LMFDB - Embedded newform 6384.2.a.ce.1.4

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			1.4
Root			$1.26848$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.53696 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.53696 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.91239 q^{11} -0.165497 q^{13} +2.53696 q^{15} +6.84768 q^{17} -1.00000 q^{19} -1.00000 q^{21} -2.16550 q^{23} +1.43617 q^{25} +1.00000 q^{27} +4.70246 q^{29} +7.07788 q^{31} -4.91239 q^{33} -2.53696 q^{35} -0.746889 q^{37} -0.165497 q^{39} +3.07392 q^{41} +2.53696 q^{45} -3.88551 q^{47} +1.00000 q^{49} +6.84768 q^{51} -0.702457 q^{53} -12.4625 q^{55} -1.00000 q^{57} +13.3349 q^{59} +6.00000 q^{61} -1.00000 q^{63} -0.419858 q^{65} +13.0439 q^{67} -2.16550 q^{69} +8.26629 q^{71} -5.05761 q^{73} +1.43617 q^{75} +4.91239 q^{77} -9.22311 q^{79} +1.00000 q^{81} -8.19623 q^{83} +17.3723 q^{85} +4.70246 q^{87} -9.38861 q^{89} +0.165497 q^{91} +7.07788 q^{93} -2.53696 q^{95} -3.65531 q^{97} -4.91239 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.53696	1.13456	0.567281	−	0.823524i	$-0.307995\pi$
$5$	2.53696	1.13456	0.567281	+	0.823524i	$0.307995\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.91239	−1.48114	−0.740570	−	0.671979i	$-0.765444\pi$
$11$	−4.91239	−1.48114	−0.740570	+	0.671979i	$0.765444\pi$
$12$	0	0
$13$	−0.165497	−0.0459005	−0.0229502	−	0.999737i	$-0.507306\pi$
$13$	−0.165497	−0.0459005	−0.0229502	+	0.999737i	$0.507306\pi$
$14$	0	0
$15$	2.53696	0.655040
$16$	0	0
$17$	6.84768	1.66081	0.830404	−	0.557162i	$-0.188110\pi$
$17$	6.84768	1.66081	0.830404	+	0.557162i	$0.188110\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−2.16550	−0.451537	−0.225769	−	0.974181i	$-0.572489\pi$
$23$	−2.16550	−0.451537	−0.225769	+	0.974181i	$0.572489\pi$
$24$	0	0
$25$	1.43617	0.287233
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.70246	0.873224	0.436612	−	0.899650i	$-0.356178\pi$
$29$	4.70246	0.873224	0.436612	+	0.899650i	$0.356178\pi$
$30$	0	0
$31$	7.07788	1.27123	0.635613	−	0.772008i	$-0.280748\pi$
$31$	7.07788	1.27123	0.635613	+	0.772008i	$0.280748\pi$
$32$	0	0
$33$	−4.91239	−0.855137
$34$	0	0
$35$	−2.53696	−0.428824
$36$	0	0
$37$	−0.746889	−0.122788	−0.0613939	−	0.998114i	$-0.519555\pi$
$37$	−0.746889	−0.122788	−0.0613939	+	0.998114i	$0.519555\pi$
$38$	0	0
$39$	−0.165497	−0.0265007
$40$	0	0
$41$	3.07392	0.480066	0.240033	−	0.970765i	$-0.422842\pi$
$41$	3.07392	0.480066	0.240033	+	0.970765i	$0.422842\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	2.53696	0.378188
$46$	0	0
$47$	−3.88551	−0.566760	−0.283380	−	0.959008i	$-0.591456\pi$
$47$	−3.88551	−0.566760	−0.283380	+	0.959008i	$0.591456\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.84768	0.958867
$52$	0	0
$53$	−0.702457	−0.0964898	−0.0482449	−	0.998836i	$-0.515363\pi$
$53$	−0.702457	−0.0964898	−0.0482449	+	0.998836i	$0.515363\pi$
$54$	0	0
$55$	−12.4625	−1.68045
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	13.3349	1.73605	0.868025	−	0.496520i	$-0.165389\pi$
$59$	13.3349	1.73605	0.868025	+	0.496520i	$0.165389\pi$
$60$	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
$65$	−0.419858	−0.0520770
$66$	0	0
$67$	13.0439	1.59357	0.796784	−	0.604264i	$-0.206533\pi$
$67$	13.0439	1.59357	0.796784	+	0.604264i	$0.206533\pi$
$68$	0	0
$69$	−2.16550	−0.260695
$70$	0	0
$71$	8.26629	0.981028	0.490514	−	0.871433i	$-0.336809\pi$
$71$	8.26629	0.981028	0.490514	+	0.871433i	$0.336809\pi$
$72$	0	0
$73$	−5.05761	−0.591949	−0.295974	−	0.955196i	$-0.595644\pi$
$73$	−5.05761	−0.591949	−0.295974	+	0.955196i	$0.595644\pi$
$74$	0	0
$75$	1.43617	0.165834
$76$	0	0
$77$	4.91239	0.559818
$78$	0	0
$79$	−9.22311	−1.03768	−0.518840	−	0.854871i	$-0.673636\pi$
$79$	−9.22311	−1.03768	−0.518840	+	0.854871i	$0.673636\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.19623	−0.899654	−0.449827	−	0.893116i	$-0.648514\pi$
$83$	−8.19623	−0.899654	−0.449827	+	0.893116i	$0.648514\pi$
$84$	0	0
$85$	17.3723	1.88429
$86$	0	0
$87$	4.70246	0.504156
$88$	0	0
$89$	−9.38861	−0.995190	−0.497595	−	0.867409i	$-0.665783\pi$
$89$	−9.38861	−0.995190	−0.497595	+	0.867409i	$0.665783\pi$
$90$	0	0
$91$	0.165497	0.0173488
$92$	0	0
$93$	7.07788	0.733942
$94$	0	0
$95$	−2.53696	−0.260287
$96$	0	0
$97$	−3.65531	−0.371141	−0.185570	−	0.982631i	$-0.559413\pi$
$97$	−3.65531	−0.371141	−0.185570	+	0.982631i	$0.559413\pi$
$98$	0	0
$99$	−4.91239	−0.493713
