LMFDB - Embedded newform 6384.2.a.bk.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} +3.46410 q^{15} +3.46410 q^{17} +1.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} +7.00000 q^{25} -1.00000 q^{27} -7.46410 q^{29} +2.92820 q^{31} -3.46410 q^{35} -2.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} +6.92820 q^{43} -3.46410 q^{45} -2.53590 q^{47} +1.00000 q^{49} -3.46410 q^{51} +3.46410 q^{53} -1.00000 q^{57} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{63} +6.92820 q^{65} -1.07180 q^{67} +4.00000 q^{69} -6.53590 q^{71} +12.9282 q^{73} -7.00000 q^{75} +1.00000 q^{81} -1.46410 q^{83} -12.0000 q^{85} +7.46410 q^{87} +11.8564 q^{89} -2.00000 q^{91} -2.92820 q^{93} -3.46410 q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{13} + 2 q^{19} - 2 q^{21} - 8 q^{23} + 14 q^{25} - 2 q^{27} - 8 q^{29} - 8 q^{31} - 4 q^{37} + 4 q^{39} + 12 q^{41} - 12 q^{47} + 2 q^{49} - 2 q^{57} - 8 q^{59} + 12 q^{61} + 2 q^{63} - 16 q^{67} + 8 q^{69} - 20 q^{71} + 12 q^{73} - 14 q^{75} + 2 q^{81} + 4 q^{83} - 24 q^{85} + 8 q^{87} - 4 q^{89} - 4 q^{91} + 8 q^{93} + 20 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^13 + 2 * q^19 - 2 * q^21 - 8 * q^23 + 14 * q^25 - 2 * q^27 - 8 * q^29 - 8 * q^31 - 4 * q^37 + 4 * q^39 + 12 * q^41 - 12 * q^47 + 2 * q^49 - 2 * q^57 - 8 * q^59 + 12 * q^61 + 2 * q^63 - 16 * q^67 + 8 * q^69 - 20 * q^71 + 12 * q^73 - 14 * q^75 + 2 * q^81 + 4 * q^83 - 24 * q^85 + 8 * q^87 - 4 * q^89 - 4 * q^91 + 8 * q^93 + 20 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.46410	−1.38605	−0.693024	−	0.720914i	$-0.743722\pi$
$29$	−7.46410	−1.38605	−0.693024	+	0.720914i	$0.743722\pi$
$30$	0	0
$31$	2.92820	0.525921	0.262960	−	0.964807i	$-0.415301\pi$
$31$	2.92820	0.525921	0.262960	+	0.964807i	$0.415301\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	6.92820	1.05654	0.528271	−	0.849076i	$-0.322841\pi$
$43$	6.92820	1.05654	0.528271	+	0.849076i	$0.322841\pi$
$44$	0	0
$45$	−3.46410	−0.516398
$46$	0	0
$47$	−2.53590	−0.369899	−0.184949	−	0.982748i	$-0.559212\pi$
$47$	−2.53590	−0.369899	−0.184949	+	0.982748i	$0.559212\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.46410	−0.485071
$52$	0	0
$53$	3.46410	0.475831	0.237915	−	0.971286i	$-0.423536\pi$
$53$	3.46410	0.475831	0.237915	+	0.971286i	$0.423536\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	6.92820	0.859338
$66$	0	0
$67$	−1.07180	−0.130941	−0.0654704	−	0.997855i	$-0.520855\pi$
$67$	−1.07180	−0.130941	−0.0654704	+	0.997855i	$0.520855\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	−6.53590	−0.775668	−0.387834	−	0.921729i	$-0.626777\pi$
$71$	−6.53590	−0.775668	−0.387834	+	0.921729i	$0.626777\pi$
$72$	0	0
$73$	12.9282	1.51313	0.756566	−	0.653917i	$-0.226876\pi$
$73$	12.9282	1.51313	0.756566	+	0.653917i	$0.226876\pi$
$74$	0	0
$75$	−7.00000	−0.808290
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.46410	−0.160706	−0.0803530	−	0.996766i	$-0.525605\pi$
$83$	−1.46410	−0.160706	−0.0803530	+	0.996766i	$0.525605\pi$
$84$	0	0
$85$	−12.0000	−1.30158
$86$	0	0
$87$	7.46410	0.800236
$88$	0	0
$89$	11.8564	1.25678	0.628388	−	0.777900i	$-0.283715\pi$
$89$	11.8564	1.25678	0.628388	+	0.777900i	$0.283715\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	−2.92820	−0.303641
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.3923	1.03407	0.517036	−	0.855963i	$-0.327035\pi$
$101$	10.3923	1.03407	0.517036	+	0.855963i	$0.327035\pi$
$102$	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
$105$	3.46410	0.338062
$106$	0	0
$107$	−8.39230	−0.811315	−0.405657	−	0.914025i	$-0.632957\pi$
$107$	−8.39230	−0.811315	−0.405657	+	0.914025i	$0.632957\pi$
$108$	0	0
$109$	0.928203	0.0889057	0.0444529	−	0.999011i	$-0.485846\pi$
$109$	0.928203	0.0889057	0.0444529	+	0.999011i	$0.485846\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	4.53590	0.426701	0.213351	−	0.976976i	$-0.431562\pi$
