LMFDB - Embedded newform 6864.2.a.bz.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.704624$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.97216 q^{5} -3.17328 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.97216 q^{5} -3.17328 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.97216 q^{15} -7.80589 q^{17} -7.17328 q^{19} -3.17328 q^{21} -3.36179 q^{23} +10.7780 q^{25} +1.00000 q^{27} -7.61738 q^{29} -3.15366 q^{31} +1.00000 q^{33} -12.6048 q^{35} +2.93731 q^{37} +1.00000 q^{39} +1.64522 q^{41} +4.15245 q^{43} +3.97216 q^{45} -0.660451 q^{47} +3.06970 q^{49} -7.80589 q^{51} +0.0696997 q^{53} +3.97216 q^{55} -7.17328 q^{57} -8.88820 q^{59} -5.47895 q^{61} -3.17328 q^{63} +3.97216 q^{65} -5.50022 q^{67} -3.36179 q^{69} -2.11881 q^{71} -14.7154 q^{73} +10.7780 q^{75} -3.17328 q^{77} -14.4990 q^{79} +1.00000 q^{81} +11.4724 q^{83} -31.0062 q^{85} -7.61738 q^{87} +0.965150 q^{89} -3.17328 q^{91} -3.15366 q^{93} -28.4934 q^{95} +7.31433 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 18 q^{19} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 10 q^{29} - 12 q^{31} + 4 q^{33} - 22 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 28 q^{43} - 6 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 18 q^{57} - 16 q^{59} - 10 q^{61} - 2 q^{63} - 10 q^{71} - 6 q^{73} + 4 q^{75} - 2 q^{77} + 8 q^{79} + 4 q^{81} + 8 q^{83} - 18 q^{85} - 10 q^{87} + 6 q^{89} - 2 q^{91} - 12 q^{93} - 22 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 - 8 * q^17 - 18 * q^19 - 2 * q^21 + 4 * q^25 + 4 * q^27 - 10 * q^29 - 12 * q^31 + 4 * q^33 - 22 * q^35 - 2 * q^37 + 4 * q^39 + 2 * q^41 - 28 * q^43 - 6 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 18 * q^57 - 16 * q^59 - 10 * q^61 - 2 * q^63 - 10 * q^71 - 6 * q^73 + 4 * q^75 - 2 * q^77 + 8 * q^79 + 4 * q^81 + 8 * q^83 - 18 * q^85 - 10 * q^87 + 6 * q^89 - 2 * q^91 - 12 * q^93 - 22 * 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
$5$	3.97216	1.77640	0.888202	−	0.459454i	$-0.151955\pi$
$5$	3.97216	1.77640	0.888202	+	0.459454i	$0.151955\pi$
$6$	0	0
$7$	−3.17328	−1.19939	−0.599693	−	0.800230i	$-0.704711\pi$
$7$	−3.17328	−1.19939	−0.599693	+	0.800230i	$0.704711\pi$
$8$	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
$15$	3.97216	1.02561
$16$	0	0
$17$	−7.80589	−1.89321	−0.946603	−	0.322401i	$-0.895510\pi$
$17$	−7.80589	−1.89321	−0.946603	+	0.322401i	$0.895510\pi$
$18$	0	0
$19$	−7.17328	−1.64566	−0.822832	−	0.568285i	$-0.807607\pi$
$19$	−7.17328	−1.64566	−0.822832	+	0.568285i	$0.807607\pi$
$20$	0	0
$21$	−3.17328	−0.692466
$22$	0	0
$23$	−3.36179	−0.700982	−0.350491	−	0.936566i	$-0.613985\pi$
$23$	−3.36179	−0.700982	−0.350491	+	0.936566i	$0.613985\pi$
$24$	0	0
$25$	10.7780	2.15561
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.61738	−1.41451	−0.707256	−	0.706958i	$-0.750067\pi$
$29$	−7.61738	−1.41451	−0.707256	+	0.706958i	$0.750067\pi$
$30$	0	0
$31$	−3.15366	−0.566414	−0.283207	−	0.959059i	$-0.591398\pi$
$31$	−3.15366	−0.566414	−0.283207	+	0.959059i	$0.591398\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−12.6048	−2.13059
$36$	0	0
$37$	2.93731	0.482891	0.241445	−	0.970414i	$-0.422379\pi$
$37$	2.93731	0.482891	0.241445	+	0.970414i	$0.422379\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	1.64522	0.256940	0.128470	−	0.991713i	$-0.458993\pi$
$41$	1.64522	0.256940	0.128470	+	0.991713i	$0.458993\pi$
$42$	0	0
$43$	4.15245	0.633242	0.316621	−	0.948552i	$-0.397452\pi$
$43$	4.15245	0.633242	0.316621	+	0.948552i	$0.397452\pi$
$44$	0	0
$45$	3.97216	0.592135
$46$	0	0
$47$	−0.660451	−0.0963367	−0.0481683	−	0.998839i	$-0.515338\pi$
$47$	−0.660451	−0.0963367	−0.0481683	+	0.998839i	$0.515338\pi$
$48$	0	0
$49$	3.06970	0.438529
$50$	0	0
$51$	−7.80589	−1.09304
$52$	0	0
$53$	0.0696997	0.00957399	0.00478700	−	0.999989i	$-0.498476\pi$
$53$	0.0696997	0.00957399	0.00478700	+	0.999989i	$0.498476\pi$
$54$	0	0
$55$	3.97216	0.535606
$56$	0	0
$57$	−7.17328	−0.950124
$58$	0	0
$59$	−8.88820	−1.15714	−0.578572	−	0.815631i	$-0.696390\pi$
$59$	−8.88820	−1.15714	−0.578572	+	0.815631i	$0.696390\pi$
$60$	0	0
$61$	−5.47895	−0.701507	−0.350754	−	0.936468i	$-0.614075\pi$
$61$	−5.47895	−0.701507	−0.350754	+	0.936468i	$0.614075\pi$
$62$	0	0
$63$	−3.17328	−0.399796
$64$	0	0
$65$	3.97216	0.492686
$66$	0	0
$67$	−5.50022	−0.671959	−0.335979	−	0.941869i	$-0.609067\pi$
$67$	−5.50022	−0.671959	−0.335979	+	0.941869i	$0.609067\pi$
$68$	0	0
$69$	−3.36179	−0.404712
$70$	0	0
$71$	−2.11881	−0.251457	−0.125728	−	0.992065i	$-0.540127\pi$
$71$	−2.11881	−0.251457	−0.125728	+	0.992065i	$0.540127\pi$
$72$	0	0
$73$	−14.7154	−1.72230	−0.861151	−	0.508349i	$-0.830256\pi$
