LMFDB - Embedded newform 8664.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8664,2,Mod(1,8664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8664, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8664 = 2^{3} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8664.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:	$69.1823883112$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	8664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.694593 q^{5} +0.305407 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.694593 q^{5} +0.305407 q^{7} +1.00000 q^{9} +4.82295 q^{11} +1.00000 q^{13} -0.694593 q^{15} -7.51754 q^{17} -0.305407 q^{21} +0.694593 q^{23} -4.51754 q^{25} -1.00000 q^{27} +10.1284 q^{29} -1.82295 q^{31} -4.82295 q^{33} +0.212134 q^{35} +6.51754 q^{37} -1.00000 q^{39} +5.38919 q^{41} +3.69459 q^{43} +0.694593 q^{45} -6.00000 q^{47} -6.90673 q^{49} +7.51754 q^{51} +5.43376 q^{53} +3.34998 q^{55} -4.08378 q^{59} +0.389185 q^{61} +0.305407 q^{63} +0.694593 q^{65} -7.82295 q^{67} -0.694593 q^{69} +10.9067 q^{71} -4.38919 q^{73} +4.51754 q^{75} +1.47296 q^{77} +14.4338 q^{79} +1.00000 q^{81} +0.739170 q^{83} -5.22163 q^{85} -10.1284 q^{87} -0.822948 q^{89} +0.305407 q^{91} +1.82295 q^{93} +10.9067 q^{97} +4.82295 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 3 q^{21} + 9 q^{25} - 3 q^{27} + 12 q^{29} + 15 q^{31} + 6 q^{33} - 24 q^{35} - 3 q^{37} - 3 q^{39} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49} - 6 q^{59} - 3 q^{61} + 3 q^{63} - 3 q^{67} + 6 q^{71} - 9 q^{73} - 9 q^{75} - 6 q^{77} + 27 q^{79} + 3 q^{81} - 12 q^{83} - 24 q^{85} - 12 q^{87} + 18 q^{89} + 3 q^{91} - 15 q^{93} + 6 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 3 * q^21 + 9 * q^25 - 3 * q^27 + 12 * q^29 + 15 * q^31 + 6 * q^33 - 24 * q^35 - 3 * q^37 - 3 * q^39 + 12 * q^41 + 9 * q^43 - 18 * q^47 + 6 * q^49 - 6 * q^59 - 3 * q^61 + 3 * q^63 - 3 * q^67 + 6 * q^71 - 9 * q^73 - 9 * q^75 - 6 * q^77 + 27 * q^79 + 3 * q^81 - 12 * q^83 - 24 * q^85 - 12 * q^87 + 18 * q^89 + 3 * q^91 - 15 * q^93 + 6 * q^97 - 6 * 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
$5$	0.694593	0.310631	0.155316	−	0.987865i	$-0.450361\pi$
$5$	0.694593	0.310631	0.155316	+	0.987865i	$0.450361\pi$
$6$	0	0
$7$	0.305407	0.115433	0.0577166	−	0.998333i	$-0.481618\pi$
$7$	0.305407	0.115433	0.0577166	+	0.998333i	$0.481618\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.82295	1.45417	0.727087	−	0.686546i	$-0.240874\pi$
$11$	4.82295	1.45417	0.727087	+	0.686546i	$0.240874\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−0.694593	−0.179343
$16$	0	0
$17$	−7.51754	−1.82327	−0.911636	−	0.410999i	$-0.865180\pi$
$17$	−7.51754	−1.82327	−0.911636	+	0.410999i	$0.865180\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−0.305407	−0.0666453
$22$	0	0
$23$	0.694593	0.144833	0.0724163	−	0.997374i	$-0.476929\pi$
$23$	0.694593	0.144833	0.0724163	+	0.997374i	$0.476929\pi$
$24$	0	0
$25$	−4.51754	−0.903508
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.1284	1.88079	0.940394	−	0.340086i	$-0.110456\pi$
$29$	10.1284	1.88079	0.940394	+	0.340086i	$0.110456\pi$
$30$	0	0
$31$	−1.82295	−0.327411	−0.163706	−	0.986509i	$-0.552345\pi$
$31$	−1.82295	−0.327411	−0.163706	+	0.986509i	$0.552345\pi$
$32$	0	0
$33$	−4.82295	−0.839568
$34$	0	0
$35$	0.212134	0.0358571
$36$	0	0
$37$	6.51754	1.07148	0.535739	−	0.844384i	$-0.320033\pi$
$37$	6.51754	1.07148	0.535739	+	0.844384i	$0.320033\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.38919	0.841649	0.420825	−	0.907142i	$-0.361741\pi$
$41$	5.38919	0.841649	0.420825	+	0.907142i	$0.361741\pi$
$42$	0	0
$43$	3.69459	0.563420	0.281710	−	0.959500i	$-0.409098\pi$
$43$	3.69459	0.563420	0.281710	+	0.959500i	$0.409098\pi$
$44$	0	0
$45$	0.694593	0.103544
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−6.90673	−0.986675
$50$	0	0
$51$	7.51754	1.05267
$52$	0	0
$53$	5.43376	0.746385	0.373192	−	0.927754i	$-0.378263\pi$
$53$	5.43376	0.746385	0.373192	+	0.927754i	$0.378263\pi$
$54$	0	0
$55$	3.34998	0.451712
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.08378	−0.531663	−0.265831	−	0.964020i	$-0.585646\pi$
$59$	−4.08378	−0.531663	−0.265831	+	0.964020i	$0.585646\pi$
$60$	0	0
$61$	0.389185	0.0498301	0.0249150	−	0.999690i	$-0.492068\pi$
$61$	0.389185	0.0498301	0.0249150	+	0.999690i	$0.492068\pi$
$62$	0	0
$63$	0.305407	0.0384777
$64$	0	0
$65$	0.694593	0.0861536
$66$	0	0
$67$	−7.82295	−0.955725	−0.477863	−	0.878435i	$-0.658588\pi$
$67$	−7.82295	−0.955725	−0.477863	+	0.878435i	$0.658588\pi$
$68$	0	0
$69$	−0.694593	−0.0836191
$70$	0	0
$71$	10.9067	1.29439	0.647195	−	0.762324i	$-0.275942\pi$
$71$	10.9067	1.29439	0.647195	+	0.762324i	$0.275942\pi$
