LMFDB - Embedded newform 605.2.j.b.9.1

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			9.1
Root			$1.42264 + 0.987975i$ of defining polynomial
Character	$\chi$	$=$	605.9
Dual form			605.2.j.b.269.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.94946 - 2.68321i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(-1.11362 + 1.93903i) q^{5} +(-2.47214 + 7.60845i) q^{9} +O(q^{10})$	$q+(-1.94946 - 2.68321i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(-1.11362 + 1.93903i) q^{5} +(-2.47214 + 7.60845i) q^{9} +6.63325i q^{12} +(7.37379 - 0.791990i) q^{15} +(1.23607 + 3.80423i) q^{16} +(4.08135 - 1.82828i) q^{20} -3.31662i q^{23} +(-2.51969 - 4.31870i) q^{25} +(15.7715 - 5.12447i) q^{27} +(1.54508 - 4.75528i) q^{31} +(12.9443 - 9.40456i) q^{36} +(-5.84839 + 8.04962i) q^{37} +(-12.0000 - 13.2665i) q^{45} +(3.89893 + 5.36641i) q^{47} +(7.79785 - 10.7328i) q^{48} +(2.16312 + 6.65740i) q^{49} +(12.6172 + 4.09957i) q^{53} +(12.1353 + 8.81678i) q^{59} +(-12.8621 - 7.38694i) q^{60} +(2.47214 - 7.60845i) q^{64} +9.94987i q^{67} +(-8.89919 + 6.46564i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(-6.67593 + 15.1800i) q^{75} +(-8.75303 - 1.83970i) q^{80} +(-25.0795 - 18.2213i) q^{81} +9.00000 q^{89} +(-3.89893 + 5.36641i) q^{92} +(-15.7715 + 5.12447i) q^{93} +(-9.46289 - 3.07468i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/605\mathbb{Z}\right)^\times$.

$n$	$122$	$486$
$\chi(n)$	$-1$	$e\left(\frac{3}{5}\right)$

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		0.309017	−	0.951057i	$-0.400000\pi$
$2$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$3$	−1.94946	−	2.68321i	−1.12552	−	1.54915i	−0.796305	−	0.604896i	$-0.793215\pi$
$3$	−1.94946	−	2.68321i	−1.12552	−	1.54915i	−0.329218	−	0.944254i	$-0.606785\pi$
$4$	−1.61803	−	1.17557i	−0.809017	−	0.587785i
$5$	−1.11362	+	1.93903i	−0.498027	+	0.867161i
$5$	−1.11362	+	1.93903i	−0.498027	+	0.867161i
$6$		0			0
$7$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$7$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$8$		0			0
$9$	−2.47214	+	7.60845i	−0.824045	+	2.53615i
$10$		0			0
$11$		0			0
$11$		0			0
$12$			6.63325i			1.91485i
$13$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$13$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$14$		0			0
$15$	7.37379	−	0.791990i	1.90390	−	0.204491i
$16$	1.23607	+	3.80423i	0.309017	+	0.951057i
$17$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$17$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$18$		0			0
$19$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$19$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$20$	4.08135	−	1.82828i	0.912617	−	0.408815i
$21$		0			0
$22$		0			0
$23$		−	3.31662i		−	0.691564i	−0.938315	−	0.345782i	$-0.887614\pi$
$23$		−	3.31662i		−	0.691564i	0.938315	−	0.345782i	$-0.112386\pi$
$24$		0			0
$25$	−2.51969	−	4.31870i	−0.503937	−	0.863740i
$26$		0			0
$27$	15.7715	−	5.12447i	3.03522	−	0.986204i
$28$		0			0
$29$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$29$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$30$		0			0
$31$	1.54508	−	4.75528i	0.277505	−	0.854074i	−0.711040	−	0.703151i	$-0.751776\pi$
$31$	1.54508	−	4.75528i	0.277505	−	0.854074i	0.988546	−	0.150923i	$-0.0482244\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	12.9443	−	9.40456i	2.15738	−	1.56743i
$37$	−5.84839	+	8.04962i	−0.961469	+	1.32335i	−0.0152291	+	0.999884i	$0.504848\pi$
$37$	−5.84839	+	8.04962i	−0.961469	+	1.32335i	−0.946240	+	0.323465i	$0.895152\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$41$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$42$		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
$47$	3.89893	+	5.36641i	0.568717	+	0.782772i	0.992402	−	0.123038i	$-0.0392637\pi$
$47$	3.89893	+	5.36641i	0.568717	+	0.782772i	−0.423685	+	0.905810i	$0.639264\pi$
$48$	7.79785	−	10.7328i	1.12552	−	1.54915i
$49$	2.16312	+	6.65740i	0.309017	+	0.951057i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	12.6172	+	4.09957i	1.73310	+	0.563120i	0.993892	−	0.110353i	$-0.0351982\pi$
$53$	12.6172	+	4.09957i	1.73310	+	0.563120i	0.739212	+	0.673473i	$0.235198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.1353	+	8.81678i	1.57988	+	1.14785i	0.916837	+	0.399262i	$0.130733\pi$
$59$	12.1353	+	8.81678i	1.57988	+	1.14785i	0.663039	+	0.748585i	$0.269267\pi$
$60$	−12.8621	−	7.38694i	−1.66049	−	0.953650i
$61$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$61$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$62$		0			0
$63$		0			0
$64$	2.47214	−	7.60845i	0.309017	−	0.951057i
