LMFDB - Embedded newform 1368.4.a.p.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,4,Mod(1,1368)] 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("1368.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,16,0,-4,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$80.7146128879$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 539x^{6} + 1968x^{5} + 81033x^{4} - 222055x^{3} - 4314559x^{2} + 4793814x + 61552224 $ x^8 - 3x^7 - 539x^6 + 1968x^5 + 81033x^4 - 222055x^3 - 4314559x^2 + 4793814*x + 61552224
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.3303$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47611 q^{5} +28.1401 q^{7} -51.2379 q^{11} -85.6249 q^{13} +52.4629 q^{17} +19.0000 q^{19} +188.807 q^{23} -118.869 q^{25} +62.6206 q^{29} -65.0821 q^{31} -69.6778 q^{35} -379.051 q^{37} +369.068 q^{41} +473.455 q^{43} +39.6326 q^{47} +448.863 q^{49} +606.393 q^{53} +126.870 q^{55} -211.869 q^{59} -89.9190 q^{61} +212.016 q^{65} +409.265 q^{67} +1009.35 q^{71} -152.918 q^{73} -1441.84 q^{77} +468.158 q^{79} -887.677 q^{83} -129.904 q^{85} -1111.30 q^{89} -2409.49 q^{91} -47.0460 q^{95} +96.4810 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} - 4 q^{7} + 16 q^{11} - 36 q^{13} - 48 q^{17} + 152 q^{19} + 88 q^{23} + 124 q^{25} + 48 q^{29} - 4 q^{31} + 72 q^{35} + 4 q^{37} + 376 q^{41} + 276 q^{43} + 384 q^{47} + 780 q^{49} + 1528 q^{53}+ \cdots + 840 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 - 4 * q^7 + 16 * q^11 - 36 * q^13 - 48 * q^17 + 152 * q^19 + 88 * q^23 + 124 * q^25 + 48 * q^29 - 4 * q^31 + 72 * q^35 + 4 * q^37 + 376 * q^41 + 276 * q^43 + 384 * q^47 + 780 * q^49 + 1528 * q^53 - 604 * q^55 + 664 * q^59 + 68 * q^61 + 1576 * q^65 - 184 * q^67 + 1704 * q^71 + 628 * q^73 + 1344 * q^77 - 1380 * q^79 + 1480 * q^83 + 796 * q^85 + 1800 * q^89 - 1864 * q^91 + 304 * q^95 + 840 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.47611	−0.221470	−0.110735	−	0.993850i	$-0.535320\pi$
$5$	−2.47611	−0.221470	−0.110735	+	0.993850i	$0.535320\pi$
$6$	0	0
$7$	28.1401	1.51942	0.759710	−	0.650262i	$-0.225341\pi$
$7$	28.1401	1.51942	0.759710	+	0.650262i	$0.225341\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−51.2379	−1.40444	−0.702218	−	0.711962i	$-0.747807\pi$
$11$	−51.2379	−1.40444	−0.702218	+	0.711962i	$0.747807\pi$
$12$	0	0
$13$	−85.6249	−1.82678	−0.913388	−	0.407090i	$-0.866543\pi$
$13$	−85.6249	−1.82678	−0.913388	+	0.407090i	$0.866543\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	52.4629	0.748477	0.374239	−	0.927332i	$-0.377904\pi$
$17$	52.4629	0.748477	0.374239	+	0.927332i	$0.377904\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	188.807	1.71169	0.855847	−	0.517230i	$-0.173037\pi$
$23$	188.807	1.71169	0.855847	+	0.517230i	$0.173037\pi$
$24$	0	0
$25$	−118.869	−0.950951
$26$	0	0
$27$	0	0
$28$	0	0
$29$	62.6206	0.400978	0.200489	−	0.979696i	$-0.435747\pi$
$29$	62.6206	0.400978	0.200489	+	0.979696i	$0.435747\pi$
$30$	0	0
$31$	−65.0821	−0.377068	−0.188534	−	0.982067i	$-0.560374\pi$
$31$	−65.0821	−0.377068	−0.188534	+	0.982067i	$0.560374\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−69.6778	−0.336506
$36$	0	0
$37$	−379.051	−1.68420	−0.842102	−	0.539318i	$-0.818682\pi$
$37$	−379.051	−1.68420	−0.842102	+	0.539318i	$0.818682\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	369.068	1.40582	0.702911	−	0.711278i	$-0.251883\pi$
$41$	369.068	1.40582	0.702911	+	0.711278i	$0.251883\pi$
$42$	0	0
$43$	473.455	1.67910	0.839549	−	0.543284i	$-0.182819\pi$
$43$	473.455	1.67910	0.839549	+	0.543284i	$0.182819\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	39.6326	0.123000	0.0615001	−	0.998107i	$-0.480412\pi$
$47$	39.6326	0.123000	0.0615001	+	0.998107i	$0.480412\pi$
$48$	0	0
$49$	448.863	1.30864
$50$	0	0
$51$	0	0
$52$	0	0
$53$	606.393	1.57159	0.785797	−	0.618484i	$-0.212253\pi$
$53$	606.393	1.57159	0.785797	+	0.618484i	$0.212253\pi$
$54$	0	0
$55$	126.870	0.311040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−211.869	−0.467509	−0.233754	−	0.972296i	$-0.575101\pi$
$59$	−211.869	−0.467509	−0.233754	+	0.972296i	$0.575101\pi$
$60$	0	0
$61$	−89.9190	−0.188737	−0.0943685	−	0.995537i	$-0.530083\pi$
$61$	−89.9190	−0.188737	−0.0943685	+	0.995537i	$0.530083\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	212.016	0.404575
$66$	0	0
$67$	409.265	0.746263	0.373131	−	0.927778i	$-0.378284\pi$
$67$	409.265	0.746263	0.373131	+	0.927778i	$0.378284\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1009.35	1.68716	0.843580	−	0.537003i	$-0.180444\pi$
