LMFDB - Embedded newform 4840.2.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4840,2,Mod(1,4840)] 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("4840.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-6,0,-7,0,5,0,0,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.25903625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 $ x^6 - 3x^5 - 7x^4 + 17x^3 + 16x^2 - 20*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.29082$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.29082 q^{3} -1.00000 q^{5} -4.66366 q^{7} +2.24785 q^{9} -5.94269 q^{13} +2.29082 q^{15} +3.30331 q^{17} +4.19195 q^{19} +10.6836 q^{21} -1.71184 q^{23} +1.00000 q^{25} +1.72304 q^{27} +7.76702 q^{29} -2.97204 q^{31} +4.66366 q^{35} -5.68500 q^{37} +13.6136 q^{39} +6.14758 q^{41} +4.42353 q^{43} -2.24785 q^{45} +4.13243 q^{47} +14.7497 q^{49} -7.56730 q^{51} +8.10104 q^{53} -9.60300 q^{57} -6.21515 q^{59} +1.52758 q^{61} -10.4832 q^{63} +5.94269 q^{65} -7.46447 q^{67} +3.92151 q^{69} -2.23747 q^{71} +7.49400 q^{73} -2.29082 q^{75} -17.5316 q^{79} -10.6907 q^{81} -11.0785 q^{83} -3.30331 q^{85} -17.7928 q^{87} +12.8171 q^{89} +27.7147 q^{91} +6.80840 q^{93} -4.19195 q^{95} -9.53760 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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
$3$	−2.29082	−1.32260	−0.661302	−	0.750119i	$-0.729996\pi$
$3$	−2.29082	−1.32260	−0.661302	+	0.750119i	$0.729996\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.66366	−1.76270	−0.881348	−	0.472467i	$-0.843363\pi$
$7$	−4.66366	−1.76270	−0.881348	+	0.472467i	$0.843363\pi$
$8$	0	0
$9$	2.24785	0.749284
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.94269	−1.64821	−0.824103	−	0.566440i	$-0.808320\pi$
$13$	−5.94269	−1.64821	−0.824103	+	0.566440i	$0.808320\pi$
$14$	0	0
$15$	2.29082	0.591487
$16$	0	0
$17$	3.30331	0.801171	0.400586	−	0.916259i	$-0.368807\pi$
$17$	3.30331	0.801171	0.400586	+	0.916259i	$0.368807\pi$
$18$	0	0
$19$	4.19195	0.961699	0.480850	−	0.876803i	$-0.340328\pi$
$19$	4.19195	0.961699	0.480850	+	0.876803i	$0.340328\pi$
$20$	0	0
$21$	10.6836	2.33135
$22$	0	0
$23$	−1.71184	−0.356942	−0.178471	−	0.983945i	$-0.557115\pi$
$23$	−1.71184	−0.356942	−0.178471	+	0.983945i	$0.557115\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.72304	0.331598
$28$	0	0
$29$	7.76702	1.44230	0.721149	−	0.692780i	$-0.243614\pi$
$29$	7.76702	1.44230	0.721149	+	0.692780i	$0.243614\pi$
$30$	0	0
$31$	−2.97204	−0.533794	−0.266897	−	0.963725i	$-0.585998\pi$
$31$	−2.97204	−0.533794	−0.266897	+	0.963725i	$0.585998\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.66366	0.788302
$36$	0	0
$37$	−5.68500	−0.934608	−0.467304	−	0.884097i	$-0.654775\pi$
$37$	−5.68500	−0.934608	−0.467304	+	0.884097i	$0.654775\pi$
$38$	0	0
$39$	13.6136	2.17993
$40$	0	0
$41$	6.14758	0.960090	0.480045	−	0.877244i	$-0.340620\pi$
$41$	6.14758	0.960090	0.480045	+	0.877244i	$0.340620\pi$
$42$	0	0
$43$	4.42353	0.674582	0.337291	−	0.941401i	$-0.390489\pi$
$43$	4.42353	0.674582	0.337291	+	0.941401i	$0.390489\pi$
$44$	0	0
$45$	−2.24785	−0.335090
$46$	0	0
$47$	4.13243	0.602776	0.301388	−	0.953502i	$-0.402550\pi$
$47$	4.13243	0.602776	0.301388	+	0.953502i	$0.402550\pi$
$48$	0	0
$49$	14.7497	2.10710
$50$	0	0
$51$	−7.56730	−1.05963
$52$	0	0
$53$	8.10104	1.11276	0.556381	−	0.830927i	$-0.312189\pi$
$53$	8.10104	1.11276	0.556381	+	0.830927i	$0.312189\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−9.60300	−1.27195
$58$	0	0
$59$	−6.21515	−0.809144	−0.404572	−	0.914506i	$-0.632580\pi$
$59$	−6.21515	−0.809144	−0.404572	+	0.914506i	$0.632580\pi$
$60$	0	0
$61$	1.52758	0.195587	0.0977934	−	0.995207i	$-0.468822\pi$
$61$	1.52758	0.195587	0.0977934	+	0.995207i	$0.468822\pi$
$62$	0	0
$63$	−10.4832	−1.32076
$64$	0	0
$65$	5.94269	0.737100
$66$	0	0
$67$	−7.46447	−0.911931	−0.455965	−	0.889998i	$-0.650706\pi$
$67$	−7.46447	−0.911931	−0.455965	+	0.889998i	$0.650706\pi$
$68$	0	0
$69$	3.92151	0.472094
$70$	0	0
$71$	−2.23747	−0.265539	−0.132770	−	0.991147i	$-0.542387\pi$
$71$	−2.23747	−0.265539	−0.132770	+	0.991147i	$0.542387\pi$
$72$	0	0
$73$	7.49400	0.877106	0.438553	−	0.898705i	$-0.355491\pi$
$73$	7.49400	0.877106	0.438553	+	0.898705i	$0.355491\pi$
$74$	0	0
$75$	−2.29082	−0.264521
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−17.5316	−1.97246	−0.986230	−	0.165378i	$-0.947116\pi$
$79$	−17.5316	−1.97246	−0.986230	+	0.165378i	$0.947116\pi$
$80$	0	0
$81$	−10.6907	−1.18786
$82$	0	0
