LMFDB - Embedded newform 8020.2.a.d.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:	$1$
Dimension:	$29$
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)

$f(q)$	$=$	$q+0.0219398 q^{3} +1.00000 q^{5} +2.04593 q^{7} -2.99952 q^{9} +O(q^{10})$	$q+0.0219398 q^{3} +1.00000 q^{5} +2.04593 q^{7} -2.99952 q^{9} -1.84171 q^{11} +5.44009 q^{13} +0.0219398 q^{15} -2.59139 q^{17} +7.53632 q^{19} +0.0448873 q^{21} -4.84220 q^{23} +1.00000 q^{25} -0.131629 q^{27} -6.87742 q^{29} -8.81233 q^{31} -0.0404068 q^{33} +2.04593 q^{35} -5.23090 q^{37} +0.119355 q^{39} -0.885990 q^{41} -8.40984 q^{43} -2.99952 q^{45} -2.13404 q^{47} -2.81419 q^{49} -0.0568547 q^{51} -4.90854 q^{53} -1.84171 q^{55} +0.165346 q^{57} -0.302551 q^{59} -3.66559 q^{61} -6.13679 q^{63} +5.44009 q^{65} -4.17949 q^{67} -0.106237 q^{69} +3.09316 q^{71} +4.77602 q^{73} +0.0219398 q^{75} -3.76800 q^{77} -13.8166 q^{79} +8.99567 q^{81} +12.0433 q^{83} -2.59139 q^{85} -0.150890 q^{87} +6.26285 q^{89} +11.1300 q^{91} -0.193341 q^{93} +7.53632 q^{95} +19.4782 q^{97} +5.52424 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * 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
$3$	0.0219398	0.0126670	0.00633349	−	0.999980i	$-0.497984\pi$
$3$	0.0219398	0.0126670	0.00633349	+	0.999980i	$0.497984\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.04593	0.773287	0.386644	−	0.922229i	$-0.373634\pi$
$7$	2.04593	0.773287	0.386644	+	0.922229i	$0.373634\pi$
$8$	0	0
$9$	−2.99952	−0.999840
$10$	0	0
$11$	−1.84171	−0.555296	−0.277648	−	0.960683i	$-0.589555\pi$
$11$	−1.84171	−0.555296	−0.277648	+	0.960683i	$0.589555\pi$
$12$	0	0
$13$	5.44009	1.50881	0.754405	−	0.656410i	$-0.227926\pi$
$13$	5.44009	1.50881	0.754405	+	0.656410i	$0.227926\pi$
$14$	0	0
$15$	0.0219398	0.00566484
$16$	0	0
$17$	−2.59139	−0.628505	−0.314252	−	0.949339i	$-0.601754\pi$
$17$	−2.59139	−0.628505	−0.314252	+	0.949339i	$0.601754\pi$
$18$	0	0
$19$	7.53632	1.72895	0.864475	−	0.502675i	$-0.167651\pi$
$19$	7.53632	1.72895	0.864475	+	0.502675i	$0.167651\pi$
$20$	0	0
$21$	0.0448873	0.00979521
$22$	0	0
$23$	−4.84220	−1.00967	−0.504835	−	0.863216i	$-0.668446\pi$
$23$	−4.84220	−1.00967	−0.504835	+	0.863216i	$0.668446\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.131629	−0.0253319
$28$	0	0
$29$	−6.87742	−1.27711	−0.638553	−	0.769578i	$-0.720467\pi$
$29$	−6.87742	−1.27711	−0.638553	+	0.769578i	$0.720467\pi$
$30$	0	0
$31$	−8.81233	−1.58274	−0.791371	−	0.611337i	$-0.790632\pi$
$31$	−8.81233	−1.58274	−0.791371	+	0.611337i	$0.790632\pi$
$32$	0	0
$33$	−0.0404068	−0.00703393
$34$	0	0
$35$	2.04593	0.345825
$36$	0	0
$37$	−5.23090	−0.859954	−0.429977	−	0.902840i	$-0.641478\pi$
$37$	−5.23090	−0.859954	−0.429977	+	0.902840i	$0.641478\pi$
$38$	0	0
$39$	0.119355	0.0191121
$40$	0	0
$41$	−0.885990	−0.138368	−0.0691842	−	0.997604i	$-0.522040\pi$
$41$	−0.885990	−0.138368	−0.0691842	+	0.997604i	$0.522040\pi$
$42$	0	0
$43$	−8.40984	−1.28249	−0.641244	−	0.767337i	$-0.721581\pi$
$43$	−8.40984	−1.28249	−0.641244	+	0.767337i	$0.721581\pi$
$44$	0	0
$45$	−2.99952	−0.447142
$46$	0	0
$47$	−2.13404	−0.311281	−0.155641	−	0.987814i	$-0.549744\pi$
$47$	−2.13404	−0.311281	−0.155641	+	0.987814i	$0.549744\pi$
$48$	0	0
$49$	−2.81419	−0.402027
$50$	0	0
$51$	−0.0568547	−0.00796125
$52$	0	0
$53$	−4.90854	−0.674240	−0.337120	−	0.941462i	$-0.609453\pi$
$53$	−4.90854	−0.674240	−0.337120	+	0.941462i	$0.609453\pi$
$54$	0	0
$55$	−1.84171	−0.248336
$56$	0	0
$57$	0.165346	0.0219006
$58$	0	0
$59$	−0.302551	−0.0393887	−0.0196944	−	0.999806i	$-0.506269\pi$
$59$	−0.302551	−0.0393887	−0.0196944	+	0.999806i	$0.506269\pi$
$60$	0	0
$61$	−3.66559	−0.469330	−0.234665	−	0.972076i	$-0.575399\pi$
$61$	−3.66559	−0.469330	−0.234665	+	0.972076i	$0.575399\pi$
$62$	0	0
$63$	−6.13679	−0.773163
$64$	0	0
$65$	5.44009	0.674760
$66$	0	0
$67$	−4.17949	−0.510606	−0.255303	−	0.966861i	$-0.582175\pi$
$67$	−4.17949	−0.510606	−0.255303	+	0.966861i	$0.582175\pi$
$68$	0	0
$69$	−0.106237	−0.0127895
$70$	0	0
$71$	3.09316	0.367091	0.183545	−	0.983011i	$-0.441243\pi$
$71$	3.09316	0.367091	0.183545	+	0.983011i	$0.441243\pi$
$72$	0	0
$73$	4.77602	0.558991	0.279496	−	0.960147i	$-0.409833\pi$
$73$	4.77602	0.558991	0.279496	+	0.960147i	$0.409833\pi$
$74$	0	0
$75$	0.0219398	0.00253339
$76$	0	0
$77$	−3.76800	−0.429404
$78$	0	0
$79$	−13.8166	−1.55448	−0.777242	−	0.629202i	$-0.783382\pi$
$79$	−13.8166	−1.55448	−0.777242	+	0.629202i	$0.783382\pi$
$80$	0	0
$81$	8.99567	0.999519
$82$	0	0
