LMFDB - Embedded newform 473.2.k.a.171.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [473,2,Mod(85,473)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(473, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("473.85");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 473 = 11 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	473.k (of order $10$, degree $4$, 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:	$3.77692401561$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.53418765625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 21x^{5} + 89x^{4} + 231x^{3} - 1210x^{2} - 1331x + 14641 $ x^8 - x^7 - 10x^6 + 21x^5 + 89x^4 + 231x^3 - 1210x^2 - 1331x + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			171.1
Root			$-3.27276 - 0.537652i$ of defining polynomial
Character	$\chi$	$=$	473.171
Dual form			473.2.k.a.343.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.61803 - 1.17557i) q^{4} +(0.927051 + 2.85317i) q^{9} +O(q^{10})$	$q+(1.61803 - 1.17557i) q^{4} +(0.927051 + 2.85317i) q^{9} +(0.500000 - 3.27872i) q^{11} +(0.786343 - 0.255498i) q^{13} +(1.23607 - 3.80423i) q^{16} +(7.84052 + 2.54754i) q^{17} -4.07338 q^{23} +(-4.04508 - 2.93893i) q^{25} +(-1.05894 - 3.25907i) q^{31} +(4.85410 + 3.52671i) q^{36} +(3.85437 - 5.30508i) q^{41} -6.55744i q^{43} +(-3.04535 - 5.89286i) q^{44} +(8.85453 + 6.43319i) q^{47} +(-2.16312 + 6.65740i) q^{49} +(0.971973 - 1.33781i) q^{52} +(4.44106 + 13.6682i) q^{53} +(-12.0909 + 8.78452i) q^{59} +(-2.47214 - 7.60845i) q^{64} -10.8717 q^{67} +(15.6810 - 5.09508i) q^{68} +(3.38266 - 1.09909i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(-4.45786 - 1.44845i) q^{83} +(-6.59086 + 4.78854i) q^{92} +(-5.87704 - 18.0877i) q^{97} +(9.81827 - 1.61296i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 6 q^{9}+O(q^{10}) $ 8 * q + 4 * q^4 - 6 * q^9	$ 8 q + 4 q^{4} - 6 q^{9} + 4 q^{11} + 15 q^{13} - 8 q^{16} - 14 q^{23} - 10 q^{25} - 18 q^{31} + 12 q^{36} + 2 q^{44} + 12 q^{47} + 14 q^{49} + 26 q^{53} - 16 q^{59} + 16 q^{64} + 30 q^{67} - 18 q^{81} + 28 q^{92} - 3 q^{97} - 3 q^{99}+O(q^{100}) $ 8 * q + 4 * q^4 - 6 * q^9 + 4 * q^11 + 15 * q^13 - 8 * q^16 - 14 * q^23 - 10 * q^25 - 18 * q^31 + 12 * q^36 + 2 * q^44 + 12 * q^47 + 14 * q^49 + 26 * q^53 - 16 * q^59 + 16 * q^64 + 30 * q^67 - 18 * q^81 + 28 * q^92 - 3 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/473\mathbb{Z}\right)^\times$.

$n$	$89$	$431$
$\chi(n)$	$-1$	$e\left(\frac{9}{10}\right)$

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		0.951057	−	0.309017i	$-0.100000\pi$
$2$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$3$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$3$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$4$	1.61803	−	1.17557i	0.809017	−	0.587785i
$5$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$5$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$6$		0			0
$7$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$7$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$8$		0			0
$9$	0.927051	+	2.85317i	0.309017	+	0.951057i
$10$		0			0
$11$	0.500000	−	3.27872i	0.150756	−	0.988571i
$11$	0.500000	−	3.27872i	0.150756	−	0.988571i
$12$		0			0
$13$	0.786343	−	0.255498i	0.218092	−	0.0708625i	−0.197933	−	0.980216i	$-0.563423\pi$
$13$	0.786343	−	0.255498i	0.218092	−	0.0708625i	0.416025	+	0.909353i	$0.363423\pi$
$14$		0			0
$15$		0			0
$16$	1.23607	−	3.80423i	0.309017	−	0.951057i
$17$	7.84052	+	2.54754i	1.90160	+	0.617869i	0.957949	+	0.286938i	$0.0926374\pi$
$17$	7.84052	+	2.54754i	1.90160	+	0.617869i	0.943655	+	0.330930i	$0.107363\pi$
$18$		0			0
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.07338			−0.849358			−0.424679	−	0.905344i	$-0.639613\pi$
$23$	−4.07338			−0.849358			−0.424679	+	0.905344i	$0.639613\pi$
$24$		0			0
$25$	−4.04508	−	2.93893i	−0.809017	−	0.587785i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$29$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$30$		0			0
$31$	−1.05894	−	3.25907i	−0.190190	−	0.585346i	0.809809	−	0.586694i	$-0.199571\pi$
$31$	−1.05894	−	3.25907i	−0.190190	−	0.585346i	−0.999999	+	0.00134811i	$0.999571\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	4.85410	+	3.52671i	0.809017	+	0.587785i
$37$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$37$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.85437	−	5.30508i	0.601951	−	0.828514i	−0.393934	−	0.919139i	$-0.628886\pi$
