LMFDB - Embedded newform 270.3.b.a.269.1

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			269.1
Root			$1.66053i$ of defining polynomial
Character	$\chi$	$=$	270.269
Dual form			270.3.b.a.269.2

$q$ -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	$-0.567419\pi$
$7$		−	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	$0.189051\pi$
$11$			12.3115i			1.11923i	−0.828753	+	0.559615i	$0.810949\pi$
$12$		0			0
$13$			17.0082i			1.30832i	0.756355	+	0.654162i	$0.226979\pi$
$13$			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	$-0.565045\pi$
$17$	−6.89949			−0.405853			−0.202926	+	0.979194i	$0.565045\pi$
$18$		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	$-0.773092\pi$
$23$	−34.7990			−1.51300			−0.756500	+	0.653994i	$0.773092\pi$
$24$		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	$0.0940060\pi$
$37$			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	$0.146254\pi$
$41$			36.3648i			0.886945i	−0.896288	+	0.443473i	$0.853746\pi$
$42$		0			0
$43$		−	6.23921i		−	0.145098i	−0.997365	−	0.0725489i	$-0.976887\pi$
$43$		−	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	$-0.359334\pi$
$47$	40.2010			0.855341			0.427670	+	0.903935i	$0.359334\pi$
$48$		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	$-0.617584\pi$
$53$	−38.2721			−0.722115			−0.361057	+	0.932544i	$0.617584\pi$
$54$		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	$0.409477\pi$
$67$			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	$-0.735200\pi$
$71$		−	104.967i		−	1.47841i	0.673478	−	0.739207i	$-0.264800\pi$
$72$		0			0
$73$		−	2.11232i		−	0.0289359i	−0.999895	−	0.0144680i	$-0.995395\pi$
$73$		−	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	$-0.392455\pi$
$83$	55.0244			0.662944			0.331472	+	0.943465i	$0.392455\pi$
$84$		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	$-0.825233\pi$
$97$		−	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	$-0.975978\pi$
$101$		−	15.2297i		−	0.150789i	0.997154	−	0.0753944i	$-0.0240216\pi$
$102$		0			0
$103$		−	77.0924i		−	0.748470i	−0.927334	−	0.374235i	$-0.877905\pi$
$103$		−	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	$-0.380563\pi$
$107$	78.4264			0.732957			0.366479	+	0.930427i	$0.380563\pi$
$108$		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	$-0.577418\pi$
$113$	−54.4264			−0.481650			−0.240825	+	0.970569i	$0.577418\pi$
$114$		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	$-0.861678\pi$
$131$		−	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	$-0.792699\pi$
$137$	−217.919			−1.59065			−0.795324	+	0.606184i	$0.792699\pi$
$138$		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	$-0.928287\pi$
$157$		−	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	$-0.438770\pi$
$167$	63.8528			0.382352			0.191176	+	0.981556i	$0.438770\pi$
$168$		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	$-0.525776\pi$
$173$	−27.9878			−0.161779			−0.0808896	+	0.996723i	$0.525776\pi$
$174$		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	$0.150911\pi$
$193$			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	$-0.748843\pi$
$197$	−277.586			−1.40906			−0.704532	+	0.709672i	$0.748843\pi$
$198$		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	$-0.678203\pi$
$227$	−241.098			−1.06210			−0.531052	+	0.847339i	$0.678203\pi$
$228$		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	$-0.673785\pi$
$233$	−241.966			−1.03848			−0.519239	+	0.854629i	$0.673785\pi$
$234$		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	$-0.363157\pi$
$257$	214.227			0.833570			0.416785	+	0.909005i	$0.363157\pi$
$258$		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	$-0.434870\pi$
