LMFDB - Embedded newform 273.2.cd.b.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.b.242.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(0.500000 + 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q+(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(0.500000 + 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{9} +(3.00000 + 1.73205i) q^{12} +(0.866025 + 3.50000i) q^{13} +(2.00000 - 3.46410i) q^{16} +(2.23205 + 8.33013i) q^{19} +(3.46410 - 3.00000i) q^{21} +(-4.33013 + 2.50000i) q^{25} +5.19615 q^{27} +(-3.46410 - 4.00000i) q^{28} +(-1.13397 - 0.303848i) q^{31} -6.00000i q^{36} +(-7.59808 + 2.03590i) q^{37} +(4.50000 - 4.33013i) q^{39} -12.1244i q^{43} -6.92820 q^{48} +(-6.50000 + 2.59808i) q^{49} +(-5.00000 - 5.19615i) q^{52} +(10.5622 - 10.5622i) q^{57} +(-3.46410 + 6.00000i) q^{61} +(-7.50000 - 2.59808i) q^{63} +8.00000i q^{64} +(15.7942 + 4.23205i) q^{67} +(-1.57180 + 5.86603i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-12.1962 - 12.1962i) q^{76} +(6.06218 - 10.5000i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-3.00000 + 8.66025i) q^{84} +(-8.66025 + 4.00000i) q^{91} +(0.526279 + 1.96410i) q^{93} +(13.9282 - 13.9282i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^7 - 6 * q^9	$ 4 q + 2 q^{7} - 6 q^{9} + 12 q^{12} + 8 q^{16} + 2 q^{19} - 8 q^{31} - 20 q^{37} + 18 q^{39} - 26 q^{49} - 20 q^{52} + 18 q^{57} - 30 q^{63} + 32 q^{67} - 34 q^{73} + 30 q^{75} - 28 q^{76} - 18 q^{81} - 12 q^{84} - 36 q^{93} + 28 q^{97}+O(q^{100}) $ 4 * q + 2 * q^7 - 6 * q^9 + 12 * q^12 + 8 * q^16 + 2 * q^19 - 8 * q^31 - 20 * q^37 + 18 * q^39 - 26 * q^49 - 20 * q^52 + 18 * q^57 - 30 * q^63 + 32 * q^67 - 34 * q^73 + 30 * q^75 - 28 * q^76 - 18 * q^81 - 12 * q^84 - 36 * 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
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		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
$19$	2.23205	+	8.33013i	0.512068	+	1.91106i	0.397360	+	0.917663i	$0.369927\pi$
$19$	2.23205	+	8.33013i	0.512068	+	1.91106i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$	3.46410	−	3.00000i	0.755929	−	0.654654i
$22$		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
$28$	−3.46410	−	4.00000i	−0.654654	−	0.755929i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−1.13397	−	0.303848i	−0.203668	−	0.0545726i	0.155543	−	0.987829i	$-0.450287\pi$
$31$	−1.13397	−	0.303848i	−0.203668	−	0.0545726i	−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		−	6.00000i		−	1.00000i
$37$	−7.59808	+	2.03590i	−1.24912	+	0.334700i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−7.59808	+	2.03590i	−1.24912	+	0.334700i	−0.427121	+	0.904194i	$0.640472\pi$
$38$		0			0
$39$	4.50000	−	4.33013i	0.720577	−	0.693375i
$40$		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	$-0.624505\pi$
$43$		−	12.1244i		−	1.84895i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		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
$52$	−5.00000	−	5.19615i	−0.693375	−	0.720577i
$53$		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
$57$	10.5622	−	10.5622i	1.39899	−	1.39899i
$58$		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	−0.997949	−	0.0640184i	$-0.979608\pi$
$61$	−3.46410	+	6.00000i	−0.443533	+	0.768221i	0.554416	+	0.832240i	$0.312942\pi$
$62$		0			0
$63$	−7.50000	−	2.59808i	−0.944911	−	0.327327i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	15.7942	+	4.23205i	1.92957	+	0.517027i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	15.7942	+	4.23205i	1.92957	+	0.517027i	0.952217	+	0.305424i	$0.0987981\pi$
$68$		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
$73$	−1.57180	+	5.86603i	−0.183965	+	0.686566i	0.810885	+	0.585206i	$0.198986\pi$
$73$	−1.57180	+	5.86603i	−0.183965	+	0.686566i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$	−12.1962	−	12.1962i	−1.39899	−	1.39899i
