LMFDB - Embedded newform 512.2.b.e.257.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [512,2,Mod(257,512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(512, 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("512.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 512 = 2^{9} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	512.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:	$4.08834058349$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			257.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	512.257
Dual form			512.2.b.e.257.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{3} -2.00000i q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.24264i q^{11} +6.00000i q^{13} -2.82843 q^{15} -4.24264i q^{19} -4.00000i q^{21} -8.48528 q^{23} +1.00000 q^{25} -5.65685i q^{27} +2.00000i q^{29} +5.65685 q^{31} -6.00000 q^{33} -5.65685i q^{35} -6.00000i q^{37} +8.48528 q^{39} -6.00000 q^{41} +4.24264i q^{43} -2.00000i q^{45} +1.00000 q^{49} +2.00000i q^{53} -8.48528 q^{55} -6.00000 q^{57} -1.41421i q^{59} +6.00000i q^{61} +2.82843 q^{63} +12.0000 q^{65} +12.7279i q^{67} +12.0000i q^{69} +8.48528 q^{71} +12.0000 q^{73} -1.41421i q^{75} -12.0000i q^{77} -5.65685 q^{79} -5.00000 q^{81} +4.24264i q^{83} +2.82843 q^{87} +12.0000 q^{89} +16.9706i q^{91} -8.00000i q^{93} -8.48528 q^{95} -8.00000 q^{97} -4.24264i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} + 4 q^{25} - 24 q^{33} - 24 q^{41} + 4 q^{49} - 24 q^{57} + 48 q^{65} + 48 q^{73} - 20 q^{81} + 48 q^{89} - 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^9 + 4 * q^25 - 24 * q^33 - 24 * q^41 + 4 * q^49 - 24 * q^57 + 48 * q^65 + 48 * q^73 - 20 * q^81 + 48 * q^89 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/512\mathbb{Z}\right)^\times$.

$n$	$5$	$511$
$\chi(n)$	$-1$	$1$

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
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 4.24264i	− 1.27920i	−0.768706	−	0.639602i	$-0.779099\pi$
$11$	− 4.24264i	− 1.27920i	0.768706	−	0.639602i	$-0.220901\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−2.82843	−0.730297
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	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
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 5.65685i	− 1.08866i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	− 5.65685i	− 0.956183i
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	8.48528	1.35873
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.24264i	0.646997i	0.946229	+	0.323498i	$0.104859\pi$
$43$	4.24264i	0.646997i	−0.946229	+	0.323498i	$0.895141\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	−8.48528	−1.14416
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	− 1.41421i	− 0.184115i	−0.995754	−	0.0920575i	$-0.970656\pi$
$59$	− 1.41421i	− 0.184115i	0.995754	−	0.0920575i	$-0.0293443\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	2.82843	0.356348
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	12.7279i	1.55496i	0.628906	+	0.777482i	$0.283503\pi$
$67$	12.7279i	1.55496i	−0.628906	+	0.777482i	$0.716497\pi$
$68$	0	0
$69$	12.0000i	1.44463i
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	0	0
$75$	− 1.41421i	− 0.163299i
$76$	0	0
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	4.24264i	0.465690i	0.972514	+	0.232845i	$0.0748035\pi$
$83$	4.24264i	0.465690i	−0.972514	+	0.232845i	$0.925196\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.82843	0.303239
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	16.9706i	1.77900i
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	−8.48528	−0.870572
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	− 4.24264i	− 0.426401i
$100$	0	0
$101$	10.0000i	0.995037i	0.867453	+	0.497519i	$0.165755\pi$
