LMFDB - Embedded newform 273.2.u.a.62.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(62,273)]

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, 5]))

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

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

chi := DirichletCharacter("273.62");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.u (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:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			62.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	273.62
Dual form			273.2.u.a.251.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-2.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	$q+(-1.50000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-2.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +3.46410i q^{12} +(-2.50000 - 2.59808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.00000 - 6.92820i) q^{19} +(4.50000 + 0.866025i) q^{21} +5.00000 q^{25} +5.19615i q^{27} +(-5.00000 + 1.73205i) q^{28} -7.00000 q^{31} +(-3.00000 - 5.19615i) q^{36} +(6.00000 - 3.46410i) q^{37} +(6.00000 + 1.73205i) q^{39} +(-6.50000 + 11.2583i) q^{43} +(6.00000 + 3.46410i) q^{48} +(1.00000 + 6.92820i) q^{49} +(-7.00000 + 1.73205i) q^{52} +13.8564i q^{57} +(7.50000 + 4.33013i) q^{61} +(-7.50000 + 2.59808i) q^{63} -8.00000 q^{64} +(-10.5000 + 6.06218i) q^{67} +17.0000 q^{73} +(-7.50000 + 4.33013i) q^{75} +(-8.00000 - 13.8564i) q^{76} +13.0000 q^{79} +(-4.50000 - 7.79423i) q^{81} +(6.00000 - 6.92820i) q^{84} +(0.500000 + 9.52628i) q^{91} +(10.5000 - 6.06218i) q^{93} +(2.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 2 q^{4} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 + 2 * q^4 - 4 * q^7 + 3 * q^9	$ 2 q - 3 q^{3} + 2 q^{4} - 4 q^{7} + 3 q^{9} - 5 q^{13} - 4 q^{16} + 8 q^{19} + 9 q^{21} + 10 q^{25} - 10 q^{28} - 14 q^{31} - 6 q^{36} + 12 q^{37} + 12 q^{39} - 13 q^{43} + 12 q^{48} + 2 q^{49} - 14 q^{52} + 15 q^{61} - 15 q^{63} - 16 q^{64} - 21 q^{67} + 34 q^{73} - 15 q^{75} - 16 q^{76} + 26 q^{79} - 9 q^{81} + 12 q^{84} + q^{91} + 21 q^{93} + 5 q^{97}+O(q^{100}) $ 2 * q - 3 * q^3 + 2 * q^4 - 4 * q^7 + 3 * q^9 - 5 * q^13 - 4 * q^16 + 8 * q^19 + 9 * q^21 + 10 * q^25 - 10 * q^28 - 14 * q^31 - 6 * q^36 + 12 * q^37 + 12 * q^39 - 13 * q^43 + 12 * q^48 + 2 * q^49 - 14 * q^52 + 15 * q^61 - 15 * q^63 - 16 * q^64 - 21 * q^67 + 34 * q^73 - 15 * q^75 - 16 * q^76 + 26 * q^79 - 9 * q^81 + 12 * q^84 + q^91 + 21 * q^93 + 5 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$92$	$106$	$157$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$	−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		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$			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	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	4.00000	−	6.92820i	0.917663	−	1.58944i	0.114708	−	0.993399i	$-0.463407\pi$
$19$	4.00000	−	6.92820i	0.917663	−	1.58944i	0.802955	−	0.596040i	$-0.203260\pi$
$20$		0			0
$21$	4.50000	+	0.866025i	0.981981	+	0.188982i
$22$		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
$25$	5.00000			1.00000
$26$		0			0
$27$			5.19615i			1.00000i
$28$	−5.00000	+	1.73205i	−0.944911	+	0.327327i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.00000	−	5.19615i	−0.500000	−	0.866025i
$37$	6.00000	−	3.46410i	0.986394	−	0.569495i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	6.00000	−	3.46410i	0.986394	−	0.569495i	0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$	6.00000	+	1.73205i	0.960769	+	0.277350i
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	−6.50000	+	11.2583i	−0.991241	+	1.71688i	−0.381246	+	0.924473i	$0.624505\pi$
$43$	−6.50000	+	11.2583i	−0.991241	+	1.71688i	−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	6.00000	+	3.46410i	0.866025	+	0.500000i