$72$	0	0
$73$	−4.38919	−0.513715	−0.256858	−	0.966449i	$-0.582687\pi$
$73$	−4.38919	−0.513715	−0.256858	+	0.966449i	$0.582687\pi$
$74$	0	0
$75$	4.51754	0.521641
$76$	0	0
$77$	1.47296	0.167860
$78$	0	0
$79$	14.4338	1.62393	0.811963	−	0.583709i	$-0.198399\pi$
$79$	14.4338	1.62393	0.811963	+	0.583709i	$0.198399\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.739170	0.0811345	0.0405672	−	0.999177i	$-0.487084\pi$
$83$	0.739170	0.0811345	0.0405672	+	0.999177i	$0.487084\pi$
$84$	0	0
$85$	−5.22163	−0.566365
$86$	0	0
$87$	−10.1284	−1.08587
$88$	0	0
$89$	−0.822948	−0.0872323	−0.0436162	−	0.999048i	$-0.513888\pi$
$89$	−0.822948	−0.0872323	−0.0436162	+	0.999048i	$0.513888\pi$
$90$	0	0
$91$	0.305407	0.0320154
$92$	0	0
$93$	1.82295	0.189031
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.9067	1.10741	0.553705	−	0.832713i	$-0.313213\pi$
$97$	10.9067	1.10741	0.553705	+	0.832713i	$0.313213\pi$
$98$	0	0
$99$	4.82295	0.484725
$100$	0	0
$101$	9.51754	0.947031	0.473515	−	0.880786i	$-0.342985\pi$
$101$	9.51754	0.947031	0.473515	+	0.880786i	$0.342985\pi$
$102$	0	0
$103$	−2.30541	−0.227159	−0.113579	−	0.993529i	$-0.536232\pi$
$103$	−2.30541	−0.227159	−0.113579	+	0.993529i	$0.536232\pi$
$104$	0	0
$105$	−0.212134	−0.0207021
$106$	0	0
$107$	9.51754	0.920095	0.460048	−	0.887894i	$-0.347832\pi$
$107$	9.51754	0.920095	0.460048	+	0.887894i	$0.347832\pi$
$108$	0	0
$109$	13.5175	1.29475	0.647373	−	0.762174i	$-0.275868\pi$
$109$	13.5175	1.29475	0.647373	+	0.762174i	$0.275868\pi$
$110$	0	0
$111$	−6.51754	−0.618618
$112$	0	0
$113$	−21.0797	−1.98301	−0.991504	−	0.130078i	$-0.958477\pi$
$113$	−21.0797	−1.98301	−0.991504	+	0.130078i	$0.958477\pi$
$114$	0	0
$115$	0.482459	0.0449895
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−2.29591	−0.210466
$120$	0	0
$121$	12.2608	1.11462
$122$	0	0
$123$	−5.38919	−0.485926
$124$	0	0
$125$	−6.61081	−0.591289
$126$	0	0
$127$	9.38919	0.833155	0.416578	−	0.909100i	$-0.363229\pi$
$127$	9.38919	0.833155	0.416578	+	0.909100i	$0.363229\pi$
$128$	0	0
$129$	−3.69459	−0.325291
$130$	0	0
$131$	−8.12836	−0.710178	−0.355089	−	0.934833i	$-0.615549\pi$
$131$	−8.12836	−0.710178	−0.355089	+	0.934833i	$0.615549\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.694593	−0.0597810
$136$	0	0
$137$	−5.34998	−0.457080	−0.228540	−	0.973535i	$-0.573395\pi$
$137$	−5.34998	−0.457080	−0.228540	+	0.973535i	$0.573395\pi$
$138$	0	0
$139$	13.9513	1.18333	0.591667	−	0.806182i	$-0.298470\pi$
$139$	13.9513	1.18333	0.591667	+	0.806182i	$0.298470\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	4.82295	0.403315
$144$	0	0
$145$	7.03508	0.584232
$146$	0	0
$147$	6.90673	0.569657
$148$	0	0
$149$	10.6946	0.876135	0.438068	−	0.898942i	$-0.355663\pi$
$149$	10.6946	0.876135	0.438068	+	0.898942i	$0.355663\pi$
$150$	0	0
$151$	−17.6459	−1.43600	−0.718001	−	0.696042i	$-0.754943\pi$
$151$	−17.6459	−1.43600	−0.718001	+	0.696042i	$0.754943\pi$
$152$	0	0
$153$	−7.51754	−0.607757
$154$	0	0
$155$	−1.26621	−0.101704
$156$	0	0
$157$	15.9067	1.26949	0.634747	−	0.772720i	$-0.281104\pi$
$157$	15.9067	1.26949	0.634747	+	0.772720i	$0.281104\pi$
$158$	0	0
$159$	−5.43376	−0.430925
$160$	0	0
$161$	0.212134	0.0167185
$162$	0	0
$163$	−2.17705	−0.170520	−0.0852599	−	0.996359i	$-0.527172\pi$
$163$	−2.17705	−0.170520	−0.0852599	+	0.996359i	$0.527172\pi$
$164$	0	0
$165$	−3.34998	−0.260796
$166$	0	0
$167$	−10.9513	−0.847437	−0.423719	−	0.905794i	$-0.639275\pi$
$167$	−10.9513	−0.847437	−0.423719	+	0.905794i	$0.639275\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.1284	−0.770045	−0.385022	−	0.922907i	$-0.625806\pi$
$173$	−10.1284	−0.770045	−0.385022	+	0.922907i	$0.625806\pi$
$174$	0	0
$175$	−1.37969	−0.104295
$176$	0	0
$177$	4.08378	0.306956
$178$	0	0
$179$	8.69459	0.649864	0.324932	−	0.945737i	$-0.394659\pi$
$179$	8.69459	0.649864	0.324932	+	0.945737i	$0.394659\pi$
$180$	0	0
$181$	7.87164	0.585095	0.292547	−	0.956251i	$-0.405497\pi$
$181$	7.87164	0.585095	0.292547	+	0.956251i	$0.405497\pi$
$182$	0	0
$183$	−0.389185	−0.0287694
$184$	0	0
$185$	4.52704	0.332834
$186$	0	0
$187$	−36.2567	−2.65135
$188$	0	0
$189$	−0.305407	−0.0222151
$190$	0	0
$191$	13.8580	1.00273	0.501366	−	0.865235i	$-0.332831\pi$
$191$	13.8580	1.00273	0.501366	+	0.865235i	$0.332831\pi$
$192$	0	0
$193$	2.09327	0.150677	0.0753386	−	0.997158i	$-0.475996\pi$
$193$	2.09327	0.150677	0.0753386	+	0.997158i	$0.475996\pi$
$194$	0	0
$195$	−0.694593	−0.0497408
$196$	0	0
$197$	14.0446	1.00063	0.500317	−	0.865842i	$-0.333217\pi$
$197$	14.0446	1.00063	0.500317	+	0.865842i	$0.333217\pi$
$198$	0	0
$199$	−21.9905	−1.55887	−0.779433	−	0.626486i	$-0.784493\pi$
$199$	−21.9905	−1.55887	−0.779433	+	0.626486i	$0.784493\pi$
$200$	0	0
