LMFDB - Embedded newform 1024.2.b.e.513.1

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:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
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)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			513.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	1024.513
Dual form			1024.2.b.e.513.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	0	0
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 1.41421i	− 0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	− 1.41421i	− 0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$	0	0
$13$	1.41421i	0.392232i	0.980581	+	0.196116i	$0.0628330\pi$
$13$	1.41421i	0.392232i	−0.980581	+	0.196116i	$0.937167\pi$
$14$	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
$19$	− 4.24264i	− 0.973329i	−0.873589	−	0.486664i	$-0.838214\pi$
$19$	− 4.24264i	− 0.973329i	0.873589	−	0.486664i	$-0.161786\pi$
$20$	0	0
$21$	− 2.82843i	− 0.617213i
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	4.24264i	0.787839i	0.919145	+	0.393919i	$0.128881\pi$
$29$	4.24264i	0.787839i	−0.919145	+	0.393919i	$0.871119\pi$
$30$	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
$35$	− 2.82843i	− 0.478091i
$36$	0	0
$37$	4.24264i	0.697486i	0.937218	+	0.348743i	$0.113391\pi$
$37$	4.24264i	0.697486i	−0.937218	+	0.348743i	$0.886609\pi$
$38$	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
$43$	− 7.07107i	− 1.07833i	−0.842201	−	0.539164i	$-0.818740\pi$
$43$	− 7.07107i	− 1.07833i	0.842201	−	0.539164i	$-0.181260\pi$
$44$	0	0
$45$	− 1.41421i	− 0.210819i
$46$	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
$51$	− 2.82843i	− 0.396059i
$52$	0	0
$53$	7.07107i	0.971286i	0.874157	+	0.485643i	$0.161414\pi$
$53$	7.07107i	0.971286i	−0.874157	+	0.485643i	$0.838586\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	− 4.24264i	− 0.552345i	−0.961108	−	0.276172i	$-0.910934\pi$
$59$	− 4.24264i	− 0.552345i	0.961108	−	0.276172i	$-0.0890661\pi$
$60$	0	0
$61$	− 12.7279i	− 1.62964i	−0.579712	−	0.814822i	$-0.696835\pi$
$61$	− 12.7279i	− 1.62964i	0.579712	−	0.814822i	$-0.303165\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	− 7.07107i	− 0.863868i	−0.901905	−	0.431934i	$-0.857831\pi$
$67$	− 7.07107i	− 0.863868i	0.901905	−	0.431934i	$-0.142169\pi$
$68$	0	0
$69$	− 8.48528i	− 1.02151i
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	− 4.24264i	− 0.489898i
$76$	0	0
$77$	− 2.82843i	− 0.322329i
$78$	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
$83$	1.41421i	0.155230i	0.996983	+	0.0776151i	$0.0247305\pi$
$83$	1.41421i	0.155230i	−0.996983	+	0.0776151i	$0.975269\pi$
$84$	0	0
$85$	− 2.82843i	− 0.306786i
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	2.82843i	0.296500i
$92$	0	0
$93$	11.3137i	1.17318i
$94$	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
$99$	− 1.41421i	− 0.142134i
$100$	0	0
$101$	15.5563i	1.54791i	0.633238	+	0.773957i	$0.281726\pi$
$101$	15.5563i	1.54791i	−0.633238	+	0.773957i	$0.718274\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	9.89949i	0.957020i	0.878082	+	0.478510i	$0.158823\pi$
$107$	9.89949i	0.957020i	−0.878082	+	0.478510i	$0.841177\pi$
$108$	0	0
$109$	− 4.24264i	− 0.406371i	−0.979140	−	0.203186i	$-0.934871\pi$
$109$	− 4.24264i	− 0.406371i	0.979140	−	0.203186i	$-0.0651295\pi$
$110$	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
$115$	− 8.48528i	− 0.791257i
$116$	0	0
$117$	1.41421i	0.130744i
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	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
$131$	15.5563i	1.35916i	0.733599	+	0.679582i	$0.237839\pi$
