LMFDB - Embedded newform 2088.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2088,4,Mod(1,2088)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2088 = 2^{3} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2088.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$123.195988092$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 $ x^5 - x^4 - 34x^3 + 74x^2 + 94*x - 198
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69859$ of defining polynomial
Character	$\chi$	$=$	2088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.3850 q^{5} +5.74609 q^{7} +O(q^{10})$	$q-16.3850 q^{5} +5.74609 q^{7} -23.3693 q^{11} -18.0318 q^{13} +24.1314 q^{17} -12.0191 q^{19} +144.669 q^{23} +143.469 q^{25} -29.0000 q^{29} +6.27664 q^{31} -94.1498 q^{35} +28.6658 q^{37} +436.340 q^{41} +495.080 q^{43} -351.351 q^{47} -309.982 q^{49} +58.0933 q^{53} +382.906 q^{55} -485.535 q^{59} +607.926 q^{61} +295.451 q^{65} -296.974 q^{67} +330.178 q^{71} -662.978 q^{73} -134.282 q^{77} +145.394 q^{79} -1077.01 q^{83} -395.393 q^{85} +851.923 q^{89} -103.612 q^{91} +196.933 q^{95} -227.768 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{5} + 32 q^{7}+O(q^{10}) $ 5 * q - 10 * q^5 + 32 * q^7	$ 5 q - 10 q^{5} + 32 q^{7} - 36 q^{11} + 26 q^{13} - 82 q^{17} + 156 q^{19} - 336 q^{23} + 151 q^{25} - 145 q^{29} + 432 q^{31} - 600 q^{35} - 18 q^{37} - 82 q^{41} + 340 q^{43} - 680 q^{47} - 115 q^{49} + 102 q^{53} + 736 q^{55} - 924 q^{59} - 618 q^{61} + 704 q^{65} + 44 q^{67} - 1032 q^{71} - 1078 q^{73} + 888 q^{77} + 200 q^{79} - 452 q^{83} - 1700 q^{85} + 1790 q^{89} - 1128 q^{91} - 1024 q^{95} - 2518 q^{97}+O(q^{100}) $ 5 * q - 10 * q^5 + 32 * q^7 - 36 * q^11 + 26 * q^13 - 82 * q^17 + 156 * q^19 - 336 * q^23 + 151 * q^25 - 145 * q^29 + 432 * q^31 - 600 * q^35 - 18 * q^37 - 82 * q^41 + 340 * q^43 - 680 * q^47 - 115 * q^49 + 102 * q^53 + 736 * q^55 - 924 * q^59 - 618 * q^61 + 704 * q^65 + 44 * q^67 - 1032 * q^71 - 1078 * q^73 + 888 * q^77 + 200 * q^79 - 452 * q^83 - 1700 * q^85 + 1790 * q^89 - 1128 * q^91 - 1024 * q^95 - 2518 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−16.3850	−1.46552	−0.732760	−	0.680487i	$-0.761768\pi$
$5$	−16.3850	−1.46552	−0.732760	+	0.680487i	$0.761768\pi$
$6$	0	0
$7$	5.74609	0.310260	0.155130	−	0.987894i	$-0.450420\pi$
$7$	5.74609	0.310260	0.155130	+	0.987894i	$0.450420\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−23.3693	−0.640554	−0.320277	−	0.947324i	$-0.603776\pi$
$11$	−23.3693	−0.640554	−0.320277	+	0.947324i	$0.603776\pi$
$12$	0	0
$13$	−18.0318	−0.384701	−0.192351	−	0.981326i	$-0.561611\pi$
$13$	−18.0318	−0.384701	−0.192351	+	0.981326i	$0.561611\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	24.1314	0.344278	0.172139	−	0.985073i	$-0.444932\pi$
$17$	24.1314	0.344278	0.172139	+	0.985073i	$0.444932\pi$
$18$	0	0
$19$	−12.0191	−0.145125	−0.0725624	−	0.997364i	$-0.523118\pi$
$19$	−12.0191	−0.145125	−0.0725624	+	0.997364i	$0.523118\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	144.669	1.31155	0.655776	−	0.754956i	$-0.272342\pi$
$23$	144.669	1.31155	0.655776	+	0.754956i	$0.272342\pi$
$24$	0	0
$25$	143.469	1.14775
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	6.27664	0.0363651	0.0181826	−	0.999835i	$-0.494212\pi$
$31$	6.27664	0.0363651	0.0181826	+	0.999835i	$0.494212\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−94.1498	−0.454692
$36$	0	0
$37$	28.6658	0.127368	0.0636842	−	0.997970i	$-0.479715\pi$
$37$	28.6658	0.127368	0.0636842	+	0.997970i	$0.479715\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	436.340	1.66207	0.831035	−	0.556220i	$-0.187749\pi$
$41$	436.340	1.66207	0.831035	+	0.556220i	$0.187749\pi$
$42$	0	0
$43$	495.080	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$43$	495.080	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−351.351	−1.09042	−0.545211	−	0.838299i	$-0.683550\pi$
$47$	−351.351	−1.09042	−0.545211	+	0.838299i	$0.683550\pi$
$48$	0	0
$49$	−309.982	−0.903739
$50$	0	0
$51$	0	0
$52$	0	0
$53$	58.0933	0.150561	0.0752804	−	0.997162i	$-0.476015\pi$
$53$	58.0933	0.150561	0.0752804	+	0.997162i	$0.476015\pi$
$54$	0	0
$55$	382.906	0.938746
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−485.535	−1.07138	−0.535689	−	0.844415i	$-0.679948\pi$
$59$	−485.535	−1.07138	−0.535689	+	0.844415i	$0.679948\pi$
$60$	0	0
$61$	607.926	1.27602	0.638008	−	0.770030i	$-0.279759\pi$
$61$	607.926	1.27602	0.638008	+	0.770030i	$0.279759\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	295.451	0.563788
$66$	0	0
$67$	−296.974	−0.541510	−0.270755	−	0.962648i	$-0.587273\pi$