$113$	4.53590	0.426701	0.213351	+	0.976976i	$0.431562\pi$
$114$	0	0
$115$	13.8564	1.29212
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	3.46410	0.317554
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	5.07180	0.450049	0.225025	−	0.974353i	$-0.427754\pi$
$127$	5.07180	0.450049	0.225025	+	0.974353i	$0.427754\pi$
$128$	0	0
$129$	−6.92820	−0.609994
$130$	0	0
$131$	−12.3923	−1.08272	−0.541360	−	0.840791i	$-0.682091\pi$
$131$	−12.3923	−1.08272	−0.541360	+	0.840791i	$0.682091\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	2.53590	0.213561
$142$	0	0
$143$	0	0
$144$	0	0
$145$	25.8564	2.14726
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	8.92820	0.731427	0.365713	−	0.930727i	$-0.380825\pi$
$149$	8.92820	0.731427	0.365713	+	0.930727i	$0.380825\pi$
$150$	0	0
$151$	−10.9282	−0.889325	−0.444662	−	0.895698i	$-0.646676\pi$
$151$	−10.9282	−0.889325	−0.444662	+	0.895698i	$0.646676\pi$
$152$	0	0
$153$	3.46410	0.280056
$154$	0	0
$155$	−10.1436	−0.814753
$156$	0	0
$157$	−15.8564	−1.26548	−0.632740	−	0.774365i	$-0.718070\pi$
$157$	−15.8564	−1.26548	−0.632740	+	0.774365i	$0.718070\pi$
$158$	0	0
$159$	−3.46410	−0.274721
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	1.07180	0.0839496	0.0419748	−	0.999119i	$-0.486635\pi$
$163$	1.07180	0.0839496	0.0419748	+	0.999119i	$0.486635\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.9282	−0.845650	−0.422825	−	0.906211i	$-0.638961\pi$
$167$	−10.9282	−0.845650	−0.422825	+	0.906211i	$0.638961\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	7.00000	0.529150
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−13.4641	−1.00635	−0.503177	−	0.864183i	$-0.667836\pi$
$179$	−13.4641	−1.00635	−0.503177	+	0.864183i	$0.667836\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	6.92820	0.509372
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−9.85641	−0.713185	−0.356592	−	0.934260i	$-0.616061\pi$
$191$	−9.85641	−0.713185	−0.356592	+	0.934260i	$0.616061\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−6.92820	−0.496139
$196$	0	0
$197$	−7.85641	−0.559746	−0.279873	−	0.960037i	$-0.590292\pi$
$197$	−7.85641	−0.559746	−0.279873	+	0.960037i	$0.590292\pi$
$198$	0	0
$199$	18.9282	1.34178	0.670892	−	0.741555i	$-0.265911\pi$
$199$	18.9282	1.34178	0.670892	+	0.741555i	$0.265911\pi$
$200$	0	0
$201$	1.07180	0.0755987
$202$	0	0
$203$	−7.46410	−0.523877
$204$	0	0
$205$	−20.7846	−1.45166
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.85641	0.127800	0.0639001	−	0.997956i	$-0.479646\pi$
$211$	1.85641	0.127800	0.0639001	+	0.997956i	$0.479646\pi$
$212$	0	0
$213$	6.53590	0.447832
$214$	0	0
$215$	−24.0000	−1.63679
$216$	0	0
$217$	2.92820	0.198779
$218$	0	0
$219$	−12.9282	−0.873607
$220$	0	0
$221$	−6.92820	−0.466041
$222$	0	0
$223$	−16.7846	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$223$	−16.7846	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	22.7846	1.50565	0.752825	−	0.658221i	$-0.228691\pi$
$229$	22.7846	1.50565	0.752825	+	0.658221i	$0.228691\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.92820	−0.584906	−0.292453	−	0.956280i	$-0.594472\pi$
$233$	−8.92820	−0.584906	−0.292453	+	0.956280i	$0.594472\pi$
$234$	0	0
$235$	8.78461	0.573045
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	−0.143594	−0.00924967	−0.00462484	−	0.999989i	$-0.501472\pi$
$241$	−0.143594	−0.00924967	−0.00462484	+	0.999989i	$0.501472\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.46410	−0.221313
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	1.46410	0.0927837
$250$	0	0
$251$	−12.3923	−0.782195	−0.391098	−	0.920349i	$-0.627905\pi$
$251$	−12.3923	−0.782195	−0.391098	+	0.920349i	$0.627905\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	12.0000	0.751469
$256$	0	0
$257$	−18.7846	−1.17175	−0.585876	−	0.810401i	$-0.699249\pi$