$131$	15.5563i	1.35916i	−0.733599	+	0.679582i	$0.762161\pi$
$132$	0	0
$133$	− 8.48528i	− 0.735767i
$134$	0	0
$135$	−8.00000	−0.688530
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	4.24264i	0.359856i	0.983680	+	0.179928i	$0.0575865\pi$
$139$	4.24264i	0.359856i	−0.983680	+	0.179928i	$0.942414\pi$
$140$	0	0
$141$	11.3137i	0.952786i
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	4.24264i	0.349927i
$148$	0	0
$149$	− 9.89949i	− 0.810998i	−0.914095	−	0.405499i	$-0.867098\pi$
$149$	− 9.89949i	− 0.810998i	0.914095	−	0.405499i	$-0.132902\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	11.3137i	0.908739i
$156$	0	0
$157$	21.2132i	1.69300i	0.532390	+	0.846499i	$0.321294\pi$
$157$	21.2132i	1.69300i	−0.532390	+	0.846499i	$0.678706\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 1.41421i	− 0.110770i	−0.998465	−	0.0553849i	$-0.982361\pi$
$163$	− 1.41421i	− 0.110770i	0.998465	−	0.0553849i	$-0.0176386\pi$
$164$	0	0
$165$	2.82843i	0.220193i
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	− 4.24264i	− 0.324443i
$172$	0	0
$173$	1.41421i	0.107521i	0.998554	+	0.0537603i	$0.0171207\pi$
$173$	1.41421i	0.107521i	−0.998554	+	0.0537603i	$0.982879\pi$
$174$	0	0
$175$	6.00000	0.453557
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	24.0416i	1.79696i	0.439019	+	0.898478i	$0.355326\pi$
$179$	24.0416i	1.79696i	−0.439019	+	0.898478i	$0.644674\pi$
$180$	0	0
$181$	12.7279i	0.946059i	0.881047	+	0.473029i	$0.156840\pi$
$181$	12.7279i	0.946059i	−0.881047	+	0.473029i	$0.843160\pi$
$182$	0	0
$183$	−18.0000	−1.33060
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	− 2.82843i	− 0.206835i
$188$	0	0
$189$	− 11.3137i	− 0.822951i
$190$	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
$195$	− 2.82843i	− 0.202548i
$196$	0	0
$197$	− 24.0416i	− 1.71290i	−0.516234	−	0.856448i	$-0.672666\pi$
$197$	− 24.0416i	− 1.71290i	0.516234	−	0.856448i	$-0.327334\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	8.48528i	0.595550i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	12.7279i	0.876226i	0.898920	+	0.438113i	$0.144353\pi$
$211$	12.7279i	0.876226i	−0.898920	+	0.438113i	$0.855647\pi$
$212$	0	0
$213$	− 14.1421i	− 0.969003i
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	5.65685i	0.382255i
$220$	0	0
$221$	2.82843i	0.190261i
$222$	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
$227$	21.2132i	1.40797i	0.710215	+	0.703985i	$0.248598\pi$
$227$	21.2132i	1.40797i	−0.710215	+	0.703985i	$0.751402\pi$
$228$	0	0
$229$	9.89949i	0.654177i	0.944994	+	0.327089i	$0.106068\pi$
$229$	9.89949i	0.654177i	−0.944994	+	0.327089i	$0.893932\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	0	0
$235$	11.3137i	0.738025i
$236$	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
$243$	− 9.89949i	− 0.635053i
$244$	0	0
$245$	4.24264i	0.271052i
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	2.00000	0.126745
$250$	0	0
$251$	29.6985i	1.87455i	0.348589	+	0.937276i	$0.386661\pi$
$251$	29.6985i	1.87455i	−0.348589	+	0.937276i	$0.613339\pi$
$252$	0	0
$253$	− 8.48528i	− 0.533465i
$254$	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
$259$	8.48528i	0.527250i
$260$	0	0
$261$	4.24264i	0.262613i
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	5.65685i	0.346194i
$268$	0	0
$269$	− 4.24264i	− 0.258678i	−0.991600	−	0.129339i	$-0.958714\pi$
$269$	− 4.24264i	− 0.258678i	0.991600	−	0.129339i	$-0.0412856\pi$
$270$	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
$275$	− 4.24264i	− 0.255841i
$276$	0	0
$277$	− 4.24264i	− 0.254916i	−0.991844	−	0.127458i	$-0.959318\pi$
