LMFDB - Embedded newform 1568.4.a.be.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,4,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1568.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1568.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:	$92.5149948890$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 98x^{4} - 8x^{3} + 1908x^{2} + 3184x + 1176 $ x^6 - 98x^4 - 8x^3 + 1908x^2 + 3184x + 1176
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$6.57148$ of defining polynomial
Character	$\chi$	$=$	1568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.57148 q^{3} -4.55478 q^{5} +4.04139 q^{9} +O(q^{10})$	$q+5.57148 q^{3} -4.55478 q^{5} +4.04139 q^{9} -65.7625 q^{11} +73.6042 q^{13} -25.3769 q^{15} +131.003 q^{17} -26.4522 q^{19} -82.1418 q^{23} -104.254 q^{25} -127.913 q^{27} +4.65118 q^{29} +80.0681 q^{31} -366.394 q^{33} -267.722 q^{37} +410.084 q^{39} +328.215 q^{41} +289.643 q^{43} -18.4077 q^{45} -119.270 q^{47} +729.882 q^{51} -150.210 q^{53} +299.534 q^{55} -147.378 q^{57} -423.187 q^{59} -502.304 q^{61} -335.251 q^{65} -390.231 q^{67} -457.651 q^{69} -803.592 q^{71} -368.456 q^{73} -580.849 q^{75} -552.799 q^{79} -821.785 q^{81} -62.3318 q^{83} -596.691 q^{85} +25.9140 q^{87} +475.368 q^{89} +446.098 q^{93} +120.484 q^{95} +11.8811 q^{97} -265.772 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 10 q^{5} + 40 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 - 10 * q^5 + 40 * q^9	$ 6 q - 6 q^{3} - 10 q^{5} + 40 q^{9} - 42 q^{11} - 8 q^{13} + 26 q^{15} + 70 q^{17} - 158 q^{19} + 158 q^{23} + 72 q^{25} - 246 q^{27} - 56 q^{29} - 2 q^{31} + 262 q^{33} - 102 q^{37} + 280 q^{39} + 48 q^{41} - 184 q^{43} - 392 q^{45} - 766 q^{47} + 394 q^{51} + 562 q^{53} - 186 q^{55} + 658 q^{57} - 854 q^{59} + 106 q^{61} + 488 q^{65} + 906 q^{67} - 1394 q^{69} - 1488 q^{71} + 202 q^{73} - 2248 q^{75} + 942 q^{79} + 674 q^{81} + 616 q^{83} + 998 q^{85} - 1656 q^{87} + 858 q^{89} - 1362 q^{93} + 2386 q^{95} + 1696 q^{97} - 5752 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 - 10 * q^5 + 40 * q^9 - 42 * q^11 - 8 * q^13 + 26 * q^15 + 70 * q^17 - 158 * q^19 + 158 * q^23 + 72 * q^25 - 246 * q^27 - 56 * q^29 - 2 * q^31 + 262 * q^33 - 102 * q^37 + 280 * q^39 + 48 * q^41 - 184 * q^43 - 392 * q^45 - 766 * q^47 + 394 * q^51 + 562 * q^53 - 186 * q^55 + 658 * q^57 - 854 * q^59 + 106 * q^61 + 488 * q^65 + 906 * q^67 - 1394 * q^69 - 1488 * q^71 + 202 * q^73 - 2248 * q^75 + 942 * q^79 + 674 * q^81 + 616 * q^83 + 998 * q^85 - 1656 * q^87 + 858 * q^89 - 1362 * q^93 + 2386 * q^95 + 1696 * q^97 - 5752 * 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
$3$	5.57148	1.07223	0.536116	−	0.844144i	$-0.319891\pi$
$3$	5.57148	1.07223	0.536116	+	0.844144i	$0.319891\pi$
$4$	0	0
$5$	−4.55478	−0.407392	−0.203696	−	0.979034i	$-0.565295\pi$
$5$	−4.55478	−0.407392	−0.203696	+	0.979034i	$0.565295\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	4.04139	0.149681
$10$	0	0
$11$	−65.7625	−1.80256	−0.901279	−	0.433240i	$-0.857370\pi$
$11$	−65.7625	−1.80256	−0.901279	+	0.433240i	$0.857370\pi$
$12$	0	0
$13$	73.6042	1.57032	0.785159	−	0.619294i	$-0.212581\pi$
$13$	73.6042	1.57032	0.785159	+	0.619294i	$0.212581\pi$
$14$	0	0
$15$	−25.3769	−0.436819
$16$	0	0
$17$	131.003	1.86900	0.934498	−	0.355967i	$-0.115849\pi$
$17$	131.003	1.86900	0.934498	+	0.355967i	$0.115849\pi$
$18$	0	0
$19$	−26.4522	−0.319397	−0.159699	−	0.987166i	$-0.551052\pi$
$19$	−26.4522	−0.319397	−0.159699	+	0.987166i	$0.551052\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−82.1418	−0.744685	−0.372342	−	0.928095i	$-0.621445\pi$
$23$	−82.1418	−0.744685	−0.372342	+	0.928095i	$0.621445\pi$
$24$	0	0
$25$	−104.254	−0.834032
$26$	0	0
$27$	−127.913	−0.911739
$28$	0	0
$29$	4.65118	0.0297828	0.0148914	−	0.999889i	$-0.495260\pi$
$29$	4.65118	0.0297828	0.0148914	+	0.999889i	$0.495260\pi$
$30$	0	0
$31$	80.0681	0.463892	0.231946	−	0.972729i	$-0.425491\pi$
$31$	80.0681	0.463892	0.231946	+	0.972729i	$0.425491\pi$
$32$	0	0
$33$	−366.394	−1.93276
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−267.722	−1.18955	−0.594774	−	0.803893i	$-0.702758\pi$
$37$	−267.722	−1.18955	−0.594774	+	0.803893i	$0.702758\pi$
$38$	0	0
$39$	410.084	1.68375
$40$	0	0
$41$	328.215	1.25021	0.625104	−	0.780541i	$-0.285057\pi$
$41$	328.215	1.25021	0.625104	+	0.780541i	$0.285057\pi$
$42$	0	0
$43$	289.643	1.02721	0.513606	−	0.858026i	$-0.328309\pi$
$43$	289.643	1.02721	0.513606	+	0.858026i	$0.328309\pi$
$44$	0	0
$45$	−18.4077	−0.0609789
$46$	0	0
$47$	−119.270	−0.370156	−0.185078	−	0.982724i	$-0.559254\pi$
$47$	−119.270	−0.370156	−0.185078	+	0.982724i	$0.559254\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	729.882	2.00400
$52$	0	0
$53$	−150.210	−0.389302	−0.194651	−	0.980873i	$-0.562357\pi$
$53$	−150.210	−0.389302	−0.194651	+	0.980873i	$0.562357\pi$
$54$	0	0
