LMFDB - Embedded newform 6864.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} +7.40312 q^{17} +5.40312 q^{19} +9.40312 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -5.40312 q^{31} -1.00000 q^{33} +7.40312 q^{37} -1.00000 q^{39} +2.00000 q^{41} +2.00000 q^{45} +8.00000 q^{47} -7.00000 q^{49} -7.40312 q^{51} -3.40312 q^{53} +2.00000 q^{55} -5.40312 q^{57} +4.00000 q^{59} -12.8062 q^{61} +2.00000 q^{65} -9.40312 q^{69} -10.8062 q^{71} +0.596876 q^{73} +1.00000 q^{75} -14.8062 q^{79} +1.00000 q^{81} +4.00000 q^{83} +14.8062 q^{85} -2.00000 q^{87} -10.0000 q^{89} +5.40312 q^{93} +10.8062 q^{95} +10.0000 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} + 6 q^{23} - 2 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{31} - 2 q^{33} + 2 q^{37} - 2 q^{39} + 4 q^{41} + 4 q^{45} + 16 q^{47} - 14 q^{49} - 2 q^{51} + 6 q^{53} + 4 q^{55} + 2 q^{57} + 8 q^{59} + 4 q^{65} - 6 q^{69} + 4 q^{71} + 14 q^{73} + 2 q^{75} - 4 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 20 q^{89} - 2 q^{93} - 4 q^{95} + 20 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 + 6 * q^23 - 2 * q^25 - 2 * q^27 + 4 * q^29 + 2 * q^31 - 2 * q^33 + 2 * q^37 - 2 * q^39 + 4 * q^41 + 4 * q^45 + 16 * q^47 - 14 * q^49 - 2 * q^51 + 6 * q^53 + 4 * q^55 + 2 * q^57 + 8 * q^59 + 4 * q^65 - 6 * q^69 + 4 * q^71 + 14 * q^73 + 2 * q^75 - 4 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 20 * q^89 - 2 * q^93 - 4 * q^95 + 20 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	7.40312	1.79552	0.897761	−	0.440484i	$-0.145193\pi$
$17$	7.40312	1.79552	0.897761	+	0.440484i	$0.145193\pi$
$18$	0	0
$19$	5.40312	1.23956	0.619781	−	0.784775i	$-0.287221\pi$
$19$	5.40312	1.23956	0.619781	+	0.784775i	$0.287221\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.40312	1.96069	0.980343	−	0.197298i	$-0.0632168\pi$
$23$	9.40312	1.96069	0.980343	+	0.197298i	$0.0632168\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−5.40312	−0.970430	−0.485215	−	0.874395i	$-0.661259\pi$
$31$	−5.40312	−0.970430	−0.485215	+	0.874395i	$0.661259\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.40312	1.21707	0.608533	−	0.793529i	$-0.291758\pi$
$37$	7.40312	1.21707	0.608533	+	0.793529i	$0.291758\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−7.40312	−1.03664
$52$	0	0
$53$	−3.40312	−0.467455	−0.233728	−	0.972302i	$-0.575092\pi$
$53$	−3.40312	−0.467455	−0.233728	+	0.972302i	$0.575092\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−5.40312	−0.715661
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−12.8062	−1.63967	−0.819836	−	0.572598i	$-0.805935\pi$
$61$	−12.8062	−1.63967	−0.819836	+	0.572598i	$0.805935\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−9.40312	−1.13200
$70$	0	0
$71$	−10.8062	−1.28247	−0.641233	−	0.767346i	$-0.721577\pi$
$71$	−10.8062	−1.28247	−0.641233	+	0.767346i	$0.721577\pi$
$72$	0	0
$73$	0.596876	0.0698590	0.0349295	−	0.999390i	$-0.488879\pi$
$73$	0.596876	0.0698590	0.0349295	+	0.999390i	$0.488879\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.8062	−1.66583	−0.832917	−	0.553399i	$-0.813331\pi$
$79$	−14.8062	−1.66583	−0.832917	+	0.553399i	$0.813331\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	14.8062	1.60596
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.40312	0.560278
$94$	0	0
$95$	10.8062	1.10870
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	18.8062	1.85303	0.926517	−	0.376252i	$-0.122787\pi$
$103$	18.8062	1.85303	0.926517	+	0.376252i	$0.122787\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.40312	0.522340	0.261170	−	0.965293i	$-0.415892\pi$
