LMFDB - Embedded newform 456.2.bg.b.73.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,2,Mod(25,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 0, 14]))

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

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

chi := DirichletCharacter("456.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.bg (of order $9$, degree $6$, 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:	$3.64117833217$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{9})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 15 x^{10} + 130 x^{9} + 117 x^{8} - 1314 x^{7} - 740 x^{6} + 7431 x^{5} + \cdots + 30837 $ x^12 - 6x^11 - 15x^10 + 130x^9 + 117x^8 - 1314x^7 - 740x^6 + 7431x^5 + 4533x^4 - 23277x^3 - 17766x^2 + 30906*x + 30837
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			73.1
Root			$2.32042 - 0.342020i$ of defining polynomial
Character	$\chi$	$=$	456.73
Dual form			456.2.bg.b.25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.939693 + 0.342020i) q^{3} +(-0.0661133 + 0.374947i) q^{5} +(-0.673648 - 1.16679i) q^{7} +(0.766044 - 0.642788i) q^{9} +(1.23135 - 2.13276i) q^{11} +(3.73590 + 1.35976i) q^{13} +(-0.0661133 - 0.374947i) q^{15} +(1.73135 + 1.45278i) q^{17} +(3.60449 - 2.45105i) q^{19} +(1.03209 + 0.866025i) q^{21} +(0.237593 + 1.34746i) q^{23} +(4.56225 + 1.66052i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.974387 - 0.817608i) q^{29} +(-0.442298 - 0.766083i) q^{31} +(-0.427643 + 2.42529i) q^{33} +(0.482023 - 0.175442i) q^{35} +9.03850 q^{37} -3.97566 q^{39} +(-4.78893 + 1.74303i) q^{41} +(0.801223 - 4.54396i) q^{43} +(0.190366 + 0.329723i) q^{45} +(-0.344142 + 0.288770i) q^{47} +(2.59240 - 4.49016i) q^{49} +(-2.12382 - 0.773005i) q^{51} +(0.438071 + 2.48443i) q^{53} +(0.718264 + 0.602695i) q^{55} +(-2.54881 + 3.53604i) q^{57} +(-2.90816 - 2.44023i) q^{59} +(-1.02048 - 5.78741i) q^{61} +(-1.26604 - 0.460802i) q^{63} +(-0.756829 + 1.31087i) q^{65} +(3.72427 - 3.12503i) q^{67} +(-0.684123 - 1.18493i) q^{69} +(-2.28478 + 12.9576i) q^{71} +(-5.59774 + 2.03741i) q^{73} -4.85504 q^{75} -3.31799 q^{77} +(-9.95937 + 3.62491i) q^{79} +(0.173648 - 0.984808i) q^{81} +(-3.62587 - 6.28020i) q^{83} +(-0.659180 + 0.553117i) q^{85} +(-0.635986 + 1.10156i) q^{87} +(-10.4353 - 3.79813i) q^{89} +(-0.930127 - 5.27501i) q^{91} +(0.677640 + 0.568608i) q^{93} +(0.680709 + 1.51354i) q^{95} +(7.12527 + 5.97881i) q^{97} +(-0.427643 - 2.42529i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{5} - 6 q^{7} - 3 q^{11} - 3 q^{13} + 3 q^{15} + 3 q^{17} - 6 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 9 q^{29} - 21 q^{31} - 6 q^{33} - 6 q^{35} + 30 q^{37} + 6 q^{39} + 15 q^{41} + 3 q^{43}+ \cdots - 6 q^{99}+O(q^{100}) $ 12 * q + 3 * q^5 - 6 * q^7 - 3 * q^11 - 3 * q^13 + 3 * q^15 + 3 * q^17 - 6 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 9 * q^29 - 21 * q^31 - 6 * q^33 - 6 * q^35 + 30 * q^37 + 6 * q^39 + 15 * q^41 + 3 * q^43 - 3 * q^45 - 18 * q^47 + 24 * q^49 - 6 * q^51 + 24 * q^53 + 60 * q^55 - 9 * q^57 - 21 * q^59 - 24 * q^61 - 6 * q^63 - 21 * q^65 + 15 * q^67 - 21 * q^71 - 3 * q^73 + 18 * q^75 - 6 * q^77 - 63 * q^79 - 12 * q^83 + 9 * q^85 - 15 * q^87 - 15 * q^89 - 3 * q^93 + 12 * q^95 - 9 * q^97 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/456\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$343$
$\chi(n)$	$e\left(\frac{2}{9}\right)$	$1$	$1$	$1$

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
$2$		0			0
$3$	−0.939693	+	0.342020i	−0.542532	+	0.197465i
$3$	−0.939693	+	0.342020i	−0.542532	+	0.197465i
$4$		0			0
$5$	−0.0661133	+	0.374947i	−0.0295668	+	0.167682i	−0.996016	−	0.0891770i	$-0.971576\pi$
$5$	−0.0661133	+	0.374947i	−0.0295668	+	0.167682i	0.966449	+	0.256859i	$0.0826874\pi$
$6$		0			0
$7$	−0.673648	−	1.16679i	−0.254615	−	0.441006i	0.710176	−	0.704024i	$-0.248615\pi$
$7$	−0.673648	−	1.16679i	−0.254615	−	0.441006i	−0.964791	+	0.263018i	$0.915282\pi$
$8$		0			0
$9$	0.766044	−	0.642788i	0.255348	−	0.214263i
$10$		0			0
$11$	1.23135	−	2.13276i	0.371266	−	0.643052i	−0.618495	−	0.785789i	$-0.712257\pi$
$11$	1.23135	−	2.13276i	0.371266	−	0.643052i	0.989761	+	0.142738i	$0.0455905\pi$
$12$		0			0
$13$	3.73590	+	1.35976i	1.03615	+	0.377128i	0.803420	−	0.595413i	$-0.203011\pi$
$13$	3.73590	+	1.35976i	1.03615	+	0.377128i	0.232731	+	0.972541i	$0.425234\pi$
$14$		0			0
$15$	−0.0661133	−	0.374947i	−0.0170704	−	0.0968110i
$16$		0			0
$17$	1.73135	+	1.45278i	0.419914	+	0.352350i	0.828130	−	0.560535i	$-0.189405\pi$
$17$	1.73135	+	1.45278i	0.419914	+	0.352350i	−0.408216	+	0.912885i	$0.633849\pi$
$18$		0			0
$19$	3.60449	−	2.45105i	0.826927	−	0.562309i
$19$	3.60449	−	2.45105i	0.826927	−	0.562309i
$20$		0			0
$21$	1.03209	+	0.866025i	0.225220	+	0.188982i
$22$		0			0
$23$	0.237593	+	1.34746i	0.0495416	+	0.280964i	0.999507	−	0.0313901i	$-0.00999342\pi$
$23$	0.237593	+	1.34746i	0.0495416	+	0.280964i	−0.949966	+	0.312355i	$0.898882\pi$
$24$		0			0
$25$	4.56225	+	1.66052i	0.912450	+	0.332105i
$26$		0			0
$27$	−0.500000	+	0.866025i	−0.0962250	+	0.166667i
$28$		0			0
$29$	0.974387	−	0.817608i	0.180939	−	0.151826i	−0.547820	−	0.836596i	$-0.684542\pi$
$29$	0.974387	−	0.817608i	0.180939	−	0.151826i	0.728759	+	0.684770i	$0.240098\pi$
$30$		0			0
$31$	−0.442298	−	0.766083i	−0.0794391	−	0.137593i	0.823569	−	0.567216i	$-0.191980\pi$
$31$	−0.442298	−	0.766083i	−0.0794391	−	0.137593i	−0.903008	+	0.429624i	$0.858646\pi$
$32$		0			0
$33$	−0.427643	+	2.42529i	−0.0744432	+	0.422188i
$34$		0			0
$35$	0.482023	−	0.175442i	0.0814768	−	0.0296551i
$36$		0			0
$37$	9.03850			1.48592			0.742960	−	0.669336i	$-0.233421\pi$
$37$	9.03850			1.48592			0.742960	+	0.669336i	$0.233421\pi$
$38$		0			0
$39$	−3.97566			−0.636615
$40$		0			0
$41$	−4.78893	+	1.74303i	−0.747905	+	0.272215i	−0.687724	−	0.725972i	$-0.741390\pi$
$41$	−4.78893	+	1.74303i	−0.747905	+	0.272215i	−0.0601812	+	0.998187i	$0.519168\pi$
$42$		0			0
$43$	0.801223	−	4.54396i	0.122185	−	0.692947i	−0.860755	−	0.509020i	$-0.830008\pi$
$43$	0.801223	−	4.54396i	0.122185	−	0.692947i	0.982940	−	0.183927i	$-0.0588810\pi$
$44$		0			0
$45$	0.190366	+	0.329723i	0.0283781	+	0.0491522i
$46$		0			0
$47$	−0.344142	+	0.288770i	−0.0501983	+	0.0421214i	−0.667542	−	0.744572i	$-0.732653\pi$
$47$	−0.344142	+	0.288770i	−0.0501983	+	0.0421214i	0.617343	+	0.786694i	$0.288209\pi$
$48$		0			0
$49$	2.59240	−	4.49016i	0.370342	−	0.641452i
$50$		0			0
$51$	−2.12382	−	0.773005i	−0.297394	−	0.108242i
$52$		0			0
$53$	0.438071	+	2.48443i	0.0601737	+	0.341262i	1.00000		0.000390770i	$-0.000124386\pi$
$53$	0.438071	+	2.48443i	0.0601737	+	0.341262i	−0.939826	+	0.341653i	$0.889013\pi$
$54$		0			0
$55$	0.718264	+	0.602695i	0.0968508	+	0.0812674i
$56$		0			0
$57$	−2.54881	+	3.53604i	−0.337598	+	0.468360i
$58$		0			0
$59$	−2.90816	−	2.44023i	−0.378610	−	0.317691i	0.433546	−	0.901131i	$-0.357262\pi$
$59$	−2.90816	−	2.44023i	−0.378610	−	0.317691i	−0.812156	+	0.583440i	$0.801706\pi$
$60$		0			0