$55$	299.534	0.734348
$56$	0	0
$57$	−147.378	−0.342468
$58$	0	0
$59$	−423.187	−0.933802	−0.466901	−	0.884310i	$-0.654630\pi$
$59$	−423.187	−0.933802	−0.466901	+	0.884310i	$0.654630\pi$
$60$	0	0
$61$	−502.304	−1.05432	−0.527160	−	0.849766i	$-0.676743\pi$
$61$	−502.304	−1.05432	−0.527160	+	0.849766i	$0.676743\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−335.251	−0.639736
$66$	0	0
$67$	−390.231	−0.711557	−0.355778	−	0.934570i	$-0.615784\pi$
$67$	−390.231	−0.711557	−0.355778	+	0.934570i	$0.615784\pi$
$68$	0	0
$69$	−457.651	−0.798475
$70$	0	0
$71$	−803.592	−1.34322	−0.671611	−	0.740904i	$-0.734397\pi$
$71$	−803.592	−1.34322	−0.671611	+	0.740904i	$0.734397\pi$
$72$	0	0
$73$	−368.456	−0.590746	−0.295373	−	0.955382i	$-0.595444\pi$
$73$	−368.456	−0.590746	−0.295373	+	0.955382i	$0.595444\pi$
$74$	0	0
$75$	−580.849	−0.894275
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−552.799	−0.787275	−0.393638	−	0.919266i	$-0.628784\pi$
$79$	−552.799	−0.787275	−0.393638	+	0.919266i	$0.628784\pi$
$80$	0	0
$81$	−821.785	−1.12728
$82$	0	0
$83$	−62.3318	−0.0824313	−0.0412157	−	0.999150i	$-0.513123\pi$
$83$	−62.3318	−0.0824313	−0.0412157	+	0.999150i	$0.513123\pi$
$84$	0	0
$85$	−596.691	−0.761415
$86$	0	0
$87$	25.9140	0.0319341
$88$	0	0
$89$	475.368	0.566167	0.283084	−	0.959095i	$-0.408643\pi$
$89$	475.368	0.566167	0.283084	+	0.959095i	$0.408643\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	446.098	0.497400
$94$	0	0
$95$	120.484	0.130120
$96$	0	0
$97$	11.8811	0.0124365	0.00621824	−	0.999981i	$-0.498021\pi$
$97$	11.8811	0.0124365	0.00621824	+	0.999981i	$0.498021\pi$
$98$	0	0
$99$	−265.772	−0.269809
$100$	0	0
$101$	734.439	0.723559	0.361779	−	0.932264i	$-0.382169\pi$
$101$	734.439	0.723559	0.361779	+	0.932264i	$0.382169\pi$
$102$	0	0
$103$	−1122.99	−1.07428	−0.537141	−	0.843492i	$-0.680496\pi$
$103$	−1122.99	−1.07428	−0.537141	+	0.843492i	$0.680496\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−404.668	−0.365614	−0.182807	−	0.983149i	$-0.558518\pi$
$107$	−404.668	−0.365614	−0.182807	+	0.983149i	$0.558518\pi$
$108$	0	0
$109$	−1945.55	−1.70963	−0.854814	−	0.518935i	$-0.826329\pi$
$109$	−1945.55	−1.70963	−0.854814	+	0.518935i	$0.826329\pi$
$110$	0	0
$111$	−1491.61	−1.27547
$112$	0	0
$113$	−1661.80	−1.38344	−0.691722	−	0.722164i	$-0.743148\pi$
$113$	−1661.80	−1.38344	−0.691722	+	0.722164i	$0.743148\pi$
$114$	0	0
$115$	374.138	0.303379
$116$	0	0
$117$	297.463	0.235047
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2993.70	2.24921
$122$	0	0
$123$	1828.64	1.34051
$124$	0	0
$125$	1044.20	0.747170
$126$	0	0
$127$	−2187.49	−1.52841	−0.764205	−	0.644973i	$-0.776869\pi$
$127$	−2187.49	−1.52841	−0.764205	+	0.644973i	$0.776869\pi$
$128$	0	0
$129$	1613.74	1.10141
$130$	0	0
$131$	90.2920	0.0602202	0.0301101	−	0.999547i	$-0.490414\pi$
$131$	90.2920	0.0602202	0.0301101	+	0.999547i	$0.490414\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	582.618	0.371435
$136$	0	0
$137$	1366.28	0.852036	0.426018	−	0.904715i	$-0.359916\pi$
$137$	1366.28	0.852036	0.426018	+	0.904715i	$0.359916\pi$
$138$	0	0
$139$	213.957	0.130558	0.0652791	−	0.997867i	$-0.479206\pi$
$139$	213.957	0.130558	0.0652791	+	0.997867i	$0.479206\pi$
$140$	0	0
$141$	−664.511	−0.396893
$142$	0	0
$143$	−4840.40	−2.83059
$144$	0	0
$145$	−21.1851	−0.0121333
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2230.70	−1.22648	−0.613240	−	0.789896i	$-0.710134\pi$
$149$	−2230.70	−1.22648	−0.613240	+	0.789896i	$0.710134\pi$
$150$	0	0
$151$	−881.041	−0.474822	−0.237411	−	0.971409i	$-0.576299\pi$
$151$	−881.041	−0.474822	−0.237411	+	0.971409i	$0.576299\pi$
$152$	0	0
$153$	529.435	0.279753
$154$	0	0
$155$	−364.693	−0.188986
$156$	0	0
$157$	−2161.63	−1.09883	−0.549417	−	0.835548i	$-0.685150\pi$
$157$	−2161.63	−1.09883	−0.549417	+	0.835548i	$0.685150\pi$
$158$	0	0
$159$	−836.894	−0.417421
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2968.98	1.42668	0.713340	−	0.700818i	$-0.247182\pi$
$163$	2968.98	1.42668	0.713340	+	0.700818i	$0.247182\pi$
$164$	0	0
$165$	1668.85	0.787391
$166$	0	0
$167$	2216.74	1.02716	0.513582	−	0.858041i	$-0.328318\pi$
$167$	2216.74	1.02716	0.513582	+	0.858041i	$0.328318\pi$
$168$	0	0
$169$	3220.58	1.46590
$170$	0	0
$171$	−106.904	−0.0478077
$172$	0	0
$173$	−744.132	−0.327025	−0.163512	−	0.986541i	$-0.552282\pi$
$173$	−744.132	−0.327025	−0.163512	+	0.986541i	$0.552282\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2357.78	−1.00125
$178$	0	0
$179$	2912.53	1.21616	0.608080	−	0.793876i	$-0.291940\pi$
$179$	2912.53	1.21616	0.608080	+	0.793876i	$0.291940\pi$
$180$	0	0
$181$	−2630.43	−1.08021	−0.540106	−	0.841597i	$-0.681616\pi$
$181$	−2630.43	−1.08021	−0.540106	+	0.841597i	$0.681616\pi$
$182$	0	0
$183$	−2798.58	−1.13048
$184$	0	0
$185$	1219.42	0.484613
$186$	0	0
$187$	−8615.10	−3.36897
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1440.54	0.545728	0.272864	−	0.962053i	$-0.412029\pi$