$101$	10.0000i	0.995037i	−0.867453	+	0.497519i	$0.834245\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	7.07107i	0.683586i	0.939775	+	0.341793i	$0.111034\pi$
$107$	7.07107i	0.683586i	−0.939775	+	0.341793i	$0.888966\pi$
$108$	0	0
$109$	− 6.00000i	− 0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	− 6.00000i	− 0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$	0	0
$111$	−8.48528	−0.805387
$112$	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
$115$	16.9706i	1.58251i
$116$	0	0
$117$	6.00000i	0.554700i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	8.48528i	0.765092i
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	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
$129$	6.00000	0.528271
$130$	0	0
$131$	1.41421i	0.123560i	0.998090	+	0.0617802i	$0.0196778\pi$
$131$	1.41421i	0.123560i	−0.998090	+	0.0617802i	$0.980322\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	−11.3137	−0.973729
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	25.4558	2.12872
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	− 1.41421i	− 0.116642i
$148$	0	0
$149$	22.0000i	1.80231i	0.433497	+	0.901155i	$0.357280\pi$
$149$	22.0000i	1.80231i	−0.433497	+	0.901155i	$0.642720\pi$
$150$	0	0
$151$	14.1421	1.15087	0.575435	−	0.817847i	$-0.304833\pi$
$151$	14.1421	1.15087	0.575435	+	0.817847i	$0.304833\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 11.3137i	− 0.908739i
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	0	0
$159$	2.82843	0.224309
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	21.2132i	1.66155i	0.556611	+	0.830773i	$0.312101\pi$
$163$	21.2132i	1.66155i	−0.556611	+	0.830773i	$0.687899\pi$
$164$	0	0
$165$	12.0000i	0.934199i
$166$	0	0
$167$	−8.48528	−0.656611	−0.328305	−	0.944572i	$-0.606478\pi$
$167$	−8.48528	−0.656611	−0.328305	+	0.944572i	$0.606478\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	− 4.24264i	− 0.324443i
$172$	0	0
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	−2.00000	−0.150329
$178$	0	0
$179$	− 18.3848i	− 1.37414i	−0.726590	−	0.687071i	$-0.758896\pi$
$179$	− 18.3848i	− 1.37414i	0.726590	−	0.687071i	$-0.241104\pi$
$180$	0	0
$181$	− 18.0000i	− 1.33793i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	− 18.0000i	− 1.33793i	0.743294	−	0.668965i	$-0.233262\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 16.0000i	− 1.16383i
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	− 16.9706i	− 1.21529i
$196$	0	0
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	−19.7990	−1.40351	−0.701757	−	0.712417i	$-0.747601\pi$
$199$	−19.7990	−1.40351	−0.701757	+	0.712417i	$0.747601\pi$
$200$	0	0
$201$	18.0000	1.26962
$202$	0	0
$203$	5.65685i	0.397033i
$204$	0	0
$205$	12.0000i	0.838116i
$206$	0	0
$207$	−8.48528	−0.589768
$208$	0	0
$209$	−18.0000	−1.24509
$210$	0	0
$211$	4.24264i	0.292075i	0.989279	+	0.146038i	$0.0466521\pi$
$211$	4.24264i	0.292075i	−0.989279	+	0.146038i	$0.953348\pi$
$212$	0	0
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	8.48528	0.578691
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	− 16.9706i	− 1.14676i
$220$	0	0
$221$	0	0
$222$	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
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 12.7279i	− 0.844782i	−0.906414	−	0.422391i	$-0.861191\pi$
$227$	− 12.7279i	− 0.844782i	0.906414	−	0.422391i	$-0.138809\pi$
$228$	0	0
$229$	− 18.0000i	− 1.18947i	−0.803921	−	0.594737i	$-0.797256\pi$
$229$	− 18.0000i	− 1.18947i	0.803921	−	0.594737i	$-0.202744\pi$
$230$	0	0
$231$	−16.9706	−1.11658
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	− 9.89949i	− 0.635053i
$244$	0	0
$245$	− 2.00000i	− 0.127775i
$246$	0	0
$247$	25.4558	1.61972
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	− 12.7279i	− 0.803379i	−0.915776	−	0.401690i	$-0.868423\pi$
$251$	− 12.7279i	− 0.803379i	0.915776	−	0.401690i	$-0.131577\pi$
$252$	0	0
$253$	36.0000i	2.26330i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	− 16.9706i	− 1.05450i
