LMFDB - Embedded newform 2268.2.t.c.1781.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(1781,2268)] 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, 3, 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("2268.1781"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == 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			1781.6
Character	$\chi$	$=$	2268.1781
Dual form			2268.2.t.c.2105.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.566658 - 0.981481i) q^{5} +(2.58723 - 0.553375i) q^{7} +(2.30092 + 1.32843i) q^{11} -1.84101i q^{13} +(-0.267475 + 0.463281i) q^{17} +(2.57178 - 1.48482i) q^{19} +(-0.839982 + 0.484964i) q^{23} +(1.85780 - 3.21780i) q^{25} +3.08254i q^{29} +(0.682710 + 0.394163i) q^{31} +(-2.00920 - 2.22575i) q^{35} +(-1.73316 - 3.00191i) q^{37} +4.45644 q^{41} +1.69893 q^{43} +(5.14286 + 8.90769i) q^{47} +(6.38755 - 2.86342i) q^{49} +(-11.9002 - 6.87059i) q^{53} -3.01107i q^{55} +(-3.64838 + 6.31917i) q^{59} +(1.74194 - 1.00571i) q^{61} +(-1.80692 + 1.04322i) q^{65} +(1.20645 - 2.08964i) q^{67} -12.3890i q^{71} +(7.05561 + 4.07356i) q^{73} +(6.68813 + 2.16370i) q^{77} +(-2.58634 - 4.47967i) q^{79} +15.1382 q^{83} +0.606268 q^{85} +(-6.52948 - 11.3094i) q^{89} +(-1.01877 - 4.76312i) q^{91} +(-2.91464 - 1.68277i) q^{95} +1.66547i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) $ 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.566658	−	0.981481i	−0.253417	−	0.438932i	0.711047	−	0.703144i	$-0.248221\pi$
$5$	−0.566658	−	0.981481i	−0.253417	−	0.438932i	−0.964464	+	0.264213i	$0.914888\pi$
$6$		0			0
$7$	2.58723	−	0.553375i	0.977882	−	0.209156i
$7$	2.58723	−	0.553375i	0.977882	−	0.209156i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.30092	+	1.32843i	0.693753	+	0.400538i	0.805016	−	0.593253i	$-0.202157\pi$
$11$	2.30092	+	1.32843i	0.693753	+	0.400538i	−0.111264	+	0.993791i	$0.535490\pi$
$12$		0			0
$13$		−	1.84101i		−	0.510604i	−0.966861	−	0.255302i	$-0.917825\pi$
$13$		−	1.84101i		−	0.510604i	0.966861	−	0.255302i	$-0.0821749\pi$
$14$		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	$-0.698910\pi$
$23$	−0.839982	+	0.484964i	−0.175148	+	0.101122i	0.409863	+	0.912147i	$0.365577\pi$
$24$		0			0
$25$	1.85780	−	3.21780i	0.371559	−	0.643560i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			3.08254i			0.572412i	0.958168	+	0.286206i	$0.0923943\pi$
$29$			3.08254i			0.572412i	−0.958168	+	0.286206i	$0.907606\pi$
$30$		0			0
$31$	0.682710	+	0.394163i	0.122618	+	0.0707937i	0.560055	−	0.828456i	$-0.310780\pi$
$31$	0.682710	+	0.394163i	0.122618	+	0.0707937i	−0.437436	+	0.899249i	$0.644113\pi$
$32$		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
$41$	4.45644			0.695979			0.347990	−	0.937498i	$-0.386864\pi$
$41$	4.45644			0.695979			0.347990	+	0.937498i	$0.386864\pi$
$42$		0			0
$43$	1.69893			0.259084			0.129542	−	0.991574i	$-0.458649\pi$
$43$	1.69893			0.259084			0.129542	+	0.991574i	$0.458649\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.14286	+	8.90769i	0.750163	+	1.29932i	0.947743	+	0.319034i	$0.103358\pi$
$47$	5.14286	+	8.90769i	0.750163	+	1.29932i	−0.197580	+	0.980287i	$0.563308\pi$
$48$		0			0
$49$	6.38755	−	2.86342i	0.912507	−	0.409060i
$50$		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
$59$	−3.64838	+	6.31917i	−0.474978	+	0.822686i	−0.999589	−	0.0286558i	$-0.990877\pi$
$59$	−3.64838	+	6.31917i	−0.474978	+	0.822686i	0.524611	+	0.851342i	$0.324211\pi$
$60$		0			0
$61$	1.74194	−	1.00571i	0.223033	−	0.128768i	−0.384321	−	0.923200i	$-0.625564\pi$
$61$	1.74194	−	1.00571i	0.223033	−	0.128768i	0.607354	+	0.794432i	$0.292231\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.80692	+	1.04322i	−0.224120	+	0.129396i
$66$		0			0
