LMFDB - Embedded newform 6864.2.a.ca.1.1

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			1.1
Root			$-3.18590$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.18590 q^{5} -1.10556 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.18590 q^{5} -1.10556 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -3.18590 q^{15} -6.29146 q^{17} -3.74404 q^{19} -1.10556 q^{21} -3.26624 q^{23} +5.14997 q^{25} +1.00000 q^{27} +3.60255 q^{29} +0.336304 q^{31} -1.00000 q^{33} +3.52221 q^{35} +6.37180 q^{37} -1.00000 q^{39} +0.894440 q^{41} -8.85808 q^{43} -3.18590 q^{45} +8.21112 q^{47} -5.77774 q^{49} -6.29146 q^{51} +5.32739 q^{53} +3.18590 q^{55} -3.74404 q^{57} +9.01028 q^{59} -14.9384 q^{61} -1.10556 q^{63} +3.18590 q^{65} -10.6021 q^{67} -3.26624 q^{69} +5.32739 q^{71} -5.03370 q^{73} +5.14997 q^{75} +1.10556 q^{77} -9.54699 q^{79} +1.00000 q^{81} +5.11627 q^{83} +20.0440 q^{85} +3.60255 q^{87} +3.66370 q^{89} +1.10556 q^{91} +0.336304 q^{93} +11.9281 q^{95} +8.13926 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.18590	−1.42478	−0.712389	−	0.701784i	$-0.752387\pi$
$5$	−3.18590	−1.42478	−0.712389	+	0.701784i	$0.752387\pi$
$6$	0	0
$7$	−1.10556	−0.417862	−0.208931	−	0.977930i	$-0.566998\pi$
$7$	−1.10556	−0.417862	−0.208931	+	0.977930i	$0.566998\pi$
$8$	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
$15$	−3.18590	−0.822596
$16$	0	0
$17$	−6.29146	−1.52590	−0.762952	−	0.646455i	$-0.776251\pi$
$17$	−6.29146	−1.52590	−0.762952	+	0.646455i	$0.776251\pi$
$18$	0	0
$19$	−3.74404	−0.858941	−0.429471	−	0.903081i	$-0.641300\pi$
$19$	−3.74404	−0.858941	−0.429471	+	0.903081i	$0.641300\pi$
$20$	0	0
$21$	−1.10556	−0.241253
$22$	0	0
$23$	−3.26624	−0.681059	−0.340529	−	0.940234i	$-0.610606\pi$
$23$	−3.26624	−0.681059	−0.340529	+	0.940234i	$0.610606\pi$
$24$	0	0
$25$	5.14997	1.02999
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.60255	0.668976	0.334488	−	0.942400i	$-0.391437\pi$
$29$	3.60255	0.668976	0.334488	+	0.942400i	$0.391437\pi$
$30$	0	0
$31$	0.336304	0.0604019	0.0302010	−	0.999544i	$-0.490385\pi$
$31$	0.336304	0.0604019	0.0302010	+	0.999544i	$0.490385\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.52221	0.595361
$36$	0	0
$37$	6.37180	1.04752	0.523759	−	0.851866i	$-0.324529\pi$
$37$	6.37180	1.04752	0.523759	+	0.851866i	$0.324529\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	0.894440	0.139688	0.0698440	−	0.997558i	$-0.477750\pi$
$41$	0.894440	0.139688	0.0698440	+	0.997558i	$0.477750\pi$
$42$	0	0
$43$	−8.85808	−1.35084	−0.675422	−	0.737431i	$-0.736039\pi$
$43$	−8.85808	−1.35084	−0.675422	+	0.737431i	$0.736039\pi$
$44$	0	0
$45$	−3.18590	−0.474926
$46$	0	0
$47$	8.21112	1.19771	0.598857	−	0.800856i	$-0.295622\pi$
$47$	8.21112	1.19771	0.598857	+	0.800856i	$0.295622\pi$
$48$	0	0
$49$	−5.77774	−0.825391
$50$	0	0
$51$	−6.29146	−0.880981
$52$	0	0
$53$	5.32739	0.731774	0.365887	−	0.930659i	$-0.380766\pi$
$53$	5.32739	0.731774	0.365887	+	0.930659i	$0.380766\pi$
$54$	0	0
$55$	3.18590	0.429587
$56$	0	0
$57$	−3.74404	−0.495910
$58$	0	0
$59$	9.01028	1.17304	0.586519	−	0.809935i	$-0.300498\pi$
$59$	9.01028	1.17304	0.586519	+	0.809935i	$0.300498\pi$
$60$	0	0
$61$	−14.9384	−1.91267	−0.956334	−	0.292275i	$-0.905588\pi$
$61$	−14.9384	−1.91267	−0.956334	+	0.292275i	$0.905588\pi$
$62$	0	0
$63$	−1.10556	−0.139287
$64$	0	0
$65$	3.18590	0.395163
$66$	0	0
$67$	−10.6021	−1.29525	−0.647627	−	0.761957i	$-0.724239\pi$
$67$	−10.6021	−1.29525	−0.647627	+	0.761957i	$0.724239\pi$
$68$	0	0
$69$	−3.26624	−0.393210
$70$	0	0
$71$	5.32739	0.632245	0.316123	−	0.948718i	$-0.397619\pi$
$71$	5.32739	0.632245	0.316123	+	0.948718i	$0.397619\pi$
$72$	0	0
$73$	−5.03370	−0.589150	−0.294575	−	0.955628i	$-0.595178\pi$
$73$	−5.03370	−0.589150	−0.294575	+	0.955628i	$0.595178\pi$
$74$	0	0
$75$	5.14997	0.594667
$76$	0	0
$77$	1.10556	0.125990
$78$	0	0
$79$	−9.54699	−1.07412	−0.537060	−	0.843544i	$-0.680465\pi$
$79$	−9.54699	−1.07412	−0.537060	+	0.843544i	$0.680465\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.11627	0.561584	0.280792	−	0.959769i	$-0.409403\pi$
$83$	5.11627	0.561584	0.280792	+	0.959769i	$0.409403\pi$
$84$	0	0
$85$	20.0440	2.17408
$86$	0	0
$87$	3.60255	0.386234
$88$	0	0
$89$	3.66370	0.388351	0.194176	−	0.980967i	$-0.437797\pi$
$89$	3.66370	0.388351	0.194176	+	0.980967i	$0.437797\pi$
$90$	0	0
$91$	1.10556	0.115894
$92$	0	0
$93$	0.336304	0.0348731
