LMFDB - Embedded newform 441.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,8,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	441.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:	$137.761796238$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -124.000 q^{4} -278.000 q^{5} +504.000 q^{8} +O(q^{10})$

$q-2.00000 q^{2} -124.000 q^{4} -278.000 q^{5} +504.000 q^{8} +556.000 q^{10} +4496.00 q^{11} +7274.00 q^{13} +14864.0 q^{16} +11382.0 q^{17} +15884.0 q^{19} +34472.0 q^{20} -8992.00 q^{22} -86100.0 q^{23} -841.000 q^{25} -14548.0 q^{26} -40702.0 q^{29} +44760.0 q^{31} -94240.0 q^{32} -22764.0 q^{34} -580962. q^{37} -31768.0 q^{38} -140112. q^{40} -171658. q^{41} -741148. q^{43} -557504. q^{44} +172200. q^{46} +1.07172e6 q^{47} +1682.00 q^{50} -901976. q^{52} +1.72178e6 q^{53} -1.24989e6 q^{55} +81404.0 q^{58} -1.55701e6 q^{59} -2.59800e6 q^{61} -89520.0 q^{62} -1.71411e6 q^{64} -2.02217e6 q^{65} -963548. q^{67} -1.41137e6 q^{68} +4.06338e6 q^{71} +5.37022e6 q^{73} +1.16192e6 q^{74} -1.96962e6 q^{76} +4.09494e6 q^{79} -4.13219e6 q^{80} +343316. q^{82} -1.34312e6 q^{83} -3.16420e6 q^{85} +1.48230e6 q^{86} +2.26598e6 q^{88} +9.08157e6 q^{89} +1.06764e7 q^{92} -2.14344e6 q^{94} -4.41575e6 q^{95} -6.48791e6 q^{97} +O(q^{100})$

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 $

$2$	−2.00000	−0.176777	−0.0883883	−	0.996086i	$-0.528172\pi$
$2$	−2.00000	−0.176777	−0.0883883	+	0.996086i	$0.528172\pi$
$3$	0	0
$3$	0	0
$4$	−124.000	−0.968750
$5$	−278.000	−0.994603	−0.497302	−	0.867578i	$-0.665676\pi$
$5$	−278.000	−0.994603	−0.497302	+	0.867578i	$0.665676\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	504.000	0.348029
$9$	0	0
$10$	556.000	0.175823
$11$	4496.00	1.01848	0.509239	−	0.860625i	$-0.329927\pi$
$11$	4496.00	1.01848	0.509239	+	0.860625i	$0.329927\pi$
$12$	0	0
$13$	7274.00	0.918272	0.459136	−	0.888366i	$-0.348159\pi$
$13$	7274.00	0.918272	0.459136	+	0.888366i	$0.348159\pi$
$14$	0	0
$15$	0	0
$16$	14864.0	0.907227
$17$	11382.0	0.561885	0.280942	−	0.959725i	$-0.409353\pi$
$17$	11382.0	0.561885	0.280942	+	0.959725i	$0.409353\pi$
$18$	0	0
$19$	15884.0	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	15884.0	0.531279	0.265639	+	0.964072i	$0.414417\pi$
$20$	34472.0	0.963522
$21$	0	0
$22$	−8992.00	−0.180043
$23$	−86100.0	−1.47556	−0.737778	−	0.675043i	$-0.764125\pi$
$23$	−86100.0	−1.47556	−0.737778	+	0.675043i	$0.764125\pi$
$24$	0	0
$25$	−841.000	−0.0107648
$26$	−14548.0	−0.162329
$27$	0	0
$28$	0	0
$29$	−40702.0	−0.309901	−0.154950	−	0.987922i	$-0.549522\pi$
$29$	−40702.0	−0.309901	−0.154950	+	0.987922i	$0.549522\pi$
$30$	0	0
$31$	44760.0	0.269851	0.134926	−	0.990856i	$-0.456920\pi$
$31$	44760.0	0.269851	0.134926	+	0.990856i	$0.456920\pi$
$32$	−94240.0	−0.508406
$33$	0	0
$34$	−22764.0	−0.0993282
$35$	0	0
$36$	0	0
$37$	−580962.	−1.88557	−0.942783	−	0.333407i	$-0.891802\pi$
$37$	−580962.	−1.88557	−0.942783	+	0.333407i	$0.891802\pi$
$38$	−31768.0	−0.0939177
$39$	0	0
$40$	−140112.	−0.346151
$41$	−171658.	−0.388974	−0.194487	−	0.980905i	$-0.562304\pi$
$41$	−171658.	−0.388974	−0.194487	+	0.980905i	$0.562304\pi$
$42$	0	0
$43$	−741148.	−1.42156	−0.710780	−	0.703414i	$-0.751658\pi$
$43$	−741148.	−1.42156	−0.710780	+	0.703414i	$0.751658\pi$
$44$	−557504.	−0.986651
$45$	0	0
$46$	172200.	0.260844
$47$	1.07172e6	1.50570	0.752851	−	0.658191i	$-0.228678\pi$
$47$	1.07172e6	1.50570	0.752851	+	0.658191i	$0.228678\pi$
$48$	0	0
$49$	0	0
$50$	1682.00	0.00190297
$51$	0	0
$52$	−901976.	−0.889576
$53$	1.72178e6	1.58859	0.794295	−	0.607533i	$-0.207841\pi$
$53$	1.72178e6	1.58859	0.794295	+	0.607533i	$0.207841\pi$
$54$	0	0
$55$	−1.24989e6	−1.01298
$56$	0	0
$57$	0	0
$58$	81404.0	0.0547832
$59$	−1.55701e6	−0.986984	−0.493492	−	0.869750i	$-0.664280\pi$
$59$	−1.55701e6	−0.986984	−0.493492	+	0.869750i	$0.664280\pi$
$60$	0	0
$61$	−2.59800e6	−1.46550	−0.732748	−	0.680501i	$-0.761762\pi$
$61$	−2.59800e6	−1.46550	−0.732748	+	0.680501i	$0.761762\pi$
$62$	−89520.0	−0.0477034
$63$	0	0
$64$	−1.71411e6	−0.817352
$65$	−2.02217e6	−0.913317
$66$	0	0
$67$	−963548.	−0.391392	−0.195696	−	0.980665i	$-0.562697\pi$
$67$	−963548.	−0.391392	−0.195696	+	0.980665i	$0.562697\pi$
$68$	−1.41137e6	−0.544326
$69$	0	0
$70$	0	0
$71$	4.06338e6	1.34736	0.673679	−	0.739024i	$-0.264713\pi$
$71$	4.06338e6	1.34736	0.673679	+	0.739024i	$0.264713\pi$
$72$	0	0
$73$	5.37022e6	1.61571	0.807853	−	0.589384i	$-0.200630\pi$
$73$	5.37022e6	1.61571	0.807853	+	0.589384i	$0.200630\pi$
$74$	1.16192e6	0.333324
$75$	0	0
$76$	−1.96962e6	−0.514676
$77$	0	0
$78$	0	0
$79$	4.09494e6	0.934442	0.467221	−	0.884141i	$-0.345255\pi$
$79$	4.09494e6	0.934442	0.467221	+	0.884141i	$0.345255\pi$
$80$	−4.13219e6	−0.902330
$81$	0	0
$82$	343316.	0.0687615