$67$	1.20645	−	2.08964i	0.147392	−	0.255290i	−0.782871	−	0.622184i	$-0.786246\pi$
$67$	1.20645	−	2.08964i	0.147392	−	0.255290i	0.930263	+	0.366894i	$0.119579\pi$
$68$		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
$77$	6.68813	+	2.16370i	0.762183	+	0.246577i
$78$		0			0
$79$	−2.58634	−	4.47967i	−0.290986	−	0.504002i	0.683057	−	0.730365i	$-0.260650\pi$
$79$	−2.58634	−	4.47967i	−0.290986	−	0.504002i	−0.974043	+	0.226363i	$0.927317\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.1382			1.66163			0.830817	−	0.556546i	$-0.187874\pi$
$83$	15.1382			1.66163			0.830817	+	0.556546i	$0.187874\pi$
$84$		0			0
$85$	0.606268			0.0657590
$86$		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
$95$	−2.91464	−	1.68277i	−0.299035	−	0.172648i
$96$		0			0
$97$			1.66547i			0.169103i	0.996419	+	0.0845515i	$0.0269458\pi$
$97$			1.66547i			0.169103i	−0.996419	+	0.0845515i	$0.973054\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.24453	+	2.15558i	−0.123835	+	0.214488i	−0.921277	−	0.388907i	$-0.872853\pi$
$101$	−1.24453	+	2.15558i	−0.123835	+	0.214488i	0.797442	+	0.603396i	$0.206186\pi$
$102$		0			0
$103$	5.75735	−	3.32401i	0.567289	−	0.327524i	−0.188777	−	0.982020i	$-0.560452\pi$
$103$	5.75735	−	3.32401i	0.567289	−	0.327524i	0.756066	+	0.654496i	$0.227119\pi$
$104$		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
$113$			0.442394i			0.0416169i	0.999783	+	0.0208085i	$0.00662402\pi$
$113$			0.442394i			0.0416169i	−0.999783	+	0.0208085i	$0.993376\pi$
$114$		0			0
$115$	0.951966	+	0.549618i	0.0887713	+	0.0512521i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.435653	+	1.34663i	−0.0399362	+	0.123445i
$120$		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	0.552564	−	0.833471i	$-0.313650\pi$
$131$	−5.09926	−	8.83218i	−0.445525	−	0.771671i	−0.998089	+	0.0617991i	$0.980316\pi$
$132$		0			0
$133$	5.83213	−	5.26472i	0.505709	−	0.456509i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.28417	+	4.20552i	0.622328	+	0.359301i	0.777775	−	0.628543i	$-0.216348\pi$
$137$	7.28417	+	4.20552i	0.622328	+	0.359301i	−0.155447	+	0.987844i	$0.549682\pi$
$138$		0			0
$139$		−	10.2046i		−	0.865541i	−0.901504	−	0.432770i	$-0.857536\pi$
$139$		−	10.2046i		−	0.865541i	0.901504	−	0.432770i	$-0.142464\pi$
$140$		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	$-0.452741\pi$
$149$	13.1633	−	7.59986i	1.07838	−	0.622605i	0.930460	+	0.366394i	$0.119408\pi$
$150$		0			0
$151$	4.20122	−	7.27673i	0.341891	−	0.592172i	−0.642893	−	0.765956i	$-0.722266\pi$
$151$	4.20122	−	7.27673i	0.341891	−	0.592172i	0.984784	+	0.173784i	$0.0555994\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	0.893423i		−	0.0717614i
$156$		0			0
$157$	16.1793	+	9.34111i	1.29125	+	0.745501i	0.978875	−	0.204459i	$-0.0655434\pi$
$157$	16.1793	+	9.34111i	1.29125	+	0.745501i	0.312371	+	0.949960i	$0.398877\pi$
$158$		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
$167$	−12.7962			−0.990199			−0.495099	−	0.868836i	$-0.664868\pi$
$167$	−12.7962			−0.990199			−0.495099	+	0.868836i	$0.664868\pi$
$168$		0			0
$169$	9.61068			0.739283
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.6999	+	18.5328i	0.813501	+	1.40903i	0.910399	+	0.413732i	$0.135775\pi$
$173$	10.6999	+	18.5328i	0.813501	+	1.40903i	−0.0968975	+	0.995294i	$0.530892\pi$
$174$		0			0
$175$	3.02590	−	9.35325i	0.228737	−	0.707040i
$176$		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
$185$	−1.96421	+	3.40212i	−0.144412	+	0.250129i
$186$		0			0
$187$	−1.23088	+	0.710647i	−0.0900106	+	0.0519676i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.6083	+	11.3209i	−1.41881	+	0.819148i	−0.996194	−	0.0871622i	$-0.972220\pi$
$191$	−19.6083	+	11.3209i	−1.41881	+	0.819148i	−0.422612	+	0.906311i	$0.638887\pi$
$192$		0			0
