LMFDB - Embedded newform 1568.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.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:	$12.5205430369$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} -1.00000 q^{9} +2.00000 q^{11} -2.82843 q^{13} -4.24264 q^{17} +4.24264 q^{19} +8.00000 q^{23} -5.00000 q^{25} +5.65685 q^{27} +6.00000 q^{29} -8.48528 q^{31} -2.82843 q^{33} -2.00000 q^{37} +4.00000 q^{39} +4.24264 q^{41} +6.00000 q^{43} +2.82843 q^{47} +6.00000 q^{51} +6.00000 q^{53} -6.00000 q^{57} -12.7279 q^{59} +5.65685 q^{61} +12.0000 q^{67} -11.3137 q^{69} -4.00000 q^{71} +1.41421 q^{73} +7.07107 q^{75} +12.0000 q^{79} -5.00000 q^{81} +9.89949 q^{83} -8.48528 q^{87} -4.24264 q^{89} +12.0000 q^{93} +18.3848 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 4 q^{11} + 16 q^{23} - 10 q^{25} + 12 q^{29} - 4 q^{37} + 8 q^{39} + 12 q^{43} + 12 q^{51} + 12 q^{53} - 12 q^{57} + 24 q^{67} - 8 q^{71} + 24 q^{79} - 10 q^{81} + 24 q^{93} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 4 * q^11 + 16 * q^23 - 10 * q^25 + 12 * q^29 - 4 * q^37 + 8 * q^39 + 12 * q^43 + 12 * q^51 + 12 * q^53 - 12 * q^57 + 24 * q^67 - 8 * q^71 + 24 * q^79 - 10 * q^81 + 24 * q^93 - 4 * 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.24264	−1.02899	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−4.24264	−1.02899	−0.514496	+	0.857493i	$0.672021\pi$
$18$	0	0
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	4.24264	0.662589	0.331295	−	0.943527i	$-0.392515\pi$
$41$	4.24264	0.662589	0.331295	+	0.943527i	$0.392515\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	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
$49$	0	0
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.00000	−0.794719
$58$	0	0
$59$	−12.7279	−1.65703	−0.828517	−	0.559964i	$-0.810815\pi$
$59$	−12.7279	−1.65703	−0.828517	+	0.559964i	$0.810815\pi$
$60$	0	0
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	−11.3137	−1.36201
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	0	0
$75$	7.07107	0.816497
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	9.89949	1.08661	0.543305	−	0.839535i	$-0.317173\pi$
$83$	9.89949	1.08661	0.543305	+	0.839535i	$0.317173\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.48528	−0.909718
$88$	0	0
$89$	−4.24264	−0.449719	−0.224860	−	0.974391i	$-0.572192\pi$
$89$	−4.24264	−0.449719	−0.224860	+	0.974391i	$0.572192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.3848	1.86669	0.933346	−	0.358979i	$-0.116875\pi$
$97$	18.3848	1.86669	0.933346	+	0.358979i	$0.116875\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	16.9706	1.68863	0.844317	−	0.535844i	$-0.180006\pi$
$101$	16.9706	1.68863	0.844317	+	0.535844i	$0.180006\pi$
$102$	0	0
$103$	8.48528	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$103$	8.48528	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$104$	0	0
$105$	0	0
$106$	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
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	2.82843	0.268462
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.82843	0.261488
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	0	0
$129$	−8.48528	−0.747087
$130$	0	0
$131$	1.41421	0.123560	0.0617802	−	0.998090i	$-0.480322\pi$
$131$	1.41421	0.123560	0.0617802	+	0.998090i	$0.480322\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	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
$139$	−12.7279	−1.07957	−0.539784	−	0.841803i	$-0.681494\pi$
$139$	−12.7279	−1.07957	−0.539784	+	0.841803i	$0.681494\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	4.24264	0.342997
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.82843	0.225733	0.112867	−	0.993610i	$-0.463997\pi$
$157$	2.82843	0.225733	0.112867	+	0.993610i	$0.463997\pi$
$158$	0	0
$159$	−8.48528	−0.672927
