LMFDB - Embedded newform 6384.2.a.bt.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$50.9764966504$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.17741$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.91852 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -4.35482 q^{13} -2.91852 q^{15} +2.91852 q^{17} +1.00000 q^{19} +1.00000 q^{21} -2.00000 q^{23} +3.51777 q^{25} -1.00000 q^{27} +7.79112 q^{29} -6.35482 q^{31} +2.00000 q^{33} -2.91852 q^{35} -3.83705 q^{37} +4.35482 q^{39} -4.00000 q^{41} -3.48223 q^{43} +2.91852 q^{45} +11.6282 q^{47} +1.00000 q^{49} -2.91852 q^{51} -0.918523 q^{53} -5.83705 q^{55} -1.00000 q^{57} +0.162955 q^{61} -1.00000 q^{63} -12.7096 q^{65} +5.83705 q^{67} +2.00000 q^{69} -0.918523 q^{71} -6.70964 q^{73} -3.51777 q^{75} +2.00000 q^{77} -10.8726 q^{79} +1.00000 q^{81} -5.79112 q^{83} +8.51777 q^{85} -7.79112 q^{87} -4.00000 q^{89} +4.35482 q^{91} +6.35482 q^{93} +2.91852 q^{95} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} + 3 q^{19} + 3 q^{21} - 6 q^{23} + 13 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 6 q^{33} + 6 q^{37} - 2 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{47} + 3 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 2 * q^13 + 3 * q^19 + 3 * q^21 - 6 * q^23 + 13 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 6 * q^33 + 6 * q^37 - 2 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^47 + 3 * q^49 + 6 * q^53 - 3 * q^57 + 18 * q^61 - 3 * q^63 - 8 * q^65 + 6 * q^69 + 6 * q^71 + 10 * q^73 - 13 * q^75 + 6 * q^77 - 20 * q^79 + 3 * q^81 + 4 * q^83 + 28 * q^85 - 2 * q^87 - 12 * q^89 - 2 * q^91 + 4 * q^93 - 6 * q^97 - 6 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.91852	1.30520	0.652601	−	0.757701i	$-0.273678\pi$
$5$	2.91852	1.30520	0.652601	+	0.757701i	$0.273678\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.35482	−1.20781	−0.603905	−	0.797056i	$-0.706389\pi$
$13$	−4.35482	−1.20781	−0.603905	+	0.797056i	$0.706389\pi$
$14$	0	0
$15$	−2.91852	−0.753559
$16$	0	0
$17$	2.91852	0.707846	0.353923	−	0.935275i	$-0.384848\pi$
$17$	2.91852	0.707846	0.353923	+	0.935275i	$0.384848\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	3.51777	0.703555
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.79112	1.44677	0.723387	−	0.690443i	$-0.242584\pi$
$29$	7.79112	1.44677	0.723387	+	0.690443i	$0.242584\pi$
$30$	0	0
$31$	−6.35482	−1.14136	−0.570680	−	0.821173i	$-0.693320\pi$
$31$	−6.35482	−1.14136	−0.570680	+	0.821173i	$0.693320\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−2.91852	−0.493320
$36$	0	0
$37$	−3.83705	−0.630806	−0.315403	−	0.948958i	$-0.602140\pi$
$37$	−3.83705	−0.630806	−0.315403	+	0.948958i	$0.602140\pi$
$38$	0	0
$39$	4.35482	0.697329
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−3.48223	−0.531034	−0.265517	−	0.964106i	$-0.585543\pi$
$43$	−3.48223	−0.531034	−0.265517	+	0.964106i	$0.585543\pi$
$44$	0	0
$45$	2.91852	0.435068
$46$	0	0
$47$	11.6282	1.69614	0.848071	−	0.529883i	$-0.177764\pi$
$47$	11.6282	1.69614	0.848071	+	0.529883i	$0.177764\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.91852	−0.408675
$52$	0	0
$53$	−0.918523	−0.126169	−0.0630844	−	0.998008i	$-0.520094\pi$
$53$	−0.918523	−0.126169	−0.0630844	+	0.998008i	$0.520094\pi$
$54$	0	0
$55$	−5.83705	−0.787067
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	0.162955	0.0208642	0.0104321	−	0.999946i	$-0.496679\pi$
$61$	0.162955	0.0208642	0.0104321	+	0.999946i	$0.496679\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−12.7096	−1.57644
$66$	0	0
$67$	5.83705	0.713109	0.356554	−	0.934275i	$-0.383951\pi$
$67$	5.83705	0.713109	0.356554	+	0.934275i	$0.383951\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	−0.918523	−0.109009	−0.0545043	−	0.998514i	$-0.517358\pi$
$71$	−0.918523	−0.109009	−0.0545043	+	0.998514i	$0.517358\pi$
$72$	0	0
$73$	−6.70964	−0.785304	−0.392652	−	0.919687i	$-0.628442\pi$
$73$	−6.70964	−0.785304	−0.392652	+	0.919687i	$0.628442\pi$
$74$	0	0
$75$	−3.51777	−0.406198
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−10.8726	−1.22326	−0.611631	−	0.791143i	$-0.709486\pi$
$79$	−10.8726	−1.22326	−0.611631	+	0.791143i	$0.709486\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.79112	−0.635658	−0.317829	−	0.948148i	$-0.602954\pi$
$83$	−5.79112	−0.635658	−0.317829	+	0.948148i	$0.602954\pi$
$84$	0	0
$85$	8.51777	0.923882
$86$	0	0
$87$	−7.79112	−0.835295