$193$	0.559276	−	0.968695i	0.0402576	−	0.0697282i	−0.845195	−	0.534459i	$-0.820515\pi$
$193$	0.559276	−	0.968695i	0.0402576	−	0.0697282i	0.885452	+	0.464731i	$0.153849\pi$
$194$		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
$203$	1.70580	+	7.97524i	0.119724	+	0.559752i
$204$		0			0
$205$	−2.52528	−	4.37391i	−0.176373	−	0.305487i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.88993			0.545758
$210$		0			0
$211$	5.64759			0.388796			0.194398	−	0.980923i	$-0.437725\pi$
$211$	5.64759			0.388796			0.194398	+	0.980923i	$0.437725\pi$
$212$		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
$221$	0.852904	+	0.492425i	0.0573725	+	0.0331241i
$222$		0			0
$223$		−	6.23782i		−	0.417715i	−0.977946	−	0.208858i	$-0.933025\pi$
$223$		−	6.23782i		−	0.417715i	0.977946	−	0.208858i	$-0.0669745\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.16442	−	14.1412i	0.541892	−	0.938584i	−0.456904	−	0.889516i	$-0.651042\pi$
$227$	8.16442	−	14.1412i	0.541892	−	0.938584i	0.998795	−	0.0490677i	$-0.0156250\pi$
$228$		0			0
$229$	−20.2825	+	11.7101i	−1.34031	+	0.773827i	−0.986852	−	0.161624i	$-0.948327\pi$
$229$	−20.2825	+	11.7101i	−1.34031	+	0.773827i	−0.353455	+	0.935451i	$0.614993\pi$
$230$		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
$239$			9.89021i			0.639744i	0.947461	+	0.319872i	$0.103640\pi$
$239$			9.89021i			0.639744i	−0.947461	+	0.319872i	$0.896360\pi$
$240$		0			0
$241$	17.5109	+	10.1099i	1.12797	+	0.651236i	0.943425	−	0.331587i	$-0.107584\pi$
$241$	17.5109	+	10.1099i	1.12797	+	0.651236i	0.184550	+	0.982823i	$0.440917\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.42995	−	4.64668i	−0.410795	−	0.296865i
$246$		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	0.717995	−	0.696049i	$-0.245060\pi$
$257$	−3.90839	−	6.76953i	−0.243798	−	0.422271i	−0.961793	+	0.273777i	$0.911727\pi$
$258$		0			0
$259$	−6.14526	−	6.80756i	−0.381848	−	0.423001i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.47396	+	0.850994i	0.0908885	+	0.0524745i	0.544755	−	0.838595i	$-0.316623\pi$
$263$	1.47396	+	0.850994i	0.0908885	+	0.0524745i	−0.453867	+	0.891070i	$0.649956\pi$
$264$		0			0
$265$			15.5731i			0.956649i
$266$		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	0.256436	+	0.966561i	$0.417452\pi$
$277$	−11.7976	+	20.4340i	−0.708849	+	1.22776i	−0.965284	+	0.261201i	$0.915881\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.4848i		−	1.10271i	−0.834270	−	0.551356i	$-0.814111\pi$
$281$		−	18.4848i		−	1.10271i	0.834270	−	0.551356i	$-0.185889\pi$
$282$		0			0
$283$	−15.3990	−	8.89059i	−0.915373	−	0.528491i	−0.0332168	−	0.999448i	$-0.510575\pi$
$283$	−15.3990	−	8.89059i	−0.915373	−	0.528491i	−0.882156	+	0.470958i	$0.843909\pi$
$284$		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
$293$	4.86712			0.284340			0.142170	−	0.989842i	$-0.454592\pi$
$293$	4.86712			0.284340			0.142170	+	0.989842i	$0.454592\pi$
$294$		0			0
$295$	8.26953			0.481471
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.892823	+	1.54642i	0.0516333	+	0.0894315i
$300$		0			0
$301$	4.39553	−	0.940145i	0.253354	−	0.0541891i
$302$		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
$311$	−8.10547	+	14.0391i	−0.459619	+	0.796084i	−0.998941	−	0.0460162i	$-0.985347\pi$
$311$	−8.10547	+	14.0391i	−0.459619	+	0.796084i	0.539322	+	0.842100i	$0.318681\pi$
$312$		0			0
$313$	−19.9011	+	11.4899i	−1.12488	+	0.649449i	−0.942642	−	0.333806i	$-0.891667\pi$
$313$	−19.9011	+	11.4899i	−1.12488	+	0.649449i	−0.182237	+	0.983255i	$0.558334\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.8322	+	10.8728i	−1.05772	+	0.610675i	−0.924801	−	0.380452i	$-0.875769\pi$
$317$	−18.8322	+	10.8728i	−1.05772	+	0.610675i	−0.132920	+	0.991127i	$0.542435\pi$
$318$		0			0
