LMFDB - Embedded newform 6864.2.a.bg.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
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			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.449490 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.449490 q^{5} +2.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.449490 q^{15} -6.44949 q^{17} -6.89898 q^{19} +2.00000 q^{21} +2.89898 q^{23} -4.79796 q^{25} +1.00000 q^{27} +1.55051 q^{29} -6.44949 q^{31} -1.00000 q^{33} +0.898979 q^{35} +2.00000 q^{37} +1.00000 q^{39} +0.898979 q^{41} +0.449490 q^{43} +0.449490 q^{45} +4.89898 q^{47} -3.00000 q^{49} -6.44949 q^{51} -6.00000 q^{53} -0.449490 q^{55} -6.89898 q^{57} -4.89898 q^{59} -11.7980 q^{61} +2.00000 q^{63} +0.449490 q^{65} +1.55051 q^{67} +2.89898 q^{69} -9.79796 q^{71} -0.898979 q^{73} -4.79796 q^{75} -2.00000 q^{77} +9.34847 q^{79} +1.00000 q^{81} -9.79796 q^{83} -2.89898 q^{85} +1.55051 q^{87} -4.44949 q^{89} +2.00000 q^{91} -6.44949 q^{93} -3.10102 q^{95} -10.0000 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 4 q^{15} - 8 q^{17} - 4 q^{19} + 4 q^{21} - 4 q^{23} + 10 q^{25} + 2 q^{27} + 8 q^{29} - 8 q^{31} - 2 q^{33} - 8 q^{35} + 4 q^{37} + 2 q^{39} - 8 q^{41} - 4 q^{43} - 4 q^{45} - 6 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} - 4 q^{57} - 4 q^{61} + 4 q^{63} - 4 q^{65} + 8 q^{67} - 4 q^{69} + 8 q^{73} + 10 q^{75} - 4 q^{77} + 4 q^{79} + 2 q^{81} + 4 q^{85} + 8 q^{87} - 4 q^{89} + 4 q^{91} - 8 q^{93} - 16 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 4 * q^15 - 8 * q^17 - 4 * q^19 + 4 * q^21 - 4 * q^23 + 10 * q^25 + 2 * q^27 + 8 * q^29 - 8 * q^31 - 2 * q^33 - 8 * q^35 + 4 * q^37 + 2 * q^39 - 8 * q^41 - 4 * q^43 - 4 * q^45 - 6 * q^49 - 8 * q^51 - 12 * q^53 + 4 * q^55 - 4 * q^57 - 4 * q^61 + 4 * q^63 - 4 * q^65 + 8 * q^67 - 4 * q^69 + 8 * q^73 + 10 * q^75 - 4 * q^77 + 4 * q^79 + 2 * q^81 + 4 * q^85 + 8 * q^87 - 4 * q^89 + 4 * q^91 - 8 * q^93 - 16 * q^95 - 20 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.449490	0.201018	0.100509	−	0.994936i	$-0.467953\pi$
$5$	0.449490	0.201018	0.100509	+	0.994936i	$0.467953\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.449490	0.116058
$16$	0	0
$17$	−6.44949	−1.56423	−0.782116	−	0.623133i	$-0.785859\pi$
$17$	−6.44949	−1.56423	−0.782116	+	0.623133i	$0.785859\pi$
$18$	0	0
$19$	−6.89898	−1.58273	−0.791367	−	0.611341i	$-0.790630\pi$
$19$	−6.89898	−1.58273	−0.791367	+	0.611341i	$0.790630\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	2.89898	0.604479	0.302240	−	0.953232i	$-0.402266\pi$
$23$	2.89898	0.604479	0.302240	+	0.953232i	$0.402266\pi$
$24$	0	0
$25$	−4.79796	−0.959592
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.55051	0.287923	0.143961	−	0.989583i	$-0.454016\pi$
$29$	1.55051	0.287923	0.143961	+	0.989583i	$0.454016\pi$
$30$	0	0
$31$	−6.44949	−1.15836	−0.579181	−	0.815199i	$-0.696628\pi$
$31$	−6.44949	−1.15836	−0.579181	+	0.815199i	$0.696628\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0.898979	0.151955
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	0.898979	0.140397	0.0701985	−	0.997533i	$-0.477637\pi$
$41$	0.898979	0.140397	0.0701985	+	0.997533i	$0.477637\pi$
$42$	0	0
$43$	0.449490	0.0685465	0.0342733	−	0.999412i	$-0.489088\pi$
$43$	0.449490	0.0685465	0.0342733	+	0.999412i	$0.489088\pi$
$44$	0	0
$45$	0.449490	0.0670060
$46$	0	0
$47$	4.89898	0.714590	0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898	0.714590	0.357295	+	0.933992i	$0.383699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−6.44949	−0.903109
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−0.449490	−0.0606092
$56$	0	0
$57$	−6.89898	−0.913792
$58$	0	0
$59$	−4.89898	−0.637793	−0.318896	−	0.947790i	$-0.603312\pi$
$59$	−4.89898	−0.637793	−0.318896	+	0.947790i	$0.603312\pi$
$60$	0	0
$61$	−11.7980	−1.51057	−0.755287	−	0.655394i	$-0.772502\pi$
$61$	−11.7980	−1.51057	−0.755287	+	0.655394i	$0.772502\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	0.449490	0.0557523
$66$	0	0
$67$	1.55051	0.189425	0.0947125	−	0.995505i	$-0.469807\pi$
$67$	1.55051	0.189425	0.0947125	+	0.995505i	$0.469807\pi$
$68$	0	0
$69$	2.89898	0.348996
$70$	0	0
$71$	−9.79796	−1.16280	−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796	−1.16280	−0.581402	+	0.813617i	$0.697496\pi$
$72$	0	0
$73$	−0.898979	−0.105218	−0.0526088	−	0.998615i	$-0.516754\pi$
$73$	−0.898979	−0.105218	−0.0526088	+	0.998615i	$0.516754\pi$
