LMFDB - Embedded newform 605.2.j.b.444.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(9,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 6]))

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

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

chi := DirichletCharacter("605.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	605.j (of order $10$, degree $4$, not 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:	$4.83094932229$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.228765625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 $ x^8 - x^7 - 2x^6 + 5x^5 + x^4 + 15x^3 - 18x^2 - 27*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{10}]$

Embedding invariants

Embedding label			444.1
Root			$-0.570223 + 1.63550i$ of defining polynomial
Character	$\chi$	$=$	605.444
Dual form			605.2.j.b.124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.15430 - 1.02489i) q^{3} +(0.618034 + 1.90211i) q^{4} +(-0.238794 + 2.22328i) q^{5} +(6.47214 + 4.70228i) q^{9} +O(q^{10})$	$q+(-3.15430 - 1.02489i) q^{3} +(0.618034 + 1.90211i) q^{4} +(-0.238794 + 2.22328i) q^{5} +(6.47214 + 4.70228i) q^{9} -6.63325i q^{12} +(3.03185 - 6.76815i) q^{15} +(-3.23607 + 2.35114i) q^{16} +(-4.37651 + 0.919850i) q^{20} +3.31662i q^{23} +(-4.88595 - 1.06181i) q^{25} +(-9.74732 - 13.4160i) q^{27} +(-4.04508 - 2.93893i) q^{31} +(-4.94427 + 15.2169i) q^{36} +(-9.46289 + 3.07468i) q^{37} +(-12.0000 + 13.2665i) q^{45} +(6.30860 + 2.04979i) q^{47} +(12.6172 - 4.09957i) q^{48} +(-5.66312 + 4.11450i) q^{49} +(-7.79785 + 10.7328i) q^{53} +(-4.63525 - 14.2658i) q^{59} +(14.7476 + 1.58398i) q^{60} +(-6.47214 - 4.70228i) q^{64} -9.94987i q^{67} +(3.39919 - 10.4616i) q^{69} +(2.42705 - 1.76336i) q^{71} +(14.3235 + 8.35685i) q^{75} +(-4.45449 - 7.75613i) q^{80} +(9.57953 + 29.4828i) q^{81} +9.00000 q^{89} +(-6.30860 + 2.04979i) q^{92} +(9.74732 + 13.4160i) q^{93} +(5.84839 - 8.04962i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} - 3 q^{5} + 16 q^{9}+O(q^{10}) $ 8 * q - 4 * q^4 - 3 * q^5 + 16 * q^9	$ 8 q - 4 q^{4} - 3 q^{5} + 16 q^{9} + 11 q^{15} - 8 q^{16} - 6 q^{20} + q^{25} - 10 q^{31} + 32 q^{36} - 96 q^{45} - 14 q^{49} + 30 q^{59} + 22 q^{60} - 16 q^{64} - 22 q^{69} + 6 q^{71} + 33 q^{75} - 12 q^{80} - 62 q^{81} + 72 q^{89}+O(q^{100}) $ 8 * q - 4 * q^4 - 3 * q^5 + 16 * q^9 + 11 * q^15 - 8 * q^16 - 6 * q^20 + q^25 - 10 * q^31 + 32 * q^36 - 96 * q^45 - 14 * q^49 + 30 * q^59 + 22 * q^60 - 16 * q^64 - 22 * q^69 + 6 * q^71 + 33 * q^75 - 12 * q^80 - 62 * q^81 + 72 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$-1$	$e\left(\frac{1}{5}\right)$

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		−0.809017	−	0.587785i	$-0.800000\pi$
$2$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$3$	−3.15430	−	1.02489i	−1.82113	−	0.591722i	−0.999773	−	0.0213149i	$-0.993215\pi$
$3$	−3.15430	−	1.02489i	−1.82113	−	0.591722i	−0.821362	−	0.570408i	$-0.806785\pi$
$4$	0.618034	+	1.90211i	0.309017	+	0.951057i
$5$	−0.238794	+	2.22328i	−0.106792	+	0.994281i
$5$	−0.238794	+	2.22328i	−0.106792	+	0.994281i
$6$		0			0
$7$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$7$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$8$		0			0
$9$	6.47214	+	4.70228i	2.15738	+	1.56743i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		−	6.63325i		−	1.91485i
$13$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$13$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$14$		0			0
$15$	3.03185	−	6.76815i	0.782821	−	1.74753i
$16$	−3.23607	+	2.35114i	−0.809017	+	0.587785i
$17$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$17$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$18$		0			0
$19$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$19$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$20$	−4.37651	+	0.919850i	−0.978618	+	0.205685i
$21$		0			0
$22$		0			0
$23$			3.31662i			0.691564i	0.938315	+	0.345782i	$0.112386\pi$
$23$			3.31662i			0.691564i	−0.938315	+	0.345782i	$0.887614\pi$
$24$		0			0
$25$	−4.88595	−	1.06181i	−0.977191	−	0.212362i
$26$		0			0
$27$	−9.74732	−	13.4160i	−1.87587	−	2.58192i
$28$		0			0
$29$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$29$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$30$		0			0
$31$	−4.04508	−	2.93893i	−0.726519	−	0.527847i	0.161942	−	0.986800i	$-0.448224\pi$
$31$	−4.04508	−	2.93893i	−0.726519	−	0.527847i	−0.888460	+	0.458954i	$0.848224\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−4.94427	+	15.2169i	−0.824045	+	2.53615i
$37$	−9.46289	+	3.07468i	−1.55569	+	0.505474i	−0.955652	−	0.294497i	$-0.904848\pi$
$37$	−9.46289	+	3.07468i	−1.55569	+	0.505474i	−0.600038	+	0.799972i	$0.704848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−12.0000	+	13.2665i	−1.78885	+	1.97765i
$46$		0			0
$47$	6.30860	+	2.04979i	0.920203	+	0.298992i	0.730550	−	0.682859i	$-0.239264\pi$
$47$	6.30860	+	2.04979i	0.920203	+	0.298992i	0.189653	+	0.981851i	$0.439264\pi$
