LMFDB - Embedded newform 1024.2.a.b.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 16)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.41421 q^{3} +1.41421 q^{5} -2.00000 q^{7} -1.00000 q^{9} +1.41421 q^{11} +1.41421 q^{13} -2.00000 q^{15} +2.00000 q^{17} -4.24264 q^{19} +2.82843 q^{21} -6.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} +4.24264 q^{29} -8.00000 q^{31} -2.00000 q^{33} -2.82843 q^{35} -4.24264 q^{37} -2.00000 q^{39} +7.07107 q^{43} -1.41421 q^{45} -8.00000 q^{47} -3.00000 q^{49} -2.82843 q^{51} -7.07107 q^{53} +2.00000 q^{55} +6.00000 q^{57} +4.24264 q^{59} -12.7279 q^{61} +2.00000 q^{63} +2.00000 q^{65} -7.07107 q^{67} +8.48528 q^{69} -10.0000 q^{71} +4.00000 q^{73} +4.24264 q^{75} -2.82843 q^{77} -5.00000 q^{81} +1.41421 q^{83} +2.82843 q^{85} -6.00000 q^{87} +4.00000 q^{89} -2.82843 q^{91} +11.3137 q^{93} -6.00000 q^{95} -2.00000 q^{97} -1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} - 12 q^{23} - 6 q^{25} - 16 q^{31} - 4 q^{33} - 4 q^{39} - 16 q^{47} - 6 q^{49} + 4 q^{55} + 12 q^{57} + 4 q^{63} + 4 q^{65} - 20 q^{71} + 8 q^{73} - 10 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^7 - 2 * q^9 - 4 * q^15 + 4 * q^17 - 12 * q^23 - 6 * q^25 - 16 * q^31 - 4 * q^33 - 4 * q^39 - 16 * q^47 - 6 * q^49 + 4 * q^55 + 12 * q^57 + 4 * q^63 + 4 * q^65 - 20 * q^71 + 8 * q^73 - 10 * q^81 - 12 * q^87 + 8 * q^89 - 12 * q^95 - 4 * q^97

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	7.07107	1.07833	0.539164	−	0.842201i	$-0.318740\pi$
$43$	7.07107	1.07833	0.539164	+	0.842201i	$0.318740\pi$
$44$	0	0
$45$	−1.41421	−0.210819
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−2.82843	−0.396059
$52$	0	0
$53$	−7.07107	−0.971286	−0.485643	−	0.874157i	$-0.661414\pi$
$53$	−7.07107	−0.971286	−0.485643	+	0.874157i	$0.661414\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	4.24264	0.552345	0.276172	−	0.961108i	$-0.410934\pi$
$59$	4.24264	0.552345	0.276172	+	0.961108i	$0.410934\pi$
$60$	0	0
$61$	−12.7279	−1.62964	−0.814822	−	0.579712i	$-0.803165\pi$
$61$	−12.7279	−1.62964	−0.814822	+	0.579712i	$0.803165\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−7.07107	−0.863868	−0.431934	−	0.901905i	$-0.642169\pi$
$67$	−7.07107	−0.863868	−0.431934	+	0.901905i	$0.642169\pi$
$68$	0	0
$69$	8.48528	1.02151
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	4.24264	0.489898
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	1.41421	0.155230	0.0776151	−	0.996983i	$-0.475269\pi$
$83$	1.41421	0.155230	0.0776151	+	0.996983i	$0.475269\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	11.3137	1.17318
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−1.41421	−0.142134
$100$	0	0
$101$	−15.5563	−1.54791	−0.773957	−	0.633238i	$-0.781726\pi$
$101$	−15.5563	−1.54791	−0.773957	+	0.633238i	$0.781726\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−9.89949	−0.957020	−0.478510	−	0.878082i	$-0.658823\pi$
$107$	−9.89949	−0.957020	−0.478510	+	0.878082i	$0.658823\pi$
$108$	0	0
$109$	−4.24264	−0.406371	−0.203186	−	0.979140i	$-0.565129\pi$
$109$	−4.24264	−0.406371	−0.203186	+	0.979140i	$0.565129\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−8.48528	−0.791257
$116$	0	0
$117$	−1.41421	−0.130744
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	15.5563	1.35916	0.679582	−	0.733599i	$-0.262161\pi$
$131$	15.5563	1.35916	0.679582	+	0.733599i	$0.262161\pi$
$132$	0	0
$133$	8.48528	0.735767
$134$	0	0
$135$	8.00000	0.688530
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−4.24264	−0.359856	−0.179928	−	0.983680i	$-0.557586\pi$
$139$	−4.24264	−0.359856	−0.179928	+	0.983680i	$0.557586\pi$
