LMFDB - Embedded newform 3072.2.a.b.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1536)
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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.41421 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.41421 q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.24264 q^{13} -1.41421 q^{15} +4.00000 q^{17} -8.00000 q^{19} -2.82843 q^{21} -5.65685 q^{23} -3.00000 q^{25} -1.00000 q^{27} -1.41421 q^{29} -2.82843 q^{31} +4.00000 q^{35} -4.24264 q^{37} +4.24264 q^{39} -4.00000 q^{41} +1.41421 q^{45} -11.3137 q^{47} +1.00000 q^{49} -4.00000 q^{51} +7.07107 q^{53} +8.00000 q^{57} -12.0000 q^{59} -1.41421 q^{61} +2.82843 q^{63} -6.00000 q^{65} +4.00000 q^{67} +5.65685 q^{69} +11.3137 q^{71} +14.0000 q^{73} +3.00000 q^{75} -8.48528 q^{79} +1.00000 q^{81} -16.0000 q^{83} +5.65685 q^{85} +1.41421 q^{87} +6.00000 q^{89} -12.0000 q^{91} +2.82843 q^{93} -11.3137 q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{17} - 16 q^{19} - 6 q^{25} - 2 q^{27} + 8 q^{35} - 8 q^{41} + 2 q^{49} - 8 q^{51} + 16 q^{57} - 24 q^{59} - 12 q^{65} + 8 q^{67} + 28 q^{73} + 6 q^{75} + 2 q^{81} - 32 q^{83} + 12 q^{89} - 24 q^{91} + 32 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 8 * q^17 - 16 * q^19 - 6 * q^25 - 2 * q^27 + 8 * q^35 - 8 * q^41 + 2 * q^49 - 8 * q^51 + 16 * q^57 - 24 * q^59 - 12 * q^65 + 8 * q^67 + 28 * q^73 + 6 * q^75 + 2 * q^81 - 32 * q^83 + 12 * q^89 - 24 * q^91 + 32 * q^97

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
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	2.82843	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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.41421	−0.262613	−0.131306	−	0.991342i	$-0.541917\pi$
$29$	−1.41421	−0.262613	−0.131306	+	0.991342i	$0.541917\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	0	0
$39$	4.24264	0.679366
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.41421	0.210819
$46$	0	0
$47$	−11.3137	−1.65027	−0.825137	−	0.564933i	$-0.808902\pi$
$47$	−11.3137	−1.65027	−0.825137	+	0.564933i	$0.808902\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	7.07107	0.971286	0.485643	−	0.874157i	$-0.338586\pi$
$53$	7.07107	0.971286	0.485643	+	0.874157i	$0.338586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−1.41421	−0.181071	−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	−1.41421	−0.181071	−0.0905357	+	0.995893i	$0.528858\pi$
$62$	0	0
$63$	2.82843	0.356348
$64$	0	0
$65$	−6.00000	−0.744208
$66$	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
$69$	5.65685	0.681005
$70$	0	0
$71$	11.3137	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$71$	11.3137	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	3.00000	0.346410
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	5.65685	0.613572
$86$	0	0
$87$	1.41421	0.151620
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	2.82843	0.293294
$94$	0	0
$95$	−11.3137	−1.16076
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.5563	1.54791	0.773957	−	0.633238i	$-0.218274\pi$
$101$	15.5563	1.54791	0.773957	+	0.633238i	$0.218274\pi$
$102$	0	0
$103$	19.7990	1.95085	0.975426	−	0.220326i	$-0.0707122\pi$
$103$	19.7990	1.95085	0.975426	+	0.220326i	$0.0707122\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	4.24264	0.406371	0.203186	−	0.979140i	$-0.434871\pi$
$109$	4.24264	0.406371	0.203186	+	0.979140i	$0.434871\pi$
$110$	0	0
$111$	4.24264	0.402694
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	−4.24264	−0.392232
$118$	0	0
$119$	11.3137	1.03713
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	8.48528	0.752947	0.376473	−	0.926427i	$-0.377137\pi$
$127$	8.48528	0.752947	0.376473	+	0.926427i	$0.377137\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−22.6274	−1.96205