$257$	−18.7846	−1.17175	−0.585876	+	0.810401i	$0.699249\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−7.46410	−0.462016
$262$	0	0
$263$	22.9282	1.41381	0.706907	−	0.707307i	$-0.250090\pi$
$263$	22.9282	1.41381	0.706907	+	0.707307i	$0.250090\pi$
$264$	0	0
$265$	−12.0000	−0.737154
$266$	0	0
$267$	−11.8564	−0.725600
$268$	0	0
$269$	−16.9282	−1.03213	−0.516065	−	0.856549i	$-0.672604\pi$
$269$	−16.9282	−1.03213	−0.516065	+	0.856549i	$0.672604\pi$
$270$	0	0
$271$	2.14359	0.130214	0.0651070	−	0.997878i	$-0.479261\pi$
$271$	2.14359	0.130214	0.0651070	+	0.997878i	$0.479261\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.8564	−1.43339	−0.716696	−	0.697385i	$-0.754347\pi$
$277$	−23.8564	−1.43339	−0.716696	+	0.697385i	$0.754347\pi$
$278$	0	0
$279$	2.92820	0.175307
$280$	0	0
$281$	−17.3205	−1.03325	−0.516627	−	0.856210i	$-0.672813\pi$
$281$	−17.3205	−1.03325	−0.516627	+	0.856210i	$0.672813\pi$
$282$	0	0
$283$	22.9282	1.36294	0.681470	−	0.731846i	$-0.261341\pi$
$283$	22.9282	1.36294	0.681470	+	0.731846i	$0.261341\pi$
$284$	0	0
$285$	3.46410	0.205196
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	13.8564	0.806751
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	6.92820	0.399335
$302$	0	0
$303$	−10.3923	−0.597022
$304$	0	0
$305$	−20.7846	−1.19012
$306$	0	0
$307$	−14.9282	−0.851998	−0.425999	−	0.904724i	$-0.640077\pi$
$307$	−14.9282	−0.851998	−0.425999	+	0.904724i	$0.640077\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−10.5359	−0.597436	−0.298718	−	0.954341i	$-0.596559\pi$
$311$	−10.5359	−0.597436	−0.298718	+	0.954341i	$0.596559\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	−3.46410	−0.195180
$316$	0	0
$317$	0.535898	0.0300991	0.0150495	−	0.999887i	$-0.495209\pi$
$317$	0.535898	0.0300991	0.0150495	+	0.999887i	$0.495209\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	8.39230	0.468413
$322$	0	0
$323$	3.46410	0.192748
$324$	0	0
$325$	−14.0000	−0.776580
$326$	0	0
$327$	−0.928203	−0.0513298
$328$	0	0
$329$	−2.53590	−0.139809
$330$	0	0
$331$	−1.07180	−0.0589113	−0.0294556	−	0.999566i	$-0.509377\pi$
$331$	−1.07180	−0.0589113	−0.0294556	+	0.999566i	$0.509377\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	3.71281	0.202853
$336$	0	0
$337$	7.07180	0.385225	0.192613	−	0.981275i	$-0.438304\pi$
$337$	7.07180	0.385225	0.192613	+	0.981275i	$0.438304\pi$
$338$	0	0
$339$	−4.53590	−0.246356
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−13.8564	−0.746004
$346$	0	0
$347$	8.78461	0.471583	0.235791	−	0.971804i	$-0.424232\pi$
$347$	8.78461	0.471583	0.235791	+	0.971804i	$0.424232\pi$
$348$	0	0
$349$	−15.8564	−0.848774	−0.424387	−	0.905481i	$-0.639510\pi$
$349$	−15.8564	−0.848774	−0.424387	+	0.905481i	$0.639510\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−12.5359	−0.667219	−0.333609	−	0.942711i	$-0.608267\pi$
$353$	−12.5359	−0.667219	−0.333609	+	0.942711i	$0.608267\pi$
$354$	0	0
$355$	22.6410	1.20166
$356$	0	0
$357$	−3.46410	−0.183340
$358$	0	0
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	−44.7846	−2.34413
$366$	0	0
$367$	10.9282	0.570448	0.285224	−	0.958461i	$-0.407932\pi$
$367$	10.9282	0.570448	0.285224	+	0.958461i	$0.407932\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	3.46410	0.179847
$372$	0	0
$373$	19.8564	1.02813	0.514063	−	0.857753i	$-0.328140\pi$
$373$	19.8564	1.02813	0.514063	+	0.857753i	$0.328140\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	14.9282	0.768842
$378$	0	0
$379$	−12.7846	−0.656701	−0.328351	−	0.944556i	$-0.606493\pi$
$379$	−12.7846	−0.656701	−0.328351	+	0.944556i	$0.606493\pi$
$380$	0	0
$381$	−5.07180	−0.259836
$382$	0	0
$383$	−5.07180	−0.259157	−0.129578	−	0.991569i	$-0.541362\pi$
$383$	−5.07180	−0.259157	−0.129578	+	0.991569i	$0.541362\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.92820	0.352180
$388$	0	0
