LMFDB - Embedded newform 2808.2.r.f.289.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(289,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.r (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,-1,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.8
Character	$\chi$	$=$	2808.289
Dual form			2808.2.r.f.2089.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.766914 - 1.32833i) q^{5} +(1.75740 + 3.04390i) q^{7} -6.38798 q^{11} +(2.60406 - 2.49377i) q^{13} +(0.550501 - 0.953495i) q^{17} +(-1.13796 + 1.97100i) q^{19} +(-1.66408 + 2.88227i) q^{23} +(1.32369 - 2.29269i) q^{25} +1.42965 q^{29} +(-1.93596 - 3.35318i) q^{31} +(2.69554 - 4.66881i) q^{35} +(1.81670 + 3.14662i) q^{37} +(5.99374 - 10.3815i) q^{41} +(-6.27640 - 10.8710i) q^{43} +(1.23298 - 2.13558i) q^{47} +(-2.67688 + 4.63648i) q^{49} +6.18395 q^{53} +(4.89903 + 8.48537i) q^{55} -4.85609 q^{59} +(-1.54959 - 2.68397i) q^{61} +(-5.30965 - 1.54655i) q^{65} +(3.80313 - 6.58721i) q^{67} +(-2.43267 + 4.21351i) q^{71} -5.40814 q^{73} +(-11.2262 - 19.4444i) q^{77} +(2.39185 - 4.14281i) q^{79} +(3.19592 - 5.53550i) q^{83} -1.68875 q^{85} +(4.25286 + 7.36618i) q^{89} +(12.1671 + 3.54395i) q^{91} +3.49086 q^{95} +(1.76312 + 3.05381i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - q^{5} + 7 q^{7} + q^{17} + 2 q^{19} + q^{23} - 23 q^{25} + 24 q^{29} + 8 q^{31} + 12 q^{35} + 18 q^{37} + 3 q^{41} + 8 q^{43} - 4 q^{47} - 23 q^{49} + 40 q^{53} - 14 q^{55} - 8 q^{59} - 3 q^{61}+ \cdots + 35 q^{97}+O(q^{100}) $ 40 * q - q^5 + 7 * q^7 + q^17 + 2 * q^19 + q^23 - 23 * q^25 + 24 * q^29 + 8 * q^31 + 12 * q^35 + 18 * q^37 + 3 * q^41 + 8 * q^43 - 4 * q^47 - 23 * q^49 + 40 * q^53 - 14 * q^55 - 8 * q^59 - 3 * q^61 + 32 * q^65 + 27 * q^67 - q^71 - 12 * q^73 + 4 * q^77 + 13 * q^79 + 15 * q^83 - 10 * q^85 - 12 * q^89 - 57 * q^91 + 2 * q^95 + 35 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.766914	−	1.32833i	−0.342974	−	0.594049i	0.642009	−	0.766697i	$-0.278101\pi$
$5$	−0.766914	−	1.32833i	−0.342974	−	0.594049i	−0.984984	+	0.172648i	$0.944768\pi$
$6$		0			0
$7$	1.75740	+	3.04390i	0.664233	+	1.15049i	0.979493	+	0.201480i	$0.0645750\pi$
$7$	1.75740	+	3.04390i	0.664233	+	1.15049i	−0.315260	+	0.949005i	$0.602092\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.38798			−1.92605			−0.963024	−	0.269414i	$-0.913170\pi$
$11$	−6.38798			−1.92605			−0.963024	+	0.269414i	$0.913170\pi$
$12$		0			0
$13$	2.60406	−	2.49377i	0.722236	−	0.691647i
$13$	2.60406	−	2.49377i	0.722236	−	0.691647i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.550501	−	0.953495i	0.133516	−	0.231256i	−0.791514	−	0.611152i	$-0.790707\pi$
$17$	0.550501	−	0.953495i	0.133516	−	0.231256i	0.925030	+	0.379895i	$0.124040\pi$
$18$		0			0
$19$	−1.13796	+	1.97100i	−0.261065	+	0.452179i	−0.966525	−	0.256571i	$-0.917407\pi$
$19$	−1.13796	+	1.97100i	−0.261065	+	0.452179i	0.705460	+	0.708750i	$0.250741\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.66408	+	2.88227i	−0.346984	+	0.600994i	−0.985712	−	0.168439i	$-0.946128\pi$
$23$	−1.66408	+	2.88227i	−0.346984	+	0.600994i	0.638728	+	0.769432i	$0.279461\pi$
$24$		0			0
$25$	1.32369	−	2.29269i	0.264737	−	0.458538i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.42965			0.265479			0.132740	−	0.991151i	$-0.457623\pi$
$29$	1.42965			0.265479			0.132740	+	0.991151i	$0.457623\pi$
$30$		0			0
$31$	−1.93596	−	3.35318i	−0.347708	−	0.602248i	0.638134	−	0.769926i	$-0.279707\pi$
$31$	−1.93596	−	3.35318i	−0.347708	−	0.602248i	−0.985842	+	0.167677i	$0.946373\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.69554	−	4.66881i	0.455630	−	0.789174i
$36$		0			0
$37$	1.81670	+	3.14662i	0.298664	+	0.517302i	0.975831	−	0.218528i	$-0.0701256\pi$
$37$	1.81670	+	3.14662i	0.298664	+	0.517302i	−0.677166	+	0.735830i	$0.736792\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5.99374	−	10.3815i	0.936065	−	1.62131i	0.163342	−	0.986570i	$-0.447773\pi$
$41$	5.99374	−	10.3815i	0.936065	−	1.62131i	0.772723	−	0.634743i	$-0.218894\pi$
$42$		0			0
$43$	−6.27640	−	10.8710i	−0.957142	−	1.65782i	−0.729389	−	0.684099i	$-0.760195\pi$
$43$	−6.27640	−	10.8710i	−0.957142	−	1.65782i	−0.227753	−	0.973719i	$-0.573138\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.23298	−	2.13558i	0.179848	−	0.311506i	−0.761980	−	0.647600i	$-0.775773\pi$
$47$	1.23298	−	2.13558i	0.179848	−	0.311506i	0.941828	+	0.336094i	$0.109106\pi$
$48$		0			0
$49$	−2.67688	+	4.63648i	−0.382411	+	0.662355i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.18395			0.849431			0.424716	−	0.905327i	$-0.360374\pi$
$53$	6.18395			0.849431			0.424716	+	0.905327i	$0.360374\pi$
$54$		0			0
$55$	4.89903	+	8.48537i	0.660585	+	1.14417i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.85609			−0.632209			−0.316104	−	0.948724i	$-0.602375\pi$
$59$	−4.85609			−0.632209			−0.316104	+	0.948724i	$0.602375\pi$
$60$		0			0
$61$	−1.54959	−	2.68397i	−0.198405	−	0.343647i	0.749607	−	0.661884i	$-0.230243\pi$
$61$	−1.54959	−	2.68397i	−0.198405	−	0.343647i	−0.948011	+	0.318237i	$0.896909\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.30965	−	1.54655i	−0.658580	−	0.191826i
$66$		0			0
$67$	3.80313	−	6.58721i	0.464626	−	0.804756i	−0.534559	−	0.845131i	$-0.679522\pi$
$67$	3.80313	−	6.58721i	0.464626	−	0.804756i	0.999185	+	0.0403756i	$0.0128554\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.43267	+	4.21351i	−0.288705	+	0.500051i	−0.973501	−	0.228683i	$-0.926558\pi$
$71$	−2.43267	+	4.21351i	−0.288705	+	0.500051i	0.684796	+	0.728735i	$0.259891\pi$
$72$		0			0
$73$	−5.40814			−0.632975			−0.316487	−	0.948597i	$-0.602504\pi$
$73$	−5.40814			−0.632975			−0.316487	+	0.948597i	$0.602504\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−11.2262	−	19.4444i	−1.27934	−	2.21589i
$78$		0			0
