LMFDB - Embedded newform 273.3.bm.a.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,3,Mod(191,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2, 4]))

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

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

chi := DirichletCharacter("273.191");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	273.bm (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.43871121704$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			191.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	273.191
Dual form			273.3.bm.a.263.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +(-2.00000 + 3.46410i) q^{4} +(5.50000 + 4.33013i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +(-2.00000 + 3.46410i) q^{4} +(5.50000 + 4.33013i) q^{7} +9.00000 q^{9} +(6.00000 - 10.3923i) q^{12} +(-11.0000 + 6.92820i) q^{13} +(-8.00000 - 13.8564i) q^{16} -37.0000 q^{19} +(-16.5000 - 12.9904i) q^{21} +(-12.5000 - 21.6506i) q^{25} -27.0000 q^{27} +(-26.0000 + 10.3923i) q^{28} +(6.50000 + 11.2583i) q^{31} +(-18.0000 + 31.1769i) q^{36} +(-23.5000 - 40.7032i) q^{37} +(33.0000 - 20.7846i) q^{39} +(11.0000 + 19.0526i) q^{43} +(24.0000 + 41.5692i) q^{48} +(11.5000 + 47.6314i) q^{49} +(-2.00000 - 51.9615i) q^{52} +111.000 q^{57} -121.000 q^{61} +(49.5000 + 38.9711i) q^{63} +64.0000 q^{64} -13.0000 q^{67} +(23.0000 + 39.8372i) q^{73} +(37.5000 + 64.9519i) q^{75} +(74.0000 - 128.172i) q^{76} +(-5.50000 + 9.52628i) q^{79} +81.0000 q^{81} +(78.0000 - 31.1769i) q^{84} +(-90.5000 - 9.52628i) q^{91} +(-19.5000 - 33.7750i) q^{93} +(-83.5000 - 144.626i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 4 q^{4} + 11 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 4 * q^4 + 11 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 4 q^{4} + 11 q^{7} + 18 q^{9} + 12 q^{12} - 22 q^{13} - 16 q^{16} - 74 q^{19} - 33 q^{21} - 25 q^{25} - 54 q^{27} - 52 q^{28} + 13 q^{31} - 36 q^{36} - 47 q^{37} + 66 q^{39} + 22 q^{43} + 48 q^{48} + 23 q^{49} - 4 q^{52} + 222 q^{57} - 242 q^{61} + 99 q^{63} + 128 q^{64} - 26 q^{67} + 46 q^{73} + 75 q^{75} + 148 q^{76} - 11 q^{79} + 162 q^{81} + 156 q^{84} - 181 q^{91} - 39 q^{93} - 167 q^{97}+O(q^{100}) $ 2 * q - 6 * q^3 - 4 * q^4 + 11 * q^7 + 18 * q^9 + 12 * q^12 - 22 * q^13 - 16 * q^16 - 74 * q^19 - 33 * q^21 - 25 * q^25 - 54 * q^27 - 52 * q^28 + 13 * q^31 - 36 * q^36 - 47 * q^37 + 66 * q^39 + 22 * q^43 + 48 * q^48 + 23 * q^49 - 4 * q^52 + 222 * q^57 - 242 * q^61 + 99 * q^63 + 128 * q^64 - 26 * q^67 + 46 * q^73 + 75 * q^75 + 148 * q^76 - 11 * q^79 + 162 * q^81 + 156 * q^84 - 181 * q^91 - 39 * q^93 - 167 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$92$	$106$	$157$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

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		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	−3.00000			−1.00000
$3$	−3.00000			−1.00000
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	5.50000	+	4.33013i	0.785714	+	0.618590i
$7$	5.50000	+	4.33013i	0.785714	+	0.618590i
$8$		0			0
$9$	9.00000			1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	6.00000	−	10.3923i	0.500000	−	0.866025i
$13$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$13$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$14$		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
$19$	−37.0000			−1.94737			−0.973684	−	0.227901i	$-0.926814\pi$
$19$	−37.0000			−1.94737			−0.973684	+	0.227901i	$0.926814\pi$
$20$		0			0
$21$	−16.5000	−	12.9904i	−0.785714	−	0.618590i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$26$		0			0
$27$	−27.0000			−1.00000
$28$	−26.0000	+	10.3923i	−0.928571	+	0.371154i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	6.50000	+	11.2583i	0.209677	+	0.363172i	0.951613	−	0.307299i	$-0.0994253\pi$
$31$	6.50000	+	11.2583i	0.209677	+	0.363172i	−0.741935	+	0.670471i	$0.766092\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−18.0000	+	31.1769i	−0.500000	+	0.866025i
$37$	−23.5000	−	40.7032i	−0.635135	−	1.10009i	−0.986486	−	0.163843i	$-0.947611\pi$
$37$	−23.5000	−	40.7032i	−0.635135	−	1.10009i	0.351351	−	0.936244i	$-0.385722\pi$
$38$		0			0
$39$	33.0000	−	20.7846i	0.846154	−	0.532939i