$107$	5.40312	0.522340	0.261170	+	0.965293i	$0.415892\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−7.40312	−0.702673
$112$	0	0
$113$	4.80625	0.452134	0.226067	−	0.974112i	$-0.427413\pi$
$113$	4.80625	0.452134	0.226067	+	0.974112i	$0.427413\pi$
$114$	0	0
$115$	18.8062	1.75369
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−6.80625	−0.603957	−0.301978	−	0.953315i	$-0.597647\pi$
$127$	−6.80625	−0.603957	−0.301978	+	0.953315i	$0.597647\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.59688	0.226890	0.113445	−	0.993544i	$-0.463811\pi$
$131$	2.59688	0.226890	0.113445	+	0.993544i	$0.463811\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.00000	−0.172133
$136$	0	0
$137$	−12.8062	−1.09411	−0.547056	−	0.837096i	$-0.684251\pi$
$137$	−12.8062	−1.09411	−0.547056	+	0.837096i	$0.684251\pi$
$138$	0	0
$139$	−18.8062	−1.59513	−0.797563	−	0.603236i	$-0.793878\pi$
$139$	−18.8062	−1.59513	−0.797563	+	0.603236i	$0.793878\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	7.00000	0.577350
$148$	0	0
$149$	−12.8062	−1.04913	−0.524564	−	0.851371i	$-0.675772\pi$
$149$	−12.8062	−1.04913	−0.524564	+	0.851371i	$0.675772\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	7.40312	0.598507
$154$	0	0
$155$	−10.8062	−0.867979
$156$	0	0
$157$	−20.8062	−1.66052	−0.830260	−	0.557377i	$-0.811808\pi$
$157$	−20.8062	−1.66052	−0.830260	+	0.557377i	$0.811808\pi$
$158$	0	0
$159$	3.40312	0.269885
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.8062	0.846411	0.423205	−	0.906034i	$-0.360905\pi$
$163$	10.8062	0.846411	0.423205	+	0.906034i	$0.360905\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.40312	0.413187
$172$	0	0
$173$	20.8062	1.58187	0.790935	−	0.611900i	$-0.209595\pi$
$173$	20.8062	1.58187	0.790935	+	0.611900i	$0.209595\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	8.80625	0.654563	0.327282	−	0.944927i	$-0.393867\pi$
$181$	8.80625	0.654563	0.327282	+	0.944927i	$0.393867\pi$
$182$	0	0
$183$	12.8062	0.946665
$184$	0	0
$185$	14.8062	1.08858
$186$	0	0
$187$	7.40312	0.541370
$188$	0	0
$189$	0	0
$190$	0	0
$191$	25.4031	1.83811	0.919053	−	0.394134i	$-0.128956\pi$
$191$	25.4031	1.83811	0.919053	+	0.394134i	$0.128956\pi$
$192$	0	0
$193$	−4.59688	−0.330890	−0.165445	−	0.986219i	$-0.552906\pi$
$193$	−4.59688	−0.330890	−0.165445	+	0.986219i	$0.552906\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−12.8062	−0.912407	−0.456204	−	0.889875i	$-0.650791\pi$
$197$	−12.8062	−0.912407	−0.456204	+	0.889875i	$0.650791\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	9.40312	0.653562
$208$	0	0
$209$	5.40312	0.373742
$210$	0	0
$211$	18.8062	1.29468	0.647338	−	0.762203i	$-0.275882\pi$
$211$	18.8062	1.29468	0.647338	+	0.762203i	$0.275882\pi$
$212$	0	0
$213$	10.8062	0.740432
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.596876	−0.0403331
$220$	0	0
$221$	7.40312	0.497988
$222$	0	0
$223$	−13.4031	−0.897540	−0.448770	−	0.893647i	$-0.648138\pi$
$223$	−13.4031	−0.897540	−0.448770	+	0.893647i	$0.648138\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−22.8062	−1.51370	−0.756852	−	0.653586i	$-0.773264\pi$
$227$	−22.8062	−1.51370	−0.756852	+	0.653586i	$0.773264\pi$
$228$	0	0
$229$	15.4031	1.01787	0.508934	−	0.860806i	$-0.330040\pi$
$229$	15.4031	1.01787	0.508934	+	0.860806i	$0.330040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.40312	−0.222946	−0.111473	−	0.993767i	$-0.535557\pi$
$233$	−3.40312	−0.222946	−0.111473	+	0.993767i	$0.535557\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	14.8062	0.961769
