LMFDB - Embedded newform 864.5.t.b.559.36

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,5,Mod(559,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("864.559");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	864.t (of order $6$, degree $2$, not 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:	$89.3116481044$
Analytic rank:	$0$
Dimension:	$88$
Relative dimension:	$44$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			559.36
Character	$\chi$	$=$	864.559
Dual form			864.5.t.b.847.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(23.2381 - 13.4165i) q^{5} +(-52.2165 - 30.1472i) q^{7} +O(q^{10})$	$q+(23.2381 - 13.4165i) q^{5} +(-52.2165 - 30.1472i) q^{7} +(92.8475 - 160.817i) q^{11} +(-166.026 + 95.8552i) q^{13} +425.178 q^{17} +114.203 q^{19} +(239.080 - 138.033i) q^{23} +(47.5051 - 82.2813i) q^{25} +(757.999 + 437.631i) q^{29} +(910.769 - 525.833i) q^{31} -1617.88 q^{35} -194.549i q^{37} +(-1420.33 - 2460.08i) q^{41} +(449.077 - 777.825i) q^{43} +(-348.382 - 201.138i) q^{47} +(617.208 + 1069.03i) q^{49} +4662.79i q^{53} -4982.75i q^{55} +(-1769.31 - 3064.53i) q^{59} +(-1797.64 - 1037.87i) q^{61} +(-2572.08 + 4454.98i) q^{65} +(-1465.53 - 2538.37i) q^{67} -3035.73i q^{71} +5570.78 q^{73} +(-9696.34 + 5598.18i) q^{77} +(-9323.76 - 5383.08i) q^{79} +(-1407.36 + 2437.61i) q^{83} +(9880.31 - 5704.40i) q^{85} +1868.54 q^{89} +11559.1 q^{91} +(2653.85 - 1532.20i) q^{95} +(1255.93 - 2175.33i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q+O(q^{10}) $ 88 * q	$ 88 q + 44 q^{11} + 1156 q^{17} - 860 q^{19} + 5998 q^{25} - 2508 q^{35} - 2348 q^{41} - 3500 q^{43} + 16462 q^{49} - 3508 q^{59} + 2502 q^{65} - 5132 q^{67} + 19004 q^{73} + 17090 q^{83} + 8272 q^{89} + 9612 q^{91} + 9980 q^{97}+O(q^{100}) $ 88 * q + 44 * q^11 + 1156 * q^17 - 860 * q^19 + 5998 * q^25 - 2508 * q^35 - 2348 * q^41 - 3500 * q^43 + 16462 * q^49 - 3508 * q^59 + 2502 * q^65 - 5132 * q^67 + 19004 * q^73 + 17090 * q^83 + 8272 * q^89 + 9612 * q^91 + 9980 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	23.2381	−	13.4165i	0.929523	−	0.536660i	0.0428621	−	0.999081i	$-0.486352\pi$
$5$	23.2381	−	13.4165i	0.929523	−	0.536660i	0.886661	+	0.462421i	$0.153019\pi$
$6$		0			0
$7$	−52.2165	−	30.1472i	−1.06564	−	0.615249i	−0.138655	−	0.990341i	$-0.544278\pi$
$7$	−52.2165	−	30.1472i	−1.06564	−	0.615249i	−0.926988	+	0.375092i	$0.877611\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	92.8475	−	160.817i	0.767335	−	1.32906i	−0.171669	−	0.985155i	$-0.554916\pi$
$11$	92.8475	−	160.817i	0.767335	−	1.32906i	0.939003	−	0.343908i	$-0.111751\pi$
$12$		0			0
$13$	−166.026	+	95.8552i	−0.982402	+	0.567190i	−0.902995	−	0.429652i	$-0.858636\pi$
$13$	−166.026	+	95.8552i	−0.982402	+	0.567190i	−0.0794078	+	0.996842i	$0.525303\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	425.178			1.47120			0.735602	−	0.677414i	$-0.236899\pi$
$17$	425.178			1.47120			0.735602	+	0.677414i	$0.236899\pi$
$18$		0			0
$19$	114.203			0.316351			0.158175	−	0.987411i	$-0.449439\pi$
$19$	114.203			0.316351			0.158175	+	0.987411i	$0.449439\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	239.080	−	138.033i	0.451946	−	0.260931i	−0.256705	−	0.966490i	$-0.582637\pi$
$23$	239.080	−	138.033i	0.451946	−	0.260931i	0.708652	+	0.705558i	$0.249304\pi$
$24$		0			0
$25$	47.5051	−	82.2813i	0.0760082	−	0.131650i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	757.999	+	437.631i	0.901306	+	0.520369i	0.877624	−	0.479350i	$-0.159128\pi$
$29$	757.999	+	437.631i	0.901306	+	0.520369i	0.0236825	+	0.999720i	$0.492461\pi$
$30$		0			0
$31$	910.769	−	525.833i	0.947730	−	0.547172i	0.0553551	−	0.998467i	$-0.482371\pi$
$31$	910.769	−	525.833i	0.947730	−	0.547172i	0.892375	+	0.451294i	$0.149038\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1617.88			−1.32072
$36$		0			0
$37$		−	194.549i		−	0.142110i	−0.997472	−	0.0710551i	$-0.977363\pi$
$37$		−	194.549i		−	0.142110i	0.997472	−	0.0710551i	$-0.0226366\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1420.33	−	2460.08i	−0.844930	−	1.46346i	−0.885682	−	0.464292i	$-0.846309\pi$
$41$	−1420.33	−	2460.08i	−0.844930	−	1.46346i	0.0407524	−	0.999169i	$-0.487025\pi$
$42$		0			0
$43$	449.077	−	777.825i	0.242876	−	0.420673i	−0.718656	−	0.695365i	$-0.755243\pi$
$43$	449.077	−	777.825i	0.242876	−	0.420673i	0.961532	+	0.274692i	$0.0885759\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−348.382	−	201.138i	−0.157710	−	0.0910540i	0.419068	−	0.907955i	$-0.362357\pi$
$47$	−348.382	−	201.138i	−0.157710	−	0.0910540i	−0.576778	+	0.816901i	$0.695690\pi$
$48$		0			0
$49$	617.208	+	1069.03i	0.257063	+	0.445246i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4662.79i			1.65995i	0.557802	+	0.829974i	$0.311645\pi$
$53$			4662.79i			1.65995i	−0.557802	+	0.829974i	$0.688355\pi$
$54$		0			0
$55$		−	4982.75i		−	1.64719i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1769.31	−	3064.53i	−0.508276	−	0.880359i	−0.999954	−	0.00958239i	$-0.996950\pi$
$59$	−1769.31	−	3064.53i	−0.508276	−	0.880359i	0.491678	−	0.870777i	$-0.336384\pi$
$60$		0			0
$61$	−1797.64	−	1037.87i	−0.483107	−	0.278922i	0.238604	−	0.971117i	$-0.423310\pi$
$61$	−1797.64	−	1037.87i	−0.483107	−	0.278922i	−0.721710	+	0.692195i	$0.756644\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2572.08	+	4454.98i	−0.608777	+	1.05443i
$66$		0			0
$67$	−1465.53	−	2538.37i	−0.326471	−	0.565465i	0.655338	−	0.755336i	$-0.272526\pi$
$67$	−1465.53	−	2538.37i	−0.326471	−	0.565465i	−0.981809	+	0.189871i	$0.939193\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3035.73i		−	0.602208i	−0.953591	−	0.301104i	$-0.902645\pi$
$71$		−	3035.73i		−	0.602208i	0.953591	−	0.301104i	$-0.0973551\pi$
$72$		0			0
$73$	5570.78			1.04537			0.522685	−	0.852526i	$-0.324930\pi$
$73$	5570.78			1.04537			0.522685	+	0.852526i	$0.324930\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9696.34	+	5598.18i	−1.63541	+	0.944204i
