LMFDB - Embedded newform 896.2.a.e.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:	$1$
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)

$f(q)$	$=$	$q+0.732051 q^{3} -2.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} -2.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} +1.46410 q^{11} -2.73205 q^{13} -2.00000 q^{15} +7.46410 q^{17} -6.19615 q^{19} +0.732051 q^{21} -8.92820 q^{23} +2.46410 q^{25} -4.00000 q^{27} -3.46410 q^{29} -2.53590 q^{31} +1.07180 q^{33} -2.73205 q^{35} -4.53590 q^{37} -2.00000 q^{39} -3.46410 q^{41} -2.53590 q^{43} +6.73205 q^{45} +1.46410 q^{47} +1.00000 q^{49} +5.46410 q^{51} -6.00000 q^{53} -4.00000 q^{55} -4.53590 q^{57} -6.19615 q^{59} +11.1244 q^{61} -2.46410 q^{63} +7.46410 q^{65} -14.9282 q^{67} -6.53590 q^{69} +13.8564 q^{71} +8.92820 q^{73} +1.80385 q^{75} +1.46410 q^{77} -8.00000 q^{79} +4.46410 q^{81} +7.26795 q^{83} -20.3923 q^{85} -2.53590 q^{87} +4.92820 q^{89} -2.73205 q^{91} -1.85641 q^{93} +16.9282 q^{95} +0.535898 q^{97} -3.60770 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	0	0
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	1.46410	0.441443	0.220722	−	0.975337i	$-0.429159\pi$
$11$	1.46410	0.441443	0.220722	+	0.975337i	$0.429159\pi$
$12$	0	0
$13$	−2.73205	−0.757735	−0.378867	−	0.925451i	$-0.623686\pi$
$13$	−2.73205	−0.757735	−0.378867	+	0.925451i	$0.623686\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	7.46410	1.81031	0.905155	−	0.425081i	$-0.139754\pi$
$17$	7.46410	1.81031	0.905155	+	0.425081i	$0.139754\pi$
$18$	0	0
$19$	−6.19615	−1.42149	−0.710747	−	0.703447i	$-0.751643\pi$
$19$	−6.19615	−1.42149	−0.710747	+	0.703447i	$0.751643\pi$
$20$	0	0
$21$	0.732051	0.159747
$22$	0	0
$23$	−8.92820	−1.86166	−0.930830	−	0.365454i	$-0.880914\pi$
$23$	−8.92820	−1.86166	−0.930830	+	0.365454i	$0.880914\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−2.53590	−0.455461	−0.227730	−	0.973724i	$-0.573130\pi$
$31$	−2.53590	−0.455461	−0.227730	+	0.973724i	$0.573130\pi$
$32$	0	0
$33$	1.07180	0.186576
$34$	0	0
$35$	−2.73205	−0.461801
$36$	0	0
$37$	−4.53590	−0.745697	−0.372849	−	0.927892i	$-0.621619\pi$
$37$	−4.53590	−0.745697	−0.372849	+	0.927892i	$0.621619\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	−2.53590	−0.386721	−0.193360	−	0.981128i	$-0.561939\pi$
$43$	−2.53590	−0.386721	−0.193360	+	0.981128i	$0.561939\pi$
$44$	0	0
$45$	6.73205	1.00355
$46$	0	0
$47$	1.46410	0.213561	0.106781	−	0.994283i	$-0.465946\pi$
$47$	1.46410	0.213561	0.106781	+	0.994283i	$0.465946\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.46410	0.765127
$52$	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
$57$	−4.53590	−0.600794
$58$	0	0
$59$	−6.19615	−0.806670	−0.403335	−	0.915052i	$-0.632149\pi$
$59$	−6.19615	−0.806670	−0.403335	+	0.915052i	$0.632149\pi$
$60$	0	0
$61$	11.1244	1.42433	0.712164	−	0.702013i	$-0.247715\pi$
$61$	11.1244	1.42433	0.712164	+	0.702013i	$0.247715\pi$
$62$	0	0
$63$	−2.46410	−0.310448
$64$	0	0
$65$	7.46410	0.925808
$66$	0	0
$67$	−14.9282	−1.82377	−0.911885	−	0.410445i	$-0.865373\pi$
$67$	−14.9282	−1.82377	−0.911885	+	0.410445i	$0.865373\pi$
$68$	0	0
$69$	−6.53590	−0.786830
$70$	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
$73$	8.92820	1.04497	0.522484	−	0.852649i	$-0.325006\pi$
$73$	8.92820	1.04497	0.522484	+	0.852649i	$0.325006\pi$
$74$	0	0
$75$	1.80385	0.208290
$76$	0	0
$77$	1.46410	0.166850
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	7.26795	0.797761	0.398881	−	0.917003i	$-0.369399\pi$
$83$	7.26795	0.797761	0.398881	+	0.917003i	$0.369399\pi$
$84$	0	0
$85$	−20.3923	−2.21186
$86$	0	0
$87$	−2.53590	−0.271877
$88$	0	0
$89$	4.92820	0.522388	0.261194	−	0.965286i	$-0.415884\pi$
$89$	4.92820	0.522388	0.261194	+	0.965286i	$0.415884\pi$
$90$	0	0
$91$	−2.73205	−0.286397
$92$	0	0
$93$	−1.85641	−0.192500
$94$	0	0
$95$	16.9282	1.73680
$96$	0	0
$97$	0.535898	0.0544122	0.0272061	−	0.999630i	$-0.491339\pi$
$97$	0.535898	0.0544122	0.0272061	+	0.999630i	$0.491339\pi$
$98$	0	0
$99$	−3.60770	−0.362587
$100$	0	0
$101$	3.80385	0.378497	0.189248	−	0.981929i	$-0.439395\pi$
$101$	3.80385	0.378497	0.189248	+	0.981929i	$0.439395\pi$
$102$	0	0
$103$	−8.39230	−0.826918	−0.413459	−	0.910523i	$-0.635680\pi$