$201$	7.82295	0.551788
$202$	0	0
$203$	3.09327	0.217105
$204$	0	0
$205$	3.74329	0.261443
$206$	0	0
$207$	0.694593	0.0482775
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−15.9513	−1.09813	−0.549067	−	0.835779i	$-0.685017\pi$
$211$	−15.9513	−1.09813	−0.549067	+	0.835779i	$0.685017\pi$
$212$	0	0
$213$	−10.9067	−0.747317
$214$	0	0
$215$	2.56624	0.175016
$216$	0	0
$217$	−0.556742	−0.0377941
$218$	0	0
$219$	4.38919	0.296594
$220$	0	0
$221$	−7.51754	−0.505685
$222$	0	0
$223$	−26.5039	−1.77483	−0.887417	−	0.460967i	$-0.847503\pi$
$223$	−26.5039	−1.77483	−0.887417	+	0.460967i	$0.847503\pi$
$224$	0	0
$225$	−4.51754	−0.301169
$226$	0	0
$227$	2.65539	0.176245	0.0881223	−	0.996110i	$-0.471913\pi$
$227$	2.65539	0.176245	0.0881223	+	0.996110i	$0.471913\pi$
$228$	0	0
$229$	21.2567	1.40468	0.702342	−	0.711840i	$-0.252138\pi$
$229$	21.2567	1.40468	0.702342	+	0.711840i	$0.252138\pi$
$230$	0	0
$231$	−1.47296	−0.0969139
$232$	0	0
$233$	13.5175	0.885564	0.442782	−	0.896629i	$-0.353992\pi$
$233$	13.5175	0.885564	0.442782	+	0.896629i	$0.353992\pi$
$234$	0	0
$235$	−4.16756	−0.271861
$236$	0	0
$237$	−14.4338	−0.937574
$238$	0	0
$239$	−24.4688	−1.58276	−0.791379	−	0.611326i	$-0.790636\pi$
$239$	−24.4688	−1.58276	−0.791379	+	0.611326i	$0.790636\pi$
$240$	0	0
$241$	11.8675	0.764455	0.382227	−	0.924068i	$-0.375157\pi$
$241$	11.8675	0.764455	0.382227	+	0.924068i	$0.375157\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.79736	−0.306492
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.739170	−0.0468430
$250$	0	0
$251$	7.38919	0.466401	0.233201	−	0.972429i	$-0.425080\pi$
$251$	7.38919	0.466401	0.233201	+	0.972429i	$0.425080\pi$
$252$	0	0
$253$	3.34998	0.210612
$254$	0	0
$255$	5.22163	0.326991
$256$	0	0
$257$	5.34461	0.333387	0.166694	−	0.986009i	$-0.446691\pi$
$257$	5.34461	0.333387	0.166694	+	0.986009i	$0.446691\pi$
$258$	0	0
$259$	1.99050	0.123684
$260$	0	0
$261$	10.1284	0.626929
$262$	0	0
$263$	−24.1284	−1.48782	−0.743909	−	0.668281i	$-0.767030\pi$
$263$	−24.1284	−1.48782	−0.743909	+	0.668281i	$0.767030\pi$
$264$	0	0
$265$	3.77425	0.231850
$266$	0	0
$267$	0.822948	0.0503636
$268$	0	0
$269$	−3.60132	−0.219576	−0.109788	−	0.993955i	$-0.535017\pi$
$269$	−3.60132	−0.219576	−0.109788	+	0.993955i	$0.535017\pi$
$270$	0	0
$271$	18.7784	1.14070	0.570352	−	0.821400i	$-0.306807\pi$
$271$	18.7784	1.14070	0.570352	+	0.821400i	$0.306807\pi$
$272$	0	0
$273$	−0.305407	−0.0184841
$274$	0	0
$275$	−21.7879	−1.31386
$276$	0	0
$277$	20.1284	1.20940	0.604698	−	0.796455i	$-0.293294\pi$
$277$	20.1284	1.20940	0.604698	+	0.796455i	$0.293294\pi$
$278$	0	0
$279$	−1.82295	−0.109137
$280$	0	0
$281$	−14.3405	−0.855482	−0.427741	−	0.903901i	$-0.640690\pi$
$281$	−14.3405	−0.855482	−0.427741	+	0.903901i	$0.640690\pi$
$282$	0	0
$283$	6.77837	0.402932	0.201466	−	0.979495i	$-0.435429\pi$
$283$	6.77837	0.402932	0.201466	+	0.979495i	$0.435429\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.64590	0.0971542
$288$	0	0
$289$	39.5134	2.32432
$290$	0	0
$291$	−10.9067	−0.639364
$292$	0	0
$293$	16.7784	0.980203	0.490101	−	0.871665i	$-0.336960\pi$
$293$	16.7784	0.980203	0.490101	+	0.871665i	$0.336960\pi$
$294$	0	0
$295$	−2.83656	−0.165151
$296$	0	0
$297$	−4.82295	−0.279856
$298$	0	0
$299$	0.694593	0.0401693
$300$	0	0
$301$	1.12836	0.0650373
$302$	0	0
$303$	−9.51754	−0.546768
$304$	0	0
$305$	0.270325	0.0154788
$306$	0	0
$307$	−17.1634	−0.979569	−0.489785	−	0.871843i	$-0.662925\pi$
$307$	−17.1634	−0.979569	−0.489785	+	0.871843i	$0.662925\pi$
$308$	0	0
$309$	2.30541	0.131150
$310$	0	0
$311$	−12.1729	−0.690264	−0.345132	−	0.938554i	$-0.612166\pi$
$311$	−12.1729	−0.690264	−0.345132	+	0.938554i	$0.612166\pi$
$312$	0	0
$313$	3.70409	0.209367	0.104684	−	0.994506i	$-0.466617\pi$
$313$	3.70409	0.209367	0.104684	+	0.994506i	$0.466617\pi$
$314$	0	0
$315$	0.212134	0.0119524
$316$	0	0
$317$	23.5621	1.32338	0.661690	−	0.749777i	$-0.269839\pi$
$317$	23.5621	1.32338	0.661690	+	0.749777i	$0.269839\pi$
$318$	0	0
$319$	48.8485	2.73499
$320$	0	0
$321$	−9.51754	−0.531217
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.51754	−0.250588
$326$	0	0
$327$	−13.5175	−0.747522
$328$	0	0
$329$	−1.83244	−0.101026
$330$	0	0
$331$	−4.30541	−0.236647	−0.118323	−	0.992975i	$-0.537752\pi$
$331$	−4.30541	−0.236647	−0.118323	+	0.992975i	$0.537752\pi$
$332$	0	0
$333$	6.51754	0.357159
$334$	0	0
$335$	−5.43376	−0.296878
$336$	0	0
$337$	19.5526	1.06510	0.532550	−	0.846399i	$-0.321234\pi$
$337$	19.5526	1.06510	0.532550	+	0.846399i	$0.321234\pi$
$338$	0	0
$339$	21.0797	1.14489
$340$	0	0
$341$	−8.79199	−0.476113
$342$	0	0
$343$	−4.24722	−0.229328
$344$	0	0
$345$	−0.482459	−0.0259747
$346$	0	0
$347$	33.6905	1.80860	0.904300	−	0.426898i	$-0.140394\pi$