$191$	1440.54	0.545728	0.272864	+	0.962053i	$0.412029\pi$
$192$	0	0
$193$	−936.251	−0.349186	−0.174593	−	0.984641i	$-0.555861\pi$
$193$	−936.251	−0.349186	−0.174593	+	0.984641i	$0.555861\pi$
$194$	0	0
$195$	−1867.85	−0.685945
$196$	0	0
$197$	4679.21	1.69228	0.846141	−	0.532959i	$-0.178920\pi$
$197$	4679.21	1.69228	0.846141	+	0.532959i	$0.178920\pi$
$198$	0	0
$199$	−2345.42	−0.835490	−0.417745	−	0.908564i	$-0.637179\pi$
$199$	−2345.42	−0.835490	−0.417745	+	0.908564i	$0.637179\pi$
$200$	0	0
$201$	−2174.16	−0.762954
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1494.95	−0.509325
$206$	0	0
$207$	−331.967	−0.111465
$208$	0	0
$209$	1739.56	0.575732
$210$	0	0
$211$	−3403.64	−1.11050	−0.555251	−	0.831683i	$-0.687378\pi$
$211$	−3403.64	−1.11050	−0.555251	+	0.831683i	$0.687378\pi$
$212$	0	0
$213$	−4477.19	−1.44025
$214$	0	0
$215$	−1319.26	−0.418478
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2052.84	−0.633417
$220$	0	0
$221$	9642.39	2.93492
$222$	0	0
$223$	−367.463	−0.110346	−0.0551729	−	0.998477i	$-0.517571\pi$
$223$	−367.463	−0.110346	−0.0551729	+	0.998477i	$0.517571\pi$
$224$	0	0
$225$	−421.331	−0.124839
$226$	0	0
$227$	2305.45	0.674089	0.337045	−	0.941489i	$-0.390573\pi$
$227$	2305.45	0.674089	0.337045	+	0.941489i	$0.390573\pi$
$228$	0	0
$229$	2867.56	0.827484	0.413742	−	0.910394i	$-0.364222\pi$
$229$	2867.56	0.827484	0.413742	+	0.910394i	$0.364222\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1628.52	−0.457888	−0.228944	−	0.973440i	$-0.573527\pi$
$233$	−1628.52	−0.457888	−0.228944	+	0.973440i	$0.573527\pi$
$234$	0	0
$235$	543.250	0.150799
$236$	0	0
$237$	−3079.91	−0.844142
$238$	0	0
$239$	−4494.70	−1.21648	−0.608238	−	0.793755i	$-0.708123\pi$
$239$	−4494.70	−1.21648	−0.608238	+	0.793755i	$0.708123\pi$
$240$	0	0
$241$	−1689.90	−0.451684	−0.225842	−	0.974164i	$-0.572513\pi$
$241$	−1689.90	−0.451684	−0.225842	+	0.974164i	$0.572513\pi$
$242$	0	0
$243$	−1124.89	−0.296963
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1946.99	−0.501555
$248$	0	0
$249$	−347.280	−0.0883855
$250$	0	0
$251$	−4402.52	−1.10711	−0.553555	−	0.832813i	$-0.686729\pi$
$251$	−4402.52	−1.10711	−0.553555	+	0.832813i	$0.686729\pi$
$252$	0	0
$253$	5401.85	1.34234
$254$	0	0
$255$	−3324.45	−0.816413
$256$	0	0
$257$	5473.75	1.32857	0.664286	−	0.747478i	$-0.268736\pi$
$257$	5473.75	1.32857	0.664286	+	0.747478i	$0.268736\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	18.7972	0.00445793
$262$	0	0
$263$	5744.71	1.34690	0.673449	−	0.739234i	$-0.264812\pi$
$263$	5744.71	1.34690	0.673449	+	0.739234i	$0.264812\pi$
$264$	0	0
$265$	684.176	0.158598
$266$	0	0
$267$	2648.50	0.607063
$268$	0	0
$269$	2033.68	0.460950	0.230475	−	0.973078i	$-0.425972\pi$
$269$	2033.68	0.460950	0.230475	+	0.973078i	$0.425972\pi$
$270$	0	0
$271$	1545.15	0.346352	0.173176	−	0.984891i	$-0.444597\pi$
$271$	1545.15	0.346352	0.173176	+	0.984891i	$0.444597\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6856.00	1.50339
$276$	0	0
$277$	−4001.94	−0.868061	−0.434031	−	0.900898i	$-0.642909\pi$
$277$	−4001.94	−0.868061	−0.434031	+	0.900898i	$0.642909\pi$
$278$	0	0
$279$	323.586	0.0694359
$280$	0	0
$281$	−6546.87	−1.38987	−0.694935	−	0.719072i	$-0.744567\pi$
$281$	−6546.87	−1.38987	−0.694935	+	0.719072i	$0.744567\pi$
$282$	0	0
$283$	−4310.93	−0.905507	−0.452753	−	0.891636i	$-0.649558\pi$
$283$	−4310.93	−0.905507	−0.452753	+	0.891636i	$0.649558\pi$
$284$	0	0
$285$	671.274	0.139519
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12248.8	2.49315
$290$	0	0
$291$	66.1951	0.0133348
$292$	0	0
$293$	357.486	0.0712783	0.0356392	−	0.999365i	$-0.488653\pi$
$293$	357.486	0.0712783	0.0356392	+	0.999365i	$0.488653\pi$
$294$	0	0
$295$	1927.53	0.380424
$296$	0	0
$297$	8411.91	1.64346
$298$	0	0
$299$	−6045.98	−1.16939
$300$	0	0
$301$	0	0
$302$	0	0
$303$	4091.91	0.775823
$304$	0	0
$305$	2287.89	0.429522
$306$	0	0
$307$	−4336.47	−0.806173	−0.403087	−	0.915162i	$-0.632063\pi$
$307$	−4336.47	−0.806173	−0.403087	+	0.915162i	$0.632063\pi$
$308$	0	0
$309$	−6256.70	−1.15188
$310$	0	0
$311$	931.059	0.169761	0.0848803	−	0.996391i	$-0.472949\pi$
$311$	931.059	0.169761	0.0848803	+	0.996391i	$0.472949\pi$
$312$	0	0
$313$	7682.05	1.38727	0.693635	−	0.720327i	$-0.256008\pi$
$313$	7682.05	1.38727	0.693635	+	0.720327i	$0.256008\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4615.08	0.817693	0.408846	−	0.912603i	$-0.365931\pi$
$317$	4615.08	0.817693	0.408846	+	0.912603i	$0.365931\pi$
$318$	0	0
$319$	−305.873	−0.0536853
$320$	0	0
$321$	−2254.60	−0.392023
$322$	0	0
$323$	−3465.32	−0.596952
$324$	0	0
$325$	−7673.53	−1.30970
$326$	0	0
$327$	−10839.6	−1.83312
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−3272.31	−0.543392	−0.271696	−	0.962383i	$-0.587584\pi$
$331$	−3272.31	−0.543392	−0.271696	+	0.962383i	$0.587584\pi$
$332$	0	0
