LMFDB - Embedded newform 5776.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5776, 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("5776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{3} -1.64575 q^{5} -3.64575 q^{7} +4.00000 q^{9} +O(q^{10})$	$q+2.64575 q^{3} -1.64575 q^{5} -3.64575 q^{7} +4.00000 q^{9} +4.64575 q^{11} -2.00000 q^{13} -4.35425 q^{15} -9.64575 q^{21} +1.64575 q^{23} -2.29150 q^{25} +2.64575 q^{27} -1.64575 q^{29} -5.64575 q^{31} +12.2915 q^{33} +6.00000 q^{35} -0.354249 q^{37} -5.29150 q^{39} +0.291503 q^{41} -11.2915 q^{43} -6.58301 q^{45} -4.35425 q^{47} +6.29150 q^{49} +12.5830 q^{53} -7.64575 q^{55} -7.93725 q^{59} +0.937254 q^{61} -14.5830 q^{63} +3.29150 q^{65} +0.645751 q^{67} +4.35425 q^{69} -2.70850 q^{71} +1.70850 q^{73} -6.06275 q^{75} -16.9373 q^{77} -4.00000 q^{79} -5.00000 q^{81} -7.93725 q^{83} -4.35425 q^{87} +7.29150 q^{91} -14.9373 q^{93} +3.70850 q^{97} +18.5830 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7} + 8 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7 + 8 * q^9	$ 2 q + 2 q^{5} - 2 q^{7} + 8 q^{9} + 4 q^{11} - 4 q^{13} - 14 q^{15} - 14 q^{21} - 2 q^{23} + 6 q^{25} + 2 q^{29} - 6 q^{31} + 14 q^{33} + 12 q^{35} - 6 q^{37} - 10 q^{41} - 12 q^{43} + 8 q^{45} - 14 q^{47} + 2 q^{49} + 4 q^{53} - 10 q^{55} - 14 q^{61} - 8 q^{63} - 4 q^{65} - 4 q^{67} + 14 q^{69} - 16 q^{71} + 14 q^{73} - 28 q^{75} - 18 q^{77} - 8 q^{79} - 10 q^{81} - 14 q^{87} + 4 q^{91} - 14 q^{93} + 18 q^{97} + 16 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 + 8 * q^9 + 4 * q^11 - 4 * q^13 - 14 * q^15 - 14 * q^21 - 2 * q^23 + 6 * q^25 + 2 * q^29 - 6 * q^31 + 14 * q^33 + 12 * q^35 - 6 * q^37 - 10 * q^41 - 12 * q^43 + 8 * q^45 - 14 * q^47 + 2 * q^49 + 4 * q^53 - 10 * q^55 - 14 * q^61 - 8 * q^63 - 4 * q^65 - 4 * q^67 + 14 * q^69 - 16 * q^71 + 14 * q^73 - 28 * q^75 - 18 * q^77 - 8 * q^79 - 10 * q^81 - 14 * q^87 + 4 * q^91 - 14 * q^93 + 18 * q^97 + 16 * 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
$3$	2.64575	1.52753	0.763763	−	0.645497i	$-0.223350\pi$
$3$	2.64575	1.52753	0.763763	+	0.645497i	$0.223350\pi$
$4$	0	0
$5$	−1.64575	−0.736002	−0.368001	−	0.929825i	$-0.619958\pi$
$5$	−1.64575	−0.736002	−0.368001	+	0.929825i	$0.619958\pi$
$6$	0	0
$7$	−3.64575	−1.37796	−0.688982	−	0.724778i	$-0.741942\pi$
$7$	−3.64575	−1.37796	−0.688982	+	0.724778i	$0.741942\pi$
$8$	0	0
$9$	4.00000	1.33333
$10$	0	0
$11$	4.64575	1.40075	0.700373	−	0.713777i	$-0.253017\pi$
$11$	4.64575	1.40075	0.700373	+	0.713777i	$0.253017\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−4.35425	−1.12426
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−9.64575	−2.10488
$22$	0	0
$23$	1.64575	0.343163	0.171581	−	0.985170i	$-0.445112\pi$
$23$	1.64575	0.343163	0.171581	+	0.985170i	$0.445112\pi$
$24$	0	0
$25$	−2.29150	−0.458301
$26$	0	0
$27$	2.64575	0.509175
$28$	0	0
$29$	−1.64575	−0.305608	−0.152804	−	0.988256i	$-0.548830\pi$
$29$	−1.64575	−0.305608	−0.152804	+	0.988256i	$0.548830\pi$
$30$	0	0
$31$	−5.64575	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$31$	−5.64575	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$32$	0	0
$33$	12.2915	2.13968
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	−0.354249	−0.0582381	−0.0291191	−	0.999576i	$-0.509270\pi$
$37$	−0.354249	−0.0582381	−0.0291191	+	0.999576i	$0.509270\pi$
$38$	0	0
$39$	−5.29150	−0.847319
$40$	0	0
$41$	0.291503	0.0455251	0.0227625	−	0.999741i	$-0.492754\pi$
$41$	0.291503	0.0455251	0.0227625	+	0.999741i	$0.492754\pi$
$42$	0	0
$43$	−11.2915	−1.72194	−0.860969	−	0.508657i	$-0.830142\pi$
$43$	−11.2915	−1.72194	−0.860969	+	0.508657i	$0.830142\pi$
$44$	0	0
$45$	−6.58301	−0.981336
$46$	0	0
$47$	−4.35425	−0.635132	−0.317566	−	0.948236i	$-0.602866\pi$
$47$	−4.35425	−0.635132	−0.317566	+	0.948236i	$0.602866\pi$
$48$	0	0
$49$	6.29150	0.898786
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.5830	1.72841	0.864204	−	0.503141i	$-0.167822\pi$
$53$	12.5830	1.72841	0.864204	+	0.503141i	$0.167822\pi$
$54$	0	0
$55$	−7.64575	−1.03095
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.93725	−1.03334	−0.516671	−	0.856184i	$-0.672829\pi$
$59$	−7.93725	−1.03334	−0.516671	+	0.856184i	$0.672829\pi$
$60$	0	0
$61$	0.937254	0.120003	0.0600015	−	0.998198i	$-0.480889\pi$
$61$	0.937254	0.120003	0.0600015	+	0.998198i	$0.480889\pi$
$62$	0	0
$63$	−14.5830	−1.83729
$64$	0	0
$65$	3.29150	0.408261
$66$	0	0
$67$	0.645751	0.0788911	0.0394455	−	0.999222i	$-0.487441\pi$
$67$	0.645751	0.0788911	0.0394455	+	0.999222i	$0.487441\pi$
$68$	0	0
$69$	4.35425	0.524190
$70$	0	0
$71$	−2.70850	−0.321440	−0.160720	−	0.987000i	$-0.551382\pi$
$71$	−2.70850	−0.321440	−0.160720	+	0.987000i	$0.551382\pi$
