LMFDB - Embedded newform 270.3.b.d.269.4

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			269.4
Root			$-1.66053i$ of defining polynomial
Character	$\chi$	$=$	270.269
Dual form			270.3.b.d.269.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(4.41421 + 2.34834i) q^{5} -13.6872i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +(4.41421 + 2.34834i) q^{5} -13.6872i q^{7} +2.82843 q^{8} +(6.24264 + 3.32106i) q^{10} -12.3115i q^{11} +17.0082i q^{13} -19.3566i q^{14} +4.00000 q^{16} +6.89949 q^{17} -7.24264 q^{19} +(8.82843 + 4.69669i) q^{20} -17.4111i q^{22} +34.7990 q^{23} +(13.9706 + 20.7322i) q^{25} +24.0532i q^{26} -27.3743i q^{28} +21.1351i q^{29} -38.2132 q^{31} +5.65685 q^{32} +9.75736 q^{34} +(32.1421 - 60.4180i) q^{35} +21.5380i q^{37} -10.2426 q^{38} +(12.4853 + 6.64212i) q^{40} -36.3648i q^{41} -6.23921i q^{43} -24.6230i q^{44} +49.2132 q^{46} -40.2010 q^{47} -138.338 q^{49} +(19.7574 + 29.3197i) q^{50} +34.0164i q^{52} +38.2721 q^{53} +(28.9117 - 54.3457i) q^{55} -38.7131i q^{56} +29.8895i q^{58} +41.6313i q^{59} -15.0589 q^{61} -54.0416 q^{62} +8.00000 q^{64} +(-39.9411 + 75.0779i) q^{65} +128.618i q^{67} +13.7990 q^{68} +(45.4558 - 85.4440i) q^{70} +104.967i q^{71} -2.11232i q^{73} +30.4593i q^{74} -14.4853 q^{76} -168.510 q^{77} -44.0955 q^{79} +(17.6569 + 9.39338i) q^{80} -51.4275i q^{82} -55.0244 q^{83} +(30.4558 + 16.2024i) q^{85} -8.82357i q^{86} -34.8222i q^{88} -68.1020i q^{89} +232.794 q^{91} +69.5980 q^{92} -56.8528 q^{94} +(-31.9706 - 17.0082i) q^{95} -101.243i q^{97} -195.640 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.41421	+	2.34834i	0.882843	+	0.469669i
$5$	4.41421	+	2.34834i	0.882843	+	0.469669i
$6$		0			0
$7$		−	13.6872i		−	1.95531i	−0.210222	−	0.977654i	$-0.567419\pi$
$7$		−	13.6872i		−	1.95531i	0.210222	−	0.977654i	$-0.432581\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$	6.24264	+	3.32106i	0.624264	+	0.332106i
$11$		−	12.3115i		−	1.11923i	−0.828753	−	0.559615i	$-0.810949\pi$
$11$		−	12.3115i		−	1.11923i	0.828753	−	0.559615i	$-0.189051\pi$
$12$		0			0
$13$			17.0082i			1.30832i	0.756355	+	0.654162i	$0.226979\pi$
$13$			17.0082i			1.30832i	−0.756355	+	0.654162i	$0.773021\pi$
$14$		−	19.3566i		−	1.38261i
$15$		0			0
$16$	4.00000			0.250000
$17$	6.89949			0.405853			0.202926	−	0.979194i	$-0.434955\pi$
$17$	6.89949			0.405853			0.202926	+	0.979194i	$0.434955\pi$
$18$		0			0
$19$	−7.24264			−0.381192			−0.190596	−	0.981669i	$-0.561042\pi$
$19$	−7.24264			−0.381192			−0.190596	+	0.981669i	$0.561042\pi$
$20$	8.82843	+	4.69669i	0.441421	+	0.234834i
$21$		0			0
$22$		−	17.4111i		−	0.791415i
$23$	34.7990			1.51300			0.756500	−	0.653994i	$-0.226908\pi$
$23$	34.7990			1.51300			0.756500	+	0.653994i	$0.226908\pi$
$24$		0			0
$25$	13.9706	+	20.7322i	0.558823	+	0.829287i
$26$			24.0532i			0.925125i
$27$		0			0
$28$		−	27.3743i		−	0.977654i
$29$			21.1351i			0.728796i	0.931243	+	0.364398i	$0.118725\pi$
$29$			21.1351i			0.728796i	−0.931243	+	0.364398i	$0.881275\pi$
$30$		0			0
$31$	−38.2132			−1.23268			−0.616342	−	0.787479i	$-0.711386\pi$
$31$	−38.2132			−1.23268			−0.616342	+	0.787479i	$0.711386\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$	9.75736			0.286981
$35$	32.1421	−	60.4180i	0.918347	−	1.72623i
$36$		0			0
$37$			21.5380i			0.582108i	0.956707	+	0.291054i	$0.0940060\pi$
$37$			21.5380i			0.582108i	−0.956707	+	0.291054i	$0.905994\pi$
$38$	−10.2426			−0.269543
$39$		0			0
$40$	12.4853	+	6.64212i	0.312132	+	0.166053i
$41$		−	36.3648i		−	0.886945i	−0.896288	−	0.443473i	$-0.853746\pi$
$41$		−	36.3648i		−	0.886945i	0.896288	−	0.443473i	$-0.146254\pi$
$42$		0			0
$43$		−	6.23921i		−	0.145098i	−0.997365	−	0.0725489i	$-0.976887\pi$
$43$		−	6.23921i		−	0.145098i	0.997365	−	0.0725489i	$-0.0231133\pi$
$44$		−	24.6230i		−	0.559615i
$45$		0			0
$46$	49.2132			1.06985
$47$	−40.2010			−0.855341			−0.427670	−	0.903935i	$-0.640666\pi$
$47$	−40.2010			−0.855341			−0.427670	+	0.903935i	$0.640666\pi$
$48$		0			0
$49$	−138.338			−2.82323
$50$	19.7574	+	29.3197i	0.395147	+	0.586395i
$51$		0			0
$52$			34.0164i			0.654162i
$53$	38.2721			0.722115			0.361057	−	0.932544i	$-0.382416\pi$
$53$	38.2721			0.722115			0.361057	+	0.932544i	$0.382416\pi$
$54$		0			0
$55$	28.9117	−	54.3457i	0.525667	−	0.988103i
$56$		−	38.7131i		−	0.691306i
$57$		0			0
$58$			29.8895i			0.515337i
$59$			41.6313i			0.705615i	0.935696	+	0.352807i	$0.114773\pi$
$59$			41.6313i			0.705615i	−0.935696	+	0.352807i	$0.885227\pi$
$60$		0			0
$61$	−15.0589			−0.246867			−0.123433	−	0.992353i	$-0.539391\pi$
$61$	−15.0589			−0.246867			−0.123433	+	0.992353i	$0.539391\pi$
$62$	−54.0416			−0.871639
$63$		0			0
$64$	8.00000			0.125000
