LMFDB - Embedded newform 201.2.f.a.164.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [201,2,Mod(38,201)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(201, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

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

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

chi := DirichletCharacter("201.38");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 201 = 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	201.f (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:	$1.60499308063$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			164.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	201.164
Dual form			201.2.f.a.38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +(1.00000 - 1.73205i) q^{4} +(3.00000 + 1.73205i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +(1.00000 - 1.73205i) q^{4} +(3.00000 + 1.73205i) q^{7} -3.00000 q^{9} +(3.00000 + 1.73205i) q^{12} +(1.50000 - 0.866025i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.00000 + 6.92820i) q^{19} +(-3.00000 + 5.19615i) q^{21} -5.00000 q^{25} -5.19615i q^{27} +(6.00000 - 3.46410i) q^{28} +(-7.50000 - 4.33013i) q^{31} +(-3.00000 + 5.19615i) q^{36} +(-5.00000 - 8.66025i) q^{37} +(1.50000 + 2.59808i) q^{39} -12.1244i q^{43} +(6.00000 - 3.46410i) q^{48} +(2.50000 + 4.33013i) q^{49} -3.46410i q^{52} +(-12.0000 + 6.92820i) q^{57} +(-7.50000 + 4.33013i) q^{61} +(-9.00000 - 5.19615i) q^{63} -8.00000 q^{64} +(-2.50000 + 7.79423i) q^{67} +(8.50000 + 14.7224i) q^{73} -8.66025i q^{75} +16.0000 q^{76} +(4.50000 + 2.59808i) q^{79} +9.00000 q^{81} +(6.00000 + 10.3923i) q^{84} +6.00000 q^{91} +(7.50000 - 12.9904i) q^{93} +(-4.50000 + 2.59808i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 6 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 + 6 * q^7 - 6 * q^9	$ 2 q + 2 q^{4} + 6 q^{7} - 6 q^{9} + 6 q^{12} + 3 q^{13} - 4 q^{16} + 8 q^{19} - 6 q^{21} - 10 q^{25} + 12 q^{28} - 15 q^{31} - 6 q^{36} - 10 q^{37} + 3 q^{39} + 12 q^{48} + 5 q^{49} - 24 q^{57} - 15 q^{61} - 18 q^{63} - 16 q^{64} - 5 q^{67} + 17 q^{73} + 32 q^{76} + 9 q^{79} + 18 q^{81} + 12 q^{84} + 12 q^{91} + 15 q^{93} - 9 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^7 - 6 * q^9 + 6 * q^12 + 3 * q^13 - 4 * q^16 + 8 * q^19 - 6 * q^21 - 10 * q^25 + 12 * q^28 - 15 * q^31 - 6 * q^36 - 10 * q^37 + 3 * q^39 + 12 * q^48 + 5 * q^49 - 24 * q^57 - 15 * q^61 - 18 * q^63 - 16 * q^64 - 5 * q^67 + 17 * q^73 + 32 * q^76 + 9 * q^79 + 18 * q^81 + 12 * q^84 + 12 * q^91 + 15 * q^93 - 9 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$			1.73205i			1.00000i
$3$			1.73205i			1.00000i
$4$	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
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.944911	−	0.327327i	$-0.106148\pi$
$7$	3.00000	+	1.73205i	1.13389	+	0.654654i	0.188982	+	0.981981i	$0.439481\pi$
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$	3.00000	+	1.73205i	0.866025	+	0.500000i
$13$	1.50000	−	0.866025i	0.416025	−	0.240192i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	1.50000	−	0.866025i	0.416025	−	0.240192i	0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	4.00000	+	6.92820i	0.917663	+	1.58944i	0.802955	+	0.596040i	$0.203260\pi$
$19$	4.00000	+	6.92820i	0.917663	+	1.58944i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$	−3.00000	+	5.19615i	−0.654654	+	1.13389i
$22$		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
$28$	6.00000	−	3.46410i	1.13389	−	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
$31$	−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
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−5.00000	−	8.66025i	−0.821995	−	1.42374i	0.0821995	−	0.996616i	$-0.473806\pi$
$38$		0			0
$39$	1.50000	+	2.59808i	0.240192	+	0.416025i
$40$		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
