LMFDB - Embedded newform 2268.2.w.j.1349.10

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			1349.10
Character	$\chi$	$=$	2268.1349
Dual form			2268.2.w.j.269.10

$q$-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 $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.440135	+	0.762336i	0.196834	+	0.340927i	0.947500	−	0.319755i	$-0.103601\pi$
$5$	0.440135	+	0.762336i	0.196834	+	0.340927i	−0.750666	+	0.660682i	$0.770267\pi$
$6$		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	−0.459362	−	0.888249i	$-0.651922\pi$
$11$	−4.83660	−	2.79241i	−1.45829	−	0.841944i	−0.998927	+	0.0463050i	$0.985255\pi$
$12$		0			0
$13$	−2.24806	−	1.29792i	−0.623500	−	0.359978i	0.154730	−	0.987957i	$-0.450549\pi$
$13$	−2.24806	−	1.29792i	−0.623500	−	0.359978i	−0.778230	+	0.627979i	$0.783882\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.35671	+	4.08193i	0.571585	+	0.990015i	0.996403	+	0.0847359i	$0.0270047\pi$
$17$	2.35671	+	4.08193i	0.571585	+	0.990015i	−0.424818	+	0.905279i	$0.639662\pi$
$18$		0			0
$19$	0.537797	+	0.310497i	0.123379	+	0.0712330i	0.560419	−	0.828209i	$-0.310640\pi$
$19$	0.537797	+	0.310497i	0.123379	+	0.0712330i	−0.437040	+	0.899442i	$0.643973\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.15847	−	1.82354i	0.658586	−	0.380235i	−0.133152	−	0.991096i	$-0.542510\pi$
$23$	3.15847	−	1.82354i	0.658586	−	0.380235i	0.791738	+	0.610861i	$0.209177\pi$
$24$		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	−0.00495283	−	0.999988i	$-0.501577\pi$
$29$	4.62362	−	2.66945i	0.858586	−	0.495705i	0.863538	+	0.504283i	$0.168243\pi$
$30$		0			0
$31$			11.0393i			1.98272i	0.131158	+	0.991361i	$0.458130\pi$
$31$			11.0393i			1.98272i	−0.131158	+	0.991361i	$0.541870\pi$
$32$		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	−0.263415	−	0.964683i	$-0.584849\pi$
$37$	4.28064	−	7.41428i	0.703732	−	1.21890i	0.967147	−	0.254218i	$-0.0818179\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.41348	−	5.91232i	0.533096	−	0.923350i	−0.466157	−	0.884702i	$-0.654362\pi$
$41$	3.41348	−	5.91232i	0.533096	−	0.923350i	0.999253	−	0.0386476i	$-0.0123050\pi$
$42$		0			0
$43$	−1.82931	−	3.16846i	−0.278967	−	0.483186i	0.692161	−	0.721743i	$-0.256659\pi$
$43$	−1.82931	−	3.16846i	−0.278967	−	0.483186i	−0.971128	+	0.238557i	$0.923325\pi$
$44$		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	−0.0387156	−	0.999250i	$-0.512327\pi$
$53$	5.87725	−	3.39323i	0.807303	−	0.466096i	0.846018	+	0.533154i	$0.178993\pi$
$54$		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	$0.0419366\pi$
$61$			2.05202i			0.262734i	−0.991334	+	0.131367i	$0.958063\pi$
$62$		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	$0.260451\pi$
$71$			12.3011i			1.45988i	−0.683514	+	0.729938i	$0.739549\pi$
$72$		0			0
$73$	−3.43707	+	1.98439i	−0.402278	+	0.232255i	−0.687466	−	0.726216i	$-0.741277\pi$
$73$	−3.43707	+	1.98439i	−0.402278	+	0.232255i	0.285188	+	0.958471i	$0.407944\pi$
$74$		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	0.756075	−	0.654485i	$-0.227115\pi$
$83$	−1.71972	−	2.97864i	−0.188764	−	0.326948i	−0.944838	+	0.327537i	$0.893781\pi$
$84$		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	−0.627080	−	0.778955i	$-0.715750\pi$
$89$	3.40619	−	5.89969i	0.361055	−	0.625366i	0.988135	+	0.153589i	$0.0490833\pi$
