LMFDB - Embedded newform 1024.2.b.h.513.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(513,1024)]

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, 1]))

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

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

chi := DirichletCharacter("1024.513");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$8.17668116698$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 256)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.6
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.2.b.h.513.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051i q^{3} +2.44949i q^{5} -3.86370 q^{7} +2.46410 q^{9} +O(q^{10})$	$q+0.732051i q^{3} +2.44949i q^{5} -3.86370 q^{7} +2.46410 q^{9} +4.73205i q^{11} -0.378937i q^{13} -1.79315 q^{15} -3.46410 q^{17} -4.73205i q^{19} -2.82843i q^{21} -1.79315 q^{23} -1.00000 q^{25} +4.00000i q^{27} +2.44949i q^{29} -5.65685 q^{31} -3.46410 q^{33} -9.46410i q^{35} -5.27792i q^{37} +0.277401 q^{39} -6.92820 q^{41} -2.19615i q^{43} +6.03579i q^{45} -9.79796 q^{47} +7.92820 q^{49} -2.53590i q^{51} +10.9348i q^{53} -11.5911 q^{55} +3.46410 q^{57} +7.26795i q^{59} -5.27792i q^{61} -9.52056 q^{63} +0.928203 q^{65} -6.19615i q^{67} -1.31268i q^{69} +1.79315 q^{71} +2.53590 q^{73} -0.732051i q^{75} -18.2832i q^{77} +4.14110 q^{79} +4.46410 q^{81} +2.19615i q^{83} -8.48528i q^{85} -1.79315 q^{87} +2.53590 q^{89} +1.46410i q^{91} -4.14110i q^{93} +11.5911 q^{95} +3.46410 q^{97} +11.6603i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} - 8 q^{25} + 8 q^{49} - 48 q^{65} + 48 q^{73} + 8 q^{81} + 48 q^{89}+O(q^{100}) $ 8 * q - 8 * q^9 - 8 * q^25 + 8 * q^49 - 48 * q^65 + 48 * q^73 + 8 * q^81 + 48 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$1$

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
$3$	0.732051i	0.422650i	0.977416	+	0.211325i	$0.0677778\pi$
$3$	0.732051i	0.422650i	−0.977416	+	0.211325i	$0.932222\pi$
$4$	0	0
$5$	2.44949i	1.09545i	0.836660	+	0.547723i	$0.184505\pi$
$5$	2.44949i	1.09545i	−0.836660	+	0.547723i	$0.815495\pi$
$6$	0	0
$7$	−3.86370	−1.46034	−0.730171	−	0.683264i	$-0.760560\pi$
$7$	−3.86370	−1.46034	−0.730171	+	0.683264i	$0.760560\pi$
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	4.73205i	1.42677i	0.700774	+	0.713384i	$0.252838\pi$
$11$	4.73205i	1.42677i	−0.700774	+	0.713384i	$0.747162\pi$
$12$	0	0
$13$	− 0.378937i	− 0.105098i	−0.998618	−	0.0525492i	$-0.983265\pi$
$13$	− 0.378937i	− 0.105098i	0.998618	−	0.0525492i	$-0.0167346\pi$
$14$	0	0
$15$	−1.79315	−0.462990
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	− 4.73205i	− 1.08561i	−0.839860	−	0.542803i	$-0.817363\pi$
$19$	− 4.73205i	− 1.08561i	0.839860	−	0.542803i	$-0.182637\pi$
$20$	0	0
$21$	− 2.82843i	− 0.617213i
$22$	0	0
$23$	−1.79315	−0.373898	−0.186949	−	0.982370i	$-0.559860\pi$
$23$	−1.79315	−0.373898	−0.186949	+	0.982370i	$0.559860\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	2.44949i	0.454859i	0.973795	+	0.227429i	$0.0730321\pi$
$29$	2.44949i	0.454859i	−0.973795	+	0.227429i	$0.926968\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	− 9.46410i	− 1.59973i
$36$	0	0
$37$	− 5.27792i	− 0.867684i	−0.900989	−	0.433842i	$-0.857158\pi$
$37$	− 5.27792i	− 0.867684i	0.900989	−	0.433842i	$-0.142842\pi$
$38$	0	0
$39$	0.277401	0.0444198
$40$	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
$43$	− 2.19615i	− 0.334910i	−0.985880	−	0.167455i	$-0.946445\pi$
$43$	− 2.19615i	− 0.334910i	0.985880	−	0.167455i	$-0.0535549\pi$
$44$	0	0
$45$	6.03579i	0.899763i
$46$	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
$49$	7.92820	1.13260
$50$	0	0
$51$	− 2.53590i	− 0.355097i
$52$	0	0
$53$	10.9348i	1.50201i	0.660299	+	0.751003i	$0.270430\pi$
$53$	10.9348i	1.50201i	−0.660299	+	0.751003i	$0.729570\pi$
$54$	0	0
$55$	−11.5911	−1.56294
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	7.26795i	0.946206i	0.881007	+	0.473103i	$0.156866\pi$
$59$	7.26795i	0.946206i	−0.881007	+	0.473103i	$0.843134\pi$
$60$	0	0
$61$	− 5.27792i	− 0.675768i	−0.941188	−	0.337884i	$-0.890289\pi$
$61$	− 5.27792i	− 0.675768i	0.941188	−	0.337884i	$-0.109711\pi$
$62$	0	0
$63$	−9.52056	−1.19948
$64$	0	0
$65$	0.928203	0.115129
$66$	0	0
$67$	− 6.19615i	− 0.756980i	−0.925605	−	0.378490i	$-0.876443\pi$
$67$	− 6.19615i	− 0.756980i	0.925605	−	0.378490i	$-0.123557\pi$
$68$	0	0
$69$	− 1.31268i	− 0.158028i
$70$	0	0
$71$	1.79315	0.212808	0.106404	−	0.994323i	$-0.466066\pi$
$71$	1.79315	0.212808	0.106404	+	0.994323i	$0.466066\pi$
$72$	0	0
$73$	2.53590	0.296804	0.148402	−	0.988927i	$-0.452587\pi$
$73$	2.53590	0.296804	0.148402	+	0.988927i	$0.452587\pi$
$74$	0	0
$75$	− 0.732051i	− 0.0845299i
$76$	0	0
$77$	− 18.2832i	− 2.08357i
$78$	0	0
$79$	4.14110	0.465911	0.232955	−	0.972487i	$-0.425160\pi$
