LMFDB - Embedded newform 8020.2.a.e.1.16

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			1.16
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)

$f(q)$	$=$	$q-0.505617 q^{3} -1.00000 q^{5} -0.615205 q^{7} -2.74435 q^{9} +O(q^{10})$	$q-0.505617 q^{3} -1.00000 q^{5} -0.615205 q^{7} -2.74435 q^{9} -0.537064 q^{11} -6.49777 q^{13} +0.505617 q^{15} -4.51606 q^{17} +4.17673 q^{19} +0.311058 q^{21} -2.96334 q^{23} +1.00000 q^{25} +2.90444 q^{27} -6.55048 q^{29} +7.16647 q^{31} +0.271549 q^{33} +0.615205 q^{35} -7.23902 q^{37} +3.28539 q^{39} -3.15175 q^{41} -8.42394 q^{43} +2.74435 q^{45} +0.179849 q^{47} -6.62152 q^{49} +2.28340 q^{51} -8.16398 q^{53} +0.537064 q^{55} -2.11183 q^{57} +5.58261 q^{59} +9.31061 q^{61} +1.68834 q^{63} +6.49777 q^{65} -6.12998 q^{67} +1.49832 q^{69} +10.8960 q^{71} -11.2034 q^{73} -0.505617 q^{75} +0.330404 q^{77} -6.93822 q^{79} +6.76452 q^{81} -1.01912 q^{83} +4.51606 q^{85} +3.31204 q^{87} +6.88952 q^{89} +3.99746 q^{91} -3.62349 q^{93} -4.17673 q^{95} -10.7258 q^{97} +1.47389 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.505617	−0.291918	−0.145959	−	0.989291i	$-0.546627\pi$
$3$	−0.505617	−0.291918	−0.145959	+	0.989291i	$0.546627\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.615205	−0.232526	−0.116263	−	0.993218i	$-0.537091\pi$
$7$	−0.615205	−0.232526	−0.116263	+	0.993218i	$0.537091\pi$
$8$	0	0
$9$	−2.74435	−0.914784
$10$	0	0
$11$	−0.537064	−0.161931	−0.0809654	−	0.996717i	$-0.525800\pi$
$11$	−0.537064	−0.161931	−0.0809654	+	0.996717i	$0.525800\pi$
$12$	0	0
$13$	−6.49777	−1.80216	−0.901078	−	0.433656i	$-0.857223\pi$
$13$	−6.49777	−1.80216	−0.901078	+	0.433656i	$0.857223\pi$
$14$	0	0
$15$	0.505617	0.130550
$16$	0	0
$17$	−4.51606	−1.09530	−0.547652	−	0.836706i	$-0.684478\pi$
$17$	−4.51606	−1.09530	−0.547652	+	0.836706i	$0.684478\pi$
$18$	0	0
$19$	4.17673	0.958207	0.479103	−	0.877758i	$-0.340962\pi$
$19$	4.17673	0.958207	0.479103	+	0.877758i	$0.340962\pi$
$20$	0	0
$21$	0.311058	0.0678785
$22$	0	0
$23$	−2.96334	−0.617899	−0.308949	−	0.951078i	$-0.599977\pi$
$23$	−2.96334	−0.617899	−0.308949	+	0.951078i	$0.599977\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.90444	0.558961
$28$	0	0
$29$	−6.55048	−1.21639	−0.608197	−	0.793786i	$-0.708107\pi$
$29$	−6.55048	−1.21639	−0.608197	+	0.793786i	$0.708107\pi$
$30$	0	0
$31$	7.16647	1.28714	0.643568	−	0.765389i	$-0.277453\pi$
$31$	7.16647	1.28714	0.643568	+	0.765389i	$0.277453\pi$
$32$	0	0
$33$	0.271549	0.0472706
$34$	0	0
$35$	0.615205	0.103989
$36$	0	0
$37$	−7.23902	−1.19009	−0.595044	−	0.803693i	$-0.702865\pi$
$37$	−7.23902	−1.19009	−0.595044	+	0.803693i	$0.702865\pi$
$38$	0	0
$39$	3.28539	0.526083
$40$	0	0
$41$	−3.15175	−0.492221	−0.246110	−	0.969242i	$-0.579153\pi$
$41$	−3.15175	−0.492221	−0.246110	+	0.969242i	$0.579153\pi$
$42$	0	0
$43$	−8.42394	−1.28464	−0.642319	−	0.766437i	$-0.722028\pi$
$43$	−8.42394	−1.28464	−0.642319	+	0.766437i	$0.722028\pi$
$44$	0	0
$45$	2.74435	0.409104
$46$	0	0
$47$	0.179849	0.0262337	0.0131168	−	0.999914i	$-0.495825\pi$
$47$	0.179849	0.0262337	0.0131168	+	0.999914i	$0.495825\pi$
$48$	0	0
$49$	−6.62152	−0.945932
$50$	0	0
$51$	2.28340	0.319739
$52$	0	0
$53$	−8.16398	−1.12141	−0.560705	−	0.828016i	$-0.689470\pi$
$53$	−8.16398	−1.12141	−0.560705	+	0.828016i	$0.689470\pi$
$54$	0	0
$55$	0.537064	0.0724177
$56$	0	0
$57$	−2.11183	−0.279718
$58$	0	0
$59$	5.58261	0.726794	0.363397	−	0.931634i	$-0.381617\pi$
$59$	5.58261	0.726794	0.363397	+	0.931634i	$0.381617\pi$
$60$	0	0
$61$	9.31061	1.19210	0.596051	−	0.802947i	$-0.296736\pi$
$61$	9.31061	1.19210	0.596051	+	0.802947i	$0.296736\pi$
$62$	0	0
$63$	1.68834	0.212711
$64$	0	0
$65$	6.49777	0.805949
$66$	0	0
$67$	−6.12998	−0.748896	−0.374448	−	0.927248i	$-0.622168\pi$
$67$	−6.12998	−0.748896	−0.374448	+	0.927248i	$0.622168\pi$
$68$	0	0
$69$	1.49832	0.180376
$70$	0	0
$71$	10.8960	1.29312	0.646558	−	0.762864i	$-0.276208\pi$
$71$	10.8960	1.29312	0.646558	+	0.762864i	$0.276208\pi$
$72$	0	0
$73$	−11.2034	−1.31126	−0.655629	−	0.755084i	$-0.727596\pi$
$73$	−11.2034	−1.31126	−0.655629	+	0.755084i	$0.727596\pi$
$74$	0	0
$75$	−0.505617	−0.0583837
$76$	0	0
$77$	0.330404	0.0376531
$78$	0	0
$79$	−6.93822	−0.780611	−0.390305	−	0.920686i	$-0.627631\pi$
$79$	−6.93822	−0.780611	−0.390305	+	0.920686i	$0.627631\pi$
$80$	0	0
$81$	6.76452	0.751613
$82$	0	0
$83$	−1.01912	−0.111863	−0.0559314	−	0.998435i	$-0.517813\pi$
$83$	−1.01912	−0.111863	−0.0559314	+	0.998435i	$0.517813\pi$