$74$	0	0
$75$	−4.79796	−0.554021
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	9.34847	1.05178	0.525892	−	0.850551i	$-0.323731\pi$
$79$	9.34847	1.05178	0.525892	+	0.850551i	$0.323731\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.79796	−1.07547	−0.537733	−	0.843115i	$-0.680719\pi$
$83$	−9.79796	−1.07547	−0.537733	+	0.843115i	$0.680719\pi$
$84$	0	0
$85$	−2.89898	−0.314438
$86$	0	0
$87$	1.55051	0.166232
$88$	0	0
$89$	−4.44949	−0.471645	−0.235822	−	0.971796i	$-0.575778\pi$
$89$	−4.44949	−0.471645	−0.235822	+	0.971796i	$0.575778\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−6.44949	−0.668781
$94$	0	0
$95$	−3.10102	−0.318158
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−9.55051	−0.950311	−0.475156	−	0.879902i	$-0.657608\pi$
$101$	−9.55051	−0.950311	−0.475156	+	0.879902i	$0.657608\pi$
$102$	0	0
$103$	−3.10102	−0.305553	−0.152776	−	0.988261i	$-0.548821\pi$
$103$	−3.10102	−0.305553	−0.152776	+	0.988261i	$0.548821\pi$
$104$	0	0
$105$	0.898979	0.0877314
$106$	0	0
$107$	−8.89898	−0.860297	−0.430148	−	0.902758i	$-0.641539\pi$
$107$	−8.89898	−0.860297	−0.430148	+	0.902758i	$0.641539\pi$
$108$	0	0
$109$	16.8990	1.61863	0.809314	−	0.587376i	$-0.199839\pi$
$109$	16.8990	1.61863	0.809314	+	0.587376i	$0.199839\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−15.7980	−1.48615	−0.743073	−	0.669210i	$-0.766633\pi$
$113$	−15.7980	−1.48615	−0.743073	+	0.669210i	$0.766633\pi$
$114$	0	0
$115$	1.30306	0.121511
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−12.8990	−1.18245
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.898979	0.0810583
$124$	0	0
$125$	−4.40408	−0.393913
$126$	0	0
$127$	13.3485	1.18449	0.592243	−	0.805760i	$-0.298243\pi$
$127$	13.3485	1.18449	0.592243	+	0.805760i	$0.298243\pi$
$128$	0	0
$129$	0.449490	0.0395754
$130$	0	0
$131$	21.7980	1.90450	0.952248	−	0.305325i	$-0.0987650\pi$
$131$	21.7980	1.90450	0.952248	+	0.305325i	$0.0987650\pi$
$132$	0	0
$133$	−13.7980	−1.19643
$134$	0	0
$135$	0.449490	0.0386859
$136$	0	0
$137$	−10.2474	−0.875499	−0.437749	−	0.899097i	$-0.644224\pi$
$137$	−10.2474	−0.875499	−0.437749	+	0.899097i	$0.644224\pi$
$138$	0	0
$139$	−2.65153	−0.224900	−0.112450	−	0.993657i	$-0.535870\pi$
$139$	−2.65153	−0.224900	−0.112450	+	0.993657i	$0.535870\pi$
$140$	0	0
$141$	4.89898	0.412568
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0.696938	0.0578776
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	8.69694	0.707747	0.353873	−	0.935293i	$-0.384864\pi$
$151$	8.69694	0.707747	0.353873	+	0.935293i	$0.384864\pi$
$152$	0	0
$153$	−6.44949	−0.521410
$154$	0	0
$155$	−2.89898	−0.232852
$156$	0	0
$157$	1.79796	0.143493	0.0717464	−	0.997423i	$-0.477143\pi$
$157$	1.79796	0.143493	0.0717464	+	0.997423i	$0.477143\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	5.79796	0.456943
$162$	0	0
$163$	23.3485	1.82879	0.914397	−	0.404819i	$-0.132666\pi$
$163$	23.3485	1.82879	0.914397	+	0.404819i	$0.132666\pi$
$164$	0	0
$165$	−0.449490	−0.0349927
$166$	0	0
$167$	21.7980	1.68678	0.843388	−	0.537304i	$-0.180557\pi$
$167$	21.7980	1.68678	0.843388	+	0.537304i	$0.180557\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.89898	−0.527578
$172$	0	0
$173$	0.247449	0.0188132	0.00940659	−	0.999956i	$-0.497006\pi$
$173$	0.247449	0.0188132	0.00940659	+	0.999956i	$0.497006\pi$
$174$	0	0
$175$	−9.59592	−0.725383
$176$	0	0
$177$	−4.89898	−0.368230
$178$	0	0
$179$	5.10102	0.381268	0.190634	−	0.981661i	$-0.438946\pi$
$179$	5.10102	0.381268	0.190634	+	0.981661i	$0.438946\pi$
$180$	0	0
$181$	15.5959	1.15924	0.579618	−	0.814889i	$-0.303202\pi$
$181$	15.5959	1.15924	0.579618	+	0.814889i	$0.303202\pi$
$182$	0	0
$183$	−11.7980	−0.872130
$184$	0	0
$185$	0.898979	0.0660943
$186$	0	0
$187$	6.44949	0.471633
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	6.69694	0.482056	0.241028	−	0.970518i	$-0.422515\pi$
$193$	6.69694	0.482056	0.241028	+	0.970518i	$0.422515\pi$
$194$	0	0
$195$	0.449490	0.0321886
$196$	0	0
$197$	−4.00000	−0.284988	−0.142494	−	0.989796i	$-0.545512\pi$
$197$	−4.00000	−0.284988	−0.142494	+	0.989796i	$0.545512\pi$
$198$	0	0
$199$	−3.10102	−0.219826	−0.109913	−	0.993941i	$-0.535057\pi$
$199$	−3.10102	−0.219826	−0.109913	+	0.993941i	$0.535057\pi$
$200$	0	0
$201$	1.55051	0.109365
$202$	0	0
$203$	3.10102	0.217649
$204$	0	0
$205$	0.404082	0.0282223
$206$	0	0
$207$	2.89898	0.201493
$208$	0	0
$209$	6.89898	0.477212
$210$	0	0