$79$	2.39185	−	4.14281i	0.269104	−	0.466102i	−0.699527	−	0.714607i	$-0.746606\pi$
$79$	2.39185	−	4.14281i	0.269104	−	0.466102i	0.968631	+	0.248505i	$0.0799391\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.19592	−	5.53550i	0.350798	−	0.607600i	−0.635591	−	0.772026i	$-0.719244\pi$
$83$	3.19592	−	5.53550i	0.350798	−	0.607600i	0.986390	+	0.164425i	$0.0525770\pi$
$84$		0			0
$85$	−1.68875			−0.183170
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.25286	+	7.36618i	0.450803	+	0.780813i	0.998436	−	0.0559053i	$-0.0178045\pi$
$89$	4.25286	+	7.36618i	0.450803	+	0.780813i	−0.547633	+	0.836718i	$0.684471\pi$
$90$		0			0
$91$	12.1671	+	3.54395i	1.27546	+	0.371507i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.49086			0.358155
$96$		0			0
$97$	1.76312	+	3.05381i	0.179017	+	0.310067i	0.941544	−	0.336889i	$-0.109375\pi$
$97$	1.76312	+	3.05381i	0.179017	+	0.310067i	−0.762527	+	0.646956i	$0.776042\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.8681			1.47943			0.739717	−	0.672918i	$-0.234959\pi$
$101$	14.8681			1.47943			0.739717	+	0.672918i	$0.234959\pi$
$102$		0			0
$103$	−5.76469	−	9.98474i	−0.568012	−	0.983826i	−0.996762	−	0.0804029i	$-0.974379\pi$
$103$	−5.76469	−	9.98474i	−0.568012	−	0.983826i	0.428750	−	0.903423i	$-0.358954\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.41074	−	7.63962i	−0.426402	−	0.738550i	0.570148	−	0.821542i	$-0.306886\pi$
$107$	−4.41074	−	7.63962i	−0.426402	−	0.738550i	−0.996550	+	0.0829919i	$0.973552\pi$
$108$		0			0
$109$	−9.26955			−0.887862			−0.443931	−	0.896061i	$-0.646416\pi$
$109$	−9.26955			−0.887862			−0.443931	+	0.896061i	$0.646416\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.72485			0.632621			0.316310	−	0.948656i	$-0.397556\pi$
$113$	6.72485			0.632621			0.316310	+	0.948656i	$0.397556\pi$
$114$		0			0
$115$	5.10481			0.476026
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.86979			0.354743
$120$		0			0
$121$	29.8063			2.70966
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.7298			−1.04914
$126$		0			0
$127$	7.61299	+	13.1861i	0.675544	+	1.17008i	0.976310	+	0.216378i	$0.0694245\pi$
$127$	7.61299	+	13.1861i	0.675544	+	1.17008i	−0.300766	+	0.953698i	$0.597242\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.89951	−	10.2183i	−0.515443	−	0.892773i	−0.999839	−	0.0179247i	$-0.994294\pi$
$131$	−5.89951	−	10.2183i	−0.515443	−	0.892773i	0.484396	−	0.874849i	$-0.339039\pi$
$132$		0			0
$133$	−7.99937			−0.693633
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.00266	−	12.1290i	−0.598277	−	1.03625i	−0.993075	−	0.117479i	$-0.962519\pi$
$137$	−7.00266	−	12.1290i	−0.598277	−	1.03625i	0.394798	−	0.918768i	$-0.370815\pi$
$138$		0			0
$139$	−8.28707			−0.702900			−0.351450	−	0.936207i	$-0.614311\pi$
$139$	−8.28707			−0.702900			−0.351450	+	0.936207i	$0.614311\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−16.6347	+	15.9301i	−1.39106	+	1.33215i
$144$		0			0
$145$	−1.09642	−	1.89905i	−0.0910526	−	0.157708i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.93701			−0.322533			−0.161266	−	0.986911i	$-0.551558\pi$
$149$	−3.93701			−0.322533			−0.161266	+	0.986911i	$0.551558\pi$
$150$		0			0
$151$	10.1395	−	17.5621i	0.825138	−	1.42918i	−0.0766753	−	0.997056i	$-0.524430\pi$
$151$	10.1395	−	17.5621i	0.825138	−	1.42918i	0.901814	−	0.432125i	$-0.142236\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.96943	+	5.14320i	−0.238510	+	0.413111i
$156$		0			0
$157$	−11.7247	−	20.3078i	−0.935735	−	1.62074i	−0.773319	−	0.634017i	$-0.781405\pi$
$157$	−11.7247	−	20.3078i	−0.935735	−	1.62074i	−0.162416	−	0.986722i	$-0.551929\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.6978			−0.921913
$162$		0			0
$163$	−1.54310	+	2.67272i	−0.120865	+	0.209344i	−0.920109	−	0.391663i	$-0.871900\pi$
$163$	−1.54310	+	2.67272i	−0.120865	+	0.209344i	0.799244	+	0.601006i	$0.205233\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.75344	+	16.8935i	−0.754744	+	1.30726i	0.190757	+	0.981637i	$0.438906\pi$
$167$	−9.75344	+	16.8935i	−0.754744	+	1.30726i	−0.945501	+	0.325618i	$0.894428\pi$
$168$		0			0
$169$	0.562232	−	12.9878i	0.0432486	−	0.999064i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.35024	−	9.26689i	−0.406771	−	0.704549i	0.587754	−	0.809039i	$-0.300012\pi$
$173$	−5.35024	−	9.26689i	−0.406771	−	0.704549i	−0.994526	+	0.104491i	$0.966679\pi$
$174$		0			0
$175$	9.30496			0.703389
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.14067	−	1.97569i	−0.0852574	−	0.147670i	0.820243	−	0.572015i	$-0.193838\pi$
$179$	−1.14067	−	1.97569i	−0.0852574	−	0.147670i	−0.905501	+	0.424344i	$0.860505\pi$
$180$		0			0
$181$	10.2756			0.763780			0.381890	−	0.924208i	$-0.375273\pi$
$181$	10.2756			0.763780			0.381890	+	0.924208i	$0.375273\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.78651	−	4.82638i	0.204868	−	0.354842i
$186$		0			0
$187$	−3.51659	+	6.09091i	−0.257158	+	0.445411i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.8139	+	20.4623i	0.854826	+	1.48060i	0.876806	+	0.480844i	$0.159670\pi$
$191$	11.8139	+	20.4623i	0.854826	+	1.48060i	−0.0219799	+	0.999758i	$0.506997\pi$
$192$		0			0
$193$	−5.51975	+	9.56048i	−0.397320	+	0.688178i	−0.993394	−	0.114751i	$-0.963393\pi$
$193$	−5.51975	+	9.56048i	−0.397320	+	0.688178i	0.596074	+	0.802929i	$0.296726\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.47677	−	7.75399i	−0.318957	−	0.552449i	0.661314	−	0.750109i	$-0.269999\pi$
$197$	−4.47677	−	7.75399i	−0.318957	−	0.552449i	−0.980271	+	0.197660i	$0.936666\pi$
$198$		0			0
$199$	11.6750	−	20.2217i	0.827620	−	1.43348i	−0.0722797	−	0.997384i	$-0.523027\pi$
$199$	11.6750	−	20.2217i	0.827620	−	1.43348i	0.899900	−	0.436096i	$-0.143639\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.51246	+	4.35171i	0.176340	+	0.305430i
$204$		0			0
$205$	−18.3867			−1.28419
