LMFDB - Embedded newform 2268.2.w.j.269.10

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)

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);

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")

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			269.10
Character	$\chi$	$=$	2268.269
Dual form			2268.2.w.j.1349.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.440135 - 0.762336i) q^{5} +(-2.46165 - 0.969691i) q^{7} +(-4.83660 + 2.79241i) q^{11} +(-2.24806 + 1.29792i) q^{13} +(2.35671 - 4.08193i) q^{17} +(0.537797 - 0.310497i) q^{19} +(3.15847 + 1.82354i) q^{23} +(2.11256 + 3.65906i) q^{25} +(4.62362 + 2.66945i) q^{29} -11.0393i q^{31} +(-1.82269 + 1.44981i) q^{35} +(4.28064 + 7.41428i) q^{37} +(3.41348 + 5.91232i) q^{41} +(-1.82931 + 3.16846i) q^{43} +7.77378 q^{47} +(5.11940 + 4.77407i) q^{49} +(5.87725 + 3.39323i) q^{53} +4.91616i q^{55} +10.7125 q^{59} -2.05202i q^{61} +2.28504i q^{65} +9.04199 q^{67} -12.3011i q^{71} +(-3.43707 - 1.98439i) q^{73} +(14.6138 - 2.18393i) q^{77} -12.0209 q^{79} +(-1.71972 + 2.97864i) q^{83} +(-2.07454 - 3.59321i) q^{85} +(3.40619 + 5.89969i) q^{89} +(6.79251 - 1.01509i) q^{91} -0.546643i q^{95} +(-14.0453 - 8.10908i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.440135	−	0.762336i	0.196834	−	0.340927i	−0.750666	−	0.660682i	$-0.770267\pi$
$5$	0.440135	−	0.762336i	0.196834	−	0.340927i	0.947500	+	0.319755i	$0.103601\pi$
$6$		0			0
$7$	−2.46165	−	0.969691i	−0.930415	−	0.366509i
$7$	−2.46165	−	0.969691i	−0.930415	−	0.366509i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.83660	+	2.79241i	−1.45829	+	0.841944i	−0.998927	−	0.0463050i	$-0.985255\pi$
$11$	−4.83660	+	2.79241i	−1.45829	+	0.841944i	−0.459362	+	0.888249i	$0.651922\pi$
$12$		0			0
$13$	−2.24806	+	1.29792i	−0.623500	+	0.359978i	−0.778230	−	0.627979i	$-0.783882\pi$
$13$	−2.24806	+	1.29792i	−0.623500	+	0.359978i	0.154730	+	0.987957i	$0.450549\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.35671	−	4.08193i	0.571585	−	0.990015i	−0.424818	−	0.905279i	$-0.639662\pi$
$17$	2.35671	−	4.08193i	0.571585	−	0.990015i	0.996403	−	0.0847359i	$-0.0270047\pi$
$18$		0			0
$19$	0.537797	−	0.310497i	0.123379	−	0.0712330i	−0.437040	−	0.899442i	$-0.643973\pi$
$19$	0.537797	−	0.310497i	0.123379	−	0.0712330i	0.560419	+	0.828209i	$0.310640\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.15847	+	1.82354i	0.658586	+	0.380235i	0.791738	−	0.610861i	$-0.209177\pi$
$23$	3.15847	+	1.82354i	0.658586	+	0.380235i	−0.133152	+	0.991096i	$0.542510\pi$
$24$		0			0
$25$	2.11256	+	3.65906i	0.422512	+	0.731813i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.62362	+	2.66945i	0.858586	+	0.495705i	0.863538	−	0.504283i	$-0.168243\pi$
$29$	4.62362	+	2.66945i	0.858586	+	0.495705i	−0.00495283	+	0.999988i	$0.501577\pi$
$30$		0			0
$31$		−	11.0393i		−	1.98272i	−0.131158	−	0.991361i	$-0.541870\pi$
$31$		−	11.0393i		−	1.98272i	0.131158	−	0.991361i	$-0.458130\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.82269	+	1.44981i	−0.308090	+	0.245062i
$36$		0			0
$37$	4.28064	+	7.41428i	0.703732	+	1.21890i	0.967147	+	0.254218i	$0.0818179\pi$
$37$	4.28064	+	7.41428i	0.703732	+	1.21890i	−0.263415	+	0.964683i	$0.584849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.41348	+	5.91232i	0.533096	+	0.923350i	0.999253	+	0.0386476i	$0.0123050\pi$
$41$	3.41348	+	5.91232i	0.533096	+	0.923350i	−0.466157	+	0.884702i	$0.654362\pi$
$42$		0			0
$43$	−1.82931	+	3.16846i	−0.278967	+	0.483186i	−0.971128	−	0.238557i	$-0.923325\pi$
$43$	−1.82931	+	3.16846i	−0.278967	+	0.483186i	0.692161	+	0.721743i	$0.256659\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.77378			1.13392			0.566961	−	0.823745i	$-0.308119\pi$
$47$	7.77378			1.13392			0.566961	+	0.823745i	$0.308119\pi$
$48$		0			0
$49$	5.11940	+	4.77407i	0.731343	+	0.682010i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.87725	+	3.39323i	0.807303	+	0.466096i	0.846018	−	0.533154i	$-0.178993\pi$
$53$	5.87725	+	3.39323i	0.807303	+	0.466096i	−0.0387156	+	0.999250i	$0.512327\pi$
$54$		0			0
$55$			4.91616i			0.662894i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	10.7125			1.39465			0.697326	−	0.716754i	$-0.254373\pi$
$59$	10.7125			1.39465			0.697326	+	0.716754i	$0.254373\pi$
$60$		0			0
$61$		−	2.05202i		−	0.262734i	−0.991334	−	0.131367i	$-0.958063\pi$
$61$		−	2.05202i		−	0.262734i	0.991334	−	0.131367i	$-0.0419366\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			2.28504i			0.283424i
$66$		0			0
$67$	9.04199			1.10465			0.552327	−	0.833627i	$-0.313740\pi$
$67$	9.04199			1.10465			0.552327	+	0.833627i	$0.313740\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	12.3011i		−	1.45988i	−0.683514	−	0.729938i	$-0.739549\pi$
$71$		−	12.3011i		−	1.45988i	0.683514	−	0.729938i	$-0.260451\pi$
$72$		0			0
$73$	−3.43707	−	1.98439i	−0.402278	−	0.232255i	0.285188	−	0.958471i	$-0.407944\pi$
$73$	−3.43707	−	1.98439i	−0.402278	−	0.232255i	−0.687466	+	0.726216i	$0.741277\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	14.6138	−	2.18393i	1.66539	−	0.248881i
$78$		0			0
$79$	−12.0209			−1.35246			−0.676228	−	0.736692i	$-0.736387\pi$
$79$	−12.0209			−1.35246			−0.676228	+	0.736692i	$0.736387\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.71972	+	2.97864i	−0.188764	+	0.326948i	−0.944838	−	0.327537i	$-0.893781\pi$
$83$	−1.71972	+	2.97864i	−0.188764	+	0.326948i	0.756075	+	0.654485i	$0.227115\pi$
$84$		0			0
$85$	−2.07454	−	3.59321i	−0.225015	−	0.389738i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.40619	+	5.89969i	0.361055	+	0.625366i	0.988135	−	0.153589i	$-0.0490833\pi$
$89$	3.40619	+	5.89969i	0.361055	+	0.625366i	−0.627080	+	0.778955i	$0.715750\pi$
$90$		0			0