$84$	0	0
$85$	4.51606	0.489835
$86$	0	0
$87$	3.31204	0.355088
$88$	0	0
$89$	6.88952	0.730288	0.365144	−	0.930951i	$-0.381020\pi$
$89$	6.88952	0.730288	0.365144	+	0.930951i	$0.381020\pi$
$90$	0	0
$91$	3.99746	0.419048
$92$	0	0
$93$	−3.62349	−0.375738
$94$	0	0
$95$	−4.17673	−0.428523
$96$	0	0
$97$	−10.7258	−1.08904	−0.544521	−	0.838747i	$-0.683289\pi$
$97$	−10.7258	−1.08904	−0.544521	+	0.838747i	$0.683289\pi$
$98$	0	0
$99$	1.47389	0.148132
$100$	0	0
$101$	−13.9967	−1.39272	−0.696361	−	0.717691i	$-0.745199\pi$
$101$	−13.9967	−1.39272	−0.696361	+	0.717691i	$0.745199\pi$
$102$	0	0
$103$	10.7966	1.06382	0.531909	−	0.846802i	$-0.321475\pi$
$103$	10.7966	1.06382	0.531909	+	0.846802i	$0.321475\pi$
$104$	0	0
$105$	−0.311058	−0.0303562
$106$	0	0
$107$	−11.3371	−1.09600	−0.547999	−	0.836479i	$-0.684610\pi$
$107$	−11.3371	−1.09600	−0.547999	+	0.836479i	$0.684610\pi$
$108$	0	0
$109$	−9.34616	−0.895200	−0.447600	−	0.894234i	$-0.647721\pi$
$109$	−9.34616	−0.895200	−0.447600	+	0.894234i	$0.647721\pi$
$110$	0	0
$111$	3.66018	0.347409
$112$	0	0
$113$	15.9060	1.49631	0.748156	−	0.663523i	$-0.230940\pi$
$113$	15.9060	1.49631	0.748156	+	0.663523i	$0.230940\pi$
$114$	0	0
$115$	2.96334	0.276333
$116$	0	0
$117$	17.8322	1.64858
$118$	0	0
$119$	2.77830	0.254686
$120$	0	0
$121$	−10.7116	−0.973778
$122$	0	0
$123$	1.59358	0.143688
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.2325	1.44040	0.720200	−	0.693767i	$-0.244050\pi$
$127$	16.2325	1.44040	0.720200	+	0.693767i	$0.244050\pi$
$128$	0	0
$129$	4.25929	0.375010
$130$	0	0
$131$	17.2209	1.50460	0.752299	−	0.658822i	$-0.228945\pi$
$131$	17.2209	1.50460	0.752299	+	0.658822i	$0.228945\pi$
$132$	0	0
$133$	−2.56954	−0.222808
$134$	0	0
$135$	−2.90444	−0.249975
$136$	0	0
$137$	−0.488927	−0.0417718	−0.0208859	−	0.999782i	$-0.506649\pi$
$137$	−0.488927	−0.0417718	−0.0208859	+	0.999782i	$0.506649\pi$
$138$	0	0
$139$	3.85107	0.326644	0.163322	−	0.986573i	$-0.447779\pi$
$139$	3.85107	0.326644	0.163322	+	0.986573i	$0.447779\pi$
$140$	0	0
$141$	−0.0909349	−0.00765810
$142$	0	0
$143$	3.48972	0.291825
$144$	0	0
$145$	6.55048	0.543988
$146$	0	0
$147$	3.34796	0.276135
$148$	0	0
$149$	−16.2227	−1.32901	−0.664506	−	0.747283i	$-0.731358\pi$
$149$	−16.2227	−1.32901	−0.664506	+	0.747283i	$0.731358\pi$
$150$	0	0
$151$	−0.862179	−0.0701632	−0.0350816	−	0.999384i	$-0.511169\pi$
$151$	−0.862179	−0.0701632	−0.0350816	+	0.999384i	$0.511169\pi$
$152$	0	0
$153$	12.3936	1.00197
$154$	0	0
$155$	−7.16647	−0.575624
$156$	0	0
$157$	−7.04611	−0.562341	−0.281170	−	0.959658i	$-0.590723\pi$
$157$	−7.04611	−0.562341	−0.281170	+	0.959658i	$0.590723\pi$
$158$	0	0
$159$	4.12785	0.327360
$160$	0	0
$161$	1.82306	0.143677
$162$	0	0
$163$	5.26671	0.412520	0.206260	−	0.978497i	$-0.433871\pi$
$163$	5.26671	0.412520	0.206260	+	0.978497i	$0.433871\pi$
$164$	0	0
$165$	−0.271549	−0.0211401
$166$	0	0
$167$	16.4482	1.27280	0.636399	−	0.771360i	$-0.280423\pi$
$167$	16.4482	1.27280	0.636399	+	0.771360i	$0.280423\pi$
$168$	0	0
$169$	29.2210	2.24777
$170$	0	0
$171$	−11.4624	−0.876552
$172$	0	0
$173$	20.1866	1.53476	0.767381	−	0.641192i	$-0.221560\pi$
$173$	20.1866	1.53476	0.767381	+	0.641192i	$0.221560\pi$
$174$	0	0
$175$	−0.615205	−0.0465051
$176$	0	0
$177$	−2.82266	−0.212164
$178$	0	0
$179$	14.1781	1.05972	0.529861	−	0.848084i	$-0.322244\pi$
$179$	14.1781	1.05972	0.529861	+	0.848084i	$0.322244\pi$
$180$	0	0
$181$	−2.19357	−0.163047	−0.0815233	−	0.996671i	$-0.525979\pi$
$181$	−2.19357	−0.163047	−0.0815233	+	0.996671i	$0.525979\pi$
$182$	0	0
$183$	−4.70761	−0.347996
$184$	0	0
$185$	7.23902	0.532224
$186$	0	0
$187$	2.42541	0.177364
$188$	0	0
$189$	−1.78683	−0.129973
$190$	0	0
$191$	14.1259	1.02211	0.511056	−	0.859548i	$-0.329255\pi$
$191$	14.1259	1.02211	0.511056	+	0.859548i	$0.329255\pi$
$192$	0	0
$193$	11.4674	0.825438	0.412719	−	0.910858i	$-0.364579\pi$
$193$	11.4674	0.825438	0.412719	+	0.910858i	$0.364579\pi$
$194$	0	0
$195$	−3.28539	−0.235271
$196$	0	0
$197$	18.8067	1.33992	0.669959	−	0.742398i	$-0.266312\pi$
$197$	18.8067	1.33992	0.669959	+	0.742398i	$0.266312\pi$
$198$	0	0
$199$	−17.3273	−1.22830	−0.614149	−	0.789190i	$-0.710501\pi$
$199$	−17.3273	−1.22830	−0.614149	+	0.789190i	$0.710501\pi$
$200$	0	0
$201$	3.09943	0.218617
$202$	0	0
$203$	4.02989	0.282843
$204$	0	0
$205$	3.15175	0.220128
$206$	0	0
$207$	8.13244	0.565244
$208$	0	0
$209$	−2.24317	−0.155163
$210$	0	0
$211$	−7.94073	−0.546662	−0.273331	−	0.961920i	$-0.588126\pi$
