LMFDB - Embedded newform 3072.2.d.d.1537.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1537,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("3072.1537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.d (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:	$24.5300435009$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 768)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1537.3
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3072.1537
Dual form			3072.2.d.d.1537.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.82843i q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.82843i q^{5} -4.24264 q^{7} -1.00000 q^{9} +4.00000i q^{11} +4.24264i q^{13} +2.82843 q^{15} +6.00000 q^{17} -2.00000i q^{19} -4.24264i q^{21} -2.82843 q^{23} -3.00000 q^{25} -1.00000i q^{27} -5.65685i q^{29} -4.24264 q^{31} -4.00000 q^{33} +12.0000i q^{35} -4.24264i q^{37} -4.24264 q^{39} +10.0000 q^{41} -6.00000i q^{43} +2.82843i q^{45} -2.82843 q^{47} +11.0000 q^{49} +6.00000i q^{51} +5.65685i q^{53} +11.3137 q^{55} +2.00000 q^{57} -4.24264i q^{61} +4.24264 q^{63} +12.0000 q^{65} -4.00000i q^{67} -2.82843i q^{69} -2.82843 q^{71} -16.0000 q^{73} -3.00000i q^{75} -16.9706i q^{77} -4.24264 q^{79} +1.00000 q^{81} -16.0000i q^{83} -16.9706i q^{85} +5.65685 q^{87} -14.0000 q^{89} -18.0000i q^{91} -4.24264i q^{93} -5.65685 q^{95} -4.00000 q^{97} -4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 24 q^{17} - 12 q^{25} - 16 q^{33} + 40 q^{41} + 44 q^{49} + 8 q^{57} + 48 q^{65} - 64 q^{73} + 4 q^{81} - 56 q^{89} - 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 + 24 * q^17 - 12 * q^25 - 16 * q^33 + 40 * q^41 + 44 * q^49 + 8 * q^57 + 48 * q^65 - 64 * q^73 + 4 * q^81 - 56 * q^89 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$.

$n$	$1025$	$2047$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	4.24264i	1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$	4.24264i	1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	− 4.24264i	− 0.925820i
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 5.65685i	− 1.05045i	−0.850963	−	0.525226i	$-0.823981\pi$
$29$	− 5.65685i	− 1.05045i	0.850963	−	0.525226i	$-0.176019\pi$
$30$	0	0
$31$	−4.24264	−0.762001	−0.381000	−	0.924575i	$-0.624420\pi$
$31$	−4.24264	−0.762001	−0.381000	+	0.924575i	$0.624420\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	12.0000i	2.02837i
$36$	0	0
$37$	− 4.24264i	− 0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$	− 4.24264i	− 0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$	0	0
$39$	−4.24264	−0.679366
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	0	0
$45$	2.82843i	0.421637i
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	6.00000i	0.840168i
$52$	0	0
$53$	5.65685i	0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$	5.65685i	0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$	0	0
$55$	11.3137	1.52554
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 4.24264i	− 0.543214i	−0.962408	−	0.271607i	$-0.912445\pi$
$61$	− 4.24264i	− 0.543214i	0.962408	−	0.271607i	$-0.0875552\pi$
$62$	0	0
$63$	4.24264	0.534522
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	− 2.82843i	− 0.340503i
$70$	0	0
$71$	−2.82843	−0.335673	−0.167836	−	0.985815i	$-0.553678\pi$
$71$	−2.82843	−0.335673	−0.167836	+	0.985815i	$0.553678\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	− 3.00000i	− 0.346410i
$76$	0	0
$77$	− 16.9706i	− 1.93398i
$78$	0	0
$79$	−4.24264	−0.477334	−0.238667	−	0.971101i	$-0.576710\pi$
$79$	−4.24264	−0.477334	−0.238667	+	0.971101i	$0.576710\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	− 16.9706i	− 1.84072i
$86$	0	0
$87$	5.65685	0.606478
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	− 18.0000i	− 1.88691i
$92$	0	0
$93$	− 4.24264i	− 0.439941i
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	0	0
$99$	− 4.00000i	− 0.402015i
$100$	0	0
$101$	− 11.3137i	− 1.12576i	−0.826540	−	0.562878i	$-0.809694\pi$
$101$	− 11.3137i	− 1.12576i	0.826540	−	0.562878i	$-0.190306\pi$
$102$	0	0