$65$		0			0
$66$		0			0
$67$			9.94987i			1.21557i	0.794101	+	0.607785i	$0.207942\pi$
$67$			9.94987i			1.21557i	−0.794101	+	0.607785i	$0.792058\pi$
$68$		0			0
$69$	−8.89919	+	6.46564i	−1.07134	+	0.778371i
$70$		0			0
$71$	−0.927051	−	2.85317i	−0.110021	−	0.338609i	0.880855	−	0.473386i	$-0.156968\pi$
$71$	−0.927051	−	2.85317i	−0.110021	−	0.338609i	−0.990876	+	0.134777i	$0.956968\pi$
$72$		0			0
$73$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$73$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$74$		0			0
$75$	−6.67593	+	15.1800i	−0.770870	+	1.75283i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$79$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$80$	−8.75303	−	1.83970i	−0.978618	−	0.205685i
$81$	−25.0795	−	18.2213i	−2.78661	−	2.02459i
$82$		0			0
$83$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$83$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$84$		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
$92$	−3.89893	+	5.36641i	−0.406491	+	0.559487i
$93$	−15.7715	+	5.12447i	−1.63543	+	0.531382i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.46289	−	3.07468i	−0.960811	−	0.312186i	−0.213710	−	0.976897i	$-0.568555\pi$
$97$	−9.46289	−	3.07468i	−0.960811	−	0.312186i	−0.747101	+	0.664711i	$0.768555\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000	+	9.94987i	−0.100000	+	0.994987i
$101$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$101$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$102$		0			0
$103$	−11.6968	+	16.0992i	−1.15252	+	1.58630i	−0.416829	+	0.908985i	$0.636859\pi$
$103$	−11.6968	+	16.0992i	−1.15252	+	1.58630i	−0.735689	+	0.677320i	$0.763141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$107$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$108$	−31.5430	−	10.2489i	−3.03522	−	0.986204i
$109$		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
$113$	1.94946	+	2.68321i	0.183390	+	0.252415i	0.890807	−	0.454382i	$-0.150140\pi$
$113$	1.94946	+	2.68321i	0.183390	+	0.252415i	−0.707417	+	0.706796i	$0.750140\pi$
$114$		0			0
$115$	6.43104	+	3.69347i	0.599698	+	0.344418i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−8.09017	+	5.87785i	−0.726519	+	0.527847i
$125$	11.1801	−	0.0763444i	0.999977	−	0.00682845i
$126$		0			0
$127$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$127$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$128$		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
$135$	−7.62699	+	36.2881i	−0.656427	+	3.12319i
$136$		0			0
$137$	−22.0801	+	7.17425i	−1.88643	+	0.612938i	−0.903613	+	0.428350i	$0.859095\pi$
$137$	−22.0801	+	7.17425i	−1.88643	+	0.612938i	−0.982816	+	0.184588i	$0.940905\pi$
$138$		0			0
$139$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$139$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$140$		0			0
$141$	6.79837	−	20.9232i	0.572526	−	1.76205i
$142$		0			0
$143$		0			0
$144$	−32.0000			−2.66667
$145$		0			0
$146$		0			0
$147$	13.6462	−	18.7824i	1.12552	−	1.54915i
$148$	18.9258	−	6.14936i	1.55569	−	0.505474i
$149$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$149$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$150$		0			0
$151$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$151$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.50000	+	8.29156i	0.602414	+	0.665994i
$156$		0			0
$157$	5.84839	+	8.04962i	0.466752	+	0.642429i	0.975892	−	0.218255i	$-0.0700363\pi$
$157$	5.84839	+	8.04962i	0.466752	+	0.642429i	−0.509140	+	0.860684i	$0.670036\pi$
$158$		0			0
$159$	−13.5967	−	41.8465i	−1.07829	−	3.31864i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	18.9258	+	6.14936i	1.48238	+	0.481655i	0.934824	−	0.355112i	$-0.115557\pi$
$163$	18.9258	+	6.14936i	1.48238	+	0.481655i	0.547558	+	0.836768i	$0.315557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$167$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$168$		0			0
$169$	−10.5172	−	7.64121i	−0.809017	−	0.587785i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$173$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	49.7494i		−	3.73939i
$178$		0			0
$179$	16.9894	−	12.3435i	1.26984	−	0.922596i	0.270648	−	0.962678i	$-0.412762\pi$
$179$	16.9894	−	12.3435i	1.26984	−	0.922596i	0.999196	+	0.0400827i	$0.0127621\pi$
$180$	3.82070	+	35.5725i	0.284778	+	2.65142i
$181$	7.72542	+	23.7764i	0.574226	+	1.76729i	0.638799	+	0.769374i	$0.279432\pi$
$181$	7.72542	+	23.7764i	0.574226	+	1.76729i	−0.0645725	+	0.997913i	$0.520568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.09556	−	20.3045i	−0.668719	−	1.49281i
