LMFDB - Embedded newform 3072.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 768)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.82843 q^{5} -4.24264 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.82843 q^{5} -4.24264 q^{7} +1.00000 q^{9} +4.00000 q^{11} +4.24264 q^{13} -2.82843 q^{15} +6.00000 q^{17} +2.00000 q^{19} +4.24264 q^{21} -2.82843 q^{23} +3.00000 q^{25} -1.00000 q^{27} -5.65685 q^{29} +4.24264 q^{31} -4.00000 q^{33} -12.0000 q^{35} +4.24264 q^{37} -4.24264 q^{39} -10.0000 q^{41} -6.00000 q^{43} +2.82843 q^{45} +2.82843 q^{47} +11.0000 q^{49} -6.00000 q^{51} -5.65685 q^{53} +11.3137 q^{55} -2.00000 q^{57} -4.24264 q^{61} -4.24264 q^{63} +12.0000 q^{65} +4.00000 q^{67} +2.82843 q^{69} -2.82843 q^{71} +16.0000 q^{73} -3.00000 q^{75} -16.9706 q^{77} +4.24264 q^{79} +1.00000 q^{81} +16.0000 q^{83} +16.9706 q^{85} +5.65685 q^{87} +14.0000 q^{89} -18.0000 q^{91} -4.24264 q^{93} +5.65685 q^{95} -4.00000 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} + 8 q^{11} + 12 q^{17} + 4 q^{19} + 6 q^{25} - 2 q^{27} - 8 q^{33} - 24 q^{35} - 20 q^{41} - 12 q^{43} + 22 q^{49} - 12 q^{51} - 4 q^{57} + 24 q^{65} + 8 q^{67} + 32 q^{73} - 6 q^{75} + 2 q^{81} + 32 q^{83} + 28 q^{89} - 36 q^{91} - 8 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 8 * q^11 + 12 * q^17 + 4 * q^19 + 6 * q^25 - 2 * q^27 - 8 * q^33 - 24 * q^35 - 20 * q^41 - 12 * q^43 + 22 * q^49 - 12 * q^51 - 4 * q^57 + 24 * q^65 + 8 * q^67 + 32 * q^73 - 6 * q^75 + 2 * q^81 + 32 * q^83 + 28 * q^89 - 36 * q^91 - 8 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	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
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	4.24264	0.925820
$22$	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
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.65685	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$30$	0	0
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	4.24264	0.697486	0.348743	−	0.937218i	$-0.386609\pi$
$37$	4.24264	0.697486	0.348743	+	0.937218i	$0.386609\pi$
$38$	0	0
$39$	−4.24264	−0.679366
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−5.65685	−0.777029	−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685	−0.777029	−0.388514	+	0.921443i	$0.627012\pi$
$54$	0	0
$55$	11.3137	1.52554
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−4.24264	−0.543214	−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264	−0.543214	−0.271607	+	0.962408i	$0.587555\pi$
$62$	0	0
$63$	−4.24264	−0.534522
$64$	0	0
$65$	12.0000	1.48842
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	2.82843	0.340503
$70$	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	$-0.114200\pi$
$73$	16.0000	1.87266	0.936329	+	0.351123i	$0.114200\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−16.9706	−1.93398
$78$	0	0
$79$	4.24264	0.477334	0.238667	−	0.971101i	$-0.423290\pi$
$79$	4.24264	0.477334	0.238667	+	0.971101i	$0.423290\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0000	1.75623	0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000	1.75623	0.878114	+	0.478451i	$0.158802\pi$
$84$	0	0
$85$	16.9706	1.84072
$86$	0	0
$87$	5.65685	0.606478
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	0	0
$93$	−4.24264	−0.439941
$94$	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
$99$	4.00000	0.402015
$100$	0	0
$101$	11.3137	1.12576	0.562878	−	0.826540i	$-0.309694\pi$
$101$	11.3137	1.12576	0.562878	+	0.826540i	$0.309694\pi$
$102$	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
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−9.89949	−0.948200	−0.474100	−	0.880471i	$-0.657226\pi$
$109$	−9.89949	−0.948200	−0.474100	+	0.880471i	$0.657226\pi$
