LMFDB - Embedded newform 2268.2.w.j.269.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(269,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.269"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.w (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,4,0,0,0,0,0,12] 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:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.6
Character	$\chi$	$=$	2268.269
Dual form			2268.2.w.j.1349.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.566658 + 0.981481i) q^{5} +(-0.814380 + 2.51730i) q^{7} +(-2.30092 + 1.32843i) q^{11} +(-1.59436 + 0.920505i) q^{13} +(-0.267475 + 0.463281i) q^{17} +(2.57178 - 1.48482i) q^{19} +(0.839982 + 0.484964i) q^{23} +(1.85780 + 3.21780i) q^{25} +(-2.66955 - 1.54127i) q^{29} -0.788326i q^{31} +(-2.00920 - 2.22575i) q^{35} +(-1.73316 - 3.00191i) q^{37} +(-2.22822 - 3.85939i) q^{41} +(-0.849465 + 1.47132i) q^{43} -10.2857 q^{47} +(-5.67357 - 4.10007i) q^{49} +(-11.9002 - 6.87059i) q^{53} -3.01107i q^{55} +7.29675 q^{59} +2.01142i q^{61} -2.08645i q^{65} -2.41291 q^{67} -12.3890i q^{71} +(7.05561 + 4.07356i) q^{73} +(-1.47025 - 6.87394i) q^{77} +5.17268 q^{79} +(-7.56910 + 13.1101i) q^{83} +(-0.303134 - 0.525044i) q^{85} +(-6.52948 - 11.3094i) q^{89} +(-1.01877 - 4.76312i) q^{91} +3.36553i q^{95} +(-1.44234 - 0.832736i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{7} + 12 q^{13} - 16 q^{25} - 4 q^{37} - 4 q^{43} + 20 q^{49} - 8 q^{67} - 36 q^{73} - 56 q^{79} + 12 q^{85} - 36 q^{91} + 12 q^{97}+O(q^{100}) $ 32 * q + 4 * q^7 + 12 * q^13 - 16 * q^25 - 4 * q^37 - 4 * q^43 + 20 * q^49 - 8 * q^67 - 36 * q^73 - 56 * q^79 + 12 * q^85 - 36 * q^91 + 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−0.566658	+	0.981481i	−0.253417	+	0.438932i	−0.964464	−	0.264213i	$-0.914888\pi$
$5$	−0.566658	+	0.981481i	−0.253417	+	0.438932i	0.711047	+	0.703144i	$0.248221\pi$
$6$		0			0
$7$	−0.814380	+	2.51730i	−0.307807	+	0.951449i
$7$	−0.814380	+	2.51730i	−0.307807	+	0.951449i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.30092	+	1.32843i	−0.693753	+	0.400538i	−0.805016	−	0.593253i	$-0.797843\pi$
$11$	−2.30092	+	1.32843i	−0.693753	+	0.400538i	0.111264	+	0.993791i	$0.464510\pi$
$12$		0			0
$13$	−1.59436	+	0.920505i	−0.442196	+	0.255302i	−0.704529	−	0.709675i	$-0.748842\pi$
$13$	−1.59436	+	0.920505i	−0.442196	+	0.255302i	0.262333	+	0.964978i	$0.415508\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.267475	+	0.463281i	−0.0648723	+	0.112362i	−0.896637	−	0.442766i	$-0.853997\pi$
$17$	−0.267475	+	0.463281i	−0.0648723	+	0.112362i	0.831765	+	0.555128i	$0.187331\pi$
$18$		0			0
$19$	2.57178	−	1.48482i	0.590006	−	0.340640i	−0.175094	−	0.984552i	$-0.556023\pi$
$19$	2.57178	−	1.48482i	0.590006	−	0.340640i	0.765100	+	0.643912i	$0.222690\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.839982	+	0.484964i	0.175148	+	0.101122i	0.585011	−	0.811025i	$-0.301090\pi$
$23$	0.839982	+	0.484964i	0.175148	+	0.101122i	−0.409863	+	0.912147i	$0.634423\pi$
$24$		0			0
$25$	1.85780	+	3.21780i	0.371559	+	0.643560i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.66955	−	1.54127i	−0.495724	−	0.286206i	0.231222	−	0.972901i	$-0.425728\pi$
$29$	−2.66955	−	1.54127i	−0.495724	−	0.286206i	−0.726946	+	0.686695i	$0.759061\pi$
$30$		0			0
$31$		−	0.788326i		−	0.141587i	−0.997491	−	0.0707937i	$-0.977447\pi$
$31$		−	0.788326i		−	0.141587i	0.997491	−	0.0707937i	$-0.0225532\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00920	−	2.22575i	−0.339618	−	0.376220i
$36$		0			0
$37$	−1.73316	−	3.00191i	−0.284929	−	0.493511i	0.687663	−	0.726030i	$-0.258637\pi$
$37$	−1.73316	−	3.00191i	−0.284929	−	0.493511i	−0.972592	+	0.232519i	$0.925303\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.22822	−	3.85939i	−0.347990	−	0.602736i	0.637903	−	0.770117i	$-0.279802\pi$
$41$	−2.22822	−	3.85939i	−0.347990	−	0.602736i	−0.985892	+	0.167381i	$0.946469\pi$
$42$		0			0
$43$	−0.849465	+	1.47132i	−0.129542	+	0.224374i	−0.923499	−	0.383600i	$-0.874684\pi$
$43$	−0.849465	+	1.47132i	−0.129542	+	0.224374i	0.793957	+	0.607974i	$0.208017\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.2857			−1.50033			−0.750163	−	0.661253i	$-0.770025\pi$
$47$	−10.2857			−1.50033			−0.750163	+	0.661253i	$0.770025\pi$
$48$		0			0
$49$	−5.67357	−	4.10007i	−0.810510	−	0.585724i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−11.9002	−	6.87059i	−1.63462	−	0.943749i	−0.982642	−	0.185511i	$-0.940606\pi$
$53$	−11.9002	−	6.87059i	−1.63462	−	0.943749i	−0.651979	−	0.758237i	$-0.726061\pi$
$54$		0			0
$55$		−	3.01107i		−	0.406013i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.29675			0.949956			0.474978	−	0.879998i	$-0.342456\pi$
$59$	7.29675			0.949956			0.474978	+	0.879998i	$0.342456\pi$
$60$		0			0
$61$			2.01142i			0.257536i	0.991675	+	0.128768i	$0.0411022\pi$
$61$			2.01142i			0.257536i	−0.991675	+	0.128768i	$0.958898\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.08645i		−	0.258792i
$66$		0			0
$67$	−2.41291			−0.294783			−0.147392	−	0.989078i	$-0.547088\pi$
$67$	−2.41291			−0.294783			−0.147392	+	0.989078i	$0.547088\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.3890i		−	1.47030i	−0.677905	−	0.735150i	$-0.737112\pi$
$71$		−	12.3890i		−	1.47030i	0.677905	−	0.735150i	$-0.262888\pi$
$72$		0			0
$73$	7.05561	+	4.07356i	0.825796	+	0.476774i	0.852411	−	0.522872i	$-0.175139\pi$
$73$	7.05561	+	4.07356i	0.825796	+	0.476774i	−0.0266149	+	0.999646i	$0.508473\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.47025	−	6.87394i	−0.167550	−	0.783358i
$78$		0			0
$79$	5.17268			0.581972			0.290986	−	0.956727i	$-0.406017\pi$
$79$	5.17268			0.581972			0.290986	+	0.956727i	$0.406017\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.56910	+	13.1101i	−0.830817	+	1.43902i	0.0665746	+	0.997781i	$0.478793\pi$