$83$	−1.34312e6	−0.257836	−0.128918	−	0.991655i	$-0.541150\pi$
$83$	−1.34312e6	−0.257836	−0.128918	+	0.991655i	$0.541150\pi$
$84$	0	0
$85$	−3.16420e6	−0.558852
$86$	1.48230e6	0.251299
$87$	0	0
$88$	2.26598e6	0.354460
$89$	9.08157e6	1.36551	0.682757	−	0.730646i	$-0.260781\pi$
$89$	9.08157e6	1.36551	0.682757	+	0.730646i	$0.260781\pi$
$90$	0	0
$91$	0	0
$92$	1.06764e7	1.42944
$93$	0	0
$94$	−2.14344e6	−0.266173
$95$	−4.41575e6	−0.528411
$96$	0	0
$97$	−6.48791e6	−0.721779	−0.360889	−	0.932609i	$-0.617527\pi$
$97$	−6.48791e6	−0.721779	−0.360889	+	0.932609i	$0.617527\pi$
$98$	0	0
$99$	0	0
$100$	104284.	0.0104284
$101$	1.82232e7	1.75994	0.879972	−	0.475026i	$-0.157561\pi$
$101$	1.82232e7	1.75994	0.879972	+	0.475026i	$0.157561\pi$
$102$	0	0
$103$	−743456.	−0.0670386	−0.0335193	−	0.999438i	$-0.510672\pi$
$103$	−743456.	−0.0670386	−0.0335193	+	0.999438i	$0.510672\pi$
$104$	3.66610e6	0.319586
$105$	0	0
$106$	−3.44356e6	−0.280826
$107$	−8.06819e6	−0.636697	−0.318349	−	0.947974i	$-0.603128\pi$
$107$	−8.06819e6	−0.636697	−0.318349	+	0.947974i	$0.603128\pi$
$108$	0	0
$109$	−1.19648e7	−0.884935	−0.442467	−	0.896785i	$-0.645897\pi$
$109$	−1.19648e7	−0.884935	−0.442467	+	0.896785i	$0.645897\pi$
$110$	2.49978e6	0.179072
$111$	0	0
$112$	0	0
$113$	−1.42952e6	−0.0932001	−0.0466000	−	0.998914i	$-0.514839\pi$
$113$	−1.42952e6	−0.0932001	−0.0466000	+	0.998914i	$0.514839\pi$
$114$	0	0
$115$	2.39358e7	1.46759
$116$	5.04705e6	0.300216
$117$	0	0
$118$	3.11402e6	0.174476
$119$	0	0
$120$	0	0
$121$	726845.	0.0372986
$122$	5.19600e6	0.259065
$123$	0	0
$124$	−5.55024e6	−0.261418
$125$	2.19525e7	1.00531
$126$	0	0
$127$	1.64521e7	0.712703	0.356352	−	0.934352i	$-0.384020\pi$
$127$	1.64521e7	0.712703	0.356352	+	0.934352i	$0.384020\pi$
$128$	1.54909e7	0.652894
$129$	0	0
$130$	4.04434e6	0.161453
$131$	−2.39749e7	−0.931768	−0.465884	−	0.884846i	$-0.654264\pi$
$131$	−2.39749e7	−0.931768	−0.465884	+	0.884846i	$0.654264\pi$
$132$	0	0
$133$	0	0
$134$	1.92710e6	0.0691889
$135$	0	0
$136$	5.73653e6	0.195552
$137$	3.20096e6	0.106355	0.0531775	−	0.998585i	$-0.483065\pi$
$137$	3.20096e6	0.106355	0.0531775	+	0.998585i	$0.483065\pi$
$138$	0	0
$139$	2.03943e7	0.644104	0.322052	−	0.946722i	$-0.395627\pi$
$139$	2.03943e7	0.644104	0.322052	+	0.946722i	$0.395627\pi$
$140$	0	0
$141$	0	0
$142$	−8.12676e6	−0.238182
$143$	3.27039e7	0.935241
$144$	0	0
$145$	1.13152e7	0.308228
$146$	−1.07404e7	−0.285619
$147$	0	0
$148$	7.20393e7	1.82664
$149$	−2.32307e7	−0.575322	−0.287661	−	0.957732i	$-0.592878\pi$
$149$	−2.32307e7	−0.575322	−0.287661	+	0.957732i	$0.592878\pi$
$150$	0	0
$151$	286088.	0.00676208	0.00338104	−	0.999994i	$-0.498924\pi$
$151$	286088.	0.00676208	0.00338104	+	0.999994i	$0.498924\pi$
$152$	8.00554e6	0.184900
$153$	0	0
$154$	0	0
$155$	−1.24433e7	−0.268395
$156$	0	0
$157$	−6.39867e7	−1.31960	−0.659798	−	0.751443i	$-0.729358\pi$
$157$	−6.39867e7	−1.31960	−0.659798	+	0.751443i	$0.729358\pi$
$158$	−8.18987e6	−0.165188
$159$	0	0
$160$	2.61987e7	0.505662
$161$	0	0
$162$	0	0
$163$	−8.08151e7	−1.46162	−0.730812	−	0.682579i	$-0.760858\pi$
$163$	−8.08151e7	−1.46162	−0.730812	+	0.682579i	$0.760858\pi$
$164$	2.12856e7	0.376819
$165$	0	0
$166$	2.68625e6	0.0455793
$167$	−1.06077e8	−1.76244	−0.881219	−	0.472708i	$-0.843277\pi$
$167$	−1.06077e8	−1.76244	−0.881219	+	0.472708i	$0.843277\pi$
$168$	0	0
$169$	−9.83744e6	−0.156776
$170$	6.32839e6	0.0987921
$171$	0	0
$172$	9.19024e7	1.37714
$173$	−5.86868e7	−0.861745	−0.430873	−	0.902413i	$-0.641794\pi$
$173$	−5.86868e7	−0.861745	−0.430873	+	0.902413i	$0.641794\pi$
$174$	0	0
$175$	0	0
$176$	6.68285e7	0.923991
$177$	0	0
$178$	−1.81631e7	−0.241391
$179$	1.19058e8	1.55157	0.775786	−	0.630996i	$-0.217353\pi$
$179$	1.19058e8	1.55157	0.775786	+	0.630996i	$0.217353\pi$
$180$	0	0
$181$	−3.20845e7	−0.402180	−0.201090	−	0.979573i	$-0.564448\pi$
$181$	−3.20845e7	−0.402180	−0.201090	+	0.979573i	$0.564448\pi$
$182$	0	0
$183$	0	0
$184$	−4.33944e7	−0.513536
$185$	1.61507e8	1.87539
$186$	0	0
$187$	5.11735e7	0.572268
$188$	−1.32893e8	−1.45865
$189$	0	0
$190$	8.83150e6	0.0934108
$191$	−5.81113e7	−0.603453	−0.301726	−	0.953395i	$-0.597563\pi$
$191$	−5.81113e7	−0.603453	−0.301726	+	0.953395i	$0.597563\pi$
$192$	0	0
$193$	6.86121e7	0.686990	0.343495	−	0.939155i	$-0.388389\pi$
$193$	6.86121e7	0.686990	0.343495	+	0.939155i	$0.388389\pi$
$194$	1.29758e7	0.127594
$195$	0	0
$196$	0	0
$197$	−1.11444e8	−1.03855	−0.519273	−	0.854608i	$-0.673797\pi$
$197$	−1.11444e8	−1.03855	−0.519273	+	0.854608i	$0.673797\pi$
$198$	0	0
$199$	1.76923e8	1.59147	0.795735	−	0.605645i	$-0.207085\pi$
$199$	1.76923e8	1.59147	0.795735	+	0.605645i	$0.207085\pi$
$200$	−423864.	−0.00374646
$201$	0	0
$202$	−3.64463e7	−0.311117
$203$	0	0
$204$	0	0
$205$	4.77209e7	0.386875