$67$	−296.974	−0.541510	−0.270755	+	0.962648i	$0.587273\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	330.178	0.551901	0.275950	−	0.961172i	$-0.411007\pi$
$71$	330.178	0.551901	0.275950	+	0.961172i	$0.411007\pi$
$72$	0	0
$73$	−662.978	−1.06296	−0.531478	−	0.847072i	$-0.678363\pi$
$73$	−662.978	−1.06296	−0.531478	+	0.847072i	$0.678363\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−134.282	−0.198738
$78$	0	0
$79$	145.394	0.207065	0.103532	−	0.994626i	$-0.466985\pi$
$79$	145.394	0.207065	0.103532	+	0.994626i	$0.466985\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1077.01	−1.42431	−0.712154	−	0.702023i	$-0.752280\pi$
$83$	−1077.01	−1.42431	−0.712154	+	0.702023i	$0.752280\pi$
$84$	0	0
$85$	−395.393	−0.504546
$86$	0	0
$87$	0	0
$88$	0	0
$89$	851.923	1.01465	0.507324	−	0.861755i	$-0.330635\pi$
$89$	851.923	1.01465	0.507324	+	0.861755i	$0.330635\pi$
$90$	0	0
$91$	−103.612	−0.119357
$92$	0	0
$93$	0	0
$94$	0	0
$95$	196.933	0.212684
$96$	0	0
$97$	−227.768	−0.238416	−0.119208	−	0.992869i	$-0.538036\pi$
$97$	−227.768	−0.238416	−0.119208	+	0.992869i	$0.538036\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	77.3495	0.0762036	0.0381018	−	0.999274i	$-0.487869\pi$
$101$	77.3495	0.0762036	0.0381018	+	0.999274i	$0.487869\pi$
$102$	0	0
$103$	−751.327	−0.718742	−0.359371	−	0.933195i	$-0.617009\pi$
$103$	−751.327	−0.718742	−0.359371	+	0.933195i	$0.617009\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	761.498	0.688007	0.344004	−	0.938968i	$-0.388217\pi$
$107$	761.498	0.688007	0.344004	+	0.938968i	$0.388217\pi$
$108$	0	0
$109$	−661.001	−0.580848	−0.290424	−	0.956898i	$-0.593796\pi$
$109$	−661.001	−0.580848	−0.290424	+	0.956898i	$0.593796\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2266.45	1.88681	0.943404	−	0.331646i	$-0.107604\pi$
$113$	2266.45	1.88681	0.943404	+	0.331646i	$0.107604\pi$
$114$	0	0
$115$	−2370.41	−1.92211
$116$	0	0
$117$	0	0
$118$	0	0
$119$	138.661	0.106815
$120$	0	0
$121$	−784.877	−0.589690
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−302.612	−0.216532
$126$	0	0
$127$	420.522	0.293821	0.146911	−	0.989150i	$-0.453067\pi$
$127$	420.522	0.293821	0.146911	+	0.989150i	$0.453067\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	208.443	0.139021	0.0695103	−	0.997581i	$-0.477856\pi$
$131$	208.443	0.139021	0.0695103	+	0.997581i	$0.477856\pi$
$132$	0	0
$133$	−69.0629	−0.0450264
$134$	0	0
$135$	0	0
$136$	0	0
$137$	188.285	0.117418	0.0587089	−	0.998275i	$-0.481302\pi$
$137$	188.285	0.117418	0.0587089	+	0.998275i	$0.481302\pi$
$138$	0	0
$139$	−1460.15	−0.890996	−0.445498	−	0.895283i	$-0.646973\pi$
$139$	−1460.15	−0.890996	−0.445498	+	0.895283i	$0.646973\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	421.390	0.246422
$144$	0	0
$145$	475.166	0.272140
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−665.203	−0.365742	−0.182871	−	0.983137i	$-0.558539\pi$
$149$	−665.203	−0.365742	−0.182871	+	0.983137i	$0.558539\pi$
$150$	0	0
$151$	3248.63	1.75079	0.875396	−	0.483406i	$-0.160601\pi$
$151$	3248.63	1.75079	0.875396	+	0.483406i	$0.160601\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−102.843	−0.0532938
$156$	0	0
$157$	1907.56	0.969681	0.484840	−	0.874603i	$-0.338878\pi$
$157$	1907.56	0.969681	0.484840	+	0.874603i	$0.338878\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	831.284	0.406921
$162$	0	0
$163$	−2961.93	−1.42329	−0.711645	−	0.702539i	$-0.752050\pi$
$163$	−2961.93	−1.42329	−0.711645	+	0.702539i	$0.752050\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2952.52	−1.36810	−0.684051	−	0.729434i	$-0.739783\pi$
$167$	−2952.52	−1.36810	−0.684051	+	0.729434i	$0.739783\pi$
$168$	0	0
$169$	−1871.85	−0.852005
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3500.71	−1.53846	−0.769232	−	0.638969i	$-0.779361\pi$
$173$	−3500.71	−1.53846	−0.769232	+	0.638969i	$0.779361\pi$
$174$	0	0
$175$	824.385	0.356101
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4562.48	−1.90511	−0.952557	−	0.304359i	$-0.901558\pi$
$179$	−4562.48	−1.90511	−0.952557	+	0.304359i	$0.901558\pi$
$180$	0	0
$181$	326.513	0.134086	0.0670430	−	0.997750i	$-0.478644\pi$
$181$	326.513	0.134086	0.0670430	+	0.997750i	$0.478644\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−469.690	−0.186661
$186$	0	0
$187$	−563.933	−0.220529
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2623.16	−0.993744	−0.496872	−	0.867824i	$-0.665518\pi$
$191$	−2623.16	−0.993744	−0.496872	+	0.867824i	$0.665518\pi$
$192$	0	0
$193$	894.480	0.333607	0.166803	−	0.985990i	$-0.446656\pi$
$193$	894.480	0.333607	0.166803	+	0.985990i	$0.446656\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2182.51	−0.789326	−0.394663	−	0.918826i	$-0.629139\pi$