$90$		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	−0.996809	−	0.0798195i	$-0.974566\pi$
$97$	−14.0453	+	8.10908i	−1.42609	+	0.823352i	−0.429279	+	0.903172i	$0.641232\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.75727	−	11.7039i	0.672373	−	1.16458i	−0.304856	−	0.952398i	$-0.598608\pi$
$101$	6.75727	−	11.7039i	0.672373	−	1.16458i	0.977229	−	0.212186i	$-0.0680583\pi$
$102$		0			0
$103$	−10.9579	+	6.32656i	−1.07972	+	0.623374i	−0.930820	−	0.365479i	$-0.880905\pi$
$103$	−10.9579	+	6.32656i	−1.07972	+	0.623374i	−0.148896	+	0.988853i	$0.547572\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.09662	−	1.21048i	−0.202688	−	0.117022i	0.395221	−	0.918586i	$-0.370668\pi$
$107$	−2.09662	−	1.21048i	−0.202688	−	0.117022i	−0.597909	+	0.801564i	$0.704001\pi$
$108$		0			0
$109$	0.286415	+	0.496086i	0.0274336	+	0.0475164i	0.879416	−	0.476054i	$-0.157933\pi$
$109$	0.286415	+	0.496086i	0.0274336	+	0.0475164i	−0.851983	+	0.523570i	$0.824600\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.18260	+	5.30158i	0.863827	+	0.498731i	0.865292	−	0.501268i	$-0.167133\pi$
$113$	9.18260	+	5.30158i	0.863827	+	0.498731i	−0.00146526	+	0.999999i	$0.500466\pi$
$114$		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	−0.999963	−	0.00860838i	$-0.997260\pi$
$131$	−5.80788	−	10.0595i	−0.507437	−	0.878906i	0.492526	−	0.870298i	$-0.336074\pi$
$132$		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	0.748427	−	0.663217i	$-0.230809\pi$
$137$	6.41744	+	3.70511i	0.548279	+	0.316549i	−0.200149	+	0.979766i	$0.564143\pi$
$138$		0			0
$139$	−2.95857	−	1.70813i	−0.250943	−	0.144882i	0.369253	−	0.929329i	$-0.379614\pi$
$139$	−2.95857	−	1.70813i	−0.250943	−	0.144882i	−0.620196	+	0.784447i	$0.712947\pi$
$140$		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	−0.0936062	−	0.995609i	$-0.529839\pi$
$149$	8.81086	−	5.08695i	0.721814	−	0.416739i	0.815420	+	0.578870i	$0.196506\pi$
$150$		0			0
$151$	2.15352	−	3.73001i	0.175251	−	0.303544i	−0.764997	−	0.644034i	$-0.777260\pi$
$151$	2.15352	−	3.73001i	0.175251	−	0.303544i	0.940248	+	0.340490i	$0.110593\pi$
$152$		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	$-0.843991\pi$
$157$		−	11.7964i		−	0.941459i	0.882278	−	0.470729i	$-0.156009\pi$
$158$		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	−0.543603	−	0.839342i	$-0.682940\pi$
$163$	5.81021	−	10.0636i	0.455090	−	0.788240i	0.998693	+	0.0511028i	$0.0162736\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.88538	−	13.6579i	0.610189	−	1.05688i	−0.381019	−	0.924567i	$-0.624427\pi$
$167$	7.88538	−	13.6579i	0.610189	−	1.05688i	0.991208	−	0.132311i	$-0.0422398\pi$
$168$		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	−0.705410	−	0.708799i	$-0.749237\pi$
$179$	−5.94403	+	3.43179i	−0.444278	+	0.256504i	0.261133	+	0.965303i	$0.415904\pi$
$180$		0			0
$181$			9.85094i			0.732215i	0.930573	+	0.366107i	$0.119310\pi$
$181$			9.85094i			0.732215i	−0.930573	+	0.366107i	$0.880690\pi$
$182$		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	$-0.774610\pi$
$191$		−	17.9768i		−	1.30076i	0.759609	−	0.650380i	$-0.225390\pi$
$192$		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	$-0.921142\pi$
$197$		−	6.88342i		−	0.490424i	0.969470	−	0.245212i	$-0.0788575\pi$
$198$		0			0
$199$	12.0894	−	6.97981i	0.856993	−	0.494785i	−0.00601088	−	0.999982i	$-0.501913\pi$
