LMFDB - Embedded newform 896.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(1,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("896.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	896.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{3} +0.732051 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})$	$q+2.73205 q^{3} +0.732051 q^{5} -1.00000 q^{7} +4.46410 q^{9} +5.46410 q^{11} +0.732051 q^{13} +2.00000 q^{15} +0.535898 q^{17} -4.19615 q^{19} -2.73205 q^{21} -4.92820 q^{23} -4.46410 q^{25} +4.00000 q^{27} +3.46410 q^{29} +9.46410 q^{31} +14.9282 q^{33} -0.732051 q^{35} -11.4641 q^{37} +2.00000 q^{39} +3.46410 q^{41} +9.46410 q^{43} +3.26795 q^{45} +5.46410 q^{47} +1.00000 q^{49} +1.46410 q^{51} -6.00000 q^{53} +4.00000 q^{55} -11.4641 q^{57} -4.19615 q^{59} -13.1244 q^{61} -4.46410 q^{63} +0.535898 q^{65} +1.07180 q^{67} -13.4641 q^{69} +13.8564 q^{71} -4.92820 q^{73} -12.1962 q^{75} -5.46410 q^{77} +8.00000 q^{79} -2.46410 q^{81} -10.7321 q^{83} +0.392305 q^{85} +9.46410 q^{87} -8.92820 q^{89} -0.732051 q^{91} +25.8564 q^{93} -3.07180 q^{95} +7.46410 q^{97} +24.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{15} + 8 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} - 2 q^{25} + 8 q^{27} + 12 q^{31} + 16 q^{33} + 2 q^{35} - 16 q^{37} + 4 q^{39} + 12 q^{43} + 10 q^{45} + 4 q^{47} + 2 q^{49} - 4 q^{51} - 12 q^{53} + 8 q^{55} - 16 q^{57} + 2 q^{59} - 2 q^{61} - 2 q^{63} + 8 q^{65} + 16 q^{67} - 20 q^{69} + 4 q^{73} - 14 q^{75} - 4 q^{77} + 16 q^{79} + 2 q^{81} - 18 q^{83} - 20 q^{85} + 12 q^{87} - 4 q^{89} + 2 q^{91} + 24 q^{93} - 20 q^{95} + 8 q^{97} + 28 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 4 * q^15 + 8 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 - 2 * q^25 + 8 * q^27 + 12 * q^31 + 16 * q^33 + 2 * q^35 - 16 * q^37 + 4 * q^39 + 12 * q^43 + 10 * q^45 + 4 * q^47 + 2 * q^49 - 4 * q^51 - 12 * q^53 + 8 * q^55 - 16 * q^57 + 2 * q^59 - 2 * q^61 - 2 * q^63 + 8 * q^65 + 16 * q^67 - 20 * q^69 + 4 * q^73 - 14 * q^75 - 4 * q^77 + 16 * q^79 + 2 * q^81 - 18 * q^83 - 20 * q^85 + 12 * q^87 - 4 * q^89 + 2 * q^91 + 24 * q^93 - 20 * q^95 + 8 * q^97 + 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	0.535898	0.129974	0.0649872	−	0.997886i	$-0.479299\pi$
$17$	0.535898	0.129974	0.0649872	+	0.997886i	$0.479299\pi$
$18$	0	0
$19$	−4.19615	−0.962663	−0.481332	−	0.876539i	$-0.659847\pi$
$19$	−4.19615	−0.962663	−0.481332	+	0.876539i	$0.659847\pi$
$20$	0	0
$21$	−2.73205	−0.596182
$22$	0	0
$23$	−4.92820	−1.02760	−0.513801	−	0.857910i	$-0.671763\pi$
$23$	−4.92820	−1.02760	−0.513801	+	0.857910i	$0.671763\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	9.46410	1.69980	0.849901	−	0.526942i	$-0.176661\pi$
$31$	9.46410	1.69980	0.849901	+	0.526942i	$0.176661\pi$
$32$	0	0
$33$	14.9282	2.59867
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	−11.4641	−1.88469	−0.942343	−	0.334648i	$-0.891383\pi$
$37$	−11.4641	−1.88469	−0.942343	+	0.334648i	$0.891383\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0	0
$43$	9.46410	1.44326	0.721631	−	0.692278i	$-0.243393\pi$
$43$	9.46410	1.44326	0.721631	+	0.692278i	$0.243393\pi$
$44$	0	0
$45$	3.26795	0.487157
$46$	0	0
$47$	5.46410	0.797021	0.398511	−	0.917164i	$-0.369527\pi$
$47$	5.46410	0.797021	0.398511	+	0.917164i	$0.369527\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.46410	0.205015
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−11.4641	−1.51846
$58$	0	0
$59$	−4.19615	−0.546293	−0.273146	−	0.961973i	$-0.588064\pi$
$59$	−4.19615	−0.546293	−0.273146	+	0.961973i	$0.588064\pi$
$60$	0	0
$61$	−13.1244	−1.68040	−0.840201	−	0.542275i	$-0.817563\pi$
$61$	−13.1244	−1.68040	−0.840201	+	0.542275i	$0.817563\pi$
$62$	0	0
$63$	−4.46410	−0.562424
$64$	0	0
$65$	0.535898	0.0664700
$66$	0	0
$67$	1.07180	0.130941	0.0654704	−	0.997855i	$-0.479145\pi$
$67$	1.07180	0.130941	0.0654704	+	0.997855i	$0.479145\pi$
$68$	0	0
$69$	−13.4641	−1.62089
$70$	0	0
$71$	13.8564	1.64445	0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564	1.64445	0.822226	+	0.569160i	$0.192732\pi$
$72$	0	0
$73$	−4.92820	−0.576803	−0.288401	−	0.957510i	$-0.593124\pi$
$73$	−4.92820	−0.576803	−0.288401	+	0.957510i	$0.593124\pi$
$74$	0	0
$75$	−12.1962	−1.40829
$76$	0	0
$77$	−5.46410	−0.622692
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−10.7321	−1.17800	−0.588998	−	0.808135i	$-0.700477\pi$
$83$	−10.7321	−1.17800	−0.588998	+	0.808135i	$0.700477\pi$
$84$	0	0
$85$	0.392305	0.0425514
$86$	0	0
