LMFDB - Embedded newform 9280.2.a.ca.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9280,2,Mod(1,9280)] 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("9280.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0,0,-4,0,0,0,8,0,0,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.1011730757$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -1.00000 q^{5} -2.44949 q^{7} -1.00000 q^{9} -1.41421 q^{11} +5.46410 q^{13} -1.41421 q^{15} +0.732051 q^{17} -2.44949 q^{19} -3.46410 q^{21} +6.31319 q^{23} +1.00000 q^{25} -5.65685 q^{27} +1.00000 q^{29} +5.27792 q^{31} -2.00000 q^{33} +2.44949 q^{35} -7.66025 q^{37} +7.72741 q^{39} -9.46410 q^{41} -3.48477 q^{43} +1.00000 q^{45} +9.14162 q^{47} -1.00000 q^{49} +1.03528 q^{51} -5.46410 q^{53} +1.41421 q^{55} -3.46410 q^{57} +8.76268 q^{59} +8.00000 q^{61} +2.44949 q^{63} -5.46410 q^{65} +3.48477 q^{67} +8.92820 q^{69} -15.1774 q^{71} -13.6603 q^{73} +1.41421 q^{75} +3.46410 q^{77} -0.656339 q^{79} -5.00000 q^{81} -10.9348 q^{83} -0.732051 q^{85} +1.41421 q^{87} +0.928203 q^{89} -13.3843 q^{91} +7.46410 q^{93} +2.44949 q^{95} +15.1244 q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{9} + 8 q^{13} - 4 q^{17} + 4 q^{25} + 4 q^{29} - 8 q^{33} + 4 q^{37} - 24 q^{41} + 4 q^{45} - 4 q^{49} - 8 q^{53} + 32 q^{61} - 8 q^{65} + 8 q^{69} - 20 q^{73} - 20 q^{81} + 4 q^{85}+ \cdots + 12 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^9 + 8 * q^13 - 4 * q^17 + 4 * q^25 + 4 * q^29 - 8 * q^33 + 4 * q^37 - 24 * q^41 + 4 * q^45 - 4 * q^49 - 8 * q^53 + 32 * q^61 - 8 * q^65 + 8 * q^69 - 20 * q^73 - 20 * q^81 + 4 * q^85 - 24 * q^89 + 16 * q^93 + 12 * 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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	0.732051	0.177548	0.0887742	−	0.996052i	$-0.471705\pi$
$17$	0.732051	0.177548	0.0887742	+	0.996052i	$0.471705\pi$
$18$	0	0
$19$	−2.44949	−0.561951	−0.280976	−	0.959715i	$-0.590658\pi$
$19$	−2.44949	−0.561951	−0.280976	+	0.959715i	$0.590658\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	6.31319	1.31639	0.658196	−	0.752847i	$-0.271320\pi$
$23$	6.31319	1.31639	0.658196	+	0.752847i	$0.271320\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.27792	0.947942	0.473971	−	0.880540i	$-0.342820\pi$
$31$	5.27792	0.947942	0.473971	+	0.880540i	$0.342820\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	2.44949	0.414039
$36$	0	0
$37$	−7.66025	−1.25934	−0.629669	−	0.776864i	$-0.716809\pi$
$37$	−7.66025	−1.25934	−0.629669	+	0.776864i	$0.716809\pi$
$38$	0	0
$39$	7.72741	1.23738
$40$	0	0
$41$	−9.46410	−1.47804	−0.739022	−	0.673681i	$-0.764712\pi$
$41$	−9.46410	−1.47804	−0.739022	+	0.673681i	$0.764712\pi$
$42$	0	0
$43$	−3.48477	−0.531422	−0.265711	−	0.964053i	$-0.585607\pi$
$43$	−3.48477	−0.531422	−0.265711	+	0.964053i	$0.585607\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	9.14162	1.33344	0.666721	−	0.745307i	$-0.267697\pi$
$47$	9.14162	1.33344	0.666721	+	0.745307i	$0.267697\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	1.03528	0.144968
$52$	0	0
$53$	−5.46410	−0.750552	−0.375276	−	0.926913i	$-0.622452\pi$
$53$	−5.46410	−0.750552	−0.375276	+	0.926913i	$0.622452\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	8.76268	1.14080	0.570402	−	0.821366i	$-0.306787\pi$
$59$	8.76268	1.14080	0.570402	+	0.821366i	$0.306787\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	2.44949	0.308607
$64$	0	0
$65$	−5.46410	−0.677738
$66$	0	0
$67$	3.48477	0.425732	0.212866	−	0.977081i	$-0.431720\pi$
$67$	3.48477	0.425732	0.212866	+	0.977081i	$0.431720\pi$
$68$	0	0
$69$	8.92820	1.07483
$70$	0	0
$71$	−15.1774	−1.80123	−0.900614	−	0.434620i	$-0.856883\pi$
$71$	−15.1774	−1.80123	−0.900614	+	0.434620i	$0.856883\pi$
$72$	0	0
$73$	−13.6603	−1.59881	−0.799406	−	0.600791i	$-0.794852\pi$
$73$	−13.6603	−1.59881	−0.799406	+	0.600791i	$0.794852\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	−0.656339	−0.0738439	−0.0369219	−	0.999318i	$-0.511755\pi$
$79$	−0.656339	−0.0738439	−0.0369219	+	0.999318i	$0.511755\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−10.9348	−1.20025	−0.600124	−	0.799907i	$-0.704882\pi$
$83$	−10.9348	−1.20025	−0.600124	+	0.799907i	$0.704882\pi$
$84$	0	0
$85$	−0.732051	−0.0794021
$86$	0	0
$87$	1.41421	0.151620
$88$	0	0
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	−13.3843	−1.40305
$92$	0	0
$93$	7.46410	0.773991
$94$	0	0
$95$	2.44949	0.251312
$96$	0	0