$277$	− 4.24264i	− 0.254916i	0.991844	−	0.127458i	$-0.0406817\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	20.0000	1.19310	0.596550	−	0.802576i	$-0.296538\pi$
$281$	20.0000	1.19310	0.596550	+	0.802576i	$0.296538\pi$
$282$	0	0
$283$	− 21.2132i	− 1.26099i	−0.776192	−	0.630497i	$-0.782851\pi$
$283$	− 21.2132i	− 1.26099i	0.776192	−	0.630497i	$-0.217149\pi$
$284$	0	0
$285$	8.48528i	0.502625i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.82843i	0.165805i
$292$	0	0
$293$	21.2132i	1.23929i	0.784883	+	0.619644i	$0.212723\pi$
$293$	21.2132i	1.23929i	−0.784883	+	0.619644i	$0.787277\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	8.48528i	0.490716i
$300$	0	0
$301$	− 14.1421i	− 0.815139i
$302$	0	0
$303$	22.0000	1.26387
$304$	0	0
$305$	−18.0000	−1.03068
$306$	0	0
$307$	7.07107i	0.403567i	0.979430	+	0.201784i	$0.0646738\pi$
$307$	7.07107i	0.403567i	−0.979430	+	0.201784i	$0.935326\pi$
$308$	0	0
$309$	8.48528i	0.482711i
$310$	0	0
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	− 2.82843i	− 0.159364i
$316$	0	0
$317$	− 7.07107i	− 0.397151i	−0.980086	−	0.198575i	$-0.936369\pi$
$317$	− 7.07107i	− 0.397151i	0.980086	−	0.198575i	$-0.0636315\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	14.0000	0.781404
$322$	0	0
$323$	− 8.48528i	− 0.472134i
$324$	0	0
$325$	4.24264i	0.235339i
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	− 1.41421i	− 0.0777322i	−0.999244	−	0.0388661i	$-0.987625\pi$
$331$	− 1.41421i	− 0.0777322i	0.999244	−	0.0388661i	$-0.0123746\pi$
$332$	0	0
$333$	4.24264i	0.232495i
$334$	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
$339$	− 8.48528i	− 0.460857i
$340$	0	0
$341$	11.3137i	0.612672i
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	18.3848i	0.986947i	0.869761	+	0.493473i	$0.164273\pi$
$347$	18.3848i	0.986947i	−0.869761	+	0.493473i	$0.835727\pi$
$348$	0	0
$349$	4.24264i	0.227103i	0.993532	+	0.113552i	$0.0362227\pi$
$349$	4.24264i	0.227103i	−0.993532	+	0.113552i	$0.963777\pi$
$350$	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
$355$	− 14.1421i	− 0.750587i
$356$	0	0
$357$	− 5.65685i	− 0.299392i
$358$	0	0
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 12.7279i	− 0.668043i
$364$	0	0
$365$	5.65685i	0.296093i
$366$	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
$371$	14.1421i	0.734223i
$372$	0	0
$373$	7.07107i	0.366126i	0.983101	+	0.183063i	$0.0586012\pi$
$373$	7.07107i	0.366126i	−0.983101	+	0.183063i	$0.941399\pi$
$374$	0	0
$375$	−16.0000	−0.826236
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	− 4.24264i	− 0.217930i	−0.994046	−	0.108965i	$-0.965246\pi$
$379$	− 4.24264i	− 0.217930i	0.994046	−	0.108965i	$-0.0347536\pi$
$380$	0	0
$381$	− 11.3137i	− 0.579619i
$382$	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
$387$	− 7.07107i	− 0.359443i
$388$	0	0
$389$	− 18.3848i	− 0.932145i	−0.884746	−	0.466073i	$-0.845669\pi$
$389$	− 18.3848i	− 0.932145i	0.884746	−	0.466073i	$-0.154331\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	22.0000	1.10975
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.07107i	0.354887i	0.984131	+	0.177443i	$0.0567827\pi$
$397$	7.07107i	0.354887i	−0.984131	+	0.177443i	$0.943217\pi$
$398$	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
$403$	− 11.3137i	− 0.563576i
$404$	0	0
$405$	7.07107i	0.351364i
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	16.0000	0.791149	0.395575	−	0.918434i	$-0.370545\pi$
$409$	16.0000	0.791149	0.395575	+	0.918434i	$0.370545\pi$
$410$	0	0
$411$	− 11.3137i	− 0.558064i
$412$	0	0
$413$	− 8.48528i	− 0.417533i
