LMFDB - Embedded newform 3856.2.a.n.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3856,2,Mod(1,3856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3856 = 2^{4} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3856.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:	$30.7903150194$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.28632$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.126224 q^{3} +0.612768 q^{5} -1.03110 q^{7} -2.98407 q^{9} +O(q^{10})$	$q+0.126224 q^{3} +0.612768 q^{5} -1.03110 q^{7} -2.98407 q^{9} -0.227935 q^{11} +3.38088 q^{13} +0.0773462 q^{15} +7.12130 q^{17} -3.40112 q^{19} -0.130150 q^{21} -6.91488 q^{23} -4.62452 q^{25} -0.755335 q^{27} +0.569431 q^{29} -4.93697 q^{31} -0.0287710 q^{33} -0.631826 q^{35} -5.37832 q^{37} +0.426749 q^{39} +10.7559 q^{41} +0.910247 q^{43} -1.82854 q^{45} +8.50333 q^{47} -5.93683 q^{49} +0.898882 q^{51} -7.76696 q^{53} -0.139671 q^{55} -0.429304 q^{57} -11.2505 q^{59} -3.65450 q^{61} +3.07688 q^{63} +2.07169 q^{65} +12.0694 q^{67} -0.872826 q^{69} +9.48630 q^{71} -7.10488 q^{73} -0.583726 q^{75} +0.235024 q^{77} -0.366201 q^{79} +8.85686 q^{81} -17.8030 q^{83} +4.36370 q^{85} +0.0718760 q^{87} -7.54236 q^{89} -3.48603 q^{91} -0.623166 q^{93} -2.08410 q^{95} -7.85505 q^{97} +0.680174 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	$ 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) $ 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * 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
$3$	0.126224	0.0728757	0.0364378	−	0.999336i	$-0.488399\pi$
$3$	0.126224	0.0728757	0.0364378	+	0.999336i	$0.488399\pi$
$4$	0	0
$5$	0.612768	0.274038	0.137019	−	0.990568i	$-0.456248\pi$
$5$	0.612768	0.274038	0.137019	+	0.990568i	$0.456248\pi$
$6$	0	0
$7$	−1.03110	−0.389720	−0.194860	−	0.980831i	$-0.562425\pi$
$7$	−1.03110	−0.389720	−0.194860	+	0.980831i	$0.562425\pi$
$8$	0	0
$9$	−2.98407	−0.994689
$10$	0	0
$11$	−0.227935	−0.0687251	−0.0343625	−	0.999409i	$-0.510940\pi$
$11$	−0.227935	−0.0687251	−0.0343625	+	0.999409i	$0.510940\pi$
$12$	0	0
$13$	3.38088	0.937686	0.468843	−	0.883281i	$-0.344671\pi$
$13$	3.38088	0.937686	0.468843	+	0.883281i	$0.344671\pi$
$14$	0	0
$15$	0.0773462	0.0199707
$16$	0	0
$17$	7.12130	1.72717	0.863585	−	0.504203i	$-0.168214\pi$
$17$	7.12130	1.72717	0.863585	+	0.504203i	$0.168214\pi$
$18$	0	0
$19$	−3.40112	−0.780270	−0.390135	−	0.920758i	$-0.627572\pi$
$19$	−3.40112	−0.780270	−0.390135	+	0.920758i	$0.627572\pi$
$20$	0	0
$21$	−0.130150	−0.0284011
$22$	0	0
$23$	−6.91488	−1.44185	−0.720926	−	0.693012i	$-0.756283\pi$
$23$	−6.91488	−1.44185	−0.720926	+	0.693012i	$0.756283\pi$
$24$	0	0
$25$	−4.62452	−0.924903
$26$	0	0
$27$	−0.755335	−0.145364
$28$	0	0
$29$	0.569431	0.105741	0.0528703	−	0.998601i	$-0.483163\pi$
$29$	0.569431	0.105741	0.0528703	+	0.998601i	$0.483163\pi$
$30$	0	0
$31$	−4.93697	−0.886706	−0.443353	−	0.896347i	$-0.646211\pi$
$31$	−4.93697	−0.886706	−0.443353	+	0.896347i	$0.646211\pi$
$32$	0	0
$33$	−0.0287710	−0.00500838
$34$	0	0
$35$	−0.631826	−0.106798
$36$	0	0
$37$	−5.37832	−0.884190	−0.442095	−	0.896968i	$-0.645765\pi$
$37$	−5.37832	−0.884190	−0.442095	+	0.896968i	$0.645765\pi$
$38$	0	0
$39$	0.426749	0.0683345
$40$	0	0
$41$	10.7559	1.67979	0.839897	−	0.542747i	$-0.182616\pi$
$41$	10.7559	1.67979	0.839897	+	0.542747i	$0.182616\pi$
$42$	0	0
$43$	0.910247	0.138811	0.0694057	−	0.997589i	$-0.477890\pi$
$43$	0.910247	0.138811	0.0694057	+	0.997589i	$0.477890\pi$
$44$	0	0
$45$	−1.82854	−0.272583
$46$	0	0
$47$	8.50333	1.24034	0.620169	−	0.784468i	$-0.287064\pi$
$47$	8.50333	1.24034	0.620169	+	0.784468i	$0.287064\pi$
$48$	0	0
$49$	−5.93683	−0.848118
$50$	0	0
$51$	0.898882	0.125869
$52$	0	0
$53$	−7.76696	−1.06687	−0.533437	−	0.845840i	$-0.679100\pi$
$53$	−7.76696	−1.06687	−0.533437	+	0.845840i	$0.679100\pi$
$54$	0	0
$55$	−0.139671	−0.0188333
$56$	0	0
$57$	−0.429304	−0.0568627
$58$	0	0
$59$	−11.2505	−1.46468	−0.732342	−	0.680937i	$-0.761573\pi$
$59$	−11.2505	−1.46468	−0.732342	+	0.680937i	$0.761573\pi$
$60$	0	0
$61$	−3.65450	−0.467910	−0.233955	−	0.972247i	$-0.575167\pi$
$61$	−3.65450	−0.467910	−0.233955	+	0.972247i	$0.575167\pi$
$62$	0	0
$63$	3.07688	0.387650
$64$	0	0
$65$	2.07169	0.256962
$66$	0	0
$67$	12.0694	1.47451	0.737257	−	0.675613i	$-0.236121\pi$
$67$	12.0694	1.47451	0.737257	+	0.675613i	$0.236121\pi$
$68$	0	0
$69$	−0.872826	−0.105076
$70$	0	0
$71$	9.48630	1.12582	0.562908	−	0.826519i	$-0.309682\pi$
$71$	9.48630	1.12582	0.562908	+	0.826519i	$0.309682\pi$
$72$	0	0
$73$	−7.10488	−0.831563	−0.415782	−	0.909464i	$-0.636492\pi$
$73$	−7.10488	−0.831563	−0.415782	+	0.909464i	$0.636492\pi$
$74$	0	0
$75$	−0.583726	−0.0674029
$76$	0	0
$77$	0.235024	0.0267835