$103$	18.3848	1.81151	0.905753	−	0.423806i	$-0.139306\pi$
$103$	18.3848	1.81151	0.905753	+	0.423806i	$0.139306\pi$
$104$	0	0
$105$	−12.0000	−1.17108
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	− 9.89949i	− 0.948200i	−0.880471	−	0.474100i	$-0.842774\pi$
$109$	− 9.89949i	− 0.948200i	0.880471	−	0.474100i	$-0.157226\pi$
$110$	0	0
$111$	4.24264	0.402694
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	8.00000i	0.746004i
$116$	0	0
$117$	− 4.24264i	− 0.392232i
$118$	0	0
$119$	−25.4558	−2.33353
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	− 5.65685i	− 0.505964i
$126$	0	0
$127$	−4.24264	−0.376473	−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264	−0.376473	−0.188237	+	0.982124i	$0.560277\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	− 20.0000i	− 1.74741i	−0.486458	−	0.873704i	$-0.661711\pi$
$131$	− 20.0000i	− 1.74741i	0.486458	−	0.873704i	$-0.338289\pi$
$132$	0	0
$133$	8.48528i	0.735767i
$134$	0	0
$135$	−2.82843	−0.243432
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	− 12.0000i	− 1.01783i	−0.860818	−	0.508913i	$-0.830047\pi$
$139$	− 12.0000i	− 1.01783i	0.860818	−	0.508913i	$-0.169953\pi$
$140$	0	0
$141$	− 2.82843i	− 0.238197i
$142$	0	0
$143$	−16.9706	−1.41915
$144$	0	0
$145$	−16.0000	−1.32873
$146$	0	0
$147$	11.0000i	0.907265i
$148$	0	0
$149$	8.48528i	0.695141i	0.937654	+	0.347571i	$0.112993\pi$
$149$	8.48528i	0.695141i	−0.937654	+	0.347571i	$0.887007\pi$
$150$	0	0
$151$	1.41421	0.115087	0.0575435	−	0.998343i	$-0.481673\pi$
$151$	1.41421	0.115087	0.0575435	+	0.998343i	$0.481673\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	12.0000i	0.963863i
$156$	0	0
$157$	− 7.07107i	− 0.564333i	−0.959366	−	0.282166i	$-0.908947\pi$
$157$	− 7.07107i	− 0.564333i	0.959366	−	0.282166i	$-0.0910530\pi$
$158$	0	0
$159$	−5.65685	−0.448618
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	0	0
$165$	11.3137i	0.880771i
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	2.00000i	0.152944i
$172$	0	0
$173$	− 2.82843i	− 0.215041i	−0.994203	−	0.107521i	$-0.965709\pi$
$173$	− 2.82843i	− 0.215041i	0.994203	−	0.107521i	$-0.0342912\pi$
$174$	0	0
$175$	12.7279	0.962140
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 1.41421i	− 0.105118i	−0.998618	−	0.0525588i	$-0.983262\pi$
$181$	− 1.41421i	− 0.105118i	0.998618	−	0.0525588i	$-0.0167377\pi$
$182$	0	0
$183$	4.24264	0.313625
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	24.0000i	1.75505i
$188$	0	0
$189$	4.24264i	0.308607i
$190$	0	0
$191$	22.6274	1.63726	0.818631	−	0.574320i	$-0.194733\pi$
$191$	22.6274	1.63726	0.818631	+	0.574320i	$0.194733\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	12.0000i	0.859338i
$196$	0	0
$197$	− 5.65685i	− 0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$	− 5.65685i	− 0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$	0	0
$199$	−7.07107	−0.501255	−0.250627	−	0.968084i	$-0.580637\pi$
$199$	−7.07107	−0.501255	−0.250627	+	0.968084i	$0.580637\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	24.0000i	1.68447i
$204$	0	0
$205$	− 28.2843i	− 1.97546i
$206$	0	0
$207$	2.82843	0.196589
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	12.0000i	0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$	12.0000i	0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	0	0
$213$	− 2.82843i	− 0.193801i
$214$	0	0
$215$	−16.9706	−1.15738
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	− 16.0000i	− 1.08118i
$220$	0	0
$221$	25.4558i	1.71235i
$222$	0	0
$223$	21.2132	1.42054	0.710271	−	0.703929i	$-0.248573\pi$
$223$	21.2132	1.42054	0.710271	+	0.703929i	$0.248573\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 15.5563i	− 1.02799i	−0.857792	−	0.513996i	$-0.828165\pi$
$229$	− 15.5563i	− 1.02799i	0.857792	−	0.513996i	$-0.171835\pi$
$230$	0	0
$231$	16.9706	1.11658
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	8.00000i	0.521862i
$236$	0	0
$237$	− 4.24264i	− 0.275589i
$238$	0	0
$239$	11.3137	0.731823	0.365911	−	0.930650i	$-0.380757\pi$
$239$	11.3137	0.731823	0.365911	+	0.930650i	$0.380757\pi$
