LMFDB - Embedded newform 8020.2.a.c.1.10

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:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-1.42561 q^{3} -1.00000 q^{5} -3.82644 q^{7} -0.967639 q^{9} +O(q^{10})$	$q-1.42561 q^{3} -1.00000 q^{5} -3.82644 q^{7} -0.967639 q^{9} +2.52224 q^{11} +7.13963 q^{13} +1.42561 q^{15} -6.37964 q^{17} -4.40687 q^{19} +5.45500 q^{21} +1.00931 q^{23} +1.00000 q^{25} +5.65630 q^{27} +2.93809 q^{29} -4.16374 q^{31} -3.59573 q^{33} +3.82644 q^{35} -11.7415 q^{37} -10.1783 q^{39} +1.67600 q^{41} +9.49358 q^{43} +0.967639 q^{45} +9.50027 q^{47} +7.64162 q^{49} +9.09487 q^{51} +5.55872 q^{53} -2.52224 q^{55} +6.28247 q^{57} +5.03660 q^{59} -6.50125 q^{61} +3.70261 q^{63} -7.13963 q^{65} -10.7038 q^{67} -1.43888 q^{69} +16.2918 q^{71} +4.97205 q^{73} -1.42561 q^{75} -9.65120 q^{77} -5.11619 q^{79} -5.16076 q^{81} -1.92072 q^{83} +6.37964 q^{85} -4.18857 q^{87} -16.4866 q^{89} -27.3194 q^{91} +5.93586 q^{93} +4.40687 q^{95} +13.7260 q^{97} -2.44062 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	−1.42561	−0.823076	−0.411538	−	0.911393i	$-0.635008\pi$
$3$	−1.42561	−0.823076	−0.411538	+	0.911393i	$0.635008\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.82644	−1.44626	−0.723129	−	0.690713i	$-0.757297\pi$
$7$	−3.82644	−1.44626	−0.723129	+	0.690713i	$0.757297\pi$
$8$	0	0
$9$	−0.967639	−0.322546
$10$	0	0
$11$	2.52224	0.760485	0.380242	−	0.924887i	$-0.375841\pi$
$11$	2.52224	0.760485	0.380242	+	0.924887i	$0.375841\pi$
$12$	0	0
$13$	7.13963	1.98018	0.990089	−	0.140442i	$-0.0448525\pi$
$13$	7.13963	1.98018	0.990089	+	0.140442i	$0.0448525\pi$
$14$	0	0
$15$	1.42561	0.368091
$16$	0	0
$17$	−6.37964	−1.54729	−0.773645	−	0.633619i	$-0.781569\pi$
$17$	−6.37964	−1.54729	−0.773645	+	0.633619i	$0.781569\pi$
$18$	0	0
$19$	−4.40687	−1.01100	−0.505502	−	0.862825i	$-0.668693\pi$
$19$	−4.40687	−1.01100	−0.505502	+	0.862825i	$0.668693\pi$
$20$	0	0
$21$	5.45500	1.19038
$22$	0	0
$23$	1.00931	0.210456	0.105228	−	0.994448i	$-0.466443\pi$
$23$	1.00931	0.210456	0.105228	+	0.994448i	$0.466443\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65630	1.08856
$28$	0	0
$29$	2.93809	0.545590	0.272795	−	0.962072i	$-0.412052\pi$
$29$	2.93809	0.545590	0.272795	+	0.962072i	$0.412052\pi$
$30$	0	0
$31$	−4.16374	−0.747830	−0.373915	−	0.927463i	$-0.621985\pi$
$31$	−4.16374	−0.747830	−0.373915	+	0.927463i	$0.621985\pi$
$32$	0	0
$33$	−3.59573	−0.625936
$34$	0	0
$35$	3.82644	0.646786
$36$	0	0
$37$	−11.7415	−1.93029	−0.965147	−	0.261708i	$-0.915714\pi$
$37$	−11.7415	−1.93029	−0.965147	+	0.261708i	$0.915714\pi$
$38$	0	0
$39$	−10.1783	−1.62984
$40$	0	0
$41$	1.67600	0.261748	0.130874	−	0.991399i	$-0.458222\pi$
$41$	1.67600	0.261748	0.130874	+	0.991399i	$0.458222\pi$