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	8.59688	0.553773	0.276887	−	0.960903i	$-0.410697\pi$
$241$	8.59688	0.553773	0.276887	+	0.960903i	$0.410697\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−14.0000	−0.894427
$246$	0	0
$247$	5.40312	0.343793
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	9.40312	0.591169
$254$	0	0
$255$	−14.8062	−0.927203
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−6.80625	−0.418105
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	−22.2094	−1.35413	−0.677065	−	0.735924i	$-0.736748\pi$
$269$	−22.2094	−1.35413	−0.677065	+	0.735924i	$0.736748\pi$
$270$	0	0
$271$	−2.80625	−0.170467	−0.0852337	−	0.996361i	$-0.527164\pi$
$271$	−2.80625	−0.170467	−0.0852337	+	0.996361i	$0.527164\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	0	0
$279$	−5.40312	−0.323477
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	−10.8062	−0.640107
$286$	0	0
$287$	0	0
$288$	0	0
$289$	37.8062	2.22390
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	24.8062	1.44920	0.724598	−	0.689172i	$-0.242025\pi$
$293$	24.8062	1.44920	0.724598	+	0.689172i	$0.242025\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	9.40312	0.543797
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−25.6125	−1.46657
$306$	0	0
$307$	−29.4031	−1.67812	−0.839062	−	0.544035i	$-0.816896\pi$
$307$	−29.4031	−1.67812	−0.839062	+	0.544035i	$0.816896\pi$
$308$	0	0
$309$	−18.8062	−1.06985
$310$	0	0
$311$	28.2094	1.59961	0.799803	−	0.600262i	$-0.204937\pi$
$311$	28.2094	1.59961	0.799803	+	0.600262i	$0.204937\pi$
$312$	0	0
$313$	−3.19375	−0.180522	−0.0902608	−	0.995918i	$-0.528770\pi$
$313$	−3.19375	−0.180522	−0.0902608	+	0.995918i	$0.528770\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	−5.40312	−0.301573
$322$	0	0
$323$	40.0000	2.22566
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−14.0000	−0.774202
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	0	0
$333$	7.40312	0.405689
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8.80625	−0.479707	−0.239853	−	0.970809i	$-0.577099\pi$
$337$	−8.80625	−0.479707	−0.239853	+	0.970809i	$0.577099\pi$
$338$	0	0
$339$	−4.80625	−0.261040
$340$	0	0
$341$	−5.40312	−0.292596
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−18.8062	−1.01249
$346$	0	0
$347$	−24.2094	−1.29963	−0.649814	−	0.760094i	$-0.725153\pi$
$347$	−24.2094	−1.29963	−0.649814	+	0.760094i	$0.725153\pi$
$348$	0	0
$349$	−4.80625	−0.257273	−0.128636	−	0.991692i	$-0.541060\pi$
$349$	−4.80625	−0.257273	−0.128636	+	0.991692i	$0.541060\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	−21.6125	−1.14707
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	10.1938	0.536513
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	1.19375	0.0624838
$366$	0	0
$367$	−18.8062	−0.981678	−0.490839	−	0.871250i	$-0.663310\pi$
$367$	−18.8062	−0.981678	−0.490839	+	0.871250i	$0.663310\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.8062	0.870195	0.435097	−	0.900383i	$-0.356714\pi$
$373$	16.8062	0.870195	0.435097	+	0.900383i	$0.356714\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	6.80625	0.348695
$382$	0	0
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.2094	0.517636	0.258818	−	0.965926i	$-0.416667\pi$
$389$	10.2094	0.517636	0.258818	+	0.965926i	$0.416667\pi$
$390$	0	0
$391$	69.6125	3.52046
$392$	0	0
$393$	−2.59688	−0.130995
$394$	0	0
$395$	−29.6125	−1.48997
$396$	0	0
$397$	12.5969	0.632219	0.316110	−	0.948723i	$-0.397623\pi$
$397$	12.5969	0.632219	0.316110	+	0.948723i	$0.397623\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.80625	0.439763	0.219882	−	0.975527i	$-0.429433\pi$
