LMFDB - Embedded newform 420.2.bv.a.317.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(53,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 6, 9, 8]))

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

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

chi := DirichletCharacter("420.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.bv (of order $12$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			317.1
Root			$-0.396143 - 1.68614i$ of defining polynomial
Character	$\chi$	$=$	420.317
Dual form			420.2.bv.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.396143 - 1.68614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(0.576028 + 2.58228i) q^{7} +(-2.68614 + 1.33591i) q^{9} +O(q^{10})$	\(q+(-0.396143 - 1.68614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(0.576028 + 2.58228i) q^{7} +(-2.68614 + 1.33591i) q^{9} +(-3.73831 + 2.15831i) q^{11} +(-2.00000 + 2.00000i) q^{13} +(3.81662 - 0.658312i) q^{15} +(-5.89662 - 1.57999i) q^{17} +(5.47036 + 3.15831i) q^{19} +(4.12590 - 1.99422i) q^{21} +(2.94831 - 0.789997i) q^{23} +(-4.96410 - 0.598076i) q^{25} +(3.31662 + 4.00000i) q^{27} -0.683375 q^{29} +(3.00000 + 5.19615i) q^{31} +(5.12012 + 5.44831i) q^{33} +(-5.84096 + 0.939764i) q^{35} +(5.46410 - 1.46410i) q^{37} +(4.16457 + 2.57999i) q^{39} +3.63325i q^{41} +(5.15831 - 5.15831i) q^{43} +(-2.62194 - 6.17458i) q^{45} +(-1.83013 - 6.83013i) q^{47} +(-6.33638 + 2.97494i) q^{49} +(-0.328185 + 10.5684i) q^{51} +(0.250133 - 0.933508i) q^{53} +(-4.31662 - 8.63325i) q^{55} +(3.15831 - 10.4749i) q^{57} +(2.84169 + 4.92195i) q^{59} +(-1.34169 + 2.32387i) q^{61} +(-4.99699 - 6.16686i) q^{63} +(-4.19615 - 4.73205i) q^{65} +(1.40616 - 5.24784i) q^{67} +(-2.50000 - 4.65831i) q^{69} +4.31662i q^{71} +(-9.06119 - 2.42794i) q^{73} +(0.958056 + 8.60710i) q^{75} +(-7.72675 - 8.41012i) q^{77} +(-10.6665 - 6.15831i) q^{79} +(5.43070 - 7.17687i) q^{81} +(7.15831 + 7.15831i) q^{83} +(4.31662 - 12.9499i) q^{85} +(0.270715 + 1.15227i) q^{87} +(-8.29156 + 14.3614i) q^{89} +(-6.31662 - 4.01251i) q^{91} +(7.57301 - 7.11684i) q^{93} +(-7.78240 + 11.7870i) q^{95} +(-2.68338 - 2.68338i) q^{97} +(7.15831 - 10.7916i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9	$ 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9} - 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{21} + 2 q^{23} - 12 q^{25} - 32 q^{29} + 24 q^{31} - 2 q^{33} + 16 q^{37} + 4 q^{39} + 28 q^{43} - 20 q^{45} + 20 q^{47} - 24 q^{51} - 16 q^{53} - 8 q^{55} + 12 q^{57} + 36 q^{59} - 24 q^{61} + 22 q^{63} + 8 q^{65} - 22 q^{67} - 20 q^{69} - 8 q^{75} + 16 q^{77} - 14 q^{81} + 44 q^{83} + 8 q^{85} - 22 q^{87} - 24 q^{91} + 12 q^{95} - 48 q^{97} + 44 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9 - 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^21 + 2 * q^23 - 12 * q^25 - 32 * q^29 + 24 * q^31 - 2 * q^33 + 16 * q^37 + 4 * q^39 + 28 * q^43 - 20 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 - 8 * q^55 + 12 * q^57 + 36 * q^59 - 24 * q^61 + 22 * q^63 + 8 * q^65 - 22 * q^67 - 20 * q^69 - 8 * q^75 + 16 * q^77 - 14 * q^81 + 44 * q^83 + 8 * q^85 - 22 * q^87 - 24 * q^91 + 12 * q^95 - 48 * q^97 + 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{4}\right)$

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
$2$		0			0
$3$	−0.396143	−	1.68614i	−0.228714	−	0.973494i
$3$	−0.396143	−	1.68614i	−0.228714	−	0.973494i
$4$		0			0
$5$	−0.133975	+	2.23205i	−0.0599153	+	0.998203i
$5$	−0.133975	+	2.23205i	−0.0599153	+	0.998203i
$6$		0			0
$7$	0.576028	+	2.58228i	0.217718	+	0.976012i
$7$	0.576028	+	2.58228i	0.217718	+	0.976012i
$8$		0			0
$9$	−2.68614	+	1.33591i	−0.895380	+	0.445302i
$10$		0			0
$11$	−3.73831	+	2.15831i	−1.12714	+	0.650756i	−0.943215	−	0.332184i	$-0.892214\pi$
$11$	−3.73831	+	2.15831i	−1.12714	+	0.650756i	−0.183927	+	0.982940i	$0.558881\pi$
$12$		0			0
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	−0.927794	−	0.373094i	$-0.878297\pi$
$13$	−2.00000	+	2.00000i	−0.554700	+	0.554700i	0.373094	+	0.927794i	$0.378297\pi$
$14$		0			0
$15$	3.81662	−	0.658312i	0.985448	−	0.169976i
$16$		0			0
$17$	−5.89662	−	1.57999i	−1.43014	−	0.383205i	−0.541071	−	0.840977i	$-0.681981\pi$
$17$	−5.89662	−	1.57999i	−1.43014	−	0.383205i	−0.889070	+	0.457772i	$0.848648\pi$
$18$		0			0
$19$	5.47036	+	3.15831i	1.25499	+	0.724567i	0.972095	−	0.234586i	$-0.0753733\pi$
$19$	5.47036	+	3.15831i	1.25499	+	0.724567i	0.282891	+	0.959152i	$0.408707\pi$
$20$		0			0
$21$	4.12590	−	1.99422i	0.900346	−	0.435174i
$22$		0			0
$23$	2.94831	−	0.789997i	0.614765	−	0.164726i	0.0620182	−	0.998075i	$-0.480246\pi$
$23$	2.94831	−	0.789997i	0.614765	−	0.164726i	0.552747	+	0.833349i	$0.313580\pi$
$24$		0			0
$25$	−4.96410	−	0.598076i	−0.992820	−	0.119615i
$26$		0			0
$27$	3.31662	+	4.00000i	0.638285	+	0.769800i
$28$		0			0
$29$	−0.683375			−0.126900			−0.0634498	−	0.997985i	$-0.520210\pi$
$29$	−0.683375			−0.126900			−0.0634498	+	0.997985i	$0.520210\pi$
$30$		0			0
$31$	3.00000	+	5.19615i	0.538816	+	0.933257i	0.998968	+	0.0454165i	$0.0144615\pi$
$31$	3.00000	+	5.19615i	0.538816	+	0.933257i	−0.460152	+	0.887840i	$0.652205\pi$
$32$		0			0
$33$	5.12012	+	5.44831i	0.891299	+	0.948429i
$34$		0			0
$35$	−5.84096	+	0.939764i	−0.987303	+	0.158849i
$36$		0			0
$37$	5.46410	−	1.46410i	0.898293	−	0.240697i	0.220010	−	0.975498i	$-0.429391\pi$
$37$	5.46410	−	1.46410i	0.898293	−	0.240697i	0.678283	+	0.734801i	$0.262724\pi$
$38$		0			0
$39$	4.16457	+	2.57999i	0.666865	+	0.413130i
$40$		0			0
$41$			3.63325i			0.567418i	0.958910	+	0.283709i	$0.0915650\pi$
$41$			3.63325i			0.567418i	−0.958910	+	0.283709i	$0.908435\pi$
$42$		0			0
$43$	5.15831	−	5.15831i	0.786635	−	0.786635i	−0.194306	−	0.980941i	$-0.562245\pi$
$43$	5.15831	−	5.15831i	0.786635	−	0.786635i	0.980941	+	0.194306i	$0.0622454\pi$
$44$		0			0
$45$	−2.62194	−	6.17458i	−0.390855	−	0.920452i
$46$		0			0
$47$	−1.83013	−	6.83013i	−0.266951	−	0.996276i	−0.961045	−	0.276392i	$-0.910861\pi$
$47$	−1.83013	−	6.83013i	−0.266951	−	0.996276i	0.694094	−	0.719885i	$-0.255805\pi$
$48$		0			0
$49$	−6.33638	+	2.97494i	−0.905198	+	0.424991i
$50$		0			0
$51$	−0.328185	+	10.5684i	−0.0459551	+	1.47988i
$52$		0			0
$53$	0.250133	−	0.933508i	0.0343584	−	0.128227i	−0.946617	−	0.322362i	$-0.895523\pi$
$53$	0.250133	−	0.933508i	0.0343584	−	0.128227i	0.980975	+	0.194135i	$0.0621899\pi$
$54$		0			0
$55$	−4.31662	−	8.63325i	−0.582054	−	1.16411i
$56$		0			0
$57$	3.15831	−	10.4749i	0.418329	−	1.38744i
$58$		0			0
$59$	2.84169	+	4.92195i	0.369956	+	0.640783i	0.989558	−	0.144133i	$-0.0460394\pi$
$59$	2.84169	+	4.92195i	0.369956	+	0.640783i	−0.619602	+	0.784916i	$0.712706\pi$
$60$		0			0
$61$	−1.34169	+	2.32387i	−0.171785	+	0.297541i	−0.939044	−	0.343797i	$-0.888287\pi$
$61$	−1.34169	+	2.32387i	−0.171785	+	0.297541i	0.767259	+	0.641338i	$0.221620\pi$
$62$		0			0
$63$	−4.99699	−	6.16686i	−0.629561	−	0.776951i
$64$		0			0
$65$	−4.19615	−	4.73205i	−0.520469	−	0.586939i
$66$		0			0