$73$	−14.7154	−1.72230	−0.861151	+	0.508349i	$0.830256\pi$
$74$	0	0
$75$	10.7780	1.24454
$76$	0	0
$77$	−3.17328	−0.361629
$78$	0	0
$79$	−14.4990	−1.63127	−0.815633	−	0.578570i	$-0.803611\pi$
$79$	−14.4990	−1.63127	−0.815633	+	0.578570i	$0.803611\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.4724	1.25926	0.629629	−	0.776896i	$-0.283207\pi$
$83$	11.4724	1.25926	0.629629	+	0.776896i	$0.283207\pi$
$84$	0	0
$85$	−31.0062	−3.36310
$86$	0	0
$87$	−7.61738	−0.816669
$88$	0	0
$89$	0.965150	0.102306	0.0511529	−	0.998691i	$-0.483710\pi$
$89$	0.965150	0.102306	0.0511529	+	0.998691i	$0.483710\pi$
$90$	0	0
$91$	−3.17328	−0.332650
$92$	0	0
$93$	−3.15366	−0.327019
$94$	0	0
$95$	−28.4934	−2.92336
$96$	0	0
$97$	7.31433	0.742658	0.371329	−	0.928501i	$-0.378902\pi$
$97$	7.31433	0.742658	0.371329	+	0.928501i	$0.378902\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	17.3480	1.72619	0.863094	−	0.505044i	$-0.168524\pi$
$101$	17.3480	1.72619	0.863094	+	0.505044i	$0.168524\pi$
$102$	0	0
$103$	−4.47194	−0.440633	−0.220317	−	0.975428i	$-0.570709\pi$
$103$	−4.47194	−0.440633	−0.220317	+	0.975428i	$0.570709\pi$
$104$	0	0
$105$	−12.6048	−1.23010
$106$	0	0
$107$	12.0837	1.16818	0.584089	−	0.811690i	$-0.301452\pi$
$107$	12.0837	1.16818	0.584089	+	0.811690i	$0.301452\pi$
$108$	0	0
$109$	−4.36880	−0.418455	−0.209228	−	0.977867i	$-0.567095\pi$
$109$	−4.36880	−0.418455	−0.209228	+	0.977867i	$0.567095\pi$
$110$	0	0
$111$	2.93731	0.278797
$112$	0	0
$113$	−8.77926	−0.825884	−0.412942	−	0.910757i	$-0.635499\pi$
$113$	−8.77926	−0.825884	−0.412942	+	0.910757i	$0.635499\pi$
$114$	0	0
$115$	−13.3536	−1.24523
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	24.7703	2.27069
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.64522	0.148344
$124$	0	0
$125$	22.9513	2.05283
$126$	0	0
$127$	−2.42186	−0.214905	−0.107453	−	0.994210i	$-0.534269\pi$
$127$	−2.42186	−0.214905	−0.107453	+	0.994210i	$0.534269\pi$
$128$	0	0
$129$	4.15245	0.365603
$130$	0	0
$131$	8.36058	0.730467	0.365233	−	0.930916i	$-0.380989\pi$
$131$	8.36058	0.730467	0.365233	+	0.930916i	$0.380989\pi$
$132$	0	0
$133$	22.7628	1.97379
$134$	0	0
$135$	3.97216	0.341869
$136$	0	0
$137$	−9.85335	−0.841828	−0.420914	−	0.907100i	$-0.638291\pi$
$137$	−9.85335	−0.841828	−0.420914	+	0.907100i	$0.638291\pi$
$138$	0	0
$139$	2.74320	0.232675	0.116338	−	0.993210i	$-0.462885\pi$
$139$	2.74320	0.232675	0.116338	+	0.993210i	$0.462885\pi$
$140$	0	0
$141$	−0.660451	−0.0556200
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−30.2574	−2.51274
$146$	0	0
$147$	3.06970	0.253185
$148$	0	0
$149$	3.99178	0.327019	0.163510	−	0.986542i	$-0.447719\pi$
$149$	3.99178	0.327019	0.163510	+	0.986542i	$0.447719\pi$
$150$	0	0
$151$	−9.18730	−0.747652	−0.373826	−	0.927499i	$-0.621954\pi$
$151$	−9.18730	−0.747652	−0.373826	+	0.927499i	$0.621954\pi$
$152$	0	0
$153$	−7.80589	−0.631069
$154$	0	0
$155$	−12.5268	−1.00618
$156$	0	0
$157$	−18.8482	−1.50425	−0.752125	−	0.659021i	$-0.770971\pi$
$157$	−18.8482	−1.50425	−0.752125	+	0.659021i	$0.770971\pi$
$158$	0	0
$159$	0.0696997	0.00552755
$160$	0	0
$161$	10.6679	0.840748
$162$	0	0
$163$	15.7027	1.22993	0.614967	−	0.788553i	$-0.289169\pi$
$163$	15.7027	1.22993	0.614967	+	0.788553i	$0.289169\pi$
$164$	0	0
$165$	3.97216	0.309232
$166$	0	0
$167$	0.709563	0.0549076	0.0274538	−	0.999623i	$-0.491260\pi$
$167$	0.709563	0.0549076	0.0274538	+	0.999623i	$0.491260\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.17328	−0.548554
$172$	0	0
$173$	6.86201	0.521709	0.260854	−	0.965378i	$-0.415996\pi$
$173$	6.86201	0.521709	0.260854	+	0.965378i	$0.415996\pi$
$174$	0	0
$175$	−34.2018	−2.58541
$176$	0	0
$177$	−8.88820	−0.668078
$178$	0	0
$179$	−10.1411	−0.757978	−0.378989	−	0.925401i	$-0.623728\pi$
$179$	−10.1411	−0.757978	−0.378989	+	0.925401i	$0.623728\pi$
$180$	0	0
$181$	3.76403	0.279778	0.139889	−	0.990167i	$-0.455325\pi$
$181$	3.76403	0.279778	0.139889	+	0.990167i	$0.455325\pi$
$182$	0	0
$183$	−5.47895	−0.405016
$184$	0	0
$185$	11.6675	0.857809
$186$	0	0
$187$	−7.80589	−0.570823
$188$	0	0
$189$	−3.17328	−0.230822
$190$	0	0
$191$	1.73716	0.125696	0.0628482	−	0.998023i	$-0.479982\pi$
$191$	1.73716	0.125696	0.0628482	+	0.998023i	$0.479982\pi$
$192$	0	0
$193$	−17.2995	−1.24525	−0.622624	−	0.782521i	$-0.713933\pi$
$193$	−17.2995	−1.24525	−0.622624	+	0.782521i	$0.713933\pi$
$194$	0	0
$195$	3.97216	0.284452
$196$	0	0
$197$	21.0062	1.49663	0.748316	−	0.663342i	$-0.230863\pi$
$197$	21.0062	1.49663	0.748316	+	0.663342i	$0.230863\pi$
$198$	0	0
$199$	23.1791	1.64312	0.821560	−	0.570121i	$-0.193104\pi$
$199$	23.1791	1.64312	0.821560	+	0.570121i	$0.193104\pi$
$200$	0	0
$201$	−5.50022	−0.387956
