LMFDB - Embedded newform 270.3.b.a.269.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,3,Mod(269,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("270.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	270.b (of order $2$, degree $1$, 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:	$7.35696713773$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.31744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 14x^{2} + 31 $ x^4 + 14*x^2 + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			269.3
Root			$-3.35300i$ of defining polynomial
Character	$\chi$	$=$	270.269
Dual form			270.3.b.a.269.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(-1.58579 - 4.74186i) q^{5} -0.813575i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +(-1.58579 - 4.74186i) q^{5} -0.813575i q^{7} +2.82843 q^{8} +(-2.24264 - 6.70601i) q^{10} -15.3762i q^{11} -5.89243i q^{13} -1.15057i q^{14} +4.00000 q^{16} +12.8995 q^{17} +1.24264 q^{19} +(-3.17157 - 9.48373i) q^{20} -21.7452i q^{22} +4.79899 q^{23} +(-19.9706 + 15.0392i) q^{25} -8.33316i q^{26} -1.62715i q^{28} -42.6768i q^{29} +4.21320 q^{31} +5.65685 q^{32} +18.2426 q^{34} +(-3.85786 + 1.29016i) q^{35} +70.3144i q^{37} +1.75736 q^{38} +(-4.48528 - 13.4120i) q^{40} -7.04300i q^{41} +41.0496i q^{43} -30.7523i q^{44} +6.78680 q^{46} +79.7990 q^{47} +48.3381 q^{49} +(-28.2426 + 21.2686i) q^{50} -11.7849i q^{52} -63.7279 q^{53} +(-72.9117 + 24.3833i) q^{55} -2.30114i q^{56} -60.3541i q^{58} +36.6448i q^{59} -82.9411 q^{61} +5.95837 q^{62} +8.00000 q^{64} +(-27.9411 + 9.34414i) q^{65} -89.0027i q^{67} +25.7990 q^{68} +(-5.45584 + 1.82456i) q^{70} +69.6982i q^{71} +89.6188i q^{73} +99.4396i q^{74} +2.48528 q^{76} -12.5097 q^{77} +134.095 q^{79} +(-6.34315 - 18.9675i) q^{80} -9.96031i q^{82} -109.024 q^{83} +(-20.4558 - 61.1677i) q^{85} +58.0529i q^{86} -43.4904i q^{88} +137.514i q^{89} -4.79394 q^{91} +9.59798 q^{92} +112.853 q^{94} +(-1.97056 - 5.89243i) q^{95} +90.6298i q^{97} +68.3604 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 12 q^{5}+O(q^{10}) $ 4 * q + 8 * q^4 - 12 * q^5	$ 4 q + 8 q^{4} - 12 q^{5} + 8 q^{10} + 16 q^{16} + 12 q^{17} - 12 q^{19} - 24 q^{20} - 60 q^{23} - 12 q^{25} - 68 q^{31} + 56 q^{34} - 72 q^{35} + 24 q^{38} + 16 q^{40} + 112 q^{46} + 240 q^{47} - 180 q^{49} - 96 q^{50} - 204 q^{53} - 88 q^{55} - 196 q^{61} + 120 q^{62} + 32 q^{64} + 24 q^{65} + 24 q^{68} + 80 q^{70} - 24 q^{76} + 312 q^{77} + 180 q^{79} - 48 q^{80} - 108 q^{83} + 20 q^{85} + 456 q^{91} - 120 q^{92} + 112 q^{94} + 60 q^{95} + 528 q^{98}+O(q^{100}) $ 4 * q + 8 * q^4 - 12 * q^5 + 8 * q^10 + 16 * q^16 + 12 * q^17 - 12 * q^19 - 24 * q^20 - 60 * q^23 - 12 * q^25 - 68 * q^31 + 56 * q^34 - 72 * q^35 + 24 * q^38 + 16 * q^40 + 112 * q^46 + 240 * q^47 - 180 * q^49 - 96 * q^50 - 204 * q^53 - 88 * q^55 - 196 * q^61 + 120 * q^62 + 32 * q^64 + 24 * q^65 + 24 * q^68 + 80 * q^70 - 24 * q^76 + 312 * q^77 + 180 * q^79 - 48 * q^80 - 108 * q^83 + 20 * q^85 + 456 * q^91 - 120 * q^92 + 112 * q^94 + 60 * q^95 + 528 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	−1.58579	−	4.74186i	−0.317157	−	0.948373i
$5$	−1.58579	−	4.74186i	−0.317157	−	0.948373i
$6$		0			0
$7$		−	0.813575i		−	0.116225i	−0.998310	−	0.0581125i	$-0.981492\pi$
$7$		−	0.813575i		−	0.116225i	0.998310	−	0.0581125i	$-0.0185082\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	−2.24264	−	6.70601i	−0.224264	−	0.670601i
$11$		−	15.3762i		−	1.39783i	−0.715203	−	0.698917i	$-0.753666\pi$
$11$		−	15.3762i		−	1.39783i	0.715203	−	0.698917i	$-0.246334\pi$
$12$		0			0
$13$		−	5.89243i		−	0.453264i	−0.973980	−	0.226632i	$-0.927229\pi$
$13$		−	5.89243i		−	0.453264i	0.973980	−	0.226632i	$-0.0727715\pi$
$14$		−	1.15057i		−	0.0821835i
$15$		0			0
$16$	4.00000			0.250000
$17$	12.8995			0.758794			0.379397	−	0.925234i	$-0.376131\pi$
$17$	12.8995			0.758794			0.379397	+	0.925234i	$0.376131\pi$
$18$		0			0
$19$	1.24264			0.0654021			0.0327011	−	0.999465i	$-0.489589\pi$
$19$	1.24264			0.0654021			0.0327011	+	0.999465i	$0.489589\pi$
$20$	−3.17157	−	9.48373i	−0.158579	−	0.474186i
$21$		0			0
$22$		−	21.7452i		−	0.988417i
$23$	4.79899			0.208652			0.104326	−	0.994543i	$-0.466732\pi$
$23$	4.79899			0.208652			0.104326	+	0.994543i	$0.466732\pi$
$24$		0			0
$25$	−19.9706	+	15.0392i	−0.798823	+	0.601567i
$26$		−	8.33316i		−	0.320506i
$27$		0			0
$28$		−	1.62715i		−	0.0581125i
$29$		−	42.6768i		−	1.47161i	−0.677192	−	0.735807i	$-0.736803\pi$
$29$		−	42.6768i		−	1.47161i	0.677192	−	0.735807i	$-0.263197\pi$
$30$		0			0
$31$	4.21320			0.135910			0.0679549	−	0.997688i	$-0.478353\pi$
$31$	4.21320			0.135910			0.0679549	+	0.997688i	$0.478353\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$	18.2426			0.536548
$35$	−3.85786	+	1.29016i	−0.110225	+	0.0368616i
$36$		0			0
$37$			70.3144i			1.90039i	0.311657	+	0.950195i	$0.399116\pi$
$37$			70.3144i			1.90039i	−0.311657	+	0.950195i	$0.600884\pi$
$38$	1.75736			0.0462463
$39$		0			0
$40$	−4.48528	−	13.4120i	−0.112132	−	0.335300i