$110$	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
$115$	−8.00000	−0.746004
$116$	0	0
$117$	4.24264	0.392232
$118$	0	0
$119$	−25.4558	−2.33353
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	10.0000	0.901670
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	4.24264	0.376473	0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264	0.376473	0.188237	+	0.982124i	$0.439723\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	−8.48528	−0.735767
$134$	0	0
$135$	−2.82843	−0.243432
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−2.82843	−0.238197
$142$	0	0
$143$	16.9706	1.41915
$144$	0	0
$145$	−16.0000	−1.32873
$146$	0	0
$147$	−11.0000	−0.907265
$148$	0	0
$149$	−8.48528	−0.695141	−0.347571	−	0.937654i	$-0.612993\pi$
$149$	−8.48528	−0.695141	−0.347571	+	0.937654i	$0.612993\pi$
$150$	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
$155$	12.0000	0.963863
$156$	0	0
$157$	−7.07107	−0.564333	−0.282166	−	0.959366i	$-0.591053\pi$
$157$	−7.07107	−0.564333	−0.282166	+	0.959366i	$0.591053\pi$
$158$	0	0
$159$	5.65685	0.448618
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	−11.3137	−0.880771
$166$	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
$171$	2.00000	0.152944
$172$	0	0
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	0	0
$175$	−12.7279	−0.962140
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	1.41421	0.105118	0.0525588	−	0.998618i	$-0.483262\pi$
$181$	1.41421	0.105118	0.0525588	+	0.998618i	$0.483262\pi$
$182$	0	0
$183$	4.24264	0.313625
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	24.0000	1.75505
$188$	0	0
$189$	4.24264	0.308607
$190$	0	0
$191$	−22.6274	−1.63726	−0.818631	−	0.574320i	$-0.805267\pi$
$191$	−22.6274	−1.63726	−0.818631	+	0.574320i	$0.805267\pi$
$192$	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
$195$	−12.0000	−0.859338
$196$	0	0
$197$	5.65685	0.403034	0.201517	−	0.979485i	$-0.435413\pi$
$197$	5.65685	0.403034	0.201517	+	0.979485i	$0.435413\pi$
$198$	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
$203$	24.0000	1.68447
$204$	0	0
$205$	−28.2843	−1.97546
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	2.82843	0.193801
$214$	0	0
$215$	−16.9706	−1.15738
$216$	0	0
$217$	−18.0000	−1.22192
$218$	0	0
$219$	−16.0000	−1.08118
$220$	0	0
$221$	25.4558	1.71235
$222$	0	0
$223$	−21.2132	−1.42054	−0.710271	−	0.703929i	$-0.751427\pi$
$223$	−21.2132	−1.42054	−0.710271	+	0.703929i	$0.751427\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	15.5563	1.02799	0.513996	−	0.857792i	$-0.328165\pi$
$229$	15.5563	1.02799	0.513996	+	0.857792i	$0.328165\pi$
$230$	0	0
$231$	16.9706	1.11658
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−4.24264	−0.275589
$238$	0	0
$239$	−11.3137	−0.731823	−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137	−0.731823	−0.365911	+	0.930650i	$0.619243\pi$
$240$	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
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	31.1127	1.98772
$246$	0	0
$247$	8.48528	0.539906
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	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
$259$	−18.0000	−1.11847
$260$	0	0
$261$	−5.65685	−0.350150
$262$	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
$267$	−14.0000	−0.856786
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	1.41421	0.0859074	0.0429537	−	0.999077i	$-0.486323\pi$
$271$	1.41421	0.0859074	0.0429537	+	0.999077i	$0.486323\pi$
$272$	0	0
$273$	18.0000	1.08941
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−1.41421	−0.0849719	−0.0424859	−	0.999097i	$-0.513528\pi$
$277$	−1.41421	−0.0849719	−0.0424859	+	0.999097i	$0.513528\pi$
$278$	0	0
$279$	4.24264	0.254000