$211$	17.3485	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$211$	17.3485	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$212$	0	0
$213$	−9.79796	−0.671345
$214$	0	0
$215$	0.202041	0.0137791
$216$	0	0
$217$	−12.8990	−0.875640
$218$	0	0
$219$	−0.898979	−0.0607474
$220$	0	0
$221$	−6.44949	−0.433840
$222$	0	0
$223$	−5.55051	−0.371690	−0.185845	−	0.982579i	$-0.559502\pi$
$223$	−5.55051	−0.371690	−0.185845	+	0.982579i	$0.559502\pi$
$224$	0	0
$225$	−4.79796	−0.319864
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	−2.89898	−0.191570	−0.0957850	−	0.995402i	$-0.530536\pi$
$229$	−2.89898	−0.191570	−0.0957850	+	0.995402i	$0.530536\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−3.75255	−0.245838	−0.122919	−	0.992417i	$-0.539226\pi$
$233$	−3.75255	−0.245838	−0.122919	+	0.992417i	$0.539226\pi$
$234$	0	0
$235$	2.20204	0.143645
$236$	0	0
$237$	9.34847	0.607248
$238$	0	0
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.34847	−0.0861505
$246$	0	0
$247$	−6.89898	−0.438972
$248$	0	0
$249$	−9.79796	−0.620920
$250$	0	0
$251$	15.5959	0.984406	0.492203	−	0.870481i	$-0.336192\pi$
$251$	15.5959	0.984406	0.492203	+	0.870481i	$0.336192\pi$
$252$	0	0
$253$	−2.89898	−0.182257
$254$	0	0
$255$	−2.89898	−0.181541
$256$	0	0
$257$	−30.8990	−1.92743	−0.963713	−	0.266942i	$-0.913987\pi$
$257$	−30.8990	−1.92743	−0.963713	+	0.266942i	$0.913987\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	1.55051	0.0959742
$262$	0	0
$263$	8.89898	0.548735	0.274367	−	0.961625i	$-0.411532\pi$
$263$	8.89898	0.548735	0.274367	+	0.961625i	$0.411532\pi$
$264$	0	0
$265$	−2.69694	−0.165672
$266$	0	0
$267$	−4.44949	−0.272304
$268$	0	0
$269$	−20.6969	−1.26191	−0.630957	−	0.775818i	$-0.717338\pi$
$269$	−20.6969	−1.26191	−0.630957	+	0.775818i	$0.717338\pi$
$270$	0	0
$271$	−26.8990	−1.63400	−0.816998	−	0.576640i	$-0.804364\pi$
$271$	−26.8990	−1.63400	−0.816998	+	0.576640i	$0.804364\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	4.79796	0.289328
$276$	0	0
$277$	11.7980	0.708871	0.354435	−	0.935081i	$-0.384673\pi$
$277$	11.7980	0.708871	0.354435	+	0.935081i	$0.384673\pi$
$278$	0	0
$279$	−6.44949	−0.386121
$280$	0	0
$281$	14.6969	0.876746	0.438373	−	0.898793i	$-0.355555\pi$
$281$	14.6969	0.876746	0.438373	+	0.898793i	$0.355555\pi$
$282$	0	0
$283$	−21.3485	−1.26903	−0.634517	−	0.772909i	$-0.718801\pi$
$283$	−21.3485	−1.26903	−0.634517	+	0.772909i	$0.718801\pi$
$284$	0	0
$285$	−3.10102	−0.183689
$286$	0	0
$287$	1.79796	0.106130
$288$	0	0
$289$	24.5959	1.44682
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−15.5959	−0.911123	−0.455562	−	0.890204i	$-0.650562\pi$
$293$	−15.5959	−0.911123	−0.455562	+	0.890204i	$0.650562\pi$
$294$	0	0
$295$	−2.20204	−0.128208
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	2.89898	0.167652
$300$	0	0
$301$	0.898979	0.0518163
$302$	0	0
$303$	−9.55051	−0.548662
$304$	0	0
$305$	−5.30306	−0.303652
$306$	0	0
$307$	−2.89898	−0.165453	−0.0827267	−	0.996572i	$-0.526363\pi$
$307$	−2.89898	−0.165453	−0.0827267	+	0.996572i	$0.526363\pi$
$308$	0	0
$309$	−3.10102	−0.176411
$310$	0	0
$311$	−1.10102	−0.0624331	−0.0312166	−	0.999513i	$-0.509938\pi$
$311$	−1.10102	−0.0624331	−0.0312166	+	0.999513i	$0.509938\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	0	0
$315$	0.898979	0.0506518
$316$	0	0
$317$	−29.8434	−1.67617	−0.838085	−	0.545539i	$-0.816325\pi$
$317$	−29.8434	−1.67617	−0.838085	+	0.545539i	$0.816325\pi$
$318$	0	0
$319$	−1.55051	−0.0868119
$320$	0	0
$321$	−8.89898	−0.496693
$322$	0	0
$323$	44.4949	2.47576
$324$	0	0
$325$	−4.79796	−0.266143
$326$	0	0
$327$	16.8990	0.934516
$328$	0	0
$329$	9.79796	0.540179
$330$	0	0
$331$	29.5505	1.62424	0.812121	−	0.583488i	$-0.198313\pi$
$331$	29.5505	1.62424	0.812121	+	0.583488i	$0.198313\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	0.696938	0.0380778
$336$	0	0
$337$	1.10102	0.0599764	0.0299882	−	0.999550i	$-0.490453\pi$
$337$	1.10102	0.0599764	0.0299882	+	0.999550i	$0.490453\pi$
$338$	0	0
$339$	−15.7980	−0.858027
$340$	0	0
$341$	6.44949	0.349259
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	1.30306	0.0701545
$346$	0	0
$347$	−32.4949	−1.74442	−0.872209	−	0.489134i	$-0.837313\pi$
$347$	−32.4949	−1.74442	−0.872209	+	0.489134i	$0.837313\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	11.1464	0.593265	0.296632	−	0.954992i	$-0.404136\pi$
$353$	11.1464	0.593265	0.296632	+	0.954992i	$0.404136\pi$
$354$	0	0
$355$	−4.40408	−0.233744