$72$	0	0
$73$	1.70850	0.199964	0.0999822	−	0.994989i	$-0.468121\pi$
$73$	1.70850	0.199964	0.0999822	+	0.994989i	$0.468121\pi$
$74$	0	0
$75$	−6.06275	−0.700066
$76$	0	0
$77$	−16.9373	−1.93018
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−7.93725	−0.871227	−0.435613	−	0.900134i	$-0.643469\pi$
$83$	−7.93725	−0.871227	−0.435613	+	0.900134i	$0.643469\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.35425	−0.466824
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	7.29150	0.764357
$92$	0	0
$93$	−14.9373	−1.54892
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.70850	0.376541	0.188270	−	0.982117i	$-0.439712\pi$
$97$	3.70850	0.376541	0.188270	+	0.982117i	$0.439712\pi$
$98$	0	0
$99$	18.5830	1.86766
$100$	0	0
$101$	−13.6458	−1.35780	−0.678902	−	0.734229i	$-0.737544\pi$
$101$	−13.6458	−1.35780	−0.678902	+	0.734229i	$0.737544\pi$
$102$	0	0
$103$	−13.2915	−1.30965	−0.654825	−	0.755780i	$-0.727258\pi$
$103$	−13.2915	−1.30965	−0.654825	+	0.755780i	$0.727258\pi$
$104$	0	0
$105$	15.8745	1.54919
$106$	0	0
$107$	15.2915	1.47829	0.739143	−	0.673549i	$-0.235231\pi$
$107$	15.2915	1.47829	0.739143	+	0.673549i	$0.235231\pi$
$108$	0	0
$109$	−14.5830	−1.39680	−0.698399	−	0.715708i	$-0.746104\pi$
$109$	−14.5830	−1.39680	−0.698399	+	0.715708i	$0.746104\pi$
$110$	0	0
$111$	−0.937254	−0.0889602
$112$	0	0
$113$	15.5830	1.46593	0.732963	−	0.680269i	$-0.238137\pi$
$113$	15.5830	1.46593	0.732963	+	0.680269i	$0.238137\pi$
$114$	0	0
$115$	−2.70850	−0.252569
$116$	0	0
$117$	−8.00000	−0.739600
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.5830	0.962091
$122$	0	0
$123$	0.771243	0.0695407
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−13.2915	−1.17943	−0.589715	−	0.807611i	$-0.700760\pi$
$127$	−13.2915	−1.17943	−0.589715	+	0.807611i	$0.700760\pi$
$128$	0	0
$129$	−29.8745	−2.63030
$130$	0	0
$131$	−1.93725	−0.169259	−0.0846293	−	0.996413i	$-0.526971\pi$
$131$	−1.93725	−0.169259	−0.0846293	+	0.996413i	$0.526971\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.35425	−0.374754
$136$	0	0
$137$	15.5830	1.33135	0.665673	−	0.746244i	$-0.268145\pi$
$137$	15.5830	1.33135	0.665673	+	0.746244i	$0.268145\pi$
$138$	0	0
$139$	−18.6458	−1.58151	−0.790756	−	0.612131i	$-0.790312\pi$
$139$	−18.6458	−1.58151	−0.790756	+	0.612131i	$0.790312\pi$
$140$	0	0
$141$	−11.5203	−0.970181
$142$	0	0
$143$	−9.29150	−0.776994
$144$	0	0
$145$	2.70850	0.224928
$146$	0	0
$147$	16.6458	1.37292
$148$	0	0
$149$	10.9373	0.896015	0.448007	−	0.894030i	$-0.352134\pi$
$149$	10.9373	0.896015	0.448007	+	0.894030i	$0.352134\pi$
$150$	0	0
$151$	12.9373	1.05282	0.526409	−	0.850231i	$-0.323538\pi$
$151$	12.9373	1.05282	0.526409	+	0.850231i	$0.323538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.29150	0.746311
$156$	0	0
$157$	−10.5830	−0.844616	−0.422308	−	0.906452i	$-0.638780\pi$
$157$	−10.5830	−0.844616	−0.422308	+	0.906452i	$0.638780\pi$
$158$	0	0
$159$	33.2915	2.64019
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	−3.93725	−0.308390	−0.154195	−	0.988040i	$-0.549278\pi$
$163$	−3.93725	−0.308390	−0.154195	+	0.988040i	$0.549278\pi$
$164$	0	0
$165$	−20.2288	−1.57481
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	8.35425	0.631522
$176$	0	0
$177$	−21.0000	−1.57846
$178$	0	0
$179$	4.06275	0.303664	0.151832	−	0.988406i	$-0.451483\pi$
$179$	4.06275	0.303664	0.151832	+	0.988406i	$0.451483\pi$
$180$	0	0
$181$	−22.2288	−1.65225	−0.826125	−	0.563487i	$-0.809460\pi$
$181$	−22.2288	−1.65225	−0.826125	+	0.563487i	$0.809460\pi$
$182$	0	0
$183$	2.47974	0.183308
$184$	0	0
$185$	0.583005	0.0428634
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−9.64575	−0.701625
$190$	0	0
$191$	−6.58301	−0.476330	−0.238165	−	0.971225i	$-0.576546\pi$
$191$	−6.58301	−0.476330	−0.238165	+	0.971225i	$0.576546\pi$
$192$	0	0
$193$	−14.5830	−1.04971	−0.524854	−	0.851192i	$-0.675880\pi$
$193$	−14.5830	−1.04971	−0.524854	+	0.851192i	$0.675880\pi$
$194$	0	0
$195$	8.70850	0.623628
$196$	0	0
$197$	−7.64575	−0.544737	−0.272369	−	0.962193i	$-0.587807\pi$
$197$	−7.64575	−0.544737	−0.272369	+	0.962193i	$0.587807\pi$
$198$	0	0
$199$	19.8745	1.40887	0.704433	−	0.709770i	$-0.251201\pi$
$199$	19.8745	1.40887	0.704433	+	0.709770i	$0.251201\pi$
$200$	0	0
$201$	1.70850	0.120508
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	−0.479741	−0.0335066
$206$	0	0
$207$	6.58301	0.457550
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−13.2915	−0.915025	−0.457512	−	0.889203i	$-0.651259\pi$
$211$	−13.2915	−0.915025	−0.457512	+	0.889203i	$0.651259\pi$
$212$	0	0
$213$	−7.16601	−0.491007
$214$	0	0
$215$	18.5830	1.26735
$216$	0	0
$217$	20.5830	1.39727
$218$	0	0
$219$	4.52026	0.305451