$333$	−1081.97	−0.178053
$334$	0	0
$335$	1777.42	0.289883
$336$	0	0
$337$	164.690	0.0266209	0.0133104	−	0.999911i	$-0.495763\pi$
$337$	164.690	0.0266209	0.0133104	+	0.999911i	$0.495763\pi$
$338$	0	0
$339$	−9258.69	−1.48337
$340$	0	0
$341$	−5265.48	−0.836192
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2084.50	0.325292
$346$	0	0
$347$	−1754.71	−0.271464	−0.135732	−	0.990746i	$-0.543339\pi$
$347$	−1754.71	−0.271464	−0.135732	+	0.990746i	$0.543339\pi$
$348$	0	0
$349$	−4237.15	−0.649884	−0.324942	−	0.945734i	$-0.605345\pi$
$349$	−4237.15	−0.649884	−0.324942	+	0.945734i	$0.605345\pi$
$350$	0	0
$351$	−9414.97	−1.43172
$352$	0	0
$353$	4624.68	0.697300	0.348650	−	0.937253i	$-0.386640\pi$
$353$	4624.68	0.697300	0.348650	+	0.937253i	$0.386640\pi$
$354$	0	0
$355$	3660.19	0.547218
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3467.29	0.509739	0.254870	−	0.966975i	$-0.417967\pi$
$359$	3467.29	0.509739	0.254870	+	0.966975i	$0.417967\pi$
$360$	0	0
$361$	−6159.28	−0.897986
$362$	0	0
$363$	16679.4	2.41168
$364$	0	0
$365$	1678.24	0.240666
$366$	0	0
$367$	8196.29	1.16578	0.582892	−	0.812549i	$-0.301921\pi$
$367$	8196.29	1.16578	0.582892	+	0.812549i	$0.301921\pi$
$368$	0	0
$369$	1326.44	0.187132
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6555.68	−0.910028	−0.455014	−	0.890484i	$-0.650366\pi$
$373$	−6555.68	−0.910028	−0.455014	+	0.890484i	$0.650366\pi$
$374$	0	0
$375$	5817.75	0.801140
$376$	0	0
$377$	342.346	0.0467685
$378$	0	0
$379$	−10883.3	−1.47503	−0.737517	−	0.675328i	$-0.764002\pi$
$379$	−10883.3	−1.47503	−0.737517	+	0.675328i	$0.764002\pi$
$380$	0	0
$381$	−12187.5	−1.63881
$382$	0	0
$383$	−14464.7	−1.92980	−0.964898	−	0.262626i	$-0.915411\pi$
$383$	−14464.7	−1.92980	−0.964898	+	0.262626i	$0.915411\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1170.56	0.153754
$388$	0	0
$389$	−1087.21	−0.141706	−0.0708529	−	0.997487i	$-0.522572\pi$
$389$	−1087.21	−0.141706	−0.0708529	+	0.997487i	$0.522572\pi$
$390$	0	0
$391$	−10760.8	−1.39181
$392$	0	0
$393$	503.060	0.0645701
$394$	0	0
$395$	2517.88	0.320730
$396$	0	0
$397$	−1361.24	−0.172088	−0.0860439	−	0.996291i	$-0.527423\pi$
$397$	−1361.24	−0.172088	−0.0860439	+	0.996291i	$0.527423\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1103.80	−0.137460	−0.0687299	−	0.997635i	$-0.521895\pi$
$401$	−1103.80	−0.137460	−0.0687299	+	0.997635i	$0.521895\pi$
$402$	0	0
$403$	5893.35	0.728458
$404$	0	0
$405$	3743.05	0.459244
$406$	0	0
$407$	17606.1	2.14423
$408$	0	0
$409$	7476.26	0.903857	0.451928	−	0.892054i	$-0.350736\pi$
$409$	7476.26	0.903857	0.451928	+	0.892054i	$0.350736\pi$
$410$	0	0
$411$	7612.19	0.913580
$412$	0	0
$413$	0	0
$414$	0	0
$415$	283.908	0.0335819
$416$	0	0
$417$	1192.06	0.139989
$418$	0	0
$419$	−1048.13	−0.122206	−0.0611030	−	0.998131i	$-0.519462\pi$
$419$	−1048.13	−0.122206	−0.0611030	+	0.998131i	$0.519462\pi$
$420$	0	0
$421$	−1023.21	−0.118452	−0.0592259	−	0.998245i	$-0.518863\pi$
$421$	−1023.21	−0.118452	−0.0592259	+	0.998245i	$0.518863\pi$
$422$	0	0
$423$	−482.017	−0.0554053
$424$	0	0
$425$	−13657.6	−1.55880
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−26968.2	−3.03505
$430$	0	0
$431$	−8098.52	−0.905086	−0.452543	−	0.891743i	$-0.649483\pi$
$431$	−8098.52	−0.905086	−0.452543	+	0.891743i	$0.649483\pi$
$432$	0	0
$433$	14919.0	1.65580	0.827901	−	0.560875i	$-0.189535\pi$
$433$	14919.0	1.65580	0.827901	+	0.560875i	$0.189535\pi$
$434$	0	0
$435$	−118.032	−0.0130097
$436$	0	0
$437$	2172.83	0.237850
$438$	0	0
$439$	13439.6	1.46114	0.730568	−	0.682840i	$-0.239256\pi$
$439$	13439.6	1.46114	0.730568	+	0.682840i	$0.239256\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1582.55	−0.169727	−0.0848637	−	0.996393i	$-0.527045\pi$
$443$	−1582.55	−0.169727	−0.0848637	+	0.996393i	$0.527045\pi$
$444$	0	0
$445$	−2165.20	−0.230652
$446$	0	0
$447$	−12428.3	−1.31507
$448$	0	0
$449$	−497.311	−0.0522707	−0.0261353	−	0.999658i	$-0.508320\pi$
$449$	−497.311	−0.0522707	−0.0261353	+	0.999658i	$0.508320\pi$
$450$	0	0
$451$	−21584.2	−2.25357
$452$	0	0
$453$	−4908.70	−0.509119
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2341.59	0.239682	0.119841	−	0.992793i	$-0.461761\pi$
$457$	2341.59	0.239682	0.119841	+	0.992793i	$0.461761\pi$
$458$	0	0
$459$	−16757.1	−1.70404
$460$	0	0
$461$	−16842.4	−1.70158	−0.850792	−	0.525503i	$-0.823877\pi$
$461$	−16842.4	−1.70158	−0.850792	+	0.525503i	$0.823877\pi$
$462$	0	0
$463$	4464.62	0.448139	0.224069	−	0.974573i	$-0.428066\pi$
$463$	4464.62	0.448139	0.224069	+	0.974573i	$0.428066\pi$
$464$	0	0
$465$	−2031.88	−0.202637
$466$	0	0
$467$	16350.7	1.62017	0.810084	−	0.586314i	$-0.199422\pi$
$467$	16350.7	1.62017	0.810084	+	0.586314i	$0.199422\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−12043.5	−1.17820
$472$	0	0
$473$	−19047.6	−1.85161
$474$	0	0
$475$	2757.74	0.266387
$476$	0	0
$477$	−607.058	−0.0582711
$478$	0	0
