LMFDB - Embedded newform 6864.2.a.ce.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:	$0$
Dimension:	$5$
Coefficient field:	5.5.46437524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 8x^{2} + 70x + 28 $ x^5 - x^4 - 20x^3 + 8x^2 + 70*x + 28
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			1.2
Root			$-1.57692$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.57692 q^{5} +5.18377 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.57692 q^{5} +5.18377 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.57692 q^{15} -4.76069 q^{17} -4.99060 q^{19} -5.18377 q^{21} -3.18377 q^{23} -2.51334 q^{25} -1.00000 q^{27} -4.23794 q^{29} +6.59745 q^{31} +1.00000 q^{33} -8.17437 q^{35} +10.3675 q^{37} -1.00000 q^{39} +7.18377 q^{41} +9.78122 q^{43} -1.57692 q^{45} -10.6971 q^{47} +19.8715 q^{49} +4.76069 q^{51} -0.329568 q^{53} +1.57692 q^{55} +4.99060 q^{57} +4.50393 q^{59} -7.02054 q^{61} +5.18377 q^{63} -1.57692 q^{65} -3.57692 q^{67} +3.18377 q^{69} +13.5245 q^{71} +9.51334 q^{73} +2.51334 q^{75} -5.18377 q^{77} -15.0714 q^{79} +1.00000 q^{81} -3.15383 q^{83} +7.50720 q^{85} +4.23794 q^{87} +13.8112 q^{89} +5.18377 q^{91} -6.59745 q^{93} +7.86975 q^{95} +8.82426 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 10 q^{19} - 3 q^{21} + 7 q^{23} + 16 q^{25} - 5 q^{27} - 3 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 6 q^{37} - 5 q^{39} + 13 q^{41} - 3 q^{43} + q^{45} - 2 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} - q^{55} + 10 q^{57} - 21 q^{59} - 15 q^{61} + 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 4 q^{71} + 19 q^{73} - 16 q^{75} - 3 q^{77} - 8 q^{79} + 5 q^{81} + 2 q^{83} + 46 q^{85} + 3 q^{87} + 12 q^{89} + 3 q^{91} - 4 q^{93} + 32 q^{97} - 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 10 * q^19 - 3 * q^21 + 7 * q^23 + 16 * q^25 - 5 * q^27 - 3 * q^29 + 4 * q^31 + 5 * q^33 - 3 * q^35 + 6 * q^37 - 5 * q^39 + 13 * q^41 - 3 * q^43 + q^45 - 2 * q^47 + 10 * q^49 - 8 * q^51 + 4 * q^53 - q^55 + 10 * q^57 - 21 * q^59 - 15 * q^61 + 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 4 * q^71 + 19 * q^73 - 16 * q^75 - 3 * q^77 - 8 * q^79 + 5 * q^81 + 2 * q^83 + 46 * q^85 + 3 * q^87 + 12 * q^89 + 3 * q^91 - 4 * q^93 + 32 * q^97 - 5 * 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
$5$	−1.57692	−0.705218	−0.352609	−	0.935771i	$-0.614705\pi$
$5$	−1.57692	−0.705218	−0.352609	+	0.935771i	$0.614705\pi$
$6$	0	0
$7$	5.18377	1.95928	0.979641	−	0.200760i	$-0.0643410\pi$
$7$	5.18377	1.95928	0.979641	+	0.200760i	$0.0643410\pi$
$8$	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
$15$	1.57692	0.407158
$16$	0	0
$17$	−4.76069	−1.15464	−0.577318	−	0.816519i	$-0.695901\pi$
$17$	−4.76069	−1.15464	−0.577318	+	0.816519i	$0.695901\pi$
$18$	0	0
$19$	−4.99060	−1.14492	−0.572461	−	0.819932i	$-0.694011\pi$
$19$	−4.99060	−1.14492	−0.572461	+	0.819932i	$0.694011\pi$
$20$	0	0
$21$	−5.18377	−1.13119
$22$	0	0
$23$	−3.18377	−0.663862	−0.331931	−	0.943304i	$-0.607700\pi$
$23$	−3.18377	−0.663862	−0.331931	+	0.943304i	$0.607700\pi$
$24$	0	0
$25$	−2.51334	−0.502668
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.23794	−0.786966	−0.393483	−	0.919332i	$-0.628730\pi$
$29$	−4.23794	−0.786966	−0.393483	+	0.919332i	$0.628730\pi$
$30$	0	0
$31$	6.59745	1.18494	0.592469	−	0.805594i	$-0.298153\pi$
$31$	6.59745	1.18494	0.592469	+	0.805594i	$0.298153\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−8.17437	−1.38172
$36$	0	0
$37$	10.3675	1.70441	0.852207	−	0.523205i	$-0.175264\pi$
$37$	10.3675	1.70441	0.852207	+	0.523205i	$0.175264\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.18377	1.12192	0.560958	−	0.827844i	$-0.310433\pi$
$41$	7.18377	1.12192	0.560958	+	0.827844i	$0.310433\pi$
$42$	0	0
$43$	9.78122	1.49162	0.745811	−	0.666157i	$-0.232062\pi$
$43$	9.78122	1.49162	0.745811	+	0.666157i	$0.232062\pi$
$44$	0	0
$45$	−1.57692	−0.235073
$46$	0	0
$47$	−10.6971	−1.56033	−0.780167	−	0.625572i	$-0.784866\pi$
$47$	−10.6971	−1.56033	−0.780167	+	0.625572i	$0.784866\pi$
$48$	0	0
$49$	19.8715	2.83878
$50$	0	0
$51$	4.76069	0.666629
$52$	0	0
$53$	−0.329568	−0.0452696	−0.0226348	−	0.999744i	$-0.507206\pi$
$53$	−0.329568	−0.0452696	−0.0226348	+	0.999744i	$0.507206\pi$
$54$	0	0
$55$	1.57692	0.212631
$56$	0	0
$57$	4.99060	0.661021
$58$	0	0
$59$	4.50393	0.586362	0.293181	−	0.956057i	$-0.405286\pi$
$59$	4.50393	0.586362	0.293181	+	0.956057i	$0.405286\pi$
$60$	0	0
$61$	−7.02054	−0.898888	−0.449444	−	0.893309i	$-0.648378\pi$
$61$	−7.02054	−0.898888	−0.449444	+	0.893309i	$0.648378\pi$
$62$	0	0
$63$	5.18377	0.653094
$64$	0	0
$65$	−1.57692	−0.195592
$66$	0	0
$67$	−3.57692	−0.436990	−0.218495	−	0.975838i	$-0.570115\pi$
