LMFDB - Embedded newform 273.2.bn.a.230.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(146,273)]

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, 3, 2]))

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

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

chi := DirichletCharacter("273.146");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.bn (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:	$2.17991597518$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			230.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	273.230
Dual form			273.2.bn.a.146.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q+(-1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +3.46410i q^{12} +(-2.50000 + 2.59808i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-3.00000 + 1.73205i) q^{19} +(4.50000 + 0.866025i) q^{21} -5.00000 q^{25} -5.19615i q^{27} +(4.00000 + 3.46410i) q^{28} -8.66025i q^{31} +(3.00000 - 5.19615i) q^{36} +(-5.00000 + 8.66025i) q^{37} +(6.00000 - 1.73205i) q^{39} +(-6.50000 - 11.2583i) q^{43} +(6.00000 - 3.46410i) q^{48} +(5.50000 - 4.33013i) q^{49} +(7.00000 + 1.73205i) q^{52} +6.00000 q^{57} +(7.50000 - 4.33013i) q^{61} +(-6.00000 - 5.19615i) q^{63} +8.00000 q^{64} +(-5.50000 + 9.52628i) q^{67} -1.73205i q^{73} +(7.50000 + 4.33013i) q^{75} +(6.00000 + 3.46410i) q^{76} -13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(4.00000 - 8.66025i) q^{91} +(-7.50000 + 12.9904i) q^{93} +(-16.5000 + 9.52628i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 2 q^{4} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 2 * q^4 - 5 * q^7 + 3 * q^9	$ 2 q - 3 q^{3} - 2 q^{4} - 5 q^{7} + 3 q^{9} - 5 q^{13} - 4 q^{16} - 6 q^{19} + 9 q^{21} - 10 q^{25} + 8 q^{28} + 6 q^{36} - 10 q^{37} + 12 q^{39} - 13 q^{43} + 12 q^{48} + 11 q^{49} + 14 q^{52} + 12 q^{57} + 15 q^{61} - 12 q^{63} + 16 q^{64} - 11 q^{67} + 15 q^{75} + 12 q^{76} - 26 q^{79} - 9 q^{81} - 6 q^{84} + 8 q^{91} - 15 q^{93} - 33 q^{97}+O(q^{100}) $ 2 * q - 3 * q^3 - 2 * q^4 - 5 * q^7 + 3 * q^9 - 5 * q^13 - 4 * q^16 - 6 * q^19 + 9 * q^21 - 10 * q^25 + 8 * q^28 + 6 * q^36 - 10 * q^37 + 12 * q^39 - 13 * q^43 + 12 * q^48 + 11 * q^49 + 14 * q^52 + 12 * q^57 + 15 * q^61 - 12 * q^63 + 16 * q^64 - 11 * q^67 + 15 * q^75 + 12 * q^76 - 26 * q^79 - 9 * q^81 - 6 * q^84 + 8 * q^91 - 15 * q^93 - 33 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$			3.46410i			1.00000i
$13$	−2.50000	+	2.59808i	−0.693375	+	0.720577i
$13$	−2.50000	+	2.59808i	−0.693375	+	0.720577i
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−3.00000	+	1.73205i	−0.688247	+	0.397360i	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−3.00000	+	1.73205i	−0.688247	+	0.397360i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$	4.50000	+	0.866025i	0.981981	+	0.188982i
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$	4.00000	+	3.46410i	0.755929	+	0.654654i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		−	8.66025i		−	1.55543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$		−	8.66025i		−	1.55543i	0.628619	−	0.777714i	$-0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	3.00000	−	5.19615i	0.500000	−	0.866025i
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	0.0821995	+	0.996616i	$0.473806\pi$
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$	6.00000	−	1.73205i	0.960769	−	0.277350i
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−6.50000	−	11.2583i	−0.991241	−	1.71688i	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−6.50000	−	11.2583i	−0.991241	−	1.71688i	−0.381246	−	0.924473i	$-0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	6.00000	−	3.46410i	0.866025	−	0.500000i