$479$	7126.74	0.679810	0.339905	−	0.940460i	$-0.389605\pi$
$479$	7126.74	0.679810	0.339905	+	0.940460i	$0.389605\pi$
$480$	0	0
$481$	−19705.5	−1.86797
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−54.1157	−0.00506653
$486$	0	0
$487$	13739.2	1.27841	0.639204	−	0.769037i	$-0.279264\pi$
$487$	13739.2	1.27841	0.639204	+	0.769037i	$0.279264\pi$
$488$	0	0
$489$	16541.6	1.52973
$490$	0	0
$491$	−10840.2	−0.996362	−0.498181	−	0.867073i	$-0.665998\pi$
$491$	−10840.2	−0.996362	−0.498181	+	0.867073i	$0.665998\pi$
$492$	0	0
$493$	609.320	0.0556640
$494$	0	0
$495$	1210.53	0.109918
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17277.3	−1.54998	−0.774989	−	0.631975i	$-0.782244\pi$
$499$	−17277.3	−1.54998	−0.774989	+	0.631975i	$0.782244\pi$
$500$	0	0
$501$	12350.5	1.10136
$502$	0	0
$503$	8368.03	0.741774	0.370887	−	0.928678i	$-0.379054\pi$
$503$	8368.03	0.741774	0.370887	+	0.928678i	$0.379054\pi$
$504$	0	0
$505$	−3345.21	−0.294772
$506$	0	0
$507$	17943.4	1.57178
$508$	0	0
$509$	−13996.1	−1.21880	−0.609398	−	0.792864i	$-0.708589\pi$
$509$	−13996.1	−1.21880	−0.609398	+	0.792864i	$0.708589\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	3383.59	0.291207
$514$	0	0
$515$	5114.96	0.437655
$516$	0	0
$517$	7843.50	0.667228
$518$	0	0
$519$	−4145.92	−0.350647
$520$	0	0
$521$	20926.1	1.75968	0.879838	−	0.475274i	$-0.157651\pi$
$521$	20926.1	1.75968	0.879838	+	0.475274i	$0.157651\pi$
$522$	0	0
$523$	−4594.88	−0.384168	−0.192084	−	0.981378i	$-0.561525\pi$
$523$	−4594.88	−0.384168	−0.192084	+	0.981378i	$0.561525\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10489.2	0.867013
$528$	0	0
$529$	−5419.72	−0.445444
$530$	0	0
$531$	−1710.27	−0.139772
$532$	0	0
$533$	24158.0	1.96322
$534$	0	0
$535$	1843.17	0.148948
$536$	0	0
$537$	16227.1	1.30401
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10980.5	0.872620	0.436310	−	0.899796i	$-0.356285\pi$
$541$	10980.5	0.872620	0.436310	+	0.899796i	$0.356285\pi$
$542$	0	0
$543$	−14655.4	−1.15824
$544$	0	0
$545$	8861.54	0.696489
$546$	0	0
$547$	20353.1	1.59092	0.795462	−	0.606003i	$-0.207228\pi$
$547$	20353.1	1.59092	0.795462	+	0.606003i	$0.207228\pi$
$548$	0	0
$549$	−2030.01	−0.157812
$550$	0	0
$551$	−123.034	−0.00951255
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6793.96	0.519617
$556$	0	0
$557$	−11568.8	−0.880047	−0.440024	−	0.897986i	$-0.645030\pi$
$557$	−11568.8	−0.880047	−0.440024	+	0.897986i	$0.645030\pi$
$558$	0	0
$559$	21318.9	1.61305
$560$	0	0
$561$	−47998.8	−3.61232
$562$	0	0
$563$	8319.75	0.622799	0.311399	−	0.950279i	$-0.399202\pi$
$563$	8319.75	0.622799	0.311399	+	0.950279i	$0.399202\pi$
$564$	0	0
$565$	7569.15	0.563604
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13679.3	1.00785	0.503926	−	0.863747i	$-0.331888\pi$
$569$	13679.3	1.00785	0.503926	+	0.863747i	$0.331888\pi$
$570$	0	0
$571$	−5961.91	−0.436949	−0.218475	−	0.975843i	$-0.570108\pi$
$571$	−5961.91	−0.436949	−0.218475	+	0.975843i	$0.570108\pi$
$572$	0	0
$573$	8025.96	0.585147
$574$	0	0
$575$	8563.61	0.621091
$576$	0	0
$577$	18245.8	1.31643	0.658217	−	0.752829i	$-0.271311\pi$
$577$	18245.8	1.31643	0.658217	+	0.752829i	$0.271311\pi$
$578$	0	0
$579$	−5216.30	−0.374408
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9878.21	0.701738
$584$	0	0
$585$	−1354.88	−0.0957563
$586$	0	0
$587$	2476.93	0.174163	0.0870816	−	0.996201i	$-0.472246\pi$
$587$	2476.93	0.174163	0.0870816	+	0.996201i	$0.472246\pi$
$588$	0	0
$589$	−2117.97	−0.148166
$590$	0	0
$591$	26070.1	1.81452
$592$	0	0
$593$	1126.12	0.0779832	0.0389916	−	0.999240i	$-0.487585\pi$
$593$	1126.12	0.0779832	0.0389916	+	0.999240i	$0.487585\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13067.5	−0.895839
$598$	0	0
$599$	−6538.01	−0.445970	−0.222985	−	0.974822i	$-0.571580\pi$
$599$	−6538.01	−0.445970	−0.222985	+	0.974822i	$0.571580\pi$
$600$	0	0
$601$	−11848.0	−0.804143	−0.402072	−	0.915608i	$-0.631710\pi$
$601$	−11848.0	−0.804143	−0.402072	+	0.915608i	$0.631710\pi$
$602$	0	0
$603$	−1577.08	−0.106507
$604$	0	0
$605$	−13635.7	−0.916312
$606$	0	0
$607$	9944.17	0.664945	0.332472	−	0.943113i	$-0.392117\pi$
$607$	9944.17	0.664945	0.332472	+	0.943113i	$0.392117\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8778.78	−0.581263
$612$	0	0
$613$	5557.74	0.366191	0.183096	−	0.983095i	$-0.441388\pi$
$613$	5557.74	0.366191	0.183096	+	0.983095i	$0.441388\pi$
$614$	0	0
$615$	−8329.07	−0.546115
$616$	0	0
$617$	18075.0	1.17937	0.589686	−	0.807633i	$-0.299252\pi$
$617$	18075.0	1.17937	0.589686	+	0.807633i	$0.299252\pi$
$618$	0	0
$619$	−4243.05	−0.275513	−0.137757	−	0.990466i	$-0.543989\pi$
$619$	−4243.05	−0.275513	−0.137757	+	0.990466i	$0.543989\pi$
$620$	0	0
$621$	10507.0	0.678958
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8275.63	0.529640
$626$	0	0
$627$	9691.93	0.617318
$628$	0	0
$629$	−35072.5	−2.22326
$630$	0	0
