LMFDB - Embedded newform 273.2.bf.a.185.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(152,273)]

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, 5, 4]))

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

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

chi := DirichletCharacter("273.152");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Embedding invariants

Embedding label			185.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	273.185
Dual form			273.2.bf.a.152.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} +(-3.00000 + 1.73205i) q^{12} +(1.00000 + 3.46410i) q^{13} +(-2.00000 + 3.46410i) q^{16} -8.66025i q^{19} +(-1.50000 + 4.33013i) q^{21} +(2.50000 - 4.33013i) q^{25} +5.19615i q^{27} +(1.00000 + 5.19615i) q^{28} +(-7.50000 - 4.33013i) q^{31} +(3.00000 + 5.19615i) q^{36} +(5.50000 - 9.52628i) q^{37} +(6.00000 - 1.73205i) q^{39} +(4.00000 - 6.92820i) q^{43} +(6.00000 + 3.46410i) q^{48} +(5.50000 + 4.33013i) q^{49} +(5.00000 - 5.19615i) q^{52} -15.0000 q^{57} +15.5885i q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +11.0000 q^{67} +(-12.0000 - 6.92820i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(-15.0000 + 8.66025i) q^{76} +(6.50000 + 11.2583i) q^{79} +9.00000 q^{81} +(9.00000 - 1.73205i) q^{84} +(0.500000 - 9.52628i) q^{91} +(-7.50000 + 12.9904i) q^{93} +(4.50000 + 2.59808i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 5 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 5 * q^7 - 6 * q^9	$ 2 q - 2 q^{4} - 5 q^{7} - 6 q^{9} - 6 q^{12} + 2 q^{13} - 4 q^{16} - 3 q^{21} + 5 q^{25} + 2 q^{28} - 15 q^{31} + 6 q^{36} + 11 q^{37} + 12 q^{39} + 8 q^{43} + 12 q^{48} + 11 q^{49} + 10 q^{52} - 30 q^{57} + 15 q^{63} + 16 q^{64} + 22 q^{67} - 24 q^{73} - 15 q^{75} - 30 q^{76} + 13 q^{79} + 18 q^{81} + 18 q^{84} + q^{91} - 15 q^{93} + 9 q^{97}+O(q^{100}) $ 2 * q - 2 * q^4 - 5 * q^7 - 6 * q^9 - 6 * q^12 + 2 * q^13 - 4 * q^16 - 3 * q^21 + 5 * q^25 + 2 * q^28 - 15 * q^31 + 6 * q^36 + 11 * q^37 + 12 * q^39 + 8 * q^43 + 12 * q^48 + 11 * q^49 + 10 * q^52 - 30 * q^57 + 15 * q^63 + 16 * q^64 + 22 * q^67 - 24 * q^73 - 15 * q^75 - 30 * q^76 + 13 * q^79 + 18 * q^81 + 18 * q^84 + q^91 - 15 * q^93 + 9 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$92$	$106$	$157$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$

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	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−3.00000	+	1.73205i	−0.866025	+	0.500000i
$13$	1.00000	+	3.46410i	0.277350	+	0.960769i
$13$	1.00000	+	3.46410i	0.277350	+	0.960769i
$14$		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
$19$		−	8.66025i		−	1.98680i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$		−	8.66025i		−	1.98680i	0.114708	−	0.993399i	$-0.463407\pi$
$20$		0			0
$21$	−1.50000	+	4.33013i	−0.327327	+	0.944911i
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$			5.19615i			1.00000i
$28$	1.00000	+	5.19615i	0.188982	+	0.981981i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−7.50000	−	4.33013i	−1.34704	−	0.777714i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−7.50000	−	4.33013i	−1.34704	−	0.777714i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	3.00000	+	5.19615i	0.500000	+	0.866025i
$37$	5.50000	−	9.52628i	0.904194	−	1.56611i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	5.50000	−	9.52628i	0.904194	−	1.56611i	0.821995	−	0.569495i	$-0.192861\pi$
$38$		0			0
$39$	6.00000	−	1.73205i	0.960769	−	0.277350i
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	6.00000	+	3.46410i	0.866025	+	0.500000i
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$	5.00000	−	5.19615i	0.693375	−	0.720577i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−15.0000			−1.98680
$58$		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
