LMFDB - Embedded newform 1024.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(1,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.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:	$8.17668116698$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 256)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	1024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -2.44949 q^{5} +1.03528 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -2.44949 q^{5} +1.03528 q^{7} +4.46410 q^{9} -1.26795 q^{11} +5.27792 q^{13} +6.69213 q^{15} +3.46410 q^{17} -1.26795 q^{19} -2.82843 q^{21} -6.69213 q^{23} +1.00000 q^{25} -4.00000 q^{27} +2.44949 q^{29} +5.65685 q^{31} +3.46410 q^{33} -2.53590 q^{35} -0.378937 q^{37} -14.4195 q^{39} -6.92820 q^{41} -8.19615 q^{43} -10.9348 q^{45} -9.79796 q^{47} -5.92820 q^{49} -9.46410 q^{51} +6.03579 q^{53} +3.10583 q^{55} +3.46410 q^{57} -10.7321 q^{59} +0.378937 q^{61} +4.62158 q^{63} -12.9282 q^{65} +4.19615 q^{67} +18.2832 q^{69} +6.69213 q^{71} -9.46410 q^{73} -2.73205 q^{75} -1.31268 q^{77} +15.4548 q^{79} -2.46410 q^{81} -8.19615 q^{83} -8.48528 q^{85} -6.69213 q^{87} -9.46410 q^{89} +5.46410 q^{91} -15.4548 q^{93} +3.10583 q^{95} -3.46410 q^{97} -5.66025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} - 12 q^{11} - 12 q^{19} + 4 q^{25} - 16 q^{27} - 24 q^{35} - 12 q^{43} + 4 q^{49} - 24 q^{51} - 36 q^{59} - 24 q^{65} - 4 q^{67} - 24 q^{73} - 4 q^{75} + 4 q^{81} - 12 q^{83} - 24 q^{89} + 8 q^{91} + 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 - 12 * q^11 - 12 * q^19 + 4 * q^25 - 16 * q^27 - 24 * q^35 - 12 * q^43 + 4 * q^49 - 24 * q^51 - 36 * q^59 - 24 * q^65 - 4 * q^67 - 24 * q^73 - 4 * q^75 + 4 * q^81 - 12 * q^83 - 24 * q^89 + 8 * q^91 + 12 * 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	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−2.44949	−1.09545	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	1.03528	0.391298	0.195649	−	0.980674i	$-0.437319\pi$
$7$	1.03528	0.391298	0.195649	+	0.980674i	$0.437319\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	5.27792	1.46383	0.731915	−	0.681396i	$-0.238627\pi$
$13$	5.27792	1.46383	0.731915	+	0.681396i	$0.238627\pi$
$14$	0	0
$15$	6.69213	1.72790
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−1.26795	−0.290887	−0.145444	−	0.989367i	$-0.546461\pi$
$19$	−1.26795	−0.290887	−0.145444	+	0.989367i	$0.546461\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	−6.69213	−1.39541	−0.697703	−	0.716387i	$-0.745794\pi$
$23$	−6.69213	−1.39541	−0.697703	+	0.716387i	$0.745794\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	2.44949	0.454859	0.227429	−	0.973795i	$-0.426968\pi$
$29$	2.44949	0.454859	0.227429	+	0.973795i	$0.426968\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	3.46410	0.603023
$34$	0	0
$35$	−2.53590	−0.428645
$36$	0	0
$37$	−0.378937	−0.0622969	−0.0311485	−	0.999515i	$-0.509916\pi$
$37$	−0.378937	−0.0622969	−0.0311485	+	0.999515i	$0.509916\pi$
$38$	0	0
$39$	−14.4195	−2.30897
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	−8.19615	−1.24990	−0.624951	−	0.780664i	$-0.714881\pi$
$43$	−8.19615	−1.24990	−0.624951	+	0.780664i	$0.714881\pi$
$44$	0	0
$45$	−10.9348	−1.63006
$46$	0	0
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	0	0
$49$	−5.92820	−0.846886
$50$	0	0
$51$	−9.46410	−1.32524
$52$	0	0
$53$	6.03579	0.829080	0.414540	−	0.910031i	$-0.363943\pi$
$53$	6.03579	0.829080	0.414540	+	0.910031i	$0.363943\pi$
$54$	0	0
$55$	3.10583	0.418790
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	−10.7321	−1.39719	−0.698597	−	0.715515i	$-0.746192\pi$
$59$	−10.7321	−1.39719	−0.698597	+	0.715515i	$0.746192\pi$
$60$	0	0
$61$	0.378937	0.0485180	0.0242590	−	0.999706i	$-0.492277\pi$
$61$	0.378937	0.0485180	0.0242590	+	0.999706i	$0.492277\pi$
$62$	0	0
$63$	4.62158	0.582264
$64$	0	0
$65$	−12.9282	−1.60355
$66$	0	0
$67$	4.19615	0.512642	0.256321	−	0.966592i	$-0.417490\pi$
$67$	4.19615	0.512642	0.256321	+	0.966592i	$0.417490\pi$
$68$	0	0
$69$	18.2832	2.20104
$70$	0	0
$71$	6.69213	0.794210	0.397105	−	0.917773i	$-0.370015\pi$
$71$	6.69213	0.794210	0.397105	+	0.917773i	$0.370015\pi$
$72$	0	0
$73$	−9.46410	−1.10769	−0.553845	−	0.832620i	$-0.686840\pi$
$73$	−9.46410	−1.10769	−0.553845	+	0.832620i	$0.686840\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	−1.31268	−0.149593
$78$	0	0
$79$	15.4548	1.73880	0.869401	−	0.494107i	$-0.164505\pi$
$79$	15.4548	1.73880	0.869401	+	0.494107i	$0.164505\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−8.19615	−0.899645	−0.449822	−	0.893118i	$-0.648513\pi$
$83$	−8.19615	−0.899645	−0.449822	+	0.893118i	$0.648513\pi$
$84$	0	0
$85$	−8.48528	−0.920358
