LMFDB - Embedded newform 6384.2.a.ce.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:	$0$
Dimension:	$5$
Coefficient field:	5.5.135076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.94486$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.88971 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.88971 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.21056 q^{11} +5.39794 q^{13} -3.88971 q^{15} -4.41108 q^{17} -1.00000 q^{19} -1.00000 q^{21} +3.39794 q^{23} +10.1299 q^{25} +1.00000 q^{27} -7.28765 q^{29} -0.187375 q^{31} -3.21056 q^{33} +3.88971 q^{35} -4.60850 q^{37} +5.39794 q^{39} -9.77943 q^{41} -3.88971 q^{45} -4.45071 q^{47} +1.00000 q^{49} -4.41108 q^{51} +11.2876 q^{53} +12.4882 q^{55} -1.00000 q^{57} +5.77156 q^{59} +6.00000 q^{61} -1.00000 q^{63} -20.9964 q^{65} -2.48173 q^{67} +3.39794 q^{69} -12.4175 q^{71} -4.08714 q^{73} +10.1299 q^{75} +3.21056 q^{77} -2.68920 q^{79} +1.00000 q^{81} -3.92935 q^{83} +17.1578 q^{85} -7.28765 q^{87} +2.70874 q^{89} -5.39794 q^{91} -0.187375 q^{93} +3.88971 q^{95} -0.227011 q^{97} -3.21056 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} - 5 q^{21} - 2 q^{23} + 19 q^{25} + 5 q^{27} + 4 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{45} + 2 q^{47} + 5 q^{49} - 2 q^{51} + 16 q^{53} + 16 q^{55} - 5 q^{57} + 12 q^{59} + 30 q^{61} - 5 q^{63} + 4 q^{65} - 18 q^{67} - 2 q^{69} + 10 q^{71} + 14 q^{73} + 19 q^{75} + 2 q^{77} + 2 q^{79} + 5 q^{81} + 6 q^{83} + 12 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{93} - 2 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 2 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 5 * q^19 - 5 * q^21 - 2 * q^23 + 19 * q^25 + 5 * q^27 + 4 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 + 2 * q^45 + 2 * q^47 + 5 * q^49 - 2 * q^51 + 16 * q^53 + 16 * q^55 - 5 * q^57 + 12 * q^59 + 30 * q^61 - 5 * q^63 + 4 * q^65 - 18 * q^67 - 2 * q^69 + 10 * q^71 + 14 * q^73 + 19 * q^75 + 2 * q^77 + 2 * q^79 + 5 * q^81 + 6 * q^83 + 12 * q^85 + 4 * q^87 + 10 * q^89 - 8 * q^91 + 4 * q^93 - 2 * q^95 + 8 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.88971	−1.73953	−0.869766	−	0.493464i	$-0.835730\pi$
$5$	−3.88971	−1.73953	−0.869766	+	0.493464i	$0.835730\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.21056	−0.968021	−0.484010	−	0.875062i	$-0.660820\pi$
$11$	−3.21056	−0.968021	−0.484010	+	0.875062i	$0.660820\pi$
$12$	0	0
$13$	5.39794	1.49712	0.748559	−	0.663068i	$-0.230746\pi$
$13$	5.39794	1.49712	0.748559	+	0.663068i	$0.230746\pi$
$14$	0	0
$15$	−3.88971	−1.00432
$16$	0	0
$17$	−4.41108	−1.06984	−0.534922	−	0.844902i	$-0.679659\pi$
$17$	−4.41108	−1.06984	−0.534922	+	0.844902i	$0.679659\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.39794	0.708519	0.354259	−	0.935147i	$-0.384733\pi$
$23$	3.39794	0.708519	0.354259	+	0.935147i	$0.384733\pi$
$24$	0	0
$25$	10.1299	2.02597
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.28765	−1.35328	−0.676641	−	0.736313i	$-0.736565\pi$
$29$	−7.28765	−1.35328	−0.676641	+	0.736313i	$0.736565\pi$
$30$	0	0
$31$	−0.187375	−0.0336535	−0.0168268	−	0.999858i	$-0.505356\pi$
$31$	−0.187375	−0.0336535	−0.0168268	+	0.999858i	$0.505356\pi$
$32$	0	0
$33$	−3.21056	−0.558887
$34$	0	0
$35$	3.88971	0.657481
$36$	0	0
$37$	−4.60850	−0.757633	−0.378816	−	0.925472i	$-0.623669\pi$
$37$	−4.60850	−0.757633	−0.378816	+	0.925472i	$0.623669\pi$
$38$	0	0
$39$	5.39794	0.864362
$40$	0	0
$41$	−9.77943	−1.52729	−0.763645	−	0.645637i	$-0.776592\pi$
$41$	−9.77943	−1.52729	−0.763645	+	0.645637i	$0.776592\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−3.88971	−0.579844
$46$	0	0
$47$	−4.45071	−0.649203	−0.324602	−	0.945851i	$-0.605230\pi$
$47$	−4.45071	−0.649203	−0.324602	+	0.945851i	$0.605230\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.41108	−0.617674
$52$	0	0
$53$	11.2876	1.55048	0.775239	−	0.631668i	$-0.217629\pi$
$53$	11.2876	1.55048	0.775239	+	0.631668i	$0.217629\pi$
$54$	0	0
$55$	12.4882	1.68390
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	5.77156	0.751394	0.375697	−	0.926743i	$-0.377403\pi$
$59$	5.77156	0.751394	0.375697	+	0.926743i	$0.377403\pi$
$60$	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
$65$	−20.9964	−2.60429
$66$	0	0
$67$	−2.48173	−0.303191	−0.151596	−	0.988443i	$-0.548441\pi$
$67$	−2.48173	−0.303191	−0.151596	+	0.988443i	$0.548441\pi$
$68$	0	0
$69$	3.39794	0.409064
$70$	0	0
$71$	−12.4175	−1.47369	−0.736844	−	0.676063i	$-0.763685\pi$
$71$	−12.4175	−1.47369	−0.736844	+	0.676063i	$0.763685\pi$
$72$	0	0
$73$	−4.08714	−0.478363	−0.239181	−	0.970975i	$-0.576879\pi$
$73$	−4.08714	−0.478363	−0.239181	+	0.970975i	$0.576879\pi$