$78$	0	0
$79$	−0.366201	−0.0412008	−0.0206004	−	0.999788i	$-0.506558\pi$
$79$	−0.366201	−0.0412008	−0.0206004	+	0.999788i	$0.506558\pi$
$80$	0	0
$81$	8.85686	0.984096
$82$	0	0
$83$	−17.8030	−1.95413	−0.977067	−	0.212932i	$-0.931699\pi$
$83$	−17.8030	−1.95413	−0.977067	+	0.212932i	$0.931699\pi$
$84$	0	0
$85$	4.36370	0.473310
$86$	0	0
$87$	0.0718760	0.00770592
$88$	0	0
$89$	−7.54236	−0.799489	−0.399744	−	0.916627i	$-0.630901\pi$
$89$	−7.54236	−0.799489	−0.399744	+	0.916627i	$0.630901\pi$
$90$	0	0
$91$	−3.48603	−0.365435
$92$	0	0
$93$	−0.623166	−0.0646193
$94$	0	0
$95$	−2.08410	−0.213824
$96$	0	0
$97$	−7.85505	−0.797560	−0.398780	−	0.917047i	$-0.630566\pi$
$97$	−7.85505	−0.797560	−0.398780	+	0.917047i	$0.630566\pi$
$98$	0	0
$99$	0.680174	0.0683601
$100$	0	0
$101$	17.4801	1.73933	0.869666	−	0.493640i	$-0.164334\pi$
$101$	17.4801	1.73933	0.869666	+	0.493640i	$0.164334\pi$
$102$	0	0
$103$	−6.10361	−0.601407	−0.300703	−	0.953718i	$-0.597221\pi$
$103$	−6.10361	−0.601407	−0.300703	+	0.953718i	$0.597221\pi$
$104$	0	0
$105$	−0.0797518	−0.00778298
$106$	0	0
$107$	−7.86203	−0.760051	−0.380026	−	0.924976i	$-0.624085\pi$
$107$	−7.86203	−0.760051	−0.380026	+	0.924976i	$0.624085\pi$
$108$	0	0
$109$	−4.17241	−0.399644	−0.199822	−	0.979832i	$-0.564036\pi$
$109$	−4.17241	−0.399644	−0.199822	+	0.979832i	$0.564036\pi$
$110$	0	0
$111$	−0.678875	−0.0644359
$112$	0	0
$113$	3.62203	0.340732	0.170366	−	0.985381i	$-0.445505\pi$
$113$	3.62203	0.340732	0.170366	+	0.985381i	$0.445505\pi$
$114$	0	0
$115$	−4.23721	−0.395122
$116$	0	0
$117$	−10.0888	−0.932706
$118$	0	0
$119$	−7.34279	−0.673113
$120$	0	0
$121$	−10.9480	−0.995277
$122$	0	0
$123$	1.35766	0.122416
$124$	0	0
$125$	−5.89759	−0.527497
$126$	0	0
$127$	−13.5446	−1.20189	−0.600943	−	0.799292i	$-0.705208\pi$
$127$	−13.5446	−1.20189	−0.600943	+	0.799292i	$0.705208\pi$
$128$	0	0
$129$	0.114895	0.0101160
$130$	0	0
$131$	−1.03003	−0.0899942	−0.0449971	−	0.998987i	$-0.514328\pi$
$131$	−1.03003	−0.0899942	−0.0449971	+	0.998987i	$0.514328\pi$
$132$	0	0
$133$	3.50690	0.304087
$134$	0	0
$135$	−0.462845	−0.0398353
$136$	0	0
$137$	−11.5780	−0.989180	−0.494590	−	0.869127i	$-0.664682\pi$
$137$	−11.5780	−0.989180	−0.494590	+	0.869127i	$0.664682\pi$
$138$	0	0
$139$	−0.110960	−0.00941151	−0.00470575	−	0.999989i	$-0.501498\pi$
$139$	−0.110960	−0.00941151	−0.00470575	+	0.999989i	$0.501498\pi$
$140$	0	0
$141$	1.07333	0.0903904
$142$	0	0
$143$	−0.770621	−0.0644425
$144$	0	0
$145$	0.348929	0.0289769
$146$	0	0
$147$	−0.749372	−0.0618072
$148$	0	0
$149$	−19.5413	−1.60089	−0.800443	−	0.599409i	$-0.795402\pi$
$149$	−19.5413	−1.60089	−0.800443	+	0.599409i	$0.795402\pi$
$150$	0	0
$151$	−17.7745	−1.44647	−0.723233	−	0.690604i	$-0.757345\pi$
$151$	−17.7745	−1.44647	−0.723233	+	0.690604i	$0.757345\pi$
$152$	0	0
$153$	−21.2505	−1.71800
$154$	0	0
$155$	−3.02522	−0.242991
$156$	0	0
$157$	6.98884	0.557770	0.278885	−	0.960325i	$-0.410035\pi$
$157$	6.98884	0.557770	0.278885	+	0.960325i	$0.410035\pi$
$158$	0	0
$159$	−0.980379	−0.0777491
$160$	0	0
$161$	7.12995	0.561918
$162$	0	0
$163$	8.87693	0.695295	0.347647	−	0.937625i	$-0.386981\pi$
$163$	8.87693	0.695295	0.347647	+	0.937625i	$0.386981\pi$
$164$	0	0
$165$	−0.0176299	−0.00137249
$166$	0	0
$167$	−4.85435	−0.375641	−0.187820	−	0.982203i	$-0.560142\pi$
$167$	−4.85435	−0.375641	−0.187820	+	0.982203i	$0.560142\pi$
$168$	0	0
$169$	−1.56968	−0.120745
$170$	0	0
$171$	10.1492	0.776126
$172$	0	0
$173$	−13.5277	−1.02849	−0.514247	−	0.857642i	$-0.671928\pi$
$173$	−13.5277	−1.02849	−0.514247	+	0.857642i	$0.671928\pi$
$174$	0	0
$175$	4.76835	0.360453
$176$	0	0
$177$	−1.42008	−0.106740
$178$	0	0
$179$	10.4800	0.783310	0.391655	−	0.920112i	$-0.371903\pi$
$179$	10.4800	0.783310	0.391655	+	0.920112i	$0.371903\pi$
$180$	0	0
$181$	−8.82686	−0.656095	−0.328048	−	0.944661i	$-0.606391\pi$
$181$	−8.82686	−0.656095	−0.328048	+	0.944661i	$0.606391\pi$
$182$	0	0
$183$	−0.461286	−0.0340993
$184$	0	0
$185$	−3.29566	−0.242302
$186$	0	0
$187$	−1.62320	−0.118700
$188$	0	0
$189$	0.778827	0.0566514
$190$	0	0
$191$	3.25920	0.235827	0.117914	−	0.993024i	$-0.462379\pi$
$191$	3.25920	0.235827	0.117914	+	0.993024i	$0.462379\pi$
$192$	0	0
$193$	16.9658	1.22123	0.610613	−	0.791929i	$-0.290923\pi$
$193$	16.9658	1.22123	0.610613	+	0.791929i	$0.290923\pi$
$194$	0	0
$195$	0.261498	0.0187262
$196$	0	0
$197$	−18.5572	−1.32214	−0.661072	−	0.750322i	$-0.729898\pi$
$197$	−18.5572	−1.32214	−0.661072	+	0.750322i	$0.729898\pi$
$198$	0	0
$199$	19.6484	1.39284	0.696420	−	0.717634i	$-0.254775\pi$
$199$	19.6484	1.39284	0.696420	+	0.717634i	$0.254775\pi$
$200$	0	0
$201$	1.52345	0.107456
$202$	0	0
$203$	−0.587141	−0.0412092
$204$	0	0
$205$	6.59088	0.460327