$103$	−8.39230	−0.826918	−0.413459	+	0.910523i	$0.635680\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	3.46410	0.331801	0.165900	−	0.986143i	$-0.446947\pi$
$109$	3.46410	0.331801	0.165900	+	0.986143i	$0.446947\pi$
$110$	0	0
$111$	−3.32051	−0.315169
$112$	0	0
$113$	0.392305	0.0369049	0.0184525	−	0.999830i	$-0.494126\pi$
$113$	0.392305	0.0369049	0.0184525	+	0.999830i	$0.494126\pi$
$114$	0	0
$115$	24.3923	2.27459
$116$	0	0
$117$	6.73205	0.622378
$118$	0	0
$119$	7.46410	0.684233
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	−2.53590	−0.228654
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−1.85641	−0.163447
$130$	0	0
$131$	2.19615	0.191879	0.0959394	−	0.995387i	$-0.469415\pi$
$131$	2.19615	0.191879	0.0959394	+	0.995387i	$0.469415\pi$
$132$	0	0
$133$	−6.19615	−0.537275
$134$	0	0
$135$	10.9282	0.940550
$136$	0	0
$137$	−20.9282	−1.78802	−0.894009	−	0.448050i	$-0.852119\pi$
$137$	−20.9282	−1.78802	−0.894009	+	0.448050i	$0.852119\pi$
$138$	0	0
$139$	−15.6603	−1.32829	−0.664143	−	0.747606i	$-0.731203\pi$
$139$	−15.6603	−1.32829	−0.664143	+	0.747606i	$0.731203\pi$
$140$	0	0
$141$	1.07180	0.0902616
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	9.46410	0.785951
$146$	0	0
$147$	0.732051	0.0603785
$148$	0	0
$149$	19.8564	1.62670	0.813350	−	0.581775i	$-0.197641\pi$
$149$	19.8564	1.62670	0.813350	+	0.581775i	$0.197641\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	−18.3923	−1.48693
$154$	0	0
$155$	6.92820	0.556487
$156$	0	0
$157$	13.2679	1.05890	0.529449	−	0.848342i	$-0.322399\pi$
$157$	13.2679	1.05890	0.529449	+	0.848342i	$0.322399\pi$
$158$	0	0
$159$	−4.39230	−0.348332
$160$	0	0
$161$	−8.92820	−0.703641
$162$	0	0
$163$	−20.3923	−1.59725	−0.798624	−	0.601830i	$-0.794439\pi$
$163$	−20.3923	−1.59725	−0.798624	+	0.601830i	$0.794439\pi$
$164$	0	0
$165$	−2.92820	−0.227960
$166$	0	0
$167$	6.53590	0.505763	0.252882	−	0.967497i	$-0.418622\pi$
$167$	6.53590	0.505763	0.252882	+	0.967497i	$0.418622\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	15.2679	1.16757
$172$	0	0
$173$	−5.66025	−0.430341	−0.215171	−	0.976576i	$-0.569031\pi$
$173$	−5.66025	−0.430341	−0.215171	+	0.976576i	$0.569031\pi$
$174$	0	0
$175$	2.46410	0.186269
$176$	0	0
$177$	−4.53590	−0.340939
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	1.66025	0.123406	0.0617029	−	0.998095i	$-0.480347\pi$
$181$	1.66025	0.123406	0.0617029	+	0.998095i	$0.480347\pi$
$182$	0	0
$183$	8.14359	0.601992
$184$	0	0
$185$	12.3923	0.911100
$186$	0	0
$187$	10.9282	0.799149
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	−10.9282	−0.790737	−0.395369	−	0.918523i	$-0.629383\pi$
$191$	−10.9282	−0.790737	−0.395369	+	0.918523i	$0.629383\pi$
$192$	0	0
$193$	−12.3923	−0.892018	−0.446009	−	0.895029i	$-0.647155\pi$
$193$	−12.3923	−0.892018	−0.446009	+	0.895029i	$0.647155\pi$
$194$	0	0
$195$	5.46410	0.391292
$196$	0	0
$197$	23.8564	1.69970	0.849849	−	0.527026i	$-0.176693\pi$
$197$	23.8564	1.69970	0.849849	+	0.527026i	$0.176693\pi$
$198$	0	0
$199$	−18.5359	−1.31397	−0.656987	−	0.753901i	$-0.728170\pi$
$199$	−18.5359	−1.31397	−0.656987	+	0.753901i	$0.728170\pi$
$200$	0	0
$201$	−10.9282	−0.770816
$202$	0	0
$203$	−3.46410	−0.243132
$204$	0	0
$205$	9.46410	0.661002
$206$	0	0
$207$	22.0000	1.52911
$208$	0	0
$209$	−9.07180	−0.627509
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	10.1436	0.695028
$214$	0	0
$215$	6.92820	0.472500
$216$	0	0
$217$	−2.53590	−0.172148
$218$	0	0
$219$	6.53590	0.441655
$220$	0	0
$221$	−20.3923	−1.37173
$222$	0	0
$223$	25.8564	1.73147	0.865737	−	0.500500i	$-0.166850\pi$
$223$	25.8564	1.73147	0.865737	+	0.500500i	$0.166850\pi$
$224$	0	0
$225$	−6.07180	−0.404786
$226$	0	0
$227$	9.80385	0.650704	0.325352	−	0.945593i	$-0.394517\pi$
$227$	9.80385	0.650704	0.325352	+	0.945593i	$0.394517\pi$
$228$	0	0
$229$	0.196152	0.0129621	0.00648106	−	0.999979i	$-0.497937\pi$
$229$	0.196152	0.0129621	0.00648106	+	0.999979i	$0.497937\pi$
$230$	0	0
$231$	1.07180	0.0705191
$232$	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
$237$	−5.85641	−0.380414
$238$	0	0
$239$	12.9282	0.836256	0.418128	−	0.908388i	$-0.362686\pi$
$239$	12.9282	0.836256	0.418128	+	0.908388i	$0.362686\pi$
$240$	0	0
$241$	19.4641	1.25379	0.626897	−	0.779103i	$-0.284325\pi$