$199$	12.0894	−	6.97981i	0.856993	−	0.494785i	0.863004	+	0.505197i	$0.168580\pi$
$200$		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	−0.819372	−	0.573263i	$-0.805677\pi$
$211$	1.26047	−	2.18320i	0.0867744	−	0.150298i	0.906146	+	0.422965i	$0.139011\pi$
$212$		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	−0.0984273	−	0.995144i	$-0.531381\pi$
$223$	10.6650	−	6.15742i	0.714179	−	0.412332i	0.812607	+	0.582813i	$0.198048\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.8806	+	22.3098i	−0.854913	+	1.48075i	0.0218124	+	0.999762i	$0.493056\pi$
$227$	−12.8806	+	22.3098i	−0.854913	+	1.48075i	−0.876726	+	0.480991i	$0.840277\pi$
$228$		0			0
$229$	5.01900	−	2.89772i	0.331665	−	0.191487i	−0.324915	−	0.945743i	$-0.605336\pi$
$229$	5.01900	−	2.89772i	0.331665	−	0.191487i	0.656580	+	0.754256i	$0.272002\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26.2280	−	15.1428i	−1.71826	−	0.992036i	−0.922107	−	0.386935i	$-0.873534\pi$
$233$	−26.2280	−	15.1428i	−1.71826	−	0.992036i	−0.796149	−	0.605101i	$-0.793133\pi$
$234$		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	0.592693	−	0.805428i	$-0.298065\pi$
$239$	2.96081	+	1.70942i	0.191519	+	0.110573i	−0.401175	+	0.916002i	$0.631398\pi$
$240$		0			0
$241$	4.24030	+	2.44814i	0.273142	+	0.157699i	0.630315	−	0.776340i	$-0.282926\pi$
$241$	4.24030	+	2.44814i	0.273142	+	0.157699i	−0.357173	+	0.934038i	$0.616259\pi$
$242$		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	0.934827	−	0.355102i	$-0.115554\pi$
$257$	2.56317	+	4.43954i	0.159886	+	0.276931i	−0.774941	+	0.632033i	$0.782221\pi$
$258$		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	0.622063	−	0.782967i	$-0.286295\pi$
$263$	4.13580	+	2.38780i	0.255024	+	0.147238i	−0.367038	+	0.930206i	$0.619628\pi$
$264$		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	0.998241	−	0.0592889i	$-0.0188833\pi$
$269$	7.34405	+	12.7203i	0.447775	+	0.775569i	−0.550466	+	0.834858i	$0.685550\pi$
$270$		0			0
$271$	9.13087	+	5.27171i	0.554661	+	0.320233i	0.751000	−	0.660303i	$-0.229572\pi$
$271$	9.13087	+	5.27171i	0.554661	+	0.320233i	−0.196339	+	0.980536i	$0.562905\pi$
$272$		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	0.0705327	+	0.997509i	$0.477530\pi$
$277$	−13.7907	+	23.8862i	−0.828602	+	1.43518i	−0.899135	+	0.437672i	$0.855803\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.7563	−	14.2930i	1.47684	−	0.852652i	0.477178	−	0.878806i	$-0.341660\pi$
$281$	24.7563	−	14.2930i	1.47684	−	0.852652i	0.999658	+	0.0261546i	$0.00832621\pi$
$282$		0			0
$283$		−	18.8385i		−	1.11983i	−0.828549	−	0.559917i	$-0.810833\pi$
$283$		−	18.8385i		−	1.11983i	0.828549	−	0.559917i	$-0.189167\pi$
$284$		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	−0.327193	+	0.944957i	$0.606103\pi$
$293$	−16.8083	+	29.1129i	−0.981954	+	1.70079i	−0.654760	+	0.755837i	$0.727230\pi$
$294$		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	$0.137604\pi$
$307$			14.6814i			0.837913i	−0.908006	+	0.418957i	$0.862396\pi$
$308$		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	$0.0557213\pi$
$313$			6.16244i			0.348322i	−0.984717	+	0.174161i	$0.944279\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.84953i		−	0.216211i	−0.994139	−	0.108105i	$-0.965522\pi$
$317$		−	3.84953i		−	0.216211i	0.994139	−	0.108105i	$-0.0344784\pi$