$240$	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
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 31.1127i	− 1.98772i
$246$	0	0
$247$	8.48528	0.539906
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	4.00000i	0.252478i	0.992000	+	0.126239i	$0.0402906\pi$
$251$	4.00000i	0.252478i	−0.992000	+	0.126239i	$0.959709\pi$
$252$	0	0
$253$	− 11.3137i	− 0.711287i
$254$	0	0
$255$	16.9706	1.06274
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	18.0000i	1.11847i
$260$	0	0
$261$	5.65685i	0.350150i
$262$	0	0
$263$	−22.6274	−1.39527	−0.697633	−	0.716455i	$-0.745763\pi$
$263$	−22.6274	−1.39527	−0.697633	+	0.716455i	$0.745763\pi$
$264$	0	0
$265$	16.0000	0.982872
$266$	0	0
$267$	− 14.0000i	− 0.856786i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−1.41421	−0.0859074	−0.0429537	−	0.999077i	$-0.513677\pi$
$271$	−1.41421	−0.0859074	−0.0429537	+	0.999077i	$0.513677\pi$
$272$	0	0
$273$	18.0000	1.08941
$274$	0	0
$275$	− 12.0000i	− 0.723627i
$276$	0	0
$277$	1.41421i	0.0849719i	0.999097	+	0.0424859i	$0.0135278\pi$
$277$	1.41421i	0.0849719i	−0.999097	+	0.0424859i	$0.986472\pi$
$278$	0	0
$279$	4.24264	0.254000
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	− 5.65685i	− 0.335083i
$286$	0	0
$287$	−42.4264	−2.50435
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 4.00000i	− 0.234484i
$292$	0	0
$293$	− 5.65685i	− 0.330477i	−0.986254	−	0.165238i	$-0.947161\pi$
$293$	− 5.65685i	− 0.330477i	0.986254	−	0.165238i	$-0.0528394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	− 12.0000i	− 0.693978i
$300$	0	0
$301$	25.4558i	1.46725i
$302$	0	0
$303$	11.3137	0.649956
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	18.3848i	1.04587i
$310$	0	0
$311$	28.2843	1.60385	0.801927	−	0.597422i	$-0.203808\pi$
$311$	28.2843	1.60385	0.801927	+	0.597422i	$0.203808\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	− 12.0000i	− 0.676123i
$316$	0	0
$317$	− 16.9706i	− 0.953162i	−0.879131	−	0.476581i	$-0.841876\pi$
$317$	− 16.9706i	− 0.953162i	0.879131	−	0.476581i	$-0.158124\pi$
$318$	0	0
$319$	22.6274	1.26689
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 12.0000i	− 0.667698i
$324$	0	0
$325$	− 12.7279i	− 0.706018i
$326$	0	0
$327$	9.89949	0.547443
$328$	0	0
$329$	12.0000	0.661581
$330$	0	0
$331$	4.00000i	0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$	4.00000i	0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$	0	0
$333$	4.24264i	0.232495i
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	− 16.9706i	− 0.919007i
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	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
$351$	4.24264	0.226455
$352$	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
$355$	8.00000i	0.424596i
$356$	0	0
$357$	− 25.4558i	− 1.34727i
$358$	0	0
$359$	−2.82843	−0.149279	−0.0746393	−	0.997211i	$-0.523781\pi$
$359$	−2.82843	−0.149279	−0.0746393	+	0.997211i	$0.523781\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	45.2548i	2.36875i
$366$	0	0
$367$	18.3848	0.959678	0.479839	−	0.877357i	$-0.340695\pi$
$367$	18.3848	0.959678	0.479839	+	0.877357i	$0.340695\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	− 24.0000i	− 1.24602i
$372$	0	0
$373$	18.3848i	0.951928i	0.879465	+	0.475964i	$0.157901\pi$
$373$	18.3848i	0.951928i	−0.879465	+	0.475964i	$0.842099\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	− 38.0000i	− 1.95193i	−0.217930	−	0.975964i	$-0.569930\pi$
$379$	− 38.0000i	− 1.95193i	0.217930	−	0.975964i	$-0.430070\pi$
$380$	0	0
$381$	− 4.24264i	− 0.217357i
$382$	0	0
$383$	−28.2843	−1.44526	−0.722629	−	0.691236i	$-0.757067\pi$
$383$	−28.2843	−1.44526	−0.722629	+	0.691236i	$0.757067\pi$
$384$	0	0
$385$	−48.0000	−2.44631
$386$	0	0
$387$	6.00000i	0.304997i
$388$	0	0
$389$	2.82843i	0.143407i	0.997426	+	0.0717035i	$0.0228435\pi$
$389$	2.82843i	0.143407i	−0.997426	+	0.0717035i	$0.977156\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	20.0000	1.00887
$394$	0	0
$395$	12.0000i	0.603786i
$396$	0	0
$397$	32.5269i	1.63248i	0.577714	+	0.816239i	$0.303945\pi$