$186$		0			0
$187$		0			0
$188$		−	13.2665i		−	0.967559i
$189$		0			0
$190$		0			0
$191$	−12.1353	−	8.81678i	−0.878076	−	0.637960i	0.0546656	−	0.998505i	$-0.482591\pi$
$191$	−12.1353	−	8.81678i	−0.878076	−	0.637960i	−0.932742	+	0.360545i	$0.882591\pi$
$192$	−25.2344	+	8.19915i	−1.82113	+	0.591722i
$193$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$193$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$194$		0			0
$195$		0			0
$196$	4.32624	−	13.3148i	0.309017	−	0.951057i
$197$		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
$201$	26.6976	−	19.3969i	1.88310	−	1.36815i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	25.2344	+	8.19915i	1.75391	+	0.569880i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$211$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$212$	−15.5957	−	21.4656i	−1.07112	−	1.47427i
$213$	−5.84839	+	8.04962i	−0.400725	+	0.551551i
$214$		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
$223$	17.5452	+	24.1489i	1.17491	+	1.61713i	0.614544	+	0.788883i	$0.289340\pi$
$223$	17.5452	+	24.1489i	1.17491	+	1.61713i	0.560368	+	0.828244i	$0.310660\pi$
$224$		0			0
$225$	39.0876	−	8.49450i	2.60584	−	0.566300i
$226$		0			0
$227$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$227$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$228$		0			0
$229$	−1.54508	+	4.75528i	−0.102102	+	0.314238i	−0.989039	−	0.147652i	$-0.952828\pi$
$229$	−1.54508	+	4.75528i	−0.102102	+	0.314238i	0.886937	+	0.461890i	$0.152828\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$233$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$234$		0			0
$235$	−14.7476	+	1.58398i	−0.962026	+	0.103327i
$236$	−9.27051	−	28.5317i	−0.603459	−	1.85726i
$237$		0			0
$238$		0			0
$239$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$239$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$240$	12.1274	+	27.0726i	0.782821	+	1.74753i
$241$		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
$245$	−15.3178	−	3.21948i	−0.978618	−	0.205685i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.34346	+	25.6785i	−0.526634	+	1.62081i	0.234427	+	0.972134i	$0.424679\pi$
$251$	−8.34346	+	25.6785i	−0.526634	+	1.62081i	−0.761061	+	0.648680i	$0.775321\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−12.9443	+	9.40456i	−0.809017	+	0.587785i
$257$	15.5957	−	21.4656i	0.972833	−	1.33899i	0.0322308	−	0.999480i	$-0.489739\pi$
$257$	15.5957	−	21.4656i	0.972833	−	1.33899i	0.940603	−	0.339510i	$-0.110261\pi$
$258$		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
$267$	−17.5452	−	24.1489i	−1.07375	−	1.47789i
$268$	11.6968	−	16.0992i	0.714494	−	0.983417i
$269$	9.27051	+	28.5317i	0.565233	+	1.73961i	0.667258	+	0.744826i	$0.267468\pi$
$269$	9.27051	+	28.5317i	0.565233	+	1.73961i	−0.102025	+	0.994782i	$0.532532\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	22.0000			1.32424
$277$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$277$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$278$		0			0
$279$	32.3607	+	23.5114i	1.93738	+	1.40759i
$280$		0			0
$281$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$281$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$282$		0			0
$283$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$283$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$284$	−1.85410	+	5.70634i	−0.110021	+	0.338609i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.7533	+	9.99235i	−0.809017	+	0.587785i
$290$		0			0
$291$	10.1976	+	31.3849i	0.597792	+	1.83981i
$292$		0			0
$293$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$293$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$294$		0			0
$295$	−30.6101	+	13.7121i	−1.78219	+	0.798348i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	28.6470	−	16.7137i	1.65394	−	0.964966i
$301$		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
$311$	9.70820	−	7.05342i	0.550502	−	0.399963i	−0.277469	−	0.960735i	$-0.589495\pi$
$311$	9.70820	−	7.05342i	0.550502	−	0.399963i	0.827970	+	0.560772i	$0.189495\pi$
$312$		0			0
$313$	28.3887	−	9.22404i	1.60462	−	0.521374i	0.636378	−	0.771377i	$-0.280432\pi$
$313$	28.3887	−	9.22404i	1.60462	−	0.521374i	0.968245	+	0.250004i	$0.0804318\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−22.0801	−	7.17425i	−1.24014	−	0.402946i	−0.385763	−	0.922598i	$-0.626062\pi$
$317$	−22.0801	−	7.17425i	−1.24014	−	0.402946i	−0.854378	+	0.519652i	$0.826062\pi$
$318$		0			0
$319$		0			0
$320$	12.0000	+	13.2665i	0.670820	+	0.741620i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	19.1591	+	58.9655i	1.06439	+	3.27586i