$61$	−1.02048	−	5.78741i	−0.130659	−	0.741002i	−0.977785	−	0.209611i	$-0.932780\pi$
$61$	−1.02048	−	5.78741i	−0.130659	−	0.741002i	0.847126	−	0.531392i	$-0.178331\pi$
$62$		0			0
$63$	−1.26604	−	0.460802i	−0.159507	−	0.0580557i
$64$		0			0
$65$	−0.756829	+	1.31087i	−0.0938731	+	0.162593i
$66$		0			0
$67$	3.72427	−	3.12503i	0.454992	−	0.381783i	−0.386293	−	0.922376i	$-0.626244\pi$
$67$	3.72427	−	3.12503i	0.454992	−	0.381783i	0.841284	+	0.540593i	$0.181800\pi$
$68$		0			0
$69$	−0.684123	−	1.18493i	−0.0823587	−	0.142649i
$70$		0			0
$71$	−2.28478	+	12.9576i	−0.271153	+	1.53779i	0.479768	+	0.877395i	$0.340721\pi$
$71$	−2.28478	+	12.9576i	−0.271153	+	1.53779i	−0.750921	+	0.660392i	$0.770390\pi$
$72$		0			0
$73$	−5.59774	+	2.03741i	−0.655166	+	0.238461i	−0.648148	−	0.761514i	$-0.724456\pi$
$73$	−5.59774	+	2.03741i	−0.655166	+	0.238461i	−0.00701818	+	0.999975i	$0.502234\pi$
$74$		0			0
$75$	−4.85504			−0.560612
$76$		0			0
$77$	−3.31799			−0.378120
$78$		0			0
$79$	−9.95937	+	3.62491i	−1.12052	+	0.407835i	−0.834841	−	0.550491i	$-0.814440\pi$
$79$	−9.95937	+	3.62491i	−1.12052	+	0.407835i	−0.285676	+	0.958326i	$0.592218\pi$
$80$		0			0
$81$	0.173648	−	0.984808i	0.0192942	−	0.109423i
$82$		0			0
$83$	−3.62587	−	6.28020i	−0.397991	−	0.689341i	0.595487	−	0.803365i	$-0.296959\pi$
$83$	−3.62587	−	6.28020i	−0.397991	−	0.689341i	−0.993478	+	0.114024i	$0.963626\pi$
$84$		0			0
$85$	−0.659180	+	0.553117i	−0.0714981	+	0.0599940i
$86$		0			0
$87$	−0.635986	+	1.10156i	−0.0681849	+	0.118100i
$88$		0			0
$89$	−10.4353	−	3.79813i	−1.10614	−	0.402601i	−0.276561	−	0.960996i	$-0.589195\pi$
$89$	−10.4353	−	3.79813i	−1.10614	−	0.402601i	−0.829575	+	0.558396i	$0.811417\pi$
$90$		0			0
$91$	−0.930127	−	5.27501i	−0.0975038	−	0.552972i
$92$		0			0
$93$	0.677640	+	0.568608i	0.0702680	+	0.0589619i
$94$		0			0
$95$	0.680709	+	1.51354i	0.0698393	+	0.155286i
$96$		0			0
$97$	7.12527	+	5.97881i	0.723461	+	0.607056i	0.928340	−	0.371731i	$-0.121236\pi$
$97$	7.12527	+	5.97881i	0.723461	+	0.607056i	−0.204879	+	0.978787i	$0.565680\pi$
$98$		0			0
$99$	−0.427643	−	2.42529i	−0.0429798	−	0.243750i
$100$		0			0
$101$	−11.6373	−	4.23563i	−1.15795	−	0.421461i	−0.309588	−	0.950871i	$-0.600191\pi$
$101$	−11.6373	−	4.23563i	−1.15795	−	0.421461i	−0.848366	+	0.529410i	$0.822413\pi$
$102$		0			0
$103$	−3.39398	+	5.87855i	−0.334419	+	0.579230i	−0.983373	−	0.181597i	$-0.941873\pi$
$103$	−3.39398	+	5.87855i	−0.334419	+	0.579230i	0.648954	+	0.760828i	$0.275207\pi$
$104$		0			0
$105$	−0.392949	+	0.329723i	−0.0383479	+	0.0321777i
$106$		0			0
$107$	0.184001	+	0.318699i	0.0177880	+	0.0308098i	0.874782	−	0.484516i	$-0.161004\pi$
$107$	0.184001	+	0.318699i	0.0177880	+	0.0308098i	−0.856994	+	0.515326i	$0.827671\pi$
$108$		0			0
$109$	−0.0992026	+	0.562606i	−0.00950189	+	0.0538879i	−0.989190	−	0.146639i	$-0.953154\pi$
$109$	−0.0992026	+	0.562606i	−0.00950189	+	0.0538879i	0.979688	+	0.200527i	$0.0642655\pi$
$110$		0			0
$111$	−8.49341	+	3.09135i	−0.806159	+	0.293418i
$112$		0			0
$113$	−3.32312			−0.312612			−0.156306	−	0.987709i	$-0.549959\pi$
$113$	−3.32312			−0.312612			−0.156306	+	0.987709i	$0.549959\pi$
$114$		0			0
$115$	−0.520934			−0.0485774
$116$		0			0
$117$	3.73590	−	1.35976i	0.345384	−	0.125709i
$118$		0			0
$119$	0.528767	−	2.99879i	0.0484720	−	0.274898i
$120$		0			0
$121$	2.46755	+	4.27393i	0.224323	+	0.388539i
$122$		0			0
$123$	3.90397	−	3.27582i	0.352009	−	0.295371i
$124$		0			0
$125$	−1.87606	+	3.24944i	−0.167800	+	0.290638i
$126$		0			0
$127$	2.31112	+	0.841180i	0.205079	+	0.0746426i	0.442517	−	0.896760i	$-0.354086\pi$
$127$	2.31112	+	0.841180i	0.205079	+	0.0746426i	−0.237438	+	0.971403i	$0.576308\pi$
$128$		0			0
$129$	0.801223	+	4.54396i	0.0705437	+	0.400073i
$130$		0			0
$131$	13.9866	+	11.7361i	1.22201	+	1.02539i	0.998717	+	0.0506382i	$0.0161255\pi$
$131$	13.9866	+	11.7361i	1.22201	+	1.02539i	0.223294	+	0.974751i	$0.428319\pi$
$132$		0			0
$133$	−5.28803	−	2.55455i	−0.458530	−	0.221508i
$134$		0			0
$135$	−0.291657	−	0.244730i	−0.0251019	−	0.0210630i
$136$		0			0
$137$	−3.06140	−	17.3621i	−0.261553	−	1.48334i	−0.778673	−	0.627430i	$-0.784107\pi$
$137$	−3.06140	−	17.3621i	−0.261553	−	1.48334i	0.517120	−	0.855913i	$-0.327004\pi$
$138$		0			0
$139$	−7.96146	−	2.89773i	−0.675282	−	0.245783i	−0.0184617	−	0.999830i	$-0.505877\pi$
$139$	−7.96146	−	2.89773i	−0.675282	−	0.245783i	−0.656821	+	0.754047i	$0.728099\pi$
$140$		0			0
$141$	0.224623	−	0.389058i	0.0189167	−	0.0327646i
$142$		0			0
$143$	7.50023	−	6.29344i	0.627201	−	0.526284i
$144$		0			0
$145$	0.242140	+	0.419399i	0.0201086	+	0.0348292i
$146$		0			0
$147$	−0.900330	+	5.10602i	−0.0742579	+	0.421138i
$148$		0			0
$149$	−16.5658	+	6.02947i	−1.35713	+	0.493953i	−0.915165	−	0.403081i	$-0.867939\pi$
$149$	−16.5658	+	6.02947i	−1.35713	+	0.493953i	−0.441962	+	0.897034i	$0.645717\pi$
$150$		0			0
$151$	9.81134			0.798435			0.399218	−	0.916856i	$-0.369282\pi$
$151$	9.81134			0.798435			0.399218	+	0.916856i	$0.369282\pi$
$152$		0			0
$153$	2.26012			0.182720
$154$		0			0
$155$	0.316483	−	0.115190i	0.0254205	−	0.00925231i
$156$		0			0
$157$	1.38929	−	7.87908i	0.110878	−	0.628819i	−0.877831	−	0.478970i	$-0.841010\pi$
$157$	1.38929	−	7.87908i	0.110878	−	0.628819i	0.988709	−	0.149849i	$-0.0478787\pi$
$158$		0			0
$159$	−1.26138	−	2.18477i	−0.100034	−	0.173263i
$160$		0			0
$161$	1.41215	−	1.18493i	0.111293	−	0.0933860i
$162$		0			0
$163$	1.38764	−	2.40347i	0.108689	−	0.188254i	−0.806551	−	0.591165i	$-0.798668\pi$
$163$	1.38764	−	2.40347i	0.108689	−	0.188254i	0.915239	+	0.402911i	$0.132001\pi$
$164$		0			0
$165$	−0.881082	−	0.320688i	−0.0685921	−	0.0249655i
$166$		0			0
$167$	3.19458	+	18.1174i	0.247204	+	1.40196i	0.815317	+	0.579014i	$0.196562\pi$
$167$	3.19458	+	18.1174i	0.247204	+	1.40196i	−0.568113	+	0.822950i	$0.692326\pi$
$168$		0			0
$169$	2.14941	+	1.80357i	0.165339	+	0.138736i
$170$		0			0
$171$	1.18570	−	4.19453i	0.0906726	−	0.320764i
$172$		0			0
$173$	13.0602	+	10.9588i	0.992951	+	0.833185i	0.985992	−	0.166791i	$-0.0533405\pi$
$173$	13.0602	+	10.9588i	0.992951	+	0.833185i	0.00695872	+	0.999976i	$0.497785\pi$
$174$		0			0
$175$	−1.13586	−	6.44181i	−0.0858633	−	0.486955i
$176$		0			0
$177$	3.56738	+	1.29842i	0.268141	+	0.0975954i
$178$		0			0
$179$	9.68946	−	16.7826i	0.724224	−	1.25439i	−0.235068	−	0.971979i	$-0.575531\pi$
$179$	9.68946	−	16.7826i	0.724224	−	1.25439i	0.959292	−	0.282415i	$-0.0911354\pi$
$180$		0			0
$181$	5.50714	−	4.62104i	0.409342	−	0.343479i	−0.414749	−	0.909936i	$-0.636131\pi$
$181$	5.50714	−	4.62104i	0.409342	−	0.343479i	0.824091	+	0.566457i	$0.191686\pi$
$182$		0			0
$183$	2.93835	+	5.08937i	0.217209	+	0.376217i
$184$		0			0
$185$	−0.597565	+	3.38896i	−0.0439339	+	0.249161i
$186$		0			0
$187$	5.23032	−	1.90368i	0.382479	−	0.139211i
$188$		0			0
$189$	1.34730			0.0980014
$190$		0			0
$191$	3.70961			0.268418			0.134209	−	0.990953i	$-0.457151\pi$
$191$	3.70961			0.268418			0.134209	+	0.990953i	$0.457151\pi$
$192$		0			0
