LMFDB - Embedded newform 147.2.o.a.101.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,2,Mod(5,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 29]))

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

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

chi := DirichletCharacter("147.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	147.o (of order $42$, degree $12$, 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:	$1.17380090971$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{42}]$

Embedding invariants

Embedding label			101.1
Root			$0.826239 + 0.563320i$ of defining polynomial
Character	$\chi$	$=$	147.101
Dual form			147.2.o.a.131.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.72721 + 0.129436i) q^{3} +(-1.46610 - 1.36035i) q^{4} +(2.34300 - 1.22896i) q^{7} +(2.96649 + 0.447127i) q^{9} +O(q^{10})$	$q+(1.72721 + 0.129436i) q^{3} +(-1.46610 - 1.36035i) q^{4} +(2.34300 - 1.22896i) q^{7} +(2.96649 + 0.447127i) q^{9} +(-2.35619 - 2.53937i) q^{12} +(-1.15450 + 0.920684i) q^{13} +(0.298920 + 3.98882i) q^{16} +(-7.54531 + 4.35628i) q^{19} +(4.20592 - 1.81940i) q^{21} +(-1.82671 - 4.65437i) q^{25} +(5.06587 + 1.15625i) q^{27} +(-5.10689 - 1.38551i) q^{28} +(4.24128 + 2.44870i) q^{31} +(-3.74094 - 4.69099i) q^{36} +(-8.50183 + 7.88854i) q^{37} +(-2.11323 + 1.44078i) q^{39} +(-5.73174 - 2.76026i) q^{43} +6.92820i q^{48} +(3.97932 - 5.75891i) q^{49} +(2.94507 + 0.220702i) q^{52} +(-13.5962 + 6.54757i) q^{57} +(10.3323 + 11.1355i) q^{61} +(7.50000 - 2.59808i) q^{63} +(4.98792 - 6.25465i) q^{64} +(7.58561 - 13.1387i) q^{67} +(9.58172 - 3.76055i) q^{73} +(-2.55265 - 8.27550i) q^{75} +(16.9883 + 3.87746i) q^{76} +(-7.35657 - 12.7420i) q^{79} +(8.60016 + 2.65280i) q^{81} +(-8.64133 - 3.05409i) q^{84} +(-1.57352 + 3.57600i) q^{91} +(7.00862 + 4.77840i) q^{93} -2.30631i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9	$ 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9} - 6 q^{12} + 4 q^{16} + 9 q^{19} + 6 q^{21} - 5 q^{25} - 8 q^{28} - 15 q^{31} + 12 q^{36} - 76 q^{37} - 66 q^{39} + 10 q^{43} + 13 q^{49} + 34 q^{52} - 18 q^{57} + 79 q^{61} + 90 q^{63} - 16 q^{64} + 11 q^{67} + 27 q^{73} + 15 q^{75} - 13 q^{79} + 9 q^{81} + 6 q^{84} - 9 q^{91} + 15 q^{93}+O(q^{100}) $ 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9 - 6 * q^12 + 4 * q^16 + 9 * q^19 + 6 * q^21 - 5 * q^25 - 8 * q^28 - 15 * q^31 + 12 * q^36 - 76 * q^37 - 66 * q^39 + 10 * q^43 + 13 * q^49 + 34 * q^52 - 18 * q^57 + 79 * q^61 + 90 * q^63 - 16 * q^64 + 11 * q^67 + 27 * q^73 + 15 * q^75 - 13 * q^79 + 9 * q^81 + 6 * q^84 - 9 * q^91 + 15 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/147\mathbb{Z}\right)^\times$.

$n$	$50$	$52$
$\chi(n)$	$-1$	$e\left(\frac{1}{42}\right)$

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		0.365341	−	0.930874i	$-0.380952\pi$
$2$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$3$	1.72721	+	0.129436i	0.997204	+	0.0747301i
$3$	1.72721	+	0.129436i	0.997204	+	0.0747301i
$4$	−1.46610	−	1.36035i	−0.733052	−	0.680173i
$5$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$5$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$6$		0			0
$7$	2.34300	−	1.22896i	0.885572	−	0.464503i
$7$	2.34300	−	1.22896i	0.885572	−	0.464503i
$8$		0			0
$9$	2.96649	+	0.447127i	0.988831	+	0.149042i
$10$		0			0
$11$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$11$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$12$	−2.35619	−	2.53937i	−0.680173	−	0.733052i
$13$	−1.15450	+	0.920684i	−0.320201	+	0.255352i	−0.770378	−	0.637588i	$-0.779932\pi$
$13$	−1.15450	+	0.920684i	−0.320201	+	0.255352i	0.450177	+	0.892939i	$0.351361\pi$
$14$		0			0
$15$		0			0
$16$	0.298920	+	3.98882i	0.0747301	+	0.997204i
$17$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$17$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$18$		0			0
$19$	−7.54531	+	4.35628i	−1.73101	+	0.999400i	−0.848190	+	0.529692i	$0.822307\pi$
$19$	−7.54531	+	4.35628i	−1.73101	+	0.999400i	−0.882822	+	0.469708i	$0.844359\pi$
$20$		0			0
$21$	4.20592	−	1.81940i	0.917808	−	0.397025i
$22$		0			0
$23$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$23$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$24$		0			0
$25$	−1.82671	−	4.65437i	−0.365341	−	0.930874i
$26$		0			0
$27$	5.06587	+	1.15625i	0.974928	+	0.222521i
$28$	−5.10689	−	1.38551i	−0.965112	−	0.261837i
$29$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$29$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$30$		0			0
$31$	4.24128	+	2.44870i	0.761756	+	0.439800i	0.829926	−	0.557874i	$-0.188383\pi$
$31$	4.24128	+	2.44870i	0.761756	+	0.439800i	−0.0681697	+	0.997674i	$0.521716\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.74094	−	4.69099i	−0.623490	−	0.781831i
$37$	−8.50183	+	7.88854i	−1.39769	+	1.29687i	−0.493497	+	0.869747i	$0.664282\pi$
$37$	−8.50183	+	7.88854i	−1.39769	+	1.29687i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$	−2.11323	+	1.44078i	−0.338388	+	0.230709i
$40$		0			0
$41$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$41$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$42$		0			0
$43$	−5.73174	−	2.76026i	−0.874082	−	0.420936i	−0.0576228	−	0.998338i	$-0.518352\pi$