$83$	−11.0785	−1.21603	−0.608014	−	0.793926i	$-0.708034\pi$
$83$	−11.0785	−1.21603	−0.608014	+	0.793926i	$0.708034\pi$
$84$	0	0
$85$	−3.30331	−0.358295
$86$	0	0
$87$	−17.7928	−1.90759
$88$	0	0
$89$	12.8171	1.35861	0.679305	−	0.733856i	$-0.262281\pi$
$89$	12.8171	1.35861	0.679305	+	0.733856i	$0.262281\pi$
$90$	0	0
$91$	27.7147	2.90529
$92$	0	0
$93$	6.80840	0.705998
$94$	0	0
$95$	−4.19195	−0.430085
$96$	0	0
$97$	−9.53760	−0.968396	−0.484198	−	0.874958i	$-0.660889\pi$
$97$	−9.53760	−0.968396	−0.484198	+	0.874958i	$0.660889\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.7259	1.06727	0.533635	−	0.845715i	$-0.320826\pi$
$101$	10.7259	1.06727	0.533635	+	0.845715i	$0.320826\pi$
$102$	0	0
$103$	2.57865	0.254082	0.127041	−	0.991897i	$-0.459452\pi$
$103$	2.57865	0.254082	0.127041	+	0.991897i	$0.459452\pi$
$104$	0	0
$105$	−10.6836	−1.04261
$106$	0	0
$107$	17.6297	1.70433	0.852164	−	0.523275i	$-0.175290\pi$
$107$	17.6297	1.70433	0.852164	+	0.523275i	$0.175290\pi$
$108$	0	0
$109$	15.3476	1.47004	0.735019	−	0.678047i	$-0.237173\pi$
$109$	15.3476	1.47004	0.735019	+	0.678047i	$0.237173\pi$
$110$	0	0
$111$	13.0233	1.23612
$112$	0	0
$113$	0.316934	0.0298147	0.0149073	−	0.999889i	$-0.495255\pi$
$113$	0.316934	0.0298147	0.0149073	+	0.999889i	$0.495255\pi$
$114$	0	0
$115$	1.71184	0.159630
$116$	0	0
$117$	−13.3583	−1.23497
$118$	0	0
$119$	−15.4055	−1.41222
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−14.0830	−1.26982
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.0671	−1.60320	−0.801599	−	0.597862i	$-0.796017\pi$
$127$	−18.0671	−1.60320	−0.801599	+	0.597862i	$0.796017\pi$
$128$	0	0
$129$	−10.1335	−0.892205
$130$	0	0
$131$	7.95176	0.694749	0.347374	−	0.937727i	$-0.387073\pi$
$131$	7.95176	0.694749	0.347374	+	0.937727i	$0.387073\pi$
$132$	0	0
$133$	−19.5498	−1.69518
$134$	0	0
$135$	−1.72304	−0.148295
$136$	0	0
$137$	−8.07748	−0.690106	−0.345053	−	0.938583i	$-0.612139\pi$
$137$	−8.07748	−0.690106	−0.345053	+	0.938583i	$0.612139\pi$
$138$	0	0
$139$	−11.3240	−0.960486	−0.480243	−	0.877135i	$-0.659452\pi$
$139$	−11.3240	−0.960486	−0.480243	+	0.877135i	$0.659452\pi$
$140$	0	0
$141$	−9.46664	−0.797235
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.76702	−0.645016
$146$	0	0
$147$	−33.7889	−2.78686
$148$	0	0
$149$	−2.35175	−0.192663	−0.0963314	−	0.995349i	$-0.530711\pi$
$149$	−2.35175	−0.192663	−0.0963314	+	0.995349i	$0.530711\pi$
$150$	0	0
$151$	−14.7268	−1.19845	−0.599227	−	0.800579i	$-0.704525\pi$
$151$	−14.7268	−1.19845	−0.599227	+	0.800579i	$0.704525\pi$
$152$	0	0
$153$	7.42536	0.600305
$154$	0	0
$155$	2.97204	0.238720
$156$	0	0
$157$	−16.4544	−1.31320	−0.656600	−	0.754239i	$-0.728006\pi$
$157$	−16.4544	−1.31320	−0.656600	+	0.754239i	$0.728006\pi$
$158$	0	0
$159$	−18.5580	−1.47175
$160$	0	0
$161$	7.98341	0.629181
$162$	0	0
$163$	5.26242	0.412184	0.206092	−	0.978533i	$-0.433925\pi$
$163$	5.26242	0.412184	0.206092	+	0.978533i	$0.433925\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.5495	1.66755	0.833776	−	0.552102i	$-0.186174\pi$
$167$	21.5495	1.66755	0.833776	+	0.552102i	$0.186174\pi$
$168$	0	0
$169$	22.3156	1.71658
$170$	0	0
$171$	9.42288	0.720586
$172$	0	0
$173$	7.08425	0.538605	0.269303	−	0.963056i	$-0.413207\pi$
$173$	7.08425	0.538605	0.269303	+	0.963056i	$0.413207\pi$
$174$	0	0
$175$	−4.66366	−0.352539
$176$	0	0
$177$	14.2378	1.07018
$178$	0	0
$179$	8.78759	0.656815	0.328408	−	0.944536i	$-0.393488\pi$
$179$	8.78759	0.656815	0.328408	+	0.944536i	$0.393488\pi$
$180$	0	0
$181$	−12.6451	−0.939904	−0.469952	−	0.882692i	$-0.655729\pi$
$181$	−12.6451	−0.939904	−0.469952	+	0.882692i	$0.655729\pi$
$182$	0	0
$183$	−3.49941	−0.258684
$184$	0	0
$185$	5.68500	0.417969
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−8.03564	−0.584507
$190$	0	0
$191$	−17.7224	−1.28235	−0.641175	−	0.767394i	$-0.721553\pi$
$191$	−17.7224	−1.28235	−0.641175	+	0.767394i	$0.721553\pi$
$192$	0	0
$193$	5.03540	0.362456	0.181228	−	0.983441i	$-0.441993\pi$
$193$	5.03540	0.362456	0.181228	+	0.983441i	$0.441993\pi$
$194$	0	0
$195$	−13.6136	−0.974892
$196$	0	0
$197$	15.0557	1.07268	0.536338	−	0.844003i	$-0.319807\pi$
$197$	15.0557	1.07268	0.536338	+	0.844003i	$0.319807\pi$
$198$	0	0
$199$	2.56534	0.181852	0.0909259	−	0.995858i	$-0.471017\pi$
$199$	2.56534	0.181852	0.0909259	+	0.995858i	$0.471017\pi$
$200$	0	0
$201$	17.0998	1.20612
$202$	0	0
$203$	−36.2227	−2.54233
$204$	0	0
$205$	−6.14758	−0.429365
$206$	0	0
$207$	−3.84795	−0.267451
