LMFDB - Embedded newform 270.3.b.d.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			$1.66053i$ of defining polynomial
Character	$\chi$	$=$	270.269
Dual form			270.3.b.d.269.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	$0.432581\pi$
$7$			13.6872i			1.95531i	−0.210222	+	0.977654i	$0.567419\pi$
$8$	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	$0.189051\pi$
$11$			12.3115i			1.11923i	−0.828753	+	0.559615i	$0.810949\pi$
$12$		0			0
$13$		−	17.0082i		−	1.30832i	−0.756355	−	0.654162i	$-0.773021\pi$
$13$		−	17.0082i		−	1.30832i	0.756355	−	0.654162i	$-0.226979\pi$
$14$			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	$-0.881275\pi$
$29$		−	21.1351i		−	0.728796i	0.931243	−	0.364398i	$-0.118725\pi$
$30$		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	$-0.905994\pi$
$37$		−	21.5380i		−	0.582108i	0.956707	−	0.291054i	$-0.0940060\pi$
$38$	−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	$0.146254\pi$
$41$			36.3648i			0.886945i	−0.896288	+	0.443473i	$0.853746\pi$
$42$		0			0
$43$			6.23921i			0.145098i	0.997365	+	0.0725489i	$0.0231133\pi$
$43$			6.23921i			0.145098i	−0.997365	+	0.0725489i	$0.976887\pi$
$44$			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	$-0.885227\pi$
$59$		−	41.6313i		−	0.705615i	0.935696	−	0.352807i	$-0.114773\pi$
$60$		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	$-0.590523\pi$
$67$		−	128.618i		−	1.91967i	0.280569	−	0.959834i	$-0.409477\pi$
$68$	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	$-0.735200\pi$
$71$		−	104.967i		−	1.47841i	0.673478	−	0.739207i	$-0.264800\pi$
$72$		0			0
$73$			2.11232i			0.0289359i	0.999895	+	0.0144680i	$0.00460546\pi$
$73$			2.11232i			0.0289359i	−0.999895	+	0.0144680i	$0.995395\pi$
$74$		−	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	$0.124970\pi$
$89$			68.1020i			0.765191i	−0.923916	+	0.382595i	$0.875030\pi$
$90$		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	$0.174767\pi$
$97$			101.243i			1.04375i	−0.853023	+	0.521873i	$0.825233\pi$
$98$	−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	$-0.975978\pi$
$101$		−	15.2297i		−	0.150789i	0.997154	−	0.0753944i	$-0.0240216\pi$
$102$		0			0
$103$			77.0924i			0.748470i	0.927334	+	0.374235i	$0.122095\pi$
$103$			77.0924i			0.748470i	−0.927334	+	0.374235i	$0.877905\pi$
$104$		−	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	$0.215649\pi$
$127$			159.215i			1.25366i	−0.779154	+	0.626832i	$0.784351\pi$
$128$	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	$-0.861678\pi$
$131$		−	110.303i		−	0.842008i	0.907059	−	0.421004i	$-0.138322\pi$
$132$		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	$-0.984870\pi$
$149$		−	14.1592i		−	0.0950281i	0.998871	−	0.0475141i	$-0.0151299\pi$
$150$		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	$0.0717131\pi$
$157$			70.1452i			0.446784i	−0.974729	+	0.223392i	$0.928287\pi$
$158$	−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	$0.00884702\pi$
$163$			9.05959i			0.0555803i	−0.999614	+	0.0277902i	$0.991153\pi$
$164$			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	$0.0187413\pi$
$179$			21.0660i			0.117687i	−0.998267	+	0.0588435i	$0.981259\pi$
$180$		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	$-0.801553\pi$
$191$		−	223.024i		−	1.16766i	0.811875	−	0.583831i	$-0.198447\pi$
$192$		0			0
$193$		−	176.224i		−	0.913075i	−0.889704	−	0.456538i	$-0.849089\pi$
$193$		−	176.224i		−	0.913075i	0.889704	−	0.456538i	$-0.150911\pi$
$194$			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	$0.118662\pi$
$223$			162.439i			0.728424i	−0.931316	+	0.364212i	$0.881338\pi$
$224$			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	$0.287583\pi$
$239$			375.458i			1.57096i	−0.618890	+	0.785478i	$0.712417\pi$
$240$		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	$-0.945493\pi$
$251$		−	85.5417i		−	0.340804i	0.985375	−	0.170402i	$-0.0545066\pi$
$252$		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	$-0.761910\pi$
$269$		−	365.927i		−	1.36032i	0.733062	−	0.680161i	$-0.238090\pi$