$61$			15.5885i			1.99590i	0.0640184	+	0.997949i	$0.479608\pi$
$61$			15.5885i			1.99590i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$	7.50000	+	2.59808i	0.944911	+	0.327327i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	11.0000			1.34386			0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000			1.34386			0.671932	+	0.740613i	$0.265465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$	−12.0000	−	6.92820i	−1.40449	−	0.810885i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−12.0000	−	6.92820i	−1.40449	−	0.810885i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	−7.50000	−	4.33013i	−0.866025	−	0.500000i
$76$	−15.0000	+	8.66025i	−1.72062	+	0.993399i
$77$		0			0
$78$		0			0
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	0.956325	+	0.292306i	$0.0944227\pi$
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	9.00000	−	1.73205i	0.981981	−	0.188982i
$85$		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
$91$	0.500000	−	9.52628i	0.0524142	−	0.998625i
$92$		0			0
$93$	−7.50000	+	12.9904i	−0.777714	+	1.34704i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.50000	+	2.59808i	0.456906	+	0.263795i	0.710742	−	0.703452i	$-0.248359\pi$
$97$	4.50000	+	2.59808i	0.456906	+	0.263795i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	3.00000	−	1.73205i	0.295599	−	0.170664i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	3.00000	−	1.73205i	0.295599	−	0.170664i	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
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	−16.5000	−	9.52628i	−1.56611	−	0.904194i
$112$	8.00000	−	6.92820i	0.755929	−	0.654654i
$113$		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
$117$	−3.00000	−	10.3923i	−0.277350	−	0.960769i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$			17.3205i			1.55543i
$125$		0			0
$126$		0			0
$127$	−10.0000	−	17.3205i	−0.887357	−	1.53695i	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−10.0000	−	17.3205i	−0.887357	−	1.53695i	−0.0443678	−	0.999015i	$-0.514127\pi$
$128$		0			0
$129$	−12.0000	−	6.92820i	−1.05654	−	0.609994i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−7.50000	+	21.6506i	−0.650332	+	1.87735i
$134$		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	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−19.5000	−	11.2583i	−1.65397	−	0.954919i	−0.678551	−	0.734553i	$-0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.00000	−	10.3923i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	7.50000	−	9.52628i	0.618590	−	0.785714i
$148$	−22.0000			−1.80839
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	0.773099	+	0.634285i	$0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−9.00000	−	8.66025i	−0.720577	−	0.693375i
$157$	−1.50000	−	0.866025i	−0.119713	−	0.0691164i	0.438948	−	0.898513i	$-0.355351\pi$
$157$	−1.50000	−	0.866025i	−0.119713	−	0.0691164i	−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	17.0000			1.33154			0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000			1.33154			0.665771	+	0.746156i	$0.268103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−11.0000	+	6.92820i	−0.846154	+	0.532939i
$170$		0			0
$171$			25.9808i			1.98680i
$172$	−16.0000			−1.21999
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−10.0000	+	8.66025i	−0.755929	+	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			19.0526i			1.41617i	0.706129	+	0.708083i	$0.250440\pi$
$181$			19.0526i			1.41617i	−0.706129	+	0.708083i	$0.749560\pi$
$182$		0			0
$183$	27.0000			1.99590
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.50000	−	12.9904i	0.327327	−	0.944911i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	13.8564i		−	1.00000i
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$	2.00000	−	13.8564i	0.142857	−	0.989743i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	22.5000	−	12.9904i	1.59498	−	0.920864i	0.602549	−	0.798082i	$-0.294152\pi$