$414$	0	0
$415$	2.00000	0.0981761
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	4.24264i	0.207267i	0.994616	+	0.103633i	$0.0330468\pi$
$419$	4.24264i	0.207267i	−0.994616	+	0.103633i	$0.966953\pi$
$420$	0	0
$421$	− 12.7279i	− 0.620321i	−0.950684	−	0.310160i	$-0.899617\pi$
$421$	− 12.7279i	− 0.620321i	0.950684	−	0.310160i	$-0.100383\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	− 25.4558i	− 1.23189i
$428$	0	0
$429$	− 2.82843i	− 0.136558i
$430$	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
$435$	− 8.48528i	− 0.406838i
$436$	0	0
$437$	− 25.4558i	− 1.21772i
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	− 21.2132i	− 1.00787i	−0.863742	−	0.503935i	$-0.831885\pi$
$443$	− 21.2132i	− 1.00787i	0.863742	−	0.503935i	$-0.168115\pi$
$444$	0	0
$445$	5.65685i	0.268161i
$446$	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
$453$	14.1421i	0.664455i
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	0	0
$459$	− 11.3137i	− 0.528079i
$460$	0	0
$461$	− 15.5563i	− 0.724531i	−0.932075	−	0.362266i	$-0.882003\pi$
$461$	− 15.5563i	− 0.724531i	0.932075	−	0.362266i	$-0.117997\pi$
$462$	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
$467$	7.07107i	0.327210i	0.986526	+	0.163605i	$0.0523123\pi$
$467$	7.07107i	0.327210i	−0.986526	+	0.163605i	$0.947688\pi$
$468$	0	0
$469$	− 14.1421i	− 0.653023i
$470$	0	0
$471$	30.0000	1.38233
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	− 12.7279i	− 0.583997i
$476$	0	0
$477$	7.07107i	0.323762i
$478$	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
$483$	− 16.9706i	− 0.772187i
$484$	0	0
$485$	2.82843i	0.128432i
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	26.8701i	1.21263i	0.795225	+	0.606314i	$0.207353\pi$
$491$	26.8701i	1.21263i	−0.795225	+	0.606314i	$0.792647\pi$
$492$	0	0
$493$	8.48528i	0.382158i
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	20.0000	0.897123
$498$	0	0
$499$	− 32.5269i	− 1.45610i	−0.685522	−	0.728052i	$-0.740426\pi$
$499$	− 32.5269i	− 1.45610i	0.685522	−	0.728052i	$-0.259574\pi$
$500$	0	0
$501$	− 2.82843i	− 0.126365i
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	22.0000	0.978987
$506$	0	0
$507$	− 15.5563i	− 0.690882i
$508$	0	0
$509$	32.5269i	1.44173i	0.693075	+	0.720865i	$0.256255\pi$
$509$	32.5269i	1.44173i	−0.693075	+	0.720865i	$0.743745\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	8.48528i	0.373906i
$516$	0	0
$517$	11.3137i	0.497576i
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	40.0000	1.75243	0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000	1.75243	0.876216	+	0.481919i	$0.160060\pi$
$522$	0	0
$523$	− 35.3553i	− 1.54598i	−0.634418	−	0.772991i	$-0.718760\pi$
$523$	− 35.3553i	− 1.54598i	0.634418	−	0.772991i	$-0.281240\pi$
$524$	0	0
$525$	− 8.48528i	− 0.370328i
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	− 4.24264i	− 0.184115i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	14.0000	0.605273
$536$	0	0
$537$	34.0000	1.46721
$538$	0	0
$539$	4.24264i	0.182743i
$540$	0	0
$541$	− 12.7279i	− 0.547216i	−0.961841	−	0.273608i	$-0.911783\pi$
$541$	− 12.7279i	− 0.547216i	0.961841	−	0.273608i	$-0.0882171\pi$
$542$	0	0
$543$	18.0000	0.772454
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	− 7.07107i	− 0.302337i	−0.988508	−	0.151169i	$-0.951696\pi$
$547$	− 7.07107i	− 0.302337i	0.988508	−	0.151169i	$-0.0483036\pi$
$548$	0	0
$549$	− 12.7279i	− 0.543214i
$550$	0	0
$551$	18.0000	0.766826
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 8.48528i	− 0.360180i
$556$	0	0
$557$	35.3553i	1.49805i	0.662540	+	0.749027i	$0.269479\pi$
