LMFDB - Embedded newform 8020.2.a.e.1.12

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			1.12
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)

$f(q)$	$=$	$q-1.36773 q^{3} -1.00000 q^{5} +2.86190 q^{7} -1.12932 q^{9} +O(q^{10})$	$q-1.36773 q^{3} -1.00000 q^{5} +2.86190 q^{7} -1.12932 q^{9} +1.14852 q^{11} +0.207337 q^{13} +1.36773 q^{15} +5.28021 q^{17} +8.39477 q^{19} -3.91430 q^{21} +7.69921 q^{23} +1.00000 q^{25} +5.64779 q^{27} +0.349148 q^{29} +7.66838 q^{31} -1.57086 q^{33} -2.86190 q^{35} -9.65526 q^{37} -0.283580 q^{39} -3.06907 q^{41} +8.05535 q^{43} +1.12932 q^{45} +5.25656 q^{47} +1.19048 q^{49} -7.22188 q^{51} -9.65930 q^{53} -1.14852 q^{55} -11.4818 q^{57} +0.336770 q^{59} -1.01160 q^{61} -3.23202 q^{63} -0.207337 q^{65} -5.12216 q^{67} -10.5304 q^{69} -1.96365 q^{71} +13.6126 q^{73} -1.36773 q^{75} +3.28695 q^{77} -11.4337 q^{79} -4.33665 q^{81} +4.88198 q^{83} -5.28021 q^{85} -0.477539 q^{87} -0.560556 q^{89} +0.593377 q^{91} -10.4882 q^{93} -8.39477 q^{95} -15.1831 q^{97} -1.29705 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.36773	−0.789657	−0.394829	−	0.918755i	$-0.629196\pi$
$3$	−1.36773	−0.789657	−0.394829	+	0.918755i	$0.629196\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.86190	1.08170	0.540849	−	0.841120i	$-0.318103\pi$
$7$	2.86190	1.08170	0.540849	+	0.841120i	$0.318103\pi$
$8$	0	0
$9$	−1.12932	−0.376441
$10$	0	0
$11$	1.14852	0.346292	0.173146	−	0.984896i	$-0.444607\pi$
$11$	1.14852	0.346292	0.173146	+	0.984896i	$0.444607\pi$
$12$	0	0
$13$	0.207337	0.0575049	0.0287524	−	0.999587i	$-0.490847\pi$
$13$	0.207337	0.0575049	0.0287524	+	0.999587i	$0.490847\pi$
$14$	0	0
$15$	1.36773	0.353145
$16$	0	0
$17$	5.28021	1.28064	0.640319	−	0.768109i	$-0.278802\pi$
$17$	5.28021	1.28064	0.640319	+	0.768109i	$0.278802\pi$
$18$	0	0
$19$	8.39477	1.92589	0.962947	−	0.269692i	$-0.0869218\pi$
$19$	8.39477	1.92589	0.962947	+	0.269692i	$0.0869218\pi$
$20$	0	0
$21$	−3.91430	−0.854170
$22$	0	0
$23$	7.69921	1.60540	0.802698	−	0.596386i	$-0.203397\pi$
$23$	7.69921	1.60540	0.802698	+	0.596386i	$0.203397\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.64779	1.08692
$28$	0	0
$29$	0.349148	0.0648351	0.0324176	−	0.999474i	$-0.489679\pi$
$29$	0.349148	0.0648351	0.0324176	+	0.999474i	$0.489679\pi$
$30$	0	0
$31$	7.66838	1.37728	0.688641	−	0.725103i	$-0.258208\pi$
$31$	7.66838	1.37728	0.688641	+	0.725103i	$0.258208\pi$
$32$	0	0
$33$	−1.57086	−0.273452
$34$	0	0
$35$	−2.86190	−0.483750
$36$	0	0
$37$	−9.65526	−1.58732	−0.793658	−	0.608364i	$-0.791826\pi$
$37$	−9.65526	−1.58732	−0.793658	+	0.608364i	$0.791826\pi$
$38$	0	0
$39$	−0.283580	−0.0454091
$40$	0	0
$41$	−3.06907	−0.479309	−0.239654	−	0.970858i	$-0.577034\pi$
$41$	−3.06907	−0.479309	−0.239654	+	0.970858i	$0.577034\pi$
$42$	0	0
$43$	8.05535	1.22843	0.614215	−	0.789139i	$-0.289473\pi$
$43$	8.05535	1.22843	0.614215	+	0.789139i	$0.289473\pi$
$44$	0	0
$45$	1.12932	0.168350
$46$	0	0
$47$	5.25656	0.766748	0.383374	−	0.923593i	$-0.374762\pi$
$47$	5.25656	0.766748	0.383374	+	0.923593i	$0.374762\pi$
$48$	0	0
$49$	1.19048	0.170069
$50$	0	0
$51$	−7.22188	−1.01127
$52$	0	0
$53$	−9.65930	−1.32681	−0.663404	−	0.748262i	$-0.730889\pi$
$53$	−9.65930	−1.32681	−0.663404	+	0.748262i	$0.730889\pi$
$54$	0	0
$55$	−1.14852	−0.154866
$56$	0	0
$57$	−11.4818	−1.52080
$58$	0	0
$59$	0.336770	0.0438437	0.0219218	−	0.999760i	$-0.493022\pi$
$59$	0.336770	0.0438437	0.0219218	+	0.999760i	$0.493022\pi$
$60$	0	0
$61$	−1.01160	−0.129522	−0.0647610	−	0.997901i	$-0.520628\pi$
$61$	−1.01160	−0.129522	−0.0647610	+	0.997901i	$0.520628\pi$
$62$	0	0
$63$	−3.23202	−0.407196
$64$	0	0
$65$	−0.207337	−0.0257170
$66$	0	0
$67$	−5.12216	−0.625771	−0.312886	−	0.949791i	$-0.601296\pi$
$67$	−5.12216	−0.625771	−0.312886	+	0.949791i	$0.601296\pi$
$68$	0	0
$69$	−10.5304	−1.26771
$70$	0	0
$71$	−1.96365	−0.233042	−0.116521	−	0.993188i	$-0.537174\pi$
$71$	−1.96365	−0.233042	−0.116521	+	0.993188i	$0.537174\pi$
$72$	0	0
$73$	13.6126	1.59323	0.796617	−	0.604484i	$-0.206621\pi$
$73$	13.6126	1.59323	0.796617	+	0.604484i	$0.206621\pi$
$74$	0	0
$75$	−1.36773	−0.157931
$76$	0	0
$77$	3.28695	0.374583
$78$	0	0
$79$	−11.4337	−1.28639	−0.643196	−	0.765702i	$-0.722392\pi$
$79$	−11.4337	−1.28639	−0.643196	+	0.765702i	$0.722392\pi$
$80$	0	0
$81$	−4.33665	−0.481851
$82$	0	0
$83$	4.88198	0.535866	0.267933	−	0.963437i	$-0.413659\pi$
$83$	4.88198	0.535866	0.267933	+	0.963437i	$0.413659\pi$
$84$	0	0