$48$	12.6172	−	4.09957i	1.82113	−	0.591722i
$49$	−5.66312	+	4.11450i	−0.809017	+	0.587785i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.79785	+	10.7328i	−1.07112	+	1.47427i	−0.202178	+	0.979349i	$0.564802\pi$
$53$	−7.79785	+	10.7328i	−1.07112	+	1.47427i	−0.868940	+	0.494918i	$0.835198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.63525	−	14.2658i	−0.603459	−	1.85726i	−0.507057	−	0.861913i	$-0.669267\pi$
$59$	−4.63525	−	14.2658i	−0.603459	−	1.85726i	−0.0964021	−	0.995342i	$-0.530733\pi$
$60$	14.7476	+	1.58398i	1.90390	+	0.204491i
$61$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$61$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$62$		0			0
$63$		0			0
$64$	−6.47214	−	4.70228i	−0.809017	−	0.587785i
$65$		0			0
$66$		0			0
$67$		−	9.94987i		−	1.21557i	−0.794101	−	0.607785i	$-0.792058\pi$
$67$		−	9.94987i		−	1.21557i	0.794101	−	0.607785i	$-0.207942\pi$
$68$		0			0
$69$	3.39919	−	10.4616i	0.409214	−	1.25943i
$70$		0			0
$71$	2.42705	−	1.76336i	0.288038	−	0.209272i	−0.434378	−	0.900731i	$-0.643032\pi$
$71$	2.42705	−	1.76336i	0.288038	−	0.209272i	0.722416	+	0.691459i	$0.243032\pi$
$72$		0			0
$73$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$73$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$74$		0			0
$75$	14.3235	+	8.35685i	1.65394	+	0.964966i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$79$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$80$	−4.45449	−	7.75613i	−0.498027	−	0.867161i
$81$	9.57953	+	29.4828i	1.06439	+	3.27586i
$82$		0			0
$83$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$83$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000			0.953998			0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000			0.953998			0.476999	+	0.878904i	$0.341725\pi$
$90$		0			0
$91$		0			0
$92$	−6.30860	+	2.04979i	−0.657717	+	0.213705i
$93$	9.74732	+	13.4160i	1.01075	+	1.39118i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.84839	−	8.04962i	0.593814	−	0.817315i	−0.401310	−	0.915942i	$-0.631445\pi$
$97$	5.84839	−	8.04962i	0.593814	−	0.817315i	0.995124	+	0.0986273i	$0.0314452\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000	−	9.94987i	−0.100000	−	0.994987i
$101$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$101$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$102$		0			0
$103$	−18.9258	+	6.14936i	−1.86481	+	0.605914i	−0.871510	+	0.490378i	$0.836859\pi$
$103$	−18.9258	+	6.14936i	−1.86481	+	0.605914i	−0.993303	+	0.115536i	$0.963141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$107$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$108$	19.4946	−	26.8321i	1.87587	−	2.58192i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	33.0000			3.13222
$112$		0			0
$113$	3.15430	+	1.02489i	0.296731	+	0.0964139i	0.453599	−	0.891206i	$-0.350140\pi$
$113$	3.15430	+	1.02489i	0.296731	+	0.0964139i	−0.156868	+	0.987620i	$0.550140\pi$
$114$		0			0
$115$	−7.37379	−	0.791990i	−0.687609	−	0.0738534i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	3.09017	−	9.51057i	0.277505	−	0.854074i
$125$	3.52744	−	10.6093i	0.315504	−	0.948924i
$126$		0			0
$127$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$127$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$128$		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
$135$	32.1552	−	18.4674i	2.76748	−	1.58942i
$136$		0			0
$137$	13.6462	+	18.7824i	1.16588	+	1.60469i	0.686617	+	0.727019i	$0.259095\pi$
$137$	13.6462	+	18.7824i	1.16588	+	1.60469i	0.479260	+	0.877673i	$0.340905\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$	−17.7984	−	12.9313i	−1.49889	−	1.08901i
$142$		0			0
$143$		0			0
$144$	−32.0000			−2.66667
$145$		0			0
$146$		0			0
$147$	22.0801	−	7.17425i	1.82113	−	0.591722i
$148$	−11.6968	−	16.0992i	−0.961469	−	1.32335i
$149$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$149$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$150$		0			0
$151$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$151$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.50000	−	8.29156i	0.602414	−	0.665994i
$156$		0			0
$157$	9.46289	+	3.07468i	0.755221	+	0.245386i	0.661226	−	0.750186i	$-0.270036\pi$
$157$	9.46289	+	3.07468i	0.755221	+	0.245386i	0.0939948	+	0.995573i	$0.470036\pi$
$158$		0			0
$159$	35.5967	−	25.8626i	2.82301	−	2.05103i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−11.6968	+	16.0992i	−0.916162	+	1.26099i	0.0488556	+	0.998806i	$0.484443\pi$
$163$	−11.6968	+	16.0992i	−0.916162	+	1.26099i	−0.965018	+	0.262184i	$0.915557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$167$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$168$		0			0
$169$	4.01722	+	12.3637i	0.309017	+	0.951057i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$173$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			49.7494i			3.73939i
$178$		0			0
$179$	−6.48936	+	19.9722i	−0.485037	+	1.49279i	0.346890	+	0.937906i	$0.387238\pi$
