LMFDB - Embedded newform 9072.2.a.bu.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9072,2,Mod(1,9072)] 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("9072.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,-3,0,0,0,6,0,3,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.74657 q^{5} -1.00000 q^{7} -0.746568 q^{11} -6.03677 q^{13} +0.543637 q^{17} +1.20293 q^{19} -7.49314 q^{23} +9.03677 q^{25} -8.03677 q^{29} -2.00000 q^{31} +3.74657 q^{35} +5.00000 q^{37} +2.79707 q^{41} -9.83384 q^{43} -4.29021 q^{47} +1.00000 q^{49} -2.45636 q^{53} +2.79707 q^{55} -14.5804 q^{59} -9.23970 q^{61} +22.6172 q^{65} -10.2397 q^{67} -8.23970 q^{71} +10.0368 q^{73} +0.746568 q^{77} -10.2397 q^{79} +2.79707 q^{83} -2.03677 q^{85} -6.54364 q^{89} +6.03677 q^{91} -4.50686 q^{95} +5.20293 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{23} + 6 q^{25} - 3 q^{29} - 6 q^{31} + 3 q^{35} + 15 q^{37} + 12 q^{41} - 12 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 3 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 + 6 * q^11 + 3 * q^13 - 3 * q^17 - 6 * q^23 + 6 * q^25 - 3 * q^29 - 6 * q^31 + 3 * q^35 + 15 * q^37 + 12 * q^41 - 12 * q^43 + 3 * q^49 - 12 * q^53 + 12 * q^55 - 18 * q^59 - 3 * q^61 + 21 * q^65 - 6 * q^67 + 9 * q^73 - 6 * q^77 - 6 * q^79 + 12 * q^83 + 15 * q^85 - 15 * q^89 - 3 * q^91 - 30 * q^95 + 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.74657	−1.67552	−0.837758	−	0.546041i	$-0.816134\pi$
$5$	−3.74657	−1.67552	−0.837758	+	0.546041i	$0.816134\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.746568	−0.225099	−0.112549	−	0.993646i	$-0.535902\pi$
$11$	−0.746568	−0.225099	−0.112549	+	0.993646i	$0.535902\pi$
$12$	0	0
$13$	−6.03677	−1.67430	−0.837150	−	0.546974i	$-0.815780\pi$
$13$	−6.03677	−1.67430	−0.837150	+	0.546974i	$0.815780\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.543637	0.131851	0.0659257	−	0.997825i	$-0.479000\pi$
$17$	0.543637	0.131851	0.0659257	+	0.997825i	$0.479000\pi$
$18$	0	0
$19$	1.20293	0.275971	0.137986	−	0.990434i	$-0.455937\pi$
$19$	1.20293	0.275971	0.137986	+	0.990434i	$0.455937\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.49314	−1.56243	−0.781213	−	0.624264i	$-0.785399\pi$
$23$	−7.49314	−1.56243	−0.781213	+	0.624264i	$0.785399\pi$
$24$	0	0
$25$	9.03677	1.80735
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.03677	−1.49239	−0.746196	−	0.665727i	$-0.768122\pi$
$29$	−8.03677	−1.49239	−0.746196	+	0.665727i	$0.768122\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.74657	0.633286
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.79707	0.436829	0.218414	−	0.975856i	$-0.429912\pi$
$41$	2.79707	0.436829	0.218414	+	0.975856i	$0.429912\pi$
$42$	0	0
$43$	−9.83384	−1.49965	−0.749823	−	0.661638i	$-0.769862\pi$
$43$	−9.83384	−1.49965	−0.749823	+	0.661638i	$0.769862\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.29021	−0.625791	−0.312895	−	0.949788i	$-0.601299\pi$
$47$	−4.29021	−0.625791	−0.312895	+	0.949788i	$0.601299\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.45636	−0.337407	−0.168704	−	0.985667i	$-0.553958\pi$
$53$	−2.45636	−0.337407	−0.168704	+	0.985667i	$0.553958\pi$
$54$	0	0
$55$	2.79707	0.377157
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.5804	−1.89821	−0.949104	−	0.314963i	$-0.898008\pi$
$59$	−14.5804	−1.89821	−0.949104	+	0.314963i	$0.898008\pi$
$60$	0	0
$61$	−9.23970	−1.18302	−0.591511	−	0.806297i	$-0.701469\pi$
$61$	−9.23970	−1.18302	−0.591511	+	0.806297i	$0.701469\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	22.6172	2.80532
$66$	0	0
$67$	−10.2397	−1.25098	−0.625490	−	0.780233i	$-0.715101\pi$
$67$	−10.2397	−1.25098	−0.625490	+	0.780233i	$0.715101\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.23970	−0.977873	−0.488937	−	0.872319i	$-0.662615\pi$
$71$	−8.23970	−0.977873	−0.488937	+	0.872319i	$0.662615\pi$
$72$	0	0
$73$	10.0368	1.17472	0.587358	−	0.809327i	$-0.300168\pi$
$73$	10.0368	1.17472	0.587358	+	0.809327i	$0.300168\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.746568	0.0850793
$78$	0	0
$79$	−10.2397	−1.15206	−0.576028	−	0.817430i	$-0.695398\pi$
$79$	−10.2397	−1.15206	−0.576028	+	0.817430i	$0.695398\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.79707	0.307018	0.153509	−	0.988147i	$-0.450943\pi$
$83$	2.79707	0.307018	0.153509	+	0.988147i	$0.450943\pi$
$84$	0	0
$85$	−2.03677	−0.220919