$94$	0	0
$95$	11.9281	1.22380
$96$	0	0
$97$	8.13926	0.826417	0.413208	−	0.910637i	$-0.364408\pi$
$97$	8.13926	0.826417	0.413208	+	0.910637i	$0.364408\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	18.5914	1.84991	0.924957	−	0.380072i	$-0.124101\pi$
$101$	18.5914	1.84991	0.924957	+	0.380072i	$0.124101\pi$
$102$	0	0
$103$	18.9384	1.86606	0.933029	−	0.359801i	$-0.117155\pi$
$103$	18.9384	1.86606	0.933029	+	0.359801i	$0.117155\pi$
$104$	0	0
$105$	3.52221	0.343732
$106$	0	0
$107$	16.0547	1.55207	0.776033	−	0.630692i	$-0.217229\pi$
$107$	16.0547	1.55207	0.776033	+	0.630692i	$0.217229\pi$
$108$	0	0
$109$	−6.41665	−0.614603	−0.307302	−	0.951612i	$-0.599426\pi$
$109$	−6.41665	−0.614603	−0.307302	+	0.951612i	$0.599426\pi$
$110$	0	0
$111$	6.37180	0.604785
$112$	0	0
$113$	−0.477794	−0.0449471	−0.0224736	−	0.999747i	$-0.507154\pi$
$113$	−0.477794	−0.0449471	−0.0224736	+	0.999747i	$0.507154\pi$
$114$	0	0
$115$	10.4059	0.970358
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	6.95559	0.637618
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.894440	0.0806489
$124$	0	0
$125$	−0.477794	−0.0427352
$126$	0	0
$127$	7.40773	0.657330	0.328665	−	0.944447i	$-0.393401\pi$
$127$	7.40773	0.657330	0.328665	+	0.944447i	$0.393401\pi$
$128$	0	0
$129$	−8.85808	−0.779910
$130$	0	0
$131$	2.19481	0.191762	0.0958809	−	0.995393i	$-0.469433\pi$
$131$	2.19481	0.191762	0.0958809	+	0.995393i	$0.469433\pi$
$132$	0	0
$133$	4.13926	0.358919
$134$	0	0
$135$	−3.18590	−0.274199
$136$	0	0
$137$	−20.5466	−1.75541	−0.877706	−	0.479200i	$-0.840927\pi$
$137$	−20.5466	−1.75541	−0.877706	+	0.479200i	$0.840927\pi$
$138$	0	0
$139$	−15.1248	−1.28286	−0.641432	−	0.767180i	$-0.721660\pi$
$139$	−15.1248	−1.28286	−0.641432	+	0.767180i	$0.721660\pi$
$140$	0	0
$141$	8.21112	0.691501
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−11.4774	−0.953143
$146$	0	0
$147$	−5.77774	−0.476540
$148$	0	0
$149$	−6.99957	−0.573427	−0.286714	−	0.958016i	$-0.592563\pi$
$149$	−6.99957	−0.573427	−0.286714	+	0.958016i	$0.592563\pi$
$150$	0	0
$151$	5.95516	0.484624	0.242312	−	0.970198i	$-0.422094\pi$
$151$	5.95516	0.484624	0.242312	+	0.970198i	$0.422094\pi$
$152$	0	0
$153$	−6.29146	−0.508635
$154$	0	0
$155$	−1.07143	−0.0860594
$156$	0	0
$157$	2.91075	0.232303	0.116151	−	0.993232i	$-0.462944\pi$
$157$	2.91075	0.232303	0.116151	+	0.993232i	$0.462944\pi$
$158$	0	0
$159$	5.32739	0.422490
$160$	0	0
$161$	3.61103	0.284589
$162$	0	0
$163$	6.06963	0.475410	0.237705	−	0.971337i	$-0.423605\pi$
$163$	6.06963	0.475410	0.237705	+	0.971337i	$0.423605\pi$
$164$	0	0
$165$	3.18590	0.248022
$166$	0	0
$167$	20.8496	1.61339	0.806695	−	0.590968i	$-0.201254\pi$
$167$	20.8496	1.61339	0.806695	+	0.590968i	$0.201254\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.74404	−0.286314
$172$	0	0
$173$	−7.81367	−0.594062	−0.297031	−	0.954868i	$-0.595996\pi$
$173$	−7.81367	−0.594062	−0.297031	+	0.954868i	$0.595996\pi$
$174$	0	0
$175$	−5.69360	−0.430396
$176$	0	0
$177$	9.01028	0.677254
$178$	0	0
$179$	6.67820	0.499152	0.249576	−	0.968355i	$-0.419709\pi$
$179$	6.67820	0.499152	0.249576	+	0.968355i	$0.419709\pi$
$180$	0	0
$181$	−12.9996	−0.966250	−0.483125	−	0.875551i	$-0.660498\pi$
$181$	−12.9996	−0.966250	−0.483125	+	0.875551i	$0.660498\pi$
$182$	0	0
$183$	−14.9384	−1.10428
$184$	0	0
$185$	−20.2999	−1.49248
$186$	0	0
$187$	6.29146	0.460077
$188$	0	0
$189$	−1.10556	−0.0804177
$190$	0	0
$191$	13.8436	1.00169	0.500843	−	0.865538i	$-0.333023\pi$
$191$	13.8436	1.00169	0.500843	+	0.865538i	$0.333023\pi$
$192$	0	0
$193$	21.2277	1.52800	0.764000	−	0.645216i	$-0.223233\pi$
$193$	21.2277	1.52800	0.764000	+	0.645216i	$0.223233\pi$
$194$	0	0
$195$	3.18590	0.228147
$196$	0	0
$197$	−23.9935	−1.70947	−0.854735	−	0.519065i	$-0.826280\pi$
$197$	−23.9935	−1.70947	−0.854735	+	0.519065i	$0.826280\pi$
$198$	0	0
$199$	25.9828	1.84187	0.920937	−	0.389711i	$-0.127425\pi$
$199$	25.9828	1.84187	0.920937	+	0.389711i	$0.127425\pi$
$200$	0	0
$201$	−10.6021	−0.747816
$202$	0	0
$203$	−3.98283	−0.279540
$204$	0	0
$205$	−2.84960	−0.199025
$206$	0	0
$207$	−3.26624	−0.227020
$208$	0	0
$209$	3.74404	0.258981
$210$	0	0
$211$	9.26401	0.637761	0.318880	−	0.947795i	$-0.396693\pi$
$211$	9.26401	0.637761	0.318880	+	0.947795i	$0.396693\pi$
$212$	0	0
$213$	5.32739	0.365027
$214$	0	0
$215$	28.2210	1.92465
$216$	0	0
$217$	−0.371804	−0.0252397
$218$	0	0
$219$	−5.03370	−0.340146