$43$	−5.73174	−	2.76026i	−0.874082	−	0.420936i	−0.816460	+	0.577402i	$0.804066\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$47$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$48$			6.92820i			1.00000i
$49$	3.97932	−	5.75891i	0.568475	−	0.822701i
$50$		0			0
$51$		0			0
$52$	2.94507	+	0.220702i	0.408407	+	0.0306059i
$53$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$53$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−13.5962	+	6.54757i	−1.80086	+	0.867247i
$58$		0			0
$59$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$59$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$60$		0			0
$61$	10.3323	+	11.1355i	1.32291	+	1.42576i	0.832240	+	0.554416i	$0.187058\pi$
$61$	10.3323	+	11.1355i	1.32291	+	1.42576i	0.490672	+	0.871344i	$0.336751\pi$
$62$		0			0
$63$	7.50000	−	2.59808i	0.944911	−	0.327327i
$64$	4.98792	−	6.25465i	0.623490	−	0.781831i
$65$		0			0
$66$		0			0
$67$	7.58561	−	13.1387i	0.926730	−	1.60514i	0.137974	−	0.990436i	$-0.455941\pi$
$67$	7.58561	−	13.1387i	0.926730	−	1.60514i	0.788756	−	0.614707i	$-0.210726\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$71$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$72$		0			0
$73$	9.58172	−	3.76055i	1.12146	−	0.440139i	0.269105	−	0.963111i	$-0.413272\pi$
$73$	9.58172	−	3.76055i	1.12146	−	0.440139i	0.852350	+	0.522972i	$0.175177\pi$
$74$		0			0
$75$	−2.55265	−	8.27550i	−0.294755	−	0.955573i
$76$	16.9883	+	3.87746i	1.94869	+	0.444775i
$77$		0			0
$78$		0			0
$79$	−7.35657	−	12.7420i	−0.827679	−	1.43358i	−0.899854	−	0.436190i	$-0.856327\pi$
$79$	−7.35657	−	12.7420i	−0.827679	−	1.43358i	0.0721754	−	0.997392i	$-0.477006\pi$
$80$		0			0
$81$	8.60016	+	2.65280i	0.955573	+	0.294755i
$82$		0			0
$83$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$83$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$84$	−8.64133	−	3.05409i	−0.942846	−	0.333228i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$89$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$90$		0			0
$91$	−1.57352	+	3.57600i	−0.164949	+	0.374867i
$92$		0			0
$93$	7.00862	+	4.77840i	0.726760	+	0.495497i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	2.30631i		−	0.234171i	−0.993122	−	0.117085i	$-0.962645\pi$
$97$		−	2.30631i		−	0.234171i	0.993122	−	0.117085i	$-0.0373551\pi$
$98$		0			0
$99$		0			0
$100$	−3.65341	+	9.30874i	−0.365341	+	0.930874i
$101$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$101$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$102$		0			0
$103$	−3.60877	+	5.29310i	−0.355583	+	0.521544i	−0.961799	−	0.273755i	$-0.911734\pi$
$103$	−3.60877	+	5.29310i	−0.355583	+	0.521544i	0.606216	+	0.795300i	$0.292687\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$107$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$108$	−5.85419	−	8.58653i	−0.563320	−	0.826239i
$109$	−9.94359	+	1.49876i	−0.952424	+	0.143555i	−0.606832	−	0.794830i	$-0.707560\pi$
$109$	−9.94359	+	1.49876i	−0.952424	+	0.143555i	−0.345592	+	0.938385i	$0.612322\pi$
$110$		0			0
$111$	−15.7055	+	12.5247i	−1.49070	+	1.18879i
$112$	5.60246	+	8.97844i	0.529383	+	0.848383i
$113$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$113$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.83648	+	2.21499i	−0.354683	+	0.204776i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.5113	−	3.24231i	0.955573	−	0.294755i
$122$		0			0
$123$		0			0
$124$	−2.88707	−	9.35966i	−0.259267	−	0.840522i
$125$		0			0
$126$		0			0
$127$	0.526730	+	2.30776i	0.0467397	+	0.204780i	0.992906	−	0.118900i	$-0.0379369\pi$
$127$	0.526730	+	2.30776i	0.0467397	+	0.204780i	−0.946166	+	0.323680i	$0.895080\pi$
$128$		0			0
$129$	−9.54263	−	5.50944i	−0.840182	−	0.485079i
$130$		0			0
$131$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$131$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$132$		0			0
$133$	−12.3250	+	19.4797i	−1.06871	+	1.68910i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$137$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$138$		0			0
$139$	−10.1199	−	21.0142i	−0.858359	−	1.78240i	−0.561493	−	0.827482i	$-0.689773\pi$
$139$	−10.1199	−	21.0142i	−0.858359	−	1.78240i	−0.296866	−	0.954919i	$-0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.896761	+	11.9664i	−0.0747301	+	0.997204i
$145$		0			0
$146$		0			0
$147$	7.61853	−	9.43176i	0.628366	−	0.777918i
$148$	23.1957			1.90668
$149$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$149$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$150$		0			0
$151$	−15.7257	−	14.5913i	−1.27974	−	1.18742i	−0.971744	−	0.236036i	$-0.924152\pi$
$151$	−15.7257	−	14.5913i	−1.27974	−	1.18742i	−0.307995	−	0.951388i	$-0.599658\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	5.05817	+	0.762397i	0.404978	+	0.0610406i
$157$	13.9707	+	20.4913i	1.11498	+	1.63538i	0.676036	+	0.736868i	$0.263696\pi$
$157$	13.9707	+	20.4913i	1.11498	+	1.63538i	0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.32488	−	17.6793i	−0.103773	−	1.38475i	−0.769544	−	0.638594i	$-0.779516\pi$
$163$	−1.32488	−	17.6793i	−0.103773	−	1.38475i	0.665771	−	0.746156i	$-0.268103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$167$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$168$		0			0