$74$	0	0
$75$	10.1299	1.16970
$76$	0	0
$77$	3.21056	0.365878
$78$	0	0
$79$	−2.68920	−0.302558	−0.151279	−	0.988491i	$-0.548339\pi$
$79$	−2.68920	−0.302558	−0.151279	+	0.988491i	$0.548339\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.92935	−0.431302	−0.215651	−	0.976470i	$-0.569187\pi$
$83$	−3.92935	−0.431302	−0.215651	+	0.976470i	$0.569187\pi$
$84$	0	0
$85$	17.1578	1.86103
$86$	0	0
$87$	−7.28765	−0.781318
$88$	0	0
$89$	2.70874	0.287126	0.143563	−	0.989641i	$-0.454144\pi$
$89$	2.70874	0.287126	0.143563	+	0.989641i	$0.454144\pi$
$90$	0	0
$91$	−5.39794	−0.565858
$92$	0	0
$93$	−0.187375	−0.0194299
$94$	0	0
$95$	3.88971	0.399076
$96$	0	0
$97$	−0.227011	−0.0230495	−0.0115248	−	0.999934i	$-0.503669\pi$
$97$	−0.227011	−0.0230495	−0.0115248	+	0.999934i	$0.503669\pi$
$98$	0	0
$99$	−3.21056	−0.322674
$100$	0	0
$101$	19.6691	1.95715	0.978576	−	0.205885i	$-0.0660073\pi$
$101$	19.6691	1.95715	0.978576	+	0.205885i	$0.0660073\pi$
$102$	0	0
$103$	14.7219	1.45059	0.725297	−	0.688436i	$-0.241703\pi$
$103$	14.7219	1.45059	0.725297	+	0.688436i	$0.241703\pi$
$104$	0	0
$105$	3.88971	0.379597
$106$	0	0
$107$	−8.95366	−0.865583	−0.432792	−	0.901494i	$-0.642471\pi$
$107$	−8.95366	−0.865583	−0.432792	+	0.901494i	$0.642471\pi$
$108$	0	0
$109$	13.5921	1.30188	0.650941	−	0.759128i	$-0.274374\pi$
$109$	13.5921	1.30188	0.650941	+	0.759128i	$0.274374\pi$
$110$	0	0
$111$	−4.60850	−0.437419
$112$	0	0
$113$	8.35047	0.785546	0.392773	−	0.919635i	$-0.371516\pi$
$113$	8.35047	0.785546	0.392773	+	0.919635i	$0.371516\pi$
$114$	0	0
$115$	−13.2170	−1.23249
$116$	0	0
$117$	5.39794	0.499039
$118$	0	0
$119$	4.41108	0.404363
$120$	0	0
$121$	−0.692290	−0.0629355
$122$	0	0
$123$	−9.77943	−0.881781
$124$	0	0
$125$	−19.9537	−1.78471
$126$	0	0
$127$	5.70565	0.506294	0.253147	−	0.967428i	$-0.418534\pi$
$127$	5.70565	0.506294	0.253147	+	0.967428i	$0.418534\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.3505	0.904325	0.452163	−	0.891936i	$-0.350653\pi$
$131$	10.3505	0.904325	0.452163	+	0.891936i	$0.350653\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−3.88971	−0.334773
$136$	0	0
$137$	16.7959	1.43497	0.717484	−	0.696575i	$-0.245294\pi$
$137$	16.7959	1.43497	0.717484	+	0.696575i	$0.245294\pi$
$138$	0	0
$139$	16.3469	1.38652	0.693261	−	0.720686i	$-0.256173\pi$
$139$	16.3469	1.38652	0.693261	+	0.720686i	$0.256173\pi$
$140$	0	0
$141$	−4.45071	−0.374818
$142$	0	0
$143$	−17.3304	−1.44924
$144$	0	0
$145$	28.3469	2.35408
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	4.82907	0.395613	0.197807	−	0.980241i	$-0.436618\pi$
$149$	4.82907	0.395613	0.197807	+	0.980241i	$0.436618\pi$
$150$	0	0
$151$	−21.2645	−1.73048	−0.865240	−	0.501358i	$-0.832834\pi$
$151$	−21.2645	−1.73048	−0.865240	+	0.501358i	$0.832834\pi$
$152$	0	0
$153$	−4.41108	−0.356614
$154$	0	0
$155$	0.728835	0.0585414
$156$	0	0
$157$	23.1377	1.84659	0.923296	−	0.384090i	$-0.125485\pi$
$157$	23.1377	1.84659	0.923296	+	0.384090i	$0.125485\pi$
$158$	0	0
$159$	11.2876	0.895169
$160$	0	0
$161$	−3.39794	−0.267795
$162$	0	0
$163$	−7.27117	−0.569522	−0.284761	−	0.958599i	$-0.591914\pi$
$163$	−7.27117	−0.569522	−0.284761	+	0.958599i	$0.591914\pi$
$164$	0	0
$165$	12.4882	0.972202
$166$	0	0
$167$	7.12986	0.551725	0.275863	−	0.961197i	$-0.411036\pi$
$167$	7.12986	0.551725	0.275863	+	0.961197i	$0.411036\pi$
$168$	0	0
$169$	16.1377	1.24136
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−3.08349	−0.234433	−0.117217	−	0.993106i	$-0.537397\pi$
$173$	−3.08349	−0.234433	−0.117217	+	0.993106i	$0.537397\pi$
$174$	0	0
$175$	−10.1299	−0.765746
$176$	0	0
$177$	5.77156	0.433817
$178$	0	0
$179$	−13.3669	−0.999091	−0.499545	−	0.866288i	$-0.666500\pi$
$179$	−13.3669	−0.999091	−0.499545	+	0.866288i	$0.666500\pi$
$180$	0	0
$181$	8.15305	0.606012	0.303006	−	0.952989i	$-0.402010\pi$
$181$	8.15305	0.606012	0.303006	+	0.952989i	$0.402010\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	17.9257	1.31793
$186$	0	0
$187$	14.1620	1.03563
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	15.5722	1.12677	0.563383	−	0.826196i	$-0.309500\pi$
$191$	15.5722	1.12677	0.563383	+	0.826196i	$0.309500\pi$
$192$	0	0
$193$	8.95727	0.644759	0.322379	−	0.946611i	$-0.395517\pi$
$193$	8.95727	0.644759	0.322379	+	0.946611i	$0.395517\pi$
$194$	0	0
$195$	−20.9964	−1.50359
$196$	0	0
$197$	4.52923	0.322694	0.161347	−	0.986898i	$-0.448416\pi$
$197$	4.52923	0.322694	0.161347	+	0.986898i	$0.448416\pi$
$198$	0	0
$199$	6.08714	0.431506	0.215753	−	0.976448i	$-0.430779\pi$
$199$	6.08714	0.431506	0.215753	+	0.976448i	$0.430779\pi$
$200$	0	0
$201$	−2.48173	−0.175048
$202$	0	0
$203$	7.28765	0.511493