$83$	12.0433	1.32193	0.660963	−	0.750419i	$-0.270148\pi$
$83$	12.0433	1.32193	0.660963	+	0.750419i	$0.270148\pi$
$84$	0	0
$85$	−2.59139	−0.281076
$86$	0	0
$87$	−0.150890	−0.0161771
$88$	0	0
$89$	6.26285	0.663860	0.331930	−	0.943304i	$-0.392300\pi$
$89$	6.26285	0.663860	0.331930	+	0.943304i	$0.392300\pi$
$90$	0	0
$91$	11.1300	1.16674
$92$	0	0
$93$	−0.193341	−0.0200485
$94$	0	0
$95$	7.53632	0.773210
$96$	0	0
$97$	19.4782	1.97771	0.988855	−	0.148884i	$-0.0475681\pi$
$97$	19.4782	1.97771	0.988855	+	0.148884i	$0.0475681\pi$
$98$	0	0
$99$	5.52424	0.555207
$100$	0	0
$101$	−18.9330	−1.88390	−0.941950	−	0.335754i	$-0.891009\pi$
$101$	−18.9330	−1.88390	−0.941950	+	0.335754i	$0.891009\pi$
$102$	0	0
$103$	1.12270	0.110622	0.0553112	−	0.998469i	$-0.482385\pi$
$103$	1.12270	0.110622	0.0553112	+	0.998469i	$0.482385\pi$
$104$	0	0
$105$	0.0448873	0.00438055
$106$	0	0
$107$	8.55332	0.826880	0.413440	−	0.910531i	$-0.364327\pi$
$107$	8.55332	0.826880	0.413440	+	0.910531i	$0.364327\pi$
$108$	0	0
$109$	−1.06282	−0.101800	−0.0508998	−	0.998704i	$-0.516209\pi$
$109$	−1.06282	−0.101800	−0.0508998	+	0.998704i	$0.516209\pi$
$110$	0	0
$111$	−0.114765	−0.0108930
$112$	0	0
$113$	4.09059	0.384810	0.192405	−	0.981316i	$-0.438371\pi$
$113$	4.09059	0.384810	0.192405	+	0.981316i	$0.438371\pi$
$114$	0	0
$115$	−4.84220	−0.451538
$116$	0	0
$117$	−16.3177	−1.50857
$118$	0	0
$119$	−5.30179	−0.486015
$120$	0	0
$121$	−7.60810	−0.691646
$122$	0	0
$123$	−0.0194385	−0.00175271
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.1939	−1.08204	−0.541018	−	0.841011i	$-0.681961\pi$
$127$	−12.1939	−1.08204	−0.541018	+	0.841011i	$0.681961\pi$
$128$	0	0
$129$	−0.184510	−0.0162452
$130$	0	0
$131$	−11.8609	−1.03629	−0.518147	−	0.855291i	$-0.673378\pi$
$131$	−11.8609	−1.03629	−0.518147	+	0.855291i	$0.673378\pi$
$132$	0	0
$133$	15.4188	1.33698
$134$	0	0
$135$	−0.131629	−0.0113288
$136$	0	0
$137$	−10.3273	−0.882319	−0.441160	−	0.897429i	$-0.645433\pi$
$137$	−10.3273	−0.882319	−0.441160	+	0.897429i	$0.645433\pi$
$138$	0	0
$139$	13.4438	1.14029	0.570143	−	0.821545i	$-0.306888\pi$
$139$	13.4438	1.14029	0.570143	+	0.821545i	$0.306888\pi$
$140$	0	0
$141$	−0.0468204	−0.00394299
$142$	0	0
$143$	−10.0191	−0.837836
$144$	0	0
$145$	−6.87742	−0.571139
$146$	0	0
$147$	−0.0617428	−0.00509246
$148$	0	0
$149$	−12.6946	−1.03998	−0.519991	−	0.854172i	$-0.674065\pi$
$149$	−12.6946	−1.03998	−0.519991	+	0.854172i	$0.674065\pi$
$150$	0	0
$151$	4.15464	0.338100	0.169050	−	0.985607i	$-0.445930\pi$
$151$	4.15464	0.338100	0.169050	+	0.985607i	$0.445930\pi$
$152$	0	0
$153$	7.77293	0.628404
$154$	0	0
$155$	−8.81233	−0.707823
$156$	0	0
$157$	6.81818	0.544150	0.272075	−	0.962276i	$-0.412290\pi$
$157$	6.81818	0.544150	0.272075	+	0.962276i	$0.412290\pi$
$158$	0	0
$159$	−0.107693	−0.00854058
$160$	0	0
$161$	−9.90679	−0.780764
$162$	0	0
$163$	−3.78191	−0.296222	−0.148111	−	0.988971i	$-0.547319\pi$
$163$	−3.78191	−0.296222	−0.148111	+	0.988971i	$0.547319\pi$
$164$	0	0
$165$	−0.0404068	−0.00314567
$166$	0	0
$167$	16.2776	1.25960	0.629800	−	0.776757i	$-0.283137\pi$
$167$	16.2776	1.25960	0.629800	+	0.776757i	$0.283137\pi$
$168$	0	0
$169$	16.5946	1.27651
$170$	0	0
$171$	−22.6053	−1.72867
$172$	0	0
$173$	−19.5713	−1.48797	−0.743987	−	0.668194i	$-0.767068\pi$
$173$	−19.5713	−1.48797	−0.743987	+	0.668194i	$0.767068\pi$
$174$	0	0
$175$	2.04593	0.154657
$176$	0	0
$177$	−0.00663791	−0.000498936 0
$178$	0	0
$179$	−3.17441	−0.237266	−0.118633	−	0.992938i	$-0.537851\pi$
$179$	−3.17441	−0.237266	−0.118633	+	0.992938i	$0.537851\pi$
$180$	0	0
$181$	21.8647	1.62519	0.812594	−	0.582830i	$-0.198055\pi$
$181$	21.8647	1.62519	0.812594	+	0.582830i	$0.198055\pi$
$182$	0	0
$183$	−0.0804224	−0.00594500
$184$	0	0
$185$	−5.23090	−0.384583
$186$	0	0
$187$	4.77259	0.349006
$188$	0	0
$189$	−0.269302	−0.0195889
$190$	0	0
$191$	−16.2566	−1.17629	−0.588145	−	0.808756i	$-0.700141\pi$
$191$	−16.2566	−1.17629	−0.588145	+	0.808756i	$0.700141\pi$
$192$	0	0
$193$	20.7057	1.49043	0.745213	−	0.666827i	$-0.232348\pi$
$193$	20.7057	1.49043	0.745213	+	0.666827i	$0.232348\pi$
$194$	0	0
$195$	0.119355	0.00854717
$196$	0	0
$197$	15.3326	1.09240	0.546200	−	0.837655i	$-0.316074\pi$
$197$	15.3326	1.09240	0.546200	+	0.837655i	$0.316074\pi$
$198$	0	0
$199$	−0.842805	−0.0597449	−0.0298724	−	0.999554i	$-0.509510\pi$
$199$	−0.842805	−0.0597449	−0.0298724	+	0.999554i	$0.509510\pi$
$200$	0	0
$201$	−0.0916974	−0.00646783
$202$	0	0
$203$	−14.0707	−0.987570
$204$	0	0
$205$	−0.885990	−0.0618802
$206$	0	0