$206$	0	0
$207$	20.6345	1.43419
$208$	0	0
$209$	0.775235	0.0536241
$210$	0	0
$211$	8.18657	0.563587	0.281794	−	0.959475i	$-0.409071\pi$
$211$	8.18657	0.563587	0.281794	+	0.959475i	$0.409071\pi$
$212$	0	0
$213$	1.19740	0.0820446
$214$	0	0
$215$	0.557770	0.0380396
$216$	0	0
$217$	5.09052	0.345567
$218$	0	0
$219$	−0.896809	−0.0606007
$220$	0	0
$221$	24.0762	1.61954
$222$	0	0
$223$	−15.5070	−1.03843	−0.519213	−	0.854645i	$-0.673775\pi$
$223$	−15.5070	−1.03843	−0.519213	+	0.854645i	$0.673775\pi$
$224$	0	0
$225$	13.7999	0.919991
$226$	0	0
$227$	27.1128	1.79954	0.899769	−	0.436366i	$-0.143735\pi$
$227$	27.1128	1.79954	0.899769	+	0.436366i	$0.143735\pi$
$228$	0	0
$229$	8.10947	0.535889	0.267944	−	0.963434i	$-0.413656\pi$
$229$	8.10947	0.535889	0.267944	+	0.963434i	$0.413656\pi$
$230$	0	0
$231$	0.0296658	0.00195187
$232$	0	0
$233$	−4.34697	−0.284780	−0.142390	−	0.989811i	$-0.545479\pi$
$233$	−4.34697	−0.284780	−0.142390	+	0.989811i	$0.545479\pi$
$234$	0	0
$235$	5.21056	0.339900
$236$	0	0
$237$	−0.0462235	−0.00300254
$238$	0	0
$239$	−18.3619	−1.18773	−0.593867	−	0.804563i	$-0.702399\pi$
$239$	−18.3619	−1.18773	−0.593867	+	0.804563i	$0.702399\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	3.38396	0.217081
$244$	0	0
$245$	−3.63790	−0.232417
$246$	0	0
$247$	−11.4988	−0.731649
$248$	0	0
$249$	−2.24717	−0.142409
$250$	0	0
$251$	−12.3629	−0.780337	−0.390169	−	0.920743i	$-0.627583\pi$
$251$	−12.3629	−0.780337	−0.390169	+	0.920743i	$0.627583\pi$
$252$	0	0
$253$	1.57614	0.0990914
$254$	0	0
$255$	0.550806	0.0344928
$256$	0	0
$257$	−12.1589	−0.758452	−0.379226	−	0.925304i	$-0.623810\pi$
$257$	−12.1589	−0.758452	−0.379226	+	0.925304i	$0.623810\pi$
$258$	0	0
$259$	5.54560	0.344587
$260$	0	0
$261$	−1.69922	−0.105179
$262$	0	0
$263$	5.76048	0.355207	0.177603	−	0.984102i	$-0.443166\pi$
$263$	5.76048	0.355207	0.177603	+	0.984102i	$0.443166\pi$
$264$	0	0
$265$	−4.75934	−0.292364
$266$	0	0
$267$	−0.952030	−0.0582633
$268$	0	0
$269$	14.4585	0.881551	0.440776	−	0.897617i	$-0.354703\pi$
$269$	14.4585	0.881551	0.440776	+	0.897617i	$0.354703\pi$
$270$	0	0
$271$	−6.81638	−0.414065	−0.207033	−	0.978334i	$-0.566381\pi$
$271$	−6.81638	−0.414065	−0.207033	+	0.978334i	$0.566381\pi$
$272$	0	0
$273$	−0.440022	−0.0266313
$274$	0	0
$275$	1.05409	0.0635640
$276$	0	0
$277$	−25.1323	−1.51005	−0.755027	−	0.655693i	$-0.772376\pi$
$277$	−25.1323	−1.51005	−0.755027	+	0.655693i	$0.772376\pi$
$278$	0	0
$279$	14.7323	0.881997
$280$	0	0
$281$	−6.03331	−0.359917	−0.179959	−	0.983674i	$-0.557596\pi$
$281$	−6.03331	−0.359917	−0.179959	+	0.983674i	$0.557596\pi$
$282$	0	0
$283$	3.47423	0.206521	0.103261	−	0.994654i	$-0.467072\pi$
$283$	3.47423	0.206521	0.103261	+	0.994654i	$0.467072\pi$
$284$	0	0
$285$	−0.263064	−0.0155825
$286$	0	0
$287$	−11.0905	−0.654649
$288$	0	0
$289$	33.7130	1.98312
$290$	0	0
$291$	−0.991499	−0.0581227
$292$	0	0
$293$	−19.7300	−1.15264	−0.576321	−	0.817224i	$-0.695512\pi$
$293$	−19.7300	−1.15264	−0.576321	+	0.817224i	$0.695512\pi$
$294$	0	0
$295$	−6.89391	−0.401379
$296$	0	0
$297$	0.172167	0.00999017
$298$	0	0
$299$	−23.3783	−1.35200
$300$	0	0
$301$	−0.938558	−0.0540976
$302$	0	0
$303$	2.20641	0.126755
$304$	0	0
$305$	−2.23936	−0.128225
$306$	0	0
$307$	0.0265060	0.00151278	0.000756388	−	1.00000i	$-0.499759\pi$
$307$	0.0265060	0.00151278	0.000756388		1.00000i	$0.499759\pi$
$308$	0	0
$309$	−0.770424	−0.0438279
$310$	0	0
$311$	−21.2822	−1.20680	−0.603400	−	0.797439i	$-0.706188\pi$
$311$	−21.2822	−1.20680	−0.603400	+	0.797439i	$0.706188\pi$
$312$	0	0
$313$	10.6916	0.604323	0.302161	−	0.953257i	$-0.402292\pi$
$313$	10.6916	0.604323	0.302161	+	0.953257i	$0.402292\pi$
$314$	0	0
$315$	1.88541	0.106231
$316$	0	0
$317$	−2.07826	−0.116727	−0.0583633	−	0.998295i	$-0.518588\pi$
$317$	−2.07826	−0.116727	−0.0583633	+	0.998295i	$0.518588\pi$
$318$	0	0
$319$	−0.129793	−0.00726703
$320$	0	0
$321$	−0.992380	−0.0553892
$322$	0	0
$323$	−24.2204	−1.34766
$324$	0	0
$325$	−15.6349	−0.867269
$326$	0	0
$327$	−0.526659	−0.0291243
$328$	0	0
$329$	−8.76780	−0.483384
$330$	0	0
$331$	28.1272	1.54601	0.773006	−	0.634399i	$-0.218752\pi$
$331$	28.1272	1.54601	0.773006	+	0.634399i	$0.218752\pi$
$332$	0	0
$333$	16.0493	0.879494
$334$	0	0
$335$	7.39575	0.404073
$336$	0	0
$337$	23.5835	1.28468	0.642338	−	0.766422i	$-0.277965\pi$
$337$	23.5835	1.28468	0.642338	+	0.766422i	$0.277965\pi$
$338$	0	0
$339$	0.457188	0.0248311
$340$	0	0
$341$	1.12531	0.0609389
$342$	0	0
$343$	13.3392	0.720249
$344$	0	0
$345$	−0.534839	−0.0287948
$346$	0	0
$347$	21.8697	1.17403	0.587013	−	0.809578i	$-0.300304\pi$
$347$	21.8697	1.17403	0.587013	+	0.809578i	$0.300304\pi$
$348$	0	0
$349$	18.8525	1.00915	0.504575	−	0.863368i	$-0.331649\pi$