$241$	19.4641	1.25379	0.626897	+	0.779103i	$0.284325\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	−2.73205	−0.174544
$246$	0	0
$247$	16.9282	1.07712
$248$	0	0
$249$	5.32051	0.337173
$250$	0	0
$251$	−26.1962	−1.65349	−0.826743	−	0.562579i	$-0.809809\pi$
$251$	−26.1962	−1.65349	−0.826743	+	0.562579i	$0.809809\pi$
$252$	0	0
$253$	−13.0718	−0.821817
$254$	0	0
$255$	−14.9282	−0.934840
$256$	0	0
$257$	−24.9282	−1.55498	−0.777489	−	0.628896i	$-0.783507\pi$
$257$	−24.9282	−1.55498	−0.777489	+	0.628896i	$0.783507\pi$
$258$	0	0
$259$	−4.53590	−0.281847
$260$	0	0
$261$	8.53590	0.528359
$262$	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
$265$	16.3923	1.00697
$266$	0	0
$267$	3.60770	0.220787
$268$	0	0
$269$	14.0526	0.856800	0.428400	−	0.903589i	$-0.359077\pi$
$269$	14.0526	0.856800	0.428400	+	0.903589i	$0.359077\pi$
$270$	0	0
$271$	−9.07180	−0.551072	−0.275536	−	0.961291i	$-0.588855\pi$
$271$	−9.07180	−0.551072	−0.275536	+	0.961291i	$0.588855\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	3.60770	0.217552
$276$	0	0
$277$	−12.9282	−0.776780	−0.388390	−	0.921495i	$-0.626969\pi$
$277$	−12.9282	−0.776780	−0.388390	+	0.921495i	$0.626969\pi$
$278$	0	0
$279$	6.24871	0.374101
$280$	0	0
$281$	−7.85641	−0.468674	−0.234337	−	0.972155i	$-0.575292\pi$
$281$	−7.85641	−0.468674	−0.234337	+	0.972155i	$0.575292\pi$
$282$	0	0
$283$	16.0526	0.954226	0.477113	−	0.878842i	$-0.341683\pi$
$283$	16.0526	0.954226	0.477113	+	0.878842i	$0.341683\pi$
$284$	0	0
$285$	12.3923	0.734057
$286$	0	0
$287$	−3.46410	−0.204479
$288$	0	0
$289$	38.7128	2.27722
$290$	0	0
$291$	0.392305	0.0229973
$292$	0	0
$293$	14.7321	0.860656	0.430328	−	0.902673i	$-0.358398\pi$
$293$	14.7321	0.860656	0.430328	+	0.902673i	$0.358398\pi$
$294$	0	0
$295$	16.9282	0.985598
$296$	0	0
$297$	−5.85641	−0.339823
$298$	0	0
$299$	24.3923	1.41064
$300$	0	0
$301$	−2.53590	−0.146167
$302$	0	0
$303$	2.78461	0.159972
$304$	0	0
$305$	−30.3923	−1.74026
$306$	0	0
$307$	26.1962	1.49509	0.747547	−	0.664209i	$-0.231232\pi$
$307$	26.1962	1.49509	0.747547	+	0.664209i	$0.231232\pi$
$308$	0	0
$309$	−6.14359	−0.349497
$310$	0	0
$311$	−10.9282	−0.619682	−0.309841	−	0.950788i	$-0.600276\pi$
$311$	−10.9282	−0.619682	−0.309841	+	0.950788i	$0.600276\pi$
$312$	0	0
$313$	12.5359	0.708571	0.354285	−	0.935137i	$-0.384724\pi$
$313$	12.5359	0.708571	0.354285	+	0.935137i	$0.384724\pi$
$314$	0	0
$315$	6.73205	0.379308
$316$	0	0
$317$	12.9282	0.726120	0.363060	−	0.931766i	$-0.381732\pi$
$317$	12.9282	0.726120	0.363060	+	0.931766i	$0.381732\pi$
$318$	0	0
$319$	−5.07180	−0.283966
$320$	0	0
$321$	11.7128	0.653745
$322$	0	0
$323$	−46.2487	−2.57335
$324$	0	0
$325$	−6.73205	−0.373427
$326$	0	0
$327$	2.53590	0.140236
$328$	0	0
$329$	1.46410	0.0807185
$330$	0	0
$331$	5.46410	0.300334	0.150167	−	0.988661i	$-0.452019\pi$
$331$	5.46410	0.300334	0.150167	+	0.988661i	$0.452019\pi$
$332$	0	0
$333$	11.1769	0.612491
$334$	0	0
$335$	40.7846	2.22830
$336$	0	0
$337$	−26.2487	−1.42986	−0.714929	−	0.699197i	$-0.753541\pi$
$337$	−26.2487	−1.42986	−0.714929	+	0.699197i	$0.753541\pi$
$338$	0	0
$339$	0.287187	0.0155979
$340$	0	0
$341$	−3.71281	−0.201060
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	17.8564	0.961357
$346$	0	0
$347$	−21.4641	−1.15225	−0.576127	−	0.817360i	$-0.695437\pi$
$347$	−21.4641	−1.15225	−0.576127	+	0.817360i	$0.695437\pi$
$348$	0	0
$349$	−23.1244	−1.23782	−0.618909	−	0.785463i	$-0.712425\pi$
$349$	−23.1244	−1.23782	−0.618909	+	0.785463i	$0.712425\pi$
$350$	0	0
$351$	10.9282	0.583304
$352$	0	0
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	−37.8564	−2.00921
$356$	0	0
$357$	5.46410	0.289191
$358$	0	0
$359$	−11.8564	−0.625757	−0.312879	−	0.949793i	$-0.601293\pi$
$359$	−11.8564	−0.625757	−0.312879	+	0.949793i	$0.601293\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	0	0
$363$	−6.48334	−0.340287
$364$	0	0
$365$	−24.3923	−1.27675
$366$	0	0
$367$	−28.7846	−1.50254	−0.751272	−	0.659992i	$-0.770559\pi$
$367$	−28.7846	−1.50254	−0.751272	+	0.659992i	$0.770559\pi$
$368$	0	0
$369$	8.53590	0.444361
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	11.8564	0.613901	0.306951	−	0.951725i	$-0.400691\pi$
$373$	11.8564	0.613901	0.306951	+	0.951725i	$0.400691\pi$
$374$	0	0
$375$	5.07180	0.261906