$179$	−6.48936	+	19.9722i	−0.485037	+	1.49279i	−0.831927	+	0.554885i	$0.812762\pi$
$180$	−32.6508	−	14.6262i	−2.43365	−	1.09017i
$181$	−20.2254	+	14.6946i	−1.50334	+	1.09224i	−0.534318	+	0.845283i	$0.679432\pi$
$181$	−20.2254	+	14.6946i	−1.50334	+	1.09224i	−0.969026	+	0.246960i	$0.920568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4.57620	−	21.7729i	−0.336449	−	1.60077i
$186$		0			0
$187$		0			0
$188$			13.2665i			0.967559i
$189$		0			0
$190$		0			0
$191$	4.63525	+	14.2658i	0.335395	+	1.03224i	0.966527	+	0.256565i	$0.0825907\pi$
$191$	4.63525	+	14.2658i	0.335395	+	1.03224i	−0.631132	+	0.775676i	$0.717409\pi$
$192$	15.5957	+	21.4656i	1.12552	+	1.54915i
$193$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$193$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$194$		0			0
$195$		0			0
$196$	−11.3262	−	8.22899i	−0.809017	−	0.587785i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	−10.1976	+	31.3849i	−0.719280	+	2.21372i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−15.5957	+	21.4656i	−1.08398	+	1.49197i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$211$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$212$	−25.2344	−	8.19915i	−1.73310	−	0.563120i
$213$	−9.46289	+	3.07468i	−0.648387	+	0.210674i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	28.3887	+	9.22404i	1.90105	+	0.617687i	0.960870	+	0.277000i	$0.0893402\pi$
$223$	28.3887	+	9.22404i	1.90105	+	0.617687i	0.940177	+	0.340687i	$0.110660\pi$
$224$		0			0
$225$	−26.6296	−	29.8473i	−1.77531	−	1.98982i
$226$		0			0
$227$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$227$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$228$		0			0
$229$	4.04508	+	2.93893i	0.267307	+	0.194210i	0.713362	−	0.700796i	$-0.247172\pi$
$229$	4.04508	+	2.93893i	0.267307	+	0.194210i	−0.446055	+	0.895005i	$0.647172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$233$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$234$		0			0
$235$	−6.06371	+	13.5363i	−0.395553	+	0.883011i
$236$	24.2705	−	17.6336i	1.57988	−	1.14785i
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$	6.10160	+	29.0305i	0.393856	+	1.87391i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$		−	53.0660i		−	3.40419i
$244$		0			0
$245$	−7.79536	−	13.5732i	−0.498027	−	0.867161i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	21.8435	+	15.8702i	1.37875	+	1.00172i	0.996996	+	0.0774530i	$0.0246788\pi$
$251$	21.8435	+	15.8702i	1.37875	+	1.00172i	0.381751	+	0.924265i	$0.375321\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.94427	−	15.2169i	0.309017	−	0.951057i
$257$	25.2344	−	8.19915i	1.57408	−	0.511449i	0.613555	−	0.789652i	$-0.289739\pi$
$257$	25.2344	−	8.19915i	1.57408	−	0.511449i	0.960522	+	0.278203i	$0.0897388\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−22.0000	−	19.8997i	−1.35145	−	1.22243i
$266$		0			0
$267$	−28.3887	−	9.22404i	−1.73736	−	0.564502i
$268$	18.9258	−	6.14936i	1.15608	−	0.375632i
$269$	−24.2705	+	17.6336i	−1.47980	+	1.07514i	−0.502178	+	0.864764i	$0.667468\pi$
$269$	−24.2705	+	17.6336i	−1.47980	+	1.07514i	−0.977621	+	0.210373i	$0.932532\pi$
$270$		0			0
$271$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$271$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	22.0000			1.32424
$277$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$277$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$278$		0			0
$279$	−12.3607	−	38.0423i	−0.740015	−	2.27753i
$280$		0			0
$281$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$281$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$282$		0			0
$283$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$283$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$284$	4.85410	+	3.52671i	0.288038	+	0.209272i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.25329	−	16.1680i	0.309017	−	0.951057i
$290$		0			0
$291$	−26.6976	+	19.3969i	−1.56504	+	1.13707i
$292$		0			0
$293$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$293$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$294$		0			0
$295$	32.8239	−	6.89888i	1.91108	−	0.401668i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−7.04326	+	32.4098i	−0.406643	+	1.87118i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	66.0000			3.75461
$310$		0			0
$311$	−3.70820	+	11.4127i	−0.210273	+	0.647154i	0.789183	+	0.614159i	$0.210505\pi$
$311$	−3.70820	+	11.4127i	−0.210273	+	0.647154i	−0.999456	+	0.0329949i	$0.989495\pi$
$312$		0			0
$313$	−17.5452	−	24.1489i	−0.991712	−	1.36497i	−0.930275	−	0.366863i	$-0.880432\pi$
$313$	−17.5452	−	24.1489i	−0.991712	−	1.36497i	−0.0614365	−	0.998111i	$-0.519568\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.6462	−	18.7824i	0.766449	−	1.05493i	−0.230201	−	0.973143i	$-0.573938\pi$
$317$	13.6462	−	18.7824i	0.766449	−	1.05493i	0.996650	−	0.0817838i	$-0.0260617\pi$
$318$		0			0
$319$		0			0