$43$		−	12.1244i		−	1.84895i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$		−	12.1244i		−	1.84895i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	6.00000	−	3.46410i	0.866025	−	0.500000i
$49$	2.50000	+	4.33013i	0.357143	+	0.618590i
$50$		0			0
$51$		0			0
$52$		−	3.46410i		−	0.480384i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−12.0000	+	6.92820i	−1.58944	+	0.917663i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−7.50000	+	4.33013i	−0.960277	+	0.554416i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−7.50000	+	4.33013i	−0.960277	+	0.554416i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$	−9.00000	−	5.19615i	−1.13389	−	0.654654i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−2.50000	+	7.79423i	−0.305424	+	0.952217i
$67$	−2.50000	+	7.79423i	−0.305424	+	0.952217i
$68$		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
$73$	8.50000	+	14.7224i	0.994850	+	1.72313i	0.585206	+	0.810885i	$0.301014\pi$
$73$	8.50000	+	14.7224i	0.994850	+	1.72313i	0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$		−	8.66025i		−	1.00000i
$76$	16.0000			1.83533
$77$		0			0
$78$		0			0
$79$	4.50000	+	2.59808i	0.506290	+	0.292306i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	4.50000	+	2.59808i	0.506290	+	0.292306i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	6.00000	+	10.3923i	0.654654	+	1.13389i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$		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	$-0.751641\pi$
$97$	−4.50000	+	2.59808i	−0.456906	+	0.263795i	0.253837	+	0.967247i	$0.418307\pi$
$98$		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
$103$	10.0000	−	17.3205i	0.985329	−	1.70664i	0.344865	−	0.938652i	$-0.387925\pi$
$103$	10.0000	−	17.3205i	0.985329	−	1.70664i	0.640464	−	0.767988i	$-0.278742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−9.00000	−	5.19615i	−0.866025	−	0.500000i
$109$			12.1244i			1.16130i	0.814152	+	0.580651i	$0.197202\pi$
$109$			12.1244i			1.16130i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$	15.0000	−	8.66025i	1.42374	−	0.821995i
$112$		−	13.8564i		−	1.30931i
$113$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$113$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.50000	+	2.59808i	−0.416025	+	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	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−0.500000	+	0.866025i	−0.0443678	+	0.0768473i	0.842989	+	0.537931i	$0.180794\pi$
$128$		0			0
$129$	21.0000			1.84895
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$			27.7128i			2.40301i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		−	22.5167i		−	1.90984i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$		−	22.5167i		−	1.90984i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		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	+	4.33013i	−0.618590	+	0.357143i
$148$	−20.0000			−1.64399
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	9.50000	+	16.4545i	0.773099	+	1.33905i	0.935857	+	0.352381i	$0.114628\pi$
$151$	9.50000	+	16.4545i	0.773099	+	1.33905i	−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	6.00000			0.480384
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	0.313304	+	0.949653i	$0.398564\pi$
$163$	−8.50000	+	14.7224i	−0.665771	+	1.15315i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−5.00000	+	8.66025i	−0.384615	+	0.666173i
$170$		0			0
$171$	−12.0000	−	20.7846i	−0.917663	−	1.58944i
$172$	−21.0000	−	12.1244i	−1.60123	−	0.924473i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−15.0000	−	8.66025i	−1.13389	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	3.50000	−	6.06218i	0.260153	−	0.450598i	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	3.50000	−	6.06218i	0.260153	−	0.450598i	0.966282	+	0.257485i	$0.0828937\pi$
$182$		0			0