$100$	0	0
$101$	0.389120	0.0387189	0.0193595	−	0.999813i	$-0.493837\pi$
$101$	0.389120	0.0387189	0.0193595	+	0.999813i	$0.493837\pi$
$102$	0	0
$103$	0.440128	0.0433671	0.0216835	−	0.999765i	$-0.493097\pi$
$103$	0.440128	0.0433671	0.0216835	+	0.999765i	$0.493097\pi$
$104$	0	0
$105$	−2.53696	−0.247582
$106$	0	0
$107$	5.46962	0.528768	0.264384	−	0.964418i	$-0.414831\pi$
$107$	5.46962	0.528768	0.264384	+	0.964418i	$0.414831\pi$
$108$	0	0
$109$	8.00396	0.766641	0.383320	−	0.923615i	$-0.374781\pi$
$109$	8.00396	0.766641	0.383320	+	0.923615i	$0.374781\pi$
$110$	0	0
$111$	−0.746889	−0.0708916
$112$	0	0
$113$	16.0210	1.50713	0.753565	−	0.657374i	$-0.228333\pi$
$113$	16.0210	1.50713	0.753565	+	0.657374i	$0.228333\pi$
$114$	0	0
$115$	−5.49378	−0.512297
$116$	0	0
$117$	−0.165497	−0.0153002
$118$	0	0
$119$	−6.84768	−0.627726
$120$	0	0
$121$	13.1315	1.19378
$122$	0	0
$123$	3.07392	0.277166
$124$	0	0
$125$	−9.04130	−0.808679
$126$	0	0
$127$	13.9660	1.23929	0.619643	−	0.784884i	$-0.287278\pi$
$127$	13.9660	1.23929	0.619643	+	0.784884i	$0.287278\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.0210	1.57450	0.787251	−	0.616632i	$-0.211503\pi$
$131$	18.0210	1.57450	0.787251	+	0.616632i	$0.211503\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	2.53696	0.218347
$136$	0	0
$137$	5.66901	0.484336	0.242168	−	0.970234i	$-0.422141\pi$
$137$	5.66901	0.484336	0.242168	+	0.970234i	$0.422141\pi$
$138$	0	0
$139$	−0.0700563	−0.00594210	−0.00297105	−	0.999996i	$-0.500946\pi$
$139$	−0.0700563	−0.00594210	−0.00297105	+	0.999996i	$0.500946\pi$
$140$	0	0
$141$	−3.88551	−0.327219
$142$	0	0
$143$	0.812983	0.0679851
$144$	0	0
$145$	11.9299	0.990728
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	13.8208	1.13224	0.566122	−	0.824321i	$-0.308443\pi$
$149$	13.8208	1.13224	0.566122	+	0.824321i	$0.308443\pi$
$150$	0	0
$151$	−3.81820	−0.310720	−0.155360	−	0.987858i	$-0.549654\pi$
$151$	−3.81820	−0.310720	−0.155360	+	0.987858i	$0.549654\pi$
$152$	0	0
$153$	6.84768	0.553602
$154$	0	0
$155$	17.9563	1.44228
$156$	0	0
$157$	−5.97261	−0.476666	−0.238333	−	0.971183i	$-0.576601\pi$
$157$	−5.97261	−0.476666	−0.238333	+	0.971183i	$0.576601\pi$
$158$	0	0
$159$	−0.702457	−0.0557084
$160$	0	0
$161$	2.16550	0.170665
$162$	0	0
$163$	9.95630	0.779838	0.389919	−	0.920849i	$-0.372503\pi$
$163$	9.95630	0.779838	0.389919	+	0.920849i	$0.372503\pi$
$164$	0	0
$165$	−12.4625	−0.970206
$166$	0	0
$167$	−1.56383	−0.121013	−0.0605066	−	0.998168i	$-0.519272\pi$
$167$	−1.56383	−0.121013	−0.0605066	+	0.998168i	$0.519272\pi$
$168$	0	0
$169$	−12.9726	−0.997893
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	23.5444	1.79005	0.895023	−	0.446021i	$-0.147159\pi$
$173$	23.5444	1.79005	0.895023	+	0.446021i	$0.147159\pi$
$174$	0	0
$175$	−1.43617	−0.108564
$176$	0	0
$177$	13.3349	1.00231
$178$	0	0
$179$	−22.7639	−1.70146	−0.850728	−	0.525606i	$-0.823839\pi$
$179$	−22.7639	−1.70146	−0.850728	+	0.525606i	$0.823839\pi$
$180$	0	0
$181$	8.42643	0.626332	0.313166	−	0.949698i	$-0.398610\pi$
$181$	8.42643	0.626332	0.313166	+	0.949698i	$0.398610\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	−1.89483	−0.139310
$186$	0	0
$187$	−33.6385	−2.45989
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	11.9497	0.864652	0.432326	−	0.901717i	$-0.357693\pi$
$191$	11.9497	0.864652	0.432326	+	0.901717i	$0.357693\pi$
$192$	0	0
$193$	18.6214	1.34040	0.670201	−	0.742180i	$-0.266208\pi$
$193$	18.6214	1.34040	0.670201	+	0.742180i	$0.266208\pi$
$194$	0	0
$195$	−0.419858	−0.0300667
$196$	0	0
$197$	−20.7195	−1.47620	−0.738102	−	0.674690i	$-0.764278\pi$
$197$	−20.7195	−1.47620	−0.738102	+	0.674690i	$0.764278\pi$
$198$	0	0
$199$	7.05761	0.500301	0.250150	−	0.968207i	$-0.419520\pi$
$199$	7.05761	0.500301	0.250150	+	0.968207i	$0.419520\pi$
$200$	0	0
$201$	13.0439	0.920047
$202$	0	0
$203$	−4.70246	−0.330048
$204$	0	0
$205$	7.79841	0.544665
$206$	0	0
$207$	−2.16550	−0.150512
$208$	0	0
$209$	4.91239	0.339797
$210$	0	0
$211$	−8.84233	−0.608731	−0.304366	−	0.952555i	$-0.598444\pi$
$211$	−8.84233	−0.608731	−0.304366	+	0.952555i	$0.598444\pi$
$212$	0	0
$213$	8.26629	0.566397
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.07788	−0.480478
$218$	0	0
$219$	−5.05761	−0.341762
$220$	0	0
$221$	−1.13327	−0.0762319
$222$	0	0
$223$	−8.28121	−0.554551	−0.277275	−	0.960791i	$-0.589431\pi$
$223$	−8.28121	−0.554551	−0.277275	+	0.960791i	$0.589431\pi$