$134$	0	0
$135$	−1.41421	−0.121716
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	11.3137	0.952786
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−12.7279	−1.04271	−0.521356	−	0.853339i	$-0.674574\pi$
$149$	−12.7279	−1.04271	−0.521356	+	0.853339i	$0.674574\pi$
$150$	0	0
$151$	−2.82843	−0.230174	−0.115087	−	0.993355i	$-0.536715\pi$
$151$	−2.82843	−0.230174	−0.115087	+	0.993355i	$0.536715\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	21.2132	1.69300	0.846499	−	0.532390i	$-0.178706\pi$
$157$	21.2132	1.69300	0.846499	+	0.532390i	$0.178706\pi$
$158$	0	0
$159$	−7.07107	−0.560772
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	0	0
$166$	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
$171$	−8.00000	−0.611775
$172$	0	0
$173$	−7.07107	−0.537603	−0.268802	−	0.963196i	$-0.586628\pi$
$173$	−7.07107	−0.537603	−0.268802	+	0.963196i	$0.586628\pi$
$174$	0	0
$175$	−8.48528	−0.641427
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	12.7279	0.946059	0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279	0.946059	0.473029	+	0.881047i	$0.343160\pi$
$182$	0	0
$183$	1.41421	0.104542
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.82843	−0.205738
$190$	0	0
$191$	5.65685	0.409316	0.204658	−	0.978834i	$-0.434392\pi$
$191$	5.65685	0.409316	0.204658	+	0.978834i	$0.434392\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	7.07107	0.503793	0.251896	−	0.967754i	$-0.418946\pi$
$197$	7.07107	0.503793	0.251896	+	0.967754i	$0.418946\pi$
$198$	0	0
$199$	−8.48528	−0.601506	−0.300753	−	0.953702i	$-0.597238\pi$
$199$	−8.48528	−0.601506	−0.300753	+	0.953702i	$0.597238\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	−4.00000	−0.280745
$204$	0	0
$205$	−5.65685	−0.395092
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	−11.3137	−0.775203
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−16.9706	−1.14156
$222$	0	0
$223$	−25.4558	−1.70465	−0.852325	−	0.523013i	$-0.824808\pi$
$223$	−25.4558	−1.70465	−0.852325	+	0.523013i	$0.824808\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	−24.0416	−1.58872	−0.794358	−	0.607450i	$-0.792192\pi$
$229$	−24.0416	−1.58872	−0.794358	+	0.607450i	$0.792192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	8.48528	0.551178
$238$	0	0
$239$	5.65685	0.365911	0.182956	−	0.983121i	$-0.441433\pi$
$239$	5.65685	0.365911	0.182956	+	0.983121i	$0.441433\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.41421	0.0903508
$246$	0	0
$247$	33.9411	2.15962
$248$	0	0
$249$	16.0000	1.01396
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	0	0
$261$	−1.41421	−0.0875376
$262$	0	0
$263$	−28.2843	−1.74408	−0.872041	−	0.489432i	$-0.837204\pi$
$263$	−28.2843	−1.74408	−0.872041	+	0.489432i	$0.837204\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−21.2132	−1.29339	−0.646696	−	0.762748i	$-0.723850\pi$
$269$	−21.2132	−1.29339	−0.646696	+	0.762748i	$0.723850\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	0	0
$273$	12.0000	0.726273
$274$	0	0
$275$	0	0
$276$	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
$279$	−2.82843	−0.169334
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	11.3137	0.670166
$286$	0	0
$287$	−11.3137	−0.667827
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	18.3848	1.07405	0.537025	−	0.843566i	$-0.319548\pi$
$293$	18.3848	1.07405	0.537025	+	0.843566i	$0.319548\pi$
$294$	0	0
$295$	−16.9706	−0.988064
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−15.5563	−0.893689
$304$	0	0
$305$	−2.00000	−0.114520
$306$	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
$309$	−19.7990	−1.12633
$310$	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
