LMFDB - Embedded newform 8020.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0559400 q^{3} -1.00000 q^{5} -0.736794 q^{7} -2.99687 q^{9} +O(q^{10})$	$q+0.0559400 q^{3} -1.00000 q^{5} -0.736794 q^{7} -2.99687 q^{9} -1.01664 q^{11} +5.70877 q^{13} -0.0559400 q^{15} +6.01629 q^{17} +0.103827 q^{19} -0.0412163 q^{21} -1.19159 q^{23} +1.00000 q^{25} -0.335465 q^{27} +8.14703 q^{29} -7.50989 q^{31} -0.0568706 q^{33} +0.736794 q^{35} +3.01658 q^{37} +0.319349 q^{39} -8.80672 q^{41} +3.83096 q^{43} +2.99687 q^{45} -3.92130 q^{47} -6.45713 q^{49} +0.336552 q^{51} +7.18506 q^{53} +1.01664 q^{55} +0.00580810 q^{57} -3.48206 q^{59} +1.70293 q^{61} +2.20808 q^{63} -5.70877 q^{65} -2.08033 q^{67} -0.0666576 q^{69} -13.5612 q^{71} +2.46931 q^{73} +0.0559400 q^{75} +0.749050 q^{77} -4.54878 q^{79} +8.97185 q^{81} +1.71175 q^{83} -6.01629 q^{85} +0.455745 q^{87} -2.35178 q^{89} -4.20619 q^{91} -0.420103 q^{93} -0.103827 q^{95} -1.04505 q^{97} +3.04672 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.0559400	0.0322970	0.0161485	−	0.999870i	$-0.494860\pi$
$3$	0.0559400	0.0322970	0.0161485	+	0.999870i	$0.494860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.736794	−0.278482	−0.139241	−	0.990259i	$-0.544466\pi$
$7$	−0.736794	−0.278482	−0.139241	+	0.990259i	$0.544466\pi$
$8$	0	0
$9$	−2.99687	−0.998957
$10$	0	0
$11$	−1.01664	−0.306527	−0.153263	−	0.988185i	$-0.548978\pi$
$11$	−1.01664	−0.306527	−0.153263	+	0.988185i	$0.548978\pi$
$12$	0	0
$13$	5.70877	1.58333	0.791664	−	0.610957i	$-0.209215\pi$
$13$	5.70877	1.58333	0.791664	+	0.610957i	$0.209215\pi$
$14$	0	0
$15$	−0.0559400	−0.0144436
$16$	0	0
$17$	6.01629	1.45917	0.729583	−	0.683892i	$-0.239714\pi$
$17$	6.01629	1.45917	0.729583	+	0.683892i	$0.239714\pi$
$18$	0	0
$19$	0.103827	0.0238196	0.0119098	−	0.999929i	$-0.496209\pi$
$19$	0.103827	0.0238196	0.0119098	+	0.999929i	$0.496209\pi$
$20$	0	0
$21$	−0.0412163	−0.00899412
$22$	0	0
$23$	−1.19159	−0.248464	−0.124232	−	0.992253i	$-0.539647\pi$
$23$	−1.19159	−0.248464	−0.124232	+	0.992253i	$0.539647\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.335465	−0.0645603
$28$	0	0
$29$	8.14703	1.51287	0.756433	−	0.654071i	$-0.226940\pi$
$29$	8.14703	1.51287	0.756433	+	0.654071i	$0.226940\pi$
$30$	0	0
$31$	−7.50989	−1.34882	−0.674408	−	0.738359i	$-0.735601\pi$
$31$	−7.50989	−1.34882	−0.674408	+	0.738359i	$0.735601\pi$
$32$	0	0
$33$	−0.0568706	−0.00989990
$34$	0	0
$35$	0.736794	0.124541
$36$	0	0
$37$	3.01658	0.495922	0.247961	−	0.968770i	$-0.420240\pi$
$37$	3.01658	0.495922	0.247961	+	0.968770i	$0.420240\pi$
$38$	0	0
$39$	0.319349	0.0511367
$40$	0	0
$41$	−8.80672	−1.37538	−0.687689	−	0.726005i	$-0.741375\pi$
$41$	−8.80672	−1.37538	−0.687689	+	0.726005i	$0.741375\pi$
$42$	0	0
$43$	3.83096	0.584217	0.292108	−	0.956385i	$-0.405643\pi$
$43$	3.83096	0.584217	0.292108	+	0.956385i	$0.405643\pi$
$44$	0	0
$45$	2.99687	0.446747
$46$	0	0
$47$	−3.92130	−0.571980	−0.285990	−	0.958233i	$-0.592322\pi$
$47$	−3.92130	−0.571980	−0.285990	+	0.958233i	$0.592322\pi$
$48$	0	0
$49$	−6.45713	−0.922448
$50$	0	0
$51$	0.336552	0.0471267
$52$	0	0
$53$	7.18506	0.986944	0.493472	−	0.869762i	$-0.335728\pi$
$53$	7.18506	0.986944	0.493472	+	0.869762i	$0.335728\pi$
$54$	0	0
$55$	1.01664	0.137083
$56$	0	0
$57$	0.00580810	0.000769302 0
$58$	0	0
$59$	−3.48206	−0.453326	−0.226663	−	0.973973i	$-0.572782\pi$
$59$	−3.48206	−0.453326	−0.226663	+	0.973973i	$0.572782\pi$
$60$	0	0
$61$	1.70293	0.218038	0.109019	−	0.994040i	$-0.465229\pi$
$61$	1.70293	0.218038	0.109019	+	0.994040i	$0.465229\pi$
$62$	0	0
$63$	2.20808	0.278191
$64$	0	0
$65$	−5.70877	−0.708086
$66$	0	0
$67$	−2.08033	−0.254152	−0.127076	−	0.991893i	$-0.540559\pi$
$67$	−2.08033	−0.254152	−0.127076	+	0.991893i	$0.540559\pi$
$68$	0	0
$69$	−0.0666576	−0.00802463
$70$	0	0
$71$	−13.5612	−1.60942	−0.804711	−	0.593667i	$-0.797680\pi$
$71$	−13.5612	−1.60942	−0.804711	+	0.593667i	$0.797680\pi$
$72$	0	0
$73$	2.46931	0.289011	0.144506	−	0.989504i	$-0.453841\pi$
$73$	2.46931	0.289011	0.144506	+	0.989504i	$0.453841\pi$
$74$	0	0
$75$	0.0559400	0.00645940
$76$	0	0
$77$	0.749050	0.0853622
$78$	0	0
$79$	−4.54878	−0.511778	−0.255889	−	0.966706i	$-0.582368\pi$
$79$	−4.54878	−0.511778	−0.255889	+	0.966706i	$0.582368\pi$
$80$	0	0
$81$	8.97185	0.996872
$82$	0	0
$83$	1.71175	0.187889	0.0939445	−	0.995577i	$-0.470052\pi$
$83$	1.71175	0.187889	0.0939445	+	0.995577i	$0.470052\pi$
$84$	0	0
$85$	−6.01629	−0.652559
$86$	0	0