$557$	35.3553i	1.49805i	−0.662540	+	0.749027i	$0.730521\pi$
$558$	0	0
$559$	10.0000	0.422955
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	− 26.8701i	− 1.13244i	−0.824255	−	0.566219i	$-0.808406\pi$
$563$	− 26.8701i	− 1.13244i	0.824255	−	0.566219i	$-0.191594\pi$
$564$	0	0
$565$	− 8.48528i	− 0.356978i
$566$	0	0
$567$	−10.0000	−0.419961
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	1.41421i	0.0591830i	0.999562	+	0.0295915i	$0.00942064\pi$
$571$	1.41421i	0.0591830i	−0.999562	+	0.0295915i	$0.990579\pi$
$572$	0	0
$573$	11.3137i	0.472637i
$574$	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
$579$	− 19.7990i	− 0.822818i
$580$	0	0
$581$	2.82843i	0.117343i
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	9.89949i	0.408596i	0.978909	+	0.204298i	$0.0654911\pi$
$587$	9.89949i	0.408596i	−0.978909	+	0.204298i	$0.934509\pi$
$588$	0	0
$589$	33.9411i	1.39852i
$590$	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
$595$	− 5.65685i	− 0.231908i
$596$	0	0
$597$	19.7990i	0.810319i
$598$	0	0
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	−20.0000	−0.815817	−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000	−0.815817	−0.407909	+	0.913023i	$0.633742\pi$
$602$	0	0
$603$	− 7.07107i	− 0.287956i
$604$	0	0
$605$	− 12.7279i	− 0.517464i
$606$	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
$611$	− 11.3137i	− 0.457704i
$612$	0	0
$613$	− 35.3553i	− 1.42799i	−0.700151	−	0.713994i	$-0.746884\pi$
$613$	− 35.3553i	− 1.42799i	0.700151	−	0.713994i	$-0.253116\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	− 24.0416i	− 0.966315i	−0.875534	−	0.483157i	$-0.839490\pi$
$619$	− 24.0416i	− 0.966315i	0.875534	−	0.483157i	$-0.160510\pi$
$620$	0	0
$621$	− 33.9411i	− 1.36201i
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	8.48528i	0.338869i
$628$	0	0
$629$	8.48528i	0.338330i
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	0	0
$633$	18.0000	0.715436
$634$	0	0
$635$	− 11.3137i	− 0.448971i
$636$	0	0
$637$	− 4.24264i	− 0.168100i
$638$	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
$643$	− 29.6985i	− 1.17119i	−0.810602	−	0.585597i	$-0.800860\pi$
$643$	− 29.6985i	− 1.17119i	0.810602	−	0.585597i	$-0.199140\pi$
$644$	0	0
$645$	14.1421i	0.556846i
$646$	0	0
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	22.6274i	0.886838i
$652$	0	0
$653$	− 26.8701i	− 1.05151i	−0.850637	−	0.525753i	$-0.823784\pi$
$653$	− 26.8701i	− 1.05151i	0.850637	−	0.525753i	$-0.176216\pi$
$654$	0	0
$655$	22.0000	0.859611
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	24.0416i	0.936529i	0.883588	+	0.468264i	$0.155121\pi$
$659$	24.0416i	0.936529i	−0.883588	+	0.468264i	$0.844879\pi$
$660$	0	0
$661$	12.7279i	0.495059i	0.968880	+	0.247529i	$0.0796187\pi$
$661$	12.7279i	0.495059i	−0.968880	+	0.247529i	$0.920381\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−12.0000	−0.465340
$666$	0	0
$667$	25.4558i	0.985654i
$668$	0	0
$669$	− 33.9411i	− 1.31224i
$670$	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
$675$	− 16.9706i	− 0.653197i
$676$	0	0
$677$	4.24264i	0.163058i	0.996671	+	0.0815290i	$0.0259803\pi$
$677$	4.24264i	0.163058i	−0.996671	+	0.0815290i	$0.974020\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	30.0000	1.14960
$682$	0	0
$683$	− 7.07107i	− 0.270567i	−0.990807	−	0.135283i	$-0.956805\pi$
$683$	− 7.07107i	− 0.270567i	0.990807	−	0.135283i	$-0.0431945\pi$
$684$	0	0
$685$	− 11.3137i	− 0.432275i
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	12.7279i	0.484193i	0.970252	+	0.242096i	$0.0778351\pi$
$691$	12.7279i	0.484193i	−0.970252	+	0.242096i	$0.922165\pi$
