LMFDB - Embedded newform 6864.2.a.ca.1.3

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:	$4$
Coefficient field:	4.4.70164.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 10x - 2 $ x^4 - x^3 - 10x^2 + 10x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.699291$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.699291 q^{5} -3.79091 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.699291 q^{5} -3.79091 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.699291 q^{15} -5.09162 q^{17} +0.538544 q^{19} -3.79091 q^{21} +7.18950 q^{23} -4.51099 q^{25} +1.00000 q^{27} -9.14116 q^{29} -1.95166 q^{31} -1.00000 q^{33} -2.65095 q^{35} -1.39858 q^{37} -1.00000 q^{39} -1.79091 q^{41} +10.8612 q^{43} +0.699291 q^{45} +13.5818 q^{47} +7.37103 q^{49} -5.09162 q^{51} +9.90332 q^{53} -0.699291 q^{55} +0.538544 q^{57} -5.72804 q^{59} +11.3514 q^{61} -3.79091 q^{63} -0.699291 q^{65} +13.3998 q^{67} +7.18950 q^{69} +9.90332 q^{71} +3.83249 q^{73} -4.51099 q^{75} +3.79091 q^{77} -1.37155 q^{79} +1.00000 q^{81} +4.32150 q^{83} -3.56053 q^{85} -9.14116 q^{87} +5.95166 q^{89} +3.79091 q^{91} -1.95166 q^{93} +0.376599 q^{95} +1.95843 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + q^{5} + q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + q^5 + q^7 + 4 * q^9	$ 4 q + 4 q^{3} + q^{5} + q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + q^{15} - 6 q^{17} + q^{21} + 9 q^{23} + q^{25} + 4 q^{27} - q^{29} + 8 q^{31} - 4 q^{33} + 7 q^{35} - 2 q^{37} - 4 q^{39} + 9 q^{41} + 5 q^{43} + q^{45} + 22 q^{47} + 9 q^{49} - 6 q^{51} + 8 q^{53} - q^{55} - q^{59} - 11 q^{61} + q^{63} - q^{65} + 13 q^{67} + 9 q^{69} + 8 q^{71} - 3 q^{73} + q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 18 q^{83} + 26 q^{85} - q^{87} + 8 q^{89} - q^{91} + 8 q^{93} + 36 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + q^5 + q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + q^15 - 6 * q^17 + q^21 + 9 * q^23 + q^25 + 4 * q^27 - q^29 + 8 * q^31 - 4 * q^33 + 7 * q^35 - 2 * q^37 - 4 * q^39 + 9 * q^41 + 5 * q^43 + q^45 + 22 * q^47 + 9 * q^49 - 6 * q^51 + 8 * q^53 - q^55 - q^59 - 11 * q^61 + q^63 - q^65 + 13 * q^67 + 9 * q^69 + 8 * q^71 - 3 * q^73 + q^75 - q^77 + 6 * q^79 + 4 * q^81 + 18 * q^83 + 26 * q^85 - q^87 + 8 * q^89 - q^91 + 8 * q^93 + 36 * q^95 + 10 * q^97 - 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.699291	0.312733	0.156366	−	0.987699i	$-0.450022\pi$
$5$	0.699291	0.312733	0.156366	+	0.987699i	$0.450022\pi$
$6$	0	0
$7$	−3.79091	−1.43283	−0.716415	−	0.697674i	$-0.754218\pi$
$7$	−3.79091	−1.43283	−0.716415	+	0.697674i	$0.754218\pi$
$8$	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
$15$	0.699291	0.180556
$16$	0	0
$17$	−5.09162	−1.23490	−0.617450	−	0.786610i	$-0.711834\pi$
$17$	−5.09162	−1.23490	−0.617450	+	0.786610i	$0.711834\pi$
$18$	0	0
$19$	0.538544	0.123550	0.0617752	−	0.998090i	$-0.480324\pi$
$19$	0.538544	0.123550	0.0617752	+	0.998090i	$0.480324\pi$
$20$	0	0
$21$	−3.79091	−0.827245
$22$	0	0
$23$	7.18950	1.49911	0.749557	−	0.661940i	$-0.230267\pi$
$23$	7.18950	1.49911	0.749557	+	0.661940i	$0.230267\pi$
$24$	0	0
$25$	−4.51099	−0.902198
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.14116	−1.69747	−0.848735	−	0.528818i	$-0.822635\pi$
$29$	−9.14116	−1.69747	−0.848735	+	0.528818i	$0.822635\pi$
$30$	0	0
$31$	−1.95166	−0.350529	−0.175264	−	0.984521i	$-0.556078\pi$
$31$	−1.95166	−0.350529	−0.175264	+	0.984521i	$0.556078\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.65095	−0.448093
$36$	0	0
$37$	−1.39858	−0.229926	−0.114963	−	0.993370i	$-0.536675\pi$
$37$	−1.39858	−0.229926	−0.114963	+	0.993370i	$0.536675\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−1.79091	−0.279694	−0.139847	−	0.990173i	$-0.544661\pi$
$41$	−1.79091	−0.279694	−0.139847	+	0.990173i	$0.544661\pi$
$42$	0	0
$43$	10.8612	1.65632	0.828161	−	0.560490i	$-0.189387\pi$
$43$	10.8612	1.65632	0.828161	+	0.560490i	$0.189387\pi$
$44$	0	0
$45$	0.699291	0.104244
$46$	0	0
$47$	13.5818	1.98111	0.990557	−	0.137104i	$-0.0437795\pi$
$47$	13.5818	1.98111	0.990557	+	0.137104i	$0.0437795\pi$
$48$	0	0
$49$	7.37103	1.05300
$50$	0	0
$51$	−5.09162	−0.712970
$52$	0	0
$53$	9.90332	1.36033	0.680163	−	0.733061i	$-0.261909\pi$
$53$	9.90332	1.36033	0.680163	+	0.733061i	$0.261909\pi$
$54$	0	0
$55$	−0.699291	−0.0942924
$56$	0	0
$57$	0.538544	0.0713318
$58$	0	0
$59$	−5.72804	−0.745727	−0.372864	−	0.927886i	$-0.621624\pi$
$59$	−5.72804	−0.745727	−0.372864	+	0.927886i	$0.621624\pi$
$60$	0	0
$61$	11.3514	1.45340	0.726702	−	0.686953i	$-0.241052\pi$
$61$	11.3514	1.45340	0.726702	+	0.686953i	$0.241052\pi$
$62$	0	0
$63$	−3.79091	−0.477610
$64$	0	0
$65$	−0.699291	−0.0867364
$66$	0	0
