LMFDB - Embedded newform 6384.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.85577$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.29966 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +5.71155 q^{13} -1.29966 q^{15} +1.29966 q^{17} +1.00000 q^{19} +1.00000 q^{21} -2.00000 q^{23} -3.31087 q^{25} -1.00000 q^{27} -10.7228 q^{29} +3.71155 q^{31} +2.00000 q^{33} -1.29966 q^{35} -0.599328 q^{37} -5.71155 q^{39} -4.00000 q^{41} -10.3109 q^{43} +1.29966 q^{45} -10.1234 q^{47} +1.00000 q^{49} -1.29966 q^{51} +0.700336 q^{53} -2.59933 q^{55} -1.00000 q^{57} +3.40067 q^{61} -1.00000 q^{63} +7.42309 q^{65} +2.59933 q^{67} +2.00000 q^{69} +0.700336 q^{71} +13.4231 q^{73} +3.31087 q^{75} +2.00000 q^{77} +6.02242 q^{79} +1.00000 q^{81} +12.7228 q^{83} +1.68913 q^{85} +10.7228 q^{87} -4.00000 q^{89} -5.71155 q^{91} -3.71155 q^{93} +1.29966 q^{95} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} + 3 q^{19} + 3 q^{21} - 6 q^{23} + 13 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 6 q^{33} + 6 q^{37} - 2 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{47} + 3 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 2 * q^13 + 3 * q^19 + 3 * q^21 - 6 * q^23 + 13 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 6 * q^33 + 6 * q^37 - 2 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^47 + 3 * q^49 + 6 * q^53 - 3 * q^57 + 18 * q^61 - 3 * q^63 - 8 * q^65 + 6 * q^69 + 6 * q^71 + 10 * q^73 - 13 * q^75 + 6 * q^77 - 20 * q^79 + 3 * q^81 + 4 * q^83 + 28 * q^85 - 2 * q^87 - 12 * q^89 - 2 * q^91 + 4 * q^93 - 6 * q^97 - 6 * q^99

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
$5$	1.29966	0.581227	0.290614	−	0.956840i	$-0.406141\pi$
$5$	1.29966	0.581227	0.290614	+	0.956840i	$0.406141\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	5.71155	1.58410	0.792049	−	0.610458i	$-0.209015\pi$
$13$	5.71155	1.58410	0.792049	+	0.610458i	$0.209015\pi$
$14$	0	0
$15$	−1.29966	−0.335572
$16$	0	0
$17$	1.29966	0.315215	0.157607	−	0.987502i	$-0.449622\pi$
$17$	1.29966	0.315215	0.157607	+	0.987502i	$0.449622\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−3.31087	−0.662175
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−10.7228	−1.99117	−0.995583	−	0.0938884i	$-0.970070\pi$
$29$	−10.7228	−1.99117	−0.995583	+	0.0938884i	$0.970070\pi$
$30$	0	0
$31$	3.71155	0.666613	0.333307	−	0.942818i	$-0.391836\pi$
$31$	3.71155	0.666613	0.333307	+	0.942818i	$0.391836\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−1.29966	−0.219683
$36$	0	0
$37$	−0.599328	−0.0985290	−0.0492645	−	0.998786i	$-0.515688\pi$
$37$	−0.599328	−0.0985290	−0.0492645	+	0.998786i	$0.515688\pi$
$38$	0	0
$39$	−5.71155	−0.914579
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−10.3109	−1.57239	−0.786197	−	0.617976i	$-0.787953\pi$
$43$	−10.3109	−1.57239	−0.786197	+	0.617976i	$0.787953\pi$
$44$	0	0
$45$	1.29966	0.193742
$46$	0	0
$47$	−10.1234	−1.47665	−0.738327	−	0.674443i	$-0.764384\pi$
$47$	−10.1234	−1.47665	−0.738327	+	0.674443i	$0.764384\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.29966	−0.181989
$52$	0	0
$53$	0.700336	0.0961985	0.0480993	−	0.998843i	$-0.484684\pi$
$53$	0.700336	0.0961985	0.0480993	+	0.998843i	$0.484684\pi$
$54$	0	0
$55$	−2.59933	−0.350493
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	3.40067	0.435411	0.217706	−	0.976014i	$-0.430143\pi$
$61$	3.40067	0.435411	0.217706	+	0.976014i	$0.430143\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	7.42309	0.920721
$66$	0	0
$67$	2.59933	0.317558	0.158779	−	0.987314i	$-0.449244\pi$
$67$	2.59933	0.317558	0.158779	+	0.987314i	$0.449244\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	0.700336	0.0831146	0.0415573	−	0.999136i	$-0.486768\pi$
$71$	0.700336	0.0831146	0.0415573	+	0.999136i	$0.486768\pi$
$72$	0	0
$73$	13.4231	1.57105	0.785527	−	0.618827i	$-0.212392\pi$
$73$	13.4231	1.57105	0.785527	+	0.618827i	$0.212392\pi$
$74$	0	0
$75$	3.31087	0.382307
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	6.02242	0.677575	0.338787	−	0.940863i	$-0.389983\pi$
$79$	6.02242	0.677575	0.338787	+	0.940863i	$0.389983\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.7228	1.39650	0.698252	−	0.715852i	$-0.253962\pi$
$83$	12.7228	1.39650	0.698252	+	0.715852i	$0.253962\pi$
$84$	0	0
$85$	1.68913	0.183212
$86$	0	0
$87$	10.7228	1.14960
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	−5.71155	−0.598733