$78$		0			0
$79$	−9323.76	−	5383.08i	−1.49395	−	0.862535i	−0.493978	−	0.869475i	$-0.664458\pi$
$79$	−9323.76	−	5383.08i	−1.49395	−	0.862535i	−0.999976	+	0.00694001i	$0.997791\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1407.36	+	2437.61i	−0.204291	+	0.353842i	−0.949906	−	0.312534i	$-0.898822\pi$
$83$	−1407.36	+	2437.61i	−0.204291	+	0.353842i	0.745616	+	0.666376i	$0.232155\pi$
$84$		0			0
$85$	9880.31	−	5704.40i	1.36752	−	0.789537i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1868.54			0.235897			0.117949	−	0.993020i	$-0.462368\pi$
$89$	1868.54			0.235897			0.117949	+	0.993020i	$0.462368\pi$
$90$		0			0
$91$	11559.1			1.39585
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2653.85	−	1532.20i	0.294055	−	0.169773i
$96$		0			0
$97$	1255.93	−	2175.33i	0.133481	−	0.231197i	−0.791535	−	0.611124i	$-0.790718\pi$
$97$	1255.93	−	2175.33i	0.133481	−	0.231197i	0.925016	+	0.379927i	$0.124051\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4729.31	−	2730.47i	−0.463612	−	0.267667i	0.249950	−	0.968259i	$-0.419586\pi$
$101$	−4729.31	−	2730.47i	−0.463612	−	0.267667i	−0.713562	+	0.700592i	$0.752919\pi$
$102$		0			0
$103$	−9762.25	+	5636.24i	−0.920186	+	0.531270i	−0.883694	−	0.468064i	$-0.844952\pi$
$103$	−9762.25	+	5636.24i	−0.920186	+	0.531270i	−0.0364916	+	0.999334i	$0.511618\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4850.33			0.423647			0.211823	−	0.977308i	$-0.432060\pi$
$107$	4850.33			0.423647			0.211823	+	0.977308i	$0.432060\pi$
$108$		0			0
$109$			17364.2i			1.46151i	0.682639	+	0.730756i	$0.260832\pi$
$109$			17364.2i			1.46151i	−0.682639	+	0.730756i	$0.739168\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6670.25	−	11553.2i	−0.522378	−	0.904786i	−0.999661	−	0.0260362i	$-0.991711\pi$
$113$	−6670.25	−	11553.2i	−0.522378	−	0.904786i	0.477283	−	0.878750i	$-0.341622\pi$
$114$		0			0
$115$	3703.83	−	6415.23i	0.280063	−	0.485083i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−22201.3	−	12817.9i	−1.56778	−	0.905157i
$120$		0			0
$121$	−9920.81	−	17183.3i	−0.677605	−	1.17365i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			14221.2i			0.910158i
$126$		0			0
$127$		−	1492.37i		−	0.0925269i	−0.998929	−	0.0462634i	$-0.985269\pi$
$127$		−	1492.37i		−	0.0925269i	0.998929	−	0.0462634i	$-0.0147314\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3213.86	−	5566.56i	−0.187277	−	0.324373i	0.757065	−	0.653340i	$-0.226633\pi$
$131$	−3213.86	−	5566.56i	−0.187277	−	0.324373i	−0.944341	+	0.328967i	$0.893299\pi$
$132$		0			0
$133$	−5963.26	−	3442.89i	−0.337117	−	0.194635i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10678.8	−	18496.2i	0.568959	−	0.985466i	−0.427710	−	0.903916i	$-0.640680\pi$
$137$	10678.8	−	18496.2i	0.568959	−	0.985466i	0.996669	−	0.0815500i	$-0.0259870\pi$
$138$		0			0
$139$	10328.6	+	17889.6i	0.534578	+	0.925916i	0.999184	+	0.0403986i	$0.0128628\pi$
$139$	10328.6	+	17889.6i	0.534578	+	0.925916i	−0.464606	+	0.885518i	$0.653804\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			35599.6i			1.74090i
$144$		0			0
$145$	23485.9			1.11705
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4499.97	−	2598.06i	0.202692	−	0.117025i	−0.395218	−	0.918587i	$-0.629331\pi$
$149$	4499.97	−	2598.06i	0.202692	−	0.117025i	0.597911	+	0.801563i	$0.295998\pi$
$150$		0			0
$151$	−17647.9	−	10189.0i	−0.773998	−	0.446868i	0.0603013	−	0.998180i	$-0.480794\pi$
$151$	−17647.9	−	10189.0i	−0.773998	−	0.446868i	−0.834299	+	0.551313i	$0.814127\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14109.7	−	24438.7i	0.587291	−	1.01722i
$156$		0			0
$157$	−7737.56	+	4467.28i	−0.313910	+	0.181236i	−0.648675	−	0.761066i	$-0.724676\pi$
$157$	−7737.56	+	4467.28i	−0.313910	+	0.181236i	0.334765	+	0.942302i	$0.391343\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16645.2			−0.642151
$162$		0			0
$163$	−38000.8			−1.43027			−0.715134	−	0.698987i	$-0.753634\pi$
$163$	−38000.8			−1.43027			−0.715134	+	0.698987i	$0.753634\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11942.4	−	6894.97i	0.428213	−	0.247229i	−0.270372	−	0.962756i	$-0.587147\pi$
$167$	11942.4	−	6894.97i	0.428213	−	0.247229i	0.698585	+	0.715527i	$0.253813\pi$
$168$		0			0
$169$	4095.92	−	7094.35i	0.143410	−	0.248393i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9687.10	−	5592.85i	−0.323669	−	0.186871i	0.329358	−	0.944205i	$-0.393168\pi$
$173$	−9687.10	−	5592.85i	−0.323669	−	0.186871i	−0.653027	+	0.757335i	$0.726501\pi$
$174$		0			0
$175$	−4961.10	+	2864.29i	−0.161995	+	0.0935279i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7133.34			0.222632			0.111316	−	0.993785i	$-0.464493\pi$
$179$	7133.34			0.222632			0.111316	+	0.993785i	$0.464493\pi$
$180$		0			0
$181$		−	26104.7i		−	0.796822i	−0.917207	−	0.398411i	$-0.869562\pi$
$181$		−	26104.7i		−	0.796822i	0.917207	−	0.398411i	$-0.130438\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2610.17	−	4520.94i	−0.0762649	−	0.132095i
$186$		0			0
$187$	39476.7	−	68375.7i	1.12891	−	1.95532i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−39532.9	−	22824.3i	−1.08366	−	0.625649i	−0.151776	−	0.988415i	$-0.548499\pi$
$191$	−39532.9	−	22824.3i	−1.08366	−	0.625649i	−0.931880	+	0.362766i	$0.881832\pi$
$192$		0			0
$193$	−6039.50	−	10460.7i	−0.162139	−	0.280832i	0.773497	−	0.633800i	$-0.218506\pi$
$193$	−6039.50	−	10460.7i	−0.162139	−	0.280832i	−0.935635	+	0.352968i	$0.885173\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	70123.2i		−	1.80688i	−0.428714	−	0.903440i	$-0.641033\pi$
$197$		−	70123.2i		−	1.80688i	0.428714	−	0.903440i	$-0.358967\pi$
$198$		0			0
$199$		−	53477.1i		−	1.35040i	−0.737636	−	0.675198i	$-0.764058\pi$
$199$		−	53477.1i		−	1.35040i	0.737636	−	0.675198i	$-0.235942\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−26386.7	−	45703.1i	−0.640314	−	1.10906i
$204$		0			0
$205$	−66011.3	−	38111.6i	−1.57076	−	0.906880i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10603.4	−	18365.7i	0.242747	−	0.420450i