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.54364	−0.693624	−0.346812	−	0.937935i	$-0.612736\pi$
$89$	−6.54364	−0.693624	−0.346812	+	0.937935i	$0.612736\pi$
$90$	0	0
$91$	6.03677	0.632826
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.50686	−0.462394
$96$	0	0
$97$	5.20293	0.528278	0.264139	−	0.964485i	$-0.414912\pi$
$97$	5.20293	0.528278	0.264139	+	0.964485i	$0.414912\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.50686	0.448450	0.224225	−	0.974537i	$-0.428015\pi$
$101$	4.50686	0.448450	0.224225	+	0.974537i	$0.428015\pi$
$102$	0	0
$103$	10.8706	1.07111	0.535557	−	0.844499i	$-0.320102\pi$
$103$	10.8706	1.07111	0.535557	+	0.844499i	$0.320102\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.32698	−0.321631	−0.160816	−	0.986984i	$-0.551412\pi$
$107$	−3.32698	−0.321631	−0.160816	+	0.986984i	$0.551412\pi$
$108$	0	0
$109$	−5.63091	−0.539343	−0.269672	−	0.962952i	$-0.586915\pi$
$109$	−5.63091	−0.539343	−0.269672	+	0.962952i	$0.586915\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.59414	−0.244036	−0.122018	−	0.992528i	$-0.538937\pi$
$113$	−2.59414	−0.244036	−0.122018	+	0.992528i	$0.538937\pi$
$114$	0	0
$115$	28.0735	2.61787
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.543637	−0.0498351
$120$	0	0
$121$	−10.4426	−0.949331
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15.1240	−1.35274
$126$	0	0
$127$	−9.83384	−0.872612	−0.436306	−	0.899798i	$-0.643714\pi$
$127$	−9.83384	−0.872612	−0.436306	+	0.899798i	$0.643714\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.37748	0.469833	0.234916	−	0.972016i	$-0.424518\pi$
$131$	5.37748	0.469833	0.234916	+	0.972016i	$0.424518\pi$
$132$	0	0
$133$	−1.20293	−0.104307
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.4931	−0.896489	−0.448245	−	0.893911i	$-0.647951\pi$
$137$	−10.4931	−0.896489	−0.448245	+	0.893911i	$0.647951\pi$
$138$	0	0
$139$	6.79707	0.576520	0.288260	−	0.957552i	$-0.406923\pi$
$139$	6.79707	0.576520	0.288260	+	0.957552i	$0.406923\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.50686	0.376883
$144$	0	0
$145$	30.1103	2.50053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.50686	−0.123447	−0.0617235	−	0.998093i	$-0.519660\pi$
$149$	−1.50686	−0.123447	−0.0617235	+	0.998093i	$0.519660\pi$
$150$	0	0
$151$	6.23970	0.507780	0.253890	−	0.967233i	$-0.418290\pi$
$151$	6.23970	0.507780	0.253890	+	0.967233i	$0.418290\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.49314	0.601863
$156$	0	0
$157$	20.5162	1.63737	0.818685	−	0.574243i	$-0.194704\pi$
$157$	20.5162	1.63737	0.818685	+	0.574243i	$0.194704\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.49314	0.590542
$162$	0	0
$163$	12.2397	0.958688	0.479344	−	0.877627i	$-0.340875\pi$
$163$	12.2397	0.958688	0.479344	+	0.877627i	$0.340875\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.7971	−1.14503	−0.572516	−	0.819894i	$-0.694032\pi$
$167$	−14.7971	−1.14503	−0.572516	+	0.819894i	$0.694032\pi$
$168$	0	0
$169$	23.4426	1.80328
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.82012	−0.138381	−0.0691904	−	0.997603i	$-0.522042\pi$
$173$	−1.82012	−0.138381	−0.0691904	+	0.997603i	$0.522042\pi$
$174$	0	0
$175$	−9.03677	−0.683116
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.08727	0.0812667	0.0406333	−	0.999174i	$-0.487062\pi$
$179$	1.08727	0.0812667	0.0406333	+	0.999174i	$0.487062\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−18.7328	−1.37727
$186$	0	0
$187$	−0.405862	−0.0296796
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.8201	0.782916	0.391458	−	0.920196i	$-0.371971\pi$
$191$	10.8201	0.782916	0.391458	+	0.920196i	$0.371971\pi$
$192$	0	0
$193$	9.63091	0.693248	0.346624	−	0.938004i	$-0.387328\pi$
$193$	9.63091	0.693248	0.346624	+	0.938004i	$0.387328\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.4794	−1.38785	−0.693925	−	0.720047i	$-0.744120\pi$
$197$	−19.4794	−1.38785	−0.693925	+	0.720047i	$0.744120\pi$
$198$	0	0
$199$	16.8706	1.19593	0.597963	−	0.801524i	$-0.295977\pi$
$199$	16.8706	1.19593	0.597963	+	0.801524i	$0.295977\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.03677	0.564071
$204$	0	0
$205$	−10.4794	−0.731914
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.898070	−0.0621208
$210$	0	0
$211$	−3.83384	−0.263933	−0.131966	−	0.991254i	$-0.542129\pi$
$211$	−3.83384	−0.263933	−0.131966	+	0.991254i	$0.542129\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	36.8432	2.51268
