LMFDB - Embedded newform 273.3.ca.b.167.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,3,Mod(20,273)]

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, 6, 11]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("273.20");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	273.ca (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:	$7.43871121704$
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			167.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	273.167
Dual form			273.3.ca.b.188.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.59808 - 1.50000i) q^{3} +(-3.46410 - 2.00000i) q^{4} +(5.50000 - 4.33013i) q^{7} +(4.50000 - 7.79423i) q^{9} +O(q^{10})$	$q+(2.59808 - 1.50000i) q^{3} +(-3.46410 - 2.00000i) q^{4} +(5.50000 - 4.33013i) q^{7} +(4.50000 - 7.79423i) q^{9} -12.0000 q^{12} +(-12.9904 + 0.500000i) q^{13} +(8.00000 + 13.8564i) q^{16} +(-9.83013 - 36.6865i) q^{19} +(7.79423 - 19.5000i) q^{21} -25.0000i q^{25} -27.0000i q^{27} +(-27.7128 + 4.00000i) q^{28} +(-23.8109 + 23.8109i) q^{31} +(-31.1769 + 18.0000i) q^{36} +(65.0788 + 17.4378i) q^{37} +(-33.0000 + 20.7846i) q^{39} +(19.5000 + 11.2583i) q^{43} +(41.5692 + 24.0000i) q^{48} +(11.5000 - 47.6314i) q^{49} +(46.0000 + 24.2487i) q^{52} +(-80.5692 - 80.5692i) q^{57} +(40.7032 + 23.5000i) q^{61} +(-9.00000 - 62.3538i) q^{63} -64.0000i q^{64} +(-22.0289 + 82.2128i) q^{67} +(86.2224 + 86.2224i) q^{73} +(-37.5000 - 64.9519i) q^{75} +(-39.3205 + 146.746i) q^{76} +157.617 q^{79} +(-40.5000 - 70.1481i) q^{81} +(-66.0000 + 51.9615i) q^{84} +(-69.2820 + 59.0000i) q^{91} +(-26.1462 + 97.5788i) q^{93} +(-13.4948 - 50.3634i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 22 q^{7} + 18 q^{9}+O(q^{10}) $ 4 * q + 22 * q^7 + 18 * q^9	$ 4 q + 22 q^{7} + 18 q^{9} - 48 q^{12} + 32 q^{16} - 22 q^{19} + 26 q^{31} + 146 q^{37} - 132 q^{39} + 78 q^{43} + 46 q^{49} + 184 q^{52} - 156 q^{57} - 36 q^{63} - 244 q^{67} + 286 q^{73} - 150 q^{75} - 88 q^{76} - 162 q^{81} - 264 q^{84} - 354 q^{93} + 334 q^{97}+O(q^{100}) $ 4 * q + 22 * q^7 + 18 * q^9 - 48 * q^12 + 32 * q^16 - 22 * q^19 + 26 * q^31 + 146 * q^37 - 132 * q^39 + 78 * q^43 + 46 * q^49 + 184 * q^52 - 156 * q^57 - 36 * q^63 - 244 * q^67 + 286 * q^73 - 150 * q^75 - 88 * q^76 - 162 * q^81 - 264 * q^84 - 354 * q^93 + 334 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$92$	$106$	$157$
$\chi(n)$	$-1$	$e\left(\frac{7}{12}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$2$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$3$	2.59808	−	1.50000i	0.866025	−	0.500000i
$3$	2.59808	−	1.50000i	0.866025	−	0.500000i
$4$	−3.46410	−	2.00000i	−0.866025	−	0.500000i
$5$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$5$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$6$		0			0
$7$	5.50000	−	4.33013i	0.785714	−	0.618590i
$7$	5.50000	−	4.33013i	0.785714	−	0.618590i
$8$		0			0
$9$	4.50000	−	7.79423i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$11$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$12$	−12.0000			−1.00000
$13$	−12.9904	+	0.500000i	−0.999260	+	0.0384615i
$13$	−12.9904	+	0.500000i	−0.999260	+	0.0384615i
$14$		0			0
$15$		0			0
$16$	8.00000	+	13.8564i	0.500000	+	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−9.83013	−	36.6865i	−0.517375	−	1.93087i	−0.289474	−	0.957186i	$-0.593480\pi$
$19$	−9.83013	−	36.6865i	−0.517375	−	1.93087i	−0.227901	−	0.973684i	$-0.573186\pi$
$20$		0			0
$21$	7.79423	−	19.5000i	0.371154	−	0.928571i
$22$		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
$25$		−	25.0000i		−	1.00000i
$26$		0			0
$27$		−	27.0000i		−	1.00000i
$28$	−27.7128	+	4.00000i	−0.989743	+	0.142857i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−23.8109	+	23.8109i	−0.768093	+	0.768093i	−0.977771	−	0.209677i	$-0.932759\pi$
$31$	−23.8109	+	23.8109i	−0.768093	+	0.768093i	0.209677	+	0.977771i	$0.432759\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−31.1769	+	18.0000i	−0.866025	+	0.500000i
$37$	65.0788	+	17.4378i	1.75889	+	0.471292i	0.986486	−	0.163843i	$-0.0523889\pi$
$37$	65.0788	+	17.4378i	1.75889	+	0.471292i	0.772401	+	0.635135i	$0.219056\pi$
$38$		0			0
$39$	−33.0000	+	20.7846i	−0.846154	+	0.532939i
$40$		0			0
$41$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$41$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$42$		0			0
$43$	19.5000	+	11.2583i	0.453488	+	0.261822i	0.709302	−	0.704904i	$-0.249010\pi$
$43$	19.5000	+	11.2583i	0.453488	+	0.261822i	−0.255814	+	0.966726i	$0.582343\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	41.5692	+	24.0000i	0.866025	+	0.500000i
$49$	11.5000	−	47.6314i	0.234694	−	0.972069i