$83$	−7.56910	+	13.1101i	−0.830817	+	1.43902i	−0.897391	+	0.441235i	$0.854540\pi$
$84$		0			0
$85$	−0.303134	−	0.525044i	−0.0328795	−	0.0569490i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.52948	−	11.3094i	−0.692123	−	1.19879i	−0.971141	−	0.238506i	$-0.923342\pi$
$89$	−6.52948	−	11.3094i	−0.692123	−	1.19879i	0.279018	−	0.960286i	$-0.409991\pi$
$90$		0			0
$91$	−1.01877	−	4.76312i	−0.106796	−	0.499311i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			3.36553i			0.345296i
$96$		0			0
$97$	−1.44234	−	0.832736i	−0.146448	−	0.0845515i	0.424986	−	0.905200i	$-0.360279\pi$
$97$	−1.44234	−	0.832736i	−0.146448	−	0.0845515i	−0.571433	+	0.820649i	$0.693612\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.24453	−	2.15558i	−0.123835	−	0.214488i	0.797442	−	0.603396i	$-0.206186\pi$
$101$	−1.24453	−	2.15558i	−0.123835	−	0.214488i	−0.921277	+	0.388907i	$0.872853\pi$
$102$		0			0
$103$	−5.75735	−	3.32401i	−0.567289	−	0.327524i	0.188777	−	0.982020i	$-0.439548\pi$
$103$	−5.75735	−	3.32401i	−0.567289	−	0.327524i	−0.756066	+	0.654496i	$0.772881\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.217160	+	0.125377i	−0.0209936	+	0.0121207i	−0.510460	−	0.859901i	$-0.670525\pi$
$107$	−0.217160	+	0.125377i	−0.0209936	+	0.0121207i	0.489466	+	0.872022i	$0.337192\pi$
$108$		0			0
$109$	1.27998	−	2.21699i	0.122600	−	0.212349i	−0.798192	−	0.602403i	$-0.794210\pi$
$109$	1.27998	−	2.21699i	0.122600	−	0.212349i	0.920792	+	0.390053i	$0.127543\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.383124	−	0.221197i	0.0360413	−	0.0208085i	−0.481871	−	0.876242i	$-0.660043\pi$
$113$	0.383124	−	0.221197i	0.0360413	−	0.0208085i	0.517912	+	0.855434i	$0.326709\pi$
$114$		0			0
$115$	−0.951966	+	0.549618i	−0.0887713	+	0.0512521i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.948389	−	1.05060i	−0.0869387	−	0.0963084i
$120$		0			0
$121$	−1.97052	+	3.41304i	−0.179138	+	0.310277i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.87753			−0.883473
$126$		0			0
$127$	17.6161			1.56317			0.781586	−	0.623797i	$-0.214411\pi$
$127$	17.6161			1.56317			0.781586	+	0.623797i	$0.214411\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.09926	+	8.83218i	−0.445525	+	0.771671i	−0.998089	−	0.0617991i	$-0.980316\pi$
$131$	−5.09926	+	8.83218i	−0.445525	+	0.771671i	0.552564	+	0.833471i	$0.313650\pi$
$132$		0			0
$133$	1.64332	+	7.68313i	0.142494	+	0.666212i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.28417	+	4.20552i	−0.622328	+	0.359301i	−0.777775	−	0.628543i	$-0.783652\pi$
$137$	−7.28417	+	4.20552i	−0.622328	+	0.359301i	0.155447	+	0.987844i	$0.450318\pi$
$138$		0			0
$139$	−8.83742	+	5.10229i	−0.749580	+	0.432770i	−0.825542	−	0.564340i	$-0.809131\pi$
$139$	−8.83742	+	5.10229i	−0.749580	+	0.432770i	0.0759620	+	0.997111i	$0.475797\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.44566	−	4.23601i	0.204517	−	0.354233i
$144$		0			0
$145$	3.02545	−	1.74674i	0.251250	−	0.145059i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−13.1633	−	7.59986i	−1.07838	−	0.622605i	−0.147923	−	0.988999i	$-0.547259\pi$
$149$	−13.1633	−	7.59986i	−1.07838	−	0.622605i	−0.930460	+	0.366394i	$0.880592\pi$
$150$		0			0
$151$	4.20122	+	7.27673i	0.341891	+	0.592172i	0.984784	−	0.173784i	$-0.0555994\pi$
$151$	4.20122	+	7.27673i	0.341891	+	0.592172i	−0.642893	+	0.765956i	$0.722266\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.773727	+	0.446711i	0.0621472	+	0.0358807i
$156$		0			0
$157$		−	18.6822i		−	1.49100i	−0.666504	−	0.745501i	$-0.732210\pi$
$157$		−	18.6822i		−	1.49100i	0.666504	−	0.745501i	$-0.267790\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.90486	+	1.71954i	−0.150124	+	0.135519i
$162$		0			0
$163$	−10.7857	−	18.6813i	−0.844800	−	1.46324i	−0.885795	−	0.464077i	$-0.846386\pi$
$163$	−10.7857	−	18.6813i	−0.844800	−	1.46324i	0.0409955	−	0.999159i	$-0.486947\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.39810	+	11.0818i	0.495099	+	0.857537i	0.999984	−	0.00564941i	$-0.00179827\pi$
$167$	6.39810	+	11.0818i	0.495099	+	0.857537i	−0.504885	+	0.863187i	$0.668465\pi$
$168$		0			0
$169$	−4.80534	+	8.32310i	−0.369642	+	0.640238i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−21.3999			−1.62700			−0.813501	−	0.581563i	$-0.802441\pi$
$173$	−21.3999			−1.62700			−0.813501	+	0.581563i	$0.802441\pi$
$174$		0			0
$175$	−9.61311	+	2.05612i	−0.726683	+	0.155428i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.36953	+	2.52275i	0.326594	+	0.188559i	0.654328	−	0.756211i	$-0.272952\pi$
$179$	4.36953	+	2.52275i	0.326594	+	0.188559i	−0.327734	+	0.944770i	$0.606285\pi$
$180$		0			0
$181$			11.6959i			0.869351i	0.900587	+	0.434675i	$0.143137\pi$
$181$			11.6959i			0.869351i	−0.900587	+	0.434675i	$0.856863\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.92843			0.288824
$186$		0			0
$187$		−	1.42129i		−	0.103935i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	22.6417i		−	1.63830i	−0.573582	−	0.819148i	$-0.694447\pi$
$191$		−	22.6417i		−	1.63830i	0.573582	−	0.819148i	$-0.305553\pi$
$192$		0			0
$193$	−1.11855			−0.0805152			−0.0402576	−	0.999189i	$-0.512818\pi$
$193$	−1.11855			−0.0805152			−0.0402576	+	0.999189i	$0.512818\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3.19138i			0.227377i	0.993516	+	0.113688i	$0.0362665\pi$
$197$			3.19138i			0.227377i	−0.993516	+	0.113688i	$0.963733\pi$
$198$		0			0
$199$	0.346408	+	0.199999i	0.0245562	+	0.0141775i	0.512228	−	0.858850i	$-0.328820\pi$
$199$	0.346408	+	0.199999i	0.0245562	+	0.0141775i	−0.487672	+	0.873027i	$0.662154\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.05386	−	5.46488i	0.424898	−	0.383560i
$204$		0			0
$205$	5.05056			0.352746
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3.94496	+	6.83288i	−0.272879	+	0.472640i
$210$		0			0
$211$	−2.82379	−	4.89095i	−0.194398	−	0.336707i	0.752305	−	0.658815i	$-0.228942\pi$