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	4.35482	0.456509
$92$	0	0
$93$	6.35482	0.658964
$94$	0	0
$95$	2.91852	0.299434
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	1.08148	0.107611	0.0538055	−	0.998551i	$-0.482865\pi$
$101$	1.08148	0.107611	0.0538055	+	0.998551i	$0.482865\pi$
$102$	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
$105$	2.91852	0.284819
$106$	0	0
$107$	1.95407	0.188907	0.0944536	−	0.995529i	$-0.469890\pi$
$107$	1.95407	0.188907	0.0944536	+	0.995529i	$0.469890\pi$
$108$	0	0
$109$	−7.83705	−0.750653	−0.375326	−	0.926893i	$-0.622469\pi$
$109$	−7.83705	−0.750653	−0.375326	+	0.926893i	$0.622469\pi$
$110$	0	0
$111$	3.83705	0.364196
$112$	0	0
$113$	−13.6282	−1.28203	−0.641015	−	0.767529i	$-0.721486\pi$
$113$	−13.6282	−1.28203	−0.641015	+	0.767529i	$0.721486\pi$
$114$	0	0
$115$	−5.83705	−0.544307
$116$	0	0
$117$	−4.35482	−0.402603
$118$	0	0
$119$	−2.91852	−0.267541
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	−4.32591	−0.386921
$126$	0	0
$127$	−9.83705	−0.872897	−0.436448	−	0.899729i	$-0.643764\pi$
$127$	−9.83705	−0.872897	−0.436448	+	0.899729i	$0.643764\pi$
$128$	0	0
$129$	3.48223	0.306593
$130$	0	0
$131$	−7.95407	−0.694950	−0.347475	−	0.937689i	$-0.612961\pi$
$131$	−7.95407	−0.694950	−0.347475	+	0.937689i	$0.612961\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−2.91852	−0.251186
$136$	0	0
$137$	2.70964	0.231500	0.115750	−	0.993278i	$-0.463073\pi$
$137$	2.70964	0.231500	0.115750	+	0.993278i	$0.463073\pi$
$138$	0	0
$139$	−20.3837	−1.72893	−0.864463	−	0.502697i	$-0.832341\pi$
$139$	−20.3837	−1.72893	−0.864463	+	0.502697i	$0.832341\pi$
$140$	0	0
$141$	−11.6282	−0.979268
$142$	0	0
$143$	8.70964	0.728337
$144$	0	0
$145$	22.7385	1.88833
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	0.872594	0.0714856	0.0357428	−	0.999361i	$-0.488620\pi$
$149$	0.872594	0.0714856	0.0357428	+	0.999361i	$0.488620\pi$
$150$	0	0
$151$	12.7096	1.03430	0.517148	−	0.855896i	$-0.326994\pi$
$151$	12.7096	1.03430	0.517148	+	0.855896i	$0.326994\pi$
$152$	0	0
$153$	2.91852	0.235949
$154$	0	0
$155$	−18.5467	−1.48971
$156$	0	0
$157$	16.5467	1.32057	0.660285	−	0.751016i	$-0.270436\pi$
$157$	16.5467	1.32057	0.660285	+	0.751016i	$0.270436\pi$
$158$	0	0
$159$	0.918523	0.0728436
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	5.22741	0.409443	0.204721	−	0.978820i	$-0.434371\pi$
$163$	5.22741	0.409443	0.204721	+	0.978820i	$0.434371\pi$
$164$	0	0
$165$	5.83705	0.454413
$166$	0	0
$167$	18.8726	1.46041	0.730203	−	0.683231i	$-0.239426\pi$
$167$	18.8726	1.46041	0.730203	+	0.683231i	$0.239426\pi$
$168$	0	0
$169$	5.96445	0.458804
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−7.67409	−0.583450	−0.291725	−	0.956502i	$-0.594229\pi$
$173$	−7.67409	−0.583450	−0.291725	+	0.956502i	$0.594229\pi$
$174$	0	0
$175$	−3.51777	−0.265919
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.95407	−0.146054	−0.0730271	−	0.997330i	$-0.523266\pi$
$179$	−1.95407	−0.146054	−0.0730271	+	0.997330i	$0.523266\pi$
$180$	0	0
$181$	−18.7096	−1.39068	−0.695338	−	0.718683i	$-0.744745\pi$
$181$	−18.7096	−1.39068	−0.695338	+	0.718683i	$0.744745\pi$
$182$	0	0
$183$	−0.162955	−0.0120460
$184$	0	0
$185$	−11.1985	−0.823330
$186$	0	0
$187$	−5.83705	−0.426847
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−10.7096	−0.774922	−0.387461	−	0.921886i	$-0.626648\pi$
$191$	−10.7096	−0.774922	−0.387461	+	0.921886i	$0.626648\pi$
$192$	0	0
$193$	−12.5467	−0.903130	−0.451565	−	0.892238i	$-0.649134\pi$
$193$	−12.5467	−0.903130	−0.451565	+	0.892238i	$0.649134\pi$
$194$	0	0
$195$	12.7096	0.910156
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−10.1630	−0.720433	−0.360216	−	0.932869i	$-0.617297\pi$
$199$	−10.1630	−0.720433	−0.360216	+	0.932869i	$0.617297\pi$
$200$	0	0
$201$	−5.83705	−0.411713
$202$	0	0
$203$	−7.79112	−0.546829
$204$	0	0
$205$	−11.6741	−0.815354
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	0.918523	0.0629361
$214$	0	0
$215$	−10.1630	−0.693108
$216$	0	0
$217$	6.35482	0.431393
$218$	0	0
$219$	6.70964	0.453395
$220$	0	0
$221$	−12.7096	−0.854943
$222$	0	0
$223$	−14.7385	−0.986966	−0.493483	−	0.869755i	$-0.664277\pi$
$223$	−14.7385	−0.986966	−0.493483	+	0.869755i	$0.664277\pi$
$224$	0	0
$225$	3.51777	0.234518
$226$	0	0
$227$	26.5467	1.76197	0.880983	−	0.473149i	$-0.156883\pi$
$227$	26.5467	1.76197	0.880983	+	0.473149i	$0.156883\pi$
$228$	0	0