$67$	−3.57692	−0.436990	−0.218495	+	0.975838i	$0.570115\pi$
$68$	0	0
$69$	3.18377	0.383281
$70$	0	0
$71$	13.5245	1.60506	0.802530	−	0.596612i	$-0.203487\pi$
$71$	13.5245	1.60506	0.802530	+	0.596612i	$0.203487\pi$
$72$	0	0
$73$	9.51334	1.11345	0.556726	−	0.830696i	$-0.312057\pi$
$73$	9.51334	1.11345	0.556726	+	0.830696i	$0.312057\pi$
$74$	0	0
$75$	2.51334	0.290215
$76$	0	0
$77$	−5.18377	−0.590745
$78$	0	0
$79$	−15.0714	−1.69567	−0.847835	−	0.530260i	$-0.822094\pi$
$79$	−15.0714	−1.69567	−0.847835	+	0.530260i	$0.822094\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.15383	−0.346178	−0.173089	−	0.984906i	$-0.555375\pi$
$83$	−3.15383	−0.346178	−0.173089	+	0.984906i	$0.555375\pi$
$84$	0	0
$85$	7.50720	0.814270
$86$	0	0
$87$	4.23794	0.454355
$88$	0	0
$89$	13.8112	1.46398	0.731990	−	0.681315i	$-0.238592\pi$
$89$	13.8112	1.46398	0.731990	+	0.681315i	$0.238592\pi$
$90$	0	0
$91$	5.18377	0.543407
$92$	0	0
$93$	−6.59745	−0.684124
$94$	0	0
$95$	7.86975	0.807419
$96$	0	0
$97$	8.82426	0.895968	0.447984	−	0.894042i	$-0.352142\pi$
$97$	8.82426	0.895968	0.447984	+	0.894042i	$0.352142\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	3.27729	0.326102	0.163051	−	0.986618i	$-0.447866\pi$
$101$	3.27729	0.326102	0.163051	+	0.986618i	$0.447866\pi$
$102$	0	0
$103$	−5.35010	−0.527161	−0.263581	−	0.964637i	$-0.584904\pi$
$103$	−5.35010	−0.527161	−0.263581	+	0.964637i	$0.584904\pi$
$104$	0	0
$105$	8.17437	0.797736
$106$	0	0
$107$	−7.34700	−0.710262	−0.355131	−	0.934817i	$-0.615564\pi$
$107$	−7.34700	−0.710262	−0.355131	+	0.934817i	$0.615564\pi$
$108$	0	0
$109$	−0.661028	−0.0633150	−0.0316575	−	0.999499i	$-0.510079\pi$
$109$	−0.661028	−0.0633150	−0.0316575	+	0.999499i	$0.510079\pi$
$110$	0	0
$111$	−10.3675	−0.984043
$112$	0	0
$113$	14.5039	1.36442	0.682208	−	0.731158i	$-0.261020\pi$
$113$	14.5039	1.36442	0.682208	+	0.731158i	$0.261020\pi$
$114$	0	0
$115$	5.02054	0.468167
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−24.6783	−2.26226
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.18377	−0.647739
$124$	0	0
$125$	11.8479	1.05971
$126$	0	0
$127$	−11.1282	−0.987470	−0.493735	−	0.869612i	$-0.664369\pi$
$127$	−11.1282	−0.987470	−0.493735	+	0.869612i	$0.664369\pi$
$128$	0	0
$129$	−9.78122	−0.861189
$130$	0	0
$131$	−16.4820	−1.44004	−0.720021	−	0.693953i	$-0.755868\pi$
$131$	−16.4820	−1.44004	−0.720021	+	0.693953i	$0.755868\pi$
$132$	0	0
$133$	−25.8701	−2.24322
$134$	0	0
$135$	1.57692	0.135719
$136$	0	0
$137$	7.77009	0.663844	0.331922	−	0.943307i	$-0.392303\pi$
$137$	7.77009	0.663844	0.331922	+	0.943307i	$0.392303\pi$
$138$	0	0
$139$	16.6335	1.41084	0.705419	−	0.708791i	$-0.250759\pi$
$139$	16.6335	1.41084	0.705419	+	0.708791i	$0.250759\pi$
$140$	0	0
$141$	10.6971	0.900859
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	6.68288	0.554983
$146$	0	0
$147$	−19.8715	−1.63897
$148$	0	0
$149$	−5.89355	−0.482818	−0.241409	−	0.970423i	$-0.577610\pi$
$149$	−5.89355	−0.482818	−0.241409	+	0.970423i	$0.577610\pi$
$150$	0	0
$151$	−3.83367	−0.311979	−0.155990	−	0.987759i	$-0.549857\pi$
$151$	−3.83367	−0.311979	−0.155990	+	0.987759i	$0.549857\pi$
$152$	0	0
$153$	−4.76069	−0.384879
$154$	0	0
$155$	−10.4036	−0.835639
$156$	0	0
$157$	5.77689	0.461046	0.230523	−	0.973067i	$-0.425956\pi$
$157$	5.77689	0.461046	0.230523	+	0.973067i	$0.425956\pi$
$158$	0	0
$159$	0.329568	0.0261364
$160$	0	0
$161$	−16.5039	−1.30069
$162$	0	0
$163$	−19.1202	−1.49761	−0.748805	−	0.662791i	$-0.769372\pi$
$163$	−19.1202	−1.49761	−0.748805	+	0.662791i	$0.769372\pi$
$164$	0	0
$165$	−1.57692	−0.122763
$166$	0	0
$167$	15.3313	1.18637	0.593186	−	0.805066i	$-0.297870\pi$
$167$	15.3313	1.18637	0.593186	+	0.805066i	$0.297870\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.99060	−0.381640
$172$	0	0
$173$	−19.7812	−1.50394	−0.751969	−	0.659198i	$-0.770896\pi$
$173$	−19.7812	−1.50394	−0.751969	+	0.659198i	$0.770896\pi$
$174$	0	0
$175$	−13.0286	−0.984867
$176$	0	0
$177$	−4.50393	−0.338536
$178$	0	0
$179$	−0.223114	−0.0166763	−0.00833817	−	0.999965i	$-0.502654\pi$
$179$	−0.223114	−0.0166763	−0.00833817	+	0.999965i	$0.502654\pi$
$180$	0	0
$181$	11.7581	0.873971	0.436986	−	0.899469i	$-0.356046\pi$
$181$	11.7581	0.873971	0.436986	+	0.899469i	$0.356046\pi$
$182$	0	0
$183$	7.02054	0.518973
$184$	0	0
$185$	−16.3487	−1.20198
$186$	0	0
$187$	4.76069	0.348136
$188$	0	0
$189$	−5.18377	−0.377064
$190$	0	0
$191$	23.6957	1.71456	0.857282	−	0.514848i	$-0.172152\pi$
$191$	23.6957	1.71456	0.857282	+	0.514848i	$0.172152\pi$
$192$	0	0
$193$	18.1444	1.30606	0.653032	−	0.757330i	$-0.273497\pi$
$193$	18.1444	1.30606	0.653032	+	0.757330i	$0.273497\pi$
$194$	0	0
$195$	1.57692	0.112925
$196$	0	0