$211$	−2.82379	−	4.89095i	−0.194398	−	0.336707i	−0.946703	+	0.322108i	$0.895609\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.962712	−	1.66747i	−0.0656564	−	0.113720i
$216$		0			0
$217$	1.98445	+	0.641996i	0.134713	+	0.0435816i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	0.984849i		−	0.0662481i
$222$		0			0
$223$	5.40211	+	3.11891i	0.361752	+	0.208858i	0.669849	−	0.742497i	$-0.266359\pi$
$223$	5.40211	+	3.11891i	0.361752	+	0.208858i	−0.308097	+	0.951355i	$0.599692\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.16442	+	14.1412i	0.541892	+	0.938584i	0.998795	+	0.0490677i	$0.0156250\pi$
$227$	8.16442	+	14.1412i	0.541892	+	0.938584i	−0.456904	+	0.889516i	$0.651042\pi$
$228$		0			0
$229$	20.2825	+	11.7101i	1.34031	+	0.773827i	0.986852	−	0.161624i	$-0.0516733\pi$
$229$	20.2825	+	11.7101i	1.34031	+	0.773827i	0.353455	+	0.935451i	$0.385007\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.9345	+	6.31304i	−0.716343	+	0.413581i	−0.813405	−	0.581698i	$-0.802389\pi$
$233$	−10.9345	+	6.31304i	−0.716343	+	0.413581i	0.0970623	+	0.995278i	$0.469055\pi$
$234$		0			0
$235$	5.82849	−	10.0952i	0.380209	−	0.658541i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.56517	−	4.94510i	0.554035	−	0.319872i	−0.196713	−	0.980461i	$-0.563027\pi$
$239$	8.56517	−	4.94510i	0.554035	−	0.319872i	0.750748	+	0.660589i	$0.229693\pi$
$240$		0			0
$241$	−17.5109	+	10.1099i	−1.12797	+	0.651236i	−0.943425	−	0.331587i	$-0.892416\pi$
$241$	−17.5109	+	10.1099i	−1.12797	+	0.651236i	−0.184550	+	0.982823i	$0.559083\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.23912	−	3.24516i	0.462490	−	0.207326i
$246$		0			0
$247$	−2.73356	+	4.73467i	−0.173932	+	0.301260i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.4354			0.658679			0.329339	−	0.944212i	$-0.393174\pi$
$251$	10.4354			0.658679			0.329339	+	0.944212i	$0.393174\pi$
$252$		0			0
$253$	−2.57697			−0.162013
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.90839	+	6.76953i	−0.243798	+	0.422271i	−0.961793	−	0.273777i	$-0.911727\pi$
$257$	−3.90839	+	6.76953i	−0.243798	+	0.422271i	0.717995	+	0.696049i	$0.245060\pi$
$258$		0			0
$259$	8.96815	−	1.91817i	0.557254	−	0.119189i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.47396	+	0.850994i	−0.0908885	+	0.0524745i	−0.544755	−	0.838595i	$-0.683377\pi$
$263$	−1.47396	+	0.850994i	−0.0908885	+	0.0524745i	0.453867	+	0.891070i	$0.350044\pi$
$264$		0			0
$265$	13.4867	−	7.78656i	0.828482	−	0.478324i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.59447	+	4.49375i	−0.158188	+	0.273989i	−0.934215	−	0.356710i	$-0.883898\pi$
$269$	−2.59447	+	4.49375i	−0.158188	+	0.273989i	0.776028	+	0.630699i	$0.217232\pi$
$270$		0			0
$271$	−22.7985	+	13.1627i	−1.38491	+	0.799579i	−0.992736	−	0.120311i	$-0.961611\pi$
$271$	−22.7985	+	13.1627i	−1.38491	+	0.799579i	−0.392176	+	0.919890i	$0.628278\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.54927	−	4.93592i	−0.515541	−	0.297647i
$276$		0			0
$277$	−11.7976	−	20.4340i	−0.708849	−	1.22776i	−0.965284	−	0.261201i	$-0.915881\pi$
$277$	−11.7976	−	20.4340i	−0.708849	−	1.22776i	0.256436	−	0.966561i	$-0.417452\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0083	+	9.24241i	0.954976	+	0.551356i	0.894623	−	0.446821i	$-0.147444\pi$
$281$	16.0083	+	9.24241i	0.954976	+	0.551356i	0.0603530	+	0.998177i	$0.480777\pi$
$282$		0			0
$283$			17.7812i			1.05698i	0.848939	+	0.528491i	$0.177242\pi$
$283$			17.7812i			1.05698i	−0.848939	+	0.528491i	$0.822758\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	11.5299	−	2.46609i	0.680586	−	0.145568i
$288$		0			0
$289$	8.35691	+	14.4746i	0.491583	+	0.851447i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.43356	−	4.21505i	−0.142170	−	0.246246i	0.786143	−	0.618044i	$-0.212075\pi$
$293$	−2.43356	−	4.21505i	−0.142170	−	0.246246i	−0.928314	+	0.371798i	$0.878741\pi$
$294$		0			0
$295$	−4.13476	+	7.16162i	−0.240735	+	0.416966i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.78565			−0.103267
$300$		0			0
$301$	−3.01195	−	3.33657i	−0.173606	−	0.192316i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.97417	−	1.13979i	−0.113041	−	0.0652641i
$306$		0			0
$307$			8.37189i			0.477809i	0.971043	+	0.238905i	$0.0767883\pi$
$307$			8.37189i			0.477809i	−0.971043	+	0.238905i	$0.923212\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.2109			0.919238			0.459619	−	0.888116i	$-0.347986\pi$
$311$	16.2109			0.919238			0.459619	+	0.888116i	$0.347986\pi$
$312$		0			0
$313$		−	22.9798i		−	1.29890i	−0.760405	−	0.649449i	$-0.775000\pi$
$313$		−	22.9798i		−	1.29890i	0.760405	−	0.649449i	$-0.225000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	21.7455i		−	1.22135i	−0.791881	−	0.610675i	$-0.790898\pi$
$317$		−	21.7455i		−	1.22135i	0.791881	−	0.610675i	$-0.209102\pi$
$318$		0			0
$319$	8.18990			0.458546
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.58861i			0.0883924i
$324$		0			0
$325$	−5.92400	−	3.42022i	−0.328604	−	0.189720i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.37648	−	25.8922i	0.461810	−	1.42748i
$330$		0			0
$331$	25.7737			1.41665			0.708325	−	0.705887i	$-0.249451\pi$
$331$	25.7737			1.41665			0.708325	+	0.705887i	$0.249451\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.36729	−	2.36822i	0.0747032	−	0.129390i
$336$		0			0
$337$	6.80663	+	11.7894i	0.370781	+	0.642211i	0.989686	−	0.143255i	$-0.0457568\pi$
$337$	6.80663	+	11.7894i	0.370781	+	0.642211i	−0.618905	+	0.785466i	$0.712423\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.04724	+	1.81387i	0.0567112	+	0.0982267i
$342$		0			0
$343$	14.9415	−	10.9431i	0.806767	−	0.590869i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			9.64908i			0.517990i	0.965879	+	0.258995i	$0.0833913\pi$
$347$			9.64908i			0.517990i	−0.965879	+	0.258995i	$0.916609\pi$
$348$		0			0