$319$	−4.09495	+	7.09266i	−0.229273	+	0.397113i
$320$		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
$329$	18.2351	+	20.2003i	1.00533	+	1.11368i
$330$		0			0
$331$	−12.8868	−	22.3207i	−0.708325	−	1.22685i	−0.965478	−	0.260484i	$-0.916118\pi$
$331$	−12.8868	−	22.3207i	−0.708325	−	1.22685i	0.257154	−	0.966371i	$-0.417215\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.73459			−0.149406
$336$		0			0
$337$	−13.6133			−0.741562			−0.370781	−	0.928720i	$-0.620910\pi$
$337$	−13.6133			−0.741562			−0.370781	+	0.928720i	$0.620910\pi$
$338$		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
$347$	−8.35635	−	4.82454i	−0.448593	−	0.258995i	0.258643	−	0.965973i	$-0.416725\pi$
$347$	−8.35635	−	4.82454i	−0.448593	−	0.258995i	−0.707236	+	0.706978i	$0.750058\pi$
$348$		0			0
$349$			23.3122i			1.24787i	0.781475	+	0.623937i	$0.214468\pi$
$349$			23.3122i			1.24787i	−0.781475	+	0.623937i	$0.785532\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.5010	−	26.8485i	0.825035	−	1.42900i	−0.0768568	−	0.997042i	$-0.524488\pi$
$353$	15.5010	−	26.8485i	0.825035	−	1.42900i	0.901892	−	0.431961i	$-0.142178\pi$
$354$		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
$365$		−	9.23326i		−	0.483291i
$366$		0			0
$367$	2.73854	+	1.58110i	0.142951	+	0.0825326i	0.569769	−	0.821805i	$-0.307032\pi$
$367$	2.73854	+	1.58110i	0.142951	+	0.0825326i	−0.426819	+	0.904337i	$0.640366\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−34.5907	−	11.1905i	−1.79586	−	0.580984i
$372$		0			0
$373$	−3.68593	−	6.38422i	−0.190850	−	0.330562i	0.754682	−	0.656091i	$-0.227791\pi$
$373$	−3.68593	−	6.38422i	−0.190850	−	0.330562i	−0.945532	+	0.325528i	$0.894458\pi$
$374$		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	−0.998704	−	0.0508905i	$-0.983794\pi$
$383$	−10.6350	−	18.4204i	−0.543425	−	0.941239i	0.455280	−	0.890349i	$-0.349539\pi$
$384$		0			0
$385$	−1.66625	−	7.79035i	−0.0849202	−	0.397033i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.17846	−	1.83508i	−0.161154	−	0.0930424i	0.417254	−	0.908790i	$-0.362993\pi$
$389$	−3.17846	−	1.83508i	−0.161154	−	0.0930424i	−0.578408	+	0.815748i	$0.696326\pi$
$390$		0			0
$391$		−	0.518863i		−	0.0262400i
$392$		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	$-0.384323\pi$
$401$	26.8868	−	15.5231i	1.34266	−	0.775186i	0.987197	+	0.159504i	$0.0509894\pi$
$402$		0			0
$403$	0.725658	−	1.25688i	0.0361476	−	0.0626095i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9.20954i		−	0.456500i
$408$		0			0
$409$	30.4429	+	17.5762i	1.50530	+	0.869087i	0.999981	+	0.00615678i	$0.00195978\pi$
$409$	30.4429	+	17.5762i	1.50530	+	0.869087i	0.505322	+	0.862931i	$0.331374\pi$
$410$		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
$419$	−29.9033			−1.46087			−0.730436	−	0.682981i	$-0.760683\pi$
$419$	−29.9033			−1.46087			−0.730436	+	0.682981i	$0.760683\pi$
$420$		0			0
$421$	−26.1683			−1.27537			−0.637683	−	0.770299i	$-0.720107\pi$
$421$	−26.1683			−1.27537			−0.637683	+	0.770299i	$0.720107\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.993829	+	1.72136i	0.0482078	+	0.0834984i
$426$		0			0
$427$	3.95027	−	3.56595i	0.191167	−	0.172569i
$428$		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
$437$	−1.44016	+	2.49444i	−0.0688924	+	0.119325i
$438$		0			0
$439$	−21.6747	+	12.5139i	−1.03447	+	0.597254i	−0.918263	−	0.395970i	$-0.870408\pi$
$439$	−21.6747	+	12.5139i	−1.03447	+	0.597254i	−0.116212	+	0.993224i	$0.537075\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.6906	+	7.32694i	−0.602950	+	0.348113i	−0.770201	−	0.637801i	$-0.779844\pi$
$443$	−12.6906	+	7.32694i	−0.602950	+	0.348113i	0.167251	+	0.985914i	$0.446511\pi$
$444$		0			0
$445$	−7.39996	+	12.8171i	−0.350792	+	0.607589i
$446$		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
