LMFDB - Embedded newform 20.10.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,10,Mod(1,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("20.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	20.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:	$10.3007167233$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	20.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-48.0000 q^{3} +625.000 q^{5} -532.000 q^{7} -17379.0 q^{9} +O(q^{10})$

$q-48.0000 q^{3} +625.000 q^{5} -532.000 q^{7} -17379.0 q^{9} -33180.0 q^{11} -99682.0 q^{13} -30000.0 q^{15} -443454. q^{17} -357244. q^{19} +25536.0 q^{21} -142956. q^{23} +390625. q^{25} +1.77898e6 q^{27} +1.52797e6 q^{29} +7.32342e6 q^{31} +1.59264e6 q^{33} -332500. q^{35} -2.66684e6 q^{37} +4.78474e6 q^{39} -7.93901e6 q^{41} -2.11745e7 q^{43} -1.08619e7 q^{45} +1.60596e7 q^{47} -4.00706e7 q^{49} +2.12858e7 q^{51} -8.78222e7 q^{53} -2.07375e7 q^{55} +1.71477e7 q^{57} +1.20625e8 q^{59} +9.35765e7 q^{61} +9.24563e6 q^{63} -6.23012e7 q^{65} +1.93622e8 q^{67} +6.86189e6 q^{69} +4.17763e8 q^{71} -4.50373e8 q^{73} -1.87500e7 q^{75} +1.76518e7 q^{77} -9.14255e7 q^{79} +2.56680e8 q^{81} -6.52637e8 q^{83} -2.77159e8 q^{85} -7.33424e7 q^{87} -1.70059e8 q^{89} +5.30308e7 q^{91} -3.51524e8 q^{93} -2.23278e8 q^{95} -1.09470e7 q^{97} +5.76635e8 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−48.0000	−0.342133	−0.171067	−	0.985259i	$-0.554721\pi$
$3$	−48.0000	−0.342133	−0.171067	+	0.985259i	$0.554721\pi$
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−532.000	−0.0837472	−0.0418736	−	0.999123i	$-0.513333\pi$
$7$	−532.000	−0.0837472	−0.0418736	+	0.999123i	$0.513333\pi$
$8$	0	0
$9$	−17379.0	−0.882945
$10$	0	0
$11$	−33180.0	−0.683297	−0.341648	−	0.939828i	$-0.610985\pi$
$11$	−33180.0	−0.683297	−0.341648	+	0.939828i	$0.610985\pi$
$12$	0	0
$13$	−99682.0	−0.967992	−0.483996	−	0.875070i	$-0.660815\pi$
$13$	−99682.0	−0.967992	−0.483996	+	0.875070i	$0.660815\pi$
$14$	0	0
$15$	−30000.0	−0.153007
$16$	0	0
$17$	−443454.	−1.28774	−0.643870	−	0.765135i	$-0.722672\pi$
$17$	−443454.	−1.28774	−0.643870	+	0.765135i	$0.722672\pi$
$18$	0	0
$19$	−357244.	−0.628889	−0.314444	−	0.949276i	$-0.601818\pi$
$19$	−357244.	−0.628889	−0.314444	+	0.949276i	$0.601818\pi$
$20$	0	0
$21$	25536.0	0.0286527
$22$	0	0
$23$	−142956.	−0.106519	−0.0532595	−	0.998581i	$-0.516961\pi$
$23$	−142956.	−0.106519	−0.0532595	+	0.998581i	$0.516961\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	1.77898e6	0.644218
$28$	0	0
$29$	1.52797e6	0.401165	0.200582	−	0.979677i	$-0.435717\pi$
$29$	1.52797e6	0.401165	0.200582	+	0.979677i	$0.435717\pi$
$30$	0	0
$31$	7.32342e6	1.42425	0.712125	−	0.702053i	$-0.247733\pi$
$31$	7.32342e6	1.42425	0.712125	+	0.702053i	$0.247733\pi$
$32$	0	0
$33$	1.59264e6	0.233779
$34$	0	0
$35$	−332500.	−0.0374529
$36$	0	0
$37$	−2.66684e6	−0.233932	−0.116966	−	0.993136i	$-0.537317\pi$
$37$	−2.66684e6	−0.233932	−0.116966	+	0.993136i	$0.537317\pi$
$38$	0	0
$39$	4.78474e6	0.331182
$40$	0	0
$41$	−7.93901e6	−0.438772	−0.219386	−	0.975638i	$-0.570405\pi$
$41$	−7.93901e6	−0.438772	−0.219386	+	0.975638i	$0.570405\pi$
$42$	0	0
$43$	−2.11745e7	−0.944508	−0.472254	−	0.881463i	$-0.656559\pi$
$43$	−2.11745e7	−0.944508	−0.472254	+	0.881463i	$0.656559\pi$
$44$	0	0
$45$	−1.08619e7	−0.394865
$46$	0	0
$47$	1.60596e7	0.480060	0.240030	−	0.970765i	$-0.422843\pi$
$47$	1.60596e7	0.480060	0.240030	+	0.970765i	$0.422843\pi$
$48$	0	0
$49$	−4.00706e7	−0.992986
$50$	0	0
$51$	2.12858e7	0.440579
$52$	0	0
$53$	−8.78222e7	−1.52884	−0.764422	−	0.644716i	$-0.776975\pi$
$53$	−8.78222e7	−1.52884	−0.764422	+	0.644716i	$0.776975\pi$
$54$	0	0
$55$	−2.07375e7	−0.305580
$56$	0	0
$57$	1.71477e7	0.215164
$58$	0	0
$59$	1.20625e8	1.29600	0.647999	−	0.761642i	$-0.275606\pi$
$59$	1.20625e8	1.29600	0.647999	+	0.761642i	$0.275606\pi$
$60$	0	0
$61$	9.35765e7	0.865332	0.432666	−	0.901554i	$-0.357573\pi$
$61$	9.35765e7	0.865332	0.432666	+	0.901554i	$0.357573\pi$
$62$	0	0
$63$	9.24563e6	0.0739442
$64$	0	0
$65$	−6.23012e7	−0.432899
$66$	0	0
$67$	1.93622e8	1.17386	0.586931	−	0.809637i	$-0.300336\pi$
$67$	1.93622e8	1.17386	0.586931	+	0.809637i	$0.300336\pi$
$68$	0	0
$69$	6.86189e6	0.0364437
$70$	0	0
$71$	4.17763e8	1.95105	0.975524	−	0.219893i	$-0.0705710\pi$
$71$	4.17763e8	1.95105	0.975524	+	0.219893i	$0.0705710\pi$
$72$	0	0
$73$	−4.50373e8	−1.85618	−0.928088	−	0.372361i	$-0.878548\pi$
$73$	−4.50373e8	−1.85618	−0.928088	+	0.372361i	$0.878548\pi$
$74$	0	0
$75$	−1.87500e7	−0.0684267
$76$	0	0
$77$	1.76518e7	0.0572242
$78$	0	0
$79$	−9.14255e7	−0.264086	−0.132043	−	0.991244i	$-0.542154\pi$
$79$	−9.14255e7	−0.264086	−0.132043	+	0.991244i	$0.542154\pi$
$80$	0	0