$202$	0	0
$203$	24.1721	1.69655
$204$	0	0
$205$	6.53507	0.456429
$206$	0	0
$207$	−3.36179	−0.233661
$208$	0	0
$209$	−7.17328	−0.496186
$210$	0	0
$211$	11.6248	0.800286	0.400143	−	0.916453i	$-0.368960\pi$
$211$	11.6248	0.800286	0.400143	+	0.916453i	$0.368960\pi$
$212$	0	0
$213$	−2.11881	−0.145179
$214$	0	0
$215$	16.4942	1.12489
$216$	0	0
$217$	10.0074	0.679350
$218$	0	0
$219$	−14.7154	−0.994372
$220$	0	0
$221$	−7.80589	−0.525081
$222$	0	0
$223$	−5.11200	−0.342325	−0.171162	−	0.985243i	$-0.554752\pi$
$223$	−5.11200	−0.342325	−0.171162	+	0.985243i	$0.554752\pi$
$224$	0	0
$225$	10.7780	0.718537
$226$	0	0
$227$	−15.2290	−1.01078	−0.505391	−	0.862891i	$-0.668652\pi$
$227$	−15.2290	−1.01078	−0.505391	+	0.862891i	$0.668652\pi$
$228$	0	0
$229$	24.4237	1.61396	0.806982	−	0.590576i	$-0.201099\pi$
$229$	24.4237	1.61396	0.806982	+	0.590576i	$0.201099\pi$
$230$	0	0
$231$	−3.17328	−0.208786
$232$	0	0
$233$	−27.4223	−1.79649	−0.898247	−	0.439491i	$-0.855159\pi$
$233$	−27.4223	−1.79649	−0.898247	+	0.439491i	$0.855159\pi$
$234$	0	0
$235$	−2.62342	−0.171133
$236$	0	0
$237$	−14.4990	−0.941812
$238$	0	0
$239$	−21.6036	−1.39742	−0.698709	−	0.715406i	$-0.746242\pi$
$239$	−21.6036	−1.39742	−0.698709	+	0.715406i	$0.746242\pi$
$240$	0	0
$241$	6.32134	0.407193	0.203597	−	0.979055i	$-0.434737\pi$
$241$	6.32134	0.407193	0.203597	+	0.979055i	$0.434737\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	12.1933	0.779004
$246$	0	0
$247$	−7.17328	−0.456425
$248$	0	0
$249$	11.4724	0.727032
$250$	0	0
$251$	9.48640	0.598776	0.299388	−	0.954131i	$-0.403217\pi$
$251$	9.48640	0.598776	0.299388	+	0.954131i	$0.403217\pi$
$252$	0	0
$253$	−3.36179	−0.211354
$254$	0	0
$255$	−31.0062	−1.94169
$256$	0	0
$257$	−11.6370	−0.725896	−0.362948	−	0.931809i	$-0.618230\pi$
$257$	−11.6370	−0.725896	−0.362948	+	0.931809i	$0.618230\pi$
$258$	0	0
$259$	−9.32090	−0.579173
$260$	0	0
$261$	−7.61738	−0.471504
$262$	0	0
$263$	9.95834	0.614057	0.307029	−	0.951700i	$-0.400665\pi$
$263$	9.95834	0.614057	0.307029	+	0.951700i	$0.400665\pi$
$264$	0	0
$265$	0.276858	0.0170073
$266$	0	0
$267$	0.965150	0.0590662
$268$	0	0
$269$	−16.4303	−1.00177	−0.500886	−	0.865513i	$-0.666992\pi$
$269$	−16.4303	−1.00177	−0.500886	+	0.865513i	$0.666992\pi$
$270$	0	0
$271$	29.1761	1.77232	0.886161	−	0.463378i	$-0.153363\pi$
$271$	29.1761	1.77232	0.886161	+	0.463378i	$0.153363\pi$
$272$	0	0
$273$	−3.17328	−0.192056
$274$	0	0
$275$	10.7780	0.649941
$276$	0	0
$277$	−4.20716	−0.252784	−0.126392	−	0.991980i	$-0.540340\pi$
$277$	−4.20716	−0.252784	−0.126392	+	0.991980i	$0.540340\pi$
$278$	0	0
$279$	−3.15366	−0.188805
$280$	0	0
$281$	−32.9113	−1.96332	−0.981662	−	0.190628i	$-0.938947\pi$
$281$	−32.9113	−1.96332	−0.981662	+	0.190628i	$0.938947\pi$
$282$	0	0
$283$	19.8690	1.18109	0.590545	−	0.807004i	$-0.298913\pi$
$283$	19.8690	1.18109	0.590545	+	0.807004i	$0.298913\pi$
$284$	0	0
$285$	−28.4934	−1.68780
$286$	0	0
$287$	−5.22074	−0.308170
$288$	0	0
$289$	43.9319	2.58423
$290$	0	0
$291$	7.31433	0.428774
$292$	0	0
$293$	7.00536	0.409257	0.204629	−	0.978840i	$-0.434401\pi$
$293$	7.00536	0.409257	0.204629	+	0.978840i	$0.434401\pi$
$294$	0	0
$295$	−35.3053	−2.05556
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.36179	−0.194417
$300$	0	0
$301$	−13.1769	−0.759502
$302$	0	0
$303$	17.3480	0.996615
$304$	0	0
$305$	−21.7633	−1.24616
$306$	0	0
$307$	8.67087	0.494873	0.247436	−	0.968904i	$-0.420412\pi$
$307$	8.67087	0.494873	0.247436	+	0.968904i	$0.420412\pi$
$308$	0	0
$309$	−4.47194	−0.254400
$310$	0	0
$311$	30.0689	1.70505	0.852526	−	0.522685i	$-0.175069\pi$
$311$	30.0689	1.70505	0.852526	+	0.522685i	$0.175069\pi$
$312$	0	0
$313$	−30.6662	−1.73336	−0.866679	−	0.498866i	$-0.833750\pi$
$313$	−30.6662	−1.73336	−0.866679	+	0.498866i	$0.833750\pi$
$314$	0	0
$315$	−12.6048	−0.710198
$316$	0	0
$317$	7.49559	0.420994	0.210497	−	0.977594i	$-0.432492\pi$
$317$	7.49559	0.420994	0.210497	+	0.977594i	$0.432492\pi$
$318$	0	0
$319$	−7.61738	−0.426491
$320$	0	0
$321$	12.0837	0.674447
$322$	0	0
$323$	55.9938	3.11558
$324$	0	0
$325$	10.7780	0.597859
$326$	0	0
$327$	−4.36880	−0.241595
$328$	0	0
$329$	2.09580	0.115545
$330$	0	0
$331$	10.8538	0.596578	0.298289	−	0.954476i	$-0.403584\pi$
$331$	10.8538	0.596578	0.298289	+	0.954476i	$0.403584\pi$
$332$	0	0
$333$	2.93731	0.160964
$334$	0	0
$335$	−21.8477	−1.19367
$336$	0	0
$337$	10.0837	0.549295	0.274648	−	0.961545i	$-0.411439\pi$
$337$	10.0837	0.549295	0.274648	+	0.961545i	$0.411439\pi$
$338$	0	0
$339$	−8.77926	−0.476824
$340$	0	0
$341$	−3.15366	−0.170780
$342$	0	0
$343$	12.4719	0.673421
$344$	0	0
$345$	−13.3536	−0.718932
$346$	0	0
$347$	0.104794	0.00562563	0.00281281	−	0.999996i	$-0.499105\pi$