$77$		0			0
$78$		0			0
$79$	6.06218	−	10.5000i	0.682048	−	1.18134i	−0.292306	−	0.956325i	$-0.594423\pi$
$79$	6.06218	−	10.5000i	0.682048	−	1.18134i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		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	$0.750000\pi$
$84$	−3.00000	+	8.66025i	−0.327327	+	0.944911i
$85$		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
$91$	−8.66025	+	4.00000i	−0.907841	+	0.419314i
$92$		0			0
$93$	0.526279	+	1.96410i	0.0545726	+	0.203668i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.9282	−	13.9282i	1.41419	−	1.41419i	0.703452	−	0.710742i	$-0.251641\pi$
$97$	13.9282	−	13.9282i	1.41419	−	1.41419i	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	$0.0545913\pi$
$103$	16.5000	+	9.52628i	1.62579	+	0.938652i	0.640464	+	0.767988i	$0.278742\pi$
$104$		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
$109$	3.33013	+	0.892305i	0.318968	+	0.0854673i	0.414751	−	0.909935i	$-0.363869\pi$
$109$	3.33013	+	0.892305i	0.318968	+	0.0854673i	−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	9.63397	+	9.63397i	0.914416	+	0.914416i
$112$	10.0000	+	3.46410i	0.944911	+	0.327327i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−10.3923	−	3.00000i	−0.960769	−	0.277350i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.52628	−	5.50000i	−0.866025	−	0.500000i
$122$		0			0
$123$		0			0
$124$	2.26795	−	0.607695i	0.203668	−	0.0545726i
$125$		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
$133$	−20.5263	+	9.96410i	−1.77985	+	0.863997i
$134$		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
$148$	11.1244	−	11.1244i	0.914416	−	0.914416i
$149$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$149$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$150$		0			0
$151$	3.70577	−	13.8301i	0.301571	−	1.12548i	−0.634285	−	0.773099i	$-0.718706\pi$
$151$	3.70577	−	13.8301i	0.301571	−	1.12548i	0.935857	−	0.352381i	$-0.114628\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−3.46410	+	12.0000i	−0.277350	+	0.960769i
$157$	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
$163$	11.0981	−	2.97372i	0.869268	−	0.232920i	0.203497	−	0.979076i	$-0.434769\pi$
$163$	11.0981	−	2.97372i	0.869268	−	0.232920i	0.665771	+	0.746156i	$0.268103\pi$
$164$		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
$171$	−24.9904	−	6.69615i	−1.91106	−	0.512068i
$172$	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
$175$	−8.66025	−	10.0000i	−0.654654	−	0.755929i
$176$		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	$0.416227\pi$
$181$			25.9808i			1.93113i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	12.0000			0.887066
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.59808	+	13.5000i	0.188982	+	0.981981i
$190$		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
$193$	6.79423	−	25.3564i	0.489059	−	1.82519i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	6.79423	−	25.3564i	0.489059	−	1.82519i	0.561041	−	0.827788i	$-0.310401\pi$
$194$		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
$201$	−7.33013	−	27.3564i	−0.517027	−	1.92957i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	13.8564	+	4.00000i	0.960769	+	0.277350i
$209$		0			0
$210$		0			0
$211$	−24.2487			−1.66935			−0.834675	−	0.550743i	$-0.814345\pi$
$211$	−24.2487			−1.66935			−0.834675	+	0.550743i	$0.814345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.222432	−	3.09808i	0.0150997	−	0.210311i
$218$		0			0
$219$	10.1603	−	2.72243i	0.686566	−	0.183965i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	19.1962	−	19.1962i	1.28547	−	1.28547i	0.347960	−	0.937509i	$-0.386874\pi$
$223$	19.1962	−	19.1962i	1.28547	−	1.28547i	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	$0.0833333\pi$
$228$	−7.73205	+	28.8564i	−0.512068	+	1.91106i