$41$	3.85437	−	5.30508i	0.601951	−	0.828514i	0.995885	+	0.0906244i	$0.0288863\pi$
$42$		0			0
$43$		−	6.55744i		−	1.00000i
$43$		−	6.55744i		−	1.00000i
$44$	−3.04535	−	5.89286i	−0.459104	−	0.888383i
$45$		0			0
$46$		0			0
$47$	8.85453	+	6.43319i	1.29157	+	0.938377i	0.999836	−	0.0181233i	$-0.00576913\pi$
$47$	8.85453	+	6.43319i	1.29157	+	0.938377i	0.291730	+	0.956501i	$0.405769\pi$
$48$		0			0
$49$	−2.16312	+	6.65740i	−0.309017	+	0.951057i
$50$		0			0
$51$		0			0
$52$	0.971973	−	1.33781i	0.134788	−	0.185520i
$53$	4.44106	+	13.6682i	0.610027	+	1.87747i	0.457607	+	0.889155i	$0.348707\pi$
$53$	4.44106	+	13.6682i	0.610027	+	1.87747i	0.152420	+	0.988316i	$0.451293\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−12.0909	+	8.78452i	−1.57410	+	1.14365i	−0.651000	+	0.759078i	$0.725650\pi$
$59$	−12.0909	+	8.78452i	−1.57410	+	1.14365i	−0.923096	+	0.384570i	$0.874350\pi$
$60$		0			0
$61$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$61$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$62$		0			0
$63$		0			0
$64$	−2.47214	−	7.60845i	−0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$	−10.8717			−1.32820			−0.664098	−	0.747646i	$-0.731184\pi$
$67$	−10.8717			−1.32820			−0.664098	+	0.747646i	$0.731184\pi$
$68$	15.6810	−	5.09508i	1.90160	−	0.617869i
$69$		0			0
$70$		0			0
$71$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$71$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$72$		0			0
$73$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$73$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.38266	−	1.09909i	0.380579	−	0.123657i	−0.112479	−	0.993654i	$-0.535879\pi$
$79$	3.38266	−	1.09909i	0.380579	−	0.123657i	0.493058	+	0.869997i	$0.335879\pi$
$80$		0			0
$81$	−7.28115	+	5.29007i	−0.809017	+	0.587785i
$82$		0			0
$83$	−4.45786	−	1.44845i	−0.489314	−	0.158988i	0.0539604	−	0.998543i	$-0.482816\pi$
$83$	−4.45786	−	1.44845i	−0.489314	−	0.158988i	−0.543274	+	0.839555i	$0.682816\pi$
$84$		0			0
$85$		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
$91$		0			0
$92$	−6.59086	+	4.78854i	−0.687145	+	0.499240i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.87704	−	18.0877i	−0.596723	−	1.83652i	−0.545956	−	0.837814i	$-0.683833\pi$
$97$	−5.87704	−	18.0877i	−0.596723	−	1.83652i	−0.0507673	−	0.998711i	$-0.516167\pi$
$98$		0			0
$99$	9.81827	−	1.61296i	0.986773	−	0.162108i
$100$	−10.0000			−1.00000
$101$	−19.1128	+	6.21014i	−1.90180	+	0.617932i	−0.945285	+	0.326245i	$0.894217\pi$
$101$	−19.1128	+	6.21014i	−1.90180	+	0.617932i	−0.956514	+	0.291687i	$0.905783\pi$
$102$		0			0
$103$	−6.08220	+	4.41898i	−0.599297	+	0.435415i	−0.845629	−	0.533771i	$-0.820775\pi$
$103$	−6.08220	+	4.41898i	−0.599297	+	0.435415i	0.246332	+	0.969185i	$0.420775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.3264	+	15.5895i	−1.09496	+	1.50709i	−0.253069	+	0.967448i	$0.581440\pi$
$107$	−11.3264	+	15.5895i	−1.09496	+	1.50709i	−0.841895	+	0.539641i	$0.818560\pi$
$108$		0			0
$109$			11.8007i			1.13031i	0.824986	+	0.565153i	$0.191183\pi$
$109$			11.8007i			1.13031i	−0.824986	+	0.565153i	$0.808817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$113$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.45796	+	2.00671i	0.134788	+	0.185520i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.5000	−	3.27872i	−0.954545	−	0.298065i
$122$		0			0
$123$		0			0
$124$	−5.54466	−	4.02843i	−0.497925	−	0.361764i
$125$		0			0
$126$		0			0
$127$	18.7095	+	6.07908i	1.66020	+	0.539431i	0.980914	−	0.194444i	$-0.0622902\pi$
$127$	18.7095	+	6.07908i	1.66020	+	0.539431i	0.679285	+	0.733875i	$0.262290\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$137$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$138$		0			0
$139$	−10.8404	−	14.9206i	−0.919472	−	1.26554i	−0.963827	−	0.266529i	$-0.914123\pi$
$139$	−10.8404	−	14.9206i	−0.919472	−	1.26554i	0.0443550	−	0.999016i	$-0.485877\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.444536	−	2.70595i	−0.0371739	−	0.226283i
$144$	12.0000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$149$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$150$		0			0
$151$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$151$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$152$		0			0
$153$			24.7320i			1.99947i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$157$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$163$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$164$		−	13.1149i		−	1.02410i