$320$	12.0000	−	13.2665i	0.670820	−	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−50.1591	+	36.4427i	−2.78661	+	2.02459i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−35.0000			−1.92377			−0.961887	−	0.273447i	$-0.911836\pi$
$331$	−35.0000			−1.92377			−0.961887	+	0.273447i	$0.911836\pi$
$332$		0			0
$333$	−75.7031	−	24.5974i	−4.14851	−	1.34793i
$334$		0			0
$335$	22.1214	+	2.37597i	1.20862	+	0.129813i
$336$		0			0
$337$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$337$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$338$		0			0
$339$	−8.89919	−	6.46564i	−0.483337	−	0.351165i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	22.4474	+	10.0555i	1.20853	+	0.541371i
$346$		0			0
$347$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$347$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			36.4829i			1.94179i	0.239511	+	0.970894i	$0.423013\pi$
$353$			36.4829i			1.94179i	−0.239511	+	0.970894i	$0.576987\pi$
$354$		0			0
$355$	3.34087	+	5.81709i	0.177315	+	0.308739i
$356$	5.56231	+	17.1190i	0.294802	+	0.907306i
$357$		0			0
$358$		0			0
$359$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$359$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$360$		0			0
$361$	15.3713	+	11.1679i	0.809017	+	0.587785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	9.46289	−	3.07468i	0.493959	−	0.160497i	−0.0514358	−	0.998676i	$-0.516380\pi$
$367$	9.46289	−	3.07468i	0.493959	−	0.160497i	0.545395	+	0.838179i	$0.316380\pi$
$368$	−7.79785	−	10.7328i	−0.406491	−	0.559487i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−19.4946	+	26.8321i	−1.01075	+	1.39118i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−22.0000	+	29.8496i	−1.13608	+	1.54143i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	20.2254	−	14.6946i	1.03891	−	0.754813i	0.0688378	−	0.997628i	$-0.478071\pi$
$379$	20.2254	−	14.6946i	1.03891	−	0.754813i	0.970073	+	0.242815i	$0.0780709\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.94946	+	2.68321i	−0.0996129	+	0.137105i	−0.855914	−	0.517119i	$-0.827005\pi$
$383$	−1.94946	+	2.68321i	−0.0996129	+	0.137105i	0.756301	+	0.654224i	$0.227005\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	18.9258	+	6.14936i	0.960811	+	0.312186i
$389$	−4.63525	−	14.2658i	−0.235017	−	0.723307i	−0.997119	−	0.0758507i	$-0.975833\pi$
$389$	−4.63525	−	14.2658i	−0.235017	−	0.723307i	0.762102	−	0.647456i	$-0.224167\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	39.7995i		−	1.99748i	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$		−	39.7995i		−	1.99748i	0.0501886	−	0.998740i	$-0.484018\pi$
$398$		0			0
$399$		0			0
$400$	18.3078	−	8.05147i	0.915388	−	0.402574i
$401$	24.2705	−	17.6336i	1.21201	−	0.880578i	0.216600	−	0.976261i	$-0.430503\pi$
$401$	24.2705	−	17.6336i	1.21201	−	0.880578i	0.995412	+	0.0956827i	$0.0305034\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−67.8360	+	14.2577i	−3.37080	+	0.708470i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$409$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$410$		0			0
$411$	−23.7943	−	73.2314i	−1.17369	−	3.61224i
$412$	−23.3936	−	32.1985i	−1.15252	−	1.58630i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	−3.09017	+	9.51057i	−0.150606	+	0.463517i	−0.997689	−	0.0679432i	$-0.978356\pi$
$421$	−3.09017	+	9.51057i	−0.150606	+	0.463517i	0.847084	+	0.531460i	$0.178356\pi$
$422$		0			0
$423$	31.1914	+	42.9313i	1.51658	+	2.08739i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$	63.0860	+	20.4979i	3.03522	+	0.986204i
$433$	−28.3887	+	9.22404i	−1.36427	+	0.443279i	−0.897467	−	0.441081i	$-0.854595\pi$
$433$	−28.3887	+	9.22404i	−1.36427	+	0.443279i	−0.466805	+	0.884360i	$0.654595\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−56.0000			−2.66667
$442$		0			0
$443$	34.6973	+	11.2738i	1.64852	+	0.535636i	0.978418	−	0.206636i	$-0.0662515\pi$
$443$	34.6973	+	11.2738i	1.64852	+	0.535636i	0.670099	+	0.742271i	$0.266252\pi$
$444$	20.3951	+	62.7697i	0.967910	+	2.97892i
$445$	−2.14915	+	20.0095i	−0.101879	+	0.948543i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	31.5517	+	22.9236i	1.48902	+	1.08183i	0.974510	+	0.224346i	$0.0720244\pi$
$449$	31.5517	+	22.9236i	1.48902	+	1.08183i	0.514505	+	0.857487i	$0.327976\pi$
$450$		0			0
$451$		0			0
$452$			6.63325i			0.312002i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$457$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$458$		0			0
$459$		0			0
$460$	−3.05080	−	14.5153i	−0.142244	−	0.676777i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$			29.8496i			1.38723i	0.720346	+	0.693615i	$0.243983\pi$
$463$			29.8496i			1.38723i	−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$	−32.1552	+	18.4674i	−1.49116	+	0.856403i
$466$		0			0
$467$	−25.3430	−	34.8817i	−1.17274	−	1.61413i	−0.642523	−	0.766267i	$-0.722112\pi$
$467$	−25.3430	−	34.8817i	−1.17274	−	1.61413i	−0.530212	−	0.847865i	$-0.677888\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−26.6976	−	19.3969i	−1.23016	−	0.893763i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−100.938	+	32.7966i	−4.62161	+	1.50165i