$206$	1.48691e6	0.0118509
$207$	0	0
$208$	1.08121e8	0.833081
$209$	7.14145e7	0.541096
$210$	0	0
$211$	1.45064e8	1.06309	0.531546	−	0.847030i	$-0.321611\pi$
$211$	1.45064e8	1.06309	0.531546	+	0.847030i	$0.321611\pi$
$212$	−2.13500e8	−1.53895
$213$	0	0
$214$	1.61364e7	0.112553
$215$	2.06039e8	1.41389
$216$	0	0
$217$	0	0
$218$	2.39295e7	0.156436
$219$	0	0
$220$	1.54986e8	0.981326
$221$	8.27927e7	0.515963
$222$	0	0
$223$	−2.16947e8	−1.31004	−0.655022	−	0.755610i	$-0.727341\pi$
$223$	−2.16947e8	−1.31004	−0.655022	+	0.755610i	$0.727341\pi$
$224$	0	0
$225$	0	0
$226$	2.85904e6	0.0164756
$227$	2.57152e8	1.45915	0.729575	−	0.683901i	$-0.239718\pi$
$227$	2.57152e8	1.45915	0.729575	+	0.683901i	$0.239718\pi$
$228$	0	0
$229$	−4.53937e7	−0.249788	−0.124894	−	0.992170i	$-0.539859\pi$
$229$	−4.53937e7	−0.249788	−0.124894	+	0.992170i	$0.539859\pi$
$230$	−4.78716e7	−0.259436
$231$	0	0
$232$	−2.05138e7	−0.107855
$233$	5.47778e7	0.283700	0.141850	−	0.989888i	$-0.454695\pi$
$233$	5.47778e7	0.283700	0.141850	+	0.989888i	$0.454695\pi$
$234$	0	0
$235$	−2.97938e8	−1.49758
$236$	1.93069e8	0.956140
$237$	0	0
$238$	0	0
$239$	4.12964e8	1.95668	0.978338	−	0.207013i	$-0.0663741\pi$
$239$	4.12964e8	1.95668	0.978338	+	0.207013i	$0.0663741\pi$
$240$	0	0
$241$	7.95356e7	0.366018	0.183009	−	0.983111i	$-0.441416\pi$
$241$	7.95356e7	0.366018	0.183009	+	0.983111i	$0.441416\pi$
$242$	−1.45369e6	−0.00659353
$243$	0	0
$244$	3.22152e8	1.41970
$245$	0	0
$246$	0	0
$247$	1.15540e8	0.487858
$248$	2.25590e7	0.0939160
$249$	0	0
$250$	−4.39051e7	−0.177715
$251$	−1.05790e7	−0.0422266	−0.0211133	−	0.999777i	$-0.506721\pi$
$251$	−1.05790e7	−0.0422266	−0.0211133	+	0.999777i	$0.506721\pi$
$252$	0	0
$253$	−3.87106e8	−1.50282
$254$	−3.29042e7	−0.125989
$255$	0	0
$256$	1.88424e8	0.701936
$257$	1.25047e8	0.459524	0.229762	−	0.973247i	$-0.426205\pi$
$257$	1.25047e8	0.459524	0.229762	+	0.973247i	$0.426205\pi$
$258$	0	0
$259$	0	0
$260$	2.50749e8	0.884775
$261$	0	0
$262$	4.79498e7	0.164715
$263$	5.25671e8	1.78184	0.890920	−	0.454160i	$-0.150060\pi$
$263$	5.25671e8	1.78184	0.890920	+	0.454160i	$0.150060\pi$
$264$	0	0
$265$	−4.78654e8	−1.58002
$266$	0	0
$267$	0	0
$268$	1.19480e8	0.379161
$269$	4.20767e8	1.31798	0.658990	−	0.752152i	$-0.270984\pi$
$269$	4.20767e8	1.31798	0.658990	+	0.752152i	$0.270984\pi$
$270$	0	0
$271$	1.01339e8	0.309303	0.154651	−	0.987969i	$-0.450575\pi$
$271$	1.01339e8	0.309303	0.154651	+	0.987969i	$0.450575\pi$
$272$	1.69182e8	0.509757
$273$	0	0
$274$	−6.40192e6	−0.0188011
$275$	−3.78114e6	−0.0109637
$276$	0	0
$277$	2.83426e8	0.801237	0.400619	−	0.916245i	$-0.368795\pi$
$277$	2.83426e8	0.801237	0.400619	+	0.916245i	$0.368795\pi$
$278$	−4.07885e7	−0.113863
$279$	0	0
$280$	0	0
$281$	5.91103e8	1.58925	0.794623	−	0.607103i	$-0.207668\pi$
$281$	5.91103e8	1.58925	0.794623	+	0.607103i	$0.207668\pi$
$282$	0	0
$283$	1.00607e8	0.263862	0.131931	−	0.991259i	$-0.457882\pi$
$283$	1.00607e8	0.263862	0.131931	+	0.991259i	$0.457882\pi$
$284$	−5.03859e8	−1.30525
$285$	0	0
$286$	−6.54078e7	−0.165329
$287$	0	0
$288$	0	0
$289$	−2.80789e8	−0.684285
$290$	−2.26303e7	−0.0544876
$291$	0	0
$292$	−6.65908e8	−1.56521
$293$	5.85152e7	0.135904	0.0679520	−	0.997689i	$-0.478354\pi$
$293$	5.85152e7	0.135904	0.0679520	+	0.997689i	$0.478354\pi$
$294$	0	0
$295$	4.32849e8	0.981657
$296$	−2.92805e8	−0.656232
$297$	0	0
$298$	4.64615e7	0.101704
$299$	−6.26291e8	−1.35496
$300$	0	0
$301$	0	0
$302$	−572176.	−0.00119538
$303$	0	0
$304$	2.36100e8	0.481990
$305$	7.22243e8	1.45759
$306$	0	0
$307$	2.35449e8	0.464421	0.232210	−	0.972666i	$-0.425404\pi$
$307$	2.35449e8	0.464421	0.232210	+	0.972666i	$0.425404\pi$
$308$	0	0
$309$	0	0
$310$	2.48866e7	0.0474459
$311$	−4.41120e8	−0.831563	−0.415782	−	0.909464i	$-0.636492\pi$
$311$	−4.41120e8	−0.831563	−0.415782	+	0.909464i	$0.636492\pi$
$312$	0	0
$313$	5.64265e8	1.04011	0.520053	−	0.854134i	$-0.325912\pi$
$313$	5.64265e8	1.04011	0.520053	+	0.854134i	$0.325912\pi$
$314$	1.27973e8	0.233274
$315$	0	0
$316$	−5.07772e8	−0.905241
$317$	−6.37212e7	−0.112351	−0.0561755	−	0.998421i	$-0.517891\pi$
$317$	−6.37212e7	−0.112351	−0.0561755	+	0.998421i	$0.517891\pi$
$318$	0	0
$319$	−1.82996e8	−0.315627
$320$	4.76523e8	0.812941
$321$	0	0
$322$	0	0
$323$	1.80792e8	0.298517
$324$	0	0
$325$	−6.11743e6	−0.00988502
$326$	1.61630e8	0.258381
$327$	0	0
$328$	−8.65156e7	−0.135374
$329$	0	0
$330$	0	0
$331$	−6.70715e7	−0.101658	−0.0508288	−	0.998707i	$-0.516186\pi$
$331$	−6.70715e7	−0.101658	−0.0508288	+	0.998707i	$0.516186\pi$
$332$	1.66547e8	0.249778
$333$	0	0
$334$	2.12154e8	0.311558
$335$	2.67866e8	0.389279
$336$	0	0
$337$	6.53041e8	0.929471	0.464735	−	0.885450i	$-0.346149\pi$
$337$	6.53041e8	0.929471	0.464735	+	0.885450i	$0.346149\pi$