$210$		0			0
$211$	−16582.5	−	28721.8i	−0.372465	−	0.645128i	0.617479	−	0.786587i	$-0.288154\pi$
$211$	−16582.5	−	28721.8i	−0.372465	−	0.645128i	−0.989944	+	0.141459i	$0.954821\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	24100.2i		−	0.521367i
$216$		0			0
$217$	−63409.5			−1.34659
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−70590.6	+	40755.5i	−1.44531	+	0.834453i
$222$		0			0
$223$	53609.6	+	30951.5i	1.07804	+	0.622404i	0.930366	−	0.366633i	$-0.119490\pi$
$223$	53609.6	+	30951.5i	1.07804	+	0.622404i	0.147669	+	0.989037i	$0.452823\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−121.231	+	209.978i	−0.00235268	+	0.00407496i	−0.867199	−	0.497961i	$-0.834082\pi$
$227$	−121.231	+	209.978i	−0.00235268	+	0.00407496i	0.864847	+	0.502036i	$0.167416\pi$
$228$		0			0
$229$	−10175.3	+	5874.70i	−0.194033	+	0.112025i	−0.593869	−	0.804562i	$-0.702400\pi$
$229$	−10175.3	+	5874.70i	−0.194033	+	0.112025i	0.399836	+	0.916587i	$0.369067\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	34752.2			0.640134			0.320067	−	0.947395i	$-0.396295\pi$
$233$	34752.2			0.640134			0.320067	+	0.947395i	$0.396295\pi$
$234$		0			0
$235$	−10794.3			−0.195460
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−41098.0	+	23728.0i	−0.719491	+	0.415398i	−0.814565	−	0.580072i	$-0.803024\pi$
$239$	−41098.0	+	23728.0i	−0.719491	+	0.415398i	0.0950746	+	0.995470i	$0.469691\pi$
$240$		0			0
$241$	34530.8	−	59809.0i	0.594527	−	1.02975i	−0.399086	−	0.916914i	$-0.630672\pi$
$241$	34530.8	−	59809.0i	0.594527	−	1.02975i	0.993613	−	0.112838i	$-0.0359942\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	28685.4	+	16561.5i	0.477891	+	0.275911i
$246$		0			0
$247$	−18960.6	+	10946.9i	−0.310784	+	0.179431i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−59187.9			−0.939475			−0.469738	−	0.882806i	$-0.655652\pi$
$251$	−59187.9			−0.939475			−0.469738	+	0.882806i	$0.655652\pi$
$252$		0			0
$253$		−	51263.9i		−	0.800887i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−26280.3	−	45518.9i	−0.397891	−	0.689168i	0.595574	−	0.803300i	$-0.296925\pi$
$257$	−26280.3	−	45518.9i	−0.397891	−	0.689168i	−0.993466	+	0.114132i	$0.963591\pi$
$258$		0			0
$259$	−5865.11	+	10158.7i	−0.0874332	+	0.151439i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	37110.2	+	21425.6i	0.536515	+	0.309757i	0.743665	−	0.668552i	$-0.233086\pi$
$263$	37110.2	+	21425.6i	0.536515	+	0.309757i	−0.207150	+	0.978309i	$0.566419\pi$
$264$		0			0
$265$	62558.4	+	108354.i	0.890828	+	1.54296i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	10584.0i		−	0.146267i	−0.997322	−	0.0731333i	$-0.976700\pi$
$269$		−	10584.0i		−	0.146267i	0.997322	−	0.0731333i	$-0.0232998\pi$
$270$		0			0
$271$			93339.2i			1.27094i	0.772125	+	0.635471i	$0.219194\pi$
$271$			93339.2i			1.27094i	−0.772125	+	0.635471i	$0.780806\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8821.46	−	15279.2i	−0.116647	−	0.202039i
$276$		0			0
$277$	−20273.5	−	11704.9i	−0.264222	−	0.152549i	0.362037	−	0.932164i	$-0.382081\pi$
$277$	−20273.5	−	11704.9i	−0.264222	−	0.152549i	−0.626259	+	0.779615i	$0.715415\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3736.40	−	6471.63i	0.0473195	−	0.0819598i	−0.841396	−	0.540420i	$-0.818265\pi$
$281$	3736.40	−	6471.63i	0.0473195	−	0.0819598i	0.888715	+	0.458460i	$0.151599\pi$
$282$		0			0
$283$	45059.8	+	78045.9i	0.562622	+	0.974490i	0.997267	+	0.0738877i	$0.0235407\pi$
$283$	45059.8	+	78045.9i	0.562622	+	0.974490i	−0.434645	+	0.900602i	$0.643126\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			171276.i			2.07937i
$288$		0			0
$289$	97255.3			1.16444
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−118256.	+	68274.9i	−1.37748	+	0.795290i	−0.991856	−	0.127365i	$-0.959348\pi$
$293$	−118256.	+	68274.9i	−1.37748	+	0.795290i	−0.385627	+	0.922655i	$0.626015\pi$
$294$		0			0
$295$	−82230.6	−	47475.8i	−0.944907	−	0.545543i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−26462.3	+	45834.0i	−0.295995	+	0.512679i
$300$		0			0
$301$	−46898.5	+	27076.9i	−0.517638	+	0.298858i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−55698.2			−0.598745
$306$		0			0
$307$	168794.			1.79094			0.895468	−	0.445126i	$-0.146841\pi$
$307$	168794.			1.79094			0.895468	+	0.445126i	$0.146841\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−68907.1	+	39783.5i	−0.712431	+	0.411323i	−0.811961	−	0.583712i	$-0.801600\pi$
$311$	−68907.1	+	39783.5i	−0.712431	+	0.411323i	0.0995292	+	0.995035i	$0.468266\pi$
$312$		0			0
$313$	93373.0	−	161727.i	0.953087	−	1.65080i	0.214401	−	0.976746i	$-0.431220\pi$
$313$	93373.0	−	161727.i	0.953087	−	1.65080i	0.738686	−	0.674049i	$-0.235446\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	95810.6	+	55316.3i	0.953443	+	0.550471i	0.894149	−	0.447770i	$-0.147782\pi$
$317$	95810.6	+	55316.3i	0.953443	+	0.550471i	0.0592945	+	0.998241i	$0.481115\pi$
$318$		0			0
$319$	140757.	−	81265.8i	1.38321	−	0.798595i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	48556.4			0.465417
$324$		0			0
$325$			18214.4i			0.172444i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12127.5	+	21005.5i	0.112042	+	0.194062i
$330$		0			0
$331$	35525.1	−	61531.2i	0.324249	−	0.561616i	−0.657111	−	0.753794i	$-0.728222\pi$
$331$	35525.1	−	61531.2i	0.324249	−	0.561616i	0.981360	+	0.192178i	$0.0615550\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−68112.2	−	39324.6i	−0.606925	−	0.350408i
$336$		0			0
$337$	−3370.32	−	5837.57i	−0.0296764	−	0.0514011i	0.850806	−	0.525480i	$-0.176114\pi$
$337$	−3370.32	−	5837.57i	−0.0296764	−	0.0514011i	−0.880482	+	0.474079i	$0.842781\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	195289.i		−	1.67946i
$342$		0			0
$343$			70338.5i			0.597868i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	43353.9	+	75091.2i	0.360055	+	0.623634i	0.987970	−	0.154648i	$-0.0494245\pi$