$67$	1.40616	−	5.24784i	0.171789	−	0.641126i	−0.825287	−	0.564713i	$-0.808987\pi$
$67$	1.40616	−	5.24784i	0.171789	−	0.641126i	0.997076	−	0.0764126i	$-0.0243466\pi$
$68$		0			0
$69$	−2.50000	−	4.65831i	−0.300965	−	0.560795i
$70$		0			0
$71$			4.31662i			0.512289i	0.966638	+	0.256145i	$0.0824523\pi$
$71$			4.31662i			0.512289i	−0.966638	+	0.256145i	$0.917548\pi$
$72$		0			0
$73$	−9.06119	−	2.42794i	−1.06053	−	0.284169i	−0.313934	−	0.949445i	$-0.601647\pi$
$73$	−9.06119	−	2.42794i	−1.06053	−	0.284169i	−0.746597	+	0.665276i	$0.768314\pi$
$74$		0			0
$75$	0.958056	+	8.60710i	0.110627	+	0.993862i
$76$		0			0
$77$	−7.72675	−	8.41012i	−0.880544	−	0.958422i
$78$		0			0
$79$	−10.6665	−	6.15831i	−1.20008	−	0.692864i	−0.239504	−	0.970895i	$-0.576985\pi$
$79$	−10.6665	−	6.15831i	−1.20008	−	0.692864i	−0.960572	+	0.278031i	$0.910318\pi$
$80$		0			0
$81$	5.43070	−	7.17687i	0.603411	−	0.797430i
$82$		0			0
$83$	7.15831	+	7.15831i	0.785727	+	0.785727i	0.980791	−	0.195064i	$-0.0624914\pi$
$83$	7.15831	+	7.15831i	0.785727	+	0.785727i	−0.195064	+	0.980791i	$0.562491\pi$
$84$		0			0
$85$	4.31662	−	12.9499i	0.468204	−	1.40461i
$86$		0			0
$87$	0.270715	+	1.15227i	0.0290237	+	0.123536i
$88$		0			0
$89$	−8.29156	+	14.3614i	−0.878904	+	1.52231i	−0.0263586	+	0.999653i	$0.508391\pi$
$89$	−8.29156	+	14.3614i	−0.878904	+	1.52231i	−0.852545	+	0.522654i	$0.824942\pi$
$90$		0			0
$91$	−6.31662	−	4.01251i	−0.662162	−	0.420626i
$92$		0			0
$93$	7.57301	−	7.11684i	0.785285	−	0.737982i
$94$		0			0
$95$	−7.78240	+	11.7870i	−0.798458	+	1.20932i
$96$		0			0
$97$	−2.68338	−	2.68338i	−0.272455	−	0.272455i	0.557632	−	0.830088i	$-0.311710\pi$
$97$	−2.68338	−	2.68338i	−0.272455	−	0.272455i	−0.830088	+	0.557632i	$0.811710\pi$
$98$		0			0
$99$	7.15831	−	10.7916i	0.719437	−	1.08459i
$100$		0			0
$101$	8.06843	−	4.65831i	0.802839	−	0.463519i	−0.0416238	−	0.999133i	$-0.513253\pi$
$101$	8.06843	−	4.65831i	0.802839	−	0.463519i	0.844463	+	0.535614i	$0.179920\pi$
$102$		0			0
$103$	4.79793	+	17.9061i	0.472754	+	1.76434i	0.629807	+	0.776752i	$0.283134\pi$
$103$	4.79793	+	17.9061i	0.472754	+	1.76434i	−0.157053	+	0.987590i	$0.550199\pi$
$104$		0			0
$105$	3.89843	+	9.47640i	0.380448	+	0.924802i
$106$		0			0
$107$	−2.62012	−	9.77844i	−0.253297	−	0.945317i	−0.969030	−	0.246943i	$-0.920574\pi$
$107$	−2.62012	−	9.77844i	−0.253297	−	0.945317i	0.715733	−	0.698374i	$-0.246093\pi$
$108$		0			0
$109$	12.6294	−	7.29156i	1.20967	−	0.698405i	0.246985	−	0.969019i	$-0.420560\pi$
$109$	12.6294	−	7.29156i	1.20967	−	0.698405i	0.962688	+	0.270614i	$0.0872268\pi$
$110$		0			0
$111$	−4.63325	−	8.63325i	−0.439769	−	0.819432i
$112$		0			0
$113$	3.63325	+	3.63325i	0.341787	+	0.341787i	0.857039	−	0.515252i	$-0.172302\pi$
$113$	3.63325	+	3.63325i	0.341787	+	0.341787i	−0.515252	+	0.857039i	$0.672302\pi$
$114$		0			0
$115$	1.36832	+	6.68662i	0.127596	+	0.623530i
$116$		0			0
$117$	2.70047	−	8.04410i	0.249658	−	0.743677i
$118$		0			0
$119$	0.683375	−	16.1369i	0.0626449	−	1.47926i
$120$		0			0
$121$	3.81662	−	6.61059i	0.346966	−	0.600963i
$122$		0			0
$123$	6.12617	−	1.43929i	0.552378	−	0.129776i
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$	−7.63325	−	7.63325i	−0.677341	−	0.677341i	0.282056	−	0.959398i	$-0.408983\pi$
$127$	−7.63325	−	7.63325i	−0.677341	−	0.677341i	−0.959398	+	0.282056i	$0.908983\pi$
$128$		0			0
$129$	−10.7411	−	6.65421i	−0.945699	−	0.585870i
$130$		0			0
$131$	8.66025	+	5.00000i	0.756650	+	0.436852i	0.828092	−	0.560593i	$-0.189427\pi$
$131$	8.66025	+	5.00000i	0.756650	+	0.436852i	−0.0714417	+	0.997445i	$0.522760\pi$
$132$		0			0
$133$	−5.00458	+	15.9453i	−0.433952	+	1.38263i
$134$		0			0
$135$	−9.37255	+	6.86698i	−0.806660	+	0.591015i
$136$		0			0
$137$	17.6899	+	4.73998i	1.51135	+	0.404964i	0.916882	−	0.399158i	$-0.130697\pi$
$137$	17.6899	+	4.73998i	1.51135	+	0.404964i	0.594464	+	0.804122i	$0.297364\pi$
$138$		0			0
$139$			12.3166i			1.04468i	0.852737	+	0.522341i	$0.174941\pi$
$139$			12.3166i			1.04468i	−0.852737	+	0.522341i	$0.825059\pi$
$140$		0			0
$141$	−10.7916	+	5.79156i	−0.908813	+	0.487738i
$142$		0			0
$143$	3.15999	−	11.7932i	0.264252	−	0.986200i
$144$		0			0
$145$	0.0915549	−	1.52533i	0.00760322	−	0.126672i
$146$		0			0
$147$	7.52628	+	9.50553i	0.620757	+	0.784003i
$148$		0			0
$149$	1.81662	−	3.14649i	0.148824	−	0.257770i	−0.781969	−	0.623317i	$-0.785785\pi$
$149$	1.81662	−	3.14649i	0.148824	−	0.257770i	0.930793	+	0.365547i	$0.119118\pi$
$150$		0			0
$151$	8.31662	+	14.4048i	0.676797	+	1.17225i	0.975940	+	0.218039i	$0.0699660\pi$
$151$	8.31662	+	14.4048i	0.676797	+	1.17225i	−0.299143	+	0.954208i	$0.596701\pi$
$152$		0			0
$153$	17.9499	−	3.63325i	1.45116	−	0.293731i
$154$		0			0
$155$	−12.0000	+	6.00000i	−0.963863	+	0.481932i
$156$		0			0
$157$	−0.481918	+	1.79854i	−0.0384613	+	0.143539i	−0.982486	−	0.186334i	$-0.940339\pi$
$157$	−0.481918	+	1.79854i	−0.0384613	+	0.143539i	0.944025	+	0.329873i	$0.107006\pi$
$158$		0			0
$159$	−1.67311	−	0.0519558i	−0.132687	−	0.00412036i
$160$		0			0
$161$	3.73831	+	7.15831i	0.294620	+	0.564154i
$162$		0			0
$163$	−0.366025	−	1.36603i	−0.0286693	−	0.106995i	0.950109	−	0.311919i	$-0.100972\pi$
$163$	−0.366025	−	1.36603i	−0.0286693	−	0.106995i	−0.978778	+	0.204924i	$0.934305\pi$
$164$		0			0
$165$	−12.8469	+	10.6984i	−1.00013	+	0.832873i
$166$		0			0
$167$	10.1082	−	10.1082i	0.782195	−	0.782195i	−0.198006	−	0.980201i	$-0.563447\pi$
$167$	10.1082	−	10.1082i	0.782195	−	0.782195i	0.980201	+	0.198006i	$0.0634465\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	−18.9134	−	1.17578i	−1.44634	−	0.0899142i
$172$		0			0
$173$	−17.6899	+	4.73998i	−1.34493	+	0.360374i	−0.858263	−	0.513210i	$-0.828456\pi$
$173$	−17.6899	+	4.73998i	−1.34493	+	0.360374i	−0.486672	+	0.873585i	$0.661789\pi$
$174$		0			0
$175$	−1.31506	−	13.1632i	−0.0994092	−	0.995047i
$176$		0			0
$177$	7.17338	−	6.74128i	0.539184	−	0.506706i
$178$		0			0
$179$	−3.63325	−	6.29297i	−0.271562	−	0.470359i	0.697700	−	0.716390i	$-0.254207\pi$
$179$	−3.63325	−	6.29297i	−0.271562	−	0.470359i	−0.969262	+	0.246031i	$0.920873\pi$
$180$		0			0
$181$	16.2665			1.20908			0.604540	−	0.796575i	$-0.293357\pi$
$181$	16.2665			1.20908			0.604540	+	0.796575i	$0.293357\pi$
$182$		0			0
$183$	4.44987	+	1.34169i	0.328944	+	0.0991804i
$184$		0			0
$185$	2.53590	+	12.3923i	0.186443	+	0.911100i
$186$		0			0
$187$	25.4535	−	6.82024i	1.86134	−	0.498746i
$188$		0			0
$189$	−8.41867	+	10.8686i	−0.612368	+	0.790573i
$190$		0			0
$191$	−3.73831	−	2.15831i	−0.270494	−	0.156170i	0.358618	−	0.933484i	$-0.383248\pi$
$191$	−3.73831	−	2.15831i	−0.270494	−	0.156170i	−0.629112	+	0.777314i	$0.716581\pi$
$192$		0			0
$193$	−8.12768	−	2.17781i	−0.585043	−	0.156762i	−0.0458564	−	0.998948i	$-0.514602\pi$
$193$	−8.12768	−	2.17781i	−0.585043	−	0.156762i	−0.539187	+	0.842186i	$0.681268\pi$
$194$		0			0
$195$	−6.31662	+	8.94987i	−0.452343	+	0.640914i
$196$		0			0
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.0712470	+	0.997459i	$0.522698\pi$