$455$	−4.09762	+	3.69896i	−0.192099	+	0.173410i
$456$		0			0
$457$	10.8976	+	18.8752i	0.509769	+	0.882945i	0.999936	+	0.0113168i	$0.00360234\pi$
$457$	10.8976	+	18.8752i	0.509769	+	0.882945i	−0.490167	+	0.871628i	$0.663064\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−26.7509			−1.24591			−0.622957	−	0.782256i	$-0.714069\pi$
$461$	−26.7509			−1.24591			−0.622957	+	0.782256i	$0.714069\pi$
$462$		0			0
$463$	15.2510			0.708773			0.354386	−	0.935099i	$-0.384690\pi$
$463$	15.2510			0.708773			0.354386	+	0.935099i	$0.384690\pi$
$464$		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
$473$	3.90910	+	2.25692i	0.179740	+	0.103773i
$474$		0			0
$475$		−	11.0339i		−	0.506272i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.4726	−	26.7993i	0.706959	−	1.22449i	−0.259020	−	0.965872i	$-0.583400\pi$
$479$	15.4726	−	26.7993i	0.706959	−	1.22449i	0.965980	−	0.258618i	$-0.0832670\pi$
$480$		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
$491$			0.121512i			0.00548376i	0.999996	+	0.00274188i	$0.000872769\pi$
$491$			0.121512i			0.00548376i	−0.999996	+	0.00274188i	$0.999127\pi$
$492$		0			0
$493$	−1.42808	−	0.824502i	−0.0643175	−	0.0371337i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−6.85574	−	32.0531i	−0.307522	−	1.43778i
$498$		0			0
$499$	−12.8339	−	22.2290i	−0.574524	−	0.995105i	−0.996093	−	0.0883085i	$-0.971854\pi$
$499$	−12.8339	−	22.2290i	−0.574524	−	0.995105i	0.421569	−	0.906796i	$-0.361479\pi$
$500$		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	−0.836127	−	0.548535i	$-0.815186\pi$
$509$	−20.1495	−	34.8999i	−0.893109	−	1.54691i	−0.0569818	−	0.998375i	$-0.518148\pi$
$510$		0			0
$511$	20.5087	+	6.63484i	0.907252	+	0.293508i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−6.52490	−	3.76715i	−0.287522	−	0.166001i
$516$		0			0
$517$			27.3278i			1.20188i
$518$		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
$533$		−	8.20436i		−	0.355370i
$534$		0			0
$535$	0.246111	+	0.142092i	0.0106403	+	0.00614318i
$536$		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
$545$	−2.90125			−0.124276
$546$		0			0
$547$	−9.75377			−0.417041			−0.208521	−	0.978018i	$-0.566865\pi$
$547$	−9.75377			−0.417041			−0.208521	+	0.978018i	$0.566865\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.57700	+	7.92759i	0.194987	+	0.337727i
$552$		0			0
$553$	−9.17040	−	10.1587i	−0.389965	−	0.431993i
$554$		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
$563$	−14.1895	+	24.5769i	−0.598014	+	1.03579i	0.395100	+	0.918638i	$0.370710\pi$
$563$	−14.1895	+	24.5769i	−0.598014	+	1.03579i	−0.993114	+	0.117153i	$0.962623\pi$
$564$		0			0
$565$	0.434201	−	0.250686i	0.0182670	−	0.0105464i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.8968	+	11.4874i	−0.834118	+	0.481578i	−0.855260	−	0.518198i	$-0.826603\pi$
$569$	−19.8968	+	11.4874i	−0.834118	+	0.481578i	0.0211427	+	0.999776i	$0.493270\pi$
$570$		0			0
$571$	−4.50245	+	7.79848i	−0.188422	+	0.326356i	−0.944724	−	0.327866i	$-0.893671\pi$
$571$	−4.50245	+	7.79848i	−0.188422	+	0.326356i	0.756302	+	0.654222i	$0.227004\pi$
$572$		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
$581$	39.1661	−	8.37711i	1.62488	−	0.347541i
$582$		0			0
$583$	−18.2543	−	31.6173i	−0.756015	−	1.30946i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	26.6794			1.10118			0.550589	−	0.834776i	$-0.314403\pi$
$587$	26.6794			1.10118			0.550589	+	0.834776i	$0.314403\pi$
$588$		0			0
$589$	2.34104			0.0964608
$590$		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
$599$	−14.7019	−	8.48812i	−0.600702	−	0.346815i	0.168616	−	0.985682i	$-0.446070\pi$
$599$	−14.7019	−	8.48812i	−0.600702	−	0.346815i	−0.769318	+	0.638867i	$0.779404\pi$
$600$		0			0