$97$	15.1244	1.53565	0.767823	−	0.640662i	$-0.221340\pi$
$97$	15.1244	1.53565	0.767823	+	0.640662i	$0.221340\pi$
$98$	0	0
$99$	1.41421	0.142134
$100$	0	0
$101$	7.46410	0.742706	0.371353	−	0.928492i	$-0.378894\pi$
$101$	7.46410	0.742706	0.371353	+	0.928492i	$0.378894\pi$
$102$	0	0
$103$	−12.7279	−1.25412	−0.627060	−	0.778971i	$-0.715742\pi$
$103$	−12.7279	−1.25412	−0.627060	+	0.778971i	$0.715742\pi$
$104$	0	0
$105$	3.46410	0.338062
$106$	0	0
$107$	6.59059	0.637137	0.318568	−	0.947900i	$-0.396798\pi$
$107$	6.59059	0.637137	0.318568	+	0.947900i	$0.396798\pi$
$108$	0	0
$109$	10.9282	1.04673	0.523366	−	0.852108i	$-0.324676\pi$
$109$	10.9282	1.04673	0.523366	+	0.852108i	$0.324676\pi$
$110$	0	0
$111$	−10.8332	−1.02825
$112$	0	0
$113$	−3.80385	−0.357836	−0.178918	−	0.983864i	$-0.557260\pi$
$113$	−3.80385	−0.357836	−0.178918	+	0.983864i	$0.557260\pi$
$114$	0	0
$115$	−6.31319	−0.588708
$116$	0	0
$117$	−5.46410	−0.505156
$118$	0	0
$119$	−1.79315	−0.164378
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	−13.3843	−1.20682
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.7985	1.31315	0.656576	−	0.754260i	$-0.272004\pi$
$127$	14.7985	1.31315	0.656576	+	0.754260i	$0.272004\pi$
$128$	0	0
$129$	−4.92820	−0.433904
$130$	0	0
$131$	7.34847	0.642039	0.321019	−	0.947073i	$-0.395975\pi$
$131$	7.34847	0.642039	0.321019	+	0.947073i	$0.395975\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	−13.1244	−1.12129	−0.560645	−	0.828056i	$-0.689447\pi$
$137$	−13.1244	−1.12129	−0.560645	+	0.828056i	$0.689447\pi$
$138$	0	0
$139$	2.07055	0.175622	0.0878110	−	0.996137i	$-0.472013\pi$
$139$	2.07055	0.175622	0.0878110	+	0.996137i	$0.472013\pi$
$140$	0	0
$141$	12.9282	1.08875
$142$	0	0
$143$	−7.72741	−0.646198
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	−1.41421	−0.116642
$148$	0	0
$149$	−0.535898	−0.0439025	−0.0219513	−	0.999759i	$-0.506988\pi$
$149$	−0.535898	−0.0439025	−0.0219513	+	0.999759i	$0.506988\pi$
$150$	0	0
$151$	−22.1469	−1.80229	−0.901146	−	0.433515i	$-0.857273\pi$
$151$	−22.1469	−1.80229	−0.901146	+	0.433515i	$0.857273\pi$
$152$	0	0
$153$	−0.732051	−0.0591828
$154$	0	0
$155$	−5.27792	−0.423932
$156$	0	0
$157$	−9.66025	−0.770972	−0.385486	−	0.922714i	$-0.625966\pi$
$157$	−9.66025	−0.770972	−0.385486	+	0.922714i	$0.625966\pi$
$158$	0	0
$159$	−7.72741	−0.612823
$160$	0	0
$161$	−15.4641	−1.21874
$162$	0	0
$163$	−4.24264	−0.332309	−0.166155	−	0.986100i	$-0.553135\pi$
$163$	−4.24264	−0.332309	−0.166155	+	0.986100i	$0.553135\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	0	0
$167$	−7.07107	−0.547176	−0.273588	−	0.961847i	$-0.588210\pi$
$167$	−7.07107	−0.547176	−0.273588	+	0.961847i	$0.588210\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	2.44949	0.187317
$172$	0	0
$173$	−21.8564	−1.66171	−0.830856	−	0.556488i	$-0.812149\pi$
$173$	−21.8564	−1.66171	−0.830856	+	0.556488i	$0.812149\pi$
$174$	0	0
$175$	−2.44949	−0.185164
$176$	0	0
$177$	12.3923	0.931463
$178$	0	0
$179$	23.9401	1.78937	0.894683	−	0.446701i	$-0.147401\pi$
$179$	23.9401	1.78937	0.894683	+	0.446701i	$0.147401\pi$
$180$	0	0
$181$	−13.4641	−1.00078	−0.500389	−	0.865800i	$-0.666810\pi$
$181$	−13.4641	−1.00078	−0.500389	+	0.865800i	$0.666810\pi$
$182$	0	0
$183$	11.3137	0.836333
$184$	0	0
$185$	7.66025	0.563193
$186$	0	0
$187$	−1.03528	−0.0757069
$188$	0	0
$189$	13.8564	1.00791
$190$	0	0
$191$	7.07107	0.511645	0.255822	−	0.966724i	$-0.417654\pi$
$191$	7.07107	0.511645	0.255822	+	0.966724i	$0.417654\pi$
$192$	0	0
$193$	−11.2679	−0.811085	−0.405542	−	0.914076i	$-0.632917\pi$
$193$	−11.2679	−0.811085	−0.405542	+	0.914076i	$0.632917\pi$
$194$	0	0
$195$	−7.72741	−0.553371
$196$	0	0
$197$	11.8564	0.844734	0.422367	−	0.906425i	$-0.361199\pi$
$197$	11.8564	0.844734	0.422367	+	0.906425i	$0.361199\pi$
$198$	0	0
$199$	23.4596	1.66301	0.831504	−	0.555518i	$-0.187480\pi$
$199$	23.4596	1.66301	0.831504	+	0.555518i	$0.187480\pi$
$200$	0	0
$201$	4.92820	0.347609
$202$	0	0
$203$	−2.44949	−0.171920
$204$	0	0
$205$	9.46410	0.661002
$206$	0	0
$207$	−6.31319	−0.438797
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	−8.38375	−0.577161	−0.288580	−	0.957456i	$-0.593183\pi$
$211$	−8.38375	−0.577161	−0.288580	+	0.957456i	$0.593183\pi$
$212$	0	0
$213$	−21.4641	−1.47070
$214$	0	0
$215$	3.48477	0.237659
$216$	0	0
$217$	−12.9282	−0.877624
$218$	0	0
$219$	−19.3185	−1.30542
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−19.6975	−1.31904	−0.659520	−	0.751687i	$-0.729240\pi$