$65$	−39.9411	+	75.0779i	−0.614479	+	1.15504i
$66$		0			0
$67$			128.618i			1.91967i	0.280569	+	0.959834i	$0.409477\pi$
$67$			128.618i			1.91967i	−0.280569	+	0.959834i	$0.590523\pi$
$68$	13.7990			0.202926
$69$		0			0
$70$	45.4558	−	85.4440i	0.649369	−	1.22063i
$71$			104.967i			1.47841i	0.673478	+	0.739207i	$0.264800\pi$
$71$			104.967i			1.47841i	−0.673478	+	0.739207i	$0.735200\pi$
$72$		0			0
$73$		−	2.11232i		−	0.0289359i	−0.999895	−	0.0144680i	$-0.995395\pi$
$73$		−	2.11232i		−	0.0289359i	0.999895	−	0.0144680i	$-0.00460546\pi$
$74$			30.4593i			0.411613i
$75$		0			0
$76$	−14.4853			−0.190596
$77$	−168.510			−2.18844
$78$		0			0
$79$	−44.0955			−0.558170			−0.279085	−	0.960266i	$-0.590031\pi$
$79$	−44.0955			−0.558170			−0.279085	+	0.960266i	$0.590031\pi$
$80$	17.6569	+	9.39338i	0.220711	+	0.117417i
$81$		0			0
$82$		−	51.4275i		−	0.627165i
$83$	−55.0244			−0.662944			−0.331472	−	0.943465i	$-0.607545\pi$
$83$	−55.0244			−0.662944			−0.331472	+	0.943465i	$0.607545\pi$
$84$		0			0
$85$	30.4558	+	16.2024i	0.358304	+	0.190616i
$86$		−	8.82357i		−	0.102600i
$87$		0			0
$88$		−	34.8222i		−	0.395707i
$89$		−	68.1020i		−	0.765191i	−0.923916	−	0.382595i	$-0.875030\pi$
$89$		−	68.1020i		−	0.765191i	0.923916	−	0.382595i	$-0.124970\pi$
$90$		0			0
$91$	232.794			2.55818
$92$	69.5980			0.756500
$93$		0			0
$94$	−56.8528			−0.604817
$95$	−31.9706	−	17.0082i	−0.336532	−	0.179034i
$96$		0			0
$97$		−	101.243i		−	1.04375i	−0.853023	−	0.521873i	$-0.825233\pi$
$97$		−	101.243i		−	1.04375i	0.853023	−	0.521873i	$-0.174767\pi$
$98$	−195.640			−1.99632
$99$		0			0
$100$	27.9411	+	41.4644i	0.279411	+	0.414644i
$101$			15.2297i			0.150789i	0.997154	+	0.0753944i	$0.0240216\pi$
$101$			15.2297i			0.150789i	−0.997154	+	0.0753944i	$0.975978\pi$
$102$		0			0
$103$		−	77.0924i		−	0.748470i	−0.927334	−	0.374235i	$-0.877905\pi$
$103$		−	77.0924i		−	0.748470i	0.927334	−	0.374235i	$-0.122095\pi$
$104$			48.1065i			0.462562i
$105$		0			0
$106$	54.1249			0.510612
$107$	−78.4264			−0.732957			−0.366479	−	0.930427i	$-0.619437\pi$
$107$	−78.4264			−0.732957			−0.366479	+	0.930427i	$0.619437\pi$
$108$		0			0
$109$	−146.279			−1.34201			−0.671006	−	0.741452i	$-0.734137\pi$
$109$	−146.279			−1.34201			−0.671006	+	0.741452i	$0.734137\pi$
$110$	40.8873	−	76.8564i	0.371703	−	0.698695i
$111$		0			0
$112$		−	54.7486i		−	0.488827i
$113$	54.4264			0.481650			0.240825	−	0.970569i	$-0.422582\pi$
$113$	54.4264			0.481650			0.240825	+	0.970569i	$0.422582\pi$
$114$		0			0
$115$	153.610	+	81.7200i	1.33574	+	0.710609i
$116$			42.2702i			0.364398i
$117$		0			0
$118$			58.8755i			0.498945i
$119$		−	94.4344i		−	0.793567i
$120$		0			0
$121$	−30.5736			−0.252674
$122$	−21.2965			−0.174561
$123$		0			0
$124$	−76.4264			−0.616342
$125$	12.9828	+	124.324i	0.103862	+	0.994592i
$126$		0			0
$127$		−	159.215i		−	1.25366i	−0.779154	−	0.626832i	$-0.784351\pi$
$127$		−	159.215i		−	1.25366i	0.779154	−	0.626832i	$-0.215649\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$	−56.4853	+	106.176i	−0.434502	+	0.816740i
$131$			110.303i			0.842008i	0.907059	+	0.421004i	$0.138322\pi$
$131$			110.303i			0.842008i	−0.907059	+	0.421004i	$0.861678\pi$
$132$		0			0
$133$			99.1311i			0.745347i
$134$			181.893i			1.35741i
$135$		0			0
$136$	19.5147			0.143491
$137$	217.919			1.59065			0.795324	−	0.606184i	$-0.207301\pi$
$137$	217.919			1.59065			0.795324	+	0.606184i	$0.207301\pi$
$138$		0			0
$139$	−106.250			−0.764387			−0.382193	−	0.924082i	$-0.624831\pi$
$139$	−106.250			−0.764387			−0.382193	+	0.924082i	$0.624831\pi$
$140$	64.2843	−	120.836i	0.459173	−	0.863114i
$141$		0			0
$142$			148.446i			1.04540i
$143$	209.397			1.46431
$144$		0			0
$145$	−49.6325	+	93.2948i	−0.342293	+	0.643413i
$146$		−	2.98728i		−	0.0204608i
$147$		0			0
$148$			43.0760i			0.291054i
$149$			14.1592i			0.0950281i	0.998871	+	0.0475141i	$0.0151299\pi$
$149$			14.1592i			0.0950281i	−0.998871	+	0.0475141i	$0.984870\pi$
$150$		0			0
$151$	73.3381			0.485683			0.242841	−	0.970066i	$-0.421921\pi$
$151$	73.3381			0.485683			0.242841	+	0.970066i	$0.421921\pi$
$152$	−20.4853			−0.134772
$153$		0			0
$154$	−238.309			−1.54746
$155$	−168.681	−	89.7377i	−1.08827	−	0.578953i
$156$		0			0
$157$		−	70.1452i		−	0.446784i	−0.974729	−	0.223392i	$-0.928287\pi$
$157$		−	70.1452i		−	0.446784i	0.974729	−	0.223392i	$-0.0717131\pi$
$158$	−62.3604			−0.394686
$159$		0			0
$160$	24.9706	+	13.2842i	0.156066	+	0.0830265i
$161$		−	476.299i		−	2.95838i
$162$		0			0
$163$		−	9.05959i		−	0.0555803i	−0.999614	−	0.0277902i	$-0.991153\pi$
$163$		−	9.05959i		−	0.0555803i	0.999614	−	0.0277902i	$-0.00884702\pi$
$164$		−	72.7295i		−	0.443473i
$165$		0			0