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	21.2132	1.19145	0.595726	−	0.803188i	$-0.296864\pi$
$317$	21.2132	1.19145	0.595726	+	0.803188i	$0.296864\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−32.0000	−1.78053
$324$	0	0
$325$	12.7279	0.706018
$326$	0	0
$327$	−4.24264	−0.234619
$328$	0	0
$329$	−32.0000	−1.76422
$330$	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
$335$	5.65685	0.309067
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	8.00000	0.430706
$346$	0	0
$347$	32.0000	1.71785	0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000	1.71785	0.858925	+	0.512101i	$0.171133\pi$
$348$	0	0
$349$	12.7279	0.681310	0.340655	−	0.940188i	$-0.389351\pi$
$349$	12.7279	0.681310	0.340655	+	0.940188i	$0.389351\pi$
$350$	0	0
$351$	4.24264	0.226455
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	−11.3137	−0.598785
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	19.7990	1.03633
$366$	0	0
$367$	−2.82843	−0.147643	−0.0738213	−	0.997271i	$-0.523519\pi$
$367$	−2.82843	−0.147643	−0.0738213	+	0.997271i	$0.523519\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	20.0000	1.03835
$372$	0	0
$373$	4.24264	0.219676	0.109838	−	0.993950i	$-0.464967\pi$
$373$	4.24264	0.219676	0.109838	+	0.993950i	$0.464967\pi$
$374$	0	0
$375$	11.3137	0.584237
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	−8.48528	−0.434714
$382$	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
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.89949	0.501924	0.250962	−	0.967997i	$-0.419253\pi$
$389$	9.89949	0.501924	0.250962	+	0.967997i	$0.419253\pi$
$390$	0	0
$391$	−22.6274	−1.14432
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	21.2132	1.06466	0.532330	−	0.846537i	$-0.321317\pi$
$397$	21.2132	1.06466	0.532330	+	0.846537i	$0.321317\pi$
$398$	0	0
$399$	22.6274	1.13279
$400$	0	0
$401$	−20.0000	−0.998752	−0.499376	−	0.866385i	$-0.666437\pi$
$401$	−20.0000	−0.998752	−0.499376	+	0.866385i	$0.666437\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	0	0
$405$	1.41421	0.0702728
$406$	0	0
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	−33.9411	−1.67013
$414$	0	0
$415$	−22.6274	−1.11074
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	−8.00000	−0.390826	−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000	−0.390826	−0.195413	+	0.980721i	$0.562605\pi$
$420$	0	0
$421$	−12.7279	−0.620321	−0.310160	−	0.950684i	$-0.600383\pi$
$421$	−12.7279	−0.620321	−0.310160	+	0.950684i	$0.600383\pi$
$422$	0	0
$423$	−11.3137	−0.550091
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.3137	0.544962	0.272481	−	0.962161i	$-0.412156\pi$
$431$	11.3137	0.544962	0.272481	+	0.962161i	$0.412156\pi$
$432$	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
$435$	2.00000	0.0958927
$436$	0	0
$437$	45.2548	2.16483
$438$	0	0
$439$	2.82843	0.134993	0.0674967	−	0.997719i	$-0.478499\pi$
$439$	2.82843	0.134993	0.0674967	+	0.997719i	$0.478499\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	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
$445$	8.48528	0.402241
$446$	0	0
$447$	12.7279	0.602010
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.82843	0.132891
$454$	0	0
$455$	−16.9706	−0.795592
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−15.5563	−0.724531	−0.362266	−	0.932075i	$-0.617997\pi$
$461$	−15.5563	−0.724531	−0.362266	+	0.932075i	$0.617997\pi$
$462$	0	0
$463$	31.1127	1.44593	0.722965	−	0.690885i	$-0.242779\pi$
$463$	31.1127	1.44593	0.722965	+	0.690885i	$0.242779\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	11.3137	0.522419
$470$	0	0
$471$	−21.2132	−0.977453
$472$	0	0
$473$	0	0
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	7.07107	0.323762
$478$	0	0
$479$	−28.2843	−1.29234	−0.646171	−	0.763193i	$-0.723631\pi$