$197$	−15.0000	+	15.0000i	−1.06871	+	1.06871i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−20.1494	+	11.6332i	−1.42835	+	0.824659i	−0.996991	−	0.0775203i	$-0.975300\pi$
$199$	−20.1494	+	11.6332i	−1.42835	+	0.824659i	−0.431361	+	0.902179i	$0.641966\pi$
$200$		0			0
$201$	−9.40564	−	0.292077i	−0.663423	−	0.0206015i
$202$		0			0
$203$	−0.393643	−	1.76467i	−0.0276284	−	0.123855i
$204$		0			0
$205$	−8.10960	−	0.486763i	−0.566399	−	0.0339970i
$206$		0			0
$207$	−6.86421	+	6.06071i	−0.477096	+	0.421249i
$208$		0			0
$209$	−27.2665			−1.88606
$210$		0			0
$211$	14.9499			1.02919			0.514596	−	0.857433i	$-0.327942\pi$
$211$	14.9499			1.02919			0.514596	+	0.857433i	$0.327942\pi$
$212$		0			0
$213$	7.27844	−	1.71000i	0.498710	−	0.117167i
$214$		0			0
$215$	10.8225	+	12.2047i	0.738091	+	0.832354i
$216$		0			0
$217$	−11.6899	+	10.7400i	−0.793559	+	0.729078i
$218$		0			0
$219$	−0.504314	+	16.2402i	−0.0340784	+	1.09741i
$220$		0			0
$221$	14.9532	−	8.63325i	1.00586	−	0.580735i
$222$		0			0
$223$	4.68338	−	4.68338i	0.313622	−	0.313622i	−0.532689	−	0.846311i	$-0.678818\pi$
$223$	4.68338	−	4.68338i	0.313622	−	0.313622i	0.846311	+	0.532689i	$0.178818\pi$
$224$		0			0
$225$	14.1332	−	5.02506i	0.942217	−	0.335004i
$226$		0			0
$227$	8.69714	+	2.33039i	0.577250	+	0.154674i	0.535619	−	0.844460i	$-0.320078\pi$
$227$	8.69714	+	2.33039i	0.577250	+	0.154674i	0.0416302	+	0.999133i	$0.486745\pi$
$228$		0			0
$229$	−6.92820	−	4.00000i	−0.457829	−	0.264327i	0.253302	−	0.967387i	$-0.418483\pi$
$229$	−6.92820	−	4.00000i	−0.457829	−	0.264327i	−0.711131	+	0.703060i	$0.751817\pi$
$230$		0			0
$231$	−11.1197	+	16.3600i	−0.731626	+	1.07641i
$232$		0			0
$233$	7.76363	−	2.08026i	0.508613	−	0.136282i	0.00461835	−	0.999989i	$-0.498530\pi$
$233$	7.76363	−	2.08026i	0.508613	−	0.136282i	0.503994	+	0.863707i	$0.331863\pi$
$234$		0			0
$235$	15.4904	−	3.16987i	1.01048	−	0.206780i
$236$		0			0
$237$	−6.15831	+	20.4248i	−0.400025	+	1.32673i
$238$		0			0
$239$	24.5330			1.58691			0.793454	−	0.608630i	$-0.208281\pi$
$239$	24.5330			1.58691			0.793454	+	0.608630i	$0.208281\pi$
$240$		0			0
$241$	−5.31662	−	9.20866i	−0.342474	−	0.593182i	0.642418	−	0.766355i	$-0.277931\pi$
$241$	−5.31662	−	9.20866i	−0.342474	−	0.593182i	−0.984891	+	0.173173i	$0.944598\pi$
$242$		0			0
$243$	−14.2525	−	6.31386i	−0.914302	−	0.405034i
$244$		0			0
$245$	−5.79130	−	14.5417i	−0.369992	−	0.929035i
$246$		0			0
$247$	−17.2573	+	4.62409i	−1.09806	+	0.294224i
$248$		0			0
$249$	9.23420	−	14.9056i	0.585194	−	0.944607i
$250$		0			0
$251$		−	17.2665i		−	1.08985i	−0.838484	−	0.544926i	$-0.816558\pi$
$251$		−	17.2665i		−	1.08985i	0.838484	−	0.544926i	$-0.183442\pi$
$252$		0			0
$253$	−9.31662	+	9.31662i	−0.585731	+	0.585731i
$254$		0			0
$255$	−23.5453	−	2.14843i	−1.47446	−	0.134540i
$256$		0			0
$257$	0.500265	+	1.86702i	0.0312057	+	0.116461i	0.979772	−	0.200118i	$-0.0641325\pi$
$257$	0.500265	+	1.86702i	0.0312057	+	0.116461i	−0.948566	+	0.316579i	$0.897466\pi$
$258$		0			0
$259$	6.92820	+	13.2665i	0.430498	+	0.824340i
$260$		0			0
$261$	1.83564	−	0.912926i	0.113623	−	0.0565087i
$262$		0			0
$263$	2.62012	−	9.77844i	0.161564	−	0.602964i	−0.836890	−	0.547371i	$-0.815628\pi$
$263$	2.62012	−	9.77844i	0.161564	−	0.602964i	0.998454	−	0.0555927i	$-0.0177048\pi$
$264$		0			0
$265$	2.05013	+	0.683375i	0.125938	+	0.0419794i
$266$		0			0
$267$	27.5000	+	8.29156i	1.68297	+	0.507435i
$268$		0			0
$269$	11.1332	+	19.2834i	0.678806	+	1.17573i	0.975341	+	0.220705i	$0.0708357\pi$
$269$	11.1332	+	19.2834i	0.678806	+	1.17573i	−0.296534	+	0.955022i	$0.595831\pi$
$270$		0			0
$271$	9.10819	−	15.7758i	0.553283	−	0.958314i	−0.444752	−	0.895654i	$-0.646708\pi$
$271$	9.10819	−	15.7758i	0.553283	−	0.958314i	0.998035	−	0.0626604i	$-0.0199585\pi$
$272$		0			0
$273$	−4.26337	+	12.2402i	−0.258031	+	0.740814i
$274$		0			0
$275$	19.8482	−	8.47829i	1.19689	−	0.511260i
$276$		0			0
$277$	0.115893	−	0.432518i	0.00696332	−	0.0259875i	−0.962357	−	0.271789i	$-0.912385\pi$
$277$	0.115893	−	0.432518i	0.00696332	−	0.0259875i	0.969320	+	0.245802i	$0.0790513\pi$
$278$		0			0
$279$	−15.0000	−	9.94987i	−0.898027	−	0.595683i
$280$		0			0
$281$			11.3668i			0.678083i	0.940771	+	0.339042i	$0.110103\pi$
$281$			11.3668i			0.678083i	−0.940771	+	0.339042i	$0.889897\pi$
$282$		0			0
$283$	−5.96509	−	1.59834i	−0.354588	−	0.0950115i	0.0771279	−	0.997021i	$-0.475425\pi$
$283$	−5.96509	−	1.59834i	−0.354588	−	0.0950115i	−0.431716	+	0.902010i	$0.642092\pi$
$284$		0			0
$285$	22.9575	+	8.45289i	1.35988	+	0.500706i
$286$		0			0
$287$	−9.38208	+	2.09285i	−0.553807	+	0.123537i
$288$		0			0
$289$	17.5513	+	10.1332i	1.03243	+	0.596074i
$290$		0			0
$291$	−3.46155	+	5.58755i	−0.202919	+	0.327548i
$292$		0			0
$293$	18.6332	+	18.6332i	1.08857	+	1.08857i	0.995676	+	0.0928897i	$0.0296104\pi$
$293$	18.6332	+	18.6332i	1.08857	+	1.08857i	0.0928897	+	0.995676i	$0.470390\pi$
$294$		0			0
$295$	−11.3668	+	5.68338i	−0.661798	+	0.330899i
$296$		0			0
$297$	−21.0318	−	7.79491i	−1.22039	−	0.452307i
$298$		0			0
$299$	−4.31662	+	7.47661i	−0.249637	+	0.432384i
$300$		0			0
$301$	16.2916	+	10.3489i	0.939030	+	0.596500i
$302$		0			0
$303$	−11.0508	−	11.7592i	−0.634853	−	0.675546i
$304$		0			0
$305$	−5.00725	−	3.30605i	−0.286714	−	0.189304i
$306$		0			0
$307$	−0.474937	−	0.474937i	−0.0271061	−	0.0271061i	0.693424	−	0.720530i	$-0.256101\pi$
$307$	−0.474937	−	0.474937i	−0.0271061	−	0.0271061i	−0.720530	+	0.693424i	$0.756101\pi$
$308$		0			0
$309$	28.2916	−	15.1834i	1.60945	−	0.863752i
$310$		0			0
$311$	3.73831	−	2.15831i	0.211980	−	0.122387i	−0.390251	−	0.920708i	$-0.627612\pi$
$311$	3.73831	−	2.15831i	0.211980	−	0.122387i	0.602231	+	0.798322i	$0.294279\pi$
$312$		0			0
$313$	4.00793	+	14.9578i	0.226542	+	0.845465i	0.981781	+	0.190016i	$0.0608540\pi$
$313$	4.00793	+	14.9578i	0.226542	+	0.845465i	−0.755239	+	0.655449i	$0.772479\pi$
$314$		0			0
$315$	14.4342	−	10.3273i	0.813276	−	0.581879i
$316$		0			0
$317$	4.73998	+	17.6899i	0.266224	+	0.993561i	0.961497	+	0.274816i	$0.0886168\pi$
$317$	4.73998	+	17.6899i	0.266224	+	0.993561i	−0.695273	+	0.718746i	$0.744717\pi$
$318$		0			0
$319$	2.55467	−	1.47494i	0.143034	−	0.0825806i
$320$		0			0
$321$	−15.4499	+	8.29156i	−0.862328	+	0.462790i
$322$		0			0
$323$	−27.2665	−	27.2665i	−1.51715	−	1.51715i
$324$		0			0
$325$	11.1244	−	8.73205i	0.617068	−	0.484367i
$326$		0			0
$327$	−17.2976	−	18.4064i	−0.956561	−	1.01787i
$328$		0			0
$329$	16.5831	−	8.66025i	0.914257	−	0.477455i
$330$		0			0
$331$	2.31662	−	4.01251i	0.127333	−	0.220548i	−0.795309	−	0.606204i	$-0.792692\pi$
$331$	2.31662	−	4.01251i	0.127333	−	0.220548i	0.922643	+	0.385656i	$0.126025\pi$
$332$		0			0
$333$	−12.7214	+	11.2323i	−0.697131	+	0.615527i
$334$		0			0
$335$	11.5251	+	3.84169i	0.629681	+	0.209894i
$336$		0			0
$337$	−17.6332	−	17.6332i	−0.960544	−	0.960544i	0.0387063	−	0.999251i	$-0.487676\pi$
$337$	−17.6332	−	17.6332i	−0.960544	−	0.960544i	−0.999251	+	0.0387063i	$0.987676\pi$
$338$		0			0
$339$	4.68688	−	7.56546i	0.254556	−	0.410899i