$199$	22.5000	−	12.9904i	1.59498	−	0.920864i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$		0			0
$201$		−	19.0526i		−	1.34386i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−14.0000	−	3.46410i	−0.970725	−	0.240192i
$209$		0			0
$210$		0			0
$211$	14.5000	+	25.1147i	0.998221	+	1.72897i	0.550743	+	0.834675i	$0.314345\pi$
$211$	14.5000	+	25.1147i	0.998221	+	1.72897i	0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	15.0000	+	17.3205i	1.01827	+	1.17579i
$218$		0			0
$219$	−12.0000	+	20.7846i	−0.810885	+	1.40449i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	9.00000	−	5.19615i	0.602685	−	0.347960i	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	9.00000	−	5.19615i	0.602685	−	0.347960i	0.770097	+	0.637927i	$0.220208\pi$
$224$		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	$0.833333\pi$
$228$	15.0000	+	25.9808i	0.993399	+	1.72062i
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−7.50000	+	4.33013i	−0.495614	+	0.286143i	0.231287	+	0.972886i	$0.425707\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		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	$0.684433\pi$
$241$	−24.0000	+	13.8564i	−1.54598	+	0.892570i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$		−	15.5885i		−	1.00000i
$244$	27.0000	−	15.5885i	1.72850	−	0.997949i
$245$		0			0
$246$		0			0
$247$	30.0000	−	8.66025i	1.90885	−	0.551039i
$248$		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	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−22.0000	+	19.0526i	−1.36701	+	1.18387i
$260$		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
$268$	−11.0000	−	19.0526i	−0.671932	−	1.16382i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	28.5000	+	16.4545i	1.73125	+	0.999539i	0.880812	+	0.473466i	$0.156997\pi$
$271$	28.5000	+	16.4545i	1.73125	+	0.999539i	0.850439	+	0.526073i	$0.176336\pi$
$272$		0			0
$273$	−16.5000	−	0.866025i	−0.998625	−	0.0524142i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	0.781094	−	0.624413i	$-0.214662\pi$
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		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
$283$		−	10.3923i		−	0.617758i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$		−	10.3923i		−	0.617758i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$		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
$291$	4.50000	−	7.79423i	0.263795	−	0.456906i
$292$			27.7128i			1.62177i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$			17.3205i			1.00000i
$301$	−16.0000	+	13.8564i	−0.922225	+	0.798670i
$302$		0			0
$303$		0			0
$304$	30.0000	+	17.3205i	1.72062	+	0.993399i
$305$		0			0
$306$		0			0
$307$		−	1.73205i		−	0.0988534i	−0.998778	−	0.0494267i	$-0.984261\pi$
$307$		−	1.73205i		−	0.0988534i	0.998778	−	0.0494267i	$-0.0157394\pi$
$308$		0			0
$309$	−3.00000	−	5.19615i	−0.170664	−	0.295599i
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	24.0000	−	13.8564i	1.35656	−	0.783210i	0.367402	−	0.930062i	$-0.380247\pi$
$313$	24.0000	−	13.8564i	1.35656	−	0.783210i	0.989158	+	0.146852i	$0.0469141\pi$
$314$		0			0
$315$		0			0
$316$	13.0000	−	22.5167i	0.731307	−	1.26666i
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−9.00000	−	15.5885i	−0.500000	−	0.866025i
$325$	17.5000	+	4.33013i	0.970725	+	0.240192i
$326$		0			0
$327$	−3.00000	−	1.73205i	−0.165900	−	0.0957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−32.0000			−1.75888			−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000			−1.75888			−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	−16.5000	+	28.5788i	−0.904194	+	1.56611i
$334$		0			0
$335$		0			0
$336$	−12.0000	−	13.8564i	−0.654654	−	0.755929i