$479$	−28.2843	−1.29234	−0.646171	+	0.763193i	$0.723631\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	22.6274	1.02746
$486$	0	0
$487$	−2.82843	−0.128168	−0.0640841	−	0.997944i	$-0.520413\pi$
$487$	−2.82843	−0.128168	−0.0640841	+	0.997944i	$0.520413\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−5.65685	−0.254772
$494$	0	0
$495$	0	0
$496$	0	0
$497$	32.0000	1.43540
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	−11.3137	−0.505459
$502$	0	0
$503$	−11.3137	−0.504453	−0.252227	−	0.967668i	$-0.581163\pi$
$503$	−11.3137	−0.504453	−0.252227	+	0.967668i	$0.581163\pi$
$504$	0	0
$505$	22.0000	0.978987
$506$	0	0
$507$	−5.00000	−0.222058
$508$	0	0
$509$	−32.5269	−1.44173	−0.720865	−	0.693075i	$-0.756255\pi$
$509$	−32.5269	−1.44173	−0.720865	+	0.693075i	$0.756255\pi$
$510$	0	0
$511$	39.5980	1.75171
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	28.0000	1.23383
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.07107	0.310385
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	0	0
$525$	8.48528	0.370328
$526$	0	0
$527$	−11.3137	−0.492833
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	16.9706	0.735077
$534$	0	0
$535$	−16.9706	−0.733701
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.24264	0.182405	0.0912027	−	0.995832i	$-0.470929\pi$
$541$	4.24264	0.182405	0.0912027	+	0.995832i	$0.470929\pi$
$542$	0	0
$543$	−12.7279	−0.546207
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	−1.41421	−0.0603572
$550$	0	0
$551$	11.3137	0.481980
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	−29.6985	−1.25837	−0.629183	−	0.777258i	$-0.716610\pi$
$557$	−29.6985	−1.25837	−0.629183	+	0.777258i	$0.716610\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	−25.4558	−1.07094
$566$	0	0
$567$	2.82843	0.118783
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	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
$573$	−5.65685	−0.236318
$574$	0	0
$575$	16.9706	0.707721
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−45.2548	−1.87749
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	0	0
$589$	22.6274	0.932346
$590$	0	0
$591$	−7.07107	−0.290865
$592$	0	0
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	16.0000	0.655936
$596$	0	0
$597$	8.48528	0.347279
$598$	0	0
$599$	−39.5980	−1.61793	−0.808965	−	0.587857i	$-0.799972\pi$
$599$	−39.5980	−1.61793	−0.808965	+	0.587857i	$0.799972\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	−15.5563	−0.632456
$606$	0	0
$607$	42.4264	1.72203	0.861017	−	0.508576i	$-0.169828\pi$
$607$	42.4264	1.72203	0.861017	+	0.508576i	$0.169828\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	15.5563	0.628315	0.314158	−	0.949371i	$-0.398278\pi$
$613$	15.5563	0.628315	0.314158	+	0.949371i	$0.398278\pi$
$614$	0	0
$615$	5.65685	0.228106
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	5.65685	0.227002
$622$	0	0
$623$	16.9706	0.679911
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.9706	−0.676661
$630$	0	0
$631$	8.48528	0.337794	0.168897	−	0.985634i	$-0.445980\pi$
$631$	8.48528	0.337794	0.168897	+	0.985634i	$0.445980\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	12.0000	0.476205
$636$	0	0
$637$	−4.24264	−0.168100
$638$	0	0
$639$	11.3137	0.447563
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.5980	−1.55676	−0.778379	−	0.627795i	$-0.783958\pi$
$647$	−39.5980	−1.55676	−0.778379	+	0.627795i	$0.783958\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−4.24264	−0.166027	−0.0830137	−	0.996548i	$-0.526455\pi$
$653$	−4.24264	−0.166027	−0.0830137	+	0.996548i	$0.526455\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−26.8701	−1.04512	−0.522562	−	0.852601i	$-0.675024\pi$
$661$	−26.8701	−1.04512	−0.522562	+	0.852601i	$0.675024\pi$