$165$		0			0
$166$		0			0
$167$	20.6855	−	6.72114i	1.60069	−	0.520097i	0.633415	−	0.773812i	$-0.281653\pi$
$167$	20.6855	−	6.72114i	1.60069	−	0.520097i	0.967279	+	0.253715i	$0.0816525\pi$
$168$		0			0
$169$	−9.96417	+	7.23939i	−0.766474	+	0.556876i
$170$		0			0
$171$		0			0
$172$	−7.70873	−	10.6102i	−0.587785	−	0.809017i
$173$	11.7820	−	16.2165i	0.895769	−	1.23292i	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	11.7820	−	16.2165i	0.895769	−	1.23292i	0.971798	−	0.235815i	$-0.0757760\pi$
$174$		0			0
$175$		0			0
$176$	−11.8550	−	5.95483i	−0.893601	−	0.448862i
$177$		0			0
$178$		0			0
$179$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$179$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$180$		0			0
$181$	−7.13578	+	21.9617i	−0.530399	+	1.63240i	0.222988	+	0.974821i	$0.428419\pi$
$181$	−7.13578	+	21.9617i	−0.530399	+	1.63240i	−0.753387	+	0.657578i	$0.771581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	12.2729	−	24.4331i	0.897485	−	1.78672i
$188$	21.8896			1.59646
$189$		0			0
$190$		0			0
$191$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$191$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$192$		0			0
$193$	20.1389	+	6.54352i	1.44963	+	0.471013i	0.924885	−	0.380247i	$-0.124161\pi$
$193$	20.1389	+	6.54352i	1.44963	+	0.471013i	0.524744	+	0.851260i	$0.324161\pi$
$194$		0			0
$195$		0			0
$196$	4.32624	+	13.3148i	0.309017	+	0.951057i
$197$			17.6160i			1.25509i	0.778581	+	0.627544i	$0.215940\pi$
$197$			17.6160i			1.25509i	−0.778581	+	0.627544i	$0.784060\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.77623	−	11.6220i	−0.262466	−	0.807787i
$208$		−	3.30724i		−	0.229316i
$209$		0			0
$210$		0			0
$211$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$211$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$212$	23.2537	+	16.8948i	1.59707	+	1.16034i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.81622			0.458509
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$		0			0
$225$	4.63525	−	14.2658i	0.309017	−	0.951057i
$226$		0			0
$227$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$227$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$228$		0			0
$229$	−7.97785	−	24.5533i	−0.527191	−	1.62253i	−0.759941	−	0.649992i	$-0.774772\pi$
$229$	−7.97785	−	24.5533i	−0.527191	−	1.62253i	0.232750	−	0.972537i	$-0.425228\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$233$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$234$		0			0
$235$		0			0
$236$	−9.23660	+	28.4273i	−0.601251	+	1.85046i
$237$		0			0
$238$		0			0
$239$	13.2703	−	18.2651i	0.858387	−	1.18147i	−0.123564	−	0.992337i	$-0.539432\pi$
$239$	13.2703	−	18.2651i	0.858387	−	1.18147i	0.981952	−	0.189132i	$-0.0605675\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6.55894	−	20.1863i	−0.413996	−	1.27415i	−0.913146	−	0.407634i	$-0.866354\pi$
$251$	−6.55894	−	20.1863i	−0.413996	−	1.27415i	0.499149	−	0.866516i	$-0.333646\pi$
$252$		0			0
$253$	−2.03669	+	13.3555i	−0.128045	+	0.839650i
$254$		0			0
$255$		0			0
$256$	−12.9443	−	9.40456i	−0.809017	−	0.587785i
$257$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$257$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−17.5909	+	12.7805i	−1.07453	+	0.780694i
$269$	5.70532	−	17.5592i	0.347860	−	1.07060i	−0.612175	−	0.790722i	$-0.709705\pi$
$269$	5.70532	−	17.5592i	0.347860	−	1.07060i	0.960035	−	0.279880i	$-0.0902946\pi$
$270$		0			0
$271$	−16.6277	−	22.8861i	−1.01006	−	1.39023i	−0.918943	−	0.394390i	$-0.870956\pi$
$271$	−16.6277	−	22.8861i	−1.01006	−	1.39023i	−0.0911185	−	0.995840i	$-0.529044\pi$
$272$	19.3828	−	26.6782i	1.17526	−	1.61760i
$273$		0			0
$274$		0			0
$275$	−11.6585	+	11.7932i	−0.703031	+	0.711159i
$276$		0			0
$277$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$277$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$278$		0			0
$279$	8.31699	−	6.04265i	0.497925	−	0.361764i
$280$		0			0
$281$	15.9675	+	5.18815i	0.952540	+	0.309499i	0.743747	−	0.668461i	$-0.233047\pi$
$281$	15.9675	+	5.18815i	0.952540	+	0.309499i	0.208792	+	0.977960i	$0.433047\pi$
$282$		0			0
$283$	3.85437	−	5.30508i	0.229118	−	0.315354i	−0.678944	−	0.734191i	$-0.737562\pi$
$283$	3.85437	−	5.30508i	0.229118	−	0.315354i	0.908062	+	0.418836i	$0.137562\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	41.2304	+	29.9557i	2.42532	+	1.76210i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.4175	−	21.2203i	−0.900698	−	1.23970i	−0.970245	−	0.242125i	$-0.922155\pi$