$140$	0	0
$141$	11.3137	0.952786
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	4.24264	0.349927
$148$	0	0
$149$	9.89949	0.810998	0.405499	−	0.914095i	$-0.367098\pi$
$149$	9.89949	0.810998	0.405499	+	0.914095i	$0.367098\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−11.3137	−0.908739
$156$	0	0
$157$	21.2132	1.69300	0.846499	−	0.532390i	$-0.178706\pi$
$157$	21.2132	1.69300	0.846499	+	0.532390i	$0.178706\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	−1.41421	−0.110770	−0.0553849	−	0.998465i	$-0.517639\pi$
$163$	−1.41421	−0.110770	−0.0553849	+	0.998465i	$0.517639\pi$
$164$	0	0
$165$	−2.82843	−0.220193
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	4.24264	0.324443
$172$	0	0
$173$	1.41421	0.107521	0.0537603	−	0.998554i	$-0.482879\pi$
$173$	1.41421	0.107521	0.0537603	+	0.998554i	$0.482879\pi$
$174$	0	0
$175$	6.00000	0.453557
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	24.0416	1.79696	0.898478	−	0.439019i	$-0.144674\pi$
$179$	24.0416	1.79696	0.898478	+	0.439019i	$0.144674\pi$
$180$	0	0
$181$	−12.7279	−0.946059	−0.473029	−	0.881047i	$-0.656840\pi$
$181$	−12.7279	−0.946059	−0.473029	+	0.881047i	$0.656840\pi$
$182$	0	0
$183$	18.0000	1.33060
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	2.82843	0.206835
$188$	0	0
$189$	−11.3137	−0.822951
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	−2.82843	−0.202548
$196$	0	0
$197$	24.0416	1.71290	0.856448	−	0.516234i	$-0.172666\pi$
$197$	24.0416	1.71290	0.856448	+	0.516234i	$0.172666\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	10.0000	0.705346
$202$	0	0
$203$	−8.48528	−0.595550
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	12.7279	0.876226	0.438113	−	0.898920i	$-0.355647\pi$
$211$	12.7279	0.876226	0.438113	+	0.898920i	$0.355647\pi$
$212$	0	0
$213$	14.1421	0.969003
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	−5.65685	−0.382255
$220$	0	0
$221$	2.82843	0.190261
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	21.2132	1.40797	0.703985	−	0.710215i	$-0.251402\pi$
$227$	21.2132	1.40797	0.703985	+	0.710215i	$0.251402\pi$
$228$	0	0
$229$	−9.89949	−0.654177	−0.327089	−	0.944994i	$-0.606068\pi$
$229$	−9.89949	−0.654177	−0.327089	+	0.944994i	$0.606068\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	−11.3137	−0.738025
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−4.24264	−0.271052
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	−2.00000	−0.126745
$250$	0	0
$251$	−29.6985	−1.87455	−0.937276	−	0.348589i	$-0.886661\pi$
$251$	−29.6985	−1.87455	−0.937276	+	0.348589i	$0.886661\pi$
$252$	0	0
$253$	−8.48528	−0.533465
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	8.48528	0.527250
$260$	0	0
$261$	−4.24264	−0.262613
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	0	0
$267$	−5.65685	−0.346194
$268$	0	0
$269$	−4.24264	−0.258678	−0.129339	−	0.991600i	$-0.541286\pi$
$269$	−4.24264	−0.258678	−0.129339	+	0.991600i	$0.541286\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	4.24264	0.254916	0.127458	−	0.991844i	$-0.459318\pi$
$277$	4.24264	0.254916	0.127458	+	0.991844i	$0.459318\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−20.0000	−1.19310	−0.596550	−	0.802576i	$-0.703462\pi$
$281$	−20.0000	−1.19310	−0.596550	+	0.802576i	$0.703462\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	0	0
$285$	8.48528	0.502625
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.82843	0.165805
$292$	0	0
$293$	−21.2132	−1.23929	−0.619644	−	0.784883i	$-0.712723\pi$
$293$	−21.2132	−1.23929	−0.619644	+	0.784883i	$0.712723\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	−8.48528	−0.490716