$197$	2.96869	0.211510	0.105755	−	0.994392i	$-0.466274\pi$
$197$	2.96869	0.211510	0.105755	+	0.994392i	$0.466274\pi$
$198$	0	0
$199$	15.6578	1.10995	0.554975	−	0.831867i	$-0.312728\pi$
$199$	15.6578	1.10995	0.554975	+	0.831867i	$0.312728\pi$
$200$	0	0
$201$	3.57692	0.252296
$202$	0	0
$203$	−21.9685	−1.54189
$204$	0	0
$205$	−11.3282	−0.791196
$206$	0	0
$207$	−3.18377	−0.221287
$208$	0	0
$209$	4.99060	0.345207
$210$	0	0
$211$	−23.1313	−1.59243	−0.796213	−	0.605016i	$-0.793167\pi$
$211$	−23.1313	−1.59243	−0.796213	+	0.605016i	$0.793167\pi$
$212$	0	0
$213$	−13.5245	−0.926681
$214$	0	0
$215$	−15.4242	−1.05192
$216$	0	0
$217$	34.1997	2.32163
$218$	0	0
$219$	−9.51334	−0.642852
$220$	0	0
$221$	−4.76069	−0.320238
$222$	0	0
$223$	18.7968	1.25872	0.629362	−	0.777112i	$-0.283316\pi$
$223$	18.7968	1.25872	0.629362	+	0.777112i	$0.283316\pi$
$224$	0	0
$225$	−2.51334	−0.167556
$226$	0	0
$227$	26.5499	1.76218	0.881091	−	0.472947i	$-0.156810\pi$
$227$	26.5499	1.76218	0.881091	+	0.472947i	$0.156810\pi$
$228$	0	0
$229$	18.5388	1.22508	0.612539	−	0.790440i	$-0.290148\pi$
$229$	18.5388	1.22508	0.612539	+	0.790440i	$0.290148\pi$
$230$	0	0
$231$	5.18377	0.341067
$232$	0	0
$233$	8.76378	0.574135	0.287067	−	0.957910i	$-0.407320\pi$
$233$	8.76378	0.574135	0.287067	+	0.957910i	$0.407320\pi$
$234$	0	0
$235$	16.8684	1.10038
$236$	0	0
$237$	15.0714	0.978996
$238$	0	0
$239$	13.5701	0.877778	0.438889	−	0.898541i	$-0.355372\pi$
$239$	13.5701	0.877778	0.438889	+	0.898541i	$0.355372\pi$
$240$	0	0
$241$	8.32957	0.536555	0.268277	−	0.963342i	$-0.413546\pi$
$241$	8.32957	0.536555	0.268277	+	0.963342i	$0.413546\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−31.3356	−2.00196
$246$	0	0
$247$	−4.99060	−0.317544
$248$	0	0
$249$	3.15383	0.199866
$250$	0	0
$251$	−15.8889	−1.00290	−0.501450	−	0.865187i	$-0.667200\pi$
$251$	−15.8889	−1.00290	−0.501450	+	0.865187i	$0.667200\pi$
$252$	0	0
$253$	3.18377	0.200162
$254$	0	0
$255$	−7.50720	−0.470119
$256$	0	0
$257$	−3.29022	−0.205239	−0.102619	−	0.994721i	$-0.532722\pi$
$257$	−3.29022	−0.205239	−0.102619	+	0.994721i	$0.532722\pi$
$258$	0	0
$259$	53.7430	3.33942
$260$	0	0
$261$	−4.23794	−0.262322
$262$	0	0
$263$	5.89165	0.363295	0.181647	−	0.983364i	$-0.441857\pi$
$263$	5.89165	0.363295	0.181647	+	0.983364i	$0.441857\pi$
$264$	0	0
$265$	0.519701	0.0319250
$266$	0	0
$267$	−13.8112	−0.845229
$268$	0	0
$269$	−25.4134	−1.54948	−0.774741	−	0.632279i	$-0.782120\pi$
$269$	−25.4134	−1.54948	−0.774741	+	0.632279i	$0.782120\pi$
$270$	0	0
$271$	−9.36124	−0.568655	−0.284327	−	0.958727i	$-0.591770\pi$
$271$	−9.36124	−0.568655	−0.284327	+	0.958727i	$0.591770\pi$
$272$	0	0
$273$	−5.18377	−0.313736
$274$	0	0
$275$	2.51334	0.151560
$276$	0	0
$277$	13.3313	0.801000	0.400500	−	0.916297i	$-0.368836\pi$
$277$	13.3313	0.801000	0.400500	+	0.916297i	$0.368836\pi$
$278$	0	0
$279$	6.59745	0.394979
$280$	0	0
$281$	−27.0190	−1.61182	−0.805909	−	0.592039i	$-0.798323\pi$
$281$	−27.0190	−1.61182	−0.805909	+	0.592039i	$0.798323\pi$
$282$	0	0
$283$	27.8192	1.65368	0.826840	−	0.562438i	$-0.190136\pi$
$283$	27.8192	1.65368	0.826840	+	0.562438i	$0.190136\pi$
$284$	0	0
$285$	−7.86975	−0.466164
$286$	0	0
$287$	37.2390	2.19815
$288$	0	0
$289$	5.66413	0.333184
$290$	0	0
$291$	−8.82426	−0.517287
$292$	0	0
$293$	7.36124	0.430048	0.215024	−	0.976609i	$-0.431017\pi$
$293$	7.36124	0.430048	0.215024	+	0.976609i	$0.431017\pi$
$294$	0	0
$295$	−7.10232	−0.413513
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.18377	−0.184122
$300$	0	0
$301$	50.7036	2.92251
$302$	0	0
$303$	−3.27729	−0.188275
$304$	0	0
$305$	11.0708	0.633912
$306$	0	0
$307$	−30.0521	−1.71517	−0.857583	−	0.514345i	$-0.828035\pi$
$307$	−30.0521	−1.71517	−0.857583	+	0.514345i	$0.828035\pi$
$308$	0	0
$309$	5.35010	0.304357
$310$	0	0
$311$	33.1492	1.87972	0.939859	−	0.341562i	$-0.110956\pi$
$311$	33.1492	1.87972	0.939859	+	0.341562i	$0.110956\pi$
$312$	0	0
$313$	−12.0491	−0.681056	−0.340528	−	0.940234i	$-0.610606\pi$
$313$	−12.0491	−0.681056	−0.340528	+	0.940234i	$0.610606\pi$
$314$	0	0
$315$	−8.17437	−0.460573
$316$	0	0
$317$	26.1281	1.46750	0.733749	−	0.679421i	$-0.237769\pi$
$317$	26.1281	1.46750	0.733749	+	0.679421i	$0.237769\pi$
$318$	0	0
$319$	4.23794	0.237279
$320$	0	0
$321$	7.34700	0.410070
$322$	0	0
$323$	23.7587	1.32197
$324$	0	0
$325$	−2.51334	−0.139415
$326$	0	0
$327$	0.661028	0.0365549
$328$	0	0
$329$	−55.4514	−3.05713
$330$	0	0
$331$	3.24735	0.178490	0.0892452	−	0.996010i	$-0.471555\pi$
$331$	3.24735	0.178490	0.0892452	+	0.996010i	$0.471555\pi$
$332$	0	0
$333$	10.3675	0.568138
$334$	0	0
$335$	5.64049	0.308173
$336$	0	0
$337$	−3.85404	−0.209943	−0.104971	−	0.994475i	$-0.533475\pi$