$293$	−15.4175	−	21.2203i	−0.900698	−	1.23970i	0.0695472	−	0.997579i	$-0.477845\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.20307	+	1.04074i	−0.185238	+	0.0601876i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	1.13053i		−	0.0645230i	−0.999479	−	0.0322615i	$-0.989729\pi$
$307$		−	1.13053i		−	0.0645230i	0.999479	−	0.0322615i	$-0.0102709\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	21.9772	+	15.9673i	1.24621	+	0.905424i	0.997996	−	0.0632807i	$-0.0201563\pi$
$311$	21.9772	+	15.9673i	1.24621	+	0.905424i	0.248214	+	0.968705i	$0.420156\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$		0			0
$315$		0			0
$316$	4.18119	−	5.75492i	0.235210	−	0.323739i
$317$	9.94106	+	30.5955i	0.558346	+	1.71841i	0.686940	+	0.726714i	$0.258954\pi$
$317$	9.94106	+	30.5955i	0.558346	+	1.71841i	−0.128594	+	0.991697i	$0.541046\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−5.56231	+	17.1190i	−0.309017	+	0.951057i
$325$	−3.93171	−	1.27749i	−0.218092	−	0.0708625i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	−8.91572	+	2.89689i	−0.489314	+	0.158988i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	13.7563	+	18.9340i	0.749355	+	1.03140i	0.998025	+	0.0628108i	$0.0200065\pi$
$337$	13.7563	+	18.9340i	0.749355	+	1.03140i	−0.248670	+	0.968588i	$0.579994\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−11.2150	+	1.84242i	−0.607328	+	0.0997725i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$347$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$348$		0			0
$349$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$349$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−17.6701			−0.940486			−0.470243	−	0.882537i	$-0.655834\pi$
$353$	−17.6701			−0.940486			−0.470243	+	0.882537i	$0.655834\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.02690	−	12.4245i	−0.476421	−	0.655738i	0.501391	−	0.865221i	$-0.332822\pi$
$359$	−9.02690	−	12.4245i	−0.476421	−	0.655738i	−0.977812	+	0.209483i	$0.932822\pi$
$360$		0			0
$361$	−5.87132	−	18.0701i	−0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.09086	+	0.792556i	−0.0569424	+	0.0413711i	−0.615892	−	0.787830i	$-0.711204\pi$
$367$	−1.09086	+	0.792556i	−0.0569424	+	0.0413711i	0.558950	+	0.829201i	$0.311204\pi$
$368$	−5.03497	+	15.4960i	−0.262466	+	0.807787i
$369$	18.7095	+	6.07908i	0.973977	+	0.316464i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	11.2053	−	34.4864i	0.575579	−	1.77145i	−0.0586213	−	0.998280i	$-0.518670\pi$
$379$	11.2053	−	34.4864i	0.575579	−	1.77145i	0.634200	−	0.773169i	$-0.281330\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$383$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	18.7095	−	6.07908i	0.951057	−	0.309017i
$388$	−30.7726	−	22.3576i	−1.56224	−	1.13503i
$389$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$389$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$	−31.9374	−	10.3771i	−1.61514	−	0.524791i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	13.9901	−	14.1519i	0.703031	−	0.711159i
$397$	39.6880			1.99188			0.995941	−	0.0900071i	$-0.0286890\pi$
$397$	39.6880			1.99188			0.995941	+	0.0900071i	$0.0286890\pi$
$398$		0			0
$399$		0			0
$400$	−16.1803	+	11.7557i	−0.809017	+	0.587785i
$401$	0.205322	−	0.631918i	0.0102533	−	0.0315565i	−0.945799	−	0.324753i	$-0.894719\pi$
$401$	0.205322	−	0.631918i	0.0102533	−	0.0315565i	0.956052	+	0.293196i	$0.0947189\pi$
$402$		0			0
$403$	−1.66537	−	2.29219i	−0.0829581	−	0.114182i
$404$	−23.6248	+	32.5167i	−1.17538	+	1.61777i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$409$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$410$		0			0
$411$		0			0
$412$	−4.64639	+	14.3001i	−0.228911	+	0.704516i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$421$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$422$		0			0
$423$	−10.1464	+	31.2274i	−0.493334	+	1.51833i
$424$		0			0
$425$	−24.2285	−	33.3477i	−1.17526	−	1.61760i
$426$		0			0
$427$		0			0
$428$			38.5392i			1.86286i
$429$		0			0
$430$		0			0
$431$	−39.0120	+	12.6758i	−1.87914	+	0.610571i	−0.891693	+	0.452640i	$0.850482\pi$
$431$	−39.0120	+	12.6758i	−1.87914	+	0.610571i	−0.987450	+	0.157930i	$0.949518\pi$
$432$		0			0
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$	13.8726	+	19.0940i	0.664377	+	0.914437i
$437$		0			0
$438$		0			0