$42$	0	0
$43$	9.49358	1.44776	0.723879	−	0.689927i	$-0.242358\pi$
$43$	9.49358	1.44776	0.723879	+	0.689927i	$0.242358\pi$
$44$	0	0
$45$	0.967639	0.144247
$46$	0	0
$47$	9.50027	1.38576	0.692878	−	0.721055i	$-0.256342\pi$
$47$	9.50027	1.38576	0.692878	+	0.721055i	$0.256342\pi$
$48$	0	0
$49$	7.64162	1.09166
$50$	0	0
$51$	9.09487	1.27354
$52$	0	0
$53$	5.55872	0.763549	0.381775	−	0.924255i	$-0.375313\pi$
$53$	5.55872	0.763549	0.381775	+	0.924255i	$0.375313\pi$
$54$	0	0
$55$	−2.52224	−0.340099
$56$	0	0
$57$	6.28247	0.832133
$58$	0	0
$59$	5.03660	0.655709	0.327855	−	0.944728i	$-0.393674\pi$
$59$	5.03660	0.655709	0.327855	+	0.944728i	$0.393674\pi$
$60$	0	0
$61$	−6.50125	−0.832400	−0.416200	−	0.909273i	$-0.636638\pi$
$61$	−6.50125	−0.832400	−0.416200	+	0.909273i	$0.636638\pi$
$62$	0	0
$63$	3.70261	0.466485
$64$	0	0
$65$	−7.13963	−0.885562
$66$	0	0
$67$	−10.7038	−1.30767	−0.653836	−	0.756636i	$-0.726841\pi$
$67$	−10.7038	−1.30767	−0.653836	+	0.756636i	$0.726841\pi$
$68$	0	0
$69$	−1.43888	−0.173221
$70$	0	0
$71$	16.2918	1.93348	0.966738	−	0.255769i	$-0.0823286\pi$
$71$	16.2918	1.93348	0.966738	+	0.255769i	$0.0823286\pi$
$72$	0	0
$73$	4.97205	0.581934	0.290967	−	0.956733i	$-0.406023\pi$
$73$	4.97205	0.581934	0.290967	+	0.956733i	$0.406023\pi$
$74$	0	0
$75$	−1.42561	−0.164615
$76$	0	0
$77$	−9.65120	−1.09986
$78$	0	0
$79$	−5.11619	−0.575617	−0.287808	−	0.957688i	$-0.592927\pi$
$79$	−5.11619	−0.575617	−0.287808	+	0.957688i	$0.592927\pi$
$80$	0	0
$81$	−5.16076	−0.573417
$82$	0	0
$83$	−1.92072	−0.210826	−0.105413	−	0.994429i	$-0.533617\pi$
$83$	−1.92072	−0.210826	−0.105413	+	0.994429i	$0.533617\pi$
$84$	0	0
$85$	6.37964	0.691969
$86$	0	0
$87$	−4.18857	−0.449062
$88$	0	0
$89$	−16.4866	−1.74757	−0.873787	−	0.486308i	$-0.838343\pi$
$89$	−16.4866	−1.74757	−0.873787	+	0.486308i	$0.838343\pi$
$90$	0	0
$91$	−27.3194	−2.86385
$92$	0	0
$93$	5.93586	0.615520
$94$	0	0
$95$	4.40687	0.452135
$96$	0	0
$97$	13.7260	1.39366	0.696832	−	0.717234i	$-0.254592\pi$
$97$	13.7260	1.39366	0.696832	+	0.717234i	$0.254592\pi$
$98$	0	0
$99$	−2.44062	−0.245292
$100$	0	0
$101$	−3.05617	−0.304100	−0.152050	−	0.988373i	$-0.548587\pi$
$101$	−3.05617	−0.304100	−0.152050	+	0.988373i	$0.548587\pi$
$102$	0	0
$103$	9.88073	0.973577	0.486788	−	0.873520i	$-0.338168\pi$
$103$	9.88073	0.973577	0.486788	+	0.873520i	$0.338168\pi$
$104$	0	0
$105$	−5.45500	−0.532354
$106$	0	0
$107$	8.34543	0.806783	0.403391	−	0.915028i	$-0.367831\pi$
$107$	8.34543	0.806783	0.403391	+	0.915028i	$0.367831\pi$
$108$	0	0
$109$	8.17421	0.782947	0.391474	−	0.920189i	$-0.371965\pi$
$109$	8.17421	0.782947	0.391474	+	0.920189i	$0.371965\pi$
$110$	0	0
$111$	16.7388	1.58878
$112$	0	0