$79$	4.14110	0.465911	0.232955	+	0.972487i	$0.425160\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	2.19615i	0.241059i	0.992710	+	0.120530i	$0.0384592\pi$
$83$	2.19615i	0.241059i	−0.992710	+	0.120530i	$0.961541\pi$
$84$	0	0
$85$	− 8.48528i	− 0.920358i
$86$	0	0
$87$	−1.79315	−0.192246
$88$	0	0
$89$	2.53590	0.268805	0.134402	−	0.990927i	$-0.457089\pi$
$89$	2.53590	0.268805	0.134402	+	0.990927i	$0.457089\pi$
$90$	0	0
$91$	1.46410i	0.153480i
$92$	0	0
$93$	− 4.14110i	− 0.429413i
$94$	0	0
$95$	11.5911	1.18922
$96$	0	0
$97$	3.46410	0.351726	0.175863	−	0.984415i	$-0.443728\pi$
$97$	3.46410	0.351726	0.175863	+	0.984415i	$0.443728\pi$
$98$	0	0
$99$	11.6603i	1.17190i
$100$	0	0
$101$	− 1.13681i	− 0.113117i	−0.998399	−	0.0565585i	$-0.981987\pi$
$101$	− 1.13681i	− 0.113117i	0.998399	−	0.0565585i	$-0.0180127\pi$
$102$	0	0
$103$	−9.52056	−0.938088	−0.469044	−	0.883175i	$-0.655402\pi$
$103$	−9.52056	−0.938088	−0.469044	+	0.883175i	$0.655402\pi$
$104$	0	0
$105$	6.92820	0.676123
$106$	0	0
$107$	18.5885i	1.79701i	0.438959	+	0.898507i	$0.355347\pi$
$107$	18.5885i	1.79701i	−0.438959	+	0.898507i	$0.644653\pi$
$108$	0	0
$109$	18.6622i	1.78751i	0.448553	+	0.893756i	$0.351940\pi$
$109$	18.6622i	1.78751i	−0.448553	+	0.893756i	$0.648060\pi$
$110$	0	0
$111$	3.86370	0.366726
$112$	0	0
$113$	−12.9282	−1.21618	−0.608092	−	0.793867i	$-0.708065\pi$
$113$	−12.9282	−1.21618	−0.608092	+	0.793867i	$0.708065\pi$
$114$	0	0
$115$	− 4.39230i	− 0.409585i
$116$	0	0
$117$	− 0.933740i	− 0.0863243i
$118$	0	0
$119$	13.3843	1.22693
$120$	0	0
$121$	−11.3923	−1.03566
$122$	0	0
$123$	− 5.07180i	− 0.457309i
$124$	0	0
$125$	9.79796i	0.876356i
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	1.60770	0.141550
$130$	0	0
$131$	− 0.339746i	− 0.0296837i	−0.999890	−	0.0148419i	$-0.995276\pi$
$131$	− 0.339746i	− 0.0296837i	0.999890	−	0.0148419i	$-0.00472449\pi$
$132$	0	0
$133$	18.2832i	1.58536i
$134$	0	0
$135$	−9.79796	−0.843274
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	− 10.1962i	− 0.864826i	−0.901676	−	0.432413i	$-0.857662\pi$
$139$	− 10.1962i	− 0.864826i	0.901676	−	0.432413i	$-0.142338\pi$
$140$	0	0
$141$	− 7.17260i	− 0.604042i
$142$	0	0
$143$	1.79315	0.149951
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	5.80385i	0.478693i
$148$	0	0
$149$	− 19.4201i	− 1.59095i	−0.605985	−	0.795476i	$-0.707221\pi$
$149$	− 19.4201i	− 1.59095i	0.605985	−	0.795476i	$-0.292779\pi$
$150$	0	0
$151$	9.52056	0.774772	0.387386	−	0.921918i	$-0.373378\pi$
$151$	9.52056	0.774772	0.387386	+	0.921918i	$0.373378\pi$
$152$	0	0
$153$	−8.53590	−0.690086
$154$	0	0
$155$	− 13.8564i	− 1.11297i
$156$	0	0
$157$	10.1769i	0.812205i	0.913828	+	0.406102i	$0.133112\pi$
$157$	10.1769i	0.812205i	−0.913828	+	0.406102i	$0.866888\pi$
$158$	0	0
$159$	−8.00481	−0.634823
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	16.7321i	1.31056i	0.755388	+	0.655278i	$0.227448\pi$
$163$	16.7321i	1.31056i	−0.755388	+	0.655278i	$0.772552\pi$
$164$	0	0
$165$	− 8.48528i	− 0.660578i
$166$	0	0
$167$	18.7637	1.45198	0.725990	−	0.687705i	$-0.241382\pi$
$167$	18.7637	1.45198	0.725990	+	0.687705i	$0.241382\pi$
$168$	0	0
$169$	12.8564	0.988954
$170$	0	0
$171$	− 11.6603i	− 0.891682i
$172$	0	0
$173$	10.9348i	0.831355i	0.909512	+	0.415678i	$0.136456\pi$
$173$	10.9348i	0.831355i	−0.909512	+	0.415678i	$0.863544\pi$
$174$	0	0
$175$	3.86370	0.292069
$176$	0	0
$177$	−5.32051	−0.399914
$178$	0	0
$179$	2.19615i	0.164148i	0.996626	+	0.0820741i	$0.0261544\pi$
$179$	2.19615i	0.164148i	−0.996626	+	0.0820741i	$0.973846\pi$
$180$	0	0
$181$	22.8033i	1.69495i	0.530832	+	0.847477i	$0.321880\pi$
$181$	22.8033i	1.69495i	−0.530832	+	0.847477i	$0.678120\pi$
$182$	0	0
$183$	3.86370	0.285613
$184$	0	0
$185$	12.9282	0.950500
$186$	0	0
$187$	− 16.3923i	− 1.19872i
$188$	0	0
$189$	− 15.4548i	− 1.12417i
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	−18.3923	−1.32391	−0.661954	−	0.749545i	$-0.730272\pi$
$193$	−18.3923	−1.32391	−0.661954	+	0.749545i	$0.730272\pi$
$194$	0	0
$195$	0.679492i	0.0486594i
$196$	0	0
$197$	2.44949i	0.174519i	0.996186	+	0.0872595i	$0.0278109\pi$
$197$	2.44949i	0.174519i	−0.996186	+	0.0872595i	$0.972189\pi$
$198$	0	0
$199$	15.7322	1.11523	0.557614	−	0.830101i	$-0.311717\pi$
$199$	15.7322	1.11523	0.557614	+	0.830101i	$0.311717\pi$
$200$	0	0
$201$	4.53590	0.319938
$202$	0	0
$203$	− 9.46410i	− 0.664250i
$204$	0	0
$205$	− 16.9706i	− 1.18528i
$206$	0	0
$207$	−4.41851	−0.307107
$208$	0	0
$209$	22.3923	1.54891
$210$	0	0
$211$	10.1962i	0.701932i	0.936388	+	0.350966i	$0.114147\pi$
$211$	10.1962i	0.701932i	−0.936388	+	0.350966i	$0.885853\pi$
$212$	0	0
$213$	1.31268i	0.0899432i
$214$	0	0
$215$	5.37945	0.366876
$216$	0	0
$217$	21.8564	1.48371
$218$	0	0
$219$	1.85641i	0.125444i
$220$	0	0
$221$	1.31268i	0.0883003i