$91$	6.79251	−	1.01509i	0.712049	−	0.106411i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	0.546643i		−	0.0560844i
$96$		0			0
$97$	−14.0453	−	8.10908i	−1.42609	−	0.823352i	−0.429279	−	0.903172i	$-0.641232\pi$
$97$	−14.0453	−	8.10908i	−1.42609	−	0.823352i	−0.996809	+	0.0798195i	$0.974566\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.75727	+	11.7039i	0.672373	+	1.16458i	0.977229	+	0.212186i	$0.0680583\pi$
$101$	6.75727	+	11.7039i	0.672373	+	1.16458i	−0.304856	+	0.952398i	$0.598608\pi$
$102$		0			0
$103$	−10.9579	−	6.32656i	−1.07972	−	0.623374i	−0.148896	−	0.988853i	$-0.547572\pi$
$103$	−10.9579	−	6.32656i	−1.07972	−	0.623374i	−0.930820	+	0.365479i	$0.880905\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.09662	+	1.21048i	−0.202688	+	0.117022i	−0.597909	−	0.801564i	$-0.704001\pi$
$107$	−2.09662	+	1.21048i	−0.202688	+	0.117022i	0.395221	+	0.918586i	$0.370668\pi$
$108$		0			0
$109$	0.286415	−	0.496086i	0.0274336	−	0.0475164i	−0.851983	−	0.523570i	$-0.824600\pi$
$109$	0.286415	−	0.496086i	0.0274336	−	0.0475164i	0.879416	+	0.476054i	$0.157933\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.18260	−	5.30158i	0.863827	−	0.498731i	−0.00146526	−	0.999999i	$-0.500466\pi$
$113$	9.18260	−	5.30158i	0.863827	−	0.498731i	0.865292	+	0.501268i	$0.167133\pi$
$114$		0			0
$115$	2.78030	−	1.60521i	0.259265	−	0.149687i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.75959	+	7.76300i	−0.894660	+	0.711633i
$120$		0			0
$121$	10.0951	−	17.4853i	0.917739	−	1.58957i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.12060			0.726329
$126$		0			0
$127$	2.85582			0.253413			0.126707	−	0.991940i	$-0.459559\pi$
$127$	2.85582			0.253413			0.126707	+	0.991940i	$0.459559\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.80788	+	10.0595i	−0.507437	+	0.878906i	0.492526	+	0.870298i	$0.336074\pi$
$131$	−5.80788	+	10.0595i	−0.507437	+	0.878906i	−0.999963	+	0.00860838i	$0.997260\pi$
$132$		0			0
$133$	−1.62495	+	0.242838i	−0.140901	+	0.0210567i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.41744	−	3.70511i	0.548279	−	0.316549i	−0.200149	−	0.979766i	$-0.564143\pi$
$137$	6.41744	−	3.70511i	0.548279	−	0.316549i	0.748427	+	0.663217i	$0.230809\pi$
$138$		0			0
$139$	−2.95857	+	1.70813i	−0.250943	+	0.144882i	−0.620196	−	0.784447i	$-0.712947\pi$
$139$	−2.95857	+	1.70813i	−0.250943	+	0.144882i	0.369253	+	0.929329i	$0.379614\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.24865	−	12.5550i	0.606162	−	1.04990i
$144$		0			0
$145$	4.07004	−	2.34984i	0.337998	−	0.195143i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.81086	+	5.08695i	0.721814	+	0.416739i	0.815420	−	0.578870i	$-0.196506\pi$
$149$	8.81086	+	5.08695i	0.721814	+	0.416739i	−0.0936062	+	0.995609i	$0.529839\pi$
$150$		0			0
$151$	2.15352	+	3.73001i	0.175251	+	0.303544i	0.940248	−	0.340490i	$-0.110593\pi$
$151$	2.15352	+	3.73001i	0.175251	+	0.303544i	−0.764997	+	0.644034i	$0.777260\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.41569	−	4.85880i	−0.675964	−	0.390268i
$156$		0			0
$157$			11.7964i			0.941459i	0.882278	+	0.470729i	$0.156009\pi$
$157$			11.7964i			0.941459i	−0.882278	+	0.470729i	$0.843991\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.00675	−	7.55165i	−0.473399	−	0.595153i
$162$		0			0
$163$	5.81021	+	10.0636i	0.455090	+	0.788240i	0.998693	−	0.0511028i	$-0.0162736\pi$
$163$	5.81021	+	10.0636i	0.455090	+	0.788240i	−0.543603	+	0.839342i	$0.682940\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.88538	+	13.6579i	0.610189	+	1.05688i	0.991208	+	0.132311i	$0.0422398\pi$
$167$	7.88538	+	13.6579i	0.610189	+	1.05688i	−0.381019	+	0.924567i	$0.624427\pi$
$168$		0			0
$169$	−3.13081	+	5.42273i	−0.240832	+	0.417133i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.77341			0.210858			0.105429	−	0.994427i	$-0.466378\pi$
$173$	2.77341			0.210858			0.105429	+	0.994427i	$0.466378\pi$
$174$		0			0
$175$	−1.65222	−	11.0559i	−0.124896	−	0.835744i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.94403	−	3.43179i	−0.444278	−	0.256504i	0.261133	−	0.965303i	$-0.415904\pi$
$179$	−5.94403	−	3.43179i	−0.444278	−	0.256504i	−0.705410	+	0.708799i	$0.749237\pi$
$180$		0			0
$181$		−	9.85094i		−	0.732215i	−0.930573	−	0.366107i	$-0.880690\pi$
$181$		−	9.85094i		−	0.732215i	0.930573	−	0.366107i	$-0.119310\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.53624			0.554075
$186$		0			0
$187$			26.3236i			1.92497i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			17.9768i			1.30076i	0.759609	+	0.650380i	$0.225390\pi$
$191$			17.9768i			1.30076i	−0.759609	+	0.650380i	$0.774610\pi$
$192$		0			0
$193$	−3.76079			−0.270708			−0.135354	−	0.990797i	$-0.543217\pi$
$193$	−3.76079			−0.270708			−0.135354	+	0.990797i	$0.543217\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.88342i			0.490424i	0.969470	+	0.245212i	$0.0788575\pi$
$197$			6.88342i			0.490424i	−0.969470	+	0.245212i	$0.921142\pi$
$198$		0			0
$199$	12.0894	+	6.97981i	0.856993	+	0.494785i	0.863004	−	0.505197i	$-0.168580\pi$
$199$	12.0894	+	6.97981i	0.856993	+	0.494785i	−0.00601088	+	0.999982i	$0.501913\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.79319	−	11.0547i	−0.617161	−	0.775890i
$204$		0			0
$205$	6.00957			0.419727
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.73407	+	3.00350i	−0.119948	+	0.207757i
$210$		0			0
$211$	1.26047	+	2.18320i	0.0867744	+	0.150298i	0.906146	−	0.422965i	$-0.139011\pi$
$211$	1.26047	+	2.18320i	0.0867744	+	0.150298i	−0.819372	+	0.573263i	$0.805677\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.61029	+	2.78910i	0.109821	+	0.190215i
$216$		0			0
$217$	−10.7047	+	27.1749i	−0.726685	+	1.84475i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			12.2353i			0.823032i
$222$		0			0
$223$	10.6650	+	6.15742i	0.714179	+	0.412332i	0.812607	−	0.582813i	$-0.198048\pi$