$397$	32.5269i	1.63248i	−0.577714	+	0.816239i	$0.696055\pi$
$398$	0	0
$399$	−8.48528	−0.424795
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	0	0
$403$	− 18.0000i	− 0.896644i
$404$	0	0
$405$	− 2.82843i	− 0.140546i
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	− 10.0000i	− 0.493264i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−45.2548	−2.22147
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	− 16.0000i	− 0.781651i	−0.920465	−	0.390826i	$-0.872190\pi$
$419$	− 16.0000i	− 0.781651i	0.920465	−	0.390826i	$-0.127810\pi$
$420$	0	0
$421$	35.3553i	1.72311i	0.507661	+	0.861557i	$0.330510\pi$
$421$	35.3553i	1.72311i	−0.507661	+	0.861557i	$0.669490\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	−18.0000	−0.873128
$426$	0	0
$427$	18.0000i	0.871081i
$428$	0	0
$429$	− 16.9706i	− 0.819346i
$430$	0	0
$431$	−36.7696	−1.77113	−0.885564	−	0.464518i	$-0.846227\pi$
$431$	−36.7696	−1.77113	−0.885564	+	0.464518i	$0.846227\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	− 16.0000i	− 0.767141i
$436$	0	0
$437$	5.65685i	0.270604i
$438$	0	0
$439$	−1.41421	−0.0674967	−0.0337484	−	0.999430i	$-0.510744\pi$
$439$	−1.41421	−0.0674967	−0.0337484	+	0.999430i	$0.510744\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	39.5980i	1.87712i
$446$	0	0
$447$	−8.48528	−0.401340
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	40.0000i	1.88353i
$452$	0	0
$453$	1.41421i	0.0664455i
$454$	0	0
$455$	−50.9117	−2.38678
$456$	0	0
$457$	−12.0000	−0.561336	−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000	−0.561336	−0.280668	+	0.959805i	$0.590556\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	8.48528i	0.395199i	0.980283	+	0.197599i	$0.0633145\pi$
$461$	8.48528i	0.395199i	−0.980283	+	0.197599i	$0.936685\pi$
$462$	0	0
$463$	12.7279	0.591517	0.295758	−	0.955263i	$-0.404428\pi$
$463$	12.7279	0.591517	0.295758	+	0.955263i	$0.404428\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	24.0000i	1.11059i	0.831654	+	0.555294i	$0.187394\pi$
$467$	24.0000i	1.11059i	−0.831654	+	0.555294i	$0.812606\pi$
$468$	0	0
$469$	16.9706i	0.783628i
$470$	0	0
$471$	7.07107	0.325818
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	6.00000i	0.275299i
$476$	0	0
$477$	− 5.65685i	− 0.259010i
$478$	0	0
$479$	−25.4558	−1.16311	−0.581554	−	0.813508i	$-0.697555\pi$
$479$	−25.4558	−1.16311	−0.581554	+	0.813508i	$0.697555\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	12.0000i	0.546019i
$484$	0	0
$485$	11.3137i	0.513729i
$486$	0	0
$487$	43.8406	1.98661	0.993304	−	0.115529i	$-0.0368564\pi$
$487$	43.8406	1.98661	0.993304	+	0.115529i	$0.0368564\pi$
$488$	0	0
$489$	10.0000	0.452216
$490$	0	0
$491$	8.00000i	0.361035i	0.983572	+	0.180517i	$0.0577772\pi$
$491$	8.00000i	0.361035i	−0.983572	+	0.180517i	$0.942223\pi$
$492$	0	0
$493$	− 33.9411i	− 1.52863i
$494$	0	0
$495$	−11.3137	−0.508513
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	− 20.0000i	− 0.895323i	−0.894203	−	0.447661i	$-0.852257\pi$
$499$	− 20.0000i	− 0.895323i	0.894203	−	0.447661i	$-0.147743\pi$
$500$	0	0
$501$	11.3137i	0.505459i
$502$	0	0
$503$	14.1421	0.630567	0.315283	−	0.948998i	$-0.397900\pi$
$503$	14.1421	0.630567	0.315283	+	0.948998i	$0.397900\pi$
$504$	0	0
$505$	−32.0000	−1.42398
$506$	0	0
$507$	− 5.00000i	− 0.222058i
$508$	0	0
$509$	− 22.6274i	− 1.00294i	−0.865174	−	0.501471i	$-0.832792\pi$
$509$	− 22.6274i	− 1.00294i	0.865174	−	0.501471i	$-0.167208\pi$
$510$	0	0
$511$	67.8823	3.00293
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	− 52.0000i	− 2.29139i
$516$	0	0
$517$	− 11.3137i	− 0.497576i
$518$	0	0
$519$	2.82843	0.124154
$520$	0	0
$521$	22.0000	0.963837	0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000	0.963837	0.481919	+	0.876216i	$0.339940\pi$
$522$	0	0
$523$	− 22.0000i	− 0.961993i	−0.876723	−	0.480996i	$-0.840275\pi$
$523$	− 22.0000i	− 0.961993i	0.876723	−	0.480996i	$-0.159725\pi$
$524$	0	0
$525$	12.7279i	0.555492i
$526$	0	0
$527$	−25.4558	−1.10887
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	42.4264i	1.83769i