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$	7.00000	+	1.73205i	0.970725	+	0.240192i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	7.50000	−	4.33013i	0.960277	−	0.554416i	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	7.50000	−	4.33013i	0.960277	−	0.554416i	0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$	−6.00000	−	5.19615i	−0.755929	−	0.654654i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−5.50000	+	9.52628i	−0.671932	+	1.16382i	0.305424	+	0.952217i	$0.401202\pi$
$67$	−5.50000	+	9.52628i	−0.671932	+	1.16382i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		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
$73$		−	1.73205i		−	0.202721i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$		−	1.73205i		−	0.202721i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$	6.00000	+	3.46410i	0.688247	+	0.397360i
$77$		0			0
$78$		0			0
$79$	−13.0000			−1.46261			−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000			−1.46261			−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−3.00000	−	8.66025i	−0.327327	−	0.944911i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	4.00000	−	8.66025i	0.419314	−	0.907841i
$92$		0			0
$93$	−7.50000	+	12.9904i	−0.777714	+	1.34704i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.5000	+	9.52628i	−1.67532	+	0.967247i	−0.710742	+	0.703452i	$0.751641\pi$
$97$	−16.5000	+	9.52628i	−1.67532	+	0.967247i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$		−	15.5885i		−	1.53598i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$		−	15.5885i		−	1.53598i	0.640464	−	0.767988i	$-0.278742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−9.00000	+	5.19615i	−0.866025	+	0.500000i
$109$	19.0000			1.81987			0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000			1.81987			0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	15.0000	−	8.66025i	1.42374	−	0.821995i
$112$	2.00000	−	10.3923i	0.188982	−	0.981981i
$113$		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
$117$	−10.5000	−	2.59808i	−0.970725	−	0.240192i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−15.0000	+	8.66025i	−1.34704	+	0.777714i
$125$		0			0
$126$		0			0
$127$	0.500000	−	0.866025i	0.0443678	−	0.0768473i	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	0.500000	−	0.866025i	0.0443678	−	0.0768473i	0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$			22.5167i			1.98248i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	6.00000	−	6.92820i	0.520266	−	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−19.5000	+	11.2583i	−1.65397	+	0.954919i	−0.678551	+	0.734553i	$0.737392\pi$
$139$	−19.5000	+	11.2583i	−1.65397	+	0.954919i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	−12.0000	+	1.73205i	−0.989743	+	0.142857i
$148$	20.0000			1.64399
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	4.00000			0.325515			0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000			0.325515			0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−9.00000	−	8.66025i	−0.720577	−	0.693375i
$157$			22.5167i			1.79703i	0.438948	+	0.898513i	$0.355351\pi$
$157$			22.5167i			1.79703i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	12.5000	+	21.6506i	0.979076	+	1.69581i	0.665771	+	0.746156i	$0.268103\pi$
$163$	12.5000	+	21.6506i	0.979076	+	1.69581i	0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−0.500000	−	12.9904i	−0.0384615	−	0.999260i
$170$		0			0
$171$	−9.00000	−	5.19615i	−0.688247	−	0.397360i
$172$	−13.0000	+	22.5167i	−0.991241	+	1.71688i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	12.5000	−	4.33013i	0.944911	−	0.327327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		0			0
$183$	−15.0000			−1.10883
$184$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−12.0000	−	6.92820i	−0.866025	−	0.500000i
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	11.5000	−	19.9186i	0.827788	−	1.43377i	0.899770	−	0.436365i	$-0.143734\pi$
$194$		0			0
$195$		0			0