$300$	0	0
$301$	−14.1421	−0.815139
$302$	0	0
$303$	22.0000	1.26387
$304$	0	0
$305$	−18.0000	−1.03068
$306$	0	0
$307$	7.07107	0.403567	0.201784	−	0.979430i	$-0.435326\pi$
$307$	7.07107	0.403567	0.201784	+	0.979430i	$0.435326\pi$
$308$	0	0
$309$	−8.48528	−0.482711
$310$	0	0
$311$	−30.0000	−1.70114	−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000	−1.70114	−0.850572	+	0.525859i	$0.823744\pi$
$312$	0	0
$313$	16.0000	0.904373	0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000	0.904373	0.452187	+	0.891923i	$0.350644\pi$
$314$	0	0
$315$	2.82843	0.159364
$316$	0	0
$317$	−7.07107	−0.397151	−0.198575	−	0.980086i	$-0.563631\pi$
$317$	−7.07107	−0.397151	−0.198575	+	0.980086i	$0.563631\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	14.0000	0.781404
$322$	0	0
$323$	−8.48528	−0.472134
$324$	0	0
$325$	−4.24264	−0.235339
$326$	0	0
$327$	6.00000	0.331801
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	1.41421	0.0777322	0.0388661	−	0.999244i	$-0.487625\pi$
$331$	1.41421	0.0777322	0.0388661	+	0.999244i	$0.487625\pi$
$332$	0	0
$333$	4.24264	0.232495
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	−8.48528	−0.460857
$340$	0	0
$341$	−11.3137	−0.612672
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	−18.3848	−0.986947	−0.493473	−	0.869761i	$-0.664273\pi$
$347$	−18.3848	−0.986947	−0.493473	+	0.869761i	$0.664273\pi$
$348$	0	0
$349$	4.24264	0.227103	0.113552	−	0.993532i	$-0.463777\pi$
$349$	4.24264	0.227103	0.113552	+	0.993532i	$0.463777\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−14.1421	−0.750587
$356$	0	0
$357$	5.65685	0.299392
$358$	0	0
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	12.7279	0.668043
$364$	0	0
$365$	5.65685	0.296093
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.1421	0.734223
$372$	0	0
$373$	−7.07107	−0.366126	−0.183063	−	0.983101i	$-0.558601\pi$
$373$	−7.07107	−0.366126	−0.183063	+	0.983101i	$0.558601\pi$
$374$	0	0
$375$	16.0000	0.826236
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	4.24264	0.217930	0.108965	−	0.994046i	$-0.465246\pi$
$379$	4.24264	0.217930	0.108965	+	0.994046i	$0.465246\pi$
$380$	0	0
$381$	−11.3137	−0.579619
$382$	0	0
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−7.07107	−0.359443
$388$	0	0
$389$	18.3848	0.932145	0.466073	−	0.884746i	$-0.345669\pi$
$389$	18.3848	0.932145	0.466073	+	0.884746i	$0.345669\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−22.0000	−1.10975
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.07107	0.354887	0.177443	−	0.984131i	$-0.443217\pi$
$397$	7.07107	0.354887	0.177443	+	0.984131i	$0.443217\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−11.3137	−0.563576
$404$	0	0
$405$	−7.07107	−0.351364
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	0	0
$411$	11.3137	0.558064
$412$	0	0
$413$	−8.48528	−0.417533
$414$	0	0
$415$	2.00000	0.0981761
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	4.24264	0.207267	0.103633	−	0.994616i	$-0.466953\pi$
$419$	4.24264	0.207267	0.103633	+	0.994616i	$0.466953\pi$
$420$	0	0
$421$	12.7279	0.620321	0.310160	−	0.950684i	$-0.399617\pi$
$421$	12.7279	0.620321	0.310160	+	0.950684i	$0.399617\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−6.00000	−0.291043
$426$	0	0
$427$	25.4558	1.23189
$428$	0	0
$429$	−2.82843	−0.136558
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−8.48528	−0.406838
$436$	0	0
$437$	25.4558	1.21772
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	21.2132	1.00787	0.503935	−	0.863742i	$-0.331885\pi$
$443$	21.2132	1.00787	0.503935	+	0.863742i	$0.331885\pi$