$224$	0	0
$225$	1.43617	0.0957444
$226$	0	0
$227$	−3.09815	−0.205632	−0.102816	−	0.994700i	$-0.532785\pi$
$227$	−3.09815	−0.205632	−0.102816	+	0.994700i	$0.532785\pi$
$228$	0	0
$229$	5.24915	0.346874	0.173437	−	0.984845i	$-0.444513\pi$
$229$	5.24915	0.346874	0.173437	+	0.984845i	$0.444513\pi$
$230$	0	0
$231$	4.91239	0.323211
$232$	0	0
$233$	−3.08184	−0.201898	−0.100949	−	0.994892i	$-0.532188\pi$
$233$	−3.08184	−0.201898	−0.100949	+	0.994892i	$0.532188\pi$
$234$	0	0
$235$	−9.85739	−0.643025
$236$	0	0
$237$	−9.22311	−0.599105
$238$	0	0
$239$	13.6055	0.880068	0.440034	−	0.897981i	$-0.354966\pi$
$239$	13.6055	0.880068	0.440034	+	0.897981i	$0.354966\pi$
$240$	0	0
$241$	19.4450	1.25256	0.626280	−	0.779598i	$-0.284577\pi$
$241$	19.4450	1.25256	0.626280	+	0.779598i	$0.284577\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.53696	0.162080
$246$	0	0
$247$	0.165497	0.0105303
$248$	0	0
$249$	−8.19623	−0.519415
$250$	0	0
$251$	3.90578	0.246531	0.123265	−	0.992374i	$-0.460663\pi$
$251$	3.90578	0.246531	0.123265	+	0.992374i	$0.460663\pi$
$252$	0	0
$253$	10.6378	0.668790
$254$	0	0
$255$	17.3723	1.08790
$256$	0	0
$257$	−23.9563	−1.49435	−0.747177	−	0.664626i	$-0.768591\pi$
$257$	−23.9563	−1.49435	−0.747177	+	0.664626i	$0.768591\pi$
$258$	0	0
$259$	0.746889	0.0464094
$260$	0	0
$261$	4.70246	0.291075
$262$	0	0
$263$	−22.3671	−1.37921	−0.689607	−	0.724184i	$-0.742217\pi$
$263$	−22.3671	−1.37921	−0.689607	+	0.724184i	$0.742217\pi$
$264$	0	0
$265$	−1.78210	−0.109474
$266$	0	0
$267$	−9.38861	−0.574573
$268$	0	0
$269$	3.98369	0.242890	0.121445	−	0.992598i	$-0.461247\pi$
$269$	3.98369	0.242890	0.121445	+	0.992598i	$0.461247\pi$
$270$	0	0
$271$	5.53432	0.336186	0.168093	−	0.985771i	$-0.446239\pi$
$271$	5.53432	0.336186	0.168093	+	0.985771i	$0.446239\pi$
$272$	0	0
$273$	0.165497	0.0100163
$274$	0	0
$275$	−7.05500	−0.425432
$276$	0	0
$277$	11.2408	0.675392	0.337696	−	0.941255i	$-0.390352\pi$
$277$	11.2408	0.675392	0.337696	+	0.941255i	$0.390352\pi$
$278$	0	0
$279$	7.07788	0.423742
$280$	0	0
$281$	−13.7764	−0.821830	−0.410915	−	0.911674i	$-0.634791\pi$
$281$	−13.7764	−0.821830	−0.410915	+	0.911674i	$0.634791\pi$
$282$	0	0
$283$	−23.3533	−1.38821	−0.694105	−	0.719874i	$-0.744199\pi$
$283$	−23.3533	−1.38821	−0.694105	+	0.719874i	$0.744199\pi$
$284$	0	0
$285$	−2.53696	−0.150277
$286$	0	0
$287$	−3.07392	−0.181448
$288$	0	0
$289$	29.8908	1.75828
$290$	0	0
$291$	−3.65531	−0.214278
$292$	0	0
$293$	0.726619	0.0424495	0.0212248	−	0.999775i	$-0.493243\pi$
$293$	0.726619	0.0424495	0.0212248	+	0.999775i	$0.493243\pi$
$294$	0	0
$295$	33.8300	1.96966
$296$	0	0
$297$	−4.91239	−0.285046
$298$	0	0
$299$	0.358382	0.0207258
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.389120	0.0223544
$304$	0	0
$305$	15.2218	0.871595
$306$	0	0
$307$	−17.3349	−0.989353	−0.494676	−	0.869077i	$-0.664713\pi$
$307$	−17.3349	−0.989353	−0.494676	+	0.869077i	$0.664713\pi$
$308$	0	0
$309$	0.440128	0.0250380
$310$	0	0
$311$	−3.56244	−0.202008	−0.101004	−	0.994886i	$-0.532205\pi$
$311$	−3.56244	−0.202008	−0.101004	+	0.994886i	$0.532205\pi$
$312$	0	0
$313$	−17.4827	−0.988180	−0.494090	−	0.869411i	$-0.664499\pi$
$313$	−17.4827	−0.988180	−0.494090	+	0.869411i	$0.664499\pi$
$314$	0	0
$315$	−2.53696	−0.142941
$316$	0	0
$317$	2.02416	0.113688	0.0568442	−	0.998383i	$-0.481896\pi$
$317$	2.02416	0.113688	0.0568442	+	0.998383i	$0.481896\pi$
$318$	0	0
$319$	−23.1003	−1.29337
$320$	0	0
$321$	5.46962	0.305284
$322$	0	0
$323$	−6.84768	−0.381015
$324$	0	0
$325$	−0.237681	−0.0131841
$326$	0	0
$327$	8.00396	0.442620
$328$	0	0
$329$	3.88551	0.214215
$330$	0	0
$331$	−1.39438	−0.0766418	−0.0383209	−	0.999265i	$-0.512201\pi$
$331$	−1.39438	−0.0766418	−0.0383209	+	0.999265i	$0.512201\pi$
$332$	0	0
$333$	−0.746889	−0.0409293
$334$	0	0
$335$	33.0919	1.80800
$336$	0	0
$337$	−5.54753	−0.302193	−0.151097	−	0.988519i	$-0.548280\pi$
$337$	−5.54753	−0.302193	−0.151097	+	0.988519i	$0.548280\pi$
$338$	0	0
$339$	16.0210	0.870142
$340$	0	0
$341$	−34.7693	−1.88286
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.49378	−0.295775
$346$	0	0
$347$	−14.0268	−0.753000	−0.376500	−	0.926417i	$-0.622873\pi$
$347$	−14.0268	−0.753000	−0.376500	+	0.926417i	$0.622873\pi$
$348$	0	0
$349$	−17.3591	−0.929211	−0.464605	−	0.885518i	$-0.653804\pi$
$349$	−17.3591	−0.929211	−0.464605	+	0.885518i	$0.653804\pi$
$350$	0	0
$351$	−0.165497	−0.00883355
$352$	0	0
