LMFDB - Embedded newform 6864.2.a.bw.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:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.699291$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.79091 q^{5} -4.65095 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.79091 q^{5} -4.65095 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.79091 q^{15} -0.860039 q^{17} -1.39858 q^{19} +4.65095 q^{21} +4.65095 q^{23} +9.37103 q^{25} -1.00000 q^{27} +5.18950 q^{29} -5.46146 q^{31} +1.00000 q^{33} +17.6314 q^{35} +0.279922 q^{37} +1.00000 q^{39} -3.72804 q^{41} +4.90957 q^{43} -3.79091 q^{45} -8.00000 q^{47} +14.6314 q^{49} +0.860039 q^{51} +6.00000 q^{53} +3.79091 q^{55} +1.39858 q^{57} +3.25237 q^{59} +5.25237 q^{61} -4.65095 q^{63} +3.79091 q^{65} +13.3727 q^{67} -4.65095 q^{69} +3.58183 q^{71} +13.6729 q^{73} -9.37103 q^{75} +4.65095 q^{77} -5.13996 q^{79} +1.00000 q^{81} -16.0991 q^{83} +3.26033 q^{85} -5.18950 q^{87} -15.5605 q^{89} +4.65095 q^{91} +5.46146 q^{93} +5.30191 q^{95} +0.279922 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{17} - 2 q^{19} + q^{21} + q^{23} + 17 q^{25} - 4 q^{27} + q^{29} - 24 q^{31} + 4 q^{33} + 17 q^{35} + 4 q^{37} + 4 q^{39} + 7 q^{41} - 3 q^{43} + q^{45} - 32 q^{47} + 5 q^{49} + 2 q^{51} + 24 q^{53} - q^{55} + 2 q^{57} - q^{59} + 7 q^{61} - q^{63} - q^{65} + 5 q^{67} - q^{69} - 18 q^{71} - q^{73} - 17 q^{75} + q^{77} - 22 q^{79} + 4 q^{81} - 22 q^{83} - 20 q^{85} - q^{87} - 22 q^{89} + q^{91} + 24 q^{93} - 14 q^{95} + 4 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 - 2 * q^17 - 2 * q^19 + q^21 + q^23 + 17 * q^25 - 4 * q^27 + q^29 - 24 * q^31 + 4 * q^33 + 17 * q^35 + 4 * q^37 + 4 * q^39 + 7 * q^41 - 3 * q^43 + q^45 - 32 * q^47 + 5 * q^49 + 2 * q^51 + 24 * q^53 - q^55 + 2 * q^57 - q^59 + 7 * q^61 - q^63 - q^65 + 5 * q^67 - q^69 - 18 * q^71 - q^73 - 17 * q^75 + q^77 - 22 * q^79 + 4 * q^81 - 22 * q^83 - 20 * q^85 - q^87 - 22 * q^89 + q^91 + 24 * q^93 - 14 * q^95 + 4 * 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
$5$	−3.79091	−1.69535	−0.847674	−	0.530517i	$-0.821998\pi$
$5$	−3.79091	−1.69535	−0.847674	+	0.530517i	$0.821998\pi$
$6$	0	0
$7$	−4.65095	−1.75790	−0.878948	−	0.476919i	$-0.841754\pi$
$7$	−4.65095	−1.75790	−0.878948	+	0.476919i	$0.841754\pi$
$8$	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
$15$	3.79091	0.978810
$16$	0	0
$17$	−0.860039	−0.208590	−0.104295	−	0.994546i	$-0.533259\pi$
$17$	−0.860039	−0.208590	−0.104295	+	0.994546i	$0.533259\pi$
$18$	0	0
$19$	−1.39858	−0.320857	−0.160428	−	0.987047i	$-0.551288\pi$
$19$	−1.39858	−0.320857	−0.160428	+	0.987047i	$0.551288\pi$
$20$	0	0
$21$	4.65095	1.01492
$22$	0	0
$23$	4.65095	0.969791	0.484895	−	0.874572i	$-0.338858\pi$
$23$	4.65095	0.969791	0.484895	+	0.874572i	$0.338858\pi$
$24$	0	0
$25$	9.37103	1.87421
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.18950	0.963665	0.481833	−	0.876263i	$-0.339971\pi$
$29$	5.18950	0.963665	0.481833	+	0.876263i	$0.339971\pi$
$30$	0	0
$31$	−5.46146	−0.980907	−0.490453	−	0.871467i	$-0.663169\pi$
$31$	−5.46146	−0.980907	−0.490453	+	0.871467i	$0.663169\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	17.6314	2.98024
$36$	0	0
$37$	0.279922	0.0460190	0.0230095	−	0.999735i	$-0.492675\pi$
$37$	0.279922	0.0460190	0.0230095	+	0.999735i	$0.492675\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.72804	−0.582222	−0.291111	−	0.956689i	$-0.594025\pi$
$41$	−3.72804	−0.582222	−0.291111	+	0.956689i	$0.594025\pi$
$42$	0	0
$43$	4.90957	0.748703	0.374352	−	0.927287i	$-0.377865\pi$
$43$	4.90957	0.748703	0.374352	+	0.927287i	$0.377865\pi$
$44$	0	0
$45$	−3.79091	−0.565116
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	14.6314	2.09019
$50$	0	0
$51$	0.860039	0.120430
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	3.79091	0.511167
$56$	0	0
$57$	1.39858	0.185247
$58$	0	0
$59$	3.25237	0.423423	0.211711	−	0.977332i	$-0.432096\pi$
$59$	3.25237	0.423423	0.211711	+	0.977332i	$0.432096\pi$
$60$	0	0
$61$	5.25237	0.672497	0.336249	−	0.941773i	$-0.390842\pi$
$61$	5.25237	0.672497	0.336249	+	0.941773i	$0.390842\pi$
$62$	0	0
$63$	−4.65095	−0.585965
$64$	0	0
$65$	3.79091	0.470205
$66$	0	0
$67$	13.3727	1.63374	0.816870	−	0.576821i	$-0.195707\pi$
$67$	13.3727	1.63374	0.816870	+	0.576821i	$0.195707\pi$
$68$	0	0
$69$	−4.65095	−0.559909
$70$	0	0
$71$	3.58183	0.425085	0.212542	−	0.977152i	$-0.431826\pi$
$71$	3.58183	0.425085	0.212542	+	0.977152i	$0.431826\pi$
$72$	0	0
$73$	13.6729	1.60030	0.800148	−	0.599802i	$-0.204754\pi$
$73$	13.6729	1.60030	0.800148	+	0.599802i	$0.204754\pi$
$74$	0	0