$260$	0	0
$261$	2.00000i	0.123797i
$262$	0	0
$263$	8.48528	0.523225	0.261612	−	0.965173i	$-0.415746\pi$
$263$	8.48528	0.523225	0.261612	+	0.965173i	$0.415746\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	− 16.9706i	− 1.03858i
$268$	0	0
$269$	− 14.0000i	− 0.853595i	−0.904347	−	0.426798i	$-0.859642\pi$
$269$	− 14.0000i	− 0.853595i	0.904347	−	0.426798i	$-0.140358\pi$
$270$	0	0
$271$	−22.6274	−1.37452	−0.687259	−	0.726413i	$-0.741186\pi$
$271$	−22.6274	−1.37452	−0.687259	+	0.726413i	$0.741186\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	− 4.24264i	− 0.255841i
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	0	0
$279$	5.65685	0.338667
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	12.7279i	0.756596i	0.925684	+	0.378298i	$0.123491\pi$
$283$	12.7279i	0.756596i	−0.925684	+	0.378298i	$0.876509\pi$
$284$	0	0
$285$	12.0000i	0.710819i
$286$	0	0
$287$	−16.9706	−1.00174
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	11.3137i	0.663221i
$292$	0	0
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	−2.82843	−0.164677
$296$	0	0
$297$	−24.0000	−1.39262
$298$	0	0
$299$	− 50.9117i	− 2.94430i
$300$	0	0
$301$	12.0000i	0.691669i
$302$	0	0
$303$	14.1421	0.812444
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	− 4.24264i	− 0.242140i	−0.992644	−	0.121070i	$-0.961367\pi$
$307$	− 4.24264i	− 0.242140i	0.992644	−	0.121070i	$-0.0386326\pi$
$308$	0	0
$309$	4.00000i	0.227552i
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	− 5.65685i	− 0.318728i
$316$	0	0
$317$	10.0000i	0.561656i	0.959758	+	0.280828i	$0.0906090\pi$
$317$	10.0000i	0.561656i	−0.959758	+	0.280828i	$0.909391\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.00000i	0.332820i
$326$	0	0
$327$	−8.48528	−0.469237
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.6985i	1.63238i	0.577786	+	0.816188i	$0.303917\pi$
$331$	29.6985i	1.63238i	−0.577786	+	0.816188i	$0.696083\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	25.4558	1.39080
$336$	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
$341$	− 24.0000i	− 1.29967i
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	24.0000	1.29212
$346$	0	0
$347$	21.2132i	1.13878i	0.822066	+	0.569392i	$0.192821\pi$
$347$	21.2132i	1.13878i	−0.822066	+	0.569392i	$0.807179\pi$
$348$	0	0
$349$	− 6.00000i	− 0.321173i	−0.987022	−	0.160586i	$-0.948662\pi$
$349$	− 6.00000i	− 0.321173i	0.987022	−	0.160586i	$-0.0513385\pi$
$350$	0	0
$351$	33.9411	1.81164
$352$	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
$355$	− 16.9706i	− 0.900704i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.89949i	0.519589i
$364$	0	0
$365$	− 24.0000i	− 1.25622i
$366$	0	0
$367$	−11.3137	−0.590571	−0.295285	−	0.955409i	$-0.595415\pi$
$367$	−11.3137	−0.590571	−0.295285	+	0.955409i	$0.595415\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	5.65685i	0.293689i
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	0	0
$375$	−16.9706	−0.876356
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	− 12.7279i	− 0.653789i	−0.945061	−	0.326895i	$-0.893998\pi$
$379$	− 12.7279i	− 0.653789i	0.945061	−	0.326895i	$-0.106002\pi$
$380$	0	0
$381$	− 8.00000i	− 0.409852i
$382$	0	0
$383$	−33.9411	−1.73431	−0.867155	−	0.498038i	$-0.834054\pi$
$383$	−33.9411	−1.73431	−0.867155	+	0.498038i	$0.834054\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	4.24264i	0.215666i
$388$	0	0
$389$	26.0000i	1.31825i	0.752032	+	0.659126i	$0.229074\pi$
$389$	26.0000i	1.31825i	−0.752032	+	0.659126i	$0.770926\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.00000	0.100887
$394$	0	0
$395$	11.3137i	0.569254i
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	−16.9706	−0.849591
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	33.9411i	1.69073i
$404$	0	0
$405$	10.0000i	0.496904i
$406$	0	0
$407$	−25.4558	−1.26180