$85$	−5.28021	−0.572719
$86$	0	0
$87$	−0.477539	−0.0511975
$88$	0	0
$89$	−0.560556	−0.0594188	−0.0297094	−	0.999559i	$-0.509458\pi$
$89$	−0.560556	−0.0594188	−0.0297094	+	0.999559i	$0.509458\pi$
$90$	0	0
$91$	0.593377	0.0622028
$92$	0	0
$93$	−10.4882	−1.08758
$94$	0	0
$95$	−8.39477	−0.861286
$96$	0	0
$97$	−15.1831	−1.54161	−0.770806	−	0.637071i	$-0.780146\pi$
$97$	−15.1831	−1.54161	−0.770806	+	0.637071i	$0.780146\pi$
$98$	0	0
$99$	−1.29705	−0.130359
$100$	0	0
$101$	17.0421	1.69575	0.847877	−	0.530192i	$-0.177880\pi$
$101$	17.0421	1.69575	0.847877	+	0.530192i	$0.177880\pi$
$102$	0	0
$103$	3.71701	0.366248	0.183124	−	0.983090i	$-0.441379\pi$
$103$	3.71701	0.366248	0.183124	+	0.983090i	$0.441379\pi$
$104$	0	0
$105$	3.91430	0.381997
$106$	0	0
$107$	0.137131	0.0132569	0.00662846	−	0.999978i	$-0.497890\pi$
$107$	0.137131	0.0132569	0.00662846	+	0.999978i	$0.497890\pi$
$108$	0	0
$109$	−7.29097	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$109$	−7.29097	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$110$	0	0
$111$	13.2058	1.25344
$112$	0	0
$113$	3.69886	0.347959	0.173980	−	0.984749i	$-0.444337\pi$
$113$	3.69886	0.347959	0.173980	+	0.984749i	$0.444337\pi$
$114$	0	0
$115$	−7.69921	−0.717955
$116$	0	0
$117$	−0.234150	−0.0216472
$118$	0	0
$119$	15.1114	1.38526
$120$	0	0
$121$	−9.68090	−0.880082
$122$	0	0
$123$	4.19766	0.378490
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−4.82719	−0.428344	−0.214172	−	0.976796i	$-0.568705\pi$
$127$	−4.82719	−0.428344	−0.214172	+	0.976796i	$0.568705\pi$
$128$	0	0
$129$	−11.0175	−0.970038
$130$	0	0
$131$	2.96402	0.258967	0.129484	−	0.991582i	$-0.458668\pi$
$131$	2.96402	0.258967	0.129484	+	0.991582i	$0.458668\pi$
$132$	0	0
$133$	24.0250	2.08323
$134$	0	0
$135$	−5.64779	−0.486084
$136$	0	0
$137$	6.01475	0.513875	0.256937	−	0.966428i	$-0.417287\pi$
$137$	6.01475	0.513875	0.256937	+	0.966428i	$0.417287\pi$
$138$	0	0
$139$	−14.0343	−1.19038	−0.595188	−	0.803586i	$-0.702923\pi$
$139$	−14.0343	−1.19038	−0.595188	+	0.803586i	$0.702923\pi$
$140$	0	0
$141$	−7.18953	−0.605468
$142$	0	0
$143$	0.238130	0.0199135
$144$	0	0
$145$	−0.349148	−0.0289951
$146$	0	0
$147$	−1.62826	−0.134296
$148$	0	0
$149$	2.68064	0.219606	0.109803	−	0.993953i	$-0.464978\pi$
$149$	2.68064	0.219606	0.109803	+	0.993953i	$0.464978\pi$
$150$	0	0
$151$	17.8522	1.45279	0.726394	−	0.687279i	$-0.241195\pi$
$151$	17.8522	1.45279	0.726394	+	0.687279i	$0.241195\pi$
$152$	0	0
$153$	−5.96306	−0.482085
$154$	0	0
$155$	−7.66838	−0.615939
$156$	0	0
$157$	2.77767	0.221682	0.110841	−	0.993838i	$-0.464646\pi$
$157$	2.77767	0.221682	0.110841	+	0.993838i	$0.464646\pi$
$158$	0	0
$159$	13.2113	1.04772
$160$	0	0
$161$	22.0344	1.73655
$162$	0	0
$163$	−12.8033	−1.00284	−0.501418	−	0.865205i	$-0.667188\pi$
$163$	−12.8033	−1.00284	−0.501418	+	0.865205i	$0.667188\pi$
$164$	0	0
$165$	1.57086	0.122291
$166$	0	0
$167$	23.9713	1.85496	0.927479	−	0.373876i	$-0.121972\pi$
$167$	23.9713	1.85496	0.927479	+	0.373876i	$0.121972\pi$
$168$	0	0
$169$	−12.9570	−0.996693
$170$	0	0
$171$	−9.48042	−0.724986
$172$	0	0
$173$	−5.30117	−0.403041	−0.201520	−	0.979484i	$-0.564588\pi$
$173$	−5.30117	−0.403041	−0.201520	+	0.979484i	$0.564588\pi$
$174$	0	0
$175$	2.86190	0.216339
$176$	0	0
$177$	−0.460609	−0.0346215
$178$	0	0
$179$	2.88054	0.215302	0.107651	−	0.994189i	$-0.465667\pi$
$179$	2.88054	0.215302	0.107651	+	0.994189i	$0.465667\pi$
$180$	0	0
$181$	18.3382	1.36307	0.681533	−	0.731787i	$-0.261314\pi$
$181$	18.3382	1.36307	0.681533	+	0.731787i	$0.261314\pi$
$182$	0	0
$183$	1.38359	0.102278
$184$	0	0
$185$	9.65526	0.709869
$186$	0	0
$187$	6.06442	0.443475
$188$	0	0
$189$	16.1634	1.17572
$190$	0	0
$191$	−1.87860	−0.135931	−0.0679655	−	0.997688i	$-0.521651\pi$
$191$	−1.87860	−0.135931	−0.0679655	+	0.997688i	$0.521651\pi$
$192$	0	0
$193$	−7.62051	−0.548536	−0.274268	−	0.961653i	$-0.588436\pi$
$193$	−7.62051	−0.548536	−0.274268	+	0.961653i	$0.588436\pi$
$194$	0	0
$195$	0.283580	0.0203076
$196$	0	0
$197$	−4.24790	−0.302650	−0.151325	−	0.988484i	$-0.548354\pi$
$197$	−4.24790	−0.302650	−0.151325	+	0.988484i	$0.548354\pi$
$198$	0	0
$199$	−18.8089	−1.33332	−0.666662	−	0.745360i	$-0.732278\pi$
$199$	−18.8089	−1.33332	−0.666662	+	0.745360i	$0.732278\pi$
$200$	0	0
$201$	7.00571	0.494145
$202$	0	0
$203$	0.999227	0.0701320
$204$	0	0
$205$	3.06907	0.214353
$206$	0	0
$207$	−8.69490	−0.604337
$208$	0	0
$209$	9.64157	0.666921
$210$	0	0