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−5.65685	−0.335083
$286$	0	0
$287$	42.4264	2.50435
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	5.65685	0.330477	0.165238	−	0.986254i	$-0.447161\pi$
$293$	5.65685	0.330477	0.165238	+	0.986254i	$0.447161\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	25.4558	1.46725
$302$	0	0
$303$	−11.3137	−0.649956
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−18.3848	−1.04587
$310$	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	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	−16.9706	−0.953162	−0.476581	−	0.879131i	$-0.658124\pi$
$317$	−16.9706	−0.953162	−0.476581	+	0.879131i	$0.658124\pi$
$318$	0	0
$319$	−22.6274	−1.26689
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	12.7279	0.706018
$326$	0	0
$327$	9.89949	0.547443
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	4.24264	0.232495
$334$	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
$339$	6.00000	0.325875
$340$	0	0
$341$	16.9706	0.919007
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	4.24264	0.227103	0.113552	−	0.993532i	$-0.463777\pi$
$349$	4.24264	0.227103	0.113552	+	0.993532i	$0.463777\pi$
$350$	0	0
$351$	−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
$355$	−8.00000	−0.424596
$356$	0	0
$357$	25.4558	1.34727
$358$	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
$363$	−5.00000	−0.262432
$364$	0	0
$365$	45.2548	2.36875
$366$	0	0
$367$	−18.3848	−0.959678	−0.479839	−	0.877357i	$-0.659305\pi$
$367$	−18.3848	−0.959678	−0.479839	+	0.877357i	$0.659305\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	−18.3848	−0.951928	−0.475964	−	0.879465i	$-0.657901\pi$
$373$	−18.3848	−0.951928	−0.475964	+	0.879465i	$0.657901\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−38.0000	−1.95193	−0.975964	−	0.217930i	$-0.930070\pi$
$379$	−38.0000	−1.95193	−0.975964	+	0.217930i	$0.930070\pi$
$380$	0	0
$381$	−4.24264	−0.217357
$382$	0	0
$383$	28.2843	1.44526	0.722629	−	0.691236i	$-0.242933\pi$
$383$	28.2843	1.44526	0.722629	+	0.691236i	$0.242933\pi$
$384$	0	0
$385$	−48.0000	−2.44631
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	−2.82843	−0.143407	−0.0717035	−	0.997426i	$-0.522844\pi$
$389$	−2.82843	−0.143407	−0.0717035	+	0.997426i	$0.522844\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	32.5269	1.63248	0.816239	−	0.577714i	$-0.196055\pi$
$397$	32.5269	1.63248	0.816239	+	0.577714i	$0.196055\pi$
$398$	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
$403$	18.0000	0.896644
$404$	0	0
$405$	2.82843	0.140546
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	0	0
$411$	−10.0000	−0.493264
$412$	0	0
$413$	0	0
$414$	0	0
$415$	45.2548	2.22147
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	−35.3553	−1.72311	−0.861557	−	0.507661i	$-0.830510\pi$
$421$	−35.3553	−1.72311	−0.861557	+	0.507661i	$0.830510\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	18.0000	0.873128
$426$	0	0
$427$	18.0000	0.871081
$428$	0	0
$429$	−16.9706	−0.819346
$430$	0	0
$431$	36.7696	1.77113	0.885564	−	0.464518i	$-0.153773\pi$
$431$	36.7696	1.77113	0.885564	+	0.464518i	$0.153773\pi$
$432$	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
$435$	16.0000	0.767141
$436$	0	0
$437$	−5.65685	−0.270604
$438$	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
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	39.5980	1.87712
$446$	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
$451$	−40.0000	−1.88353
$452$	0	0
$453$	−1.41421	−0.0664455
$454$	0	0
$455$	−50.9117	−2.38678
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	8.48528	0.395199	0.197599	−	0.980283i	$-0.436685\pi$
$461$	8.48528	0.395199	0.197599	+	0.980283i	$0.436685\pi$
$462$	0	0