$229$	6.80150	0.449456	0.224728	−	0.974422i	$-0.427851\pi$
$229$	6.80150	0.449456	0.224728	+	0.974422i	$0.427851\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	21.2563	1.39255	0.696274	−	0.717776i	$-0.254840\pi$
$233$	21.2563	1.39255	0.696274	+	0.717776i	$0.254840\pi$
$234$	0	0
$235$	33.9371	2.21381
$236$	0	0
$237$	10.8726	0.706251
$238$	0	0
$239$	24.5467	1.58779	0.793896	−	0.608053i	$-0.208049\pi$
$239$	24.5467	1.58779	0.793896	+	0.608053i	$0.208049\pi$
$240$	0	0
$241$	−0.964452	−0.0621258	−0.0310629	−	0.999517i	$-0.509889\pi$
$241$	−0.964452	−0.0621258	−0.0310629	+	0.999517i	$0.509889\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.91852	0.186458
$246$	0	0
$247$	−4.35482	−0.277091
$248$	0	0
$249$	5.79112	0.366997
$250$	0	0
$251$	23.6282	1.49140	0.745698	−	0.666284i	$-0.232116\pi$
$251$	23.6282	1.49140	0.745698	+	0.666284i	$0.232116\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	−8.51777	−0.533404
$256$	0	0
$257$	−5.12741	−0.319839	−0.159919	−	0.987130i	$-0.551123\pi$
$257$	−5.12741	−0.319839	−0.159919	+	0.987130i	$0.551123\pi$
$258$	0	0
$259$	3.83705	0.238422
$260$	0	0
$261$	7.79112	0.482258
$262$	0	0
$263$	3.83705	0.236602	0.118301	−	0.992978i	$-0.462255\pi$
$263$	3.83705	0.236602	0.118301	+	0.992978i	$0.462255\pi$
$264$	0	0
$265$	−2.68073	−0.164676
$266$	0	0
$267$	4.00000	0.244796
$268$	0	0
$269$	−2.16295	−0.131878	−0.0659388	−	0.997824i	$-0.521004\pi$
$269$	−2.16295	−0.131878	−0.0659388	+	0.997824i	$0.521004\pi$
$270$	0	0
$271$	−20.8015	−1.26360	−0.631800	−	0.775131i	$-0.717684\pi$
$271$	−20.8015	−1.26360	−0.631800	+	0.775131i	$0.717684\pi$
$272$	0	0
$273$	−4.35482	−0.263566
$274$	0	0
$275$	−7.03555	−0.424260
$276$	0	0
$277$	23.6111	1.41866	0.709328	−	0.704879i	$-0.248999\pi$
$277$	23.6111	1.41866	0.709328	+	0.704879i	$0.248999\pi$
$278$	0	0
$279$	−6.35482	−0.380453
$280$	0	0
$281$	−7.08148	−0.422446	−0.211223	−	0.977438i	$-0.567745\pi$
$281$	−7.08148	−0.422446	−0.211223	+	0.977438i	$0.567745\pi$
$282$	0	0
$283$	−10.8726	−0.646309	−0.323154	−	0.946346i	$-0.604743\pi$
$283$	−10.8726	−0.646309	−0.323154	+	0.946346i	$0.604743\pi$
$284$	0	0
$285$	−2.91852	−0.172878
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−8.48223	−0.498954
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	−22.6385	−1.32256	−0.661279	−	0.750140i	$-0.729986\pi$
$293$	−22.6385	−1.32256	−0.661279	+	0.750140i	$0.729986\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	8.70964	0.503691
$300$	0	0
$301$	3.48223	0.200712
$302$	0	0
$303$	−1.08148	−0.0621293
$304$	0	0
$305$	0.475587	0.0272320
$306$	0	0
$307$	33.7030	1.92353	0.961766	−	0.273873i	$-0.0883047\pi$
$307$	33.7030	1.92353	0.961766	+	0.273873i	$0.0883047\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	11.7200	0.664581	0.332291	−	0.943177i	$-0.392179\pi$
$311$	11.7200	0.664581	0.332291	+	0.943177i	$0.392179\pi$
$312$	0	0
$313$	−1.19850	−0.0677434	−0.0338717	−	0.999426i	$-0.510784\pi$
$313$	−1.19850	−0.0677434	−0.0338717	+	0.999426i	$0.510784\pi$
$314$	0	0
$315$	−2.91852	−0.164440
$316$	0	0
$317$	−31.0474	−1.74380	−0.871899	−	0.489686i	$-0.837111\pi$
$317$	−31.0474	−1.74380	−0.871899	+	0.489686i	$0.837111\pi$
$318$	0	0
$319$	−15.5822	−0.872438
$320$	0	0
$321$	−1.95407	−0.109066
$322$	0	0
$323$	2.91852	0.162391
$324$	0	0
$325$	−15.3193	−0.849760
$326$	0	0
$327$	7.83705	0.433390
$328$	0	0
$329$	−11.6282	−0.641081
$330$	0	0
$331$	−0.325910	−0.0179136	−0.00895681	−	0.999960i	$-0.502851\pi$
$331$	−0.325910	−0.0179136	−0.00895681	+	0.999960i	$0.502851\pi$
$332$	0	0
$333$	−3.83705	−0.210269
$334$	0	0
$335$	17.0355	0.930751
$336$	0	0
$337$	15.8370	0.862699	0.431349	−	0.902185i	$-0.358038\pi$
$337$	15.8370	0.862699	0.431349	+	0.902185i	$0.358038\pi$
$338$	0	0
$339$	13.6282	0.740180
$340$	0	0
$341$	12.7096	0.688266
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.83705	0.314256
$346$	0	0
$347$	−12.8726	−0.691037	−0.345519	−	0.938412i	$-0.612297\pi$
$347$	−12.8726	−0.691037	−0.345519	+	0.938412i	$0.612297\pi$
$348$	0	0
$349$	−20.6385	−1.10476	−0.552378	−	0.833594i	$-0.686279\pi$
$349$	−20.6385	−1.10476	−0.552378	+	0.833594i	$0.686279\pi$
$350$	0	0
$351$	4.35482	0.232443
$352$	0	0
$353$	−18.9185	−1.00693	−0.503466	−	0.864015i	$-0.667942\pi$
$353$	−18.9185	−1.00693	−0.503466	+	0.864015i	$0.667942\pi$
$354$	0	0
$355$	−2.68073	−0.142278
$356$	0	0
$357$	2.91852	0.154465
$358$	0	0
$359$	−24.6385	−1.30037	−0.650186	−	0.759775i	$-0.725309\pi$