$347$	43353.9	+	75091.2i	0.360055	+	0.623634i	−0.627914	+	0.778283i	$0.716091\pi$
$348$		0			0
$349$	−108745.	−	62783.7i	−0.892805	−	0.515461i	−0.0179461	−	0.999839i	$-0.505713\pi$
$349$	−108745.	−	62783.7i	−0.892805	−	0.515461i	−0.874859	+	0.484378i	$0.839046\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−24176.0	+	41874.1i	−0.194015	+	0.336044i	−0.946577	−	0.322477i	$-0.895484\pi$
$353$	−24176.0	+	41874.1i	−0.194015	+	0.336044i	0.752562	+	0.658521i	$0.228818\pi$
$354$		0			0
$355$	−40728.9	−	70544.5i	−0.323181	−	0.559766i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			59745.2i			0.463568i	0.972767	+	0.231784i	$0.0744563\pi$
$359$			59745.2i			0.463568i	−0.972767	+	0.231784i	$0.925544\pi$
$360$		0			0
$361$	−117279.			−0.899922
$362$		0			0
$363$		0			0
$364$		0			0
$365$	129454.	−	74740.4i	0.971696	−	0.561009i
$366$		0			0
$367$	−145654.	−	84093.5i	−1.08141	−	0.624353i	−0.150135	−	0.988666i	$-0.547971\pi$
$367$	−145654.	−	84093.5i	−1.08141	−	0.624353i	−0.931277	+	0.364312i	$0.881304\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	140570.	−	243475.i	1.02128	−	1.76891i
$372$		0			0
$373$	233844.	−	135010.i	1.68077	−	0.970393i	0.719620	−	0.694368i	$-0.244316\pi$
$373$	233844.	−	135010.i	1.68077	−	0.970393i	0.961150	−	0.276026i	$-0.0890175\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−167797.			−1.18059
$378$		0			0
$379$	−110475.			−0.769102			−0.384551	−	0.923104i	$-0.625644\pi$
$379$	−110475.			−0.769102			−0.384551	+	0.923104i	$0.625644\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	174715.	−	100872.i	1.19106	−	0.687659i	0.232513	−	0.972593i	$-0.425305\pi$
$383$	174715.	−	100872.i	1.19106	−	0.687659i	0.958547	+	0.284934i	$0.0919718\pi$
$384$		0			0
$385$	−150216.	+	260182.i	−1.01343	+	1.75532i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	28286.6	+	16331.2i	0.186931	+	0.107925i	0.590545	−	0.807005i	$-0.298913\pi$
$389$	28286.6	+	16331.2i	0.186931	+	0.107925i	−0.403614	+	0.914929i	$0.632246\pi$
$390$		0			0
$391$	101651.	−	58688.5i	0.664905	−	0.383883i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−288888.			−1.85155
$396$		0			0
$397$			157999.i			1.00247i	0.865310	+	0.501237i	$0.167122\pi$
$397$			157999.i			1.00247i	−0.865310	+	0.501237i	$0.832878\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	71882.8	+	124505.i	0.447029	+	0.774277i	0.998191	−	0.0601210i	$-0.0191487\pi$
$401$	71882.8	+	124505.i	0.447029	+	0.774277i	−0.551162	+	0.834398i	$0.685815\pi$
$402$		0			0
$403$	−100808.	+	174604.i	−0.620702	+	1.07509i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−31286.7	−	18063.4i	−0.188873	−	0.109046i
$408$		0			0
$409$	−6867.24	−	11894.4i	−0.0410521	−	0.0711044i	0.844769	−	0.535131i	$-0.179738\pi$
$409$	−6867.24	−	11894.4i	−0.0410521	−	0.0711044i	−0.885821	+	0.464026i	$0.846404\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			213359.i			1.25086i
$414$		0			0
$415$			75527.3i			0.438538i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	135489.	+	234673.i	0.771748	+	1.33671i	0.936604	+	0.350389i	$0.113951\pi$
$419$	135489.	+	234673.i	0.771748	+	1.33671i	−0.164857	+	0.986318i	$0.552716\pi$
$420$		0			0
$421$	206716.	+	119347.i	1.16630	+	0.673362i	0.952805	−	0.303582i	$-0.0981828\pi$
$421$	206716.	+	119347.i	1.16630	+	0.673362i	0.213493	+	0.976945i	$0.431516\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	20198.1	−	34984.2i	0.111824	−	0.193684i
$426$		0			0
$427$	62577.6	+	108388.i	0.343213	+	0.594462i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	208255.i		−	1.12109i	−0.828124	−	0.560545i	$-0.810592\pi$
$431$		−	208255.i		−	1.12109i	0.828124	−	0.560545i	$-0.189408\pi$
$432$		0			0
$433$	−275527.			−1.46957			−0.734783	−	0.678303i	$-0.762716\pi$
$433$	−275527.			−1.46957			−0.734783	+	0.678303i	$0.762716\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	27303.5	−	15763.7i	0.142974	−	0.0825458i
$438$		0			0
$439$	−5755.89	−	3323.16i	−0.0298664	−	0.0172434i	0.484992	−	0.874518i	$-0.338822\pi$
$439$	−5755.89	−	3323.16i	−0.0298664	−	0.0172434i	−0.514859	+	0.857275i	$0.672156\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−122953.	+	212960.i	−0.626514	+	1.08515i	0.361732	+	0.932282i	$0.382186\pi$
$443$	−122953.	+	212960.i	−0.626514	+	1.08515i	−0.988246	+	0.152872i	$0.951148\pi$
$444$		0			0
$445$	43421.3	−	25069.3i	0.219272	−	0.126597i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	282931.			1.40342			0.701711	−	0.712462i	$-0.252420\pi$
$449$	282931.			1.40342			0.701711	+	0.712462i	$0.252420\pi$
$450$		0			0
$451$	−527495.			−2.59338
$452$		0			0
$453$		0			0
$454$		0			0
$455$	268610.	−	155082.i	1.29748	−	0.749099i
$456$		0			0
$457$	108539.	−	187995.i	0.519702	−	0.900150i	−0.480036	−	0.877249i	$-0.659376\pi$
$457$	108539.	−	187995.i	0.519702	−	0.900150i	0.999738	−	0.0229008i	$-0.00729020\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−86601.0	−	49999.1i	−0.407494	−	0.235267i	0.282219	−	0.959350i	$-0.408930\pi$
$461$	−86601.0	−	49999.1i	−0.407494	−	0.235267i	−0.689712	+	0.724084i	$0.742263\pi$
$462$		0			0
$463$	−6135.09	+	3542.10i	−0.0286193	+	0.0165234i	−0.514241	−	0.857645i	$-0.671926\pi$
$463$	−6135.09	+	3542.10i	−0.0286193	+	0.0165234i	0.485622	+	0.874169i	$0.338593\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	429771.			1.97062			0.985312	−	0.170765i	$-0.0546240\pi$
$467$	429771.			1.97062			0.985312	+	0.170765i	$0.0546240\pi$
$468$		0			0
$469$			176727.i			0.803445i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−83391.4	−	144438.i	−0.372734	−	0.645594i
$474$		0			0
$475$	5425.21	−	9396.74i	0.0240453	−	0.0416476i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	49730.6	+	28712.0i	0.216747	+	0.125139i	0.604443	−	0.796648i	$-0.293396\pi$
$479$	49730.6	+	28712.0i	0.216747	+	0.125139i	−0.387696	+	0.921787i	$0.626729\pi$
$480$		0			0