$349$	18.8525	1.00915	0.504575	+	0.863368i	$0.331649\pi$
$350$	0	0
$351$	−2.55369	−0.136306
$352$	0	0
$353$	−11.3768	−0.605526	−0.302763	−	0.953066i	$-0.597909\pi$
$353$	−11.3768	−0.605526	−0.302763	+	0.953066i	$0.597909\pi$
$354$	0	0
$355$	5.81289	0.308516
$356$	0	0
$357$	−0.926839	−0.0490535
$358$	0	0
$359$	0.146567	0.00773548	0.00386774	−	0.999993i	$-0.498769\pi$
$359$	0.146567	0.00773548	0.00386774	+	0.999993i	$0.498769\pi$
$360$	0	0
$361$	−7.43238	−0.391178
$362$	0	0
$363$	−1.38191	−0.0725315
$364$	0	0
$365$	−4.35364	−0.227880
$366$	0	0
$367$	−32.6647	−1.70508	−0.852541	−	0.522660i	$-0.824940\pi$
$367$	−32.6647	−1.70508	−0.852541	+	0.522660i	$0.824940\pi$
$368$	0	0
$369$	−32.0964	−1.67087
$370$	0	0
$371$	8.00852	0.415782
$372$	0	0
$373$	−17.7530	−0.919218	−0.459609	−	0.888121i	$-0.652010\pi$
$373$	−17.7530	−0.919218	−0.459609	+	0.888121i	$0.652010\pi$
$374$	0	0
$375$	−0.744420	−0.0384417
$376$	0	0
$377$	1.92517	0.0991515
$378$	0	0
$379$	4.82435	0.247810	0.123905	−	0.992294i	$-0.460458\pi$
$379$	4.82435	0.247810	0.123905	+	0.992294i	$0.460458\pi$
$380$	0	0
$381$	−1.70965	−0.0875882
$382$	0	0
$383$	−14.7127	−0.751783	−0.375892	−	0.926664i	$-0.622664\pi$
$383$	−14.7127	−0.751783	−0.375892	+	0.926664i	$0.622664\pi$
$384$	0	0
$385$	0.144015	0.00733970
$386$	0	0
$387$	−2.71624	−0.138074
$388$	0	0
$389$	4.90322	0.248603	0.124301	−	0.992244i	$-0.460331\pi$
$389$	4.90322	0.248603	0.124301	+	0.992244i	$0.460331\pi$
$390$	0	0
$391$	−49.2430	−2.49032
$392$	0	0
$393$	−0.130015	−0.00655839
$394$	0	0
$395$	−0.224396	−0.0112906
$396$	0	0
$397$	−14.8959	−0.747606	−0.373803	−	0.927508i	$-0.621946\pi$
$397$	−14.8959	−0.747606	−0.373803	+	0.927508i	$0.621946\pi$
$398$	0	0
$399$	0.442656	0.0221605
$400$	0	0
$401$	6.72128	0.335645	0.167822	−	0.985817i	$-0.446327\pi$
$401$	6.72128	0.335645	0.167822	+	0.985817i	$0.446327\pi$
$402$	0	0
$403$	−16.6913	−0.831452
$404$	0	0
$405$	5.42720	0.269680
$406$	0	0
$407$	1.22591	0.0607660
$408$	0	0
$409$	−2.59935	−0.128529	−0.0642647	−	0.997933i	$-0.520470\pi$
$409$	−2.59935	−0.128529	−0.0642647	+	0.997933i	$0.520470\pi$
$410$	0	0
$411$	−1.46143	−0.0720871
$412$	0	0
$413$	11.6004	0.570817
$414$	0	0
$415$	−10.9091	−0.535507
$416$	0	0
$417$	−0.0140059	−0.000685870 0
$418$	0	0
$419$	13.1622	0.643018	0.321509	−	0.946907i	$-0.395810\pi$
$419$	13.1622	0.643018	0.321509	+	0.946907i	$0.395810\pi$
$420$	0	0
$421$	−6.53715	−0.318601	−0.159301	−	0.987230i	$-0.550924\pi$
$421$	−6.53715	−0.318601	−0.159301	+	0.987230i	$0.550924\pi$
$422$	0	0
$423$	−25.3745	−1.23375
$424$	0	0
$425$	−32.9326	−1.59746
$426$	0	0
$427$	3.76816	0.182354
$428$	0	0
$429$	−0.0972711	−0.00469629
$430$	0	0
$431$	−10.8319	−0.521755	−0.260878	−	0.965372i	$-0.584012\pi$
$431$	−10.8319	−0.521755	−0.260878	+	0.965372i	$0.584012\pi$
$432$	0	0
$433$	−25.6037	−1.23044	−0.615218	−	0.788357i	$-0.710932\pi$
$433$	−25.6037	−1.23044	−0.615218	+	0.788357i	$0.710932\pi$
$434$	0	0
$435$	0.0440433	0.00211171
$436$	0	0
$437$	23.5183	1.12503
$438$	0	0
$439$	−14.6842	−0.700839	−0.350419	−	0.936593i	$-0.613961\pi$
$439$	−14.6842	−0.700839	−0.350419	+	0.936593i	$0.613961\pi$
$440$	0	0
$441$	17.7159	0.843614
$442$	0	0
$443$	15.2687	0.725436	0.362718	−	0.931899i	$-0.381849\pi$
$443$	15.2687	0.725436	0.362718	+	0.931899i	$0.381849\pi$
$444$	0	0
$445$	−4.62171	−0.219090
$446$	0	0
$447$	−2.46659	−0.116666
$448$	0	0
$449$	31.2160	1.47317	0.736587	−	0.676343i	$-0.236436\pi$
$449$	31.2160	1.47317	0.736587	+	0.676343i	$0.236436\pi$
$450$	0	0
$451$	−2.45165	−0.115444
$452$	0	0
$453$	−2.24357	−0.105412
$454$	0	0
$455$	−2.13612	−0.100143
$456$	0	0
$457$	9.43632	0.441412	0.220706	−	0.975340i	$-0.429164\pi$
$457$	9.43632	0.441412	0.220706	+	0.975340i	$0.429164\pi$
$458$	0	0
$459$	−5.37897	−0.251069
$460$	0	0
$461$	8.65525	0.403115	0.201557	−	0.979477i	$-0.435400\pi$
$461$	8.65525	0.403115	0.201557	+	0.979477i	$0.435400\pi$
$462$	0	0
$463$	12.5415	0.582853	0.291426	−	0.956593i	$-0.405870\pi$
$463$	12.5415	0.582853	0.291426	+	0.956593i	$0.405870\pi$
$464$	0	0
$465$	−0.381856	−0.0177081
$466$	0	0
$467$	5.28477	0.244550	0.122275	−	0.992496i	$-0.460981\pi$
$467$	5.28477	0.244550	0.122275	+	0.992496i	$0.460981\pi$
$468$	0	0
$469$	−12.4448	−0.574647
$470$	0	0
$471$	0.882161	0.0406479
$472$	0	0
$473$	−0.207477	−0.00953982
$474$	0	0
$475$	15.7285	0.721675
$476$	0	0
$477$	23.1771	1.06121
$478$	0	0
$479$	29.0596	1.32777	0.663884	−	0.747835i	$-0.268907\pi$
$479$	29.0596	1.32777	0.663884	+	0.747835i	$0.268907\pi$
$480$	0	0
$481$	−18.1834	−0.829093
$482$	0	0
$483$	0.899973	0.0409502
$484$	0	0
$485$	−4.81332	−0.218562
$486$	0	0
$487$	36.2854	1.64425	0.822124	−	0.569309i	$-0.192789\pi$
$487$	36.2854	1.64425	0.822124	+	0.569309i	$0.192789\pi$