$222$	0	0
$223$	−5.65685	−0.378811	−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685	−0.378811	−0.189405	+	0.981899i	$0.560656\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	− 21.1244i	− 1.40207i	−0.713126	−	0.701036i	$-0.752721\pi$
$227$	− 21.1244i	− 1.40207i	0.713126	−	0.701036i	$-0.247279\pi$
$228$	0	0
$229$	− 13.0053i	− 0.859416i	−0.902968	−	0.429708i	$-0.858617\pi$
$229$	− 13.0053i	− 0.859416i	0.902968	−	0.429708i	$-0.141383\pi$
$230$	0	0
$231$	13.3843	0.880620
$232$	0	0
$233$	2.53590	0.166132	0.0830661	−	0.996544i	$-0.473529\pi$
$233$	2.53590	0.166132	0.0830661	+	0.996544i	$0.473529\pi$
$234$	0	0
$235$	− 24.0000i	− 1.56559i
$236$	0	0
$237$	3.03150i	0.196917i
$238$	0	0
$239$	26.7685	1.73151	0.865756	−	0.500467i	$-0.166838\pi$
$239$	26.7685	1.73151	0.865756	+	0.500467i	$0.166838\pi$
$240$	0	0
$241$	2.39230	0.154102	0.0770510	−	0.997027i	$-0.475450\pi$
$241$	2.39230	0.154102	0.0770510	+	0.997027i	$0.475450\pi$
$242$	0	0
$243$	15.2679i	0.979439i
$244$	0	0
$245$	19.4201i	1.24070i
$246$	0	0
$247$	−1.79315	−0.114095
$248$	0	0
$249$	−1.60770	−0.101884
$250$	0	0
$251$	21.1244i	1.33336i	0.745345	+	0.666679i	$0.232285\pi$
$251$	21.1244i	1.33336i	−0.745345	+	0.666679i	$0.767715\pi$
$252$	0	0
$253$	− 8.48528i	− 0.533465i
$254$	0	0
$255$	6.21166	0.388989
$256$	0	0
$257$	12.9282	0.806439	0.403220	−	0.915103i	$-0.367891\pi$
$257$	12.9282	0.806439	0.403220	+	0.915103i	$0.367891\pi$
$258$	0	0
$259$	20.3923i	1.26712i
$260$	0	0
$261$	6.03579i	0.373606i
$262$	0	0
$263$	21.3891	1.31891	0.659453	−	0.751746i	$-0.270788\pi$
$263$	21.3891	1.31891	0.659453	+	0.751746i	$0.270788\pi$
$264$	0	0
$265$	−26.7846	−1.64537
$266$	0	0
$267$	1.85641i	0.113610i
$268$	0	0
$269$	7.34847i	0.448044i	0.974584	+	0.224022i	$0.0719188\pi$
$269$	7.34847i	0.448044i	−0.974584	+	0.224022i	$0.928081\pi$
$270$	0	0
$271$	1.51575	0.0920752	0.0460376	−	0.998940i	$-0.485341\pi$
$271$	1.51575	0.0920752	0.0460376	+	0.998940i	$0.485341\pi$
$272$	0	0
$273$	−1.07180	−0.0648681
$274$	0	0
$275$	− 4.73205i	− 0.285353i
$276$	0	0
$277$	− 4.52004i	− 0.271583i	−0.990737	−	0.135792i	$-0.956642\pi$
$277$	− 4.52004i	− 0.271583i	0.990737	−	0.135792i	$-0.0433577\pi$
$278$	0	0
$279$	−13.9391	−0.834510
$280$	0	0
$281$	−16.3923	−0.977883	−0.488941	−	0.872317i	$-0.662617\pi$
$281$	−16.3923	−0.977883	−0.488941	+	0.872317i	$0.662617\pi$
$282$	0	0
$283$	− 1.80385i	− 0.107228i	−0.998562	−	0.0536138i	$-0.982926\pi$
$283$	− 1.80385i	− 0.107228i	0.998562	−	0.0536138i	$-0.0170740\pi$
$284$	0	0
$285$	8.48528i	0.502625i
$286$	0	0
$287$	26.7685	1.58010
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	2.53590i	0.148657i
$292$	0	0
$293$	− 17.1464i	− 1.00171i	−0.865533	−	0.500853i	$-0.833020\pi$
$293$	− 17.1464i	− 1.00171i	0.865533	−	0.500853i	$-0.166980\pi$
$294$	0	0
$295$	−17.8028	−1.03652
$296$	0	0
$297$	−18.9282	−1.09833
$298$	0	0
$299$	0.679492i	0.0392960i
$300$	0	0
$301$	8.48528i	0.489083i
$302$	0	0
$303$	0.832204	0.0478089
$304$	0	0
$305$	12.9282	0.740267
$306$	0	0
$307$	30.9808i	1.76817i	0.467330	+	0.884083i	$0.345216\pi$
$307$	30.9808i	1.76817i	−0.467330	+	0.884083i	$0.654784\pi$
$308$	0	0
$309$	− 6.96953i	− 0.396483i
$310$	0	0
$311$	15.1774	0.860632	0.430316	−	0.902678i	$-0.358402\pi$
$311$	15.1774	0.860632	0.430316	+	0.902678i	$0.358402\pi$
$312$	0	0
$313$	−12.7846	−0.722629	−0.361314	−	0.932444i	$-0.617672\pi$
$313$	−12.7846	−0.722629	−0.361314	+	0.932444i	$0.617672\pi$
$314$	0	0
$315$	− 23.3205i	− 1.31396i
$316$	0	0
$317$	9.62209i	0.540431i	0.962800	+	0.270215i	$0.0870949\pi$
$317$	9.62209i	0.540431i	−0.962800	+	0.270215i	$0.912905\pi$
$318$	0	0
$319$	−11.5911	−0.648978
$320$	0	0
$321$	−13.6077	−0.759507
$322$	0	0
$323$	16.3923i	0.912092i
$324$	0	0
$325$	0.378937i	0.0210197i
$326$	0	0
$327$	−13.6617	−0.755492
$328$	0	0
$329$	37.8564	2.08709
$330$	0	0
$331$	26.5885i	1.46143i	0.682681	+	0.730717i	$0.260814\pi$
$331$	26.5885i	1.46143i	−0.682681	+	0.730717i	$0.739186\pi$
$332$	0	0
$333$	− 13.0053i	− 0.712687i
$334$	0	0
$335$	15.1774	0.829231
$336$	0	0
$337$	−21.7128	−1.18277	−0.591386	−	0.806389i	$-0.701419\pi$
$337$	−21.7128	−1.18277	−0.591386	+	0.806389i	$0.701419\pi$
$338$	0	0
$339$	− 9.46410i	− 0.514019i
$340$	0	0
$341$	− 26.7685i	− 1.44960i
$342$	0	0
$343$	−3.58630	−0.193642
$344$	0	0
$345$	3.21539	0.173111
$346$	0	0
$347$	7.26795i	0.390164i	0.980787	+	0.195082i	$0.0624973\pi$
$347$	7.26795i	0.390164i	−0.980787	+	0.195082i	$0.937503\pi$
$348$	0	0
$349$	− 5.83272i	− 0.312218i	−0.987740	−	0.156109i	$-0.950105\pi$
$349$	− 5.83272i	− 0.312218i	0.987740	−	0.156109i	$-0.0498952\pi$
$350$	0	0
$351$	1.51575	0.0809047
$352$	0	0
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	4.39230i	0.233119i
$356$	0	0
$357$	9.79796i	0.518563i
$358$	0	0