$349$	−20.1890	−	11.6561i	−1.08069	−	0.623937i	−0.149608	−	0.988745i	$-0.547801\pi$
$349$	−20.1890	−	11.6561i	−1.08069	−	0.623937i	−0.931083	+	0.364808i	$0.881135\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.5010	+	26.8485i	0.825035	+	1.42900i	0.901892	+	0.431961i	$0.142178\pi$
$353$	15.5010	+	26.8485i	0.825035	+	1.42900i	−0.0768568	+	0.997042i	$0.524488\pi$
$354$		0			0
$355$	12.1595	+	7.02031i	0.645361	+	0.372599i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.6898	−	9.05853i	0.828077	−	0.478091i	−0.0251166	−	0.999685i	$-0.507996\pi$
$359$	15.6898	−	9.05853i	0.828077	−	0.478091i	0.853194	+	0.521594i	$0.174662\pi$
$360$		0			0
$361$	−5.09064	+	8.81725i	−0.267929	+	0.464066i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.99624	+	4.61663i	−0.418542	+	0.241645i
$366$		0			0
$367$	−2.73854	+	1.58110i	−0.142951	+	0.0825326i	−0.569769	−	0.821805i	$-0.692968\pi$
$367$	−2.73854	+	1.58110i	−0.142951	+	0.0825326i	0.426819	+	0.904337i	$0.359634\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	26.9866	−	24.3611i	1.40108	−	1.26477i
$372$		0			0
$373$	−3.68593	+	6.38422i	−0.190850	+	0.330562i	−0.945532	−	0.325528i	$-0.894458\pi$
$373$	−3.68593	+	6.38422i	−0.190850	+	0.330562i	0.754682	+	0.656091i	$0.227791\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	5.67498			0.292276
$378$		0			0
$379$	−17.7598			−0.912259			−0.456130	−	0.889913i	$-0.650765\pi$
$379$	−17.7598			−0.912259			−0.456130	+	0.889913i	$0.650765\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.6350	+	18.4204i	−0.543425	+	0.941239i	0.455280	+	0.890349i	$0.349539\pi$
$383$	−10.6350	+	18.4204i	−0.543425	+	0.941239i	−0.998704	+	0.0508905i	$0.983794\pi$
$384$		0			0
$385$	7.57977	+	2.45216i	0.386301	+	0.124974i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.17846	−	1.83508i	0.161154	−	0.0930424i	−0.417254	−	0.908790i	$-0.637007\pi$
$389$	3.17846	−	1.83508i	0.161154	−	0.0930424i	0.578408	+	0.815748i	$0.303674\pi$
$390$		0			0
$391$	−0.449349	+	0.259432i	−0.0227245	+	0.0131200i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.93114	+	5.07688i	−0.147482	+	0.255446i
$396$		0			0
$397$	−16.4664	+	9.50687i	−0.826424	+	0.477136i	−0.852627	−	0.522521i	$-0.824992\pi$
$397$	−16.4664	+	9.50687i	−0.826424	+	0.477136i	0.0262027	+	0.999657i	$0.491658\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−26.8868	−	15.5231i	−1.34266	−	0.775186i	−0.355464	−	0.934690i	$-0.615677\pi$
$401$	−26.8868	−	15.5231i	−1.34266	−	0.775186i	−0.987197	+	0.159504i	$0.949011\pi$
$402$		0			0
$403$	0.725658	+	1.25688i	0.0361476	+	0.0626095i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.97569	+	4.60477i	0.395340	+	0.228250i
$408$		0			0
$409$		−	35.1524i		−	1.73817i	−0.494659	−	0.869087i	$-0.664707\pi$
$409$		−	35.1524i		−	1.73817i	0.494659	−	0.869087i	$-0.335293\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.94233	+	18.3681i	−0.292403	+	0.903835i
$414$		0			0
$415$	−8.57819	−	14.8579i	−0.421087	−	0.729344i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	14.9516	+	25.8970i	0.730436	+	1.26515i	0.956697	+	0.291085i	$0.0940163\pi$
$419$	14.9516	+	25.8970i	0.730436	+	1.26515i	−0.226261	+	0.974067i	$0.572650\pi$
$420$		0			0
$421$	13.0842	−	22.6624i	0.637683	−	1.10450i	−0.348257	−	0.937399i	$-0.613226\pi$
$421$	13.0842	−	22.6624i	0.637683	−	1.10450i	0.985940	−	0.167100i	$-0.0534402\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.98766			−0.0964156
$426$		0			0
$427$	−5.06334	−	1.63806i	−0.245032	−	0.0792713i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	34.8031	+	20.0936i	1.67641	+	0.967875i	0.963921	+	0.266189i	$0.0857646\pi$
$431$	34.8031	+	20.0936i	1.67641	+	0.967875i	0.712487	+	0.701685i	$0.247569\pi$
$432$		0			0
$433$			15.4379i			0.741896i	0.928654	+	0.370948i	$0.120967\pi$
$433$			15.4379i			0.741896i	−0.928654	+	0.370948i	$0.879033\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.88033			0.137785
$438$		0			0
$439$		−	25.0277i		−	1.19451i	−0.802052	−	0.597254i	$-0.796258\pi$
$439$		−	25.0277i		−	1.19451i	0.802052	−	0.597254i	$-0.203742\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	14.6539i		−	0.696227i	−0.937452	−	0.348113i	$-0.886822\pi$
$443$		−	14.6539i		−	0.696227i	0.937452	−	0.348113i	$-0.113178\pi$
$444$		0			0
$445$	14.7999			0.701584
$446$		0			0
$447$		0			0
$448$		0			0
$449$			31.1359i			1.46939i	0.678395	+	0.734697i	$0.262676\pi$
$449$			31.1359i			1.46939i	−0.678395	+	0.734697i	$0.737324\pi$
$450$		0			0
$451$	10.2539	+	5.92009i	0.482838	+	0.278766i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.25221	+	1.69916i	0.246227	+	0.0796578i
$456$		0			0
$457$	−21.7952			−1.01954			−0.509769	−	0.860312i	$-0.670269\pi$
$457$	−21.7952			−1.01954			−0.509769	+	0.860312i	$0.670269\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.3754	−	23.1670i	0.622957	−	1.07899i	−0.365976	−	0.930624i	$-0.619265\pi$
$461$	13.3754	−	23.1670i	0.622957	−	1.07899i	0.988932	−	0.148368i	$-0.0474020\pi$
$462$		0			0
$463$	−7.62549	−	13.2077i	−0.354386	−	0.613815i	0.632626	−	0.774457i	$-0.281977\pi$
$463$	−7.62549	−	13.2077i	−0.354386	−	0.613815i	−0.987013	+	0.160642i	$0.948644\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	17.9707	+	31.1262i	0.831585	+	1.44035i	0.896781	+	0.442475i	$0.145899\pi$
$467$	17.9707	+	31.1262i	0.831585	+	1.44035i	−0.0651959	+	0.997872i	$0.520767\pi$
$468$		0			0
$469$	1.96502	−	6.07400i	0.0907363	−	0.280471i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4.51383i		−	0.207546i
$474$		0			0
$475$	9.55568	+	5.51697i	0.438445	+	0.253136i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.4726	+	26.7993i	0.706959	+	1.22449i	0.965980	+	0.258618i	$0.0832670\pi$
$479$	15.4726	+	26.7993i	0.706959	+	1.22449i	−0.259020	+	0.965872i	$0.583400\pi$
$480$		0			0
$481$	5.52655	+	3.19076i	0.251989	+	0.145486i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.63463	−	0.943753i	0.0742247	−	0.0428536i