$86$	0	0
$87$	−6.69213	−0.717472
$88$	0	0
$89$	−9.46410	−1.00319	−0.501596	−	0.865102i	$-0.667254\pi$
$89$	−9.46410	−1.00319	−0.501596	+	0.865102i	$0.667254\pi$
$90$	0	0
$91$	5.46410	0.572793
$92$	0	0
$93$	−15.4548	−1.60259
$94$	0	0
$95$	3.10583	0.318651
$96$	0	0
$97$	−3.46410	−0.351726	−0.175863	−	0.984415i	$-0.556272\pi$
$97$	−3.46410	−0.351726	−0.175863	+	0.984415i	$0.556272\pi$
$98$	0	0
$99$	−5.66025	−0.568877
$100$	0	0
$101$	−15.8338	−1.57552	−0.787759	−	0.615984i	$-0.788759\pi$
$101$	−15.8338	−1.57552	−0.787759	+	0.615984i	$0.788759\pi$
$102$	0	0
$103$	−4.62158	−0.455378	−0.227689	−	0.973734i	$-0.573117\pi$
$103$	−4.62158	−0.455378	−0.227689	+	0.973734i	$0.573117\pi$
$104$	0	0
$105$	6.92820	0.676123
$106$	0	0
$107$	12.5885	1.21697	0.608486	−	0.793565i	$-0.291777\pi$
$107$	12.5885	1.21697	0.608486	+	0.793565i	$0.291777\pi$
$108$	0	0
$109$	−3.96524	−0.379801	−0.189901	−	0.981803i	$-0.560817\pi$
$109$	−3.96524	−0.379801	−0.189901	+	0.981803i	$0.560817\pi$
$110$	0	0
$111$	1.03528	0.0982641
$112$	0	0
$113$	0.928203	0.0873180	0.0436590	−	0.999046i	$-0.486098\pi$
$113$	0.928203	0.0873180	0.0436590	+	0.999046i	$0.486098\pi$
$114$	0	0
$115$	16.3923	1.52859
$116$	0	0
$117$	23.5612	2.17823
$118$	0	0
$119$	3.58630	0.328756
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	18.9282	1.70670
$124$	0	0
$125$	9.79796	0.876356
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	0	0
$129$	22.3923	1.97153
$130$	0	0
$131$	−17.6603	−1.54298	−0.771492	−	0.636239i	$-0.780489\pi$
$131$	−17.6603	−1.54298	−0.771492	+	0.636239i	$0.780489\pi$
$132$	0	0
$133$	−1.31268	−0.113824
$134$	0	0
$135$	9.79796	0.843274
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−0.196152	−0.0166374	−0.00831872	−	0.999965i	$-0.502648\pi$
$139$	−0.196152	−0.0166374	−0.00831872	+	0.999965i	$0.502648\pi$
$140$	0	0
$141$	26.7685	2.25432
$142$	0	0
$143$	−6.69213	−0.559624
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	16.1962	1.33584
$148$	0	0
$149$	−14.5211	−1.18961	−0.594806	−	0.803869i	$-0.702771\pi$
$149$	−14.5211	−1.18961	−0.594806	+	0.803869i	$0.702771\pi$
$150$	0	0
$151$	4.62158	0.376099	0.188049	−	0.982160i	$-0.439784\pi$
$151$	4.62158	0.376099	0.188049	+	0.982160i	$0.439784\pi$
$152$	0	0
$153$	15.4641	1.25020
$154$	0	0
$155$	−13.8564	−1.11297
$156$	0	0
$157$	4.52004	0.360739	0.180369	−	0.983599i	$-0.442271\pi$
$157$	4.52004	0.360739	0.180369	+	0.983599i	$0.442271\pi$
$158$	0	0
$159$	−16.4901	−1.30775
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	13.2679	1.03923	0.519613	−	0.854402i	$-0.326076\pi$
$163$	13.2679	1.03923	0.519613	+	0.854402i	$0.326076\pi$
$164$	0	0
$165$	−8.48528	−0.660578
$166$	0	0
$167$	23.6627	1.83107	0.915537	−	0.402234i	$-0.131766\pi$
$167$	23.6627	1.83107	0.915537	+	0.402234i	$0.131766\pi$
$168$	0	0
$169$	14.8564	1.14280
$170$	0	0
$171$	−5.66025	−0.432850
$172$	0	0
$173$	−6.03579	−0.458893	−0.229446	−	0.973321i	$-0.573692\pi$
$173$	−6.03579	−0.458893	−0.229446	+	0.973321i	$0.573692\pi$
$174$	0	0
$175$	1.03528	0.0782595
$176$	0	0
$177$	29.3205	2.20386
$178$	0	0
$179$	−8.19615	−0.612609	−0.306305	−	0.951934i	$-0.599093\pi$
$179$	−8.19615	−0.612609	−0.306305	+	0.951934i	$0.599093\pi$
$180$	0	0
$181$	−11.4896	−0.854013	−0.427007	−	0.904248i	$-0.640432\pi$
$181$	−11.4896	−0.854013	−0.427007	+	0.904248i	$0.640432\pi$
$182$	0	0
$183$	−1.03528	−0.0765298
$184$	0	0
$185$	0.928203	0.0682429
$186$	0	0
$187$	−4.39230	−0.321197
$188$	0	0
$189$	−4.14110	−0.301221
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	2.39230	0.172202	0.0861009	−	0.996286i	$-0.472559\pi$
$193$	2.39230	0.172202	0.0861009	+	0.996286i	$0.472559\pi$
$194$	0	0
$195$	35.3205	2.52935
$196$	0	0
$197$	−2.44949	−0.174519	−0.0872595	−	0.996186i	$-0.527811\pi$
$197$	−2.44949	−0.174519	−0.0872595	+	0.996186i	$0.527811\pi$
$198$	0	0
$199$	−18.5606	−1.31573	−0.657865	−	0.753136i	$-0.728540\pi$
$199$	−18.5606	−1.31573	−0.657865	+	0.753136i	$0.728540\pi$
$200$	0	0
$201$	−11.4641	−0.808615
$202$	0	0
$203$	2.53590	0.177985
$204$	0	0
$205$	16.9706	1.18528
$206$	0	0
$207$	−29.8744	−2.07641
$208$	0	0
$209$	1.60770	0.111207
$210$	0	0
$211$	−0.196152	−0.0135037	−0.00675184	−	0.999977i	$-0.502149\pi$
$211$	−0.196152	−0.0135037	−0.00675184	+	0.999977i	$0.502149\pi$
$212$	0	0
$213$	−18.2832	−1.25275
$214$	0	0
$215$	20.0764	1.36920
$216$	0	0
$217$	5.85641	0.397559
$218$	0	0
$219$	25.8564	1.74721
$220$	0	0
$221$	18.2832	1.22986
$222$	0	0
$223$	5.65685	0.378811	0.189405	−	0.981899i	$-0.439344\pi$
$223$	5.65685	0.378811	0.189405	+	0.981899i	$0.439344\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	3.12436	0.207371	0.103685	−	0.994610i	$-0.466936\pi$