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		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
$57$			13.8564i			1.83533i
$58$		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
$61$	7.50000	+	4.33013i	0.960277	+	0.554416i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	7.50000	+	4.33013i	0.960277	+	0.554416i	0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$	−7.50000	+	2.59808i	−0.944911	+	0.327327i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−10.5000	+	6.06218i	−1.28278	+	0.740613i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−10.5000	+	6.06218i	−1.28278	+	0.740613i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	17.0000			1.98970			0.994850	−	0.101361i	$-0.0323196\pi$
$73$	17.0000			1.98970			0.994850	+	0.101361i	$0.0323196\pi$
$74$		0			0
$75$	−7.50000	+	4.33013i	−0.866025	+	0.500000i
$76$	−8.00000	−	13.8564i	−0.917663	−	1.58944i
$77$		0			0
$78$		0			0
$79$	13.0000			1.46261			0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000			1.46261			0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	6.00000	−	6.92820i	0.654654	−	0.755929i
$85$		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
$91$	0.500000	+	9.52628i	0.0524142	+	0.998625i
$92$		0			0
$93$	10.5000	−	6.06218i	1.08880	−	0.628619i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	0.964579	+	0.263795i	$0.0849741\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	−	8.66025i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		−	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	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	9.00000	+	5.19615i	0.866025	+	0.500000i
$109$		−	8.66025i		−	0.829502i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$		−	8.66025i		−	0.829502i	0.909935	−	0.414751i	$-0.136131\pi$
$110$		0			0
$111$	−6.00000	+	10.3923i	−0.569495	+	0.986394i
$112$	−2.00000	+	10.3923i	−0.188982	+	0.981981i
$113$		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
$117$	−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
$124$	−7.00000	+	12.1244i	−0.628619	+	1.08880i
$125$		0			0
$126$		0			0
$127$	−0.500000	−	0.866025i	−0.0443678	−	0.0768473i	0.842989	−	0.537931i	$-0.180794\pi$
$127$	−0.500000	−	0.866025i	−0.0443678	−	0.0768473i	−0.887357	+	0.461084i	$0.847461\pi$
$128$		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
$133$	−20.0000	+	6.92820i	−1.73422	+	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	19.5000	+	11.2583i	1.65397	+	0.954919i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	19.5000	+	11.2583i	1.65397	+	0.954919i	0.678551	+	0.734553i	$0.262608\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$	−7.50000	−	9.52628i	−0.618590	−	0.785714i
$148$		−	13.8564i		−	1.13899i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$		−	24.2487i		−	1.97333i	−0.162758	−	0.986666i	$-0.552039\pi$
$151$		−	24.2487i		−	1.97333i	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
$163$	−4.50000	−	2.59808i	−0.352467	−	0.203497i	0.313304	−	0.949653i	$-0.398564\pi$
$163$	−4.50000	−	2.59808i	−0.352467	−	0.203497i	−0.665771	+	0.746156i	$0.731897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−0.500000	+	12.9904i	−0.0384615	+	0.999260i
$170$		0			0
$171$	−12.0000	−	20.7846i	−0.917663	−	1.58944i
$172$	13.0000	+	22.5167i	0.991241	+	1.71688i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		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
$189$	9.00000	−	10.3923i	0.654654	−	0.755929i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	12.0000	−	6.92820i	0.866025	−	0.500000i
$193$	13.5000	−	7.79423i	0.971751	−	0.561041i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	13.5000	−	7.79423i	0.971751	−	0.561041i	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		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$	19.5000	+	11.2583i	1.38232	+	0.798082i	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	19.5000	+	11.2583i	1.38232	+	0.798082i	0.389885	+	0.920864i	$0.372515\pi$