$169$	−2.40756	+	10.5482i	−0.185197	+	0.811400i
$170$		0			0
$171$	−24.3309	+	9.54918i	−1.86063	+	0.730244i
$172$	4.64842	+	11.8440i	0.354439	+	0.903095i
$173$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$173$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$174$		0			0
$175$	−10.0000	−	8.66025i	−0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$179$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$180$		0			0
$181$	−15.9264	−	12.7009i	−1.18380	−	0.944051i	−0.184554	−	0.982822i	$-0.559084\pi$
$181$	−15.9264	−	12.7009i	−1.18380	−	0.944051i	−0.999248	+	0.0387714i	$0.987656\pi$
$182$		0			0
$183$	16.4046	+	20.5708i	1.21267	+	1.52063i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.2903	−	3.51664i	0.966730	−	0.255798i
$190$		0			0
$191$		0			0		−0.826239	−	0.563320i	$-0.809524\pi$
$191$		0			0		0.826239	+	0.563320i	$0.190476\pi$
$192$	9.42475	−	10.1575i	0.680173	−	0.733052i
$193$	−2.07636	+	27.7071i	−0.149460	+	1.99440i	−0.0719816	+	0.997406i	$0.522932\pi$
$193$	−2.07636	+	27.7071i	−0.149460	+	1.99440i	−0.0774780	+	0.996994i	$0.524687\pi$
$194$		0			0
$195$		0			0
$196$	−13.6682	+	3.02990i	−0.976300	+	0.216421i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−19.6763	−	1.47453i	−1.39481	−	0.104527i	−0.643993	−	0.765032i	$-0.722723\pi$
$199$	−19.6763	−	1.47453i	−1.39481	−	0.104527i	−0.750822	+	0.660505i	$0.770342\pi$
$200$		0			0
$201$	14.8025	−	21.7113i	1.04409	−	1.53140i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4.01754	−	4.32988i	−0.278566	−	0.300223i
$209$		0			0
$210$		0			0
$211$	12.5199	−	15.6995i	0.861908	−	1.08080i	−0.134050	−	0.990975i	$-0.542798\pi$
$211$	12.5199	−	15.6995i	0.861908	−	1.08080i	0.995958	−	0.0898234i	$-0.0286303\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	12.9467	+	0.524964i	0.878878	+	0.0356369i
$218$		0			0
$219$	17.0364	−	5.25502i	1.15121	−	0.355102i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	24.3590	+	5.55979i	1.63120	+	0.372311i	0.937509	−	0.347960i	$-0.113126\pi$
$223$	24.3590	+	5.55979i	1.63120	+	0.372311i	0.693693	+	0.720271i	$0.255983\pi$
$224$		0			0
$225$	−3.33781	−	14.6239i	−0.222521	−	0.974928i
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$	28.8404	+	8.89607i	1.91000	+	0.589157i
$229$	23.7648	−	1.78092i	1.57042	−	0.117687i	0.739144	−	0.673548i	$-0.235230\pi$
$229$	23.7648	−	1.78092i	1.57042	−	0.117687i	0.831275	+	0.555861i	$0.187611\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$233$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−11.0571	−	22.9602i	−0.718233	−	1.49143i
$238$		0			0
$239$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$239$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$240$		0			0
$241$	−12.8512	+	13.8503i	−0.827818	+	0.892176i	−0.995605	−	0.0936477i	$-0.970147\pi$
$241$	−12.8512	+	13.8503i	−0.827818	+	0.892176i	0.167787	+	0.985823i	$0.446338\pi$
$242$		0			0
$243$	14.5109	+	5.69510i	0.930874	+	0.365341i
$244$		−	30.3813i		−	1.94496i
$245$		0			0
$246$		0			0
$247$	4.70030	−	11.9762i	0.299073	−	0.762026i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$251$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$252$	−14.5301	−	6.39354i	−0.915308	−	0.402755i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−15.8213	+	2.38468i	−0.988831	+	0.149042i
$257$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$257$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$258$		0			0
$259$	−10.2251	+	28.9313i	−0.635358	+	1.79770i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−28.9944	+	8.94359i	−1.77112	+	0.546317i
$269$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$269$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$270$		0			0
$271$	8.18063	+	26.5210i	0.496938	+	1.61103i	0.762534	+	0.646949i	$0.223955\pi$
$271$	8.18063	+	26.5210i	0.496938	+	1.61103i	−0.265596	+	0.964084i	$0.585569\pi$
$272$		0			0
$273$	−3.18065	+	5.97282i	−0.192502	+	0.361492i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	21.6623	+	6.68194i	1.30156	+	0.401479i	0.866616	−	0.498976i	$-0.166291\pi$
$277$	21.6623	+	6.68194i	1.30156	+	0.401479i	0.434948	+	0.900456i	$0.356767\pi$
$278$		0			0
$279$	11.4868	+	9.16045i	0.687699	+	0.548422i
$280$		0			0
$281$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$281$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$282$		0			0
$283$	−0.0742884	−	0.492871i	−0.00441599	−	0.0292982i	0.986514	−	0.163675i	$-0.0523348\pi$
$283$	−0.0742884	−	0.492871i	−0.00441599	−	0.0292982i	−0.990930	+	0.134377i	$0.957097\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−14.0461	−	9.57644i	−0.826239	−	0.563320i
$290$		0			0
$291$	0.298521	−	3.98348i	0.0174996	−	0.233516i
$292$	−19.1634	−	7.52110i	−1.12146	−	0.440139i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−7.51509	+	15.6052i	−0.433884	+	0.900969i
$301$	−16.8217	+	0.576772i	−0.969589	+	0.0332446i
$302$		0			0
$303$		0			0
$304$	−19.6319	−	28.7946i	−1.12596	−	1.65149i
$305$		0			0
$306$		0			0
$307$	−22.2384	+	17.7346i	−1.26921	+	1.01216i	−0.270436	+	0.962738i	$0.587168\pi$
$307$	−22.2384	+	17.7346i	−1.26921	+	1.01216i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$		0			0
$309$	−6.91822	+	8.67517i	−0.393564	+	0.493513i
$310$		0			0