$340$		0			0
$341$	−22.4298	−	12.9499i	−1.21464	−	0.701275i
$342$		0			0
$343$	−11.3321	−	14.6487i	−0.611874	−	0.790955i
$344$		0			0
$345$	10.7325	−	4.95603i	0.577820	−	0.266824i
$346$		0			0
$347$	29.3353	+	7.86037i	1.57480	+	0.421967i	0.937312	−	0.348492i	$-0.113306\pi$
$347$	29.3353	+	7.86037i	1.57480	+	0.421967i	0.637490	+	0.770459i	$0.279973\pi$
$348$		0			0
$349$		−	11.0000i		−	0.588817i	−0.955680	−	0.294408i	$-0.904877\pi$
$349$		−	11.0000i		−	0.588817i	0.955680	−	0.294408i	$-0.0951225\pi$
$350$		0			0
$351$	−14.6332	−	1.36675i	−0.781065	−	0.0729517i
$352$		0			0
$353$	1.32986	−	4.96311i	0.0707814	−	0.264160i	−0.921462	−	0.388468i	$-0.873004\pi$
$353$	1.32986	−	4.96311i	0.0707814	−	0.264160i	0.992244	+	0.124308i	$0.0396711\pi$
$354$		0			0
$355$	−9.63493	−	0.578318i	−0.511369	−	0.0306939i
$356$		0			0
$357$	−27.4797	+	5.24025i	−1.45438	+	0.277343i
$358$		0			0
$359$	1.36675	−	2.36728i	0.0721343	−	0.124940i	−0.827702	−	0.561168i	$-0.810352\pi$
$359$	1.36675	−	2.36728i	0.0721343	−	0.124940i	0.899837	+	0.436227i	$0.143686\pi$
$360$		0			0
$361$	10.4499	+	18.0997i	0.549993	+	0.952616i
$362$		0			0
$363$	−12.6583	−	3.81662i	−0.664389	−	0.200321i
$364$		0			0
$365$	6.63325	−	19.8997i	0.347200	−	1.04160i
$366$		0			0
$367$	3.81575	−	14.2406i	0.199180	−	0.743351i	−0.791965	−	0.610567i	$-0.790942\pi$
$367$	3.81575	−	14.2406i	0.199180	−	0.743351i	0.991145	−	0.132784i	$-0.0423917\pi$
$368$		0			0
$369$	−4.85368	−	9.75942i	−0.252673	−	0.508055i
$370$		0			0
$371$	2.55467	+	0.108187i	0.132632	+	0.00561678i
$372$		0			0
$373$	0.847944	+	3.16457i	0.0439049	+	0.163855i	0.984398	−	0.175959i	$-0.0563026\pi$
$373$	0.847944	+	3.16457i	0.0439049	+	0.163855i	−0.940493	+	0.339814i	$0.889636\pi$
$374$		0			0
$375$	−19.3398	+	0.985297i	−0.998705	+	0.0508805i
$376$		0			0
$377$	1.36675	−	1.36675i	0.0703912	−	0.0703912i
$378$		0			0
$379$			8.21637i			0.422047i	0.977481	+	0.211023i	$0.0676796\pi$
$379$			8.21637i			0.422047i	−0.977481	+	0.211023i	$0.932320\pi$
$380$		0			0
$381$	−9.84687	+	15.8946i	−0.504470	+	0.814305i
$382$		0			0
$383$	18.7712	−	5.02971i	0.959161	−	0.257006i	0.254916	−	0.966963i	$-0.417952\pi$
$383$	18.7712	−	5.02971i	0.959161	−	0.257006i	0.704245	+	0.709957i	$0.251286\pi$
$384$		0			0
$385$	19.8070	−	16.1197i	1.00946	−	0.821538i
$386$		0			0
$387$	−6.96493	+	20.7470i	−0.354047	+	1.05463i
$388$		0			0
$389$	−4.31662	−	7.47661i	−0.218862	−	0.379079i	0.735599	−	0.677418i	$-0.236901\pi$
$389$	−4.31662	−	7.47661i	−0.218862	−	0.379079i	−0.954460	+	0.298338i	$0.903568\pi$
$390$		0			0
$391$	−18.6332			−0.942324
$392$		0			0
$393$	5.00000	−	16.5831i	0.252217	−	0.836508i
$394$		0			0
$395$	15.1747	−	22.9831i	0.763522	−	1.15641i
$396$		0			0
$397$	−26.8195	+	7.18627i	−1.34603	+	0.360668i	−0.858669	−	0.512530i	$-0.828708\pi$
$397$	−26.8195	+	7.18627i	−1.34603	+	0.360668i	−0.487364	+	0.873199i	$0.662041\pi$
$398$		0			0
$399$	28.8685	+	2.12180i	1.44523	+	0.106223i
$400$		0			0
$401$	−34.0492	−	19.6583i	−1.70034	−	0.981689i	−0.945412	−	0.325876i	$-0.894341\pi$
$401$	−34.0492	−	19.6583i	−1.70034	−	0.981689i	−0.754923	−	0.655813i	$-0.772326\pi$
$402$		0			0
$403$	−16.3923	−	4.39230i	−0.816559	−	0.218796i
$404$		0			0
$405$	15.2916	+	13.0831i	0.759844	+	0.650106i
$406$		0			0
$407$	−17.2665	+	17.2665i	−0.855869	+	0.855869i
$408$		0			0
$409$	22.4733	−	12.9749i	1.11123	−	0.641569i	0.172083	−	0.985082i	$-0.444950\pi$
$409$	22.4733	−	12.9749i	1.11123	−	0.641569i	0.939148	+	0.343513i	$0.111617\pi$
$410$		0			0
$411$	0.984556	−	31.7053i	0.0485646	−	1.56391i
$412$		0			0
$413$	−11.0730	+	10.1732i	−0.544865	+	0.500592i
$414$		0			0
$415$	−16.9367	+	15.0187i	−0.831392	+	0.737238i
$416$		0			0
$417$	20.7676	−	4.87915i	1.01699	−	0.238933i
$418$		0			0
$419$	27.2665			1.33206			0.666028	−	0.745927i	$-0.267993\pi$
$419$	27.2665			1.33206			0.666028	+	0.745927i	$0.267993\pi$
$420$		0			0
$421$	14.6834			0.715624			0.357812	−	0.933794i	$-0.383523\pi$
$421$	14.6834			0.715624			0.357812	+	0.933794i	$0.383523\pi$
$422$		0			0
$423$	14.0404	+	15.9018i	0.682667	+	0.773172i
$424$		0			0
$425$	28.3265	+	11.3699i	1.37404	+	0.551520i
$426$		0			0
$427$	−6.77375	−	2.12600i	−0.327805	−	0.102885i
$428$		0			0
$429$	−21.1369	−	0.656371i	−1.02050	−	0.0316899i
$430$		0			0
$431$	−4.92195	+	2.84169i	−0.237082	+	0.136879i	−0.613835	−	0.789435i	$-0.710374\pi$
$431$	−4.92195	+	2.84169i	−0.237082	+	0.136879i	0.376753	+	0.926314i	$0.377041\pi$
$432$		0			0
$433$	5.94987	−	5.94987i	0.285933	−	0.285933i	−0.549537	−	0.835470i	$-0.685196\pi$
$433$	5.94987	−	5.94987i	0.285933	−	0.285933i	0.835470	+	0.549537i	$0.185196\pi$
$434$		0			0
$435$	−2.60819	+	0.449874i	−0.125053	+	0.0215698i
$436$		0			0
$437$	18.6234	+	4.99012i	0.890876	+	0.238710i
$438$		0			0
$439$	−23.5267	−	13.5831i	−1.12287	−	0.648287i	−0.180735	−	0.983532i	$-0.557848\pi$
$439$	−23.5267	−	13.5831i	−1.12287	−	0.648287i	−0.942131	+	0.335245i	$0.891181\pi$
$440$		0			0
$441$	13.0462	−	16.4559i	0.621246	−	0.783615i
$442$		0			0
$443$	−8.84493	+	2.36999i	−0.420235	+	0.112602i	−0.462739	−	0.886495i	$-0.653133\pi$
$443$	−8.84493	+	2.36999i	−0.420235	+	0.112602i	0.0425037	+	0.999096i	$0.486467\pi$
$444$		0			0
$445$	−30.9445	−	20.4313i	−1.46691	−	0.968534i
$446$		0			0
$447$	−6.02506	−	1.81662i	−0.284976	−	0.0859234i
$448$		0			0
$449$	−30.8997			−1.45825			−0.729125	−	0.684381i	$-0.760073\pi$
$449$	−30.8997			−1.45825			−0.729125	+	0.684381i	$0.760073\pi$
$450$		0			0
$451$	−7.84169	−	13.5822i	−0.369251	−	0.639561i
$452$		0			0
$453$	20.9940	−	19.7294i	0.986383	−	0.926967i
$454$		0			0
$455$	9.80240	−	13.5615i	0.459543	−	0.635771i
$456$		0			0
$457$	7.62669	−	2.04357i	0.356761	−	0.0955939i	−0.0759866	−	0.997109i	$-0.524211\pi$
$457$	7.62669	−	2.04357i	0.356761	−	0.0955939i	0.432748	+	0.901515i	$0.357544\pi$
$458$		0			0
$459$	−13.2369	−	28.8267i	−0.617845	−	1.34552i
$460$		0			0
$461$		−	18.6332i		−	0.867837i	−0.900952	−	0.433918i	$-0.857131\pi$
$461$		−	18.6332i		−	0.867837i	0.900952	−	0.433918i	$-0.142869\pi$
$462$		0			0
$463$	5.84169	−	5.84169i	0.271486	−	0.271486i	−0.558212	−	0.829698i	$-0.688512\pi$
$463$	5.84169	−	5.84169i	0.271486	−	0.271486i	0.829698	+	0.558212i	$0.188512\pi$
$464$		0			0
$465$	14.8706	+	17.8568i	0.689606	+	0.828091i
$466$		0			0
$467$	4.20012	+	15.6751i	0.194358	+	0.725355i	0.992432	+	0.122795i	$0.0391858\pi$
$467$	4.20012	+	15.6751i	0.194358	+	0.725355i	−0.798074	+	0.602560i	$0.794147\pi$
$468$		0			0
$469$	14.3614	+	0.608187i	0.663148	+	0.0280835i
$470$		0			0
$471$	3.22351	+	0.100101i	0.148531	+	0.00461240i
$472$		0			0
$473$	−8.15010	+	30.4166i	−0.374742	+	1.39856i
$474$		0			0
$475$	−25.2665	−	18.9499i	−1.15931	−	0.869480i
$476$		0			0
$477$	0.575188	+	2.84169i	0.0263361	+	0.130112i
$478$		0			0
$479$	8.63325	+	14.9532i	0.394463	+	0.683230i	0.993033	−	0.117841i	$-0.0375972\pi$
$479$	8.63325	+	14.9532i	0.394463	+	0.683230i	−0.598569	+	0.801071i	$0.704264\pi$
$480$		0			0
$481$	−8.00000	+	13.8564i	−0.364769	+	0.631798i