$337$	34.0000			1.85210			0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000			1.85210			0.926049	+	0.377403i	$0.123183\pi$
$338$		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
$349$	−4.50000	+	2.59808i	−0.240879	+	0.139072i	−0.615581	−	0.788074i	$-0.711079\pi$
$349$	−4.50000	+	2.59808i	−0.240879	+	0.139072i	0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$	−18.0000	+	5.19615i	−0.960769	+	0.277350i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−56.0000			−2.94737
$362$		0			0
$363$		−	19.0526i		−	1.00000i
$364$	−17.0000	+	8.66025i	−0.891042	+	0.453921i
$365$		0			0
$366$		0			0
$367$			38.1051i			1.98907i	0.104399	+	0.994535i	$0.466708\pi$
$367$			38.1051i			1.98907i	−0.104399	+	0.994535i	$0.533292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	30.0000			1.55543
$373$	−13.0000			−0.673114			−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000			−0.673114			−0.336557	+	0.941663i	$0.609263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−18.5000	−	32.0429i	−0.950281	−	1.64594i	−0.744815	−	0.667271i	$-0.767462\pi$
$379$	−18.5000	−	32.0429i	−0.950281	−	1.64594i	−0.205466	−	0.978664i	$-0.565871\pi$
$380$		0			0
$381$	−30.0000	+	17.3205i	−1.53695	+	0.887357i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−12.0000	+	20.7846i	−0.609994	+	1.05654i
$388$		−	10.3923i		−	0.527589i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	20.7846i		−	1.04315i	−0.853206	−	0.521575i	$-0.825345\pi$
$397$		−	20.7846i		−	1.04315i	0.853206	−	0.521575i	$-0.174655\pi$
$398$		0			0
$399$	37.5000	+	12.9904i	1.87735	+	0.650332i
$400$	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
$403$	7.50000	−	30.3109i	0.373602	−	1.50989i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	12.0000	−	6.92820i	0.593362	−	0.342578i	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	12.0000	−	6.92820i	0.593362	−	0.342578i	0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$	−6.00000	−	3.46410i	−0.295599	−	0.170664i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−19.5000	+	33.7750i	−0.954919	+	1.65397i
$418$		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
$421$	−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
$427$	13.5000	−	38.9711i	0.653311	−	1.88595i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−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
$436$	−4.00000			−0.191565
$437$		0			0
$438$		0			0
$439$	7.50000	+	4.33013i	0.357955	+	0.206666i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	7.50000	+	4.33013i	0.357955	+	0.206666i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$	−16.5000	−	12.9904i	−0.785714	−	0.618590i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$			38.1051i			1.80839i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−20.0000	−	6.92820i	−0.944911	−	0.327327i
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		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	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$	−15.0000	+	15.5885i	−0.693375	+	0.720577i
$469$	−27.5000	−	9.52628i	−1.26983	−	0.439883i
$470$		0			0
$471$	−1.50000	+	2.59808i	−0.0691164	+	0.119713i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−37.5000	−	21.6506i	−1.72062	−	0.993399i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	38.5000	+	9.52628i	1.75545	+	0.434361i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−9.50000	−	16.4545i	−0.430486	−	0.745624i	0.566429	−	0.824110i	$-0.308325\pi$
$487$	−9.50000	−	16.4545i	−0.430486	−	0.745624i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$		−	29.4449i		−	1.33154i
$490$		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
$496$	30.0000	−	17.3205i	1.34704	−	0.777714i
$497$		0			0
$498$		0			0
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	0.962472	−	0.271380i	$-0.0874801\pi$
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	−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