$87$	0.455745	0.0488610
$88$	0	0
$89$	−2.35178	−0.249288	−0.124644	−	0.992202i	$-0.539779\pi$
$89$	−2.35178	−0.249288	−0.124644	+	0.992202i	$0.539779\pi$
$90$	0	0
$91$	−4.20619	−0.440928
$92$	0	0
$93$	−0.420103	−0.0435627
$94$	0	0
$95$	−0.103827	−0.0106525
$96$	0	0
$97$	−1.04505	−0.106109	−0.0530545	−	0.998592i	$-0.516896\pi$
$97$	−1.04505	−0.106109	−0.0530545	+	0.998592i	$0.516896\pi$
$98$	0	0
$99$	3.04672	0.306207
$100$	0	0
$101$	4.52263	0.450018	0.225009	−	0.974357i	$-0.427759\pi$
$101$	4.52263	0.450018	0.225009	+	0.974357i	$0.427759\pi$
$102$	0	0
$103$	−16.7773	−1.65312	−0.826560	−	0.562848i	$-0.809706\pi$
$103$	−16.7773	−1.65312	−0.826560	+	0.562848i	$0.809706\pi$
$104$	0	0
$105$	0.0412163	0.00402229
$106$	0	0
$107$	7.81776	0.755772	0.377886	−	0.925852i	$-0.376651\pi$
$107$	7.81776	0.755772	0.377886	+	0.925852i	$0.376651\pi$
$108$	0	0
$109$	17.2065	1.64808	0.824042	−	0.566528i	$-0.191714\pi$
$109$	17.2065	1.64808	0.824042	+	0.566528i	$0.191714\pi$
$110$	0	0
$111$	0.168747	0.0160168
$112$	0	0
$113$	11.2847	1.06157	0.530787	−	0.847506i	$-0.321897\pi$
$113$	11.2847	1.06157	0.530787	+	0.847506i	$0.321897\pi$
$114$	0	0
$115$	1.19159	0.111116
$116$	0	0
$117$	−17.1084	−1.58168
$118$	0	0
$119$	−4.43277	−0.406351
$120$	0	0
$121$	−9.96645	−0.906041
$122$	0	0
$123$	−0.492648	−0.0444206
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	17.7658	1.57646	0.788228	−	0.615383i	$-0.210999\pi$
$127$	17.7658	1.57646	0.788228	+	0.615383i	$0.210999\pi$
$128$	0	0
$129$	0.214304	0.0188684
$130$	0	0
$131$	10.2815	0.898295	0.449147	−	0.893458i	$-0.351728\pi$
$131$	10.2815	0.898295	0.449147	+	0.893458i	$0.351728\pi$
$132$	0	0
$133$	−0.0764994	−0.00663334
$134$	0	0
$135$	0.335465	0.0288722
$136$	0	0
$137$	21.8045	1.86288	0.931442	−	0.363891i	$-0.118552\pi$
$137$	21.8045	1.86288	0.931442	+	0.363891i	$0.118552\pi$
$138$	0	0
$139$	−17.3451	−1.47119	−0.735597	−	0.677420i	$-0.763098\pi$
$139$	−17.3451	−1.47119	−0.735597	+	0.677420i	$0.763098\pi$
$140$	0	0
$141$	−0.219357	−0.0184732
$142$	0	0
$143$	−5.80374	−0.485333
$144$	0	0
$145$	−8.14703	−0.676574
$146$	0	0
$147$	−0.361212	−0.0297923
$148$	0	0
$149$	−23.3219	−1.91060	−0.955300	−	0.295637i	$-0.904468\pi$
$149$	−23.3219	−1.91060	−0.955300	+	0.295637i	$0.904468\pi$
$150$	0	0
$151$	18.1396	1.47618	0.738090	−	0.674702i	$-0.235728\pi$
$151$	18.1396	1.47618	0.738090	+	0.674702i	$0.235728\pi$
$152$	0	0
$153$	−18.0301	−1.45764
$154$	0	0
$155$	7.50989	0.603209
$156$	0	0
$157$	8.94620	0.713985	0.356992	−	0.934107i	$-0.383802\pi$
$157$	8.94620	0.713985	0.356992	+	0.934107i	$0.383802\pi$
$158$	0	0
$159$	0.401932	0.0318753
$160$	0	0
$161$	0.877957	0.0691927
$162$	0	0
$163$	20.1606	1.57910	0.789551	−	0.613684i	$-0.210313\pi$
$163$	20.1606	1.57910	0.789551	+	0.613684i	$0.210313\pi$
$164$	0	0
$165$	0.0568706	0.00442737
$166$	0	0
$167$	−12.3962	−0.959245	−0.479623	−	0.877475i	$-0.659226\pi$
$167$	−12.3962	−0.959245	−0.479623	+	0.877475i	$0.659226\pi$
$168$	0	0
$169$	19.5901	1.50693
$170$	0	0
$171$	−0.311157	−0.0237948
$172$	0	0
$173$	9.96530	0.757647	0.378824	−	0.925469i	$-0.376329\pi$
$173$	9.96530	0.757647	0.378824	+	0.925469i	$0.376329\pi$
$174$	0	0
$175$	−0.736794	−0.0556964
$176$	0	0
$177$	−0.194786	−0.0146410
$178$	0	0
$179$	19.7309	1.47476	0.737379	−	0.675479i	$-0.236063\pi$
$179$	19.7309	1.47476	0.737379	+	0.675479i	$0.236063\pi$
$180$	0	0
$181$	3.91272	0.290831	0.145415	−	0.989371i	$-0.453548\pi$
$181$	3.91272	0.290831	0.145415	+	0.989371i	$0.453548\pi$
$182$	0	0
$183$	0.0952619	0.00704196
$184$	0	0
$185$	−3.01658	−0.221783
$186$	0	0
$187$	−6.11638	−0.447274
$188$	0	0
$189$	0.247169	0.0179789
$190$	0	0
$191$	−3.35144	−0.242501	−0.121251	−	0.992622i	$-0.538690\pi$
$191$	−3.35144	−0.242501	−0.121251	+	0.992622i	$0.538690\pi$
$192$	0	0
$193$	11.9446	0.859788	0.429894	−	0.902879i	$-0.358551\pi$
$193$	11.9446	0.859788	0.429894	+	0.902879i	$0.358551\pi$
$194$	0	0
$195$	−0.319349	−0.0228690
$196$	0	0
$197$	13.5167	0.963025	0.481512	−	0.876439i	$-0.340088\pi$
$197$	13.5167	0.963025	0.481512	+	0.876439i	$0.340088\pi$
$198$	0	0
$199$	−9.56861	−0.678301	−0.339150	−	0.940732i	$-0.610140\pi$
$199$	−9.56861	−0.678301	−0.339150	+	0.940732i	$0.610140\pi$
$200$	0	0
$201$	−0.116373	−0.00820835
$202$	0	0
$203$	−6.00268	−0.421306
$204$	0	0
$205$	8.80672	0.615088
$206$	0	0
$207$	3.57104	0.248205
$208$	0	0
$209$	−0.105555	−0.00730136
$210$	0	0
$211$	9.44074	0.649927	0.324964	−	0.945727i	$-0.394648\pi$