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−18.8118	−1.25973	−0.629864	−	0.776705i	$-0.716890\pi$
$223$	−18.8118	−1.25973	−0.629864	+	0.776705i	$0.716890\pi$
$224$	0	0
$225$	−9.16601	−0.611067
$226$	0	0
$227$	−7.35425	−0.488119	−0.244059	−	0.969760i	$-0.578479\pi$
$227$	−7.35425	−0.488119	−0.244059	+	0.969760i	$0.578479\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	0	0
$231$	−44.8118	−2.94840
$232$	0	0
$233$	18.8745	1.23651	0.618255	−	0.785978i	$-0.287840\pi$
$233$	18.8745	1.23651	0.618255	+	0.785978i	$0.287840\pi$
$234$	0	0
$235$	7.16601	0.467459
$236$	0	0
$237$	−10.5830	−0.687440
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	7.58301	0.488464	0.244232	−	0.969717i	$-0.421464\pi$
$241$	7.58301	0.488464	0.244232	+	0.969717i	$0.421464\pi$
$242$	0	0
$243$	−21.1660	−1.35780
$244$	0	0
$245$	−10.3542	−0.661509
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−21.0000	−1.33082
$250$	0	0
$251$	−29.2288	−1.84490	−0.922451	−	0.386113i	$-0.873817\pi$
$251$	−29.2288	−1.84490	−0.922451	+	0.386113i	$0.873817\pi$
$252$	0	0
$253$	7.64575	0.480684
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.2915	−0.766723	−0.383361	−	0.923598i	$-0.625234\pi$
$257$	−12.2915	−0.766723	−0.383361	+	0.923598i	$0.625234\pi$
$258$	0	0
$259$	1.29150	0.0802501
$260$	0	0
$261$	−6.58301	−0.407478
$262$	0	0
$263$	10.9373	0.674420	0.337210	−	0.941429i	$-0.390517\pi$
$263$	10.9373	0.674420	0.337210	+	0.941429i	$0.390517\pi$
$264$	0	0
$265$	−20.7085	−1.27211
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−12.3542	−0.750467	−0.375234	−	0.926930i	$-0.622438\pi$
$271$	−12.3542	−0.750467	−0.375234	+	0.926930i	$0.622438\pi$
$272$	0	0
$273$	19.2915	1.16757
$274$	0	0
$275$	−10.6458	−0.641963
$276$	0	0
$277$	−27.5203	−1.65353	−0.826766	−	0.562546i	$-0.809822\pi$
$277$	−27.5203	−1.65353	−0.826766	+	0.562546i	$0.809822\pi$
$278$	0	0
$279$	−22.5830	−1.35201
$280$	0	0
$281$	−25.4575	−1.51867	−0.759334	−	0.650701i	$-0.774475\pi$
$281$	−25.4575	−1.51867	−0.759334	+	0.650701i	$0.774475\pi$
$282$	0	0
$283$	−30.6458	−1.82170	−0.910850	−	0.412737i	$-0.864573\pi$
$283$	−30.6458	−1.82170	−0.910850	+	0.412737i	$0.864573\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.06275	−0.0627319
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	9.81176	0.575176
$292$	0	0
$293$	−28.9373	−1.69053	−0.845266	−	0.534345i	$-0.820558\pi$
$293$	−28.9373	−1.69053	−0.845266	+	0.534345i	$0.820558\pi$
$294$	0	0
$295$	13.0627	0.760542
$296$	0	0
$297$	12.2915	0.713225
$298$	0	0
$299$	−3.29150	−0.190353
$300$	0	0
$301$	41.1660	2.37277
$302$	0	0
$303$	−36.1033	−2.07408
$304$	0	0
$305$	−1.54249	−0.0883225
$306$	0	0
$307$	0.645751	0.0368550	0.0184275	−	0.999830i	$-0.494134\pi$
$307$	0.645751	0.0368550	0.0184275	+	0.999830i	$0.494134\pi$
$308$	0	0
$309$	−35.1660	−2.00052
$310$	0	0
$311$	−13.6458	−0.773780	−0.386890	−	0.922126i	$-0.626451\pi$
$311$	−13.6458	−0.773780	−0.386890	+	0.922126i	$0.626451\pi$
$312$	0	0
$313$	8.87451	0.501617	0.250808	−	0.968037i	$-0.419304\pi$
$313$	8.87451	0.501617	0.250808	+	0.968037i	$0.419304\pi$
$314$	0	0
$315$	24.0000	1.35225
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−7.64575	−0.428080
$320$	0	0
$321$	40.4575	2.25812
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.58301	0.254219
$326$	0	0
$327$	−38.5830	−2.13365
$328$	0	0
$329$	15.8745	0.875190
$330$	0	0
$331$	19.8118	1.08895	0.544476	−	0.838776i	$-0.316728\pi$
$331$	19.8118	1.08895	0.544476	+	0.838776i	$0.316728\pi$
$332$	0	0
$333$	−1.41699	−0.0776508
$334$	0	0
$335$	−1.06275	−0.0580640
$336$	0	0
$337$	9.70850	0.528856	0.264428	−	0.964405i	$-0.414817\pi$
$337$	9.70850	0.528856	0.264428	+	0.964405i	$0.414817\pi$
$338$	0	0
$339$	41.2288	2.23924
$340$	0	0
$341$	−26.2288	−1.42037
$342$	0	0
$343$	2.58301	0.139469
$344$	0	0
$345$	−7.16601	−0.385805
$346$	0	0
$347$	23.2288	1.24698	0.623492	−	0.781829i	$-0.285713\pi$
$347$	23.2288	1.24698	0.623492	+	0.781829i	$0.285713\pi$
$348$	0	0
$349$	21.1660	1.13299	0.566495	−	0.824065i	$-0.308299\pi$
$349$	21.1660	1.13299	0.566495	+	0.824065i	$0.308299\pi$
$350$	0	0
$351$	−5.29150	−0.282440
$352$	0	0
$353$	12.8745	0.685241	0.342620	−	0.939474i	$-0.388686\pi$
$353$	12.8745	0.685241	0.342620	+	0.939474i	$0.388686\pi$
$354$	0	0
$355$	4.45751	0.236580
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.93725	0.260578	0.130289	−	0.991476i	$-0.458409\pi$
$359$	4.93725	0.260578	0.130289	+	0.991476i	$0.458409\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	28.0000	1.46962
$364$	0	0
$365$	−2.81176	−0.147174
$366$	0	0
$367$	−16.2288	−0.847134	−0.423567	−	0.905865i	$-0.639222\pi$
$367$	−16.2288	−0.847134	−0.423567	+	0.905865i	$0.639222\pi$
$368$	0	0
$369$	1.16601	0.0607001
$370$	0	0