$211$	−11.0341	−0.759620	−0.379810	−	0.925065i	$-0.624011\pi$
$211$	−11.0341	−0.759620	−0.379810	+	0.925065i	$0.624011\pi$
$212$	0	0
$213$	2.68573	0.184023
$214$	0	0
$215$	−8.05535	−0.549370
$216$	0	0
$217$	21.9462	1.48980
$218$	0	0
$219$	−18.6183	−1.25811
$220$	0	0
$221$	1.09478	0.0736429
$222$	0	0
$223$	23.8522	1.59726	0.798629	−	0.601823i	$-0.205559\pi$
$223$	23.8522	1.59726	0.798629	+	0.601823i	$0.205559\pi$
$224$	0	0
$225$	−1.12932	−0.0752883
$226$	0	0
$227$	−11.1485	−0.739951	−0.369975	−	0.929042i	$-0.620634\pi$
$227$	−11.1485	−0.739951	−0.369975	+	0.929042i	$0.620634\pi$
$228$	0	0
$229$	17.3979	1.14968	0.574842	−	0.818265i	$-0.305064\pi$
$229$	17.3979	1.14968	0.574842	+	0.818265i	$0.305064\pi$
$230$	0	0
$231$	−4.49565	−0.295792
$232$	0	0
$233$	10.5436	0.690736	0.345368	−	0.938467i	$-0.387754\pi$
$233$	10.5436	0.690736	0.345368	+	0.938467i	$0.387754\pi$
$234$	0	0
$235$	−5.25656	−0.342900
$236$	0	0
$237$	15.6382	1.01581
$238$	0	0
$239$	17.1394	1.10865	0.554327	−	0.832299i	$-0.312976\pi$
$239$	17.1394	1.10865	0.554327	+	0.832299i	$0.312976\pi$
$240$	0	0
$241$	15.8772	1.02274	0.511372	−	0.859360i	$-0.329138\pi$
$241$	15.8772	1.02274	0.511372	+	0.859360i	$0.329138\pi$
$242$	0	0
$243$	−11.0120	−0.706420
$244$	0	0
$245$	−1.19048	−0.0760573
$246$	0	0
$247$	1.74054	0.110748
$248$	0	0
$249$	−6.67721	−0.423151
$250$	0	0
$251$	20.7863	1.31202	0.656010	−	0.754753i	$-0.272243\pi$
$251$	20.7863	1.31202	0.656010	+	0.754753i	$0.272243\pi$
$252$	0	0
$253$	8.84269	0.555935
$254$	0	0
$255$	7.22188	0.452252
$256$	0	0
$257$	1.11340	0.0694519	0.0347260	−	0.999397i	$-0.488944\pi$
$257$	1.11340	0.0694519	0.0347260	+	0.999397i	$0.488944\pi$
$258$	0	0
$259$	−27.6324	−1.71700
$260$	0	0
$261$	−0.394301	−0.0244066
$262$	0	0
$263$	−0.463126	−0.0285576	−0.0142788	−	0.999898i	$-0.504545\pi$
$263$	−0.463126	−0.0285576	−0.0142788	+	0.999898i	$0.504545\pi$
$264$	0	0
$265$	9.65930	0.593366
$266$	0	0
$267$	0.766687	0.0469205
$268$	0	0
$269$	−14.0184	−0.854715	−0.427358	−	0.904083i	$-0.640555\pi$
$269$	−14.0184	−0.854715	−0.427358	+	0.904083i	$0.640555\pi$
$270$	0	0
$271$	24.5593	1.49187	0.745937	−	0.666017i	$-0.232002\pi$
$271$	24.5593	1.49187	0.745937	+	0.666017i	$0.232002\pi$
$272$	0	0
$273$	−0.811578	−0.0491189
$274$	0	0
$275$	1.14852	0.0692584
$276$	0	0
$277$	2.55926	0.153771	0.0768855	−	0.997040i	$-0.475502\pi$
$277$	2.55926	0.153771	0.0768855	+	0.997040i	$0.475502\pi$
$278$	0	0
$279$	−8.66009	−0.518466
$280$	0	0
$281$	−32.2824	−1.92581	−0.962904	−	0.269842i	$-0.913028\pi$
$281$	−32.2824	−1.92581	−0.962904	+	0.269842i	$0.913028\pi$
$282$	0	0
$283$	−21.9693	−1.30594	−0.652970	−	0.757384i	$-0.726477\pi$
$283$	−21.9693	−1.30594	−0.652970	+	0.757384i	$0.726477\pi$
$284$	0	0
$285$	11.4818	0.680121
$286$	0	0
$287$	−8.78339	−0.518467
$288$	0	0
$289$	10.8806	0.640034
$290$	0	0
$291$	20.7663	1.21734
$292$	0	0
$293$	−13.7387	−0.802622	−0.401311	−	0.915942i	$-0.631445\pi$
$293$	−13.7387	−0.802622	−0.401311	+	0.915942i	$0.631445\pi$
$294$	0	0
$295$	−0.336770	−0.0196075
$296$	0	0
$297$	6.48660	0.376390
$298$	0	0
$299$	1.59633	0.0923180
$300$	0	0
$301$	23.0536	1.32879
$302$	0	0
$303$	−23.3090	−1.33907
$304$	0	0
$305$	1.01160	0.0579240
$306$	0	0
$307$	3.22204	0.183892	0.0919459	−	0.995764i	$-0.470691\pi$
$307$	3.22204	0.183892	0.0919459	+	0.995764i	$0.470691\pi$
$308$	0	0
$309$	−5.08386	−0.289210
$310$	0	0
$311$	−28.7152	−1.62829	−0.814144	−	0.580663i	$-0.802793\pi$
$311$	−28.7152	−1.62829	−0.814144	+	0.580663i	$0.802793\pi$
$312$	0	0
$313$	−6.88909	−0.389395	−0.194697	−	0.980863i	$-0.562372\pi$
$313$	−6.88909	−0.389395	−0.194697	+	0.980863i	$0.562372\pi$
$314$	0	0
$315$	3.23202	0.182103
$316$	0	0
$317$	−9.29310	−0.521953	−0.260976	−	0.965345i	$-0.584044\pi$
$317$	−9.29310	−0.521953	−0.260976	+	0.965345i	$0.584044\pi$
$318$	0	0
$319$	0.401003	0.0224519
$320$	0	0
$321$	−0.187557	−0.0104684
$322$	0	0
$323$	44.3261	2.46637
$324$	0	0
$325$	0.207337	0.0115010
$326$	0	0
$327$	9.97206	0.551456
$328$	0	0
$329$	15.0438	0.829389
$330$	0	0
$331$	3.93408	0.216237	0.108118	−	0.994138i	$-0.465517\pi$
$331$	3.93408	0.216237	0.108118	+	0.994138i	$0.465517\pi$
$332$	0	0
$333$	10.9039	0.597531
$334$	0	0
$335$	5.12216	0.279853
$336$	0	0
$337$	3.24690	0.176870	0.0884350	−	0.996082i	$-0.471813\pi$
$337$	3.24690	0.176870	0.0884350	+	0.996082i	$0.471813\pi$
$338$	0	0
$339$	−5.05902	−0.274768
$340$	0	0
$341$	8.80729	0.476941
$342$	0	0
$343$	−16.6263	−0.897734
$344$	0	0