$200$		0			0
$201$	10.5000	−	18.1865i	0.740613	−	1.28278i
$202$		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	$-0.0189888\pi$
$211$	6.50000	+	11.2583i	0.447478	+	0.775055i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	14.0000	+	12.1244i	0.950382	+	0.823055i
$218$		0			0
$219$	−25.5000	+	14.7224i	−1.72313	+	0.994850i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−14.0000	−	24.2487i	−0.937509	−	1.62381i	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−14.0000	−	24.2487i	−0.937509	−	1.62381i	−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		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	24.0000	+	13.8564i	1.58944	+	0.917663i
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−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			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−7.00000	+	12.1244i	−0.450910	+	0.780998i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−7.00000	+	12.1244i	−0.450910	+	0.780998i	0.547533	+	0.836784i	$0.315567\pi$
$242$		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
$247$	−28.0000	+	6.92820i	−1.78160	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$	−3.00000	+	15.5885i	−0.188982	+	0.981981i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−18.0000	−	3.46410i	−1.11847	−	0.215249i
$260$		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
$268$			24.2487i			1.48123i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	14.5000	+	25.1147i	0.880812	+	1.52561i	0.850439	+	0.526073i	$0.176336\pi$
$271$	14.5000	+	25.1147i	0.880812	+	1.52561i	0.0303728	+	0.999539i	$0.490331\pi$
$272$		0			0
$273$	−9.00000	−	13.8564i	−0.544705	−	0.838628i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.0000	+	22.5167i	−0.781094	+	1.35290i	0.150210	+	0.988654i	$0.452005\pi$
$277$	−13.0000	+	22.5167i	−0.781094	+	1.35290i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$	−10.5000	+	18.1865i	−0.628619	+	1.08880i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	−19.5000	+	11.2583i	−1.15915	+	0.669238i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−19.5000	+	11.2583i	−1.15915	+	0.669238i	−0.208053	+	0.978117i	$0.566713\pi$
$284$		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
$291$			8.66025i			0.507673i
$292$	17.0000	−	29.4449i	0.994850	−	1.72313i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$			17.3205i			1.00000i
$301$	32.5000	−	11.2583i	1.87327	−	0.648919i
$302$		0			0
$303$		0			0
$304$	−32.0000			−1.83533
$305$		0			0
$306$		0			0
$307$	35.0000			1.99756			0.998778	−	0.0494267i	$-0.0157394\pi$
$307$	35.0000			1.99756			0.998778	+	0.0494267i	$0.0157394\pi$
$308$		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	$-0.619753\pi$
$313$		−	32.9090i		−	1.86012i	0.367402	−	0.930062i	$-0.380247\pi$
$314$		0			0
$315$		0			0
$316$	13.0000	−	22.5167i	0.731307	−	1.26666i
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		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
$327$	7.50000	+	12.9904i	0.414751	+	0.718370i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−16.5000	−	9.52628i	−0.906922	−	0.523612i	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−16.5000	−	9.52628i	−0.906922	−	0.523612i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$		−	20.7846i		−	1.13899i
$334$		0			0
$335$		0			0
$336$	−6.00000	−	17.3205i	−0.327327	−	0.944911i
$337$	29.0000			1.57973			0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000			1.57973			0.789865	+	0.613280i	$0.210150\pi$
$338$		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	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	0.374701	−	0.927146i	$-0.377745\pi$
$350$		0			0