$486$		0			0
$487$	−7.67907	+	13.3005i	−0.347972	+	0.602705i	−0.985889	−	0.167400i	$-0.946463\pi$
$487$	−7.67907	+	13.3005i	−0.347972	+	0.602705i	0.637917	+	0.770105i	$0.279796\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.105233	−	0.0607560i	0.00474908	−	0.00274188i	−0.497624	−	0.867393i	$-0.665794\pi$
$491$	0.105233	−	0.0607560i	0.00474908	−	0.00274188i	0.502373	+	0.864651i	$0.332461\pi$
$492$		0			0
$493$	1.42808	−	0.824502i	0.0643175	−	0.0371337i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	31.1867	+	10.0893i	1.39891	+	0.452568i
$498$		0			0
$499$	−12.8339	+	22.2290i	−0.574524	+	0.995105i	0.421569	+	0.906796i	$0.361479\pi$
$499$	−12.8339	+	22.2290i	−0.574524	+	0.995105i	−0.996093	+	0.0883085i	$0.971854\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.44680			0.153685			0.0768425	−	0.997043i	$-0.475516\pi$
$503$	3.44680			0.153685			0.0768425	+	0.997043i	$0.475516\pi$
$504$		0			0
$505$	2.82088			0.125528
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−20.1495	+	34.8999i	−0.893109	+	1.54691i	−0.0569818	+	0.998375i	$0.518148\pi$
$509$	−20.1495	+	34.8999i	−0.893109	+	1.54691i	−0.836127	+	0.548535i	$0.815186\pi$
$510$		0			0
$511$	−16.0003	+	14.4436i	−0.707811	+	0.638949i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.52490	−	3.76715i	0.287522	−	0.166001i
$516$		0			0
$517$	23.6666	−	13.6639i	1.04086	−	0.600938i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.12248	+	3.67624i	−0.0929874	+	0.161059i	−0.908767	−	0.417304i	$-0.862975\pi$
$521$	−2.12248	+	3.67624i	−0.0929874	+	0.161059i	0.815779	+	0.578363i	$0.196308\pi$
$522$		0			0
$523$	−15.9490	+	9.20818i	−0.697403	+	0.402646i	−0.806379	−	0.591399i	$-0.798576\pi$
$523$	−15.9490	+	9.20818i	−0.697403	+	0.402646i	0.108977	+	0.994044i	$0.465243\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.365216	+	0.210858i	0.0159091	+	0.00918510i
$528$		0			0
$529$	−11.0296	−	19.1039i	−0.479549	−	0.830603i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	7.10518	+	4.10218i	0.307760	+	0.177685i
$534$		0			0
$535$		−	0.284184i		−	0.0122864i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.5011	+	1.89695i	0.796899	+	0.0817075i
$540$		0			0
$541$	−11.7817	−	20.4065i	−0.506535	−	0.877344i	−0.999971	−	0.00756235i	$-0.997593\pi$
$541$	−11.7817	−	20.4065i	−0.506535	−	0.877344i	0.493437	−	0.869782i	$-0.335741\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.45062	+	2.51256i	0.0621379	+	0.107626i
$546$		0			0
$547$	4.87688	−	8.44701i	0.208521	−	0.361168i	−0.742728	−	0.669593i	$-0.766469\pi$
$547$	4.87688	−	8.44701i	0.208521	−	0.361168i	0.951249	+	0.308425i	$0.0998019\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−9.15400			−0.389973
$552$		0			0
$553$	−4.21252	+	13.0212i	−0.179135	+	0.553716i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.22763	+	4.75023i	0.348616	+	0.201274i	0.664076	−	0.747666i	$-0.268825\pi$
$557$	8.22763	+	4.75023i	0.348616	+	0.201274i	−0.315460	+	0.948939i	$0.602159\pi$
$558$		0			0
$559$		−	3.12775i		−	0.132290i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.3789			1.19603			0.598014	−	0.801485i	$-0.295957\pi$
$563$	28.3789			1.19603			0.598014	+	0.801485i	$0.295957\pi$
$564$		0			0
$565$			0.501372i			0.0210929i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	22.9749i		−	0.963156i	−0.876403	−	0.481578i	$-0.840064\pi$
$569$		−	22.9749i		−	0.963156i	0.876403	−	0.481578i	$-0.159936\pi$
$570$		0			0
$571$	9.00491			0.376844			0.188422	−	0.982088i	$-0.439663\pi$
$571$	9.00491			0.376844			0.188422	+	0.982088i	$0.439663\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			3.60386i			0.150291i
$576$		0			0
$577$	31.7024	+	18.3034i	1.31979	+	0.761980i	0.983694	−	0.179848i	$-0.0575607\pi$
$577$	31.7024	+	18.3034i	1.31979	+	0.761980i	0.336094	+	0.941828i	$0.390894\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−26.8378	−	29.7303i	−1.11342	−	1.23342i
$582$		0			0
$583$	36.5086			1.51203
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.3397	+	23.1051i	−0.550589	+	0.953649i	0.447643	+	0.894212i	$0.352264\pi$
$587$	−13.3397	+	23.1051i	−0.550589	+	0.953649i	−0.998232	+	0.0594361i	$0.981070\pi$
$588$		0			0
$589$	−1.17052	−	2.02740i	−0.0482304	−	0.0835375i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.167630	−	0.290343i	−0.00688372	−	0.0119230i	0.862563	−	0.505950i	$-0.168858\pi$
$593$	−0.167630	−	0.290343i	−0.00688372	−	0.0119230i	−0.869447	+	0.494027i	$0.835525\pi$
$594$		0			0
$595$	1.56856	−	0.335494i	0.0643046	−	0.0137539i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			16.9762i			0.693630i	0.937933	+	0.346815i	$0.112737\pi$
$599$			16.9762i			0.693630i	−0.937933	+	0.346815i	$0.887263\pi$
$600$		0			0
$601$	−14.5160	−	8.38082i	−0.592120	−	0.341861i	0.173815	−	0.984778i	$-0.444391\pi$
$601$	−14.5160	−	8.38082i	−0.592120	−	0.341861i	−0.765936	+	0.642917i	$0.777724\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.23322	−	3.86806i	−0.0907935	−	0.157259i
$606$		0			0
$607$	36.0575	+	20.8178i	1.46353	+	0.844968i	0.999172	−	0.0406801i	$-0.0129525\pi$
$607$	36.0575	+	20.8178i	1.46353	+	0.844968i	0.464356	+	0.885649i	$0.346286\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16.3991	−	9.46805i	0.663439	−	0.383036i
$612$		0			0
$613$	−12.8608	+	22.2756i	−0.519444	+	0.899704i	0.480300	+	0.877104i	$0.340528\pi$
$613$	−12.8608	+	22.2756i	−0.519444	+	0.899704i	−0.999745	+	0.0225996i	$0.992806\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	37.0394	−	21.3847i	1.49115	−	0.860915i	0.491199	−	0.871047i	$-0.336559\pi$
$617$	37.0394	−	21.3847i	1.49115	−	0.860915i	0.999949	+	0.0101324i	$0.00322529\pi$
$618$		0			0
$619$	−23.9473	+	13.8260i	−0.962522	+	0.555712i	−0.896948	−	0.442136i	$-0.854221\pi$