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	11.0000	+	19.0526i	0.255814	+	0.443083i	0.965116	−	0.261822i	$-0.0843232\pi$
$43$	11.0000	+	19.0526i	0.255814	+	0.443083i	−0.709302	+	0.704904i	$0.750990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	24.0000	+	41.5692i	0.500000	+	0.866025i
$49$	11.5000	+	47.6314i	0.234694	+	0.972069i
$50$		0			0
$51$		0			0
$52$	−2.00000	−	51.9615i	−0.0384615	−	0.999260i
$53$		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
$57$	111.000			1.94737
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	−121.000			−1.98361			−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−121.000			−1.98361			−0.991803	+	0.127774i	$0.959217\pi$
$62$		0			0
$63$	49.5000	+	38.9711i	0.785714	+	0.618590i
$64$	64.0000			1.00000
$65$		0			0
$66$		0			0
$67$	−13.0000			−0.194030			−0.0970149	−	0.995283i	$-0.530929\pi$
$67$	−13.0000			−0.194030			−0.0970149	+	0.995283i	$0.530929\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	23.0000	+	39.8372i	0.315068	+	0.545715i	0.979452	−	0.201677i	$-0.0646392\pi$
$73$	23.0000	+	39.8372i	0.315068	+	0.545715i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$	37.5000	+	64.9519i	0.500000	+	0.866025i
$76$	74.0000	−	128.172i	0.973684	−	1.68647i
$77$		0			0
$78$		0			0
$79$	−5.50000	+	9.52628i	−0.0696203	+	0.120586i	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	−5.50000	+	9.52628i	−0.0696203	+	0.120586i	0.829114	+	0.559080i	$0.188845\pi$
$80$		0			0
$81$	81.0000			1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	78.0000	−	31.1769i	0.928571	−	0.371154i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$	−90.5000	−	9.52628i	−0.994505	−	0.104684i
$92$		0			0
$93$	−19.5000	−	33.7750i	−0.209677	−	0.363172i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−83.5000	−	144.626i	−0.860825	−	1.49099i	−0.871134	−	0.491045i	$-0.836615\pi$
$97$	−83.5000	−	144.626i	−0.860825	−	1.49099i	0.0103093	−	0.999947i	$-0.496718\pi$
$98$		0			0
$99$		0			0
$100$	100.000			1.00000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	−97.0000	+	168.009i	−0.941748	+	1.63115i	−0.179612	+	0.983738i	$0.557484\pi$
$103$	−97.0000	+	168.009i	−0.941748	+	1.63115i	−0.762136	+	0.647417i	$0.775849\pi$
$104$		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
$109$	107.000	+	185.329i	0.981651	+	1.70027i	0.655963	+	0.754793i	$0.272263\pi$
$109$	107.000	+	185.329i	0.981651	+	1.70027i	0.325688	+	0.945477i	$0.394404\pi$
$110$		0			0
$111$	70.5000	+	122.110i	0.635135	+	1.10009i
$112$	16.0000	−	110.851i	0.142857	−	0.989743i
$113$		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
$117$	−99.0000	+	62.3538i	−0.846154	+	0.532939i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	121.000			1.00000
$122$		0			0
$123$		0			0
$124$	−52.0000			−0.419355
$125$		0			0
$126$		0			0
$127$	−73.0000	+	126.440i	−0.574803	+	0.995588i	0.421260	+	0.906940i	$0.361588\pi$
$127$	−73.0000	+	126.440i	−0.574803	+	0.995588i	−0.996063	+	0.0886483i	$0.971745\pi$
$128$		0			0
$129$	−33.0000	−	57.1577i	−0.255814	−	0.443083i
$130$		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
$133$	−203.500	−	160.215i	−1.53008	−	1.20462i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	114.500	+	198.320i	0.823741	+	1.42676i	0.902878	+	0.429898i	$0.141450\pi$
$139$	114.500	+	198.320i	0.823741	+	1.42676i	−0.0791367	+	0.996864i	$0.525216\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−72.0000	−	124.708i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	−34.5000	−	142.894i	−0.234694	−	0.972069i
$148$	188.000			1.27027
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	143.000	+	247.683i	0.947020	+	1.64029i	0.751656	+	0.659556i	$0.229256\pi$
$151$	143.000	+	247.683i	0.947020	+	1.64029i	0.195364	+	0.980731i	$0.437411\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	6.00000	+	155.885i	0.0384615	+	0.999260i
$157$	−155.500	−	269.334i	−0.990446	−	1.71550i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	−155.500	−	269.334i	−0.990446	−	1.71550i	−0.375796	−	0.926702i	$-0.622631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−37.0000			−0.226994			−0.113497	−	0.993538i	$-0.536205\pi$
$163$	−37.0000			−0.226994			−0.113497	+	0.993538i	$0.536205\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	73.0000	−	152.420i	0.431953	−	0.901896i