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	0	0
$538$	0	0
$539$	44.0000i	1.89521i
$540$	0	0
$541$	− 41.0122i	− 1.76325i	−0.471949	−	0.881626i	$-0.656449\pi$
$541$	− 41.0122i	− 1.76325i	0.471949	−	0.881626i	$-0.343551\pi$
$542$	0	0
$543$	1.41421	0.0606897
$544$	0	0
$545$	−28.0000	−1.19939
$546$	0	0
$547$	34.0000i	1.45374i	0.686778	+	0.726868i	$0.259025\pi$
$547$	34.0000i	1.45374i	−0.686778	+	0.726868i	$0.740975\pi$
$548$	0	0
$549$	4.24264i	0.181071i
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	18.0000	0.765438
$554$	0	0
$555$	− 12.0000i	− 0.509372i
$556$	0	0
$557$	− 19.7990i	− 0.838910i	−0.907776	−	0.419455i	$-0.862221\pi$
$557$	− 19.7990i	− 0.838910i	0.907776	−	0.419455i	$-0.137779\pi$
$558$	0	0
$559$	25.4558	1.07667
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	16.9706i	0.713957i
$566$	0	0
$567$	−4.24264	−0.178174
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	− 20.0000i	− 0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$	− 20.0000i	− 0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$	0	0
$573$	22.6274i	0.945274i
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	0	0
$579$	18.0000i	0.748054i
$580$	0	0
$581$	67.8823i	2.81623i
$582$	0	0
$583$	−22.6274	−0.937132
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	− 24.0000i	− 0.990586i	−0.868726	−	0.495293i	$-0.835061\pi$
$587$	− 24.0000i	− 0.990586i	0.868726	−	0.495293i	$-0.164939\pi$
$588$	0	0
$589$	8.48528i	0.349630i
$590$	0	0
$591$	5.65685	0.232692
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	72.0000i	2.95171i
$596$	0	0
$597$	− 7.07107i	− 0.289400i
$598$	0	0
$599$	−48.0833	−1.96463	−0.982314	−	0.187239i	$-0.940046\pi$
$599$	−48.0833	−1.96463	−0.982314	+	0.187239i	$0.940046\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	14.1421i	0.574960i
$606$	0	0
$607$	−4.24264	−0.172203	−0.0861017	−	0.996286i	$-0.527441\pi$
$607$	−4.24264	−0.172203	−0.0861017	+	0.996286i	$0.527441\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	− 12.0000i	− 0.485468i
$612$	0	0
$613$	18.3848i	0.742554i	0.928522	+	0.371277i	$0.121080\pi$
$613$	18.3848i	0.742554i	−0.928522	+	0.371277i	$0.878920\pi$
$614$	0	0
$615$	28.2843	1.14053
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	− 36.0000i	− 1.44696i	−0.690344	−	0.723481i	$-0.742541\pi$
$619$	− 36.0000i	− 1.44696i	0.690344	−	0.723481i	$-0.257459\pi$
$620$	0	0
$621$	2.82843i	0.113501i
$622$	0	0
$623$	59.3970	2.37969
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	8.00000i	0.319489i
$628$	0	0
$629$	− 25.4558i	− 1.01499i
$630$	0	0
$631$	−35.3553	−1.40747	−0.703737	−	0.710461i	$-0.748487\pi$
$631$	−35.3553	−1.40747	−0.703737	+	0.710461i	$0.748487\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	12.0000i	0.476205i
$636$	0	0
$637$	46.6690i	1.84909i
$638$	0	0
$639$	2.82843	0.111891
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	− 16.9706i	− 0.668215i
$646$	0	0
$647$	−14.1421	−0.555985	−0.277992	−	0.960583i	$-0.589669\pi$
$647$	−14.1421	−0.555985	−0.277992	+	0.960583i	$0.589669\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	18.0000i	0.705476i
$652$	0	0
$653$	− 8.48528i	− 0.332055i	−0.986121	−	0.166027i	$-0.946906\pi$
$653$	− 8.48528i	− 0.332055i	0.986121	−	0.166027i	$-0.0530940\pi$
$654$	0	0
$655$	−56.5685	−2.21032
$656$	0	0
$657$	16.0000	0.624219
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 12.7279i	− 0.495059i	−0.968880	−	0.247529i	$-0.920381\pi$
$661$	− 12.7279i	− 0.495059i	0.968880	−	0.247529i	$-0.0796187\pi$
$662$	0	0
$663$	−25.4558	−0.988623
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	21.2132i	0.820150i
$670$	0	0
$671$	16.9706	0.655141
$672$	0	0
$673$	−40.0000	−1.54189	−0.770943	−	0.636904i	$-0.780215\pi$
$673$	−40.0000	−1.54189	−0.770943	+	0.636904i	$0.780215\pi$
$674$	0	0
$675$	3.00000i	0.115470i
$676$	0	0
$677$	36.7696i	1.41317i	0.707629	+	0.706584i	$0.249765\pi$
$677$	36.7696i	1.41317i	−0.707629	+	0.706584i	$0.750235\pi$
$678$	0	0
$679$	16.9706	0.651270
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	28.2843i	1.08069i
$686$	0	0