$353$	−14.0694	−0.748841	−0.374420	−	0.927259i	$-0.622158\pi$
$353$	−14.0694	−0.748841	−0.374420	+	0.927259i	$0.622158\pi$
$354$	0	0
$355$	20.9712	1.11304
$356$	0	0
$357$	−6.84768	−0.362418
$358$	0	0
$359$	−14.1381	−0.746181	−0.373090	−	0.927795i	$-0.621702\pi$
$359$	−14.1381	−0.746181	−0.373090	+	0.927795i	$0.621702\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	13.1315	0.689227
$364$	0	0
$365$	−12.8310	−0.671603
$366$	0	0
$367$	−16.3925	−0.855680	−0.427840	−	0.903855i	$-0.640725\pi$
$367$	−16.3925	−0.855680	−0.427840	+	0.903855i	$0.640725\pi$
$368$	0	0
$369$	3.07392	0.160022
$370$	0	0
$371$	0.702457	0.0364697
$372$	0	0
$373$	25.0779	1.29848	0.649242	−	0.760582i	$-0.275086\pi$
$373$	25.0779	1.29848	0.649242	+	0.760582i	$0.275086\pi$
$374$	0	0
$375$	−9.04130	−0.466891
$376$	0	0
$377$	−0.778240	−0.0400814
$378$	0	0
$379$	9.73329	0.499966	0.249983	−	0.968250i	$-0.419575\pi$
$379$	9.73329	0.499966	0.249983	+	0.968250i	$0.419575\pi$
$380$	0	0
$381$	13.9660	0.715502
$382$	0	0
$383$	−36.0420	−1.84166	−0.920830	−	0.389963i	$-0.872488\pi$
$383$	−36.0420	−1.84166	−0.920830	+	0.389963i	$0.872488\pi$
$384$	0	0
$385$	12.4625	0.635149
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.9915	0.557292	0.278646	−	0.960394i	$-0.410114\pi$
$389$	10.9915	0.557292	0.278646	+	0.960394i	$0.410114\pi$
$390$	0	0
$391$	−14.8286	−0.749916
$392$	0	0
$393$	18.0210	0.909039
$394$	0	0
$395$	−23.3987	−1.17731
$396$	0	0
$397$	13.3185	0.668439	0.334219	−	0.942495i	$-0.391527\pi$
$397$	13.3185	0.668439	0.334219	+	0.942495i	$0.391527\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	6.27422	0.313319	0.156660	−	0.987653i	$-0.449927\pi$
$401$	6.27422	0.313319	0.156660	+	0.987653i	$0.449927\pi$
$402$	0	0
$403$	−1.17137	−0.0583499
$404$	0	0
$405$	2.53696	0.126063
$406$	0	0
$407$	3.66901	0.181866
$408$	0	0
$409$	−38.0315	−1.88054	−0.940268	−	0.340436i	$-0.889425\pi$
$409$	−38.0315	−1.88054	−0.940268	+	0.340436i	$0.889425\pi$
$410$	0	0
$411$	5.66901	0.279631
$412$	0	0
$413$	−13.3349	−0.656165
$414$	0	0
$415$	−20.7935	−1.02071
$416$	0	0
$417$	−0.0700563	−0.00343067
$418$	0	0
$419$	29.0598	1.41966	0.709832	−	0.704371i	$-0.248771\pi$
$419$	29.0598	1.41966	0.709832	+	0.704371i	$0.248771\pi$
$420$	0	0
$421$	−27.1042	−1.32098	−0.660490	−	0.750835i	$-0.729651\pi$
$421$	−27.1042	−1.32098	−0.660490	+	0.750835i	$0.729651\pi$
$422$	0	0
$423$	−3.88551	−0.188920
$424$	0	0
$425$	9.83441	0.477039
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	0.812983	0.0392512
$430$	0	0
$431$	−13.0092	−0.626632	−0.313316	−	0.949649i	$-0.601440\pi$
$431$	−13.0092	−0.626632	−0.313316	+	0.949649i	$0.601440\pi$
$432$	0	0
$433$	−14.4651	−0.695150	−0.347575	−	0.937652i	$-0.612995\pi$
$433$	−14.4651	−0.695150	−0.347575	+	0.937652i	$0.612995\pi$
$434$	0	0
$435$	11.9299	0.571997
$436$	0	0
$437$	2.16550	0.103590
$438$	0	0
$439$	28.8753	1.37814	0.689071	−	0.724694i	$-0.258019\pi$
$439$	28.8753	1.37814	0.689071	+	0.724694i	$0.258019\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	16.7556	0.796082	0.398041	−	0.917368i	$-0.369690\pi$
$443$	16.7556	0.796082	0.398041	+	0.917368i	$0.369690\pi$
$444$	0	0
$445$	−23.8185	−1.12911
$446$	0	0
$447$	13.8208	0.653702
$448$	0	0
$449$	10.2468	0.483578	0.241789	−	0.970329i	$-0.422266\pi$
$449$	10.2468	0.483578	0.241789	+	0.970329i	$0.422266\pi$
$450$	0	0
$451$	−15.1003	−0.711044
$452$	0	0
$453$	−3.81820	−0.179394
$454$	0	0
$455$	0.419858	0.0196833
$456$	0	0
$457$	32.5403	1.52217	0.761086	−	0.648651i	$-0.224667\pi$
$457$	32.5403	1.52217	0.761086	+	0.648651i	$0.224667\pi$
$458$	0	0
$459$	6.84768	0.319622
$460$	0	0
$461$	−35.2199	−1.64035	−0.820177	−	0.572110i	$-0.806125\pi$
$461$	−35.2199	−1.64035	−0.820177	+	0.572110i	$0.806125\pi$
$462$	0	0
$463$	29.2755	1.36055	0.680274	−	0.732958i	$-0.261861\pi$
$463$	29.2755	1.36055	0.680274	+	0.732958i	$0.261861\pi$
$464$	0	0
$465$	17.9563	0.832704
$466$	0	0
$467$	39.6196	1.83338	0.916688	−	0.399604i	$-0.130852\pi$
$467$	39.6196	1.83338	0.916688	+	0.399604i	$0.130852\pi$
$468$	0	0
$469$	−13.0439	−0.602312
$470$	0	0
$471$	−5.97261	−0.275203
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.43617	−0.0658958
$476$	0	0
$477$	−0.702457	−0.0321633
$478$	0	0
$479$	16.1760	0.739099	0.369549	−	0.929211i	$-0.379512\pi$
$479$	16.1760	0.739099	0.369549	+	0.929211i	$0.379512\pi$
$480$	0	0
$481$	0.123608	0.00563602
$482$	0	0
$483$	2.16550	0.0985335