$75$	−9.37103	−1.08207
$76$	0	0
$77$	4.65095	0.530025
$78$	0	0
$79$	−5.13996	−0.578291	−0.289145	−	0.957285i	$-0.593371\pi$
$79$	−5.13996	−0.578291	−0.289145	+	0.957285i	$0.593371\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.0991	−1.76710	−0.883551	−	0.468334i	$-0.844854\pi$
$83$	−16.0991	−1.76710	−0.883551	+	0.468334i	$0.844854\pi$
$84$	0	0
$85$	3.26033	0.353633
$86$	0	0
$87$	−5.18950	−0.556372
$88$	0	0
$89$	−15.5605	−1.64941	−0.824706	−	0.565561i	$-0.808660\pi$
$89$	−15.5605	−1.64941	−0.824706	+	0.565561i	$0.808660\pi$
$90$	0	0
$91$	4.65095	0.487552
$92$	0	0
$93$	5.46146	0.566327
$94$	0	0
$95$	5.30191	0.543964
$96$	0	0
$97$	0.279922	0.0284218	0.0142109	−	0.999899i	$-0.495476\pi$
$97$	0.279922	0.0284218	0.0142109	+	0.999899i	$0.495476\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	13.5190	1.34519	0.672593	−	0.740012i	$-0.265180\pi$
$101$	13.5190	1.34519	0.672593	+	0.740012i	$0.265180\pi$
$102$	0	0
$103$	2.69253	0.265302	0.132651	−	0.991163i	$-0.457651\pi$
$103$	2.69253	0.265302	0.132651	+	0.991163i	$0.457651\pi$
$104$	0	0
$105$	−17.6314	−1.72064
$106$	0	0
$107$	13.1141	1.26779	0.633895	−	0.773419i	$-0.281455\pi$
$107$	13.1141	1.26779	0.633895	+	0.773419i	$0.281455\pi$
$108$	0	0
$109$	−0.0415727	−0.00398195	−0.00199097	−	0.999998i	$-0.500634\pi$
$109$	−0.0415727	−0.00398195	−0.00199097	+	0.999998i	$0.500634\pi$
$110$	0	0
$111$	−0.279922	−0.0265691
$112$	0	0
$113$	6.32946	0.595425	0.297713	−	0.954656i	$-0.403776\pi$
$113$	6.32946	0.595425	0.297713	+	0.954656i	$0.403776\pi$
$114$	0	0
$115$	−17.6314	−1.64413
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.72804	0.336146
$124$	0	0
$125$	−16.5702	−1.48208
$126$	0	0
$127$	−19.2265	−1.70608	−0.853039	−	0.521847i	$-0.825243\pi$
$127$	−19.2265	−1.70608	−0.853039	+	0.521847i	$0.825243\pi$
$128$	0	0
$129$	−4.90957	−0.432264
$130$	0	0
$131$	18.5668	1.62219	0.811093	−	0.584917i	$-0.198873\pi$
$131$	18.5668	1.62219	0.811093	+	0.584917i	$0.198873\pi$
$132$	0	0
$133$	6.50474	0.564033
$134$	0	0
$135$	3.79091	0.326270
$136$	0	0
$137$	12.3452	1.05472	0.527360	−	0.849642i	$-0.323182\pi$
$137$	12.3452	1.05472	0.527360	+	0.849642i	$0.323182\pi$
$138$	0	0
$139$	−15.5190	−1.31630	−0.658150	−	0.752887i	$-0.728661\pi$
$139$	−15.5190	−1.31630	−0.658150	+	0.752887i	$0.728661\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−19.6729	−1.63375
$146$	0	0
$147$	−14.6314	−1.20677
$148$	0	0
$149$	−3.52433	−0.288724	−0.144362	−	0.989525i	$-0.546113\pi$
$149$	−3.52433	−0.288724	−0.144362	+	0.989525i	$0.546113\pi$
$150$	0	0
$151$	−11.9033	−0.968679	−0.484339	−	0.874880i	$-0.660940\pi$
$151$	−11.9033	−0.968679	−0.484339	+	0.874880i	$0.660940\pi$
$152$	0	0
$153$	−0.860039	−0.0695300
$154$	0	0
$155$	20.7039	1.66298
$156$	0	0
$157$	−21.0670	−1.68133	−0.840664	−	0.541557i	$-0.817835\pi$
$157$	−21.0670	−1.68133	−0.840664	+	0.541557i	$0.817835\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−21.6314	−1.70479
$162$	0	0
$163$	0.588079	0.0460619	0.0230310	−	0.999735i	$-0.492668\pi$
$163$	0.588079	0.0460619	0.0230310	+	0.999735i	$0.492668\pi$
$164$	0	0
$165$	−3.79091	−0.295122
$166$	0	0
$167$	−16.4285	−1.27128	−0.635639	−	0.771987i	$-0.719263\pi$
$167$	−16.4285	−1.27128	−0.635639	+	0.771987i	$0.719263\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.39858	−0.106952
$172$	0	0
$173$	15.0512	1.14433	0.572163	−	0.820140i	$-0.306105\pi$
$173$	15.0512	1.14433	0.572163	+	0.820140i	$0.306105\pi$
$174$	0	0
$175$	−43.5842	−3.29466
$176$	0	0
$177$	−3.25237	−0.244463
$178$	0	0
$179$	−20.7871	−1.55370	−0.776849	−	0.629687i	$-0.783183\pi$
$179$	−20.7871	−1.55370	−0.776849	+	0.629687i	$0.783183\pi$
$180$	0	0
$181$	14.4632	1.07504	0.537519	−	0.843251i	$-0.319361\pi$
$181$	14.4632	1.07504	0.537519	+	0.843251i	$0.319361\pi$
$182$	0	0
$183$	−5.25237	−0.388266
$184$	0	0
$185$	−1.06116	−0.0780182
$186$	0	0
$187$	0.860039	0.0628923
$188$	0	0
$189$	4.65095	0.338307
$190$	0	0
$191$	7.76961	0.562189	0.281095	−	0.959680i	$-0.409302\pi$
$191$	7.76961	0.562189	0.281095	+	0.959680i	$0.409302\pi$
$192$	0	0
$193$	9.34348	0.672558	0.336279	−	0.941762i	$-0.390831\pi$
$193$	9.34348	0.672558	0.336279	+	0.941762i	$0.390831\pi$
$194$	0	0
$195$	−3.79091	−0.271473
$196$	0	0
$197$	20.2823	1.44506	0.722528	−	0.691342i	$-0.242980\pi$
$197$	20.2823	1.44506	0.722528	+	0.691342i	$0.242980\pi$
$198$	0	0
$199$	−15.2524	−1.08121	−0.540606	−	0.841276i	$-0.681805\pi$
$199$	−15.2524	−1.08121	−0.540606	+	0.841276i	$0.681805\pi$
$200$	0	0
$201$	−13.3727	−0.943241
$202$	0	0
$203$	−24.1361	−1.69402
$204$	0	0
$205$	14.1327	0.987069
$206$	0	0
$207$	4.65095	0.323264
$208$	0	0