$204$	0	0
$205$	38.0392	2.65677
$206$	0	0
$207$	3.39794	0.236173
$208$	0	0
$209$	3.21056	0.222079
$210$	0	0
$211$	−23.5574	−1.62176	−0.810880	−	0.585212i	$-0.801011\pi$
$211$	−23.5574	−1.62176	−0.810880	+	0.585212i	$0.801011\pi$
$212$	0	0
$213$	−12.4175	−0.850834
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.187375	0.0127198
$218$	0	0
$219$	−4.08714	−0.276183
$220$	0	0
$221$	−23.8107	−1.60168
$222$	0	0
$223$	−7.27648	−0.487269	−0.243635	−	0.969867i	$-0.578340\pi$
$223$	−7.27648	−0.487269	−0.243635	+	0.969867i	$0.578340\pi$
$224$	0	0
$225$	10.1299	0.675324
$226$	0	0
$227$	10.4619	0.694380	0.347190	−	0.937795i	$-0.387136\pi$
$227$	10.4619	0.694380	0.347190	+	0.937795i	$0.387136\pi$
$228$	0	0
$229$	−4.20055	−0.277580	−0.138790	−	0.990322i	$-0.544321\pi$
$229$	−4.20055	−0.277580	−0.138790	+	0.990322i	$0.544321\pi$
$230$	0	0
$231$	3.21056	0.211239
$232$	0	0
$233$	−1.40468	−0.0920233	−0.0460117	−	0.998941i	$-0.514651\pi$
$233$	−1.40468	−0.0920233	−0.0460117	+	0.998941i	$0.514651\pi$
$234$	0	0
$235$	17.3120	1.12931
$236$	0	0
$237$	−2.68920	−0.174682
$238$	0	0
$239$	20.2994	1.31306	0.656528	−	0.754301i	$-0.272024\pi$
$239$	20.2994	1.31306	0.656528	+	0.754301i	$0.272024\pi$
$240$	0	0
$241$	−23.6245	−1.52179	−0.760893	−	0.648878i	$-0.775239\pi$
$241$	−23.6245	−1.52179	−0.760893	+	0.648878i	$0.775239\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.88971	−0.248505
$246$	0	0
$247$	−5.39794	−0.343463
$248$	0	0
$249$	−3.92935	−0.249012
$250$	0	0
$251$	−1.82380	−0.115117	−0.0575585	−	0.998342i	$-0.518332\pi$
$251$	−1.82380	−0.115117	−0.0575585	+	0.998342i	$0.518332\pi$
$252$	0	0
$253$	−10.9093	−0.685861
$254$	0	0
$255$	17.1578	1.07446
$256$	0	0
$257$	−6.72883	−0.419733	−0.209867	−	0.977730i	$-0.567303\pi$
$257$	−6.72883	−0.419733	−0.209867	+	0.977730i	$0.567303\pi$
$258$	0	0
$259$	4.60850	0.286358
$260$	0	0
$261$	−7.28765	−0.451094
$262$	0	0
$263$	13.4371	0.828566	0.414283	−	0.910148i	$-0.364032\pi$
$263$	13.4371	0.828566	0.414283	+	0.910148i	$0.364032\pi$
$264$	0	0
$265$	−43.9057	−2.69711
$266$	0	0
$267$	2.70874	0.165772
$268$	0	0
$269$	15.8666	0.967401	0.483701	−	0.875234i	$-0.339292\pi$
$269$	15.8666	0.967401	0.483701	+	0.875234i	$0.339292\pi$
$270$	0	0
$271$	0.667979	0.0405768	0.0202884	−	0.999794i	$-0.493542\pi$
$271$	0.667979	0.0405768	0.0202884	+	0.999794i	$0.493542\pi$
$272$	0	0
$273$	−5.39794	−0.326698
$274$	0	0
$275$	−32.5226	−1.96118
$276$	0	0
$277$	24.8501	1.49310	0.746549	−	0.665330i	$-0.231709\pi$
$277$	24.8501	1.49310	0.746549	+	0.665330i	$0.231709\pi$
$278$	0	0
$279$	−0.187375	−0.0112178
$280$	0	0
$281$	11.0671	0.660206	0.330103	−	0.943945i	$-0.392916\pi$
$281$	11.0671	0.660206	0.330103	+	0.943945i	$0.392916\pi$
$282$	0	0
$283$	29.0306	1.72569	0.862844	−	0.505470i	$-0.168681\pi$
$283$	29.0306	1.72569	0.862844	+	0.505470i	$0.168681\pi$
$284$	0	0
$285$	3.88971	0.230407
$286$	0	0
$287$	9.77943	0.577261
$288$	0	0
$289$	2.45760	0.144565
$290$	0	0
$291$	−0.227011	−0.0133076
$292$	0	0
$293$	10.8830	0.635792	0.317896	−	0.948126i	$-0.397024\pi$
$293$	10.8830	0.635792	0.317896	+	0.948126i	$0.397024\pi$
$294$	0	0
$295$	−22.4497	−1.30707
$296$	0	0
$297$	−3.21056	−0.186296
$298$	0	0
$299$	18.3419	1.06074
$300$	0	0
$301$	0	0
$302$	0	0
$303$	19.6691	1.12996
$304$	0	0
$305$	−23.3383	−1.33635
$306$	0	0
$307$	−9.77156	−0.557693	−0.278846	−	0.960336i	$-0.589952\pi$
$307$	−9.77156	−0.557693	−0.278846	+	0.960336i	$0.589952\pi$
$308$	0	0
$309$	14.7219	0.837500
$310$	0	0
$311$	−26.4307	−1.49875	−0.749373	−	0.662148i	$-0.769645\pi$
$311$	−26.4307	−1.49875	−0.749373	+	0.662148i	$0.769645\pi$
$312$	0	0
$313$	15.7873	0.892350	0.446175	−	0.894946i	$-0.352786\pi$
$313$	15.7873	0.892350	0.446175	+	0.894946i	$0.352786\pi$
$314$	0	0
$315$	3.88971	0.219160
$316$	0	0
$317$	24.1707	1.35756	0.678780	−	0.734342i	$-0.262509\pi$
$317$	24.1707	1.35756	0.678780	+	0.734342i	$0.262509\pi$
$318$	0	0
$319$	23.3975	1.31001
$320$	0	0
$321$	−8.95366	−0.499745
$322$	0	0
$323$	4.41108	0.245439
$324$	0	0
$325$	54.6804	3.03312
$326$	0	0
$327$	13.5921	0.751642
$328$	0	0
$329$	4.45071	0.245376
$330$	0	0
$331$	7.32398	0.402562	0.201281	−	0.979534i	$-0.435490\pi$
$331$	7.32398	0.402562	0.201281	+	0.979534i	$0.435490\pi$
$332$	0	0
$333$	−4.60850	−0.252544
$334$	0	0
$335$	9.65321	0.527411
$336$	0	0
$337$	−8.73670	−0.475918	−0.237959	−	0.971275i	$-0.576478\pi$
$337$	−8.73670	−0.475918	−0.237959	+	0.971275i	$0.576478\pi$
$338$	0	0
$339$	8.35047	0.453535
$340$	0	0
$341$	0.601579	0.0325773
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−13.2170	−0.711579
$346$	0	0
$347$	13.1179	0.704205	0.352102	−	0.935961i	$-0.385467\pi$