$376$	0	0
$377$	9.46410	0.487426
$378$	0	0
$379$	−38.2487	−1.96470	−0.982352	−	0.187041i	$-0.940110\pi$
$379$	−38.2487	−1.96470	−0.982352	+	0.187041i	$0.940110\pi$
$380$	0	0
$381$	−1.46410	−0.0750082
$382$	0	0
$383$	−33.4641	−1.70994	−0.854968	−	0.518681i	$-0.826423\pi$
$383$	−33.4641	−1.70994	−0.854968	+	0.518681i	$0.826423\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	6.24871	0.317640
$388$	0	0
$389$	0.535898	0.0271711	0.0135856	−	0.999908i	$-0.495675\pi$
$389$	0.535898	0.0271711	0.0135856	+	0.999908i	$0.495675\pi$
$390$	0	0
$391$	−66.6410	−3.37018
$392$	0	0
$393$	1.60770	0.0810975
$394$	0	0
$395$	21.8564	1.09972
$396$	0	0
$397$	−28.9808	−1.45450	−0.727251	−	0.686371i	$-0.759203\pi$
$397$	−28.9808	−1.45450	−0.727251	+	0.686371i	$0.759203\pi$
$398$	0	0
$399$	−4.53590	−0.227079
$400$	0	0
$401$	−30.2487	−1.51055	−0.755274	−	0.655409i	$-0.772496\pi$
$401$	−30.2487	−1.51055	−0.755274	+	0.655409i	$0.772496\pi$
$402$	0	0
$403$	6.92820	0.345118
$404$	0	0
$405$	−12.1962	−0.606032
$406$	0	0
$407$	−6.64102	−0.329183
$408$	0	0
$409$	24.5359	1.21322	0.606611	−	0.794999i	$-0.292529\pi$
$409$	24.5359	1.21322	0.606611	+	0.794999i	$0.292529\pi$
$410$	0	0
$411$	−15.3205	−0.755705
$412$	0	0
$413$	−6.19615	−0.304893
$414$	0	0
$415$	−19.8564	−0.974713
$416$	0	0
$417$	−11.4641	−0.561399
$418$	0	0
$419$	−18.1962	−0.888940	−0.444470	−	0.895794i	$-0.646608\pi$
$419$	−18.1962	−0.888940	−0.444470	+	0.895794i	$0.646608\pi$
$420$	0	0
$421$	−20.9282	−1.01998	−0.509989	−	0.860181i	$-0.670351\pi$
$421$	−20.9282	−1.01998	−0.509989	+	0.860181i	$0.670351\pi$
$422$	0	0
$423$	−3.60770	−0.175412
$424$	0	0
$425$	18.3923	0.892158
$426$	0	0
$427$	11.1244	0.538345
$428$	0	0
$429$	−2.92820	−0.141375
$430$	0	0
$431$	−10.7846	−0.519476	−0.259738	−	0.965679i	$-0.583636\pi$
$431$	−10.7846	−0.519476	−0.259738	+	0.965679i	$0.583636\pi$
$432$	0	0
$433$	−12.2487	−0.588636	−0.294318	−	0.955708i	$-0.595092\pi$
$433$	−12.2487	−0.588636	−0.294318	+	0.955708i	$0.595092\pi$
$434$	0	0
$435$	6.92820	0.332182
$436$	0	0
$437$	55.3205	2.64634
$438$	0	0
$439$	15.7128	0.749932	0.374966	−	0.927039i	$-0.377654\pi$
$439$	15.7128	0.749932	0.374966	+	0.927039i	$0.377654\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	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
$445$	−13.4641	−0.638260
$446$	0	0
$447$	14.5359	0.687524
$448$	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
$451$	−5.07180	−0.238822
$452$	0	0
$453$	−1.46410	−0.0687895
$454$	0	0
$455$	7.46410	0.349922
$456$	0	0
$457$	9.46410	0.442712	0.221356	−	0.975193i	$-0.428952\pi$
$457$	9.46410	0.442712	0.221356	+	0.975193i	$0.428952\pi$
$458$	0	0
$459$	−29.8564	−1.39358
$460$	0	0
$461$	−14.3397	−0.667869	−0.333934	−	0.942596i	$-0.608376\pi$
$461$	−14.3397	−0.667869	−0.333934	+	0.942596i	$0.608376\pi$
$462$	0	0
$463$	−35.7128	−1.65972	−0.829858	−	0.557975i	$-0.811578\pi$
$463$	−35.7128	−1.65972	−0.829858	+	0.557975i	$0.811578\pi$
$464$	0	0
$465$	5.07180	0.235199
$466$	0	0
$467$	24.4449	1.13117	0.565587	−	0.824689i	$-0.308650\pi$
$467$	24.4449	1.13117	0.565587	+	0.824689i	$0.308650\pi$
$468$	0	0
$469$	−14.9282	−0.689320
$470$	0	0
$471$	9.71281	0.447543
$472$	0	0
$473$	−3.71281	−0.170715
$474$	0	0
$475$	−15.2679	−0.700542
$476$	0	0
$477$	14.7846	0.676941
$478$	0	0
$479$	16.3923	0.748984	0.374492	−	0.927230i	$-0.377817\pi$
$479$	16.3923	0.748984	0.374492	+	0.927230i	$0.377817\pi$
$480$	0	0
$481$	12.3923	0.565040
$482$	0	0
$483$	−6.53590	−0.297394
$484$	0	0
$485$	−1.46410	−0.0664814
$486$	0	0
$487$	32.6410	1.47911	0.739553	−	0.673099i	$-0.235037\pi$
$487$	32.6410	1.47911	0.739553	+	0.673099i	$0.235037\pi$
$488$	0	0
$489$	−14.9282	−0.675077
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−25.8564	−1.16451
$494$	0	0
$495$	9.85641	0.443013
$496$	0	0
$497$	13.8564	0.621545
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	4.78461	0.213761
$502$	0	0
$503$	−9.07180	−0.404491	−0.202246	−	0.979335i	$-0.564824\pi$
$503$	−9.07180	−0.404491	−0.202246	+	0.979335i	$0.564824\pi$
$504$	0	0
$505$	−10.3923	−0.462451
$506$	0	0
$507$	−4.05256	−0.179980
$508$	0	0
$509$	−9.26795	−0.410795	−0.205397	−	0.978679i	$-0.565849\pi$
$509$	−9.26795	−0.410795	−0.205397	+	0.978679i	$0.565849\pi$
$510$	0	0