$207$	14.5243	1.00951
$208$	0	0
$209$	−13.8797	−0.960080
$210$	0	0
$211$	15.3875	1.05932	0.529660	−	0.848210i	$-0.322320\pi$
$211$	15.3875	1.05932	0.529660	+	0.848210i	$0.322320\pi$
$212$	0	0
$213$	0.0678635	0.00464993
$214$	0	0
$215$	−8.40984	−0.573546
$216$	0	0
$217$	−18.0294	−1.22391
$218$	0	0
$219$	0.104785	0.00708073
$220$	0	0
$221$	−14.0974	−0.948294
$222$	0	0
$223$	−14.4104	−0.964989	−0.482495	−	0.875899i	$-0.660269\pi$
$223$	−14.4104	−0.964989	−0.482495	+	0.875899i	$0.660269\pi$
$224$	0	0
$225$	−2.99952	−0.199968
$226$	0	0
$227$	−15.0392	−0.998185	−0.499092	−	0.866549i	$-0.666333\pi$
$227$	−15.0392	−0.998185	−0.499092	+	0.866549i	$0.666333\pi$
$228$	0	0
$229$	−22.7907	−1.50605	−0.753026	−	0.657991i	$-0.771406\pi$
$229$	−22.7907	−1.50605	−0.753026	+	0.657991i	$0.771406\pi$
$230$	0	0
$231$	−0.0826694	−0.00543925
$232$	0	0
$233$	−15.4468	−1.01195	−0.505976	−	0.862547i	$-0.668868\pi$
$233$	−15.4468	−1.01195	−0.505976	+	0.862547i	$0.668868\pi$
$234$	0	0
$235$	−2.13404	−0.139209
$236$	0	0
$237$	−0.303133	−0.0196906
$238$	0	0
$239$	−1.75914	−0.113789	−0.0568945	−	0.998380i	$-0.518120\pi$
$239$	−1.75914	−0.113789	−0.0568945	+	0.998380i	$0.518120\pi$
$240$	0	0
$241$	−24.4596	−1.57558	−0.787791	−	0.615942i	$-0.788775\pi$
$241$	−24.4596	−1.57558	−0.787791	+	0.615942i	$0.788775\pi$
$242$	0	0
$243$	0.592249	0.0379928
$244$	0	0
$245$	−2.81419	−0.179792
$246$	0	0
$247$	40.9983	2.60866
$248$	0	0
$249$	0.264228	0.0167448
$250$	0	0
$251$	−10.4111	−0.657144	−0.328572	−	0.944479i	$-0.606567\pi$
$251$	−10.4111	−0.657144	−0.328572	+	0.944479i	$0.606567\pi$
$252$	0	0
$253$	8.91793	0.560666
$254$	0	0
$255$	−0.0568547	−0.00356038
$256$	0	0
$257$	−17.9205	−1.11785	−0.558926	−	0.829217i	$-0.688787\pi$
$257$	−17.9205	−1.11785	−0.558926	+	0.829217i	$0.688787\pi$
$258$	0	0
$259$	−10.7020	−0.664992
$260$	0	0
$261$	20.6290	1.27690
$262$	0	0
$263$	0.0497916	0.00307028	0.00153514	−	0.999999i	$-0.499511\pi$
$263$	0.0497916	0.00307028	0.00153514	+	0.999999i	$0.499511\pi$
$264$	0	0
$265$	−4.90854	−0.301529
$266$	0	0
$267$	0.137406	0.00840910
$268$	0	0
$269$	−27.8806	−1.69991	−0.849954	−	0.526857i	$-0.823370\pi$
$269$	−27.8806	−1.69991	−0.849954	+	0.526857i	$0.823370\pi$
$270$	0	0
$271$	14.9487	0.908070	0.454035	−	0.890984i	$-0.349984\pi$
$271$	14.9487	0.908070	0.454035	+	0.890984i	$0.349984\pi$
$272$	0	0
$273$	0.244191	0.0147791
$274$	0	0
$275$	−1.84171	−0.111059
$276$	0	0
$277$	20.3926	1.22527	0.612636	−	0.790365i	$-0.290109\pi$
$277$	20.3926	1.22527	0.612636	+	0.790365i	$0.290109\pi$
$278$	0	0
$279$	26.4328	1.58249
$280$	0	0
$281$	−2.10343	−0.125480	−0.0627400	−	0.998030i	$-0.519984\pi$
$281$	−2.10343	−0.125480	−0.0627400	+	0.998030i	$0.519984\pi$
$282$	0	0
$283$	−25.7498	−1.53067	−0.765333	−	0.643635i	$-0.777425\pi$
$283$	−25.7498	−1.53067	−0.765333	+	0.643635i	$0.777425\pi$
$284$	0	0
$285$	0.165346	0.00979424
$286$	0	0
$287$	−1.81267	−0.106999
$288$	0	0
$289$	−10.2847	−0.604982
$290$	0	0
$291$	0.427348	0.0250516
$292$	0	0
$293$	−11.5498	−0.674746	−0.337373	−	0.941371i	$-0.609538\pi$
$293$	−11.5498	−0.674746	−0.337373	+	0.941371i	$0.609538\pi$
$294$	0	0
$295$	−0.302551	−0.0176152
$296$	0	0
$297$	0.242422	0.0140667
$298$	0	0
$299$	−26.3420	−1.52340
$300$	0	0
$301$	−17.2059	−0.991732
$302$	0	0
$303$	−0.415386	−0.0238633
$304$	0	0
$305$	−3.66559	−0.209891
$306$	0	0
$307$	13.6695	0.780160	0.390080	−	0.920781i	$-0.372447\pi$
$307$	13.6695	0.780160	0.390080	+	0.920781i	$0.372447\pi$
$308$	0	0
$309$	0.0246318	0.00140125
$310$	0	0
$311$	18.7669	1.06417	0.532087	−	0.846689i	$-0.321408\pi$
$311$	18.7669	1.06417	0.532087	+	0.846689i	$0.321408\pi$
$312$	0	0
$313$	4.37992	0.247568	0.123784	−	0.992309i	$-0.460497\pi$
$313$	4.37992	0.247568	0.123784	+	0.992309i	$0.460497\pi$
$314$	0	0
$315$	−6.13679	−0.345769
$316$	0	0
$317$	−23.5602	−1.32327	−0.661636	−	0.749825i	$-0.730138\pi$
$317$	−23.5602	−1.32327	−0.661636	+	0.749825i	$0.730138\pi$
$318$	0	0
$319$	12.6662	0.709172
$320$	0	0
$321$	0.187658	0.0104741
$322$	0	0
$323$	−19.5296	−1.08665
$324$	0	0
$325$	5.44009	0.301762
$326$	0	0
$327$	−0.0233181	−0.00128949
$328$	0	0
$329$	−4.36608	−0.240710
$330$	0	0
$331$	12.0821	0.664090	0.332045	−	0.943264i	$-0.392261\pi$
$331$	12.0821	0.664090	0.332045	+	0.943264i	$0.392261\pi$
$332$	0	0
$333$	15.6902	0.859816
$334$	0	0
$335$	−4.17949	−0.228350
$336$	0	0
$337$	−5.00144	−0.272446	−0.136223	−	0.990678i	$-0.543496\pi$
$337$	−5.00144	−0.272446	−0.136223	+	0.990678i	$0.543496\pi$
$338$	0	0