$67$	13.3998	1.63704	0.818522	−	0.574475i	$-0.194794\pi$
$67$	13.3998	1.63704	0.818522	+	0.574475i	$0.194794\pi$
$68$	0	0
$69$	7.18950	0.865514
$70$	0	0
$71$	9.90332	1.17531	0.587654	−	0.809112i	$-0.300052\pi$
$71$	9.90332	1.17531	0.587654	+	0.809112i	$0.300052\pi$
$72$	0	0
$73$	3.83249	0.448559	0.224279	−	0.974525i	$-0.427997\pi$
$73$	3.83249	0.448559	0.224279	+	0.974525i	$0.427997\pi$
$74$	0	0
$75$	−4.51099	−0.520884
$76$	0	0
$77$	3.79091	0.432015
$78$	0	0
$79$	−1.37155	−0.154311	−0.0771555	−	0.997019i	$-0.524584\pi$
$79$	−1.37155	−0.154311	−0.0771555	+	0.997019i	$0.524584\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.32150	0.474346	0.237173	−	0.971467i	$-0.423779\pi$
$83$	4.32150	0.474346	0.237173	+	0.971467i	$0.423779\pi$
$84$	0	0
$85$	−3.56053	−0.386193
$86$	0	0
$87$	−9.14116	−0.980035
$88$	0	0
$89$	5.95166	0.630875	0.315437	−	0.948946i	$-0.397849\pi$
$89$	5.95166	0.630875	0.315437	+	0.948946i	$0.397849\pi$
$90$	0	0
$91$	3.79091	0.397396
$92$	0	0
$93$	−1.95166	−0.202378
$94$	0	0
$95$	0.376599	0.0386382
$96$	0	0
$97$	1.95843	0.198848	0.0994241	−	0.995045i	$-0.468300\pi$
$97$	1.95843	0.198848	0.0994241	+	0.995045i	$0.468300\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−1.93036	−0.192078	−0.0960390	−	0.995378i	$-0.530617\pi$
$101$	−1.93036	−0.192078	−0.0960390	+	0.995378i	$0.530617\pi$
$102$	0	0
$103$	−7.35144	−0.724359	−0.362180	−	0.932108i	$-0.617967\pi$
$103$	−7.35144	−0.724359	−0.362180	+	0.932108i	$0.617967\pi$
$104$	0	0
$105$	−2.65095	−0.258707
$106$	0	0
$107$	−11.0299	−1.06631	−0.533153	−	0.846019i	$-0.678993\pi$
$107$	−11.0299	−1.06631	−0.533153	+	0.846019i	$0.678993\pi$
$108$	0	0
$109$	2.44187	0.233888	0.116944	−	0.993138i	$-0.462690\pi$
$109$	2.44187	0.233888	0.116944	+	0.993138i	$0.462690\pi$
$110$	0	0
$111$	−1.39858	−0.132748
$112$	0	0
$113$	−6.65095	−0.625669	−0.312835	−	0.949808i	$-0.601279\pi$
$113$	−6.65095	−0.625669	−0.312835	+	0.949808i	$0.601279\pi$
$114$	0	0
$115$	5.02755	0.468822
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	19.3019	1.76940
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.79091	−0.161481
$124$	0	0
$125$	−6.65095	−0.594879
$126$	0	0
$127$	5.41312	0.480337	0.240168	−	0.970731i	$-0.422797\pi$
$127$	5.41312	0.480337	0.240168	+	0.970731i	$0.422797\pi$
$128$	0	0
$129$	10.8612	0.956279
$130$	0	0
$131$	−8.55428	−0.747391	−0.373695	−	0.927552i	$-0.621909\pi$
$131$	−8.55428	−0.747391	−0.373695	+	0.927552i	$0.621909\pi$
$132$	0	0
$133$	−2.04157	−0.177027
$134$	0	0
$135$	0.699291	0.0601854
$136$	0	0
$137$	−1.11292	−0.0950835	−0.0475418	−	0.998869i	$-0.515139\pi$
$137$	−1.11292	−0.0950835	−0.0475418	+	0.998869i	$0.515139\pi$
$138$	0	0
$139$	3.79211	0.321643	0.160821	−	0.986984i	$-0.448586\pi$
$139$	3.79211	0.321643	0.160821	+	0.986984i	$0.448586\pi$
$140$	0	0
$141$	13.5818	1.14380
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−6.39233	−0.530854
$146$	0	0
$147$	7.37103	0.607952
$148$	0	0
$149$	4.25862	0.348880	0.174440	−	0.984668i	$-0.444189\pi$
$149$	4.25862	0.348880	0.174440	+	0.984668i	$0.444189\pi$
$150$	0	0
$151$	7.04328	0.573174	0.286587	−	0.958054i	$-0.407479\pi$
$151$	7.04328	0.573174	0.286587	+	0.958054i	$0.407479\pi$
$152$	0	0
$153$	−5.09162	−0.411633
$154$	0	0
$155$	−1.36478	−0.109622
$156$	0	0
$157$	16.3452	1.30449	0.652244	−	0.758009i	$-0.273828\pi$
$157$	16.3452	1.30449	0.652244	+	0.758009i	$0.273828\pi$
$158$	0	0
$159$	9.90332	0.785385
$160$	0	0
$161$	−27.2548	−2.14798
$162$	0	0
$163$	2.97921	0.233350	0.116675	−	0.993170i	$-0.462776\pi$
$163$	2.97921	0.233350	0.116675	+	0.993170i	$0.462776\pi$
$164$	0	0
$165$	−0.699291	−0.0544397
$166$	0	0
$167$	19.2524	1.48979	0.744897	−	0.667180i	$-0.232499\pi$
$167$	19.2524	1.48979	0.744897	+	0.667180i	$0.232499\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.538544	0.0411835
$172$	0	0
$173$	−0.440670	−0.0335035	−0.0167518	−	0.999860i	$-0.505333\pi$
$173$	−0.440670	−0.0335035	−0.0167518	+	0.999860i	$0.505333\pi$
$174$	0	0
$175$	17.1008	1.29270
$176$	0	0
$177$	−5.72804	−0.430546
$178$	0	0
$179$	21.7022	1.62210	0.811049	−	0.584978i	$-0.198897\pi$
$179$	21.7022	1.62210	0.811049	+	0.584978i	$0.198897\pi$
$180$	0	0
$181$	−1.74138	−0.129436	−0.0647178	−	0.997904i	$-0.520615\pi$
$181$	−1.74138	−0.129436	−0.0647178	+	0.997904i	$0.520615\pi$
$182$	0	0
$183$	11.3514	0.839123
$184$	0	0
$185$	−0.978016	−0.0719052
$186$	0	0
$187$	5.09162	0.372336
$188$	0	0
$189$	−3.79091	−0.275748
$190$	0	0
$191$	−18.6118	−1.34670	−0.673350	−	0.739324i	$-0.735145\pi$
$191$	−18.6118	−1.34670	−0.673350	+	0.739324i	$0.735145\pi$
$192$	0	0
$193$	−20.9040	−1.50470	−0.752352	−	0.658762i	$-0.771081\pi$
$193$	−20.9040	−1.50470	−0.752352	+	0.658762i	$0.771081\pi$
$194$	0	0
