LMFDB - Embedded newform 9464.2.a.bf.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] 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("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,-4,0,-6,0,-4,0,10,0,0,0,-6,0,2] 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:	$75.5704204729$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.276564$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.276564 q^{3} +1.57603 q^{5} -1.00000 q^{7} -2.92351 q^{9} -2.22298 q^{11} -0.435874 q^{15} +3.14016 q^{17} +0.136405 q^{19} +0.276564 q^{21} +7.39724 q^{23} -2.51612 q^{25} +1.63823 q^{27} -6.70623 q^{29} +1.91004 q^{31} +0.614797 q^{33} -1.57603 q^{35} -1.31429 q^{37} +8.30023 q^{41} -4.75496 q^{43} -4.60755 q^{45} +5.92050 q^{47} +1.00000 q^{49} -0.868455 q^{51} -1.59894 q^{53} -3.50350 q^{55} -0.0377246 q^{57} -9.82982 q^{59} -2.05506 q^{61} +2.92351 q^{63} -8.04343 q^{67} -2.04581 q^{69} +1.82979 q^{71} +9.37056 q^{73} +0.695868 q^{75} +2.22298 q^{77} +6.91298 q^{79} +8.31746 q^{81} -5.95108 q^{83} +4.94900 q^{85} +1.85470 q^{87} +1.13368 q^{89} -0.528248 q^{93} +0.214978 q^{95} -10.0696 q^{97} +6.49892 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 4 q^{5} - 6 q^{7} - 4 q^{9} + 10 q^{11} - 6 q^{15} + 2 q^{17} + 14 q^{19} - 2 q^{21} - 6 q^{23} - 2 q^{25} + 2 q^{27} - 20 q^{29} - 4 q^{31} + 4 q^{33} + 4 q^{35} - 4 q^{37} + 10 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 - 4 * q^9 + 10 * q^11 - 6 * q^15 + 2 * q^17 + 14 * q^19 - 2 * q^21 - 6 * q^23 - 2 * q^25 + 2 * q^27 - 20 * q^29 - 4 * q^31 + 4 * q^33 + 4 * q^35 - 4 * q^37 + 10 * q^41 - 12 * q^43 + 6 * q^45 - 6 * q^47 + 6 * q^49 - 8 * q^51 + 10 * q^53 - 24 * q^55 - 8 * q^59 + 4 * q^61 + 4 * q^63 - 18 * q^67 + 12 * q^71 + 14 * q^73 - 10 * q^75 - 10 * q^77 - 10 * q^79 - 18 * q^81 + 28 * q^83 + 20 * q^85 - 32 * q^87 - 6 * q^89 + 12 * q^93 - 26 * q^95 + 8 * q^97 + 2 * q^99

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
$3$	−0.276564	−0.159674	−0.0798372	−	0.996808i	$-0.525440\pi$
$3$	−0.276564	−0.159674	−0.0798372	+	0.996808i	$0.525440\pi$
$4$	0	0
$5$	1.57603	0.704824	0.352412	−	0.935845i	$-0.385362\pi$
$5$	1.57603	0.704824	0.352412	+	0.935845i	$0.385362\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.92351	−0.974504
$10$	0	0
$11$	−2.22298	−0.670254	−0.335127	−	0.942173i	$-0.608779\pi$
$11$	−2.22298	−0.670254	−0.335127	+	0.942173i	$0.608779\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.435874	−0.112542
$16$	0	0
$17$	3.14016	0.761601	0.380800	−	0.924657i	$-0.375649\pi$
$17$	3.14016	0.761601	0.380800	+	0.924657i	$0.375649\pi$
$18$	0	0
$19$	0.136405	0.0312933	0.0156467	−	0.999878i	$-0.495019\pi$
$19$	0.136405	0.0312933	0.0156467	+	0.999878i	$0.495019\pi$
$20$	0	0
$21$	0.276564	0.0603512
$22$	0	0
$23$	7.39724	1.54243	0.771216	−	0.636574i	$-0.219649\pi$
$23$	7.39724	1.54243	0.771216	+	0.636574i	$0.219649\pi$
$24$	0	0
$25$	−2.51612	−0.503223
$26$	0	0
$27$	1.63823	0.315278
$28$	0	0
$29$	−6.70623	−1.24532	−0.622658	−	0.782494i	$-0.713947\pi$
$29$	−6.70623	−1.24532	−0.622658	+	0.782494i	$0.713947\pi$
$30$	0	0
$31$	1.91004	0.343053	0.171526	−	0.985180i	$-0.445130\pi$
$31$	1.91004	0.343053	0.171526	+	0.985180i	$0.445130\pi$
$32$	0	0
$33$	0.614797	0.107022
$34$	0	0
$35$	−1.57603	−0.266398
$36$	0	0
$37$	−1.31429	−0.216068	−0.108034	−	0.994147i	$-0.534455\pi$
$37$	−1.31429	−0.216068	−0.108034	+	0.994147i	$0.534455\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.30023	1.29628	0.648139	−	0.761522i	$-0.275547\pi$
$41$	8.30023	1.29628	0.648139	+	0.761522i	$0.275547\pi$
$42$	0	0
$43$	−4.75496	−0.725124	−0.362562	−	0.931960i	$-0.618098\pi$
$43$	−4.75496	−0.725124	−0.362562	+	0.931960i	$0.618098\pi$
$44$	0	0
$45$	−4.60755	−0.686854
$46$	0	0
$47$	5.92050	0.863594	0.431797	−	0.901971i	$-0.357880\pi$
$47$	5.92050	0.863594	0.431797	+	0.901971i	$0.357880\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.868455	−0.121608
$52$	0	0
$53$	−1.59894	−0.219631	−0.109816	−	0.993952i	$-0.535026\pi$
$53$	−1.59894	−0.219631	−0.109816	+	0.993952i	$0.535026\pi$
$54$	0	0
$55$	−3.50350	−0.472411
$56$	0	0
$57$	−0.0377246	−0.00499674
$58$	0	0
$59$	−9.82982	−1.27973	−0.639867	−	0.768486i	$-0.721011\pi$
$59$	−9.82982	−1.27973	−0.639867	+	0.768486i	$0.721011\pi$
$60$	0	0
$61$	−2.05506	−0.263123	−0.131562	−	0.991308i	$-0.541999\pi$
$61$	−2.05506	−0.263123	−0.131562	+	0.991308i	$0.541999\pi$
$62$	0	0
$63$	2.92351	0.368328
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.04343	−0.982661	−0.491331	−	0.870973i	$-0.663489\pi$
$67$	−8.04343	−0.982661	−0.491331	+	0.870973i	$0.663489\pi$
$68$	0	0
$69$	−2.04581	−0.246287
$70$	0	0
$71$	1.82979	0.217157	0.108578	−	0.994088i	$-0.465370\pi$