$339$	0.0897469	0.00487438
$340$	0	0
$341$	16.2298	0.878891
$342$	0	0
$343$	−20.0791	−1.08417
$344$	0	0
$345$	−0.106237	−0.00571962
$346$	0	0
$347$	24.6710	1.32441	0.662204	−	0.749324i	$-0.269621\pi$
$347$	24.6710	1.32441	0.662204	+	0.749324i	$0.269621\pi$
$348$	0	0
$349$	−16.2346	−0.869017	−0.434508	−	0.900668i	$-0.643078\pi$
$349$	−16.2346	−0.869017	−0.434508	+	0.900668i	$0.643078\pi$
$350$	0	0
$351$	−0.716071	−0.0382210
$352$	0	0
$353$	30.7001	1.63400	0.817000	−	0.576637i	$-0.195635\pi$
$353$	30.7001	1.63400	0.817000	+	0.576637i	$0.195635\pi$
$354$	0	0
$355$	3.09316	0.164168
$356$	0	0
$357$	−0.116321	−0.00615634
$358$	0	0
$359$	27.8788	1.47139	0.735694	−	0.677314i	$-0.236856\pi$
$359$	27.8788	1.47139	0.735694	+	0.677314i	$0.236856\pi$
$360$	0	0
$361$	37.7961	1.98927
$362$	0	0
$363$	−0.166921	−0.00876106
$364$	0	0
$365$	4.77602	0.249989
$366$	0	0
$367$	15.9209	0.831065	0.415533	−	0.909578i	$-0.363595\pi$
$367$	15.9209	0.831065	0.415533	+	0.909578i	$0.363595\pi$
$368$	0	0
$369$	2.65754	0.138346
$370$	0	0
$371$	−10.0425	−0.521381
$372$	0	0
$373$	11.8158	0.611801	0.305900	−	0.952064i	$-0.401043\pi$
$373$	11.8158	0.611801	0.305900	+	0.952064i	$0.401043\pi$
$374$	0	0
$375$	0.0219398	0.00113297
$376$	0	0
$377$	−37.4138	−1.92691
$378$	0	0
$379$	−26.3118	−1.35155	−0.675773	−	0.737110i	$-0.736190\pi$
$379$	−26.3118	−1.35155	−0.675773	+	0.737110i	$0.736190\pi$
$380$	0	0
$381$	−0.267533	−0.0137061
$382$	0	0
$383$	−14.6496	−0.748559	−0.374280	−	0.927316i	$-0.622110\pi$
$383$	−14.6496	−0.748559	−0.374280	+	0.927316i	$0.622110\pi$
$384$	0	0
$385$	−3.76800	−0.192035
$386$	0	0
$387$	25.2255	1.28228
$388$	0	0
$389$	3.16822	0.160635	0.0803175	−	0.996769i	$-0.474407\pi$
$389$	3.16822	0.160635	0.0803175	+	0.996769i	$0.474407\pi$
$390$	0	0
$391$	12.5480	0.634582
$392$	0	0
$393$	−0.260227	−0.0131267
$394$	0	0
$395$	−13.8166	−0.695186
$396$	0	0
$397$	7.06988	0.354827	0.177414	−	0.984136i	$-0.443227\pi$
$397$	7.06988	0.354827	0.177414	+	0.984136i	$0.443227\pi$
$398$	0	0
$399$	0.338285	0.0169354
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−47.9399	−2.38805
$404$	0	0
$405$	8.99567	0.446998
$406$	0	0
$407$	9.63380	0.477530
$408$	0	0
$409$	23.8568	1.17964	0.589820	−	0.807535i	$-0.299199\pi$
$409$	23.8568	1.17964	0.589820	+	0.807535i	$0.299199\pi$
$410$	0	0
$411$	−0.226579	−0.0111763
$412$	0	0
$413$	−0.618996	−0.0304588
$414$	0	0
$415$	12.0433	0.591183
$416$	0	0
$417$	0.294954	0.0144440
$418$	0	0
$419$	−31.0625	−1.51750	−0.758752	−	0.651380i	$-0.774191\pi$
$419$	−31.0625	−1.51750	−0.758752	+	0.651380i	$0.774191\pi$
$420$	0	0
$421$	−29.7851	−1.45163	−0.725817	−	0.687887i	$-0.758538\pi$
$421$	−29.7851	−1.45163	−0.725817	+	0.687887i	$0.758538\pi$
$422$	0	0
$423$	6.40108	0.311231
$424$	0	0
$425$	−2.59139	−0.125701
$426$	0	0
$427$	−7.49952	−0.362927
$428$	0	0
$429$	−0.219817	−0.0106129
$430$	0	0
$431$	5.84500	0.281544	0.140772	−	0.990042i	$-0.455042\pi$
$431$	5.84500	0.281544	0.140772	+	0.990042i	$0.455042\pi$
$432$	0	0
$433$	−10.8988	−0.523761	−0.261881	−	0.965100i	$-0.584343\pi$
$433$	−10.8988	−0.523761	−0.261881	+	0.965100i	$0.584343\pi$
$434$	0	0
$435$	−0.150890	−0.00723460
$436$	0	0
$437$	−36.4924	−1.74567
$438$	0	0
$439$	−2.14507	−0.102379	−0.0511894	−	0.998689i	$-0.516301\pi$
$439$	−2.14507	−0.102379	−0.0511894	+	0.998689i	$0.516301\pi$
$440$	0	0
$441$	8.44121	0.401962
$442$	0	0
$443$	−1.84614	−0.0877129	−0.0438565	−	0.999038i	$-0.513964\pi$
$443$	−1.84614	−0.0877129	−0.0438565	+	0.999038i	$0.513964\pi$
$444$	0	0
$445$	6.26285	0.296887
$446$	0	0
$447$	−0.278517	−0.0131734
$448$	0	0
$449$	10.1003	0.476661	0.238331	−	0.971184i	$-0.423400\pi$
$449$	10.1003	0.476661	0.238331	+	0.971184i	$0.423400\pi$
$450$	0	0
$451$	1.63174	0.0768355
$452$	0	0
$453$	0.0911522	0.00428271
$454$	0	0
$455$	11.1300	0.521783
$456$	0	0
$457$	−1.30097	−0.0608569	−0.0304285	−	0.999537i	$-0.509687\pi$
$457$	−1.30097	−0.0608569	−0.0304285	+	0.999537i	$0.509687\pi$
$458$	0	0
$459$	0.341101	0.0159212
$460$	0	0
$461$	13.8178	0.643560	0.321780	−	0.946814i	$-0.395719\pi$
$461$	13.8178	0.643560	0.321780	+	0.946814i	$0.395719\pi$
$462$	0	0
$463$	24.1765	1.12358	0.561788	−	0.827281i	$-0.310114\pi$
$463$	24.1765	1.12358	0.561788	+	0.827281i	$0.310114\pi$
$464$	0	0
$465$	−0.193341	−0.00896598
$466$	0	0
$467$	3.55747	0.164620	0.0823099	−	0.996607i	$-0.473770\pi$
$467$	3.55747	0.164620	0.0823099	+	0.996607i	$0.473770\pi$
$468$	0	0
$469$	−8.55093	−0.394845
$470$	0	0
$471$	0.149590	0.00689273
$472$	0	0