$87$	9.46410	1.01466
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	−0.732051	−0.0767398
$92$	0	0
$93$	25.8564	2.68118
$94$	0	0
$95$	−3.07180	−0.315160
$96$	0	0
$97$	7.46410	0.757865	0.378932	−	0.925424i	$-0.376291\pi$
$97$	7.46410	0.757865	0.378932	+	0.925424i	$0.376291\pi$
$98$	0	0
$99$	24.3923	2.45152
$100$	0	0
$101$	14.1962	1.41257	0.706285	−	0.707928i	$-0.250370\pi$
$101$	14.1962	1.41257	0.706285	+	0.707928i	$0.250370\pi$
$102$	0	0
$103$	−12.3923	−1.22105	−0.610525	−	0.791997i	$-0.709042\pi$
$103$	−12.3923	−1.22105	−0.610525	+	0.791997i	$0.709042\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−16.0000	−1.54678	−0.773389	−	0.633932i	$-0.781440\pi$
$107$	−16.0000	−1.54678	−0.773389	+	0.633932i	$0.781440\pi$
$108$	0	0
$109$	−3.46410	−0.331801	−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410	−0.331801	−0.165900	+	0.986143i	$0.553053\pi$
$110$	0	0
$111$	−31.3205	−2.97281
$112$	0	0
$113$	−20.3923	−1.91835	−0.959173	−	0.282819i	$-0.908730\pi$
$113$	−20.3923	−1.91835	−0.959173	+	0.282819i	$0.908730\pi$
$114$	0	0
$115$	−3.60770	−0.336419
$116$	0	0
$117$	3.26795	0.302122
$118$	0	0
$119$	−0.535898	−0.0491257
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	9.46410	0.853349
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	25.8564	2.27653
$130$	0	0
$131$	8.19615	0.716101	0.358051	−	0.933702i	$-0.383442\pi$
$131$	8.19615	0.716101	0.358051	+	0.933702i	$0.383442\pi$
$132$	0	0
$133$	4.19615	0.363853
$134$	0	0
$135$	2.92820	0.252020
$136$	0	0
$137$	−7.07180	−0.604184	−0.302092	−	0.953279i	$-0.597685\pi$
$137$	−7.07180	−0.604184	−0.302092	+	0.953279i	$0.597685\pi$
$138$	0	0
$139$	−1.66025	−0.140821	−0.0704105	−	0.997518i	$-0.522431\pi$
$139$	−1.66025	−0.140821	−0.0704105	+	0.997518i	$0.522431\pi$
$140$	0	0
$141$	14.9282	1.25718
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	2.53590	0.210595
$146$	0	0
$147$	2.73205	0.225336
$148$	0	0
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	2.39230	0.193406
$154$	0	0
$155$	6.92820	0.556487
$156$	0	0
$157$	16.7321	1.33536	0.667682	−	0.744447i	$-0.267287\pi$
$157$	16.7321	1.33536	0.667682	+	0.744447i	$0.267287\pi$
$158$	0	0
$159$	−16.3923	−1.29999
$160$	0	0
$161$	4.92820	0.388397
$162$	0	0
$163$	−0.392305	−0.0307277	−0.0153638	−	0.999882i	$-0.504891\pi$
$163$	−0.392305	−0.0307277	−0.0153638	+	0.999882i	$0.504891\pi$
$164$	0	0
$165$	10.9282	0.850759
$166$	0	0
$167$	−13.4641	−1.04188	−0.520942	−	0.853592i	$-0.674419\pi$
$167$	−13.4641	−1.04188	−0.520942	+	0.853592i	$0.674419\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	−18.7321	−1.43248
$172$	0	0
$173$	11.6603	0.886513	0.443256	−	0.896395i	$-0.353823\pi$
$173$	11.6603	0.886513	0.443256	+	0.896395i	$0.353823\pi$
$174$	0	0
$175$	4.46410	0.337454
$176$	0	0
$177$	−11.4641	−0.861695
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−15.6603	−1.16402	−0.582009	−	0.813183i	$-0.697733\pi$
$181$	−15.6603	−1.16402	−0.582009	+	0.813183i	$0.697733\pi$
$182$	0	0
$183$	−35.8564	−2.65058
$184$	0	0
$185$	−8.39230	−0.617015
$186$	0	0
$187$	2.92820	0.214131
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−2.92820	−0.211877	−0.105939	−	0.994373i	$-0.533785\pi$
$191$	−2.92820	−0.211877	−0.105939	+	0.994373i	$0.533785\pi$
$192$	0	0
$193$	8.39230	0.604091	0.302046	−	0.953293i	$-0.402330\pi$
$193$	8.39230	0.604091	0.302046	+	0.953293i	$0.402330\pi$
$194$	0	0
$195$	1.46410	0.104846
$196$	0	0
$197$	−3.85641	−0.274758	−0.137379	−	0.990519i	$-0.543868\pi$
$197$	−3.85641	−0.274758	−0.137379	+	0.990519i	$0.543868\pi$
$198$	0	0
$199$	25.4641	1.80510	0.902551	−	0.430583i	$-0.141692\pi$
$199$	25.4641	1.80510	0.902551	+	0.430583i	$0.141692\pi$
$200$	0	0
$201$	2.92820	0.206540
$202$	0	0
$203$	−3.46410	−0.243132
$204$	0	0
$205$	2.53590	0.177115
$206$	0	0
$207$	−22.0000	−1.52911
$208$	0	0
$209$	−22.9282	−1.58598
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	37.8564	2.59388
$214$	0	0
$215$	6.92820	0.472500
$216$	0	0
$217$	−9.46410	−0.642465
$218$	0	0
$219$	−13.4641	−0.909820
$220$	0	0
$221$	0.392305	0.0263893
$222$	0	0
$223$	1.85641	0.124314	0.0621571	−	0.998066i	$-0.480202\pi$
$223$	1.85641	0.124314	0.0621571	+	0.998066i	$0.480202\pi$
$224$	0	0
$225$	−19.9282	−1.32855
$226$	0	0
$227$	−20.1962	−1.34047	−0.670233	−	0.742151i	$-0.733806\pi$
$227$	−20.1962	−1.34047	−0.670233	+	0.742151i	$0.733806\pi$
$228$	0	0
$229$	−10.1962	−0.673781	−0.336890	−	0.941544i	$-0.609375\pi$
$229$	−10.1962	−0.673781	−0.336890	+	0.941544i	$0.609375\pi$