$347$	33.6905	1.80860	0.904300	+	0.426898i	$0.140394\pi$
$348$	0	0
$349$	12.5567	0.672147	0.336073	−	0.941836i	$-0.390901\pi$
$349$	12.5567	0.672147	0.336073	+	0.941836i	$0.390901\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−5.65951	−0.301225	−0.150613	−	0.988593i	$-0.548125\pi$
$353$	−5.65951	−0.301225	−0.150613	+	0.988593i	$0.548125\pi$
$354$	0	0
$355$	7.57573	0.402078
$356$	0	0
$357$	2.29591	0.121513
$358$	0	0
$359$	22.7784	1.20220	0.601098	−	0.799175i	$-0.294730\pi$
$359$	22.7784	1.20220	0.601098	+	0.799175i	$0.294730\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−12.2608	−0.643527
$364$	0	0
$365$	−3.04870	−0.159576
$366$	0	0
$367$	−27.1147	−1.41538	−0.707689	−	0.706524i	$-0.750262\pi$
$367$	−27.1147	−1.41538	−0.707689	+	0.706524i	$0.750262\pi$
$368$	0	0
$369$	5.38919	0.280550
$370$	0	0
$371$	1.65951	0.0861575
$372$	0	0
$373$	−13.9418	−0.721879	−0.360940	−	0.932589i	$-0.617544\pi$
$373$	−13.9418	−0.721879	−0.360940	+	0.932589i	$0.617544\pi$
$374$	0	0
$375$	6.61081	0.341381
$376$	0	0
$377$	10.1284	0.521637
$378$	0	0
$379$	1.08378	0.0556699	0.0278350	−	0.999613i	$-0.491139\pi$
$379$	1.08378	0.0556699	0.0278350	+	0.999613i	$0.491139\pi$
$380$	0	0
$381$	−9.38919	−0.481023
$382$	0	0
$383$	33.0797	1.69029	0.845146	−	0.534536i	$-0.179514\pi$
$383$	33.0797	1.69029	0.845146	+	0.534536i	$0.179514\pi$
$384$	0	0
$385$	1.02311	0.0521425
$386$	0	0
$387$	3.69459	0.187807
$388$	0	0
$389$	−2.82295	−0.143129	−0.0715646	−	0.997436i	$-0.522799\pi$
$389$	−2.82295	−0.143129	−0.0715646	+	0.997436i	$0.522799\pi$
$390$	0	0
$391$	−5.22163	−0.264069
$392$	0	0
$393$	8.12836	0.410021
$394$	0	0
$395$	10.0256	0.504442
$396$	0	0
$397$	−29.5134	−1.48124	−0.740618	−	0.671926i	$-0.765467\pi$
$397$	−29.5134	−1.48124	−0.740618	+	0.671926i	$0.765467\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.3013	1.11367	0.556837	−	0.830622i	$-0.312015\pi$
$401$	22.3013	1.11367	0.556837	+	0.830622i	$0.312015\pi$
$402$	0	0
$403$	−1.82295	−0.0908075
$404$	0	0
$405$	0.694593	0.0345146
$406$	0	0
$407$	31.4338	1.55811
$408$	0	0
$409$	29.7743	1.47224	0.736121	−	0.676850i	$-0.236655\pi$
$409$	29.7743	1.47224	0.736121	+	0.676850i	$0.236655\pi$
$410$	0	0
$411$	5.34998	0.263895
$412$	0	0
$413$	−1.24722	−0.0613715
$414$	0	0
$415$	0.513422	0.0252029
$416$	0	0
$417$	−13.9513	−0.683198
$418$	0	0
$419$	−40.1147	−1.95973	−0.979867	−	0.199653i	$-0.936019\pi$
$419$	−40.1147	−1.95973	−0.979867	+	0.199653i	$0.936019\pi$
$420$	0	0
$421$	34.0310	1.65857	0.829284	−	0.558828i	$-0.188749\pi$
$421$	34.0310	1.65857	0.829284	+	0.558828i	$0.188749\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	33.9608	1.64734
$426$	0	0
$427$	0.118860	0.00575204
$428$	0	0
$429$	−4.82295	−0.232854
$430$	0	0
$431$	10.2567	0.494048	0.247024	−	0.969009i	$-0.420547\pi$
$431$	10.2567	0.494048	0.247024	+	0.969009i	$0.420547\pi$
$432$	0	0
$433$	31.9959	1.53762	0.768812	−	0.639475i	$-0.220848\pi$
$433$	31.9959	1.53762	0.768812	+	0.639475i	$0.220848\pi$
$434$	0	0
$435$	−7.03508	−0.337306
$436$	0	0
$437$	0	0
$438$	0	0
$439$	12.1189	0.578402	0.289201	−	0.957268i	$-0.406610\pi$
$439$	12.1189	0.578402	0.289201	+	0.957268i	$0.406610\pi$
$440$	0	0
$441$	−6.90673	−0.328892
$442$	0	0
$443$	4.90673	0.233126	0.116563	−	0.993183i	$-0.462812\pi$
$443$	4.90673	0.233126	0.116563	+	0.993183i	$0.462812\pi$
$444$	0	0
$445$	−0.571614	−0.0270971
$446$	0	0
$447$	−10.6946	−0.505837
$448$	0	0
$449$	−6.12836	−0.289215	−0.144607	−	0.989489i	$-0.546192\pi$
$449$	−6.12836	−0.289215	−0.144607	+	0.989489i	$0.546192\pi$
$450$	0	0
$451$	25.9918	1.22390
$452$	0	0
$453$	17.6459	0.829077
$454$	0	0
$455$	0.212134	0.00994498
$456$	0	0
$457$	29.9959	1.40315	0.701574	−	0.712597i	$-0.252481\pi$
$457$	29.9959	1.40315	0.701574	+	0.712597i	$0.252481\pi$
$458$	0	0
$459$	7.51754	0.350889
$460$	0	0
$461$	−17.0797	−0.795479	−0.397740	−	0.917498i	$-0.630205\pi$
$461$	−17.0797	−0.795479	−0.397740	+	0.917498i	$0.630205\pi$
$462$	0	0
$463$	28.3756	1.31872	0.659362	−	0.751825i	$-0.270826\pi$
$463$	28.3756	1.31872	0.659362	+	0.751825i	$0.270826\pi$
$464$	0	0
$465$	1.26621	0.0587189
$466$	0	0
$467$	6.61081	0.305912	0.152956	−	0.988233i	$-0.451121\pi$
$467$	6.61081	0.305912	0.152956	+	0.988233i	$0.451121\pi$
$468$	0	0
$469$	−2.38919	−0.110322
$470$	0	0
$471$	−15.9067	−0.732943
$472$	0	0
$473$	17.8188	0.819311
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.43376	0.248795
$478$	0	0
$479$	10.9067	0.498341	0.249171	−	0.968460i	$-0.419842\pi$
$479$	10.9067	0.498341	0.249171	+	0.968460i	$0.419842\pi$
$480$	0	0
$481$	6.51754	0.297174
$482$	0	0
$483$	−0.212134	−0.00965242
$484$	0	0
$485$	7.57573	0.343996
$486$	0	0
$487$	11.8324	0.536179	0.268090	−	0.963394i	$-0.413608\pi$
