LMFDB - Embedded newform 2268.2.t.c.2105.10

Error: table mf_hecke_newspace_traces does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(1781,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.1781"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2105.10
Character	$\chi$	$=$	2268.2105
Dual form			2268.2.t.c.1781.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.440135 - 0.762336i) q^{5} +(0.391046 - 2.61669i) q^{7} +(4.83660 - 2.79241i) q^{11} +2.59584i q^{13} +(2.35671 + 4.08193i) q^{17} +(0.537797 + 0.310497i) q^{19} +(-3.15847 - 1.82354i) q^{23} +(2.11256 + 3.65906i) q^{25} +5.33890i q^{29} +(9.56034 - 5.51967i) q^{31} +(-1.82269 - 1.44981i) q^{35} +(4.28064 - 7.41428i) q^{37} -6.82696 q^{41} +3.65862 q^{43} +(-3.88689 + 6.73229i) q^{47} +(-6.69417 - 2.04650i) q^{49} +(5.87725 - 3.39323i) q^{53} -4.91616i q^{55} +(-5.35626 - 9.27732i) q^{59} +(-1.77710 - 1.02601i) q^{61} +(1.97890 + 1.14252i) q^{65} +(-4.52099 - 7.83059i) q^{67} +12.3011i q^{71} +(-3.43707 + 1.98439i) q^{73} +(-5.41555 - 13.7479i) q^{77} +(6.01045 - 10.4104i) q^{79} +3.43944 q^{83} +4.14908 q^{85} +(3.40619 - 5.89969i) q^{89} +(6.79251 + 1.01509i) q^{91} +(0.473407 - 0.273322i) q^{95} -16.2182i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) $ 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.440135	−	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
$7$	0.391046	−	2.61669i	0.147802	−	0.989017i
$7$	0.391046	−	2.61669i	0.147802	−	0.989017i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.83660	−	2.79241i	1.45829	−	0.841944i	0.459362	−	0.888249i	$-0.348078\pi$
$11$	4.83660	−	2.79241i	1.45829	−	0.841944i	0.998927	+	0.0463050i	$0.0147446\pi$
$12$		0			0
$13$			2.59584i			0.719956i	0.932961	+	0.359978i	$0.117216\pi$
$13$			2.59584i			0.719956i	−0.932961	+	0.359978i	$0.882784\pi$
$14$		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	$-0.457490\pi$
$23$	−3.15847	−	1.82354i	−0.658586	−	0.380235i	−0.791738	+	0.610861i	$0.790823\pi$
$24$		0			0
$25$	2.11256	+	3.65906i	0.422512	+	0.731813i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.33890i			0.991409i	0.868491	+	0.495705i	$0.165090\pi$
$29$			5.33890i			0.991409i	−0.868491	+	0.495705i	$0.834910\pi$
$30$		0			0
$31$	9.56034	−	5.51967i	1.71709	−	0.991361i	0.792965	−	0.609267i	$-0.208536\pi$
$31$	9.56034	−	5.51967i	1.71709	−	0.991361i	0.924123	−	0.382094i	$-0.124797\pi$
$32$		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
$41$	−6.82696			−1.06619			−0.533096	−	0.846055i	$-0.678972\pi$
$41$	−6.82696			−1.06619			−0.533096	+	0.846055i	$0.678972\pi$
$42$		0			0
$43$	3.65862			0.557935			0.278967	−	0.960301i	$-0.410008\pi$
$43$	3.65862			0.557935			0.278967	+	0.960301i	$0.410008\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.88689	+	6.73229i	−0.566961	+	0.982005i	0.429903	+	0.902875i	$0.358548\pi$
$47$	−3.88689	+	6.73229i	−0.566961	+	0.982005i	−0.996864	+	0.0791303i	$0.974786\pi$
$48$		0			0
$49$	−6.69417	−	2.04650i	−0.956309	−	0.292357i
$50$		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
$59$	−5.35626	−	9.27732i	−0.697326	−	1.20780i	−0.969390	−	0.245525i	$-0.921040\pi$
$59$	−5.35626	−	9.27732i	−0.697326	−	1.20780i	0.272064	−	0.962279i	$-0.412294\pi$
$60$		0			0
$61$	−1.77710	−	1.02601i	−0.227534	−	0.131367i	0.381900	−	0.924204i	$-0.375270\pi$