$211$	9.44074	0.649927	0.324964	+	0.945727i	$0.394648\pi$
$212$	0	0
$213$	−0.758616	−0.0519795
$214$	0	0
$215$	−3.83096	−0.261270
$216$	0	0
$217$	5.53324	0.375621
$218$	0	0
$219$	0.138133	0.00933418
$220$	0	0
$221$	34.3456	2.31034
$222$	0	0
$223$	−21.6260	−1.44818	−0.724091	−	0.689704i	$-0.757740\pi$
$223$	−21.6260	−1.44818	−0.724091	+	0.689704i	$0.757740\pi$
$224$	0	0
$225$	−2.99687	−0.199791
$226$	0	0
$227$	29.1353	1.93378	0.966890	−	0.255193i	$-0.0821390\pi$
$227$	29.1353	1.93378	0.966890	+	0.255193i	$0.0821390\pi$
$228$	0	0
$229$	5.71119	0.377406	0.188703	−	0.982034i	$-0.439572\pi$
$229$	5.71119	0.377406	0.188703	+	0.982034i	$0.439572\pi$
$230$	0	0
$231$	0.0419019	0.00275694
$232$	0	0
$233$	11.1998	0.733720	0.366860	−	0.930276i	$-0.380433\pi$
$233$	11.1998	0.733720	0.366860	+	0.930276i	$0.380433\pi$
$234$	0	0
$235$	3.92130	0.255797
$236$	0	0
$237$	−0.254459	−0.0165289
$238$	0	0
$239$	18.1608	1.17473	0.587363	−	0.809324i	$-0.300166\pi$
$239$	18.1608	1.17473	0.587363	+	0.809324i	$0.300166\pi$
$240$	0	0
$241$	10.8460	0.698655	0.349328	−	0.937001i	$-0.386410\pi$
$241$	10.8460	0.698655	0.349328	+	0.937001i	$0.386410\pi$
$242$	0	0
$243$	1.50828	0.0967562
$244$	0	0
$245$	6.45713	0.412531
$246$	0	0
$247$	0.592727	0.0377143
$248$	0	0
$249$	0.0957554	0.00606825
$250$	0	0
$251$	20.5014	1.29403	0.647017	−	0.762476i	$-0.276016\pi$
$251$	20.5014	1.29403	0.647017	+	0.762476i	$0.276016\pi$
$252$	0	0
$253$	1.21141	0.0761609
$254$	0	0
$255$	−0.336552	−0.0210757
$256$	0	0
$257$	14.4795	0.903207	0.451604	−	0.892219i	$-0.350852\pi$
$257$	14.4795	0.903207	0.451604	+	0.892219i	$0.350852\pi$
$258$	0	0
$259$	−2.22259	−0.138105
$260$	0	0
$261$	−24.4156	−1.51129
$262$	0	0
$263$	−15.4668	−0.953723	−0.476862	−	0.878978i	$-0.658226\pi$
$263$	−15.4668	−0.953723	−0.476862	+	0.878978i	$0.658226\pi$
$264$	0	0
$265$	−7.18506	−0.441375
$266$	0	0
$267$	−0.131558	−0.00805124
$268$	0	0
$269$	−6.47071	−0.394526	−0.197263	−	0.980351i	$-0.563205\pi$
$269$	−6.47071	−0.394526	−0.197263	+	0.980351i	$0.563205\pi$
$270$	0	0
$271$	8.24884	0.501081	0.250541	−	0.968106i	$-0.419392\pi$
$271$	8.24884	0.501081	0.250541	+	0.968106i	$0.419392\pi$
$272$	0	0
$273$	−0.235294	−0.0142406
$274$	0	0
$275$	−1.01664	−0.0613054
$276$	0	0
$277$	16.0199	0.962544	0.481272	−	0.876571i	$-0.340175\pi$
$277$	16.0199	0.962544	0.481272	+	0.876571i	$0.340175\pi$
$278$	0	0
$279$	22.5062	1.34741
$280$	0	0
$281$	−26.0162	−1.55200	−0.775998	−	0.630735i	$-0.782753\pi$
$281$	−26.0162	−1.55200	−0.775998	+	0.630735i	$0.782753\pi$
$282$	0	0
$283$	13.6094	0.808995	0.404497	−	0.914539i	$-0.367446\pi$
$283$	13.6094	0.808995	0.404497	+	0.914539i	$0.367446\pi$
$284$	0	0
$285$	−0.00580810	−0.000344042 0
$286$	0	0
$287$	6.48874	0.383018
$288$	0	0
$289$	19.1958	1.12916
$290$	0	0
$291$	−0.0584603	−0.00342700
$292$	0	0
$293$	−12.3136	−0.719371	−0.359685	−	0.933074i	$-0.617116\pi$
$293$	−12.3136	−0.719371	−0.359685	+	0.933074i	$0.617116\pi$
$294$	0	0
$295$	3.48206	0.202733
$296$	0	0
$297$	0.341045	0.0197895
$298$	0	0
$299$	−6.80252	−0.393400
$300$	0	0
$301$	−2.82263	−0.162694
$302$	0	0
$303$	0.252996	0.0145342
$304$	0	0
$305$	−1.70293	−0.0975094
$306$	0	0
$307$	32.9682	1.88160	0.940798	−	0.338966i	$-0.110077\pi$
$307$	32.9682	1.88160	0.940798	+	0.338966i	$0.110077\pi$
$308$	0	0
$309$	−0.938525	−0.0533908
$310$	0	0
$311$	23.5413	1.33490	0.667452	−	0.744653i	$-0.267385\pi$
$311$	23.5413	1.33490	0.667452	+	0.744653i	$0.267385\pi$
$312$	0	0
$313$	−17.2127	−0.972917	−0.486459	−	0.873704i	$-0.661712\pi$
$313$	−17.2127	−0.972917	−0.486459	+	0.873704i	$0.661712\pi$
$314$	0	0
$315$	−2.20808	−0.124411
$316$	0	0
$317$	14.4649	0.812429	0.406214	−	0.913778i	$-0.366849\pi$
$317$	14.4649	0.812429	0.406214	+	0.913778i	$0.366849\pi$
$318$	0	0
$319$	−8.28256	−0.463734
$320$	0	0
$321$	0.437326	0.0244091
$322$	0	0
$323$	0.624656	0.0347568
$324$	0	0
$325$	5.70877	0.316666
$326$	0	0
$327$	0.962532	0.0532282
$328$	0	0
$329$	2.88919	0.159286
$330$	0	0
$331$	6.01677	0.330711	0.165356	−	0.986234i	$-0.447123\pi$
$331$	6.01677	0.330711	0.165356	+	0.986234i	$0.447123\pi$
$332$	0	0
$333$	−9.04029	−0.495405
$334$	0	0
$335$	2.08033	0.113660
$336$	0	0
$337$	−32.7882	−1.78609	−0.893043	−	0.449972i	$-0.851434\pi$
$337$	−32.7882	−1.78609	−0.893043	+	0.449972i	$0.851434\pi$
$338$	0	0
$339$	0.631265	0.0342856
$340$	0	0
$341$	7.63482	0.413448
$342$	0	0
$343$	9.91513	0.535367
$344$	0	0
$345$	0.0666576	0.00358872
$346$	0	0