$311$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$311$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$312$		0			0
$313$	30.5523	−	17.6394i	1.72692	−	0.997036i	0.824989	−	0.565149i	$-0.191181\pi$
$313$	30.5523	−	17.6394i	1.72692	−	0.997036i	0.901928	−	0.431887i	$-0.142152\pi$
$314$		0			0
$315$		0			0
$316$	−6.54797	+	28.6885i	−0.368352	+	1.61385i
$317$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$317$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$318$		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
$325$	6.39414	+	3.69166i	0.354683	+	0.204776i
$326$		0			0
$327$	−17.3686	+	1.30160i	−0.960488	+	0.0719786i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	20.3892	−	18.9184i	1.12069	−	1.03985i	0.121810	−	0.992553i	$-0.461130\pi$
$331$	20.3892	−	18.9184i	1.12069	−	1.03985i	0.998881	−	0.0472962i	$-0.0150605\pi$
$332$		0			0
$333$	−28.7478	+	19.5999i	−1.57537	+	1.07407i
$334$		0			0
$335$		0			0
$336$	8.51447	+	16.2328i	0.464503	+	0.885572i
$337$	−18.2775	−	8.80199i	−0.995640	−	0.479475i	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−18.2775	−	8.80199i	−0.995640	−	0.479475i	−0.859457	+	0.511208i	$0.829198\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	2.24611	−	18.3836i	0.121279	−	0.992619i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$347$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$348$		0			0
$349$	−14.1203	+	29.3211i	−0.755842	+	1.56952i	0.0646791	+	0.997906i	$0.479398\pi$
$349$	−14.1203	+	29.3211i	−0.755842	+	1.56952i	−0.820521	+	0.571616i	$0.806317\pi$
$350$		0			0
$351$	−6.91310	+	3.32917i	−0.368994	+	0.177698i
$352$		0			0
$353$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$353$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$359$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$360$		0			0
$361$	28.4544	−	49.2845i	1.49760	−	2.59392i
$362$		0			0
$363$	18.5749	−	4.23959i	0.974928	−	0.222521i
$364$	7.17153	−	3.10226i	0.375890	−	0.162603i
$365$		0			0
$366$		0			0
$367$	27.2100	−	10.6792i	1.42035	−	0.557447i	0.473983	−	0.880534i	$-0.342816\pi$
$367$	27.2100	−	10.6792i	1.42035	−	0.557447i	0.946369	+	0.323087i	$0.104721\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.77509	−	16.5398i	−0.195729	−	0.857547i
$373$	18.2715	+	31.6471i	0.946062	+	1.63863i	0.753611	+	0.657320i	$0.228310\pi$
$373$	18.2715	+	31.6471i	0.946062	+	1.63863i	0.192451	+	0.981307i	$0.438356\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	20.2935	+	25.4473i	1.04241	+	1.30714i	0.950281	+	0.311393i	$0.100796\pi$
$379$	20.2935	+	25.4473i	1.04241	+	1.30714i	0.0921284	+	0.995747i	$0.470633\pi$
$380$		0			0
$381$	0.611065	+	4.05415i	0.0313058	+	0.207700i
$382$		0			0
$383$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$383$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−15.7690	−	10.7511i	−0.801582	−	0.546510i
$388$	−3.13738	+	3.38129i	−0.159276	+	0.171659i
$389$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$389$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.8336	+	24.6904i	−0.844856	+	1.23918i	0.124181	+	0.992260i	$0.460370\pi$
$397$	−16.8336	+	24.6904i	−0.844856	+	1.23918i	−0.969037	+	0.246916i	$0.920583\pi$
$398$		0			0
$399$	−23.8092	+	32.0501i	−1.19195	+	1.60451i
$400$	18.0194	−	8.67767i	0.900969	−	0.433884i
$401$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$401$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$402$		0			0
$403$	−7.15104	+	1.07785i	−0.356219	+	0.0536914i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.62444	−	14.9921i	0.228664	−	0.741310i	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	4.62444	−	14.9921i	0.228664	−	0.741310i	0.995090	−	0.0989775i	$-0.0315572\pi$
$410$		0			0
$411$		0			0
$412$	12.4913	−	2.85105i	0.615401	−	0.140461i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−14.7592	−	37.6058i	−0.722760	−	1.84156i
$418$		0			0
$419$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$419$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$420$		0			0
$421$	−7.32096	−	32.0752i	−0.356802	−	1.56325i	−0.761107	−	0.648626i	$-0.775344\pi$
$421$	−7.32096	−	32.0752i	−0.356802	−	1.56325i	0.404305	−	0.914624i	$-0.367513\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	37.8937	+	13.3927i	1.83380	+	0.648117i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.826239	−	0.563320i	$-0.190476\pi$
$431$		0			0		−0.826239	+	0.563320i	$0.809524\pi$
$432$	−3.09779	+	20.5525i	−0.149042	+	0.988831i
$433$	−5.78158	−	12.0056i	−0.277845	−	0.576952i	0.714615	−	0.699518i	$-0.246602\pi$
$433$	−5.78158	−	12.0056i	−0.277845	−	0.576952i	−0.992460	+	0.122566i	$0.960888\pi$
$434$		0			0
$435$		0			0
$436$	16.6172	+	11.3294i	0.795818	+	0.542579i
$437$		0			0
$438$		0			0
$439$	−14.8388	−	5.82379i	−0.708216	−	0.277954i	−0.0162329	−	0.999868i	$-0.505167\pi$
$439$	−14.8388	−	5.82379i	−0.708216	−	0.277954i	−0.691983	+	0.721914i	$0.743263\pi$
$440$		0			0
$441$	14.3796	−	15.3045i	0.684743	−	0.728785i
$442$		0			0
$443$		0			0		0.365341	−	0.930874i	$-0.380952\pi$
$443$		0			0		−0.365341	+	0.930874i	$0.619048\pi$
$444$	40.0638	+	3.00237i	1.90134	+	0.142486i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	4.00000	−	20.7846i	0.188982	−	0.981981i