$223$	10.6650	+	6.15742i	0.714179	+	0.412332i	−0.0984273	+	0.995144i	$0.531381\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.8806	−	22.3098i	−0.854913	−	1.48075i	−0.876726	−	0.480991i	$-0.840277\pi$
$227$	−12.8806	−	22.3098i	−0.854913	−	1.48075i	0.0218124	−	0.999762i	$-0.493056\pi$
$228$		0			0
$229$	5.01900	+	2.89772i	0.331665	+	0.191487i	0.656580	−	0.754256i	$-0.272002\pi$
$229$	5.01900	+	2.89772i	0.331665	+	0.191487i	−0.324915	+	0.945743i	$0.605336\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26.2280	+	15.1428i	−1.71826	+	0.992036i	−0.796149	+	0.605101i	$0.793133\pi$
$233$	−26.2280	+	15.1428i	−1.71826	+	0.992036i	−0.922107	+	0.386935i	$0.873534\pi$
$234$		0			0
$235$	3.42151	−	5.92623i	0.223195	−	0.386585i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.96081	−	1.70942i	0.191519	−	0.110573i	−0.401175	−	0.916002i	$-0.631398\pi$
$239$	2.96081	−	1.70942i	0.191519	−	0.110573i	0.592693	+	0.805428i	$0.298065\pi$
$240$		0			0
$241$	4.24030	−	2.44814i	0.273142	−	0.157699i	−0.357173	−	0.934038i	$-0.616259\pi$
$241$	4.24030	−	2.44814i	0.273142	−	0.157699i	0.630315	+	0.776340i	$0.282926\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.89268	−	1.80147i	0.376469	−	0.115092i
$246$		0			0
$247$	−0.806001	+	1.39603i	−0.0512846	+	0.0888275i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.2710			1.72133			0.860667	−	0.509169i	$-0.170047\pi$
$251$	27.2710			1.72133			0.860667	+	0.509169i	$0.170047\pi$
$252$		0			0
$253$	−20.3683			−1.28054
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.56317	−	4.43954i	0.159886	−	0.276931i	−0.774941	−	0.632033i	$-0.782221\pi$
$257$	2.56317	−	4.43954i	0.159886	−	0.276931i	0.934827	+	0.355102i	$0.115554\pi$
$258$		0			0
$259$	−3.34785	−	22.4022i	−0.208026	−	1.39201i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.13580	−	2.38780i	0.255024	−	0.147238i	−0.367038	−	0.930206i	$-0.619628\pi$
$263$	4.13580	−	2.38780i	0.255024	−	0.147238i	0.622063	+	0.782967i	$0.286295\pi$
$264$		0			0
$265$	5.17357	−	2.98696i	0.317810	−	0.183488i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.34405	−	12.7203i	0.447775	−	0.775569i	−0.550466	−	0.834858i	$-0.685550\pi$
$269$	7.34405	−	12.7203i	0.447775	−	0.775569i	0.998241	+	0.0592889i	$0.0188833\pi$
$270$		0			0
$271$	9.13087	−	5.27171i	0.554661	−	0.320233i	−0.196339	−	0.980536i	$-0.562905\pi$
$271$	9.13087	−	5.27171i	0.554661	−	0.320233i	0.751000	+	0.660303i	$0.229572\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−20.4352	−	11.7983i	−1.23229	−	0.711464i
$276$		0			0
$277$	−13.7907	−	23.8862i	−0.828602	−	1.43518i	−0.899135	−	0.437672i	$-0.855803\pi$
$277$	−13.7907	−	23.8862i	−0.828602	−	1.43518i	0.0705327	−	0.997509i	$-0.477530\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.7563	+	14.2930i	1.47684	+	0.852652i	0.999658	−	0.0261546i	$-0.00832621\pi$
$281$	24.7563	+	14.2930i	1.47684	+	0.852652i	0.477178	+	0.878806i	$0.341660\pi$
$282$		0			0
$283$			18.8385i			1.11983i	0.828549	+	0.559917i	$0.189167\pi$
$283$			18.8385i			1.11983i	−0.828549	+	0.559917i	$0.810833\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.66966	−	17.8641i	−0.157585	−	1.05448i
$288$		0			0
$289$	−2.60813	−	4.51741i	−0.153419	−	0.265730i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.8083	−	29.1129i	−0.981954	−	1.70079i	−0.654760	−	0.755837i	$-0.727230\pi$
$293$	−16.8083	−	29.1129i	−0.981954	−	1.70079i	−0.327193	−	0.944957i	$-0.606103\pi$
$294$		0			0
$295$	4.71496	−	8.16655i	0.274516	−	0.475475i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.46723			−0.547504
$300$		0			0
$301$	7.57554	−	6.02576i	0.436647	−	0.347319i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.56433	−	0.903164i	−0.0895731	−	0.0517150i
$306$		0			0
$307$		−	14.6814i		−	0.837913i	−0.908006	−	0.418957i	$-0.862396\pi$
$307$		−	14.6814i		−	0.837913i	0.908006	−	0.418957i	$-0.137604\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.9147			0.675620			0.337810	−	0.941214i	$-0.390314\pi$
$311$	11.9147			0.675620			0.337810	+	0.941214i	$0.390314\pi$
$312$		0			0
$313$		−	6.16244i		−	0.348322i	−0.984717	−	0.174161i	$-0.944279\pi$
$313$		−	6.16244i		−	0.348322i	0.984717	−	0.174161i	$-0.0557213\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			3.84953i			0.216211i	0.994139	+	0.108105i	$0.0344784\pi$
$317$			3.84953i			0.216211i	−0.994139	+	0.108105i	$0.965522\pi$
$318$		0			0
$319$	−29.8168			−1.66942
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.92700i		−	0.162863i
$324$		0			0
$325$	−9.49834	−	5.48387i	−0.526873	−	0.304190i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−19.1363	−	7.53816i	−1.05502	−	0.415592i
$330$		0			0
$331$	−17.6316			−0.969118			−0.484559	−	0.874759i	$-0.661020\pi$
$331$	−17.6316			−0.969118			−0.484559	+	0.874759i	$0.661020\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.97970	−	6.89304i	0.217434	−	0.376607i
$336$		0			0
$337$	9.83450	+	17.0339i	0.535720	+	0.927894i	0.999128	+	0.0417488i	$0.0132929\pi$
$337$	9.83450	+	17.0339i	0.535720	+	0.927894i	−0.463409	+	0.886145i	$0.653374\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	30.8264	+	53.3928i	1.66934	+	2.89138i
$342$		0			0
$343$	−7.97278	−	16.7163i	−0.430490	−	0.902595i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			14.5284i			0.779923i	0.920831	+	0.389962i	$0.127512\pi$
$347$			14.5284i			0.779923i	−0.920831	+	0.389962i	$0.872488\pi$
$348$		0			0
$349$	26.9608	+	15.5658i	1.44318	+	0.833218i	0.998060	−	0.0622544i	$-0.0198290\pi$
$349$	26.9608	+	15.5658i	1.44318	+	0.833218i	0.445116	+	0.895473i	$0.353162\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.11432	−	7.12621i	−0.218983	−	0.379290i	0.735514	−	0.677509i	$-0.236941\pi$