$347$	0.104794	0.00562563	0.00281281	+	0.999996i	$0.499105\pi$
$348$	0	0
$349$	−12.0566	−0.645373	−0.322686	−	0.946506i	$-0.604586\pi$
$349$	−12.0566	−0.645373	−0.322686	+	0.946506i	$0.604586\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	6.16812	0.328296	0.164148	−	0.986436i	$-0.447513\pi$
$353$	6.16812	0.328296	0.164148	+	0.986436i	$0.447513\pi$
$354$	0	0
$355$	−8.41626	−0.446689
$356$	0	0
$357$	24.7703	1.31098
$358$	0	0
$359$	−8.67612	−0.457908	−0.228954	−	0.973437i	$-0.573531\pi$
$359$	−8.67612	−0.457908	−0.228954	+	0.973437i	$0.573531\pi$
$360$	0	0
$361$	32.4559	1.70821
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−58.4517	−3.05950
$366$	0	0
$367$	−19.9443	−1.04108	−0.520542	−	0.853836i	$-0.674270\pi$
$367$	−19.9443	−1.04108	−0.520542	+	0.853836i	$0.674270\pi$
$368$	0	0
$369$	1.64522	0.0856467
$370$	0	0
$371$	−0.221177	−0.0114829
$372$	0	0
$373$	−5.56730	−0.288264	−0.144132	−	0.989558i	$-0.546039\pi$
$373$	−5.56730	−0.288264	−0.144132	+	0.989558i	$0.546039\pi$
$374$	0	0
$375$	22.9513	1.18520
$376$	0	0
$377$	−7.61738	−0.392315
$378$	0	0
$379$	−8.64663	−0.444147	−0.222074	−	0.975030i	$-0.571283\pi$
$379$	−8.64663	−0.444147	−0.222074	+	0.975030i	$0.571283\pi$
$380$	0	0
$381$	−2.42186	−0.124075
$382$	0	0
$383$	−8.41626	−0.430051	−0.215025	−	0.976608i	$-0.568983\pi$
$383$	−8.41626	−0.430051	−0.215025	+	0.976608i	$0.568983\pi$
$384$	0	0
$385$	−12.6048	−0.642399
$386$	0	0
$387$	4.15245	0.211081
$388$	0	0
$389$	33.8330	1.71540	0.857699	−	0.514151i	$-0.171893\pi$
$389$	33.8330	1.71540	0.857699	+	0.514151i	$0.171893\pi$
$390$	0	0
$391$	26.2418	1.32710
$392$	0	0
$393$	8.36058	0.421735
$394$	0	0
$395$	−57.5924	−2.89779
$396$	0	0
$397$	−25.5660	−1.28312	−0.641560	−	0.767073i	$-0.721712\pi$
$397$	−25.5660	−1.28312	−0.641560	+	0.767073i	$0.721712\pi$
$398$	0	0
$399$	22.7628	1.13957
$400$	0	0
$401$	0.122757	0.00613020	0.00306510	−	0.999995i	$-0.499024\pi$
$401$	0.122757	0.00613020	0.00306510	+	0.999995i	$0.499024\pi$
$402$	0	0
$403$	−3.15366	−0.157095
$404$	0	0
$405$	3.97216	0.197378
$406$	0	0
$407$	2.93731	0.145597
$408$	0	0
$409$	−12.4637	−0.616291	−0.308146	−	0.951339i	$-0.599708\pi$
$409$	−12.4637	−0.616291	−0.308146	+	0.951339i	$0.599708\pi$
$410$	0	0
$411$	−9.85335	−0.486030
$412$	0	0
$413$	28.2047	1.38786
$414$	0	0
$415$	45.5701	2.23695
$416$	0	0
$417$	2.74320	0.134335
$418$	0	0
$419$	15.4324	0.753921	0.376960	−	0.926229i	$-0.376969\pi$
$419$	15.4324	0.753921	0.376960	+	0.926229i	$0.376969\pi$
$420$	0	0
$421$	−2.61134	−0.127269	−0.0636344	−	0.997973i	$-0.520269\pi$
$421$	−2.61134	−0.127269	−0.0636344	+	0.997973i	$0.520269\pi$
$422$	0	0
$423$	−0.660451	−0.0321122
$424$	0	0
$425$	−84.1322	−4.08101
$426$	0	0
$427$	17.3862	0.841379
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	2.31852	0.111679	0.0558396	−	0.998440i	$-0.482216\pi$
$431$	2.31852	0.111679	0.0558396	+	0.998440i	$0.482216\pi$
$432$	0	0
$433$	−17.0842	−0.821012	−0.410506	−	0.911858i	$-0.634648\pi$
$433$	−17.0842	−0.821012	−0.410506	+	0.911858i	$0.634648\pi$
$434$	0	0
$435$	−30.2574	−1.45073
$436$	0	0
$437$	24.1151	1.15358
$438$	0	0
$439$	−1.62395	−0.0775068	−0.0387534	−	0.999249i	$-0.512339\pi$
$439$	−1.62395	−0.0775068	−0.0387534	+	0.999249i	$0.512339\pi$
$440$	0	0
$441$	3.06970	0.146176
$442$	0	0
$443$	−2.30208	−0.109375	−0.0546874	−	0.998504i	$-0.517416\pi$
$443$	−2.30208	−0.109375	−0.0546874	+	0.998504i	$0.517416\pi$
$444$	0	0
$445$	3.83373	0.181736
$446$	0	0
$447$	3.99178	0.188805
$448$	0	0
$449$	−39.4726	−1.86283	−0.931413	−	0.363964i	$-0.881423\pi$
$449$	−39.4726	−1.86283	−0.931413	+	0.363964i	$0.881423\pi$
$450$	0	0
$451$	1.64522	0.0774703
$452$	0	0
$453$	−9.18730	−0.431657
$454$	0	0
$455$	−12.6048	−0.590921
$456$	0	0
$457$	19.9191	0.931776	0.465888	−	0.884844i	$-0.345735\pi$
$457$	19.9191	0.931776	0.465888	+	0.884844i	$0.345735\pi$
$458$	0	0
$459$	−7.80589	−0.364348
$460$	0	0
$461$	31.7855	1.48040	0.740199	−	0.672388i	$-0.234731\pi$
$461$	31.7855	1.48040	0.740199	+	0.672388i	$0.234731\pi$
$462$	0	0
$463$	5.75561	0.267486	0.133743	−	0.991016i	$-0.457300\pi$
$463$	5.75561	0.267486	0.133743	+	0.991016i	$0.457300\pi$
$464$	0	0
$465$	−12.5268	−0.580919
$466$	0	0
$467$	4.38745	0.203027	0.101513	−	0.994834i	$-0.467632\pi$
$467$	4.38745	0.203027	0.101513	+	0.994834i	$0.467632\pi$
$468$	0	0
$469$	17.4537	0.805938
$470$	0	0
$471$	−18.8482	−0.868479
$472$	0	0
$473$	4.15245	0.190930
$474$	0	0
$475$	−77.3139	−3.54741
$476$	0	0
$477$	0.0696997	0.00319133
$478$	0	0
$479$	−27.0062	−1.23395	−0.616973	−	0.786984i	$-0.711641\pi$
$479$	−27.0062	−1.23395	−0.616973	+	0.786984i	$0.711641\pi$
$480$	0	0
$481$	2.93731	0.133930