$92$	0	0
$93$	−3.71155	−0.384869
$94$	0	0
$95$	1.29966	0.133343
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	2.70034	0.268693	0.134347	−	0.990934i	$-0.457106\pi$
$101$	2.70034	0.268693	0.134347	+	0.990934i	$0.457106\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	1.29966	0.126834
$106$	0	0
$107$	−13.3221	−1.28789	−0.643947	−	0.765070i	$-0.722704\pi$
$107$	−13.3221	−1.28789	−0.643947	+	0.765070i	$0.722704\pi$
$108$	0	0
$109$	−4.59933	−0.440536	−0.220268	−	0.975439i	$-0.570693\pi$
$109$	−4.59933	−0.440536	−0.220268	+	0.975439i	$0.570693\pi$
$110$	0	0
$111$	0.599328	0.0568857
$112$	0	0
$113$	8.12343	0.764188	0.382094	−	0.924124i	$-0.375203\pi$
$113$	8.12343	0.764188	0.382094	+	0.924124i	$0.375203\pi$
$114$	0	0
$115$	−2.59933	−0.242389
$116$	0	0
$117$	5.71155	0.528033
$118$	0	0
$119$	−1.29966	−0.119140
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	−10.8013	−0.966102
$126$	0	0
$127$	−6.59933	−0.585596	−0.292798	−	0.956174i	$-0.594586\pi$
$127$	−6.59933	−0.585596	−0.292798	+	0.956174i	$0.594586\pi$
$128$	0	0
$129$	10.3109	0.907822
$130$	0	0
$131$	7.32208	0.639733	0.319867	−	0.947463i	$-0.396362\pi$
$131$	7.32208	0.639733	0.319867	+	0.947463i	$0.396362\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−1.29966	−0.111857
$136$	0	0
$137$	−17.4231	−1.48856	−0.744278	−	0.667870i	$-0.767206\pi$
$137$	−17.4231	−1.48856	−0.744278	+	0.667870i	$0.767206\pi$
$138$	0	0
$139$	6.22443	0.527950	0.263975	−	0.964530i	$-0.414966\pi$
$139$	6.22443	0.527950	0.263975	+	0.964530i	$0.414966\pi$
$140$	0	0
$141$	10.1234	0.852546
$142$	0	0
$143$	−11.4231	−0.955247
$144$	0	0
$145$	−13.9360	−1.15732
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−16.0224	−1.31261	−0.656304	−	0.754497i	$-0.727881\pi$
$149$	−16.0224	−1.31261	−0.656304	+	0.754497i	$0.727881\pi$
$150$	0	0
$151$	−7.42309	−0.604082	−0.302041	−	0.953295i	$-0.597668\pi$
$151$	−7.42309	−0.604082	−0.302041	+	0.953295i	$0.597668\pi$
$152$	0	0
$153$	1.29966	0.105072
$154$	0	0
$155$	4.82376	0.387454
$156$	0	0
$157$	−6.82376	−0.544595	−0.272298	−	0.962213i	$-0.587784\pi$
$157$	−6.82376	−0.544595	−0.272298	+	0.962213i	$0.587784\pi$
$158$	0	0
$159$	−0.700336	−0.0555402
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−21.7340	−1.70234	−0.851168	−	0.524894i	$-0.824105\pi$
$163$	−21.7340	−1.70234	−0.851168	+	0.524894i	$0.824105\pi$
$164$	0	0
$165$	2.59933	0.202357
$166$	0	0
$167$	1.97758	0.153030	0.0765149	−	0.997068i	$-0.475621\pi$
$167$	1.97758	0.153030	0.0765149	+	0.997068i	$0.475621\pi$
$168$	0	0
$169$	19.6217	1.50937
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−1.19866	−0.0911322	−0.0455661	−	0.998961i	$-0.514509\pi$
$173$	−1.19866	−0.0911322	−0.0455661	+	0.998961i	$0.514509\pi$
$174$	0	0
$175$	3.31087	0.250278
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.3221	0.995739	0.497870	−	0.867252i	$-0.334116\pi$
$179$	13.3221	0.995739	0.497870	+	0.867252i	$0.334116\pi$
$180$	0	0
$181$	1.42309	0.105777	0.0528887	−	0.998600i	$-0.483157\pi$
$181$	1.42309	0.105777	0.0528887	+	0.998600i	$0.483157\pi$
$182$	0	0
$183$	−3.40067	−0.251385
$184$	0	0
$185$	−0.778925	−0.0572677
$186$	0	0
$187$	−2.59933	−0.190082
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	9.42309	0.681831	0.340915	−	0.940094i	$-0.389263\pi$
$191$	9.42309	0.681831	0.340915	+	0.940094i	$0.389263\pi$
$192$	0	0
$193$	10.8238	0.779111	0.389556	−	0.921003i	$-0.372629\pi$
$193$	10.8238	0.779111	0.389556	+	0.921003i	$0.372629\pi$
$194$	0	0
$195$	−7.42309	−0.531579
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−13.4007	−0.949948	−0.474974	−	0.880000i	$-0.657543\pi$
$199$	−13.4007	−0.949948	−0.474974	+	0.880000i	$0.657543\pi$
$200$	0	0
$201$	−2.59933	−0.183342
$202$	0	0
$203$	10.7228	0.752590
$204$	0	0
$205$	−5.19866	−0.363090
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	−0.700336	−0.0479862
$214$	0	0
$215$	−13.4007	−0.913918
$216$	0	0
$217$	−3.71155	−0.251956
$218$	0	0
$219$	−13.4231	−0.907048
$220$	0	0
$221$	7.42309	0.499331
$222$	0	0
$223$	21.9360	1.46894	0.734471	−	0.678640i	$-0.237430\pi$
$223$	21.9360	1.46894	0.734471	+	0.678640i	$0.237430\pi$
$224$	0	0
$225$	−3.31087	−0.220725
$226$	0	0
$227$	3.17624	0.210814	0.105407	−	0.994429i	$-0.466385\pi$
$227$	3.17624	0.210814	0.105407	+	0.994429i	$0.466385\pi$
$228$	0	0
$229$	17.2211	1.13800	0.569000	−	0.822337i	$-0.307330\pi$