$211$	−7.94073	−0.546662	−0.273331	+	0.961920i	$0.588126\pi$
$212$	0	0
$213$	−5.50921	−0.377485
$214$	0	0
$215$	8.42394	0.574508
$216$	0	0
$217$	−4.40885	−0.299292
$218$	0	0
$219$	5.66463	0.382780
$220$	0	0
$221$	29.3443	1.97391
$222$	0	0
$223$	−22.1671	−1.48442	−0.742210	−	0.670168i	$-0.766222\pi$
$223$	−22.1671	−1.48442	−0.742210	+	0.670168i	$0.766222\pi$
$224$	0	0
$225$	−2.74435	−0.182957
$226$	0	0
$227$	13.8291	0.917868	0.458934	−	0.888470i	$-0.348231\pi$
$227$	13.8291	0.917868	0.458934	+	0.888470i	$0.348231\pi$
$228$	0	0
$229$	18.6182	1.23033	0.615164	−	0.788399i	$-0.289090\pi$
$229$	18.6182	1.23033	0.615164	+	0.788399i	$0.289090\pi$
$230$	0	0
$231$	−0.167058	−0.0109916
$232$	0	0
$233$	−16.5748	−1.08585	−0.542924	−	0.839782i	$-0.682683\pi$
$233$	−16.5748	−1.08585	−0.542924	+	0.839782i	$0.682683\pi$
$234$	0	0
$235$	−0.179849	−0.0117321
$236$	0	0
$237$	3.50808	0.227875
$238$	0	0
$239$	−8.44916	−0.546531	−0.273265	−	0.961939i	$-0.588104\pi$
$239$	−8.44916	−0.546531	−0.273265	+	0.961939i	$0.588104\pi$
$240$	0	0
$241$	−4.74406	−0.305592	−0.152796	−	0.988258i	$-0.548828\pi$
$241$	−4.74406	−0.305592	−0.152796	+	0.988258i	$0.548828\pi$
$242$	0	0
$243$	−12.1336	−0.778370
$244$	0	0
$245$	6.62152	0.423034
$246$	0	0
$247$	−27.1394	−1.72684
$248$	0	0
$249$	0.515284	0.0326548
$250$	0	0
$251$	−14.3554	−0.906103	−0.453051	−	0.891484i	$-0.649665\pi$
$251$	−14.3554	−0.906103	−0.453051	+	0.891484i	$0.649665\pi$
$252$	0	0
$253$	1.59150	0.100057
$254$	0	0
$255$	−2.28340	−0.142992
$256$	0	0
$257$	−22.4918	−1.40300	−0.701499	−	0.712671i	$-0.747485\pi$
$257$	−22.4918	−1.40300	−0.701499	+	0.712671i	$0.747485\pi$
$258$	0	0
$259$	4.45348	0.276726
$260$	0	0
$261$	17.9768	1.11274
$262$	0	0
$263$	−24.2872	−1.49761	−0.748805	−	0.662790i	$-0.769372\pi$
$263$	−24.2872	−1.49761	−0.748805	+	0.662790i	$0.769372\pi$
$264$	0	0
$265$	8.16398	0.501510
$266$	0	0
$267$	−3.48346	−0.213185
$268$	0	0
$269$	−13.5741	−0.827626	−0.413813	−	0.910362i	$-0.635803\pi$
$269$	−13.5741	−0.827626	−0.413813	+	0.910362i	$0.635803\pi$
$270$	0	0
$271$	16.5813	1.00724	0.503621	−	0.863924i	$-0.332001\pi$
$271$	16.5813	1.00724	0.503621	+	0.863924i	$0.332001\pi$
$272$	0	0
$273$	−2.02119	−0.122328
$274$	0	0
$275$	−0.537064	−0.0323862
$276$	0	0
$277$	5.90693	0.354913	0.177457	−	0.984129i	$-0.443213\pi$
$277$	5.90693	0.354913	0.177457	+	0.984129i	$0.443213\pi$
$278$	0	0
$279$	−19.6673	−1.17745
$280$	0	0
$281$	7.06068	0.421205	0.210602	−	0.977572i	$-0.432457\pi$
$281$	7.06068	0.421205	0.210602	+	0.977572i	$0.432457\pi$
$282$	0	0
$283$	12.2084	0.725715	0.362857	−	0.931845i	$-0.381801\pi$
$283$	12.2084	0.725715	0.362857	+	0.931845i	$0.381801\pi$
$284$	0	0
$285$	2.11183	0.125094
$286$	0	0
$287$	1.93897	0.114454
$288$	0	0
$289$	3.39476	0.199691
$290$	0	0
$291$	5.42316	0.317911
$292$	0	0
$293$	6.04282	0.353026	0.176513	−	0.984298i	$-0.443518\pi$
$293$	6.04282	0.353026	0.176513	+	0.984298i	$0.443518\pi$
$294$	0	0
$295$	−5.58261	−0.325032
$296$	0	0
$297$	−1.55987	−0.0905130
$298$	0	0
$299$	19.2551	1.11355
$300$	0	0
$301$	5.18245	0.298711
$302$	0	0
$303$	7.07697	0.406561
$304$	0	0
$305$	−9.31061	−0.533124
$306$	0	0
$307$	−1.76272	−0.100604	−0.0503018	−	0.998734i	$-0.516018\pi$
$307$	−1.76272	−0.100604	−0.0503018	+	0.998734i	$0.516018\pi$
$308$	0	0
$309$	−5.45894	−0.310548
$310$	0	0
$311$	28.3049	1.60502	0.802512	−	0.596636i	$-0.203496\pi$
$311$	28.3049	1.60502	0.802512	+	0.596636i	$0.203496\pi$
$312$	0	0
$313$	−6.27025	−0.354415	−0.177208	−	0.984173i	$-0.556706\pi$
$313$	−6.27025	−0.354415	−0.177208	+	0.984173i	$0.556706\pi$
$314$	0	0
$315$	−1.68834	−0.0951271
$316$	0	0
$317$	15.3021	0.859451	0.429726	−	0.902960i	$-0.358610\pi$
$317$	15.3021	0.859451	0.429726	+	0.902960i	$0.358610\pi$
$318$	0	0
$319$	3.51803	0.196972
$320$	0	0
$321$	5.73223	0.319942
$322$	0	0
$323$	−18.8623	−1.04953
$324$	0	0
$325$	−6.49777	−0.360431
$326$	0	0
$327$	4.72558	0.261325
$328$	0	0
$329$	−0.110644	−0.00610000
$330$	0	0
$331$	−13.8663	−0.762160	−0.381080	−	0.924542i	$-0.624448\pi$
$331$	−13.8663	−0.762160	−0.381080	+	0.924542i	$0.624448\pi$
$332$	0	0
$333$	19.8664	1.08867
$334$	0	0
$335$	6.12998	0.334917
$336$	0	0
$337$	−27.8375	−1.51641	−0.758204	−	0.652018i	$-0.773923\pi$
$337$	−27.8375	−1.51641	−0.758204	+	0.652018i	$0.773923\pi$
$338$	0	0
$339$	−8.04236	−0.436801
$340$	0	0
$341$	−3.84885	−0.208427
$342$	0	0
$343$	8.38003	0.452479
$344$	0	0
$345$	−1.49832	−0.0806666
$346$	0	0