$488$	0	0
$489$	1.12048	0.0506701
$490$	0	0
$491$	3.99282	0.180193	0.0900967	−	0.995933i	$-0.471282\pi$
$491$	3.99282	0.180193	0.0900967	+	0.995933i	$0.471282\pi$
$492$	0	0
$493$	4.05509	0.182632
$494$	0	0
$495$	0.416789	0.0187333
$496$	0	0
$497$	−9.78134	−0.438753
$498$	0	0
$499$	−9.25262	−0.414204	−0.207102	−	0.978319i	$-0.566403\pi$
$499$	−9.25262	−0.414204	−0.207102	+	0.978319i	$0.566403\pi$
$500$	0	0
$501$	−0.612737	−0.0273751
$502$	0	0
$503$	−30.0225	−1.33864	−0.669319	−	0.742975i	$-0.733414\pi$
$503$	−30.0225	−1.33864	−0.669319	+	0.742975i	$0.733414\pi$
$504$	0	0
$505$	10.7112	0.476643
$506$	0	0
$507$	−0.198132	−0.00879934
$508$	0	0
$509$	8.01192	0.355122	0.177561	−	0.984110i	$-0.443179\pi$
$509$	8.01192	0.355122	0.177561	+	0.984110i	$0.443179\pi$
$510$	0	0
$511$	7.32586	0.324077
$512$	0	0
$513$	2.56898	0.113423
$514$	0	0
$515$	−3.74009	−0.164808
$516$	0	0
$517$	−1.93821	−0.0852423
$518$	0	0
$519$	−1.70753	−0.0749521
$520$	0	0
$521$	−10.1940	−0.446608	−0.223304	−	0.974749i	$-0.571684\pi$
$521$	−10.1940	−0.446608	−0.223304	+	0.974749i	$0.571684\pi$
$522$	0	0
$523$	41.0217	1.79375	0.896877	−	0.442281i	$-0.145830\pi$
$523$	41.0217	1.79375	0.896877	+	0.442281i	$0.145830\pi$
$524$	0	0
$525$	0.601882	0.0262683
$526$	0	0
$527$	−35.1577	−1.53149
$528$	0	0
$529$	24.8156	1.07894
$530$	0	0
$531$	33.5721	1.45691
$532$	0	0
$533$	36.3644	1.57512
$534$	0	0
$535$	−4.81760	−0.208283
$536$	0	0
$537$	1.32283	0.0570842
$538$	0	0
$539$	1.35321	0.0582870
$540$	0	0
$541$	−25.9044	−1.11372	−0.556858	−	0.830608i	$-0.687993\pi$
$541$	−25.9044	−1.11372	−0.556858	+	0.830608i	$0.687993\pi$
$542$	0	0
$543$	−1.11416	−0.0478134
$544$	0	0
$545$	−2.55671	−0.109518
$546$	0	0
$547$	43.6488	1.86629	0.933145	−	0.359501i	$-0.117053\pi$
$547$	43.6488	1.86629	0.933145	+	0.359501i	$0.117053\pi$
$548$	0	0
$549$	10.9053	0.465425
$550$	0	0
$551$	−1.93670	−0.0825063
$552$	0	0
$553$	0.377591	0.0160568
$554$	0	0
$555$	−0.415992	−0.0176579
$556$	0	0
$557$	22.1122	0.936925	0.468463	−	0.883483i	$-0.344808\pi$
$557$	22.1122	0.936925	0.468463	+	0.883483i	$0.344808\pi$
$558$	0	0
$559$	3.07743	0.130162
$560$	0	0
$561$	−0.204887	−0.00865033
$562$	0	0
$563$	−19.0461	−0.802697	−0.401348	−	0.915925i	$-0.631458\pi$
$563$	−19.0461	−0.802697	−0.401348	+	0.915925i	$0.631458\pi$
$564$	0	0
$565$	2.21946	0.0933735
$566$	0	0
$567$	−9.13233	−0.383522
$568$	0	0
$569$	−24.1421	−1.01209	−0.506045	−	0.862507i	$-0.668893\pi$
$569$	−24.1421	−1.01209	−0.506045	+	0.862507i	$0.668893\pi$
$570$	0	0
$571$	37.2649	1.55949	0.779745	−	0.626098i	$-0.215349\pi$
$571$	37.2649	1.55949	0.779745	+	0.626098i	$0.215349\pi$
$572$	0	0
$573$	0.411390	0.0171861
$574$	0	0
$575$	31.9780	1.33357
$576$	0	0
$577$	−6.94471	−0.289112	−0.144556	−	0.989497i	$-0.546175\pi$
$577$	−6.94471	−0.289112	−0.144556	+	0.989497i	$0.546175\pi$
$578$	0	0
$579$	2.14150	0.0889977
$580$	0	0
$581$	18.3567	0.761565
$582$	0	0
$583$	1.77036	0.0733209
$584$	0	0
$585$	−6.18206	−0.255597
$586$	0	0
$587$	33.6812	1.39017	0.695085	−	0.718927i	$-0.255367\pi$
$587$	33.6812	1.39017	0.695085	+	0.718927i	$0.255367\pi$
$588$	0	0
$589$	16.7912	0.691871
$590$	0	0
$591$	−2.34237	−0.0963522
$592$	0	0
$593$	−25.1034	−1.03087	−0.515437	−	0.856927i	$-0.672370\pi$
$593$	−25.1034	−1.03087	−0.515437	+	0.856927i	$0.672370\pi$
$594$	0	0
$595$	−4.49942	−0.184458
$596$	0	0
$597$	2.48011	0.101504
$598$	0	0
$599$	5.93752	0.242601	0.121300	−	0.992616i	$-0.461294\pi$
$599$	5.93752	0.242601	0.121300	+	0.992616i	$0.461294\pi$
$600$	0	0
$601$	−29.7235	−1.21245	−0.606223	−	0.795294i	$-0.707316\pi$
$601$	−29.7235	−1.21245	−0.606223	+	0.795294i	$0.707316\pi$
$602$	0	0
$603$	−36.0159	−1.46668
$604$	0	0
$605$	−6.70861	−0.272744
$606$	0	0
$607$	−16.5792	−0.672929	−0.336464	−	0.941696i	$-0.609231\pi$
$607$	−16.5792	−0.672929	−0.336464	+	0.941696i	$0.609231\pi$
$608$	0	0
$609$	−0.0741115	−0.00300315
$610$	0	0
$611$	28.7487	1.16305
$612$	0	0
$613$	39.4836	1.59473	0.797364	−	0.603499i	$-0.206227\pi$
$613$	39.4836	1.59473	0.797364	+	0.603499i	$0.206227\pi$
$614$	0	0
$615$	0.831930	0.0335466
$616$	0	0
$617$	38.4323	1.54723	0.773613	−	0.633658i	$-0.218447\pi$
$617$	38.4323	1.54723	0.773613	+	0.633658i	$0.218447\pi$
$618$	0	0
$619$	−14.9244	−0.599861	−0.299931	−	0.953961i	$-0.596964\pi$
$619$	−14.9244	−0.599861	−0.299931	+	0.953961i	$0.596964\pi$
$620$	0	0
$621$	5.22305	0.209594
$622$	0	0
$623$	7.77695	0.311577
$624$	0	0
$625$	19.5087	0.780349
$626$	0	0
$627$	0.0978535	0.00390789
$628$	0	0
$629$	−38.3006	−1.52715
$630$	0	0
$631$	−30.8867	−1.22958	−0.614791	−	0.788690i	$-0.710759\pi$
$631$	−30.8867	−1.22958	−0.614791	+	0.788690i	$0.710759\pi$
$632$	0	0
$633$	1.03334	0.0410718
$634$	0	0