$223$	−19.6975	−1.31904	−0.659520	+	0.751687i	$0.729240\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−14.0406	−0.931907	−0.465954	−	0.884809i	$-0.654289\pi$
$227$	−14.0406	−0.931907	−0.465954	+	0.884809i	$0.654289\pi$
$228$	0	0
$229$	−26.7846	−1.76998	−0.884988	−	0.465613i	$-0.845834\pi$
$229$	−26.7846	−1.76998	−0.884988	+	0.465613i	$0.845834\pi$
$230$	0	0
$231$	4.89898	0.322329
$232$	0	0
$233$	−15.4641	−1.01309	−0.506543	−	0.862214i	$-0.669077\pi$
$233$	−15.4641	−1.01309	−0.506543	+	0.862214i	$0.669077\pi$
$234$	0	0
$235$	−9.14162	−0.596334
$236$	0	0
$237$	−0.928203	−0.0602933
$238$	0	0
$239$	−27.8038	−1.79848	−0.899239	−	0.437457i	$-0.855879\pi$
$239$	−27.8038	−1.79848	−0.899239	+	0.437457i	$0.855879\pi$
$240$	0	0
$241$	0.928203	0.0597908	0.0298954	−	0.999553i	$-0.490483\pi$
$241$	0.928203	0.0597908	0.0298954	+	0.999553i	$0.490483\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−13.3843	−0.851620
$248$	0	0
$249$	−15.4641	−0.979998
$250$	0	0
$251$	5.00052	0.315630	0.157815	−	0.987469i	$-0.449555\pi$
$251$	5.00052	0.315630	0.157815	+	0.987469i	$0.449555\pi$
$252$	0	0
$253$	−8.92820	−0.561311
$254$	0	0
$255$	−1.03528	−0.0648315
$256$	0	0
$257$	−24.7846	−1.54602	−0.773011	−	0.634393i	$-0.781250\pi$
$257$	−24.7846	−1.54602	−0.773011	+	0.634393i	$0.781250\pi$
$258$	0	0
$259$	18.7637	1.16592
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−23.2838	−1.43574	−0.717869	−	0.696178i	$-0.754882\pi$
$263$	−23.2838	−1.43574	−0.717869	+	0.696178i	$0.754882\pi$
$264$	0	0
$265$	5.46410	0.335657
$266$	0	0
$267$	1.31268	0.0803346
$268$	0	0
$269$	−5.46410	−0.333152	−0.166576	−	0.986029i	$-0.553271\pi$
$269$	−5.46410	−0.333152	−0.166576	+	0.986029i	$0.553271\pi$
$270$	0	0
$271$	−0.101536	−0.00616787	−0.00308394	−	0.999995i	$-0.500982\pi$
$271$	−0.101536	−0.00616787	−0.00308394	+	0.999995i	$0.500982\pi$
$272$	0	0
$273$	−18.9282	−1.14559
$274$	0	0
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	19.3205	1.16086	0.580428	−	0.814311i	$-0.302885\pi$
$277$	19.3205	1.16086	0.580428	+	0.814311i	$0.302885\pi$
$278$	0	0
$279$	−5.27792	−0.315981
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	5.75839	0.342301	0.171150	−	0.985245i	$-0.445252\pi$
$283$	5.75839	0.342301	0.171150	+	0.985245i	$0.445252\pi$
$284$	0	0
$285$	3.46410	0.205196
$286$	0	0
$287$	23.1822	1.36840
$288$	0	0
$289$	−16.4641	−0.968477
$290$	0	0
$291$	21.3891	1.25385
$292$	0	0
$293$	−1.26795	−0.0740744	−0.0370372	−	0.999314i	$-0.511792\pi$
$293$	−1.26795	−0.0740744	−0.0370372	+	0.999314i	$0.511792\pi$
$294$	0	0
$295$	−8.76268	−0.510183
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	34.4959	1.99495
$300$	0	0
$301$	8.53590	0.492001
$302$	0	0
$303$	10.5558	0.606417
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−16.3142	−0.931102	−0.465551	−	0.885021i	$-0.654144\pi$
$307$	−16.3142	−0.931102	−0.465551	+	0.885021i	$0.654144\pi$
$308$	0	0
$309$	−18.0000	−1.02398
$310$	0	0
$311$	−4.52004	−0.256308	−0.128154	−	0.991754i	$-0.540905\pi$
$311$	−4.52004	−0.256308	−0.128154	+	0.991754i	$0.540905\pi$
$312$	0	0
$313$	12.3923	0.700454	0.350227	−	0.936665i	$-0.386104\pi$
$313$	12.3923	0.700454	0.350227	+	0.936665i	$0.386104\pi$
$314$	0	0
$315$	−2.44949	−0.138013
$316$	0	0
$317$	−10.5885	−0.594707	−0.297354	−	0.954767i	$-0.596104\pi$
$317$	−10.5885	−0.594707	−0.297354	+	0.954767i	$0.596104\pi$
$318$	0	0
$319$	−1.41421	−0.0791808
$320$	0	0
$321$	9.32051	0.520220
$322$	0	0
$323$	−1.79315	−0.0997736
$324$	0	0
$325$	5.46410	0.303094
$326$	0	0
$327$	15.4548	0.854653
$328$	0	0
$329$	−22.3923	−1.23453
$330$	0	0
$331$	−32.6012	−1.79193	−0.895963	−	0.444128i	$-0.853513\pi$
$331$	−32.6012	−1.79193	−0.895963	+	0.444128i	$0.853513\pi$
$332$	0	0
$333$	7.66025	0.419779
$334$	0	0
$335$	−3.48477	−0.190393
$336$	0	0
$337$	−4.58846	−0.249949	−0.124975	−	0.992160i	$-0.539885\pi$
$337$	−4.58846	−0.249949	−0.124975	+	0.992160i	$0.539885\pi$
$338$	0	0
$339$	−5.37945	−0.292172
$340$	0	0
$341$	−7.46410	−0.404204
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	−8.92820	−0.480678
$346$	0	0
$347$	−18.6622	−1.00184	−0.500919	−	0.865494i	$-0.667005\pi$
$347$	−18.6622	−1.00184	−0.500919	+	0.865494i	$0.667005\pi$
$348$	0	0
$349$	−19.8564	−1.06289	−0.531445	−	0.847093i	$-0.678351\pi$
$349$	−19.8564	−1.06289	−0.531445	+	0.847093i	$0.678351\pi$
$350$	0	0
$351$	−30.9096	−1.64983
$352$	0	0
$353$	−27.3205	−1.45412	−0.727062	−	0.686572i	$-0.759115\pi$