$351$	13.5000	−	12.9904i	0.720577	−	0.693375i
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−22.5000	−	38.9711i	−1.18421	−	2.05111i
$362$		0			0
$363$			19.0526i			1.00000i
$364$	17.0000	+	8.66025i	0.891042	+	0.453921i
$365$		0			0
$366$		0			0
$367$	19.5000	−	11.2583i	1.01789	−	0.587680i	0.104399	−	0.994535i	$-0.466708\pi$
$367$	19.5000	−	11.2583i	1.01789	−	0.587680i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		−	24.2487i		−	1.25724i
$373$	−6.50000	+	11.2583i	−0.336557	+	0.582934i	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−6.50000	+	11.2583i	−0.336557	+	0.582934i	0.647225	+	0.762299i	$0.275929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−10.5000	+	6.06218i	−0.539349	+	0.311393i	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−10.5000	+	6.06218i	−0.539349	+	0.311393i	0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$	1.50000	+	0.866025i	0.0768473	+	0.0443678i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	19.5000	+	33.7750i	0.991241	+	1.71688i
$388$	−5.00000	−	8.66025i	−0.253837	−	0.439658i
$389$		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
$397$	−0.500000	+	0.866025i	−0.0250943	+	0.0434646i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$	−0.500000	+	0.866025i	−0.0250943	+	0.0434646i	0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$	24.0000	−	27.7128i	1.20150	−	1.38738i
$400$	−10.0000	−	17.3205i	−0.500000	−	0.866025i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	17.5000	+	18.1865i	0.871737	+	0.905936i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−15.5000	+	26.8468i	−0.766426	+	1.32749i	0.173064	+	0.984911i	$0.444633\pi$
$409$	−15.5000	+	26.8468i	−0.766426	+	1.32749i	−0.939490	+	0.342578i	$0.888700\pi$
$410$		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	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$		−	36.3731i		−	1.77271i	−0.463002	−	0.886357i	$-0.653228\pi$
$421$		−	36.3731i		−	1.77271i	0.463002	−	0.886357i	$-0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.50000	−	21.6506i	−0.362950	−	1.04775i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	18.0000	−	10.3923i	0.866025	−	0.500000i
$433$	−19.5000	−	11.2583i	−0.937110	−	0.541041i	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−19.5000	−	11.2583i	−0.937110	−	0.541041i	−0.889053	+	0.457804i	$0.848636\pi$
$434$		0			0
$435$		0			0
$436$	−15.0000	−	8.66025i	−0.718370	−	0.414751i
$437$		0			0
$438$		0			0
$439$	−34.5000	+	19.9186i	−1.64660	+	0.950662i	−0.668184	+	0.743996i	$0.732928\pi$
$439$	−34.5000	+	19.9186i	−1.64660	+	0.950662i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		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			$\pi$
$444$	12.0000	+	20.7846i	0.569495	+	0.986394i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	16.0000	+	13.8564i	0.755929	+	0.654654i
$449$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$449$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	21.0000	+	36.3731i	0.986666	+	1.70896i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.5000	+	6.06218i	−0.491169	+	0.283577i	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−10.5000	+	6.06218i	−0.491169	+	0.283577i	0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$			1.73205i			0.0804952i	0.999190	+	0.0402476i	$0.0128147\pi$
$463$			1.73205i			0.0804952i	−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
$469$	31.5000	+	6.06218i	1.45453	+	0.279925i
$470$		0			0
$471$	−19.5000	−	33.7750i	−0.898513	−	1.55627i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	20.0000	−	34.6410i	0.917663	−	1.58944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	−24.0000	−	6.92820i	−1.09431	−	0.315899i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−3.00000	−	1.73205i	−0.135943	−	0.0784867i	0.430486	−	0.902597i	$-0.358342\pi$