$170$		0			0
$171$	−333.000			−1.94737
$172$	−88.0000			−0.511628
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	25.0000	−	173.205i	0.142857	−	0.989743i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−1.00000			−0.00552486			−0.00276243	−	0.999996i	$-0.500879\pi$
$181$	−1.00000			−0.00552486			−0.00276243	+	0.999996i	$0.500879\pi$
$182$		0			0
$183$	363.000			1.98361
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−148.500	−	116.913i	−0.785714	−	0.618590i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−192.000			−1.00000
$193$	−382.000			−1.97927			−0.989637	−	0.143590i	$-0.954135\pi$
$193$	−382.000			−1.97927			−0.989637	+	0.143590i	$0.954135\pi$
$194$		0			0
$195$		0			0
$196$	−188.000	−	55.4256i	−0.959184	−	0.282784i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	138.500	−	239.889i	0.695980	−	1.20547i	−0.273869	−	0.961767i	$-0.588304\pi$
$199$	138.500	−	239.889i	0.695980	−	1.20547i	0.969849	−	0.243706i	$-0.0783631\pi$
$200$		0			0
$201$	39.0000			0.194030
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	184.000	+	96.9948i	0.884615	+	0.466321i
$209$		0			0
$210$		0			0
$211$	−209.500	+	362.865i	−0.992891	+	1.71974i	−0.393365	+	0.919382i	$0.628689\pi$
$211$	−209.500	+	362.865i	−0.992891	+	1.71974i	−0.599526	+	0.800355i	$0.704644\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.0000	+	90.0666i	−0.0599078	+	0.415054i
$218$		0			0
$219$	−69.0000	−	119.512i	−0.315068	−	0.545715i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−169.000	+	292.717i	−0.757848	+	1.31263i	0.186099	+	0.982531i	$0.440416\pi$
$223$	−169.000	+	292.717i	−0.757848	+	1.31263i	−0.943946	+	0.330099i	$0.892918\pi$
$224$		0			0
$225$	−112.500	−	194.856i	−0.500000	−	0.866025i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−222.000	+	384.515i	−0.973684	+	1.68647i
$229$	−191.500	+	331.688i	−0.836245	+	1.44842i	0.0567686	+	0.998387i	$0.481920\pi$
$229$	−191.500	+	331.688i	−0.836245	+	1.44842i	−0.893013	+	0.450031i	$0.851413\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	16.5000	−	28.5788i	0.0696203	−	0.120586i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	143.000	−	247.683i	0.593361	−	1.02773i	−0.400415	−	0.916334i	$-0.631134\pi$
$241$	143.000	−	247.683i	0.593361	−	1.02773i	0.993776	−	0.111397i	$-0.0355327\pi$
$242$		0			0
$243$	−243.000			−1.00000
$244$	242.000	−	419.156i	0.991803	−	1.71785i
$245$		0			0
$246$		0			0
$247$	407.000	−	256.344i	1.64777	−	1.03783i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$	−234.000	+	93.5307i	−0.928571	+	0.371154i
$253$		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
$259$	47.0000	−	325.626i	0.181467	−	1.25724i
$260$		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
$268$	26.0000	−	45.0333i	0.0970149	−	0.168035i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	270.500	+	468.520i	0.998155	+	1.72886i	0.551661	+	0.834069i	$0.313994\pi$
$271$	270.500	+	468.520i	0.998155	+	1.72886i	0.446494	+	0.894786i	$0.352672\pi$
$272$		0			0
$273$	271.500	+	28.5788i	0.994505	+	0.104684i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	264.500	−	458.127i	0.954874	−	1.65389i	0.220217	−	0.975451i	$-0.429324\pi$
$277$	264.500	−	458.127i	0.954874	−	1.65389i	0.734657	−	0.678439i	$-0.237343\pi$
$278$		0			0
$279$	58.5000	+	101.325i	0.209677	+	0.363172i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	458.000			1.61837			0.809187	−	0.587551i	$-0.199908\pi$
$283$	458.000			1.61837			0.809187	+	0.587551i	$0.199908\pi$
$284$		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
$291$	250.500	+	433.879i	0.860825	+	1.49099i
$292$	−184.000			−0.630137
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−300.000			−1.00000
$301$	−22.0000	+	152.420i	−0.0730897	+	0.506380i
$302$		0			0
$303$		0			0
$304$	296.000	+	512.687i	0.973684	+	1.68647i
$305$		0			0
$306$		0			0
$307$	611.000			1.99023			0.995114	−	0.0987325i	$-0.0314788\pi$
$307$	611.000			1.99023			0.995114	+	0.0987325i	$0.0314788\pi$
$308$		0			0
$309$	291.000	−	504.027i	0.941748	−	1.63115i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	71.0000	−	122.976i	0.226837	−	0.392893i	−0.730032	−	0.683413i	$-0.760495\pi$
$313$	71.0000	−	122.976i	0.226837	−	0.392893i	0.956869	+	0.290520i	$0.0938282\pi$