$353$	−27.3205	−1.45412	−0.727062	+	0.686572i	$0.759115\pi$
$354$	0	0
$355$	15.1774	0.805533
$356$	0	0
$357$	−2.53590	−0.134214
$358$	0	0
$359$	18.6622	0.984952	0.492476	−	0.870326i	$-0.336092\pi$
$359$	18.6622	0.984952	0.492476	+	0.870326i	$0.336092\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	−12.7279	−0.668043
$364$	0	0
$365$	13.6603	0.715010
$366$	0	0
$367$	−21.7680	−1.13628	−0.568140	−	0.822932i	$-0.692337\pi$
$367$	−21.7680	−1.13628	−0.568140	+	0.822932i	$0.692337\pi$
$368$	0	0
$369$	9.46410	0.492681
$370$	0	0
$371$	13.3843	0.694876
$372$	0	0
$373$	32.2487	1.66977	0.834887	−	0.550421i	$-0.185533\pi$
$373$	32.2487	1.66977	0.834887	+	0.550421i	$0.185533\pi$
$374$	0	0
$375$	−1.41421	−0.0730297
$376$	0	0
$377$	5.46410	0.281416
$378$	0	0
$379$	8.58682	0.441075	0.220538	−	0.975378i	$-0.429219\pi$
$379$	8.58682	0.441075	0.220538	+	0.975378i	$0.429219\pi$
$380$	0	0
$381$	20.9282	1.07218
$382$	0	0
$383$	−3.96524	−0.202614	−0.101307	−	0.994855i	$-0.532302\pi$
$383$	−3.96524	−0.202614	−0.101307	+	0.994855i	$0.532302\pi$
$384$	0	0
$385$	−3.46410	−0.176547
$386$	0	0
$387$	3.48477	0.177141
$388$	0	0
$389$	−6.53590	−0.331383	−0.165692	−	0.986178i	$-0.552986\pi$
$389$	−6.53590	−0.331383	−0.165692	+	0.986178i	$0.552986\pi$
$390$	0	0
$391$	4.62158	0.233723
$392$	0	0
$393$	10.3923	0.524222
$394$	0	0
$395$	0.656339	0.0330240
$396$	0	0
$397$	7.85641	0.394302	0.197151	−	0.980373i	$-0.436831\pi$
$397$	7.85641	0.394302	0.197151	+	0.980373i	$0.436831\pi$
$398$	0	0
$399$	8.48528	0.424795
$400$	0	0
$401$	−35.3205	−1.76382	−0.881911	−	0.471416i	$-0.843743\pi$
$401$	−35.3205	−1.76382	−0.881911	+	0.471416i	$0.843743\pi$
$402$	0	0
$403$	28.8391	1.43658
$404$	0	0
$405$	5.00000	0.248452
$406$	0	0
$407$	10.8332	0.536984
$408$	0	0
$409$	−20.7846	−1.02773	−0.513866	−	0.857870i	$-0.671787\pi$
$409$	−20.7846	−1.02773	−0.513866	+	0.857870i	$0.671787\pi$
$410$	0	0
$411$	−18.5606	−0.915529
$412$	0	0
$413$	−21.4641	−1.05618
$414$	0	0
$415$	10.9348	0.536767
$416$	0	0
$417$	2.92820	0.143395
$418$	0	0
$419$	1.23835	0.0604973	0.0302486	−	0.999542i	$-0.490370\pi$
$419$	1.23835	0.0604973	0.0302486	+	0.999542i	$0.490370\pi$
$420$	0	0
$421$	−12.3923	−0.603964	−0.301982	−	0.953314i	$-0.597648\pi$
$421$	−12.3923	−0.603964	−0.301982	+	0.953314i	$0.597648\pi$
$422$	0	0
$423$	−9.14162	−0.444481
$424$	0	0
$425$	0.732051	0.0355097
$426$	0	0
$427$	−19.5959	−0.948313
$428$	0	0
$429$	−10.9282	−0.527619
$430$	0	0
$431$	24.6980	1.18966	0.594830	−	0.803852i	$-0.297219\pi$
$431$	24.6980	1.18966	0.594830	+	0.803852i	$0.297219\pi$
$432$	0	0
$433$	−13.8038	−0.663371	−0.331685	−	0.943390i	$-0.607617\pi$
$433$	−13.8038	−0.663371	−0.331685	+	0.943390i	$0.607617\pi$
$434$	0	0
$435$	−1.41421	−0.0678064
$436$	0	0
$437$	−15.4641	−0.739748
$438$	0	0
$439$	11.5911	0.553213	0.276607	−	0.960983i	$-0.410790\pi$
$439$	11.5911	0.553213	0.276607	+	0.960983i	$0.410790\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	9.89949	0.470339	0.235170	−	0.971954i	$-0.424435\pi$
$443$	9.89949	0.470339	0.235170	+	0.971954i	$0.424435\pi$
$444$	0	0
$445$	−0.928203	−0.0440011
$446$	0	0
$447$	−0.757875	−0.0358462
$448$	0	0
$449$	−17.0718	−0.805668	−0.402834	−	0.915273i	$-0.631975\pi$
$449$	−17.0718	−0.805668	−0.402834	+	0.915273i	$0.631975\pi$
$450$	0	0
$451$	13.3843	0.630240
$452$	0	0
$453$	−31.3205	−1.47157
$454$	0	0
$455$	13.3843	0.627464
$456$	0	0
$457$	1.46410	0.0684878	0.0342439	−	0.999414i	$-0.489098\pi$
$457$	1.46410	0.0684878	0.0342439	+	0.999414i	$0.489098\pi$
$458$	0	0
$459$	−4.14110	−0.193290
$460$	0	0
$461$	−5.32051	−0.247801	−0.123900	−	0.992295i	$-0.539540\pi$
$461$	−5.32051	−0.247801	−0.123900	+	0.992295i	$0.539540\pi$
$462$	0	0
$463$	35.9101	1.66889	0.834443	−	0.551094i	$-0.185790\pi$
$463$	35.9101	1.66889	0.834443	+	0.551094i	$0.185790\pi$
$464$	0	0
$465$	−7.46410	−0.346139
$466$	0	0
$467$	5.00052	0.231396	0.115698	−	0.993284i	$-0.463089\pi$
$467$	5.00052	0.231396	0.115698	+	0.993284i	$0.463089\pi$
$468$	0	0
$469$	−8.53590	−0.394151
$470$	0	0
$471$	−13.6617	−0.629496
$472$	0	0
$473$	4.92820	0.226599
$474$	0	0
$475$	−2.44949	−0.112390
$476$	0	0
$477$	5.46410	0.250184
$478$	0	0
$479$	−4.79744	−0.219201	−0.109600	−	0.993976i	$-0.534957\pi$
$479$	−4.79744	−0.219201	−0.109600	+	0.993976i	$0.534957\pi$
$480$	0	0
$481$	−41.8564	−1.90849
$482$	0	0
$483$	−21.8695	−0.995099
$484$	0	0
$485$	−15.1244	−0.686762
$486$	0	0