$61$	−1.77710	−	1.02601i	−0.227534	−	0.131367i	−0.609434	+	0.792837i	$0.708603\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.97890	+	1.14252i	0.245453	+	0.141712i
$66$		0			0
$67$	−4.52099	−	7.83059i	−0.552327	−	0.956659i	−0.998106	−	0.0615158i	$-0.980407\pi$
$67$	−4.52099	−	7.83059i	−0.552327	−	0.956659i	0.445779	−	0.895143i	$-0.352927\pi$
$68$		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
$77$	−5.41555	−	13.7479i	−0.617159	−	1.56671i
$78$		0			0
$79$	6.01045	−	10.4104i	0.676228	−	1.17126i	−0.299880	−	0.953977i	$-0.596947\pi$
$79$	6.01045	−	10.4104i	0.676228	−	1.17126i	0.976108	−	0.217285i	$-0.0697200\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.43944			0.377527			0.188764	−	0.982023i	$-0.439552\pi$
$83$	3.43944			0.377527			0.188764	+	0.982023i	$0.439552\pi$
$84$		0			0
$85$	4.14908			0.450031
$86$		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
$95$	0.473407	−	0.273322i	0.0485705	−	0.0280422i
$96$		0			0
$97$		−	16.2182i		−	1.64670i	−0.567530	−	0.823352i	$-0.692101\pi$
$97$		−	16.2182i		−	1.64670i	0.567530	−	0.823352i	$-0.307899\pi$
$98$		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	0.930820	−	0.365479i	$-0.119095\pi$
$103$	10.9579	+	6.32656i	1.07972	+	0.623374i	0.148896	+	0.988853i	$0.452428\pi$
$104$		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
$113$		−	10.6032i		−	0.997461i	−0.866757	−	0.498731i	$-0.833800\pi$
$113$		−	10.6032i		−	0.997461i	0.866757	−	0.498731i	$-0.166200\pi$
$114$		0			0
$115$	−2.78030	+	1.60521i	−0.259265	+	0.149687i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.6028	−	4.57055i	1.06362	−	0.418982i
$120$		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
$133$	1.02278	−	1.28583i	0.0886863	−	0.111496i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.41744	+	3.70511i	−0.548279	+	0.316549i	−0.748427	−	0.663217i	$-0.769191\pi$
$137$	−6.41744	+	3.70511i	−0.548279	+	0.316549i	0.200149	+	0.979766i	$0.435857\pi$
$138$		0			0
$139$			3.41627i			0.289764i	0.989449	+	0.144882i	$0.0462802\pi$
$139$			3.41627i			0.289764i	−0.989449	+	0.144882i	$0.953720\pi$
$140$		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	$-0.470161\pi$
$149$	−8.81086	−	5.08695i	−0.721814	−	0.416739i	−0.815420	+	0.578870i	$0.803494\pi$
$150$		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
$155$		−	9.71760i		−	0.780536i
$156$		0			0
$157$	−10.2160	+	5.89822i	−0.815327	+	0.470729i	−0.848802	−	0.528710i	$-0.822676\pi$
$157$	−10.2160	+	5.89822i	−0.815327	+	0.470729i	0.0334752	+	0.999440i	$0.489343\pi$
$158$		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
$167$	−15.7708			−1.22038			−0.610189	−	0.792256i	$-0.708907\pi$
$167$	−15.7708			−1.22038			−0.610189	+	0.792256i	$0.708907\pi$
$168$		0			0
$169$	6.26163			0.481664
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.38670	+	2.40184i	−0.105429	+	0.182609i	−0.913913	−	0.405909i	$-0.866955\pi$
$173$	−1.38670	+	2.40184i	−0.105429	+	0.182609i	0.808484	+	0.588518i	$0.200288\pi$
$174$		0			0
$175$	10.4008	−	4.09706i	0.786223	−	0.309709i
$176$		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
$185$	−3.76812	−	6.52657i	−0.277038	−	0.479843i
$186$		0			0
$187$	22.7969	+	13.1618i	1.66707	+	0.962485i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	15.5684	+	8.98842i	1.12649	+	0.650380i	0.943050	−	0.332651i	$-0.107943\pi$
$191$	15.5684	+	8.98842i	1.12649	+	0.650380i	0.183440	+	0.983031i	$0.441277\pi$
$192$		0			0