$230$	0	0
$231$	−14.9282	−0.982204
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	21.8564	1.41973
$238$	0	0
$239$	0.928203	0.0600405	0.0300202	−	0.999549i	$-0.490443\pi$
$239$	0.928203	0.0600405	0.0300202	+	0.999549i	$0.490443\pi$
$240$	0	0
$241$	12.5359	0.807508	0.403754	−	0.914868i	$-0.367705\pi$
$241$	12.5359	0.807508	0.403754	+	0.914868i	$0.367705\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	0.732051	0.0467690
$246$	0	0
$247$	−3.07180	−0.195454
$248$	0	0
$249$	−29.3205	−1.85811
$250$	0	0
$251$	15.8038	0.997530	0.498765	−	0.866737i	$-0.333787\pi$
$251$	15.8038	0.997530	0.498765	+	0.866737i	$0.333787\pi$
$252$	0	0
$253$	−26.9282	−1.69296
$254$	0	0
$255$	1.07180	0.0671185
$256$	0	0
$257$	−11.0718	−0.690640	−0.345320	−	0.938485i	$-0.612230\pi$
$257$	−11.0718	−0.690640	−0.345320	+	0.938485i	$0.612230\pi$
$258$	0	0
$259$	11.4641	0.712345
$260$	0	0
$261$	15.4641	0.957204
$262$	0	0
$263$	−13.8564	−0.854423	−0.427211	−	0.904152i	$-0.640504\pi$
$263$	−13.8564	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$264$	0	0
$265$	−4.39230	−0.269817
$266$	0	0
$267$	−24.3923	−1.49278
$268$	0	0
$269$	−24.0526	−1.46651	−0.733255	−	0.679954i	$-0.762000\pi$
$269$	−24.0526	−1.46651	−0.733255	+	0.679954i	$0.762000\pi$
$270$	0	0
$271$	22.9282	1.39279	0.696395	−	0.717659i	$-0.254786\pi$
$271$	22.9282	1.39279	0.696395	+	0.717659i	$0.254786\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−24.3923	−1.47091
$276$	0	0
$277$	0.928203	0.0557703	0.0278852	−	0.999611i	$-0.491123\pi$
$277$	0.928203	0.0557703	0.0278852	+	0.999611i	$0.491123\pi$
$278$	0	0
$279$	42.2487	2.52936
$280$	0	0
$281$	19.8564	1.18453	0.592267	−	0.805742i	$-0.298233\pi$
$281$	19.8564	1.18453	0.592267	+	0.805742i	$0.298233\pi$
$282$	0	0
$283$	22.0526	1.31089	0.655444	−	0.755244i	$-0.272481\pi$
$283$	22.0526	1.31089	0.655444	+	0.755244i	$0.272481\pi$
$284$	0	0
$285$	−8.39230	−0.497117
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	−16.7128	−0.983107
$290$	0	0
$291$	20.3923	1.19542
$292$	0	0
$293$	11.2679	0.658281	0.329140	−	0.944281i	$-0.393241\pi$
$293$	11.2679	0.658281	0.329140	+	0.944281i	$0.393241\pi$
$294$	0	0
$295$	−3.07180	−0.178847
$296$	0	0
$297$	21.8564	1.26824
$298$	0	0
$299$	−3.60770	−0.208638
$300$	0	0
$301$	−9.46410	−0.545502
$302$	0	0
$303$	38.7846	2.22812
$304$	0	0
$305$	−9.60770	−0.550135
$306$	0	0
$307$	−15.8038	−0.901973	−0.450987	−	0.892531i	$-0.648928\pi$
$307$	−15.8038	−0.901973	−0.450987	+	0.892531i	$0.648928\pi$
$308$	0	0
$309$	−33.8564	−1.92602
$310$	0	0
$311$	−2.92820	−0.166043	−0.0830216	−	0.996548i	$-0.526457\pi$
$311$	−2.92820	−0.166043	−0.0830216	+	0.996548i	$0.526457\pi$
$312$	0	0
$313$	19.4641	1.10018	0.550088	−	0.835107i	$-0.314594\pi$
$313$	19.4641	1.10018	0.550088	+	0.835107i	$0.314594\pi$
$314$	0	0
$315$	−3.26795	−0.184128
$316$	0	0
$317$	−0.928203	−0.0521331	−0.0260665	−	0.999660i	$-0.508298\pi$
$317$	−0.928203	−0.0521331	−0.0260665	+	0.999660i	$0.508298\pi$
$318$	0	0
$319$	18.9282	1.05978
$320$	0	0
$321$	−43.7128	−2.43981
$322$	0	0
$323$	−2.24871	−0.125122
$324$	0	0
$325$	−3.26795	−0.181273
$326$	0	0
$327$	−9.46410	−0.523366
$328$	0	0
$329$	−5.46410	−0.301246
$330$	0	0
$331$	1.46410	0.0804743	0.0402372	−	0.999190i	$-0.487189\pi$
$331$	1.46410	0.0804743	0.0402372	+	0.999190i	$0.487189\pi$
$332$	0	0
$333$	−51.1769	−2.80448
$334$	0	0
$335$	0.784610	0.0428678
$336$	0	0
$337$	22.2487	1.21196	0.605982	−	0.795478i	$-0.292780\pi$
$337$	22.2487	1.21196	0.605982	+	0.795478i	$0.292780\pi$
$338$	0	0
$339$	−55.7128	−3.02590
$340$	0	0
$341$	51.7128	2.80041
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−9.85641	−0.530651
$346$	0	0
$347$	14.5359	0.780328	0.390164	−	0.920745i	$-0.372418\pi$
$347$	14.5359	0.780328	0.390164	+	0.920745i	$0.372418\pi$
$348$	0	0
$349$	1.12436	0.0601854	0.0300927	−	0.999547i	$-0.490420\pi$
$349$	1.12436	0.0601854	0.0300927	+	0.999547i	$0.490420\pi$
$350$	0	0
$351$	2.92820	0.156296
$352$	0	0
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	0	0
$355$	10.1436	0.538366
$356$	0	0
$357$	−1.46410	−0.0774885
$358$	0	0
$359$	−15.8564	−0.836869	−0.418435	−	0.908247i	$-0.637421\pi$
$359$	−15.8564	−0.836869	−0.418435	+	0.908247i	$0.637421\pi$
$360$	0	0
$361$	−1.39230	−0.0732792
$362$	0	0
$363$	51.5167	2.70392
$364$	0	0
$365$	−3.60770	−0.188835
$366$	0	0
$367$	−12.7846	−0.667351	−0.333676	−	0.942688i	$-0.608289\pi$
$367$	−12.7846	−0.667351	−0.333676	+	0.942688i	$0.608289\pi$
$368$	0	0
$369$	15.4641	0.805029
