LMFDB - Embedded newform 1296.3.q.o.1025.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,3,Mod(593,1296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1296.593"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,8,0,0,0,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{8} $
Twist minimal:	no (minimal twist has level 162)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1025.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1296.1025
Dual form			1296.3.q.o.593.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.01910 - 2.89778i) q^{5} +(-4.19615 - 7.26795i) q^{7} +(-12.7279 + 7.34847i) q^{11} +(10.5981 - 18.3564i) q^{13} +7.76457i q^{17} -24.3923 q^{19} +(-12.7279 - 7.34847i) q^{23} +(4.29423 + 7.43782i) q^{25} +(30.7387 - 17.7470i) q^{29} +(4.00000 - 6.92820i) q^{31} +48.6381i q^{35} -60.5692 q^{37} +(29.1301 + 16.8183i) q^{41} +(4.58846 + 7.94744i) q^{43} +(14.6969 - 8.48528i) q^{47} +(-10.7154 + 18.5596i) q^{49} +25.7605i q^{53} +85.1769 q^{55} +(53.4083 + 30.8353i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-106.386 + 61.4217i) q^{65} +(10.5885 - 18.3397i) q^{67} +101.214i q^{71} +40.4115 q^{73} +(106.817 + 61.6706i) q^{77} +(49.3731 + 85.5167i) q^{79} +(-89.6231 + 51.7439i) q^{83} +(22.5000 - 38.9711i) q^{85} +134.130i q^{89} -177.885 q^{91} +(122.427 + 70.6835i) q^{95} +(37.5692 + 65.0718i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} + 64 q^{13} - 112 q^{19} - 28 q^{25} + 32 q^{31} - 152 q^{37} - 88 q^{43} - 252 q^{49} + 432 q^{55} + 52 q^{61} - 40 q^{67} + 448 q^{73} + 104 q^{79} + 180 q^{85} - 176 q^{91} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 64 * q^13 - 112 * q^19 - 28 * q^25 + 32 * q^31 - 152 * q^37 - 88 * q^43 - 252 * q^49 + 432 * q^55 + 52 * q^61 - 40 * q^67 + 448 * q^73 + 104 * q^79 + 180 * q^85 - 176 * q^91 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	−5.01910	−	2.89778i	−1.00382	−	0.579555i	−0.0944434	−	0.995530i	$-0.530107\pi$
$5$	−5.01910	−	2.89778i	−1.00382	−	0.579555i	−0.909376	+	0.415975i	$0.863440\pi$
$6$		0			0
$7$	−4.19615	−	7.26795i	−0.599450	−	1.03828i	−0.992902	−	0.118933i	$-0.962053\pi$
$7$	−4.19615	−	7.26795i	−0.599450	−	1.03828i	0.393452	−	0.919345i	$-0.371281\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−12.7279	+	7.34847i	−1.15708	+	0.668043i	−0.950603	−	0.310408i	$-0.899534\pi$
$11$	−12.7279	+	7.34847i	−1.15708	+	0.668043i	−0.206480	+	0.978451i	$0.566201\pi$
$12$		0			0
$13$	10.5981	−	18.3564i	0.815237	−	1.41203i	−0.0939212	−	0.995580i	$-0.529940\pi$
$13$	10.5981	−	18.3564i	0.815237	−	1.41203i	0.909158	−	0.416452i	$-0.136726\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			7.76457i			0.456739i	0.973574	+	0.228370i	$0.0733395\pi$
$17$			7.76457i			0.456739i	−0.973574	+	0.228370i	$0.926660\pi$
$18$		0			0
$19$	−24.3923			−1.28381			−0.641903	−	0.766786i	$-0.721855\pi$
$19$	−24.3923			−1.28381			−0.641903	+	0.766786i	$0.721855\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.7279	−	7.34847i	−0.553388	−	0.319499i	0.197099	−	0.980384i	$-0.436848\pi$
$23$	−12.7279	−	7.34847i	−0.553388	−	0.319499i	−0.750487	+	0.660885i	$0.770181\pi$
$24$		0			0
$25$	4.29423	+	7.43782i	0.171769	+	0.297513i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	30.7387	−	17.7470i	1.05996	−	0.611966i	0.134536	−	0.990909i	$-0.457046\pi$
$29$	30.7387	−	17.7470i	1.05996	−	0.611966i	0.925420	+	0.378942i	$0.123712\pi$
$30$		0			0
$31$	4.00000	−	6.92820i	0.129032	−	0.223490i	−0.794270	−	0.607565i	$-0.792146\pi$
$31$	4.00000	−	6.92820i	0.129032	−	0.223490i	0.923302	+	0.384075i	$0.125480\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			48.6381i			1.38966i
$36$		0			0
$37$	−60.5692			−1.63701			−0.818503	−	0.574502i	$-0.805196\pi$
$37$	−60.5692			−1.63701			−0.818503	+	0.574502i	$0.805196\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	29.1301	+	16.8183i	0.710490	+	0.410201i	0.811242	−	0.584710i	$-0.198792\pi$
$41$	29.1301	+	16.8183i	0.710490	+	0.410201i	−0.100753	+	0.994912i	$0.532125\pi$
$42$		0			0
$43$	4.58846	+	7.94744i	0.106708	+	0.184824i	0.914435	−	0.404733i	$-0.132636\pi$
$43$	4.58846	+	7.94744i	0.106708	+	0.184824i	−0.807727	+	0.589557i	$0.799302\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	14.6969	−	8.48528i	0.312701	−	0.180538i	−0.335434	−	0.942064i	$-0.608883\pi$
$47$	14.6969	−	8.48528i	0.312701	−	0.180538i	0.648134	+	0.761526i	$0.275549\pi$
$48$		0			0
$49$	−10.7154	+	18.5596i	−0.218681	+	0.378767i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			25.7605i			0.486046i	0.970020	+	0.243023i	$0.0781391\pi$
$53$			25.7605i			0.486046i	−0.970020	+	0.243023i	$0.921861\pi$
$54$		0			0
$55$	85.1769			1.54867
$56$		0			0
$57$		0			0
$58$		0			0
$59$	53.4083	+	30.8353i	0.905225	+	0.522632i	0.878892	−	0.477021i	$-0.158283\pi$
$59$	53.4083	+	30.8353i	0.905225	+	0.522632i	0.0263336	+	0.999653i	$0.491617\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.106557	+	0.184563i	0.914373	−	0.404872i	$-0.132684\pi$
$61$	6.50000	+	11.2583i	0.106557	+	0.184563i	−0.807816	+	0.589435i	$0.799351\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−106.386	+	61.4217i	−1.63670	+	0.944950i
$66$		0			0
$67$	10.5885	−	18.3397i	0.158037	−	0.273728i	−0.776124	−	0.630580i	$-0.782817\pi$
$67$	10.5885	−	18.3397i	0.158037	−	0.273728i	0.934161	+	0.356853i	$0.116150\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			101.214i			1.42555i	0.701392	+	0.712776i	$0.252562\pi$
$71$			101.214i			1.42555i	−0.701392	+	0.712776i	$0.747438\pi$
$72$		0			0
$73$	40.4115			0.553583			0.276791	−	0.960930i	$-0.410729\pi$
$73$	40.4115			0.553583			0.276791	+	0.960930i	$0.410729\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	106.817	+	61.6706i	1.38723	+	0.800917i
$78$		0			0
$79$	49.3731	+	85.5167i	0.624976	+	1.08249i	0.988546	+	0.150923i	$0.0482244\pi$
