LMFDB - Embedded newform 2268.2.t.c.2105.7

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			2105.7
Character	$\chi$	$=$	2268.2105
Dual form			2268.2.t.c.1781.7

$q$-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	−0.947500	−	0.319755i	$-0.896399\pi$
$5$	−0.440135	+	0.762336i	−0.196834	+	0.340927i	0.750666	+	0.660682i	$0.229733\pi$
$6$		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	−0.998927	−	0.0463050i	$-0.985255\pi$
$11$	−4.83660	+	2.79241i	−1.45829	+	0.841944i	−0.459362	+	0.888249i	$0.651922\pi$
$12$		0			0
$13$			2.59584i			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	$-0.972995\pi$
$17$	−2.35671	−	4.08193i	−0.571585	−	0.990015i	0.424818	−	0.905279i	$-0.360338\pi$
$18$		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	0.791738	−	0.610861i	$-0.209177\pi$
$23$	3.15847	+	1.82354i	0.658586	+	0.380235i	−0.133152	+	0.991096i	$0.542510\pi$
$24$		0			0
$25$	2.11256	+	3.65906i	0.422512	+	0.731813i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.33890i		−	0.991409i	−0.868491	−	0.495705i	$-0.834910\pi$
$29$		−	5.33890i		−	0.991409i	0.868491	−	0.495705i	$-0.165090\pi$
$30$		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	$-0.321028\pi$
$41$	6.82696			1.06619			0.533096	+	0.846055i	$0.321028\pi$
$42$		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	$-0.641452\pi$
$47$	3.88689	−	6.73229i	0.566961	−	0.982005i	0.996864	−	0.0791303i	$-0.0252143\pi$
$48$		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	−0.846018	−	0.533154i	$-0.821007\pi$
$53$	−5.87725	+	3.39323i	−0.807303	+	0.466096i	0.0387156	+	0.999250i	$0.487673\pi$
$54$		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	$0.0789605\pi$
$59$	5.35626	+	9.27732i	0.697326	+	1.20780i	−0.272064	+	0.962279i	$0.587706\pi$
$60$		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	$-0.739549\pi$
$71$		−	12.3011i		−	1.45988i	0.683514	−	0.729938i	$-0.260451\pi$
$72$		0			0
$73$	−3.43707	+	1.98439i	−0.402278	+	0.232255i	−0.687466	−	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	$-0.560448\pi$
$83$	−3.43944			−0.377527			−0.188764	+	0.982023i	$0.560448\pi$
$84$		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	−0.988135	−	0.153589i	$-0.950917\pi$
$89$	−3.40619	+	5.89969i	−0.361055	+	0.625366i	0.627080	+	0.778955i	$0.284250\pi$
$90$		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	$-0.931942\pi$
$101$	−6.75727	−	11.7039i	−0.672373	−	1.16458i	0.304856	−	0.952398i	$-0.401392\pi$
$102$		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	0.597909	−	0.801564i	$-0.295999\pi$
$107$	2.09662	+	1.21048i	0.202688	+	0.117022i	−0.395221	+	0.918586i	$0.629332\pi$
$108$		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	$0.166200\pi$
$113$			10.6032i			0.997461i	−0.866757	+	0.498731i	$0.833800\pi$
$114$		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	$-0.663926\pi$
$131$	5.80788	−	10.0595i	0.507437	−	0.878906i	0.999963	−	0.00860838i	$-0.00274017\pi$
$132$		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	−0.200149	−	0.979766i	$-0.564143\pi$
$137$	6.41744	−	3.70511i	0.548279	−	0.316549i	0.748427	+	0.663217i	$0.230809\pi$
$138$		0			0
$139$			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	0.815420	−	0.578870i	$-0.196506\pi$
$149$	8.81086	+	5.08695i	0.721814	+	0.416739i	−0.0936062	+	0.995609i	$0.529839\pi$
$150$		0			0
$151$	2.15352	+	3.73001i	0.175251	+	0.303544i	0.940248	−	0.340490i	$-0.110593\pi$
$151$	2.15352	+	3.73001i	0.175251	+	0.303544i	−0.764997	+	0.644034i	$0.777260\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			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	$-0.291093\pi$
$167$	15.7708			1.22038			0.610189	+	0.792256i	$0.291093\pi$
$168$		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	−0.808484	−	0.588518i	$-0.799712\pi$
$173$	1.38670	−	2.40184i	0.105429	−	0.182609i	0.913913	+	0.405909i	$0.133045\pi$
$174$		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	−0.261133	−	0.965303i	$-0.584096\pi$
$179$	5.94403	−	3.43179i	0.444278	−	0.256504i	0.705410	+	0.708799i	$0.250763\pi$