$507$	12.0000	+	19.0526i	0.532939	+	0.846154i
$508$	−20.0000	+	34.6410i	−0.887357	+	1.53695i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	24.0000	+	27.7128i	1.06170	+	1.22594i
$512$		0			0
$513$	45.0000			1.98680
$514$		0			0
$515$		0			0
$516$			27.7128i			1.21999i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−13.5000	−	7.79423i	−0.590314	−	0.340818i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	−13.5000	−	7.79423i	−0.590314	−	0.340818i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$	15.0000	+	17.3205i	0.654654	+	0.755929i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	−	19.9186i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	45.0000	−	8.66025i	1.95100	−	0.375470i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	23.0000	+	39.8372i	0.988847	+	1.71273i	0.623404	+	0.781900i	$0.285749\pi$
$541$	23.0000	+	39.8372i	0.988847	+	1.71273i	0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	33.0000			1.41617
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.00000			−0.0427569			−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000			−0.0427569			−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$		−	46.7654i		−	1.99590i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−6.50000	−	33.7750i	−0.276408	−	1.43626i
$554$		0			0
$555$		0			0
$556$			45.0333i			1.90984i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	28.0000	+	6.92820i	1.18427	+	0.293032i
$560$		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
$567$	−22.5000	−	7.79423i	−0.944911	−	0.327327i
$568$		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
$571$	−23.5000	+	40.7032i	−0.983444	+	1.70338i	−0.334790	+	0.942293i	$0.608665\pi$
$571$	−23.5000	+	40.7032i	−0.983444	+	1.70338i	−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	40.5000	+	23.3827i	1.68604	+	0.973434i	0.957503	+	0.288425i	$0.0931316\pi$
$577$	40.5000	+	23.3827i	1.68604	+	0.973434i	0.728535	+	0.685009i	$0.240202\pi$
$578$		0			0
$579$			3.46410i			0.143963i
$580$		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	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$	−24.0000	−	3.46410i	−0.989743	−	0.142857i
$589$	−37.5000	+	64.9519i	−1.54516	+	2.67630i
$590$		0			0
$591$		0			0
$592$	22.0000	+	38.1051i	0.904194	+	1.56611i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−22.5000	−	38.9711i	−0.920864	−	1.59498i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	37.5000	−	21.6506i	1.52966	−	0.883148i	0.530281	−	0.847822i	$-0.322086\pi$
$601$	37.5000	−	21.6506i	1.52966	−	0.883148i	0.999376	−	0.0353259i	$-0.0112469\pi$
$602$		0			0
$603$	−33.0000			−1.34386
$604$	8.00000			0.325515
$605$		0			0
$606$		0			0
$607$			39.8372i			1.61694i	0.588537	+	0.808470i	$0.299704\pi$
$607$			39.8372i			1.61694i	−0.588537	+	0.808470i	$0.700296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	47.0000			1.89831			0.949156	−	0.314806i	$-0.101939\pi$
$613$	47.0000			1.89831			0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	−33.0000	−	19.0526i	−1.32638	−	0.765787i	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−33.0000	−	19.0526i	−1.32638	−	0.765787i	−0.984738	+	0.174042i	$0.944317\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−6.00000	+	24.2487i	−0.240192	+	0.970725i
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$			3.46410i			0.138233i
$629$		0			0
$630$		0			0
$631$	21.5000	−	37.2391i	0.855901	−	1.48246i	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	21.5000	−	37.2391i	0.855901	−	1.48246i	0.875806	−	0.482663i	$-0.160330\pi$
$632$		0			0
$633$	43.5000	−	25.1147i	1.72897	−	0.998221i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−9.50000	+	23.3827i	−0.376404	+	0.926456i
$638$		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