$473$	15.4885	0.712161
$474$	0	0
$475$	7.53632	0.345790
$476$	0	0
$477$	14.7233	0.674131
$478$	0	0
$479$	14.9293	0.682135	0.341068	−	0.940039i	$-0.389212\pi$
$479$	14.9293	0.682135	0.341068	+	0.940039i	$0.389212\pi$
$480$	0	0
$481$	−28.4565	−1.29751
$482$	0	0
$483$	−0.217353	−0.00988992
$484$	0	0
$485$	19.4782	0.884458
$486$	0	0
$487$	−20.7960	−0.942356	−0.471178	−	0.882038i	$-0.656171\pi$
$487$	−20.7960	−0.942356	−0.471178	+	0.882038i	$0.656171\pi$
$488$	0	0
$489$	−0.0829746	−0.00375224
$490$	0	0
$491$	31.4680	1.42013	0.710065	−	0.704136i	$-0.248666\pi$
$491$	31.4680	1.42013	0.710065	+	0.704136i	$0.248666\pi$
$492$	0	0
$493$	17.8221	0.802667
$494$	0	0
$495$	5.52424	0.248296
$496$	0	0
$497$	6.32838	0.283867
$498$	0	0
$499$	5.88848	0.263604	0.131802	−	0.991276i	$-0.457924\pi$
$499$	5.88848	0.263604	0.131802	+	0.991276i	$0.457924\pi$
$500$	0	0
$501$	0.357128	0.0159553
$502$	0	0
$503$	−30.7129	−1.36942	−0.684711	−	0.728815i	$-0.740071\pi$
$503$	−30.7129	−1.36942	−0.684711	+	0.728815i	$0.740071\pi$
$504$	0	0
$505$	−18.9330	−0.842505
$506$	0	0
$507$	0.364082	0.0161695
$508$	0	0
$509$	−10.6710	−0.472985	−0.236492	−	0.971633i	$-0.575998\pi$
$509$	−10.6710	−0.472985	−0.236492	+	0.971633i	$0.575998\pi$
$510$	0	0
$511$	9.77139	0.432261
$512$	0	0
$513$	−0.991995	−0.0437976
$514$	0	0
$515$	1.12270	0.0494719
$516$	0	0
$517$	3.93027	0.172853
$518$	0	0
$519$	−0.429390	−0.0188481
$520$	0	0
$521$	10.0391	0.439823	0.219912	−	0.975520i	$-0.429423\pi$
$521$	10.0391	0.439823	0.219912	+	0.975520i	$0.429423\pi$
$522$	0	0
$523$	−3.53849	−0.154727	−0.0773637	−	0.997003i	$-0.524650\pi$
$523$	−3.53849	−0.154727	−0.0773637	+	0.997003i	$0.524650\pi$
$524$	0	0
$525$	0.0448873	0.00195904
$526$	0	0
$527$	22.8362	0.994760
$528$	0	0
$529$	0.446928	0.0194316
$530$	0	0
$531$	0.907506	0.0393824
$532$	0	0
$533$	−4.81986	−0.208772
$534$	0	0
$535$	8.55332	0.369792
$536$	0	0
$537$	−0.0696460	−0.00300545
$538$	0	0
$539$	5.18292	0.223244
$540$	0	0
$541$	3.24677	0.139589	0.0697947	−	0.997561i	$-0.477766\pi$
$541$	3.24677	0.139589	0.0697947	+	0.997561i	$0.477766\pi$
$542$	0	0
$543$	0.479708	0.0205862
$544$	0	0
$545$	−1.06282	−0.0455261
$546$	0	0
$547$	20.7396	0.886763	0.443382	−	0.896333i	$-0.353779\pi$
$547$	20.7396	0.886763	0.443382	+	0.896333i	$0.353779\pi$
$548$	0	0
$549$	10.9950	0.469255
$550$	0	0
$551$	−51.8305	−2.20805
$552$	0	0
$553$	−28.2676	−1.20206
$554$	0	0
$555$	−0.114765	−0.00487151
$556$	0	0
$557$	−22.4585	−0.951599	−0.475799	−	0.879554i	$-0.657841\pi$
$557$	−22.4585	−0.951599	−0.475799	+	0.879554i	$0.657841\pi$
$558$	0	0
$559$	−45.7503	−1.93503
$560$	0	0
$561$	0.104710	0.00442086
$562$	0	0
$563$	−10.5097	−0.442932	−0.221466	−	0.975168i	$-0.571084\pi$
$563$	−10.5097	−0.442932	−0.221466	+	0.975168i	$0.571084\pi$
$564$	0	0
$565$	4.09059	0.172092
$566$	0	0
$567$	18.4045	0.772915
$568$	0	0
$569$	−31.5091	−1.32093	−0.660464	−	0.750857i	$-0.729641\pi$
$569$	−31.5091	−1.32093	−0.660464	+	0.750857i	$0.729641\pi$
$570$	0	0
$571$	31.0046	1.29750	0.648752	−	0.761000i	$-0.275291\pi$
$571$	31.0046	1.29750	0.648752	+	0.761000i	$0.275291\pi$
$572$	0	0
$573$	−0.356668	−0.0149000
$574$	0	0
$575$	−4.84220	−0.201934
$576$	0	0
$577$	−9.27298	−0.386039	−0.193020	−	0.981195i	$-0.561828\pi$
$577$	−9.27298	−0.386039	−0.193020	+	0.981195i	$0.561828\pi$
$578$	0	0
$579$	0.454279	0.0188792
$580$	0	0
$581$	24.6397	1.02223
$582$	0	0
$583$	9.04010	0.374403
$584$	0	0
$585$	−16.3177	−0.674652
$586$	0	0
$587$	−26.1248	−1.07828	−0.539142	−	0.842215i	$-0.681251\pi$
$587$	−26.1248	−1.07828	−0.539142	+	0.842215i	$0.681251\pi$
$588$	0	0
$589$	−66.4126	−2.73648
$590$	0	0
$591$	0.336394	0.0138374
$592$	0	0
$593$	−3.07816	−0.126405	−0.0632025	−	0.998001i	$-0.520131\pi$
$593$	−3.07816	−0.126405	−0.0632025	+	0.998001i	$0.520131\pi$
$594$	0	0
$595$	−5.30179	−0.217352
$596$	0	0
$597$	−0.0184910	−0.000756787 0
$598$	0	0
$599$	31.3033	1.27902	0.639510	−	0.768783i	$-0.279137\pi$
$599$	31.3033	1.27902	0.639510	+	0.768783i	$0.279137\pi$
$600$	0	0
$601$	16.0338	0.654031	0.327015	−	0.945019i	$-0.393957\pi$
$601$	16.0338	0.654031	0.327015	+	0.945019i	$0.393957\pi$
$602$	0	0
$603$	12.5365	0.510524
$604$	0	0
$605$	−7.60810	−0.309313
$606$	0	0
$607$	−3.90245	−0.158395	−0.0791977	−	0.996859i	$-0.525236\pi$
$607$	−3.90245	−0.158395	−0.0791977	+	0.996859i	$0.525236\pi$
$608$	0	0
$609$	−0.308709	−0.0125095
$610$	0	0
$611$	−11.6093	−0.469664
$612$	0	0
$613$	−9.06438	−0.366107	−0.183053	−	0.983103i	$-0.558598\pi$