$371$	−45.8745	−2.38169
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	31.7490	1.63951
$376$	0	0
$377$	3.29150	0.169521
$378$	0	0
$379$	10.7085	0.550059	0.275029	−	0.961436i	$-0.411312\pi$
$379$	10.7085	0.550059	0.275029	+	0.961436i	$0.411312\pi$
$380$	0	0
$381$	−35.1660	−1.80161
$382$	0	0
$383$	5.52026	0.282072	0.141036	−	0.990004i	$-0.454957\pi$
$383$	5.52026	0.282072	0.141036	+	0.990004i	$0.454957\pi$
$384$	0	0
$385$	27.8745	1.42062
$386$	0	0
$387$	−45.1660	−2.29592
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−5.12549	−0.258547
$394$	0	0
$395$	6.58301	0.331227
$396$	0	0
$397$	21.0627	1.05711	0.528554	−	0.848899i	$-0.322734\pi$
$397$	21.0627	1.05711	0.528554	+	0.848899i	$0.322734\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.5830	−1.37743	−0.688715	−	0.725032i	$-0.741825\pi$
$401$	−27.5830	−1.37743	−0.688715	+	0.725032i	$0.741825\pi$
$402$	0	0
$403$	11.2915	0.562470
$404$	0	0
$405$	8.22876	0.408890
$406$	0	0
$407$	−1.64575	−0.0815769
$408$	0	0
$409$	7.58301	0.374955	0.187478	−	0.982269i	$-0.439969\pi$
$409$	7.58301	0.374955	0.187478	+	0.982269i	$0.439969\pi$
$410$	0	0
$411$	41.2288	2.03366
$412$	0	0
$413$	28.9373	1.42391
$414$	0	0
$415$	13.0627	0.641225
$416$	0	0
$417$	−49.3320	−2.41580
$418$	0	0
$419$	31.7490	1.55104	0.775520	−	0.631322i	$-0.217488\pi$
$419$	31.7490	1.55104	0.775520	+	0.631322i	$0.217488\pi$
$420$	0	0
$421$	24.8118	1.20925	0.604626	−	0.796510i	$-0.293323\pi$
$421$	24.8118	1.20925	0.604626	+	0.796510i	$0.293323\pi$
$422$	0	0
$423$	−17.4170	−0.846843
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.41699	−0.165360
$428$	0	0
$429$	−24.5830	−1.18688
$430$	0	0
$431$	27.8745	1.34267	0.671334	−	0.741155i	$-0.265722\pi$
$431$	27.8745	1.34267	0.671334	+	0.741155i	$0.265722\pi$
$432$	0	0
$433$	−17.8745	−0.858994	−0.429497	−	0.903068i	$-0.641309\pi$
$433$	−17.8745	−0.858994	−0.429497	+	0.903068i	$0.641309\pi$
$434$	0	0
$435$	7.16601	0.343584
$436$	0	0
$437$	0	0
$438$	0	0
$439$	10.8118	0.516017	0.258009	−	0.966143i	$-0.416934\pi$
$439$	10.8118	0.516017	0.258009	+	0.966143i	$0.416934\pi$
$440$	0	0
$441$	25.1660	1.19838
$442$	0	0
$443$	−10.6458	−0.505795	−0.252897	−	0.967493i	$-0.581384\pi$
$443$	−10.6458	−0.505795	−0.252897	+	0.967493i	$0.581384\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.9373	1.36869
$448$	0	0
$449$	24.2915	1.14639	0.573193	−	0.819420i	$-0.305704\pi$
$449$	24.2915	1.14639	0.573193	+	0.819420i	$0.305704\pi$
$450$	0	0
$451$	1.35425	0.0637691
$452$	0	0
$453$	34.2288	1.60821
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	32.8745	1.53780	0.768902	−	0.639366i	$-0.220803\pi$
$457$	32.8745	1.53780	0.768902	+	0.639366i	$0.220803\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.1660	0.892650	0.446325	−	0.894871i	$-0.352733\pi$
$461$	19.1660	0.892650	0.446325	+	0.894871i	$0.352733\pi$
$462$	0	0
$463$	38.4575	1.78727	0.893636	−	0.448792i	$-0.148146\pi$
$463$	38.4575	1.78727	0.893636	+	0.448792i	$0.148146\pi$
$464$	0	0
$465$	24.5830	1.14001
$466$	0	0
$467$	19.3542	0.895608	0.447804	−	0.894132i	$-0.352206\pi$
$467$	19.3542	0.895608	0.447804	+	0.894132i	$0.352206\pi$
$468$	0	0
$469$	−2.35425	−0.108709
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	−52.4575	−2.41200
$474$	0	0
$475$	0	0
$476$	0	0
$477$	50.3320	2.30454
$478$	0	0
$479$	−6.58301	−0.300785	−0.150393	−	0.988626i	$-0.548054\pi$
$479$	−6.58301	−0.300785	−0.150393	+	0.988626i	$0.548054\pi$
$480$	0	0
$481$	0.708497	0.0323047
$482$	0	0
$483$	−15.8745	−0.722315
$484$	0	0
$485$	−6.10326	−0.277135
$486$	0	0
$487$	4.22876	0.191623	0.0958116	−	0.995399i	$-0.469455\pi$
$487$	4.22876	0.191623	0.0958116	+	0.995399i	$0.469455\pi$
$488$	0	0
$489$	−10.4170	−0.471073
$490$	0	0
$491$	−39.2915	−1.77320	−0.886600	−	0.462536i	$-0.846939\pi$
$491$	−39.2915	−1.77320	−0.886600	+	0.462536i	$0.846939\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−30.5830	−1.37460
$496$	0	0
$497$	9.87451	0.442932
$498$	0	0
$499$	4.77124	0.213590	0.106795	−	0.994281i	$-0.465941\pi$
$499$	4.77124	0.213590	0.106795	+	0.994281i	$0.465941\pi$
$500$	0	0
$501$	−31.7490	−1.41844
$502$	0	0
$503$	40.9373	1.82530	0.912651	−	0.408740i	$-0.134032\pi$
$503$	40.9373	1.82530	0.912651	+	0.408740i	$0.134032\pi$
$504$	0	0
$505$	22.4575	0.999346
$506$	0	0
$507$	−23.8118	−1.05752
$508$	0	0
$509$	31.7490	1.40725	0.703625	−	0.710571i	$-0.251563\pi$
$509$	31.7490	1.40725	0.703625	+	0.710571i	$0.251563\pi$
$510$	0	0
$511$	−6.22876	−0.275544
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.8745	0.963906
$516$	0	0
$517$	−20.2288	−0.889660
$518$	0	0
$519$	15.8745	0.696814
$520$	0	0
$521$	11.7085	0.512959	0.256479	−	0.966550i	$-0.417437\pi$