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	− 8.48528i	− 0.418548i
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	8.48528	0.416526
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	12.7279i	0.621800i	0.950443	+	0.310900i	$0.100630\pi$
$419$	12.7279i	0.621800i	−0.950443	+	0.310900i	$0.899370\pi$
$420$	0	0
$421$	6.00000i	0.292422i	0.989253	+	0.146211i	$0.0467079\pi$
$421$	6.00000i	0.292422i	−0.989253	+	0.146211i	$0.953292\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.9706i	0.821263i
$428$	0	0
$429$	− 36.0000i	− 1.73810i
$430$	0	0
$431$	−16.9706	−0.817443	−0.408722	−	0.912659i	$-0.634025\pi$
$431$	−16.9706	−0.817443	−0.408722	+	0.912659i	$0.634025\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	− 5.65685i	− 0.271225i
$436$	0	0
$437$	36.0000i	1.72211i
$438$	0	0
$439$	−2.82843	−0.134993	−0.0674967	−	0.997719i	$-0.521501\pi$
$439$	−2.82843	−0.134993	−0.0674967	+	0.997719i	$0.521501\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	12.7279i	0.604722i	0.953194	+	0.302361i	$0.0977748\pi$
$443$	12.7279i	0.604722i	−0.953194	+	0.302361i	$0.902225\pi$
$444$	0	0
$445$	− 24.0000i	− 1.13771i
$446$	0	0
$447$	31.1127	1.47158
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	25.4558i	1.19867i
$452$	0	0
$453$	− 20.0000i	− 0.939682i
$454$	0	0
$455$	33.9411	1.59118
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.0000i	1.21094i	0.795868	+	0.605470i	$0.207015\pi$
$461$	26.0000i	1.21094i	−0.795868	+	0.605470i	$0.792985\pi$
$462$	0	0
$463$	5.65685	0.262896	0.131448	−	0.991323i	$-0.458037\pi$
$463$	5.65685	0.262896	0.131448	+	0.991323i	$0.458037\pi$
$464$	0	0
$465$	−16.0000	−0.741982
$466$	0	0
$467$	− 4.24264i	− 0.196326i	−0.995170	−	0.0981630i	$-0.968703\pi$
$467$	− 4.24264i	− 0.196326i	0.995170	−	0.0981630i	$-0.0312967\pi$
$468$	0	0
$469$	36.0000i	1.66233i
$470$	0	0
$471$	8.48528	0.390981
$472$	0	0
$473$	18.0000	0.827641
$474$	0	0
$475$	− 4.24264i	− 0.194666i
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	−16.9706	−0.775405	−0.387702	−	0.921785i	$-0.626731\pi$
$479$	−16.9706	−0.775405	−0.387702	+	0.921785i	$0.626731\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	33.9411i	1.54437i
$484$	0	0
$485$	16.0000i	0.726523i
$486$	0	0
$487$	36.7696	1.66619	0.833094	−	0.553132i	$-0.186567\pi$
$487$	36.7696	1.66619	0.833094	+	0.553132i	$0.186567\pi$
$488$	0	0
$489$	30.0000	1.35665
$490$	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
$493$	0	0
$494$	0	0
$495$	−8.48528	−0.381385
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	4.24264i	0.189927i	0.995481	+	0.0949633i	$0.0302734\pi$
$499$	4.24264i	0.189927i	−0.995481	+	0.0949633i	$0.969727\pi$
$500$	0	0
$501$	12.0000i	0.536120i
$502$	0	0
$503$	−42.4264	−1.89170	−0.945850	−	0.324604i	$-0.894769\pi$
$503$	−42.4264	−1.89170	−0.945850	+	0.324604i	$0.894769\pi$
$504$	0	0
$505$	20.0000	0.889988
$506$	0	0
$507$	32.5269i	1.44457i
$508$	0	0
$509$	− 34.0000i	− 1.50702i	−0.657434	−	0.753512i	$-0.728358\pi$
$509$	− 34.0000i	− 1.50702i	0.657434	−	0.753512i	$-0.271642\pi$
$510$	0	0
$511$	33.9411	1.50147
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	5.65685i	0.249271i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	19.7990	0.869079
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	− 4.24264i	− 0.185518i	−0.995689	−	0.0927589i	$-0.970431\pi$
$523$	− 4.24264i	− 0.185518i	0.995689	−	0.0927589i	$-0.0295686\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	− 1.41421i	− 0.0613716i
$532$	0	0
$533$	− 36.0000i	− 1.55933i
$534$	0	0
$535$	14.1421	0.611418
$536$	0	0
$537$	−26.0000	−1.12198
$538$	0	0
$539$	− 4.24264i	− 0.182743i
$540$	0	0
$541$	− 42.0000i	− 1.80572i	−0.429934	−	0.902861i	$-0.641463\pi$
$541$	− 42.0000i	− 1.80572i	0.429934	−	0.902861i	$-0.358537\pi$
$542$	0	0
$543$	−25.4558	−1.09241
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	− 21.2132i	− 0.907011i	−0.891253	−	0.453506i	$-0.850173\pi$