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	−15.8564	−0.821014	−0.410507	−	0.911858i	$-0.634648\pi$
$373$	−15.8564	−0.821014	−0.410507	+	0.911858i	$0.634648\pi$
$374$	0	0
$375$	−18.9282	−0.977448
$376$	0	0
$377$	2.53590	0.130605
$378$	0	0
$379$	−10.2487	−0.526441	−0.263220	−	0.964736i	$-0.584785\pi$
$379$	−10.2487	−0.526441	−0.263220	+	0.964736i	$0.584785\pi$
$380$	0	0
$381$	5.46410	0.279934
$382$	0	0
$383$	26.5359	1.35592	0.677961	−	0.735098i	$-0.262864\pi$
$383$	26.5359	1.35592	0.677961	+	0.735098i	$0.262864\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	42.2487	2.14762
$388$	0	0
$389$	7.46410	0.378445	0.189222	−	0.981934i	$-0.439403\pi$
$389$	7.46410	0.378445	0.189222	+	0.981934i	$0.439403\pi$
$390$	0	0
$391$	−2.64102	−0.133562
$392$	0	0
$393$	22.3923	1.12954
$394$	0	0
$395$	5.85641	0.294668
$396$	0	0
$397$	22.9808	1.15337	0.576686	−	0.816966i	$-0.304346\pi$
$397$	22.9808	1.15337	0.576686	+	0.816966i	$0.304346\pi$
$398$	0	0
$399$	11.4641	0.573923
$400$	0	0
$401$	18.2487	0.911297	0.455649	−	0.890160i	$-0.349407\pi$
$401$	18.2487	0.911297	0.455649	+	0.890160i	$0.349407\pi$
$402$	0	0
$403$	6.92820	0.345118
$404$	0	0
$405$	−1.80385	−0.0896339
$406$	0	0
$407$	−62.6410	−3.10500
$408$	0	0
$409$	31.4641	1.55580	0.777900	−	0.628388i	$-0.216285\pi$
$409$	31.4641	1.55580	0.777900	+	0.628388i	$0.216285\pi$
$410$	0	0
$411$	−19.3205	−0.953010
$412$	0	0
$413$	4.19615	0.206479
$414$	0	0
$415$	−7.85641	−0.385656
$416$	0	0
$417$	−4.53590	−0.222124
$418$	0	0
$419$	7.80385	0.381243	0.190621	−	0.981664i	$-0.438950\pi$
$419$	7.80385	0.381243	0.190621	+	0.981664i	$0.438950\pi$
$420$	0	0
$421$	−7.07180	−0.344658	−0.172329	−	0.985039i	$-0.555129\pi$
$421$	−7.07180	−0.344658	−0.172329	+	0.985039i	$0.555129\pi$
$422$	0	0
$423$	24.3923	1.18599
$424$	0	0
$425$	−2.39230	−0.116044
$426$	0	0
$427$	13.1244	0.635132
$428$	0	0
$429$	10.9282	0.527619
$430$	0	0
$431$	−30.7846	−1.48284	−0.741421	−	0.671040i	$-0.765848\pi$
$431$	−30.7846	−1.48284	−0.741421	+	0.671040i	$0.765848\pi$
$432$	0	0
$433$	36.2487	1.74200	0.871001	−	0.491281i	$-0.163471\pi$
$433$	36.2487	1.74200	0.871001	+	0.491281i	$0.163471\pi$
$434$	0	0
$435$	6.92820	0.332182
$436$	0	0
$437$	20.6795	0.989234
$438$	0	0
$439$	39.7128	1.89539	0.947695	−	0.319179i	$-0.103407\pi$
$439$	39.7128	1.89539	0.947695	+	0.319179i	$0.103407\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	0	0
$443$	6.92820	0.329169	0.164584	−	0.986363i	$-0.447372\pi$
$443$	6.92820	0.329169	0.164584	+	0.986363i	$0.447372\pi$
$444$	0	0
$445$	−6.53590	−0.309831
$446$	0	0
$447$	−21.4641	−1.01522
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	18.9282	0.891294
$452$	0	0
$453$	5.46410	0.256726
$454$	0	0
$455$	−0.535898	−0.0251233
$456$	0	0
$457$	2.53590	0.118624	0.0593122	−	0.998239i	$-0.481109\pi$
$457$	2.53590	0.118624	0.0593122	+	0.998239i	$0.481109\pi$
$458$	0	0
$459$	2.14359	0.100054
$460$	0	0
$461$	−31.6603	−1.47457	−0.737283	−	0.675585i	$-0.763891\pi$
$461$	−31.6603	−1.47457	−0.737283	+	0.675585i	$0.763891\pi$
$462$	0	0
$463$	−19.7128	−0.916132	−0.458066	−	0.888918i	$-0.651458\pi$
$463$	−19.7128	−0.916132	−0.458066	+	0.888918i	$0.651458\pi$
$464$	0	0
$465$	18.9282	0.877774
$466$	0	0
$467$	34.4449	1.59392	0.796959	−	0.604033i	$-0.206441\pi$
$467$	34.4449	1.59392	0.796959	+	0.604033i	$0.206441\pi$
$468$	0	0
$469$	−1.07180	−0.0494910
$470$	0	0
$471$	45.7128	2.10634
$472$	0	0
$473$	51.7128	2.37776
$474$	0	0
$475$	18.7321	0.859485
$476$	0	0
$477$	−26.7846	−1.22638
$478$	0	0
$479$	4.39230	0.200690	0.100345	−	0.994953i	$-0.468005\pi$
$479$	4.39230	0.200690	0.100345	+	0.994953i	$0.468005\pi$
$480$	0	0
$481$	−8.39230	−0.382656
$482$	0	0
$483$	13.4641	0.612638
$484$	0	0
$485$	5.46410	0.248112
$486$	0	0
$487$	36.6410	1.66036	0.830181	−	0.557493i	$-0.188237\pi$
$487$	36.6410	1.66036	0.830181	+	0.557493i	$0.188237\pi$
$488$	0	0
$489$	−1.07180	−0.0484683
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	1.85641	0.0836083
$494$	0	0
$495$	17.8564	0.802586
$496$	0	0
$497$	−13.8564	−0.621545
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	−36.7846	−1.64342
$502$	0	0
$503$	22.9282	1.02232	0.511159	−	0.859486i	$-0.329216\pi$
$503$	22.9282	1.02232	0.511159	+	0.859486i	$0.329216\pi$
$504$	0	0
$505$	10.3923	0.462451
$506$	0	0
$507$	−34.0526	−1.51233
$508$	0	0
$509$	−12.7321	−0.564338	−0.282169	−	0.959365i	$-0.591054\pi$
$509$	−12.7321	−0.564338	−0.282169	+	0.959365i	$0.591054\pi$
$510$	0	0
$511$	4.92820	0.218011
$512$	0	0