$613$	−9.06438	−0.366107	−0.183053	+	0.983103i	$0.558598\pi$
$614$	0	0
$615$	−0.0194385	−0.000783835 0
$616$	0	0
$617$	−27.4684	−1.10583	−0.552917	−	0.833236i	$-0.686485\pi$
$617$	−27.4684	−1.10583	−0.552917	+	0.833236i	$0.686485\pi$
$618$	0	0
$619$	16.5220	0.664073	0.332037	−	0.943266i	$-0.392264\pi$
$619$	16.5220	0.664073	0.332037	+	0.943266i	$0.392264\pi$
$620$	0	0
$621$	0.637372	0.0255769
$622$	0	0
$623$	12.8133	0.513355
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.304519	−0.0121613
$628$	0	0
$629$	13.5553	0.540485
$630$	0	0
$631$	−39.9897	−1.59197	−0.795983	−	0.605319i	$-0.793045\pi$
$631$	−39.9897	−1.59197	−0.795983	+	0.605319i	$0.793045\pi$
$632$	0	0
$633$	0.337599	0.0134184
$634$	0	0
$635$	−12.1939	−0.483901
$636$	0	0
$637$	−15.3094	−0.606582
$638$	0	0
$639$	−9.27800	−0.367032
$640$	0	0
$641$	24.6329	0.972941	0.486470	−	0.873697i	$-0.338284\pi$
$641$	24.6329	0.972941	0.486470	+	0.873697i	$0.338284\pi$
$642$	0	0
$643$	−34.5755	−1.36352	−0.681762	−	0.731574i	$-0.738786\pi$
$643$	−34.5755	−1.36352	−0.681762	+	0.731574i	$0.738786\pi$
$644$	0	0
$645$	−0.184510	−0.00726509
$646$	0	0
$647$	3.85525	0.151565	0.0757827	−	0.997124i	$-0.475854\pi$
$647$	3.85525	0.151565	0.0757827	+	0.997124i	$0.475854\pi$
$648$	0	0
$649$	0.557211	0.0218724
$650$	0	0
$651$	−0.395562	−0.0155033
$652$	0	0
$653$	1.94615	0.0761587	0.0380793	−	0.999275i	$-0.487876\pi$
$653$	1.94615	0.0761587	0.0380793	+	0.999275i	$0.487876\pi$
$654$	0	0
$655$	−11.8609	−0.463445
$656$	0	0
$657$	−14.3258	−0.558902
$658$	0	0
$659$	−28.5561	−1.11239	−0.556193	−	0.831053i	$-0.687739\pi$
$659$	−28.5561	−1.11239	−0.556193	+	0.831053i	$0.687739\pi$
$660$	0	0
$661$	1.25338	0.0487507	0.0243754	−	0.999703i	$-0.492240\pi$
$661$	1.25338	0.0487507	0.0243754	+	0.999703i	$0.492240\pi$
$662$	0	0
$663$	−0.309295	−0.0120120
$664$	0	0
$665$	15.4188	0.597914
$666$	0	0
$667$	33.3019	1.28945
$668$	0	0
$669$	−0.316161	−0.0122235
$670$	0	0
$671$	6.75095	0.260617
$672$	0	0
$673$	−27.0433	−1.04244	−0.521222	−	0.853421i	$-0.674524\pi$
$673$	−27.0433	−1.04244	−0.521222	+	0.853421i	$0.674524\pi$
$674$	0	0
$675$	−0.131629	−0.00506638
$676$	0	0
$677$	−8.22111	−0.315963	−0.157982	−	0.987442i	$-0.550499\pi$
$677$	−8.22111	−0.315963	−0.157982	+	0.987442i	$0.550499\pi$
$678$	0	0
$679$	39.8509	1.52934
$680$	0	0
$681$	−0.329957	−0.0126440
$682$	0	0
$683$	−13.8236	−0.528947	−0.264474	−	0.964393i	$-0.585198\pi$
$683$	−13.8236	−0.528947	−0.264474	+	0.964393i	$0.585198\pi$
$684$	0	0
$685$	−10.3273	−0.394585
$686$	0	0
$687$	−0.500024	−0.0190771
$688$	0	0
$689$	−26.7029	−1.01730
$690$	0	0
$691$	12.1822	0.463431	0.231716	−	0.972784i	$-0.425566\pi$
$691$	12.1822	0.463431	0.231716	+	0.972784i	$0.425566\pi$
$692$	0	0
$693$	11.3022	0.429335
$694$	0	0
$695$	13.4438	0.509952
$696$	0	0
$697$	2.29595	0.0869652
$698$	0	0
$699$	−0.338900	−0.0128184
$700$	0	0
$701$	−29.1247	−1.10003	−0.550013	−	0.835156i	$-0.685377\pi$
$701$	−29.1247	−1.10003	−0.550013	+	0.835156i	$0.685377\pi$
$702$	0	0
$703$	−39.4217	−1.48682
$704$	0	0
$705$	−0.0468204	−0.00176336
$706$	0	0
$707$	−38.7354	−1.45680
$708$	0	0
$709$	−50.1985	−1.88524	−0.942622	−	0.333863i	$-0.891648\pi$
$709$	−50.1985	−1.88524	−0.942622	+	0.333863i	$0.891648\pi$
$710$	0	0
$711$	41.4430	1.55423
$712$	0	0
$713$	42.6711	1.59804
$714$	0	0
$715$	−10.0191	−0.374692
$716$	0	0
$717$	−0.0385952	−0.00144136
$718$	0	0
$719$	2.97634	0.110999	0.0554993	−	0.998459i	$-0.482325\pi$
$719$	2.97634	0.110999	0.0554993	+	0.998459i	$0.482325\pi$
$720$	0	0
$721$	2.29695	0.0855429
$722$	0	0
$723$	−0.536640	−0.0199579
$724$	0	0
$725$	−6.87742	−0.255421
$726$	0	0
$727$	26.9880	1.00093	0.500464	−	0.865757i	$-0.333163\pi$
$727$	26.9880	1.00093	0.500464	+	0.865757i	$0.333163\pi$
$728$	0	0
$729$	−26.9740	−0.999037
$730$	0	0
$731$	21.7932	0.806050
$732$	0	0
$733$	−15.2544	−0.563435	−0.281718	−	0.959497i	$-0.590904\pi$
$733$	−15.2544	−0.563435	−0.281718	+	0.959497i	$0.590904\pi$
$734$	0	0
$735$	−0.0617428	−0.00227742
$736$	0	0
$737$	7.69741	0.283538
$738$	0	0
$739$	−2.58149	−0.0949615	−0.0474808	−	0.998872i	$-0.515119\pi$
$739$	−2.58149	−0.0949615	−0.0474808	+	0.998872i	$0.515119\pi$
$740$	0	0
$741$	0.899496	0.0330438
$742$	0	0
$743$	−5.20358	−0.190901	−0.0954505	−	0.995434i	$-0.530429\pi$
$743$	−5.20358	−0.190901	−0.0954505	+	0.995434i	$0.530429\pi$
$744$	0	0
$745$	−12.6946	−0.465094
$746$	0	0
$747$	−36.1241	−1.32171
$748$	0	0
$749$	17.4995	0.639416
$750$	0	0
$751$	12.3248	0.449739	0.224869	−	0.974389i	$-0.427804\pi$