$193$	1.88040	+	3.25694i	0.135354	+	0.234440i	0.925733	−	0.378179i	$-0.123450\pi$
$193$	1.88040	+	3.25694i	0.135354	+	0.234440i	−0.790379	+	0.612619i	$0.790116\pi$
$194$		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
$203$	13.9703	+	2.08776i	0.980521	+	0.146532i
$204$		0			0
$205$	−3.00479	+	5.20444i	−0.209863	+	0.363494i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.46815			0.239897
$210$		0			0
$211$	−2.52094			−0.173549			−0.0867744	−	0.996228i	$-0.527656\pi$
$211$	−2.52094			−0.173549			−0.0867744	+	0.996228i	$0.527656\pi$
$212$		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
$221$	−10.5960	+	6.11763i	−0.712767	+	0.411516i
$222$		0			0
$223$			12.3148i			0.824663i	0.911034	+	0.412332i	$0.135285\pi$
$223$			12.3148i			0.824663i	−0.911034	+	0.412332i	$0.864715\pi$
$224$		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	0.324915	−	0.945743i	$-0.394664\pi$
$229$	−5.01900	−	2.89772i	−0.331665	−	0.191487i	−0.656580	+	0.754256i	$0.727998\pi$
$230$		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
$239$		−	3.41885i		−	0.221147i	−0.993868	−	0.110573i	$-0.964731\pi$
$239$		−	3.41885i		−	0.221147i	0.993868	−	0.110573i	$-0.0352688\pi$
$240$		0			0
$241$	−4.24030	+	2.44814i	−0.273142	+	0.157699i	−0.630315	−	0.776340i	$-0.717074\pi$
$241$	−4.24030	+	2.44814i	−0.273142	+	0.157699i	0.357173	+	0.934038i	$0.383741\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50646	+	4.20247i	−0.287907	+	0.268486i
$246$		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
$259$	−17.7270	−	14.1004i	−1.10150	−	0.876159i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4.13580	+	2.38780i	−0.255024	+	0.147238i	−0.622063	−	0.782967i	$-0.713705\pi$
$263$	−4.13580	+	2.38780i	−0.255024	+	0.147238i	0.367038	+	0.930206i	$0.380372\pi$
$264$		0			0
$265$		−	5.97392i		−	0.366975i
$266$		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	−0.899135	−	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
$281$			28.5861i			1.70530i	0.522479	+	0.852652i	$0.325007\pi$
$281$			28.5861i			1.70530i	−0.522479	+	0.852652i	$0.674993\pi$
$282$		0			0
$283$	−16.3146	+	9.41927i	−0.969805	+	0.559917i	−0.899177	−	0.437585i	$-0.855834\pi$
$283$	−16.3146	+	9.41927i	−0.969805	+	0.559917i	−0.0706283	+	0.997503i	$0.522500\pi$
$284$		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
$293$	33.6167			1.96391			0.981954	−	0.189121i	$-0.0605638\pi$
$293$	33.6167			1.96391			0.981954	+	0.189121i	$0.0605638\pi$
$294$		0			0
$295$	−9.42992			−0.549031
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.73362	−	8.19886i	0.273752	−	0.474153i
$300$		0			0
$301$	1.43069	−	9.57350i	0.0824637	−	0.551807i
$302$		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
$311$	−5.95735	−	10.3184i	−0.337810	−	0.585104i	0.646210	−	0.763159i	$-0.276353\pi$
$311$	−5.95735	−	10.3184i	−0.337810	−	0.585104i	−0.984021	+	0.178055i	$0.943020\pi$
$312$		0			0
$313$	−5.33683	−	3.08122i	−0.301655	−	0.174161i	0.341531	−	0.939871i	$-0.389055\pi$
$313$	−5.33683	−	3.08122i	−0.301655	−	0.174161i	−0.643186	+	0.765710i	$0.722388\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3.33379	+	1.92476i	0.187244	+	0.108105i	0.590692	−	0.806897i	$-0.298855\pi$
$317$	3.33379	+	1.92476i	0.187244	+	0.108105i	−0.403448	+	0.915003i	$0.632188\pi$
$318$		0			0
$319$	14.9084	+	25.8221i	0.834711	+	1.44576i
$320$		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