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−9.71897	−0.669082	−0.334541	−	0.942381i	$-0.608581\pi$
$211$	−9.71897	−0.669082	−0.334541	+	0.942381i	$0.608581\pi$
$212$	0	0
$213$	5.12565	0.351204
$214$	0	0
$215$	−4.42353	−0.301682
$216$	0	0
$217$	13.8606	0.940916
$218$	0	0
$219$	−17.1674	−1.16006
$220$	0	0
$221$	−19.6306	−1.32050
$222$	0	0
$223$	0.543504	0.0363957	0.0181979	−	0.999834i	$-0.494207\pi$
$223$	0.543504	0.0363957	0.0181979	+	0.999834i	$0.494207\pi$
$224$	0	0
$225$	2.24785	0.149857
$226$	0	0
$227$	−5.19727	−0.344955	−0.172478	−	0.985013i	$-0.555177\pi$
$227$	−5.19727	−0.344955	−0.172478	+	0.985013i	$0.555177\pi$
$228$	0	0
$229$	−4.09116	−0.270352	−0.135176	−	0.990822i	$-0.543160\pi$
$229$	−4.09116	−0.270352	−0.135176	+	0.990822i	$0.543160\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.9446	−0.913541	−0.456771	−	0.889584i	$-0.650994\pi$
$233$	−13.9446	−0.913541	−0.456771	+	0.889584i	$0.650994\pi$
$234$	0	0
$235$	−4.13243	−0.269570
$236$	0	0
$237$	40.1618	2.60879
$238$	0	0
$239$	−0.523568	−0.0338668	−0.0169334	−	0.999857i	$-0.505390\pi$
$239$	−0.523568	−0.0338668	−0.0169334	+	0.999857i	$0.505390\pi$
$240$	0	0
$241$	6.38011	0.410979	0.205489	−	0.978659i	$-0.434121\pi$
$241$	6.38011	0.410979	0.205489	+	0.978659i	$0.434121\pi$
$242$	0	0
$243$	19.3214	1.23947
$244$	0	0
$245$	−14.7497	−0.942323
$246$	0	0
$247$	−24.9115	−1.58508
$248$	0	0
$249$	25.3789	1.60832
$250$	0	0
$251$	13.6413	0.861030	0.430515	−	0.902583i	$-0.358332\pi$
$251$	13.6413	0.861030	0.430515	+	0.902583i	$0.358332\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	7.56730	0.473882
$256$	0	0
$257$	−2.40705	−0.150148	−0.0750738	−	0.997178i	$-0.523919\pi$
$257$	−2.40705	−0.150148	−0.0750738	+	0.997178i	$0.523919\pi$
$258$	0	0
$259$	26.5129	1.64743
$260$	0	0
$261$	17.4591	1.08069
$262$	0	0
$263$	11.7841	0.726641	0.363321	−	0.931664i	$-0.381643\pi$
$263$	11.7841	0.726641	0.363321	+	0.931664i	$0.381643\pi$
$264$	0	0
$265$	−8.10104	−0.497643
$266$	0	0
$267$	−29.3617	−1.79691
$268$	0	0
$269$	−15.1436	−0.923324	−0.461662	−	0.887056i	$-0.652747\pi$
$269$	−15.1436	−0.923324	−0.461662	+	0.887056i	$0.652747\pi$
$270$	0	0
$271$	10.5739	0.642316	0.321158	−	0.947026i	$-0.395928\pi$
$271$	10.5739	0.642316	0.321158	+	0.947026i	$0.395928\pi$
$272$	0	0
$273$	−63.4893	−3.84255
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.1516	1.51121	0.755606	−	0.655026i	$-0.227342\pi$
$277$	25.1516	1.51121	0.755606	+	0.655026i	$0.227342\pi$
$278$	0	0
$279$	−6.68070	−0.399963
$280$	0	0
$281$	0.322966	0.0192666	0.00963328	−	0.999954i	$-0.496934\pi$
$281$	0.322966	0.0192666	0.00963328	+	0.999954i	$0.496934\pi$
$282$	0	0
$283$	−21.3723	−1.27045	−0.635227	−	0.772325i	$-0.719094\pi$
$283$	−21.3723	−1.27045	−0.635227	+	0.772325i	$0.719094\pi$
$284$	0	0
$285$	9.60300	0.568832
$286$	0	0
$287$	−28.6702	−1.69235
$288$	0	0
$289$	−6.08811	−0.358124
$290$	0	0
$291$	21.8489	1.28081
$292$	0	0
$293$	10.5001	0.613424	0.306712	−	0.951802i	$-0.400771\pi$
$293$	10.5001	0.613424	0.306712	+	0.951802i	$0.400771\pi$
$294$	0	0
$295$	6.21515	0.361860
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.1729	0.588315
$300$	0	0
$301$	−20.6298	−1.18908
$302$	0	0
$303$	−24.5712	−1.41158
$304$	0	0
$305$	−1.52758	−0.0874691
$306$	0	0
$307$	29.0598	1.65853	0.829266	−	0.558854i	$-0.188759\pi$
$307$	29.0598	1.65853	0.829266	+	0.558854i	$0.188759\pi$
$308$	0	0
$309$	−5.90722	−0.336050
$310$	0	0
$311$	−19.9483	−1.13116	−0.565582	−	0.824692i	$-0.691348\pi$
$311$	−19.9483	−1.13116	−0.565582	+	0.824692i	$0.691348\pi$
$312$	0	0
$313$	17.4800	0.988030	0.494015	−	0.869453i	$-0.335529\pi$
$313$	17.4800	0.988030	0.494015	+	0.869453i	$0.335529\pi$
$314$	0	0
$315$	10.4832	0.590662
$316$	0	0
$317$	−29.8295	−1.67539	−0.837696	−	0.546136i	$-0.816098\pi$
$317$	−29.8295	−1.67539	−0.837696	+	0.546136i	$0.816098\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−40.3865	−2.25415
$322$	0	0
$323$	13.8473	0.770486
$324$	0	0
$325$	−5.94269	−0.329641
$326$	0	0
$327$	−35.1587	−1.94428
$328$	0	0
$329$	−19.2722	−1.06251
$330$	0	0
$331$	29.4740	1.62004	0.810018	−	0.586405i	$-0.199457\pi$
$331$	29.4740	1.62004	0.810018	+	0.586405i	$0.199457\pi$
$332$	0	0
$333$	−12.7790	−0.700287
$334$	0	0
$335$	7.46447	0.407828
$336$	0	0
$337$	−20.9682	−1.14221	−0.571106	−	0.820877i	$-0.693485\pi$
$337$	−20.9682	−1.14221	−0.571106	+	0.820877i	$0.693485\pi$
$338$	0	0
$339$	−0.726039	−0.0394330
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−36.1419	−1.95148
$344$	0	0