$359$	−17.8028	−0.939594	−0.469797	−	0.882774i	$-0.655673\pi$
$359$	−17.8028	−0.939594	−0.469797	+	0.882774i	$0.655673\pi$
$360$	0	0
$361$	−3.39230	−0.178542
$362$	0	0
$363$	− 8.33975i	− 0.437723i
$364$	0	0
$365$	6.21166i	0.325133i
$366$	0	0
$367$	−32.4254	−1.69259	−0.846295	−	0.532714i	$-0.821172\pi$
$367$	−32.4254	−1.69259	−0.846295	+	0.532714i	$0.821172\pi$
$368$	0	0
$369$	−17.0718	−0.888722
$370$	0	0
$371$	− 42.2487i	− 2.19344i
$372$	0	0
$373$	− 19.9749i	− 1.03426i	−0.855907	−	0.517129i	$-0.827001\pi$
$373$	− 19.9749i	− 1.03426i	0.855907	−	0.517129i	$-0.172999\pi$
$374$	0	0
$375$	−7.17260	−0.370392
$376$	0	0
$377$	0.928203	0.0478049
$378$	0	0
$379$	− 2.87564i	− 0.147712i	−0.997269	−	0.0738560i	$-0.976469\pi$
$379$	− 2.87564i	− 0.147712i	0.997269	−	0.0738560i	$-0.0235305\pi$
$380$	0	0
$381$	4.14110i	0.212155i
$382$	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
$385$	44.7846	2.28244
$386$	0	0
$387$	− 5.41154i	− 0.275084i
$388$	0	0
$389$	6.03579i	0.306027i	0.988224	+	0.153013i	$0.0488978\pi$
$389$	6.03579i	0.306027i	−0.988224	+	0.153013i	$0.951102\pi$
$390$	0	0
$391$	6.21166	0.314137
$392$	0	0
$393$	0.248711	0.0125458
$394$	0	0
$395$	10.1436i	0.510380i
$396$	0	0
$397$	− 27.7023i	− 1.39034i	−0.718847	−	0.695168i	$-0.755330\pi$
$397$	− 27.7023i	− 1.39034i	0.718847	−	0.695168i	$-0.244670\pi$
$398$	0	0
$399$	−13.3843	−0.670051
$400$	0	0
$401$	−13.6077	−0.679536	−0.339768	−	0.940509i	$-0.610349\pi$
$401$	−13.6077	−0.679536	−0.339768	+	0.940509i	$0.610349\pi$
$402$	0	0
$403$	2.14359i	0.106780i
$404$	0	0
$405$	10.9348i	0.543353i
$406$	0	0
$407$	24.9754	1.23798
$408$	0	0
$409$	−17.0718	−0.844146	−0.422073	−	0.906562i	$-0.638697\pi$
$409$	−17.0718	−0.844146	−0.422073	+	0.906562i	$0.638697\pi$
$410$	0	0
$411$	− 8.78461i	− 0.433313i
$412$	0	0
$413$	− 28.0812i	− 1.38179i
$414$	0	0
$415$	−5.37945	−0.264067
$416$	0	0
$417$	7.46410	0.365519
$418$	0	0
$419$	− 14.1962i	− 0.693527i	−0.937953	−	0.346764i	$-0.887281\pi$
$419$	− 14.1962i	− 0.693527i	0.937953	−	0.346764i	$-0.112719\pi$
$420$	0	0
$421$	26.1865i	1.27625i	0.769932	+	0.638126i	$0.220290\pi$
$421$	26.1865i	1.27625i	−0.769932	+	0.638126i	$0.779710\pi$
$422$	0	0
$423$	−24.1432	−1.17388
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	20.3923i	0.986853i
$428$	0	0
$429$	1.31268i	0.0633767i
$430$	0	0
$431$	26.7685	1.28939	0.644697	−	0.764438i	$-0.276983\pi$
$431$	26.7685	1.28939	0.644697	+	0.764438i	$0.276983\pi$
$432$	0	0
$433$	34.3923	1.65279	0.826394	−	0.563092i	$-0.190388\pi$
$433$	34.3923	1.65279	0.826394	+	0.563092i	$0.190388\pi$
$434$	0	0
$435$	− 4.39230i	− 0.210595i
$436$	0	0
$437$	8.48528i	0.405906i
$438$	0	0
$439$	12.1459	0.579693	0.289846	−	0.957073i	$-0.406396\pi$
$439$	12.1459	0.579693	0.289846	+	0.957073i	$0.406396\pi$
$440$	0	0
$441$	19.5359	0.930281
$442$	0	0
$443$	14.1962i	0.674480i	0.941419	+	0.337240i	$0.109493\pi$
$443$	14.1962i	0.674480i	−0.941419	+	0.337240i	$0.890507\pi$
$444$	0	0
$445$	6.21166i	0.294461i
$446$	0	0
$447$	14.2165	0.672416
$448$	0	0
$449$	−24.2487	−1.14437	−0.572184	−	0.820125i	$-0.693904\pi$
$449$	−24.2487	−1.14437	−0.572184	+	0.820125i	$0.693904\pi$
$450$	0	0
$451$	− 32.7846i	− 1.54377i
$452$	0	0
$453$	6.96953i	0.327457i
$454$	0	0
$455$	−3.58630	−0.168128
$456$	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
$459$	− 13.8564i	− 0.646762i
$460$	0	0
$461$	34.1170i	1.58899i	0.607273	+	0.794493i	$0.292263\pi$
$461$	34.1170i	1.58899i	−0.607273	+	0.794493i	$0.707737\pi$
$462$	0	0
$463$	−38.0822	−1.76983	−0.884916	−	0.465751i	$-0.845784\pi$
$463$	−38.0822	−1.76983	−0.884916	+	0.465751i	$0.845784\pi$
$464$	0	0
$465$	10.1436	0.470398
$466$	0	0
$467$	− 28.7321i	− 1.32956i	−0.747039	−	0.664780i	$-0.768525\pi$
$467$	− 28.7321i	− 1.32956i	0.747039	−	0.664780i	$-0.231475\pi$
$468$	0	0
$469$	23.9401i	1.10545i
$470$	0	0
$471$	−7.45001	−0.343278
$472$	0	0
$473$	10.3923	0.477839
$474$	0	0
$475$	4.73205i	0.217121i
$476$	0	0
$477$	26.9444i	1.23370i
$478$	0	0
$479$	−36.5665	−1.67077	−0.835383	−	0.549669i	$-0.814754\pi$
$479$	−36.5665	−1.67077	−0.835383	+	0.549669i	$0.814754\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	5.07180i	0.230775i
$484$	0	0
$485$	8.48528i	0.385297i
$486$	0	0
$487$	27.0459	1.22557	0.612784	−	0.790251i	$-0.290050\pi$
$487$	27.0459	1.22557	0.612784	+	0.790251i	$0.290050\pi$
$488$	0	0
$489$	−12.2487	−0.553906
$490$	0	0
$491$	21.8038i	0.983994i	0.870597	+	0.491997i	$0.163733\pi$
$491$	21.8038i	0.983994i	−0.870597	+	0.491997i	$0.836267\pi$
$492$	0	0
$493$	− 8.48528i	− 0.382158i
$494$	0	0
$495$	−28.5617	−1.28375
$496$	0	0
$497$	−6.92820	−0.310772
$498$	0	0
$499$	− 5.80385i	− 0.259816i	−0.991526	−	0.129908i	$-0.958532\pi$
$499$	− 5.80385i	− 0.259816i	0.991526	−	0.129908i	$-0.0414682\pi$
$500$	0	0
$501$	13.7360i	0.613679i
$502$	0	0