$478$		0			0
$479$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$479$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	16.5000	+	14.9248i	0.749226	+	0.677701i
$486$		0			0
$487$	−9.46289	−	3.07468i	−0.428805	−	0.139327i	0.0866600	−	0.996238i	$-0.472381\pi$
$487$	−9.46289	−	3.07468i	−0.428805	−	0.139327i	−0.515465	+	0.856911i	$0.672381\pi$
$488$		0			0
$489$	53.3951	−	38.7938i	2.41461	−	1.75432i
$490$		0			0
$491$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$491$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	20.0000			0.898027
$497$		0			0
$498$		0			0
$499$	12.3607	+	38.0423i	0.553340	+	1.70301i	0.700287	+	0.713861i	$0.253055\pi$
$499$	12.3607	+	38.0423i	0.553340	+	1.70301i	−0.146947	+	0.989144i	$0.546945\pi$
$500$	22.3602	+	0.152689i	0.999977	+	0.00682845i
$501$		0			0
$502$		0			0
$503$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$503$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	43.1161i		−	1.91485i
$508$		0			0
$509$	−13.9058	+	42.7975i	−0.616362	+	1.89697i	−0.238275	+	0.971198i	$0.576582\pi$
$509$	−13.9058	+	42.7975i	−0.616362	+	1.89697i	−0.378087	+	0.925770i	$0.623418\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.15239	−	43.5458i	−0.403303	−	1.91886i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.63525	+	14.2658i	0.203074	+	0.624998i	0.999787	+	0.0206400i	$0.00657038\pi$
$521$	4.63525	+	14.2658i	0.203074	+	0.624998i	−0.796713	+	0.604358i	$0.793430\pi$
$522$		0			0
$523$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$523$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	12.0000			0.521739
$530$		0			0
$531$	37.0820	−	114.127i	1.60922	−	4.95268i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	40.9387	−	56.3473i	1.76664	−	2.43157i
$538$		0			0
$539$		0			0
$540$	55.0000	+	49.7494i	2.36682	+	2.14087i
$541$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$541$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$542$		0			0
$543$	78.8574	−	25.6223i	3.38410	−	1.09956i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$547$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$548$	−27.2925	+	37.5649i	−1.16588	+	1.60469i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−7.88020	+	73.3683i	−0.334496	+	3.11431i
$556$		0			0
$557$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$557$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$563$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$564$	13.5967	−	41.8465i	0.572526	−	1.76205i
$565$	−3.03185	+	6.76815i	−0.127551	+	0.284738i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$569$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		−	49.7494i		−	2.07831i
$574$		0			0
$575$	3.52163	−	16.2049i	0.146862	−	0.675790i
$576$	−19.7771	−	60.8676i	−0.824045	−	2.53615i
$577$	5.84839	+	8.04962i	0.243472	+	0.335110i	0.913212	−	0.407486i	$-0.133594\pi$
$577$	5.84839	+	8.04962i	0.243472	+	0.335110i	−0.669740	+	0.742596i	$0.733594\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.30860	−	2.04979i	0.260384	−	0.0846038i	−0.175916	−	0.984405i	$-0.556289\pi$
$587$	6.30860	−	2.04979i	0.260384	−	0.0846038i	0.436300	+	0.899801i	$0.356289\pi$
$588$	27.2925	+	37.5649i	1.12552	+	1.54915i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	23.3936	−	32.1985i	0.961469	−	1.32335i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	63.0860	+	20.4979i	2.58194	+	0.838922i
$598$		0			0
$599$	−29.1246	+	21.1603i	−1.19000	+	0.864585i	−0.993264	−	0.115872i	$-0.963034\pi$
$599$	−29.1246	+	21.1603i	−1.19000	+	0.864585i	−0.196735	+	0.980457i	$0.563034\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$		0			0
$603$	46.7871	−	64.3969i	1.90532	−	2.62245i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$607$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$613$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	26.5330i		−	1.06818i	−0.845428	−	0.534089i	$-0.820655\pi$
$617$		−	26.5330i		−	1.06818i	0.845428	−	0.534089i	$-0.179345\pi$
$618$		0			0
$619$	0.309017	−	0.951057i	0.0124204	−	0.0382262i	−0.944654	−	0.328068i	$-0.893603\pi$
$619$	0.309017	−	0.951057i	0.0124204	−	0.0382262i	0.957075	+	0.289841i	$0.0936026\pi$
$620$	20.4067	+	9.14138i	0.819555	+	0.367127i
$621$	44.4959	−	32.3282i	1.78556	−	1.29729i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	22.7451	+	10.3759i	0.909804	+	0.415037i
$626$		0			0
$627$		0			0
$628$			19.8997i			0.794086i
$629$		0			0
$630$		0			0
$631$	2.16312	+	6.65740i	0.0861124	+	0.265027i	0.984836	−	0.173489i	$-0.0555042\pi$
$631$	2.16312	+	6.65740i	0.0861124	+	0.265027i	−0.898723	+	0.438516i	$0.855504\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	71.1935	+	51.7251i	2.82301	+	2.05103i
$637$		0			0
$638$		0			0
$639$	24.0000			0.949425
$640$		0			0
$641$	13.9058	−	42.7975i	0.549245	−	1.69040i	−0.161433	−	0.986884i	$-0.551612\pi$