$662$	0	0
$663$	16.9706	0.659082
$664$	0	0
$665$	−32.0000	−1.24091
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	25.4558	0.984180
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	3.00000	0.115470
$676$	0	0
$677$	32.5269	1.25011	0.625055	−	0.780580i	$-0.285076\pi$
$677$	32.5269	1.25011	0.625055	+	0.780580i	$0.285076\pi$
$678$	0	0
$679$	45.2548	1.73672
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	32.0000	1.22445	0.612223	−	0.790685i	$-0.290275\pi$
$683$	32.0000	1.22445	0.612223	+	0.790685i	$0.290275\pi$
$684$	0	0
$685$	−16.9706	−0.648412
$686$	0	0
$687$	24.0416	0.917245
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.65685	−0.214577
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	46.6690	1.76267	0.881333	−	0.472496i	$-0.156647\pi$
$701$	46.6690	1.76267	0.881333	+	0.472496i	$0.156647\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	44.0000	1.65479
$708$	0	0
$709$	35.3553	1.32780	0.663899	−	0.747822i	$-0.268901\pi$
$709$	35.3553	1.32780	0.663899	+	0.747822i	$0.268901\pi$
$710$	0	0
$711$	−8.48528	−0.318223
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.65685	−0.211259
$718$	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
$721$	56.0000	2.08555
$722$	0	0
$723$	−8.00000	−0.297523
$724$	0	0
$725$	4.24264	0.157568
$726$	0	0
$727$	−42.4264	−1.57351	−0.786754	−	0.617266i	$-0.788240\pi$
$727$	−42.4264	−1.57351	−0.786754	+	0.617266i	$0.788240\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−4.24264	−0.156706	−0.0783528	−	0.996926i	$-0.524966\pi$
$733$	−4.24264	−0.156706	−0.0783528	+	0.996926i	$0.524966\pi$
$734$	0	0
$735$	−1.41421	−0.0521641
$736$	0	0
$737$	0	0
$738$	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
$741$	−33.9411	−1.24686
$742$	0	0
$743$	−11.3137	−0.415060	−0.207530	−	0.978229i	$-0.566542\pi$
$743$	−11.3137	−0.415060	−0.207530	+	0.978229i	$0.566542\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	−16.0000	−0.585409
$748$	0	0
$749$	−33.9411	−1.24018
$750$	0	0
$751$	8.48528	0.309632	0.154816	−	0.987943i	$-0.450521\pi$
$751$	8.48528	0.309632	0.154816	+	0.987943i	$0.450521\pi$
$752$	0	0
$753$	−8.00000	−0.291536
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−21.2132	−0.771007	−0.385503	−	0.922706i	$-0.625972\pi$
$757$	−21.2132	−0.771007	−0.385503	+	0.922706i	$0.625972\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	12.0000	0.434429
$764$	0	0
$765$	5.65685	0.204524
$766$	0	0
$767$	50.9117	1.83831
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	−18.3848	−0.661254	−0.330627	−	0.943761i	$-0.607260\pi$
$773$	−18.3848	−0.661254	−0.330627	+	0.943761i	$0.607260\pi$
$774$	0	0
$775$	8.48528	0.304800
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	32.0000	1.14652
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.41421	0.0505399
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	0	0
$789$	28.2843	1.00695
$790$	0	0
$791$	−50.9117	−1.81021
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	−10.0000	−0.354663
$796$	0	0
$797$	−49.4975	−1.75329	−0.876645	−	0.481137i	$-0.840224\pi$
$797$	−49.4975	−1.75329	−0.876645	+	0.481137i	$0.840224\pi$
$798$	0	0
$799$	−45.2548	−1.60100
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−22.6274	−0.797512
$806$	0	0
$807$	21.2132	0.746740
$808$	0	0
$809$	4.00000	0.140633	0.0703163	−	0.997525i	$-0.477599\pi$
$809$	4.00000	0.140633	0.0703163	+	0.997525i	$0.477599\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	0	0
$813$	−8.48528	−0.297592
$814$	0	0
$815$	−22.6274	−0.792604
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	7.07107	0.246782	0.123391	−	0.992358i	$-0.460623\pi$
$821$	7.07107	0.246782	0.123391	+	0.992358i	$0.460623\pi$
$822$	0	0