$263$	106.877			0.406377			0.203189	+	0.979140i	$0.434870\pi$
$264$		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	$0.133840\pi$
$277$			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	$0.0359886\pi$
$281$			63.4053i			0.225642i	−0.993615	+	0.112821i	$0.964011\pi$
$282$		0			0
$283$		−	363.106i		−	1.28306i	−0.767097	−	0.641531i	$-0.778300\pi$
$283$		−	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	$-0.373127\pi$
$293$	227.434			0.776224			0.388112	+	0.921612i	$0.373127\pi$
$294$		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	$0.213604\pi$
$311$			386.769i			1.24363i	−0.783164	+	0.621815i	$0.786396\pi$
$312$		0			0
$313$			376.403i			1.20256i	0.799037	+	0.601282i	$0.205343\pi$
$313$			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	$-0.689417\pi$
$317$	−355.399			−1.12113			−0.560566	+	0.828110i	$0.689417\pi$
$318$		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	$-0.250856\pi$
$347$	489.411			1.41041			0.705204	+	0.709005i	$0.250856\pi$
$348$		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	$-0.384116\pi$
$353$	251.387			0.712144			0.356072	+	0.934459i	$0.384116\pi$
$354$		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	$0.154276\pi$
$367$			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	$-0.768680\pi$
$383$	−572.480			−1.49473			−0.747363	+	0.664416i	$0.768680\pi$
$384$		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	$-0.841458\pi$
$397$		−	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	$-0.855820\pi$
$401$		−	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	$-0.754962\pi$
$433$		−	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	$-0.664809\pi$
$443$	−438.515			−0.989875			−0.494938	+	0.868929i	$0.664809\pi$
$444$		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	$0.275010\pi$
$457$			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	$0.00451888\pi$
$461$			13.0887i			0.0283920i	−0.999899	+	0.0141960i	$0.995481\pi$
$462$		0			0
$463$			234.281i			0.506007i	0.967465	+	0.253004i	$0.0814185\pi$
$463$			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	$-0.207273\pi$
$467$	742.882			1.59075			0.795377	+	0.606115i	$0.207273\pi$
$468$		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	$-0.983549\pi$
$487$		−	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	$0.297901\pi$
$491$			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	$-0.358237\pi$
$503$	433.368			0.861566			0.430783	+	0.902456i	$0.358237\pi$
$504$		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	$-0.949317\pi$
$523$		−	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	$-0.533396\pi$
$557$	−116.662			−0.209447			−0.104723	+	0.994501i	$0.533396\pi$
$558$		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	$-0.657249\pi$
$563$	−533.907			−0.948324			−0.474162	+	0.880438i	$0.657249\pi$
$564$		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	$-0.989703\pi$
$577$		−	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	$-0.348441\pi$
$587$	538.103			0.916699			0.458350	+	0.888772i	$0.348441\pi$
$588$		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	$-0.232208\pi$
$593$	884.169			1.49101			0.745505	+	0.666500i	$0.232208\pi$
$594$		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	$-0.943282\pi$
$607$		−	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	$-0.483116\pi$
$617$	65.4234			0.106035			0.0530174	+	0.998594i	$0.483116\pi$
$618$		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	$0.00931861\pi$
$643$			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	$-0.319411\pi$
$647$	695.382			1.07478			0.537389	+	0.843334i	$0.319411\pi$
$648$		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	$-0.249719\pi$