$635$	−8.29967	−0.329362
$636$	0	0
$637$	−20.0717	−0.795269
$638$	0	0
$639$	−28.3077	−1.11984
$640$	0	0
$641$	31.6649	1.25069	0.625344	−	0.780349i	$-0.284959\pi$
$641$	31.6649	1.25069	0.625344	+	0.780349i	$0.284959\pi$
$642$	0	0
$643$	17.2630	0.680786	0.340393	−	0.940283i	$-0.389440\pi$
$643$	17.2630	0.680786	0.340393	+	0.940283i	$0.389440\pi$
$644$	0	0
$645$	0.0704041	0.00277216
$646$	0	0
$647$	−35.2693	−1.38658	−0.693289	−	0.720659i	$-0.743839\pi$
$647$	−35.2693	−1.38658	−0.693289	+	0.720659i	$0.743839\pi$
$648$	0	0
$649$	2.56437	0.100661
$650$	0	0
$651$	0.642548	0.0251834
$652$	0	0
$653$	−38.0371	−1.48851	−0.744253	−	0.667898i	$-0.767194\pi$
$653$	−38.0371	−1.48851	−0.744253	+	0.667898i	$0.767194\pi$
$654$	0	0
$655$	−0.631169	−0.0246618
$656$	0	0
$657$	21.2014	0.827147
$658$	0	0
$659$	7.14619	0.278376	0.139188	−	0.990266i	$-0.455551\pi$
$659$	7.14619	0.278376	0.139188	+	0.990266i	$0.455551\pi$
$660$	0	0
$661$	33.6588	1.30917	0.654587	−	0.755986i	$-0.272842\pi$
$661$	33.6588	1.30917	0.654587	+	0.755986i	$0.272842\pi$
$662$	0	0
$663$	3.03901	0.118025
$664$	0	0
$665$	2.14892	0.0833314
$666$	0	0
$667$	−3.93754	−0.152462
$668$	0	0
$669$	−1.95736	−0.0756759
$670$	0	0
$671$	0.832989	0.0321572
$672$	0	0
$673$	−4.10894	−0.158388	−0.0791939	−	0.996859i	$-0.525235\pi$
$673$	−4.10894	−0.158388	−0.0791939	+	0.996859i	$0.525235\pi$
$674$	0	0
$675$	3.49306	0.134448
$676$	0	0
$677$	−26.3822	−1.01395	−0.506976	−	0.861960i	$-0.669237\pi$
$677$	−26.3822	−1.01395	−0.506976	+	0.861960i	$0.669237\pi$
$678$	0	0
$679$	8.09936	0.310825
$680$	0	0
$681$	3.42229	0.131143
$682$	0	0
$683$	18.1548	0.694676	0.347338	−	0.937740i	$-0.387086\pi$
$683$	18.1548	0.694676	0.347338	+	0.937740i	$0.387086\pi$
$684$	0	0
$685$	−7.09465	−0.271073
$686$	0	0
$687$	1.02361	0.0390532
$688$	0	0
$689$	−26.2591	−1.00039
$690$	0	0
$691$	−28.1899	−1.07239	−0.536196	−	0.844093i	$-0.680139\pi$
$691$	−28.1899	−1.07239	−0.536196	+	0.844093i	$0.680139\pi$
$692$	0	0
$693$	−0.701329	−0.0266413
$694$	0	0
$695$	−0.0679927	−0.00257911
$696$	0	0
$697$	76.5962	2.90129
$698$	0	0
$699$	−0.548694	−0.0207535
$700$	0	0
$701$	39.3669	1.48687	0.743434	−	0.668809i	$-0.233196\pi$
$701$	39.3669	1.48687	0.743434	+	0.668809i	$0.233196\pi$
$702$	0	0
$703$	18.2923	0.689907
$704$	0	0
$705$	0.657700	0.0247704
$706$	0	0
$707$	−18.0237	−0.677853
$708$	0	0
$709$	48.4112	1.81812	0.909060	−	0.416664i	$-0.136801\pi$
$709$	48.4112	1.81812	0.909060	+	0.416664i	$0.136801\pi$
$710$	0	0
$711$	1.09277	0.0409820
$712$	0	0
$713$	34.1386	1.27850
$714$	0	0
$715$	−0.472211	−0.0176597
$716$	0	0
$717$	−2.31772	−0.0865569
$718$	0	0
$719$	10.8564	0.404877	0.202438	−	0.979295i	$-0.435113\pi$
$719$	10.8564	0.404877	0.202438	+	0.979295i	$0.435113\pi$
$720$	0	0
$721$	6.29345	0.234380
$722$	0	0
$723$	0.126224	0.00469433
$724$	0	0
$725$	−2.63334	−0.0977999
$726$	0	0
$727$	37.5110	1.39121	0.695604	−	0.718426i	$-0.255137\pi$
$727$	37.5110	1.39121	0.695604	+	0.718426i	$0.255137\pi$
$728$	0	0
$729$	−26.1434	−0.968276
$730$	0	0
$731$	6.48215	0.239751
$732$	0	0
$733$	−1.31177	−0.0484512	−0.0242256	−	0.999707i	$-0.507712\pi$
$733$	−1.31177	−0.0484512	−0.0242256	+	0.999707i	$0.507712\pi$
$734$	0	0
$735$	−0.459191	−0.0169375
$736$	0	0
$737$	−2.75105	−0.101336
$738$	0	0
$739$	−52.6696	−1.93748	−0.968742	−	0.248072i	$-0.920203\pi$
$739$	−52.6696	−1.93748	−0.968742	+	0.248072i	$0.920203\pi$
$740$	0	0
$741$	−1.45142	−0.0533194
$742$	0	0
$743$	−44.6512	−1.63809	−0.819046	−	0.573728i	$-0.805497\pi$
$743$	−44.6512	−1.63809	−0.819046	+	0.573728i	$0.805497\pi$
$744$	0	0
$745$	−11.9743	−0.438704
$746$	0	0
$747$	53.1254	1.94376
$748$	0	0
$749$	8.10656	0.296207
$750$	0	0
$751$	30.0001	1.09472	0.547360	−	0.836897i	$-0.315633\pi$
$751$	30.0001	1.09472	0.547360	+	0.836897i	$0.315633\pi$
$752$	0	0
$753$	−1.56049	−0.0568676
$754$	0	0
$755$	−10.8916	−0.396386
$756$	0	0
$757$	10.8998	0.396160	0.198080	−	0.980186i	$-0.436529\pi$
$757$	10.8998	0.396160	0.198080	+	0.980186i	$0.436529\pi$
$758$	0	0
$759$	0.198948	0.00722135
$760$	0	0
$761$	54.3944	1.97180	0.985898	−	0.167348i	$-0.0535203\pi$
$761$	54.3944	1.97180	0.985898	+	0.167348i	$0.0535203\pi$
$762$	0	0
$763$	4.30218	0.155749
$764$	0	0
$765$	−13.0216	−0.470796
$766$	0	0
$767$	−38.0364	−1.37341
$768$	0	0
$769$	−16.9359	−0.610723	−0.305361	−	0.952237i	$-0.598777\pi$
$769$	−16.9359	−0.610723	−0.305361	+	0.952237i	$0.598777\pi$
$770$	0	0
$771$	−1.53475	−0.0552727
$772$	0	0
$773$	−29.5026	−1.06113	−0.530567	−	0.847643i	$-0.678021\pi$
$773$	−29.5026	−1.06113	−0.530567	+	0.847643i	$0.678021\pi$
$774$	0	0
$775$	22.8311	0.820117
$776$	0	0
$777$	0.699989	0.0251120
$778$	0	0
$779$	−36.5822	−1.31069
$780$	0	0
$781$	−2.16226	−0.0773718
$782$	0	0