$347$	13.1179	0.704205	0.352102	+	0.935961i	$0.385467\pi$
$348$	0	0
$349$	−9.08910	−0.486529	−0.243264	−	0.969960i	$-0.578218\pi$
$349$	−9.08910	−0.486529	−0.243264	+	0.969960i	$0.578218\pi$
$350$	0	0
$351$	5.39794	0.288121
$352$	0	0
$353$	35.7494	1.90275	0.951373	−	0.308041i	$-0.0996735\pi$
$353$	35.7494	1.90275	0.951373	+	0.308041i	$0.0996735\pi$
$354$	0	0
$355$	48.3006	2.56353
$356$	0	0
$357$	4.41108	0.233459
$358$	0	0
$359$	20.5357	1.08383	0.541915	−	0.840433i	$-0.317699\pi$
$359$	20.5357	1.08383	0.541915	+	0.840433i	$0.317699\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.692290	−0.0363358
$364$	0	0
$365$	15.8978	0.832128
$366$	0	0
$367$	−7.85870	−0.410221	−0.205110	−	0.978739i	$-0.565755\pi$
$367$	−7.85870	−0.410221	−0.205110	+	0.978739i	$0.565755\pi$
$368$	0	0
$369$	−9.77943	−0.509097
$370$	0	0
$371$	−11.2876	−0.586026
$372$	0	0
$373$	17.8126	0.922303	0.461151	−	0.887321i	$-0.347437\pi$
$373$	17.8126	0.922303	0.461151	+	0.887321i	$0.347437\pi$
$374$	0	0
$375$	−19.9537	−1.03040
$376$	0	0
$377$	−39.3383	−2.02602
$378$	0	0
$379$	1.06425	0.0546668	0.0273334	−	0.999626i	$-0.491298\pi$
$379$	1.06425	0.0546668	0.0273334	+	0.999626i	$0.491298\pi$
$380$	0	0
$381$	5.70565	0.292309
$382$	0	0
$383$	−20.7009	−1.05777	−0.528884	−	0.848694i	$-0.677390\pi$
$383$	−20.7009	−1.05777	−0.528884	+	0.848694i	$0.677390\pi$
$384$	0	0
$385$	−12.4882	−0.636456
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.0260	1.62379	0.811893	−	0.583807i	$-0.198437\pi$
$389$	32.0260	1.62379	0.811893	+	0.583807i	$0.198437\pi$
$390$	0	0
$391$	−14.9886	−0.758004
$392$	0	0
$393$	10.3505	0.522112
$394$	0	0
$395$	10.4602	0.526310
$396$	0	0
$397$	17.6381	0.885232	0.442616	−	0.896711i	$-0.354050\pi$
$397$	17.6381	0.885232	0.442616	+	0.896711i	$0.354050\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−3.23341	−0.161469	−0.0807345	−	0.996736i	$-0.525727\pi$
$401$	−3.23341	−0.161469	−0.0807345	+	0.996736i	$0.525727\pi$
$402$	0	0
$403$	−1.01144	−0.0503833
$404$	0	0
$405$	−3.88971	−0.193281
$406$	0	0
$407$	14.7959	0.733404
$408$	0	0
$409$	−37.9523	−1.87662	−0.938309	−	0.345797i	$-0.887609\pi$
$409$	−37.9523	−1.87662	−0.938309	+	0.345797i	$0.887609\pi$
$410$	0	0
$411$	16.7959	0.828479
$412$	0	0
$413$	−5.77156	−0.284000
$414$	0	0
$415$	15.2840	0.750264
$416$	0	0
$417$	16.3469	0.800509
$418$	0	0
$419$	−27.7016	−1.35331	−0.676655	−	0.736301i	$-0.736571\pi$
$419$	−27.7016	−1.35331	−0.676655	+	0.736301i	$0.736571\pi$
$420$	0	0
$421$	13.8054	0.672834	0.336417	−	0.941713i	$-0.390785\pi$
$421$	13.8054	0.672834	0.336417	+	0.941713i	$0.390785\pi$
$422$	0	0
$423$	−4.45071	−0.216401
$424$	0	0
$425$	−44.6836	−2.16747
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	−17.3304	−0.836720
$430$	0	0
$431$	9.40106	0.452833	0.226417	−	0.974031i	$-0.427299\pi$
$431$	9.40106	0.452833	0.226417	+	0.974031i	$0.427299\pi$
$432$	0	0
$433$	36.9236	1.77443	0.887217	−	0.461352i	$-0.152635\pi$
$433$	36.9236	1.77443	0.887217	+	0.461352i	$0.152635\pi$
$434$	0	0
$435$	28.3469	1.35913
$436$	0	0
$437$	−3.39794	−0.162545
$438$	0	0
$439$	−10.9040	−0.520418	−0.260209	−	0.965552i	$-0.583792\pi$
$439$	−10.9040	−0.520418	−0.260209	+	0.965552i	$0.583792\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−33.1704	−1.57598	−0.787988	−	0.615691i	$-0.788877\pi$
$443$	−33.1704	−1.57598	−0.787988	+	0.615691i	$0.788877\pi$
$444$	0	0
$445$	−10.5362	−0.499465
$446$	0	0
$447$	4.82907	0.228407
$448$	0	0
$449$	−28.3711	−1.33892	−0.669458	−	0.742850i	$-0.733474\pi$
$449$	−28.3711	−1.33892	−0.669458	+	0.742850i	$0.733474\pi$
$450$	0	0
$451$	31.3975	1.47845
$452$	0	0
$453$	−21.2645	−0.999093
$454$	0	0
$455$	20.9964	0.984328
$456$	0	0
$457$	−1.70015	−0.0795298	−0.0397649	−	0.999209i	$-0.512661\pi$
$457$	−1.70015	−0.0795298	−0.0397649	+	0.999209i	$0.512661\pi$
$458$	0	0
$459$	−4.41108	−0.205891
$460$	0	0
$461$	−21.7221	−1.01170	−0.505850	−	0.862621i	$-0.668821\pi$
$461$	−21.7221	−1.01170	−0.505850	+	0.862621i	$0.668821\pi$
$462$	0	0
$463$	−13.8186	−0.642204	−0.321102	−	0.947045i	$-0.604053\pi$
$463$	−13.8186	−0.642204	−0.321102	+	0.947045i	$0.604053\pi$
$464$	0	0
$465$	0.728835	0.0337989
$466$	0	0
$467$	−33.4481	−1.54779	−0.773896	−	0.633312i	$-0.781695\pi$
$467$	−33.4481	−1.54779	−0.773896	+	0.633312i	$0.781695\pi$
$468$	0	0
$469$	2.48173	0.114596
$470$	0	0
$471$	23.1377	1.06613
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−10.1299	−0.464790
$476$	0	0
$477$	11.2876	0.516826
$478$	0	0
$479$	18.2039	0.831756	0.415878	−	0.909421i	$-0.363474\pi$
$479$	18.2039	0.831756	0.415878	+	0.909421i	$0.363474\pi$
$480$	0	0
$481$	−24.8764	−1.13427
$482$	0	0
