LMFDB - Embedded newform 6384.2.a.bj.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:	$0$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.46410 q^{11} -4.00000 q^{13} -0.732051 q^{15} -6.19615 q^{17} +1.00000 q^{19} -1.00000 q^{21} +0.535898 q^{23} -4.46410 q^{25} -1.00000 q^{27} +4.19615 q^{29} +8.92820 q^{31} -3.46410 q^{33} +0.732051 q^{35} -0.535898 q^{37} +4.00000 q^{39} -2.53590 q^{41} -3.46410 q^{43} +0.732051 q^{45} +0.732051 q^{47} +1.00000 q^{49} +6.19615 q^{51} +5.26795 q^{53} +2.53590 q^{55} -1.00000 q^{57} -2.92820 q^{59} +6.39230 q^{61} +1.00000 q^{63} -2.92820 q^{65} +5.46410 q^{67} -0.535898 q^{69} +0.196152 q^{71} +11.8564 q^{73} +4.46410 q^{75} +3.46410 q^{77} +6.53590 q^{79} +1.00000 q^{81} +4.73205 q^{83} -4.53590 q^{85} -4.19615 q^{87} +2.53590 q^{89} -4.00000 q^{91} -8.92820 q^{93} +0.732051 q^{95} -6.00000 q^{97} +3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} - 2 q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 14 q^{53} + 12 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} + 2 q^{63} + 8 q^{65} + 4 q^{67} - 8 q^{69} - 10 q^{71} - 4 q^{73} + 2 q^{75} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 16 q^{85} + 2 q^{87} + 12 q^{89} - 8 q^{91} - 4 q^{93} - 2 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 2 * q^17 + 2 * q^19 - 2 * q^21 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 - 2 * q^35 - 8 * q^37 + 8 * q^39 - 12 * q^41 - 2 * q^45 - 2 * q^47 + 2 * q^49 + 2 * q^51 + 14 * q^53 + 12 * q^55 - 2 * q^57 + 8 * q^59 - 8 * q^61 + 2 * q^63 + 8 * q^65 + 4 * q^67 - 8 * q^69 - 10 * q^71 - 4 * q^73 + 2 * q^75 + 20 * q^79 + 2 * q^81 + 6 * q^83 - 16 * q^85 + 2 * q^87 + 12 * q^89 - 8 * q^91 - 4 * q^93 - 2 * q^95 - 12 * 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
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	−6.19615	−1.50279	−0.751394	−	0.659854i	$-0.770618\pi$
$17$	−6.19615	−1.50279	−0.751394	+	0.659854i	$0.770618\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.19615	0.779206	0.389603	−	0.920983i	$-0.372612\pi$
$29$	4.19615	0.779206	0.389603	+	0.920983i	$0.372612\pi$
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	0.732051	0.123739
$36$	0	0
$37$	−0.535898	−0.0881012	−0.0440506	−	0.999029i	$-0.514026\pi$
$37$	−0.535898	−0.0881012	−0.0440506	+	0.999029i	$0.514026\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−2.53590	−0.396041	−0.198020	−	0.980198i	$-0.563451\pi$
$41$	−2.53590	−0.396041	−0.198020	+	0.980198i	$0.563451\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	0.732051	0.109128
$46$	0	0
$47$	0.732051	0.106781	0.0533903	−	0.998574i	$-0.482997\pi$
$47$	0.732051	0.106781	0.0533903	+	0.998574i	$0.482997\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.19615	0.867635
$52$	0	0
$53$	5.26795	0.723608	0.361804	−	0.932254i	$-0.382161\pi$
$53$	5.26795	0.723608	0.361804	+	0.932254i	$0.382161\pi$
$54$	0	0
$55$	2.53590	0.341940
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−2.92820	−0.381220	−0.190610	−	0.981666i	$-0.561047\pi$
$59$	−2.92820	−0.381220	−0.190610	+	0.981666i	$0.561047\pi$
$60$	0	0
$61$	6.39230	0.818451	0.409225	−	0.912433i	$-0.365799\pi$
$61$	6.39230	0.818451	0.409225	+	0.912433i	$0.365799\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−2.92820	−0.363199
$66$	0	0
$67$	5.46410	0.667546	0.333773	−	0.942653i	$-0.391678\pi$
$67$	5.46410	0.667546	0.333773	+	0.942653i	$0.391678\pi$
$68$	0	0
$69$	−0.535898	−0.0645146
$70$	0	0
$71$	0.196152	0.0232790	0.0116395	−	0.999932i	$-0.496295\pi$
$71$	0.196152	0.0232790	0.0116395	+	0.999932i	$0.496295\pi$
$72$	0	0
$73$	11.8564	1.38769	0.693844	−	0.720126i	$-0.255916\pi$
$73$	11.8564	1.38769	0.693844	+	0.720126i	$0.255916\pi$
$74$	0	0
$75$	4.46410	0.515470
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	6.53590	0.735346	0.367673	−	0.929955i	$-0.380155\pi$
$79$	6.53590	0.735346	0.367673	+	0.929955i	$0.380155\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.73205	0.519410	0.259705	−	0.965688i	$-0.416375\pi$
$83$	4.73205	0.519410	0.259705	+	0.965688i	$0.416375\pi$
$84$	0	0
$85$	−4.53590	−0.491987
$86$	0	0
$87$	−4.19615	−0.449875
$88$	0	0
$89$	2.53590	0.268805	0.134402	−	0.990927i	$-0.457089\pi$
$89$	2.53590	0.268805	0.134402	+	0.990927i	$0.457089\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−8.92820	−0.925812