$356$	0	0
$357$	−12.8990	−0.682686
$358$	0	0
$359$	23.7980	1.25601	0.628004	−	0.778210i	$-0.283872\pi$
$359$	23.7980	1.25601	0.628004	+	0.778210i	$0.283872\pi$
$360$	0	0
$361$	28.5959	1.50505
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−0.404082	−0.0211506
$366$	0	0
$367$	−25.3939	−1.32555	−0.662775	−	0.748819i	$-0.730621\pi$
$367$	−25.3939	−1.32555	−0.662775	+	0.748819i	$0.730621\pi$
$368$	0	0
$369$	0.898979	0.0467990
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	0.696938	0.0360861	0.0180431	−	0.999837i	$-0.494256\pi$
$373$	0.696938	0.0360861	0.0180431	+	0.999837i	$0.494256\pi$
$374$	0	0
$375$	−4.40408	−0.227426
$376$	0	0
$377$	1.55051	0.0798553
$378$	0	0
$379$	14.0454	0.721464	0.360732	−	0.932669i	$-0.382527\pi$
$379$	14.0454	0.721464	0.360732	+	0.932669i	$0.382527\pi$
$380$	0	0
$381$	13.3485	0.683863
$382$	0	0
$383$	16.8990	0.863498	0.431749	−	0.901994i	$-0.357897\pi$
$383$	16.8990	0.863498	0.431749	+	0.901994i	$0.357897\pi$
$384$	0	0
$385$	−0.898979	−0.0458162
$386$	0	0
$387$	0.449490	0.0228488
$388$	0	0
$389$	−16.6969	−0.846568	−0.423284	−	0.905997i	$-0.639123\pi$
$389$	−16.6969	−0.846568	−0.423284	+	0.905997i	$0.639123\pi$
$390$	0	0
$391$	−18.6969	−0.945545
$392$	0	0
$393$	21.7980	1.09956
$394$	0	0
$395$	4.20204	0.211428
$396$	0	0
$397$	−15.7980	−0.792877	−0.396438	−	0.918061i	$-0.629754\pi$
$397$	−15.7980	−0.792877	−0.396438	+	0.918061i	$0.629754\pi$
$398$	0	0
$399$	−13.7980	−0.690762
$400$	0	0
$401$	−22.2474	−1.11098	−0.555492	−	0.831522i	$-0.687470\pi$
$401$	−22.2474	−1.11098	−0.555492	+	0.831522i	$0.687470\pi$
$402$	0	0
$403$	−6.44949	−0.321272
$404$	0	0
$405$	0.449490	0.0223353
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−15.1010	−0.746697	−0.373349	−	0.927691i	$-0.621790\pi$
$409$	−15.1010	−0.746697	−0.373349	+	0.927691i	$0.621790\pi$
$410$	0	0
$411$	−10.2474	−0.505469
$412$	0	0
$413$	−9.79796	−0.482126
$414$	0	0
$415$	−4.40408	−0.216188
$416$	0	0
$417$	−2.65153	−0.129846
$418$	0	0
$419$	0.696938	0.0340477	0.0170238	−	0.999855i	$-0.494581\pi$
$419$	0.696938	0.0340477	0.0170238	+	0.999855i	$0.494581\pi$
$420$	0	0
$421$	−4.20204	−0.204795	−0.102397	−	0.994744i	$-0.532651\pi$
$421$	−4.20204	−0.204795	−0.102397	+	0.994744i	$0.532651\pi$
$422$	0	0
$423$	4.89898	0.238197
$424$	0	0
$425$	30.9444	1.50102
$426$	0	0
$427$	−23.5959	−1.14189
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−1.59592	−0.0766949	−0.0383475	−	0.999264i	$-0.512209\pi$
$433$	−1.59592	−0.0766949	−0.0383475	+	0.999264i	$0.512209\pi$
$434$	0	0
$435$	0.696938	0.0334156
$436$	0	0
$437$	−20.0000	−0.956730
$438$	0	0
$439$	14.6515	0.699279	0.349640	−	0.936884i	$-0.386304\pi$
$439$	14.6515	0.699279	0.349640	+	0.936884i	$0.386304\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	6.24745	0.294835	0.147418	−	0.989074i	$-0.452904\pi$
$449$	6.24745	0.294835	0.147418	+	0.989074i	$0.452904\pi$
$450$	0	0
$451$	−0.898979	−0.0423313
$452$	0	0
$453$	8.69694	0.408618
$454$	0	0
$455$	0.898979	0.0421448
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	−6.44949	−0.301036
$460$	0	0
$461$	−20.8990	−0.973363	−0.486681	−	0.873580i	$-0.661793\pi$
$461$	−20.8990	−0.973363	−0.486681	+	0.873580i	$0.661793\pi$
$462$	0	0
$463$	−30.0454	−1.39633	−0.698164	−	0.715938i	$-0.745999\pi$
$463$	−30.0454	−1.39633	−0.698164	+	0.715938i	$0.745999\pi$
$464$	0	0
$465$	−2.89898	−0.134437
$466$	0	0
$467$	3.30306	0.152847	0.0764237	−	0.997075i	$-0.475650\pi$
$467$	3.30306	0.152847	0.0764237	+	0.997075i	$0.475650\pi$
$468$	0	0
$469$	3.10102	0.143192
$470$	0	0
$471$	1.79796	0.0828456
$472$	0	0
$473$	−0.449490	−0.0206676
$474$	0	0
$475$	33.1010	1.51878
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−0.202041	−0.00923149	−0.00461575	−	0.999989i	$-0.501469\pi$
$479$	−0.202041	−0.00923149	−0.00461575	+	0.999989i	$0.501469\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	5.79796	0.263816
$484$	0	0
$485$	−4.49490	−0.204103
$486$	0	0
$487$	−1.55051	−0.0702603	−0.0351302	−	0.999383i	$-0.511185\pi$
$487$	−1.55051	−0.0702603	−0.0351302	+	0.999383i	$0.511185\pi$
$488$	0	0
$489$	23.3485	1.05585
$490$	0	0
$491$	−9.30306	−0.419841	−0.209921	−	0.977718i	$-0.567321\pi$
$491$	−9.30306	−0.419841	−0.209921	+	0.977718i	$0.567321\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	0	0
$495$	−0.449490	−0.0202031
$496$	0	0
$497$	−19.5959	−0.878997
$498$	0	0
$499$	5.14643	0.230386	0.115193	−	0.993343i	$-0.463251\pi$