$209$	1.39858	0.0967420
$210$	0	0
$211$	−19.0017	−1.30813	−0.654066	−	0.756438i	$-0.726938\pi$
$211$	−19.0017	−1.30813	−0.654066	+	0.756438i	$0.726938\pi$
$212$	0	0
$213$	−3.58183	−0.245423
$214$	0	0
$215$	−18.6118	−1.26931
$216$	0	0
$217$	25.4010	1.72433
$218$	0	0
$219$	−13.6729	−0.923931
$220$	0	0
$221$	0.860039	0.0578525
$222$	0	0
$223$	−23.9395	−1.60311	−0.801554	−	0.597922i	$-0.795993\pi$
$223$	−23.9395	−1.60311	−0.801554	+	0.597922i	$0.795993\pi$
$224$	0	0
$225$	9.37103	0.624735
$226$	0	0
$227$	26.9412	1.78815	0.894076	−	0.447915i	$-0.147833\pi$
$227$	26.9412	1.78815	0.894076	+	0.447915i	$0.147833\pi$
$228$	0	0
$229$	−24.2744	−1.60409	−0.802047	−	0.597261i	$-0.796256\pi$
$229$	−24.2744	−1.60409	−0.802047	+	0.597261i	$0.796256\pi$
$230$	0	0
$231$	−4.65095	−0.306010
$232$	0	0
$233$	15.2390	0.998342	0.499171	−	0.866503i	$-0.333638\pi$
$233$	15.2390	0.998342	0.499171	+	0.866503i	$0.333638\pi$
$234$	0	0
$235$	30.3273	1.97834
$236$	0	0
$237$	5.13996	0.333876
$238$	0	0
$239$	−3.71554	−0.240338	−0.120169	−	0.992753i	$-0.538344\pi$
$239$	−3.71554	−0.240338	−0.120169	+	0.992753i	$0.538344\pi$
$240$	0	0
$241$	−1.35701	−0.0874127	−0.0437063	−	0.999044i	$-0.513917\pi$
$241$	−1.35701	−0.0874127	−0.0437063	+	0.999044i	$0.513917\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−55.4662	−3.54361
$246$	0	0
$247$	1.39858	0.0889897
$248$	0	0
$249$	16.0991	1.02024
$250$	0	0
$251$	5.30191	0.334653	0.167327	−	0.985902i	$-0.446487\pi$
$251$	5.30191	0.334653	0.167327	+	0.985902i	$0.446487\pi$
$252$	0	0
$253$	−4.65095	−0.292403
$254$	0	0
$255$	−3.26033	−0.204170
$256$	0	0
$257$	7.11412	0.443767	0.221883	−	0.975073i	$-0.428780\pi$
$257$	7.11412	0.443767	0.221883	+	0.975073i	$0.428780\pi$
$258$	0	0
$259$	−1.30191	−0.0808965
$260$	0	0
$261$	5.18950	0.321222
$262$	0	0
$263$	−12.8837	−0.794445	−0.397223	−	0.917722i	$-0.630026\pi$
$263$	−12.8837	−0.794445	−0.397223	+	0.917722i	$0.630026\pi$
$264$	0	0
$265$	−22.7455	−1.39724
$266$	0	0
$267$	15.5605	0.952289
$268$	0	0
$269$	−19.8191	−1.20839	−0.604197	−	0.796835i	$-0.706506\pi$
$269$	−19.8191	−1.20839	−0.604197	+	0.796835i	$0.706506\pi$
$270$	0	0
$271$	17.4977	1.06291	0.531453	−	0.847088i	$-0.321646\pi$
$271$	17.4977	1.06291	0.531453	+	0.847088i	$0.321646\pi$
$272$	0	0
$273$	−4.65095	−0.281488
$274$	0	0
$275$	−9.37103	−0.565094
$276$	0	0
$277$	−10.9884	−0.660227	−0.330114	−	0.943941i	$-0.607087\pi$
$277$	−10.9884	−0.660227	−0.330114	+	0.943941i	$0.607087\pi$
$278$	0	0
$279$	−5.46146	−0.326969
$280$	0	0
$281$	7.53469	0.449482	0.224741	−	0.974419i	$-0.427846\pi$
$281$	7.53469	0.449482	0.224741	+	0.974419i	$0.427846\pi$
$282$	0	0
$283$	−0.727358	−0.0432369	−0.0216185	−	0.999766i	$-0.506882\pi$
$283$	−0.727358	−0.0432369	−0.0216185	+	0.999766i	$0.506882\pi$
$284$	0	0
$285$	−5.30191	−0.314058
$286$	0	0
$287$	17.3389	1.02349
$288$	0	0
$289$	−16.2603	−0.956490
$290$	0	0
$291$	−0.279922	−0.0164093
$292$	0	0
$293$	7.06356	0.412657	0.206329	−	0.978483i	$-0.433848\pi$
$293$	7.06356	0.412657	0.206329	+	0.978483i	$0.433848\pi$
$294$	0	0
$295$	−12.3295	−0.717849
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−4.65095	−0.268972
$300$	0	0
$301$	−22.8342	−1.31614
$302$	0	0
$303$	−13.5190	−0.776644
$304$	0	0
$305$	−19.9113	−1.14012
$306$	0	0
$307$	18.9679	1.08256	0.541278	−	0.840844i	$-0.317941\pi$
$307$	18.9679	1.08256	0.541278	+	0.840844i	$0.317941\pi$
$308$	0	0
$309$	−2.69253	−0.153172
$310$	0	0
$311$	−21.3469	−1.21047	−0.605236	−	0.796046i	$-0.706921\pi$
$311$	−21.3469	−1.21047	−0.605236	+	0.796046i	$0.706921\pi$
$312$	0	0
$313$	−0.831805	−0.0470164	−0.0235082	−	0.999724i	$-0.507484\pi$
$313$	−0.831805	−0.0470164	−0.0235082	+	0.999724i	$0.507484\pi$
$314$	0	0
$315$	17.6314	0.993415
$316$	0	0
$317$	−16.2565	−0.913055	−0.456527	−	0.889709i	$-0.650907\pi$
$317$	−16.2565	−0.913055	−0.456527	+	0.889709i	$0.650907\pi$
$318$	0	0
$319$	−5.18950	−0.290556
$320$	0	0
$321$	−13.1141	−0.731959
$322$	0	0
$323$	1.20284	0.0669275
$324$	0	0
$325$	−9.37103	−0.519811
$326$	0	0
$327$	0.0415727	0.00229898
$328$	0	0
$329$	37.2076	2.05132
$330$	0	0
$331$	17.4985	0.961804	0.480902	−	0.876774i	$-0.340309\pi$
$331$	17.4985	0.961804	0.480902	+	0.876774i	$0.340309\pi$
$332$	0	0
$333$	0.279922	0.0153397
$334$	0	0
$335$	−50.6949	−2.76976
$336$	0	0
$337$	−14.3239	−0.780272	−0.390136	−	0.920757i	$-0.627572\pi$
$337$	−14.3239	−0.780272	−0.390136	+	0.920757i	$0.627572\pi$
$338$	0	0
$339$	−6.32946	−0.343769
$340$	0	0
$341$	5.46146	0.295754
$342$	0	0
$343$	−35.4931	−1.91645
$344$	0	0
$345$	17.6314	0.949241
$346$	0	0
$347$	13.0345	0.699728	0.349864	−	0.936801i	$-0.386228\pi$