$314$		0			0
$315$		0			0
$316$	−22.0000	−	38.1051i	−0.0696203	−	0.120586i
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−162.000	+	280.592i	−0.500000	+	0.866025i
$325$	287.500	+	151.554i	0.884615	+	0.466321i
$326$		0			0
$327$	−321.000	−	555.988i	−0.981651	−	1.70027i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	362.000			1.09366			0.546828	−	0.837245i	$-0.315835\pi$
$331$	362.000			1.09366			0.546828	+	0.837245i	$0.315835\pi$
$332$		0			0
$333$	−211.500	−	366.329i	−0.635135	−	1.10009i
$334$		0			0
$335$		0			0
$336$	−48.0000	+	332.554i	−0.142857	+	0.989743i
$337$	482.000			1.43027			0.715134	−	0.698988i	$-0.246366\pi$
$337$	482.000			1.43027			0.715134	+	0.698988i	$0.246366\pi$
$338$		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
$349$	−335.500	+	581.103i	−0.961318	+	1.66505i	−0.242120	+	0.970246i	$0.577843\pi$
$349$	−335.500	+	581.103i	−0.961318	+	1.66505i	−0.719198	+	0.694805i	$0.755490\pi$
$350$		0			0
$351$	297.000	−	187.061i	0.846154	−	0.532939i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	1008.00			2.79224
$362$		0			0
$363$	−363.000			−1.00000
$364$	214.000	−	294.449i	0.587912	−	0.808925i
$365$		0			0
$366$		0			0
$367$	−718.000			−1.95640			−0.978202	−	0.207657i	$-0.933416\pi$
$367$	−718.000			−1.95640			−0.978202	+	0.207657i	$0.933416\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	156.000			0.419355
$373$	−577.000			−1.54692			−0.773458	−	0.633847i	$-0.781475\pi$
$373$	−577.000			−1.54692			−0.773458	+	0.633847i	$0.781475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−305.500	+	529.142i	−0.806069	+	1.39615i	0.109499	+	0.993987i	$0.465075\pi$
$379$	−305.500	+	529.142i	−0.806069	+	1.39615i	−0.915567	+	0.402165i	$0.868258\pi$
$380$		0			0
$381$	219.000	−	379.319i	0.574803	−	0.995588i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	99.0000	+	171.473i	0.255814	+	0.443083i
$388$	668.000			1.72165
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	362.000			0.911839			0.455919	−	0.890021i	$-0.349311\pi$
$397$	362.000			0.911839			0.455919	+	0.890021i	$0.349311\pi$
$398$		0			0
$399$	610.500	+	480.644i	1.53008	+	1.20462i
$400$	−200.000	+	346.410i	−0.500000	+	0.866025i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−149.500	−	78.8083i	−0.370968	−	0.195554i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−313.000	+	542.132i	−0.765281	+	1.32551i	0.174817	+	0.984601i	$0.444067\pi$
$409$	−313.000	+	542.132i	−0.765281	+	1.32551i	−0.940098	+	0.340905i	$0.889267\pi$
$410$		0			0
$411$		0			0
$412$	−388.000	−	672.036i	−0.941748	−	1.63115i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−343.500	−	594.959i	−0.823741	−	1.42676i
$418$		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
$421$	−481.000			−1.14252			−0.571259	−	0.820770i	$-0.693545\pi$
$421$	−481.000			−1.14252			−0.571259	+	0.820770i	$0.693545\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−665.500	−	523.945i	−1.55855	−	1.22704i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	216.000	+	374.123i	0.500000	+	0.866025i
$433$	−179.500	−	310.903i	−0.414550	−	0.718021i	0.580831	−	0.814024i	$-0.302728\pi$
$433$	−179.500	−	310.903i	−0.414550	−	0.718021i	−0.995381	+	0.0960028i	$0.969394\pi$
$434$		0			0
$435$		0			0
$436$	−856.000			−1.96330
$437$		0			0
$438$		0			0
$439$	−401.500	−	695.418i	−0.914579	−	1.58410i	−0.807517	−	0.589844i	$-0.799189\pi$
$439$	−401.500	−	695.418i	−0.914579	−	1.58410i	−0.107062	−	0.994252i	$-0.534144\pi$
$440$		0			0
$441$	103.500	+	428.683i	0.234694	+	0.972069i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−564.000			−1.27027
$445$		0			0
$446$		0			0
$447$		0			0
$448$	352.000	+	277.128i	0.785714	+	0.618590i
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−429.000	−	743.050i	−0.947020	−	1.64029i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−383.500	−	664.241i	−0.839168	−	1.45348i	−0.890591	−	0.454805i	$-0.849709\pi$
$457$	−383.500	−	664.241i	−0.839168	−	1.45348i	0.0514223	−	0.998677i	$-0.483625\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	−526.000			−1.13607			−0.568035	−	0.823005i	$-0.692296\pi$
$463$	−526.000			−1.13607			−0.568035	+	0.823005i	$0.692296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	−18.0000	−	467.654i	−0.0384615	−	0.999260i