$499$	5.14643	0.230386	0.115193	+	0.993343i	$0.463251\pi$
$500$	0	0
$501$	21.7980	0.973861
$502$	0	0
$503$	−18.6969	−0.833655	−0.416828	−	0.908986i	$-0.636858\pi$
$503$	−18.6969	−0.833655	−0.416828	+	0.908986i	$0.636858\pi$
$504$	0	0
$505$	−4.29286	−0.191030
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	16.4495	0.729111	0.364555	−	0.931182i	$-0.381221\pi$
$509$	16.4495	0.729111	0.364555	+	0.931182i	$0.381221\pi$
$510$	0	0
$511$	−1.79796	−0.0795370
$512$	0	0
$513$	−6.89898	−0.304597
$514$	0	0
$515$	−1.39388	−0.0614216
$516$	0	0
$517$	−4.89898	−0.215457
$518$	0	0
$519$	0.247449	0.0108618
$520$	0	0
$521$	−7.79796	−0.341635	−0.170817	−	0.985303i	$-0.554641\pi$
$521$	−7.79796	−0.341635	−0.170817	+	0.985303i	$0.554641\pi$
$522$	0	0
$523$	29.3485	1.28332	0.641659	−	0.766990i	$-0.278246\pi$
$523$	29.3485	1.28332	0.641659	+	0.766990i	$0.278246\pi$
$524$	0	0
$525$	−9.59592	−0.418800
$526$	0	0
$527$	41.5959	1.81195
$528$	0	0
$529$	−14.5959	−0.634605
$530$	0	0
$531$	−4.89898	−0.212598
$532$	0	0
$533$	0.898979	0.0389391
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	5.10102	0.220125
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	−3.79796	−0.163287	−0.0816435	−	0.996662i	$-0.526017\pi$
$541$	−3.79796	−0.163287	−0.0816435	+	0.996662i	$0.526017\pi$
$542$	0	0
$543$	15.5959	0.669285
$544$	0	0
$545$	7.59592	0.325373
$546$	0	0
$547$	−34.7423	−1.48548	−0.742738	−	0.669582i	$-0.766473\pi$
$547$	−34.7423	−1.48548	−0.742738	+	0.669582i	$0.766473\pi$
$548$	0	0
$549$	−11.7980	−0.503525
$550$	0	0
$551$	−10.6969	−0.455705
$552$	0	0
$553$	18.6969	0.795075
$554$	0	0
$555$	0.898979	0.0381596
$556$	0	0
$557$	15.5959	0.660820	0.330410	−	0.943837i	$-0.392813\pi$
$557$	15.5959	0.660820	0.330410	+	0.943837i	$0.392813\pi$
$558$	0	0
$559$	0.449490	0.0190114
$560$	0	0
$561$	6.44949	0.272298
$562$	0	0
$563$	−25.7980	−1.08725	−0.543627	−	0.839327i	$-0.682949\pi$
$563$	−25.7980	−1.08725	−0.543627	+	0.839327i	$0.682949\pi$
$564$	0	0
$565$	−7.10102	−0.298742
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	12.6515	0.530380	0.265190	−	0.964196i	$-0.414565\pi$
$569$	12.6515	0.530380	0.265190	+	0.964196i	$0.414565\pi$
$570$	0	0
$571$	29.3485	1.22820	0.614098	−	0.789230i	$-0.289520\pi$
$571$	29.3485	1.22820	0.614098	+	0.789230i	$0.289520\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.9092	−0.580053
$576$	0	0
$577$	23.3939	0.973900	0.486950	−	0.873430i	$-0.338109\pi$
$577$	23.3939	0.973900	0.486950	+	0.873430i	$0.338109\pi$
$578$	0	0
$579$	6.69694	0.278315
$580$	0	0
$581$	−19.5959	−0.812976
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	0.449490	0.0185841
$586$	0	0
$587$	−0.898979	−0.0371049	−0.0185524	−	0.999828i	$-0.505906\pi$
$587$	−0.898979	−0.0371049	−0.0185524	+	0.999828i	$0.505906\pi$
$588$	0	0
$589$	44.4949	1.83338
$590$	0	0
$591$	−4.00000	−0.164538
$592$	0	0
$593$	27.5959	1.13323	0.566614	−	0.823983i	$-0.308253\pi$
$593$	27.5959	1.13323	0.566614	+	0.823983i	$0.308253\pi$
$594$	0	0
$595$	−5.79796	−0.237693
$596$	0	0
$597$	−3.10102	−0.126916
$598$	0	0
$599$	−39.1918	−1.60134	−0.800668	−	0.599109i	$-0.795522\pi$
$599$	−39.1918	−1.60134	−0.800668	+	0.599109i	$0.795522\pi$
$600$	0	0
$601$	−11.7980	−0.481249	−0.240624	−	0.970618i	$-0.577352\pi$
$601$	−11.7980	−0.481249	−0.240624	+	0.970618i	$0.577352\pi$
$602$	0	0
$603$	1.55051	0.0631417
$604$	0	0
$605$	0.449490	0.0182744
$606$	0	0
$607$	−8.85357	−0.359355	−0.179678	−	0.983726i	$-0.557505\pi$
$607$	−8.85357	−0.359355	−0.179678	+	0.983726i	$0.557505\pi$
$608$	0	0
$609$	3.10102	0.125660
$610$	0	0
$611$	4.89898	0.198191
$612$	0	0
$613$	40.4949	1.63557	0.817787	−	0.575521i	$-0.195201\pi$
$613$	40.4949	1.63557	0.817787	+	0.575521i	$0.195201\pi$
$614$	0	0
$615$	0.404082	0.0162942
$616$	0	0
$617$	22.6515	0.911916	0.455958	−	0.890001i	$-0.349297\pi$
$617$	22.6515	0.911916	0.455958	+	0.890001i	$0.349297\pi$
$618$	0	0
$619$	−42.9444	−1.72608	−0.863040	−	0.505135i	$-0.831443\pi$
$619$	−42.9444	−1.72608	−0.863040	+	0.505135i	$0.831443\pi$
$620$	0	0
$621$	2.89898	0.116332
$622$	0	0
$623$	−8.89898	−0.356530
$624$	0	0
$625$	22.0102	0.880408
$626$	0	0
$627$	6.89898	0.275519
$628$	0	0
$629$	−12.8990	−0.514316
$630$	0	0
$631$	29.1464	1.16030	0.580150	−	0.814509i	$-0.302994\pi$
$631$	29.1464	1.16030	0.580150	+	0.814509i	$0.302994\pi$
$632$	0	0
$633$	17.3485	0.689540
$634$	0	0
$635$	6.00000	0.238103
$636$	0	0
$637$	−3.00000	−0.118864
$638$	0	0
$639$	−9.79796	−0.387601