$347$	13.0345	0.699728	0.349864	+	0.936801i	$0.386228\pi$
$348$	0	0
$349$	−4.69809	−0.251483	−0.125742	−	0.992063i	$-0.540131\pi$
$349$	−4.69809	−0.251483	−0.125742	+	0.992063i	$0.540131\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	8.86243	0.471700	0.235850	−	0.971790i	$-0.424213\pi$
$353$	8.86243	0.471700	0.235850	+	0.971790i	$0.424213\pi$
$354$	0	0
$355$	−13.5784	−0.720667
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	19.7155	1.04055	0.520273	−	0.854000i	$-0.325830\pi$
$359$	19.7155	1.04055	0.520273	+	0.854000i	$0.325830\pi$
$360$	0	0
$361$	−17.0440	−0.897051
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−51.8329	−2.71306
$366$	0	0
$367$	−11.4402	−0.597171	−0.298586	−	0.954383i	$-0.596515\pi$
$367$	−11.4402	−0.597171	−0.298586	+	0.954383i	$0.596515\pi$
$368$	0	0
$369$	−3.72804	−0.194074
$370$	0	0
$371$	−27.9057	−1.44879
$372$	0	0
$373$	−25.3514	−1.31265	−0.656324	−	0.754479i	$-0.727890\pi$
$373$	−25.3514	−1.31265	−0.656324	+	0.754479i	$0.727890\pi$
$374$	0	0
$375$	16.5702	0.855682
$376$	0	0
$377$	−5.18950	−0.267273
$378$	0	0
$379$	24.3060	1.24852	0.624258	−	0.781218i	$-0.285401\pi$
$379$	24.3060	1.24852	0.624258	+	0.781218i	$0.285401\pi$
$380$	0	0
$381$	19.2265	0.985005
$382$	0	0
$383$	25.5427	1.30517	0.652584	−	0.757716i	$-0.273685\pi$
$383$	25.5427	1.30517	0.652584	+	0.757716i	$0.273685\pi$
$384$	0	0
$385$	−17.6314	−0.898578
$386$	0	0
$387$	4.90957	0.249568
$388$	0	0
$389$	0.922913	0.0467935	0.0233968	−	0.999726i	$-0.492552\pi$
$389$	0.922913	0.0467935	0.0233968	+	0.999726i	$0.492552\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	−18.5668	−0.936570
$394$	0	0
$395$	19.4852	0.980404
$396$	0	0
$397$	−5.15330	−0.258637	−0.129318	−	0.991603i	$-0.541279\pi$
$397$	−5.15330	−0.258637	−0.129318	+	0.991603i	$0.541279\pi$
$398$	0	0
$399$	−6.50474	−0.325644
$400$	0	0
$401$	9.71470	0.485129	0.242565	−	0.970135i	$-0.422011\pi$
$401$	9.71470	0.485129	0.242565	+	0.970135i	$0.422011\pi$
$402$	0	0
$403$	5.46146	0.272055
$404$	0	0
$405$	−3.79091	−0.188372
$406$	0	0
$407$	−0.279922	−0.0138752
$408$	0	0
$409$	−5.58979	−0.276397	−0.138199	−	0.990405i	$-0.544131\pi$
$409$	−5.58979	−0.276397	−0.138199	+	0.990405i	$0.544131\pi$
$410$	0	0
$411$	−12.3452	−0.608943
$412$	0	0
$413$	−15.1266	−0.744332
$414$	0	0
$415$	61.0302	2.99585
$416$	0	0
$417$	15.5190	0.759966
$418$	0	0
$419$	8.22242	0.401692	0.200846	−	0.979623i	$-0.435631\pi$
$419$	8.22242	0.401692	0.200846	+	0.979623i	$0.435631\pi$
$420$	0	0
$421$	−12.2744	−0.598215	−0.299108	−	0.954219i	$-0.596689\pi$
$421$	−12.2744	−0.598215	−0.299108	+	0.954219i	$0.596689\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	−8.05945	−0.390941
$426$	0	0
$427$	−24.4285	−1.18218
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	30.5207	1.47013	0.735064	−	0.677997i	$-0.237152\pi$
$431$	30.5207	1.47013	0.735064	+	0.677997i	$0.237152\pi$
$432$	0	0
$433$	9.12662	0.438597	0.219299	−	0.975658i	$-0.429623\pi$
$433$	9.12662	0.438597	0.219299	+	0.975658i	$0.429623\pi$
$434$	0	0
$435$	19.6729	0.943245
$436$	0	0
$437$	−6.50474	−0.311164
$438$	0	0
$439$	−35.9686	−1.71669	−0.858344	−	0.513075i	$-0.828506\pi$
$439$	−35.9686	−1.71669	−0.858344	+	0.513075i	$0.828506\pi$
$440$	0	0
$441$	14.6314	0.696732
$442$	0	0
$443$	−16.7580	−0.796196	−0.398098	−	0.917343i	$-0.630330\pi$
$443$	−16.7580	−0.796196	−0.398098	+	0.917343i	$0.630330\pi$
$444$	0	0
$445$	58.9886	2.79633
$446$	0	0
$447$	3.52433	0.166695
$448$	0	0
$449$	8.80596	0.415579	0.207790	−	0.978174i	$-0.433373\pi$
$449$	8.80596	0.415579	0.207790	+	0.978174i	$0.433373\pi$
$450$	0	0
$451$	3.72804	0.175547
$452$	0	0
$453$	11.9033	0.559267
$454$	0	0
$455$	−17.6314	−0.826571
$456$	0	0
$457$	−28.3734	−1.32725	−0.663626	−	0.748064i	$-0.730984\pi$
$457$	−28.3734	−1.32725	−0.663626	+	0.748064i	$0.730984\pi$
$458$	0	0
$459$	0.860039	0.0401432
$460$	0	0
$461$	5.44255	0.253485	0.126742	−	0.991936i	$-0.459548\pi$
$461$	5.44255	0.253485	0.126742	+	0.991936i	$0.459548\pi$
$462$	0	0
$463$	38.6850	1.79784	0.898922	−	0.438108i	$-0.144351\pi$
$463$	38.6850	1.79784	0.898922	+	0.438108i	$0.144351\pi$
$464$	0	0
$465$	−20.7039	−0.960121
$466$	0	0
$467$	−10.3766	−0.480172	−0.240086	−	0.970752i	$-0.577176\pi$
$467$	−10.3766	−0.480172	−0.240086	+	0.970752i	$0.577176\pi$
$468$	0	0
$469$	−62.1960	−2.87194
$470$	0	0
$471$	21.0670	0.970715
$472$	0	0
$473$	−4.90957	−0.225742
$474$	0	0
$475$	−13.1062	−0.601352
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	38.0679	1.73936	0.869682	−	0.493612i	$-0.164324\pi$
$479$	38.0679	1.73936	0.869682	+	0.493612i	$0.164324\pi$