$71$	1009.35	1.68716	0.843580	+	0.537003i	$0.180444\pi$
$72$	0	0
$73$	−152.918	−0.245174	−0.122587	−	0.992458i	$-0.539119\pi$
$73$	−152.918	−0.245174	−0.122587	+	0.992458i	$0.539119\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1441.84	−2.13393
$78$	0	0
$79$	468.158	0.666732	0.333366	−	0.942797i	$-0.391816\pi$
$79$	468.158	0.666732	0.333366	+	0.942797i	$0.391816\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−887.677	−1.17392	−0.586959	−	0.809617i	$-0.699675\pi$
$83$	−887.677	−1.17392	−0.586959	+	0.809617i	$0.699675\pi$
$84$	0	0
$85$	−129.904	−0.165765
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1111.30	−1.32357	−0.661785	−	0.749694i	$-0.730201\pi$
$89$	−1111.30	−1.32357	−0.661785	+	0.749694i	$0.730201\pi$
$90$	0	0
$91$	−2409.49	−2.77564
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−47.0460	−0.0508086
$96$	0	0
$97$	96.4810	0.100991	0.0504957	−	0.998724i	$-0.483920\pi$
$97$	96.4810	0.100991	0.0504957	+	0.998724i	$0.483920\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−285.530	−0.281300	−0.140650	−	0.990059i	$-0.544919\pi$
$101$	−285.530	−0.281300	−0.140650	+	0.990059i	$0.544919\pi$
$102$	0	0
$103$	−232.959	−0.222856	−0.111428	−	0.993773i	$-0.535542\pi$
$103$	−232.959	−0.222856	−0.111428	+	0.993773i	$0.535542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1410.12	1.27403	0.637016	−	0.770851i	$-0.280169\pi$
$107$	1410.12	1.27403	0.637016	+	0.770851i	$0.280169\pi$
$108$	0	0
$109$	710.925	0.624718	0.312359	−	0.949964i	$-0.398881\pi$
$109$	710.925	0.624718	0.312359	+	0.949964i	$0.398881\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	614.621	0.511670	0.255835	−	0.966721i	$-0.417650\pi$
$113$	614.621	0.511670	0.255835	+	0.966721i	$0.417650\pi$
$114$	0	0
$115$	−467.506	−0.379088
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1476.31	1.13725
$120$	0	0
$121$	1294.32	0.972443
$122$	0	0
$123$	0	0
$124$	0	0
$125$	603.845	0.432076
$126$	0	0
$127$	107.588	0.0751725	0.0375862	−	0.999293i	$-0.488033\pi$
$127$	107.588	0.0751725	0.0375862	+	0.999293i	$0.488033\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−895.942	−0.597548	−0.298774	−	0.954324i	$-0.596578\pi$
$131$	−895.942	−0.597548	−0.298774	+	0.954324i	$0.596578\pi$
$132$	0	0
$133$	534.661	0.348579
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1067.62	0.665790	0.332895	−	0.942964i	$-0.391974\pi$
$137$	1067.62	0.665790	0.332895	+	0.942964i	$0.391974\pi$
$138$	0	0
$139$	−2011.64	−1.22752	−0.613759	−	0.789494i	$-0.710343\pi$
$139$	−2011.64	−1.22752	−0.613759	+	0.789494i	$0.710343\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4387.24	2.56559
$144$	0	0
$145$	−155.055	−0.0888044
$146$	0	0
$147$	0	0
$148$	0	0
$149$	912.553	0.501740	0.250870	−	0.968021i	$-0.419283\pi$
$149$	912.553	0.501740	0.250870	+	0.968021i	$0.419283\pi$
$150$	0	0
$151$	3199.56	1.72435	0.862173	−	0.506614i	$-0.169103\pi$
$151$	3199.56	1.72435	0.862173	+	0.506614i	$0.169103\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	161.150	0.0835090
$156$	0	0
$157$	−3085.87	−1.56866	−0.784328	−	0.620346i	$-0.786992\pi$
$157$	−3085.87	−1.56866	−0.784328	+	0.620346i	$0.786992\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5313.04	2.60078
$162$	0	0
$163$	554.596	0.266499	0.133249	−	0.991083i	$-0.457459\pi$
$163$	554.596	0.266499	0.133249	+	0.991083i	$0.457459\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1540.72	0.713919	0.356960	−	0.934120i	$-0.383813\pi$
$167$	1540.72	0.713919	0.356960	+	0.934120i	$0.383813\pi$
$168$	0	0
$169$	5134.63	2.33711
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4360.49	1.91631	0.958155	−	0.286250i	$-0.0924089\pi$
$173$	4360.49	1.91631	0.958155	+	0.286250i	$0.0924089\pi$
$174$	0	0
$175$	−3344.98	−1.44489
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2165.72	0.904322	0.452161	−	0.891936i	$-0.350653\pi$
$179$	2165.72	0.904322	0.452161	+	0.891936i	$0.350653\pi$
$180$	0	0
$181$	2754.58	1.13120	0.565598	−	0.824681i	$-0.308645\pi$
$181$	2754.58	1.13120	0.565598	+	0.824681i	$0.308645\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	938.570	0.373000
$186$	0	0
$187$	−2688.09	−1.05119
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3344.89	−1.26716	−0.633580	−	0.773677i	$-0.718415\pi$
$191$	−3344.89	−1.26716	−0.633580	+	0.773677i	$0.718415\pi$
$192$	0	0
$193$	2990.63	1.11539	0.557695	−	0.830046i	$-0.311686\pi$
$193$	2990.63	1.11539	0.557695	+	0.830046i	$0.311686\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3932.24	1.42213	0.711067	−	0.703124i	$-0.248212\pi$