$71$	1.82979	0.217157	0.108578	+	0.994088i	$0.465370\pi$
$72$	0	0
$73$	9.37056	1.09674	0.548371	−	0.836235i	$-0.315248\pi$
$73$	9.37056	1.09674	0.548371	+	0.836235i	$0.315248\pi$
$74$	0	0
$75$	0.695868	0.0803519
$76$	0	0
$77$	2.22298	0.253332
$78$	0	0
$79$	6.91298	0.777771	0.388885	−	0.921286i	$-0.372860\pi$
$79$	6.91298	0.777771	0.388885	+	0.921286i	$0.372860\pi$
$80$	0	0
$81$	8.31746	0.924162
$82$	0	0
$83$	−5.95108	−0.653216	−0.326608	−	0.945160i	$-0.605906\pi$
$83$	−5.95108	−0.653216	−0.326608	+	0.945160i	$0.605906\pi$
$84$	0	0
$85$	4.94900	0.536794
$86$	0	0
$87$	1.85470	0.198845
$88$	0	0
$89$	1.13368	0.120169	0.0600847	−	0.998193i	$-0.480863\pi$
$89$	1.13368	0.120169	0.0600847	+	0.998193i	$0.480863\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.528248	−0.0547767
$94$	0	0
$95$	0.214978	0.0220563
$96$	0	0
$97$	−10.0696	−1.02241	−0.511205	−	0.859459i	$-0.670801\pi$
$97$	−10.0696	−1.02241	−0.511205	+	0.859459i	$0.670801\pi$
$98$	0	0
$99$	6.49892	0.653166
$100$	0	0
$101$	−1.15040	−0.114469	−0.0572343	−	0.998361i	$-0.518228\pi$
$101$	−1.15040	−0.114469	−0.0572343	+	0.998361i	$0.518228\pi$
$102$	0	0
$103$	18.3301	1.80612	0.903060	−	0.429515i	$-0.141315\pi$
$103$	18.3301	1.80612	0.903060	+	0.429515i	$0.141315\pi$
$104$	0	0
$105$	0.435874	0.0425370
$106$	0	0
$107$	−14.6609	−1.41732	−0.708661	−	0.705549i	$-0.750700\pi$
$107$	−14.6609	−1.41732	−0.708661	+	0.705549i	$0.750700\pi$
$108$	0	0
$109$	−0.304515	−0.0291673	−0.0145836	−	0.999894i	$-0.504642\pi$
$109$	−0.304515	−0.0291673	−0.0145836	+	0.999894i	$0.504642\pi$
$110$	0	0
$111$	0.363485	0.0345005
$112$	0	0
$113$	−13.5341	−1.27318	−0.636590	−	0.771203i	$-0.719656\pi$
$113$	−13.5341	−1.27318	−0.636590	+	0.771203i	$0.719656\pi$
$114$	0	0
$115$	11.6583	1.08714
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.14016	−0.287858
$120$	0	0
$121$	−6.05835	−0.550759
$122$	0	0
$123$	−2.29555	−0.206982
$124$	0	0
$125$	−11.8457	−1.05951
$126$	0	0
$127$	0.272899	0.0242159	0.0121079	−	0.999927i	$-0.496146\pi$
$127$	0.272899	0.0242159	0.0121079	+	0.999927i	$0.496146\pi$
$128$	0	0
$129$	1.31505	0.115784
$130$	0	0
$131$	11.1530	0.974440	0.487220	−	0.873279i	$-0.338011\pi$
$131$	11.1530	0.974440	0.487220	+	0.873279i	$0.338011\pi$
$132$	0	0
$133$	−0.136405	−0.0118278
$134$	0	0
$135$	2.58191	0.222215
$136$	0	0
$137$	−18.6128	−1.59020	−0.795100	−	0.606479i	$-0.792582\pi$
$137$	−18.6128	−1.59020	−0.795100	+	0.606479i	$0.792582\pi$
$138$	0	0
$139$	16.3576	1.38744	0.693718	−	0.720247i	$-0.255971\pi$
$139$	16.3576	1.38744	0.693718	+	0.720247i	$0.255971\pi$
$140$	0	0
$141$	−1.63740	−0.137894
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.5693	−0.877729
$146$	0	0
$147$	−0.276564	−0.0228106
$148$	0	0
$149$	−9.72396	−0.796618	−0.398309	−	0.917251i	$-0.630403\pi$
$149$	−9.72396	−0.796618	−0.398309	+	0.917251i	$0.630403\pi$
$150$	0	0
$151$	−4.62076	−0.376032	−0.188016	−	0.982166i	$-0.560206\pi$
$151$	−4.62076	−0.376032	−0.188016	+	0.982166i	$0.560206\pi$
$152$	0	0
$153$	−9.18030	−0.742183
$154$	0	0
$155$	3.01028	0.241792
$156$	0	0
$157$	−10.6284	−0.848239	−0.424119	−	0.905606i	$-0.639416\pi$
$157$	−10.6284	−0.848239	−0.424119	+	0.905606i	$0.639416\pi$
$158$	0	0
$159$	0.442209	0.0350695
$160$	0	0
$161$	−7.39724	−0.582984
$162$	0	0
$163$	0.980916	0.0768312	0.0384156	−	0.999262i	$-0.487769\pi$
$163$	0.980916	0.0768312	0.0384156	+	0.999262i	$0.487769\pi$
$164$	0	0
$165$	0.968941	0.0754320
$166$	0	0
$167$	12.5182	0.968687	0.484343	−	0.874878i	$-0.339059\pi$
$167$	12.5182	0.968687	0.484343	+	0.874878i	$0.339059\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−0.398780	−0.0304955
$172$	0	0
$173$	−15.2987	−1.16314	−0.581571	−	0.813496i	$-0.697562\pi$
$173$	−15.2987	−1.16314	−0.581571	+	0.813496i	$0.697562\pi$
$174$	0	0
$175$	2.51612	0.190201
$176$	0	0
$177$	2.71858	0.204341
$178$	0	0
$179$	−14.3684	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$179$	−14.3684	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$180$	0	0
$181$	−19.0430	−1.41546	−0.707728	−	0.706485i	$-0.750280\pi$
$181$	−19.0430	−1.41546	−0.707728	+	0.706485i	$0.750280\pi$
$182$	0	0
$183$	0.568355	0.0420140
$184$	0	0
$185$	−2.07136	−0.152290
$186$	0	0
$187$	−6.98052	−0.510466
$188$	0	0
$189$	−1.63823	−0.119164
$190$	0	0
$191$	−23.2985	−1.68582	−0.842910	−	0.538054i	$-0.819160\pi$
$191$	−23.2985	−1.68582	−0.842910	+	0.538054i	$0.819160\pi$
$192$	0	0
$193$	25.9932	1.87103	0.935516	−	0.353285i	$-0.114935\pi$
$193$	25.9932	1.87103	0.935516	+	0.353285i	$0.114935\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.9477	−0.922482	−0.461241	−	0.887275i	$-0.652596\pi$
$197$	−12.9477	−0.922482	−0.461241	+	0.887275i	$0.652596\pi$