$511$	8.92820	0.394960
$512$	0	0
$513$	24.7846	1.09427
$514$	0	0
$515$	22.9282	1.01034
$516$	0	0
$517$	2.14359	0.0942751
$518$	0	0
$519$	−4.14359	−0.181884
$520$	0	0
$521$	30.3923	1.33151	0.665756	−	0.746170i	$-0.268109\pi$
$521$	30.3923	1.33151	0.665756	+	0.746170i	$0.268109\pi$
$522$	0	0
$523$	2.58846	0.113185	0.0565927	−	0.998397i	$-0.481976\pi$
$523$	2.58846	0.113185	0.0565927	+	0.998397i	$0.481976\pi$
$524$	0	0
$525$	1.80385	0.0787264
$526$	0	0
$527$	−18.9282	−0.824525
$528$	0	0
$529$	56.7128	2.46577
$530$	0	0
$531$	15.2679	0.662573
$532$	0	0
$533$	9.46410	0.409936
$534$	0	0
$535$	−43.7128	−1.88987
$536$	0	0
$537$	8.78461	0.379084
$538$	0	0
$539$	1.46410	0.0630633
$540$	0	0
$541$	−34.7846	−1.49551	−0.747754	−	0.663976i	$-0.768868\pi$
$541$	−34.7846	−1.49551	−0.747754	+	0.663976i	$0.768868\pi$
$542$	0	0
$543$	1.21539	0.0521574
$544$	0	0
$545$	−9.46410	−0.405398
$546$	0	0
$547$	−3.60770	−0.154254	−0.0771270	−	0.997021i	$-0.524575\pi$
$547$	−3.60770	−0.154254	−0.0771270	+	0.997021i	$0.524575\pi$
$548$	0	0
$549$	−27.4115	−1.16990
$550$	0	0
$551$	21.4641	0.914401
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	9.07180	0.385076
$556$	0	0
$557$	−25.7128	−1.08949	−0.544743	−	0.838603i	$-0.683373\pi$
$557$	−25.7128	−1.08949	−0.544743	+	0.838603i	$0.683373\pi$
$558$	0	0
$559$	6.92820	0.293032
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	9.12436	0.384546	0.192273	−	0.981342i	$-0.438414\pi$
$563$	9.12436	0.384546	0.192273	+	0.981342i	$0.438414\pi$
$564$	0	0
$565$	−1.07180	−0.0450908
$566$	0	0
$567$	4.46410	0.187475
$568$	0	0
$569$	−22.2487	−0.932714	−0.466357	−	0.884596i	$-0.654434\pi$
$569$	−22.2487	−0.932714	−0.466357	+	0.884596i	$0.654434\pi$
$570$	0	0
$571$	−32.3923	−1.35558	−0.677788	−	0.735257i	$-0.737061\pi$
$571$	−32.3923	−1.35558	−0.677788	+	0.735257i	$0.737061\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	−22.0000	−0.917463
$576$	0	0
$577$	8.92820	0.371686	0.185843	−	0.982579i	$-0.440498\pi$
$577$	8.92820	0.371686	0.185843	+	0.982579i	$0.440498\pi$
$578$	0	0
$579$	−9.07180	−0.377011
$580$	0	0
$581$	7.26795	0.301525
$582$	0	0
$583$	−8.78461	−0.363821
$584$	0	0
$585$	−18.3923	−0.760428
$586$	0	0
$587$	−10.5885	−0.437032	−0.218516	−	0.975833i	$-0.570122\pi$
$587$	−10.5885	−0.437032	−0.218516	+	0.975833i	$0.570122\pi$
$588$	0	0
$589$	15.7128	0.647435
$590$	0	0
$591$	17.4641	0.718377
$592$	0	0
$593$	7.85641	0.322624	0.161312	−	0.986903i	$-0.448427\pi$
$593$	7.85641	0.322624	0.161312	+	0.986903i	$0.448427\pi$
$594$	0	0
$595$	−20.3923	−0.836003
$596$	0	0
$597$	−13.5692	−0.555351
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−0.928203	−0.0378622	−0.0189311	−	0.999821i	$-0.506026\pi$
$601$	−0.928203	−0.0378622	−0.0189311	+	0.999821i	$0.506026\pi$
$602$	0	0
$603$	36.7846	1.49799
$604$	0	0
$605$	24.1962	0.983713
$606$	0	0
$607$	26.9282	1.09298	0.546491	−	0.837465i	$-0.315963\pi$
$607$	26.9282	1.09298	0.546491	+	0.837465i	$0.315963\pi$
$608$	0	0
$609$	−2.53590	−0.102760
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	21.3205	0.861127	0.430564	−	0.902560i	$-0.358315\pi$
$613$	21.3205	0.861127	0.430564	+	0.902560i	$0.358315\pi$
$614$	0	0
$615$	6.92820	0.279372
$616$	0	0
$617$	15.3205	0.616780	0.308390	−	0.951260i	$-0.400210\pi$
$617$	15.3205	0.616780	0.308390	+	0.951260i	$0.400210\pi$
$618$	0	0
$619$	−47.3731	−1.90408	−0.952042	−	0.305967i	$-0.901020\pi$
$619$	−47.3731	−1.90408	−0.952042	+	0.305967i	$0.901020\pi$
$620$	0	0
$621$	35.7128	1.43311
$622$	0	0
$623$	4.92820	0.197444
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	−6.64102	−0.265217
$628$	0	0
$629$	−33.8564	−1.34994
$630$	0	0
$631$	21.0718	0.838855	0.419427	−	0.907789i	$-0.362231\pi$
$631$	21.0718	0.838855	0.419427	+	0.907789i	$0.362231\pi$
$632$	0	0
$633$	−2.92820	−0.116386
$634$	0	0
$635$	5.46410	0.216836
$636$	0	0
$637$	−2.73205	−0.108248
$638$	0	0
$639$	−34.1436	−1.35070
$640$	0	0
$641$	−12.6795	−0.500810	−0.250405	−	0.968141i	$-0.580564\pi$
$641$	−12.6795	−0.500810	−0.250405	+	0.968141i	$0.580564\pi$
$642$	0	0
$643$	19.2679	0.759854	0.379927	−	0.925017i	$-0.375949\pi$
$643$	19.2679	0.759854	0.379927	+	0.925017i	$0.375949\pi$
$644$	0	0
$645$	5.07180	0.199702
$646$	0	0