$270$		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	$-0.866160\pi$
$277$		−	226.137i		−	0.816380i	0.912897	−	0.408190i	$-0.133840\pi$
$278$	−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	$0.0359886\pi$
$281$			63.4053i			0.225642i	−0.993615	+	0.112821i	$0.964011\pi$
$282$		0			0
$283$			363.106i			1.28306i	0.767097	+	0.641531i	$0.221700\pi$
$283$			363.106i			1.28306i	−0.767097	+	0.641531i	$0.778300\pi$
$284$		−	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	$0.186904\pi$
$307$			340.164i			1.10803i	−0.832508	+	0.554013i	$0.813096\pi$
$308$	−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	$0.213604\pi$
$311$			386.769i			1.24363i	−0.783164	+	0.621815i	$0.786396\pi$
$312$		0			0
$313$		−	376.403i		−	1.20256i	−0.799037	−	0.601282i	$-0.794657\pi$
$313$		−	376.403i		−	1.20256i	0.799037	−	0.601282i	$-0.205343\pi$
$314$			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	$0.206421\pi$
$337$			407.086i			1.20797i	−0.796995	+	0.603985i	$0.793579\pi$
$338$	−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	$0.256844\pi$
$359$			518.500i			1.44429i	−0.691742	+	0.722145i	$0.743156\pi$
$360$		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	$-0.845724\pi$
$367$		−	341.983i		−	0.931834i	0.884828	−	0.465917i	$-0.154276\pi$
$368$	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	$-0.952762\pi$
$373$		−	110.303i		−	0.295719i	0.989008	−	0.147859i	$-0.0472383\pi$
$374$			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	$0.378125\pi$
$389$			721.666i			1.85518i	−0.373595	+	0.927592i	$0.621875\pi$
$390$		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	$0.158542\pi$
$397$			379.321i			0.955468i	−0.878505	+	0.477734i	$0.841458\pi$
$398$	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	$-0.855820\pi$
$401$		−	350.974i		−	0.875246i	0.899159	−	0.437623i	$-0.144180\pi$
$402$		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	$-0.907640\pi$
$419$		−	239.755i		−	0.572208i	0.958199	−	0.286104i	$-0.0923603\pi$
$420$		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	$-0.782589\pi$
$431$		−	544.039i		−	1.26227i	0.775673	−	0.631135i	$-0.217411\pi$
$432$		0			0
$433$			602.735i			1.39200i	0.718043	+	0.695999i	$0.245038\pi$
$433$			602.735i			1.39200i	−0.718043	+	0.695999i	$0.754962\pi$
$434$		−	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	$0.268538\pi$
$449$			670.866i			1.49413i	−0.664749	+	0.747067i	$0.731462\pi$
$450$		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	$-0.724990\pi$
$457$		−	695.029i		−	1.52085i	0.649425	−	0.760425i	$-0.275010\pi$
$458$	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	$0.00451888\pi$
$461$			13.0887i			0.0283920i	−0.999899	+	0.0141960i	$0.995481\pi$
$462$		0			0
$463$		−	234.281i		−	0.506007i	−0.967465	−	0.253004i	$-0.918582\pi$
$463$		−	234.281i		−	0.506007i	0.967465	−	0.253004i	$-0.0814185\pi$
$464$		−	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	$0.154675\pi$
$479$			447.411i			0.934052i	−0.884244	+	0.467026i	$0.845325\pi$
$480$		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	$0.0164511\pi$
$487$			50.3166i			0.103319i	−0.998665	+	0.0516597i	$0.983549\pi$
$488$	−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	$0.297901\pi$
$491$			790.631i			1.61025i	−0.593107	+	0.805124i	$0.702099\pi$
$492$		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	$0.393657\pi$
$509$			961.715i			1.88942i	−0.327907	+	0.944710i	$0.606343\pi$
$510$		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	$0.00180400\pi$
$521$			5.90542i			0.0113348i	−0.999984	+	0.00566739i	$0.998196\pi$
$522$		0			0
$523$			165.846i			0.317104i	0.987351	+	0.158552i	$0.0506826\pi$
$523$			165.846i			0.317104i	−0.987351	+	0.158552i	$0.949317\pi$
$524$		−	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	$0.0293251\pi$
$547$			100.645i			0.183995i	−0.995759	+	0.0919973i	$0.970675\pi$
$548$	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	$-0.968439\pi$