$641$	13.9058	−	42.7975i	0.549245	−	1.69040i	0.710678	−	0.703518i	$-0.248388\pi$
$642$		0			0
$643$	17.5452	+	24.1489i	0.691914	+	0.952338i	1.00000		0.000979141i	$0.000311670\pi$
$643$	17.5452	+	24.1489i	0.691914	+	0.952338i	−0.308086	+	0.951359i	$0.599688\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−25.3430	+	34.8817i	−0.996337	+	1.37134i	−0.0687910	+	0.997631i	$0.521914\pi$
$647$	−25.3430	+	34.8817i	−0.996337	+	1.37134i	−0.927546	+	0.373709i	$0.878086\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−37.8516	−	12.2987i	−1.48238	−	0.481655i
$653$	−3.15430	+	1.02489i	−0.123437	+	0.0401072i	−0.370084	−	0.928998i	$-0.620671\pi$
$653$	−3.15430	+	1.02489i	−0.123437	+	0.0401072i	0.246647	+	0.969105i	$0.420671\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	13.0000			0.505641			0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000			0.505641			0.252821	+	0.967513i	$0.418642\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−80.0927	−	58.1907i	−3.09656	−	2.24978i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$673$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$674$		0			0
$675$	33.3796	+	75.8999i	1.28478	+	2.92139i
$676$	−21.0344	+	15.2824i	−0.809017	+	0.587785i
$677$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$677$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			46.4327i			1.77670i	0.459167	+	0.888350i	$0.348148\pi$
$683$			46.4327i			1.77670i	−0.459167	+	0.888350i	$0.651852\pi$
$684$		0			0
$685$	−45.0173	+	25.8543i	−1.72002	+	0.987842i
$686$		0			0
$687$	−9.74732	−	13.4160i	−0.371883	−	0.511853i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−13.7533	−	9.99235i	−0.523200	−	0.380127i	0.294608	−	0.955618i	$-0.404811\pi$
$691$	−13.7533	−	9.99235i	−0.523200	−	0.380127i	−0.817808	+	0.575491i	$0.804811\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$701$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	33.0000	−	36.4829i	1.24285	−	1.37402i
$706$		0			0
$707$		0			0
$708$	−94.6289	+	30.7468i	−3.55637	+	1.15554i
$709$	15.3713	−	11.1679i	0.577282	−	0.419420i	−0.260461	−	0.965484i	$-0.583875\pi$
$709$	15.3713	−	11.1679i	0.577282	−	0.419420i	0.837743	+	0.546064i	$0.183875\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.74732	−	13.4160i	0.365040	−	0.502434i
$714$		0			0
$715$		0			0
$716$	−42.0000			−1.56961
$717$		0			0
$718$		0			0
$719$	−15.7599	−	48.5039i	−0.587744	−	1.80889i	−0.587961	−	0.808890i	$-0.700069\pi$
$719$	−15.7599	−	48.5039i	−0.587744	−	1.80889i	0.000216702	−	1.00000i	$-0.499931\pi$
$720$	7.64141	−	71.1450i	0.284778	−	2.65142i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−40.4508	−	29.3893i	−1.50334	−	1.09224i
$725$		0			0
$726$		0			0
$727$			9.94987i			0.369020i	0.982831	+	0.184510i	$0.0590699\pi$
$727$			9.94987i			0.369020i	−0.982831	+	0.184510i	$0.940930\pi$
$728$		0			0
$729$	−25.6484	+	78.9377i	−0.949941	+	2.92362i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$733$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$734$		0			0
$735$	10.6778	+	50.8034i	0.393856	+	1.87391i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$739$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$740$	38.5862	−	22.1608i	1.41846	−	0.814648i
$741$		0			0
$742$		0			0
$743$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$743$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−7.10739	+	21.8743i	−0.259352	+	0.798205i	0.733588	+	0.679594i	$0.237844\pi$
$751$	−7.10739	+	21.8743i	−0.259352	+	0.798205i	−0.992941	+	0.118611i	$0.962156\pi$
$752$	−25.2344	+	8.19915i	−0.920203	+	0.298992i
$753$	−52.6355	−	72.4466i	−1.91814	−	2.64010i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−23.3936	+	32.1985i	−0.850253	+	1.17027i	0.133554	+	0.991042i	$0.457361\pi$
$757$	−23.3936	+	32.1985i	−0.850253	+	1.17027i	−0.983807	+	0.179232i	$0.942639\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$761$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$762$		0			0
$763$		0			0
$764$	−24.2705	+	17.6336i	−0.878076	+	0.637960i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−31.1914	+	42.9313i	−1.12552	+	1.54915i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−88.0000			−3.16924
$772$		0			0
$773$	12.6172	+	4.09957i	0.453809	+	0.147451i	0.526998	−	0.849867i	$-0.323318\pi$
$773$	12.6172	+	4.09957i	0.453809	+	0.147451i	−0.0731890	+	0.997318i	$0.523318\pi$
$774$		0			0
$775$	16.6435	+	18.6546i	0.597853	+	0.670092i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	8.65248	−	26.6296i	0.309017	−	0.951057i
$785$	−9.09556	+	20.3045i	−0.324634	+	0.724697i
$786$		0			0
$787$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$787$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	48.9994	+	85.3174i	1.73783	+	3.02590i
$796$	−12.3607	−	38.0423i	−0.438113	−	1.34837i
$797$	33.1409	+	45.6145i	1.17391	+	1.61575i	0.629940	+	0.776644i	$0.283079\pi$