$503$	−40.9850	−1.82743	−0.913715	−	0.406355i	$-0.866800\pi$
$503$	−40.9850	−1.82743	−0.913715	+	0.406355i	$0.866800\pi$
$504$	0	0
$505$	2.78461	0.123914
$506$	0	0
$507$	9.41154i	0.417981i
$508$	0	0
$509$	− 13.5601i	− 0.601042i	−0.953775	−	0.300521i	$-0.902839\pi$
$509$	− 13.5601i	− 0.601042i	0.953775	−	0.300521i	$-0.0971606\pi$
$510$	0	0
$511$	−9.79796	−0.433436
$512$	0	0
$513$	18.9282	0.835701
$514$	0	0
$515$	− 23.3205i	− 1.02762i
$516$	0	0
$517$	− 46.3644i	− 2.03911i
$518$	0	0
$519$	−8.00481	−0.351372
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	− 19.2679i	− 0.842529i	−0.906938	−	0.421264i	$-0.861586\pi$
$523$	− 19.2679i	− 0.842529i	0.906938	−	0.421264i	$-0.138414\pi$
$524$	0	0
$525$	2.82843i	0.123443i
$526$	0	0
$527$	19.5959	0.853612
$528$	0	0
$529$	−19.7846	−0.860200
$530$	0	0
$531$	17.9090i	0.777183i
$532$	0	0
$533$	2.62536i	0.113717i
$534$	0	0
$535$	−45.5322	−1.96853
$536$	0	0
$537$	−1.60770	−0.0693772
$538$	0	0
$539$	37.5167i	1.61596i
$540$	0	0
$541$	14.3180i	0.615579i	0.951454	+	0.307789i	$0.0995892\pi$
$541$	14.3180i	0.615579i	−0.951454	+	0.307789i	$0.900411\pi$
$542$	0	0
$543$	−16.6932	−0.716372
$544$	0	0
$545$	−45.7128	−1.95812
$546$	0	0
$547$	− 0.339746i	− 0.0145265i	−0.999974	−	0.00726324i	$-0.997688\pi$
$547$	− 0.339746i	− 0.0145265i	0.999974	−	0.00726324i	$-0.00231198\pi$
$548$	0	0
$549$	− 13.0053i	− 0.555054i
$550$	0	0
$551$	11.5911	0.493798
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	9.46410i	0.401729i
$556$	0	0
$557$	− 15.8338i	− 0.670898i	−0.942058	−	0.335449i	$-0.891112\pi$
$557$	− 15.8338i	− 0.670898i	0.942058	−	0.335449i	$-0.108888\pi$
$558$	0	0
$559$	−0.832204	−0.0351985
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	− 14.8756i	− 0.626934i	−0.949599	−	0.313467i	$-0.898510\pi$
$563$	− 14.8756i	− 0.626934i	0.949599	−	0.313467i	$-0.101490\pi$
$564$	0	0
$565$	− 31.6675i	− 1.33226i
$566$	0	0
$567$	−17.2480	−0.724346
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	30.1962i	1.26367i	0.775103	+	0.631835i	$0.217698\pi$
$571$	30.1962i	1.26367i	−0.775103	+	0.631835i	$0.782302\pi$
$572$	0	0
$573$	12.4233i	0.518991i
$574$	0	0
$575$	1.79315	0.0747796
$576$	0	0
$577$	−43.8564	−1.82577	−0.912883	−	0.408221i	$-0.866149\pi$
$577$	−43.8564	−1.82577	−0.912883	+	0.408221i	$0.866149\pi$
$578$	0	0
$579$	− 13.4641i	− 0.559549i
$580$	0	0
$581$	− 8.48528i	− 0.352029i
$582$	0	0
$583$	−51.7439	−2.14301
$584$	0	0
$585$	2.28719	0.0945635
$586$	0	0
$587$	− 9.12436i	− 0.376602i	−0.982111	−	0.188301i	$-0.939702\pi$
$587$	− 9.12436i	− 0.376602i	0.982111	−	0.188301i	$-0.0602981\pi$
$588$	0	0
$589$	26.7685i	1.10298i
$590$	0	0
$591$	−1.79315	−0.0737604
$592$	0	0
$593$	19.8564	0.815405	0.407702	−	0.913115i	$-0.366330\pi$
$593$	19.8564	0.815405	0.407702	+	0.913115i	$0.366330\pi$
$594$	0	0
$595$	32.7846i	1.34404i
$596$	0	0
$597$	11.5168i	0.471350i
$598$	0	0
$599$	−4.41851	−0.180535	−0.0902676	−	0.995918i	$-0.528772\pi$
$599$	−4.41851	−0.180535	−0.0902676	+	0.995918i	$0.528772\pi$
$600$	0	0
$601$	40.3923	1.64764	0.823818	−	0.566854i	$-0.191840\pi$
$601$	40.3923	1.64764	0.823818	+	0.566854i	$0.191840\pi$
$602$	0	0
$603$	− 15.2679i	− 0.621759i
$604$	0	0
$605$	− 27.9053i	− 1.13451i
$606$	0	0
$607$	8.28221	0.336165	0.168082	−	0.985773i	$-0.446243\pi$
$607$	8.28221	0.336165	0.168082	+	0.985773i	$0.446243\pi$
$608$	0	0
$609$	6.92820	0.280745
$610$	0	0
$611$	3.71281i	0.150204i
$612$	0	0
$613$	− 21.2875i	− 0.859795i	−0.902878	−	0.429898i	$-0.858550\pi$
$613$	− 21.2875i	− 0.859795i	0.902878	−	0.429898i	$-0.141450\pi$
$614$	0	0
$615$	12.4233	0.500956
$616$	0	0
$617$	11.3205	0.455746	0.227873	−	0.973691i	$-0.426823\pi$
$617$	11.3205	0.455746	0.227873	+	0.973691i	$0.426823\pi$
$618$	0	0
$619$	26.5885i	1.06868i	0.845270	+	0.534340i	$0.179440\pi$
$619$	26.5885i	1.06868i	−0.845270	+	0.534340i	$0.820560\pi$
$620$	0	0
$621$	− 7.17260i	− 0.287827i
$622$	0	0
$623$	−9.79796	−0.392547
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	16.3923i	0.654646i
$628$	0	0
$629$	18.2832i	0.729001i
$630$	0	0
$631$	1.23835	0.0492979	0.0246489	−	0.999696i	$-0.492153\pi$
$631$	1.23835	0.0492979	0.0246489	+	0.999696i	$0.492153\pi$
$632$	0	0
$633$	−7.46410	−0.296671
$634$	0	0
$635$	13.8564i	0.549875i
$636$	0	0
$637$	− 3.00429i	− 0.119034i
$638$	0	0
$639$	4.41851	0.174793
$640$	0	0
$641$	13.6077	0.537472	0.268736	−	0.963214i	$-0.413394\pi$
$641$	13.6077	0.537472	0.268736	+	0.963214i	$0.413394\pi$
$642$	0	0
$643$	9.80385i	0.386626i	0.981137	+	0.193313i	$0.0619233\pi$
$643$	9.80385i	0.386626i	−0.981137	+	0.193313i	$0.938077\pi$
$644$	0	0
$645$	3.93803i	0.155060i
$646$	0	0
$647$	1.79315	0.0704960	0.0352480	−	0.999379i	$-0.488778\pi$
$647$	1.79315	0.0704960	0.0352480	+	0.999379i	$0.488778\pi$
$648$	0	0
$649$	−34.3923	−1.35002