$389$	−20.9282	−1.06110	−0.530551	−	0.847653i	$-0.678015\pi$
$389$	−20.9282	−1.06110	−0.530551	+	0.847653i	$0.678015\pi$
$390$	0	0
$391$	−13.8564	−0.700749
$392$	0	0
$393$	12.3923	0.625109
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.8564	1.39807	0.699036	−	0.715086i	$-0.253612\pi$
$397$	27.8564	1.39807	0.699036	+	0.715086i	$0.253612\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−14.3923	−0.718717	−0.359359	−	0.933200i	$-0.617005\pi$
$401$	−14.3923	−0.718717	−0.359359	+	0.933200i	$0.617005\pi$
$402$	0	0
$403$	−5.85641	−0.291728
$404$	0	0
$405$	−3.46410	−0.172133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	5.07180	0.248965
$416$	0	0
$417$	6.92820	0.339276
$418$	0	0
$419$	−9.46410	−0.462352	−0.231176	−	0.972912i	$-0.574257\pi$
$419$	−9.46410	−0.462352	−0.231176	+	0.972912i	$0.574257\pi$
$420$	0	0
$421$	−28.9282	−1.40987	−0.704937	−	0.709270i	$-0.749025\pi$
$421$	−28.9282	−1.40987	−0.704937	+	0.709270i	$0.749025\pi$
$422$	0	0
$423$	−2.53590	−0.123300
$424$	0	0
$425$	24.2487	1.17624
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.5359	−1.47086	−0.735431	−	0.677599i	$-0.763020\pi$
$431$	−30.5359	−1.47086	−0.735431	+	0.677599i	$0.763020\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−25.8564	−1.23972
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	40.7846	1.94654	0.973272	−	0.229657i	$-0.0737605\pi$
$439$	40.7846	1.94654	0.973272	+	0.229657i	$0.0737605\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−24.7846	−1.17755	−0.588776	−	0.808296i	$-0.700390\pi$
$443$	−24.7846	−1.17755	−0.588776	+	0.808296i	$0.700390\pi$
$444$	0	0
$445$	−41.0718	−1.94699
$446$	0	0
$447$	−8.92820	−0.422290
$448$	0	0
$449$	−0.535898	−0.0252906	−0.0126453	−	0.999920i	$-0.504025\pi$
$449$	−0.535898	−0.0252906	−0.0126453	+	0.999920i	$0.504025\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	10.9282	0.513452
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	23.8564	1.11596	0.557978	−	0.829856i	$-0.311577\pi$
$457$	23.8564	1.11596	0.557978	+	0.829856i	$0.311577\pi$
$458$	0	0
$459$	−3.46410	−0.161690
$460$	0	0
$461$	−19.4641	−0.906534	−0.453267	−	0.891375i	$-0.649742\pi$
$461$	−19.4641	−0.906534	−0.453267	+	0.891375i	$0.649742\pi$
$462$	0	0
$463$	−35.7128	−1.65972	−0.829858	−	0.557975i	$-0.811578\pi$
$463$	−35.7128	−1.65972	−0.829858	+	0.557975i	$0.811578\pi$
$464$	0	0
$465$	10.1436	0.470398
$466$	0	0
$467$	−30.5359	−1.41303	−0.706516	−	0.707697i	$-0.749734\pi$
$467$	−30.5359	−1.41303	−0.706516	+	0.707697i	$0.749734\pi$
$468$	0	0
$469$	−1.07180	−0.0494910
$470$	0	0
$471$	15.8564	0.730625
$472$	0	0
$473$	0	0
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	3.46410	0.158610
$478$	0	0
$479$	−29.4641	−1.34625	−0.673125	−	0.739529i	$-0.735048\pi$
$479$	−29.4641	−1.34625	−0.673125	+	0.739529i	$0.735048\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−34.6410	−1.57297
$486$	0	0
$487$	−27.7128	−1.25579	−0.627894	−	0.778299i	$-0.716083\pi$
$487$	−27.7128	−1.25579	−0.627894	+	0.778299i	$0.716083\pi$
$488$	0	0
$489$	−1.07180	−0.0484683
$490$	0	0
$491$	−18.9282	−0.854218	−0.427109	−	0.904200i	$-0.640468\pi$
$491$	−18.9282	−0.854218	−0.427109	+	0.904200i	$0.640468\pi$
$492$	0	0
$493$	−25.8564	−1.16451
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.53590	−0.293175
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	10.9282	0.488236
$502$	0	0
$503$	25.1769	1.12258	0.561292	−	0.827618i	$-0.310305\pi$
$503$	25.1769	1.12258	0.561292	+	0.827618i	$0.310305\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	42.7846	1.89639	0.948197	−	0.317682i	$-0.102905\pi$
$509$	42.7846	1.89639	0.948197	+	0.317682i	$0.102905\pi$
$510$	0	0
$511$	12.9282	0.571910
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	27.7128	1.22117
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	14.7846	0.647726	0.323863	−	0.946104i	$-0.395018\pi$