$329$	16.0964	+	12.8034i	0.887422	+	0.705876i
$330$		0			0
$331$	8.81578	−	15.2694i	0.484559	−	0.839281i	−0.515284	−	0.857020i	$-0.672313\pi$
$331$	8.81578	−	15.2694i	0.484559	−	0.839281i	0.999843	+	0.0177388i	$0.00564674\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.95939			−0.434868
$336$		0			0
$337$	−19.6690			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$337$	−19.6690			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$338$		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
$347$	−12.5819	+	7.26418i	−0.675433	+	0.389962i	−0.798132	−	0.602482i	$-0.794178\pi$
$347$	−12.5819	+	7.26418i	−0.675433	+	0.389962i	0.122699	+	0.992444i	$0.460845\pi$
$348$		0			0
$349$			31.1316i			1.66644i	0.552944	+	0.833218i	$0.313504\pi$
$349$			31.1316i			1.66644i	−0.552944	+	0.833218i	$0.686496\pi$
$350$		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	−0.349557	−	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
$365$			3.49360i			0.182863i
$366$		0			0
$367$	0.491228	−	0.283610i	0.0256419	−	0.0148043i	−0.487124	−	0.873333i	$-0.661954\pi$
$367$	0.491228	−	0.283610i	0.0256419	−	0.0148043i	0.512766	+	0.858528i	$0.328621\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.58077	−	16.7059i	−0.341657	−	0.867326i
$372$		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
$385$	−12.8641	−	1.92244i	−0.655614	−	0.0979768i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−18.2236	+	10.5214i	−0.923976	+	0.533458i	−0.884901	−	0.465779i	$-0.845774\pi$
$389$	−18.2236	+	10.5214i	−0.923976	+	0.533458i	−0.0390745	+	0.999236i	$0.512441\pi$
$390$		0			0
$391$		−	17.1902i		−	0.869346i
$392$		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	0.993117	+	0.117129i	$0.0373690\pi$
$401$	31.8620	+	18.3955i	1.59111	+	0.918629i	0.597995	+	0.801500i	$0.295964\pi$
$402$		0			0
$403$	14.3282	+	24.8171i	0.713736	+	1.23623i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	47.8132i		−	2.37001i
$408$		0			0
$409$	−29.1585	+	16.8347i	−1.44180	+	0.832422i	−0.997970	−	0.0636901i	$-0.979713\pi$
$409$	−29.1585	+	16.8347i	−1.44180	+	0.832422i	−0.443828	+	0.896112i	$0.646380\pi$
$410$		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
$419$	−9.19382			−0.449148			−0.224574	−	0.974457i	$-0.572099\pi$
$419$	−9.19382			−0.449148			−0.224574	+	0.974457i	$0.572099\pi$
$420$		0			0
$421$	−8.60745			−0.419502			−0.209751	−	0.977755i	$-0.567265\pi$
$421$	−8.60745			−0.419502			−0.209751	+	0.977755i	$0.567265\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−9.95738	+	17.2467i	−0.483004	+	0.836587i
$426$		0			0
$427$	−3.37968	+	4.24890i	−0.163554	+	0.205619i
$428$		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
$437$	−1.13241	−	1.96139i	−0.0541705	−	0.0938260i
$438$		0			0
$439$	22.0686	+	12.7413i	1.05328	+	0.608110i	0.923565	−	0.383441i	$-0.125261\pi$
$439$	22.0686	+	12.7413i	1.05328	+	0.608110i	0.129713	+	0.991552i	$0.458594\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.81934	−	2.78244i	−0.228974	−	0.132198i	0.381125	−	0.924524i	$-0.375537\pi$
$443$	−4.81934	−	2.78244i	−0.228974	−	0.132198i	−0.610098	+	0.792326i	$0.708870\pi$
$444$		0			0
$445$	−2.99837	−	5.19332i	−0.142136	−	0.246187i
$446$		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
$455$	3.76346	−	4.73140i	0.176434	−	0.221811i
$456$		0			0