$487$	11.8324	0.536179	0.268090	+	0.963394i	$0.413608\pi$
$488$	0	0
$489$	2.17705	0.0984497
$490$	0	0
$491$	−14.9067	−0.672731	−0.336366	−	0.941731i	$-0.609198\pi$
$491$	−14.9067	−0.672731	−0.336366	+	0.941731i	$0.609198\pi$
$492$	0	0
$493$	−76.1403	−3.42919
$494$	0	0
$495$	3.34998	0.150571
$496$	0	0
$497$	3.33099	0.149415
$498$	0	0
$499$	39.8229	1.78272	0.891360	−	0.453296i	$-0.149752\pi$
$499$	39.8229	1.78272	0.891360	+	0.453296i	$0.149752\pi$
$500$	0	0
$501$	10.9513	0.489268
$502$	0	0
$503$	−10.1284	−0.451601	−0.225801	−	0.974174i	$-0.572500\pi$
$503$	−10.1284	−0.451601	−0.225801	+	0.974174i	$0.572500\pi$
$504$	0	0
$505$	6.61081	0.294177
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	25.0351	1.10966	0.554830	−	0.831964i	$-0.312783\pi$
$509$	25.0351	1.10966	0.554830	+	0.831964i	$0.312783\pi$
$510$	0	0
$511$	−1.34049	−0.0592998
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.60132	−0.0705626
$516$	0	0
$517$	−28.9377	−1.27268
$518$	0	0
$519$	10.1284	0.444585
$520$	0	0
$521$	23.9162	1.04779	0.523894	−	0.851783i	$-0.324479\pi$
$521$	23.9162	1.04779	0.523894	+	0.851783i	$0.324479\pi$
$522$	0	0
$523$	−36.8580	−1.61169	−0.805845	−	0.592127i	$-0.798288\pi$
$523$	−36.8580	−1.61169	−0.805845	+	0.592127i	$0.798288\pi$
$524$	0	0
$525$	1.37969	0.0602146
$526$	0	0
$527$	13.7041	0.596959
$528$	0	0
$529$	−22.5175	−0.979024
$530$	0	0
$531$	−4.08378	−0.177221
$532$	0	0
$533$	5.38919	0.233432
$534$	0	0
$535$	6.61081	0.285810
$536$	0	0
$537$	−8.69459	−0.375199
$538$	0	0
$539$	−33.3108	−1.43480
$540$	0	0
$541$	22.4201	0.963917	0.481959	−	0.876194i	$-0.339926\pi$
$541$	22.4201	0.963917	0.481959	+	0.876194i	$0.339926\pi$
$542$	0	0
$543$	−7.87164	−0.337805
$544$	0	0
$545$	9.38919	0.402189
$546$	0	0
$547$	−6.37557	−0.272600	−0.136300	−	0.990668i	$-0.543521\pi$
$547$	−6.37557	−0.272600	−0.136300	+	0.990668i	$0.543521\pi$
$548$	0	0
$549$	0.389185	0.0166100
$550$	0	0
$551$	0	0
$552$	0	0
$553$	4.40818	0.187455
$554$	0	0
$555$	−4.52704	−0.192162
$556$	0	0
$557$	−17.9418	−0.760219	−0.380109	−	0.924942i	$-0.624114\pi$
$557$	−17.9418	−0.760219	−0.380109	+	0.924942i	$0.624114\pi$
$558$	0	0
$559$	3.69459	0.156265
$560$	0	0
$561$	36.2567	1.53076
$562$	0	0
$563$	−17.5175	−0.738276	−0.369138	−	0.929375i	$-0.620347\pi$
$563$	−17.5175	−0.738276	−0.369138	+	0.929375i	$0.620347\pi$
$564$	0	0
$565$	−14.6418	−0.615984
$566$	0	0
$567$	0.305407	0.0128259
$568$	0	0
$569$	12.6810	0.531614	0.265807	−	0.964026i	$-0.414362\pi$
$569$	12.6810	0.531614	0.265807	+	0.964026i	$0.414362\pi$
$570$	0	0
$571$	39.0256	1.63317	0.816585	−	0.577225i	$-0.195865\pi$
$571$	39.0256	1.63317	0.816585	+	0.577225i	$0.195865\pi$
$572$	0	0
$573$	−13.8580	−0.578928
$574$	0	0
$575$	−3.13785	−0.130857
$576$	0	0
$577$	28.2959	1.17797	0.588987	−	0.808142i	$-0.299527\pi$
$577$	28.2959	1.17797	0.588987	+	0.808142i	$0.299527\pi$
$578$	0	0
$579$	−2.09327	−0.0869935
$580$	0	0
$581$	0.225748	0.00936560
$582$	0	0
$583$	26.2068	1.08537
$584$	0	0
$585$	0.694593	0.0287179
$586$	0	0
$587$	25.7297	1.06198	0.530989	−	0.847379i	$-0.321821\pi$
$587$	25.7297	1.06198	0.530989	+	0.847379i	$0.321821\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−14.0446	−0.577717
$592$	0	0
$593$	−12.9905	−0.533456	−0.266728	−	0.963772i	$-0.585943\pi$
$593$	−12.9905	−0.533456	−0.266728	+	0.963772i	$0.585943\pi$
$594$	0	0
$595$	−1.59472	−0.0653773
$596$	0	0
$597$	21.9905	0.900011
$598$	0	0
$599$	−10.3797	−0.424103	−0.212051	−	0.977259i	$-0.568014\pi$
$599$	−10.3797	−0.424103	−0.212051	+	0.977259i	$0.568014\pi$
$600$	0	0
$601$	−7.29591	−0.297606	−0.148803	−	0.988867i	$-0.547542\pi$
$601$	−7.29591	−0.297606	−0.148803	+	0.988867i	$0.547542\pi$
$602$	0	0
$603$	−7.82295	−0.318575
$604$	0	0
$605$	8.51628	0.346236
$606$	0	0
$607$	39.1147	1.58762	0.793809	−	0.608167i	$-0.208095\pi$
$607$	39.1147	1.58762	0.793809	+	0.608167i	$0.208095\pi$
$608$	0	0
$609$	−3.09327	−0.125346
$610$	0	0
$611$	−6.00000	−0.242734
$612$	0	0
$613$	40.6418	1.64151	0.820753	−	0.571283i	$-0.193554\pi$
$613$	40.6418	1.64151	0.820753	+	0.571283i	$0.193554\pi$
$614$	0	0
$615$	−3.74329	−0.150944
$616$	0	0
$617$	−16.7648	−0.674924	−0.337462	−	0.941339i	$-0.609568\pi$
$617$	−16.7648	−0.674924	−0.337462	+	0.941339i	$0.609568\pi$
$618$	0	0
$619$	−14.0797	−0.565909	−0.282955	−	0.959133i	$-0.591315\pi$
$619$	−14.0797	−0.565909	−0.282955	+	0.959133i	$0.591315\pi$
$620$	0	0
$621$	−0.694593	−0.0278730
$622$	0	0
$623$	−0.251334	−0.0100695
$624$	0	0
$625$	17.9959	0.719835
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−48.9959	−1.95359
$630$	0	0
$631$	20.8972	0.831906	0.415953	−	0.909386i	$-0.363448\pi$
$631$	20.8972	0.831906	0.415953	+	0.909386i	$0.363448\pi$