$197$	3932.24	1.42213	0.711067	+	0.703124i	$0.248212\pi$
$198$	0	0
$199$	1102.76	0.392827	0.196413	−	0.980521i	$-0.437071\pi$
$199$	1102.76	0.392827	0.196413	+	0.980521i	$0.437071\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1762.15	0.609254
$204$	0	0
$205$	−913.851	−0.311347
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−973.520	−0.322200
$210$	0	0
$211$	−1495.50	−0.487935	−0.243967	−	0.969783i	$-0.578449\pi$
$211$	−1495.50	−0.487935	−0.243967	+	0.969783i	$0.578449\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1172.32	−0.371869
$216$	0	0
$217$	−1831.42	−0.572924
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4492.13	−1.36730
$222$	0	0
$223$	−120.410	−0.0361580	−0.0180790	−	0.999837i	$-0.505755\pi$
$223$	−120.410	−0.0361580	−0.0180790	+	0.999837i	$0.505755\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−231.451	−0.0676738	−0.0338369	−	0.999427i	$-0.510773\pi$
$227$	−231.451	−0.0676738	−0.0338369	+	0.999427i	$0.510773\pi$
$228$	0	0
$229$	1859.19	0.536501	0.268251	−	0.963349i	$-0.413554\pi$
$229$	1859.19	0.536501	0.268251	+	0.963349i	$0.413554\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4642.50	−1.30532	−0.652661	−	0.757650i	$-0.726347\pi$
$233$	−4642.50	−1.30532	−0.652661	+	0.757650i	$0.726347\pi$
$234$	0	0
$235$	−98.1346	−0.0272408
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1771.01	0.479319	0.239659	−	0.970857i	$-0.422964\pi$
$239$	1771.01	0.479319	0.239659	+	0.970857i	$0.422964\pi$
$240$	0	0
$241$	6467.63	1.72870	0.864350	−	0.502891i	$-0.167730\pi$
$241$	6467.63	1.72870	0.864350	+	0.502891i	$0.167730\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1111.43	−0.289824
$246$	0	0
$247$	−1626.87	−0.419091
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4160.27	1.04619	0.523095	−	0.852274i	$-0.324777\pi$
$251$	4160.27	1.04619	0.523095	+	0.852274i	$0.324777\pi$
$252$	0	0
$253$	−9674.06	−2.40397
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7137.04	1.73228	0.866141	−	0.499799i	$-0.166593\pi$
$257$	7137.04	1.73228	0.866141	+	0.499799i	$0.166593\pi$
$258$	0	0
$259$	−10666.5	−2.55901
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7957.12	−1.86562	−0.932808	−	0.360374i	$-0.882649\pi$
$263$	−7957.12	−1.86562	−0.932808	+	0.360374i	$0.882649\pi$
$264$	0	0
$265$	−1501.49	−0.348060
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6002.30	1.36047	0.680235	−	0.732994i	$-0.261878\pi$
$269$	6002.30	1.36047	0.680235	+	0.732994i	$0.261878\pi$
$270$	0	0
$271$	3319.69	0.744122	0.372061	−	0.928208i	$-0.378651\pi$
$271$	3319.69	0.744122	0.372061	+	0.928208i	$0.378651\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6090.59	1.33555
$276$	0	0
$277$	230.983	0.0501027	0.0250513	−	0.999686i	$-0.492025\pi$
$277$	230.983	0.0501027	0.0250513	+	0.999686i	$0.492025\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3004.05	0.637746	0.318873	−	0.947798i	$-0.396696\pi$
$281$	3004.05	0.637746	0.318873	+	0.947798i	$0.396696\pi$
$282$	0	0
$283$	−1765.79	−0.370902	−0.185451	−	0.982654i	$-0.559375\pi$
$283$	−1765.79	−0.370902	−0.185451	+	0.982654i	$0.559375\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10385.6	2.13604
$288$	0	0
$289$	−2160.65	−0.439781
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6494.10	−1.29485	−0.647423	−	0.762131i	$-0.724153\pi$
$293$	−6494.10	−1.29485	−0.647423	+	0.762131i	$0.724153\pi$
$294$	0	0
$295$	524.610	0.103539
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16166.6	−3.12688
$300$	0	0
$301$	13323.1	2.55126
$302$	0	0
$303$	0	0
$304$	0	0
$305$	222.649	0.0417995
$306$	0	0
$307$	−1305.94	−0.242782	−0.121391	−	0.992605i	$-0.538735\pi$
$307$	−1305.94	−0.242782	−0.121391	+	0.992605i	$0.538735\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8956.85	1.63311	0.816553	−	0.577270i	$-0.195882\pi$
$311$	8956.85	1.63311	0.816553	+	0.577270i	$0.195882\pi$
$312$	0	0
$313$	80.0024	0.0144473	0.00722365	−	0.999974i	$-0.497701\pi$
$313$	80.0024	0.0144473	0.00722365	+	0.999974i	$0.497701\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	778.706	0.137970	0.0689850	−	0.997618i	$-0.478024\pi$
$317$	778.706	0.137970	0.0689850	+	0.997618i	$0.478024\pi$
$318$	0	0
$319$	−3208.55	−0.563148
$320$	0	0
$321$	0	0
$322$	0	0
$323$	996.795	0.171713
$324$	0	0
$325$	10178.1	1.73717
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1115.26	0.186889
$330$	0	0
$331$	−3945.35	−0.655154	−0.327577	−	0.944824i	$-0.606232\pi$
$331$	−3945.35	−0.655154	−0.327577	+	0.944824i	$0.606232\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1013.38	−0.165275