$521$	14.7846	0.647726	0.323863	+	0.946104i	$0.395018\pi$
$522$	0	0
$523$	−9.07180	−0.396682	−0.198341	−	0.980133i	$-0.563555\pi$
$523$	−9.07180	−0.396682	−0.198341	+	0.980133i	$0.563555\pi$
$524$	0	0
$525$	−7.00000	−0.305505
$526$	0	0
$527$	10.1436	0.441862
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	29.0718	1.25688
$536$	0	0
$537$	13.4641	0.581019
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−3.21539	−0.137732
$546$	0	0
$547$	15.7128	0.671831	0.335916	−	0.941892i	$-0.390954\pi$
$547$	15.7128	0.671831	0.335916	+	0.941892i	$0.390954\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−7.46410	−0.317981
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.92820	−0.294086
$556$	0	0
$557$	22.7846	0.965415	0.482707	−	0.875782i	$-0.339653\pi$
$557$	22.7846	0.965415	0.482707	+	0.875782i	$0.339653\pi$
$558$	0	0
$559$	−13.8564	−0.586064
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.9282	0.629149	0.314574	−	0.949233i	$-0.398138\pi$
$563$	14.9282	0.629149	0.314574	+	0.949233i	$0.398138\pi$
$564$	0	0
$565$	−15.7128	−0.661043
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	20.5359	0.860910	0.430455	−	0.902612i	$-0.358353\pi$
$569$	20.5359	0.860910	0.430455	+	0.902612i	$0.358353\pi$
$570$	0	0
$571$	−15.7128	−0.657561	−0.328780	−	0.944406i	$-0.606638\pi$
$571$	−15.7128	−0.657561	−0.328780	+	0.944406i	$0.606638\pi$
$572$	0	0
$573$	9.85641	0.411757
$574$	0	0
$575$	−28.0000	−1.16768
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	−1.46410	−0.0607412
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.92820	0.286446
$586$	0	0
$587$	−12.3923	−0.511485	−0.255743	−	0.966745i	$-0.582320\pi$
$587$	−12.3923	−0.511485	−0.255743	+	0.966745i	$0.582320\pi$
$588$	0	0
$589$	2.92820	0.120655
$590$	0	0
$591$	7.85641	0.323169
$592$	0	0
$593$	8.53590	0.350527	0.175264	−	0.984522i	$-0.443922\pi$
$593$	8.53590	0.350527	0.175264	+	0.984522i	$0.443922\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	0	0
$597$	−18.9282	−0.774680
$598$	0	0
$599$	36.3923	1.48695	0.743475	−	0.668764i	$-0.233176\pi$
$599$	36.3923	1.48695	0.743475	+	0.668764i	$0.233176\pi$
$600$	0	0
$601$	−17.7128	−0.722521	−0.361260	−	0.932465i	$-0.617653\pi$
$601$	−17.7128	−0.722521	−0.361260	+	0.932465i	$0.617653\pi$
$602$	0	0
$603$	−1.07180	−0.0436469
$604$	0	0
$605$	38.1051	1.54919
$606$	0	0
$607$	−35.7128	−1.44954	−0.724769	−	0.688992i	$-0.758054\pi$
$607$	−35.7128	−1.44954	−0.724769	+	0.688992i	$0.758054\pi$
$608$	0	0
$609$	7.46410	0.302461
$610$	0	0
$611$	5.07180	0.205183
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	20.7846	0.838116
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	25.0718	1.00772	0.503860	−	0.863785i	$-0.331913\pi$
$619$	25.0718	1.00772	0.503860	+	0.863785i	$0.331913\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	11.8564	0.475017
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6.92820	−0.276246
$630$	0	0
$631$	−40.7846	−1.62361	−0.811805	−	0.583929i	$-0.801515\pi$
$631$	−40.7846	−1.62361	−0.811805	+	0.583929i	$0.801515\pi$
$632$	0	0
$633$	−1.85641	−0.0737855
$634$	0	0
$635$	−17.5692	−0.697213
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−6.53590	−0.258556
$640$	0	0
$641$	19.1769	0.757443	0.378721	−	0.925511i	$-0.376364\pi$
$641$	19.1769	0.757443	0.378721	+	0.925511i	$0.376364\pi$
$642$	0	0
$643$	−44.7846	−1.76613	−0.883066	−	0.469248i	$-0.844525\pi$
$643$	−44.7846	−1.76613	−0.883066	+	0.469248i	$0.844525\pi$
$644$	0	0
$645$	24.0000	0.944999
$646$	0	0
$647$	−6.24871	−0.245662	−0.122831	−	0.992428i	$-0.539197\pi$
$647$	−6.24871	−0.245662	−0.122831	+	0.992428i	$0.539197\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.92820	−0.114765
$652$	0	0
$653$	9.71281	0.380092	0.190046	−	0.981775i	$-0.439136\pi$
$653$	9.71281	0.380092	0.190046	+	0.981775i	$0.439136\pi$