$229$	17.2211	1.13800	0.569000	+	0.822337i	$0.307330\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−22.2469	−1.45744	−0.728720	−	0.684812i	$-0.759884\pi$
$233$	−22.2469	−1.45744	−0.728720	+	0.684812i	$0.759884\pi$
$234$	0	0
$235$	−13.1571	−0.858272
$236$	0	0
$237$	−6.02242	−0.391198
$238$	0	0
$239$	1.17624	0.0760845	0.0380423	−	0.999276i	$-0.487888\pi$
$239$	1.17624	0.0760845	0.0380423	+	0.999276i	$0.487888\pi$
$240$	0	0
$241$	−14.6217	−0.941869	−0.470935	−	0.882168i	$-0.656083\pi$
$241$	−14.6217	−0.941869	−0.470935	+	0.882168i	$0.656083\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.29966	0.0830325
$246$	0	0
$247$	5.71155	0.363417
$248$	0	0
$249$	−12.7228	−0.806272
$250$	0	0
$251$	1.87657	0.118448	0.0592242	−	0.998245i	$-0.481137\pi$
$251$	1.87657	0.118448	0.0592242	+	0.998245i	$0.481137\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	−1.68913	−0.105777
$256$	0	0
$257$	−22.0224	−1.37372	−0.686860	−	0.726789i	$-0.741012\pi$
$257$	−22.0224	−1.37372	−0.686860	+	0.726789i	$0.741012\pi$
$258$	0	0
$259$	0.599328	0.0372404
$260$	0	0
$261$	−10.7228	−0.663722
$262$	0	0
$263$	0.599328	0.0369562	0.0184781	−	0.999829i	$-0.494118\pi$
$263$	0.599328	0.0369562	0.0184781	+	0.999829i	$0.494118\pi$
$264$	0	0
$265$	0.910201	0.0559132
$266$	0	0
$267$	4.00000	0.244796
$268$	0	0
$269$	−5.40067	−0.329285	−0.164642	−	0.986353i	$-0.552647\pi$
$269$	−5.40067	−0.329285	−0.164642	+	0.986353i	$0.552647\pi$
$270$	0	0
$271$	−31.2211	−1.89655	−0.948273	−	0.317457i	$-0.897171\pi$
$271$	−31.2211	−1.89655	−0.948273	+	0.317457i	$0.897171\pi$
$272$	0	0
$273$	5.71155	0.345678
$274$	0	0
$275$	6.62175	0.399306
$276$	0	0
$277$	−29.9584	−1.80003	−0.900013	−	0.435863i	$-0.856443\pi$
$277$	−29.9584	−1.80003	−0.900013	+	0.435863i	$0.856443\pi$
$278$	0	0
$279$	3.71155	0.222204
$280$	0	0
$281$	−8.70034	−0.519019	−0.259509	−	0.965741i	$-0.583561\pi$
$281$	−8.70034	−0.519019	−0.259509	+	0.965741i	$0.583561\pi$
$282$	0	0
$283$	6.02242	0.357996	0.178998	−	0.983849i	$-0.442714\pi$
$283$	6.02242	0.357996	0.178998	+	0.983849i	$0.442714\pi$
$284$	0	0
$285$	−1.29966	−0.0769855
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−15.3109	−0.900640
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−29.8204	−1.74213	−0.871063	−	0.491171i	$-0.836569\pi$
$293$	−29.8204	−1.74213	−0.871063	+	0.491171i	$0.836569\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	−11.4231	−0.660614
$300$	0	0
$301$	10.3109	0.594309
$302$	0	0
$303$	−2.70034	−0.155130
$304$	0	0
$305$	4.41973	0.253073
$306$	0	0
$307$	10.6858	0.609869	0.304934	−	0.952373i	$-0.401365\pi$
$307$	10.6858	0.609869	0.304934	+	0.952373i	$0.401365\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	20.5207	1.16362	0.581812	−	0.813323i	$-0.302344\pi$
$311$	20.5207	1.16362	0.581812	+	0.813323i	$0.302344\pi$
$312$	0	0
$313$	9.22107	0.521206	0.260603	−	0.965446i	$-0.416079\pi$
$313$	9.22107	0.521206	0.260603	+	0.965446i	$0.416079\pi$
$314$	0	0
$315$	−1.29966	−0.0732278
$316$	0	0
$317$	30.9696	1.73943	0.869713	−	0.493557i	$-0.164304\pi$
$317$	30.9696	1.73943	0.869713	+	0.493557i	$0.164304\pi$
$318$	0	0
$319$	21.4455	1.20072
$320$	0	0
$321$	13.3221	0.743566
$322$	0	0
$323$	1.29966	0.0723152
$324$	0	0
$325$	−18.9102	−1.04895
$326$	0	0
$327$	4.59933	0.254343
$328$	0	0
$329$	10.1234	0.558123
$330$	0	0
$331$	−6.80134	−0.373836	−0.186918	−	0.982376i	$-0.559850\pi$
$331$	−6.80134	−0.373836	−0.186918	+	0.982376i	$0.559850\pi$
$332$	0	0
$333$	−0.599328	−0.0328430
$334$	0	0
$335$	3.37825	0.184574
$336$	0	0
$337$	12.5993	0.686329	0.343165	−	0.939275i	$-0.388501\pi$
$337$	12.5993	0.686329	0.343165	+	0.939275i	$0.388501\pi$
$338$	0	0
$339$	−8.12343	−0.441204
$340$	0	0
$341$	−7.42309	−0.401983
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.59933	0.139943
$346$	0	0
$347$	4.02242	0.215935	0.107967	−	0.994154i	$-0.465566\pi$
$347$	4.02242	0.215935	0.107967	+	0.994154i	$0.465566\pi$
$348$	0	0
$349$	−27.8204	−1.48919	−0.744596	−	0.667515i	$-0.767358\pi$
$349$	−27.8204	−1.48919	−0.744596	+	0.667515i	$0.767358\pi$
$350$	0	0
$351$	−5.71155	−0.304860
$352$	0	0
$353$	−17.2997	−0.920768	−0.460384	−	0.887720i	$-0.652288\pi$
$353$	−17.2997	−0.920768	−0.460384	+	0.887720i	$0.652288\pi$
$354$	0	0
$355$	0.910201	0.0483085
$356$	0	0
$357$	1.29966	0.0687855
$358$	0	0
$359$	−31.8204	−1.67942	−0.839708	−	0.543038i	$-0.817274\pi$
$359$	−31.8204	−1.67942	−0.839708	+	0.543038i	$0.817274\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	17.4455	0.913140