$347$	6.43477	0.345437	0.172718	−	0.984971i	$-0.444745\pi$
$347$	6.43477	0.345437	0.172718	+	0.984971i	$0.444745\pi$
$348$	0	0
$349$	24.6651	1.32029	0.660147	−	0.751137i	$-0.270494\pi$
$349$	24.6651	1.32029	0.660147	+	0.751137i	$0.270494\pi$
$350$	0	0
$351$	−18.8724	−1.00733
$352$	0	0
$353$	10.2314	0.544560	0.272280	−	0.962218i	$-0.412222\pi$
$353$	10.2314	0.544560	0.272280	+	0.962218i	$0.412222\pi$
$354$	0	0
$355$	−10.8960	−0.578299
$356$	0	0
$357$	−1.40476	−0.0743476
$358$	0	0
$359$	−22.1303	−1.16799	−0.583996	−	0.811757i	$-0.698511\pi$
$359$	−22.1303	−1.16799	−0.583996	+	0.811757i	$0.698511\pi$
$360$	0	0
$361$	−1.55495	−0.0818395
$362$	0	0
$363$	5.41595	0.284264
$364$	0	0
$365$	11.2034	0.586412
$366$	0	0
$367$	20.6627	1.07858	0.539291	−	0.842119i	$-0.318692\pi$
$367$	20.6627	1.07858	0.539291	+	0.842119i	$0.318692\pi$
$368$	0	0
$369$	8.64951	0.450276
$370$	0	0
$371$	5.02252	0.260756
$372$	0	0
$373$	−0.788638	−0.0408341	−0.0204171	−	0.999792i	$-0.506499\pi$
$373$	−0.788638	−0.0408341	−0.0204171	+	0.999792i	$0.506499\pi$
$374$	0	0
$375$	0.505617	0.0261100
$376$	0	0
$377$	42.5635	2.19213
$378$	0	0
$379$	5.03528	0.258645	0.129323	−	0.991603i	$-0.458720\pi$
$379$	5.03528	0.258645	0.129323	+	0.991603i	$0.458720\pi$
$380$	0	0
$381$	−8.20743	−0.420479
$382$	0	0
$383$	14.3722	0.734383	0.367192	−	0.930145i	$-0.380319\pi$
$383$	14.3722	0.734383	0.367192	+	0.930145i	$0.380319\pi$
$384$	0	0
$385$	−0.330404	−0.0168390
$386$	0	0
$387$	23.1182	1.17517
$388$	0	0
$389$	11.3558	0.575760	0.287880	−	0.957667i	$-0.407050\pi$
$389$	11.3558	0.575760	0.287880	+	0.957667i	$0.407050\pi$
$390$	0	0
$391$	13.3826	0.676787
$392$	0	0
$393$	−8.70720	−0.439220
$394$	0	0
$395$	6.93822	0.349100
$396$	0	0
$397$	−33.4862	−1.68062	−0.840312	−	0.542103i	$-0.817628\pi$
$397$	−33.4862	−1.68062	−0.840312	+	0.542103i	$0.817628\pi$
$398$	0	0
$399$	1.29921	0.0650417
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−46.5660	−2.31962
$404$	0	0
$405$	−6.76452	−0.336131
$406$	0	0
$407$	3.88782	0.192712
$408$	0	0
$409$	11.6243	0.574783	0.287392	−	0.957813i	$-0.407212\pi$
$409$	11.6243	0.574783	0.287392	+	0.957813i	$0.407212\pi$
$410$	0	0
$411$	0.247210	0.0121940
$412$	0	0
$413$	−3.43445	−0.168998
$414$	0	0
$415$	1.01912	0.0500266
$416$	0	0
$417$	−1.94717	−0.0953532
$418$	0	0
$419$	−28.6656	−1.40041	−0.700203	−	0.713944i	$-0.746907\pi$
$419$	−28.6656	−1.40041	−0.700203	+	0.713944i	$0.746907\pi$
$420$	0	0
$421$	−1.49809	−0.0730126	−0.0365063	−	0.999333i	$-0.511623\pi$
$421$	−1.49809	−0.0730126	−0.0365063	+	0.999333i	$0.511623\pi$
$422$	0	0
$423$	−0.493569	−0.0239981
$424$	0	0
$425$	−4.51606	−0.219061
$426$	0	0
$427$	−5.72793	−0.277194
$428$	0	0
$429$	−1.76446	−0.0851890
$430$	0	0
$431$	14.6939	0.707781	0.353890	−	0.935287i	$-0.384859\pi$
$431$	14.6939	0.707781	0.353890	+	0.935287i	$0.384859\pi$
$432$	0	0
$433$	21.9111	1.05298	0.526489	−	0.850182i	$-0.323508\pi$
$433$	21.9111	1.05298	0.526489	+	0.850182i	$0.323508\pi$
$434$	0	0
$435$	−3.31204	−0.158800
$436$	0	0
$437$	−12.3771	−0.592075
$438$	0	0
$439$	24.3491	1.16212	0.581059	−	0.813861i	$-0.302638\pi$
$439$	24.3491	1.16212	0.581059	+	0.813861i	$0.302638\pi$
$440$	0	0
$441$	18.1718	0.865323
$442$	0	0
$443$	−20.5538	−0.976540	−0.488270	−	0.872693i	$-0.662372\pi$
$443$	−20.5538	−0.976540	−0.488270	+	0.872693i	$0.662372\pi$
$444$	0	0
$445$	−6.88952	−0.326595
$446$	0	0
$447$	8.20246	0.387963
$448$	0	0
$449$	−15.6621	−0.739139	−0.369570	−	0.929203i	$-0.620495\pi$
$449$	−15.6621	−0.739139	−0.369570	+	0.929203i	$0.620495\pi$
$450$	0	0
$451$	1.69269	0.0797058
$452$	0	0
$453$	0.435933	0.0204819
$454$	0	0
$455$	−3.99746	−0.187404
$456$	0	0
$457$	19.2802	0.901890	0.450945	−	0.892552i	$-0.351087\pi$
$457$	19.2802	0.901890	0.450945	+	0.892552i	$0.351087\pi$
$458$	0	0
$459$	−13.1166	−0.612232
$460$	0	0
$461$	9.56171	0.445333	0.222667	−	0.974895i	$-0.428524\pi$
$461$	9.56171	0.445333	0.222667	+	0.974895i	$0.428524\pi$
$462$	0	0
$463$	−6.66391	−0.309698	−0.154849	−	0.987938i	$-0.549489\pi$
$463$	−6.66391	−0.309698	−0.154849	+	0.987938i	$0.549489\pi$
$464$	0	0
$465$	3.62349	0.168035
$466$	0	0
$467$	13.0200	0.602492	0.301246	−	0.953546i	$-0.402597\pi$
$467$	13.0200	0.602492	0.301246	+	0.953546i	$0.402597\pi$
$468$	0	0
$469$	3.77119	0.174138
$470$	0	0
$471$	3.56263	0.164158
$472$	0	0
$473$	4.52419	0.208023
$474$	0	0
$475$	4.17673	0.191641
$476$	0	0
$477$	22.4048	1.02585
$478$	0	0