$653$	924.296			1.41546			0.707731	+	0.706482i	$0.249719\pi$
$654$		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	$-0.997595\pi$
$673$		−	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	$-0.640112\pi$
$677$	−576.936			−0.852195			−0.426098	+	0.904677i	$0.640112\pi$
$678$		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	$-0.374176\pi$
$683$	526.009			0.770145			0.385073	+	0.922886i	$0.374176\pi$
$684$		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	$0.118237\pi$
$727$			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	$-0.844685\pi$
$743$	−1312.59			−1.76660			−0.883302	+	0.468804i	$0.844685\pi$
$744$		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	$0.112130\pi$
$761$			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	$-0.390230\pi$
$773$	522.640			0.676119			0.338059	+	0.941125i	$0.390230\pi$
$774$		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
$789$		0			0
$790$	−275.272	−	146.444i	−0.348446	−	0.185372i
$791$			744.942i			0.941773i
$792$		0			0
$793$		−	256.125i		−	0.322982i
$794$			536.441i			0.675618i
$795$		0			0
$796$	143.147			0.179833
$797$	−613.606			−0.769895			−0.384947	−	0.922939i	$-0.625780\pi$
$797$	−613.606			−0.769895			−0.384947	+	0.922939i	$0.625780\pi$
$798$		0			0
$799$	−277.367			−0.347142
$800$	−79.0294	−	117.279i	−0.0987868	−	0.146599i
$801$		0			0
$802$			496.352i			0.618892i
$803$	26.0059			0.0323859
$804$		0			0
$805$	1118.51	−	2102.49i	1.38946	−	2.61178i
$806$			919.151i			1.14039i
$807$		0			0
$808$			43.0760i			0.0533119i
$809$			545.471i			0.674254i	0.941459	+	0.337127i	$0.109455\pi$
$809$			545.471i			0.674254i	−0.941459	+	0.337127i	$0.890545\pi$
$810$		0			0
$811$	−293.192			−0.361519			−0.180759	−	0.983527i	$-0.557856\pi$
$811$	−293.192			−0.361519			−0.180759	+	0.983527i	$0.557856\pi$
$812$	−578.558			−0.712510
$813$		0			0
$814$	375.001			0.460689
$815$	−21.2750	+	39.9910i	−0.0261043	+	0.0490687i
$816$		0			0
$817$			45.1883i			0.0553101i
$818$	754.379			0.922224
$819$		0			0
$820$	170.794	−	321.044i	0.208285	−	0.391517i
$821$			853.691i			1.03982i	0.854221	+	0.519909i	$0.174034\pi$
$821$			853.691i			1.03982i	−0.854221	+	0.519909i	$0.825966\pi$
$822$		0			0
$823$		−	202.475i		−	0.246021i	−0.992405	−	0.123010i	$-0.960745\pi$
$823$		−	202.475i		−	0.246021i	0.992405	−	0.123010i	$-0.0392548\pi$
$824$			218.050i			0.264624i
$825$		0			0
$826$	805.838			0.975590
$827$	15.3734			0.0185894			0.00929471	−	0.999957i	$-0.497041\pi$
$827$	15.3734			0.0185894			0.00929471	+	0.999957i	$0.497041\pi$
$828$		0			0
$829$	−1135.22			−1.36938			−0.684692	−	0.728833i	$-0.740063\pi$
$829$	−1135.22			−1.36938			−0.684692	+	0.728833i	$0.740063\pi$
$830$	343.497	+	182.739i	0.413852	+	0.220168i
$831$		0			0
$832$			136.066i			0.163540i
$833$	954.463			1.14581
$834$		0			0
$835$	−281.860	−	149.948i	−0.337557	−	0.179579i
$836$		−	178.336i		−	0.213320i
$837$		0			0
$838$			339.065i			0.404612i
$839$			248.510i			0.296197i	0.988973	+	0.148099i	$0.0473153\pi$
$839$			248.510i			0.296197i	−0.988973	+	0.148099i	$0.952685\pi$
$840$		0			0
$841$	394.308			0.468856
$842$	−600.958			−0.713726
$843$		0			0
$844$	638.985			0.757091
$845$	530.938	+	282.457i	0.628329	+	0.334269i
$846$		0			0
$847$			418.465i			0.494056i
$848$	−153.088			−0.180529
$849$		0			0
$850$	136.316	+	202.291i	0.160372	+	0.237990i
$851$		−	749.501i		−	0.880730i
$852$		0			0
$853$			20.1219i			0.0235895i	0.999930	+	0.0117948i	$0.00375448\pi$
$853$			20.1219i			0.0235895i	−0.999930	+	0.0117948i	$0.996246\pi$
$854$		−	291.488i		−	0.341321i
$855$		0			0
$856$	−221.823			−0.259139
$857$	−185.567			−0.216531			−0.108266	−	0.994122i	$-0.534530\pi$