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.26925	−	12.5907i	0.502825	−	0.870918i
$210$		0			0
$211$	0.0859803	−	0.148922i	0.00591913	−	0.0102522i	−0.863051	−	0.505117i	$-0.831449\pi$
$211$	0.0859803	−	0.148922i	0.00591913	−	0.0102522i	0.868970	+	0.494865i	$0.164783\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.62691	+	16.6743i	−0.656550	+	1.13718i
$216$		0			0
$217$	6.80449	−	11.7857i	0.461919	−	0.800066i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.944261	−	3.85578i	−0.0635179	−	0.259368i
$222$		0			0
$223$	−1.62625			−0.108902			−0.0544510	−	0.998516i	$-0.517341\pi$
$223$	−1.62625			−0.108902			−0.0544510	+	0.998516i	$0.517341\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−11.2694	−	19.5192i	−0.747979	−	1.29554i	−0.948790	−	0.315908i	$-0.897691\pi$
$227$	−11.2694	−	19.5192i	−0.747979	−	1.29554i	0.200811	−	0.979630i	$-0.435642\pi$
$228$		0			0
$229$	−12.0818	−	20.9263i	−0.798389	−	1.38285i	−0.920665	−	0.390354i	$-0.872353\pi$
$229$	−12.0818	−	20.9263i	−0.798389	−	1.38285i	0.122276	−	0.992496i	$-0.460981\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.4919			−1.08042			−0.540211	−	0.841530i	$-0.681656\pi$
$233$	−16.4919			−1.08042			−0.540211	+	0.841530i	$0.681656\pi$
$234$		0			0
$235$	−3.78235			−0.246733
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.24014	+	9.07618i	0.338956	+	0.587089i	0.984237	−	0.176856i	$-0.0565928\pi$
$239$	5.24014	+	9.07618i	0.338956	+	0.587089i	−0.645280	+	0.763946i	$0.723259\pi$
$240$		0			0
$241$	−11.5900	−	20.0745i	−0.746577	−	1.29311i	−0.949454	−	0.313905i	$-0.898363\pi$
$241$	−11.5900	−	20.0745i	−0.746577	−	1.29311i	0.202877	−	0.979204i	$-0.434971\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	8.21173			0.524628
$246$		0			0
$247$	1.95191	+	7.97040i	0.124197	+	0.507145i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.1392	−	17.5616i	0.639979	−	1.10848i	−0.345458	−	0.938434i	$-0.612276\pi$
$251$	10.1392	−	17.5616i	0.639979	−	1.10848i	0.985437	−	0.170042i	$-0.0543904\pi$
$252$		0			0
$253$	10.6301	−	18.4119i	0.668308	−	1.15754i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.362989	+	0.628715i	−0.0226426	+	0.0392182i	−0.877125	−	0.480263i	$-0.840541\pi$
$257$	−0.362989	+	0.628715i	−0.0226426	+	0.0392182i	0.854482	+	0.519481i	$0.173875\pi$
$258$		0			0
$259$	−6.38534	+	11.0597i	−0.396765	+	0.687218i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.62127			0.593273			0.296637	−	0.954990i	$-0.404135\pi$
$263$	9.62127			0.593273			0.296637	+	0.954990i	$0.404135\pi$
$264$		0			0
$265$	−4.74256	−	8.21435i	−0.291333	−	0.504604i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.12405	−	10.6072i	0.373390	−	0.646731i	−0.616695	−	0.787202i	$-0.711529\pi$
$269$	6.12405	−	10.6072i	0.373390	−	0.646731i	0.990085	+	0.140472i	$0.0448620\pi$
$270$		0			0
$271$	2.64462	+	4.58062i	0.160650	+	0.278253i	0.935102	−	0.354379i	$-0.115308\pi$
$271$	2.64462	+	4.58062i	0.160650	+	0.278253i	−0.774452	+	0.632632i	$0.781974\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.45568	+	14.6457i	−0.509897	+	0.883167i
$276$		0			0
$277$	−2.96456	−	5.13476i	−0.178123	−	0.308518i	0.763115	−	0.646263i	$-0.223669\pi$
$277$	−2.96456	−	5.13476i	−0.178123	−	0.308518i	−0.941238	+	0.337745i	$0.890336\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.2879	+	28.2115i	−0.971654	+	1.68295i	−0.281094	+	0.959680i	$0.590697\pi$
$281$	−16.2879	+	28.2115i	−0.971654	+	1.68295i	−0.690561	+	0.723275i	$0.742636\pi$
$282$		0			0
$283$	−9.27398	+	16.0630i	−0.551281	+	0.954847i	0.446902	+	0.894583i	$0.352527\pi$
$283$	−9.27398	+	16.0630i	−0.551281	+	0.954847i	−0.998183	+	0.0602634i	$0.980806\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	42.1335			2.48706
$288$		0			0
$289$	7.89390	+	13.6726i	0.464347	+	0.804273i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.9848			1.05069			0.525343	−	0.850891i	$-0.323937\pi$
$293$	17.9848			1.05069			0.525343	+	0.850891i	$0.323937\pi$
$294$		0			0
$295$	3.72420	+	6.45051i	0.216831	+	0.375563i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.85435	+	11.6554i	0.165071	+	0.674050i
$300$		0			0
$301$	22.0602	−	38.2094i	1.27153	−	2.20236i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.37680	+	4.11675i	−0.136095	+	0.235724i
$306$		0			0
$307$	14.0831			0.803768			0.401884	−	0.915691i	$-0.368356\pi$
$307$	14.0831			0.803768			0.401884	+	0.915691i	$0.368356\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−15.3440	−	26.5767i	−0.870081	−	1.50702i	−0.861912	−	0.507058i	$-0.830733\pi$
$311$	−15.3440	−	26.5767i	−0.870081	−	1.50702i	−0.00816904	−	0.999967i	$-0.502600\pi$
$312$		0			0
$313$	−5.48343	+	9.49757i	−0.309942	+	0.536835i	−0.978349	−	0.206961i	$-0.933643\pi$
$313$	−5.48343	+	9.49757i	−0.309942	+	0.536835i	0.668408	+	0.743795i	$0.266976\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.15353	−	1.99797i	0.0647885	−	0.112217i	−0.831812	−	0.555058i	$-0.812696\pi$
$317$	1.15353	−	1.99797i	0.0647885	−	0.112217i	0.896600	+	0.442841i	$0.146029\pi$
$318$		0			0
$319$	−9.13258			−0.511326
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.25289	+	2.17007i	0.0697128	+	0.120746i
$324$		0			0
$325$	−2.27049	−	9.27127i	−0.125944	−	0.514278i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.66730			0.477844
$330$		0			0
$331$	16.8249	+	29.1416i	0.924780	+	1.60177i	0.791914	+	0.610632i	$0.209085\pi$
$331$	16.8249	+	29.1416i	0.924780	+	1.60177i	0.132866	+	0.991134i	$0.457582\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−11.6667			−0.637419
$336$		0			0
$337$	6.88906	+	11.9322i	0.375271	+	0.649988i	0.990368	−	0.138464i	$-0.0442164\pi$
$337$	6.88906	+	11.9322i	0.375271	+	0.649988i	−0.615097	+	0.788452i	$0.710883\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.3669	+	21.4200i	0.669703	+	1.15996i