$647$	11.3205	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$647$	11.3205	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$648$	0	0
$649$	−9.07180	−0.356099
$650$	0	0
$651$	−1.85641	−0.0727583
$652$	0	0
$653$	−27.4641	−1.07475	−0.537377	−	0.843342i	$-0.680585\pi$
$653$	−27.4641	−1.07475	−0.537377	+	0.843342i	$0.680585\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	−23.3205	−0.908438	−0.454219	−	0.890890i	$-0.650082\pi$
$659$	−23.3205	−0.908438	−0.454219	+	0.890890i	$0.650082\pi$
$660$	0	0
$661$	28.5885	1.11196	0.555981	−	0.831195i	$-0.312343\pi$
$661$	28.5885	1.11196	0.555981	+	0.831195i	$0.312343\pi$
$662$	0	0
$663$	−14.9282	−0.579763
$664$	0	0
$665$	16.9282	0.656448
$666$	0	0
$667$	30.9282	1.19754
$668$	0	0
$669$	18.9282	0.731807
$670$	0	0
$671$	16.2872	0.628760
$672$	0	0
$673$	3.07180	0.118409	0.0592045	−	0.998246i	$-0.481144\pi$
$673$	3.07180	0.118409	0.0592045	+	0.998246i	$0.481144\pi$
$674$	0	0
$675$	−9.85641	−0.379373
$676$	0	0
$677$	39.5167	1.51875	0.759374	−	0.650654i	$-0.225505\pi$
$677$	39.5167	1.51875	0.759374	+	0.650654i	$0.225505\pi$
$678$	0	0
$679$	0.535898	0.0205659
$680$	0	0
$681$	7.17691	0.275020
$682$	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
$685$	57.1769	2.18462
$686$	0	0
$687$	0.143594	0.00547844
$688$	0	0
$689$	16.3923	0.624497
$690$	0	0
$691$	2.19615	0.0835456	0.0417728	−	0.999127i	$-0.486699\pi$
$691$	2.19615	0.0835456	0.0417728	+	0.999127i	$0.486699\pi$
$692$	0	0
$693$	−3.60770	−0.137045
$694$	0	0
$695$	42.7846	1.62291
$696$	0	0
$697$	−25.8564	−0.979381
$698$	0	0
$699$	13.1769	0.498397
$700$	0	0
$701$	−0.535898	−0.0202406	−0.0101203	−	0.999949i	$-0.503221\pi$
$701$	−0.535898	−0.0202406	−0.0101203	+	0.999949i	$0.503221\pi$
$702$	0	0
$703$	28.1051	1.06000
$704$	0	0
$705$	−2.92820	−0.110283
$706$	0	0
$707$	3.80385	0.143058
$708$	0	0
$709$	−36.5359	−1.37213	−0.686067	−	0.727538i	$-0.740664\pi$
$709$	−36.5359	−1.37213	−0.686067	+	0.727538i	$0.740664\pi$
$710$	0	0
$711$	19.7128	0.739288
$712$	0	0
$713$	22.6410	0.847913
$714$	0	0
$715$	10.9282	0.408692
$716$	0	0
$717$	9.46410	0.353443
$718$	0	0
$719$	−42.2487	−1.57561	−0.787806	−	0.615924i	$-0.788783\pi$
$719$	−42.2487	−1.57561	−0.787806	+	0.615924i	$0.788783\pi$
$720$	0	0
$721$	−8.39230	−0.312546
$722$	0	0
$723$	14.2487	0.529915
$724$	0	0
$725$	−8.53590	−0.317015
$726$	0	0
$727$	−38.5359	−1.42922	−0.714609	−	0.699524i	$-0.753395\pi$
$727$	−38.5359	−1.42922	−0.714609	+	0.699524i	$0.753395\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−18.9282	−0.700085
$732$	0	0
$733$	−0.588457	−0.0217352	−0.0108676	−	0.999941i	$-0.503459\pi$
$733$	−0.588457	−0.0217352	−0.0108676	+	0.999941i	$0.503459\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	−21.8564	−0.805091
$738$	0	0
$739$	23.3205	0.857859	0.428929	−	0.903338i	$-0.358891\pi$
$739$	23.3205	0.857859	0.428929	+	0.903338i	$0.358891\pi$
$740$	0	0
$741$	12.3923	0.455243
$742$	0	0
$743$	29.7128	1.09006	0.545029	−	0.838417i	$-0.316519\pi$
$743$	29.7128	1.09006	0.545029	+	0.838417i	$0.316519\pi$
$744$	0	0
$745$	−54.2487	−1.98752
$746$	0	0
$747$	−17.9090	−0.655255
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−4.92820	−0.179833	−0.0899163	−	0.995949i	$-0.528660\pi$
$751$	−4.92820	−0.179833	−0.0899163	+	0.995949i	$0.528660\pi$
$752$	0	0
$753$	−19.1769	−0.698846
$754$	0	0
$755$	5.46410	0.198859
$756$	0	0
$757$	−0.535898	−0.0194776	−0.00973878	−	0.999953i	$-0.503100\pi$
$757$	−0.535898	−0.0194776	−0.00973878	+	0.999953i	$0.503100\pi$
$758$	0	0
$759$	−9.56922	−0.347341
$760$	0	0
$761$	−1.32051	−0.0478684	−0.0239342	−	0.999714i	$-0.507619\pi$
$761$	−1.32051	−0.0478684	−0.0239342	+	0.999714i	$0.507619\pi$
$762$	0	0
$763$	3.46410	0.125409
$764$	0	0
$765$	50.2487	1.81675
$766$	0	0
$767$	16.9282	0.611242
$768$	0	0
$769$	40.5359	1.46176	0.730881	−	0.682505i	$-0.239109\pi$
$769$	40.5359	1.46176	0.730881	+	0.682505i	$0.239109\pi$
$770$	0	0
$771$	−18.2487	−0.657211
$772$	0	0
$773$	−43.5167	−1.56519	−0.782593	−	0.622534i	$-0.786103\pi$
$773$	−43.5167	−1.56519	−0.782593	+	0.622534i	$0.786103\pi$
$774$	0	0
$775$	−6.24871	−0.224460
$776$	0	0
$777$	−3.32051	−0.119123
$778$	0	0
$779$	21.4641	0.769031
$780$	0	0
$781$	20.2872	0.725933
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	−36.2487	−1.29377