$196$	−13.0000	−	5.19615i	−0.928571	−	0.371154i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	19.5000	−	11.2583i	1.38232	−	0.798082i	0.389885	−	0.920864i	$-0.372515\pi$
$199$	19.5000	−	11.2583i	1.38232	−	0.798082i	0.992434	+	0.122782i	$0.0391815\pi$
$200$		0			0
$201$	16.5000	−	9.52628i	1.16382	−	0.671932i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4.00000	−	13.8564i	−0.277350	−	0.960769i
$209$		0			0
$210$		0			0
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−6.50000	+	11.2583i	−0.447478	+	0.775055i	0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	7.50000	+	21.6506i	0.509133	+	1.46974i
$218$		0			0
$219$	−1.50000	+	2.59808i	−0.101361	+	0.175562i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	9.00000	+	5.19615i	0.602685	+	0.347960i	0.770097	−	0.637927i	$-0.220208\pi$
$223$	9.00000	+	5.19615i	0.602685	+	0.347960i	−0.167412	+	0.985887i	$0.553541\pi$
$224$		0			0
$225$	−7.50000	−	12.9904i	−0.500000	−	0.866025i
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	−6.00000	−	10.3923i	−0.397360	−	0.688247i
$229$			20.7846i			1.37349i	0.726900	+	0.686743i	$0.240960\pi$
$229$			20.7846i			1.37349i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	19.5000	+	11.2583i	1.26666	+	0.731307i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	24.0000	−	13.8564i	1.54598	−	0.892570i	0.547533	−	0.836784i	$-0.315567\pi$
$241$	24.0000	−	13.8564i	1.54598	−	0.892570i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$	−15.0000	−	8.66025i	−0.960277	−	0.554416i
$245$		0			0
$246$		0			0
$247$	3.00000	−	12.1244i	0.190885	−	0.771454i
$248$		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	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	5.00000	−	25.9808i	0.310685	−	1.61437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	22.0000			1.34386
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	13.5000	+	7.79423i	0.820067	+	0.473466i	0.850439	−	0.526073i	$-0.176336\pi$
$271$	13.5000	+	7.79423i	0.820067	+	0.473466i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$		0			0
$273$	−13.5000	+	9.52628i	−0.817057	+	0.576557i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.0000	−	22.5167i	−0.781094	−	1.35290i	−0.931305	−	0.364241i	$-0.881328\pi$
$277$	−13.0000	−	22.5167i	−0.781094	−	1.35290i	0.150210	−	0.988654i	$-0.452005\pi$
$278$		0			0
$279$	22.5000	−	12.9904i	1.34704	−	0.777714i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−19.5000	−	11.2583i	−1.15915	−	0.669238i	−0.208053	−	0.978117i	$-0.566713\pi$
$283$	−19.5000	−	11.2583i	−1.15915	−	0.669238i	−0.951101	+	0.308879i	$0.900046\pi$
$284$		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
$291$	33.0000			1.93449
$292$	−3.00000	+	1.73205i	−0.175562	+	0.101361i
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	17.3205i		−	1.00000i
$301$	26.0000	+	22.5167i	1.49862	+	1.29784i
$302$		0			0
$303$		0			0
$304$		−	13.8564i		−	0.794719i
$305$		0			0
$306$		0			0
$307$			1.73205i			0.0988534i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$			1.73205i			0.0988534i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$		0			0
$309$	−13.5000	+	23.3827i	−0.767988	+	1.33019i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			32.9090i			1.86012i	0.367402	+	0.930062i	$0.380247\pi$
$313$			32.9090i			1.86012i	−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$		0			0
$316$	13.0000	+	22.5167i	0.731307	+	1.26666i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	18.0000			1.00000
$325$	12.5000	−	12.9904i	0.693375	−	0.720577i
$326$		0			0
$327$	−28.5000	−	16.4545i	−1.57605	−	0.909935i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−15.5000	−	26.8468i	−0.851957	−	1.47563i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−15.5000	−	26.8468i	−0.851957	−	1.47563i	0.0274825	−	0.999622i	$-0.491251\pi$