$81$	2.56680e8	0.662536
$82$	0	0
$83$	−6.52637e8	−1.50946	−0.754728	−	0.656038i	$-0.772231\pi$
$83$	−6.52637e8	−1.50946	−0.754728	+	0.656038i	$0.772231\pi$
$84$	0	0
$85$	−2.77159e8	−0.575895
$86$	0	0
$87$	−7.33424e7	−0.137252
$88$	0	0
$89$	−1.70059e8	−0.287306	−0.143653	−	0.989628i	$-0.545885\pi$
$89$	−1.70059e8	−0.287306	−0.143653	+	0.989628i	$0.545885\pi$
$90$	0	0
$91$	5.30308e7	0.0810666
$92$	0	0
$93$	−3.51524e8	−0.487283
$94$	0	0
$95$	−2.23278e8	−0.281248
$96$	0	0
$97$	−1.09470e7	−0.0125552	−0.00627759	−	0.999980i	$-0.501998\pi$
$97$	−1.09470e7	−0.0125552	−0.00627759	+	0.999980i	$0.501998\pi$
$98$	0	0
$99$	5.76635e8	0.603313
$100$	0	0
$101$	1.09963e9	1.05148	0.525739	−	0.850646i	$-0.323789\pi$
$101$	1.09963e9	1.05148	0.525739	+	0.850646i	$0.323789\pi$
$102$	0	0
$103$	−1.35878e9	−1.18955	−0.594773	−	0.803894i	$-0.702758\pi$
$103$	−1.35878e9	−1.18955	−0.594773	+	0.803894i	$0.702758\pi$
$104$	0	0
$105$	1.59600e7	0.0128139
$106$	0	0
$107$	1.55988e9	1.15044	0.575220	−	0.817999i	$-0.304916\pi$
$107$	1.55988e9	1.15044	0.575220	+	0.817999i	$0.304916\pi$
$108$	0	0
$109$	1.55615e8	0.105593	0.0527963	−	0.998605i	$-0.483187\pi$
$109$	1.55615e8	0.105593	0.0527963	+	0.998605i	$0.483187\pi$
$110$	0	0
$111$	1.28008e8	0.0800360
$112$	0	0
$113$	2.29946e9	1.32670	0.663349	−	0.748310i	$-0.269134\pi$
$113$	2.29946e9	1.32670	0.663349	+	0.748310i	$0.269134\pi$
$114$	0	0
$115$	−8.93475e7	−0.0476367
$116$	0	0
$117$	1.73237e9	0.854683
$118$	0	0
$119$	2.35918e8	0.107845
$120$	0	0
$121$	−1.25704e9	−0.533106
$122$	0	0
$123$	3.81073e8	0.150119
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−5.39204e9	−1.83923	−0.919616	−	0.392819i	$-0.871500\pi$
$127$	−5.39204e9	−1.83923	−0.919616	+	0.392819i	$0.871500\pi$
$128$	0	0
$129$	1.01638e9	0.323148
$130$	0	0
$131$	−2.66643e9	−0.791061	−0.395531	−	0.918453i	$-0.629439\pi$
$131$	−2.66643e9	−0.791061	−0.395531	+	0.918453i	$0.629439\pi$
$132$	0	0
$133$	1.90054e8	0.0526677
$134$	0	0
$135$	1.11186e9	0.288103
$136$	0	0
$137$	−2.05588e9	−0.498604	−0.249302	−	0.968426i	$-0.580201\pi$
$137$	−2.05588e9	−0.498604	−0.249302	+	0.968426i	$0.580201\pi$
$138$	0	0
$139$	−6.76177e9	−1.53636	−0.768181	−	0.640232i	$-0.778838\pi$
$139$	−6.76177e9	−1.53636	−0.768181	+	0.640232i	$0.778838\pi$
$140$	0	0
$141$	−7.70863e8	−0.164245
$142$	0	0
$143$	3.30745e9	0.661426
$144$	0	0
$145$	9.54979e8	0.179406
$146$	0	0
$147$	1.92339e9	0.339734
$148$	0	0
$149$	1.30009e9	0.216091	0.108045	−	0.994146i	$-0.465541\pi$
$149$	1.30009e9	0.216091	0.108045	+	0.994146i	$0.465541\pi$
$150$	0	0
$151$	−9.20235e9	−1.44046	−0.720232	−	0.693733i	$-0.755965\pi$
$151$	−9.20235e9	−1.44046	−0.720232	+	0.693733i	$0.755965\pi$
$152$	0	0
$153$	7.70679e9	1.13700
$154$	0	0
$155$	4.57714e9	0.636944
$156$	0	0
$157$	−6.77345e9	−0.889737	−0.444868	−	0.895596i	$-0.646750\pi$
$157$	−6.77345e9	−0.889737	−0.444868	+	0.895596i	$0.646750\pi$
$158$	0	0
$159$	4.21547e9	0.523069
$160$	0	0
$161$	7.60526e7	0.00892067
$162$	0	0
$163$	1.15450e10	1.28100	0.640500	−	0.767958i	$-0.278727\pi$
$163$	1.15450e10	1.28100	0.640500	+	0.767958i	$0.278727\pi$
$164$	0	0
$165$	9.95400e8	0.104549
$166$	0	0
$167$	3.99897e9	0.397854	0.198927	−	0.980014i	$-0.436254\pi$
$167$	3.99897e9	0.397854	0.198927	+	0.980014i	$0.436254\pi$
$168$	0	0
$169$	−6.67998e8	−0.0629920
$170$	0	0
$171$	6.20854e9	0.555274
$172$	0	0
$173$	1.04311e10	0.885369	0.442685	−	0.896677i	$-0.354026\pi$
$173$	1.04311e10	0.885369	0.442685	+	0.896677i	$0.354026\pi$
$174$	0	0
$175$	−2.07812e8	−0.0167494
$176$	0	0
$177$	−5.79001e9	−0.443404
$178$	0	0
$179$	3.86256e9	0.281214	0.140607	−	0.990065i	$-0.455095\pi$
$179$	3.86256e9	0.281214	0.140607	+	0.990065i	$0.455095\pi$
$180$	0	0
$181$	−2.00994e10	−1.39197	−0.695983	−	0.718058i	$-0.745031\pi$
$181$	−2.00994e10	−1.39197	−0.695983	+	0.718058i	$0.745031\pi$
$182$	0	0
$183$	−4.49167e9	−0.296059
$184$	0	0
$185$	−1.66678e9	−0.104618
$186$	0	0
$187$	1.47138e10	0.879909
$188$	0	0
$189$	−9.46415e8	−0.0539515
$190$	0	0
$191$	−6.09262e9	−0.331249	−0.165624	−	0.986189i	$-0.552964\pi$
$191$	−6.09262e9	−0.331249	−0.165624	+	0.986189i	$0.552964\pi$
$192$	0	0
$193$	−1.74370e10	−0.904614	−0.452307	−	0.891862i	$-0.649399\pi$
$193$	−1.74370e10	−0.904614	−0.452307	+	0.891862i	$0.649399\pi$
$194$	0	0
$195$	2.99046e9	0.148109
$196$	0	0
$197$	−1.76183e10	−0.833424	−0.416712	−	0.909039i	$-0.636818\pi$
$197$	−1.76183e10	−0.833424	−0.416712	+	0.909039i	$0.636818\pi$
$198$	0	0
$199$	−2.72118e10	−1.23004	−0.615020	−	0.788512i	$-0.710852\pi$
$199$	−2.72118e10	−1.23004	−0.615020	+	0.788512i	$0.710852\pi$
$200$	0	0
$201$	−9.29384e9	−0.401618
$202$	0	0
$203$	−8.12878e8	−0.0335964
$204$	0	0
$205$	−4.96188e9	−0.196225
$206$	0	0