$166$	−77.8162			−0.468772
$167$	−63.8528			−0.382352			−0.191176	−	0.981556i	$-0.561230\pi$
$167$	−63.8528			−0.382352			−0.191176	+	0.981556i	$0.561230\pi$
$168$		0			0
$169$	−120.279			−0.711711
$170$	43.0711	+	22.9136i	0.253359	+	0.134786i
$171$		0			0
$172$		−	12.4784i		−	0.0725489i
$173$	27.9878			0.161779			0.0808896	−	0.996723i	$-0.474224\pi$
$173$	27.9878			0.161779			0.0808896	+	0.996723i	$0.474224\pi$
$174$		0			0
$175$	283.765	−	191.217i	1.62151	−	1.09267i
$176$		−	49.2461i		−	0.279807i
$177$		0			0
$178$		−	96.3107i		−	0.541072i
$179$		−	21.0660i		−	0.117687i	−0.998267	−	0.0588435i	$-0.981259\pi$
$179$		−	21.0660i		−	0.117687i	0.998267	−	0.0588435i	$-0.0187413\pi$
$180$		0			0
$181$	53.6030			0.296149			0.148075	−	0.988976i	$-0.452692\pi$
$181$	53.6030			0.296149			0.148075	+	0.988976i	$0.452692\pi$
$182$	329.220			1.80890
$183$		0			0
$184$	98.4264			0.534926
$185$	−50.5786	+	95.0734i	−0.273398	+	0.513910i
$186$		0			0
$187$		−	84.9433i		−	0.454242i
$188$	−80.4020			−0.427670
$189$		0			0
$190$	−45.2132	−	24.0532i	−0.237964	−	0.126596i
$191$			223.024i			1.16766i	0.811875	+	0.583831i	$0.198447\pi$
$191$			223.024i			1.16766i	−0.811875	+	0.583831i	$0.801553\pi$
$192$		0			0
$193$			176.224i			0.913075i	0.889704	+	0.456538i	$0.150911\pi$
$193$			176.224i			0.913075i	−0.889704	+	0.456538i	$0.849089\pi$
$194$		−	143.180i		−	0.738040i
$195$		0			0
$196$	−276.676			−1.41161
$197$	277.586			1.40906			0.704532	−	0.709672i	$-0.251157\pi$
$197$	277.586			1.40906			0.704532	+	0.709672i	$0.251157\pi$
$198$		0			0
$199$	71.5736			0.359666			0.179833	−	0.983697i	$-0.442444\pi$
$199$	71.5736			0.359666			0.179833	+	0.983697i	$0.442444\pi$
$200$	39.5147	+	58.6395i	0.197574	+	0.293197i
$201$		0			0
$202$			21.5380i			0.106624i
$203$	289.279			1.42502
$204$		0			0
$205$	85.3970	−	160.522i	0.416571	−	0.783033i
$206$		−	109.025i		−	0.529248i
$207$		0			0
$208$			68.0328i			0.327081i
$209$			89.1679i			0.426641i
$210$		0			0
$211$	319.492			1.51418			0.757091	−	0.653309i	$-0.226620\pi$
$211$	319.492			1.51418			0.757091	+	0.653309i	$0.226620\pi$
$212$	76.5442			0.361057
$213$		0			0
$214$	−110.912			−0.518279
$215$	14.6518	−	27.5412i	0.0681479	−	0.128099i
$216$		0			0
$217$			523.030i			2.41028i
$218$	−206.870			−0.948945
$219$		0			0
$220$	57.8234	−	108.691i	0.262834	−	0.494052i
$221$			117.348i			0.530987i
$222$		0			0
$223$		−	162.439i		−	0.728424i	−0.931316	−	0.364212i	$-0.881338\pi$
$223$		−	162.439i		−	0.728424i	0.931316	−	0.364212i	$-0.118662\pi$
$224$		−	77.4262i		−	0.345653i
$225$		0			0
$226$	76.9706			0.340578
$227$	241.098			1.06210			0.531052	−	0.847339i	$-0.321797\pi$
$227$	241.098			1.06210			0.531052	+	0.847339i	$0.321797\pi$
$228$		0			0
$229$	102.220			0.446377			0.223189	−	0.974775i	$-0.428353\pi$
$229$	102.220			0.446377			0.223189	+	0.974775i	$0.428353\pi$
$230$	217.238	+	115.570i	0.944511	+	0.502476i
$231$		0			0
$232$			59.7791i			0.257668i
$233$	241.966			1.03848			0.519239	−	0.854629i	$-0.326215\pi$
$233$	241.966			1.03848			0.519239	+	0.854629i	$0.326215\pi$
$234$		0			0
$235$	−177.456	−	94.4058i	−0.755131	−	0.401727i
$236$			83.2625i			0.352807i
$237$		0			0
$238$		−	133.550i		−	0.561136i
$239$		−	375.458i		−	1.57096i	−0.618890	−	0.785478i	$-0.712417\pi$
$239$		−	375.458i		−	1.57096i	0.618890	−	0.785478i	$-0.287583\pi$
$240$		0			0
$241$	−318.279			−1.32066			−0.660330	−	0.750975i	$-0.729584\pi$
$241$	−318.279			−1.32066			−0.660330	+	0.750975i	$0.729584\pi$
$242$	−43.2376			−0.178668
$243$		0			0
$244$	−30.1177			−0.123433
$245$	−610.654	−	324.865i	−2.49246	−	1.32598i
$246$		0			0
$247$		−	123.184i		−	0.498722i
$248$	−108.083			−0.435820
$249$		0			0
$250$	18.3604	+	175.821i	0.0734416	+	0.703283i
$251$			85.5417i			0.340804i	0.985375	+	0.170402i	$0.0545066\pi$
$251$			85.5417i			0.340804i	−0.985375	+	0.170402i	$0.945493\pi$
$252$		0			0
$253$		−	428.429i		−	1.69339i
$254$		−	225.164i		−	0.886474i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−214.227			−0.833570			−0.416785	−	0.909005i	$-0.636843\pi$
$257$	−214.227			−0.833570			−0.416785	+	0.909005i	$0.636843\pi$
$258$		0			0
$259$	294.794			1.13820
$260$	−79.8823	+	150.156i	−0.307239	+	0.577522i
$261$		0			0
$262$			155.992i			0.595389i
$263$	−106.877			−0.406377			−0.203189	−	0.979140i	$-0.565130\pi$
$263$	−106.877			−0.406377			−0.203189	+	0.979140i	$0.565130\pi$
$264$		0			0
$265$	168.941	+	89.8760i	0.637514	+	0.339155i
$266$			140.193i			0.527040i
$267$		0			0
$268$			257.235i			0.959834i
$269$			365.927i			1.36032i	0.733062	+	0.680161i	$0.238090\pi$
$269$			365.927i			1.36032i	−0.733062	+	0.680161i	$0.761910\pi$
$270$		0			0