$229$	−24.8923	+	6.66987i	−1.64493	+	0.440758i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−24.8923	+	6.66987i	−1.64493	+	0.440758i	−0.686743	+	0.726900i	$0.740960\pi$
$230$		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
$241$	7.63397	−	28.4904i	0.491748	−	1.83523i	−0.0557856	−	0.998443i	$-0.517766\pi$
$241$	7.63397	−	28.4904i	0.491748	−	1.83523i	0.547533	−	0.836784i	$-0.315567\pi$
$242$		0			0
$243$	−7.79423	+	13.5000i	−0.500000	+	0.866025i
$244$		−	13.8564i		−	0.887066i
$245$		0			0
$246$		0			0
$247$	−27.2224	+	15.0263i	−1.73212	+	0.956099i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	15.5885	−	3.00000i	0.981981	−	0.188982i
$253$		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
$259$	−9.08846	−	18.7224i	−0.564729	−	1.16336i
$260$		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
$268$	−31.5885	+	8.46410i	−1.92957	+	0.517027i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	7.29423	−	1.95448i	0.443093	−	0.118726i	−0.0303728	−	0.999539i	$-0.509669\pi$
$271$	7.29423	−	1.95448i	0.443093	−	0.118726i	0.473466	+	0.880812i	$0.343003\pi$
$272$		0			0
$273$	13.5000	+	9.52628i	0.817057	+	0.576557i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−10.5000	+	6.06218i	−0.630884	+	0.364241i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−10.5000	+	6.06218i	−0.630884	+	0.364241i	0.150210	+	0.988654i	$0.452005\pi$
$278$		0			0
$279$	2.49038	−	2.49038i	0.149095	−	0.149095i
$280$		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
$291$	−32.9545	−	8.83013i	−1.93183	−	0.517631i
$292$	−3.14359	−	11.7321i	−0.183965	−	0.686566i
$293$		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
$301$	31.5000	−	6.06218i	1.81563	−	0.349418i
$302$		0			0
$303$		0			0
$304$	33.3205	+	8.92820i	1.91106	+	0.512068i
$305$		0			0
$306$		0			0
$307$	18.3660	+	18.3660i	1.04820	+	1.04820i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$	18.3660	+	18.3660i	1.04820	+	1.04820i	0.0494267	+	0.998778i	$0.484261\pi$
$308$		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	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	2.59808	−	4.50000i	0.146852	−	0.254355i	0.930062	+	0.367402i	$0.119753\pi$
$314$		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
$325$	−12.5000	−	12.9904i	−0.693375	−	0.720577i
$326$		0			0
$327$	−1.54552	−	5.76795i	−0.0854673	−	0.318968i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.83975	+	25.5263i	0.375946	+	1.40305i	0.851957	+	0.523612i	$0.175416\pi$
$331$	6.83975	+	25.5263i	0.375946	+	1.40305i	−0.476011	+	0.879440i	$0.657918\pi$
$332$		0			0
$333$	6.10770	−	22.7942i	0.334700	−	1.24912i
$334$		0			0
$335$		0			0
$336$	−3.46410	−	18.0000i	−0.188982	−	0.981981i
$337$			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
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		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
$349$	10.3205	+	10.3205i	0.552444	+	0.552444i	0.927146	−	0.374701i	$-0.122255\pi$
$349$	10.3205	+	10.3205i	0.552444	+	0.552444i	−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
$361$	−47.9545	+	27.6865i	−2.52392	+	1.45719i
$362$		0			0
$363$			19.0526i			1.00000i
$364$	11.0000	−	15.5885i	0.576557	−	0.817057i
$365$		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
$372$	−2.87564	−	2.87564i	−0.149095	−	0.149095i
$373$	18.1865	−	31.5000i	0.941663	−	1.63101i	0.179364	−	0.983783i	$-0.442596\pi$
$373$	18.1865	−	31.5000i	0.941663	−	1.63101i	0.762299	−	0.647225i	$-0.224071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−24.5622	+	24.5622i	−1.26167	+	1.26167i	−0.311393	+	0.950281i	$0.600796\pi$
$379$	−24.5622	+	24.5622i	−1.26167	+	1.26167i	−0.950281	+	0.311393i	$0.899204\pi$
$380$		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
$388$	−10.1962	+	38.0526i	−0.517631	+	1.93183i
$389$		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