$487$	−22.8033	−1.03332	−0.516658	−	0.856192i	$-0.672824\pi$
$487$	−22.8033	−1.03332	−0.516658	+	0.856192i	$0.672824\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	−15.2789	−0.689529	−0.344765	−	0.938689i	$-0.612041\pi$
$491$	−15.2789	−0.689529	−0.344765	+	0.938689i	$0.612041\pi$
$492$	0	0
$493$	0.732051	0.0329699
$494$	0	0
$495$	−1.41421	−0.0635642
$496$	0	0
$497$	37.1769	1.66761
$498$	0	0
$499$	−8.28221	−0.370763	−0.185381	−	0.982667i	$-0.559352\pi$
$499$	−8.28221	−0.370763	−0.185381	+	0.982667i	$0.559352\pi$
$500$	0	0
$501$	−10.0000	−0.446767
$502$	0	0
$503$	−36.6680	−1.63495	−0.817473	−	0.575967i	$-0.804626\pi$
$503$	−36.6680	−1.63495	−0.817473	+	0.575967i	$0.804626\pi$
$504$	0	0
$505$	−7.46410	−0.332148
$506$	0	0
$507$	23.8386	1.05871
$508$	0	0
$509$	−21.8564	−0.968768	−0.484384	−	0.874855i	$-0.660956\pi$
$509$	−21.8564	−0.968768	−0.484384	+	0.874855i	$0.660956\pi$
$510$	0	0
$511$	33.4607	1.48021
$512$	0	0
$513$	13.8564	0.611775
$514$	0	0
$515$	12.7279	0.560859
$516$	0	0
$517$	−12.9282	−0.568582
$518$	0	0
$519$	−30.9096	−1.35678
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	34.9492	1.52822	0.764111	−	0.645085i	$-0.223178\pi$
$523$	34.9492	1.52822	0.764111	+	0.645085i	$0.223178\pi$
$524$	0	0
$525$	−3.46410	−0.151186
$526$	0	0
$527$	3.86370	0.168306
$528$	0	0
$529$	16.8564	0.732887
$530$	0	0
$531$	−8.76268	−0.380268
$532$	0	0
$533$	−51.7128	−2.23993
$534$	0	0
$535$	−6.59059	−0.284936
$536$	0	0
$537$	33.8564	1.46101
$538$	0	0
$539$	1.41421	0.0609145
$540$	0	0
$541$	41.7128	1.79337	0.896687	−	0.442665i	$-0.145967\pi$
$541$	41.7128	1.79337	0.896687	+	0.442665i	$0.145967\pi$
$542$	0	0
$543$	−19.0411	−0.817132
$544$	0	0
$545$	−10.9282	−0.468113
$546$	0	0
$547$	30.5307	1.30540	0.652699	−	0.757617i	$-0.273637\pi$
$547$	30.5307	1.30540	0.652699	+	0.757617i	$0.273637\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−2.44949	−0.104352
$552$	0	0
$553$	1.60770	0.0683662
$554$	0	0
$555$	10.8332	0.459845
$556$	0	0
$557$	5.32051	0.225437	0.112719	−	0.993627i	$-0.464044\pi$
$557$	5.32051	0.225437	0.112719	+	0.993627i	$0.464044\pi$
$558$	0	0
$559$	−19.0411	−0.805353
$560$	0	0
$561$	−1.46410	−0.0618144
$562$	0	0
$563$	9.89949	0.417214	0.208607	−	0.978000i	$-0.433107\pi$
$563$	9.89949	0.417214	0.208607	+	0.978000i	$0.433107\pi$
$564$	0	0
$565$	3.80385	0.160029
$566$	0	0
$567$	12.2474	0.514344
$568$	0	0
$569$	40.1051	1.68129	0.840647	−	0.541583i	$-0.182175\pi$
$569$	40.1051	1.68129	0.840647	+	0.541583i	$0.182175\pi$
$570$	0	0
$571$	−11.8685	−0.496682	−0.248341	−	0.968673i	$-0.579885\pi$
$571$	−11.8685	−0.496682	−0.248341	+	0.968673i	$0.579885\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	0	0
$575$	6.31319	0.263278
$576$	0	0
$577$	−17.1244	−0.712896	−0.356448	−	0.934315i	$-0.616012\pi$
$577$	−17.1244	−0.712896	−0.356448	+	0.934315i	$0.616012\pi$
$578$	0	0
$579$	−15.9353	−0.662248
$580$	0	0
$581$	26.7846	1.11121
$582$	0	0
$583$	7.72741	0.320036
$584$	0	0
$585$	5.46410	0.225913
$586$	0	0
$587$	2.17209	0.0896517	0.0448258	−	0.998995i	$-0.485727\pi$
$587$	2.17209	0.0896517	0.0448258	+	0.998995i	$0.485727\pi$
$588$	0	0
$589$	−12.9282	−0.532697
$590$	0	0
$591$	16.7675	0.689722
$592$	0	0
$593$	22.9282	0.941548	0.470774	−	0.882254i	$-0.343975\pi$
$593$	22.9282	0.941548	0.470774	+	0.882254i	$0.343975\pi$
$594$	0	0
$595$	1.79315	0.0735120
$596$	0	0
$597$	33.1769	1.35784
$598$	0	0
$599$	33.0817	1.35168	0.675841	−	0.737047i	$-0.263781\pi$
$599$	33.0817	1.35168	0.675841	+	0.737047i	$0.263781\pi$
$600$	0	0
$601$	−10.5359	−0.429768	−0.214884	−	0.976640i	$-0.568937\pi$
$601$	−10.5359	−0.429768	−0.214884	+	0.976640i	$0.568937\pi$
$602$	0	0
$603$	−3.48477	−0.141911
$604$	0	0
$605$	9.00000	0.365902
$606$	0	0
$607$	33.2848	1.35099	0.675494	−	0.737366i	$-0.263930\pi$
$607$	33.2848	1.35099	0.675494	+	0.737366i	$0.263930\pi$
$608$	0	0
$609$	−3.46410	−0.140372
$610$	0	0
$611$	49.9507	2.02079
$612$	0	0
$613$	28.9282	1.16840	0.584200	−	0.811610i	$-0.301409\pi$
$613$	28.9282	1.16840	0.584200	+	0.811610i	$0.301409\pi$
$614$	0	0
$615$	13.3843	0.539705
$616$	0	0
$617$	43.1244	1.73612	0.868061	−	0.496458i	$-0.165366\pi$
$617$	43.1244	1.73612	0.868061	+	0.496458i	$0.165366\pi$
$618$	0	0
$619$	25.9091	1.04138	0.520688	−	0.853747i	$-0.325676\pi$
$619$	25.9091	1.04138	0.520688	+	0.853747i	$0.325676\pi$
$620$	0	0
$621$	−35.7128	−1.43311
$622$	0	0
$623$	−2.27362	−0.0910908
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.89898	0.195646