$338$	1.96749e7	0.0277143
$339$	0	0
$340$	3.92360e8	0.541388
$341$	2.01241e8	0.274838
$342$	0	0
$343$	0	0
$344$	−3.73539e8	−0.494744
$345$	0	0
$346$	1.17374e8	0.152337
$347$	−5.42120e8	−0.696534	−0.348267	−	0.937395i	$-0.613230\pi$
$347$	−5.42120e8	−0.696534	−0.348267	+	0.937395i	$0.613230\pi$
$348$	0	0
$349$	1.12783e9	1.42022	0.710108	−	0.704093i	$-0.248646\pi$
$349$	1.12783e9	1.42022	0.710108	+	0.704093i	$0.248646\pi$
$350$	0	0
$351$	0	0
$352$	−4.23703e8	−0.517800
$353$	1.34844e9	1.63162	0.815810	−	0.578320i	$-0.196292\pi$
$353$	1.34844e9	1.63162	0.815810	+	0.578320i	$0.196292\pi$
$354$	0	0
$355$	−1.12962e9	−1.34009
$356$	−1.12612e9	−1.32284
$357$	0	0
$358$	−2.38116e8	−0.274282
$359$	−1.80002e8	−0.205327	−0.102664	−	0.994716i	$-0.532737\pi$
$359$	−1.80002e8	−0.205327	−0.102664	+	0.994716i	$0.532737\pi$
$360$	0	0
$361$	−6.41570e8	−0.717743
$362$	6.41690e7	0.0710960
$363$	0	0
$364$	0	0
$365$	−1.49292e9	−1.60699
$366$	0	0
$367$	−1.32409e9	−1.39826	−0.699129	−	0.714995i	$-0.746429\pi$
$367$	−1.32409e9	−1.39826	−0.699129	+	0.714995i	$0.746429\pi$
$368$	−1.27979e9	−1.33866
$369$	0	0
$370$	−3.23015e8	−0.331525
$371$	0	0
$372$	0	0
$373$	−6.21114e8	−0.619713	−0.309856	−	0.950783i	$-0.600281\pi$
$373$	−6.21114e8	−0.619713	−0.309856	+	0.950783i	$0.600281\pi$
$374$	−1.02347e8	−0.101164
$375$	0	0
$376$	5.40147e8	0.524028
$377$	−2.96066e8	−0.284573
$378$	0	0
$379$	1.39116e9	1.31262	0.656309	−	0.754492i	$-0.272117\pi$
$379$	1.39116e9	1.31262	0.656309	+	0.754492i	$0.272117\pi$
$380$	5.47553e8	0.511898
$381$	0	0
$382$	1.16223e8	0.106676
$383$	2.78241e8	0.253061	0.126531	−	0.991963i	$-0.459616\pi$
$383$	2.78241e8	0.253061	0.126531	+	0.991963i	$0.459616\pi$
$384$	0	0
$385$	0	0
$386$	−1.37224e8	−0.121444
$387$	0	0
$388$	8.04501e8	0.699223
$389$	7.81513e8	0.673151	0.336576	−	0.941656i	$-0.390731\pi$
$389$	7.81513e8	0.673151	0.336576	+	0.941656i	$0.390731\pi$
$390$	0	0
$391$	−9.79990e8	−0.829093
$392$	0	0
$393$	0	0
$394$	2.22889e8	0.183591
$395$	−1.13839e9	−0.929399
$396$	0	0
$397$	2.24696e9	1.80230	0.901152	−	0.433503i	$-0.142723\pi$
$397$	2.24696e9	1.80230	0.901152	+	0.433503i	$0.142723\pi$
$398$	−3.53846e8	−0.281335
$399$	0	0
$400$	−1.25006e7	−0.00976611
$401$	3.48219e8	0.269679	0.134839	−	0.990867i	$-0.456948\pi$
$401$	3.48219e8	0.269679	0.134839	+	0.990867i	$0.456948\pi$
$402$	0	0
$403$	3.25584e8	0.247797
$404$	−2.25967e9	−1.70495
$405$	0	0
$406$	0	0
$407$	−2.61201e9	−1.92041
$408$	0	0
$409$	−6.71432e8	−0.485256	−0.242628	−	0.970119i	$-0.578009\pi$
$409$	−6.71432e8	−0.485256	−0.242628	+	0.970119i	$0.578009\pi$
$410$	−9.54418e7	−0.0683904
$411$	0	0
$412$	9.21885e7	0.0649437
$413$	0	0
$414$	0	0
$415$	3.73388e8	0.256444
$416$	−6.85502e8	−0.466855
$417$	0	0
$418$	−1.42829e8	−0.0956531
$419$	−4.22351e8	−0.280495	−0.140247	−	0.990117i	$-0.544790\pi$
$419$	−4.22351e8	−0.280495	−0.140247	+	0.990117i	$0.544790\pi$
$420$	0	0
$421$	−8.47125e8	−0.553299	−0.276650	−	0.960971i	$-0.589224\pi$
$421$	−8.47125e8	−0.553299	−0.276650	+	0.960971i	$0.589224\pi$
$422$	−2.90128e8	−0.187930
$423$	0	0
$424$	8.67776e8	0.552875
$425$	−9.57226e6	−0.00604858
$426$	0	0
$427$	0	0
$428$	1.00046e9	0.616801
$429$	0	0
$430$	−4.12078e8	−0.249943
$431$	−8.28495e7	−0.0498447	−0.0249224	−	0.999689i	$-0.507934\pi$
$431$	−8.28495e7	−0.0498447	−0.0249224	+	0.999689i	$0.507934\pi$
$432$	0	0
$433$	6.80201e8	0.402652	0.201326	−	0.979524i	$-0.435475\pi$
$433$	6.80201e8	0.402652	0.201326	+	0.979524i	$0.435475\pi$
$434$	0	0
$435$	0	0
$436$	1.48363e9	0.857281
$437$	−1.36761e9	−0.783931
$438$	0	0
$439$	1.12411e9	0.634136	0.317068	−	0.948403i	$-0.397302\pi$
$439$	1.12411e9	0.634136	0.317068	+	0.948403i	$0.397302\pi$
$440$	−6.29944e8	−0.352547
$441$	0	0
$442$	−1.65585e8	−0.0912103
$443$	2.03339e9	1.11124	0.555621	−	0.831436i	$-0.312481\pi$
$443$	2.03339e9	1.11124	0.555621	+	0.831436i	$0.312481\pi$
$444$	0	0
$445$	−2.52468e9	−1.35814
$446$	4.33893e8	0.231585
$447$	0	0
$448$	0	0
$449$	−5.64995e8	−0.294566	−0.147283	−	0.989094i	$-0.547053\pi$
$449$	−5.64995e8	−0.294566	−0.147283	+	0.989094i	$0.547053\pi$
$450$	0	0
$451$	−7.71774e8	−0.396162
$452$	1.77261e8	0.0902876
$453$	0	0
$454$	−5.14304e8	−0.257944
$455$	0	0
$456$	0	0
$457$	2.00935e9	0.984800	0.492400	−	0.870369i	$-0.336120\pi$
$457$	2.00935e9	0.984800	0.492400	+	0.870369i	$0.336120\pi$
$458$	9.07874e7	0.0441567
$459$	0	0
$460$	−2.96804e9	−1.42173
$461$	2.49143e9	1.18439	0.592196	−	0.805794i	$-0.298261\pi$
$461$	2.49143e9	1.18439	0.592196	+	0.805794i	$0.298261\pi$
$462$	0	0
$463$	7.88856e8	0.369372	0.184686	−	0.982798i	$-0.440873\pi$
$463$	7.88856e8	0.369372	0.184686	+	0.982798i	$0.440873\pi$
$464$	−6.04995e8	−0.281150
$465$	0	0
$466$	−1.09556e8	−0.0501515
$467$	−1.19949e9	−0.544991	−0.272495	−	0.962157i	$-0.587849\pi$