$484$	0	0
$485$	−9.27338	−0.421083
$486$	0	0
$487$	−6.41851	−0.290850	−0.145425	−	0.989369i	$-0.546455\pi$
$487$	−6.41851	−0.290850	−0.145425	+	0.989369i	$0.546455\pi$
$488$	0	0
$489$	9.95630	0.450240
$490$	0	0
$491$	−35.1217	−1.58502	−0.792510	−	0.609859i	$-0.791226\pi$
$491$	−35.1217	−1.58502	−0.792510	+	0.609859i	$0.791226\pi$
$492$	0	0
$493$	32.2009	1.45026
$494$	0	0
$495$	−12.4625	−0.560149
$496$	0	0
$497$	−8.26629	−0.370794
$498$	0	0
$499$	−12.9712	−0.580673	−0.290336	−	0.956925i	$-0.593767\pi$
$499$	−12.9712	−0.580673	−0.290336	+	0.956925i	$0.593767\pi$
$500$	0	0
$501$	−1.56383	−0.0698670
$502$	0	0
$503$	29.2225	1.30297	0.651483	−	0.758663i	$-0.274147\pi$
$503$	29.2225	1.30297	0.651483	+	0.758663i	$0.274147\pi$
$504$	0	0
$505$	0.987182	0.0439290
$506$	0	0
$507$	−12.9726	−0.576134
$508$	0	0
$509$	−36.9414	−1.63740	−0.818698	−	0.574224i	$-0.805304\pi$
$509$	−36.9414	−1.63740	−0.818698	+	0.574224i	$0.805304\pi$
$510$	0	0
$511$	5.05761	0.223736
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	1.11659	0.0492027
$516$	0	0
$517$	19.0871	0.839451
$518$	0	0
$519$	23.5444	1.03348
$520$	0	0
$521$	−12.6298	−0.553323	−0.276661	−	0.960967i	$-0.589228\pi$
$521$	−12.6298	−0.553323	−0.276661	+	0.960967i	$0.589228\pi$
$522$	0	0
$523$	−26.5778	−1.16216	−0.581082	−	0.813845i	$-0.697370\pi$
$523$	−26.5778	−1.16216	−0.581082	+	0.813845i	$0.697370\pi$
$524$	0	0
$525$	−1.43617	−0.0626794
$526$	0	0
$527$	48.4671	2.11126
$528$	0	0
$529$	−18.3106	−0.796114
$530$	0	0
$531$	13.3349	0.578683
$532$	0	0
$533$	−0.508723	−0.0220352
$534$	0	0
$535$	13.8762	0.599920
$536$	0	0
$537$	−22.7639	−0.982336
$538$	0	0
$539$	−4.91239	−0.211591
$540$	0	0
$541$	38.4757	1.65420	0.827099	−	0.562056i	$-0.189989\pi$
$541$	38.4757	1.65420	0.827099	+	0.562056i	$0.189989\pi$
$542$	0	0
$543$	8.42643	0.361613
$544$	0	0
$545$	20.3057	0.869802
$546$	0	0
$547$	22.7993	0.974827	0.487414	−	0.873171i	$-0.337940\pi$
$547$	22.7993	0.974827	0.487414	+	0.873171i	$0.337940\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	−4.70246	−0.200331
$552$	0	0
$553$	9.22311	0.392206
$554$	0	0
$555$	−1.89483	−0.0804309
$556$	0	0
$557$	−7.45470	−0.315866	−0.157933	−	0.987450i	$-0.550483\pi$
$557$	−7.45470	−0.315866	−0.157933	+	0.987450i	$0.550483\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−33.6385	−1.42022
$562$	0	0
$563$	27.0576	1.14034	0.570171	−	0.821526i	$-0.306877\pi$
$563$	27.0576	1.14034	0.570171	+	0.821526i	$0.306877\pi$
$564$	0	0
$565$	40.6446	1.70993
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−26.3256	−1.10363	−0.551814	−	0.833967i	$-0.686064\pi$
$569$	−26.3256	−1.10363	−0.551814	+	0.833967i	$0.686064\pi$
$570$	0	0
$571$	2.96875	0.124238	0.0621191	−	0.998069i	$-0.480214\pi$
$571$	2.96875	0.124238	0.0621191	+	0.998069i	$0.480214\pi$
$572$	0	0
$573$	11.9497	0.499207
$574$	0	0
$575$	−3.11001	−0.129696
$576$	0	0
$577$	−24.1637	−1.00595	−0.502974	−	0.864302i	$-0.667761\pi$
$577$	−24.1637	−1.00595	−0.502974	+	0.864302i	$0.667761\pi$
$578$	0	0
$579$	18.6214	0.773881
$580$	0	0
$581$	8.19623	0.340037
$582$	0	0
$583$	3.45074	0.142915
$584$	0	0
$585$	−0.419858	−0.0173590
$586$	0	0
$587$	−22.2226	−0.917225	−0.458612	−	0.888636i	$-0.651653\pi$
$587$	−22.2226	−0.917225	−0.458612	+	0.888636i	$0.651653\pi$
$588$	0	0
$589$	−7.07788	−0.291639
$590$	0	0
$591$	−20.7195	−0.852286
$592$	0	0
$593$	22.9576	0.942757	0.471378	−	0.881931i	$-0.343757\pi$
$593$	22.9576	0.942757	0.471378	+	0.881931i	$0.343757\pi$
$594$	0	0
$595$	−17.3723	−0.712195
$596$	0	0
$597$	7.05761	0.288849
$598$	0	0
$599$	26.2847	1.07396	0.536982	−	0.843593i	$-0.319564\pi$
$599$	26.2847	1.07396	0.536982	+	0.843593i	$0.319564\pi$
$600$	0	0
$601$	−12.9229	−0.527136	−0.263568	−	0.964641i	$-0.584899\pi$
$601$	−12.9229	−0.527136	−0.263568	+	0.964641i	$0.584899\pi$
$602$	0	0
$603$	13.0439	0.531189
$604$	0	0
$605$	33.3142	1.35441
$606$	0	0
$607$	36.4127	1.47795	0.738974	−	0.673734i	$-0.235311\pi$
$607$	36.4127	1.47795	0.738974	+	0.673734i	$0.235311\pi$
$608$	0	0
$609$	−4.70246	−0.190553
$610$	0	0
$611$	0.643039	0.0260146
$612$	0	0
$613$	35.5444	1.43562	0.717812	−	0.696237i	$-0.245144\pi$
$613$	35.5444	1.43562	0.717812	+	0.696237i	$0.245144\pi$
$614$	0	0
$615$	7.79841	0.314462
$616$	0	0
$617$	−14.3231	−0.576625	−0.288313	−	0.957536i	$-0.593094\pi$
$617$	−14.3231	−0.576625	−0.288313	+	0.957536i	$0.593094\pi$
$618$	0	0