$347$	9.96304	0.534844	0.267422	−	0.963580i	$-0.413828\pi$
$347$	9.96304	0.534844	0.267422	+	0.963580i	$0.413828\pi$
$348$	0	0
$349$	−7.01301	−0.375398	−0.187699	−	0.982227i	$-0.560103\pi$
$349$	−7.01301	−0.375398	−0.187699	+	0.982227i	$0.560103\pi$
$350$	0	0
$351$	−1.91509	−0.102220
$352$	0	0
$353$	−24.2147	−1.28882	−0.644408	−	0.764681i	$-0.722896\pi$
$353$	−24.2147	−1.28882	−0.644408	+	0.764681i	$0.722896\pi$
$354$	0	0
$355$	13.5612	0.719755
$356$	0	0
$357$	−0.247969	−0.0131239
$358$	0	0
$359$	21.1319	1.11530	0.557649	−	0.830077i	$-0.311704\pi$
$359$	21.1319	1.11530	0.557649	+	0.830077i	$0.311704\pi$
$360$	0	0
$361$	−18.9892	−0.999433
$362$	0	0
$363$	−0.557524	−0.0292624
$364$	0	0
$365$	−2.46931	−0.129250
$366$	0	0
$367$	23.6940	1.23682	0.618408	−	0.785857i	$-0.287778\pi$
$367$	23.6940	1.23682	0.618408	+	0.785857i	$0.287778\pi$
$368$	0	0
$369$	26.3926	1.37394
$370$	0	0
$371$	−5.29391	−0.274846
$372$	0	0
$373$	22.8102	1.18107	0.590535	−	0.807012i	$-0.298917\pi$
$373$	22.8102	1.18107	0.590535	+	0.807012i	$0.298917\pi$
$374$	0	0
$375$	−0.0559400	−0.00288873
$376$	0	0
$377$	46.5095	2.39536
$378$	0	0
$379$	18.6051	0.955680	0.477840	−	0.878447i	$-0.341420\pi$
$379$	18.6051	0.955680	0.477840	+	0.878447i	$0.341420\pi$
$380$	0	0
$381$	0.993817	0.0509148
$382$	0	0
$383$	−6.10391	−0.311895	−0.155948	−	0.987765i	$-0.549843\pi$
$383$	−6.10391	−0.311895	−0.155948	+	0.987765i	$0.549843\pi$
$384$	0	0
$385$	−0.749050	−0.0381751
$386$	0	0
$387$	−11.4809	−0.583607
$388$	0	0
$389$	−10.3571	−0.525127	−0.262564	−	0.964915i	$-0.584568\pi$
$389$	−10.3571	−0.525127	−0.262564	+	0.964915i	$0.584568\pi$
$390$	0	0
$391$	−7.16896	−0.362550
$392$	0	0
$393$	0.575145	0.0290122
$394$	0	0
$395$	4.54878	0.228874
$396$	0	0
$397$	1.62955	0.0817849	0.0408925	−	0.999164i	$-0.486980\pi$
$397$	1.62955	0.0817849	0.0408925	+	0.999164i	$0.486980\pi$
$398$	0	0
$399$	−0.00427938	−0.000214237 0
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−42.8722	−2.13562
$404$	0	0
$405$	−8.97185	−0.445815
$406$	0	0
$407$	−3.06676	−0.152013
$408$	0	0
$409$	−33.4646	−1.65472	−0.827358	−	0.561675i	$-0.810157\pi$
$409$	−33.4646	−1.65472	−0.827358	+	0.561675i	$0.810157\pi$
$410$	0	0
$411$	1.21974	0.0601655
$412$	0	0
$413$	2.56556	0.126243
$414$	0	0
$415$	−1.71175	−0.0840265
$416$	0	0
$417$	−0.970286	−0.0475151
$418$	0	0
$419$	−31.1323	−1.52091	−0.760455	−	0.649391i	$-0.775024\pi$
$419$	−31.1323	−1.52091	−0.760455	+	0.649391i	$0.775024\pi$
$420$	0	0
$421$	−23.8723	−1.16347	−0.581733	−	0.813380i	$-0.697625\pi$
$421$	−23.8723	−1.16347	−0.581733	+	0.813380i	$0.697625\pi$
$422$	0	0
$423$	11.7516	0.571383
$424$	0	0
$425$	6.01629	0.291833
$426$	0	0
$427$	−1.25471	−0.0607196
$428$	0	0
$429$	−0.324661	−0.0156748
$430$	0	0
$431$	15.0817	0.726459	0.363230	−	0.931700i	$-0.381674\pi$
$431$	15.0817	0.726459	0.363230	+	0.931700i	$0.381674\pi$
$432$	0	0
$433$	2.92172	0.140409	0.0702045	−	0.997533i	$-0.477635\pi$
$433$	2.92172	0.140409	0.0702045	+	0.997533i	$0.477635\pi$
$434$	0	0
$435$	−0.455745	−0.0218513
$436$	0	0
$437$	−0.123720	−0.00591832
$438$	0	0
$439$	7.59620	0.362547	0.181273	−	0.983433i	$-0.441978\pi$
$439$	7.59620	0.362547	0.181273	+	0.983433i	$0.441978\pi$
$440$	0	0
$441$	19.3512	0.921486
$442$	0	0
$443$	−23.1114	−1.09806	−0.549028	−	0.835804i	$-0.685002\pi$
$443$	−23.1114	−1.09806	−0.549028	+	0.835804i	$0.685002\pi$
$444$	0	0
$445$	2.35178	0.111485
$446$	0	0
$447$	−1.30462	−0.0617066
$448$	0	0
$449$	3.83065	0.180780	0.0903898	−	0.995906i	$-0.471189\pi$
$449$	3.83065	0.180780	0.0903898	+	0.995906i	$0.471189\pi$
$450$	0	0
$451$	8.95322	0.421591
$452$	0	0
$453$	1.01473	0.0476762
$454$	0	0
$455$	4.20619	0.197189
$456$	0	0
$457$	26.9786	1.26200	0.631002	−	0.775781i	$-0.282644\pi$
$457$	26.9786	1.26200	0.631002	+	0.775781i	$0.282644\pi$
$458$	0	0
$459$	−2.01826	−0.0942041
$460$	0	0
$461$	−9.80040	−0.456450	−0.228225	−	0.973608i	$-0.573292\pi$
$461$	−9.80040	−0.456450	−0.228225	+	0.973608i	$0.573292\pi$
$462$	0	0
$463$	−3.02387	−0.140531	−0.0702655	−	0.997528i	$-0.522385\pi$
$463$	−3.02387	−0.140531	−0.0702655	+	0.997528i	$0.522385\pi$
$464$	0	0
$465$	0.420103	0.0194818
$466$	0	0
$467$	7.84086	0.362832	0.181416	−	0.983406i	$-0.441932\pi$
$467$	7.84086	0.362832	0.181416	+	0.983406i	$0.441932\pi$
$468$	0	0
$469$	1.53277	0.0707768
$470$	0	0
$471$	0.500451	0.0230595
$472$	0	0
$473$	−3.89469	−0.179078
$474$	0	0
$475$	0.103827	0.00476393
$476$	0	0
$477$	−21.5327	−0.985914