$345$	10.5304	0.566938
$346$	0	0
$347$	25.5386	1.37099	0.685493	−	0.728079i	$-0.259587\pi$
$347$	25.5386	1.37099	0.685493	+	0.728079i	$0.259587\pi$
$348$	0	0
$349$	−15.8849	−0.850300	−0.425150	−	0.905123i	$-0.639779\pi$
$349$	−15.8849	−0.850300	−0.425150	+	0.905123i	$0.639779\pi$
$350$	0	0
$351$	1.17099	0.0625030
$352$	0	0
$353$	20.4959	1.09088	0.545442	−	0.838148i	$-0.316362\pi$
$353$	20.4959	1.09088	0.545442	+	0.838148i	$0.316362\pi$
$354$	0	0
$355$	1.96365	0.104220
$356$	0	0
$357$	−20.6683	−1.09388
$358$	0	0
$359$	21.4150	1.13024	0.565121	−	0.825008i	$-0.308830\pi$
$359$	21.4150	1.13024	0.565121	+	0.825008i	$0.308830\pi$
$360$	0	0
$361$	51.4722	2.70907
$362$	0	0
$363$	13.2408	0.694963
$364$	0	0
$365$	−13.6126	−0.712516
$366$	0	0
$367$	−26.1799	−1.36658	−0.683291	−	0.730146i	$-0.739452\pi$
$367$	−26.1799	−1.36658	−0.683291	+	0.730146i	$0.739452\pi$
$368$	0	0
$369$	3.46598	0.180432
$370$	0	0
$371$	−27.6440	−1.43520
$372$	0	0
$373$	−36.2946	−1.87926	−0.939632	−	0.342188i	$-0.888832\pi$
$373$	−36.2946	−1.87926	−0.939632	+	0.342188i	$0.888832\pi$
$374$	0	0
$375$	1.36773	0.0706291
$376$	0	0
$377$	0.0723911	0.00372833
$378$	0	0
$379$	−2.56644	−0.131829	−0.0659145	−	0.997825i	$-0.520996\pi$
$379$	−2.56644	−0.131829	−0.0659145	+	0.997825i	$0.520996\pi$
$380$	0	0
$381$	6.60227	0.338245
$382$	0	0
$383$	−2.60925	−0.133327	−0.0666633	−	0.997776i	$-0.521235\pi$
$383$	−2.60925	−0.133327	−0.0666633	+	0.997776i	$0.521235\pi$
$384$	0	0
$385$	−3.28695	−0.167519
$386$	0	0
$387$	−9.09710	−0.462432
$388$	0	0
$389$	−35.8015	−1.81521	−0.907603	−	0.419829i	$-0.862090\pi$
$389$	−35.8015	−1.81521	−0.907603	+	0.419829i	$0.862090\pi$
$390$	0	0
$391$	40.6534	2.05593
$392$	0	0
$393$	−4.05396	−0.204495
$394$	0	0
$395$	11.4337	0.575292
$396$	0	0
$397$	−17.2945	−0.867985	−0.433992	−	0.900917i	$-0.642896\pi$
$397$	−17.2945	−0.867985	−0.433992	+	0.900917i	$0.642896\pi$
$398$	0	0
$399$	−32.8597	−1.64504
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	1.58994	0.0792004
$404$	0	0
$405$	4.33665	0.215490
$406$	0	0
$407$	−11.0893	−0.549674
$408$	0	0
$409$	18.6456	0.921967	0.460983	−	0.887409i	$-0.347497\pi$
$409$	18.6456	0.921967	0.460983	+	0.887409i	$0.347497\pi$
$410$	0	0
$411$	−8.22653	−0.405785
$412$	0	0
$413$	0.963801	0.0474256
$414$	0	0
$415$	−4.88198	−0.239647
$416$	0	0
$417$	19.1951	0.939990
$418$	0	0
$419$	2.65192	0.129555	0.0647775	−	0.997900i	$-0.479366\pi$
$419$	2.65192	0.129555	0.0647775	+	0.997900i	$0.479366\pi$
$420$	0	0
$421$	−6.89753	−0.336165	−0.168083	−	0.985773i	$-0.553758\pi$
$421$	−6.89753	−0.336165	−0.168083	+	0.985773i	$0.553758\pi$
$422$	0	0
$423$	−5.93636	−0.288636
$424$	0	0
$425$	5.28021	0.256128
$426$	0	0
$427$	−2.89510	−0.140104
$428$	0	0
$429$	−0.325697	−0.0157248
$430$	0	0
$431$	25.1785	1.21281	0.606404	−	0.795157i	$-0.292612\pi$
$431$	25.1785	1.21281	0.606404	+	0.795157i	$0.292612\pi$
$432$	0	0
$433$	16.7174	0.803388	0.401694	−	0.915774i	$-0.368422\pi$
$433$	16.7174	0.803388	0.401694	+	0.915774i	$0.368422\pi$
$434$	0	0
$435$	0.477539	0.0228962
$436$	0	0
$437$	64.6331	3.09182
$438$	0	0
$439$	15.1203	0.721650	0.360825	−	0.932633i	$-0.382495\pi$
$439$	15.1203	0.721650	0.360825	+	0.932633i	$0.382495\pi$
$440$	0	0
$441$	−1.34444	−0.0640211
$442$	0	0
$443$	−33.8464	−1.60809	−0.804045	−	0.594568i	$-0.797323\pi$
$443$	−33.8464	−1.60809	−0.804045	+	0.594568i	$0.797323\pi$
$444$	0	0
$445$	0.560556	0.0265729
$446$	0	0
$447$	−3.66638	−0.173414
$448$	0	0
$449$	−0.931838	−0.0439762	−0.0219881	−	0.999758i	$-0.507000\pi$
$449$	−0.931838	−0.0439762	−0.0219881	+	0.999758i	$0.507000\pi$
$450$	0	0
$451$	−3.52489	−0.165981
$452$	0	0
$453$	−24.4169	−1.14720
$454$	0	0
$455$	−0.593377	−0.0278180
$456$	0	0
$457$	−29.9505	−1.40102	−0.700512	−	0.713641i	$-0.747045\pi$
$457$	−29.9505	−1.40102	−0.700512	+	0.713641i	$0.747045\pi$
$458$	0	0
$459$	29.8215	1.39195
$460$	0	0
$461$	10.1219	0.471424	0.235712	−	0.971823i	$-0.424258\pi$
$461$	10.1219	0.471424	0.235712	+	0.971823i	$0.424258\pi$
$462$	0	0
$463$	0.0165686	0.000770006 0	0.000385003	−	1.00000i	$-0.499877\pi$
$463$	0.0165686	0.000770006 0	0.000385003		1.00000i	$0.499877\pi$
$464$	0	0
$465$	10.4882	0.486381
$466$	0	0
$467$	−9.77756	−0.452452	−0.226226	−	0.974075i	$-0.572639\pi$
$467$	−9.77756	−0.452452	−0.226226	+	0.974075i	$0.572639\pi$
$468$	0	0
$469$	−14.6591	−0.676895
$470$	0	0
$471$	−3.79909	−0.175053
$472$	0	0
$473$	9.25173	0.425395
$474$	0	0
$475$	8.39477	0.385179