$195$	−0.699291	−0.0500773
$196$	0	0
$197$	18.1228	1.29119	0.645597	−	0.763678i	$-0.276609\pi$
$197$	18.1228	1.29119	0.645597	+	0.763678i	$0.276609\pi$
$198$	0	0
$199$	−12.6533	−0.896972	−0.448486	−	0.893790i	$-0.648037\pi$
$199$	−12.6533	−0.896972	−0.448486	+	0.893790i	$0.648037\pi$
$200$	0	0
$201$	13.3998	0.945148
$202$	0	0
$203$	34.6533	2.43219
$204$	0	0
$205$	−1.25237	−0.0874694
$206$	0	0
$207$	7.18950	0.499705
$208$	0	0
$209$	−0.538544	−0.0372518
$210$	0	0
$211$	−15.8337	−1.09004	−0.545018	−	0.838424i	$-0.683477\pi$
$211$	−15.8337	−1.09004	−0.545018	+	0.838424i	$0.683477\pi$
$212$	0	0
$213$	9.90332	0.678565
$214$	0	0
$215$	7.59517	0.517986
$216$	0	0
$217$	7.39858	0.502249
$218$	0	0
$219$	3.83249	0.258975
$220$	0	0
$221$	5.09162	0.342500
$222$	0	0
$223$	−4.65215	−0.311531	−0.155766	−	0.987794i	$-0.549784\pi$
$223$	−4.65215	−0.311531	−0.155766	+	0.987794i	$0.549784\pi$
$224$	0	0
$225$	−4.51099	−0.300733
$226$	0	0
$227$	5.84045	0.387644	0.193822	−	0.981037i	$-0.437912\pi$
$227$	5.84045	0.387644	0.193822	+	0.981037i	$0.437912\pi$
$228$	0	0
$229$	15.9113	1.05145	0.525724	−	0.850655i	$-0.323795\pi$
$229$	15.9113	1.05145	0.525724	+	0.850655i	$0.323795\pi$
$230$	0	0
$231$	3.79091	0.249424
$232$	0	0
$233$	−8.90838	−0.583607	−0.291804	−	0.956478i	$-0.594255\pi$
$233$	−8.90838	−0.583607	−0.291804	+	0.956478i	$0.594255\pi$
$234$	0	0
$235$	9.49765	0.619559
$236$	0	0
$237$	−1.37155	−0.0890914
$238$	0	0
$239$	−14.1699	−0.916575	−0.458288	−	0.888804i	$-0.651537\pi$
$239$	−14.1699	−0.916575	−0.458288	+	0.888804i	$0.651537\pi$
$240$	0	0
$241$	−6.46317	−0.416329	−0.208165	−	0.978094i	$-0.566749\pi$
$241$	−6.46317	−0.416329	−0.208165	+	0.978094i	$0.566749\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.15450	0.329309
$246$	0	0
$247$	−0.538544	−0.0342667
$248$	0	0
$249$	4.32150	0.273864
$250$	0	0
$251$	11.1187	0.701804	0.350902	−	0.936412i	$-0.385875\pi$
$251$	11.1187	0.701804	0.350902	+	0.936412i	$0.385875\pi$
$252$	0	0
$253$	−7.18950	−0.452000
$254$	0	0
$255$	−3.56053	−0.222969
$256$	0	0
$257$	3.63136	0.226518	0.113259	−	0.993565i	$-0.463871\pi$
$257$	3.63136	0.226518	0.113259	+	0.993565i	$0.463871\pi$
$258$	0	0
$259$	5.30191	0.329444
$260$	0	0
$261$	−9.14116	−0.565824
$262$	0	0
$263$	18.7005	1.15312	0.576561	−	0.817054i	$-0.304394\pi$
$263$	18.7005	1.15312	0.576561	+	0.817054i	$0.304394\pi$
$264$	0	0
$265$	6.92531	0.425418
$266$	0	0
$267$	5.95166	0.364236
$268$	0	0
$269$	−16.9253	−1.03195	−0.515977	−	0.856602i	$-0.672571\pi$
$269$	−16.9253	−1.03195	−0.515977	+	0.856602i	$0.672571\pi$
$270$	0	0
$271$	17.7438	1.07786	0.538929	−	0.842351i	$-0.318829\pi$
$271$	17.7438	1.07786	0.538929	+	0.842351i	$0.318829\pi$
$272$	0	0
$273$	3.79091	0.229437
$274$	0	0
$275$	4.51099	0.272023
$276$	0	0
$277$	18.6533	1.12077	0.560386	−	0.828232i	$-0.310653\pi$
$277$	18.6533	1.12077	0.560386	+	0.828232i	$0.310653\pi$
$278$	0	0
$279$	−1.95166	−0.116843
$280$	0	0
$281$	−8.26658	−0.493143	−0.246572	−	0.969125i	$-0.579304\pi$
$281$	−8.26658	−0.493143	−0.246572	+	0.969125i	$0.579304\pi$
$282$	0	0
$283$	3.92582	0.233366	0.116683	−	0.993169i	$-0.462774\pi$
$283$	3.92582	0.233366	0.116683	+	0.993169i	$0.462774\pi$
$284$	0	0
$285$	0.376599	0.0223078
$286$	0	0
$287$	6.78920	0.400754
$288$	0	0
$289$	8.92462	0.524978
$290$	0	0
$291$	1.95843	0.114805
$292$	0	0
$293$	−16.3036	−0.952468	−0.476234	−	0.879319i	$-0.657998\pi$
$293$	−16.3036	−0.952468	−0.476234	+	0.879319i	$0.657998\pi$
$294$	0	0
$295$	−4.00557	−0.233213
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−7.18950	−0.415779
$300$	0	0
$301$	−41.1740	−2.37323
$302$	0	0
$303$	−1.93036	−0.110896
$304$	0	0
$305$	7.93796	0.454526
$306$	0	0
$307$	−6.77586	−0.386719	−0.193359	−	0.981128i	$-0.561938\pi$
$307$	−6.77586	−0.386719	−0.193359	+	0.981128i	$0.561938\pi$
$308$	0	0
$309$	−7.35144	−0.418209
$310$	0	0
$311$	13.8820	0.787177	0.393589	−	0.919287i	$-0.371233\pi$
$311$	13.8820	0.787177	0.393589	+	0.919287i	$0.371233\pi$
$312$	0	0
$313$	−19.1344	−1.08154	−0.540770	−	0.841171i	$-0.681867\pi$
$313$	−19.1344	−1.08154	−0.540770	+	0.841171i	$0.681867\pi$
$314$	0	0
$315$	−2.65095	−0.149364
$316$	0	0
$317$	16.1429	0.906674	0.453337	−	0.891339i	$-0.350233\pi$
$317$	16.1429	0.906674	0.453337	+	0.891339i	$0.350233\pi$
$318$	0	0
$319$	9.14116	0.511807
$320$	0	0
$321$	−11.0299	−0.615632
$322$	0	0
$323$	−2.74206	−0.152572
$324$	0	0
$325$	4.51099	0.250225
$326$	0	0
$327$	2.44187	0.135036
$328$	0	0
$329$	−51.4875	−2.83860
$330$	0	0
$331$	−7.00829	−0.385210	−0.192605	−	0.981276i	$-0.561694\pi$
$331$	−7.00829	−0.385210	−0.192605	+	0.981276i	$0.561694\pi$
$332$	0	0
$333$	−1.39858	−0.0766418
$334$	0	0
$335$	9.37035	0.511957
$336$	0	0