$463$	−12.7279	−0.591517	−0.295758	−	0.955263i	$-0.595572\pi$
$463$	−12.7279	−0.591517	−0.295758	+	0.955263i	$0.595572\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−16.9706	−0.783628
$470$	0	0
$471$	7.07107	0.325818
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	6.00000	0.275299
$476$	0	0
$477$	−5.65685	−0.259010
$478$	0	0
$479$	25.4558	1.16311	0.581554	−	0.813508i	$-0.302445\pi$
$479$	25.4558	1.16311	0.581554	+	0.813508i	$0.302445\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	−12.0000	−0.546019
$484$	0	0
$485$	−11.3137	−0.513729
$486$	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
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−33.9411	−1.52863
$494$	0	0
$495$	11.3137	0.508513
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	−11.3137	−0.505459
$502$	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
$507$	−5.00000	−0.222058
$508$	0	0
$509$	−22.6274	−1.00294	−0.501471	−	0.865174i	$-0.667208\pi$
$509$	−22.6274	−1.00294	−0.501471	+	0.865174i	$0.667208\pi$
$510$	0	0
$511$	−67.8823	−3.00293
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	52.0000	2.29139
$516$	0	0
$517$	11.3137	0.497576
$518$	0	0
$519$	2.82843	0.124154
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	12.7279	0.555492
$526$	0	0
$527$	25.4558	1.10887
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−42.4264	−1.83769
$534$	0	0
$535$	11.3137	0.489134
$536$	0	0
$537$	0	0
$538$	0	0
$539$	44.0000	1.89521
$540$	0	0
$541$	−41.0122	−1.76325	−0.881626	−	0.471949i	$-0.843551\pi$
$541$	−41.0122	−1.76325	−0.881626	+	0.471949i	$0.843551\pi$
$542$	0	0
$543$	−1.41421	−0.0606897
$544$	0	0
$545$	−28.0000	−1.19939
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	0	0
$549$	−4.24264	−0.181071
$550$	0	0
$551$	−11.3137	−0.481980
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	−19.7990	−0.838910	−0.419455	−	0.907776i	$-0.637779\pi$
$557$	−19.7990	−0.838910	−0.419455	+	0.907776i	$0.637779\pi$
$558$	0	0
$559$	−25.4558	−1.07667
$560$	0	0
$561$	−24.0000	−1.01328
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−16.9706	−0.713957
$566$	0	0
$567$	−4.24264	−0.178174
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	22.6274	0.945274
$574$	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
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−67.8823	−2.81623
$582$	0	0
$583$	−22.6274	−0.937132
$584$	0	0
$585$	12.0000	0.496139
$586$	0	0
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	8.48528	0.349630
$590$	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
$595$	−72.0000	−2.95171
$596$	0	0
$597$	7.07107	0.289400
$598$	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	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	14.1421	0.574960
$606$	0	0
$607$	4.24264	0.172203	0.0861017	−	0.996286i	$-0.472559\pi$
$607$	4.24264	0.172203	0.0861017	+	0.996286i	$0.472559\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−18.3848	−0.742554	−0.371277	−	0.928522i	$-0.621080\pi$
$613$	−18.3848	−0.742554	−0.371277	+	0.928522i	$0.621080\pi$
$614$	0	0
$615$	28.2843	1.14053
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	2.82843	0.113501
$622$	0	0
$623$	−59.3970	−2.37969
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	25.4558	1.01499
$630$	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
$635$	12.0000	0.476205
$636$	0	0
$637$	46.6690	1.84909
$638$	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
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	0	0
$645$	16.9706	0.668215
$646$	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
$651$	18.0000	0.705476
$652$	0	0
$653$	−8.48528	−0.332055	−0.166027	−	0.986121i	$-0.553094\pi$
$653$	−8.48528	−0.332055	−0.166027	+	0.986121i	$0.553094\pi$