$631$	29769.9	1.87816	0.939082	−	0.343692i	$-0.111678\pi$
$631$	29769.9	1.87816	0.939082	+	0.343692i	$0.111678\pi$
$632$	0	0
$633$	−18963.3	−1.19072
$634$	0	0
$635$	9963.54	0.622663
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3247.63	−0.201055
$640$	0	0
$641$	8138.64	0.501493	0.250746	−	0.968053i	$-0.419324\pi$
$641$	8138.64	0.501493	0.250746	+	0.968053i	$0.419324\pi$
$642$	0	0
$643$	−14788.0	−0.906971	−0.453486	−	0.891264i	$-0.649820\pi$
$643$	−14788.0	−0.906971	−0.453486	+	0.891264i	$0.649820\pi$
$644$	0	0
$645$	−7350.24	−0.448706
$646$	0	0
$647$	−28355.0	−1.72295	−0.861477	−	0.507797i	$-0.830460\pi$
$647$	−28355.0	−1.72295	−0.861477	+	0.507797i	$0.830460\pi$
$648$	0	0
$649$	27829.9	1.68323
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1185.89	0.0710679	0.0355340	−	0.999368i	$-0.488687\pi$
$653$	1185.89	0.0710679	0.0355340	+	0.999368i	$0.488687\pi$
$654$	0	0
$655$	−411.261	−0.0245333
$656$	0	0
$657$	−1489.07	−0.0884236
$658$	0	0
$659$	7804.32	0.461325	0.230662	−	0.973034i	$-0.425911\pi$
$659$	7804.32	0.461325	0.230662	+	0.973034i	$0.425911\pi$
$660$	0	0
$661$	−8689.83	−0.511339	−0.255670	−	0.966764i	$-0.582296\pi$
$661$	−8689.83	−0.511339	−0.255670	+	0.966764i	$0.582296\pi$
$662$	0	0
$663$	53722.4	3.14691
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−382.056	−0.0221788
$668$	0	0
$669$	−2047.31	−0.118316
$670$	0	0
$671$	33032.8	1.90047
$672$	0	0
$673$	31681.3	1.81460	0.907300	−	0.420483i	$-0.138139\pi$
$673$	31681.3	1.81460	0.907300	+	0.420483i	$0.138139\pi$
$674$	0	0
$675$	13335.5	0.760419
$676$	0	0
$677$	18567.5	1.05407	0.527036	−	0.849843i	$-0.323303\pi$
$677$	18567.5	1.05407	0.527036	+	0.849843i	$0.323303\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	12844.8	0.722780
$682$	0	0
$683$	−13887.9	−0.778046	−0.389023	−	0.921228i	$-0.627187\pi$
$683$	−13887.9	−0.778046	−0.389023	+	0.921228i	$0.627187\pi$
$684$	0	0
$685$	−6223.10	−0.347113
$686$	0	0
$687$	15976.6	0.887255
$688$	0	0
$689$	−11056.1	−0.611327
$690$	0	0
$691$	10225.0	0.562917	0.281459	−	0.959573i	$-0.409182\pi$
$691$	10225.0	0.562917	0.281459	+	0.959573i	$0.409182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−974.527	−0.0531884
$696$	0	0
$697$	42997.2	2.33664
$698$	0	0
$699$	−9073.27	−0.490962
$700$	0	0
$701$	−17766.2	−0.957232	−0.478616	−	0.878024i	$-0.658861\pi$
$701$	−17766.2	−0.957232	−0.478616	+	0.878024i	$0.658861\pi$
$702$	0	0
$703$	7081.84	0.379938
$704$	0	0
$705$	3026.70	0.161691
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15169.5	−0.803532	−0.401766	−	0.915742i	$-0.631603\pi$
$709$	−15169.5	−0.803532	−0.401766	+	0.915742i	$0.631603\pi$
$710$	0	0
$711$	−2234.08	−0.117840
$712$	0	0
$713$	−6576.94	−0.345453
$714$	0	0
$715$	22047.0	1.15316
$716$	0	0
$717$	−25042.1	−1.30434
$718$	0	0
$719$	5711.56	0.296252	0.148126	−	0.988968i	$-0.452676\pi$
$719$	5711.56	0.296252	0.148126	+	0.988968i	$0.452676\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9415.23	−0.484310
$724$	0	0
$725$	−484.904	−0.0248398
$726$	0	0
$727$	−38137.1	−1.94557	−0.972783	−	0.231719i	$-0.925565\pi$
$727$	−38137.1	−1.94557	−0.972783	+	0.231719i	$0.925565\pi$
$728$	0	0
$729$	15920.9	0.808864
$730$	0	0
$731$	37944.2	1.91986
$732$	0	0
$733$	−29567.6	−1.48991	−0.744956	−	0.667114i	$-0.767529\pi$
$733$	−29567.6	−1.48991	−0.744956	+	0.667114i	$0.767529\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25662.6	1.28262
$738$	0	0
$739$	−3010.57	−0.149859	−0.0749293	−	0.997189i	$-0.523873\pi$
$739$	−3010.57	−0.149859	−0.0749293	+	0.997189i	$0.523873\pi$
$740$	0	0
$741$	−10847.6	−0.537783
$742$	0	0
$743$	−10229.6	−0.505100	−0.252550	−	0.967584i	$-0.581269\pi$
$743$	−10229.6	−0.505100	−0.252550	+	0.967584i	$0.581269\pi$
$744$	0	0
$745$	10160.3	0.499659
$746$	0	0
$747$	−251.907	−0.0123384
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29261.9	−1.42181	−0.710906	−	0.703287i	$-0.751715\pi$
$751$	−29261.9	−1.42181	−0.710906	+	0.703287i	$0.751715\pi$
$752$	0	0
$753$	−24528.5	−1.18708
$754$	0	0
$755$	4012.95	0.193439
$756$	0	0
$757$	10735.1	0.515420	0.257710	−	0.966222i	$-0.417032\pi$
$757$	10735.1	0.515420	0.257710	+	0.966222i	$0.417032\pi$
$758$	0	0
$759$	30096.3	1.43930
$760$	0	0
$761$	3655.02	0.174105	0.0870527	−	0.996204i	$-0.472255\pi$
$761$	3655.02	0.174105	0.0870527	+	0.996204i	$0.472255\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2411.46	−0.113969
$766$	0	0
$767$	−31148.4	−1.46637
$768$	0	0
$769$	−24785.4	−1.16227	−0.581133	−	0.813808i	$-0.697390\pi$
$769$	−24785.4	−1.16227	−0.581133	+	0.813808i	$0.697390\pi$
$770$	0	0
$771$	30496.9	1.42454
$772$	0	0
$773$	−9923.99	−0.461761	−0.230880	−	0.972982i	$-0.574161\pi$
$773$	−9923.99	−0.461761	−0.230880	+	0.972982i	$0.574161\pi$
$774$	0	0
$775$	−8347.41	−0.386901
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8681.99	−0.399313
$780$	0	0
$781$	52846.2	2.42124
$782$	0	0
$783$	−594.948	−0.0271542