$444$	0	0
$445$	5.65685	0.268161
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−14.1421	−0.664455
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	0	0
$459$	11.3137	0.528079
$460$	0	0
$461$	−15.5563	−0.724531	−0.362266	−	0.932075i	$-0.617997\pi$
$461$	−15.5563	−0.724531	−0.362266	+	0.932075i	$0.617997\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	0	0
$467$	7.07107	0.327210	0.163605	−	0.986526i	$-0.447688\pi$
$467$	7.07107	0.327210	0.163605	+	0.986526i	$0.447688\pi$
$468$	0	0
$469$	14.1421	0.653023
$470$	0	0
$471$	−30.0000	−1.38233
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	12.7279	0.583997
$476$	0	0
$477$	7.07107	0.323762
$478$	0	0
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	−16.9706	−0.772187
$484$	0	0
$485$	−2.82843	−0.128432
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	2.00000	0.0904431
$490$	0	0
$491$	−26.8701	−1.21263	−0.606314	−	0.795225i	$-0.707353\pi$
$491$	−26.8701	−1.21263	−0.606314	+	0.795225i	$0.707353\pi$
$492$	0	0
$493$	8.48528	0.382158
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	20.0000	0.897123
$498$	0	0
$499$	−32.5269	−1.45610	−0.728052	−	0.685522i	$-0.759574\pi$
$499$	−32.5269	−1.45610	−0.728052	+	0.685522i	$0.759574\pi$
$500$	0	0
$501$	2.82843	0.126365
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−22.0000	−0.978987
$506$	0	0
$507$	15.5563	0.690882
$508$	0	0
$509$	32.5269	1.44173	0.720865	−	0.693075i	$-0.243745\pi$
$509$	32.5269	1.44173	0.720865	+	0.693075i	$0.243745\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	8.48528	0.373906
$516$	0	0
$517$	−11.3137	−0.497576
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−40.0000	−1.75243	−0.876216	−	0.481919i	$-0.839940\pi$
$521$	−40.0000	−1.75243	−0.876216	+	0.481919i	$0.839940\pi$
$522$	0	0
$523$	35.3553	1.54598	0.772991	−	0.634418i	$-0.218760\pi$
$523$	35.3553	1.54598	0.772991	+	0.634418i	$0.218760\pi$
$524$	0	0
$525$	−8.48528	−0.370328
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−4.24264	−0.184115
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−14.0000	−0.605273
$536$	0	0
$537$	−34.0000	−1.46721
$538$	0	0
$539$	−4.24264	−0.182743
$540$	0	0
$541$	−12.7279	−0.547216	−0.273608	−	0.961841i	$-0.588217\pi$
$541$	−12.7279	−0.547216	−0.273608	+	0.961841i	$0.588217\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−7.07107	−0.302337	−0.151169	−	0.988508i	$-0.548304\pi$
$547$	−7.07107	−0.302337	−0.151169	+	0.988508i	$0.548304\pi$
$548$	0	0
$549$	12.7279	0.543214
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.48528	0.360180
$556$	0	0
$557$	35.3553	1.49805	0.749027	−	0.662540i	$-0.230521\pi$
$557$	35.3553	1.49805	0.749027	+	0.662540i	$0.230521\pi$
$558$	0	0
$559$	10.0000	0.422955
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−26.8701	−1.13244	−0.566219	−	0.824255i	$-0.691594\pi$
$563$	−26.8701	−1.13244	−0.566219	+	0.824255i	$0.691594\pi$
$564$	0	0
$565$	8.48528	0.356978
$566$	0	0
$567$	10.0000	0.419961
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−1.41421	−0.0591830	−0.0295915	−	0.999562i	$-0.509421\pi$
$571$	−1.41421	−0.0591830	−0.0295915	+	0.999562i	$0.509421\pi$
$572$	0	0
$573$	11.3137	0.472637
$574$	0	0
$575$	18.0000	0.750652
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	−19.7990	−0.822818
$580$	0	0
$581$	−2.82843	−0.117343
$582$	0	0
$583$	−10.0000	−0.414158
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−9.89949	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$587$	−9.89949	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$588$	0	0
$589$	33.9411	1.39852
$590$	0	0
$591$	−34.0000	−1.39857