$478$	0	0
$479$	18.4791	0.844330	0.422165	−	0.906519i	$-0.361270\pi$
$479$	18.4791	0.844330	0.422165	+	0.906519i	$0.361270\pi$
$480$	0	0
$481$	17.2209	0.785207
$482$	0	0
$483$	0.0491129	0.00223471
$484$	0	0
$485$	1.04505	0.0474534
$486$	0	0
$487$	−29.3130	−1.32830	−0.664150	−	0.747600i	$-0.731206\pi$
$487$	−29.3130	−1.32830	−0.664150	+	0.747600i	$0.731206\pi$
$488$	0	0
$489$	1.12779	0.0510003
$490$	0	0
$491$	22.6576	1.02252	0.511261	−	0.859425i	$-0.329178\pi$
$491$	22.6576	1.02252	0.511261	+	0.859425i	$0.329178\pi$
$492$	0	0
$493$	49.0149	2.20752
$494$	0	0
$495$	−3.04672	−0.136940
$496$	0	0
$497$	9.99183	0.448195
$498$	0	0
$499$	−36.1842	−1.61983	−0.809915	−	0.586548i	$-0.800487\pi$
$499$	−36.1842	−1.61983	−0.809915	+	0.586548i	$0.800487\pi$
$500$	0	0
$501$	−0.693442	−0.0309807
$502$	0	0
$503$	3.57506	0.159404	0.0797021	−	0.996819i	$-0.474603\pi$
$503$	3.57506	0.159404	0.0797021	+	0.996819i	$0.474603\pi$
$504$	0	0
$505$	−4.52263	−0.201254
$506$	0	0
$507$	1.09587	0.0486692
$508$	0	0
$509$	21.8702	0.969379	0.484689	−	0.874686i	$-0.338933\pi$
$509$	21.8702	0.969379	0.484689	+	0.874686i	$0.338933\pi$
$510$	0	0
$511$	−1.81937	−0.0804843
$512$	0	0
$513$	−0.0348305	−0.00153780
$514$	0	0
$515$	16.7773	0.739298
$516$	0	0
$517$	3.98653	0.175327
$518$	0	0
$519$	0.557459	0.0244697
$520$	0	0
$521$	−18.0355	−0.790148	−0.395074	−	0.918649i	$-0.629281\pi$
$521$	−18.0355	−0.790148	−0.395074	+	0.918649i	$0.629281\pi$
$522$	0	0
$523$	−19.0996	−0.835167	−0.417583	−	0.908639i	$-0.637123\pi$
$523$	−19.0996	−0.835167	−0.417583	+	0.908639i	$0.637123\pi$
$524$	0	0
$525$	−0.0412163	−0.00179882
$526$	0	0
$527$	−45.1817	−1.96815
$528$	0	0
$529$	−21.5801	−0.938266
$530$	0	0
$531$	10.4353	0.452853
$532$	0	0
$533$	−50.2755	−2.17768
$534$	0	0
$535$	−7.81776	−0.337991
$536$	0	0
$537$	1.10375	0.0476303
$538$	0	0
$539$	6.56455	0.282755
$540$	0	0
$541$	3.17926	0.136687	0.0683436	−	0.997662i	$-0.478229\pi$
$541$	3.17926	0.136687	0.0683436	+	0.997662i	$0.478229\pi$
$542$	0	0
$543$	0.218878	0.00939295
$544$	0	0
$545$	−17.2065	−0.737046
$546$	0	0
$547$	−38.3493	−1.63970	−0.819850	−	0.572579i	$-0.805943\pi$
$547$	−38.3493	−1.63970	−0.819850	+	0.572579i	$0.805943\pi$
$548$	0	0
$549$	−5.10346	−0.217810
$550$	0	0
$551$	0.845885	0.0360359
$552$	0	0
$553$	3.35151	0.142521
$554$	0	0
$555$	−0.168747	−0.00716292
$556$	0	0
$557$	−14.3375	−0.607499	−0.303749	−	0.952752i	$-0.598239\pi$
$557$	−14.3375	−0.607499	−0.303749	+	0.952752i	$0.598239\pi$
$558$	0	0
$559$	21.8701	0.925006
$560$	0	0
$561$	−0.342150	−0.0144456
$562$	0	0
$563$	−22.0942	−0.931161	−0.465580	−	0.885006i	$-0.654154\pi$
$563$	−22.0942	−0.931161	−0.465580	+	0.885006i	$0.654154\pi$
$564$	0	0
$565$	−11.2847	−0.474750
$566$	0	0
$567$	−6.61040	−0.277611
$568$	0	0
$569$	24.3343	1.02015	0.510073	−	0.860131i	$-0.329618\pi$
$569$	24.3343	1.02015	0.510073	+	0.860131i	$0.329618\pi$
$570$	0	0
$571$	1.08525	0.0454162	0.0227081	−	0.999742i	$-0.492771\pi$
$571$	1.08525	0.0454162	0.0227081	+	0.999742i	$0.492771\pi$
$572$	0	0
$573$	−0.187479	−0.00783206
$574$	0	0
$575$	−1.19159	−0.0496928
$576$	0	0
$577$	−12.6179	−0.525288	−0.262644	−	0.964893i	$-0.584595\pi$
$577$	−12.6179	−0.525288	−0.262644	+	0.964893i	$0.584595\pi$
$578$	0	0
$579$	0.668178	0.0277685
$580$	0	0
$581$	−1.26121	−0.0523237
$582$	0	0
$583$	−7.30458	−0.302525
$584$	0	0
$585$	17.1084	0.707347
$586$	0	0
$587$	7.70942	0.318202	0.159101	−	0.987262i	$-0.449140\pi$
$587$	7.70942	0.318202	0.159101	+	0.987262i	$0.449140\pi$
$588$	0	0
$589$	−0.779732	−0.0321283
$590$	0	0
$591$	0.756124	0.0311028
$592$	0	0
$593$	31.9697	1.31284	0.656418	−	0.754397i	$-0.272071\pi$
$593$	31.9697	1.31284	0.656418	+	0.754397i	$0.272071\pi$
$594$	0	0
$595$	4.43277	0.181726
$596$	0	0
$597$	−0.535268	−0.0219071
$598$	0	0
$599$	32.9848	1.34772	0.673862	−	0.738857i	$-0.264634\pi$
$599$	32.9848	1.34772	0.673862	+	0.738857i	$0.264634\pi$
$600$	0	0
$601$	32.0845	1.30875	0.654376	−	0.756169i	$-0.272931\pi$
$601$	32.0845	1.30875	0.654376	+	0.756169i	$0.272931\pi$
$602$	0	0
$603$	6.23447	0.253887
$604$	0	0
$605$	9.96645	0.405194
$606$	0	0
$607$	−20.6391	−0.837717	−0.418858	−	0.908052i	$-0.637570\pi$
$607$	−20.6391	−0.837717	−0.418858	+	0.908052i	$0.637570\pi$
$608$	0	0
$609$	−0.335790	−0.0136069
$610$	0	0
$611$	−22.3858	−0.905632
$612$	0	0
$613$	−38.3962	−1.55081	−0.775403	−	0.631466i	$-0.782453\pi$
$613$	−38.3962	−1.55081	−0.775403	+	0.631466i	$0.782453\pi$
$614$	0	0
$615$	0.492648	0.0198655