$227$	3.12436	0.207371	0.103685	+	0.994610i	$0.466936\pi$
$228$	0	0
$229$	1.69161	0.111785	0.0558925	−	0.998437i	$-0.482200\pi$
$229$	1.69161	0.111785	0.0558925	+	0.998437i	$0.482200\pi$
$230$	0	0
$231$	3.58630	0.235961
$232$	0	0
$233$	−9.46410	−0.620014	−0.310007	−	0.950734i	$-0.600331\pi$
$233$	−9.46410	−0.620014	−0.310007	+	0.950734i	$0.600331\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	−42.2233	−2.74270
$238$	0	0
$239$	−7.17260	−0.463957	−0.231979	−	0.972721i	$-0.574520\pi$
$239$	−7.17260	−0.463957	−0.231979	+	0.972721i	$0.574520\pi$
$240$	0	0
$241$	−18.3923	−1.18475	−0.592376	−	0.805661i	$-0.701810\pi$
$241$	−18.3923	−1.18475	−0.592376	+	0.805661i	$0.701810\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	14.5211	0.927717
$246$	0	0
$247$	−6.69213	−0.425810
$248$	0	0
$249$	22.3923	1.41905
$250$	0	0
$251$	3.12436	0.197208	0.0986038	−	0.995127i	$-0.468562\pi$
$251$	3.12436	0.197208	0.0986038	+	0.995127i	$0.468562\pi$
$252$	0	0
$253$	8.48528	0.533465
$254$	0	0
$255$	23.1822	1.45173
$256$	0	0
$257$	−0.928203	−0.0578997	−0.0289499	−	0.999581i	$-0.509216\pi$
$257$	−0.928203	−0.0578997	−0.0289499	+	0.999581i	$0.509216\pi$
$258$	0	0
$259$	−0.392305	−0.0243766
$260$	0	0
$261$	10.9348	0.676845
$262$	0	0
$263$	−12.9038	−0.795682	−0.397841	−	0.917454i	$-0.630240\pi$
$263$	−12.9038	−0.795682	−0.397841	+	0.917454i	$0.630240\pi$
$264$	0	0
$265$	−14.7846	−0.908211
$266$	0	0
$267$	25.8564	1.58239
$268$	0	0
$269$	7.34847	0.448044	0.224022	−	0.974584i	$-0.428081\pi$
$269$	7.34847	0.448044	0.224022	+	0.974584i	$0.428081\pi$
$270$	0	0
$271$	−21.1117	−1.28244	−0.641221	−	0.767356i	$-0.721572\pi$
$271$	−21.1117	−1.28244	−0.641221	+	0.767356i	$0.721572\pi$
$272$	0	0
$273$	−14.9282	−0.903496
$274$	0	0
$275$	−1.26795	−0.0764602
$276$	0	0
$277$	10.1769	0.611470	0.305735	−	0.952117i	$-0.401098\pi$
$277$	10.1769	0.611470	0.305735	+	0.952117i	$0.401098\pi$
$278$	0	0
$279$	25.2528	1.51184
$280$	0	0
$281$	−4.39230	−0.262023	−0.131011	−	0.991381i	$-0.541822\pi$
$281$	−4.39230	−0.262023	−0.131011	+	0.991381i	$0.541822\pi$
$282$	0	0
$283$	12.1962	0.724986	0.362493	−	0.931986i	$-0.381926\pi$
$283$	12.1962	0.724986	0.362493	+	0.931986i	$0.381926\pi$
$284$	0	0
$285$	−8.48528	−0.502625
$286$	0	0
$287$	−7.17260	−0.423385
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	9.46410	0.554795
$292$	0	0
$293$	17.1464	1.00171	0.500853	−	0.865533i	$-0.333020\pi$
$293$	17.1464	1.00171	0.500853	+	0.865533i	$0.333020\pi$
$294$	0	0
$295$	26.2880	1.53055
$296$	0	0
$297$	5.07180	0.294295
$298$	0	0
$299$	−35.3205	−2.04264
$300$	0	0
$301$	−8.48528	−0.489083
$302$	0	0
$303$	43.2586	2.48514
$304$	0	0
$305$	−0.928203	−0.0531488
$306$	0	0
$307$	−20.9808	−1.19744	−0.598718	−	0.800960i	$-0.704323\pi$
$307$	−20.9808	−1.19744	−0.598718	+	0.800960i	$0.704323\pi$
$308$	0	0
$309$	12.6264	0.718290
$310$	0	0
$311$	10.2784	0.582836	0.291418	−	0.956596i	$-0.405873\pi$
$311$	10.2784	0.582836	0.291418	+	0.956596i	$0.405873\pi$
$312$	0	0
$313$	−28.7846	−1.62700	−0.813501	−	0.581563i	$-0.802441\pi$
$313$	−28.7846	−1.62700	−0.813501	+	0.581563i	$0.802441\pi$
$314$	0	0
$315$	−11.3205	−0.637838
$316$	0	0
$317$	−24.3190	−1.36589	−0.682946	−	0.730468i	$-0.739302\pi$
$317$	−24.3190	−1.36589	−0.682946	+	0.730468i	$0.739302\pi$
$318$	0	0
$319$	−3.10583	−0.173893
$320$	0	0
$321$	−34.3923	−1.91959
$322$	0	0
$323$	−4.39230	−0.244394
$324$	0	0
$325$	5.27792	0.292766
$326$	0	0
$327$	10.8332	0.599079
$328$	0	0
$329$	−10.1436	−0.559234
$330$	0	0
$331$	4.58846	0.252204	0.126102	−	0.992017i	$-0.459753\pi$
$331$	4.58846	0.252204	0.126102	+	0.992017i	$0.459753\pi$
$332$	0	0
$333$	−1.69161	−0.0926999
$334$	0	0
$335$	−10.2784	−0.561571
$336$	0	0
$337$	33.7128	1.83645	0.918227	−	0.396055i	$-0.129621\pi$
$337$	33.7128	1.83645	0.918227	+	0.396055i	$0.129621\pi$
$338$	0	0
$339$	−2.53590	−0.137731
$340$	0	0
$341$	−7.17260	−0.388418
$342$	0	0
$343$	−13.3843	−0.722682
$344$	0	0
$345$	−44.7846	−2.41112
$346$	0	0
$347$	−10.7321	−0.576127	−0.288063	−	0.957611i	$-0.593011\pi$
$347$	−10.7321	−0.576127	−0.288063	+	0.957611i	$0.593011\pi$
$348$	0	0
$349$	−28.4601	−1.52344	−0.761718	−	0.647909i	$-0.775644\pi$
$349$	−28.4601	−1.52344	−0.761718	+	0.647909i	$0.775644\pi$
$350$	0	0
$351$	−21.1117	−1.12686
$352$	0	0
$353$	12.9282	0.688099	0.344049	−	0.938952i	$-0.388201\pi$
$353$	12.9282	0.688099	0.344049	+	0.938952i	$0.388201\pi$
$354$	0	0
$355$	−16.3923	−0.870013
$356$	0	0
$357$	−9.79796	−0.518563
$358$	0	0
$359$	26.2880	1.38743	0.693715	−	0.720250i	$-0.255973\pi$
$359$	26.2880	1.38743	0.693715	+	0.720250i	$0.255973\pi$
$360$	0	0
$361$	−17.3923	−0.915384
$362$	0	0
$363$	25.6603	1.34681
$364$	0	0
$365$	23.1822	1.21341