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.28181	−0.220759
$222$	0	0
$223$	−12.4794	−0.835683	−0.417842	−	0.908520i	$-0.637213\pi$
$223$	−12.4794	−0.835683	−0.417842	+	0.908520i	$0.637213\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.8569	−1.45069	−0.725346	−	0.688384i	$-0.758320\pi$
$227$	−21.8569	−1.45069	−0.725346	+	0.688384i	$0.758320\pi$
$228$	0	0
$229$	19.2397	1.27140	0.635698	−	0.771938i	$-0.280712\pi$
$229$	19.2397	1.27140	0.635698	+	0.771938i	$0.280712\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.4564	0.750531	0.375266	−	0.926917i	$-0.377551\pi$
$233$	11.4564	0.750531	0.375266	+	0.926917i	$0.377551\pi$
$234$	0	0
$235$	16.0735	1.04852
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.16616	−0.269486	−0.134743	−	0.990881i	$-0.543021\pi$
$239$	−4.16616	−0.269486	−0.134743	+	0.990881i	$0.543021\pi$
$240$	0	0
$241$	13.2397	0.852844	0.426422	−	0.904524i	$-0.359774\pi$
$241$	13.2397	0.852844	0.426422	+	0.904524i	$0.359774\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.74657	−0.239359
$246$	0	0
$247$	−7.26182	−0.462059
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.8706	−0.812386	−0.406193	−	0.913787i	$-0.633144\pi$
$251$	−12.8706	−0.812386	−0.406193	+	0.913787i	$0.633144\pi$
$252$	0	0
$253$	5.59414	0.351700
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.137775	−0.00859416	−0.00429708	−	0.999991i	$-0.501368\pi$
$257$	−0.137775	−0.00859416	−0.00429708	+	0.999991i	$0.501368\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.2534	0.693916	0.346958	−	0.937881i	$-0.387215\pi$
$263$	11.2534	0.693916	0.346958	+	0.937881i	$0.387215\pi$
$264$	0	0
$265$	9.20293	0.565332
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.45636	−0.332680	−0.166340	−	0.986068i	$-0.553195\pi$
$269$	−5.45636	−0.332680	−0.166340	+	0.986068i	$0.553195\pi$
$270$	0	0
$271$	3.12938	0.190097	0.0950483	−	0.995473i	$-0.469699\pi$
$271$	3.12938	0.190097	0.0950483	+	0.995473i	$0.469699\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.74657	−0.406833
$276$	0	0
$277$	−8.63091	−0.518581	−0.259291	−	0.965799i	$-0.583489\pi$
$277$	−8.63091	−0.518581	−0.259291	+	0.965799i	$0.583489\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.0735	0.779902	0.389951	−	0.920836i	$-0.372492\pi$
$281$	13.0735	0.779902	0.389951	+	0.920836i	$0.372492\pi$
$282$	0	0
$283$	13.6088	0.808959	0.404479	−	0.914547i	$-0.367453\pi$
$283$	13.6088	0.808959	0.404479	+	0.914547i	$0.367453\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.79707	−0.165106
$288$	0	0
$289$	−16.7045	−0.982615
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.8338	−0.632920	−0.316460	−	0.948606i	$-0.602494\pi$
$293$	−10.8338	−0.632920	−0.316460	+	0.948606i	$0.602494\pi$
$294$	0	0
$295$	54.6265	3.18048
$296$	0	0
$297$	0	0
$298$	0	0
$299$	45.2344	2.61597
$300$	0	0
$301$	9.83384	0.566813
$302$	0	0
$303$	0	0
$304$	0	0
$305$	34.6172	1.98217
$306$	0	0
$307$	−18.0735	−1.03151	−0.515756	−	0.856736i	$-0.672489\pi$
$307$	−18.0735	−1.03151	−0.515756	+	0.856736i	$0.672489\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.29021	0.243275	0.121638	−	0.992575i	$-0.461185\pi$
$311$	4.29021	0.243275	0.121638	+	0.992575i	$0.461185\pi$
$312$	0	0
$313$	5.55736	0.314121	0.157060	−	0.987589i	$-0.449798\pi$
$313$	5.55736	0.314121	0.157060	+	0.987589i	$0.449798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.5299	1.54623	0.773117	−	0.634264i	$-0.218697\pi$
$317$	27.5299	1.54623	0.773117	+	0.634264i	$0.218697\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.653958	0.0363872
$324$	0	0
$325$	−54.5530	−3.02605
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.29021	0.236527
$330$	0	0
$331$	−16.1471	−0.887525	−0.443762	−	0.896145i	$-0.646357\pi$
$331$	−16.1471	−0.887525	−0.443762	+	0.896145i	$0.646357\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	38.3638	2.09604
$336$	0	0
$337$	−1.11032	−0.0604830	−0.0302415	−	0.999543i	$-0.509628\pi$
$337$	−1.11032	−0.0604830	−0.0302415	+	0.999543i	$0.509628\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.49314	0.0808579
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−28.8201	−1.54714	−0.773572	−	0.633708i	$-0.781532\pi$
$347$	−28.8201	−1.54714	−0.773572	+	0.633708i	$0.781532\pi$
$348$	0	0
$349$	−23.2765	−1.24596	−0.622981	−	0.782237i	$-0.714078\pi$
$349$	−23.2765	−1.24596	−0.622981	+	0.782237i	$0.714078\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.36375	−0.125810	−0.0629049	−	0.998020i	$-0.520036\pi$