$797$	33.1409	+	45.6145i	1.17391	+	1.61575i	0.543970	+	0.839105i	$0.316921\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	58.2492	+	42.3205i	2.05814	+	1.49532i
$802$		0			0
$803$		0			0
$804$	−66.0000			−2.32764
$805$		0			0
$806$		0			0
$807$	94.6289	−	30.7468i	3.33110	−	1.08234i
$808$		0			0
$809$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$809$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−33.0000	−	29.8496i	−1.15594	−	1.04559i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$821$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$822$		0			0
$823$	−17.5452	+	24.1489i	−0.611586	+	0.841776i	−0.996707	−	0.0810902i	$-0.974160\pi$
$823$	−17.5452	+	24.1489i	−0.611586	+	0.841776i	0.385121	+	0.922866i	$0.374160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$827$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$828$	−50.4688	−	16.3983i	−1.75391	−	0.569880i
$829$	−8.96149	−	27.5806i	−0.311246	−	0.957915i	−0.977272	−	0.211987i	$-0.932006\pi$
$829$	−8.96149	−	27.5806i	−0.311246	−	0.957915i	0.666027	−	0.745928i	$-0.267994\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$			82.9156i			2.86598i
$838$		0			0
$839$	−13.9058	+	42.7975i	−0.480080	+	1.47754i	0.358901	+	0.933376i	$0.383151\pi$
$839$	−13.9058	+	42.7975i	−0.480080	+	1.47754i	−0.838982	+	0.544160i	$0.816849\pi$
$840$		0			0
$841$	23.4615	−	17.0458i	0.809017	−	0.587785i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−28.4473	+	5.97903i	−0.978618	+	0.205685i
$846$		0			0
$847$		0			0
$848$		−	53.0660i		−	1.82229i
$849$		0			0
$850$		0			0
$851$	−10.1976	−	31.3849i	−0.349568	−	1.07586i
$852$	−11.6968	−	16.0992i	−0.400725	−	0.551551i
$853$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$853$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	31.0000			1.05771			0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000			1.05771			0.528853	+	0.848713i	$0.322622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−27.2925	−	37.5649i	−0.929047	−	1.27872i	−0.960230	−	0.279210i	$-0.909928\pi$
$863$	−27.2925	−	37.5649i	−0.929047	−	1.27872i	0.0311832	−	0.999514i	$-0.490072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−33.1409	+	45.6145i	−1.12552	+	1.54915i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	75.7031	−	24.5974i	2.56216	−	0.832497i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$877$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	57.0000			1.92038			0.960189	−	0.279350i	$-0.0901189\pi$
$881$	57.0000			1.92038			0.960189	+	0.279350i	$0.0901189\pi$
$882$		0			0
$883$	−18.9258	−	6.14936i	−0.636903	−	0.206942i	−0.0272727	−	0.999628i	$-0.508682\pi$
$883$	−18.9258	−	6.14936i	−0.636903	−	0.206942i	−0.609631	+	0.792686i	$0.708682\pi$
$884$		0			0
$885$	−110.607	−	11.8798i	−3.71801	−	0.399337i
$886$		0			0
$887$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$887$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$			59.6992i			1.99888i
$893$		0			0
$894$		0			0
$895$	−42.8542	−	19.1969i	−1.43246	−	0.641682i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	40.3150	−	69.0992i	1.34383	−	2.30331i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−27.8406	−	48.4758i	−0.925452	−	1.61139i
$906$		0			0
$907$	35.0903	+	48.2977i	1.16516	+	1.60370i	0.690030	+	0.723781i	$0.257597\pi$
$907$	35.0903	+	48.2977i	1.16516	+	1.60370i	0.475126	+	0.879918i	$0.342403\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−48.5410	−	35.2671i	−1.60824	−	1.16845i	−0.868706	−	0.495327i	$-0.835048\pi$
$911$	−48.5410	−	35.2671i	−1.60824	−	1.16845i	−0.739529	−	0.673124i	$-0.764952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−3.09017	+	9.51057i	−0.102102	+	0.314238i
$917$		0			0
$918$		0			0
$919$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	49.5000	−	4.97494i	1.62755	−	0.163575i
$926$		0			0
$927$	−151.406	−	49.1949i	−4.97283	−	1.61577i
$928$		0			0
$929$	−24.2705	+	17.6336i	−0.796290	+	0.578538i	−0.909823	−	0.414996i	$-0.863783\pi$
$929$	−24.2705	+	17.6336i	−0.796290	+	0.578538i	0.113534	+	0.993534i	$0.463783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	23.3936	−	32.1985i	0.765871	−	1.05413i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$937$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$938$		0			0
$939$	30.5927	+	94.1546i	0.998354	+	3.07262i
$940$	−29.4952	−	3.16796i	−0.962026	−	0.103327i
$941$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$941$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$942$		0			0
$943$		0			0
$944$	48.5410	+	35.2671i	1.57988	+	1.14785i
$945$		0			0
$946$		0			0
$947$		−	23.2164i		−	0.754431i	−0.926126	−	0.377215i	$-0.876882\pi$
$947$		−	23.2164i		−	0.754431i	0.926126	−	0.377215i	$-0.123118\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−62.2943	+	45.2595i	−2.02003	+	1.46764i
$952$		0			0
$953$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$953$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$954$		0			0