$467$	−1.19949e9	−0.544991	−0.272495	+	0.962157i	$0.587849\pi$
$468$	0	0
$469$	0	0
$470$	5.95876e8	0.264736
$471$	0	0
$472$	−7.84734e8	−0.343499
$473$	−3.33220e9	−1.44783
$474$	0	0
$475$	−1.33584e7	−0.00571911
$476$	0	0
$477$	0	0
$478$	−8.25927e8	−0.345895
$479$	−3.06254e9	−1.27323	−0.636616	−	0.771181i	$-0.719666\pi$
$479$	−3.06254e9	−1.27323	−0.636616	+	0.771181i	$0.719666\pi$
$480$	0	0
$481$	−4.22592e9	−1.73146
$482$	−1.59071e8	−0.0647034
$483$	0	0
$484$	−9.01288e7	−0.0361331
$485$	1.80364e9	0.717883
$486$	0	0
$487$	−6.96199e8	−0.273138	−0.136569	−	0.990631i	$-0.543608\pi$
$487$	−6.96199e8	−0.273138	−0.136569	+	0.990631i	$0.543608\pi$
$488$	−1.30939e9	−0.510035
$489$	0	0
$490$	0	0
$491$	3.19001e9	1.21621	0.608103	−	0.793858i	$-0.291931\pi$
$491$	3.19001e9	1.21621	0.608103	+	0.793858i	$0.291931\pi$
$492$	0	0
$493$	−4.63270e8	−0.174129
$494$	−2.31080e8	−0.0862420
$495$	0	0
$496$	6.65313e8	0.244816
$497$	0	0
$498$	0	0
$499$	−3.41005e9	−1.22860	−0.614298	−	0.789074i	$-0.710561\pi$
$499$	−3.41005e9	−1.22860	−0.614298	+	0.789074i	$0.710561\pi$
$500$	−2.72212e9	−0.973894
$501$	0	0
$502$	2.11580e7	0.00746467
$503$	5.44883e9	1.90904	0.954521	−	0.298145i	$-0.0963679\pi$
$503$	5.44883e9	1.90904	0.954521	+	0.298145i	$0.0963679\pi$
$504$	0	0
$505$	−5.06604e9	−1.75045
$506$	7.74211e8	0.265664
$507$	0	0
$508$	−2.04006e9	−0.690431
$509$	4.08025e9	1.37143	0.685716	−	0.727869i	$-0.259489\pi$
$509$	4.08025e9	1.37143	0.685716	+	0.727869i	$0.259489\pi$
$510$	0	0
$511$	0	0
$512$	−2.35969e9	−0.776980
$513$	0	0
$514$	−2.50095e8	−0.0812332
$515$	2.06681e8	0.0666768
$516$	0	0
$517$	4.81845e9	1.53352
$518$	0	0
$519$	0	0
$520$	−1.01917e9	−0.317861
$521$	−2.91322e9	−0.902488	−0.451244	−	0.892401i	$-0.649020\pi$
$521$	−2.91322e9	−0.902488	−0.451244	+	0.892401i	$0.649020\pi$
$522$	0	0
$523$	1.10998e9	0.339281	0.169640	−	0.985506i	$-0.445739\pi$
$523$	1.10998e9	0.339281	0.169640	+	0.985506i	$0.445739\pi$
$524$	2.97289e9	0.902650
$525$	0	0
$526$	−1.05134e9	−0.314988
$527$	5.09458e8	0.151625
$528$	0	0
$529$	4.00838e9	1.17727
$530$	9.57309e8	0.279310
$531$	0	0
$532$	0	0
$533$	−1.24864e9	−0.357184
$534$	0	0
$535$	2.24296e9	0.633261
$536$	−4.85628e8	−0.136216
$537$	0	0
$538$	−8.41534e8	−0.232988
$539$	0	0
$540$	0	0
$541$	5.38334e9	1.46171	0.730856	−	0.682532i	$-0.239121\pi$
$541$	5.38334e9	1.46171	0.730856	+	0.682532i	$0.239121\pi$
$542$	−2.02678e8	−0.0546775
$543$	0	0
$544$	−1.07264e9	−0.285665
$545$	3.32620e9	0.880159
$546$	0	0
$547$	−6.38630e9	−1.66837	−0.834187	−	0.551481i	$-0.814063\pi$
$547$	−6.38630e9	−1.66837	−0.834187	+	0.551481i	$0.814063\pi$
$548$	−3.96919e8	−0.103031
$549$	0	0
$550$	7.56227e6	0.00193813
$551$	−6.46511e8	−0.164644
$552$	0	0
$553$	0	0
$554$	−5.66853e8	−0.141640
$555$	0	0
$556$	−2.52889e9	−0.623976
$557$	−2.69921e9	−0.661825	−0.330913	−	0.943661i	$-0.607357\pi$
$557$	−2.69921e9	−0.661825	−0.330913	+	0.943661i	$0.607357\pi$
$558$	0	0
$559$	−5.39111e9	−1.30538
$560$	0	0
$561$	0	0
$562$	−1.18221e9	−0.280942
$563$	−3.49039e9	−0.824317	−0.412159	−	0.911112i	$-0.635225\pi$
$563$	−3.49039e9	−0.824317	−0.412159	+	0.911112i	$0.635225\pi$
$564$	0	0
$565$	3.97407e8	0.0926971
$566$	−2.01215e8	−0.0466447
$567$	0	0
$568$	2.04794e9	0.468920
$569$	−5.15262e9	−1.17256	−0.586280	−	0.810108i	$-0.699408\pi$
$569$	−5.15262e9	−1.17256	−0.586280	+	0.810108i	$0.699408\pi$
$570$	0	0
$571$	−3.06051e9	−0.687967	−0.343984	−	0.938976i	$-0.611776\pi$
$571$	−3.06051e9	−0.687967	−0.343984	+	0.938976i	$0.611776\pi$
$572$	−4.05528e9	−0.906015
$573$	0	0
$574$	0	0
$575$	7.24101e7	0.0158841
$576$	0	0
$577$	−3.45107e9	−0.747891	−0.373945	−	0.927451i	$-0.621995\pi$
$577$	−3.45107e9	−0.747891	−0.373945	+	0.927451i	$0.621995\pi$
$578$	5.61577e8	0.120966
$579$	0	0
$580$	−1.40308e9	−0.298596
$581$	0	0
$582$	0	0
$583$	7.74111e9	1.61794
$584$	2.70659e9	0.562313
$585$	0	0
$586$	−1.17030e8	−0.0240247
$587$	−5.67214e9	−1.15748	−0.578740	−	0.815512i	$-0.696456\pi$
$587$	−5.67214e9	−1.15748	−0.578740	+	0.815512i	$0.696456\pi$
$588$	0	0
$589$	7.10968e8	0.143366
$590$	−8.65699e8	−0.173534
$591$	0	0
$592$	−8.63542e9	−1.71064
$593$	4.06124e9	0.799774	0.399887	−	0.916564i	$-0.369049\pi$
$593$	4.06124e9	0.799774	0.399887	+	0.916564i	$0.369049\pi$
$594$	0	0
$595$	0	0
$596$	2.88061e9	0.557343
$597$	0	0
$598$	1.25258e9	0.239526
$599$	−2.75544e9	−0.523837	−0.261919	−	0.965090i	$-0.584355\pi$
$599$	−2.75544e9	−0.523837	−0.261919	+	0.965090i	$0.584355\pi$
$600$	0	0
$601$	3.94831e9	0.741910	0.370955	−	0.928651i	$-0.379030\pi$
$601$	3.94831e9	0.741910	0.370955	+	0.928651i	$0.379030\pi$
$602$	0	0
$603$	0	0
$604$	−3.54749e7	−0.00655077
$605$	−2.02063e8	−0.0370973
$606$	0	0
$607$	3.86042e9	0.700607	0.350304	−	0.936636i	$-0.386078\pi$