$482$		0			0
$483$	10.5890	−	9.13903i	0.481817	−	0.415840i
$484$		0			0
$485$	6.34893	−	5.62993i	0.288290	−	0.255642i
$486$		0			0
$487$	−5.45369	+	20.3534i	−0.247130	+	0.922302i	0.725171	+	0.688569i	$0.241761\pi$
$487$	−5.45369	+	20.3534i	−0.247130	+	0.922302i	−0.972301	+	0.233733i	$0.924906\pi$
$488$		0			0
$489$	−2.15831	+	1.15831i	−0.0976023	+	0.0523807i
$490$		0			0
$491$			35.8997i			1.62013i	0.586338	+	0.810066i	$0.300569\pi$
$491$			35.8997i			1.62013i	−0.586338	+	0.810066i	$0.699431\pi$
$492$		0			0
$493$	4.02960	+	1.07973i	0.181484	+	0.0486285i
$494$		0			0
$495$	23.1283	+	17.4235i	1.03954	+	0.783129i
$496$		0			0
$497$	−11.1468	+	2.48650i	−0.500000	+	0.111535i
$498$		0			0
$499$	30.2675	+	17.4749i	1.35496	+	0.782286i	0.988939	−	0.148322i	$-0.0473871\pi$
$499$	30.2675	+	17.4749i	1.35496	+	0.782286i	0.366019	+	0.930607i	$0.380720\pi$
$500$		0			0
$501$	−21.0481	−	13.0395i	−0.940361	−	0.582563i
$502$		0			0
$503$	−20.7916	−	20.7916i	−0.927050	−	0.927050i	0.0704644	−	0.997514i	$-0.477552\pi$
$503$	−20.7916	−	20.7916i	−0.927050	−	0.927050i	−0.997514	+	0.0704644i	$0.977552\pi$
$504$		0			0
$505$	9.31662	+	18.6332i	0.414584	+	0.829169i
$506$		0			0
$507$	8.43070	−	1.98072i	0.374421	−	0.0879668i
$508$		0			0
$509$	2.60819	−	4.51751i	0.115606	−	0.200235i	−0.802416	−	0.596765i	$-0.796452\pi$
$509$	2.60819	−	4.51751i	0.115606	−	0.200235i	0.918022	+	0.396530i	$0.129786\pi$
$510$		0			0
$511$	1.05013	−	24.7971i	0.0464548	−	1.09696i
$512$		0			0
$513$	5.50987	+	32.3564i	0.243267	+	1.42857i
$514$		0			0
$515$	−40.6102	+	8.31026i	−1.78950	+	0.366194i
$516$		0			0
$517$	21.5831	+	21.5831i	0.949225	+	0.949225i
$518$		0			0
$519$	15.0000	+	27.9499i	0.658427	+	1.22686i
$520$		0			0
$521$	−6.29297	+	3.63325i	−0.275700	+	0.159176i	−0.631475	−	0.775396i	$-0.717550\pi$
$521$	−6.29297	+	3.63325i	−0.275700	+	0.159176i	0.355775	+	0.934572i	$0.384217\pi$
$522$		0			0
$523$	−2.06191	−	7.69516i	−0.0901611	−	0.336486i	0.906080	−	0.423106i	$-0.139060\pi$
$523$	−2.06191	−	7.69516i	−0.0901611	−	0.336486i	−0.996241	+	0.0866198i	$0.972393\pi$
$524$		0			0
$525$	−21.6741	+	7.43190i	−0.945935	+	0.324355i
$526$		0			0
$527$	−9.47997	−	35.3797i	−0.412954	−	1.54116i
$528$		0			0
$529$	−11.8502	+	6.84169i	−0.515224	+	0.297465i
$530$		0			0
$531$	−14.2084	−	9.42481i	−0.616594	−	0.409002i
$532$		0			0
$533$	−7.26650	−	7.26650i	−0.314747	−	0.314747i
$534$		0			0
$535$	22.1770	−	4.53819i	0.958795	−	0.196203i
$536$		0			0
$537$	−9.17155	+	8.61909i	−0.395782	+	0.371941i
$538$		0			0
$539$	17.2665	−	24.7971i	0.743721	−	1.06809i
$540$		0			0
$541$	4.13325	−	7.15900i	0.177702	−	0.307789i	−0.763391	−	0.645937i	$-0.776467\pi$
$541$	4.13325	−	7.15900i	0.177702	−	0.307789i	0.941093	+	0.338147i	$0.109800\pi$
$542$		0			0
$543$	−6.44387	−	27.4276i	−0.276533	−	1.17703i
$544$		0			0
$545$	14.5831	+	29.1662i	0.624672	+	1.24934i
$546$		0			0
$547$	−7.10819	−	7.10819i	−0.303924	−	0.303924i	0.538623	−	0.842547i	$-0.318945\pi$
$547$	−7.10819	−	7.10819i	−0.303924	−	0.303924i	−0.842547	+	0.538623i	$0.818945\pi$
$548$		0			0
$549$	0.499485	−	8.03461i	0.0213175	−	0.342909i
$550$		0			0
$551$	−3.73831	−	2.15831i	−0.159257	−	0.0919472i
$552$		0			0
$553$	9.75830	−	31.0913i	0.414965	−	1.32214i
$554$		0			0
$555$	19.8906	−	9.18501i	0.844309	−	0.389882i
$556$		0			0
$557$	0.933508	+	0.250133i	0.0395540	+	0.0105985i	0.278542	−	0.960424i	$-0.410149\pi$
$557$	0.933508	+	0.250133i	0.0395540	+	0.0105985i	−0.238988	+	0.971023i	$0.576816\pi$
$558$		0			0
$559$			20.6332i			0.872693i
$560$		0			0
$561$	−21.5831	−	40.2164i	−0.911240	−	1.69794i
$562$		0			0
$563$	−10.5201	+	39.2615i	−0.443369	+	1.65468i	0.276838	+	0.960917i	$0.410714\pi$
$563$	−10.5201	+	39.2615i	−0.443369	+	1.65468i	−0.720207	+	0.693760i	$0.755953\pi$
$564$		0			0
$565$	−8.59636	+	7.62283i	−0.361652	+	0.320695i
$566$		0			0
$567$	21.6610	+	9.88954i	0.909675	+	0.415322i
$568$		0			0
$569$	5.68338	−	9.84389i	0.238259	−	0.412678i	−0.721955	−	0.691940i	$-0.756756\pi$
$569$	5.68338	−	9.84389i	0.238259	−	0.412678i	0.960215	+	0.279262i	$0.0900898\pi$
$570$		0			0
$571$	−0.525063	−	0.909435i	−0.0219732	−	0.0380587i	0.854830	−	0.518909i	$-0.173662\pi$
$571$	−0.525063	−	0.909435i	−0.0219732	−	0.0380587i	−0.876803	+	0.480850i	$0.840328\pi$
$572$		0			0
$573$	−2.15831	+	7.15831i	−0.0901648	+	0.299043i
$574$		0			0
$575$	−15.1082	+	2.15831i	−0.630055	+	0.0900078i
$576$		0			0
$577$	−6.95448	+	25.9545i	−0.289519	+	1.08050i	0.655955	+	0.754800i	$0.272266\pi$
$577$	−6.95448	+	25.9545i	−0.289519	+	1.08050i	−0.945474	+	0.325699i	$0.894400\pi$
$578$		0			0
$579$	−0.452358	+	14.5671i	−0.0187994	+	0.605389i
$580$		0			0
$581$	−14.3614	+	22.6082i	−0.595812	+	0.937946i
$582$		0			0
$583$	1.07973	+	4.02960i	0.0447178	+	0.166889i
$584$		0			0
$585$	17.5930	+	7.10528i	0.727383	+	0.293767i
$586$		0			0
$587$	−13.6332	+	13.6332i	−0.562704	+	0.562704i	−0.930075	−	0.367370i	$-0.880258\pi$
$587$	−13.6332	+	13.6332i	−0.562704	+	0.562704i	0.367370	+	0.930075i	$0.380258\pi$
$588$		0			0
$589$			37.8997i			1.56163i
$590$		0			0
$591$	31.2343	+	19.3500i	1.28481	+	0.795951i
$592$		0			0
$593$	8.99271	−	2.40959i	0.369287	−	0.0989500i	−0.0694030	−	0.997589i	$-0.522109\pi$
$593$	8.99271	−	2.40959i	0.369287	−	0.0989500i	0.438690	+	0.898639i	$0.355443\pi$
$594$		0			0
$595$	35.9268	+	3.68726i	1.47285	+	0.151163i
$596$		0			0
$597$	27.5973	+	29.3663i	1.12948	+	1.20188i
$598$		0			0
$599$	−12.2665	−	21.2462i	−0.501196	−	0.868096i	−0.999999	−	0.00138107i	$-0.999560\pi$
$599$	−12.2665	−	21.2462i	−0.501196	−	0.868096i	0.498803	−	0.866715i	$-0.333773\pi$
$600$		0			0
$601$	48.5330			1.97970			0.989851	−	0.142108i	$-0.0453881\pi$
$601$	48.5330			1.97970			0.989851	+	0.142108i	$0.0453881\pi$
$602$		0			0
$603$	3.23350	+	15.9749i	0.131678	+	0.650550i
$604$		0			0
$605$	14.2438	+	9.40455i	0.579094	+	0.382349i
$606$		0			0
$607$	15.2425	−	4.08423i	0.618676	−	0.165774i	0.0641499	−	0.997940i	$-0.479566\pi$
$607$	15.2425	−	4.08423i	0.618676	−	0.165774i	0.554526	+	0.832167i	$0.312900\pi$
$608$		0			0
$609$	−2.81954	+	1.36280i	−0.114254	+	0.0552235i
$610$		0			0
$611$	17.3205	+	10.0000i	0.700713	+	0.404557i
$612$		0			0
$613$	−15.3903	−	4.12382i	−0.621609	−	0.166560i	−0.0657501	−	0.997836i	$-0.520944\pi$
$613$	−15.3903	−	4.12382i	−0.621609	−	0.166560i	−0.555859	+	0.831276i	$0.687611\pi$
$614$		0			0
$615$	2.39181	+	13.8668i	0.0964472	+	0.559161i
$616$		0			0
$617$	−31.5831	+	31.5831i	−1.27149	+	1.27149i	−0.326182	+	0.945307i	$0.605762\pi$
$617$	−31.5831	+	31.5831i	−1.27149	+	1.27149i	−0.945307	+	0.326182i	$0.894238\pi$
$618$		0			0
$619$	−10.5797	+	6.10819i	−0.425234	+	0.245509i	−0.697314	−	0.716766i	$-0.745622\pi$
$619$	−10.5797	+	6.10819i	−0.425234	+	0.245509i	0.272080	+	0.962275i	$0.412288\pi$
$620$		0			0
$621$	12.9384	+	9.17311i	0.519201	+	0.368104i
$622$		0			0
$623$	−41.8614	−	13.1386i	−1.67714	−	0.526387i
$624$		0			0
$625$	24.2846	+	5.93782i	0.971384	+	0.237513i
$626$		0			0