$50$		0			0
$51$		0			0
$52$	46.0000	+	24.2487i	0.884615	+	0.466321i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−80.5692	−	80.5692i	−1.41350	−	1.41350i
$58$		0			0
$59$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$59$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$60$		0			0
$61$	40.7032	+	23.5000i	0.667265	+	0.385246i	0.795040	−	0.606557i	$-0.207450\pi$
$61$	40.7032	+	23.5000i	0.667265	+	0.385246i	−0.127774	+	0.991803i	$0.540783\pi$
$62$		0			0
$63$	−9.00000	−	62.3538i	−0.142857	−	0.989743i
$64$		−	64.0000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−22.0289	+	82.2128i	−0.328789	+	1.22706i	0.581659	+	0.813433i	$0.302404\pi$
$67$	−22.0289	+	82.2128i	−0.328789	+	1.22706i	−0.910448	+	0.413624i	$0.864263\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$71$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$72$		0			0
$73$	86.2224	+	86.2224i	1.18113	+	1.18113i	0.979452	+	0.201677i	$0.0646392\pi$
$73$	86.2224	+	86.2224i	1.18113	+	1.18113i	0.201677	+	0.979452i	$0.435361\pi$
$74$		0			0
$75$	−37.5000	−	64.9519i	−0.500000	−	0.866025i
$76$	−39.3205	+	146.746i	−0.517375	+	1.93087i
$77$		0			0
$78$		0			0
$79$	157.617			1.99515			0.997574	−	0.0696203i	$-0.0221788\pi$
$79$	157.617			1.99515			0.997574	+	0.0696203i	$0.0221788\pi$
$80$		0			0
$81$	−40.5000	−	70.1481i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$	−66.0000	+	51.9615i	−0.785714	+	0.618590i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$89$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$90$		0			0
$91$	−69.2820	+	59.0000i	−0.761341	+	0.648352i
$92$		0			0
$93$	−26.1462	+	97.5788i	−0.281142	+	1.04923i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−13.4948	−	50.3634i	−0.139122	−	0.519211i	−0.999947	−	0.0103093i	$-0.996718\pi$
$97$	−13.4948	−	50.3634i	−0.139122	−	0.519211i	0.860825	−	0.508902i	$-0.169948\pi$
$98$		0			0
$99$		0			0
$100$	−50.0000	+	86.6025i	−0.500000	+	0.866025i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−37.0000			−0.359223			−0.179612	−	0.983738i	$-0.557484\pi$
$103$	−37.0000			−0.359223			−0.179612	+	0.983738i	$0.557484\pi$
$104$		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	$0.333333\pi$
$108$	−54.0000	+	93.5307i	−0.500000	+	0.866025i
$109$	153.772	+	153.772i	1.41076	+	1.41076i	0.754793	+	0.655963i	$0.227737\pi$
$109$	153.772	+	153.772i	1.41076	+	1.41076i	0.655963	+	0.754793i	$0.272263\pi$
$110$		0			0
$111$	195.237	−	52.3135i	1.75889	−	0.471292i
$112$	104.000	+	41.5692i	0.928571	+	0.371154i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−54.5596	+	103.500i	−0.466321	+	0.884615i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	104.789	+	60.5000i	0.866025	+	0.500000i
$122$		0			0
$123$		0			0
$124$	130.105	−	34.8616i	1.04923	−	0.281142i
$125$		0			0
$126$		0			0
$127$	−219.104	+	126.500i	−1.72523	+	0.996063i	−0.818292	+	0.574803i	$0.805079\pi$
$127$	−219.104	+	126.500i	−1.72523	+	0.996063i	−0.906940	+	0.421260i	$0.861588\pi$
$128$		0			0
$129$	67.5500			0.523643
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−212.923	−	159.210i	−1.60093	−	1.19707i
$134$		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
$139$	−136.500	−	78.8083i	−0.982014	−	0.566966i	−0.0791367	−	0.996864i	$-0.525216\pi$
$139$	−136.500	−	78.8083i	−0.982014	−	0.566966i	−0.902878	+	0.429898i	$0.858550\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	144.000			1.00000
$145$		0			0
$146$		0			0
$147$	−41.5692	−	141.000i	−0.282784	−	0.959184i
$148$	−190.564	−	190.564i	−1.28760	−	1.28760i
$149$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$149$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$150$		0			0
$151$	191.497	−	191.497i	1.26819	−	1.26819i	0.321175	−	0.947020i	$-0.395922\pi$
$151$	191.497	−	191.497i	1.26819	−	1.26819i	0.947020	−	0.321175i	$-0.104078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	155.885	−	6.00000i	0.999260	−	0.0384615i
$157$		−	247.683i		−	1.57760i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$		−	247.683i		−	1.57760i	0.614650	−	0.788800i	$-0.289297\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	30.9468	+	115.495i	0.189857	+	0.708557i	0.993538	+	0.113497i	$0.0362052\pi$
$163$	30.9468	+	115.495i	0.189857	+	0.708557i	−0.803681	+	0.595060i	$0.797128\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$167$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$168$		0			0
$169$	168.500	−	12.9904i	0.997041	−	0.0768662i
$170$		0			0
$171$	−330.179	−	88.4711i	−1.93087	−	0.517375i
$172$	−45.0333	−	78.0000i	−0.261822	−	0.453488i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	−108.253	−	137.500i	−0.618590	−	0.785714i