$607$	3.86042e9	0.700607	0.350304	+	0.936636i	$0.386078\pi$
$608$	−1.49691e9	−0.270105
$609$	0	0
$610$	−1.44449e9	−0.257667
$611$	7.79569e9	1.38264
$612$	0	0
$613$	8.88340e9	1.55764	0.778821	−	0.627246i	$-0.215818\pi$
$613$	8.88340e9	1.55764	0.778821	+	0.627246i	$0.215818\pi$
$614$	−4.70897e8	−0.0820988
$615$	0	0
$616$	0	0
$617$	−4.68037e9	−0.802199	−0.401100	−	0.916034i	$-0.631372\pi$
$617$	−4.68037e9	−0.802199	−0.401100	+	0.916034i	$0.631372\pi$
$618$	0	0
$619$	1.96100e9	0.332323	0.166161	−	0.986099i	$-0.446863\pi$
$619$	1.96100e9	0.332323	0.166161	+	0.986099i	$0.446863\pi$
$620$	1.54297e9	0.260007
$621$	0	0
$622$	8.82240e8	0.147001
$623$	0	0
$624$	0	0
$625$	−6.03711e9	−0.989119
$626$	−1.12853e9	−0.183867
$627$	0	0
$628$	7.93436e9	1.27836
$629$	−6.61251e9	−1.05947
$630$	0	0
$631$	6.87456e9	1.08929	0.544643	−	0.838668i	$-0.316665\pi$
$631$	6.87456e9	1.08929	0.544643	+	0.838668i	$0.316665\pi$
$632$	2.06385e9	0.325213
$633$	0	0
$634$	1.27442e8	0.0198610
$635$	−4.57369e9	−0.708857
$636$	0	0
$637$	0	0
$638$	3.65992e8	0.0557956
$639$	0	0
$640$	−4.30648e9	−0.649371
$641$	7.52780e9	1.12892	0.564462	−	0.825459i	$-0.309084\pi$
$641$	7.52780e9	1.12892	0.564462	+	0.825459i	$0.309084\pi$
$642$	0	0
$643$	−2.93245e9	−0.435003	−0.217502	−	0.976060i	$-0.569791\pi$
$643$	−2.93245e9	−0.435003	−0.217502	+	0.976060i	$0.569791\pi$
$644$	0	0
$645$	0	0
$646$	−3.61583e8	−0.0527709
$647$	−6.56642e9	−0.953155	−0.476578	−	0.879132i	$-0.658123\pi$
$647$	−6.56642e9	−0.953155	−0.476578	+	0.879132i	$0.658123\pi$
$648$	0	0
$649$	−7.00033e9	−1.00522
$650$	1.22349e7	0.00174744
$651$	0	0
$652$	1.00211e10	1.41595
$653$	6.46865e9	0.909112	0.454556	−	0.890718i	$-0.349798\pi$
$653$	6.46865e9	0.909112	0.454556	+	0.890718i	$0.349798\pi$
$654$	0	0
$655$	6.66503e9	0.926739
$656$	−2.55152e9	−0.352888
$657$	0	0
$658$	0	0
$659$	−8.04313e9	−1.09478	−0.547389	−	0.836878i	$-0.684378\pi$
$659$	−8.04313e9	−1.09478	−0.547389	+	0.836878i	$0.684378\pi$
$660$	0	0
$661$	7.04726e9	0.949107	0.474553	−	0.880227i	$-0.342610\pi$
$661$	7.04726e9	0.949107	0.474553	+	0.880227i	$0.342610\pi$
$662$	1.34143e8	0.0179707
$663$	0	0
$664$	−6.76934e8	−0.0897343
$665$	0	0
$666$	0	0
$667$	3.50444e9	0.457276
$668$	1.31536e10	1.70736
$669$	0	0
$670$	−5.35733e8	−0.0688155
$671$	−1.16806e10	−1.49258
$672$	0	0
$673$	2.71888e9	0.343825	0.171912	−	0.985112i	$-0.445005\pi$
$673$	2.71888e9	0.343825	0.171912	+	0.985112i	$0.445005\pi$
$674$	−1.30608e9	−0.164309
$675$	0	0
$676$	1.21984e9	0.151876
$677$	8.16879e9	1.01181	0.505903	−	0.862590i	$-0.331159\pi$
$677$	8.16879e9	1.01181	0.505903	+	0.862590i	$0.331159\pi$
$678$	0	0
$679$	0	0
$680$	−1.59475e9	−0.194497
$681$	0	0
$682$	−4.02482e8	−0.0485849
$683$	6.83926e9	0.821366	0.410683	−	0.911778i	$-0.365290\pi$
$683$	6.83926e9	0.821366	0.410683	+	0.911778i	$0.365290\pi$
$684$	0	0
$685$	−8.89866e8	−0.105781
$686$	0	0
$687$	0	0
$688$	−1.10164e10	−1.28968
$689$	1.25242e10	1.45876
$690$	0	0
$691$	5.42614e9	0.625630	0.312815	−	0.949814i	$-0.398728\pi$
$691$	5.42614e9	0.625630	0.312815	+	0.949814i	$0.398728\pi$
$692$	7.27716e9	0.834816
$693$	0	0
$694$	1.08424e9	0.123131
$695$	−5.66960e9	−0.640628
$696$	0	0
$697$	−1.95381e9	−0.218559
$698$	−2.25566e9	−0.251061
$699$	0	0
$700$	0	0
$701$	−3.59886e9	−0.394595	−0.197298	−	0.980344i	$-0.563217\pi$
$701$	−3.59886e9	−0.394595	−0.197298	+	0.980344i	$0.563217\pi$
$702$	0	0
$703$	−9.22800e9	−1.00176
$704$	−7.70665e9	−0.832456
$705$	0	0
$706$	−2.69687e9	−0.288432
$707$	0	0
$708$	0	0
$709$	−6.26847e9	−0.660541	−0.330271	−	0.943886i	$-0.607140\pi$
$709$	−6.26847e9	−0.660541	−0.330271	+	0.943886i	$0.607140\pi$
$710$	2.25924e9	0.236896
$711$	0	0
$712$	4.57711e9	0.475239
$713$	−3.85384e9	−0.398180
$714$	0	0
$715$	−9.09169e9	−0.930193
$716$	−1.47632e10	−1.50309
$717$	0	0
$718$	3.60004e8	0.0362971
$719$	9.10238e9	0.913280	0.456640	−	0.889652i	$-0.349053\pi$
$719$	9.10238e9	0.913280	0.456640	+	0.889652i	$0.349053\pi$
$720$	0	0
$721$	0	0
$722$	1.28314e9	0.126880
$723$	0	0
$724$	3.97848e9	0.389612
$725$	3.42304e7	0.00333602
$726$	0	0
$727$	1.25445e10	1.21083	0.605415	−	0.795910i	$-0.293007\pi$
$727$	1.25445e10	1.21083	0.605415	+	0.795910i	$0.293007\pi$
$728$	0	0
$729$	0	0
$730$	2.98584e9	0.284078
$731$	−8.43575e9	−0.798753
$732$	0	0
$733$	3.99755e8	0.0374912	0.0187456	−	0.999824i	$-0.494033\pi$
$733$	3.99755e8	0.0374912	0.0187456	+	0.999824i	$0.494033\pi$
$734$	2.64819e9	0.247180
$735$	0	0
$736$	8.11406e9	0.750181
$737$	−4.33211e9	−0.398624
$738$	0	0
$739$	−3.96698e9	−0.361580	−0.180790	−	0.983522i	$-0.557865\pi$
$739$	−3.96698e9	−0.361580	−0.180790	+	0.983522i	$0.557865\pi$
$740$	−2.00269e10	−1.81678
$741$	0	0
$742$	0	0
$743$	1.06112e10	0.949082	0.474541	−	0.880233i	$-0.342614\pi$
