625958	−	0.779857i	$-0.715292\pi$
$691$	−12.4545	−	46.4808i	−0.473791	−	1.76821i	0.152167	−	0.988355i	$-0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−8.66025	+	25.0000i	−0.327327	+	0.944911i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	5.91858	+	10.2513i	0.223224	+	0.386635i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18.0981	+	4.84936i	−0.679688	+	0.182122i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−18.0981	+	4.84936i	−0.679688	+	0.182122i	−0.0975728	+	0.995228i	$0.531108\pi$
$710$		0			0
$711$	−18.1865	−	31.5000i	−0.682048	−	1.18134i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	47.6314	+	16.5000i	1.77389	+	0.614492i
$722$		0			0
$723$	−16.2224	+	4.34679i	−0.603319	+	0.161659i
$724$	25.9808	+	45.0000i	0.965567	+	1.67241i
$725$		0			0
$726$		0			0
$727$		−	49.0000i		−	1.81731i	−0.417548	−	0.908655i	$-0.637111\pi$
$727$		−	49.0000i		−	1.81731i	0.417548	−	0.908655i	$-0.362889\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	−20.7846	+	12.0000i	−0.768221	+	0.443533i
$733$	−11.1077	−	41.4545i	−0.410272	−	1.53116i	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−11.1077	−	41.4545i	−0.410272	−	1.53116i	0.383849	−	0.923396i	$-0.374598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−11.9186	+	44.4808i	−0.438432	+	1.63625i	0.294285	+	0.955718i	$0.404919\pi$
$739$	−11.9186	+	44.4808i	−0.438432	+	1.63625i	−0.732717	+	0.680534i	$0.761748\pi$
$740$		0			0
$741$	−4.11474	−	6.82051i	−0.151159	−	0.250558i
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−31.5000	−	18.1865i	−1.14945	−	0.663636i	−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−31.5000	−	18.1865i	−1.14945	−	0.663636i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−20.7846	+	18.0000i	−0.755929	+	0.654654i
$757$	−48.4974			−1.76267			−0.881334	−	0.472493i	$-0.843354\pi$
$757$	−48.4974			−1.76267			−0.881334	+	0.472493i	$0.843354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$761$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$762$		0			0
$763$	−54.3468	−	3.90192i	−1.96749	−	0.141259i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	13.8564	−	24.0000i	0.500000	−	0.866025i
$769$	−11.5096	+	11.5096i	−0.415047	+	0.415047i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−11.5096	+	11.5096i	−0.415047	+	0.415047i	0.468445	+	0.883493i	$0.344814\pi$
$770$		0			0
$771$		0			0
$772$	4.71281	+	17.5885i	0.169618	+	0.633022i
$773$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$773$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$	−53.4808	+	14.3301i	−1.92109	+	0.514753i
$776$		0			0
$777$	−35.2128	−	23.8468i	−1.26325	−	0.855499i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	4.00000	−	27.7128i	0.142857	−	0.989743i
$785$		0			0
$786$		0			0
$787$	10.8827	−	40.6147i	0.387926	−	1.44776i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	10.8827	−	40.6147i	0.387926	−	1.44776i	0.833503	−	0.552515i	$-0.186332\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	18.0000	+	17.3205i	0.639199	+	0.615069i
$794$		0			0
$795$		0			0
$796$	−28.0000	+	48.4974i	−0.992434	+	1.71895i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.94744	+	1.94744i	0.0686810	+	0.0686810i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	−22.8038	+	22.8038i	−0.800751	+	0.800751i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−22.8038	+	22.8038i	−0.800751	+	0.800751i	0.182462	+	0.983213i	$0.441593\pi$
$812$		0			0
$813$	39.2487	+	39.2487i	1.37651	+	1.37651i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−14.9378	+	4.00258i	−0.522608	+	0.140032i
$818$		0			0
$819$	24.0000	+	15.5885i	0.838628	+	0.544705i
$820$		0			0
$821$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$821$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$822$		0			0