$482$	0	0
$483$	10.6679	0.485406
$484$	0	0
$485$	29.0537	1.31926
$486$	0	0
$487$	0.800534	0.0362757	0.0181378	−	0.999835i	$-0.494226\pi$
$487$	0.800534	0.0362757	0.0181378	+	0.999835i	$0.494226\pi$
$488$	0	0
$489$	15.7027	0.710103
$490$	0	0
$491$	−9.26985	−0.418342	−0.209171	−	0.977879i	$-0.567077\pi$
$491$	−9.26985	−0.418342	−0.209171	+	0.977879i	$0.567077\pi$
$492$	0	0
$493$	59.4604	2.67796
$494$	0	0
$495$	3.97216	0.178535
$496$	0	0
$497$	6.72358	0.301594
$498$	0	0
$499$	−14.1116	−0.631720	−0.315860	−	0.948806i	$-0.602293\pi$
$499$	−14.1116	−0.631720	−0.315860	+	0.948806i	$0.602293\pi$
$500$	0	0
$501$	0.709563	0.0317009
$502$	0	0
$503$	15.2512	0.680017	0.340009	−	0.940422i	$-0.389570\pi$
$503$	15.2512	0.680017	0.340009	+	0.940422i	$0.389570\pi$
$504$	0	0
$505$	68.9089	3.06641
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	7.69993	0.341293	0.170647	−	0.985332i	$-0.445414\pi$
$509$	7.69993	0.341293	0.170647	+	0.985332i	$0.445414\pi$
$510$	0	0
$511$	46.6959	2.06571
$512$	0	0
$513$	−7.17328	−0.316708
$514$	0	0
$515$	−17.7633	−0.782743
$516$	0	0
$517$	−0.660451	−0.0290466
$518$	0	0
$519$	6.86201	0.301209
$520$	0	0
$521$	−7.30776	−0.320159	−0.160079	−	0.987104i	$-0.551175\pi$
$521$	−7.30776	−0.320159	−0.160079	+	0.987104i	$0.551175\pi$
$522$	0	0
$523$	−21.5716	−0.943259	−0.471630	−	0.881797i	$-0.656334\pi$
$523$	−21.5716	−0.943259	−0.471630	+	0.881797i	$0.656334\pi$
$524$	0	0
$525$	−34.2018	−1.49269
$526$	0	0
$527$	24.6171	1.07234
$528$	0	0
$529$	−11.6984	−0.508625
$530$	0	0
$531$	−8.88820	−0.385715
$532$	0	0
$533$	1.64522	0.0712623
$534$	0	0
$535$	47.9984	2.07515
$536$	0	0
$537$	−10.1411	−0.437619
$538$	0	0
$539$	3.06970	0.132221
$540$	0	0
$541$	18.1394	0.779874	0.389937	−	0.920842i	$-0.372497\pi$
$541$	18.1394	0.779874	0.389937	+	0.920842i	$0.372497\pi$
$542$	0	0
$543$	3.76403	0.161530
$544$	0	0
$545$	−17.3536	−0.743345
$546$	0	0
$547$	−35.5757	−1.52111	−0.760554	−	0.649275i	$-0.775072\pi$
$547$	−35.5757	−1.52111	−0.760554	+	0.649275i	$0.775072\pi$
$548$	0	0
$549$	−5.47895	−0.233836
$550$	0	0
$551$	54.6416	2.32781
$552$	0	0
$553$	46.0094	1.95652
$554$	0	0
$555$	11.6675	0.495256
$556$	0	0
$557$	37.4670	1.58753	0.793763	−	0.608227i	$-0.208119\pi$
$557$	37.4670	1.58753	0.793763	+	0.608227i	$0.208119\pi$
$558$	0	0
$559$	4.15245	0.175630
$560$	0	0
$561$	−7.80589	−0.329565
$562$	0	0
$563$	−19.4447	−0.819498	−0.409749	−	0.912198i	$-0.634384\pi$
$563$	−19.4447	−0.819498	−0.409749	+	0.912198i	$0.634384\pi$
$564$	0	0
$565$	−34.8726	−1.46710
$566$	0	0
$567$	−3.17328	−0.133265
$568$	0	0
$569$	2.59635	0.108845	0.0544223	−	0.998518i	$-0.482668\pi$
$569$	2.59635	0.108845	0.0544223	+	0.998518i	$0.482668\pi$
$570$	0	0
$571$	15.3427	0.642073	0.321036	−	0.947067i	$-0.395969\pi$
$571$	15.3427	0.642073	0.321036	+	0.947067i	$0.395969\pi$
$572$	0	0
$573$	1.73716	0.0725709
$574$	0	0
$575$	−36.2335	−1.51104
$576$	0	0
$577$	9.29044	0.386766	0.193383	−	0.981123i	$-0.438054\pi$
$577$	9.29044	0.386766	0.193383	+	0.981123i	$0.438054\pi$
$578$	0	0
$579$	−17.2995	−0.718944
$580$	0	0
$581$	−36.4051	−1.51034
$582$	0	0
$583$	0.0696997	0.00288667
$584$	0	0
$585$	3.97216	0.164229
$586$	0	0
$587$	−20.1819	−0.832998	−0.416499	−	0.909136i	$-0.636743\pi$
$587$	−20.1819	−0.832998	−0.416499	+	0.909136i	$0.636743\pi$
$588$	0	0
$589$	22.6221	0.932127
$590$	0	0
$591$	21.0062	0.864081
$592$	0	0
$593$	28.8800	1.18596	0.592979	−	0.805218i	$-0.297952\pi$
$593$	28.8800	1.18596	0.592979	+	0.805218i	$0.297952\pi$
$594$	0	0
$595$	98.3914	4.03366
$596$	0	0
$597$	23.1791	0.948656
$598$	0	0
$599$	−22.1515	−0.905085	−0.452542	−	0.891743i	$-0.649483\pi$
$599$	−22.1515	−0.905085	−0.452542	+	0.891743i	$0.649483\pi$
$600$	0	0
$601$	−39.9307	−1.62881	−0.814403	−	0.580299i	$-0.802936\pi$
$601$	−39.9307	−1.62881	−0.814403	+	0.580299i	$0.802936\pi$
$602$	0	0
$603$	−5.50022	−0.223986
$604$	0	0
$605$	3.97216	0.161491
$606$	0	0
$607$	−10.2320	−0.415305	−0.207653	−	0.978203i	$-0.566582\pi$
$607$	−10.2320	−0.415305	−0.207653	+	0.978203i	$0.566582\pi$
$608$	0	0
$609$	24.1721	0.979501
$610$	0	0
$611$	−0.660451	−0.0267190
$612$	0	0
$613$	17.2013	0.694755	0.347377	−	0.937725i	$-0.387072\pi$
$613$	17.2013	0.694755	0.347377	+	0.937725i	$0.387072\pi$
$614$	0	0
$615$	6.53507	0.263519
$616$	0	0
$617$	−1.30426	−0.0525075	−0.0262538	−	0.999655i	$-0.508358\pi$
$617$	−1.30426	−0.0525075	−0.0262538	+	0.999655i	$0.508358\pi$
$618$	0	0
$619$	20.3698	0.818730	0.409365	−	0.912371i	$-0.365750\pi$
$619$	20.3698	0.818730	0.409365	+	0.912371i	$0.365750\pi$
$620$	0	0
$621$	−3.36179	−0.134904
$622$	0	0
$623$	−3.06269	−0.122704
$624$	0	0
$625$	37.2761	1.49104
$626$	0	0
$627$	−7.17328	−0.286473