$692$	0	0
$693$	− 2.82843i	− 0.107443i
$694$	0	0
$695$	6.00000	0.227593
$696$	0	0
$697$	0	0
$698$	0	0
$699$	5.65685i	0.213962i
$700$	0	0
$701$	43.8406i	1.65584i	0.560848	+	0.827919i	$0.310475\pi$
$701$	43.8406i	1.65584i	−0.560848	+	0.827919i	$0.689525\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	31.1127i	1.17011i
$708$	0	0
$709$	38.1838i	1.43402i	0.697062	+	0.717011i	$0.254490\pi$
$709$	38.1838i	1.43402i	−0.697062	+	0.717011i	$0.745510\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	− 2.82843i	− 0.105777i
$716$	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
$723$	− 25.4558i	− 0.946713i
$724$	0	0
$725$	12.7279i	0.472703i
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	− 14.1421i	− 0.523066i
$732$	0	0
$733$	− 29.6985i	− 1.09694i	−0.836171	−	0.548469i	$-0.815211\pi$
$733$	− 29.6985i	− 1.09694i	0.836171	−	0.548469i	$-0.184789\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	32.5269i	1.19652i	0.801301	+	0.598261i	$0.204141\pi$
$739$	32.5269i	1.19652i	−0.801301	+	0.598261i	$0.795859\pi$
$740$	0	0
$741$	− 8.48528i	− 0.311715i
$742$	0	0
$743$	−46.0000	−1.68758	−0.843788	−	0.536676i	$-0.819680\pi$
$743$	−46.0000	−1.68758	−0.843788	+	0.536676i	$0.819680\pi$
$744$	0	0
$745$	−14.0000	−0.512920
$746$	0	0
$747$	1.41421i	0.0517434i
$748$	0	0
$749$	19.7990i	0.723439i
$750$	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
$755$	14.1421i	0.514685i
$756$	0	0
$757$	− 32.5269i	− 1.18221i	−0.806594	−	0.591105i	$-0.798692\pi$
$757$	− 32.5269i	− 1.18221i	0.806594	−	0.591105i	$-0.201308\pi$
$758$	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
$763$	− 8.48528i	− 0.307188i
$764$	0	0
$765$	− 2.82843i	− 0.102262i
$766$	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
$771$	31.1127i	1.12050i
$772$	0	0
$773$	− 7.07107i	− 0.254329i	−0.991882	−	0.127164i	$-0.959412\pi$
$773$	− 7.07107i	− 0.254329i	0.991882	−	0.127164i	$-0.0405876\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 14.1421i	− 0.506045i
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	− 21.2132i	− 0.756169i	−0.925771	−	0.378085i	$-0.876583\pi$
$787$	− 21.2132i	− 0.756169i	0.925771	−	0.378085i	$-0.123417\pi$
$788$	0	0
$789$	8.48528i	0.302084i
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	− 14.1421i	− 0.501570i
$796$	0	0
$797$	− 35.3553i	− 1.25235i	−0.779682	−	0.626175i	$-0.784619\pi$
$797$	− 35.3553i	− 1.25235i	0.779682	−	0.626175i	$-0.215381\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	5.65685i	0.199626i
$804$	0	0
$805$	− 16.9706i	− 0.598134i
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	55.1543i	1.93673i	0.249537	+	0.968365i	$0.419722\pi$
$811$	55.1543i	1.93673i	−0.249537	+	0.968365i	$0.580278\pi$
$812$	0	0
$813$	− 11.3137i	− 0.396789i
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	2.82843i	0.0988332i
$820$	0	0
$821$	− 15.5563i	− 0.542920i	−0.962450	−	0.271460i	$-0.912493\pi$
$821$	− 15.5563i	− 0.542920i	0.962450	−	0.271460i	$-0.0875065\pi$
$822$	0	0
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	46.6690i	1.62284i	0.584462	+	0.811421i	$0.301305\pi$
$827$	46.6690i	1.62284i	−0.584462	+	0.811421i	$0.698695\pi$
$828$	0	0
$829$	32.5269i	1.12971i	0.825191	+	0.564853i	$0.191067\pi$
$829$	32.5269i	1.12971i	−0.825191	+	0.564853i	$0.808933\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	− 2.82843i	− 0.0978818i
$836$	0	0
$837$	45.2548i	1.56424i
$838$	0	0
$839$	−14.0000	−0.483334	−0.241667	−	0.970359i	$-0.577694\pi$
$839$	−14.0000	−0.483334	−0.241667	+	0.970359i	$0.577694\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	− 28.2843i	− 0.974162i