$79$	49.3731	+	85.5167i	0.624976	+	1.08249i	−0.363570	+	0.931567i	$0.618442\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−89.6231	+	51.7439i	−1.07980	+	0.623420i	−0.930841	−	0.365424i	$-0.880924\pi$
$83$	−89.6231	+	51.7439i	−1.07980	+	0.623420i	−0.148954	+	0.988844i	$0.547591\pi$
$84$		0			0
$85$	22.5000	−	38.9711i	0.264706	−	0.458484i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			134.130i			1.50708i	0.657403	+	0.753539i	$0.271655\pi$
$89$			134.130i			1.50708i	−0.657403	+	0.753539i	$0.728345\pi$
$90$		0			0
$91$	−177.885			−1.95478
$92$		0			0
$93$		0			0
$94$		0			0
$95$	122.427	+	70.6835i	1.28871	+	0.744037i
$96$		0			0
$97$	37.5692	+	65.0718i	0.387312	+	0.670843i	0.992087	−	0.125553i	$-0.0400705\pi$
$97$	37.5692	+	65.0718i	0.387312	+	0.670843i	−0.604775	+	0.796396i	$0.706737\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−25.1920	+	14.5446i	−0.249426	+	0.144006i	−0.619501	−	0.784995i	$-0.712665\pi$
$101$	−25.1920	+	14.5446i	−0.249426	+	0.144006i	0.370075	+	0.929002i	$0.379332\pi$
$102$		0			0
$103$	48.3538	−	83.7513i	0.469455	−	0.813119i	−0.529936	−	0.848038i	$-0.677784\pi$
$103$	48.3538	−	83.7513i	0.469455	−	0.813119i	0.999390	+	0.0349186i	$0.0111172\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	177.582i		−	1.65964i	−0.558030	−	0.829821i	$-0.688442\pi$
$107$		−	177.582i		−	1.65964i	0.558030	−	0.829821i	$-0.311558\pi$
$108$		0			0
$109$	61.9423			0.568278			0.284139	−	0.958783i	$-0.408292\pi$
$109$	61.9423			0.568278			0.284139	+	0.958783i	$0.408292\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−95.0991	−	54.9055i	−0.841585	−	0.485889i	0.0162179	−	0.999868i	$-0.494837\pi$
$113$	−95.0991	−	54.9055i	−0.841585	−	0.485889i	−0.857803	+	0.513979i	$0.828171\pi$
$114$		0			0
$115$	42.5885	+	73.7654i	0.370334	+	0.641438i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	56.4325	−	32.5813i	0.474223	−	0.273793i
$120$		0			0
$121$	47.5000	−	82.2724i	0.392562	−	0.679937i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			95.1140i			0.760912i
$126$		0			0
$127$	−141.177			−1.11163			−0.555815	−	0.831306i	$-0.687594\pi$
$127$	−141.177			−1.11163			−0.555815	+	0.831306i	$0.687594\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−46.0598	−	26.5927i	−0.351602	−	0.202997i	0.313789	−	0.949493i	$-0.398402\pi$
$131$	−46.0598	−	26.5927i	−0.351602	−	0.202997i	−0.665391	+	0.746495i	$0.731735\pi$
$132$		0			0
$133$	102.354	+	177.282i	0.769578	+	1.33295i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.41555	−	0.817267i	0.0103325	−	0.00596545i	−0.494825	−	0.868993i	$-0.664768\pi$
$137$	1.41555	−	0.817267i	0.0103325	−	0.00596545i	0.505157	+	0.863027i	$0.331434\pi$
$138$		0			0
$139$	−9.60770	+	16.6410i	−0.0691201	+	0.119720i	−0.898514	−	0.438944i	$-0.855353\pi$
$139$	−9.60770	+	16.6410i	−0.0691201	+	0.119720i	0.829394	+	0.558664i	$0.188686\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			311.519i			2.17845i
$144$		0			0
$145$	−205.708			−1.41867
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−81.6504	−	47.1409i	−0.547989	−	0.316382i	0.200321	−	0.979730i	$-0.435801\pi$
$149$	−81.6504	−	47.1409i	−0.547989	−	0.316382i	−0.748311	+	0.663348i	$0.769135\pi$
$150$		0			0
$151$	16.0000	+	27.7128i	0.105960	+	0.183529i	0.914130	−	0.405421i	$-0.132875\pi$
$151$	16.0000	+	27.7128i	0.105960	+	0.183529i	−0.808170	+	0.588949i	$0.799542\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−40.1528	+	23.1822i	−0.259050	+	0.149563i
$156$		0			0
$157$	−0.146171	+	0.253175i	−0.000931025	+	0.00161258i	−0.866491	−	0.499193i	$-0.833630\pi$
$157$	−0.146171	+	0.253175i	−0.000931025	+	0.00161258i	0.865560	+	0.500806i	$0.166963\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			123.341i			0.766094i
$162$		0			0
$163$	−28.7846			−0.176593			−0.0882963	−	0.996094i	$-0.528142\pi$
$163$	−28.7846			−0.176593			−0.0882963	+	0.996094i	$0.528142\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	48.5564	+	28.0341i	0.290757	+	0.167869i	0.638283	−	0.769802i	$-0.279645\pi$
$167$	48.5564	+	28.0341i	0.290757	+	0.167869i	−0.347526	+	0.937670i	$0.612978\pi$
$168$		0			0
$169$	−140.138	−	242.727i	−0.829222	−	1.43625i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−191.350	+	110.476i	−1.10607	+	0.638589i	−0.937808	−	0.347154i	$-0.887148\pi$
$173$	−191.350	+	110.476i	−1.10607	+	0.638589i	−0.168260	+	0.985743i	$0.553815\pi$
$174$		0			0
$175$	36.0385	−	62.4205i	0.205934	−	0.356688i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			248.347i			1.38741i	0.720258	+	0.693706i	$0.244023\pi$
$179$			248.347i			1.38741i	−0.720258	+	0.693706i	$0.755977\pi$
$180$		0			0
$181$	−158.277			−0.874458			−0.437229	−	0.899350i	$-0.644040\pi$
$181$	−158.277			−0.874458			−0.437229	+	0.899350i	$0.644040\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	304.003	+	175.516i	1.64326	+	0.948736i
$186$		0			0
$187$	−57.0577	−	98.8269i	−0.305121	−	0.528486i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−29.9215	+	17.2752i	−0.156657	+	0.0904459i	−0.576279	−	0.817253i	$-0.695496\pi$
$191$	−29.9215	+	17.2752i	−0.156657	+	0.0904459i	0.419622	+	0.907699i	$0.362163\pi$
$192$		0			0
$193$	27.5000	−	47.6314i	0.142487	−	0.246795i	−0.785946	−	0.618296i	$-0.787823\pi$
$193$	27.5000	−	47.6314i	0.142487	−	0.246795i	0.928433	+	0.371501i	$0.121157\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			35.7170i			0.181305i	0.995883	+	0.0906524i	$0.0288952\pi$
$197$			35.7170i			0.181305i	−0.995883	+	0.0906524i	$0.971105\pi$
$198$		0			0
$199$	375.138			1.88512			0.942559	−	0.334040i	$-0.108412\pi$
$199$	375.138			1.88512			0.942559	+	0.334040i	$0.108412\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−257.969	−	148.938i	−1.27078	−	0.733687i
$204$		0			0