$687$	15.5563	0.593512
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	22.0000i	0.836919i	0.908235	+	0.418460i	$0.137430\pi$
$691$	22.0000i	0.836919i	−0.908235	+	0.418460i	$0.862570\pi$
$692$	0	0
$693$	16.9706i	0.644658i
$694$	0	0
$695$	−33.9411	−1.28746
$696$	0	0
$697$	60.0000	2.27266
$698$	0	0
$699$	− 18.0000i	− 0.680823i
$700$	0	0
$701$	5.65685i	0.213656i	0.994277	+	0.106828i	$0.0340695\pi$
$701$	5.65685i	0.213656i	−0.994277	+	0.106828i	$0.965931\pi$
$702$	0	0
$703$	−8.48528	−0.320028
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	48.0000i	1.80523i
$708$	0	0
$709$	− 18.3848i	− 0.690455i	−0.938519	−	0.345227i	$-0.887802\pi$
$709$	− 18.3848i	− 0.690455i	0.938519	−	0.345227i	$-0.112198\pi$
$710$	0	0
$711$	4.24264	0.159111
$712$	0	0
$713$	12.0000	0.449404
$714$	0	0
$715$	48.0000i	1.79510i
$716$	0	0
$717$	11.3137i	0.422518i
$718$	0	0
$719$	8.48528	0.316448	0.158224	−	0.987403i	$-0.449423\pi$
$719$	8.48528	0.316448	0.158224	+	0.987403i	$0.449423\pi$
$720$	0	0
$721$	−78.0000	−2.90487
$722$	0	0
$723$	18.0000i	0.669427i
$724$	0	0
$725$	16.9706i	0.630271i
$726$	0	0
$727$	26.8701	0.996555	0.498278	−	0.867018i	$-0.333966\pi$
$727$	26.8701	0.996555	0.498278	+	0.867018i	$0.333966\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 36.0000i	− 1.33151i
$732$	0	0
$733$	35.3553i	1.30588i	0.757410	+	0.652940i	$0.226464\pi$
$733$	35.3553i	1.30588i	−0.757410	+	0.652940i	$0.773536\pi$
$734$	0	0
$735$	31.1127	1.14761
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	− 36.0000i	− 1.32428i	−0.749380	−	0.662141i	$-0.769648\pi$
$739$	− 36.0000i	− 1.32428i	0.749380	−	0.662141i	$-0.230352\pi$
$740$	0	0
$741$	8.48528i	0.311715i
$742$	0	0
$743$	22.6274	0.830119	0.415060	−	0.909794i	$-0.363761\pi$
$743$	22.6274	0.830119	0.415060	+	0.909794i	$0.363761\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	0	0
$747$	16.0000i	0.585409i
$748$	0	0
$749$	− 16.9706i	− 0.620091i
$750$	0	0
$751$	−12.7279	−0.464448	−0.232224	−	0.972662i	$-0.574600\pi$
$751$	−12.7279	−0.464448	−0.232224	+	0.972662i	$0.574600\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	− 4.00000i	− 0.145575i
$756$	0	0
$757$	38.1838i	1.38781i	0.720065	+	0.693906i	$0.244112\pi$
$757$	38.1838i	1.38781i	−0.720065	+	0.693906i	$0.755888\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	42.0000i	1.52050i
$764$	0	0
$765$	16.9706i	0.613572i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	− 14.0000i	− 0.504198i
$772$	0	0
$773$	5.65685i	0.203463i	0.994812	+	0.101731i	$0.0324382\pi$
$773$	5.65685i	0.203463i	−0.994812	+	0.101731i	$0.967562\pi$
$774$	0	0
$775$	12.7279	0.457200
$776$	0	0
$777$	−18.0000	−0.645746
$778$	0	0
$779$	− 20.0000i	− 0.716574i
$780$	0	0
$781$	− 11.3137i	− 0.404836i
$782$	0	0
$783$	−5.65685	−0.202159
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	− 38.0000i	− 1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$	− 38.0000i	− 1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$	0	0
$789$	− 22.6274i	− 0.805557i
$790$	0	0
$791$	25.4558	0.905106
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	16.0000i	0.567462i
$796$	0	0
$797$	31.1127i	1.10207i	0.834483	+	0.551034i	$0.185767\pi$
$797$	31.1127i	1.10207i	−0.834483	+	0.551034i	$0.814233\pi$
$798$	0	0
$799$	−16.9706	−0.600375
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	− 64.0000i	− 2.25851i
$804$	0	0
$805$	− 33.9411i	− 1.19627i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	− 2.00000i	− 0.0702295i	−0.999383	−	0.0351147i	$-0.988820\pi$
$811$	− 2.00000i	− 0.0702295i	0.999383	−	0.0351147i	$-0.0111797\pi$
$812$	0	0
$813$	− 1.41421i	− 0.0495986i
$814$	0	0
$815$	−28.2843	−0.990755
$816$	0	0
$817$	−12.0000	−0.419827
$818$	0	0
$819$	18.0000i	0.628971i
$820$	0	0
$821$	− 28.2843i	− 0.987128i	−0.869710	−	0.493564i	$-0.835694\pi$
$821$	− 28.2843i	− 0.987128i	0.869710	−	0.493564i	$-0.164306\pi$
$822$	0	0
$823$	4.24264	0.147889	0.0739446	−	0.997262i	$-0.476441\pi$
$823$	4.24264	0.147889	0.0739446	+	0.997262i	$0.476441\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	0	0