$336$	0	0
$337$	−1809.80	−0.292540	−0.146270	−	0.989245i	$-0.546727\pi$
$337$	−1809.80	−0.292540	−0.146270	+	0.989245i	$0.546727\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3334.67	0.529568
$342$	0	0
$343$	2979.00	0.468953
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4047.45	0.626163	0.313081	−	0.949726i	$-0.398639\pi$
$347$	4047.45	0.626163	0.313081	+	0.949726i	$0.398639\pi$
$348$	0	0
$349$	3495.47	0.536126	0.268063	−	0.963401i	$-0.413616\pi$
$349$	3495.47	0.536126	0.268063	+	0.963401i	$0.413616\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4448.20	−0.670691	−0.335346	−	0.942095i	$-0.608853\pi$
$353$	−4448.20	−0.670691	−0.335346	+	0.942095i	$0.608853\pi$
$354$	0	0
$355$	−2499.27	−0.373655
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12243.1	−1.79991	−0.899956	−	0.435981i	$-0.856402\pi$
$359$	−12243.1	−1.79991	−0.899956	+	0.435981i	$0.856402\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	378.641	0.0542985
$366$	0	0
$367$	−7434.33	−1.05741	−0.528704	−	0.848806i	$-0.677322\pi$
$367$	−7434.33	−1.05741	−0.528704	+	0.848806i	$0.677322\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	17063.9	2.38791
$372$	0	0
$373$	2714.80	0.376855	0.188427	−	0.982087i	$-0.439661\pi$
$373$	2714.80	0.376855	0.188427	+	0.982087i	$0.439661\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5361.89	−0.732497
$378$	0	0
$379$	8717.91	1.18155	0.590777	−	0.806835i	$-0.298821\pi$
$379$	8717.91	1.18155	0.590777	+	0.806835i	$0.298821\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8259.66	−1.10196	−0.550978	−	0.834520i	$-0.685745\pi$
$383$	−8259.66	−1.10196	−0.550978	+	0.834520i	$0.685745\pi$
$384$	0	0
$385$	3570.14	0.472601
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10093.1	1.31553	0.657766	−	0.753222i	$-0.271502\pi$
$389$	10093.1	1.31553	0.657766	+	0.753222i	$0.271502\pi$
$390$	0	0
$391$	9905.35	1.28116
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1159.21	−0.147661
$396$	0	0
$397$	8750.50	1.10623	0.553117	−	0.833104i	$-0.313438\pi$
$397$	8750.50	1.10623	0.553117	+	0.833104i	$0.313438\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5055.29	0.629549	0.314774	−	0.949167i	$-0.398071\pi$
$401$	5055.29	0.629549	0.314774	+	0.949167i	$0.398071\pi$
$402$	0	0
$403$	5572.65	0.688818
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19421.8	2.36536
$408$	0	0
$409$	−8261.53	−0.998793	−0.499396	−	0.866374i	$-0.666445\pi$
$409$	−8261.53	−0.998793	−0.499396	+	0.866374i	$0.666445\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5962.01	−0.710342
$414$	0	0
$415$	2197.98	0.259987
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14078.5	−1.64148	−0.820740	−	0.571302i	$-0.806439\pi$
$419$	−14078.5	−1.64148	−0.820740	+	0.571302i	$0.806439\pi$
$420$	0	0
$421$	−14481.8	−1.67648	−0.838242	−	0.545298i	$-0.816417\pi$
$421$	−14481.8	−1.67648	−0.838242	+	0.545298i	$0.816417\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6236.20	−0.711766
$426$	0	0
$427$	−2530.33	−0.286771
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9861.18	−1.10208	−0.551040	−	0.834479i	$-0.685769\pi$
$431$	−9861.18	−1.10208	−0.551040	+	0.834479i	$0.685769\pi$
$432$	0	0
$433$	−10770.5	−1.19537	−0.597687	−	0.801730i	$-0.703913\pi$
$433$	−10770.5	−1.19537	−0.597687	+	0.801730i	$0.703913\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3587.33	0.392689
$438$	0	0
$439$	3733.78	0.405931	0.202966	−	0.979186i	$-0.434942\pi$
$439$	3733.78	0.405931	0.202966	+	0.979186i	$0.434942\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4241.25	0.454871	0.227436	−	0.973793i	$-0.426966\pi$
$443$	4241.25	0.454871	0.227436	+	0.973793i	$0.426966\pi$
$444$	0	0
$445$	2751.70	0.293131
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5802.88	−0.609921	−0.304961	−	0.952365i	$-0.598643\pi$
$449$	−5802.88	−0.609921	−0.304961	+	0.952365i	$0.598643\pi$
$450$	0	0
$451$	−18910.3	−1.97439
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5966.16	0.614720
$456$	0	0
$457$	−16392.8	−1.67795	−0.838976	−	0.544169i	$-0.816845\pi$
$457$	−16392.8	−1.67795	−0.838976	+	0.544169i	$0.816845\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13821.3	−1.39636	−0.698179	−	0.715923i	$-0.746006\pi$
$461$	−13821.3	−1.39636	−0.698179	+	0.715923i	$0.746006\pi$
$462$	0	0
$463$	−14142.5	−1.41956	−0.709782	−	0.704421i	$-0.751207\pi$
$463$	−14142.5	−1.41956	−0.709782	+	0.704421i	$0.751207\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11543.3	1.14382	0.571908	−	0.820318i	$-0.306203\pi$