$197$	−2182.51	−0.789326	−0.394663	+	0.918826i	$0.629139\pi$
$198$	0	0
$199$	−635.874	−0.226512	−0.113256	−	0.993566i	$-0.536128\pi$
$199$	−635.874	−0.226512	−0.113256	+	0.993566i	$0.536128\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−166.637	−0.0576138
$204$	0	0
$205$	−7149.44	−2.43580
$206$	0	0
$207$	0	0
$208$	0	0
$209$	280.878	0.0929604
$210$	0	0
$211$	−745.334	−0.243180	−0.121590	−	0.992580i	$-0.538799\pi$
$211$	−745.334	−0.243180	−0.121590	+	0.992580i	$0.538799\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8111.90	−2.57315
$216$	0	0
$217$	36.0661	0.0112826
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−435.132	−0.132444
$222$	0	0
$223$	159.305	0.0478380	0.0239190	−	0.999714i	$-0.492386\pi$
$223$	159.305	0.0478380	0.0239190	+	0.999714i	$0.492386\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3699.17	1.08160	0.540798	−	0.841153i	$-0.318122\pi$
$227$	3699.17	1.08160	0.540798	+	0.841153i	$0.318122\pi$
$228$	0	0
$229$	−6713.39	−1.93726	−0.968632	−	0.248499i	$-0.920063\pi$
$229$	−6713.39	−1.93726	−0.968632	+	0.248499i	$0.920063\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1250.99	−0.351740	−0.175870	−	0.984413i	$-0.556274\pi$
$233$	−1250.99	−0.351740	−0.175870	+	0.984413i	$0.556274\pi$
$234$	0	0
$235$	5756.89	1.59803
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3073.10	0.831726	0.415863	−	0.909427i	$-0.363480\pi$
$239$	3073.10	0.831726	0.415863	+	0.909427i	$0.363480\pi$
$240$	0	0
$241$	4745.39	1.26837	0.634185	−	0.773181i	$-0.281336\pi$
$241$	4745.39	1.26837	0.634185	+	0.773181i	$0.281336\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5079.07	1.32445
$246$	0	0
$247$	216.726	0.0558298
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1919.33	−0.482658	−0.241329	−	0.970443i	$-0.577583\pi$
$251$	−1919.33	−0.482658	−0.241329	+	0.970443i	$0.577583\pi$
$252$	0	0
$253$	−3380.82	−0.840120
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4680.45	−1.13602	−0.568012	−	0.823020i	$-0.692287\pi$
$257$	−4680.45	−1.13602	−0.568012	+	0.823020i	$0.692287\pi$
$258$	0	0
$259$	164.716	0.0395173
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4404.71	1.03272	0.516361	−	0.856371i	$-0.327286\pi$
$263$	4404.71	1.03272	0.516361	+	0.856371i	$0.327286\pi$
$264$	0	0
$265$	−951.859	−0.220650
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−8244.86	−1.86877	−0.934383	−	0.356271i	$-0.884048\pi$
$269$	−8244.86	−1.86877	−0.934383	+	0.356271i	$0.884048\pi$
$270$	0	0
$271$	−1050.02	−0.235366	−0.117683	−	0.993051i	$-0.537547\pi$
$271$	−1050.02	−0.235366	−0.117683	+	0.993051i	$0.537547\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3352.76	−0.735197
$276$	0	0
$277$	3630.09	0.787405	0.393703	−	0.919238i	$-0.371194\pi$
$277$	3630.09	0.787405	0.393703	+	0.919238i	$0.371194\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6718.58	−1.42632	−0.713162	−	0.700999i	$-0.752738\pi$
$281$	−6718.58	−1.42632	−0.713162	+	0.700999i	$0.752738\pi$
$282$	0	0
$283$	−1096.86	−0.230394	−0.115197	−	0.993343i	$-0.536750\pi$
$283$	−1096.86	−0.230394	−0.115197	+	0.993343i	$0.536750\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2507.25	0.515673
$288$	0	0
$289$	−4330.68	−0.881473
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1991.65	0.397111	0.198556	−	0.980090i	$-0.436375\pi$
$293$	1991.65	0.397111	0.198556	+	0.980090i	$0.436375\pi$
$294$	0	0
$295$	7955.50	1.57013
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2608.65	−0.504556
$300$	0	0
$301$	2844.78	0.544751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9960.88	−1.87003
$306$	0	0
$307$	−3847.15	−0.715207	−0.357604	−	0.933873i	$-0.616406\pi$
$307$	−3847.15	−0.715207	−0.357604	+	0.933873i	$0.616406\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8281.96	−1.51005	−0.755027	−	0.655693i	$-0.772376\pi$
$311$	−8281.96	−1.51005	−0.755027	+	0.655693i	$0.772376\pi$
$312$	0	0
$313$	−8930.42	−1.61271	−0.806354	−	0.591434i	$-0.798562\pi$
$313$	−8930.42	−1.61271	−0.806354	+	0.591434i	$0.798562\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2716.45	0.481297	0.240649	−	0.970612i	$-0.422640\pi$
$317$	2716.45	0.481297	0.240649	+	0.970612i	$0.422640\pi$
$318$	0	0
$319$	677.709	0.118948
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−290.038	−0.0499633
$324$	0	0
$325$	−2587.00	−0.441541
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2018.89	−0.338314
$330$	0	0
$331$	6445.84	1.07038	0.535189	−	0.844732i	$-0.320240\pi$
$331$	6445.84	1.07038	0.535189	+	0.844732i	$0.320240\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4865.92	0.793593
$336$	0	0
$337$	6898.90	1.11515	0.557577	−	0.830125i	$-0.311731\pi$