$337$	−3.85404	−0.209943	−0.104971	+	0.994475i	$0.533475\pi$
$338$	0	0
$339$	−14.5039	−0.787746
$340$	0	0
$341$	−6.59745	−0.357272
$342$	0	0
$343$	66.7228	3.60269
$344$	0	0
$345$	−5.02054	−0.270297
$346$	0	0
$347$	6.38635	0.342837	0.171419	−	0.985198i	$-0.445165\pi$
$347$	6.38635	0.342837	0.171419	+	0.985198i	$0.445165\pi$
$348$	0	0
$349$	2.51350	0.134545	0.0672723	−	0.997735i	$-0.478570\pi$
$349$	2.51350	0.134545	0.0672723	+	0.997735i	$0.478570\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	31.0030	1.65012	0.825061	−	0.565044i	$-0.191141\pi$
$353$	31.0030	1.65012	0.825061	+	0.565044i	$0.191141\pi$
$354$	0	0
$355$	−21.3269	−1.13192
$356$	0	0
$357$	24.6783	1.30611
$358$	0	0
$359$	37.4726	1.97773	0.988865	−	0.148817i	$-0.0475466\pi$
$359$	37.4726	1.97773	0.988865	+	0.148817i	$0.0475466\pi$
$360$	0	0
$361$	5.90605	0.310845
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−15.0017	−0.785226
$366$	0	0
$367$	−13.6411	−0.712061	−0.356031	−	0.934474i	$-0.615870\pi$
$367$	−13.6411	−0.712061	−0.356031	+	0.934474i	$0.615870\pi$
$368$	0	0
$369$	7.18377	0.373972
$370$	0	0
$371$	−1.70840	−0.0886960
$372$	0	0
$373$	−12.8116	−0.663359	−0.331680	−	0.943392i	$-0.607615\pi$
$373$	−12.8116	−0.663359	−0.331680	+	0.943392i	$0.607615\pi$
$374$	0	0
$375$	−11.8479	−0.611823
$376$	0	0
$377$	−4.23794	−0.218265
$378$	0	0
$379$	18.5052	0.950547	0.475273	−	0.879838i	$-0.342349\pi$
$379$	18.5052	0.950547	0.475273	+	0.879838i	$0.342349\pi$
$380$	0	0
$381$	11.1282	0.570116
$382$	0	0
$383$	−7.48030	−0.382225	−0.191113	−	0.981568i	$-0.561210\pi$
$383$	−7.48030	−0.382225	−0.191113	+	0.981568i	$0.561210\pi$
$384$	0	0
$385$	8.17437	0.416604
$386$	0	0
$387$	9.78122	0.497207
$388$	0	0
$389$	−20.0790	−1.01805	−0.509024	−	0.860752i	$-0.669994\pi$
$389$	−20.0790	−1.01805	−0.509024	+	0.860752i	$0.669994\pi$
$390$	0	0
$391$	15.1569	0.766519
$392$	0	0
$393$	16.4820	0.831408
$394$	0	0
$395$	23.7664	1.19582
$396$	0	0
$397$	−7.59789	−0.381327	−0.190663	−	0.981655i	$-0.561064\pi$
$397$	−7.59789	−0.381327	−0.190663	+	0.981655i	$0.561064\pi$
$398$	0	0
$399$	25.8701	1.29513
$400$	0	0
$401$	12.5595	0.627190	0.313595	−	0.949557i	$-0.398467\pi$
$401$	12.5595	0.627190	0.313595	+	0.949557i	$0.398467\pi$
$402$	0	0
$403$	6.59745	0.328642
$404$	0	0
$405$	−1.57692	−0.0783575
$406$	0	0
$407$	−10.3675	−0.513900
$408$	0	0
$409$	−2.83539	−0.140201	−0.0701006	−	0.997540i	$-0.522332\pi$
$409$	−2.83539	−0.140201	−0.0701006	+	0.997540i	$0.522332\pi$
$410$	0	0
$411$	−7.77009	−0.383270
$412$	0	0
$413$	23.3474	1.14885
$414$	0	0
$415$	4.97332	0.244131
$416$	0	0
$417$	−16.6335	−0.814548
$418$	0	0
$419$	−13.8716	−0.677674	−0.338837	−	0.940845i	$-0.610034\pi$
$419$	−13.8716	−0.677674	−0.338837	+	0.940845i	$0.610034\pi$
$420$	0	0
$421$	1.86671	0.0909777	0.0454888	−	0.998965i	$-0.485515\pi$
$421$	1.86671	0.0909777	0.0454888	+	0.998965i	$0.485515\pi$
$422$	0	0
$423$	−10.6971	−0.520111
$424$	0	0
$425$	11.9652	0.580398
$426$	0	0
$427$	−36.3928	−1.76117
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	11.6297	0.560184	0.280092	−	0.959973i	$-0.409635\pi$
$431$	11.6297	0.560184	0.280092	+	0.959973i	$0.409635\pi$
$432$	0	0
$433$	30.1201	1.44748	0.723739	−	0.690074i	$-0.242422\pi$
$433$	30.1201	1.44748	0.723739	+	0.690074i	$0.242422\pi$
$434$	0	0
$435$	−6.68288	−0.320419
$436$	0	0
$437$	15.8889	0.760070
$438$	0	0
$439$	−33.3878	−1.59351	−0.796756	−	0.604301i	$-0.793452\pi$
$439$	−33.3878	−1.59351	−0.796756	+	0.604301i	$0.793452\pi$
$440$	0	0
$441$	19.8715	0.946261
$442$	0	0
$443$	10.3515	0.491813	0.245907	−	0.969294i	$-0.420914\pi$
$443$	10.3515	0.491813	0.245907	+	0.969294i	$0.420914\pi$
$444$	0	0
$445$	−21.7790	−1.03243
$446$	0	0
$447$	5.89355	0.278755
$448$	0	0
$449$	−15.6398	−0.738090	−0.369045	−	0.929412i	$-0.620315\pi$
$449$	−15.6398	−0.738090	−0.369045	+	0.929412i	$0.620315\pi$
$450$	0	0
$451$	−7.18377	−0.338271
$452$	0	0
$453$	3.83367	0.180121
$454$	0	0
$455$	−8.17437	−0.383220
$456$	0	0
$457$	10.4820	0.490329	0.245164	−	0.969482i	$-0.421158\pi$
$457$	10.4820	0.490329	0.245164	+	0.969482i	$0.421158\pi$
$458$	0	0
$459$	4.76069	0.222210
$460$	0	0
$461$	16.4901	0.768019	0.384009	−	0.923329i	$-0.374543\pi$
$461$	16.4901	0.768019	0.384009	+	0.923329i	$0.374543\pi$
$462$	0	0
$463$	−18.5567	−0.862405	−0.431202	−	0.902255i	$-0.641910\pi$
$463$	−18.5567	−0.862405	−0.431202	+	0.902255i	$0.641910\pi$
$464$	0	0
$465$	10.4036	0.482456
$466$	0	0
$467$	1.66854	0.0772108	0.0386054	−	0.999255i	$-0.487708\pi$
$467$	1.66854	0.0772108	0.0386054	+	0.999255i	$0.487708\pi$
$468$	0	0
$469$	−18.5419	−0.856186
$470$	0	0
$471$	−5.77689	−0.266185
$472$	0	0
$473$	−9.78122	−0.449741
$474$	0	0
$475$	12.5431	0.575515
$476$	0	0