$220$	0	0
$221$	6.29146	0.423210
$222$	0	0
$223$	17.7525	1.18880	0.594398	−	0.804171i	$-0.297390\pi$
$223$	17.7525	1.18880	0.594398	+	0.804171i	$0.297390\pi$
$224$	0	0
$225$	5.14997	0.343331
$226$	0	0
$227$	−10.7884	−0.716055	−0.358027	−	0.933711i	$-0.616551\pi$
$227$	−10.7884	−0.716055	−0.358027	+	0.933711i	$0.616551\pi$
$228$	0	0
$229$	3.57264	0.236087	0.118043	−	0.993008i	$-0.462338\pi$
$229$	3.57264	0.236087	0.118043	+	0.993008i	$0.462338\pi$
$230$	0	0
$231$	1.10556	0.0727405
$232$	0	0
$233$	−7.70854	−0.505003	−0.252502	−	0.967596i	$-0.581253\pi$
$233$	−7.70854	−0.505003	−0.252502	+	0.967596i	$0.581253\pi$
$234$	0	0
$235$	−26.1598	−1.70648
$236$	0	0
$237$	−9.54699	−0.620144
$238$	0	0
$239$	9.42693	0.609777	0.304889	−	0.952388i	$-0.401381\pi$
$239$	9.42693	0.609777	0.304889	+	0.952388i	$0.401381\pi$
$240$	0	0
$241$	−15.8385	−1.02024	−0.510122	−	0.860102i	$-0.670400\pi$
$241$	−15.8385	−1.02024	−0.510122	+	0.860102i	$0.670400\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	18.4073	1.17600
$246$	0	0
$247$	3.74404	0.238227
$248$	0	0
$249$	5.11627	0.324231
$250$	0	0
$251$	−3.62734	−0.228955	−0.114478	−	0.993426i	$-0.536519\pi$
$251$	−3.62734	−0.228955	−0.114478	+	0.993426i	$0.536519\pi$
$252$	0	0
$253$	3.26624	0.205347
$254$	0	0
$255$	20.0440	1.25520
$256$	0	0
$257$	−15.6829	−0.978272	−0.489136	−	0.872208i	$-0.662688\pi$
$257$	−15.6829	−0.978272	−0.489136	+	0.872208i	$0.662688\pi$
$258$	0	0
$259$	−7.04441	−0.437718
$260$	0	0
$261$	3.60255	0.222992
$262$	0	0
$263$	−1.41622	−0.0873276	−0.0436638	−	0.999046i	$-0.513903\pi$
$263$	−1.41622	−0.0873276	−0.0436638	+	0.999046i	$0.513903\pi$
$264$	0	0
$265$	−16.9726	−1.04262
$266$	0	0
$267$	3.66370	0.224215
$268$	0	0
$269$	6.97255	0.425124	0.212562	−	0.977148i	$-0.431819\pi$
$269$	6.97255	0.425124	0.212562	+	0.977148i	$0.431819\pi$
$270$	0	0
$271$	−3.46106	−0.210244	−0.105122	−	0.994459i	$-0.533523\pi$
$271$	−3.46106	−0.210244	−0.105122	+	0.994459i	$0.533523\pi$
$272$	0	0
$273$	1.10556	0.0669115
$274$	0	0
$275$	−5.14997	−0.310555
$276$	0	0
$277$	−19.9828	−1.20065	−0.600326	−	0.799755i	$-0.704963\pi$
$277$	−19.9828	−1.20065	−0.600326	+	0.799755i	$0.704963\pi$
$278$	0	0
$279$	0.336304	0.0201340
$280$	0	0
$281$	10.7543	0.641549	0.320774	−	0.947156i	$-0.396057\pi$
$281$	10.7543	0.641549	0.320774	+	0.947156i	$0.396057\pi$
$282$	0	0
$283$	1.35218	0.0803788	0.0401894	−	0.999192i	$-0.487204\pi$
$283$	1.35218	0.0803788	0.0401894	+	0.999192i	$0.487204\pi$
$284$	0	0
$285$	11.9281	0.706562
$286$	0	0
$287$	−0.988857	−0.0583704
$288$	0	0
$289$	22.5825	1.32838
$290$	0	0
$291$	8.13926	0.477132
$292$	0	0
$293$	−9.05001	−0.528707	−0.264353	−	0.964426i	$-0.585159\pi$
$293$	−9.05001	−0.528707	−0.264353	+	0.964426i	$0.585159\pi$
$294$	0	0
$295$	−28.7059	−1.67132
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	3.26624	0.188892
$300$	0	0
$301$	9.79314	0.564467
$302$	0	0
$303$	18.5914	1.06805
$304$	0	0
$305$	47.5923	2.72513
$306$	0	0
$307$	26.9987	1.54090	0.770449	−	0.637502i	$-0.220032\pi$
$307$	26.9987	1.54090	0.770449	+	0.637502i	$0.220032\pi$
$308$	0	0
$309$	18.9384	1.07737
$310$	0	0
$311$	−8.92771	−0.506244	−0.253122	−	0.967434i	$-0.581457\pi$
$311$	−8.92771	−0.506244	−0.253122	+	0.967434i	$0.581457\pi$
$312$	0	0
$313$	2.07811	0.117462	0.0587309	−	0.998274i	$-0.481295\pi$
$313$	2.07811	0.117462	0.0587309	+	0.998274i	$0.481295\pi$
$314$	0	0
$315$	3.52221	0.198454
$316$	0	0
$317$	8.49187	0.476951	0.238475	−	0.971149i	$-0.423352\pi$
$317$	8.49187	0.476951	0.238475	+	0.971149i	$0.423352\pi$
$318$	0	0
$319$	−3.60255	−0.201704
$320$	0	0
$321$	16.0547	0.896086
$322$	0	0
$323$	23.5555	1.31066
$324$	0	0
$325$	−5.14997	−0.285669
$326$	0	0
$327$	−6.41665	−0.354841
$328$	0	0
$329$	−9.07789	−0.500480
$330$	0	0
$331$	−29.6287	−1.62854	−0.814270	−	0.580486i	$-0.802863\pi$
$331$	−29.6287	−1.62854	−0.814270	+	0.580486i	$0.802863\pi$
$332$	0	0
$333$	6.37180	0.349173
$334$	0	0
$335$	33.7773	1.84545
$336$	0	0
$337$	27.8385	1.51646	0.758229	−	0.651989i	$-0.226065\pi$
$337$	27.8385	1.51646	0.758229	+	0.651989i	$0.226065\pi$
$338$	0	0
$339$	−0.477794	−0.0259502
$340$	0	0
$341$	−0.336304	−0.0182119
$342$	0	0
$343$	14.1266	0.762762
$344$	0	0
$345$	10.4059	0.560237
$346$	0	0
$347$	8.69317	0.466674	0.233337	−	0.972396i	$-0.425035\pi$
$347$	8.69317	0.466674	0.233337	+	0.972396i	$0.425035\pi$
$348$	0	0