$857$	−185.567			−0.216531			−0.108266	+	0.994122i	$0.534530\pi$
$858$		0			0
$859$	−1098.48			−1.27879			−0.639393	−	0.768880i	$-0.720814\pi$
$859$	−1098.48			−1.27879			−0.639393	+	0.768880i	$0.720814\pi$
$860$	−29.3036	+	55.0824i	−0.0340740	+	0.0640493i
$861$		0			0
$862$			769.387i			0.892560i
$863$	−32.5736			−0.0377446			−0.0188723	−	0.999822i	$-0.506008\pi$
$863$	−32.5736			−0.0377446			−0.0188723	+	0.999822i	$0.506008\pi$
$864$		0			0
$865$	123.544	+	65.7250i	0.142826	+	0.0759827i
$866$			852.396i			0.984292i
$867$		0			0
$868$			1046.06i			1.20514i
$869$		−	542.882i		−	0.624721i
$870$		0			0
$871$	−2187.56			−2.51155
$872$	413.740			0.474473
$873$		0			0
$874$	−356.434			−0.407819
$875$	−1701.64	+	177.697i	−1.94473	+	0.203082i
$876$		0			0
$877$			1001.29i			1.14172i	0.821048	+	0.570859i	$0.193390\pi$
$877$			1001.29i			1.14172i	−0.821048	+	0.570859i	$0.806610\pi$
$878$	−568.919			−0.647971
$879$		0			0
$880$	115.647	−	217.383i	0.131417	−	0.247026i
$881$		−	1664.18i		−	1.88897i	−0.328559	−	0.944483i	$-0.606563\pi$
$881$		−	1664.18i		−	1.88897i	0.328559	−	0.944483i	$-0.393437\pi$
$882$		0			0
$883$			221.400i			0.250736i	0.992110	+	0.125368i	$0.0400112\pi$
$883$			221.400i			0.250736i	−0.992110	+	0.125368i	$0.959989\pi$
$884$		−	234.696i		−	0.265493i
$885$		0			0
$886$	620.153			0.699947
$887$	−1659.54			−1.87096			−0.935481	−	0.353378i	$-0.885033\pi$
$887$	−1659.54			−1.87096			−0.935481	+	0.353378i	$0.885033\pi$
$888$		0			0
$889$	−2179.20			−2.45130
$890$	−226.171	+	425.136i	−0.254124	+	0.477681i
$891$		0			0
$892$		−	324.877i		−	0.364212i
$893$	−291.161			−0.326049
$894$		0			0
$895$	49.4701	−	92.9897i	0.0552739	−	0.103899i
$896$			154.852i			0.172826i
$897$		0			0
$898$		−	948.748i		−	1.05651i
$899$			807.640i			0.898376i
$900$		0			0
$901$	264.058			0.293072
$902$	633.151			0.701942
$903$		0			0
$904$	153.941			0.170289
$905$	−236.615	−	125.878i	−0.261453	−	0.139092i
$906$		0			0
$907$			168.494i			0.185771i	0.995677	+	0.0928854i	$0.0296090\pi$
$907$			168.494i			0.185771i	−0.995677	+	0.0928854i	$0.970391\pi$
$908$	−482.195			−0.531052
$909$		0			0
$910$	1453.25	+	773.123i	1.59698	+	0.849585i
$911$		−	1446.53i		−	1.58785i	−0.608019	−	0.793923i	$-0.708035\pi$
$911$		−	1446.53i		−	1.58785i	0.608019	−	0.793923i	$-0.291965\pi$
$912$		0			0
$913$			677.434i			0.741987i
$914$		−	982.919i		−	1.07540i
$915$		0			0
$916$	204.441			0.223189
$917$	−1509.73			−1.64638
$918$		0			0
$919$	−1279.60			−1.39239			−0.696193	−	0.717855i	$-0.745124\pi$
$919$	−1279.60			−1.39239			−0.696193	+	0.717855i	$0.745124\pi$
$920$	−434.475	−	231.139i	−0.472256	−	0.251238i
$921$		0			0
$922$		−	18.5102i		−	0.0200762i
$923$	1785.31			1.93424
$924$		0			0
$925$	−446.530	+	300.898i	−0.482735	+	0.325295i
$926$		−	331.324i		−	0.357801i
$927$		0			0
$928$			119.558i			0.128834i
$929$			983.265i			1.05841i	0.848493	+	0.529206i	$0.177510\pi$
$929$			983.265i			1.05841i	−0.848493	+	0.529206i	$0.822490\pi$
$930$		0			0
$931$	1001.93			1.07619
$932$	−483.931			−0.519239
$933$		0			0
$934$	−1050.59			−1.12483
$935$	−199.476	+	374.958i	−0.213343	+	0.401024i
$936$		0			0
$937$			964.938i			1.02982i	0.857245	+	0.514908i	$0.172174\pi$
$937$			964.938i			1.02982i	−0.857245	+	0.514908i	$0.827826\pi$
$938$	−2489.60			−2.65415
$939$		0			0
$940$	−354.912	−	188.812i	−0.377566	−	0.200863i
$941$			1029.89i			1.09446i	0.836983	+	0.547230i	$0.184318\pi$
$941$			1029.89i			1.09446i	−0.836983	+	0.547230i	$0.815682\pi$
$942$		0			0
$943$		−	1265.46i		−	1.34195i
$944$		−	166.525i		−	0.176404i
$945$		0			0
$946$	−108.632			−0.114833
$947$	−186.868			−0.197326			−0.0986631	−	0.995121i	$-0.531457\pi$