$547$	− 21.2132i	− 0.907011i	0.891253	−	0.453506i	$-0.149827\pi$
$548$	0	0
$549$	6.00000i	0.256074i
$550$	0	0
$551$	8.48528	0.361485
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	16.9706i	0.720360i
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	−25.4558	−1.07667
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.6985i	1.25164i	0.779967	+	0.625821i	$0.215236\pi$
$563$	29.6985i	1.25164i	−0.779967	+	0.625821i	$0.784764\pi$
$564$	0	0
$565$	− 12.0000i	− 0.504844i
$566$	0	0
$567$	−14.1421	−0.593914
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	12.7279i	0.532647i	0.963884	+	0.266323i	$0.0858089\pi$
$571$	12.7279i	0.532647i	−0.963884	+	0.266323i	$0.914191\pi$
$572$	0	0
$573$	24.0000i	1.00261i
$574$	0	0
$575$	−8.48528	−0.353861
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	− 33.9411i	− 1.41055i
$580$	0	0
$581$	12.0000i	0.497844i
$582$	0	0
$583$	8.48528	0.351424
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	− 15.5563i	− 0.642079i	−0.947066	−	0.321040i	$-0.895968\pi$
$587$	− 15.5563i	− 0.642079i	0.947066	−	0.321040i	$-0.104032\pi$
$588$	0	0
$589$	− 24.0000i	− 0.988903i
$590$	0	0
$591$	−14.1421	−0.581730
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.0000i	1.14596i
$598$	0	0
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	0	0
$601$	12.0000	0.489490	0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000	0.489490	0.244745	+	0.969587i	$0.421296\pi$
$602$	0	0
$603$	12.7279i	0.518321i
$604$	0	0
$605$	14.0000i	0.569181i
$606$	0	0
$607$	22.6274	0.918419	0.459209	−	0.888328i	$-0.348133\pi$
$607$	22.6274	0.918419	0.459209	+	0.888328i	$0.348133\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	0	0
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	16.9706	0.684319
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	− 38.1838i	− 1.53474i	−0.641207	−	0.767368i	$-0.721566\pi$
$619$	− 38.1838i	− 1.53474i	0.641207	−	0.767368i	$-0.278434\pi$
$620$	0	0
$621$	48.0000i	1.92617i
$622$	0	0
$623$	33.9411	1.35982
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	25.4558i	1.01661i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.1127	−1.23858	−0.619288	−	0.785164i	$-0.712579\pi$
$631$	−31.1127	−1.23858	−0.619288	+	0.785164i	$0.712579\pi$
$632$	0	0
$633$	6.00000	0.238479
$634$	0	0
$635$	− 11.3137i	− 0.448971i
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	8.48528	0.335673
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	12.7279i	0.501940i	0.967995	+	0.250970i	$0.0807496\pi$
$643$	12.7279i	0.501940i	−0.967995	+	0.250970i	$0.919250\pi$
$644$	0	0
$645$	− 12.0000i	− 0.472500i
$646$	0	0
$647$	−25.4558	−1.00077	−0.500386	−	0.865802i	$-0.666809\pi$
$647$	−25.4558	−1.00077	−0.500386	+	0.865802i	$0.666809\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	− 22.6274i	− 0.886838i
$652$	0	0
$653$	2.00000i	0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$	2.00000i	0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$	0	0
$655$	2.82843	0.110516
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	− 41.0122i	− 1.59761i	−0.601591	−	0.798804i	$-0.705466\pi$
$659$	− 41.0122i	− 1.59761i	0.601591	−	0.798804i	$-0.294534\pi$
$660$	0	0
$661$	− 18.0000i	− 0.700119i	−0.936727	−	0.350059i	$-0.886161\pi$
$661$	− 18.0000i	− 0.700119i	0.936727	−	0.350059i	$-0.113839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	− 16.9706i	− 0.657103i
$668$	0	0
$669$	8.00000i	0.309298i
$670$	0	0
$671$	25.4558	0.982712
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	− 5.65685i	− 0.217732i
$676$	0	0
$677$	− 38.0000i	− 1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$	− 38.0000i	− 1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$	0	0
$679$	−22.6274	−0.868361
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	38.1838i	1.46106i	0.682880	+	0.730531i	$0.260727\pi$