$619$	−23.9473	+	13.8260i	−0.962522	+	0.555712i	−0.0655735	+	0.997848i	$0.520888\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	33.7866	−	7.22650i	1.35363	−	0.289524i
$624$		0			0
$625$	−3.69180	+	6.39439i	−0.147672	+	0.255776i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.85430			0.0739359
$630$		0			0
$631$	2.23244			0.0888720			0.0444360	−	0.999012i	$-0.485851\pi$
$631$	2.23244			0.0888720			0.0444360	+	0.999012i	$0.485851\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.98229	+	17.2898i	−0.396135	+	0.686126i
$636$		0			0
$637$	12.8199	+	1.31444i	0.507941	+	0.0520802i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.4181	−	13.5205i	0.924960	−	0.534026i	0.0397458	−	0.999210i	$-0.487345\pi$
$641$	23.4181	−	13.5205i	0.924960	−	0.534026i	0.885214	+	0.465184i	$0.154012\pi$
$642$		0			0
$643$	10.4874	−	6.05488i	0.413581	−	0.238781i	−0.278746	−	0.960365i	$-0.589919\pi$
$643$	10.4874	−	6.05488i	0.413581	−	0.238781i	0.692327	+	0.721584i	$0.256585\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.40271	+	5.89366i	−0.133774	+	0.231704i	−0.925129	−	0.379654i	$-0.876043\pi$
$647$	−3.40271	+	5.89366i	−0.133774	+	0.231704i	0.791354	+	0.611358i	$0.209376\pi$
$648$		0			0
$649$	−16.7892	+	9.69326i	−0.659034	+	0.380494i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−39.7689	−	22.9606i	−1.55628	−	0.898516i	−0.997608	−	0.0691245i	$-0.977979\pi$
$653$	−39.7689	−	22.9606i	−1.55628	−	0.898516i	−0.558668	−	0.829392i	$-0.688687\pi$
$654$		0			0
$655$	−5.77908	−	10.0097i	−0.225807	−	0.391110i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.9825	+	7.49548i	0.505728	+	0.291982i	0.731076	−	0.682296i	$-0.239018\pi$
$659$	12.9825	+	7.49548i	0.505728	+	0.291982i	−0.225348	+	0.974278i	$0.572352\pi$
$660$		0			0
$661$			33.0959i			1.28728i	0.765328	+	0.643640i	$0.222577\pi$
$661$			33.0959i			1.28728i	−0.765328	+	0.643640i	$0.777423\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.47205	−	2.74082i	−0.328532	−	0.106285i
$666$		0			0
$667$	−1.49492	−	2.58927i	−0.0578835	−	0.100257i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2.67204	−	4.62811i	−0.103153	−	0.178666i
$672$		0			0
$673$	−18.6158	+	32.2436i	−0.717588	+	1.24290i	0.244365	+	0.969683i	$0.421420\pi$
$673$	−18.6158	+	32.2436i	−0.717588	+	1.24290i	−0.961953	+	0.273215i	$0.911913\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−40.2923			−1.54856			−0.774280	−	0.632844i	$-0.781888\pi$
$677$	−40.2923			−1.54856			−0.774280	+	0.632844i	$0.781888\pi$
$678$		0			0
$679$	3.27086	−	2.95264i	0.125524	−	0.113312i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.2066	−	7.04746i	−0.467071	−	0.269664i	0.247942	−	0.968775i	$-0.420246\pi$
$683$	−12.2066	−	7.04746i	−0.467071	−	0.269664i	−0.715013	+	0.699111i	$0.753579\pi$
$684$		0			0
$685$		−	9.53236i		−	0.364213i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	25.2977			0.963764
$690$		0			0
$691$		−	37.7579i		−	1.43638i	−0.695849	−	0.718188i	$-0.744972\pi$
$691$		−	37.7579i		−	1.43638i	0.695849	−	0.718188i	$-0.255028\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	11.5650i		−	0.438686i
$696$		0			0
$697$	2.38398			0.0902995
$698$		0			0
$699$		0			0
$700$		0			0
$701$			13.0860i			0.494252i	0.968983	+	0.247126i	$0.0794861\pi$
$701$			13.0860i			0.494252i	−0.968983	+	0.247126i	$0.920514\pi$
$702$		0			0
$703$	−8.91458	−	5.14683i	−0.336220	−	0.194116i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	6.43975	−	1.37738i	0.242192	−	0.0518017i
$708$		0			0
$709$	22.5555			0.847089			0.423544	−	0.905875i	$-0.360786\pi$
$709$	22.5555			0.847089			0.423544	+	0.905875i	$0.360786\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.382310	−	0.662180i	0.0143176	−	0.0247988i
$714$		0			0
$715$	2.77171	+	4.80074i	0.103656	+	0.179538i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.27786	+	9.14152i	0.196831	+	0.340921i	0.947499	−	0.319758i	$-0.103602\pi$
$719$	5.27786	+	9.14152i	0.196831	+	0.340921i	−0.750668	+	0.660679i	$0.770268\pi$
$720$		0			0
$721$	13.0562	−	11.7860i	0.486238	−	0.438932i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	11.4534i		−	0.425370i
$726$		0			0
$727$	−21.9441	−	12.6694i	−0.813861	−	0.469883i	0.0344341	−	0.999407i	$-0.489037\pi$
$727$	−21.9441	−	12.6694i	−0.813861	−	0.469883i	−0.848295	+	0.529524i	$0.822370\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.454422	−	0.787081i	−0.0168074	−	0.0291112i
$732$		0			0
$733$	−3.31425	−	1.91348i	−0.122415	−	0.0706762i	0.437542	−	0.899198i	$-0.355849\pi$
$733$	−3.31425	−	1.91348i	−0.122415	−	0.0706762i	−0.559957	+	0.828522i	$0.689182\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.55190	−	3.20539i	0.204507	−	0.118072i
$738$		0			0
$739$	−18.1289	+	31.4003i	−0.666884	+	1.15508i	0.311887	+	0.950119i	$0.399039\pi$
$739$	−18.1289	+	31.4003i	−0.666884	+	1.15508i	−0.978771	+	0.204958i	$0.934294\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.3598	−	7.13594i	0.453438	−	0.261792i	−0.255843	−	0.966718i	$-0.582353\pi$
$743$	12.3598	−	7.13594i	0.453438	−	0.261792i	0.709281	+	0.704926i	$0.249020\pi$
$744$		0			0
$745$	14.9182	−	8.61305i	0.546562	−	0.315558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.138761	−	0.648761i	−0.00507023	−	0.0237052i
$750$		0			0
$751$	−14.6005	+	25.2887i	−0.532778	+	0.922799i	0.466489	+	0.884527i	$0.345519\pi$
$751$	−14.6005	+	25.2887i	−0.532778	+	0.922799i	−0.999267	+	0.0382721i	$0.987815\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9.52263			−0.346564
$756$		0			0
$757$	−33.2051			−1.20686			−0.603431	−	0.797415i	$-0.706200\pi$
$757$	−33.2051			−1.20686			−0.603431	+	0.797415i	$0.706200\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.46846	−	12.9358i	0.270731	−	0.468921i	−0.698318	−	0.715788i	$-0.746068\pi$
$761$	7.46846	−	12.9358i	0.270731	−	0.468921i	0.969049	+	0.246867i	$0.0794011\pi$
$762$		0			0