$353$	−4.11432	−	7.12621i	−0.218983	−	0.379290i	−0.954497	+	0.298219i	$0.903607\pi$
$354$		0			0
$355$	−9.37760	−	5.41416i	−0.497711	−	0.287354i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.3084	+	14.6118i	−1.33573	+	0.771183i	−0.986171	−	0.165732i	$-0.947001\pi$
$359$	−25.3084	+	14.6118i	−1.33573	+	0.771183i	−0.349557	+	0.936915i	$0.613668\pi$
$360$		0			0
$361$	−9.30718	+	16.1205i	−0.489852	+	0.848448i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.02555	+	1.74680i	−0.158364	+	0.0914317i
$366$		0			0
$367$	−0.491228	+	0.283610i	−0.0256419	+	0.0148043i	−0.512766	−	0.858528i	$-0.671379\pi$
$367$	−0.491228	+	0.283610i	−0.0256419	+	0.0148043i	0.487124	+	0.873333i	$0.338046\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.1773	−	14.0521i	−0.580298	−	0.729546i
$372$		0			0
$373$	−0.901927	+	1.56218i	−0.0467000	+	0.0808868i	−0.888431	−	0.459011i	$-0.848204\pi$
$373$	−0.901927	+	1.56218i	−0.0467000	+	0.0808868i	0.841731	+	0.539898i	$0.181537\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13.8589			−0.713771
$378$		0			0
$379$	−2.38270			−0.122391			−0.0611956	−	0.998126i	$-0.519491\pi$
$379$	−2.38270			−0.122391			−0.0611956	+	0.998126i	$0.519491\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.6264	+	20.1375i	−0.594080	+	1.02898i	0.399596	+	0.916691i	$0.369150\pi$
$383$	−11.6264	+	20.1375i	−0.594080	+	1.02898i	−0.993676	+	0.112286i	$0.964183\pi$
$384$		0			0
$385$	4.76715	−	12.1018i	0.242956	−	0.616767i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.2236	−	10.5214i	0.923976	−	0.533458i	0.0390745	−	0.999236i	$-0.487559\pi$
$389$	18.2236	−	10.5214i	0.923976	−	0.533458i	0.884901	+	0.465779i	$0.154226\pi$
$390$		0			0
$391$	14.8872	−	8.59510i	0.752876	−	0.434673i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.29082	+	9.16397i	−0.266210	+	0.461089i
$396$		0			0
$397$	23.7334	−	13.7025i	1.19114	−	0.687708i	0.232578	−	0.972578i	$-0.425284\pi$
$397$	23.7334	−	13.7025i	1.19114	−	0.687708i	0.958566	+	0.284870i	$0.0919505\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.8620	−	18.3955i	−1.59111	−	0.918629i	−0.993117	−	0.117129i	$-0.962631\pi$
$401$	−31.8620	−	18.3955i	−1.59111	−	0.918629i	−0.597995	−	0.801500i	$-0.704036\pi$
$402$		0			0
$403$	14.3282	+	24.8171i	0.713736	+	1.23623i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−41.4075	−	23.9066i	−2.05249	−	1.18501i
$408$		0			0
$409$			33.6694i			1.66484i	0.554142	+	0.832422i	$0.313046\pi$
$409$			33.6694i			1.66484i	−0.554142	+	0.832422i	$0.686954\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−26.3705	−	10.3878i	−1.29761	−	0.511152i
$414$		0			0
$415$	1.51382	+	2.62201i	0.0743104	+	0.128709i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.59691	+	7.96209i	0.224574	+	0.388973i	0.956192	−	0.292742i	$-0.0945677\pi$
$419$	4.59691	+	7.96209i	0.224574	+	0.388973i	−0.731618	+	0.681715i	$0.761234\pi$
$420$		0			0
$421$	4.30373	−	7.45427i	0.209751	−	0.363299i	−0.741885	−	0.670527i	$-0.766068\pi$
$421$	4.30373	−	7.45427i	0.209751	−	0.363299i	0.951636	+	0.307228i	$0.0994014\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	19.9148			0.966007
$426$		0			0
$427$	−1.98982	+	5.05134i	−0.0962942	+	0.244451i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.90554	+	3.98691i	0.332628	+	0.192043i	0.657007	−	0.753884i	$-0.271822\pi$
$431$	6.90554	+	3.98691i	0.332628	+	0.192043i	−0.324379	+	0.945927i	$0.605155\pi$
$432$		0			0
$433$		−	1.42915i		−	0.0686806i	−0.999410	−	0.0343403i	$-0.989067\pi$
$433$		−	1.42915i		−	0.0686806i	0.999410	−	0.0343403i	$-0.0109330\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.26482			0.108341
$438$		0			0
$439$			25.4827i			1.21622i	0.793852	+	0.608110i	$0.208072\pi$
$439$			25.4827i			1.21622i	−0.793852	+	0.608110i	$0.791928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	5.56489i		−	0.264396i	−0.991223	−	0.132198i	$-0.957797\pi$
$443$		−	5.56489i		−	0.264396i	0.991223	−	0.132198i	$-0.0422035\pi$
$444$		0			0
$445$	5.99673			0.284272
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	26.4132i		−	1.24652i	−0.782016	−	0.623259i	$-0.785808\pi$
$449$		−	26.4132i		−	1.24652i	0.782016	−	0.623259i	$-0.214192\pi$
$450$		0			0
$451$	−33.0193	−	19.0637i	−1.55482	−	0.897674i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.21578	−	5.62496i	0.103877	−	0.263702i
$456$		0			0
$457$	22.3618			1.04604			0.523021	−	0.852320i	$-0.324805\pi$
$457$	22.3618			1.04604			0.523021	+	0.852320i	$0.324805\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.2503	−	22.9501i	0.617126	−	1.06889i	−0.372881	−	0.927879i	$-0.621630\pi$
$461$	13.2503	−	22.9501i	0.617126	−	1.06889i	0.990007	−	0.141015i	$-0.0450365\pi$
$462$		0			0
$463$	7.24376	+	12.5466i	0.336646	+	0.583088i	0.983800	−	0.179271i	$-0.0573740\pi$
$463$	7.24376	+	12.5466i	0.336646	+	0.583088i	−0.647153	+	0.762360i	$0.724041\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.21034	−	12.4887i	−0.333655	−	0.577907i	0.649571	−	0.760301i	$-0.274949\pi$
$467$	−7.21034	−	12.4887i	−0.333655	−	0.577907i	−0.983226	+	0.182394i	$0.941615\pi$
$468$		0			0
$469$	−22.2582	−	8.76793i	−1.02779	−	0.404865i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	20.4328i		−	0.939500i
$474$		0			0
$475$	2.27226	+	1.31189i	0.104258	+	0.0601936i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.6311	−	25.3418i	−0.668513	−	1.15790i	−0.978320	−	0.207099i	$-0.933598\pi$
$479$	−14.6311	−	25.3418i	−0.668513	−	1.15790i	0.309807	−	0.950799i	$-0.399736\pi$
$480$		0			0
$481$	−19.2463	−	11.1118i	−0.877554	−	0.506656i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.3637	+	7.13818i	−0.561407	+	0.324128i
$486$		0			0
$487$	−3.23843	+	5.60913i	−0.146747	+	0.254174i	−0.930024	−	0.367500i	$-0.880214\pi$