$397$	10.8923	−	2.91858i	0.546669	−	0.146480i	0.0250943	−	0.999685i	$-0.492011\pi$
$397$	10.8923	−	2.91858i	0.546669	−	0.146480i	0.521575	+	0.853206i	$0.325345\pi$
$398$		0			0
$399$	32.7224	+	22.1603i	1.63817	+	1.10940i
$400$			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
$403$	0.0814157	−	4.23205i	0.00405561	−	0.210813i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	22.4282	+	6.00962i	1.10900	+	0.297157i	0.766426	−	0.642333i	$-0.222033\pi$
$409$	22.4282	+	6.00962i	1.10900	+	0.297157i	0.342578	+	0.939490i	$0.388700\pi$
$410$		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
$421$	−8.68653	+	8.68653i	−0.423356	+	0.423356i	−0.886357	−	0.463002i	$-0.846772\pi$
$421$	−8.68653	+	8.68653i	−0.423356	+	0.423356i	0.463002	+	0.886357i	$0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−17.3205	−	6.00000i	−0.838198	−	0.290360i
$428$		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
$436$	−6.66025	+	1.78461i	−0.318968	+	0.0854673i
$437$		0			0
$438$		0			0
$439$	−27.0000	−	15.5885i	−1.28864	−	0.743996i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−27.0000	−	15.5885i	−1.28864	−	0.743996i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		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	$0.666667\pi$
$444$	−26.3205	−	7.05256i	−1.24912	−	0.334700i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−20.7846	+	4.00000i	−0.981981	+	0.188982i
$449$		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
$453$	−23.9545	+	6.41858i	−1.12548	+	0.301571i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.28461	+	19.7224i	0.247204	+	0.922576i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	5.28461	+	19.7224i	0.247204	+	0.922576i	−0.725059	+	0.688686i	$0.758188\pi$
$458$		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
$463$	29.6865	+	29.6865i	1.37965	+	1.37965i	0.845200	+	0.534450i	$0.179481\pi$
$463$	29.6865	+	29.6865i	1.37965	+	1.37965i	0.534450	+	0.845200i	$0.320519\pi$
$464$		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
$469$	−3.09808	+	43.1506i	−0.143056	+	1.99251i
$470$		0			0
$471$	12.1244	−	21.0000i	0.558661	−	0.967629i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−30.4904	−	30.4904i	−1.39899	−	1.39899i
$476$		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
$481$	−13.7058	−	24.8301i	−0.624929	−	1.13216i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−7.76795	−	2.08142i	−0.351999	−	0.0943179i	0.0784867	−	0.996915i	$-0.474991\pi$
$487$	−7.76795	−	2.08142i	−0.351999	−	0.0943179i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	−14.0718	−	14.0718i	−0.636349	−	0.636349i
$490$		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
$496$	−3.32051	+	3.32051i	−0.149095	+	0.149095i
$497$		0			0
$498$		0			0
$499$	−5.65064	−	21.0885i	−0.252957	−	0.944049i	−0.969216	−	0.246214i	$-0.920813\pi$
$499$	−5.65064	−	21.0885i	−0.252957	−	0.944049i	0.716258	−	0.697835i	$-0.245853\pi$
$500$		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
$507$	19.0526	+	12.0000i	0.846154	+	0.532939i
$508$	−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
$511$	−16.0263	−	1.15064i	−0.708961	−	0.0509011i
$512$		0			0
$513$	11.5981	+	43.2846i	0.512068	+	1.91106i
$514$		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
$525$	−7.50000	+	21.6506i	−0.327327	+	0.944911i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	25.5885	−	37.7846i	1.10940	−	1.63817i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−5.03590	+	1.34936i	−0.216510	+	0.0580137i	−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−5.03590	+	1.34936i	−0.216510	+	0.0580137i	0.148933	+	0.988847i	$0.452416\pi$
$542$		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
$553$	30.3109	+	10.5000i	1.28895	+	0.446505i
$554$		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
