LMFDB - Embedded newform 273.2.bw.a.254.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(11,273)]

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, 8, 7]))

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

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

chi := DirichletCharacter("273.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.bw (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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			254.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	273.254
Dual form			273.2.bw.a.158.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +(3.00000 + 1.73205i) q^{12} +(3.46410 + 1.00000i) q^{13} +(2.00000 + 3.46410i) q^{16} +(4.83013 + 4.83013i) q^{19} +(-1.50000 - 4.33013i) q^{21} +(4.33013 - 2.50000i) q^{25} -5.19615 q^{27} +(1.00000 - 5.19615i) q^{28} +(-2.86603 + 10.6962i) q^{31} +(-5.19615 - 3.00000i) q^{36} +(-11.0622 - 2.96410i) q^{37} +(-6.00000 - 1.73205i) q^{39} +(9.00000 - 5.19615i) q^{43} +(-3.46410 - 6.00000i) q^{48} +(-5.50000 + 4.33013i) q^{49} +(-5.00000 - 5.19615i) q^{52} +(-8.36603 - 8.36603i) q^{57} +15.5885 q^{61} +(2.59808 + 7.50000i) q^{63} -8.00000i q^{64} +(0.562178 + 0.562178i) q^{67} +(2.63397 + 0.705771i) q^{73} +(-7.50000 + 4.33013i) q^{75} +(-3.53590 - 13.1962i) q^{76} +(-6.06218 + 10.5000i) q^{79} +9.00000 q^{81} +(-1.73205 + 9.00000i) q^{84} +(0.500000 + 9.52628i) q^{91} +(4.96410 - 18.5263i) q^{93} +(-4.42820 + 16.5263i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 12 q^{12} + 8 q^{16} + 2 q^{19} - 6 q^{21} + 4 q^{28} - 8 q^{31} - 20 q^{37} - 24 q^{39} + 36 q^{43} - 22 q^{49} - 20 q^{52} - 30 q^{57} - 22 q^{67} + 14 q^{73} - 30 q^{75} - 28 q^{76} + 36 q^{81} + 2 q^{91} + 6 q^{93} + 10 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 + 12 * q^12 + 8 * q^16 + 2 * q^19 - 6 * q^21 + 4 * q^28 - 8 * q^31 - 20 * q^37 - 24 * q^39 + 36 * q^43 - 22 * q^49 - 20 * q^52 - 30 * q^57 - 22 * q^67 + 14 * q^73 - 30 * q^75 - 28 * q^76 + 36 * q^81 + 2 * q^91 + 6 * q^93 + 10 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$92$	$106$	$157$
$\chi(n)$	$-1$	$e\left(\frac{11}{12}\right)$	$e\left(\frac{1}{3}\right)$

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	$-0.416667\pi$
$2$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$3$	−1.73205			−1.00000
$3$	−1.73205			−1.00000
$4$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$5$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$5$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$6$		0			0
$7$	0.866025	+	2.50000i	0.327327	+	0.944911i
$7$	0.866025	+	2.50000i	0.327327	+	0.944911i
$8$		0			0
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$	3.00000	+	1.73205i	0.866025	+	0.500000i
$13$	3.46410	+	1.00000i	0.960769	+	0.277350i
$13$	3.46410	+	1.00000i	0.960769	+	0.277350i
$14$		0			0
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	4.83013	+	4.83013i	1.10811	+	1.10811i	0.993399	+	0.114708i	$0.0365932\pi$
$19$	4.83013	+	4.83013i	1.10811	+	1.10811i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$	−1.50000	−	4.33013i	−0.327327	−	0.944911i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	4.33013	−	2.50000i	0.866025	−	0.500000i
$26$		0			0
$27$	−5.19615			−1.00000
$28$	1.00000	−	5.19615i	0.188982	−	0.981981i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	−2.86603	+	10.6962i	−0.514753	+	1.92109i	−0.155543	+	0.987829i	$0.549713\pi$
$31$	−2.86603	+	10.6962i	−0.514753	+	1.92109i	−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.19615	−	3.00000i	−0.866025	−	0.500000i
$37$	−11.0622	−	2.96410i	−1.81861	−	0.487295i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−11.0622	−	2.96410i	−1.81861	−	0.487295i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$		0			0
$39$	−6.00000	−	1.73205i	−0.960769	−	0.277350i
$40$		0			0
$41$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$41$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$42$		0			0
$43$	9.00000	−	5.19615i	1.37249	−	0.792406i	0.381246	−	0.924473i	$-0.375495\pi$
$43$	9.00000	−	5.19615i	1.37249	−	0.792406i	0.991241	+	0.132068i	$0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$47$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$48$	−3.46410	−	6.00000i	−0.500000	−	0.866025i
$49$	−5.50000	+	4.33013i	−0.785714	+	0.618590i