$337$	6898.90	1.11515	0.557577	+	0.830125i	$0.311731\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−146.681	−0.0232938
$342$	0	0
$343$	−3752.10	−0.590653
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4463.85	−0.690583	−0.345291	−	0.938496i	$-0.612220\pi$
$347$	−4463.85	−0.690583	−0.345291	+	0.938496i	$0.612220\pi$
$348$	0	0
$349$	−9789.78	−1.50153	−0.750766	−	0.660568i	$-0.770315\pi$
$349$	−9789.78	−1.50153	−0.750766	+	0.660568i	$0.770315\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3601.80	0.543072	0.271536	−	0.962428i	$-0.412468\pi$
$353$	3601.80	0.543072	0.271536	+	0.962428i	$0.412468\pi$
$354$	0	0
$355$	−5409.98	−0.808822
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11426.7	1.67988	0.839940	−	0.542679i	$-0.182590\pi$
$359$	11426.7	1.67988	0.839940	+	0.542679i	$0.182590\pi$
$360$	0	0
$361$	−6714.54	−0.978939
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10862.9	1.55778
$366$	0	0
$367$	6545.41	0.930975	0.465487	−	0.885055i	$-0.345879\pi$
$367$	6545.41	0.930975	0.465487	+	0.885055i	$0.345879\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	333.809	0.0467129
$372$	0	0
$373$	12488.0	1.73352	0.866761	−	0.498724i	$-0.166198\pi$
$373$	12488.0	1.73352	0.866761	+	0.498724i	$0.166198\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	522.922	0.0714373
$378$	0	0
$379$	10431.7	1.41383	0.706915	−	0.707298i	$-0.250086\pi$
$379$	10431.7	1.41383	0.706915	+	0.707298i	$0.250086\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9500.35	1.26748	0.633741	−	0.773545i	$-0.281519\pi$
$383$	9500.35	1.26748	0.633741	+	0.773545i	$0.281519\pi$
$384$	0	0
$385$	2200.21	0.291255
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12956.1	−1.68869	−0.844345	−	0.535800i	$-0.820010\pi$
$389$	−12956.1	−1.68869	−0.844345	+	0.535800i	$0.820010\pi$
$390$	0	0
$391$	3491.08	0.451538
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2382.28	−0.303457
$396$	0	0
$397$	−167.727	−0.0212040	−0.0106020	−	0.999944i	$-0.503375\pi$
$397$	−167.727	−0.0212040	−0.0106020	+	0.999944i	$0.503375\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13225.9	−1.64705	−0.823526	−	0.567279i	$-0.807996\pi$
$401$	−13225.9	−1.64705	−0.823526	+	0.567279i	$0.807996\pi$
$402$	0	0
$403$	−113.179	−0.0139897
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−669.899	−0.0815864
$408$	0	0
$409$	−674.413	−0.0815344	−0.0407672	−	0.999169i	$-0.512980\pi$
$409$	−674.413	−0.0815344	−0.0407672	+	0.999169i	$0.512980\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2789.93	−0.332405
$414$	0	0
$415$	17646.9	2.08735
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12355.6	1.44059	0.720296	−	0.693667i	$-0.244006\pi$
$419$	12355.6	1.44059	0.720296	+	0.693667i	$0.244006\pi$
$420$	0	0
$421$	−8788.18	−1.01736	−0.508682	−	0.860955i	$-0.669867\pi$
$421$	−8788.18	−1.01736	−0.508682	+	0.860955i	$0.669867\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3462.10	0.395145
$426$	0	0
$427$	3493.20	0.395896
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15171.7	−1.69559	−0.847793	−	0.530328i	$-0.822069\pi$
$431$	−15171.7	−1.69559	−0.847793	+	0.530328i	$0.822069\pi$
$432$	0	0
$433$	3884.76	0.431154	0.215577	−	0.976487i	$-0.430837\pi$
$433$	3884.76	0.431154	0.215577	+	0.976487i	$0.430837\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1738.80	−0.190339
$438$	0	0
$439$	14998.9	1.63066	0.815329	−	0.578998i	$-0.196556\pi$
$439$	14998.9	1.63066	0.815329	+	0.578998i	$0.196556\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4945.82	−0.530435	−0.265218	−	0.964189i	$-0.585444\pi$
$443$	−4945.82	−0.530435	−0.265218	+	0.964189i	$0.585444\pi$
$444$	0	0
$445$	−13958.8	−1.48699
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13041.0	−1.37070	−0.685351	−	0.728213i	$-0.740351\pi$
$449$	−13041.0	−1.37070	−0.685351	+	0.728213i	$0.740351\pi$
$450$	0	0
$451$	−10196.9	−1.06465
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1697.69	0.174921
$456$	0	0
$457$	3833.64	0.392408	0.196204	−	0.980563i	$-0.437139\pi$
$457$	3833.64	0.392408	0.196204	+	0.980563i	$0.437139\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2806.70	0.283560	0.141780	−	0.989898i	$-0.454717\pi$
$461$	2806.70	0.283560	0.141780	+	0.989898i	$0.454717\pi$
$462$	0	0
$463$	−4240.08	−0.425600	−0.212800	−	0.977096i	$-0.568258\pi$
$463$	−4240.08	−0.425600	−0.212800	+	0.977096i	$0.568258\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7538.50	−0.746981	−0.373491	−	0.927634i	$-0.621839\pi$
$467$	−7538.50	−0.746981	−0.373491	+	0.927634i	$0.621839\pi$
$468$	0	0
$469$	−1706.44	−0.168009
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11569.7	−1.12468
$474$	0	0