$751$	12.3248	0.449739	0.224869	+	0.974389i	$0.427804\pi$
$752$	0	0
$753$	−0.228418	−0.00832403
$754$	0	0
$755$	4.15464	0.151203
$756$	0	0
$757$	35.4343	1.28788	0.643941	−	0.765075i	$-0.277298\pi$
$757$	35.4343	1.28788	0.643941	+	0.765075i	$0.277298\pi$
$758$	0	0
$759$	0.195658	0.00710194
$760$	0	0
$761$	12.3534	0.447810	0.223905	−	0.974611i	$-0.428119\pi$
$761$	12.3534	0.447810	0.223905	+	0.974611i	$0.428119\pi$
$762$	0	0
$763$	−2.17445	−0.0787203
$764$	0	0
$765$	7.77293	0.281031
$766$	0	0
$767$	−1.64590	−0.0594301
$768$	0	0
$769$	19.9050	0.717791	0.358896	−	0.933378i	$-0.383153\pi$
$769$	19.9050	0.717791	0.358896	+	0.933378i	$0.383153\pi$
$770$	0	0
$771$	−0.393174	−0.0141598
$772$	0	0
$773$	−17.7311	−0.637745	−0.318873	−	0.947798i	$-0.603304\pi$
$773$	−17.7311	−0.637745	−0.318873	+	0.947798i	$0.603304\pi$
$774$	0	0
$775$	−8.81233	−0.316548
$776$	0	0
$777$	−0.234801	−0.00842343
$778$	0	0
$779$	−6.67711	−0.239232
$780$	0	0
$781$	−5.69671	−0.203844
$782$	0	0
$783$	0.905265	0.0323515
$784$	0	0
$785$	6.81818	0.243351
$786$	0	0
$787$	16.0543	0.572275	0.286138	−	0.958189i	$-0.407629\pi$
$787$	16.0543	0.572275	0.286138	+	0.958189i	$0.407629\pi$
$788$	0	0
$789$	0.00109242	3.88912e−5 0
$790$	0	0
$791$	8.36905	0.297569
$792$	0	0
$793$	−19.9411	−0.708130
$794$	0	0
$795$	−0.107693	−0.00381946
$796$	0	0
$797$	−5.22606	−0.185116	−0.0925582	−	0.995707i	$-0.529504\pi$
$797$	−5.22606	−0.185116	−0.0925582	+	0.995707i	$0.529504\pi$
$798$	0	0
$799$	5.53012	0.195642
$800$	0	0
$801$	−18.7855	−0.663754
$802$	0	0
$803$	−8.79605	−0.310406
$804$	0	0
$805$	−9.90679	−0.349168
$806$	0	0
$807$	−0.611695	−0.0215327
$808$	0	0
$809$	20.4182	0.717867	0.358934	−	0.933363i	$-0.383140\pi$
$809$	20.4182	0.717867	0.358934	+	0.933363i	$0.383140\pi$
$810$	0	0
$811$	22.0912	0.775728	0.387864	−	0.921717i	$-0.373213\pi$
$811$	22.0912	0.775728	0.387864	+	0.921717i	$0.373213\pi$
$812$	0	0
$813$	0.327973	0.0115025
$814$	0	0
$815$	−3.78191	−0.132475
$816$	0	0
$817$	−63.3792	−2.21736
$818$	0	0
$819$	−33.3847	−1.16656
$820$	0	0
$821$	−3.74444	−0.130682	−0.0653409	−	0.997863i	$-0.520813\pi$
$821$	−3.74444	−0.130682	−0.0653409	+	0.997863i	$0.520813\pi$
$822$	0	0
$823$	21.0826	0.734892	0.367446	−	0.930045i	$-0.380232\pi$
$823$	21.0826	0.734892	0.367446	+	0.930045i	$0.380232\pi$
$824$	0	0
$825$	−0.0404068	−0.00140679
$826$	0	0
$827$	47.4943	1.65154	0.825769	−	0.564008i	$-0.190741\pi$
$827$	47.4943	1.65154	0.825769	+	0.564008i	$0.190741\pi$
$828$	0	0
$829$	21.1371	0.734123	0.367062	−	0.930197i	$-0.380364\pi$
$829$	21.1371	0.734123	0.367062	+	0.930197i	$0.380364\pi$
$830$	0	0
$831$	0.447410	0.0155205
$832$	0	0
$833$	7.29266	0.252676
$834$	0	0
$835$	16.2776	0.563310
$836$	0	0
$837$	1.15995	0.0400939
$838$	0	0
$839$	38.6495	1.33433	0.667164	−	0.744911i	$-0.267508\pi$
$839$	38.6495	1.33433	0.667164	+	0.744911i	$0.267508\pi$
$840$	0	0
$841$	18.2990	0.630999
$842$	0	0
$843$	−0.0461489	−0.00158945
$844$	0	0
$845$	16.5946	0.570871
$846$	0	0
$847$	−15.5656	−0.534841
$848$	0	0
$849$	−0.564946	−0.0193889
$850$	0	0
$851$	25.3291	0.868269
$852$	0	0
$853$	−35.4525	−1.21387	−0.606936	−	0.794751i	$-0.707601\pi$
$853$	−35.4525	−1.21387	−0.606936	+	0.794751i	$0.707601\pi$
$854$	0	0
$855$	−22.6053	−0.773086
$856$	0	0
$857$	2.35888	0.0805777	0.0402888	−	0.999188i	$-0.487172\pi$
$857$	2.35888	0.0805777	0.0402888	+	0.999188i	$0.487172\pi$
$858$	0	0
$859$	16.9290	0.577611	0.288805	−	0.957388i	$-0.406742\pi$
$859$	16.9290	0.577611	0.288805	+	0.957388i	$0.406742\pi$
$860$	0	0
$861$	−0.0397697	−0.00135535
$862$	0	0
$863$	−44.3947	−1.51121	−0.755607	−	0.655025i	$-0.772658\pi$
$863$	−44.3947	−1.51121	−0.755607	+	0.655025i	$0.772658\pi$
$864$	0	0
$865$	−19.5713	−0.665442
$866$	0	0
$867$	−0.225645	−0.00766329
$868$	0	0
$869$	25.4461	0.863199
$870$	0	0
$871$	−22.7368	−0.770407
$872$	0	0
$873$	−58.4252	−1.97739
$874$	0	0
$875$	2.04593	0.0691649
$876$	0	0
$877$	17.4787	0.590213	0.295107	−	0.955464i	$-0.404645\pi$
$877$	17.4787	0.590213	0.295107	+	0.955464i	$0.404645\pi$
$878$	0	0
$879$	−0.253401	−0.00854699
$880$	0	0
$881$	−42.5977	−1.43515	−0.717577	−	0.696479i	$-0.754749\pi$
$881$	−42.5977	−1.43515	−0.717577	+	0.696479i	$0.754749\pi$
$882$	0	0
$883$	8.28664	0.278867	0.139434	−	0.990231i	$-0.455472\pi$
$883$	8.28664	0.278867	0.139434	+	0.990231i	$0.455472\pi$
$884$	0	0
$885$	−0.00663791	−0.000223131 0
$886$	0	0
$887$	−9.48125	−0.318349	−0.159175	−	0.987250i	$-0.550883\pi$
$887$	−9.48125	−0.318349	−0.159175	+	0.987250i	$0.550883\pi$
$888$	0	0