$50$		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
$57$	−8.36603	−	8.36603i	−1.10811	−	1.10811i
$58$		0			0
$59$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$59$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$60$		0			0
$61$	15.5885			1.99590			0.997949	−	0.0640184i	$-0.0203916\pi$
$61$	15.5885			1.99590			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
$67$	0.562178	+	0.562178i	0.0686810	+	0.0686810i	0.740613	−	0.671932i	$-0.234535\pi$
$67$	0.562178	+	0.562178i	0.0686810	+	0.0686810i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$71$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$72$		0			0
$73$	2.63397	+	0.705771i	0.308283	+	0.0826043i	0.409644	−	0.912245i	$-0.365653\pi$
$73$	2.63397	+	0.705771i	0.308283	+	0.0826043i	−0.101361	+	0.994850i	$0.532320\pi$
$74$		0			0
$75$	−7.50000	+	4.33013i	−0.866025	+	0.500000i
$76$	−3.53590	−	13.1962i	−0.405595	−	1.51370i
$77$		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
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$	−1.73205	+	9.00000i	−0.188982	+	0.981981i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$89$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$90$		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
$97$	−4.42820	+	16.5263i	−0.449616	+	1.67799i	0.253837	+	0.967247i	$0.418307\pi$
$97$	−4.42820	+	16.5263i	−0.449616	+	1.67799i	−0.703452	+	0.710742i	$0.748359\pi$
$98$		0			0
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	0.344865	−	0.938652i	$-0.387925\pi$
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	9.00000	+	5.19615i	0.866025	+	0.500000i
$109$	3.43782	−	12.8301i	0.329284	−	1.22890i	−0.580651	−	0.814152i	$-0.697202\pi$
$109$	3.43782	−	12.8301i	0.329284	−	1.22890i	0.909935	−	0.414751i	$-0.136131\pi$
$110$		0			0
$111$	19.1603	+	5.13397i	1.81861	+	0.487295i
$112$	−6.92820	+	8.00000i	−0.654654	+	0.755929i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		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
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$	15.6603	−	15.6603i	1.40633	−	1.40633i
$125$		0			0
$126$		0			0
$127$	−17.3205	−	10.0000i	−1.53695	−	0.887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$	−17.3205	−	10.0000i	−1.53695	−	0.887357i	−0.537931	−	0.842989i	$-0.680794\pi$
$128$		0			0
$129$	−15.5885	+	9.00000i	−1.37249	+	0.792406i
$130$		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
$133$	−7.89230	+	16.2583i	−0.684350	+	1.40978i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$137$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$138$		0			0
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		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
$148$	16.1962	+	16.1962i	1.33132	+	1.33132i
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$		0			0
$151$	19.2942	+	5.16987i	1.57014	+	0.420718i	0.935857	−	0.352381i	$-0.114628\pi$
$151$	19.2942	+	5.16987i	1.57014	+	0.420718i	0.634285	+	0.773099i	$0.281294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	8.66025	+	9.00000i	0.693375	+	0.720577i
$157$	−12.5000	−	21.6506i	−0.997609	−	1.72791i	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−12.5000	−	21.6506i	−0.997609	−	1.72791i	−0.438948	−	0.898513i	$-0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.02628	+	1.02628i	−0.0803844	+	0.0803844i	−0.746156	−	0.665771i	$-0.768103\pi$
$163$	−1.02628	+	1.02628i	−0.0803844	+	0.0803844i	0.665771	+	0.746156i	$0.268103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$167$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$168$		0			0
$169$	11.0000	+	6.92820i	0.846154	+	0.532939i
$170$		0			0
$171$	14.4904	+	14.4904i	1.10811	+	1.10811i
$172$	−20.7846			−1.58481
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	10.0000	+	8.66025i	0.755929	+	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		−	19.0526i		−	1.41617i	−0.706129	−	0.708083i	$-0.749560\pi$
$181$		−	19.0526i		−	1.41617i	0.706129	−	0.708083i	$-0.250440\pi$
$182$		0			0
$183$	−27.0000			−1.99590
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−4.50000	−	12.9904i	−0.327327	−	0.944911i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			13.8564i			1.00000i
$193$	12.8564	−	12.8564i	0.925424	−	0.925424i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	12.8564	−	12.8564i	0.925424	−	0.925424i	0.997406	+	0.0719816i	$0.0229323\pi$