$401$	8.80625	0.439763	0.219882	+	0.975527i	$0.429433\pi$
$402$	0	0
$403$	−5.40312	−0.269149
$404$	0	0
$405$	2.00000	0.0993808
$406$	0	0
$407$	7.40312	0.366959
$408$	0	0
$409$	35.4031	1.75057	0.875286	−	0.483606i	$-0.160673\pi$
$409$	35.4031	1.75057	0.875286	+	0.483606i	$0.160673\pi$
$410$	0	0
$411$	12.8062	0.631686
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	18.8062	0.920946
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−6.20937	−0.302626	−0.151313	−	0.988486i	$-0.548350\pi$
$421$	−6.20937	−0.302626	−0.151313	+	0.988486i	$0.548350\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−7.40312	−0.359104
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−29.6125	−1.42638	−0.713192	−	0.700969i	$-0.752751\pi$
$431$	−29.6125	−1.42638	−0.713192	+	0.700969i	$0.752751\pi$
$432$	0	0
$433$	−38.4187	−1.84629	−0.923144	−	0.384455i	$-0.874389\pi$
$433$	−38.4187	−1.84629	−0.923144	+	0.384455i	$0.874389\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	0	0
$437$	50.8062	2.43039
$438$	0	0
$439$	6.80625	0.324845	0.162422	−	0.986721i	$-0.448069\pi$
$439$	6.80625	0.324845	0.162422	+	0.986721i	$0.448069\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	22.8062	1.08356	0.541779	−	0.840521i	$-0.317751\pi$
$443$	22.8062	1.08356	0.541779	+	0.840521i	$0.317751\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	12.8062	0.605715
$448$	0	0
$449$	32.8062	1.54822	0.774111	−	0.633050i	$-0.218197\pi$
$449$	32.8062	1.54822	0.774111	+	0.633050i	$0.218197\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.40312	0.159191	0.0795957	−	0.996827i	$-0.474637\pi$
$457$	3.40312	0.159191	0.0795957	+	0.996827i	$0.474637\pi$
$458$	0	0
$459$	−7.40312	−0.345548
$460$	0	0
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	0	0
$463$	−10.5969	−0.492479	−0.246239	−	0.969209i	$-0.579195\pi$
$463$	−10.5969	−0.492479	−0.246239	+	0.969209i	$0.579195\pi$
$464$	0	0
$465$	10.8062	0.501128
$466$	0	0
$467$	−30.8062	−1.42554	−0.712772	−	0.701396i	$-0.752560\pi$
$467$	−30.8062	−1.42554	−0.712772	+	0.701396i	$0.752560\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	20.8062	0.958701
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.40312	−0.247912
$476$	0	0
$477$	−3.40312	−0.155818
$478$	0	0
$479$	37.6125	1.71856	0.859279	−	0.511506i	$-0.170912\pi$
$479$	37.6125	1.71856	0.859279	+	0.511506i	$0.170912\pi$
$480$	0	0
$481$	7.40312	0.337553
$482$	0	0
$483$	0	0
$484$	0	0
$485$	20.0000	0.908153
$486$	0	0
$487$	−10.5969	−0.480190	−0.240095	−	0.970749i	$-0.577179\pi$
$487$	−10.5969	−0.480190	−0.240095	+	0.970749i	$0.577179\pi$
$488$	0	0
$489$	−10.8062	−0.488675
$490$	0	0
$491$	16.2094	0.731519	0.365759	−	0.930709i	$-0.380809\pi$
$491$	16.2094	0.731519	0.365759	+	0.930709i	$0.380809\pi$
$492$	0	0
$493$	14.8062	0.666840
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.40312	−0.238554
$514$	0	0
$515$	37.6125	1.65740
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−20.8062	−0.913293
$520$	0	0
$521$	31.6125	1.38497	0.692484	−	0.721433i	$-0.256516\pi$
$521$	31.6125	1.38497	0.692484	+	0.721433i	$0.256516\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−40.0000	−1.74243
$528$	0	0
$529$	65.4187	2.84429
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	2.00000	0.0866296
$534$	0	0
$535$	10.8062	0.467195
$536$	0	0
$537$	−20.0000	−0.863064
$538$	0	0
$539$	−7.00000	−0.301511
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	−8.80625	−0.377912
$544$	0	0
$545$	28.0000	1.19939
$546$	0	0
$547$	13.1938	0.564124	0.282062	−	0.959396i	$-0.408982\pi$