$619$	−2.83934	−0.114123	−0.0570614	−	0.998371i	$-0.518173\pi$
$619$	−2.83934	−0.114123	−0.0570614	+	0.998371i	$0.518173\pi$
$620$	0	0
$621$	−2.16550	−0.0868984
$622$	0	0
$623$	9.38861	0.376147
$624$	0	0
$625$	−30.1183	−1.20473
$626$	0	0
$627$	4.91239	0.196182
$628$	0	0
$629$	−5.11446	−0.203927
$630$	0	0
$631$	15.4827	0.616356	0.308178	−	0.951329i	$-0.400281\pi$
$631$	15.4827	0.616356	0.308178	+	0.951329i	$0.400281\pi$
$632$	0	0
$633$	−8.84233	−0.351451
$634$	0	0
$635$	35.4313	1.40605
$636$	0	0
$637$	−0.165497	−0.00655721
$638$	0	0
$639$	8.26629	0.327009
$640$	0	0
$641$	−14.0321	−0.554234	−0.277117	−	0.960836i	$-0.589379\pi$
$641$	−14.0321	−0.554234	−0.277117	+	0.960836i	$0.589379\pi$
$642$	0	0
$643$	2.02636	0.0799118	0.0399559	−	0.999201i	$-0.487278\pi$
$643$	2.02636	0.0799118	0.0399559	+	0.999201i	$0.487278\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.9250	1.49099	0.745493	−	0.666513i	$-0.232214\pi$
$647$	37.9250	1.49099	0.745493	+	0.666513i	$0.232214\pi$
$648$	0	0
$649$	−65.5060	−2.57133
$650$	0	0
$651$	−7.07788	−0.277404
$652$	0	0
$653$	−41.0100	−1.60484	−0.802422	−	0.596757i	$-0.796455\pi$
$653$	−41.0100	−1.60484	−0.802422	+	0.596757i	$0.796455\pi$
$654$	0	0
$655$	45.7186	1.78637
$656$	0	0
$657$	−5.05761	−0.197316
$658$	0	0
$659$	−0.389898	−0.0151883	−0.00759414	−	0.999971i	$-0.502417\pi$
$659$	−0.389898	−0.0151883	−0.00759414	+	0.999971i	$0.502417\pi$
$660$	0	0
$661$	24.1734	0.940237	0.470119	−	0.882603i	$-0.344211\pi$
$661$	24.1734	0.940237	0.470119	+	0.882603i	$0.344211\pi$
$662$	0	0
$663$	−1.13327	−0.0440125
$664$	0	0
$665$	2.53696	0.0983791
$666$	0	0
$667$	−10.1832	−0.394293
$668$	0	0
$669$	−8.28121	−0.320170
$670$	0	0
$671$	−29.4743	−1.13784
$672$	0	0
$673$	16.3387	0.629811	0.314906	−	0.949123i	$-0.398027\pi$
$673$	16.3387	0.629811	0.314906	+	0.949123i	$0.398027\pi$
$674$	0	0
$675$	1.43617	0.0552780
$676$	0	0
$677$	−11.6932	−0.449408	−0.224704	−	0.974427i	$-0.572141\pi$
$677$	−11.6932	−0.449408	−0.224704	+	0.974427i	$0.572141\pi$
$678$	0	0
$679$	3.65531	0.140278
$680$	0	0
$681$	−3.09815	−0.118721
$682$	0	0
$683$	14.3115	0.547613	0.273806	−	0.961785i	$-0.411717\pi$
$683$	14.3115	0.547613	0.273806	+	0.961785i	$0.411717\pi$
$684$	0	0
$685$	14.3820	0.549510
$686$	0	0
$687$	5.24915	0.200268
$688$	0	0
$689$	0.116254	0.00442893
$690$	0	0
$691$	46.4109	1.76555	0.882777	−	0.469792i	$-0.155671\pi$
$691$	46.4109	1.76555	0.882777	+	0.469792i	$0.155671\pi$
$692$	0	0
$693$	4.91239	0.186606
$694$	0	0
$695$	−0.177730	−0.00674169
$696$	0	0
$697$	21.0492	0.797296
$698$	0	0
$699$	−3.08184	−0.116566
$700$	0	0
$701$	−23.8825	−0.902029	−0.451014	−	0.892517i	$-0.648938\pi$
$701$	−23.8825	−0.902029	−0.451014	+	0.892517i	$0.648938\pi$
$702$	0	0
$703$	0.746889	0.0281695
$704$	0	0
$705$	−9.85739	−0.371251
$706$	0	0
$707$	−0.389120	−0.0146344
$708$	0	0
$709$	42.8362	1.60875	0.804373	−	0.594124i	$-0.202501\pi$
$709$	42.8362	1.60875	0.804373	+	0.594124i	$0.202501\pi$
$710$	0	0
$711$	−9.22311	−0.345894
$712$	0	0
$713$	−15.3271	−0.574006
$714$	0	0
$715$	2.06251	0.0771333
$716$	0	0
$717$	13.6055	0.508108
$718$	0	0
$719$	−4.07918	−0.152128	−0.0760638	−	0.997103i	$-0.524235\pi$
$719$	−4.07918	−0.152128	−0.0760638	+	0.997103i	$0.524235\pi$
$720$	0	0
$721$	−0.440128	−0.0163912
$722$	0	0
$723$	19.4450	0.723166
$724$	0	0
$725$	6.75351	0.250819
$726$	0	0
$727$	50.0525	1.85635	0.928173	−	0.372150i	$-0.121379\pi$
$727$	50.0525	1.85635	0.928173	+	0.372150i	$0.121379\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	19.5879	0.723494	0.361747	−	0.932276i	$-0.382180\pi$
$733$	19.5879	0.723494	0.361747	+	0.932276i	$0.382180\pi$
$734$	0	0
$735$	2.53696	0.0935772
$736$	0	0
$737$	−64.0768	−2.36030
$738$	0	0
$739$	0.530454	0.0195131	0.00975653	−	0.999952i	$-0.496894\pi$
$739$	0.530454	0.0195131	0.00975653	+	0.999952i	$0.496894\pi$
$740$	0	0
$741$	0.165497	0.00607967
$742$	0	0
$743$	−41.3402	−1.51663	−0.758313	−	0.651891i	$-0.773976\pi$
$743$	−41.3402	−1.51663	−0.758313	+	0.651891i	$0.773976\pi$
$744$	0	0
$745$	35.0628	1.28460
$746$	0	0
$747$	−8.19623	−0.299885
$748$	0	0
$749$	−5.46962	−0.199855
$750$	0	0
$751$	47.7555	1.74262	0.871311	−	0.490731i	$-0.163270\pi$
$751$	47.7555	1.74262	0.871311	+	0.490731i	$0.163270\pi$
$752$	0	0
$753$	3.90578	0.142335
$754$	0	0
$755$	−9.68661	−0.352532
$756$	0	0
$757$	−7.75259	−0.281773	−0.140886	−	0.990026i	$-0.544995\pi$
$757$	−7.75259	−0.281773	−0.140886	+	0.990026i	$0.544995\pi$