$183$	−7.50000	−	12.9904i	−0.554416	−	0.960277i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	9.00000	−	15.5885i	0.654654	−	1.13389i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		−	13.8564i		−	1.00000i
$193$	23.0000			1.65558			0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000			1.65558			0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$		0			0
$196$	10.0000			0.714286
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	0.992434	−	0.122782i	$-0.0391815\pi$
$199$	5.50000	+	9.52628i	0.389885	+	0.675300i	−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$	−13.5000	−	4.33013i	−0.952217	−	0.305424i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−6.00000	−	3.46410i	−0.416025	−	0.240192i
$209$		0			0
$210$		0			0
$211$	−6.50000	−	11.2583i	−0.447478	−	0.775055i	0.550743	−	0.834675i	$-0.314345\pi$
$211$	−6.50000	−	11.2583i	−0.447478	−	0.775055i	−0.998221	+	0.0596196i	$0.981011\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−15.0000	−	25.9808i	−1.01827	−	1.76369i
$218$		0			0
$219$	−25.5000	+	14.7224i	−1.72313	+	0.994850i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	23.0000			1.54019			0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000			1.54019			0.770097	+	0.637927i	$0.220208\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$			27.7128i			1.83533i
$229$	25.5000	−	14.7224i	1.68509	−	0.972886i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	25.5000	−	14.7224i	1.68509	−	0.972886i	0.958187	−	0.286143i	$-0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.50000	+	7.79423i	−0.292306	+	0.506290i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−31.0000			−1.99689			−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−31.0000			−1.99689			−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$			15.5885i			1.00000i
$244$			17.3205i			1.10883i
$245$		0			0
$246$		0			0
$247$	12.0000	+	6.92820i	0.763542	+	0.440831i
$248$		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	$0.833333\pi$
$252$	−18.0000	+	10.3923i	−1.13389	+	0.654654i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		−	34.6410i		−	2.15249i
$260$		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	+	12.1244i	0.671932	+	0.740613i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$			32.9090i			1.99908i	0.0303728	+	0.999539i	$0.490331\pi$
$271$			32.9090i			1.99908i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$		0			0
$273$			10.3923i			0.628971i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.00000			−0.300421			−0.150210	−	0.988654i	$-0.547995\pi$
$277$	−5.00000			−0.300421			−0.150210	+	0.988654i	$0.547995\pi$
$278$		0			0
$279$	22.5000	+	12.9904i	1.34704	+	0.777714i
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$	32.0000			1.90220			0.951101	−	0.308879i	$-0.0999539\pi$
$283$	32.0000			1.90220			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
$292$	34.0000			1.98970
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−15.0000	−	8.66025i	−0.866025	−	0.500000i
$301$	21.0000	−	36.3731i	1.21042	−	2.09651i
$302$		0			0
$303$		0			0
$304$	16.0000	−	27.7128i	0.917663	−	1.58944i
$305$		0			0
$306$		0			0
$307$	−17.5000	−	30.3109i	−0.998778	−	1.72993i	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−17.5000	−	30.3109i	−0.998778	−	1.72993i	−0.456584	−	0.889680i	$-0.650927\pi$
$308$		0			0
$309$	30.0000	+	17.3205i	1.70664	+	0.985329i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			27.7128i			1.56642i	0.621757	+	0.783210i	$0.286419\pi$
$313$			27.7128i			1.56642i	−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$		0			0
$316$	9.00000	−	5.19615i	0.506290	−	0.292306i
$317$		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
$325$	−7.50000	+	4.33013i	−0.416025	+	0.240192i
$326$		0			0
$327$	−21.0000			−1.16130