$366$	0	0
$367$	12.8295	0.669692	0.334846	−	0.942273i	$-0.391316\pi$
$367$	12.8295	0.669692	0.334846	+	0.942273i	$0.391316\pi$
$368$	0	0
$369$	−30.9282	−1.61006
$370$	0	0
$371$	6.24871	0.324417
$372$	0	0
$373$	14.3180	0.741358	0.370679	−	0.928761i	$-0.379125\pi$
$373$	14.3180	0.741358	0.370679	+	0.928761i	$0.379125\pi$
$374$	0	0
$375$	−26.7685	−1.38232
$376$	0	0
$377$	12.9282	0.665836
$378$	0	0
$379$	27.1244	1.39328	0.696642	−	0.717419i	$-0.254676\pi$
$379$	27.1244	1.39328	0.696642	+	0.717419i	$0.254676\pi$
$380$	0	0
$381$	15.4548	0.791774
$382$	0	0
$383$	−19.5959	−1.00130	−0.500652	−	0.865648i	$-0.666906\pi$
$383$	−19.5959	−1.00130	−0.500652	+	0.865648i	$0.666906\pi$
$384$	0	0
$385$	3.21539	0.163871
$386$	0	0
$387$	−36.5885	−1.85990
$388$	0	0
$389$	10.9348	0.554415	0.277207	−	0.960810i	$-0.410591\pi$
$389$	10.9348	0.554415	0.277207	+	0.960810i	$0.410591\pi$
$390$	0	0
$391$	−23.1822	−1.17238
$392$	0	0
$393$	48.2487	2.43383
$394$	0	0
$395$	−37.8564	−1.90476
$396$	0	0
$397$	−16.3886	−0.822518	−0.411259	−	0.911519i	$-0.634911\pi$
$397$	−16.3886	−0.822518	−0.411259	+	0.911519i	$0.634911\pi$
$398$	0	0
$399$	3.58630	0.179540
$400$	0	0
$401$	−34.3923	−1.71747	−0.858735	−	0.512420i	$-0.828749\pi$
$401$	−34.3923	−1.71747	−0.858735	+	0.512420i	$0.828749\pi$
$402$	0	0
$403$	29.8564	1.48725
$404$	0	0
$405$	6.03579	0.299921
$406$	0	0
$407$	0.480473	0.0238162
$408$	0	0
$409$	30.9282	1.52930	0.764651	−	0.644445i	$-0.222912\pi$
$409$	30.9282	1.52930	0.764651	+	0.644445i	$0.222912\pi$
$410$	0	0
$411$	−32.7846	−1.61715
$412$	0	0
$413$	−11.1106	−0.546719
$414$	0	0
$415$	20.0764	0.985511
$416$	0	0
$417$	0.535898	0.0262431
$418$	0	0
$419$	−3.80385	−0.185830	−0.0929151	−	0.995674i	$-0.529619\pi$
$419$	−3.80385	−0.185830	−0.0929151	+	0.995674i	$0.529619\pi$
$420$	0	0
$421$	−37.5002	−1.82765	−0.913824	−	0.406109i	$-0.866885\pi$
$421$	−37.5002	−1.82765	−0.913824	+	0.406109i	$0.866885\pi$
$422$	0	0
$423$	−43.7391	−2.12667
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	0.392305	0.0189850
$428$	0	0
$429$	18.2832	0.882723
$430$	0	0
$431$	−7.17260	−0.345492	−0.172746	−	0.984966i	$-0.555264\pi$
$431$	−7.17260	−0.345492	−0.172746	+	0.984966i	$0.555264\pi$
$432$	0	0
$433$	13.6077	0.653944	0.326972	−	0.945034i	$-0.393972\pi$
$433$	13.6077	0.653944	0.326972	+	0.945034i	$0.393972\pi$
$434$	0	0
$435$	16.3923	0.785951
$436$	0	0
$437$	8.48528	0.405906
$438$	0	0
$439$	−31.9449	−1.52465	−0.762324	−	0.647196i	$-0.775941\pi$
$439$	−31.9449	−1.52465	−0.762324	+	0.647196i	$0.775941\pi$
$440$	0	0
$441$	−26.4641	−1.26020
$442$	0	0
$443$	−3.80385	−0.180726	−0.0903631	−	0.995909i	$-0.528803\pi$
$443$	−3.80385	−0.180726	−0.0903631	+	0.995909i	$0.528803\pi$
$444$	0	0
$445$	23.1822	1.09894
$446$	0	0
$447$	39.6723	1.87644
$448$	0	0
$449$	24.2487	1.14437	0.572184	−	0.820125i	$-0.306096\pi$
$449$	24.2487	1.14437	0.572184	+	0.820125i	$0.306096\pi$
$450$	0	0
$451$	8.78461	0.413651
$452$	0	0
$453$	−12.6264	−0.593239
$454$	0	0
$455$	−13.3843	−0.627464
$456$	0	0
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	0	0
$459$	−13.8564	−0.646762
$460$	0	0
$461$	0.175865	0.00819087	0.00409544	−	0.999992i	$-0.498696\pi$
$461$	0.175865	0.00819087	0.00409544	+	0.999992i	$0.498696\pi$
$462$	0	0
$463$	18.4863	0.859132	0.429566	−	0.903036i	$-0.358667\pi$
$463$	18.4863	0.859132	0.429566	+	0.903036i	$0.358667\pi$
$464$	0	0
$465$	37.8564	1.75555
$466$	0	0
$467$	−25.2679	−1.16926	−0.584631	−	0.811300i	$-0.698761\pi$
$467$	−25.2679	−1.16926	−0.584631	+	0.811300i	$0.698761\pi$
$468$	0	0
$469$	4.34418	0.200595
$470$	0	0
$471$	−12.3490	−0.569011
$472$	0	0
$473$	10.3923	0.477839
$474$	0	0
$475$	−1.26795	−0.0581775
$476$	0	0
$477$	26.9444	1.23370
$478$	0	0
$479$	−2.62536	−0.119956	−0.0599778	−	0.998200i	$-0.519103\pi$
$479$	−2.62536	−0.119956	−0.0599778	+	0.998200i	$0.519103\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	18.9282	0.861263
$484$	0	0
$485$	8.48528	0.385297
$486$	0	0
$487$	−7.24693	−0.328390	−0.164195	−	0.986428i	$-0.552503\pi$
$487$	−7.24693	−0.328390	−0.164195	+	0.986428i	$0.552503\pi$
$488$	0	0
$489$	−36.2487	−1.63922
$490$	0	0
$491$	−32.1962	−1.45299	−0.726496	−	0.687171i	$-0.758852\pi$
$491$	−32.1962	−1.45299	−0.726496	+	0.687171i	$0.758852\pi$
$492$	0	0
$493$	8.48528	0.382158
$494$	0	0
$495$	13.8647	0.623173
$496$	0	0
$497$	6.92820	0.310772
$498$	0	0
$499$	−16.1962	−0.725039	−0.362520	−	0.931976i	$-0.618083\pi$
$499$	−16.1962	−0.725039	−0.362520	+	0.931976i	$0.618083\pi$
$500$	0	0
$501$	−64.6477	−2.88825
$502$	0	0
$503$	32.4997	1.44909	0.724545	−	0.689227i	$-0.242050\pi$
$503$	32.4997	1.44909	0.724545	+	0.689227i	$0.242050\pi$
$504$	0	0
$505$	38.7846	1.72589