$193$	−5.36246	+	1.95178i	−0.385998	+	0.140492i	−0.527728	−	0.849414i	$-0.676956\pi$
$193$	−5.36246	+	1.95178i	−0.385998	+	0.140492i	0.141730	+	0.989905i	$0.454734\pi$
$194$		0			0
$195$	0.262844	−	1.49066i	0.0188227	−	0.106749i
$196$		0			0
$197$	1.26059	+	2.18341i	0.0898135	+	0.155562i	0.907432	−	0.420199i	$-0.138040\pi$
$197$	1.26059	+	2.18341i	0.0898135	+	0.155562i	−0.817619	+	0.575760i	$0.804706\pi$
$198$		0			0
$199$	−15.1511	+	12.7133i	−1.07403	+	0.901222i	−0.995412	−	0.0956826i	$-0.969497\pi$
$199$	−15.1511	+	12.7133i	−1.07403	+	0.901222i	−0.0786225	+	0.996904i	$0.525052\pi$
$200$		0			0
$201$	−2.43084	+	4.21034i	−0.171458	+	0.296975i
$202$		0			0
$203$	−1.61037	−	0.586128i	−0.113026	−	0.0411381i
$204$		0			0
$205$	−0.336932	−	1.91083i	−0.0235323	−	0.133458i
$206$		0			0
$207$	1.04814	+	0.879491i	0.0728505	+	0.0611288i
$208$		0			0
$209$	−0.789109	−	10.7056i	−0.0545838	−	0.740523i
$210$		0			0
$211$	8.03996	+	6.74633i	0.553494	+	0.464437i	0.876122	−	0.482089i	$-0.160122\pi$
$211$	8.03996	+	6.74633i	0.553494	+	0.464437i	−0.322628	+	0.946526i	$0.604566\pi$
$212$		0			0
$213$	−2.28478	−	12.9576i	−0.156550	−	0.887842i
$214$		0			0
$215$	1.65077	+	0.600833i	0.112582	+	0.0409765i
$216$		0			0
$217$	−0.595907	+	1.03214i	−0.0404528	+	0.0700663i
$218$		0			0
$219$	4.56332	−	3.82908i	0.308361	−	0.258745i
$220$		0			0
$221$	4.49273	+	7.78163i	0.302213	+	0.523449i
$222$		0			0
$223$	−3.34341	+	18.9614i	−0.223891	+	1.26975i	0.640902	+	0.767623i	$0.278561\pi$
$223$	−3.34341	+	18.9614i	−0.223891	+	1.26975i	−0.864793	+	0.502128i	$0.832550\pi$
$224$		0			0
$225$	4.56225	−	1.66052i	0.304150	−	0.110702i
$226$		0			0
$227$	26.5078			1.75938			0.879691	−	0.475546i	$-0.157749\pi$
$227$	26.5078			1.75938			0.879691	+	0.475546i	$0.157749\pi$
$228$		0			0
$229$	−27.8314			−1.83915			−0.919575	−	0.392915i	$-0.871467\pi$
$229$	−27.8314			−1.83915			−0.919575	+	0.392915i	$0.871467\pi$
$230$		0			0
$231$	3.11789	−	1.13482i	0.205142	−	0.0746656i
$232$		0			0
$233$	4.54912	−	25.7994i	0.298023	−	1.69017i	−0.356632	−	0.934245i	$-0.616075\pi$
$233$	4.54912	−	25.7994i	0.298023	−	1.69017i	0.654655	−	0.755928i	$-0.272814\pi$
$234$		0			0
$235$	−0.0855211	−	0.148127i	−0.00557878	−	0.00966273i
$236$		0			0
$237$	8.11895	−	6.81261i	0.527383	−	0.442527i
$238$		0			0
$239$	−8.24111	+	14.2740i	−0.533073	+	0.923309i	0.466181	+	0.884689i	$0.345630\pi$
$239$	−8.24111	+	14.2740i	−0.533073	+	0.923309i	−0.999254	+	0.0386200i	$0.987704\pi$
$240$		0			0
$241$	−0.632703	−	0.230285i	−0.0407560	−	0.0148340i	0.321562	−	0.946889i	$-0.395792\pi$
$241$	−0.632703	−	0.230285i	−0.0407560	−	0.0148340i	−0.362318	+	0.932055i	$0.618014\pi$
$242$		0			0
$243$	0.173648	+	0.984808i	0.0111395	+	0.0631754i
$244$		0			0
$245$	1.51218	+	1.26887i	0.0966098	+	0.0810653i
$246$		0			0
$247$	16.7988	−	4.25564i	1.06888	−	0.270780i
$248$		0			0
$249$	5.55516	+	4.66133i	0.352044	+	0.295400i
$250$		0			0
$251$	−1.46953	−	8.33412i	−0.0927559	−	0.526045i	−0.995412	−	0.0956822i	$-0.969497\pi$
$251$	−1.46953	−	8.33412i	−0.0927559	−	0.526045i	0.902656	−	0.430363i	$-0.141614\pi$
$252$		0			0
$253$	3.16637	+	1.15246i	0.199068	+	0.0724547i
$254$		0			0
$255$	0.430249	−	0.745213i	0.0269432	−	0.0466671i
$256$		0			0
$257$	8.50138	−	7.13351i	0.530302	−	0.444976i	−0.337904	−	0.941181i	$-0.609718\pi$
$257$	8.50138	−	7.13351i	0.530302	−	0.444976i	0.868206	+	0.496205i	$0.165273\pi$
$258$		0			0
$259$	−6.08877	−	10.5461i	−0.378338	−	0.655300i
$260$		0			0
$261$	0.220876	−	1.25265i	0.0136719	−	0.0775370i
$262$		0			0
$263$	10.6055	−	3.86009i	0.653963	−	0.238023i	0.00633530	−	0.999980i	$-0.497983\pi$
$263$	10.6055	−	3.86009i	0.653963	−	0.238023i	0.647628	+	0.761957i	$0.275761\pi$
$264$		0			0
$265$	−0.960491			−0.0590025
$266$		0			0
$267$	11.1050			0.679614
$268$		0			0
$269$	0.147327	−	0.0536227i	0.00898270	−	0.00326944i	−0.337525	−	0.941317i	$-0.609590\pi$
$269$	0.147327	−	0.0536227i	0.00898270	−	0.00326944i	0.346508	+	0.938047i	$0.387367\pi$
$270$		0			0
$271$	−1.19859	+	6.79754i	−0.0728091	+	0.412921i	0.926518	+	0.376250i	$0.122787\pi$
$271$	−1.19859	+	6.79754i	−0.0728091	+	0.412921i	−0.999327	+	0.0366713i	$0.988325\pi$
$272$		0			0
$273$	2.67819	+	4.63877i	0.162092	+	0.280751i
$274$		0			0
$275$	9.15922	−	7.68550i	0.552322	−	0.463453i
$276$		0			0
$277$	1.61927	−	2.80466i	0.0972927	−	0.168516i	−0.813270	−	0.581886i	$-0.802315\pi$
$277$	1.61927	−	2.80466i	0.0972927	−	0.168516i	0.910563	+	0.413370i	$0.135648\pi$
$278$		0			0
$279$	−0.831249	−	0.302550i	−0.0497656	−	0.0181132i
$280$		0			0
$281$	−2.93725	−	16.6580i	−0.175222	−	0.993732i	−0.937888	−	0.346939i	$-0.887221\pi$
$281$	−2.93725	−	16.6580i	−0.175222	−	0.993732i	0.762666	−	0.646793i	$-0.223890\pi$
$282$		0			0
$283$	13.9724	+	11.7242i	0.830572	+	0.696932i	0.955422	−	0.295243i	$-0.0954007\pi$
$283$	13.9724	+	11.7242i	0.830572	+	0.696932i	−0.124850	+	0.992176i	$0.539845\pi$
$284$		0			0
$285$	−1.15732	−	1.18945i	−0.0685537	−	0.0704568i
$286$		0			0
$287$	5.25981	+	4.41350i	0.310477	+	0.260521i
$288$		0			0
$289$	−2.06500	−	11.7112i	−0.121471	−	0.688895i
$290$		0			0
$291$	−8.74043	−	3.18126i	−0.512373	−	0.186489i
$292$		0			0
$293$	−13.5260	+	23.4277i	−0.790198	+	1.36866i	0.135646	+	0.990757i	$0.456689\pi$
$293$	−13.5260	+	23.4277i	−0.790198	+	1.36866i	−0.925844	+	0.377906i	$0.876644\pi$
$294$		0			0
$295$	1.10723	−	0.929074i	0.0644653	−	0.0540928i
$296$		0			0
$297$	1.23135	+	2.13276i	0.0714502	+	0.123755i
$298$		0			0
$299$	−0.944590	+	5.35703i	−0.0546270	+	0.309805i
$300$		0			0
$301$	−5.84160	+	2.12617i	−0.336704	+	0.122550i
$302$		0			0
$303$	12.3841			0.711451
$304$		0			0
$305$	2.23744			0.128116
$306$		0			0
$307$	−31.4727	+	11.4551i	−1.79624	+	0.653779i	−0.797519	+	0.603294i	$0.793855\pi$
$307$	−31.4727	+	11.4551i	−1.79624	+	0.653779i	−0.998725	+	0.0504852i	$0.983923\pi$
$308$		0			0
$309$	1.17872	−	6.68484i	0.0670549	−	0.380287i
$310$		0			0
$311$	−2.04662	−	3.54485i	−0.116053	−	0.201010i	0.802147	−	0.597127i	$-0.203691\pi$
$311$	−2.04662	−	3.54485i	−0.116053	−	0.201010i	−0.918200	+	0.396116i	$0.870358\pi$
$312$		0			0
$313$	−13.8463	+	11.6184i	−0.782640	+	0.656713i	−0.943912	−	0.330197i	$-0.892885\pi$
$313$	−13.8463	+	11.6184i	−0.782640	+	0.656713i	0.161272	+	0.986910i	$0.448440\pi$
$314$		0			0
$315$	0.256479	−	0.444235i	0.0144510	−	0.0250298i
$316$		0			0
$317$	10.7174	+	3.90080i	0.601948	+	0.219091i	0.624976	−	0.780644i	$-0.285109\pi$
$317$	10.7174	+	3.90080i	0.601948	+	0.219091i	−0.0230286	+	0.999735i	$0.507331\pi$
$318$		0			0
$319$	−0.543950	−	3.08490i	−0.0304554	−	0.172721i
$320$		0			0
$321$	−0.281906	−	0.236547i	−0.0157344	−	0.0132028i
$322$		0			0
$323$	9.80146	+	0.992894i	0.545368	+	0.0552461i
$324$		0			0
$325$	14.7862	+	12.4071i	0.820190	+	0.688221i
$326$		0			0
$327$	−0.0992026	−	0.562606i	−0.00548592	−	0.0311122i
$328$		0			0
$329$	0.568765	+	0.207014i	0.0313570	+	0.0114130i
$330$		0			0
$331$	8.64354	−	14.9711i	0.475092	−	0.822883i	−0.524501	−	0.851410i	$-0.675748\pi$