$94$	0	0
$95$	0.732051	0.0751068
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	3.46410	0.348155
$100$	0	0
$101$	−5.80385	−0.577504	−0.288752	−	0.957404i	$-0.593240\pi$
$101$	−5.80385	−0.577504	−0.288752	+	0.957404i	$0.593240\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	0	0
$105$	−0.732051	−0.0714408
$106$	0	0
$107$	5.66025	0.547197	0.273599	−	0.961844i	$-0.411786\pi$
$107$	5.66025	0.547197	0.273599	+	0.961844i	$0.411786\pi$
$108$	0	0
$109$	−3.46410	−0.331801	−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410	−0.331801	−0.165900	+	0.986143i	$0.553053\pi$
$110$	0	0
$111$	0.535898	0.0508652
$112$	0	0
$113$	13.2679	1.24814	0.624072	−	0.781367i	$-0.285477\pi$
$113$	13.2679	1.24814	0.624072	+	0.781367i	$0.285477\pi$
$114$	0	0
$115$	0.392305	0.0365826
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−6.19615	−0.568000
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.53590	0.228654
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	12.3923	1.09964	0.549820	−	0.835283i	$-0.314696\pi$
$127$	12.3923	1.09964	0.549820	+	0.835283i	$0.314696\pi$
$128$	0	0
$129$	3.46410	0.304997
$130$	0	0
$131$	−1.80385	−0.157603	−0.0788014	−	0.996890i	$-0.525109\pi$
$131$	−1.80385	−0.157603	−0.0788014	+	0.996890i	$0.525109\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−0.732051	−0.0630049
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	9.07180	0.769460	0.384730	−	0.923029i	$-0.374295\pi$
$139$	9.07180	0.769460	0.384730	+	0.923029i	$0.374295\pi$
$140$	0	0
$141$	−0.732051	−0.0616498
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	3.07180	0.255099
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	10.3923	0.851371	0.425685	−	0.904871i	$-0.360033\pi$
$149$	10.3923	0.851371	0.425685	+	0.904871i	$0.360033\pi$
$150$	0	0
$151$	14.9282	1.21484	0.607420	−	0.794381i	$-0.292205\pi$
$151$	14.9282	1.21484	0.607420	+	0.794381i	$0.292205\pi$
$152$	0	0
$153$	−6.19615	−0.500929
$154$	0	0
$155$	6.53590	0.524976
$156$	0	0
$157$	−14.3923	−1.14863	−0.574315	−	0.818634i	$-0.694732\pi$
$157$	−14.3923	−1.14863	−0.574315	+	0.818634i	$0.694732\pi$
$158$	0	0
$159$	−5.26795	−0.417776
$160$	0	0
$161$	0.535898	0.0422347
$162$	0	0
$163$	−8.53590	−0.668583	−0.334292	−	0.942470i	$-0.608497\pi$
$163$	−8.53590	−0.668583	−0.334292	+	0.942470i	$0.608497\pi$
$164$	0	0
$165$	−2.53590	−0.197419
$166$	0	0
$167$	−12.3923	−0.958945	−0.479473	−	0.877557i	$-0.659172\pi$
$167$	−12.3923	−0.958945	−0.479473	+	0.877557i	$0.659172\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−0.392305	−0.0298264	−0.0149132	−	0.999889i	$-0.504747\pi$
$173$	−0.392305	−0.0298264	−0.0149132	+	0.999889i	$0.504747\pi$
$174$	0	0
$175$	−4.46410	−0.337454
$176$	0	0
$177$	2.92820	0.220097
$178$	0	0
$179$	2.33975	0.174881	0.0874404	−	0.996170i	$-0.472131\pi$
$179$	2.33975	0.174881	0.0874404	+	0.996170i	$0.472131\pi$
$180$	0	0
$181$	−22.7846	−1.69357	−0.846783	−	0.531938i	$-0.821464\pi$
$181$	−22.7846	−1.69357	−0.846783	+	0.531938i	$0.821464\pi$
$182$	0	0
$183$	−6.39230	−0.472533
$184$	0	0
$185$	−0.392305	−0.0288428
$186$	0	0
$187$	−21.4641	−1.56961
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−7.46410	−0.540083	−0.270042	−	0.962849i	$-0.587037\pi$
$191$	−7.46410	−0.540083	−0.270042	+	0.962849i	$0.587037\pi$
$192$	0	0
$193$	−14.3923	−1.03598	−0.517990	−	0.855386i	$-0.673320\pi$
$193$	−14.3923	−1.03598	−0.517990	+	0.855386i	$0.673320\pi$
$194$	0	0
$195$	2.92820	0.209693
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	15.3205	1.08604	0.543021	−	0.839719i	$-0.317280\pi$
$199$	15.3205	1.08604	0.543021	+	0.839719i	$0.317280\pi$
$200$	0	0
$201$	−5.46410	−0.385408
$202$	0	0
$203$	4.19615	0.294512
$204$	0	0
$205$	−1.85641	−0.129657
$206$	0	0
$207$	0.535898	0.0372475
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	−0.196152	−0.0134401
$214$	0	0
$215$	−2.53590	−0.172947
$216$	0	0
$217$	8.92820	0.606086
$218$	0	0
$219$	−11.8564	−0.801182
$220$	0	0
$221$	24.7846	1.66719
$222$	0	0
$223$	−8.92820	−0.597877	−0.298938	−	0.954272i	$-0.596632\pi$
$223$	−8.92820	−0.597877	−0.298938	+	0.954272i	$0.596632\pi$
$224$	0	0
$225$	−4.46410	−0.297607
$226$	0	0
$227$	−8.39230	−0.557017	−0.278508	−	0.960434i	$-0.589840\pi$
$227$	−8.39230	−0.557017	−0.278508	+	0.960434i	$0.589840\pi$
$228$	0	0