$176$		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
$181$	−314.000			−1.73481			−0.867403	−	0.497606i	$-0.834213\pi$
$181$	−314.000			−1.73481			−0.867403	+	0.497606i	$0.834213\pi$
$182$		0			0
$183$	141.000			0.770492
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−116.913	−	148.500i	−0.618590	−	0.785714i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−96.0000	−	166.277i	−0.500000	−	0.866025i
$193$	147.213	+	39.4456i	0.762761	+	0.204381i	0.619171	−	0.785256i	$-0.287469\pi$
$193$	147.213	+	39.4456i	0.762761	+	0.204381i	0.143590	+	0.989637i	$0.454135\pi$
$194$		0			0
$195$		0			0
$196$	−135.100	+	142.000i	−0.689286	+	0.724490i
$197$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$197$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$198$		0			0
$199$	191.392	−	331.500i	0.961767	−	1.66583i	0.243706	−	0.969849i	$-0.421637\pi$
$199$	191.392	−	331.500i	0.961767	−	1.66583i	0.718061	−	0.695980i	$-0.245030\pi$
$200$		0			0
$201$	66.0866	+	246.638i	0.328789	+	1.22706i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−110.851	−	176.000i	−0.532939	−	0.846154i
$209$		0			0
$210$		0			0
$211$	−168.875	−	292.500i	−0.800355	−	1.38626i	−0.919382	−	0.393365i	$-0.871311\pi$
$211$	−168.875	−	292.500i	−0.800355	−	1.38626i	0.119027	−	0.992891i	$-0.462022\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−27.8557	+	234.064i	−0.128367	+	1.07864i
$218$		0			0
$219$	353.346	+	94.6788i	1.61345	+	0.432323i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	32.1122	+	8.60443i	0.144001	+	0.0385849i	0.330099	−	0.943946i	$-0.392918\pi$
$223$	32.1122	+	8.60443i	0.144001	+	0.0385849i	−0.186099	+	0.982531i	$0.559584\pi$
$224$		0			0
$225$	−194.856	−	112.500i	−0.866025	−	0.500000i
$226$		0			0
$227$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$227$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$228$	117.962	+	440.238i	0.517375	+	1.93087i
$229$	241.631	+	241.631i	1.05516	+	1.05516i	0.998387	+	0.0567686i	$0.0180797\pi$
$229$	241.631	+	241.631i	1.05516	+	1.05516i	0.0567686	+	0.998387i	$0.481920\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	409.500	−	236.425i	1.72785	−	0.997574i
$238$		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
$241$	−460.336	+	123.347i	−1.91011	+	0.511812i	−0.916334	+	0.400415i	$0.868866\pi$
$241$	−460.336	+	123.347i	−1.91011	+	0.511812i	−0.993776	+	0.111397i	$0.964467\pi$
$242$		0			0
$243$	−210.444	−	121.500i	−0.866025	−	0.500000i
$244$	−94.0000	−	162.813i	−0.385246	−	0.667265i
$245$		0			0
$246$		0			0
$247$	146.040	+	471.657i	0.591257	+	1.90954i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$	−93.5307	+	234.000i	−0.371154	+	0.928571i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	433.442	−	185.892i	1.67352	−	0.717728i
$260$		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
$268$	240.736	−	240.736i	0.898268	−	0.898268i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−28.0129	+	104.546i	−0.103369	+	0.385777i	−0.998155	−	0.0607176i	$-0.980661\pi$
$271$	−28.0129	+	104.546i	−0.103369	+	0.385777i	0.894786	+	0.446494i	$0.147328\pi$
$272$		0			0
$273$	−91.5000	+	257.210i	−0.335165	+	0.942160i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−468.000	−	270.200i	−1.68953	−	0.975451i	−0.954874	−	0.297012i	$-0.904010\pi$
$277$	−468.000	−	270.200i	−1.68953	−	0.975451i	−0.734657	−	0.678439i	$-0.762657\pi$
$278$		0			0
$279$	78.4385	+	292.737i	0.281142	+	1.04923i
$280$		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
$283$	281.458	+	487.500i	0.994552	+	1.72261i	0.587551	+	0.809187i	$0.300092\pi$
$283$	281.458	+	487.500i	0.994552	+	1.72261i	0.407001	+	0.913428i	$0.366574\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−144.500	−	250.281i	−0.500000	−	0.866025i
$290$		0			0
$291$	−110.606	−	110.606i	−0.380089	−	0.380089i
$292$	−126.238	−	471.128i	−0.432323	−	1.61345i
$293$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$293$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$			300.000i			1.00000i
$301$	156.000	−	22.5167i	0.518272	−	0.0748062i
$302$		0			0
$303$		0			0
$304$	429.703	−	429.703i	1.41350	−	1.41350i
$305$		0			0
$306$		0			0
$307$	−275.189	−	275.189i	−0.896381	−	0.896381i	0.0987325	−	0.995114i	$-0.468521\pi$
$307$	−275.189	−	275.189i	−0.896381	−	0.896381i	−0.995114	+	0.0987325i	$0.968521\pi$
$308$		0			0
$309$	−96.1288	+	55.5000i	−0.311097	+	0.179612i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			457.000i			1.46006i	0.683413	+	0.730032i	$0.260495\pi$
$313$			457.000i			1.46006i	−0.683413	+	0.730032i	$0.739505\pi$