$479$	34.1377	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$479$	34.1377	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$480$	0	0
$481$	47.0375	2.14473
$482$	0	0
$483$	−0.921771	−0.0419420
$484$	0	0
$485$	10.7258	0.487034
$486$	0	0
$487$	31.3634	1.42121	0.710606	−	0.703590i	$-0.248421\pi$
$487$	31.3634	1.42121	0.710606	+	0.703590i	$0.248421\pi$
$488$	0	0
$489$	−2.66294	−0.120422
$490$	0	0
$491$	−10.9110	−0.492405	−0.246202	−	0.969218i	$-0.579183\pi$
$491$	−10.9110	−0.492405	−0.246202	+	0.969218i	$0.579183\pi$
$492$	0	0
$493$	29.5823	1.33232
$494$	0	0
$495$	−1.47389	−0.0662465
$496$	0	0
$497$	−6.70327	−0.300683
$498$	0	0
$499$	−15.3909	−0.688990	−0.344495	−	0.938788i	$-0.611950\pi$
$499$	−15.3909	−0.688990	−0.344495	+	0.938788i	$0.611950\pi$
$500$	0	0
$501$	−8.31649	−0.371553
$502$	0	0
$503$	−0.226039	−0.0100786	−0.00503930	−	0.999987i	$-0.501604\pi$
$503$	−0.226039	−0.0100786	−0.00503930	+	0.999987i	$0.501604\pi$
$504$	0	0
$505$	13.9967	0.622845
$506$	0	0
$507$	−14.7747	−0.656165
$508$	0	0
$509$	−19.0630	−0.844953	−0.422476	−	0.906374i	$-0.638839\pi$
$509$	−19.0630	−0.844953	−0.422476	+	0.906374i	$0.638839\pi$
$510$	0	0
$511$	6.89238	0.304901
$512$	0	0
$513$	12.1311	0.535600
$514$	0	0
$515$	−10.7966	−0.475754
$516$	0	0
$517$	−0.0965905	−0.00424804
$518$	0	0
$519$	−10.2067	−0.448025
$520$	0	0
$521$	−34.8591	−1.52720	−0.763602	−	0.645687i	$-0.776571\pi$
$521$	−34.8591	−1.52720	−0.763602	+	0.645687i	$0.776571\pi$
$522$	0	0
$523$	−29.7851	−1.30241	−0.651206	−	0.758901i	$-0.725737\pi$
$523$	−29.7851	−1.30241	−0.651206	+	0.758901i	$0.725737\pi$
$524$	0	0
$525$	0.311058	0.0135757
$526$	0	0
$527$	−32.3642	−1.40980
$528$	0	0
$529$	−14.2186	−0.618201
$530$	0	0
$531$	−15.3206	−0.664859
$532$	0	0
$533$	20.4794	0.887059
$534$	0	0
$535$	11.3371	0.490145
$536$	0	0
$537$	−7.16870	−0.309352
$538$	0	0
$539$	3.55618	0.153176
$540$	0	0
$541$	11.3465	0.487825	0.243913	−	0.969797i	$-0.421569\pi$
$541$	11.3465	0.487825	0.243913	+	0.969797i	$0.421569\pi$
$542$	0	0
$543$	1.10911	0.0475963
$544$	0	0
$545$	9.34616	0.400345
$546$	0	0
$547$	37.4443	1.60100	0.800501	−	0.599331i	$-0.204567\pi$
$547$	37.4443	1.60100	0.800501	+	0.599331i	$0.204567\pi$
$548$	0	0
$549$	−25.5516	−1.09052
$550$	0	0
$551$	−27.3596	−1.16556
$552$	0	0
$553$	4.26843	0.181512
$554$	0	0
$555$	−3.66018	−0.155366
$556$	0	0
$557$	31.3391	1.32788	0.663941	−	0.747785i	$-0.268883\pi$
$557$	31.3391	1.32788	0.663941	+	0.747785i	$0.268883\pi$
$558$	0	0
$559$	54.7368	2.31512
$560$	0	0
$561$	−1.22633	−0.0517757
$562$	0	0
$563$	6.26310	0.263958	0.131979	−	0.991252i	$-0.457867\pi$
$563$	6.26310	0.263958	0.131979	+	0.991252i	$0.457867\pi$
$564$	0	0
$565$	−15.9060	−0.669171
$566$	0	0
$567$	−4.16156	−0.174769
$568$	0	0
$569$	3.83261	0.160671	0.0803356	−	0.996768i	$-0.474401\pi$
$569$	3.83261	0.160671	0.0803356	+	0.996768i	$0.474401\pi$
$570$	0	0
$571$	−18.5117	−0.774689	−0.387345	−	0.921935i	$-0.626608\pi$
$571$	−18.5117	−0.774689	−0.387345	+	0.921935i	$0.626608\pi$
$572$	0	0
$573$	−7.14228	−0.298373
$574$	0	0
$575$	−2.96334	−0.123580
$576$	0	0
$577$	−21.3229	−0.887686	−0.443843	−	0.896105i	$-0.646385\pi$
$577$	−21.3229	−0.887686	−0.443843	+	0.896105i	$0.646385\pi$
$578$	0	0
$579$	−5.79810	−0.240961
$580$	0	0
$581$	0.626967	0.0260110
$582$	0	0
$583$	4.38458	0.181591
$584$	0	0
$585$	−17.8322	−0.737269
$586$	0	0
$587$	−10.1300	−0.418112	−0.209056	−	0.977904i	$-0.567039\pi$
$587$	−10.1300	−0.418112	−0.209056	+	0.977904i	$0.567039\pi$
$588$	0	0
$589$	29.9324	1.23334
$590$	0	0
$591$	−9.50897	−0.391147
$592$	0	0
$593$	−10.3150	−0.423585	−0.211793	−	0.977315i	$-0.567930\pi$
$593$	−10.3150	−0.423585	−0.211793	+	0.977315i	$0.567930\pi$
$594$	0	0
$595$	−2.77830	−0.113899
$596$	0	0
$597$	8.76097	0.358563
$598$	0	0
$599$	−37.0006	−1.51180	−0.755902	−	0.654685i	$-0.772801\pi$
$599$	−37.0006	−1.51180	−0.755902	+	0.654685i	$0.772801\pi$
$600$	0	0
$601$	16.5461	0.674931	0.337465	−	0.941338i	$-0.390430\pi$
$601$	16.5461	0.674931	0.337465	+	0.941338i	$0.390430\pi$
$602$	0	0
$603$	16.8228	0.685078
$604$	0	0
$605$	10.7116	0.435487
$606$	0	0
$607$	0.172200	0.00698940	0.00349470	−	0.999994i	$-0.498888\pi$
$607$	0.172200	0.00698940	0.00349470	+	0.999994i	$0.498888\pi$
$608$	0	0
$609$	−2.03758	−0.0825669
$610$	0	0
$611$	−1.16862	−0.0472772
$612$	0	0
$613$	5.00814	0.202277	0.101138	−	0.994872i	$-0.467752\pi$
$613$	5.00814	0.202277	0.101138	+	0.994872i	$0.467752\pi$
$614$	0	0
$615$	−1.59358	−0.0642594
$616$	0	0