$349$	6.28298	0.336320	0.168160	−	0.985760i	$-0.446217\pi$
$349$	6.28298	0.336320	0.168160	+	0.985760i	$0.446217\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	10.6689	0.567846	0.283923	−	0.958847i	$-0.408364\pi$
$353$	10.6689	0.567846	0.283923	+	0.958847i	$0.408364\pi$
$354$	0	0
$355$	−16.9726	−0.900809
$356$	0	0
$357$	6.95559	0.368129
$358$	0	0
$359$	9.08413	0.479442	0.239721	−	0.970842i	$-0.422944\pi$
$359$	9.08413	0.479442	0.239721	+	0.970842i	$0.422944\pi$
$360$	0	0
$361$	−4.98218	−0.262220
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	16.0369	0.839408
$366$	0	0
$367$	−18.5821	−0.969976	−0.484988	−	0.874521i	$-0.661176\pi$
$367$	−18.5821	−0.969976	−0.484988	+	0.874521i	$0.661176\pi$
$368$	0	0
$369$	0.894440	0.0465627
$370$	0	0
$371$	−5.88975	−0.305781
$372$	0	0
$373$	−22.4769	−1.16381	−0.581906	−	0.813256i	$-0.697693\pi$
$373$	−22.4769	−1.16381	−0.581906	+	0.813256i	$0.697693\pi$
$374$	0	0
$375$	−0.477794	−0.0246732
$376$	0	0
$377$	−3.60255	−0.185541
$378$	0	0
$379$	−4.85183	−0.249222	−0.124611	−	0.992206i	$-0.539768\pi$
$379$	−4.85183	−0.249222	−0.124611	+	0.992206i	$0.539768\pi$
$380$	0	0
$381$	7.40773	0.379510
$382$	0	0
$383$	21.7162	1.10964	0.554822	−	0.831969i	$-0.312786\pi$
$383$	21.7162	1.10964	0.554822	+	0.831969i	$0.312786\pi$
$384$	0	0
$385$	−3.52221	−0.179508
$386$	0	0
$387$	−8.85808	−0.450281
$388$	0	0
$389$	−1.64876	−0.0835955	−0.0417977	−	0.999126i	$-0.513309\pi$
$389$	−1.64876	−0.0835955	−0.0417977	+	0.999126i	$0.513309\pi$
$390$	0	0
$391$	20.5494	1.03923
$392$	0	0
$393$	2.19481	0.110714
$394$	0	0
$395$	30.4158	1.53038
$396$	0	0
$397$	20.0547	1.00652	0.503258	−	0.864136i	$-0.332134\pi$
$397$	20.0547	1.00652	0.503258	+	0.864136i	$0.332134\pi$
$398$	0	0
$399$	4.13926	0.207222
$400$	0	0
$401$	14.2680	0.712512	0.356256	−	0.934388i	$-0.384053\pi$
$401$	14.2680	0.712512	0.356256	+	0.934388i	$0.384053\pi$
$402$	0	0
$403$	−0.336304	−0.0167525
$404$	0	0
$405$	−3.18590	−0.158309
$406$	0	0
$407$	−6.37180	−0.315839
$408$	0	0
$409$	39.4979	1.95305	0.976523	−	0.215411i	$-0.0691092\pi$
$409$	39.4979	1.95305	0.976523	+	0.215411i	$0.0691092\pi$
$410$	0	0
$411$	−20.5466	−1.01349
$412$	0	0
$413$	−9.96141	−0.490169
$414$	0	0
$415$	−16.2999	−0.800133
$416$	0	0
$417$	−15.1248	−0.740662
$418$	0	0
$419$	−18.3439	−0.896159	−0.448080	−	0.893994i	$-0.647892\pi$
$419$	−18.3439	−0.896159	−0.448080	+	0.893994i	$0.647892\pi$
$420$	0	0
$421$	28.0154	1.36539	0.682695	−	0.730704i	$-0.260808\pi$
$421$	28.0154	1.36539	0.682695	+	0.730704i	$0.260808\pi$
$422$	0	0
$423$	8.21112	0.399238
$424$	0	0
$425$	−32.4008	−1.57167
$426$	0	0
$427$	16.5153	0.799232
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	18.7445	0.902889	0.451445	−	0.892299i	$-0.350909\pi$
$431$	18.7445	0.902889	0.451445	+	0.892299i	$0.350909\pi$
$432$	0	0
$433$	−32.9170	−1.58189	−0.790945	−	0.611887i	$-0.790411\pi$
$433$	−32.9170	−1.58189	−0.790945	+	0.611887i	$0.790411\pi$
$434$	0	0
$435$	−11.4774	−0.550297
$436$	0	0
$437$	12.2289	0.584990
$438$	0	0
$439$	24.1804	1.15407	0.577033	−	0.816721i	$-0.304211\pi$
$439$	24.1804	1.15407	0.577033	+	0.816721i	$0.304211\pi$
$440$	0	0
$441$	−5.77774	−0.275130
$442$	0	0
$443$	−8.74361	−0.415421	−0.207711	−	0.978190i	$-0.566601\pi$
$443$	−8.74361	−0.415421	−0.207711	+	0.978190i	$0.566601\pi$
$444$	0	0
$445$	−11.6722	−0.553314
$446$	0	0
$447$	−6.99957	−0.331068
$448$	0	0
$449$	30.7969	1.45340	0.726699	−	0.686957i	$-0.241054\pi$
$449$	30.7969	1.45340	0.726699	+	0.686957i	$0.241054\pi$
$450$	0	0
$451$	−0.894440	−0.0421175
$452$	0	0
$453$	5.95516	0.279798
$454$	0	0
$455$	−3.52221	−0.165124
$456$	0	0
$457$	−21.6820	−1.01424	−0.507121	−	0.861875i	$-0.669290\pi$
$457$	−21.6820	−1.01424	−0.507121	+	0.861875i	$0.669290\pi$
$458$	0	0
$459$	−6.29146	−0.293660
$460$	0	0
$461$	38.2542	1.78168	0.890839	−	0.454320i	$-0.150118\pi$
$461$	38.2542	1.78168	0.890839	+	0.454320i	$0.150118\pi$
$462$	0	0
$463$	6.64071	0.308620	0.154310	−	0.988022i	$-0.450685\pi$
$463$	6.64071	0.308620	0.154310	+	0.988022i	$0.450685\pi$
$464$	0	0
$465$	−1.07143	−0.0496864
$466$	0	0
$467$	−12.4158	−0.574534	−0.287267	−	0.957851i	$-0.592747\pi$
$467$	−12.4158	−0.574534	−0.287267	+	0.957851i	$0.592747\pi$
$468$	0	0
$469$	11.7213	0.541238
$470$	0	0
$471$	2.91075	0.134120
$472$	0	0
$473$	8.85808	0.407295
$474$	0	0
$475$	−19.2817	−0.884705
$476$	0	0
$477$	5.32739	0.243925
$478$	0	0