$632$	0	0
$633$	15.9513	0.634007
$634$	0	0
$635$	6.52166	0.258804
$636$	0	0
$637$	−6.90673	−0.273654
$638$	0	0
$639$	10.9067	0.431463
$640$	0	0
$641$	3.38919	0.133865	0.0669324	−	0.997758i	$-0.478679\pi$
$641$	3.38919	0.133865	0.0669324	+	0.997758i	$0.478679\pi$
$642$	0	0
$643$	−17.6554	−0.696261	−0.348130	−	0.937446i	$-0.613183\pi$
$643$	−17.6554	−0.696261	−0.348130	+	0.937446i	$0.613183\pi$
$644$	0	0
$645$	−2.56624	−0.101045
$646$	0	0
$647$	−16.6500	−0.654580	−0.327290	−	0.944924i	$-0.606135\pi$
$647$	−16.6500	−0.654580	−0.327290	+	0.944924i	$0.606135\pi$
$648$	0	0
$649$	−19.6959	−0.773130
$650$	0	0
$651$	0.556742	0.0218204
$652$	0	0
$653$	−35.5877	−1.39265	−0.696327	−	0.717724i	$-0.745184\pi$
$653$	−35.5877	−1.39265	−0.696327	+	0.717724i	$0.745184\pi$
$654$	0	0
$655$	−5.64590	−0.220603
$656$	0	0
$657$	−4.38919	−0.171238
$658$	0	0
$659$	40.9121	1.59371	0.796855	−	0.604171i	$-0.206496\pi$
$659$	40.9121	1.59371	0.796855	+	0.604171i	$0.206496\pi$
$660$	0	0
$661$	−1.29179	−0.0502449	−0.0251225	−	0.999684i	$-0.507998\pi$
$661$	−1.29179	−0.0502449	−0.0251225	+	0.999684i	$0.507998\pi$
$662$	0	0
$663$	7.51754	0.291957
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.03508	0.272399
$668$	0	0
$669$	26.5039	1.02470
$670$	0	0
$671$	1.87702	0.0724616
$672$	0	0
$673$	7.99588	0.308219	0.154109	−	0.988054i	$-0.450749\pi$
$673$	7.99588	0.308219	0.154109	+	0.988054i	$0.450749\pi$
$674$	0	0
$675$	4.51754	0.173880
$676$	0	0
$677$	46.7701	1.79752	0.898761	−	0.438439i	$-0.144468\pi$
$677$	46.7701	1.79752	0.898761	+	0.438439i	$0.144468\pi$
$678$	0	0
$679$	3.33099	0.127832
$680$	0	0
$681$	−2.65539	−0.101755
$682$	0	0
$683$	−5.69047	−0.217740	−0.108870	−	0.994056i	$-0.534723\pi$
$683$	−5.69047	−0.217740	−0.108870	+	0.994056i	$0.534723\pi$
$684$	0	0
$685$	−3.71606	−0.141983
$686$	0	0
$687$	−21.2567	−0.810994
$688$	0	0
$689$	5.43376	0.207010
$690$	0	0
$691$	−28.8485	−1.09745	−0.548725	−	0.836003i	$-0.684887\pi$
$691$	−28.8485	−1.09745	−0.548725	+	0.836003i	$0.684887\pi$
$692$	0	0
$693$	1.47296	0.0559533
$694$	0	0
$695$	9.69047	0.367581
$696$	0	0
$697$	−40.5134	−1.53456
$698$	0	0
$699$	−13.5175	−0.511280
$700$	0	0
$701$	−45.5431	−1.72014	−0.860070	−	0.510176i	$-0.829580\pi$
$701$	−45.5431	−1.72014	−0.860070	+	0.510176i	$0.829580\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4.16756	0.156959
$706$	0	0
$707$	2.90673	0.109319
$708$	0	0
$709$	−30.2026	−1.13428	−0.567142	−	0.823620i	$-0.691951\pi$
$709$	−30.2026	−1.13428	−0.567142	+	0.823620i	$0.691951\pi$
$710$	0	0
$711$	14.4338	0.541308
$712$	0	0
$713$	−1.26621	−0.0474198
$714$	0	0
$715$	3.34998	0.125282
$716$	0	0
$717$	24.4688	0.913806
$718$	0	0
$719$	−36.1539	−1.34831	−0.674157	−	0.738588i	$-0.735493\pi$
$719$	−36.1539	−1.34831	−0.674157	+	0.738588i	$0.735493\pi$
$720$	0	0
$721$	−0.704088	−0.0262216
$722$	0	0
$723$	−11.8675	−0.441358
$724$	0	0
$725$	−45.7553	−1.69931
$726$	0	0
$727$	−7.30129	−0.270790	−0.135395	−	0.990792i	$-0.543230\pi$
$727$	−7.30129	−0.270790	−0.135395	+	0.990792i	$0.543230\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.7743	−1.02727
$732$	0	0
$733$	51.4986	1.90214	0.951071	−	0.308972i	$-0.0999849\pi$
$733$	51.4986	1.90214	0.951071	+	0.308972i	$0.0999849\pi$
$734$	0	0
$735$	4.79736	0.176953
$736$	0	0
$737$	−37.7297	−1.38979
$738$	0	0
$739$	32.6905	1.20254	0.601269	−	0.799046i	$-0.294662\pi$
$739$	32.6905	1.20254	0.601269	+	0.799046i	$0.294662\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.5931	−1.67265	−0.836324	−	0.548236i	$-0.815300\pi$
$743$	−45.5931	−1.67265	−0.836324	+	0.548236i	$0.815300\pi$
$744$	0	0
$745$	7.42839	0.272155
$746$	0	0
$747$	0.739170	0.0270448
$748$	0	0
$749$	2.90673	0.106209
$750$	0	0
$751$	−18.3054	−0.667974	−0.333987	−	0.942578i	$-0.608394\pi$
$751$	−18.3054	−0.667974	−0.333987	+	0.942578i	$0.608394\pi$
$752$	0	0
$753$	−7.38919	−0.269277
$754$	0	0
$755$	−12.2567	−0.446067
$756$	0	0
$757$	−31.4243	−1.14213	−0.571067	−	0.820903i	$-0.693470\pi$
$757$	−31.4243	−1.14213	−0.571067	+	0.820903i	$0.693470\pi$
$758$	0	0
$759$	−3.34998	−0.121597
$760$	0	0
$761$	38.5580	1.39773	0.698863	−	0.715255i	$-0.253690\pi$
$761$	38.5580	1.39773	0.698863	+	0.715255i	$0.253690\pi$
$762$	0	0
$763$	4.12836	0.149457
$764$	0	0
$765$	−5.22163	−0.188788
$766$	0	0
$767$	−4.08378	−0.147457
$768$	0	0
$769$	14.7433	0.531657	0.265828	−	0.964020i	$-0.414355\pi$
$769$	14.7433	0.531657	0.265828	+	0.964020i	$0.414355\pi$
$770$	0	0
$771$	−5.34461	−0.192481
$772$	0	0
$773$	−3.90261	−0.140367	−0.0701835	−	0.997534i	$-0.522358\pi$
$773$	−3.90261	−0.140367	−0.0701835	+	0.997534i	$0.522358\pi$
$774$	0	0
$775$	8.23524	0.295819
$776$	0	0
$777$	−1.99050	−0.0714090
$778$	0	0
$779$	0	0
$780$	0	0