$449$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$449$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−25.2729	−	27.2377i	−1.18742	−	1.27974i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.56513	+	20.8852i	0.0732136	+	0.976967i	0.905974	+	0.423333i	$0.139140\pi$
$457$	1.56513	+	20.8852i	0.0732136	+	0.976967i	−0.832760	+	0.553634i	$0.813241\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$461$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$462$		0			0
$463$	1.09939	−	4.81673i	0.0510929	−	0.223852i	−0.942935	−	0.332976i	$-0.891947\pi$
$463$	1.09939	−	4.81673i	0.0510929	−	0.223852i	0.994028	+	0.109123i	$0.0348044\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$467$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$468$	8.63784	+	1.97153i	0.399284	+	0.0911340i
$469$	1.62624	−	40.1063i	0.0750926	−	1.85194i
$470$		0			0
$471$	21.4780	+	37.2010i	0.989654	+	1.71413i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	34.0588	+	27.1610i	1.56272	+	1.24623i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$479$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$480$		0			0
$481$	2.55251	−	16.9348i	0.116385	−	0.772162i
$482$		0			0
$483$		0			0
$484$	−19.8213	−	9.54544i	−0.900969	−	0.433884i
$485$		0			0
$486$		0			0
$487$	−3.06574	+	40.9094i	−0.138922	+	1.85378i	0.298353	+	0.954455i	$0.403563\pi$
$487$	−3.06574	+	40.9094i	−0.138922	+	1.85378i	−0.437275	+	0.899328i	$0.644056\pi$
$488$		0			0
$489$		−	30.7073i		−	1.38863i
$490$		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
$496$	−8.49962	+	17.6496i	−0.381644	+	0.792493i
$497$		0			0
$498$		0			0
$499$	−2.22682	−	0.335639i	−0.0996861	−	0.0150253i	0.0990097	−	0.995086i	$-0.468433\pi$
$499$	−2.22682	−	0.335639i	−0.0996861	−	0.0150253i	−0.198696	+	0.980061i	$0.563671\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$503$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.52367	+	17.9073i	−0.245315	+	0.795292i
$508$	2.36710	−	4.09994i	0.105023	−	0.181906i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	17.8284	−	20.5865i	0.788684	−	0.910693i
$512$		0			0
$513$	−43.2605	+	13.3441i	−1.91000	+	0.589157i
$514$		0			0
$515$		0			0
$516$	6.49574	+	21.0587i	0.285959	+	0.927057i
$517$		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
$523$	−40.5608	+	3.03961i	−1.77360	+	0.132913i	−0.920979	−	0.389613i	$-0.872609\pi$
$523$	−40.5608	+	3.03961i	−1.77360	+	0.132913i	−0.852623	+	0.522526i	$0.824990\pi$
$524$		0			0
$525$	−16.1511	−	16.2524i	−0.704893	−	0.709314i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	19.0035	−	12.9564i	0.826239	−	0.563320i
$530$		0			0
$531$		0			0
$532$	44.5688	−	11.7930i	1.93230	−	0.511290i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.78362	+	9.64051i	−0.162671	+	0.414478i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−3.78362	+	9.64051i	−0.162671	+	0.414478i	0.826177	+	0.563411i	$0.190511\pi$
$542$		0			0
$543$	−25.8643	−	23.9986i	−1.10994	−	1.02988i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−29.3190	+	14.1193i	−1.25359	+	0.603696i	−0.938471	−	0.345357i	$-0.887758\pi$
$547$	−29.3190	+	14.1193i	−1.25359	+	0.603696i	−0.315116	+	0.949053i	$0.602044\pi$
$548$		0			0
$549$	25.6716	+	37.6533i	1.09564	+	1.60701i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−32.8958	−	20.8135i	−1.39887	−	0.885081i
$554$		0			0
$555$		0			0
$556$	−13.7497	+	44.5756i	−0.583119	+	1.89042i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	9.15863	−	2.09040i	0.387369	−	0.0884144i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$563$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	23.4104	−	4.35372i	0.983143	−	0.182839i
$568$		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
$571$	−27.2940	−	8.41908i	−1.14222	−	0.352328i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−27.2940	−	8.41908i	−1.14222	−	0.352328i	−0.807429	+	0.589965i	$0.799141\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	17.5932	−	16.3241i	0.733052	−	0.680173i
$577$	−2.15575	−	14.3025i	−0.0897451	−	0.595420i	−0.987823	−	0.155583i	$-0.950274\pi$
$577$	−2.15575	−	14.3025i	−0.0897451	−	0.595420i	0.898078	−	0.439837i	$-0.144964\pi$
$578$		0			0
$579$	−7.17261	+	47.5871i	−0.298083	+	1.97765i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−24.0000	+	3.46410i	−0.989743	+	0.142857i
$589$	−42.6690			−1.75815
$590$		0			0
$591$		0			0
$592$	−34.0073	−	31.5542i	−1.39769	−	1.29687i
$593$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$593$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−33.7942	−	5.09365i	−1.38310	−	0.208469i
$598$		0			0
$599$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$599$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$600$		0			0
$601$	−1.35417	+	1.07992i	−0.0552378	+	0.0440507i	−0.650720	−	0.759318i	$-0.725533\pi$
$601$	−1.35417	+	1.07992i	−0.0552378	+	0.0440507i	0.595482	+	0.803369i	$0.296961\pi$
$602$		0			0
$603$	28.3773	−	35.5840i	1.15561	−	1.44909i
$604$	3.20627	+	42.7848i	0.130461	+	1.74089i
$605$		0			0
$606$		0			0
$607$	−4.50000	+	2.59808i	−0.182649	+	0.105453i	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−4.50000	+	2.59808i	−0.182649	+	0.105453i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	18.0868	+	46.0845i	0.730520	+	1.86133i	0.384856	+	0.922977i	$0.374251\pi$