$342$		0			0
$343$	5.78622			0.312427
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.32520			−0.339555			−0.169777	−	0.985482i	$-0.554305\pi$
$347$	−6.32520			−0.339555			−0.169777	+	0.985482i	$0.554305\pi$
$348$		0			0
$349$	13.3337			0.713736			0.356868	−	0.934155i	$-0.383845\pi$
$349$	13.3337			0.713736			0.356868	+	0.934155i	$0.383845\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	25.2494			1.34389			0.671945	−	0.740601i	$-0.265459\pi$
$353$	25.2494			1.34389			0.671945	+	0.740601i	$0.265459\pi$
$354$		0			0
$355$	7.46259			0.396073
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.22423			−0.0646123			−0.0323062	−	0.999478i	$-0.510285\pi$
$359$	−1.22423			−0.0646123			−0.0323062	+	0.999478i	$0.510285\pi$
$360$		0			0
$361$	6.91010	+	11.9687i	0.363690	+	0.629929i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.14758	+	7.18381i	0.217094	+	0.376018i
$366$		0			0
$367$	−14.2000			−0.741233			−0.370617	−	0.928786i	$-0.620854\pi$
$367$	−14.2000			−0.741233			−0.370617	+	0.928786i	$0.620854\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.8676	+	18.8233i	0.564220	+	0.977258i
$372$		0			0
$373$	−16.7767			−0.868666			−0.434333	−	0.900752i	$-0.643016\pi$
$373$	−16.7767			−0.868666			−0.434333	+	0.900752i	$0.643016\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.72289	−	3.56522i	0.191739	−	0.183618i
$378$		0			0
$379$	18.3934	+	31.8583i	0.944807	+	1.63645i	0.756138	+	0.654413i	$0.227084\pi$
$379$	18.3934	+	31.8583i	0.944807	+	1.63645i	0.188669	+	0.982041i	$0.439583\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.3260			−0.783121			−0.391561	−	0.920152i	$-0.628065\pi$
$383$	−15.3260			−0.783121			−0.391561	+	0.920152i	$0.628065\pi$
$384$		0			0
$385$	−17.2191	+	29.8243i	−0.877565	+	1.51999i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.10671	+	7.11304i	−0.208219	+	0.360645i	−0.951153	−	0.308718i	$-0.900100\pi$
$389$	−4.10671	+	7.11304i	−0.208219	+	0.360645i	0.742935	+	0.669364i	$0.233433\pi$
$390$		0			0
$391$	1.83215	+	3.17338i	0.0926558	+	0.160485i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.33737			−0.369183
$396$		0			0
$397$	−12.3499	+	21.3906i	−0.619822	+	1.07356i	0.369695	+	0.929153i	$0.379462\pi$
$397$	−12.3499	+	21.3906i	−0.619822	+	1.07356i	−0.989518	+	0.144411i	$0.953871\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.92365	−	11.9921i	0.345750	−	0.598857i	−0.639739	−	0.768592i	$-0.720958\pi$
$401$	6.92365	−	11.9921i	0.345750	−	0.598857i	0.985490	+	0.169735i	$0.0542911\pi$
$402$		0			0
$403$	−13.4034	−	3.90404i	−0.667671	−	0.194474i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11.6051	−	20.1006i	−0.575242	−	0.996349i
$408$		0			0
$409$	18.6508			0.922222			0.461111	−	0.887342i	$-0.347451\pi$
$409$	18.6508			0.922222			0.461111	+	0.887342i	$0.347451\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8.53407	−	14.7814i	−0.419934	−	0.727347i
$414$		0			0
$415$	−9.80399			−0.481259
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.87761	+	13.6444i	−0.384846	+	0.666574i	−0.991748	−	0.128203i	$-0.959079\pi$
$419$	−7.87761	+	13.6444i	−0.384846	+	0.666574i	0.606901	+	0.794777i	$0.292412\pi$
$420$		0			0
$421$	−7.16989	+	12.4186i	−0.349439	+	0.605246i	−0.986150	−	0.165856i	$-0.946961\pi$
$421$	−7.16989	+	12.4186i	−0.349439	+	0.605246i	0.636711	+	0.771103i	$0.280295\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.45738	−	2.52426i	−0.0706933	−	0.122444i
$426$		0			0
$427$	5.44648	−	9.43359i	0.263574	−	0.456523i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.49274	−	7.78165i	−0.216408	−	0.374829i	0.737300	−	0.675566i	$-0.236101\pi$
$431$	−4.49274	−	7.78165i	−0.216408	−	0.374829i	−0.953707	+	0.300737i	$0.902767\pi$
$432$		0			0
$433$	−13.8669	+	24.0182i	−0.666401	+	1.15424i	0.312503	+	0.949917i	$0.398833\pi$
$433$	−13.8669	+	24.0182i	−0.666401	+	1.15424i	−0.978904	+	0.204323i	$0.934501\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.78730	−	6.55979i	−0.181171	−	0.313797i
$438$		0			0
$439$	5.95194			0.284071			0.142035	−	0.989862i	$-0.454635\pi$
$439$	5.95194			0.284071			0.142035	+	0.989862i	$0.454635\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.66685	−	9.81526i	0.269240	−	0.466337i	−0.699426	−	0.714705i	$-0.746561\pi$
$443$	5.66685	−	9.81526i	0.269240	−	0.466337i	0.968666	+	0.248368i	$0.0798942\pi$
$444$		0			0
$445$	6.52316	−	11.2984i	0.309227	−	0.535598i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.47394	+	11.2132i	−0.305524	+	0.529183i	−0.977378	−	0.211501i	$-0.932165\pi$
$449$	−6.47394	+	11.2132i	−0.305524	+	0.529183i	0.671854	+	0.740684i	$0.265498\pi$
$450$		0			0
$451$	−38.2879	+	66.3166i	−1.80291	+	3.12273i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.62360	−	18.8799i	−0.216758	−	0.885104i
$456$		0			0
$457$	−19.3479			−0.905058			−0.452529	−	0.891750i	$-0.649478\pi$
$457$	−19.3479			−0.905058			−0.452529	+	0.891750i	$0.649478\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.3986	+	30.1353i	0.810334	+	1.40354i	0.912631	+	0.408785i	$0.134047\pi$
$461$	17.3986	+	30.1353i	0.810334	+	1.40354i	−0.102297	+	0.994754i	$0.532619\pi$
$462$		0			0
$463$	13.0277	+	22.5647i	0.605450	+	1.04867i	0.991980	+	0.126394i	$0.0403402\pi$
$463$	13.0277	+	22.5647i	0.605450	+	1.04867i	−0.386530	+	0.922277i	$0.626326\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	39.3170			1.81937			0.909686	−	0.415296i	$-0.136322\pi$
$467$	39.3170			1.81937			0.909686	+	0.415296i	$0.136322\pi$
$468$		0			0
$469$	26.7344			1.23448
$470$		0			0
$471$		0			0
$472$		0			0
$473$	40.0935	+	69.4440i	1.84350	+	3.19304i
$474$		0			0
$475$	3.01260	+	5.21797i	0.138228	+	0.239417i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.7294			−1.12991			−0.564957	−	0.825120i	$-0.691107\pi$
$479$	−24.7294			−1.12991			−0.564957	+	0.825120i	$0.691107\pi$