$359$	−24.6385	−1.30037	−0.650186	+	0.759775i	$0.725309\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−19.5822	−1.02498
$366$	0	0
$367$	32.2919	1.68562	0.842811	−	0.538210i	$-0.180899\pi$
$367$	32.2919	1.68562	0.842811	+	0.538210i	$0.180899\pi$
$368$	0	0
$369$	−4.00000	−0.208232
$370$	0	0
$371$	0.918523	0.0476873
$372$	0	0
$373$	−7.74519	−0.401031	−0.200515	−	0.979691i	$-0.564262\pi$
$373$	−7.74519	−0.401031	−0.200515	+	0.979691i	$0.564262\pi$
$374$	0	0
$375$	4.32591	0.223389
$376$	0	0
$377$	−33.9289	−1.74743
$378$	0	0
$379$	−22.4548	−1.15343	−0.576713	−	0.816946i	$-0.695665\pi$
$379$	−22.4548	−1.15343	−0.576713	+	0.816946i	$0.695665\pi$
$380$	0	0
$381$	9.83705	0.503967
$382$	0	0
$383$	−18.1630	−0.928084	−0.464042	−	0.885813i	$-0.653601\pi$
$383$	−18.1630	−0.928084	−0.464042	+	0.885813i	$0.653601\pi$
$384$	0	0
$385$	5.83705	0.297483
$386$	0	0
$387$	−3.48223	−0.177011
$388$	0	0
$389$	−9.90814	−0.502363	−0.251181	−	0.967940i	$-0.580819\pi$
$389$	−9.90814	−0.502363	−0.251181	+	0.967940i	$0.580819\pi$
$390$	0	0
$391$	−5.83705	−0.295192
$392$	0	0
$393$	7.95407	0.401230
$394$	0	0
$395$	−31.7319	−1.59661
$396$	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
$401$	−20.9185	−1.04462	−0.522311	−	0.852755i	$-0.674930\pi$
$401$	−20.9185	−1.04462	−0.522311	+	0.852755i	$0.674930\pi$
$402$	0	0
$403$	27.6741	1.37854
$404$	0	0
$405$	2.91852	0.145023
$406$	0	0
$407$	7.67409	0.380391
$408$	0	0
$409$	−20.0289	−0.990366	−0.495183	−	0.868789i	$-0.664899\pi$
$409$	−20.0289	−0.990366	−0.495183	+	0.868789i	$0.664899\pi$
$410$	0	0
$411$	−2.70964	−0.133657
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−16.9015	−0.829662
$416$	0	0
$417$	20.3837	0.998196
$418$	0	0
$419$	23.7200	1.15880	0.579399	−	0.815044i	$-0.303287\pi$
$419$	23.7200	1.15880	0.579399	+	0.815044i	$0.303287\pi$
$420$	0	0
$421$	−2.70964	−0.132060	−0.0660299	−	0.997818i	$-0.521033\pi$
$421$	−2.70964	−0.132060	−0.0660299	+	0.997818i	$0.521033\pi$
$422$	0	0
$423$	11.6282	0.565381
$424$	0	0
$425$	10.2667	0.498008
$426$	0	0
$427$	−0.162955	−0.00788594
$428$	0	0
$429$	−8.70964	−0.420505
$430$	0	0
$431$	−31.1393	−1.49993	−0.749964	−	0.661479i	$-0.769929\pi$
$431$	−31.1393	−1.49993	−0.749964	+	0.661479i	$0.769929\pi$
$432$	0	0
$433$	−24.6385	−1.18405	−0.592026	−	0.805919i	$-0.701672\pi$
$433$	−24.6385	−1.18405	−0.592026	+	0.805919i	$0.701672\pi$
$434$	0	0
$435$	−22.7385	−1.09023
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	20.1000	0.959321	0.479660	−	0.877454i	$-0.340760\pi$
$439$	20.1000	0.959321	0.479660	+	0.877454i	$0.340760\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−27.8370	−1.32258	−0.661289	−	0.750131i	$-0.729990\pi$
$443$	−27.8370	−1.32258	−0.661289	+	0.750131i	$0.729990\pi$
$444$	0	0
$445$	−11.6741	−0.553405
$446$	0	0
$447$	−0.872594	−0.0412723
$448$	0	0
$449$	4.82666	0.227784	0.113892	−	0.993493i	$-0.463668\pi$
$449$	4.82666	0.227784	0.113892	+	0.993493i	$0.463668\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	−12.7096	−0.597151
$454$	0	0
$455$	12.7096	0.595837
$456$	0	0
$457$	1.48223	0.0693356	0.0346678	−	0.999399i	$-0.488963\pi$
$457$	1.48223	0.0693356	0.0346678	+	0.999399i	$0.488963\pi$
$458$	0	0
$459$	−2.91852	−0.136225
$460$	0	0
$461$	−32.3378	−1.50612	−0.753061	−	0.657951i	$-0.771423\pi$
$461$	−32.3378	−1.50612	−0.753061	+	0.657951i	$0.771423\pi$
$462$	0	0
$463$	−29.4193	−1.36723	−0.683615	−	0.729843i	$-0.739593\pi$
$463$	−29.4193	−1.36723	−0.683615	+	0.729843i	$0.739593\pi$
$464$	0	0
$465$	18.5467	0.860082
$466$	0	0
$467$	13.4652	0.623095	0.311548	−	0.950231i	$-0.399153\pi$
$467$	13.4652	0.623095	0.311548	+	0.950231i	$0.399153\pi$
$468$	0	0
$469$	−5.83705	−0.269530
$470$	0	0
$471$	−16.5467	−0.762431
$472$	0	0
$473$	6.96445	0.320226
$474$	0	0
$475$	3.51777	0.161407
$476$	0	0
$477$	−0.918523	−0.0420563
$478$	0	0
$479$	−13.6993	−0.625935	−0.312968	−	0.949764i	$-0.601323\pi$
$479$	−13.6993	−0.625935	−0.312968	+	0.949764i	$0.601323\pi$
$480$	0	0
$481$	16.7096	0.761894
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	−5.83705	−0.265047
$486$	0	0
$487$	25.0355	1.13447	0.567234	−	0.823556i	$-0.308013\pi$
$487$	25.0355	1.13447	0.567234	+	0.823556i	$0.308013\pi$
$488$	0	0
$489$	−5.22741	−0.236392
$490$	0	0
$491$	−10.8015	−0.487465	−0.243732	−	0.969843i	$-0.578372\pi$
$491$	−10.8015	−0.487465	−0.243732	+	0.969843i	$0.578372\pi$
$492$	0	0
$493$	22.7385	1.02409