$318$		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	−0.463409	−	0.886145i	$-0.653374\pi$
$337$	9.83450	−	17.0339i	0.535720	−	0.927894i	0.999128	−	0.0417488i	$-0.0132929\pi$
$338$		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	$-0.872488\pi$
$347$		−	14.5284i		−	0.779923i	0.920831	−	0.389962i	$-0.127512\pi$
$348$		0			0
$349$	26.9608	−	15.5658i	1.44318	−	0.833218i	0.445116	−	0.895473i	$-0.353162\pi$
$349$	26.9608	−	15.5658i	1.44318	−	0.833218i	0.998060	+	0.0622544i	$0.0198290\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.11432	+	7.12621i	−0.218983	+	0.379290i	−0.954497	−	0.298219i	$-0.903607\pi$
$353$	−4.11432	+	7.12621i	−0.218983	+	0.379290i	0.735514	+	0.677509i	$0.236941\pi$
$354$		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	−0.349557	−	0.936915i	$-0.613668\pi$
$359$	−25.3084	−	14.6118i	−1.33573	−	0.771183i	−0.986171	+	0.165732i	$0.947001\pi$
$360$		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	0.487124	−	0.873333i	$-0.338046\pi$
$367$	−0.491228	−	0.283610i	−0.0256419	−	0.0148043i	−0.512766	+	0.858528i	$0.671379\pi$
$368$		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	0.841731	−	0.539898i	$-0.181537\pi$
$373$	−0.901927	−	1.56218i	−0.0467000	−	0.0808868i	−0.888431	+	0.459011i	$0.848204\pi$
$374$		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	−0.993676	−	0.112286i	$-0.964183\pi$
$383$	−11.6264	−	20.1375i	−0.594080	−	1.02898i	0.399596	−	0.916691i	$-0.369150\pi$
$384$		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	0.884901	−	0.465779i	$-0.154226\pi$
$389$	18.2236	+	10.5214i	0.923976	+	0.533458i	0.0390745	+	0.999236i	$0.487559\pi$
$390$		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	0.958566	−	0.284870i	$-0.0919505\pi$
$397$	23.7334	+	13.7025i	1.19114	+	0.687708i	0.232578	+	0.972578i	$0.425284\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−31.8620	+	18.3955i	−1.59111	+	0.918629i	−0.597995	+	0.801500i	$0.704036\pi$
$401$	−31.8620	+	18.3955i	−1.59111	+	0.918629i	−0.993117	+	0.117129i	$0.962631\pi$
$402$		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	$-0.686954\pi$
$409$		−	33.6694i		−	1.66484i	0.554142	−	0.832422i	$-0.313046\pi$
$410$		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	−0.731618	−	0.681715i	$-0.761234\pi$
$419$	4.59691	−	7.96209i	0.224574	−	0.388973i	0.956192	+	0.292742i	$0.0945677\pi$
$420$		0			0
$421$	4.30373	+	7.45427i	0.209751	+	0.363299i	0.951636	−	0.307228i	$-0.0994014\pi$
$421$	4.30373	+	7.45427i	0.209751	+	0.363299i	−0.741885	+	0.670527i	$0.766068\pi$
$422$		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	−0.324379	−	0.945927i	$-0.605155\pi$
$431$	6.90554	−	3.98691i	0.332628	−	0.192043i	0.657007	+	0.753884i	$0.271822\pi$
$432$		0			0
$433$			1.42915i			0.0686806i	0.999410	+	0.0343403i	$0.0109330\pi$
$433$			1.42915i			0.0686806i	−0.999410	+	0.0343403i	$0.989067\pi$
$434$		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	$-0.791928\pi$
$439$		−	25.4827i		−	1.21622i	0.793852	−	0.608110i	$-0.208072\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			5.56489i			0.264396i	0.991223	+	0.132198i	$0.0422035\pi$
$443$			5.56489i			0.264396i	−0.991223	+	0.132198i	$0.957797\pi$
$444$		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	$0.214192\pi$
$449$			26.4132i			1.24652i	−0.782016	+	0.623259i	$0.785808\pi$