$271$	−92.8894			−0.342765			−0.171383	−	0.985205i	$-0.554823\pi$
$271$	−92.8894			−0.342765			−0.171383	+	0.985205i	$0.554823\pi$
$272$	27.5980			0.101463
$273$		0			0
$274$	308.184			1.12476
$275$	255.245	−	171.999i	0.928163	−	0.625451i
$276$		0			0
$277$			226.137i			0.816380i	0.912897	+	0.408190i	$0.133840\pi$
$277$			226.137i			0.816380i	−0.912897	+	0.408190i	$0.866160\pi$
$278$	−150.260			−0.540503
$279$		0			0
$280$	90.9117	−	170.888i	0.324685	−	0.610314i
$281$		−	63.4053i		−	0.225642i	−0.993615	−	0.112821i	$-0.964011\pi$
$281$		−	63.4053i		−	0.225642i	0.993615	−	0.112821i	$-0.0359886\pi$
$282$		0			0
$283$		−	363.106i		−	1.28306i	−0.767097	−	0.641531i	$-0.778300\pi$
$283$		−	363.106i		−	1.28306i	0.767097	−	0.641531i	$-0.221700\pi$
$284$			209.935i			0.739207i
$285$		0			0
$286$	296.132			1.03543
$287$	−497.730			−1.73425
$288$		0			0
$289$	−241.397			−0.835284
$290$	−70.1909	+	131.939i	−0.242038	+	0.454961i
$291$		0			0
$292$		−	4.22465i		−	0.0144680i
$293$	−227.434			−0.776224			−0.388112	−	0.921612i	$-0.626873\pi$
$293$	−227.434			−0.776224			−0.388112	+	0.921612i	$0.626873\pi$
$294$		0			0
$295$	−97.7645	+	183.769i	−0.331405	+	0.622947i
$296$			60.9187i			0.205806i
$297$		0			0
$298$			20.0241i			0.0671950i
$299$			591.869i			1.97949i
$300$		0			0
$301$	−85.3970			−0.283711
$302$	103.716			0.343430
$303$		0			0
$304$	−28.9706			−0.0952979
$305$	−66.4731	−	35.3634i	−0.217945	−	0.115946i
$306$		0			0
$307$		−	340.164i		−	1.10803i	−0.832508	−	0.554013i	$-0.813096\pi$
$307$		−	340.164i		−	1.10803i	0.832508	−	0.554013i	$-0.186904\pi$
$308$	−337.019			−1.09422
$309$		0			0
$310$	−238.551	−	126.908i	−0.769520	−	0.409382i
$311$		−	386.769i		−	1.24363i	−0.783164	−	0.621815i	$-0.786396\pi$
$311$		−	386.769i		−	1.24363i	0.783164	−	0.621815i	$-0.213604\pi$
$312$		0			0
$313$			376.403i			1.20256i	0.799037	+	0.601282i	$0.205343\pi$
$313$			376.403i			1.20256i	−0.799037	+	0.601282i	$0.794657\pi$
$314$		−	99.2002i		−	0.315924i
$315$		0			0
$316$	−88.1909			−0.279085
$317$	355.399			1.12113			0.560566	−	0.828110i	$-0.310583\pi$
$317$	355.399			1.12113			0.560566	+	0.828110i	$0.310583\pi$
$318$		0			0
$319$	260.205			0.815690
$320$	35.3137	+	18.7868i	0.110355	+	0.0587086i
$321$		0			0
$322$		−	673.589i		−	2.09189i
$323$	−49.9706			−0.154708
$324$		0			0
$325$	−352.617	+	237.614i	−1.08498	+	0.731121i
$326$		−	12.8122i		−	0.0393012i
$327$		0			0
$328$		−	102.855i		−	0.313583i
$329$			550.237i			1.67245i
$330$		0			0
$331$	−444.191			−1.34197			−0.670983	−	0.741473i	$-0.734128\pi$
$331$	−444.191			−1.34197			−0.670983	+	0.741473i	$0.734128\pi$
$332$	−110.049			−0.331472
$333$		0			0
$334$	−90.3015			−0.270364
$335$	−302.039	+	567.746i	−0.901608	+	1.69476i
$336$		0			0
$337$		−	407.086i		−	1.20797i	−0.796995	−	0.603985i	$-0.793579\pi$
$337$		−	407.086i		−	1.20797i	0.796995	−	0.603985i	$-0.206421\pi$
$338$	−170.101			−0.503256
$339$		0			0
$340$	60.9117	+	32.4048i	0.179152	+	0.0953082i
$341$			470.463i			1.37966i
$342$		0			0
$343$			1222.78i			3.56497i
$344$		−	17.6471i		−	0.0512998i
$345$		0			0
$346$	39.5807			0.114395
$347$	−489.411			−1.41041			−0.705204	−	0.709005i	$-0.749144\pi$
$347$	−489.411			−1.41041			−0.705204	+	0.709005i	$0.749144\pi$
$348$		0			0
$349$	81.7939			0.234367			0.117183	−	0.993110i	$-0.462613\pi$
$349$	81.7939			0.234367			0.117183	+	0.993110i	$0.462613\pi$
$350$	401.304	−	270.422i	1.14658	−	0.772634i
$351$		0			0
$352$		−	69.6445i		−	0.197854i
$353$	−251.387			−0.712144			−0.356072	−	0.934459i	$-0.615884\pi$
$353$	−251.387			−0.712144			−0.356072	+	0.934459i	$0.615884\pi$
$354$		0			0
$355$	−246.500	+	463.349i	−0.694365	+	1.30521i
$356$		−	136.204i		−	0.382595i
$357$		0			0
$358$		−	29.7918i		−	0.0832172i
$359$		−	518.500i		−	1.44429i	−0.691742	−	0.722145i	$-0.743156\pi$
$359$		−	518.500i		−	1.44429i	0.691742	−	0.722145i	$-0.256844\pi$
$360$		0			0
$361$	−308.544			−0.854693
$362$	75.8061			0.209409
$363$		0			0
$364$	465.588			1.27909
$365$	4.96046	−	9.32425i	0.0135903	−	0.0255459i
$366$		0			0
$367$			341.983i			0.931834i	0.884828	+	0.465917i	$0.154276\pi$
$367$			341.983i			0.931834i	−0.884828	+	0.465917i	$0.845724\pi$
$368$	139.196			0.378250
$369$		0			0
$370$	−71.5290	+	134.454i	−0.193322	+	0.363389i
$371$		−	523.836i		−	1.41196i
$372$		0			0
$373$			110.303i			0.295719i	0.989008	+	0.147859i	$0.0472383\pi$
$373$			110.303i			0.295719i	−0.989008	+	0.147859i	$0.952762\pi$
$374$		−	120.128i		−	0.321198i
$375$		0			0
$376$	−113.706			−0.302409
$377$	−359.470			−0.953502
$378$		0			0
$379$	324.345			0.855792			0.427896	−	0.903828i	$-0.359255\pi$
$379$	324.345			0.855792			0.427896	+	0.903828i	$0.359255\pi$
$380$	−63.9411	−	34.0164i	−0.168266	−	0.0895169i