$457$	−11.1809	+	19.3659i	−0.523021	+	0.905899i	0.476620	+	0.879109i	$0.341862\pi$
$457$	−11.1809	+	19.3659i	−0.523021	+	0.905899i	−0.999641	+	0.0267896i	$0.991472\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−26.5005			−1.23425			−0.617126	−	0.786864i	$-0.711703\pi$
$461$	−26.5005			−1.23425			−0.617126	+	0.786864i	$0.711703\pi$
$462$		0			0
$463$	−14.4875			−0.673293			−0.336646	−	0.941631i	$-0.609293\pi$
$463$	−14.4875			−0.673293			−0.336646	+	0.941631i	$0.609293\pi$
$464$		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
$473$	17.6953	−	10.2164i	0.813631	−	0.469750i
$474$		0			0
$475$			2.62378i			0.120387i
$476$		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	0.783276	−	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
$491$			35.5282i			1.60337i	0.597750	+	0.801683i	$0.296062\pi$
$491$			35.5282i			1.60337i	−0.597750	+	0.801683i	$0.703938\pi$
$492$		0			0
$493$	−21.7930	+	12.5822i	−0.981510	+	0.566675i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	32.1883	+	4.81031i	1.44384	+	0.215772i
$498$		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
$511$	3.84849	+	9.76974i	0.170247	+	0.432188i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9.64593	−	5.56908i	0.425050	−	0.245403i
$516$		0			0
$517$			43.4152i			1.90940i
$518$		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
$533$		−	17.7217i		−	0.767612i
$534$		0			0
$535$	−1.84559	+	1.06555i	−0.0797920	+	0.0460679i
$536$		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
$545$	0.504246			0.0215995
$546$		0			0
$547$	−15.3929			−0.658154			−0.329077	−	0.944303i	$-0.606738\pi$
$547$	−15.3929			−0.658154			−0.329077	+	0.944303i	$0.606738\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.65771	+	2.87125i	−0.0706210	+	0.122319i
$552$		0			0
$553$	−24.8905	−	19.7984i	−1.05845	−	0.841915i
$554$		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
$563$	9.73637	+	16.8639i	0.410339	+	0.710728i	0.994927	−	0.100603i	$-0.0320771\pi$
$563$	9.73637	+	16.8639i	0.410339	+	0.710728i	−0.584588	+	0.811331i	$0.698744\pi$
$564$		0			0
$565$	−8.08317	−	4.66682i	−0.340062	−	0.196335i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.73683	+	1.58011i	0.114734	+	0.0662417i	0.556269	−	0.831002i	$-0.312233\pi$
$569$	2.73683	+	1.58011i	0.114734	+	0.0662417i	−0.441535	+	0.897244i	$0.645566\pi$
$570$		0			0
$571$	11.5293	+	19.9693i	0.482486	+	0.835690i	0.999798	−	0.0201068i	$-0.00640061\pi$
$571$	11.5293	+	19.9693i	0.482486	+	0.835690i	−0.517312	+	0.855797i	$0.673067\pi$
$572$		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
$581$	1.34498	−	8.99995i	0.0557992	−	0.373381i
$582$		0			0
$583$	18.9506	−	32.8234i	0.784854	−	1.35941i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.0485			0.414746			0.207373	−	0.978262i	$-0.433509\pi$
$587$	10.0485			0.414746			0.207373	+	0.978262i	$0.433509\pi$
$588$		0			0
$589$	6.85537			0.282470
$590$		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
$599$	31.4823	−	18.1763i	1.28633	−	0.742665i	0.308335	−	0.951278i	$-0.400228\pi$
$599$	31.4823	−	18.1763i	1.28633	−	0.742665i	0.977998	+	0.208613i	$0.0668948\pi$
$600$		0			0
$601$			3.14659i			0.128352i	0.997939	+	0.0641760i	$0.0204419\pi$
$601$			3.14659i			0.128352i	−0.997939	+	0.0641760i	$0.979558\pi$
$602$		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	$-0.372713\pi$
$607$	−5.26589	−	3.04026i	−0.213736	−	0.123400i	−0.603046	+	0.797706i	$0.706047\pi$