$325$		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
$333$	−46.7871	−	64.3969i	−2.56392	−	3.52893i
$334$		0			0
$335$	−19.2931	−	11.0804i	−1.05410	−	0.605388i
$336$		0			0
$337$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$337$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$338$		0			0
$339$	3.39919	−	10.4616i	0.184618	−	0.568197i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2.62673	−	24.4561i	−0.141419	−	1.31667i
$346$		0			0
$347$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$347$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$348$		0			0
$349$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$349$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	36.4829i		−	1.94179i	−0.239511	−	0.970894i	$-0.576987\pi$
$353$		−	36.4829i		−	1.94179i	0.239511	−	0.970894i	$-0.423013\pi$
$354$		0			0
$355$	6.56477	+	1.37978i	0.348422	+	0.0732309i
$356$	−14.5623	−	10.5801i	−0.771801	−	0.560746i
$357$		0			0
$358$		0			0
$359$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$359$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$360$		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
$367$	5.84839	−	8.04962i	0.305283	−	0.420187i	−0.628620	−	0.777713i	$-0.716380\pi$
$367$	5.84839	−	8.04962i	0.305283	−	0.420187i	0.933903	+	0.357526i	$0.116380\pi$
$368$	12.6172	−	4.09957i	0.657717	−	0.213705i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	31.5430	+	10.2489i	1.63543	+	0.531382i
$373$		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
$379$	−7.72542	−	23.7764i	−0.396828	−	1.22131i	−0.927528	−	0.373753i	$-0.878071\pi$
$379$	−7.72542	−	23.7764i	−0.396828	−	1.22131i	0.530700	−	0.847560i	$-0.321929\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.15430	+	1.02489i	0.161177	+	0.0523696i	0.388494	−	0.921451i	$-0.372995\pi$
$383$	3.15430	+	1.02489i	0.161177	+	0.0523696i	−0.227317	+	0.973821i	$0.572995\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	11.6968	+	16.0992i	0.593814	+	0.817315i
$389$	12.1353	+	8.81678i	0.615282	+	0.447028i	0.851270	−	0.524727i	$-0.175833\pi$
$389$	12.1353	+	8.81678i	0.615282	+	0.447028i	−0.235988	+	0.971756i	$0.575833\pi$
$390$		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	$0.484018\pi$
$397$			39.7995i			1.99748i	−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$		0			0
$400$	13.3148	−	14.9237i	0.665741	−	0.746183i
$401$	−9.27051	−	28.5317i	−0.462947	−	1.42480i	−0.861546	−	0.507679i	$-0.830503\pi$
$401$	−9.27051	−	28.5317i	−0.462947	−	1.42480i	0.398599	−	0.917125i	$-0.369497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	63.2609	−	28.3383i	3.14346	−	1.40814i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$		0			0
$411$	62.2943	+	45.2595i	3.07275	+	2.23248i
$412$	37.8516	−	12.2987i	1.86481	−	0.605914i
$413$		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
$421$	8.09017	−	5.87785i	0.394291	−	0.286469i	−0.372921	−	0.927863i	$-0.621644\pi$
$421$	8.09017	−	5.87785i	0.394291	−	0.286469i	0.767211	+	0.641394i	$0.221644\pi$
$422$		0			0
$423$	−50.4688	+	16.3983i	−2.45388	+	0.797312i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$431$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$432$	38.9893	+	53.6641i	1.87587	+	2.58192i
$433$	−17.5452	+	24.1489i	−0.843167	+	1.16052i	0.142160	+	0.989844i	$0.454595\pi$
$433$	−17.5452	+	24.1489i	−0.843167	+	1.16052i	−0.985327	+	0.170676i	$0.945405\pi$
$434$		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
$443$	21.4441	+	29.5153i	1.01884	+	1.40231i	0.913014	+	0.407928i	$0.133748\pi$
$443$	21.4441	+	29.5153i	1.01884	+	1.40231i	0.105825	+	0.994385i	$0.466252\pi$
$444$	−53.3951	−	38.7938i	−2.53402	−	1.84107i
$445$	−10.0226	+	17.4513i	−0.475117	+	0.827270i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.0517	+	37.0912i	−0.568753	+	1.75044i	0.0877747	+	0.996140i	$0.472024\pi$
$449$	−12.0517	+	37.0912i	−0.568753	+	1.75044i	−0.656528	+	0.754302i	$0.727976\pi$
$450$		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		−0.309017	−	0.951057i	$-0.600000\pi$
$457$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$458$		0			0
$459$		0			0
$460$	−6.06371	−	13.5363i	−0.282722	−	0.631133i
$461$		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	$-0.756017\pi$
$463$		−	29.8496i		−	1.38723i	0.720346	−	0.693615i	$-0.243983\pi$
$464$		0			0
$465$	7.62699	−	36.2881i	0.353693	−	1.68282i
$466$		0			0
$467$	41.0059	−	13.3236i	1.89753	−	0.616543i	0.927313	−	0.374286i	$-0.122112\pi$