$823$	−21.0000	+	12.1244i	−0.732014	+	0.422628i	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−21.0000	+	12.1244i	−0.732014	+	0.422628i	0.0871445	+	0.996196i	$0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−6.06218	+	3.50000i	−0.210548	+	0.121560i	−0.601566	−	0.798823i	$-0.705456\pi$
$829$	−6.06218	+	3.50000i	−0.210548	+	0.121560i	0.391018	+	0.920383i	$0.372123\pi$
$830$		0			0
$831$	18.1865	+	10.5000i	0.630884	+	0.364241i
$832$	−28.0000	−	6.92820i	−0.970725	−	0.240192i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−55.5788	−	14.8923i	−1.92109	−	0.514753i
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$	−42.0000	+	24.2487i	−1.44570	+	0.834675i
$845$		0			0
$846$		0			0
$847$	27.5000	+	9.52628i	0.944911	+	0.327327i
$848$		0			0
$849$	−10.5000	−	6.06218i	−0.360359	−	0.208053i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−5.88269	−	5.88269i	−0.201419	−	0.201419i	0.599189	−	0.800608i	$-0.295490\pi$
$853$	−5.88269	−	5.88269i	−0.201419	−	0.201419i	−0.800608	+	0.599189i	$0.795490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	8.66025	−	15.0000i	0.295484	−	0.511793i	−0.679613	−	0.733571i	$-0.737852\pi$
$859$	8.66025	−	15.0000i	0.295484	−	0.511793i	0.975097	+	0.221777i	$0.0711857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$863$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	29.4449			1.00000
$868$	52.7128	−	25.5885i	1.78919	−	0.868529i
$869$		0			0
$870$		0			0
$871$	1.38526	−	2.50962i	0.0469379	−	0.0850352i
$872$		0			0
$873$	0.0788383	+	0.294229i	0.00266827	+	0.00995813i
$874$		0			0
$875$		0			0
$876$	−39.1244	+	39.1244i	−1.32189	+	1.32189i
$877$	−15.0981	−	56.3468i	−0.509826	−	1.90270i	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	−15.0981	−	56.3468i	−0.509826	−	1.90270i	−0.0877308	−	0.996144i	$-0.527962\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		−	47.0000i		−	1.58168i	−0.612026	−	0.790838i	$-0.709645\pi$
$883$		−	47.0000i		−	1.58168i	0.612026	−	0.790838i	$-0.290355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−9.50000	−	49.3634i	−0.318620	−	1.65560i
$890$		0			0
$891$		0			0
$892$	−6.44486	+	24.0526i	−0.215790	+	0.805339i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	15.0000	+	25.9808i	0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	42.0000	−	36.3731i	1.39767	−	1.21042i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−51.0955	+	29.5000i	−1.69660	+	0.979531i	−0.747653	+	0.664089i	$0.768820\pi$
$907$	−51.0955	+	29.5000i	−1.69660	+	0.979531i	−0.948945	+	0.315442i	$0.897847\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	2.28719	+	8.53590i	0.0757363	+	0.282652i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−22.4449	+	22.4449i	−0.741599	+	0.741599i
$917$		0			0
$918$		0			0
$919$	−30.3109	+	52.5000i	−0.999864	+	1.73182i	−0.485648	+	0.874154i	$0.661416\pi$
$919$	−30.3109	+	52.5000i	−0.999864	+	1.73182i	−0.514216	+	0.857661i	$0.671917\pi$
$920$		0			0
$921$	−10.5455	+	39.3564i	−0.347487	+	1.29684i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−32.8109	+	32.8109i	−1.07882	+	1.07882i
$926$		0			0
$927$	49.5000	−	28.5788i	1.62579	−	0.938652i
$928$		0			0
$929$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$929$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$930$		0			0
$931$	5.34679	+	7.15064i	0.175234	+	0.234353i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	50.2295			1.64093			0.820463	−	0.571700i	$-0.193716\pi$
$937$	50.2295			1.64093			0.820463	+	0.571700i	$0.193716\pi$
$938$		0			0
$939$	−9.00000			−0.293704
$940$		0			0
$941$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$941$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$947$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$948$	36.3731	−	21.0000i	1.18134	−	0.682048i
$949$	50.4186	+	27.8301i	1.63666	+	0.903404i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	79.3468	+	45.8109i	2.55957	+	1.47777i