$506$	0	0
$507$	−40.5885	−1.80260
$508$	0	0
$509$	−30.5307	−1.35325	−0.676625	−	0.736328i	$-0.736558\pi$
$509$	−30.5307	−1.35325	−0.676625	+	0.736328i	$0.736558\pi$
$510$	0	0
$511$	−9.79796	−0.433436
$512$	0	0
$513$	5.07180	0.223925
$514$	0	0
$515$	11.3205	0.498841
$516$	0	0
$517$	12.4233	0.546377
$518$	0	0
$519$	16.4901	0.723835
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	22.7321	0.994003	0.497002	−	0.867750i	$-0.334434\pi$
$523$	22.7321	0.994003	0.497002	+	0.867750i	$0.334434\pi$
$524$	0	0
$525$	−2.82843	−0.123443
$526$	0	0
$527$	19.5959	0.853612
$528$	0	0
$529$	21.7846	0.947157
$530$	0	0
$531$	−47.9090	−2.07907
$532$	0	0
$533$	−36.5665	−1.58387
$534$	0	0
$535$	−30.8353	−1.33313
$536$	0	0
$537$	22.3923	0.966299
$538$	0	0
$539$	7.51666	0.323765
$540$	0	0
$541$	19.9749	0.858786	0.429393	−	0.903118i	$-0.358728\pi$
$541$	19.9749	0.858786	0.429393	+	0.903118i	$0.358728\pi$
$542$	0	0
$543$	31.3901	1.34708
$544$	0	0
$545$	9.71281	0.416051
$546$	0	0
$547$	−17.6603	−0.755098	−0.377549	−	0.925990i	$-0.623233\pi$
$547$	−17.6603	−0.755098	−0.377549	+	0.925990i	$0.623233\pi$
$548$	0	0
$549$	1.69161	0.0721964
$550$	0	0
$551$	−3.10583	−0.132313
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	−2.53590	−0.107643
$556$	0	0
$557$	1.13681	0.0481683	0.0240841	−	0.999710i	$-0.492333\pi$
$557$	1.13681	0.0481683	0.0240841	+	0.999710i	$0.492333\pi$
$558$	0	0
$559$	−43.2586	−1.82964
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−39.1244	−1.64889	−0.824447	−	0.565938i	$-0.808514\pi$
$563$	−39.1244	−1.64889	−0.824447	+	0.565938i	$0.808514\pi$
$564$	0	0
$565$	−2.27362	−0.0956521
$566$	0	0
$567$	−2.55103	−0.107133
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−19.8038	−0.828765	−0.414383	−	0.910103i	$-0.636002\pi$
$571$	−19.8038	−0.828765	−0.414383	+	0.910103i	$0.636002\pi$
$572$	0	0
$573$	46.3644	1.93690
$574$	0	0
$575$	−6.69213	−0.279081
$576$	0	0
$577$	−16.1436	−0.672067	−0.336033	−	0.941850i	$-0.609085\pi$
$577$	−16.1436	−0.672067	−0.336033	+	0.941850i	$0.609085\pi$
$578$	0	0
$579$	−6.53590	−0.271623
$580$	0	0
$581$	−8.48528	−0.352029
$582$	0	0
$583$	−7.65308	−0.316958
$584$	0	0
$585$	−57.7128	−2.38613
$586$	0	0
$587$	−15.1244	−0.624249	−0.312124	−	0.950041i	$-0.601041\pi$
$587$	−15.1244	−0.624249	−0.312124	+	0.950041i	$0.601041\pi$
$588$	0	0
$589$	−7.17260	−0.295542
$590$	0	0
$591$	6.69213	0.275277
$592$	0	0
$593$	−7.85641	−0.322624	−0.161312	−	0.986903i	$-0.551573\pi$
$593$	−7.85641	−0.322624	−0.161312	+	0.986903i	$0.551573\pi$
$594$	0	0
$595$	−8.78461	−0.360134
$596$	0	0
$597$	50.7086	2.07537
$598$	0	0
$599$	29.8744	1.22063	0.610316	−	0.792158i	$-0.291042\pi$
$599$	29.8744	1.22063	0.610316	+	0.792158i	$0.291042\pi$
$600$	0	0
$601$	−19.6077	−0.799815	−0.399907	−	0.916556i	$-0.630958\pi$
$601$	−19.6077	−0.799815	−0.399907	+	0.916556i	$0.630958\pi$
$602$	0	0
$603$	18.7321	0.762828
$604$	0	0
$605$	23.0064	0.935341
$606$	0	0
$607$	30.9096	1.25458	0.627292	−	0.778785i	$-0.284163\pi$
$607$	30.9096	1.25458	0.627292	+	0.778785i	$0.284163\pi$
$608$	0	0
$609$	−6.92820	−0.280745
$610$	0	0
$611$	−51.7128	−2.09208
$612$	0	0
$613$	32.6012	1.31675	0.658376	−	0.752689i	$-0.271244\pi$
$613$	32.6012	1.31675	0.658376	+	0.752689i	$0.271244\pi$
$614$	0	0
$615$	−46.3644	−1.86959
$616$	0	0
$617$	23.3205	0.938848	0.469424	−	0.882973i	$-0.344462\pi$
$617$	23.3205	0.938848	0.469424	+	0.882973i	$0.344462\pi$
$618$	0	0
$619$	4.58846	0.184426	0.0922128	−	0.995739i	$-0.470606\pi$
$619$	4.58846	0.184426	0.0922128	+	0.995739i	$0.470606\pi$
$620$	0	0
$621$	26.7685	1.07418
$622$	0	0
$623$	−9.79796	−0.392547
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−4.39230	−0.175412
$628$	0	0
$629$	−1.31268	−0.0523399
$630$	0	0
$631$	35.5312	1.41447	0.707237	−	0.706976i	$-0.249941\pi$
$631$	35.5312	1.41447	0.707237	+	0.706976i	$0.249941\pi$
$632$	0	0
$633$	0.535898	0.0213000
$634$	0	0
$635$	13.8564	0.549875
$636$	0	0
$637$	−31.2886	−1.23970
$638$	0	0
$639$	29.8744	1.18181
$640$	0	0
$641$	34.3923	1.35841	0.679207	−	0.733947i	$-0.262324\pi$
$641$	34.3923	1.35841	0.679207	+	0.733947i	$0.262324\pi$
$642$	0	0
$643$	20.1962	0.796459	0.398229	−	0.917286i	$-0.369625\pi$
$643$	20.1962	0.796459	0.398229	+	0.917286i	$0.369625\pi$
$644$	0	0
$645$	−54.8497	−2.15971
$646$	0	0
$647$	6.69213	0.263095	0.131547	−	0.991310i	$-0.458005\pi$
$647$	6.69213	0.263095	0.131547	+	0.991310i	$0.458005\pi$
$648$	0	0
$649$	13.6077	0.534149
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	−46.1886	−1.80750	−0.903749	−	0.428062i	$-0.859196\pi$
$653$	−46.1886	−1.80750	−0.903749	+	0.428062i	$0.859196\pi$
$654$	0	0
$655$	43.2586	1.69025
$656$	0	0
$657$	−42.2487	−1.64828
$658$	0	0