$477$	−0.329568	−0.0150899
$478$	0	0
$479$	17.6300	0.805535	0.402768	−	0.915302i	$-0.368048\pi$
$479$	17.6300	0.805535	0.402768	+	0.915302i	$0.368048\pi$
$480$	0	0
$481$	10.3675	0.472719
$482$	0	0
$483$	16.5039	0.750955
$484$	0	0
$485$	−13.9151	−0.631853
$486$	0	0
$487$	−42.5055	−1.92611	−0.963055	−	0.269306i	$-0.913206\pi$
$487$	−42.5055	−1.92611	−0.963055	+	0.269306i	$0.913206\pi$
$488$	0	0
$489$	19.1202	0.864645
$490$	0	0
$491$	9.97195	0.450028	0.225014	−	0.974356i	$-0.427757\pi$
$491$	9.97195	0.450028	0.225014	+	0.974356i	$0.427757\pi$
$492$	0	0
$493$	20.1755	0.908660
$494$	0	0
$495$	1.57692	0.0708771
$496$	0	0
$497$	70.1077	3.14476
$498$	0	0
$499$	−38.6419	−1.72985	−0.864925	−	0.501901i	$-0.832634\pi$
$499$	−38.6419	−1.72985	−0.864925	+	0.501901i	$0.832634\pi$
$500$	0	0
$501$	−15.3313	−0.684952
$502$	0	0
$503$	−2.88724	−0.128736	−0.0643679	−	0.997926i	$-0.520503\pi$
$503$	−2.88724	−0.128736	−0.0643679	+	0.997926i	$0.520503\pi$
$504$	0	0
$505$	−5.16800	−0.229973
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	15.1529	0.671640	0.335820	−	0.941926i	$-0.390987\pi$
$509$	15.1529	0.671640	0.335820	+	0.941926i	$0.390987\pi$
$510$	0	0
$511$	49.3150	2.18157
$512$	0	0
$513$	4.99060	0.220340
$514$	0	0
$515$	8.43666	0.371764
$516$	0	0
$517$	10.6971	0.470458
$518$	0	0
$519$	19.7812	0.868299
$520$	0	0
$521$	24.7443	1.08407	0.542034	−	0.840356i	$-0.317654\pi$
$521$	24.7443	1.08407	0.542034	+	0.840356i	$0.317654\pi$
$522$	0	0
$523$	21.5661	0.943021	0.471511	−	0.881860i	$-0.343709\pi$
$523$	21.5661	0.943021	0.471511	+	0.881860i	$0.343709\pi$
$524$	0	0
$525$	13.0286	0.568613
$526$	0	0
$527$	−31.4084	−1.36817
$528$	0	0
$529$	−12.8636	−0.559287
$530$	0	0
$531$	4.50393	0.195454
$532$	0	0
$533$	7.18377	0.311164
$534$	0	0
$535$	11.5856	0.500889
$536$	0	0
$537$	0.223114	0.00962809
$538$	0	0
$539$	−19.8715	−0.855925
$540$	0	0
$541$	34.7351	1.49338	0.746689	−	0.665173i	$-0.231642\pi$
$541$	34.7351	1.49338	0.746689	+	0.665173i	$0.231642\pi$
$542$	0	0
$543$	−11.7581	−0.504587
$544$	0	0
$545$	1.04239	0.0446509
$546$	0	0
$547$	−9.20474	−0.393566	−0.196783	−	0.980447i	$-0.563049\pi$
$547$	−9.20474	−0.393566	−0.196783	+	0.980447i	$0.563049\pi$
$548$	0	0
$549$	−7.02054	−0.299629
$550$	0	0
$551$	21.1499	0.901014
$552$	0	0
$553$	−78.1269	−3.32229
$554$	0	0
$555$	16.3487	0.693965
$556$	0	0
$557$	4.18514	0.177330	0.0886651	−	0.996061i	$-0.471740\pi$
$557$	4.18514	0.177330	0.0886651	+	0.996061i	$0.471740\pi$
$558$	0	0
$559$	9.78122	0.413702
$560$	0	0
$561$	−4.76069	−0.200996
$562$	0	0
$563$	−5.21681	−0.219862	−0.109931	−	0.993939i	$-0.535063\pi$
$563$	−5.21681	−0.219862	−0.109931	+	0.993939i	$0.535063\pi$
$564$	0	0
$565$	−22.8715	−0.962210
$566$	0	0
$567$	5.18377	0.217698
$568$	0	0
$569$	−18.5925	−0.779436	−0.389718	−	0.920934i	$-0.627428\pi$
$569$	−18.5925	−0.779436	−0.389718	+	0.920934i	$0.627428\pi$
$570$	0	0
$571$	7.67288	0.321100	0.160550	−	0.987028i	$-0.448673\pi$
$571$	7.67288	0.321100	0.160550	+	0.987028i	$0.448673\pi$
$572$	0	0
$573$	−23.6957	−0.989904
$574$	0	0
$575$	8.00189	0.333702
$576$	0	0
$577$	24.3675	1.01443	0.507217	−	0.861818i	$-0.330674\pi$
$577$	24.3675	1.01443	0.507217	+	0.861818i	$0.330674\pi$
$578$	0	0
$579$	−18.1444	−0.754057
$580$	0	0
$581$	−16.3487	−0.678260
$582$	0	0
$583$	0.329568	0.0136493
$584$	0	0
$585$	−1.57692	−0.0651974
$586$	0	0
$587$	−16.5416	−0.682743	−0.341371	−	0.939928i	$-0.610891\pi$
$587$	−16.5416	−0.682743	−0.341371	+	0.939928i	$0.610891\pi$
$588$	0	0
$589$	−32.9252	−1.35666
$590$	0	0
$591$	−2.96869	−0.122116
$592$	0	0
$593$	−22.3694	−0.918603	−0.459301	−	0.888281i	$-0.651900\pi$
$593$	−22.3694	−0.918603	−0.459301	+	0.888281i	$0.651900\pi$
$594$	0	0
$595$	38.9156	1.59538
$596$	0	0
$597$	−15.6578	−0.640830
$598$	0	0
$599$	−28.4308	−1.16165	−0.580826	−	0.814028i	$-0.697270\pi$
$599$	−28.4308	−1.16165	−0.580826	+	0.814028i	$0.697270\pi$
$600$	0	0
$601$	1.61365	0.0658222	0.0329111	−	0.999458i	$-0.489522\pi$
$601$	1.61365	0.0658222	0.0329111	+	0.999458i	$0.489522\pi$
$602$	0	0
$603$	−3.57692	−0.145663
$604$	0	0
$605$	−1.57692	−0.0641107
$606$	0	0
$607$	5.56924	0.226048	0.113024	−	0.993592i	$-0.463946\pi$
$607$	5.56924	0.226048	0.113024	+	0.993592i	$0.463946\pi$
$608$	0	0
$609$	21.9685	0.890210
$610$	0	0
$611$	−10.6971	−0.432759
$612$	0	0
$613$	6.35337	0.256610	0.128305	−	0.991735i	$-0.459046\pi$
$613$	6.35337	0.256610	0.128305	+	0.991735i	$0.459046\pi$
$614$	0	0
$615$	11.3282	0.456797
$616$	0	0
$617$	4.70270	0.189323	0.0946617	−	0.995509i	$-0.469823\pi$
$617$	4.70270	0.189323	0.0946617	+	0.995509i	$0.469823\pi$
$618$	0	0
$619$	11.4467	0.460080	0.230040	−	0.973181i	$-0.426114\pi$