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.1421	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$167$	14.1421	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−4.24264	−0.324443
$172$	0	0
$173$	8.48528	0.645124	0.322562	−	0.946548i	$-0.395456\pi$
$173$	8.48528	0.645124	0.322562	+	0.946548i	$0.395456\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.0000	1.35296
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−14.1421	−1.05118	−0.525588	−	0.850739i	$-0.676155\pi$
$181$	−14.1421	−1.05118	−0.525588	+	0.850739i	$0.676155\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.48528	−0.620505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	25.4558	1.80452	0.902258	−	0.431196i	$-0.141908\pi$
$199$	25.4558	1.80452	0.902258	+	0.431196i	$0.141908\pi$
$200$	0	0
$201$	−16.9706	−1.19701
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	8.48528	0.586939
$210$	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
$213$	5.65685	0.387601
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−16.9706	−1.13643	−0.568216	−	0.822879i	$-0.692366\pi$
$223$	−16.9706	−1.13643	−0.568216	+	0.822879i	$0.692366\pi$
$224$	0	0
$225$	5.00000	0.333333
$226$	0	0
$227$	12.7279	0.844782	0.422391	−	0.906414i	$-0.361191\pi$
$227$	12.7279	0.844782	0.422391	+	0.906414i	$0.361191\pi$
$228$	0	0
$229$	−2.82843	−0.186908	−0.0934539	−	0.995624i	$-0.529791\pi$
$229$	−2.82843	−0.186908	−0.0934539	+	0.995624i	$0.529791\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.9706	−1.10236
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−18.3848	−1.18427	−0.592134	−	0.805840i	$-0.701714\pi$
$241$	−18.3848	−1.18427	−0.592134	+	0.805840i	$0.701714\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	−14.0000	−0.887214
$250$	0	0
$251$	−12.7279	−0.803379	−0.401690	−	0.915776i	$-0.631577\pi$
$251$	−12.7279	−0.803379	−0.401690	+	0.915776i	$0.631577\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.2132	1.32324	0.661622	−	0.749838i	$-0.269869\pi$
$257$	21.2132	1.32324	0.661622	+	0.749838i	$0.269869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−20.0000	−1.23325	−0.616626	−	0.787256i	$-0.711501\pi$
$263$	−20.0000	−1.23325	−0.616626	+	0.787256i	$0.711501\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	−25.4558	−1.55207	−0.776035	−	0.630690i	$-0.782772\pi$
$269$	−25.4558	−1.55207	−0.776035	+	0.630690i	$0.782772\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.0000	−0.603023
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	0	0
$279$	8.48528	0.508001
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	21.2132	1.26099	0.630497	−	0.776192i	$-0.282851\pi$
$283$	21.2132	1.26099	0.630497	+	0.776192i	$0.282851\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−26.0000	−1.52415
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	11.3137	0.656488
$298$	0	0
$299$	−22.6274	−1.30858
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−21.2132	−1.21070	−0.605351	−	0.795959i	$-0.706967\pi$
$307$	−21.2132	−1.21070	−0.605351	+	0.795959i	$0.706967\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	−18.3848	−1.03917	−0.519584	−	0.854419i	$-0.673913\pi$
$313$	−18.3848	−1.03917	−0.519584	+	0.854419i	$0.673913\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−5.65685	−0.315735
$322$	0	0
$323$	−18.0000	−1.00155
$324$	0	0
$325$	14.1421	0.784465
$326$	0	0
$327$	−25.4558	−1.40771
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	16.9706	0.921714
$340$	0	0
$341$	−16.9706	−0.919007
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	19.7990	1.05982	0.529908	−	0.848055i	$-0.322227\pi$
$349$	19.7990	1.05982	0.529908	+	0.848055i	$0.322227\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	−21.2132	−1.12906	−0.564532	−	0.825411i	$-0.690943\pi$
$353$	−21.2132	−1.12906	−0.564532	+	0.825411i	$0.690943\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	9.89949	0.519589