$229$	2.39230	0.158088	0.0790440	−	0.996871i	$-0.474813\pi$
$229$	2.39230	0.158088	0.0790440	+	0.996871i	$0.474813\pi$
$230$	0	0
$231$	−3.46410	−0.227921
$232$	0	0
$233$	−17.3205	−1.13470	−0.567352	−	0.823475i	$-0.692032\pi$
$233$	−17.3205	−1.13470	−0.567352	+	0.823475i	$0.692032\pi$
$234$	0	0
$235$	0.535898	0.0349582
$236$	0	0
$237$	−6.53590	−0.424552
$238$	0	0
$239$	16.9282	1.09499	0.547497	−	0.836808i	$-0.315581\pi$
$239$	16.9282	1.09499	0.547497	+	0.836808i	$0.315581\pi$
$240$	0	0
$241$	25.7128	1.65631	0.828154	−	0.560501i	$-0.189391\pi$
$241$	25.7128	1.65631	0.828154	+	0.560501i	$0.189391\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.732051	0.0467690
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−4.73205	−0.299882
$250$	0	0
$251$	17.8038	1.12377	0.561884	−	0.827216i	$-0.310077\pi$
$251$	17.8038	1.12377	0.561884	+	0.827216i	$0.310077\pi$
$252$	0	0
$253$	1.85641	0.116711
$254$	0	0
$255$	4.53590	0.284049
$256$	0	0
$257$	18.9282	1.18071	0.590354	−	0.807144i	$-0.298988\pi$
$257$	18.9282	1.18071	0.590354	+	0.807144i	$0.298988\pi$
$258$	0	0
$259$	−0.535898	−0.0332991
$260$	0	0
$261$	4.19615	0.259735
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	3.85641	0.236897
$266$	0	0
$267$	−2.53590	−0.155194
$268$	0	0
$269$	−13.8564	−0.844840	−0.422420	−	0.906400i	$-0.638819\pi$
$269$	−13.8564	−0.844840	−0.422420	+	0.906400i	$0.638819\pi$
$270$	0	0
$271$	15.3205	0.930655	0.465327	−	0.885139i	$-0.345937\pi$
$271$	15.3205	0.930655	0.465327	+	0.885139i	$0.345937\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	−15.4641	−0.932520
$276$	0	0
$277$	−8.39230	−0.504245	−0.252122	−	0.967695i	$-0.581129\pi$
$277$	−8.39230	−0.504245	−0.252122	+	0.967695i	$0.581129\pi$
$278$	0	0
$279$	8.92820	0.534518
$280$	0	0
$281$	21.6603	1.29214	0.646071	−	0.763277i	$-0.276411\pi$
$281$	21.6603	1.29214	0.646071	+	0.763277i	$0.276411\pi$
$282$	0	0
$283$	1.46410	0.0870318	0.0435159	−	0.999053i	$-0.486144\pi$
$283$	1.46410	0.0870318	0.0435159	+	0.999053i	$0.486144\pi$
$284$	0	0
$285$	−0.732051	−0.0433629
$286$	0	0
$287$	−2.53590	−0.149689
$288$	0	0
$289$	21.3923	1.25837
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−9.46410	−0.552899	−0.276449	−	0.961028i	$-0.589158\pi$
$293$	−9.46410	−0.552899	−0.276449	+	0.961028i	$0.589158\pi$
$294$	0	0
$295$	−2.14359	−0.124805
$296$	0	0
$297$	−3.46410	−0.201008
$298$	0	0
$299$	−2.14359	−0.123967
$300$	0	0
$301$	−3.46410	−0.199667
$302$	0	0
$303$	5.80385	0.333422
$304$	0	0
$305$	4.67949	0.267947
$306$	0	0
$307$	0.143594	0.00819532	0.00409766	−	0.999992i	$-0.498696\pi$
$307$	0.143594	0.00819532	0.00409766	+	0.999992i	$0.498696\pi$
$308$	0	0
$309$	2.92820	0.166580
$310$	0	0
$311$	11.2679	0.638947	0.319473	−	0.947595i	$-0.396494\pi$
$311$	11.2679	0.638947	0.319473	+	0.947595i	$0.396494\pi$
$312$	0	0
$313$	−9.60770	−0.543059	−0.271530	−	0.962430i	$-0.587529\pi$
$313$	−9.60770	−0.543059	−0.271530	+	0.962430i	$0.587529\pi$
$314$	0	0
$315$	0.732051	0.0412464
$316$	0	0
$317$	27.1244	1.52346	0.761728	−	0.647897i	$-0.224351\pi$
$317$	27.1244	1.52346	0.761728	+	0.647897i	$0.224351\pi$
$318$	0	0
$319$	14.5359	0.813854
$320$	0	0
$321$	−5.66025	−0.315925
$322$	0	0
$323$	−6.19615	−0.344763
$324$	0	0
$325$	17.8564	0.990495
$326$	0	0
$327$	3.46410	0.191565
$328$	0	0
$329$	0.732051	0.0403593
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−0.535898	−0.0293671
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	13.6077	0.741258	0.370629	−	0.928781i	$-0.379142\pi$
$337$	13.6077	0.741258	0.370629	+	0.928781i	$0.379142\pi$
$338$	0	0
$339$	−13.2679	−0.720616
$340$	0	0
$341$	30.9282	1.67486
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.392305	−0.0211210
$346$	0	0
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	4.14359	0.221801	0.110901	−	0.993831i	$-0.464626\pi$
$349$	4.14359	0.221801	0.110901	+	0.993831i	$0.464626\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	32.0526	1.70599	0.852993	−	0.521923i	$-0.174785\pi$
$353$	32.0526	1.70599	0.852993	+	0.521923i	$0.174785\pi$
$354$	0	0
$355$	0.143594	0.00762115
$356$	0	0
$357$	6.19615	0.327935
$358$	0	0
$359$	−18.3923	−0.970709	−0.485354	−	0.874318i	$-0.661309\pi$
$359$	−18.3923	−0.970709	−0.485354	+	0.874318i	$0.661309\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	8.67949	0.454305