$481$	18648.5	+	32300.2i	0.0806036	+	0.139609i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	67400.6i		−	0.286537i
$486$		0			0
$487$			265572.i			1.11976i	0.828574	+	0.559879i	$0.189152\pi$
$487$			265572.i			1.11976i	−0.828574	+	0.559879i	$0.810848\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−77229.5	−	133765.i	−0.320347	−	0.554857i	0.660213	−	0.751079i	$-0.270466\pi$
$491$	−77229.5	−	133765.i	−0.320347	−	0.554857i	−0.980560	+	0.196222i	$0.937133\pi$
$492$		0			0
$493$	322284.	+	186071.i	1.32601	+	0.765570i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−91518.7	+	158515.i	−0.370508	+	0.641738i
$498$		0			0
$499$	98724.4	+	170996.i	0.396482	+	0.686727i	0.993289	−	0.115658i	$-0.0368976\pi$
$499$	98724.4	+	170996.i	0.396482	+	0.686727i	−0.596807	+	0.802385i	$0.703564\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	343592.i		−	1.35802i	−0.734127	−	0.679012i	$-0.762408\pi$
$503$		−	343592.i		−	1.35802i	0.734127	−	0.679012i	$-0.237592\pi$
$504$		0			0
$505$	−146533.			−0.574584
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−73009.5	+	42152.1i	−0.281802	+	0.162698i	−0.634239	−	0.773137i	$-0.718686\pi$
$509$	−73009.5	+	42152.1i	−0.281802	+	0.162698i	0.352437	+	0.935836i	$0.385353\pi$
$510$		0			0
$511$	−290887.	−	167943.i	−1.11399	−	0.643163i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−151237.	+	261951.i	−0.570222	+	0.987654i
$516$		0			0
$517$	−64692.7	+	37350.4i	−0.242033	+	0.139738i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	397536.			1.46454			0.732270	−	0.681015i	$-0.238461\pi$
$521$	397536.			1.46454			0.732270	+	0.681015i	$0.238461\pi$
$522$		0			0
$523$	286663.			1.04802			0.524008	−	0.851714i	$-0.324436\pi$
$523$	286663.			1.04802			0.524008	+	0.851714i	$0.324436\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	387239.	−	223572.i	1.39430	−	0.805002i
$528$		0			0
$529$	−101814.	+	176348.i	−0.363830	+	0.630171i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	471622.	+	272291.i	1.66012	+	0.958472i
$534$		0			0
$535$	112712.	−	65074.5i	0.393789	−	0.227354i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	229225.			0.789012
$540$		0			0
$541$			412153.i			1.40820i	0.710101	+	0.704100i	$0.248649\pi$
$541$			412153.i			1.40820i	−0.710101	+	0.704100i	$0.751351\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	232967.	+	403511.i	0.784335	+	1.35851i
$546$		0			0
$547$	168911.	−	292563.i	0.564527	−	0.977789i	−0.432567	−	0.901602i	$-0.642392\pi$
$547$	168911.	−	292563.i	0.564527	−	0.977789i	0.997094	−	0.0761869i	$-0.0242746\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	86565.4	+	49978.6i	0.285129	+	0.164619i
$552$		0			0
$553$	324569.	+	562171.i	1.06135	+	1.83831i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			251395.i			0.810302i	0.914250	+	0.405151i	$0.132781\pi$
$557$			251395.i			0.810302i	−0.914250	+	0.405151i	$0.867219\pi$
$558$		0			0
$559$			172186.i			0.551027i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−32923.4	−	57025.0i	−0.103870	−	0.179907i	0.809406	−	0.587249i	$-0.199789\pi$
$563$	−32923.4	−	57025.0i	−0.103870	−	0.179907i	−0.913276	+	0.407342i	$0.866456\pi$
$564$		0			0
$565$	−310007.	−	178983.i	−0.971125	−	0.560679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−59792.5	+	103564.i	−0.184681	+	0.319877i	−0.943469	−	0.331461i	$-0.892459\pi$
$569$	−59792.5	+	103564.i	−0.184681	+	0.319877i	0.758788	+	0.651338i	$0.225792\pi$
$570$		0			0
$571$	−14325.3	−	24812.2i	−0.0439372	−	0.0761014i	0.843220	−	0.537568i	$-0.180657\pi$
$571$	−14325.3	−	24812.2i	−0.0439372	−	0.0761014i	−0.887158	+	0.461466i	$0.847323\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	26229.0i		−	0.0793317i
$576$		0			0
$577$	−101252.			−0.304126			−0.152063	−	0.988371i	$-0.548592\pi$
$577$	−101252.			−0.304126			−0.152063	+	0.988371i	$0.548592\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	146975.	−	84855.8i	0.435401	−	0.251379i
$582$		0			0
$583$	749854.	+	432929.i	2.20617	+	1.27374i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−36792.0	+	63725.7i	−0.106777	+	0.184943i	−0.914463	−	0.404670i	$-0.867386\pi$
$587$	−36792.0	+	63725.7i	−0.106777	+	0.184943i	0.807686	+	0.589613i	$0.200720\pi$
$588$		0			0
$589$	104012.	−	60051.5i	0.299815	−	0.173098i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−295880.			−0.841407			−0.420704	−	0.907198i	$-0.638217\pi$
$593$	−295880.			−0.841407			−0.420704	+	0.907198i	$0.638217\pi$
$594$		0			0
$595$	−687887.			−1.94305
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−528443.	+	305097.i	−1.47280	+	0.850323i	−0.999532	−	0.0305942i	$-0.990260\pi$
$599$	−528443.	+	305097.i	−1.47280	+	0.850323i	−0.473271	+	0.880917i	$0.656927\pi$
$600$		0			0
$601$	56375.3	−	97644.9i	0.156077	−	0.270334i	−0.777373	−	0.629039i	$-0.783448\pi$
$601$	56375.3	−	97644.9i	0.156077	−	0.270334i	0.933451	+	0.358705i	$0.116782\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−461081.	−	266205.i	−1.25970	−	0.727287i
$606$		0			0
$607$	121995.	−	70434.1i	0.331105	−	0.191164i	−0.325226	−	0.945636i	$-0.605440\pi$
$607$	121995.	−	70434.1i	0.331105	−	0.191164i	0.656332	+	0.754472i	$0.272107\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	77120.6			0.206580
$612$		0			0
$613$			471219.i			1.25401i	0.779014	+	0.627006i	$0.215720\pi$
$613$			471219.i			1.25401i	−0.779014	+	0.627006i	$0.784280\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	189422.	+	328089.i	0.497577	+	0.861828i	0.999996	−	0.00279589i	$-0.000889962\pi$
$617$	189422.	+	328089.i	0.497577	+	0.861828i	−0.502419	+	0.864624i	$0.667557\pi$
$618$		0			0
$619$	−90319.8	+	156439.i	−0.235723	+	0.408284i	−0.959483	−	0.281768i	$-0.909079\pi$
$619$	−90319.8	+	156439.i	−0.235723	+	0.408284i	0.723760	+	0.690052i	$0.242412\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−97568.6	−	56331.3i	−0.251382	−	0.145135i
$624$		0			0
$625$	220490.	+	381899.i	0.564454	+	0.977663i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	82717.9i		−	0.209073i