$41$		−	7.04300i		−	0.171781i	−0.996305	−	0.0858903i	$-0.972627\pi$
$41$		−	7.04300i		−	0.171781i	0.996305	−	0.0858903i	$-0.0273735\pi$
$42$		0			0
$43$			41.0496i			0.954643i	0.878729	+	0.477321i	$0.158392\pi$
$43$			41.0496i			0.954643i	−0.878729	+	0.477321i	$0.841608\pi$
$44$		−	30.7523i		−	0.698917i
$45$		0			0
$46$	6.78680			0.147539
$47$	79.7990			1.69785			0.848925	−	0.528513i	$-0.177250\pi$
$47$	79.7990			1.69785			0.848925	+	0.528513i	$0.177250\pi$
$48$		0			0
$49$	48.3381			0.986492
$50$	−28.2426	+	21.2686i	−0.564853	+	0.425372i
$51$		0			0
$52$		−	11.7849i		−	0.226632i
$53$	−63.7279			−1.20241			−0.601207	−	0.799093i	$-0.705313\pi$
$53$	−63.7279			−1.20241			−0.601207	+	0.799093i	$0.705313\pi$
$54$		0			0
$55$	−72.9117	+	24.3833i	−1.32567	+	0.443333i
$56$		−	2.30114i		−	0.0410918i
$57$		0			0
$58$		−	60.3541i		−	1.04059i
$59$			36.6448i			0.621098i	0.950557	+	0.310549i	$0.100513\pi$
$59$			36.6448i			0.621098i	−0.950557	+	0.310549i	$0.899487\pi$
$60$		0			0
$61$	−82.9411			−1.35969			−0.679845	−	0.733356i	$-0.737953\pi$
$61$	−82.9411			−1.35969			−0.679845	+	0.733356i	$0.737953\pi$
$62$	5.95837			0.0961027
$63$		0			0
$64$	8.00000			0.125000
$65$	−27.9411	+	9.34414i	−0.429863	+	0.143756i
$66$		0			0
$67$		−	89.0027i		−	1.32840i	−0.747556	−	0.664199i	$-0.768773\pi$
$67$		−	89.0027i		−	1.32840i	0.747556	−	0.664199i	$-0.231227\pi$
$68$	25.7990			0.379397
$69$		0			0
$70$	−5.45584	+	1.82456i	−0.0779406	+	0.0260651i
$71$			69.6982i			0.981665i	0.871254	+	0.490833i	$0.163307\pi$
$71$			69.6982i			0.981665i	−0.871254	+	0.490833i	$0.836693\pi$
$72$		0			0
$73$			89.6188i			1.22766i	0.789440	+	0.613828i	$0.210371\pi$
$73$			89.6188i			1.22766i	−0.789440	+	0.613828i	$0.789629\pi$
$74$			99.4396i			1.34378i
$75$		0			0
$76$	2.48528			0.0327011
$77$	−12.5097			−0.162463
$78$		0			0
$79$	134.095			1.69741			0.848705	−	0.528866i	$-0.177383\pi$
$79$	134.095			1.69741			0.848705	+	0.528866i	$0.177383\pi$
$80$	−6.34315	−	18.9675i	−0.0792893	−	0.237093i
$81$		0			0
$82$		−	9.96031i		−	0.121467i
$83$	−109.024			−1.31355			−0.656773	−	0.754088i	$-0.728079\pi$
$83$	−109.024			−1.31355			−0.656773	+	0.754088i	$0.728079\pi$
$84$		0			0
$85$	−20.4558	−	61.1677i	−0.240657	−	0.719620i
$86$			58.0529i			0.675034i
$87$		0			0
$88$		−	43.4904i		−	0.494209i
$89$			137.514i			1.54510i	0.634953	+	0.772551i	$0.281020\pi$
$89$			137.514i			1.54510i	−0.634953	+	0.772551i	$0.718980\pi$
$90$		0			0
$91$	−4.79394			−0.0526807
$92$	9.59798			0.104326
$93$		0			0
$94$	112.853			1.20056
$95$	−1.97056	−	5.89243i	−0.0207428	−	0.0620256i
$96$		0			0
$97$			90.6298i			0.934328i	0.884171	+	0.467164i	$0.154724\pi$
$97$			90.6298i			0.934328i	−0.884171	+	0.467164i	$0.845276\pi$
$98$	68.3604			0.697555
$99$		0			0
$100$	−39.9411	+	30.0783i	−0.399411	+	0.300783i
$101$			49.7198i			0.492275i	0.969235	+	0.246138i	$0.0791615\pi$
$101$			49.7198i			0.492275i	−0.969235	+	0.246138i	$0.920839\pi$
$102$		0			0
$103$		−	128.844i		−	1.25091i	−0.780259	−	0.625456i	$-0.784913\pi$
$103$		−	128.844i		−	1.25091i	0.780259	−	0.625456i	$-0.215087\pi$
$104$		−	16.6663i		−	0.160253i
$105$		0			0
$106$	−90.1249			−0.850235
$107$	−6.42641			−0.0600599			−0.0300299	−	0.999549i	$-0.509560\pi$
$107$	−6.42641			−0.0600599			−0.0300299	+	0.999549i	$0.509560\pi$
$108$		0			0
$109$	108.279			0.993387			0.496694	−	0.867926i	$-0.334547\pi$
$109$	108.279			0.993387			0.496694	+	0.867926i	$0.334547\pi$
$110$	−103.113	+	34.4832i	−0.937388	+	0.313484i
$111$		0			0
$112$		−	3.25430i		−	0.0290563i
$113$	30.4264			0.269260			0.134630	−	0.990896i	$-0.457015\pi$
$113$	30.4264			0.269260			0.134630	+	0.990896i	$0.457015\pi$
$114$		0			0
$115$	−7.61017	−	22.7562i	−0.0661754	−	0.197880i
$116$		−	85.3536i		−	0.735807i
$117$		0			0
$118$			51.8235i			0.439182i
$119$		−	10.4947i		−	0.0881909i
$120$		0			0
$121$	−115.426			−0.953937
$122$	−117.296			−0.961446
$123$		0			0
$124$	8.42641			0.0679549
$125$	102.983	+	70.8488i	0.823862	+	0.566790i
$126$		0			0
$127$		−	133.725i		−	1.05296i	−0.850189	−	0.526478i	$-0.823512\pi$
$127$		−	133.725i		−	1.05296i	0.850189	−	0.526478i	$-0.176488\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$	−39.5147	+	13.2146i	−0.303959	+	0.101651i
$131$		−	61.7838i		−	0.471632i	−0.971798	−	0.235816i	$-0.924224\pi$
$131$		−	61.7838i		−	0.471632i	0.971798	−	0.235816i	$-0.0757763\pi$
$132$		0			0
$133$		−	1.01098i		−	0.00760137i
$134$		−	125.869i		−	0.939319i
$135$		0			0
$136$	36.4853			0.268274
$137$	163.919			1.19649			0.598244	−	0.801314i	$-0.295865\pi$
$137$	163.919			1.19649			0.598244	+	0.801314i	$0.295865\pi$
$138$		0			0
$139$	182.250			1.31115			0.655575	−	0.755130i	$-0.272426\pi$
$139$	182.250			1.31115			0.655575	+	0.755130i	$0.272426\pi$
$140$	−7.71573	+	2.58031i	−0.0551123	+	0.0184308i
$141$		0			0
$142$			98.5682i			0.694142i
$143$	−90.6030			−0.633588
$144$		0			0
$145$	−202.368	+	67.6763i	−1.39564	+	0.466733i
$146$			126.740i			0.868084i
$147$		0			0
$148$			140.629i			0.950195i