$521$	11.7085	0.512959	0.256479	+	0.966550i	$0.417437\pi$
$522$	0	0
$523$	−1.87451	−0.0819665	−0.0409833	−	0.999160i	$-0.513049\pi$
$523$	−1.87451	−0.0819665	−0.0409833	+	0.999160i	$0.513049\pi$
$524$	0	0
$525$	22.1033	0.964666
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−20.2915	−0.882239
$530$	0	0
$531$	−31.7490	−1.37779
$532$	0	0
$533$	−0.583005	−0.0252528
$534$	0	0
$535$	−25.1660	−1.08802
$536$	0	0
$537$	10.7490	0.463854
$538$	0	0
$539$	29.2288	1.25897
$540$	0	0
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	0	0
$543$	−58.8118	−2.52385
$544$	0	0
$545$	24.0000	1.02805
$546$	0	0
$547$	11.2915	0.482790	0.241395	−	0.970427i	$-0.422395\pi$
$547$	11.2915	0.482790	0.241395	+	0.970427i	$0.422395\pi$
$548$	0	0
$549$	3.74902	0.160004
$550$	0	0
$551$	0	0
$552$	0	0
$553$	14.5830	0.620132
$554$	0	0
$555$	1.54249	0.0654749
$556$	0	0
$557$	−5.41699	−0.229525	−0.114763	−	0.993393i	$-0.536611\pi$
$557$	−5.41699	−0.229525	−0.114763	+	0.993393i	$0.536611\pi$
$558$	0	0
$559$	22.5830	0.955159
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.0627	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$563$	10.0627	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$564$	0	0
$565$	−25.6458	−1.07892
$566$	0	0
$567$	18.2288	0.765536
$568$	0	0
$569$	−6.58301	−0.275974	−0.137987	−	0.990434i	$-0.544063\pi$
$569$	−6.58301	−0.275974	−0.137987	+	0.990434i	$0.544063\pi$
$570$	0	0
$571$	−7.81176	−0.326912	−0.163456	−	0.986551i	$-0.552264\pi$
$571$	−7.81176	−0.326912	−0.163456	+	0.986551i	$0.552264\pi$
$572$	0	0
$573$	−17.4170	−0.727605
$574$	0	0
$575$	−3.77124	−0.157272
$576$	0	0
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	0	0
$579$	−38.5830	−1.60345
$580$	0	0
$581$	28.9373	1.20052
$582$	0	0
$583$	58.4575	2.42106
$584$	0	0
$585$	13.1660	0.544348
$586$	0	0
$587$	14.1255	0.583021	0.291511	−	0.956568i	$-0.405842\pi$
$587$	14.1255	0.583021	0.291511	+	0.956568i	$0.405842\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−20.2288	−0.832100
$592$	0	0
$593$	29.7085	1.21998	0.609991	−	0.792408i	$-0.291173\pi$
$593$	29.7085	1.21998	0.609991	+	0.792408i	$0.291173\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	52.5830	2.15208
$598$	0	0
$599$	−16.9373	−0.692037	−0.346019	−	0.938228i	$-0.612467\pi$
$599$	−16.9373	−0.692037	−0.346019	+	0.938228i	$0.612467\pi$
$600$	0	0
$601$	−10.4170	−0.424918	−0.212459	−	0.977170i	$-0.568147\pi$
$601$	−10.4170	−0.424918	−0.212459	+	0.977170i	$0.568147\pi$
$602$	0	0
$603$	2.58301	0.105188
$604$	0	0
$605$	−17.4170	−0.708102
$606$	0	0
$607$	6.93725	0.281574	0.140787	−	0.990040i	$-0.455037\pi$
$607$	6.93725	0.281574	0.140787	+	0.990040i	$0.455037\pi$
$608$	0	0
$609$	15.8745	0.643268
$610$	0	0
$611$	8.70850	0.352308
$612$	0	0
$613$	7.41699	0.299570	0.149785	−	0.988719i	$-0.452142\pi$
$613$	7.41699	0.299570	0.149785	+	0.988719i	$0.452142\pi$
$614$	0	0
$615$	−1.26927	−0.0511821
$616$	0	0
$617$	−30.8745	−1.24296	−0.621480	−	0.783430i	$-0.713468\pi$
$617$	−30.8745	−1.24296	−0.621480	+	0.783430i	$0.713468\pi$
$618$	0	0
$619$	8.45751	0.339936	0.169968	−	0.985450i	$-0.445634\pi$
$619$	8.45751	0.339936	0.169968	+	0.985450i	$0.445634\pi$
$620$	0	0
$621$	4.35425	0.174730
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.29150	−0.331660
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	24.8118	0.987741	0.493870	−	0.869536i	$-0.335582\pi$
$631$	24.8118	0.987741	0.493870	+	0.869536i	$0.335582\pi$
$632$	0	0
$633$	−35.1660	−1.39772
$634$	0	0
$635$	21.8745	0.868063
$636$	0	0
$637$	−12.5830	−0.498557
$638$	0	0
$639$	−10.8340	−0.428586
$640$	0	0
$641$	−12.8745	−0.508512	−0.254256	−	0.967137i	$-0.581831\pi$
$641$	−12.8745	−0.508512	−0.254256	+	0.967137i	$0.581831\pi$
$642$	0	0
$643$	6.52026	0.257134	0.128567	−	0.991701i	$-0.458962\pi$
$643$	6.52026	0.257134	0.128567	+	0.991701i	$0.458962\pi$
$644$	0	0
$645$	49.1660	1.93591
$646$	0	0
$647$	22.4575	0.882896	0.441448	−	0.897287i	$-0.354465\pi$
$647$	22.4575	0.882896	0.441448	+	0.897287i	$0.354465\pi$
$648$	0	0
$649$	−36.8745	−1.44745
$650$	0	0
$651$	54.4575	2.13436
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	3.18824	0.124575
$656$	0	0
$657$	6.83399	0.266619
$658$	0	0
$659$	−18.5830	−0.723891	−0.361946	−	0.932199i	$-0.617887\pi$
$659$	−18.5830	−0.723891	−0.361946	+	0.932199i	$0.617887\pi$
$660$	0	0
$661$	−16.2288	−0.631225	−0.315613	−	0.948888i	$-0.602210\pi$
$661$	−16.2288	−0.631225	−0.315613	+	0.948888i	$0.602210\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.70850	−0.104873
$668$	0	0
$669$	−49.7712	−1.92427
$670$	0	0
$671$	4.35425	0.168094
$672$	0	0
$673$	13.8745	0.534823	0.267411	−	0.963582i	$-0.413832\pi$
$673$	13.8745	0.534823	0.267411	+	0.963582i	$0.413832\pi$
$674$	0	0