$823$	−25.4558	−0.887335	−0.443667	−	0.896191i	$-0.646323\pi$
$823$	−25.4558	−0.887335	−0.443667	+	0.896191i	$0.646323\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	−52.3259	−1.81735	−0.908677	−	0.417500i	$-0.862906\pi$
$829$	−52.3259	−1.81735	−0.908677	+	0.417500i	$0.862906\pi$
$830$	0	0
$831$	1.41421	0.0490585
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	2.82843	0.0977647
$838$	0	0
$839$	45.2548	1.56237	0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548	1.56237	0.781185	+	0.624299i	$0.214615\pi$
$840$	0	0
$841$	−27.0000	−0.931034
$842$	0	0
$843$	22.0000	0.757720
$844$	0	0
$845$	7.07107	0.243252
$846$	0	0
$847$	−31.1127	−1.06904
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	15.5563	0.532639	0.266320	−	0.963885i	$-0.414192\pi$
$853$	15.5563	0.532639	0.266320	+	0.963885i	$0.414192\pi$
$854$	0	0
$855$	−11.3137	−0.386921
$856$	0	0
$857$	−20.0000	−0.683187	−0.341593	−	0.939848i	$-0.610967\pi$
$857$	−20.0000	−0.683187	−0.341593	+	0.939848i	$0.610967\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	11.3137	0.385570
$862$	0	0
$863$	−5.65685	−0.192562	−0.0962808	−	0.995354i	$-0.530695\pi$
$863$	−5.65685	−0.192562	−0.0962808	+	0.995354i	$0.530695\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.9706	−0.575026
$872$	0	0
$873$	16.0000	0.541518
$874$	0	0
$875$	−32.0000	−1.08180
$876$	0	0
$877$	32.5269	1.09836	0.549178	−	0.835705i	$-0.314941\pi$
$877$	32.5269	1.09836	0.549178	+	0.835705i	$0.314941\pi$
$878$	0	0
$879$	−18.3848	−0.620103
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	0	0
$885$	16.9706	0.570459
$886$	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
$889$	24.0000	0.804934
$890$	0	0
$891$	0	0
$892$	0	0
$893$	90.5097	3.02879
$894$	0	0
$895$	5.65685	0.189088
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	4.00000	0.133407
$900$	0	0
$901$	28.2843	0.942286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	0	0
$909$	15.5563	0.515972
$910$	0	0
$911$	5.65685	0.187420	0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685	0.187420	0.0937100	+	0.995600i	$0.470127\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	−11.3137	−0.373612
$918$	0	0
$919$	−2.82843	−0.0933012	−0.0466506	−	0.998911i	$-0.514855\pi$
$919$	−2.82843	−0.0933012	−0.0466506	+	0.998911i	$0.514855\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	−48.0000	−1.57994
$924$	0	0
$925$	12.7279	0.418491
$926$	0	0
$927$	19.7990	0.650284
$928$	0	0
$929$	−20.0000	−0.656179	−0.328089	−	0.944647i	$-0.606405\pi$
$929$	−20.0000	−0.656179	−0.328089	+	0.944647i	$0.606405\pi$
$930$	0	0
$931$	−8.00000	−0.262189
$932$	0	0
$933$	−28.2843	−0.925985
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42.0000	−1.37208	−0.686040	−	0.727564i	$-0.740653\pi$
$937$	−42.0000	−1.37208	−0.686040	+	0.727564i	$0.740653\pi$
$938$	0	0
$939$	16.0000	0.522140
$940$	0	0
$941$	52.3259	1.70578	0.852888	−	0.522094i	$-0.174849\pi$
$941$	52.3259	1.70578	0.852888	+	0.522094i	$0.174849\pi$
$942$	0	0
$943$	22.6274	0.736850
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	−59.3970	−1.92811
$950$	0	0
$951$	−21.2132	−0.687885
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−33.9411	−1.09602
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−42.4264	−1.36434	−0.682171	−	0.731193i	$-0.738964\pi$
$967$	−42.4264	−1.36434	−0.682171	+	0.731193i	$0.738964\pi$
$968$	0	0
$969$	32.0000	1.02799
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	−11.3137	−0.362701
$974$	0	0
$975$	−12.7279	−0.407620
$976$	0	0
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.24264	0.135457
$982$	0	0
$983$	−16.9706	−0.541277	−0.270638	−	0.962681i	$-0.587235\pi$
$983$	−16.9706	−0.541277	−0.270638	+	0.962681i	$0.587235\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	32.0000	1.01857