$513$	−16.7846	−0.741059
$514$	0	0
$515$	−9.07180	−0.399751
$516$	0	0
$517$	29.8564	1.31308
$518$	0	0
$519$	31.8564	1.39834
$520$	0	0
$521$	9.60770	0.420921	0.210460	−	0.977602i	$-0.432504\pi$
$521$	9.60770	0.420921	0.210460	+	0.977602i	$0.432504\pi$
$522$	0	0
$523$	28.5885	1.25009	0.625043	−	0.780590i	$-0.285081\pi$
$523$	28.5885	1.25009	0.625043	+	0.780590i	$0.285081\pi$
$524$	0	0
$525$	12.1962	0.532284
$526$	0	0
$527$	5.07180	0.220931
$528$	0	0
$529$	1.28719	0.0559647
$530$	0	0
$531$	−18.7321	−0.812902
$532$	0	0
$533$	2.53590	0.109842
$534$	0	0
$535$	−11.7128	−0.506389
$536$	0	0
$537$	−32.7846	−1.41476
$538$	0	0
$539$	5.46410	0.235356
$540$	0	0
$541$	6.78461	0.291693	0.145847	−	0.989307i	$-0.453409\pi$
$541$	6.78461	0.291693	0.145847	+	0.989307i	$0.453409\pi$
$542$	0	0
$543$	−42.7846	−1.83606
$544$	0	0
$545$	−2.53590	−0.108626
$546$	0	0
$547$	24.3923	1.04294	0.521470	−	0.853270i	$-0.325384\pi$
$547$	24.3923	1.04294	0.521470	+	0.853270i	$0.325384\pi$
$548$	0	0
$549$	−58.5885	−2.50049
$550$	0	0
$551$	−14.5359	−0.619250
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	−22.9282	−0.973248
$556$	0	0
$557$	29.7128	1.25897	0.629486	−	0.777012i	$-0.283265\pi$
$557$	29.7128	1.25897	0.629486	+	0.777012i	$0.283265\pi$
$558$	0	0
$559$	6.92820	0.293032
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	15.1244	0.637416	0.318708	−	0.947853i	$-0.396751\pi$
$563$	15.1244	0.637416	0.318708	+	0.947853i	$0.396751\pi$
$564$	0	0
$565$	−14.9282	−0.628034
$566$	0	0
$567$	2.46410	0.103483
$568$	0	0
$569$	26.2487	1.10040	0.550202	−	0.835032i	$-0.314551\pi$
$569$	26.2487	1.10040	0.550202	+	0.835032i	$0.314551\pi$
$570$	0	0
$571$	11.6077	0.485767	0.242883	−	0.970055i	$-0.421907\pi$
$571$	11.6077	0.485767	0.242883	+	0.970055i	$0.421907\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	22.0000	0.917463
$576$	0	0
$577$	−4.92820	−0.205164	−0.102582	−	0.994725i	$-0.532710\pi$
$577$	−4.92820	−0.205164	−0.102582	+	0.994725i	$0.532710\pi$
$578$	0	0
$579$	22.9282	0.952864
$580$	0	0
$581$	10.7321	0.445240
$582$	0	0
$583$	−32.7846	−1.35780
$584$	0	0
$585$	2.39230	0.0989096
$586$	0	0
$587$	−20.5885	−0.849777	−0.424888	−	0.905246i	$-0.639687\pi$
$587$	−20.5885	−0.849777	−0.424888	+	0.905246i	$0.639687\pi$
$588$	0	0
$589$	−39.7128	−1.63634
$590$	0	0
$591$	−10.5359	−0.433389
$592$	0	0
$593$	−19.8564	−0.815405	−0.407702	−	0.913115i	$-0.633670\pi$
$593$	−19.8564	−0.815405	−0.407702	+	0.913115i	$0.633670\pi$
$594$	0	0
$595$	−0.392305	−0.0160829
$596$	0	0
$597$	69.5692	2.84728
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	12.9282	0.527352	0.263676	−	0.964611i	$-0.415065\pi$
$601$	12.9282	0.527352	0.263676	+	0.964611i	$0.415065\pi$
$602$	0	0
$603$	4.78461	0.194844
$604$	0	0
$605$	13.8038	0.561206
$606$	0	0
$607$	−13.0718	−0.530568	−0.265284	−	0.964170i	$-0.585466\pi$
$607$	−13.0718	−0.530568	−0.265284	+	0.964170i	$0.585466\pi$
$608$	0	0
$609$	−9.46410	−0.383505
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	−13.3205	−0.538010	−0.269005	−	0.963139i	$-0.586695\pi$
$613$	−13.3205	−0.538010	−0.269005	+	0.963139i	$0.586695\pi$
$614$	0	0
$615$	6.92820	0.279372
$616$	0	0
$617$	−19.3205	−0.777814	−0.388907	−	0.921277i	$-0.627147\pi$
$617$	−19.3205	−0.777814	−0.388907	+	0.921277i	$0.627147\pi$
$618$	0	0
$619$	−25.3731	−1.01983	−0.509915	−	0.860225i	$-0.670323\pi$
$619$	−25.3731	−1.01983	−0.509915	+	0.860225i	$0.670323\pi$
$620$	0	0
$621$	−19.7128	−0.791048
$622$	0	0
$623$	8.92820	0.357701
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	−62.6410	−2.50164
$628$	0	0
$629$	−6.14359	−0.244961
$630$	0	0
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	0	0
$633$	10.9282	0.434357
$634$	0	0
$635$	1.46410	0.0581011
$636$	0	0
$637$	0.732051	0.0290049
$638$	0	0
$639$	61.8564	2.44700
$640$	0	0
$641$	−47.3205	−1.86905	−0.934524	−	0.355901i	$-0.884174\pi$
$641$	−47.3205	−1.86905	−0.934524	+	0.355901i	$0.884174\pi$
$642$	0	0
$643$	−22.7321	−0.896465	−0.448232	−	0.893917i	$-0.647946\pi$
$643$	−22.7321	−0.896465	−0.448232	+	0.893917i	$0.647946\pi$
$644$	0	0
$645$	18.9282	0.745297
$646$	0	0
$647$	23.3205	0.916824	0.458412	−	0.888740i	$-0.348418\pi$
$647$	23.3205	0.916824	0.458412	+	0.888740i	$0.348418\pi$
$648$	0	0
$649$	−22.9282	−0.900011
$650$	0	0
$651$	−25.8564	−1.01339
$652$	0	0
$653$	−20.5359	−0.803632	−0.401816	−	0.915720i	$-0.631621\pi$
$653$	−20.5359	−0.803632	−0.401816	+	0.915720i	$0.631621\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	−11.3205	−0.440984	−0.220492	−	0.975389i	$-0.570766\pi$