$889$	−24.9479	−0.836724
$890$	0	0
$891$	−16.5674	−0.555029
$892$	0	0
$893$	−16.0828	−0.538190
$894$	0	0
$895$	−3.17441	−0.106109
$896$	0	0
$897$	−0.577940	−0.0192968
$898$	0	0
$899$	60.6061	2.02133
$900$	0	0
$901$	12.7199	0.423763
$902$	0	0
$903$	−0.377495	−0.0125622
$904$	0	0
$905$	21.8647	0.726806
$906$	0	0
$907$	−50.9179	−1.69070	−0.845350	−	0.534213i	$-0.820608\pi$
$907$	−50.9179	−1.69070	−0.845350	+	0.534213i	$0.820608\pi$
$908$	0	0
$909$	56.7898	1.88360
$910$	0	0
$911$	−8.52551	−0.282463	−0.141231	−	0.989977i	$-0.545106\pi$
$911$	−8.52551	−0.282463	−0.141231	+	0.989977i	$0.545106\pi$
$912$	0	0
$913$	−22.1803	−0.734060
$914$	0	0
$915$	−0.0804224	−0.00265868
$916$	0	0
$917$	−24.2666	−0.801354
$918$	0	0
$919$	27.1263	0.894815	0.447408	−	0.894330i	$-0.352347\pi$
$919$	27.1263	0.894815	0.447408	+	0.894330i	$0.352347\pi$
$920$	0	0
$921$	0.299907	0.00988226
$922$	0	0
$923$	16.8271	0.553870
$924$	0	0
$925$	−5.23090	−0.171991
$926$	0	0
$927$	−3.36755	−0.110605
$928$	0	0
$929$	−4.03906	−0.132517	−0.0662587	−	0.997802i	$-0.521106\pi$
$929$	−4.03906	−0.132517	−0.0662587	+	0.997802i	$0.521106\pi$
$930$	0	0
$931$	−21.2086	−0.695084
$932$	0	0
$933$	0.411743	0.0134799
$934$	0	0
$935$	4.77259	0.156080
$936$	0	0
$937$	31.6305	1.03332	0.516662	−	0.856190i	$-0.327174\pi$
$937$	31.6305	1.03332	0.516662	+	0.856190i	$0.327174\pi$
$938$	0	0
$939$	0.0960947	0.00313593
$940$	0	0
$941$	6.69573	0.218275	0.109137	−	0.994027i	$-0.465191\pi$
$941$	6.69573	0.218275	0.109137	+	0.994027i	$0.465191\pi$
$942$	0	0
$943$	4.29014	0.139706
$944$	0	0
$945$	−0.269302	−0.00876040
$946$	0	0
$947$	28.0827	0.912566	0.456283	−	0.889835i	$-0.349180\pi$
$947$	28.0827	0.912566	0.456283	+	0.889835i	$0.349180\pi$
$948$	0	0
$949$	25.9820	0.843411
$950$	0	0
$951$	−0.516907	−0.0167619
$952$	0	0
$953$	−56.4528	−1.82869	−0.914343	−	0.404942i	$-0.867292\pi$
$953$	−56.4528	−1.82869	−0.914343	+	0.404942i	$0.867292\pi$
$954$	0	0
$955$	−16.2566	−0.526053
$956$	0	0
$957$	0.277895	0.00898307
$958$	0	0
$959$	−21.1289	−0.682286
$960$	0	0
$961$	46.6572	1.50507
$962$	0	0
$963$	−25.6558	−0.826748
$964$	0	0
$965$	20.7057	0.666539
$966$	0	0
$967$	3.12119	0.100371	0.0501853	−	0.998740i	$-0.484019\pi$
$967$	3.12119	0.100371	0.0501853	+	0.998740i	$0.484019\pi$
$968$	0	0
$969$	−0.428475	−0.0137646
$970$	0	0
$971$	45.8642	1.47185	0.735926	−	0.677062i	$-0.236747\pi$
$971$	45.8642	1.47185	0.735926	+	0.677062i	$0.236747\pi$
$972$	0	0
$973$	27.5050	0.881769
$974$	0	0
$975$	0.119355	0.00382241
$976$	0	0
$977$	40.8678	1.30748	0.653739	−	0.756720i	$-0.273199\pi$
$977$	40.8678	1.30748	0.653739	+	0.756720i	$0.273199\pi$
$978$	0	0
$979$	−11.5343	−0.368639
$980$	0	0
$981$	3.18794	0.101783
$982$	0	0
$983$	46.5199	1.48376	0.741878	−	0.670535i	$-0.233935\pi$
$983$	46.5199	1.48376	0.741878	+	0.670535i	$0.233935\pi$
$984$	0	0
$985$	15.3326	0.488536
$986$	0	0
$987$	−0.0957911	−0.00304906
$988$	0	0
$989$	40.7221	1.29489
$990$	0	0
$991$	18.5952	0.590695	0.295347	−	0.955390i	$-0.404565\pi$
$991$	18.5952	0.590695	0.295347	+	0.955390i	$0.404565\pi$
$992$	0	0
$993$	0.265079	0.00841201
$994$	0	0
$995$	−0.842805	−0.0267187
$996$	0	0
$997$	−25.0484	−0.793292	−0.396646	−	0.917972i	$-0.629826\pi$
$997$	−25.0484	−0.793292	−0.396646	+	0.917972i	$0.629826\pi$
$998$	0	0
$999$	0.688535	0.0217843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.16	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.16	✓	29	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)

\(f(q)\)	\(=\)	\(q+0.0219398 q^{3} +1.00000 q^{5} +2.04593 q^{7} -2.99952 q^{9} +O(q^{10})\)	\(q+0.0219398 q^{3} +1.00000 q^{5} +2.04593 q^{7} -2.99952 q^{9} -1.84171 q^{11} +5.44009 q^{13} +0.0219398 q^{15} -2.59139 q^{17} +7.53632 q^{19} +0.0448873 q^{21} -4.84220 q^{23} +1.00000 q^{25} -0.131629 q^{27} -6.87742 q^{29} -8.81233 q^{31} -0.0404068 q^{33} +2.04593 q^{35} -5.23090 q^{37} +0.119355 q^{39} -0.885990 q^{41} -8.40984 q^{43} -2.99952 q^{45} -2.13404 q^{47} -2.81419 q^{49} -0.0568547 q^{51} -4.90854 q^{53} -1.84171 q^{55} +0.165346 q^{57} -0.302551 q^{59} -3.66559 q^{61} -6.13679 q^{63} +5.44009 q^{65} -4.17949 q^{67} -0.106237 q^{69} +3.09316 q^{71} +4.77602 q^{73} +0.0219398 q^{75} -3.76800 q^{77} -13.8166 q^{79} +8.99567 q^{81} +12.0433 q^{83} -2.59139 q^{85} -0.150890 q^{87} +6.26285 q^{89} +11.1300 q^{91} -0.193341 q^{93} +7.53632 q^{95} +19.4782 q^{97} +5.52424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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