$487$	−3.23843	+	5.60913i	−0.146747	+	0.254174i	0.783276	+	0.621674i	$0.213547\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.7683	+	17.7641i	−1.38856	+	0.801683i	−0.993153	−	0.116825i	$-0.962728\pi$
$491$	−30.7683	+	17.7641i	−1.38856	+	0.801683i	−0.395403	+	0.918508i	$0.629395\pi$
$492$		0			0
$493$	21.7930	−	12.5822i	0.981510	−	0.566675i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.9283	+	30.2810i	−0.535057	+	1.35829i
$498$		0			0
$499$	16.0108	−	27.7315i	0.716742	−	1.24143i	−0.245542	−	0.969386i	$-0.578966\pi$
$499$	16.0108	−	27.7315i	0.716742	−	1.24143i	0.962284	−	0.272047i	$-0.0877006\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−11.2012			−0.499437			−0.249718	−	0.968319i	$-0.580338\pi$
$503$	−11.2012			−0.499437			−0.249718	+	0.968319i	$0.580338\pi$
$504$		0			0
$505$	11.8964			0.529385
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.9860	+	32.8848i	−0.841541	+	1.45759i	0.0470504	+	0.998893i	$0.485018\pi$
$509$	−18.9860	+	32.8848i	−0.841541	+	1.45759i	−0.888591	+	0.458699i	$0.848315\pi$
$510$		0			0
$511$	6.53659	+	8.21776i	0.289162	+	0.363532i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.64593	+	5.56908i	−0.425050	+	0.245403i
$516$		0			0
$517$	−37.5987	+	21.7076i	−1.65359	+	0.954699i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.318682	−	0.551974i	0.0139617	−	0.0241824i	−0.858960	−	0.512042i	$-0.828889\pi$
$521$	0.318682	−	0.551974i	0.0139617	−	0.0241824i	0.872922	+	0.487860i	$0.162222\pi$
$522$		0			0
$523$	15.4557	−	8.92337i	0.675832	−	0.390192i	−0.122451	−	0.992475i	$-0.539075\pi$
$523$	15.4557	−	8.92337i	0.675832	−	0.390192i	0.798283	+	0.602283i	$0.205742\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−45.0618	−	26.0165i	−1.96292	−	1.13330i
$528$		0			0
$529$	−4.84940	−	8.39940i	−0.210843	−	0.365191i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.3474	−	8.86084i	−0.664771	−	0.383806i
$534$		0			0
$535$			2.13111i			0.0921358i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−38.0917	−	8.79479i	−1.64072	−	0.378818i
$540$		0			0
$541$	6.65241	+	11.5223i	0.286009	+	0.495383i	0.972853	−	0.231422i	$-0.0743378\pi$
$541$	6.65241	+	11.5223i	0.286009	+	0.495383i	−0.686844	+	0.726805i	$0.741004\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.252123	−	0.436690i	−0.0107998	−	0.0187057i
$546$		0			0
$547$	7.69646	−	13.3307i	0.329077	−	0.569978i	−0.653252	−	0.757141i	$-0.726596\pi$
$547$	7.69646	−	13.3307i	0.329077	−	0.569978i	0.982329	+	0.187162i	$0.0599291\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.31543			0.141242
$552$		0			0
$553$	29.5912	+	11.6565i	1.25835	+	0.495687i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	18.1119	+	10.4569i	0.767426	+	0.443074i	0.831956	−	0.554842i	$-0.187221\pi$
$557$	18.1119	+	10.4569i	0.767426	+	0.443074i	−0.0645293	+	0.997916i	$0.520555\pi$
$558$		0			0
$559$		−	9.49719i		−	0.401688i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.4727			−0.820678			−0.410339	−	0.911933i	$-0.634590\pi$
$563$	−19.4727			−0.820678			−0.410339	+	0.911933i	$0.634590\pi$
$564$		0			0
$565$		−	9.33364i		−	0.392669i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			3.16022i			0.132483i	0.997804	+	0.0662417i	$0.0211008\pi$
$569$			3.16022i			0.132483i	−0.997804	+	0.0662417i	$0.978899\pi$
$570$		0			0
$571$	−23.0586			−0.964972			−0.482486	−	0.875904i	$-0.660266\pi$
$571$	−23.0586			−0.964972			−0.482486	+	0.875904i	$0.660266\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			15.4094i			0.642615i
$576$		0			0
$577$	−37.6033	−	21.7103i	−1.56545	−	0.903810i	−0.996689	−	0.0813040i	$-0.974092\pi$
$577$	−37.6033	−	21.7103i	−1.56545	−	0.903810i	−0.568756	−	0.822506i	$-0.692575\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.12170	−	5.66476i	0.295458	−	0.235014i
$582$		0			0
$583$	−37.9012			−1.56971
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.02425	+	8.70226i	−0.207373	+	0.359181i	−0.950886	−	0.309541i	$-0.899825\pi$
$587$	−5.02425	+	8.70226i	−0.207373	+	0.359181i	0.743513	+	0.668721i	$0.233158\pi$
$588$		0			0
$589$	−3.42768	−	5.93692i	−0.141235	−	0.244627i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	18.9110	+	32.7547i	0.776580	+	1.34508i	0.933902	+	0.357528i	$0.116380\pi$
$593$	18.9110	+	32.7547i	0.776580	+	1.34508i	−0.157322	+	0.987547i	$0.550286\pi$
$594$		0			0
$595$	1.62248	+	10.8569i	0.0665152	+	0.445088i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	36.3527i		−	1.48533i	−0.669663	−	0.742665i	$-0.733561\pi$
$599$		−	36.3527i		−	1.48533i	0.669663	−	0.742665i	$-0.266439\pi$
$600$		0			0
$601$	2.72502	+	1.57329i	0.111156	+	0.0641760i	0.554547	−	0.832152i	$-0.312891\pi$
$601$	2.72502	+	1.57329i	0.111156	+	0.0641760i	−0.443391	+	0.896328i	$0.646225\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.88645	−	15.3918i	−0.361285	−	0.625765i
$606$		0			0
$607$	5.26589	+	3.04026i	0.213736	+	0.123400i	0.603046	−	0.797706i	$-0.293953\pi$
$607$	5.26589	+	3.04026i	0.213736	+	0.123400i	−0.389311	+	0.921107i	$0.627287\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−17.4759	+	10.0897i	−0.707000	+	0.408187i
$612$		0			0
$613$	23.6217	−	40.9140i	0.954072	−	1.65250i	0.217593	−	0.976040i	$-0.430179\pi$
$613$	23.6217	−	40.9140i	0.954072	−	1.65250i	0.736478	−	0.676461i	$-0.236487\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.4011	−	14.0880i	0.982350	−	0.567160i	0.0793713	−	0.996845i	$-0.474709\pi$
$617$	24.4011	−	14.0880i	0.982350	−	0.567160i	0.902979	+	0.429685i	$0.141375\pi$
$618$		0			0
$619$	30.8831	−	17.8304i	1.24130	−	0.716663i	0.271939	−	0.962314i	$-0.412335\pi$
$619$	30.8831	−	17.8304i	1.24130	−	0.716663i	0.969358	+	0.245651i	$0.0790018\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.66395	−	17.8259i	−0.106729	−	0.714179i