$758$	0	0
$759$	10.6378	0.386126
$760$	0	0
$761$	−17.3618	−0.629366	−0.314683	−	0.949197i	$-0.601898\pi$
$761$	−17.3618	−0.629366	−0.314683	+	0.949197i	$0.601898\pi$
$762$	0	0
$763$	−8.00396	−0.289763
$764$	0	0
$765$	17.3723	0.628097
$766$	0	0
$767$	−2.20687	−0.0796856
$768$	0	0
$769$	−18.6026	−0.670828	−0.335414	−	0.942071i	$-0.608876\pi$
$769$	−18.6026	−0.670828	−0.335414	+	0.942071i	$0.608876\pi$
$770$	0	0
$771$	−23.9563	−0.862765
$772$	0	0
$773$	3.59122	0.129167	0.0645837	−	0.997912i	$-0.479428\pi$
$773$	3.59122	0.129167	0.0645837	+	0.997912i	$0.479428\pi$
$774$	0	0
$775$	10.1650	0.365138
$776$	0	0
$777$	0.746889	0.0267945
$778$	0	0
$779$	−3.07392	−0.110135
$780$	0	0
$781$	−40.6072	−1.45304
$782$	0	0
$783$	4.70246	0.168052
$784$	0	0
$785$	−15.1523	−0.540808
$786$	0	0
$787$	−40.4157	−1.44066	−0.720332	−	0.693630i	$-0.756010\pi$
$787$	−40.4157	−1.44066	−0.720332	+	0.693630i	$0.756010\pi$
$788$	0	0
$789$	−22.3671	−0.796289
$790$	0	0
$791$	−16.0210	−0.569641
$792$	0	0
$793$	−0.992979	−0.0352617
$794$	0	0
$795$	−1.78210	−0.0632047
$796$	0	0
$797$	16.7124	0.591983	0.295991	−	0.955191i	$-0.404350\pi$
$797$	16.7124	0.591983	0.295991	+	0.955191i	$0.404350\pi$
$798$	0	0
$799$	−26.6067	−0.941279
$800$	0	0
$801$	−9.38861	−0.331730
$802$	0	0
$803$	24.8449	0.876759
$804$	0	0
$805$	5.49378	0.193630
$806$	0	0
$807$	3.98369	0.140233
$808$	0	0
$809$	−33.0324	−1.16136	−0.580678	−	0.814133i	$-0.697212\pi$
$809$	−33.0324	−1.16136	−0.580678	+	0.814133i	$0.697212\pi$
$810$	0	0
$811$	−9.55804	−0.335628	−0.167814	−	0.985819i	$-0.553671\pi$
$811$	−9.55804	−0.335628	−0.167814	+	0.985819i	$0.553671\pi$
$812$	0	0
$813$	5.53432	0.194097
$814$	0	0
$815$	25.2587	0.884775
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0.165497	0.00578292
$820$	0	0
$821$	32.8489	1.14643	0.573217	−	0.819403i	$-0.305695\pi$
$821$	32.8489	1.14643	0.573217	+	0.819403i	$0.305695\pi$
$822$	0	0
$823$	−12.2716	−0.427763	−0.213881	−	0.976860i	$-0.568611\pi$
$823$	−12.2716	−0.427763	−0.213881	+	0.976860i	$0.568611\pi$
$824$	0	0
$825$	−7.05500	−0.245624
$826$	0	0
$827$	−34.9302	−1.21464	−0.607321	−	0.794456i	$-0.707756\pi$
$827$	−34.9302	−1.21464	−0.607321	+	0.794456i	$0.707756\pi$
$828$	0	0
$829$	−13.7099	−0.476163	−0.238082	−	0.971245i	$-0.576519\pi$
$829$	−13.7099	−0.476163	−0.238082	+	0.971245i	$0.576519\pi$
$830$	0	0
$831$	11.2408	0.389938
$832$	0	0
$833$	6.84768	0.237258
$834$	0	0
$835$	−3.96739	−0.137297
$836$	0	0
$837$	7.07788	0.244647
$838$	0	0
$839$	−48.2116	−1.66445	−0.832225	−	0.554438i	$-0.812933\pi$
$839$	−48.2116	−1.66445	−0.832225	+	0.554438i	$0.812933\pi$
$840$	0	0
$841$	−6.88690	−0.237479
$842$	0	0
$843$	−13.7764	−0.474484
$844$	0	0
$845$	−32.9110	−1.13217
$846$	0	0
$847$	−13.1315	−0.451205
$848$	0	0
$849$	−23.3533	−0.801483
$850$	0	0
$851$	1.61739	0.0554433
$852$	0	0
$853$	1.38648	0.0474721	0.0237360	−	0.999718i	$-0.492444\pi$
$853$	1.38648	0.0474721	0.0237360	+	0.999718i	$0.492444\pi$
$854$	0	0
$855$	−2.53696	−0.0867622
$856$	0	0
$857$	−12.1984	−0.416690	−0.208345	−	0.978055i	$-0.566808\pi$
$857$	−12.1984	−0.416690	−0.208345	+	0.978055i	$0.566808\pi$
$858$	0	0
$859$	−20.7293	−0.707273	−0.353637	−	0.935383i	$-0.615055\pi$
$859$	−20.7293	−0.707273	−0.353637	+	0.935383i	$0.615055\pi$
$860$	0	0
$861$	−3.07392	−0.104759
$862$	0	0
$863$	−46.1843	−1.57213	−0.786066	−	0.618142i	$-0.787886\pi$
$863$	−46.1843	−1.57213	−0.786066	+	0.618142i	$0.787886\pi$
$864$	0	0
$865$	59.7311	2.03092
$866$	0	0
$867$	29.8908	1.01514
$868$	0	0
$869$	45.3075	1.53695
$870$	0	0
$871$	−2.15872	−0.0731456
$872$	0	0
$873$	−3.65531	−0.123714
$874$	0	0
$875$	9.04130	0.305652
$876$	0	0
$877$	0.232940	0.00786582	0.00393291	−	0.999992i	$-0.498748\pi$
$877$	0.232940	0.00786582	0.00393291	+	0.999992i	$0.498748\pi$
$878$	0	0
$879$	0.726619	0.0245083
$880$	0	0
$881$	−57.8591	−1.94932	−0.974661	−	0.223686i	$-0.928191\pi$
$881$	−57.8591	−1.94932	−0.974661	+	0.223686i	$0.928191\pi$
$882$	0	0
$883$	−46.6825	−1.57099	−0.785496	−	0.618866i	$-0.787592\pi$
$883$	−46.6825	−1.57099	−0.785496	+	0.618866i	$0.787592\pi$
$884$	0	0
$885$	33.8300	1.13718
$886$	0	0
$887$	−53.0508	−1.78127	−0.890636	−	0.454718i	$-0.849740\pi$
$887$	−53.0508	−1.78127	−0.890636	+	0.454718i	$0.849740\pi$
$888$	0	0
$889$	−13.9660	−0.468406
$890$	0	0
$891$	−4.91239	−0.164571
$892$	0	0
$893$	3.88551	0.130024
$894$	0	0
$895$	−57.7512	−1.93041
$896$	0	0