$450$		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	0.990007	+	0.141015i	$0.0450365\pi$
$461$	13.2503	+	22.9501i	0.617126	+	1.06889i	−0.372881	+	0.927879i	$0.621630\pi$
$462$		0			0
$463$	7.24376	−	12.5466i	0.336646	−	0.583088i	−0.647153	−	0.762360i	$-0.724041\pi$
$463$	7.24376	−	12.5466i	0.336646	−	0.583088i	0.983800	+	0.179271i	$0.0573740\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.21034	+	12.4887i	−0.333655	+	0.577907i	−0.983226	−	0.182394i	$-0.941615\pi$
$467$	−7.21034	+	12.4887i	−0.333655	+	0.577907i	0.649571	+	0.760301i	$0.274949\pi$
$468$		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	0.309807	+	0.950799i	$0.399736\pi$
$479$	−14.6311	+	25.3418i	−0.668513	+	1.15790i	−0.978320	+	0.207099i	$0.933598\pi$
$480$		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	0.783276	−	0.621674i	$-0.213547\pi$
$487$	−3.23843	−	5.60913i	−0.146747	−	0.254174i	−0.930024	+	0.367500i	$0.880214\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.7683	−	17.7641i	−1.38856	−	0.801683i	−0.395403	−	0.918508i	$-0.629395\pi$
$491$	−30.7683	−	17.7641i	−1.38856	−	0.801683i	−0.993153	+	0.116825i	$0.962728\pi$
$492$		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	0.962284	+	0.272047i	$0.0877006\pi$
$499$	16.0108	+	27.7315i	0.716742	+	1.24143i	−0.245542	+	0.969386i	$0.578966\pi$
$500$		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	−0.888591	−	0.458699i	$-0.848315\pi$
$509$	−18.9860	−	32.8848i	−0.841541	−	1.45759i	0.0470504	−	0.998893i	$-0.485018\pi$
$510$		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	0.872922	−	0.487860i	$-0.162222\pi$
$521$	0.318682	+	0.551974i	0.0139617	+	0.0241824i	−0.858960	+	0.512042i	$0.828889\pi$
$522$		0			0
$523$	15.4557	+	8.92337i	0.675832	+	0.390192i	0.798283	−	0.602283i	$-0.205742\pi$
$523$	15.4557	+	8.92337i	0.675832	+	0.390192i	−0.122451	+	0.992475i	$0.539075\pi$
$524$		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	−0.686844	−	0.726805i	$-0.741004\pi$
$541$	6.65241	−	11.5223i	0.286009	−	0.495383i	0.972853	+	0.231422i	$0.0743378\pi$
$542$		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	0.982329	−	0.187162i	$-0.0599291\pi$
$547$	7.69646	+	13.3307i	0.329077	+	0.569978i	−0.653252	+	0.757141i	$0.726596\pi$
$548$		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	−0.0645293	−	0.997916i	$-0.520555\pi$
$557$	18.1119	−	10.4569i	0.767426	−	0.443074i	0.831956	+	0.554842i	$0.187221\pi$
$558$		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	$-0.978899\pi$
$569$		−	3.16022i		−	0.132483i	0.997804	−	0.0662417i	$-0.0211008\pi$
$570$		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	−0.568756	+	0.822506i	$0.692575\pi$
$577$	−37.6033	+	21.7103i	−1.56545	+	0.903810i	−0.996689	+	0.0813040i	$0.974092\pi$
$578$		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	0.743513	−	0.668721i	$-0.233158\pi$
$587$	−5.02425	−	8.70226i	−0.207373	−	0.359181i	−0.950886	+	0.309541i	$0.899825\pi$
$588$		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	−0.157322	−	0.987547i	$-0.550286\pi$
$593$	18.9110	−	32.7547i	0.776580	−	1.34508i	0.933902	−	0.357528i	$-0.116380\pi$
$594$		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	$0.266439\pi$
$599$			36.3527i			1.48533i	−0.669663	+	0.742665i	$0.733561\pi$
$600$		0			0