$332$		0			0
$333$	−30.0000			−1.64399
$334$		0			0
$335$		0			0
$336$	−12.0000	+	13.8564i	−0.654654	+	0.755929i
$337$	−29.0000			−1.57973			−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000			−1.57973			−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−25.5000	−	14.7224i	−1.36498	−	0.788074i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−25.5000	−	14.7224i	−1.36498	−	0.788074i	−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$	13.5000	+	12.9904i	0.720577	+	0.693375i
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−3.50000	+	6.06218i	−0.184211	+	0.319062i
$362$		0			0
$363$			19.0526i			1.00000i
$364$	−19.0000	+	1.73205i	−0.995871	+	0.0907841i
$365$		0			0
$366$		0			0
$367$	−19.5000	−	11.2583i	−1.01789	−	0.587680i	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−19.5000	−	11.2583i	−1.01789	−	0.587680i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	30.0000			1.55543
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−18.5000	+	32.0429i	−0.950281	+	1.64594i	−0.205466	+	0.978664i	$0.565871\pi$
$379$	−18.5000	+	32.0429i	−0.950281	+	1.64594i	−0.744815	+	0.667271i	$0.767462\pi$
$380$		0			0
$381$	−1.50000	+	0.866025i	−0.0768473	+	0.0443678i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	19.5000	−	33.7750i	0.991241	−	1.71688i
$388$	33.0000	+	19.0526i	1.67532	+	0.967247i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−34.5000	+	19.9186i	−1.73151	+	0.999685i	−0.853206	+	0.521575i	$0.825345\pi$
$397$	−34.5000	+	19.9186i	−1.73151	+	0.999685i	−0.878300	+	0.478110i	$0.841322\pi$
$398$		0			0
$399$	−15.0000	+	5.19615i	−0.750939	+	0.260133i
$400$	10.0000	−	17.3205i	0.500000	−	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	22.5000	+	21.6506i	1.12080	+	1.07849i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−22.5000	+	12.9904i	−1.11255	+	0.642333i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−22.5000	+	12.9904i	−1.11255	+	0.642333i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$	−27.0000	+	15.5885i	−1.33019	+	0.767988i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	39.0000			1.90984
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−15.0000	+	17.3205i	−0.725901	+	0.838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	18.0000	+	10.3923i	0.866025	+	0.500000i
$433$	−19.5000	+	11.2583i	−0.937110	+	0.541041i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−19.5000	+	11.2583i	−0.937110	+	0.541041i	−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$	−19.0000	−	32.9090i	−0.909935	−	1.57605i
$437$		0			0
$438$		0			0
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	19.5000	+	7.79423i	0.928571	+	0.371154i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−30.0000	−	17.3205i	−1.42374	−	0.821995i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−20.0000	+	6.92820i	−0.944911	+	0.327327i
$449$		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
$453$	−6.00000	−	3.46410i	−0.281905	−	0.162758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.5000	−	35.5070i	0.958950	−	1.66095i	0.233890	−	0.972263i	$-0.424854\pi$
$457$	20.5000	−	35.5070i	0.958950	−	1.66095i	0.725059	−	0.688686i	$-0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	−43.0000			−1.99838			−0.999190	−	0.0402476i	$-0.987185\pi$
$463$	−43.0000			−1.99838			−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	6.00000	+	20.7846i	0.277350	+	0.960769i
$469$	5.50000	−	28.5788i	0.253966	−	1.31965i
$470$		0			0
$471$	19.5000	−	33.7750i	0.898513	−	1.55627i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	15.0000	−	8.66025i	0.688247	−	0.397360i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−10.0000	−	34.6410i	−0.455961	−	1.57949i
$482$		0			0
$483$		0			0
$484$	−11.0000	+	19.0526i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	22.0000	+	38.1051i	0.996915	+	1.72671i	0.566429	+	0.824110i	$0.308325\pi$