$366$	0	0
$367$	14.2487	0.743777	0.371888	−	0.928277i	$-0.378710\pi$
$367$	14.2487	0.743777	0.371888	+	0.928277i	$0.378710\pi$
$368$	0	0
$369$	−2.53590	−0.132014
$370$	0	0
$371$	5.26795	0.273498
$372$	0	0
$373$	−18.7846	−0.972630	−0.486315	−	0.873784i	$-0.661659\pi$
$373$	−18.7846	−0.972630	−0.486315	+	0.873784i	$0.661659\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	−16.7846	−0.864451
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	−12.3923	−0.634877
$382$	0	0
$383$	28.3923	1.45078	0.725390	−	0.688339i	$-0.241660\pi$
$383$	28.3923	1.45078	0.725390	+	0.688339i	$0.241660\pi$
$384$	0	0
$385$	2.53590	0.129241
$386$	0	0
$387$	−3.46410	−0.176090
$388$	0	0
$389$	35.1769	1.78354	0.891770	−	0.452489i	$-0.149464\pi$
$389$	35.1769	1.78354	0.891770	+	0.452489i	$0.149464\pi$
$390$	0	0
$391$	−3.32051	−0.167925
$392$	0	0
$393$	1.80385	0.0909921
$394$	0	0
$395$	4.78461	0.240740
$396$	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
$401$	34.0526	1.70050	0.850252	−	0.526376i	$-0.176450\pi$
$401$	34.0526	1.70050	0.850252	+	0.526376i	$0.176450\pi$
$402$	0	0
$403$	−35.7128	−1.77898
$404$	0	0
$405$	0.732051	0.0363759
$406$	0	0
$407$	−1.85641	−0.0920187
$408$	0	0
$409$	5.07180	0.250784	0.125392	−	0.992107i	$-0.459981\pi$
$409$	5.07180	0.250784	0.125392	+	0.992107i	$0.459981\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	0	0
$413$	−2.92820	−0.144087
$414$	0	0
$415$	3.46410	0.170046
$416$	0	0
$417$	−9.07180	−0.444248
$418$	0	0
$419$	28.0526	1.37046	0.685229	−	0.728328i	$-0.259702\pi$
$419$	28.0526	1.37046	0.685229	+	0.728328i	$0.259702\pi$
$420$	0	0
$421$	28.9282	1.40987	0.704937	−	0.709270i	$-0.250975\pi$
$421$	28.9282	1.40987	0.704937	+	0.709270i	$0.250975\pi$
$422$	0	0
$423$	0.732051	0.0355935
$424$	0	0
$425$	27.6603	1.34172
$426$	0	0
$427$	6.39230	0.309345
$428$	0	0
$429$	13.8564	0.668994
$430$	0	0
$431$	31.5167	1.51810	0.759052	−	0.651030i	$-0.225663\pi$
$431$	31.5167	1.51810	0.759052	+	0.651030i	$0.225663\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	−3.07180	−0.147281
$436$	0	0
$437$	0.535898	0.0256355
$438$	0	0
$439$	−8.92820	−0.426120	−0.213060	−	0.977039i	$-0.568343\pi$
$439$	−8.92820	−0.426120	−0.213060	+	0.977039i	$0.568343\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	11.0718	0.526037	0.263018	−	0.964791i	$-0.415282\pi$
$443$	11.0718	0.526037	0.263018	+	0.964791i	$0.415282\pi$
$444$	0	0
$445$	1.85641	0.0880021
$446$	0	0
$447$	−10.3923	−0.491539
$448$	0	0
$449$	32.1962	1.51943	0.759715	−	0.650256i	$-0.225338\pi$
$449$	32.1962	1.51943	0.759715	+	0.650256i	$0.225338\pi$
$450$	0	0
$451$	−8.78461	−0.413651
$452$	0	0
$453$	−14.9282	−0.701388
$454$	0	0
$455$	−2.92820	−0.137276
$456$	0	0
$457$	3.60770	0.168761	0.0843804	−	0.996434i	$-0.473109\pi$
$457$	3.60770	0.168761	0.0843804	+	0.996434i	$0.473109\pi$
$458$	0	0
$459$	6.19615	0.289212
$460$	0	0
$461$	−35.6603	−1.66086	−0.830432	−	0.557120i	$-0.811906\pi$
$461$	−35.6603	−1.66086	−0.830432	+	0.557120i	$0.811906\pi$
$462$	0	0
$463$	−25.8564	−1.20165	−0.600825	−	0.799381i	$-0.705161\pi$
$463$	−25.8564	−1.20165	−0.600825	+	0.799381i	$0.705161\pi$
$464$	0	0
$465$	−6.53590	−0.303095
$466$	0	0
$467$	−36.4449	−1.68647	−0.843234	−	0.537547i	$-0.819351\pi$
$467$	−36.4449	−1.68647	−0.843234	+	0.537547i	$0.819351\pi$
$468$	0	0
$469$	5.46410	0.252309
$470$	0	0
$471$	14.3923	0.663162
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	−4.46410	−0.204827
$476$	0	0
$477$	5.26795	0.241203
$478$	0	0
$479$	18.9808	0.867253	0.433627	−	0.901093i	$-0.357234\pi$
$479$	18.9808	0.867253	0.433627	+	0.901093i	$0.357234\pi$
$480$	0	0
$481$	2.14359	0.0977395
$482$	0	0
$483$	−0.535898	−0.0243842
$484$	0	0
$485$	−4.39230	−0.199444
$486$	0	0
$487$	−2.92820	−0.132690	−0.0663448	−	0.997797i	$-0.521134\pi$
$487$	−2.92820	−0.132690	−0.0663448	+	0.997797i	$0.521134\pi$
$488$	0	0
$489$	8.53590	0.386007
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	−26.0000	−1.17098
$494$	0	0
$495$	2.53590	0.113980
$496$	0	0
$497$	0.196152	0.00879864
$498$	0	0
$499$	−6.92820	−0.310149	−0.155074	−	0.987903i	$-0.549562\pi$
$499$	−6.92820	−0.310149	−0.155074	+	0.987903i	$0.549562\pi$
$500$	0	0
$501$	12.3923	0.553647
$502$	0	0
$503$	15.2679	0.680764	0.340382	−	0.940287i	$-0.389444\pi$
$503$	15.2679	0.680764	0.340382	+	0.940287i	$0.389444\pi$