$487$	−3.00000	−	1.73205i	−0.135943	−	0.0784867i	−0.566429	+	0.824110i	$0.691675\pi$
$488$		0			0
$489$	9.00000			0.406994
$490$		0			0
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	14.0000	+	24.2487i	0.628619	+	1.08880i
$497$		0			0
$498$		0			0
$499$			31.1769i			1.39567i	0.716258	+	0.697835i	$0.245853\pi$
$499$			31.1769i			1.39567i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		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
$507$	−10.5000	−	19.9186i	−0.466321	−	0.884615i
$508$	−2.00000			−0.0887357
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−34.0000	−	29.4449i	−1.50407	−	1.30256i
$512$		0			0
$513$	36.0000	+	20.7846i	1.58944	+	0.917663i
$514$		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	0.765222	−	0.643767i	$-0.222629\pi$
$523$	39.0000	−	22.5167i	1.70535	−	0.984585i	0.940129	−	0.340818i	$-0.110704\pi$
$524$		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
$532$	−8.00000	+	41.5692i	−0.346844	+	1.80225i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$			43.3013i			1.86167i	0.365444	+	0.930834i	$0.380917\pi$
$541$			43.3013i			1.86167i	−0.365444	+	0.930834i	$0.619083\pi$
$542$		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
$553$	−26.0000	−	22.5167i	−1.10563	−	0.957506i
$554$		0			0
$555$		0			0
$556$	39.0000	−	22.5167i	1.65397	−	0.954919i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		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	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		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	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−12.0000	+	20.7846i	−0.500000	+	0.866025i
$577$	46.0000			1.91501			0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000			1.91501			0.957503	+	0.288425i	$0.0931316\pi$
$578$		0			0
$579$	−13.5000	+	23.3827i	−0.561041	+	0.971751i
$580$		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	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$	−24.0000	+	3.46410i	−0.989743	+	0.142857i
$589$	−28.0000	+	48.4974i	−1.15372	+	1.99830i
$590$		0			0
$591$		0			0
$592$	−24.0000	−	13.8564i	−0.986394	−	0.569495i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		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	−0.999376	−	0.0353259i	$-0.988753\pi$
$601$	−36.0000	+	20.7846i	−1.46847	+	0.847822i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$			36.3731i			1.48123i
$604$	−42.0000	−	24.2487i	−1.70896	−	0.986666i
$605$		0			0
$606$		0			0
$607$	−39.0000	−	22.5167i	−1.58296	−	0.913923i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−39.0000	−	22.5167i	−1.58296	−	0.913923i	−0.588537	−	0.808470i	$-0.700296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−28.5000	+	16.4545i	−1.15110	+	0.664590i	−0.949156	−	0.314806i	$-0.898061\pi$
$613$	−28.5000	+	16.4545i	−1.15110	+	0.664590i	−0.201948	+	0.979396i	$0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$	−17.0000			−0.683288			−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000			−0.683288			−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
$631$	43.5000	+	25.1147i	1.73171	+	0.999802i	0.875806	+	0.482663i	$0.160330\pi$
$631$	43.5000	+	25.1147i	1.73171	+	0.999802i	0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$	−19.5000	−	11.2583i	−0.775055	−	0.447478i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.5000	−	19.9186i	0.614132	−	0.789203i
$638$		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
$643$	−23.5000	+	40.7032i	−0.926750	+	1.60518i	−0.138027	+	0.990429i	$0.544076\pi$
$643$	−23.5000	+	40.7032i	−0.926750	+	1.60518i	−0.788723	+	0.614749i	$0.789257\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−31.5000	−	6.06218i	−1.23458	−	0.237595i
$652$	−9.00000	+	5.19615i	−0.352467	+	0.203497i
$653$		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