$616$	0	0
$617$	−9.14875	−0.368315	−0.184157	−	0.982897i	$-0.558956\pi$
$617$	−9.14875	−0.368315	−0.184157	+	0.982897i	$0.558956\pi$
$618$	0	0
$619$	−31.3787	−1.26122	−0.630608	−	0.776102i	$-0.717194\pi$
$619$	−31.3787	−1.26122	−0.630608	+	0.776102i	$0.717194\pi$
$620$	0	0
$621$	0.399737	0.0160409
$622$	0	0
$623$	1.73277	0.0694221
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.00590472	−0.000235812 0
$628$	0	0
$629$	18.1486	0.723632
$630$	0	0
$631$	4.69929	0.187076	0.0935379	−	0.995616i	$-0.470182\pi$
$631$	4.69929	0.187076	0.0935379	+	0.995616i	$0.470182\pi$
$632$	0	0
$633$	0.528115	0.0209907
$634$	0	0
$635$	−17.7658	−0.705013
$636$	0	0
$637$	−36.8623	−1.46054
$638$	0	0
$639$	40.6413	1.60774
$640$	0	0
$641$	44.2553	1.74798	0.873990	−	0.485944i	$-0.161524\pi$
$641$	44.2553	1.74798	0.873990	+	0.485944i	$0.161524\pi$
$642$	0	0
$643$	6.27061	0.247289	0.123644	−	0.992327i	$-0.460542\pi$
$643$	6.27061	0.247289	0.123644	+	0.992327i	$0.460542\pi$
$644$	0	0
$645$	−0.214304	−0.00843822
$646$	0	0
$647$	10.3866	0.408339	0.204170	−	0.978936i	$-0.434551\pi$
$647$	10.3866	0.408339	0.204170	+	0.978936i	$0.434551\pi$
$648$	0	0
$649$	3.53998	0.138957
$650$	0	0
$651$	0.309530	0.0121314
$652$	0	0
$653$	43.0204	1.68352	0.841759	−	0.539853i	$-0.181520\pi$
$653$	43.0204	1.68352	0.841759	+	0.539853i	$0.181520\pi$
$654$	0	0
$655$	−10.2815	−0.401730
$656$	0	0
$657$	−7.40021	−0.288710
$658$	0	0
$659$	−11.3272	−0.441244	−0.220622	−	0.975359i	$-0.570809\pi$
$659$	−11.3272	−0.441244	−0.220622	+	0.975359i	$0.570809\pi$
$660$	0	0
$661$	−10.4963	−0.408259	−0.204129	−	0.978944i	$-0.565436\pi$
$661$	−10.4963	−0.408259	−0.204129	+	0.978944i	$0.565436\pi$
$662$	0	0
$663$	1.92130	0.0746169
$664$	0	0
$665$	0.0764994	0.00296652
$666$	0	0
$667$	−9.70793	−0.375892
$668$	0	0
$669$	−1.20976	−0.0467719
$670$	0	0
$671$	−1.73126	−0.0668344
$672$	0	0
$673$	−32.1530	−1.23941	−0.619703	−	0.784836i	$-0.712747\pi$
$673$	−32.1530	−1.23941	−0.619703	+	0.784836i	$0.712747\pi$
$674$	0	0
$675$	−0.335465	−0.0129121
$676$	0	0
$677$	24.4900	0.941228	0.470614	−	0.882339i	$-0.344032\pi$
$677$	24.4900	0.941228	0.470614	+	0.882339i	$0.344032\pi$
$678$	0	0
$679$	0.769988	0.0295494
$680$	0	0
$681$	1.62983	0.0624553
$682$	0	0
$683$	−5.75146	−0.220074	−0.110037	−	0.993928i	$-0.535097\pi$
$683$	−5.75146	−0.220074	−0.110037	+	0.993928i	$0.535097\pi$
$684$	0	0
$685$	−21.8045	−0.833107
$686$	0	0
$687$	0.319484	0.0121891
$688$	0	0
$689$	41.0179	1.56266
$690$	0	0
$691$	−14.5568	−0.553767	−0.276884	−	0.960903i	$-0.589302\pi$
$691$	−14.5568	−0.553767	−0.276884	+	0.960903i	$0.589302\pi$
$692$	0	0
$693$	−2.24481	−0.0852732
$694$	0	0
$695$	17.3451	0.657938
$696$	0	0
$697$	−52.9838	−2.00691
$698$	0	0
$699$	0.626514	0.0236970
$700$	0	0
$701$	23.0125	0.869172	0.434586	−	0.900630i	$-0.356895\pi$
$701$	23.0125	0.869172	0.434586	+	0.900630i	$0.356895\pi$
$702$	0	0
$703$	0.313203	0.0118127
$704$	0	0
$705$	0.219357	0.00826148
$706$	0	0
$707$	−3.33224	−0.125322
$708$	0	0
$709$	40.4728	1.51999	0.759993	−	0.649931i	$-0.225202\pi$
$709$	40.4728	1.51999	0.759993	+	0.649931i	$0.225202\pi$
$710$	0	0
$711$	13.6321	0.511244
$712$	0	0
$713$	8.94871	0.335132
$714$	0	0
$715$	5.80374	0.217047
$716$	0	0
$717$	1.01592	0.0379401
$718$	0	0
$719$	36.5112	1.36164	0.680818	−	0.732452i	$-0.261624\pi$
$719$	36.5112	1.36164	0.680818	+	0.732452i	$0.261624\pi$
$720$	0	0
$721$	12.3614	0.460364
$722$	0	0
$723$	0.606728	0.0225645
$724$	0	0
$725$	8.14703	0.302573
$726$	0	0
$727$	26.3001	0.975415	0.487708	−	0.873007i	$-0.337833\pi$
$727$	26.3001	0.975415	0.487708	+	0.873007i	$0.337833\pi$
$728$	0	0
$729$	−26.8312	−0.993747
$730$	0	0
$731$	23.0482	0.852469
$732$	0	0
$733$	51.0130	1.88421	0.942104	−	0.335320i	$-0.108844\pi$
$733$	51.0130	1.88421	0.942104	+	0.335320i	$0.108844\pi$
$734$	0	0
$735$	0.361212	0.0133235
$736$	0	0
$737$	2.11493	0.0779045
$738$	0	0
$739$	−35.4298	−1.30331	−0.651654	−	0.758517i	$-0.725924\pi$
$739$	−35.4298	−1.30331	−0.651654	+	0.758517i	$0.725924\pi$
$740$	0	0
$741$	0.0331571	0.00121806
$742$	0	0
$743$	−26.3987	−0.968475	−0.484238	−	0.874937i	$-0.660903\pi$
$743$	−26.3987	−0.968475	−0.484238	+	0.874937i	$0.660903\pi$
$744$	0	0
$745$	23.3219	0.854447
$746$	0	0
$747$	−5.12990	−0.187693
$748$	0	0
$749$	−5.76008	−0.210469
$750$	0	0
$751$	−18.4592	−0.673586	−0.336793	−	0.941579i	$-0.609342\pi$
$751$	−18.4592	−0.673586	−0.336793	+	0.941579i	$0.609342\pi$
$752$	0	0