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.9411	1.77171	0.885856	−	0.463960i	$-0.153572\pi$
$367$	33.9411	1.77171	0.885856	+	0.463960i	$0.153572\pi$
$368$	0	0
$369$	−4.24264	−0.220863
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.9706	−0.874028
$378$	0	0
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	−16.9706	−0.869428
$382$	0	0
$383$	−25.4558	−1.30073	−0.650366	−	0.759621i	$-0.725385\pi$
$383$	−25.4558	−1.30073	−0.650366	+	0.759621i	$0.725385\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−33.9411	−1.71648
$392$	0	0
$393$	−2.00000	−0.100887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	36.7696	1.84541	0.922705	−	0.385506i	$-0.125973\pi$
$397$	36.7696	1.84541	0.922705	+	0.385506i	$0.125973\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.00000	−0.198273
$408$	0	0
$409$	35.3553	1.74821	0.874105	−	0.485738i	$-0.161449\pi$
$409$	35.3553	1.74821	0.874105	+	0.485738i	$0.161449\pi$
$410$	0	0
$411$	16.9706	0.837096
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.0000	0.881464
$418$	0	0
$419$	12.7279	0.621800	0.310900	−	0.950443i	$-0.399370\pi$
$419$	12.7279	0.621800	0.310900	+	0.950443i	$0.399370\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	−2.82843	−0.137523
$424$	0	0
$425$	21.2132	1.02899
$426$	0	0
$427$	0	0
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	0	0
$433$	1.41421	0.0679628	0.0339814	−	0.999422i	$-0.489181\pi$
$433$	1.41421	0.0679628	0.0339814	+	0.999422i	$0.489181\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	33.9411	1.62362
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.48528	0.401340
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	8.48528	0.399556
$452$	0	0
$453$	−16.9706	−0.797347
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	0	0
$459$	−24.0000	−1.12022
$460$	0	0
$461$	25.4558	1.18560	0.592798	−	0.805351i	$-0.298023\pi$
$461$	25.4558	1.18560	0.592798	+	0.805351i	$0.298023\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.7279	−0.588978	−0.294489	−	0.955655i	$-0.595149\pi$
$467$	−12.7279	−0.588978	−0.294489	+	0.955655i	$0.595149\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	12.0000	0.551761
$474$	0	0
$475$	−21.2132	−0.973329
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−2.82843	−0.129234	−0.0646171	−	0.997910i	$-0.520583\pi$
$479$	−2.82843	−0.129234	−0.0646171	+	0.997910i	$0.520583\pi$
$480$	0	0
$481$	5.65685	0.257930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−8.48528	−0.383718
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−25.4558	−1.14647
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−20.0000	−0.893534
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.07107	0.314037
$508$	0	0
$509$	−42.4264	−1.88052	−0.940259	−	0.340461i	$-0.889417\pi$
$509$	−42.4264	−1.88052	−0.940259	+	0.340461i	$0.889417\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	24.0000	1.05963
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	29.6985	1.30111	0.650557	−	0.759457i	$-0.274535\pi$
$521$	29.6985	1.30111	0.650557	+	0.759457i	$0.274535\pi$
$522$	0	0
$523$	12.7279	0.556553	0.278277	−	0.960501i	$-0.410237\pi$
$523$	12.7279	0.556553	0.278277	+	0.960501i	$0.410237\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	36.0000	1.56818
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	12.7279	0.552345
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.65685	0.244111
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	0	0
$546$	0	0
$547$	42.0000	1.79579	0.897895	−	0.440209i	$-0.145096\pi$
$547$	42.0000	1.79579	0.897895	+	0.440209i	$0.145096\pi$
$548$	0	0
$549$	−5.65685	−0.241429
$550$	0	0
$551$	25.4558	1.08446
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−16.9706	−0.717778