$331$	8.64354	−	14.9711i	0.475092	−	0.822883i	0.999593	+	0.0285264i	$0.00908146\pi$
$332$		0			0
$333$	6.92389	−	5.80983i	0.379427	−	0.318377i
$334$		0			0
$335$	0.925499	+	1.60301i	0.0505654	+	0.0875819i
$336$		0			0
$337$	−4.47465	+	25.3770i	−0.243750	+	1.38237i	0.579628	+	0.814881i	$0.303198\pi$
$337$	−4.47465	+	25.3770i	−0.243750	+	1.38237i	−0.823378	+	0.567494i	$0.807913\pi$
$338$		0			0
$339$	3.12271	−	1.13657i	0.169602	−	0.0617302i
$340$		0			0
$341$	−2.17850			−0.117972
$342$		0			0
$343$	−16.4165			−0.886409
$344$		0			0
$345$	0.489518	−	0.178170i	0.0263548	−	0.00959235i
$346$		0			0
$347$	−1.39588	+	7.91645i	−0.0749350	+	0.424977i	0.924143	+	0.382047i	$0.124781\pi$
$347$	−1.39588	+	7.91645i	−0.0749350	+	0.424977i	−0.999078	+	0.0429308i	$0.986330\pi$
$348$		0			0
$349$	8.24297	+	14.2772i	0.441236	+	0.764243i	0.997782	−	0.0665738i	$-0.0212068\pi$
$349$	8.24297	+	14.2772i	0.441236	+	0.764243i	−0.556545	+	0.830817i	$0.687873\pi$
$350$		0			0
$351$	−3.04553	+	2.55550i	−0.162558	+	0.136403i
$352$		0			0
$353$	4.32145	−	7.48498i	0.230008	−	0.398385i	−0.727802	−	0.685787i	$-0.759458\pi$
$353$	4.32145	−	7.48498i	0.230008	−	0.398385i	0.957810	+	0.287402i	$0.0927915\pi$
$354$		0			0
$355$	−4.70737	−	1.71334i	−0.249841	−	0.0909348i
$356$		0			0
$357$	0.528767	+	2.99879i	0.0279853	+	0.158713i
$358$		0			0
$359$	−7.84445	−	6.58228i	−0.414014	−	0.347399i	0.411866	−	0.911244i	$-0.364877\pi$
$359$	−7.84445	−	6.58228i	−0.414014	−	0.347399i	−0.825881	+	0.563845i	$0.809322\pi$
$360$		0			0
$361$	6.98472	−	17.6696i	0.367617	−	0.929977i
$362$		0			0
$363$	−3.78051	−	3.17223i	−0.198425	−	0.166499i
$364$		0			0
$365$	−0.393837	−	2.23356i	−0.0206144	−	0.116910i
$366$		0			0
$367$	−1.09089	−	0.397053i	−0.0569442	−	0.0207260i	0.313391	−	0.949624i	$-0.398535\pi$
$367$	−1.09089	−	0.397053i	−0.0569442	−	0.0207260i	−0.370335	+	0.928898i	$0.620757\pi$
$368$		0			0
$369$	−2.54814	+	4.41350i	−0.132651	+	0.229758i
$370$		0			0
$371$	2.60370	−	2.18477i	0.135178	−	0.113427i
$372$		0			0
$373$	−14.7127	−	25.4832i	−0.761796	−	1.31947i	−0.941924	−	0.335826i	$-0.890985\pi$
$373$	−14.7127	−	25.4832i	−0.761796	−	1.31947i	0.180129	−	0.983643i	$-0.442349\pi$
$374$		0			0
$375$	0.651550	−	3.69512i	0.0336459	−	0.190815i
$376$		0			0
$377$	4.75196	−	1.72957i	0.244738	−	0.0890774i
$378$		0			0
$379$	−22.0345			−1.13184			−0.565918	−	0.824462i	$-0.691478\pi$
$379$	−22.0345			−1.13184			−0.565918	+	0.824462i	$0.691478\pi$
$380$		0			0
$381$	−2.45944			−0.126001
$382$		0			0
$383$	−32.7699	+	11.9273i	−1.67446	+	0.609455i	−0.992535	−	0.121964i	$-0.961081\pi$
$383$	−32.7699	+	11.9273i	−1.67446	+	0.609455i	−0.681929	+	0.731419i	$0.738859\pi$
$384$		0			0
$385$	0.219363	−	1.24407i	0.0111798	−	0.0634037i
$386$		0			0
$387$	−2.30703	−	3.99589i	−0.117273	−	0.203123i
$388$		0			0
$389$	11.3131	−	9.49279i	0.573595	−	0.481304i	−0.309241	−	0.950984i	$-0.600075\pi$
$389$	11.3131	−	9.49279i	0.573595	−	0.481304i	0.882837	+	0.469680i	$0.155631\pi$
$390$		0			0
$391$	−1.54620	+	2.67809i	−0.0781945	+	0.135437i
$392$		0			0
$393$	−17.1571	−	6.24466i	−0.865459	−	0.315001i
$394$		0			0
$395$	−0.700705	−	3.97390i	−0.0352563	−	0.199948i
$396$		0			0
$397$	9.18898	+	7.71047i	0.461181	+	0.386977i	0.843565	−	0.537026i	$-0.180452\pi$
$397$	9.18898	+	7.71047i	0.461181	+	0.386977i	−0.382384	+	0.924003i	$0.624897\pi$
$398$		0			0
$399$	5.84283	+	0.591882i	0.292507	+	0.0296311i
$400$		0			0
$401$	16.8468	+	14.1362i	0.841289	+	0.705926i	0.957853	−	0.287258i	$-0.0927436\pi$
$401$	16.8468	+	14.1362i	0.841289	+	0.705926i	−0.116564	+	0.993183i	$0.537188\pi$
$402$		0			0
$403$	−0.610695	−	3.46342i	−0.0304209	−	0.172525i
$404$		0			0
$405$	0.357771	+	0.130218i	0.0177778	+	0.00647058i
$406$		0			0
$407$	11.1296	−	19.2769i	0.551671	−	0.955523i
$408$		0			0
$409$	−22.2117	+	18.6379i	−1.09830	+	0.921583i	−0.997310	−	0.0733049i	$-0.976645\pi$
$409$	−22.2117	+	18.6379i	−1.09830	+	0.921583i	−0.100990	+	0.994887i	$0.532201\pi$
$410$		0			0
$411$	8.81496	+	15.2680i	0.434810	+	0.753113i
$412$		0			0
$413$	−0.888173	+	5.03708i	−0.0437041	+	0.247858i
$414$		0			0
$415$	2.59446	−	0.944307i	0.127357	−	0.0463542i
$416$		0			0
$417$	8.47241			0.414896
$418$		0			0
$419$	−32.8592			−1.60527			−0.802637	−	0.596467i	$-0.796571\pi$
$419$	−32.8592			−1.60527			−0.802637	+	0.596467i	$0.796571\pi$
$420$		0			0
$421$	−23.4268	+	8.52665i	−1.14175	+	0.415564i	−0.842546	−	0.538625i	$-0.818944\pi$
$421$	−23.4268	+	8.52665i	−1.14175	+	0.415564i	−0.299206	+	0.954188i	$0.596722\pi$
$422$		0			0
$423$	−0.0780107	+	0.442421i	−0.00379301	+	0.0215112i
$424$		0			0
$425$	5.48648	+	9.50287i	0.266134	+	0.460957i
$426$		0			0
$427$	−6.06527	+	5.08937i	−0.293519	+	0.246292i
$428$		0			0
$429$	−4.89543	+	8.47913i	−0.236353	+	0.409376i
$430$		0			0
$431$	−30.9262	−	11.2562i	−1.48966	−	0.542193i	−0.536302	−	0.844026i	$-0.680179\pi$
$431$	−30.9262	−	11.2562i	−1.48966	−	0.542193i	−0.953361	+	0.301833i	$0.902401\pi$
$432$		0			0
$433$	−0.774040	−	4.38980i	−0.0371980	−	0.210960i	0.960544	−	0.278129i	$-0.0897144\pi$
$433$	−0.774040	−	4.38980i	−0.0371980	−	0.210960i	−0.997742	+	0.0671691i	$0.978603\pi$
$434$		0			0
$435$	−0.370980	−	0.311289i	−0.0177871	−	0.0149252i
$436$		0			0
$437$	4.15909	+	4.27455i	0.198956	+	0.204479i
$438$		0			0
$439$	−17.6624	−	14.8205i	−0.842981	−	0.707345i	0.115251	−	0.993336i	$-0.463233\pi$
$439$	−17.6624	−	14.8205i	−0.842981	−	0.707345i	−0.958232	+	0.285991i	$0.907677\pi$
$440$		0			0
$441$	−0.900330	−	5.10602i	−0.0428728	−	0.243144i
$442$		0			0
$443$	−27.2678	−	9.92467i	−1.29553	−	0.471535i	−0.399994	−	0.916518i	$-0.630988\pi$
$443$	−27.2678	−	9.92467i	−1.29553	−	0.471535i	−0.895539	+	0.444982i	$0.853210\pi$
$444$		0			0
$445$	2.11401	−	3.66157i	0.100214	−	0.173575i
$446$		0			0
$447$	13.5046	−	11.3317i	0.638745	−	0.535971i
$448$		0			0
$449$	12.9082	+	22.3577i	0.609178	+	1.05513i	0.991376	+	0.131047i	$0.0418338\pi$
$449$	12.9082	+	22.3577i	0.609178	+	1.05513i	−0.382198	+	0.924080i	$0.624833\pi$
$450$		0			0
$451$	−2.17939	+	12.3599i	−0.102623	+	0.582006i
$452$		0			0
$453$	−9.21964	+	3.35567i	−0.433176	+	0.157663i
$454$		0			0
$455$	2.03935			0.0956061
$456$		0			0
$457$	−38.8110			−1.81550			−0.907750	−	0.419512i	$-0.862201\pi$
$457$	−38.8110			−1.81550			−0.907750	+	0.419512i	$0.862201\pi$
$458$		0			0
$459$	−2.12382	+	0.773005i	−0.0991312	+	0.0360808i
$460$		0			0
$461$	2.94335	−	16.6926i	0.137086	−	0.777451i	−0.836299	−	0.548273i	$-0.815285\pi$
$461$	2.94335	−	16.6926i	0.137086	−	0.777451i	0.973385	−	0.229177i	$-0.0736036\pi$
$462$		0			0
$463$	−3.42570	−	5.93349i	−0.159206	−	0.275752i	0.775377	−	0.631499i	$-0.217560\pi$
$463$	−3.42570	−	5.93349i	−0.159206	−	0.275752i	−0.934582	+	0.355746i	$0.884227\pi$
$464$		0			0
$465$	−0.257999	+	0.216487i	−0.0119644	+	0.0100393i
$466$		0			0
$467$	−9.98984	+	17.3029i	−0.462275	+	0.800684i	−0.999074	−	0.0430265i	$-0.986300\pi$
$467$	−9.98984	+	17.3029i	−0.462275	+	0.800684i	0.536799	+	0.843710i	$0.319633\pi$
$468$		0			0