$617$	23.9433	0.963920	0.481960	−	0.876193i	$-0.339925\pi$
$617$	23.9433	0.963920	0.481960	+	0.876193i	$0.339925\pi$
$618$	0	0
$619$	−2.52131	−0.101340	−0.0506701	−	0.998715i	$-0.516136\pi$
$619$	−2.52131	−0.101340	−0.0506701	+	0.998715i	$0.516136\pi$
$620$	0	0
$621$	−8.60685	−0.345381
$622$	0	0
$623$	−4.23847	−0.169811
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.13419	0.0452950
$628$	0	0
$629$	32.6918	1.30351
$630$	0	0
$631$	35.3875	1.40875	0.704377	−	0.709826i	$-0.251227\pi$
$631$	35.3875	1.40875	0.704377	+	0.709826i	$0.251227\pi$
$632$	0	0
$633$	4.01497	0.159581
$634$	0	0
$635$	−16.2325	−0.644166
$636$	0	0
$637$	43.0251	1.70472
$638$	0	0
$639$	−29.9024	−1.18292
$640$	0	0
$641$	5.96637	0.235658	0.117829	−	0.993034i	$-0.462407\pi$
$641$	5.96637	0.235658	0.117829	+	0.993034i	$0.462407\pi$
$642$	0	0
$643$	37.6553	1.48498	0.742490	−	0.669858i	$-0.233645\pi$
$643$	37.6553	1.48498	0.742490	+	0.669858i	$0.233645\pi$
$644$	0	0
$645$	−4.25929	−0.167709
$646$	0	0
$647$	23.6700	0.930565	0.465283	−	0.885162i	$-0.345953\pi$
$647$	23.6700	0.930565	0.465283	+	0.885162i	$0.345953\pi$
$648$	0	0
$649$	−2.99822	−0.117690
$650$	0	0
$651$	2.22919	0.0873688
$652$	0	0
$653$	−11.1944	−0.438070	−0.219035	−	0.975717i	$-0.570291\pi$
$653$	−11.1944	−0.438070	−0.219035	+	0.975717i	$0.570291\pi$
$654$	0	0
$655$	−17.2209	−0.672877
$656$	0	0
$657$	30.7460	1.19952
$658$	0	0
$659$	17.2987	0.673861	0.336931	−	0.941529i	$-0.390611\pi$
$659$	17.2987	0.673861	0.336931	+	0.941529i	$0.390611\pi$
$660$	0	0
$661$	39.8466	1.54985	0.774927	−	0.632051i	$-0.217787\pi$
$661$	39.8466	1.54985	0.774927	+	0.632051i	$0.217787\pi$
$662$	0	0
$663$	−14.8370	−0.576221
$664$	0	0
$665$	2.56954	0.0996426
$666$	0	0
$667$	19.4113	0.751608
$668$	0	0
$669$	11.2081	0.433329
$670$	0	0
$671$	−5.00039	−0.193038
$672$	0	0
$673$	−9.99869	−0.385421	−0.192711	−	0.981256i	$-0.561728\pi$
$673$	−9.99869	−0.385421	−0.192711	+	0.981256i	$0.561728\pi$
$674$	0	0
$675$	2.90444	0.111792
$676$	0	0
$677$	−3.47077	−0.133392	−0.0666962	−	0.997773i	$-0.521246\pi$
$677$	−3.47077	−0.133392	−0.0666962	+	0.997773i	$0.521246\pi$
$678$	0	0
$679$	6.59858	0.253230
$680$	0	0
$681$	−6.99223	−0.267943
$682$	0	0
$683$	3.03081	0.115971	0.0579854	−	0.998317i	$-0.481532\pi$
$683$	3.03081	0.115971	0.0579854	+	0.998317i	$0.481532\pi$
$684$	0	0
$685$	0.488927	0.0186809
$686$	0	0
$687$	−9.41371	−0.359155
$688$	0	0
$689$	53.0477	2.02096
$690$	0	0
$691$	−22.3043	−0.848495	−0.424248	−	0.905546i	$-0.639461\pi$
$691$	−22.3043	−0.848495	−0.424248	+	0.905546i	$0.639461\pi$
$692$	0	0
$693$	−0.906746	−0.0344444
$694$	0	0
$695$	−3.85107	−0.146079
$696$	0	0
$697$	14.2335	0.539132
$698$	0	0
$699$	8.38049	0.316979
$700$	0	0
$701$	−26.9270	−1.01702	−0.508510	−	0.861056i	$-0.669804\pi$
$701$	−26.9270	−1.01702	−0.508510	+	0.861056i	$0.669804\pi$
$702$	0	0
$703$	−30.2354	−1.14035
$704$	0	0
$705$	0.0909349	0.00342480
$706$	0	0
$707$	8.61083	0.323844
$708$	0	0
$709$	24.2584	0.911042	0.455521	−	0.890225i	$-0.349453\pi$
$709$	24.2584	0.911042	0.455521	+	0.890225i	$0.349453\pi$
$710$	0	0
$711$	19.0409	0.714090
$712$	0	0
$713$	−21.2367	−0.795319
$714$	0	0
$715$	−3.48972	−0.130508
$716$	0	0
$717$	4.27204	0.159542
$718$	0	0
$719$	−5.09754	−0.190106	−0.0950531	−	0.995472i	$-0.530302\pi$
$719$	−5.09754	−0.190106	−0.0950531	+	0.995472i	$0.530302\pi$
$720$	0	0
$721$	−6.64211	−0.247365
$722$	0	0
$723$	2.39868	0.0892078
$724$	0	0
$725$	−6.55048	−0.243279
$726$	0	0
$727$	−26.1658	−0.970434	−0.485217	−	0.874394i	$-0.661259\pi$
$727$	−26.1658	−0.970434	−0.485217	+	0.874394i	$0.661259\pi$
$728$	0	0
$729$	−14.1586	−0.524392
$730$	0	0
$731$	38.0430	1.40707
$732$	0	0
$733$	−33.1297	−1.22367	−0.611837	−	0.790984i	$-0.709569\pi$
$733$	−33.1297	−1.22367	−0.611837	+	0.790984i	$0.709569\pi$
$734$	0	0
$735$	−3.34796	−0.123491
$736$	0	0
$737$	3.29219	0.121269
$738$	0	0
$739$	2.60619	0.0958703	0.0479351	−	0.998850i	$-0.484736\pi$
$739$	2.60619	0.0958703	0.0479351	+	0.998850i	$0.484736\pi$
$740$	0	0
$741$	13.7222	0.504096
$742$	0	0
$743$	3.38286	0.124105	0.0620525	−	0.998073i	$-0.480235\pi$
$743$	3.38286	0.124105	0.0620525	+	0.998073i	$0.480235\pi$
$744$	0	0
$745$	16.2227	0.594352
$746$	0	0
$747$	2.79682	0.102330
$748$	0	0
$749$	6.97464	0.254848
$750$	0	0
$751$	−15.3919	−0.561657	−0.280829	−	0.959758i	$-0.590609\pi$
$751$	−15.3919	−0.561657	−0.280829	+	0.959758i	$0.590609\pi$
$752$	0	0
$753$	7.25832	0.264508
$754$	0	0
$755$	0.862179	0.0313779
$756$	0	0