$947$	−186.868			−0.197326			−0.0986631	+	0.995121i	$0.531457\pi$
$948$		0			0
$949$	35.9268			0.0378576
$950$	143.095	+	212.352i	0.150627	+	0.223529i
$951$		0			0
$952$		−	267.101i		−	0.280568i
$953$	−857.170			−0.899444			−0.449722	−	0.893169i	$-0.648477\pi$
$953$	−857.170			−0.899444			−0.449722	+	0.893169i	$0.648477\pi$
$954$		0			0
$955$	−523.736	+	984.474i	−0.548415	+	1.03086i
$956$			750.917i			0.785478i
$957$		0			0
$958$		−	632.734i		−	0.660474i
$959$			2982.69i			3.11021i
$960$		0			0
$961$	499.249			0.519510
$962$	518.059			0.538523
$963$		0			0
$964$	−636.558			−0.660330
$965$	413.833	−	777.888i	0.428843	−	0.806102i
$966$		0			0
$967$		−	1234.04i		−	1.27615i	−0.769972	−	0.638077i	$-0.779730\pi$
$967$		−	1234.04i		−	1.27615i	0.769972	−	0.638077i	$-0.220270\pi$
$968$	86.4752			0.0893339
$969$		0			0
$970$	336.235	−	632.026i	0.346635	−	0.651574i
$971$		−	1313.38i		−	1.35261i	−0.736624	−	0.676303i	$-0.763581\pi$
$971$		−	1313.38i		−	1.35261i	0.736624	−	0.676303i	$-0.236419\pi$
$972$		0			0
$973$			1454.26i			1.49461i
$974$			71.1584i			0.0730579i
$975$		0			0
$976$	−60.2355			−0.0617167
$977$	109.486			0.112064			0.0560318	−	0.998429i	$-0.482155\pi$
$977$	109.486			0.112064			0.0560318	+	0.998429i	$0.482155\pi$
$978$		0			0
$979$	−838.439			−0.856424
$980$	1221.31	+	649.731i	1.24623	+	0.662991i
$981$		0			0
$982$		−	1118.12i		−	1.13862i
$983$	−341.995			−0.347909			−0.173955	−	0.984754i	$-0.555655\pi$
$983$	−341.995			−0.347909			−0.173955	+	0.984754i	$0.555655\pi$
$984$		0			0
$985$	1225.32	+	651.867i	1.24398	+	0.661794i
$986$		−	206.223i		−	0.209151i
$987$		0			0
$988$		−	246.369i		−	0.249361i
$989$			217.118i			0.219533i
$990$		0			0
$991$	140.258			0.141532			0.0707658	−	0.997493i	$-0.477456\pi$
$991$	140.258			0.141532			0.0707658	+	0.997493i	$0.477456\pi$
$992$	216.167			0.217910
$993$		0			0
$994$	2031.81			2.04407
$995$	−315.941	−	168.079i	−0.317529	−	0.168924i
$996$		0			0
$997$			619.634i			0.621498i	0.950492	+	0.310749i	$0.100580\pi$
$997$			619.634i			0.621498i	−0.950492	+	0.310749i	$0.899420\pi$
$998$	849.079			0.850781
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.3.b.a.269.1	✓	4
3.2	odd	2		270.3.b.d.269.4	yes	4
4.3	odd	2		2160.3.c.g.1889.1		4
5.2	odd	4		1350.3.d.o.701.4		8
5.3	odd	4		1350.3.d.o.701.5		8
5.4	even	2		270.3.b.d.269.3	yes	4
9.2	odd	6		810.3.j.a.539.1		8
9.4	even	3		810.3.j.f.269.3		8
9.5	odd	6		810.3.j.a.269.2		8
9.7	even	3		810.3.j.f.539.4		8
12.11	even	2		2160.3.c.m.1889.4		4
15.2	even	4		1350.3.d.o.701.8		8
15.8	even	4		1350.3.d.o.701.1		8
15.14	odd	2	inner	270.3.b.a.269.2	yes	4
20.19	odd	2		2160.3.c.m.1889.3		4
45.4	even	6		810.3.j.a.269.1		8
45.14	odd	6		810.3.j.f.269.4		8
45.29	odd	6		810.3.j.f.539.3		8
45.34	even	6		810.3.j.a.539.2		8
60.59	even	2		2160.3.c.g.1889.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.b.a.269.1	✓	4	1.1	even	1	trivial
270.3.b.a.269.2	yes	4	15.14	odd	2	inner
270.3.b.d.269.3	yes	4	5.4	even	2
270.3.b.d.269.4	yes	4	3.2	odd	2
810.3.j.a.269.1		8	45.4	even	6
810.3.j.a.269.2		8	9.5	odd	6
810.3.j.a.539.1		8	9.2	odd	6
810.3.j.a.539.2		8	45.34	even	6
810.3.j.f.269.3		8	9.4	even	3
810.3.j.f.269.4		8	45.14	odd	6
810.3.j.f.539.3		8	45.29	odd	6
810.3.j.f.539.4		8	9.7	even	3
1350.3.d.o.701.1		8	15.8	even	4
1350.3.d.o.701.4		8	5.2	odd	4
1350.3.d.o.701.5		8	5.3	odd	4
1350.3.d.o.701.8		8	15.2	even	4
2160.3.c.g.1889.1		4	4.3	odd	2
2160.3.c.g.1889.2		4	60.59	even	2
2160.3.c.m.1889.3		4	20.19	odd	2
2160.3.c.m.1889.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	270.3.b.a.269.1

Level	$270$
Weight	$3$
Character	270.269
Analytic conductor	$7.357$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$