$337$	18.4632	1.00575	0.502876	−	0.864358i	$-0.332275\pi$
$337$	18.4632	1.00575	0.502876	+	0.864358i	$0.332275\pi$
$338$	0	0
$339$	−6.65095	−0.361230
$340$	0	0
$341$	1.95166	0.105688
$342$	0	0
$343$	−1.40655	−0.0759463
$344$	0	0
$345$	5.02755	0.270674
$346$	0	0
$347$	−25.3594	−1.36136	−0.680682	−	0.732579i	$-0.738316\pi$
$347$	−25.3594	−1.36136	−0.680682	+	0.732579i	$0.738316\pi$
$348$	0	0
$349$	23.2052	1.24215	0.621074	−	0.783752i	$-0.286697\pi$
$349$	23.2052	1.24215	0.621074	+	0.783752i	$0.286697\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	21.2986	1.13361	0.566804	−	0.823853i	$-0.308180\pi$
$353$	21.2986	1.13361	0.566804	+	0.823853i	$0.308180\pi$
$354$	0	0
$355$	6.92531	0.367557
$356$	0	0
$357$	19.3019	1.02157
$358$	0	0
$359$	18.7298	0.988519	0.494259	−	0.869315i	$-0.335439\pi$
$359$	18.7298	0.988519	0.494259	+	0.869315i	$0.335439\pi$
$360$	0	0
$361$	−18.7100	−0.984735
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	2.68002	0.140279
$366$	0	0
$367$	6.33400	0.330632	0.165316	−	0.986241i	$-0.447136\pi$
$367$	6.33400	0.330632	0.165316	+	0.986241i	$0.447136\pi$
$368$	0	0
$369$	−1.79091	−0.0932313
$370$	0	0
$371$	−37.5427	−1.94912
$372$	0	0
$373$	−6.13371	−0.317591	−0.158796	−	0.987311i	$-0.550761\pi$
$373$	−6.13371	−0.317591	−0.158796	+	0.987311i	$0.550761\pi$
$374$	0	0
$375$	−6.65095	−0.343454
$376$	0	0
$377$	9.14116	0.470794
$378$	0	0
$379$	−17.8966	−0.919284	−0.459642	−	0.888104i	$-0.652022\pi$
$379$	−17.8966	−0.919284	−0.459642	+	0.888104i	$0.652022\pi$
$380$	0	0
$381$	5.41312	0.277322
$382$	0	0
$383$	−17.7225	−0.905576	−0.452788	−	0.891618i	$-0.649571\pi$
$383$	−17.7225	−0.905576	−0.452788	+	0.891618i	$0.649571\pi$
$384$	0	0
$385$	2.65095	0.135105
$386$	0	0
$387$	10.8612	0.552108
$388$	0	0
$389$	20.0575	1.01696	0.508478	−	0.861075i	$-0.330208\pi$
$389$	20.0575	1.01696	0.508478	+	0.861075i	$0.330208\pi$
$390$	0	0
$391$	−36.6062	−1.85126
$392$	0	0
$393$	−8.55428	−0.431506
$394$	0	0
$395$	−0.959110	−0.0482580
$396$	0	0
$397$	−7.02995	−0.352823	−0.176411	−	0.984317i	$-0.556449\pi$
$397$	−7.02995	−0.352823	−0.176411	+	0.984317i	$0.556449\pi$
$398$	0	0
$399$	−2.04157	−0.102206
$400$	0	0
$401$	7.19607	0.359355	0.179677	−	0.983726i	$-0.442495\pi$
$401$	7.19607	0.359355	0.179677	+	0.983726i	$0.442495\pi$
$402$	0	0
$403$	1.95166	0.0972192
$404$	0	0
$405$	0.699291	0.0347481
$406$	0	0
$407$	1.39858	0.0693252
$408$	0	0
$409$	4.93625	0.244082	0.122041	−	0.992525i	$-0.461056\pi$
$409$	4.93625	0.244082	0.122041	+	0.992525i	$0.461056\pi$
$410$	0	0
$411$	−1.11292	−0.0548965
$412$	0	0
$413$	21.7145	1.06850
$414$	0	0
$415$	3.02198	0.148343
$416$	0	0
$417$	3.79211	0.185700
$418$	0	0
$419$	24.5825	1.20093	0.600467	−	0.799649i	$-0.294981\pi$
$419$	24.5825	1.20093	0.600467	+	0.799649i	$0.294981\pi$
$420$	0	0
$421$	21.6189	1.05364	0.526819	−	0.849977i	$-0.323384\pi$
$421$	21.6189	1.05364	0.526819	+	0.849977i	$0.323384\pi$
$422$	0	0
$423$	13.5818	0.660371
$424$	0	0
$425$	22.9683	1.11412
$426$	0	0
$427$	−43.0323	−2.08248
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	25.7201	1.23889	0.619446	−	0.785039i	$-0.287357\pi$
$431$	25.7201	1.23889	0.619446	+	0.785039i	$0.287357\pi$
$432$	0	0
$433$	−13.5874	−0.652969	−0.326484	−	0.945203i	$-0.605864\pi$
$433$	−13.5874	−0.652969	−0.326484	+	0.945203i	$0.605864\pi$
$434$	0	0
$435$	−6.39233	−0.306489
$436$	0	0
$437$	3.87186	0.185216
$438$	0	0
$439$	32.1170	1.53286	0.766431	−	0.642327i	$-0.222031\pi$
$439$	32.1170	1.53286	0.766431	+	0.642327i	$0.222031\pi$
$440$	0	0
$441$	7.37103	0.351001
$442$	0	0
$443$	6.79716	0.322943	0.161472	−	0.986877i	$-0.448376\pi$
$443$	6.79716	0.322943	0.161472	+	0.986877i	$0.448376\pi$
$444$	0	0
$445$	4.16194	0.197295
$446$	0	0
$447$	4.25862	0.201426
$448$	0	0
$449$	−3.95406	−0.186603	−0.0933017	−	0.995638i	$-0.529742\pi$
$449$	−3.95406	−0.186603	−0.0933017	+	0.995638i	$0.529742\pi$
$450$	0	0
$451$	1.79091	0.0843309
$452$	0	0
$453$	7.04328	0.330922
$454$	0	0
$455$	2.65095	0.124279
$456$	0	0
$457$	20.1486	0.942512	0.471256	−	0.881996i	$-0.343801\pi$
$457$	20.1486	0.942512	0.471256	+	0.881996i	$0.343801\pi$
$458$	0	0
$459$	−5.09162	−0.237657
$460$	0	0
$461$	−2.49594	−0.116248	−0.0581238	−	0.998309i	$-0.518512\pi$
$461$	−2.49594	−0.116248	−0.0581238	+	0.998309i	$0.518512\pi$
$462$	0	0
$463$	14.3147	0.665262	0.332631	−	0.943057i	$-0.392064\pi$
$463$	14.3147	0.665262	0.332631	+	0.943057i	$0.392064\pi$
$464$	0	0
$465$	−1.36478	−0.0632902
$466$	0	0
$467$	18.9591	0.877323	0.438661	−	0.898652i	$-0.355453\pi$
$467$	18.9591	0.877323	0.438661	+	0.898652i	$0.355453\pi$
$468$	0	0
$469$	−50.7974	−2.34561
$470$	0	0
$471$	16.3452	0.753147
$472$	0	0
$473$	−10.8612	−0.499400
$474$	0	0
$475$	−2.42937	−0.111467
$476$	0	0