$753$	1.14685	0.0417934
$754$	0	0
$755$	−18.1396	−0.660168
$756$	0	0
$757$	−7.41926	−0.269658	−0.134829	−	0.990869i	$-0.543048\pi$
$757$	−7.41926	−0.269658	−0.134829	+	0.990869i	$0.543048\pi$
$758$	0	0
$759$	0.0677664	0.00245977
$760$	0	0
$761$	30.2932	1.09813	0.549063	−	0.835781i	$-0.314985\pi$
$761$	30.2932	1.09813	0.549063	+	0.835781i	$0.314985\pi$
$762$	0	0
$763$	−12.6776	−0.458962
$764$	0	0
$765$	18.0301	0.651878
$766$	0	0
$767$	−19.8783	−0.717763
$768$	0	0
$769$	−30.0369	−1.08316	−0.541580	−	0.840649i	$-0.682174\pi$
$769$	−30.0369	−1.08316	−0.541580	+	0.840649i	$0.682174\pi$
$770$	0	0
$771$	0.809984	0.0291709
$772$	0	0
$773$	10.9803	0.394933	0.197466	−	0.980310i	$-0.436729\pi$
$773$	10.9803	0.394933	0.197466	+	0.980310i	$0.436729\pi$
$774$	0	0
$775$	−7.50989	−0.269763
$776$	0	0
$777$	−0.124332	−0.00446038
$778$	0	0
$779$	−0.914379	−0.0327610
$780$	0	0
$781$	13.7868	0.493331
$782$	0	0
$783$	−2.73304	−0.0976710
$784$	0	0
$785$	−8.94620	−0.319304
$786$	0	0
$787$	−8.95648	−0.319264	−0.159632	−	0.987177i	$-0.551031\pi$
$787$	−8.95648	−0.319264	−0.159632	+	0.987177i	$0.551031\pi$
$788$	0	0
$789$	−0.865213	−0.0308024
$790$	0	0
$791$	−8.31448	−0.295629
$792$	0	0
$793$	9.72163	0.345225
$794$	0	0
$795$	−0.401932	−0.0142551
$796$	0	0
$797$	33.0978	1.17239	0.586193	−	0.810172i	$-0.300626\pi$
$797$	33.0978	1.17239	0.586193	+	0.810172i	$0.300626\pi$
$798$	0	0
$799$	−23.5917	−0.834614
$800$	0	0
$801$	7.04797	0.249028
$802$	0	0
$803$	−2.51039	−0.0885897
$804$	0	0
$805$	−0.877957	−0.0309439
$806$	0	0
$807$	−0.361971	−0.0127420
$808$	0	0
$809$	22.9087	0.805428	0.402714	−	0.915326i	$-0.368067\pi$
$809$	22.9087	0.805428	0.402714	+	0.915326i	$0.368067\pi$
$810$	0	0
$811$	30.3711	1.06647	0.533237	−	0.845966i	$-0.320975\pi$
$811$	30.3711	1.06647	0.533237	+	0.845966i	$0.320975\pi$
$812$	0	0
$813$	0.461440	0.0161834
$814$	0	0
$815$	−20.1606	−0.706196
$816$	0	0
$817$	0.397759	0.0139158
$818$	0	0
$819$	12.6054	0.440468
$820$	0	0
$821$	48.5877	1.69572	0.847862	−	0.530217i	$-0.177890\pi$
$821$	48.5877	1.69572	0.847862	+	0.530217i	$0.177890\pi$
$822$	0	0
$823$	46.4484	1.61909	0.809545	−	0.587057i	$-0.199714\pi$
$823$	46.4484	1.61909	0.809545	+	0.587057i	$0.199714\pi$
$824$	0	0
$825$	−0.0568706	−0.00197998
$826$	0	0
$827$	13.7294	0.477418	0.238709	−	0.971091i	$-0.423276\pi$
$827$	13.7294	0.477418	0.238709	+	0.971091i	$0.423276\pi$
$828$	0	0
$829$	−2.50366	−0.0869558	−0.0434779	−	0.999054i	$-0.513844\pi$
$829$	−2.50366	−0.0869558	−0.0434779	+	0.999054i	$0.513844\pi$
$830$	0	0
$831$	0.896155	0.0310873
$832$	0	0
$833$	−38.8480	−1.34600
$834$	0	0
$835$	12.3962	0.428987
$836$	0	0
$837$	2.51931	0.0870799
$838$	0	0
$839$	19.4060	0.669969	0.334985	−	0.942224i	$-0.391269\pi$
$839$	19.4060	0.669969	0.334985	+	0.942224i	$0.391269\pi$
$840$	0	0
$841$	37.3741	1.28876
$842$	0	0
$843$	−1.45535	−0.0501248
$844$	0	0
$845$	−19.5901	−0.673918
$846$	0	0
$847$	7.34322	0.252316
$848$	0	0
$849$	0.761310	0.0261281
$850$	0	0
$851$	−3.59452	−0.123219
$852$	0	0
$853$	−13.2015	−0.452011	−0.226005	−	0.974126i	$-0.572567\pi$
$853$	−13.2015	−0.452011	−0.226005	+	0.974126i	$0.572567\pi$
$854$	0	0
$855$	0.311157	0.0106414
$856$	0	0
$857$	23.6956	0.809426	0.404713	−	0.914444i	$-0.367372\pi$
$857$	23.6956	0.809426	0.404713	+	0.914444i	$0.367372\pi$
$858$	0	0
$859$	33.4871	1.14257	0.571283	−	0.820753i	$-0.306446\pi$
$859$	33.4871	1.14257	0.571283	+	0.820753i	$0.306446\pi$
$860$	0	0
$861$	0.362980	0.0123703
$862$	0	0
$863$	−39.8110	−1.35518	−0.677591	−	0.735439i	$-0.736976\pi$
$863$	−39.8110	−1.35518	−0.677591	+	0.735439i	$0.736976\pi$
$864$	0	0
$865$	−9.96530	−0.338830
$866$	0	0
$867$	1.07381	0.0364686
$868$	0	0
$869$	4.62445	0.156874
$870$	0	0
$871$	−11.8761	−0.402406
$872$	0	0
$873$	3.13189	0.105998
$874$	0	0
$875$	0.736794	0.0249082
$876$	0	0
$877$	−35.0401	−1.18322	−0.591610	−	0.806224i	$-0.701508\pi$
$877$	−35.0401	−1.18322	−0.591610	+	0.806224i	$0.701508\pi$
$878$	0	0
$879$	−0.688826	−0.0232335
$880$	0	0
$881$	4.43645	0.149468	0.0747339	−	0.997204i	$-0.476189\pi$
$881$	4.43645	0.149468	0.0747339	+	0.997204i	$0.476189\pi$
$882$	0	0
$883$	−7.11394	−0.239403	−0.119702	−	0.992810i	$-0.538194\pi$
$883$	−7.11394	−0.239403	−0.119702	+	0.992810i	$0.538194\pi$
$884$	0	0
$885$	0.194786	0.00654768
$886$	0	0
$887$	−34.5936	−1.16154	−0.580769	−	0.814068i	$-0.697248\pi$
$887$	−34.5936	−1.16154	−0.580769	+	0.814068i	$0.697248\pi$
$888$	0	0
$889$	−13.0897	−0.439015
$890$	0	0