$469$	−71.5000	−	56.2917i	−0.152452	−	0.120025i
$470$		0			0
$471$	466.500	+	808.002i	0.990446	+	1.71550i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	462.500	+	801.073i	0.973684	+	1.68647i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	540.500	+	284.922i	1.12370	+	0.592354i
$482$		0			0
$483$		0			0
$484$	−242.000	+	419.156i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	306.500	−	530.874i	0.629363	−	1.09009i	−0.358316	−	0.933600i	$-0.616649\pi$
$487$	306.500	−	530.874i	0.629363	−	1.09009i	0.987680	−	0.156489i	$-0.0500176\pi$
$488$		0			0
$489$	111.000			0.226994
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	104.000	−	180.133i	0.209677	−	0.363172i
$497$		0			0
$498$		0			0
$499$	438.500	−	759.504i	0.878758	−	1.52205i	0.0260521	−	0.999661i	$-0.491706\pi$
$499$	438.500	−	759.504i	0.878758	−	1.52205i	0.852705	−	0.522392i	$-0.174960\pi$
$500$		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
$507$	−219.000	+	457.261i	−0.431953	+	0.901896i
$508$	−292.000	−	505.759i	−0.574803	−	0.995588i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−46.0000	+	318.697i	−0.0900196	+	0.623674i
$512$		0			0
$513$	999.000			1.94737
$514$		0			0
$515$		0			0
$516$	264.000			0.511628
$517$		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
$523$	−401.500	−	695.418i	−0.767686	−	1.32967i	−0.938815	−	0.344423i	$-0.888075\pi$
$523$	−401.500	−	695.418i	−0.767686	−	1.32967i	0.171128	−	0.985249i	$-0.445259\pi$
$524$		0			0
$525$	−75.0000	+	519.615i	−0.142857	+	0.989743i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	962.000	−	384.515i	1.80827	−	0.722773i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−517.000	+	895.470i	−0.955638	+	1.65521i	−0.222736	+	0.974879i	$0.571499\pi$
$541$	−517.000	+	895.470i	−0.955638	+	1.65521i	−0.732902	+	0.680334i	$0.761835\pi$
$542$		0			0
$543$	3.00000			0.00552486
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1093.00			−1.99817			−0.999086	−	0.0427471i	$-0.986389\pi$
$547$	−1093.00			−1.99817			−0.999086	+	0.0427471i	$0.986389\pi$
$548$		0			0
$549$	−1089.00			−1.98361
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−71.5000	+	28.5788i	−0.129295	+	0.0516796i
$554$		0			0
$555$		0			0
$556$	−916.000			−1.64748
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	−253.000	−	133.368i	−0.452594	−	0.238583i
$560$		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
$567$	445.500	+	350.740i	0.785714	+	0.618590i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−533.500	−	924.049i	−0.934326	−	1.61830i	−0.775832	−	0.630940i	$-0.782670\pi$
$571$	−533.500	−	924.049i	−0.934326	−	1.61830i	−0.158494	−	0.987360i	$-0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	576.000			1.00000
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.833622	+	0.552335i	$0.186263\pi$
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.0615251	+	0.998106i	$0.480404\pi$
$578$		0			0
$579$	1146.00			1.97927
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	564.000	+	166.277i	0.959184	+	0.282784i
$589$	−240.500	−	416.558i	−0.408319	−	0.707230i
$590$		0			0
$591$		0			0
$592$	−376.000	+	651.251i	−0.635135	+	1.10009i
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−415.500	+	719.667i	−0.695980	+	1.20547i
$598$		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
$601$	336.500	−	582.835i	0.559900	−	0.969776i	−0.437604	−	0.899168i	$-0.644173\pi$
$601$	336.500	−	582.835i	0.559900	−	0.969776i	0.997504	−	0.0706077i	$-0.0224939\pi$
$602$		0			0
$603$	−117.000			−0.194030
$604$	−1144.00			−1.89404
$605$		0			0
$606$		0			0
$607$	−373.000			−0.614498			−0.307249	−	0.951629i	$-0.599408\pi$
$607$	−373.000			−0.614498			−0.307249	+	0.951629i	$0.599408\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	983.000			1.60359			0.801794	−	0.597600i	$-0.203879\pi$
$613$	983.000			1.60359			0.801794	+	0.597600i	$0.203879\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	107.000	+	185.329i	0.172859	+	0.299401i	0.939418	−	0.342773i	$-0.111366\pi$
$619$	107.000	+	185.329i	0.172859	+	0.299401i	−0.766559	+	0.642174i	$0.778033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−552.000	−	290.985i	−0.884615	−	0.466321i