$675$	−6.06275	−0.233355
$676$	0	0
$677$	11.4170	0.438791	0.219395	−	0.975636i	$-0.429592\pi$
$677$	11.4170	0.438791	0.219395	+	0.975636i	$0.429592\pi$
$678$	0	0
$679$	−13.5203	−0.518860
$680$	0	0
$681$	−19.4575	−0.745614
$682$	0	0
$683$	−5.41699	−0.207276	−0.103638	−	0.994615i	$-0.533048\pi$
$683$	−5.41699	−0.207276	−0.103638	+	0.994615i	$0.533048\pi$
$684$	0	0
$685$	−25.6458	−0.979874
$686$	0	0
$687$	52.9150	2.01883
$688$	0	0
$689$	−25.1660	−0.958749
$690$	0	0
$691$	−2.58301	−0.0982622	−0.0491311	−	0.998792i	$-0.515645\pi$
$691$	−2.58301	−0.0982622	−0.0491311	+	0.998792i	$0.515645\pi$
$692$	0	0
$693$	−67.7490	−2.57357
$694$	0	0
$695$	30.6863	1.16400
$696$	0	0
$697$	0	0
$698$	0	0
$699$	49.9373	1.88880
$700$	0	0
$701$	10.3542	0.391075	0.195537	−	0.980696i	$-0.437355\pi$
$701$	10.3542	0.391075	0.195537	+	0.980696i	$0.437355\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	18.9595	0.714055
$706$	0	0
$707$	49.7490	1.87100
$708$	0	0
$709$	3.64575	0.136919	0.0684595	−	0.997654i	$-0.478192\pi$
$709$	3.64575	0.136919	0.0684595	+	0.997654i	$0.478192\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	−9.29150	−0.347970
$714$	0	0
$715$	15.2915	0.571870
$716$	0	0
$717$	31.7490	1.18569
$718$	0	0
$719$	−2.70850	−0.101010	−0.0505050	−	0.998724i	$-0.516083\pi$
$719$	−2.70850	−0.101010	−0.0505050	+	0.998724i	$0.516083\pi$
$720$	0	0
$721$	48.4575	1.80465
$722$	0	0
$723$	20.0627	0.746142
$724$	0	0
$725$	3.77124	0.140060
$726$	0	0
$727$	−1.41699	−0.0525534	−0.0262767	−	0.999655i	$-0.508365\pi$
$727$	−1.41699	−0.0525534	−0.0262767	+	0.999655i	$0.508365\pi$
$728$	0	0
$729$	−41.0000	−1.51852
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−16.1033	−0.594788	−0.297394	−	0.954755i	$-0.596117\pi$
$733$	−16.1033	−0.594788	−0.297394	+	0.954755i	$0.596117\pi$
$734$	0	0
$735$	−27.3948	−1.01047
$736$	0	0
$737$	3.00000	0.110506
$738$	0	0
$739$	33.8118	1.24379	0.621893	−	0.783102i	$-0.286364\pi$
$739$	33.8118	1.24379	0.621893	+	0.783102i	$0.286364\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−47.5203	−1.74335	−0.871675	−	0.490085i	$-0.836966\pi$
$743$	−47.5203	−1.74335	−0.871675	+	0.490085i	$0.836966\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	−31.7490	−1.16164
$748$	0	0
$749$	−55.7490	−2.03702
$750$	0	0
$751$	−7.87451	−0.287345	−0.143672	−	0.989625i	$-0.545891\pi$
$751$	−7.87451	−0.287345	−0.143672	+	0.989625i	$0.545891\pi$
$752$	0	0
$753$	−77.3320	−2.81814
$754$	0	0
$755$	−21.2915	−0.774877
$756$	0	0
$757$	−4.58301	−0.166572	−0.0832861	−	0.996526i	$-0.526542\pi$
$757$	−4.58301	−0.166572	−0.0832861	+	0.996526i	$0.526542\pi$
$758$	0	0
$759$	20.2288	0.734257
$760$	0	0
$761$	42.8745	1.55420	0.777100	−	0.629377i	$-0.216690\pi$
$761$	42.8745	1.55420	0.777100	+	0.629377i	$0.216690\pi$
$762$	0	0
$763$	53.1660	1.92474
$764$	0	0
$765$	0	0
$766$	0	0
$767$	15.8745	0.573195
$768$	0	0
$769$	35.2915	1.27264	0.636322	−	0.771424i	$-0.280455\pi$
$769$	35.2915	1.27264	0.636322	+	0.771424i	$0.280455\pi$
$770$	0	0
$771$	−32.5203	−1.17119
$772$	0	0
$773$	4.93725	0.177581	0.0887903	−	0.996050i	$-0.471700\pi$
$773$	4.93725	0.177581	0.0887903	+	0.996050i	$0.471700\pi$
$774$	0	0
$775$	12.9373	0.464720
$776$	0	0
$777$	3.41699	0.122584
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−12.5830	−0.450255
$782$	0	0
$783$	−4.35425	−0.155608
$784$	0	0
$785$	17.4170	0.621639
$786$	0	0
$787$	−42.5203	−1.51568	−0.757842	−	0.652438i	$-0.773746\pi$
$787$	−42.5203	−1.51568	−0.757842	+	0.652438i	$0.773746\pi$
$788$	0	0
$789$	28.9373	1.03019
$790$	0	0
$791$	−56.8118	−2.01999
$792$	0	0
$793$	−1.87451	−0.0665657
$794$	0	0
$795$	−54.7895	−1.94318
$796$	0	0
$797$	44.8118	1.58731	0.793657	−	0.608365i	$-0.208174\pi$
$797$	44.8118	1.58731	0.793657	+	0.608365i	$0.208174\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.93725	0.280100
$804$	0	0
$805$	9.87451	0.348031
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	−31.2915	−1.09879	−0.549397	−	0.835562i	$-0.685142\pi$
$811$	−31.2915	−1.09879	−0.549397	+	0.835562i	$0.685142\pi$
$812$	0	0
$813$	−32.6863	−1.14636
$814$	0	0
$815$	6.47974	0.226975
$816$	0	0
$817$	0	0
$818$	0	0
$819$	29.1660	1.01914
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	31.8745	1.11108	0.555538	−	0.831491i	$-0.312512\pi$
$823$	31.8745	1.11108	0.555538	+	0.831491i	$0.312512\pi$
$824$	0	0
$825$	−28.1660	−0.980615
$826$	0	0
$827$	−52.6458	−1.83067	−0.915336	−	0.402691i	$-0.868075\pi$
$827$	−52.6458	−1.83067	−0.915336	+	0.402691i	$0.868075\pi$
$828$	0	0
$829$	17.1660	0.596200	0.298100	−	0.954535i	$-0.403647\pi$
$829$	17.1660	0.596200	0.298100	+	0.954535i	$0.403647\pi$
$830$	0	0
$831$	−72.8118	−2.52581
$832$	0	0
$833$	0	0
$834$	0	0
$835$	19.7490	0.683443