$480$		0			0
$481$	12.5778	+	3.66355i	0.573496	+	0.167044i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.70432	−	4.68401i	0.122797	−	0.212690i
$486$		0			0
$487$	15.5768	−	26.9798i	0.705853	−	1.22257i	−0.260530	−	0.965466i	$-0.583897\pi$
$487$	15.5768	−	26.9798i	0.705853	−	1.22257i	0.966383	−	0.257107i	$-0.0827694\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.824435	+	1.42796i	−0.0372062	+	0.0644431i	−0.884029	−	0.467432i	$-0.845179\pi$
$491$	−0.824435	+	1.42796i	−0.0372062	+	0.0644431i	0.846823	+	0.531875i	$0.178513\pi$
$492$		0			0
$493$	0.787023	−	1.36316i	0.0354457	−	0.0613938i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−17.1006			−0.767069
$498$		0			0
$499$	−21.3882	−	37.0455i	−0.957468	−	1.65838i	−0.728618	−	0.684921i	$-0.759837\pi$
$499$	−21.3882	−	37.0455i	−0.957468	−	1.65838i	−0.228850	−	0.973462i	$-0.573496\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.6416	−	34.0203i	0.875777	−	1.51689i	0.0198439	−	0.999803i	$-0.493683\pi$
$503$	19.6416	−	34.0203i	0.875777	−	1.51689i	0.855933	−	0.517087i	$-0.172984\pi$
$504$		0			0
$505$	−11.4026	−	19.7498i	−0.507408	−	0.878856i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.02885	−	10.4423i	0.267224	−	0.462846i	−0.700920	−	0.713240i	$-0.747227\pi$
$509$	6.02885	−	10.4423i	0.267224	−	0.462846i	0.968144	+	0.250394i	$0.0805602\pi$
$510$		0			0
$511$	−9.50424	−	16.4618i	−0.420443	−	0.728228i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.84205	+	15.3149i	−0.389627	+	0.674854i
$516$		0			0
$517$	−7.87623	+	13.6420i	−0.346396	+	0.599976i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.6196			−0.728116			−0.364058	−	0.931376i	$-0.618609\pi$
$521$	−16.6196			−0.728116			−0.364058	+	0.931376i	$0.618609\pi$
$522$		0			0
$523$	−0.621458	−	1.07640i	−0.0271745	−	0.0470676i	0.852118	−	0.523349i	$-0.175318\pi$
$523$	−0.621458	−	1.07640i	−0.0271745	−	0.0470676i	−0.879293	+	0.476282i	$0.841984\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.26298			−0.185698
$528$		0			0
$529$	5.96170	+	10.3260i	0.259204	+	0.448955i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−10.2809	−	41.9809i	−0.445316	−	1.81840i
$534$		0			0
$535$	−6.76531	+	11.7179i	−0.292490	+	0.506607i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	17.0998	−	29.6178i	0.736542	−	1.27573i
$540$		0			0
$541$	−41.8105			−1.79757			−0.898787	−	0.438386i	$-0.855550\pi$
$541$	−41.8105			−1.79757			−0.898787	+	0.438386i	$0.855550\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.10895	+	12.3131i	0.304514	+	0.527433i
$546$		0			0
$547$	−0.266912	+	0.462305i	−0.0114123	+	0.0197668i	−0.871675	−	0.490084i	$-0.836966\pi$
$547$	−0.266912	+	0.462305i	−0.0114123	+	0.0197668i	0.860263	+	0.509851i	$0.170299\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.62688	+	2.81784i	−0.0693075	+	0.120044i
$552$		0			0
$553$	16.8137			0.714991
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.96368	−	6.86530i	−0.167947	−	0.290892i	0.769751	−	0.638344i	$-0.220380\pi$
$557$	−3.96368	−	6.86530i	−0.167947	−	0.290892i	−0.937698	+	0.347452i	$0.887047\pi$
$558$		0			0
$559$	−43.4540	−	12.6569i	−1.83791	−	0.535331i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−31.1732			−1.31379			−0.656897	−	0.753980i	$-0.728131\pi$
$563$	−31.1732			−1.31379			−0.656897	+	0.753980i	$0.728131\pi$
$564$		0			0
$565$	−5.15738	−	8.93284i	−0.216973	−	0.375808i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.0506			−0.882488			−0.441244	−	0.897387i	$-0.645463\pi$
$569$	−21.0506			−0.882488			−0.441244	+	0.897387i	$0.645463\pi$
$570$		0			0
$571$	−2.80025	−	4.85017i	−0.117187	−	0.202973i	0.801465	−	0.598042i	$-0.204054\pi$
$571$	−2.80025	−	4.85017i	−0.117187	−	0.202973i	−0.918652	+	0.395068i	$0.870721\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.40543	+	7.63043i	0.183719	+	0.318211i
$576$		0			0
$577$	−30.6676			−1.27671			−0.638355	−	0.769742i	$-0.720385\pi$
$577$	−30.6676			−1.27671			−0.638355	+	0.769742i	$0.720385\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	22.4660			0.932047
$582$		0			0
$583$	−39.5030			−1.63605
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.4657			1.29873			0.649364	−	0.760477i	$-0.275035\pi$
$587$	31.4657			1.29873			0.649364	+	0.760477i	$0.275035\pi$
$588$		0			0
$589$	8.81215			0.363098
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.583862			−0.0239763			−0.0119882	−	0.999928i	$-0.503816\pi$
$593$	−0.583862			−0.0239763			−0.0119882	+	0.999928i	$0.503816\pi$
$594$		0			0
$595$	−2.96779	−	5.14037i	−0.121668	−	0.210735i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.5346	+	21.7106i	0.512150	+	0.887069i	0.999901	+	0.0140866i	$0.00448405\pi$
$599$	12.5346	+	21.7106i	0.512150	+	0.887069i	−0.487751	+	0.872983i	$0.662183\pi$
$600$		0			0
$601$	15.5243			0.633249			0.316625	−	0.948551i	$-0.397450\pi$
$601$	15.5243			0.633249			0.316625	+	0.948551i	$0.397450\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.8589	−	39.5927i	−0.929345	−	1.60967i
$606$		0			0
$607$	30.8883			1.25372			0.626858	−	0.779134i	$-0.284341\pi$
$607$	30.8883			1.25372			0.626858	+	0.779134i	$0.284341\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.11490	−	8.63592i	−0.0855595	−	0.349372i
$612$		0			0
$613$	15.1558	+	26.2506i	0.612138	+	1.06025i	0.990879	+	0.134751i	$0.0430235\pi$
$613$	15.1558	+	26.2506i	0.612138	+	1.06025i	−0.378742	+	0.925502i	$0.623643\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.9498			−0.481083			−0.240541	−	0.970639i	$-0.577325\pi$
$617$	−11.9498			−0.481083			−0.240541	+	0.970639i	$0.577325\pi$
$618$		0			0
$619$	−11.3093	+	19.5884i	−0.454561	+	0.787323i	−0.998663	−	0.0516965i	$-0.983537\pi$
$619$	−11.3093	+	19.5884i	−0.454561	+	0.787323i	0.544102	+	0.839019i	$0.316870\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.9479	+	25.8906i	−0.598876	+	1.03728i