$567$	18.0000	−	15.5885i	0.755929	−	0.654654i
$568$		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
$577$	−1.42820	−	0.382686i	−0.0594569	−	0.0159314i	0.228968	−	0.973434i	$-0.426465\pi$
$577$	−1.42820	−	0.382686i	−0.0594569	−	0.0159314i	−0.288425	+	0.957503i	$0.593132\pi$
$578$		0			0
$579$	−43.9186	+	11.7679i	−1.82519	+	0.489059i
$580$		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
$589$		−	10.1244i		−	0.417167i
$590$		0			0
$591$		0			0
$592$	−8.14359	+	30.3923i	−0.334700	+	1.24912i
$593$		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	$-0.511247\pi$
$601$	−1.73205			−0.0706518			−0.0353259	+	0.999376i	$0.511247\pi$
$602$		0			0
$603$	−34.6865	+	34.6865i	−1.41254	+	1.41254i
$604$	7.41154	+	27.6603i	0.301571	+	1.12548i
$605$		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
$613$	39.9545	+	10.7058i	1.61375	+	0.432402i	0.949156	−	0.314806i	$-0.101939\pi$
$613$	39.9545	+	10.7058i	1.61375	+	0.432402i	0.664590	+	0.747208i	$0.268606\pi$
$614$		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
$619$	7.38269	−	27.5526i	0.296735	−	1.10743i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	7.38269	−	27.5526i	0.296735	−	1.10743i	0.939829	−	0.341644i	$-0.110984\pi$
$620$		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
$631$	−34.1244	−	34.1244i	−1.35847	−	1.35847i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−34.1244	−	34.1244i	−1.35847	−	1.35847i	−0.482663	−	0.875806i	$-0.660330\pi$
$632$		0			0
$633$	21.0000	+	36.3731i	0.834675	+	1.44570i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−14.7224	−	20.5000i	−0.583324	−	0.812240i
$638$		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
$643$	28.6147	−	28.6147i	1.12846	−	1.12846i	0.138027	−	0.990429i	$-0.455924\pi$
$643$	28.6147	−	28.6147i	1.12846	−	1.12846i	0.990429	−	0.138027i	$-0.0440759\pi$
$644$		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
$651$	−4.83975	+	2.34936i	−0.189685	+	0.0920789i
$652$	−16.2487	+	16.2487i	−0.636349	+	0.636349i
$653$		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
$657$	−12.8827	−	12.8827i	−0.502601	−	0.502601i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−6.11474	+	22.8205i	−0.237836	+	0.887615i	0.739014	+	0.673690i	$0.235292\pi$
$661$	−6.11474	+	22.8205i	−0.237836	+	0.887615i	−0.976850	+	0.213925i	$0.931375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−45.4186	−	12.1699i	−1.75598	−	0.470514i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	36.3731i		−	1.40208i	−0.713123	−	0.701039i	$-0.752720\pi$
$673$		−	36.3731i		−	1.40208i	0.713123	−	0.701039i	$-0.247280\pi$
$674$		0			0
$675$	−22.5000	+	12.9904i	−0.866025	+	0.500000i
$676$	13.8564	−	22.0000i	0.532939	−	0.846154i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	43.1506	+	29.2224i	1.65597	+	1.12145i
$680$		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	$0.583333\pi$
$684$	49.9808	−	13.3923i	1.91106	−	0.512068i
$685$		0			0
$686$		0			0
$687$	31.5622	+	31.5622i	1.20417	+	1.20417i
$688$	−42.0000	−	24.2487i	−1.60123	−	0.924473i
$689$		0			0
$690$		0			0
$691$	−5.48076	−	20.4545i	−0.208498	−	0.778125i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$	−5.48076	−	20.4545i	−0.208498	−	0.778125i	0.779857	−	0.625958i	$-0.215292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	25.0000	+	8.66025i	0.944911	+	0.327327i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−33.9186	−	58.7487i	−1.27926	−	2.21575i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	48.1506	−	12.9019i	1.80834	−	0.484542i	0.813107	−	0.582115i	$-0.197775\pi$
$709$	48.1506	−	12.9019i	1.80834	−	0.484542i	0.995228	+	0.0975728i	$0.0311079\pi$
$710$		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
$721$	−16.5000	+	47.6314i	−0.614492	+	1.77389i
$722$		0			0
$723$	−49.3468	+	13.2224i	−1.83523	+	0.491748i