$366$	0	0
$367$	−24.8686	−1.29813	−0.649065	−	0.760733i	$-0.724840\pi$
$367$	−24.8686	−1.29813	−0.649065	+	0.760733i	$0.724840\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	−0.700336	−0.0363596
$372$	0	0
$373$	26.0448	1.34855	0.674275	−	0.738480i	$-0.264456\pi$
$373$	26.0448	1.34855	0.674275	+	0.738480i	$0.264456\pi$
$374$	0	0
$375$	10.8013	0.557779
$376$	0	0
$377$	−61.2435	−3.15420
$378$	0	0
$379$	31.4679	1.61640	0.808199	−	0.588909i	$-0.200442\pi$
$379$	31.4679	1.61640	0.808199	+	0.588909i	$0.200442\pi$
$380$	0	0
$381$	6.59933	0.338094
$382$	0	0
$383$	−21.4007	−1.09352	−0.546762	−	0.837288i	$-0.684140\pi$
$383$	−21.4007	−1.09352	−0.546762	+	0.837288i	$0.684140\pi$
$384$	0	0
$385$	2.59933	0.132474
$386$	0	0
$387$	−10.3109	−0.524131
$388$	0	0
$389$	20.6442	1.04670	0.523350	−	0.852118i	$-0.324682\pi$
$389$	20.6442	1.04670	0.523350	+	0.852118i	$0.324682\pi$
$390$	0	0
$391$	−2.59933	−0.131454
$392$	0	0
$393$	−7.32208	−0.369350
$394$	0	0
$395$	7.82712	0.393825
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−19.2997	−0.963779	−0.481890	−	0.876232i	$-0.660049\pi$
$401$	−19.2997	−0.963779	−0.481890	+	0.876232i	$0.660049\pi$
$402$	0	0
$403$	21.1987	1.05598
$404$	0	0
$405$	1.29966	0.0645808
$406$	0	0
$407$	1.19866	0.0594152
$408$	0	0
$409$	−3.48711	−0.172427	−0.0862133	−	0.996277i	$-0.527477\pi$
$409$	−3.48711	−0.172427	−0.0862133	+	0.996277i	$0.527477\pi$
$410$	0	0
$411$	17.4231	0.859418
$412$	0	0
$413$	0	0
$414$	0	0
$415$	16.5353	0.811686
$416$	0	0
$417$	−6.22443	−0.304812
$418$	0	0
$419$	32.5207	1.58874	0.794371	−	0.607433i	$-0.207801\pi$
$419$	32.5207	1.58874	0.794371	+	0.607433i	$0.207801\pi$
$420$	0	0
$421$	17.4231	0.849149	0.424575	−	0.905393i	$-0.360424\pi$
$421$	17.4231	0.849149	0.424575	+	0.905393i	$0.360424\pi$
$422$	0	0
$423$	−10.1234	−0.492218
$424$	0	0
$425$	−4.30302	−0.208727
$426$	0	0
$427$	−3.40067	−0.164570
$428$	0	0
$429$	11.4231	0.551512
$430$	0	0
$431$	0.325441	0.0156760	0.00783798	−	0.999969i	$-0.497505\pi$
$431$	0.325441	0.0156760	0.00783798	+	0.999969i	$0.497505\pi$
$432$	0	0
$433$	−31.8204	−1.52919	−0.764595	−	0.644510i	$-0.777061\pi$
$433$	−31.8204	−1.52919	−0.764595	+	0.644510i	$0.777061\pi$
$434$	0	0
$435$	13.9360	0.668179
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	−23.7564	−1.13383	−0.566915	−	0.823776i	$-0.691863\pi$
$439$	−23.7564	−1.13383	−0.566915	+	0.823776i	$0.691863\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−24.5993	−1.16875	−0.584375	−	0.811484i	$-0.698660\pi$
$443$	−24.5993	−1.16875	−0.584375	+	0.811484i	$0.698660\pi$
$444$	0	0
$445$	−5.19866	−0.246440
$446$	0	0
$447$	16.0224	0.757834
$448$	0	0
$449$	−27.3445	−1.29047	−0.645233	−	0.763986i	$-0.723240\pi$
$449$	−27.3445	−1.29047	−0.645233	+	0.763986i	$0.723240\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	7.42309	0.348767
$454$	0	0
$455$	−7.42309	−0.348000
$456$	0	0
$457$	8.31087	0.388766	0.194383	−	0.980926i	$-0.437730\pi$
$457$	8.31087	0.388766	0.194383	+	0.980926i	$0.437730\pi$
$458$	0	0
$459$	−1.29966	−0.0606631
$460$	0	0
$461$	9.54652	0.444626	0.222313	−	0.974975i	$-0.428639\pi$
$461$	9.54652	0.444626	0.222313	+	0.974975i	$0.428639\pi$
$462$	0	0
$463$	10.8462	0.504065	0.252032	−	0.967719i	$-0.418901\pi$
$463$	10.8462	0.504065	0.252032	+	0.967719i	$0.418901\pi$
$464$	0	0
$465$	−4.82376	−0.223697
$466$	0	0
$467$	−11.5241	−0.533272	−0.266636	−	0.963797i	$-0.585912\pi$
$467$	−11.5241	−0.533272	−0.266636	+	0.963797i	$0.585912\pi$
$468$	0	0
$469$	−2.59933	−0.120026
$470$	0	0
$471$	6.82376	0.314422
$472$	0	0
$473$	20.6217	0.948189
$474$	0	0
$475$	−3.31087	−0.151913
$476$	0	0
$477$	0.700336	0.0320662
$478$	0	0
$479$	35.3669	1.61596	0.807978	−	0.589213i	$-0.200562\pi$
$479$	35.3669	1.61596	0.807978	+	0.589213i	$0.200562\pi$
$480$	0	0
$481$	−3.42309	−0.156079
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	−2.59933	−0.118029
$486$	0	0
$487$	11.3783	0.515598	0.257799	−	0.966199i	$-0.417003\pi$
$487$	11.3783	0.515598	0.257799	+	0.966199i	$0.417003\pi$
$488$	0	0
$489$	21.7340	0.982844
$490$	0	0
$491$	−21.2211	−0.957694	−0.478847	−	0.877898i	$-0.658945\pi$
$491$	−21.2211	−0.957694	−0.478847	+	0.877898i	$0.658945\pi$
$492$	0	0
$493$	−13.9360	−0.627645
$494$	0	0
$495$	−2.59933	−0.116831
$496$	0	0
$497$	−0.700336	−0.0314144
$498$	0	0
$499$	−37.2435	−1.66725	−0.833624	−	0.552333i	$-0.813738\pi$