$194$		0			0
$195$		0			0
$196$	13.8564	−	2.00000i	0.989743	−	0.142857i
$197$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$197$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$	−9.52628	−	5.50000i	−0.675300	−	0.389885i	0.122782	−	0.992434i	$-0.460818\pi$
$199$	−9.52628	−	5.50000i	−0.675300	−	0.389885i	−0.798082	+	0.602549i	$0.794152\pi$
$200$		0			0
$201$	−0.973721	−	0.973721i	−0.0686810	−	0.0686810i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	3.46410	+	14.0000i	0.240192	+	0.970725i
$209$		0			0
$210$		0			0
$211$	−0.866025	+	1.50000i	−0.0596196	+	0.103264i	−0.894295	−	0.447478i	$-0.852322\pi$
$211$	−0.866025	+	1.50000i	−0.0596196	+	0.103264i	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
$219$	−4.56218	−	1.22243i	−0.308283	−	0.0826043i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.0263	+	3.22243i	−0.805339	+	0.215790i	−0.637927	−	0.770097i	$-0.720208\pi$
$223$	−12.0263	+	3.22243i	−0.805339	+	0.215790i	−0.167412	+	0.985887i	$0.553541\pi$
$224$		0			0
$225$	12.9904	−	7.50000i	0.866025	−	0.500000i
$226$		0			0
$227$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$227$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$228$	6.12436	+	22.8564i	0.405595	+	1.51370i
$229$	−3.72243	−	13.8923i	−0.245985	−	0.918029i	−0.972886	−	0.231287i	$-0.925707\pi$
$229$	−3.72243	−	13.8923i	−0.245985	−	0.918029i	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
$237$	10.5000	−	18.1865i	0.682048	−	1.18134i
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−2.50962	−	9.36603i	−0.161659	−	0.603319i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−2.50962	−	9.36603i	−0.161659	−	0.603319i	0.836784	−	0.547533i	$-0.184433\pi$
$242$		0			0
$243$	−15.5885			−1.00000
$244$	−27.0000	−	15.5885i	−1.72850	−	0.997949i
$245$		0			0
$246$		0			0
$247$	11.9019	+	21.5622i	0.757301	+	1.37197i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$	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	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−2.16987	−	30.2224i	−0.134829	−	1.87793i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.411543	−	1.53590i	−0.0251390	−	0.0938199i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−21.7942	−	5.83975i	−1.32391	−	0.354739i	−0.473466	−	0.880812i	$-0.656997\pi$
$271$	−21.7942	−	5.83975i	−1.32391	−	0.354739i	−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$	−0.866025	−	16.5000i	−0.0524142	−	0.998625i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−28.5000	−	16.4545i	−1.71240	−	0.988654i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−28.5000	−	16.4545i	−1.71240	−	0.988654i	−0.781094	−	0.624413i	$-0.785338\pi$
$278$		0			0
$279$	−8.59808	+	32.0885i	−0.514753	+	1.92109i
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$		−	32.0000i		−	1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$		−	32.0000i		−	1.90220i	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
$291$	7.66987	−	28.6244i	0.449616	−	1.67799i
$292$	−3.85641	−	3.85641i	−0.225679	−	0.225679i
$293$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$293$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	17.3205			1.00000
$301$	20.7846	+	18.0000i	1.19800	+	1.03750i
$302$		0			0
$303$		0			0
$304$	−7.07180	+	26.3923i	−0.405595	+	1.51370i
$305$		0			0
$306$		0			0
$307$	16.6340	−	16.6340i	0.949351	−	0.949351i	−0.0494267	−	0.998778i	$-0.515739\pi$
$307$	16.6340	−	16.6340i	0.949351	−	0.949351i	0.998778	+	0.0494267i	$0.0157394\pi$
$308$		0			0
$309$	5.19615	+	3.00000i	0.295599	+	0.170664i
$310$		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
$313$	−13.8564	+	24.0000i	−0.783210	+	1.35656i	0.146852	+	0.989158i	$0.453086\pi$
$313$	−13.8564	+	24.0000i	−0.783210	+	1.35656i	−0.930062	+	0.367402i	$0.880247\pi$
$314$		0			0
$315$		0			0
$316$	21.0000	−	12.1244i	1.18134	−	0.682048i
$317$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$317$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$318$		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
$325$	17.5000	−	4.33013i	0.970725	−	0.240192i
$326$		0			0
$327$	−5.95448	+	22.2224i	−0.329284	+	1.22890i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.33975	−	7.33975i	−0.403429	−	0.403429i	0.476011	−	0.879440i	$-0.342082\pi$