$724$	−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
$733$	8.54552	+	31.8923i	0.315636	+	1.17797i	0.923396	+	0.383849i	$0.125402\pi$
$733$	8.54552	+	31.8923i	0.315636	+	1.17797i	−0.607760	+	0.794121i	$0.707932\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−7.48076	+	27.9186i	−0.275184	+	1.02700i	0.680534	+	0.732717i	$0.261748\pi$
$739$	−7.48076	+	27.9186i	−0.275184	+	1.02700i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$	46.1147	+	27.8205i	1.69407	+	1.02201i
$742$		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	0.948753	−	0.316017i	$-0.102346\pi$
$751$	31.5000	+	18.1865i	1.14945	+	0.663636i	0.200698	+	0.979653i	$0.435679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−18.0000	−	20.7846i	−0.654654	−	0.755929i
$757$	48.4974			1.76267			0.881334	−	0.472493i	$-0.156646\pi$
$757$	48.4974			1.76267			0.881334	+	0.472493i	$0.156646\pi$
$758$		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
$763$	−0.653212	+	9.09808i	−0.0236479	+	0.329372i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−13.8564	+	24.0000i	−0.500000	+	0.866025i
$769$	−37.4904	+	37.4904i	−1.35194	+	1.35194i	−0.468445	+	0.883493i	$0.655186\pi$
$769$	−37.4904	+	37.4904i	−1.35194	+	1.35194i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$	13.5885	+	50.7128i	0.489059	+	1.82519i
$773$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$773$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$	5.66987	−	1.51924i	0.203668	−	0.0545726i
$776$		0			0
$777$	−20.2128	+	29.8468i	−0.725131	+	1.07075i
$778$		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
$787$	9.61474	−	35.8827i	0.342728	−	1.27908i	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	9.61474	−	35.8827i	0.342728	−	1.27908i	0.895244	−	0.445577i	$-0.147001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−24.0000	−	6.92820i	−0.852265	−	0.246028i
$794$		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
$804$	40.0526	+	40.0526i	1.41254	+	1.41254i
$805$		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
$811$	−33.1962	+	33.1962i	−1.16567	+	1.16567i	−0.182462	+	0.983213i	$0.558407\pi$
$811$	−33.1962	+	33.1962i	−1.16567	+	1.16567i	−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$	−9.24871	−	9.24871i	−0.324366	−	0.324366i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	100.997	−	27.0622i	3.53345	−	0.946786i
$818$		0			0
$819$	2.59808	−	28.5000i	0.0907841	−	0.995871i
$820$		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	−0.0871445	−	0.996196i	$-0.527774\pi$
$823$	21.0000	−	12.1244i	0.732014	−	0.422628i	0.819159	+	0.573567i	$0.194441\pi$
$824$		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
$837$	−5.89230	−	1.57884i	−0.203668	−	0.0545726i
$838$		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
$847$	9.52628	−	27.5000i	0.327327	−	0.944911i
$848$		0			0
$849$	10.5000	+	6.06218i	0.360359	+	0.208053i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	40.8827	+	40.8827i	1.39980	+	1.39980i	0.800608	+	0.599189i	$0.204510\pi$
$853$	40.8827	+	40.8827i	1.39980	+	1.39980i	0.599189	+	0.800608i	$0.295490\pi$
$854$		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	−0.975097	−	0.221777i	$-0.928814\pi$
$859$	−8.66025	+	15.0000i	−0.295484	+	0.511793i	0.679613	+	0.733571i	$0.262148\pi$
$860$		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
$868$	2.71281	+	5.58846i	0.0920789	+	0.189685i
$869$		0			0
$870$		0			0
$871$	−1.13397	+	58.9449i	−0.0384233	+	1.99727i
$872$		0			0
$873$	15.2942	+	57.0788i	0.517631	+	1.93183i
$874$		0			0
$875$		0			0
$876$	−14.8756	+	14.8756i	−0.502601	+	0.502601i
$877$	2.65321	+	9.90192i	0.0895926	+	0.334364i	0.996144	−	0.0877308i	$-0.0279615\pi$
$877$	2.65321	+	9.90192i	0.0895926	+	0.334364i	−0.906552	+	0.422095i	$0.861295\pi$
$878$		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