$628$	0	0
$629$	−22.9283	−0.914212
$630$	0	0
$631$	−6.67184	−0.265602	−0.132801	−	0.991143i	$-0.542397\pi$
$631$	−6.67184	−0.265602	−0.132801	+	0.991143i	$0.542397\pi$
$632$	0	0
$633$	11.6248	0.462045
$634$	0	0
$635$	−9.62000	−0.381758
$636$	0	0
$637$	3.06970	0.121626
$638$	0	0
$639$	−2.11881	−0.0838189
$640$	0	0
$641$	−17.9836	−0.710308	−0.355154	−	0.934808i	$-0.615572\pi$
$641$	−17.9836	−0.710308	−0.355154	+	0.934808i	$0.615572\pi$
$642$	0	0
$643$	13.9557	0.550360	0.275180	−	0.961393i	$-0.411263\pi$
$643$	13.9557	0.550360	0.275180	+	0.961393i	$0.411263\pi$
$644$	0	0
$645$	16.4942	0.649458
$646$	0	0
$647$	−23.7380	−0.933239	−0.466619	−	0.884458i	$-0.654528\pi$
$647$	−23.7380	−0.933239	−0.466619	+	0.884458i	$0.654528\pi$
$648$	0	0
$649$	−8.88820	−0.348892
$650$	0	0
$651$	10.0074	0.392223
$652$	0	0
$653$	−10.8325	−0.423909	−0.211955	−	0.977280i	$-0.567983\pi$
$653$	−10.8325	−0.423909	−0.211955	+	0.977280i	$0.567983\pi$
$654$	0	0
$655$	33.2095	1.29760
$656$	0	0
$657$	−14.7154	−0.574101
$658$	0	0
$659$	31.7315	1.23608	0.618041	−	0.786146i	$-0.287926\pi$
$659$	31.7315	1.23608	0.618041	+	0.786146i	$0.287926\pi$
$660$	0	0
$661$	−15.1225	−0.588198	−0.294099	−	0.955775i	$-0.595019\pi$
$661$	−15.1225	−0.588198	−0.294099	+	0.955775i	$0.595019\pi$
$662$	0	0
$663$	−7.80589	−0.303156
$664$	0	0
$665$	90.4175	3.50624
$666$	0	0
$667$	25.6080	0.991547
$668$	0	0
$669$	−5.11200	−0.197641
$670$	0	0
$671$	−5.47895	−0.211512
$672$	0	0
$673$	10.4209	0.401696	0.200848	−	0.979622i	$-0.435630\pi$
$673$	10.4209	0.401696	0.200848	+	0.979622i	$0.435630\pi$
$674$	0	0
$675$	10.7780	0.414847
$676$	0	0
$677$	−22.5229	−0.865625	−0.432813	−	0.901484i	$-0.642479\pi$
$677$	−22.5229	−0.865625	−0.432813	+	0.901484i	$0.642479\pi$
$678$	0	0
$679$	−23.2104	−0.890734
$680$	0	0
$681$	−15.2290	−0.583575
$682$	0	0
$683$	−40.6164	−1.55414	−0.777071	−	0.629413i	$-0.783295\pi$
$683$	−40.6164	−1.55414	−0.777071	+	0.629413i	$0.783295\pi$
$684$	0	0
$685$	−39.1391	−1.49543
$686$	0	0
$687$	24.4237	0.931823
$688$	0	0
$689$	0.0696997	0.00265535
$690$	0	0
$691$	4.00725	0.152443	0.0762215	−	0.997091i	$-0.475714\pi$
$691$	4.00725	0.152443	0.0762215	+	0.997091i	$0.475714\pi$
$692$	0	0
$693$	−3.17328	−0.120543
$694$	0	0
$695$	10.8964	0.413325
$696$	0	0
$697$	−12.8424	−0.486440
$698$	0	0
$699$	−27.4223	−1.03721
$700$	0	0
$701$	−18.0018	−0.679920	−0.339960	−	0.940440i	$-0.610414\pi$
$701$	−18.0018	−0.679920	−0.339960	+	0.940440i	$0.610414\pi$
$702$	0	0
$703$	−21.0701	−0.794675
$704$	0	0
$705$	−2.62342	−0.0988036
$706$	0	0
$707$	−55.0499	−2.07037
$708$	0	0
$709$	−44.1696	−1.65883	−0.829413	−	0.558636i	$-0.811325\pi$
$709$	−44.1696	−1.65883	−0.829413	+	0.558636i	$0.811325\pi$
$710$	0	0
$711$	−14.4990	−0.543755
$712$	0	0
$713$	10.6019	0.397046
$714$	0	0
$715$	3.97216	0.148550
$716$	0	0
$717$	−21.6036	−0.806800
$718$	0	0
$719$	−36.6931	−1.36842	−0.684211	−	0.729284i	$-0.739853\pi$
$719$	−36.6931	−1.36842	−0.684211	+	0.729284i	$0.739853\pi$
$720$	0	0
$721$	14.1907	0.528490
$722$	0	0
$723$	6.32134	0.235093
$724$	0	0
$725$	−82.1005	−3.04913
$726$	0	0
$727$	−3.89984	−0.144637	−0.0723184	−	0.997382i	$-0.523040\pi$
$727$	−3.89984	−0.144637	−0.0723184	+	0.997382i	$0.523040\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.4135	−1.19886
$732$	0	0
$733$	3.01986	0.111541	0.0557706	−	0.998444i	$-0.482238\pi$
$733$	3.01986	0.111541	0.0557706	+	0.998444i	$0.482238\pi$
$734$	0	0
$735$	12.1933	0.449758
$736$	0	0
$737$	−5.50022	−0.202603
$738$	0	0
$739$	−4.95298	−0.182198	−0.0910992	−	0.995842i	$-0.529038\pi$
$739$	−4.95298	−0.182198	−0.0910992	+	0.995842i	$0.529038\pi$
$740$	0	0
$741$	−7.17328	−0.263517
$742$	0	0
$743$	34.8891	1.27996	0.639978	−	0.768393i	$-0.278943\pi$
$743$	34.8891	1.27996	0.639978	+	0.768393i	$0.278943\pi$
$744$	0	0
$745$	15.8560	0.580918
$746$	0	0
$747$	11.4724	0.419752
$748$	0	0
$749$	−38.3450	−1.40110
$750$	0	0
$751$	−21.1539	−0.771915	−0.385958	−	0.922517i	$-0.626129\pi$
$751$	−21.1539	−0.771915	−0.385958	+	0.922517i	$0.626129\pi$
$752$	0	0
$753$	9.48640	0.345704
$754$	0	0
$755$	−36.4934	−1.32813
$756$	0	0
$757$	−19.5264	−0.709699	−0.354850	−	0.934923i	$-0.615468\pi$
$757$	−19.5264	−0.709699	−0.354850	+	0.934923i	$0.615468\pi$
$758$	0	0
$759$	−3.36179	−0.122025
$760$	0	0
$761$	22.2546	0.806729	0.403365	−	0.915039i	$-0.367841\pi$
$761$	22.2546	0.806729	0.403365	+	0.915039i	$0.367841\pi$
$762$	0	0
$763$	13.8634	0.501889
$764$	0	0
$765$	−31.0062	−1.12103
$766$	0	0
$767$	−8.88820	−0.320934
$768$	0	0
$769$	−46.4424	−1.67475	−0.837377	−	0.546626i	$-0.815912\pi$
$769$	−46.4424	−1.67475	−0.837377	+	0.546626i	$0.815912\pi$
$770$	0	0
$771$	−11.6370	−0.419096