$345$	−3.92151	−0.211127
$346$	0	0
$347$	7.27059	0.390306	0.195153	−	0.980773i	$-0.437480\pi$
$347$	7.27059	0.390306	0.195153	+	0.980773i	$0.437480\pi$
$348$	0	0
$349$	−19.2432	−1.03006	−0.515032	−	0.857171i	$-0.672220\pi$
$349$	−19.2432	−1.03006	−0.515032	+	0.857171i	$0.672220\pi$
$350$	0	0
$351$	−10.2395	−0.546542
$352$	0	0
$353$	14.5882	0.776454	0.388227	−	0.921564i	$-0.373088\pi$
$353$	14.5882	0.776454	0.388227	+	0.921564i	$0.373088\pi$
$354$	0	0
$355$	2.23747	0.118753
$356$	0	0
$357$	35.2913	1.86781
$358$	0	0
$359$	−2.56617	−0.135437	−0.0677187	−	0.997704i	$-0.521572\pi$
$359$	−2.56617	−0.135437	−0.0677187	+	0.997704i	$0.521572\pi$
$360$	0	0
$361$	−1.42756	−0.0751348
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.49400	−0.392254
$366$	0	0
$367$	−21.7911	−1.13748	−0.568742	−	0.822516i	$-0.692570\pi$
$367$	−21.7911	−1.13748	−0.568742	+	0.822516i	$0.692570\pi$
$368$	0	0
$369$	13.8188	0.719380
$370$	0	0
$371$	−37.7804	−1.96146
$372$	0	0
$373$	−19.7684	−1.02357	−0.511783	−	0.859115i	$-0.671015\pi$
$373$	−19.7684	−1.02357	−0.511783	+	0.859115i	$0.671015\pi$
$374$	0	0
$375$	2.29082	0.118297
$376$	0	0
$377$	−46.1570	−2.37721
$378$	0	0
$379$	−13.5393	−0.695467	−0.347733	−	0.937594i	$-0.613049\pi$
$379$	−13.5393	−0.695467	−0.347733	+	0.937594i	$0.613049\pi$
$380$	0	0
$381$	41.3885	2.12040
$382$	0	0
$383$	14.3937	0.735483	0.367742	−	0.929928i	$-0.380131\pi$
$383$	14.3937	0.735483	0.367742	+	0.929928i	$0.380131\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.94343	0.505453
$388$	0	0
$389$	−15.2204	−0.771705	−0.385853	−	0.922560i	$-0.626093\pi$
$389$	−15.2204	−0.771705	−0.385853	+	0.922560i	$0.626093\pi$
$390$	0	0
$391$	−5.65473	−0.285972
$392$	0	0
$393$	−18.2160	−0.918878
$394$	0	0
$395$	17.5316	0.882111
$396$	0	0
$397$	−37.0013	−1.85704	−0.928520	−	0.371282i	$-0.878918\pi$
$397$	−37.0013	−1.85704	−0.928520	+	0.371282i	$0.878918\pi$
$398$	0	0
$399$	44.7851	2.24206
$400$	0	0
$401$	−24.6104	−1.22898	−0.614492	−	0.788923i	$-0.710639\pi$
$401$	−24.6104	−1.22898	−0.614492	+	0.788923i	$0.710639\pi$
$402$	0	0
$403$	17.6619	0.879802
$404$	0	0
$405$	10.6907	0.531226
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−18.3346	−0.906586	−0.453293	−	0.891362i	$-0.649751\pi$
$409$	−18.3346	−0.906586	−0.453293	+	0.891362i	$0.649751\pi$
$410$	0	0
$411$	18.5040	0.912737
$412$	0	0
$413$	28.9853	1.42628
$414$	0	0
$415$	11.0785	0.543824
$416$	0	0
$417$	25.9411	1.27034
$418$	0	0
$419$	23.6524	1.15550	0.577748	−	0.816215i	$-0.303932\pi$
$419$	23.6524	1.15550	0.577748	+	0.816215i	$0.303932\pi$
$420$	0	0
$421$	11.6921	0.569840	0.284920	−	0.958551i	$-0.408033\pi$
$421$	11.6921	0.569840	0.284920	+	0.958551i	$0.408033\pi$
$422$	0	0
$423$	9.28908	0.451651
$424$	0	0
$425$	3.30331	0.160234
$426$	0	0
$427$	−7.12412	−0.344760
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.42118	0.261129	0.130565	−	0.991440i	$-0.458321\pi$
$431$	5.42118	0.261129	0.130565	+	0.991440i	$0.458321\pi$
$432$	0	0
$433$	−14.0482	−0.675116	−0.337558	−	0.941305i	$-0.609601\pi$
$433$	−14.0482	−0.675116	−0.337558	+	0.941305i	$0.609601\pi$
$434$	0	0
$435$	17.7928	0.853101
$436$	0	0
$437$	−7.17593	−0.343271
$438$	0	0
$439$	−13.4133	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$439$	−13.4133	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$440$	0	0
$441$	33.1551	1.57881
$442$	0	0
$443$	−28.1807	−1.33891	−0.669453	−	0.742855i	$-0.733471\pi$
$443$	−28.1807	−1.33891	−0.669453	+	0.742855i	$0.733471\pi$
$444$	0	0
$445$	−12.8171	−0.607589
$446$	0	0
$447$	5.38743	0.254817
$448$	0	0
$449$	−37.5023	−1.76984	−0.884921	−	0.465741i	$-0.845788\pi$
$449$	−37.5023	−1.76984	−0.884921	+	0.465741i	$0.845788\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	33.7365	1.58508
$454$	0	0
$455$	−27.7147	−1.29928
$456$	0	0
$457$	−15.5343	−0.726666	−0.363333	−	0.931659i	$-0.618361\pi$
$457$	−15.5343	−0.726666	−0.363333	+	0.931659i	$0.618361\pi$
$458$	0	0
$459$	5.69173	0.265667
$460$	0	0
$461$	20.4407	0.952018	0.476009	−	0.879440i	$-0.342083\pi$
$461$	20.4407	0.952018	0.476009	+	0.879440i	$0.342083\pi$
$462$	0	0
$463$	−9.64734	−0.448350	−0.224175	−	0.974549i	$-0.571969\pi$
$463$	−9.64734	−0.448350	−0.224175	+	0.974549i	$0.571969\pi$
$464$	0	0
$465$	−6.80840	−0.315732
$466$	0	0
$467$	13.9773	0.646790	0.323395	−	0.946264i	$-0.395176\pi$
$467$	13.9773	0.646790	0.323395	+	0.946264i	$0.395176\pi$
$468$	0	0
$469$	34.8117	1.60746
$470$	0	0
$471$	37.6939	1.73685