$608$		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
$617$		−	28.1759i		−	1.13432i	−0.823608	−	0.567160i	$-0.808042\pi$
$617$		−	28.1759i		−	1.13432i	0.823608	−	0.567160i	$-0.191958\pi$
$618$		0			0
$619$	−30.8831	+	17.8304i	−1.24130	+	0.716663i	−0.969358	−	0.245651i	$-0.920998\pi$
$619$	−30.8831	+	17.8304i	−1.24130	+	0.716663i	−0.271939	+	0.962314i	$0.587665\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.1057	−	11.2200i	−0.565133	−	0.449520i
$624$		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
$637$	5.31237	−	17.3770i	0.210484	−	0.688500i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.6617	−	6.15551i	0.421110	−	0.243128i	−0.274442	−	0.961604i	$-0.588493\pi$
$641$	10.6617	−	6.15551i	0.421110	−	0.243128i	0.695552	+	0.718476i	$0.255160\pi$
$642$		0			0
$643$			45.3728i			1.78933i	0.446739	+	0.894664i	$0.352585\pi$
$643$			45.3728i			1.78933i	−0.446739	+	0.894664i	$0.647415\pi$
$644$		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	$-0.274951\pi$
$653$	8.07258	+	4.66071i	0.315905	+	0.182388i	−0.333661	+	0.942693i	$0.608284\pi$
$654$		0			0
$655$	5.11250	+	8.85511i	0.199762	+	0.345998i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			11.8913i			0.463218i	0.972809	+	0.231609i	$0.0743990\pi$
$659$			11.8913i			0.463218i	−0.972809	+	0.231609i	$0.925601\pi$
$660$		0			0
$661$	−8.32726	+	4.80775i	−0.323893	+	0.187000i	−0.653126	−	0.757249i	$-0.726543\pi$
$661$	−8.32726	+	4.80775i	−0.323893	+	0.187000i	0.329234	+	0.944249i	$0.393210\pi$
$662$		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
$671$	−11.4601			−0.442414
$672$		0			0
$673$	−22.5795			−0.870377			−0.435188	−	0.900339i	$-0.643318\pi$
$673$	−22.5795			−0.870377			−0.435188	+	0.900339i	$0.643318\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.4679	+	37.1834i	−0.825077	+	1.42907i	0.0767843	+	0.997048i	$0.475535\pi$
$677$	−21.4679	+	37.1834i	−0.825077	+	1.42907i	−0.901861	+	0.432027i	$0.857799\pi$
$678$		0			0
$679$	−42.4380	−	6.34205i	−1.62862	−	0.243386i
$680$		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
$689$	8.80828	+	15.2564i	0.335569	+	0.581222i
$690$		0			0
$691$	5.60142	+	3.23398i	0.213088	+	0.123026i	0.602746	−	0.797933i	$-0.294073\pi$
$691$	5.60142	+	3.23398i	0.213088	+	0.123026i	−0.389658	+	0.920960i	$0.627407\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.60434	+	1.50362i	0.0987884	+	0.0570355i
$696$		0			0
$697$	−16.0891	−	27.8672i	−0.609420	−	1.05555i
$698$		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
$707$	33.2680	−	13.1049i	1.25117	−	0.492861i
$708$		0			0
$709$	17.4171	−	30.1673i	0.654113	−	1.13296i	−0.328002	−	0.944677i	$-0.606375\pi$
$709$	17.4171	−	30.1673i	0.654113	−	1.13296i	0.982115	−	0.188280i	$-0.0602913\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−40.2614			−1.50780
$714$		0			0
$715$	12.7615			0.477255
$716$		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
$725$	−19.5354	+	11.2788i	−0.725526	+	0.418883i
$726$		0			0
$727$			5.97941i			0.221764i	0.993834	+	0.110882i	$0.0353676\pi$
$727$			5.97941i			0.221764i	−0.993834	+	0.110882i	$0.964632\pi$
$728$		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	$-0.238299\pi$
$733$	13.7934	+	7.96362i	0.509471	+	0.294143i	−0.223145	+	0.974785i	$0.571632\pi$
$734$		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