$624$		0			0
$625$	−6.98865	+	12.1047i	−0.279546	+	0.484188i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	40.3528			1.60897
$630$		0			0
$631$	12.7686			0.508311			0.254155	−	0.967163i	$-0.418203\pi$
$631$	12.7686			0.508311			0.254155	+	0.967163i	$0.418203\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.25695	−	2.17710i	0.0498805	−	0.0863956i
$636$		0			0
$637$	−17.7051	−	4.08783i	−0.701501	−	0.161966i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.6617	+	6.15551i	−0.421110	+	0.243128i	−0.695552	−	0.718476i	$-0.744840\pi$
$641$	−10.6617	+	6.15551i	−0.421110	+	0.243128i	0.274442	+	0.961604i	$0.411507\pi$
$642$		0			0
$643$	−39.2940	+	22.6864i	−1.54960	+	0.894664i	−0.551432	+	0.834220i	$0.685919\pi$
$643$	−39.2940	+	22.6864i	−1.54960	+	0.894664i	−0.998172	+	0.0604447i	$0.980748\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	23.9646	−	41.5079i	0.942145	−	1.63184i	0.180776	−	0.983524i	$-0.442139\pi$
$647$	23.9646	−	41.5079i	0.942145	−	1.63184i	0.761369	−	0.648319i	$-0.224527\pi$
$648$		0			0
$649$	−51.8122	+	29.9138i	−2.03381	+	1.17422i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.07258	−	4.66071i	−0.315905	−	0.182388i	0.333661	−	0.942693i	$-0.391716\pi$
$653$	−8.07258	−	4.66071i	−0.315905	−	0.182388i	−0.649566	+	0.760305i	$0.725049\pi$
$654$		0			0
$655$	5.11250	+	8.85511i	0.199762	+	0.345998i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.2981	+	5.94563i	0.401158	+	0.231609i	0.686984	−	0.726673i	$-0.258934\pi$
$659$	10.2981	+	5.94563i	0.401158	+	0.231609i	−0.285825	+	0.958282i	$0.592268\pi$
$660$		0			0
$661$			9.61549i			0.373999i	0.982360	+	0.187000i	$0.0598763\pi$
$661$			9.61549i			0.373999i	−0.982360	+	0.187000i	$0.940124\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.530075	+	1.34564i	−0.0205554	+	0.0521818i
$666$		0			0
$667$	9.73571	+	16.8627i	0.376968	+	0.652928i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.73007	+	9.92478i	0.221207	+	0.383142i
$672$		0			0
$673$	11.2898	−	19.5544i	0.435188	−	0.753768i	−0.562123	−	0.827054i	$-0.690015\pi$
$673$	11.2898	−	19.5544i	0.435188	−	0.753768i	0.997311	+	0.0732855i	$0.0233484\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	42.9357			1.65015			0.825077	−	0.565021i	$-0.191132\pi$
$677$	42.9357			1.65015			0.825077	+	0.565021i	$0.191132\pi$
$678$		0			0
$679$	26.7114	+	33.5813i	1.02509	+	1.28873i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.66677	−	0.962311i	−0.0637772	−	0.0368218i	0.467772	−	0.883849i	$-0.345057\pi$
$683$	−1.66677	−	0.962311i	−0.0637772	−	0.0368218i	−0.531550	+	0.847027i	$0.678390\pi$
$684$		0			0
$685$		−	6.52300i		−	0.249231i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−17.6166			−0.671138
$690$		0			0
$691$			6.46796i			0.246053i	0.992403	+	0.123026i	$0.0392600\pi$
$691$			6.46796i			0.246053i	−0.992403	+	0.123026i	$0.960740\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			3.00724i			0.114071i
$696$		0			0
$697$	32.1783			1.21884
$698$		0			0
$699$		0			0
$700$		0			0
$701$			26.1067i			0.986036i	0.870019	+	0.493018i	$0.164106\pi$
$701$			26.1067i			0.986036i	−0.870019	+	0.493018i	$0.835894\pi$
$702$		0			0
$703$	4.60423	+	2.65825i	0.173652	+	0.100258i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.28481	−	35.3634i	−0.198756	−	1.32998i
$708$		0			0
$709$	−34.8342			−1.30823			−0.654113	−	0.756397i	$-0.726958\pi$
$709$	−34.8342			−1.30823			−0.654113	+	0.756397i	$0.726958\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	20.1307	−	34.8674i	0.753900	−	1.30579i
$714$		0			0
$715$	−6.38077	−	11.0518i	−0.238627	−	0.413315i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.5506	+	32.1306i	0.691822	+	1.19827i	0.971240	+	0.238101i	$0.0765250\pi$
$719$	18.5506	+	32.1306i	0.691822	+	1.19827i	−0.279418	+	0.960169i	$0.590142\pi$
$720$		0			0
$721$	20.8397	+	26.1995i	0.776111	+	0.975721i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			22.5575i			0.837765i
$726$		0			0
$727$	5.17832	+	2.98970i	0.192053	+	0.110882i	0.592943	−	0.805244i	$-0.297966\pi$
$727$	5.17832	+	2.98970i	0.192053	+	0.110882i	−0.400890	+	0.916126i	$0.631299\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.62230	+	14.9343i	0.318907	+	0.552364i
$732$		0			0
$733$	−13.7934	−	7.96362i	−0.509471	−	0.294143i	0.223145	−	0.974785i	$-0.428368\pi$
$733$	−13.7934	−	7.96362i	−0.509471	−	0.294143i	−0.732616	+	0.680642i	$0.761701\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−43.7325	+	25.2490i	−1.61091	+	0.930057i
$738$		0			0
$739$	3.41719	−	5.91875i	0.125704	−	0.217725i	−0.796304	−	0.604896i	$-0.793215\pi$
$739$	3.41719	−	5.91875i	0.125704	−	0.217725i	0.922008	+	0.387171i	$0.126548\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.7681	+	11.9905i	−0.761907	+	0.439887i	−0.829980	−	0.557793i	$-0.811648\pi$
$743$	−20.7681	+	11.9905i	−0.761907	+	0.439887i	0.0680728	+	0.997680i	$0.478315\pi$
$744$		0			0
$745$	7.75594	−	4.47789i	0.284156	−	0.164057i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.33493	−	0.946711i	0.231473	−	0.0345921i
$750$		0			0
$751$	−3.13906	+	5.43702i	−0.114546	+	0.198400i	−0.917598	−	0.397509i	$-0.869875\pi$
$751$	−3.13906	+	5.43702i	−0.114546	+	0.198400i	0.803052	+	0.595909i	$0.203208\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	3.79137			0.137982
$756$		0			0
$757$	−43.9573			−1.59766			−0.798828	−	0.601559i	$-0.794546\pi$
$757$	−43.9573			−1.59766			−0.798828	+	0.601559i	$0.794546\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.85711	+	13.6089i	−0.284820	+	0.493323i	−0.972566	−	0.232629i	$-0.925267\pi$
$761$	−7.85711	+	13.6089i	−0.284820	+	0.493323i	0.687745	+	0.725952i	$0.258601\pi$
$762$		0			0
$763$	−1.18610	+	0.943454i	−0.0429398	+	0.0341553i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.0824	+	13.9040i	−0.869566	+	0.502044i