$504$	0	0
$505$	−4.24871	−0.189065
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	−8.78461	−0.389371	−0.194685	−	0.980866i	$-0.562369\pi$
$509$	−8.78461	−0.389371	−0.194685	+	0.980866i	$0.562369\pi$
$510$	0	0
$511$	11.8564	0.524497
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−2.14359	−0.0944580
$516$	0	0
$517$	2.53590	0.111529
$518$	0	0
$519$	0.392305	0.0172203
$520$	0	0
$521$	−16.0000	−0.700973	−0.350486	−	0.936568i	$-0.613984\pi$
$521$	−16.0000	−0.700973	−0.350486	+	0.936568i	$0.613984\pi$
$522$	0	0
$523$	−18.7846	−0.821394	−0.410697	−	0.911772i	$-0.634715\pi$
$523$	−18.7846	−0.821394	−0.410697	+	0.911772i	$0.634715\pi$
$524$	0	0
$525$	4.46410	0.194829
$526$	0	0
$527$	−55.3205	−2.40980
$528$	0	0
$529$	−22.7128	−0.987514
$530$	0	0
$531$	−2.92820	−0.127073
$532$	0	0
$533$	10.1436	0.439368
$534$	0	0
$535$	4.14359	0.179143
$536$	0	0
$537$	−2.33975	−0.100967
$538$	0	0
$539$	3.46410	0.149209
$540$	0	0
$541$	11.8564	0.509747	0.254873	−	0.966974i	$-0.417966\pi$
$541$	11.8564	0.509747	0.254873	+	0.966974i	$0.417966\pi$
$542$	0	0
$543$	22.7846	0.977781
$544$	0	0
$545$	−2.53590	−0.108626
$546$	0	0
$547$	−3.32051	−0.141975	−0.0709873	−	0.997477i	$-0.522615\pi$
$547$	−3.32051	−0.141975	−0.0709873	+	0.997477i	$0.522615\pi$
$548$	0	0
$549$	6.39230	0.272817
$550$	0	0
$551$	4.19615	0.178762
$552$	0	0
$553$	6.53590	0.277935
$554$	0	0
$555$	0.392305	0.0166524
$556$	0	0
$557$	−14.3923	−0.609822	−0.304911	−	0.952381i	$-0.598627\pi$
$557$	−14.3923	−0.609822	−0.304911	+	0.952381i	$0.598627\pi$
$558$	0	0
$559$	13.8564	0.586064
$560$	0	0
$561$	21.4641	0.906215
$562$	0	0
$563$	27.3205	1.15142	0.575711	−	0.817653i	$-0.304725\pi$
$563$	27.3205	1.15142	0.575711	+	0.817653i	$0.304725\pi$
$564$	0	0
$565$	9.71281	0.408621
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−27.9090	−1.17000	−0.585002	−	0.811032i	$-0.698906\pi$
$569$	−27.9090	−1.17000	−0.585002	+	0.811032i	$0.698906\pi$
$570$	0	0
$571$	2.14359	0.0897066	0.0448533	−	0.998994i	$-0.485718\pi$
$571$	2.14359	0.0897066	0.0448533	+	0.998994i	$0.485718\pi$
$572$	0	0
$573$	7.46410	0.311817
$574$	0	0
$575$	−2.39230	−0.0997660
$576$	0	0
$577$	−27.1769	−1.13139	−0.565695	−	0.824615i	$-0.691392\pi$
$577$	−27.1769	−1.13139	−0.565695	+	0.824615i	$0.691392\pi$
$578$	0	0
$579$	14.3923	0.598124
$580$	0	0
$581$	4.73205	0.196319
$582$	0	0
$583$	18.2487	0.755784
$584$	0	0
$585$	−2.92820	−0.121066
$586$	0	0
$587$	37.5167	1.54848	0.774239	−	0.632893i	$-0.218133\pi$
$587$	37.5167	1.54848	0.774239	+	0.632893i	$0.218133\pi$
$588$	0	0
$589$	8.92820	0.367880
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	17.5167	0.719323	0.359662	−	0.933083i	$-0.382892\pi$
$593$	17.5167	0.719323	0.359662	+	0.933083i	$0.382892\pi$
$594$	0	0
$595$	−4.53590	−0.185954
$596$	0	0
$597$	−15.3205	−0.627027
$598$	0	0
$599$	5.26795	0.215243	0.107621	−	0.994192i	$-0.465677\pi$
$599$	5.26795	0.215243	0.107621	+	0.994192i	$0.465677\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	5.46410	0.222515
$604$	0	0
$605$	0.732051	0.0297621
$606$	0	0
$607$	32.7846	1.33069	0.665343	−	0.746538i	$-0.268285\pi$
$607$	32.7846	1.33069	0.665343	+	0.746538i	$0.268285\pi$
$608$	0	0
$609$	−4.19615	−0.170037
$610$	0	0
$611$	−2.92820	−0.118462
$612$	0	0
$613$	−20.3923	−0.823637	−0.411819	−	0.911266i	$-0.635106\pi$
$613$	−20.3923	−0.823637	−0.411819	+	0.911266i	$0.635106\pi$
$614$	0	0
$615$	1.85641	0.0748575
$616$	0	0
$617$	−29.7128	−1.19619	−0.598096	−	0.801424i	$-0.704076\pi$
$617$	−29.7128	−1.19619	−0.598096	+	0.801424i	$0.704076\pi$
$618$	0	0
$619$	−44.7846	−1.80005	−0.900023	−	0.435843i	$-0.856450\pi$
$619$	−44.7846	−1.80005	−0.900023	+	0.435843i	$0.856450\pi$
$620$	0	0
$621$	−0.535898	−0.0215049
$622$	0	0
$623$	2.53590	0.101599
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	−3.46410	−0.138343
$628$	0	0
$629$	3.32051	0.132397
$630$	0	0
$631$	−12.2487	−0.487613	−0.243807	−	0.969824i	$-0.578396\pi$
$631$	−12.2487	−0.487613	−0.243807	+	0.969824i	$0.578396\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	9.07180	0.360003
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0.196152	0.00775967
$640$	0	0
$641$	22.4449	0.886519	0.443259	−	0.896393i	$-0.353822\pi$
$641$	22.4449	0.886519	0.443259	+	0.896393i	$0.353822\pi$
$642$	0	0
$643$	32.1051	1.26610	0.633051	−	0.774110i	$-0.281802\pi$