$659$	19.5167	0.760261	0.380131	−	0.924933i	$-0.375879\pi$
$659$	19.5167	0.760261	0.380131	+	0.924933i	$0.375879\pi$
$660$	0	0
$661$	30.7338	1.19540	0.597702	−	0.801718i	$-0.296080\pi$
$661$	30.7338	1.19540	0.597702	+	0.801718i	$0.296080\pi$
$662$	0	0
$663$	−49.9507	−1.93993
$664$	0	0
$665$	3.21539	0.124687
$666$	0	0
$667$	−16.3923	−0.634713
$668$	0	0
$669$	−15.4548	−0.597518
$670$	0	0
$671$	−0.480473	−0.0185485
$672$	0	0
$673$	6.67949	0.257475	0.128738	−	0.991679i	$-0.458907\pi$
$673$	6.67949	0.257475	0.128738	+	0.991679i	$0.458907\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	44.8759	1.72472	0.862360	−	0.506295i	$-0.168985\pi$
$677$	44.8759	1.72472	0.862360	+	0.506295i	$0.168985\pi$
$678$	0	0
$679$	−3.58630	−0.137630
$680$	0	0
$681$	−8.53590	−0.327096
$682$	0	0
$683$	29.6603	1.13492	0.567459	−	0.823402i	$-0.307927\pi$
$683$	29.6603	1.13492	0.567459	+	0.823402i	$0.307927\pi$
$684$	0	0
$685$	−29.3939	−1.12308
$686$	0	0
$687$	−4.62158	−0.176324
$688$	0	0
$689$	31.8564	1.21363
$690$	0	0
$691$	33.3731	1.26957	0.634786	−	0.772688i	$-0.281088\pi$
$691$	33.3731	1.26957	0.634786	+	0.772688i	$0.281088\pi$
$692$	0	0
$693$	−5.85993	−0.222600
$694$	0	0
$695$	0.480473	0.0182254
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	25.8564	0.977979
$700$	0	0
$701$	−37.7033	−1.42403	−0.712017	−	0.702162i	$-0.752218\pi$
$701$	−37.7033	−1.42403	−0.712017	+	0.702162i	$0.752218\pi$
$702$	0	0
$703$	0.480473	0.0181214
$704$	0	0
$705$	−65.5692	−2.46948
$706$	0	0
$707$	−16.3923	−0.616496
$708$	0	0
$709$	−12.8023	−0.480799	−0.240399	−	0.970674i	$-0.577278\pi$
$709$	−12.8023	−0.480799	−0.240399	+	0.970674i	$0.577278\pi$
$710$	0	0
$711$	68.9919	2.58740
$712$	0	0
$713$	−37.8564	−1.41773
$714$	0	0
$715$	16.3923	0.613037
$716$	0	0
$717$	19.5959	0.731823
$718$	0	0
$719$	46.3644	1.72910	0.864551	−	0.502545i	$-0.167603\pi$
$719$	46.3644	1.72910	0.864551	+	0.502545i	$0.167603\pi$
$720$	0	0
$721$	−4.78461	−0.178188
$722$	0	0
$723$	50.2487	1.86877
$724$	0	0
$725$	2.44949	0.0909718
$726$	0	0
$727$	−20.6312	−0.765169	−0.382584	−	0.923921i	$-0.624966\pi$
$727$	−20.6312	−0.765169	−0.382584	+	0.923921i	$0.624966\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−28.3923	−1.05013
$732$	0	0
$733$	−35.6327	−1.31613	−0.658063	−	0.752963i	$-0.728624\pi$
$733$	−35.6327	−1.31613	−0.658063	+	0.752963i	$0.728624\pi$
$734$	0	0
$735$	−39.6723	−1.46334
$736$	0	0
$737$	−5.32051	−0.195983
$738$	0	0
$739$	48.9808	1.80179	0.900893	−	0.434041i	$-0.142913\pi$
$739$	48.9808	1.80179	0.900893	+	0.434041i	$0.142913\pi$
$740$	0	0
$741$	18.2832	0.671652
$742$	0	0
$743$	−10.2784	−0.377079	−0.188540	−	0.982066i	$-0.560375\pi$
$743$	−10.2784	−0.377079	−0.188540	+	0.982066i	$0.560375\pi$
$744$	0	0
$745$	35.5692	1.30316
$746$	0	0
$747$	−36.5885	−1.33870
$748$	0	0
$749$	13.0325	0.476198
$750$	0	0
$751$	−15.4548	−0.563954	−0.281977	−	0.959421i	$-0.590990\pi$
$751$	−15.4548	−0.563954	−0.281977	+	0.959421i	$0.590990\pi$
$752$	0	0
$753$	−8.53590	−0.311065
$754$	0	0
$755$	−11.3205	−0.411995
$756$	0	0
$757$	−25.8348	−0.938981	−0.469491	−	0.882937i	$-0.655562\pi$
$757$	−25.8348	−0.938981	−0.469491	+	0.882937i	$0.655562\pi$
$758$	0	0
$759$	−23.1822	−0.841461
$760$	0	0
$761$	−34.6410	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$761$	−34.6410	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$762$	0	0
$763$	−4.10512	−0.148615
$764$	0	0
$765$	−37.8792	−1.36952
$766$	0	0
$767$	−56.6429	−2.04526
$768$	0	0
$769$	18.3923	0.663243	0.331622	−	0.943412i	$-0.392404\pi$
$769$	18.3923	0.663243	0.331622	+	0.943412i	$0.392404\pi$
$770$	0	0
$771$	2.53590	0.0913281
$772$	0	0
$773$	3.76217	0.135316	0.0676579	−	0.997709i	$-0.478447\pi$
$773$	3.76217	0.135316	0.0676579	+	0.997709i	$0.478447\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	1.07180	0.0384505
$778$	0	0
$779$	8.78461	0.314741
$780$	0	0
$781$	−8.48528	−0.303627
$782$	0	0
$783$	−9.79796	−0.350150
$784$	0	0
$785$	−11.0718	−0.395169
$786$	0	0
$787$	−18.3397	−0.653741	−0.326871	−	0.945069i	$-0.605994\pi$
$787$	−18.3397	−0.653741	−0.326871	+	0.945069i	$0.605994\pi$
$788$	0	0
$789$	35.2538	1.25507
$790$	0	0
$791$	0.960947	0.0341673
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	40.3923	1.43257
$796$	0	0
$797$	3.41044	0.120804	0.0604019	−	0.998174i	$-0.480762\pi$
$797$	3.41044	0.120804	0.0604019	+	0.998174i	$0.480762\pi$
$798$	0	0
$799$	−33.9411	−1.20075
$800$	0	0
$801$	−42.2487	−1.49278
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	16.9706	0.598134
$806$	0	0
$807$	−20.0764	−0.706722
$808$	0	0
$809$	30.9282	1.08738	0.543689	−	0.839287i	$-0.317027\pi$
$809$	30.9282	1.08738	0.543689	+	0.839287i	$0.317027\pi$