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−31.5000	−	18.1865i	−1.73140	−	0.999622i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−31.5000	−	18.1865i	−1.73140	−	0.999622i	−0.851957	−	0.523612i	$-0.824584\pi$
$332$		0			0
$333$	15.0000	+	25.9808i	0.821995	+	1.42374i
$334$		0			0
$335$		0			0
$336$	24.0000			1.30931
$337$	−19.5000	+	11.2583i	−1.06223	+	0.613280i	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−19.5000	+	11.2583i	−1.06223	+	0.613280i	−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	6.92820i		−	0.374088i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	37.0000			1.98056			0.990282	−	0.139072i	$-0.0444119\pi$
$349$	37.0000			1.98056			0.990282	+	0.139072i	$0.0444119\pi$
$350$		0			0
$351$	−4.50000	−	7.79423i	−0.240192	−	0.416025i
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		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
$361$	−22.5000	+	38.9711i	−1.18421	+	2.05111i
$362$		0			0
$363$	−16.5000	+	9.52628i	−0.866025	+	0.500000i
$364$	6.00000	−	10.3923i	0.314485	−	0.544705i
$365$		0			0
$366$		0			0
$367$	33.0000	+	19.0526i	1.72259	+	0.994535i	0.913493	+	0.406855i	$0.133375\pi$
$367$	33.0000	+	19.0526i	1.72259	+	0.994535i	0.809093	+	0.587680i	$0.199959\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−15.0000	−	25.9808i	−0.777714	−	1.34704i
$373$	−31.5000	−	18.1865i	−1.63101	−	0.941663i	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−31.5000	−	18.1865i	−1.63101	−	0.941663i	−0.647225	−	0.762299i	$-0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	22.5000	−	12.9904i	1.15575	−	0.667271i	0.205466	−	0.978664i	$-0.434129\pi$
$379$	22.5000	−	12.9904i	1.15575	−	0.667271i	0.950281	+	0.311393i	$0.100796\pi$
$380$		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
$387$			36.3731i			1.84895i
$388$			10.3923i			0.527589i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.00000			0.0501886			0.0250943	−	0.999685i	$-0.492011\pi$
$397$	1.00000			0.0501886			0.0250943	+	0.999685i	$0.492011\pi$
$398$		0			0
$399$	−48.0000			−2.40301
$400$	10.0000	+	17.3205i	0.500000	+	0.866025i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−15.0000			−0.747203
$404$		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	$-0.444633\pi$
$409$	−12.0000	−	6.92820i	−0.593362	−	0.342578i	−0.766426	+	0.642333i	$0.777967\pi$
$410$		0			0
$411$		0			0
$412$	−20.0000	−	34.6410i	−0.985329	−	1.70664i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	39.0000			1.90984
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	−9.50000	−	16.4545i	−0.463002	−	0.801942i	0.536107	−	0.844150i	$-0.319894\pi$
$421$	−9.50000	−	16.4545i	−0.463002	−	0.801942i	−0.999109	+	0.0422075i	$0.986561\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−30.0000			−1.45180
$428$		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
$433$	−36.0000	−	20.7846i	−1.73005	−	0.998845i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$	−36.0000	−	20.7846i	−1.73005	−	0.998845i	−0.840996	−	0.541041i	$-0.818030\pi$
$434$		0			0
$435$		0			0
$436$	21.0000	+	12.1244i	1.00572	+	0.580651i
$437$		0			0
$438$		0			0
$439$	20.5000	+	35.5070i	0.978412	+	1.69466i	0.668184	+	0.743996i	$0.267072\pi$
$439$	20.5000	+	35.5070i	0.978412	+	1.69466i	0.310228	+	0.950662i	$0.399595\pi$
$440$		0			0
$441$	−7.50000	−	12.9904i	−0.357143	−	0.618590i
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		−	34.6410i		−	1.64399i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−24.0000	−	13.8564i	−1.13389	−	0.654654i
$449$		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
$453$	−28.5000	+	16.4545i	−1.33905	+	0.773099i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	15.5000	−	26.8468i	0.725059	−	1.25584i	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	15.5000	−	26.8468i	0.725059	−	1.25584i	0.958950	−	0.283577i	$-0.0915211\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	33.0000	−	19.0526i	1.53364	−	0.885448i	0.534450	−	0.845200i	$-0.320519\pi$