$479$	−6.34990	−0.290134	−0.145067	−	0.989422i	$-0.546340\pi$
$479$	−6.34990	−0.290134	−0.145067	+	0.989422i	$0.546340\pi$
$480$	0	0
$481$	−6.37180	−0.290529
$482$	0	0
$483$	3.61103	0.164307
$484$	0	0
$485$	−25.9309	−1.17746
$486$	0	0
$487$	−16.9201	−0.766722	−0.383361	−	0.923599i	$-0.625234\pi$
$487$	−16.9201	−0.766722	−0.383361	+	0.923599i	$0.625234\pi$
$488$	0	0
$489$	6.06963	0.274478
$490$	0	0
$491$	−11.6231	−0.524542	−0.262271	−	0.964994i	$-0.584471\pi$
$491$	−11.6231	−0.524542	−0.262271	+	0.964994i	$0.584471\pi$
$492$	0	0
$493$	−22.6653	−1.02079
$494$	0	0
$495$	3.18590	0.143196
$496$	0	0
$497$	−5.88975	−0.264191
$498$	0	0
$499$	8.35261	0.373914	0.186957	−	0.982368i	$-0.440137\pi$
$499$	8.35261	0.373914	0.186957	+	0.982368i	$0.440137\pi$
$500$	0	0
$501$	20.8496	0.931491
$502$	0	0
$503$	−1.23857	−0.0552251	−0.0276125	−	0.999619i	$-0.508790\pi$
$503$	−1.23857	−0.0552251	−0.0276125	+	0.999619i	$0.508790\pi$
$504$	0	0
$505$	−59.2304	−2.63572
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	39.0576	1.73120	0.865599	−	0.500737i	$-0.166938\pi$
$509$	39.0576	1.73120	0.865599	+	0.500737i	$0.166938\pi$
$510$	0	0
$511$	5.56506	0.246184
$512$	0	0
$513$	−3.74404	−0.165303
$514$	0	0
$515$	−60.3360	−2.65872
$516$	0	0
$517$	−8.21112	−0.361125
$518$	0	0
$519$	−7.81367	−0.342982
$520$	0	0
$521$	−7.82661	−0.342890	−0.171445	−	0.985194i	$-0.554844\pi$
$521$	−7.82661	−0.342890	−0.171445	+	0.985194i	$0.554844\pi$
$522$	0	0
$523$	−13.1033	−0.572969	−0.286484	−	0.958085i	$-0.592487\pi$
$523$	−13.1033	−0.572969	−0.286484	+	0.958085i	$0.592487\pi$
$524$	0	0
$525$	−5.69360	−0.248489
$526$	0	0
$527$	−2.11584	−0.0921675
$528$	0	0
$529$	−12.3317	−0.536159
$530$	0	0
$531$	9.01028	0.391013
$532$	0	0
$533$	−0.894440	−0.0387425
$534$	0	0
$535$	−51.1487	−2.21135
$536$	0	0
$537$	6.67820	0.288186
$538$	0	0
$539$	5.77774	0.248865
$540$	0	0
$541$	11.9451	0.513560	0.256780	−	0.966470i	$-0.417338\pi$
$541$	11.9451	0.513560	0.256780	+	0.966470i	$0.417338\pi$
$542$	0	0
$543$	−12.9996	−0.557865
$544$	0	0
$545$	20.4428	0.875674
$546$	0	0
$547$	27.2504	1.16515	0.582573	−	0.812779i	$-0.302046\pi$
$547$	27.2504	1.16515	0.582573	+	0.812779i	$0.302046\pi$
$548$	0	0
$549$	−14.9384	−0.637556
$550$	0	0
$551$	−13.4881	−0.574611
$552$	0	0
$553$	10.5548	0.448835
$554$	0	0
$555$	−20.2999	−0.861685
$556$	0	0
$557$	−25.2987	−1.07194	−0.535969	−	0.844238i	$-0.680054\pi$
$557$	−25.2987	−1.07194	−0.535969	+	0.844238i	$0.680054\pi$
$558$	0	0
$559$	8.85808	0.374657
$560$	0	0
$561$	6.29146	0.265626
$562$	0	0
$563$	37.6778	1.58793	0.793964	−	0.607964i	$-0.208014\pi$
$563$	37.6778	1.58793	0.793964	+	0.607964i	$0.208014\pi$
$564$	0	0
$565$	1.52221	0.0640397
$566$	0	0
$567$	−1.10556	−0.0464292
$568$	0	0
$569$	−15.5854	−0.653373	−0.326687	−	0.945133i	$-0.605932\pi$
$569$	−15.5854	−0.653373	−0.326687	+	0.945133i	$0.605932\pi$
$570$	0	0
$571$	−4.18719	−0.175229	−0.0876143	−	0.996154i	$-0.527924\pi$
$571$	−4.18719	−0.175229	−0.0876143	+	0.996154i	$0.527924\pi$
$572$	0	0
$573$	13.8436	0.578324
$574$	0	0
$575$	−16.8211	−0.701487
$576$	0	0
$577$	33.8760	1.41028	0.705138	−	0.709070i	$-0.250885\pi$
$577$	33.8760	1.41028	0.705138	+	0.709070i	$0.250885\pi$
$578$	0	0
$579$	21.2277	0.882191
$580$	0	0
$581$	−5.65635	−0.234665
$582$	0	0
$583$	−5.32739	−0.220638
$584$	0	0
$585$	3.18590	0.131721
$586$	0	0
$587$	−17.7539	−0.732781	−0.366391	−	0.930461i	$-0.619407\pi$
$587$	−17.7539	−0.732781	−0.366391	+	0.930461i	$0.619407\pi$
$588$	0	0
$589$	−1.25913	−0.0518817
$590$	0	0
$591$	−23.9935	−0.986963
$592$	0	0
$593$	−12.6278	−0.518560	−0.259280	−	0.965802i	$-0.583485\pi$
$593$	−12.6278	−0.518560	−0.259280	+	0.965802i	$0.583485\pi$
$594$	0	0
$595$	−22.1598	−0.908464
$596$	0	0
$597$	25.9828	1.06341
$598$	0	0
$599$	27.0777	1.10636	0.553182	−	0.833060i	$-0.313413\pi$
$599$	27.0777	1.10636	0.553182	+	0.833060i	$0.313413\pi$
$600$	0	0
$601$	11.2770	0.459997	0.229998	−	0.973191i	$-0.426128\pi$
$601$	11.2770	0.459997	0.229998	+	0.973191i	$0.426128\pi$
$602$	0	0
$603$	−10.6021	−0.431752
$604$	0	0
$605$	−3.18590	−0.129525
$606$	0	0
$607$	11.5854	0.470236	0.235118	−	0.971967i	$-0.424452\pi$
$607$	11.5854	0.470236	0.235118	+	0.971967i	$0.424452\pi$
$608$	0	0
$609$	−3.98283	−0.161393
$610$	0	0
$611$	−8.21112	−0.332186
$612$	0	0
$613$	−20.5167	−0.828660	−0.414330	−	0.910127i	$-0.635984\pi$
$613$	−20.5167	−0.828660	−0.414330	+	0.910127i	$0.635984\pi$