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.19195	0.192340
$476$	0	0
$477$	18.2099	0.833775
$478$	0	0
$479$	−28.7475	−1.31351	−0.656754	−	0.754105i	$-0.728071\pi$
$479$	−28.7475	−1.31351	−0.656754	+	0.754105i	$0.728071\pi$
$480$	0	0
$481$	33.7842	1.54043
$482$	0	0
$483$	−18.2886	−0.832158
$484$	0	0
$485$	9.53760	0.433080
$486$	0	0
$487$	10.2994	0.466709	0.233355	−	0.972392i	$-0.425030\pi$
$487$	10.2994	0.466709	0.233355	+	0.972392i	$0.425030\pi$
$488$	0	0
$489$	−12.0552	−0.545157
$490$	0	0
$491$	7.35859	0.332088	0.166044	−	0.986118i	$-0.446901\pi$
$491$	7.35859	0.332088	0.166044	+	0.986118i	$0.446901\pi$
$492$	0	0
$493$	25.6569	1.15553
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.4348	0.468065
$498$	0	0
$499$	2.83645	0.126977	0.0634885	−	0.997983i	$-0.479777\pi$
$499$	2.83645	0.126977	0.0634885	+	0.997983i	$0.479777\pi$
$500$	0	0
$501$	−49.3661	−2.20551
$502$	0	0
$503$	13.8353	0.616887	0.308444	−	0.951243i	$-0.400192\pi$
$503$	13.8353	0.616887	0.308444	+	0.951243i	$0.400192\pi$
$504$	0	0
$505$	−10.7259	−0.477297
$506$	0	0
$507$	−51.1209	−2.27036
$508$	0	0
$509$	24.9515	1.10596	0.552978	−	0.833196i	$-0.313491\pi$
$509$	24.9515	1.10596	0.552978	+	0.833196i	$0.313491\pi$
$510$	0	0
$511$	−34.9494	−1.54607
$512$	0	0
$513$	7.22288	0.318898
$514$	0	0
$515$	−2.57865	−0.113629
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−16.2287	−0.712362
$520$	0	0
$521$	1.72550	0.0755955	0.0377978	−	0.999285i	$-0.487966\pi$
$521$	1.72550	0.0755955	0.0377978	+	0.999285i	$0.487966\pi$
$522$	0	0
$523$	14.9448	0.653489	0.326745	−	0.945113i	$-0.394048\pi$
$523$	14.9448	0.653489	0.326745	+	0.945113i	$0.394048\pi$
$524$	0	0
$525$	10.6836	0.466270
$526$	0	0
$527$	−9.81758	−0.427660
$528$	0	0
$529$	−20.0696	−0.872592
$530$	0	0
$531$	−13.9707	−0.606279
$532$	0	0
$533$	−36.5331	−1.58243
$534$	0	0
$535$	−17.6297	−0.762199
$536$	0	0
$537$	−20.1308	−0.868707
$538$	0	0
$539$	0	0
$540$	0	0
$541$	40.0200	1.72059	0.860297	−	0.509793i	$-0.170278\pi$
$541$	40.0200	1.72059	0.860297	+	0.509793i	$0.170278\pi$
$542$	0	0
$543$	28.9677	1.24312
$544$	0	0
$545$	−15.3476	−0.657421
$546$	0	0
$547$	−22.9990	−0.983368	−0.491684	−	0.870774i	$-0.663618\pi$
$547$	−22.9990	−0.983368	−0.491684	+	0.870774i	$0.663618\pi$
$548$	0	0
$549$	3.43378	0.146550
$550$	0	0
$551$	32.5589	1.38706
$552$	0	0
$553$	81.7614	3.47685
$554$	0	0
$555$	−13.0233	−0.552808
$556$	0	0
$557$	−11.9261	−0.505323	−0.252662	−	0.967555i	$-0.581306\pi$
$557$	−11.9261	−0.505323	−0.252662	+	0.967555i	$0.581306\pi$
$558$	0	0
$559$	−26.2877	−1.11185
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.2951	1.65609	0.828046	−	0.560660i	$-0.189453\pi$
$563$	39.2951	1.65609	0.828046	+	0.560660i	$0.189453\pi$
$564$	0	0
$565$	−0.316934	−0.0133335
$566$	0	0
$567$	49.8578	2.09383
$568$	0	0
$569$	17.1895	0.720621	0.360310	−	0.932832i	$-0.382671\pi$
$569$	17.1895	0.720621	0.360310	+	0.932832i	$0.382671\pi$
$570$	0	0
$571$	26.4297	1.10605	0.553025	−	0.833165i	$-0.313473\pi$
$571$	26.4297	1.10605	0.553025	+	0.833165i	$0.313473\pi$
$572$	0	0
$573$	40.5989	1.69604
$574$	0	0
$575$	−1.71184	−0.0713885
$576$	0	0
$577$	10.3838	0.432284	0.216142	−	0.976362i	$-0.430653\pi$
$577$	10.3838	0.432284	0.216142	+	0.976362i	$0.430653\pi$
$578$	0	0
$579$	−11.5352	−0.479386
$580$	0	0
$581$	51.6665	2.14349
$582$	0	0
$583$	0	0
$584$	0	0
$585$	13.3583	0.552297
$586$	0	0
$587$	−21.2848	−0.878517	−0.439258	−	0.898361i	$-0.644759\pi$
$587$	−21.2848	−0.878517	−0.439258	+	0.898361i	$0.644759\pi$
$588$	0	0
$589$	−12.4586	−0.513349
$590$	0	0
$591$	−34.4900	−1.41873
$592$	0	0
$593$	6.74652	0.277046	0.138523	−	0.990359i	$-0.455764\pi$
$593$	6.74652	0.277046	0.138523	+	0.990359i	$0.455764\pi$
$594$	0	0
$595$	15.4055	0.631565
$596$	0	0
$597$	−5.87672	−0.240518
$598$	0	0
$599$	−4.09390	−0.167272	−0.0836361	−	0.996496i	$-0.526653\pi$
$599$	−4.09390	−0.167272	−0.0836361	+	0.996496i	$0.526653\pi$
$600$	0	0
$601$	−2.46697	−0.100630	−0.0503149	−	0.998733i	$-0.516022\pi$
$601$	−2.46697	−0.100630	−0.0503149	+	0.998733i	$0.516022\pi$
$602$	0	0
$603$	−16.7790	−0.683295
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−27.9423	−1.13414	−0.567072	−	0.823668i	$-0.691924\pi$
$607$	−27.9423	−1.13414	−0.567072	+	0.823668i	$0.691924\pi$
$608$	0	0
$609$	82.9796	3.36250
$610$	0	0
$611$	−24.5577	−0.993499
$612$	0	0
$613$	38.9146	1.57174	0.785872	−	0.618389i	$-0.212214\pi$
$613$	38.9146	1.57174	0.785872	+	0.618389i	$0.212214\pi$
$614$	0	0
$615$	14.0830	0.567881
$616$	0	0