$480$	0	0
$481$	−0.279922	−0.0127634
$482$	0	0
$483$	21.6314	0.984261
$484$	0	0
$485$	−1.06116	−0.0481849
$486$	0	0
$487$	−7.96620	−0.360983	−0.180491	−	0.983577i	$-0.557769\pi$
$487$	−7.96620	−0.360983	−0.180491	+	0.983577i	$0.557769\pi$
$488$	0	0
$489$	−0.588079	−0.0265939
$490$	0	0
$491$	−26.6658	−1.20341	−0.601706	−	0.798717i	$-0.705512\pi$
$491$	−26.6658	−1.20341	−0.601706	+	0.798717i	$0.705512\pi$
$492$	0	0
$493$	−4.46317	−0.201011
$494$	0	0
$495$	3.79091	0.170389
$496$	0	0
$497$	−16.6589	−0.747254
$498$	0	0
$499$	12.4623	0.557891	0.278945	−	0.960307i	$-0.410015\pi$
$499$	12.4623	0.557891	0.278945	+	0.960307i	$0.410015\pi$
$500$	0	0
$501$	16.4285	0.733973
$502$	0	0
$503$	10.2799	0.458359	0.229180	−	0.973384i	$-0.426396\pi$
$503$	10.2799	0.458359	0.229180	+	0.973384i	$0.426396\pi$
$504$	0	0
$505$	−51.2492	−2.28056
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−21.7414	−0.963670	−0.481835	−	0.876262i	$-0.660029\pi$
$509$	−21.7414	−0.963670	−0.481835	+	0.876262i	$0.660029\pi$
$510$	0	0
$511$	−63.5922	−2.81315
$512$	0	0
$513$	1.39858	0.0617489
$514$	0	0
$515$	−10.2071	−0.449780
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−15.0512	−0.660676
$520$	0	0
$521$	39.8295	1.74496	0.872481	−	0.488648i	$-0.162510\pi$
$521$	39.8295	1.74496	0.872481	+	0.488648i	$0.162510\pi$
$522$	0	0
$523$	18.3993	0.804544	0.402272	−	0.915520i	$-0.368221\pi$
$523$	18.3993	0.804544	0.402272	+	0.915520i	$0.368221\pi$
$524$	0	0
$525$	43.5842	1.90217
$526$	0	0
$527$	4.69706	0.204607
$528$	0	0
$529$	−1.36864	−0.0595059
$530$	0	0
$531$	3.25237	0.141141
$532$	0	0
$533$	3.72804	0.161479
$534$	0	0
$535$	−49.7145	−2.14935
$536$	0	0
$537$	20.7871	0.897028
$538$	0	0
$539$	−14.6314	−0.630217
$540$	0	0
$541$	18.1981	0.782399	0.391200	−	0.920306i	$-0.372060\pi$
$541$	18.1981	0.782399	0.391200	+	0.920306i	$0.372060\pi$
$542$	0	0
$543$	−14.4632	−0.620674
$544$	0	0
$545$	0.157599	0.00675079
$546$	0	0
$547$	21.1503	0.904322	0.452161	−	0.891936i	$-0.350653\pi$
$547$	21.1503	0.904322	0.452161	+	0.891936i	$0.350653\pi$
$548$	0	0
$549$	5.25237	0.224166
$550$	0	0
$551$	−7.25794	−0.309199
$552$	0	0
$553$	23.9057	1.01657
$554$	0	0
$555$	1.06116	0.0450438
$556$	0	0
$557$	−19.3469	−0.819755	−0.409877	−	0.912141i	$-0.634428\pi$
$557$	−19.3469	−0.819755	−0.409877	+	0.912141i	$0.634428\pi$
$558$	0	0
$559$	−4.90957	−0.207653
$560$	0	0
$561$	−0.860039	−0.0363109
$562$	0	0
$563$	24.8411	1.04693	0.523464	−	0.852048i	$-0.324639\pi$
$563$	24.8411	1.04693	0.523464	+	0.852048i	$0.324639\pi$
$564$	0	0
$565$	−23.9944	−1.00945
$566$	0	0
$567$	−4.65095	−0.195322
$568$	0	0
$569$	−17.0017	−0.712749	−0.356374	−	0.934343i	$-0.615987\pi$
$569$	−17.0017	−0.712749	−0.356374	+	0.934343i	$0.615987\pi$
$570$	0	0
$571$	34.7196	1.45297	0.726486	−	0.687181i	$-0.241152\pi$
$571$	34.7196	1.45297	0.726486	+	0.687181i	$0.241152\pi$
$572$	0	0
$573$	−7.76961	−0.324580
$574$	0	0
$575$	43.5842	1.81759
$576$	0	0
$577$	−23.6934	−0.986369	−0.493185	−	0.869925i	$-0.664167\pi$
$577$	−23.6934	−0.986369	−0.493185	+	0.869925i	$0.664167\pi$
$578$	0	0
$579$	−9.34348	−0.388302
$580$	0	0
$581$	74.8760	3.10638
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	3.79091	0.156735
$586$	0	0
$587$	29.5888	1.22126	0.610629	−	0.791916i	$-0.290917\pi$
$587$	29.5888	1.22126	0.610629	+	0.791916i	$0.290917\pi$
$588$	0	0
$589$	7.63830	0.314731
$590$	0	0
$591$	−20.2823	−0.834303
$592$	0	0
$593$	9.77758	0.401517	0.200758	−	0.979641i	$-0.435659\pi$
$593$	9.77758	0.401517	0.200758	+	0.979641i	$0.435659\pi$
$594$	0	0
$595$	−15.1637	−0.621649
$596$	0	0
$597$	15.2524	0.624238
$598$	0	0
$599$	−40.0530	−1.63652	−0.818260	−	0.574849i	$-0.805061\pi$
$599$	−40.0530	−1.63652	−0.818260	+	0.574849i	$0.805061\pi$
$600$	0	0
$601$	2.46556	0.100572	0.0502862	−	0.998735i	$-0.483987\pi$
$601$	2.46556	0.100572	0.0502862	+	0.998735i	$0.483987\pi$
$602$	0	0
$603$	13.3727	0.544580
$604$	0	0
$605$	−3.79091	−0.154123
$606$	0	0
$607$	8.94661	0.363132	0.181566	−	0.983379i	$-0.441883\pi$
$607$	8.94661	0.363132	0.181566	+	0.983379i	$0.441883\pi$
$608$	0	0
$609$	24.1361	0.978044
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	44.9727	1.81643	0.908215	−	0.418504i	$-0.137445\pi$
$613$	44.9727	1.81643	0.908215	+	0.418504i	$0.137445\pi$
$614$	0	0
$615$	−14.1327	−0.569885
$616$	0	0
$617$	6.38437	0.257025	0.128513	−	0.991708i	$-0.458980\pi$
$617$	6.38437	0.257025	0.128513	+	0.991708i	$0.458980\pi$
$618$	0	0
$619$	−22.7173	−0.913083	−0.456542	−	0.889702i	$-0.650912\pi$
$619$	−22.7173	−0.913083	−0.456542	+	0.889702i	$0.650912\pi$
$620$	0	0
$621$	−4.65095	−0.186636
$622$	0	0