$772$	0	0
$773$	−49.6587	−1.78610	−0.893049	−	0.449960i	$-0.851438\pi$
$773$	−49.6587	−1.78610	−0.893049	+	0.449960i	$0.851438\pi$
$774$	0	0
$775$	−33.9903	−1.22097
$776$	0	0
$777$	−9.32090	−0.334385
$778$	0	0
$779$	−11.8016	−0.422837
$780$	0	0
$781$	−2.11881	−0.0758170
$782$	0	0
$783$	−7.61738	−0.272223
$784$	0	0
$785$	−74.8680	−2.67215
$786$	0	0
$787$	−4.95298	−0.176555	−0.0882774	−	0.996096i	$-0.528136\pi$
$787$	−4.95298	−0.176555	−0.0882774	+	0.996096i	$0.528136\pi$
$788$	0	0
$789$	9.95834	0.354526
$790$	0	0
$791$	27.8591	0.990554
$792$	0	0
$793$	−5.47895	−0.194563
$794$	0	0
$795$	0.276858	0.00981915
$796$	0	0
$797$	31.3325	1.10985	0.554927	−	0.831899i	$-0.312746\pi$
$797$	31.3325	1.10985	0.554927	+	0.831899i	$0.312746\pi$
$798$	0	0
$799$	5.15541	0.182385
$800$	0	0
$801$	0.965150	0.0341019
$802$	0	0
$803$	−14.7154	−0.519294
$804$	0	0
$805$	42.3746	1.49351
$806$	0	0
$807$	−16.4303	−0.578373
$808$	0	0
$809$	−41.9985	−1.47659	−0.738295	−	0.674478i	$-0.764369\pi$
$809$	−41.9985	−1.47659	−0.738295	+	0.674478i	$0.764369\pi$
$810$	0	0
$811$	−45.2186	−1.58784	−0.793921	−	0.608021i	$-0.791963\pi$
$811$	−45.2186	−1.58784	−0.793921	+	0.608021i	$0.791963\pi$
$812$	0	0
$813$	29.1761	1.02325
$814$	0	0
$815$	62.3738	2.18486
$816$	0	0
$817$	−29.7867	−1.04210
$818$	0	0
$819$	−3.17328	−0.110883
$820$	0	0
$821$	21.2850	0.742853	0.371426	−	0.928462i	$-0.378869\pi$
$821$	21.2850	0.742853	0.371426	+	0.928462i	$0.378869\pi$
$822$	0	0
$823$	−18.9467	−0.660440	−0.330220	−	0.943904i	$-0.607123\pi$
$823$	−18.9467	−0.660440	−0.330220	+	0.943904i	$0.607123\pi$
$824$	0	0
$825$	10.7780	0.375243
$826$	0	0
$827$	46.8411	1.62883	0.814413	−	0.580286i	$-0.197059\pi$
$827$	46.8411	1.62883	0.814413	+	0.580286i	$0.197059\pi$
$828$	0	0
$829$	4.69224	0.162968	0.0814841	−	0.996675i	$-0.474034\pi$
$829$	4.69224	0.162968	0.0814841	+	0.996675i	$0.474034\pi$
$830$	0	0
$831$	−4.20716	−0.145945
$832$	0	0
$833$	−23.9617	−0.830225
$834$	0	0
$835$	2.81850	0.0975381
$836$	0	0
$837$	−3.15366	−0.109006
$838$	0	0
$839$	0.234316	0.00808948	0.00404474	−	0.999992i	$-0.498713\pi$
$839$	0.234316	0.00808948	0.00404474	+	0.999992i	$0.498713\pi$
$840$	0	0
$841$	29.0244	1.00084
$842$	0	0
$843$	−32.9113	−1.13353
$844$	0	0
$845$	3.97216	0.136646
$846$	0	0
$847$	−3.17328	−0.109035
$848$	0	0
$849$	19.8690	0.681903
$850$	0	0
$851$	−9.87462	−0.338498
$852$	0	0
$853$	22.8770	0.783294	0.391647	−	0.920116i	$-0.371906\pi$
$853$	22.8770	0.783294	0.391647	+	0.920116i	$0.371906\pi$
$854$	0	0
$855$	−28.4934	−0.974454
$856$	0	0
$857$	25.3545	0.866094	0.433047	−	0.901371i	$-0.357438\pi$
$857$	25.3545	0.866094	0.433047	+	0.901371i	$0.357438\pi$
$858$	0	0
$859$	−46.2580	−1.57830	−0.789151	−	0.614199i	$-0.789479\pi$
$859$	−46.2580	−1.57830	−0.789151	+	0.614199i	$0.789479\pi$
$860$	0	0
$861$	−5.22074	−0.177922
$862$	0	0
$863$	48.4921	1.65069	0.825345	−	0.564629i	$-0.190981\pi$
$863$	48.4921	1.65069	0.825345	+	0.564629i	$0.190981\pi$
$864$	0	0
$865$	27.2570	0.926766
$866$	0	0
$867$	43.9319	1.49201
$868$	0	0
$869$	−14.4990	−0.491845
$870$	0	0
$871$	−5.50022	−0.186368
$872$	0	0
$873$	7.31433	0.247553
$874$	0	0
$875$	−72.8310	−2.46214
$876$	0	0
$877$	−32.7628	−1.10632	−0.553161	−	0.833074i	$-0.686579\pi$
$877$	−32.7628	−1.10632	−0.553161	+	0.833074i	$0.686579\pi$
$878$	0	0
$879$	7.00536	0.236285
$880$	0	0
$881$	−21.7773	−0.733695	−0.366847	−	0.930281i	$-0.619563\pi$
$881$	−21.7773	−0.733695	−0.366847	+	0.930281i	$0.619563\pi$
$882$	0	0
$883$	34.3137	1.15475	0.577373	−	0.816480i	$-0.304078\pi$
$883$	34.3137	1.15475	0.577373	+	0.816480i	$0.304078\pi$
$884$	0	0
$885$	−35.3053	−1.18678
$886$	0	0
$887$	−48.9064	−1.64211	−0.821057	−	0.570846i	$-0.806615\pi$
$887$	−48.9064	−1.64211	−0.821057	+	0.570846i	$0.806615\pi$
$888$	0	0
$889$	7.68523	0.257754
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	4.73760	0.158538
$894$	0	0
$895$	−40.2819	−1.34647
$896$	0	0
$897$	−3.36179	−0.112247
$898$	0	0
$899$	24.0226	0.801200
$900$	0	0
$901$	−0.544068	−0.0181255
$902$	0	0
$903$	−13.1769	−0.438499
$904$	0	0
$905$	14.9513	0.496999
$906$	0	0
$907$	49.9956	1.66008	0.830039	−	0.557706i	$-0.188318\pi$
$907$	49.9956	1.66008	0.830039	+	0.557706i	$0.188318\pi$
$908$	0	0
$909$	17.3480	0.575396
$910$	0	0
$911$	8.52762	0.282533	0.141266	−	0.989972i	$-0.454883\pi$
$911$	8.52762	0.282533	0.141266	+	0.989972i	$0.454883\pi$
$912$	0	0
$913$	11.4724	0.379680
$914$	0	0
$915$	−21.7633	−0.719471
$916$	0	0
$917$	−26.5304	−0.876112
$918$	0	0
$919$	−1.49112	−0.0491874	−0.0245937	−	0.999698i	$-0.507829\pi$
$919$	−1.49112	−0.0491874	−0.0245937	+	0.999698i	$0.507829\pi$
$920$	0	0
$921$	8.67087	0.285715
$922$	0	0
$923$	−2.11881	−0.0697415