$205$	−97.4711	−	168.825i	−0.475469	−	0.823536i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	310.463	−	179.246i	1.48547	−	0.857637i
$210$		0			0
$211$	153.727	−	266.263i	0.728563	−	1.26191i	−0.228927	−	0.973444i	$-0.573522\pi$
$211$	153.727	−	266.263i	0.728563	−	1.26191i	0.957490	−	0.288465i	$-0.0931450\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	53.1853i		−	0.247374i
$216$		0			0
$217$	−67.1384			−0.309394
$218$		0			0
$219$		0			0
$220$		0			0
$221$	142.530	+	82.2895i	0.644930	+	0.372351i
$222$		0			0
$223$	−169.296	−	293.229i	−0.759175	−	1.31493i	−0.943272	−	0.332022i	$-0.892269\pi$
$223$	−169.296	−	293.229i	−0.759175	−	1.31493i	0.184096	−	0.982908i	$-0.441064\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−107.344	+	61.9752i	−0.472882	+	0.273019i	−0.717445	−	0.696615i	$-0.754689\pi$
$227$	−107.344	+	61.9752i	−0.472882	+	0.273019i	0.244563	+	0.969633i	$0.421355\pi$
$228$		0			0
$229$	−37.0289	+	64.1359i	−0.161698	+	0.280069i	−0.935478	−	0.353386i	$-0.885030\pi$
$229$	−37.0289	+	64.1359i	−0.161698	+	0.280069i	0.773780	+	0.633455i	$0.218364\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	273.223i		−	1.17263i	−0.810082	−	0.586316i	$-0.800578\pi$
$233$		−	273.223i		−	1.17263i	0.810082	−	0.586316i	$-0.199422\pi$
$234$		0			0
$235$	−98.3538			−0.418527
$236$		0			0
$237$		0			0
$238$		0			0
$239$	392.210	+	226.443i	1.64105	+	0.947459i	0.980462	+	0.196708i	$0.0630252\pi$
$239$	392.210	+	226.443i	1.64105	+	0.947459i	0.660585	+	0.750751i	$0.270308\pi$
$240$		0			0
$241$	191.344	+	331.418i	0.793959	+	1.37518i	0.923498	+	0.383604i	$0.125317\pi$
$241$	191.344	+	331.418i	0.793959	+	1.37518i	−0.129538	+	0.991574i	$0.541350\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	107.563	−	62.1016i	0.439033	−	0.253476i
$246$		0			0
$247$	−258.512	+	447.755i	−1.04661	+	1.81277i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	73.9307i		−	0.294544i	−0.989096	−	0.147272i	$-0.952951\pi$
$251$		−	73.9307i		−	0.294544i	0.989096	−	0.147272i	$-0.0470493\pi$
$252$		0			0
$253$	216.000			0.853755
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−138.398	−	79.9044i	−0.538515	−	0.310912i	0.205962	−	0.978560i	$-0.433968\pi$
$257$	−138.398	−	79.9044i	−0.538515	−	0.310912i	−0.744477	+	0.667648i	$0.767301\pi$
$258$		0			0
$259$	254.158	+	440.214i	0.981304	+	1.69967i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	224.392	−	129.553i	0.853202	−	0.492596i	−0.00852798	−	0.999964i	$-0.502715\pi$
$263$	224.392	−	129.553i	0.853202	−	0.492596i	0.861730	+	0.507367i	$0.169381\pi$
$264$		0			0
$265$	74.6481	−	129.294i	0.281691	−	0.487903i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	24.8168i		−	0.0922556i	−0.998936	−	0.0461278i	$-0.985312\pi$
$269$		−	24.8168i		−	0.0922556i	0.998936	−	0.0461278i	$-0.0146881\pi$
$270$		0			0
$271$	−98.1154			−0.362050			−0.181025	−	0.983479i	$-0.557941\pi$
$271$	−98.1154			−0.362050			−0.181025	+	0.983479i	$0.557941\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−109.313	−	63.1120i	−0.397503	−	0.229498i
$276$		0			0
$277$	0.707658	+	1.22570i	0.00255472	+	0.00442491i	0.867300	−	0.497786i	$-0.165853\pi$
$277$	0.707658	+	1.22570i	0.00255472	+	0.00442491i	−0.864745	+	0.502211i	$0.832520\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	87.2230	−	50.3582i	0.310402	−	0.179211i	−0.336704	−	0.941610i	$-0.609312\pi$
$281$	87.2230	−	50.3582i	0.310402	−	0.179211i	0.647106	+	0.762400i	$0.275979\pi$
$282$		0			0
$283$	−23.6462	+	40.9564i	−0.0835554	+	0.144722i	−0.904775	−	0.425890i	$-0.859961\pi$
$283$	−23.6462	+	40.9564i	−0.0835554	+	0.144722i	0.821219	+	0.570613i	$0.193294\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	282.288i		−	0.983582i
$288$		0			0
$289$	228.711			0.791389
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−288.920	−	166.808i	−0.986074	−	0.569310i	−0.0819755	−	0.996634i	$-0.526123\pi$
$293$	−288.920	−	166.808i	−0.986074	−	0.569310i	−0.904098	+	0.427324i	$0.859456\pi$
$294$		0			0
$295$	−178.708	−	309.531i	−0.605789	−	1.04926i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−269.783	+	155.759i	−0.902284	+	0.520934i
$300$		0			0
$301$	38.5077	−	66.6973i	0.127933	−	0.221586i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	75.3422i		−	0.247024i
$306$		0			0
$307$	−10.3538			−0.0337258			−0.0168629	−	0.999858i	$-0.505368\pi$
$307$	−10.3538			−0.0337258			−0.0168629	+	0.999858i	$0.505368\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.26229	+	4.77024i	0.0265669	+	0.0153384i	0.513225	−	0.858254i	$-0.328451\pi$
$311$	8.26229	+	4.77024i	0.0265669	+	0.0153384i	−0.486658	+	0.873593i	$0.661784\pi$
$312$		0			0
$313$	239.638	+	415.066i	0.765618	+	1.32609i	0.939919	+	0.341397i	$0.110900\pi$
$313$	239.638	+	415.066i	0.765618	+	1.32609i	−0.174301	+	0.984692i	$0.555767\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−155.592	+	89.8311i	−0.490827	+	0.283379i	−0.724917	−	0.688836i	$-0.758122\pi$
$317$	−155.592	+	89.8311i	−0.490827	+	0.283379i	0.234091	+	0.972215i	$0.424789\pi$
$318$		0			0
$319$	−260.827	+	451.765i	−0.817639	+	1.41619i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	189.396i		−	0.586365i
$324$		0			0
$325$	182.042			0.560130
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−123.341	−	71.2111i	−0.374897	−	0.216447i
$330$		0			0
$331$	147.727	+	255.870i	0.446305	+	0.773023i	0.998142	−	0.0609292i	$-0.0194064\pi$
$331$	147.727	+	255.870i	0.446305	+	0.773023i	−0.551837	+	0.833952i	$0.686073\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−106.289	+	61.3660i	−0.317281	+	0.183182i
$336$		0			0
$337$	−244.631	+	423.713i	−0.725907	+	1.25731i	0.232692	+	0.972550i	$0.425246\pi$
$337$	−244.631	+	423.713i	−0.725907	+	1.25731i	−0.958600	+	0.284758i	$0.908087\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			117.576i			0.344796i
$342$		0			0
$343$	−231.369			−0.674546