$463$	33.0000	−	19.0526i	1.53364	−	0.885448i	0.999190	−	0.0402476i	$-0.0128147\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$			10.3923i			0.480384i
$469$	−21.0000	+	19.0526i	−0.969690	+	0.879765i
$470$		0			0
$471$	21.0000	−	12.1244i	0.967629	−	0.558661i
$472$		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
$481$	−15.0000	−	8.66025i	−0.683941	−	0.394874i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	31.5000	−	18.1865i	1.42740	−	0.824110i	0.430486	−	0.902597i	$-0.358342\pi$
$487$	31.5000	−	18.1865i	1.42740	−	0.824110i	0.996915	+	0.0784867i	$0.0250088\pi$
$488$		0			0
$489$	−25.5000	−	14.7224i	−1.15315	−	0.665771i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$			34.6410i			1.55543i
$497$		0			0
$498$		0			0
$499$	−27.0000	+	15.5885i	−1.20869	+	0.697835i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−27.0000	+	15.5885i	−1.20869	+	0.697835i	−0.246214	+	0.969216i	$0.579187\pi$
$500$		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
$507$	−15.0000	−	8.66025i	−0.666173	−	0.384615i
$508$	1.00000	+	1.73205i	0.0443678	+	0.0768473i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$			58.8897i			2.60513i
$512$		0			0
$513$	36.0000	−	20.7846i	1.58944	−	0.917663i
$514$		0			0
$515$		0			0
$516$	21.0000	−	36.3731i	0.924473	−	1.60123i
$517$		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
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	0.940129	−	0.340818i	$-0.110704\pi$
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$	15.0000	−	25.9808i	0.654654	−	1.13389i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	48.0000	+	27.7128i	2.08106	+	1.20150i
$533$		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
$543$	10.5000	+	6.06218i	0.450598	+	0.260153i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	19.5000	+	11.2583i	0.833760	+	0.481371i	0.855138	−	0.518400i	$-0.173472\pi$
$547$	19.5000	+	11.2583i	0.833760	+	0.481371i	−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$	22.5000	−	12.9904i	0.960277	−	0.554416i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	9.00000	+	15.5885i	0.382719	+	0.662889i
$554$		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
$559$	−10.5000	−	18.1865i	−0.444103	−	0.769208i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	27.0000	+	15.5885i	1.13389	+	0.654654i
$568$		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	$-0.906868\pi$
$577$	−40.5000	−	23.3827i	−1.68604	−	0.973434i	−0.728535	−	0.685009i	$-0.759798\pi$
$578$		0			0
$579$			39.8372i			1.65558i
$580$		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	$0.833333\pi$
$588$			17.3205i			0.714286i
$589$		−	69.2820i		−	2.85472i
$590$		0			0
$591$		0			0
$592$	−20.0000	+	34.6410i	−0.821995	+	1.42374i
$593$		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
$597$	−16.5000	+	9.52628i	−0.675300	+	0.389885i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	11.5000	+	19.9186i	0.469095	+	0.812496i	0.999376	−	0.0353259i	$-0.0112469\pi$
$601$	11.5000	+	19.9186i	0.469095	+	0.812496i	−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	7.50000	−	23.3827i	0.305424	−	0.952217i
$604$	38.0000			1.54620
$605$		0			0
$606$		0			0
$607$	−14.5000	+	25.1147i	−0.588537	+	1.01938i	0.405887	+	0.913923i	$0.366962\pi$
$607$	−14.5000	+	25.1147i	−0.588537	+	1.01938i	−0.994424	+	0.105453i	$0.966371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	24.5000	+	42.4352i	0.984738	+	1.70562i	0.643094	+	0.765787i	$0.277650\pi$
$619$	24.5000	+	42.4352i	0.984738	+	1.70562i	0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	6.00000	−	10.3923i	0.240192	−	0.416025i
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$	−28.0000			−1.11732
$629$		0			0
$630$		0			0