$381$		0			0
$382$			315.403i			0.825662i
$383$	572.480			1.49473			0.747363	−	0.664416i	$-0.231320\pi$
$383$	572.480			1.49473			0.747363	+	0.664416i	$0.231320\pi$
$384$		0			0
$385$	−743.838	−	395.719i	−1.93205	−	1.02784i
$386$			249.218i			0.645642i
$387$		0			0
$388$		−	202.487i		−	0.521873i
$389$		−	721.666i		−	1.85518i	−0.373595	−	0.927592i	$-0.621875\pi$
$389$		−	721.666i		−	1.85518i	0.373595	−	0.927592i	$-0.378125\pi$
$390$		0			0
$391$	240.095			0.614055
$392$	−391.279			−0.998161
$393$		0			0
$394$	392.566			0.996359
$395$	−194.647	−	103.551i	−0.492777	−	0.262155i
$396$		0			0
$397$		−	379.321i		−	0.955468i	−0.878505	−	0.477734i	$-0.841458\pi$
$397$		−	379.321i		−	0.955468i	0.878505	−	0.477734i	$-0.158542\pi$
$398$	101.220			0.254322
$399$		0			0
$400$	55.8823	+	82.9287i	0.139706	+	0.207322i
$401$			350.974i			0.875246i	0.899159	+	0.437623i	$0.144180\pi$
$401$			350.974i			0.875246i	−0.899159	+	0.437623i	$0.855820\pi$
$402$		0			0
$403$		−	649.938i		−	1.61275i
$404$			30.4593i			0.0753944i
$405$		0			0
$406$	409.103			1.00764
$407$	265.166			0.651513
$408$		0			0
$409$	−533.426			−1.30422			−0.652111	−	0.758124i	$-0.726116\pi$
$409$	−533.426			−1.30422			−0.652111	+	0.758124i	$0.726116\pi$
$410$	120.770	−	227.012i	0.294560	−	0.553688i
$411$		0			0
$412$		−	154.185i		−	0.374235i
$413$	569.813			1.37969
$414$		0			0
$415$	−242.889	−	129.216i	−0.585276	−	0.311364i
$416$			96.2130i			0.231281i
$417$		0			0
$418$			126.103i			0.301681i
$419$			239.755i			0.572208i	0.958199	+	0.286104i	$0.0923603\pi$
$419$			239.755i			0.572208i	−0.958199	+	0.286104i	$0.907640\pi$
$420$		0			0
$421$	424.941			1.00936			0.504681	−	0.863306i	$-0.331610\pi$
$421$	424.941			1.00936			0.504681	+	0.863306i	$0.331610\pi$
$422$	451.831			1.07069
$423$		0			0
$424$	108.250			0.255306
$425$	96.3898	+	143.042i	0.226800	+	0.336568i
$426$		0			0
$427$			206.113i			0.482700i
$428$	−156.853			−0.366479
$429$		0			0
$430$	20.7208	−	38.9491i	0.0481879	−	0.0905794i
$431$			544.039i			1.26227i	0.775673	+	0.631135i	$0.217411\pi$
$431$			544.039i			1.26227i	−0.775673	+	0.631135i	$0.782589\pi$
$432$		0			0
$433$		−	602.735i		−	1.39200i	−0.718043	−	0.695999i	$-0.754962\pi$
$433$		−	602.735i		−	1.39200i	0.718043	−	0.695999i	$-0.245038\pi$
$434$			739.676i			1.70432i
$435$		0			0
$436$	−292.558			−0.671006
$437$	−252.037			−0.576743
$438$		0			0
$439$	402.286			0.916370			0.458185	−	0.888857i	$-0.348500\pi$
$439$	402.286			0.916370			0.458185	+	0.888857i	$0.348500\pi$
$440$	81.7746	−	153.713i	0.185851	−	0.349347i
$441$		0			0
$442$			165.955i			0.375464i
$443$	438.515			0.989875			0.494938	−	0.868929i	$-0.335191\pi$
$443$	438.515			0.989875			0.494938	+	0.868929i	$0.335191\pi$
$444$		0			0
$445$	159.927	−	300.617i	0.359386	−	0.675543i
$446$		−	229.723i		−	0.515074i
$447$		0			0
$448$		−	109.497i		−	0.244413i
$449$		−	670.866i		−	1.49413i	−0.664749	−	0.747067i	$-0.731462\pi$
$449$		−	670.866i		−	1.49413i	0.664749	−	0.747067i	$-0.268538\pi$
$450$		0			0
$451$	−447.706			−0.992695
$452$	108.853			0.240825
$453$		0			0
$454$	340.963			0.751021
$455$	1027.60	+	546.680i	2.25847	+	1.20149i
$456$		0			0
$457$			695.029i			1.52085i	0.649425	+	0.760425i	$0.275010\pi$
$457$			695.029i			1.52085i	−0.649425	+	0.760425i	$0.724990\pi$
$458$	144.561			0.315636
$459$		0			0
$460$	307.220	+	163.440i	0.667870	+	0.355304i
$461$		−	13.0887i		−	0.0283920i	−0.999899	−	0.0141960i	$-0.995481\pi$
$461$		−	13.0887i		−	0.0283920i	0.999899	−	0.0141960i	$-0.00451888\pi$
$462$		0			0
$463$			234.281i			0.506007i	0.967465	+	0.253004i	$0.0814185\pi$
$463$			234.281i			0.506007i	−0.967465	+	0.253004i	$0.918582\pi$
$464$			84.5404i			0.182199i
$465$		0			0
$466$	342.191			0.734315
$467$	−742.882			−1.59075			−0.795377	−	0.606115i	$-0.792727\pi$
$467$	−742.882			−1.59075			−0.795377	+	0.606115i	$0.792727\pi$
$468$		0			0
$469$	1760.41			3.75354
$470$	−250.960	−	133.510i	−0.533958	−	0.284064i
$471$		0			0
$472$			117.751i			0.249472i
$473$	−76.8141			−0.162398
$474$		0			0
$475$	−101.184	−	150.156i	−0.213018	−	0.316117i
$476$		−	188.869i		−	0.396783i
$477$		0			0
$478$		−	530.978i		−	1.11083i
$479$		−	447.411i		−	0.934052i	−0.884244	−	0.467026i	$-0.845325\pi$
$479$		−	447.411i		−	0.934052i	0.884244	−	0.467026i	$-0.154675\pi$
$480$		0			0
$481$	−366.323			−0.761586
$482$	−450.115			−0.933848
$483$		0			0
$484$	−61.1472			−0.126337
$485$	237.754	−	446.910i	0.490215	−	0.921464i
$486$		0			0
$487$		−	50.3166i		−	0.103319i	−0.998665	−	0.0516597i	$-0.983549\pi$
$487$		−	50.3166i		−	0.103319i	0.998665	−	0.0516597i	$-0.0164511\pi$
$488$	−42.5929			−0.0872806
$489$		0			0
$490$	−863.595	−	459.429i	−1.76244	−	0.937610i