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−577.363	−	333.341i	−1.66387	−	0.960637i	−0.970840	−	0.239730i	$-0.922941\pi$
$347$	−577.363	−	333.341i	−1.66387	−	0.960637i	−0.693032	−	0.720907i	$-0.743726\pi$
$348$		0			0
$349$	−255.985	−	443.378i	−0.733480	−	1.27042i	−0.955387	−	0.295357i	$-0.904561\pi$
$349$	−255.985	−	443.378i	−0.733480	−	1.27042i	0.221907	−	0.975068i	$-0.428772\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−508.081	+	293.340i	−1.43932	+	0.830993i	−0.997802	−	0.0662588i	$-0.978894\pi$
$353$	−508.081	+	293.340i	−1.43932	+	0.830993i	−0.441519	+	0.897252i	$0.645560\pi$
$354$		0			0
$355$	293.296	−	508.004i	0.826186	−	1.43100i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	534.573i		−	1.48906i	−0.667589	−	0.744530i	$-0.732674\pi$
$359$		−	534.573i		−	1.48906i	0.667589	−	0.744530i	$-0.267326\pi$
$360$		0			0
$361$	233.985			0.648157
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−202.829	−	117.104i	−0.555697	−	0.320832i
$366$		0			0
$367$	7.64617	+	13.2436i	0.0208343	+	0.0360860i	0.876255	−	0.481848i	$-0.160034\pi$
$367$	7.64617	+	13.2436i	0.0208343	+	0.0360860i	−0.855420	+	0.517934i	$0.826701\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	187.226	−	108.095i	0.504651	−	0.291361i
$372$		0			0
$373$	−326.492	+	565.501i	−0.875314	+	1.51609i	−0.0188869	+	0.999822i	$0.506012\pi$
$373$	−326.492	+	565.501i	−0.875314	+	1.51609i	−0.856427	+	0.516267i	$0.827321\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	752.337i		−	1.99559i
$378$		0			0
$379$	655.215			1.72880			0.864400	−	0.502804i	$-0.167698\pi$
$379$	655.215			1.72880			0.864400	+	0.502804i	$0.167698\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	259.410	+	149.771i	0.677311	+	0.391046i	0.798841	−	0.601542i	$-0.205447\pi$
$383$	259.410	+	149.771i	0.677311	+	0.391046i	−0.121530	+	0.992588i	$0.538780\pi$
$384$		0			0
$385$	−357.415	−	619.061i	−0.928351	−	1.60795i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	402.319	−	232.279i	1.03424	−	0.597119i	0.116043	−	0.993244i	$-0.462979\pi$
$389$	402.319	−	232.279i	1.03424	−	0.597119i	0.918196	+	0.396126i	$0.129646\pi$
$390$		0			0
$391$	57.0577	−	98.8269i	0.145928	−	0.252754i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	572.289i		−	1.44883i
$396$		0			0
$397$	−185.708			−0.467777			−0.233889	−	0.972263i	$-0.575145\pi$
$397$	−185.708			−0.467777			−0.233889	+	0.972263i	$0.575145\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−54.4376	−	31.4296i	−0.135755	−	0.0783780i	0.430585	−	0.902550i	$-0.358307\pi$
$401$	−54.4376	−	31.4296i	−0.135755	−	0.0783780i	−0.566339	+	0.824172i	$0.691641\pi$
$402$		0			0
$403$	−84.7846	−	146.851i	−0.210384	−	0.364395i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	770.920	−	445.091i	1.89415	−	1.09359i
$408$		0			0
$409$	163.640	−	283.433i	0.400099	−	0.692991i	−0.593639	−	0.804732i	$-0.702309\pi$
$409$	163.640	−	283.433i	0.400099	−	0.692991i	0.993737	+	0.111741i	$0.0356425\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	517.558i		−	1.25317i
$414$		0			0
$415$	599.769			1.44523
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−562.808	−	324.937i	−1.34322	−	0.775507i	−0.355939	−	0.934509i	$-0.615839\pi$
$419$	−562.808	−	324.937i	−1.34322	−	0.775507i	−0.987278	+	0.159003i	$0.949172\pi$
$420$		0			0
$421$	−1.65956	−	2.87445i	−0.00394196	−	0.00682767i	0.864048	−	0.503410i	$-0.167921\pi$
$421$	−1.65956	−	2.87445i	−0.00394196	−	0.00682767i	−0.867990	+	0.496582i	$0.834588\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−57.7515	+	33.3428i	−0.135886	+	0.0784538i
$426$		0			0
$427$	54.5500	−	94.4833i	0.127752	−	0.221272i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			803.502i			1.86427i	0.362107	+	0.932136i	$0.382057\pi$
$431$			803.502i			1.86427i	−0.362107	+	0.932136i	$0.617943\pi$
$432$		0			0
$433$	−93.1230			−0.215065			−0.107532	−	0.994202i	$-0.534295\pi$
$433$	−93.1230			−0.215065			−0.107532	+	0.994202i	$0.534295\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	310.463	+	179.246i	0.710442	+	0.410174i
$438$		0			0
$439$	−236.000	−	408.764i	−0.537585	−	0.931125i	−0.999033	−	0.0439580i	$-0.986003\pi$
$439$	−236.000	−	408.764i	−0.537585	−	0.931125i	0.461448	−	0.887167i	$-0.347330\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	493.365	−	284.844i	1.11369	−	0.642989i	0.173908	−	0.984762i	$-0.444361\pi$
$443$	493.365	−	284.844i	1.11369	−	0.642989i	0.939783	+	0.341773i	$0.111027\pi$
$444$		0			0
$445$	388.679	−	673.211i	0.873436	−	1.51283i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			559.115i			1.24524i	0.782523	+	0.622622i	$0.213933\pi$
$449$			559.115i			1.24524i	−0.782523	+	0.622622i	$0.786067\pi$
$450$		0			0
$451$	−494.354			−1.09613
$452$		0			0
$453$		0			0
$454$		0			0
$455$	892.820	+	515.470i	1.96224	+	1.13290i
$456$		0			0
$457$	−302.148	−	523.336i	−0.661155	−	1.14515i	−0.980312	−	0.197453i	$-0.936733\pi$
$457$	−302.148	−	523.336i	−0.661155	−	1.14515i	0.319157	−	0.947702i	$-0.396600\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.44669	−	2.56730i	0.00964575	−	0.00556897i	−0.495169	−	0.868796i	$-0.664894\pi$
$461$	4.44669	−	2.56730i	0.00964575	−	0.00556897i	0.504815	+	0.863227i	$0.331561\pi$
$462$		0			0
$463$	−80.7461	+	139.856i	−0.174398	+	0.302066i	−0.939953	−	0.341305i	$-0.889131\pi$
$463$	−80.7461	+	139.856i	−0.174398	+	0.302066i	0.765555	+	0.643370i	$0.222464\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	503.025i		−	1.07714i	−0.842581	−	0.538570i	$-0.818965\pi$
$467$		−	503.025i		−	1.07714i	0.842581	−	0.538570i	$-0.181035\pi$
$468$		0			0
$469$	−177.723			−0.378941
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−116.803	−	67.4363i	−0.246941	−	0.142571i
$474$		0			0
$475$	−104.746	−	181.426i	−0.220518	−	0.381949i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	185.153	−	106.898i	0.386541	−	0.223170i	−0.294119	−	0.955769i	$-0.595026\pi$