$836$	0	0
$837$	−14.9373	−0.516307
$838$	0	0
$839$	−41.5203	−1.43344	−0.716719	−	0.697362i	$-0.754357\pi$
$839$	−41.5203	−1.43344	−0.716719	+	0.697362i	$0.754357\pi$
$840$	0	0
$841$	−26.2915	−0.906604
$842$	0	0
$843$	−67.3542	−2.31980
$844$	0	0
$845$	14.8118	0.509540
$846$	0	0
$847$	−38.5830	−1.32573
$848$	0	0
$849$	−81.0810	−2.78269
$850$	0	0
$851$	−0.583005	−0.0199852
$852$	0	0
$853$	8.58301	0.293877	0.146938	−	0.989146i	$-0.453058\pi$
$853$	8.58301	0.293877	0.146938	+	0.989146i	$0.453058\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	−13.2288	−0.451359	−0.225680	−	0.974202i	$-0.572460\pi$
$859$	−13.2288	−0.451359	−0.225680	+	0.974202i	$0.572460\pi$
$860$	0	0
$861$	−2.81176	−0.0958246
$862$	0	0
$863$	−31.0627	−1.05739	−0.528694	−	0.848812i	$-0.677318\pi$
$863$	−31.0627	−1.05739	−0.528694	+	0.848812i	$0.677318\pi$
$864$	0	0
$865$	−9.87451	−0.335743
$866$	0	0
$867$	−44.9778	−1.52753
$868$	0	0
$869$	−18.5830	−0.630385
$870$	0	0
$871$	−1.29150	−0.0437609
$872$	0	0
$873$	14.8340	0.502054
$874$	0	0
$875$	−43.7490	−1.47899
$876$	0	0
$877$	41.6458	1.40628	0.703139	−	0.711053i	$-0.251781\pi$
$877$	41.6458	1.40628	0.703139	+	0.711053i	$0.251781\pi$
$878$	0	0
$879$	−76.5608	−2.58233
$880$	0	0
$881$	−36.8745	−1.24233	−0.621167	−	0.783678i	$-0.713341\pi$
$881$	−36.8745	−1.24233	−0.621167	+	0.783678i	$0.713341\pi$
$882$	0	0
$883$	28.3948	0.955560	0.477780	−	0.878480i	$-0.341442\pi$
$883$	28.3948	0.955560	0.477780	+	0.878480i	$0.341442\pi$
$884$	0	0
$885$	34.5608	1.16175
$886$	0	0
$887$	−17.4170	−0.584805	−0.292403	−	0.956295i	$-0.594455\pi$
$887$	−17.4170	−0.584805	−0.292403	+	0.956295i	$0.594455\pi$
$888$	0	0
$889$	48.4575	1.62521
$890$	0	0
$891$	−23.2288	−0.778193
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−6.68627	−0.223497
$896$	0	0
$897$	−8.70850	−0.290768
$898$	0	0
$899$	9.29150	0.309889
$900$	0	0
$901$	0	0
$902$	0	0
$903$	108.915	3.62447
$904$	0	0
$905$	36.5830	1.21606
$906$	0	0
$907$	39.9373	1.32609	0.663047	−	0.748577i	$-0.269263\pi$
$907$	39.9373	1.32609	0.663047	+	0.748577i	$0.269263\pi$
$908$	0	0
$909$	−54.5830	−1.81040
$910$	0	0
$911$	16.9373	0.561156	0.280578	−	0.959831i	$-0.409474\pi$
$911$	16.9373	0.561156	0.280578	+	0.959831i	$0.409474\pi$
$912$	0	0
$913$	−36.8745	−1.22037
$914$	0	0
$915$	−4.08104	−0.134915
$916$	0	0
$917$	7.06275	0.233232
$918$	0	0
$919$	19.8745	0.655600	0.327800	−	0.944747i	$-0.393693\pi$
$919$	19.8745	0.655600	0.327800	+	0.944747i	$0.393693\pi$
$920$	0	0
$921$	1.70850	0.0562969
$922$	0	0
$923$	5.41699	0.178303
$924$	0	0
$925$	0.811762	0.0266906
$926$	0	0
$927$	−53.1660	−1.74620
$928$	0	0
$929$	−9.58301	−0.314408	−0.157204	−	0.987566i	$-0.550248\pi$
$929$	−9.58301	−0.314408	−0.157204	+	0.987566i	$0.550248\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−36.1033	−1.18197
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.12549	0.232780	0.116390	−	0.993204i	$-0.462868\pi$
$937$	7.12549	0.232780	0.116390	+	0.993204i	$0.462868\pi$
$938$	0	0
$939$	23.4797	0.766232
$940$	0	0
$941$	−16.8340	−0.548772	−0.274386	−	0.961620i	$-0.588475\pi$
$941$	−16.8340	−0.548772	−0.274386	+	0.961620i	$0.588475\pi$
$942$	0	0
$943$	0.479741	0.0156225
$944$	0	0
$945$	15.8745	0.516398
$946$	0	0
$947$	7.74902	0.251809	0.125905	−	0.992042i	$-0.459817\pi$
$947$	7.74902	0.251809	0.125905	+	0.992042i	$0.459817\pi$
$948$	0	0
$949$	−3.41699	−0.110920
$950$	0	0
$951$	15.8745	0.514766
$952$	0	0
$953$	13.4575	0.435932	0.217966	−	0.975956i	$-0.430058\pi$
$953$	13.4575	0.435932	0.217966	+	0.975956i	$0.430058\pi$
$954$	0	0
$955$	10.8340	0.350580
$956$	0	0
$957$	−20.2288	−0.653903
$958$	0	0
$959$	−56.8118	−1.83455
$960$	0	0
$961$	0.874508	0.0282099
$962$	0	0
$963$	61.1660	1.97105
$964$	0	0
$965$	24.0000	0.772587
$966$	0	0
$967$	13.2915	0.427426	0.213713	−	0.976897i	$-0.431444\pi$
$967$	13.2915	0.427426	0.213713	+	0.976897i	$0.431444\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−54.3948	−1.74561	−0.872806	−	0.488068i	$-0.837702\pi$
$971$	−54.3948	−1.74561	−0.872806	+	0.488068i	$0.837702\pi$
$972$	0	0
$973$	67.9778	2.17927
$974$	0	0
$975$	12.1255	0.388327
$976$	0	0
$977$	7.45751	0.238587	0.119293	−	0.992859i	$-0.461937\pi$
$977$	7.45751	0.238587	0.119293	+	0.992859i	$0.461937\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−58.3320	−1.86240
$982$	0	0
$983$	31.7490	1.01264	0.506318	−	0.862347i	$-0.331006\pi$
$983$	31.7490	1.01264	0.506318	+	0.862347i	$0.331006\pi$
$984$	0	0
$985$	12.5830	0.400928
$986$	0	0
$987$	42.0000	1.33687
$988$	0	0
$989$	−18.5830	−0.590905
$990$	0	0
$991$	−2.83399	−0.0900246	−0.0450123	−	0.998986i	$-0.514333\pi$
$991$	−2.83399	−0.0900246	−0.0450123	+	0.998986i	$0.514333\pi$
$992$	0	0