$617$	−0.639294	−0.0257370	−0.0128685	−	0.999917i	$-0.504096\pi$
$617$	−0.639294	−0.0257370	−0.0128685	+	0.999917i	$0.504096\pi$
$618$	0	0
$619$	−19.2927	−0.775440	−0.387720	−	0.921777i	$-0.626737\pi$
$619$	−19.2927	−0.775440	−0.387720	+	0.921777i	$0.626737\pi$
$620$	0	0
$621$	−2.94955	−0.118361
$622$	0	0
$623$	−59.7746	−2.39482
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18.7793	−0.748781
$630$	0	0
$631$	16.3865	0.652336	0.326168	−	0.945312i	$-0.394242\pi$
$631$	16.3865	0.652336	0.326168	+	0.945312i	$0.394242\pi$
$632$	0	0
$633$	22.2644	0.884931
$634$	0	0
$635$	18.0671	0.716972
$636$	0	0
$637$	−87.6528	−3.47293
$638$	0	0
$639$	−5.02951	−0.198964
$640$	0	0
$641$	−9.13171	−0.360681	−0.180341	−	0.983604i	$-0.557720\pi$
$641$	−9.13171	−0.360681	−0.180341	+	0.983604i	$0.557720\pi$
$642$	0	0
$643$	39.5059	1.55796	0.778981	−	0.627047i	$-0.215737\pi$
$643$	39.5059	1.55796	0.778981	+	0.627047i	$0.215737\pi$
$644$	0	0
$645$	10.1335	0.399006
$646$	0	0
$647$	4.26785	0.167786	0.0838932	−	0.996475i	$-0.473265\pi$
$647$	4.26785	0.167786	0.0838932	+	0.996475i	$0.473265\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−31.7520	−1.24446
$652$	0	0
$653$	−4.36846	−0.170951	−0.0854755	−	0.996340i	$-0.527241\pi$
$653$	−4.36846	−0.170951	−0.0854755	+	0.996340i	$0.527241\pi$
$654$	0	0
$655$	−7.95176	−0.310701
$656$	0	0
$657$	16.8454	0.657201
$658$	0	0
$659$	0.146275	0.00569808	0.00284904	−	0.999996i	$-0.499093\pi$
$659$	0.146275	0.00569808	0.00284904	+	0.999996i	$0.499093\pi$
$660$	0	0
$661$	41.7505	1.62391	0.811953	−	0.583723i	$-0.198405\pi$
$661$	41.7505	1.62391	0.811953	+	0.583723i	$0.198405\pi$
$662$	0	0
$663$	44.9701	1.74649
$664$	0	0
$665$	19.5498	0.758109
$666$	0	0
$667$	−13.2959	−0.514818
$668$	0	0
$669$	−1.24507	−0.0481372
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.58869	−0.0997866	−0.0498933	−	0.998755i	$-0.515888\pi$
$673$	−2.58869	−0.0997866	−0.0498933	+	0.998755i	$0.515888\pi$
$674$	0	0
$675$	1.72304	0.0663197
$676$	0	0
$677$	−45.0415	−1.73108	−0.865542	−	0.500836i	$-0.833026\pi$
$677$	−45.0415	−1.73108	−0.865542	+	0.500836i	$0.833026\pi$
$678$	0	0
$679$	44.4801	1.70699
$680$	0	0
$681$	11.9060	0.456239
$682$	0	0
$683$	−34.4943	−1.31989	−0.659944	−	0.751314i	$-0.729420\pi$
$683$	−34.4943	−1.31989	−0.659944	+	0.751314i	$0.729420\pi$
$684$	0	0
$685$	8.07748	0.308625
$686$	0	0
$687$	9.37212	0.357569
$688$	0	0
$689$	−48.1420	−1.83406
$690$	0	0
$691$	22.2062	0.844764	0.422382	−	0.906418i	$-0.361194\pi$
$691$	22.2062	0.844764	0.422382	+	0.906418i	$0.361194\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.3240	0.429542
$696$	0	0
$697$	20.3074	0.769197
$698$	0	0
$699$	31.9446	1.20825
$700$	0	0
$701$	−8.01026	−0.302543	−0.151272	−	0.988492i	$-0.548337\pi$
$701$	−8.01026	−0.302543	−0.151272	+	0.988492i	$0.548337\pi$
$702$	0	0
$703$	−23.8312	−0.898812
$704$	0	0
$705$	9.46664	0.356534
$706$	0	0
$707$	−50.0220	−1.88127
$708$	0	0
$709$	12.7422	0.478544	0.239272	−	0.970953i	$-0.423091\pi$
$709$	12.7422	0.478544	0.239272	+	0.970953i	$0.423091\pi$
$710$	0	0
$711$	−39.4085	−1.47793
$712$	0	0
$713$	5.08764	0.190534
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.19940	0.0447924
$718$	0	0
$719$	−6.15517	−0.229549	−0.114775	−	0.993392i	$-0.536615\pi$
$719$	−6.15517	−0.229549	−0.114775	+	0.993392i	$0.536615\pi$
$720$	0	0
$721$	−12.0259	−0.447869
$722$	0	0
$723$	−14.6157	−0.543563
$724$	0	0
$725$	7.76702	0.288460
$726$	0	0
$727$	−9.77647	−0.362589	−0.181295	−	0.983429i	$-0.558029\pi$
$727$	−9.77647	−0.362589	−0.181295	+	0.983429i	$0.558029\pi$
$728$	0	0
$729$	−12.1897	−0.451469
$730$	0	0
$731$	14.6123	0.540455
$732$	0	0
$733$	22.1966	0.819849	0.409925	−	0.912119i	$-0.365555\pi$
$733$	22.1966	0.819849	0.409925	+	0.912119i	$0.365555\pi$
$734$	0	0
$735$	33.7889	1.24632
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−20.7597	−0.763657	−0.381828	−	0.924233i	$-0.624705\pi$
$739$	−20.7597	−0.763657	−0.381828	+	0.924233i	$0.624705\pi$
$740$	0	0
$741$	57.0676	2.09643
$742$	0	0
$743$	34.6946	1.27282	0.636410	−	0.771351i	$-0.280419\pi$
$743$	34.6946	1.27282	0.636410	+	0.771351i	$0.280419\pi$
$744$	0	0
$745$	2.35175	0.0861614
$746$	0	0
$747$	−24.9029	−0.911150
$748$	0	0
$749$	−82.2189	−3.00421
$750$	0	0
$751$	−20.6367	−0.753043	−0.376521	−	0.926408i	$-0.622880\pi$
$751$	−20.6367	−0.753043	−0.376521	+	0.926408i	$0.622880\pi$
$752$	0	0
$753$	−31.2497	−1.13880
$754$	0	0
$755$	14.7268	0.535965
$756$	0	0
$757$	−23.6794	−0.860642	−0.430321	−	0.902676i	$-0.641600\pi$