$757$	−18.0167	−0.654827	−0.327413	−	0.944881i	$-0.606177\pi$
$757$	−18.0167	−0.654827	−0.327413	+	0.944881i	$0.606177\pi$
$758$	0	0
$759$	−0.804691	−0.0292084
$760$	0	0
$761$	19.6357	0.711793	0.355896	−	0.934525i	$-0.384176\pi$
$761$	19.6357	0.711793	0.355896	+	0.934525i	$0.384176\pi$
$762$	0	0
$763$	5.74980	0.208157
$764$	0	0
$765$	−12.3936	−0.448093
$766$	0	0
$767$	−36.2745	−1.30980
$768$	0	0
$769$	−9.09017	−0.327800	−0.163900	−	0.986477i	$-0.552407\pi$
$769$	−9.09017	−0.327800	−0.163900	+	0.986477i	$0.552407\pi$
$770$	0	0
$771$	11.3722	0.409561
$772$	0	0
$773$	10.6669	0.383662	0.191831	−	0.981428i	$-0.438557\pi$
$773$	10.6669	0.383662	0.191831	+	0.981428i	$0.438557\pi$
$774$	0	0
$775$	7.16647	0.257427
$776$	0	0
$777$	−2.25176	−0.0807814
$778$	0	0
$779$	−13.1640	−0.471650
$780$	0	0
$781$	−5.85185	−0.209396
$782$	0	0
$783$	−19.0255	−0.679916
$784$	0	0
$785$	7.04611	0.251486
$786$	0	0
$787$	−46.1642	−1.64558	−0.822788	−	0.568348i	$-0.807583\pi$
$787$	−46.1642	−1.64558	−0.822788	+	0.568348i	$0.807583\pi$
$788$	0	0
$789$	12.2800	0.437180
$790$	0	0
$791$	−9.78546	−0.347931
$792$	0	0
$793$	−60.4982	−2.14835
$794$	0	0
$795$	−4.12785	−0.146400
$796$	0	0
$797$	13.0503	0.462265	0.231133	−	0.972922i	$-0.425757\pi$
$797$	13.0503	0.462265	0.231133	+	0.972922i	$0.425757\pi$
$798$	0	0
$799$	−0.812209	−0.0287339
$800$	0	0
$801$	−18.9073	−0.668056
$802$	0	0
$803$	6.01693	0.212333
$804$	0	0
$805$	−1.82306	−0.0642544
$806$	0	0
$807$	6.86329	0.241599
$808$	0	0
$809$	16.9284	0.595170	0.297585	−	0.954695i	$-0.403819\pi$
$809$	16.9284	0.595170	0.297585	+	0.954695i	$0.403819\pi$
$810$	0	0
$811$	30.1239	1.05779	0.528896	−	0.848687i	$-0.322606\pi$
$811$	30.1239	1.05779	0.528896	+	0.848687i	$0.322606\pi$
$812$	0	0
$813$	−8.38380	−0.294033
$814$	0	0
$815$	−5.26671	−0.184485
$816$	0	0
$817$	−35.1845	−1.23095
$818$	0	0
$819$	−10.9704	−0.383338
$820$	0	0
$821$	−22.1272	−0.772244	−0.386122	−	0.922448i	$-0.626186\pi$
$821$	−22.1272	−0.772244	−0.386122	+	0.922448i	$0.626186\pi$
$822$	0	0
$823$	−7.91226	−0.275804	−0.137902	−	0.990446i	$-0.544036\pi$
$823$	−7.91226	−0.275804	−0.137902	+	0.990446i	$0.544036\pi$
$824$	0	0
$825$	0.271549	0.00945412
$826$	0	0
$827$	45.9731	1.59864	0.799320	−	0.600905i	$-0.205193\pi$
$827$	45.9731	1.59864	0.799320	+	0.600905i	$0.205193\pi$
$828$	0	0
$829$	−7.34120	−0.254971	−0.127485	−	0.991840i	$-0.540691\pi$
$829$	−7.34120	−0.254971	−0.127485	+	0.991840i	$0.540691\pi$
$830$	0	0
$831$	−2.98665	−0.103606
$832$	0	0
$833$	29.9032	1.03608
$834$	0	0
$835$	−16.4482	−0.569213
$836$	0	0
$837$	20.8146	0.719458
$838$	0	0
$839$	−1.03211	−0.0356324	−0.0178162	−	0.999841i	$-0.505671\pi$
$839$	−1.03211	−0.0356324	−0.0178162	+	0.999841i	$0.505671\pi$
$840$	0	0
$841$	13.9088	0.479612
$842$	0	0
$843$	−3.57000	−0.122957
$844$	0	0
$845$	−29.2210	−1.00523
$846$	0	0
$847$	6.58981	0.226428
$848$	0	0
$849$	−6.17279	−0.211850
$850$	0	0
$851$	21.4517	0.735354
$852$	0	0
$853$	−33.8256	−1.15817	−0.579083	−	0.815269i	$-0.696589\pi$
$853$	−33.8256	−1.15817	−0.579083	+	0.815269i	$0.696589\pi$
$854$	0	0
$855$	11.4624	0.392006
$856$	0	0
$857$	17.8958	0.611310	0.305655	−	0.952142i	$-0.401125\pi$
$857$	17.8958	0.611310	0.305655	+	0.952142i	$0.401125\pi$
$858$	0	0
$859$	−57.2293	−1.95264	−0.976320	−	0.216332i	$-0.930591\pi$
$859$	−57.2293	−1.95264	−0.976320	+	0.216332i	$0.930591\pi$
$860$	0	0
$861$	−0.980379	−0.0334112
$862$	0	0
$863$	11.7346	0.399452	0.199726	−	0.979852i	$-0.435995\pi$
$863$	11.7346	0.399452	0.199726	+	0.979852i	$0.435995\pi$
$864$	0	0
$865$	−20.1866	−0.686366
$866$	0	0
$867$	−1.71645	−0.0582936
$868$	0	0
$869$	3.72627	0.126405
$870$	0	0
$871$	39.8312	1.34963
$872$	0	0
$873$	29.4354	0.996238
$874$	0	0
$875$	0.615205	0.0207977
$876$	0	0
$877$	0.723755	0.0244395	0.0122197	−	0.999925i	$-0.496110\pi$
$877$	0.723755	0.0244395	0.0122197	+	0.999925i	$0.496110\pi$
$878$	0	0
$879$	−3.05536	−0.103055
$880$	0	0
$881$	19.4127	0.654032	0.327016	−	0.945019i	$-0.393957\pi$
$881$	19.4127	0.654032	0.327016	+	0.945019i	$0.393957\pi$
$882$	0	0
$883$	−43.6838	−1.47008	−0.735038	−	0.678025i	$-0.762836\pi$
$883$	−43.6838	−1.47008	−0.735038	+	0.678025i	$0.762836\pi$
$884$	0	0
$885$	2.82266	0.0948828
$886$	0	0
$887$	−32.9596	−1.10668	−0.553338	−	0.832957i	$-0.686646\pi$
$887$	−32.9596	−1.10668	−0.553338	+	0.832957i	$0.686646\pi$
$888$	0	0
$889$	−9.98630	−0.334930
$890$	0	0
$891$	−3.63298	−0.121709
$892$	0	0
$893$	0.751181	0.0251373
$894$	0	0
$895$	−14.1781	−0.473922