$627$	10.8014	+	45.9752i	0.431368	+	1.83607i
$628$		0			0
$629$	−34.5330			−1.37692
$630$		0			0
$631$	−2.31662			−0.0922234			−0.0461117	−	0.998936i	$-0.514683\pi$
$631$	−2.31662			−0.0922234			−0.0461117	+	0.998936i	$0.514683\pi$
$632$		0			0
$633$	−5.92230	−	25.2076i	−0.235390	−	1.00191i
$634$		0			0
$635$	18.0605	−	16.0151i	0.716708	−	0.635541i
$636$		0			0
$637$	6.72289	−	18.6226i	0.266371	−	0.737856i
$638$		0			0
$639$	−5.76661	−	11.5951i	−0.228124	−	0.458694i
$640$		0			0
$641$	35.4202	−	20.4499i	1.39901	−	0.807721i	0.404725	−	0.914438i	$-0.367367\pi$
$641$	35.4202	−	20.4499i	1.39901	−	0.807721i	0.994289	+	0.106717i	$0.0340339\pi$
$642$		0			0
$643$	−22.8997	+	22.8997i	−0.903078	+	0.903078i	−0.995701	−	0.0926233i	$-0.970475\pi$
$643$	−22.8997	+	22.8997i	−0.903078	+	0.903078i	0.0926233	+	0.995701i	$0.470475\pi$
$644$		0			0
$645$	16.2916	−	23.0831i	0.641480	−	0.908897i
$646$		0			0
$647$	4.81533	+	1.29026i	0.189310	+	0.0507255i	0.352228	−	0.935914i	$-0.385424\pi$
$647$	4.81533	+	1.29026i	0.189310	+	0.0507255i	−0.162918	+	0.986640i	$0.552091\pi$
$648$		0			0
$649$	−21.2462	−	12.2665i	−0.833986	−	0.481502i
$650$		0			0
$651$	22.7400	+	15.4562i	0.891250	+	0.605775i
$652$		0			0
$653$	8.05921	−	2.15946i	0.315381	−	0.0845061i	−0.0976562	−	0.995220i	$-0.531135\pi$
$653$	8.05921	−	2.15946i	0.315381	−	0.0845061i	0.413037	+	0.910714i	$0.364468\pi$
$654$		0			0
$655$	−12.3205	+	18.6603i	−0.481402	+	0.729116i
$656$		0			0
$657$	27.5831	−	5.58312i	1.07612	−	0.217818i
$658$		0			0
$659$	−22.9499			−0.894000			−0.447000	−	0.894534i	$-0.647508\pi$
$659$	−22.9499			−0.894000			−0.447000	+	0.894534i	$0.647508\pi$
$660$		0			0
$661$	2.65831	+	4.60433i	0.103396	+	0.179088i	0.913082	−	0.407776i	$-0.133696\pi$
$661$	2.65831	+	4.60433i	0.103396	+	0.179088i	−0.809686	+	0.586864i	$0.800362\pi$
$662$		0			0
$663$	−20.4805	−	21.7932i	−0.795397	−	0.846379i
$664$		0			0
$665$	−34.9202	−	13.3067i	−1.35415	−	0.516013i
$666$		0			0
$667$	−2.01480	+	0.539864i	−0.0780134	+	0.0209036i
$668$		0			0
$669$	−9.75212	−	6.04154i	−0.377039	−	0.233579i
$670$		0			0
$671$		−	11.5831i		−	0.447162i
$672$		0			0
$673$	−19.2665	+	19.2665i	−0.742669	+	0.742669i	−0.973091	−	0.230422i	$-0.925989\pi$
$673$	−19.2665	+	19.2665i	−0.742669	+	0.742669i	0.230422	+	0.973091i	$0.425989\pi$
$674$		0			0
$675$	−14.0718	−	21.8400i	−0.541622	−	0.840622i
$676$		0			0
$677$	−8.97970	−	33.5127i	−0.345118	−	1.28800i	−0.892474	−	0.451098i	$-0.851032\pi$
$677$	−8.97970	−	33.5127i	−0.345118	−	1.28800i	0.547356	−	0.836900i	$-0.315634\pi$
$678$		0			0
$679$	5.38354	−	8.47494i	0.206601	−	0.325238i
$680$		0			0
$681$	0.484053	−	15.5878i	0.0185489	−	0.597325i
$682$		0			0
$683$	−11.2705	+	42.0621i	−0.431253	+	1.60946i	0.318623	+	0.947882i	$0.396780\pi$
$683$	−11.2705	+	42.0621i	−0.431253	+	1.60946i	−0.749876	+	0.661578i	$0.769887\pi$
$684$		0			0
$685$	−12.9499	+	38.8496i	−0.494789	+	1.48437i
$686$		0			0
$687$	−4.00000	+	13.2665i	−0.152610	+	0.506149i
$688$		0			0
$689$	1.36675	+	2.36728i	0.0520690	+	0.0901862i
$690$		0			0
$691$	7.42481	−	12.8602i	0.282453	−	0.489223i	−0.689535	−	0.724252i	$-0.742185\pi$
$691$	7.42481	−	12.8602i	0.282453	−	0.489223i	0.971988	+	0.235029i	$0.0755185\pi$
$692$		0			0
$693$	31.9903	+	12.2686i	1.21521	+	0.466044i
$694$		0			0
$695$	−27.4913	−	1.65011i	−1.04281	−	0.0625924i
$696$		0			0
$697$	5.74051	−	21.4239i	0.217437	−	0.811488i
$698$		0			0
$699$	−6.58312	−	12.2665i	−0.248997	−	0.463962i
$700$		0			0
$701$		−	6.36675i		−	0.240469i	−0.992746	−	0.120234i	$-0.961635\pi$
$701$		−	6.36675i		−	0.240469i	0.992746	−	0.120234i	$-0.0383646\pi$
$702$		0			0
$703$	34.5147	+	9.24818i	1.30175	+	0.348802i
$704$		0			0
$705$	−11.4813	−	24.8632i	−0.432409	−	0.936404i
$706$		0			0
$707$	16.6767	+	18.1517i	0.627193	+	0.682664i
$708$		0			0
$709$	−37.7007	−	21.7665i	−1.41588	−	0.817458i	−0.419945	−	0.907549i	$-0.637951\pi$
$709$	−37.7007	−	21.7665i	−1.41588	−	0.817458i	−0.995934	+	0.0900914i	$0.971284\pi$
$710$		0			0
$711$	36.8787	+	2.29262i	1.38306	+	0.0859801i
$712$		0			0
$713$	12.9499	+	12.9499i	0.484977	+	0.484977i
$714$		0			0
$715$	25.8997	+	8.63325i	0.968596	+	0.322865i
$716$		0			0
$717$	−9.71859	−	41.3661i	−0.362947	−	1.54484i
$718$		0			0
$719$	−10.7916	+	18.6915i	−0.402457	+	0.697077i	−0.994022	−	0.109181i	$-0.965177\pi$
$719$	−10.7916	+	18.6915i	−0.402457	+	0.697077i	0.591565	+	0.806258i	$0.298510\pi$
$720$		0			0
$721$	−43.4749	+	22.7040i	−1.61909	+	0.845543i
$722$		0			0
$723$	−13.4210	+	12.6125i	−0.499131	+	0.469065i
$724$		0			0
$725$	3.39234	+	0.408710i	0.125988	+	0.0151791i
$726$		0			0
$727$	−24.7916	−	24.7916i	−0.919468	−	0.919468i	0.0775225	−	0.996991i	$-0.475299\pi$
$727$	−24.7916	−	24.7916i	−0.919468	−	0.919468i	−0.996991	+	0.0775225i	$0.975299\pi$
$728$		0			0
$729$	−5.00000	+	26.5330i	−0.185185	+	0.982704i
$730$		0			0
$731$	−38.5667	+	22.2665i	−1.42644	+	0.823556i
$732$		0			0
$733$	−5.22190	−	19.4884i	−0.192875	−	0.719820i	−0.992807	−	0.119729i	$-0.961797\pi$
$733$	−5.22190	−	19.4884i	−0.192875	−	0.719820i	0.799931	−	0.600092i	$-0.204869\pi$
$734$		0			0
$735$	−22.2252	+	15.5255i	−0.819787	+	0.572668i
$736$		0			0
$737$	6.06984	+	22.6530i	0.223586	+	0.834433i
$738$		0			0
$739$	0.822615	−	0.474937i	0.0302604	−	0.0174708i	−0.484793	−	0.874629i	$-0.661105\pi$
$739$	0.822615	−	0.474937i	0.0302604	−	0.0174708i	0.515054	+	0.857158i	$0.327772\pi$
$740$		0			0
$741$	14.6332	+	27.2665i	0.537566	+	1.00166i
$742$		0			0
$743$	2.84169	+	2.84169i	0.104251	+	0.104251i	0.757309	−	0.653057i	$-0.226514\pi$
$743$	2.84169	+	2.84169i	0.104251	+	0.104251i	−0.653057	+	0.757309i	$0.726514\pi$
$744$		0			0
$745$	6.77974	+	4.47635i	0.248390	+	0.164001i
$746$		0			0
$747$	−28.7911	−	9.66539i	−1.05341	−	0.353638i
$748$		0			0
$749$	23.7414	−	12.3986i	0.867493	−	0.453034i
$750$		0			0
$751$	0.0501256	−	0.0868201i	0.00182911	−	0.00316811i	−0.865109	−	0.501583i	$-0.832751\pi$
$751$	0.0501256	−	0.0868201i	0.00182911	−	0.00316811i	0.866939	+	0.498415i	$0.166084\pi$
$752$		0			0
$753$	−29.1137	+	6.84001i	−1.06096	+	0.249264i
$754$		0			0
$755$	−33.2665	+	16.6332i	−1.21069	+	0.605346i
$756$		0			0
$757$	−7.68338	−	7.68338i	−0.279257	−	0.279257i	0.553555	−	0.832812i	$-0.313271\pi$
$757$	−7.68338	−	7.68338i	−0.279257	−	0.279257i	−0.832812	+	0.553555i	$0.813271\pi$
$758$		0			0
$759$	19.3999	+	12.0184i	0.704170	+	0.436241i
$760$		0			0
$761$	−17.3205	−	10.0000i	−0.627868	−	0.362500i	0.152058	−	0.988372i	$-0.451410\pi$
$761$	−17.3205	−	10.0000i	−0.627868	−	0.362500i	−0.779926	+	0.625872i	$0.784743\pi$
$762$		0			0
$763$	26.1038	+	28.4124i	0.945019	+	1.02860i
$764$		0			0
$765$	5.70477	+	40.5518i	0.206256	+	1.46615i
$766$		0			0
$767$	−15.5273	−	4.16052i	−0.560657	−	0.150228i
$768$		0			0
$769$		−	6.53300i		−	0.235586i	−0.993038	−	0.117793i	$-0.962418\pi$
$769$		−	6.53300i		−	0.235586i	0.993038	−	0.117793i	$-0.0375819\pi$
$770$		0			0
$771$	2.94987	−	1.58312i	0.106237	−	0.0570148i
$772$		0			0