$113$	10.7614	1.01235	0.506175	−	0.862431i	$-0.331059\pi$
$113$	10.7614	1.01235	0.506175	+	0.862431i	$0.331059\pi$
$114$	0	0
$115$	−1.00931	−0.0941189
$116$	0	0
$117$	−6.90859	−0.638699
$118$	0	0
$119$	24.4113	2.23778
$120$	0	0
$121$	−4.63829	−0.421663
$122$	0	0
$123$	−2.38933	−0.215438
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.22352	−0.640984	−0.320492	−	0.947251i	$-0.603848\pi$
$127$	−7.22352	−0.640984	−0.320492	+	0.947251i	$0.603848\pi$
$128$	0	0
$129$	−13.5341	−1.19161
$130$	0	0
$131$	13.7818	1.20412	0.602059	−	0.798452i	$-0.294347\pi$
$131$	13.7818	1.20412	0.602059	+	0.798452i	$0.294347\pi$
$132$	0	0
$133$	16.8626	1.46217
$134$	0	0
$135$	−5.65630	−0.486817
$136$	0	0
$137$	−1.52083	−0.129934	−0.0649668	−	0.997887i	$-0.520694\pi$
$137$	−1.52083	−0.129934	−0.0649668	+	0.997887i	$0.520694\pi$
$138$	0	0
$139$	2.77428	0.235311	0.117656	−	0.993054i	$-0.462462\pi$
$139$	2.77428	0.235311	0.117656	+	0.993054i	$0.462462\pi$
$140$	0	0
$141$	−13.5437	−1.14058
$142$	0	0
$143$	18.0079	1.50589
$144$	0	0
$145$	−2.93809	−0.243995
$146$	0	0
$147$	−10.8940	−0.898519
$148$	0	0
$149$	−19.2544	−1.57738	−0.788689	−	0.614792i	$-0.789240\pi$
$149$	−19.2544	−1.57738	−0.788689	+	0.614792i	$0.789240\pi$
$150$	0	0
$151$	13.9250	1.13320	0.566600	−	0.823993i	$-0.308259\pi$
$151$	13.9250	1.13320	0.566600	+	0.823993i	$0.308259\pi$
$152$	0	0
$153$	6.17319	0.499073
$154$	0	0
$155$	4.16374	0.334440
$156$	0	0
$157$	8.34856	0.666288	0.333144	−	0.942876i	$-0.391891\pi$
$157$	8.34856	0.666288	0.333144	+	0.942876i	$0.391891\pi$
$158$	0	0
$159$	−7.92456	−0.628459
$160$	0	0
$161$	−3.86207	−0.304374
$162$	0	0
$163$	8.93608	0.699928	0.349964	−	0.936763i	$-0.386194\pi$
$163$	8.93608	0.699928	0.349964	+	0.936763i	$0.386194\pi$
$164$	0	0
$165$	3.59573	0.279927
$166$	0	0
$167$	10.9435	0.846832	0.423416	−	0.905935i	$-0.360831\pi$
$167$	10.9435	0.846832	0.423416	+	0.905935i	$0.360831\pi$
$168$	0	0
$169$	37.9743	2.92110
$170$	0	0
$171$	4.26426	0.326096
$172$	0	0
$173$	−5.17402	−0.393373	−0.196687	−	0.980466i	$-0.563018\pi$
$173$	−5.17402	−0.393373	−0.196687	+	0.980466i	$0.563018\pi$
$174$	0	0
$175$	−3.82644	−0.289251
$176$	0	0
$177$	−7.18022	−0.539699
$178$	0	0
$179$	−24.7530	−1.85013	−0.925064	−	0.379811i	$-0.875989\pi$
$179$	−24.7530	−1.85013	−0.925064	+	0.379811i	$0.875989\pi$
$180$	0	0
$181$	−3.83998	−0.285424	−0.142712	−	0.989764i	$-0.545582\pi$
$181$	−3.83998	−0.285424	−0.142712	+	0.989764i	$0.545582\pi$
$182$	0	0
$183$	9.26824	0.685128
$184$	0	0
$185$	11.7415	0.863254
$186$	0	0
$187$	−16.0910	−1.17669
$188$	0	0
$189$	−21.6435	−1.57433
$190$	0	0
$191$	8.31951	0.601979	0.300989	−	0.953627i	$-0.402683\pi$
$191$	8.31951	0.601979	0.300989	+	0.953627i	$0.402683\pi$
$192$	0	0