$198$	0	0
$199$	−4.28129	−0.303493	−0.151746	−	0.988419i	$-0.548490\pi$
$199$	−4.28129	−0.303493	−0.151746	+	0.988419i	$0.548490\pi$
$200$	0	0
$201$	2.22452	0.156906
$202$	0	0
$203$	6.70623	0.470685
$204$	0	0
$205$	13.0814	0.913648
$206$	0	0
$207$	−21.6259	−1.50311
$208$	0	0
$209$	−0.303225	−0.0209745
$210$	0	0
$211$	−16.4636	−1.13340	−0.566701	−	0.823923i	$-0.691781\pi$
$211$	−16.4636	−1.13340	−0.566701	+	0.823923i	$0.691781\pi$
$212$	0	0
$213$	−0.506056	−0.0346744
$214$	0	0
$215$	−7.49397	−0.511085
$216$	0	0
$217$	−1.91004	−0.129662
$218$	0	0
$219$	−2.59156	−0.175121
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.12024	0.0750169	0.0375085	−	0.999296i	$-0.488058\pi$
$223$	1.12024	0.0750169	0.0375085	+	0.999296i	$0.488058\pi$
$224$	0	0
$225$	7.35590	0.490393
$226$	0	0
$227$	23.7935	1.57923	0.789617	−	0.613600i	$-0.210279\pi$
$227$	23.7935	1.57923	0.789617	+	0.613600i	$0.210279\pi$
$228$	0	0
$229$	−1.54558	−0.102135	−0.0510673	−	0.998695i	$-0.516262\pi$
$229$	−1.54558	−0.102135	−0.0510673	+	0.998695i	$0.516262\pi$
$230$	0	0
$231$	−0.614797	−0.0404507
$232$	0	0
$233$	−10.6215	−0.695839	−0.347919	−	0.937524i	$-0.613112\pi$
$233$	−10.6215	−0.695839	−0.347919	+	0.937524i	$0.613112\pi$
$234$	0	0
$235$	9.33091	0.608681
$236$	0	0
$237$	−1.91188	−0.124190
$238$	0	0
$239$	0.811352	0.0524820	0.0262410	−	0.999656i	$-0.491646\pi$
$239$	0.811352	0.0524820	0.0262410	+	0.999656i	$0.491646\pi$
$240$	0	0
$241$	−11.9045	−0.766838	−0.383419	−	0.923574i	$-0.625254\pi$
$241$	−11.9045	−0.766838	−0.383419	+	0.923574i	$0.625254\pi$
$242$	0	0
$243$	−7.21500	−0.462843
$244$	0	0
$245$	1.57603	0.100689
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.64586	0.104302
$250$	0	0
$251$	1.47621	0.0931776	0.0465888	−	0.998914i	$-0.485165\pi$
$251$	1.47621	0.0931776	0.0465888	+	0.998914i	$0.485165\pi$
$252$	0	0
$253$	−16.4439	−1.03382
$254$	0	0
$255$	−1.36872	−0.0857123
$256$	0	0
$257$	−18.7557	−1.16995	−0.584976	−	0.811051i	$-0.698896\pi$
$257$	−18.7557	−1.16995	−0.584976	+	0.811051i	$0.698896\pi$
$258$	0	0
$259$	1.31429	0.0816659
$260$	0	0
$261$	19.6058	1.21357
$262$	0	0
$263$	−4.85409	−0.299316	−0.149658	−	0.988738i	$-0.547817\pi$
$263$	−4.85409	−0.299316	−0.149658	+	0.988738i	$0.547817\pi$
$264$	0	0
$265$	−2.51998	−0.154801
$266$	0	0
$267$	−0.313534	−0.0191880
$268$	0	0
$269$	15.5091	0.945604	0.472802	−	0.881169i	$-0.343243\pi$
$269$	15.5091	0.945604	0.472802	+	0.881169i	$0.343243\pi$
$270$	0	0
$271$	14.1843	0.861637	0.430818	−	0.902439i	$-0.358225\pi$
$271$	14.1843	0.861637	0.430818	+	0.902439i	$0.358225\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.59328	0.337288
$276$	0	0
$277$	−28.7375	−1.72667	−0.863334	−	0.504633i	$-0.831628\pi$
$277$	−28.7375	−1.72667	−0.863334	+	0.504633i	$0.831628\pi$
$278$	0	0
$279$	−5.58402	−0.334306
$280$	0	0
$281$	−16.0016	−0.954573	−0.477286	−	0.878748i	$-0.658380\pi$
$281$	−16.0016	−0.954573	−0.477286	+	0.878748i	$0.658380\pi$
$282$	0	0
$283$	−8.05297	−0.478700	−0.239350	−	0.970933i	$-0.576934\pi$
$283$	−8.05297	−0.478700	−0.239350	+	0.970933i	$0.576934\pi$
$284$	0	0
$285$	−0.0594552	−0.00352182
$286$	0	0
$287$	−8.30023	−0.489947
$288$	0	0
$289$	−7.13940	−0.419965
$290$	0	0
$291$	2.78488	0.163253
$292$	0	0
$293$	23.1051	1.34982	0.674908	−	0.737902i	$-0.264183\pi$
$293$	23.1051	1.34982	0.674908	+	0.737902i	$0.264183\pi$
$294$	0	0
$295$	−15.4921	−0.901987
$296$	0	0
$297$	−3.64176	−0.211316
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.75496	0.274071
$302$	0	0
$303$	0.318158	0.0182777
$304$	0	0
$305$	−3.23884	−0.185455
$306$	0	0
$307$	4.92149	0.280884	0.140442	−	0.990089i	$-0.455148\pi$
$307$	4.92149	0.280884	0.140442	+	0.990089i	$0.455148\pi$
$308$	0	0
$309$	−5.06945	−0.288391
$310$	0	0
$311$	−4.37793	−0.248250	−0.124125	−	0.992267i	$-0.539612\pi$
$311$	−4.37793	−0.248250	−0.124125	+	0.992267i	$0.539612\pi$
$312$	0	0
$313$	−20.2804	−1.14631	−0.573157	−	0.819445i	$-0.694282\pi$
$313$	−20.2804	−1.14631	−0.573157	+	0.819445i	$0.694282\pi$
$314$	0	0
$315$	4.60755	0.259606
$316$	0	0
$317$	−5.44830	−0.306007	−0.153004	−	0.988226i	$-0.548895\pi$
$317$	−5.44830	−0.306007	−0.153004	+	0.988226i	$0.548895\pi$
$318$	0	0
$319$	14.9078	0.834679
$320$	0	0
$321$	4.05468	0.226310
$322$	0	0
$323$	0.428332	0.0238330
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.0842180	0.00465727
$328$	0	0
$329$	−5.92050	−0.326408
$330$	0	0
$331$	−33.8099	−1.85836	−0.929181	−	0.369626i	$-0.879486\pi$
$331$	−33.8099	−1.85836	−0.929181	+	0.369626i	$0.879486\pi$
$332$	0	0
$333$	3.84234	0.210559
$334$	0	0
$335$	−12.6767	−0.692603
$336$	0	0
$337$	−30.6901	−1.67180	−0.835899	−	0.548883i	$-0.815053\pi$