$499$	−37.2435	−1.66725	−0.833624	+	0.552333i	$0.813738\pi$
$500$	0	0
$501$	−1.97758	−0.0883518
$502$	0	0
$503$	0.924770	0.0412334	0.0206167	−	0.999787i	$-0.493437\pi$
$503$	0.924770	0.0412334	0.0206167	+	0.999787i	$0.493437\pi$
$504$	0	0
$505$	3.50953	0.156172
$506$	0	0
$507$	−19.6217	−0.871432
$508$	0	0
$509$	−20.2469	−0.897426	−0.448713	−	0.893676i	$-0.648117\pi$
$509$	−20.2469	−0.897426	−0.448713	+	0.893676i	$0.648117\pi$
$510$	0	0
$511$	−13.4231	−0.593803
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−10.3973	−0.458160
$516$	0	0
$517$	20.2469	0.890456
$518$	0	0
$519$	1.19866	0.0526152
$520$	0	0
$521$	−44.2469	−1.93849	−0.969245	−	0.246098i	$-0.920851\pi$
$521$	−44.2469	−1.93849	−0.969245	+	0.246098i	$0.920851\pi$
$522$	0	0
$523$	37.9808	1.66079	0.830393	−	0.557179i	$-0.188116\pi$
$523$	37.9808	1.66079	0.830393	+	0.557179i	$0.188116\pi$
$524$	0	0
$525$	−3.31087	−0.144498
$526$	0	0
$527$	4.82376	0.210126
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−22.8462	−0.989578
$534$	0	0
$535$	−17.3142	−0.748560
$536$	0	0
$537$	−13.3221	−0.574890
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−9.46793	−0.407058	−0.203529	−	0.979069i	$-0.565241\pi$
$541$	−9.46793	−0.407058	−0.203529	+	0.979069i	$0.565241\pi$
$542$	0	0
$543$	−1.42309	−0.0610706
$544$	0	0
$545$	−5.97758	−0.256051
$546$	0	0
$547$	11.1762	0.477861	0.238931	−	0.971037i	$-0.423203\pi$
$547$	11.1762	0.477861	0.238931	+	0.971037i	$0.423203\pi$
$548$	0	0
$549$	3.40067	0.145137
$550$	0	0
$551$	−10.7228	−0.456805
$552$	0	0
$553$	−6.02242	−0.256099
$554$	0	0
$555$	0.778925	0.0330635
$556$	0	0
$557$	36.0224	1.52632	0.763159	−	0.646210i	$-0.223647\pi$
$557$	36.0224	1.52632	0.763159	+	0.646210i	$0.223647\pi$
$558$	0	0
$559$	−58.8910	−2.49082
$560$	0	0
$561$	2.59933	0.109744
$562$	0	0
$563$	10.0224	0.422395	0.211197	−	0.977443i	$-0.432264\pi$
$563$	10.0224	0.422395	0.211197	+	0.977443i	$0.432264\pi$
$564$	0	0
$565$	10.5577	0.444167
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−11.0976	−0.465238	−0.232619	−	0.972568i	$-0.574729\pi$
$569$	−11.0976	−0.465238	−0.232619	+	0.972568i	$0.574729\pi$
$570$	0	0
$571$	28.0448	1.17364	0.586820	−	0.809717i	$-0.300380\pi$
$571$	28.0448	1.17364	0.586820	+	0.809717i	$0.300380\pi$
$572$	0	0
$573$	−9.42309	−0.393655
$574$	0	0
$575$	6.62175	0.276146
$576$	0	0
$577$	−17.1762	−0.715056	−0.357528	−	0.933902i	$-0.616380\pi$
$577$	−17.1762	−0.715056	−0.357528	+	0.933902i	$0.616380\pi$
$578$	0	0
$579$	−10.8238	−0.449820
$580$	0	0
$581$	−12.7228	−0.527829
$582$	0	0
$583$	−1.40067	−0.0580099
$584$	0	0
$585$	7.42309	0.306907
$586$	0	0
$587$	−16.5207	−0.681884	−0.340942	−	0.940084i	$-0.610746\pi$
$587$	−16.5207	−0.681884	−0.340942	+	0.940084i	$0.610746\pi$
$588$	0	0
$589$	3.71155	0.152932
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	36.1907	1.48617	0.743087	−	0.669195i	$-0.233361\pi$
$593$	36.1907	1.48617	0.743087	+	0.669195i	$0.233361\pi$
$594$	0	0
$595$	−1.68913	−0.0692474
$596$	0	0
$597$	13.4007	0.548453
$598$	0	0
$599$	−4.32544	−0.176733	−0.0883664	−	0.996088i	$-0.528165\pi$
$599$	−4.32544	−0.176733	−0.0883664	+	0.996088i	$0.528165\pi$
$600$	0	0
$601$	−31.6475	−1.29093	−0.645465	−	0.763790i	$-0.723336\pi$
$601$	−31.6475	−1.29093	−0.645465	+	0.763790i	$0.723336\pi$
$602$	0	0
$603$	2.59933	0.105853
$604$	0	0
$605$	−9.09765	−0.369872
$606$	0	0
$607$	14.8462	0.602588	0.301294	−	0.953531i	$-0.402581\pi$
$607$	14.8462	0.602588	0.301294	+	0.953531i	$0.402581\pi$
$608$	0	0
$609$	−10.7228	−0.434508
$610$	0	0
$611$	−57.8204	−2.33916
$612$	0	0
$613$	12.8878	0.520533	0.260266	−	0.965537i	$-0.416190\pi$
$613$	12.8878	0.520533	0.260266	+	0.965537i	$0.416190\pi$
$614$	0	0
$615$	5.19866	0.209630
$616$	0	0
$617$	3.60269	0.145039	0.0725194	−	0.997367i	$-0.476896\pi$
$617$	3.60269	0.145039	0.0725194	+	0.997367i	$0.476896\pi$
$618$	0	0
$619$	−6.22443	−0.250181	−0.125091	−	0.992145i	$-0.539922\pi$
$619$	−6.22443	−0.250181	−0.125091	+	0.992145i	$0.539922\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	2.51625	0.100650
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−0.778925	−0.0310578
$630$	0	0
$631$	8.08644	0.321916	0.160958	−	0.986961i	$-0.448542\pi$
$631$	8.08644	0.321916	0.160958	+	0.986961i	$0.448542\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	−8.57691	−0.340364
$636$	0	0
$637$	5.71155	0.226300
$638$	0	0
$639$	0.700336	0.0277049
$640$	0	0