$469$	−6.15511	−	2.24028i	−0.284217	−	0.103446i
$470$		0			0
$471$	1.38929	+	7.87908i	0.0640153	+	0.363049i
$472$		0			0
$473$	−8.70459	−	7.30402i	−0.400238	−	0.335839i
$474$		0			0
$475$	20.5146	−	5.19695i	0.941275	−	0.238452i
$476$		0			0
$477$	1.93254	+	1.62159i	0.0884849	+	0.0742477i
$478$		0			0
$479$	−6.38671	−	36.2208i	−0.291816	−	1.65497i	−0.679868	−	0.733334i	$-0.737963\pi$
$479$	−6.38671	−	36.2208i	−0.291816	−	1.65497i	0.388052	−	0.921637i	$-0.373148\pi$
$480$		0			0
$481$	33.7669	+	12.2901i	1.53964	+	0.560382i
$482$		0			0
$483$	−0.921716	+	1.59646i	−0.0419395	+	0.0726414i
$484$		0			0
$485$	−2.71281	+	2.27632i	−0.123183	+	0.103362i
$486$		0			0
$487$	−3.54193	−	6.13481i	−0.160500	−	0.277995i	0.774548	−	0.632515i	$-0.217977\pi$
$487$	−3.54193	−	6.13481i	−0.160500	−	0.277995i	−0.935048	+	0.354521i	$0.884644\pi$
$488$		0			0
$489$	−0.481924	+	2.73313i	−0.0217934	+	0.123596i
$490$		0			0
$491$	−25.0470	+	9.11638i	−1.13036	+	0.411416i	−0.838422	−	0.545022i	$-0.816521\pi$
$491$	−25.0470	+	9.11638i	−1.13036	+	0.411416i	−0.291935	+	0.956438i	$0.594299\pi$
$492$		0			0
$493$	2.87481			0.129475
$494$		0			0
$495$	0.937628			0.0421432
$496$		0			0
$497$	16.6580	−	6.06302i	0.747213	−	0.271963i
$498$		0			0
$499$	3.27127	−	18.5523i	0.146442	−	0.830514i	−0.819756	−	0.572713i	$-0.805891\pi$
$499$	3.27127	−	18.5523i	0.146442	−	0.830514i	0.966198	−	0.257801i	$-0.0829979\pi$
$500$		0			0
$501$	−9.19843	−	15.9322i	−0.410956	−	0.711796i
$502$		0			0
$503$	−0.885495	+	0.743019i	−0.0394823	+	0.0331296i	−0.662315	−	0.749225i	$-0.730426\pi$
$503$	−0.885495	+	0.743019i	−0.0394823	+	0.0331296i	0.622833	+	0.782355i	$0.285982\pi$
$504$		0			0
$505$	2.35752	−	4.08334i	0.104908	−	0.181706i
$506$		0			0
$507$	−2.63664	−	0.959660i	−0.117097	−	0.0426200i
$508$		0			0
$509$	6.26735	+	35.5439i	0.277795	+	1.57546i	0.729940	+	0.683512i	$0.239548\pi$
$509$	6.26735	+	35.5439i	0.277795	+	1.57546i	−0.452144	+	0.891945i	$0.649341\pi$
$510$		0			0
$511$	6.14815	+	5.15891i	0.271978	+	0.228217i
$512$		0			0
$513$	0.320424	+	4.34711i	0.0141471	+	0.191929i
$514$		0			0
$515$	−1.97976	−	1.66121i	−0.0872386	−	0.0732019i
$516$		0			0
$517$	0.192117	+	1.08955i	0.00844930	+	0.0479183i
$518$		0			0
$519$	−16.0207	−	5.83107i	−0.703233	−	0.255956i
$520$		0			0
$521$	−11.2605	+	19.5037i	−0.493329	+	0.854472i	−0.999970	−	0.00768552i	$-0.997554\pi$
$521$	−11.2605	+	19.5037i	−0.493329	+	0.854472i	0.506641	+	0.862157i	$0.330887\pi$
$522$		0			0
$523$	1.04699	−	0.878530i	0.0457817	−	0.0384154i	−0.619610	−	0.784910i	$-0.712709\pi$
$523$	1.04699	−	0.878530i	0.0457817	−	0.0384154i	0.665391	+	0.746495i	$0.268265\pi$
$524$		0			0
$525$	3.27059	+	5.66483i	0.142740	+	0.247233i
$526$		0			0
$527$	0.347173	−	1.96892i	0.0151231	−	0.0857674i
$528$		0			0
$529$	19.8537	−	7.22617i	0.863206	−	0.314181i
$530$		0			0
$531$	−3.79633			−0.164747
$532$		0			0
$533$	−20.2610			−0.877603
$534$		0			0
$535$	−0.131660	+	0.0479204i	−0.00569217	+	0.00207178i
$536$		0			0
$537$	−3.36511	+	19.0845i	−0.145215	+	0.823558i
$538$		0			0
$539$	−6.38429	−	11.0579i	−0.274991	−	0.476298i
$540$		0			0
$541$	25.5561	−	21.4441i	1.09874	−	0.921953i	0.101401	−	0.994846i	$-0.467668\pi$
$541$	25.5561	−	21.4441i	1.09874	−	0.921953i	0.997340	+	0.0728928i	$0.0232231\pi$
$542$		0			0
$543$	−3.59453	+	6.22591i	−0.154256	+	0.267179i
$544$		0			0
$545$	−0.204389	−	0.0743915i	−0.00875507	−	0.00318658i
$546$		0			0
$547$	4.82414	+	27.3590i	0.206265	+	1.16979i	0.895436	+	0.445189i	$0.146864\pi$
$547$	4.82414	+	27.3590i	0.206265	+	1.16979i	−0.689171	+	0.724598i	$0.742025\pi$
$548$		0			0
$549$	−4.50181	−	3.77747i	−0.192132	−	0.161218i
$550$		0			0
$551$	1.50817	−	5.33533i	0.0642504	−	0.227293i
$552$		0			0
$553$	10.9386	+	9.17861i	0.465158	+	0.390314i
$554$		0			0
$555$	−0.597565	−	3.38896i	−0.0253652	−	0.143853i
$556$		0			0
$557$	27.8939	+	10.1525i	1.18190	+	0.430177i	0.856873	−	0.515527i	$-0.172404\pi$
$557$	27.8939	+	10.1525i	1.18190	+	0.430177i	0.325028	+	0.945704i	$0.394626\pi$
$558$		0			0
$559$	9.17196	−	15.8863i	0.387932	−	0.671919i
$560$		0			0
$561$	−4.26380	+	3.57775i	−0.180018	+	0.151053i
$562$		0			0
$563$	4.03778	+	6.99365i	0.170172	+	0.294747i	0.938480	−	0.345334i	$-0.112234\pi$
$563$	4.03778	+	6.99365i	0.170172	+	0.294747i	−0.768308	+	0.640081i	$0.778901\pi$
$564$		0			0
$565$	0.219702	−	1.24599i	0.00924295	−	0.0524194i
$566$		0			0
$567$	−1.26604	+	0.460802i	−0.0531689	+	0.0193519i
$568$		0			0
$569$	14.4830			0.607158			0.303579	−	0.952806i	$-0.401818\pi$
$569$	14.4830			0.607158			0.303579	+	0.952806i	$0.401818\pi$
$570$		0			0
$571$	4.96083			0.207604			0.103802	−	0.994598i	$-0.466899\pi$
$571$	4.96083			0.207604			0.103802	+	0.994598i	$0.466899\pi$
$572$		0			0
$573$	−3.48589	+	1.26876i	−0.145625	+	0.0530032i
$574$		0			0
$575$	−1.15353	+	6.54197i	−0.0481053	+	0.272819i
$576$		0			0
$577$	9.57974	+	16.5926i	0.398810	+	0.690759i	0.993579	−	0.113138i	$-0.0360901\pi$
$577$	9.57974	+	16.5926i	0.398810	+	0.690759i	−0.594770	+	0.803896i	$0.702757\pi$
$578$		0			0
$579$	4.37152	−	3.66814i	0.181674	−	0.152443i
$580$		0			0
$581$	−4.88513	+	8.46129i	−0.202669	+	0.351033i
$582$		0			0
$583$	5.83810	+	2.12490i	0.241790	+	0.0880042i
$584$		0			0
$585$	0.262844	+	1.49066i	0.0108673	+	0.0616313i
$586$		0			0
$587$	19.3953	+	16.2746i	0.800531	+	0.671726i	0.948328	−	0.317292i	$-0.102774\pi$
$587$	19.3953	+	16.2746i	0.800531	+	0.671726i	−0.147796	+	0.989018i	$0.547218\pi$
$588$		0			0
$589$	−3.47197	−	1.67725i	−0.143060	−	0.0691097i
$590$		0			0
$591$	−1.93134	−	1.62059i	−0.0794447	−	0.0666620i
$592$		0			0
$593$	−7.71732	−	43.7671i	−0.316912	−	1.79730i	−0.561289	−	0.827620i	$-0.689694\pi$
$593$	−7.71732	−	43.7671i	−0.316912	−	1.79730i	0.244377	−	0.969680i	$-0.421417\pi$
$594$		0			0
$595$	1.08943	+	0.396520i	0.0446622	+	0.0162557i
$596$		0			0
$597$	9.88919	−	17.1286i	0.404738	−	0.701026i
$598$		0			0
$599$	17.8151	−	14.9487i	0.727907	−	0.610787i	−0.201653	−	0.979457i	$-0.564631\pi$
$599$	17.8151	−	14.9487i	0.727907	−	0.610787i	0.929560	+	0.368670i	$0.120187\pi$
$600$		0			0
$601$	−9.26240	−	16.0430i	−0.377821	−	0.654406i	0.612924	−	0.790142i	$-0.289993\pi$
$601$	−9.26240	−	16.0430i	−0.377821	−	0.654406i	−0.990745	+	0.135736i	$0.956660\pi$
$602$		0			0
$603$	0.844223	−	4.78783i	0.0343794	−	0.194975i
$604$		0			0
$605$	−1.76564	+	0.642639i	−0.0717834	+	0.0261270i
$606$		0			0
$607$	20.6956			0.840009			0.420004	−	0.907522i	$-0.362029\pi$
$607$	20.6956			0.840009			0.420004	+	0.907522i	$0.362029\pi$
$608$		0			0
$609$	1.71372			0.0694436
$610$		0			0
$611$	−1.67834	+	0.610864i	−0.0678982	+	0.0247129i
$612$		0			0
$613$	3.66809	−	20.8028i	0.148153	−	0.840216i	−0.816629	−	0.577163i	$-0.804160\pi$
$613$	3.66809	−	20.8028i	0.148153	−	0.840216i	0.964782	−	0.263053i	$-0.0847293\pi$
$614$		0			0
$615$	0.970156	+	1.68036i	0.0391205	+	0.0677586i
$616$		0			0
$617$	6.50853	−	5.46131i	0.262024	−	0.219864i	−0.502305	−	0.864690i	$-0.667515\pi$