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−25.4558	−0.808632	−0.404316	−	0.914619i	$-0.632490\pi$
$991$	−25.4558	−0.808632	−0.404316	+	0.914619i	$0.632490\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	7.07107	0.223943	0.111971	−	0.993711i	$-0.464283\pi$
$997$	7.07107	0.223943	0.111971	+	0.993711i	$0.464283\pi$
$998$	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.b.1.2		2
3.2	odd	2		9216.2.a.h.1.1		2
4.3	odd	2		3072.2.a.h.1.2		2
8.3	odd	2	inner	3072.2.a.b.1.1		2
8.5	even	2		3072.2.a.h.1.1		2
12.11	even	2		9216.2.a.i.1.1		2
16.3	odd	4		3072.2.d.c.1537.3		4
16.5	even	4		3072.2.d.c.1537.4		4
16.11	odd	4		3072.2.d.c.1537.2		4
16.13	even	4		3072.2.d.c.1537.1		4
24.5	odd	2		9216.2.a.i.1.2		2
24.11	even	2		9216.2.a.h.1.2		2
32.3	odd	8		1536.2.j.b.1153.2	yes	4
32.5	even	8		1536.2.j.c.385.2	yes	4
32.11	odd	8		1536.2.j.b.385.2	yes	4
32.13	even	8		1536.2.j.c.1153.2	yes	4
32.19	odd	8		1536.2.j.c.1153.1	yes	4
32.21	even	8		1536.2.j.b.385.1	✓	4
32.27	odd	8		1536.2.j.c.385.1	yes	4
32.29	even	8		1536.2.j.b.1153.1	yes	4
96.5	odd	8		4608.2.k.y.3457.2		4
96.11	even	8		4608.2.k.bb.3457.1		4
96.29	odd	8		4608.2.k.bb.1153.1		4
96.35	even	8		4608.2.k.bb.1153.2		4
96.53	odd	8		4608.2.k.bb.3457.2		4
96.59	even	8		4608.2.k.y.3457.1		4
96.77	odd	8		4608.2.k.y.1153.1		4
96.83	even	8		4608.2.k.y.1153.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.b.385.1	✓	4	32.21	even	8
1536.2.j.b.385.2	yes	4	32.11	odd	8
1536.2.j.b.1153.1	yes	4	32.29	even	8
1536.2.j.b.1153.2	yes	4	32.3	odd	8
1536.2.j.c.385.1	yes	4	32.27	odd	8
1536.2.j.c.385.2	yes	4	32.5	even	8
1536.2.j.c.1153.1	yes	4	32.19	odd	8
1536.2.j.c.1153.2	yes	4	32.13	even	8
3072.2.a.b.1.1		2	8.3	odd	2	inner
3072.2.a.b.1.2		2	1.1	even	1	trivial
3072.2.a.h.1.1		2	8.5	even	2
3072.2.a.h.1.2		2	4.3	odd	2
3072.2.d.c.1537.1		4	16.13	even	4
3072.2.d.c.1537.2		4	16.11	odd	4
3072.2.d.c.1537.3		4	16.3	odd	4
3072.2.d.c.1537.4		4	16.5	even	4
4608.2.k.y.1153.1		4	96.77	odd	8
4608.2.k.y.1153.2		4	96.83	even	8
4608.2.k.y.3457.1		4	96.59	even	8
4608.2.k.y.3457.2		4	96.5	odd	8
4608.2.k.bb.1153.1		4	96.29	odd	8
4608.2.k.bb.1153.2		4	96.35	even	8
4608.2.k.bb.3457.1		4	96.11	even	8
4608.2.k.bb.3457.2		4	96.53	odd	8
9216.2.a.h.1.1		2	3.2	odd	2
9216.2.a.h.1.2		2	24.11	even	2
9216.2.a.i.1.1		2	12.11	even	2
9216.2.a.i.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.41421 q^{5} +2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.41421 q^{5} +2.82843 q^{7} +1.00000 q^{9} -4.24264 q^{13} -1.41421 q^{15} +4.00000 q^{17} -8.00000 q^{19} -2.82843 q^{21} -5.65685 q^{23} -3.00000 q^{25} -1.00000 q^{27} -1.41421 q^{29} -2.82843 q^{31} +4.00000 q^{35} -4.24264 q^{37} +4.24264 q^{39} -4.00000 q^{41} +1.41421 q^{45} -11.3137 q^{47} +1.00000 q^{49} -4.00000 q^{51} +7.07107 q^{53} +8.00000 q^{57} -12.0000 q^{59} -1.41421 q^{61} +2.82843 q^{63} -6.00000 q^{65} +4.00000 q^{67} +5.65685 q^{69} +11.3137 q^{71} +14.0000 q^{73} +3.00000 q^{75} -8.48528 q^{79} +1.00000 q^{81} -16.0000 q^{83} +5.65685 q^{85} +1.41421 q^{87} +6.00000 q^{89} -12.0000 q^{91} +2.82843 q^{93} -11.3137 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 8 q^{17} - 16 q^{19} - 6 q^{25} - 2 q^{27} + 8 q^{35} - 8 q^{41} + 2 q^{49} - 8 q^{51} + 16 q^{57} - 24 q^{59} - 12 q^{65} + 8 q^{67} + 28 q^{73} + 6 q^{75} + 2 q^{81} - 32 q^{83} + 12 q^{89} - 24 q^{91} + 32 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 8 * q^17 - 16 * q^19 - 6 * q^25 - 2 * q^27 + 8 * q^35 - 8 * q^41 + 2 * q^49 - 8 * q^51 + 16 * q^57 - 24 * q^59 - 12 * q^65 + 8 * q^67 + 28 * q^73 + 6 * q^75 + 2 * q^81 - 32 * q^83 + 12 * q^89 - 24 * q^91 + 32 * q^97

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