$467$	11543.3	1.14382	0.571908	+	0.820318i	$0.306203\pi$
$468$	0	0
$469$	11516.7	1.13389
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24258.8	−2.35819
$474$	0	0
$475$	−2258.51	−0.218163
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4919.91	−0.469303	−0.234652	−	0.972080i	$-0.575395\pi$
$479$	−4919.91	−0.469303	−0.234652	+	0.972080i	$0.575395\pi$
$480$	0	0
$481$	32456.2	3.07666
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−238.897	−0.0223665
$486$	0	0
$487$	−713.007	−0.0663438	−0.0331719	−	0.999450i	$-0.510561\pi$
$487$	−713.007	−0.0663438	−0.0331719	+	0.999450i	$0.510561\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1797.78	0.165240	0.0826200	−	0.996581i	$-0.473671\pi$
$491$	1797.78	0.165240	0.0826200	+	0.996581i	$0.473671\pi$
$492$	0	0
$493$	3285.26	0.300123
$494$	0	0
$495$	0	0
$496$	0	0
$497$	28403.3	2.56351
$498$	0	0
$499$	−21065.3	−1.88980	−0.944902	−	0.327354i	$-0.893843\pi$
$499$	−21065.3	−1.88980	−0.944902	+	0.327354i	$0.893843\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13516.4	−1.19815	−0.599073	−	0.800694i	$-0.704464\pi$
$503$	−13516.4	−1.19815	−0.599073	+	0.800694i	$0.704464\pi$
$504$	0	0
$505$	707.002	0.0622993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18973.3	1.65221	0.826107	−	0.563514i	$-0.190551\pi$
$509$	18973.3	1.65221	0.826107	+	0.563514i	$0.190551\pi$
$510$	0	0
$511$	−4303.12	−0.372522
$512$	0	0
$513$	0	0
$514$	0	0
$515$	576.832	0.0493558
$516$	0	0
$517$	−2030.69	−0.172746
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1267.80	0.106609	0.0533044	−	0.998578i	$-0.483025\pi$
$521$	1267.80	0.106609	0.0533044	+	0.998578i	$0.483025\pi$
$522$	0	0
$523$	21227.9	1.77482	0.887411	−	0.460980i	$-0.152502\pi$
$523$	21227.9	1.77482	0.887411	+	0.460980i	$0.152502\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3414.40	−0.282227
$528$	0	0
$529$	23481.0	1.92989
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−31601.4	−2.56812
$534$	0	0
$535$	−3491.61	−0.282159
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−22998.8	−1.83790
$540$	0	0
$541$	−1011.95	−0.0804196	−0.0402098	−	0.999191i	$-0.512803\pi$
$541$	−1011.95	−0.0804196	−0.0402098	+	0.999191i	$0.512803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1760.33	−0.138356
$546$	0	0
$547$	6173.81	0.482583	0.241292	−	0.970453i	$-0.422429\pi$
$547$	6173.81	0.482583	0.241292	+	0.970453i	$0.422429\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1189.79	0.0919906
$552$	0	0
$553$	13174.0	1.01305
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10935.3	−0.831858	−0.415929	−	0.909397i	$-0.636544\pi$
$557$	−10935.3	−0.831858	−0.415929	+	0.909397i	$0.636544\pi$
$558$	0	0
$559$	−40539.6	−3.06734
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13858.3	−1.03741	−0.518703	−	0.854954i	$-0.673585\pi$
$563$	−13858.3	−1.03741	−0.518703	+	0.854954i	$0.673585\pi$
$564$	0	0
$565$	−1521.87	−0.113319
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10779.8	0.794221	0.397111	−	0.917771i	$-0.370013\pi$
$569$	10779.8	0.794221	0.397111	+	0.917771i	$0.370013\pi$
$570$	0	0
$571$	12487.2	0.915191	0.457596	−	0.889160i	$-0.348711\pi$
$571$	12487.2	0.915191	0.457596	+	0.889160i	$0.348711\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−22443.3	−1.62774
$576$	0	0
$577$	13051.9	0.941693	0.470846	−	0.882215i	$-0.343949\pi$
$577$	13051.9	0.941693	0.470846	+	0.882215i	$0.343949\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−24979.3	−1.78368
$582$	0	0
$583$	−31070.3	−2.20720
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10506.4	−0.738751	−0.369376	−	0.929280i	$-0.620428\pi$
$587$	−10506.4	−0.738751	−0.369376	+	0.929280i	$0.620428\pi$
$588$	0	0
$589$	−1236.56	−0.0865053
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19773.0	1.36927	0.684636	−	0.728885i	$-0.259961\pi$
$593$	19773.0	1.36927	0.684636	+	0.728885i	$0.259961\pi$
$594$	0	0
$595$	−3655.50	−0.251867
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10160.8	−0.693088	−0.346544	−	0.938034i	$-0.612645\pi$
$599$	−10160.8	−0.693088	−0.346544	+	0.938034i	$0.612645\pi$
$600$	0	0
$601$	9904.74	0.672251	0.336126	−	0.941817i	$-0.390883\pi$
$601$	9904.74	0.672251	0.336126	+	0.941817i	$0.390883\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3204.88	−0.215367
$606$	0	0
$607$	−16036.9	−1.07235	−0.536177	−	0.844106i	$-0.680132\pi$
$607$	−16036.9	−1.07235	−0.536177	+	0.844106i	$0.680132\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3393.54	−0.224694
$612$	0	0
$613$	5909.00	0.389335	0.194667	−	0.980869i	$-0.437637\pi$