$314$		0			0
$315$		0			0
$316$	−546.000	−	315.233i	−1.72785	−	0.997574i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$			324.000i			1.00000i
$325$	12.5000	+	324.760i	0.0384615	+	0.999260i
$326$		0			0
$327$	630.171	+	168.854i	1.92713	+	0.516373i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	199.187	−	53.3719i	0.601772	−	0.161244i	0.0549442	−	0.998489i	$-0.482502\pi$
$331$	199.187	−	53.3719i	0.601772	−	0.161244i	0.546828	+	0.837245i	$0.315835\pi$
$332$		0			0
$333$	428.769	−	428.769i	1.28760	−	1.28760i
$334$		0			0
$335$		0			0
$336$	332.554	−	48.0000i	0.989743	−	0.142857i
$337$			167.000i			0.495549i	0.968818	+	0.247774i	$0.0796992\pi$
$337$			167.000i			0.495549i	−0.968818	+	0.247774i	$0.920301\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−143.000	−	311.769i	−0.416910	−	0.908948i
$344$		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
$349$	−154.871	+	577.987i	−0.443757	+	1.65612i	0.275441	+	0.961318i	$0.411176\pi$
$349$	−154.871	+	577.987i	−0.443757	+	1.65612i	−0.719198	+	0.694805i	$0.755490\pi$
$350$		0			0
$351$	13.5000	+	350.740i	0.0384615	+	0.999260i
$352$		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		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$	−936.635	+	540.767i	−2.59456	+	1.49797i
$362$		0			0
$363$	363.000			1.00000
$364$	358.000	−	65.8179i	0.983516	−	0.180818i
$365$		0			0
$366$		0			0
$367$	604.500	−	349.008i	1.64714	−	0.950976i	0.668937	−	0.743319i	$-0.266749\pi$
$367$	604.500	−	349.008i	1.64714	−	0.950976i	0.978202	−	0.207657i	$-0.0665839\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	285.731	−	285.731i	0.768093	−	0.768093i
$373$	−236.425	+	409.500i	−0.633847	+	1.09786i	0.352911	+	0.935657i	$0.385192\pi$
$373$	−236.425	+	409.500i	−0.633847	+	1.09786i	−0.986758	+	0.162198i	$0.948142\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−193.920	+	723.721i	−0.511664	+	1.90955i	−0.109499	+	0.993987i	$0.534925\pi$
$379$	−193.920	+	723.721i	−0.511664	+	1.90955i	−0.402165	+	0.915567i	$0.631742\pi$
$380$		0			0
$381$	−379.500	+	657.313i	−0.996063	+	1.72523i
$382$		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
$387$	175.500	−	101.325i	0.453488	−	0.261822i
$388$	−53.9794	+	201.454i	−0.139122	+	0.519211i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−514.420	+	137.838i	−1.29577	+	0.347200i	−0.839848	−	0.542821i	$-0.817356\pi$
$397$	−514.420	+	137.838i	−1.29577	+	0.347200i	−0.455919	+	0.890021i	$0.650689\pi$
$398$		0			0
$399$	−792.006	−	94.2558i	−1.98498	−	0.236230i
$400$	346.410	−	200.000i	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
$403$	297.407	−	321.218i	0.737983	−	0.797067i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	121.228	+	452.430i	0.296402	+	1.10619i	0.940098	+	0.340905i	$0.110733\pi$
$409$	121.228	+	452.430i	0.296402	+	1.10619i	−0.643696	+	0.765281i	$0.722600\pi$
$410$		0			0
$411$		0			0
$412$	128.172	+	74.0000i	0.311097	+	0.179612i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−472.850			−1.13393
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	586.044	+	586.044i	1.39203	+	1.39203i	0.820770	+	0.571259i	$0.193545\pi$
$421$	586.044	+	586.044i	1.39203	+	1.39203i	0.571259	+	0.820770i	$0.306455\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	325.626	−	47.0000i	0.762589	−	0.110070i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$431$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$432$	374.123	−	216.000i	0.866025	−	0.500000i
$433$	394.042	−	682.500i	0.910027	−	1.57621i	0.0960028	−	0.995381i	$-0.469394\pi$
$433$	394.042	−	682.500i	0.910027	−	1.57621i	0.814024	−	0.580831i	$-0.197272\pi$
$434$		0			0
$435$		0			0
$436$	−225.138	−	840.228i	−0.516373	−	1.92713i
$437$		0			0
$438$		0			0
$439$	−354.500	−	614.012i	−0.807517	−	1.39866i	−0.914579	−	0.404408i	$-0.867478\pi$
$439$	−354.500	−	614.012i	−0.807517	−	1.39866i	0.107062	−	0.994252i	$-0.465856\pi$
$440$		0			0
$441$	−319.500	−	303.975i	−0.724490	−	0.689286i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−780.946	−	209.254i	−1.75889	−	0.471292i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−277.128	−	352.000i	−0.618590	−	0.785714i
$449$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$449$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	210.279	−	784.771i	0.464192	−	1.73239i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−49.3954	+	184.346i	−0.108086	+	0.403383i	−0.998677	−	0.0514223i	$-0.983625\pi$
$457$	−49.3954	+	184.346i	−0.108086	+	0.403383i	0.890591	+	0.454805i	$0.150291\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$461$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$462$		0			0