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	34.4495	1.35856	0.679278	−	0.733881i	$-0.262293\pi$
$643$	34.4495	1.35856	0.679278	+	0.733881i	$0.262293\pi$
$644$	0	0
$645$	0.202041	0.00795536
$646$	0	0
$647$	−37.3939	−1.47010	−0.735052	−	0.678010i	$-0.762843\pi$
$647$	−37.3939	−1.47010	−0.735052	+	0.678010i	$0.762843\pi$
$648$	0	0
$649$	4.89898	0.192302
$650$	0	0
$651$	−12.8990	−0.505551
$652$	0	0
$653$	20.2929	0.794121	0.397060	−	0.917792i	$-0.370030\pi$
$653$	20.2929	0.794121	0.397060	+	0.917792i	$0.370030\pi$
$654$	0	0
$655$	9.79796	0.382838
$656$	0	0
$657$	−0.898979	−0.0350725
$658$	0	0
$659$	17.7980	0.693310	0.346655	−	0.937993i	$-0.387317\pi$
$659$	17.7980	0.693310	0.346655	+	0.937993i	$0.387317\pi$
$660$	0	0
$661$	6.89898	0.268339	0.134170	−	0.990958i	$-0.457163\pi$
$661$	6.89898	0.268339	0.134170	+	0.990958i	$0.457163\pi$
$662$	0	0
$663$	−6.44949	−0.250477
$664$	0	0
$665$	−6.20204	−0.240505
$666$	0	0
$667$	4.49490	0.174043
$668$	0	0
$669$	−5.55051	−0.214595
$670$	0	0
$671$	11.7980	0.455455
$672$	0	0
$673$	22.4949	0.867115	0.433557	−	0.901126i	$-0.357258\pi$
$673$	22.4949	0.867115	0.433557	+	0.901126i	$0.357258\pi$
$674$	0	0
$675$	−4.79796	−0.184674
$676$	0	0
$677$	48.2474	1.85430	0.927150	−	0.374690i	$-0.122251\pi$
$677$	48.2474	1.85430	0.927150	+	0.374690i	$0.122251\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	−13.3939	−0.512502	−0.256251	−	0.966610i	$-0.582487\pi$
$683$	−13.3939	−0.512502	−0.256251	+	0.966610i	$0.582487\pi$
$684$	0	0
$685$	−4.60612	−0.175991
$686$	0	0
$687$	−2.89898	−0.110603
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−39.8434	−1.51571	−0.757857	−	0.652421i	$-0.773753\pi$
$691$	−39.8434	−1.51571	−0.757857	+	0.652421i	$0.773753\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−1.19184	−0.0452089
$696$	0	0
$697$	−5.79796	−0.219613
$698$	0	0
$699$	−3.75255	−0.141935
$700$	0	0
$701$	26.4495	0.998983	0.499492	−	0.866319i	$-0.333520\pi$
$701$	26.4495	0.998983	0.499492	+	0.866319i	$0.333520\pi$
$702$	0	0
$703$	−13.7980	−0.520400
$704$	0	0
$705$	2.20204	0.0829337
$706$	0	0
$707$	−19.1010	−0.718368
$708$	0	0
$709$	22.8990	0.859989	0.429995	−	0.902831i	$-0.358515\pi$
$709$	22.8990	0.859989	0.429995	+	0.902831i	$0.358515\pi$
$710$	0	0
$711$	9.34847	0.350595
$712$	0	0
$713$	−18.6969	−0.700206
$714$	0	0
$715$	−0.449490	−0.0168100
$716$	0	0
$717$	−22.0000	−0.821605
$718$	0	0
$719$	−31.1918	−1.16326	−0.581630	−	0.813454i	$-0.697585\pi$
$719$	−31.1918	−1.16326	−0.581630	+	0.813454i	$0.697585\pi$
$720$	0	0
$721$	−6.20204	−0.230976
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	−7.43928	−0.276288
$726$	0	0
$727$	−5.30306	−0.196680	−0.0983398	−	0.995153i	$-0.531353\pi$
$727$	−5.30306	−0.196680	−0.0983398	+	0.995153i	$0.531353\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.89898	−0.107223
$732$	0	0
$733$	52.0908	1.92402	0.962009	−	0.273017i	$-0.0880217\pi$
$733$	52.0908	1.92402	0.962009	+	0.273017i	$0.0880217\pi$
$734$	0	0
$735$	−1.34847	−0.0497390
$736$	0	0
$737$	−1.55051	−0.0571138
$738$	0	0
$739$	−27.3939	−1.00770	−0.503850	−	0.863791i	$-0.668084\pi$
$739$	−27.3939	−1.00770	−0.503850	+	0.863791i	$0.668084\pi$
$740$	0	0
$741$	−6.89898	−0.253440
$742$	0	0
$743$	3.59592	0.131921	0.0659607	−	0.997822i	$-0.478989\pi$
$743$	3.59592	0.131921	0.0659607	+	0.997822i	$0.478989\pi$
$744$	0	0
$745$	−5.39388	−0.197616
$746$	0	0
$747$	−9.79796	−0.358489
$748$	0	0
$749$	−17.7980	−0.650323
$750$	0	0
$751$	15.5959	0.569103	0.284552	−	0.958661i	$-0.408155\pi$
$751$	15.5959	0.569103	0.284552	+	0.958661i	$0.408155\pi$
$752$	0	0
$753$	15.5959	0.568347
$754$	0	0
$755$	3.90918	0.142270
$756$	0	0
$757$	9.79796	0.356113	0.178056	−	0.984020i	$-0.443019\pi$
$757$	9.79796	0.356113	0.178056	+	0.984020i	$0.443019\pi$
$758$	0	0
$759$	−2.89898	−0.105226
$760$	0	0
$761$	−37.7980	−1.37017	−0.685087	−	0.728461i	$-0.740236\pi$
$761$	−37.7980	−1.37017	−0.685087	+	0.728461i	$0.740236\pi$
$762$	0	0
$763$	33.7980	1.22357
$764$	0	0
$765$	−2.89898	−0.104813
$766$	0	0
$767$	−4.89898	−0.176892
$768$	0	0
$769$	19.7980	0.713933	0.356966	−	0.934117i	$-0.383811\pi$
$769$	19.7980	0.713933	0.356966	+	0.934117i	$0.383811\pi$
$770$	0	0
$771$	−30.8990	−1.11280
$772$	0	0
$773$	−32.0454	−1.15259	−0.576297	−	0.817241i	$-0.695503\pi$
$773$	−32.0454	−1.15259	−0.576297	+	0.817241i	$0.695503\pi$
$774$	0	0
$775$	30.9444	1.11156
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−6.20204	−0.222211