$773$	2.58053	−	9.63065i	0.0928150	−	0.346390i	−0.903864	−	0.427820i	$-0.859282\pi$
$773$	2.58053	−	9.63065i	0.0928150	−	0.346390i	0.996679	+	0.0814293i	$0.0259485\pi$
$774$		0			0
$775$	−11.7846	−	27.5885i	−0.423316	−	0.991007i
$776$		0			0
$777$	19.6246	−	16.9374i	0.704029	−	0.607624i
$778$		0			0
$779$	−11.4749	+	19.8752i	−0.411132	+	0.712102i
$780$		0			0
$781$	−9.31662	−	16.1369i	−0.333375	−	0.577423i
$782$		0			0
$783$	−2.26650	−	2.73350i	−0.0809981	−	0.0976874i
$784$		0			0
$785$	−3.94987	−	1.31662i	−0.140977	−	0.0469924i
$786$		0			0
$787$	−2.02231	+	7.54738i	−0.0720877	+	0.269035i	−0.992557	−	0.121780i	$-0.961140\pi$
$787$	−2.02231	+	7.54738i	−0.0720877	+	0.269035i	0.920469	+	0.390814i	$0.127807\pi$
$788$		0			0
$789$	−17.5258	−	0.544234i	−0.623934	−	0.0193752i
$790$		0			0
$791$	−7.28923	+	11.4749i	−0.259175	+	0.408002i
$792$		0			0
$793$	−1.96437	−	7.33112i	−0.0697567	−	0.260336i
$794$		0			0
$795$	0.340123	−	3.72751i	0.0120629	−	0.132201i
$796$		0			0
$797$	28.6332	−	28.6332i	1.01424	−	1.01424i	0.0143446	−	0.999897i	$-0.495434\pi$
$797$	28.6332	−	28.6332i	1.01424	−	1.01424i	0.999897	−	0.0143446i	$-0.00456619\pi$
$798$		0			0
$799$			43.1662i			1.52711i
$800$		0			0
$801$	3.08679	−	49.6535i	0.109066	−	1.75442i
$802$		0			0
$803$	39.1137	−	10.4805i	1.38029	−	0.369849i
$804$		0			0
$805$	−16.4786	+	7.38506i	−0.580793	+	0.260289i
$806$		0			0
$807$	28.1041	−	26.4112i	0.989311	−	0.929718i
$808$		0			0
$809$	−22.6082	−	39.1585i	−0.794862	−	1.37674i	−0.922927	−	0.384975i	$-0.874210\pi$
$809$	−22.6082	−	39.1585i	−0.794862	−	1.37674i	0.128066	−	0.991766i	$-0.459123\pi$
$810$		0			0
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$	−30.2084	−	9.10819i	−1.05946	−	0.319438i
$814$		0			0
$815$	3.09808	−	0.633975i	0.108521	−	0.0222072i
$816$		0			0
$817$	44.5094	−	11.9263i	1.55719	−	0.417247i
$818$		0			0
$819$	22.3277	+	2.33975i	0.780193	+	0.0817573i
$820$		0			0
$821$	−14.9532	−	8.63325i	−0.521871	−	0.301302i	0.215829	−	0.976431i	$-0.430755\pi$
$821$	−14.9532	−	8.63325i	−0.521871	−	0.301302i	−0.737700	+	0.675129i	$0.764088\pi$
$822$		0			0
$823$	51.1917	+	13.7168i	1.78443	+	0.478137i	0.991380	−	0.131018i	$-0.0418247\pi$
$823$	51.1917	+	13.7168i	1.78443	+	0.478137i	0.793051	+	0.609155i	$0.208491\pi$
$824$		0			0
$825$	−22.1583	−	30.1082i	−0.771453	−	1.04823i
$826$		0			0
$827$	16.4749	−	16.4749i	0.572890	−	0.572890i	−0.360045	−	0.932935i	$-0.617239\pi$
$827$	16.4749	−	16.4749i	0.572890	−	0.572890i	0.932935	+	0.360045i	$0.117239\pi$
$828$		0			0
$829$	43.1276	−	24.8997i	1.49788	−	0.864804i	0.497887	−	0.867242i	$-0.334109\pi$
$829$	43.1276	−	24.8997i	1.49788	−	0.864804i	0.999997	+	0.00243762i	$0.000775920\pi$
$830$		0			0
$831$	−0.775196	−	0.0240724i	−0.0268912	−	0.000835063i
$832$		0			0
$833$	42.0636	−	7.53062i	1.45742	−	0.260921i
$834$		0			0
$835$	21.2077	+	23.9162i	0.733924	+	0.827655i
$836$		0			0
$837$	−10.8347	+	29.2337i	−0.374503	+	1.01046i
$838$		0			0
$839$	27.0501			0.933874			0.466937	−	0.884291i	$-0.345357\pi$
$839$	27.0501			0.933874			0.466937	+	0.884291i	$0.345357\pi$
$840$		0			0
$841$	−28.5330			−0.983896
$842$		0			0
$843$	19.1659	−	4.50286i	0.660110	−	0.155087i
$844$		0			0
$845$	−11.1603	−	0.669873i	−0.383924	−	0.0230443i
$846$		0			0
$847$	19.2689	+	6.04772i	0.662087	+	0.207802i
$848$		0			0
$849$	−0.331996	+	10.6912i	−0.0113941	+	0.366920i
$850$		0			0
$851$	14.9532	−	8.63325i	0.512590	−	0.295944i
$852$		0			0
$853$	−5.63325	+	5.63325i	−0.192879	+	0.192879i	−0.796939	−	0.604060i	$-0.793549\pi$
$853$	−5.63325	+	5.63325i	−0.192879	+	0.192879i	0.604060	+	0.796939i	$0.293549\pi$
$854$		0			0
$855$	5.15831	−	42.0581i	0.176411	−	1.43836i
$856$		0			0
$857$	−14.5938	−	3.91039i	−0.498513	−	0.133576i	0.000796575	−	1.00000i	$-0.499746\pi$
$857$	−14.5938	−	3.91039i	−0.498513	−	0.133576i	−0.499310	+	0.866423i	$0.666413\pi$
$858$		0			0
$859$	−38.1051	−	22.0000i	−1.30013	−	0.750630i	−0.319704	−	0.947518i	$-0.603583\pi$
$859$	−38.1051	−	22.0000i	−1.30013	−	0.750630i	−0.980426	+	0.196887i	$0.936917\pi$
$860$		0			0
$861$	7.24550	+	14.9904i	0.246926	+	0.510873i
$862$		0			0
$863$	1.71923	−	0.460666i	0.0585233	−	0.0156813i	−0.229439	−	0.973323i	$-0.573689\pi$
$863$	1.71923	−	0.460666i	0.0585233	−	0.0156813i	0.287962	+	0.957642i	$0.407022\pi$
$864$		0			0
$865$	−8.20989	−	40.1197i	−0.279145	−	1.36411i
$866$		0			0
$867$	10.1332	−	33.6082i	0.344143	−	1.14139i
$868$		0			0
$869$	53.1662			1.80354
$870$		0			0
$871$	7.68338	+	13.3080i	0.260341	+	0.450924i
$872$		0			0
$873$	10.7927	+	3.62318i	0.365276	+	0.122626i
$874$		0			0
$875$	29.5572	−	1.17174i	0.999215	−	0.0396121i
$876$		0			0
$877$	−35.4482	+	9.49831i	−1.19700	+	0.320735i	−0.801649	−	0.597795i	$-0.796044\pi$
$877$	−35.4482	+	9.49831i	−1.19700	+	0.320735i	−0.395351	+	0.918530i	$0.629377\pi$
$878$		0			0
$879$	24.0368	−	38.7997i	0.810742	−	1.30868i
$880$		0			0
$881$			19.5330i			0.658083i	0.944315	+	0.329042i	$0.106726\pi$
$881$			19.5330i			0.658083i	−0.944315	+	0.329042i	$0.893274\pi$
$882$		0			0
$883$	20.2665	−	20.2665i	0.682022	−	0.682022i	−0.278434	−	0.960456i	$-0.589815\pi$
$883$	20.2665	−	20.2665i	0.682022	−	0.682022i	0.960456	+	0.278434i	$0.0898152\pi$
$884$		0			0
$885$	14.0858	+	16.9145i	0.473490	+	0.568575i
$886$		0			0
$887$	−2.36999	−	8.84493i	−0.0795765	−	0.296984i	0.914655	−	0.404234i	$-0.132462\pi$
$887$	−2.36999	−	8.84493i	−0.0795765	−	0.296984i	−0.994232	+	0.107251i	$0.965795\pi$
$888$		0			0
$889$	15.3143	−	24.1082i	0.513624	−	0.808563i
$890$		0			0
$891$	−4.81171	+	38.5505i	−0.161198	+	1.29149i
$892$		0			0
$893$	11.5602	−	43.1433i	0.386848	−	1.44374i
$894$		0			0
$895$	14.5330	−	7.26650i	0.485785	−	0.242892i
$896$		0			0
$897$	14.3166	+	4.31662i	0.478018	+	0.144128i
$898$		0			0
$899$	−2.05013	−	3.55092i	−0.0683755	−	0.118430i
$900$		0			0
$901$	−2.94987	+	5.10933i	−0.0982746	+	0.170217i
$902$		0			0
$903$	10.9959	−	31.5695i	0.365921	−	1.05057i
$904$		0			0
$905$	−2.17930	+	36.3077i	−0.0724423	+	1.20691i
$906$		0			0
$907$	−8.32394	+	31.0654i	−0.276392	+	1.03151i	0.678511	+	0.734590i	$0.262626\pi$
$907$	−8.32394	+	31.0654i	−0.276392	+	1.03151i	−0.954903	+	0.296919i	$0.904041\pi$
$908$		0			0
$909$	−15.4499	+	23.2916i	−0.512440	+	0.772532i
$910$		0			0
$911$		−	7.05013i		−	0.233581i	−0.993157	−	0.116791i	$-0.962739\pi$
$911$		−	7.05013i		−	0.233581i	0.993157	−	0.116791i	$-0.0372606\pi$
$912$		0			0
$913$	−42.2098	−	11.3101i	−1.39694	−	0.374309i
$914$		0			0
$915$	−3.59089	+	9.75259i	−0.118711	+	0.322411i
$916$		0			0
$917$	−7.92287	+	25.2434i	−0.261636	+	0.833610i
$918$		0			0
$919$	−7.11559	−	4.10819i	−0.234722	−	0.135517i	0.378027	−	0.925795i	$-0.376603\pi$
$919$	−7.11559	−	4.10819i	−0.234722	−	0.135517i	−0.612748	+	0.790278i	$0.709936\pi$
$920$		0			0
$921$	−0.612668	+	0.988954i	−0.0201881	+	0.0325871i
$922$		0			0
$923$	−8.63325	−	8.63325i	−0.284167	−	0.284167i
$924$		0			0
$925$	−28.0000	+	4.00000i	−0.920634	+	0.131519i
$926$		0			0
$927$	−36.8088	−	41.6888i	−1.20896	−	1.36924i
$928$		0			0