$683$	38.1838i	1.46106i	−0.682880	+	0.730531i	$0.739273\pi$
$684$	0	0
$685$	− 12.0000i	− 0.458496i
$686$	0	0
$687$	−25.4558	−0.971201
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	− 29.6985i	− 1.12978i	−0.825165	−	0.564892i	$-0.808918\pi$
$691$	− 29.6985i	− 1.12978i	0.825165	−	0.564892i	$-0.191082\pi$
$692$	0	0
$693$	− 12.0000i	− 0.455842i
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 16.9706i	− 0.641886i
$700$	0	0
$701$	46.0000i	1.73740i	0.495342	+	0.868698i	$0.335043\pi$
$701$	46.0000i	1.73740i	−0.495342	+	0.868698i	$0.664957\pi$
$702$	0	0
$703$	−25.4558	−0.960085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.2843i	1.06374i
$708$	0	0
$709$	18.0000i	0.676004i	0.941145	+	0.338002i	$0.109751\pi$
$709$	18.0000i	0.676004i	−0.941145	+	0.338002i	$0.890249\pi$
$710$	0	0
$711$	−5.65685	−0.212149
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	− 50.9117i	− 1.90399i
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	16.9706	0.632895	0.316448	−	0.948610i	$-0.397510\pi$
$719$	16.9706	0.632895	0.316448	+	0.948610i	$0.397510\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−2.82843	−0.104901	−0.0524503	−	0.998624i	$-0.516703\pi$
$727$	−2.82843	−0.104901	−0.0524503	+	0.998624i	$0.516703\pi$
$728$	0	0
$729$	−29.0000	−1.07407
$730$	0	0
$731$	0	0
$732$	0	0
$733$	42.0000i	1.55131i	0.631160	+	0.775653i	$0.282579\pi$
$733$	42.0000i	1.55131i	−0.631160	+	0.775653i	$0.717421\pi$
$734$	0	0
$735$	−2.82843	−0.104328
$736$	0	0
$737$	54.0000	1.98912
$738$	0	0
$739$	− 46.6690i	− 1.71675i	−0.513024	−	0.858374i	$-0.671475\pi$
$739$	− 46.6690i	− 1.71675i	0.513024	−	0.858374i	$-0.328525\pi$
$740$	0	0
$741$	− 36.0000i	− 1.32249i
$742$	0	0
$743$	25.4558	0.933884	0.466942	−	0.884288i	$-0.345356\pi$
$743$	25.4558	0.933884	0.466942	+	0.884288i	$0.345356\pi$
$744$	0	0
$745$	44.0000	1.61204
$746$	0	0
$747$	4.24264i	0.155230i
$748$	0	0
$749$	20.0000i	0.730784i
$750$	0	0
$751$	−5.65685	−0.206422	−0.103211	−	0.994660i	$-0.532912\pi$
$751$	−5.65685	−0.206422	−0.103211	+	0.994660i	$0.532912\pi$
$752$	0	0
$753$	−18.0000	−0.655956
$754$	0	0
$755$	− 28.2843i	− 1.02937i
$756$	0	0
$757$	− 42.0000i	− 1.52652i	−0.646094	−	0.763258i	$-0.723599\pi$
$757$	− 42.0000i	− 1.52652i	0.646094	−	0.763258i	$-0.276401\pi$
$758$	0	0
$759$	50.9117	1.84798
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	− 16.9706i	− 0.614376i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.48528	0.306386
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	− 8.48528i	− 0.305590i
$772$	0	0
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	25.4558i	0.912050i
$780$	0	0
$781$	− 36.0000i	− 1.28818i
$782$	0	0
$783$	11.3137	0.404319
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	− 29.6985i	− 1.05864i	−0.848423	−	0.529318i	$-0.822448\pi$
$787$	− 29.6985i	− 1.05864i	0.848423	−	0.529318i	$-0.177552\pi$
$788$	0	0
$789$	− 12.0000i	− 0.427211i
$790$	0	0
$791$	16.9706	0.603404
$792$	0	0
$793$	−36.0000	−1.27840
$794$	0	0
$795$	− 5.65685i	− 0.200628i
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	− 50.9117i	− 1.79663i
$804$	0	0
$805$	48.0000i	1.69178i
$806$	0	0
$807$	−19.7990	−0.696957
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	− 4.24264i	− 0.148979i	−0.997222	−	0.0744896i	$-0.976267\pi$
$811$	− 4.24264i	− 0.148979i	0.997222	−	0.0744896i	$-0.0237328\pi$
$812$	0	0
$813$	32.0000i	1.12229i
$814$	0	0
$815$	42.4264	1.48613
$816$	0	0
$817$	18.0000	0.629740
$818$	0	0
$819$	16.9706i	0.592999i
$820$	0	0
$821$	− 38.0000i	− 1.32621i	−0.748527	−	0.663105i	$-0.769238\pi$
$821$	− 38.0000i	− 1.32621i	0.748527	−	0.663105i	$-0.230762\pi$
$822$	0	0
$823$	19.7990	0.690149	0.345075	−	0.938575i	$-0.387854\pi$
$823$	19.7990	0.690149	0.345075	+	0.938575i	$0.387854\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	− 43.8406i	− 1.52449i	−0.647290	−	0.762244i	$-0.724098\pi$