$149$		−	189.535i		−	1.27205i	−0.771670	−	0.636023i	$-0.780578\pi$
$149$		−	189.535i		−	1.27205i	0.771670	−	0.636023i	$-0.219422\pi$
$150$		0			0
$151$	−113.338			−0.750583			−0.375292	−	0.926907i	$-0.622457\pi$
$151$	−113.338			−0.750583			−0.375292	+	0.926907i	$0.622457\pi$
$152$	3.51472			0.0231231
$153$		0			0
$154$	−17.6913			−0.114879
$155$	−6.68124	−	19.9784i	−0.0431048	−	0.128893i
$156$		0			0
$157$			113.189i			0.720946i	0.932770	+	0.360473i	$0.117385\pi$
$157$			113.189i			0.720946i	−0.932770	+	0.360473i	$0.882615\pi$
$158$	189.640			1.20025
$159$		0			0
$160$	−8.97056	−	26.8240i	−0.0560660	−	0.167650i
$161$		−	3.90434i		−	0.0242506i
$162$		0			0
$163$		−	152.414i		−	0.935053i	−0.883979	−	0.467527i	$-0.845145\pi$
$163$		−	152.414i		−	0.935053i	0.883979	−	0.467527i	$-0.154855\pi$
$164$		−	14.0860i		−	0.0858903i
$165$		0			0
$166$	−154.184			−0.928818
$167$	−105.853			−0.633849			−0.316925	−	0.948451i	$-0.602650\pi$
$167$	−105.853			−0.633849			−0.316925	+	0.948451i	$0.602650\pi$
$168$		0			0
$169$	134.279			0.794552
$170$	−28.9289	−	86.5041i	−0.170170	−	0.508848i
$171$		0			0
$172$			82.0993i			0.477321i
$173$	−110.012			−0.635909			−0.317954	−	0.948106i	$-0.602996\pi$
$173$	−110.012			−0.635909			−0.317954	+	0.948106i	$0.602996\pi$
$174$		0			0
$175$	12.2355	+	16.2476i	0.0699171	+	0.0928432i
$176$		−	61.5047i		−	0.349458i
$177$		0			0
$178$			194.474i			1.09255i
$179$		−	118.407i		−	0.661492i	−0.943720	−	0.330746i	$-0.892700\pi$
$179$		−	118.407i		−	0.661492i	0.943720	−	0.330746i	$-0.107300\pi$
$180$		0			0
$181$	172.397			0.952469			0.476235	−	0.879318i	$-0.342001\pi$
$181$	172.397			0.952469			0.476235	+	0.879318i	$0.342001\pi$
$182$	−6.77965			−0.0372508
$183$		0			0
$184$	13.5736			0.0737695
$185$	333.421	−	111.504i	1.80228	−	0.602722i
$186$		0			0
$187$		−	198.345i		−	1.06067i
$188$	159.598			0.848925
$189$		0			0
$190$	−2.78680	−	8.33316i	−0.0146674	−	0.0438587i
$191$		−	289.393i		−	1.51515i	−0.652749	−	0.757574i	$-0.726384\pi$
$191$		−	289.393i		−	1.51515i	0.652749	−	0.757574i	$-0.273616\pi$
$192$		0			0
$193$			127.833i			0.662347i	0.943570	+	0.331173i	$0.107445\pi$
$193$			127.833i			0.662347i	−0.943570	+	0.331173i	$0.892555\pi$
$194$			128.170i			0.660670i
$195$		0			0
$196$	96.6762			0.493246
$197$	−280.414			−1.42342			−0.711711	−	0.702472i	$-0.752080\pi$
$197$	−280.414			−1.42342			−0.711711	+	0.702472i	$0.752080\pi$
$198$		0			0
$199$	156.426			0.786062			0.393031	−	0.919525i	$-0.371426\pi$
$199$	156.426			0.786062			0.393031	+	0.919525i	$0.371426\pi$
$200$	−56.4853	+	42.5372i	−0.282426	+	0.212686i
$201$		0			0
$202$			70.3144i			0.348091i
$203$	−34.7208			−0.171038
$204$		0			0
$205$	−33.3970	+	11.1687i	−0.162912	+	0.0544815i
$206$		−	182.213i		−	0.884528i
$207$		0			0
$208$		−	23.5697i		−	0.113316i
$209$		−	19.1070i		−	0.0914213i
$210$		0			0
$211$	22.5076			0.106671			0.0533355	−	0.998577i	$-0.483015\pi$
$211$	22.5076			0.106671			0.0533355	+	0.998577i	$0.483015\pi$
$212$	−127.456			−0.601207
$213$		0			0
$214$	−9.08831			−0.0424687
$215$	194.652	−	65.0959i	0.905357	−	0.302772i
$216$		0			0
$217$		−	3.42776i		−	0.0157961i
$218$	153.130			0.702431
$219$		0			0
$220$	−145.823	+	48.7666i	−0.662834	+	0.221666i
$221$		−	76.0094i		−	0.343934i
$222$		0			0
$223$		−	354.826i		−	1.59115i	−0.605856	−	0.795575i	$-0.707169\pi$
$223$		−	354.826i		−	1.59115i	0.605856	−	0.795575i	$-0.292831\pi$
$224$		−	4.60228i		−	0.0205459i
$225$		0			0
$226$	43.0294			0.190396
$227$	415.098			1.82862			0.914312	−	0.405011i	$-0.132732\pi$
$227$	415.098			1.82862			0.914312	+	0.405011i	$0.132732\pi$
$228$		0			0
$229$	−220.220			−0.961661			−0.480830	−	0.876814i	$-0.659665\pi$
$229$	−220.220			−0.961661			−0.480830	+	0.876814i	$0.659665\pi$
$230$	−10.7624	−	32.1821i	−0.0467931	−	0.139922i
$231$		0			0
$232$		−	120.708i		−	0.520294i
$233$	−10.0345			−0.0430665			−0.0215332	−	0.999768i	$-0.506855\pi$
$233$	−10.0345			−0.0430665			−0.0215332	+	0.999768i	$0.506855\pi$
$234$		0			0
$235$	−126.544	−	378.396i	−0.538486	−	1.61020i
$236$			73.2895i			0.310549i
$237$		0			0
$238$		−	14.8418i		−	0.0623603i
$239$			114.363i			0.478507i	0.970957	+	0.239253i	$0.0769027\pi$
$239$			114.363i			0.478507i	−0.970957	+	0.239253i	$0.923097\pi$
$240$		0			0
$241$	−63.7208			−0.264402			−0.132201	−	0.991223i	$-0.542204\pi$
$241$	−63.7208			−0.264402			−0.132201	+	0.991223i	$0.542204\pi$
$242$	−163.238			−0.674535
$243$		0			0
$244$	−165.882			−0.679845
$245$	−76.6539	−	229.213i	−0.312873	−	0.935562i
$246$		0			0
$247$		−	7.32218i		−	0.0296444i
$248$	11.9167			0.0480514
$249$		0			0
$250$	145.640	+	100.195i	0.582558	+	0.400781i
$251$			229.631i			0.914866i	0.889244	+	0.457433i	$0.151231\pi$
$251$			229.631i			0.914866i	−0.889244	+	0.457433i	$0.848769\pi$
$252$		0			0
$253$		−	73.7901i		−	0.291660i
$254$		−	189.116i		−	0.744552i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−388.227			−1.51061			−0.755306	−	0.655372i	$-0.772512\pi$
$257$	−388.227			−1.51061			−0.755306	+	0.655372i	$0.772512\pi$