$743$	1.06112e10	0.949082	0.474541	+	0.880233i	$0.342614\pi$
$744$	0	0
$745$	6.45814e9	0.572217
$746$	1.24223e9	0.109551
$747$	0	0
$748$	−6.34551e9	−0.554384
$749$	0	0
$750$	0	0
$751$	6.59069e9	0.567795	0.283897	−	0.958855i	$-0.408373\pi$
$751$	6.59069e9	0.567795	0.283897	+	0.958855i	$0.408373\pi$
$752$	1.59300e10	1.36601
$753$	0	0
$754$	5.92133e8	0.0503059
$755$	−7.95325e7	−0.00672559
$756$	0	0
$757$	2.35834e8	0.0197593	0.00987965	−	0.999951i	$-0.496855\pi$
$757$	2.35834e8	0.0197593	0.00987965	+	0.999951i	$0.496855\pi$
$758$	−2.78231e9	−0.232040
$759$	0	0
$760$	−2.22554e9	−0.183903
$761$	−3.07283e9	−0.252750	−0.126375	−	0.991983i	$-0.540334\pi$
$761$	−3.07283e9	−0.252750	−0.126375	+	0.991983i	$0.540334\pi$
$762$	0	0
$763$	0	0
$764$	7.20580e9	0.584595
$765$	0	0
$766$	−5.56483e8	−0.0447354
$767$	−1.13257e10	−0.906320
$768$	0	0
$769$	1.52466e10	1.20901	0.604507	−	0.796600i	$-0.293370\pi$
$769$	1.52466e10	1.20901	0.604507	+	0.796600i	$0.293370\pi$
$770$	0	0
$771$	0	0
$772$	−8.50790e9	−0.665521
$773$	1.24849e10	0.972202	0.486101	−	0.873903i	$-0.338419\pi$
$773$	1.24849e10	0.972202	0.486101	+	0.873903i	$0.338419\pi$
$774$	0	0
$775$	−3.76432e7	−0.00290489
$776$	−3.26991e9	−0.251200
$777$	0	0
$778$	−1.56303e9	−0.118997
$779$	−2.72662e9	−0.206654
$780$	0	0
$781$	1.82690e10	1.37226
$782$	1.95998e9	0.146564
$783$	0	0
$784$	0	0
$785$	1.77883e10	1.31247
$786$	0	0
$787$	6.26674e9	0.458279	0.229140	−	0.973394i	$-0.426409\pi$
$787$	6.26674e9	0.458279	0.229140	+	0.973394i	$0.426409\pi$
$788$	1.38191e10	1.00609
$789$	0	0
$790$	2.27678e9	0.164296
$791$	0	0
$792$	0	0
$793$	−1.88978e10	−1.34572
$794$	−4.49392e9	−0.318605
$795$	0	0
$796$	−2.19385e10	−1.54174
$797$	2.60872e10	1.82525	0.912626	−	0.408795i	$-0.134051\pi$
$797$	2.60872e10	1.82525	0.912626	+	0.408795i	$0.134051\pi$
$798$	0	0
$799$	1.21983e10	0.846031
$800$	7.92558e7	0.00547288
$801$	0	0
$802$	−6.96437e8	−0.0476729
$803$	2.41445e10	1.64556
$804$	0	0
$805$	0	0
$806$	−6.51168e8	−0.0438047
$807$	0	0
$808$	9.18447e9	0.612512
$809$	−2.64946e9	−0.175929	−0.0879645	−	0.996124i	$-0.528036\pi$
$809$	−2.64946e9	−0.175929	−0.0879645	+	0.996124i	$0.528036\pi$
$810$	0	0
$811$	−9.94761e9	−0.654856	−0.327428	−	0.944876i	$-0.606182\pi$
$811$	−9.94761e9	−0.654856	−0.327428	+	0.944876i	$0.606182\pi$
$812$	0	0
$813$	0	0
$814$	5.22401e9	0.339483
$815$	2.24666e10	1.45374
$816$	0	0
$817$	−1.17724e10	−0.755245
$818$	1.34286e9	0.0857819
$819$	0	0
$820$	−5.91739e9	−0.374785
$821$	2.20915e10	1.39324	0.696619	−	0.717442i	$-0.254687\pi$
$821$	2.20915e10	1.39324	0.696619	+	0.717442i	$0.254687\pi$
$822$	0	0
$823$	−4.40068e9	−0.275182	−0.137591	−	0.990489i	$-0.543936\pi$
$823$	−4.40068e9	−0.275182	−0.137591	+	0.990489i	$0.543936\pi$
$824$	−3.74702e8	−0.0233314
$825$	0	0
$826$	0	0
$827$	1.00062e10	0.615180	0.307590	−	0.951519i	$-0.400478\pi$
$827$	1.00062e10	0.615180	0.307590	+	0.951519i	$0.400478\pi$
$828$	0	0
$829$	1.86029e10	1.13407	0.567035	−	0.823694i	$-0.308090\pi$
$829$	1.86029e10	1.13407	0.567035	+	0.823694i	$0.308090\pi$
$830$	−7.46777e8	−0.0453333
$831$	0	0
$832$	−1.24685e10	−0.750552
$833$	0	0
$834$	0	0
$835$	2.94894e10	1.75293
$836$	−8.85539e9	−0.524187
$837$	0	0
$838$	8.44702e8	0.0495849
$839$	−1.52096e9	−0.0889103	−0.0444551	−	0.999011i	$-0.514155\pi$
$839$	−1.52096e9	−0.0889103	−0.0444551	+	0.999011i	$0.514155\pi$
$840$	0	0
$841$	−1.55932e10	−0.903961
$842$	1.69425e9	0.0978105
$843$	0	0
$844$	−1.79879e10	−1.02987
$845$	2.73481e9	0.155930
$846$	0	0
$847$	0	0
$848$	2.55925e10	1.44121
$849$	0	0
$850$	1.91445e7	0.00106925
$851$	5.00208e10	2.78226
$852$	0	0
$853$	−1.94410e10	−1.07250	−0.536250	−	0.844059i	$-0.680159\pi$
$853$	−1.94410e10	−1.07250	−0.536250	+	0.844059i	$0.680159\pi$
$854$	0	0
$855$	0	0
$856$	−4.06637e9	−0.221589
$857$	3.66298e9	0.198793	0.0993966	−	0.995048i	$-0.468309\pi$
$857$	3.66298e9	0.198793	0.0993966	+	0.995048i	$0.468309\pi$
$858$	0	0
$859$	2.10549e10	1.13338	0.566691	−	0.823930i	$-0.308223\pi$
$859$	2.10549e10	1.13338	0.566691	+	0.823930i	$0.308223\pi$
$860$	−2.55489e10	−1.36970
$861$	0	0
$862$	1.65699e8	0.00881139
$863$	−3.17256e10	−1.68024	−0.840121	−	0.542399i	$-0.817516\pi$
$863$	−3.17256e10	−1.68024	−0.840121	+	0.542399i	$0.817516\pi$
$864$	0	0
$865$	1.63149e10	0.857095
$866$	−1.36040e9	−0.0711794
$867$	0	0
$868$	0	0
$869$	1.84108e10	0.951709
$870$	0	0
$871$	−7.00885e9	−0.359404
$872$	−6.03024e9	−0.307983
$873$	0	0
$874$	2.73522e9	0.138581
$875$	0	0
$876$	0	0
$877$	−6.44796e8	−0.0322793	−0.0161396	−	0.999870i	$-0.505138\pi$
$877$	−6.44796e8	−0.0322793	−0.0161396	+	0.999870i	$0.505138\pi$
$878$	−2.24822e9	−0.112101
$879$	0	0
$880$	−1.85783e10	−0.919004
$881$	−1.95481e10	−0.963137	−0.481568	−	0.876409i	$-0.659933\pi$
$881$	−1.95481e10	−0.963137	−0.481568	+	0.876409i	$0.659933\pi$