$491$		−	790.631i		−	1.61025i	−0.593107	−	0.805124i	$-0.702099\pi$
$491$		−	790.631i		−	1.61025i	0.593107	−	0.805124i	$-0.297901\pi$
$492$		0			0
$493$			145.821i			0.295784i
$494$		−	174.209i		−	0.352650i
$495$		0			0
$496$	−152.853			−0.308171
$497$	1436.70			2.89075
$498$		0			0
$499$	−600.390			−1.20319			−0.601593	−	0.798803i	$-0.705467\pi$
$499$	−600.390			−1.20319			−0.601593	+	0.798803i	$0.705467\pi$
$500$	25.9655	+	248.648i	0.0519310	+	0.497296i
$501$		0			0
$502$			120.974i			0.240985i
$503$	−433.368			−0.861566			−0.430783	−	0.902456i	$-0.641763\pi$
$503$	−433.368			−0.861566			−0.430783	+	0.902456i	$0.641763\pi$
$504$		0			0
$505$	−35.7645	+	67.2270i	−0.0708208	+	0.133123i
$506$		−	605.889i		−	1.19741i
$507$		0			0
$508$		−	318.431i		−	0.626832i
$509$		−	961.715i		−	1.88942i	−0.327907	−	0.944710i	$-0.606343\pi$
$509$		−	961.715i		−	1.88942i	0.327907	−	0.944710i	$-0.393657\pi$
$510$		0			0
$511$	−28.9117			−0.0565786
$512$	22.6274			0.0441942
$513$		0			0
$514$	−302.963			−0.589423
$515$	181.040	−	340.302i	0.351533	−	0.660781i
$516$		0			0
$517$			494.936i			0.957322i
$518$	416.902			0.804829
$519$		0			0
$520$	−112.971	+	212.352i	−0.217251	+	0.408370i
$521$		−	5.90542i		−	0.0113348i	−0.999984	−	0.00566739i	$-0.998196\pi$
$521$		−	5.90542i		−	0.0113348i	0.999984	−	0.00566739i	$-0.00180400\pi$
$522$		0			0
$523$		−	165.846i		−	0.317104i	−0.987351	−	0.158552i	$-0.949317\pi$
$523$		−	165.846i		−	0.317104i	0.987351	−	0.158552i	$-0.0506826\pi$
$524$			220.606i			0.421004i
$525$		0			0
$526$	−151.147			−0.287352
$527$	−263.652			−0.500288
$528$		0			0
$529$	681.970			1.28917
$530$	238.919	+	127.104i	0.450790	+	0.239819i
$531$		0			0
$532$			198.262i			0.372673i
$533$	618.500			1.16041
$534$		0			0
$535$	−346.191	−	184.172i	−0.647086	−	0.344247i
$536$			363.786i			0.678705i
$537$		0			0
$538$			517.499i			0.961894i
$539$			1703.15i			3.15984i
$540$		0			0
$541$	216.985			0.401081			0.200541	−	0.979685i	$-0.435730\pi$
$541$	216.985			0.401081			0.200541	+	0.979685i	$0.435730\pi$
$542$	−131.365			−0.242372
$543$		0			0
$544$	39.0294			0.0717453
$545$	−645.708	−	343.514i	−1.18478	−	0.630301i
$546$		0			0
$547$		−	100.645i		−	0.183995i	−0.995759	−	0.0919973i	$-0.970675\pi$
$547$		−	100.645i		−	0.183995i	0.995759	−	0.0919973i	$-0.0293251\pi$
$548$	435.838			0.795324
$549$		0			0
$550$	360.971	−	243.243i	0.656310	−	0.442260i
$551$		−	153.074i		−	0.277811i
$552$		0			0
$553$			603.541i			1.09139i
$554$			319.806i			0.577268i
$555$		0			0
$556$	−212.500			−0.382193
$557$	116.662			0.209447			0.104723	−	0.994501i	$-0.466604\pi$
$557$	116.662			0.209447			0.104723	+	0.994501i	$0.466604\pi$
$558$		0			0
$559$	106.118			0.189835
$560$	128.569	−	241.672i	0.229587	−	0.431557i
$561$		0			0
$562$		−	89.6686i		−	0.159553i
$563$	533.907			0.948324			0.474162	−	0.880438i	$-0.342751\pi$
$563$	533.907			0.948324			0.474162	+	0.880438i	$0.342751\pi$
$564$		0			0
$565$	240.250	+	127.812i	0.425221	+	0.226216i
$566$		−	513.510i		−	0.907262i
$567$		0			0
$568$			296.893i			0.522698i
$569$			112.651i			0.197981i	0.995088	+	0.0989907i	$0.0315614\pi$
$569$			112.651i			0.197981i	−0.995088	+	0.0989907i	$0.968439\pi$
$570$		0			0
$571$	−861.535			−1.50882			−0.754409	−	0.656404i	$-0.772077\pi$
$571$	−861.535			−1.50882			−0.754409	+	0.656404i	$0.772077\pi$
$572$	418.794			0.732157
$573$		0			0
$574$	−703.897			−1.22630
$575$	486.161	+	721.459i	0.845498	+	1.25471i
$576$		0			0
$577$		−	37.3256i		−	0.0646891i	−0.999477	−	0.0323446i	$-0.989703\pi$
$577$		−	37.3256i		−	0.0646891i	0.999477	−	0.0323446i	$-0.0102974\pi$
$578$	−341.387			−0.590635
$579$		0			0
$580$	−99.2649	+	186.590i	−0.171146	+	0.321706i
$581$			753.127i			1.29626i
$582$		0			0
$583$		−	471.188i		−	0.808212i
$584$		−	5.97455i		−	0.0102304i
$585$		0			0
$586$	−321.640			−0.548873
$587$	−538.103			−0.916699			−0.458350	−	0.888772i	$-0.651559\pi$
$587$	−538.103			−0.916699			−0.458350	+	0.888772i	$0.651559\pi$
$588$		0			0
$589$	276.765			0.469889
$590$	−138.260	+	259.889i	−0.234339	+	0.440490i
$591$		0			0
$592$			86.1520i			0.145527i
$593$	−884.169			−1.49101			−0.745505	−	0.666500i	$-0.767792\pi$
$593$	−884.169			−1.49101			−0.745505	+	0.666500i	$0.767792\pi$
$594$		0			0
$595$	221.765	−	416.854i	0.372713	−	0.700595i
$596$			28.3184i			0.0475141i
$597$		0			0
$598$			837.029i			1.39971i
$599$			44.7736i			0.0747472i	0.999301	+	0.0373736i	$0.0118992\pi$
$599$			44.7736i			0.0747472i	−0.999301	+	0.0373736i	$0.988101\pi$
$600$		0			0
$601$	394.574			0.656528			0.328264	−	0.944586i	$-0.393536\pi$
$601$	394.574			0.656528			0.328264	+	0.944586i	$0.393536\pi$
$602$	−120.770			−0.200614