$844$	0	0
$845$	− 15.5563i	− 0.535155i
$846$	0	0
$847$	18.0000	0.618487
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	25.4558i	0.872615i
$852$	0	0
$853$	7.07107i	0.242109i	0.992646	+	0.121054i	$0.0386275\pi$
$853$	7.07107i	0.242109i	−0.992646	+	0.121054i	$0.961372\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	−8.00000	−0.273275	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000	−0.273275	−0.136637	+	0.990621i	$0.543630\pi$
$858$	0	0
$859$	− 4.24264i	− 0.144757i	−0.997377	−	0.0723785i	$-0.976941\pi$
$859$	− 4.24264i	− 0.144757i	0.997377	−	0.0723785i	$-0.0230590\pi$
$860$	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
$867$	18.3848i	0.624380i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	− 22.6274i	− 0.764946i
$876$	0	0
$877$	7.07107i	0.238773i	0.992848	+	0.119386i	$0.0380928\pi$
$877$	7.07107i	0.238773i	−0.992848	+	0.119386i	$0.961907\pi$
$878$	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
$883$	29.6985i	0.999434i	0.866189	+	0.499717i	$0.166563\pi$
$883$	29.6985i	0.999434i	−0.866189	+	0.499717i	$0.833437\pi$
$884$	0	0
$885$	8.48528i	0.285230i
$886$	0	0
$887$	−2.00000	−0.0671534	−0.0335767	−	0.999436i	$-0.510690\pi$
$887$	−2.00000	−0.0671534	−0.0335767	+	0.999436i	$0.510690\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	7.07107i	0.236890i
$892$	0	0
$893$	33.9411i	1.13580i
$894$	0	0
$895$	34.0000	1.13649
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	− 33.9411i	− 1.13200i
$900$	0	0
$901$	14.1421i	0.471143i
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	38.1838i	1.26787i	0.773386	+	0.633936i	$0.218562\pi$
$907$	38.1838i	1.26787i	−0.773386	+	0.633936i	$0.781438\pi$
$908$	0	0
$909$	15.5563i	0.515972i
$910$	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
$915$	25.4558i	0.841544i
$916$	0	0
$917$	31.1127i	1.02743i
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	0	0
$923$	14.1421i	0.465494i
$924$	0	0
$925$	12.7279i	0.418491i
$926$	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
$931$	12.7279i	0.417141i
$932$	0	0
$933$	− 42.4264i	− 1.38898i
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	0	0
$939$	22.6274i	0.738418i
$940$	0	0
$941$	41.0122i	1.33696i	0.743730	+	0.668480i	$0.233055\pi$
$941$	41.0122i	1.33696i	−0.743730	+	0.668480i	$0.766945\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	7.07107i	0.229779i	0.993378	+	0.114889i	$0.0366514\pi$
$947$	7.07107i	0.229779i	−0.993378	+	0.114889i	$0.963349\pi$
$948$	0	0
$949$	− 5.65685i	− 0.183629i
$950$	0	0
$951$	−10.0000	−0.324272
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	11.3137i	0.366103i
$956$	0	0
$957$	− 8.48528i	− 0.274290i
$958$	0	0
$959$	16.0000	0.516667
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	9.89949i	0.319007i
$964$	0	0
$965$	− 19.7990i	− 0.637352i
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	26.8701i	0.862301i	0.902280	+	0.431151i	$0.141892\pi$
$971$	26.8701i	0.862301i	−0.902280	+	0.431151i	$0.858108\pi$
$972$	0	0
$973$	8.48528i	0.272026i
$974$	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
$979$	5.65685i	0.180794i
$980$	0	0
$981$	− 4.24264i	− 0.135457i
$982$	0	0
$983$	−34.0000	−1.08443	−0.542216	−	0.840239i	$-0.682414\pi$
$983$	−34.0000	−1.08443	−0.542216	+	0.840239i	$0.682414\pi$
$984$	0	0
$985$	−34.0000	−1.08333
$986$	0	0
$987$	22.6274i	0.720239i
$988$	0	0
$989$	− 42.4264i	− 1.34908i
$990$	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
$995$	19.7990i	0.627670i
$996$	0	0
$997$	− 52.3259i	− 1.65718i	−0.559857	−	0.828589i	$-0.689144\pi$