$659$	−11.3205	−0.440984	−0.220492	+	0.975389i	$0.570766\pi$
$660$	0	0
$661$	−2.58846	−0.100679	−0.0503397	−	0.998732i	$-0.516030\pi$
$661$	−2.58846	−0.100679	−0.0503397	+	0.998732i	$0.516030\pi$
$662$	0	0
$663$	1.07180	0.0416251
$664$	0	0
$665$	3.07180	0.119119
$666$	0	0
$667$	−17.0718	−0.661023
$668$	0	0
$669$	5.07180	0.196087
$670$	0	0
$671$	−71.7128	−2.76844
$672$	0	0
$673$	16.9282	0.652534	0.326267	−	0.945278i	$-0.394209\pi$
$673$	16.9282	0.652534	0.326267	+	0.945278i	$0.394209\pi$
$674$	0	0
$675$	−17.8564	−0.687293
$676$	0	0
$677$	−5.51666	−0.212022	−0.106011	−	0.994365i	$-0.533808\pi$
$677$	−5.51666	−0.212022	−0.106011	+	0.994365i	$0.533808\pi$
$678$	0	0
$679$	−7.46410	−0.286446
$680$	0	0
$681$	−55.1769	−2.11438
$682$	0	0
$683$	−13.8564	−0.530201	−0.265100	−	0.964221i	$-0.585405\pi$
$683$	−13.8564	−0.530201	−0.265100	+	0.964221i	$0.585405\pi$
$684$	0	0
$685$	−5.17691	−0.197800
$686$	0	0
$687$	−27.8564	−1.06279
$688$	0	0
$689$	−4.39230	−0.167333
$690$	0	0
$691$	8.19615	0.311796	0.155898	−	0.987773i	$-0.450173\pi$
$691$	8.19615	0.311796	0.155898	+	0.987773i	$0.450173\pi$
$692$	0	0
$693$	−24.3923	−0.926587
$694$	0	0
$695$	−1.21539	−0.0461024
$696$	0	0
$697$	1.85641	0.0703164
$698$	0	0
$699$	49.1769	1.86004
$700$	0	0
$701$	−7.46410	−0.281915	−0.140958	−	0.990016i	$-0.545018\pi$
$701$	−7.46410	−0.281915	−0.140958	+	0.990016i	$0.545018\pi$
$702$	0	0
$703$	48.1051	1.81432
$704$	0	0
$705$	10.9282	0.411580
$706$	0	0
$707$	−14.1962	−0.533901
$708$	0	0
$709$	−43.4641	−1.63233	−0.816164	−	0.577820i	$-0.803904\pi$
$709$	−43.4641	−1.63233	−0.816164	+	0.577820i	$0.803904\pi$
$710$	0	0
$711$	35.7128	1.33934
$712$	0	0
$713$	−46.6410	−1.74672
$714$	0	0
$715$	2.92820	0.109509
$716$	0	0
$717$	2.53590	0.0947049
$718$	0	0
$719$	−6.24871	−0.233038	−0.116519	−	0.993188i	$-0.537174\pi$
$719$	−6.24871	−0.233038	−0.116519	+	0.993188i	$0.537174\pi$
$720$	0	0
$721$	12.3923	0.461514
$722$	0	0
$723$	34.2487	1.27372
$724$	0	0
$725$	−15.4641	−0.574322
$726$	0	0
$727$	45.4641	1.68617	0.843085	−	0.537780i	$-0.180737\pi$
$727$	45.4641	1.68617	0.843085	+	0.537780i	$0.180737\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	5.07180	0.187587
$732$	0	0
$733$	30.5885	1.12981	0.564905	−	0.825156i	$-0.308913\pi$
$733$	30.5885	1.12981	0.564905	+	0.825156i	$0.308913\pi$
$734$	0	0
$735$	2.00000	0.0737711
$736$	0	0
$737$	5.85641	0.215724
$738$	0	0
$739$	11.3205	0.416432	0.208216	−	0.978083i	$-0.433234\pi$
$739$	11.3205	0.416432	0.208216	+	0.978083i	$0.433234\pi$
$740$	0	0
$741$	−8.39230	−0.308299
$742$	0	0
$743$	25.7128	0.943312	0.471656	−	0.881783i	$-0.343656\pi$
$743$	25.7128	0.943312	0.471656	+	0.881783i	$0.343656\pi$
$744$	0	0
$745$	−5.75129	−0.210711
$746$	0	0
$747$	−47.9090	−1.75290
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−8.92820	−0.325795	−0.162897	−	0.986643i	$-0.552084\pi$
$751$	−8.92820	−0.325795	−0.162897	+	0.986643i	$0.552084\pi$
$752$	0	0
$753$	43.1769	1.57345
$754$	0	0
$755$	1.46410	0.0532841
$756$	0	0
$757$	−7.46410	−0.271287	−0.135644	−	0.990758i	$-0.543310\pi$
$757$	−7.46410	−0.271287	−0.135644	+	0.990758i	$0.543310\pi$
$758$	0	0
$759$	−73.5692	−2.67039
$760$	0	0
$761$	33.3205	1.20787	0.603934	−	0.797035i	$-0.293599\pi$
$761$	33.3205	1.20787	0.603934	+	0.797035i	$0.293599\pi$
$762$	0	0
$763$	3.46410	0.125409
$764$	0	0
$765$	1.75129	0.0633180
$766$	0	0
$767$	−3.07180	−0.110916
$768$	0	0
$769$	47.4641	1.71160	0.855800	−	0.517307i	$-0.173066\pi$
$769$	47.4641	1.71160	0.855800	+	0.517307i	$0.173066\pi$
$770$	0	0
$771$	−30.2487	−1.08938
$772$	0	0
$773$	1.51666	0.0545505	0.0272752	−	0.999628i	$-0.491317\pi$
$773$	1.51666	0.0545505	0.0272752	+	0.999628i	$0.491317\pi$
$774$	0	0
$775$	−42.2487	−1.51762
$776$	0	0
$777$	31.3205	1.12362
$778$	0	0
$779$	−14.5359	−0.520803
$780$	0	0
$781$	75.7128	2.70922
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	12.2487	0.437175
$786$	0	0
$787$	−9.26795	−0.330367	−0.165183	−	0.986263i	$-0.552822\pi$
$787$	−9.26795	−0.330367	−0.165183	+	0.986263i	$0.552822\pi$
$788$	0	0
$789$	−37.8564	−1.34772
$790$	0	0
$791$	20.3923	0.725067
$792$	0	0
$793$	−9.60770	−0.341179
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	11.6603	0.413027	0.206514	−	0.978444i	$-0.433788\pi$
$797$	11.6603	0.413027	0.206514	+	0.978444i	$0.433788\pi$
$798$	0	0
$799$	2.92820	0.103592
$800$	0	0
$801$	−39.8564	−1.40826
$802$	0	0
$803$	−26.9282	−0.950276
$804$	0	0
$805$	3.60770	0.127155
$806$	0	0