$641$	−20.1683	−0.796598	−0.398299	−	0.917256i	$-0.630399\pi$
$641$	−20.1683	−0.796598	−0.398299	+	0.917256i	$0.630399\pi$
$642$	0	0
$643$	−20.8686	−0.822977	−0.411489	−	0.911415i	$-0.634991\pi$
$643$	−20.8686	−0.822977	−0.411489	+	0.911415i	$0.634991\pi$
$644$	0	0
$645$	13.4007	0.527651
$646$	0	0
$647$	5.87657	0.231032	0.115516	−	0.993306i	$-0.463148\pi$
$647$	5.87657	0.231032	0.115516	+	0.993306i	$0.463148\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	3.71155	0.145467
$652$	0	0
$653$	19.1987	0.751301	0.375651	−	0.926761i	$-0.377419\pi$
$653$	19.1987	0.751301	0.375651	+	0.926761i	$0.377419\pi$
$654$	0	0
$655$	9.51625	0.371831
$656$	0	0
$657$	13.4231	0.523685
$658$	0	0
$659$	−41.9438	−1.63390	−0.816950	−	0.576709i	$-0.804337\pi$
$659$	−41.9438	−1.63390	−0.816950	+	0.576709i	$0.804337\pi$
$660$	0	0
$661$	26.7822	1.04171	0.520853	−	0.853647i	$-0.325614\pi$
$661$	26.7822	1.04171	0.520853	+	0.853647i	$0.325614\pi$
$662$	0	0
$663$	−7.42309	−0.288289
$664$	0	0
$665$	−1.29966	−0.0503988
$666$	0	0
$667$	21.4455	0.830373
$668$	0	0
$669$	−21.9360	−0.848094
$670$	0	0
$671$	−6.80134	−0.262563
$672$	0	0
$673$	25.1762	0.970473	0.485236	−	0.874383i	$-0.338734\pi$
$673$	25.1762	0.970473	0.485236	+	0.874383i	$0.338734\pi$
$674$	0	0
$675$	3.31087	0.127436
$676$	0	0
$677$	−31.8428	−1.22382	−0.611910	−	0.790928i	$-0.709598\pi$
$677$	−31.8428	−1.22382	−0.611910	+	0.790928i	$0.709598\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−3.17624	−0.121714
$682$	0	0
$683$	−38.7228	−1.48169	−0.740843	−	0.671679i	$-0.765574\pi$
$683$	−38.7228	−1.48169	−0.740843	+	0.671679i	$0.765574\pi$
$684$	0	0
$685$	−22.6442	−0.865189
$686$	0	0
$687$	−17.2211	−0.657025
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	30.0224	1.14211	0.571053	−	0.820913i	$-0.306535\pi$
$691$	30.0224	1.14211	0.571053	+	0.820913i	$0.306535\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	8.08967	0.306859
$696$	0	0
$697$	−5.19866	−0.196913
$698$	0	0
$699$	22.2469	0.841453
$700$	0	0
$701$	47.6475	1.79962	0.899811	−	0.436280i	$-0.143704\pi$
$701$	47.6475	1.79962	0.899811	+	0.436280i	$0.143704\pi$
$702$	0	0
$703$	−0.599328	−0.0226041
$704$	0	0
$705$	13.1571	0.495523
$706$	0	0
$707$	−2.70034	−0.101557
$708$	0	0
$709$	25.5095	0.958030	0.479015	−	0.877807i	$-0.340994\pi$
$709$	25.5095	0.958030	0.479015	+	0.877807i	$0.340994\pi$
$710$	0	0
$711$	6.02242	0.225858
$712$	0	0
$713$	−7.42309	−0.277997
$714$	0	0
$715$	−14.8462	−0.555216
$716$	0	0
$717$	−1.17624	−0.0439274
$718$	0	0
$719$	−9.67456	−0.360800	−0.180400	−	0.983593i	$-0.557739\pi$
$719$	−9.67456	−0.360800	−0.180400	+	0.983593i	$0.557739\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	14.6217	0.543789
$724$	0	0
$725$	35.5017	1.31850
$726$	0	0
$727$	45.4903	1.68714	0.843572	−	0.537016i	$-0.180449\pi$
$727$	45.4903	1.68714	0.843572	+	0.537016i	$0.180449\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.4007	−0.495642
$732$	0	0
$733$	15.7756	0.582684	0.291342	−	0.956619i	$-0.405898\pi$
$733$	15.7756	0.582684	0.291342	+	0.956619i	$0.405898\pi$
$734$	0	0
$735$	−1.29966	−0.0479388
$736$	0	0
$737$	−5.19866	−0.191495
$738$	0	0
$739$	50.3557	1.85236	0.926182	−	0.377076i	$-0.123070\pi$
$739$	50.3557	1.85236	0.926182	+	0.377076i	$0.123070\pi$
$740$	0	0
$741$	−5.71155	−0.209819
$742$	0	0
$743$	−29.7709	−1.09219	−0.546095	−	0.837723i	$-0.683886\pi$
$743$	−29.7709	−1.09219	−0.546095	+	0.837723i	$0.683886\pi$
$744$	0	0
$745$	−20.8238	−0.762924
$746$	0	0
$747$	12.7228	0.465501
$748$	0	0
$749$	13.3221	0.486778
$750$	0	0
$751$	−25.8204	−0.942200	−0.471100	−	0.882080i	$-0.656143\pi$
$751$	−25.8204	−0.942200	−0.471100	+	0.882080i	$0.656143\pi$
$752$	0	0
$753$	−1.87657	−0.0683862
$754$	0	0
$755$	−9.64752	−0.351109
$756$	0	0
$757$	−21.8720	−0.794950	−0.397475	−	0.917613i	$-0.630113\pi$
$757$	−21.8720	−0.794950	−0.397475	+	0.917613i	$0.630113\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	28.1458	1.02029	0.510143	−	0.860090i	$-0.329592\pi$
$761$	28.1458	1.02029	0.510143	+	0.860090i	$0.329592\pi$
$762$	0	0
$763$	4.59933	0.166507
$764$	0	0
$765$	1.68913	0.0610705
$766$	0	0
$767$	0	0
$768$	0	0
$769$	30.8238	1.11153	0.555767	−	0.831338i	$-0.312425\pi$
$769$	30.8238	1.11153	0.555767	+	0.831338i	$0.312425\pi$
$770$	0	0
$771$	22.0224	0.793118
$772$	0	0
$773$	−18.8462	−0.677850	−0.338925	−	0.940813i	$-0.610063\pi$
$773$	−18.8462	−0.677850	−0.338925	+	0.940813i	$0.610063\pi$