$207$	2.48443e9	0.0940504
$208$	0	0
$209$	1.18534e10	0.429717
$210$	0	0
$211$	4.40383e10	1.52953	0.764767	−	0.644307i	$-0.222854\pi$
$211$	4.40383e10	1.52953	0.764767	+	0.644307i	$0.222854\pi$
$212$	0	0
$213$	−2.00526e10	−0.667519
$214$	0	0
$215$	−1.32341e10	−0.422397
$216$	0	0
$217$	−3.89606e9	−0.119277
$218$	0	0
$219$	2.16179e10	0.635060
$220$	0	0
$221$	4.42044e10	1.24652
$222$	0	0
$223$	−2.47524e10	−0.670262	−0.335131	−	0.942172i	$-0.608781\pi$
$223$	−2.47524e10	−0.670262	−0.335131	+	0.942172i	$0.608781\pi$
$224$	0	0
$225$	−6.78867e9	−0.176589
$226$	0	0
$227$	5.22331e10	1.30566	0.652829	−	0.757505i	$-0.273582\pi$
$227$	5.22331e10	1.30566	0.652829	+	0.757505i	$0.273582\pi$
$228$	0	0
$229$	4.85187e10	1.16587	0.582935	−	0.812519i	$-0.301904\pi$
$229$	4.85187e10	1.16587	0.582935	+	0.812519i	$0.301904\pi$
$230$	0	0
$231$	−8.47284e8	−0.0195783
$232$	0	0
$233$	−1.55978e10	−0.346705	−0.173353	−	0.984860i	$-0.555460\pi$
$233$	−1.55978e10	−0.346705	−0.173353	+	0.984860i	$0.555460\pi$
$234$	0	0
$235$	1.00373e10	0.214689
$236$	0	0
$237$	4.38842e9	0.0903526
$238$	0	0
$239$	−7.87748e10	−1.56170	−0.780849	−	0.624720i	$-0.785213\pi$
$239$	−7.87748e10	−1.56170	−0.780849	+	0.624720i	$0.785213\pi$
$240$	0	0
$241$	9.12330e10	1.74211	0.871054	−	0.491187i	$-0.163437\pi$
$241$	9.12330e10	1.74211	0.871054	+	0.491187i	$0.163437\pi$
$242$	0	0
$243$	−4.73362e10	−0.870894
$244$	0	0
$245$	−2.50441e10	−0.444077
$246$	0	0
$247$	3.56108e10	0.608759
$248$	0	0
$249$	3.13266e10	0.516436
$250$	0	0
$251$	−1.22779e9	−0.0195251	−0.00976255	−	0.999952i	$-0.503108\pi$
$251$	−1.22779e9	−0.0195251	−0.00976255	+	0.999952i	$0.503108\pi$
$252$	0	0
$253$	4.74328e9	0.0727841
$254$	0	0
$255$	1.33036e10	0.197033
$256$	0	0
$257$	−5.64457e10	−0.807108	−0.403554	−	0.914956i	$-0.632225\pi$
$257$	−5.64457e10	−0.807108	−0.403554	+	0.914956i	$0.632225\pi$
$258$	0	0
$259$	1.41876e9	0.0195912
$260$	0	0
$261$	−2.65545e10	−0.354206
$262$	0	0
$263$	−1.40883e11	−1.81576	−0.907878	−	0.419235i	$-0.862298\pi$
$263$	−1.40883e11	−1.81576	−0.907878	+	0.419235i	$0.862298\pi$
$264$	0	0
$265$	−5.48889e10	−0.683720
$266$	0	0
$267$	8.16284e9	0.0982971
$268$	0	0
$269$	4.62184e10	0.538183	0.269091	−	0.963115i	$-0.413277\pi$
$269$	4.62184e10	0.538183	0.269091	+	0.963115i	$0.413277\pi$
$270$	0	0
$271$	−4.88883e10	−0.550609	−0.275304	−	0.961357i	$-0.588779\pi$
$271$	−4.88883e10	−0.550609	−0.275304	+	0.961357i	$0.588779\pi$
$272$	0	0
$273$	−2.54548e9	−0.0277356
$274$	0	0
$275$	−1.29609e10	−0.136659
$276$	0	0
$277$	1.60734e10	0.164040	0.0820200	−	0.996631i	$-0.473863\pi$
$277$	1.60734e10	0.164040	0.0820200	+	0.996631i	$0.473863\pi$
$278$	0	0
$279$	−1.27274e11	−1.25753
$280$	0	0
$281$	−1.46267e10	−0.139948	−0.0699741	−	0.997549i	$-0.522292\pi$
$281$	−1.46267e10	−0.139948	−0.0699741	+	0.997549i	$0.522292\pi$
$282$	0	0
$283$	9.24491e10	0.856769	0.428385	−	0.903597i	$-0.359083\pi$
$283$	9.24491e10	0.856769	0.428385	+	0.903597i	$0.359083\pi$
$284$	0	0
$285$	1.07173e10	0.0962242
$286$	0	0
$287$	4.22356e9	0.0367460
$288$	0	0
$289$	7.80636e10	0.658276
$290$	0	0
$291$	5.25457e8	0.00429555
$292$	0	0
$293$	1.48756e11	1.17915	0.589575	−	0.807714i	$-0.299295\pi$
$293$	1.48756e11	1.17915	0.589575	+	0.807714i	$0.299295\pi$
$294$	0	0
$295$	7.53908e10	0.579588
$296$	0	0
$297$	−5.90264e10	−0.440192
$298$	0	0
$299$	1.42501e10	0.103110
$300$	0	0
$301$	1.12648e10	0.0790999
$302$	0	0
$303$	−5.27822e10	−0.359746
$304$	0	0
$305$	5.84853e10	0.386988
$306$	0	0
$307$	2.45184e11	1.57532	0.787661	−	0.616108i	$-0.211292\pi$
$307$	2.45184e11	1.57532	0.787661	+	0.616108i	$0.211292\pi$
$308$	0	0
$309$	6.52213e10	0.406983
$310$	0	0
$311$	−9.29598e10	−0.563473	−0.281737	−	0.959492i	$-0.590910\pi$
$311$	−9.29598e10	−0.563473	−0.281737	+	0.959492i	$0.590910\pi$
$312$	0	0
$313$	2.44768e11	1.44147	0.720733	−	0.693212i	$-0.243805\pi$
$313$	2.44768e11	1.44147	0.720733	+	0.693212i	$0.243805\pi$
$314$	0	0
$315$	5.77852e9	0.0330688
$316$	0	0
$317$	−2.80185e11	−1.55840	−0.779199	−	0.626776i	$-0.784374\pi$
$317$	−2.80185e11	−1.55840	−0.779199	+	0.626776i	$0.784374\pi$
$318$	0	0
$319$	−5.06979e10	−0.274114
$320$	0	0
$321$	−7.48742e10	−0.393604
$322$	0	0
$323$	1.58421e11	0.809845
$324$	0	0
$325$	−3.89383e10	−0.193598
$326$	0	0
$327$	−7.46954e9	−0.0361267
$328$	0	0
$329$	−8.54373e9	−0.0402037
$330$	0	0
$331$	−4.43208e10	−0.202946	−0.101473	−	0.994838i	$-0.532356\pi$
$331$	−4.43208e10	−0.202946	−0.101473	+	0.994838i	$0.532356\pi$
$332$	0	0
$333$	4.63470e10	0.206549
$334$	0	0
$335$	1.21014e11	0.524967
$336$	0	0
$337$	−4.12784e11	−1.74337	−0.871683	−	0.490070i	$-0.836971\pi$
$337$	−4.12784e11	−1.74337	−0.871683	+	0.490070i	$0.836971\pi$
$338$	0	0
$339$	−1.10374e11	−0.453908