$654$	0	0
$655$	56.5685	2.21032
$656$	0	0
$657$	16.0000	0.624219
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	12.7279	0.495059	0.247529	−	0.968880i	$-0.420381\pi$
$661$	12.7279	0.495059	0.247529	+	0.968880i	$0.420381\pi$
$662$	0	0
$663$	−25.4558	−0.988623
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	21.2132	0.820150
$670$	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
$675$	−3.00000	−0.115470
$676$	0	0
$677$	−36.7696	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$677$	−36.7696	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$678$	0	0
$679$	16.9706	0.651270
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	28.2843	1.08069
$686$	0	0
$687$	−15.5563	−0.593512
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−22.0000	−0.836919	−0.418460	−	0.908235i	$-0.637430\pi$
$691$	−22.0000	−0.836919	−0.418460	+	0.908235i	$0.637430\pi$
$692$	0	0
$693$	−16.9706	−0.644658
$694$	0	0
$695$	−33.9411	−1.28746
$696$	0	0
$697$	−60.0000	−2.27266
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	5.65685	0.213656	0.106828	−	0.994277i	$-0.465931\pi$
$701$	5.65685	0.213656	0.106828	+	0.994277i	$0.465931\pi$
$702$	0	0
$703$	8.48528	0.320028
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	−48.0000	−1.80523
$708$	0	0
$709$	18.3848	0.690455	0.345227	−	0.938519i	$-0.387802\pi$
$709$	18.3848	0.690455	0.345227	+	0.938519i	$0.387802\pi$
$710$	0	0
$711$	4.24264	0.159111
$712$	0	0
$713$	−12.0000	−0.449404
$714$	0	0
$715$	48.0000	1.79510
$716$	0	0
$717$	11.3137	0.422518
$718$	0	0
$719$	−8.48528	−0.316448	−0.158224	−	0.987403i	$-0.550577\pi$
$719$	−8.48528	−0.316448	−0.158224	+	0.987403i	$0.550577\pi$
$720$	0	0
$721$	−78.0000	−2.90487
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	−16.9706	−0.630271
$726$	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
$731$	−36.0000	−1.33151
$732$	0	0
$733$	35.3553	1.30588	0.652940	−	0.757410i	$-0.273536\pi$
$733$	35.3553	1.30588	0.652940	+	0.757410i	$0.273536\pi$
$734$	0	0
$735$	−31.1127	−1.14761
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	−8.48528	−0.311715
$742$	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
$747$	16.0000	0.585409
$748$	0	0
$749$	−16.9706	−0.620091
$750$	0	0
$751$	12.7279	0.464448	0.232224	−	0.972662i	$-0.425400\pi$
$751$	12.7279	0.464448	0.232224	+	0.972662i	$0.425400\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	−38.1838	−1.38781	−0.693906	−	0.720065i	$-0.744112\pi$
$757$	−38.1838	−1.38781	−0.693906	+	0.720065i	$0.744112\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	0	0
$763$	42.0000	1.52050
$764$	0	0
$765$	16.9706	0.613572
$766$	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
$771$	14.0000	0.504198
$772$	0	0
$773$	−5.65685	−0.203463	−0.101731	−	0.994812i	$-0.532438\pi$
$773$	−5.65685	−0.203463	−0.101731	+	0.994812i	$0.532438\pi$
$774$	0	0
$775$	12.7279	0.457200
$776$	0	0
$777$	18.0000	0.645746
$778$	0	0
$779$	−20.0000	−0.716574
$780$	0	0
$781$	−11.3137	−0.404836
$782$	0	0
$783$	5.65685	0.202159
$784$	0	0
$785$	−20.0000	−0.713831
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	0	0
$789$	22.6274	0.805557
$790$	0	0
$791$	25.4558	0.905106
$792$	0	0
$793$	−18.0000	−0.639199
$794$	0	0
$795$	16.0000	0.567462
$796$	0	0
$797$	31.1127	1.10207	0.551034	−	0.834483i	$-0.314233\pi$
$797$	31.1127	1.10207	0.551034	+	0.834483i	$0.314233\pi$
$798$	0	0
$799$	16.9706	0.600375
$800$	0	0
$801$	14.0000	0.494666
$802$	0	0
$803$	64.0000	2.25851
$804$	0	0
$805$	33.9411	1.19627
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	−1.41421	−0.0495986
$814$	0	0
$815$	28.2843	0.990755
$816$	0	0
$817$	−12.0000	−0.419827
$818$	0	0
$819$	−18.0000	−0.628971