$613$	18.0868	+	46.0845i	0.730520	+	1.86133i	0.345664	+	0.938358i	$0.387654\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$617$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$618$		0			0
$619$	7.50000	+	4.33013i	0.301450	+	0.174042i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	7.50000	+	4.33013i	0.301450	+	0.174042i	−0.341644	+	0.939829i	$0.610984\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−6.37869	−	7.99862i	−0.255352	−	0.320201i
$625$	−18.3263	+	17.0043i	−0.733052	+	0.680173i
$626$		0			0
$627$		0			0
$628$	7.39269	−	49.0473i	0.295001	−	1.95720i
$629$		0			0
$630$		0			0
$631$	−12.4783	−	6.00921i	−0.496752	−	0.239223i	0.168695	−	0.985668i	$-0.446045\pi$
$631$	−12.4783	−	6.00921i	−0.496752	−	0.239223i	−0.665447	+	0.746445i	$0.731759\pi$
$632$		0			0
$633$	23.6566	−	25.4958i	0.940266	−	1.01337i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.707998	+	10.3124i	0.0280519	+	0.408591i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.733052	−	0.680173i	$-0.761905\pi$
$641$		0			0		0.733052	+	0.680173i	$0.238095\pi$
$642$		0			0
$643$	−18.3177	+	38.0371i	−0.722380	+	1.50004i	0.138027	+	0.990429i	$0.455924\pi$
$643$	−18.3177	+	38.0371i	−0.722380	+	1.50004i	−0.860406	+	0.509609i	$0.829790\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$647$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	22.2937	+	2.58249i	0.873757	+	0.101216i
$652$	−22.1075	+	27.7220i	−0.865798	+	1.08568i
$653$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$653$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	30.1055	−	6.87139i	1.17453	−	0.268079i
$658$		0			0
$659$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$659$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$660$		0			0
$661$	−14.5109	+	5.69510i	−0.564408	+	0.221514i	−0.630352	−	0.776310i	$-0.717089\pi$
$661$	−14.5109	+	5.69510i	−0.564408	+	0.221514i	0.0659439	+	0.997823i	$0.478994\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	41.3535	+	12.7559i	1.59882	+	0.493170i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−10.9448	−	13.7244i	−0.421893	−	0.529037i	0.524778	−	0.851239i	$-0.324148\pi$
$673$	−10.9448	−	13.7244i	−0.421893	−	0.529037i	−0.946671	+	0.322202i	$0.895577\pi$
$674$		0			0
$675$	−3.87223	−	25.6906i	−0.149042	−	0.988831i
$676$	17.8789	−	12.1896i	0.687651	−	0.468833i
$677$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$677$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$678$		0			0
$679$	−2.83436	−	5.40370i	−0.108773	−	0.207375i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$683$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$684$	48.6618	+	19.0984i	1.86063	+	0.730244i
$685$		0			0
$686$		0			0
$687$	41.2772			1.57482
$688$	9.29684	−	23.6880i	0.354439	−	0.903095i
$689$		0			0
$690$		0			0
$691$	−10.7327	+	15.7420i	−0.408291	+	0.598852i	−0.974220	−	0.225602i	$-0.927565\pi$
$691$	−10.7327	+	15.7420i	−0.408291	+	0.598852i	0.565929	+	0.824454i	$0.308518\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.88010	+	26.3003i	0.108858	+	0.994057i
$701$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$701$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$702$		0			0
$703$	29.7841	−	96.5578i	1.12333	−	3.64175i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	49.3397	−	15.2193i	1.85299	−	0.571572i	0.854335	−	0.519723i	$-0.173965\pi$
$709$	49.3397	−	15.2193i	1.85299	−	0.571572i	0.998655	−	0.0518481i	$-0.0165112\pi$
$710$		0			0
$711$	−16.1259	−	41.0882i	−0.604770	−	1.54093i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$719$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$720$		0			0
$721$	−1.95037	+	16.8368i	−0.0726355	+	0.627034i
$722$		0			0
$723$	−23.9894	+	22.2589i	−0.892176	+	0.827818i
$724$	6.07218	+	40.2863i	0.225671	+	1.49723i
$725$		0			0
$726$		0			0
$727$	9.76961	+	20.2868i	0.362335	+	0.752396i	0.999837	−	0.0180524i	$-0.00574658\pi$
$727$	9.76961	+	20.2868i	0.362335	+	0.752396i	−0.637502	+	0.770449i	$0.720032\pi$
$728$		0			0
$729$	24.3262	+	11.7149i	0.900969	+	0.433884i
$730$		0			0
$731$		0			0
$732$	3.93245	−	52.4749i	0.145347	−	1.93953i
$733$	−36.2577	−	14.2301i	−1.33921	−	0.525601i	−0.415814	−	0.909450i	$-0.636503\pi$
$733$	−36.2577	−	14.2301i	−1.33921	−	0.525601i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	38.8605	+	36.0573i	1.42951	+	1.32639i	0.865779	+	0.500426i	$0.166823\pi$
$739$	38.8605	+	36.0573i	1.42951	+	1.32639i	0.563727	+	0.825961i	$0.309367\pi$
$740$		0			0
$741$	9.66855	−	20.0770i	0.355183	−	0.737545i
$742$		0			0
$743$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$743$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−4.03548	−	53.8497i	−0.147257	−	1.96500i	−0.244233	−	0.969717i	$-0.578536\pi$
$751$	−4.03548	−	53.8497i	−0.147257	−	1.96500i	0.0969764	−	0.995287i	$-0.469083\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−24.2689	−	12.9237i	−0.882650	−	0.470030i
$757$	−0.530256	+	2.32320i	−0.0192725	+	0.0844382i	−0.983650	−	0.180093i	$-0.942360\pi$
$757$	−0.530256	+	2.32320i	−0.0192725	+	0.0844382i	0.964377	+	0.264531i	$0.0852172\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.294755	−	0.955573i	$-0.595238\pi$