$643$	32.1051	1.26610	0.633051	+	0.774110i	$0.281802\pi$
$644$	0	0
$645$	2.53590	0.0998509
$646$	0	0
$647$	−34.9808	−1.37524	−0.687618	−	0.726073i	$-0.741344\pi$
$647$	−34.9808	−1.37524	−0.687618	+	0.726073i	$0.741344\pi$
$648$	0	0
$649$	−10.1436	−0.398171
$650$	0	0
$651$	−8.92820	−0.349924
$652$	0	0
$653$	25.7128	1.00622	0.503110	−	0.864222i	$-0.332189\pi$
$653$	25.7128	1.00622	0.503110	+	0.864222i	$0.332189\pi$
$654$	0	0
$655$	−1.32051	−0.0515965
$656$	0	0
$657$	11.8564	0.462562
$658$	0	0
$659$	1.94744	0.0758615	0.0379308	−	0.999280i	$-0.487923\pi$
$659$	1.94744	0.0758615	0.0379308	+	0.999280i	$0.487923\pi$
$660$	0	0
$661$	−12.0000	−0.466746	−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000	−0.466746	−0.233373	+	0.972387i	$0.574976\pi$
$662$	0	0
$663$	−24.7846	−0.962554
$664$	0	0
$665$	0.732051	0.0283877
$666$	0	0
$667$	2.24871	0.0870704
$668$	0	0
$669$	8.92820	0.345184
$670$	0	0
$671$	22.1436	0.854844
$672$	0	0
$673$	16.2487	0.626342	0.313171	−	0.949697i	$-0.398609\pi$
$673$	16.2487	0.626342	0.313171	+	0.949697i	$0.398609\pi$
$674$	0	0
$675$	4.46410	0.171823
$676$	0	0
$677$	30.9282	1.18867	0.594334	−	0.804219i	$-0.297416\pi$
$677$	30.9282	1.18867	0.594334	+	0.804219i	$0.297416\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	8.39230	0.321594
$682$	0	0
$683$	−8.58846	−0.328628	−0.164314	−	0.986408i	$-0.552541\pi$
$683$	−8.58846	−0.328628	−0.164314	+	0.986408i	$0.552541\pi$
$684$	0	0
$685$	13.1769	0.503464
$686$	0	0
$687$	−2.39230	−0.0912721
$688$	0	0
$689$	−21.0718	−0.802772
$690$	0	0
$691$	−34.5359	−1.31381	−0.656904	−	0.753974i	$-0.728134\pi$
$691$	−34.5359	−1.31381	−0.656904	+	0.753974i	$0.728134\pi$
$692$	0	0
$693$	3.46410	0.131590
$694$	0	0
$695$	6.64102	0.251908
$696$	0	0
$697$	15.7128	0.595165
$698$	0	0
$699$	17.3205	0.655122
$700$	0	0
$701$	−50.7846	−1.91811	−0.959054	−	0.283223i	$-0.908596\pi$
$701$	−50.7846	−1.91811	−0.959054	+	0.283223i	$0.908596\pi$
$702$	0	0
$703$	−0.535898	−0.0202118
$704$	0	0
$705$	−0.535898	−0.0201831
$706$	0	0
$707$	−5.80385	−0.218276
$708$	0	0
$709$	−43.3205	−1.62694	−0.813468	−	0.581610i	$-0.802423\pi$
$709$	−43.3205	−1.62694	−0.813468	+	0.581610i	$0.802423\pi$
$710$	0	0
$711$	6.53590	0.245115
$712$	0	0
$713$	4.78461	0.179185
$714$	0	0
$715$	−10.1436	−0.379349
$716$	0	0
$717$	−16.9282	−0.632195
$718$	0	0
$719$	−40.0526	−1.49371	−0.746854	−	0.664988i	$-0.768437\pi$
$719$	−40.0526	−1.49371	−0.746854	+	0.664988i	$0.768437\pi$
$720$	0	0
$721$	−2.92820	−0.109052
$722$	0	0
$723$	−25.7128	−0.956270
$724$	0	0
$725$	−18.7321	−0.695691
$726$	0	0
$727$	−13.4641	−0.499356	−0.249678	−	0.968329i	$-0.580325\pi$
$727$	−13.4641	−0.499356	−0.249678	+	0.968329i	$0.580325\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	21.4641	0.793878
$732$	0	0
$733$	−52.6410	−1.94434	−0.972170	−	0.234276i	$-0.924728\pi$
$733$	−52.6410	−1.94434	−0.972170	+	0.234276i	$0.924728\pi$
$734$	0	0
$735$	−0.732051	−0.0270021
$736$	0	0
$737$	18.9282	0.697229
$738$	0	0
$739$	23.4641	0.863141	0.431570	−	0.902079i	$-0.357960\pi$
$739$	23.4641	0.863141	0.431570	+	0.902079i	$0.357960\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	20.5885	0.755317	0.377659	−	0.925945i	$-0.376729\pi$
$743$	20.5885	0.755317	0.377659	+	0.925945i	$0.376729\pi$
$744$	0	0
$745$	7.60770	0.278724
$746$	0	0
$747$	4.73205	0.173137
$748$	0	0
$749$	5.66025	0.206821
$750$	0	0
$751$	−12.7846	−0.466517	−0.233259	−	0.972415i	$-0.574939\pi$
$751$	−12.7846	−0.466517	−0.233259	+	0.972415i	$0.574939\pi$
$752$	0	0
$753$	−17.8038	−0.648808
$754$	0	0
$755$	10.9282	0.397718
$756$	0	0
$757$	4.92820	0.179119	0.0895593	−	0.995981i	$-0.471454\pi$
$757$	4.92820	0.179119	0.0895593	+	0.995981i	$0.471454\pi$
$758$	0	0
$759$	−1.85641	−0.0673833
$760$	0	0
$761$	1.80385	0.0653894	0.0326947	−	0.999465i	$-0.489591\pi$
$761$	1.80385	0.0653894	0.0326947	+	0.999465i	$0.489591\pi$
$762$	0	0
$763$	−3.46410	−0.125409
$764$	0	0
$765$	−4.53590	−0.163996
$766$	0	0
$767$	11.7128	0.422925
$768$	0	0
$769$	−53.3205	−1.92279	−0.961393	−	0.275178i	$-0.911263\pi$
$769$	−53.3205	−1.92279	−0.961393	+	0.275178i	$0.911263\pi$
$770$	0	0
$771$	−18.9282	−0.681683
$772$	0	0
$773$	−45.4641	−1.63523	−0.817615	−	0.575765i	$-0.804704\pi$
$773$	−45.4641	−1.63523	−0.817615	+	0.575765i	$0.804704\pi$
$774$	0	0