$483$	−3.39794	−0.154611
$484$	0	0
$485$	0.883009	0.0400954
$486$	0	0
$487$	5.03105	0.227979	0.113989	−	0.993482i	$-0.463637\pi$
$487$	5.03105	0.227979	0.113989	+	0.993482i	$0.463637\pi$
$488$	0	0
$489$	−7.27117	−0.328813
$490$	0	0
$491$	−10.3063	−0.465116	−0.232558	−	0.972583i	$-0.574710\pi$
$491$	−10.3063	−0.465116	−0.232558	+	0.972583i	$0.574710\pi$
$492$	0	0
$493$	32.1464	1.44780
$494$	0	0
$495$	12.4882	0.561301
$496$	0	0
$497$	12.4175	0.557002
$498$	0	0
$499$	−40.3006	−1.80410	−0.902050	−	0.431631i	$-0.857938\pi$
$499$	−40.3006	−1.80410	−0.902050	+	0.431631i	$0.857938\pi$
$500$	0	0
$501$	7.12986	0.318539
$502$	0	0
$503$	−10.7133	−0.477682	−0.238841	−	0.971059i	$-0.576767\pi$
$503$	−10.7133	−0.477682	−0.238841	+	0.971059i	$0.576767\pi$
$504$	0	0
$505$	−76.5073	−3.40453
$506$	0	0
$507$	16.1377	0.716702
$508$	0	0
$509$	24.8429	1.10114	0.550571	−	0.834788i	$-0.314410\pi$
$509$	24.8429	1.10114	0.550571	+	0.834788i	$0.314410\pi$
$510$	0	0
$511$	4.08714	0.180804
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−57.2640	−2.52335
$516$	0	0
$517$	14.2893	0.628442
$518$	0	0
$519$	−3.08349	−0.135350
$520$	0	0
$521$	20.0934	0.880307	0.440154	−	0.897922i	$-0.354924\pi$
$521$	20.0934	0.880307	0.440154	+	0.897922i	$0.354924\pi$
$522$	0	0
$523$	0.313894	0.0137256	0.00686281	−	0.999976i	$-0.497815\pi$
$523$	0.313894	0.0137256	0.00686281	+	0.999976i	$0.497815\pi$
$524$	0	0
$525$	−10.1299	−0.442103
$526$	0	0
$527$	0.826525	0.0360040
$528$	0	0
$529$	−11.4540	−0.498001
$530$	0	0
$531$	5.77156	0.250465
$532$	0	0
$533$	−52.7887	−2.28653
$534$	0	0
$535$	34.8272	1.50571
$536$	0	0
$537$	−13.3669	−0.576825
$538$	0	0
$539$	−3.21056	−0.138289
$540$	0	0
$541$	−28.1749	−1.21133	−0.605667	−	0.795718i	$-0.707094\pi$
$541$	−28.1749	−1.21133	−0.605667	+	0.795718i	$0.707094\pi$
$542$	0	0
$543$	8.15305	0.349881
$544$	0	0
$545$	−52.8692	−2.26467
$546$	0	0
$547$	−9.89928	−0.423262	−0.211631	−	0.977350i	$-0.567878\pi$
$547$	−9.89928	−0.423262	−0.211631	+	0.977350i	$0.567878\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	7.28765	0.310464
$552$	0	0
$553$	2.68920	0.114356
$554$	0	0
$555$	17.9257	0.760905
$556$	0	0
$557$	26.6477	1.12910	0.564549	−	0.825400i	$-0.309050\pi$
$557$	26.6477	1.12910	0.564549	+	0.825400i	$0.309050\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	14.1620	0.597922
$562$	0	0
$563$	26.0871	1.09944	0.549721	−	0.835348i	$-0.314734\pi$
$563$	26.0871	1.09944	0.549721	+	0.835348i	$0.314734\pi$
$564$	0	0
$565$	−32.4809	−1.36648
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−41.1726	−1.72605	−0.863023	−	0.505164i	$-0.831432\pi$
$569$	−41.1726	−1.72605	−0.863023	+	0.505164i	$0.831432\pi$
$570$	0	0
$571$	−29.7052	−1.24312	−0.621561	−	0.783366i	$-0.713501\pi$
$571$	−29.7052	−1.24312	−0.621561	+	0.783366i	$0.713501\pi$
$572$	0	0
$573$	15.5722	0.650538
$574$	0	0
$575$	34.4206	1.43544
$576$	0	0
$577$	−20.8094	−0.866305	−0.433152	−	0.901321i	$-0.642599\pi$
$577$	−20.8094	−0.866305	−0.433152	+	0.901321i	$0.642599\pi$
$578$	0	0
$579$	8.95727	0.372251
$580$	0	0
$581$	3.92935	0.163017
$582$	0	0
$583$	−36.2397	−1.50090
$584$	0	0
$585$	−20.9964	−0.868095
$586$	0	0
$587$	15.6887	0.647541	0.323771	−	0.946136i	$-0.395049\pi$
$587$	15.6887	0.647541	0.323771	+	0.946136i	$0.395049\pi$
$588$	0	0
$589$	0.187375	0.00772065
$590$	0	0
$591$	4.52923	0.186307
$592$	0	0
$593$	12.8786	0.528860	0.264430	−	0.964405i	$-0.414816\pi$
$593$	12.8786	0.528860	0.264430	+	0.964405i	$0.414816\pi$
$594$	0	0
$595$	−17.1578	−0.703402
$596$	0	0
$597$	6.08714	0.249130
$598$	0	0
$599$	−39.2196	−1.60247	−0.801236	−	0.598349i	$-0.795824\pi$
$599$	−39.2196	−1.60247	−0.801236	+	0.598349i	$0.795824\pi$
$600$	0	0
$601$	44.7757	1.82644	0.913219	−	0.407470i	$-0.133589\pi$
$601$	44.7757	1.82644	0.913219	+	0.407470i	$0.133589\pi$
$602$	0	0
$603$	−2.48173	−0.101064
$604$	0	0
$605$	2.69281	0.109478
$606$	0	0
$607$	21.5842	0.876075	0.438038	−	0.898957i	$-0.355674\pi$
$607$	21.5842	0.876075	0.438038	+	0.898957i	$0.355674\pi$
$608$	0	0
$609$	7.28765	0.295310
$610$	0	0
$611$	−24.0247	−0.971934
$612$	0	0
$613$	8.91651	0.360134	0.180067	−	0.983654i	$-0.442368\pi$
$613$	8.91651	0.360134	0.180067	+	0.983654i	$0.442368\pi$
$614$	0	0
$615$	38.0392	1.53389
$616$	0	0
$617$	7.97997	0.321262	0.160631	−	0.987015i	$-0.448647\pi$
$617$	7.97997	0.321262	0.160631	+	0.987015i	$0.448647\pi$
$618$	0	0
$619$	48.9484	1.96740	0.983702	−	0.179807i	$-0.0575472\pi$
$619$	48.9484	1.96740	0.983702	+	0.179807i	$0.0575472\pi$
$620$	0	0
$621$	3.39794	0.136355
$622$	0	0
$623$	−2.70874	−0.108523
$624$	0	0
$625$	26.9648	1.07859
$626$	0	0
$627$	3.21056	0.128218
$628$	0	0