$630$		0			0
$631$		−	115341.i		−	0.289685i	−0.989455	−	0.144842i	$-0.953733\pi$
$631$		−	115341.i		−	0.289685i	0.989455	−	0.144842i	$-0.0462675\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−20022.3	−	34679.7i	−0.0496555	−	0.0860058i
$636$		0			0
$637$	−204945.	−	118325.i	−0.505078	−	0.291607i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−71144.9	+	123227.i	−0.173152	+	0.299908i	−0.939520	−	0.342493i	$-0.888729\pi$
$641$	−71144.9	+	123227.i	−0.173152	+	0.299908i	0.766368	+	0.642402i	$0.222062\pi$
$642$		0			0
$643$	150535.	+	260734.i	0.364095	+	0.630631i	0.988630	−	0.150365i	$-0.0480450\pi$
$643$	150535.	+	260734.i	0.364095	+	0.630631i	−0.624535	+	0.780996i	$0.714712\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			562947.i			1.34480i	0.740186	+	0.672402i	$0.234737\pi$
$647$			562947.i			1.34480i	−0.740186	+	0.672402i	$0.765263\pi$
$648$		0			0
$649$	−657103.			−1.56007
$650$		0			0
$651$		0			0
$652$		0			0
$653$	658714.	−	380308.i	1.54479	−	0.891887i	0.546267	−	0.837611i	$-0.316048\pi$
$653$	658714.	−	380308.i	1.54479	−	0.891887i	0.998526	−	0.0542753i	$-0.0172848\pi$
$654$		0			0
$655$	−149368.	−	86237.4i	−0.348156	−	0.201008i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−207982.	+	360235.i	−0.478911	+	0.829497i	−0.999708	−	0.0241831i	$-0.992302\pi$
$659$	−207982.	+	360235.i	−0.478911	+	0.829497i	0.520797	+	0.853681i	$0.325635\pi$
$660$		0			0
$661$	317798.	−	183481.i	0.727358	−	0.419940i	−0.0900969	−	0.995933i	$-0.528718\pi$
$661$	317798.	−	183481.i	0.727358	−	0.419940i	0.817455	+	0.575993i	$0.195384\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−184766.			−0.417810
$666$		0			0
$667$	241629.			0.543123
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−333813.	+	192727.i	−0.741409	+	0.428053i
$672$		0			0
$673$	208572.	−	361257.i	0.460496	−	0.797602i	−0.538490	−	0.842632i	$-0.681005\pi$
$673$	208572.	−	361257.i	0.460496	−	0.797602i	0.998986	+	0.0450300i	$0.0143383\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−270633.	−	156250.i	−0.590477	−	0.340912i	0.174809	−	0.984602i	$-0.444069\pi$
$677$	−270633.	−	156250.i	−0.590477	−	0.340912i	−0.765286	+	0.643690i	$0.777403\pi$
$678$		0			0
$679$	−131160.	+	75725.3i	−0.284487	+	0.164249i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−140348.			−0.300860			−0.150430	−	0.988621i	$-0.548066\pi$
$683$	−140348.			−0.300860			−0.150430	+	0.988621i	$0.548066\pi$
$684$		0			0
$685$		−	573088.i		−	1.22135i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−446953.	−	774145.i	−0.941506	−	1.63074i
$690$		0			0
$691$	−6583.82	+	11403.5i	−0.0137887	+	0.0238827i	−0.872837	−	0.488011i	$-0.837723\pi$
$691$	−6583.82	+	11403.5i	−0.0137887	+	0.0238827i	0.859049	+	0.511894i	$0.171056\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	480033.	+	277147.i	0.993805	+	0.573773i
$696$		0			0
$697$	−603892.	−	1.04597e6i	−1.24306	−	2.15305i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	143500.i		−	0.292022i	−0.989283	−	0.146011i	$-0.953357\pi$
$701$		−	143500.i		−	0.292022i	0.989283	−	0.146011i	$-0.0466434\pi$
$702$		0			0
$703$		−	22218.0i		−	0.0449567i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	164632.	+	285151.i	0.329363	+	0.570474i
$708$		0			0
$709$	−386361.	−	223066.i	−0.768602	−	0.443752i	0.0637739	−	0.997964i	$-0.479686\pi$
$709$	−386361.	−	223066.i	−0.768602	−	0.443752i	−0.832376	+	0.554212i	$0.813020\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	145164.	−	251432.i	0.285549	−	0.494585i
$714$		0			0
$715$	477623.	+	827267.i	0.934271	+	1.61820i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	316954.i		−	0.613110i	−0.951853	−	0.306555i	$-0.900824\pi$
$719$		−	316954.i		−	0.613110i	0.951853	−	0.306555i	$-0.0991763\pi$
$720$		0			0
$721$	679667.			1.30745
$722$		0			0
$723$		0			0
$724$		0			0
$725$	72017.6	−	41579.4i	0.137013	−	0.0791047i
$726$		0			0
$727$	−429578.	−	248017.i	−0.812779	−	0.469258i	0.0351407	−	0.999382i	$-0.488812\pi$
$727$	−429578.	−	248017.i	−0.812779	−	0.469258i	−0.847920	+	0.530124i	$0.822145\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	190938.	−	330714.i	0.357320	−	0.618896i
$732$		0			0
$733$	607822.	−	350926.i	1.13128	−	0.653143i	0.187021	−	0.982356i	$-0.440117\pi$
$733$	607822.	−	350926.i	1.13128	−	0.653143i	0.944256	+	0.329213i	$0.106783\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−544283.			−1.00205
$738$		0			0
$739$	−591401.			−1.08291			−0.541456	−	0.840729i	$-0.682127\pi$
$739$	−591401.			−1.08291			−0.541456	+	0.840729i	$0.682127\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	76816.9	−	44350.2i	0.139149	−	0.0803375i	−0.428810	−	0.903395i	$-0.641067\pi$
$743$	76816.9	−	44350.2i	0.139149	−	0.0803375i	0.567958	+	0.823057i	$0.307734\pi$
$744$		0			0
$745$	69713.8	−	120748.i	0.125605	−	0.217554i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−253267.	−	146224.i	−0.451456	−	0.260648i
$750$		0			0
$751$	692048.	−	399554.i	1.22703	−	0.708428i	0.260626	−	0.965440i	$-0.416071\pi$
$751$	692048.	−	399554.i	1.22703	−	0.708428i	0.966408	+	0.257011i	$0.0827378\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−546805.			−0.959264
$756$		0			0
$757$		−	1.07119e6i		−	1.86927i	−0.355603	−	0.934637i	$-0.615725\pi$
$757$		−	1.07119e6i		−	1.86927i	0.355603	−	0.934637i	$-0.384275\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−173781.	−	300998.i	−0.300078	−	0.519750i	0.676075	−	0.736832i	$-0.263679\pi$
$761$	−173781.	−	300998.i	−0.300078	−	0.519750i	−0.976153	+	0.217082i	$0.930346\pi$
$762$		0			0
$763$	523483.	−	906699.i	0.899194	−	1.55745i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	587502.	+	339195.i	0.998662	+	0.576578i
$768$		0			0
$769$	307338.	+	532324.i	0.519712	+	0.900168i	0.999737	+	0.0229135i	$0.00729422\pi$