$439$			37.6633i			1.79757i	0.438389	+	0.898785i	$0.355549\pi$
$439$			37.6633i			1.79757i	−0.438389	+	0.898785i	$0.644451\pi$
$440$		0			0
$441$	−21.0000			−1.00000
$442$		0			0
$443$	−19.9438	−	14.4901i	−0.947561	−	0.688443i	0.00266799	−	0.999996i	$-0.499151\pi$
$443$	−19.9438	−	14.4901i	−0.947561	−	0.688443i	−0.950229	+	0.311553i	$0.899151\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$449$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$450$		0			0
$451$	−15.4667	−	15.2899i	−0.728298	−	0.719974i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$457$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	24.2305i		−	1.12853i	−0.825595	−	0.564263i	$-0.809160\pi$
$461$		−	24.2305i		−	1.12853i	0.825595	−	0.564263i	$-0.190840\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$467$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$468$	4.71806	+	1.53299i	0.218092	+	0.0708625i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−21.5000	−	3.27872i	−0.988571	−	0.150756i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−34.8806	+	25.3422i	−1.59707	+	1.16034i
$478$		0			0
$479$	35.8667	+	11.6538i	1.63879	+	0.532475i	0.976268	−	0.216568i	$-0.0694862\pi$
$479$	35.8667	+	11.6538i	1.63879	+	0.532475i	0.662522	+	0.749043i	$0.269486\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−20.8437	+	7.03841i	−0.947442	+	0.319928i
$485$		0			0
$486$		0			0
$487$	−25.2726	−	18.3616i	−1.14521	−	0.832044i	−0.157373	−	0.987539i	$-0.550303\pi$
$487$	−25.2726	−	18.3616i	−1.14521	−	0.832044i	−0.987837	+	0.155495i	$0.950303\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$491$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−13.7071			−0.615469
$497$		0			0
$498$		0			0
$499$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$499$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$503$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	37.4190	−	12.1582i	1.66020	−	0.539431i
$509$	32.9772	+	23.9593i	1.46169	+	1.06198i	0.982920	+	0.184035i	$0.0589161\pi$
$509$	32.9772	+	23.9593i	1.46169	+	1.06198i	0.478767	+	0.877942i	$0.341084\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	25.5199	−	25.8149i	1.12236	−	1.13534i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$521$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$523$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	28.2505i		−	1.23061i
$528$		0			0
$529$	−6.40761			−0.278592
$530$		0			0
$531$	−36.2726	−	26.3536i	−1.57410	−	1.14365i
$532$		0			0
$533$	1.67541	−	5.15639i	0.0725702	−	0.223348i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.7462	+	10.4210i	0.893601	+	0.448862i
$540$		0			0
$541$	−31.1825	+	10.1318i	−1.34064	+	0.435600i	−0.889533	−	0.456870i	$-0.848970\pi$
$541$	−31.1825	+	10.1318i	−1.34064	+	0.435600i	−0.451106	+	0.892470i	$0.648970\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−18.7183	+	25.7635i	−0.800337	+	1.10157i	0.192406	+	0.981315i	$0.438371\pi$
$547$	−18.7183	+	25.7635i	−0.800337	+	1.10157i	−0.992743	+	0.120254i	$0.961629\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−35.0803	−	11.3983i	−1.48774	−	0.483395i
$557$	26.9806	+	37.1356i	1.14320	+	1.57348i	0.760137	+	0.649763i	$0.225132\pi$
$557$	26.9806	+	37.1356i	1.14320	+	1.57348i	0.383066	+	0.923721i	$0.374868\pi$
$558$		0			0
$559$	−1.67541	−	5.15639i	−0.0708625	−	0.218092i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	40.5847	−	13.1868i	1.71044	−	0.555756i	0.720035	−	0.693938i	$-0.244126\pi$
$563$	40.5847	−	13.1868i	1.71044	−	0.555756i	0.990407	+	0.138182i	$0.0441257\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−26.3191	+	36.2252i	−1.10335	+	1.51864i	−0.272494	+	0.962157i	$0.587849\pi$
$569$	−26.3191	+	36.2252i	−1.10335	+	1.51864i	−0.830861	+	0.556480i	$0.812151\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$	−3.90030	−	3.85573i	−0.163080	−	0.161216i
$573$		0			0
$574$		0			0
$575$	16.4772	+	11.9714i	0.687145	+	0.499240i
$576$	19.4164	−	14.1068i	0.809017	−	0.587785i
$577$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$577$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	47.0347	−	7.72691i	1.94798	−	0.320016i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.1136	−	46.5150i	0.617526	−	1.90055i	0.270226	−	0.962797i	$-0.412902\pi$
$599$	15.1136	−	46.5150i	0.617526	−	1.90055i	0.347301	−	0.937754i	$-0.387098\pi$
$600$		0			0
$601$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$601$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$602$		0			0