$331$	−7.33975	−	7.33975i	−0.403429	−	0.403429i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	−33.1865	−	8.89230i	−1.81861	−	0.487295i
$334$		0			0
$335$		0			0
$336$	12.0000	−	13.8564i	0.654654	−	0.755929i
$337$			34.0000i			1.85210i	0.377403	+	0.926049i	$0.376817\pi$
$337$			34.0000i			1.85210i	−0.377403	+	0.926049i	$0.623183\pi$
$338$		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	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	7.72243	+	28.8205i	0.413372	+	1.54273i	0.788074	+	0.615581i	$0.211079\pi$
$349$	7.72243	+	28.8205i	0.413372	+	1.54273i	−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	−18.0000	−	5.19615i	−0.960769	−	0.277350i
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$359$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$360$		0			0
$361$			27.6603i			1.45580i
$362$		0			0
$363$			19.0526i			1.00000i
$364$	8.66025	−	17.0000i	0.453921	−	0.891042i
$365$		0			0
$366$		0			0
$367$	4.00000			0.208798			0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000			0.208798			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
$373$	36.3731			1.88333			0.941663	−	0.336557i	$-0.109263\pi$
$373$	36.3731			1.88333			0.941663	+	0.336557i	$0.109263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	16.9904	−	4.55256i	0.872737	−	0.233849i	0.205466	−	0.978664i	$-0.434129\pi$
$379$	16.9904	−	4.55256i	0.872737	−	0.233849i	0.667271	+	0.744815i	$0.267462\pi$
$380$		0			0
$381$	30.0000	+	17.3205i	1.53695	+	0.887357i
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	27.0000	−	15.5885i	1.37249	−	0.792406i
$388$	24.1962	−	24.1962i	1.22837	−	1.22837i
$389$		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
$397$	6.60770	−	6.60770i	0.331631	−	0.331631i	−0.521575	−	0.853206i	$-0.674655\pi$
$397$	6.60770	−	6.60770i	0.331631	−	0.331631i	0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$	13.6699	−	28.1603i	0.684350	−	1.40978i
$400$	17.3205	+	10.0000i	0.866025	+	0.500000i
$401$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$401$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$402$		0			0
$403$	−20.6244	+	34.1865i	−1.02737	+	1.70295i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	35.4186	−	9.49038i	1.75134	−	0.469269i	0.766426	−	0.642333i	$-0.222033\pi$
$409$	35.4186	−	9.49038i	1.75134	−	0.469269i	0.984911	+	0.173064i	$0.0553667\pi$
$410$		0			0
$411$		0			0
$412$	3.46410	+	6.00000i	0.170664	+	0.295599i
$413$		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		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	27.6865	+	27.6865i	1.34936	+	1.34936i	0.886357	+	0.463002i	$0.153228\pi$
$421$	27.6865	+	27.6865i	1.34936	+	1.34936i	0.463002	+	0.886357i	$0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.5000	+	38.9711i	0.653311	+	1.88595i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	−10.3923	−	18.0000i	−0.500000	−	0.866025i
$433$	−30.3109	+	17.5000i	−1.45665	+	0.840996i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	−30.3109	+	17.5000i	−1.45665	+	0.840996i	−0.457804	+	0.889053i	$0.651364\pi$
$434$		0			0
$435$		0			0
$436$	−18.7846	+	18.7846i	−0.899620	+	0.899620i
$437$		0			0
$438$		0			0
$439$	7.50000	−	4.33013i	0.357955	−	0.206666i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	7.50000	−	4.33013i	0.357955	−	0.206666i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	−16.5000	+	12.9904i	−0.785714	+	0.618590i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−28.0526	−	28.0526i	−1.33132	−	1.33132i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	20.0000	−	6.92820i	0.944911	−	0.327327i
$449$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$449$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$450$		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	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	9.72243	−	36.2846i	0.454796	−	1.69732i	0.688686	−	0.725059i	$-0.258188\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$461$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$462$		0			0
$463$	29.0526	−	29.0526i	1.35019	−	1.35019i	0.464739	−	0.885448i	$-0.346148\pi$
$463$	29.0526	−	29.0526i	1.35019	−	1.35019i	0.885448	−	0.464739i	$-0.153852\pi$
$464$		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	$0.166667\pi$
$468$	−15.0000	−	15.5885i	−0.693375	−	0.720577i