$619$	11.4467	0.460080	0.230040	+	0.973181i	$0.426114\pi$
$620$	0	0
$621$	3.18377	0.127760
$622$	0	0
$623$	71.5939	2.86835
$624$	0	0
$625$	−6.11644	−0.244658
$626$	0	0
$627$	−4.99060	−0.199305
$628$	0	0
$629$	−49.3566	−1.96798
$630$	0	0
$631$	−29.8976	−1.19020	−0.595102	−	0.803650i	$-0.702888\pi$
$631$	−29.8976	−1.19020	−0.595102	+	0.803650i	$0.702888\pi$
$632$	0	0
$633$	23.1313	0.919388
$634$	0	0
$635$	17.5483	0.696382
$636$	0	0
$637$	19.8715	0.787337
$638$	0	0
$639$	13.5245	0.535020
$640$	0	0
$641$	−28.1451	−1.11166	−0.555832	−	0.831295i	$-0.687600\pi$
$641$	−28.1451	−1.11166	−0.555832	+	0.831295i	$0.687600\pi$
$642$	0	0
$643$	5.04058	0.198781	0.0993905	−	0.995049i	$-0.468311\pi$
$643$	5.04058	0.198781	0.0993905	+	0.995049i	$0.468311\pi$
$644$	0	0
$645$	15.4242	0.607326
$646$	0	0
$647$	4.96990	0.195387	0.0976934	−	0.995217i	$-0.468854\pi$
$647$	4.96990	0.195387	0.0976934	+	0.995217i	$0.468854\pi$
$648$	0	0
$649$	−4.50393	−0.176795
$650$	0	0
$651$	−34.1997	−1.34039
$652$	0	0
$653$	1.51827	0.0594146	0.0297073	−	0.999559i	$-0.490542\pi$
$653$	1.51827	0.0594146	0.0297073	+	0.999559i	$0.490542\pi$
$654$	0	0
$655$	25.9908	1.01554
$656$	0	0
$657$	9.51334	0.371151
$658$	0	0
$659$	4.82736	0.188047	0.0940237	−	0.995570i	$-0.470027\pi$
$659$	4.82736	0.188047	0.0940237	+	0.995570i	$0.470027\pi$
$660$	0	0
$661$	23.4161	0.910782	0.455391	−	0.890292i	$-0.349500\pi$
$661$	23.4161	0.910782	0.455391	+	0.890292i	$0.349500\pi$
$662$	0	0
$663$	4.76069	0.184890
$664$	0	0
$665$	40.7950	1.58196
$666$	0	0
$667$	13.4926	0.522437
$668$	0	0
$669$	−18.7968	−0.726725
$670$	0	0
$671$	7.02054	0.271025
$672$	0	0
$673$	−11.8948	−0.458509	−0.229255	−	0.973367i	$-0.573629\pi$
$673$	−11.8948	−0.458509	−0.229255	+	0.973367i	$0.573629\pi$
$674$	0	0
$675$	2.51334	0.0967384
$676$	0	0
$677$	47.4069	1.82200	0.910998	−	0.412410i	$-0.135313\pi$
$677$	47.4069	1.82200	0.910998	+	0.412410i	$0.135313\pi$
$678$	0	0
$679$	45.7430	1.75545
$680$	0	0
$681$	−26.5499	−1.01740
$682$	0	0
$683$	7.78802	0.298000	0.149000	−	0.988837i	$-0.452395\pi$
$683$	7.78802	0.298000	0.149000	+	0.988837i	$0.452395\pi$
$684$	0	0
$685$	−12.2528	−0.468154
$686$	0	0
$687$	−18.5388	−0.707300
$688$	0	0
$689$	−0.329568	−0.0125555
$690$	0	0
$691$	38.9995	1.48361	0.741806	−	0.670614i	$-0.233969\pi$
$691$	38.9995	1.48361	0.741806	+	0.670614i	$0.233969\pi$
$692$	0	0
$693$	−5.18377	−0.196915
$694$	0	0
$695$	−26.2297	−0.994948
$696$	0	0
$697$	−34.1997	−1.29540
$698$	0	0
$699$	−8.76378	−0.331477
$700$	0	0
$701$	−10.8455	−0.409629	−0.204815	−	0.978801i	$-0.565659\pi$
$701$	−10.8455	−0.409629	−0.204815	+	0.978801i	$0.565659\pi$
$702$	0	0
$703$	−51.7402	−1.95142
$704$	0	0
$705$	−16.8684	−0.635302
$706$	0	0
$707$	16.9887	0.638926
$708$	0	0
$709$	35.6504	1.33888	0.669439	−	0.742867i	$-0.266535\pi$
$709$	35.6504	1.33888	0.669439	+	0.742867i	$0.266535\pi$
$710$	0	0
$711$	−15.0714	−0.565223
$712$	0	0
$713$	−21.0048	−0.786635
$714$	0	0
$715$	1.57692	0.0589733
$716$	0	0
$717$	−13.5701	−0.506785
$718$	0	0
$719$	17.8479	0.665614	0.332807	−	0.942995i	$-0.392004\pi$
$719$	17.8479	0.665614	0.332807	+	0.942995i	$0.392004\pi$
$720$	0	0
$721$	−27.7337	−1.03286
$722$	0	0
$723$	−8.32957	−0.309780
$724$	0	0
$725$	10.6514	0.395583
$726$	0	0
$727$	−22.8780	−0.848497	−0.424249	−	0.905546i	$-0.639462\pi$
$727$	−22.8780	−0.848497	−0.424249	+	0.905546i	$0.639462\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−46.5653	−1.72228
$732$	0	0
$733$	43.4372	1.60439	0.802194	−	0.597063i	$-0.203666\pi$
$733$	43.4372	1.60439	0.802194	+	0.597063i	$0.203666\pi$
$734$	0	0
$735$	31.3356	1.15583
$736$	0	0
$737$	3.57692	0.131757
$738$	0	0
$739$	−41.9985	−1.54494	−0.772469	−	0.635052i	$-0.780979\pi$
$739$	−41.9985	−1.54494	−0.772469	+	0.635052i	$0.780979\pi$
$740$	0	0
$741$	4.99060	0.183334
$742$	0	0
$743$	−31.1764	−1.14375	−0.571875	−	0.820340i	$-0.693784\pi$
$743$	−31.1764	−1.14375	−0.571875	+	0.820340i	$0.693784\pi$
$744$	0	0
$745$	9.29362	0.340492
$746$	0	0
$747$	−3.15383	−0.115393
$748$	0	0
$749$	−38.0852	−1.39160
$750$	0	0
$751$	53.9204	1.96758	0.983792	−	0.179314i	$-0.0573877\pi$
$751$	53.9204	1.96758	0.983792	+	0.179314i	$0.0573877\pi$
$752$	0	0
$753$	15.8889	0.579024
$754$	0	0
$755$	6.04537	0.220013
$756$	0	0
$757$	5.78308	0.210190	0.105095	−	0.994462i	$-0.466485\pi$
$757$	5.78308	0.210190	0.105095	+	0.994462i	$0.466485\pi$
$758$	0	0
$759$	−3.18377	−0.115564
$760$	0	0
$761$	−0.742757	−0.0269249	−0.0134625	−	0.999909i	$-0.504285\pi$
$761$	−0.742757	−0.0269249	−0.0134625	+	0.999909i	$0.504285\pi$
$762$	0	0
$763$	−3.42662	−0.124052
$764$	0	0
$765$	7.50720	0.271423
$766$	0	0
$767$	4.50393	0.162628
$768$	0	0
$769$	−7.22638	−0.260590	−0.130295	−	0.991475i	$-0.541592\pi$