$993$	52.4170	1.66340
$994$	0	0
$995$	−32.7085	−1.03693
$996$	0	0
$997$	16.2288	0.513970	0.256985	−	0.966415i	$-0.417271\pi$
$997$	16.2288	0.513970	0.256985	+	0.966415i	$0.417271\pi$
$998$	0	0
$999$	−0.937254	−0.0296534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.z.1.2		2
4.3	odd	2		722.2.a.g.1.1		2
12.11	even	2		6498.2.a.bg.1.2		2
19.8	odd	6		304.2.i.e.273.2		4
19.12	odd	6		304.2.i.e.49.2		4
19.18	odd	2		5776.2.a.ba.1.1		2
57.8	even	6		2736.2.s.v.577.1		4
57.50	even	6		2736.2.s.v.1873.1		4
76.3	even	18		722.2.e.n.389.2		12
76.7	odd	6		722.2.c.j.429.2		4
76.11	odd	6		722.2.c.j.653.2		4
76.15	even	18		722.2.e.n.415.1		12
76.23	odd	18		722.2.e.o.415.2		12
76.27	even	6		38.2.c.b.7.1	✓	4
76.31	even	6		38.2.c.b.11.1	yes	4
76.35	odd	18		722.2.e.o.389.1		12
76.43	odd	18		722.2.e.o.595.2		12
76.47	odd	18		722.2.e.o.423.1		12
76.51	even	18		722.2.e.n.245.2		12
76.55	odd	18		722.2.e.o.99.1		12
76.59	even	18		722.2.e.n.99.2		12
76.63	odd	18		722.2.e.o.245.1		12
76.67	even	18		722.2.e.n.423.2		12
76.71	even	18		722.2.e.n.595.1		12
76.75	even	2		722.2.a.j.1.2		2
152.27	even	6		1216.2.i.l.577.2		4
152.69	odd	6		1216.2.i.k.961.1		4
152.107	even	6		1216.2.i.l.961.2		4
152.141	odd	6		1216.2.i.k.577.1		4
228.107	odd	6		342.2.g.f.163.1		4
228.179	odd	6		342.2.g.f.235.1		4
228.227	odd	2		6498.2.a.ba.1.2		2
380.27	odd	12		950.2.j.g.349.3		8
380.103	odd	12		950.2.j.g.349.2		8
380.107	odd	12		950.2.j.g.49.2		8
380.179	even	6		950.2.e.k.501.2		4
380.183	odd	12		950.2.j.g.49.3		8
380.259	even	6		950.2.e.k.201.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.c.b.7.1	✓	4	76.27	even	6
38.2.c.b.11.1	yes	4	76.31	even	6
304.2.i.e.49.2		4	19.12	odd	6
304.2.i.e.273.2		4	19.8	odd	6
342.2.g.f.163.1		4	228.107	odd	6
342.2.g.f.235.1		4	228.179	odd	6
722.2.a.g.1.1		2	4.3	odd	2
722.2.a.j.1.2		2	76.75	even	2
722.2.c.j.429.2		4	76.7	odd	6
722.2.c.j.653.2		4	76.11	odd	6
722.2.e.n.99.2		12	76.59	even	18
722.2.e.n.245.2		12	76.51	even	18
722.2.e.n.389.2		12	76.3	even	18
722.2.e.n.415.1		12	76.15	even	18
722.2.e.n.423.2		12	76.67	even	18
722.2.e.n.595.1		12	76.71	even	18
722.2.e.o.99.1		12	76.55	odd	18
722.2.e.o.245.1		12	76.63	odd	18
722.2.e.o.389.1		12	76.35	odd	18
722.2.e.o.415.2		12	76.23	odd	18
722.2.e.o.423.1		12	76.47	odd	18
722.2.e.o.595.2		12	76.43	odd	18
950.2.e.k.201.2		4	380.259	even	6
950.2.e.k.501.2		4	380.179	even	6
950.2.j.g.49.2		8	380.107	odd	12
950.2.j.g.49.3		8	380.183	odd	12
950.2.j.g.349.2		8	380.103	odd	12
950.2.j.g.349.3		8	380.27	odd	12
1216.2.i.k.577.1		4	152.141	odd	6
1216.2.i.k.961.1		4	152.69	odd	6
1216.2.i.l.577.2		4	152.27	even	6
1216.2.i.l.961.2		4	152.107	even	6
2736.2.s.v.577.1		4	57.8	even	6
2736.2.s.v.1873.1		4	57.50	even	6
5776.2.a.z.1.2		2	1.1	even	1	trivial
5776.2.a.ba.1.1		2	19.18	odd	2
6498.2.a.ba.1.2		2	228.227	odd	2
6498.2.a.bg.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{3} -1.64575 q^{5} -3.64575 q^{7} +4.00000 q^{9} +O(q^{10})\)	\(q+2.64575 q^{3} -1.64575 q^{5} -3.64575 q^{7} +4.00000 q^{9} +4.64575 q^{11} -2.00000 q^{13} -4.35425 q^{15} -9.64575 q^{21} +1.64575 q^{23} -2.29150 q^{25} +2.64575 q^{27} -1.64575 q^{29} -5.64575 q^{31} +12.2915 q^{33} +6.00000 q^{35} -0.354249 q^{37} -5.29150 q^{39} +0.291503 q^{41} -11.2915 q^{43} -6.58301 q^{45} -4.35425 q^{47} +6.29150 q^{49} +12.5830 q^{53} -7.64575 q^{55} -7.93725 q^{59} +0.937254 q^{61} -14.5830 q^{63} +3.29150 q^{65} +0.645751 q^{67} +4.35425 q^{69} -2.70850 q^{71} +1.70850 q^{73} -6.06275 q^{75} -16.9373 q^{77} -4.00000 q^{79} -5.00000 q^{81} -7.93725 q^{83} -4.35425 q^{87} +7.29150 q^{91} -14.9373 q^{93} +3.70850 q^{97} +18.5830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7} + 8 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7 + 8 * q^9	\( 2 q + 2 q^{5} - 2 q^{7} + 8 q^{9} + 4 q^{11} - 4 q^{13} - 14 q^{15} - 14 q^{21} - 2 q^{23} + 6 q^{25} + 2 q^{29} - 6 q^{31} + 14 q^{33} + 12 q^{35} - 6 q^{37} - 10 q^{41} - 12 q^{43} + 8 q^{45} - 14 q^{47} + 2 q^{49} + 4 q^{53} - 10 q^{55} - 14 q^{61} - 8 q^{63} - 4 q^{65} - 4 q^{67} + 14 q^{69} - 16 q^{71} + 14 q^{73} - 28 q^{75} - 18 q^{77} - 8 q^{79} - 10 q^{81} - 14 q^{87} + 4 q^{91} - 14 q^{93} + 18 q^{97} + 16 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 + 8 * q^9 + 4 * q^11 - 4 * q^13 - 14 * q^15 - 14 * q^21 - 2 * q^23 + 6 * q^25 + 2 * q^29 - 6 * q^31 + 14 * q^33 + 12 * q^35 - 6 * q^37 - 10 * q^41 - 12 * q^43 + 8 * q^45 - 14 * q^47 + 2 * q^49 + 4 * q^53 - 10 * q^55 - 14 * q^61 - 8 * q^63 - 4 * q^65 - 4 * q^67 + 14 * q^69 - 16 * q^71 + 14 * q^73 - 28 * q^75 - 18 * q^77 - 8 * q^79 - 10 * q^81 - 14 * q^87 + 4 * q^91 - 14 * q^93 + 18 * q^97 + 16 * q^99

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