$654$	0	0
$655$	42.9282	1.67734
$656$	0	0
$657$	12.9282	0.504377
$658$	0	0
$659$	44.1051	1.71809	0.859046	−	0.511899i	$-0.171058\pi$
$659$	44.1051	1.71809	0.859046	+	0.511899i	$0.171058\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	6.92820	0.269069
$664$	0	0
$665$	−3.46410	−0.134332
$666$	0	0
$667$	29.8564	1.15604
$668$	0	0
$669$	16.7846	0.648931
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	−7.00000	−0.269430
$676$	0	0
$677$	42.7846	1.64435	0.822173	−	0.569238i	$-0.192762\pi$
$677$	42.7846	1.64435	0.822173	+	0.569238i	$0.192762\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	6.92820	0.265489
$682$	0	0
$683$	−15.6077	−0.597212	−0.298606	−	0.954376i	$-0.596522\pi$
$683$	−15.6077	−0.597212	−0.298606	+	0.954376i	$0.596522\pi$
$684$	0	0
$685$	−6.92820	−0.264713
$686$	0	0
$687$	−22.7846	−0.869287
$688$	0	0
$689$	−6.92820	−0.263944
$690$	0	0
$691$	17.8564	0.679290	0.339645	−	0.940554i	$-0.389693\pi$
$691$	17.8564	0.679290	0.339645	+	0.940554i	$0.389693\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.0000	0.910372
$696$	0	0
$697$	20.7846	0.787273
$698$	0	0
$699$	8.92820	0.337696
$700$	0	0
$701$	19.8564	0.749966	0.374983	−	0.927032i	$-0.377649\pi$
$701$	19.8564	0.749966	0.374983	+	0.927032i	$0.377649\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	−8.78461	−0.330848
$706$	0	0
$707$	10.3923	0.390843
$708$	0	0
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.7128	−0.438648
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.92820	0.258738
$718$	0	0
$719$	22.2487	0.829737	0.414868	−	0.909881i	$-0.363828\pi$
$719$	22.2487	0.829737	0.414868	+	0.909881i	$0.363828\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0.143594	0.00534030
$724$	0	0
$725$	−52.2487	−1.94047
$726$	0	0
$727$	21.8564	0.810609	0.405305	−	0.914182i	$-0.367165\pi$
$727$	21.8564	0.810609	0.405305	+	0.914182i	$0.367165\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	−18.7846	−0.693825	−0.346913	−	0.937897i	$-0.612770\pi$
$733$	−18.7846	−0.693825	−0.346913	+	0.937897i	$0.612770\pi$
$734$	0	0
$735$	3.46410	0.127775
$736$	0	0
$737$	0	0
$738$	0	0
$739$	11.2154	0.412565	0.206282	−	0.978492i	$-0.433863\pi$
$739$	11.2154	0.412565	0.206282	+	0.978492i	$0.433863\pi$
$740$	0	0
$741$	2.00000	0.0734718
$742$	0	0
$743$	−28.3923	−1.04161	−0.520806	−	0.853675i	$-0.674369\pi$
$743$	−28.3923	−1.04161	−0.520806	+	0.853675i	$0.674369\pi$
$744$	0	0
$745$	−30.9282	−1.13312
$746$	0	0
$747$	−1.46410	−0.0535687
$748$	0	0
$749$	−8.39230	−0.306648
$750$	0	0
$751$	−18.9282	−0.690700	−0.345350	−	0.938474i	$-0.612240\pi$
$751$	−18.9282	−0.690700	−0.345350	+	0.938474i	$0.612240\pi$
$752$	0	0
$753$	12.3923	0.451601
$754$	0	0
$755$	37.8564	1.37774
$756$	0	0
$757$	39.5692	1.43817	0.719084	−	0.694923i	$-0.244562\pi$
$757$	39.5692	1.43817	0.719084	+	0.694923i	$0.244562\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−51.1769	−1.85516	−0.927581	−	0.373622i	$-0.878116\pi$
$761$	−51.1769	−1.85516	−0.927581	+	0.373622i	$0.878116\pi$
$762$	0	0
$763$	0.928203	0.0336032
$764$	0	0
$765$	−12.0000	−0.433861
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−31.5692	−1.13842	−0.569208	−	0.822194i	$-0.692750\pi$
$769$	−31.5692	−1.13842	−0.569208	+	0.822194i	$0.692750\pi$
$770$	0	0
$771$	18.7846	0.676511
$772$	0	0
$773$	15.8564	0.570315	0.285158	−	0.958481i	$-0.407954\pi$
$773$	15.8564	0.570315	0.285158	+	0.958481i	$0.407954\pi$
$774$	0	0
$775$	20.4974	0.736289
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	7.46410	0.266745
$784$	0	0
$785$	54.9282	1.96047
$786$	0	0
$787$	4.78461	0.170553	0.0852765	−	0.996357i	$-0.472823\pi$
$787$	4.78461	0.170553	0.0852765	+	0.996357i	$0.472823\pi$
$788$	0	0
$789$	−22.9282	−0.816266
$790$	0	0
$791$	4.53590	0.161278
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	−41.7128	−1.47754	−0.738772	−	0.673956i	$-0.764594\pi$