$625$	−312.500	+	541.266i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	1244.00			1.98089
$629$		0			0
$630$		0			0
$631$	−293.500	−	508.357i	−0.465135	−	0.805637i	0.534073	−	0.845438i	$-0.320661\pi$
$631$	−293.500	−	508.357i	−0.465135	−	0.805637i	−0.999208	+	0.0398015i	$0.987327\pi$
$632$		0			0
$633$	628.500	−	1088.59i	0.992891	−	1.71974i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−456.500	−	444.271i	−0.716641	−	0.697443i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−157.000	−	271.932i	−0.244168	−	0.422911i	0.717729	−	0.696322i	$-0.245181\pi$
$643$	−157.000	−	271.932i	−0.244168	−	0.422911i	−0.961897	+	0.273411i	$0.911848\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	39.0000	−	270.200i	0.0599078	−	0.415054i
$652$	74.0000	−	128.172i	0.113497	−	0.196582i
$653$		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
$657$	207.000	+	358.535i	0.315068	+	0.545715i
$658$		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
$661$	1079.00			1.63238			0.816188	−	0.577787i	$-0.196084\pi$
$661$	1079.00			1.63238			0.816188	+	0.577787i	$0.196084\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	507.000	−	878.150i	0.757848	−	1.31263i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−577.000	+	999.393i	−0.857355	+	1.48498i	0.0170877	+	0.999854i	$0.494561\pi$
$673$	−577.000	+	999.393i	−0.857355	+	1.48498i	−0.874443	+	0.485129i	$0.838773\pi$
$674$		0			0
$675$	337.500	+	584.567i	0.500000	+	0.866025i
$676$	382.000	+	557.720i	0.565089	+	0.825030i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	167.000	−	1157.01i	0.245950	−	1.70399i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	666.000	−	1153.55i	0.973684	−	1.68647i
$685$		0			0
$686$		0			0
$687$	574.500	−	995.063i	0.836245	−	1.44842i
$688$	176.000	−	304.841i	0.255814	−	0.443083i
$689$		0			0
$690$		0			0
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.737337	+	0.675525i	$0.236083\pi$
$691$	659.000	+	1141.42i	0.953690	+	1.65184i	0.216353	+	0.976315i	$0.430584\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	550.000	+	433.013i	0.785714	+	0.618590i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	869.500	+	1506.02i	1.23684	+	2.14227i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−457.000			−0.644570			−0.322285	−	0.946643i	$-0.604451\pi$
$709$	−457.000			−0.644570			−0.322285	+	0.946643i	$0.604451\pi$
$710$		0			0
$711$	−49.5000	+	85.7365i	−0.0696203	+	0.120586i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−1261.00	+	504.027i	−1.74896	+	0.699066i
$722$		0			0
$723$	−429.000	+	743.050i	−0.593361	+	1.02773i
$724$	2.00000	−	3.46410i	0.00276243	−	0.00478467i
$725$		0			0
$726$		0			0
$727$	947.000			1.30261			0.651307	−	0.758815i	$-0.274221\pi$
$727$	947.000			1.30261			0.651307	+	0.758815i	$0.274221\pi$
$728$		0			0
$729$	729.000			1.00000
$730$		0			0
$731$		0			0
$732$	−726.000	+	1257.47i	−0.991803	+	1.71785i
$733$	708.500	−	1227.16i	0.966576	−	1.67416i	0.261255	−	0.965270i	$-0.415864\pi$
$733$	708.500	−	1227.16i	0.966576	−	1.67416i	0.705321	−	0.708888i	$-0.250803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−109.000			−0.147497			−0.0737483	−	0.997277i	$-0.523496\pi$
$739$	−109.000			−0.147497			−0.0737483	+	0.997277i	$0.523496\pi$
$740$		0			0
$741$	−1221.00	+	769.031i	−1.64777	+	1.03783i
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	690.500	+	1195.98i	0.919441	+	1.59252i	0.800266	+	0.599645i	$0.204691\pi$
$751$	690.500	+	1195.98i	0.919441	+	1.59252i	0.119174	+	0.992873i	$0.461975\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	702.000	−	280.592i	0.928571	−	0.371154i
$757$	419.000	−	725.729i	0.553501	−	0.958691i	−0.444518	−	0.895770i	$-0.646625\pi$
$757$	419.000	−	725.729i	0.553501	−	0.958691i	0.998018	−	0.0629213i	$-0.0200417\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−214.000	+	1482.64i	−0.280472	+	1.94317i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	384.000	−	665.108i	0.500000	−	0.866025i
$769$	−335.500	+	581.103i	−0.436281	+	0.755661i	−0.997399	−	0.0720749i	$-0.977038\pi$
$769$	−335.500	+	581.103i	−0.436281	+	0.755661i	0.561118	+	0.827736i	$0.310371\pi$
$770$		0			0
$771$		0			0
$772$	764.000	−	1323.29i	0.989637	−	1.71410i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	162.500	−	281.458i	0.209677	−	0.363172i