$337$	−30.6901	−1.67180	−0.835899	+	0.548883i	$0.815053\pi$
$338$	0	0
$339$	3.74304	0.203294
$340$	0	0
$341$	−4.24598	−0.229933
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.22427	−0.173589
$346$	0	0
$347$	−21.5291	−1.15575	−0.577873	−	0.816127i	$-0.696117\pi$
$347$	−21.5291	−1.15575	−0.577873	+	0.816127i	$0.696117\pi$
$348$	0	0
$349$	25.0583	1.34134	0.670671	−	0.741755i	$-0.266006\pi$
$349$	25.0583	1.34134	0.670671	+	0.741755i	$0.266006\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.35237	0.391327	0.195664	−	0.980671i	$-0.437314\pi$
$353$	7.35237	0.391327	0.195664	+	0.980671i	$0.437314\pi$
$354$	0	0
$355$	2.88382	0.153057
$356$	0	0
$357$	0.868455	0.0459635
$358$	0	0
$359$	24.7243	1.30490	0.652449	−	0.757833i	$-0.273742\pi$
$359$	24.7243	1.30490	0.652449	+	0.757833i	$0.273742\pi$
$360$	0	0
$361$	−18.9814	−0.999021
$362$	0	0
$363$	1.67552	0.0879421
$364$	0	0
$365$	14.7683	0.773009
$366$	0	0
$367$	−20.8860	−1.09024	−0.545121	−	0.838357i	$-0.683516\pi$
$367$	−20.8860	−1.09024	−0.545121	+	0.838357i	$0.683516\pi$
$368$	0	0
$369$	−24.2658	−1.26323
$370$	0	0
$371$	1.59894	0.0830128
$372$	0	0
$373$	37.1443	1.92326	0.961630	−	0.274351i	$-0.0884629\pi$
$373$	37.1443	1.92326	0.961630	+	0.274351i	$0.0884629\pi$
$374$	0	0
$375$	3.27608	0.169176
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−5.08176	−0.261033	−0.130516	−	0.991446i	$-0.541663\pi$
$379$	−5.08176	−0.261033	−0.130516	+	0.991446i	$0.541663\pi$
$380$	0	0
$381$	−0.0754740	−0.00386665
$382$	0	0
$383$	−30.3874	−1.55272	−0.776361	−	0.630288i	$-0.782937\pi$
$383$	−30.3874	−1.55272	−0.776361	+	0.630288i	$0.782937\pi$
$384$	0	0
$385$	3.50350	0.178555
$386$	0	0
$387$	13.9012	0.706636
$388$	0	0
$389$	10.3158	0.523032	0.261516	−	0.965199i	$-0.415778\pi$
$389$	10.3158	0.523032	0.261516	+	0.965199i	$0.415778\pi$
$390$	0	0
$391$	23.2285	1.17472
$392$	0	0
$393$	−3.08451	−0.155593
$394$	0	0
$395$	10.8951	0.548191
$396$	0	0
$397$	−14.2274	−0.714054	−0.357027	−	0.934094i	$-0.616210\pi$
$397$	−14.2274	−0.714054	−0.357027	+	0.934094i	$0.616210\pi$
$398$	0	0
$399$	0.0377246	0.00188859
$400$	0	0
$401$	13.0172	0.650046	0.325023	−	0.945706i	$-0.394628\pi$
$401$	13.0172	0.650046	0.325023	+	0.945706i	$0.394628\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	13.1086	0.651372
$406$	0	0
$407$	2.92164	0.144820
$408$	0	0
$409$	−3.49856	−0.172993	−0.0864963	−	0.996252i	$-0.527567\pi$
$409$	−3.49856	−0.172993	−0.0864963	+	0.996252i	$0.527567\pi$
$410$	0	0
$411$	5.14763	0.253914
$412$	0	0
$413$	9.82982	0.483694
$414$	0	0
$415$	−9.37911	−0.460402
$416$	0	0
$417$	−4.52393	−0.221538
$418$	0	0
$419$	−5.06349	−0.247368	−0.123684	−	0.992322i	$-0.539471\pi$
$419$	−5.06349	−0.247368	−0.123684	+	0.992322i	$0.539471\pi$
$420$	0	0
$421$	3.98914	0.194419	0.0972095	−	0.995264i	$-0.469008\pi$
$421$	3.98914	0.194419	0.0972095	+	0.995264i	$0.469008\pi$
$422$	0	0
$423$	−17.3087	−0.841575
$424$	0	0
$425$	−7.90101	−0.383255
$426$	0	0
$427$	2.05506	0.0994512
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.2322	1.40806	0.704032	−	0.710169i	$-0.251381\pi$
$431$	29.2322	1.40806	0.704032	+	0.710169i	$0.251381\pi$
$432$	0	0
$433$	19.1666	0.921089	0.460545	−	0.887637i	$-0.347654\pi$
$433$	19.1666	0.921089	0.460545	+	0.887637i	$0.347654\pi$
$434$	0	0
$435$	2.92308	0.140151
$436$	0	0
$437$	1.00902	0.0482678
$438$	0	0
$439$	3.86109	0.184280	0.0921398	−	0.995746i	$-0.470629\pi$
$439$	3.86109	0.184280	0.0921398	+	0.995746i	$0.470629\pi$
$440$	0	0
$441$	−2.92351	−0.139215
$442$	0	0
$443$	12.5435	0.595958	0.297979	−	0.954572i	$-0.403688\pi$
$443$	12.5435	0.595958	0.297979	+	0.954572i	$0.403688\pi$
$444$	0	0
$445$	1.78671	0.0846982
$446$	0	0
$447$	2.68930	0.127199
$448$	0	0
$449$	−12.7738	−0.602833	−0.301416	−	0.953493i	$-0.597459\pi$
$449$	−12.7738	−0.602833	−0.301416	+	0.953493i	$0.597459\pi$
$450$	0	0
$451$	−18.4513	−0.868836
$452$	0	0
$453$	1.27794	0.0600427
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.7198	0.735340	0.367670	−	0.929956i	$-0.380156\pi$
$457$	15.7198	0.735340	0.367670	+	0.929956i	$0.380156\pi$
$458$	0	0
$459$	5.14431	0.240116
$460$	0	0
$461$	−11.5691	−0.538824	−0.269412	−	0.963025i	$-0.586829\pi$
$461$	−11.5691	−0.538824	−0.269412	+	0.963025i	$0.586829\pi$
$462$	0	0
$463$	2.10201	0.0976887	0.0488444	−	0.998806i	$-0.484446\pi$
$463$	2.10201	0.0976887	0.0488444	+	0.998806i	$0.484446\pi$
$464$	0	0
$465$	−0.832536	−0.0386080
$466$	0	0
$467$	23.7277	1.09799	0.548994	−	0.835826i	$-0.315011\pi$
$467$	23.7277	1.09799	0.548994	+	0.835826i	$0.315011\pi$
$468$	0	0
$469$	8.04343	0.371411
$470$	0	0
$471$	2.93943	0.135442
$472$	0	0
$473$	10.5702	0.486018
$474$	0	0
$475$	−0.343210	−0.0157475
$476$	0	0
$477$	4.67452	0.214032