$467$	41.0059	−	13.3236i	1.89753	−	0.616543i	0.970212	−	0.242257i	$-0.0778878\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	10.1976	−	31.3849i	0.469879	−	1.44614i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−62.3828	+	85.8626i	−2.85631	+	3.93138i
$478$		0			0
$479$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$479$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$480$		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
$487$	−5.84839	−	8.04962i	−0.265016	−	0.364763i	0.655683	−	0.755036i	$-0.272381\pi$
$487$	−5.84839	−	8.04962i	−0.265016	−	0.364763i	−0.920699	+	0.390273i	$0.872381\pi$
$488$		0			0
$489$	−20.3951	−	62.7697i	−0.922299	−	2.83855i
$490$		0			0
$491$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$491$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	20.0000			0.898027
$497$		0			0
$498$		0			0
$499$	−32.3607	−	23.5114i	−1.44866	−	1.05252i	−0.986141	−	0.165907i	$-0.946945\pi$
$499$	−32.3607	−	23.5114i	−1.44866	−	1.05252i	−0.462522	−	0.886608i	$-0.653055\pi$
$500$	−18.1795	−	13.0194i	−0.813012	−	0.582247i
$501$		0			0
$502$		0			0
$503$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$503$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			43.1161i			1.91485i
$508$		0			0
$509$	36.4058	−	26.4503i	1.61366	−	1.17239i	0.763624	−	0.645661i	$-0.223418\pi$
$509$	36.4058	−	26.4503i	1.61366	−	1.17239i	0.850033	−	0.526730i	$-0.176582\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−18.1911	−	40.6089i	−0.801596	−	1.78944i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.1353	−	8.81678i	−0.531655	−	0.386270i	0.289321	−	0.957232i	$-0.406570\pi$
$521$	−12.1353	−	8.81678i	−0.531655	−	0.386270i	−0.820977	+	0.570962i	$0.806570\pi$
$522$		0			0
$523$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$523$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	12.0000			0.521739
$530$		0			0
$531$	−97.0820	+	70.5342i	−4.21300	+	3.06092i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−66.2402	−	21.5228i	−2.85848	−	0.928776i
$538$		0			0
$539$		0			0
$540$	55.0000	−	49.7494i	2.36682	−	2.14087i
$541$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$541$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$542$		0			0
$543$	48.7366	−	67.0801i	2.09149	−	2.87868i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$547$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$548$	44.1602	+	14.3485i	1.88643	+	0.612938i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−36.7496	+	63.9880i	−1.55993	+	2.71614i
$556$		0			0
$557$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$557$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$563$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$564$	−35.5967	+	25.8626i	−1.49889	+	1.08901i
$565$	−7.37379	+	0.791990i	−0.310218	+	0.0333193i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$569$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$570$		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
$575$	−14.3235	+	8.35685i	−0.597332	+	0.348505i
$576$	51.7771	+	37.6183i	2.15738	+	1.56743i
$577$	−9.46289	+	3.07468i	−0.393945	+	0.128001i	−0.499290	−	0.866435i	$-0.666406\pi$
$577$	−9.46289	+	3.07468i	−0.393945	+	0.128001i	0.105344	+	0.994436i	$0.466406\pi$
$578$		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
$587$	3.89893	−	5.36641i	0.160926	−	0.221496i	−0.720938	−	0.693000i	$-0.756289\pi$
$587$	3.89893	−	5.36641i	0.160926	−	0.221496i	0.881864	+	0.471504i	$0.156289\pi$
$588$	−44.1602	+	14.3485i	−1.82113	+	0.591722i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−37.8516	−	12.2987i	−1.55569	−	0.505474i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	38.9893	+	53.6641i	1.59572	+	2.19633i
$598$		0			0
$599$	11.1246	+	34.2380i	0.454539	+	1.39893i	0.871675	+	0.490084i	$0.163034\pi$
$599$	11.1246	+	34.2380i	0.454539	+	1.39893i	−0.417136	+	0.908844i	$0.636966\pi$
$600$		0			0
$601$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$601$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$602$		0			0
$603$	−75.7031	−	24.5974i	−3.08287	−	1.00169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$607$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$613$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26.5330i			1.06818i	0.845428	+	0.534089i	$0.179345\pi$
$617$			26.5330i			1.06818i	−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	−0.809017	+	0.587785i	−0.0325171	+	0.0236251i	−0.603925	−	0.797041i	$-0.706397\pi$
$619$	−0.809017	+	0.587785i	−0.0325171	+	0.0236251i	0.571408	+	0.820666i	$0.306397\pi$
$620$	−2.38794	−	22.2328i	−0.0959019	−	0.892891i