$479$	185.153	−	106.898i	0.386541	−	0.223170i	0.680660	+	0.732599i	$0.261693\pi$
$480$		0			0
$481$	−641.917	+	1111.83i	−1.33455	+	2.31150i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	435.469i		−	0.897874i
$486$		0			0
$487$	−8.63071			−0.0177222			−0.00886110	−	0.999961i	$-0.502821\pi$
$487$	−8.63071			−0.0177222			−0.00886110	+	0.999961i	$0.502821\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−798.873	−	461.229i	−1.62703	−	0.939367i	−0.984972	−	0.172713i	$-0.944747\pi$
$491$	−798.873	−	461.229i	−1.62703	−	0.939367i	−0.642060	−	0.766654i	$-0.721920\pi$
$492$		0			0
$493$	137.798	+	238.673i	0.279509	+	0.484124i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	735.619	−	424.710i	1.48012	−	0.854547i
$498$		0			0
$499$	−217.296	+	376.368i	−0.435463	+	0.754244i	−0.997333	−	0.0729811i	$-0.976749\pi$
$499$	−217.296	+	376.368i	−0.435463	+	0.754244i	0.561870	+	0.827225i	$0.310082\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			144.087i			0.286454i	0.989690	+	0.143227i	$0.0457480\pi$
$503$			144.087i			0.286454i	−0.989690	+	0.143227i	$0.954252\pi$
$504$		0			0
$505$	168.588			0.333839
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−603.856	−	348.636i	−1.18636	−	0.684943i	−0.228880	−	0.973455i	$-0.573506\pi$
$509$	−603.856	−	348.636i	−1.18636	−	0.684943i	−0.957477	+	0.288511i	$0.906840\pi$
$510$		0			0
$511$	−169.573	−	293.709i	−0.331845	−	0.574773i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−485.385	+	280.237i	−0.942496	+	0.544150i
$516$		0			0
$517$	−124.708	+	216.000i	−0.241214	+	0.417795i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			426.962i			0.819504i	0.912197	+	0.409752i	$0.134385\pi$
$521$			426.962i			0.819504i	−0.912197	+	0.409752i	$0.865615\pi$
$522$		0			0
$523$	−179.762			−0.343712			−0.171856	−	0.985122i	$-0.554976\pi$
$523$	−179.762			−0.343712			−0.171856	+	0.985122i	$0.554976\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	53.7945	+	31.0583i	0.102077	+	0.0589341i
$528$		0			0
$529$	−156.500	−	271.066i	−0.295841	−	0.512412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	617.446	−	356.482i	1.15843	−	0.668822i
$534$		0			0
$535$	−514.592	+	891.300i	−0.961855	+	1.66598i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	314.967i		−	0.584354i
$540$		0			0
$541$	708.734			1.31005			0.655023	−	0.755609i	$-0.272659\pi$
$541$	708.734			1.31005			0.655023	+	0.755609i	$0.272659\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−310.894	−	179.495i	−0.570448	−	0.329349i
$546$		0			0
$547$	−98.1154	−	169.941i	−0.179370	−	0.310678i	0.762295	−	0.647230i	$-0.224073\pi$
$547$	−98.1154	−	169.941i	−0.179370	−	0.310678i	−0.941665	+	0.336552i	$0.890739\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−749.789	+	432.891i	−1.36078	+	0.785646i
$552$		0			0
$553$	414.354	−	717.682i	0.749284	−	1.29780i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	353.610i		−	0.634848i	−0.948284	−	0.317424i	$-0.897182\pi$
$557$		−	353.610i		−	0.634848i	0.948284	−	0.317424i	$-0.102818\pi$
$558$		0			0
$559$	194.515			0.347970
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−323.050	−	186.513i	−0.573801	−	0.331284i	0.184865	−	0.982764i	$-0.440815\pi$
$563$	−323.050	−	186.513i	−0.573801	−	0.331284i	−0.758666	+	0.651480i	$0.774149\pi$
$564$		0			0
$565$	318.208	+	551.152i	0.563199	+	0.975490i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−52.0634	+	30.0588i	−0.0914999	+	0.0528275i	−0.545052	−	0.838402i	$-0.683490\pi$
$569$	−52.0634	+	30.0588i	−0.0914999	+	0.0528275i	0.453552	+	0.891230i	$0.350157\pi$
$570$		0			0
$571$	43.1000	−	74.6513i	0.0754815	−	0.130738i	−0.825814	−	0.563943i	$-0.809284\pi$
$571$	43.1000	−	74.6513i	0.0754815	−	0.130738i	0.901296	+	0.433205i	$0.142617\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	126.224i		−	0.219520i
$576$		0			0
$577$	709.123			1.22898			0.614491	−	0.788924i	$-0.289361\pi$
$577$	709.123			1.22898			0.614491	+	0.788924i	$0.289361\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	752.144	+	434.251i	1.29457	+	0.747419i
$582$		0			0
$583$	−189.300	−	327.877i	−0.324700	−	0.562396i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−833.363	+	481.143i	−1.41970	+	0.819664i	−0.996272	−	0.0862666i	$-0.972506\pi$
$587$	−833.363	+	481.143i	−1.41970	+	0.819664i	−0.423427	+	0.905930i	$0.639173\pi$
$588$		0			0
$589$	−97.5692	+	168.995i	−0.165652	+	0.286918i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			104.350i			0.175969i	0.996122	+	0.0879847i	$0.0280427\pi$
$593$			104.350i			0.175969i	−0.996122	+	0.0879847i	$0.971957\pi$
$594$		0			0
$595$	−377.654			−0.634712
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−247.738	−	143.031i	−0.413585	−	0.238784i	0.278744	−	0.960366i	$-0.410082\pi$
$599$	−247.738	−	143.031i	−0.413585	−	0.238784i	−0.692329	+	0.721582i	$0.743415\pi$
$600$		0			0
$601$	140.208	+	242.847i	0.233291	+	0.404071i	0.958775	−	0.284168i	$-0.0917173\pi$
$601$	140.208	+	242.847i	0.233291	+	0.404071i	−0.725484	+	0.688239i	$0.758384\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−476.814	+	275.289i	−0.788123	+	0.455023i
$606$		0			0
$607$	368.865	−	638.894i	0.607686	−	1.05254i	−0.383935	−	0.923360i	$-0.625431\pi$
$607$	368.865	−	638.894i	0.607686	−	1.05254i	0.991621	−	0.129183i	$-0.0412354\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	359.711i		−	0.588724i
$612$		0			0
$613$	−679.415			−1.10834			−0.554172	−	0.832402i	$-0.686965\pi$
$613$	−679.415			−1.10834			−0.554172	+	0.832402i	$0.686965\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−409.326	−	236.325i	−0.663414	−	0.383022i	0.130163	−	0.991493i	$-0.458450\pi$
$617$	−409.326	−	236.325i	−0.663414	−	0.383022i	−0.793576	+	0.608471i	$0.791783\pi$
$618$		0			0