$631$	21.0000	−	12.1244i	0.835997	−	0.482663i	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	21.0000	−	12.1244i	0.835997	−	0.482663i	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
$637$	7.50000	+	4.33013i	0.297161	+	0.171566i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	40.0000			1.57745			0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000			1.57745			0.788723	+	0.614749i	$0.210743\pi$
$644$		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
$651$	45.0000	−	25.9808i	1.76369	−	1.01827i
$652$	17.0000	+	29.4449i	0.665771	+	1.15315i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		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
$661$			15.5885i			0.606321i	0.952940	+	0.303160i	$0.0980418\pi$
$661$			15.5885i			0.606321i	−0.952940	+	0.303160i	$0.901958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			39.8372i			1.54019i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	50.2295i		−	1.93620i	−0.250557	−	0.968102i	$-0.580614\pi$
$673$		−	50.2295i		−	1.93620i	0.250557	−	0.968102i	$-0.419386\pi$
$674$		0			0
$675$			25.9808i			1.00000i
$676$	10.0000	+	17.3205i	0.384615	+	0.666173i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−18.0000			−0.690777
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$	−48.0000			−1.83533
$685$		0			0
$686$		0			0
$687$	25.5000	+	44.1673i	0.972886	+	1.68509i
$688$	−42.0000	+	24.2487i	−1.60123	+	0.924473i
$689$		0			0
$690$		0			0
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	4.00000	−	6.92820i	0.152167	−	0.263561i	0.932024	+	0.362397i	$0.118041\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−30.0000	+	17.3205i	−1.13389	+	0.654654i
$701$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$701$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$702$		0			0
$703$	40.0000	−	69.2820i	1.50863	−	2.61302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.0000	−	19.0526i	−0.413114	−	0.715534i	0.582115	−	0.813107i	$-0.302225\pi$
$709$	−11.0000	−	19.0526i	−0.413114	−	0.715534i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$	−13.5000	−	7.79423i	−0.506290	−	0.292306i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	60.0000	−	34.6410i	2.23452	−	1.29010i
$722$		0			0
$723$		−	53.6936i		−	1.99689i
$724$	−7.00000	−	12.1244i	−0.260153	−	0.450598i
$725$		0			0
$726$		0			0
$727$	−46.5000	+	26.8468i	−1.72459	+	0.995692i	−0.815935	+	0.578144i	$0.803777\pi$
$727$	−46.5000	+	26.8468i	−1.72459	+	0.995692i	−0.908655	+	0.417548i	$0.862889\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	−30.0000			−1.10883
$733$	18.0000	+	10.3923i	0.664845	+	0.383849i	0.794121	−	0.607760i	$-0.207932\pi$
$733$	18.0000	+	10.3923i	0.664845	+	0.383849i	−0.129275	+	0.991609i	$0.541265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−10.5000	+	6.06218i	−0.386249	+	0.223001i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−10.5000	+	6.06218i	−0.386249	+	0.223001i	0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$	−12.0000	+	20.7846i	−0.440831	+	0.763542i
$742$		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
$751$	−52.0000			−1.89751			−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−52.0000			−1.89751			−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−18.0000	−	31.1769i	−0.654654	−	1.13389i
$757$	1.50000	+	0.866025i	0.0545184	+	0.0314762i	0.527011	−	0.849858i	$-0.323312\pi$
$757$	1.50000	+	0.866025i	0.0545184	+	0.0314762i	−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−21.0000	+	36.3731i	−0.760251	+	1.31679i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.0000	−	13.8564i	−0.866025	−	0.500000i
$769$	−22.5000	−	12.9904i	−0.811371	−	0.468445i	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	−22.5000	−	12.9904i	−0.811371	−	0.468445i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$	23.0000	−	39.8372i	0.827788	−	1.43377i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	37.5000	+	21.6506i	1.34704	+	0.777714i