$353$	−2.36375	−0.125810	−0.0629049	+	0.998020i	$0.520036\pi$
$354$	0	0
$355$	30.8706	1.63844
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.32698	−0.175591	−0.0877956	−	0.996139i	$-0.527982\pi$
$359$	−3.32698	−0.175591	−0.0877956	+	0.996139i	$0.527982\pi$
$360$	0	0
$361$	−17.5530	−0.923840
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−37.6035	−1.96825
$366$	0	0
$367$	−5.66769	−0.295851	−0.147925	−	0.988999i	$-0.547260\pi$
$367$	−5.66769	−0.295851	−0.147925	+	0.988999i	$0.547260\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.45636	0.127528
$372$	0	0
$373$	23.5162	1.21762	0.608811	−	0.793315i	$-0.291647\pi$
$373$	23.5162	1.21762	0.608811	+	0.793315i	$0.291647\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.5162	2.49871
$378$	0	0
$379$	−36.7927	−1.88991	−0.944956	−	0.327197i	$-0.893896\pi$
$379$	−36.7927	−1.88991	−0.944956	+	0.327197i	$0.893896\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.9863	1.37893	0.689467	−	0.724317i	$-0.257845\pi$
$383$	26.9863	1.37893	0.689467	+	0.724317i	$0.257845\pi$
$384$	0	0
$385$	−2.79707	−0.142552
$386$	0	0
$387$	0	0
$388$	0	0
$389$	8.98627	0.455622	0.227811	−	0.973705i	$-0.426843\pi$
$389$	8.98627	0.455622	0.227811	+	0.973705i	$0.426843\pi$
$390$	0	0
$391$	−4.07355	−0.206008
$392$	0	0
$393$	0	0
$394$	0	0
$395$	38.3638	1.93029
$396$	0	0
$397$	28.5015	1.43045	0.715225	−	0.698894i	$-0.246324\pi$
$397$	28.5015	1.43045	0.715225	+	0.698894i	$0.246324\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7.50686	−0.374875	−0.187437	−	0.982277i	$-0.560018\pi$
$401$	−7.50686	−0.374875	−0.187437	+	0.982277i	$0.560018\pi$
$402$	0	0
$403$	12.0735	0.601426
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.73284	−0.185030
$408$	0	0
$409$	−27.2397	−1.34692	−0.673458	−	0.739225i	$-0.735192\pi$
$409$	−27.2397	−1.34692	−0.673458	+	0.739225i	$0.735192\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.5804	0.717455
$414$	0	0
$415$	−10.4794	−0.514414
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.6823	0.961545	0.480773	−	0.876845i	$-0.340356\pi$
$419$	19.6823	0.961545	0.480773	+	0.876845i	$0.340356\pi$
$420$	0	0
$421$	21.4794	1.04684	0.523421	−	0.852074i	$-0.324655\pi$
$421$	21.4794	1.04684	0.523421	+	0.852074i	$0.324655\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.91273	0.238302
$426$	0	0
$427$	9.23970	0.447141
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.17455	−0.104744	−0.0523722	−	0.998628i	$-0.516678\pi$
$431$	−2.17455	−0.104744	−0.0523722	+	0.998628i	$0.516678\pi$
$432$	0	0
$433$	35.3133	1.69705	0.848523	−	0.529158i	$-0.177492\pi$
$433$	35.3133	1.69705	0.848523	+	0.529158i	$0.177492\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.01373	−0.431185
$438$	0	0
$439$	19.2029	0.916506	0.458253	−	0.888822i	$-0.348475\pi$
$439$	19.2029	0.916506	0.458253	+	0.888822i	$0.348475\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.31859	0.252694	0.126347	−	0.991986i	$-0.459675\pi$
$443$	5.31859	0.252694	0.126347	+	0.991986i	$0.459675\pi$
$444$	0	0
$445$	24.5162	1.16218
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.36909	0.347769	0.173884	−	0.984766i	$-0.444368\pi$
$449$	7.36909	0.347769	0.173884	+	0.984766i	$0.444368\pi$
$450$	0	0
$451$	−2.08820	−0.0983296
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−22.6172	−1.06031
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	30.4648	1.41889	0.709443	−	0.704763i	$-0.248947\pi$
$461$	30.4648	1.41889	0.709443	+	0.704763i	$0.248947\pi$
$462$	0	0
$463$	27.5015	1.27810	0.639052	−	0.769163i	$-0.279327\pi$
$463$	27.5015	1.27810	0.639052	+	0.769163i	$0.279327\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−39.1755	−1.81282	−0.906412	−	0.422394i	$-0.861190\pi$
$467$	−39.1755	−1.81282	−0.906412	+	0.422394i	$0.861190\pi$
$468$	0	0
$469$	10.2397	0.472826
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.34163	0.337569
$474$	0	0
$475$	10.8706	0.498778
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−20.9863	−0.958887	−0.479444	−	0.877573i	$-0.659162\pi$
$479$	−20.9863	−0.958887	−0.479444	+	0.877573i	$0.659162\pi$
$480$	0	0
$481$	−30.1839	−1.37627
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−19.4931	−0.885138
$486$	0	0
$487$	−28.6456	−1.29805	−0.649027	−	0.760765i	$-0.724824\pi$