$781$	52.6026	1.88227
$782$	0	0
$783$	−10.1284	−0.361958
$784$	0	0
$785$	11.0487	0.394345
$786$	0	0
$787$	−10.7879	−0.384546	−0.192273	−	0.981341i	$-0.561586\pi$
$787$	−10.7879	−0.384546	−0.192273	+	0.981341i	$0.561586\pi$
$788$	0	0
$789$	24.1284	0.858992
$790$	0	0
$791$	−6.43788	−0.228905
$792$	0	0
$793$	0.389185	0.0138204
$794$	0	0
$795$	−3.77425	−0.133859
$796$	0	0
$797$	−48.5526	−1.71982	−0.859911	−	0.510444i	$-0.829481\pi$
$797$	−48.5526	−1.71982	−0.859911	+	0.510444i	$0.829481\pi$
$798$	0	0
$799$	45.1052	1.59571
$800$	0	0
$801$	−0.822948	−0.0290774
$802$	0	0
$803$	−21.1688	−0.747031
$804$	0	0
$805$	0.147347	0.00519328
$806$	0	0
$807$	3.60132	0.126772
$808$	0	0
$809$	−35.8972	−1.26208	−0.631040	−	0.775751i	$-0.717372\pi$
$809$	−35.8972	−1.26208	−0.631040	+	0.775751i	$0.717372\pi$
$810$	0	0
$811$	−6.35410	−0.223123	−0.111561	−	0.993758i	$-0.535585\pi$
$811$	−6.35410	−0.223123	−0.111561	+	0.993758i	$0.535585\pi$
$812$	0	0
$813$	−18.7784	−0.658586
$814$	0	0
$815$	−1.51216	−0.0529688
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.305407	0.0106718
$820$	0	0
$821$	−13.8716	−0.484124	−0.242062	−	0.970261i	$-0.577824\pi$
$821$	−13.8716	−0.484124	−0.242062	+	0.970261i	$0.577824\pi$
$822$	0	0
$823$	5.90261	0.205752	0.102876	−	0.994694i	$-0.467196\pi$
$823$	5.90261	0.205752	0.102876	+	0.994694i	$0.467196\pi$
$824$	0	0
$825$	21.7879	0.758556
$826$	0	0
$827$	23.9418	0.832538	0.416269	−	0.909241i	$-0.363337\pi$
$827$	23.9418	0.832538	0.416269	+	0.909241i	$0.363337\pi$
$828$	0	0
$829$	20.4593	0.710583	0.355291	−	0.934756i	$-0.384382\pi$
$829$	20.4593	0.710583	0.355291	+	0.934756i	$0.384382\pi$
$830$	0	0
$831$	−20.1284	−0.698245
$832$	0	0
$833$	51.9216	1.79898
$834$	0	0
$835$	−7.60670	−0.263241
$836$	0	0
$837$	1.82295	0.0630103
$838$	0	0
$839$	−28.7256	−0.991716	−0.495858	−	0.868404i	$-0.665146\pi$
$839$	−28.7256	−0.991716	−0.495858	+	0.868404i	$0.665146\pi$
$840$	0	0
$841$	73.5836	2.53737
$842$	0	0
$843$	14.3405	0.493913
$844$	0	0
$845$	−8.33511	−0.286737
$846$	0	0
$847$	3.74455	0.128664
$848$	0	0
$849$	−6.77837	−0.232633
$850$	0	0
$851$	4.52704	0.155185
$852$	0	0
$853$	−2.44738	−0.0837966	−0.0418983	−	0.999122i	$-0.513341\pi$
$853$	−2.44738	−0.0837966	−0.0418983	+	0.999122i	$0.513341\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.0702	−0.685584	−0.342792	−	0.939411i	$-0.611373\pi$
$857$	−20.0702	−0.685584	−0.342792	+	0.939411i	$0.611373\pi$
$858$	0	0
$859$	45.0256	1.53625	0.768127	−	0.640298i	$-0.221189\pi$
$859$	45.0256	1.53625	0.768127	+	0.640298i	$0.221189\pi$
$860$	0	0
$861$	−1.64590	−0.0560920
$862$	0	0
$863$	−18.5918	−0.632873	−0.316437	−	0.948614i	$-0.602486\pi$
$863$	−18.5918	−0.632873	−0.316437	+	0.948614i	$0.602486\pi$
$864$	0	0
$865$	−7.03508	−0.239200
$866$	0	0
$867$	−39.5134	−1.34195
$868$	0	0
$869$	69.6133	2.36147
$870$	0	0
$871$	−7.82295	−0.265070
$872$	0	0
$873$	10.9067	0.369137
$874$	0	0
$875$	−2.01899	−0.0682544
$876$	0	0
$877$	7.73917	0.261333	0.130667	−	0.991426i	$-0.458288\pi$
$877$	7.73917	0.261333	0.130667	+	0.991426i	$0.458288\pi$
$878$	0	0
$879$	−16.7784	−0.565920
$880$	0	0
$881$	15.0487	0.507003	0.253502	−	0.967335i	$-0.418418\pi$
$881$	15.0487	0.507003	0.253502	+	0.967335i	$0.418418\pi$
$882$	0	0
$883$	−0.562118	−0.0189168	−0.00945839	−	0.999955i	$-0.503011\pi$
$883$	−0.562118	−0.0189168	−0.00945839	+	0.999955i	$0.503011\pi$
$884$	0	0
$885$	2.83656	0.0953500
$886$	0	0
$887$	2.86215	0.0961016	0.0480508	−	0.998845i	$-0.484699\pi$
$887$	2.86215	0.0961016	0.0480508	+	0.998845i	$0.484699\pi$
$888$	0	0
$889$	2.86753	0.0961737
$890$	0	0
$891$	4.82295	0.161575
$892$	0	0
$893$	0	0
$894$	0	0
$895$	6.03920	0.201868
$896$	0	0
$897$	−0.694593	−0.0231918
$898$	0	0
$899$	−18.4635	−0.615791
$900$	0	0
$901$	−40.8485	−1.36086
$902$	0	0
$903$	−1.12836	−0.0375493
$904$	0	0
$905$	5.46759	0.181749
$906$	0	0
$907$	−34.0702	−1.13128	−0.565641	−	0.824652i	$-0.691371\pi$
$907$	−34.0702	−1.13128	−0.565641	+	0.824652i	$0.691371\pi$
$908$	0	0
$909$	9.51754	0.315677
$910$	0	0
$911$	−5.90261	−0.195562	−0.0977811	−	0.995208i	$-0.531174\pi$
$911$	−5.90261	−0.195562	−0.0977811	+	0.995208i	$0.531174\pi$
$912$	0	0
$913$	3.56498	0.117984
$914$	0	0
$915$	−0.270325	−0.00893668
$916$	0	0
$917$	−2.48246	−0.0819780
$918$	0	0
$919$	−0.394562	−0.0130154	−0.00650770	−	0.999979i	$-0.502071\pi$
$919$	−0.394562	−0.0130154	−0.00650770	+	0.999979i	$0.502071\pi$
$920$	0	0
$921$	17.1634	0.565554
$922$	0	0
$923$	10.9067	0.358999
$924$	0	0
$925$	−29.4433	−0.968088
$926$	0	0
$927$	−2.30541	−0.0757195
$928$	0	0
$929$	−33.7998	−1.10894	−0.554468	−	0.832205i	$-0.687078\pi$
$929$	−33.7998	−1.10894	−0.554468	+	0.832205i	$0.687078\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.1729	0.398524