$478$	0	0
$479$	37.5392	1.71521	0.857605	−	0.514310i	$-0.171952\pi$
$479$	37.5392	1.71521	0.857605	+	0.514310i	$0.171952\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.04581	0.0930876
$484$	0	0
$485$	−15.8700	−0.720619
$486$	0	0
$487$	34.1880	1.54921	0.774603	−	0.632447i	$-0.217949\pi$
$487$	34.1880	1.54921	0.774603	+	0.632447i	$0.217949\pi$
$488$	0	0
$489$	−0.271286	−0.0122680
$490$	0	0
$491$	9.61356	0.433854	0.216927	−	0.976188i	$-0.430397\pi$
$491$	9.61356	0.433854	0.216927	+	0.976188i	$0.430397\pi$
$492$	0	0
$493$	−21.0586	−0.948434
$494$	0	0
$495$	10.2425	0.460367
$496$	0	0
$497$	−1.82979	−0.0820775
$498$	0	0
$499$	14.4923	0.648766	0.324383	−	0.945926i	$-0.394843\pi$
$499$	14.4923	0.648766	0.324383	+	0.945926i	$0.394843\pi$
$500$	0	0
$501$	−3.46208	−0.154674
$502$	0	0
$503$	−26.2264	−1.16938	−0.584689	−	0.811258i	$-0.698783\pi$
$503$	−26.2264	−1.16938	−0.584689	+	0.811258i	$0.698783\pi$
$504$	0	0
$505$	−1.81306	−0.0806802
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.4554	−1.08397	−0.541984	−	0.840389i	$-0.682327\pi$
$509$	−24.4554	−1.08397	−0.541984	+	0.840389i	$0.682327\pi$
$510$	0	0
$511$	−9.37056	−0.414529
$512$	0	0
$513$	0.223462	0.00986609
$514$	0	0
$515$	28.8889	1.27300
$516$	0	0
$517$	−13.1612	−0.578827
$518$	0	0
$519$	4.23108	0.185724
$520$	0	0
$521$	24.7193	1.08297	0.541486	−	0.840710i	$-0.317862\pi$
$521$	24.7193	1.08297	0.541486	+	0.840710i	$0.317862\pi$
$522$	0	0
$523$	−32.7879	−1.43372	−0.716858	−	0.697219i	$-0.754421\pi$
$523$	−32.7879	−1.43372	−0.716858	+	0.697219i	$0.754421\pi$
$524$	0	0
$525$	−0.695868	−0.0303702
$526$	0	0
$527$	5.99782	0.261269
$528$	0	0
$529$	31.7192	1.37910
$530$	0	0
$531$	28.7376	1.24711
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−23.1061	−0.998962
$536$	0	0
$537$	3.97378	0.171481
$538$	0	0
$539$	−2.22298	−0.0957506
$540$	0	0
$541$	16.4928	0.709081	0.354541	−	0.935041i	$-0.384637\pi$
$541$	16.4928	0.709081	0.354541	+	0.935041i	$0.384637\pi$
$542$	0	0
$543$	5.26661	0.226012
$544$	0	0
$545$	−0.479926	−0.0205578
$546$	0	0
$547$	−42.3299	−1.80990	−0.904948	−	0.425522i	$-0.860091\pi$
$547$	−42.3299	−1.80990	−0.904948	+	0.425522i	$0.860091\pi$
$548$	0	0
$549$	6.00798	0.256415
$550$	0	0
$551$	−0.914761	−0.0389701
$552$	0	0
$553$	−6.91298	−0.293970
$554$	0	0
$555$	0.572865	0.0243168
$556$	0	0
$557$	2.18577	0.0926142	0.0463071	−	0.998927i	$-0.485255\pi$
$557$	2.18577	0.0926142	0.0463071	+	0.998927i	$0.485255\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.93056	0.0815083
$562$	0	0
$563$	12.9619	0.546281	0.273140	−	0.961974i	$-0.411938\pi$
$563$	12.9619	0.546281	0.273140	+	0.961974i	$0.411938\pi$
$564$	0	0
$565$	−21.3302	−0.897367
$566$	0	0
$567$	−8.31746	−0.349301
$568$	0	0
$569$	−21.4158	−0.897798	−0.448899	−	0.893583i	$-0.648184\pi$
$569$	−21.4158	−0.897798	−0.448899	+	0.893583i	$0.648184\pi$
$570$	0	0
$571$	−22.1427	−0.926644	−0.463322	−	0.886190i	$-0.653343\pi$
$571$	−22.1427	−0.926644	−0.463322	+	0.886190i	$0.653343\pi$
$572$	0	0
$573$	6.44353	0.269182
$574$	0	0
$575$	−18.6123	−0.776188
$576$	0	0
$577$	14.2201	0.591989	0.295995	−	0.955190i	$-0.404349\pi$
$577$	14.2201	0.591989	0.295995	+	0.955190i	$0.404349\pi$
$578$	0	0
$579$	−7.18879	−0.298756
$580$	0	0
$581$	5.95108	0.246893
$582$	0	0
$583$	3.55441	0.147209
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.9528	−1.44266	−0.721329	−	0.692593i	$-0.756468\pi$
$587$	−34.9528	−1.44266	−0.721329	+	0.692593i	$0.756468\pi$
$588$	0	0
$589$	0.260538	0.0107353
$590$	0	0
$591$	3.58086	0.147297
$592$	0	0
$593$	−39.1300	−1.60688	−0.803438	−	0.595389i	$-0.796998\pi$
$593$	−39.1300	−1.60688	−0.803438	+	0.595389i	$0.796998\pi$
$594$	0	0
$595$	−4.94900	−0.202889
$596$	0	0
$597$	1.18405	0.0484600
$598$	0	0
$599$	13.8757	0.566946	0.283473	−	0.958980i	$-0.408513\pi$
$599$	13.8757	0.566946	0.283473	+	0.958980i	$0.408513\pi$
$600$	0	0
$601$	−17.8128	−0.726598	−0.363299	−	0.931673i	$-0.618350\pi$
$601$	−17.8128	−0.726598	−0.363299	+	0.931673i	$0.618350\pi$
$602$	0	0
$603$	23.5151	0.957607
$604$	0	0
$605$	−9.54817	−0.388188
$606$	0	0
$607$	31.0141	1.25882	0.629411	−	0.777072i	$-0.283296\pi$
$607$	31.0141	1.25882	0.629411	+	0.777072i	$0.283296\pi$
$608$	0	0
$609$	−1.85470	−0.0751564
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−21.3918	−0.864005	−0.432003	−	0.901872i	$-0.642193\pi$
$613$	−21.3918	−0.864005	−0.432003	+	0.901872i	$0.642193\pi$
$614$	0	0
$615$	−3.61786	−0.145886
$616$	0	0
$617$	−29.8127	−1.20021	−0.600107	−	0.799919i	$-0.704875\pi$
$617$	−29.8127	−1.20021	−0.600107	+	0.799919i	$0.704875\pi$
$618$	0	0
$619$	31.0495	1.24799	0.623993	−	0.781430i	$-0.285510\pi$
$619$	31.0495	1.24799	0.623993	+	0.781430i	$0.285510\pi$