$897$	0.358382	0.0119660
$898$	0	0
$899$	33.2834	1.11006
$900$	0	0
$901$	−4.81020	−0.160251
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.3775	0.710613
$906$	0	0
$907$	−4.33361	−0.143895	−0.0719475	−	0.997408i	$-0.522921\pi$
$907$	−4.33361	−0.143895	−0.0719475	+	0.997408i	$0.522921\pi$
$908$	0	0
$909$	0.389120	0.0129063
$910$	0	0
$911$	−58.7959	−1.94799	−0.973997	−	0.226559i	$-0.927252\pi$
$911$	−58.7959	−1.94799	−0.973997	+	0.226559i	$0.927252\pi$
$912$	0	0
$913$	40.2631	1.33251
$914$	0	0
$915$	15.2218	0.503216
$916$	0	0
$917$	−18.0210	−0.595106
$918$	0	0
$919$	−4.31153	−0.142224	−0.0711121	−	0.997468i	$-0.522655\pi$
$919$	−4.31153	−0.142224	−0.0711121	+	0.997468i	$0.522655\pi$
$920$	0	0
$921$	−17.3349	−0.571203
$922$	0	0
$923$	−1.36804	−0.0450297
$924$	0	0
$925$	−1.07266	−0.0352687
$926$	0	0
$927$	0.440128	0.0144557
$928$	0	0
$929$	−51.2454	−1.68131	−0.840653	−	0.541574i	$-0.817829\pi$
$929$	−51.2454	−1.68131	−0.840653	+	0.541574i	$0.817829\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−3.56244	−0.116629
$934$	0	0
$935$	−85.3394	−2.79090
$936$	0	0
$937$	−7.97577	−0.260557	−0.130278	−	0.991477i	$-0.541587\pi$
$937$	−7.97577	−0.260557	−0.130278	+	0.991477i	$0.541587\pi$
$938$	0	0
$939$	−17.4827	−0.570526
$940$	0	0
$941$	36.9087	1.20319	0.601595	−	0.798801i	$-0.294532\pi$
$941$	36.9087	1.20319	0.601595	+	0.798801i	$0.294532\pi$
$942$	0	0
$943$	−6.65656	−0.216768
$944$	0	0
$945$	−2.53696	−0.0825273
$946$	0	0
$947$	−39.3586	−1.27898	−0.639491	−	0.768798i	$-0.720855\pi$
$947$	−39.3586	−1.27898	−0.639491	+	0.768798i	$0.720855\pi$
$948$	0	0
$949$	0.837018	0.0271707
$950$	0	0
$951$	2.02416	0.0656380
$952$	0	0
$953$	−39.4075	−1.27653	−0.638267	−	0.769815i	$-0.720348\pi$
$953$	−39.4075	−1.27653	−0.638267	+	0.769815i	$0.720348\pi$
$954$	0	0
$955$	30.3160	0.981002
$956$	0	0
$957$	−23.1003	−0.746726
$958$	0	0
$959$	−5.66901	−0.183062
$960$	0	0
$961$	19.0964	0.616013
$962$	0	0
$963$	5.46962	0.176256
$964$	0	0
$965$	47.2419	1.52077
$966$	0	0
$967$	48.0636	1.54562	0.772811	−	0.634637i	$-0.218850\pi$
$967$	48.0636	1.54562	0.772811	+	0.634637i	$0.218850\pi$
$968$	0	0
$969$	−6.84768	−0.219979
$970$	0	0
$971$	−24.1041	−0.773539	−0.386769	−	0.922176i	$-0.626409\pi$
$971$	−24.1041	−0.773539	−0.386769	+	0.922176i	$0.626409\pi$
$972$	0	0
$973$	0.0700563	0.00224590
$974$	0	0
$975$	−0.237681	−0.00761187
$976$	0	0
$977$	33.0840	1.05845	0.529226	−	0.848481i	$-0.322482\pi$
$977$	33.0840	1.05845	0.529226	+	0.848481i	$0.322482\pi$
$978$	0	0
$979$	46.1205	1.47402
$980$	0	0
$981$	8.00396	0.255547
$982$	0	0
$983$	15.2724	0.487112	0.243556	−	0.969887i	$-0.421686\pi$
$983$	15.2724	0.487112	0.243556	+	0.969887i	$0.421686\pi$
$984$	0	0
$985$	−52.5645	−1.67485
$986$	0	0
$987$	3.88551	0.123677
$988$	0	0
$989$	0	0
$990$	0	0
$991$	13.5946	0.431848	0.215924	−	0.976410i	$-0.430724\pi$
$991$	13.5946	0.431848	0.215924	+	0.976410i	$0.430724\pi$
$992$	0	0
$993$	−1.39438	−0.0442492
$994$	0	0
$995$	17.9049	0.567623
$996$	0	0
$997$	5.15506	0.163262	0.0816312	−	0.996663i	$-0.473987\pi$
$997$	5.15506	0.163262	0.0816312	+	0.996663i	$0.473987\pi$
$998$	0	0
$999$	−0.746889	−0.0236305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.ce.1.4		5
4.3	odd	2		3192.2.a.ba.1.4	✓	5
12.11	even	2		9576.2.a.co.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.ba.1.4	✓	5	4.3	odd	2
6384.2.a.ce.1.4		5	1.1	even	1	trivial
9576.2.a.co.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.53696 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.53696 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.91239 q^{11} -0.165497 q^{13} +2.53696 q^{15} +6.84768 q^{17} -1.00000 q^{19} -1.00000 q^{21} -2.16550 q^{23} +1.43617 q^{25} +1.00000 q^{27} +4.70246 q^{29} +7.07788 q^{31} -4.91239 q^{33} -2.53696 q^{35} -0.746889 q^{37} -0.165497 q^{39} +3.07392 q^{41} +2.53696 q^{45} -3.88551 q^{47} +1.00000 q^{49} +6.84768 q^{51} -0.702457 q^{53} -12.4625 q^{55} -1.00000 q^{57} +13.3349 q^{59} +6.00000 q^{61} -1.00000 q^{63} -0.419858 q^{65} +13.0439 q^{67} -2.16550 q^{69} +8.26629 q^{71} -5.05761 q^{73} +1.43617 q^{75} +4.91239 q^{77} -9.22311 q^{79} +1.00000 q^{81} -8.19623 q^{83} +17.3723 q^{85} +4.70246 q^{87} -9.38861 q^{89} +0.165497 q^{91} +7.07788 q^{93} -2.53696 q^{95} -3.65531 q^{97} -4.91239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 2 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more