$624$		0			0
$625$	2.37728	+	4.11756i	0.0950911	+	0.164703i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.00039			0.159506
$630$		0			0
$631$	2.92924	−	5.07359i	0.116611	−	0.201976i	−0.801811	−	0.597577i	$-0.796130\pi$
$631$	2.92924	−	5.07359i	0.116611	−	0.201976i	0.918423	+	0.395601i	$0.129464\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	11.6770	−	20.2252i	0.463388	−	0.802612i
$636$		0			0
$637$	4.59158	+	18.7492i	0.181925	+	0.742870i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.94265	−	12.0250i	−0.274218	−	0.474960i	0.695719	−	0.718314i	$-0.255086\pi$
$641$	−6.94265	−	12.0250i	−0.274218	−	0.474960i	−0.969938	+	0.243354i	$0.921752\pi$
$642$		0			0
$643$	27.9811			1.10347			0.551734	−	0.834020i	$-0.313966\pi$
$643$	27.9811			1.10347			0.551734	+	0.834020i	$0.313966\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13.4477	−	23.2920i	−0.528682	−	0.915705i	−0.999441	−	0.0334425i	$-0.989353\pi$
$647$	−13.4477	−	23.2920i	−0.528682	−	0.915705i	0.470758	−	0.882262i	$-0.343980\pi$
$648$		0			0
$649$	31.0206			1.21767
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.69819	+	16.7978i	−0.379519	+	0.657347i	−0.990992	−	0.133918i	$-0.957244\pi$
$653$	−9.69819	+	16.7978i	−0.379519	+	0.657347i	0.611473	+	0.791265i	$0.290577\pi$
$654$		0			0
$655$	−9.04884	+	15.6730i	−0.353567	+	0.612397i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.9251	+	34.5112i	0.776171	+	1.34437i	0.934134	+	0.356922i	$0.116174\pi$
$659$	19.9251	+	34.5112i	0.776171	+	1.34437i	−0.157963	+	0.987445i	$0.550493\pi$
$660$		0			0
$661$	−5.73725	+	9.93720i	−0.223153	+	0.386512i	−0.955764	−	0.294136i	$-0.904968\pi$
$661$	−5.73725	+	9.93720i	−0.223153	+	0.386512i	0.732611	+	0.680648i	$0.238302\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.13482	+	10.6258i	0.237898	+	0.412052i
$666$		0			0
$667$	−2.37905	+	4.12063i	−0.0921171	+	0.159551i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9.89875	+	17.1451i	0.382137	+	0.661881i
$672$		0			0
$673$	−36.7086			−1.41501			−0.707506	−	0.706707i	$-0.750180\pi$
$673$	−36.7086			−1.41501			−0.707506	+	0.706707i	$0.750180\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.57943	−	6.19976i	0.137569	−	0.238276i	−0.789007	−	0.614384i	$-0.789405\pi$
$677$	3.57943	−	6.19976i	0.137569	−	0.238276i	0.926576	+	0.376108i	$0.122738\pi$
$678$		0			0
$679$	−6.19698	+	10.7335i	−0.237818	+	0.411914i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−13.2982	+	23.0331i	−0.508841	+	0.881338i	0.491107	+	0.871099i	$0.336592\pi$
$683$	−13.2982	+	23.0331i	−0.508841	+	0.881338i	−0.999948	+	0.0102387i	$0.996741\pi$
$684$		0			0
$685$	−10.7409	+	18.6037i	−0.410387	+	0.710812i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.1034	−	15.4214i	0.613490	−	0.587507i
$690$		0			0
$691$	−31.1162			−1.18372			−0.591858	−	0.806042i	$-0.701605\pi$
$691$	−31.1162			−1.18372			−0.591858	+	0.806042i	$0.701605\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.35547	+	11.0080i	0.241077	+	0.417557i
$696$		0			0
$697$	−6.59912	−	11.4300i	−0.249959	−	0.432942i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	38.5419			1.45571			0.727854	−	0.685732i	$-0.240518\pi$
$701$	38.5419			1.45571			0.727854	+	0.685732i	$0.240518\pi$
$702$		0			0
$703$	−8.26933			−0.311884
$704$		0			0
$705$		0			0
$706$		0			0
$707$	26.1292	+	45.2570i	0.982688	+	1.70207i
$708$		0			0
$709$	−0.920397	−	1.59417i	−0.0345662	−	0.0598705i	0.848225	−	0.529636i	$-0.177672\pi$
$709$	−0.920397	−	1.59417i	−0.0345662	−	0.0598705i	−0.882791	+	0.469766i	$0.844338\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.8863			0.482597
$714$		0			0
$715$	33.9179	+	9.87934i	1.26846	+	0.369466i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.15519	+	15.8573i	−0.341431	+	0.591376i	−0.984699	−	0.174265i	$-0.944245\pi$
$719$	−9.15519	+	15.8573i	−0.341431	+	0.591376i	0.643268	+	0.765641i	$0.277578\pi$
$720$		0			0
$721$	20.2617	−	35.0943i	0.754585	−	1.30698i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.89241	−	3.27775i	0.0702823	−	0.121732i
$726$		0			0
$727$	−11.6183	+	20.1235i	−0.430900	+	0.746340i	−0.996951	−	0.0780300i	$-0.975137\pi$
$727$	−11.6183	+	20.1235i	−0.430900	+	0.746340i	0.566051	+	0.824370i	$0.308470\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−13.8206			−0.511175
$732$		0			0
$733$	−9.62013	−	16.6626i	−0.355328	−	0.615446i	0.631846	−	0.775094i	$-0.282297\pi$
$733$	−9.62013	−	16.6626i	−0.355328	−	0.615446i	−0.987174	+	0.159648i	$0.948964\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.2943	+	42.0790i	−0.894892	+	1.55000i
$738$		0			0
$739$	−1.98535	−	3.43873i	−0.0730324	−	0.126496i	0.827197	−	0.561913i	$-0.189934\pi$
$739$	−1.98535	−	3.43873i	−0.0730324	−	0.126496i	−0.900229	+	0.435417i	$0.856601\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.87589	+	4.98118i	−0.105506	+	0.182742i	−0.913945	−	0.405838i	$-0.866980\pi$
$743$	−2.87589	+	4.98118i	−0.105506	+	0.182742i	0.808439	+	0.588580i	$0.200313\pi$
$744$		0			0
$745$	3.01935	+	5.22966i	0.110620	+	0.191600i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.5028	−	26.8517i	0.566460	−	0.981138i
$750$		0			0
$751$	22.6801	−	39.2831i	0.827610	−	1.43346i	−0.0722986	−	0.997383i	$-0.523033\pi$
$751$	22.6801	−	39.2831i	0.827610	−	1.43346i	0.899908	−	0.436079i	$-0.143633\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−31.1044			−1.13200
$756$		0			0
$757$	6.54346	+	11.3336i	0.237826	+	0.411927i	0.960090	−	0.279691i	$-0.0902319\pi$
$757$	6.54346	+	11.3336i	0.237826	+	0.411927i	−0.722264	+	0.691617i	$0.756899\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.0183			0.979411			0.489706	−	0.871888i	$-0.337104\pi$
$761$	27.0183			0.979411			0.489706	+	0.871888i	$0.337104\pi$