$768$		0			0
$769$	29.3970	−	16.9724i	1.06008	−	0.612040i	0.134628	−	0.990896i	$-0.457016\pi$
$769$	29.3970	−	16.9724i	1.06008	−	0.612040i	0.925455	+	0.378857i	$0.123683\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24.4346	−	42.3220i	0.878852	−	1.52222i	0.0262509	−	0.999655i	$-0.491643\pi$
$773$	24.4346	−	42.3220i	0.878852	−	1.52222i	0.852602	−	0.522562i	$-0.175024\pi$
$774$		0			0
$775$	40.3936	−	23.3213i	1.45098	−	0.837725i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3.67152	+	2.11975i	0.131546	+	0.0759481i
$780$		0			0
$781$	34.3498	+	59.4956i	1.22913	+	2.12892i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.99286	+	5.19203i	0.320969	+	0.185312i
$786$		0			0
$787$		−	36.3244i		−	1.29483i	−0.762139	−	0.647413i	$-0.775851\pi$
$787$		−	36.3244i		−	1.29483i	0.762139	−	0.647413i	$-0.224149\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−27.7452	+	4.14633i	−0.986506	+	0.147426i
$792$		0			0
$793$	2.66335	+	4.61306i	0.0945783	+	0.163814i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.0153	+	24.2753i	0.496449	+	0.859875i	0.999992	−	0.00409568i	$-0.00130370\pi$
$797$	14.0153	+	24.2753i	0.496449	+	0.859875i	−0.503543	+	0.863970i	$0.667970\pi$
$798$		0			0
$799$	18.3205	−	31.7321i	0.648133	−	1.12260i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	22.1650			0.782184
$804$		0			0
$805$	−8.40068	+	1.25542i	−0.296085	+	0.0442478i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−10.9591	−	6.32725i	−0.385302	−	0.222454i	0.294820	−	0.955553i	$-0.404740\pi$
$809$	−10.9591	−	6.32725i	−0.385302	−	0.222454i	−0.680123	+	0.733098i	$0.738074\pi$
$810$		0			0
$811$			33.7585i			1.18542i	0.805415	+	0.592711i	$0.201943\pi$
$811$			33.7585i			1.18542i	−0.805415	+	0.592711i	$0.798057\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.2291			0.358310
$816$		0			0
$817$			2.27199i			0.0794867i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			5.87082i			0.204893i	0.994739	+	0.102447i	$0.0326671\pi$
$821$			5.87082i			0.204893i	−0.994739	+	0.102447i	$0.967333\pi$
$822$		0			0
$823$	30.9017			1.07717			0.538583	−	0.842572i	$-0.318960\pi$
$823$	30.9017			1.07717			0.538583	+	0.842572i	$0.318960\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	44.0586i		−	1.53207i	−0.642800	−	0.766034i	$-0.722227\pi$
$827$		−	44.0586i		−	1.53207i	0.642800	−	0.766034i	$-0.277773\pi$
$828$		0			0
$829$	5.47912	+	3.16337i	0.190298	+	0.109868i	0.592122	−	0.805848i	$-0.298290\pi$
$829$	5.47912	+	3.16337i	0.190298	+	0.109868i	−0.401824	+	0.915717i	$0.631624\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	31.5524	−	9.64598i	1.09322	−	0.334213i
$834$		0			0
$835$	13.8825			0.480425
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.29837	+	9.17704i	−0.182920	+	0.316827i	−0.942874	−	0.333151i	$-0.891888\pi$
$839$	−5.29837	+	9.17704i	−0.182920	+	0.316827i	0.759954	+	0.649977i	$0.225222\pi$
$840$		0			0
$841$	−0.248064	−	0.429659i	−0.00855392	−	0.0148158i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.75596	+	4.77347i	0.0948080	+	0.164212i
$846$		0			0
$847$	−41.8060	+	33.2534i	−1.43647	+	1.14260i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			31.2237i			1.07033i
$852$		0			0
$853$	19.0713	+	11.0108i	0.652989	+	0.377003i	0.789600	−	0.613621i	$-0.210288\pi$
$853$	19.0713	+	11.0108i	0.652989	+	0.377003i	−0.136611	+	0.990625i	$0.543621\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.76205	+	3.05197i	0.0601906	+	0.104253i	0.894551	−	0.446967i	$-0.147496\pi$
$857$	1.76205	+	3.05197i	0.0601906	+	0.104253i	−0.834360	+	0.551220i	$0.814162\pi$
$858$		0			0
$859$	−32.0730	−	18.5174i	−1.09432	−	0.631805i	−0.159595	−	0.987183i	$-0.551019\pi$
$859$	−32.0730	−	18.5174i	−1.09432	−	0.631805i	−0.934723	+	0.355378i	$0.884352\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.5411	+	7.81797i	−0.460945	+	0.266127i	−0.712442	−	0.701731i	$-0.752411\pi$
$863$	−13.5411	+	7.81797i	−0.460945	+	0.266127i	0.251496	+	0.967858i	$0.419077\pi$
$864$		0			0
$865$	1.22067	−	2.11427i	0.0415042	−	0.0718873i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	58.1403	−	33.5673i	1.97227	−	1.13869i
$870$		0			0
$871$	−20.3269	+	11.7358i	−0.688752	+	0.397651i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.9900	−	7.87447i	−0.675787	−	0.266206i
$876$		0			0
$877$	1.92641	−	3.33664i	0.0650502	−	0.112670i	−0.831666	−	0.555276i	$-0.812613\pi$
$877$	1.92641	−	3.33664i	0.0650502	−	0.112670i	0.896716	+	0.442606i	$0.145946\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−44.3493			−1.49417			−0.747084	−	0.664730i	$-0.768547\pi$
$881$	−44.3493			−1.49417			−0.747084	+	0.664730i	$0.768547\pi$
$882$		0			0
$883$	−20.5569			−0.691795			−0.345897	−	0.938272i	$-0.612425\pi$
$883$	−20.5569			−0.691795			−0.345897	+	0.938272i	$0.612425\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2.05890	−	3.56612i	0.0691311	−	0.119739i	−0.829388	−	0.558673i	$-0.811311\pi$
$887$	2.05890	−	3.56612i	0.0691311	−	0.119739i	0.898519	+	0.438934i	$0.144644\pi$
$888$		0			0
$889$	−7.03003	−	2.76927i	−0.235780	−	0.0928782i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.18072	−	2.41374i	0.139902	−	0.0807726i
$894$		0			0
$895$	−5.23235	+	3.02090i	−0.174898	+	0.100978i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	29.4690	−	51.0417i	0.982845	−	1.70234i
$900$		0			0
$901$	27.7019	−	15.9937i	0.922884	−	0.532828i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.50973	−	4.33575i	−0.249632	−	0.144125i
$906$		0			0
$907$	−21.1401	−	36.6158i	−0.701946	−	1.21581i	−0.967782	−	0.251789i	$-0.918981\pi$
$907$	−21.1401	−	36.6158i	−0.701946	−	1.21581i	0.265836	−	0.964018i	$-0.414352\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.6408	−	9.60760i	−0.551336	−	0.318314i	0.198325	−	0.980136i	$-0.436450\pi$