$487$	−28.6456	−1.29805	−0.649027	+	0.760765i	$0.724824\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.1471	−1.45078	−0.725389	−	0.688339i	$-0.758340\pi$
$491$	−32.1471	−1.45078	−0.725389	+	0.688339i	$0.758340\pi$
$492$	0	0
$493$	−4.36909	−0.196774
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.23970	0.369601
$498$	0	0
$499$	19.2618	0.862278	0.431139	−	0.902286i	$-0.358112\pi$
$499$	19.2618	0.862278	0.431139	+	0.902286i	$0.358112\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.2206	1.61500	0.807499	−	0.589869i	$-0.200820\pi$
$503$	36.2206	1.61500	0.807499	+	0.589869i	$0.200820\pi$
$504$	0	0
$505$	−16.8853	−0.751385
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8.98627	−0.398310	−0.199155	−	0.979968i	$-0.563820\pi$
$509$	−8.98627	−0.398310	−0.199155	+	0.979968i	$0.563820\pi$
$510$	0	0
$511$	−10.0368	−0.444001
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−40.7275	−1.79467
$516$	0	0
$517$	3.20293	0.140865
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.5667	−0.769610	−0.384805	−	0.922998i	$-0.625731\pi$
$521$	−17.5667	−0.769610	−0.384805	+	0.922998i	$0.625731\pi$
$522$	0	0
$523$	25.6088	1.11979	0.559897	−	0.828562i	$-0.310841\pi$
$523$	25.6088	1.11979	0.559897	+	0.828562i	$0.310841\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.08727	−0.0473624
$528$	0	0
$529$	33.1471	1.44118
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16.8853	−0.731382
$534$	0	0
$535$	12.4648	0.538898
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.746568	−0.0321570
$540$	0	0
$541$	−41.0735	−1.76589	−0.882945	−	0.469477i	$-0.844443\pi$
$541$	−41.0735	−1.76589	−0.882945	+	0.469477i	$0.844443\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	21.0966	0.903679
$546$	0	0
$547$	36.6456	1.56685	0.783426	−	0.621486i	$-0.213471\pi$
$547$	36.6456	1.56685	0.783426	+	0.621486i	$0.213471\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.66769	−0.411857
$552$	0	0
$553$	10.2397	0.435437
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.9127	0.589501	0.294751	−	0.955574i	$-0.404763\pi$
$557$	13.9127	0.589501	0.294751	+	0.955574i	$0.404763\pi$
$558$	0	0
$559$	59.3647	2.51086
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.681412	0.0287181	0.0143590	−	0.999897i	$-0.495429\pi$
$563$	0.681412	0.0287181	0.0143590	+	0.999897i	$0.495429\pi$
$564$	0	0
$565$	9.71911	0.408886
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.0598	−1.42786	−0.713931	−	0.700216i	$-0.753087\pi$
$569$	−34.0598	−1.42786	−0.713931	+	0.700216i	$0.753087\pi$
$570$	0	0
$571$	−17.6677	−0.739370	−0.369685	−	0.929157i	$-0.620534\pi$
$571$	−17.6677	−0.739370	−0.369685	+	0.929157i	$0.620534\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−67.7138	−2.82386
$576$	0	0
$577$	−12.5015	−0.520445	−0.260223	−	0.965549i	$-0.583796\pi$
$577$	−12.5015	−0.520445	−0.260223	+	0.965549i	$0.583796\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.79707	−0.116042
$582$	0	0
$583$	1.83384	0.0759500
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.3647	1.87240	0.936200	−	0.351467i	$-0.114317\pi$
$587$	45.3647	1.87240	0.936200	+	0.351467i	$0.114317\pi$
$588$	0	0
$589$	−2.40586	−0.0991318
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.93577	0.408013	0.204007	−	0.978970i	$-0.434604\pi$
$593$	9.93577	0.408013	0.204007	+	0.978970i	$0.434604\pi$
$594$	0	0
$595$	2.03677	0.0834996
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.3407	−0.749381	−0.374690	−	0.927150i	$-0.622251\pi$
$599$	−18.3407	−0.749381	−0.374690	+	0.927150i	$0.622251\pi$
$600$	0	0
$601$	2.35443	0.0960393	0.0480197	−	0.998846i	$-0.484709\pi$
$601$	2.35443	0.0960393	0.0480197	+	0.998846i	$0.484709\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	39.1240	1.59062
$606$	0	0
$607$	41.6823	1.69183	0.845917	−	0.533315i	$-0.179054\pi$
$607$	41.6823	1.69183	0.845917	+	0.533315i	$0.179054\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	25.8990	1.04776
$612$	0	0
$613$	21.5897	0.872001	0.436000	−	0.899946i	$-0.356395\pi$
$613$	21.5897	0.872001	0.436000	+	0.899946i	$0.356395\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.819187	0.0329792	0.0164896	−	0.999864i	$-0.494751\pi$
$617$	0.819187	0.0329792	0.0164896	+	0.999864i	$0.494751\pi$
$618$	0	0
$619$	−31.3500	−1.26006	−0.630032	−	0.776569i	$-0.716958\pi$
$619$	−31.3500	−1.26006	−0.630032	+	0.776569i	$0.716958\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.54364	0.262165
$624$	0	0