$775$	−39.8564	−1.43168
$776$	0	0
$777$	0.535898	0.0192252
$778$	0	0
$779$	−2.53590	−0.0908580
$780$	0	0
$781$	0.679492	0.0243141
$782$	0	0
$783$	−4.19615	−0.149958
$784$	0	0
$785$	−10.5359	−0.376042
$786$	0	0
$787$	42.7846	1.52511	0.762553	−	0.646925i	$-0.223945\pi$
$787$	42.7846	1.52511	0.762553	+	0.646925i	$0.223945\pi$
$788$	0	0
$789$	−2.00000	−0.0712019
$790$	0	0
$791$	13.2679	0.471754
$792$	0	0
$793$	−25.5692	−0.907990
$794$	0	0
$795$	−3.85641	−0.136773
$796$	0	0
$797$	21.1769	0.750125	0.375062	−	0.927000i	$-0.377621\pi$
$797$	21.1769	0.750125	0.375062	+	0.927000i	$0.377621\pi$
$798$	0	0
$799$	−4.53590	−0.160469
$800$	0	0
$801$	2.53590	0.0896016
$802$	0	0
$803$	41.0718	1.44939
$804$	0	0
$805$	0.392305	0.0138269
$806$	0	0
$807$	13.8564	0.487769
$808$	0	0
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	0	0
$811$	46.9282	1.64787	0.823936	−	0.566683i	$-0.191773\pi$
$811$	46.9282	1.64787	0.823936	+	0.566683i	$0.191773\pi$
$812$	0	0
$813$	−15.3205	−0.537314
$814$	0	0
$815$	−6.24871	−0.218883
$816$	0	0
$817$	−3.46410	−0.121194
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−57.0333	−1.99048	−0.995238	−	0.0974715i	$-0.968925\pi$
$821$	−57.0333	−1.99048	−0.995238	+	0.0974715i	$0.968925\pi$
$822$	0	0
$823$	44.5359	1.55242	0.776212	−	0.630472i	$-0.217139\pi$
$823$	44.5359	1.55242	0.776212	+	0.630472i	$0.217139\pi$
$824$	0	0
$825$	15.4641	0.538391
$826$	0	0
$827$	26.0526	0.905936	0.452968	−	0.891527i	$-0.350365\pi$
$827$	26.0526	0.905936	0.452968	+	0.891527i	$0.350365\pi$
$828$	0	0
$829$	−7.07180	−0.245614	−0.122807	−	0.992431i	$-0.539190\pi$
$829$	−7.07180	−0.245614	−0.122807	+	0.992431i	$0.539190\pi$
$830$	0	0
$831$	8.39230	0.291126
$832$	0	0
$833$	−6.19615	−0.214684
$834$	0	0
$835$	−9.07180	−0.313942
$836$	0	0
$837$	−8.92820	−0.308604
$838$	0	0
$839$	30.9282	1.06776	0.533880	−	0.845560i	$-0.320733\pi$
$839$	30.9282	1.06776	0.533880	+	0.845560i	$0.320733\pi$
$840$	0	0
$841$	−11.3923	−0.392838
$842$	0	0
$843$	−21.6603	−0.746019
$844$	0	0
$845$	2.19615	0.0755499
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−1.46410	−0.0502478
$850$	0	0
$851$	−0.287187	−0.00984465
$852$	0	0
$853$	−0.143594	−0.00491655	−0.00245827	−	0.999997i	$-0.500782\pi$
$853$	−0.143594	−0.00491655	−0.00245827	+	0.999997i	$0.500782\pi$
$854$	0	0
$855$	0.732051	0.0250356
$856$	0	0
$857$	−0.679492	−0.0232110	−0.0116055	−	0.999933i	$-0.503694\pi$
$857$	−0.679492	−0.0232110	−0.0116055	+	0.999933i	$0.503694\pi$
$858$	0	0
$859$	49.5692	1.69128	0.845640	−	0.533754i	$-0.179219\pi$
$859$	49.5692	1.69128	0.845640	+	0.533754i	$0.179219\pi$
$860$	0	0
$861$	2.53590	0.0864232
$862$	0	0
$863$	−22.7321	−0.773808	−0.386904	−	0.922120i	$-0.626456\pi$
$863$	−22.7321	−0.773808	−0.386904	+	0.922120i	$0.626456\pi$
$864$	0	0
$865$	−0.287187	−0.00976465
$866$	0	0
$867$	−21.3923	−0.726521
$868$	0	0
$869$	22.6410	0.768044
$870$	0	0
$871$	−21.8564	−0.740576
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	28.6410	0.967138	0.483569	−	0.875306i	$-0.339340\pi$
$877$	28.6410	0.967138	0.483569	+	0.875306i	$0.339340\pi$
$878$	0	0
$879$	9.46410	0.319216
$880$	0	0
$881$	31.2679	1.05344	0.526722	−	0.850038i	$-0.323421\pi$
$881$	31.2679	1.05344	0.526722	+	0.850038i	$0.323421\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	2.14359	0.0720561
$886$	0	0
$887$	18.2487	0.612732	0.306366	−	0.951914i	$-0.400887\pi$
$887$	18.2487	0.612732	0.306366	+	0.951914i	$0.400887\pi$
$888$	0	0
$889$	12.3923	0.415625
$890$	0	0
$891$	3.46410	0.116052
$892$	0	0
$893$	0.732051	0.0244971
$894$	0	0
$895$	1.71281	0.0572530
$896$	0	0
$897$	2.14359	0.0715725
$898$	0	0
$899$	37.4641	1.24950
$900$	0	0
$901$	−32.6410	−1.08743
$902$	0	0
$903$	3.46410	0.115278
$904$	0	0
$905$	−16.6795	−0.554445
$906$	0	0
$907$	−46.9282	−1.55823	−0.779113	−	0.626884i	$-0.784330\pi$
$907$	−46.9282	−1.55823	−0.779113	+	0.626884i	$0.784330\pi$
$908$	0	0
$909$	−5.80385	−0.192501
$910$	0	0
$911$	29.3731	0.973173	0.486587	−	0.873632i	$-0.338242\pi$
$911$	29.3731	0.973173	0.486587	+	0.873632i	$0.338242\pi$
$912$	0	0
$913$	16.3923	0.542506
$914$	0	0
$915$	−4.67949	−0.154699
$916$	0	0
$917$	−1.80385	−0.0595683
$918$	0	0
$919$	−51.7128	−1.70585	−0.852924	−	0.522035i	$-0.825173\pi$
$919$	−51.7128	−1.70585	−0.852924	+	0.522035i	$0.825173\pi$
$920$	0	0
$921$	−0.143594	−0.00473157