$997$	− 52.3259i	− 1.65718i	0.559857	−	0.828589i	$-0.310856\pi$
$998$	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.b.e.513.1		2
4.3	odd	2		1024.2.b.b.513.2		2
8.3	odd	2		1024.2.b.b.513.1		2
8.5	even	2	inner	1024.2.b.e.513.2		2
16.3	odd	4		1024.2.a.e.1.1		2
16.5	even	4		1024.2.a.b.1.1		2
16.11	odd	4		1024.2.a.e.1.2		2
16.13	even	4		1024.2.a.b.1.2		2
32.3	odd	8		128.2.e.a.97.1		2
32.5	even	8		128.2.e.b.33.1		2
32.11	odd	8		64.2.e.a.17.1		2
32.13	even	8		16.2.e.a.5.1	✓	2
32.19	odd	8		64.2.e.a.49.1		2
32.21	even	8		16.2.e.a.13.1	yes	2
32.27	odd	8		128.2.e.a.33.1		2
32.29	even	8		128.2.e.b.97.1		2
48.5	odd	4		9216.2.a.d.1.1		2
48.11	even	4		9216.2.a.s.1.1		2
48.29	odd	4		9216.2.a.d.1.2		2
48.35	even	4		9216.2.a.s.1.2		2
96.5	odd	8		1152.2.k.b.289.1		2
96.11	even	8		576.2.k.a.145.1		2
96.29	odd	8		1152.2.k.b.865.1		2
96.35	even	8		1152.2.k.a.865.1		2
96.53	odd	8		144.2.k.a.109.1		2
96.59	even	8		1152.2.k.a.289.1		2
96.77	odd	8		144.2.k.a.37.1		2
96.83	even	8		576.2.k.a.433.1		2
160.13	odd	8		400.2.q.a.149.1		2
160.19	odd	8		1600.2.l.a.1201.1		2
160.43	even	8		1600.2.q.a.849.1		2
160.53	odd	8		400.2.q.b.349.1		2
160.77	odd	8		400.2.q.b.149.1		2
160.83	even	8		1600.2.q.b.49.1		2
160.107	even	8		1600.2.q.b.849.1		2
160.109	even	8		400.2.l.c.101.1		2
160.117	odd	8		400.2.q.a.349.1		2
160.139	odd	8		1600.2.l.a.401.1		2
160.147	even	8		1600.2.q.a.49.1		2
160.149	even	8		400.2.l.c.301.1		2
224.13	odd	8		784.2.m.b.197.1		2
224.45	odd	24		784.2.x.c.373.1		4
224.53	even	24		784.2.x.f.765.1		4
224.109	even	24		784.2.x.f.373.1		4
224.117	odd	24		784.2.x.c.557.1		4
224.149	even	24		784.2.x.f.557.1		4
224.173	odd	24		784.2.x.c.165.1		4
224.181	odd	8		784.2.m.b.589.1		2
224.205	even	24		784.2.x.f.165.1		4
224.213	odd	24		784.2.x.c.765.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.2.e.a.5.1	✓	2	32.13	even	8
16.2.e.a.13.1	yes	2	32.21	even	8
64.2.e.a.17.1		2	32.11	odd	8
64.2.e.a.49.1		2	32.19	odd	8
128.2.e.a.33.1		2	32.27	odd	8
128.2.e.a.97.1		2	32.3	odd	8
128.2.e.b.33.1		2	32.5	even	8
128.2.e.b.97.1		2	32.29	even	8
144.2.k.a.37.1		2	96.77	odd	8
144.2.k.a.109.1		2	96.53	odd	8
400.2.l.c.101.1		2	160.109	even	8
400.2.l.c.301.1		2	160.149	even	8
400.2.q.a.149.1		2	160.13	odd	8
400.2.q.a.349.1		2	160.117	odd	8
400.2.q.b.149.1		2	160.77	odd	8
400.2.q.b.349.1		2	160.53	odd	8
576.2.k.a.145.1		2	96.11	even	8
576.2.k.a.433.1		2	96.83	even	8
784.2.m.b.197.1		2	224.13	odd	8
784.2.m.b.589.1		2	224.181	odd	8
784.2.x.c.165.1		4	224.173	odd	24
784.2.x.c.373.1		4	224.45	odd	24
784.2.x.c.557.1		4	224.117	odd	24
784.2.x.c.765.1		4	224.213	odd	24
784.2.x.f.165.1		4	224.205	even	24
784.2.x.f.373.1		4	224.109	even	24
784.2.x.f.557.1		4	224.149	even	24
784.2.x.f.765.1		4	224.53	even	24
1024.2.a.b.1.1		2	16.5	even	4
1024.2.a.b.1.2		2	16.13	even	4
1024.2.a.e.1.1		2	16.3	odd	4
1024.2.a.e.1.2		2	16.11	odd	4
1024.2.b.b.513.1		2	8.3	odd	2
1024.2.b.b.513.2		2	4.3	odd	2
1024.2.b.e.513.1		2	1.1	even	1	trivial
1024.2.b.e.513.2		2	8.5	even	2	inner
1152.2.k.a.289.1		2	96.59	even	8
1152.2.k.a.865.1		2	96.35	even	8
1152.2.k.b.289.1		2	96.5	odd	8
1152.2.k.b.865.1		2	96.29	odd	8
1600.2.l.a.401.1		2	160.139	odd	8
1600.2.l.a.1201.1		2	160.19	odd	8
1600.2.q.a.49.1		2	160.147	even	8
1600.2.q.a.849.1		2	160.43	even	8
1600.2.q.b.49.1		2	160.83	even	8
1600.2.q.b.849.1		2	160.107	even	8
9216.2.a.d.1.1		2	48.5	odd	4
9216.2.a.d.1.2		2	48.29	odd	4
9216.2.a.s.1.1		2	48.11	even	4
9216.2.a.s.1.2		2	48.35	even	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)

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