$657$	25.5000	−	44.1673i	0.994850	−	1.72313i
$658$		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
$661$	24.5000	+	42.4352i	0.952940	+	1.65054i	0.739014	+	0.673690i	$0.235292\pi$
$661$	24.5000	+	42.4352i	0.952940	+	1.65054i	0.213925	+	0.976850i	$0.431375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	42.0000	+	24.2487i	1.62381	+	0.937509i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−6.50000	−	11.2583i	−0.250557	−	0.433977i	0.713123	−	0.701039i	$-0.247280\pi$
$673$	−6.50000	−	11.2583i	−0.250557	−	0.433977i	−0.963679	+	0.267063i	$0.913947\pi$
$674$		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
$679$	−12.5000	+	4.33013i	−0.479706	+	0.166175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$	−48.0000			−1.83533
$685$		0			0
$686$		0			0
$687$	33.0000	−	19.0526i	1.25903	−	0.726900i
$688$	52.0000			1.98248
$689$		0			0
$690$		0			0
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	0.932024	+	0.362397i	$0.118041\pi$
$691$	20.5000	+	35.5070i	0.779857	+	1.35075i	−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−25.0000	+	8.66025i	−0.944911	+	0.327327i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		−	55.4256i		−	2.09042i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.50000	+	2.59808i	0.169001	+	0.0975728i	0.582115	−	0.813107i	$-0.302225\pi$
$709$	4.50000	+	2.59808i	0.169001	+	0.0975728i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		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	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−27.0000	+	31.1769i	−1.00553	+	1.16109i
$722$		0			0
$723$		−	24.2487i		−	0.901819i
$724$	−12.0000	−	6.92820i	−0.445976	−	0.257485i
$725$		0			0
$726$		0			0
$727$			22.5167i			0.835097i	0.908655	+	0.417548i	$0.137111\pi$
$727$			22.5167i			0.835097i	−0.908655	+	0.417548i	$0.862889\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	−15.0000	+	25.9808i	−0.554416	+	0.960277i
$733$	−7.00000			−0.258551			−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−7.00000			−0.258551			−0.129275	+	0.991609i	$0.541265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	45.0000	−	25.9808i	1.65535	−	0.955718i	0.680534	−	0.732717i	$-0.261748\pi$
$739$	45.0000	−	25.9808i	1.65535	−	0.955718i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$	36.0000	−	34.6410i	1.32249	−	1.27257i
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		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	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	−26.0000	−	45.0333i	−0.948753	−	1.64329i	−0.200698	−	0.979653i	$-0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−9.00000	−	25.9808i	−0.327327	−	0.944911i
$757$	−13.0000	−	22.5167i	−0.472493	−	0.818382i	0.527011	−	0.849858i	$-0.323312\pi$
$757$	−13.0000	−	22.5167i	−0.472493	−	0.818382i	−0.999505	+	0.0314762i	$0.989979\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−15.0000	+	17.3205i	−0.543036	+	0.627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	27.7128i		−	1.00000i
$769$	−1.00000	−	1.73205i	−0.0360609	−	0.0624593i	0.847432	−	0.530904i	$-0.178148\pi$
$769$	−1.00000	−	1.73205i	−0.0360609	−	0.0624593i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$		−	31.1769i		−	1.12208i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	−35.0000			−1.25724
$776$		0			0
$777$	30.0000	−	10.3923i	1.07624	−	0.372822i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	22.0000	−	17.3205i	0.785714	−	0.618590i
$785$		0			0
$786$		0			0
$787$	−12.5000	+	21.6506i	−0.445577	+	0.771762i	−0.998092	−	0.0617409i	$-0.980335\pi$
$787$	−12.5000	+	21.6506i	−0.445577	+	0.771762i	0.552515	+	0.833503i	$0.313668\pi$
$788$		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	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−21.0000	−	36.3731i	−0.740613	−	1.28278i