$827$	− 43.8406i	− 1.52449i	0.647290	−	0.762244i	$-0.275902\pi$
$828$	0	0
$829$	− 18.0000i	− 0.625166i	−0.949890	−	0.312583i	$-0.898806\pi$
$829$	− 18.0000i	− 0.625166i	0.949890	−	0.312583i	$-0.101194\pi$
$830$	0	0
$831$	−8.48528	−0.294351
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.9706i	0.587291i
$836$	0	0
$837$	− 32.0000i	− 1.10608i
$838$	0	0
$839$	−8.48528	−0.292944	−0.146472	−	0.989215i	$-0.546792\pi$
$839$	−8.48528	−0.292944	−0.146472	+	0.989215i	$0.546792\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	16.9706i	0.584497i
$844$	0	0
$845$	46.0000i	1.58245i
$846$	0	0
$847$	−19.7990	−0.680301
$848$	0	0
$849$	18.0000	0.617758
$850$	0	0
$851$	50.9117i	1.74523i
$852$	0	0
$853$	18.0000i	0.616308i	0.951336	+	0.308154i	$0.0997113\pi$
$853$	18.0000i	0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$854$	0	0
$855$	−8.48528	−0.290191
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	− 46.6690i	− 1.59233i	−0.605081	−	0.796164i	$-0.706859\pi$
$859$	− 46.6690i	− 1.59233i	0.605081	−	0.796164i	$-0.293141\pi$
$860$	0	0
$861$	24.0000i	0.817918i
$862$	0	0
$863$	−50.9117	−1.73305	−0.866527	−	0.499130i	$-0.833653\pi$
$863$	−50.9117	−1.73305	−0.866527	+	0.499130i	$0.833653\pi$
$864$	0	0
$865$	28.0000	0.952029
$866$	0	0
$867$	24.0416i	0.816497i
$868$	0	0
$869$	24.0000i	0.814144i
$870$	0	0
$871$	−76.3675	−2.58762
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	− 33.9411i	− 1.14742i
$876$	0	0
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	0	0
$879$	−14.1421	−0.477002
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	− 38.1838i	− 1.28499i	−0.766292	−	0.642493i	$-0.777900\pi$
$883$	− 38.1838i	− 1.28499i	0.766292	−	0.642493i	$-0.222100\pi$
$884$	0	0
$885$	4.00000i	0.134459i
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	21.2132i	0.710669i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−36.7696	−1.22907
$896$	0	0
$897$	−72.0000	−2.40401
$898$	0	0
$899$	11.3137i	0.377333i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	16.9706	0.564745
$904$	0	0
$905$	−36.0000	−1.19668
$906$	0	0
$907$	4.24264i	0.140875i	0.997516	+	0.0704373i	$0.0224395\pi$
$907$	4.24264i	0.140875i	−0.997516	+	0.0704373i	$0.977561\pi$
$908$	0	0
$909$	10.0000i	0.331679i
$910$	0	0
$911$	−33.9411	−1.12452	−0.562260	−	0.826961i	$-0.690068\pi$
$911$	−33.9411	−1.12452	−0.562260	+	0.826961i	$0.690068\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	0	0
$915$	− 16.9706i	− 0.561029i
$916$	0	0
$917$	4.00000i	0.132092i
$918$	0	0
$919$	2.82843	0.0933012	0.0466506	−	0.998911i	$-0.485145\pi$
$919$	2.82843	0.0933012	0.0466506	+	0.998911i	$0.485145\pi$
$920$	0	0
$921$	−6.00000	−0.197707
$922$	0	0
$923$	50.9117i	1.67578i
$924$	0	0
$925$	− 6.00000i	− 0.197279i
$926$	0	0
$927$	−2.82843	−0.0928977
$928$	0	0
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	0	0
$931$	− 4.24264i	− 0.139047i
$932$	0	0
$933$	− 12.0000i	− 0.392862i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.0000	−0.653372	−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000	−0.653372	−0.326686	+	0.945133i	$0.605932\pi$
$938$	0	0
$939$	8.48528i	0.276907i
$940$	0	0
$941$	− 22.0000i	− 0.717180i	−0.933495	−	0.358590i	$-0.883258\pi$
$941$	− 22.0000i	− 0.717180i	0.933495	−	0.358590i	$-0.116742\pi$
$942$	0	0
$943$	50.9117	1.65791
$944$	0	0
$945$	−32.0000	−1.04096
$946$	0	0
$947$	41.0122i	1.33272i	0.745631	+	0.666359i	$0.232148\pi$
$947$	41.0122i	1.33272i	−0.745631	+	0.666359i	$0.767852\pi$
$948$	0	0
$949$	72.0000i	2.33722i
$950$	0	0
$951$	14.1421	0.458590
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	33.9411i	1.09831i
$956$	0	0
$957$	− 12.0000i	− 0.387905i
$958$	0	0
$959$	16.9706	0.548008
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	7.07107i	0.227862i
$964$	0	0
$965$	− 48.0000i	− 1.54517i
$966$	0	0
$967$	−31.1127	−1.00052	−0.500258	−	0.865876i	$-0.666762\pi$