$820$	0	0
$821$	28.2843	0.987128	0.493564	−	0.869710i	$-0.335694\pi$
$821$	28.2843	0.987128	0.493564	+	0.869710i	$0.335694\pi$
$822$	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
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−21.2132	−0.736765	−0.368383	−	0.929674i	$-0.620088\pi$
$829$	−21.2132	−0.736765	−0.368383	+	0.929674i	$0.620088\pi$
$830$	0	0
$831$	1.41421	0.0490585
$832$	0	0
$833$	66.0000	2.28676
$834$	0	0
$835$	32.0000	1.10741
$836$	0	0
$837$	−4.24264	−0.146647
$838$	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
$843$	−6.00000	−0.206651
$844$	0	0
$845$	14.1421	0.486504
$846$	0	0
$847$	−21.2132	−0.728894
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	21.2132	0.726326	0.363163	−	0.931726i	$-0.381697\pi$
$853$	21.2132	0.726326	0.363163	+	0.931726i	$0.381697\pi$
$854$	0	0
$855$	5.65685	0.193460
$856$	0	0
$857$	30.0000	1.02478	0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000	1.02478	0.512390	+	0.858753i	$0.328760\pi$
$858$	0	0
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	0	0
$861$	−42.4264	−1.44589
$862$	0	0
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	−19.0000	−0.645274
$868$	0	0
$869$	16.9706	0.575687
$870$	0	0
$871$	16.9706	0.575026
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	−15.5563	−0.525301	−0.262650	−	0.964891i	$-0.584597\pi$
$877$	−15.5563	−0.525301	−0.262650	+	0.964891i	$0.584597\pi$
$878$	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
$883$	34.0000	1.14419	0.572096	−	0.820187i	$-0.306131\pi$
$883$	34.0000	1.14419	0.572096	+	0.820187i	$0.306131\pi$
$884$	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
$891$	4.00000	0.134005
$892$	0	0
$893$	5.65685	0.189299
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	0	0
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−33.9411	−1.13074
$902$	0	0
$903$	−25.4558	−0.847117
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	−58.0000	−1.92586	−0.962929	−	0.269754i	$-0.913058\pi$
$907$	−58.0000	−1.92586	−0.962929	+	0.269754i	$0.913058\pi$
$908$	0	0
$909$	11.3137	0.375252
$910$	0	0
$911$	22.6274	0.749680	0.374840	−	0.927090i	$-0.377698\pi$
$911$	22.6274	0.749680	0.374840	+	0.927090i	$0.377698\pi$
$912$	0	0
$913$	64.0000	2.11809
$914$	0	0
$915$	12.0000	0.396708
$916$	0	0
$917$	−84.8528	−2.80209
$918$	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
$923$	−12.0000	−0.394985
$924$	0	0
$925$	12.7279	0.418491
$926$	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
$931$	22.0000	0.721021
$932$	0	0
$933$	−28.2843	−0.925985
$934$	0	0
$935$	67.8823	2.21999
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	36.7696	1.19865	0.599327	−	0.800505i	$-0.295435\pi$
$941$	36.7696	1.19865	0.599327	+	0.800505i	$0.295435\pi$
$942$	0	0
$943$	28.2843	0.921063
$944$	0	0
$945$	12.0000	0.390360
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	67.8823	2.20355
$950$	0	0
$951$	16.9706	0.550308
$952$	0	0
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	0	0
$955$	−64.0000	−2.07099
$956$	0	0
$957$	22.6274	0.731441
$958$	0	0
$959$	−42.4264	−1.37002
$960$	0	0
$961$	−13.0000	−0.419355
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	50.9117	1.63891
$966$	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
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	0	0
$973$	50.9117	1.63215
$974$	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
$979$	56.0000	1.78977
$980$	0	0
$981$	−9.89949	−0.316067
$982$	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
$987$	12.0000	0.381964
$988$	0	0
$989$	16.9706	0.539633
$990$	0	0
$991$	46.6690	1.48249	0.741246	−	0.671234i	$-0.234235\pi$
$991$	46.6690	1.48249	0.741246	+	0.671234i	$0.234235\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−20.0000	−0.634043