$621$	−16.9959	−	52.3081i	−0.682023	−	2.09905i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.3024	+	21.7635i	−0.492095	+	0.870542i
$626$		0			0
$627$		0			0
$628$		−	19.8997i		−	0.794086i
$629$		0			0
$630$		0			0
$631$	−5.66312	−	4.11450i	−0.225445	−	0.163796i	0.469329	−	0.883023i	$-0.344496\pi$
$631$	−5.66312	−	4.11450i	−0.225445	−	0.163796i	−0.694774	+	0.719228i	$0.744496\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−27.1935	+	83.6930i	−1.07829	+	3.31864i
$637$		0			0
$638$		0			0
$639$	24.0000			0.949425
$640$		0			0
$641$	−36.4058	+	26.4503i	−1.43794	+	1.04473i	−0.449474	+	0.893294i	$0.648388\pi$
$641$	−36.4058	+	26.4503i	−1.43794	+	1.04473i	−0.988468	+	0.151432i	$0.951612\pi$
$642$		0			0
$643$	−28.3887	+	9.22404i	−1.11954	+	0.363761i	−0.809592	−	0.586993i	$-0.800312\pi$
$643$	−28.3887	+	9.22404i	−1.11954	+	0.363761i	−0.309948	+	0.950753i	$0.600312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	41.0059	+	13.3236i	1.61211	+	0.523805i	0.970061	−	0.242861i	$-0.0780858\pi$
$647$	41.0059	+	13.3236i	1.61211	+	0.523805i	0.642046	+	0.766666i	$0.278086\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−23.3936	−	32.1985i	−0.916162	−	1.26099i
$653$	−1.94946	+	2.68321i	−0.0762884	+	0.105002i	−0.845456	−	0.534045i	$-0.820671\pi$
$653$	−1.94946	+	2.68321i	−0.0762884	+	0.105002i	0.769167	+	0.639047i	$0.220671\pi$
$654$		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
$669$	30.5927	−	94.1546i	1.18278	−	3.64023i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$673$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$674$		0			0
$675$	−61.8702	−	55.2003i	−2.38139	−	2.12466i
$676$	8.03444	+	24.7275i	0.309017	+	0.951057i
$677$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$677$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	46.4327i		−	1.77670i	−0.459167	−	0.888350i	$-0.651852\pi$
$683$		−	46.4327i		−	1.77670i	0.459167	−	0.888350i	$-0.348148\pi$
$684$		0			0
$685$	10.6778	−	50.8034i	0.407977	−	1.94110i
$686$		0			0
$687$	15.7715	−	5.12447i	0.601720	−	0.195511i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	5.25329	−	16.1680i	0.199845	−	0.615058i	−0.800041	−	0.599945i	$-0.795189\pi$
$691$	5.25329	−	16.1680i	0.199845	−	0.615058i	0.999886	−	0.0151132i	$-0.00481087\pi$
$692$		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		−0.587785	−	0.809017i	$-0.700000\pi$
$701$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	33.0000	+	36.4829i	1.24285	+	1.37402i
$706$		0			0
$707$		0			0
$708$	−58.4839	+	80.4962i	−2.19796	+	3.02523i
$709$	−5.87132	−	18.0701i	−0.220502	−	0.678636i	−0.998717	−	0.0506378i	$-0.983875\pi$
$709$	−5.87132	−	18.0701i	−0.220502	−	0.678636i	0.778215	−	0.627998i	$-0.216125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15.7715	−	5.12447i	−0.590647	−	0.191913i
$714$		0			0
$715$		0			0
$716$	−42.0000			−1.56961
$717$		0			0
$718$		0			0
$719$	41.2599	+	29.9770i	1.53873	+	1.11796i	0.951123	+	0.308811i	$0.0999310\pi$
$719$	41.2599	+	29.9770i	1.53873	+	1.11796i	0.587610	+	0.809144i	$0.300069\pi$
$720$	35.6359	−	62.0490i	1.32807	−	2.31243i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	15.4508	−	47.5528i	0.574226	−	1.76729i
$725$		0			0
$726$		0			0
$727$		−	9.94987i		−	0.369020i	−0.982831	−	0.184510i	$-0.940930\pi$
$727$		−	9.94987i		−	0.369020i	0.982831	−	0.184510i	$-0.0590699\pi$
$728$		0			0
$729$	67.1484	−	48.7862i	2.48698	−	1.80690i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$733$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$734$		0			0
$735$	21.2230	+	47.3771i	0.782821	+	1.74753i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$739$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$740$	−9.15239	+	43.5458i	−0.336449	+	1.60077i
$741$		0			0
$742$		0			0
$743$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$743$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18.6074	−	13.5191i	0.678993	−	0.493318i	−0.194030	−	0.980996i	$-0.562156\pi$
$751$	18.6074	−	13.5191i	0.678993	−	0.493318i	0.873024	+	0.487678i	$0.162156\pi$
$752$	−15.5957	+	21.4656i	−0.568717	+	0.782772i
$753$	85.1660	−	27.6721i	3.10362	−	1.00843i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	37.8516	+	12.2987i	1.37574	+	0.447005i	0.901266	−	0.433266i	$-0.142639\pi$
$757$	37.8516	+	12.2987i	1.37574	+	0.447005i	0.474473	+	0.880270i	$0.342639\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$761$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$762$		0			0
$763$		0			0
$764$	9.27051	+	28.5317i	0.335395	+	1.03224i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	50.4688	+	16.3983i	1.82113	+	0.591722i