$784$	0	0
$785$	9845.76	0.447657
$786$	0	0
$787$	32983.7	1.49396	0.746978	−	0.664849i	$-0.231504\pi$
$787$	32983.7	1.49396	0.746978	+	0.664849i	$0.231504\pi$
$788$	0	0
$789$	32006.6	1.44419
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−36971.7	−1.65562
$794$	0	0
$795$	3811.87	0.170054
$796$	0	0
$797$	14832.3	0.659207	0.329603	−	0.944119i	$-0.393085\pi$
$797$	14832.3	0.659207	0.329603	+	0.944119i	$0.393085\pi$
$798$	0	0
$799$	−15624.8	−0.691820
$800$	0	0
$801$	1921.15	0.0847445
$802$	0	0
$803$	24230.6	1.06485
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11330.6	0.494245
$808$	0	0
$809$	4499.54	0.195544	0.0977722	−	0.995209i	$-0.468828\pi$
$809$	4499.54	0.195544	0.0977722	+	0.995209i	$0.468828\pi$
$810$	0	0
$811$	24307.3	1.05246	0.526229	−	0.850343i	$-0.323605\pi$
$811$	24307.3	1.05246	0.526229	+	0.850343i	$0.323605\pi$
$812$	0	0
$813$	8608.79	0.371370
$814$	0	0
$815$	−13523.1	−0.581218
$816$	0	0
$817$	−7661.69	−0.328089
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1067.79	0.0453913	0.0226957	−	0.999742i	$-0.492775\pi$
$821$	1067.79	0.0453913	0.0226957	+	0.999742i	$0.492775\pi$
$822$	0	0
$823$	−17505.9	−0.741456	−0.370728	−	0.928741i	$-0.620892\pi$
$823$	−17505.9	−0.741456	−0.370728	+	0.928741i	$0.620892\pi$
$824$	0	0
$825$	38198.1	1.61198
$826$	0	0
$827$	−32525.9	−1.36764	−0.683818	−	0.729653i	$-0.739682\pi$
$827$	−32525.9	−1.36764	−0.683818	+	0.729653i	$0.739682\pi$
$828$	0	0
$829$	35924.1	1.50506	0.752532	−	0.658556i	$-0.228833\pi$
$829$	35924.1	1.50506	0.752532	+	0.658556i	$0.228833\pi$
$830$	0	0
$831$	−22296.7	−0.930763
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−10096.8	−0.418459
$836$	0	0
$837$	−10241.8	−0.422948
$838$	0	0
$839$	11760.4	0.483927	0.241964	−	0.970285i	$-0.422209\pi$
$839$	11760.4	0.483927	0.241964	+	0.970285i	$0.422209\pi$
$840$	0	0
$841$	−24367.4	−0.999113
$842$	0	0
$843$	−36475.8	−1.49026
$844$	0	0
$845$	−14669.1	−0.597196
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−24018.3	−0.970913
$850$	0	0
$851$	21991.2	0.885838
$852$	0	0
$853$	31670.4	1.27125	0.635624	−	0.771999i	$-0.280743\pi$
$853$	31670.4	1.27125	0.635624	+	0.771999i	$0.280743\pi$
$854$	0	0
$855$	486.922	0.0194765
$856$	0	0
$857$	20698.3	0.825017	0.412508	−	0.910954i	$-0.364653\pi$
$857$	20698.3	0.825017	0.412508	+	0.910954i	$0.364653\pi$
$858$	0	0
$859$	23653.0	0.939498	0.469749	−	0.882800i	$-0.344344\pi$
$859$	23653.0	0.939498	0.469749	+	0.882800i	$0.344344\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32390.0	−1.27760	−0.638799	−	0.769373i	$-0.720569\pi$
$863$	−32390.0	−1.27760	−0.638799	+	0.769373i	$0.720569\pi$
$864$	0	0
$865$	3389.36	0.133227
$866$	0	0
$867$	68244.2	2.67323
$868$	0	0
$869$	36353.4	1.41911
$870$	0	0
$871$	−28722.7	−1.11737
$872$	0	0
$873$	48.0160	0.00186151
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38404.6	1.47871	0.739356	−	0.673315i	$-0.235130\pi$
$877$	38404.6	1.47871	0.739356	+	0.673315i	$0.235130\pi$
$878$	0	0
$879$	1991.73	0.0764269
$880$	0	0
$881$	4438.83	0.169748	0.0848740	−	0.996392i	$-0.472951\pi$
$881$	4438.83	0.169748	0.0848740	+	0.996392i	$0.472951\pi$
$882$	0	0
$883$	29750.2	1.13383	0.566917	−	0.823775i	$-0.308136\pi$
$883$	29750.2	1.13383	0.566917	+	0.823775i	$0.308136\pi$
$884$	0	0
$885$	10739.2	0.407902
$886$	0	0
$887$	37496.8	1.41941	0.709707	−	0.704497i	$-0.248827\pi$
$887$	37496.8	1.41941	0.709707	+	0.704497i	$0.248827\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	54042.6	2.03198
$892$	0	0
$893$	3154.95	0.118227
$894$	0	0
$895$	−13265.9	−0.495454
$896$	0	0
$897$	−33685.1	−1.25386
$898$	0	0
$899$	372.411	0.0138160
$900$	0	0
$901$	−19678.0	−0.727603
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11981.1	0.440070
$906$	0	0
$907$	−35564.5	−1.30199	−0.650993	−	0.759084i	$-0.725647\pi$
$907$	−35564.5	−1.30199	−0.650993	+	0.759084i	$0.725647\pi$
$908$	0	0
$909$	2968.15	0.108303
$910$	0	0
$911$	19554.7	0.711170	0.355585	−	0.934644i	$-0.384282\pi$
$911$	19554.7	0.711170	0.355585	+	0.934644i	$0.384282\pi$
$912$	0	0
$913$	4099.09	0.148587
$914$	0	0
$915$	12746.9	0.460547
$916$	0	0
$917$	0	0
$918$	0	0
$919$	19898.7	0.714253	0.357126	−	0.934056i	$-0.383757\pi$
$919$	19898.7	0.714253	0.357126	+	0.934056i	$0.383757\pi$
$920$	0	0
$921$	−24160.5	−0.864404
$922$	0	0
$923$	−59147.7	−2.10929
$924$	0	0
$925$	27911.1	0.992120
$926$	0	0
$927$	−4538.42	−0.160800
$928$	0	0
$929$	−7009.95	−0.247566	−0.123783	−	0.992309i	$-0.539503\pi$
$929$	−7009.95	−0.247566	−0.123783	+	0.992309i	$0.539503\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	5187.38	0.182023
$934$	0	0
$935$	39239.9	1.37249
$936$	0	0
$937$	−813.410	−0.0283596	−0.0141798	−	0.999899i	$-0.504514\pi$
$937$	−813.410	−0.0283596	−0.0141798	+	0.999899i	$0.504514\pi$
$938$	0	0
$939$	42800.4	1.48747
$940$	0	0
$941$	−14246.3	−0.493535	−0.246767	−	0.969075i	$-0.579368\pi$