$463$	424.261	−	424.261i	0.916330	−	0.916330i	−0.0804300	−	0.996760i	$-0.525629\pi$
$463$	424.261	−	424.261i	0.916330	−	0.916330i	0.996760	+	0.0804300i	$0.0256293\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	396.000	−	249.415i	0.846154	−	0.532939i
$469$	234.833	+	547.558i	0.500710	+	1.16750i
$470$		0			0
$471$	−371.525	−	643.500i	−0.788800	−	1.36624i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−917.163	+	245.753i	−1.93087	+	0.517375i
$476$		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
$481$	−854.118	−	193.985i	−1.77571	−	0.403294i
$482$		0			0
$483$		0			0
$484$	−242.000	−	419.156i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	761.163	−	203.953i	1.56296	−	0.418795i	0.629363	−	0.777111i	$-0.283316\pi$
$487$	761.163	−	203.953i	1.56296	−	0.418795i	0.933600	+	0.358316i	$0.116649\pi$
$488$		0			0
$489$	253.644	+	253.644i	0.518700	+	0.518700i
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−520.420	−	139.446i	−1.04923	−	0.281142i
$497$		0			0
$498$		0			0
$499$	−511.831	−	511.831i	−1.02571	−	1.02571i	−0.999661	−	0.0260521i	$-0.991706\pi$
$499$	−511.831	−	511.831i	−1.02571	−	1.02571i	−0.0260521	−	0.999661i	$-0.508294\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	418.290	−	286.500i	0.825030	−	0.565089i
$508$	1012.00			1.99213
$509$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$509$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$510$		0			0
$511$	847.577	+	100.869i	1.65866	+	0.197396i
$512$		0			0
$513$	−990.536	+	265.413i	−1.93087	+	0.517375i
$514$		0			0
$515$		0			0
$516$	−234.000	−	135.100i	−0.453488	−	0.261822i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−312.000	+	180.133i	−0.596558	+	0.344423i	−0.767686	−	0.640826i	$-0.778592\pi$
$523$	−312.000	+	180.133i	−0.596558	+	0.344423i	0.171128	+	0.985249i	$0.445259\pi$
$524$		0			0
$525$	−487.500	−	194.856i	−0.928571	−	0.371154i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	264.500	−	458.127i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	419.167	+	977.367i	0.787907	+	1.83716i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	28.4392	−	28.4392i	0.0525678	−	0.0525678i	−0.680334	−	0.732902i	$-0.738165\pi$
$541$	28.4392	−	28.4392i	0.0525678	−	0.0525678i	0.732902	+	0.680334i	$0.238165\pi$
$542$		0			0
$543$	−815.796	+	471.000i	−1.50239	+	0.867403i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	587.000			1.07313			0.536563	−	0.843860i	$-0.319722\pi$
$547$	587.000			1.07313			0.536563	+	0.843860i	$0.319722\pi$
$548$		0			0
$549$	366.329	−	211.500i	0.667265	−	0.385246i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	866.891	−	682.500i	1.56762	−	1.23418i
$554$		0			0
$555$		0			0
$556$	315.233	+	546.000i	0.566966	+	0.982014i
$557$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$557$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$558$		0			0
$559$	−258.942	−	136.500i	−0.463223	−	0.244186i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−526.500	−	210.444i	−0.928571	−	0.371154i
$568$		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
$571$		−	886.000i		−	1.55166i	−0.630940	−	0.775832i	$-0.717330\pi$
$571$		−	886.000i		−	1.55166i	0.630940	−	0.775832i	$-0.282670\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−498.831	−	288.000i	−0.866025	−	0.500000i
$577$	162.303	−	162.303i	0.281287	−	0.281287i	−0.552335	−	0.833622i	$-0.686263\pi$
$577$	162.303	−	162.303i	0.281287	−	0.281287i	0.833622	+	0.552335i	$0.186263\pi$
$578$		0			0
$579$	441.638	−	118.337i	0.762761	−	0.204381i
$580$		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	$-0.916667\pi$
$587$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$588$	−138.000	+	571.577i	−0.234694	+	0.972069i
$589$	1107.60	+	639.475i	1.88048	+	1.08570i
$590$		0			0
$591$		0			0
$592$	279.005	+	1041.26i	0.471292	+	1.75889i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1148.35i		−	1.92353i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−455.529	+	263.000i	−0.757952	+	0.437604i	−0.828560	−	0.559900i	$-0.810839\pi$
$601$	−455.529	+	263.000i	−0.757952	+	0.437604i	0.0706077	+	0.997504i	$0.477506\pi$
$602$		0			0
$603$	541.656	+	541.656i	0.898268	+	0.898268i
$604$	−1046.36	+	280.372i	−1.73239	+	0.464192i
$605$		0			0
$606$		0			0
$607$	780.000	+	450.333i	1.28501	+	0.741900i	0.977759	−	0.209729i	$-0.0672583\pi$
$607$	780.000	+	450.333i	1.28501	+	0.741900i	0.307249	+	0.951629i	$0.400592\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−196.671	+	733.987i	−0.320834	+	1.19737i	0.597600	+	0.801794i	$0.296121\pi$