$769$	307338.	+	532324.i	0.519712	+	0.900168i	−0.480025	+	0.877255i	$0.659372\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			558543.i			0.934755i	0.884058	+	0.467378i	$0.154801\pi$
$773$			558543.i			0.934755i	−0.884058	+	0.467378i	$0.845199\pi$
$774$		0			0
$775$		−	99919.0i		−	0.166358i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−162205.	−	280947.i	−0.267294	−	0.462967i
$780$		0			0
$781$	−488195.	−	281860.i	−0.800371	−	0.462095i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−119871.	+	207622.i	−0.194524	+	0.336925i
$786$		0			0
$787$	−171055.	−	296276.i	−0.276176	−	0.478351i	0.694255	−	0.719729i	$-0.255734\pi$
$787$	−171055.	−	296276.i	−0.276176	−	0.478351i	−0.970431	+	0.241378i	$0.922401\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			804358.i			1.28557i
$792$		0			0
$793$	397940.			0.632807
$794$		0			0
$795$		0			0
$796$		0			0
$797$	639205.	−	369045.i	1.00629	−	0.580982i	0.0961872	−	0.995363i	$-0.469335\pi$
$797$	639205.	−	369045.i	1.00629	−	0.580982i	0.910103	+	0.414381i	$0.136002\pi$
$798$		0			0
$799$	−148124.	−	85519.6i	−0.232024	−	0.133959i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	517233.	−	895874.i	0.802149	−	1.38936i
$804$		0			0
$805$	−386802.	+	223320.i	−0.596894	+	0.344617i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	869283.			1.32820			0.664101	−	0.747643i	$-0.268815\pi$
$809$	869283.			1.32820			0.664101	+	0.747643i	$0.268815\pi$
$810$		0			0
$811$	524099.			0.796841			0.398420	−	0.917203i	$-0.369559\pi$
$811$	524099.			0.796841			0.398420	+	0.917203i	$0.369559\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−883065.	+	509838.i	−1.32947	+	0.767568i
$816$		0			0
$817$	51285.8	−	88829.7i	0.0768340	−	0.133080i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−176424.	−	101858.i	−0.261741	−	0.151116i	0.363388	−	0.931638i	$-0.381620\pi$
$821$	−176424.	−	101858.i	−0.261741	−	0.151116i	−0.625128	+	0.780522i	$0.714953\pi$
$822$		0			0
$823$	−840341.	+	485171.i	−1.24067	+	0.716300i	−0.969230	−	0.246156i	$-0.920832\pi$
$823$	−840341.	+	485171.i	−1.24067	+	0.716300i	−0.271438	+	0.962456i	$0.587499\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−876265.			−1.28122			−0.640611	−	0.767865i	$-0.721319\pi$
$827$	−876265.			−1.28122			−0.640611	+	0.767865i	$0.721319\pi$
$828$		0			0
$829$			728747.i			1.06040i	0.847874	+	0.530198i	$0.177882\pi$
$829$			728747.i			1.06040i	−0.847874	+	0.530198i	$0.822118\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	262423.	+	454530.i	0.378192	+	0.655047i
$834$		0			0
$835$	185013.	−	320451.i	0.265356	−	0.459610i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−261880.	−	151197.i	−0.372031	−	0.214792i	0.302314	−	0.953208i	$-0.402241\pi$
$839$	−261880.	−	151197.i	−0.372031	−	0.214792i	−0.674345	+	0.738416i	$0.735574\pi$
$840$		0			0
$841$	29400.8	+	50923.6i	0.0415687	+	0.0719991i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	219812.i		−	0.307849i
$846$		0			0
$847$			1.19634e6i			1.66758i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−26854.1	−	46512.7i	−0.0370810	−	0.0642262i
$852$		0			0
$853$	−613667.	−	354301.i	−0.843402	−	0.486938i	0.0150171	−	0.999887i	$-0.495220\pi$
$853$	−613667.	−	354301.i	−0.843402	−	0.486938i	−0.858419	+	0.512949i	$0.828553\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	506749.	−	877715.i	0.689972	−	1.19507i	−0.281875	−	0.959451i	$-0.590956\pi$
$857$	506749.	−	877715.i	0.689972	−	1.19507i	0.971847	−	0.235615i	$-0.0757103\pi$
$858$		0			0
$859$	−187918.	−	325483.i	−0.254672	−	0.441105i	0.710134	−	0.704066i	$-0.248634\pi$
$859$	−187918.	−	325483.i	−0.254672	−	0.441105i	−0.964806	+	0.262961i	$0.915301\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	771077.i		−	1.03532i	−0.855585	−	0.517662i	$-0.826802\pi$
$863$		−	771077.i		−	1.03532i	0.855585	−	0.517662i	$-0.173198\pi$
$864$		0			0
$865$	−300146.			−0.401144
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.73138e6	+	999610.i	−2.29272	+	1.32371i
$870$		0			0
$871$	486632.	+	280957.i	0.641453	+	0.370343i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	428730.	−	742582.i	0.559974	−	0.969903i
$876$		0			0
$877$	121036.	−	69880.4i	0.157368	−	0.0908566i	−0.419248	−	0.907872i	$-0.637706\pi$
$877$	121036.	−	69880.4i	0.157368	−	0.0908566i	0.576616	+	0.817015i	$0.304373\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−764399.			−0.984846			−0.492423	−	0.870356i	$-0.663889\pi$
$881$	−764399.			−0.984846			−0.492423	+	0.870356i	$0.663889\pi$
$882$		0			0
$883$	276974.			0.355237			0.177618	−	0.984099i	$-0.443161\pi$
$883$	276974.			0.355237			0.177618	+	0.984099i	$0.443161\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−284311.	+	164147.i	−0.361365	+	0.208634i	−0.669679	−	0.742650i	$-0.733568\pi$
$887$	−284311.	+	164147.i	−0.361365	+	0.208634i	0.308314	+	0.951285i	$0.400235\pi$
$888$		0			0
$889$	−44990.7	+	77926.1i	−0.0569271	+	0.0986006i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−39786.1	−	22970.5i	−0.0498917	−	0.0288050i
$894$		0			0
$895$	165765.	−	95704.5i	0.206941	−	0.119478i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	920482.			1.13893
$900$		0			0
$901$			1.98252e6i			2.44212i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−350234.	−	606623.i	−0.427623	−	0.740664i
$906$		0			0
$907$	−2523.63	+	4371.06i	−0.00306769	+	0.00531339i	−0.867555	−	0.497341i	$-0.834310\pi$
$907$	−2523.63	+	4371.06i	−0.00306769	+	0.00531339i	0.864487	+	0.502654i	$0.167643\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	124609.	+	71943.0i	0.150145	+	0.0866865i	0.573191	−	0.819422i	$-0.305705\pi$
$911$	124609.	+	71943.0i	0.150145	+	0.0866865i	−0.423045	+	0.906109i	$0.639039\pi$
$912$		0			0
$913$	261339.	+	452653.i	0.313518	+	0.543030i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			387555.i			0.460887i
$918$		0			0
$919$		−	1.50889e6i		−	1.78660i	−0.449466	−	0.893298i	$-0.648386\pi$
$919$		−	1.50889e6i		−	1.78660i	0.449466	−	0.893298i	$-0.351614\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	290990.	+	504010.i	0.341566	+	0.591610i