$614$	0	0
$615$	−2.84960	−0.114907
$616$	0	0
$617$	40.8175	1.64325	0.821625	−	0.570028i	$-0.193068\pi$
$617$	40.8175	1.64325	0.821625	+	0.570028i	$0.193068\pi$
$618$	0	0
$619$	22.7619	0.914880	0.457440	−	0.889241i	$-0.348767\pi$
$619$	22.7619	0.914880	0.457440	+	0.889241i	$0.348767\pi$
$620$	0	0
$621$	−3.26624	−0.131070
$622$	0	0
$623$	−4.05044	−0.162277
$624$	0	0
$625$	−24.2277	−0.969106
$626$	0	0
$627$	3.74404	0.149522
$628$	0	0
$629$	−40.0880	−1.59841
$630$	0	0
$631$	26.2167	1.04367	0.521836	−	0.853046i	$-0.325247\pi$
$631$	26.2167	1.04367	0.521836	+	0.853046i	$0.325247\pi$
$632$	0	0
$633$	9.26401	0.368211
$634$	0	0
$635$	−23.6003	−0.936550
$636$	0	0
$637$	5.77774	0.228922
$638$	0	0
$639$	5.32739	0.210748
$640$	0	0
$641$	−28.7947	−1.13732	−0.568661	−	0.822572i	$-0.692538\pi$
$641$	−28.7947	−1.13732	−0.568661	+	0.822572i	$0.692538\pi$
$642$	0	0
$643$	−48.5251	−1.91364	−0.956822	−	0.290673i	$-0.906121\pi$
$643$	−48.5251	−1.91364	−0.956822	+	0.290673i	$0.906121\pi$
$644$	0	0
$645$	28.2210	1.11120
$646$	0	0
$647$	8.72665	0.343080	0.171540	−	0.985177i	$-0.445126\pi$
$647$	8.72665	0.343080	0.171540	+	0.985177i	$0.445126\pi$
$648$	0	0
$649$	−9.01028	−0.353685
$650$	0	0
$651$	−0.371804	−0.0145721
$652$	0	0
$653$	−43.5881	−1.70573	−0.852867	−	0.522128i	$-0.825138\pi$
$653$	−43.5881	−1.70573	−0.852867	+	0.522128i	$0.825138\pi$
$654$	0	0
$655$	−6.99246	−0.273218
$656$	0	0
$657$	−5.03370	−0.196383
$658$	0	0
$659$	−27.9665	−1.08942	−0.544711	−	0.838624i	$-0.683360\pi$
$659$	−27.9665	−1.08942	−0.544711	+	0.838624i	$0.683360\pi$
$660$	0	0
$661$	−21.4154	−0.832961	−0.416480	−	0.909145i	$-0.636737\pi$
$661$	−21.4154	−0.832961	−0.416480	+	0.909145i	$0.636737\pi$
$662$	0	0
$663$	6.29146	0.244340
$664$	0	0
$665$	−13.1873	−0.511381
$666$	0	0
$667$	−11.7668	−0.455612
$668$	0	0
$669$	17.7525	0.686352
$670$	0	0
$671$	14.9384	0.576691
$672$	0	0
$673$	−36.2486	−1.39728	−0.698641	−	0.715472i	$-0.746212\pi$
$673$	−36.2486	−1.39728	−0.698641	+	0.715472i	$0.746212\pi$
$674$	0	0
$675$	5.14997	0.198222
$676$	0	0
$677$	38.1469	1.46610	0.733052	−	0.680173i	$-0.238095\pi$
$677$	38.1469	1.46610	0.733052	+	0.680173i	$0.238095\pi$
$678$	0	0
$679$	−8.99844	−0.345328
$680$	0	0
$681$	−10.7884	−0.413414
$682$	0	0
$683$	−2.58804	−0.0990287	−0.0495143	−	0.998773i	$-0.515767\pi$
$683$	−2.58804	−0.0990287	−0.0495143	+	0.998773i	$0.515767\pi$
$684$	0	0
$685$	65.4593	2.50107
$686$	0	0
$687$	3.57264	0.136305
$688$	0	0
$689$	−5.32739	−0.202957
$690$	0	0
$691$	3.55345	0.135180	0.0675898	−	0.997713i	$-0.478469\pi$
$691$	3.55345	0.135180	0.0675898	+	0.997713i	$0.478469\pi$
$692$	0	0
$693$	1.10556	0.0419968
$694$	0	0
$695$	48.1860	1.82780
$696$	0	0
$697$	−5.62734	−0.213151
$698$	0	0
$699$	−7.70854	−0.291564
$700$	0	0
$701$	−27.9342	−1.05506	−0.527531	−	0.849536i	$-0.676882\pi$
$701$	−27.9342	−1.05506	−0.527531	+	0.849536i	$0.676882\pi$
$702$	0	0
$703$	−23.8563	−0.899757
$704$	0	0
$705$	−26.1598	−0.985236
$706$	0	0
$707$	−20.5539	−0.773009
$708$	0	0
$709$	−43.1701	−1.62129	−0.810644	−	0.585540i	$-0.800883\pi$
$709$	−43.1701	−1.62129	−0.810644	+	0.585540i	$0.800883\pi$
$710$	0	0
$711$	−9.54699	−0.358040
$712$	0	0
$713$	−1.09845	−0.0411373
$714$	0	0
$715$	−3.18590	−0.119146
$716$	0	0
$717$	9.42693	0.352055
$718$	0	0
$719$	16.4385	0.613054	0.306527	−	0.951862i	$-0.400833\pi$
$719$	16.4385	0.613054	0.306527	+	0.951862i	$0.400833\pi$
$720$	0	0
$721$	−20.9376	−0.779755
$722$	0	0
$723$	−15.8385	−0.589038
$724$	0	0
$725$	18.5530	0.689042
$726$	0	0
$727$	−50.8137	−1.88458	−0.942289	−	0.334801i	$-0.891331\pi$
$727$	−50.8137	−1.88458	−0.942289	+	0.334801i	$0.891331\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	55.7303	2.06126
$732$	0	0
$733$	−15.8842	−0.586695	−0.293347	−	0.956006i	$-0.594769\pi$
$733$	−15.8842	−0.586695	−0.293347	+	0.956006i	$0.594769\pi$
$734$	0	0
$735$	18.4073	0.678964
$736$	0	0
$737$	10.6021	0.390534
$738$	0	0
$739$	−35.7994	−1.31690	−0.658451	−	0.752624i	$-0.728788\pi$
$739$	−35.7994	−1.31690	−0.658451	+	0.752624i	$0.728788\pi$
$740$	0	0
$741$	3.74404	0.137541
$742$	0	0
$743$	−4.19842	−0.154025	−0.0770125	−	0.997030i	$-0.524538\pi$
$743$	−4.19842	−0.154025	−0.0770125	+	0.997030i	$0.524538\pi$
$744$	0	0
$745$	22.2999	0.817007
$746$	0	0
$747$	5.11627	0.187195
$748$	0	0
$749$	−17.7494	−0.648550
$750$	0	0
$751$	45.6428	1.66553	0.832764	−	0.553628i	$-0.186757\pi$