$603$	−10.0787	−	31.0189i	−0.410435	−	1.26319i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$607$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.60636	+	2.79638i	0.348176	+	0.113129i
$612$	29.0742	+	40.0172i	1.17526	+	1.61760i
$613$	26.9836	−	37.1398i	1.08986	−	1.50006i	0.241677	−	0.970357i	$-0.422303\pi$
$613$	26.9836	−	37.1398i	1.08986	−	1.50006i	0.848182	−	0.529705i	$-0.177697\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.6092			−0.467370			−0.233685	−	0.972312i	$-0.575078\pi$
$617$	−11.6092			−0.467370			−0.233685	+	0.972312i	$0.575078\pi$
$618$		0			0
$619$	37.6529	+	27.3564i	1.51340	+	1.09955i	0.964641	+	0.263566i	$0.0848986\pi$
$619$	37.6529	+	27.3564i	1.51340	+	1.09955i	0.548756	+	0.835982i	$0.315101\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.72542	+	23.7764i	0.309017	+	0.951057i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			5.78767i			0.229316i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$641$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$642$		0			0
$643$	0.853613	−	2.62715i	0.0336632	−	0.103605i	−0.932813	−	0.360361i	$-0.882654\pi$
$643$	0.853613	−	2.62715i	0.0336632	−	0.103605i	0.966476	+	0.256756i	$0.0826536\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$		0			0
$649$	22.7566	+	44.0348i	0.893273	+	1.72852i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$653$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$	−15.4175	−	21.2203i	−0.601951	−	0.828514i
$657$		0			0
$658$		0			0
$659$		−	36.0588i		−	1.40465i	−0.711855	−	0.702326i	$-0.752145\pi$
$659$		−	36.0588i		−	1.40465i	0.711855	−	0.702326i	$-0.247855\pi$
$660$		0			0
$661$	9.52337			0.370416			0.185208	−	0.982699i	$-0.440704\pi$
$661$	9.52337			0.370416			0.185208	+	0.982699i	$0.440704\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	25.5687	−	35.1923i	0.989284	−	1.36163i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$673$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$674$		0			0
$675$		0			0
$676$	−7.61195	+	23.4272i	−0.292767	+	0.901045i
$677$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$677$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−47.2060			−1.80629			−0.903143	−	0.429339i	$-0.858747\pi$
$683$	−47.2060			−1.80629			−0.903143	+	0.429339i	$0.858747\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−24.9460	−	8.10544i	−0.951057	−	0.309017i
$689$	6.98440	+	9.61320i	0.266084	+	0.366234i
$690$		0			0
$691$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$691$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$692$		−	40.0895i		−	1.52397i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	43.7351	−	31.7754i	1.65659	−	1.20358i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−3.41962	+	4.70670i	−0.129157	+	0.177770i	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−3.41962	+	4.70670i	−0.129157	+	0.177770i	0.739541	+	0.673112i	$0.235043\pi$
$702$		0			0
$703$		0			0
$704$	−26.1820	+	4.30121i	−0.986773	+	0.162108i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	10.9120	−	33.5837i	0.409809	−	1.26126i	−0.507003	−	0.861944i	$-0.669247\pi$
$709$	10.9120	−	33.5837i	0.409809	−	1.26126i	0.916812	−	0.399319i	$-0.130753\pi$
$710$		0			0
$711$	6.27179	+	8.63238i	0.235210	+	0.323739i
$712$		0			0
$713$	4.31344	+	13.2754i	0.161540	+	0.497168i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−23.0909	+	16.7765i	−0.861144	+	0.625658i	−0.928196	−	0.372092i	$-0.878641\pi$
$719$	−23.0909	+	16.7765i	−0.861144	+	0.625658i	0.0670521	+	0.997749i	$0.478641\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	14.2716	+	43.9234i	0.530399	+	1.63240i
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−21.8435	−	15.8702i	−0.809017	−	0.587785i
$730$		0			0
$731$	16.7053	−	51.4137i	0.617869	−	1.90160i
$732$		0			0
$733$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$733$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.43587	+	35.6454i	−0.200233	+	1.31302i
$738$		0			0
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$743$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	14.0618i		−	0.514495i
$748$	−8.86481	−	53.9612i	−0.324129	−	1.97302i
$749$		0			0
$750$		0			0