$476$	0	0
$477$	10.9085	0.499465
$478$	0	0
$479$	−10.1109	−0.461981	−0.230991	−	0.972956i	$-0.574197\pi$
$479$	−10.1109	−0.461981	−0.230991	+	0.972956i	$0.574197\pi$
$480$	0	0
$481$	−2.00189	−0.0912784
$482$	0	0
$483$	−30.1370	−1.37128
$484$	0	0
$485$	15.1831	0.689429
$486$	0	0
$487$	27.9909	1.26839	0.634194	−	0.773174i	$-0.281332\pi$
$487$	27.9909	1.26839	0.634194	+	0.773174i	$0.281332\pi$
$488$	0	0
$489$	17.5115	0.791896
$490$	0	0
$491$	25.1004	1.13276	0.566382	−	0.824143i	$-0.308343\pi$
$491$	25.1004	1.13276	0.566382	+	0.824143i	$0.308343\pi$
$492$	0	0
$493$	1.84357	0.0830303
$494$	0	0
$495$	1.29705	0.0582981
$496$	0	0
$497$	−5.61976	−0.252081
$498$	0	0
$499$	14.2919	0.639794	0.319897	−	0.947452i	$-0.396352\pi$
$499$	14.2919	0.639794	0.319897	+	0.947452i	$0.396352\pi$
$500$	0	0
$501$	−32.7862	−1.46478
$502$	0	0
$503$	37.2904	1.66270	0.831348	−	0.555752i	$-0.187569\pi$
$503$	37.2904	1.66270	0.831348	+	0.555752i	$0.187569\pi$
$504$	0	0
$505$	−17.0421	−0.758365
$506$	0	0
$507$	17.7216	0.787046
$508$	0	0
$509$	24.9762	1.10705	0.553525	−	0.832832i	$-0.313282\pi$
$509$	24.9762	1.10705	0.553525	+	0.832832i	$0.313282\pi$
$510$	0	0
$511$	38.9579	1.72340
$512$	0	0
$513$	47.4119	2.09329
$514$	0	0
$515$	−3.71701	−0.163791
$516$	0	0
$517$	6.03726	0.265518
$518$	0	0
$519$	7.25056	0.318264
$520$	0	0
$521$	21.8889	0.958968	0.479484	−	0.877551i	$-0.340824\pi$
$521$	21.8889	0.958968	0.479484	+	0.877551i	$0.340824\pi$
$522$	0	0
$523$	−9.92845	−0.434141	−0.217070	−	0.976156i	$-0.569650\pi$
$523$	−9.92845	−0.434141	−0.217070	+	0.976156i	$0.569650\pi$
$524$	0	0
$525$	−3.91430	−0.170834
$526$	0	0
$527$	40.4906	1.76380
$528$	0	0
$529$	36.2778	1.57729
$530$	0	0
$531$	−0.380322	−0.0165046
$532$	0	0
$533$	−0.636332	−0.0275626
$534$	0	0
$535$	−0.137131	−0.00592867
$536$	0	0
$537$	−3.93980	−0.170015
$538$	0	0
$539$	1.36730	0.0588936
$540$	0	0
$541$	20.1205	0.865049	0.432524	−	0.901622i	$-0.357623\pi$
$541$	20.1205	0.865049	0.432524	+	0.901622i	$0.357623\pi$
$542$	0	0
$543$	−25.0816	−1.07636
$544$	0	0
$545$	7.29097	0.312311
$546$	0	0
$547$	−40.1895	−1.71838	−0.859190	−	0.511657i	$-0.829032\pi$
$547$	−40.1895	−1.71838	−0.859190	+	0.511657i	$0.829032\pi$
$548$	0	0
$549$	1.14242	0.0487574
$550$	0	0
$551$	2.93102	0.124866
$552$	0	0
$553$	−32.7221	−1.39149
$554$	0	0
$555$	−13.2058	−0.560553
$556$	0	0
$557$	37.5866	1.59260	0.796299	−	0.604904i	$-0.206788\pi$
$557$	37.5866	1.59260	0.796299	+	0.604904i	$0.206788\pi$
$558$	0	0
$559$	1.67017	0.0706406
$560$	0	0
$561$	−8.29447	−0.350193
$562$	0	0
$563$	−8.23872	−0.347221	−0.173610	−	0.984814i	$-0.555543\pi$
$563$	−8.23872	−0.347221	−0.173610	+	0.984814i	$0.555543\pi$
$564$	0	0
$565$	−3.69886	−0.155612
$566$	0	0
$567$	−12.4111	−0.521216
$568$	0	0
$569$	−16.9338	−0.709901	−0.354951	−	0.934885i	$-0.615502\pi$
$569$	−16.9338	−0.709901	−0.354951	+	0.934885i	$0.615502\pi$
$570$	0	0
$571$	31.9560	1.33732	0.668659	−	0.743570i	$-0.266869\pi$
$571$	31.9560	1.33732	0.668659	+	0.743570i	$0.266869\pi$
$572$	0	0
$573$	2.56942	0.107339
$574$	0	0
$575$	7.69921	0.321079
$576$	0	0
$577$	−1.10426	−0.0459709	−0.0229854	−	0.999736i	$-0.507317\pi$
$577$	−1.10426	−0.0459709	−0.0229854	+	0.999736i	$0.507317\pi$
$578$	0	0
$579$	10.4228	0.433156
$580$	0	0
$581$	13.9717	0.579645
$582$	0	0
$583$	−11.0939	−0.459462
$584$	0	0
$585$	0.234150	0.00968092
$586$	0	0
$587$	27.7320	1.14462	0.572312	−	0.820036i	$-0.306047\pi$
$587$	27.7320	1.14462	0.572312	+	0.820036i	$0.306047\pi$
$588$	0	0
$589$	64.3743	2.65250
$590$	0	0
$591$	5.80996	0.238990
$592$	0	0
$593$	−18.6278	−0.764951	−0.382476	−	0.923966i	$-0.624928\pi$
$593$	−18.6278	−0.764951	−0.382476	+	0.923966i	$0.624928\pi$
$594$	0	0
$595$	−15.1114	−0.619508
$596$	0	0
$597$	25.7254	1.05287
$598$	0	0
$599$	−5.55623	−0.227021	−0.113511	−	0.993537i	$-0.536210\pi$
$599$	−5.55623	−0.227021	−0.113511	+	0.993537i	$0.536210\pi$
$600$	0	0
$601$	−25.7683	−1.05111	−0.525556	−	0.850759i	$-0.676143\pi$
$601$	−25.7683	−1.05111	−0.525556	+	0.850759i	$0.676143\pi$
$602$	0	0
$603$	5.78458	0.235566
$604$	0	0
$605$	9.68090	0.393585
$606$	0	0
$607$	3.85022	0.156276	0.0781378	−	0.996943i	$-0.475103\pi$
$607$	3.85022	0.156276	0.0781378	+	0.996943i	$0.475103\pi$
$608$	0	0
$609$	−1.36667	−0.0553802
$610$	0	0
$611$	1.08988	0.0440917
$612$	0	0
$613$	−9.85235	−0.397933	−0.198966	−	0.980006i	$-0.563758\pi$
$613$	−9.85235	−0.397933	−0.198966	+	0.980006i	$0.563758\pi$
$614$	0	0
$615$	−4.19766	−0.169266