$547$	13.1938	0.564124	0.282062	+	0.959396i	$0.408982\pi$
$548$	0	0
$549$	−12.8062	−0.546557
$550$	0	0
$551$	10.8062	0.460362
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−14.8062	−0.628490
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.40312	−0.312560
$562$	0	0
$563$	8.20937	0.345984	0.172992	−	0.984923i	$-0.444657\pi$
$563$	8.20937	0.345984	0.172992	+	0.984923i	$0.444657\pi$
$564$	0	0
$565$	9.61250	0.404401
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.79063	−0.242756	−0.121378	−	0.992606i	$-0.538731\pi$
$569$	−5.79063	−0.242756	−0.121378	+	0.992606i	$0.538731\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	−25.4031	−1.06123
$574$	0	0
$575$	−9.40312	−0.392137
$576$	0	0
$577$	36.8062	1.53226	0.766132	−	0.642683i	$-0.222179\pi$
$577$	36.8062	1.53226	0.766132	+	0.642683i	$0.222179\pi$
$578$	0	0
$579$	4.59688	0.191040
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.40312	−0.140943
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	9.61250	0.396750	0.198375	−	0.980126i	$-0.436434\pi$
$587$	9.61250	0.396750	0.198375	+	0.980126i	$0.436434\pi$
$588$	0	0
$589$	−29.1938	−1.20291
$590$	0	0
$591$	12.8062	0.526779
$592$	0	0
$593$	−0.806248	−0.0331087	−0.0165543	−	0.999863i	$-0.505270\pi$
$593$	−0.806248	−0.0331087	−0.0165543	+	0.999863i	$0.505270\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−17.4031	−0.711072	−0.355536	−	0.934663i	$-0.615702\pi$
$599$	−17.4031	−0.711072	−0.355536	+	0.934663i	$0.615702\pi$
$600$	0	0
$601$	−24.8062	−1.01187	−0.505934	−	0.862572i	$-0.668852\pi$
$601$	−24.8062	−1.01187	−0.505934	+	0.862572i	$0.668852\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	36.4187	1.47819	0.739096	−	0.673601i	$-0.235253\pi$
$607$	36.4187	1.47819	0.739096	+	0.673601i	$0.235253\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−15.6125	−0.630583	−0.315291	−	0.948995i	$-0.602102\pi$
$613$	−15.6125	−0.630583	−0.315291	+	0.948995i	$0.602102\pi$
$614$	0	0
$615$	−4.00000	−0.161296
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	−5.19375	−0.208754	−0.104377	−	0.994538i	$-0.533285\pi$
$619$	−5.19375	−0.208754	−0.104377	+	0.994538i	$0.533285\pi$
$620$	0	0
$621$	−9.40312	−0.377334
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−5.40312	−0.215780
$628$	0	0
$629$	54.8062	2.18527
$630$	0	0
$631$	−27.0156	−1.07547	−0.537737	−	0.843112i	$-0.680721\pi$
$631$	−27.0156	−1.07547	−0.537737	+	0.843112i	$0.680721\pi$
$632$	0	0
$633$	−18.8062	−0.747481
$634$	0	0
$635$	−13.6125	−0.540195
$636$	0	0
$637$	−7.00000	−0.277350
$638$	0	0
$639$	−10.8062	−0.427489
$640$	0	0
$641$	−32.8062	−1.29577	−0.647884	−	0.761739i	$-0.724346\pi$
$641$	−32.8062	−1.29577	−0.647884	+	0.761739i	$0.724346\pi$
$642$	0	0
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−36.2094	−1.42354	−0.711769	−	0.702414i	$-0.752106\pi$
$647$	−36.2094	−1.42354	−0.711769	+	0.702414i	$0.752106\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.0156	−0.978937	−0.489468	−	0.872021i	$-0.662809\pi$
$653$	−25.0156	−0.978937	−0.489468	+	0.872021i	$0.662809\pi$
$654$	0	0
$655$	5.19375	0.202937
$656$	0	0
$657$	0.596876	0.0232863
$658$	0	0
$659$	32.2094	1.25470	0.627350	−	0.778738i	$-0.284140\pi$
$659$	32.2094	1.25470	0.627350	+	0.778738i	$0.284140\pi$
$660$	0	0
$661$	7.40312	0.287948	0.143974	−	0.989581i	$-0.454012\pi$
$661$	7.40312	0.287948	0.143974	+	0.989581i	$0.454012\pi$
$662$	0	0
$663$	−7.40312	−0.287514
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.8062	0.728181
$668$	0	0
$669$	13.4031	0.518195
$670$	0	0
$671$	−12.8062	−0.494380
$672$	0	0
$673$	23.6125	0.910195	0.455097	−	0.890442i	$-0.349604\pi$