$617$	6.50853	−	5.46131i	0.262024	−	0.219864i	0.764329	+	0.644826i	$0.223070\pi$
$618$		0			0
$619$	6.46435	−	11.1966i	0.259824	−	0.450029i	−0.706370	−	0.707842i	$-0.749669\pi$
$619$	6.46435	−	11.1966i	0.259824	−	0.450029i	0.966195	+	0.257813i	$0.0830020\pi$
$620$		0			0
$621$	−1.28573	−	0.467967i	−0.0515946	−	0.0187789i
$622$		0			0
$623$	2.59807	+	14.7344i	0.104090	+	0.590321i
$624$		0			0
$625$	17.5016	+	14.6856i	0.700062	+	0.587422i
$626$		0			0
$627$	4.40306	+	9.79010i	0.175841	+	0.390979i
$628$		0			0
$629$	15.6488	+	13.1309i	0.623958	+	0.523563i
$630$		0			0
$631$	6.92594	+	39.2789i	0.275717	+	1.56367i	0.736674	+	0.676248i	$0.236395\pi$
$631$	6.92594	+	39.2789i	0.275717	+	1.56367i	−0.460957	+	0.887422i	$0.652494\pi$
$632$		0			0
$633$	−9.86247	−	3.58965i	−0.391998	−	0.142676i
$634$		0			0
$635$	−0.468194	+	0.810936i	−0.0185797	+	0.0321810i
$636$		0			0
$637$	15.7904	−	13.2498i	0.625640	−	0.524974i
$638$		0			0
$639$	6.57876	+	11.3947i	0.260252	+	0.450769i
$640$		0			0
$641$	4.80291	−	27.2387i	0.189704	−	1.07586i	−0.730058	−	0.683385i	$-0.760507\pi$
$641$	4.80291	−	27.2387i	0.189704	−	1.07586i	0.919762	−	0.392477i	$-0.128382\pi$
$642$		0			0
$643$	16.3210	−	5.94037i	0.643638	−	0.234265i	0.000481773	−	1.00000i	$-0.499847\pi$
$643$	16.3210	−	5.94037i	0.643638	−	0.234265i	0.643157	+	0.765735i	$0.277624\pi$
$644$		0			0
$645$	−1.75672			−0.0691707
$646$		0			0
$647$	40.1725			1.57934			0.789672	−	0.613529i	$-0.210251\pi$
$647$	40.1725			1.57934			0.789672	+	0.613529i	$0.210251\pi$
$648$		0			0
$649$	−8.78540	+	3.19762i	−0.344857	+	0.125518i
$650$		0			0
$651$	0.206956	−	1.17371i	0.00811126	−	0.0460012i
$652$		0			0
$653$	14.8008	+	25.6358i	0.579200	+	1.00320i	0.995571	+	0.0940097i	$0.0299685\pi$
$653$	14.8008	+	25.6358i	0.579200	+	1.00320i	−0.416371	+	0.909195i	$0.636698\pi$
$654$		0			0
$655$	−5.32513	+	4.46831i	−0.208070	+	0.174591i
$656$		0			0
$657$	−2.97850	+	5.15891i	−0.116202	+	0.201268i
$658$		0			0
$659$	−34.6195	−	12.6005i	−1.34858	−	0.490844i	−0.436078	−	0.899909i	$-0.643633\pi$
$659$	−34.6195	−	12.6005i	−1.34858	−	0.490844i	−0.912505	+	0.409065i	$0.865855\pi$
$660$		0			0
$661$	−7.66349	−	43.4618i	−0.298075	−	1.69047i	−0.654434	−	0.756119i	$-0.727093\pi$
$661$	−7.66349	−	43.4618i	−0.298075	−	1.69047i	0.356359	−	0.934349i	$-0.384018\pi$
$662$		0			0
$663$	−6.88326	−	5.77574i	−0.267323	−	0.224311i
$664$		0			0
$665$	1.30743	−	1.81384i	0.0507000	−	0.0703378i
$666$		0			0
$667$	1.33320	+	1.11869i	0.0516217	+	0.0433158i
$668$		0			0
$669$	−3.34341	−	18.9614i	−0.129264	−	0.733091i
$670$		0			0
$671$	−13.5997	−	4.94990i	−0.525012	−	0.191089i
$672$		0			0
$673$	−7.23427	+	12.5301i	−0.278861	+	0.483001i	−0.971102	−	0.238665i	$-0.923290\pi$
$673$	−7.23427	+	12.5301i	−0.278861	+	0.483001i	0.692241	+	0.721666i	$0.256623\pi$
$674$		0			0
$675$	−3.71918	+	3.12076i	−0.143151	+	0.120118i
$676$		0			0
$677$	19.5277	+	33.8230i	0.750512	+	1.29992i	0.947575	+	0.319534i	$0.103526\pi$
$677$	19.5277	+	33.8230i	0.750512	+	1.29992i	−0.197063	+	0.980391i	$0.563140\pi$
$678$		0			0
$679$	2.17611	−	12.3413i	0.0835114	−	0.473617i
$680$		0			0
$681$	−24.9092	+	9.06619i	−0.954521	+	0.347417i
$682$		0			0
$683$	−8.27403			−0.316597			−0.158298	−	0.987391i	$-0.550601\pi$
$683$	−8.27403			−0.316597			−0.158298	+	0.987391i	$0.550601\pi$
$684$		0			0
$685$	6.71227			0.256463
$686$		0			0
$687$	26.1530	−	9.51890i	0.997797	−	0.363169i
$688$		0			0
$689$	−1.74162	+	9.87723i	−0.0663505	+	0.376292i
$690$		0			0
$691$	14.7934	+	25.6230i	0.562769	+	0.974744i	0.997253	+	0.0740648i	$0.0235972\pi$
$691$	14.7934	+	25.6230i	0.562769	+	0.974744i	−0.434485	+	0.900679i	$0.643069\pi$
$692$		0			0
$693$	−2.54173	+	2.13276i	−0.0965522	+	0.0810169i
$694$		0			0
$695$	1.61286	−	2.79355i	0.0611792	−	0.105965i
$696$		0			0
$697$	−10.8235	−	3.93945i	−0.409971	−	0.149217i
$698$		0			0
$699$	4.54912	+	25.7994i	0.172064	+	0.975822i
$700$		0			0
$701$	−19.3279	−	16.2180i	−0.730003	−	0.612545i	0.200130	−	0.979769i	$-0.435864\pi$
$701$	−19.3279	−	16.2180i	−0.730003	−	0.612545i	−0.930132	+	0.367224i	$0.880308\pi$
$702$		0			0
$703$	32.5792	−	22.1538i	1.22875	−	0.835546i
$704$		0			0
$705$	0.131026	+	0.109944i	0.00493472	+	0.00414072i
$706$		0			0
$707$	2.89734	+	16.4316i	0.108966	+	0.617975i
$708$		0			0
$709$	25.7381	+	9.36791i	0.966616	+	0.351819i	0.776623	−	0.629966i	$-0.216931\pi$
$709$	25.7381	+	9.36791i	0.966616	+	0.351819i	0.189993	+	0.981785i	$0.439153\pi$
$710$		0			0
$711$	−5.29927	+	9.17861i	−0.198738	+	0.344225i
$712$		0			0
$713$	0.927178	−	0.777995i	0.0347231	−	0.0291361i
$714$		0			0
$715$	1.86384	+	3.22827i	0.0697038	+	0.120731i
$716$		0			0
$717$	2.86211	−	16.2318i	0.106887	−	0.606188i
$718$		0			0
$719$	−26.5095	+	9.64866i	−0.988636	+	0.359834i	−0.785192	−	0.619252i	$-0.787436\pi$
$719$	−26.5095	+	9.64866i	−0.988636	+	0.359834i	−0.203444	+	0.979087i	$0.565214\pi$
$720$		0			0
$721$	9.14540			0.340592
$722$		0			0
$723$	0.673308			0.0250406
$724$		0			0
$725$	5.80305	−	2.11214i	0.215520	−	0.0784429i
$726$		0			0
$727$	6.21951	−	35.2726i	0.230669	−	1.30819i	−0.620877	−	0.783908i	$-0.713223\pi$
$727$	6.21951	−	35.2726i	0.230669	−	1.30819i	0.851546	−	0.524280i	$-0.175665\pi$
$728$		0			0
$729$	−0.500000	−	0.866025i	−0.0185185	−	0.0320750i
$730$		0			0
$731$	7.98855	−	6.70319i	0.295467	−	0.247926i
$732$		0			0
$733$	−1.08170	+	1.87357i	−0.0399536	+	0.0692017i	−0.885311	−	0.465000i	$-0.846054\pi$
$733$	−1.08170	+	1.87357i	−0.0399536	+	0.0692017i	0.845357	+	0.534202i	$0.179388\pi$
$734$		0			0
$735$	−1.85497	−	0.675153i	−0.0684215	−	0.0249034i
$736$		0			0
$737$	−2.07907	−	11.7910i	−0.0765835	−	0.434326i
$738$		0			0
$739$	17.8185	+	14.9515i	0.655464	+	0.549999i	0.908723	−	0.417399i	$-0.137058\pi$
$739$	17.8185	+	14.9515i	0.655464	+	0.549999i	−0.253260	+	0.967398i	$0.581503\pi$
$740$		0			0
$741$	−14.3302	+	9.74453i	−0.526434	+	0.357974i
$742$		0			0
$743$	2.44021	+	2.04758i	0.0895227	+	0.0751185i	0.686451	−	0.727176i	$-0.259168\pi$
$743$	2.44021	+	2.04758i	0.0895227	+	0.0751185i	−0.596928	+	0.802295i	$0.703612\pi$
$744$		0			0
$745$	−1.16551	−	6.60995i	−0.0427010	−	0.242170i
$746$		0			0
$747$	−6.81441	−	2.48024i	−0.249326	−	0.0907474i
$748$		0			0
$749$	0.247904	−	0.429382i	0.00905820	−	0.0156893i
$750$		0			0
$751$	39.2402	−	32.9264i	1.43189	−	1.20150i	0.487312	−	0.873228i	$-0.337977\pi$
$751$	39.2402	−	32.9264i	1.43189	−	1.20150i	0.944582	−	0.328274i	$-0.106467\pi$
$752$		0			0
$753$	4.23134	+	7.32890i	0.154199	+	0.267080i
$754$		0			0
$755$	−0.648660	+	3.67874i	−0.0236072	+	0.133883i
$756$		0			0
$757$	11.7569	−	4.27916i	0.427312	−	0.155529i	−0.119406	−	0.992845i	$-0.538099\pi$
$757$	11.7569	−	4.27916i	0.427312	−	0.155529i	0.546718	+	0.837317i	$0.315877\pi$
$758$		0			0
$759$	−3.36958			−0.122308
$760$		0			0
$761$	−41.3258			−1.49806			−0.749030	−	0.662536i	$-0.769480\pi$
$761$	−41.3258			−1.49806			−0.749030	+	0.662536i	$0.769480\pi$
$762$		0			0
$763$	0.723272	−	0.263250i	0.0261842	−	0.00953028i
$764$		0			0