$475$	−1724.37	−0.166567
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14810.8	1.41279	0.706393	−	0.707820i	$-0.250321\pi$
$479$	14810.8	1.41279	0.706393	+	0.707820i	$0.250321\pi$
$480$	0	0
$481$	−516.896	−0.0489988
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3731.98	0.349403
$486$	0	0
$487$	2951.06	0.274590	0.137295	−	0.990530i	$-0.456159\pi$
$487$	2951.06	0.274590	0.137295	+	0.990530i	$0.456159\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19549.3	−1.79683	−0.898417	−	0.439143i	$-0.855282\pi$
$491$	−19549.3	−1.79683	−0.898417	+	0.439143i	$0.855282\pi$
$492$	0	0
$493$	−699.810	−0.0639308
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1897.23	0.171233
$498$	0	0
$499$	−5713.30	−0.512550	−0.256275	−	0.966604i	$-0.582495\pi$
$499$	−5713.30	−0.512550	−0.256275	+	0.966604i	$0.582495\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7379.92	−0.654184	−0.327092	−	0.944993i	$-0.606069\pi$
$503$	−7379.92	−0.654184	−0.327092	+	0.944993i	$0.606069\pi$
$504$	0	0
$505$	−1267.37	−0.111678
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4642.94	−0.404312	−0.202156	−	0.979353i	$-0.564795\pi$
$509$	−4642.94	−0.404312	−0.202156	+	0.979353i	$0.564795\pi$
$510$	0	0
$511$	−3809.53	−0.329792
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12310.5	1.05333
$516$	0	0
$517$	8210.81	0.698474
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11184.4	0.940494	0.470247	−	0.882535i	$-0.344165\pi$
$521$	11184.4	0.940494	0.470247	+	0.882535i	$0.344165\pi$
$522$	0	0
$523$	10752.9	0.899025	0.449512	−	0.893274i	$-0.351598\pi$
$523$	10752.9	0.899025	0.449512	+	0.893274i	$0.351598\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	151.464	0.0125197
$528$	0	0
$529$	8762.26	0.720166
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7867.99	−0.639401
$534$	0	0
$535$	−12477.2	−1.00829
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7244.06	0.578894
$540$	0	0
$541$	−20175.9	−1.60338	−0.801691	−	0.597738i	$-0.796066\pi$
$541$	−20175.9	−1.60338	−0.801691	+	0.597738i	$0.796066\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10830.5	0.851245
$546$	0	0
$547$	−9678.22	−0.756510	−0.378255	−	0.925702i	$-0.623476\pi$
$547$	−9678.22	−0.756510	−0.378255	+	0.925702i	$0.623476\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	348.554	0.0269490
$552$	0	0
$553$	835.447	0.0642438
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19870.3	−1.51155	−0.755775	−	0.654832i	$-0.772740\pi$
$557$	−19870.3	−1.51155	−0.755775	+	0.654832i	$0.772740\pi$
$558$	0	0
$559$	−8927.18	−0.675455
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9424.02	−0.705462	−0.352731	−	0.935725i	$-0.614747\pi$
$563$	−9424.02	−0.705462	−0.352731	+	0.935725i	$0.614747\pi$
$564$	0	0
$565$	−37135.8	−2.76516
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14765.7	1.08789	0.543945	−	0.839121i	$-0.316930\pi$
$569$	14765.7	1.08789	0.543945	+	0.839121i	$0.316930\pi$
$570$	0	0
$571$	−10061.1	−0.737380	−0.368690	−	0.929552i	$-0.620194\pi$
$571$	−10061.1	−0.737380	−0.368690	+	0.929552i	$0.620194\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	20755.6	1.50533
$576$	0	0
$577$	−26307.2	−1.89806	−0.949032	−	0.315179i	$-0.897935\pi$
$577$	−26307.2	−1.89806	−0.949032	+	0.315179i	$0.897935\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6188.62	−0.441905
$582$	0	0
$583$	−1357.60	−0.0964424
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2053.70	−0.144404	−0.0722021	−	0.997390i	$-0.523003\pi$
$587$	−2053.70	−0.144404	−0.0722021	+	0.997390i	$0.523003\pi$
$588$	0	0
$589$	−75.4397	−0.00527748
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12767.4	−0.884140	−0.442070	−	0.896981i	$-0.645756\pi$
$593$	−12767.4	−0.884140	−0.442070	+	0.896981i	$0.645756\pi$
$594$	0	0
$595$	−2271.96	−0.156540
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22284.1	1.52004	0.760021	−	0.649899i	$-0.225189\pi$
$599$	22284.1	1.52004	0.760021	+	0.649899i	$0.225189\pi$
$600$	0	0
$601$	1145.05	0.0777166	0.0388583	−	0.999245i	$-0.487628\pi$
$601$	1145.05	0.0777166	0.0388583	+	0.999245i	$0.487628\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12860.2	0.864203
$606$	0	0
$607$	5726.01	0.382886	0.191443	−	0.981504i	$-0.438683\pi$
$607$	5726.01	0.382886	0.191443	+	0.981504i	$0.438683\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6335.49	0.419487
$612$	0	0
$613$	17973.2	1.18423	0.592113	−	0.805855i	$-0.298294\pi$
$613$	17973.2	1.18423	0.592113	+	0.805855i	$0.298294\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−28779.6	−1.87783	−0.938916	−	0.344146i	$-0.888168\pi$
$617$	−28779.6	−1.87783	−0.938916	+	0.344146i	$0.888168\pi$
$618$	0	0