$810$	0	0
$811$	−4.98076	−0.174898	−0.0874491	−	0.996169i	$-0.527872\pi$
$811$	−4.98076	−0.174898	−0.0874491	+	0.996169i	$0.527872\pi$
$812$	0	0
$813$	57.6781	2.02286
$814$	0	0
$815$	−32.4997	−1.13842
$816$	0	0
$817$	10.3923	0.363581
$818$	0	0
$819$	24.3923	0.852336
$820$	0	0
$821$	45.2276	1.57846	0.789228	−	0.614101i	$-0.210481\pi$
$821$	45.2276	1.57846	0.789228	+	0.614101i	$0.210481\pi$
$822$	0	0
$823$	18.5606	0.646983	0.323492	−	0.946231i	$-0.395143\pi$
$823$	18.5606	0.646983	0.323492	+	0.946231i	$0.395143\pi$
$824$	0	0
$825$	3.46410	0.120605
$826$	0	0
$827$	6.33975	0.220455	0.110227	−	0.993906i	$-0.464842\pi$
$827$	6.33975	0.220455	0.110227	+	0.993906i	$0.464842\pi$
$828$	0	0
$829$	31.2886	1.08670	0.543348	−	0.839507i	$-0.317156\pi$
$829$	31.2886	1.08670	0.543348	+	0.839507i	$0.317156\pi$
$830$	0	0
$831$	−27.8038	−0.964503
$832$	0	0
$833$	−20.5359	−0.711527
$834$	0	0
$835$	−57.9615	−2.00584
$836$	0	0
$837$	−22.6274	−0.782118
$838$	0	0
$839$	4.06678	0.140401	0.0702003	−	0.997533i	$-0.477636\pi$
$839$	4.06678	0.140401	0.0702003	+	0.997533i	$0.477636\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	0	0
$843$	12.0000	0.413302
$844$	0	0
$845$	−36.3906	−1.25188
$846$	0	0
$847$	−9.72363	−0.334108
$848$	0	0
$849$	−33.3205	−1.14356
$850$	0	0
$851$	2.53590	0.0869295
$852$	0	0
$853$	48.2591	1.65236	0.826181	−	0.563406i	$-0.190509\pi$
$853$	48.2591	1.65236	0.826181	+	0.563406i	$0.190509\pi$
$854$	0	0
$855$	13.8647	0.474164
$856$	0	0
$857$	1.85641	0.0634136	0.0317068	−	0.999497i	$-0.489906\pi$
$857$	1.85641	0.0634136	0.0317068	+	0.999497i	$0.489906\pi$
$858$	0	0
$859$	−38.4449	−1.31172	−0.655861	−	0.754882i	$-0.727694\pi$
$859$	−38.4449	−1.31172	−0.655861	+	0.754882i	$0.727694\pi$
$860$	0	0
$861$	19.5959	0.667827
$862$	0	0
$863$	36.5665	1.24474	0.622369	−	0.782724i	$-0.286170\pi$
$863$	36.5665	1.24474	0.622369	+	0.782724i	$0.286170\pi$
$864$	0	0
$865$	14.7846	0.502692
$866$	0	0
$867$	13.6603	0.463927
$868$	0	0
$869$	−19.5959	−0.664746
$870$	0	0
$871$	22.1469	0.750421
$872$	0	0
$873$	−15.4641	−0.523381
$874$	0	0
$875$	10.1436	0.342916
$876$	0	0
$877$	11.4896	0.387975	0.193988	−	0.981004i	$-0.437858\pi$
$877$	11.4896	0.387975	0.193988	+	0.981004i	$0.437858\pi$
$878$	0	0
$879$	−46.8449	−1.58004
$880$	0	0
$881$	−4.14359	−0.139601	−0.0698006	−	0.997561i	$-0.522236\pi$
$881$	−4.14359	−0.139601	−0.0698006	+	0.997561i	$0.522236\pi$
$882$	0	0
$883$	32.8756	1.10635	0.553177	−	0.833064i	$-0.313415\pi$
$883$	32.8756	1.10635	0.553177	+	0.833064i	$0.313415\pi$
$884$	0	0
$885$	−71.8203	−2.41421
$886$	0	0
$887$	10.2784	0.345116	0.172558	−	0.984999i	$-0.444797\pi$
$887$	10.2784	0.345116	0.172558	+	0.984999i	$0.444797\pi$
$888$	0	0
$889$	−5.85641	−0.196418
$890$	0	0
$891$	3.12436	0.104670
$892$	0	0
$893$	12.4233	0.415730
$894$	0	0
$895$	20.0764	0.671080
$896$	0	0
$897$	96.4974	3.22196
$898$	0	0
$899$	13.8564	0.462137
$900$	0	0
$901$	20.9086	0.696566
$902$	0	0
$903$	23.1822	0.771456
$904$	0	0
$905$	28.1436	0.935525
$906$	0	0
$907$	−35.9090	−1.19234	−0.596169	−	0.802859i	$-0.703311\pi$
$907$	−35.9090	−1.19234	−0.596169	+	0.802859i	$0.703311\pi$
$908$	0	0
$909$	−70.6835	−2.34442
$910$	0	0
$911$	−43.7391	−1.44914	−0.724570	−	0.689201i	$-0.757962\pi$
$911$	−43.7391	−1.44914	−0.724570	+	0.689201i	$0.757962\pi$
$912$	0	0
$913$	10.3923	0.343935
$914$	0	0
$915$	2.53590	0.0838342
$916$	0	0
$917$	−18.2832	−0.603766
$918$	0	0
$919$	55.1271	1.81848	0.909238	−	0.416277i	$-0.136665\pi$
$919$	55.1271	1.81848	0.909238	+	0.416277i	$0.136665\pi$
$920$	0	0
$921$	57.3205	1.88877
$922$	0	0
$923$	35.3205	1.16259
$924$	0	0
$925$	−0.378937	−0.0124594
$926$	0	0
$927$	−20.6312	−0.677617
$928$	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
$931$	7.51666	0.246349
$932$	0	0
$933$	−28.0812	−0.919337
$934$	0	0
$935$	10.7589	0.351854
$936$	0	0
$937$	−53.1769	−1.73721	−0.868607	−	0.495502i	$-0.834984\pi$
$937$	−53.1769	−1.73721	−0.868607	+	0.495502i	$0.834984\pi$
$938$	0	0
$939$	78.6410	2.56635
$940$	0	0
$941$	41.2896	1.34600	0.673001	−	0.739641i	$-0.265005\pi$
$941$	41.2896	1.34600	0.673001	+	0.739641i	$0.265005\pi$
$942$	0	0
$943$	46.3644	1.50983
$944$	0	0
$945$	10.1436	0.329971
$946$	0	0
$947$	22.7321	0.738692	0.369346	−	0.929292i	$-0.379582\pi$
$947$	22.7321	0.738692	0.369346	+	0.929292i	$0.379582\pi$
$948$	0	0
$949$	−49.9507	−1.62147
$950$	0	0
$951$	66.4408	2.15449
$952$	0	0
$953$	29.5692	0.957841	0.478920	−	0.877858i	$-0.341028\pi$
$953$	29.5692	0.957841	0.478920	+	0.877858i	$0.341028\pi$
$954$	0	0
$955$	41.5692	1.34515
$956$	0	0
$957$	8.48528	0.274290
$958$	0	0
$959$	12.4233	0.401170
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	56.1962	1.81090