$650$	0	0
$651$	16.0000i	0.627089i
$652$	0	0
$653$	21.6937i	0.848939i	0.905442	+	0.424470i	$0.139539\pi$
$653$	21.6937i	0.848939i	−0.905442	+	0.424470i	$0.860461\pi$
$654$	0	0
$655$	0.832204	0.0325169
$656$	0	0
$657$	6.24871	0.243785
$658$	0	0
$659$	− 25.5167i	− 0.993988i	−0.867754	−	0.496994i	$-0.834437\pi$
$659$	− 25.5167i	− 0.993988i	0.867754	−	0.496994i	$-0.165563\pi$
$660$	0	0
$661$	25.8348i	1.00486i	0.864619	+	0.502428i	$0.167560\pi$
$661$	25.8348i	1.00486i	−0.864619	+	0.502428i	$0.832440\pi$
$662$	0	0
$663$	−0.960947	−0.0373201
$664$	0	0
$665$	−44.7846	−1.73667
$666$	0	0
$667$	− 4.39230i	− 0.170071i
$668$	0	0
$669$	− 4.14110i	− 0.160104i
$670$	0	0
$671$	24.9754	0.964163
$672$	0	0
$673$	41.3205	1.59279	0.796394	−	0.604778i	$-0.206738\pi$
$673$	41.3205	1.59279	0.796394	+	0.604778i	$0.206738\pi$
$674$	0	0
$675$	− 4.00000i	− 0.153960i
$676$	0	0
$677$	39.9769i	1.53644i	0.640187	+	0.768219i	$0.278857\pi$
$677$	39.9769i	1.53644i	−0.640187	+	0.768219i	$0.721143\pi$
$678$	0	0
$679$	−13.3843	−0.513641
$680$	0	0
$681$	15.4641	0.592586
$682$	0	0
$683$	− 12.3397i	− 0.472167i	−0.971733	−	0.236084i	$-0.924136\pi$
$683$	− 12.3397i	− 0.472167i	0.971733	−	0.236084i	$-0.0758639\pi$
$684$	0	0
$685$	− 29.3939i	− 1.12308i
$686$	0	0
$687$	9.52056	0.363232
$688$	0	0
$689$	4.14359	0.157858
$690$	0	0
$691$	− 39.3731i	− 1.49782i	−0.662671	−	0.748911i	$-0.730577\pi$
$691$	− 39.3731i	− 1.49782i	0.662671	−	0.748911i	$-0.269423\pi$
$692$	0	0
$693$	− 45.0518i	− 1.71137i
$694$	0	0
$695$	24.9754	0.947370
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	1.85641i	0.0702157i
$700$	0	0
$701$	13.2084i	0.498874i	0.968391	+	0.249437i	$0.0802456\pi$
$701$	13.2084i	0.498874i	−0.968391	+	0.249437i	$0.919754\pi$
$702$	0	0
$703$	−24.9754	−0.941964
$704$	0	0
$705$	17.5692	0.661695
$706$	0	0
$707$	4.39230i	0.165190i
$708$	0	0
$709$	41.0865i	1.54304i	0.636207	+	0.771518i	$0.280502\pi$
$709$	41.0865i	1.54304i	−0.636207	+	0.771518i	$0.719498\pi$
$710$	0	0
$711$	10.2041	0.382684
$712$	0	0
$713$	10.1436	0.379881
$714$	0	0
$715$	4.39230i	0.164263i
$716$	0	0
$717$	19.5959i	0.731823i
$718$	0	0
$719$	12.4233	0.463311	0.231656	−	0.972798i	$-0.425586\pi$
$719$	12.4233	0.463311	0.231656	+	0.972798i	$0.425586\pi$
$720$	0	0
$721$	36.7846	1.36993
$722$	0	0
$723$	1.75129i	0.0651311i
$724$	0	0
$725$	− 2.44949i	− 0.0909718i
$726$	0	0
$727$	23.4596	0.870069	0.435035	−	0.900414i	$-0.356736\pi$
$727$	23.4596	0.870069	0.435035	+	0.900414i	$0.356736\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	7.60770i	0.281381i
$732$	0	0
$733$	20.9358i	0.773281i	0.922230	+	0.386641i	$0.126365\pi$
$733$	20.9358i	0.773281i	−0.922230	+	0.386641i	$0.873635\pi$
$734$	0	0
$735$	−14.2165	−0.524382
$736$	0	0
$737$	29.3205	1.08003
$738$	0	0
$739$	− 2.98076i	− 0.109649i	−0.998496	−	0.0548246i	$-0.982540\pi$
$739$	− 2.98076i	− 0.109649i	0.998496	−	0.0548246i	$-0.0174600\pi$
$740$	0	0
$741$	− 1.31268i	− 0.0482224i
$742$	0	0
$743$	−15.1774	−0.556805	−0.278403	−	0.960464i	$-0.589805\pi$
$743$	−15.1774	−0.556805	−0.278403	+	0.960464i	$0.589805\pi$
$744$	0	0
$745$	47.5692	1.74280
$746$	0	0
$747$	5.41154i	0.197998i
$748$	0	0
$749$	− 71.8203i	− 2.62426i
$750$	0	0
$751$	−4.14110	−0.151111	−0.0755555	−	0.997142i	$-0.524073\pi$
$751$	−4.14110	−0.151111	−0.0755555	+	0.997142i	$0.524073\pi$
$752$	0	0
$753$	−15.4641	−0.563543
$754$	0	0
$755$	23.3205i	0.848720i
$756$	0	0
$757$	− 30.7338i	− 1.11704i	−0.829492	−	0.558519i	$-0.811370\pi$
$757$	− 30.7338i	− 1.11704i	0.829492	−	0.558519i	$-0.188630\pi$
$758$	0	0
$759$	6.21166	0.225469
$760$	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
$763$	− 72.1051i	− 2.61038i
$764$	0	0
$765$	− 20.9086i	− 0.755952i
$766$	0	0
$767$	2.75410	0.0994447
$768$	0	0
$769$	−2.39230	−0.0862687	−0.0431344	−	0.999069i	$-0.513734\pi$
$769$	−2.39230	−0.0862687	−0.0431344	+	0.999069i	$0.513734\pi$
$770$	0	0
$771$	9.46410i	0.340841i
$772$	0	0
$773$	− 20.7327i	− 0.745704i	−0.927891	−	0.372852i	$-0.878380\pi$
$773$	− 20.7327i	− 0.745704i	0.927891	−	0.372852i	$-0.121620\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	−14.9282	−0.535546
$778$	0	0
$779$	32.7846i	1.17463i
$780$	0	0
$781$	8.48528i	0.303627i
$782$	0	0
$783$	−9.79796	−0.350150
$784$	0	0
$785$	−24.9282	−0.889726
$786$	0	0
$787$	− 35.6603i	− 1.27115i	−0.772039	−	0.635575i	$-0.780763\pi$
$787$	− 35.6603i	− 1.27115i	0.772039	−	0.635575i	$-0.219237\pi$
$788$	0	0
$789$	15.6579i	0.557435i
$790$	0	0
$791$	49.9507	1.77604
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	− 19.6077i	− 0.695413i
$796$	0	0
$797$	− 47.5013i	− 1.68258i	−0.540584	−	0.841290i	$-0.681797\pi$
$797$	− 47.5013i	− 1.68258i	0.540584	−	0.841290i	$-0.318203\pi$
$798$	0	0
$799$	33.9411	1.20075
$800$	0	0
$801$	6.24871	0.220787
$802$	0	0
$803$	12.0000i	0.423471i