$613$	5909.00	0.389335	0.194667	+	0.980869i	$0.437637\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12235.7	−0.798365	−0.399182	−	0.916872i	$-0.630706\pi$
$617$	−12235.7	−0.798365	−0.399182	+	0.916872i	$0.630706\pi$
$618$	0	0
$619$	11228.8	0.729115	0.364557	−	0.931181i	$-0.381220\pi$
$619$	11228.8	0.729115	0.364557	+	0.931181i	$0.381220\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−31272.1	−2.01106
$624$	0	0
$625$	13363.4	0.855259
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19886.1	−1.26059
$630$	0	0
$631$	21093.7	1.33079	0.665393	−	0.746493i	$-0.268264\pi$
$631$	21093.7	1.33079	0.665393	+	0.746493i	$0.268264\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−266.400	−0.0166484
$636$	0	0
$637$	−38433.9	−2.39059
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1722.83	0.106159	0.0530793	−	0.998590i	$-0.483096\pi$
$641$	1722.83	0.106159	0.0530793	+	0.998590i	$0.483096\pi$
$642$	0	0
$643$	−31747.9	−1.94714	−0.973572	−	0.228378i	$-0.926658\pi$
$643$	−31747.9	−1.94714	−0.973572	+	0.228378i	$0.926658\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9414.68	−0.572070	−0.286035	−	0.958219i	$-0.592337\pi$
$647$	−9414.68	−0.572070	−0.286035	+	0.958219i	$0.592337\pi$
$648$	0	0
$649$	10855.7	0.656586
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4579.65	0.274450	0.137225	−	0.990540i	$-0.456182\pi$
$653$	4579.65	0.274450	0.137225	+	0.990540i	$0.456182\pi$
$654$	0	0
$655$	2218.45	0.132339
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8323.79	0.492032	0.246016	−	0.969266i	$-0.420878\pi$
$659$	8323.79	0.492032	0.246016	+	0.969266i	$0.420878\pi$
$660$	0	0
$661$	9447.38	0.555916	0.277958	−	0.960593i	$-0.410342\pi$
$661$	9447.38	0.555916	0.277958	+	0.960593i	$0.410342\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1323.88	−0.0771997
$666$	0	0
$667$	11823.2	0.686351
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4607.26	0.265069
$672$	0	0
$673$	19267.7	1.10359	0.551794	−	0.833981i	$-0.313944\pi$
$673$	19267.7	1.10359	0.551794	+	0.833981i	$0.313944\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7399.49	−0.420067	−0.210033	−	0.977694i	$-0.567357\pi$
$677$	−7399.49	−0.420067	−0.210033	+	0.977694i	$0.567357\pi$
$678$	0	0
$679$	2714.98	0.153448
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14876.3	0.833418	0.416709	−	0.909040i	$-0.363183\pi$
$683$	14876.3	0.833418	0.416709	+	0.909040i	$0.363183\pi$
$684$	0	0
$685$	−2643.55	−0.147452
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−51922.4	−2.87095
$690$	0	0
$691$	27915.9	1.53686	0.768431	−	0.639933i	$-0.221038\pi$
$691$	27915.9	1.53686	0.768431	+	0.639933i	$0.221038\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4981.03	0.271858
$696$	0	0
$697$	19362.4	1.05223
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16788.1	0.904531	0.452265	−	0.891883i	$-0.350616\pi$
$701$	16788.1	0.904531	0.452265	+	0.891883i	$0.350616\pi$
$702$	0	0
$703$	−7201.96	−0.386383
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8034.82	−0.427412
$708$	0	0
$709$	−9342.88	−0.494893	−0.247447	−	0.968902i	$-0.579591\pi$
$709$	−9342.88	−0.494893	−0.247447	+	0.968902i	$0.579591\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12288.0	−0.645424
$714$	0	0
$715$	−10863.3	−0.568201
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29345.1	1.52210	0.761049	−	0.648695i	$-0.224685\pi$
$719$	29345.1	1.52210	0.761049	+	0.648695i	$0.224685\pi$
$720$	0	0
$721$	−6555.49	−0.338612
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7443.65	−0.381310
$726$	0	0
$727$	15766.3	0.804318	0.402159	−	0.915570i	$-0.368260\pi$
$727$	15766.3	0.804318	0.402159	+	0.915570i	$0.368260\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24838.8	1.25677
$732$	0	0
$733$	−34455.3	−1.73620	−0.868101	−	0.496387i	$-0.834660\pi$
$733$	−34455.3	−1.73620	−0.868101	+	0.496387i	$0.834660\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20969.8	−1.04808
$738$	0	0
$739$	−14894.7	−0.741421	−0.370710	−	0.928749i	$-0.620886\pi$
$739$	−14894.7	−0.741421	−0.370710	+	0.928749i	$0.620886\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11306.5	0.558273	0.279137	−	0.960251i	$-0.409952\pi$
$743$	11306.5	0.558273	0.279137	+	0.960251i	$0.409952\pi$
$744$	0	0
$745$	−2259.58	−0.111120
$746$	0	0
$747$	0	0
$748$	0	0
$749$	39680.9	1.93579
$750$	0	0
$751$	−10834.6	−0.526447	−0.263223	−	0.964735i	$-0.584786\pi$
$751$	−10834.6	−0.526447	−0.263223	+	0.964735i	$0.584786\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7922.44	−0.381890