$763$	4.53844	+	5.02757i	0.164303	+	0.182010i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.6337	+	6.71669i	−0.420067	+	0.242526i
$768$		0			0
$769$	27.1703	−	15.6868i	0.979788	−	0.565681i	0.0775816	−	0.996986i	$-0.475280\pi$
$769$	27.1703	−	15.6868i	0.979788	−	0.565681i	0.902206	+	0.431305i	$0.141947\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.464799	+	0.805055i	−0.0167176	+	0.0289558i	−0.874263	−	0.485452i	$-0.838655\pi$
$773$	−0.464799	+	0.805055i	−0.0167176	+	0.0289558i	0.857546	+	0.514408i	$0.171988\pi$
$774$		0			0
$775$	2.53667	−	1.46455i	0.0911200	−	0.0526082i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.4610	−	6.61700i	−0.410632	−	0.237079i
$780$		0			0
$781$	16.4579	+	28.5060i	0.588911	+	1.02002i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.3362	+	10.5864i	0.654448	+	0.377846i
$786$		0			0
$787$		−	40.3516i		−	1.43838i	−0.694814	−	0.719189i	$-0.744513\pi$
$787$		−	40.3516i		−	1.43838i	0.694814	−	0.719189i	$-0.255487\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.244810	+	1.14458i	0.00870443	+	0.0406964i
$792$		0			0
$793$	−1.85152	−	3.20693i	−0.0657495	−	0.113881i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.2121	+	19.4199i	0.397152	+	0.687888i	0.993373	−	0.114932i	$-0.0366652\pi$
$797$	11.2121	+	19.4199i	0.397152	+	0.687888i	−0.596221	+	0.802820i	$0.703332\pi$
$798$		0			0
$799$	2.75117	−	4.76517i	0.0973296	−	0.168580i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.6458			−0.763864
$804$		0			0
$805$	−0.608290	−	2.84398i	−0.0214394	−	0.100237i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.2271	−	10.5234i	−0.640832	−	0.369985i	0.144103	−	0.989563i	$-0.453970\pi$
$809$	−18.2271	−	10.5234i	−0.640832	−	0.369985i	−0.784935	+	0.619578i	$0.787304\pi$
$810$		0			0
$811$		−	31.5558i		−	1.10808i	−0.832492	−	0.554038i	$-0.813086\pi$
$811$		−	31.5558i		−	1.10808i	0.832492	−	0.554038i	$-0.186914\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	24.4472			0.856347
$816$		0			0
$817$			5.04520i			0.176509i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	54.6328i		−	1.90670i	−0.301871	−	0.953349i	$-0.597611\pi$
$821$		−	54.6328i		−	1.90670i	0.301871	−	0.953349i	$-0.402389\pi$
$822$		0			0
$823$	−11.7749			−0.410448			−0.205224	−	0.978715i	$-0.565792\pi$
$823$	−11.7749			−0.410448			−0.205224	+	0.978715i	$0.565792\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	27.9059i		−	0.970383i	−0.874408	−	0.485192i	$-0.838750\pi$
$827$		−	27.9059i		−	0.970383i	0.874408	−	0.485192i	$-0.161250\pi$
$828$		0			0
$829$	21.7374	+	12.5501i	0.754972	+	0.435883i	0.827488	−	0.561484i	$-0.189769\pi$
$829$	21.7374	+	12.5501i	0.754972	+	0.435883i	−0.0725156	+	0.997367i	$0.523103\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.41702	−	1.53179i	0.118393	−	0.0530733i
$834$		0			0
$835$	−14.5021			−0.501867
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−22.6654	+	39.2576i	−0.782497	+	1.35532i	0.147987	+	0.988989i	$0.452721\pi$
$839$	−22.6654	+	39.2576i	−0.782497	+	1.35532i	−0.930483	+	0.366335i	$0.880613\pi$
$840$		0			0
$841$	−9.74899	−	16.8857i	−0.336172	−	0.582267i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.44597	−	9.43270i	−0.187347	−	0.324495i
$846$		0			0
$847$	−6.98689	−	7.73990i	−0.240072	−	0.265946i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	3.36207i		−	0.115250i
$852$		0			0
$853$	−20.8404	−	12.0322i	−0.713560	−	0.411974i	0.0988175	−	0.995106i	$-0.468494\pi$
$853$	−20.8404	−	12.0322i	−0.713560	−	0.411974i	−0.812378	+	0.583131i	$0.801827\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.3869	+	43.9714i	0.867200	+	1.50203i	0.864846	+	0.502037i	$0.167416\pi$
$857$	25.3869	+	43.9714i	0.867200	+	1.50203i	0.00235426	+	0.999997i	$0.499251\pi$
$858$		0			0
$859$	−6.78222	−	3.91572i	−0.231406	−	0.133603i	0.379814	−	0.925063i	$-0.375988\pi$
$859$	−6.78222	−	3.91572i	−0.231406	−	0.133603i	−0.611221	+	0.791460i	$0.709321\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.68312	+	2.12645i	−0.125375	+	0.0723852i	−0.561376	−	0.827561i	$-0.689728\pi$
$863$	−3.68312	+	2.12645i	−0.125375	+	0.0723852i	0.436001	+	0.899946i	$0.356394\pi$
$864$		0			0
$865$	12.1264	−	21.0036i	0.412311	−	0.714143i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−11.9019	+	6.87156i	−0.403744	+	0.233102i
$870$		0			0
$871$	3.84704	−	2.22109i	0.130352	−	0.0752588i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.04406	−	24.8647i	0.271939	−	0.840579i
$876$		0			0
$877$	−17.0535	+	29.5375i	−0.575855	+	0.997411i	0.420093	+	0.907481i	$0.361997\pi$
$877$	−17.0535	+	29.5375i	−0.575855	+	0.997411i	−0.995948	+	0.0899294i	$0.971336\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	8.87180			0.298898			0.149449	−	0.988769i	$-0.452250\pi$
$881$	8.87180			0.298898			0.149449	+	0.988769i	$0.452250\pi$
$882$		0			0
$883$	43.8791			1.47665			0.738325	−	0.674446i	$-0.235617\pi$
$883$	43.8791			1.47665			0.738325	+	0.674446i	$0.235617\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.3699	+	31.8176i	−0.616801	+	1.06833i	0.373265	+	0.927725i	$0.378238\pi$
$887$	−18.3699	+	31.8176i	−0.616801	+	1.06833i	−0.990066	+	0.140605i	$0.955095\pi$
$888$		0			0
$889$	−14.3462	+	44.3449i	−0.481155	+	1.48728i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−26.4526	+	15.2724i	−0.885202	+	0.511071i
$894$		0			0
$895$	−4.95206	+	2.85907i	−0.165529	+	0.0955683i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.21502	+	2.10448i	−0.0405232	+	0.0701883i
$900$		0			0
$901$	6.36603	−	3.67543i	0.212083	−	0.122446i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−11.4793	−	6.62759i	−0.381585	−	0.220308i
$906$		0			0
$907$	8.32897	+	14.4262i	0.276559	+	0.479014i	0.970527	−	0.240991i	$-0.0774726\pi$
$907$	8.32897	+	14.4262i	0.276559	+	0.479014i	−0.693968	+	0.720006i	$0.744139\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−29.0963	−	16.7988i	−0.964004	−	0.556568i	−0.0666011	−	0.997780i	$-0.521215\pi$