$643$	−27.0000	−	15.5885i	−1.06478	−	0.614749i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−27.0000	−	15.5885i	−1.06478	−	0.614749i	−0.926750	+	0.375680i	$0.877409\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	30.0000	−	25.9808i	1.17579	−	1.01827i
$652$	−17.0000	−	29.4449i	−0.665771	−	1.15315i
$653$		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
$657$	36.0000	+	20.7846i	1.40449	+	0.810885i
$658$		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
$661$		−	15.5885i		−	0.606321i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$		−	15.5885i		−	0.606321i	0.952940	−	0.303160i	$-0.0980418\pi$
$662$		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
$673$	25.0000	+	43.3013i	0.963679	+	1.66914i	0.713123	+	0.701039i	$0.247280\pi$
$673$	25.0000	+	43.3013i	0.963679	+	1.66914i	0.250557	+	0.968102i	$0.419386\pi$
$674$		0			0
$675$	22.5000	+	12.9904i	0.866025	+	0.500000i
$676$	23.0000	+	12.1244i	0.884615	+	0.466321i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−9.00000	−	10.3923i	−0.345388	−	0.398820i
$680$		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	$0.333333\pi$
$684$	45.0000	−	25.9808i	1.72062	−	0.993399i
$685$		0			0
$686$		0			0
$687$	7.50000	+	12.9904i	0.286143	+	0.495614i
$688$	16.0000	+	27.7128i	0.609994	+	1.05654i
$689$		0			0
$690$		0			0
$691$	−45.0000	−	25.9808i	−1.71188	−	0.988355i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−45.0000	−	25.9808i	−1.71188	−	0.988355i	−0.779857	−	0.625958i	$-0.784708\pi$
$692$		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
$703$	−82.5000	−	47.6314i	−3.11155	−	1.79645i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	31.0000			1.16423			0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000			1.16423			0.582115	+	0.813107i	$0.302225\pi$
$710$		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			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−9.00000	+	1.73205i	−0.335178	+	0.0645049i
$722$		0			0
$723$	24.0000	+	41.5692i	0.892570	+	1.54598i
$724$	33.0000	−	19.0526i	1.22644	−	0.708083i
$725$		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
$732$	−27.0000	−	46.7654i	−0.997949	−	1.72850i
$733$	46.5000	−	26.8468i	1.71752	−	0.991609i	0.794121	−	0.607760i	$-0.207932\pi$
$733$	46.5000	−	26.8468i	1.71752	−	0.991609i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−37.0000			−1.36107			−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000			−1.36107			−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$	−15.0000	−	51.9615i	−0.551039	−	1.90885i
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	−0.748056	−	0.663636i	$-0.769012\pi$
$751$	5.50000	−	9.52628i	0.200698	−	0.347619i	0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−27.0000	+	5.19615i	−0.981981	+	0.188982i
$757$	13.0000	+	22.5167i	0.472493	+	0.818382i	0.999505	−	0.0314762i	$-0.0100208\pi$
$757$	13.0000	+	22.5167i	0.472493	+	0.818382i	−0.527011	+	0.849858i	$0.676688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−4.00000	+	3.46410i	−0.144810	+	0.125409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.0000	+	13.8564i	−0.866025	+	0.500000i
$769$	−25.5000	+	14.7224i	−0.919554	+	0.530904i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−25.5000	+	14.7224i	−0.919554	+	0.530904i	−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$	2.00000	+	3.46410i	0.0719816	+	0.124676i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−37.5000	+	21.6506i	−1.34704	+	0.777714i
$776$		0			0
$777$	33.0000	+	38.1051i	1.18387	+	1.36701i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−26.0000	+	10.3923i	−0.928571	+	0.371154i
$785$		0			0
$786$		0			0
$787$	−40.5000	+	23.3827i	−1.44367	+	0.833503i	−0.998092	−	0.0617409i	$-0.980335\pi$