$620$	0	0
$621$	12.1184	0.486294
$622$	0	0
$623$	−1.13368	−0.0454197
$624$	0	0
$625$	−6.08857	−0.243543
$626$	0	0
$627$	0.0838611	0.00334909
$628$	0	0
$629$	−4.12708	−0.164557
$630$	0	0
$631$	−9.26400	−0.368794	−0.184397	−	0.982852i	$-0.559033\pi$
$631$	−9.26400	−0.368794	−0.184397	+	0.982852i	$0.559033\pi$
$632$	0	0
$633$	4.55325	0.180975
$634$	0	0
$635$	0.430098	0.0170679
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.34943	−0.211620
$640$	0	0
$641$	7.77540	0.307110	0.153555	−	0.988140i	$-0.450928\pi$
$641$	7.77540	0.307110	0.153555	+	0.988140i	$0.450928\pi$
$642$	0	0
$643$	33.1937	1.30903	0.654516	−	0.756048i	$-0.272873\pi$
$643$	33.1937	1.30903	0.654516	+	0.756048i	$0.272873\pi$
$644$	0	0
$645$	2.07256	0.0816071
$646$	0	0
$647$	−35.6197	−1.40036	−0.700178	−	0.713968i	$-0.746896\pi$
$647$	−35.6197	−1.40036	−0.700178	+	0.713968i	$0.746896\pi$
$648$	0	0
$649$	21.8515	0.857747
$650$	0	0
$651$	0.528248	0.0207037
$652$	0	0
$653$	7.15438	0.279972	0.139986	−	0.990153i	$-0.455294\pi$
$653$	7.15438	0.279972	0.139986	+	0.990153i	$0.455294\pi$
$654$	0	0
$655$	17.5775	0.686809
$656$	0	0
$657$	−27.3950	−1.06878
$658$	0	0
$659$	−16.7681	−0.653194	−0.326597	−	0.945164i	$-0.605902\pi$
$659$	−16.7681	−0.653194	−0.326597	+	0.945164i	$0.605902\pi$
$660$	0	0
$661$	−28.5287	−1.10964	−0.554819	−	0.831971i	$-0.687213\pi$
$661$	−28.5287	−1.10964	−0.554819	+	0.831971i	$0.687213\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.214978	−0.00833650
$666$	0	0
$667$	−49.6076	−1.92082
$668$	0	0
$669$	−0.309819	−0.0119783
$670$	0	0
$671$	4.56835	0.176359
$672$	0	0
$673$	−38.4895	−1.48366	−0.741830	−	0.670588i	$-0.766042\pi$
$673$	−38.4895	−1.48366	−0.741830	+	0.670588i	$0.766042\pi$
$674$	0	0
$675$	−4.12198	−0.158655
$676$	0	0
$677$	−3.30956	−0.127197	−0.0635983	−	0.997976i	$-0.520258\pi$
$677$	−3.30956	−0.127197	−0.0635983	+	0.997976i	$0.520258\pi$
$678$	0	0
$679$	10.0696	0.386435
$680$	0	0
$681$	−6.58044	−0.252163
$682$	0	0
$683$	26.4438	1.01184	0.505921	−	0.862580i	$-0.331153\pi$
$683$	26.4438	1.01184	0.505921	+	0.862580i	$0.331153\pi$
$684$	0	0
$685$	−29.3344	−1.12081
$686$	0	0
$687$	0.427451	0.0163083
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.39755	0.243374	0.121687	−	0.992569i	$-0.461170\pi$
$691$	6.39755	0.243374	0.121687	+	0.992569i	$0.461170\pi$
$692$	0	0
$693$	−6.49892	−0.246873
$694$	0	0
$695$	25.7802	0.977898
$696$	0	0
$697$	26.0641	0.987246
$698$	0	0
$699$	2.93753	0.111108
$700$	0	0
$701$	−5.58236	−0.210843	−0.105421	−	0.994428i	$-0.533619\pi$
$701$	−5.58236	−0.210843	−0.105421	+	0.994428i	$0.533619\pi$
$702$	0	0
$703$	−0.179275	−0.00676148
$704$	0	0
$705$	−2.58059	−0.0971908
$706$	0	0
$707$	1.15040	0.0432651
$708$	0	0
$709$	−44.5771	−1.67413	−0.837064	−	0.547105i	$-0.815730\pi$
$709$	−44.5771	−1.67413	−0.837064	+	0.547105i	$0.815730\pi$
$710$	0	0
$711$	−20.2102	−0.757941
$712$	0	0
$713$	14.1290	0.529136
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.224391	−0.00838003
$718$	0	0
$719$	29.3603	1.09495	0.547477	−	0.836821i	$-0.315588\pi$
$719$	29.3603	1.09495	0.547477	+	0.836821i	$0.315588\pi$
$720$	0	0
$721$	−18.3301	−0.682649
$722$	0	0
$723$	3.29237	0.122444
$724$	0	0
$725$	16.8737	0.626672
$726$	0	0
$727$	39.9756	1.48261	0.741307	−	0.671166i	$-0.234206\pi$
$727$	39.9756	1.48261	0.741307	+	0.671166i	$0.234206\pi$
$728$	0	0
$729$	−22.9570	−0.850258
$730$	0	0
$731$	−14.9313	−0.552255
$732$	0	0
$733$	−49.7640	−1.83807	−0.919037	−	0.394171i	$-0.871032\pi$
$733$	−49.7640	−1.83807	−0.919037	+	0.394171i	$0.871032\pi$
$734$	0	0
$735$	−0.435874	−0.0160775
$736$	0	0
$737$	17.8804	0.658633
$738$	0	0
$739$	28.7646	1.05812	0.529061	−	0.848584i	$-0.322544\pi$
$739$	28.7646	1.05812	0.529061	+	0.848584i	$0.322544\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−50.9933	−1.87076	−0.935381	−	0.353642i	$-0.884943\pi$
$743$	−50.9933	−1.87076	−0.935381	+	0.353642i	$0.884943\pi$
$744$	0	0
$745$	−15.3253	−0.561475
$746$	0	0
$747$	17.3981	0.636562
$748$	0	0
$749$	14.6609	0.535697
$750$	0	0
$751$	8.24908	0.301013	0.150507	−	0.988609i	$-0.451910\pi$
$751$	8.24908	0.301013	0.150507	+	0.988609i	$0.451910\pi$
$752$	0	0
$753$	−0.408267	−0.0148781
$754$	0	0
$755$	−7.28247	−0.265036
$756$	0	0
$757$	−24.9361	−0.906317	−0.453159	−	0.891430i	$-0.649703\pi$
$757$	−24.9361	−0.906317	−0.453159	+	0.891430i	$0.649703\pi$
$758$	0	0
$759$	4.54780	0.165075
$760$	0	0
$761$	31.4535	1.14019	0.570094	−	0.821579i	$-0.306907\pi$
$761$	31.4535	1.14019	0.570094	+	0.821579i	$0.306907\pi$
$762$	0	0
$763$	0.304515	0.0110242
$764$	0	0
$765$	−14.4685	−0.523108
$766$	0	0
$767$	0	0
$768$	0	0
$769$	8.02969	0.289558	0.144779	−	0.989464i	$-0.453753\pi$