$922$	0	0
$923$	−0.784610	−0.0258257
$924$	0	0
$925$	2.39230	0.0786585
$926$	0	0
$927$	−2.92820	−0.0961748
$928$	0	0
$929$	−4.33975	−0.142382	−0.0711912	−	0.997463i	$-0.522680\pi$
$929$	−4.33975	−0.142382	−0.0711912	+	0.997463i	$0.522680\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−11.2679	−0.368896
$934$	0	0
$935$	−15.7128	−0.513864
$936$	0	0
$937$	20.9282	0.683695	0.341847	−	0.939756i	$-0.388947\pi$
$937$	20.9282	0.683695	0.341847	+	0.939756i	$0.388947\pi$
$938$	0	0
$939$	9.60770	0.313535
$940$	0	0
$941$	−50.6410	−1.65085	−0.825425	−	0.564512i	$-0.809064\pi$
$941$	−50.6410	−1.65085	−0.825425	+	0.564512i	$0.809064\pi$
$942$	0	0
$943$	−1.35898	−0.0442546
$944$	0	0
$945$	−0.732051	−0.0238136
$946$	0	0
$947$	−55.1769	−1.79301	−0.896504	−	0.443035i	$-0.853902\pi$
$947$	−55.1769	−1.79301	−0.896504	+	0.443035i	$0.853902\pi$
$948$	0	0
$949$	−47.4256	−1.53950
$950$	0	0
$951$	−27.1244	−0.879567
$952$	0	0
$953$	−2.73205	−0.0884998	−0.0442499	−	0.999020i	$-0.514090\pi$
$953$	−2.73205	−0.0884998	−0.0442499	+	0.999020i	$0.514090\pi$
$954$	0	0
$955$	−5.46410	−0.176814
$956$	0	0
$957$	−14.5359	−0.469879
$958$	0	0
$959$	18.0000	0.581250
$960$	0	0
$961$	48.7128	1.57138
$962$	0	0
$963$	5.66025	0.182399
$964$	0	0
$965$	−10.5359	−0.339163
$966$	0	0
$967$	6.39230	0.205563	0.102781	−	0.994704i	$-0.467226\pi$
$967$	6.39230	0.205563	0.102781	+	0.994704i	$0.467226\pi$
$968$	0	0
$969$	6.19615	0.199049
$970$	0	0
$971$	39.7128	1.27444	0.637222	−	0.770680i	$-0.280083\pi$
$971$	39.7128	1.27444	0.637222	+	0.770680i	$0.280083\pi$
$972$	0	0
$973$	9.07180	0.290828
$974$	0	0
$975$	−17.8564	−0.571863
$976$	0	0
$977$	49.2679	1.57622	0.788111	−	0.615534i	$-0.211059\pi$
$977$	49.2679	1.57622	0.788111	+	0.615534i	$0.211059\pi$
$978$	0	0
$979$	8.78461	0.280757
$980$	0	0
$981$	−3.46410	−0.110600
$982$	0	0
$983$	47.7128	1.52180	0.760901	−	0.648868i	$-0.224757\pi$
$983$	47.7128	1.52180	0.760901	+	0.648868i	$0.224757\pi$
$984$	0	0
$985$	7.32051	0.233251
$986$	0	0
$987$	−0.732051	−0.0233014
$988$	0	0
$989$	−1.85641	−0.0590303
$990$	0	0
$991$	46.6410	1.48160	0.740800	−	0.671725i	$-0.234446\pi$
$991$	46.6410	1.48160	0.740800	+	0.671725i	$0.234446\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	11.2154	0.355552
$996$	0	0
$997$	−25.3205	−0.801909	−0.400954	−	0.916098i	$-0.631321\pi$
$997$	−25.3205	−0.801909	−0.400954	+	0.916098i	$0.631321\pi$
$998$	0	0
$999$	0.535898	0.0169551

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.i.1.2	✓	2	4.3	odd	2
4788.2.a.j.1.1		2	12.11	even	2
6384.2.a.bj.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.46410 q^{11} -4.00000 q^{13} -0.732051 q^{15} -6.19615 q^{17} +1.00000 q^{19} -1.00000 q^{21} +0.535898 q^{23} -4.46410 q^{25} -1.00000 q^{27} +4.19615 q^{29} +8.92820 q^{31} -3.46410 q^{33} +0.732051 q^{35} -0.535898 q^{37} +4.00000 q^{39} -2.53590 q^{41} -3.46410 q^{43} +0.732051 q^{45} +0.732051 q^{47} +1.00000 q^{49} +6.19615 q^{51} +5.26795 q^{53} +2.53590 q^{55} -1.00000 q^{57} -2.92820 q^{59} +6.39230 q^{61} +1.00000 q^{63} -2.92820 q^{65} +5.46410 q^{67} -0.535898 q^{69} +0.196152 q^{71} +11.8564 q^{73} +4.46410 q^{75} +3.46410 q^{77} +6.53590 q^{79} +1.00000 q^{81} +4.73205 q^{83} -4.53590 q^{85} -4.19615 q^{87} +2.53590 q^{89} -4.00000 q^{91} -8.92820 q^{93} +0.732051 q^{95} -6.00000 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} - 2 q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 14 q^{53} + 12 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} + 2 q^{63} + 8 q^{65} + 4 q^{67} - 8 q^{69} - 10 q^{71} - 4 q^{73} + 2 q^{75} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 16 q^{85} + 2 q^{87} + 12 q^{89} - 8 q^{91} - 4 q^{93} - 2 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 8 * q^13 + 2 * q^15 - 2 * q^17 + 2 * q^19 - 2 * q^21 + 8 * q^23 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 - 2 * q^35 - 8 * q^37 + 8 * q^39 - 12 * q^41 - 2 * q^45 - 2 * q^47 + 2 * q^49 + 2 * q^51 + 14 * q^53 + 12 * q^55 - 2 * q^57 + 8 * q^59 - 8 * q^61 + 2 * q^63 + 8 * q^65 + 4 * q^67 - 8 * q^69 - 10 * q^71 - 4 * q^73 + 2 * q^75 + 20 * q^79 + 2 * q^81 + 6 * q^83 - 16 * q^85 + 2 * q^87 + 12 * q^89 - 8 * q^91 - 4 * q^93 - 2 * q^95 - 12 * q^97

Introduction

L-functions

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