$477$	9.90332	0.453442
$478$	0	0
$479$	37.1402	1.69698	0.848490	−	0.529212i	$-0.177512\pi$
$479$	37.1402	1.69698	0.848490	+	0.529212i	$0.177512\pi$
$480$	0	0
$481$	1.39858	0.0637699
$482$	0	0
$483$	−27.2548	−1.24013
$484$	0	0
$485$	1.36951	0.0621863
$486$	0	0
$487$	−34.7488	−1.57462	−0.787310	−	0.616558i	$-0.788527\pi$
$487$	−34.7488	−1.57462	−0.787310	+	0.616558i	$0.788527\pi$
$488$	0	0
$489$	2.97921	0.134725
$490$	0	0
$491$	−42.4735	−1.91680	−0.958402	−	0.285423i	$-0.907866\pi$
$491$	−42.4735	−1.91680	−0.958402	+	0.285423i	$0.907866\pi$
$492$	0	0
$493$	46.5433	2.09621
$494$	0	0
$495$	−0.699291	−0.0314308
$496$	0	0
$497$	−37.5427	−1.68402
$498$	0	0
$499$	22.1844	0.993112	0.496556	−	0.868005i	$-0.334598\pi$
$499$	22.1844	0.993112	0.496556	+	0.868005i	$0.334598\pi$
$500$	0	0
$501$	19.2524	0.860132
$502$	0	0
$503$	−30.5071	−1.36025	−0.680123	−	0.733098i	$-0.738074\pi$
$503$	−30.5071	−1.36025	−0.680123	+	0.733098i	$0.738074\pi$
$504$	0	0
$505$	−1.34988	−0.0600691
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	5.67277	0.251441	0.125721	−	0.992066i	$-0.459876\pi$
$509$	5.67277	0.251441	0.125721	+	0.992066i	$0.459876\pi$
$510$	0	0
$511$	−14.5286	−0.642709
$512$	0	0
$513$	0.538544	0.0237773
$514$	0	0
$515$	−5.14080	−0.226531
$516$	0	0
$517$	−13.5818	−0.597328
$518$	0	0
$519$	−0.440670	−0.0193433
$520$	0	0
$521$	−11.6154	−0.508882	−0.254441	−	0.967088i	$-0.581891\pi$
$521$	−11.6154	−0.508882	−0.254441	+	0.967088i	$0.581891\pi$
$522$	0	0
$523$	−1.14673	−0.0501429	−0.0250714	−	0.999686i	$-0.507981\pi$
$523$	−1.14673	−0.0501429	−0.0250714	+	0.999686i	$0.507981\pi$
$524$	0	0
$525$	17.1008	0.746339
$526$	0	0
$527$	9.93713	0.432868
$528$	0	0
$529$	28.6889	1.24734
$530$	0	0
$531$	−5.72804	−0.248576
$532$	0	0
$533$	1.79091	0.0775731
$534$	0	0
$535$	−7.71314	−0.333468
$536$	0	0
$537$	21.7022	0.936519
$538$	0	0
$539$	−7.37103	−0.317493
$540$	0	0
$541$	−35.8506	−1.54134	−0.770669	−	0.637235i	$-0.780078\pi$
$541$	−35.8506	−1.54134	−0.770669	+	0.637235i	$0.780078\pi$
$542$	0	0
$543$	−1.74138	−0.0747297
$544$	0	0
$545$	1.70758	0.0731445
$546$	0	0
$547$	−29.7159	−1.27056	−0.635280	−	0.772282i	$-0.719115\pi$
$547$	−29.7159	−1.27056	−0.635280	+	0.772282i	$0.719115\pi$
$548$	0	0
$549$	11.3514	0.484468
$550$	0	0
$551$	−4.92291	−0.209723
$552$	0	0
$553$	5.19941	0.221101
$554$	0	0
$555$	−0.978016	−0.0415145
$556$	0	0
$557$	27.7978	1.17783	0.588916	−	0.808194i	$-0.299555\pi$
$557$	27.7978	1.17783	0.588916	+	0.808194i	$0.299555\pi$
$558$	0	0
$559$	−10.8612	−0.459381
$560$	0	0
$561$	5.09162	0.214968
$562$	0	0
$563$	41.4436	1.74664	0.873319	−	0.487148i	$-0.161963\pi$
$563$	41.4436	1.74664	0.873319	+	0.487148i	$0.161963\pi$
$564$	0	0
$565$	−4.65095	−0.195667
$566$	0	0
$567$	−3.79091	−0.159203
$568$	0	0
$569$	35.7945	1.50058	0.750292	−	0.661107i	$-0.229913\pi$
$569$	35.7945	1.50058	0.750292	+	0.661107i	$0.229913\pi$
$570$	0	0
$571$	−34.0766	−1.42606	−0.713030	−	0.701133i	$-0.752678\pi$
$571$	−34.0766	−1.42606	−0.713030	+	0.701133i	$0.752678\pi$
$572$	0	0
$573$	−18.6118	−0.777518
$574$	0	0
$575$	−32.4318	−1.35250
$576$	0	0
$577$	−41.2201	−1.71602	−0.858008	−	0.513636i	$-0.828298\pi$
$577$	−41.2201	−1.71602	−0.858008	+	0.513636i	$0.828298\pi$
$578$	0	0
$579$	−20.9040	−0.868741
$580$	0	0
$581$	−16.3824	−0.679657
$582$	0	0
$583$	−9.90332	−0.410154
$584$	0	0
$585$	−0.699291	−0.0289121
$586$	0	0
$587$	12.5252	0.516971	0.258485	−	0.966015i	$-0.416777\pi$
$587$	12.5252	0.516971	0.258485	+	0.966015i	$0.416777\pi$
$588$	0	0
$589$	−1.05106	−0.0433080
$590$	0	0
$591$	18.1228	0.745471
$592$	0	0
$593$	−9.13996	−0.375333	−0.187667	−	0.982233i	$-0.560092\pi$
$593$	−9.13996	−0.375333	−0.187667	+	0.982233i	$0.560092\pi$
$594$	0	0
$595$	13.4977	0.553350
$596$	0	0
$597$	−12.6533	−0.517867
$598$	0	0
$599$	−5.39301	−0.220353	−0.110176	−	0.993912i	$-0.535142\pi$
$599$	−5.39301	−0.220353	−0.110176	+	0.993912i	$0.535142\pi$
$600$	0	0
$601$	−2.65892	−0.108459	−0.0542297	−	0.998528i	$-0.517270\pi$
$601$	−2.65892	−0.108459	−0.0542297	+	0.998528i	$0.517270\pi$
$602$	0	0
$603$	13.3998	0.545681
$604$	0	0
$605$	0.699291	0.0284302
$606$	0	0
$607$	−39.7945	−1.61521	−0.807605	−	0.589724i	$-0.799236\pi$
$607$	−39.7945	−1.61521	−0.807605	+	0.589724i	$0.799236\pi$
$608$	0	0
$609$	34.6533	1.40422
$610$	0	0
$611$	−13.5818	−0.549462
$612$	0	0
$613$	−26.1654	−1.05681	−0.528405	−	0.848993i	$-0.677210\pi$
$613$	−26.1654	−1.05681	−0.528405	+	0.848993i	$0.677210\pi$
$614$	0	0
$615$	−1.25237	−0.0505005
$616$	0	0
$617$	−23.4101	−0.942457	−0.471228	−	0.882011i	$-0.656189\pi$
$617$	−23.4101	−0.942457	−0.471228	+	0.882011i	$0.656189\pi$
$618$	0	0
$619$	−36.8974	−1.48303	−0.741517	−	0.670935i	$-0.765893\pi$