$941$	−14246.3	−0.493535	−0.246767	+	0.969075i	$0.579368\pi$
$942$	0	0
$943$	−26960.2	−0.931011
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34106.3	1.17033	0.585167	−	0.810913i	$-0.301029\pi$
$947$	34106.3	1.17033	0.585167	+	0.810913i	$0.301029\pi$
$948$	0	0
$949$	−27119.9	−0.927660
$950$	0	0
$951$	25712.8	0.876756
$952$	0	0
$953$	10497.7	0.356826	0.178413	−	0.983956i	$-0.442904\pi$
$953$	10497.7	0.356826	0.178413	+	0.983956i	$0.442904\pi$
$954$	0	0
$955$	−6561.37	−0.222326
$956$	0	0
$957$	−1704.17	−0.0575631
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23380.1	−0.784804
$962$	0	0
$963$	−1635.42	−0.0547255
$964$	0	0
$965$	4264.42	0.142256
$966$	0	0
$967$	−47342.5	−1.57439	−0.787193	−	0.616707i	$-0.788466\pi$
$967$	−47342.5	−1.57439	−0.787193	+	0.616707i	$0.788466\pi$
$968$	0	0
$969$	−19307.0	−0.640071
$970$	0	0
$971$	1634.28	0.0540129	0.0270065	−	0.999635i	$-0.491403\pi$
$971$	1634.28	0.0540129	0.0270065	+	0.999635i	$0.491403\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−42752.9	−1.40430
$976$	0	0
$977$	25063.5	0.820729	0.410364	−	0.911922i	$-0.365402\pi$
$977$	25063.5	0.820729	0.410364	+	0.911922i	$0.365402\pi$
$978$	0	0
$979$	−31261.4	−1.02055
$980$	0	0
$981$	−7862.71	−0.255899
$982$	0	0
$983$	59124.2	1.91838	0.959191	−	0.282758i	$-0.0912495\pi$
$983$	59124.2	1.91838	0.959191	+	0.282758i	$0.0912495\pi$
$984$	0	0
$985$	−21312.8	−0.689423
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23791.8	−0.764950
$990$	0	0
$991$	38975.1	1.24933	0.624664	−	0.780894i	$-0.285236\pi$
$991$	38975.1	1.24933	0.624664	+	0.780894i	$0.285236\pi$
$992$	0	0
$993$	−18231.6	−0.582642
$994$	0	0
$995$	10682.9	0.340372
$996$	0	0
$997$	44303.7	1.40734	0.703668	−	0.710529i	$-0.251544\pi$
$997$	44303.7	1.40734	0.703668	+	0.710529i	$0.251544\pi$
$998$	0	0
$999$	34245.3	1.08456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.4.a.be.1.5		6
4.3	odd	2		1568.4.a.bi.1.2		6
7.2	even	3		224.4.i.e.193.2	yes	12
7.4	even	3		224.4.i.e.65.2	yes	12
7.6	odd	2		1568.4.a.bj.1.2		6
28.11	odd	6		224.4.i.c.65.5	✓	12
28.23	odd	6		224.4.i.c.193.5	yes	12
28.27	even	2		1568.4.a.bf.1.5		6
56.11	odd	6		448.4.i.q.65.2		12
56.37	even	6		448.4.i.o.193.5		12
56.51	odd	6		448.4.i.q.193.2		12
56.53	even	6		448.4.i.o.65.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.4.i.c.65.5	✓	12	28.11	odd	6
224.4.i.c.193.5	yes	12	28.23	odd	6
224.4.i.e.65.2	yes	12	7.4	even	3
224.4.i.e.193.2	yes	12	7.2	even	3
448.4.i.o.65.5		12	56.53	even	6
448.4.i.o.193.5		12	56.37	even	6
448.4.i.q.65.2		12	56.11	odd	6
448.4.i.q.193.2		12	56.51	odd	6
1568.4.a.be.1.5		6	1.1	even	1	trivial
1568.4.a.bf.1.5		6	28.27	even	2
1568.4.a.bi.1.2		6	4.3	odd	2
1568.4.a.bj.1.2		6	7.6	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1568.a (trivial)

\(f(q)\)	\(=\)	\(q+5.57148 q^{3} -4.55478 q^{5} +4.04139 q^{9} +O(q^{10})\)	\(q+5.57148 q^{3} -4.55478 q^{5} +4.04139 q^{9} -65.7625 q^{11} +73.6042 q^{13} -25.3769 q^{15} +131.003 q^{17} -26.4522 q^{19} -82.1418 q^{23} -104.254 q^{25} -127.913 q^{27} +4.65118 q^{29} +80.0681 q^{31} -366.394 q^{33} -267.722 q^{37} +410.084 q^{39} +328.215 q^{41} +289.643 q^{43} -18.4077 q^{45} -119.270 q^{47} +729.882 q^{51} -150.210 q^{53} +299.534 q^{55} -147.378 q^{57} -423.187 q^{59} -502.304 q^{61} -335.251 q^{65} -390.231 q^{67} -457.651 q^{69} -803.592 q^{71} -368.456 q^{73} -580.849 q^{75} -552.799 q^{79} -821.785 q^{81} -62.3318 q^{83} -596.691 q^{85} +25.9140 q^{87} +475.368 q^{89} +446.098 q^{93} +120.484 q^{95} +11.8811 q^{97} -265.772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 10 q^{5} + 40 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 10 * q^5 + 40 * q^9	\( 6 q - 6 q^{3} - 10 q^{5} + 40 q^{9} - 42 q^{11} - 8 q^{13} + 26 q^{15} + 70 q^{17} - 158 q^{19} + 158 q^{23} + 72 q^{25} - 246 q^{27} - 56 q^{29} - 2 q^{31} + 262 q^{33} - 102 q^{37} + 280 q^{39} + 48 q^{41} - 184 q^{43} - 392 q^{45} - 766 q^{47} + 394 q^{51} + 562 q^{53} - 186 q^{55} + 658 q^{57} - 854 q^{59} + 106 q^{61} + 488 q^{65} + 906 q^{67} - 1394 q^{69} - 1488 q^{71} + 202 q^{73} - 2248 q^{75} + 942 q^{79} + 674 q^{81} + 616 q^{83} + 998 q^{85} - 1656 q^{87} + 858 q^{89} - 1362 q^{93} + 2386 q^{95} + 1696 q^{97} - 5752 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 - 10 * q^5 + 40 * q^9 - 42 * q^11 - 8 * q^13 + 26 * q^15 + 70 * q^17 - 158 * q^19 + 158 * q^23 + 72 * q^25 - 246 * q^27 - 56 * q^29 - 2 * q^31 + 262 * q^33 - 102 * q^37 + 280 * q^39 + 48 * q^41 - 184 * q^43 - 392 * q^45 - 766 * q^47 + 394 * q^51 + 562 * q^53 - 186 * q^55 + 658 * q^57 - 854 * q^59 + 106 * q^61 + 488 * q^65 + 906 * q^67 - 1394 * q^69 - 1488 * q^71 + 202 * q^73 - 2248 * q^75 + 942 * q^79 + 674 * q^81 + 616 * q^83 + 998 * q^85 - 1656 * q^87 + 858 * q^89 - 1362 * q^93 + 2386 * q^95 + 1696 * q^97 - 5752 * q^99

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