$889$	−49.3634	+	9.50000i	−1.65560	+	0.318620i
$890$		0			0
$891$		0			0
$892$	−14.0526	+	52.4449i	−0.470514	+	1.75598i
$893$		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
$903$	−36.3731	−	42.0000i	−1.21042	−	1.39767i
$904$		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			$\pi$
$912$	−15.4641	−	57.7128i	−0.512068	−	1.91106i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	36.4449	−	36.4449i	1.20417	−	1.20417i
$917$		0			0
$918$		0			0
$919$	30.3109	−	52.5000i	0.999864	−	1.73182i	0.485648	−	0.874154i	$-0.338584\pi$
$919$	30.3109	−	52.5000i	0.999864	−	1.73182i	0.514216	−	0.857661i	$-0.328083\pi$
$920$		0			0
$921$	11.6436	−	43.4545i	0.383669	−	1.43187i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	27.8109	−	27.8109i	0.914416	−	0.914416i
$926$		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
$931$	−36.1506	−	48.3468i	−1.18479	−	1.58450i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−50.2295			−1.64093			−0.820463	−	0.571700i	$-0.806284\pi$
$937$	−50.2295			−1.64093			−0.820463	+	0.571700i	$0.806284\pi$
$938$		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
$949$	−21.8923	−	0.421162i	−0.710654	−	0.0136715i
$950$		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
$961$	−25.6532	−	14.8109i	−0.827523	−	0.477771i
$962$		0			0
$963$		0			0
$964$	15.2679	+	56.9808i	0.491748	+	1.83523i
$965$		0			0
$966$		0			0
$967$	36.5622	−	36.5622i	1.17576	−	1.17576i	0.194946	−	0.980814i	$-0.437547\pi$
$967$	36.5622	−	36.5622i	1.17576	−	1.17576i	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
$973$	−3.50000	−	18.1865i	−0.112205	−	0.583033i
$974$		0			0
$975$	−8.66025	+	30.0000i	−0.277350	+	0.960769i
$976$	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
$981$	−7.31347	+	7.31347i	−0.233501	+	0.233501i
$982$		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
$988$	32.1244	−	53.2487i	1.02201	−	1.69407i
$989$		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
$993$	32.3660	−	32.3660i	1.02710	−	1.02710i
$994$		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
$999$	−39.4808	+	10.5788i	−1.24912	+	0.334700i

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(0.500000 + 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(0.500000 + 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{9} +(3.00000 + 1.73205i) q^{12} +(0.866025 + 3.50000i) q^{13} +(2.00000 - 3.46410i) q^{16} +(2.23205 + 8.33013i) q^{19} +(3.46410 - 3.00000i) q^{21} +(-4.33013 + 2.50000i) q^{25} +5.19615 q^{27} +(-3.46410 - 4.00000i) q^{28} +(-1.13397 - 0.303848i) q^{31} -6.00000i q^{36} +(-7.59808 + 2.03590i) q^{37} +(4.50000 - 4.33013i) q^{39} -12.1244i q^{43} -6.92820 q^{48} +(-6.50000 + 2.59808i) q^{49} +(-5.00000 - 5.19615i) q^{52} +(10.5622 - 10.5622i) q^{57} +(-3.46410 + 6.00000i) q^{61} +(-7.50000 - 2.59808i) q^{63} +8.00000i q^{64} +(15.7942 + 4.23205i) q^{67} +(-1.57180 + 5.86603i) q^{73} +(7.50000 + 4.33013i) q^{75} +(-12.1962 - 12.1962i) q^{76} +(6.06218 - 10.5000i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-3.00000 + 8.66025i) q^{84} +(-8.66025 + 4.00000i) q^{91} +(0.526279 + 1.96410i) q^{93} +(13.9282 - 13.9282i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^7 - 6 * q^9	\( 4 q + 2 q^{7} - 6 q^{9} + 12 q^{12} + 8 q^{16} + 2 q^{19} - 8 q^{31} - 20 q^{37} + 18 q^{39} - 26 q^{49} - 20 q^{52} + 18 q^{57} - 30 q^{63} + 32 q^{67} - 34 q^{73} + 30 q^{75} - 28 q^{76} - 18 q^{81} - 12 q^{84} - 36 q^{93} + 28 q^{97}+O(q^{100}) \) 4 * q + 2 * q^7 - 6 * q^9 + 12 * q^12 + 8 * q^16 + 2 * q^19 - 8 * q^31 - 20 * q^37 + 18 * q^39 - 26 * q^49 - 20 * q^52 + 18 * q^57 - 30 * q^63 + 32 * q^67 - 34 * q^73 + 30 * q^75 - 28 * q^76 - 18 * q^81 - 12 * q^84 - 36 * q^93 + 28 * q^97

Introduction

L-functions

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