$469$	−0.918584	+	1.89230i	−0.0424163	+	0.0873785i
$470$		0			0
$471$	21.6506	+	37.5000i	0.997609	+	1.72791i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	32.9904	+	8.83975i	1.51370	+	0.405595i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$	−35.3564	−	21.3301i	−1.61211	−	0.972570i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−14.2321	+	3.81347i	−0.644916	+	0.172805i	−0.566429	−	0.824110i	$-0.691675\pi$
$487$	−14.2321	+	3.81347i	−0.644916	+	0.172805i	−0.0784867	+	0.996915i	$0.525009\pi$
$488$		0			0
$489$	1.77757	−	1.77757i	0.0803844	−	0.0803844i
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−42.7846	+	11.4641i	−1.92109	+	0.514753i
$497$		0			0
$498$		0			0
$499$	−37.0885	+	9.93782i	−1.66031	+	0.444878i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−37.0885	+	9.93782i	−1.66031	+	0.444878i	−0.697835	+	0.716258i	$0.745853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−19.0526	−	12.0000i	−0.846154	−	0.532939i
$508$	20.0000	+	34.6410i	0.887357	+	1.53695i
$509$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$509$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$510$		0			0
$511$	0.516660	+	7.19615i	0.0228557	+	0.318339i
$512$		0			0
$513$	−25.0981	−	25.0981i	−1.10811	−	1.10811i
$514$		0			0
$515$		0			0
$516$	36.0000			1.58481
$517$		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
$523$	−21.5000	−	37.2391i	−0.940129	−	1.62835i	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−21.5000	−	37.2391i	−0.940129	−	1.62835i	−0.174908	−	0.984585i	$-0.555963\pi$
$524$		0			0
$525$	−17.3205	−	15.0000i	−0.755929	−	0.654654i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	29.9282	−	20.2679i	1.29755	−	0.878727i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.15064	+	26.6865i	0.307430	+	1.14734i	0.930834	+	0.365444i	$0.119083\pi$
$541$	7.15064	+	26.6865i	0.307430	+	1.14734i	−0.623404	+	0.781900i	$0.714251\pi$
$542$		0			0
$543$			33.0000i			1.41617i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.00000			−0.0427569			−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000			−0.0427569			−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$	46.7654			1.99590
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−31.5000	−	6.06218i	−1.33952	−	0.257790i
$554$		0			0
$555$		0			0
$556$		−	14.0000i		−	0.593732i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$	36.3731	−	9.00000i	1.53842	−	0.380659i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.79423	+	22.5000i	0.327327	+	0.944911i
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	40.7032	−	23.5000i	1.70338	−	0.983444i	0.761083	−	0.648655i	$-0.224668\pi$
$571$	40.7032	−	23.5000i	1.70338	−	0.983444i	0.942293	−	0.334790i	$-0.108665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		−	24.0000i		−	1.00000i
$577$	−6.54552	+	24.4282i	−0.272493	+	1.01696i	0.685009	+	0.728535i	$0.259798\pi$
$577$	−6.54552	+	24.4282i	−0.272493	+	1.01696i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	−22.2679	+	22.2679i	−0.925424	+	0.925424i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$	−24.0000	+	3.46410i	−0.989743	+	0.142857i
$589$	−65.5070	+	37.8205i	−2.69917	+	1.55837i
$590$		0			0
$591$		0			0
$592$	−11.8564	−	44.2487i	−0.487295	−	1.81861i
$593$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$593$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	16.5000	+	9.52628i	0.675300	+	0.389885i
$598$		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
$601$	21.6506	−	37.5000i	0.883148	−	1.52966i	0.0353259	−	0.999376i	$-0.488753\pi$
$601$	21.6506	−	37.5000i	0.883148	−	1.52966i	0.847822	−	0.530281i	$-0.177914\pi$
$602$		0			0
$603$	1.68653	+	1.68653i	0.0686810	+	0.0686810i
$604$	−28.2487	−	28.2487i	−1.14942	−	1.14942i
$605$		0			0
$606$		0			0
$607$	−29.0000			−1.17707			−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000			−1.17707			−0.588537	+	0.808470i	$0.700296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	15.7058	+	15.7058i	0.634350	+	0.634350i	0.949156	−	0.314806i	$-0.101939\pi$
$613$	15.7058	+	15.7058i	0.634350	+	0.634350i	−0.314806	+	0.949156i	$0.601939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$617$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$618$		0			0
$619$	−4.16987	−	1.11731i	−0.167601	−	0.0449086i	0.174042	−	0.984738i	$-0.444317\pi$