$757$	−23.6794	−0.860642	−0.430321	+	0.902676i	$0.641600\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.5039	−1.68576	−0.842882	−	0.538098i	$-0.819143\pi$
$761$	−46.5039	−1.68576	−0.842882	+	0.538098i	$0.819143\pi$
$762$	0	0
$763$	−71.5761	−2.59123
$764$	0	0
$765$	−7.42536	−0.268465
$766$	0	0
$767$	36.9347	1.33364
$768$	0	0
$769$	−8.48780	−0.306078	−0.153039	−	0.988220i	$-0.548906\pi$
$769$	−8.48780	−0.306078	−0.153039	+	0.988220i	$0.548906\pi$
$770$	0	0
$771$	5.51412	0.198586
$772$	0	0
$773$	44.8095	1.61168	0.805842	−	0.592130i	$-0.201713\pi$
$773$	44.8095	1.61168	0.805842	+	0.592130i	$0.201713\pi$
$774$	0	0
$775$	−2.97204	−0.106759
$776$	0	0
$777$	−60.7362	−2.17890
$778$	0	0
$779$	25.7703	0.923318
$780$	0	0
$781$	0	0
$782$	0	0
$783$	13.3828	0.478264
$784$	0	0
$785$	16.4544	0.587281
$786$	0	0
$787$	−4.50207	−0.160481	−0.0802407	−	0.996776i	$-0.525569\pi$
$787$	−4.50207	−0.160481	−0.0802407	+	0.996776i	$0.525569\pi$
$788$	0	0
$789$	−26.9953	−0.961059
$790$	0	0
$791$	−1.47807	−0.0525542
$792$	0	0
$793$	−9.07795	−0.322367
$794$	0	0
$795$	18.5580	0.658185
$796$	0	0
$797$	34.7713	1.23166	0.615832	−	0.787878i	$-0.288820\pi$
$797$	34.7713	1.23166	0.615832	+	0.787878i	$0.288820\pi$
$798$	0	0
$799$	13.6507	0.482927
$800$	0	0
$801$	28.8110	1.01799
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−7.98341	−0.281378
$806$	0	0
$807$	34.6913	1.22119
$808$	0	0
$809$	−53.5499	−1.88271	−0.941357	−	0.337413i	$-0.890448\pi$
$809$	−53.5499	−1.88271	−0.941357	+	0.337413i	$0.890448\pi$
$810$	0	0
$811$	17.5535	0.616387	0.308193	−	0.951324i	$-0.400276\pi$
$811$	17.5535	0.616387	0.308193	+	0.951324i	$0.400276\pi$
$812$	0	0
$813$	−24.2228	−0.849530
$814$	0	0
$815$	−5.26242	−0.184334
$816$	0	0
$817$	18.5432	0.648744
$818$	0	0
$819$	62.2985	2.17688
$820$	0	0
$821$	−12.3663	−0.431586	−0.215793	−	0.976439i	$-0.569234\pi$
$821$	−12.3663	−0.431586	−0.215793	+	0.976439i	$0.569234\pi$
$822$	0	0
$823$	−14.0358	−0.489258	−0.244629	−	0.969617i	$-0.578666\pi$
$823$	−14.0358	−0.489258	−0.244629	+	0.969617i	$0.578666\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.8085	0.897450	0.448725	−	0.893670i	$-0.351878\pi$
$827$	25.8085	0.897450	0.448725	+	0.893670i	$0.351878\pi$
$828$	0	0
$829$	14.0081	0.486522	0.243261	−	0.969961i	$-0.421783\pi$
$829$	14.0081	0.486522	0.243261	+	0.969961i	$0.421783\pi$
$830$	0	0
$831$	−57.6177	−1.99874
$832$	0	0
$833$	48.7229	1.68815
$834$	0	0
$835$	−21.5495	−0.745752
$836$	0	0
$837$	−5.12093	−0.177005
$838$	0	0
$839$	6.80179	0.234824	0.117412	−	0.993083i	$-0.462540\pi$
$839$	6.80179	0.234824	0.117412	+	0.993083i	$0.462540\pi$
$840$	0	0
$841$	31.3265	1.08023
$842$	0	0
$843$	−0.739857	−0.0254820
$844$	0	0
$845$	−22.3156	−0.767679
$846$	0	0
$847$	0	0
$848$	0	0
$849$	48.9602	1.68031
$850$	0	0
$851$	9.73178	0.333601
$852$	0	0
$853$	46.8963	1.60570	0.802849	−	0.596183i	$-0.203317\pi$
$853$	46.8963	1.60570	0.802849	+	0.596183i	$0.203317\pi$
$854$	0	0
$855$	−9.42288	−0.322256
$856$	0	0
$857$	2.79309	0.0954100	0.0477050	−	0.998861i	$-0.484809\pi$
$857$	2.79309	0.0954100	0.0477050	+	0.998861i	$0.484809\pi$
$858$	0	0
$859$	5.83184	0.198980	0.0994898	−	0.995039i	$-0.468279\pi$
$859$	5.83184	0.198980	0.0994898	+	0.995039i	$0.468279\pi$
$860$	0	0
$861$	65.6782	2.23831
$862$	0	0
$863$	44.8495	1.52670	0.763348	−	0.645987i	$-0.223554\pi$
$863$	44.8495	1.52670	0.763348	+	0.645987i	$0.223554\pi$
$864$	0	0
$865$	−7.08425	−0.240872
$866$	0	0
$867$	13.9468	0.473657
$868$	0	0
$869$	0	0
$870$	0	0
$871$	44.3591	1.50305
$872$	0	0
$873$	−21.4391	−0.725604
$874$	0	0
$875$	4.66366	0.157660
$876$	0	0
$877$	15.8686	0.535846	0.267923	−	0.963440i	$-0.413663\pi$
$877$	15.8686	0.535846	0.267923	+	0.963440i	$0.413663\pi$
$878$	0	0
$879$	−24.0539	−0.811318
$880$	0	0
$881$	−30.1411	−1.01548	−0.507740	−	0.861510i	$-0.669519\pi$
$881$	−30.1411	−1.01548	−0.507740	+	0.861510i	$0.669519\pi$
$882$	0	0
$883$	−42.7708	−1.43935	−0.719676	−	0.694310i	$-0.755710\pi$
$883$	−42.7708	−1.43935	−0.719676	+	0.694310i	$0.755710\pi$
$884$	0	0
$885$	−14.2378	−0.478598
$886$	0	0
$887$	−6.66076	−0.223647	−0.111823	−	0.993728i	$-0.535669\pi$
$887$	−6.66076	−0.223647	−0.111823	+	0.993728i	$0.535669\pi$
$888$	0	0
$889$	84.2588	2.82595
$890$	0	0
$891$	0	0
$892$	0	0
$893$	17.3229	0.579689
$894$	0	0
$895$	−8.78759	−0.293737
$896$	0	0
$897$	−23.3043	−0.778108
$898$	0	0
$899$	−23.0839	−0.769890
$900$	0	0
$901$	26.7603	0.891514
$902$	0	0
$903$	47.2592	1.57269