$619$	−20981.6	−1.36239	−0.681197	−	0.732100i	$-0.738540\pi$
$619$	−20981.6	−1.36239	−0.681197	+	0.732100i	$0.738540\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4895.22	0.314804
$624$	0	0
$625$	−12975.3	−0.830419
$626$	0	0
$627$	0	0
$628$	0	0
$629$	691.746	0.0438501
$630$	0	0
$631$	618.760	0.0390372	0.0195186	−	0.999809i	$-0.493787\pi$
$631$	618.760	0.0390372	0.0195186	+	0.999809i	$0.493787\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6890.26	−0.430601
$636$	0	0
$637$	5589.54	0.347670
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21528.5	1.32656	0.663280	−	0.748371i	$-0.269164\pi$
$641$	21528.5	1.32656	0.663280	+	0.748371i	$0.269164\pi$
$642$	0	0
$643$	8142.79	0.499410	0.249705	−	0.968322i	$-0.419666\pi$
$643$	8142.79	0.499410	0.249705	+	0.968322i	$0.419666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12779.6	−0.776538	−0.388269	−	0.921546i	$-0.626927\pi$
$647$	−12779.6	−0.776538	−0.388269	+	0.921546i	$0.626927\pi$
$648$	0	0
$649$	11346.6	0.686276
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19463.5	1.16641	0.583205	−	0.812325i	$-0.301799\pi$
$653$	19463.5	1.16641	0.583205	+	0.812325i	$0.301799\pi$
$654$	0	0
$655$	−3415.33	−0.203738
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23009.3	1.36011	0.680056	−	0.733160i	$-0.261955\pi$
$659$	23009.3	1.36011	0.680056	+	0.733160i	$0.261955\pi$
$660$	0	0
$661$	4135.64	0.243355	0.121678	−	0.992570i	$-0.461173\pi$
$661$	4135.64	0.243355	0.121678	+	0.992570i	$0.461173\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1131.60	0.0659871
$666$	0	0
$667$	−4195.42	−0.243549
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−14206.8	−0.817357
$672$	0	0
$673$	19479.7	1.11573	0.557865	−	0.829931i	$-0.311621\pi$
$673$	19479.7	1.11573	0.557865	+	0.829931i	$0.311621\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5334.44	−0.302835	−0.151417	−	0.988470i	$-0.548384\pi$
$677$	−5334.44	−0.302835	−0.151417	+	0.988470i	$0.548384\pi$
$678$	0	0
$679$	−1308.78	−0.0739708
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8454.59	−0.473654	−0.236827	−	0.971552i	$-0.576107\pi$
$683$	−8454.59	−0.473654	−0.236827	+	0.971552i	$0.576107\pi$
$684$	0	0
$685$	−3085.05	−0.172078
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1047.53	−0.0579210
$690$	0	0
$691$	10176.2	0.560231	0.280115	−	0.959966i	$-0.409627\pi$
$691$	10176.2	0.560231	0.280115	+	0.959966i	$0.409627\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23924.6	1.30577
$696$	0	0
$697$	10529.5	0.572214
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20411.6	1.09976	0.549882	−	0.835243i	$-0.314673\pi$
$701$	20411.6	1.09976	0.549882	+	0.835243i	$0.314673\pi$
$702$	0	0
$703$	−344.538	−0.0184843
$704$	0	0
$705$	0	0
$706$	0	0
$707$	444.457	0.0236429
$708$	0	0
$709$	8169.75	0.432752	0.216376	−	0.976310i	$-0.430576\pi$
$709$	8169.75	0.432752	0.216376	+	0.976310i	$0.430576\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	908.039	0.0476947
$714$	0	0
$715$	−6904.48	−0.361137
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2017.01	−0.104620	−0.0523101	−	0.998631i	$-0.516658\pi$
$719$	−2017.01	−0.104620	−0.0523101	+	0.998631i	$0.516658\pi$
$720$	0	0
$721$	−4317.19	−0.222997
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4160.60	−0.213132
$726$	0	0
$727$	9836.46	0.501807	0.250904	−	0.968012i	$-0.419272\pi$
$727$	9836.46	0.501807	0.250904	+	0.968012i	$0.419272\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11947.0	0.604480
$732$	0	0
$733$	14843.6	0.747969	0.373984	−	0.927435i	$-0.377991\pi$
$733$	14843.6	0.747969	0.373984	+	0.927435i	$0.377991\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6940.06	0.346866
$738$	0	0
$739$	2834.15	0.141077	0.0705385	−	0.997509i	$-0.477528\pi$
$739$	2834.15	0.141077	0.0705385	+	0.997509i	$0.477528\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22076.2	1.09004	0.545019	−	0.838424i	$-0.316523\pi$
$743$	22076.2	1.09004	0.545019	+	0.838424i	$0.316523\pi$
$744$	0	0
$745$	10899.4	0.536002
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4375.63	0.213461
$750$	0	0
$751$	24518.8	1.19135	0.595675	−	0.803225i	$-0.296885\pi$
$751$	24518.8	1.19135	0.595675	+	0.803225i	$0.296885\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−53228.8	−2.56582
$756$	0	0
$757$	−19834.9	−0.952326	−0.476163	−	0.879357i	$-0.657973\pi$
$757$	−19834.9	−0.952326	−0.476163	+	0.879357i	$0.657973\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17719.2	−0.844048	−0.422024	−	0.906585i	$-0.638680\pi$
$761$	−17719.2	−0.844048	−0.422024	+	0.906585i	$0.638680\pi$
$762$	0	0
$763$	−3798.17	−0.180214
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8755.07	0.412161