$601$			16.7616i			0.683722i	0.939751	+	0.341861i	$0.111057\pi$
$601$			16.7616i			0.683722i	−0.939751	+	0.341861i	$0.888943\pi$
$602$		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	$-0.987048\pi$
$607$	−36.0575	+	20.8178i	−1.46353	+	0.844968i	−0.464356	+	0.885649i	$0.653714\pi$
$608$		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
$617$			42.7694i			1.72183i	0.508749	+	0.860915i	$0.330108\pi$
$617$			42.7694i			1.72183i	−0.508749	+	0.860915i	$0.669892\pi$
$618$		0			0
$619$	23.9473	+	13.8260i	0.962522	+	0.555712i	0.896948	−	0.442136i	$-0.145779\pi$
$619$	23.9473	+	13.8260i	0.962522	+	0.555712i	0.0655735	+	0.997848i	$0.479112\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−23.1516	−	25.6468i	−0.927550	−	1.02752i
$624$		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
$637$	−5.27159	−	11.7595i	−0.208868	−	0.465930i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−23.4181	−	13.5205i	−0.924960	−	0.534026i	−0.0397458	−	0.999210i	$-0.512655\pi$
$641$	−23.4181	−	13.5205i	−0.924960	−	0.534026i	−0.885214	+	0.465184i	$0.845988\pi$
$642$		0			0
$643$			12.1098i			0.477563i	0.971073	+	0.238781i	$0.0767479\pi$
$643$			12.1098i			0.477563i	−0.971073	+	0.238781i	$0.923252\pi$
$644$		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	0.558668	−	0.829392i	$-0.311313\pi$
$653$	39.7689	−	22.9606i	1.55628	−	0.898516i	0.997608	−	0.0691245i	$-0.0220206\pi$
$654$		0			0
$655$	−5.77908	+	10.0097i	−0.225807	+	0.391110i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	14.9910i		−	0.583965i	−0.956424	−	0.291982i	$-0.905685\pi$
$659$		−	14.9910i		−	0.583965i	0.956424	−	0.291982i	$-0.0943148\pi$
$660$		0			0
$661$	−28.6619	−	16.5479i	−1.11482	−	0.643640i	−0.174745	−	0.984614i	$-0.555910\pi$
$661$	−28.6619	−	16.5479i	−1.11482	−	0.643640i	−0.940073	+	0.340974i	$0.889243\pi$
$662$		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
$671$	5.34408			0.206306
$672$		0			0
$673$	37.2317			1.43518			0.717588	−	0.696468i	$-0.245246\pi$
$673$	37.2317			1.43518			0.717588	+	0.696468i	$0.245246\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.1462	+	34.8942i	0.774280	+	1.34109i	0.935198	+	0.354124i	$0.115221\pi$
$677$	20.1462	+	34.8942i	0.774280	+	1.34109i	−0.160919	+	0.986968i	$0.551446\pi$
$678$		0			0
$679$	0.921631	+	4.30896i	0.0353689	+	0.165363i
$680$		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
$689$	−12.6488	+	21.9084i	−0.481882	+	0.834644i
$690$		0			0
$691$	−32.6993	+	18.8789i	−1.24394	+	0.718188i	−0.969894	−	0.243528i	$-0.921695\pi$
$691$	−32.6993	+	18.8789i	−1.24394	+	0.718188i	−0.274045	+	0.961717i	$0.588362\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.0156	+	5.78251i	−0.379913	+	0.219343i
$696$		0			0
$697$	−1.19199	+	2.06458i	−0.0451498	+	0.0782017i
$698$		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
$707$	−2.02703	+	6.26568i	−0.0762344	+	0.235645i
$708$		0			0
$709$	−11.2777	−	19.5336i	−0.423544	−	0.733600i	0.572739	−	0.819738i	$-0.305881\pi$
$709$	−11.2777	−	19.5336i	−0.423544	−	0.733600i	−0.996283	+	0.0861375i	$0.972548\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.764619			−0.0286352
$714$		0			0
$715$	−5.54342			−0.207312
$716$		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
$725$	9.91898	+	5.72672i	0.368382	+	0.212685i
$726$		0			0
$727$			25.3388i			0.939765i	0.882729	+	0.469883i	$0.155704\pi$
$727$			25.3388i			0.939765i	−0.882729	+	0.469883i	$0.844296\pi$
$728$		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	$-0.644151\pi$
$733$	3.31425	−	1.91348i	0.122415	−	0.0706762i	0.559957	+	0.828522i	$0.310818\pi$
$734$		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
$743$			14.2719i			0.523585i	0.965124	+	0.261792i	$0.0843136\pi$
$743$			14.2719i			0.523585i	−0.965124	+	0.261792i	$0.915686\pi$