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	−5.65685	−0.231908
$596$	0	0
$597$	−19.7990	−0.810319
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	0	0
$601$	20.0000	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$601$	20.0000	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$602$	0	0
$603$	7.07107	0.287956
$604$	0	0
$605$	−12.7279	−0.517464
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−11.3137	−0.457704
$612$	0	0
$613$	35.3553	1.42799	0.713994	−	0.700151i	$-0.246884\pi$
$613$	35.3553	1.42799	0.713994	+	0.700151i	$0.246884\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	24.0416	0.966315	0.483157	−	0.875534i	$-0.339490\pi$
$619$	24.0416	0.966315	0.483157	+	0.875534i	$0.339490\pi$
$620$	0	0
$621$	−33.9411	−1.36201
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	8.48528	0.338869
$628$	0	0
$629$	−8.48528	−0.338330
$630$	0	0
$631$	10.0000	0.398094	0.199047	−	0.979990i	$-0.436215\pi$
$631$	10.0000	0.398094	0.199047	+	0.979990i	$0.436215\pi$
$632$	0	0
$633$	−18.0000	−0.715436
$634$	0	0
$635$	11.3137	0.448971
$636$	0	0
$637$	−4.24264	−0.168100
$638$	0	0
$639$	10.0000	0.395594
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−29.6985	−1.17119	−0.585597	−	0.810602i	$-0.699140\pi$
$643$	−29.6985	−1.17119	−0.585597	+	0.810602i	$0.699140\pi$
$644$	0	0
$645$	−14.1421	−0.556846
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	−22.6274	−0.886838
$652$	0	0
$653$	−26.8701	−1.05151	−0.525753	−	0.850637i	$-0.676216\pi$
$653$	−26.8701	−1.05151	−0.525753	+	0.850637i	$0.676216\pi$
$654$	0	0
$655$	22.0000	0.859611
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	24.0416	0.936529	0.468264	−	0.883588i	$-0.344879\pi$
$659$	24.0416	0.936529	0.468264	+	0.883588i	$0.344879\pi$
$660$	0	0
$661$	−12.7279	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$661$	−12.7279	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	−25.4558	−0.985654
$668$	0	0
$669$	−33.9411	−1.31224
$670$	0	0
$671$	−18.0000	−0.694882
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	−16.9706	−0.653197
$676$	0	0
$677$	−4.24264	−0.163058	−0.0815290	−	0.996671i	$-0.525980\pi$
$677$	−4.24264	−0.163058	−0.0815290	+	0.996671i	$0.525980\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	−30.0000	−1.14960
$682$	0	0
$683$	7.07107	0.270567	0.135283	−	0.990807i	$-0.456805\pi$
$683$	7.07107	0.270567	0.135283	+	0.990807i	$0.456805\pi$
$684$	0	0
$685$	−11.3137	−0.432275
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	12.7279	0.484193	0.242096	−	0.970252i	$-0.422165\pi$
$691$	12.7279	0.484193	0.242096	+	0.970252i	$0.422165\pi$
$692$	0	0
$693$	2.82843	0.107443
$694$	0	0
$695$	−6.00000	−0.227593
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−5.65685	−0.213962
$700$	0	0
$701$	43.8406	1.65584	0.827919	−	0.560848i	$-0.189525\pi$
$701$	43.8406	1.65584	0.827919	+	0.560848i	$0.189525\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	31.1127	1.17011
$708$	0	0
$709$	−38.1838	−1.43402	−0.717011	−	0.697062i	$-0.754490\pi$
$709$	−38.1838	−1.43402	−0.717011	+	0.697062i	$0.754490\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	48.0000	1.79761
$714$	0	0
$715$	2.82843	0.105777
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−12.0000	−0.446903
$722$	0	0
$723$	−25.4558	−0.946713
$724$	0	0
$725$	−12.7279	−0.472703
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	14.1421	0.523066
$732$	0	0
$733$	−29.6985	−1.09694	−0.548469	−	0.836171i	$-0.684789\pi$
$733$	−29.6985	−1.09694	−0.548469	+	0.836171i	$0.684789\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	32.5269	1.19652	0.598261	−	0.801301i	$-0.295859\pi$