$896$	0	0
$897$	−9.73571	−0.325066
$898$	0	0
$899$	−46.9438	−1.56566
$900$	0	0
$901$	36.8690	1.22828
$902$	0	0
$903$	−2.62034	−0.0871994
$904$	0	0
$905$	2.19357	0.0729167
$906$	0	0
$907$	16.1708	0.536944	0.268472	−	0.963287i	$-0.413481\pi$
$907$	16.1708	0.536944	0.268472	+	0.963287i	$0.413481\pi$
$908$	0	0
$909$	38.4118	1.27404
$910$	0	0
$911$	11.3390	0.375679	0.187840	−	0.982200i	$-0.439851\pi$
$911$	11.3390	0.375679	0.187840	+	0.982200i	$0.439851\pi$
$912$	0	0
$913$	0.547332	0.0181140
$914$	0	0
$915$	4.70761	0.155629
$916$	0	0
$917$	−10.5944	−0.349858
$918$	0	0
$919$	20.2219	0.667060	0.333530	−	0.942739i	$-0.391760\pi$
$919$	20.2219	0.667060	0.333530	+	0.942739i	$0.391760\pi$
$920$	0	0
$921$	0.891261	0.0293681
$922$	0	0
$923$	−70.7997	−2.33040
$924$	0	0
$925$	−7.23902	−0.238018
$926$	0	0
$927$	−29.6296	−0.973163
$928$	0	0
$929$	−50.0179	−1.64103	−0.820517	−	0.571623i	$-0.806314\pi$
$929$	−50.0179	−1.64103	−0.820517	+	0.571623i	$0.806314\pi$
$930$	0	0
$931$	−27.6563	−0.906398
$932$	0	0
$933$	−14.3115	−0.468536
$934$	0	0
$935$	−2.42541	−0.0793194
$936$	0	0
$937$	9.58749	0.313210	0.156605	−	0.987661i	$-0.449945\pi$
$937$	9.58749	0.313210	0.156605	+	0.987661i	$0.449945\pi$
$938$	0	0
$939$	3.17035	0.103460
$940$	0	0
$941$	5.63009	0.183536	0.0917678	−	0.995780i	$-0.470748\pi$
$941$	5.63009	0.183536	0.0917678	+	0.995780i	$0.470748\pi$
$942$	0	0
$943$	9.33971	0.304143
$944$	0	0
$945$	1.78683	0.0581255
$946$	0	0
$947$	18.1295	0.589130	0.294565	−	0.955631i	$-0.404825\pi$
$947$	18.1295	0.589130	0.294565	+	0.955631i	$0.404825\pi$
$948$	0	0
$949$	72.7970	2.36309
$950$	0	0
$951$	−7.73701	−0.250890
$952$	0	0
$953$	9.26882	0.300246	0.150123	−	0.988667i	$-0.452033\pi$
$953$	9.26882	0.300246	0.150123	+	0.988667i	$0.452033\pi$
$954$	0	0
$955$	−14.1259	−0.457102
$956$	0	0
$957$	−1.77877	−0.0574996
$958$	0	0
$959$	0.300790	0.00971302
$960$	0	0
$961$	20.3582	0.656718
$962$	0	0
$963$	31.1130	1.00260
$964$	0	0
$965$	−11.4674	−0.369147
$966$	0	0
$967$	32.5517	1.04679	0.523395	−	0.852090i	$-0.324665\pi$
$967$	32.5517	1.04679	0.523395	+	0.852090i	$0.324665\pi$
$968$	0	0
$969$	9.53712	0.306377
$970$	0	0
$971$	51.5837	1.65540	0.827700	−	0.561171i	$-0.189649\pi$
$971$	51.5837	1.65540	0.827700	+	0.561171i	$0.189649\pi$
$972$	0	0
$973$	−2.36920	−0.0759530
$974$	0	0
$975$	3.28539	0.105217
$976$	0	0
$977$	−19.2808	−0.616848	−0.308424	−	0.951249i	$-0.599802\pi$
$977$	−19.2808	−0.616848	−0.308424	+	0.951249i	$0.599802\pi$
$978$	0	0
$979$	−3.70012	−0.118256
$980$	0	0
$981$	25.6491	0.818914
$982$	0	0
$983$	−23.6387	−0.753956	−0.376978	−	0.926222i	$-0.623037\pi$
$983$	−23.6387	−0.753956	−0.376978	+	0.926222i	$0.623037\pi$
$984$	0	0
$985$	−18.8067	−0.599230
$986$	0	0
$987$	0.0559436	0.00178070
$988$	0	0
$989$	24.9630	0.793777
$990$	0	0
$991$	−13.3985	−0.425618	−0.212809	−	0.977094i	$-0.568261\pi$
$991$	−13.3985	−0.425618	−0.212809	+	0.977094i	$0.568261\pi$
$992$	0	0
$993$	7.01104	0.222489
$994$	0	0
$995$	17.3273	0.549311
$996$	0	0
$997$	29.7231	0.941339	0.470669	−	0.882310i	$-0.344013\pi$
$997$	29.7231	0.941339	0.470669	+	0.882310i	$0.344013\pi$
$998$	0	0
$999$	−21.0253	−0.665212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.16	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.16	✓	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)

\(f(q)\)	\(=\)	\(q-0.505617 q^{3} -1.00000 q^{5} -0.615205 q^{7} -2.74435 q^{9} +O(q^{10})\)	\(q-0.505617 q^{3} -1.00000 q^{5} -0.615205 q^{7} -2.74435 q^{9} -0.537064 q^{11} -6.49777 q^{13} +0.505617 q^{15} -4.51606 q^{17} +4.17673 q^{19} +0.311058 q^{21} -2.96334 q^{23} +1.00000 q^{25} +2.90444 q^{27} -6.55048 q^{29} +7.16647 q^{31} +0.271549 q^{33} +0.615205 q^{35} -7.23902 q^{37} +3.28539 q^{39} -3.15175 q^{41} -8.42394 q^{43} +2.74435 q^{45} +0.179849 q^{47} -6.62152 q^{49} +2.28340 q^{51} -8.16398 q^{53} +0.537064 q^{55} -2.11183 q^{57} +5.58261 q^{59} +9.31061 q^{61} +1.68834 q^{63} +6.49777 q^{65} -6.12998 q^{67} +1.49832 q^{69} +10.8960 q^{71} -11.2034 q^{73} -0.505617 q^{75} +0.330404 q^{77} -6.93822 q^{79} +6.76452 q^{81} -1.01912 q^{83} +4.51606 q^{85} +3.31204 q^{87} +6.88952 q^{89} +3.99746 q^{91} -3.62349 q^{93} -4.17673 q^{95} -10.7258 q^{97} +1.47389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 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

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Groups

Database

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