$769$	−7.22638	−0.260590	−0.130295	+	0.991475i	$0.541592\pi$
$770$	0	0
$771$	3.29022	0.118495
$772$	0	0
$773$	20.7219	0.745314	0.372657	−	0.927969i	$-0.378447\pi$
$773$	20.7219	0.745314	0.372657	+	0.927969i	$0.378447\pi$
$774$	0	0
$775$	−16.5816	−0.595630
$776$	0	0
$777$	−53.7430	−1.92802
$778$	0	0
$779$	−35.8513	−1.28451
$780$	0	0
$781$	−13.5245	−0.483944
$782$	0	0
$783$	4.23794	0.151452
$784$	0	0
$785$	−9.10966	−0.325138
$786$	0	0
$787$	19.3581	0.690043	0.345022	−	0.938595i	$-0.387872\pi$
$787$	19.3581	0.690043	0.345022	+	0.938595i	$0.387872\pi$
$788$	0	0
$789$	−5.89165	−0.209748
$790$	0	0
$791$	75.1851	2.67327
$792$	0	0
$793$	−7.02054	−0.249307
$794$	0	0
$795$	−0.519701	−0.0184319
$796$	0	0
$797$	−14.2435	−0.504530	−0.252265	−	0.967658i	$-0.581175\pi$
$797$	−14.2435	−0.504530	−0.252265	+	0.967658i	$0.581175\pi$
$798$	0	0
$799$	50.9256	1.80162
$800$	0	0
$801$	13.8112	0.487993
$802$	0	0
$803$	−9.51334	−0.335718
$804$	0	0
$805$	26.0253	0.917271
$806$	0	0
$807$	25.4134	0.894593
$808$	0	0
$809$	−21.8957	−0.769812	−0.384906	−	0.922956i	$-0.625766\pi$
$809$	−21.8957	−0.769812	−0.384906	+	0.922956i	$0.625766\pi$
$810$	0	0
$811$	8.64186	0.303457	0.151728	−	0.988422i	$-0.451516\pi$
$811$	8.64186	0.303457	0.151728	+	0.988422i	$0.451516\pi$
$812$	0	0
$813$	9.36124	0.328313
$814$	0	0
$815$	30.1509	1.05614
$816$	0	0
$817$	−48.8141	−1.70779
$818$	0	0
$819$	5.18377	0.181136
$820$	0	0
$821$	−0.274680	−0.00958638	−0.00479319	−	0.999989i	$-0.501526\pi$
$821$	−0.274680	−0.00958638	−0.00479319	+	0.999989i	$0.501526\pi$
$822$	0	0
$823$	14.2562	0.496938	0.248469	−	0.968640i	$-0.420073\pi$
$823$	14.2562	0.496938	0.248469	+	0.968640i	$0.420073\pi$
$824$	0	0
$825$	−2.51334	−0.0875032
$826$	0	0
$827$	43.9065	1.52678	0.763390	−	0.645938i	$-0.223533\pi$
$827$	43.9065	1.52678	0.763390	+	0.645938i	$0.223533\pi$
$828$	0	0
$829$	−33.7457	−1.17204	−0.586018	−	0.810298i	$-0.699305\pi$
$829$	−33.7457	−1.17204	−0.586018	+	0.810298i	$0.699305\pi$
$830$	0	0
$831$	−13.3313	−0.462458
$832$	0	0
$833$	−94.6018	−3.27776
$834$	0	0
$835$	−24.1762	−0.836650
$836$	0	0
$837$	−6.59745	−0.228041
$838$	0	0
$839$	−54.7669	−1.89076	−0.945381	−	0.325966i	$-0.894310\pi$
$839$	−54.7669	−1.89076	−0.945381	+	0.325966i	$0.894310\pi$
$840$	0	0
$841$	−11.0398	−0.380684
$842$	0	0
$843$	27.0190	0.930584
$844$	0	0
$845$	−1.57692	−0.0542475
$846$	0	0
$847$	5.18377	0.178116
$848$	0	0
$849$	−27.8192	−0.954752
$850$	0	0
$851$	−33.0079	−1.13150
$852$	0	0
$853$	−7.08688	−0.242650	−0.121325	−	0.992613i	$-0.538714\pi$
$853$	−7.08688	−0.242650	−0.121325	+	0.992613i	$0.538714\pi$
$854$	0	0
$855$	7.86975	0.269140
$856$	0	0
$857$	39.6571	1.35466	0.677331	−	0.735679i	$-0.263137\pi$
$857$	39.6571	1.35466	0.677331	+	0.735679i	$0.263137\pi$
$858$	0	0
$859$	−5.52137	−0.188387	−0.0941934	−	0.995554i	$-0.530027\pi$
$859$	−5.52137	−0.188387	−0.0941934	+	0.995554i	$0.530027\pi$
$860$	0	0
$861$	−37.2390	−1.26910
$862$	0	0
$863$	−6.55457	−0.223120	−0.111560	−	0.993758i	$-0.535585\pi$
$863$	−6.55457	−0.223120	−0.111560	+	0.993758i	$0.535585\pi$
$864$	0	0
$865$	31.1933	1.06060
$866$	0	0
$867$	−5.66413	−0.192364
$868$	0	0
$869$	15.0714	0.511264
$870$	0	0
$871$	−3.57692	−0.121199
$872$	0	0
$873$	8.82426	0.298656
$874$	0	0
$875$	61.4168	2.07627
$876$	0	0
$877$	−3.01526	−0.101818	−0.0509091	−	0.998703i	$-0.516212\pi$
$877$	−3.01526	−0.101818	−0.0509091	+	0.998703i	$0.516212\pi$
$878$	0	0
$879$	−7.36124	−0.248288
$880$	0	0
$881$	−56.0852	−1.88956	−0.944779	−	0.327708i	$-0.893724\pi$
$881$	−56.0852	−1.88956	−0.944779	+	0.327708i	$0.893724\pi$
$882$	0	0
$883$	−28.8671	−0.971456	−0.485728	−	0.874110i	$-0.661445\pi$
$883$	−28.8671	−0.971456	−0.485728	+	0.874110i	$0.661445\pi$
$884$	0	0
$885$	7.10232	0.238742
$886$	0	0
$887$	−40.0951	−1.34626	−0.673131	−	0.739523i	$-0.735051\pi$
$887$	−40.0951	−1.34626	−0.673131	+	0.739523i	$0.735051\pi$
$888$	0	0
$889$	−57.6862	−1.93473
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	53.3849	1.78646
$894$	0	0
$895$	0.351832	0.0117605
$896$	0	0
$897$	3.18377	0.106303
$898$	0	0
$899$	−27.9596	−0.932506
$900$	0	0
$901$	1.56897	0.0522700
$902$	0	0
$903$	−50.7036	−1.68731
$904$	0	0
$905$	−18.5415	−0.616340
$906$	0	0
$907$	55.6507	1.84785	0.923925	−	0.382573i	$-0.124962\pi$
$907$	55.6507	1.84785	0.923925	+	0.382573i	$0.124962\pi$
$908$	0	0
$909$	3.27729	0.108701
$910$	0	0
$911$	−12.3867	−0.410390	−0.205195	−	0.978721i	$-0.565783\pi$
$911$	−12.3867	−0.410390	−0.205195	+	0.978721i	$0.565783\pi$
$912$	0	0
$913$	3.15383	0.104377
$914$	0	0
$915$	−11.0708	−0.365989
$916$	0	0
$917$	−85.4390	−2.82145
$918$	0	0
$919$	−3.73747	−0.123288	−0.0616438	−	0.998098i	$-0.519634\pi$