$967$	−31.1127	−1.00052	−0.500258	+	0.865876i	$0.666762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 29.6985i	− 0.953070i	−0.879156	−	0.476535i	$-0.841893\pi$
$971$	− 29.6985i	− 0.953070i	0.879156	−	0.476535i	$-0.158107\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	0	0
$975$	8.48528	0.271746
$976$	0	0
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	0	0
$979$	− 50.9117i	− 1.62714i
$980$	0	0
$981$	− 6.00000i	− 0.191565i
$982$	0	0
$983$	42.4264	1.35319	0.676596	−	0.736354i	$-0.263454\pi$
$983$	42.4264	1.35319	0.676596	+	0.736354i	$0.263454\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 36.0000i	− 1.14473i
$990$	0	0
$991$	45.2548	1.43757	0.718784	−	0.695234i	$-0.244699\pi$
$991$	45.2548	1.43757	0.718784	+	0.695234i	$0.244699\pi$
$992$	0	0
$993$	42.0000	1.33283
$994$	0	0
$995$	39.5980i	1.25534i
$996$	0	0
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	0	0
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	512.2.b.e.257.1		4
3.2	odd	2		4608.2.d.j.2305.4		4
4.3	odd	2	inner	512.2.b.e.257.3		4
8.3	odd	2	inner	512.2.b.e.257.2		4
8.5	even	2	inner	512.2.b.e.257.4		4
12.11	even	2		4608.2.d.j.2305.3		4
16.3	odd	4		512.2.a.b.1.1	✓	2
16.5	even	4		512.2.a.e.1.1	yes	2
16.11	odd	4		512.2.a.e.1.2	yes	2
16.13	even	4		512.2.a.b.1.2	yes	2
24.5	odd	2		4608.2.d.j.2305.2		4
24.11	even	2		4608.2.d.j.2305.1		4
32.3	odd	8		1024.2.e.h.769.2		4
32.5	even	8		1024.2.e.n.257.2		4
32.11	odd	8		1024.2.e.n.257.1		4
32.13	even	8		1024.2.e.h.769.1		4
32.19	odd	8		1024.2.e.n.769.1		4
32.21	even	8		1024.2.e.h.257.1		4
32.27	odd	8		1024.2.e.h.257.2		4
32.29	even	8		1024.2.e.n.769.2		4
48.5	odd	4		4608.2.a.c.1.1		2
48.11	even	4		4608.2.a.c.1.2		2
48.29	odd	4		4608.2.a.p.1.1		2
48.35	even	4		4608.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.b.1.1	✓	2	16.3	odd	4
512.2.a.b.1.2	yes	2	16.13	even	4
512.2.a.e.1.1	yes	2	16.5	even	4
512.2.a.e.1.2	yes	2	16.11	odd	4
512.2.b.e.257.1		4	1.1	even	1	trivial
512.2.b.e.257.2		4	8.3	odd	2	inner
512.2.b.e.257.3		4	4.3	odd	2	inner
512.2.b.e.257.4		4	8.5	even	2	inner
1024.2.e.h.257.1		4	32.21	even	8
1024.2.e.h.257.2		4	32.27	odd	8
1024.2.e.h.769.1		4	32.13	even	8
1024.2.e.h.769.2		4	32.3	odd	8
1024.2.e.n.257.1		4	32.11	odd	8
1024.2.e.n.257.2		4	32.5	even	8
1024.2.e.n.769.1		4	32.19	odd	8
1024.2.e.n.769.2		4	32.29	even	8
4608.2.a.c.1.1		2	48.5	odd	4
4608.2.a.c.1.2		2	48.11	even	4
4608.2.a.p.1.1		2	48.29	odd	4
4608.2.a.p.1.2		2	48.35	even	4
4608.2.d.j.2305.1		4	24.11	even	2
4608.2.d.j.2305.2		4	24.5	odd	2
4608.2.d.j.2305.3		4	12.11	even	2
4608.2.d.j.2305.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 512 = 2^{9} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	512.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{3} -2.00000i q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.24264i q^{11} +6.00000i q^{13} -2.82843 q^{15} -4.24264i q^{19} -4.00000i q^{21} -8.48528 q^{23} +1.00000 q^{25} -5.65685i q^{27} +2.00000i q^{29} +5.65685 q^{31} -6.00000 q^{33} -5.65685i q^{35} -6.00000i q^{37} +8.48528 q^{39} -6.00000 q^{41} +4.24264i q^{43} -2.00000i q^{45} +1.00000 q^{49} +2.00000i q^{53} -8.48528 q^{55} -6.00000 q^{57} -1.41421i q^{59} +6.00000i q^{61} +2.82843 q^{63} +12.0000 q^{65} +12.7279i q^{67} +12.0000i q^{69} +8.48528 q^{71} +12.0000 q^{73} -1.41421i q^{75} -12.0000i q^{77} -5.65685 q^{79} -5.00000 q^{81} +4.24264i q^{83} +2.82843 q^{87} +12.0000 q^{89} +16.9706i q^{91} -8.00000i q^{93} -8.48528 q^{95} -8.00000 q^{97} -4.24264i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} + 4 q^{25} - 24 q^{33} - 24 q^{41} + 4 q^{49} - 24 q^{57} + 48 q^{65} + 48 q^{73} - 20 q^{81} + 48 q^{89} - 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^9 + 4 * q^25 - 24 * q^33 - 24 * q^41 + 4 * q^49 - 24 * q^57 + 48 * q^65 + 48 * q^73 - 20 * q^81 + 48 * q^89 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more