$180$		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	−0.183440	−	0.983031i	$-0.558723\pi$
$191$	−15.5684	−	8.98842i	−1.12649	−	0.650380i	−0.943050	+	0.332651i	$0.892057\pi$
$192$		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	$0.0788575\pi$
$197$			6.88342i			0.490424i	−0.969470	+	0.245212i	$0.921142\pi$
$198$		0			0
$199$	12.0894	−	6.97981i	0.856993	−	0.494785i	−0.00601088	−	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	$0.159723\pi$
$227$	12.8806	+	22.3098i	0.854913	+	1.48075i	−0.0218124	+	0.999762i	$0.506944\pi$
$228$		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	$0.126466\pi$
$233$	26.2280	+	15.1428i	1.71826	+	0.992036i	0.796149	+	0.605101i	$0.206867\pi$
$234$		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	$0.0352688\pi$
$239$			3.41885i			0.221147i	−0.993868	+	0.110573i	$0.964731\pi$
$240$		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	$-0.829953\pi$
$251$	−27.2710			−1.72133			−0.860667	+	0.509169i	$0.829953\pi$
$252$		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	$-0.884446\pi$
$257$	−2.56317	+	4.43954i	−0.159886	+	0.276931i	0.774941	+	0.632033i	$0.217779\pi$
$258$		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	−0.367038	−	0.930206i	$-0.619628\pi$
$263$	4.13580	−	2.38780i	0.255024	−	0.147238i	0.622063	+	0.782967i	$0.286295\pi$
$264$		0			0
$265$		−	5.97392i		−	0.366975i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−7.34405	−	12.7203i	−0.447775	−	0.775569i	0.550466	−	0.834858i	$-0.314450\pi$
$269$	−7.34405	−	12.7203i	−0.447775	−	0.775569i	−0.998241	+	0.0592889i	$0.981117\pi$
$270$		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	$-0.674993\pi$
$281$		−	28.5861i		−	1.70530i	0.522479	−	0.852652i	$-0.325007\pi$
$282$		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	$-0.939436\pi$
$293$	−33.6167			−1.96391			−0.981954	+	0.189121i	$0.939436\pi$
$294$		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	0.984021	−	0.178055i	$-0.0569805\pi$
$311$	5.95735	+	10.3184i	0.337810	+	0.585104i	−0.646210	+	0.763159i	$0.723647\pi$
$312$		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	0.403448	−	0.915003i	$-0.367812\pi$
$317$	−3.33379	−	1.92476i	−0.187244	−	0.108105i	−0.590692	+	0.806897i	$0.701145\pi$
$318$		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	−0.122699	−	0.992444i	$-0.539155\pi$
$347$	12.5819	−	7.26418i	0.675433	−	0.389962i	0.798132	+	0.602482i	$0.205822\pi$
$348$		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	0.954497	−	0.298219i	$-0.0963927\pi$
$353$	4.11432	+	7.12621i	0.218983	+	0.379290i	−0.735514	+	0.677509i	$0.763059\pi$
$354$		0			0
$355$	9.37760	+	5.41416i	0.497711	+	0.287354i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	25.3084	+	14.6118i	1.33573	+	0.771183i	0.986171	−	0.165732i	$-0.0529987\pi$
$359$	25.3084	+	14.6118i	1.33573	+	0.771183i	0.349557	+	0.936915i	$0.386332\pi$
$360$		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	$-0.630850\pi$
$383$	11.6264	−	20.1375i	0.594080	−	1.02898i	0.993676	−	0.112286i	$-0.0358171\pi$
$384$		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	0.0390745	−	0.999236i	$-0.487559\pi$
$389$	18.2236	−	10.5214i	0.923976	−	0.533458i	0.884901	+	0.465779i	$0.154226\pi$
$390$		0			0
$391$		−	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	$-0.962631\pi$
$401$	−31.8620	−	18.3955i	−1.59111	−	0.918629i	−0.597995	−	0.801500i	$-0.704036\pi$
$402$		0			0
$403$	14.3282	+	24.8171i	0.713736	+	1.23623i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			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	$-0.427901\pi$
$419$	9.19382			0.449148			0.224574	+	0.974457i	$0.427901\pi$
$420$		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	−0.657007	−	0.753884i	$-0.728178\pi$
$431$	−6.90554	+	3.98691i	−0.332628	+	0.192043i	0.324379	+	0.945927i	$0.394845\pi$
$432$		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	0.610098	−	0.792326i	$-0.291130\pi$
$443$	4.81934	+	2.78244i	0.228974	+	0.132198i	−0.381125	+	0.924524i	$0.624463\pi$
$444$		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	$-0.785808\pi$