$628$	0	0
$629$	−5.60770	−0.223593
$630$	0	0
$631$	5.65685	0.225196	0.112598	−	0.993641i	$-0.464083\pi$
$631$	5.65685	0.225196	0.112598	+	0.993641i	$0.464083\pi$
$632$	0	0
$633$	−11.8564	−0.471250
$634$	0	0
$635$	−14.7985	−0.587260
$636$	0	0
$637$	−5.46410	−0.216496
$638$	0	0
$639$	15.1774	0.600409
$640$	0	0
$641$	−5.46410	−0.215819	−0.107910	−	0.994161i	$-0.534416\pi$
$641$	−5.46410	−0.215819	−0.107910	+	0.994161i	$0.534416\pi$
$642$	0	0
$643$	33.6365	1.32649	0.663247	−	0.748400i	$-0.269178\pi$
$643$	33.6365	1.32649	0.663247	+	0.748400i	$0.269178\pi$
$644$	0	0
$645$	4.92820	0.194048
$646$	0	0
$647$	−0.175865	−0.00691398	−0.00345699	−	0.999994i	$-0.501100\pi$
$647$	−0.175865	−0.00691398	−0.00345699	+	0.999994i	$0.501100\pi$
$648$	0	0
$649$	−12.3923	−0.486441
$650$	0	0
$651$	−18.2832	−0.716577
$652$	0	0
$653$	−18.3397	−0.717690	−0.358845	−	0.933397i	$-0.616829\pi$
$653$	−18.3397	−0.717690	−0.358845	+	0.933397i	$0.616829\pi$
$654$	0	0
$655$	−7.34847	−0.287128
$656$	0	0
$657$	13.6603	0.532937
$658$	0	0
$659$	−10.1769	−0.396436	−0.198218	−	0.980158i	$-0.563515\pi$
$659$	−10.1769	−0.396436	−0.198218	+	0.980158i	$0.563515\pi$
$660$	0	0
$661$	26.9282	1.04739	0.523693	−	0.851907i	$-0.324554\pi$
$661$	26.9282	1.04739	0.523693	+	0.851907i	$0.324554\pi$
$662$	0	0
$663$	5.65685	0.219694
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	6.31319	0.244448
$668$	0	0
$669$	−27.8564	−1.07699
$670$	0	0
$671$	−11.3137	−0.436761
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	−13.2679	−0.509929	−0.254964	−	0.966950i	$-0.582064\pi$
$677$	−13.2679	−0.509929	−0.254964	+	0.966950i	$0.582064\pi$
$678$	0	0
$679$	−37.0470	−1.42173
$680$	0	0
$681$	−19.8564	−0.760899
$682$	0	0
$683$	−5.75839	−0.220339	−0.110169	−	0.993913i	$-0.535139\pi$
$683$	−5.75839	−0.220339	−0.110169	+	0.993913i	$0.535139\pi$
$684$	0	0
$685$	13.1244	0.501456
$686$	0	0
$687$	−37.8792	−1.44518
$688$	0	0
$689$	−29.8564	−1.13744
$690$	0	0
$691$	−24.2175	−0.921277	−0.460638	−	0.887588i	$-0.652380\pi$
$691$	−24.2175	−0.921277	−0.460638	+	0.887588i	$0.652380\pi$
$692$	0	0
$693$	−3.46410	−0.131590
$694$	0	0
$695$	−2.07055	−0.0785405
$696$	0	0
$697$	−6.92820	−0.262424
$698$	0	0
$699$	−21.8695	−0.827182
$700$	0	0
$701$	18.9282	0.714908	0.357454	−	0.933931i	$-0.383645\pi$
$701$	18.9282	0.714908	0.357454	+	0.933931i	$0.383645\pi$
$702$	0	0
$703$	18.7637	0.707687
$704$	0	0
$705$	−12.9282	−0.486904
$706$	0	0
$707$	−18.2832	−0.687612
$708$	0	0
$709$	30.6410	1.15075	0.575374	−	0.817891i	$-0.304857\pi$
$709$	30.6410	1.15075	0.575374	+	0.817891i	$0.304857\pi$
$710$	0	0
$711$	0.656339	0.0246146
$712$	0	0
$713$	33.3205	1.24786
$714$	0	0
$715$	7.72741	0.288989
$716$	0	0
$717$	−39.3205	−1.46845
$718$	0	0
$719$	44.0165	1.64154	0.820769	−	0.571260i	$-0.193545\pi$
$719$	44.0165	1.64154	0.820769	+	0.571260i	$0.193545\pi$
$720$	0	0
$721$	31.1769	1.16109
$722$	0	0
$723$	1.31268	0.0488190
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	43.4345	1.61090	0.805448	−	0.592667i	$-0.201925\pi$
$727$	43.4345	1.61090	0.805448	+	0.592667i	$0.201925\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−2.55103	−0.0943531
$732$	0	0
$733$	34.9808	1.29204	0.646022	−	0.763319i	$-0.276431\pi$
$733$	34.9808	1.29204	0.646022	+	0.763319i	$0.276431\pi$
$734$	0	0
$735$	1.41421	0.0521641
$736$	0	0
$737$	−4.92820	−0.181533
$738$	0	0
$739$	43.3601	1.59503	0.797514	−	0.603300i	$-0.206148\pi$
$739$	43.3601	1.59503	0.797514	+	0.603300i	$0.206148\pi$
$740$	0	0
$741$	−18.9282	−0.695345
$742$	0	0
$743$	−46.2629	−1.69722	−0.848611	−	0.529018i	$-0.822560\pi$
$743$	−46.2629	−1.69722	−0.848611	+	0.529018i	$0.822560\pi$
$744$	0	0
$745$	0.535898	0.0196338
$746$	0	0
$747$	10.9348	0.400082
$748$	0	0
$749$	−16.1436	−0.589874
$750$	0	0
$751$	−46.1886	−1.68545	−0.842723	−	0.538348i	$-0.819049\pi$
$751$	−46.1886	−1.68545	−0.842723	+	0.538348i	$0.819049\pi$
$752$	0	0
$753$	7.07180	0.257711
$754$	0	0
$755$	22.1469	0.806010
$756$	0	0
$757$	−6.73205	−0.244681	−0.122340	−	0.992488i	$-0.539040\pi$
$757$	−6.73205	−0.244681	−0.122340	+	0.992488i	$0.539040\pi$
$758$	0	0
$759$	−12.6264	−0.458309
$760$	0	0
$761$	44.6410	1.61824	0.809118	−	0.587647i	$-0.199945\pi$
$761$	44.6410	1.61824	0.809118	+	0.587647i	$0.199945\pi$
$762$	0	0
$763$	−26.7685	−0.969086
$764$	0	0
$765$	0.732051	0.0264674
$766$	0	0
$767$	47.8802	1.72885
$768$	0	0
$769$	−14.3923	−0.519000	−0.259500	−	0.965743i	$-0.583558\pi$