$756$	0	0
$757$	7142.38	0.342925	0.171463	−	0.985191i	$-0.445151\pi$
$757$	7142.38	0.342925	0.171463	+	0.985191i	$0.445151\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−14832.5	−0.706541	−0.353270	−	0.935521i	$-0.614930\pi$
$761$	−14832.5	−0.706541	−0.353270	+	0.935521i	$0.614930\pi$
$762$	0	0
$763$	20005.5	0.949209
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18141.3	0.854033
$768$	0	0
$769$	−8149.07	−0.382137	−0.191068	−	0.981577i	$-0.561195\pi$
$769$	−8149.07	−0.382137	−0.191068	+	0.981577i	$0.561195\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2001.12	−0.0931118	−0.0465559	−	0.998916i	$-0.514825\pi$
$773$	−2001.12	−0.0931118	−0.0465559	+	0.998916i	$0.514825\pi$
$774$	0	0
$775$	7736.24	0.358573
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7012.29	0.322518
$780$	0	0
$781$	−51717.2	−2.36951
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7640.94	0.347410
$786$	0	0
$787$	38254.0	1.73267	0.866333	−	0.499467i	$-0.166471\pi$
$787$	38254.0	1.73267	0.866333	+	0.499467i	$0.166471\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17295.5	0.777442
$792$	0	0
$793$	7699.31	0.344780
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18101.6	0.804506	0.402253	−	0.915529i	$-0.368227\pi$
$797$	18101.6	0.804506	0.402253	+	0.915529i	$0.368227\pi$
$798$	0	0
$799$	2079.24	0.0920629
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7835.18	0.344331
$804$	0	0
$805$	−13155.6	−0.575994
$806$	0	0
$807$	0	0
$808$	0	0
$809$	25248.6	1.09727	0.548635	−	0.836062i	$-0.315148\pi$
$809$	25248.6	1.09727	0.548635	+	0.836062i	$0.315148\pi$
$810$	0	0
$811$	−20572.4	−0.890746	−0.445373	−	0.895345i	$-0.646929\pi$
$811$	−20572.4	−0.890746	−0.445373	+	0.895345i	$0.646929\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1373.24	−0.0590214
$816$	0	0
$817$	8995.64	0.385211
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43349.9	1.84278	0.921389	−	0.388642i	$-0.127056\pi$
$821$	43349.9	1.84278	0.921389	+	0.388642i	$0.127056\pi$
$822$	0	0
$823$	12573.6	0.532548	0.266274	−	0.963897i	$-0.414207\pi$
$823$	12573.6	0.532548	0.266274	+	0.963897i	$0.414207\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5285.33	−0.222236	−0.111118	−	0.993807i	$-0.535443\pi$
$827$	−5285.33	−0.222236	−0.111118	+	0.993807i	$0.535443\pi$
$828$	0	0
$829$	−28452.7	−1.19204	−0.596022	−	0.802968i	$-0.703253\pi$
$829$	−28452.7	−1.19204	−0.596022	+	0.802968i	$0.703253\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23548.7	0.979487
$834$	0	0
$835$	−3814.99	−0.158111
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32855.8	−1.35198	−0.675989	−	0.736912i	$-0.736283\pi$
$839$	−32855.8	−1.35198	−0.675989	+	0.736912i	$0.736283\pi$
$840$	0	0
$841$	−20467.7	−0.839217
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12713.9	−0.517599
$846$	0	0
$847$	36422.3	1.47755
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−71567.4	−2.88284
$852$	0	0
$853$	−22262.6	−0.893617	−0.446809	−	0.894630i	$-0.647440\pi$
$853$	−22262.6	−0.893617	−0.446809	+	0.894630i	$0.647440\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18649.5	−0.743354	−0.371677	−	0.928362i	$-0.621217\pi$
$857$	−18649.5	−0.743354	−0.371677	+	0.928362i	$0.621217\pi$
$858$	0	0
$859$	8204.35	0.325878	0.162939	−	0.986636i	$-0.447903\pi$
$859$	8204.35	0.325878	0.162939	+	0.986636i	$0.447903\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3545.82	0.139862	0.0699311	−	0.997552i	$-0.477722\pi$
$863$	3545.82	0.139862	0.0699311	+	0.997552i	$0.477722\pi$
$864$	0	0
$865$	−10797.0	−0.424404
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23987.4	−0.936383
$870$	0	0
$871$	−35043.3	−1.36326
$872$	0	0
$873$	0	0
$874$	0	0
$875$	16992.2	0.656506
$876$	0	0
$877$	−19963.5	−0.768666	−0.384333	−	0.923195i	$-0.625569\pi$
$877$	−19963.5	−0.768666	−0.384333	+	0.923195i	$0.625569\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13827.8	−0.528799	−0.264400	−	0.964413i	$-0.585174\pi$
$881$	−13827.8	−0.528799	−0.264400	+	0.964413i	$0.585174\pi$
$882$	0	0
$883$	44684.3	1.70300	0.851499	−	0.524356i	$-0.175694\pi$
$883$	44684.3	1.70300	0.851499	+	0.524356i	$0.175694\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30588.4	−1.15790	−0.578951	−	0.815362i	$-0.696538\pi$
$887$	−30588.4	−1.15790	−0.578951	+	0.815362i	$0.696538\pi$
$888$	0	0
$889$	3027.54	0.114219
$890$	0	0
$891$	0	0
$892$	0	0
$893$	753.020	0.0282182
$894$	0	0
$895$	−5362.56	−0.200280
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4075.48	−0.151196
$900$	0	0
$901$	31813.1	1.17630