$487$	22.0000	+	38.1051i	0.996915	+	1.72671i	0.430486	+	0.902597i	$0.358342\pi$
$488$		0			0
$489$		−	43.3013i		−	1.95815i
$490$		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
$496$	30.0000	+	17.3205i	1.34704	+	0.777714i
$497$		0			0
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.5000	+	19.9186i	−0.466321	+	0.884615i
$508$	−2.00000			−0.0887357
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	1.50000	+	4.33013i	0.0663561	+	0.191554i
$512$		0			0
$513$	9.00000	+	15.5885i	0.397360	+	0.688247i
$514$		0			0
$515$		0			0
$516$	39.0000	−	22.5167i	1.71688	−	0.991241i
$517$		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
$523$	−39.0000	−	22.5167i	−1.70535	−	0.984585i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−39.0000	−	22.5167i	−1.70535	−	0.984585i	−0.765222	−	0.643767i	$-0.777371\pi$
$524$		0			0
$525$	−22.5000	−	4.33013i	−0.981981	−	0.188982i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−18.0000	−	3.46410i	−0.780399	−	0.150188i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	−6.00000	+	10.3923i	−0.257485	+	0.445976i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	41.0000			1.75303			0.876517	−	0.481371i	$-0.159861\pi$
$547$	41.0000			1.75303			0.876517	+	0.481371i	$0.159861\pi$
$548$		0			0
$549$	22.5000	+	12.9904i	0.960277	+	0.554416i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	32.5000	−	11.2583i	1.38204	−	0.478753i
$554$		0			0
$555$		0			0
$556$	39.0000	+	22.5167i	1.65397	+	0.954919i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	45.5000	+	11.2583i	1.92444	+	0.476177i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.50000	−	23.3827i	0.188982	−	0.981981i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−16.0000			−0.669579			−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000			−0.669579			−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	+	20.7846i	0.500000	+	0.866025i
$577$		−	13.8564i		−	0.576850i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$		−	13.8564i		−	0.576850i	0.957503	−	0.288425i	$-0.0931316\pi$
$578$		0			0
$579$	−34.5000	+	19.9186i	−1.43377	+	0.827788i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$	15.0000	+	19.0526i	0.618590	+	0.785714i
$589$	15.0000	+	25.9808i	0.618064	+	1.07052i
$590$		0			0
$591$		0			0
$592$	−20.0000	−	34.6410i	−0.821995	−	1.42374i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−39.0000			−1.59616
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−36.0000	−	20.7846i	−1.46847	−	0.847822i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−36.0000	−	20.7846i	−1.46847	−	0.847822i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$	−33.0000			−1.34386
$604$	−4.00000	−	6.92820i	−0.162758	−	0.281905i
$605$		0			0
$606$		0			0
$607$	39.0000	−	22.5167i	1.58296	−	0.913923i	0.588537	−	0.808470i	$-0.299704\pi$
$607$	39.0000	−	22.5167i	1.58296	−	0.913923i	0.994424	−	0.105453i	$-0.0336291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	18.5000	−	32.0429i	0.747208	−	1.29420i	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	18.5000	−	32.0429i	0.747208	−	1.29420i	0.949156	−	0.314806i	$-0.101939\pi$
$614$		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
$619$			46.7654i			1.87966i	0.341644	+	0.939829i	$0.389016\pi$
$619$			46.7654i			1.87966i	−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
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$	39.0000	−	22.5167i	1.55627	−	0.898513i
$629$		0			0
$630$		0			0
$631$	0.500000	+	0.866025i	0.0199047	+	0.0344759i	0.875806	−	0.482663i	$-0.160330\pi$