$340$	0	0
$341$	−2.42991e11	−0.973185
$342$	0	0
$343$	4.27857e10	0.166907
$344$	0	0
$345$	4.28868e9	0.0162981
$346$	0	0
$347$	2.70962e10	0.100329	0.0501643	−	0.998741i	$-0.484025\pi$
$347$	2.70962e10	0.100329	0.0501643	+	0.998741i	$0.484025\pi$
$348$	0	0
$349$	−1.88319e11	−0.679486	−0.339743	−	0.940518i	$-0.610340\pi$
$349$	−1.88319e11	−0.679486	−0.339743	+	0.940518i	$0.610340\pi$
$350$	0	0
$351$	−1.77332e11	−0.623598
$352$	0	0
$353$	1.01309e11	0.347267	0.173634	−	0.984810i	$-0.444449\pi$
$353$	1.01309e11	0.347267	0.173634	+	0.984810i	$0.444449\pi$
$354$	0	0
$355$	2.61102e11	0.872535
$356$	0	0
$357$	−1.13240e10	−0.0368973
$358$	0	0
$359$	−7.19992e10	−0.228772	−0.114386	−	0.993436i	$-0.536490\pi$
$359$	−7.19992e10	−0.228772	−0.114386	+	0.993436i	$0.536490\pi$
$360$	0	0
$361$	−1.95064e11	−0.604499
$362$	0	0
$363$	6.03377e10	0.182393
$364$	0	0
$365$	−2.81483e11	−0.830107
$366$	0	0
$367$	3.43665e11	0.988868	0.494434	−	0.869215i	$-0.335375\pi$
$367$	3.43665e11	0.988868	0.494434	+	0.869215i	$0.335375\pi$
$368$	0	0
$369$	1.37972e11	0.387412
$370$	0	0
$371$	4.67214e10	0.128036
$372$	0	0
$373$	−1.42007e11	−0.379857	−0.189928	−	0.981798i	$-0.560826\pi$
$373$	−1.42007e11	−0.379857	−0.189928	+	0.981798i	$0.560826\pi$
$374$	0	0
$375$	−1.17188e10	−0.0306013
$376$	0	0
$377$	−1.52311e11	−0.388324
$378$	0	0
$379$	−7.90435e10	−0.196784	−0.0983919	−	0.995148i	$-0.531370\pi$
$379$	−7.90435e10	−0.196784	−0.0983919	+	0.995148i	$0.531370\pi$
$380$	0	0
$381$	2.58818e11	0.629263
$382$	0	0
$383$	2.91620e11	0.692504	0.346252	−	0.938141i	$-0.387454\pi$
$383$	2.91620e11	0.692504	0.346252	+	0.938141i	$0.387454\pi$
$384$	0	0
$385$	1.10324e10	0.0255914
$386$	0	0
$387$	3.67992e11	0.833948
$388$	0	0
$389$	8.47185e11	1.87588	0.937940	−	0.346798i	$-0.112731\pi$
$389$	8.47185e11	1.87588	0.937940	+	0.346798i	$0.112731\pi$
$390$	0	0
$391$	6.33944e10	0.137169
$392$	0	0
$393$	1.27989e11	0.270649
$394$	0	0
$395$	−5.71409e10	−0.118103
$396$	0	0
$397$	−8.56081e10	−0.172965	−0.0864824	−	0.996253i	$-0.527563\pi$
$397$	−8.56081e10	−0.172965	−0.0864824	+	0.996253i	$0.527563\pi$
$398$	0	0
$399$	−9.12258e9	−0.0180194
$400$	0	0
$401$	6.05161e11	1.16875	0.584374	−	0.811484i	$-0.301340\pi$
$401$	6.05161e11	1.16875	0.584374	+	0.811484i	$0.301340\pi$
$402$	0	0
$403$	−7.30013e11	−1.37866
$404$	0	0
$405$	1.60425e11	0.296295
$406$	0	0
$407$	8.84858e10	0.159845
$408$	0	0
$409$	−2.09247e11	−0.369747	−0.184873	−	0.982762i	$-0.559187\pi$
$409$	−2.09247e11	−0.369747	−0.184873	+	0.982762i	$0.559187\pi$
$410$	0	0
$411$	9.86825e10	0.170589
$412$	0	0
$413$	−6.41726e10	−0.108536
$414$	0	0
$415$	−4.07898e11	−0.675049
$416$	0	0
$417$	3.24565e11	0.525641
$418$	0	0
$419$	−5.40288e11	−0.856371	−0.428186	−	0.903691i	$-0.640847\pi$
$419$	−5.40288e11	−0.856371	−0.428186	+	0.903691i	$0.640847\pi$
$420$	0	0
$421$	−3.58323e11	−0.555911	−0.277955	−	0.960594i	$-0.589657\pi$
$421$	−3.58323e11	−0.555911	−0.277955	+	0.960594i	$0.589657\pi$
$422$	0	0
$423$	−2.79100e11	−0.423866
$424$	0	0
$425$	−1.73224e11	−0.257548
$426$	0	0
$427$	−4.97827e10	−0.0724692
$428$	0	0
$429$	−1.58758e11	−0.226296
$430$	0	0
$431$	5.83729e11	0.814824	0.407412	−	0.913245i	$-0.366431\pi$
$431$	5.83729e11	0.814824	0.407412	+	0.913245i	$0.366431\pi$
$432$	0	0
$433$	−1.25754e12	−1.71919	−0.859597	−	0.510973i	$-0.829285\pi$
$433$	−1.25754e12	−1.71919	−0.859597	+	0.510973i	$0.829285\pi$
$434$	0	0
$435$	−4.58390e10	−0.0613809
$436$	0	0
$437$	5.10702e10	0.0669886
$438$	0	0
$439$	−2.14988e11	−0.276264	−0.138132	−	0.990414i	$-0.544110\pi$
$439$	−2.14988e11	−0.276264	−0.138132	+	0.990414i	$0.544110\pi$
$440$	0	0
$441$	6.96387e11	0.876752
$442$	0	0
$443$	−1.50998e12	−1.86275	−0.931374	−	0.364065i	$-0.881389\pi$
$443$	−1.50998e12	−1.86275	−0.931374	+	0.364065i	$0.881389\pi$
$444$	0	0
$445$	−1.06287e11	−0.128487
$446$	0	0
$447$	−6.24044e10	−0.0739318
$448$	0	0
$449$	1.06639e12	1.23824	0.619122	−	0.785295i	$-0.287489\pi$
$449$	1.06639e12	1.23824	0.619122	+	0.785295i	$0.287489\pi$
$450$	0	0
$451$	2.63416e11	0.299812
$452$	0	0
$453$	4.41713e11	0.492831
$454$	0	0
$455$	3.31443e10	0.0362541
$456$	0	0
$457$	−3.89934e11	−0.418184	−0.209092	−	0.977896i	$-0.567051\pi$
$457$	−3.89934e11	−0.418184	−0.209092	+	0.977896i	$0.567051\pi$
$458$	0	0
$459$	−7.88894e11	−0.829586
$460$	0	0
$461$	5.49112e11	0.566249	0.283124	−	0.959083i	$-0.408629\pi$
$461$	5.49112e11	0.566249	0.283124	+	0.959083i	$0.408629\pi$
$462$	0	0
$463$	1.99371e11	0.201627	0.100813	−	0.994905i	$-0.467855\pi$
$463$	1.99371e11	0.201627	0.100813	+	0.994905i	$0.467855\pi$
$464$	0	0
$465$	−2.19702e11	−0.217920
$466$	0	0
$467$	−7.93835e11	−0.772332	−0.386166	−	0.922429i	$-0.626201\pi$