$762$		0			0
$763$	−16.2903	−	28.2156i	−0.589747	−	1.02147i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.6455	+	12.1100i	−0.456604	+	0.437265i
$768$		0			0
$769$	−23.1288	+	40.0602i	−0.834045	+	1.44461i	0.0607607	+	0.998152i	$0.480647\pi$
$769$	−23.1288	+	40.0602i	−0.834045	+	1.44461i	−0.894806	+	0.446456i	$0.852686\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20.9744	+	36.3287i	−0.754397	+	1.30665i	0.191276	+	0.981536i	$0.438737\pi$
$773$	−20.9744	+	36.3287i	−0.754397	+	1.30665i	−0.945673	+	0.325118i	$0.894596\pi$
$774$		0			0
$775$	−10.2504			−0.368205
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.6413	+	23.6273i	0.488749	+	0.846537i
$780$		0			0
$781$	15.5398	−	26.9158i	0.556059	−	0.963123i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−17.9837	+	31.1487i	−0.641866	+	1.11174i
$786$		0			0
$787$	41.0216			1.46226			0.731131	−	0.682237i	$-0.238993\pi$
$787$	41.0216			1.46226			0.731131	+	0.682237i	$0.238993\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	11.8182	+	20.4698i	0.420207	+	0.727821i
$792$		0			0
$793$	−10.7284	−	3.12489i	−0.380977	−	0.110968i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−16.5621			−0.586660			−0.293330	−	0.956011i	$-0.594764\pi$
$797$	−16.5621			−0.586660			−0.293330	+	0.956011i	$0.594764\pi$
$798$		0			0
$799$	−1.35751	−	2.35127i	−0.0480252	−	0.0831821i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	34.5471			1.21914
$804$		0			0
$805$	8.97117	+	15.5385i	0.316192	+	0.547661i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.0672	+	43.4176i	0.881314	+	1.52648i	0.849880	+	0.526976i	$0.176674\pi$
$809$	25.0672	+	43.4176i	0.881314	+	1.52648i	0.0314339	+	0.999506i	$0.489993\pi$
$810$		0			0
$811$	−15.2618			−0.535914			−0.267957	−	0.963431i	$-0.586349\pi$
$811$	−15.2618			−0.535914			−0.267957	+	0.963431i	$0.586349\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.73369			0.165814
$816$		0			0
$817$	28.5691			0.999507
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.12150			0.178742			0.0893708	−	0.995998i	$-0.471514\pi$
$821$	5.12150			0.178742			0.0893708	+	0.995998i	$0.471514\pi$
$822$		0			0
$823$	−14.9031			−0.519490			−0.259745	−	0.965677i	$-0.583638\pi$
$823$	−14.9031			−0.519490			−0.259745	+	0.965677i	$0.583638\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.5879			0.646364			0.323182	−	0.946337i	$-0.395247\pi$
$827$	18.5879			0.646364			0.323182	+	0.946337i	$0.395247\pi$
$828$		0			0
$829$	−10.4298	−	18.0650i	−0.362243	−	0.627422i	0.626087	−	0.779753i	$-0.284655\pi$
$829$	−10.4298	−	18.0650i	−0.362243	−	0.627422i	−0.988330	+	0.152331i	$0.951322\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.94724	+	5.10477i	0.102116	+	0.176870i
$834$		0			0
$835$	29.9202			1.03543
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.0229	+	31.2165i	0.622219	+	1.07771i	0.989072	+	0.147436i	$0.0471019\pi$
$839$	18.0229	+	31.2165i	0.622219	+	1.07771i	−0.366853	+	0.930279i	$0.619565\pi$
$840$		0			0
$841$	−26.9561			−0.929521
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.6834	+	9.21372i	−0.608326	+	0.316962i
$846$		0			0
$847$	52.3814	+	90.7273i	1.79985	+	3.11743i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.0925			−0.414527
$852$		0			0
$853$	6.97776	−	12.0858i	0.238914	−	0.413811i	−0.721489	−	0.692426i	$-0.756542\pi$
$853$	6.97776	−	12.0858i	0.238914	−	0.413811i	0.960403	+	0.278615i	$0.0898753\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−8.73939	+	15.1371i	−0.298532	+	0.517073i	−0.975800	−	0.218664i	$-0.929830\pi$
$857$	−8.73939	+	15.1371i	−0.298532	+	0.517073i	0.677268	+	0.735736i	$0.263164\pi$
$858$		0			0
$859$	−4.17576	−	7.23263i	−0.142475	−	0.246774i	0.785953	−	0.618286i	$-0.212173\pi$
$859$	−4.17576	−	7.23263i	−0.142475	−	0.246774i	−0.928428	+	0.371512i	$0.878839\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.11501			−0.140077			−0.0700384	−	0.997544i	$-0.522312\pi$
$863$	−4.11501			−0.140077			−0.0700384	+	0.997544i	$0.522312\pi$
$864$		0			0
$865$	−8.20635	+	14.2138i	−0.279024	+	0.483284i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.2791	+	26.4642i	−0.518308	+	0.897735i
$870$		0			0
$871$	−6.52342	−	26.6376i	−0.221038	−	0.902581i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20.6138	−	35.7042i	−0.696874	−	1.20702i
$876$		0			0
$877$	10.6198			0.358606			0.179303	−	0.983794i	$-0.442616\pi$
$877$	10.6198			0.358606			0.179303	+	0.983794i	$0.442616\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.0048	−	25.9891i	−0.505525	−	0.875595i	−0.999980	−	0.00639173i	$-0.997965\pi$
$881$	−15.0048	−	25.9891i	−0.505525	−	0.875595i	0.494454	−	0.869204i	$-0.335368\pi$
$882$		0			0
$883$	21.3674			0.719072			0.359536	−	0.933131i	$-0.382935\pi$
$883$	21.3674			0.719072			0.359536	+	0.933131i	$0.382935\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.1581	−	34.9148i	0.676841	−	1.17232i	−0.299086	−	0.954226i	$-0.596682\pi$
$887$	20.1581	−	34.9148i	0.676841	−	1.17232i	0.975927	−	0.218097i	$-0.0699850\pi$
$888$		0			0
$889$	−26.7581	+	46.3464i	−0.897437	+	1.55441i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.80615	+	4.86039i	0.0939042	+	0.162647i
$894$		0			0
$895$	−1.74959	+	3.03037i	−0.0584822	+	0.101294i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.76774	−	4.79387i	−0.0923094	−	0.159884i
$900$		0			0
$901$	3.40427	−	5.89637i	0.113413	−	0.196436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−7.88051	−	13.6494i	−0.261957	−	0.453723i
$906$		0			0
$907$	14.7911			0.491131			0.245566	−	0.969380i	$-0.421026\pi$
$907$	14.7911			0.491131			0.245566	+	0.969380i	$0.421026\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.1529	−	27.9777i	0.535170	−	0.926942i	−0.463985	−	0.885843i	$-0.653581\pi$
$911$	16.1529	−	27.9777i	0.535170	−	0.926942i	0.999155	−	0.0410987i	$-0.0130858\pi$