$601$	2.72502	−	1.57329i	0.111156	−	0.0641760i	−0.443391	−	0.896328i	$-0.646225\pi$
$601$	2.72502	−	1.57329i	0.111156	−	0.0641760i	0.554547	+	0.832152i	$0.312891\pi$
$602$		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	−0.389311	−	0.921107i	$-0.627287\pi$
$607$	5.26589	−	3.04026i	0.213736	−	0.123400i	0.603046	+	0.797706i	$0.293953\pi$
$608$		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	0.736478	+	0.676461i	$0.236487\pi$
$613$	23.6217	+	40.9140i	0.954072	+	1.65250i	0.217593	+	0.976040i	$0.430179\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.4011	+	14.0880i	0.982350	+	0.567160i	0.902979	−	0.429685i	$-0.141375\pi$
$617$	24.4011	+	14.0880i	0.982350	+	0.567160i	0.0793713	+	0.996845i	$0.474709\pi$
$618$		0			0
$619$	30.8831	+	17.8304i	1.24130	+	0.716663i	0.969358	−	0.245651i	$-0.0790018\pi$
$619$	30.8831	+	17.8304i	1.24130	+	0.716663i	0.271939	+	0.962314i	$0.412335\pi$
$620$		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	0.274442	−	0.961604i	$-0.411507\pi$
$641$	−10.6617	−	6.15551i	−0.421110	−	0.243128i	−0.695552	+	0.718476i	$0.744840\pi$
$642$		0			0
$643$	−39.2940	−	22.6864i	−1.54960	−	0.894664i	−0.998172	−	0.0604447i	$-0.980748\pi$
$643$	−39.2940	−	22.6864i	−1.54960	−	0.894664i	−0.551432	−	0.834220i	$-0.685919\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	23.9646	+	41.5079i	0.942145	+	1.63184i	0.761369	+	0.648319i	$0.224527\pi$
$647$	23.9646	+	41.5079i	0.942145	+	1.63184i	0.180776	+	0.983524i	$0.442139\pi$
$648$		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	−0.649566	−	0.760305i	$-0.725049\pi$
$653$	−8.07258	+	4.66071i	−0.315905	+	0.182388i	0.333661	+	0.942693i	$0.391716\pi$
$654$		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	−0.285825	−	0.958282i	$-0.592268\pi$
$659$	10.2981	−	5.94563i	0.401158	−	0.231609i	0.686984	+	0.726673i	$0.258934\pi$
$660$		0			0
$661$		−	9.61549i		−	0.373999i	−0.982360	−	0.187000i	$-0.940124\pi$
$661$		−	9.61549i		−	0.373999i	0.982360	−	0.187000i	$-0.0598763\pi$
$662$		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	0.997311	−	0.0732855i	$-0.0233484\pi$
$673$	11.2898	+	19.5544i	0.435188	+	0.753768i	−0.562123	+	0.827054i	$0.690015\pi$
$674$		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	−0.531550	−	0.847027i	$-0.678390\pi$
$683$	−1.66677	+	0.962311i	−0.0637772	+	0.0368218i	0.467772	+	0.883849i	$0.345057\pi$
$684$		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	$-0.960740\pi$
$691$		−	6.46796i		−	0.246053i	0.992403	−	0.123026i	$-0.0392600\pi$
$692$		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	$-0.835894\pi$
$701$		−	26.1067i		−	0.986036i	0.870019	−	0.493018i	$-0.164106\pi$
$702$		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	−0.279418	−	0.960169i	$-0.590142\pi$
$719$	18.5506	−	32.1306i	0.691822	−	1.19827i	0.971240	−	0.238101i	$-0.0765250\pi$
$720$		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	−0.400890	−	0.916126i	$-0.631299\pi$
$727$	5.17832	−	2.98970i	0.192053	−	0.110882i	0.592943	+	0.805244i	$0.297966\pi$
$728$		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	−0.732616	−	0.680642i	$-0.761701\pi$
$733$	−13.7934	+	7.96362i	−0.509471	+	0.294143i	0.223145	+	0.974785i	$0.428368\pi$
$734$		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	0.922008	−	0.387171i	$-0.126548\pi$