$193$	−21.0278	−1.51361	−0.756806	−	0.653639i	$-0.773241\pi$
$193$	−21.0278	−1.51361	−0.756806	+	0.653639i	$0.773241\pi$
$194$	0	0
$195$	10.1783	0.728885
$196$	0	0
$197$	−26.1768	−1.86502	−0.932510	−	0.361145i	$-0.882386\pi$
$197$	−26.1768	−1.86502	−0.932510	+	0.361145i	$0.882386\pi$
$198$	0	0
$199$	−13.6712	−0.969127	−0.484563	−	0.874756i	$-0.661022\pi$
$199$	−13.6712	−0.969127	−0.484563	+	0.874756i	$0.661022\pi$
$200$	0	0
$201$	15.2594	1.07631
$202$	0	0
$203$	−11.2424	−0.789064
$204$	0	0
$205$	−1.67600	−0.117057
$206$	0	0
$207$	−0.976650	−0.0678819
$208$	0	0
$209$	−11.1152	−0.768853
$210$	0	0
$211$	10.1118	0.696127	0.348063	−	0.937471i	$-0.386839\pi$
$211$	10.1118	0.696127	0.348063	+	0.937471i	$0.386839\pi$
$212$	0	0
$213$	−23.2257	−1.59140
$214$	0	0
$215$	−9.49358	−0.647457
$216$	0	0
$217$	15.9323	1.08155
$218$	0	0
$219$	−7.08819	−0.478976
$220$	0	0
$221$	−45.5483	−3.06391
$222$	0	0
$223$	−2.15048	−0.144007	−0.0720033	−	0.997404i	$-0.522939\pi$
$223$	−2.15048	−0.144007	−0.0720033	+	0.997404i	$0.522939\pi$
$224$	0	0
$225$	−0.967639	−0.0645093
$226$	0	0
$227$	3.18146	0.211161	0.105580	−	0.994411i	$-0.466330\pi$
$227$	3.18146	0.211161	0.105580	+	0.994411i	$0.466330\pi$
$228$	0	0
$229$	−20.2860	−1.34054	−0.670268	−	0.742119i	$-0.733821\pi$
$229$	−20.2860	−1.34054	−0.670268	+	0.742119i	$0.733821\pi$
$230$	0	0
$231$	13.7588	0.905265
$232$	0	0
$233$	−21.3700	−1.39999	−0.699996	−	0.714147i	$-0.746815\pi$
$233$	−21.3700	−1.39999	−0.699996	+	0.714147i	$0.746815\pi$
$234$	0	0
$235$	−9.50027	−0.619729
$236$	0	0
$237$	7.29369	0.473776
$238$	0	0
$239$	−30.4733	−1.97116	−0.985578	−	0.169223i	$-0.945874\pi$
$239$	−30.4733	−1.97116	−0.985578	+	0.169223i	$0.945874\pi$
$240$	0	0
$241$	−15.2679	−0.983492	−0.491746	−	0.870739i	$-0.663641\pi$
$241$	−15.2679	−0.983492	−0.491746	+	0.870739i	$0.663641\pi$
$242$	0	0
$243$	−9.61169	−0.616590
$244$	0	0
$245$	−7.64162	−0.488205
$246$	0	0
$247$	−31.4634	−2.00197
$248$	0	0
$249$	2.73820	0.173526
$250$	0	0
$251$	−16.8775	−1.06530	−0.532648	−	0.846337i	$-0.678803\pi$
$251$	−16.8775	−1.06530	−0.532648	+	0.846337i	$0.678803\pi$
$252$	0	0
$253$	2.54573	0.160049
$254$	0	0
$255$	−9.09487	−0.569543
$256$	0	0
$257$	5.33868	0.333017	0.166509	−	0.986040i	$-0.446751\pi$
$257$	5.33868	0.333017	0.166509	+	0.986040i	$0.446751\pi$
$258$	0	0
$259$	44.9282	2.79170
$260$	0	0
$261$	−2.84301	−0.175978
$262$	0	0
$263$	16.3133	1.00592	0.502960	−	0.864310i	$-0.332244\pi$
$263$	16.3133	1.00592	0.502960	+	0.864310i	$0.332244\pi$
$264$	0	0
$265$	−5.55872	−0.341470
$266$	0	0
$267$	23.5034	1.43839
$268$	0	0
$269$	−6.27216	−0.382420	−0.191210	−	0.981549i	$-0.561241\pi$
$269$	−6.27216	−0.382420	−0.191210	+	0.981549i	$0.561241\pi$
$270$	0	0