$616$	0	0
$617$	−28.0417	−1.12892	−0.564459	−	0.825461i	$-0.690915\pi$
$617$	−28.0417	−1.12892	−0.564459	+	0.825461i	$0.690915\pi$
$618$	0	0
$619$	38.0014	1.52741	0.763703	−	0.645567i	$-0.223379\pi$
$619$	38.0014	1.52741	0.763703	+	0.645567i	$0.223379\pi$
$620$	0	0
$621$	43.4835	1.74493
$622$	0	0
$623$	−1.60426	−0.0642732
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−13.1870	−0.526639
$628$	0	0
$629$	−50.9818	−2.03278
$630$	0	0
$631$	−13.7052	−0.545594	−0.272797	−	0.962072i	$-0.587949\pi$
$631$	−13.7052	−0.545594	−0.272797	+	0.962072i	$0.587949\pi$
$632$	0	0
$633$	15.0917	0.599839
$634$	0	0
$635$	4.82719	0.191561
$636$	0	0
$637$	0.246831	0.00977981
$638$	0	0
$639$	2.21759	0.0877266
$640$	0	0
$641$	23.6561	0.934360	0.467180	−	0.884162i	$-0.345270\pi$
$641$	23.6561	0.934360	0.467180	+	0.884162i	$0.345270\pi$
$642$	0	0
$643$	−23.5650	−0.929311	−0.464656	−	0.885491i	$-0.653822\pi$
$643$	−23.5650	−0.929311	−0.464656	+	0.885491i	$0.653822\pi$
$644$	0	0
$645$	11.0175	0.433814
$646$	0	0
$647$	−15.3586	−0.603807	−0.301904	−	0.953338i	$-0.597622\pi$
$647$	−15.3586	−0.603807	−0.301904	+	0.953338i	$0.597622\pi$
$648$	0	0
$649$	0.386787	0.0151827
$650$	0	0
$651$	−30.0163	−1.17643
$652$	0	0
$653$	29.2299	1.14386	0.571928	−	0.820304i	$-0.306196\pi$
$653$	29.2299	1.14386	0.571928	+	0.820304i	$0.306196\pi$
$654$	0	0
$655$	−2.96402	−0.115814
$656$	0	0
$657$	−15.3730	−0.599759
$658$	0	0
$659$	17.6491	0.687513	0.343756	−	0.939059i	$-0.388301\pi$
$659$	17.6491	0.687513	0.343756	+	0.939059i	$0.388301\pi$
$660$	0	0
$661$	−35.1643	−1.36773	−0.683867	−	0.729607i	$-0.739703\pi$
$661$	−35.1643	−1.36773	−0.683867	+	0.729607i	$0.739703\pi$
$662$	0	0
$663$	−1.49736	−0.0581527
$664$	0	0
$665$	−24.0250	−0.931651
$666$	0	0
$667$	2.68816	0.104086
$668$	0	0
$669$	−32.6232	−1.26129
$670$	0	0
$671$	−1.16184	−0.0448524
$672$	0	0
$673$	22.6303	0.872335	0.436167	−	0.899866i	$-0.356336\pi$
$673$	22.6303	0.872335	0.436167	+	0.899866i	$0.356336\pi$
$674$	0	0
$675$	5.64779	0.217383
$676$	0	0
$677$	3.87321	0.148860	0.0744299	−	0.997226i	$-0.476286\pi$
$677$	3.87321	0.148860	0.0744299	+	0.997226i	$0.476286\pi$
$678$	0	0
$679$	−43.4526	−1.66756
$680$	0	0
$681$	15.2481	0.584307
$682$	0	0
$683$	−29.9167	−1.14473	−0.572365	−	0.819999i	$-0.693974\pi$
$683$	−29.9167	−1.14473	−0.572365	+	0.819999i	$0.693974\pi$
$684$	0	0
$685$	−6.01475	−0.229812
$686$	0	0
$687$	−23.7955	−0.907856
$688$	0	0
$689$	−2.00273	−0.0762978
$690$	0	0
$691$	30.1097	1.14543	0.572714	−	0.819755i	$-0.305891\pi$
$691$	30.1097	1.14543	0.572714	+	0.819755i	$0.305891\pi$
$692$	0	0
$693$	−3.71203	−0.141009
$694$	0	0
$695$	14.0343	0.532353
$696$	0	0
$697$	−16.2053	−0.613821
$698$	0	0
$699$	−14.4208	−0.545444
$700$	0	0
$701$	45.7171	1.72671	0.863355	−	0.504596i	$-0.168359\pi$
$701$	45.7171	1.72671	0.863355	+	0.504596i	$0.168359\pi$
$702$	0	0
$703$	−81.0538	−3.05700
$704$	0	0
$705$	7.18953	0.270774
$706$	0	0
$707$	48.7729	1.83429
$708$	0	0
$709$	−21.5815	−0.810510	−0.405255	−	0.914204i	$-0.632817\pi$
$709$	−21.5815	−0.810510	−0.405255	+	0.914204i	$0.632817\pi$
$710$	0	0
$711$	12.9124	0.484251
$712$	0	0
$713$	59.0404	2.21108
$714$	0	0
$715$	−0.238130	−0.00890557
$716$	0	0
$717$	−23.4420	−0.875457
$718$	0	0
$719$	−6.38248	−0.238027	−0.119013	−	0.992893i	$-0.537973\pi$
$719$	−6.38248	−0.238027	−0.119013	+	0.992893i	$0.537973\pi$
$720$	0	0
$721$	10.6377	0.396170
$722$	0	0
$723$	−21.7157	−0.807617
$724$	0	0
$725$	0.349148	0.0129670
$726$	0	0
$727$	5.41162	0.200706	0.100353	−	0.994952i	$-0.468003\pi$
$727$	5.41162	0.200706	0.100353	+	0.994952i	$0.468003\pi$
$728$	0	0
$729$	28.0714	1.03968
$730$	0	0
$731$	42.5339	1.57317
$732$	0	0
$733$	27.1131	1.00144	0.500722	−	0.865608i	$-0.333068\pi$
$733$	27.1131	1.00144	0.500722	+	0.865608i	$0.333068\pi$
$734$	0	0
$735$	1.62826	0.0600592
$736$	0	0
$737$	−5.88290	−0.216699
$738$	0	0
$739$	28.5892	1.05167	0.525836	−	0.850586i	$-0.323753\pi$
$739$	28.5892	1.05167	0.525836	+	0.850586i	$0.323753\pi$
$740$	0	0
$741$	−2.38059	−0.0874531
$742$	0	0
$743$	−32.4426	−1.19020	−0.595102	−	0.803650i	$-0.702888\pi$
$743$	−32.4426	−1.19020	−0.595102	+	0.803650i	$0.702888\pi$
$744$	0	0
$745$	−2.68064	−0.0982109
$746$	0	0
$747$	−5.51333	−0.201722
$748$	0	0
$749$	0.392454	0.0143400
$750$	0	0
$751$	10.8554	0.396119	0.198060	−	0.980190i	$-0.436536\pi$
$751$	10.8554	0.396119	0.198060	+	0.980190i	$0.436536\pi$
$752$	0	0
$753$	−28.4300	−1.03605
$754$	0	0