$619$	443.177	+	767.605i	0.715956	+	1.24007i	0.962590	+	0.270964i	$0.0873423\pi$
$619$	443.177	+	767.605i	0.715956	+	1.24007i	−0.246633	+	0.969109i	$0.579324\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	974.850	−	562.830i	1.56477	−	0.903419i
$624$		0			0
$625$	382.975	−	663.332i	0.612760	−	1.06133i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	470.294i		−	0.747685i
$630$		0			0
$631$	729.108			1.15548			0.577740	−	0.816221i	$-0.303935\pi$
$631$	729.108			1.15548			0.577740	+	0.816221i	$0.303935\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	708.581	+	409.099i	1.11588	+	0.644251i
$636$		0			0
$637$	227.125	+	393.392i	0.356554	+	0.617570i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−753.611	+	435.098i	−1.17568	+	0.678780i	−0.955011	−	0.296569i	$-0.904157\pi$
$641$	−753.611	+	435.098i	−1.17568	+	0.678780i	−0.220669	+	0.975349i	$0.570824\pi$
$642$		0			0
$643$	−309.061	+	535.310i	−0.480656	+	0.832520i	−0.999754	−	0.0221949i	$-0.992935\pi$
$643$	−309.061	+	535.310i	−0.480656	+	0.832520i	0.519098	+	0.854715i	$0.326268\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			425.439i			0.657556i	0.944407	+	0.328778i	$0.106637\pi$
$647$			425.439i			0.657556i	−0.944407	+	0.328778i	$0.893363\pi$
$648$		0			0
$649$	−906.369			−1.39656
$650$		0			0
$651$		0			0
$652$		0			0
$653$	163.513	+	94.4042i	0.250403	+	0.144570i	0.619949	−	0.784642i	$-0.287153\pi$
$653$	163.513	+	94.4042i	0.250403	+	0.144570i	−0.369546	+	0.929212i	$0.620487\pi$
$654$		0			0
$655$	154.119	+	266.942i	0.235296	+	0.407545i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−571.984	+	330.235i	−0.867958	+	0.501116i	−0.866669	−	0.498884i	$-0.833744\pi$
$659$	−571.984	+	330.235i	−0.867958	+	0.501116i	−0.00128859	+	0.999999i	$0.500410\pi$
$660$		0			0
$661$	294.915	−	510.808i	0.446165	−	0.772781i	−0.551967	−	0.833866i	$-0.686123\pi$
$661$	294.915	−	510.808i	0.446165	−	0.772781i	0.998133	+	0.0610847i	$0.0194560\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1186.39i		−	1.78405i
$666$		0			0
$667$	−521.654			−0.782090
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−165.463	−	95.5301i	−0.246592	−	0.142370i
$672$		0			0
$673$	195.715	+	338.989i	0.290810	+	0.503698i	0.974002	−	0.226541i	$-0.0727418\pi$
$673$	195.715	+	338.989i	0.290810	+	0.503698i	−0.683191	+	0.730240i	$0.739408\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	600.728	−	346.830i	0.887338	−	0.512305i	0.0142672	−	0.999898i	$-0.495458\pi$
$677$	600.728	−	346.830i	0.887338	−	0.512305i	0.873071	+	0.487593i	$0.162125\pi$
$678$		0			0
$679$	315.292	−	546.102i	0.464348	−	0.804274i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			678.170i			0.992928i	0.868057	+	0.496464i	$0.165368\pi$
$683$			678.170i			0.992928i	−0.868057	+	0.496464i	$0.834632\pi$
$684$		0			0
$685$	−9.47303			−0.0138292
$686$		0			0
$687$		0			0
$688$		0			0
$689$	472.869	+	273.011i	0.686313	+	0.396243i
$690$		0			0
$691$	337.888	+	585.240i	0.488985	+	0.846946i	0.999920	−	0.0126731i	$-0.00403409\pi$
$691$	337.888	+	585.240i	0.488985	+	0.846946i	−0.510935	+	0.859619i	$0.670701\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	96.4439	−	55.6819i	0.138768	−	0.0801179i
$696$		0			0
$697$	−130.587	+	226.183i	−0.187355	+	0.324509i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			797.260i			1.13732i	0.822573	+	0.568659i	$0.192538\pi$
$701$			797.260i			1.13732i	−0.822573	+	0.568659i	$0.807462\pi$
$702$		0			0
$703$	1477.42			2.10160
$704$		0			0
$705$		0			0
$706$		0			0
$707$	211.419	+	122.063i	0.299037	+	0.172649i
$708$		0			0
$709$	−86.7520	−	150.259i	−0.122358	−	0.211931i	0.798339	−	0.602208i	$-0.205712\pi$
$709$	−86.7520	−	150.259i	−0.122358	−	0.211931i	−0.920697	+	0.390278i	$0.872379\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−101.823	+	58.7878i	−0.142810	+	0.0824513i
$714$		0			0
$715$	902.711	−	1563.54i	1.26253	−	2.18677i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	188.177i		−	0.261721i	−0.991401	−	0.130860i	$-0.958226\pi$
$719$		−	188.177i		−	0.261721i	0.991401	−	0.130860i	$-0.0417740\pi$
$720$		0			0
$721$	−811.600			−1.12566
$722$		0			0
$723$		0			0
$724$		0			0
$725$	263.998	+	152.420i	0.364136	+	0.210234i
$726$		0			0
$727$	355.888	+	616.417i	0.489530	+	0.847891i	0.999927	−	0.0120478i	$-0.00383501\pi$
$727$	355.888	+	616.417i	0.489530	+	0.847891i	−0.510397	+	0.859939i	$0.670502\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−61.7085	+	35.6274i	−0.0844165	+	0.0487379i
$732$		0			0
$733$	−613.415	+	1062.47i	−0.836856	+	1.44948i	0.0556546	+	0.998450i	$0.482275\pi$
$733$	−613.415	+	1062.47i	−0.836856	+	1.44948i	−0.892510	+	0.451027i	$0.851058\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			311.236i			0.422301i
$738$		0			0
$739$	−741.892			−1.00391			−0.501957	−	0.864893i	$-0.667386\pi$
$739$	−741.892			−1.00391			−0.501957	+	0.864893i	$0.667386\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	677.218	+	390.992i	0.911464	+	0.526234i	0.880902	−	0.473299i	$-0.156937\pi$
$743$	677.218	+	390.992i	0.911464	+	0.526234i	0.0305622	+	0.999533i	$0.490270\pi$
$744$		0			0
$745$	273.208	+	473.210i	0.366722	+	0.635181i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1290.65	+	745.160i	−1.72317	+	0.994873i
$750$		0			0
$751$	−516.665	+	894.891i	−0.687970	+	1.19160i	0.284524	+	0.958669i	$0.408165\pi$
$751$	−516.665	+	894.891i	−0.687970	+	1.19160i	−0.972494	+	0.232930i	$0.925169\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	185.458i		−	0.245639i
$756$		0			0
$757$	473.877			0.625993			0.312997	−	0.949754i	$-0.398667\pi$
$757$	473.877			0.625993			0.312997	+	0.949754i	$0.398667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	814.458	+	470.227i	1.07025	+	0.617907i	0.928249	−	0.371960i	$-0.121314\pi$
$761$	814.458	+	470.227i	1.07025	+	0.617907i	0.141998	+	0.989867i	$0.454647\pi$