$739$	3.41719	+	5.91875i	0.125704	+	0.217725i	−0.796304	+	0.604896i	$0.793215\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.7681	−	11.9905i	−0.761907	−	0.439887i	0.0680728	−	0.997680i	$-0.478315\pi$
$743$	−20.7681	−	11.9905i	−0.761907	−	0.439887i	−0.829980	+	0.557793i	$0.811648\pi$
$744$		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	0.803052	−	0.595909i	$-0.203208\pi$
$751$	−3.13906	−	5.43702i	−0.114546	−	0.198400i	−0.917598	+	0.397509i	$0.869875\pi$
$752$		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	0.687745	−	0.725952i	$-0.258601\pi$
$761$	−7.85711	−	13.6089i	−0.284820	−	0.493323i	−0.972566	+	0.232629i	$0.925267\pi$
$762$		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	0.925455	−	0.378857i	$-0.123683\pi$
$769$	29.3970	+	16.9724i	1.06008	+	0.612040i	0.134628	+	0.990896i	$0.457016\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24.4346	+	42.3220i	0.878852	+	1.52222i	0.852602	+	0.522562i	$0.175024\pi$
$773$	24.4346	+	42.3220i	0.878852	+	1.52222i	0.0262509	+	0.999655i	$0.491643\pi$
$774$		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	$0.224149\pi$
$787$			36.3244i			1.29483i	−0.762139	+	0.647413i	$0.775851\pi$
$788$		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	−0.503543	−	0.863970i	$-0.667970\pi$
$797$	14.0153	−	24.2753i	0.496449	−	0.859875i	0.999992	+	0.00409568i	$0.00130370\pi$
$798$		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	−0.680123	−	0.733098i	$-0.738074\pi$
$809$	−10.9591	+	6.32725i	−0.385302	+	0.222454i	0.294820	+	0.955553i	$0.404740\pi$
$810$		0			0
$811$		−	33.7585i		−	1.18542i	−0.805415	−	0.592711i	$-0.798057\pi$
$811$		−	33.7585i		−	1.18542i	0.805415	−	0.592711i	$-0.201943\pi$
$812$		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	$-0.967333\pi$
$821$		−	5.87082i		−	0.204893i	0.994739	−	0.102447i	$-0.0326671\pi$
$822$		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	$0.277773\pi$
$827$			44.0586i			1.53207i	−0.642800	+	0.766034i	$0.722227\pi$
$828$		0			0
$829$	5.47912	−	3.16337i	0.190298	−	0.109868i	−0.401824	−	0.915717i	$-0.631624\pi$
$829$	5.47912	−	3.16337i	0.190298	−	0.109868i	0.592122	+	0.805848i	$0.298290\pi$
$830$		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	0.759954	−	0.649977i	$-0.225222\pi$
$839$	−5.29837	−	9.17704i	−0.182920	−	0.316827i	−0.942874	+	0.333151i	$0.891888\pi$
$840$		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	−0.136611	−	0.990625i	$-0.543621\pi$
$853$	19.0713	−	11.0108i	0.652989	−	0.377003i	0.789600	+	0.613621i	$0.210288\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.76205	−	3.05197i	0.0601906	−	0.104253i	−0.834360	−	0.551220i	$-0.814162\pi$
$857$	1.76205	−	3.05197i	0.0601906	−	0.104253i	0.894551	+	0.446967i	$0.147496\pi$
$858$		0			0
$859$	−32.0730	+	18.5174i	−1.09432	+	0.631805i	−0.934723	−	0.355378i	$-0.884352\pi$
$859$	−32.0730	+	18.5174i	−1.09432	+	0.631805i	−0.159595	+	0.987183i	$0.551019\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.5411	−	7.81797i	−0.460945	−	0.266127i	0.251496	−	0.967858i	$-0.419077\pi$
$863$	−13.5411	−	7.81797i	−0.460945	−	0.266127i	−0.712442	+	0.701731i	$0.752411\pi$
$864$		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	0.896716	−	0.442606i	$-0.145946\pi$
$877$	1.92641	+	3.33664i	0.0650502	+	0.112670i	−0.831666	+	0.555276i	$0.812613\pi$