$625$	11.4794	0.459176
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.71819	0.108381
$630$	0	0
$631$	1.66769	0.0663895	0.0331947	−	0.999449i	$-0.489432\pi$
$631$	1.66769	0.0663895	0.0331947	+	0.999449i	$0.489432\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	36.8432	1.46208
$636$	0	0
$637$	−6.03677	−0.239186
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.2481	0.602264	0.301132	−	0.953583i	$-0.402636\pi$
$641$	15.2481	0.602264	0.301132	+	0.953583i	$0.402636\pi$
$642$	0	0
$643$	22.6265	0.892302	0.446151	−	0.894958i	$-0.352794\pi$
$643$	22.6265	0.892302	0.446151	+	0.894958i	$0.352794\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−42.0040	−1.65135	−0.825673	−	0.564148i	$-0.809205\pi$
$647$	−42.0040	−1.65135	−0.825673	+	0.564148i	$0.809205\pi$
$648$	0	0
$649$	10.8853	0.427284
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22.3554	−0.874833	−0.437416	−	0.899259i	$-0.644106\pi$
$653$	−22.3554	−0.874833	−0.437416	+	0.899259i	$0.644106\pi$
$654$	0	0
$655$	−20.1471	−0.787212
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31.4280	1.22426	0.612130	−	0.790757i	$-0.290313\pi$
$659$	31.4280	1.22426	0.612130	+	0.790757i	$0.290313\pi$
$660$	0	0
$661$	−24.4426	−0.950708	−0.475354	−	0.879795i	$-0.657680\pi$
$661$	−24.4426	−0.950708	−0.475354	+	0.879795i	$0.657680\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.50686	0.174769
$666$	0	0
$667$	60.2206	2.33175
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.89807	0.266297
$672$	0	0
$673$	39.4794	1.52182	0.760910	−	0.648858i	$-0.224753\pi$
$673$	39.4794	1.52182	0.760910	+	0.648858i	$0.224753\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	42.3784	1.62873	0.814367	−	0.580350i	$-0.197084\pi$
$677$	42.3784	1.62873	0.814367	+	0.580350i	$0.197084\pi$
$678$	0	0
$679$	−5.20293	−0.199670
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22.4143	−0.857658	−0.428829	−	0.903386i	$-0.641074\pi$
$683$	−22.4143	−0.857658	−0.428829	+	0.903386i	$0.641074\pi$
$684$	0	0
$685$	39.3133	1.50208
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.8285	0.564921
$690$	0	0
$691$	−12.4794	−0.474739	−0.237370	−	0.971419i	$-0.576285\pi$
$691$	−12.4794	−0.474739	−0.237370	+	0.971419i	$0.576285\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.4657	−0.965968
$696$	0	0
$697$	1.52059	0.0575965
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	0	0
$703$	6.01466	0.226847
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.50686	−0.169498
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14.9863	0.561240
$714$	0	0
$715$	−16.8853	−0.631473
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.65396	0.248151	0.124075	−	0.992273i	$-0.460404\pi$
$719$	6.65396	0.248151	0.124075	+	0.992273i	$0.460404\pi$
$720$	0	0
$721$	−10.8706	−0.404843
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−72.6265	−2.69728
$726$	0	0
$727$	−36.9442	−1.37018	−0.685092	−	0.728457i	$-0.740238\pi$
$727$	−36.9442	−1.37018	−0.685092	+	0.728457i	$0.740238\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.34604	−0.197731
$732$	0	0
$733$	−38.1324	−1.40845	−0.704227	−	0.709975i	$-0.748706\pi$
$733$	−38.1324	−1.40845	−0.704227	+	0.709975i	$0.748706\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.64464	0.281594
$738$	0	0
$739$	6.23970	0.229531	0.114766	−	0.993393i	$-0.463388\pi$
$739$	6.23970	0.229531	0.114766	+	0.993393i	$0.463388\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.0957	0.407060	0.203530	−	0.979069i	$-0.434758\pi$
$743$	11.0957	0.407060	0.203530	+	0.979069i	$0.434758\pi$
$744$	0	0
$745$	5.64557	0.206838
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.32698	0.121565
$750$	0	0
$751$	−36.3868	−1.32777	−0.663887	−	0.747833i	$-0.731094\pi$
$751$	−36.3868	−1.32777	−0.663887	+	0.747833i	$0.731094\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−23.3775	−0.850794
$756$	0	0
$757$	−4.40586	−0.160134	−0.0800669	−	0.996789i	$-0.525513\pi$
$757$	−4.40586	−0.160134	−0.0800669	+	0.996789i	$0.525513\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	47.2123	1.71144	0.855721	−	0.517437i	$-0.173114\pi$
$761$	47.2123	1.71144	0.855721	+	0.517437i	$0.173114\pi$
$762$	0	0
$763$	5.63091	0.203853
$764$	0	0
$765$	0	0
$766$	0	0
$767$	88.0186	3.17817
$768$	0	0
$769$	20.7603	0.748635	0.374318	−	0.927301i	$-0.377877\pi$
$769$	20.7603	0.748635	0.374318	+	0.927301i	$0.377877\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−0.976953	−0.0351386	−0.0175693	−	0.999846i	$-0.505593\pi$