$964$	0	0
$965$	−5.85993	−0.188638
$966$	0	0
$967$	−7.24693	−0.233046	−0.116523	−	0.993188i	$-0.537175\pi$
$967$	−7.24693	−0.233046	−0.116523	+	0.993188i	$0.537175\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	33.3731	1.07099	0.535496	−	0.844538i	$-0.320125\pi$
$971$	33.3731	1.07099	0.535496	+	0.844538i	$0.320125\pi$
$972$	0	0
$973$	−0.203072	−0.00651019
$974$	0	0
$975$	−14.4195	−0.461795
$976$	0	0
$977$	51.4641	1.64648	0.823241	−	0.567692i	$-0.192163\pi$
$977$	51.4641	1.64648	0.823241	+	0.567692i	$0.192163\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	−17.7012	−0.565157
$982$	0	0
$983$	−57.6038	−1.83728	−0.918638	−	0.395100i	$-0.870710\pi$
$983$	−57.6038	−1.83728	−0.918638	+	0.395100i	$0.870710\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	27.7128	0.882109
$988$	0	0
$989$	54.8497	1.74412
$990$	0	0
$991$	16.5644	0.526186	0.263093	−	0.964770i	$-0.415257\pi$
$991$	16.5644	0.526186	0.263093	+	0.964770i	$0.415257\pi$
$992$	0	0
$993$	−12.5359	−0.397815
$994$	0	0
$995$	45.4641	1.44131
$996$	0	0
$997$	6.79367	0.215158	0.107579	−	0.994197i	$-0.465690\pi$
$997$	6.79367	0.215158	0.107579	+	0.994197i	$0.465690\pi$
$998$	0	0
$999$	1.51575	0.0479562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.a.g.1.1		4
3.2	odd	2		9216.2.a.bk.1.3		4
4.3	odd	2		1024.2.a.j.1.3		4
8.3	odd	2	inner	1024.2.a.g.1.2		4
8.5	even	2		1024.2.a.j.1.4		4
12.11	even	2		9216.2.a.bb.1.4		4
16.3	odd	4		1024.2.b.h.513.8		8
16.5	even	4		1024.2.b.h.513.7		8
16.11	odd	4		1024.2.b.h.513.1		8
16.13	even	4		1024.2.b.h.513.2		8
24.5	odd	2		9216.2.a.bb.1.1		4
24.11	even	2		9216.2.a.bk.1.2		4
32.3	odd	8		256.2.e.a.65.4	yes	8
32.5	even	8		256.2.e.b.193.4	yes	8
32.11	odd	8		256.2.e.a.193.4	yes	8
32.13	even	8		256.2.e.b.65.4	yes	8
32.19	odd	8		256.2.e.b.65.1	yes	8
32.21	even	8		256.2.e.a.193.1	yes	8
32.27	odd	8		256.2.e.b.193.1	yes	8
32.29	even	8		256.2.e.a.65.1	✓	8
96.5	odd	8		2304.2.k.k.1729.4		8
96.11	even	8		2304.2.k.f.1729.1		8
96.29	odd	8		2304.2.k.f.577.1		8
96.35	even	8		2304.2.k.f.577.2		8
96.53	odd	8		2304.2.k.f.1729.2		8
96.59	even	8		2304.2.k.k.1729.3		8
96.77	odd	8		2304.2.k.k.577.3		8
96.83	even	8		2304.2.k.k.577.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.e.a.65.1	✓	8	32.29	even	8
256.2.e.a.65.4	yes	8	32.3	odd	8
256.2.e.a.193.1	yes	8	32.21	even	8
256.2.e.a.193.4	yes	8	32.11	odd	8
256.2.e.b.65.1	yes	8	32.19	odd	8
256.2.e.b.65.4	yes	8	32.13	even	8
256.2.e.b.193.1	yes	8	32.27	odd	8
256.2.e.b.193.4	yes	8	32.5	even	8
1024.2.a.g.1.1		4	1.1	even	1	trivial
1024.2.a.g.1.2		4	8.3	odd	2	inner
1024.2.a.j.1.3		4	4.3	odd	2
1024.2.a.j.1.4		4	8.5	even	2
1024.2.b.h.513.1		8	16.11	odd	4
1024.2.b.h.513.2		8	16.13	even	4
1024.2.b.h.513.7		8	16.5	even	4
1024.2.b.h.513.8		8	16.3	odd	4
2304.2.k.f.577.1		8	96.29	odd	8
2304.2.k.f.577.2		8	96.35	even	8
2304.2.k.f.1729.1		8	96.11	even	8
2304.2.k.f.1729.2		8	96.53	odd	8
2304.2.k.k.577.3		8	96.77	odd	8
2304.2.k.k.577.4		8	96.83	even	8
2304.2.k.k.1729.3		8	96.59	even	8
2304.2.k.k.1729.4		8	96.5	odd	8
9216.2.a.bb.1.1		4	24.5	odd	2
9216.2.a.bb.1.4		4	12.11	even	2
9216.2.a.bk.1.2		4	24.11	even	2
9216.2.a.bk.1.3		4	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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -2.44949 q^{5} +1.03528 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -2.44949 q^{5} +1.03528 q^{7} +4.46410 q^{9} -1.26795 q^{11} +5.27792 q^{13} +6.69213 q^{15} +3.46410 q^{17} -1.26795 q^{19} -2.82843 q^{21} -6.69213 q^{23} +1.00000 q^{25} -4.00000 q^{27} +2.44949 q^{29} +5.65685 q^{31} +3.46410 q^{33} -2.53590 q^{35} -0.378937 q^{37} -14.4195 q^{39} -6.92820 q^{41} -8.19615 q^{43} -10.9348 q^{45} -9.79796 q^{47} -5.92820 q^{49} -9.46410 q^{51} +6.03579 q^{53} +3.10583 q^{55} +3.46410 q^{57} -10.7321 q^{59} +0.378937 q^{61} +4.62158 q^{63} -12.9282 q^{65} +4.19615 q^{67} +18.2832 q^{69} +6.69213 q^{71} -9.46410 q^{73} -2.73205 q^{75} -1.31268 q^{77} +15.4548 q^{79} -2.46410 q^{81} -8.19615 q^{83} -8.48528 q^{85} -6.69213 q^{87} -9.46410 q^{89} +5.46410 q^{91} -15.4548 q^{93} +3.10583 q^{95} -3.46410 q^{97} -5.66025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} - 12 q^{11} - 12 q^{19} + 4 q^{25} - 16 q^{27} - 24 q^{35} - 12 q^{43} + 4 q^{49} - 24 q^{51} - 36 q^{59} - 24 q^{65} - 4 q^{67} - 24 q^{73} - 4 q^{75} + 4 q^{81} - 12 q^{83} - 24 q^{89} + 8 q^{91} + 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 - 12 * q^11 - 12 * q^19 + 4 * q^25 - 16 * q^27 - 24 * q^35 - 12 * q^43 + 4 * q^49 - 24 * q^51 - 36 * q^59 - 24 * q^65 - 4 * q^67 - 24 * q^73 - 4 * q^75 + 4 * q^81 - 12 * q^83 - 24 * q^89 + 8 * q^91 + 12 * q^99

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