$780$	0	0
$781$	9.79796	0.350599
$782$	0	0
$783$	1.55051	0.0554107
$784$	0	0
$785$	0.808164	0.0288446
$786$	0	0
$787$	−31.7980	−1.13347	−0.566737	−	0.823898i	$-0.691795\pi$
$787$	−31.7980	−1.13347	−0.566737	+	0.823898i	$0.691795\pi$
$788$	0	0
$789$	8.89898	0.316812
$790$	0	0
$791$	−31.5959	−1.12342
$792$	0	0
$793$	−11.7980	−0.418958
$794$	0	0
$795$	−2.69694	−0.0956506
$796$	0	0
$797$	−5.10102	−0.180687	−0.0903437	−	0.995911i	$-0.528797\pi$
$797$	−5.10102	−0.180687	−0.0903437	+	0.995911i	$0.528797\pi$
$798$	0	0
$799$	−31.5959	−1.11778
$800$	0	0
$801$	−4.44949	−0.157215
$802$	0	0
$803$	0.898979	0.0317243
$804$	0	0
$805$	2.60612	0.0918538
$806$	0	0
$807$	−20.6969	−0.728567
$808$	0	0
$809$	−1.55051	−0.0545130	−0.0272565	−	0.999628i	$-0.508677\pi$
$809$	−1.55051	−0.0545130	−0.0272565	+	0.999628i	$0.508677\pi$
$810$	0	0
$811$	−1.10102	−0.0386621	−0.0193310	−	0.999813i	$-0.506154\pi$
$811$	−1.10102	−0.0386621	−0.0193310	+	0.999813i	$0.506154\pi$
$812$	0	0
$813$	−26.8990	−0.943388
$814$	0	0
$815$	10.4949	0.367620
$816$	0	0
$817$	−3.10102	−0.108491
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	12.4949	0.436075	0.218037	−	0.975940i	$-0.430035\pi$
$821$	12.4949	0.436075	0.218037	+	0.975940i	$0.430035\pi$
$822$	0	0
$823$	23.1010	0.805251	0.402625	−	0.915365i	$-0.368098\pi$
$823$	23.1010	0.805251	0.402625	+	0.915365i	$0.368098\pi$
$824$	0	0
$825$	4.79796	0.167043
$826$	0	0
$827$	−4.20204	−0.146119	−0.0730596	−	0.997328i	$-0.523276\pi$
$827$	−4.20204	−0.146119	−0.0730596	+	0.997328i	$0.523276\pi$
$828$	0	0
$829$	33.5959	1.16683	0.583417	−	0.812173i	$-0.301715\pi$
$829$	33.5959	1.16683	0.583417	+	0.812173i	$0.301715\pi$
$830$	0	0
$831$	11.7980	0.409267
$832$	0	0
$833$	19.3485	0.670385
$834$	0	0
$835$	9.79796	0.339072
$836$	0	0
$837$	−6.44949	−0.222927
$838$	0	0
$839$	29.3939	1.01479	0.507395	−	0.861714i	$-0.330609\pi$
$839$	29.3939	1.01479	0.507395	+	0.861714i	$0.330609\pi$
$840$	0	0
$841$	−26.5959	−0.917101
$842$	0	0
$843$	14.6969	0.506189
$844$	0	0
$845$	0.449490	0.0154629
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−21.3485	−0.732678
$850$	0	0
$851$	5.79796	0.198751
$852$	0	0
$853$	−29.1918	−0.999509	−0.499755	−	0.866167i	$-0.666577\pi$
$853$	−29.1918	−0.999509	−0.499755	+	0.866167i	$0.666577\pi$
$854$	0	0
$855$	−3.10102	−0.106053
$856$	0	0
$857$	−43.8434	−1.49766	−0.748830	−	0.662762i	$-0.769384\pi$
$857$	−43.8434	−1.49766	−0.748830	+	0.662762i	$0.769384\pi$
$858$	0	0
$859$	36.0908	1.23140	0.615701	−	0.787980i	$-0.288873\pi$
$859$	36.0908	1.23140	0.615701	+	0.787980i	$0.288873\pi$
$860$	0	0
$861$	1.79796	0.0612743
$862$	0	0
$863$	7.10102	0.241722	0.120861	−	0.992669i	$-0.461435\pi$
$863$	7.10102	0.241722	0.120861	+	0.992669i	$0.461435\pi$
$864$	0	0
$865$	0.111226	0.00378179
$866$	0	0
$867$	24.5959	0.835321
$868$	0	0
$869$	−9.34847	−0.317125
$870$	0	0
$871$	1.55051	0.0525370
$872$	0	0
$873$	−10.0000	−0.338449
$874$	0	0
$875$	−8.80816	−0.297770
$876$	0	0
$877$	33.5959	1.13445	0.567227	−	0.823562i	$-0.308016\pi$
$877$	33.5959	1.13445	0.567227	+	0.823562i	$0.308016\pi$
$878$	0	0
$879$	−15.5959	−0.526037
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	−0.898979	−0.0302531	−0.0151265	−	0.999886i	$-0.504815\pi$
$883$	−0.898979	−0.0302531	−0.0151265	+	0.999886i	$0.504815\pi$
$884$	0	0
$885$	−2.20204	−0.0740208
$886$	0	0
$887$	13.7980	0.463290	0.231645	−	0.972800i	$-0.425589\pi$
$887$	13.7980	0.463290	0.231645	+	0.972800i	$0.425589\pi$
$888$	0	0
$889$	26.6969	0.895387
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−33.7980	−1.13101
$894$	0	0
$895$	2.29286	0.0766417
$896$	0	0
$897$	2.89898	0.0967941
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	38.6969	1.28918
$902$	0	0
$903$	0.898979	0.0299162
$904$	0	0
$905$	7.01021	0.233027
$906$	0	0
$907$	31.1918	1.03571	0.517854	−	0.855469i	$-0.326731\pi$
$907$	31.1918	1.03571	0.517854	+	0.855469i	$0.326731\pi$
$908$	0	0
$909$	−9.55051	−0.316770
$910$	0	0
$911$	−17.7980	−0.589673	−0.294836	−	0.955548i	$-0.595265\pi$
$911$	−17.7980	−0.589673	−0.294836	+	0.955548i	$0.595265\pi$
$912$	0	0
$913$	9.79796	0.324265
$914$	0	0
$915$	−5.30306	−0.175314
$916$	0	0
$917$	43.5959	1.43966
$918$	0	0
$919$	5.75255	0.189759	0.0948796	−	0.995489i	$-0.469753\pi$
$919$	5.75255	0.189759	0.0948796	+	0.995489i	$0.469753\pi$
$920$	0	0
$921$	−2.89898	−0.0955246
$922$	0	0
$923$	−9.79796	−0.322504
$924$	0	0
$925$	−9.59592	−0.315512