$804$	0	0
$805$	16.9706i	0.598134i
$806$	0	0
$807$	−5.37945	−0.189366
$808$	0	0
$809$	−17.0718	−0.600212	−0.300106	−	0.953906i	$-0.597022\pi$
$809$	−17.0718	−0.600212	−0.300106	+	0.953906i	$0.597022\pi$
$810$	0	0
$811$	− 46.9808i	− 1.64972i	−0.565339	−	0.824859i	$-0.691255\pi$
$811$	− 46.9808i	− 1.64972i	0.565339	−	0.824859i	$-0.308745\pi$
$812$	0	0
$813$	1.10961i	0.0389156i
$814$	0	0
$815$	−40.9850	−1.43564
$816$	0	0
$817$	−10.3923	−0.363581
$818$	0	0
$819$	3.60770i	0.126063i
$820$	0	0
$821$	− 28.2571i	− 0.986178i	−0.869979	−	0.493089i	$-0.835868\pi$
$821$	− 28.2571i	− 0.986178i	0.869979	−	0.493089i	$-0.164132\pi$
$822$	0	0
$823$	−15.7322	−0.548391	−0.274195	−	0.961674i	$-0.588411\pi$
$823$	−15.7322	−0.548391	−0.274195	+	0.961674i	$0.588411\pi$
$824$	0	0
$825$	3.46410	0.120605
$826$	0	0
$827$	− 23.6603i	− 0.822748i	−0.911467	−	0.411374i	$-0.865049\pi$
$827$	− 23.6603i	− 0.822748i	0.911467	−	0.411374i	$-0.134951\pi$
$828$	0	0
$829$	3.00429i	0.104343i	0.998638	+	0.0521717i	$0.0166143\pi$
$829$	3.00429i	0.104343i	−0.998638	+	0.0521717i	$0.983386\pi$
$830$	0	0
$831$	3.30890	0.114784
$832$	0	0
$833$	−27.4641	−0.951575
$834$	0	0
$835$	45.9615i	1.59056i
$836$	0	0
$837$	− 22.6274i	− 0.782118i
$838$	0	0
$839$	38.3596	1.32432	0.662161	−	0.749362i	$-0.269640\pi$
$839$	38.3596	1.32432	0.662161	+	0.749362i	$0.269640\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	− 12.0000i	− 0.413302i
$844$	0	0
$845$	31.4916i	1.08335i
$846$	0	0
$847$	44.0165	1.51242
$848$	0	0
$849$	1.32051	0.0453197
$850$	0	0
$851$	9.46410i	0.324425i
$852$	0	0
$853$	13.9663i	0.478196i	0.970995	+	0.239098i	$0.0768517\pi$
$853$	13.9663i	0.478196i	−0.970995	+	0.239098i	$0.923148\pi$
$854$	0	0
$855$	28.5617	0.976789
$856$	0	0
$857$	25.8564	0.883238	0.441619	−	0.897203i	$-0.354404\pi$
$857$	25.8564	0.883238	0.441619	+	0.897203i	$0.354404\pi$
$858$	0	0
$859$	− 20.4449i	− 0.697570i	−0.937203	−	0.348785i	$-0.886594\pi$
$859$	− 20.4449i	− 0.697570i	0.937203	−	0.348785i	$-0.113406\pi$
$860$	0	0
$861$	19.5959i	0.667827i
$862$	0	0
$863$	2.62536	0.0893681	0.0446841	−	0.999001i	$-0.485772\pi$
$863$	2.62536	0.0893681	0.0446841	+	0.999001i	$0.485772\pi$
$864$	0	0
$865$	−26.7846	−0.910704
$866$	0	0
$867$	− 3.66025i	− 0.124309i
$868$	0	0
$869$	19.5959i	0.664746i
$870$	0	0
$871$	−2.34795	−0.0795574
$872$	0	0
$873$	8.53590	0.288896
$874$	0	0
$875$	− 37.8564i	− 1.27978i
$876$	0	0
$877$	22.8033i	0.770012i	0.922914	+	0.385006i	$0.125801\pi$
$877$	22.8033i	0.770012i	−0.922914	+	0.385006i	$0.874199\pi$
$878$	0	0
$879$	12.5521	0.423370
$880$	0	0
$881$	−31.8564	−1.07327	−0.536635	−	0.843815i	$-0.680305\pi$
$881$	−31.8564	−1.07327	−0.536635	+	0.843815i	$0.680305\pi$
$882$	0	0
$883$	57.1244i	1.92239i	0.275874	+	0.961194i	$0.411033\pi$
$883$	57.1244i	1.92239i	−0.275874	+	0.961194i	$0.588967\pi$
$884$	0	0
$885$	− 13.0325i	− 0.438084i
$886$	0	0
$887$	15.1774	0.509608	0.254804	−	0.966993i	$-0.417989\pi$
$887$	15.1774	0.509608	0.254804	+	0.966993i	$0.417989\pi$
$888$	0	0
$889$	−21.8564	−0.733040
$890$	0	0
$891$	21.1244i	0.707693i
$892$	0	0
$893$	46.3644i	1.55153i
$894$	0	0
$895$	−5.37945	−0.179815
$896$	0	0
$897$	−0.497423	−0.0166085
$898$	0	0
$899$	− 13.8564i	− 0.462137i
$900$	0	0
$901$	− 37.8792i	− 1.26194i
$902$	0	0
$903$	−6.21166	−0.206711
$904$	0	0
$905$	−55.8564	−1.85673
$906$	0	0
$907$	− 29.9090i	− 0.993111i	−0.868005	−	0.496555i	$-0.834598\pi$
$907$	− 29.9090i	− 0.993111i	0.868005	−	0.496555i	$-0.165402\pi$
$908$	0	0
$909$	− 2.80122i	− 0.0929106i
$910$	0	0
$911$	24.1432	0.799899	0.399949	−	0.916537i	$-0.369028\pi$
$911$	24.1432	0.799899	0.399949	+	0.916537i	$0.369028\pi$
$912$	0	0
$913$	−10.3923	−0.343935
$914$	0	0
$915$	9.46410i	0.312874i
$916$	0	0
$917$	1.31268i	0.0433484i
$918$	0	0
$919$	−18.3576	−0.605560	−0.302780	−	0.953060i	$-0.597915\pi$
$919$	−18.3576	−0.605560	−0.302780	+	0.953060i	$0.597915\pi$
$920$	0	0
$921$	−22.6795	−0.747315
$922$	0	0
$923$	− 0.679492i	− 0.0223657i
$924$	0	0
$925$	5.27792i	0.173537i
$926$	0	0
$927$	−23.4596	−0.770515
$928$	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
$931$	− 37.5167i	− 1.22956i
$932$	0	0
$933$	11.1106i	0.363746i
$934$	0	0
$935$	40.1528	1.31314
$936$	0	0
$937$	−9.17691	−0.299797	−0.149898	−	0.988701i	$-0.547895\pi$
$937$	−9.17691	−0.299797	−0.149898	+	0.988701i	$0.547895\pi$
$938$	0	0
$939$	− 9.35898i	− 0.305419i
$940$	0	0
$941$	− 26.5927i	− 0.866896i	−0.901179	−	0.433448i	$-0.857297\pi$
$941$	− 26.5927i	− 0.866896i	0.901179	−	0.433448i	$-0.142703\pi$
$942$	0	0
$943$	12.4233	0.404559
$944$	0	0
$945$	37.8564	1.23147
$946$	0	0
$947$	19.2679i	0.626124i	0.949733	+	0.313062i	$0.101355\pi$
$947$	19.2679i	0.626124i	−0.949733	+	0.313062i	$0.898645\pi$
$948$	0	0
$949$	− 0.960947i	− 0.0311936i
$950$	0	0
$951$	−7.04386	−0.228413