$912$		0			0
$913$	−20.4155	+	35.3607i	−0.675654	+	1.17027i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	20.7356	−	35.9150i	0.684748	−	1.18602i
$918$		0			0
$919$	18.5950	−	32.2075i	0.613392	−	1.06243i	−0.377272	−	0.926103i	$-0.623138\pi$
$919$	18.5950	−	32.2075i	0.613392	−	1.06243i	0.990664	−	0.136324i	$-0.0435289\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.17270	+	17.0387i	0.137346	+	0.560837i
$924$		0			0
$925$	9.61899			0.316270
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.60222	+	13.1674i	0.249421	+	0.432009i	0.963365	−	0.268193i	$-0.0864265\pi$
$929$	7.60222	+	13.1674i	0.249421	+	0.432009i	−0.713944	+	0.700202i	$0.753093\pi$
$930$		0			0
$931$	−6.09234	−	10.5522i	−0.199668	−	0.345836i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	10.7877			0.352795
$936$		0			0
$937$	56.1934			1.83576			0.917880	−	0.396859i	$-0.129900\pi$
$937$	56.1934			1.83576			0.917880	+	0.396859i	$0.129900\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.6111	−	27.0393i	−0.508909	−	0.881455i	−0.999947	−	0.0103174i	$-0.996716\pi$
$941$	−15.6111	−	27.0393i	−0.508909	−	0.881455i	0.491038	−	0.871138i	$-0.336618\pi$
$942$		0			0
$943$	19.9481	+	34.5511i	0.649599	+	1.12514i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	40.6727			1.32169			0.660843	−	0.750524i	$-0.270199\pi$
$947$	40.6727			1.32169			0.660843	+	0.750524i	$0.270199\pi$
$948$		0			0
$949$	−14.0831	+	13.4867i	−0.457157	+	0.437795i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.35525	+	2.34736i	−0.0439008	+	0.0760384i	−0.887141	−	0.461499i	$-0.847312\pi$
$953$	−1.35525	+	2.34736i	−0.0439008	+	0.0760384i	0.843240	+	0.537537i	$0.180645\pi$
$954$		0			0
$955$	18.1205	−	31.3857i	0.586367	−	1.01562i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.6129	−	42.6307i	0.794791	−	1.37662i
$960$		0			0
$961$	8.00414	−	13.8636i	0.258198	−	0.447212i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.9327			0.545082
$966$		0			0
$967$	−2.04491	−	3.54189i	−0.0657600	−	0.113900i	0.831271	−	0.555868i	$-0.187614\pi$
$967$	−2.04491	−	3.54189i	−0.0657600	−	0.113900i	−0.897031	+	0.441968i	$0.854280\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.04999	−	10.4789i	0.194153	−	0.336283i	−0.752469	−	0.658627i	$-0.771137\pi$
$971$	6.04999	−	10.4789i	0.194153	−	0.336283i	0.946623	+	0.322344i	$0.104471\pi$
$972$		0			0
$973$	−14.5637	−	25.2250i	−0.466890	−	0.808677i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.82475	−	6.62467i	0.122365	−	0.211942i	−0.798335	−	0.602214i	$-0.794286\pi$
$977$	3.82475	−	6.62467i	0.122365	−	0.211942i	0.920700	+	0.390272i	$0.127619\pi$
$978$		0			0
$979$	−27.1672	−	47.0550i	−0.868268	−	1.50388i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−25.0980	+	43.4711i	−0.800503	+	1.38651i	0.118783	+	0.992920i	$0.462101\pi$
$983$	−25.0980	+	43.4711i	−0.800503	+	1.38651i	−0.919286	+	0.393591i	$0.871233\pi$
$984$		0			0
$985$	−6.86659	+	11.8933i	−0.218788	+	0.378952i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	41.7776			1.32845
$990$		0			0
$991$	−21.7547	−	37.6803i	−0.691062	−	1.19695i	−0.971490	−	0.237079i	$-0.923810\pi$
$991$	−21.7547	−	37.6803i	−0.691062	−	1.19695i	0.280428	−	0.959875i	$-0.409523\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−35.8149			−1.13541
$996$		0			0
$997$	15.1014	+	26.1564i	0.478266	+	0.828381i	0.999690	−	0.0249170i	$-0.00793213\pi$
$997$	15.1014	+	26.1564i	0.478266	+	0.828381i	−0.521423	+	0.853298i	$0.674599\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.r.f.289.8		40
3.2	odd	2		936.2.r.f.601.11	✓	40
9.4	even	3		2808.2.s.f.1225.8		40
9.5	odd	6		936.2.s.f.913.17	yes	40
13.9	even	3		2808.2.s.f.1153.8		40
39.35	odd	6		936.2.s.f.529.17	yes	40
117.22	even	3	inner	2808.2.r.f.2089.8		40
117.113	odd	6		936.2.r.f.841.11	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.r.f.601.11	✓	40	3.2	odd	2
936.2.r.f.841.11	yes	40	117.113	odd	6
936.2.s.f.529.17	yes	40	39.35	odd	6
936.2.s.f.913.17	yes	40	9.5	odd	6
2808.2.r.f.289.8		40	1.1	even	1	trivial
2808.2.r.f.2089.8		40	117.22	even	3	inner
2808.2.s.f.1153.8		40	13.9	even	3
2808.2.s.f.1225.8		40	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.r (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.766914 - 1.32833i) q^{5} +(1.75740 + 3.04390i) q^{7} -6.38798 q^{11} +(2.60406 - 2.49377i) q^{13} +(0.550501 - 0.953495i) q^{17} +(-1.13796 + 1.97100i) q^{19} +(-1.66408 + 2.88227i) q^{23} +(1.32369 - 2.29269i) q^{25} +1.42965 q^{29} +(-1.93596 - 3.35318i) q^{31} +(2.69554 - 4.66881i) q^{35} +(1.81670 + 3.14662i) q^{37} +(5.99374 - 10.3815i) q^{41} +(-6.27640 - 10.8710i) q^{43} +(1.23298 - 2.13558i) q^{47} +(-2.67688 + 4.63648i) q^{49} +6.18395 q^{53} +(4.89903 + 8.48537i) q^{55} -4.85609 q^{59} +(-1.54959 - 2.68397i) q^{61} +(-5.30965 - 1.54655i) q^{65} +(3.80313 - 6.58721i) q^{67} +(-2.43267 + 4.21351i) q^{71} -5.40814 q^{73} +(-11.2262 - 19.4444i) q^{77} +(2.39185 - 4.14281i) q^{79} +(3.19592 - 5.53550i) q^{83} -1.68875 q^{85} +(4.25286 + 7.36618i) q^{89} +(12.1671 + 3.54395i) q^{91} +3.49086 q^{95} +(1.76312 + 3.05381i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - q^{5} + 7 q^{7} + q^{17} + 2 q^{19} + q^{23} - 23 q^{25} + 24 q^{29} + 8 q^{31} + 12 q^{35} + 18 q^{37} + 3 q^{41} + 8 q^{43} - 4 q^{47} - 23 q^{49} + 40 q^{53} - 14 q^{55} - 8 q^{59} - 3 q^{61}+ \cdots + 35 q^{97}+O(q^{100}) \) 40 * q - q^5 + 7 * q^7 + q^17 + 2 * q^19 + q^23 - 23 * q^25 + 24 * q^29 + 8 * q^31 + 12 * q^35 + 18 * q^37 + 3 * q^41 + 8 * q^43 - 4 * q^47 - 23 * q^49 + 40 * q^53 - 14 * q^55 - 8 * q^59 - 3 * q^61 + 32 * q^65 + 27 * q^67 - q^71 - 12 * q^73 + 4 * q^77 + 13 * q^79 + 15 * q^83 - 10 * q^85 - 12 * q^89 - 57 * q^91 + 2 * q^95 + 35 * q^97

Introduction

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