$768$	0	0
$769$	−17318.8	−0.812134	−0.406067	−	0.913843i	$-0.633100\pi$
$769$	−17318.8	−0.812134	−0.406067	+	0.913843i	$0.633100\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6232.89	−0.290015	−0.145007	−	0.989431i	$-0.546321\pi$
$773$	−6232.89	−0.290015	−0.145007	+	0.989431i	$0.546321\pi$
$774$	0	0
$775$	900.502	0.0417381
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5244.42	−0.241208
$780$	0	0
$781$	−7716.02	−0.353522
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−31255.4	−1.42109
$786$	0	0
$787$	−18625.8	−0.843634	−0.421817	−	0.906681i	$-0.638607\pi$
$787$	−18625.8	−0.843634	−0.421817	+	0.906681i	$0.638607\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13023.2	0.585400
$792$	0	0
$793$	−10962.0	−0.490885
$794$	0	0
$795$	0	0
$796$	0	0
$797$	39096.7	1.73761	0.868805	−	0.495155i	$-0.164889\pi$
$797$	39096.7	1.73761	0.868805	+	0.495155i	$0.164889\pi$
$798$	0	0
$799$	−8478.59	−0.375408
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15493.3	0.680881
$804$	0	0
$805$	−13620.6	−0.596352
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13821.7	0.600672	0.300336	−	0.953833i	$-0.402901\pi$
$809$	13821.7	0.600672	0.300336	+	0.953833i	$0.402901\pi$
$810$	0	0
$811$	−7106.62	−0.307703	−0.153852	−	0.988094i	$-0.549168\pi$
$811$	−7106.62	−0.307703	−0.153852	+	0.988094i	$0.549168\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	48531.3	2.08586
$816$	0	0
$817$	−5950.42	−0.254809
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22099.3	0.939427	0.469714	−	0.882819i	$-0.344357\pi$
$821$	22099.3	0.939427	0.469714	+	0.882819i	$0.344357\pi$
$822$	0	0
$823$	−11652.0	−0.493517	−0.246758	−	0.969077i	$-0.579365\pi$
$823$	−11652.0	−0.493517	−0.246758	+	0.969077i	$0.579365\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14728.5	0.619298	0.309649	−	0.950851i	$-0.399788\pi$
$827$	14728.5	0.619298	0.309649	+	0.950851i	$0.399788\pi$
$828$	0	0
$829$	−11640.5	−0.487684	−0.243842	−	0.969815i	$-0.578408\pi$
$829$	−11640.5	−0.487684	−0.243842	+	0.969815i	$0.578408\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7480.31	−0.311137
$834$	0	0
$835$	48377.1	2.00498
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−17270.0	−0.710638	−0.355319	−	0.934745i	$-0.615628\pi$
$839$	−17270.0	−0.710638	−0.355319	+	0.934745i	$0.615628\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	30670.4	1.24863
$846$	0	0
$847$	−4509.98	−0.182957
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4147.07	0.167050
$852$	0	0
$853$	12968.3	0.520547	0.260274	−	0.965535i	$-0.416187\pi$
$853$	12968.3	0.520547	0.260274	+	0.965535i	$0.416187\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9597.62	0.382554	0.191277	−	0.981536i	$-0.438737\pi$
$857$	9597.62	0.382554	0.191277	+	0.981536i	$0.438737\pi$
$858$	0	0
$859$	18055.5	0.717164	0.358582	−	0.933498i	$-0.383260\pi$
$859$	18055.5	0.717164	0.358582	+	0.933498i	$0.383260\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1068.84	0.0421594	0.0210797	−	0.999778i	$-0.493290\pi$
$863$	1068.84	0.0421594	0.0210797	+	0.999778i	$0.493290\pi$
$864$	0	0
$865$	57359.3	2.25465
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3397.75	−0.132636
$870$	0	0
$871$	5354.97	0.208319
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1738.84	−0.0671810
$876$	0	0
$877$	50968.4	1.96246	0.981231	−	0.192837i	$-0.0617688\pi$
$877$	50968.4	1.96246	0.981231	+	0.192837i	$0.0617688\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11100.3	0.424495	0.212247	−	0.977216i	$-0.431922\pi$
$881$	11100.3	0.424495	0.212247	+	0.977216i	$0.431922\pi$
$882$	0	0
$883$	24959.0	0.951232	0.475616	−	0.879653i	$-0.342225\pi$
$883$	24959.0	0.951232	0.475616	+	0.879653i	$0.342225\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22444.1	0.849604	0.424802	−	0.905286i	$-0.360344\pi$
$887$	22444.1	0.849604	0.424802	+	0.905286i	$0.360344\pi$
$888$	0	0
$889$	2416.36	0.0911609
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4222.93	0.158247
$894$	0	0
$895$	74756.3	2.79199
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−182.023	−0.00675283
$900$	0	0
$901$	1401.87	0.0518347
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5349.93	−0.196506
$906$	0	0
$907$	18929.6	0.692994	0.346497	−	0.938051i	$-0.387371\pi$
$907$	18929.6	0.692994	0.346497	+	0.938051i	$0.387371\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32210.8	1.17145	0.585726	−	0.810509i	$-0.300810\pi$
$911$	32210.8	1.17145	0.585726	+	0.810509i	$0.300810\pi$
$912$	0	0
$913$	25169.0	0.912347
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1197.73	0.0431325
$918$	0	0