$762$		0			0
$763$	−259.919	−	450.193i	−0.340654	−	0.590031i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1132.05	−	653.590i	1.47595	−	0.852138i
$768$		0			0
$769$	−490.408	+	849.411i	−0.637721	+	1.10457i	0.348210	+	0.937416i	$0.386789\pi$
$769$	−490.408	+	849.411i	−0.637721	+	1.10457i	−0.985932	+	0.167149i	$0.946544\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1042.20i			1.34825i	0.738618	+	0.674124i	$0.235479\pi$
$773$			1042.20i			1.34825i	−0.738618	+	0.674124i	$0.764521\pi$
$774$		0			0
$775$	68.7077			0.0886550
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−710.550	−	410.236i	−0.912131	−	0.526619i
$780$		0			0
$781$	−743.769	−	1288.25i	−0.952329	−	1.64948i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.46729	−	0.847142i	0.00186916	−	0.00107916i
$786$		0			0
$787$	−72.3501	+	125.314i	−0.0919315	+	0.159230i	−0.908324	−	0.418268i	$-0.862637\pi$
$787$	−72.3501	+	125.314i	−0.0919315	+	0.159230i	0.816392	+	0.577498i	$0.195971\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			921.567i			1.16507i
$792$		0			0
$793$	275.550			0.347478
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−134.866	−	77.8647i	−0.169217	−	0.0976972i	0.413000	−	0.910731i	$-0.364481\pi$
$797$	−134.866	−	77.8647i	−0.169217	−	0.0976972i	−0.582216	+	0.813034i	$0.697814\pi$
$798$		0			0
$799$	65.8846	+	114.115i	0.0824588	+	0.142823i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−514.355	+	296.963i	−0.640542	+	0.369817i
$804$		0			0
$805$	357.415	−	619.061i	0.443994	−	0.769020i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1216.92i			1.50422i	0.659035	+	0.752112i	$0.270965\pi$
$809$			1216.92i			1.50422i	−0.659035	+	0.752112i	$0.729035\pi$
$810$		0			0
$811$	−1243.31			−1.53305			−0.766527	−	0.642212i	$-0.778017\pi$
$811$	−1243.31			−1.53305			−0.766527	+	0.642212i	$0.778017\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	144.473	+	83.4114i	0.177267	+	0.102345i
$816$		0			0
$817$	−111.923	−	193.856i	−0.136993	−	0.237278i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.5055	+	6.64270i	−0.0140140	+	0.00809098i	−0.506991	−	0.861952i	$-0.669242\pi$
$821$	−11.5055	+	6.64270i	−0.0140140	+	0.00809098i	0.492977	+	0.870043i	$0.335909\pi$
$822$		0			0
$823$	−107.100	+	185.503i	−0.130134	+	0.225398i	−0.923728	−	0.383049i	$-0.874874\pi$
$823$	−107.100	+	185.503i	−0.130134	+	0.225398i	0.793594	+	0.608447i	$0.208207\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1221.27i			1.47675i	0.674391	+	0.738374i	$0.264406\pi$
$827$			1221.27i			1.47675i	−0.674391	+	0.738374i	$0.735594\pi$
$828$		0			0
$829$	−358.431			−0.432365			−0.216183	−	0.976353i	$-0.569361\pi$
$829$	−358.431			−0.432365			−0.216183	+	0.976353i	$0.569361\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−144.107	−	83.2004i	−0.172998	−	0.0998804i
$834$		0			0
$835$	−162.473	−	281.412i	−0.194578	−	0.337020i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−283.707	+	163.799i	−0.338150	+	0.195231i	−0.659453	−	0.751745i	$-0.729212\pi$
$839$	−283.707	+	163.799i	−0.338150	+	0.195231i	0.321304	+	0.946976i	$0.395879\pi$
$840$		0			0
$841$	209.413	−	362.715i	0.249005	−	0.431290i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1624.36i			1.92232i
$846$		0			0
$847$	−797.269			−0.941286
$848$		0			0
$849$		0			0
$850$		0			0
$851$	770.920	+	445.091i	0.905899	+	0.523021i
$852$		0			0
$853$	26.4153	+	45.7527i	0.0309675	+	0.0536374i	0.881094	−	0.472942i	$-0.156808\pi$
$853$	26.4153	+	45.7527i	0.0309675	+	0.0536374i	−0.850126	+	0.526579i	$0.823474\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1081.16	−	624.211i	1.26157	−	0.728367i	0.288191	−	0.957573i	$-0.406946\pi$
$857$	1081.16	−	624.211i	1.26157	−	0.728367i	0.973378	+	0.229206i	$0.0736128\pi$
$858$		0			0
$859$	−223.369	+	386.887i	−0.260034	+	0.450392i	−0.966251	−	0.257604i	$-0.917067\pi$
$859$	−223.369	+	386.887i	−0.260034	+	0.450392i	0.706217	+	0.707996i	$0.250400\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			635.297i			0.736150i	0.929796	+	0.368075i	$0.119983\pi$
$863$			635.297i			0.736150i	−0.929796	+	0.368075i	$0.880017\pi$
$864$		0			0
$865$	1280.54			1.48039
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1256.83	−	725.633i	−1.44630	−	0.835021i
$870$		0			0
$871$	−224.435	−	388.732i	−0.257675	−	0.446305i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	691.284	−	399.113i	0.790039	−	0.456129i
$876$		0			0
$877$	394.346	−	683.027i	0.449653	−	0.778823i	−0.548710	−	0.836013i	$-0.684881\pi$
$877$	394.346	−	683.027i	0.449653	−	0.778823i	0.998363	+	0.0571902i	$0.0182141\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			558.506i			0.633945i	0.948435	+	0.316972i	$0.102666\pi$
$881$			558.506i			0.633945i	−0.948435	+	0.316972i	$0.897334\pi$
$882$		0			0
$883$	−313.338			−0.354857			−0.177428	−	0.984134i	$-0.556778\pi$
$883$	−313.338			−0.354857			−0.177428	+	0.984134i	$0.556778\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	517.624	+	298.850i	0.583567	+	0.336923i	0.762550	−	0.646930i	$-0.223947\pi$
$887$	517.624	+	298.850i	0.583567	+	0.336923i	−0.178983	+	0.983852i	$0.557281\pi$
$888$		0			0
$889$	592.400	+	1026.07i	0.666367	+	1.15418i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−358.492	+	206.976i	−0.401447	+	0.231776i
$894$		0			0
$895$	719.654	−	1246.48i	0.804082	−	1.39271i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	283.952i		−	0.315854i
$900$		0			0
$901$	−200.019			−0.221997
$902$		0			0
$903$		0			0
$904$		0			0
$905$	794.407	+	458.651i	0.877798	+	0.506797i
$906$		0			0
$907$	543.254	+	940.943i	0.598957	+	1.03742i	0.992975	+	0.118321i	$0.0377512\pi$
$907$	543.254	+	940.943i	0.598957	+	1.03742i	−0.394019	+	0.919103i	$0.628915\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−615.123	+	355.142i	−0.675218	+	0.389837i	−0.798051	−	0.602590i	$-0.794135\pi$