$449$		−	26.4132i		−	1.24652i	0.782016	−	0.623259i	$-0.214192\pi$
$450$		0			0
$451$	−33.0193	+	19.0637i	−1.55482	+	0.897674i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−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	$-0.288297\pi$
$461$	26.5005			1.23425			0.617126	+	0.786864i	$0.288297\pi$
$462$		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	−0.649571	−	0.760301i	$-0.725051\pi$
$467$	7.21034	−	12.4887i	0.333655	−	0.577907i	0.983226	+	0.182394i	$0.0583846\pi$
$468$		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	$0.0664022\pi$
$479$	14.6311	+	25.3418i	0.668513	+	1.15790i	−0.309807	+	0.950799i	$0.600264\pi$
$480$		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	$-0.703938\pi$
$491$		−	35.5282i		−	1.60337i	0.597750	−	0.801683i	$-0.296062\pi$
$492$		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	$-0.419662\pi$
$503$	11.2012			0.499437			0.249718	+	0.968319i	$0.419662\pi$
$504$		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	$-0.514982\pi$
$509$	18.9860	−	32.8848i	0.841541	−	1.45759i	0.888591	−	0.458699i	$-0.151685\pi$
$510$		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	0.858960	−	0.512042i	$-0.171111\pi$
$521$	−0.318682	−	0.551974i	−0.0139617	−	0.0241824i	−0.872922	+	0.487860i	$0.837778\pi$
$522$		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	−0.831956	−	0.554842i	$-0.812779\pi$
$557$	−18.1119	+	10.4569i	−0.767426	+	0.443074i	0.0645293	+	0.997916i	$0.479445\pi$
$558$		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	0.584588	−	0.811331i	$-0.301256\pi$
$563$	−9.73637	−	16.8639i	−0.410339	−	0.710728i	−0.994927	+	0.100603i	$0.967923\pi$
$564$		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	0.441535	−	0.897244i	$-0.354434\pi$
$569$	−2.73683	−	1.58011i	−0.114734	−	0.0662417i	−0.556269	+	0.831002i	$0.687767\pi$
$570$		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	$-0.566491\pi$
$587$	−10.0485			−0.414746			−0.207373	+	0.978262i	$0.566491\pi$
$588$		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	$0.449714\pi$
$593$	−18.9110	+	32.7547i	−0.776580	+	1.34508i	−0.933902	+	0.357528i	$0.883620\pi$
$594$		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	−0.977998	−	0.208613i	$-0.933105\pi$
$599$	−31.4823	+	18.1763i	−1.28633	+	0.742665i	−0.308335	+	0.951278i	$0.599772\pi$
$600$		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	$0.191958\pi$
$617$			28.1759i			1.13432i	−0.823608	+	0.567160i	$0.808042\pi$
$618$		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	−0.695552	−	0.718476i	$-0.744840\pi$
$641$	−10.6617	+	6.15551i	−0.421110	+	0.243128i	0.274442	+	0.961604i	$0.411507\pi$
$642$		0			0
$643$			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	$-0.775473\pi$
$647$	−23.9646	−	41.5079i	−0.942145	−	1.63184i	−0.180776	−	0.983524i	$-0.557861\pi$
$648$		0			0
$649$	−51.8122	−	29.9138i	−2.03381	−	1.17422i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8.07258	−	4.66071i	−0.315905	−	0.182388i	0.333661	−	0.942693i	$-0.391716\pi$
$653$	−8.07258	−	4.66071i	−0.315905	−	0.182388i	−0.649566	+	0.760305i	$0.725049\pi$
$654$		0			0
$655$	5.11250	+	8.85511i	0.199762	+	0.345998i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	11.8913i		−	0.463218i	−0.972809	−	0.231609i	$-0.925601\pi$
$659$		−	11.8913i		−	0.463218i	0.972809	−	0.231609i	$-0.0743990\pi$
$660$		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	$-0.524465\pi$
$677$	21.4679	−	37.1834i	0.825077	−	1.42907i	0.901861	−	0.432027i	$-0.142201\pi$
$678$		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	−0.467772	−	0.883849i	$-0.654943\pi$
$683$	1.66677	−	0.962311i	0.0637772	−	0.0368218i	0.531550	+	0.847027i	$0.321610\pi$
$684$		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	$0.164106\pi$
$701$			26.1067i			0.986036i	−0.870019	+	0.493018i	$0.835894\pi$
$702$		0			0
$703$	4.60423	−	2.65825i	0.173652	−	0.100258i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−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	$0.409858\pi$