$919$	−13885.5	−0.498411	−0.249206	−	0.968451i	$-0.580169\pi$
$919$	−13885.5	−0.498411	−0.249206	+	0.968451i	$0.580169\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5953.70	−0.212317
$924$	0	0
$925$	4112.65	0.146187
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−46561.6	−1.64439	−0.822194	−	0.569207i	$-0.807250\pi$
$929$	−46561.6	−1.64439	−0.822194	+	0.569207i	$0.807250\pi$
$930$	0	0
$931$	3725.71	0.131155
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9240.05	0.323189
$936$	0	0
$937$	−3046.54	−0.106218	−0.0531088	−	0.998589i	$-0.516913\pi$
$937$	−3046.54	−0.106218	−0.0531088	+	0.998589i	$0.516913\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24897.4	−0.862521	−0.431261	−	0.902227i	$-0.641931\pi$
$941$	−24897.4	−0.862521	−0.431261	+	0.902227i	$0.641931\pi$
$942$	0	0
$943$	63125.1	2.17989
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2787.52	−0.0956519	−0.0478259	−	0.998856i	$-0.515229\pi$
$947$	−2787.52	−0.0956519	−0.0478259	+	0.998856i	$0.515229\pi$
$948$	0	0
$949$	11954.7	0.408920
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45400.9	−1.54321	−0.771605	−	0.636102i	$-0.780546\pi$
$953$	−45400.9	−1.54321	−0.771605	+	0.636102i	$0.780546\pi$
$954$	0	0
$955$	42980.5	1.45635
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1081.90	0.0364300
$960$	0	0
$961$	−29751.6	−0.998678
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14656.1	−0.488907
$966$	0	0
$967$	−43256.9	−1.43852	−0.719259	−	0.694742i	$-0.755519\pi$
$967$	−43256.9	−1.43852	−0.719259	+	0.694742i	$0.755519\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35734.1	−1.18101	−0.590505	−	0.807034i	$-0.701072\pi$
$971$	−35734.1	−1.18101	−0.590505	+	0.807034i	$0.701072\pi$
$972$	0	0
$973$	−8390.16	−0.276440
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15675.2	0.513299	0.256650	−	0.966505i	$-0.417381\pi$
$977$	15675.2	0.513299	0.256650	+	0.966505i	$0.417381\pi$
$978$	0	0
$979$	−19908.8	−0.649937
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17324.1	−0.562109	−0.281055	−	0.959692i	$-0.590684\pi$
$983$	−17324.1	−0.562109	−0.281055	+	0.959692i	$0.590684\pi$
$984$	0	0
$985$	35760.4	1.15677
$986$	0	0
$987$	0	0
$988$	0	0
$989$	71623.0	2.30281
$990$	0	0
$991$	22853.9	0.732572	0.366286	−	0.930502i	$-0.380629\pi$
$991$	22853.9	0.732572	0.366286	+	0.930502i	$0.380629\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10418.8	0.331958
$996$	0	0
$997$	−20703.7	−0.657667	−0.328834	−	0.944388i	$-0.606656\pi$
$997$	−20703.7	−0.657667	−0.328834	+	0.944388i	$0.606656\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2088.4.a.f.1.1		5
3.2	odd	2		232.4.a.d.1.4	✓	5
12.11	even	2		464.4.a.m.1.2		5
24.5	odd	2		1856.4.a.z.1.2		5
24.11	even	2		1856.4.a.ba.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.d.1.4	✓	5	3.2	odd	2
464.4.a.m.1.2		5	12.11	even	2
1856.4.a.z.1.2		5	24.5	odd	2
1856.4.a.ba.1.4		5	24.11	even	2
2088.4.a.f.1.1		5	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2088 = 2^{3} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2088.a (trivial)

\(f(q)\)	\(=\)	\(q-16.3850 q^{5} +5.74609 q^{7} +O(q^{10})\)	\(q-16.3850 q^{5} +5.74609 q^{7} -23.3693 q^{11} -18.0318 q^{13} +24.1314 q^{17} -12.0191 q^{19} +144.669 q^{23} +143.469 q^{25} -29.0000 q^{29} +6.27664 q^{31} -94.1498 q^{35} +28.6658 q^{37} +436.340 q^{41} +495.080 q^{43} -351.351 q^{47} -309.982 q^{49} +58.0933 q^{53} +382.906 q^{55} -485.535 q^{59} +607.926 q^{61} +295.451 q^{65} -296.974 q^{67} +330.178 q^{71} -662.978 q^{73} -134.282 q^{77} +145.394 q^{79} -1077.01 q^{83} -395.393 q^{85} +851.923 q^{89} -103.612 q^{91} +196.933 q^{95} -227.768 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{5} + 32 q^{7}+O(q^{10}) \) 5 * q - 10 * q^5 + 32 * q^7	\( 5 q - 10 q^{5} + 32 q^{7} - 36 q^{11} + 26 q^{13} - 82 q^{17} + 156 q^{19} - 336 q^{23} + 151 q^{25} - 145 q^{29} + 432 q^{31} - 600 q^{35} - 18 q^{37} - 82 q^{41} + 340 q^{43} - 680 q^{47} - 115 q^{49} + 102 q^{53} + 736 q^{55} - 924 q^{59} - 618 q^{61} + 704 q^{65} + 44 q^{67} - 1032 q^{71} - 1078 q^{73} + 888 q^{77} + 200 q^{79} - 452 q^{83} - 1700 q^{85} + 1790 q^{89} - 1128 q^{91} - 1024 q^{95} - 2518 q^{97}+O(q^{100}) \) 5 * q - 10 * q^5 + 32 * q^7 - 36 * q^11 + 26 * q^13 - 82 * q^17 + 156 * q^19 - 336 * q^23 + 151 * q^25 - 145 * q^29 + 432 * q^31 - 600 * q^35 - 18 * q^37 - 82 * q^41 + 340 * q^43 - 680 * q^47 - 115 * q^49 + 102 * q^53 + 736 * q^55 - 924 * q^59 - 618 * q^61 + 704 * q^65 + 44 * q^67 - 1032 * q^71 - 1078 * q^73 + 888 * q^77 + 200 * q^79 - 452 * q^83 - 1700 * q^85 + 1790 * q^89 - 1128 * q^91 - 1024 * q^95 - 2518 * q^97

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