$787$	−40.5000	+	23.3827i	−1.44367	+	0.833503i	−0.445577	+	0.895244i	$0.647001\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−54.0000	+	15.5885i	−1.91760	+	0.553562i
$794$		0			0
$795$		0			0
$796$	−45.0000	−	25.9808i	−1.59498	−	0.920864i
$797$		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
$804$	−33.0000	+	19.0526i	−1.16382	+	0.671932i
$805$		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
$811$			43.3013i			1.52051i	0.649623	+	0.760257i	$0.274927\pi$
$811$			43.3013i			1.52051i	−0.649623	+	0.760257i	$0.725073\pi$
$812$		0			0
$813$	28.5000	−	49.3634i	0.999539	−	1.73125i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−60.0000	−	34.6410i	−2.09913	−	1.21194i
$818$		0			0
$819$	−1.50000	+	28.5788i	−0.0524142	+	0.998625i
$820$		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
$829$		−	34.6410i		−	1.20313i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$		−	34.6410i		−	1.20313i	0.798823	−	0.601566i	$-0.205456\pi$
$830$		0			0
$831$	−7.50000	+	4.33013i	−0.260172	+	0.150210i
$832$	8.00000	+	27.7128i	0.277350	+	0.960769i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	22.5000	−	38.9711i	0.777714	−	1.34704i
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	29.0000	−	50.2295i	0.998221	−	1.72897i
$845$		0			0
$846$		0			0
$847$	−27.5000	−	9.52628i	−0.944911	−	0.327327i
$848$		0			0
$849$	−18.0000			−0.617758
$850$		0			0
$851$		0			0
$852$		0			0
$853$			46.7654i			1.60122i	0.599189	+	0.800608i	$0.295490\pi$
$853$			46.7654i			1.60122i	−0.599189	+	0.800608i	$0.704510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	34.5000	−	19.9186i	1.17712	−	0.679613i	0.221777	−	0.975097i	$-0.428814\pi$
$859$	34.5000	−	19.9186i	1.17712	−	0.679613i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−25.5000	−	14.7224i	−0.866025	−	0.500000i
$868$	15.0000	−	43.3013i	0.509133	−	1.46974i
$869$		0			0
$870$		0			0
$871$	11.0000	+	38.1051i	0.372721	+	1.29114i
$872$		0			0
$873$	−13.5000	−	7.79423i	−0.456906	−	0.263795i
$874$		0			0
$875$		0			0
$876$	48.0000			1.62177
$877$	−34.0000			−1.14810			−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000			−1.14810			−0.574049	+	0.818821i	$0.694628\pi$
$878$		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
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$		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
$889$	10.0000	+	51.9615i	0.335389	+	1.74273i
$890$		0			0
$891$		0			0
$892$	−18.0000	−	10.3923i	−0.602685	−	0.347960i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	30.0000			1.00000
$901$		0			0
$902$		0			0
$903$	24.0000	+	27.7128i	0.798670	+	0.922225i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	19.0000			0.630885			0.315442	−	0.948945i	$-0.397847\pi$
$907$	19.0000			0.630885			0.315442	+	0.948945i	$0.397847\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	30.0000	−	51.9615i	0.993399	−	1.72062i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	15.0000	+	8.66025i	0.495614	+	0.286143i
$917$		0			0
$918$		0			0
$919$	−53.0000			−1.74831			−0.874154	−	0.485648i	$-0.838584\pi$
$919$	−53.0000			−1.74831			−0.874154	+	0.485648i	$0.838584\pi$
$920$		0			0
$921$	−3.00000			−0.0988534
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−27.5000	−	47.6314i	−0.904194	−	1.56611i
$926$		0			0
$927$	−9.00000	+	5.19615i	−0.295599	+	0.170664i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	37.5000	−	47.6314i	1.22901	−	1.56106i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	5.19615i		−	0.169751i	−0.996392	−	0.0848755i	$-0.972951\pi$
$937$		−	5.19615i		−	0.169751i	0.996392	−	0.0848755i	$-0.0270492\pi$
$938$		0			0
$939$	−24.0000	−	41.5692i	−0.783210	−	1.35656i
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		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	$0.666667\pi$
$948$	−39.0000	−	22.5167i	−1.26666	−	0.731307i
$949$	12.0000	−	48.4974i	0.389536	−	1.57429i