$751$	45.6428	1.66553	0.832764	+	0.553628i	$0.186757\pi$
$752$	0	0
$753$	−3.62734	−0.132187
$754$	0	0
$755$	−18.9726	−0.690482
$756$	0	0
$757$	16.8987	0.614194	0.307097	−	0.951678i	$-0.400642\pi$
$757$	16.8987	0.614194	0.307097	+	0.951678i	$0.400642\pi$
$758$	0	0
$759$	3.26624	0.118557
$760$	0	0
$761$	42.1995	1.52973	0.764866	−	0.644189i	$-0.222805\pi$
$761$	42.1995	1.52973	0.764866	+	0.644189i	$0.222805\pi$
$762$	0	0
$763$	7.09399	0.256820
$764$	0	0
$765$	20.0440	0.724692
$766$	0	0
$767$	−9.01028	−0.325342
$768$	0	0
$769$	40.3836	1.45627	0.728136	−	0.685433i	$-0.240387\pi$
$769$	40.3836	1.45627	0.728136	+	0.685433i	$0.240387\pi$
$770$	0	0
$771$	−15.6829	−0.564805
$772$	0	0
$773$	−15.3790	−0.553144	−0.276572	−	0.960993i	$-0.589198\pi$
$773$	−15.3790	−0.553144	−0.276572	+	0.960993i	$0.589198\pi$
$774$	0	0
$775$	1.73195	0.0622136
$776$	0	0
$777$	−7.04441	−0.252717
$778$	0	0
$779$	−3.34882	−0.119984
$780$	0	0
$781$	−5.32739	−0.190629
$782$	0	0
$783$	3.60255	0.128745
$784$	0	0
$785$	−9.27335	−0.330980
$786$	0	0
$787$	11.2535	0.401145	0.200573	−	0.979679i	$-0.435720\pi$
$787$	11.2535	0.401145	0.200573	+	0.979679i	$0.435720\pi$
$788$	0	0
$789$	−1.41622	−0.0504186
$790$	0	0
$791$	0.528230	0.0187817
$792$	0	0
$793$	14.9384	0.530479
$794$	0	0
$795$	−16.9726	−0.601954
$796$	0	0
$797$	−37.3770	−1.32396	−0.661980	−	0.749521i	$-0.730284\pi$
$797$	−37.3770	−1.32396	−0.661980	+	0.749521i	$0.730284\pi$
$798$	0	0
$799$	−51.6599	−1.82760
$800$	0	0
$801$	3.66370	0.129450
$802$	0	0
$803$	5.03370	0.177635
$804$	0	0
$805$	−11.5044	−0.405476
$806$	0	0
$807$	6.97255	0.245445
$808$	0	0
$809$	−1.96165	−0.0689679	−0.0344839	−	0.999405i	$-0.510979\pi$
$809$	−1.96165	−0.0689679	−0.0344839	+	0.999405i	$0.510979\pi$
$810$	0	0
$811$	−42.4542	−1.49077	−0.745384	−	0.666636i	$-0.767734\pi$
$811$	−42.4542	−1.49077	−0.745384	+	0.666636i	$0.767734\pi$
$812$	0	0
$813$	−3.46106	−0.121385
$814$	0	0
$815$	−19.3372	−0.677354
$816$	0	0
$817$	33.1650	1.16030
$818$	0	0
$819$	1.10556	0.0386314
$820$	0	0
$821$	−37.3714	−1.30427	−0.652135	−	0.758103i	$-0.726126\pi$
$821$	−37.3714	−1.30427	−0.652135	+	0.758103i	$0.726126\pi$
$822$	0	0
$823$	−44.9206	−1.56583	−0.782917	−	0.622126i	$-0.786269\pi$
$823$	−44.9206	−1.56583	−0.782917	+	0.622126i	$0.786269\pi$
$824$	0	0
$825$	−5.14997	−0.179299
$826$	0	0
$827$	52.1860	1.81468	0.907342	−	0.420393i	$-0.138108\pi$
$827$	52.1860	1.81468	0.907342	+	0.420393i	$0.138108\pi$
$828$	0	0
$829$	36.2308	1.25835	0.629174	−	0.777264i	$-0.283393\pi$
$829$	36.2308	1.25835	0.629174	+	0.777264i	$0.283393\pi$
$830$	0	0
$831$	−19.9828	−0.693197
$832$	0	0
$833$	36.3504	1.25947
$834$	0	0
$835$	−66.4248	−2.29872
$836$	0	0
$837$	0.336304	0.0116244
$838$	0	0
$839$	−3.73398	−0.128911	−0.0644557	−	0.997921i	$-0.520531\pi$
$839$	−3.73398	−0.128911	−0.0644557	+	0.997921i	$0.520531\pi$
$840$	0	0
$841$	−16.0216	−0.552471
$842$	0	0
$843$	10.7543	0.370398
$844$	0	0
$845$	−3.18590	−0.109598
$846$	0	0
$847$	−1.10556	−0.0379875
$848$	0	0
$849$	1.35218	0.0464067
$850$	0	0
$851$	−20.8119	−0.713422
$852$	0	0
$853$	3.46579	0.118666	0.0593332	−	0.998238i	$-0.481103\pi$
$853$	3.46579	0.118666	0.0593332	+	0.998238i	$0.481103\pi$
$854$	0	0
$855$	11.9281	0.407934
$856$	0	0
$857$	0.996683	0.0340460	0.0170230	−	0.999855i	$-0.494581\pi$
$857$	0.996683	0.0340460	0.0170230	+	0.999855i	$0.494581\pi$
$858$	0	0
$859$	40.1590	1.37021	0.685103	−	0.728446i	$-0.259757\pi$
$859$	40.1590	1.37021	0.685103	+	0.728446i	$0.259757\pi$
$860$	0	0
$861$	−0.988857	−0.0337002
$862$	0	0
$863$	24.3044	0.827332	0.413666	−	0.910429i	$-0.364248\pi$
$863$	24.3044	0.827332	0.413666	+	0.910429i	$0.364248\pi$
$864$	0	0
$865$	24.8936	0.846407
$866$	0	0
$867$	22.5825	0.766942
$868$	0	0
$869$	9.54699	0.323860
$870$	0	0
$871$	10.6021	0.359239
$872$	0	0
$873$	8.13926	0.275472
$874$	0	0
$875$	0.528230	0.0178574
$876$	0	0
$877$	33.5377	1.13249	0.566243	−	0.824238i	$-0.308396\pi$
$877$	33.5377	1.13249	0.566243	+	0.824238i	$0.308396\pi$
$878$	0	0
$879$	−9.05001	−0.305249
$880$	0	0
$881$	−5.10357	−0.171944	−0.0859718	−	0.996298i	$-0.527399\pi$
$881$	−5.10357	−0.171944	−0.0859718	+	0.996298i	$0.527399\pi$
$882$	0	0
$883$	44.6196	1.50157	0.750784	−	0.660547i	$-0.229676\pi$
$883$	44.6196	1.50157	0.750784	+	0.660547i	$0.229676\pi$
$884$	0	0
$885$	−28.7059	−0.964937
$886$	0	0
$887$	−50.4419	−1.69367	−0.846837	−	0.531852i	$-0.821496\pi$