$773$	−0.976953	−0.0351386	−0.0175693	+	0.999846i	$0.505593\pi$
$774$	0	0
$775$	−18.0735	−0.649221
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.36468	0.120552
$780$	0	0
$781$	6.15150	0.220118
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−76.8653	−2.74344
$786$	0	0
$787$	−2.87062	−0.102326	−0.0511632	−	0.998690i	$-0.516293\pi$
$787$	−2.87062	−0.102326	−0.0511632	+	0.998690i	$0.516293\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.59414	0.0922369
$792$	0	0
$793$	55.7780	1.98074
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.22505	−0.0433935	−0.0216967	−	0.999765i	$-0.506907\pi$
$797$	−1.22505	−0.0433935	−0.0216967	+	0.999765i	$0.506907\pi$
$798$	0	0
$799$	−2.33231	−0.0825114
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.49314	−0.264427
$804$	0	0
$805$	−28.0735	−0.989463
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.4333	0.542607	0.271303	−	0.962494i	$-0.412545\pi$
$809$	15.4333	0.542607	0.271303	+	0.962494i	$0.412545\pi$
$810$	0	0
$811$	2.47941	0.0870638	0.0435319	−	0.999052i	$-0.486139\pi$
$811$	2.47941	0.0870638	0.0435319	+	0.999052i	$0.486139\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−45.8569	−1.60630
$816$	0	0
$817$	−11.8294	−0.413860
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.8990	1.00858	0.504291	−	0.863534i	$-0.331754\pi$
$821$	28.8990	1.00858	0.504291	+	0.863534i	$0.331754\pi$
$822$	0	0
$823$	24.9588	0.870010	0.435005	−	0.900428i	$-0.356747\pi$
$823$	24.9588	0.870010	0.435005	+	0.900428i	$0.356747\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.7191	0.442287	0.221143	−	0.975241i	$-0.429021\pi$
$827$	12.7191	0.442287	0.221143	+	0.975241i	$0.429021\pi$
$828$	0	0
$829$	−6.33231	−0.219930	−0.109965	−	0.993935i	$-0.535074\pi$
$829$	−6.33231	−0.219930	−0.109965	+	0.993935i	$0.535074\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.543637	0.0188359
$834$	0	0
$835$	55.4382	1.91852
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.4236	−1.56820	−0.784098	−	0.620637i	$-0.786874\pi$
$839$	−45.4236	−1.56820	−0.784098	+	0.620637i	$0.786874\pi$
$840$	0	0
$841$	35.5897	1.22723
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−87.8294	−3.02142
$846$	0	0
$847$	10.4426	0.358813
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−37.4657	−1.28431
$852$	0	0
$853$	−24.0882	−0.824764	−0.412382	−	0.911011i	$-0.635303\pi$
$853$	−24.0882	−0.824764	−0.412382	+	0.911011i	$0.635303\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.8622	−0.610162	−0.305081	−	0.952326i	$-0.598684\pi$
$857$	−17.8622	−0.610162	−0.305081	+	0.952326i	$0.598684\pi$
$858$	0	0
$859$	−33.7412	−1.15124	−0.575618	−	0.817719i	$-0.695238\pi$
$859$	−33.7412	−1.15124	−0.575618	+	0.817719i	$0.695238\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	54.1555	1.84347	0.921737	−	0.387815i	$-0.126770\pi$
$863$	54.1555	1.84347	0.921737	+	0.387815i	$0.126770\pi$
$864$	0	0
$865$	6.81919	0.231859
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.64464	0.259327
$870$	0	0
$871$	61.8148	2.09451
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15.1240	0.511286
$876$	0	0
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−41.3500	−1.39312	−0.696559	−	0.717500i	$-0.745287\pi$
$881$	−41.3500	−1.39312	−0.696559	+	0.717500i	$0.745287\pi$
$882$	0	0
$883$	−52.4603	−1.76543	−0.882716	−	0.469908i	$-0.844287\pi$
$883$	−52.4603	−1.76543	−0.882716	+	0.469908i	$0.844287\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.2344	−0.712980	−0.356490	−	0.934299i	$-0.616027\pi$
$887$	−21.2344	−0.712980	−0.356490	+	0.934299i	$0.616027\pi$
$888$	0	0
$889$	9.83384	0.329816
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.16082	−0.172700
$894$	0	0
$895$	−4.07355	−0.136164
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.0735	0.536083
$900$	0	0
$901$	−1.33537	−0.0444876
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.49314	−0.249080
$906$	0	0
$907$	−40.2397	−1.33614	−0.668069	−	0.744100i	$-0.732879\pi$
$907$	−40.2397	−1.33614	−0.668069	+	0.744100i	$0.732879\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.811724	−0.0268936	−0.0134468	−	0.999910i	$-0.504280\pi$
$911$	−0.811724	−0.0268936	−0.0134468	+	0.999910i	$0.504280\pi$
$912$	0	0
$913$	−2.08820	−0.0691094
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.37748	−0.177580
$918$	0	0