$769$	8.02969	0.289558	0.144779	+	0.989464i	$0.453753\pi$
$770$	0	0
$771$	5.18717	0.186811
$772$	0	0
$773$	−6.14041	−0.220855	−0.110428	−	0.993884i	$-0.535222\pi$
$773$	−6.14041	−0.220855	−0.110428	+	0.993884i	$0.535222\pi$
$774$	0	0
$775$	−4.80588	−0.172632
$776$	0	0
$777$	−0.363485	−0.0130400
$778$	0	0
$779$	1.13219	0.0405649
$780$	0	0
$781$	−4.06760	−0.145550
$782$	0	0
$783$	−10.9864	−0.392620
$784$	0	0
$785$	−16.7507	−0.597859
$786$	0	0
$787$	21.5467	0.768058	0.384029	−	0.923321i	$-0.374536\pi$
$787$	21.5467	0.768058	0.384029	+	0.923321i	$0.374536\pi$
$788$	0	0
$789$	1.34247	0.0477930
$790$	0	0
$791$	13.5341	0.481217
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.696937	0.0247178
$796$	0	0
$797$	−27.1682	−0.962348	−0.481174	−	0.876625i	$-0.659789\pi$
$797$	−27.1682	−0.962348	−0.481174	+	0.876625i	$0.659789\pi$
$798$	0	0
$799$	18.5913	0.657713
$800$	0	0
$801$	−3.31431	−0.117106
$802$	0	0
$803$	−20.8306	−0.735096
$804$	0	0
$805$	−11.6583	−0.410901
$806$	0	0
$807$	−4.28925	−0.150989
$808$	0	0
$809$	−22.4138	−0.788026	−0.394013	−	0.919105i	$-0.628913\pi$
$809$	−22.4138	−0.788026	−0.394013	+	0.919105i	$0.628913\pi$
$810$	0	0
$811$	−12.5372	−0.440242	−0.220121	−	0.975473i	$-0.570645\pi$
$811$	−12.5372	−0.440242	−0.220121	+	0.975473i	$0.570645\pi$
$812$	0	0
$813$	−3.92288	−0.137581
$814$	0	0
$815$	1.54596	0.0541525
$816$	0	0
$817$	−0.648598	−0.0226916
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.2933	0.882741	0.441370	−	0.897325i	$-0.354492\pi$
$821$	25.2933	0.882741	0.441370	+	0.897325i	$0.354492\pi$
$822$	0	0
$823$	−22.1814	−0.773194	−0.386597	−	0.922249i	$-0.626350\pi$
$823$	−22.1814	−0.773194	−0.386597	+	0.922249i	$0.626350\pi$
$824$	0	0
$825$	−1.54690	−0.0538562
$826$	0	0
$827$	−25.9044	−0.900784	−0.450392	−	0.892831i	$-0.648716\pi$
$827$	−25.9044	−0.900784	−0.450392	+	0.892831i	$0.648716\pi$
$828$	0	0
$829$	41.3884	1.43748	0.718739	−	0.695280i	$-0.244720\pi$
$829$	41.3884	1.43748	0.718739	+	0.695280i	$0.244720\pi$
$830$	0	0
$831$	7.94776	0.275705
$832$	0	0
$833$	3.14016	0.108800
$834$	0	0
$835$	19.7291	0.682754
$836$	0	0
$837$	3.12908	0.108157
$838$	0	0
$839$	−13.2374	−0.457006	−0.228503	−	0.973543i	$-0.573383\pi$
$839$	−13.2374	−0.457006	−0.228503	+	0.973543i	$0.573383\pi$
$840$	0	0
$841$	15.9736	0.550813
$842$	0	0
$843$	4.42545	0.152421
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.05835	0.208167
$848$	0	0
$849$	2.22716	0.0764361
$850$	0	0
$851$	−9.72211	−0.333270
$852$	0	0
$853$	28.5605	0.977893	0.488946	−	0.872314i	$-0.337381\pi$
$853$	28.5605	0.977893	0.488946	+	0.872314i	$0.337381\pi$
$854$	0	0
$855$	−0.628491	−0.0214940
$856$	0	0
$857$	−17.0293	−0.581711	−0.290855	−	0.956767i	$-0.593940\pi$
$857$	−17.0293	−0.581711	−0.290855	+	0.956767i	$0.593940\pi$
$858$	0	0
$859$	7.70031	0.262731	0.131366	−	0.991334i	$-0.458064\pi$
$859$	7.70031	0.262731	0.131366	+	0.991334i	$0.458064\pi$
$860$	0	0
$861$	2.29555	0.0782320
$862$	0	0
$863$	−47.5912	−1.62002	−0.810012	−	0.586413i	$-0.800540\pi$
$863$	−47.5912	−1.62002	−0.810012	+	0.586413i	$0.800540\pi$
$864$	0	0
$865$	−24.1113	−0.819810
$866$	0	0
$867$	1.97450	0.0670576
$868$	0	0
$869$	−15.3674	−0.521304
$870$	0	0
$871$	0	0
$872$	0	0
$873$	29.4385	0.996343
$874$	0	0
$875$	11.8457	0.400456
$876$	0	0
$877$	29.9394	1.01098	0.505491	−	0.862832i	$-0.331311\pi$
$877$	29.9394	1.01098	0.505491	+	0.862832i	$0.331311\pi$
$878$	0	0
$879$	−6.39005	−0.215531
$880$	0	0
$881$	20.0812	0.676552	0.338276	−	0.941047i	$-0.390156\pi$
$881$	20.0812	0.676552	0.338276	+	0.941047i	$0.390156\pi$
$882$	0	0
$883$	12.3122	0.414339	0.207170	−	0.978305i	$-0.433575\pi$
$883$	12.3122	0.414339	0.207170	+	0.978305i	$0.433575\pi$
$884$	0	0
$885$	4.28457	0.144024
$886$	0	0
$887$	−16.5630	−0.556130	−0.278065	−	0.960562i	$-0.589693\pi$
$887$	−16.5630	−0.556130	−0.278065	+	0.960562i	$0.589693\pi$
$888$	0	0
$889$	−0.272899	−0.00915274
$890$	0	0
$891$	−18.4896	−0.619424
$892$	0	0
$893$	0.807583	0.0270247
$894$	0	0
$895$	−22.6451	−0.756941
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−12.8092	−0.427209
$900$	0	0
$901$	−5.02093	−0.167271
$902$	0	0
$903$	−1.31505	−0.0437621
$904$	0	0
$905$	−30.0124	−0.997647
$906$	0	0
$907$	21.9305	0.728191	0.364095	−	0.931362i	$-0.381378\pi$
$907$	21.9305	0.728191	0.364095	+	0.931362i	$0.381378\pi$
$908$	0	0
$909$	3.36320	0.111550
$910$	0	0
$911$	25.1692	0.833894	0.416947	−	0.908931i	$-0.363100\pi$
$911$	25.1692	0.833894	0.416947	+	0.908931i	$0.363100\pi$
$912$	0	0
$913$	13.2292	0.437821
$914$	0	0
$915$	0.895747	0.0296125
$916$	0	0
$917$	−11.1530	−0.368304
$918$	0	0
$919$	−47.7606	−1.57548	−0.787738	−	0.616010i	$-0.788748\pi$