$887$	−50.4419	−1.69367	−0.846837	+	0.531852i	$0.821496\pi$
$888$	0	0
$889$	−8.18969	−0.274674
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−30.7427	−1.02877
$894$	0	0
$895$	−21.2761	−0.711181
$896$	0	0
$897$	3.26624	0.109057
$898$	0	0
$899$	1.21155	0.0404075
$900$	0	0
$901$	−33.5171	−1.11662
$902$	0	0
$903$	9.79314	0.325895
$904$	0	0
$905$	41.4154	1.37669
$906$	0	0
$907$	41.5939	1.38110	0.690551	−	0.723284i	$-0.257368\pi$
$907$	41.5939	1.38110	0.690551	+	0.723284i	$0.257368\pi$
$908$	0	0
$909$	18.5914	0.616638
$910$	0	0
$911$	10.8155	0.358332	0.179166	−	0.983819i	$-0.442660\pi$
$911$	10.8155	0.358332	0.179166	+	0.983819i	$0.442660\pi$
$912$	0	0
$913$	−5.11627	−0.169324
$914$	0	0
$915$	47.5923	1.57335
$916$	0	0
$917$	−2.42650	−0.0801300
$918$	0	0
$919$	−12.4231	−0.409801	−0.204901	−	0.978783i	$-0.565687\pi$
$919$	−12.4231	−0.409801	−0.204901	+	0.978783i	$0.565687\pi$
$920$	0	0
$921$	26.9987	0.889638
$922$	0	0
$923$	−5.32739	−0.175353
$924$	0	0
$925$	32.8146	1.07894
$926$	0	0
$927$	18.9384	0.622019
$928$	0	0
$929$	−17.4481	−0.572454	−0.286227	−	0.958162i	$-0.592401\pi$
$929$	−17.4481	−0.572454	−0.286227	+	0.958162i	$0.592401\pi$
$930$	0	0
$931$	21.6321	0.708962
$932$	0	0
$933$	−8.92771	−0.292280
$934$	0	0
$935$	−20.0440	−0.655508
$936$	0	0
$937$	15.4582	0.504998	0.252499	−	0.967597i	$-0.418748\pi$
$937$	15.4582	0.504998	0.252499	+	0.967597i	$0.418748\pi$
$938$	0	0
$939$	2.07811	0.0678166
$940$	0	0
$941$	11.7432	0.382817	0.191408	−	0.981510i	$-0.438695\pi$
$941$	11.7432	0.382817	0.191408	+	0.981510i	$0.438695\pi$
$942$	0	0
$943$	−2.92146	−0.0951358
$944$	0	0
$945$	3.52221	0.114577
$946$	0	0
$947$	37.4238	1.21611	0.608055	−	0.793895i	$-0.291950\pi$
$947$	37.4238	1.21611	0.608055	+	0.793895i	$0.291950\pi$
$948$	0	0
$949$	5.03370	0.163401
$950$	0	0
$951$	8.49187	0.275368
$952$	0	0
$953$	2.71010	0.0877887	0.0438944	−	0.999036i	$-0.486024\pi$
$953$	2.71010	0.0877887	0.0438944	+	0.999036i	$0.486024\pi$
$954$	0	0
$955$	−44.1043	−1.42718
$956$	0	0
$957$	−3.60255	−0.116454
$958$	0	0
$959$	22.7155	0.733520
$960$	0	0
$961$	−30.8869	−0.996352
$962$	0	0
$963$	16.0547	0.517355
$964$	0	0
$965$	−67.6292	−2.17706
$966$	0	0
$967$	−23.7559	−0.763938	−0.381969	−	0.924175i	$-0.624754\pi$
$967$	−23.7559	−0.763938	−0.381969	+	0.924175i	$0.624754\pi$
$968$	0	0
$969$	23.5555	0.756711
$970$	0	0
$971$	−18.6286	−0.597821	−0.298911	−	0.954281i	$-0.596623\pi$
$971$	−18.6286	−0.597821	−0.298911	+	0.954281i	$0.596623\pi$
$972$	0	0
$973$	16.7213	0.536061
$974$	0	0
$975$	−5.14997	−0.164931
$976$	0	0
$977$	−28.7287	−0.919112	−0.459556	−	0.888149i	$-0.651991\pi$
$977$	−28.7287	−0.919112	−0.459556	+	0.888149i	$0.651991\pi$
$978$	0	0
$979$	−3.66370	−0.117092
$980$	0	0
$981$	−6.41665	−0.204868
$982$	0	0
$983$	−25.6863	−0.819265	−0.409633	−	0.912251i	$-0.634343\pi$
$983$	−25.6863	−0.819265	−0.409633	+	0.912251i	$0.634343\pi$
$984$	0	0
$985$	76.4411	2.43562
$986$	0	0
$987$	−9.07789	−0.288952
$988$	0	0
$989$	28.9326	0.920005
$990$	0	0
$991$	8.58804	0.272808	0.136404	−	0.990653i	$-0.456445\pi$
$991$	8.58804	0.272808	0.136404	+	0.990653i	$0.456445\pi$
$992$	0	0
$993$	−29.6287	−0.940239
$994$	0	0
$995$	−82.7788	−2.62426
$996$	0	0
$997$	12.8104	0.405708	0.202854	−	0.979209i	$-0.434978\pi$
$997$	12.8104	0.405708	0.202854	+	0.979209i	$0.434978\pi$
$998$	0	0
$999$	6.37180	0.201595

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.18590 q^{5} -1.10556 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.18590 q^{5} -1.10556 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -3.18590 q^{15} -6.29146 q^{17} -3.74404 q^{19} -1.10556 q^{21} -3.26624 q^{23} +5.14997 q^{25} +1.00000 q^{27} +3.60255 q^{29} +0.336304 q^{31} -1.00000 q^{33} +3.52221 q^{35} +6.37180 q^{37} -1.00000 q^{39} +0.894440 q^{41} -8.85808 q^{43} -3.18590 q^{45} +8.21112 q^{47} -5.77774 q^{49} -6.29146 q^{51} +5.32739 q^{53} +3.18590 q^{55} -3.74404 q^{57} +9.01028 q^{59} -14.9384 q^{61} -1.10556 q^{63} +3.18590 q^{65} -10.6021 q^{67} -3.26624 q^{69} +5.32739 q^{71} -5.03370 q^{73} +5.14997 q^{75} +1.10556 q^{77} -9.54699 q^{79} +1.00000 q^{81} +5.11627 q^{83} +20.0440 q^{85} +3.60255 q^{87} +3.66370 q^{89} +1.10556 q^{91} +0.336304 q^{93} +11.9281 q^{95} +8.13926 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) 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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