$919$	−17.7603	−0.585858	−0.292929	−	0.956134i	$-0.594630\pi$
$919$	−17.7603	−0.585858	−0.292929	+	0.956134i	$0.594630\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	49.7412	1.63725
$924$	0	0
$925$	45.1839	1.48564
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.59947	−0.0524770	−0.0262385	−	0.999656i	$-0.508353\pi$
$929$	−1.59947	−0.0524770	−0.0262385	+	0.999656i	$0.508353\pi$
$930$	0	0
$931$	1.20293	0.0394245
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.52059	0.0497286
$936$	0	0
$937$	2.70140	0.0882510	0.0441255	−	0.999026i	$-0.485950\pi$
$937$	2.70140	0.0882510	0.0441255	+	0.999026i	$0.485950\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−36.7917	−1.19938	−0.599688	−	0.800234i	$-0.704709\pi$
$941$	−36.7917	−1.19938	−0.599688	+	0.800234i	$0.704709\pi$
$942$	0	0
$943$	−20.9588	−0.682513
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.3784	−0.792192	−0.396096	−	0.918209i	$-0.629635\pi$
$947$	−24.3784	−0.792192	−0.396096	+	0.918209i	$0.629635\pi$
$948$	0	0
$949$	−60.5897	−1.96683
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.63091	0.0528304	0.0264152	−	0.999651i	$-0.491591\pi$
$953$	1.63091	0.0528304	0.0264152	+	0.999651i	$0.491591\pi$
$954$	0	0
$955$	−40.5383	−1.31179
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.4931	0.338841
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−36.0829	−1.16155
$966$	0	0
$967$	−48.0735	−1.54594	−0.772971	−	0.634442i	$-0.781230\pi$
$967$	−48.0735	−1.54594	−0.772971	+	0.634442i	$0.781230\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−10.3491	−0.332118	−0.166059	−	0.986116i	$-0.553104\pi$
$971$	−10.3491	−0.332118	−0.166059	+	0.986116i	$0.553104\pi$
$972$	0	0
$973$	−6.79707	−0.217904
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11.1608	−0.357066	−0.178533	−	0.983934i	$-0.557135\pi$
$977$	−11.1608	−0.357066	−0.178533	+	0.983934i	$0.557135\pi$
$978$	0	0
$979$	4.88527	0.156134
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.3775	−1.31974	−0.659868	−	0.751381i	$-0.729388\pi$
$983$	−41.3775	−1.31974	−0.659868	+	0.751381i	$0.729388\pi$
$984$	0	0
$985$	72.9809	2.32537
$986$	0	0
$987$	0	0
$988$	0	0
$989$	73.6863	2.34309
$990$	0	0
$991$	−0.977882	−0.0310634	−0.0155317	−	0.999879i	$-0.504944\pi$
$991$	−0.977882	−0.0310634	−0.0155317	+	0.999879i	$0.504944\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−63.2069	−2.00379
$996$	0	0
$997$	−28.1103	−0.890263	−0.445131	−	0.895465i	$-0.646843\pi$
$997$	−28.1103	−0.890263	−0.445131	+	0.895465i	$0.646843\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.bu.1.1		3
3.2	odd	2		9072.2.a.cb.1.3		3
4.3	odd	2		567.2.a.e.1.1	✓	3
12.11	even	2		567.2.a.f.1.3	yes	3
28.27	even	2		3969.2.a.o.1.1		3
36.7	odd	6		567.2.f.m.190.3		6
36.11	even	6		567.2.f.l.190.1		6
36.23	even	6		567.2.f.l.379.1		6
36.31	odd	6		567.2.f.m.379.3		6
84.83	odd	2		3969.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.a.e.1.1	✓	3	4.3	odd	2
567.2.a.f.1.3	yes	3	12.11	even	2
567.2.f.l.190.1		6	36.11	even	6
567.2.f.l.379.1		6	36.23	even	6
567.2.f.m.190.3		6	36.7	odd	6
567.2.f.m.379.3		6	36.31	odd	6
3969.2.a.n.1.3		3	84.83	odd	2
3969.2.a.o.1.1		3	28.27	even	2
9072.2.a.bu.1.1		3	1.1	even	1	trivial
9072.2.a.cb.1.3		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-3.74657 q^{5} -1.00000 q^{7} -0.746568 q^{11} -6.03677 q^{13} +0.543637 q^{17} +1.20293 q^{19} -7.49314 q^{23} +9.03677 q^{25} -8.03677 q^{29} -2.00000 q^{31} +3.74657 q^{35} +5.00000 q^{37} +2.79707 q^{41} -9.83384 q^{43} -4.29021 q^{47} +1.00000 q^{49} -2.45636 q^{53} +2.79707 q^{55} -14.5804 q^{59} -9.23970 q^{61} +22.6172 q^{65} -10.2397 q^{67} -8.23970 q^{71} +10.0368 q^{73} +0.746568 q^{77} -10.2397 q^{79} +2.79707 q^{83} -2.03677 q^{85} -6.54364 q^{89} +6.03677 q^{91} -4.50686 q^{95} +5.20293 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{23} + 6 q^{25} - 3 q^{29} - 6 q^{31} + 3 q^{35} + 15 q^{37} + 12 q^{41} - 12 q^{43} + 3 q^{49} - 12 q^{53} + 12 q^{55} - 18 q^{59} - 3 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 + 6 * q^11 + 3 * q^13 - 3 * q^17 - 6 * q^23 + 6 * q^25 - 3 * q^29 - 6 * q^31 + 3 * q^35 + 15 * q^37 + 12 * q^41 - 12 * q^43 + 3 * q^49 - 12 * q^53 + 12 * q^55 - 18 * q^59 - 3 * q^61 + 21 * q^65 - 6 * q^67 + 9 * q^73 - 6 * q^77 - 6 * q^79 + 12 * q^83 + 15 * q^85 - 15 * q^89 - 3 * q^91 - 30 * q^95 + 12 * q^97

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