$258$		0			0
$259$	57.2061			0.220873
$260$	−55.8823	+	18.6883i	−0.214932	+	0.0718780i
$261$		0			0
$262$		−	87.3755i		−	0.333494i
$263$	−226.877			−0.862651			−0.431325	−	0.902196i	$-0.641954\pi$
$263$	−226.877			−0.862651			−0.431325	+	0.902196i	$0.641954\pi$
$264$		0			0
$265$	101.059	+	302.189i	0.381354	+	1.14034i
$266$		−	1.42974i		−	0.00537498i
$267$		0			0
$268$		−	178.005i		−	0.664199i
$269$			226.772i			0.843019i	0.906824	+	0.421509i	$0.138500\pi$
$269$			226.772i			0.843019i	−0.906824	+	0.421509i	$0.861500\pi$
$270$		0			0
$271$	322.889			1.19147			0.595737	−	0.803180i	$-0.296860\pi$
$271$	322.889			1.19147			0.595737	+	0.803180i	$0.296860\pi$
$272$	51.5980			0.189698
$273$		0			0
$274$	231.816			0.846045
$275$	231.245	+	307.071i	0.840890	+	1.11662i
$276$		0			0
$277$		−	200.564i		−	0.724058i	−0.932167	−	0.362029i	$-0.882084\pi$
$277$		−	200.564i		−	0.724058i	0.932167	−	0.362029i	$-0.117916\pi$
$278$	257.740			0.927123
$279$		0			0
$280$	−10.9117	+	3.64911i	−0.0389703	+	0.0130326i
$281$			128.030i			0.455624i	0.973705	+	0.227812i	$0.0731572\pi$
$281$			128.030i			0.455624i	−0.973705	+	0.227812i	$0.926843\pi$
$282$		0			0
$283$			420.235i			1.48493i	0.669885	+	0.742465i	$0.266344\pi$
$283$			420.235i			1.48493i	−0.669885	+	0.742465i	$0.733656\pi$
$284$			139.396i			0.490833i
$285$		0			0
$286$	−128.132			−0.448014
$287$	−5.73001			−0.0199652
$288$		0			0
$289$	−122.603			−0.424232
$290$	−286.191	+	95.7087i	−0.986865	+	0.330030i
$291$		0			0
$292$			179.238i			0.613828i
$293$	−137.434			−0.469056			−0.234528	−	0.972109i	$-0.575355\pi$
$293$	−137.434			−0.469056			−0.234528	+	0.972109i	$0.575355\pi$
$294$		0			0
$295$	173.765	−	58.1108i	0.589032	−	0.196986i
$296$			198.879i			0.671889i
$297$		0			0
$298$		−	268.043i		−	0.899473i
$299$		−	28.2777i		−	0.0945744i
$300$		0			0
$301$	33.3970			0.110953
$302$	−160.284			−0.530743
$303$		0			0
$304$	4.97056			0.0163505
$305$	131.527	+	393.296i	0.431236	+	1.28949i
$306$		0			0
$307$			117.849i			0.383872i	0.981407	+	0.191936i	$0.0614766\pi$
$307$			117.849i			0.383872i	−0.981407	+	0.191936i	$0.938523\pi$
$308$	−25.0193			−0.0812316
$309$		0			0
$310$	−9.44870	−	28.2538i	−0.0304797	−	0.0911412i
$311$			499.326i			1.60555i	0.596283	+	0.802774i	$0.296644\pi$
$311$			499.326i			1.60555i	−0.596283	+	0.802774i	$0.703356\pi$
$312$		0			0
$313$			491.806i			1.57127i	0.618693	+	0.785633i	$0.287662\pi$
$313$			491.806i			1.57127i	−0.618693	+	0.785633i	$0.712338\pi$
$314$			160.073i			0.509786i
$315$		0			0
$316$	268.191			0.848705
$317$	241.399			0.761511			0.380756	−	0.924676i	$-0.375664\pi$
$317$	241.399			0.761511			0.380756	+	0.924676i	$0.375664\pi$
$318$		0			0
$319$	−656.205			−2.05707
$320$	−12.6863	−	37.9349i	−0.0396447	−	0.118547i
$321$		0			0
$322$		−	5.52157i		−	0.0171477i
$323$	16.0294			0.0496267
$324$		0			0
$325$	88.6173	+	117.675i	0.272669	+	0.362078i
$326$		−	215.545i		−	0.661182i
$327$		0			0
$328$		−	19.9206i		−	0.0607336i
$329$		−	64.9225i		−	0.197333i
$330$		0			0
$331$	−87.8091			−0.265284			−0.132642	−	0.991164i	$-0.542346\pi$
$331$	−87.8091			−0.265284			−0.132642	+	0.991164i	$0.542346\pi$
$332$	−218.049			−0.656773
$333$		0			0
$334$	−149.698			−0.448199
$335$	−422.039	+	141.139i	−1.25982	+	0.421311i
$336$		0			0
$337$			452.138i			1.34166i	0.741613	+	0.670828i	$0.234061\pi$
$337$			452.138i			1.34166i	−0.741613	+	0.670828i	$0.765939\pi$
$338$	189.899			0.561833
$339$		0			0
$340$	−40.9117	−	122.335i	−0.120328	−	0.359810i
$341$		−	64.7829i		−	0.189979i
$342$		0			0
$343$		−	79.1919i		−	0.230880i
$344$			116.106i			0.337517i
$345$		0			0
$346$	−155.581			−0.449655
$347$	−189.411			−0.545854			−0.272927	−	0.962035i	$-0.587992\pi$
$347$	−189.411			−0.545854			−0.272927	+	0.962035i	$0.587992\pi$
$348$		0			0
$349$	−155.794			−0.446401			−0.223200	−	0.974773i	$-0.571650\pi$
$349$	−155.794			−0.446401			−0.223200	+	0.974773i	$0.571650\pi$
$350$	17.3036	+	22.9775i	0.0494389	+	0.0656500i
$351$		0			0
$352$		−	86.9807i		−	0.247104i
$353$	−263.387			−0.746138			−0.373069	−	0.927804i	$-0.621695\pi$
$353$	−263.387			−0.746138			−0.373069	+	0.927804i	$0.621695\pi$
$354$		0			0
$355$	330.500	−	110.527i	0.930985	−	0.311342i
$356$			275.028i			0.772551i
$357$		0			0
$358$		−	167.453i		−	0.467745i
$359$		−	79.6346i		−	0.221823i	−0.993830	−	0.110912i	$-0.964623\pi$
$359$		−	79.6346i		−	0.221823i	0.993830	−	0.110912i	$-0.0353771\pi$
$360$		0			0
$361$	−359.456			−0.995723
$362$	243.806			0.673498
$363$		0			0
$364$	−9.58788			−0.0263403
$365$	424.960	−	142.116i	1.16428	−	0.389360i
$366$		0			0
$367$			475.953i			1.29688i	0.761268	+	0.648438i	$0.224577\pi$
$367$			475.953i			1.29688i	−0.761268	+	0.648438i	$0.775423\pi$
$368$	19.1960			0.0521629
$369$		0			0
$370$	471.529	−	157.690i	1.27440	−	0.426189i
$371$			51.8475i			0.139751i
$372$		0			0
$373$			61.7838i			0.165640i	0.996565	+	0.0828201i	$0.0263927\pi$
$373$			61.7838i			0.165640i	−0.996565	+	0.0828201i	$0.973607\pi$