$755$	−17.8522	−0.649706
$756$	0	0
$757$	32.8430	1.19370	0.596850	−	0.802353i	$-0.296419\pi$
$757$	32.8430	1.19370	0.596850	+	0.802353i	$0.296419\pi$
$758$	0	0
$759$	−12.0944	−0.438998
$760$	0	0
$761$	43.7627	1.58640	0.793198	−	0.608964i	$-0.208414\pi$
$761$	43.7627	1.58640	0.793198	+	0.608964i	$0.208414\pi$
$762$	0	0
$763$	−20.8660	−0.755402
$764$	0	0
$765$	5.96306	0.215595
$766$	0	0
$767$	0.0698247	0.00252122
$768$	0	0
$769$	9.13604	0.329454	0.164727	−	0.986339i	$-0.447326\pi$
$769$	9.13604	0.329454	0.164727	+	0.986339i	$0.447326\pi$
$770$	0	0
$771$	−1.52283	−0.0548432
$772$	0	0
$773$	−1.47035	−0.0528847	−0.0264424	−	0.999650i	$-0.508418\pi$
$773$	−1.47035	−0.0528847	−0.0264424	+	0.999650i	$0.508418\pi$
$774$	0	0
$775$	7.66838	0.275456
$776$	0	0
$777$	37.7936	1.35584
$778$	0	0
$779$	−25.7642	−0.923098
$780$	0	0
$781$	−2.25529	−0.0807005
$782$	0	0
$783$	1.97191	0.0704704
$784$	0	0
$785$	−2.77767	−0.0991392
$786$	0	0
$787$	−1.55189	−0.0553189	−0.0276595	−	0.999617i	$-0.508805\pi$
$787$	−1.55189	−0.0553189	−0.0276595	+	0.999617i	$0.508805\pi$
$788$	0	0
$789$	0.633429	0.0225507
$790$	0	0
$791$	10.5858	0.376387
$792$	0	0
$793$	−0.209742	−0.00744814
$794$	0	0
$795$	−13.2113	−0.468556
$796$	0	0
$797$	6.88623	0.243923	0.121961	−	0.992535i	$-0.461082\pi$
$797$	6.88623	0.243923	0.121961	+	0.992535i	$0.461082\pi$
$798$	0	0
$799$	27.7557	0.981927
$800$	0	0
$801$	0.633049	0.0223677
$802$	0	0
$803$	15.6343	0.551724
$804$	0	0
$805$	−22.0344	−0.776610
$806$	0	0
$807$	19.1733	0.674932
$808$	0	0
$809$	24.4957	0.861223	0.430612	−	0.902537i	$-0.358298\pi$
$809$	24.4957	0.861223	0.430612	+	0.902537i	$0.358298\pi$
$810$	0	0
$811$	−25.5715	−0.897938	−0.448969	−	0.893547i	$-0.648209\pi$
$811$	−25.5715	−0.897938	−0.448969	+	0.893547i	$0.648209\pi$
$812$	0	0
$813$	−33.5905	−1.17807
$814$	0	0
$815$	12.8033	0.448482
$816$	0	0
$817$	67.6228	2.36582
$818$	0	0
$819$	−0.670115	−0.0234157
$820$	0	0
$821$	−20.4057	−0.712164	−0.356082	−	0.934455i	$-0.615888\pi$
$821$	−20.4057	−0.712164	−0.356082	+	0.934455i	$0.615888\pi$
$822$	0	0
$823$	4.81550	0.167858	0.0839289	−	0.996472i	$-0.473253\pi$
$823$	4.81550	0.167858	0.0839289	+	0.996472i	$0.473253\pi$
$824$	0	0
$825$	−1.57086	−0.0546904
$826$	0	0
$827$	4.51043	0.156843	0.0784215	−	0.996920i	$-0.475012\pi$
$827$	4.51043	0.156843	0.0784215	+	0.996920i	$0.475012\pi$
$828$	0	0
$829$	−16.5459	−0.574662	−0.287331	−	0.957831i	$-0.592768\pi$
$829$	−16.5459	−0.574662	−0.287331	+	0.957831i	$0.592768\pi$
$830$	0	0
$831$	−3.50037	−0.121426
$832$	0	0
$833$	6.28600	0.217797
$834$	0	0
$835$	−23.9713	−0.829562
$836$	0	0
$837$	43.3094	1.49699
$838$	0	0
$839$	33.3962	1.15296	0.576482	−	0.817110i	$-0.304425\pi$
$839$	33.3962	1.15296	0.576482	+	0.817110i	$0.304425\pi$
$840$	0	0
$841$	−28.8781	−0.995796
$842$	0	0
$843$	44.1536	1.52073
$844$	0	0
$845$	12.9570	0.445735
$846$	0	0
$847$	−27.7058	−0.951982
$848$	0	0
$849$	30.0480	1.03125
$850$	0	0
$851$	−74.3379	−2.54827
$852$	0	0
$853$	−9.29976	−0.318418	−0.159209	−	0.987245i	$-0.550894\pi$
$853$	−9.29976	−0.318418	−0.159209	+	0.987245i	$0.550894\pi$
$854$	0	0
$855$	9.48042	0.324224
$856$	0	0
$857$	−48.2211	−1.64720	−0.823601	−	0.567170i	$-0.808038\pi$
$857$	−48.2211	−1.64720	−0.823601	+	0.567170i	$0.808038\pi$
$858$	0	0
$859$	−27.8522	−0.950304	−0.475152	−	0.879904i	$-0.657607\pi$
$859$	−27.8522	−0.950304	−0.475152	+	0.879904i	$0.657607\pi$
$860$	0	0
$861$	12.0133	0.409411
$862$	0	0
$863$	56.8168	1.93407	0.967033	−	0.254650i	$-0.0819602\pi$
$863$	56.8168	1.93407	0.967033	+	0.254650i	$0.0819602\pi$
$864$	0	0
$865$	5.30117	0.180245
$866$	0	0
$867$	−14.8817	−0.505408
$868$	0	0
$869$	−13.1318	−0.445467
$870$	0	0
$871$	−1.06201	−0.0359849
$872$	0	0
$873$	17.1466	0.580326
$874$	0	0
$875$	−2.86190	−0.0967500
$876$	0	0
$877$	−14.4871	−0.489194	−0.244597	−	0.969625i	$-0.578656\pi$
$877$	−14.4871	−0.489194	−0.244597	+	0.969625i	$0.578656\pi$
$878$	0	0
$879$	18.7907	0.633796
$880$	0	0
$881$	−40.5865	−1.36739	−0.683697	−	0.729766i	$-0.739629\pi$
$881$	−40.5865	−1.36739	−0.683697	+	0.729766i	$0.739629\pi$
$882$	0	0
$883$	−0.186761	−0.00628500	−0.00314250	−	0.999995i	$-0.501000\pi$
$883$	−0.186761	−0.00628500	−0.00314250	+	0.999995i	$0.501000\pi$
$884$	0	0
$885$	0.460609	0.0154832
$886$	0	0
$887$	38.1132	1.27972	0.639858	−	0.768494i	$-0.278993\pi$
$887$	38.1132	1.27972	0.639858	+	0.768494i	$0.278993\pi$
$888$	0	0
$889$	−13.8149	−0.463338
$890$	0	0
$891$	−4.98073	−0.166861