$619$	−36.8974	−1.48303	−0.741517	+	0.670935i	$0.765893\pi$
$620$	0	0
$621$	7.18950	0.288505
$622$	0	0
$623$	−22.5622	−0.903937
$624$	0	0
$625$	17.9040	0.716160
$626$	0	0
$627$	−0.538544	−0.0215074
$628$	0	0
$629$	7.12105	0.283935
$630$	0	0
$631$	−21.8834	−0.871165	−0.435582	−	0.900149i	$-0.643458\pi$
$631$	−21.8834	−0.871165	−0.435582	+	0.900149i	$0.643458\pi$
$632$	0	0
$633$	−15.8337	−0.629332
$634$	0	0
$635$	3.78535	0.150217
$636$	0	0
$637$	−7.37103	−0.292051
$638$	0	0
$639$	9.90332	0.391769
$640$	0	0
$641$	20.5982	0.813582	0.406791	−	0.913521i	$-0.366648\pi$
$641$	20.5982	0.813582	0.406791	+	0.913521i	$0.366648\pi$
$642$	0	0
$643$	−36.0518	−1.42174	−0.710871	−	0.703322i	$-0.751699\pi$
$643$	−36.0518	−1.42174	−0.710871	+	0.703322i	$0.751699\pi$
$644$	0	0
$645$	7.59517	0.299059
$646$	0	0
$647$	29.4300	1.15701	0.578507	−	0.815677i	$-0.303635\pi$
$647$	29.4300	1.15701	0.578507	+	0.815677i	$0.303635\pi$
$648$	0	0
$649$	5.72804	0.224845
$650$	0	0
$651$	7.39858	0.289973
$652$	0	0
$653$	−49.5301	−1.93826	−0.969132	−	0.246541i	$-0.920706\pi$
$653$	−49.5301	−1.93826	−0.969132	+	0.246541i	$0.920706\pi$
$654$	0	0
$655$	−5.98193	−0.233733
$656$	0	0
$657$	3.83249	0.149520
$658$	0	0
$659$	26.7895	1.04357	0.521784	−	0.853077i	$-0.325267\pi$
$659$	26.7895	1.04357	0.521784	+	0.853077i	$0.325267\pi$
$660$	0	0
$661$	21.2177	0.825274	0.412637	−	0.910896i	$-0.364608\pi$
$661$	21.2177	0.825274	0.412637	+	0.910896i	$0.364608\pi$
$662$	0	0
$663$	5.09162	0.197742
$664$	0	0
$665$	−1.42765	−0.0553620
$666$	0	0
$667$	−65.7203	−2.54470
$668$	0	0
$669$	−4.65215	−0.179863
$670$	0	0
$671$	−11.3514	−0.438218
$672$	0	0
$673$	24.1015	0.929043	0.464522	−	0.885562i	$-0.346226\pi$
$673$	24.1015	0.929043	0.464522	+	0.885562i	$0.346226\pi$
$674$	0	0
$675$	−4.51099	−0.173628
$676$	0	0
$677$	−8.67242	−0.333308	−0.166654	−	0.986015i	$-0.553296\pi$
$677$	−8.67242	−0.333308	−0.166654	+	0.986015i	$0.553296\pi$
$678$	0	0
$679$	−7.42423	−0.284916
$680$	0	0
$681$	5.84045	0.223807
$682$	0	0
$683$	22.8917	0.875926	0.437963	−	0.898993i	$-0.355700\pi$
$683$	22.8917	0.875926	0.437963	+	0.898993i	$0.355700\pi$
$684$	0	0
$685$	−0.778258	−0.0297357
$686$	0	0
$687$	15.9113	0.607053
$688$	0	0
$689$	−9.90332	−0.377287
$690$	0	0
$691$	37.4943	1.42635	0.713175	−	0.700986i	$-0.247256\pi$
$691$	37.4943	1.42635	0.713175	+	0.700986i	$0.247256\pi$
$692$	0	0
$693$	3.79091	0.144005
$694$	0	0
$695$	2.65179	0.100588
$696$	0	0
$697$	9.11866	0.345394
$698$	0	0
$699$	−8.90838	−0.336946
$700$	0	0
$701$	−5.59183	−0.211200	−0.105600	−	0.994409i	$-0.533676\pi$
$701$	−5.59183	−0.211200	−0.105600	+	0.994409i	$0.533676\pi$
$702$	0	0
$703$	−0.753198	−0.0284074
$704$	0	0
$705$	9.49765	0.357702
$706$	0	0
$707$	7.31783	0.275215
$708$	0	0
$709$	7.22569	0.271367	0.135683	−	0.990752i	$-0.456677\pi$
$709$	7.22569	0.271367	0.135683	+	0.990752i	$0.456677\pi$
$710$	0	0
$711$	−1.37155	−0.0514370
$712$	0	0
$713$	−14.0315	−0.525483
$714$	0	0
$715$	0.699291	0.0261520
$716$	0	0
$717$	−14.1699	−0.529185
$718$	0	0
$719$	43.2998	1.61481	0.807404	−	0.589999i	$-0.200872\pi$
$719$	43.2998	1.61481	0.807404	+	0.589999i	$0.200872\pi$
$720$	0	0
$721$	27.8687	1.03788
$722$	0	0
$723$	−6.46317	−0.240368
$724$	0	0
$725$	41.2357	1.53146
$726$	0	0
$727$	−1.79175	−0.0664524	−0.0332262	−	0.999448i	$-0.510578\pi$
$727$	−1.79175	−0.0664524	−0.0332262	+	0.999448i	$0.510578\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−55.3013	−2.04539
$732$	0	0
$733$	−27.9371	−1.03188	−0.515941	−	0.856624i	$-0.672557\pi$
$733$	−27.9371	−1.03188	−0.515941	+	0.856624i	$0.672557\pi$
$734$	0	0
$735$	5.15450	0.190126
$736$	0	0
$737$	−13.3998	−0.493587
$738$	0	0
$739$	47.9318	1.76320	0.881600	−	0.471997i	$-0.156467\pi$
$739$	47.9318	1.76320	0.881600	+	0.471997i	$0.156467\pi$
$740$	0	0
$741$	−0.538544	−0.0197839
$742$	0	0
$743$	−0.216856	−0.00795568	−0.00397784	−	0.999992i	$-0.501266\pi$
$743$	−0.216856	−0.00795568	−0.00397784	+	0.999992i	$0.501266\pi$
$744$	0	0
$745$	2.97802	0.109106
$746$	0	0
$747$	4.32150	0.158115
$748$	0	0
$749$	41.8136	1.52783
$750$	0	0
$751$	24.5002	0.894025	0.447013	−	0.894528i	$-0.352488\pi$
$751$	24.5002	0.894025	0.447013	+	0.894528i	$0.352488\pi$
$752$	0	0
$753$	11.1187	0.405186
$754$	0	0
$755$	4.92531	0.179250
$756$	0	0
$757$	−31.3831	−1.14064	−0.570319	−	0.821423i	$-0.693180\pi$
$757$	−31.3831	−1.14064	−0.570319	+	0.821423i	$0.693180\pi$
$758$	0	0
$759$	−7.18950	−0.260962
$760$	0	0
$761$	28.5340	1.03436	0.517178	−	0.855878i	$-0.326982\pi$
$761$	28.5340	1.03436	0.517178	+	0.855878i	$0.326982\pi$
$762$	0	0
$763$	−9.25691	−0.335123
$764$	0	0
$765$	−3.56053	−0.128731
$766$	0	0
$767$	5.72804	0.206828
$768$	0	0
$769$	19.4491	0.701354	0.350677	−	0.936496i	$-0.385951\pi$