$631$	0.500000	+	0.866025i	0.0199047	+	0.0344759i	−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$	19.5000	−	11.2583i	0.775055	−	0.447478i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.50000	+	25.1147i	−0.0990536	+	0.995082i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−16.5000	+	9.52628i	−0.650696	+	0.375680i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−16.5000	+	9.52628i	−0.650696	+	0.375680i	0.138027	+	0.990429i	$0.455924\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	7.50000	−	38.9711i	0.293948	−	1.52740i
$652$	25.0000	−	43.3013i	0.979076	−	1.69581i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	4.50000	−	2.59808i	0.175562	−	0.101361i
$658$		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
$661$	−13.5000	−	7.79423i	−0.525089	−	0.303160i	0.213925	−	0.976850i	$-0.431375\pi$
$661$	−13.5000	−	7.79423i	−0.525089	−	0.303160i	−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−9.00000	−	15.5885i	−0.347960	−	0.602685i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−6.50000	+	11.2583i	−0.250557	+	0.433977i	−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−6.50000	+	11.2583i	−0.250557	+	0.433977i	0.713123	+	0.701039i	$0.247280\pi$
$674$		0			0
$675$			25.9808i			1.00000i
$676$	−22.0000	+	13.8564i	−0.846154	+	0.532939i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	33.0000	−	38.1051i	1.26642	−	1.46234i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$			20.7846i			0.794719i
$685$		0			0
$686$		0			0
$687$	18.0000	−	31.1769i	0.686743	−	1.18947i
$688$	52.0000			1.98248
$689$		0			0
$690$		0			0
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−20.0000	−	17.3205i	−0.755929	−	0.654654i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		−	34.6410i		−	1.30651i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.5000	+	45.8993i	0.995228	+	1.72379i	0.582115	+	0.813107i	$0.302225\pi$
$709$	26.5000	+	45.8993i	0.995228	+	1.72379i	0.413114	+	0.910679i	$0.364441\pi$
$710$		0			0
$711$	−19.5000	−	33.7750i	−0.731307	−	1.26666i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	13.5000	+	38.9711i	0.502766	+	1.45136i
$722$		0			0
$723$	−48.0000			−1.78514
$724$	−12.0000	+	6.92820i	−0.445976	+	0.257485i
$725$		0			0
$726$		0			0
$727$		−	22.5167i		−	0.835097i	−0.908655	−	0.417548i	$-0.862889\pi$
$727$		−	22.5167i		−	0.835097i	0.908655	−	0.417548i	$-0.137111\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	15.0000	+	25.9808i	0.554416	+	0.960277i
$733$		−	53.6936i		−	1.98322i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$		−	53.6936i		−	1.98322i	0.129275	−	0.991609i	$-0.458735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	8.00000	−	13.8564i	0.294285	−	0.509716i	0.974818	+	0.223001i	$0.0715853\pi$
$740$		0			0
$741$	−15.0000	+	15.5885i	−0.551039	+	0.572656i
$742$		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
$751$	−26.0000	+	45.0333i	−0.948753	+	1.64329i	−0.200698	+	0.979653i	$0.564321\pi$
$751$	−26.0000	+	45.0333i	−0.948753	+	1.64329i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	18.0000	−	20.7846i	0.654654	−	0.755929i
$757$	13.0000	−	22.5167i	0.472493	−	0.818382i	−0.527011	−	0.849858i	$-0.676688\pi$
$757$	13.0000	−	22.5167i	0.472493	−	0.818382i	0.999505	+	0.0314762i	$0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−47.5000	+	16.4545i	−1.71962	+	0.595692i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$			27.7128i			1.00000i
$769$	48.0000	+	27.7128i	1.73092	+	0.999350i	0.883493	+	0.468445i	$0.155186\pi$
$769$	48.0000	+	27.7128i	1.73092	+	0.999350i	0.847432	+	0.530904i	$0.178148\pi$
$770$		0			0
$771$		0			0
$772$	−46.0000			−1.65558
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$			43.3013i			1.55543i
$776$		0			0
$777$	−30.0000	+	34.6410i	−1.07624	+	1.24274i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	4.00000	+	27.7128i	0.142857	+	0.989743i
$785$		0			0
$786$		0			0
$787$	−43.5000	+	25.1147i	−1.55061	+	0.895244i	−0.552515	+	0.833503i	$0.686332\pi$