$603$		0			0
$604$	146.676			0.242841
$605$	−134.958	−	71.7973i	−0.223072	−	0.118673i
$606$		0			0
$607$		−	215.173i		−	0.354485i	−0.984167	−	0.177243i	$-0.943282\pi$
$607$		−	215.173i		−	0.354485i	0.984167	−	0.177243i	$-0.0567178\pi$
$608$	−40.9706			−0.0673858
$609$		0			0
$610$	−94.0071	−	50.0114i	−0.154110	−	0.0819859i
$611$		−	683.747i		−	1.11906i
$612$		0			0
$613$		−	333.937i		−	0.544758i	−0.962190	−	0.272379i	$-0.912189\pi$
$613$		−	333.937i		−	0.544758i	0.962190	−	0.272379i	$-0.0878105\pi$
$614$		−	481.065i		−	0.783493i
$615$		0			0
$616$	−476.617			−0.773729
$617$	−65.4234			−0.106035			−0.0530174	−	0.998594i	$-0.516884\pi$
$617$	−65.4234			−0.106035			−0.0530174	+	0.998594i	$0.516884\pi$
$618$		0			0
$619$	−19.9268			−0.0321920			−0.0160960	−	0.999870i	$-0.505124\pi$
$619$	−19.9268			−0.0321920			−0.0160960	+	0.999870i	$0.505124\pi$
$620$	−337.362	−	179.475i	−0.544133	−	0.289477i
$621$		0			0
$622$		−	546.973i		−	0.879379i
$623$	−932.122			−1.49618
$624$		0			0
$625$	−234.647	+	579.281i	−0.375435	+	0.926849i
$626$			532.314i			0.850341i
$627$		0			0
$628$		−	140.290i		−	0.223392i
$629$			148.601i			0.236250i
$630$		0			0
$631$	−210.625			−0.333796			−0.166898	−	0.985974i	$-0.553375\pi$
$631$	−210.625			−0.333796			−0.166898	+	0.985974i	$0.553375\pi$
$632$	−124.721			−0.197343
$633$		0			0
$634$	502.610			0.792761
$635$	373.892	−	702.811i	0.588807	−	1.10679i
$636$		0			0
$637$		−	2352.88i		−	3.69369i
$638$	367.986			0.576780
$639$		0			0
$640$	49.9411	+	26.5685i	0.0780330	+	0.0415132i
$641$			1177.76i			1.83738i	0.394976	+	0.918692i	$0.370753\pi$
$641$			1177.76i			1.83738i	−0.394976	+	0.918692i	$0.629247\pi$
$642$		0			0
$643$			37.6426i			0.0585422i	0.999572	+	0.0292711i	$0.00931861\pi$
$643$			37.6426i			0.0585422i	−0.999572	+	0.0292711i	$0.990681\pi$
$644$		−	952.598i		−	1.47919i
$645$		0			0
$646$	−70.6690			−0.109395
$647$	−695.382			−1.07478			−0.537389	−	0.843334i	$-0.680589\pi$
$647$	−695.382			−1.07478			−0.537389	+	0.843334i	$0.680589\pi$
$648$		0			0
$649$	512.544			0.789744
$650$	−498.676	+	336.037i	−0.767194	+	0.516980i
$651$		0			0
$652$		−	18.1192i		−	0.0277902i
$653$	−924.296			−1.41546			−0.707731	−	0.706482i	$-0.750281\pi$
$653$	−924.296			−1.41546			−0.707731	+	0.706482i	$0.750281\pi$
$654$		0			0
$655$	−259.029	+	486.901i	−0.395465	+	0.743361i
$656$		−	145.459i		−	0.221736i
$657$		0			0
$658$			778.153i			1.18260i
$659$		−	67.6704i		−	0.102687i	−0.998681	−	0.0513433i	$-0.983650\pi$
$659$		−	67.6704i		−	0.102687i	0.998681	−	0.0513433i	$-0.0163503\pi$
$660$		0			0
$661$	1032.87			1.56258			0.781291	−	0.624167i	$-0.214561\pi$
$661$	1032.87			1.56258			0.781291	+	0.624167i	$0.214561\pi$
$662$	−628.181			−0.948914
$663$		0			0
$664$	−155.632			−0.234386
$665$	−232.794	+	437.586i	−0.350066	+	0.658024i
$666$		0			0
$667$			735.480i			1.10267i
$668$	−127.706			−0.191176
$669$		0			0
$670$	−427.147	+	802.914i	−0.637533	+	1.19838i
$671$			185.398i			0.276301i
$672$		0			0
$673$		−	10.1706i		−	0.0151123i	−0.999971	−	0.00755614i	$-0.997595\pi$
$673$		−	10.1706i		−	0.0151123i	0.999971	−	0.00755614i	$-0.00240522\pi$
$674$		−	575.707i		−	0.854164i
$675$		0			0
$676$	−240.558			−0.355856
$677$	576.936			0.852195			0.426098	−	0.904677i	$-0.359888\pi$
$677$	576.936			0.852195			0.426098	+	0.904677i	$0.359888\pi$
$678$		0			0
$679$	−1385.73			−2.04085
$680$	86.1421	+	45.8273i	0.126680	+	0.0673930i
$681$		0			0
$682$			665.335i			0.975564i
$683$	−526.009			−0.770145			−0.385073	−	0.922886i	$-0.625824\pi$
$683$	−526.009			−0.770145			−0.385073	+	0.922886i	$0.625824\pi$
$684$		0			0
$685$	961.940	+	511.748i	1.40429	+	0.747078i
$686$			1729.28i			2.52081i
$687$		0			0
$688$		−	24.9568i		−	0.0362745i
$689$			650.940i			0.944760i
$690$		0			0
$691$	422.904			0.612017			0.306008	−	0.952029i	$-0.401006\pi$
$691$	422.904			0.612017			0.306008	+	0.952029i	$0.401006\pi$
$692$	55.9756			0.0808896
$693$		0			0
$694$	−692.132			−0.997308
$695$	−469.009	−	249.511i	−0.674833	−	0.359009i
$696$		0			0
$697$		−	250.899i		−	0.359969i
$698$	115.674			0.165722
$699$		0			0
$700$	567.529	−	382.434i	0.810756	−	0.546335i
$701$		−	881.146i		−	1.25698i	−0.777816	−	0.628492i	$-0.783672\pi$
$701$		−	881.146i		−	1.25698i	0.777816	−	0.628492i	$-0.216328\pi$
$702$		0			0
$703$		−	155.992i		−	0.221895i
$704$		−	98.4922i		−	0.139904i
$705$		0			0
$706$	−355.515			−0.503562
$707$	208.451			0.294838
$708$		0			0
$709$	101.427			0.143057			0.0715284	−	0.997439i	$-0.477212\pi$
$709$	101.427			0.143057			0.0715284	+	0.997439i	$0.477212\pi$
$710$	−348.603	+	655.274i	−0.490990	+	0.922921i
$711$		0			0
$712$		−	192.621i		−	0.270536i
$713$	−1329.78			−1.86505