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−12.7279	−0.536418	−0.268209	−	0.963361i	$-0.586432\pi$
$563$	−12.7279	−0.536418	−0.268209	+	0.963361i	$0.586432\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	42.0000	1.75765	0.878823	−	0.477149i	$-0.158330\pi$
$571$	42.0000	1.75765	0.878823	+	0.477149i	$0.158330\pi$
$572$	0	0
$573$	28.2843	1.18159
$574$	0	0
$575$	−40.0000	−1.66812
$576$	0	0
$577$	−1.41421	−0.0588745	−0.0294372	−	0.999567i	$-0.509372\pi$
$577$	−1.41421	−0.0588745	−0.0294372	+	0.999567i	$0.509372\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.1838	1.57601	0.788006	−	0.615667i	$-0.211113\pi$
$587$	38.1838	1.57601	0.788006	+	0.615667i	$0.211113\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	0	0
$591$	−25.4558	−1.04711
$592$	0	0
$593$	29.6985	1.21957	0.609785	−	0.792567i	$-0.291256\pi$
$593$	29.6985	1.21957	0.609785	+	0.792567i	$0.291256\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−36.0000	−1.47338
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−18.3848	−0.749931	−0.374965	−	0.927039i	$-0.622345\pi$
$601$	−18.3848	−0.749931	−0.374965	+	0.927039i	$0.622345\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.9706	−0.688814	−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706	−0.688814	−0.344407	+	0.938820i	$0.611920\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	−4.24264	−0.170526	−0.0852631	−	0.996358i	$-0.527173\pi$
$619$	−4.24264	−0.170526	−0.0852631	+	0.996358i	$0.527173\pi$
$620$	0	0
$621$	45.2548	1.81601
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	8.48528	0.338330
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	0	0
$633$	16.9706	0.674519
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	4.00000	0.158238
$640$	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
$643$	−12.7279	−0.501940	−0.250970	−	0.967995i	$-0.580750\pi$
$643$	−12.7279	−0.501940	−0.250970	+	0.967995i	$0.580750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−19.7990	−0.778379	−0.389189	−	0.921158i	$-0.627245\pi$
$647$	−19.7990	−0.778379	−0.389189	+	0.921158i	$0.627245\pi$
$648$	0	0
$649$	−25.4558	−0.999229
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.41421	−0.0551737
$658$	0	0
$659$	34.0000	1.32445	0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000	1.32445	0.662226	+	0.749304i	$0.269612\pi$
$660$	0	0
$661$	−22.6274	−0.880105	−0.440052	−	0.897972i	$-0.645040\pi$
$661$	−22.6274	−0.880105	−0.440052	+	0.897972i	$0.645040\pi$
$662$	0	0
$663$	−16.9706	−0.659082
$664$	0	0
$665$	0	0
$666$	0	0
$667$	48.0000	1.85857
$668$	0	0
$669$	24.0000	0.927894
$670$	0	0
$671$	11.3137	0.436761
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	−28.2843	−1.08866
$676$	0	0
$677$	−42.4264	−1.63058	−0.815290	−	0.579053i	$-0.803422\pi$
$677$	−42.4264	−1.63058	−0.815290	+	0.579053i	$0.803422\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.00000	0.152610
$688$	0	0
$689$	−16.9706	−0.646527
$690$	0	0
$691$	46.6690	1.77537	0.887687	−	0.460447i	$-0.152311\pi$
$691$	46.6690	1.77537	0.887687	+	0.460447i	$0.152311\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	33.9411	1.28377
$700$	0	0
$701$	42.0000	1.58632	0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000	1.58632	0.793159	+	0.609015i	$0.208435\pi$
$702$	0	0
$703$	−8.48528	−0.320028
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	0	0
$713$	−67.8823	−2.54221
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.3137	−0.422518
$718$	0	0
$719$	25.4558	0.949343	0.474671	−	0.880163i	$-0.342567\pi$
$719$	25.4558	0.949343	0.474671	+	0.880163i	$0.342567\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	26.0000	0.966950
$724$	0	0
$725$	−30.0000	−1.11417
$726$	0	0
$727$	−8.48528	−0.314702	−0.157351	−	0.987543i	$-0.550295\pi$
$727$	−8.48528	−0.314702	−0.157351	+	0.987543i	$0.550295\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−25.4558	−0.941518