$739$	32.5269	1.19652	0.598261	+	0.801301i	$0.295859\pi$
$740$	0	0
$741$	8.48528	0.311715
$742$	0	0
$743$	46.0000	1.68758	0.843788	−	0.536676i	$-0.180320\pi$
$743$	46.0000	1.68758	0.843788	+	0.536676i	$0.180320\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	0	0
$747$	−1.41421	−0.0517434
$748$	0	0
$749$	19.7990	0.723439
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	42.0000	1.53057
$754$	0	0
$755$	14.1421	0.514685
$756$	0	0
$757$	32.5269	1.18221	0.591105	−	0.806594i	$-0.298692\pi$
$757$	32.5269	1.18221	0.591105	+	0.806594i	$0.298692\pi$
$758$	0	0
$759$	12.0000	0.435572
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	8.48528	0.307188
$764$	0	0
$765$	−2.82843	−0.102262
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	31.1127	1.12050
$772$	0	0
$773$	7.07107	0.254329	0.127164	−	0.991882i	$-0.459412\pi$
$773$	7.07107	0.254329	0.127164	+	0.991882i	$0.459412\pi$
$774$	0	0
$775$	24.0000	0.862105
$776$	0	0
$777$	−12.0000	−0.430498
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−14.1421	−0.506045
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	−21.2132	−0.756169	−0.378085	−	0.925771i	$-0.623417\pi$
$787$	−21.2132	−0.756169	−0.378085	+	0.925771i	$0.623417\pi$
$788$	0	0
$789$	−8.48528	−0.302084
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−18.0000	−0.639199
$794$	0	0
$795$	14.1421	0.501570
$796$	0	0
$797$	−35.3553	−1.25235	−0.626175	−	0.779682i	$-0.715381\pi$
$797$	−35.3553	−1.25235	−0.626175	+	0.779682i	$0.715381\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	5.65685	0.199626
$804$	0	0
$805$	16.9706	0.598134
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	−55.1543	−1.93673	−0.968365	−	0.249537i	$-0.919722\pi$
$811$	−55.1543	−1.93673	−0.968365	+	0.249537i	$0.919722\pi$
$812$	0	0
$813$	−11.3137	−0.396789
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	15.5563	0.542920	0.271460	−	0.962450i	$-0.412493\pi$
$821$	15.5563	0.542920	0.271460	+	0.962450i	$0.412493\pi$
$822$	0	0
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−46.6690	−1.62284	−0.811421	−	0.584462i	$-0.801305\pi$
$827$	−46.6690	−1.62284	−0.811421	+	0.584462i	$0.801305\pi$
$828$	0	0
$829$	32.5269	1.12971	0.564853	−	0.825191i	$-0.308933\pi$
$829$	32.5269	1.12971	0.564853	+	0.825191i	$0.308933\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−2.82843	−0.0978818
$836$	0	0
$837$	−45.2548	−1.56424
$838$	0	0
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	28.2843	0.974162
$844$	0	0
$845$	−15.5563	−0.535155
$846$	0	0
$847$	18.0000	0.618487
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	25.4558	0.872615
$852$	0	0
$853$	−7.07107	−0.242109	−0.121054	−	0.992646i	$-0.538628\pi$
$853$	−7.07107	−0.242109	−0.121054	+	0.992646i	$0.538628\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	0	0
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	0	0
$859$	4.24264	0.144757	0.0723785	−	0.997377i	$-0.476941\pi$
$859$	4.24264	0.144757	0.0723785	+	0.997377i	$0.476941\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	18.3848	0.624380
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−10.0000	−0.338837
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	22.6274	0.764946
$876$	0	0
$877$	7.07107	0.238773	0.119386	−	0.992848i	$-0.461907\pi$
$877$	7.07107	0.238773	0.119386	+	0.992848i	$0.461907\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	29.6985	0.999434	0.499717	−	0.866189i	$-0.333437\pi$
$883$	29.6985	0.999434	0.499717	+	0.866189i	$0.333437\pi$
$884$	0	0
$885$	−8.48528	−0.285230
$886$	0	0
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−7.07107	−0.236890