$891$	−9.12109	−0.305568
$892$	0	0
$893$	−0.407138	−0.0136244
$894$	0	0
$895$	−19.7309	−0.659532
$896$	0	0
$897$	−0.380533	−0.0127056
$898$	0	0
$899$	−61.1833	−2.04058
$900$	0	0
$901$	43.2274	1.44011
$902$	0	0
$903$	−0.157898	−0.00525452
$904$	0	0
$905$	−3.91272	−0.130063
$906$	0	0
$907$	13.1407	0.436331	0.218166	−	0.975912i	$-0.429993\pi$
$907$	13.1407	0.436331	0.218166	+	0.975912i	$0.429993\pi$
$908$	0	0
$909$	−13.5537	−0.449549
$910$	0	0
$911$	37.3408	1.23715	0.618577	−	0.785724i	$-0.287709\pi$
$911$	37.3408	1.23715	0.618577	+	0.785724i	$0.287709\pi$
$912$	0	0
$913$	−1.74023	−0.0575931
$914$	0	0
$915$	−0.0952619	−0.00314926
$916$	0	0
$917$	−7.57531	−0.250159
$918$	0	0
$919$	−43.2123	−1.42544	−0.712720	−	0.701448i	$-0.752537\pi$
$919$	−43.2123	−1.42544	−0.712720	+	0.701448i	$0.752537\pi$
$920$	0	0
$921$	1.84424	0.0607699
$922$	0	0
$923$	−77.4180	−2.54824
$924$	0	0
$925$	3.01658	0.0991844
$926$	0	0
$927$	50.2795	1.65140
$928$	0	0
$929$	−15.6096	−0.512135	−0.256068	−	0.966659i	$-0.582427\pi$
$929$	−15.6096	−0.512135	−0.256068	+	0.966659i	$0.582427\pi$
$930$	0	0
$931$	−0.670427	−0.0219724
$932$	0	0
$933$	1.31690	0.0431134
$934$	0	0
$935$	6.11638	0.200027
$936$	0	0
$937$	51.5634	1.68450	0.842251	−	0.539085i	$-0.181230\pi$
$937$	51.5634	1.68450	0.842251	+	0.539085i	$0.181230\pi$
$938$	0	0
$939$	−0.962877	−0.0314223
$940$	0	0
$941$	−17.9777	−0.586056	−0.293028	−	0.956104i	$-0.594663\pi$
$941$	−17.9777	−0.586056	−0.293028	+	0.956104i	$0.594663\pi$
$942$	0	0
$943$	10.4940	0.341732
$944$	0	0
$945$	−0.247169	−0.00804039
$946$	0	0
$947$	27.0820	0.880048	0.440024	−	0.897986i	$-0.354970\pi$
$947$	27.0820	0.880048	0.440024	+	0.897986i	$0.354970\pi$
$948$	0	0
$949$	14.0967	0.457599
$950$	0	0
$951$	0.809166	0.0262390
$952$	0	0
$953$	37.1480	1.20334	0.601672	−	0.798744i	$-0.294502\pi$
$953$	37.1480	1.20334	0.601672	+	0.798744i	$0.294502\pi$
$954$	0	0
$955$	3.35144	0.108450
$956$	0	0
$957$	−0.463326	−0.0149772
$958$	0	0
$959$	−16.0654	−0.518779
$960$	0	0
$961$	25.3984	0.819305
$962$	0	0
$963$	−23.4288	−0.754983
$964$	0	0
$965$	−11.9446	−0.384509
$966$	0	0
$967$	−7.15390	−0.230054	−0.115027	−	0.993362i	$-0.536695\pi$
$967$	−7.15390	−0.230054	−0.115027	+	0.993362i	$0.536695\pi$
$968$	0	0
$969$	0.0349433	0.00112254
$970$	0	0
$971$	24.6987	0.792620	0.396310	−	0.918117i	$-0.370291\pi$
$971$	24.6987	0.792620	0.396310	+	0.918117i	$0.370291\pi$
$972$	0	0
$973$	12.7798	0.409701
$974$	0	0
$975$	0.319349	0.0102273
$976$	0	0
$977$	−16.1866	−0.517855	−0.258927	−	0.965897i	$-0.583369\pi$
$977$	−16.1866	−0.517855	−0.258927	+	0.965897i	$0.583369\pi$
$978$	0	0
$979$	2.39090	0.0764134
$980$	0	0
$981$	−51.5657	−1.64637
$982$	0	0
$983$	−7.98250	−0.254602	−0.127301	−	0.991864i	$-0.540631\pi$
$983$	−7.98250	−0.254602	−0.127301	+	0.991864i	$0.540631\pi$
$984$	0	0
$985$	−13.5167	−0.430678
$986$	0	0
$987$	0.161621	0.00514446
$988$	0	0
$989$	−4.56494	−0.145157
$990$	0	0
$991$	32.4057	1.02940	0.514701	−	0.857370i	$-0.327903\pi$
$991$	32.4057	1.02940	0.514701	+	0.857370i	$0.327903\pi$
$992$	0	0
$993$	0.336578	0.0106810
$994$	0	0
$995$	9.56861	0.303345
$996$	0	0
$997$	−28.6635	−0.907781	−0.453891	−	0.891057i	$-0.649964\pi$
$997$	−28.6635	−0.907781	−0.453891	+	0.891057i	$0.649964\pi$
$998$	0	0
$999$	−1.01196	−0.0320169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.19	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.19	✓	35	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0559400 q^{3} -1.00000 q^{5} -0.736794 q^{7} -2.99687 q^{9} +O(q^{10})\)	\(q+0.0559400 q^{3} -1.00000 q^{5} -0.736794 q^{7} -2.99687 q^{9} -1.01664 q^{11} +5.70877 q^{13} -0.0559400 q^{15} +6.01629 q^{17} +0.103827 q^{19} -0.0412163 q^{21} -1.19159 q^{23} +1.00000 q^{25} -0.335465 q^{27} +8.14703 q^{29} -7.50989 q^{31} -0.0568706 q^{33} +0.736794 q^{35} +3.01658 q^{37} +0.319349 q^{39} -8.80672 q^{41} +3.83096 q^{43} +2.99687 q^{45} -3.92130 q^{47} -6.45713 q^{49} +0.336552 q^{51} +7.18506 q^{53} +1.01664 q^{55} +0.00580810 q^{57} -3.48206 q^{59} +1.70293 q^{61} +2.20808 q^{63} -5.70877 q^{65} -2.08033 q^{67} -0.0666576 q^{69} -13.5612 q^{71} +2.46931 q^{73} +0.0559400 q^{75} +0.749050 q^{77} -4.54878 q^{79} +8.97185 q^{81} +1.71175 q^{83} -6.01629 q^{85} +0.455745 q^{87} -2.35178 q^{89} -4.20619 q^{91} -0.420103 q^{93} -0.103827 q^{95} -1.04505 q^{97} +3.04672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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