$769$	−14.3923	−0.519000	−0.259500	+	0.965743i	$0.583558\pi$
$770$	0	0
$771$	−35.0507	−1.26232
$772$	0	0
$773$	−17.4115	−0.626250	−0.313125	−	0.949712i	$-0.601376\pi$
$773$	−17.4115	−0.626250	−0.313125	+	0.949712i	$0.601376\pi$
$774$	0	0
$775$	5.27792	0.189588
$776$	0	0
$777$	26.5359	0.951970
$778$	0	0
$779$	23.1822	0.830589
$780$	0	0
$781$	21.4641	0.768046
$782$	0	0
$783$	−5.65685	−0.202159
$784$	0	0
$785$	9.66025	0.344789
$786$	0	0
$787$	−50.8845	−1.81384	−0.906918	−	0.421307i	$-0.861571\pi$
$787$	−50.8845	−1.81384	−0.906918	+	0.421307i	$0.861571\pi$
$788$	0	0
$789$	−32.9282	−1.17228
$790$	0	0
$791$	9.31749	0.331292
$792$	0	0
$793$	43.7128	1.55229
$794$	0	0
$795$	7.72741	0.274063
$796$	0	0
$797$	−7.51666	−0.266254	−0.133127	−	0.991099i	$-0.542502\pi$
$797$	−7.51666	−0.266254	−0.133127	+	0.991099i	$0.542502\pi$
$798$	0	0
$799$	6.69213	0.236751
$800$	0	0
$801$	−0.928203	−0.0327964
$802$	0	0
$803$	19.3185	0.681736
$804$	0	0
$805$	15.4641	0.545038
$806$	0	0
$807$	−7.72741	−0.272018
$808$	0	0
$809$	−27.8564	−0.979379	−0.489690	−	0.871897i	$-0.662890\pi$
$809$	−27.8564	−0.979379	−0.489690	+	0.871897i	$0.662890\pi$
$810$	0	0
$811$	11.1106	0.390147	0.195074	−	0.980789i	$-0.437505\pi$
$811$	11.1106	0.390147	0.195074	+	0.980789i	$0.437505\pi$
$812$	0	0
$813$	−0.143594	−0.00503605
$814$	0	0
$815$	4.24264	0.148613
$816$	0	0
$817$	8.53590	0.298633
$818$	0	0
$819$	13.3843	0.467684
$820$	0	0
$821$	−36.2487	−1.26509	−0.632544	−	0.774524i	$-0.717989\pi$
$821$	−36.2487	−1.26509	−0.632544	+	0.774524i	$0.717989\pi$
$822$	0	0
$823$	27.4249	0.955971	0.477985	−	0.878368i	$-0.341367\pi$
$823$	27.4249	0.955971	0.477985	+	0.878368i	$0.341367\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	42.1218	1.46472	0.732359	−	0.680918i	$-0.238419\pi$
$827$	42.1218	1.46472	0.732359	+	0.680918i	$0.238419\pi$
$828$	0	0
$829$	15.6077	0.542078	0.271039	−	0.962568i	$-0.412633\pi$
$829$	15.6077	0.542078	0.271039	+	0.962568i	$0.412633\pi$
$830$	0	0
$831$	27.3233	0.947836
$832$	0	0
$833$	−0.732051	−0.0253641
$834$	0	0
$835$	7.07107	0.244704
$836$	0	0
$837$	−29.8564	−1.03199
$838$	0	0
$839$	3.68784	0.127318	0.0636592	−	0.997972i	$-0.479723\pi$
$839$	3.68784	0.127318	0.0636592	+	0.997972i	$0.479723\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.9706	−0.584497
$844$	0	0
$845$	−16.8564	−0.579878
$846$	0	0
$847$	22.0454	0.757489
$848$	0	0
$849$	8.14359	0.279487
$850$	0	0
$851$	−48.3607	−1.65778
$852$	0	0
$853$	36.0526	1.23442	0.617208	−	0.786800i	$-0.288264\pi$
$853$	36.0526	1.23442	0.617208	+	0.786800i	$0.288264\pi$
$854$	0	0
$855$	−2.44949	−0.0837708
$856$	0	0
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	0	0
$859$	−7.07107	−0.241262	−0.120631	−	0.992697i	$-0.538492\pi$
$859$	−7.07107	−0.241262	−0.120631	+	0.992697i	$0.538492\pi$
$860$	0	0
$861$	32.7846	1.11730
$862$	0	0
$863$	37.1485	1.26455	0.632275	−	0.774744i	$-0.282121\pi$
$863$	37.1485	1.26455	0.632275	+	0.774744i	$0.282121\pi$
$864$	0	0
$865$	21.8564	0.743140
$866$	0	0
$867$	−23.2838	−0.790758
$868$	0	0
$869$	0.928203	0.0314871
$870$	0	0
$871$	19.0411	0.645184
$872$	0	0
$873$	−15.1244	−0.511882
$874$	0	0
$875$	2.44949	0.0828079
$876$	0	0
$877$	−14.5359	−0.490842	−0.245421	−	0.969417i	$-0.578926\pi$
$877$	−14.5359	−0.490842	−0.245421	+	0.969417i	$0.578926\pi$
$878$	0	0
$879$	−1.79315	−0.0604815
$880$	0	0
$881$	−46.6410	−1.57138	−0.785688	−	0.618623i	$-0.787691\pi$
$881$	−46.6410	−1.57138	−0.785688	+	0.618623i	$0.787691\pi$
$882$	0	0
$883$	−3.76217	−0.126607	−0.0633035	−	0.997994i	$-0.520164\pi$
$883$	−3.76217	−0.126607	−0.0633035	+	0.997994i	$0.520164\pi$
$884$	0	0
$885$	−12.3923	−0.416563
$886$	0	0
$887$	−26.1122	−0.876761	−0.438381	−	0.898789i	$-0.644448\pi$
$887$	−26.1122	−0.876761	−0.438381	+	0.898789i	$0.644448\pi$
$888$	0	0
$889$	−36.2487	−1.21574
$890$	0	0
$891$	7.07107	0.236890
$892$	0	0
$893$	−22.3923	−0.749330
$894$	0	0
$895$	−23.9401	−0.800229
$896$	0	0
$897$	48.7846	1.62887
$898$	0	0
$899$	5.27792	0.176028
$900$	0	0
$901$	−4.00000	−0.133259
$902$	0	0
$903$	12.0716	0.401717
$904$	0	0
$905$	13.4641	0.447562
$906$	0	0
$907$	52.8807	1.75587	0.877937	−	0.478775i	$-0.158919\pi$
$907$	52.8807	1.75587	0.877937	+	0.478775i	$0.158919\pi$
$908$	0	0
$909$	−7.46410	−0.247569
$910$	0	0
$911$	38.8129	1.28593	0.642964	−	0.765896i	$-0.277704\pi$
$911$	38.8129	1.28593	0.642964	+	0.765896i	$0.277704\pi$
$912$	0	0
$913$	15.4641	0.511787
$914$	0	0