$774$	0	0
$775$	−12.2885	−0.441414
$776$	0	0
$777$	−0.599328	−0.0215008
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	−1.40067	−0.0501200
$782$	0	0
$783$	10.7228	0.383200
$784$	0	0
$785$	−8.86860	−0.316534
$786$	0	0
$787$	−10.6858	−0.380906	−0.190453	−	0.981696i	$-0.560996\pi$
$787$	−10.6858	−0.380906	−0.190453	+	0.981696i	$0.560996\pi$
$788$	0	0
$789$	−0.599328	−0.0213366
$790$	0	0
$791$	−8.12343	−0.288836
$792$	0	0
$793$	19.4231	0.689734
$794$	0	0
$795$	−0.910201	−0.0322815
$796$	0	0
$797$	−39.8720	−1.41234	−0.706169	−	0.708044i	$-0.749578\pi$
$797$	−39.8720	−1.41234	−0.706169	+	0.708044i	$0.749578\pi$
$798$	0	0
$799$	−13.1571	−0.465463
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	−26.8462	−0.947381
$804$	0	0
$805$	2.59933	0.0916143
$806$	0	0
$807$	5.40067	0.190113
$808$	0	0
$809$	30.4489	1.07053	0.535263	−	0.844686i	$-0.320213\pi$
$809$	30.4489	1.07053	0.535263	+	0.844686i	$0.320213\pi$
$810$	0	0
$811$	0.448867	0.0157619	0.00788093	−	0.999969i	$-0.497491\pi$
$811$	0.448867	0.0157619	0.00788093	+	0.999969i	$0.497491\pi$
$812$	0	0
$813$	31.2211	1.09497
$814$	0	0
$815$	−28.2469	−0.989444
$816$	0	0
$817$	−10.3109	−0.360732
$818$	0	0
$819$	−5.71155	−0.199578
$820$	0	0
$821$	−21.0482	−0.734587	−0.367294	−	0.930105i	$-0.619716\pi$
$821$	−21.0482	−0.734587	−0.367294	+	0.930105i	$0.619716\pi$
$822$	0	0
$823$	−35.9584	−1.25343	−0.626715	−	0.779248i	$-0.715601\pi$
$823$	−35.9584	−1.25343	−0.626715	+	0.779248i	$0.715601\pi$
$824$	0	0
$825$	−6.62175	−0.230540
$826$	0	0
$827$	−23.3445	−0.811768	−0.405884	−	0.913925i	$-0.633036\pi$
$827$	−23.3445	−0.811768	−0.405884	+	0.913925i	$0.633036\pi$
$828$	0	0
$829$	32.4421	1.12676	0.563381	−	0.826197i	$-0.309500\pi$
$829$	32.4421	1.12676	0.563381	+	0.826197i	$0.309500\pi$
$830$	0	0
$831$	29.9584	1.03925
$832$	0	0
$833$	1.29966	0.0450307
$834$	0	0
$835$	2.57019	0.0889452
$836$	0	0
$837$	−3.71155	−0.128290
$838$	0	0
$839$	8.04484	0.277739	0.138869	−	0.990311i	$-0.455653\pi$
$839$	8.04484	0.277739	0.138869	+	0.990311i	$0.455653\pi$
$840$	0	0
$841$	85.9775	2.96474
$842$	0	0
$843$	8.70034	0.299655
$844$	0	0
$845$	25.5017	0.877284
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−6.02242	−0.206689
$850$	0	0
$851$	1.19866	0.0410894
$852$	0	0
$853$	−24.2693	−0.830964	−0.415482	−	0.909601i	$-0.636387\pi$
$853$	−24.2693	−0.830964	−0.415482	+	0.909601i	$0.636387\pi$
$854$	0	0
$855$	1.29966	0.0444476
$856$	0	0
$857$	−17.7756	−0.607202	−0.303601	−	0.952799i	$-0.598189\pi$
$857$	−17.7756	−0.607202	−0.303601	+	0.952799i	$0.598189\pi$
$858$	0	0
$859$	−9.02578	−0.307956	−0.153978	−	0.988074i	$-0.549208\pi$
$859$	−9.02578	−0.307956	−0.153978	+	0.988074i	$0.549208\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	−42.3187	−1.44055	−0.720273	−	0.693691i	$-0.755983\pi$
$863$	−42.3187	−1.44055	−0.720273	+	0.693691i	$0.755983\pi$
$864$	0	0
$865$	−1.55785	−0.0529685
$866$	0	0
$867$	15.3109	0.519985
$868$	0	0
$869$	−12.0448	−0.408593
$870$	0	0
$871$	14.8462	0.503044
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	10.8013	0.365152
$876$	0	0
$877$	29.5960	0.999385	0.499692	−	0.866203i	$-0.333446\pi$
$877$	29.5960	0.999385	0.499692	+	0.866203i	$0.333446\pi$
$878$	0	0
$879$	29.8204	1.00582
$880$	0	0
$881$	16.3479	0.550773	0.275387	−	0.961334i	$-0.411194\pi$
$881$	16.3479	0.550773	0.275387	+	0.961334i	$0.411194\pi$
$882$	0	0
$883$	−21.6475	−0.728497	−0.364249	−	0.931302i	$-0.618674\pi$
$883$	−21.6475	−0.728497	−0.364249	+	0.931302i	$0.618674\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.2211	−0.645381	−0.322690	−	0.946505i	$-0.604587\pi$
$887$	−19.2211	−0.645381	−0.322690	+	0.946505i	$0.604587\pi$
$888$	0	0
$889$	6.59933	0.221334
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−10.1234	−0.338768
$894$	0	0
$895$	17.3142	0.578751
$896$	0	0
$897$	11.4231	0.381406
$898$	0	0
$899$	−39.7980	−1.32734
$900$	0	0
$901$	0.910201	0.0303232
$902$	0	0
$903$	−10.3109	−0.343124
$904$	0	0
$905$	1.84954	0.0614808
$906$	0	0
$907$	25.7756	0.855864	0.427932	−	0.903811i	$-0.359242\pi$
$907$	25.7756	0.855864	0.427932	+	0.903811i	$0.359242\pi$
$908$	0	0
$909$	2.70034	0.0895645
$910$	0	0
$911$	8.49832	0.281562	0.140781	−	0.990041i	$-0.455039\pi$
$911$	8.49832	0.281562	0.140781	+	0.990041i	$0.455039\pi$
$912$	0	0
$913$	−25.4455	−0.842123
$914$	0	0
$915$	−4.41973	−0.146112
$916$	0	0
$917$	−7.32208	−0.241796
$918$	0	0