$892$	0	0
$893$	33.9411	1.13580
$894$	0	0
$895$	34.0000	1.13649
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	−33.9411	−1.13200
$900$	0	0
$901$	−14.1421	−0.471143
$902$	0	0
$903$	20.0000	0.665558
$904$	0	0
$905$	−18.0000	−0.598340
$906$	0	0
$907$	−38.1838	−1.26787	−0.633936	−	0.773386i	$-0.718562\pi$
$907$	−38.1838	−1.26787	−0.633936	+	0.773386i	$0.718562\pi$
$908$	0	0
$909$	15.5563	0.515972
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	0	0
$913$	2.00000	0.0661903
$914$	0	0
$915$	25.4558	0.841544
$916$	0	0
$917$	−31.1127	−1.02743
$918$	0	0
$919$	26.0000	0.857661	0.428830	−	0.903385i	$-0.358926\pi$
$919$	26.0000	0.857661	0.428830	+	0.903385i	$0.358926\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	0	0
$923$	−14.1421	−0.465494
$924$	0	0
$925$	12.7279	0.418491
$926$	0	0
$927$	−6.00000	−0.197066
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	12.7279	0.417141
$932$	0	0
$933$	42.4264	1.38898
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	−22.6274	−0.738418
$940$	0	0
$941$	41.0122	1.33696	0.668480	−	0.743730i	$-0.266945\pi$
$941$	41.0122	1.33696	0.668480	+	0.743730i	$0.266945\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	7.07107	0.229779	0.114889	−	0.993378i	$-0.463349\pi$
$947$	7.07107	0.229779	0.114889	+	0.993378i	$0.463349\pi$
$948$	0	0
$949$	5.65685	0.183629
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	0	0
$955$	−11.3137	−0.366103
$956$	0	0
$957$	−8.48528	−0.274290
$958$	0	0
$959$	16.0000	0.516667
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	9.89949	0.319007
$964$	0	0
$965$	19.7990	0.637352
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−26.8701	−0.862301	−0.431151	−	0.902280i	$-0.641892\pi$
$971$	−26.8701	−0.862301	−0.431151	+	0.902280i	$0.641892\pi$
$972$	0	0
$973$	8.48528	0.272026
$974$	0	0
$975$	6.00000	0.192154
$976$	0	0
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	0	0
$979$	5.65685	0.180794
$980$	0	0
$981$	4.24264	0.135457
$982$	0	0
$983$	34.0000	1.08443	0.542216	−	0.840239i	$-0.317586\pi$
$983$	34.0000	1.08443	0.542216	+	0.840239i	$0.317586\pi$
$984$	0	0
$985$	34.0000	1.08333
$986$	0	0
$987$	−22.6274	−0.720239
$988$	0	0
$989$	−42.4264	−1.34908
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	−2.00000	−0.0634681
$994$	0	0
$995$	19.7990	0.627670
$996$	0	0
$997$	52.3259	1.65718	0.828589	−	0.559857i	$-0.189144\pi$
$997$	52.3259	1.65718	0.828589	+	0.559857i	$0.189144\pi$
$998$	0	0
$999$	−24.0000	−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.a.b.1.1		2
3.2	odd	2		9216.2.a.d.1.1		2
4.3	odd	2		1024.2.a.e.1.2		2
8.3	odd	2		1024.2.a.e.1.1		2
8.5	even	2	inner	1024.2.a.b.1.2		2
12.11	even	2		9216.2.a.s.1.1		2
16.3	odd	4		1024.2.b.b.513.2		2
16.5	even	4		1024.2.b.e.513.2		2
16.11	odd	4		1024.2.b.b.513.1		2
16.13	even	4		1024.2.b.e.513.1		2
24.5	odd	2		9216.2.a.d.1.2		2
24.11	even	2		9216.2.a.s.1.2		2
32.3	odd	8		64.2.e.a.17.1		2
32.5	even	8		128.2.e.b.97.1		2
32.11	odd	8		64.2.e.a.49.1		2
32.13	even	8		128.2.e.b.33.1		2
32.19	odd	8		128.2.e.a.33.1		2
32.21	even	8		16.2.e.a.5.1	✓	2
32.27	odd	8		128.2.e.a.97.1		2
32.29	even	8		16.2.e.a.13.1	yes	2
96.5	odd	8		1152.2.k.b.865.1		2
96.11	even	8		576.2.k.a.433.1		2
96.29	odd	8		144.2.k.a.109.1		2
96.35	even	8		576.2.k.a.145.1		2
96.53	odd	8		144.2.k.a.37.1		2
96.59	even	8		1152.2.k.a.865.1		2
96.77	odd	8		1152.2.k.b.289.1		2
96.83	even	8		1152.2.k.a.289.1		2
160.3	even	8		1600.2.q.a.849.1		2
160.29	even	8		400.2.l.c.301.1		2
160.43	even	8		1600.2.q.b.49.1		2
160.53	odd	8		400.2.q.a.149.1		2
160.67	even	8		1600.2.q.b.849.1		2
160.93	odd	8		400.2.q.b.349.1		2