$744$		0			0
$745$	−14.9182	−	8.61305i	−0.546562	−	0.315558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.492462	+	0.444551i	−0.0179942	+	0.0162435i
$750$		0			0
$751$	−14.6005	−	25.2887i	−0.532778	−	0.922799i	−0.999267	−	0.0382721i	$-0.987815\pi$
$751$	−14.6005	−	25.2887i	−0.532778	−	0.922799i	0.466489	−	0.884527i	$-0.345519\pi$
$752$		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	0.969049	−	0.246867i	$-0.0794011\pi$
$761$	7.46846	+	12.9358i	0.270731	+	0.468921i	−0.698318	+	0.715788i	$0.746068\pi$
$762$		0			0
$763$	2.08478	−	6.44419i	0.0754742	−	0.233295i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	11.6337	+	6.71669i	0.420067	+	0.242526i
$768$		0			0
$769$			31.3736i			1.13136i	0.824624	+	0.565681i	$0.191387\pi$
$769$			31.3736i			1.13136i	−0.824624	+	0.565681i	$0.808613\pi$
$770$		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
$785$		−	21.1729i		−	0.755692i
$786$		0			0
$787$	34.9455	+	20.1758i	1.24567	+	0.719189i	0.970243	−	0.242132i	$-0.0778467\pi$
$787$	34.9455	+	20.1758i	1.24567	+	0.719189i	0.275429	+	0.961321i	$0.411180\pi$
$788$		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
$797$	−22.4242			−0.794304			−0.397152	−	0.917753i	$-0.630002\pi$
$797$	−22.4242			−0.794304			−0.397152	+	0.917753i	$0.630002\pi$
$798$		0			0
$799$	−5.50235			−0.194659
$800$		0			0
$801$		0			0
$802$		0			0
$803$	10.8229	+	18.7458i	0.381932	+	0.661526i
$804$		0			0
$805$	2.76710	+	0.895195i	0.0975275	+	0.0315515i
$806$		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
$815$	−12.2236	+	21.1719i	−0.428174	+	0.741619i
$816$		0			0
$817$	4.36927	−	2.52260i	0.152861	−	0.0882545i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−47.3134	+	27.3164i	−1.65125	+	0.953349i	−0.674688	+	0.738103i	$0.735722\pi$
$821$	−47.3134	+	27.3164i	−1.65125	+	0.953349i	−0.976560	+	0.215246i	$0.930945\pi$
$822$		0			0
$823$	5.88746	−	10.1974i	0.205224	−	0.355458i	−0.744980	−	0.667087i	$-0.767541\pi$
$823$	5.88746	−	10.1974i	0.205224	−	0.355458i	0.950204	+	0.311628i	$0.100874\pi$
$824$		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
$833$	−0.381944	+	3.72512i	−0.0132336	+	0.129068i
$834$		0			0
$835$	7.25107	+	12.5592i	0.250934	+	0.434630i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	45.3308			1.56499			0.782497	−	0.622655i	$-0.213946\pi$
$839$	45.3308			1.56499			0.782497	+	0.622655i	$0.213946\pi$
$840$		0			0
$841$	19.4980			0.672344
$842$		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
$851$	2.91164	+	1.68104i	0.0998097	+	0.0576252i
$852$		0			0
$853$			24.0644i			0.823949i	0.911195	+	0.411974i	$0.135161\pi$
$853$			24.0644i			0.823949i	−0.911195	+	0.411974i	$0.864839\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	25.3869	−	43.9714i	0.867200	−	1.50203i	0.00235426	−	0.999997i	$-0.499251\pi$
$857$	25.3869	−	43.9714i	0.867200	−	1.50203i	0.864846	−	0.502037i	$-0.167416\pi$
$858$		0			0
$859$	6.78222	−	3.91572i	0.231406	−	0.133603i	−0.379814	−	0.925063i	$-0.624012\pi$
$859$	6.78222	−	3.91572i	0.231406	−	0.133603i	0.611221	+	0.791460i	$0.290679\pi$
$860$		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
$869$		−	13.7431i		−	0.466204i
$870$		0			0
$871$	−3.84704	−	2.22109i	−0.130352	−	0.0752588i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25.5555	+	5.46598i	−0.863932	+	0.184784i
$876$		0			0
$877$	−17.0535	−	29.5375i	−0.575855	−	0.997411i	−0.995948	−	0.0899294i	$-0.971336\pi$
$877$	−17.0535	−	29.5375i	−0.575855	−	0.997411i	0.420093	−	0.907481i	$-0.361997\pi$
$878$		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	−0.990066	−	0.140605i	$-0.955095\pi$
$887$	−18.3699	−	31.8176i	−0.616801	−	1.06833i	0.373265	−	0.927725i	$-0.378238\pi$
$888$		0			0