$915$	−11.3137	−0.374020
$916$	0	0
$917$	−18.0000	−0.594412
$918$	0	0
$919$	43.2586	1.42697	0.713485	−	0.700671i	$-0.247116\pi$
$919$	43.2586	1.42697	0.713485	+	0.700671i	$0.247116\pi$
$920$	0	0
$921$	−23.0718	−0.760242
$922$	0	0
$923$	−82.9309	−2.72970
$924$	0	0
$925$	−7.66025	−0.251868
$926$	0	0
$927$	12.7279	0.418040
$928$	0	0
$929$	−10.3923	−0.340960	−0.170480	−	0.985361i	$-0.554532\pi$
$929$	−10.3923	−0.340960	−0.170480	+	0.985361i	$0.554532\pi$
$930$	0	0
$931$	2.44949	0.0802788
$932$	0	0
$933$	−6.39230	−0.209275
$934$	0	0
$935$	1.03528	0.0338572
$936$	0	0
$937$	33.0718	1.08041	0.540204	−	0.841534i	$-0.318347\pi$
$937$	33.0718	1.08041	0.540204	+	0.841534i	$0.318347\pi$
$938$	0	0
$939$	17.5254	0.571919
$940$	0	0
$941$	−19.8564	−0.647300	−0.323650	−	0.946177i	$-0.604910\pi$
$941$	−19.8564	−0.647300	−0.323650	+	0.946177i	$0.604910\pi$
$942$	0	0
$943$	−59.7487	−1.94569
$944$	0	0
$945$	−13.8564	−0.450749
$946$	0	0
$947$	2.37516	0.0771823	0.0385912	−	0.999255i	$-0.487713\pi$
$947$	2.37516	0.0771823	0.0385912	+	0.999255i	$0.487713\pi$
$948$	0	0
$949$	−74.6410	−2.42295
$950$	0	0
$951$	−14.9743	−0.485576
$952$	0	0
$953$	−23.8564	−0.772785	−0.386392	−	0.922334i	$-0.626279\pi$
$953$	−23.8564	−0.772785	−0.386392	+	0.922334i	$0.626279\pi$
$954$	0	0
$955$	−7.07107	−0.228814
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	0	0
$959$	32.1480	1.03811
$960$	0	0
$961$	−3.14359	−0.101406
$962$	0	0
$963$	−6.59059	−0.212379
$964$	0	0
$965$	11.2679	0.362728
$966$	0	0
$967$	38.5355	1.23922	0.619609	−	0.784911i	$-0.287291\pi$
$967$	38.5355	1.23922	0.619609	+	0.784911i	$0.287291\pi$
$968$	0	0
$969$	−2.53590	−0.0814648
$970$	0	0
$971$	20.7327	0.665345	0.332672	−	0.943042i	$-0.392050\pi$
$971$	20.7327	0.665345	0.332672	+	0.943042i	$0.392050\pi$
$972$	0	0
$973$	−5.07180	−0.162594
$974$	0	0
$975$	7.72741	0.247475
$976$	0	0
$977$	20.0000	0.639857	0.319928	−	0.947442i	$-0.396341\pi$
$977$	20.0000	0.639857	0.319928	+	0.947442i	$0.396341\pi$
$978$	0	0
$979$	−1.31268	−0.0419534
$980$	0	0
$981$	−10.9282	−0.348911
$982$	0	0
$983$	−11.2122	−0.357613	−0.178806	−	0.983884i	$-0.557224\pi$
$983$	−11.2122	−0.357613	−0.178806	+	0.983884i	$0.557224\pi$
$984$	0	0
$985$	−11.8564	−0.377777
$986$	0	0
$987$	−31.6675	−1.00799
$988$	0	0
$989$	−22.0000	−0.699559
$990$	0	0
$991$	−34.0155	−1.08054	−0.540268	−	0.841493i	$-0.681677\pi$
$991$	−34.0155	−1.08054	−0.540268	+	0.841493i	$0.681677\pi$
$992$	0	0
$993$	−46.1051	−1.46310
$994$	0	0
$995$	−23.4596	−0.743720
$996$	0	0
$997$	−20.9808	−0.664467	−0.332234	−	0.943197i	$-0.607802\pi$
$997$	−20.9808	−0.664467	−0.332234	+	0.943197i	$0.607802\pi$
$998$	0	0
$999$	43.3329	1.37099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.ca.1.3		4
4.3	odd	2	inner	9280.2.a.ca.1.2		4
8.3	odd	2		4640.2.a.q.1.4	yes	4
8.5	even	2		4640.2.a.q.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.q.1.1	✓	4	8.5	even	2
4640.2.a.q.1.4	yes	4	8.3	odd	2
9280.2.a.ca.1.2		4	4.3	odd	2	inner
9280.2.a.ca.1.3		4	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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -1.00000 q^{5} -2.44949 q^{7} -1.00000 q^{9} -1.41421 q^{11} +5.46410 q^{13} -1.41421 q^{15} +0.732051 q^{17} -2.44949 q^{19} -3.46410 q^{21} +6.31319 q^{23} +1.00000 q^{25} -5.65685 q^{27} +1.00000 q^{29} +5.27792 q^{31} -2.00000 q^{33} +2.44949 q^{35} -7.66025 q^{37} +7.72741 q^{39} -9.46410 q^{41} -3.48477 q^{43} +1.00000 q^{45} +9.14162 q^{47} -1.00000 q^{49} +1.03528 q^{51} -5.46410 q^{53} +1.41421 q^{55} -3.46410 q^{57} +8.76268 q^{59} +8.00000 q^{61} +2.44949 q^{63} -5.46410 q^{65} +3.48477 q^{67} +8.92820 q^{69} -15.1774 q^{71} -13.6603 q^{73} +1.41421 q^{75} +3.46410 q^{77} -0.656339 q^{79} -5.00000 q^{81} -10.9348 q^{83} -0.732051 q^{85} +1.41421 q^{87} +0.928203 q^{89} -13.3843 q^{91} +7.46410 q^{93} +2.44949 q^{95} +15.1244 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{9} + 8 q^{13} - 4 q^{17} + 4 q^{25} + 4 q^{29} - 8 q^{33} + 4 q^{37} - 24 q^{41} + 4 q^{45} - 4 q^{49} - 8 q^{53} + 32 q^{61} - 8 q^{65} + 8 q^{69} - 20 q^{73} - 20 q^{81} + 4 q^{85}+ \cdots + 12 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^9 + 8 * q^13 - 4 * q^17 + 4 * q^25 + 4 * q^29 - 8 * q^33 + 4 * q^37 - 24 * q^41 + 4 * q^45 - 4 * q^49 - 8 * q^53 + 32 * q^61 - 8 * q^65 + 8 * q^69 - 20 * q^73 - 20 * q^81 + 4 * q^85 - 24 * q^89 + 16 * q^93 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more