$761$		0			0		0.294755	+	0.955573i	$0.404762\pi$
$762$		0			0
$763$	−21.4560	+	15.7319i	−0.776758	+	0.569531i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−27.6353	+	2.07098i	−0.997204	+	0.0747301i
$769$	20.3126	+	16.1987i	0.732490	+	0.584142i	0.917094	−	0.398672i	$-0.130529\pi$
$769$	20.3126	+	16.1987i	0.732490	+	0.584142i	−0.184603	+	0.982813i	$0.559100\pi$
$770$		0			0
$771$		0			0
$772$	40.7354	−	37.7969i	1.46610	−	1.36034i
$773$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$773$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$774$		0			0
$775$	3.64960	−	24.2135i	0.131098	−	0.869776i
$776$		0			0
$777$	−21.4057	+	48.6468i	−0.767923	+	1.74519i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	24.1607	+	14.1513i	0.862882	+	0.505405i
$785$		0			0
$786$		0			0
$787$	−55.2120	−	4.13757i	−1.96809	−	0.147488i	−0.971948	−	0.235195i	$-0.924427\pi$
$787$	−55.2120	−	4.13757i	−1.96809	−	0.147488i	−0.996146	+	0.0877066i	$0.972046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−22.1809	−	3.34324i	−0.787668	−	0.118722i
$794$		0			0
$795$		0			0
$796$	26.8416	+	28.9284i	0.951375	+	1.02534i
$797$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$797$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−51.2370	+	11.6945i	−1.80699	+	0.412433i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.365341	−	0.930874i	$-0.619048\pi$
$809$		0			0		0.365341	+	0.930874i	$0.380952\pi$
$810$		0			0
$811$	−32.8167	−	7.49020i	−1.15235	−	0.263016i	−0.396672	−	0.917960i	$-0.629835\pi$
$811$	−32.8167	−	7.49020i	−1.15235	−	0.263016i	−0.755677	+	0.654944i	$0.772692\pi$
$812$		0			0
$813$	10.6969	+	46.8661i	0.375156	+	1.64366i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	55.2722	−	4.14208i	1.93373	−	0.144913i
$818$		0			0
$819$	−6.26675	+	9.90461i	−0.218978	+	0.346095i
$820$		0			0
$821$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$821$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$822$		0			0
$823$	−11.1237	+	7.58401i	−0.387748	+	0.264362i	−0.741467	−	0.670990i	$-0.765869\pi$
$823$	−11.1237	+	7.58401i	−0.387748	+	0.264362i	0.353719	+	0.935352i	$0.384917\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$827$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$828$		0			0
$829$	4.00913	−	4.32082i	0.139243	−	0.150068i	−0.659580	−	0.751634i	$-0.729266\pi$
$829$	4.00913	−	4.32082i	0.139243	−	0.150068i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$	36.5505	+	14.3450i	1.26792	+	0.497622i
$832$			11.8133i			0.409552i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	18.6545	+	17.3088i	0.644793	+	0.598280i
$838$		0			0
$839$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$839$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$840$		0			0
$841$	−26.1281	+	12.5826i	−0.900969	+	0.433884i
$842$		0			0
$843$		0			0
$844$	−39.7123	+	5.98566i	−1.36695	+	0.206035i
$845$		0			0
$846$		0			0
$847$	20.6433	−	20.5147i	0.709314	−	0.704893i
$848$		0			0
$849$	−0.0645160	−	0.860906i	−0.00221418	−	0.0295462i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.74866	+	0.399120i	−0.0598730	+	0.0136656i	−0.252353	−	0.967635i	$-0.581204\pi$
$853$	−1.74866	+	0.399120i	−0.0598730	+	0.0136656i	0.192480	+	0.981301i	$0.438347\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$857$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$858$		0			0
$859$	−16.0882	−	52.1568i	−0.548924	−	1.77957i	−0.622893	−	0.782307i	$-0.714043\pi$
$859$	−16.0882	−	52.1568i	−0.548924	−	1.77957i	0.0739693	−	0.997261i	$-0.476433\pi$
$860$		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
$867$	−23.0209	−	18.3586i	−0.781831	−	0.623490i
$868$	−18.2670	−	18.3816i	−0.620024	−	0.623913i
$869$		0			0
$870$		0			0
$871$	3.33896	+	22.1525i	0.113136	+	0.750610i
$872$		0			0
$873$	1.03121	−	6.84166i	0.0349013	−	0.231555i
$874$		0			0
$875$		0			0
$876$	−32.1257	−	15.4709i	−1.08543	−	0.522715i
$877$	13.8132	+	9.41767i	0.466438	+	0.318012i	0.773632	−	0.633635i	$-0.218438\pi$
$877$	13.8132	+	9.41767i	0.466438	+	0.318012i	−0.307194	+	0.951647i	$0.599390\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	58.1272			1.95614			0.978068	−	0.208284i	$-0.0667879\pi$
$883$	58.1272			1.95614			0.978068	+	0.208284i	$0.0667879\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$887$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$888$		0			0
$889$	4.07026	+	4.75975i	0.136512	+	0.159637i
$890$		0			0
$891$		0			0
$892$	−28.1496	−	41.2879i	−0.942520	−	1.38242i
$893$		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
$903$	−29.1293	−	1.18114i	−0.969362	−	0.0393058i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	17.4986	+	44.5857i	0.581031	+	1.48044i	0.854960	+	0.518694i	$0.173581\pi$
$907$	17.4986	+	44.5857i	0.581031	+	1.48044i	−0.273929	+	0.961750i	$0.588323\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$911$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$912$	−30.1812	−	52.2754i	−0.999400	−	1.73101i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−37.2643	−	29.7173i	−1.23125	−	0.981886i
$917$		0			0
$918$		0			0
$919$	−18.6209	+	17.2776i	−0.614246	+	0.569937i	−0.924721	−	0.380645i	$-0.875702\pi$