$769$	19.4491	0.701354	0.350677	+	0.936496i	$0.385951\pi$
$770$	0	0
$771$	3.63136	0.130780
$772$	0	0
$773$	44.2881	1.59293	0.796465	−	0.604684i	$-0.206701\pi$
$773$	44.2881	1.59293	0.796465	+	0.604684i	$0.206701\pi$
$774$	0	0
$775$	8.80393	0.316247
$776$	0	0
$777$	5.30191	0.190205
$778$	0	0
$779$	−0.964485	−0.0345563
$780$	0	0
$781$	−9.90332	−0.354369
$782$	0	0
$783$	−9.14116	−0.326678
$784$	0	0
$785$	11.4300	0.407956
$786$	0	0
$787$	−8.55447	−0.304934	−0.152467	−	0.988309i	$-0.548722\pi$
$787$	−8.55447	−0.304934	−0.152467	+	0.988309i	$0.548722\pi$
$788$	0	0
$789$	18.7005	0.665755
$790$	0	0
$791$	25.2132	0.896478
$792$	0	0
$793$	−11.3514	−0.403102
$794$	0	0
$795$	6.92531	0.245615
$796$	0	0
$797$	−37.9483	−1.34420	−0.672099	−	0.740461i	$-0.734607\pi$
$797$	−37.9483	−1.34420	−0.672099	+	0.740461i	$0.734607\pi$
$798$	0	0
$799$	−69.1536	−2.44648
$800$	0	0
$801$	5.95166	0.210292
$802$	0	0
$803$	−3.83249	−0.135246
$804$	0	0
$805$	−19.0590	−0.671742
$806$	0	0
$807$	−16.9253	−0.595799
$808$	0	0
$809$	27.9047	0.981077	0.490539	−	0.871419i	$-0.336800\pi$
$809$	27.9047	0.981077	0.490539	+	0.871419i	$0.336800\pi$
$810$	0	0
$811$	32.1252	1.12807	0.564033	−	0.825752i	$-0.309249\pi$
$811$	32.1252	1.12807	0.564033	+	0.825752i	$0.309249\pi$
$812$	0	0
$813$	17.7438	0.622301
$814$	0	0
$815$	2.08334	0.0729761
$816$	0	0
$817$	5.84925	0.204639
$818$	0	0
$819$	3.79091	0.132465
$820$	0	0
$821$	−18.3428	−0.640168	−0.320084	−	0.947389i	$-0.603711\pi$
$821$	−18.3428	−0.640168	−0.320084	+	0.947389i	$0.603711\pi$
$822$	0	0
$823$	−32.3585	−1.12795	−0.563974	−	0.825793i	$-0.690728\pi$
$823$	−32.3585	−1.12795	−0.563974	+	0.825793i	$0.690728\pi$
$824$	0	0
$825$	4.51099	0.157053
$826$	0	0
$827$	6.65179	0.231305	0.115653	−	0.993290i	$-0.463104\pi$
$827$	6.65179	0.231305	0.115653	+	0.993290i	$0.463104\pi$
$828$	0	0
$829$	−10.3915	−0.360912	−0.180456	−	0.983583i	$-0.557757\pi$
$829$	−10.3915	−0.360912	−0.180456	+	0.983583i	$0.557757\pi$
$830$	0	0
$831$	18.6533	0.647077
$832$	0	0
$833$	−37.5305	−1.30036
$834$	0	0
$835$	13.4630	0.465907
$836$	0	0
$837$	−1.95166	−0.0674593
$838$	0	0
$839$	49.4324	1.70660	0.853299	−	0.521422i	$-0.174598\pi$
$839$	49.4324	1.70660	0.853299	+	0.521422i	$0.174598\pi$
$840$	0	0
$841$	54.5608	1.88141
$842$	0	0
$843$	−8.26658	−0.284716
$844$	0	0
$845$	0.699291	0.0240563
$846$	0	0
$847$	−3.79091	−0.130257
$848$	0	0
$849$	3.92582	0.134734
$850$	0	0
$851$	−10.0551	−0.344685
$852$	0	0
$853$	−20.6555	−0.707231	−0.353615	−	0.935391i	$-0.615048\pi$
$853$	−20.6555	−0.707231	−0.353615	+	0.935391i	$0.615048\pi$
$854$	0	0
$855$	0.376599	0.0128794
$856$	0	0
$857$	27.4605	0.938033	0.469017	−	0.883189i	$-0.344608\pi$
$857$	27.4605	0.938033	0.469017	+	0.883189i	$0.344608\pi$
$858$	0	0
$859$	−18.0149	−0.614660	−0.307330	−	0.951603i	$-0.599436\pi$
$859$	−18.0149	−0.614660	−0.307330	+	0.951603i	$0.599436\pi$
$860$	0	0
$861$	6.78920	0.231375
$862$	0	0
$863$	34.2664	1.16644	0.583221	−	0.812314i	$-0.301792\pi$
$863$	34.2664	1.16644	0.583221	+	0.812314i	$0.301792\pi$
$864$	0	0
$865$	−0.308157	−0.0104776
$866$	0	0
$867$	8.92462	0.303096
$868$	0	0
$869$	1.37155	0.0465265
$870$	0	0
$871$	−13.3998	−0.454034
$872$	0	0
$873$	1.95843	0.0662827
$874$	0	0
$875$	25.2132	0.852361
$876$	0	0
$877$	20.9679	0.708036	0.354018	−	0.935239i	$-0.384815\pi$
$877$	20.9679	0.708036	0.354018	+	0.935239i	$0.384815\pi$
$878$	0	0
$879$	−16.3036	−0.549907
$880$	0	0
$881$	5.04348	0.169919	0.0849595	−	0.996384i	$-0.472924\pi$
$881$	5.04348	0.169919	0.0849595	+	0.996384i	$0.472924\pi$
$882$	0	0
$883$	−46.0173	−1.54861	−0.774303	−	0.632816i	$-0.781899\pi$
$883$	−46.0173	−1.54861	−0.774303	+	0.632816i	$0.781899\pi$
$884$	0	0
$885$	−4.00557	−0.134646
$886$	0	0
$887$	−9.19033	−0.308581	−0.154291	−	0.988026i	$-0.549309\pi$
$887$	−9.19033	−0.308581	−0.154291	+	0.988026i	$0.549309\pi$
$888$	0	0
$889$	−20.5207	−0.688241
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	7.31441	0.244767
$894$	0	0
$895$	15.1762	0.507283
$896$	0	0
$897$	−7.18950	−0.240050
$898$	0	0
$899$	17.8404	0.595012
$900$	0	0
$901$	−50.4240	−1.67987
$902$	0	0
$903$	−41.1740	−1.37019
$904$	0	0
$905$	−1.21773	−0.0404787
$906$	0	0
$907$	−27.9081	−0.926674	−0.463337	−	0.886182i	$-0.653348\pi$
$907$	−27.9081	−0.926674	−0.463337	+	0.886182i	$0.653348\pi$
$908$	0	0
$909$	−1.93036	−0.0640260
$910$	0	0
$911$	6.82624	0.226163	0.113082	−	0.993586i	$-0.463928\pi$
$911$	6.82624	0.226163	0.113082	+	0.993586i	$0.463928\pi$
$912$	0	0
$913$	−4.32150	−0.143021
$914$	0	0
$915$	7.93796	0.262421
$916$	0	0
$917$	32.4285	1.07088
$918$	0	0
$919$	27.3899	0.903508	0.451754	−	0.892142i	$-0.350798\pi$
$919$	27.3899	0.903508	0.451754	+	0.892142i	$0.350798\pi$
$920$	0	0