$783$	−0.430111	−0.0153709
$784$	0	0
$785$	4.28253	0.152850
$786$	0	0
$787$	5.96992	0.212805	0.106402	−	0.994323i	$-0.466067\pi$
$787$	5.96992	0.212805	0.106402	+	0.994323i	$0.466067\pi$
$788$	0	0
$789$	0.727113	0.0258859
$790$	0	0
$791$	−3.73468	−0.132790
$792$	0	0
$793$	−12.3554	−0.438753
$794$	0	0
$795$	−0.600744	−0.0213062
$796$	0	0
$797$	19.1226	0.677357	0.338679	−	0.940902i	$-0.390020\pi$
$797$	19.1226	0.677357	0.338679	+	0.940902i	$0.390020\pi$
$798$	0	0
$799$	60.5548	2.14227
$800$	0	0
$801$	22.5069	0.795243
$802$	0	0
$803$	1.61945	0.0571492
$804$	0	0
$805$	4.36900	0.153987
$806$	0	0
$807$	1.82502	0.0642436
$808$	0	0
$809$	34.3768	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$809$	34.3768	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$810$	0	0
$811$	−18.2250	−0.639967	−0.319984	−	0.947423i	$-0.603677\pi$
$811$	−18.2250	−0.639967	−0.319984	+	0.947423i	$0.603677\pi$
$812$	0	0
$813$	−0.860393	−0.0301753
$814$	0	0
$815$	5.43949	0.190537
$816$	0	0
$817$	−3.09586	−0.108310
$818$	0	0
$819$	10.4025	0.363494
$820$	0	0
$821$	36.2130	1.26384	0.631920	−	0.775033i	$-0.282267\pi$
$821$	36.2130	1.26384	0.631920	+	0.775033i	$0.282267\pi$
$822$	0	0
$823$	42.2212	1.47174	0.735869	−	0.677124i	$-0.236774\pi$
$823$	42.2212	1.47174	0.735869	+	0.677124i	$0.236774\pi$
$824$	0	0
$825$	0.133052	0.00463227
$826$	0	0
$827$	−15.1566	−0.527048	−0.263524	−	0.964653i	$-0.584885\pi$
$827$	−15.1566	−0.527048	−0.263524	+	0.964653i	$0.584885\pi$
$828$	0	0
$829$	−3.49294	−0.121315	−0.0606575	−	0.998159i	$-0.519320\pi$
$829$	−3.49294	−0.121315	−0.0606575	+	0.998159i	$0.519320\pi$
$830$	0	0
$831$	−3.17231	−0.110046
$832$	0	0
$833$	−42.2780	−1.46484
$834$	0	0
$835$	−2.97459	−0.102940
$836$	0	0
$837$	3.72907	0.128895
$838$	0	0
$839$	28.5020	0.983999	0.491999	−	0.870596i	$-0.336266\pi$
$839$	28.5020	0.983999	0.491999	+	0.870596i	$0.336266\pi$
$840$	0	0
$841$	−28.6757	−0.988819
$842$	0	0
$843$	−0.761550	−0.0262292
$844$	0	0
$845$	−0.961849	−0.0330886
$846$	0	0
$847$	11.2886	0.387879
$848$	0	0
$849$	0.438532	0.0150504
$850$	0	0
$851$	37.1904	1.27487
$852$	0	0
$853$	5.52236	0.189082	0.0945410	−	0.995521i	$-0.469862\pi$
$853$	5.52236	0.189082	0.0945410	+	0.995521i	$0.469862\pi$
$854$	0	0
$855$	6.21908	0.212688
$856$	0	0
$857$	−17.0172	−0.581298	−0.290649	−	0.956830i	$-0.593871\pi$
$857$	−17.0172	−0.581298	−0.290649	+	0.956830i	$0.593871\pi$
$858$	0	0
$859$	1.16642	0.0397979	0.0198989	−	0.999802i	$-0.493666\pi$
$859$	1.16642	0.0397979	0.0198989	+	0.999802i	$0.493666\pi$
$860$	0	0
$861$	−1.39989	−0.0477080
$862$	0	0
$863$	8.50057	0.289363	0.144681	−	0.989478i	$-0.453784\pi$
$863$	8.50057	0.289363	0.144681	+	0.989478i	$0.453784\pi$
$864$	0	0
$865$	−8.28935	−0.281846
$866$	0	0
$867$	4.25540	0.144521
$868$	0	0
$869$	0.0834701	0.00283153
$870$	0	0
$871$	40.8052	1.38263
$872$	0	0
$873$	23.4400	0.793324
$874$	0	0
$875$	6.08102	0.205576
$876$	0	0
$877$	7.37310	0.248972	0.124486	−	0.992221i	$-0.460272\pi$
$877$	7.37310	0.248972	0.124486	+	0.992221i	$0.460272\pi$
$878$	0	0
$879$	−2.49041	−0.0839995
$880$	0	0
$881$	−48.8726	−1.64656	−0.823279	−	0.567637i	$-0.807858\pi$
$881$	−48.8726	−1.64656	−0.823279	+	0.567637i	$0.807858\pi$
$882$	0	0
$883$	26.8006	0.901911	0.450956	−	0.892546i	$-0.351083\pi$
$883$	26.8006	0.901911	0.450956	+	0.892546i	$0.351083\pi$
$884$	0	0
$885$	−0.870180	−0.0292508
$886$	0	0
$887$	3.15346	0.105883	0.0529414	−	0.998598i	$-0.483140\pi$
$887$	3.15346	0.105883	0.0529414	+	0.998598i	$0.483140\pi$
$888$	0	0
$889$	13.9658	0.468399
$890$	0	0
$891$	−2.01879	−0.0676320
$892$	0	0
$893$	−28.9208	−0.967799
$894$	0	0
$895$	6.42179	0.214657
$896$	0	0
$897$	−2.95092	−0.0985282
$898$	0	0
$899$	−2.81126	−0.0937609
$900$	0	0
$901$	−55.3109	−1.84267
$902$	0	0
$903$	−0.118469	−0.00394240
$904$	0	0
$905$	−5.40881	−0.179795
$906$	0	0
$907$	25.7319	0.854414	0.427207	−	0.904154i	$-0.359498\pi$
$907$	25.7319	0.854414	0.427207	+	0.904154i	$0.359498\pi$
$908$	0	0
$909$	−52.1617	−1.73010
$910$	0	0
$911$	−10.3710	−0.343606	−0.171803	−	0.985131i	$-0.554959\pi$
$911$	−10.3710	−0.343606	−0.171803	+	0.985131i	$0.554959\pi$
$912$	0	0
$913$	4.05793	0.134298
$914$	0	0
$915$	−0.282661	−0.00934450
$916$	0	0
$917$	1.06207	0.0350725
$918$	0	0
$919$	33.5553	1.10689	0.553444	−	0.832887i	$-0.313313\pi$
$919$	33.5553	1.10689	0.553444	+	0.832887i	$0.313313\pi$
$920$	0	0
$921$	0.00334570	0.000110244 0
$922$	0	0
$923$	32.0720	1.05566
$924$	0	0
$925$	24.8721	0.817790
$926$	0	0
$927$	18.2136	0.598213
$928$	0	0
$929$	11.6222	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$929$	11.6222	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$930$	0	0
$931$	20.1919	0.661762
$932$	0	0
$933$	−2.68633	−0.0879464