$889$	45.5769	−	9.74829i	1.52860	−	0.326947i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26.4526	+	15.2724i	0.885202	+	0.511071i
$894$		0			0
$895$		−	5.71815i		−	0.191137i
$896$		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	−0.693968	−	0.720006i	$-0.744139\pi$
$907$	8.32897	−	14.4262i	0.276559	−	0.479014i	0.970527	+	0.240991i	$0.0774726\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			33.5975i			1.11314i	0.830802	+	0.556568i	$0.187882\pi$
$911$			33.5975i			1.11314i	−0.830802	+	0.556568i	$0.812118\pi$
$912$		0			0
$913$	34.8317	+	20.1101i	1.15276	+	0.665548i
$914$		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
$923$	−22.8082			−0.750741
$924$		0			0
$925$	−12.8794			−0.423472
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21.8590	+	37.8609i	0.717170	+	1.24217i	0.962117	+	0.272638i	$0.0878961\pi$
$929$	21.8590	+	37.8609i	0.717170	+	1.24217i	−0.244947	+	0.969536i	$0.578771\pi$
$930$		0			0
$931$	12.1757	−	16.8484i	0.399043	−	0.552185i
$932$		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
$941$	11.0516	−	19.1420i	0.360273	−	0.624011i	−0.627733	−	0.778429i	$-0.716017\pi$
$941$	11.0516	−	19.1420i	0.360273	−	0.624011i	0.988006	+	0.154418i	$0.0493502\pi$
$942$		0			0
$943$	−3.74333	+	2.16121i	−0.121900	+	0.0703788i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.0288642	−	0.0166647i	0.000937959	−	0.000541531i	−0.499531	−	0.866296i	$-0.666494\pi$
$947$	0.0288642	−	0.0166647i	0.000937959	−	0.000541531i	0.500469	+	0.865755i	$0.333161\pi$
$948$		0			0
$949$	7.49946	−	12.9894i	0.243443	−	0.421655i
$950$		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
$959$	21.1731	+	6.84977i	0.683714	+	0.221191i
$960$		0			0
$961$	−15.1893	−	26.3086i	−0.489976	−	0.848664i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.26767			−0.0408079
$966$		0			0
$967$	52.9254			1.70197			0.850983	−	0.525193i	$-0.176007\pi$
$967$	52.9254			1.70197			0.850983	+	0.525193i	$0.176007\pi$
$968$		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
$977$	−25.2685	−	14.5888i	−0.808412	−	0.466737i	0.0379920	−	0.999278i	$-0.487904\pi$
$977$	−25.2685	−	14.5888i	−0.808412	−	0.466737i	−0.846404	+	0.532541i	$0.821237\pi$
$978$		0			0
$979$		−	34.6959i		−	1.10889i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.40545	−	14.5587i	0.268092	−	0.464349i	−0.700277	−	0.713871i	$-0.746940\pi$
$983$	8.40545	−	14.5587i	0.268092	−	0.464349i	0.968369	+	0.249522i	$0.0802735\pi$
$984$		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
$995$		−	0.453324i		−	0.0143713i
$996$		0			0
$997$	15.4850	+	8.94029i	0.490416	+	0.283142i	0.724747	−	0.689015i	$-0.241957\pi$
$997$	15.4850	+	8.94029i	0.490416	+	0.283142i	−0.234331	+	0.972157i	$0.575290\pi$
$998$		0			0
$999$		0			0

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-0.566658 - 0.981481i) q^{5} +(2.58723 - 0.553375i) q^{7} +(2.30092 + 1.32843i) q^{11} -1.84101i q^{13} +(-0.267475 + 0.463281i) q^{17} +(2.57178 - 1.48482i) q^{19} +(-0.839982 + 0.484964i) q^{23} +(1.85780 - 3.21780i) q^{25} +3.08254i q^{29} +(0.682710 + 0.394163i) q^{31} +(-2.00920 - 2.22575i) q^{35} +(-1.73316 - 3.00191i) q^{37} +4.45644 q^{41} +1.69893 q^{43} +(5.14286 + 8.90769i) q^{47} +(6.38755 - 2.86342i) q^{49} +(-11.9002 - 6.87059i) q^{53} -3.01107i q^{55} +(-3.64838 + 6.31917i) q^{59} +(1.74194 - 1.00571i) q^{61} +(-1.80692 + 1.04322i) q^{65} +(1.20645 - 2.08964i) q^{67} -12.3890i q^{71} +(7.05561 + 4.07356i) q^{73} +(6.68813 + 2.16370i) q^{77} +(-2.58634 - 4.47967i) q^{79} +15.1382 q^{83} +0.606268 q^{85} +(-6.52948 - 11.3094i) q^{89} +(-1.01877 - 4.76312i) q^{91} +(-2.91464 - 1.68277i) q^{95} +1.66547i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) \) 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more