$629$	20.3284	0.810548
$630$	0	0
$631$	−17.7873	−0.708101	−0.354050	−	0.935226i	$-0.615196\pi$
$631$	−17.7873	−0.708101	−0.354050	+	0.935226i	$0.615196\pi$
$632$	0	0
$633$	−23.5574	−0.936324
$634$	0	0
$635$	−22.1933	−0.880715
$636$	0	0
$637$	5.39794	0.213874
$638$	0	0
$639$	−12.4175	−0.491229
$640$	0	0
$641$	−47.3548	−1.87040	−0.935200	−	0.354119i	$-0.884781\pi$
$641$	−47.3548	−1.87040	−0.935200	+	0.354119i	$0.884781\pi$
$642$	0	0
$643$	−31.6180	−1.24689	−0.623447	−	0.781866i	$-0.714268\pi$
$643$	−31.6180	−1.24689	−0.623447	+	0.781866i	$0.714268\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−36.8541	−1.44888	−0.724441	−	0.689336i	$-0.757902\pi$
$647$	−36.8541	−1.44888	−0.724441	+	0.689336i	$0.757902\pi$
$648$	0	0
$649$	−18.5300	−0.727365
$650$	0	0
$651$	0.187375	0.00734380
$652$	0	0
$653$	−17.2239	−0.674024	−0.337012	−	0.941500i	$-0.609416\pi$
$653$	−17.2239	−0.674024	−0.337012	+	0.941500i	$0.609416\pi$
$654$	0	0
$655$	−40.2604	−1.57310
$656$	0	0
$657$	−4.08714	−0.159454
$658$	0	0
$659$	45.2939	1.76440	0.882200	−	0.470875i	$-0.156062\pi$
$659$	45.2939	1.76440	0.882200	+	0.470875i	$0.156062\pi$
$660$	0	0
$661$	29.7862	1.15855	0.579274	−	0.815133i	$-0.303336\pi$
$661$	29.7862	1.15855	0.579274	+	0.815133i	$0.303336\pi$
$662$	0	0
$663$	−23.8107	−0.924732
$664$	0	0
$665$	−3.88971	−0.150837
$666$	0	0
$667$	−24.7630	−0.958826
$668$	0	0
$669$	−7.27648	−0.281325
$670$	0	0
$671$	−19.2634	−0.743654
$672$	0	0
$673$	12.3390	0.475634	0.237817	−	0.971310i	$-0.423568\pi$
$673$	12.3390	0.475634	0.237817	+	0.971310i	$0.423568\pi$
$674$	0	0
$675$	10.1299	0.389899
$676$	0	0
$677$	−22.1134	−0.849888	−0.424944	−	0.905220i	$-0.639706\pi$
$677$	−22.1134	−0.849888	−0.424944	+	0.905220i	$0.639706\pi$
$678$	0	0
$679$	0.227011	0.00871190
$680$	0	0
$681$	10.4619	0.400900
$682$	0	0
$683$	8.10362	0.310076	0.155038	−	0.987908i	$-0.450450\pi$
$683$	8.10362	0.310076	0.155038	+	0.987908i	$0.450450\pi$
$684$	0	0
$685$	−65.3311	−2.49617
$686$	0	0
$687$	−4.20055	−0.160261
$688$	0	0
$689$	60.9300	2.32125
$690$	0	0
$691$	−6.94343	−0.264141	−0.132070	−	0.991240i	$-0.542162\pi$
$691$	−6.94343	−0.264141	−0.132070	+	0.991240i	$0.542162\pi$
$692$	0	0
$693$	3.21056	0.121959
$694$	0	0
$695$	−63.5846	−2.41190
$696$	0	0
$697$	43.1378	1.63396
$698$	0	0
$699$	−1.40468	−0.0531297
$700$	0	0
$701$	−21.5329	−0.813285	−0.406643	−	0.913587i	$-0.633301\pi$
$701$	−21.5329	−0.813285	−0.406643	+	0.913587i	$0.633301\pi$
$702$	0	0
$703$	4.60850	0.173813
$704$	0	0
$705$	17.3120	0.652007
$706$	0	0
$707$	−19.6691	−0.739734
$708$	0	0
$709$	−38.7686	−1.45599	−0.727993	−	0.685584i	$-0.759547\pi$
$709$	−38.7686	−1.45599	−0.727993	+	0.685584i	$0.759547\pi$
$710$	0	0
$711$	−2.68920	−0.100853
$712$	0	0
$713$	−0.636688	−0.0238442
$714$	0	0
$715$	67.4103	2.52100
$716$	0	0
$717$	20.2994	0.758094
$718$	0	0
$719$	36.7725	1.37138	0.685692	−	0.727892i	$-0.259500\pi$
$719$	36.7725	1.37138	0.685692	+	0.727892i	$0.259500\pi$
$720$	0	0
$721$	−14.7219	−0.548273
$722$	0	0
$723$	−23.6245	−0.878603
$724$	0	0
$725$	−73.8229	−2.74171
$726$	0	0
$727$	−21.2853	−0.789427	−0.394714	−	0.918804i	$-0.629156\pi$
$727$	−21.2853	−0.789427	−0.394714	+	0.918804i	$0.629156\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	6.13845	0.226729	0.113364	−	0.993553i	$-0.463837\pi$
$733$	6.13845	0.226729	0.113364	+	0.993553i	$0.463837\pi$
$734$	0	0
$735$	−3.88971	−0.143474
$736$	0	0
$737$	7.96774	0.293496
$738$	0	0
$739$	−7.89946	−0.290586	−0.145293	−	0.989389i	$-0.546413\pi$
$739$	−7.89946	−0.290586	−0.145293	+	0.989389i	$0.546413\pi$
$740$	0	0
$741$	−5.39794	−0.198298
$742$	0	0
$743$	−7.80306	−0.286267	−0.143133	−	0.989703i	$-0.545718\pi$
$743$	−7.80306	−0.286267	−0.143133	+	0.989703i	$0.545718\pi$
$744$	0	0
$745$	−18.7837	−0.688182
$746$	0	0
$747$	−3.92935	−0.143767
$748$	0	0
$749$	8.95366	0.327160
$750$	0	0
$751$	−4.19505	−0.153080	−0.0765399	−	0.997067i	$-0.524387\pi$
$751$	−4.19505	−0.153080	−0.0765399	+	0.997067i	$0.524387\pi$
$752$	0	0
$753$	−1.82380	−0.0664628
$754$	0	0
$755$	82.7128	3.01023
$756$	0	0
$757$	−53.7036	−1.95189	−0.975944	−	0.218019i	$-0.930040\pi$
$757$	−53.7036	−1.95189	−0.975944	+	0.218019i	$0.930040\pi$
$758$	0	0
$759$	−10.9093	−0.395982
$760$	0	0
$761$	−9.55602	−0.346406	−0.173203	−	0.984886i	$-0.555412\pi$
$761$	−9.55602	−0.346406	−0.173203	+	0.984886i	$0.555412\pi$
$762$	0	0
$763$	−13.5921	−0.492065
$764$	0	0
$765$	17.1578	0.620342
$766$	0	0
$767$	31.1545	1.12493
$768$	0	0
$769$	39.1819	1.41294	0.706468	−	0.707745i	$-0.250288\pi$
$769$	39.1819	1.41294	0.706468	+	0.707745i	$0.250288\pi$
$770$	0	0
$771$	−6.72883	−0.242333
$772$	0	0