$776$		0			0
$777$	−141.000	+	976.877i	−0.181467	+	1.25724i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	568.000	−	540.400i	0.724490	−	0.689286i
$785$		0			0
$786$		0			0
$787$	306.500	−	530.874i	0.389454	−	0.674553i	−0.602922	−	0.797800i	$-0.705997\pi$
$787$	306.500	−	530.874i	0.389454	−	0.674553i	0.992376	+	0.123246i	$0.0393305\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1331.00	−	838.313i	1.67844	−	1.05714i
$794$		0			0
$795$		0			0
$796$	554.000	+	959.556i	0.695980	+	1.20547i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−78.0000	+	135.100i	−0.0970149	+	0.168035i
$805$		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
$811$	−253.000			−0.311961			−0.155980	−	0.987760i	$-0.549854\pi$
$811$	−253.000			−0.311961			−0.155980	+	0.987760i	$0.549854\pi$
$812$		0			0
$813$	−811.500	−	1405.56i	−0.998155	−	1.72886i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−407.000	−	704.945i	−0.498164	−	0.862845i
$818$		0			0
$819$	−814.500	−	85.7365i	−0.994505	−	0.104684i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−529.000	+	916.255i	−0.642770	+	1.11331i	0.342041	+	0.939685i	$0.388882\pi$
$823$	−529.000	+	916.255i	−0.642770	+	1.11331i	−0.984812	+	0.173626i	$0.944452\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	458.000			0.552473			0.276236	−	0.961090i	$-0.410913\pi$
$829$	458.000			0.552473			0.276236	+	0.961090i	$0.410913\pi$
$830$		0			0
$831$	−793.500	+	1374.38i	−0.954874	+	1.65389i
$832$	−704.000	+	443.405i	−0.846154	+	0.532939i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−175.500	−	303.975i	−0.209677	−	0.363172i
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	−838.000	−	1451.46i	−0.992891	−	1.71974i
$845$		0			0
$846$		0			0
$847$	665.500	+	523.945i	0.785714	+	0.618590i
$848$		0			0
$849$	−1374.00			−1.61837
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−481.000			−0.563892			−0.281946	−	0.959430i	$-0.590980\pi$
$853$	−481.000			−0.563892			−0.281946	+	0.959430i	$0.590980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	−65.5000	+	113.449i	−0.0762515	+	0.132071i	−0.901630	−	0.432509i	$-0.857629\pi$
$859$	−65.5000	+	113.449i	−0.0762515	+	0.132071i	0.825378	+	0.564580i	$0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	433.500	+	750.844i	0.500000	+	0.866025i
$868$	−286.000	−	225.167i	−0.329493	−	0.259409i
$869$		0			0
$870$		0			0
$871$	143.000	−	90.0666i	0.164179	−	0.103406i
$872$		0			0
$873$	−751.500	−	1301.64i	−0.860825	−	1.49099i
$874$		0			0
$875$		0			0
$876$	552.000			0.630137
$877$	−598.000			−0.681870			−0.340935	−	0.940087i	$-0.610744\pi$
$877$	−598.000			−0.681870			−0.340935	+	0.940087i	$0.610744\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	−1702.00			−1.92752			−0.963760	−	0.266771i	$-0.914043\pi$
$883$	−1702.00			−1.92752			−0.963760	+	0.266771i	$0.914043\pi$
$884$		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
$889$	−949.000	+	379.319i	−1.06749	+	0.426681i
$890$		0			0
$891$		0			0
$892$	−676.000	−	1170.87i	−0.757848	−	1.31263i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	900.000			1.00000
$901$		0			0
$902$		0			0
$903$	66.0000	−	457.261i	0.0730897	−	0.506380i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1453.00			−1.60198			−0.800992	−	0.598675i	$-0.795694\pi$
$907$	−1453.00			−1.60198			−0.800992	+	0.598675i	$0.795694\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−888.000	−	1538.06i	−0.973684	−	1.68647i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−766.000	−	1326.75i	−0.836245	−	1.44842i
$917$		0			0
$918$		0			0
$919$	971.000			1.05658			0.528292	−	0.849063i	$-0.322833\pi$
$919$	971.000			1.05658			0.528292	+	0.849063i	$0.322833\pi$
$920$		0			0
$921$	−1833.00			−1.99023
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−587.500	+	1017.58i	−0.635135	+	1.10009i
$926$		0			0
$927$	−873.000	+	1512.08i	−0.941748	+	1.63115i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−425.500	−	1762.36i	−0.457035	−	1.89298i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1847.00			1.97118			0.985592	−	0.169138i	$-0.0540985\pi$
$937$	1847.00			1.97118			0.985592	+	0.169138i	$0.0540985\pi$
$938$		0			0
$939$	−213.000	+	368.927i	−0.226837	+	0.392893i