$623$	72.3713	2.89949
$624$	0	0
$625$	15.9611	0.638442
$626$	0	0
$627$	−1.39858	−0.0558540
$628$	0	0
$629$	−0.240744	−0.00959910
$630$	0	0
$631$	−9.35163	−0.372283	−0.186141	−	0.982523i	$-0.559598\pi$
$631$	−9.35163	−0.372283	−0.186141	+	0.982523i	$0.559598\pi$
$632$	0	0
$633$	19.0017	0.755250
$634$	0	0
$635$	72.8861	2.89240
$636$	0	0
$637$	−14.6314	−0.579716
$638$	0	0
$639$	3.58183	0.141695
$640$	0	0
$641$	−42.5276	−1.67974	−0.839870	−	0.542788i	$-0.817369\pi$
$641$	−42.5276	−1.67974	−0.839870	+	0.542788i	$0.817369\pi$
$642$	0	0
$643$	−8.38437	−0.330647	−0.165324	−	0.986239i	$-0.552867\pi$
$643$	−8.38437	−0.330647	−0.165324	+	0.986239i	$0.552867\pi$
$644$	0	0
$645$	18.6118	0.732838
$646$	0	0
$647$	−29.4010	−1.15587	−0.577936	−	0.816082i	$-0.696142\pi$
$647$	−29.4010	−1.15587	−0.577936	+	0.816082i	$0.696142\pi$
$648$	0	0
$649$	−3.25237	−0.127667
$650$	0	0
$651$	−25.4010	−0.995543
$652$	0	0
$653$	6.51724	0.255039	0.127520	−	0.991836i	$-0.459298\pi$
$653$	6.51724	0.255039	0.127520	+	0.991836i	$0.459298\pi$
$654$	0	0
$655$	−70.3851	−2.75017
$656$	0	0
$657$	13.6729	0.533432
$658$	0	0
$659$	−32.8885	−1.28115	−0.640577	−	0.767894i	$-0.721305\pi$
$659$	−32.8885	−1.28115	−0.640577	+	0.767894i	$0.721305\pi$
$660$	0	0
$661$	9.28940	0.361316	0.180658	−	0.983546i	$-0.442177\pi$
$661$	9.28940	0.361316	0.180658	+	0.983546i	$0.442177\pi$
$662$	0	0
$663$	−0.860039	−0.0334011
$664$	0	0
$665$	−24.6589	−0.956232
$666$	0	0
$667$	24.1361	0.934554
$668$	0	0
$669$	23.9395	0.925555
$670$	0	0
$671$	−5.25237	−0.202766
$672$	0	0
$673$	−37.7124	−1.45370	−0.726852	−	0.686794i	$-0.759018\pi$
$673$	−37.7124	−1.45370	−0.726852	+	0.686794i	$0.759018\pi$
$674$	0	0
$675$	−9.37103	−0.360691
$676$	0	0
$677$	44.2485	1.70061	0.850304	−	0.526291i	$-0.176418\pi$
$677$	44.2485	1.70061	0.850304	+	0.526291i	$0.176418\pi$
$678$	0	0
$679$	−1.30191	−0.0499626
$680$	0	0
$681$	−26.9412	−1.03239
$682$	0	0
$683$	39.4505	1.50953	0.754766	−	0.655994i	$-0.227750\pi$
$683$	39.4505	1.50953	0.754766	+	0.655994i	$0.227750\pi$
$684$	0	0
$685$	−46.7996	−1.78812
$686$	0	0
$687$	24.2744	0.926124
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	41.9429	1.59558	0.797792	−	0.602932i	$-0.206001\pi$
$691$	41.9429	1.59558	0.797792	+	0.602932i	$0.206001\pi$
$692$	0	0
$693$	4.65095	0.176675
$694$	0	0
$695$	58.8310	2.23159
$696$	0	0
$697$	3.20626	0.121446
$698$	0	0
$699$	−15.2390	−0.576393
$700$	0	0
$701$	3.34367	0.126289	0.0631444	−	0.998004i	$-0.479887\pi$
$701$	3.34367	0.126289	0.0631444	+	0.998004i	$0.479887\pi$
$702$	0	0
$703$	−0.391495	−0.0147655
$704$	0	0
$705$	−30.3273	−1.14219
$706$	0	0
$707$	−62.8760	−2.36470
$708$	0	0
$709$	16.3734	0.614917	0.307458	−	0.951562i	$-0.400521\pi$
$709$	16.3734	0.614917	0.307458	+	0.951562i	$0.400521\pi$
$710$	0	0
$711$	−5.13996	−0.192764
$712$	0	0
$713$	−25.4010	−0.951274
$714$	0	0
$715$	−3.79091	−0.141772
$716$	0	0
$717$	3.71554	0.138759
$718$	0	0
$719$	29.2558	1.09106	0.545529	−	0.838092i	$-0.316329\pi$
$719$	29.2558	1.09106	0.545529	+	0.838092i	$0.316329\pi$
$720$	0	0
$721$	−12.5228	−0.466374
$722$	0	0
$723$	1.35701	0.0504677
$724$	0	0
$725$	48.6309	1.80611
$726$	0	0
$727$	18.2799	0.677965	0.338982	−	0.940793i	$-0.389917\pi$
$727$	18.2799	0.677965	0.338982	+	0.940793i	$0.389917\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.22242	−0.156172
$732$	0	0
$733$	38.0183	1.40424	0.702119	−	0.712059i	$-0.252237\pi$
$733$	38.0183	1.40424	0.702119	+	0.712059i	$0.252237\pi$
$734$	0	0
$735$	55.4662	2.04590
$736$	0	0
$737$	−13.3727	−0.492591
$738$	0	0
$739$	−27.0244	−0.994108	−0.497054	−	0.867720i	$-0.665585\pi$
$739$	−27.0244	−0.994108	−0.497054	+	0.867720i	$0.665585\pi$
$740$	0	0
$741$	−1.39858	−0.0513782
$742$	0	0
$743$	−2.83420	−0.103977	−0.0519883	−	0.998648i	$-0.516556\pi$
$743$	−2.83420	−0.103977	−0.0519883	+	0.998648i	$0.516556\pi$
$744$	0	0
$745$	13.3604	0.489488
$746$	0	0
$747$	−16.0991	−0.589034
$748$	0	0
$749$	−60.9932	−2.22864
$750$	0	0
$751$	0.0370341	0.00135140	0.000675698	−	1.00000i	$-0.499785\pi$
$751$	0.0370341	0.00135140	0.000675698		1.00000i	$0.499785\pi$
$752$	0	0
$753$	−5.30191	−0.193212
$754$	0	0
$755$	45.1245	1.64225
$756$	0	0
$757$	45.1501	1.64101	0.820505	−	0.571640i	$-0.193693\pi$
$757$	45.1501	1.64101	0.820505	+	0.571640i	$0.193693\pi$
$758$	0	0
$759$	4.65095	0.168819
$760$	0	0
$761$	11.5773	0.419676	0.209838	−	0.977736i	$-0.432706\pi$
$761$	11.5773	0.419676	0.209838	+	0.977736i	$0.432706\pi$
$762$	0	0
$763$	0.193353	0.00699984
$764$	0	0
$765$	3.26033	0.117878
$766$	0	0
$767$	−3.25237	−0.117436
$768$	0	0
$769$	−17.1155	−0.617200	−0.308600	−	0.951192i	$-0.599860\pi$