$743$			23.9809i			0.879775i	0.898053	+	0.439887i	$0.144982\pi$
$743$			23.9809i			0.879775i	−0.898053	+	0.439887i	$0.855018\pi$
$744$		0			0
$745$	−7.75594	+	4.47789i	−0.284156	+	0.164057i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−3.98734	+	5.01286i	−0.145694	+	0.183166i
$750$		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
$763$	1.41011	−	0.555469i	0.0510493	−	0.0201093i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.0824	−	13.9040i	0.869566	−	0.502044i
$768$		0			0
$769$		−	33.9448i		−	1.22408i	−0.790827	−	0.612040i	$-0.790349\pi$
$769$		−	33.9448i		−	1.22408i	0.790827	−	0.612040i	$-0.209651\pi$
$770$		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
$785$			10.3841i			0.370623i
$786$		0			0
$787$	31.4579	−	18.1622i	1.12135	−	0.647413i	0.179607	−	0.983738i	$-0.442517\pi$
$787$	31.4579	−	18.1622i	1.12135	−	0.647413i	0.941746	+	0.336325i	$0.109184\pi$
$788$		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
$797$	−28.0307			−0.992898			−0.496449	−	0.868066i	$-0.665363\pi$
$797$	−28.0307			−0.992898			−0.496449	+	0.868066i	$0.665363\pi$
$798$		0			0
$799$	−36.6410			−1.29627
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.0825	+	19.1954i	−0.391092	+	0.677391i
$804$		0			0
$805$	3.11311	+	7.90291i	0.109723	+	0.278541i
$806$		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
$815$	−5.11455	−	8.85866i	−0.179155	−	0.310305i
$816$		0			0
$817$	1.96760	+	1.13599i	0.0688375	+	0.0397434i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.08428	+	2.93541i	0.177443	+	0.102447i	0.586091	−	0.810246i	$-0.300666\pi$
$821$	5.08428	+	2.93541i	0.177443	+	0.102447i	−0.408648	+	0.912692i	$0.634000\pi$
$822$		0			0
$823$	−15.4509	−	26.7617i	−0.538583	−	0.932853i	−0.998981	−	0.0451404i	$-0.985626\pi$
$823$	−15.4509	−	26.7617i	−0.538583	−	0.932853i	0.460398	−	0.887713i	$-0.347707\pi$
$824$		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
$833$	−7.42252	−	32.1481i	−0.257175	−	1.11387i
$834$		0			0
$835$	−6.94127	+	12.0226i	−0.240212	+	0.416060i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.5967			0.365840			0.182920	−	0.983128i	$-0.441445\pi$
$839$	10.5967			0.365840			0.182920	+	0.983128i	$0.441445\pi$
$840$		0			0
$841$	0.496127			0.0171078
$842$		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
$851$	−27.0405	+	15.6118i	−0.926936	+	0.535167i
$852$		0			0
$853$			22.0216i			0.754007i	0.926212	+	0.377003i	$0.123045\pi$
$853$			22.0216i			0.754007i	−0.926212	+	0.377003i	$0.876955\pi$
$854$		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	0.934723	−	0.355378i	$-0.115648\pi$
$859$	32.0730	+	18.5174i	1.09432	+	0.631805i	0.159595	+	0.987183i	$0.448981\pi$
$860$		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
$869$		−	67.1346i		−	2.27738i
$870$		0			0
$871$	20.3269	−	11.7358i	0.688752	−	0.397651i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.17553	−	21.2491i	0.107353	−	0.718352i
$876$		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
$889$	1.11676	−	7.47282i	0.0374549	−	0.250630i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4.18072	+	2.41374i	−0.139902	+	0.0807726i
$894$		0			0
$895$			6.04180i			0.201955i
$896$		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
$911$		−	19.2152i		−	0.636628i	−0.947985	−	0.318314i	$-0.896883\pi$
$911$		−	19.2152i		−	0.636628i	0.947985	−	0.318314i	$-0.103117\pi$
$912$		0			0