$765$	−0.149424	+	0.847425i	−0.00540243	+	0.0306387i
$766$		0			0
$767$	−7.54646	−	13.0708i	−0.272487	−	0.471961i
$768$		0			0
$769$	−9.11104	+	7.64507i	−0.328552	+	0.275688i	−0.792110	−	0.610379i	$-0.791017\pi$
$769$	−9.11104	+	7.64507i	−0.328552	+	0.275688i	0.463557	+	0.886067i	$0.346573\pi$
$770$		0			0
$771$	−5.54888	+	9.61095i	−0.199838	+	0.346130i
$772$		0			0
$773$	−3.55618	−	1.29434i	−0.127907	−	0.0465543i	0.277274	−	0.960791i	$-0.410569\pi$
$773$	−3.55618	−	1.29434i	−0.127907	−	0.0465543i	−0.405181	+	0.914237i	$0.632791\pi$
$774$		0			0
$775$	−0.745776	−	4.22951i	−0.0267891	−	0.151928i
$776$		0			0
$777$	9.32853	+	7.82757i	0.334659	+	0.280812i
$778$		0			0
$779$	−12.9894	+	18.0206i	−0.465394	+	0.645656i
$780$		0			0
$781$	24.8221	+	20.8283i	0.888206	+	0.745294i
$782$		0			0
$783$	0.220876	+	1.25265i	0.00789345	+	0.0447660i
$784$		0			0
$785$	2.86239	+	1.04182i	0.102163	+	0.0371843i
$786$		0			0
$787$	14.4410	−	25.0126i	0.514767	−	0.891603i	−0.485086	−	0.874466i	$-0.661212\pi$
$787$	14.4410	−	25.0126i	0.514767	−	0.891603i	0.999853	−	0.0171362i	$-0.00545490\pi$
$788$		0			0
$789$	−8.64568	+	7.25459i	−0.307794	+	0.258270i
$790$		0			0
$791$	2.23861	+	3.87739i	0.0795958	+	0.137864i
$792$		0			0
$793$	4.05707	−	23.0088i	0.144071	−	0.817066i
$794$		0			0
$795$	0.902567	−	0.328507i	0.0320107	−	0.0116510i
$796$		0			0
$797$	6.10268			0.216168			0.108084	−	0.994142i	$-0.465528\pi$
$797$	6.10268			0.216168			0.108084	+	0.994142i	$0.465528\pi$
$798$		0			0
$799$	−1.01535			−0.0359204
$800$		0			0
$801$	−10.4353	+	3.79813i	−0.368712	+	0.134200i
$802$		0			0
$803$	−2.54747	+	14.4474i	−0.0898982	+	0.509838i
$804$		0			0
$805$	0.350926	+	0.607822i	0.0123685	+	0.0214229i
$806$		0			0
$807$	−0.120102	+	0.100778i	−0.00422780	+	0.00354755i
$808$		0			0
$809$	6.57482	−	11.3879i	0.231158	−	0.400378i	−0.726991	−	0.686647i	$-0.759082\pi$
$809$	6.57482	−	11.3879i	0.231158	−	0.400378i	0.958149	+	0.286269i	$0.0924151\pi$
$810$		0			0
$811$	3.65410	+	1.32998i	0.128313	+	0.0467021i	0.405379	−	0.914149i	$-0.367140\pi$
$811$	3.65410	+	1.32998i	0.128313	+	0.0467021i	−0.277066	+	0.960851i	$0.589362\pi$
$812$		0			0
$813$	−1.19859	−	6.79754i	−0.0420364	−	0.238400i
$814$		0			0
$815$	0.809434	+	0.679195i	0.0283532	+	0.0237912i
$816$		0			0
$817$	−8.24946	−	18.3425i	−0.288612	−	0.641723i
$818$		0			0
$819$	−4.10323	−	3.44302i	−0.143379	−	0.120309i
$820$		0			0
$821$	−3.32963	−	18.8833i	−0.116205	−	0.659032i	−0.986147	−	0.165877i	$-0.946955\pi$
$821$	−3.32963	−	18.8833i	−0.116205	−	0.659032i	0.869941	−	0.493155i	$-0.164156\pi$
$822$		0			0
$823$	18.7839	+	6.83678i	0.654766	+	0.238315i	0.647975	−	0.761662i	$-0.275616\pi$
$823$	18.7839	+	6.83678i	0.654766	+	0.238315i	0.00679075	+	0.999977i	$0.497838\pi$
$824$		0			0
$825$	−5.97826	+	10.3546i	−0.208136	+	0.360502i
$826$		0			0
$827$	−1.28973	+	1.08222i	−0.0448485	+	0.0376323i	−0.664936	−	0.746900i	$-0.731541\pi$
$827$	−1.28973	+	1.08222i	−0.0448485	+	0.0376323i	0.620088	+	0.784532i	$0.287097\pi$
$828$		0			0
$829$	−13.6454	−	23.6345i	−0.473925	−	0.820862i	0.525630	−	0.850713i	$-0.323830\pi$
$829$	−13.6454	−	23.6345i	−0.473925	−	0.820862i	−0.999554	+	0.0298519i	$0.990496\pi$
$830$		0			0
$831$	−0.562367	+	3.18934i	−0.0195083	+	0.110637i
$832$		0			0
$833$	11.0115	−	4.00787i	0.381527	−	0.138865i
$834$		0			0
$835$	−7.00427			−0.242393
$836$		0			0
$837$	0.884596			0.0305761
$838$		0			0
$839$	−3.71621	+	1.35259i	−0.128298	+	0.0466966i	−0.405371	−	0.914152i	$-0.632858\pi$
$839$	−3.71621	+	1.35259i	−0.128298	+	0.0466966i	0.277073	+	0.960849i	$0.410635\pi$
$840$		0			0
$841$	−4.75485	+	26.9661i	−0.163960	+	0.929865i
$842$		0			0
$843$	8.45748	+	14.6488i	0.291291	+	0.504531i
$844$		0			0
$845$	−0.818349	+	0.686676i	−0.0281521	+	0.0236224i
$846$		0			0
$847$	3.32453	−	5.75825i	0.114232	−	0.197856i
$848$		0			0
$849$	−17.1397	−	6.23833i	−0.588232	−	0.214099i
$850$		0			0
$851$	2.14749	+	12.1790i	0.0736149	+	0.417491i
$852$		0			0
$853$	−36.3166	−	30.4732i	−1.24346	−	1.04338i	−0.997246	−	0.0741679i	$-0.976370\pi$
$853$	−36.3166	−	30.4732i	−1.24346	−	1.04338i	−0.246211	−	0.969216i	$-0.579186\pi$
$854$		0			0
$855$	1.49434	+	0.721889i	0.0511053	+	0.0246881i
$856$		0			0
$857$	−42.0573	−	35.2903i	−1.43665	−	1.20549i	−0.941647	−	0.336603i	$-0.890722\pi$
$857$	−42.0573	−	35.2903i	−1.43665	−	1.20549i	−0.495004	−	0.868891i	$-0.664833\pi$
$858$		0			0
$859$	6.84643	+	38.8281i	0.233597	+	1.32480i	0.845548	+	0.533900i	$0.179274\pi$
$859$	6.84643	+	38.8281i	0.233597	+	1.32480i	−0.611950	+	0.790896i	$0.709615\pi$
$860$		0			0
$861$	−6.45211	−	2.34838i	−0.219887	−	0.0800324i
$862$		0			0
$863$	10.4441	−	18.0897i	0.355522	−	0.615782i	−0.631685	−	0.775225i	$-0.717637\pi$
$863$	10.4441	−	18.0897i	0.355522	−	0.615782i	0.987207	+	0.159443i	$0.0509699\pi$
$864$		0			0
$865$	−4.97244	+	4.17237i	−0.169068	+	0.141865i
$866$		0			0
$867$	5.94594	+	10.2987i	0.201935	+	0.349761i
$868$		0			0
$869$	−4.53240	+	25.7045i	−0.153751	+	0.871965i
$870$		0			0
$871$	18.1628	−	6.61070i	0.615422	−	0.223995i
$872$		0			0
$873$	9.30137			0.314804
$874$		0			0
$875$	5.05523			0.170898
$876$		0			0
$877$	−46.6746	+	16.9882i	−1.57609	+	0.573650i	−0.974349	−	0.225041i	$-0.927748\pi$
$877$	−46.6746	+	16.9882i	−1.57609	+	0.573650i	−0.601741	+	0.798691i	$0.705526\pi$
$878$		0			0
$879$	4.69753	−	26.6410i	0.158444	−	0.898580i
$880$		0			0
$881$	7.52754	+	13.0381i	0.253609	+	0.439264i	0.964517	−	0.264021i	$-0.0850489\pi$
$881$	7.52754	+	13.0381i	0.253609	+	0.439264i	−0.710908	+	0.703285i	$0.751716\pi$
$882$		0			0
$883$	−44.2202	+	37.1051i	−1.48813	+	1.24869i	−0.591189	+	0.806533i	$0.701341\pi$
$883$	−44.2202	+	37.1051i	−1.48813	+	1.24869i	−0.896939	+	0.442155i	$0.854214\pi$
$884$		0			0
$885$	−0.722692	+	1.25174i	−0.0242930	+	0.0420767i
$886$		0			0
$887$	3.68881	+	1.34262i	0.123858	+	0.0450807i	0.403206	−	0.915109i	$-0.367896\pi$
$887$	3.68881	+	1.34262i	0.123858	+	0.0450807i	−0.279347	+	0.960190i	$0.590118\pi$
$888$		0			0
$889$	−0.575401	−	3.26326i	−0.0192983	−	0.109446i
$890$		0			0
$891$	−1.88654	−	1.58299i	−0.0632014	−	0.0530323i
$892$		0			0
$893$	−0.532670	+	1.88438i	−0.0178251	+	0.0630583i
$894$		0			0
$895$	5.65201	+	4.74260i	0.188926	+	0.158527i
$896$		0			0
$897$	−0.944590	−	5.35703i	−0.0315389	−	0.178866i
$898$		0			0
$899$	−1.05733	−	0.384835i	−0.0352638	−	0.0128350i
$900$		0			0
$901$	−2.85086	+	4.93783i	−0.0949758	+	0.164503i
$902$		0			0
$903$	4.76212	−	3.99589i	0.158473	−	0.132975i
$904$		0			0
$905$	1.36855	+	2.37040i	0.0454922	+	0.0787948i
$906$		0			0
$907$	−6.22365	+	35.2961i	−0.206653	+	1.17199i	0.688164	+	0.725555i	$0.258417\pi$
$907$	−6.22365	+	35.2961i	−0.206653	+	1.17199i	−0.894817	+	0.446432i	$0.852694\pi$
$908$		0			0
$909$	−11.6373	+	4.23563i	−0.385985	+	0.140487i
$910$		0			0
$911$	11.0762			0.366969			0.183485	−	0.983023i	$-0.441262\pi$
$911$	11.0762			0.366969			0.183485	+	0.983023i	$0.441262\pi$
$912$		0			0
$913$	−17.8589			−0.591043
$914$		0			0