$776$		0			0
$777$	60.0000			2.15249
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	10.0000	−	17.3205i	0.357143	−	0.618590i
$785$		0			0
$786$		0			0
$787$	−3.00000	+	1.73205i	−0.106938	+	0.0617409i	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−3.00000	+	1.73205i	−0.106938	+	0.0617409i	0.445577	+	0.895244i	$0.352999\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.50000	+	12.9904i	−0.266333	+	0.461302i
$794$		0			0
$795$		0			0
$796$	22.0000			0.779769
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−21.0000	+	19.0526i	−0.740613	+	0.671932i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−37.5000	−	21.6506i	−1.31680	−	0.760257i	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	−37.5000	−	21.6506i	−1.31680	−	0.760257i	−0.983213	+	0.182462i	$0.941593\pi$
$812$		0			0
$813$	−57.0000			−1.99908
$814$		0			0
$815$		0			0
$816$		0			0
$817$	84.0000	−	48.4974i	2.93879	−	1.69671i
$818$		0			0
$819$	−18.0000			−0.628971
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	2.50000	+	4.33013i	0.0871445	+	0.150939i	0.906303	−	0.422628i	$-0.138892\pi$
$823$	2.50000	+	4.33013i	0.0871445	+	0.150939i	−0.819159	+	0.573567i	$0.805559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$827$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$	−53.0000			−1.84077			−0.920383	−	0.391018i	$-0.872123\pi$
$829$	−53.0000			−1.84077			−0.920383	+	0.391018i	$0.872123\pi$
$830$		0			0
$831$		−	8.66025i		−	0.300421i
$832$	−12.0000	+	6.92820i	−0.416025	+	0.240192i
$833$		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		0.500000	−	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
$847$			38.1051i			1.30931i
$848$		0			0
$849$			55.4256i			1.90220i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−11.5000	−	19.9186i	−0.393753	−	0.681999i	0.599189	−	0.800608i	$-0.295490\pi$
$853$	−11.5000	−	19.9186i	−0.393753	−	0.681999i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		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
$859$	21.5000	−	37.2391i	0.733571	−	1.27058i	−0.221777	−	0.975097i	$-0.571186\pi$
$859$	21.5000	−	37.2391i	0.733571	−	1.27058i	0.955348	−	0.295484i	$-0.0954809\pi$
$860$		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
$868$	−60.0000			−2.03653
$869$		0			0
$870$		0			0
$871$	3.00000	+	13.8564i	0.101651	+	0.469506i
$872$		0			0
$873$	13.5000	−	7.79423i	0.456906	−	0.263795i
$874$		0			0
$875$		0			0
$876$			58.8897i			1.98970i
$877$	−12.5000	−	21.6506i	−0.422095	−	0.731090i	0.574049	−	0.818821i	$-0.305372\pi$
$877$	−12.5000	−	21.6506i	−0.422095	−	0.731090i	−0.996144	+	0.0877308i	$0.972038\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	51.0000	+	29.4449i	1.71629	+	0.990899i	0.925449	+	0.378873i	$0.123688\pi$
$883$	51.0000	+	29.4449i	1.71629	+	0.990899i	0.790838	+	0.612026i	$0.209645\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−3.00000	+	1.73205i	−0.100617	+	0.0580911i
$890$		0			0
$891$		0			0
$892$	23.0000	−	39.8372i	0.770097	−	1.33385i
$893$		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
$903$	63.0000	+	36.3731i	2.09651	+	1.21042i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	20.0000	−	34.6410i	0.664089	−	1.15024i	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	20.0000	−	34.6410i	0.664089	−	1.15024i	0.979531	−	0.201291i	$-0.0645138\pi$
$908$		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
$916$		−	58.8897i		−	1.94577i
$917$		0			0
$918$		0			0
$919$	27.0000	−	15.5885i	0.890648	−	0.514216i	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	27.0000	−	15.5885i	0.890648	−	0.514216i	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
$925$	25.0000	+	43.3013i	0.821995	+	1.42374i
$926$		0			0
$927$	−30.0000	+	51.9615i	−0.985329	+	1.70664i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−20.0000	+	34.6410i	−0.655474	+	1.13531i