$919$	−41.2435	−1.36050	−0.680249	−	0.732981i	$-0.738128\pi$
$919$	−41.2435	−1.36050	−0.680249	+	0.732981i	$0.738128\pi$
$920$	0	0
$921$	−10.6858	−0.352108
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	1.98430	0.0652434
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−21.5914	−0.708389	−0.354195	−	0.935172i	$-0.615245\pi$
$929$	−21.5914	−0.708389	−0.354195	+	0.935172i	$0.615245\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−20.5207	−0.671819
$934$	0	0
$935$	−3.37825	−0.110481
$936$	0	0
$937$	38.0448	1.24287	0.621435	−	0.783465i	$-0.286550\pi$
$937$	38.0448	1.24287	0.621435	+	0.783465i	$0.286550\pi$
$938$	0	0
$939$	−9.22107	−0.300918
$940$	0	0
$941$	12.4197	0.404872	0.202436	−	0.979296i	$-0.435114\pi$
$941$	12.4197	0.404872	0.202436	+	0.979296i	$0.435114\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	1.29966	0.0422781
$946$	0	0
$947$	−42.6666	−1.38648	−0.693239	−	0.720708i	$-0.743817\pi$
$947$	−42.6666	−1.38648	−0.693239	+	0.720708i	$0.743817\pi$
$948$	0	0
$949$	76.6666	2.48870
$950$	0	0
$951$	−30.9696	−1.00426
$952$	0	0
$953$	32.7003	1.05927	0.529634	−	0.848226i	$-0.322329\pi$
$953$	32.7003	1.05927	0.529634	+	0.848226i	$0.322329\pi$
$954$	0	0
$955$	12.2469	0.396299
$956$	0	0
$957$	−21.4455	−0.693235
$958$	0	0
$959$	17.4231	0.562621
$960$	0	0
$961$	−17.2244	−0.555627
$962$	0	0
$963$	−13.3221	−0.429298
$964$	0	0
$965$	14.0673	0.452841
$966$	0	0
$967$	25.1571	0.808996	0.404498	−	0.914539i	$-0.367446\pi$
$967$	25.1571	0.808996	0.404498	+	0.914539i	$0.367446\pi$
$968$	0	0
$969$	−1.29966	−0.0417512
$970$	0	0
$971$	28.0897	0.901441	0.450720	−	0.892665i	$-0.351167\pi$
$971$	28.0897	0.901441	0.450720	+	0.892665i	$0.351167\pi$
$972$	0	0
$973$	−6.22443	−0.199546
$974$	0	0
$975$	18.9102	0.605611
$976$	0	0
$977$	−14.1010	−0.451131	−0.225566	−	0.974228i	$-0.572423\pi$
$977$	−14.1010	−0.451131	−0.225566	+	0.974228i	$0.572423\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	−4.59933	−0.146845
$982$	0	0
$983$	39.4231	1.25740	0.628701	−	0.777647i	$-0.283587\pi$
$983$	39.4231	1.25740	0.628701	+	0.777647i	$0.283587\pi$
$984$	0	0
$985$	−7.79798	−0.248464
$986$	0	0
$987$	−10.1234	−0.322232
$988$	0	0
$989$	20.6217	0.655733
$990$	0	0
$991$	43.4679	1.38080	0.690402	−	0.723426i	$-0.257434\pi$
$991$	43.4679	1.38080	0.690402	+	0.723426i	$0.257434\pi$
$992$	0	0
$993$	6.80134	0.215834
$994$	0	0
$995$	−17.4164	−0.552136
$996$	0	0
$997$	31.6184	1.00136	0.500682	−	0.865631i	$-0.333083\pi$
$997$	31.6184	1.00136	0.500682	+	0.865631i	$0.333083\pi$
$998$	0	0
$999$	0.599328	0.0189619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bt.1.2		3
4.3	odd	2		1596.2.a.j.1.2	✓	3
12.11	even	2		4788.2.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.j.1.2	✓	3	4.3	odd	2
4788.2.a.p.1.2		3	12.11	even	2
6384.2.a.bt.1.2		3	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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.29966 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +5.71155 q^{13} -1.29966 q^{15} +1.29966 q^{17} +1.00000 q^{19} +1.00000 q^{21} -2.00000 q^{23} -3.31087 q^{25} -1.00000 q^{27} -10.7228 q^{29} +3.71155 q^{31} +2.00000 q^{33} -1.29966 q^{35} -0.599328 q^{37} -5.71155 q^{39} -4.00000 q^{41} -10.3109 q^{43} +1.29966 q^{45} -10.1234 q^{47} +1.00000 q^{49} -1.29966 q^{51} +0.700336 q^{53} -2.59933 q^{55} -1.00000 q^{57} +3.40067 q^{61} -1.00000 q^{63} +7.42309 q^{65} +2.59933 q^{67} +2.00000 q^{69} +0.700336 q^{71} +13.4231 q^{73} +3.31087 q^{75} +2.00000 q^{77} +6.02242 q^{79} +1.00000 q^{81} +12.7228 q^{83} +1.68913 q^{85} +10.7228 q^{87} -4.00000 q^{89} -5.71155 q^{91} -3.71155 q^{93} +1.29966 q^{95} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} + 3 q^{19} + 3 q^{21} - 6 q^{23} + 13 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 6 q^{33} + 6 q^{37} - 2 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{47} + 3 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 2 * q^13 + 3 * q^19 + 3 * q^21 - 6 * q^23 + 13 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 6 * q^33 + 6 * q^37 - 2 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^47 + 3 * q^49 + 6 * q^53 - 3 * q^57 + 18 * q^61 - 3 * q^63 - 8 * q^65 + 6 * q^69 + 6 * q^71 + 10 * q^73 - 13 * q^75 + 6 * q^77 - 20 * q^79 + 3 * q^81 + 4 * q^83 + 28 * q^85 - 2 * q^87 - 12 * q^89 - 2 * q^91 + 4 * q^93 - 6 * q^97 - 6 * q^99

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