$805$		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
$811$	−19.0000			−0.667180			−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−19.0000			−0.667180			−0.333590	+	0.942718i	$0.608260\pi$
$812$		0			0
$813$	−43.5000	−	25.1147i	−1.52561	−	0.880812i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	52.0000	+	90.0666i	1.81925	+	3.15103i
$818$		0			0
$819$	25.5000	+	12.9904i	0.891042	+	0.453921i
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$	26.0000	−	45.0333i	0.906303	−	1.56976i	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	26.0000	−	45.0333i	0.906303	−	1.56976i	0.819159	−	0.573567i	$-0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−19.5000	+	11.2583i	−0.677263	+	0.391018i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−19.5000	+	11.2583i	−0.677263	+	0.391018i	0.121560	+	0.992584i	$0.461210\pi$
$830$		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
$837$		−	36.3731i		−	1.25724i
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		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
$847$	−27.5000	+	9.52628i	−0.944911	+	0.327327i
$848$		0			0
$849$	19.5000	−	33.7750i	0.669238	−	1.15915i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−35.0000			−1.19838			−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000			−1.19838			−0.599189	+	0.800608i	$0.704510\pi$
$854$		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	$-0.571186\pi$
$859$		−	57.1577i		−	1.95019i	0.221777	−	0.975097i	$-0.428814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−25.5000	−	14.7224i	−0.866025	−	0.500000i
$868$	35.0000	−	12.1244i	1.18798	−	0.411527i
$869$		0			0
$870$		0			0
$871$	42.0000	+	12.1244i	1.42312	+	0.410818i
$872$		0			0
$873$	−7.50000	−	12.9904i	−0.253837	−	0.439658i
$874$		0			0
$875$		0			0
$876$			58.8897i			1.98970i
$877$	42.0000	+	24.2487i	1.41824	+	0.818821i	0.996144	−	0.0877308i	$-0.0279615\pi$
$877$	42.0000	+	24.2487i	1.41824	+	0.818821i	0.422095	+	0.906552i	$0.361295\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$		0			0
$883$	−55.0000			−1.85090			−0.925449	−	0.378873i	$-0.876312\pi$
$883$	−55.0000			−1.85090			−0.925449	+	0.378873i	$0.876312\pi$
$884$		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
$889$	−0.500000	+	2.59808i	−0.0167695	+	0.0871367i
$890$		0			0
$891$		0			0
$892$	−56.0000			−1.87502
$893$		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
$903$	−39.0000	+	45.0333i	−1.29784	+	1.49862i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	20.0000	+	34.6410i	0.664089	+	1.15024i	0.979531	+	0.201291i	$0.0645138\pi$
$907$	20.0000	+	34.6410i	0.664089	+	1.15024i	−0.315442	+	0.948945i	$0.602153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	48.0000	−	27.7128i	1.58944	−	0.917663i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−22.0000	+	38.1051i	−0.726900	+	1.25903i
$917$		0			0
$918$		0			0
$919$	26.0000	−	45.0333i	0.857661	−	1.48551i	−0.0164935	−	0.999864i	$-0.505250\pi$
$919$	26.0000	−	45.0333i	0.857661	−	1.48551i	0.874154	−	0.485648i	$-0.161416\pi$
$920$		0			0
$921$	−52.5000	+	30.3109i	−1.72993	+	0.998778i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	30.0000	−	17.3205i	0.986394	−	0.569495i
$926$		0			0
$927$	−40.5000	−	23.3827i	−1.33019	−	0.767988i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	52.0000	+	20.7846i	1.70423	+	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			55.4256i			1.81068i	0.424691	+	0.905338i	$0.360383\pi$
$937$			55.4256i			1.81068i	−0.424691	+	0.905338i	$0.639617\pi$
$938$		0			0
$939$	28.5000	+	49.3634i	0.930062	+	1.61092i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$			45.0333i			1.46261i
$949$	−42.5000	−	44.1673i	−1.37961	−	1.43373i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$953$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	18.0000			0.580645
$962$		0			0