$619$	−4.16987	−	1.11731i	−0.167601	−	0.0449086i	−0.341644	+	0.939829i	$0.610984\pi$
$620$		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
$628$			50.0000i			1.99522i
$629$		0			0
$630$		0			0
$631$	−47.1147	−	12.6244i	−1.87561	−	0.502568i	−0.999802	−	0.0199047i	$-0.993664\pi$
$631$	−47.1147	−	12.6244i	−1.87561	−	0.502568i	−0.875806	−	0.482663i	$-0.839670\pi$
$632$		0			0
$633$	1.50000	−	2.59808i	0.0596196	−	0.103264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−23.3827	+	9.50000i	−0.926456	+	0.376404i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	13.0263	−	48.6147i	0.513706	−	1.91718i	0.138027	−	0.990429i	$-0.455924\pi$
$643$	13.0263	−	48.6147i	0.513706	−	1.91718i	0.375680	−	0.926750i	$-0.377409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	50.6147	−	3.63397i	1.98375	−	0.142427i
$652$	2.80385	−	0.751289i	0.109807	−	0.0294227i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	7.90192	+	2.11731i	0.308283	+	0.0826043i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−32.2942	+	32.2942i	−1.25610	+	1.25610i	−0.303160	+	0.952940i	$0.598042\pi$
$661$	−32.2942	+	32.2942i	−1.25610	+	1.25610i	−0.952940	+	0.303160i	$0.901958\pi$
$662$		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
$673$	12.0000	+	6.92820i	0.462566	+	0.267063i	0.713123	−	0.701039i	$-0.247280\pi$
$673$	12.0000	+	6.92820i	0.462566	+	0.267063i	−0.250557	+	0.968102i	$0.580614\pi$
$674$		0			0
$675$	−22.5000	+	12.9904i	−0.866025	+	0.500000i
$676$	−12.1244	−	23.0000i	−0.466321	−	0.884615i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−45.1506	+	3.24167i	−1.73272	+	0.124404i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$683$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$684$	−10.6077	−	39.5885i	−0.405595	−	1.51370i
$685$		0			0
$686$		0			0
$687$	6.44744	+	24.0622i	0.245985	+	0.918029i
$688$	36.0000	+	20.7846i	1.37249	+	0.792406i
$689$		0			0
$690$		0			0
$691$	10.9737	−	40.9545i	0.417460	−	1.55798i	−0.362397	−	0.932024i	$-0.618041\pi$
$691$	10.9737	−	40.9545i	0.417460	−	1.55798i	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
$700$	−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
$703$	−39.1147	−	67.7487i	−1.47524	−	2.55519i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−37.1506	+	37.1506i	−1.39522	+	1.39522i	−0.582115	+	0.813107i	$0.697775\pi$
$709$	−37.1506	+	37.1506i	−1.39522	+	1.39522i	−0.813107	+	0.582115i	$0.802225\pi$
$710$		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			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	1.73205	−	9.00000i	0.0645049	−	0.335178i
$722$		0			0
$723$	4.34679	+	16.2224i	0.161659	+	0.603319i
$724$	−19.0526	+	33.0000i	−0.708083	+	1.22644i
$725$		0			0
$726$		0			0
$727$			49.0000i			1.81731i	0.417548	+	0.908655i	$0.362889\pi$
$727$			49.0000i			1.81731i	−0.417548	+	0.908655i	$0.637111\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	46.7654	+	27.0000i	1.72850	+	0.997949i
$733$	41.4545	−	11.1077i	1.53116	−	0.410272i	0.607760	−	0.794121i	$-0.292068\pi$
$733$	41.4545	−	11.1077i	1.53116	−	0.410272i	0.923396	+	0.383849i	$0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.41858	+	1.41858i	−0.0521835	+	0.0521835i	−0.732717	−	0.680534i	$-0.761748\pi$
$739$	−1.41858	+	1.41858i	−0.0521835	+	0.0521835i	0.680534	+	0.732717i	$0.261748\pi$
$740$		0			0
$741$	−20.6147	−	37.3468i	−0.757301	−	1.37197i
$742$		0			0
$743$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$743$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	46.5000	−	26.8468i	1.69681	−	0.979653i	0.748056	−	0.663636i	$-0.230988\pi$
$751$	46.5000	−	26.8468i	1.69681	−	0.979653i	0.948753	−	0.316017i	$-0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−5.19615	+	27.0000i	−0.188982	+	0.981981i
$757$	24.2487	−	42.0000i	0.881334	−	1.52652i	0.0314762	−	0.999505i	$-0.489979\pi$
$757$	24.2487	−	42.0000i	0.881334	−	1.52652i	0.849858	−	0.527011i	$-0.176688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$761$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$762$		0			0
$763$	35.0526	−	2.51666i	1.26899	−	0.0911092i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	13.8564	−	24.0000i	0.500000	−	0.866025i
$769$	−52.2128	+	13.9904i	−1.88284	+	0.504506i	−0.883493	+	0.468445i	$0.844814\pi$