$467$	−7.93835e11	−0.772332	−0.386166	+	0.922429i	$0.626201\pi$
$468$	0	0
$469$	−1.03007e11	−0.0983078
$470$	0	0
$471$	3.25126e11	0.304409
$472$	0	0
$473$	7.02571e11	0.645379
$474$	0	0
$475$	−1.39548e11	−0.125778
$476$	0	0
$477$	1.52626e12	1.34988
$478$	0	0
$479$	1.77175e12	1.53778	0.768888	−	0.639384i	$-0.220810\pi$
$479$	1.77175e12	1.53778	0.768888	+	0.639384i	$0.220810\pi$
$480$	0	0
$481$	2.65836e11	0.226444
$482$	0	0
$483$	−3.65052e9	−0.00305206
$484$	0	0
$485$	−6.84189e9	−0.00561485
$486$	0	0
$487$	1.62132e12	1.30613	0.653067	−	0.757300i	$-0.273482\pi$
$487$	1.62132e12	1.30613	0.653067	+	0.757300i	$0.273482\pi$
$488$	0	0
$489$	−5.54159e11	−0.438273
$490$	0	0
$491$	−9.07687e11	−0.704806	−0.352403	−	0.935848i	$-0.614635\pi$
$491$	−9.07687e11	−0.704806	−0.352403	+	0.935848i	$0.614635\pi$
$492$	0	0
$493$	−6.77583e11	−0.516596
$494$	0	0
$495$	3.60397e11	0.269810
$496$	0	0
$497$	−2.22250e11	−0.163395
$498$	0	0
$499$	−2.68633e11	−0.193958	−0.0969789	−	0.995286i	$-0.530918\pi$
$499$	−2.68633e11	−0.193958	−0.0969789	+	0.995286i	$0.530918\pi$
$500$	0	0
$501$	−1.91950e11	−0.136119
$502$	0	0
$503$	1.30260e12	0.907307	0.453653	−	0.891178i	$-0.350120\pi$
$503$	1.30260e12	0.907307	0.453653	+	0.891178i	$0.350120\pi$
$504$	0	0
$505$	6.87268e11	0.470235
$506$	0	0
$507$	3.20639e10	0.0215517
$508$	0	0
$509$	3.45798e11	0.228345	0.114173	−	0.993461i	$-0.463578\pi$
$509$	3.45798e11	0.228345	0.114173	+	0.993461i	$0.463578\pi$
$510$	0	0
$511$	2.39598e11	0.155450
$512$	0	0
$513$	−6.35529e11	−0.405142
$514$	0	0
$515$	−8.49236e11	−0.531981
$516$	0	0
$517$	−5.32859e11	−0.328023
$518$	0	0
$519$	−5.00695e11	−0.302914
$520$	0	0
$521$	−1.02611e12	−0.610134	−0.305067	−	0.952331i	$-0.598679\pi$
$521$	−1.02611e12	−0.610134	−0.305067	+	0.952331i	$0.598679\pi$
$522$	0	0
$523$	1.97289e12	1.15304	0.576521	−	0.817082i	$-0.304410\pi$
$523$	1.97289e12	1.15304	0.576521	+	0.817082i	$0.304410\pi$
$524$	0	0
$525$	9.97500e9	0.00573055
$526$	0	0
$527$	−3.24760e12	−1.83406
$528$	0	0
$529$	−1.78072e12	−0.988654
$530$	0	0
$531$	−2.09635e12	−1.14429
$532$	0	0
$533$	7.91377e11	0.424728
$534$	0	0
$535$	9.74924e11	0.514492
$536$	0	0
$537$	−1.85403e11	−0.0962128
$538$	0	0
$539$	1.32954e12	0.678504
$540$	0	0
$541$	−3.19857e12	−1.60534	−0.802672	−	0.596421i	$-0.796589\pi$
$541$	−3.19857e12	−1.60534	−0.802672	+	0.596421i	$0.796589\pi$
$542$	0	0
$543$	9.64769e11	0.476238
$544$	0	0
$545$	9.72596e10	0.0472224
$546$	0	0
$547$	−2.95150e11	−0.140961	−0.0704805	−	0.997513i	$-0.522453\pi$
$547$	−2.95150e11	−0.140961	−0.0704805	+	0.997513i	$0.522453\pi$
$548$	0	0
$549$	−1.62627e12	−0.764040
$550$	0	0
$551$	−5.45857e11	−0.252288
$552$	0	0
$553$	4.86384e10	0.0221165
$554$	0	0
$555$	8.00053e10	0.0357932
$556$	0	0
$557$	−7.66147e11	−0.337259	−0.168629	−	0.985680i	$-0.553934\pi$
$557$	−7.66147e11	−0.337259	−0.168629	+	0.985680i	$0.553934\pi$
$558$	0	0
$559$	2.11072e12	0.914276
$560$	0	0
$561$	−7.06263e11	−0.301046
$562$	0	0
$563$	1.82771e12	0.766689	0.383345	−	0.923605i	$-0.374772\pi$
$563$	1.82771e12	0.766689	0.383345	+	0.923605i	$0.374772\pi$
$564$	0	0
$565$	1.43716e12	0.593317
$566$	0	0
$567$	−1.36554e11	−0.0554856
$568$	0	0
$569$	−1.82583e12	−0.730224	−0.365112	−	0.930964i	$-0.618969\pi$
$569$	−1.82583e12	−0.730224	−0.365112	+	0.930964i	$0.618969\pi$
$570$	0	0
$571$	2.72562e12	1.07301	0.536505	−	0.843897i	$-0.319744\pi$
$571$	2.72562e12	1.07301	0.536505	+	0.843897i	$0.319744\pi$
$572$	0	0
$573$	2.92446e11	0.113331
$574$	0	0
$575$	−5.58422e10	−0.0213038
$576$	0	0
$577$	4.11938e12	1.54718	0.773590	−	0.633687i	$-0.218459\pi$
$577$	4.11938e12	1.54718	0.773590	+	0.633687i	$0.218459\pi$
$578$	0	0
$579$	8.36975e11	0.309499
$580$	0	0
$581$	3.47203e11	0.126413
$582$	0	0
$583$	2.91394e12	1.04465
$584$	0	0
$585$	1.08273e12	0.382226
$586$	0	0
$587$	−5.27713e12	−1.83454	−0.917268	−	0.398271i	$-0.869610\pi$
$587$	−5.27713e12	−1.83454	−0.917268	+	0.398271i	$0.869610\pi$
$588$	0	0
$589$	−2.61625e12	−0.895694
$590$	0	0
$591$	8.45678e11	0.285142
$592$	0	0
$593$	3.36836e12	1.11859	0.559297	−	0.828967i	$-0.311071\pi$
$593$	3.36836e12	1.11859	0.559297	+	0.828967i	$0.311071\pi$
$594$	0	0
$595$	1.47448e11	0.0482296
$596$	0	0
$597$	1.30617e12	0.420838
$598$	0	0
$599$	2.92713e12	0.929013	0.464507	−	0.885570i	$-0.346232\pi$
$599$	2.92713e12	0.929013	0.464507	+	0.885570i	$0.346232\pi$
$600$	0	0
$601$	−1.00636e12	−0.314644	−0.157322	−	0.987547i	$-0.550286\pi$
$601$	−1.00636e12	−0.314644	−0.157322	+	0.987547i	$0.550286\pi$
$602$	0	0
$603$	−3.36495e12	−1.03646
$604$	0	0
$605$	−7.85647e11	−0.238412
$606$	0	0
$607$	3.31911e12	0.992367	0.496183	−	0.868218i	$-0.334734\pi$
$607$	3.31911e12	0.992367	0.496183	+	0.868218i	$0.334734\pi$