$904$	0	0
$905$	12.6451	0.420338
$906$	0	0
$907$	−37.3310	−1.23955	−0.619777	−	0.784778i	$-0.712777\pi$
$907$	−37.3310	−1.23955	−0.619777	+	0.784778i	$0.712777\pi$
$908$	0	0
$909$	24.1103	0.799688
$910$	0	0
$911$	−14.7247	−0.487852	−0.243926	−	0.969794i	$-0.578435\pi$
$911$	−14.7247	−0.487852	−0.243926	+	0.969794i	$0.578435\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.49941	0.115687
$916$	0	0
$917$	−37.0843	−1.22463
$918$	0	0
$919$	34.0203	1.12222	0.561112	−	0.827740i	$-0.310374\pi$
$919$	34.0203	1.12222	0.561112	+	0.827740i	$0.310374\pi$
$920$	0	0
$921$	−66.5708	−2.19358
$922$	0	0
$923$	13.2966	0.437664
$924$	0	0
$925$	−5.68500	−0.186922
$926$	0	0
$927$	5.79643	0.190380
$928$	0	0
$929$	18.6992	0.613502	0.306751	−	0.951790i	$-0.400758\pi$
$929$	18.6992	0.613502	0.306751	+	0.951790i	$0.400758\pi$
$930$	0	0
$931$	61.8299	2.02639
$932$	0	0
$933$	45.6979	1.49608
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.122357	−0.00399723	−0.00199862	−	0.999998i	$-0.500636\pi$
$937$	−0.122357	−0.00399723	−0.00199862	+	0.999998i	$0.500636\pi$
$938$	0	0
$939$	−40.0436	−1.30677
$940$	0	0
$941$	−29.8311	−0.972465	−0.486233	−	0.873829i	$-0.661629\pi$
$941$	−29.8311	−0.972465	−0.486233	+	0.873829i	$0.661629\pi$
$942$	0	0
$943$	−10.5236	−0.342697
$944$	0	0
$945$	8.03564	0.261399
$946$	0	0
$947$	−49.0368	−1.59348	−0.796741	−	0.604320i	$-0.793445\pi$
$947$	−49.0368	−1.59348	−0.796741	+	0.604320i	$0.793445\pi$
$948$	0	0
$949$	−44.5345	−1.44565
$950$	0	0
$951$	68.3340	2.21588
$952$	0	0
$953$	−33.5129	−1.08559	−0.542794	−	0.839866i	$-0.682634\pi$
$953$	−33.5129	−1.08559	−0.542794	+	0.839866i	$0.682634\pi$
$954$	0	0
$955$	17.7224	0.573485
$956$	0	0
$957$	0	0
$958$	0	0
$959$	37.6706	1.21645
$960$	0	0
$961$	−22.1670	−0.715064
$962$	0	0
$963$	39.6290	1.27703
$964$	0	0
$965$	−5.03540	−0.162095
$966$	0	0
$967$	−52.5565	−1.69010	−0.845052	−	0.534685i	$-0.820430\pi$
$967$	−52.5565	−1.69010	−0.845052	+	0.534685i	$0.820430\pi$
$968$	0	0
$969$	−31.7217	−1.01905
$970$	0	0
$971$	4.28868	0.137630	0.0688152	−	0.997629i	$-0.478078\pi$
$971$	4.28868	0.137630	0.0688152	+	0.997629i	$0.478078\pi$
$972$	0	0
$973$	52.8111	1.69305
$974$	0	0
$975$	13.6136	0.435985
$976$	0	0
$977$	−21.5573	−0.689678	−0.344839	−	0.938662i	$-0.612066\pi$
$977$	−21.5573	−0.689678	−0.344839	+	0.938662i	$0.612066\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	34.4992	1.10148
$982$	0	0
$983$	26.3667	0.840966	0.420483	−	0.907300i	$-0.361861\pi$
$983$	26.3667	0.840966	0.420483	+	0.907300i	$0.361861\pi$
$984$	0	0
$985$	−15.0557	−0.479716
$986$	0	0
$987$	44.1491	1.40528
$988$	0	0
$989$	−7.57235	−0.240787
$990$	0	0
$991$	−49.3520	−1.56772	−0.783858	−	0.620940i	$-0.786751\pi$
$991$	−49.3520	−1.56772	−0.783858	+	0.620940i	$0.786751\pi$
$992$	0	0
$993$	−67.5195	−2.14267
$994$	0	0
$995$	−2.56534	−0.0813266
$996$	0	0
$997$	2.83009	0.0896300	0.0448150	−	0.998995i	$-0.485730\pi$
$997$	2.83009	0.0896300	0.0448150	+	0.998995i	$0.485730\pi$
$998$	0	0
$999$	−9.79545	−0.309914

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.ba.1.2		6
4.3	odd	2		9680.2.a.dd.1.5		6
11.7	odd	10		440.2.y.c.401.3	yes	12
11.8	odd	10		440.2.y.c.361.3	✓	12
11.10	odd	2		4840.2.a.bb.1.2		6
44.7	even	10		880.2.bo.i.401.1		12
44.19	even	10		880.2.bo.i.801.1		12
44.43	even	2		9680.2.a.dc.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.c.361.3	✓	12	11.8	odd	10
440.2.y.c.401.3	yes	12	11.7	odd	10
880.2.bo.i.401.1		12	44.7	even	10
880.2.bo.i.801.1		12	44.19	even	10
4840.2.a.ba.1.2		6	1.1	even	1	trivial
4840.2.a.bb.1.2		6	11.10	odd	2
9680.2.a.dc.1.5		6	44.43	even	2
9680.2.a.dd.1.5		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q-2.29082 q^{3} -1.00000 q^{5} -4.66366 q^{7} +2.24785 q^{9} -5.94269 q^{13} +2.29082 q^{15} +3.30331 q^{17} +4.19195 q^{19} +10.6836 q^{21} -1.71184 q^{23} +1.00000 q^{25} +1.72304 q^{27} +7.76702 q^{29} -2.97204 q^{31} +4.66366 q^{35} -5.68500 q^{37} +13.6136 q^{39} +6.14758 q^{41} +4.42353 q^{43} -2.24785 q^{45} +4.13243 q^{47} +14.7497 q^{49} -7.56730 q^{51} +8.10104 q^{53} -9.60300 q^{57} -6.21515 q^{59} +1.52758 q^{61} -10.4832 q^{63} +5.94269 q^{65} -7.46447 q^{67} +3.92151 q^{69} -2.23747 q^{71} +7.49400 q^{73} -2.29082 q^{75} -17.5316 q^{79} -10.6907 q^{81} -11.0785 q^{83} -3.30331 q^{85} -17.7928 q^{87} +12.8171 q^{89} +27.7147 q^{91} +6.80840 q^{93} -4.19195 q^{95} -9.53760 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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