$878$		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	0.898519	−	0.438934i	$-0.144644\pi$
$887$	2.05890	+	3.56612i	0.0691311	+	0.119739i	−0.829388	+	0.558673i	$0.811311\pi$
$888$		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	0.265836	+	0.964018i	$0.414352\pi$
$907$	−21.1401	+	36.6158i	−0.701946	+	1.21581i	−0.967782	+	0.251789i	$0.918981\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.6408	+	9.60760i	−0.551336	+	0.318314i	−0.749661	−	0.661822i	$-0.769783\pi$
$911$	−16.6408	+	9.60760i	−0.551336	+	0.318314i	0.198325	+	0.980136i	$0.436450\pi$
$912$		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	−0.749330	−	0.662197i	$-0.769624\pi$
$919$	6.02706	−	10.4392i	0.198814	−	0.344357i	0.948144	+	0.317840i	$0.102958\pi$
$920$		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	$0.222629\pi$
$937$			39.4119i			1.28753i	−0.765222	+	0.643766i	$0.777371\pi$
$938$		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	$-0.838733\pi$
$947$		−	29.8649i		−	0.970478i	0.874382	−	0.485239i	$-0.161267\pi$
$948$		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	$0.118652\pi$
$953$			22.4851i			0.728365i	−0.931328	+	0.364183i	$0.881348\pi$
$954$		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	−0.833613	−	0.552348i	$-0.813732\pi$
$967$	1.91372	−	3.31466i	0.0615411	−	0.106592i	0.895154	+	0.445756i	$0.147065\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	29.4317	−	50.9773i	0.944509	−	1.63594i	0.187779	−	0.982211i	$-0.439871\pi$
$971$	29.4317	−	50.9773i	0.944509	−	1.63594i	0.756731	−	0.653727i	$-0.226795\pi$
$972$		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	$0.0542387\pi$
$977$			10.6006i			0.339145i	−0.985518	+	0.169572i	$0.945761\pi$
$978$		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	−0.479089	−	0.877766i	$-0.659033\pi$
$983$	16.3230	−	28.2723i	0.520624	−	0.901747i	0.999712	−	0.0239802i	$-0.00763386\pi$
$984$		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	0.999730	+	0.0232498i	$0.00740131\pi$
$991$	16.3697	+	28.3531i	0.520000	+	0.900666i	−0.479730	+	0.877416i	$0.659265\pi$
$992$		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	0.970126	−	0.242601i	$-0.0780004\pi$
$997$	39.3141	+	22.6980i	1.24509	+	0.718854i	0.274965	+	0.961454i	$0.411334\pi$
$998$		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.1349.10		32
3.2	odd	2	inner	2268.2.w.j.1349.7		32
7.3	odd	6		2268.2.bm.j.1025.10		32
9.2	odd	6		2268.2.bm.j.593.10		32
9.4	even	3		2268.2.t.c.2105.10	yes	32
9.5	odd	6		2268.2.t.c.2105.7	yes	32
9.7	even	3		2268.2.bm.j.593.7		32
21.17	even	6		2268.2.bm.j.1025.7		32
63.31	odd	6		2268.2.t.c.1781.7	✓	32
63.38	even	6	inner	2268.2.w.j.269.10		32
63.52	odd	6	inner	2268.2.w.j.269.7		32
63.59	even	6		2268.2.t.c.1781.10	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.t.c.1781.7	✓	32	63.31	odd	6
2268.2.t.c.1781.10	yes	32	63.59	even	6
2268.2.t.c.2105.7	yes	32	9.5	odd	6
2268.2.t.c.2105.10	yes	32	9.4	even	3
2268.2.w.j.269.7		32	63.52	odd	6	inner
2268.2.w.j.269.10		32	63.38	even	6	inner
2268.2.w.j.1349.7		32	3.2	odd	2	inner
2268.2.w.j.1349.10		32	1.1	even	1	trivial
2268.2.bm.j.593.7		32	9.7	even	3
2268.2.bm.j.593.10		32	9.2	odd	6
2268.2.bm.j.1025.7		32	21.17	even	6
2268.2.bm.j.1025.10		32	7.3	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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