$773$	24.0079	0.863503	0.431751	−	0.901993i	$-0.357896\pi$
$773$	24.0079	0.863503	0.431751	+	0.901993i	$0.357896\pi$
$774$	0	0
$775$	−1.89808	−0.0681811
$776$	0	0
$777$	4.60850	0.165329
$778$	0	0
$779$	9.77943	0.350384
$780$	0	0
$781$	39.8672	1.42656
$782$	0	0
$783$	−7.28765	−0.260439
$784$	0	0
$785$	−89.9991	−3.21221
$786$	0	0
$787$	31.5795	1.12569	0.562844	−	0.826563i	$-0.309707\pi$
$787$	31.5795	1.12569	0.562844	+	0.826563i	$0.309707\pi$
$788$	0	0
$789$	13.4371	0.478373
$790$	0	0
$791$	−8.35047	−0.296909
$792$	0	0
$793$	32.3876	1.15012
$794$	0	0
$795$	−43.9057	−1.55718
$796$	0	0
$797$	−43.9415	−1.55649	−0.778243	−	0.627963i	$-0.783889\pi$
$797$	−43.9415	−1.55649	−0.778243	+	0.627963i	$0.783889\pi$
$798$	0	0
$799$	19.6324	0.694546
$800$	0	0
$801$	2.70874	0.0957086
$802$	0	0
$803$	13.1220	0.463065
$804$	0	0
$805$	13.2170	0.465838
$806$	0	0
$807$	15.8666	0.558529
$808$	0	0
$809$	29.9862	1.05426	0.527129	−	0.849786i	$-0.323269\pi$
$809$	29.9862	1.05426	0.527129	+	0.849786i	$0.323269\pi$
$810$	0	0
$811$	43.2495	1.51870	0.759348	−	0.650684i	$-0.225518\pi$
$811$	43.2495	1.51870	0.759348	+	0.650684i	$0.225518\pi$
$812$	0	0
$813$	0.667979	0.0234270
$814$	0	0
$815$	28.2827	0.990701
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−5.39794	−0.188619
$820$	0	0
$821$	26.7141	0.932327	0.466163	−	0.884699i	$-0.345636\pi$
$821$	26.7141	0.932327	0.466163	+	0.884699i	$0.345636\pi$
$822$	0	0
$823$	34.3860	1.19862	0.599311	−	0.800517i	$-0.295441\pi$
$823$	34.3860	1.19862	0.599311	+	0.800517i	$0.295441\pi$
$824$	0	0
$825$	−32.5226	−1.13229
$826$	0	0
$827$	44.9941	1.56460	0.782298	−	0.622904i	$-0.214047\pi$
$827$	44.9941	1.56460	0.782298	+	0.622904i	$0.214047\pi$
$828$	0	0
$829$	18.4814	0.641886	0.320943	−	0.947098i	$-0.396000\pi$
$829$	18.4814	0.641886	0.320943	+	0.947098i	$0.396000\pi$
$830$	0	0
$831$	24.8501	0.862041
$832$	0	0
$833$	−4.41108	−0.152835
$834$	0	0
$835$	−27.7331	−0.959744
$836$	0	0
$837$	−0.187375	−0.00647662
$838$	0	0
$839$	3.79062	0.130867	0.0654334	−	0.997857i	$-0.479157\pi$
$839$	3.79062	0.130867	0.0654334	+	0.997857i	$0.479157\pi$
$840$	0	0
$841$	24.1098	0.831374
$842$	0	0
$843$	11.0671	0.381170
$844$	0	0
$845$	−62.7711	−2.15939
$846$	0	0
$847$	0.692290	0.0237874
$848$	0	0
$849$	29.0306	0.996326
$850$	0	0
$851$	−15.6594	−0.536797
$852$	0	0
$853$	22.2268	0.761032	0.380516	−	0.924774i	$-0.375746\pi$
$853$	22.2268	0.761032	0.380516	+	0.924774i	$0.375746\pi$
$854$	0	0
$855$	3.88971	0.133025
$856$	0	0
$857$	47.8593	1.63484	0.817422	−	0.576039i	$-0.195403\pi$
$857$	47.8593	1.63484	0.817422	+	0.576039i	$0.195403\pi$
$858$	0	0
$859$	−45.1825	−1.54161	−0.770803	−	0.637073i	$-0.780145\pi$
$859$	−45.1825	−1.54161	−0.770803	+	0.637073i	$0.780145\pi$
$860$	0	0
$861$	9.77943	0.333282
$862$	0	0
$863$	57.7815	1.96690	0.983452	−	0.181169i	$-0.0579881\pi$
$863$	57.7815	1.96690	0.983452	+	0.181169i	$0.0579881\pi$
$864$	0	0
$865$	11.9939	0.407804
$866$	0	0
$867$	2.45760	0.0834644
$868$	0	0
$869$	8.63384	0.292883
$870$	0	0
$871$	−13.3962	−0.453913
$872$	0	0
$873$	−0.227011	−0.00768318
$874$	0	0
$875$	19.9537	0.674558
$876$	0	0
$877$	4.69062	0.158391	0.0791955	−	0.996859i	$-0.474765\pi$
$877$	4.69062	0.158391	0.0791955	+	0.996859i	$0.474765\pi$
$878$	0	0
$879$	10.8830	0.367075
$880$	0	0
$881$	31.6008	1.06466	0.532329	−	0.846537i	$-0.321317\pi$
$881$	31.6008	1.06466	0.532329	+	0.846537i	$0.321317\pi$
$882$	0	0
$883$	53.3294	1.79468	0.897339	−	0.441341i	$-0.145497\pi$
$883$	53.3294	1.79468	0.897339	+	0.441341i	$0.145497\pi$
$884$	0	0
$885$	−22.4497	−0.754639
$886$	0	0
$887$	54.7883	1.83961	0.919805	−	0.392375i	$-0.128346\pi$
$887$	54.7883	1.83961	0.919805	+	0.392375i	$0.128346\pi$
$888$	0	0
$889$	−5.70565	−0.191361
$890$	0	0
$891$	−3.21056	−0.107558
$892$	0	0
$893$	4.45071	0.148937
$894$	0	0
$895$	51.9935	1.73795
$896$	0	0
$897$	18.3419	0.612417
$898$	0	0
$899$	1.36552	0.0455427
$900$	0	0
$901$	−49.7907	−1.65877
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.7130	−1.05418
$906$	0	0
$907$	33.2313	1.10343	0.551714	−	0.834034i	$-0.313974\pi$
$907$	33.2313	1.10343	0.551714	+	0.834034i	$0.313974\pi$
$908$	0	0
$909$	19.6691	0.652384
$910$	0	0
$911$	−55.8193	−1.84938	−0.924689	−	0.380724i	$-0.875675\pi$
$911$	−55.8193	−1.84938	−0.924689	+	0.380724i	$0.875675\pi$
$912$	0	0
$913$	12.6154	0.417509
$914$	0	0
$915$	−23.3383	−0.771540
$916$	0	0
$917$	−10.3505	−0.341803
$918$	0	0
$919$	24.7495	0.816411	0.408205	−	0.912890i	$-0.366155\pi$
$919$	24.7495	0.816411	0.408205	+	0.912890i	$0.366155\pi$
$920$	0	0
$921$	−9.77156	−0.321984
$922$	0	0
$923$	−67.0290	−2.20629
$924$	0	0
$925$	−46.6835	−1.53494
$926$	0	0
$927$	14.7219	0.483531