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	66.0000	+	114.315i	0.0696203	+	0.120586i
$949$	−529.000	−	278.860i	−0.557429	−	0.293846i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	396.000	−	685.892i	0.412071	−	0.713727i
$962$		0			0
$963$		0			0
$964$	572.000	+	990.733i	0.593361	+	1.02773i
$965$		0			0
$966$		0			0
$967$	−253.000			−0.261634			−0.130817	−	0.991407i	$-0.541760\pi$
$967$	−253.000			−0.261634			−0.130817	+	0.991407i	$0.541760\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	486.000	−	841.777i	0.500000	−	0.866025i
$973$	−229.000	+	1586.56i	−0.235355	+	1.63058i
$974$		0			0
$975$	−862.500	−	454.663i	−0.884615	−	0.466321i
$976$	968.000	+	1676.63i	0.991803	+	1.71785i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	963.000	+	1667.96i	0.981651	+	1.70027i
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	74.0000	+	1922.58i	0.0748988	+	1.94593i
$989$		0			0
$990$		0			0
$991$	1739.00			1.75479			0.877397	−	0.479766i	$-0.159278\pi$
$991$	1739.00			1.75479			0.877397	+	0.479766i	$0.159278\pi$
$992$		0			0
$993$	−1086.00			−1.09366
$994$		0			0
$995$		0			0
$996$		0			0
$997$	947.000	−	1640.25i	0.949850	−	1.64519i	0.204112	−	0.978947i	$-0.434569\pi$
$997$	947.000	−	1640.25i	0.949850	−	1.64519i	0.745737	−	0.666240i	$-0.232097\pi$
$998$		0			0
$999$	634.500	+	1098.99i	0.635135	+	1.10009i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.3.bm.a.191.1	yes	2
3.2	odd	2	CM	273.3.bm.a.191.1	yes	2
7.4	even	3		273.3.s.a.74.1	✓	2
13.3	even	3		273.3.s.a.107.1	yes	2
21.11	odd	6		273.3.s.a.74.1	✓	2
39.29	odd	6		273.3.s.a.107.1	yes	2
91.81	even	3	inner	273.3.bm.a.263.1	yes	2
273.263	odd	6	inner	273.3.bm.a.263.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.3.s.a.74.1	✓	2	7.4	even	3
273.3.s.a.74.1	✓	2	21.11	odd	6
273.3.s.a.107.1	yes	2	13.3	even	3
273.3.s.a.107.1	yes	2	39.29	odd	6
273.3.bm.a.191.1	yes	2	1.1	even	1	trivial
273.3.bm.a.191.1	yes	2	3.2	odd	2	CM
273.3.bm.a.263.1	yes	2	91.81	even	3	inner
273.3.bm.a.263.1	yes	2	273.263	odd	6	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.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +(-2.00000 + 3.46410i) q^{4} +(5.50000 + 4.33013i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +(-2.00000 + 3.46410i) q^{4} +(5.50000 + 4.33013i) q^{7} +9.00000 q^{9} +(6.00000 - 10.3923i) q^{12} +(-11.0000 + 6.92820i) q^{13} +(-8.00000 - 13.8564i) q^{16} -37.0000 q^{19} +(-16.5000 - 12.9904i) q^{21} +(-12.5000 - 21.6506i) q^{25} -27.0000 q^{27} +(-26.0000 + 10.3923i) q^{28} +(6.50000 + 11.2583i) q^{31} +(-18.0000 + 31.1769i) q^{36} +(-23.5000 - 40.7032i) q^{37} +(33.0000 - 20.7846i) q^{39} +(11.0000 + 19.0526i) q^{43} +(24.0000 + 41.5692i) q^{48} +(11.5000 + 47.6314i) q^{49} +(-2.00000 - 51.9615i) q^{52} +111.000 q^{57} -121.000 q^{61} +(49.5000 + 38.9711i) q^{63} +64.0000 q^{64} -13.0000 q^{67} +(23.0000 + 39.8372i) q^{73} +(37.5000 + 64.9519i) q^{75} +(74.0000 - 128.172i) q^{76} +(-5.50000 + 9.52628i) q^{79} +81.0000 q^{81} +(78.0000 - 31.1769i) q^{84} +(-90.5000 - 9.52628i) q^{91} +(-19.5000 - 33.7750i) q^{93} +(-83.5000 - 144.626i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 4 q^{4} + 11 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 4 * q^4 + 11 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 4 q^{4} + 11 q^{7} + 18 q^{9} + 12 q^{12} - 22 q^{13} - 16 q^{16} - 74 q^{19} - 33 q^{21} - 25 q^{25} - 54 q^{27} - 52 q^{28} + 13 q^{31} - 36 q^{36} - 47 q^{37} + 66 q^{39} + 22 q^{43} + 48 q^{48} + 23 q^{49} - 4 q^{52} + 222 q^{57} - 242 q^{61} + 99 q^{63} + 128 q^{64} - 26 q^{67} + 46 q^{73} + 75 q^{75} + 148 q^{76} - 11 q^{79} + 162 q^{81} + 156 q^{84} - 181 q^{91} - 39 q^{93} - 167 q^{97}+O(q^{100}) \) 2 * q - 6 * q^3 - 4 * q^4 + 11 * q^7 + 18 * q^9 + 12 * q^12 - 22 * q^13 - 16 * q^16 - 74 * q^19 - 33 * q^21 - 25 * q^25 - 54 * q^27 - 52 * q^28 + 13 * q^31 - 36 * q^36 - 47 * q^37 + 66 * q^39 + 22 * q^43 + 48 * q^48 + 23 * q^49 - 4 * q^52 + 222 * q^57 - 242 * q^61 + 99 * q^63 + 128 * q^64 - 26 * q^67 + 46 * q^73 + 75 * q^75 + 148 * q^76 - 11 * q^79 + 162 * q^81 + 156 * q^84 - 181 * q^91 - 39 * q^93 - 167 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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