$769$	−52.2128	+	13.9904i	−1.88284	+	0.504506i	−0.999350	+	0.0360609i	$0.988519\pi$
$770$		0			0
$771$		0			0
$772$	−35.1244	+	9.41154i	−1.26415	+	0.338729i
$773$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$773$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$774$		0			0
$775$	14.3301	+	53.4808i	0.514753	+	1.92109i
$776$		0			0
$777$	3.75833	+	52.3468i	0.134829	+	1.87793i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.0000	−	10.3923i	−0.928571	−	0.371154i
$785$		0			0
$786$		0			0
$787$	2.88526	+	10.7679i	0.102849	+	0.383836i	0.998092	−	0.0617409i	$-0.0196653\pi$
$787$	2.88526	+	10.7679i	0.102849	+	0.383836i	−0.895244	+	0.445577i	$0.852999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	54.0000	+	15.5885i	1.91760	+	0.553562i
$794$		0			0
$795$		0			0
$796$	11.0000	+	19.0526i	0.389885	+	0.675300i
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.712813	+	2.66025i	0.0251390	+	0.0938199i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	40.1506	+	40.1506i	1.40988	+	1.40988i	0.760257	+	0.649623i	$0.225073\pi$
$811$	40.1506	+	40.1506i	1.40988	+	1.40988i	0.649623	+	0.760257i	$0.274927\pi$
$812$		0			0
$813$	37.7487	+	10.1147i	1.32391	+	0.354739i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	68.5692	+	18.3731i	2.39893	+	0.642792i
$818$		0			0
$819$	1.50000	+	28.5788i	0.0524142	+	0.998625i
$820$		0			0
$821$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$821$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$822$		0			0
$823$	−21.0000	−	12.1244i	−0.732014	−	0.422628i	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	−21.0000	−	12.1244i	−0.732014	−	0.422628i	−0.819159	+	0.573567i	$0.805559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$			46.0000i			1.59765i	0.601566	+	0.798823i	$0.294544\pi$
$829$			46.0000i			1.59765i	−0.601566	+	0.798823i	$0.705456\pi$
$830$		0			0
$831$	49.3634	+	28.5000i	1.71240	+	0.988654i
$832$	8.00000	−	27.7128i	0.277350	−	0.960769i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	14.8923	−	55.5788i	0.514753	−	1.92109i
$838$		0			0
$839$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$839$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	3.00000	−	1.73205i	0.103264	−	0.0596196i
$845$		0			0
$846$		0			0
$847$	27.5000	−	9.52628i	0.944911	−	0.327327i
$848$		0			0
$849$			55.4256i			1.90220i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−5.88269	+	5.88269i	−0.201419	+	0.201419i	−0.800608	−	0.599189i	$-0.795490\pi$
$853$	−5.88269	+	5.88269i	−0.201419	+	0.201419i	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
$859$	19.9186	−	34.5000i	0.679613	−	1.17712i	−0.295484	−	0.955348i	$-0.595481\pi$
$859$	19.9186	−	34.5000i	0.679613	−	1.17712i	0.975097	−	0.221777i	$-0.0711857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$863$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−14.7224	−	25.5000i	−0.500000	−	0.866025i
$868$	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
$873$	−13.2846	+	49.5788i	−0.449616	+	1.67799i
$874$		0			0
$875$		0			0
$876$	6.67949	+	6.67949i	0.225679	+	0.225679i
$877$	−41.2487	−	41.2487i	−1.39287	−	1.39287i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	−41.2487	−	41.2487i	−1.39287	−	1.39287i	−0.574049	−	0.818821i	$-0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$		0			0
$883$			8.00000i			0.269221i	0.990899	+	0.134611i	$0.0429784\pi$
$883$			8.00000i			0.269221i	−0.990899	+	0.134611i	$0.957022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	10.0000	−	51.9615i	0.335389	−	1.74273i
$890$		0			0
$891$		0			0
$892$	24.0526	+	6.44486i	0.805339	+	0.215790i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−30.0000			−1.00000
$901$		0			0
$902$		0			0
$903$	−36.0000	−	31.1769i	−1.19800	−	1.03750i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	19.0000i		−	0.630885i	−0.948945	−	0.315442i	$-0.897847\pi$
$907$		−	19.0000i		−	0.630885i	0.948945	−	0.315442i	$-0.102153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	12.2487	−	45.7128i	0.405595	−	1.51370i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−7.44486	+	27.7846i	−0.245985	+	0.918029i
$917$		0			0
$918$		0			0
$919$	−29.4449			−0.971296			−0.485648	−	0.874154i	$-0.661416\pi$
$919$	−29.4449			−0.971296			−0.485648	+	0.874154i	$0.661416\pi$
$920$		0			0