$911$	−29.0963	−	16.7988i	−0.964004	−	0.556568i	−0.897403	+	0.441212i	$0.854549\pi$
$912$		0			0
$913$		−	40.2202i		−	1.33110i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−18.0805	−	20.0291i	−0.597071	−	0.661420i
$918$		0			0
$919$	−5.96310	−	10.3284i	−0.196705	−	0.340702i	0.750753	−	0.660583i	$-0.229691\pi$
$919$	−5.96310	−	10.3284i	−0.196705	−	0.340702i	−0.947458	+	0.319880i	$0.896357\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	11.4041	+	19.7525i	0.375371	+	0.650161i
$924$		0			0
$925$	6.43970	−	11.1539i	0.211736	−	0.366738i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−43.7180			−1.43434			−0.717170	−	0.696899i	$-0.754563\pi$
$929$	−43.7180			−1.43434			−0.717170	+	0.696899i	$0.754563\pi$
$930$		0			0
$931$	−20.6790	−	2.12026i	−0.677727	−	0.0694886i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.39497	+	0.805388i	0.0456205	+	0.0263390i
$936$		0			0
$937$			47.1189i			1.53931i	0.638461	+	0.769654i	$0.279571\pi$
$937$			47.1189i			1.53931i	−0.638461	+	0.769654i	$0.720429\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22.1033			−0.720546			−0.360273	−	0.932847i	$-0.617316\pi$
$941$	−22.1033			−0.720546			−0.360273	+	0.932847i	$0.617316\pi$
$942$		0			0
$943$		−	4.32243i		−	0.140758i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			0.0333295i			0.00108306i	1.00000		0.000541531i	$0.000172375\pi$
$947$			0.0333295i			0.00108306i	−1.00000		0.000541531i	$0.999828\pi$
$948$		0			0
$949$	−14.9989			−0.486885
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21.6371i			0.700893i	0.936583	+	0.350446i	$0.113970\pi$
$953$			21.6371i			0.700893i	−0.936583	+	0.350446i	$0.886030\pi$
$954$		0			0
$955$	22.2224	+	12.8301i	0.719100	+	0.415173i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.65446	−	21.7613i	−0.150300	−	0.702709i
$960$		0			0
$961$	30.3785			0.979953
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.633837	−	1.09784i	0.0204039	−	0.0353406i
$966$		0			0
$967$	−26.4627	−	45.8347i	−0.850983	−	1.47395i	−0.880322	−	0.474376i	$-0.842674\pi$
$967$	−26.4627	−	45.8347i	−0.850983	−	1.47395i	0.0293390	−	0.999570i	$-0.490660\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.18246	−	14.1724i	−0.262588	−	0.454815i	0.704341	−	0.709862i	$-0.251243\pi$
$971$	−8.18246	−	14.1724i	−0.262588	−	0.454815i	−0.966929	+	0.255047i	$0.917909\pi$
$972$		0			0
$973$	−5.64696	−	26.4016i	−0.181033	−	0.846397i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			29.1776i			0.933474i	0.884396	+	0.466737i	$0.154571\pi$
$977$			29.1776i			0.933474i	−0.884396	+	0.466737i	$0.845429\pi$
$978$		0			0
$979$	30.0476	+	17.3480i	0.960324	+	0.554444i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.40545	+	14.5587i	0.268092	+	0.464349i	0.968369	−	0.249522i	$-0.0802735\pi$
$983$	8.40545	+	14.5587i	0.268092	+	0.464349i	−0.700277	+	0.713871i	$0.746940\pi$
$984$		0			0
$985$	−3.13228	−	1.80842i	−0.0998027	−	0.0576211i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.42707	+	0.823919i	−0.0453782	+	0.0261991i
$990$		0			0
$991$	24.5748	−	42.5649i	0.780645	−	1.35212i	−0.150921	−	0.988546i	$-0.548224\pi$
$991$	24.5748	−	42.5649i	0.780645	−	1.35212i	0.931566	−	0.363571i	$-0.118443\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.392590	+	0.226662i	−0.0124459	+	0.00718567i
$996$		0			0
$997$	−15.4850	+	8.94029i	−0.490416	+	0.283142i	−0.724747	−	0.689015i	$-0.758043\pi$
$997$	−15.4850	+	8.94029i	−0.490416	+	0.283142i	0.234331	+	0.972157i	$0.424710\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.w.j.269.6		32
3.2	odd	2	inner	2268.2.w.j.269.11		32
7.5	odd	6		2268.2.bm.j.593.6		32
9.2	odd	6		2268.2.t.c.1781.11	yes	32
9.4	even	3		2268.2.bm.j.1025.11		32
9.5	odd	6		2268.2.bm.j.1025.6		32
9.7	even	3		2268.2.t.c.1781.6	✓	32
21.5	even	6		2268.2.bm.j.593.11		32
63.5	even	6	inner	2268.2.w.j.1349.6		32
63.40	odd	6	inner	2268.2.w.j.1349.11		32
63.47	even	6		2268.2.t.c.2105.6	yes	32
63.61	odd	6		2268.2.t.c.2105.11	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.6	✓	32	9.7	even	3
2268.2.t.c.1781.11	yes	32	9.2	odd	6
2268.2.t.c.2105.6	yes	32	63.47	even	6
2268.2.t.c.2105.11	yes	32	63.61	odd	6
2268.2.w.j.269.6		32	1.1	even	1	trivial
2268.2.w.j.269.11		32	3.2	odd	2	inner
2268.2.w.j.1349.6		32	63.5	even	6	inner
2268.2.w.j.1349.11		32	63.40	odd	6	inner
2268.2.bm.j.593.6		32	7.5	odd	6
2268.2.bm.j.593.11		32	21.5	even	6
2268.2.bm.j.1025.6		32	9.5	odd	6
2268.2.bm.j.1025.11		32	9.4	even	3

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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.566658 + 0.981481i) q^{5} +(-0.814380 + 2.51730i) q^{7} +(-2.30092 + 1.32843i) q^{11} +(-1.59436 + 0.920505i) q^{13} +(-0.267475 + 0.463281i) q^{17} +(2.57178 - 1.48482i) q^{19} +(0.839982 + 0.484964i) q^{23} +(1.85780 + 3.21780i) q^{25} +(-2.66955 - 1.54127i) q^{29} -0.788326i q^{31} +(-2.00920 - 2.22575i) q^{35} +(-1.73316 - 3.00191i) q^{37} +(-2.22822 - 3.85939i) q^{41} +(-0.849465 + 1.47132i) q^{43} -10.2857 q^{47} +(-5.67357 - 4.10007i) q^{49} +(-11.9002 - 6.87059i) q^{53} -3.01107i q^{55} +7.29675 q^{59} +2.01142i q^{61} -2.08645i q^{65} -2.41291 q^{67} -12.3890i q^{71} +(7.05561 + 4.07356i) q^{73} +(-1.47025 - 6.87394i) q^{77} +5.17268 q^{79} +(-7.56910 + 13.1101i) q^{83} +(-0.303134 - 0.525044i) q^{85} +(-6.52948 - 11.3094i) q^{89} +(-1.01877 - 4.76312i) q^{91} +3.36553i q^{95} +(-1.44234 - 0.832736i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} + 12 q^{13} - 16 q^{25} - 4 q^{37} - 4 q^{43} + 20 q^{49} - 8 q^{67} - 36 q^{73} - 56 q^{79} + 12 q^{85} - 36 q^{91} + 12 q^{97}+O(q^{100}) \) 32 * q + 4 * q^7 + 12 * q^13 - 16 * q^25 - 4 * q^37 - 4 * q^43 + 20 * q^49 - 8 * q^67 - 36 * q^73 - 56 * q^79 + 12 * q^85 - 36 * q^91 + 12 * q^97

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