$963$		0			0
$964$	14.0000	+	24.2487i	0.450910	+	0.780998i
$965$		0			0
$966$		0			0
$967$		−	58.8897i		−	1.89377i	−0.321578	−	0.946883i	$-0.604213\pi$
$967$		−	58.8897i		−	1.89377i	0.321578	−	0.946883i	$-0.395787\pi$
$968$		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	$0.166667\pi$
$972$	27.0000	−	15.5885i	0.866025	−	0.500000i
$973$	−19.5000	−	56.2917i	−0.625141	−	1.80463i
$974$		0			0
$975$	30.0000	+	8.66025i	0.960769	+	0.277350i
$976$		−	34.6410i		−	1.10883i
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−22.5000	−	12.9904i	−0.718370	−	0.414751i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−16.0000	+	55.4256i	−0.509028	+	1.76332i
$989$		0			0
$990$		0			0
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	0.968864	+	0.247592i	$0.0796392\pi$
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$	33.0000			1.04722
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−19.5000	−	11.2583i	−0.617571	−	0.356555i	0.158352	−	0.987383i	$-0.449382\pi$
$997$	−19.5000	−	11.2583i	−0.617571	−	0.356555i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$	18.0000	+	31.1769i	0.569495	+	0.986394i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.u.a.62.1	✓	2
3.2	odd	2	CM	273.2.u.a.62.1	✓	2
7.6	odd	2		273.2.u.b.62.1	yes	2
13.4	even	6		273.2.u.b.251.1	yes	2
21.20	even	2		273.2.u.b.62.1	yes	2
39.17	odd	6		273.2.u.b.251.1	yes	2
91.69	odd	6	inner	273.2.u.a.251.1	yes	2
273.251	even	6	inner	273.2.u.a.251.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.u.a.62.1	✓	2	1.1	even	1	trivial
273.2.u.a.62.1	✓	2	3.2	odd	2	CM
273.2.u.a.251.1	yes	2	91.69	odd	6	inner
273.2.u.a.251.1	yes	2	273.251	even	6	inner
273.2.u.b.62.1	yes	2	7.6	odd	2
273.2.u.b.62.1	yes	2	21.20	even	2
273.2.u.b.251.1	yes	2	13.4	even	6
273.2.u.b.251.1	yes	2	39.17	odd	6

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

\(f(q)\)	\(=\)	\(q+(-1.50000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-2.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{4} +(-2.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +3.46410i q^{12} +(-2.50000 - 2.59808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.00000 - 6.92820i) q^{19} +(4.50000 + 0.866025i) q^{21} +5.00000 q^{25} +5.19615i q^{27} +(-5.00000 + 1.73205i) q^{28} -7.00000 q^{31} +(-3.00000 - 5.19615i) q^{36} +(6.00000 - 3.46410i) q^{37} +(6.00000 + 1.73205i) q^{39} +(-6.50000 + 11.2583i) q^{43} +(6.00000 + 3.46410i) q^{48} +(1.00000 + 6.92820i) q^{49} +(-7.00000 + 1.73205i) q^{52} +13.8564i q^{57} +(7.50000 + 4.33013i) q^{61} +(-7.50000 + 2.59808i) q^{63} -8.00000 q^{64} +(-10.5000 + 6.06218i) q^{67} +17.0000 q^{73} +(-7.50000 + 4.33013i) q^{75} +(-8.00000 - 13.8564i) q^{76} +13.0000 q^{79} +(-4.50000 - 7.79423i) q^{81} +(6.00000 - 6.92820i) q^{84} +(0.500000 + 9.52628i) q^{91} +(10.5000 - 6.06218i) q^{93} +(2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 2 q^{4} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 + 2 * q^4 - 4 * q^7 + 3 * q^9	\( 2 q - 3 q^{3} + 2 q^{4} - 4 q^{7} + 3 q^{9} - 5 q^{13} - 4 q^{16} + 8 q^{19} + 9 q^{21} + 10 q^{25} - 10 q^{28} - 14 q^{31} - 6 q^{36} + 12 q^{37} + 12 q^{39} - 13 q^{43} + 12 q^{48} + 2 q^{49} - 14 q^{52} + 15 q^{61} - 15 q^{63} - 16 q^{64} - 21 q^{67} + 34 q^{73} - 15 q^{75} - 16 q^{76} + 26 q^{79} - 9 q^{81} + 12 q^{84} + q^{91} + 21 q^{93} + 5 q^{97}+O(q^{100}) \) 2 * q - 3 * q^3 + 2 * q^4 - 4 * q^7 + 3 * q^9 - 5 * q^13 - 4 * q^16 + 8 * q^19 + 9 * q^21 + 10 * q^25 - 10 * q^28 - 14 * q^31 - 6 * q^36 + 12 * q^37 + 12 * q^39 - 13 * q^43 + 12 * q^48 + 2 * q^49 - 14 * q^52 + 15 * q^61 - 15 * q^63 - 16 * q^64 - 21 * q^67 + 34 * q^73 - 15 * q^75 - 16 * q^76 + 26 * q^79 - 9 * q^81 + 12 * q^84 + q^91 + 21 * q^93 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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