$911$	−16.6408	−	9.60760i	−0.551336	−	0.318314i	−0.749661	+	0.661822i	$0.769783\pi$
$912$		0			0
$913$		−	19.2087i		−	0.635714i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	24.0516	−	19.1312i	0.794253	−	0.631767i
$918$		0			0
$919$	6.02706	+	10.4392i	0.198814	+	0.344357i	0.948144	−	0.317840i	$-0.102958\pi$
$919$	6.02706	+	10.4392i	0.198814	+	0.344357i	−0.749330	+	0.662197i	$0.769624\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	15.9659	+	27.6537i	0.525523	+	0.910232i
$924$		0			0
$925$	−18.0862	+	31.3263i	−0.594671	+	1.03000i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−36.5814			−1.20020			−0.600099	−	0.799926i	$-0.704872\pi$
$929$	−36.5814			−1.20020			−0.600099	+	0.799926i	$0.704872\pi$
$930$		0			0
$931$	4.23553	+	0.977921i	0.138814	+	0.0320501i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	20.0674	+	11.5859i	0.656275	+	0.378901i
$936$		0			0
$937$		−	39.4119i		−	1.28753i	−0.765222	−	0.643766i	$-0.777371\pi$
$937$		−	39.4119i		−	1.28753i	0.765222	−	0.643766i	$-0.222629\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.4350			0.405371			0.202685	−	0.979244i	$-0.435033\pi$
$941$	12.4350			0.405371			0.202685	+	0.979244i	$0.435033\pi$
$942$		0			0
$943$			24.8985i			0.810807i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			29.8649i			0.970478i	0.874382	+	0.485239i	$0.161267\pi$
$947$			29.8649i			0.970478i	−0.874382	+	0.485239i	$0.838733\pi$
$948$		0			0
$949$	10.3023			0.334427
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	22.4851i		−	0.728365i	−0.931328	−	0.364183i	$-0.881348\pi$
$953$		−	22.4851i		−	0.728365i	0.931328	−	0.364183i	$-0.118652\pi$
$954$		0			0
$955$	13.7044	+	7.91224i	0.443464	+	0.256034i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−19.3903	+	2.89774i	−0.626144	+	0.0935729i
$960$		0			0
$961$	−90.8669			−2.93119
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.65526	+	2.86699i	−0.0532846	+	0.0922916i
$966$		0			0
$967$	1.91372	+	3.31466i	0.0615411	+	0.106592i	0.895154	−	0.445756i	$-0.147065\pi$
$967$	1.91372	+	3.31466i	0.0615411	+	0.106592i	−0.833613	+	0.552348i	$0.813732\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	29.4317	+	50.9773i	0.944509	+	1.63594i	0.756731	+	0.653727i	$0.226795\pi$
$971$	29.4317	+	50.9773i	0.944509	+	1.63594i	0.187779	+	0.982211i	$0.439871\pi$
$972$		0			0
$973$	8.93932	−	1.33592i	0.286581	−	0.0428276i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	10.6006i		−	0.339145i	−0.985518	−	0.169572i	$-0.945761\pi$
$977$		−	10.6006i		−	0.339145i	0.985518	−	0.169572i	$-0.0542387\pi$
$978$		0			0
$979$	−32.9487	−	19.0230i	−1.05305	−	0.607976i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16.3230	+	28.2723i	0.520624	+	0.901747i	0.999712	+	0.0239802i	$0.00763386\pi$
$983$	16.3230	+	28.2723i	0.520624	+	0.901747i	−0.479089	+	0.877766i	$0.659033\pi$
$984$		0			0
$985$	5.24749	+	3.02964i	0.167199	+	0.0965323i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−11.5556	+	6.67165i	−0.367448	+	0.212146i
$990$		0			0
$991$	16.3697	−	28.3531i	0.520000	−	0.900666i	−0.479730	−	0.877416i	$-0.659265\pi$
$991$	16.3697	−	28.3531i	0.520000	−	0.900666i	0.999730	−	0.0232498i	$-0.00740131\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	10.6419	−	6.14412i	0.337372	−	0.194782i
$996$		0			0
$997$	39.3141	−	22.6980i	1.24509	−	0.718854i	0.274965	−	0.961454i	$-0.411334\pi$
$997$	39.3141	−	22.6980i	1.24509	−	0.718854i	0.970126	+	0.242601i	$0.0780004\pi$
$998$		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.269.10		32
3.2	odd	2	inner	2268.2.w.j.269.7		32
7.5	odd	6		2268.2.bm.j.593.10		32
9.2	odd	6		2268.2.t.c.1781.7	✓	32
9.4	even	3		2268.2.bm.j.1025.7		32
9.5	odd	6		2268.2.bm.j.1025.10		32
9.7	even	3		2268.2.t.c.1781.10	yes	32
21.5	even	6		2268.2.bm.j.593.7		32
63.5	even	6	inner	2268.2.w.j.1349.10		32
63.40	odd	6	inner	2268.2.w.j.1349.7		32
63.47	even	6		2268.2.t.c.2105.10	yes	32
63.61	odd	6		2268.2.t.c.2105.7	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.7	✓	32	9.2	odd	6
2268.2.t.c.1781.10	yes	32	9.7	even	3
2268.2.t.c.2105.7	yes	32	63.61	odd	6
2268.2.t.c.2105.10	yes	32	63.47	even	6
2268.2.w.j.269.7		32	3.2	odd	2	inner
2268.2.w.j.269.10		32	1.1	even	1	trivial
2268.2.w.j.1349.7		32	63.40	odd	6	inner
2268.2.w.j.1349.10		32	63.5	even	6	inner
2268.2.bm.j.593.7		32	21.5	even	6
2268.2.bm.j.593.10		32	7.5	odd	6
2268.2.bm.j.1025.7		32	9.4	even	3
2268.2.bm.j.1025.10		32	9.5	odd	6

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)

\(f(q)\)	\(=\)	\(q+(0.440135 - 0.762336i) q^{5} +(-2.46165 - 0.969691i) q^{7} +(-4.83660 + 2.79241i) q^{11} +(-2.24806 + 1.29792i) q^{13} +(2.35671 - 4.08193i) q^{17} +(0.537797 - 0.310497i) q^{19} +(3.15847 + 1.82354i) q^{23} +(2.11256 + 3.65906i) q^{25} +(4.62362 + 2.66945i) q^{29} -11.0393i q^{31} +(-1.82269 + 1.44981i) q^{35} +(4.28064 + 7.41428i) q^{37} +(3.41348 + 5.91232i) q^{41} +(-1.82931 + 3.16846i) q^{43} +7.77378 q^{47} +(5.11940 + 4.77407i) q^{49} +(5.87725 + 3.39323i) q^{53} +4.91616i q^{55} +10.7125 q^{59} -2.05202i q^{61} +2.28504i q^{65} +9.04199 q^{67} -12.3011i q^{71} +(-3.43707 - 1.98439i) q^{73} +(14.6138 - 2.18393i) q^{77} -12.0209 q^{79} +(-1.71972 + 2.97864i) q^{83} +(-2.07454 - 3.59321i) q^{85} +(3.40619 + 5.89969i) q^{89} +(6.79251 - 1.01509i) q^{91} -0.546643i q^{95} +(-14.0453 - 8.10908i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{7} + 12 q^{13} - 16 q^{25} - 4 q^{37} - 4 q^{43} + 20 q^{49} - 8 q^{67} - 36 q^{73} - 56 q^{79} + 12 q^{85} - 36 q^{91} + 12 q^{97}+O(q^{100}) \) 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

Introduction

L-functions

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