$786$	0	0
$787$	12.7321	0.453849	0.226924	−	0.973912i	$-0.427133\pi$
$787$	12.7321	0.453849	0.226924	+	0.973912i	$0.427133\pi$
$788$	0	0
$789$	−10.1436	−0.361121
$790$	0	0
$791$	0.392305	0.0139488
$792$	0	0
$793$	−30.3923	−1.07926
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	−5.66025	−0.200496	−0.100248	−	0.994962i	$-0.531964\pi$
$797$	−5.66025	−0.200496	−0.100248	+	0.994962i	$0.531964\pi$
$798$	0	0
$799$	10.9282	0.386612
$800$	0	0
$801$	−12.1436	−0.429073
$802$	0	0
$803$	13.0718	0.461294
$804$	0	0
$805$	24.3923	0.859716
$806$	0	0
$807$	10.2872	0.362126
$808$	0	0
$809$	9.46410	0.332740	0.166370	−	0.986063i	$-0.446795\pi$
$809$	9.46410	0.332740	0.166370	+	0.986063i	$0.446795\pi$
$810$	0	0
$811$	34.9808	1.22834	0.614170	−	0.789173i	$-0.289491\pi$
$811$	34.9808	1.22834	0.614170	+	0.789173i	$0.289491\pi$
$812$	0	0
$813$	−6.64102	−0.232911
$814$	0	0
$815$	55.7128	1.95153
$816$	0	0
$817$	15.7128	0.549722
$818$	0	0
$819$	6.73205	0.235237
$820$	0	0
$821$	−5.71281	−0.199379	−0.0996893	−	0.995019i	$-0.531785\pi$
$821$	−5.71281	−0.199379	−0.0996893	+	0.995019i	$0.531785\pi$
$822$	0	0
$823$	−0.784610	−0.0273498	−0.0136749	−	0.999906i	$-0.504353\pi$
$823$	−0.784610	−0.0273498	−0.0136749	+	0.999906i	$0.504353\pi$
$824$	0	0
$825$	2.64102	0.0919484
$826$	0	0
$827$	34.9282	1.21457	0.607286	−	0.794483i	$-0.292258\pi$
$827$	34.9282	1.21457	0.607286	+	0.794483i	$0.292258\pi$
$828$	0	0
$829$	29.2679	1.01652	0.508259	−	0.861204i	$-0.330289\pi$
$829$	29.2679	1.01652	0.508259	+	0.861204i	$0.330289\pi$
$830$	0	0
$831$	−9.46410	−0.328306
$832$	0	0
$833$	7.46410	0.258616
$834$	0	0
$835$	−17.8564	−0.617946
$836$	0	0
$837$	10.1436	0.350614
$838$	0	0
$839$	2.53590	0.0875489	0.0437745	−	0.999041i	$-0.486062\pi$
$839$	2.53590	0.0875489	0.0437745	+	0.999041i	$0.486062\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−5.75129	−0.198085
$844$	0	0
$845$	15.1244	0.520294
$846$	0	0
$847$	−8.85641	−0.304310
$848$	0	0
$849$	11.7513	0.403303
$850$	0	0
$851$	40.4974	1.38823
$852$	0	0
$853$	−11.5167	−0.394323	−0.197161	−	0.980371i	$-0.563172\pi$
$853$	−11.5167	−0.394323	−0.197161	+	0.980371i	$0.563172\pi$
$854$	0	0
$855$	−41.7128	−1.42655
$856$	0	0
$857$	−20.5359	−0.701493	−0.350746	−	0.936470i	$-0.614072\pi$
$857$	−20.5359	−0.701493	−0.350746	+	0.936470i	$0.614072\pi$
$858$	0	0
$859$	3.66025	0.124886	0.0624431	−	0.998049i	$-0.480111\pi$
$859$	3.66025	0.124886	0.0624431	+	0.998049i	$0.480111\pi$
$860$	0	0
$861$	−2.53590	−0.0864232
$862$	0	0
$863$	21.0718	0.717292	0.358646	−	0.933474i	$-0.383238\pi$
$863$	21.0718	0.717292	0.358646	+	0.933474i	$0.383238\pi$
$864$	0	0
$865$	15.4641	0.525795
$866$	0	0
$867$	28.3397	0.962468
$868$	0	0
$869$	−11.7128	−0.397330
$870$	0	0
$871$	40.7846	1.38193
$872$	0	0
$873$	−1.32051	−0.0446924
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	34.1051	1.15165	0.575824	−	0.817574i	$-0.304681\pi$
$877$	34.1051	1.15165	0.575824	+	0.817574i	$0.304681\pi$
$878$	0	0
$879$	10.7846	0.363756
$880$	0	0
$881$	−13.7128	−0.461996	−0.230998	−	0.972954i	$-0.574199\pi$
$881$	−13.7128	−0.461996	−0.230998	+	0.972954i	$0.574199\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	12.3923	0.416563
$886$	0	0
$887$	33.4641	1.12361	0.561807	−	0.827268i	$-0.310106\pi$
$887$	33.4641	1.12361	0.561807	+	0.827268i	$0.310106\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	6.53590	0.218961
$892$	0	0
$893$	−9.07180	−0.303576
$894$	0	0
$895$	−32.7846	−1.09587
$896$	0	0
$897$	17.8564	0.596208
$898$	0	0
$899$	8.78461	0.292983
$900$	0	0
$901$	−44.7846	−1.49199
$902$	0	0
$903$	−1.85641	−0.0617773
$904$	0	0
$905$	−4.53590	−0.150778
$906$	0	0
$907$	42.6410	1.41587	0.707936	−	0.706277i	$-0.249627\pi$
$907$	42.6410	1.41587	0.707936	+	0.706277i	$0.249627\pi$
$908$	0	0
$909$	−9.37307	−0.310885
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	10.6410	0.352166
$914$	0	0
$915$	−22.2487	−0.735520
$916$	0	0
$917$	2.19615	0.0725233
$918$	0	0
$919$	−8.78461	−0.289778	−0.144889	−	0.989448i	$-0.546282\pi$
$919$	−8.78461	−0.289778	−0.144889	+	0.989448i	$0.546282\pi$
$920$	0	0
$921$	19.1769	0.631901
$922$	0	0
$923$	−37.8564	−1.24606
$924$	0	0
$925$	−11.1769	−0.367495
$926$	0	0
$927$	20.6795	0.679204
$928$	0	0