$374$		−	280.502i		−	0.750005i
$375$		0			0
$376$	225.706			0.600281
$377$	−251.470			−0.667030
$378$		0			0
$379$	−142.345			−0.375581			−0.187791	−	0.982209i	$-0.560133\pi$
$379$	−142.345			−0.375581			−0.187791	+	0.982209i	$0.560133\pi$
$380$	−3.94113	−	11.7849i	−0.0103714	−	0.0310128i
$381$		0			0
$382$		−	409.264i		−	1.07137i
$383$	−357.520			−0.933472			−0.466736	−	0.884397i	$-0.654570\pi$
$383$	−357.520			−0.933472			−0.466736	+	0.884397i	$0.654570\pi$
$384$		0			0
$385$	19.8377	+	59.3192i	0.0515264	+	0.154076i
$386$			180.783i			0.468350i
$387$		0			0
$388$			181.260i			0.467164i
$389$			411.079i			1.05676i	0.849009	+	0.528379i	$0.177200\pi$
$389$			411.079i			1.05676i	−0.849009	+	0.528379i	$0.822800\pi$
$390$		0			0
$391$	61.9045			0.158324
$392$	136.721			0.348777
$393$		0			0
$394$	−396.566			−1.00651
$395$	−212.647	−	635.863i	−0.538346	−	1.60978i
$396$		0			0
$397$		−	457.462i		−	1.15230i	−0.817345	−	0.576149i	$-0.804555\pi$
$397$		−	457.462i		−	1.15230i	0.817345	−	0.576149i	$-0.195445\pi$
$398$	221.220			0.555830
$399$		0			0
$400$	−79.8823	+	60.1567i	−0.199706	+	0.150392i
$401$		−	467.283i		−	1.16529i	−0.812725	−	0.582647i	$-0.802017\pi$
$401$		−	467.283i		−	1.16529i	0.812725	−	0.582647i	$-0.197983\pi$
$402$		0			0
$403$		−	24.8260i		−	0.0616030i
$404$			99.4396i			0.246138i
$405$		0			0
$406$	−49.1026			−0.120942
$407$	1081.17			2.65643
$408$		0			0
$409$	−448.574			−1.09676			−0.548378	−	0.836230i	$-0.684755\pi$
$409$	−448.574			−1.09676			−0.548378	+	0.836230i	$0.684755\pi$
$410$	−47.2304	+	15.7949i	−0.115196	+	0.0385242i
$411$		0			0
$412$		−	257.688i		−	0.625456i
$413$	29.8133			0.0721871
$414$		0			0
$415$	172.889	+	516.979i	0.416601	+	1.24573i
$416$		−	33.3326i		−	0.0801265i
$417$		0			0
$418$		−	27.0214i		−	0.0646446i
$419$			360.834i			0.861180i	0.902548	+	0.430590i	$0.141694\pi$
$419$			360.834i			0.861180i	−0.902548	+	0.430590i	$0.858306\pi$
$420$		0			0
$421$	357.059			0.848121			0.424060	−	0.905634i	$-0.360604\pi$
$421$	357.059			0.848121			0.424060	+	0.905634i	$0.360604\pi$
$422$	31.8305			0.0754278
$423$		0			0
$424$	−180.250			−0.425117
$425$	−257.610	+	193.998i	−0.606142	+	0.456465i
$426$		0			0
$427$			67.4789i			0.158030i
$428$	−12.8528			−0.0300299
$429$		0			0
$430$	275.279	−	92.0596i	0.640184	−	0.214092i
$431$		−	655.947i		−	1.52192i	−0.648800	−	0.760959i	$-0.724729\pi$
$431$		−	655.947i		−	1.52192i	0.648800	−	0.760959i	$-0.275271\pi$
$432$		0			0
$433$			164.372i			0.379612i	0.981822	+	0.189806i	$0.0607859\pi$
$433$			164.372i			0.379612i	−0.981822	+	0.189806i	$0.939214\pi$
$434$		−	4.84758i		−	0.0111695i
$435$		0			0
$436$	216.558			0.496694
$437$	5.96342			0.0136463
$438$		0			0
$439$	−132.286			−0.301336			−0.150668	−	0.988584i	$-0.548142\pi$
$439$	−132.286			−0.301336			−0.150668	+	0.988584i	$0.548142\pi$
$440$	−206.225	+	68.9664i	−0.468694	+	0.156742i
$441$		0			0
$442$		−	107.494i		−	0.243198i
$443$	−455.485			−1.02818			−0.514092	−	0.857735i	$-0.671871\pi$
$443$	−455.485			−1.02818			−0.514092	+	0.857735i	$0.671871\pi$
$444$		0			0
$445$	652.073	−	218.068i	1.46533	−	0.490040i
$446$		−	501.800i		−	1.12511i
$447$		0			0
$448$		−	6.50860i		−	0.0145281i
$449$		−	415.749i		−	0.925944i	−0.886373	−	0.462972i	$-0.846783\pi$
$449$		−	415.749i		−	0.925944i	0.886373	−	0.462972i	$-0.153217\pi$
$450$		0			0
$451$	−108.294			−0.240121
$452$	60.8528			0.134630
$453$		0			0
$454$	587.037			1.29303
$455$	7.60216	+	22.7322i	0.0167081	+	0.0499609i
$456$		0			0
$457$			303.643i			0.664426i	0.943204	+	0.332213i	$0.107795\pi$
$457$			303.643i			0.664426i	−0.943204	+	0.332213i	$0.892205\pi$
$458$	−311.439			−0.679997
$459$		0			0
$460$	−15.2203	−	45.5123i	−0.0330877	−	0.0989398i
$461$			428.790i			0.930130i	0.885277	+	0.465065i	$0.153969\pi$
$461$			428.790i			0.930130i	−0.885277	+	0.465065i	$0.846031\pi$
$462$		0			0
$463$		−	814.484i		−	1.75914i	−0.475765	−	0.879572i	$-0.657829\pi$
$463$		−	814.484i		−	1.75914i	0.475765	−	0.879572i	$-0.342171\pi$
$464$		−	170.707i		−	0.367903i
$465$		0			0
$466$	−14.1909			−0.0304526
$467$	607.118			1.30004			0.650019	−	0.759918i	$-0.274761\pi$
$467$	607.118			1.30004			0.650019	+	0.759918i	$0.274761\pi$
$468$		0			0
$469$	−72.4104			−0.154393
$470$	−178.960	−	535.133i	−0.380767	−	1.13858i
$471$		0			0
$472$			103.647i			0.219591i
$473$	631.186			1.33443
$474$		0			0
$475$	−24.8162	+	18.6883i	−0.0522447	+	0.0393438i
$476$		−	20.9894i		−	0.0440954i
$477$		0			0
$478$			161.734i			0.338355i
$479$		−	343.889i		−	0.717931i	−0.933351	−	0.358965i	$-0.883130\pi$
$479$		−	343.889i		−	0.717931i	0.933351	−	0.358965i	$-0.116870\pi$
$480$		0			0
$481$	414.323			0.861378
$482$	−90.1148			−0.186960
$483$		0			0
$484$	−230.853			−0.476969
$485$	429.754	−	143.720i	0.886092	−	0.296329i
$486$		0			0
$487$			300.759i			0.617576i	0.951131	+	0.308788i	$0.0999233\pi$
$487$			300.759i			0.617576i	−0.951131	+	0.308788i	$0.900077\pi$
$488$	−234.593			−0.480723
$489$		0			0
$490$	−108.405	−	324.156i	−0.221235	−	0.661542i