$955$	−32.8239	+	6.89888i	−1.06216	+	0.223242i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−51.4483	+	29.5478i	−1.66049	+	0.953650i
$961$	−1.85410	−	5.70634i	−0.0598097	−	0.184075i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13.9058	−	42.7975i	0.446257	−	1.37344i	−0.434842	−	0.900507i	$-0.643196\pi$
$971$	13.9058	−	42.7975i	0.446257	−	1.37344i	0.881099	−	0.472932i	$-0.156804\pi$
$972$	100.938	−	32.7966i	3.23757	−	1.05195i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	33.1409	−	45.6145i	1.06027	−	1.45934i	0.180718	−	0.983535i	$-0.442158\pi$
$977$	33.1409	−	45.6145i	1.06027	−	1.45934i	0.879552	−	0.475802i	$-0.157842\pi$
$978$		0			0
$979$		0			0
$980$	21.0000	−	23.2164i	0.670820	−	0.741620i
$981$		0			0
$982$		0			0
$983$	−34.6973	+	11.2738i	−1.10667	+	0.359579i	−0.804666	−	0.593727i	$-0.797656\pi$
$983$	−34.6973	+	11.2738i	−1.10667	+	0.359579i	−0.302005	+	0.953306i	$0.597656\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$		0			0
$993$	110.400	+	35.8713i	3.50345	+	1.13834i
$994$		0			0
$995$	4.77588	−	44.4656i	0.151406	−	1.40965i
$996$		0			0
$997$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$997$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$998$		0			0
$999$	133.488	+	96.9846i	4.22337	+	3.06846i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.2.j.b.444.1		8
5.4	even	2	inner	605.2.j.b.444.2		8
11.2	odd	10	inner	605.2.j.b.269.1		8
11.3	even	5	inner	605.2.j.b.124.2		8
11.4	even	5	inner	605.2.j.b.9.2		8
11.5	even	5		605.2.b.a.364.1	✓	2
11.6	odd	10		605.2.b.a.364.1	✓	2
11.7	odd	10	inner	605.2.j.b.9.2		8
11.8	odd	10	inner	605.2.j.b.124.2		8
11.9	even	5	inner	605.2.j.b.269.1		8
11.10	odd	2	CM	605.2.j.b.444.1		8
55.4	even	10	inner	605.2.j.b.9.1		8
55.9	even	10	inner	605.2.j.b.269.2		8
55.14	even	10	inner	605.2.j.b.124.1		8
55.17	even	20		3025.2.a.k.1.1		2
55.19	odd	10	inner	605.2.j.b.124.1		8
55.24	odd	10	inner	605.2.j.b.269.2		8
55.27	odd	20		3025.2.a.k.1.1		2
55.28	even	20		3025.2.a.k.1.2		2
55.29	odd	10	inner	605.2.j.b.9.1		8
55.38	odd	20		3025.2.a.k.1.2		2
55.39	odd	10		605.2.b.a.364.2	yes	2
55.49	even	10		605.2.b.a.364.2	yes	2
55.54	odd	2	inner	605.2.j.b.444.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.a.364.1	✓	2	11.5	even	5
605.2.b.a.364.1	✓	2	11.6	odd	10
605.2.b.a.364.2	yes	2	55.39	odd	10
605.2.b.a.364.2	yes	2	55.49	even	10
605.2.j.b.9.1		8	55.4	even	10	inner
605.2.j.b.9.1		8	55.29	odd	10	inner
605.2.j.b.9.2		8	11.4	even	5	inner
605.2.j.b.9.2		8	11.7	odd	10	inner
605.2.j.b.124.1		8	55.14	even	10	inner
605.2.j.b.124.1		8	55.19	odd	10	inner
605.2.j.b.124.2		8	11.3	even	5	inner
605.2.j.b.124.2		8	11.8	odd	10	inner
605.2.j.b.269.1		8	11.2	odd	10	inner
605.2.j.b.269.1		8	11.9	even	5	inner
605.2.j.b.269.2		8	55.9	even	10	inner
605.2.j.b.269.2		8	55.24	odd	10	inner
605.2.j.b.444.1		8	1.1	even	1	trivial
605.2.j.b.444.1		8	11.10	odd	2	CM
605.2.j.b.444.2		8	5.4	even	2	inner
605.2.j.b.444.2		8	55.54	odd	2	inner
3025.2.a.k.1.1		2	55.17	even	20
3025.2.a.k.1.1		2	55.27	odd	20
3025.2.a.k.1.2		2	55.28	even	20
3025.2.a.k.1.2		2	55.38	odd	20

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:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	605.j (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.15430 - 1.02489i) q^{3} +(0.618034 + 1.90211i) q^{4} +(-0.238794 + 2.22328i) q^{5} +(6.47214 + 4.70228i) q^{9} +O(q^{10})\)	\(q+(-3.15430 - 1.02489i) q^{3} +(0.618034 + 1.90211i) q^{4} +(-0.238794 + 2.22328i) q^{5} +(6.47214 + 4.70228i) q^{9} -6.63325i q^{12} +(3.03185 - 6.76815i) q^{15} +(-3.23607 + 2.35114i) q^{16} +(-4.37651 + 0.919850i) q^{20} +3.31662i q^{23} +(-4.88595 - 1.06181i) q^{25} +(-9.74732 - 13.4160i) q^{27} +(-4.04508 - 2.93893i) q^{31} +(-4.94427 + 15.2169i) q^{36} +(-9.46289 + 3.07468i) q^{37} +(-12.0000 + 13.2665i) q^{45} +(6.30860 + 2.04979i) q^{47} +(12.6172 - 4.09957i) q^{48} +(-5.66312 + 4.11450i) q^{49} +(-7.79785 + 10.7328i) q^{53} +(-4.63525 - 14.2658i) q^{59} +(14.7476 + 1.58398i) q^{60} +(-6.47214 - 4.70228i) q^{64} -9.94987i q^{67} +(3.39919 - 10.4616i) q^{69} +(2.42705 - 1.76336i) q^{71} +(14.3235 + 8.35685i) q^{75} +(-4.45449 - 7.75613i) q^{80} +(9.57953 + 29.4828i) q^{81} +9.00000 q^{89} +(-6.30860 + 2.04979i) q^{92} +(9.74732 + 13.4160i) q^{93} +(5.84839 - 8.04962i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} - 3 q^{5} + 16 q^{9}+O(q^{10}) \) 8 * q - 4 * q^4 - 3 * q^5 + 16 * q^9	\( 8 q - 4 q^{4} - 3 q^{5} + 16 q^{9} + 11 q^{15} - 8 q^{16} - 6 q^{20} + q^{25} - 10 q^{31} + 32 q^{36} - 96 q^{45} - 14 q^{49} + 30 q^{59} + 22 q^{60} - 16 q^{64} - 22 q^{69} + 6 q^{71} + 33 q^{75} - 12 q^{80} - 62 q^{81} + 72 q^{89}+O(q^{100}) \) 8 * q - 4 * q^4 - 3 * q^5 + 16 * q^9 + 11 * q^15 - 8 * q^16 - 6 * q^20 + q^25 - 10 * q^31 + 32 * q^36 - 96 * q^45 - 14 * q^49 + 30 * q^59 + 22 * q^60 - 16 * q^64 - 22 * q^69 + 6 * q^71 + 33 * q^75 - 12 * q^80 - 62 * q^81 + 72 * q^89

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