$962$		0			0
$963$		0			0
$964$	−5.01924	−	18.7321i	−0.161659	−	0.603319i
$965$		0			0
$966$		0			0
$967$	24.4378	−	24.4378i	0.785867	−	0.785867i	−0.194946	−	0.980814i	$-0.562453\pi$
$967$	24.4378	−	24.4378i	0.785867	−	0.785867i	0.980814	+	0.194946i	$0.0624533\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$			31.1769i			1.00000i
$973$	18.1865	−	3.50000i	0.583033	−	0.112205i
$974$		0			0
$975$	21.6506	−	22.5000i	0.693375	−	0.720577i
$976$	−13.8564	−	24.0000i	−0.443533	−	0.768221i
$977$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$977$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−43.6865	+	43.6865i	−1.39480	+	1.39480i
$982$		0			0
$983$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$983$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	7.87564	−	4.75129i	0.250558	−	0.151159i
$989$		0			0
$990$		0			0
$991$	−8.50000	−	14.7224i	−0.270011	−	0.467673i	0.698853	−	0.715265i	$-0.253694\pi$
$991$	−8.50000	−	14.7224i	−0.270011	−	0.467673i	−0.968864	+	0.247592i	$0.920361\pi$
$992$		0			0
$993$	30.6340	−	30.6340i	0.972140	−	0.972140i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.5000	−	42.4352i	−0.775923	−	1.34394i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−24.5000	−	42.4352i	−0.775923	−	1.34394i	0.158352	−	0.987383i	$-0.449382\pi$
$998$		0			0
$999$	−46.5788	+	12.4808i	−1.47369	+	0.394874i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.cd.a.44.1	✓	4
3.2	odd	2	CM	273.2.cd.a.44.1	✓	4
7.4	even	3		273.2.cd.b.200.1	yes	4
13.8	odd	4		273.2.cd.b.86.1	yes	4
21.11	odd	6		273.2.cd.b.200.1	yes	4
39.8	even	4		273.2.cd.b.86.1	yes	4
91.60	odd	12	inner	273.2.cd.a.242.1	yes	4
273.242	even	12	inner	273.2.cd.a.242.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.cd.a.44.1	✓	4	1.1	even	1	trivial
273.2.cd.a.44.1	✓	4	3.2	odd	2	CM
273.2.cd.a.242.1	yes	4	91.60	odd	12	inner
273.2.cd.a.242.1	yes	4	273.242	even	12	inner
273.2.cd.b.86.1	yes	4	13.8	odd	4
273.2.cd.b.86.1	yes	4	39.8	even	4
273.2.cd.b.200.1	yes	4	7.4	even	3
273.2.cd.b.200.1	yes	4	21.11	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:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.cd (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(0.866025 + 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-2.59808 + 0.500000i) q^{7} +(-1.50000 + 2.59808i) q^{9} +(-3.00000 - 1.73205i) q^{12} +(-0.866025 + 3.50000i) q^{13} +(2.00000 - 3.46410i) q^{16} +(0.330127 + 1.23205i) q^{19} +(-3.00000 - 3.46410i) q^{21} +(-4.33013 + 2.50000i) q^{25} -5.19615 q^{27} +(4.00000 - 3.46410i) q^{28} +(10.6962 + 2.86603i) q^{31} -6.00000i q^{36} +(8.96410 - 2.40192i) q^{37} +(-6.00000 + 1.73205i) q^{39} +12.1244i q^{43} +6.92820 q^{48} +(6.50000 - 2.59808i) q^{49} +(-2.00000 - 6.92820i) q^{52} +(-1.56218 + 1.56218i) q^{57} +(3.46410 - 6.00000i) q^{61} +(2.59808 - 7.50000i) q^{63} +8.00000i q^{64} +(-0.767949 - 0.205771i) q^{67} +(4.13397 - 15.4282i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(-1.80385 - 1.80385i) q^{76} +(-6.06218 + 10.5000i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(8.66025 + 3.00000i) q^{84} +(0.500000 - 9.52628i) q^{91} +(4.96410 + 18.5263i) q^{93} +(0.0717968 - 0.0717968i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} - 12 q^{12} + 8 q^{16} - 16 q^{19} - 12 q^{21} + 16 q^{28} + 22 q^{31} + 22 q^{37} - 24 q^{39} + 26 q^{49} - 8 q^{52} + 18 q^{57} - 10 q^{67} + 20 q^{73} - 30 q^{75} - 28 q^{76} - 18 q^{81} + 2 q^{91} + 6 q^{93} + 28 q^{97}+O(q^{100}) \) 4 * q - 6 * q^9 - 12 * q^12 + 8 * q^16 - 16 * q^19 - 12 * q^21 + 16 * q^28 + 22 * q^31 + 22 * q^37 - 24 * q^39 + 26 * q^49 - 8 * q^52 + 18 * q^57 - 10 * q^67 + 20 * q^73 - 30 * q^75 - 28 * q^76 - 18 * q^81 + 2 * q^91 + 6 * q^93 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more