$829$	− 21.2132i	− 0.736765i	−0.929674	−	0.368383i	$-0.879912\pi$
$829$	− 21.2132i	− 0.736765i	0.929674	−	0.368383i	$-0.120088\pi$
$830$	0	0
$831$	−1.41421	−0.0490585
$832$	0	0
$833$	66.0000	2.28676
$834$	0	0
$835$	− 32.0000i	− 1.10741i
$836$	0	0
$837$	4.24264i	0.146647i
$838$	0	0
$839$	−31.1127	−1.07413	−0.537065	−	0.843541i	$-0.680467\pi$
$839$	−31.1127	−1.07413	−0.537065	+	0.843541i	$0.680467\pi$
$840$	0	0
$841$	−3.00000	−0.103448
$842$	0	0
$843$	− 6.00000i	− 0.206651i
$844$	0	0
$845$	14.1421i	0.486504i
$846$	0	0
$847$	21.2132	0.728894
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	12.0000i	0.411355i
$852$	0	0
$853$	− 21.2132i	− 0.726326i	−0.931726	−	0.363163i	$-0.881697\pi$
$853$	− 21.2132i	− 0.726326i	0.931726	−	0.363163i	$-0.118303\pi$
$854$	0	0
$855$	5.65685	0.193460
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	− 10.0000i	− 0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$859$	− 10.0000i	− 0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$860$	0	0
$861$	− 42.4264i	− 1.44589i
$862$	0	0
$863$	−28.2843	−0.962808	−0.481404	−	0.876499i	$-0.659873\pi$
$863$	−28.2843	−0.962808	−0.481404	+	0.876499i	$0.659873\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	− 16.9706i	− 0.575687i
$870$	0	0
$871$	16.9706	0.575026
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	24.0000i	0.811348i
$876$	0	0
$877$	− 15.5563i	− 0.525301i	−0.964891	−	0.262650i	$-0.915403\pi$
$877$	− 15.5563i	− 0.525301i	0.964891	−	0.262650i	$-0.0845966\pi$
$878$	0	0
$879$	5.65685	0.190801
$880$	0	0
$881$	58.0000	1.95407	0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000	1.95407	0.977035	+	0.213080i	$0.0683494\pi$
$882$	0	0
$883$	− 34.0000i	− 1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$	− 34.0000i	− 1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.9706	−0.569816	−0.284908	−	0.958555i	$-0.591963\pi$
$887$	−16.9706	−0.569816	−0.284908	+	0.958555i	$0.591963\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	5.65685i	0.189299i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	24.0000i	0.800445i
$900$	0	0
$901$	33.9411i	1.13074i
$902$	0	0
$903$	−25.4558	−0.847117
$904$	0	0
$905$	−4.00000	−0.132964
$906$	0	0
$907$	− 58.0000i	− 1.92586i	−0.269754	−	0.962929i	$-0.586942\pi$
$907$	− 58.0000i	− 1.92586i	0.269754	−	0.962929i	$-0.413058\pi$
$908$	0	0
$909$	11.3137i	0.375252i
$910$	0	0
$911$	−22.6274	−0.749680	−0.374840	−	0.927090i	$-0.622302\pi$
$911$	−22.6274	−0.749680	−0.374840	+	0.927090i	$0.622302\pi$
$912$	0	0
$913$	64.0000	2.11809
$914$	0	0
$915$	− 12.0000i	− 0.396708i
$916$	0	0
$917$	84.8528i	2.80209i
$918$	0	0
$919$	−18.3848	−0.606458	−0.303229	−	0.952918i	$-0.598065\pi$
$919$	−18.3848	−0.606458	−0.303229	+	0.952918i	$0.598065\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	12.7279i	0.418491i
$926$	0	0
$927$	−18.3848	−0.603835
$928$	0	0
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	0	0
$931$	− 22.0000i	− 0.721021i
$932$	0	0
$933$	28.2843i	0.925985i
$934$	0	0
$935$	67.8823	2.21999
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	− 26.0000i	− 0.848478i
$940$	0	0
$941$	36.7696i	1.19865i	0.800505	+	0.599327i	$0.204565\pi$
$941$	36.7696i	1.19865i	−0.800505	+	0.599327i	$0.795435\pi$
$942$	0	0
$943$	−28.2843	−0.921063
$944$	0	0
$945$	12.0000	0.390360
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	− 67.8823i	− 2.20355i
$950$	0	0
$951$	16.9706	0.550308
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	− 64.0000i	− 2.07099i
$956$	0	0
$957$	22.6274i	0.731441i
$958$	0	0
$959$	42.4264	1.37002
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	− 4.00000i	− 0.128898i
$964$	0	0
$965$	− 50.9117i	− 1.63891i
$966$	0	0
$967$	35.3553	1.13695	0.568476	−	0.822700i	$-0.307533\pi$
$967$	35.3553	1.13695	0.568476	+	0.822700i	$0.307533\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	32.0000i	1.02693i	0.858111	+	0.513464i	$0.171638\pi$
$971$	32.0000i	1.02693i	−0.858111	+	0.513464i	$0.828362\pi$
$972$	0	0
$973$	50.9117i	1.63215i
$974$	0	0
$975$	12.7279	0.407620