$751$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$751$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$752$	35.4181	−	25.7328i	1.29157	−	0.938377i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$757$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$761$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.26313	+	9.99684i	−0.262256	+	0.360965i
$768$		0			0
$769$		−	53.0208i		−	1.91198i	−0.293403	−	0.955989i	$-0.594788\pi$
$769$		−	53.0208i		−	1.91198i	0.293403	−	0.955989i	$-0.405212\pi$
$770$		0			0
$771$		0			0
$772$	40.2778	−	13.0870i	1.44963	−	0.471013i
$773$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$773$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$774$		0			0
$775$	−5.29468	+	16.2953i	−0.190190	+	0.585346i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	22.6525	+	16.4580i	0.809017	+	0.587785i
$785$		0			0
$786$		0			0
$787$	24.6172	+	7.99863i	0.877510	+	0.285120i	0.712923	−	0.701242i	$-0.247371\pi$
$787$	24.6172	+	7.99863i	0.877510	+	0.285120i	0.164587	+	0.986363i	$0.447371\pi$
$788$	20.7089	+	28.5033i	0.737722	+	1.01539i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.93416	+	9.03040i	−0.103933	+	0.319873i	−0.989479	−	0.144679i	$-0.953785\pi$
$797$	−2.93416	+	9.03040i	−0.103933	+	0.319873i	0.885545	+	0.464553i	$0.153785\pi$
$798$		0			0
$799$	53.0353	+	72.9968i	1.87625	+	2.58244i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−49.8920	−	16.2109i	−1.75411	−	0.569944i	−0.757545	−	0.652783i	$-0.773601\pi$
$809$	−49.8920	−	16.2109i	−1.75411	−	0.569944i	−0.996563	+	0.0828389i	$0.973601\pi$
$810$		0			0
$811$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$811$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−31.8293	−	43.8093i	−1.11085	−	1.52896i	−0.820156	−	0.572140i	$-0.806113\pi$
$821$	−31.8293	−	43.8093i	−1.11085	−	1.52896i	−0.290696	−	0.956816i	$-0.593887\pi$
$822$		0			0
$823$	15.4411	+	47.5227i	0.538242	+	1.65654i	0.736539	+	0.676396i	$0.236459\pi$
$823$	15.4411	+	47.5227i	0.538242	+	1.65654i	−0.198297	+	0.980142i	$0.563541\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21.2141	+	6.89288i	−0.737686	+	0.239689i	−0.653674	−	0.756776i	$-0.726773\pi$
$827$	−21.2141	+	6.89288i	−0.737686	+	0.239689i	−0.0840119	+	0.996465i	$0.526773\pi$
$828$	−19.7726	−	14.3656i	−0.687145	−	0.499240i
$829$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$829$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$830$		0			0
$831$		0			0
$832$	−3.88789	−	5.35122i	−0.134788	−	0.185520i
$833$	−33.9199	+	46.6868i	−1.17526	+	1.61760i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$839$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$840$		0			0
$841$	−8.96149	+	27.5806i	−0.309017	+	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	57.4863			1.97409
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	18.7095	+	6.07908i	0.640601	+	0.208144i	0.611265	−	0.791426i	$-0.290661\pi$
$853$	18.7095	+	6.07908i	0.640601	+	0.208144i	0.0293354	+	0.999570i	$0.490661\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	27.1582i		−	0.927707i	−0.885912	−	0.463854i	$-0.846466\pi$
$857$		−	27.1582i		−	0.927707i	0.885912	−	0.463854i	$-0.153534\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$863$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.91228	−	11.6403i	−0.0648698	−	0.394871i
$870$		0			0
$871$	−8.54892	+	2.77771i	−0.289669	+	0.0941192i
$872$		0			0
$873$	46.1589	−	33.5364i	1.56224	−	1.13503i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.92450	−	10.9071i	0.267591	−	0.368308i	−0.653983	−	0.756509i	$-0.726903\pi$
$877$	7.92450	−	10.9071i	0.267591	−	0.368308i	0.921575	+	0.388201i	$0.126903\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.4685			−0.824365			−0.412182	−	0.911101i	$-0.635233\pi$
$881$	−24.4685			−0.824365			−0.412182	+	0.911101i	$0.635233\pi$
$882$		0			0
$883$	−32.6458	−	23.7186i	−1.09862	−	0.798193i	−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−32.6458	−	23.7186i	−1.09862	−	0.798193i	−0.980834	+	0.194846i	$0.937579\pi$
$884$	11.0289	−	8.01295i	0.370941	−	0.269505i
$885$		0			0
$886$		0			0
$887$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$887$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	13.7041	+	26.5179i	0.459104	+	0.888383i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−9.27051	−	28.5317i	−0.309017	−	0.951057i
$901$			118.479i			3.94712i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	17.2144	−	52.9806i	0.571596	−	1.75919i	−0.0758905	−	0.997116i	$-0.524180\pi$
$907$	17.2144	−	52.9806i	0.571596	−	1.75919i	0.647487	−	0.762077i	$-0.275820\pi$