$673$	23.6125	0.910195	0.455097	+	0.890442i	$0.349604\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	20.8062	0.799649	0.399825	−	0.916592i	$-0.369071\pi$
$677$	20.8062	0.799649	0.399825	+	0.916592i	$0.369071\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	22.8062	0.873937
$682$	0	0
$683$	25.6125	0.980035	0.490017	−	0.871713i	$-0.336990\pi$
$683$	25.6125	0.980035	0.490017	+	0.871713i	$0.336990\pi$
$684$	0	0
$685$	−25.6125	−0.978603
$686$	0	0
$687$	−15.4031	−0.587666
$688$	0	0
$689$	−3.40312	−0.129649
$690$	0	0
$691$	21.1938	0.806248	0.403124	−	0.915145i	$-0.367924\pi$
$691$	21.1938	0.806248	0.403124	+	0.915145i	$0.367924\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−37.6125	−1.42672
$696$	0	0
$697$	14.8062	0.560827
$698$	0	0
$699$	3.40312	0.128718
$700$	0	0
$701$	−0.806248	−0.0304516	−0.0152258	−	0.999884i	$-0.504847\pi$
$701$	−0.806248	−0.0304516	−0.0152258	+	0.999884i	$0.504847\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−38.2094	−1.43498	−0.717492	−	0.696567i	$-0.754710\pi$
$709$	−38.2094	−1.43498	−0.717492	+	0.696567i	$0.754710\pi$
$710$	0	0
$711$	−14.8062	−0.555278
$712$	0	0
$713$	−50.8062	−1.90271
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.0156	−1.15669	−0.578344	−	0.815793i	$-0.696301\pi$
$719$	−31.0156	−1.15669	−0.578344	+	0.815793i	$0.696301\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8.59688	−0.319721
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−18.8062	−0.697485	−0.348743	−	0.937219i	$-0.613391\pi$
$727$	−18.8062	−0.697485	−0.348743	+	0.937219i	$0.613391\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	35.6125	1.31538	0.657689	−	0.753290i	$-0.271534\pi$
$733$	35.6125	1.31538	0.657689	+	0.753290i	$0.271534\pi$
$734$	0	0
$735$	14.0000	0.516398
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.2094	0.596271	0.298136	−	0.954523i	$-0.403635\pi$
$739$	16.2094	0.596271	0.298136	+	0.954523i	$0.403635\pi$
$740$	0	0
$741$	−5.40312	−0.198489
$742$	0	0
$743$	−37.6125	−1.37987	−0.689934	−	0.723872i	$-0.742361\pi$
$743$	−37.6125	−1.37987	−0.689934	+	0.723872i	$0.742361\pi$
$744$	0	0
$745$	−25.6125	−0.938369
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−28.8062	−1.04698	−0.523490	−	0.852032i	$-0.675370\pi$
$757$	−28.8062	−1.04698	−0.523490	+	0.852032i	$0.675370\pi$
$758$	0	0
$759$	−9.40312	−0.341312
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	14.8062	0.535321
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	11.4031	0.411207	0.205604	−	0.978635i	$-0.434084\pi$
$769$	11.4031	0.411207	0.205604	+	0.978635i	$0.434084\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	−11.1938	−0.402611	−0.201306	−	0.979528i	$-0.564518\pi$
$773$	−11.1938	−0.402611	−0.201306	+	0.979528i	$0.564518\pi$
$774$	0	0
$775$	5.40312	0.194086
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.8062	0.387174
$780$	0	0
$781$	−10.8062	−0.386678
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−41.6125	−1.48521
$786$	0	0
$787$	29.4031	1.04811	0.524054	−	0.851685i	$-0.324419\pi$
$787$	29.4031	1.04811	0.524054	+	0.851685i	$0.324419\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12.8062	−0.454763
$794$	0	0
$795$	6.80625	0.241393
$796$	0	0
$797$	18.2094	0.645009	0.322505	−	0.946568i	$-0.395475\pi$
$797$	18.2094	0.645009	0.322505	+	0.946568i	$0.395475\pi$
$798$	0	0
$799$	59.2250	2.09523
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	0.596876	0.0210633
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.2094	0.781807
$808$	0	0
$809$	−27.4031	−0.963443	−0.481721	−	0.876324i	$-0.659988\pi$