$952$	0	0
$953$	53.5692	1.73528	0.867639	−	0.497195i	$-0.165637\pi$
$953$	53.5692	1.73528	0.867639	+	0.497195i	$0.165637\pi$
$954$	0	0
$955$	41.5692i	1.34515i
$956$	0	0
$957$	− 8.48528i	− 0.274290i
$958$	0	0
$959$	46.3644	1.49719
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	45.8038i	1.47601i
$964$	0	0
$965$	− 45.0518i	− 1.45027i
$966$	0	0
$967$	27.0459	0.869738	0.434869	−	0.900494i	$-0.356795\pi$
$967$	27.0459	0.869738	0.434869	+	0.900494i	$0.356795\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	39.3731i	1.26354i	0.775155	+	0.631771i	$0.217672\pi$
$971$	39.3731i	1.26354i	−0.775155	+	0.631771i	$0.782328\pi$
$972$	0	0
$973$	39.3949i	1.26294i
$974$	0	0
$975$	−0.277401	−0.00888396
$976$	0	0
$977$	44.5359	1.42483	0.712415	−	0.701759i	$-0.247601\pi$
$977$	44.5359	1.42483	0.712415	+	0.701759i	$0.247601\pi$
$978$	0	0
$979$	12.0000i	0.383522i
$980$	0	0
$981$	45.9855i	1.46820i
$982$	0	0
$983$	−52.7048	−1.68102	−0.840512	−	0.541793i	$-0.817745\pi$
$983$	−52.7048	−1.68102	−0.840512	+	0.541793i	$0.817745\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	27.7128i	0.882109i
$988$	0	0
$989$	3.93803i	0.125222i
$990$	0	0
$991$	61.8193	1.96375	0.981877	−	0.189521i	$-0.0606936\pi$
$991$	61.8193	1.96375	0.981877	+	0.189521i	$0.0606936\pi$
$992$	0	0
$993$	−19.4641	−0.617675
$994$	0	0
$995$	38.5359i	1.22167i
$996$	0	0
$997$	21.4906i	0.680614i	0.940314	+	0.340307i	$0.110531\pi$
$997$	21.4906i	0.680614i	−0.940314	+	0.340307i	$0.889469\pi$
$998$	0	0
$999$	21.1117	0.667944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.b.h.513.6		8
4.3	odd	2	inner	1024.2.b.h.513.4		8
8.3	odd	2	inner	1024.2.b.h.513.5		8
8.5	even	2	inner	1024.2.b.h.513.3		8
16.3	odd	4		1024.2.a.g.1.4		4
16.5	even	4		1024.2.a.g.1.3		4
16.11	odd	4		1024.2.a.j.1.1		4
16.13	even	4		1024.2.a.j.1.2		4
32.3	odd	8		256.2.e.a.193.3	yes	8
32.5	even	8		256.2.e.a.65.2	✓	8
32.11	odd	8		256.2.e.b.65.2	yes	8
32.13	even	8		256.2.e.b.193.3	yes	8
32.19	odd	8		256.2.e.b.193.2	yes	8
32.21	even	8		256.2.e.b.65.3	yes	8
32.27	odd	8		256.2.e.a.65.3	yes	8
32.29	even	8		256.2.e.a.193.2	yes	8
48.5	odd	4		9216.2.a.bk.1.4		4
48.11	even	4		9216.2.a.bb.1.3		4
48.29	odd	4		9216.2.a.bb.1.2		4
48.35	even	4		9216.2.a.bk.1.1		4
96.5	odd	8		2304.2.k.f.577.3		8
96.11	even	8		2304.2.k.k.577.2		8
96.29	odd	8		2304.2.k.f.1729.4		8
96.35	even	8		2304.2.k.f.1729.3		8
96.53	odd	8		2304.2.k.k.577.1		8
96.59	even	8		2304.2.k.f.577.4		8
96.77	odd	8		2304.2.k.k.1729.2		8
96.83	even	8		2304.2.k.k.1729.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.e.a.65.2	✓	8	32.5	even	8
256.2.e.a.65.3	yes	8	32.27	odd	8
256.2.e.a.193.2	yes	8	32.29	even	8
256.2.e.a.193.3	yes	8	32.3	odd	8
256.2.e.b.65.2	yes	8	32.11	odd	8
256.2.e.b.65.3	yes	8	32.21	even	8
256.2.e.b.193.2	yes	8	32.19	odd	8
256.2.e.b.193.3	yes	8	32.13	even	8
1024.2.a.g.1.3		4	16.5	even	4
1024.2.a.g.1.4		4	16.3	odd	4
1024.2.a.j.1.1		4	16.11	odd	4
1024.2.a.j.1.2		4	16.13	even	4
1024.2.b.h.513.3		8	8.5	even	2	inner
1024.2.b.h.513.4		8	4.3	odd	2	inner
1024.2.b.h.513.5		8	8.3	odd	2	inner
1024.2.b.h.513.6		8	1.1	even	1	trivial
2304.2.k.f.577.3		8	96.5	odd	8
2304.2.k.f.577.4		8	96.59	even	8
2304.2.k.f.1729.3		8	96.35	even	8
2304.2.k.f.1729.4		8	96.29	odd	8
2304.2.k.k.577.1		8	96.53	odd	8
2304.2.k.k.577.2		8	96.11	even	8
2304.2.k.k.1729.1		8	96.83	even	8
2304.2.k.k.1729.2		8	96.77	odd	8
9216.2.a.bb.1.2		4	48.29	odd	4
9216.2.a.bb.1.3		4	48.11	even	4
9216.2.a.bk.1.1		4	48.35	even	4
9216.2.a.bk.1.4		4	48.5	odd	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.732051i q^{3} +2.44949i q^{5} -3.86370 q^{7} +2.46410 q^{9} +O(q^{10})\)	\(q+0.732051i q^{3} +2.44949i q^{5} -3.86370 q^{7} +2.46410 q^{9} +4.73205i q^{11} -0.378937i q^{13} -1.79315 q^{15} -3.46410 q^{17} -4.73205i q^{19} -2.82843i q^{21} -1.79315 q^{23} -1.00000 q^{25} +4.00000i q^{27} +2.44949i q^{29} -5.65685 q^{31} -3.46410 q^{33} -9.46410i q^{35} -5.27792i q^{37} +0.277401 q^{39} -6.92820 q^{41} -2.19615i q^{43} +6.03579i q^{45} -9.79796 q^{47} +7.92820 q^{49} -2.53590i q^{51} +10.9348i q^{53} -11.5911 q^{55} +3.46410 q^{57} +7.26795i q^{59} -5.27792i q^{61} -9.52056 q^{63} +0.928203 q^{65} -6.19615i q^{67} -1.31268i q^{69} +1.79315 q^{71} +2.53590 q^{73} -0.732051i q^{75} -18.2832i q^{77} +4.14110 q^{79} +4.46410 q^{81} +2.19615i q^{83} -8.48528i q^{85} -1.79315 q^{87} +2.53590 q^{89} +1.46410i q^{91} -4.14110i q^{93} +11.5911 q^{95} +3.46410 q^{97} +11.6603i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} - 8 q^{25} + 8 q^{49} - 48 q^{65} + 48 q^{73} + 8 q^{81} + 48 q^{89}+O(q^{100}) \) 8 * q - 8 * q^9 - 8 * q^25 + 8 * q^49 - 48 * q^65 + 48 * q^73 + 8 * q^81 + 48 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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