$929$	−20.4499	+	35.4202i	−0.670939	+	1.16210i	0.306700	+	0.951806i	$0.400775\pi$
$929$	−20.4499	+	35.4202i	−0.670939	+	1.16210i	−0.977638	+	0.210293i	$0.932558\pi$
$930$		0			0
$931$	−44.0581	−	3.73831i	−1.44394	−	0.122518i
$932$		0			0
$933$	−5.12012	−	5.44831i	−0.167625	−	0.178370i
$934$		0			0
$935$	11.8130	+	57.7272i	0.386327	+	1.88788i
$936$		0			0
$937$	−14.0000	−	14.0000i	−0.457360	−	0.457360i	0.440428	−	0.897788i	$-0.354827\pi$
$937$	−14.0000	−	14.0000i	−0.457360	−	0.457360i	−0.897788	+	0.440428i	$0.854827\pi$
$938$		0			0
$939$	23.6332	−	12.6834i	0.771242	−	0.413906i
$940$		0			0
$941$	−6.29297	+	3.63325i	−0.205145	+	0.118441i	−0.599053	−	0.800709i	$-0.704456\pi$
$941$	−6.29297	+	3.63325i	−0.205145	+	0.118441i	0.393908	+	0.919150i	$0.371123\pi$
$942$		0			0
$943$	2.87026	+	10.7119i	0.0934684	+	0.348829i
$944$		0			0
$945$	−23.1313	−	20.2470i	−0.752462	−	0.658635i
$946$		0			0
$947$	−15.1809	−	56.6558i	−0.493312	−	1.84107i	−0.539284	−	0.842124i	$-0.681305\pi$
$947$	−15.1809	−	56.6558i	−0.493312	−	1.84107i	0.0459717	−	0.998943i	$-0.485362\pi$
$948$		0			0
$949$	22.9783	−	13.2665i	0.745906	−	0.430649i
$950$		0			0
$951$	27.9499	−	15.0000i	0.906337	−	0.486408i
$952$		0			0
$953$	20.0000	+	20.0000i	0.647864	+	0.647864i	0.952476	−	0.304613i	$-0.0985270\pi$
$953$	20.0000	+	20.0000i	0.647864	+	0.647864i	−0.304613	+	0.952476i	$0.598527\pi$
$954$		0			0
$955$	5.31830	−	8.05493i	0.172096	−	0.260651i
$956$		0			0
$957$	−3.49897	−	3.72324i	−0.113106	−	0.120355i
$958$		0			0
$959$	−2.05013	+	48.4106i	−0.0662020	+	1.56326i
$960$		0			0
$961$	−2.50000	+	4.33013i	−0.0806452	+	0.139682i
$962$		0			0
$963$	20.1011	+	22.7660i	0.647749	+	0.733625i
$964$		0			0
$965$	5.94987	−	17.8496i	0.191533	−	0.574600i
$966$		0			0
$967$	−3.89181	−	3.89181i	−0.125152	−	0.125152i	0.641756	−	0.766909i	$-0.278206\pi$
$967$	−3.89181	−	3.89181i	−0.125152	−	0.125152i	−0.766909	+	0.641756i	$0.778206\pi$
$968$		0			0
$969$	−35.1737	+	56.7766i	−1.12994	+	1.82393i
$970$		0			0
$971$	−22.4298	−	12.9499i	−0.719808	−	0.415581i	0.0948741	−	0.995489i	$-0.469755\pi$
$971$	−22.4298	−	12.9499i	−0.719808	−	0.415581i	−0.814682	+	0.579908i	$0.803088\pi$
$972$		0			0
$973$	−31.8050	+	7.09472i	−1.01962	+	0.227446i
$974$		0			0
$975$	−19.1303	−	15.2981i	−0.612660	−	0.489931i
$976$		0			0
$977$	−6.83013	−	1.83013i	−0.218515	−	0.0585510i	0.147900	−	0.989002i	$-0.452748\pi$
$977$	−6.83013	−	1.83013i	−0.218515	−	0.0585510i	−0.366416	+	0.930451i	$0.619415\pi$
$978$		0			0
$979$		−	71.5831i		−	2.28781i
$980$		0			0
$981$	−24.1834	+	36.4578i	−0.772116	+	1.16401i
$982$		0			0
$983$	2.11986	−	7.91142i	0.0676130	−	0.252335i	−0.923844	−	0.382769i	$-0.874970\pi$
$983$	2.11986	−	7.91142i	0.0676130	−	0.252335i	0.991457	+	0.130434i	$0.0416371\pi$
$984$		0			0
$985$	−31.4711	−	35.4904i	−1.00275	−	1.13082i
$986$		0			0
$987$	−21.1717	−	24.5308i	−0.673903	−	0.780823i
$988$		0			0
$989$	11.1332	−	19.2834i	0.354017	−	0.613175i
$990$		0			0
$991$	0.733501	+	1.27046i	0.0233004	+	0.0403575i	0.877440	−	0.479685i	$-0.159249\pi$
$991$	0.733501	+	1.27046i	0.0233004	+	0.0403575i	−0.854140	+	0.520043i	$0.825916\pi$
$992$		0			0
$993$	−7.68338	−	2.31662i	−0.243825	−	0.0735159i
$994$		0			0
$995$	−23.2665	−	46.5330i	−0.737598	−	1.47520i
$996$		0			0
$997$	−6.45422	+	24.0875i	−0.204407	+	0.762858i	0.785222	+	0.619214i	$0.212549\pi$
$997$	−6.45422	+	24.0875i	−0.204407	+	0.762858i	−0.989629	+	0.143644i	$0.954118\pi$
$998$		0			0
$999$	23.9788	+	17.0005i	0.758655	+	0.537873i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.bv.a.317.1	yes	8
3.2	odd	2		420.2.bv.b.317.1	yes	8
5.3	odd	4		420.2.bv.b.233.2	yes	8
7.4	even	3	inner	420.2.bv.a.137.1	yes	8
15.8	even	4	inner	420.2.bv.a.233.1	yes	8
21.11	odd	6		420.2.bv.b.137.2	yes	8
35.18	odd	12		420.2.bv.b.53.1	yes	8
105.53	even	12	inner	420.2.bv.a.53.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bv.a.53.1	✓	8	105.53	even	12	inner
420.2.bv.a.137.1	yes	8	7.4	even	3	inner
420.2.bv.a.233.1	yes	8	15.8	even	4	inner
420.2.bv.a.317.1	yes	8	1.1	even	1	trivial
420.2.bv.b.53.1	yes	8	35.18	odd	12
420.2.bv.b.137.2	yes	8	21.11	odd	6
420.2.bv.b.233.2	yes	8	5.3	odd	4
420.2.bv.b.317.1	yes	8	3.2	odd	2

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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.bv (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.396143 - 1.68614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(0.576028 + 2.58228i) q^{7} +(-2.68614 + 1.33591i) q^{9} +O(q^{10})\)	\(q+(-0.396143 - 1.68614i) q^{3} +(-0.133975 + 2.23205i) q^{5} +(0.576028 + 2.58228i) q^{7} +(-2.68614 + 1.33591i) q^{9} +(-3.73831 + 2.15831i) q^{11} +(-2.00000 + 2.00000i) q^{13} +(3.81662 - 0.658312i) q^{15} +(-5.89662 - 1.57999i) q^{17} +(5.47036 + 3.15831i) q^{19} +(4.12590 - 1.99422i) q^{21} +(2.94831 - 0.789997i) q^{23} +(-4.96410 - 0.598076i) q^{25} +(3.31662 + 4.00000i) q^{27} -0.683375 q^{29} +(3.00000 + 5.19615i) q^{31} +(5.12012 + 5.44831i) q^{33} +(-5.84096 + 0.939764i) q^{35} +(5.46410 - 1.46410i) q^{37} +(4.16457 + 2.57999i) q^{39} +3.63325i q^{41} +(5.15831 - 5.15831i) q^{43} +(-2.62194 - 6.17458i) q^{45} +(-1.83013 - 6.83013i) q^{47} +(-6.33638 + 2.97494i) q^{49} +(-0.328185 + 10.5684i) q^{51} +(0.250133 - 0.933508i) q^{53} +(-4.31662 - 8.63325i) q^{55} +(3.15831 - 10.4749i) q^{57} +(2.84169 + 4.92195i) q^{59} +(-1.34169 + 2.32387i) q^{61} +(-4.99699 - 6.16686i) q^{63} +(-4.19615 - 4.73205i) q^{65} +(1.40616 - 5.24784i) q^{67} +(-2.50000 - 4.65831i) q^{69} +4.31662i q^{71} +(-9.06119 - 2.42794i) q^{73} +(0.958056 + 8.60710i) q^{75} +(-7.72675 - 8.41012i) q^{77} +(-10.6665 - 6.15831i) q^{79} +(5.43070 - 7.17687i) q^{81} +(7.15831 + 7.15831i) q^{83} +(4.31662 - 12.9499i) q^{85} +(0.270715 + 1.15227i) q^{87} +(-8.29156 + 14.3614i) q^{89} +(-6.31662 - 4.01251i) q^{91} +(7.57301 - 7.11684i) q^{93} +(-7.78240 + 11.7870i) q^{95} +(-2.68338 - 2.68338i) q^{97} +(7.15831 - 10.7916i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9	\( 8 q - 8 q^{5} + 6 q^{7} - 10 q^{9} - 16 q^{13} + 4 q^{15} - 4 q^{17} + 14 q^{21} + 2 q^{23} - 12 q^{25} - 32 q^{29} + 24 q^{31} - 2 q^{33} + 16 q^{37} + 4 q^{39} + 28 q^{43} - 20 q^{45} + 20 q^{47} - 24 q^{51} - 16 q^{53} - 8 q^{55} + 12 q^{57} + 36 q^{59} - 24 q^{61} + 22 q^{63} + 8 q^{65} - 22 q^{67} - 20 q^{69} - 8 q^{75} + 16 q^{77} - 14 q^{81} + 44 q^{83} + 8 q^{85} - 22 q^{87} - 24 q^{91} + 12 q^{95} - 48 q^{97} + 44 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 + 6 * q^7 - 10 * q^9 - 16 * q^13 + 4 * q^15 - 4 * q^17 + 14 * q^21 + 2 * q^23 - 12 * q^25 - 32 * q^29 + 24 * q^31 - 2 * q^33 + 16 * q^37 + 4 * q^39 + 28 * q^43 - 20 * q^45 + 20 * q^47 - 24 * q^51 - 16 * q^53 - 8 * q^55 + 12 * q^57 + 36 * q^59 - 24 * q^61 + 22 * q^63 + 8 * q^65 - 22 * q^67 - 20 * q^69 - 8 * q^75 + 16 * q^77 - 14 * q^81 + 44 * q^83 + 8 * q^85 - 22 * q^87 - 24 * q^91 + 12 * q^95 - 48 * q^97 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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