$732$	0	0
$733$	−28.2843	−1.04470	−0.522352	−	0.852730i	$-0.674945\pi$
$733$	−28.2843	−1.04470	−0.522352	+	0.852730i	$0.674945\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	0	0
$741$	16.9706	0.623429
$742$	0	0
$743$	44.0000	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.89949	−0.362204
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	−22.6274	−0.821323
$760$	0	0
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.0000	1.29988
$768$	0	0
$769$	−35.3553	−1.27495	−0.637473	−	0.770473i	$-0.720020\pi$
$769$	−35.3553	−1.27495	−0.637473	+	0.770473i	$0.720020\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	42.4264	1.52400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18.0000	0.644917
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	33.9411	1.21296
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−12.7279	−0.453701	−0.226851	−	0.973930i	$-0.572843\pi$
$787$	−12.7279	−0.453701	−0.226851	+	0.973930i	$0.572843\pi$
$788$	0	0
$789$	28.2843	1.00695
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−16.0000	−0.568177
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.4264	1.50282	0.751410	−	0.659835i	$-0.229374\pi$
$797$	42.4264	1.50282	0.751410	+	0.659835i	$0.229374\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	4.24264	0.149906
$802$	0	0
$803$	2.82843	0.0998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	36.0000	1.26726
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	−29.6985	−1.04285	−0.521427	−	0.853296i	$-0.674600\pi$
$811$	−29.6985	−1.04285	−0.521427	+	0.853296i	$0.674600\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25.4558	0.890587
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	14.1421	0.492366
$826$	0	0
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	−39.5980	−1.37529	−0.687647	−	0.726045i	$-0.741356\pi$
$829$	−39.5980	−1.37529	−0.687647	+	0.726045i	$0.741356\pi$
$830$	0	0
$831$	25.4558	0.883053
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−48.0000	−1.65912
$838$	0	0
$839$	48.0833	1.66002	0.830009	−	0.557750i	$-0.188335\pi$
$839$	48.0833	1.66002	0.830009	+	0.557750i	$0.188335\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	−19.7990	−0.677905	−0.338952	−	0.940804i	$-0.610073\pi$
$853$	−19.7990	−0.677905	−0.338952	+	0.940804i	$0.610073\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−12.7279	−0.434778	−0.217389	−	0.976085i	$-0.569754\pi$
$857$	−12.7279	−0.434778	−0.217389	+	0.976085i	$0.569754\pi$
$858$	0	0
$859$	−4.24264	−0.144757	−0.0723785	−	0.997377i	$-0.523059\pi$
$859$	−4.24264	−0.144757	−0.0723785	+	0.997377i	$0.523059\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.41421	−0.0480292
$868$	0	0
$869$	24.0000	0.814144
$870$	0	0
$871$	−33.9411	−1.15005
$872$	0	0
$873$	−18.3848	−0.622230
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.24264	0.142938	0.0714691	−	0.997443i	$-0.477231\pi$
$881$	4.24264	0.142938	0.0714691	+	0.997443i	$0.477231\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.7696	−1.23460	−0.617300	−	0.786728i	$-0.711774\pi$
$887$	−36.7696	−1.23460	−0.617300	+	0.786728i	$0.711774\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−10.0000	−0.335013
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	−50.9117	−1.69800
$900$	0	0
$901$	−25.4558	−0.848057
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	0	0
$909$	−16.9706	−0.562878
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	19.7990	0.655251
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	30.0000	0.988534
$922$	0	0
$923$	11.3137	0.372395
$924$	0	0
$925$	10.0000	0.328798
$926$	0	0
$927$	−8.48528	−0.278693
$928$	0	0
$929$	−4.24264	−0.139197	−0.0695983	−	0.997575i	$-0.522172\pi$