$928$	0	0
$929$	21.4232	0.702873	0.351437	−	0.936212i	$-0.385693\pi$
$929$	21.4232	0.702873	0.351437	+	0.936212i	$0.385693\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−26.4307	−0.865302
$934$	0	0
$935$	−55.0863	−1.80151
$936$	0	0
$937$	−8.68246	−0.283644	−0.141822	−	0.989892i	$-0.545296\pi$
$937$	−8.68246	−0.283644	−0.141822	+	0.989892i	$0.545296\pi$
$938$	0	0
$939$	15.7873	0.515199
$940$	0	0
$941$	−1.10977	−0.0361774	−0.0180887	−	0.999836i	$-0.505758\pi$
$941$	−1.10977	−0.0361774	−0.0180887	+	0.999836i	$0.505758\pi$
$942$	0	0
$943$	−33.2299	−1.08211
$944$	0	0
$945$	3.88971	0.126532
$946$	0	0
$947$	−24.5890	−0.799034	−0.399517	−	0.916726i	$-0.630822\pi$
$947$	−24.5890	−0.799034	−0.399517	+	0.916726i	$0.630822\pi$
$948$	0	0
$949$	−22.0621	−0.716166
$950$	0	0
$951$	24.1707	0.783787
$952$	0	0
$953$	−52.5773	−1.70315	−0.851573	−	0.524236i	$-0.824351\pi$
$953$	−52.5773	−1.70315	−0.851573	+	0.524236i	$0.824351\pi$
$954$	0	0
$955$	−60.5714	−1.96004
$956$	0	0
$957$	23.3975	0.756332
$958$	0	0
$959$	−16.7959	−0.542367
$960$	0	0
$961$	−30.9649	−0.998867
$962$	0	0
$963$	−8.95366	−0.288528
$964$	0	0
$965$	−34.8412	−1.12158
$966$	0	0
$967$	17.7190	0.569805	0.284902	−	0.958557i	$-0.408039\pi$
$967$	17.7190	0.569805	0.284902	+	0.958557i	$0.408039\pi$
$968$	0	0
$969$	4.41108	0.141704
$970$	0	0
$971$	18.8300	0.604284	0.302142	−	0.953263i	$-0.402298\pi$
$971$	18.8300	0.604284	0.302142	+	0.953263i	$0.402298\pi$
$972$	0	0
$973$	−16.3469	−0.524056
$974$	0	0
$975$	54.6804	1.75117
$976$	0	0
$977$	−24.3840	−0.780114	−0.390057	−	0.920791i	$-0.627545\pi$
$977$	−24.3840	−0.780114	−0.390057	+	0.920791i	$0.627545\pi$
$978$	0	0
$979$	−8.69658	−0.277944
$980$	0	0
$981$	13.5921	0.433961
$982$	0	0
$983$	−57.6388	−1.83839	−0.919196	−	0.393801i	$-0.871160\pi$
$983$	−57.6388	−1.83839	−0.919196	+	0.393801i	$0.871160\pi$
$984$	0	0
$985$	−17.6174	−0.561337
$986$	0	0
$987$	4.45071	0.141668
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.6557	−0.529086	−0.264543	−	0.964374i	$-0.585221\pi$
$991$	−16.6557	−0.529086	−0.264543	+	0.964374i	$0.585221\pi$
$992$	0	0
$993$	7.32398	0.232419
$994$	0	0
$995$	−23.6772	−0.750618
$996$	0	0
$997$	16.8780	0.534532	0.267266	−	0.963623i	$-0.413880\pi$
$997$	16.8780	0.534532	0.267266	+	0.963623i	$0.413880\pi$
$998$	0	0
$999$	−4.60850	−0.145806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.ce.1.1		5
4.3	odd	2		3192.2.a.ba.1.1	✓	5
12.11	even	2		9576.2.a.co.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.ba.1.1	✓	5	4.3	odd	2
6384.2.a.ce.1.1		5	1.1	even	1	trivial
9576.2.a.co.1.5		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.88971 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.88971 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.21056 q^{11} +5.39794 q^{13} -3.88971 q^{15} -4.41108 q^{17} -1.00000 q^{19} -1.00000 q^{21} +3.39794 q^{23} +10.1299 q^{25} +1.00000 q^{27} -7.28765 q^{29} -0.187375 q^{31} -3.21056 q^{33} +3.88971 q^{35} -4.60850 q^{37} +5.39794 q^{39} -9.77943 q^{41} -3.88971 q^{45} -4.45071 q^{47} +1.00000 q^{49} -4.41108 q^{51} +11.2876 q^{53} +12.4882 q^{55} -1.00000 q^{57} +5.77156 q^{59} +6.00000 q^{61} -1.00000 q^{63} -20.9964 q^{65} -2.48173 q^{67} +3.39794 q^{69} -12.4175 q^{71} -4.08714 q^{73} +10.1299 q^{75} +3.21056 q^{77} -2.68920 q^{79} +1.00000 q^{81} -3.92935 q^{83} +17.1578 q^{85} -7.28765 q^{87} +2.70874 q^{89} -5.39794 q^{91} -0.187375 q^{93} +3.88971 q^{95} -0.227011 q^{97} -3.21056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + 8 q^{13} + 2 q^{15} - 2 q^{17} - 5 q^{19} - 5 q^{21} - 2 q^{23} + 19 q^{25} + 5 q^{27} + 4 q^{29} + 4 q^{31} - 2 q^{33} - 2 q^{35} + 10 q^{37} + 8 q^{39} - 6 q^{41} + 2 q^{45} + 2 q^{47} + 5 q^{49} - 2 q^{51} + 16 q^{53} + 16 q^{55} - 5 q^{57} + 12 q^{59} + 30 q^{61} - 5 q^{63} + 4 q^{65} - 18 q^{67} - 2 q^{69} + 10 q^{71} + 14 q^{73} + 19 q^{75} + 2 q^{77} + 2 q^{79} + 5 q^{81} + 6 q^{83} + 12 q^{85} + 4 q^{87} + 10 q^{89} - 8 q^{91} + 4 q^{93} - 2 q^{95} + 8 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 2 * q^5 - 5 * q^7 + 5 * q^9 - 2 * q^11 + 8 * q^13 + 2 * q^15 - 2 * q^17 - 5 * q^19 - 5 * q^21 - 2 * q^23 + 19 * q^25 + 5 * q^27 + 4 * q^29 + 4 * q^31 - 2 * q^33 - 2 * q^35 + 10 * q^37 + 8 * q^39 - 6 * q^41 + 2 * q^45 + 2 * q^47 + 5 * q^49 - 2 * q^51 + 16 * q^53 + 16 * q^55 - 5 * q^57 + 12 * q^59 + 30 * q^61 - 5 * q^63 + 4 * q^65 - 18 * q^67 - 2 * q^69 + 10 * q^71 + 14 * q^73 + 19 * q^75 + 2 * q^77 + 2 * q^79 + 5 * q^81 + 6 * q^83 + 12 * q^85 + 4 * q^87 + 10 * q^89 - 8 * q^91 + 4 * q^93 - 2 * q^95 + 8 * q^97 - 2 * q^99

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