$809$	−27.4031	−0.963443	−0.481721	+	0.876324i	$0.659988\pi$
$810$	0	0
$811$	−24.2094	−0.850106	−0.425053	−	0.905168i	$-0.639745\pi$
$811$	−24.2094	−0.850106	−0.425053	+	0.905168i	$0.639745\pi$
$812$	0	0
$813$	2.80625	0.0984194
$814$	0	0
$815$	21.6125	0.757053
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.61250	0.126077	0.0630385	−	0.998011i	$-0.479921\pi$
$821$	3.61250	0.126077	0.0630385	+	0.998011i	$0.479921\pi$
$822$	0	0
$823$	−5.61250	−0.195639	−0.0978197	−	0.995204i	$-0.531187\pi$
$823$	−5.61250	−0.195639	−0.0978197	+	0.995204i	$0.531187\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	6.80625	0.236676	0.118338	−	0.992973i	$-0.462243\pi$
$827$	6.80625	0.236676	0.118338	+	0.992973i	$0.462243\pi$
$828$	0	0
$829$	−7.61250	−0.264393	−0.132196	−	0.991224i	$-0.542203\pi$
$829$	−7.61250	−0.264393	−0.132196	+	0.991224i	$0.542203\pi$
$830$	0	0
$831$	−22.0000	−0.763172
$832$	0	0
$833$	−51.8219	−1.79552
$834$	0	0
$835$	32.0000	1.10741
$836$	0	0
$837$	5.40312	0.186759
$838$	0	0
$839$	−21.6125	−0.746146	−0.373073	−	0.927802i	$-0.621696\pi$
$839$	−21.6125	−0.746146	−0.373073	+	0.927802i	$0.621696\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	14.0000	0.482186
$844$	0	0
$845$	2.00000	0.0688021
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	69.6125	2.38629
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	10.8062	0.369566
$856$	0	0
$857$	4.59688	0.157026	0.0785131	−	0.996913i	$-0.474983\pi$
$857$	4.59688	0.157026	0.0785131	+	0.996913i	$0.474983\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.4187	0.831224	0.415612	−	0.909542i	$-0.363567\pi$
$863$	24.4187	0.831224	0.415612	+	0.909542i	$0.363567\pi$
$864$	0	0
$865$	41.6125	1.41487
$866$	0	0
$867$	−37.8062	−1.28397
$868$	0	0
$869$	−14.8062	−0.502268
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.4187	1.29731	0.648654	−	0.761083i	$-0.275332\pi$
$877$	38.4187	1.29731	0.648654	+	0.761083i	$0.275332\pi$
$878$	0	0
$879$	−24.8062	−0.836694
$880$	0	0
$881$	−27.1938	−0.916181	−0.458090	−	0.888906i	$-0.651466\pi$
$881$	−27.1938	−0.916181	−0.458090	+	0.888906i	$0.651466\pi$
$882$	0	0
$883$	−1.61250	−0.0542648	−0.0271324	−	0.999632i	$-0.508638\pi$
$883$	−1.61250	−0.0542648	−0.0271324	+	0.999632i	$0.508638\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	−40.4187	−1.35713	−0.678564	−	0.734541i	$-0.737398\pi$
$887$	−40.4187	−1.35713	−0.678564	+	0.734541i	$0.737398\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	43.2250	1.44647
$894$	0	0
$895$	40.0000	1.33705
$896$	0	0
$897$	−9.40312	−0.313961
$898$	0	0
$899$	−10.8062	−0.360409
$900$	0	0
$901$	−25.1938	−0.839326
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.6125	0.585459
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−28.2094	−0.934618	−0.467309	−	0.884094i	$-0.654776\pi$
$911$	−28.2094	−0.934618	−0.467309	+	0.884094i	$0.654776\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	25.6125	0.846723
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.8062	−0.488413	−0.244207	−	0.969723i	$-0.578527\pi$
$919$	−14.8062	−0.488413	−0.244207	+	0.969723i	$0.578527\pi$
$920$	0	0
$921$	29.4031	0.968866
$922$	0	0
$923$	−10.8062	−0.355692
$924$	0	0
$925$	−7.40312	−0.243413
$926$	0	0
$927$	18.8062	0.617678
$928$	0	0
$929$	−20.8062	−0.682631	−0.341315	−	0.939949i	$-0.610872\pi$
$929$	−20.8062	−0.682631	−0.341315	+	0.939949i	$0.610872\pi$
$930$	0	0
$931$	−37.8219	−1.23956
$932$	0	0
$933$	−28.2094	−0.923533
$934$	0	0
$935$	14.8062	0.484216
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	3.19375	0.104224
$940$	0	0