$719$	−18.5506	+	32.1306i	−0.691822	+	1.19827i	−0.971240	+	0.238101i	$0.923475\pi$
$720$		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	$-0.855018\pi$
$743$		−	23.9809i		−	0.879775i	0.898053	−	0.439887i	$-0.144982\pi$
$744$		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	−0.687745	−	0.725952i	$-0.741399\pi$
$761$	7.85711	−	13.6089i	0.284820	−	0.493323i	0.972566	+	0.232629i	$0.0747328\pi$
$762$		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	$-0.824976\pi$
$773$	−24.4346	−	42.3220i	−0.878852	−	1.52222i	−0.0262509	−	0.999655i	$-0.508357\pi$
$774$		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	$-0.334637\pi$
$797$	28.0307			0.992898			0.496449	+	0.868066i	$0.334637\pi$
$798$		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	−0.294820	−	0.955553i	$-0.595260\pi$
$809$	10.9591	−	6.32725i	0.385302	−	0.222454i	0.680123	+	0.733098i	$0.261926\pi$
$810$		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	0.408648	−	0.912692i	$-0.366000\pi$
$821$	−5.08428	−	2.93541i	−0.177443	−	0.102447i	−0.586091	+	0.810246i	$0.699334\pi$
$822$		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	$-0.722227\pi$
$827$		−	44.0586i		−	1.53207i	0.642800	−	0.766034i	$-0.277773\pi$
$828$		0			0
$829$	5.47912	−	3.16337i	0.190298	−	0.109868i	−0.401824	−	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	$-0.558555\pi$
$839$	−10.5967			−0.365840			−0.182920	+	0.983128i	$0.558555\pi$
$840$		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	0.834360	−	0.551220i	$-0.185838\pi$
$857$	−1.76205	−	3.05197i	−0.0601906	−	0.104253i	−0.894551	+	0.446967i	$0.852504\pi$
$858$		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	0.712442	−	0.701731i	$-0.247589\pi$
$863$	13.5411	+	7.81797i	0.460945	+	0.266127i	−0.251496	+	0.967858i	$0.580923\pi$
$864$		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	$-0.231453\pi$
$881$	44.3493			1.49417			0.747084	+	0.664730i	$0.231453\pi$
$882$		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	$-0.855356\pi$
$887$	−2.05890	+	3.56612i	−0.0691311	+	0.119739i	0.829388	+	0.558673i	$0.188689\pi$
$888$		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	$0.103117\pi$
$911$			19.2152i			0.636628i	−0.947985	+	0.318314i	$0.896883\pi$
$912$		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	$0.371539\pi$
$929$	−18.2907	+	31.6805i	−0.600099	+	1.03940i	−0.992806	+	0.119738i	$0.961794\pi$
$930$		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	0.949393	−	0.314091i	$-0.101700\pi$
$941$	6.21752	+	10.7691i	0.202685	+	0.351061i	−0.746707	+	0.665153i	$0.768366\pi$
$942$		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	0.0169614	−	0.999856i	$-0.494601\pi$
$947$	−25.8637	−	14.9324i	−0.840459	−	0.485239i	−0.857420	+	0.514617i	$0.827934\pi$
$948$		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	$-0.881348\pi$
$953$		−	22.4851i		−	0.728365i	0.931328	−	0.364183i	$-0.118652\pi$
$954$		0			0
$955$	13.7044	−	7.91224i	0.443464	−	0.256034i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−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	$0.560129\pi$
$971$	−29.4317	+	50.9773i	−0.944509	+	1.63594i	−0.756731	+	0.653727i	$0.773205\pi$
$972$		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	−0.639613	−	0.768697i	$-0.720905\pi$
$977$	−9.18043	+	5.30032i	−0.293708	+	0.169572i	0.345905	+	0.938270i	$0.387572\pi$
$978$		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	$-0.992366\pi$
$983$	−16.3230	−	28.2723i	−0.520624	−	0.901747i	0.479089	−	0.877766i	$-0.340967\pi$
$984$		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.2105.7	yes	32
3.2	odd	2	inner	2268.2.t.c.2105.10	yes	32
7.3	odd	6	inner	2268.2.t.c.1781.10	yes	32
9.2	odd	6		2268.2.w.j.1349.10		32
9.4	even	3		2268.2.bm.j.593.10		32
9.5	odd	6		2268.2.bm.j.593.7		32
9.7	even	3		2268.2.w.j.1349.7		32
21.17	even	6	inner	2268.2.t.c.1781.7	✓	32
63.31	odd	6		2268.2.w.j.269.10		32
63.38	even	6		2268.2.bm.j.1025.10		32
63.52	odd	6		2268.2.bm.j.1025.7		32
63.59	even	6		2268.2.w.j.269.7		32

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