$934$	0	0
$935$	−25.1836	−0.823593
$936$	0	0
$937$	41.7202	1.36294	0.681469	−	0.731847i	$-0.261341\pi$
$937$	41.7202	1.36294	0.681469	+	0.731847i	$0.261341\pi$
$938$	0	0
$939$	−3.70409	−0.120878
$940$	0	0
$941$	9.47834	0.308985	0.154493	−	0.987994i	$-0.450626\pi$
$941$	9.47834	0.308985	0.154493	+	0.987994i	$0.450626\pi$
$942$	0	0
$943$	3.74329	0.121898
$944$	0	0
$945$	−0.212134	−0.00690071
$946$	0	0
$947$	7.42015	0.241122	0.120561	−	0.992706i	$-0.461531\pi$
$947$	7.42015	0.241122	0.120561	+	0.992706i	$0.461531\pi$
$948$	0	0
$949$	−4.38919	−0.142479
$950$	0	0
$951$	−23.5621	−0.764054
$952$	0	0
$953$	1.81883	0.0589177	0.0294588	−	0.999566i	$-0.490622\pi$
$953$	1.81883	0.0589177	0.0294588	+	0.999566i	$0.490622\pi$
$954$	0	0
$955$	9.62569	0.311480
$956$	0	0
$957$	−48.8485	−1.57905
$958$	0	0
$959$	−1.63392	−0.0527622
$960$	0	0
$961$	−27.6769	−0.892802
$962$	0	0
$963$	9.51754	0.306698
$964$	0	0
$965$	1.45397	0.0468050
$966$	0	0
$967$	5.90135	0.189775	0.0948873	−	0.995488i	$-0.469751\pi$
$967$	5.90135	0.189775	0.0948873	+	0.995488i	$0.469751\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45.3702	1.45600	0.727999	−	0.685578i	$-0.240450\pi$
$971$	45.3702	1.45600	0.727999	+	0.685578i	$0.240450\pi$
$972$	0	0
$973$	4.26083	0.136596
$974$	0	0
$975$	4.51754	0.144677
$976$	0	0
$977$	−36.2377	−1.15935	−0.579674	−	0.814849i	$-0.696820\pi$
$977$	−36.2377	−1.15935	−0.579674	+	0.814849i	$0.696820\pi$
$978$	0	0
$979$	−3.96904	−0.126851
$980$	0	0
$981$	13.5175	0.431582
$982$	0	0
$983$	−8.65539	−0.276064	−0.138032	−	0.990428i	$-0.544078\pi$
$983$	−8.65539	−0.276064	−0.138032	+	0.990428i	$0.544078\pi$
$984$	0	0
$985$	9.75526	0.310828
$986$	0	0
$987$	1.83244	0.0583273
$988$	0	0
$989$	2.56624	0.0816016
$990$	0	0
$991$	59.8040	1.89974	0.949868	−	0.312652i	$-0.101217\pi$
$991$	59.8040	1.89974	0.949868	+	0.312652i	$0.101217\pi$
$992$	0	0
$993$	4.30541	0.136628
$994$	0	0
$995$	−15.2744	−0.484232
$996$	0	0
$997$	−15.0500	−0.476637	−0.238318	−	0.971187i	$-0.576596\pi$
$997$	−15.0500	−0.476637	−0.238318	+	0.971187i	$0.576596\pi$
$998$	0	0
$999$	−6.51754	−0.206206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8664.2.a.x.1.2		3
19.7	even	3		456.2.q.f.49.2	✓	6
19.11	even	3		456.2.q.f.121.2	yes	6
19.18	odd	2		8664.2.a.z.1.2		3
57.11	odd	6		1368.2.s.j.577.2		6
57.26	odd	6		1368.2.s.j.505.2		6
76.7	odd	6		912.2.q.k.49.2		6
76.11	odd	6		912.2.q.k.577.2		6
228.11	even	6		2736.2.s.x.577.2		6
228.83	even	6		2736.2.s.x.1873.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.q.f.49.2	✓	6	19.7	even	3
456.2.q.f.121.2	yes	6	19.11	even	3
912.2.q.k.49.2		6	76.7	odd	6
912.2.q.k.577.2		6	76.11	odd	6
1368.2.s.j.505.2		6	57.26	odd	6
1368.2.s.j.577.2		6	57.11	odd	6
2736.2.s.x.577.2		6	228.11	even	6
2736.2.s.x.1873.2		6	228.83	even	6
8664.2.a.x.1.2		3	1.1	even	1	trivial
8664.2.a.z.1.2		3	19.18	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8664.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.694593 q^{5} +0.305407 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.694593 q^{5} +0.305407 q^{7} +1.00000 q^{9} +4.82295 q^{11} +1.00000 q^{13} -0.694593 q^{15} -7.51754 q^{17} -0.305407 q^{21} +0.694593 q^{23} -4.51754 q^{25} -1.00000 q^{27} +10.1284 q^{29} -1.82295 q^{31} -4.82295 q^{33} +0.212134 q^{35} +6.51754 q^{37} -1.00000 q^{39} +5.38919 q^{41} +3.69459 q^{43} +0.694593 q^{45} -6.00000 q^{47} -6.90673 q^{49} +7.51754 q^{51} +5.43376 q^{53} +3.34998 q^{55} -4.08378 q^{59} +0.389185 q^{61} +0.305407 q^{63} +0.694593 q^{65} -7.82295 q^{67} -0.694593 q^{69} +10.9067 q^{71} -4.38919 q^{73} +4.51754 q^{75} +1.47296 q^{77} +14.4338 q^{79} +1.00000 q^{81} +0.739170 q^{83} -5.22163 q^{85} -10.1284 q^{87} -0.822948 q^{89} +0.305407 q^{91} +1.82295 q^{93} +10.9067 q^{97} +4.82295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 3 q^{13} - 3 q^{21} + 9 q^{25} - 3 q^{27} + 12 q^{29} + 15 q^{31} + 6 q^{33} - 24 q^{35} - 3 q^{37} - 3 q^{39} + 12 q^{41} + 9 q^{43} - 18 q^{47} + 6 q^{49} - 6 q^{59} - 3 q^{61} + 3 q^{63} - 3 q^{67} + 6 q^{71} - 9 q^{73} - 9 q^{75} - 6 q^{77} + 27 q^{79} + 3 q^{81} - 12 q^{83} - 24 q^{85} - 12 q^{87} + 18 q^{89} + 3 q^{91} - 15 q^{93} + 6 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 - 6 * q^11 + 3 * q^13 - 3 * q^21 + 9 * q^25 - 3 * q^27 + 12 * q^29 + 15 * q^31 + 6 * q^33 - 24 * q^35 - 3 * q^37 - 3 * q^39 + 12 * q^41 + 9 * q^43 - 18 * q^47 + 6 * q^49 - 6 * q^59 - 3 * q^61 + 3 * q^63 - 3 * q^67 + 6 * q^71 - 9 * q^73 - 9 * q^75 - 6 * q^77 + 27 * q^79 + 3 * q^81 - 12 * q^83 - 24 * q^85 - 12 * q^87 + 18 * q^89 + 3 * q^91 - 15 * q^93 + 6 * q^97 - 6 * q^99

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