$911$	−615.123	+	355.142i	−0.675218	+	0.389837i	0.122833	+	0.992427i	$0.460802\pi$
$912$		0			0
$913$	760.477	−	1317.18i	0.832943	−	1.44270i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			446.347i			0.486747i
$918$		0			0
$919$	−1540.01			−1.67574			−0.837871	−	0.545868i	$-0.816200\pi$
$919$	−1540.01			−1.67574			−0.837871	+	0.545868i	$0.816200\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1857.93	+	1072.68i	2.01292	+	1.16216i
$924$		0			0
$925$	−260.098	−	450.503i	−0.281187	−	0.487030i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1370.57	+	791.297i	−1.47531	+	0.851773i	−0.999613	−	0.0278334i	$-0.991139\pi$
$929$	−1370.57	+	791.297i	−1.47531	+	0.851773i	−0.475702	+	0.879607i	$0.657806\pi$
$930$		0			0
$931$	261.373	−	452.711i	0.280744	−	0.486264i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			661.362i			0.707339i
$936$		0			0
$937$	−1747.40			−1.86489			−0.932444	−	0.361315i	$-0.882328\pi$
$937$	−1747.40			−1.86489			−0.932444	+	0.361315i	$0.882328\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	318.332	+	183.789i	0.338292	+	0.195313i	0.659516	−	0.751690i	$-0.270761\pi$
$941$	318.332	+	183.789i	0.338292	+	0.195313i	−0.321225	+	0.947003i	$0.604094\pi$
$942$		0			0
$943$	−247.177	−	428.123i	−0.262118	−	0.454001i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	164.691	−	95.0841i	0.173908	−	0.100406i	−0.410519	−	0.911852i	$-0.634653\pi$
$947$	164.691	−	95.0841i	0.173908	−	0.100406i	0.584427	+	0.811446i	$0.301319\pi$
$948$		0			0
$949$	428.285	−	741.811i	0.451301	−	0.781676i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	861.754i		−	0.904254i	−0.891954	−	0.452127i	$-0.850665\pi$
$953$		−	861.754i		−	0.904254i	0.891954	−	0.452127i	$-0.149335\pi$
$954$		0			0
$955$	200.238			0.209674
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.8797	−	6.85875i	−0.0123876	−	0.00715198i
$960$		0			0
$961$	448.500	+	776.825i	0.466701	+	0.808350i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−276.050	+	159.378i	−0.286063	+	0.165158i
$966$		0			0
$967$	637.657	−	1104.46i	0.659418	−	1.14215i	−0.321348	−	0.946961i	$-0.604136\pi$
$967$	637.657	−	1104.46i	0.659418	−	1.14215i	0.980766	−	0.195185i	$-0.0625307\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1238.69i		−	1.27568i	−0.770168	−	0.637841i	$-0.779828\pi$
$971$		−	1238.69i		−	1.27568i	0.770168	−	0.637841i	$-0.220172\pi$
$972$		0			0
$973$	161.261			0.165736
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−16.0159	−	9.24680i	−0.0163930	−	0.00946448i	0.491781	−	0.870719i	$-0.336346\pi$
$977$	−16.0159	−	9.24680i	−0.0163930	−	0.00946448i	−0.508174	+	0.861254i	$0.669679\pi$
$978$		0			0
$979$	−985.650	−	1707.20i	−1.00679	−	1.74382i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−570.401	+	329.321i	−0.580266	+	0.335017i	−0.761239	−	0.648471i	$-0.775409\pi$
$983$	−570.401	+	329.321i	−0.580266	+	0.335017i	0.180973	+	0.983488i	$0.442075\pi$
$984$		0			0
$985$	103.500	−	179.267i	0.105076	−	0.181997i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	134.873i		−	0.136373i
$990$		0			0
$991$	−608.484			−0.614010			−0.307005	−	0.951708i	$-0.599327\pi$
$991$	−608.484			−0.614010			−0.307005	+	0.951708i	$0.599327\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1882.86	−	1087.07i	−1.89232	−	1.09253i
$996$		0			0
$997$	308.623	+	534.551i	0.309552	+	0.536159i	0.978264	−	0.207362i	$-0.0664877\pi$
$997$	308.623	+	534.551i	0.309552	+	0.536159i	−0.668713	+	0.743521i	$0.733154\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.q.o.1025.1		8
3.2	odd	2	inner	1296.3.q.o.1025.4		8
4.3	odd	2		162.3.d.c.53.1		8
9.2	odd	6	inner	1296.3.q.o.593.1		8
9.4	even	3		1296.3.e.d.161.4		4
9.5	odd	6		1296.3.e.d.161.1		4
9.7	even	3	inner	1296.3.q.o.593.4		8
12.11	even	2		162.3.d.c.53.4		8
36.7	odd	6		162.3.d.c.107.4		8
36.11	even	6		162.3.d.c.107.1		8
36.23	even	6		162.3.b.b.161.3	yes	4
36.31	odd	6		162.3.b.b.161.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.b.b.161.2	✓	4	36.31	odd	6
162.3.b.b.161.3	yes	4	36.23	even	6
162.3.d.c.53.1		8	4.3	odd	2
162.3.d.c.53.4		8	12.11	even	2
162.3.d.c.107.1		8	36.11	even	6
162.3.d.c.107.4		8	36.7	odd	6
1296.3.e.d.161.1		4	9.5	odd	6
1296.3.e.d.161.4		4	9.4	even	3
1296.3.q.o.593.1		8	9.2	odd	6	inner
1296.3.q.o.593.4		8	9.7	even	3	inner
1296.3.q.o.1025.1		8	1.1	even	1	trivial
1296.3.q.o.1025.4		8	3.2	odd	2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.01910 - 2.89778i) q^{5} +(-4.19615 - 7.26795i) q^{7} +(-12.7279 + 7.34847i) q^{11} +(10.5981 - 18.3564i) q^{13} +7.76457i q^{17} -24.3923 q^{19} +(-12.7279 - 7.34847i) q^{23} +(4.29423 + 7.43782i) q^{25} +(30.7387 - 17.7470i) q^{29} +(4.00000 - 6.92820i) q^{31} +48.6381i q^{35} -60.5692 q^{37} +(29.1301 + 16.8183i) q^{41} +(4.58846 + 7.94744i) q^{43} +(14.6969 - 8.48528i) q^{47} +(-10.7154 + 18.5596i) q^{49} +25.7605i q^{53} +85.1769 q^{55} +(53.4083 + 30.8353i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-106.386 + 61.4217i) q^{65} +(10.5885 - 18.3397i) q^{67} +101.214i q^{71} +40.4115 q^{73} +(106.817 + 61.6706i) q^{77} +(49.3731 + 85.5167i) q^{79} +(-89.6231 + 51.7439i) q^{83} +(22.5000 - 38.9711i) q^{85} +134.130i q^{89} -177.885 q^{91} +(122.427 + 70.6835i) q^{95} +(37.5692 + 65.0718i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} + 64 q^{13} - 112 q^{19} - 28 q^{25} + 32 q^{31} - 152 q^{37} - 88 q^{43} - 252 q^{49} + 432 q^{55} + 52 q^{61} - 40 q^{67} + 448 q^{73} + 104 q^{79} + 180 q^{85} - 176 q^{91} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 64 * q^13 - 112 * q^19 - 28 * q^25 + 32 * q^31 - 152 * q^37 - 88 * q^43 - 252 * q^49 + 432 * q^55 + 52 * q^61 - 40 * q^67 + 448 * q^73 + 104 * q^79 + 180 * q^85 - 176 * q^91 - 32 * q^97

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