160.99	odd	8		1600.2.l.a.401.1		2
160.107	even	8		1600.2.q.a.49.1		2
160.117	odd	8		400.2.q.b.149.1		2
160.139	odd	8		1600.2.l.a.1201.1		2
160.149	even	8		400.2.l.c.101.1		2
160.157	odd	8		400.2.q.a.349.1		2
224.53	even	24		784.2.x.f.373.1		4
224.61	odd	24		784.2.x.c.557.1		4
224.93	even	24		784.2.x.f.557.1		4
224.117	odd	24		784.2.x.c.165.1		4
224.125	odd	8		784.2.m.b.589.1		2
224.149	even	24		784.2.x.f.165.1		4
224.157	odd	24		784.2.x.c.765.1		4
224.181	odd	8		784.2.m.b.197.1		2
224.213	odd	24		784.2.x.c.373.1		4
224.221	even	24		784.2.x.f.765.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.2.e.a.5.1	✓	2	32.21	even	8
16.2.e.a.13.1	yes	2	32.29	even	8
64.2.e.a.17.1		2	32.3	odd	8
64.2.e.a.49.1		2	32.11	odd	8
128.2.e.a.33.1		2	32.19	odd	8
128.2.e.a.97.1		2	32.27	odd	8
128.2.e.b.33.1		2	32.13	even	8
128.2.e.b.97.1		2	32.5	even	8
144.2.k.a.37.1		2	96.53	odd	8
144.2.k.a.109.1		2	96.29	odd	8
400.2.l.c.101.1		2	160.149	even	8
400.2.l.c.301.1		2	160.29	even	8
400.2.q.a.149.1		2	160.53	odd	8
400.2.q.a.349.1		2	160.157	odd	8
400.2.q.b.149.1		2	160.117	odd	8
400.2.q.b.349.1		2	160.93	odd	8
576.2.k.a.145.1		2	96.35	even	8
576.2.k.a.433.1		2	96.11	even	8
784.2.m.b.197.1		2	224.181	odd	8
784.2.m.b.589.1		2	224.125	odd	8
784.2.x.c.165.1		4	224.117	odd	24
784.2.x.c.373.1		4	224.213	odd	24
784.2.x.c.557.1		4	224.61	odd	24
784.2.x.c.765.1		4	224.157	odd	24
784.2.x.f.165.1		4	224.149	even	24
784.2.x.f.373.1		4	224.53	even	24
784.2.x.f.557.1		4	224.93	even	24
784.2.x.f.765.1		4	224.221	even	24
1024.2.a.b.1.1		2	1.1	even	1	trivial
1024.2.a.b.1.2		2	8.5	even	2	inner
1024.2.a.e.1.1		2	8.3	odd	2
1024.2.a.e.1.2		2	4.3	odd	2
1024.2.b.b.513.1		2	16.11	odd	4
1024.2.b.b.513.2		2	16.3	odd	4
1024.2.b.e.513.1		2	16.13	even	4
1024.2.b.e.513.2		2	16.5	even	4
1152.2.k.a.289.1		2	96.83	even	8
1152.2.k.a.865.1		2	96.59	even	8
1152.2.k.b.289.1		2	96.77	odd	8
1152.2.k.b.865.1		2	96.5	odd	8
1600.2.l.a.401.1		2	160.99	odd	8
1600.2.l.a.1201.1		2	160.139	odd	8
1600.2.q.a.49.1		2	160.107	even	8
1600.2.q.a.849.1		2	160.3	even	8
1600.2.q.b.49.1		2	160.43	even	8
1600.2.q.b.849.1		2	160.67	even	8
9216.2.a.d.1.1		2	3.2	odd	2
9216.2.a.d.1.2		2	24.5	odd	2
9216.2.a.s.1.1		2	12.11	even	2
9216.2.a.s.1.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +1.41421 q^{5} -2.00000 q^{7} -1.00000 q^{9} +1.41421 q^{11} +1.41421 q^{13} -2.00000 q^{15} +2.00000 q^{17} -4.24264 q^{19} +2.82843 q^{21} -6.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} +4.24264 q^{29} -8.00000 q^{31} -2.00000 q^{33} -2.82843 q^{35} -4.24264 q^{37} -2.00000 q^{39} +7.07107 q^{43} -1.41421 q^{45} -8.00000 q^{47} -3.00000 q^{49} -2.82843 q^{51} -7.07107 q^{53} +2.00000 q^{55} +6.00000 q^{57} +4.24264 q^{59} -12.7279 q^{61} +2.00000 q^{63} +2.00000 q^{65} -7.07107 q^{67} +8.48528 q^{69} -10.0000 q^{71} +4.00000 q^{73} +4.24264 q^{75} -2.82843 q^{77} -5.00000 q^{81} +1.41421 q^{83} +2.82843 q^{85} -6.00000 q^{87} +4.00000 q^{89} -2.82843 q^{91} +11.3137 q^{93} -6.00000 q^{95} -2.00000 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} - 12 q^{23} - 6 q^{25} - 16 q^{31} - 4 q^{33} - 4 q^{39} - 16 q^{47} - 6 q^{49} + 4 q^{55} + 12 q^{57} + 4 q^{63} + 4 q^{65} - 20 q^{71} + 8 q^{73} - 10 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^7 - 2 * q^9 - 4 * q^15 + 4 * q^17 - 12 * q^23 - 6 * q^25 - 16 * q^31 - 4 * q^33 - 4 * q^39 - 16 * q^47 - 6 * q^49 + 4 * q^55 + 12 * q^57 + 4 * q^63 + 4 * q^65 - 20 * q^71 + 8 * q^73 - 10 * q^81 - 12 * q^87 + 8 * q^89 - 12 * q^95 - 4 * q^97

Introduction

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