$797$	−41.7128	−1.47754	−0.738772	+	0.673956i	$0.764594\pi$
$798$	0	0
$799$	−8.78461	−0.310777
$800$	0	0
$801$	11.8564	0.418926
$802$	0	0
$803$	0	0
$804$	0	0
$805$	13.8564	0.488374
$806$	0	0
$807$	16.9282	0.595901
$808$	0	0
$809$	15.8564	0.557482	0.278741	−	0.960366i	$-0.410083\pi$
$809$	15.8564	0.557482	0.278741	+	0.960366i	$0.410083\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	−2.14359	−0.0751791
$814$	0	0
$815$	−3.71281	−0.130054
$816$	0	0
$817$	6.92820	0.242387
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−44.9282	−1.56801	−0.784003	−	0.620758i	$-0.786825\pi$
$821$	−44.9282	−1.56801	−0.784003	+	0.620758i	$0.786825\pi$
$822$	0	0
$823$	2.92820	0.102071	0.0510354	−	0.998697i	$-0.483748\pi$
$823$	2.92820	0.102071	0.0510354	+	0.998697i	$0.483748\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.3205	0.671840	0.335920	−	0.941891i	$-0.390953\pi$
$827$	19.3205	0.671840	0.335920	+	0.941891i	$0.390953\pi$
$828$	0	0
$829$	−47.8564	−1.66212	−0.831061	−	0.556182i	$-0.812266\pi$
$829$	−47.8564	−1.66212	−0.831061	+	0.556182i	$0.812266\pi$
$830$	0	0
$831$	23.8564	0.827570
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	37.8564	1.31007
$836$	0	0
$837$	−2.92820	−0.101214
$838$	0	0
$839$	37.8564	1.30695	0.653474	−	0.756949i	$-0.273311\pi$
$839$	37.8564	1.30695	0.653474	+	0.756949i	$0.273311\pi$
$840$	0	0
$841$	26.7128	0.921131
$842$	0	0
$843$	17.3205	0.596550
$844$	0	0
$845$	31.1769	1.07252
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	−22.9282	−0.786894
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	47.5692	1.62874	0.814370	−	0.580347i	$-0.197083\pi$
$853$	47.5692	1.62874	0.814370	+	0.580347i	$0.197083\pi$
$854$	0	0
$855$	−3.46410	−0.118470
$856$	0	0
$857$	−4.14359	−0.141542	−0.0707712	−	0.997493i	$-0.522546\pi$
$857$	−4.14359	−0.141542	−0.0707712	+	0.997493i	$0.522546\pi$
$858$	0	0
$859$	−31.7128	−1.08203	−0.541014	−	0.841014i	$-0.681959\pi$
$859$	−31.7128	−1.08203	−0.541014	+	0.841014i	$0.681959\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	−7.32051	−0.249193	−0.124596	−	0.992207i	$-0.539764\pi$
$863$	−7.32051	−0.249193	−0.124596	+	0.992207i	$0.539764\pi$
$864$	0	0
$865$	−6.92820	−0.235566
$866$	0	0
$867$	5.00000	0.169809
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.14359	0.0726329
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	22.0000	0.742042
$880$	0	0
$881$	41.3205	1.39212	0.696062	−	0.717982i	$-0.254934\pi$
$881$	41.3205	1.39212	0.696062	+	0.717982i	$0.254934\pi$
$882$	0	0
$883$	33.8564	1.13936	0.569679	−	0.821867i	$-0.307067\pi$
$883$	33.8564	1.13936	0.569679	+	0.821867i	$0.307067\pi$
$884$	0	0
$885$	−13.8564	−0.465778
$886$	0	0
$887$	22.6410	0.760211	0.380105	−	0.924943i	$-0.375888\pi$
$887$	22.6410	0.760211	0.380105	+	0.924943i	$0.375888\pi$
$888$	0	0
$889$	5.07180	0.170103
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.53590	−0.0848606
$894$	0	0
$895$	46.6410	1.55904
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	−21.8564	−0.728952
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−6.92820	−0.230556
$904$	0	0
$905$	6.92820	0.230301
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	10.3923	0.344691
$910$	0	0
$911$	−57.4641	−1.90387	−0.951935	−	0.306299i	$-0.900909\pi$
$911$	−57.4641	−1.90387	−0.951935	+	0.306299i	$0.900909\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	20.7846	0.687118
$916$	0	0
$917$	−12.3923	−0.409230
$918$	0	0
$919$	51.7128	1.70585	0.852924	−	0.522035i	$-0.174827\pi$
$919$	51.7128	1.70585	0.852924	+	0.522035i	$0.174827\pi$
$920$	0	0
$921$	14.9282	0.491901
$922$	0	0
$923$	13.0718	0.430263
$924$	0	0
$925$	−14.0000	−0.460317
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	27.4641	0.901068	0.450534	−	0.892759i	$-0.351234\pi$