$807$	−65.7128	−2.31320
$808$	0	0
$809$	2.53590	0.0891574	0.0445787	−	0.999006i	$-0.485805\pi$
$809$	2.53590	0.0891574	0.0445787	+	0.999006i	$0.485805\pi$
$810$	0	0
$811$	16.9808	0.596275	0.298138	−	0.954523i	$-0.403635\pi$
$811$	16.9808	0.596275	0.298138	+	0.954523i	$0.403635\pi$
$812$	0	0
$813$	62.6410	2.19692
$814$	0	0
$815$	−0.287187	−0.0100597
$816$	0	0
$817$	−39.7128	−1.38938
$818$	0	0
$819$	−3.26795	−0.114191
$820$	0	0
$821$	49.7128	1.73499	0.867495	−	0.497447i	$-0.165729\pi$
$821$	49.7128	1.73499	0.867495	+	0.497447i	$0.165729\pi$
$822$	0	0
$823$	−40.7846	−1.42166	−0.710831	−	0.703363i	$-0.751681\pi$
$823$	−40.7846	−1.42166	−0.710831	+	0.703363i	$0.751681\pi$
$824$	0	0
$825$	−66.6410	−2.32014
$826$	0	0
$827$	−21.0718	−0.732738	−0.366369	−	0.930470i	$-0.619399\pi$
$827$	−21.0718	−0.732738	−0.366369	+	0.930470i	$0.619399\pi$
$828$	0	0
$829$	32.7321	1.13683	0.568416	−	0.822742i	$-0.307557\pi$
$829$	32.7321	1.13683	0.568416	+	0.822742i	$0.307557\pi$
$830$	0	0
$831$	2.53590	0.0879693
$832$	0	0
$833$	0.535898	0.0185678
$834$	0	0
$835$	−9.85641	−0.341095
$836$	0	0
$837$	37.8564	1.30851
$838$	0	0
$839$	−9.46410	−0.326737	−0.163369	−	0.986565i	$-0.552236\pi$
$839$	−9.46410	−0.326737	−0.163369	+	0.986565i	$0.552236\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	54.2487	1.86842
$844$	0	0
$845$	−9.12436	−0.313887
$846$	0	0
$847$	−18.8564	−0.647914
$848$	0	0
$849$	60.2487	2.06773
$850$	0	0
$851$	56.4974	1.93671
$852$	0	0
$853$	33.5167	1.14759	0.573794	−	0.818999i	$-0.305471\pi$
$853$	33.5167	1.14759	0.573794	+	0.818999i	$0.305471\pi$
$854$	0	0
$855$	−13.7128	−0.468968
$856$	0	0
$857$	−27.4641	−0.938156	−0.469078	−	0.883157i	$-0.655414\pi$
$857$	−27.4641	−0.938156	−0.469078	+	0.883157i	$0.655414\pi$
$858$	0	0
$859$	13.6603	0.466082	0.233041	−	0.972467i	$-0.425132\pi$
$859$	13.6603	0.466082	0.233041	+	0.972467i	$0.425132\pi$
$860$	0	0
$861$	−9.46410	−0.322536
$862$	0	0
$863$	−34.9282	−1.18897	−0.594485	−	0.804107i	$-0.702644\pi$
$863$	−34.9282	−1.18897	−0.594485	+	0.804107i	$0.702644\pi$
$864$	0	0
$865$	8.53590	0.290229
$866$	0	0
$867$	−45.6603	−1.55070
$868$	0	0
$869$	43.7128	1.48286
$870$	0	0
$871$	0.784610	0.0265855
$872$	0	0
$873$	33.3205	1.12773
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	−42.1051	−1.42179	−0.710894	−	0.703299i	$-0.751710\pi$
$877$	−42.1051	−1.42179	−0.710894	+	0.703299i	$0.751710\pi$
$878$	0	0
$879$	30.7846	1.03834
$880$	0	0
$881$	41.7128	1.40534	0.702670	−	0.711516i	$-0.251991\pi$
$881$	41.7128	1.40534	0.702670	+	0.711516i	$0.251991\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	−8.39230	−0.282104
$886$	0	0
$887$	−26.5359	−0.890988	−0.445494	−	0.895285i	$-0.646972\pi$
$887$	−26.5359	−0.890988	−0.445494	+	0.895285i	$0.646972\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−13.4641	−0.451064
$892$	0	0
$893$	−22.9282	−0.767263
$894$	0	0
$895$	−8.78461	−0.293637
$896$	0	0
$897$	−9.85641	−0.329096
$898$	0	0
$899$	32.7846	1.09343
$900$	0	0
$901$	−3.21539	−0.107120
$902$	0	0
$903$	−25.8564	−0.860447
$904$	0	0
$905$	−11.4641	−0.381080
$906$	0	0
$907$	26.6410	0.884600	0.442300	−	0.896867i	$-0.354163\pi$
$907$	26.6410	0.884600	0.442300	+	0.896867i	$0.354163\pi$
$908$	0	0
$909$	63.3731	2.10195
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	−58.6410	−1.94073
$914$	0	0
$915$	−26.2487	−0.867756
$916$	0	0
$917$	−8.19615	−0.270661
$918$	0	0
$919$	−32.7846	−1.08146	−0.540732	−	0.841195i	$-0.681853\pi$
$919$	−32.7846	−1.08146	−0.540732	+	0.841195i	$0.681853\pi$
$920$	0	0
$921$	−43.1769	−1.42273
$922$	0	0
$923$	10.1436	0.333880
$924$	0	0
$925$	51.1769	1.68269
$926$	0	0
$927$	−55.3205	−1.81696
$928$	0	0
$929$	3.46410	0.113653	0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410	0.113653	0.0568267	+	0.998384i	$0.481902\pi$
$930$	0	0
$931$	−4.19615	−0.137523
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	2.14359	0.0701030
$936$	0	0
$937$	39.5692	1.29267	0.646335	−	0.763054i	$-0.276301\pi$
$937$	39.5692	1.29267	0.646335	+	0.763054i	$0.276301\pi$
$938$	0	0
$939$	53.1769	1.73536
$940$	0	0
$941$	9.90897	0.323023	0.161512	−	0.986871i	$-0.448363\pi$
$941$	9.90897	0.323023	0.161512	+	0.986871i	$0.448363\pi$
$942$	0	0
$943$	−17.0718	−0.555934
$944$	0	0
$945$	−2.92820	−0.0952545
$946$	0	0
$947$	−27.3205	−0.887797	−0.443899	−	0.896077i	$-0.646405\pi$
$947$	−27.3205	−0.887797	−0.443899	+	0.896077i	$0.646405\pi$
$948$	0	0
$949$	−3.60770	−0.117111
$950$	0	0
$951$	−2.53590	−0.0822321