$714$		0			0
$715$	924.323	+	491.736i	1.29276	+	0.687743i
$716$		−	42.1319i		−	0.0588435i
$717$		0			0
$718$		−	733.270i		−	1.02127i
$719$		−	36.2956i		−	0.0504807i	−0.999681	−	0.0252404i	$-0.991965\pi$
$719$		−	36.2956i		−	0.0504807i	0.999681	−	0.0252404i	$-0.00803511\pi$
$720$		0			0
$721$	−1055.18			−1.46349
$722$	−436.347			−0.604359
$723$		0			0
$724$	107.206			0.148075
$725$	−438.177	+	295.269i	−0.604382	+	0.407268i
$726$		0			0
$727$			527.755i			0.725936i	0.931802	+	0.362968i	$0.118237\pi$
$727$			527.755i			0.725936i	−0.931802	+	0.362968i	$0.881763\pi$
$728$	658.441			0.904452
$729$		0			0
$730$	7.01515	−	13.1865i	0.00960980	−	0.0180637i
$731$		−	43.0474i		−	0.0588883i
$732$		0			0
$733$		−	1305.93i		−	1.78163i	−0.454370	−	0.890813i	$-0.650136\pi$
$733$		−	1305.93i		−	1.78163i	0.454370	−	0.890813i	$-0.349864\pi$
$734$			483.637i			0.658906i
$735$		0			0
$736$	196.853			0.267463
$737$	1583.48			2.14855
$738$		0			0
$739$	754.875			1.02148			0.510741	−	0.859735i	$-0.329371\pi$
$739$	754.875			1.02148			0.510741	+	0.859735i	$0.329371\pi$
$740$	−101.157	+	190.147i	−0.136699	+	0.256955i
$741$		0			0
$742$		−	740.816i		−	0.998404i
$743$	1312.59			1.76660			0.883302	−	0.468804i	$-0.155315\pi$
$743$	1312.59			1.76660			0.883302	+	0.468804i	$0.155315\pi$
$744$		0			0
$745$	−33.2506	+	62.5017i	−0.0446317	+	0.0838949i
$746$			155.992i			0.209105i
$747$		0			0
$748$		−	169.887i		−	0.227121i
$749$			1073.43i			1.43316i
$750$		0			0
$751$	1232.43			1.64106			0.820528	−	0.571607i	$-0.193680\pi$
$751$	1232.43			1.64106			0.820528	+	0.571607i	$0.193680\pi$
$752$	−160.804			−0.213835
$753$		0			0
$754$	−508.368			−0.674227
$755$	323.730	+	172.223i	0.428781	+	0.228110i
$756$		0			0
$757$		−	401.164i		−	0.529939i	−0.964257	−	0.264970i	$-0.914638\pi$
$757$		−	401.164i		−	0.529939i	0.964257	−	0.264970i	$-0.0853619\pi$
$758$	458.693			0.605136
$759$		0			0
$760$	−90.4264	−	48.1065i	−0.118982	−	0.0632980i
$761$		−	525.130i		−	0.690053i	−0.938593	−	0.345027i	$-0.887870\pi$
$761$		−	525.130i		−	0.690053i	0.938593	−	0.345027i	$-0.112130\pi$
$762$		0			0
$763$			2002.15i			2.62404i
$764$			446.047i			0.583831i
$765$		0			0
$766$	809.609			1.05693
$767$	−708.073			−0.923172
$768$		0			0
$769$	−542.750			−0.705787			−0.352894	−	0.935663i	$-0.614802\pi$
$769$	−542.750			−0.705787			−0.352894	+	0.935663i	$0.614802\pi$
$770$	−1051.95	−	559.631i	−1.36616	−	0.726793i
$771$		0			0
$772$			352.447i			0.456538i
$773$	−522.640			−0.676119			−0.338059	−	0.941125i	$-0.609770\pi$
$773$	−522.640			−0.676119			−0.338059	+	0.941125i	$0.609770\pi$
$774$		0			0
$775$	−533.860	−	792.243i	−0.688852	−	1.02225i
$776$		−	286.360i		−	0.369020i
$777$		0			0
$778$		−	1020.59i		−	1.31181i
$779$			263.377i			0.338096i
$780$		0			0
$781$	1292.31			1.65468
$782$	339.546			0.434202
$783$		0			0
$784$	−553.352			−0.705807
$785$	164.725	−	309.636i	0.209841	−	0.394440i
$786$		0			0
$787$			828.884i			1.05322i	0.850107	+	0.526610i	$0.176537\pi$
$787$			828.884i			1.05322i	−0.850107	+	0.526610i	$0.823463\pi$
$788$	555.172			0.704532

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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(4.41421 + 2.34834i) q^{5} -13.6872i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(4.41421 + 2.34834i) q^{5} -13.6872i q^{7} +2.82843 q^{8} +(6.24264 + 3.32106i) q^{10} -12.3115i q^{11} +17.0082i q^{13} -19.3566i q^{14} +4.00000 q^{16} +6.89949 q^{17} -7.24264 q^{19} +(8.82843 + 4.69669i) q^{20} -17.4111i q^{22} +34.7990 q^{23} +(13.9706 + 20.7322i) q^{25} +24.0532i q^{26} -27.3743i q^{28} +21.1351i q^{29} -38.2132 q^{31} +5.65685 q^{32} +9.75736 q^{34} +(32.1421 - 60.4180i) q^{35} +21.5380i q^{37} -10.2426 q^{38} +(12.4853 + 6.64212i) q^{40} -36.3648i q^{41} -6.23921i q^{43} -24.6230i q^{44} +49.2132 q^{46} -40.2010 q^{47} -138.338 q^{49} +(19.7574 + 29.3197i) q^{50} +34.0164i q^{52} +38.2721 q^{53} +(28.9117 - 54.3457i) q^{55} -38.7131i q^{56} +29.8895i q^{58} +41.6313i q^{59} -15.0589 q^{61} -54.0416 q^{62} +8.00000 q^{64} +(-39.9411 + 75.0779i) q^{65} +128.618i q^{67} +13.7990 q^{68} +(45.4558 - 85.4440i) q^{70} +104.967i q^{71} -2.11232i q^{73} +30.4593i q^{74} -14.4853 q^{76} -168.510 q^{77} -44.0955 q^{79} +(17.6569 + 9.39338i) q^{80} -51.4275i q^{82} -55.0244 q^{83} +(30.4558 + 16.2024i) q^{85} -8.82357i q^{86} -34.8222i q^{88} -68.1020i q^{89} +232.794 q^{91} +69.5980 q^{92} -56.8528 q^{94} +(-31.9706 - 17.0082i) q^{95} -101.243i q^{97} -195.640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 12 q^{5}+O(q^{10}) \) 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

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