$882$	0	0
$883$	−1.35038e9	−0.0660078	−0.0330039	−	0.999455i	$-0.510507\pi$
$883$	−1.35038e9	−0.0660078	−0.0330039	+	0.999455i	$0.510507\pi$
$884$	−1.02663e10	−0.499840
$885$	0	0
$886$	−4.06679e9	−0.196442
$887$	2.74149e10	1.31903	0.659514	−	0.751692i	$-0.270762\pi$
$887$	2.74149e10	1.31903	0.659514	+	0.751692i	$0.270762\pi$
$888$	0	0
$889$	0	0
$890$	5.04936e9	0.240088
$891$	0	0
$892$	2.69014e10	1.26911
$893$	1.70232e10	0.799947
$894$	0	0
$895$	−3.30981e10	−1.54320
$896$	0	0
$897$	0	0
$898$	1.12999e9	0.0520724
$899$	−1.82182e9	−0.0836271
$900$	0	0
$901$	1.95973e10	0.892604
$902$	1.54355e9	0.0700322
$903$	0	0
$904$	−7.20479e8	−0.0324363
$905$	8.91949e9	0.400009
$906$	0	0
$907$	−1.99635e10	−0.888403	−0.444202	−	0.895927i	$-0.646513\pi$
$907$	−1.99635e10	−0.888403	−0.444202	+	0.895927i	$0.646513\pi$
$908$	−3.18869e10	−1.41355
$909$	0	0
$910$	0	0
$911$	−9.43544e9	−0.413474	−0.206737	−	0.978397i	$-0.566284\pi$
$911$	−9.43544e9	−0.413474	−0.206737	+	0.978397i	$0.566284\pi$
$912$	0	0
$913$	−6.03869e9	−0.262600
$914$	−4.01869e9	−0.174090
$915$	0	0
$916$	5.62882e9	0.241982
$917$	0	0
$918$	0	0
$919$	−8.84171e9	−0.375779	−0.187889	−	0.982190i	$-0.560165\pi$
$919$	−8.84171e9	−0.375779	−0.187889	+	0.982190i	$0.560165\pi$
$920$	1.20636e10	0.510765
$921$	0	0
$922$	−4.98286e9	−0.209373
$923$	2.95570e10	1.23724
$924$	0	0
$925$	4.88589e8	0.0202977
$926$	−1.57771e9	−0.0652964
$927$	0	0
$928$	3.83576e9	0.157555
$929$	3.21564e10	1.31587	0.657933	−	0.753076i	$-0.271431\pi$
$929$	3.21564e10	1.31587	0.657933	+	0.753076i	$0.271431\pi$
$930$	0	0
$931$	0	0
$932$	−6.79245e9	−0.274834
$933$	0	0
$934$	2.39899e9	0.0963416
$935$	−1.42262e10	−0.569179
$936$	0	0
$937$	−7.27287e9	−0.288813	−0.144407	−	0.989518i	$-0.546127\pi$
$937$	−7.27287e9	−0.288813	−0.144407	+	0.989518i	$0.546127\pi$
$938$	0	0
$939$	0	0
$940$	3.69443e10	1.45078
$941$	−1.50928e10	−0.590482	−0.295241	−	0.955423i	$-0.595400\pi$
$941$	−1.50928e10	−0.590482	−0.295241	+	0.955423i	$0.595400\pi$
$942$	0	0
$943$	1.47798e10	0.573953
$944$	−2.31434e10	−0.895418
$945$	0	0
$946$	6.66440e9	0.255942
$947$	−4.21375e10	−1.61229	−0.806147	−	0.591715i	$-0.798451\pi$
$947$	−4.21375e10	−1.61229	−0.806147	+	0.591715i	$0.798451\pi$
$948$	0	0
$949$	3.90630e10	1.48366
$950$	2.67169e7	0.00101100
$951$	0	0
$952$	0	0
$953$	−3.37120e10	−1.26171	−0.630855	−	0.775901i	$-0.717296\pi$
$953$	−3.37120e10	−1.26171	−0.630855	+	0.775901i	$0.717296\pi$
$954$	0	0
$955$	1.61549e10	0.600196
$956$	−5.12075e10	−1.89553
$957$	0	0
$958$	6.12508e9	0.225078
$959$	0	0
$960$	0	0
$961$	−2.55092e10	−0.927180
$962$	8.45184e9	0.306082
$963$	0	0
$964$	−9.86242e9	−0.354580
$965$	−1.90742e10	−0.683282
$966$	0	0
$967$	9.54991e9	0.339630	0.169815	−	0.985476i	$-0.445683\pi$
$967$	9.54991e9	0.339630	0.169815	+	0.985476i	$0.445683\pi$
$968$	3.66330e8	0.0129810
$969$	0	0
$970$	−3.60728e9	−0.126905
$971$	−6.18234e9	−0.216713	−0.108357	−	0.994112i	$-0.534559\pi$
$971$	−6.18234e9	−0.216713	−0.108357	+	0.994112i	$0.534559\pi$
$972$	0	0
$973$	0	0
$974$	1.39240e9	0.0482844
$975$	0	0
$976$	−3.86166e10	−1.32954
$977$	2.09339e10	0.718157	0.359079	−	0.933307i	$-0.383091\pi$
$977$	2.09339e10	0.718157	0.359079	+	0.933307i	$0.383091\pi$
$978$	0	0
$979$	4.08308e10	1.39075
$980$	0	0
$981$	0	0
$982$	−6.38003e9	−0.214997
$983$	2.71634e9	0.0912111	0.0456055	−	0.998960i	$-0.485478\pi$
$983$	2.71634e9	0.0912111	0.0456055	+	0.998960i	$0.485478\pi$
$984$	0	0
$985$	3.09815e10	1.03294
$986$	9.26540e8	0.0307819
$987$	0	0
$988$	−1.43270e10	−0.472613
$989$	6.38128e10	2.09759
$990$	0	0
$991$	−2.51840e10	−0.821991	−0.410995	−	0.911637i	$-0.634819\pi$
$991$	−2.51840e10	−0.821991	−0.410995	+	0.911637i	$0.634819\pi$
$992$	−4.21818e9	−0.137194
$993$	0	0
$994$	0	0
$995$	−4.91846e10	−1.58288
$996$	0	0
$997$	−2.62842e10	−0.839966	−0.419983	−	0.907532i	$-0.637964\pi$
$997$	−2.62842e10	−0.839966	−0.419983	+	0.907532i	$0.637964\pi$
$998$	6.82010e9	0.217187
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.8.a.b.1.1		1
3.2	odd	2		147.8.a.a.1.1		1
7.6	odd	2		63.8.a.a.1.1		1
21.2	odd	6		147.8.e.d.67.1		2
21.5	even	6		147.8.e.c.67.1		2
21.11	odd	6		147.8.e.d.79.1		2
21.17	even	6		147.8.e.c.79.1		2
21.20	even	2		21.8.a.a.1.1	✓	1
84.83	odd	2		336.8.a.b.1.1		1
105.104	even	2		525.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.8.a.a.1.1	✓	1	21.20	even	2
63.8.a.a.1.1		1	7.6	odd	2
147.8.a.a.1.1		1	3.2	odd	2
147.8.e.c.67.1		2	21.5	even	6
147.8.e.c.79.1		2	21.17	even	6
147.8.e.d.67.1		2	21.2	odd	6
147.8.e.d.79.1		2	21.11	odd	6
336.8.a.b.1.1		1	84.83	odd	2
441.8.a.b.1.1		1	1.1	even	1	trivial
525.8.a.b.1.1		1	105.104	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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