$915$	−2.10251	+	0.765250i	−0.0695068	+	0.0252984i
$916$		0			0
$917$	4.27160	−	24.2254i	0.141061	−	0.799994i
$918$		0			0
$919$	−9.26624	−	16.0496i	−0.305665	−	0.529427i	0.671744	−	0.740783i	$-0.265545\pi$
$919$	−9.26624	−	16.0496i	−0.305665	−	0.529427i	−0.977409	+	0.211356i	$0.932212\pi$
$920$		0			0
$921$	25.6568	−	21.5286i	0.845421	−	0.709392i
$922$		0			0
$923$	−26.1549	+	45.3016i	−0.860899	+	1.49112i
$924$		0			0
$925$	41.2359	+	15.0086i	1.35583	+	0.493481i
$926$		0			0
$927$	1.17872	+	6.68484i	0.0387142	+	0.219559i
$928$		0			0
$929$	17.5287	+	14.7083i	0.575097	+	0.482564i	0.883333	−	0.468746i	$-0.155294\pi$
$929$	17.5287	+	14.7083i	0.575097	+	0.482564i	−0.308236	+	0.951310i	$0.599739\pi$
$930$		0			0
$931$	−1.66133	−	22.5388i	−0.0544480	−	0.738681i
$932$		0			0
$933$	3.13561	+	2.63109i	0.102655	+	0.0861379i
$934$		0			0
$935$	0.367986	+	2.08695i	0.0120344	+	0.0682507i
$936$		0			0
$937$	29.2000	+	10.6279i	0.953924	+	0.347200i	0.771650	−	0.636048i	$-0.219432\pi$
$937$	29.2000	+	10.6279i	0.953924	+	0.347200i	0.182274	+	0.983248i	$0.441654\pi$
$938$		0			0
$939$	9.03754	−	15.6535i	0.294929	−	0.510832i
$940$		0			0
$941$	−25.8945	+	21.7281i	−0.844136	+	0.708314i	−0.958490	−	0.285126i	$-0.907965\pi$
$941$	−25.8945	+	21.7281i	−0.844136	+	0.708314i	0.114354	+	0.993440i	$0.463520\pi$
$942$		0			0
$943$	−3.48648	−	6.03875i	−0.113535	−	0.196649i
$944$		0			0
$945$	−0.0890743	+	0.505165i	−0.00289759	+	0.0164330i
$946$		0			0
$947$	23.4144	−	8.52216i	0.760867	−	0.276933i	0.0676959	−	0.997706i	$-0.478435\pi$
$947$	23.4144	−	8.52216i	0.760867	−	0.276933i	0.693171	+	0.720773i	$0.256213\pi$
$948$		0			0
$949$	−23.6830			−0.768782
$950$		0			0
$951$	−11.4052			−0.369839
$952$		0			0
$953$	30.6539	−	11.1571i	0.992978	−	0.361414i	0.206105	−	0.978530i	$-0.433921\pi$
$953$	30.6539	−	11.1571i	0.992978	−	0.361414i	0.786873	+	0.617116i	$0.211699\pi$
$954$		0			0
$955$	−0.245254	+	1.39091i	−0.00793625	+	0.0450087i
$956$		0			0
$957$	1.56624	+	2.71281i	0.0506294	+	0.0876928i
$958$		0			0
$959$	−18.1956	+	15.2680i	−0.587568	+	0.493028i
$960$		0			0
$961$	15.1087	−	26.1691i	0.487379	−	0.844165i
$962$		0			0
$963$	0.345809	+	0.125864i	0.0111435	+	0.00405591i
$964$		0			0
$965$	−0.377283	−	2.13968i	−0.0121452	−	0.0688787i
$966$		0			0
$967$	−17.0726	−	14.3256i	−0.549018	−	0.460680i	0.325591	−	0.945511i	$-0.394437\pi$
$967$	−17.0726	−	14.3256i	−0.549018	−	0.460680i	−0.874608	+	0.484830i	$0.838881\pi$
$968$		0			0
$969$	−9.54995	+	2.41928i	−0.306789	+	0.0777185i
$970$		0			0
$971$	−4.47457	−	3.75461i	−0.143596	−	0.120491i	0.568160	−	0.822918i	$-0.307655\pi$
$971$	−4.47457	−	3.75461i	−0.143596	−	0.120491i	−0.711756	+	0.702427i	$0.752100\pi$
$972$		0			0
$973$	1.98217	+	11.2414i	0.0635454	+	0.360384i
$974$		0			0
$975$	−18.1379	−	6.60167i	−0.580879	−	0.211423i
$976$		0			0
$977$	−8.37965	+	14.5140i	−0.268089	+	0.464344i	−0.968368	−	0.249525i	$-0.919725\pi$
$977$	−8.37965	+	14.5140i	−0.268089	+	0.464344i	0.700279	+	0.713869i	$0.253059\pi$
$978$		0			0
$979$	−20.9500	+	17.5791i	−0.669564	+	0.561831i
$980$		0			0
$981$	0.285643	+	0.494748i	0.00911987	+	0.0157961i
$982$		0			0
$983$	7.12791	−	40.4244i	0.227345	−	1.28934i	−0.630806	−	0.775941i	$-0.717275\pi$
$983$	7.12791	−	40.4244i	0.227345	−	1.28934i	0.858151	−	0.513398i	$-0.171613\pi$
$984$		0			0
$985$	−0.902006	+	0.328303i	−0.0287403	+	0.0104606i
$986$		0			0
$987$	−0.605267			−0.0192659
$988$		0			0
$989$	6.31316			0.200747
$990$		0			0
$991$	−15.6614	+	5.70028i	−0.497500	+	0.181075i	−0.578569	−	0.815633i	$-0.696389\pi$
$991$	−15.6614	+	5.70028i	−0.497500	+	0.181075i	0.0810690	+	0.996708i	$0.474167\pi$
$992$		0			0
$993$	−3.00187	+	17.0245i	−0.0952615	+	0.540255i
$994$		0			0
$995$	−3.76513	−	6.52139i	−0.119363	−	0.206742i
$996$		0			0
$997$	−20.5932	+	17.2797i	−0.652192	+	0.547254i	−0.907735	−	0.419543i	$-0.862190\pi$
$997$	−20.5932	+	17.2797i	−0.652192	+	0.547254i	0.255543	+	0.966798i	$0.417746\pi$
$998$		0			0
$999$	−4.51925	+	7.82757i	−0.142983	+	0.247653i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.bg.b.73.1	yes	12
4.3	odd	2		912.2.bo.i.529.1		12
19.5	even	9		8664.2.a.bk.1.3		6
19.6	even	9	inner	456.2.bg.b.25.1	✓	12
19.14	odd	18		8664.2.a.bh.1.3		6
76.63	odd	18		912.2.bo.i.481.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bg.b.25.1	✓	12	19.6	even	9	inner
456.2.bg.b.73.1	yes	12	1.1	even	1	trivial
912.2.bo.i.481.1		12	76.63	odd	18
912.2.bo.i.529.1		12	4.3	odd	2
8664.2.a.bh.1.3		6	19.14	odd	18
8664.2.a.bk.1.3		6	19.5	even	9

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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.bg (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.939693 + 0.342020i) q^{3} +(-0.0661133 + 0.374947i) q^{5} +(-0.673648 - 1.16679i) q^{7} +(0.766044 - 0.642788i) q^{9} +(1.23135 - 2.13276i) q^{11} +(3.73590 + 1.35976i) q^{13} +(-0.0661133 - 0.374947i) q^{15} +(1.73135 + 1.45278i) q^{17} +(3.60449 - 2.45105i) q^{19} +(1.03209 + 0.866025i) q^{21} +(0.237593 + 1.34746i) q^{23} +(4.56225 + 1.66052i) q^{25} +(-0.500000 + 0.866025i) q^{27} +(0.974387 - 0.817608i) q^{29} +(-0.442298 - 0.766083i) q^{31} +(-0.427643 + 2.42529i) q^{33} +(0.482023 - 0.175442i) q^{35} +9.03850 q^{37} -3.97566 q^{39} +(-4.78893 + 1.74303i) q^{41} +(0.801223 - 4.54396i) q^{43} +(0.190366 + 0.329723i) q^{45} +(-0.344142 + 0.288770i) q^{47} +(2.59240 - 4.49016i) q^{49} +(-2.12382 - 0.773005i) q^{51} +(0.438071 + 2.48443i) q^{53} +(0.718264 + 0.602695i) q^{55} +(-2.54881 + 3.53604i) q^{57} +(-2.90816 - 2.44023i) q^{59} +(-1.02048 - 5.78741i) q^{61} +(-1.26604 - 0.460802i) q^{63} +(-0.756829 + 1.31087i) q^{65} +(3.72427 - 3.12503i) q^{67} +(-0.684123 - 1.18493i) q^{69} +(-2.28478 + 12.9576i) q^{71} +(-5.59774 + 2.03741i) q^{73} -4.85504 q^{75} -3.31799 q^{77} +(-9.95937 + 3.62491i) q^{79} +(0.173648 - 0.984808i) q^{81} +(-3.62587 - 6.28020i) q^{83} +(-0.659180 + 0.553117i) q^{85} +(-0.635986 + 1.10156i) q^{87} +(-10.4353 - 3.79813i) q^{89} +(-0.930127 - 5.27501i) q^{91} +(0.677640 + 0.568608i) q^{93} +(0.680709 + 1.51354i) q^{95} +(7.12527 + 5.97881i) q^{97} +(-0.427643 - 2.42529i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{5} - 6 q^{7} - 3 q^{11} - 3 q^{13} + 3 q^{15} + 3 q^{17} - 6 q^{21} - 12 q^{23} + 9 q^{25} - 6 q^{27} + 9 q^{29} - 21 q^{31} - 6 q^{33} - 6 q^{35} + 30 q^{37} + 6 q^{39} + 15 q^{41} + 3 q^{43}+ \cdots - 6 q^{99}+O(q^{100}) \) 12 * q + 3 * q^5 - 6 * q^7 - 3 * q^11 - 3 * q^13 + 3 * q^15 + 3 * q^17 - 6 * q^21 - 12 * q^23 + 9 * q^25 - 6 * q^27 + 9 * q^29 - 21 * q^31 - 6 * q^33 - 6 * q^35 + 30 * q^37 + 6 * q^39 + 15 * q^41 + 3 * q^43 - 3 * q^45 - 18 * q^47 + 24 * q^49 - 6 * q^51 + 24 * q^53 + 60 * q^55 - 9 * q^57 - 21 * q^59 - 24 * q^61 - 6 * q^63 - 21 * q^65 + 15 * q^67 - 21 * q^71 - 3 * q^73 + 18 * q^75 - 6 * q^77 - 63 * q^79 - 12 * q^83 + 9 * q^85 - 15 * q^87 - 15 * q^89 - 3 * q^93 + 12 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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