$491$		−	173.914i		−	0.354203i	−0.984193	−	0.177101i	$-0.943328\pi$
$491$		−	173.914i		−	0.354203i	0.984193	−	0.177101i	$-0.0566720\pi$
$492$		0			0
$493$		−	550.509i		−	1.11665i
$494$		−	10.3551i		−	0.0209618i
$495$		0			0
$496$	16.8528			0.0339774
$497$	56.7048			0.114094
$498$		0			0
$499$	−761.610			−1.52627			−0.763136	−	0.646237i	$-0.776342\pi$
$499$	−761.610			−1.52627			−0.763136	+	0.646237i	$0.776342\pi$
$500$	205.966	+	141.698i	0.411931	+	0.283395i
$501$		0			0
$502$			324.748i			0.646908i
$503$	280.632			0.557917			0.278959	−	0.960303i	$-0.410011\pi$
$503$	280.632			0.557917			0.278959	+	0.960303i	$0.410011\pi$
$504$		0			0
$505$	235.765	−	78.8450i	0.466860	−	0.156129i
$506$		−	104.355i		−	0.206235i
$507$		0			0
$508$		−	267.451i		−	0.526478i
$509$		−	633.176i		−	1.24396i	−0.783032	−	0.621981i	$-0.786328\pi$
$509$		−	633.176i		−	1.24396i	0.783032	−	0.621981i	$-0.213672\pi$
$510$		0			0
$511$	72.9117			0.142684
$512$	22.6274			0.0441942
$513$		0			0
$514$	−549.037			−1.06816
$515$	−610.960	+	204.319i	−1.18633	+	0.396736i
$516$		0			0
$517$		−	1227.00i		−	2.37331i
$518$	80.9016			0.156181
$519$		0			0
$520$	−79.0294	+	26.4292i	−0.151980	+	0.0508254i
$521$			92.3966i			0.177345i	0.996061	+	0.0886723i	$0.0282624\pi$
$521$			92.3966i			0.177345i	−0.996061	+	0.0886723i	$0.971738\pi$
$522$		0			0
$523$			818.552i			1.56511i	0.622582	+	0.782554i	$0.286084\pi$
$523$			818.552i			1.56511i	−0.622582	+	0.782554i	$0.713916\pi$
$524$		−	123.568i		−	0.235816i
$525$		0			0
$526$	−320.853			−0.609986
$527$	54.3482			0.103128
$528$		0			0
$529$	−505.970			−0.956464
$530$	142.919	+	427.360i	0.269658	+	0.806340i
$531$		0			0
$532$		−	2.02196i		−	0.00380068i
$533$	−41.5004			−0.0778620
$534$		0			0
$535$	10.1909	+	30.4732i	0.0190484	+	0.0569592i
$536$		−	251.738i		−	0.469660i
$537$		0			0
$538$			320.704i			0.596104i
$539$		−	743.254i		−	1.37895i
$540$		0			0
$541$	−376.985			−0.696830			−0.348415	−	0.937340i	$-0.613280\pi$
$541$	−376.985			−0.696830			−0.348415	+	0.937340i	$0.613280\pi$
$542$	456.635			0.842499
$543$		0			0
$544$	72.9706			0.134137
$545$	−171.708	−	513.445i	−0.315060	−	0.942102i
$546$		0			0
$547$		−	337.346i		−	0.616721i	−0.951270	−	0.308360i	$-0.900220\pi$
$547$		−	337.346i		−	0.616721i	0.951270	−	0.308360i	$-0.0997803\pi$
$548$	327.838			0.598244
$549$		0			0
$550$	327.029	+	434.263i	0.594599	+	0.789570i
$551$		−	53.0319i		−	0.0962467i
$552$		0			0
$553$		−	109.097i		−	0.197282i
$554$		−	283.640i		−	0.511986i
$555$		0			0
$556$	364.500			0.655575
$557$	−303.338			−0.544593			−0.272296	−	0.962213i	$-0.587783\pi$
$557$	−303.338			−0.544593			−0.272296	+	0.962213i	$0.587783\pi$
$558$		0			0
$559$	241.882			0.432705
$560$	−15.4315	+	5.16063i	−0.0275562	+	0.00921541i
$561$		0			0
$562$			181.062i			0.322175i
$563$	−234.093			−0.415796			−0.207898	−	0.978150i	$-0.566662\pi$
$563$	−234.093			−0.415796			−0.207898	+	0.978150i	$0.566662\pi$
$564$		0			0
$565$	−48.2498	−	144.278i	−0.0853978	−	0.255359i
$566$			594.302i			1.05000i
$567$		0			0
$568$			197.136i			0.347071i
$569$		−	66.5257i		−	0.116917i	−0.998290	−	0.0584584i	$-0.981381\pi$
$569$		−	66.5257i		−	0.116917i	0.998290	−	0.0584584i	$-0.0186185\pi$
$570$		0			0
$571$	1115.54			1.95365			0.976826	−	0.214035i	$-0.0686607\pi$
$571$	1115.54			1.95365			0.976826	+	0.214035i	$0.0686607\pi$
$572$	−181.206			−0.316794
$573$		0			0
$574$	−8.10346			−0.0141175
$575$	−95.8385	+	72.1728i	−0.166676	+	0.125518i
$576$		0			0
$577$			957.356i			1.65920i	0.558361	+	0.829598i	$0.311430\pi$
$577$			957.356i			1.65920i	−0.558361	+	0.829598i	$0.688570\pi$
$578$	−173.387			−0.299977
$579$		0			0
$580$	−404.735	+	135.353i	−0.697819	+	0.233366i
$581$			88.6996i			0.152667i
$582$		0			0
$583$			979.891i			1.68077i
$584$			253.480i			0.434042i
$585$		0			0
$586$	−194.360			−0.331673
$587$	79.8974			0.136111			0.0680557	−	0.997682i	$-0.478320\pi$
$587$	79.8974			0.136111			0.0680557	+	0.997682i	$0.478320\pi$
$588$		0			0
$589$	5.23550			0.00888879
$590$	245.740	−	82.1810i	0.416509	−	0.139290i
$591$		0			0
$592$			281.258i			0.475097i
$593$	213.831			0.360593			0.180296	−	0.983612i	$-0.442294\pi$
$593$	213.831			0.360593			0.180296	+	0.983612i	$0.442294\pi$
$594$		0			0
$595$	−49.7645	+	16.6424i	−0.0836378	+	0.0279704i
$596$		−	379.070i		−	0.636023i
$597$		0			0
$598$		−	39.9908i		−	0.0668742i
$599$			915.493i			1.52837i	0.644998	+	0.764184i	$0.276858\pi$
$599$			915.493i			1.52837i	−0.644998	+	0.764184i	$0.723142\pi$
$600$		0			0
$601$	479.426			0.797714			0.398857	−	0.917013i	$-0.369407\pi$
$601$	479.426			0.797714			0.398857	+	0.917013i	$0.369407\pi$
$602$	47.2304			0.0784559
$603$		0			0
$604$	−226.676			−0.375292
$605$	183.042	+	547.336i	0.302548	+	0.904688i
$606$		0			0
$607$		−	219.893i		−	0.362261i	−0.983459	−	0.181131i	$-0.942024\pi$
$607$		−	219.893i		−	0.362261i	0.983459	−	0.181131i	$-0.0579757\pi$
$608$	7.02944			0.0115616
$609$		0			0
$610$	186.007	+	556.204i	0.304930	+	0.911810i
$611$		−	470.210i		−	0.769575i