$494$	0	0
$495$	−5.83705	−0.262356
$496$	0	0
$497$	0.918523	0.0412014
$498$	0	0
$499$	−9.92890	−0.444479	−0.222239	−	0.974992i	$-0.571337\pi$
$499$	−9.92890	−0.444479	−0.222239	+	0.974992i	$0.571337\pi$
$500$	0	0
$501$	−18.8726	−0.843165
$502$	0	0
$503$	−27.3023	−1.21735	−0.608674	−	0.793421i	$-0.708298\pi$
$503$	−27.3023	−1.21735	−0.608674	+	0.793421i	$0.708298\pi$
$504$	0	0
$505$	3.15632	0.140454
$506$	0	0
$507$	−5.96445	−0.264891
$508$	0	0
$509$	23.2563	1.03082	0.515409	−	0.856944i	$-0.327640\pi$
$509$	23.2563	1.03082	0.515409	+	0.856944i	$0.327640\pi$
$510$	0	0
$511$	6.70964	0.296817
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−23.3482	−1.02884
$516$	0	0
$517$	−23.2563	−1.02281
$518$	0	0
$519$	7.67409	0.336855
$520$	0	0
$521$	−0.743677	−0.0325811	−0.0162905	−	0.999867i	$-0.505186\pi$
$521$	−0.743677	−0.0325811	−0.0162905	+	0.999867i	$0.505186\pi$
$522$	0	0
$523$	−32.4837	−1.42041	−0.710207	−	0.703993i	$-0.751399\pi$
$523$	−32.4837	−1.42041	−0.710207	+	0.703993i	$0.751399\pi$
$524$	0	0
$525$	3.51777	0.153528
$526$	0	0
$527$	−18.5467	−0.807906
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.4193	0.754513
$534$	0	0
$535$	5.70300	0.246562
$536$	0	0
$537$	1.95407	0.0843244
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	44.4548	1.91126	0.955631	−	0.294566i	$-0.0951751\pi$
$541$	44.4548	1.91126	0.955631	+	0.294566i	$0.0951751\pi$
$542$	0	0
$543$	18.7096	0.802907
$544$	0	0
$545$	−22.8726	−0.979754
$546$	0	0
$547$	34.5467	1.47711	0.738555	−	0.674193i	$-0.235509\pi$
$547$	34.5467	1.47711	0.738555	+	0.674193i	$0.235509\pi$
$548$	0	0
$549$	0.162955	0.00695474
$550$	0	0
$551$	7.79112	0.331913
$552$	0	0
$553$	10.8726	0.462350
$554$	0	0
$555$	11.1985	0.475350
$556$	0	0
$557$	19.1274	0.810454	0.405227	−	0.914216i	$-0.367192\pi$
$557$	19.1274	0.810454	0.405227	+	0.914216i	$0.367192\pi$
$558$	0	0
$559$	15.1645	0.641389
$560$	0	0
$561$	5.83705	0.246440
$562$	0	0
$563$	−6.87259	−0.289645	−0.144823	−	0.989458i	$-0.546261\pi$
$563$	−6.87259	−0.289645	−0.144823	+	0.989458i	$0.546261\pi$
$564$	0	0
$565$	−39.7741	−1.67331
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−22.4297	−0.940300	−0.470150	−	0.882586i	$-0.655800\pi$
$569$	−22.4297	−0.940300	−0.470150	+	0.882586i	$0.655800\pi$
$570$	0	0
$571$	−5.74519	−0.240429	−0.120214	−	0.992748i	$-0.538358\pi$
$571$	−5.74519	−0.240429	−0.120214	+	0.992748i	$0.538358\pi$
$572$	0	0
$573$	10.7096	0.447402
$574$	0	0
$575$	−7.03555	−0.293403
$576$	0	0
$577$	−40.5467	−1.68798	−0.843990	−	0.536358i	$-0.819800\pi$
$577$	−40.5467	−1.68798	−0.843990	+	0.536358i	$0.819800\pi$
$578$	0	0
$579$	12.5467	0.521422
$580$	0	0
$581$	5.79112	0.240256
$582$	0	0
$583$	1.83705	0.0760826
$584$	0	0
$585$	−12.7096	−0.525479
$586$	0	0
$587$	−7.72002	−0.318639	−0.159320	−	0.987227i	$-0.550930\pi$
$587$	−7.72002	−0.318639	−0.159320	+	0.987227i	$0.550930\pi$
$588$	0	0
$589$	−6.35482	−0.261846
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	−36.2459	−1.48844	−0.744221	−	0.667933i	$-0.767179\pi$
$593$	−36.2459	−1.48844	−0.744221	+	0.667933i	$0.767179\pi$
$594$	0	0
$595$	−8.51777	−0.349195
$596$	0	0
$597$	10.1630	0.415942
$598$	0	0
$599$	27.1393	1.10888	0.554441	−	0.832223i	$-0.312932\pi$
$599$	27.1393	1.10888	0.554441	+	0.832223i	$0.312932\pi$
$600$	0	0
$601$	15.0934	0.615671	0.307836	−	0.951440i	$-0.400395\pi$
$601$	15.0934	0.615671	0.307836	+	0.951440i	$0.400395\pi$
$602$	0	0
$603$	5.83705	0.237703
$604$	0	0
$605$	−20.4297	−0.830584
$606$	0	0
$607$	−25.4193	−1.03174	−0.515868	−	0.856668i	$-0.672531\pi$
$607$	−25.4193	−1.03174	−0.515868	+	0.856668i	$0.672531\pi$
$608$	0	0
$609$	7.79112	0.315712
$610$	0	0
$611$	−50.6385	−2.04862
$612$	0	0
$613$	26.1919	1.05788	0.528940	−	0.848659i	$-0.322590\pi$
$613$	26.1919	1.05788	0.528940	+	0.848659i	$0.322590\pi$
$614$	0	0
$615$	11.6741	0.470745
$616$	0	0
$617$	−9.34818	−0.376344	−0.188172	−	0.982136i	$-0.560256\pi$
$617$	−9.34818	−0.376344	−0.188172	+	0.982136i	$0.560256\pi$
$618$	0	0
$619$	20.3837	0.819291	0.409646	−	0.912245i	$-0.365652\pi$
$619$	20.3837	0.819291	0.409646	+	0.912245i	$0.365652\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	−30.2141	−1.20857
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	−11.1985	−0.446514
$630$	0	0
$631$	27.8660	1.10933	0.554663	−	0.832075i	$-0.312847\pi$
$631$	27.8660	1.10933	0.554663	+	0.832075i	$0.312847\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	−28.7096	−1.13931