$926$	0	0
$927$	−3.10102	−0.101851
$928$	0	0
$929$	−42.2474	−1.38609	−0.693047	−	0.720892i	$-0.743732\pi$
$929$	−42.2474	−1.38609	−0.693047	+	0.720892i	$0.743732\pi$
$930$	0	0
$931$	20.6969	0.678315
$932$	0	0
$933$	−1.10102	−0.0360458
$934$	0	0
$935$	2.89898	0.0948068
$936$	0	0
$937$	−44.2929	−1.44698	−0.723492	−	0.690332i	$-0.757464\pi$
$937$	−44.2929	−1.44698	−0.723492	+	0.690332i	$0.757464\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	0	0
$941$	−42.2929	−1.37871	−0.689354	−	0.724425i	$-0.742105\pi$
$941$	−42.2929	−1.37871	−0.689354	+	0.724425i	$0.742105\pi$
$942$	0	0
$943$	2.60612	0.0848670
$944$	0	0
$945$	0.898979	0.0292438
$946$	0	0
$947$	3.10102	0.100770	0.0503848	−	0.998730i	$-0.483955\pi$
$947$	3.10102	0.100770	0.0503848	+	0.998730i	$0.483955\pi$
$948$	0	0
$949$	−0.898979	−0.0291821
$950$	0	0
$951$	−29.8434	−0.967737
$952$	0	0
$953$	−14.4495	−0.468065	−0.234032	−	0.972229i	$-0.575192\pi$
$953$	−14.4495	−0.468065	−0.234032	+	0.972229i	$0.575192\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.55051	−0.0501209
$958$	0	0
$959$	−20.4949	−0.661815
$960$	0	0
$961$	10.5959	0.341804
$962$	0	0
$963$	−8.89898	−0.286766
$964$	0	0
$965$	3.01021	0.0969019
$966$	0	0
$967$	−45.1918	−1.45327	−0.726636	−	0.687023i	$-0.758917\pi$
$967$	−45.1918	−1.45327	−0.726636	+	0.687023i	$0.758917\pi$
$968$	0	0
$969$	44.4949	1.42938
$970$	0	0
$971$	50.8990	1.63343	0.816713	−	0.577044i	$-0.195794\pi$
$971$	50.8990	1.63343	0.816713	+	0.577044i	$0.195794\pi$
$972$	0	0
$973$	−5.30306	−0.170008
$974$	0	0
$975$	−4.79796	−0.153658
$976$	0	0
$977$	56.0454	1.79305	0.896526	−	0.442992i	$-0.146083\pi$
$977$	56.0454	1.79305	0.896526	+	0.442992i	$0.146083\pi$
$978$	0	0
$979$	4.44949	0.142206
$980$	0	0
$981$	16.8990	0.539543
$982$	0	0
$983$	4.40408	0.140468	0.0702342	−	0.997531i	$-0.477625\pi$
$983$	4.40408	0.140468	0.0702342	+	0.997531i	$0.477625\pi$
$984$	0	0
$985$	−1.79796	−0.0572877
$986$	0	0
$987$	9.79796	0.311872
$988$	0	0
$989$	1.30306	0.0414349
$990$	0	0
$991$	−58.2929	−1.85173	−0.925867	−	0.377850i	$-0.876663\pi$
$991$	−58.2929	−1.85173	−0.925867	+	0.377850i	$0.876663\pi$
$992$	0	0
$993$	29.5505	0.937757
$994$	0	0
$995$	−1.39388	−0.0441889
$996$	0	0
$997$	48.2929	1.52945	0.764725	−	0.644357i	$-0.222875\pi$
$997$	48.2929	1.52945	0.764725	+	0.644357i	$0.222875\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bg.1.2		2
4.3	odd	2		3432.2.a.j.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.j.1.2	✓	2	4.3	odd	2
6864.2.a.bg.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.449490 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.449490 q^{5} +2.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.449490 q^{15} -6.44949 q^{17} -6.89898 q^{19} +2.00000 q^{21} +2.89898 q^{23} -4.79796 q^{25} +1.00000 q^{27} +1.55051 q^{29} -6.44949 q^{31} -1.00000 q^{33} +0.898979 q^{35} +2.00000 q^{37} +1.00000 q^{39} +0.898979 q^{41} +0.449490 q^{43} +0.449490 q^{45} +4.89898 q^{47} -3.00000 q^{49} -6.44949 q^{51} -6.00000 q^{53} -0.449490 q^{55} -6.89898 q^{57} -4.89898 q^{59} -11.7980 q^{61} +2.00000 q^{63} +0.449490 q^{65} +1.55051 q^{67} +2.89898 q^{69} -9.79796 q^{71} -0.898979 q^{73} -4.79796 q^{75} -2.00000 q^{77} +9.34847 q^{79} +1.00000 q^{81} -9.79796 q^{83} -2.89898 q^{85} +1.55051 q^{87} -4.44949 q^{89} +2.00000 q^{91} -6.44949 q^{93} -3.10102 q^{95} -10.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 4 q^{15} - 8 q^{17} - 4 q^{19} + 4 q^{21} - 4 q^{23} + 10 q^{25} + 2 q^{27} + 8 q^{29} - 8 q^{31} - 2 q^{33} - 8 q^{35} + 4 q^{37} + 2 q^{39} - 8 q^{41} - 4 q^{43} - 4 q^{45} - 6 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} - 4 q^{57} - 4 q^{61} + 4 q^{63} - 4 q^{65} + 8 q^{67} - 4 q^{69} + 8 q^{73} + 10 q^{75} - 4 q^{77} + 4 q^{79} + 2 q^{81} + 4 q^{85} + 8 q^{87} - 4 q^{89} + 4 q^{91} - 8 q^{93} - 16 q^{95} - 20 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 4 * q^15 - 8 * q^17 - 4 * q^19 + 4 * q^21 - 4 * q^23 + 10 * q^25 + 2 * q^27 + 8 * q^29 - 8 * q^31 - 2 * q^33 - 8 * q^35 + 4 * q^37 + 2 * q^39 - 8 * q^41 - 4 * q^43 - 4 * q^45 - 6 * q^49 - 8 * q^51 - 12 * q^53 + 4 * q^55 - 4 * q^57 - 4 * q^61 + 4 * q^63 - 4 * q^65 + 8 * q^67 - 4 * q^69 + 8 * q^73 + 10 * q^75 - 4 * q^77 + 4 * q^79 + 2 * q^81 + 4 * q^85 + 8 * q^87 - 4 * q^89 + 4 * q^91 - 8 * q^93 - 16 * q^95 - 20 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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