$612$		0			0
$613$		−	862.066i		−	1.40631i	−0.711038	−	0.703154i	$-0.751775\pi$
$613$		−	862.066i		−	1.40631i	0.711038	−	0.703154i	$-0.248225\pi$
$614$			166.663i			0.271438i
$615$		0			0
$616$	−35.3827			−0.0574394
$617$	−695.423			−1.12710			−0.563552	−	0.826080i	$-0.690566\pi$
$617$	−695.423			−1.12710			−0.563552	+	0.826080i	$0.690566\pi$
$618$		0			0
$619$	−512.073			−0.827259			−0.413629	−	0.910445i	$-0.635739\pi$
$619$	−512.073			−0.827259			−0.413629	+	0.910445i	$0.635739\pi$
$620$	−13.3625	−	39.9569i	−0.0215524	−	0.0644466i
$621$		0			0
$622$			706.153i			1.13529i
$623$	111.878			0.179580
$624$		0			0
$625$	172.647	−	600.681i	0.276235	−	0.961090i
$626$			695.519i			1.11105i
$627$		0			0
$628$			226.377i			0.360473i
$629$			907.020i			1.44200i
$630$		0			0
$631$	−643.375			−1.01961			−0.509806	−	0.860290i	$-0.670283\pi$
$631$	−643.375			−1.01961			−0.509806	+	0.860290i	$0.670283\pi$
$632$	379.279			0.600125
$633$		0			0
$634$	341.390			0.538470
$635$	−634.108	+	212.060i	−0.998595	+	0.333953i
$636$		0			0
$637$		−	284.829i		−	0.447141i
$638$	−928.014			−1.45457
$639$		0			0
$640$	−17.9411	−	53.6481i	−0.0280330	−	0.0838251i
$641$			196.925i			0.307215i	0.988132	+	0.153608i	$0.0490892\pi$
$641$			196.925i			0.307215i	−0.988132	+	0.153608i	$0.950911\pi$
$642$		0			0
$643$			236.954i			0.368513i	0.982878	+	0.184256i	$0.0589877\pi$
$643$			236.954i			0.368513i	−0.982878	+	0.184256i	$0.941012\pi$
$644$		−	7.80868i		−	0.0121253i
$645$		0			0
$646$	22.6690			0.0350914
$647$	−17.3818			−0.0268653			−0.0134326	−	0.999910i	$-0.504276\pi$
$647$	−17.3818			−0.0268653			−0.0134326	+	0.999910i	$0.504276\pi$
$648$		0			0
$649$	563.456			0.868191
$650$	125.324	+	166.418i	0.192806	+	0.256028i
$651$		0			0
$652$		−	304.827i		−	0.467527i
$653$	785.704			1.20322			0.601611	−	0.798789i	$-0.294526\pi$
$653$	785.704			1.20322			0.601611	+	0.798789i	$0.294526\pi$
$654$		0			0
$655$	−292.971	+	97.9760i	−0.447283	+	0.149582i
$656$		−	28.1720i		−	0.0429451i
$657$		0			0
$658$		−	91.8143i		−	0.139535i
$659$			498.767i			0.756855i	0.925631	+	0.378427i	$0.123535\pi$
$659$			498.767i			0.756855i	−0.925631	+	0.378427i	$0.876465\pi$
$660$		0			0
$661$	303.133			0.458597			0.229299	−	0.973356i	$-0.426357\pi$
$661$	303.133			0.458597			0.229299	+	0.973356i	$0.426357\pi$
$662$	−124.181			−0.187584
$663$		0			0
$664$	−308.368			−0.464409
$665$	−4.79394	+	1.60320i	−0.00720893	+	0.00241083i
$666$		0			0
$667$		−	204.805i		−	0.307055i
$668$	−211.706			−0.316925
$669$		0			0
$670$	−596.853	+	199.601i	−0.890825	+	0.297912i
$671$			1275.32i			1.90062i
$672$		0			0
$673$		−	463.133i		−	0.688163i	−0.938940	−	0.344081i	$-0.888190\pi$
$673$		−	463.133i		−	0.688163i	0.938940	−	0.344081i	$-0.111810\pi$
$674$			639.420i			0.948694i
$675$		0			0
$676$	268.558			0.397276
$677$	−311.064			−0.459474			−0.229737	−	0.973253i	$-0.573787\pi$
$677$	−311.064			−0.459474			−0.229737	+	0.973253i	$0.573787\pi$
$678$		0			0
$679$	73.7342			0.108592
$680$	−57.8579	−	173.008i	−0.0850851	−	0.254424i
$681$		0			0
$682$		−	91.6169i		−	0.134336i
$683$	−232.009			−0.339691			−0.169846	−	0.985471i	$-0.554327\pi$
$683$	−232.009			−0.339691			−0.169846	+	0.985471i	$0.554327\pi$
$684$		0			0
$685$	−259.940	−	777.281i	−0.379475	−	1.13472i
$686$		−	111.994i		−	0.163257i
$687$		0			0
$688$			164.199i			0.238661i
$689$			375.513i			0.545011i
$690$		0			0
$691$	−552.904			−0.800150			−0.400075	−	0.916482i	$-0.631016\pi$
$691$	−552.904			−0.800150			−0.400075	+	0.916482i	$0.631016\pi$
$692$	−220.024			−0.317954
$693$		0			0
$694$	−267.868			−0.385977
$695$	−289.009	−	864.204i	−0.415841	−	1.24346i
$696$		0			0
$697$		−	90.8512i		−	0.130346i
$698$	−220.326			−0.315653
$699$		0			0
$700$	24.4710	+	32.4951i	0.0349586	+	0.0464216i
$701$			250.274i			0.357024i	0.983938	+	0.178512i	$0.0571284\pi$
$701$			250.274i			0.357024i	−0.983938	+	0.178512i	$0.942872\pi$
$702$		0			0
$703$			87.3755i			0.124290i
$704$		−	123.009i		−	0.174729i
$705$		0			0
$706$	−372.485			−0.527600
$707$	40.4508			0.0572147
$708$		0			0
$709$	1170.57			1.65102			0.825510	−	0.564388i	$-0.190888\pi$
$709$	1170.57			1.65102			0.825510	+	0.564388i	$0.190888\pi$
$710$	467.397	−	156.308i	0.658306	−	0.220152i
$711$		0			0
$712$			388.949i			0.546276i
$713$	20.2191			0.0283578
$714$		0			0
$715$	143.677	+	429.627i	0.200947	+	0.600877i
$716$		−	236.814i		−	0.330746i
$717$		0			0
$718$		−	112.620i		−	0.156853i
$719$		−	168.127i		−	0.233834i	−0.993142	−	0.116917i	$-0.962699\pi$
$719$		−	168.127i		−	0.233834i	0.993142	−	0.116917i	$-0.0373012\pi$
$720$		0			0
$721$	−104.824			−0.145387
$722$	−508.347			−0.704082
$723$		0			0
$724$	344.794			0.476235
$725$	641.823	+	852.279i	0.885274	+	1.17556i
$726$		0			0
$727$		−	382.835i		−	0.526595i	−0.964715	−	0.263298i	$-0.915190\pi$
$727$		−	382.835i		−	0.526595i	0.964715	−	0.263298i	$-0.0848101\pi$
$728$	−13.5593			−0.0186254
$729$		0			0
$730$	600.985	−	200.983i	0.823267	−	0.275319i
$731$			529.520i			0.724377i