$636$	0	0
$637$	−4.35482	−0.172544
$638$	0	0
$639$	−0.918523	−0.0363362
$640$	0	0
$641$	35.3733	1.39716	0.698582	−	0.715530i	$-0.253815\pi$
$641$	35.3733	1.39716	0.698582	+	0.715530i	$0.253815\pi$
$642$	0	0
$643$	36.2919	1.43121	0.715606	−	0.698504i	$-0.246151\pi$
$643$	36.2919	1.43121	0.715606	+	0.698504i	$0.246151\pi$
$644$	0	0
$645$	10.1630	0.400166
$646$	0	0
$647$	27.6282	1.08618	0.543088	−	0.839676i	$-0.317255\pi$
$647$	27.6282	1.08618	0.543088	+	0.839676i	$0.317255\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−6.35482	−0.249065
$652$	0	0
$653$	25.6741	1.00470	0.502352	−	0.864663i	$-0.332468\pi$
$653$	25.6741	1.00470	0.502352	+	0.864663i	$0.332468\pi$
$654$	0	0
$655$	−23.2141	−0.907051
$656$	0	0
$657$	−6.70964	−0.261768
$658$	0	0
$659$	−13.0104	−0.506812	−0.253406	−	0.967360i	$-0.581551\pi$
$659$	−13.0104	−0.506812	−0.253406	+	0.967360i	$0.581551\pi$
$660$	0	0
$661$	−50.1578	−1.95091	−0.975457	−	0.220192i	$-0.929332\pi$
$661$	−50.1578	−1.95091	−0.975457	+	0.220192i	$0.929332\pi$
$662$	0	0
$663$	12.7096	0.493601
$664$	0	0
$665$	−2.91852	−0.113175
$666$	0	0
$667$	−15.5822	−0.603346
$668$	0	0
$669$	14.7385	0.569825
$670$	0	0
$671$	−0.325910	−0.0125816
$672$	0	0
$673$	48.5467	1.87134	0.935669	−	0.352880i	$-0.114798\pi$
$673$	48.5467	1.87134	0.935669	+	0.352880i	$0.114798\pi$
$674$	0	0
$675$	−3.51777	−0.135399
$676$	0	0
$677$	−7.76595	−0.298470	−0.149235	−	0.988802i	$-0.547681\pi$
$677$	−7.76595	−0.298470	−0.149235	+	0.988802i	$0.547681\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−26.5467	−1.01727
$682$	0	0
$683$	−20.2089	−0.773271	−0.386636	−	0.922233i	$-0.626363\pi$
$683$	−20.2089	−0.773271	−0.386636	+	0.922233i	$0.626363\pi$
$684$	0	0
$685$	7.90814	0.302155
$686$	0	0
$687$	−6.80150	−0.259493
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	13.1274	0.499390	0.249695	−	0.968325i	$-0.419670\pi$
$691$	13.1274	0.499390	0.249695	+	0.968325i	$0.419670\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	−59.4904	−2.25660
$696$	0	0
$697$	−11.6741	−0.442188
$698$	0	0
$699$	−21.2563	−0.803988
$700$	0	0
$701$	0.906632	0.0342430	0.0171215	−	0.999853i	$-0.494550\pi$
$701$	0.906632	0.0342430	0.0171215	+	0.999853i	$0.494550\pi$
$702$	0	0
$703$	−3.83705	−0.144717
$704$	0	0
$705$	−33.9371	−1.27814
$706$	0	0
$707$	−1.08148	−0.0406731
$708$	0	0
$709$	25.1563	0.944765	0.472383	−	0.881394i	$-0.343394\pi$
$709$	25.1563	0.944765	0.472383	+	0.881394i	$0.343394\pi$
$710$	0	0
$711$	−10.8726	−0.407754
$712$	0	0
$713$	12.7096	0.475980
$714$	0	0
$715$	25.4193	0.950627
$716$	0	0
$717$	−24.5467	−0.916713
$718$	0	0
$719$	−41.1393	−1.53424	−0.767118	−	0.641505i	$-0.778310\pi$
$719$	−41.1393	−1.53424	−0.767118	+	0.641505i	$0.778310\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0.964452	0.0358683
$724$	0	0
$725$	27.4074	1.01788
$726$	0	0
$727$	−25.3274	−0.939342	−0.469671	−	0.882842i	$-0.655627\pi$
$727$	−25.3274	−0.939342	−0.469671	+	0.882842i	$0.655627\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.1630	−0.375890
$732$	0	0
$733$	42.3837	1.56548	0.782739	−	0.622350i	$-0.213822\pi$
$733$	42.3837	1.56548	0.782739	+	0.622350i	$0.213822\pi$
$734$	0	0
$735$	−2.91852	−0.107651
$736$	0	0
$737$	−11.6741	−0.430021
$738$	0	0
$739$	9.73704	0.358183	0.179091	−	0.983832i	$-0.442684\pi$
$739$	9.73704	0.358183	0.179091	+	0.983832i	$0.442684\pi$
$740$	0	0
$741$	4.35482	0.159978
$742$	0	0
$743$	38.7215	1.42056	0.710278	−	0.703922i	$-0.248569\pi$
$743$	38.7215	1.42056	0.710278	+	0.703922i	$0.248569\pi$
$744$	0	0
$745$	2.54668	0.0933033
$746$	0	0
$747$	−5.79112	−0.211886
$748$	0	0
$749$	−1.95407	−0.0714002
$750$	0	0
$751$	−18.6385	−0.680130	−0.340065	−	0.940402i	$-0.610449\pi$
$751$	−18.6385	−0.680130	−0.340065	+	0.940402i	$0.610449\pi$
$752$	0	0
$753$	−23.6282	−0.861058
$754$	0	0
$755$	37.0934	1.34997
$756$	0	0
$757$	51.4771	1.87097	0.935483	−	0.353370i	$-0.114964\pi$
$757$	51.4771	1.87097	0.935483	+	0.353370i	$0.114964\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−10.5008	−0.380652	−0.190326	−	0.981721i	$-0.560955\pi$
$761$	−10.5008	−0.380652	−0.190326	+	0.981721i	$0.560955\pi$
$762$	0	0
$763$	7.83705	0.283720
$764$	0	0
$765$	8.51777	0.307961
$766$	0	0
$767$	0	0
$768$	0	0
$769$	7.45332	0.268773	0.134387	−	0.990929i	$-0.457094\pi$
$769$	7.45332	0.268773	0.134387	+	0.990929i	$0.457094\pi$
$770$	0	0
$771$	5.12741	0.184659