$919$	−3.73747	−0.123288	−0.0616438	+	0.998098i	$0.519634\pi$
$920$	0	0
$921$	30.0521	0.990252
$922$	0	0
$923$	13.5245	0.445163
$924$	0	0
$925$	−26.0571	−0.856753
$926$	0	0
$927$	−5.35010	−0.175720
$928$	0	0
$929$	11.0511	0.362574	0.181287	−	0.983430i	$-0.441974\pi$
$929$	11.0511	0.362574	0.181287	+	0.983430i	$0.441974\pi$
$930$	0	0
$931$	−99.1705	−3.25018
$932$	0	0
$933$	−33.1492	−1.08526
$934$	0	0
$935$	−7.50720	−0.245512
$936$	0	0
$937$	−3.68185	−0.120281	−0.0601403	−	0.998190i	$-0.519155\pi$
$937$	−3.68185	−0.120281	−0.0601403	+	0.998190i	$0.519155\pi$
$938$	0	0
$939$	12.0491	0.393208
$940$	0	0
$941$	22.9175	0.747089	0.373544	−	0.927612i	$-0.378142\pi$
$941$	22.9175	0.747089	0.373544	+	0.927612i	$0.378142\pi$
$942$	0	0
$943$	−22.8715	−0.744798
$944$	0	0
$945$	8.17437	0.265912
$946$	0	0
$947$	−21.7115	−0.705529	−0.352765	−	0.935712i	$-0.614758\pi$
$947$	−21.7115	−0.705529	−0.352765	+	0.935712i	$0.614758\pi$
$948$	0	0
$949$	9.51334	0.308816
$950$	0	0
$951$	−26.1281	−0.847260
$952$	0	0
$953$	47.3690	1.53443	0.767216	−	0.641389i	$-0.221642\pi$
$953$	47.3690	1.53443	0.767216	+	0.641389i	$0.221642\pi$
$954$	0	0
$955$	−37.3662	−1.20914
$956$	0	0
$957$	−4.23794	−0.136993
$958$	0	0
$959$	40.2784	1.30066
$960$	0	0
$961$	12.5264	0.404076
$962$	0	0
$963$	−7.34700	−0.236754
$964$	0	0
$965$	−28.6122	−0.921060
$966$	0	0
$967$	−16.1846	−0.520461	−0.260230	−	0.965547i	$-0.583799\pi$
$967$	−16.1846	−0.520461	−0.260230	+	0.965547i	$0.583799\pi$
$968$	0	0
$969$	−23.7587	−0.763238
$970$	0	0
$971$	−18.0173	−0.578202	−0.289101	−	0.957299i	$-0.593356\pi$
$971$	−18.0173	−0.578202	−0.289101	+	0.957299i	$0.593356\pi$
$972$	0	0
$973$	86.2244	2.76423
$974$	0	0
$975$	2.51334	0.0804912
$976$	0	0
$977$	−36.1948	−1.15797	−0.578987	−	0.815337i	$-0.696552\pi$
$977$	−36.1948	−1.15797	−0.578987	+	0.815337i	$0.696552\pi$
$978$	0	0
$979$	−13.8112	−0.441407
$980$	0	0
$981$	−0.661028	−0.0211050
$982$	0	0
$983$	23.7179	0.756485	0.378242	−	0.925707i	$-0.376529\pi$
$983$	23.7179	0.756485	0.378242	+	0.925707i	$0.376529\pi$
$984$	0	0
$985$	−4.68137	−0.149161
$986$	0	0
$987$	55.4514	1.76504
$988$	0	0
$989$	−31.1412	−0.990231
$990$	0	0
$991$	10.1901	0.323698	0.161849	−	0.986816i	$-0.448254\pi$
$991$	10.1901	0.323698	0.161849	+	0.986816i	$0.448254\pi$
$992$	0	0
$993$	−3.24735	−0.103051
$994$	0	0
$995$	−24.6910	−0.782756
$996$	0	0
$997$	12.0765	0.382467	0.191234	−	0.981545i	$-0.438751\pi$
$997$	12.0765	0.382467	0.191234	+	0.981545i	$0.438751\pi$
$998$	0	0
$999$	−10.3675	−0.328014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.ce.1.2		5
4.3	odd	2		3432.2.a.x.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.x.1.2	✓	5	4.3	odd	2
6864.2.a.ce.1.2		5	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.57692 q^{5} +5.18377 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.57692 q^{5} +5.18377 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.57692 q^{15} -4.76069 q^{17} -4.99060 q^{19} -5.18377 q^{21} -3.18377 q^{23} -2.51334 q^{25} -1.00000 q^{27} -4.23794 q^{29} +6.59745 q^{31} +1.00000 q^{33} -8.17437 q^{35} +10.3675 q^{37} -1.00000 q^{39} +7.18377 q^{41} +9.78122 q^{43} -1.57692 q^{45} -10.6971 q^{47} +19.8715 q^{49} +4.76069 q^{51} -0.329568 q^{53} +1.57692 q^{55} +4.99060 q^{57} +4.50393 q^{59} -7.02054 q^{61} +5.18377 q^{63} -1.57692 q^{65} -3.57692 q^{67} +3.18377 q^{69} +13.5245 q^{71} +9.51334 q^{73} +2.51334 q^{75} -5.18377 q^{77} -15.0714 q^{79} +1.00000 q^{81} -3.15383 q^{83} +7.50720 q^{85} +4.23794 q^{87} +13.8112 q^{89} +5.18377 q^{91} -6.59745 q^{93} +7.86975 q^{95} +8.82426 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 10 q^{19} - 3 q^{21} + 7 q^{23} + 16 q^{25} - 5 q^{27} - 3 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 6 q^{37} - 5 q^{39} + 13 q^{41} - 3 q^{43} + q^{45} - 2 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} - q^{55} + 10 q^{57} - 21 q^{59} - 15 q^{61} + 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 4 q^{71} + 19 q^{73} - 16 q^{75} - 3 q^{77} - 8 q^{79} + 5 q^{81} + 2 q^{83} + 46 q^{85} + 3 q^{87} + 12 q^{89} + 3 q^{91} - 4 q^{93} + 32 q^{97} - 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 10 * q^19 - 3 * q^21 + 7 * q^23 + 16 * q^25 - 5 * q^27 - 3 * q^29 + 4 * q^31 + 5 * q^33 - 3 * q^35 + 6 * q^37 - 5 * q^39 + 13 * q^41 - 3 * q^43 + q^45 - 2 * q^47 + 10 * q^49 - 8 * q^51 + 4 * q^53 - q^55 + 10 * q^57 - 21 * q^59 - 15 * q^61 + 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 4 * q^71 + 19 * q^73 - 16 * q^75 - 3 * q^77 - 8 * q^79 + 5 * q^81 + 2 * q^83 + 46 * q^85 + 3 * q^87 + 12 * q^89 + 3 * q^91 - 4 * q^93 + 32 * q^97 - 5 * q^99

Introduction

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