$921$	−6.77586	−0.223272
$922$	0	0
$923$	−9.90332	−0.325972
$924$	0	0
$925$	6.30899	0.207438
$926$	0	0
$927$	−7.35144	−0.241453
$928$	0	0
$929$	14.9185	0.489461	0.244731	−	0.969591i	$-0.421300\pi$
$929$	14.9185	0.489461	0.244731	+	0.969591i	$0.421300\pi$
$930$	0	0
$931$	3.96962	0.130099
$932$	0	0
$933$	13.8820	0.454477
$934$	0	0
$935$	3.56053	0.116442
$936$	0	0
$937$	−41.0954	−1.34253	−0.671264	−	0.741218i	$-0.734248\pi$
$937$	−41.0954	−1.34253	−0.671264	+	0.741218i	$0.734248\pi$
$938$	0	0
$939$	−19.1344	−0.624427
$940$	0	0
$941$	−15.0558	−0.490805	−0.245402	−	0.969421i	$-0.578920\pi$
$941$	−15.0558	−0.490805	−0.245402	+	0.969421i	$0.578920\pi$
$942$	0	0
$943$	−12.8758	−0.419293
$944$	0	0
$945$	−2.65095	−0.0862355
$946$	0	0
$947$	49.7394	1.61631	0.808157	−	0.588967i	$-0.200465\pi$
$947$	49.7394	1.61631	0.808157	+	0.588967i	$0.200465\pi$
$948$	0	0
$949$	−3.83249	−0.124408
$950$	0	0
$951$	16.1429	0.523468
$952$	0	0
$953$	5.48415	0.177649	0.0888245	−	0.996047i	$-0.471689\pi$
$953$	5.48415	0.177649	0.0888245	+	0.996047i	$0.471689\pi$
$954$	0	0
$955$	−13.0151	−0.421157
$956$	0	0
$957$	9.14116	0.295492
$958$	0	0
$959$	4.21900	0.136239
$960$	0	0
$961$	−27.1910	−0.877130
$962$	0	0
$963$	−11.0299	−0.355435
$964$	0	0
$965$	−14.6180	−0.470570
$966$	0	0
$967$	−6.30919	−0.202890	−0.101445	−	0.994841i	$-0.532347\pi$
$967$	−6.30919	−0.202890	−0.101445	+	0.994841i	$0.532347\pi$
$968$	0	0
$969$	−2.74206	−0.0880877
$970$	0	0
$971$	−37.6572	−1.20848	−0.604239	−	0.796803i	$-0.706523\pi$
$971$	−37.6572	−1.20848	−0.604239	+	0.796803i	$0.706523\pi$
$972$	0	0
$973$	−14.3756	−0.460860
$974$	0	0
$975$	4.51099	0.144467
$976$	0	0
$977$	10.8063	0.345725	0.172862	−	0.984946i	$-0.444698\pi$
$977$	10.8063	0.345725	0.172862	+	0.984946i	$0.444698\pi$
$978$	0	0
$979$	−5.95166	−0.190216
$980$	0	0
$981$	2.44187	0.0779628
$982$	0	0
$983$	61.7408	1.96923	0.984613	−	0.174751i	$-0.0559119\pi$
$983$	61.7408	1.96923	0.984613	+	0.174751i	$0.0559119\pi$
$984$	0	0
$985$	12.6731	0.403798
$986$	0	0
$987$	−51.4875	−1.63887
$988$	0	0
$989$	78.0868	2.48302
$990$	0	0
$991$	−16.8917	−0.536582	−0.268291	−	0.963338i	$-0.586459\pi$
$991$	−16.8917	−0.536582	−0.268291	+	0.963338i	$0.586459\pi$
$992$	0	0
$993$	−7.00829	−0.222401
$994$	0	0
$995$	−8.84837	−0.280512
$996$	0	0
$997$	31.9012	1.01032	0.505160	−	0.863026i	$-0.331433\pi$
$997$	31.9012	1.01032	0.505160	+	0.863026i	$0.331433\pi$
$998$	0	0
$999$	−1.39858	−0.0442492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.ca.1.3		4
4.3	odd	2		3432.2.a.r.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.r.1.3	✓	4	4.3	odd	2
6864.2.a.ca.1.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.699291 q^{5} -3.79091 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.699291 q^{5} -3.79091 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.699291 q^{15} -5.09162 q^{17} +0.538544 q^{19} -3.79091 q^{21} +7.18950 q^{23} -4.51099 q^{25} +1.00000 q^{27} -9.14116 q^{29} -1.95166 q^{31} -1.00000 q^{33} -2.65095 q^{35} -1.39858 q^{37} -1.00000 q^{39} -1.79091 q^{41} +10.8612 q^{43} +0.699291 q^{45} +13.5818 q^{47} +7.37103 q^{49} -5.09162 q^{51} +9.90332 q^{53} -0.699291 q^{55} +0.538544 q^{57} -5.72804 q^{59} +11.3514 q^{61} -3.79091 q^{63} -0.699291 q^{65} +13.3998 q^{67} +7.18950 q^{69} +9.90332 q^{71} +3.83249 q^{73} -4.51099 q^{75} +3.79091 q^{77} -1.37155 q^{79} +1.00000 q^{81} +4.32150 q^{83} -3.56053 q^{85} -9.14116 q^{87} +5.95166 q^{89} +3.79091 q^{91} -1.95166 q^{93} +0.376599 q^{95} +1.95843 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + q^5 + q^7 + 4 * q^9	\( 4 q + 4 q^{3} + q^{5} + q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} + q^{15} - 6 q^{17} + q^{21} + 9 q^{23} + q^{25} + 4 q^{27} - q^{29} + 8 q^{31} - 4 q^{33} + 7 q^{35} - 2 q^{37} - 4 q^{39} + 9 q^{41} + 5 q^{43} + q^{45} + 22 q^{47} + 9 q^{49} - 6 q^{51} + 8 q^{53} - q^{55} - q^{59} - 11 q^{61} + q^{63} - q^{65} + 13 q^{67} + 9 q^{69} + 8 q^{71} - 3 q^{73} + q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 18 q^{83} + 26 q^{85} - q^{87} + 8 q^{89} - q^{91} + 8 q^{93} + 36 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + q^5 + q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 + q^15 - 6 * q^17 + q^21 + 9 * q^23 + q^25 + 4 * q^27 - q^29 + 8 * q^31 - 4 * q^33 + 7 * q^35 - 2 * q^37 - 4 * q^39 + 9 * q^41 + 5 * q^43 + q^45 + 22 * q^47 + 9 * q^49 - 6 * q^51 + 8 * q^53 - q^55 - q^59 - 11 * q^61 + q^63 - q^65 + 13 * q^67 + 9 * q^69 + 8 * q^71 - 3 * q^73 + q^75 - q^77 + 6 * q^79 + 4 * q^81 + 18 * q^83 + 26 * q^85 - q^87 + 8 * q^89 - q^91 + 8 * q^93 + 36 * q^95 + 10 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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