$929$	27.4641	0.901068	0.450534	+	0.892759i	$0.351234\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	10.5359	0.344930
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.71281	−0.0559552	−0.0279776	−	0.999609i	$-0.508907\pi$
$937$	−1.71281	−0.0559552	−0.0279776	+	0.999609i	$0.508907\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	9.21539	0.300413	0.150207	−	0.988655i	$-0.452006\pi$
$941$	9.21539	0.300413	0.150207	+	0.988655i	$0.452006\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	3.46410	0.112687
$946$	0	0
$947$	−29.8564	−0.970203	−0.485101	−	0.874458i	$-0.661217\pi$
$947$	−29.8564	−0.970203	−0.485101	+	0.874458i	$0.661217\pi$
$948$	0	0
$949$	−25.8564	−0.839334
$950$	0	0
$951$	−0.535898	−0.0173777
$952$	0	0
$953$	1.60770	0.0520784	0.0260392	−	0.999661i	$-0.491711\pi$
$953$	1.60770	0.0520784	0.0260392	+	0.999661i	$0.491711\pi$
$954$	0	0
$955$	34.1436	1.10486
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.00000	0.0645834
$960$	0	0
$961$	−22.4256	−0.723407
$962$	0	0
$963$	−8.39230	−0.270438
$964$	0	0
$965$	48.4974	1.56119
$966$	0	0
$967$	−2.92820	−0.0941647	−0.0470823	−	0.998891i	$-0.514992\pi$
$967$	−2.92820	−0.0941647	−0.0470823	+	0.998891i	$0.514992\pi$
$968$	0	0
$969$	−3.46410	−0.111283
$970$	0	0
$971$	−33.8564	−1.08650	−0.543252	−	0.839570i	$-0.682807\pi$
$971$	−33.8564	−1.08650	−0.543252	+	0.839570i	$0.682807\pi$
$972$	0	0
$973$	−6.92820	−0.222108
$974$	0	0
$975$	14.0000	0.448359
$976$	0	0
$977$	19.1769	0.613524	0.306762	−	0.951786i	$-0.400754\pi$
$977$	19.1769	0.613524	0.306762	+	0.951786i	$0.400754\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0.928203	0.0296352
$982$	0	0
$983$	40.0000	1.27580	0.637901	−	0.770118i	$-0.279803\pi$
$983$	40.0000	1.27580	0.637901	+	0.770118i	$0.279803\pi$
$984$	0	0
$985$	27.2154	0.867154
$986$	0	0
$987$	2.53590	0.0807185
$988$	0	0
$989$	−27.7128	−0.881216
$990$	0	0
$991$	−25.5692	−0.812233	−0.406117	−	0.913821i	$-0.633117\pi$
$991$	−25.5692	−0.812233	−0.406117	+	0.913821i	$0.633117\pi$
$992$	0	0
$993$	1.07180	0.0340124
$994$	0	0
$995$	−65.5692	−2.07868
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bk.1.1		2
4.3	odd	2		3192.2.a.t.1.1	✓	2
12.11	even	2		9576.2.a.bk.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.t.1.1	✓	2	4.3	odd	2
6384.2.a.bk.1.1		2	1.1	even	1	trivial
9576.2.a.bk.1.2		2	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} +3.46410 q^{15} +3.46410 q^{17} +1.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} +7.00000 q^{25} -1.00000 q^{27} -7.46410 q^{29} +2.92820 q^{31} -3.46410 q^{35} -2.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} +6.92820 q^{43} -3.46410 q^{45} -2.53590 q^{47} +1.00000 q^{49} -3.46410 q^{51} +3.46410 q^{53} -1.00000 q^{57} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{63} +6.92820 q^{65} -1.07180 q^{67} +4.00000 q^{69} -6.53590 q^{71} +12.9282 q^{73} -7.00000 q^{75} +1.00000 q^{81} -1.46410 q^{83} -12.0000 q^{85} +7.46410 q^{87} +11.8564 q^{89} -2.00000 q^{91} -2.92820 q^{93} -3.46410 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{13} + 2 q^{19} - 2 q^{21} - 8 q^{23} + 14 q^{25} - 2 q^{27} - 8 q^{29} - 8 q^{31} - 4 q^{37} + 4 q^{39} + 12 q^{41} - 12 q^{47} + 2 q^{49} - 2 q^{57} - 8 q^{59} + 12 q^{61} + 2 q^{63} - 16 q^{67} + 8 q^{69} - 20 q^{71} + 12 q^{73} - 14 q^{75} + 2 q^{81} + 4 q^{83} - 24 q^{85} + 8 q^{87} - 4 q^{89} - 4 q^{91} + 8 q^{93} + 20 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^13 + 2 * q^19 - 2 * q^21 - 8 * q^23 + 14 * q^25 - 2 * q^27 - 8 * q^29 - 8 * q^31 - 4 * q^37 + 4 * q^39 + 12 * q^41 - 12 * q^47 + 2 * q^49 - 2 * q^57 - 8 * q^59 + 12 * q^61 + 2 * q^63 - 16 * q^67 + 8 * q^69 - 20 * q^71 + 12 * q^73 - 14 * q^75 + 2 * q^81 + 4 * q^83 - 24 * q^85 + 8 * q^87 - 4 * q^89 - 4 * q^91 + 8 * q^93 + 20 * q^97

Introduction

L-functions

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