$924$	0	0
$925$	31.6585	1.04092
$926$	0	0
$927$	−4.47194	−0.146878
$928$	0	0
$929$	−31.1447	−1.02182	−0.510912	−	0.859633i	$-0.670692\pi$
$929$	−31.1447	−1.02182	−0.510912	+	0.859633i	$0.670692\pi$
$930$	0	0
$931$	−22.0198	−0.721670
$932$	0	0
$933$	30.0689	0.984412
$934$	0	0
$935$	−31.0062	−1.01401
$936$	0	0
$937$	−2.88357	−0.0942021	−0.0471010	−	0.998890i	$-0.514998\pi$
$937$	−2.88357	−0.0942021	−0.0471010	+	0.998890i	$0.514998\pi$
$938$	0	0
$939$	−30.6662	−1.00076
$940$	0	0
$941$	1.78462	0.0581769	0.0290884	−	0.999577i	$-0.490740\pi$
$941$	1.78462	0.0581769	0.0290884	+	0.999577i	$0.490740\pi$
$942$	0	0
$943$	−5.53088	−0.180110
$944$	0	0
$945$	−12.6048	−0.410033
$946$	0	0
$947$	−17.9518	−0.583354	−0.291677	−	0.956517i	$-0.594213\pi$
$947$	−17.9518	−0.583354	−0.291677	+	0.956517i	$0.594213\pi$
$948$	0	0
$949$	−14.7154	−0.477681
$950$	0	0
$951$	7.49559	0.243061
$952$	0	0
$953$	−33.2857	−1.07823	−0.539115	−	0.842232i	$-0.681241\pi$
$953$	−33.2857	−1.07823	−0.539115	+	0.842232i	$0.681241\pi$
$954$	0	0
$955$	6.90027	0.223288
$956$	0	0
$957$	−7.61738	−0.246235
$958$	0	0
$959$	31.2674	1.00968
$960$	0	0
$961$	−21.0544	−0.679175
$962$	0	0
$963$	12.0837	0.389392
$964$	0	0
$965$	−68.7165	−2.21206
$966$	0	0
$967$	−22.4525	−0.722024	−0.361012	−	0.932561i	$-0.617569\pi$
$967$	−22.4525	−0.722024	−0.361012	+	0.932561i	$0.617569\pi$
$968$	0	0
$969$	55.9938	1.79878
$970$	0	0
$971$	−36.2108	−1.16206	−0.581029	−	0.813883i	$-0.697350\pi$
$971$	−36.2108	−1.16206	−0.581029	+	0.813883i	$0.697350\pi$
$972$	0	0
$973$	−8.70493	−0.279067
$974$	0	0
$975$	10.7780	0.345174
$976$	0	0
$977$	−32.7929	−1.04914	−0.524568	−	0.851368i	$-0.675773\pi$
$977$	−32.7929	−1.04914	−0.524568	+	0.851368i	$0.675773\pi$
$978$	0	0
$979$	0.965150	0.0308463
$980$	0	0
$981$	−4.36880	−0.139485
$982$	0	0
$983$	8.04448	0.256579	0.128290	−	0.991737i	$-0.459051\pi$
$983$	8.04448	0.256579	0.128290	+	0.991737i	$0.459051\pi$
$984$	0	0
$985$	83.4401	2.65862
$986$	0	0
$987$	2.09580	0.0667099
$988$	0	0
$989$	−13.9597	−0.443891
$990$	0	0
$991$	26.1679	0.831251	0.415626	−	0.909536i	$-0.363563\pi$
$991$	26.1679	0.831251	0.415626	+	0.909536i	$0.363563\pi$
$992$	0	0
$993$	10.8538	0.344435
$994$	0	0
$995$	92.0710	2.91885
$996$	0	0
$997$	48.1911	1.52623	0.763114	−	0.646264i	$-0.223670\pi$
$997$	48.1911	1.52623	0.763114	+	0.646264i	$0.223670\pi$
$998$	0	0
$999$	2.93731	0.0929324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bz.1.4		4
4.3	odd	2		429.2.a.h.1.2	✓	4
12.11	even	2		1287.2.a.m.1.3		4
44.43	even	2		4719.2.a.z.1.3		4
52.51	odd	2		5577.2.a.m.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.h.1.2	✓	4	4.3	odd	2
1287.2.a.m.1.3		4	12.11	even	2
4719.2.a.z.1.3		4	44.43	even	2
5577.2.a.m.1.3		4	52.51	odd	2
6864.2.a.bz.1.4		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.97216 q^{5} -3.17328 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.97216 q^{5} -3.17328 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.97216 q^{15} -7.80589 q^{17} -7.17328 q^{19} -3.17328 q^{21} -3.36179 q^{23} +10.7780 q^{25} +1.00000 q^{27} -7.61738 q^{29} -3.15366 q^{31} +1.00000 q^{33} -12.6048 q^{35} +2.93731 q^{37} +1.00000 q^{39} +1.64522 q^{41} +4.15245 q^{43} +3.97216 q^{45} -0.660451 q^{47} +3.06970 q^{49} -7.80589 q^{51} +0.0696997 q^{53} +3.97216 q^{55} -7.17328 q^{57} -8.88820 q^{59} -5.47895 q^{61} -3.17328 q^{63} +3.97216 q^{65} -5.50022 q^{67} -3.36179 q^{69} -2.11881 q^{71} -14.7154 q^{73} +10.7780 q^{75} -3.17328 q^{77} -14.4990 q^{79} +1.00000 q^{81} +11.4724 q^{83} -31.0062 q^{85} -7.61738 q^{87} +0.965150 q^{89} -3.17328 q^{91} -3.15366 q^{93} -28.4934 q^{95} +7.31433 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 18 q^{19} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 10 q^{29} - 12 q^{31} + 4 q^{33} - 22 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 28 q^{43} - 6 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 18 q^{57} - 16 q^{59} - 10 q^{61} - 2 q^{63} - 10 q^{71} - 6 q^{73} + 4 q^{75} - 2 q^{77} + 8 q^{79} + 4 q^{81} + 8 q^{83} - 18 q^{85} - 10 q^{87} + 6 q^{89} - 2 q^{91} - 12 q^{93} - 22 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 - 8 * q^17 - 18 * q^19 - 2 * q^21 + 4 * q^25 + 4 * q^27 - 10 * q^29 - 12 * q^31 + 4 * q^33 - 22 * q^35 - 2 * q^37 + 4 * q^39 + 2 * q^41 - 28 * q^43 - 6 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 18 * q^57 - 16 * q^59 - 10 * q^61 - 2 * q^63 - 10 * q^71 - 6 * q^73 + 4 * q^75 - 2 * q^77 + 8 * q^79 + 4 * q^81 + 8 * q^83 - 18 * q^85 - 10 * q^87 + 6 * q^89 - 2 * q^91 - 12 * q^93 - 22 * q^95 + 10 * q^97 + 4 * q^99

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