$924$		0			0
$925$	−16007.7	−	9242.07i	−0.0187088	−	0.0108015i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−489589.	+	847994.i	−0.567284	+	0.982565i	0.429549	+	0.903043i	$0.358672\pi$
$929$	−489589.	+	847994.i	−0.567284	+	0.982565i	−0.996833	+	0.0795211i	$0.974661\pi$
$930$		0			0
$931$	70486.7	+	122087.i	0.0813220	+	0.140854i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2.11856e6i		−	2.42335i
$936$		0			0
$937$	75553.9			0.0860553			0.0430276	−	0.999074i	$-0.486300\pi$
$937$	75553.9			0.0860553			0.0430276	+	0.999074i	$0.486300\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	33992.9	−	19625.8i	0.0383892	−	0.0221640i	−0.480683	−	0.876895i	$-0.659611\pi$
$941$	33992.9	−	19625.8i	0.0383892	−	0.0221640i	0.519072	+	0.854731i	$0.326278\pi$
$942$		0			0
$943$	−679142.	−	392103.i	−0.763726	−	0.440937i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−313680.	+	543310.i	−0.349773	+	0.605825i	−0.986209	−	0.165504i	$-0.947075\pi$
$947$	−313680.	+	543310.i	−0.349773	+	0.605825i	0.636436	+	0.771330i	$0.280408\pi$
$948$		0			0
$949$	−924894.	+	533988.i	−1.02697	+	0.592924i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	884152.			0.973512			0.486756	−	0.873538i	$-0.338180\pi$
$953$	884152.			0.973512			0.486756	+	0.873538i	$0.338180\pi$
$954$		0			0
$955$	−1.22489e6			−1.34304
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.11522e6	+	643871.i	−1.21261	+	0.700103i
$960$		0			0
$961$	91239.2	−	158031.i	0.0987949	−	0.171118i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−280693.	−	162058.i	−0.301423	−	0.174027i
$966$		0			0
$967$	415020.	−	239612.i	0.443829	−	0.256245i	−0.261391	−	0.965233i	$-0.584181\pi$
$967$	415020.	−	239612.i	0.443829	−	0.256245i	0.705220	+	0.708988i	$0.250848\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	37374.6			0.0396404			0.0198202	−	0.999804i	$-0.493691\pi$
$971$	37374.6			0.0396404			0.0198202	+	0.999804i	$0.493691\pi$
$972$		0			0
$973$		−	1.24551e6i		−	1.31559i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	380919.	+	659770.i	0.399064	+	0.691200i	0.993611	−	0.112861i	$-0.0360016\pi$
$977$	380919.	+	659770.i	0.399064	+	0.691200i	−0.594546	+	0.804061i	$0.702668\pi$
$978$		0			0
$979$	173489.	−	300492.i	0.181012	−	0.313522i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−958351.	−	553304.i	−0.991785	−	0.572607i	−0.0859774	−	0.996297i	$-0.527401\pi$
$983$	−958351.	−	553304.i	−0.991785	−	0.572607i	−0.905807	+	0.423690i	$0.860735\pi$
$984$		0			0
$985$	−940809.	−	1.62953e6i	−0.969681	−	1.67954i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	247949.i		−	0.253496i
$990$		0			0
$991$			306890.i			0.312489i	0.987718	+	0.156245i	$0.0499388\pi$
$991$			306890.i			0.312489i	−0.987718	+	0.156245i	$0.950061\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−717475.	−	1.24270e6i	−0.724704	−	1.25522i
$996$		0			0
$997$	13764.4	+	7946.86i	0.0138473	+	0.00799476i	0.506908	−	0.862000i	$-0.330788\pi$
$997$	13764.4	+	7946.86i	0.0138473	+	0.00799476i	−0.493060	+	0.869995i	$0.664122\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.5.t.b.559.36		88
3.2	odd	2		288.5.t.b.79.3		88
4.3	odd	2		216.5.p.b.19.5		88
8.3	odd	2	inner	864.5.t.b.559.9		88
8.5	even	2		216.5.p.b.19.27		88
9.4	even	3	inner	864.5.t.b.847.9		88
9.5	odd	6		288.5.t.b.175.4		88
12.11	even	2		72.5.p.b.43.40	yes	88
24.5	odd	2		72.5.p.b.43.18	✓	88
24.11	even	2		288.5.t.b.79.4		88
36.23	even	6		72.5.p.b.67.18	yes	88
36.31	odd	6		216.5.p.b.91.27		88
72.5	odd	6		72.5.p.b.67.40	yes	88
72.13	even	6		216.5.p.b.91.5		88
72.59	even	6		288.5.t.b.175.3		88
72.67	odd	6	inner	864.5.t.b.847.36		88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.5.p.b.43.18	✓	88	24.5	odd	2
72.5.p.b.43.40	yes	88	12.11	even	2
72.5.p.b.67.18	yes	88	36.23	even	6
72.5.p.b.67.40	yes	88	72.5	odd	6
216.5.p.b.19.5		88	4.3	odd	2
216.5.p.b.19.27		88	8.5	even	2
216.5.p.b.91.5		88	72.13	even	6
216.5.p.b.91.27		88	36.31	odd	6
288.5.t.b.79.3		88	3.2	odd	2
288.5.t.b.79.4		88	24.11	even	2
288.5.t.b.175.3		88	72.59	even	6
288.5.t.b.175.4		88	9.5	odd	6
864.5.t.b.559.9		88	8.3	odd	2	inner
864.5.t.b.559.36		88	1.1	even	1	trivial
864.5.t.b.847.9		88	9.4	even	3	inner
864.5.t.b.847.36		88	72.67	odd	6	inner

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	864.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(23.2381 - 13.4165i) q^{5} +(-52.2165 - 30.1472i) q^{7} +O(q^{10})\)	\(q+(23.2381 - 13.4165i) q^{5} +(-52.2165 - 30.1472i) q^{7} +(92.8475 - 160.817i) q^{11} +(-166.026 + 95.8552i) q^{13} +425.178 q^{17} +114.203 q^{19} +(239.080 - 138.033i) q^{23} +(47.5051 - 82.2813i) q^{25} +(757.999 + 437.631i) q^{29} +(910.769 - 525.833i) q^{31} -1617.88 q^{35} -194.549i q^{37} +(-1420.33 - 2460.08i) q^{41} +(449.077 - 777.825i) q^{43} +(-348.382 - 201.138i) q^{47} +(617.208 + 1069.03i) q^{49} +4662.79i q^{53} -4982.75i q^{55} +(-1769.31 - 3064.53i) q^{59} +(-1797.64 - 1037.87i) q^{61} +(-2572.08 + 4454.98i) q^{65} +(-1465.53 - 2538.37i) q^{67} -3035.73i q^{71} +5570.78 q^{73} +(-9696.34 + 5598.18i) q^{77} +(-9323.76 - 5383.08i) q^{79} +(-1407.36 + 2437.61i) q^{83} +(9880.31 - 5704.40i) q^{85} +1868.54 q^{89} +11559.1 q^{91} +(2653.85 - 1532.20i) q^{95} +(1255.93 - 2175.33i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q+O(q^{10}) \) 88 * q	\( 88 q + 44 q^{11} + 1156 q^{17} - 860 q^{19} + 5998 q^{25} - 2508 q^{35} - 2348 q^{41} - 3500 q^{43} + 16462 q^{49} - 3508 q^{59} + 2502 q^{65} - 5132 q^{67} + 19004 q^{73} + 17090 q^{83} + 8272 q^{89} + 9612 q^{91} + 9980 q^{97}+O(q^{100}) \) 88 * q + 44 * q^11 + 1156 * q^17 - 860 * q^19 + 5998 * q^25 - 2508 * q^35 - 2348 * q^41 - 3500 * q^43 + 16462 * q^49 - 3508 * q^59 + 2502 * q^65 - 5132 * q^67 + 19004 * q^73 + 17090 * q^83 + 8272 * q^89 + 9612 * q^91 + 9980 * q^97

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