$996$	0	0
$997$	−46.6690	−1.47802	−0.739012	−	0.673693i	$-0.764707\pi$
$997$	−46.6690	−1.47802	−0.739012	+	0.673693i	$0.764707\pi$
$998$	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.a.d.1.2		2
3.2	odd	2		9216.2.a.e.1.1		2
4.3	odd	2		3072.2.a.f.1.2		2
8.3	odd	2	inner	3072.2.a.d.1.1		2
8.5	even	2		3072.2.a.f.1.1		2
12.11	even	2		9216.2.a.q.1.1		2
16.3	odd	4		3072.2.d.d.1537.3		4
16.5	even	4		3072.2.d.d.1537.4		4
16.11	odd	4		3072.2.d.d.1537.2		4
16.13	even	4		3072.2.d.d.1537.1		4
24.5	odd	2		9216.2.a.q.1.2		2
24.11	even	2		9216.2.a.e.1.2		2
32.3	odd	8		768.2.j.a.577.2	yes	4
32.5	even	8		768.2.j.d.193.2	yes	4
32.11	odd	8		768.2.j.a.193.2	yes	4
32.13	even	8		768.2.j.d.577.2	yes	4
32.19	odd	8		768.2.j.d.577.1	yes	4
32.21	even	8		768.2.j.a.193.1	✓	4
32.27	odd	8		768.2.j.d.193.1	yes	4
32.29	even	8		768.2.j.a.577.1	yes	4
96.5	odd	8		2304.2.k.a.1729.1		4
96.11	even	8		2304.2.k.d.1729.2		4
96.29	odd	8		2304.2.k.d.577.2		4
96.35	even	8		2304.2.k.d.577.1		4
96.53	odd	8		2304.2.k.d.1729.1		4
96.59	even	8		2304.2.k.a.1729.2		4
96.77	odd	8		2304.2.k.a.577.2		4
96.83	even	8		2304.2.k.a.577.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.a.193.1	✓	4	32.21	even	8
768.2.j.a.193.2	yes	4	32.11	odd	8
768.2.j.a.577.1	yes	4	32.29	even	8
768.2.j.a.577.2	yes	4	32.3	odd	8
768.2.j.d.193.1	yes	4	32.27	odd	8
768.2.j.d.193.2	yes	4	32.5	even	8
768.2.j.d.577.1	yes	4	32.19	odd	8
768.2.j.d.577.2	yes	4	32.13	even	8
2304.2.k.a.577.1		4	96.83	even	8
2304.2.k.a.577.2		4	96.77	odd	8
2304.2.k.a.1729.1		4	96.5	odd	8
2304.2.k.a.1729.2		4	96.59	even	8
2304.2.k.d.577.1		4	96.35	even	8
2304.2.k.d.577.2		4	96.29	odd	8
2304.2.k.d.1729.1		4	96.53	odd	8
2304.2.k.d.1729.2		4	96.11	even	8
3072.2.a.d.1.1		2	8.3	odd	2	inner
3072.2.a.d.1.2		2	1.1	even	1	trivial
3072.2.a.f.1.1		2	8.5	even	2
3072.2.a.f.1.2		2	4.3	odd	2
3072.2.d.d.1537.1		4	16.13	even	4
3072.2.d.d.1537.2		4	16.11	odd	4
3072.2.d.d.1537.3		4	16.3	odd	4
3072.2.d.d.1537.4		4	16.5	even	4
9216.2.a.e.1.1		2	3.2	odd	2
9216.2.a.e.1.2		2	24.11	even	2
9216.2.a.q.1.1		2	12.11	even	2
9216.2.a.q.1.2		2	24.5	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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.82843 q^{5} -4.24264 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.82843 q^{5} -4.24264 q^{7} +1.00000 q^{9} +4.00000 q^{11} +4.24264 q^{13} -2.82843 q^{15} +6.00000 q^{17} +2.00000 q^{19} +4.24264 q^{21} -2.82843 q^{23} +3.00000 q^{25} -1.00000 q^{27} -5.65685 q^{29} +4.24264 q^{31} -4.00000 q^{33} -12.0000 q^{35} +4.24264 q^{37} -4.24264 q^{39} -10.0000 q^{41} -6.00000 q^{43} +2.82843 q^{45} +2.82843 q^{47} +11.0000 q^{49} -6.00000 q^{51} -5.65685 q^{53} +11.3137 q^{55} -2.00000 q^{57} -4.24264 q^{61} -4.24264 q^{63} +12.0000 q^{65} +4.00000 q^{67} +2.82843 q^{69} -2.82843 q^{71} +16.0000 q^{73} -3.00000 q^{75} -16.9706 q^{77} +4.24264 q^{79} +1.00000 q^{81} +16.0000 q^{83} +16.9706 q^{85} +5.65685 q^{87} +14.0000 q^{89} -18.0000 q^{91} -4.24264 q^{93} +5.65685 q^{95} -4.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 8 q^{11} + 12 q^{17} + 4 q^{19} + 6 q^{25} - 2 q^{27} - 8 q^{33} - 24 q^{35} - 20 q^{41} - 12 q^{43} + 22 q^{49} - 12 q^{51} - 4 q^{57} + 24 q^{65} + 8 q^{67} + 32 q^{73} - 6 q^{75} + 2 q^{81} + 32 q^{83} + 28 q^{89} - 36 q^{91} - 8 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 8 * q^11 + 12 * q^17 + 4 * q^19 + 6 * q^25 - 2 * q^27 - 8 * q^33 - 24 * q^35 - 20 * q^41 - 12 * q^43 + 22 * q^49 - 12 * q^51 - 4 * q^57 + 24 * q^65 + 8 * q^67 + 32 * q^73 - 6 * q^75 + 2 * q^81 + 32 * q^83 + 28 * q^89 - 36 * q^91 - 8 * q^97 + 8 * q^99

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