$613$	−196.671	+	733.987i	−0.320834	+	1.19737i	−0.918434	+	0.395574i	$0.870546\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$617$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$618$		0			0
$619$	76.9943	+	76.9943i	0.124385	+	0.124385i	0.766559	−	0.642174i	$-0.221967\pi$
$619$	76.9943	+	76.9943i	0.124385	+	0.124385i	−0.642174	+	0.766559i	$0.721967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−552.000	−	290.985i	−0.884615	−	0.466321i
$625$	−625.000			−1.00000
$626$		0			0
$627$		0			0
$628$	−495.367	+	858.000i	−0.788800	+	1.36624i
$629$		0			0
$630$		0			0
$631$	221.586	+	826.972i	0.351167	+	1.31057i	0.885240	+	0.465135i	$0.153994\pi$
$631$	221.586	+	826.972i	0.351167	+	1.31057i	−0.534073	+	0.845438i	$0.679339\pi$
$632$		0			0
$633$	−877.500	−	506.625i	−1.38626	−	0.800355i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−125.574	+	624.500i	−0.197133	+	0.980377i
$638$		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
$643$	5.03829	+	18.8032i	0.00783560	+	0.0292429i	0.969733	−	0.244168i	$-0.0785148\pi$
$643$	5.03829	+	18.8032i	0.00783560	+	0.0292429i	−0.961897	+	0.273411i	$0.911848\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	278.725	+	649.900i	0.428149	+	0.998310i
$652$	123.787	−	461.979i	0.189857	−	0.708557i
$653$		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
$657$	1060.04	−	284.036i	1.61345	−	0.432323i
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−337.262	+	1258.68i	−0.510230	+	1.90420i	−0.0922844	+	0.995733i	$0.529417\pi$
$661$	−337.262	+	1258.68i	−0.510230	+	1.90420i	−0.417946	+	0.908472i	$0.637250\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	96.3365	−	25.8133i	0.144001	−	0.0385849i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−565.500	+	326.492i	−0.840267	+	0.485129i	−0.857355	−	0.514725i	$-0.827894\pi$
$673$	−565.500	+	326.492i	−0.840267	+	0.485129i	0.0170877	+	0.999854i	$0.494561\pi$
$674$		0			0
$675$	−675.000			−1.00000
$676$	−609.682	−	292.000i	−0.901896	−	0.431953i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−292.302	−	218.565i	−0.430489	−	0.321892i
$680$		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	$0.416667\pi$
$684$	966.831	+	966.831i	1.41350	+	1.41350i
$685$		0			0
$686$		0			0
$687$	990.221	+	265.329i	1.44137	+	0.386214i
$688$			360.267i			0.523643i
$689$		0			0
$690$		0			0
$691$	717.346	+	192.212i	1.03813	+	0.278165i	0.737337	−	0.675525i	$-0.236083\pi$
$691$	717.346	+	192.212i	1.03813	+	0.278165i	0.300790	+	0.953690i	$0.402750\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	100.000	+	692.820i	0.142857	+	0.989743i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		−	2558.93i		−	3.64002i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−304.972	−	1138.17i	−0.430143	−	1.60532i	−0.752428	−	0.658674i	$-0.771118\pi$
$709$	−304.972	−	1138.17i	−0.430143	−	1.60532i	0.322285	−	0.946643i	$-0.395549\pi$
$710$		0			0
$711$	709.275	−	1228.50i	0.997574	−	1.72785i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	−203.500	+	160.215i	−0.282247	+	0.222212i
$722$		0			0
$723$	−1010.97	+	1010.97i	−1.39830	+	1.39830i
$724$	1087.73	+	628.000i	1.50239	+	0.867403i
$725$		0			0
$726$		0			0
$727$	1103.32			1.51763			0.758815	−	0.651307i	$-0.225779\pi$
$727$	1103.32			1.51763			0.758815	+	0.651307i	$0.225779\pi$
$728$		0			0
$729$	−729.000			−1.00000
$730$		0			0
$731$		0			0
$732$	−488.438	−	282.000i	−0.667265	−	0.385246i
$733$	520.572	−	520.572i	0.710194	−	0.710194i	−0.256381	−	0.966576i	$-0.582530\pi$
$733$	520.572	−	520.572i	0.710194	−	0.710194i	0.966576	+	0.256381i	$0.0825303\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1402.49	−	375.795i	−1.89782	−	0.508519i	−0.997277	−	0.0737483i	$-0.976504\pi$
$739$	−1402.49	−	375.795i	−1.89782	−	0.508519i	−0.900541	−	0.434771i	$-0.856829\pi$
$740$		0			0
$741$	1086.91	+	1006.34i	1.46681	+	1.35808i
$742$		0			0
$743$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$743$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	780.000	−	450.333i	1.03862	−	0.599645i	0.119174	−	0.992873i	$-0.461975\pi$
$751$	780.000	−	450.333i	1.03862	−	0.599645i	0.919441	+	0.393229i	$0.128642\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	108.000	+	748.246i	0.142857	+	0.989743i
$757$	630.466	+	1092.00i	0.832849	+	1.44254i	0.895770	+	0.444518i	$0.146625\pi$
$757$	630.466	+	1092.00i	0.832849	+	1.44254i	−0.0629213	+	0.998018i	$0.520042\pi$
$758$		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
$763$	1511.60	+	179.894i	1.98113	+	0.235772i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$			768.000i			1.00000i
$769$	−1123.45	−	301.029i	−1.46093	−	0.391455i	−0.561118	−	0.827736i	$-0.689629\pi$