$769$	−17.1155	−0.617200	−0.308600	+	0.951192i	$0.599860\pi$
$770$	0	0
$771$	−7.11412	−0.256209
$772$	0	0
$773$	14.4994	0.521506	0.260753	−	0.965406i	$-0.416029\pi$
$773$	14.4994	0.521506	0.260753	+	0.965406i	$0.416029\pi$
$774$	0	0
$775$	−51.1795	−1.83842
$776$	0	0
$777$	1.30191	0.0467056
$778$	0	0
$779$	5.21397	0.186810
$780$	0	0
$781$	−3.58183	−0.128168
$782$	0	0
$783$	−5.18950	−0.185457
$784$	0	0
$785$	79.8631	2.85044
$786$	0	0
$787$	−28.3422	−1.01029	−0.505145	−	0.863034i	$-0.668561\pi$
$787$	−28.3422	−1.01029	−0.505145	+	0.863034i	$0.668561\pi$
$788$	0	0
$789$	12.8837	0.458673
$790$	0	0
$791$	−29.4380	−1.04670
$792$	0	0
$793$	−5.25237	−0.186517
$794$	0	0
$795$	22.7455	0.806699
$796$	0	0
$797$	−46.8726	−1.66031	−0.830156	−	0.557531i	$-0.811749\pi$
$797$	−46.8726	−1.66031	−0.830156	+	0.557531i	$0.811749\pi$
$798$	0	0
$799$	6.88031	0.243408
$800$	0	0
$801$	−15.5605	−0.549804
$802$	0	0
$803$	−13.6729	−0.482507
$804$	0	0
$805$	82.0026	2.89021
$806$	0	0
$807$	19.8191	0.697667
$808$	0	0
$809$	−41.0017	−1.44154	−0.720772	−	0.693173i	$-0.756212\pi$
$809$	−41.0017	−1.44154	−0.720772	+	0.693173i	$0.756212\pi$
$810$	0	0
$811$	−17.4145	−0.611506	−0.305753	−	0.952111i	$-0.598908\pi$
$811$	−17.4145	−0.611506	−0.305753	+	0.952111i	$0.598908\pi$
$812$	0	0
$813$	−17.4977	−0.613669
$814$	0	0
$815$	−2.22936	−0.0780910
$816$	0	0
$817$	−6.86644	−0.240226
$818$	0	0
$819$	4.65095	0.162517
$820$	0	0
$821$	−11.2164	−0.391454	−0.195727	−	0.980658i	$-0.562707\pi$
$821$	−11.2164	−0.391454	−0.195727	+	0.980658i	$0.562707\pi$
$822$	0	0
$823$	0.945771	0.0329675	0.0164838	−	0.999864i	$-0.494753\pi$
$823$	0.945771	0.0329675	0.0164838	+	0.999864i	$0.494753\pi$
$824$	0	0
$825$	9.37103	0.326257
$826$	0	0
$827$	23.2177	0.807360	0.403680	−	0.914900i	$-0.367731\pi$
$827$	23.2177	0.807360	0.403680	+	0.914900i	$0.367731\pi$
$828$	0	0
$829$	32.6038	1.13238	0.566189	−	0.824276i	$-0.308417\pi$
$829$	32.6038	1.13238	0.566189	+	0.824276i	$0.308417\pi$
$830$	0	0
$831$	10.9884	0.381182
$832$	0	0
$833$	−12.5835	−0.435994
$834$	0	0
$835$	62.2791	2.15526
$836$	0	0
$837$	5.46146	0.188776
$838$	0	0
$839$	19.5392	0.674569	0.337284	−	0.941403i	$-0.390492\pi$
$839$	19.5392	0.674569	0.337284	+	0.941403i	$0.390492\pi$
$840$	0	0
$841$	−2.06912	−0.0713491
$842$	0	0
$843$	−7.53469	−0.259508
$844$	0	0
$845$	−3.79091	−0.130411
$846$	0	0
$847$	−4.65095	−0.159809
$848$	0	0
$849$	0.727358	0.0249629
$850$	0	0
$851$	1.30191	0.0446288
$852$	0	0
$853$	−20.9814	−0.718391	−0.359195	−	0.933262i	$-0.616949\pi$
$853$	−20.9814	−0.718391	−0.359195	+	0.933262i	$0.616949\pi$
$854$	0	0
$855$	5.30191	0.181321
$856$	0	0
$857$	−27.6146	−0.943297	−0.471648	−	0.881787i	$-0.656341\pi$
$857$	−27.6146	−0.943297	−0.471648	+	0.881787i	$0.656341\pi$
$858$	0	0
$859$	−41.3334	−1.41028	−0.705138	−	0.709070i	$-0.749115\pi$
$859$	−41.3334	−1.41028	−0.705138	+	0.709070i	$0.749115\pi$
$860$	0	0
$861$	−17.3389	−0.590909
$862$	0	0
$863$	43.4875	1.48033	0.740167	−	0.672423i	$-0.234747\pi$
$863$	43.4875	1.48033	0.740167	+	0.672423i	$0.234747\pi$
$864$	0	0
$865$	−57.0580	−1.94003
$866$	0	0
$867$	16.2603	0.552230
$868$	0	0
$869$	5.13996	0.174361
$870$	0	0
$871$	−13.3727	−0.453118
$872$	0	0
$873$	0.279922	0.00947394
$874$	0	0
$875$	77.0672	2.60535
$876$	0	0
$877$	−33.6542	−1.13642	−0.568211	−	0.822883i	$-0.692364\pi$
$877$	−33.6542	−1.13642	−0.568211	+	0.822883i	$0.692364\pi$
$878$	0	0
$879$	−7.06356	−0.238248
$880$	0	0
$881$	14.9724	0.504435	0.252217	−	0.967671i	$-0.418840\pi$
$881$	14.9724	0.504435	0.252217	+	0.967671i	$0.418840\pi$
$882$	0	0
$883$	−27.8631	−0.937668	−0.468834	−	0.883286i	$-0.655326\pi$
$883$	−27.8631	−0.937668	−0.468834	+	0.883286i	$0.655326\pi$
$884$	0	0
$885$	12.3295	0.414450
$886$	0	0
$887$	−36.9669	−1.24123	−0.620613	−	0.784117i	$-0.713116\pi$
$887$	−36.9669	−1.24123	−0.620613	+	0.784117i	$0.713116\pi$
$888$	0	0
$889$	89.4217	2.99911
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	11.1887	0.374414
$894$	0	0
$895$	78.8020	2.63406
$896$	0	0
$897$	4.65095	0.155291
$898$	0	0
$899$	−28.3422	−0.945266
$900$	0	0
$901$	−5.16023	−0.171912
$902$	0	0
$903$	22.8342	0.759875
$904$	0	0
$905$	−54.8286	−1.82257
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	13.5190	0.448395
$910$	0	0
$911$	−49.5741	−1.64246	−0.821232	−	0.570595i	$-0.806713\pi$
$911$	−49.5741	−1.64246	−0.821232	+	0.570595i	$0.806713\pi$
$912$	0	0
$913$	16.0991	0.532801
$914$	0	0
$915$	19.9113	0.658247
$916$	0	0
$917$	−86.3532	−2.85163
$918$	0	0
$919$	−1.38978	−0.0458447	−0.0229223	−	0.999737i	$-0.507297\pi$