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	−2.14359	−0.0693651
$956$	0	0
$957$	51.7128	1.67164
$958$	0	0
$959$	7.07180	0.228360
$960$	0	0
$961$	58.5692	1.88933
$962$	0	0
$963$	−71.4256	−2.30166
$964$	0	0
$965$	6.14359	0.197769
$966$	0	0
$967$	36.6410	1.17830	0.589148	−	0.808025i	$-0.299464\pi$
$967$	36.6410	1.17830	0.589148	+	0.808025i	$0.299464\pi$
$968$	0	0
$969$	−6.14359	−0.197361
$970$	0	0
$971$	−28.1962	−0.904858	−0.452429	−	0.891801i	$-0.649442\pi$
$971$	−28.1962	−0.904858	−0.452429	+	0.891801i	$0.649442\pi$
$972$	0	0
$973$	1.66025	0.0532253
$974$	0	0
$975$	−8.92820	−0.285931
$976$	0	0
$977$	−52.6410	−1.68414	−0.842068	−	0.539372i	$-0.818662\pi$
$977$	−52.6410	−1.68414	−0.842068	+	0.539372i	$0.818662\pi$
$978$	0	0
$979$	−48.7846	−1.55916
$980$	0	0
$981$	−15.4641	−0.493731
$982$	0	0
$983$	20.3923	0.650414	0.325207	−	0.945643i	$-0.394566\pi$
$983$	20.3923	0.650414	0.325207	+	0.945643i	$0.394566\pi$
$984$	0	0
$985$	−2.82309	−0.0899510
$986$	0	0
$987$	−14.9282	−0.475170
$988$	0	0
$989$	−46.6410	−1.48310
$990$	0	0
$991$	2.92820	0.0930174	0.0465087	−	0.998918i	$-0.485190\pi$
$991$	2.92820	0.0930174	0.0465087	+	0.998918i	$0.485190\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	18.6410	0.590960
$996$	0	0
$997$	−36.3397	−1.15089	−0.575446	−	0.817840i	$-0.695171\pi$
$997$	−36.3397	−1.15089	−0.575446	+	0.817840i	$0.695171\pi$
$998$	0	0
$999$	−45.8564	−1.45083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.a.g.1.2	yes	2
3.2	odd	2		8064.2.a.bm.1.1		2
4.3	odd	2		896.2.a.e.1.1	✓	2
7.6	odd	2		6272.2.a.j.1.1		2
8.3	odd	2		896.2.a.h.1.2	yes	2
8.5	even	2		896.2.a.f.1.1	yes	2
12.11	even	2		8064.2.a.br.1.1		2
16.3	odd	4		1792.2.b.l.897.1		4
16.5	even	4		1792.2.b.n.897.1		4
16.11	odd	4		1792.2.b.l.897.4		4
16.13	even	4		1792.2.b.n.897.4		4
24.5	odd	2		8064.2.a.be.1.2		2
24.11	even	2		8064.2.a.bf.1.2		2
28.27	even	2		6272.2.a.t.1.2		2
56.13	odd	2		6272.2.a.s.1.2		2
56.27	even	2		6272.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.e.1.1	✓	2	4.3	odd	2
896.2.a.f.1.1	yes	2	8.5	even	2
896.2.a.g.1.2	yes	2	1.1	even	1	trivial
896.2.a.h.1.2	yes	2	8.3	odd	2
1792.2.b.l.897.1		4	16.3	odd	4
1792.2.b.l.897.4		4	16.11	odd	4
1792.2.b.n.897.1		4	16.5	even	4
1792.2.b.n.897.4		4	16.13	even	4
6272.2.a.i.1.1		2	56.27	even	2
6272.2.a.j.1.1		2	7.6	odd	2
6272.2.a.s.1.2		2	56.13	odd	2
6272.2.a.t.1.2		2	28.27	even	2
8064.2.a.be.1.2		2	24.5	odd	2
8064.2.a.bf.1.2		2	24.11	even	2
8064.2.a.bm.1.1		2	3.2	odd	2
8064.2.a.br.1.1		2	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} +0.732051 q^{5} -1.00000 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q+2.73205 q^{3} +0.732051 q^{5} -1.00000 q^{7} +4.46410 q^{9} +5.46410 q^{11} +0.732051 q^{13} +2.00000 q^{15} +0.535898 q^{17} -4.19615 q^{19} -2.73205 q^{21} -4.92820 q^{23} -4.46410 q^{25} +4.00000 q^{27} +3.46410 q^{29} +9.46410 q^{31} +14.9282 q^{33} -0.732051 q^{35} -11.4641 q^{37} +2.00000 q^{39} +3.46410 q^{41} +9.46410 q^{43} +3.26795 q^{45} +5.46410 q^{47} +1.00000 q^{49} +1.46410 q^{51} -6.00000 q^{53} +4.00000 q^{55} -11.4641 q^{57} -4.19615 q^{59} -13.1244 q^{61} -4.46410 q^{63} +0.535898 q^{65} +1.07180 q^{67} -13.4641 q^{69} +13.8564 q^{71} -4.92820 q^{73} -12.1962 q^{75} -5.46410 q^{77} +8.00000 q^{79} -2.46410 q^{81} -10.7321 q^{83} +0.392305 q^{85} +9.46410 q^{87} -8.92820 q^{89} -0.732051 q^{91} +25.8564 q^{93} -3.07180 q^{95} +7.46410 q^{97} +24.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{15} + 8 q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} - 2 q^{25} + 8 q^{27} + 12 q^{31} + 16 q^{33} + 2 q^{35} - 16 q^{37} + 4 q^{39} + 12 q^{43} + 10 q^{45} + 4 q^{47} + 2 q^{49} - 4 q^{51} - 12 q^{53} + 8 q^{55} - 16 q^{57} + 2 q^{59} - 2 q^{61} - 2 q^{63} + 8 q^{65} + 16 q^{67} - 20 q^{69} + 4 q^{73} - 14 q^{75} - 4 q^{77} + 16 q^{79} + 2 q^{81} - 18 q^{83} - 20 q^{85} + 12 q^{87} - 4 q^{89} + 2 q^{91} + 24 q^{93} - 20 q^{95} + 8 q^{97} + 28 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 4 * q^15 + 8 * q^17 + 2 * q^19 - 2 * q^21 + 4 * q^23 - 2 * q^25 + 8 * q^27 + 12 * q^31 + 16 * q^33 + 2 * q^35 - 16 * q^37 + 4 * q^39 + 12 * q^43 + 10 * q^45 + 4 * q^47 + 2 * q^49 - 4 * q^51 - 12 * q^53 + 8 * q^55 - 16 * q^57 + 2 * q^59 - 2 * q^61 - 2 * q^63 + 8 * q^65 + 16 * q^67 - 20 * q^69 + 4 * q^73 - 14 * q^75 - 4 * q^77 + 16 * q^79 + 2 * q^81 - 18 * q^83 - 20 * q^85 + 12 * q^87 - 4 * q^89 + 2 * q^91 + 24 * q^93 - 20 * q^95 + 8 * q^97 + 28 * q^99

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