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6820.63	−0.250525
$906$	0	0
$907$	33005.5	1.20830	0.604150	−	0.796870i	$-0.293513\pi$
$907$	33005.5	1.20830	0.604150	+	0.796870i	$0.293513\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34832.2	1.26679	0.633393	−	0.773830i	$-0.281662\pi$
$911$	34832.2	1.26679	0.633393	+	0.773830i	$0.281662\pi$
$912$	0	0
$913$	45482.7	1.64869
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−25211.9	−0.907927
$918$	0	0
$919$	113.784	0.00408420	0.00204210	−	0.999998i	$-0.499350\pi$
$919$	113.784	0.00408420	0.00204210	+	0.999998i	$0.499350\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−86426.0	−3.08206
$924$	0	0
$925$	45057.3	1.60160
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15492.6	0.547142	0.273571	−	0.961852i	$-0.411795\pi$
$929$	15492.6	0.547142	0.273571	+	0.961852i	$0.411795\pi$
$930$	0	0
$931$	8528.40	0.300222
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6655.99	0.232807
$936$	0	0
$937$	−20431.1	−0.712332	−0.356166	−	0.934423i	$-0.615916\pi$
$937$	−20431.1	−0.712332	−0.356166	+	0.934423i	$0.615916\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31926.0	−1.10601	−0.553005	−	0.833178i	$-0.686519\pi$
$941$	−31926.0	−1.10601	−0.553005	+	0.833178i	$0.686519\pi$
$942$	0	0
$943$	69682.5	2.40634
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14080.3	0.483154	0.241577	−	0.970382i	$-0.422335\pi$
$947$	14080.3	0.483154	0.241577	+	0.970382i	$0.422335\pi$
$948$	0	0
$949$	13093.6	0.447877
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38816.4	−1.31940	−0.659698	−	0.751530i	$-0.729316\pi$
$953$	−38816.4	−1.31940	−0.659698	+	0.751530i	$0.729316\pi$
$954$	0	0
$955$	8282.30	0.280638
$956$	0	0
$957$	0	0
$958$	0	0
$959$	30043.0	1.01162
$960$	0	0
$961$	−25555.3	−0.857820
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−7405.12	−0.247025
$966$	0	0
$967$	−43661.6	−1.45198	−0.725989	−	0.687706i	$-0.758618\pi$
$967$	−43661.6	−1.45198	−0.725989	+	0.687706i	$0.758618\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19910.9	−0.658055	−0.329028	−	0.944320i	$-0.606721\pi$
$971$	−19910.9	−0.658055	−0.329028	+	0.944320i	$0.606721\pi$
$972$	0	0
$973$	−56607.6	−1.86512
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8546.12	0.279851	0.139926	−	0.990162i	$-0.455314\pi$
$977$	8546.12	0.279851	0.139926	+	0.990162i	$0.455314\pi$
$978$	0	0
$979$	56940.7	1.85887
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8800.15	−0.285535	−0.142768	−	0.989756i	$-0.545600\pi$
$983$	−8800.15	−0.285535	−0.142768	+	0.989756i	$0.545600\pi$
$984$	0	0
$985$	−9736.64	−0.314960
$986$	0	0
$987$	0	0
$988$	0	0
$989$	89391.5	2.87410
$990$	0	0
$991$	−20253.1	−0.649205	−0.324603	−	0.945850i	$-0.605231\pi$
$991$	−20253.1	−0.649205	−0.324603	+	0.945850i	$0.605231\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2730.55	−0.0869992
$996$	0	0
$997$	13804.4	0.438504	0.219252	−	0.975668i	$-0.429638\pi$
$997$	13804.4	0.438504	0.219252	+	0.975668i	$0.429638\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.p.1.3	yes	8
3.2	odd	2		1368.4.a.o.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.4.a.o.1.6	✓	8	3.2	odd	2
1368.4.a.p.1.3	yes	8	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47611 q^{5} +28.1401 q^{7} -51.2379 q^{11} -85.6249 q^{13} +52.4629 q^{17} +19.0000 q^{19} +188.807 q^{23} -118.869 q^{25} +62.6206 q^{29} -65.0821 q^{31} -69.6778 q^{35} -379.051 q^{37} +369.068 q^{41} +473.455 q^{43} +39.6326 q^{47} +448.863 q^{49} +606.393 q^{53} +126.870 q^{55} -211.869 q^{59} -89.9190 q^{61} +212.016 q^{65} +409.265 q^{67} +1009.35 q^{71} -152.918 q^{73} -1441.84 q^{77} +468.158 q^{79} -887.677 q^{83} -129.904 q^{85} -1111.30 q^{89} -2409.49 q^{91} -47.0460 q^{95} +96.4810 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 4 q^{7} + 16 q^{11} - 36 q^{13} - 48 q^{17} + 152 q^{19} + 88 q^{23} + 124 q^{25} + 48 q^{29} - 4 q^{31} + 72 q^{35} + 4 q^{37} + 376 q^{41} + 276 q^{43} + 384 q^{47} + 780 q^{49} + 1528 q^{53}+ \cdots + 840 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 - 4 * q^7 + 16 * q^11 - 36 * q^13 - 48 * q^17 + 152 * q^19 + 88 * q^23 + 124 * q^25 + 48 * q^29 - 4 * q^31 + 72 * q^35 + 4 * q^37 + 376 * q^41 + 276 * q^43 + 384 * q^47 + 780 * q^49 + 1528 * q^53 - 604 * q^55 + 664 * q^59 + 68 * q^61 + 1576 * q^65 - 184 * q^67 + 1704 * q^71 + 628 * q^73 + 1344 * q^77 - 1380 * q^79 + 1480 * q^83 + 796 * q^85 + 1800 * q^89 - 1864 * q^91 + 304 * q^95 + 840 * q^97

Introduction

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