$787$	−43.5000	+	25.1147i	−1.55061	+	0.895244i	−0.998092	+	0.0617409i	$0.980335\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.50000	+	30.3109i	−0.266333	+	1.07637i
$794$		0			0
$795$		0			0
$796$	−39.0000	−	22.5167i	−1.38232	−	0.798082i
$797$		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
$804$	−33.0000	−	19.0526i	−1.16382	−	0.671932i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$			53.6936i			1.88544i	0.333590	+	0.942718i	$0.391740\pi$
$811$			53.6936i			1.88544i	−0.333590	+	0.942718i	$0.608260\pi$
$812$		0			0
$813$	−13.5000	−	23.3827i	−0.473466	−	0.820067i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	39.0000	+	22.5167i	1.36444	+	0.787758i
$818$		0			0
$819$	28.5000	−	2.59808i	0.995871	−	0.0907841i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	26.0000	+	45.0333i	0.906303	+	1.56976i	0.819159	+	0.573567i	$0.194441\pi$
$823$	26.0000	+	45.0333i	0.906303	+	1.56976i	0.0871445	+	0.996196i	$0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−19.5000	−	11.2583i	−0.677263	−	0.391018i	0.121560	−	0.992584i	$-0.461210\pi$
$829$	−19.5000	−	11.2583i	−0.677263	−	0.391018i	−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$			45.0333i			1.56219i
$832$	−20.0000	+	20.7846i	−0.693375	+	0.720577i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−45.0000			−1.55543
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	26.0000			0.894957
$845$		0			0
$846$		0			0
$847$	22.0000	+	19.0526i	0.755929	+	0.654654i
$848$		0			0
$849$	19.5000	+	33.7750i	0.669238	+	1.15915i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	46.7654i		−	1.60122i	−0.599189	−	0.800608i	$-0.704510\pi$
$853$		−	46.7654i		−	1.60122i	0.599189	−	0.800608i	$-0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			57.1577i			1.95019i	0.221777	+	0.975097i	$0.428814\pi$
$859$			57.1577i			1.95019i	−0.221777	+	0.975097i	$0.571186\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−25.5000	+	14.7224i	−0.866025	+	0.500000i
$868$	30.0000	−	34.6410i	1.01827	−	1.17579i
$869$		0			0
$870$		0			0
$871$	−11.0000	−	38.1051i	−0.372721	−	1.29114i
$872$		0			0
$873$	−49.5000	−	28.5788i	−1.67532	−	0.967247i
$874$		0			0
$875$		0			0
$876$	6.00000			0.202721
$877$	17.0000	+	29.4449i	0.574049	+	0.994282i	0.996144	+	0.0877308i	$0.0279615\pi$
$877$	17.0000	+	29.4449i	0.574049	+	0.994282i	−0.422095	+	0.906552i	$0.638705\pi$
$878$		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
$883$	55.0000			1.85090			0.925449	−	0.378873i	$-0.123688\pi$
$883$	55.0000			1.85090			0.925449	+	0.378873i	$0.123688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−0.500000	+	2.59808i	−0.0167695	+	0.0871367i
$890$		0			0
$891$		0			0
$892$		−	20.7846i		−	0.695920i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−15.0000	+	25.9808i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−19.5000	−	56.2917i	−0.648919	−	1.87327i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	0.315442	+	0.948945i	$0.397847\pi$
$907$	−20.0000	+	34.6410i	−0.664089	+	1.15024i	−0.979531	+	0.201291i	$0.935486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−12.0000	+	20.7846i	−0.397360	+	0.688247i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	36.0000	−	20.7846i	1.18947	−	0.686743i
$917$		0			0
$918$		0			0
$919$	−26.0000	−	45.0333i	−0.857661	−	1.48551i	−0.874154	−	0.485648i	$-0.838584\pi$
$919$	−26.0000	−	45.0333i	−0.857661	−	1.48551i	0.0164935	−	0.999864i	$-0.494750\pi$
$920$		0			0
$921$	1.50000	−	2.59808i	0.0494267	−	0.0856095i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	25.0000	−	43.3013i	0.821995	−	1.42374i
$926$		0			0
$927$	40.5000	−	23.3827i	1.33019	−	0.767988i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−9.00000	+	22.5167i	−0.294963	+	0.737954i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	55.4256i		−	1.81068i	−0.424691	−	0.905338i	$-0.639617\pi$