$892$	0	0
$893$	44.1276	1.47667
$894$	0	0
$895$	−2.88054	−0.0962860
$896$	0	0
$897$	−2.18334	−0.0728996
$898$	0	0
$899$	2.67740	0.0892962
$900$	0	0
$901$	−51.0031	−1.69916
$902$	0	0
$903$	−31.5311	−1.04929
$904$	0	0
$905$	−18.3382	−0.609582
$906$	0	0
$907$	−29.9492	−0.994448	−0.497224	−	0.867622i	$-0.665647\pi$
$907$	−29.9492	−0.994448	−0.497224	+	0.867622i	$0.665647\pi$
$908$	0	0
$909$	−19.2461	−0.638352
$910$	0	0
$911$	37.9601	1.25768	0.628838	−	0.777537i	$-0.283531\pi$
$911$	37.9601	1.25768	0.628838	+	0.777537i	$0.283531\pi$
$912$	0	0
$913$	5.60705	0.185566
$914$	0	0
$915$	−1.38359	−0.0457401
$916$	0	0
$917$	8.48272	0.280124
$918$	0	0
$919$	37.2720	1.22949	0.614745	−	0.788726i	$-0.289259\pi$
$919$	37.2720	1.22949	0.614745	+	0.788726i	$0.289259\pi$
$920$	0	0
$921$	−4.40687	−0.145211
$922$	0	0
$923$	−0.407136	−0.0134010
$924$	0	0
$925$	−9.65526	−0.317463
$926$	0	0
$927$	−4.19771	−0.137871
$928$	0	0
$929$	14.2446	0.467351	0.233676	−	0.972315i	$-0.424925\pi$
$929$	14.2446	0.467351	0.233676	+	0.972315i	$0.424925\pi$
$930$	0	0
$931$	9.99385	0.327535
$932$	0	0
$933$	39.2745	1.28579
$934$	0	0
$935$	−6.06442	−0.198328
$936$	0	0
$937$	3.49996	0.114339	0.0571694	−	0.998364i	$-0.481792\pi$
$937$	3.49996	0.114339	0.0571694	+	0.998364i	$0.481792\pi$
$938$	0	0
$939$	9.42239	0.307488
$940$	0	0
$941$	9.16903	0.298902	0.149451	−	0.988769i	$-0.452249\pi$
$941$	9.16903	0.298902	0.149451	+	0.988769i	$0.452249\pi$
$942$	0	0
$943$	−23.6294	−0.769480
$944$	0	0
$945$	−16.1634	−0.525796
$946$	0	0
$947$	−6.18999	−0.201148	−0.100574	−	0.994930i	$-0.532068\pi$
$947$	−6.18999	−0.201148	−0.100574	+	0.994930i	$0.532068\pi$
$948$	0	0
$949$	2.82239	0.0916187
$950$	0	0
$951$	12.7104	0.412164
$952$	0	0
$953$	−35.6420	−1.15456	−0.577279	−	0.816547i	$-0.695885\pi$
$953$	−35.6420	−1.15456	−0.577279	+	0.816547i	$0.695885\pi$
$954$	0	0
$955$	1.87860	0.0607902
$956$	0	0
$957$	−0.548463	−0.0177293
$958$	0	0
$959$	17.2136	0.555857
$960$	0	0
$961$	27.8041	0.896905
$962$	0	0
$963$	−0.154865	−0.00499045
$964$	0	0
$965$	7.62051	0.245313
$966$	0	0
$967$	−41.9580	−1.34928	−0.674640	−	0.738147i	$-0.735701\pi$
$967$	−41.9580	−1.34928	−0.674640	+	0.738147i	$0.735701\pi$
$968$	0	0
$969$	−60.6260	−1.94759
$970$	0	0
$971$	10.9168	0.350338	0.175169	−	0.984538i	$-0.443953\pi$
$971$	10.9168	0.350338	0.175169	+	0.984538i	$0.443953\pi$
$972$	0	0
$973$	−40.1649	−1.28763
$974$	0	0
$975$	−0.283580	−0.00908183
$976$	0	0
$977$	−53.1236	−1.69957	−0.849787	−	0.527127i	$-0.823269\pi$
$977$	−53.1236	−1.69957	−0.849787	+	0.527127i	$0.823269\pi$
$978$	0	0
$979$	−0.643810	−0.0205762
$980$	0	0
$981$	8.23387	0.262887
$982$	0	0
$983$	37.9947	1.21184	0.605921	−	0.795525i	$-0.292805\pi$
$983$	37.9947	1.21184	0.605921	+	0.795525i	$0.292805\pi$
$984$	0	0
$985$	4.24790	0.135349
$986$	0	0
$987$	−20.5757	−0.654933
$988$	0	0
$989$	62.0198	1.97211
$990$	0	0
$991$	−60.1044	−1.90928	−0.954639	−	0.297765i	$-0.903759\pi$
$991$	−60.1044	−1.90928	−0.954639	+	0.297765i	$0.903759\pi$
$992$	0	0
$993$	−5.38075	−0.170753
$994$	0	0
$995$	18.8089	0.596281
$996$	0	0
$997$	5.29846	0.167804	0.0839019	−	0.996474i	$-0.473262\pi$
$997$	5.29846	0.167804	0.0839019	+	0.996474i	$0.473262\pi$
$998$	0	0
$999$	−54.5309	−1.72528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.12	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-1.36773 q^{3} -1.00000 q^{5} +2.86190 q^{7} -1.12932 q^{9} +O(q^{10})\)	\(q-1.36773 q^{3} -1.00000 q^{5} +2.86190 q^{7} -1.12932 q^{9} +1.14852 q^{11} +0.207337 q^{13} +1.36773 q^{15} +5.28021 q^{17} +8.39477 q^{19} -3.91430 q^{21} +7.69921 q^{23} +1.00000 q^{25} +5.64779 q^{27} +0.349148 q^{29} +7.66838 q^{31} -1.57086 q^{33} -2.86190 q^{35} -9.65526 q^{37} -0.283580 q^{39} -3.06907 q^{41} +8.05535 q^{43} +1.12932 q^{45} +5.25656 q^{47} +1.19048 q^{49} -7.22188 q^{51} -9.65930 q^{53} -1.14852 q^{55} -11.4818 q^{57} +0.336770 q^{59} -1.01160 q^{61} -3.23202 q^{63} -0.207337 q^{65} -5.12216 q^{67} -10.5304 q^{69} -1.96365 q^{71} +13.6126 q^{73} -1.36773 q^{75} +3.28695 q^{77} -11.4337 q^{79} -4.33665 q^{81} +4.88198 q^{83} -5.28021 q^{85} -0.477539 q^{87} -0.560556 q^{89} +0.593377 q^{91} -10.4882 q^{93} -8.39477 q^{95} -15.1831 q^{97} -1.29705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 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

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Database

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