$919$	−1.38978	−0.0458447	−0.0229223	+	0.999737i	$0.507297\pi$
$920$	0	0
$921$	−18.9679	−0.625014
$922$	0	0
$923$	−3.58183	−0.117897
$924$	0	0
$925$	2.62316	0.0862490
$926$	0	0
$927$	2.69253	0.0884341
$928$	0	0
$929$	−0.203518	−0.00667719	−0.00333860	−	0.999994i	$-0.501063\pi$
$929$	−0.203518	−0.00667719	−0.00333860	+	0.999994i	$0.501063\pi$
$930$	0	0
$931$	−20.4632	−0.670653
$932$	0	0
$933$	21.3469	0.698866
$934$	0	0
$935$	−3.26033	−0.106624
$936$	0	0
$937$	4.89145	0.159797	0.0798983	−	0.996803i	$-0.474540\pi$
$937$	4.89145	0.159797	0.0798983	+	0.996803i	$0.474540\pi$
$938$	0	0
$939$	0.831805	0.0271449
$940$	0	0
$941$	−58.5670	−1.90923	−0.954615	−	0.297842i	$-0.903733\pi$
$941$	−58.5670	−1.90923	−0.954615	+	0.297842i	$0.903733\pi$
$942$	0	0
$943$	−17.3389	−0.564634
$944$	0	0
$945$	−17.6314	−0.573548
$946$	0	0
$947$	−4.56463	−0.148331	−0.0741653	−	0.997246i	$-0.523629\pi$
$947$	−4.56463	−0.148331	−0.0741653	+	0.997246i	$0.523629\pi$
$948$	0	0
$949$	−13.6729	−0.443842
$950$	0	0
$951$	16.2565	0.527152
$952$	0	0
$953$	−1.29414	−0.0419212	−0.0209606	−	0.999780i	$-0.506672\pi$
$953$	−1.29414	−0.0419212	−0.0209606	+	0.999780i	$0.506672\pi$
$954$	0	0
$955$	−29.4539	−0.953107
$956$	0	0
$957$	5.18950	0.167753
$958$	0	0
$959$	−57.4169	−1.85409
$960$	0	0
$961$	−1.17249	−0.0378224
$962$	0	0
$963$	13.1141	0.422597
$964$	0	0
$965$	−35.4203	−1.14022
$966$	0	0
$967$	14.8051	0.476101	0.238050	−	0.971253i	$-0.423492\pi$
$967$	14.8051	0.476101	0.238050	+	0.971253i	$0.423492\pi$
$968$	0	0
$969$	−1.20284	−0.0386406
$970$	0	0
$971$	−22.4805	−0.721432	−0.360716	−	0.932676i	$-0.617468\pi$
$971$	−22.4805	−0.721432	−0.360716	+	0.932676i	$0.617468\pi$
$972$	0	0
$973$	72.1779	2.31392
$974$	0	0
$975$	9.37103	0.300113
$976$	0	0
$977$	−34.2901	−1.09704	−0.548519	−	0.836138i	$-0.684808\pi$
$977$	−34.2901	−1.09704	−0.548519	+	0.836138i	$0.684808\pi$
$978$	0	0
$979$	15.5605	0.497317
$980$	0	0
$981$	−0.0415727	−0.00132732
$982$	0	0
$983$	−38.2282	−1.21929	−0.609646	−	0.792674i	$-0.708688\pi$
$983$	−38.2282	−1.21929	−0.609646	+	0.792674i	$0.708688\pi$
$984$	0	0
$985$	−76.8885	−2.44987
$986$	0	0
$987$	−37.2076	−1.18433
$988$	0	0
$989$	22.8342	0.726085
$990$	0	0
$991$	33.6628	1.06934	0.534668	−	0.845063i	$-0.320437\pi$
$991$	33.6628	1.06934	0.534668	+	0.845063i	$0.320437\pi$
$992$	0	0
$993$	−17.4985	−0.555298
$994$	0	0
$995$	57.8204	1.83303
$996$	0	0
$997$	−4.32603	−0.137007	−0.0685034	−	0.997651i	$-0.521822\pi$
$997$	−4.32603	−0.137007	−0.0685034	+	0.997651i	$0.521822\pi$
$998$	0	0
$999$	−0.279922	−0.00885635

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.79091 q^{5} -4.65095 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.79091 q^{5} -4.65095 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.79091 q^{15} -0.860039 q^{17} -1.39858 q^{19} +4.65095 q^{21} +4.65095 q^{23} +9.37103 q^{25} -1.00000 q^{27} +5.18950 q^{29} -5.46146 q^{31} +1.00000 q^{33} +17.6314 q^{35} +0.279922 q^{37} +1.00000 q^{39} -3.72804 q^{41} +4.90957 q^{43} -3.79091 q^{45} -8.00000 q^{47} +14.6314 q^{49} +0.860039 q^{51} +6.00000 q^{53} +3.79091 q^{55} +1.39858 q^{57} +3.25237 q^{59} +5.25237 q^{61} -4.65095 q^{63} +3.79091 q^{65} +13.3727 q^{67} -4.65095 q^{69} +3.58183 q^{71} +13.6729 q^{73} -9.37103 q^{75} +4.65095 q^{77} -5.13996 q^{79} +1.00000 q^{81} -16.0991 q^{83} +3.26033 q^{85} -5.18950 q^{87} -15.5605 q^{89} +4.65095 q^{91} +5.46146 q^{93} +5.30191 q^{95} +0.279922 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9	\( 4 q - 4 q^{3} + q^{5} - q^{7} + 4 q^{9} - 4 q^{11} - 4 q^{13} - q^{15} - 2 q^{17} - 2 q^{19} + q^{21} + q^{23} + 17 q^{25} - 4 q^{27} + q^{29} - 24 q^{31} + 4 q^{33} + 17 q^{35} + 4 q^{37} + 4 q^{39} + 7 q^{41} - 3 q^{43} + q^{45} - 32 q^{47} + 5 q^{49} + 2 q^{51} + 24 q^{53} - q^{55} + 2 q^{57} - q^{59} + 7 q^{61} - q^{63} - q^{65} + 5 q^{67} - q^{69} - 18 q^{71} - q^{73} - 17 q^{75} + q^{77} - 22 q^{79} + 4 q^{81} - 22 q^{83} - 20 q^{85} - q^{87} - 22 q^{89} + q^{91} + 24 q^{93} - 14 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + q^5 - q^7 + 4 * q^9 - 4 * q^11 - 4 * q^13 - q^15 - 2 * q^17 - 2 * q^19 + q^21 + q^23 + 17 * q^25 - 4 * q^27 + q^29 - 24 * q^31 + 4 * q^33 + 17 * q^35 + 4 * q^37 + 4 * q^39 + 7 * q^41 - 3 * q^43 + q^45 - 32 * q^47 + 5 * q^49 + 2 * q^51 + 24 * q^53 - q^55 + 2 * q^57 - q^59 + 7 * q^61 - q^63 - q^65 + 5 * q^67 - q^69 - 18 * q^71 - q^73 - 17 * q^75 + q^77 - 22 * q^79 + 4 * q^81 - 22 * q^83 - 20 * q^85 - q^87 - 22 * q^89 + q^91 + 24 * q^93 - 14 * q^95 + 4 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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