$732$		0			0
$733$		−	1403.08i		−	1.91416i	−0.289819	−	0.957082i	$-0.593595\pi$
$733$		−	1403.08i		−	1.91416i	0.289819	−	0.957082i	$-0.406405\pi$
$734$			673.099i			0.917029i
$735$		0			0
$736$	27.1472			0.0368848
$737$	−1368.52			−1.85688
$738$		0			0
$739$	899.125			1.21668			0.608339	−	0.793677i	$-0.291836\pi$
$739$	899.125			1.21668			0.608339	+	0.793677i	$0.291836\pi$
$740$	666.843	−	223.007i	0.901139	−	0.301361i
$741$		0			0
$742$			73.3234i			0.0988186i
$743$	316.587			0.426093			0.213046	−	0.977042i	$-0.431661\pi$
$743$	316.587			0.426093			0.213046	+	0.977042i	$0.431661\pi$
$744$		0			0
$745$	−898.749	+	300.562i	−1.20637	+	0.403439i
$746$			87.3755i			0.117125i
$747$		0			0
$748$		−	396.689i		−	0.530334i
$749$			5.22837i			0.00698046i
$750$		0			0
$751$	−286.433			−0.381402			−0.190701	−	0.981648i	$-0.561076\pi$
$751$	−286.433			−0.381402			−0.190701	+	0.981648i	$0.561076\pi$
$752$	319.196			0.424463
$753$		0			0
$754$	−355.632			−0.471661
$755$	179.730	+	537.434i	0.238053	+	0.711833i
$756$		0			0
$757$		−	783.221i		−	1.03464i	−0.855793	−	0.517319i	$-0.826930\pi$
$757$		−	783.221i		−	1.03464i	0.855793	−	0.517319i	$-0.173070\pi$
$758$	−201.307			−0.265576
$759$		0			0
$760$	−5.57359	−	16.6663i	−0.00733368	−	0.0219294i
$761$		−	1031.91i		−	1.35599i	−0.735064	−	0.677997i	$-0.762848\pi$
$761$		−	1031.91i		−	1.35599i	0.735064	−	0.677997i	$-0.237152\pi$
$762$		0			0
$763$		−	88.0933i		−	0.115456i
$764$		−	578.787i		−	0.757574i
$765$		0			0
$766$	−505.609			−0.660064
$767$	215.927			0.281521
$768$		0			0
$769$	−831.250			−1.08095			−0.540475	−	0.841360i	$-0.681755\pi$
$769$	−831.250			−1.08095			−0.540475	+	0.841360i	$0.681755\pi$
$770$	28.0547	+	83.8900i	0.0364347	+	0.108948i
$771$		0			0
$772$			255.666i			0.331173i
$773$	395.360			0.511462			0.255731	−	0.966748i	$-0.417684\pi$
$773$	395.360			0.511462			0.255731	+	0.966748i	$0.417684\pi$
$774$		0			0
$775$	−84.1400	+	63.3631i	−0.108568	+	0.0817588i
$776$			256.340i			0.330335i
$777$		0			0
$778$			581.353i			0.747240i
$779$		−	8.75192i		−	0.0112348i
$780$		0			0
$781$	1071.69			1.37220
$782$	87.5462			0.111952
$783$		0			0
$784$	193.352			0.246623
$785$	536.725	−	179.493i	0.683726	−	0.228653i
$786$		0			0
$787$			573.929i			0.729262i	0.931152	+	0.364631i	$0.118805\pi$
$787$			573.929i			0.729262i	−0.931152	+	0.364631i	$0.881195\pi$
$788$	−560.828			−0.711711

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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	270.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(-1.58579 - 4.74186i) q^{5} -0.813575i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(-1.58579 - 4.74186i) q^{5} -0.813575i q^{7} +2.82843 q^{8} +(-2.24264 - 6.70601i) q^{10} -15.3762i q^{11} -5.89243i q^{13} -1.15057i q^{14} +4.00000 q^{16} +12.8995 q^{17} +1.24264 q^{19} +(-3.17157 - 9.48373i) q^{20} -21.7452i q^{22} +4.79899 q^{23} +(-19.9706 + 15.0392i) q^{25} -8.33316i q^{26} -1.62715i q^{28} -42.6768i q^{29} +4.21320 q^{31} +5.65685 q^{32} +18.2426 q^{34} +(-3.85786 + 1.29016i) q^{35} +70.3144i q^{37} +1.75736 q^{38} +(-4.48528 - 13.4120i) q^{40} -7.04300i q^{41} +41.0496i q^{43} -30.7523i q^{44} +6.78680 q^{46} +79.7990 q^{47} +48.3381 q^{49} +(-28.2426 + 21.2686i) q^{50} -11.7849i q^{52} -63.7279 q^{53} +(-72.9117 + 24.3833i) q^{55} -2.30114i q^{56} -60.3541i q^{58} +36.6448i q^{59} -82.9411 q^{61} +5.95837 q^{62} +8.00000 q^{64} +(-27.9411 + 9.34414i) q^{65} -89.0027i q^{67} +25.7990 q^{68} +(-5.45584 + 1.82456i) q^{70} +69.6982i q^{71} +89.6188i q^{73} +99.4396i q^{74} +2.48528 q^{76} -12.5097 q^{77} +134.095 q^{79} +(-6.34315 - 18.9675i) q^{80} -9.96031i q^{82} -109.024 q^{83} +(-20.4558 - 61.1677i) q^{85} +58.0529i q^{86} -43.4904i q^{88} +137.514i q^{89} -4.79394 q^{91} +9.59798 q^{92} +112.853 q^{94} +(-1.97056 - 5.89243i) q^{95} +90.6298i q^{97} +68.3604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 12 q^{5}+O(q^{10}) \) 4 * q + 8 * q^4 - 12 * q^5	\( 4 q + 8 q^{4} - 12 q^{5} + 8 q^{10} + 16 q^{16} + 12 q^{17} - 12 q^{19} - 24 q^{20} - 60 q^{23} - 12 q^{25} - 68 q^{31} + 56 q^{34} - 72 q^{35} + 24 q^{38} + 16 q^{40} + 112 q^{46} + 240 q^{47} - 180 q^{49} - 96 q^{50} - 204 q^{53} - 88 q^{55} - 196 q^{61} + 120 q^{62} + 32 q^{64} + 24 q^{65} + 24 q^{68} + 80 q^{70} - 24 q^{76} + 312 q^{77} + 180 q^{79} - 48 q^{80} - 108 q^{83} + 20 q^{85} + 456 q^{91} - 120 q^{92} + 112 q^{94} + 60 q^{95} + 528 q^{98}+O(q^{100}) \) 4 * q + 8 * q^4 - 12 * q^5 + 8 * q^10 + 16 * q^16 + 12 * q^17 - 12 * q^19 - 24 * q^20 - 60 * q^23 - 12 * q^25 - 68 * q^31 + 56 * q^34 - 72 * q^35 + 24 * q^38 + 16 * q^40 + 112 * q^46 + 240 * q^47 - 180 * q^49 - 96 * q^50 - 204 * q^53 - 88 * q^55 - 196 * q^61 + 120 * q^62 + 32 * q^64 + 24 * q^65 + 24 * q^68 + 80 * q^70 - 24 * q^76 + 312 * q^77 + 180 * q^79 - 48 * q^80 - 108 * q^83 + 20 * q^85 + 456 * q^91 - 120 * q^92 + 112 * q^94 + 60 * q^95 + 528 * q^98

Introduction

L-functions

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