$569$		−	112.651i		−	0.197981i	0.995088	−	0.0989907i	$-0.0315614\pi$
$570$		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	$0.0102974\pi$
$577$			37.3256i			0.0646891i	−0.999477	+	0.0323446i	$0.989703\pi$
$578$	−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	$-0.988101\pi$
$599$		−	44.7736i		−	0.0747472i	0.999301	−	0.0373736i	$-0.0118992\pi$
$600$		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	$0.0567178\pi$
$607$			215.173i			0.354485i	−0.984167	+	0.177243i	$0.943282\pi$
$608$	−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	$0.0878105\pi$
$613$			333.937i			0.544758i	−0.962190	+	0.272379i	$0.912189\pi$
$614$			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	$-0.629247\pi$
$641$		−	1177.76i		−	1.83738i	0.394976	−	0.918692i	$-0.370753\pi$
$642$		0			0
$643$		−	37.6426i		−	0.0585422i	−0.999572	−	0.0292711i	$-0.990681\pi$
$643$		−	37.6426i		−	0.0585422i	0.999572	−	0.0292711i	$-0.00931861\pi$
$644$			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	$0.0163503\pi$
$659$			67.6704i			0.102687i	−0.998681	+	0.0513433i	$0.983650\pi$
$660$		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	$0.00240522\pi$
$673$			10.1706i			0.0151123i	−0.999971	+	0.00755614i	$0.997595\pi$
$674$			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	$0.216328\pi$
$701$			881.146i			1.25698i	−0.777816	+	0.628492i	$0.783672\pi$
$702$		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	$0.00803511\pi$
$719$			36.2956i			0.0504807i	−0.999681	+	0.0252404i	$0.991965\pi$
$720$		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	$-0.881763\pi$
$727$		−	527.755i		−	0.725936i	0.931802	−	0.362968i	$-0.118237\pi$
$728$	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	$0.349864\pi$
$733$			1305.93i			1.78163i	−0.454370	+	0.890813i	$0.650136\pi$
$734$		−	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	$0.0853619\pi$
$757$			401.164i			0.529939i	−0.964257	+	0.264970i	$0.914638\pi$
$758$	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	$0.112130\pi$
$761$			525.130i			0.690053i	−0.938593	+	0.345027i	$0.887870\pi$
$762$		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	$-0.823463\pi$
$787$		−	828.884i		−	1.05322i	0.850107	−	0.526610i	$-0.176537\pi$
$788$	555.172			0.704532
$789$		0			0
$790$	−275.272	+	146.444i	−0.348446	+	0.185372i

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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(4.41421 - 2.34834i) q^{5} +13.6872i q^{7} +2.82843 q^{8} +(6.24264 - 3.32106i) q^{10} +12.3115i q^{11} -17.0082i q^{13} +19.3566i q^{14} +4.00000 q^{16} +6.89949 q^{17} -7.24264 q^{19} +(8.82843 - 4.69669i) q^{20} +17.4111i q^{22} +34.7990 q^{23} +(13.9706 - 20.7322i) q^{25} -24.0532i q^{26} +27.3743i q^{28} -21.1351i q^{29} -38.2132 q^{31} +5.65685 q^{32} +9.75736 q^{34} +(32.1421 + 60.4180i) q^{35} -21.5380i q^{37} -10.2426 q^{38} +(12.4853 - 6.64212i) q^{40} +36.3648i q^{41} +6.23921i q^{43} +24.6230i q^{44} +49.2132 q^{46} -40.2010 q^{47} -138.338 q^{49} +(19.7574 - 29.3197i) q^{50} -34.0164i q^{52} +38.2721 q^{53} +(28.9117 + 54.3457i) q^{55} +38.7131i q^{56} -29.8895i q^{58} -41.6313i q^{59} -15.0589 q^{61} -54.0416 q^{62} +8.00000 q^{64} +(-39.9411 - 75.0779i) q^{65} -128.618i q^{67} +13.7990 q^{68} +(45.4558 + 85.4440i) q^{70} -104.967i q^{71} +2.11232i q^{73} -30.4593i q^{74} -14.4853 q^{76} -168.510 q^{77} -44.0955 q^{79} +(17.6569 - 9.39338i) q^{80} +51.4275i q^{82} -55.0244 q^{83} +(30.4558 - 16.2024i) q^{85} +8.82357i q^{86} +34.8222i q^{88} +68.1020i q^{89} +232.794 q^{91} +69.5980 q^{92} -56.8528 q^{94} +(-31.9706 + 17.0082i) q^{95} +101.243i q^{97} -195.640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 12 q^{5} + 8 q^{10} + 16 q^{16} - 12 q^{17} - 12 q^{19} + 24 q^{20} + 60 q^{23} - 12 q^{25} - 68 q^{31} + 56 q^{34} + 72 q^{35} - 24 q^{38} + 16 q^{40} + 112 q^{46} - 240 q^{47} - 180 q^{49}+ \cdots - 528 q^{98}+O(q^{100}) \) 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

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