$772$	0	0
$773$	21.4193	0.770398	0.385199	−	0.922834i	$-0.374133\pi$
$773$	21.4193	0.770398	0.385199	+	0.922834i	$0.374133\pi$
$774$	0	0
$775$	−22.3548	−0.803009
$776$	0	0
$777$	−3.83705	−0.137653
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	1.83705	0.0657346
$782$	0	0
$783$	−7.79112	−0.278432
$784$	0	0
$785$	48.2919	1.72361
$786$	0	0
$787$	−33.7030	−1.20138	−0.600691	−	0.799481i	$-0.705108\pi$
$787$	−33.7030	−1.20138	−0.600691	+	0.799481i	$0.705108\pi$
$788$	0	0
$789$	−3.83705	−0.136602
$790$	0	0
$791$	13.6282	0.484562
$792$	0	0
$793$	−0.709639	−0.0252000
$794$	0	0
$795$	2.68073	0.0950757
$796$	0	0
$797$	33.4771	1.18582	0.592910	−	0.805269i	$-0.297979\pi$
$797$	33.4771	1.18582	0.592910	+	0.805269i	$0.297979\pi$
$798$	0	0
$799$	33.9371	1.20061
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	13.4193	0.473556
$804$	0	0
$805$	5.83705	0.205729
$806$	0	0
$807$	2.16295	0.0761396
$808$	0	0
$809$	−22.7675	−0.800461	−0.400231	−	0.916414i	$-0.631070\pi$
$809$	−22.7675	−0.800461	−0.400231	+	0.916414i	$0.631070\pi$
$810$	0	0
$811$	−52.7675	−1.85292	−0.926458	−	0.376398i	$-0.877163\pi$
$811$	−52.7675	−1.85292	−0.926458	+	0.376398i	$0.877163\pi$
$812$	0	0
$813$	20.8015	0.729540
$814$	0	0
$815$	15.2563	0.534406
$816$	0	0
$817$	−3.48223	−0.121828
$818$	0	0
$819$	4.35482	0.152170
$820$	0	0
$821$	28.9304	1.00968	0.504839	−	0.863213i	$-0.331552\pi$
$821$	28.9304	1.00968	0.504839	+	0.863213i	$0.331552\pi$
$822$	0	0
$823$	17.6111	0.613886	0.306943	−	0.951728i	$-0.400694\pi$
$823$	17.6111	0.613886	0.306943	+	0.951728i	$0.400694\pi$
$824$	0	0
$825$	7.03555	0.244946
$826$	0	0
$827$	8.82666	0.306933	0.153467	−	0.988154i	$-0.450956\pi$
$827$	8.82666	0.306933	0.153467	+	0.988154i	$0.450956\pi$
$828$	0	0
$829$	11.6030	0.402989	0.201494	−	0.979490i	$-0.435420\pi$
$829$	11.6030	0.402989	0.201494	+	0.979490i	$0.435420\pi$
$830$	0	0
$831$	−23.6111	−0.819061
$832$	0	0
$833$	2.91852	0.101121
$834$	0	0
$835$	55.0801	1.90613
$836$	0	0
$837$	6.35482	0.219655
$838$	0	0
$839$	−25.7452	−0.888823	−0.444411	−	0.895823i	$-0.646587\pi$
$839$	−25.7452	−0.888823	−0.444411	+	0.895823i	$0.646587\pi$
$840$	0	0
$841$	31.7015	1.09315
$842$	0	0
$843$	7.08148	0.243899
$844$	0	0
$845$	17.4074	0.598832
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	10.8726	0.373146
$850$	0	0
$851$	7.67409	0.263064
$852$	0	0
$853$	36.1289	1.23703	0.618515	−	0.785773i	$-0.287734\pi$
$853$	36.1289	1.23703	0.618515	+	0.785773i	$0.287734\pi$
$854$	0	0
$855$	2.91852	0.0998114
$856$	0	0
$857$	−44.3837	−1.51612	−0.758060	−	0.652185i	$-0.773852\pi$
$857$	−44.3837	−1.51612	−0.758060	+	0.652185i	$0.773852\pi$
$858$	0	0
$859$	24.0578	0.820842	0.410421	−	0.911896i	$-0.365382\pi$
$859$	24.0578	0.820842	0.410421	+	0.911896i	$0.365382\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	−43.2312	−1.47161	−0.735803	−	0.677196i	$-0.763195\pi$
$863$	−43.2312	−1.47161	−0.735803	+	0.677196i	$0.763195\pi$
$864$	0	0
$865$	−22.3970	−0.761521
$866$	0	0
$867$	8.48223	0.288071
$868$	0	0
$869$	21.7452	0.737655
$870$	0	0
$871$	−25.4193	−0.861299
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	4.32591	0.146242
$876$	0	0
$877$	49.0223	1.65536	0.827682	−	0.561197i	$-0.189659\pi$
$877$	49.0223	1.65536	0.827682	+	0.561197i	$0.189659\pi$
$878$	0	0
$879$	22.6385	0.763579
$880$	0	0
$881$	−32.0119	−1.07851	−0.539254	−	0.842143i	$-0.681294\pi$
$881$	−32.0119	−1.07851	−0.539254	+	0.842143i	$0.681294\pi$
$882$	0	0
$883$	25.0934	0.844459	0.422230	−	0.906489i	$-0.361248\pi$
$883$	25.0934	0.844459	0.422230	+	0.906489i	$0.361248\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−8.80150	−0.295525	−0.147763	−	0.989023i	$-0.547207\pi$
$887$	−8.80150	−0.295525	−0.147763	+	0.989023i	$0.547207\pi$
$888$	0	0
$889$	9.83705	0.329924
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	11.6282	0.389122
$894$	0	0
$895$	−5.70300	−0.190630
$896$	0	0
$897$	−8.70964	−0.290806
$898$	0	0
$899$	−49.5111	−1.65129
$900$	0	0
$901$	−2.68073	−0.0893080
$902$	0	0
$903$	−3.48223	−0.115881
$904$	0	0
$905$	−54.6045	−1.81512
$906$	0	0
$907$	52.3837	1.73937	0.869687	−	0.493604i	$-0.164321\pi$
$907$	52.3837	1.73937	0.869687	+	0.493604i	$0.164321\pi$
$908$	0	0
$909$	1.08148	0.0358703
$910$	0	0
$911$	16.5926	0.549738	0.274869	−	0.961482i	$-0.411366\pi$
$911$	16.5926	0.549738	0.274869	+	0.961482i	$0.411366\pi$
$912$	0	0
$913$	11.5822	0.383316
$914$	0	0
$915$	−0.475587	−0.0157224