$919$	−18.6209	+	17.2776i	−0.614246	+	0.569937i	0.310476	+	0.950581i	$0.399512\pi$
$920$		0			0
$921$	−40.7059	+	27.7528i	−1.34130	+	0.914486i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	52.2465	+	25.1606i	1.71785	+	0.827275i
$926$		0			0
$927$	−13.0721	+	14.0884i	−0.429343	+	0.462722i
$928$		0			0
$929$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$929$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$930$		0			0
$931$	−4.93778	+	60.7878i	−0.161829	+	1.99224i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	25.4610	−	52.8704i	0.831775	−	1.72720i	0.159099	−	0.987263i	$-0.449141\pi$
$937$	25.4610	−	52.8704i	0.831775	−	1.72720i	0.672677	−	0.739936i	$-0.265144\pi$
$938$		0			0
$939$	55.0533	−	26.5123i	1.79660	−	0.865195i
$940$		0			0
$941$		0			0		−0.563320	−	0.826239i	$-0.690476\pi$
$941$		0			0		0.563320	+	0.826239i	$0.309524\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.0747301	−	0.997204i	$-0.523810\pi$
$947$		0			0		0.0747301	+	0.997204i	$0.476190\pi$
$948$	−15.0230	+	48.7035i	−0.487925	+	1.58182i
$949$	−7.59983	+	13.1633i	−0.246701	+	0.427299i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$953$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−3.50770	−	6.07551i	−0.113152	−	0.195984i
$962$		0			0
$963$		0			0
$964$	37.6824	−	2.82390i	1.21367	−	0.0909518i
$965$		0			0
$966$		0			0
$967$	−17.8029	−	22.3242i	−0.572503	−	0.717897i	0.408310	−	0.912843i	$-0.366118\pi$
$967$	−17.8029	−	22.3242i	−0.572503	−	0.717897i	−0.980814	+	0.194946i	$0.937547\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$971$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$972$	−13.5272	−	28.0894i	−0.433884	−	0.900969i
$973$	−49.5365	−	36.7994i	−1.58807	−	1.17973i
$974$		0			0
$975$	10.5662	+	7.20389i	0.338388	+	0.230709i
$976$	−41.3291	+	44.5422i	−1.32291	+	1.42576i
$977$		0			0		0.0747301	−	0.997204i	$-0.476190\pi$
$977$		0			0		−0.0747301	+	0.997204i	$0.523810\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−30.1677			−0.963181
$982$		0			0
$983$		0			0		−0.997204	−	0.0747301i	$-0.976190\pi$
$983$		0			0		0.997204	+	0.0747301i	$0.0238095\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−23.1829	+	11.1643i	−0.737545	+	0.355183i
$989$		0			0
$990$		0			0
$991$	57.3476	−	8.64376i	1.82171	−	0.274578i	0.852842	−	0.522170i	$-0.174877\pi$
$991$	57.3476	−	8.64376i	1.82171	−	0.274578i	0.968864	+	0.247592i	$0.0796392\pi$
$992$		0			0
$993$	37.6651	−	30.0369i	1.19527	−	0.953193i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−11.7422	+	38.0673i	−0.371880	+	1.20560i	0.555511	+	0.831509i	$0.312523\pi$
$997$	−11.7422	+	38.0673i	−0.371880	+	1.20560i	−0.927390	+	0.374095i	$0.877953\pi$
$998$		0			0
$999$	−52.1903	+	30.1321i	−1.65123	+	0.953338i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.2.o.a.101.1	✓	12
3.2	odd	2	CM	147.2.o.a.101.1	✓	12
49.33	odd	42	inner	147.2.o.a.131.1	yes	12
147.131	even	42	inner	147.2.o.a.131.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.o.a.101.1	✓	12	1.1	even	1	trivial
147.2.o.a.101.1	✓	12	3.2	odd	2	CM
147.2.o.a.131.1	yes	12	49.33	odd	42	inner
147.2.o.a.131.1	yes	12	147.131	even	42	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.o (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(1.72721 + 0.129436i) q^{3} +(-1.46610 - 1.36035i) q^{4} +(2.34300 - 1.22896i) q^{7} +(2.96649 + 0.447127i) q^{9} +O(q^{10})\)	\(q+(1.72721 + 0.129436i) q^{3} +(-1.46610 - 1.36035i) q^{4} +(2.34300 - 1.22896i) q^{7} +(2.96649 + 0.447127i) q^{9} +(-2.35619 - 2.53937i) q^{12} +(-1.15450 + 0.920684i) q^{13} +(0.298920 + 3.98882i) q^{16} +(-7.54531 + 4.35628i) q^{19} +(4.20592 - 1.81940i) q^{21} +(-1.82671 - 4.65437i) q^{25} +(5.06587 + 1.15625i) q^{27} +(-5.10689 - 1.38551i) q^{28} +(4.24128 + 2.44870i) q^{31} +(-3.74094 - 4.69099i) q^{36} +(-8.50183 + 7.88854i) q^{37} +(-2.11323 + 1.44078i) q^{39} +(-5.73174 - 2.76026i) q^{43} +6.92820i q^{48} +(3.97932 - 5.75891i) q^{49} +(2.94507 + 0.220702i) q^{52} +(-13.5962 + 6.54757i) q^{57} +(10.3323 + 11.1355i) q^{61} +(7.50000 - 2.59808i) q^{63} +(4.98792 - 6.25465i) q^{64} +(7.58561 - 13.1387i) q^{67} +(9.58172 - 3.76055i) q^{73} +(-2.55265 - 8.27550i) q^{75} +(16.9883 + 3.87746i) q^{76} +(-7.35657 - 12.7420i) q^{79} +(8.60016 + 2.65280i) q^{81} +(-8.64133 - 3.05409i) q^{84} +(-1.57352 + 3.57600i) q^{91} +(7.00862 + 4.77840i) q^{93} -2.30631i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9	\( 12 q + 3 q^{3} + 2 q^{4} - q^{7} - 3 q^{9} - 6 q^{12} + 4 q^{16} + 9 q^{19} + 6 q^{21} - 5 q^{25} - 8 q^{28} - 15 q^{31} + 12 q^{36} - 76 q^{37} - 66 q^{39} + 10 q^{43} + 13 q^{49} + 34 q^{52} - 18 q^{57} + 79 q^{61} + 90 q^{63} - 16 q^{64} + 11 q^{67} + 27 q^{73} + 15 q^{75} - 13 q^{79} + 9 q^{81} + 6 q^{84} - 9 q^{91} + 15 q^{93}+O(q^{100}) \) 12 * q + 3 * q^3 + 2 * q^4 - q^7 - 3 * q^9 - 6 * q^12 + 4 * q^16 + 9 * q^19 + 6 * q^21 - 5 * q^25 - 8 * q^28 - 15 * q^31 + 12 * q^36 - 76 * q^37 - 66 * q^39 + 10 * q^43 + 13 * q^49 + 34 * q^52 - 18 * q^57 + 79 * q^61 + 90 * q^63 - 16 * q^64 + 11 * q^67 + 27 * q^73 + 15 * q^75 - 13 * q^79 + 9 * q^81 + 6 * q^84 - 9 * q^91 + 15 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

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