$934$	0	0
$935$	−0.994642	−0.0325283
$936$	0	0
$937$	46.3339	1.51366	0.756831	−	0.653611i	$-0.226747\pi$
$937$	46.3339	1.51366	0.756831	+	0.653611i	$0.226747\pi$
$938$	0	0
$939$	1.34954	0.0440404
$940$	0	0
$941$	−7.22995	−0.235690	−0.117845	−	0.993032i	$-0.537599\pi$
$941$	−7.22995	−0.235690	−0.117845	+	0.993032i	$0.537599\pi$
$942$	0	0
$943$	−74.3759	−2.42201
$944$	0	0
$945$	0.477240	0.0155246
$946$	0	0
$947$	−30.3453	−0.986091	−0.493045	−	0.870004i	$-0.664116\pi$
$947$	−30.3453	−0.986091	−0.493045	+	0.870004i	$0.664116\pi$
$948$	0	0
$949$	−24.0207	−0.779745
$950$	0	0
$951$	−0.262327	−0.00850653
$952$	0	0
$953$	4.89634	0.158608	0.0793040	−	0.996850i	$-0.474730\pi$
$953$	4.89634	0.158608	0.0793040	+	0.996850i	$0.474730\pi$
$954$	0	0
$955$	1.99713	0.0646256
$956$	0	0
$957$	−0.0163831	−0.000529590 0
$958$	0	0
$959$	11.9382	0.385503
$960$	0	0
$961$	−6.62631	−0.213752
$962$	0	0
$963$	23.4608	0.756015
$964$	0	0
$965$	10.3961	0.334662
$966$	0	0
$967$	−25.6038	−0.823363	−0.411681	−	0.911328i	$-0.635058\pi$
$967$	−25.6038	−0.823363	−0.411681	+	0.911328i	$0.635058\pi$
$968$	0	0
$969$	−3.05720	−0.0982116
$970$	0	0
$971$	−19.2440	−0.617570	−0.308785	−	0.951132i	$-0.599922\pi$
$971$	−19.2440	−0.617570	−0.308785	+	0.951132i	$0.599922\pi$
$972$	0	0
$973$	0.114411	0.00366785
$974$	0	0
$975$	−1.97351	−0.0632028
$976$	0	0
$977$	14.0956	0.450957	0.225479	−	0.974248i	$-0.427605\pi$
$977$	14.0956	0.450957	0.225479	+	0.974248i	$0.427605\pi$
$978$	0	0
$979$	1.71917	0.0549449
$980$	0	0
$981$	12.4507	0.397521
$982$	0	0
$983$	50.6189	1.61449	0.807246	−	0.590216i	$-0.200957\pi$
$983$	50.6189	1.61449	0.807246	+	0.590216i	$0.200957\pi$
$984$	0	0
$985$	−11.3712	−0.362318
$986$	0	0
$987$	−1.10671	−0.0352270
$988$	0	0
$989$	−6.29425	−0.200145
$990$	0	0
$991$	−15.7226	−0.499444	−0.249722	−	0.968318i	$-0.580339\pi$
$991$	−15.7226	−0.499444	−0.249722	+	0.968318i	$0.580339\pi$
$992$	0	0
$993$	3.55034	0.112667
$994$	0	0
$995$	12.0399	0.381691
$996$	0	0
$997$	−9.08507	−0.287727	−0.143864	−	0.989598i	$-0.545953\pi$
$997$	−9.08507	−0.287727	−0.143864	+	0.989598i	$0.545953\pi$
$998$	0	0
$999$	4.06243	0.128530

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.n.1.7		12
4.3	odd	2		241.2.a.b.1.4	✓	12
12.11	even	2		2169.2.a.h.1.9		12
20.19	odd	2		6025.2.a.h.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.4	✓	12	4.3	odd	2
2169.2.a.h.1.9		12	12.11	even	2
3856.2.a.n.1.7		12	1.1	even	1	trivial
6025.2.a.h.1.9		12	20.19	odd	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:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q+0.126224 q^{3} +0.612768 q^{5} -1.03110 q^{7} -2.98407 q^{9} +O(q^{10})\)	\(q+0.126224 q^{3} +0.612768 q^{5} -1.03110 q^{7} -2.98407 q^{9} -0.227935 q^{11} +3.38088 q^{13} +0.0773462 q^{15} +7.12130 q^{17} -3.40112 q^{19} -0.130150 q^{21} -6.91488 q^{23} -4.62452 q^{25} -0.755335 q^{27} +0.569431 q^{29} -4.93697 q^{31} -0.0287710 q^{33} -0.631826 q^{35} -5.37832 q^{37} +0.426749 q^{39} +10.7559 q^{41} +0.910247 q^{43} -1.82854 q^{45} +8.50333 q^{47} -5.93683 q^{49} +0.898882 q^{51} -7.76696 q^{53} -0.139671 q^{55} -0.429304 q^{57} -11.2505 q^{59} -3.65450 q^{61} +3.07688 q^{63} +2.07169 q^{65} +12.0694 q^{67} -0.872826 q^{69} +9.48630 q^{71} -7.10488 q^{73} -0.583726 q^{75} +0.235024 q^{77} -0.366201 q^{79} +8.85686 q^{81} -17.8030 q^{83} +4.36370 q^{85} +0.0718760 q^{87} -7.54236 q^{89} -3.48603 q^{91} -0.623166 q^{93} -2.08410 q^{95} -7.85505 q^{97} +0.680174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9	\( 12 q - q^{3} + 6 q^{5} - 3 q^{7} + 15 q^{9} - 22 q^{11} - 5 q^{13} - 13 q^{15} - 4 q^{17} + 6 q^{19} - 14 q^{21} - 32 q^{23} + 4 q^{25} + 5 q^{27} + 6 q^{29} - 8 q^{31} - 24 q^{33} - 15 q^{35} - 8 q^{37} - 31 q^{39} - q^{41} + 2 q^{43} - 15 q^{45} - 34 q^{47} - 9 q^{49} + 3 q^{51} + 5 q^{53} + 3 q^{55} - 22 q^{57} - 26 q^{59} - 26 q^{61} + 4 q^{63} - 25 q^{65} - 6 q^{67} - 2 q^{69} - 94 q^{71} - 22 q^{73} - 7 q^{77} - 9 q^{79} + 4 q^{81} + 8 q^{83} + 4 q^{85} - 4 q^{87} - 3 q^{89} + 20 q^{91} + 12 q^{93} - 33 q^{95} - 29 q^{97} - 36 q^{99}+O(q^{100}) \) 12 * q - q^3 + 6 * q^5 - 3 * q^7 + 15 * q^9 - 22 * q^11 - 5 * q^13 - 13 * q^15 - 4 * q^17 + 6 * q^19 - 14 * q^21 - 32 * q^23 + 4 * q^25 + 5 * q^27 + 6 * q^29 - 8 * q^31 - 24 * q^33 - 15 * q^35 - 8 * q^37 - 31 * q^39 - q^41 + 2 * q^43 - 15 * q^45 - 34 * q^47 - 9 * q^49 + 3 * q^51 + 5 * q^53 + 3 * q^55 - 22 * q^57 - 26 * q^59 - 26 * q^61 + 4 * q^63 - 25 * q^65 - 6 * q^67 - 2 * q^69 - 94 * q^71 - 22 * q^73 - 7 * q^77 - 9 * q^79 + 4 * q^81 + 8 * q^83 + 4 * q^85 - 4 * q^87 - 3 * q^89 + 20 * q^91 + 12 * q^93 - 33 * q^95 - 29 * q^97 - 36 * q^99

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