$976$	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
$979$	− 56.0000i	− 1.78977i
$980$	0	0
$981$	9.89949i	0.316067i
$982$	0	0
$983$	−22.6274	−0.721703	−0.360851	−	0.932623i	$-0.617514\pi$
$983$	−22.6274	−0.721703	−0.360851	+	0.932623i	$0.617514\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	12.0000i	0.381964i
$988$	0	0
$989$	16.9706i	0.539633i
$990$	0	0
$991$	−46.6690	−1.48249	−0.741246	−	0.671234i	$-0.765765\pi$
$991$	−46.6690	−1.48249	−0.741246	+	0.671234i	$0.765765\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	20.0000i	0.634043i
$996$	0	0
$997$	46.6690i	1.47802i	0.673693	+	0.739012i	$0.264707\pi$
$997$	46.6690i	1.47802i	−0.673693	+	0.739012i	$0.735293\pi$
$998$	0	0
$999$	−4.24264	−0.134231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.d.d.1537.3		4
4.3	odd	2	inner	3072.2.d.d.1537.1		4
8.3	odd	2	inner	3072.2.d.d.1537.4		4
8.5	even	2	inner	3072.2.d.d.1537.2		4
16.3	odd	4		3072.2.a.f.1.1		2
16.5	even	4		3072.2.a.f.1.2		2
16.11	odd	4		3072.2.a.d.1.2		2
16.13	even	4		3072.2.a.d.1.1		2
32.3	odd	8		768.2.j.d.193.2	yes	4
32.5	even	8		768.2.j.d.577.1	yes	4
32.11	odd	8		768.2.j.a.577.1	yes	4
32.13	even	8		768.2.j.a.193.2	yes	4
32.19	odd	8		768.2.j.a.193.1	✓	4
32.21	even	8		768.2.j.a.577.2	yes	4
32.27	odd	8		768.2.j.d.577.2	yes	4
32.29	even	8		768.2.j.d.193.1	yes	4
48.5	odd	4		9216.2.a.q.1.1		2
48.11	even	4		9216.2.a.e.1.1		2
48.29	odd	4		9216.2.a.e.1.2		2
48.35	even	4		9216.2.a.q.1.2		2
96.5	odd	8		2304.2.k.a.577.1		4
96.11	even	8		2304.2.k.d.577.2		4
96.29	odd	8		2304.2.k.a.1729.2		4
96.35	even	8		2304.2.k.a.1729.1		4
96.53	odd	8		2304.2.k.d.577.1		4
96.59	even	8		2304.2.k.a.577.2		4
96.77	odd	8		2304.2.k.d.1729.2		4
96.83	even	8		2304.2.k.d.1729.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.a.193.1	✓	4	32.19	odd	8
768.2.j.a.193.2	yes	4	32.13	even	8
768.2.j.a.577.1	yes	4	32.11	odd	8
768.2.j.a.577.2	yes	4	32.21	even	8
768.2.j.d.193.1	yes	4	32.29	even	8
768.2.j.d.193.2	yes	4	32.3	odd	8
768.2.j.d.577.1	yes	4	32.5	even	8
768.2.j.d.577.2	yes	4	32.27	odd	8
2304.2.k.a.577.1		4	96.5	odd	8
2304.2.k.a.577.2		4	96.59	even	8
2304.2.k.a.1729.1		4	96.35	even	8
2304.2.k.a.1729.2		4	96.29	odd	8
2304.2.k.d.577.1		4	96.53	odd	8
2304.2.k.d.577.2		4	96.11	even	8
2304.2.k.d.1729.1		4	96.83	even	8
2304.2.k.d.1729.2		4	96.77	odd	8
3072.2.a.d.1.1		2	16.13	even	4
3072.2.a.d.1.2		2	16.11	odd	4
3072.2.a.f.1.1		2	16.3	odd	4
3072.2.a.f.1.2		2	16.5	even	4
3072.2.d.d.1537.1		4	4.3	odd	2	inner
3072.2.d.d.1537.2		4	8.5	even	2	inner
3072.2.d.d.1537.3		4	1.1	even	1	trivial
3072.2.d.d.1537.4		4	8.3	odd	2	inner
9216.2.a.e.1.1		2	48.11	even	4
9216.2.a.e.1.2		2	48.29	odd	4
9216.2.a.q.1.1		2	48.5	odd	4
9216.2.a.q.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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.82843i q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.82843i q^{5} -4.24264 q^{7} -1.00000 q^{9} +4.00000i q^{11} +4.24264i q^{13} +2.82843 q^{15} +6.00000 q^{17} -2.00000i q^{19} -4.24264i q^{21} -2.82843 q^{23} -3.00000 q^{25} -1.00000i q^{27} -5.65685i q^{29} -4.24264 q^{31} -4.00000 q^{33} +12.0000i q^{35} -4.24264i q^{37} -4.24264 q^{39} +10.0000 q^{41} -6.00000i q^{43} +2.82843i q^{45} -2.82843 q^{47} +11.0000 q^{49} +6.00000i q^{51} +5.65685i q^{53} +11.3137 q^{55} +2.00000 q^{57} -4.24264i q^{61} +4.24264 q^{63} +12.0000 q^{65} -4.00000i q^{67} -2.82843i q^{69} -2.82843 q^{71} -16.0000 q^{73} -3.00000i q^{75} -16.9706i q^{77} -4.24264 q^{79} +1.00000 q^{81} -16.0000i q^{83} -16.9706i q^{85} +5.65685 q^{87} -14.0000 q^{89} -18.0000i q^{91} -4.24264i q^{93} -5.65685 q^{95} -4.00000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 24 q^{17} - 12 q^{25} - 16 q^{33} + 40 q^{41} + 44 q^{49} + 8 q^{57} + 48 q^{65} - 64 q^{73} + 4 q^{81} - 56 q^{89} - 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 + 24 * q^17 - 12 * q^25 - 16 * q^33 + 40 * q^41 + 44 * q^49 + 8 * q^57 + 48 * q^65 - 64 * q^73 + 4 * q^81 - 56 * q^89 - 16 * q^97

Introduction

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