$769$	−1123.45	−	301.029i	−1.46093	−	0.391455i	−0.899811	+	0.436281i	$0.856295\pi$
$770$		0			0
$771$		0			0
$772$	−431.069	−	431.069i	−0.558380	−	0.558380i
$773$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$773$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$774$		0			0
$775$	595.272	+	595.272i	0.768093	+	0.768093i
$776$		0			0
$777$	847.277	−	1133.12i	1.09045	−	1.45833i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	752.000	−	221.703i	0.959184	−	0.282784i
$785$		0			0
$786$		0			0
$787$	−209.505	+	56.1367i	−0.266207	+	0.0713300i	−0.389454	−	0.921046i	$-0.627336\pi$
$787$	−209.505	+	56.1367i	−0.266207	+	0.0713300i	0.123246	+	0.992376i	$0.460669\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−540.500	−	284.922i	−0.681589	−	0.359297i
$794$		0			0
$795$		0			0
$796$	−1326.00	+	765.566i	−1.66583	+	0.961767i
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	264.346	−	986.554i	0.328789	−	1.22706i
$805$		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
$811$	−1140.59	+	1140.59i	−1.40640	+	1.40640i	−0.628963	+	0.777435i	$0.716520\pi$
$811$	−1140.59	+	1140.59i	−1.40640	+	1.40640i	−0.777435	+	0.628963i	$0.783480\pi$
$812$		0			0
$813$	84.0387	+	313.637i	0.103369	+	0.385777i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	221.342	−	826.058i	0.270920	−	1.01109i
$818$		0			0
$819$	148.090	+	805.500i	0.180818	+	0.983516i
$820$		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
$823$	−1092.00	−	630.466i	−1.32685	−	0.766059i	−0.342041	−	0.939685i	$-0.611118\pi$
$823$	−1092.00	−	630.466i	−1.32685	−	0.766059i	−0.984812	+	0.173626i	$0.944452\pi$
$824$		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
$829$	−596.692	−	1033.50i	−0.719773	−	1.24668i	−0.961090	−	0.276236i	$-0.910913\pi$
$829$	−596.692	−	1033.50i	−0.719773	−	1.24668i	0.241317	−	0.970446i	$-0.422421\pi$
$830$		0			0
$831$	−1621.20			−1.95090
$832$	32.0000	+	831.384i	0.0384615	+	0.999260i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	642.894	+	642.894i	0.768093	+	0.768093i
$838$		0			0
$839$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$839$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$840$		0			0
$841$	−420.500	+	728.327i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$			1351.00i			1.60071i
$845$		0			0
$846$		0			0
$847$	838.313	−	121.000i	0.989743	−	0.142857i
$848$		0			0
$849$	1462.50	+	844.375i	1.72261	+	0.994552i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1058.89	−	1058.89i	−1.24138	−	1.24138i	−0.959430	−	0.281946i	$-0.909020\pi$
$853$	−1058.89	−	1058.89i	−1.24138	−	1.24138i	−0.281946	−	0.959430i	$-0.590980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		−	1549.00i		−	1.80326i	−0.432509	−	0.901630i	$-0.642371\pi$
$859$		−	1549.00i		−	1.80326i	0.432509	−	0.901630i	$-0.357629\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−750.844	−	433.500i	−0.866025	−	0.500000i
$868$	564.623	−	755.110i	0.650488	−	0.869943i
$869$		0			0
$870$		0			0
$871$	245.057	−	1078.99i	0.281351	−	1.23879i
$872$		0			0
$873$	−453.271	−	121.454i	−0.519211	−	0.139122i
$874$		0			0
$875$		0			0
$876$	−1034.67	−	1034.67i	−1.18113	−	1.18113i
$877$	1534.67	−	411.214i	1.74991	−	0.468887i	0.765302	−	0.643672i	$-0.222590\pi$
$877$	1534.67	−	411.214i	1.74991	−	0.468887i	0.984607	+	0.174785i	$0.0559231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$		−	1259.00i		−	1.42582i	−0.701255	−	0.712911i	$-0.747377\pi$
$883$		−	1259.00i		−	1.42582i	0.701255	−	0.712911i	$-0.252623\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−657.313	+	1644.50i	−0.739385	+	1.84983i
$890$		0			0
$891$		0			0
$892$	−94.0309	−	94.0309i	−0.105416	−	0.105416i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	450.000	+	779.423i	0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	371.525	−	292.500i	0.411434	−	0.323920i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	185.329	−	107.000i	0.204332	−	0.117971i	−0.394342	−	0.918964i	$-0.629028\pi$
$907$	185.329	−	107.000i	0.204332	−	0.117971i	0.598675	+	0.800992i	$0.295694\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	471.846	−	1760.95i	0.517375	−	1.93087i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−353.772	−	1320.29i	−0.386214	−	1.44137i
$917$		0			0
$918$		0			0
$919$	−810.600	+	1404.00i	−0.882045	+	1.52775i	−0.0329825	+	0.999456i	$0.510501\pi$
$919$	−810.600	+	1404.00i	−0.882045	+	1.52775i	−0.849063	+	0.528292i	$0.822833\pi$
$920$		0			0
$921$	−1127.75	−	302.179i	−1.22448	−	0.328098i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	435.946	−	1626.97i	0.471292	−	1.75889i