$929$	−4.24264	−0.139197	−0.0695983	+	0.997575i	$0.522172\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−16.0000	−0.523816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.0416	0.785406	0.392703	−	0.919665i	$-0.371540\pi$
$937$	24.0416	0.785406	0.392703	+	0.919665i	$0.371540\pi$
$938$	0	0
$939$	26.0000	0.848478
$940$	0	0
$941$	−33.9411	−1.10645	−0.553225	−	0.833032i	$-0.686603\pi$
$941$	−33.9411	−1.10645	−0.553225	+	0.833032i	$0.686603\pi$
$942$	0	0
$943$	33.9411	1.10528
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.0000	1.10485	0.552426	−	0.833562i	$-0.313702\pi$
$947$	34.0000	1.10485	0.552426	+	0.833562i	$0.313702\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	−25.4558	−0.825462
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−16.9706	−0.548580
$958$	0	0
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	0	0
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	25.4558	0.817760
$970$	0	0
$971$	43.8406	1.40691	0.703456	−	0.710739i	$-0.251639\pi$
$971$	43.8406	1.40691	0.703456	+	0.710739i	$0.251639\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−20.0000	−0.640513
$976$	0	0
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	0	0
$979$	−8.48528	−0.271191
$980$	0	0
$981$	−18.0000	−0.574696
$982$	0	0
$983$	−25.4558	−0.811915	−0.405958	−	0.913892i	$-0.633062\pi$
$983$	−25.4558	−0.811915	−0.405958	+	0.913892i	$0.633062\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	0	0
$993$	25.4558	0.807817
$994$	0	0
$995$	0	0
$996$	0	0
$997$	56.5685	1.79154	0.895772	−	0.444514i	$-0.146624\pi$
$997$	56.5685	1.79154	0.895772	+	0.444514i	$0.146624\pi$
$998$	0	0
$999$	−11.3137	−0.357950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.a.r.1.1	yes	2
4.3	odd	2		1568.2.a.q.1.2	yes	2
7.2	even	3		1568.2.i.q.1537.2		4
7.3	odd	6		1568.2.i.q.961.1		4
7.4	even	3		1568.2.i.q.961.2		4
7.5	odd	6		1568.2.i.q.1537.1		4
7.6	odd	2	inner	1568.2.a.r.1.2	yes	2
8.3	odd	2		3136.2.a.bo.1.1		2
8.5	even	2		3136.2.a.bl.1.2		2
28.3	even	6		1568.2.i.r.961.2		4
28.11	odd	6		1568.2.i.r.961.1		4
28.19	even	6		1568.2.i.r.1537.2		4
28.23	odd	6		1568.2.i.r.1537.1		4
28.27	even	2		1568.2.a.q.1.1	✓	2
56.13	odd	2		3136.2.a.bl.1.1		2
56.27	even	2		3136.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1568.2.a.q.1.1	✓	2	28.27	even	2
1568.2.a.q.1.2	yes	2	4.3	odd	2
1568.2.a.r.1.1	yes	2	1.1	even	1	trivial
1568.2.a.r.1.2	yes	2	7.6	odd	2	inner
1568.2.i.q.961.1		4	7.3	odd	6
1568.2.i.q.961.2		4	7.4	even	3
1568.2.i.q.1537.1		4	7.5	odd	6
1568.2.i.q.1537.2		4	7.2	even	3
1568.2.i.r.961.1		4	28.11	odd	6
1568.2.i.r.961.2		4	28.3	even	6
1568.2.i.r.1537.1		4	28.23	odd	6
1568.2.i.r.1537.2		4	28.19	even	6
3136.2.a.bl.1.1		2	56.13	odd	2
3136.2.a.bl.1.2		2	8.5	even	2
3136.2.a.bo.1.1		2	8.3	odd	2
3136.2.a.bo.1.2		2	56.27	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} -1.00000 q^{9} +2.00000 q^{11} -2.82843 q^{13} -4.24264 q^{17} +4.24264 q^{19} +8.00000 q^{23} -5.00000 q^{25} +5.65685 q^{27} +6.00000 q^{29} -8.48528 q^{31} -2.82843 q^{33} -2.00000 q^{37} +4.00000 q^{39} +4.24264 q^{41} +6.00000 q^{43} +2.82843 q^{47} +6.00000 q^{51} +6.00000 q^{53} -6.00000 q^{57} -12.7279 q^{59} +5.65685 q^{61} +12.0000 q^{67} -11.3137 q^{69} -4.00000 q^{71} +1.41421 q^{73} +7.07107 q^{75} +12.0000 q^{79} -5.00000 q^{81} +9.89949 q^{83} -8.48528 q^{87} -4.24264 q^{89} +12.0000 q^{93} +18.3848 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 4 q^{11} + 16 q^{23} - 10 q^{25} + 12 q^{29} - 4 q^{37} + 8 q^{39} + 12 q^{43} + 12 q^{51} + 12 q^{53} - 12 q^{57} + 24 q^{67} - 8 q^{71} + 24 q^{79} - 10 q^{81} + 24 q^{93} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 4 * q^11 + 16 * q^23 - 10 * q^25 + 12 * q^29 - 4 * q^37 + 8 * q^39 + 12 * q^43 + 12 * q^51 + 12 * q^53 - 12 * q^57 + 24 * q^67 - 8 * q^71 + 24 * q^79 - 10 * q^81 + 24 * q^93 - 4 * q^99

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