$908$		0			0
$909$	−35.4372	−	48.7751i	−1.17538	−	1.61777i
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$	−6.97798	+	13.8918i	−0.230937	+	0.459753i
$914$		0			0
$915$		0			0
$916$	−41.7726	−	30.3496i	−1.38020	−	1.00278i
$917$		0			0
$918$		0			0
$919$	55.7658	+	18.1194i	1.83955	+	0.597705i	0.998378	+	0.0569297i	$0.0181311\pi$
$919$	55.7658	+	18.1194i	1.83955	+	0.597705i	0.841167	+	0.540775i	$0.181869\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−18.2466	−	13.2569i	−0.599297	−	0.435415i
$928$		0			0
$929$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$929$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$937$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−29.0546	−	9.44041i	−0.947153	−	0.307749i	−0.205595	−	0.978637i	$-0.565913\pi$
$941$	−29.0546	−	9.44041i	−0.947153	−	0.307749i	−0.741558	+	0.670888i	$0.765913\pi$
$942$		0			0
$943$	−15.7003	+	21.6096i	−0.511271	+	0.703705i
$944$	18.4732	+	56.8546i	0.601251	+	1.85046i
$945$		0			0
$946$		0			0
$947$	6.18914			0.201120			0.100560	−	0.994931i	$-0.467937\pi$
$947$	6.18914			0.201120			0.100560	+	0.994931i	$0.467937\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$953$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$954$		0			0
$955$		0			0
$956$		−	45.1537i		−	1.46038i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5793	−	11.3191i	0.502560	−	0.365131i
$962$		0			0
$963$	−54.9795	−	17.8639i	−1.77169	−	0.575657i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			50.5946i			1.62701i	0.581557	+	0.813506i	$0.302444\pi$
$967$			50.5946i			1.62701i	−0.581557	+	0.813506i	$0.697556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	38.4772	+	27.9553i	1.23479	+	0.897128i	0.997240	−	0.0742473i	$-0.0236554\pi$
$971$	38.4772	+	27.9553i	1.23479	+	0.897128i	0.237551	+	0.971375i	$0.423655\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	12.1708	+	37.4577i	0.389377	+	1.19838i	0.933255	+	0.359215i	$0.116955\pi$
$977$	12.1708	+	37.4577i	0.389377	+	1.19838i	−0.543878	+	0.839164i	$0.683045\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−33.6695	+	10.9399i	−1.07498	+	0.349284i
$982$		0			0
$983$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$983$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$			26.7109i			0.849358i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$997$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	473.2.k.a.171.1	✓	8
11.2	odd	10	inner	473.2.k.a.343.2	yes	8
43.42	odd	2	CM	473.2.k.a.171.1	✓	8
473.343	even	10	inner	473.2.k.a.343.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
473.2.k.a.171.1	✓	8	1.1	even	1	trivial
473.2.k.a.171.1	✓	8	43.42	odd	2	CM
473.2.k.a.343.2	yes	8	11.2	odd	10	inner
473.2.k.a.343.2	yes	8	473.343	even	10	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:	\( N \)	\(=\)	\( 473 = 11 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	473.k (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(1.61803 - 1.17557i) q^{4} +(0.927051 + 2.85317i) q^{9} +O(q^{10})\)	\(q+(1.61803 - 1.17557i) q^{4} +(0.927051 + 2.85317i) q^{9} +(0.500000 - 3.27872i) q^{11} +(0.786343 - 0.255498i) q^{13} +(1.23607 - 3.80423i) q^{16} +(7.84052 + 2.54754i) q^{17} -4.07338 q^{23} +(-4.04508 - 2.93893i) q^{25} +(-1.05894 - 3.25907i) q^{31} +(4.85410 + 3.52671i) q^{36} +(3.85437 - 5.30508i) q^{41} -6.55744i q^{43} +(-3.04535 - 5.89286i) q^{44} +(8.85453 + 6.43319i) q^{47} +(-2.16312 + 6.65740i) q^{49} +(0.971973 - 1.33781i) q^{52} +(4.44106 + 13.6682i) q^{53} +(-12.0909 + 8.78452i) q^{59} +(-2.47214 - 7.60845i) q^{64} -10.8717 q^{67} +(15.6810 - 5.09508i) q^{68} +(3.38266 - 1.09909i) q^{79} +(-7.28115 + 5.29007i) q^{81} +(-4.45786 - 1.44845i) q^{83} +(-6.59086 + 4.78854i) q^{92} +(-5.87704 - 18.0877i) q^{97} +(9.81827 - 1.61296i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 6 q^{9}+O(q^{10}) \) 8 * q + 4 * q^4 - 6 * q^9	\( 8 q + 4 q^{4} - 6 q^{9} + 4 q^{11} + 15 q^{13} - 8 q^{16} - 14 q^{23} - 10 q^{25} - 18 q^{31} + 12 q^{36} + 2 q^{44} + 12 q^{47} + 14 q^{49} + 26 q^{53} - 16 q^{59} + 16 q^{64} + 30 q^{67} - 18 q^{81} + 28 q^{92} - 3 q^{97} - 3 q^{99}+O(q^{100}) \) 8 * q + 4 * q^4 - 6 * q^9 + 4 * q^11 + 15 * q^13 - 8 * q^16 - 14 * q^23 - 10 * q^25 - 18 * q^31 + 12 * q^36 + 2 * q^44 + 12 * q^47 + 14 * q^49 + 26 * q^53 - 16 * q^59 + 16 * q^64 + 30 * q^67 - 18 * q^81 + 28 * q^92 - 3 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more