$932$		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
$939$	−48.0000			−1.56642
$940$		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		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$	9.00000	+	15.5885i	0.292306	+	0.506290i
$949$	25.5000	+	14.7224i	0.827765	+	0.477910i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		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
$964$	−31.0000	+	53.6936i	−0.998443	+	1.72935i
$965$		0			0
$966$		0			0
$967$	20.5000	−	35.5070i	0.659236	−	1.14183i	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	20.5000	−	35.5070i	0.659236	−	1.14183i	0.980814	−	0.194946i	$-0.0624533\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	27.0000	+	15.5885i	0.866025	+	0.500000i
$973$	39.0000	−	67.5500i	1.25028	−	2.16555i
$974$		0			0
$975$	−7.50000	−	12.9904i	−0.240192	−	0.416025i
$976$	30.0000	+	17.3205i	0.960277	+	0.554416i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	36.3731i		−	1.16130i
$982$		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	−	13.8564i	0.763542	−	0.440831i
$989$		0			0
$990$		0			0
$991$		−	45.0333i		−	1.43053i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$		−	45.0333i		−	1.43053i	0.698853	−	0.715265i	$-0.253694\pi$
$992$		0			0
$993$	31.5000	−	54.5596i	0.999622	−	1.73140i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	10.0000			0.316703			0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000			0.316703			0.158352	+	0.987383i	$0.449382\pi$
$998$		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	201.2.f.a.164.1	yes	2
3.2	odd	2	CM	201.2.f.a.164.1	yes	2
67.38	odd	6	inner	201.2.f.a.38.1	✓	2
201.38	even	6	inner	201.2.f.a.38.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.f.a.38.1	✓	2	67.38	odd	6	inner
201.2.f.a.38.1	✓	2	201.38	even	6	inner
201.2.f.a.164.1	yes	2	1.1	even	1	trivial
201.2.f.a.164.1	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:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +(1.00000 - 1.73205i) q^{4} +(3.00000 + 1.73205i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +(1.00000 - 1.73205i) q^{4} +(3.00000 + 1.73205i) q^{7} -3.00000 q^{9} +(3.00000 + 1.73205i) q^{12} +(1.50000 - 0.866025i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(4.00000 + 6.92820i) q^{19} +(-3.00000 + 5.19615i) q^{21} -5.00000 q^{25} -5.19615i q^{27} +(6.00000 - 3.46410i) q^{28} +(-7.50000 - 4.33013i) q^{31} +(-3.00000 + 5.19615i) q^{36} +(-5.00000 - 8.66025i) q^{37} +(1.50000 + 2.59808i) q^{39} -12.1244i q^{43} +(6.00000 - 3.46410i) q^{48} +(2.50000 + 4.33013i) q^{49} -3.46410i q^{52} +(-12.0000 + 6.92820i) q^{57} +(-7.50000 + 4.33013i) q^{61} +(-9.00000 - 5.19615i) q^{63} -8.00000 q^{64} +(-2.50000 + 7.79423i) q^{67} +(8.50000 + 14.7224i) q^{73} -8.66025i q^{75} +16.0000 q^{76} +(4.50000 + 2.59808i) q^{79} +9.00000 q^{81} +(6.00000 + 10.3923i) q^{84} +6.00000 q^{91} +(7.50000 - 12.9904i) q^{93} +(-4.50000 + 2.59808i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 + 6 * q^7 - 6 * q^9	\( 2 q + 2 q^{4} + 6 q^{7} - 6 q^{9} + 6 q^{12} + 3 q^{13} - 4 q^{16} + 8 q^{19} - 6 q^{21} - 10 q^{25} + 12 q^{28} - 15 q^{31} - 6 q^{36} - 10 q^{37} + 3 q^{39} + 12 q^{48} + 5 q^{49} - 24 q^{57} - 15 q^{61} - 18 q^{63} - 16 q^{64} - 5 q^{67} + 17 q^{73} + 32 q^{76} + 9 q^{79} + 18 q^{81} + 12 q^{84} + 12 q^{91} + 15 q^{93} - 9 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^7 - 6 * q^9 + 6 * q^12 + 3 * q^13 - 4 * q^16 + 8 * q^19 - 6 * q^21 - 10 * q^25 + 12 * q^28 - 15 * q^31 - 6 * q^36 - 10 * q^37 + 3 * q^39 + 12 * q^48 + 5 * q^49 - 24 * q^57 - 15 * q^61 - 18 * q^63 - 16 * q^64 - 5 * q^67 + 17 * q^73 + 32 * q^76 + 9 * q^79 + 18 * q^81 + 12 * q^84 + 12 * q^91 + 15 * q^93 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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