$941$	−31.6125	−1.03054	−0.515269	−	0.857029i	$-0.672308\pi$
$941$	−31.6125	−1.03054	−0.515269	+	0.857029i	$0.672308\pi$
$942$	0	0
$943$	18.8062	0.612416
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.6125	−0.572329	−0.286165	−	0.958180i	$-0.592380\pi$
$947$	−17.6125	−0.572329	−0.286165	+	0.958180i	$0.592380\pi$
$948$	0	0
$949$	0.596876	0.0193754
$950$	0	0
$951$	−26.0000	−0.843108
$952$	0	0
$953$	−5.79063	−0.187577	−0.0937884	−	0.995592i	$-0.529898\pi$
$953$	−5.79063	−0.187577	−0.0937884	+	0.995592i	$0.529898\pi$
$954$	0	0
$955$	50.8062	1.64405
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.80625	−0.0582661
$962$	0	0
$963$	5.40312	0.174113
$964$	0	0
$965$	−9.19375	−0.295957
$966$	0	0
$967$	21.6125	0.695011	0.347506	−	0.937678i	$-0.387029\pi$
$967$	21.6125	0.695011	0.347506	+	0.937678i	$0.387029\pi$
$968$	0	0
$969$	−40.0000	−1.28499
$970$	0	0
$971$	14.8062	0.475155	0.237578	−	0.971369i	$-0.423647\pi$
$971$	14.8062	0.475155	0.237578	+	0.971369i	$0.423647\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−47.6125	−1.52326	−0.761629	−	0.648013i	$-0.775600\pi$
$977$	−47.6125	−1.52326	−0.761629	+	0.648013i	$0.775600\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	−40.4187	−1.28916	−0.644579	−	0.764538i	$-0.722967\pi$
$983$	−40.4187	−1.28916	−0.644579	+	0.764538i	$0.722967\pi$
$984$	0	0
$985$	−25.6125	−0.816082
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	16.0000	0.507745
$994$	0	0
$995$	−32.0000	−1.01447
$996$	0	0
$997$	−26.4187	−0.836690	−0.418345	−	0.908288i	$-0.637390\pi$
$997$	−26.4187	−0.836690	−0.418345	+	0.908288i	$0.637390\pi$
$998$	0	0
$999$	−7.40312	−0.234224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bf.1.2		2
4.3	odd	2		858.2.a.q.1.2	✓	2
12.11	even	2		2574.2.a.z.1.1		2
44.43	even	2		9438.2.a.bs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.q.1.2	✓	2	4.3	odd	2
2574.2.a.z.1.1		2	12.11	even	2
6864.2.a.bf.1.2		2	1.1	even	1	trivial
9438.2.a.bs.1.1		2	44.43	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.00000 q^{15} +7.40312 q^{17} +5.40312 q^{19} +9.40312 q^{23} -1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} -5.40312 q^{31} -1.00000 q^{33} +7.40312 q^{37} -1.00000 q^{39} +2.00000 q^{41} +2.00000 q^{45} +8.00000 q^{47} -7.00000 q^{49} -7.40312 q^{51} -3.40312 q^{53} +2.00000 q^{55} -5.40312 q^{57} +4.00000 q^{59} -12.8062 q^{61} +2.00000 q^{65} -9.40312 q^{69} -10.8062 q^{71} +0.596876 q^{73} +1.00000 q^{75} -14.8062 q^{79} +1.00000 q^{81} +4.00000 q^{83} +14.8062 q^{85} -2.00000 q^{87} -10.0000 q^{89} +5.40312 q^{93} +10.8062 q^{95} +10.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} - 2 q^{19} + 6 q^{23} - 2 q^{25} - 2 q^{27} + 4 q^{29} + 2 q^{31} - 2 q^{33} + 2 q^{37} - 2 q^{39} + 4 q^{41} + 4 q^{45} + 16 q^{47} - 14 q^{49} - 2 q^{51} + 6 q^{53} + 4 q^{55} + 2 q^{57} + 8 q^{59} + 4 q^{65} - 6 q^{69} + 4 q^{71} + 14 q^{73} + 2 q^{75} - 4 q^{79} + 2 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 20 q^{89} - 2 q^{93} - 4 q^{95} + 20 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 2 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 - 2 * q^19 + 6 * q^23 - 2 * q^25 - 2 * q^27 + 4 * q^29 + 2 * q^31 - 2 * q^33 + 2 * q^37 - 2 * q^39 + 4 * q^41 + 4 * q^45 + 16 * q^47 - 14 * q^49 - 2 * q^51 + 6 * q^53 + 4 * q^55 + 2 * q^57 + 8 * q^59 + 4 * q^65 - 6 * q^69 + 4 * q^71 + 14 * q^73 + 2 * q^75 - 4 * q^79 + 2 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 20 * q^89 - 2 * q^93 - 4 * q^95 + 20 * q^97 + 2 * q^99

Introduction

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