$913$	16.6352	−	9.60433i	0.550544	−	0.317857i
$914$		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
$923$	−31.9317			−1.05105
$924$		0			0
$925$	36.1724			1.18934
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.2907	−	31.6805i	0.600099	−	1.03940i	−0.392706	−	0.919664i	$-0.628461\pi$
$929$	18.2907	−	31.6805i	0.600099	−	1.03940i	0.992806	−	0.119738i	$-0.0382055\pi$
$930$		0			0
$931$	−2.96467	−	3.17912i	−0.0971632	−	0.104191i
$932$		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
$941$	−6.21752	−	10.7691i	−0.202685	−	0.351061i	0.746707	−	0.665153i	$-0.231634\pi$
$941$	−6.21752	−	10.7691i	−0.202685	−	0.351061i	−0.949393	+	0.314091i	$0.898300\pi$
$942$		0			0
$943$	21.5627	+	12.4492i	0.702179	+	0.405403i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	25.8637	+	14.9324i	0.840459	+	0.485239i	0.857420	−	0.514617i	$-0.172066\pi$
$947$	25.8637	+	14.9324i	0.840459	+	0.485239i	−0.0169614	+	0.999856i	$0.505399\pi$
$948$		0			0
$949$	−5.15116	−	8.92206i	−0.167214	−	0.289622i
$950$		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
$959$	7.18562	+	18.2413i	0.232036	+	0.589043i
$960$		0			0
$961$	45.4334	−	78.6930i	1.46559	−	2.53848i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.31051			0.106569
$966$		0			0
$967$	−3.82744			−0.123082			−0.0615411	−	0.998105i	$-0.519602\pi$
$967$	−3.82744			−0.123082			−0.0615411	+	0.998105i	$0.519602\pi$
$968$		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
$977$	9.18043	−	5.30032i	0.293708	−	0.169572i	−0.345905	−	0.938270i	$-0.612428\pi$
$977$	9.18043	−	5.30032i	0.293708	−	0.169572i	0.639613	+	0.768697i	$0.279095\pi$
$978$		0			0
$979$		−	38.0459i		−	1.21595i
$980$		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	0.999730	+	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
$995$		−	12.2882i		−	0.389563i
$996$		0			0
$997$	−39.3141	+	22.6980i	−1.24509	+	0.718854i	−0.970126	−	0.242601i	$-0.922000\pi$
$997$	−39.3141	+	22.6980i	−1.24509	+	0.718854i	−0.274965	+	0.961454i	$0.588666\pi$
$998$		0			0
$999$		0			0

Twists

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

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

\(f(q)\)	\(=\)	\(q+(0.440135 - 0.762336i) q^{5} +(0.391046 - 2.61669i) q^{7} +(4.83660 - 2.79241i) q^{11} +2.59584i q^{13} +(2.35671 + 4.08193i) q^{17} +(0.537797 + 0.310497i) q^{19} +(-3.15847 - 1.82354i) q^{23} +(2.11256 + 3.65906i) q^{25} +5.33890i q^{29} +(9.56034 - 5.51967i) q^{31} +(-1.82269 - 1.44981i) q^{35} +(4.28064 - 7.41428i) q^{37} -6.82696 q^{41} +3.65862 q^{43} +(-3.88689 + 6.73229i) q^{47} +(-6.69417 - 2.04650i) q^{49} +(5.87725 - 3.39323i) q^{53} -4.91616i q^{55} +(-5.35626 - 9.27732i) q^{59} +(-1.77710 - 1.02601i) q^{61} +(1.97890 + 1.14252i) q^{65} +(-4.52099 - 7.83059i) q^{67} +12.3011i q^{71} +(-3.43707 + 1.98439i) q^{73} +(-5.41555 - 13.7479i) q^{77} +(6.01045 - 10.4104i) q^{79} +3.43944 q^{83} +4.14908 q^{85} +(3.40619 - 5.89969i) q^{89} +(6.79251 + 1.01509i) q^{91} +(0.473407 - 0.273322i) q^{95} -16.2182i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 16 q^{25} + 24 q^{31} - 4 q^{37} + 8 q^{43} - 4 q^{49} + 12 q^{61} + 4 q^{67} - 36 q^{73} + 28 q^{79} - 24 q^{85} - 36 q^{91}+O(q^{100}) \) 32 * q - 8 * q^7 - 16 * q^25 + 24 * q^31 - 4 * q^37 + 8 * q^43 - 4 * q^49 + 12 * q^61 + 4 * q^67 - 36 * q^73 + 28 * q^79 - 24 * q^85 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more