$769$		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
$773$	7.79785	+	10.7328i	0.280469	+	0.386033i	0.925889	−	0.377795i	$-0.123318\pi$
$773$	7.79785	+	10.7328i	0.280469	+	0.386033i	−0.645420	+	0.763828i	$0.723318\pi$
$774$		0			0
$775$	−24.4298	+	5.30906i	−0.877543	+	0.190707i
$776$		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
$785$	−22.1214	+	2.37597i	−0.789545	+	0.0848020i
$786$		0			0
$787$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$787$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	96.2833	+	20.2367i	3.41482	+	0.717722i
$796$	32.3607	+	23.5114i	1.14699	+	0.833340i
$797$	−53.6231	+	17.4232i	−1.89943	+	0.617161i	−0.933294	+	0.359113i	$0.883079\pi$
$797$	−53.6231	+	17.4232i	−1.89943	+	0.617161i	−0.966132	+	0.258048i	$0.916921\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−22.2492	+	68.4761i	−0.786138	+	2.41948i
$802$		0			0
$803$		0			0
$804$	−66.0000			−2.32764
$805$		0			0
$806$		0			0
$807$	58.4839	−	80.4962i	2.05873	−	2.83360i
$808$		0			0
$809$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$809$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$810$		0			0
$811$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$811$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$812$		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		0.587785	−	0.809017i	$-0.300000\pi$
$821$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$822$		0			0
$823$	28.3887	+	9.22404i	0.989567	+	0.321530i	0.758689	−	0.651453i	$-0.225840\pi$
$823$	28.3887	+	9.22404i	0.989567	+	0.321530i	0.230878	+	0.972983i	$0.425840\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$827$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$828$	−31.1914	−	42.9313i	−1.08398	−	1.49197i
$829$	23.4615	+	17.0458i	0.814851	+	0.592024i	0.915233	−	0.402925i	$-0.132006\pi$
$829$	23.4615	+	17.0458i	0.814851	+	0.592024i	−0.100382	+	0.994949i	$0.532006\pi$
$830$		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
$839$	36.4058	−	26.4503i	1.25687	−	0.913167i	0.258267	−	0.966074i	$-0.416849\pi$
$839$	36.4058	−	26.4503i	1.25687	−	0.913167i	0.998599	+	0.0529065i	$0.0168485\pi$
$840$		0			0
$841$	−8.96149	−	27.5806i	−0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	26.5288	−	11.8838i	0.912617	−	0.408815i
$846$		0			0
$847$		0			0
$848$			53.0660i			1.82229i
$849$		0			0
$850$		0			0
$851$	26.6976	+	19.3969i	0.915181	+	0.664918i
$852$	18.9258	−	6.14936i	0.648387	−	0.210674i
$853$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$853$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$854$		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
$863$	44.1602	−	14.3485i	1.50323	−	0.488429i	0.562272	−	0.826953i	$-0.309928\pi$
$863$	44.1602	−	14.3485i	1.50323	−	0.488429i	0.940958	+	0.338524i	$0.109928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	53.6231	+	17.4232i	1.82113	+	0.591722i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	46.7871	−	64.3969i	1.58350	−	2.17951i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$877$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$878$		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
$883$	−11.6968	−	16.0992i	−0.393628	−	0.541782i	0.565503	−	0.824747i	$-0.308682\pi$
$883$	−11.6968	−	16.0992i	−0.393628	−	0.541782i	−0.959130	+	0.282964i	$0.908682\pi$
$884$		0			0
$885$	96.4656	+	55.4021i	3.24266	+	1.86232i
$886$		0			0
$887$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$887$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		−	59.6992i		−	1.99888i
$893$		0			0
$894$		0			0
$895$	5.01467	+	46.6889i	0.167622	+	1.56064i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−73.2310	−	32.2059i	−2.44103	−	1.07353i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−54.7064	−	11.4981i	−1.81850	−	0.382211i
$906$		0			0
$907$	−56.7774	+	18.4481i	−1.88526	+	0.612558i	−0.901588	+	0.432597i	$0.857597\pi$
$907$	−56.7774	+	18.4481i	−1.88526	+	0.612558i	−0.983674	+	0.179962i	$0.942403\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.5410	−	57.0634i	0.614291	−	1.89059i	0.202639	−	0.979253i	$-0.435048\pi$
$911$	18.5410	−	57.0634i	0.614291	−	1.89059i	0.411652	−	0.911341i	$-0.364952\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	8.09017	−	5.87785i	0.267307	−	0.194210i
$917$		0			0
$918$		0			0
$919$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$919$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$920$		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
$927$	−93.5742	−	128.794i	−3.07338	−	4.23015i
$928$		0			0