$950$		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
$961$	22.0000	+	38.1051i	0.709677	+	1.22920i
$962$		0			0
$963$		0			0
$964$	48.0000	+	27.7128i	1.54598	+	0.892570i
$965$		0			0
$966$		0			0
$967$	41.0000			1.31847			0.659236	−	0.751936i	$-0.270880\pi$
$967$	41.0000			1.31847			0.659236	+	0.751936i	$0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−27.0000	+	15.5885i	−0.866025	+	0.500000i
$973$	39.0000	+	45.0333i	1.25028	+	1.44370i
$974$		0			0
$975$	7.50000	−	30.3109i	0.240192	−	0.970725i
$976$	−54.0000	−	31.1769i	−1.72850	−	0.997949i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−3.00000	+	5.19615i	−0.0957826	+	0.165900i
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−45.0000	−	43.3013i	−1.43164	−	1.37760i
$989$		0			0
$990$		0			0
$991$	61.0000			1.93773			0.968864	−	0.247592i	$-0.0796392\pi$
$991$	61.0000			1.93773			0.968864	+	0.247592i	$0.0796392\pi$
$992$		0			0
$993$			55.4256i			1.75888i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	54.0000	−	31.1769i	1.71020	−	0.987383i	0.775923	−	0.630828i	$-0.217285\pi$
$997$	54.0000	−	31.1769i	1.71020	−	0.987383i	0.934274	−	0.356555i	$-0.116049\pi$
$998$		0			0
$999$	49.5000	+	28.5788i	1.56611	+	0.904194i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	273.2.bf.a.185.1	yes	2
3.2	odd	2	CM	273.2.bf.a.185.1	yes	2
7.5	odd	6		273.2.r.a.68.1	✓	2
13.9	even	3		273.2.r.a.269.1	yes	2
21.5	even	6		273.2.r.a.68.1	✓	2
39.35	odd	6		273.2.r.a.269.1	yes	2
91.61	odd	6	inner	273.2.bf.a.152.1	yes	2
273.152	even	6	inner	273.2.bf.a.152.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.r.a.68.1	✓	2	7.5	odd	6
273.2.r.a.68.1	✓	2	21.5	even	6
273.2.r.a.269.1	yes	2	13.9	even	3
273.2.r.a.269.1	yes	2	39.35	odd	6
273.2.bf.a.152.1	yes	2	91.61	odd	6	inner
273.2.bf.a.152.1	yes	2	273.152	even	6	inner
273.2.bf.a.185.1	yes	2	1.1	even	1	trivial
273.2.bf.a.185.1	yes	2	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +(-1.00000 - 1.73205i) q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} +(-3.00000 + 1.73205i) q^{12} +(1.00000 + 3.46410i) q^{13} +(-2.00000 + 3.46410i) q^{16} -8.66025i q^{19} +(-1.50000 + 4.33013i) q^{21} +(2.50000 - 4.33013i) q^{25} +5.19615i q^{27} +(1.00000 + 5.19615i) q^{28} +(-7.50000 - 4.33013i) q^{31} +(3.00000 + 5.19615i) q^{36} +(5.50000 - 9.52628i) q^{37} +(6.00000 - 1.73205i) q^{39} +(4.00000 - 6.92820i) q^{43} +(6.00000 + 3.46410i) q^{48} +(5.50000 + 4.33013i) q^{49} +(5.00000 - 5.19615i) q^{52} -15.0000 q^{57} +15.5885i q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +11.0000 q^{67} +(-12.0000 - 6.92820i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(-15.0000 + 8.66025i) q^{76} +(6.50000 + 11.2583i) q^{79} +9.00000 q^{81} +(9.00000 - 1.73205i) q^{84} +(0.500000 - 9.52628i) q^{91} +(-7.50000 + 12.9904i) q^{93} +(4.50000 + 2.59808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 5 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 5 * q^7 - 6 * q^9	\( 2 q - 2 q^{4} - 5 q^{7} - 6 q^{9} - 6 q^{12} + 2 q^{13} - 4 q^{16} - 3 q^{21} + 5 q^{25} + 2 q^{28} - 15 q^{31} + 6 q^{36} + 11 q^{37} + 12 q^{39} + 8 q^{43} + 12 q^{48} + 11 q^{49} + 10 q^{52} - 30 q^{57} + 15 q^{63} + 16 q^{64} + 22 q^{67} - 24 q^{73} - 15 q^{75} - 30 q^{76} + 13 q^{79} + 18 q^{81} + 18 q^{84} + q^{91} - 15 q^{93} + 9 q^{97}+O(q^{100}) \) 2 * q - 2 * q^4 - 5 * q^7 - 6 * q^9 - 6 * q^12 + 2 * q^13 - 4 * q^16 - 3 * q^21 + 5 * q^25 + 2 * q^28 - 15 * q^31 + 6 * q^36 + 11 * q^37 + 12 * q^39 + 8 * q^43 + 12 * q^48 + 11 * q^49 + 10 * q^52 - 30 * q^57 + 15 * q^63 + 16 * q^64 + 22 * q^67 - 24 * q^73 - 15 * q^75 - 30 * q^76 + 13 * q^79 + 18 * q^81 + 18 * q^84 + q^91 - 15 * q^93 + 9 * q^97

Introduction

L-functions

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Database

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