$937$		−	55.4256i		−	1.81068i	0.424691	−	0.905338i	$-0.360383\pi$
$938$		0			0
$939$	28.5000	−	49.3634i	0.930062	−	1.61092i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		−	45.0333i		−	1.46261i
$949$	4.50000	+	4.33013i	0.146076	+	0.140562i
$950$		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
$961$	−44.0000			−1.41935
$962$		0			0
$963$		0			0
$964$	−48.0000	−	27.7128i	−1.54598	−	0.892570i
$965$		0			0
$966$		0			0
$967$	20.0000			0.643157			0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000			0.643157			0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$	−27.0000	−	15.5885i	−0.866025	−	0.500000i
$973$	39.0000	−	45.0333i	1.25028	−	1.44370i
$974$		0			0
$975$	−30.0000	+	8.66025i	−0.960769	+	0.277350i
$976$			34.6410i			1.10883i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	28.5000	+	49.3634i	0.909935	+	1.57605i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−24.0000	+	6.92820i	−0.763542	+	0.220416i
$989$		0			0
$990$		0			0
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	22.0000	−	38.1051i	0.698853	−	1.21045i	0.968864	−	0.247592i	$-0.0796392\pi$
$992$		0			0
$993$			53.6936i			1.70391i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	45.0000	+	25.9808i	1.42374	+	0.821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bn.a.230.1	yes	2
3.2	odd	2	CM	273.2.bn.a.230.1	yes	2
7.6	odd	2		273.2.bn.b.230.1	yes	2
13.3	even	3		273.2.bn.b.146.1	yes	2
21.20	even	2		273.2.bn.b.230.1	yes	2
39.29	odd	6		273.2.bn.b.146.1	yes	2
91.55	odd	6	inner	273.2.bn.a.146.1	✓	2
273.146	even	6	inner	273.2.bn.a.146.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.bn.a.146.1	✓	2	91.55	odd	6	inner
273.2.bn.a.146.1	✓	2	273.146	even	6	inner
273.2.bn.a.230.1	yes	2	1.1	even	1	trivial
273.2.bn.a.230.1	yes	2	3.2	odd	2	CM
273.2.bn.b.146.1	yes	2	13.3	even	3
273.2.bn.b.146.1	yes	2	39.29	odd	6
273.2.bn.b.230.1	yes	2	7.6	odd	2
273.2.bn.b.230.1	yes	2	21.20	even	2

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.bn (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 + 0.866025i) q^{7} +(1.50000 + 2.59808i) q^{9} +3.46410i q^{12} +(-2.50000 + 2.59808i) q^{13} +(-2.00000 + 3.46410i) q^{16} +(-3.00000 + 1.73205i) q^{19} +(4.50000 + 0.866025i) q^{21} -5.00000 q^{25} -5.19615i q^{27} +(4.00000 + 3.46410i) q^{28} -8.66025i q^{31} +(3.00000 - 5.19615i) q^{36} +(-5.00000 + 8.66025i) q^{37} +(6.00000 - 1.73205i) q^{39} +(-6.50000 - 11.2583i) q^{43} +(6.00000 - 3.46410i) q^{48} +(5.50000 - 4.33013i) q^{49} +(7.00000 + 1.73205i) q^{52} +6.00000 q^{57} +(7.50000 - 4.33013i) q^{61} +(-6.00000 - 5.19615i) q^{63} +8.00000 q^{64} +(-5.50000 + 9.52628i) q^{67} -1.73205i q^{73} +(7.50000 + 4.33013i) q^{75} +(6.00000 + 3.46410i) q^{76} -13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(4.00000 - 8.66025i) q^{91} +(-7.50000 + 12.9904i) q^{93} +(-16.5000 + 9.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 2 q^{4} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 2 * q^4 - 5 * q^7 + 3 * q^9	\( 2 q - 3 q^{3} - 2 q^{4} - 5 q^{7} + 3 q^{9} - 5 q^{13} - 4 q^{16} - 6 q^{19} + 9 q^{21} - 10 q^{25} + 8 q^{28} + 6 q^{36} - 10 q^{37} + 12 q^{39} - 13 q^{43} + 12 q^{48} + 11 q^{49} + 14 q^{52} + 12 q^{57} + 15 q^{61} - 12 q^{63} + 16 q^{64} - 11 q^{67} + 15 q^{75} + 12 q^{76} - 26 q^{79} - 9 q^{81} - 6 q^{84} + 8 q^{91} - 15 q^{93} - 33 q^{97}+O(q^{100}) \) 2 * q - 3 * q^3 - 2 * q^4 - 5 * q^7 + 3 * q^9 - 5 * q^13 - 4 * q^16 - 6 * q^19 + 9 * q^21 - 10 * q^25 + 8 * q^28 + 6 * q^36 - 10 * q^37 + 12 * q^39 - 13 * q^43 + 12 * q^48 + 11 * q^49 + 14 * q^52 + 12 * q^57 + 15 * q^61 - 12 * q^63 + 16 * q^64 - 11 * q^67 + 15 * q^75 + 12 * q^76 - 26 * q^79 - 9 * q^81 - 6 * q^84 + 8 * q^91 - 15 * q^93 - 33 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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