$608$	0	0
$609$	3.90181e10	0.0114945
$610$	0	0
$611$	−1.60086e12	−0.464694
$612$	0	0
$613$	9.69366e11	0.277278	0.138639	−	0.990343i	$-0.455727\pi$
$613$	9.69366e11	0.277278	0.138639	+	0.990343i	$0.455727\pi$
$614$	0	0
$615$	2.38170e11	0.0671351
$616$	0	0
$617$	−3.76924e12	−1.04706	−0.523529	−	0.852008i	$-0.675385\pi$
$617$	−3.76924e12	−1.04706	−0.523529	+	0.852008i	$0.675385\pi$
$618$	0	0
$619$	3.72454e12	1.01968	0.509841	−	0.860269i	$-0.329704\pi$
$619$	3.72454e12	1.01968	0.509841	+	0.860269i	$0.329704\pi$
$620$	0	0
$621$	−2.54315e11	−0.0686215
$622$	0	0
$623$	9.04715e10	0.0240611
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−5.68961e11	−0.147021
$628$	0	0
$629$	1.18262e12	0.301244
$630$	0	0
$631$	−1.86229e12	−0.467644	−0.233822	−	0.972279i	$-0.575123\pi$
$631$	−1.86229e12	−0.467644	−0.233822	+	0.972279i	$0.575123\pi$
$632$	0	0
$633$	−2.11384e12	−0.523305
$634$	0	0
$635$	−3.37003e12	−0.822529
$636$	0	0
$637$	3.99432e12	0.961203
$638$	0	0
$639$	−7.26031e12	−1.72267
$640$	0	0
$641$	4.04105e12	0.945437	0.472719	−	0.881213i	$-0.343273\pi$
$641$	4.04105e12	0.945437	0.472719	+	0.881213i	$0.343273\pi$
$642$	0	0
$643$	−4.43752e12	−1.02374	−0.511871	−	0.859062i	$-0.671047\pi$
$643$	−4.43752e12	−1.02374	−0.511871	+	0.859062i	$0.671047\pi$
$644$	0	0
$645$	6.35236e11	0.144516
$646$	0	0
$647$	−6.01835e12	−1.35023	−0.675116	−	0.737712i	$-0.735906\pi$
$647$	−6.01835e12	−1.35023	−0.675116	+	0.737712i	$0.735906\pi$
$648$	0	0
$649$	−4.00234e12	−0.885551
$650$	0	0
$651$	1.87011e11	0.0408086
$652$	0	0
$653$	−7.43135e12	−1.59940	−0.799702	−	0.600397i	$-0.795009\pi$
$653$	−7.43135e12	−1.59940	−0.799702	+	0.600397i	$0.795009\pi$
$654$	0	0
$655$	−1.66652e12	−0.353773
$656$	0	0
$657$	7.82703e12	1.63890
$658$	0	0
$659$	3.40289e12	0.702851	0.351425	−	0.936216i	$-0.385697\pi$
$659$	3.40289e12	0.702851	0.351425	+	0.936216i	$0.385697\pi$
$660$	0	0
$661$	4.85283e12	0.988754	0.494377	−	0.869247i	$-0.335396\pi$
$661$	4.85283e12	0.988754	0.494377	+	0.869247i	$0.335396\pi$
$662$	0	0
$663$	−2.12181e12	−0.426477
$664$	0	0
$665$	1.18784e11	0.0235537
$666$	0	0
$667$	−2.18432e11	−0.0427317
$668$	0	0
$669$	1.18811e12	0.229319
$670$	0	0
$671$	−3.10487e12	−0.591278
$672$	0	0
$673$	−3.90740e12	−0.734210	−0.367105	−	0.930180i	$-0.619651\pi$
$673$	−3.90740e12	−0.734210	−0.367105	+	0.930180i	$0.619651\pi$
$674$	0	0
$675$	6.94912e11	0.128844
$676$	0	0
$677$	4.28234e12	0.783488	0.391744	−	0.920074i	$-0.371872\pi$
$677$	4.28234e12	0.783488	0.391744	+	0.920074i	$0.371872\pi$
$678$	0	0
$679$	5.82382e9	0.00105146
$680$	0	0
$681$	−2.50719e12	−0.446710
$682$	0	0
$683$	3.05978e12	0.538018	0.269009	−	0.963138i	$-0.413304\pi$
$683$	3.05978e12	0.538018	0.269009	+	0.963138i	$0.413304\pi$
$684$	0	0
$685$	−1.28493e12	−0.222983
$686$	0	0
$687$	−2.32890e12	−0.398883
$688$	0	0
$689$	8.75430e12	1.47991
$690$	0	0
$691$	1.14399e13	1.90885	0.954424	−	0.298454i	$-0.0964709\pi$
$691$	1.14399e13	1.90885	0.954424	+	0.298454i	$0.0964709\pi$
$692$	0	0
$693$	−3.06770e11	−0.0505258
$694$	0	0
$695$	−4.22611e12	−0.687082
$696$	0	0
$697$	3.52059e12	0.565025
$698$	0	0
$699$	7.48692e11	0.118620
$700$	0	0
$701$	7.76862e12	1.21510	0.607551	−	0.794281i	$-0.292152\pi$
$701$	7.76862e12	1.21510	0.607551	+	0.794281i	$0.292152\pi$
$702$	0	0
$703$	9.52713e11	0.147117
$704$	0	0
$705$	−4.81789e11	−0.0734524
$706$	0	0
$707$	−5.85002e11	−0.0880583
$708$	0	0
$709$	−6.47245e12	−0.961967	−0.480984	−	0.876730i	$-0.659720\pi$
$709$	−6.47245e12	−0.961967	−0.480984	+	0.876730i	$0.659720\pi$
$710$	0	0
$711$	1.58888e12	0.233173
$712$	0	0
$713$	−1.04693e12	−0.151710
$714$	0	0
$715$	2.06716e12	0.295798
$716$	0	0
$717$	3.78119e12	0.534309
$718$	0	0
$719$	6.99561e12	0.976215	0.488108	−	0.872783i	$-0.337687\pi$
$719$	6.99561e12	0.976215	0.488108	+	0.872783i	$0.337687\pi$
$720$	0	0
$721$	7.22870e11	0.0996211
$722$	0	0
$723$	−4.37918e12	−0.596033
$724$	0	0
$725$	5.96862e11	0.0802329
$726$	0	0
$727$	−2.68044e12	−0.355878	−0.177939	−	0.984041i	$-0.556943\pi$
$727$	−2.68044e12	−0.355878	−0.177939	+	0.984041i	$0.556943\pi$
$728$	0	0
$729$	−2.78009e12	−0.364574
$730$	0	0
$731$	9.38993e12	1.21628
$732$	0	0
$733$	−8.18530e12	−1.04729	−0.523645	−	0.851937i	$-0.675428\pi$
$733$	−8.18530e12	−1.04729	−0.523645	+	0.851937i	$0.675428\pi$
$734$	0	0
$735$	1.20212e12	0.151934
$736$	0	0
$737$	−6.42437e12	−0.802097
$738$	0	0
$739$	−3.29514e12	−0.406418	−0.203209	−	0.979135i	$-0.565137\pi$
$739$	−3.29514e12	−0.406418	−0.203209	+	0.979135i	$0.565137\pi$
$740$	0	0
$741$	−1.70932e12	−0.208277
$742$	0	0
$743$	−1.28635e12	−0.154849	−0.0774247	−	0.996998i	$-0.524670\pi$
$743$	−1.28635e12	−0.154849	−0.0774247	+	0.996998i	$0.524670\pi$
$744$	0	0