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	−6.19615	−0.203071
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	−29.8564	−0.976409
$936$	0	0
$937$	−43.5692	−1.42334	−0.711672	−	0.702512i	$-0.752062\pi$
$937$	−43.5692	−1.42334	−0.711672	+	0.702512i	$0.752062\pi$
$938$	0	0
$939$	9.17691	0.299477
$940$	0	0
$941$	−55.9090	−1.82258	−0.911290	−	0.411765i	$-0.864912\pi$
$941$	−55.9090	−1.82258	−0.911290	+	0.411765i	$0.864912\pi$
$942$	0	0
$943$	30.9282	1.00716
$944$	0	0
$945$	10.9282	0.355494
$946$	0	0
$947$	−7.32051	−0.237885	−0.118942	−	0.992901i	$-0.537950\pi$
$947$	−7.32051	−0.237885	−0.118942	+	0.992901i	$0.537950\pi$
$948$	0	0
$949$	−24.3923	−0.791808
$950$	0	0
$951$	9.46410	0.306895
$952$	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
$955$	29.8564	0.966131
$956$	0	0
$957$	−3.71281	−0.120018
$958$	0	0
$959$	−20.9282	−0.675807
$960$	0	0
$961$	−24.5692	−0.792555
$962$	0	0
$963$	−39.4256	−1.27047
$964$	0	0
$965$	33.8564	1.08988
$966$	0	0
$967$	32.6410	1.04966	0.524832	−	0.851206i	$-0.324128\pi$
$967$	32.6410	1.04966	0.524832	+	0.851206i	$0.324128\pi$
$968$	0	0
$969$	−33.8564	−1.08762
$970$	0	0
$971$	17.8038	0.571353	0.285676	−	0.958326i	$-0.407782\pi$
$971$	17.8038	0.571353	0.285676	+	0.958326i	$0.407782\pi$
$972$	0	0
$973$	−15.6603	−0.502045
$974$	0	0
$975$	−4.92820	−0.157829
$976$	0	0
$977$	16.6410	0.532393	0.266197	−	0.963919i	$-0.414233\pi$
$977$	16.6410	0.532393	0.266197	+	0.963919i	$0.414233\pi$
$978$	0	0
$979$	7.21539	0.230605
$980$	0	0
$981$	−8.53590	−0.272530
$982$	0	0
$983$	0.392305	0.0125126	0.00625629	−	0.999980i	$-0.498009\pi$
$983$	0.392305	0.0125126	0.00625629	+	0.999980i	$0.498009\pi$
$984$	0	0
$985$	−65.1769	−2.07671
$986$	0	0
$987$	1.07180	0.0341157
$988$	0	0
$989$	22.6410	0.719942
$990$	0	0
$991$	10.9282	0.347146	0.173573	−	0.984821i	$-0.444469\pi$
$991$	10.9282	0.347146	0.173573	+	0.984821i	$0.444469\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	50.6410	1.60543
$996$	0	0
$997$	−53.6603	−1.69944	−0.849719	−	0.527236i	$-0.823228\pi$
$997$	−53.6603	−1.69944	−0.849719	+	0.527236i	$0.823228\pi$
$998$	0	0
$999$	18.1436	0.574038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.a.e.1.2	✓	2
3.2	odd	2		8064.2.a.br.1.2		2
4.3	odd	2		896.2.a.g.1.1	yes	2
7.6	odd	2		6272.2.a.t.1.1		2
8.3	odd	2		896.2.a.f.1.2	yes	2
8.5	even	2		896.2.a.h.1.1	yes	2
12.11	even	2		8064.2.a.bm.1.2		2
16.3	odd	4		1792.2.b.n.897.2		4
16.5	even	4		1792.2.b.l.897.2		4
16.11	odd	4		1792.2.b.n.897.3		4
16.13	even	4		1792.2.b.l.897.3		4
24.5	odd	2		8064.2.a.bf.1.1		2
24.11	even	2		8064.2.a.be.1.1		2
28.27	even	2		6272.2.a.j.1.2		2
56.13	odd	2		6272.2.a.i.1.2		2
56.27	even	2		6272.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.e.1.2	✓	2	1.1	even	1	trivial
896.2.a.f.1.2	yes	2	8.3	odd	2
896.2.a.g.1.1	yes	2	4.3	odd	2
896.2.a.h.1.1	yes	2	8.5	even	2
1792.2.b.l.897.2		4	16.5	even	4
1792.2.b.l.897.3		4	16.13	even	4
1792.2.b.n.897.2		4	16.3	odd	4
1792.2.b.n.897.3		4	16.11	odd	4
6272.2.a.i.1.2		2	56.13	odd	2
6272.2.a.j.1.2		2	28.27	even	2
6272.2.a.s.1.1		2	56.27	even	2
6272.2.a.t.1.1		2	7.6	odd	2
8064.2.a.be.1.1		2	24.11	even	2
8064.2.a.bf.1.1		2	24.5	odd	2
8064.2.a.bm.1.2		2	12.11	even	2
8064.2.a.br.1.2		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -2.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} -2.73205 q^{5} +1.00000 q^{7} -2.46410 q^{9} +1.46410 q^{11} -2.73205 q^{13} -2.00000 q^{15} +7.46410 q^{17} -6.19615 q^{19} +0.732051 q^{21} -8.92820 q^{23} +2.46410 q^{25} -4.00000 q^{27} -3.46410 q^{29} -2.53590 q^{31} +1.07180 q^{33} -2.73205 q^{35} -4.53590 q^{37} -2.00000 q^{39} -3.46410 q^{41} -2.53590 q^{43} +6.73205 q^{45} +1.46410 q^{47} +1.00000 q^{49} +5.46410 q^{51} -6.00000 q^{53} -4.00000 q^{55} -4.53590 q^{57} -6.19615 q^{59} +11.1244 q^{61} -2.46410 q^{63} +7.46410 q^{65} -14.9282 q^{67} -6.53590 q^{69} +13.8564 q^{71} +8.92820 q^{73} +1.80385 q^{75} +1.46410 q^{77} -8.00000 q^{79} +4.46410 q^{81} +7.26795 q^{83} -20.3923 q^{85} -2.53590 q^{87} +4.92820 q^{89} -2.73205 q^{91} -1.85641 q^{93} +16.9282 q^{95} +0.535898 q^{97} -3.60770 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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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