$926$		0			0
$927$	−166.500	+	288.386i	−0.179612	+	0.311097i
$928$		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
$931$	−1860.48	+	46.3275i	−1.99836	+	0.0497611i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1198.00i		−	1.27855i	−0.768979	−	0.639274i	$-0.779235\pi$
$937$		−	1198.00i		−	1.27855i	0.768979	−	0.639274i	$-0.220765\pi$
$938$		0			0
$939$	685.500	+	1187.32i	0.730032	+	1.26445i
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		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	$0.0833333\pi$
$948$	−1891.40			−1.99515
$949$	−1163.17	−	1076.95i	−1.22568	−	1.13483i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	172.917i		−	0.179934i
$962$		0			0
$963$		0			0
$964$	1841.35	+	493.387i	1.91011	+	0.511812i
$965$		0			0
$966$		0			0
$967$	178.103	+	178.103i	0.184181	+	0.184181i	0.793175	−	0.608994i	$-0.208427\pi$
$967$	178.103	+	178.103i	0.184181	+	0.184181i	−0.608994	+	0.793175i	$0.708427\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$	486.000	+	841.777i	0.500000	+	0.866025i
$973$	−1092.00	+	157.617i	−1.12230	+	0.161990i
$974$		0			0
$975$	519.615	+	825.000i	0.532939	+	0.846154i
$976$			752.000i			0.770492i
$977$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$977$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1890.51	−	506.561i	1.92713	−	0.516373i
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	437.415	−	1925.95i	0.442728	−	1.94934i
$989$		0			0
$990$		0			0
$991$	−23.0000	−	39.8372i	−0.0232089	−	0.0401990i	0.854188	−	0.519965i	$-0.174055\pi$
$991$	−23.0000	−	39.8372i	−0.0232089	−	0.0401990i	−0.877397	+	0.479766i	$0.840722\pi$
$992$		0			0
$993$	437.444	−	437.444i	0.440528	−	0.440528i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1150.50	+	664.241i	1.15396	+	0.666240i	0.949850	−	0.312707i	$-0.101236\pi$
$997$	1150.50	+	664.241i	1.15396	+	0.666240i	0.204112	+	0.978947i	$0.434569\pi$
$998$		0			0
$999$	470.821	−	1757.13i	0.471292	−	1.75889i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.3.ca.b.167.1	yes	4
3.2	odd	2	CM	273.3.ca.b.167.1	yes	4
7.6	odd	2		273.3.ca.a.167.1	✓	4
13.6	odd	12		273.3.ca.a.188.1	yes	4
21.20	even	2		273.3.ca.a.167.1	✓	4
39.32	even	12		273.3.ca.a.188.1	yes	4
91.6	even	12	inner	273.3.ca.b.188.1	yes	4
273.188	odd	12	inner	273.3.ca.b.188.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.3.ca.a.167.1	✓	4	7.6	odd	2
273.3.ca.a.167.1	✓	4	21.20	even	2
273.3.ca.a.188.1	yes	4	13.6	odd	12
273.3.ca.a.188.1	yes	4	39.32	even	12
273.3.ca.b.167.1	yes	4	1.1	even	1	trivial
273.3.ca.b.167.1	yes	4	3.2	odd	2	CM
273.3.ca.b.188.1	yes	4	91.6	even	12	inner
273.3.ca.b.188.1	yes	4	273.188	odd	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	273.ca (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(2.59808 - 1.50000i) q^{3} +(-3.46410 - 2.00000i) q^{4} +(5.50000 - 4.33013i) q^{7} +(4.50000 - 7.79423i) q^{9} +O(q^{10})\)	\(q+(2.59808 - 1.50000i) q^{3} +(-3.46410 - 2.00000i) q^{4} +(5.50000 - 4.33013i) q^{7} +(4.50000 - 7.79423i) q^{9} -12.0000 q^{12} +(-12.9904 + 0.500000i) q^{13} +(8.00000 + 13.8564i) q^{16} +(-9.83013 - 36.6865i) q^{19} +(7.79423 - 19.5000i) q^{21} -25.0000i q^{25} -27.0000i q^{27} +(-27.7128 + 4.00000i) q^{28} +(-23.8109 + 23.8109i) q^{31} +(-31.1769 + 18.0000i) q^{36} +(65.0788 + 17.4378i) q^{37} +(-33.0000 + 20.7846i) q^{39} +(19.5000 + 11.2583i) q^{43} +(41.5692 + 24.0000i) q^{48} +(11.5000 - 47.6314i) q^{49} +(46.0000 + 24.2487i) q^{52} +(-80.5692 - 80.5692i) q^{57} +(40.7032 + 23.5000i) q^{61} +(-9.00000 - 62.3538i) q^{63} -64.0000i q^{64} +(-22.0289 + 82.2128i) q^{67} +(86.2224 + 86.2224i) q^{73} +(-37.5000 - 64.9519i) q^{75} +(-39.3205 + 146.746i) q^{76} +157.617 q^{79} +(-40.5000 - 70.1481i) q^{81} +(-66.0000 + 51.9615i) q^{84} +(-69.2820 + 59.0000i) q^{91} +(-26.1462 + 97.5788i) q^{93} +(-13.4948 - 50.3634i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 22 q^{7} + 18 q^{9}+O(q^{10}) \) 4 * q + 22 * q^7 + 18 * q^9	\( 4 q + 22 q^{7} + 18 q^{9} - 48 q^{12} + 32 q^{16} - 22 q^{19} + 26 q^{31} + 146 q^{37} - 132 q^{39} + 78 q^{43} + 46 q^{49} + 184 q^{52} - 156 q^{57} - 36 q^{63} - 244 q^{67} + 286 q^{73} - 150 q^{75} - 88 q^{76} - 162 q^{81} - 264 q^{84} - 354 q^{93} + 334 q^{97}+O(q^{100}) \) 4 * q + 22 * q^7 + 18 * q^9 - 48 * q^12 + 32 * q^16 - 22 * q^19 + 26 * q^31 + 146 * q^37 - 132 * q^39 + 78 * q^43 + 46 * q^49 + 184 * q^52 - 156 * q^57 - 36 * q^63 - 244 * q^67 + 286 * q^73 - 150 * q^75 - 88 * q^76 - 162 * q^81 - 264 * q^84 - 354 * q^93 + 334 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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