$745$	8.12558e11	0.0966387
$746$	0	0
$747$	1.13422e13	1.33277
$748$	0	0
$749$	−8.29856e11	−0.0963461
$750$	0	0
$751$	−1.41009e13	−1.61759	−0.808794	−	0.588092i	$-0.799879\pi$
$751$	−1.41009e13	−1.61759	−0.808794	+	0.588092i	$0.799879\pi$
$752$	0	0
$753$	5.89340e10	0.00668019
$754$	0	0
$755$	−5.75147e12	−0.644195
$756$	0	0
$757$	−4.94222e12	−0.547004	−0.273502	−	0.961871i	$-0.588182\pi$
$757$	−4.94222e12	−0.547004	−0.273502	+	0.961871i	$0.588182\pi$
$758$	0	0
$759$	−2.27677e11	−0.0249019
$760$	0	0
$761$	−6.65861e12	−0.719701	−0.359851	−	0.933010i	$-0.617172\pi$
$761$	−6.65861e12	−0.719701	−0.359851	+	0.933010i	$0.617172\pi$
$762$	0	0
$763$	−8.27874e10	−0.00884308
$764$	0	0
$765$	4.81674e12	0.508484
$766$	0	0
$767$	−1.20242e13	−1.25451
$768$	0	0
$769$	−1.20797e13	−1.24563	−0.622814	−	0.782370i	$-0.714011\pi$
$769$	−1.20797e13	−1.24563	−0.622814	+	0.782370i	$0.714011\pi$
$770$	0	0
$771$	2.70939e12	0.276139
$772$	0	0
$773$	1.84847e13	1.86211	0.931054	−	0.364881i	$-0.118890\pi$
$773$	1.84847e13	1.86211	0.931054	+	0.364881i	$0.118890\pi$
$774$	0	0
$775$	2.86071e12	0.284850
$776$	0	0
$777$	−6.81005e10	−0.00670279
$778$	0	0
$779$	2.83617e12	0.275939
$780$	0	0
$781$	−1.38614e13	−1.33314
$782$	0	0
$783$	2.71821e12	0.258438
$784$	0	0
$785$	−4.23341e12	−0.397902
$786$	0	0
$787$	1.27372e13	1.18355	0.591776	−	0.806102i	$-0.298427\pi$
$787$	1.27372e13	1.18355	0.591776	+	0.806102i	$0.298427\pi$
$788$	0	0
$789$	6.76238e12	0.621231
$790$	0	0
$791$	−1.22331e12	−0.111107
$792$	0	0
$793$	−9.32790e12	−0.837634
$794$	0	0
$795$	2.63467e12	0.233923
$796$	0	0
$797$	−2.26895e13	−1.99188	−0.995938	−	0.0900464i	$-0.971298\pi$
$797$	−2.26895e13	−1.99188	−0.995938	+	0.0900464i	$0.971298\pi$
$798$	0	0
$799$	−7.12171e12	−0.618193
$800$	0	0
$801$	2.95546e12	0.253676
$802$	0	0
$803$	1.49434e13	1.26832
$804$	0	0
$805$	4.75329e10	0.00398945
$806$	0	0
$807$	−2.21848e12	−0.184130
$808$	0	0
$809$	1.68841e13	1.38583	0.692915	−	0.721020i	$-0.256326\pi$
$809$	1.68841e13	1.38583	0.692915	+	0.721020i	$0.256326\pi$
$810$	0	0
$811$	−3.72696e12	−0.302524	−0.151262	−	0.988494i	$-0.548334\pi$
$811$	−3.72696e12	−0.302524	−0.151262	+	0.988494i	$0.548334\pi$
$812$	0	0
$813$	2.34664e12	0.188382
$814$	0	0
$815$	7.21561e12	0.572880
$816$	0	0
$817$	7.56447e12	0.593990
$818$	0	0
$819$	−9.21623e11	−0.0715773
$820$	0	0
$821$	−6.87951e12	−0.528461	−0.264231	−	0.964460i	$-0.585118\pi$
$821$	−6.87951e12	−0.528461	−0.264231	+	0.964460i	$0.585118\pi$
$822$	0	0
$823$	1.62514e13	1.23479	0.617395	−	0.786654i	$-0.288188\pi$
$823$	1.62514e13	1.23479	0.617395	+	0.786654i	$0.288188\pi$
$824$	0	0
$825$	6.22125e11	0.0467557
$826$	0	0
$827$	1.01160e13	0.752028	0.376014	−	0.926614i	$-0.377295\pi$
$827$	1.01160e13	0.752028	0.376014	+	0.926614i	$0.377295\pi$
$828$	0	0
$829$	−4.52458e11	−0.0332723	−0.0166362	−	0.999862i	$-0.505296\pi$
$829$	−4.52458e11	−0.0332723	−0.0166362	+	0.999862i	$0.505296\pi$
$830$	0	0
$831$	−7.71525e11	−0.0561236
$832$	0	0
$833$	1.77695e13	1.27871
$834$	0	0
$835$	2.49935e12	0.177926
$836$	0	0
$837$	1.30282e13	0.917528
$838$	0	0
$839$	−2.43486e12	−0.169647	−0.0848233	−	0.996396i	$-0.527033\pi$
$839$	−2.43486e12	−0.169647	−0.0848233	+	0.996396i	$0.527033\pi$
$840$	0	0
$841$	−1.21725e13	−0.839067
$842$	0	0
$843$	7.02081e11	0.0478810
$844$	0	0
$845$	−4.17499e11	−0.0281709
$846$	0	0
$847$	6.68743e11	0.0446461
$848$	0	0
$849$	−4.43756e12	−0.293129
$850$	0	0
$851$	3.81241e11	0.0249182
$852$	0	0
$853$	6.22737e12	0.402748	0.201374	−	0.979514i	$-0.435459\pi$
$853$	6.22737e12	0.402748	0.201374	+	0.979514i	$0.435459\pi$
$854$	0	0
$855$	3.88034e12	0.248326
$856$	0	0
$857$	−1.37768e13	−0.872439	−0.436220	−	0.899840i	$-0.643683\pi$
$857$	−1.37768e13	−0.872439	−0.436220	+	0.899840i	$0.643683\pi$
$858$	0	0
$859$	−5.05248e12	−0.316618	−0.158309	−	0.987390i	$-0.550604\pi$
$859$	−5.05248e12	−0.316618	−0.158309	+	0.987390i	$0.550604\pi$
$860$	0	0
$861$	−2.02731e11	−0.0125720
$862$	0	0
$863$	8.53324e12	0.523679	0.261840	−	0.965111i	$-0.415671\pi$
$863$	8.53324e12	0.523679	0.261840	+	0.965111i	$0.415671\pi$
$864$	0	0
$865$	6.51946e12	0.395949
$866$	0	0
$867$	−3.74705e12	−0.225218
$868$	0	0
$869$	3.03350e12	0.180449
$870$	0	0
$871$	−1.93006e13	−1.13629
$872$	0	0
$873$	1.90248e11	0.0110855
$874$	0	0
$875$	−1.29883e11	−0.00749058
$876$	0	0
$877$	−2.54542e13	−1.45298	−0.726491	−	0.687175i	$-0.758850\pi$
$877$	−2.54542e13	−1.45298	−0.726491	+	0.687175i	$0.758850\pi$
$878$	0	0
$879$	−7.14027e12	−0.403427
$880$	0	0
$881$	9.53558e12	0.533280	0.266640	−	0.963796i	$-0.414087\pi$
$881$	9.53558e12	0.533280	0.266640	+	0.963796i	$0.414087\pi$
$882$	0	0
$883$	−2.59298e13	−1.43541	−0.717704	−	0.696349i	$-0.754807\pi$