$921$	−28.8109	+	28.8109i	−0.949351	+	0.949351i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−55.3109	+	14.8205i	−1.81861	+	0.487295i
$926$		0			0
$927$	−9.00000	−	5.19615i	−0.295599	−	0.170664i
$928$		0			0
$929$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$929$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$	−47.4808	−	5.65064i	−1.55612	−	0.185192i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	5.19615			0.169751			0.0848755	−	0.996392i	$-0.472951\pi$
$937$	5.19615			0.169751			0.0848755	+	0.996392i	$0.472951\pi$
$938$		0			0
$939$	24.0000	−	41.5692i	0.783210	−	1.35656i
$940$		0			0
$941$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$941$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$947$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$948$	−36.3731	+	21.0000i	−1.18134	+	0.682048i
$949$	8.41858	+	5.07884i	0.273279	+	0.164866i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		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
$967$	−43.8827	−	43.8827i	−1.41117	−	1.41117i	−0.751936	−	0.659236i	$-0.770880\pi$
$967$	−43.8827	−	43.8827i	−1.41117	−	1.41117i	−0.659236	−	0.751936i	$-0.729120\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	27.0000	+	15.5885i	0.866025	+	0.500000i
$973$	−12.1244	+	14.0000i	−0.388689	+	0.448819i
$974$		0			0
$975$	−30.3109	+	7.50000i	−0.970725	+	0.240192i
$976$	31.1769	+	54.0000i	0.997949	+	1.72850i
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	10.3135	−	38.4904i	0.329284	−	1.22890i
$982$		0			0
$983$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$983$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.947441	−	49.2487i	0.0301421	−	1.56681i
$989$		0			0
$990$		0			0
$991$	−61.0000			−1.93773			−0.968864	−	0.247592i	$-0.920361\pi$
$991$	−61.0000			−1.93773			−0.968864	+	0.247592i	$0.920361\pi$
$992$		0			0
$993$	12.7128	+	12.7128i	0.403429	+	0.403429i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	57.4808	+	15.4019i	1.81861	+	0.487295i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bw.a.254.1	yes	4
3.2	odd	2	CM	273.2.bw.a.254.1	yes	4
7.4	even	3		273.2.bv.a.137.1	yes	4
13.2	odd	12		273.2.bv.a.2.1	✓	4
21.11	odd	6		273.2.bv.a.137.1	yes	4
39.2	even	12		273.2.bv.a.2.1	✓	4
91.67	odd	12	inner	273.2.bw.a.158.1	yes	4
273.158	even	12	inner	273.2.bw.a.158.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bv.a.2.1	✓	4	13.2	odd	12
273.2.bv.a.2.1	✓	4	39.2	even	12
273.2.bv.a.137.1	yes	4	7.4	even	3
273.2.bv.a.137.1	yes	4	21.11	odd	6
273.2.bw.a.158.1	yes	4	91.67	odd	12	inner
273.2.bw.a.158.1	yes	4	273.158	even	12	inner
273.2.bw.a.254.1	yes	4	1.1	even	1	trivial
273.2.bw.a.254.1	yes	4	3.2	odd	2	CM

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.bw (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +(-1.73205 - 1.00000i) q^{4} +(0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +(3.00000 + 1.73205i) q^{12} +(3.46410 + 1.00000i) q^{13} +(2.00000 + 3.46410i) q^{16} +(4.83013 + 4.83013i) q^{19} +(-1.50000 - 4.33013i) q^{21} +(4.33013 - 2.50000i) q^{25} -5.19615 q^{27} +(1.00000 - 5.19615i) q^{28} +(-2.86603 + 10.6962i) q^{31} +(-5.19615 - 3.00000i) q^{36} +(-11.0622 - 2.96410i) q^{37} +(-6.00000 - 1.73205i) q^{39} +(9.00000 - 5.19615i) q^{43} +(-3.46410 - 6.00000i) q^{48} +(-5.50000 + 4.33013i) q^{49} +(-5.00000 - 5.19615i) q^{52} +(-8.36603 - 8.36603i) q^{57} +15.5885 q^{61} +(2.59808 + 7.50000i) q^{63} -8.00000i q^{64} +(0.562178 + 0.562178i) q^{67} +(2.63397 + 0.705771i) q^{73} +(-7.50000 + 4.33013i) q^{75} +(-3.53590 - 13.1962i) q^{76} +(-6.06218 + 10.5000i) q^{79} +9.00000 q^{81} +(-1.73205 + 9.00000i) q^{84} +(0.500000 + 9.52628i) q^{91} +(4.96410 - 18.5263i) q^{93} +(-4.42820 + 16.5263i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 12 q^{12} + 8 q^{16} + 2 q^{19} - 6 q^{21} + 4 q^{28} - 8 q^{31} - 20 q^{37} - 24 q^{39} + 36 q^{43} - 22 q^{49} - 20 q^{52} - 30 q^{57} - 22 q^{67} + 14 q^{73} - 30 q^{75} - 28 q^{76} + 36 q^{81} + 2 q^{91} + 6 q^{93} + 10 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 + 12 * q^12 + 8 * q^16 + 2 * q^19 - 6 * q^21 + 4 * q^28 - 8 * q^31 - 20 * q^37 - 24 * q^39 + 36 * q^43 - 22 * q^49 - 20 * q^52 - 30 * q^57 - 22 * q^67 + 14 * q^73 - 30 * q^75 - 28 * q^76 + 36 * q^81 + 2 * q^91 + 6 * q^93 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more