$916$	0	0
$917$	7.95407	0.262667
$918$	0	0
$919$	−13.9289	−0.459472	−0.229736	−	0.973253i	$-0.573786\pi$
$919$	−13.9289	−0.459472	−0.229736	+	0.973253i	$0.573786\pi$
$920$	0	0
$921$	−33.7030	−1.11055
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	−13.4979	−0.443807
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	54.0830	1.77441	0.887203	−	0.461380i	$-0.152646\pi$
$929$	54.0830	1.77441	0.887203	+	0.461380i	$0.152646\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−11.7200	−0.383696
$934$	0	0
$935$	−17.0355	−0.557122
$936$	0	0
$937$	4.25481	0.138999	0.0694993	−	0.997582i	$-0.477860\pi$
$937$	4.25481	0.138999	0.0694993	+	0.997582i	$0.477860\pi$
$938$	0	0
$939$	1.19850	0.0391117
$940$	0	0
$941$	8.47559	0.276296	0.138148	−	0.990412i	$-0.455885\pi$
$941$	8.47559	0.276296	0.138148	+	0.990412i	$0.455885\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	2.91852	0.0949395
$946$	0	0
$947$	4.78074	0.155353	0.0776765	−	0.996979i	$-0.475250\pi$
$947$	4.78074	0.155353	0.0776765	+	0.996979i	$0.475250\pi$
$948$	0	0
$949$	29.2193	0.948497
$950$	0	0
$951$	31.0474	1.00678
$952$	0	0
$953$	31.0815	1.00683	0.503414	−	0.864045i	$-0.332077\pi$
$953$	31.0815	1.00683	0.503414	+	0.864045i	$0.332077\pi$
$954$	0	0
$955$	−31.2563	−1.01143
$956$	0	0
$957$	15.5822	0.503702
$958$	0	0
$959$	−2.70964	−0.0874988
$960$	0	0
$961$	9.38373	0.302701
$962$	0	0
$963$	1.95407	0.0629691
$964$	0	0
$965$	−36.6178	−1.17877
$966$	0	0
$967$	−21.9371	−0.705448	−0.352724	−	0.935727i	$-0.614745\pi$
$967$	−21.9371	−0.705448	−0.352724	+	0.935727i	$0.614745\pi$
$968$	0	0
$969$	−2.91852	−0.0937565
$970$	0	0
$971$	−39.4904	−1.26731	−0.633653	−	0.773617i	$-0.718445\pi$
$971$	−39.4904	−1.26731	−0.633653	+	0.773617i	$0.718445\pi$
$972$	0	0
$973$	20.3837	0.653472
$974$	0	0
$975$	15.3193	0.490609
$976$	0	0
$977$	−9.24443	−0.295756	−0.147878	−	0.989006i	$-0.547244\pi$
$977$	−9.24443	−0.295756	−0.147878	+	0.989006i	$0.547244\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	−7.83705	−0.250218
$982$	0	0
$983$	19.2904	0.615267	0.307633	−	0.951505i	$-0.400463\pi$
$983$	19.2904	0.615267	0.307633	+	0.951505i	$0.400463\pi$
$984$	0	0
$985$	−17.5111	−0.557951
$986$	0	0
$987$	11.6282	0.370128
$988$	0	0
$989$	6.96445	0.221457
$990$	0	0
$991$	−10.4548	−0.332108	−0.166054	−	0.986117i	$-0.553103\pi$
$991$	−10.4548	−0.332108	−0.166054	+	0.986117i	$0.553103\pi$
$992$	0	0
$993$	0.325910	0.0103424
$994$	0	0
$995$	−29.6608	−0.940311
$996$	0	0
$997$	34.1497	1.08153	0.540766	−	0.841173i	$-0.318135\pi$
$997$	34.1497	1.08153	0.540766	+	0.841173i	$0.318135\pi$
$998$	0	0
$999$	3.83705	0.121399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bt.1.3		3
4.3	odd	2		1596.2.a.j.1.3	✓	3
12.11	even	2		4788.2.a.p.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.j.1.3	✓	3	4.3	odd	2
4788.2.a.p.1.1		3	12.11	even	2
6384.2.a.bt.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.91852 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -4.35482 q^{13} -2.91852 q^{15} +2.91852 q^{17} +1.00000 q^{19} +1.00000 q^{21} -2.00000 q^{23} +3.51777 q^{25} -1.00000 q^{27} +7.79112 q^{29} -6.35482 q^{31} +2.00000 q^{33} -2.91852 q^{35} -3.83705 q^{37} +4.35482 q^{39} -4.00000 q^{41} -3.48223 q^{43} +2.91852 q^{45} +11.6282 q^{47} +1.00000 q^{49} -2.91852 q^{51} -0.918523 q^{53} -5.83705 q^{55} -1.00000 q^{57} +0.162955 q^{61} -1.00000 q^{63} -12.7096 q^{65} +5.83705 q^{67} +2.00000 q^{69} -0.918523 q^{71} -6.70964 q^{73} -3.51777 q^{75} +2.00000 q^{77} -10.8726 q^{79} +1.00000 q^{81} -5.79112 q^{83} +8.51777 q^{85} -7.79112 q^{87} -4.00000 q^{89} +4.35482 q^{91} +6.35482 q^{93} +2.91852 q^{95} -2.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} + 3 q^{19} + 3 q^{21} - 6 q^{23} + 13 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 6 q^{33} + 6 q^{37} - 2 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{47} + 3 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 2 * q^13 + 3 * q^19 + 3 * q^21 - 6 * q^23 + 13 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 6 * q^33 + 6 * q^37 - 2 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^47 + 3 * q^49 + 6 * q^53 - 3 * q^57 + 18 * q^61 - 3 * q^63 - 8 * q^65 + 6 * q^69 + 6 * q^71 + 10 * q^73 - 13 * q^75 + 6 * q^77 - 20 * q^79 + 3 * q^81 + 4 * q^83 + 28 * q^85 - 2 * q^87 - 12 * q^89 - 2 * q^91 + 4 * q^93 - 6 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more