$883$	−2.59298e13	−1.43541	−0.717704	+	0.696349i	$0.754807\pi$
$884$	0	0
$885$	−3.61876e12	−0.198296
$886$	0	0
$887$	−1.76744e13	−0.958711	−0.479356	−	0.877621i	$-0.659130\pi$
$887$	−1.76744e13	−0.958711	−0.479356	+	0.877621i	$0.659130\pi$
$888$	0	0
$889$	2.86857e12	0.154031
$890$	0	0
$891$	−8.51664e12	−0.452709
$892$	0	0
$893$	−5.73721e12	−0.301904
$894$	0	0
$895$	2.41410e12	0.125763
$896$	0	0
$897$	−6.84007e11	−0.0352772
$898$	0	0
$899$	1.11899e13	0.571359
$900$	0	0
$901$	3.89451e13	1.96875
$902$	0	0
$903$	−5.40713e11	−0.0270627
$904$	0	0
$905$	−1.25621e13	−0.622506
$906$	0	0
$907$	1.33097e13	0.653035	0.326517	−	0.945191i	$-0.394125\pi$
$907$	1.33097e13	0.653035	0.326517	+	0.945191i	$0.394125\pi$
$908$	0	0
$909$	−1.91104e13	−0.928396
$910$	0	0
$911$	−1.58647e13	−0.763133	−0.381566	−	0.924341i	$-0.624615\pi$
$911$	−1.58647e13	−0.763133	−0.381566	+	0.924341i	$0.624615\pi$
$912$	0	0
$913$	2.16545e13	1.03141
$914$	0	0
$915$	−2.80730e12	−0.132402
$916$	0	0
$917$	1.41854e12	0.0662492
$918$	0	0
$919$	−2.95447e13	−1.36634	−0.683170	−	0.730259i	$-0.739399\pi$
$919$	−2.95447e13	−1.36634	−0.683170	+	0.730259i	$0.739399\pi$
$920$	0	0
$921$	−1.17688e13	−0.538971
$922$	0	0
$923$	−4.16435e13	−1.88860
$924$	0	0
$925$	−1.04174e12	−0.0467864
$926$	0	0
$927$	2.36142e13	1.05030
$928$	0	0
$929$	2.97113e13	1.30873	0.654366	−	0.756178i	$-0.272936\pi$
$929$	2.97113e13	1.30873	0.654366	+	0.756178i	$0.272936\pi$
$930$	0	0
$931$	1.43150e13	0.624478
$932$	0	0
$933$	4.46207e12	0.192783
$934$	0	0
$935$	9.19613e12	0.393507
$936$	0	0
$937$	8.00724e12	0.339355	0.169678	−	0.985500i	$-0.445727\pi$
$937$	8.00724e12	0.339355	0.169678	+	0.985500i	$0.445727\pi$
$938$	0	0
$939$	−1.17489e13	−0.493174
$940$	0	0
$941$	3.12033e12	0.129732	0.0648660	−	0.997894i	$-0.479338\pi$
$941$	3.12033e12	0.129732	0.0648660	+	0.997894i	$0.479338\pi$
$942$	0	0
$943$	1.13493e12	0.0467376
$944$	0	0
$945$	−5.91510e11	−0.0241278
$946$	0	0
$947$	1.95498e12	0.0789891	0.0394945	−	0.999220i	$-0.487425\pi$
$947$	1.95498e12	0.0789891	0.0394945	+	0.999220i	$0.487425\pi$
$948$	0	0
$949$	4.48941e13	1.79676
$950$	0	0
$951$	1.34489e13	0.533180
$952$	0	0
$953$	−1.30190e13	−0.511280	−0.255640	−	0.966772i	$-0.582286\pi$
$953$	−1.30190e13	−0.511280	−0.255640	+	0.966772i	$0.582286\pi$
$954$	0	0
$955$	−3.80789e12	−0.148139
$956$	0	0
$957$	2.43350e12	0.0937837
$958$	0	0
$959$	1.09373e12	0.0417567
$960$	0	0
$961$	2.71928e13	1.02849
$962$	0	0
$963$	−2.71091e13	−1.01577
$964$	0	0
$965$	−1.08981e13	−0.404555
$966$	0	0
$967$	3.66621e13	1.34834	0.674169	−	0.738577i	$-0.264502\pi$
$967$	3.66621e13	1.34834	0.674169	+	0.738577i	$0.264502\pi$
$968$	0	0
$969$	−7.60422e12	−0.277075
$970$	0	0
$971$	3.36686e13	1.21545	0.607727	−	0.794146i	$-0.292082\pi$
$971$	3.36686e13	1.21545	0.607727	+	0.794146i	$0.292082\pi$
$972$	0	0
$973$	3.59726e12	0.128666
$974$	0	0
$975$	1.86904e12	0.0662365
$976$	0	0
$977$	−1.32241e13	−0.464344	−0.232172	−	0.972675i	$-0.574583\pi$
$977$	−1.32241e13	−0.464344	−0.232172	+	0.972675i	$0.574583\pi$
$978$	0	0
$979$	5.64256e12	0.196315
$980$	0	0
$981$	−2.70444e12	−0.0932324
$982$	0	0
$983$	7.76523e12	0.265255	0.132627	−	0.991166i	$-0.457659\pi$
$983$	7.76523e12	0.265255	0.132627	+	0.991166i	$0.457659\pi$
$984$	0	0
$985$	−1.10114e13	−0.372718
$986$	0	0
$987$	4.10099e11	0.0137550
$988$	0	0
$989$	3.02702e12	0.100608
$990$	0	0
$991$	7.58853e12	0.249935	0.124967	−	0.992161i	$-0.460117\pi$
$991$	7.58853e12	0.249935	0.124967	+	0.992161i	$0.460117\pi$
$992$	0	0
$993$	2.12740e12	0.0694348
$994$	0	0
$995$	−1.70074e13	−0.550090
$996$	0	0
$997$	1.59077e13	0.509894	0.254947	−	0.966955i	$-0.417942\pi$
$997$	1.59077e13	0.509894	0.254947	+	0.966955i	$0.417942\pi$
$998$	0	0
$999$	−4.74425e12	−0.150703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.10.a.a.1.1	✓	1
3.2	odd	2		180.10.a.b.1.1		1
4.3	odd	2		80.10.a.c.1.1		1
5.2	odd	4		100.10.c.b.49.2		2
5.3	odd	4		100.10.c.b.49.1		2
5.4	even	2		100.10.a.b.1.1		1
8.3	odd	2		320.10.a.d.1.1		1
8.5	even	2		320.10.a.g.1.1		1
20.3	even	4		400.10.c.h.49.2		2
20.7	even	4		400.10.c.h.49.1		2
20.19	odd	2		400.10.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.a.a.1.1	✓	1	1.1	even	1	trivial
80.10.a.c.1.1		1	4.3	odd	2
100.10.a.b.1.1		1	5.4	even	2
100.10.c.b.49.1		2	5.3	odd	4
100.10.c.b.49.2		2	5.2	odd	4
180.10.a.b.1.1		1	3.2	odd	2
320.10.a.d.1.1		1	8.3	odd	2
320.10.a.g.1.1		1	8.5	even	2
400.10.a.e.1.1		1	20.19	odd	2
400.10.c.h.49.1		2	20.7	even	4
400.10.c.h.49.2		2	20.3	even	4

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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	20.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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