$919$	−47.7606	−1.57548	−0.787738	+	0.616010i	$0.788748\pi$
$920$	0	0
$921$	−1.36111	−0.0448500
$922$	0	0
$923$	0	0
$924$	0	0
$925$	3.30690	0.108730
$926$	0	0
$927$	−53.5883	−1.76007
$928$	0	0
$929$	−58.3477	−1.91433	−0.957163	−	0.289551i	$-0.906494\pi$
$929$	−58.3477	−1.91433	−0.957163	+	0.289551i	$0.906494\pi$
$930$	0	0
$931$	0.136405	0.00447048
$932$	0	0
$933$	1.21078	0.0396391
$934$	0	0
$935$	−11.0015	−0.359789
$936$	0	0
$937$	−6.46313	−0.211141	−0.105571	−	0.994412i	$-0.533667\pi$
$937$	−6.46313	−0.211141	−0.105571	+	0.994412i	$0.533667\pi$
$938$	0	0
$939$	5.60882	0.183037
$940$	0	0
$941$	2.77909	0.0905957	0.0452979	−	0.998974i	$-0.485576\pi$
$941$	2.77909	0.0905957	0.0452979	+	0.998974i	$0.485576\pi$
$942$	0	0
$943$	61.3988	1.99942
$944$	0	0
$945$	−2.58191	−0.0839894
$946$	0	0
$947$	56.7044	1.84265	0.921323	−	0.388797i	$-0.127109\pi$
$947$	56.7044	1.84265	0.921323	+	0.388797i	$0.127109\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.50680	0.0488615
$952$	0	0
$953$	−8.27003	−0.267893	−0.133946	−	0.990989i	$-0.542765\pi$
$953$	−8.27003	−0.267893	−0.133946	+	0.990989i	$0.542765\pi$
$954$	0	0
$955$	−36.7192	−1.18821
$956$	0	0
$957$	−4.12297	−0.133277
$958$	0	0
$959$	18.6128	0.601039
$960$	0	0
$961$	−27.3518	−0.882315
$962$	0	0
$963$	42.8613	1.38119
$964$	0	0
$965$	40.9662	1.31875
$966$	0	0
$967$	−13.7191	−0.441176	−0.220588	−	0.975367i	$-0.570798\pi$
$967$	−13.7191	−0.441176	−0.220588	+	0.975367i	$0.570798\pi$
$968$	0	0
$969$	−0.118461	−0.00380552
$970$	0	0
$971$	−51.0197	−1.63730	−0.818651	−	0.574292i	$-0.805278\pi$
$971$	−51.0197	−1.63730	−0.818651	+	0.574292i	$0.805278\pi$
$972$	0	0
$973$	−16.3576	−0.524401
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.3908	1.03627	0.518137	−	0.855297i	$-0.326626\pi$
$977$	32.3908	1.03627	0.518137	+	0.855297i	$0.326626\pi$
$978$	0	0
$979$	−2.52014	−0.0805440
$980$	0	0
$981$	0.890254	0.0284236
$982$	0	0
$983$	−7.96805	−0.254141	−0.127071	−	0.991894i	$-0.540557\pi$
$983$	−7.96805	−0.254141	−0.127071	+	0.991894i	$0.540557\pi$
$984$	0	0
$985$	−20.4059	−0.650188
$986$	0	0
$987$	1.63740	0.0521189
$988$	0	0
$989$	−35.1736	−1.11845
$990$	0	0
$991$	−2.33295	−0.0741087	−0.0370543	−	0.999313i	$-0.511797\pi$
$991$	−2.33295	−0.0741087	−0.0370543	+	0.999313i	$0.511797\pi$
$992$	0	0
$993$	9.35061	0.296733
$994$	0	0
$995$	−6.74746	−0.213909
$996$	0	0
$997$	−44.0418	−1.39482	−0.697409	−	0.716674i	$-0.745664\pi$
$997$	−44.0418	−1.39482	−0.697409	+	0.716674i	$0.745664\pi$
$998$	0	0
$999$	−2.15311	−0.0681213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.bf.1.3		6
13.2	odd	12		728.2.bm.b.225.4	✓	12
13.7	odd	12		728.2.bm.b.673.4	yes	12
13.12	even	2		9464.2.a.bg.1.3		6
52.7	even	12		1456.2.cc.e.673.3		12
52.15	even	12		1456.2.cc.e.225.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.bm.b.225.4	✓	12	13.2	odd	12
728.2.bm.b.673.4	yes	12	13.7	odd	12
1456.2.cc.e.225.3		12	52.15	even	12
1456.2.cc.e.673.3		12	52.7	even	12
9464.2.a.bf.1.3		6	1.1	even	1	trivial
9464.2.a.bg.1.3		6	13.12	even	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:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q-0.276564 q^{3} +1.57603 q^{5} -1.00000 q^{7} -2.92351 q^{9} -2.22298 q^{11} -0.435874 q^{15} +3.14016 q^{17} +0.136405 q^{19} +0.276564 q^{21} +7.39724 q^{23} -2.51612 q^{25} +1.63823 q^{27} -6.70623 q^{29} +1.91004 q^{31} +0.614797 q^{33} -1.57603 q^{35} -1.31429 q^{37} +8.30023 q^{41} -4.75496 q^{43} -4.60755 q^{45} +5.92050 q^{47} +1.00000 q^{49} -0.868455 q^{51} -1.59894 q^{53} -3.50350 q^{55} -0.0377246 q^{57} -9.82982 q^{59} -2.05506 q^{61} +2.92351 q^{63} -8.04343 q^{67} -2.04581 q^{69} +1.82979 q^{71} +9.37056 q^{73} +0.695868 q^{75} +2.22298 q^{77} +6.91298 q^{79} +8.31746 q^{81} -5.95108 q^{83} +4.94900 q^{85} +1.85470 q^{87} +1.13368 q^{89} -0.528248 q^{93} +0.214978 q^{95} -10.0696 q^{97} +6.49892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 4 q^{5} - 6 q^{7} - 4 q^{9} + 10 q^{11} - 6 q^{15} + 2 q^{17} + 14 q^{19} - 2 q^{21} - 6 q^{23} - 2 q^{25} + 2 q^{27} - 20 q^{29} - 4 q^{31} + 4 q^{33} + 4 q^{35} - 4 q^{37} + 10 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 4 * q^5 - 6 * q^7 - 4 * q^9 + 10 * q^11 - 6 * q^15 + 2 * q^17 + 14 * q^19 - 2 * q^21 - 6 * q^23 - 2 * q^25 + 2 * q^27 - 20 * q^29 - 4 * q^31 + 4 * q^33 + 4 * q^35 - 4 * q^37 + 10 * q^41 - 12 * q^43 + 6 * q^45 - 6 * q^47 + 6 * q^49 - 8 * q^51 + 10 * q^53 - 24 * q^55 - 8 * q^59 + 4 * q^61 + 4 * q^63 - 18 * q^67 + 12 * q^71 + 14 * q^73 - 10 * q^75 - 10 * q^77 - 10 * q^79 - 18 * q^81 + 28 * q^83 + 20 * q^85 - 32 * q^87 - 6 * q^89 + 12 * q^93 - 26 * q^95 + 8 * q^97 + 2 * q^99

Introduction

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