$929$	9.27051	+	28.5317i	0.304156	+	0.936095i	0.979991	+	0.199042i	$0.0637830\pi$
$929$	9.27051	+	28.5317i	0.304156	+	0.936095i	−0.675835	+	0.737053i	$0.736217\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−37.8516	−	12.2987i	−1.23920	−	0.402642i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$937$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$938$		0			0
$939$	−80.0927	−	58.1907i	−2.61373	−	1.89898i
$940$	25.7242	+	14.7739i	0.839030	+	0.481871i
$941$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$941$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$942$		0			0
$943$		0			0
$944$	−18.5410	+	57.0634i	−0.603459	+	1.85726i
$945$		0			0
$946$		0			0
$947$			23.2164i			0.754431i	0.926126	+	0.377215i	$0.123118\pi$
$947$			23.2164i			0.754431i	−0.926126	+	0.377215i	$0.876882\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	23.7943	+	73.2314i	0.771583	+	2.37469i
$952$		0			0
$953$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$953$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$954$		0			0
$955$	30.6101	−	13.7121i	0.990520	−	0.443712i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	12.2032	−	58.0610i	0.393856	−	1.87391i
$961$	4.85410	+	3.52671i	0.156584	+	0.113765i
$962$		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
$971$	−36.4058	+	26.4503i	−1.16832	+	0.848832i	−0.990806	−	0.135287i	$-0.956804\pi$
$971$	−36.4058	+	26.4503i	−1.16832	+	0.848832i	−0.177510	+	0.984119i	$0.556804\pi$
$972$	62.3828	−	85.8626i	2.00093	−	2.75404i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−53.6231	−	17.4232i	−1.71555	−	0.557417i	−0.724311	−	0.689473i	$-0.757842\pi$
$977$	−53.6231	−	17.4232i	−1.71555	−	0.557417i	−0.991242	+	0.132056i	$0.957842\pi$
$978$		0			0
$979$		0			0
$980$	21.0000	+	23.2164i	0.670820	+	0.741620i
$981$		0			0
$982$		0			0
$983$	−21.4441	+	29.5153i	−0.683960	+	0.941391i	−0.999973	−	0.00736431i	$-0.997656\pi$
$983$	−21.4441	+	29.5153i	−0.683960	+	0.941391i	0.316012	+	0.948755i	$0.397656\pi$
$984$		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
$993$	68.2312	+	93.9122i	2.16525	+	2.98021i
$994$		0			0
$995$	22.2725	−	38.7806i	0.706085	−	1.22943i
$996$		0			0
$997$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$997$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$998$		0			0
$999$	−50.9878	+	156.924i	−1.61318	+	4.96487i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.b.a.364.1	✓	2	55.4	even	10
605.2.b.a.364.1	✓	2	55.29	odd	10
605.2.b.a.364.2	yes	2	11.4	even	5
605.2.b.a.364.2	yes	2	11.7	odd	10
605.2.j.b.9.1		8	1.1	even	1	trivial
605.2.j.b.9.1		8	11.10	odd	2	CM
605.2.j.b.9.2		8	5.4	even	2	inner
605.2.j.b.9.2		8	55.54	odd	2	inner
605.2.j.b.124.1		8	11.2	odd	10	inner
605.2.j.b.124.1		8	11.9	even	5	inner
605.2.j.b.124.2		8	55.9	even	10	inner
605.2.j.b.124.2		8	55.24	odd	10	inner
605.2.j.b.269.1		8	55.39	odd	10	inner
605.2.j.b.269.1		8	55.49	even	10	inner
605.2.j.b.269.2		8	11.5	even	5	inner
605.2.j.b.269.2		8	11.6	odd	10	inner
605.2.j.b.444.1		8	55.14	even	10	inner
605.2.j.b.444.1		8	55.19	odd	10	inner
605.2.j.b.444.2		8	11.3	even	5	inner
605.2.j.b.444.2		8	11.8	odd	10	inner
3025.2.a.k.1.1		2	55.18	even	20
3025.2.a.k.1.1		2	55.48	odd	20
3025.2.a.k.1.2		2	55.7	even	20
3025.2.a.k.1.2		2	55.37	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)

\(f(q)\)	\(=\)	\(q+(-1.94946 - 2.68321i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(-1.11362 + 1.93903i) q^{5} +(-2.47214 + 7.60845i) q^{9} +O(q^{10})\)	\(q+(-1.94946 - 2.68321i) q^{3} +(-1.61803 - 1.17557i) q^{4} +(-1.11362 + 1.93903i) q^{5} +(-2.47214 + 7.60845i) q^{9} +6.63325i q^{12} +(7.37379 - 0.791990i) q^{15} +(1.23607 + 3.80423i) q^{16} +(4.08135 - 1.82828i) q^{20} -3.31662i q^{23} +(-2.51969 - 4.31870i) q^{25} +(15.7715 - 5.12447i) q^{27} +(1.54508 - 4.75528i) q^{31} +(12.9443 - 9.40456i) q^{36} +(-5.84839 + 8.04962i) q^{37} +(-12.0000 - 13.2665i) q^{45} +(3.89893 + 5.36641i) q^{47} +(7.79785 - 10.7328i) q^{48} +(2.16312 + 6.65740i) q^{49} +(12.6172 + 4.09957i) q^{53} +(12.1353 + 8.81678i) q^{59} +(-12.8621 - 7.38694i) q^{60} +(2.47214 - 7.60845i) q^{64} +9.94987i q^{67} +(-8.89919 + 6.46564i) q^{69} +(-0.927051 - 2.85317i) q^{71} +(-6.67593 + 15.1800i) q^{75} +(-8.75303 - 1.83970i) q^{80} +(-25.0795 - 18.2213i) q^{81} +9.00000 q^{89} +(-3.89893 + 5.36641i) q^{92} +(-15.7715 + 5.12447i) q^{93} +(-9.46289 - 3.07468i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} - 3 q^{5} + 16 q^{9}+O(q^{10}) \) 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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