$271$	24.4756	1.48679	0.743393	−	0.668855i	$-0.233215\pi$
$271$	24.4756	1.48679	0.743393	+	0.668855i	$0.233215\pi$
$272$	0	0
$273$	38.9467	2.35716
$274$	0	0
$275$	2.52224	0.152097
$276$	0	0
$277$	−6.12014	−0.367724	−0.183862	−	0.982952i	$-0.558860\pi$
$277$	−6.12014	−0.367724	−0.183862	+	0.982952i	$0.558860\pi$
$278$	0	0
$279$	4.02900	0.241210
$280$	0	0
$281$	−10.7234	−0.639705	−0.319852	−	0.947467i	$-0.603633\pi$
$281$	−10.7234	−0.639705	−0.319852	+	0.947467i	$0.603633\pi$
$282$	0	0
$283$	−17.9150	−1.06494	−0.532470	−	0.846449i	$-0.678736\pi$
$283$	−17.9150	−1.06494	−0.532470	+	0.846449i	$0.678736\pi$
$284$	0	0
$285$	−6.28247	−0.372141
$286$	0	0
$287$	−6.41313	−0.378555
$288$	0	0
$289$	23.6998	1.39411
$290$	0	0
$291$	−19.5679	−1.14709
$292$	0	0
$293$	23.8503	1.39335	0.696675	−	0.717387i	$-0.254662\pi$
$293$	23.8503	1.39335	0.696675	+	0.717387i	$0.254662\pi$
$294$	0	0
$295$	−5.03660	−0.293242
$296$	0	0
$297$	14.2666	0.827830
$298$	0	0
$299$	7.20612	0.416741
$300$	0	0
$301$	−36.3266	−2.09383
$302$	0	0
$303$	4.35690	0.250297
$304$	0	0
$305$	6.50125	0.372260
$306$	0	0
$307$	−28.1123	−1.60446	−0.802228	−	0.597018i	$-0.796352\pi$
$307$	−28.1123	−1.60446	−0.802228	+	0.597018i	$0.796352\pi$
$308$	0	0
$309$	−14.0861	−0.801328
$310$	0	0
$311$	−18.6160	−1.05562	−0.527809	−	0.849363i	$-0.676986\pi$
$311$	−18.6160	−1.05562	−0.527809	+	0.849363i	$0.676986\pi$
$312$	0	0
$313$	−8.33454	−0.471096	−0.235548	−	0.971863i	$-0.575689\pi$
$313$	−8.33454	−0.471096	−0.235548	+	0.971863i	$0.575689\pi$
$314$	0	0
$315$	−3.70261	−0.208619
$316$	0	0
$317$	15.7243	0.883164	0.441582	−	0.897221i	$-0.354417\pi$
$317$	15.7243	0.883164	0.441582	+	0.897221i	$0.354417\pi$
$318$	0	0
$319$	7.41058	0.414913
$320$	0	0
$321$	−11.8973	−0.664043
$322$	0	0
$323$	28.1142	1.56432
$324$	0	0
$325$	7.13963	0.396036
$326$	0	0
$327$	−11.6532	−0.644425
$328$	0	0
$329$	−36.3522	−2.00416
$330$	0	0
$331$	−17.8283	−0.979934	−0.489967	−	0.871741i	$-0.662991\pi$
$331$	−17.8283	−0.979934	−0.489967	+	0.871741i	$0.662991\pi$
$332$	0	0
$333$	11.3616	0.622609
$334$	0	0
$335$	10.7038	0.584809
$336$	0	0
$337$	−26.5384	−1.44564	−0.722819	−	0.691038i	$-0.757154\pi$
$337$	−26.5384	−1.44564	−0.722819	+	0.691038i	$0.757154\pi$
$338$	0	0
$339$	−15.3416	−0.833240
$340$	0	0
$341$	−10.5020	−0.568713
$342$	0	0
$343$	−2.45513	−0.132565
$344$	0	0
$345$	1.43888	0.0774670
$346$	0	0
$347$	18.1241	0.972950	0.486475	−	0.873694i	$-0.338282\pi$
$347$	18.1241	0.972950	0.486475	+	0.873694i	$0.338282\pi$
$348$	0	0
$349$	27.1814	1.45499	0.727493	−	0.686115i	$-0.240686\pi$
$349$	27.1814	1.45499	0.727493	+	0.686115i	$0.240686\pi$
$350$	0	0
$351$	40.3839	2.15553
$352$	0	0
$353$	15.6912	0.835158	0.417579	−	0.908641i	$-0.362879\pi$
$353$	15.6912	0.835158	0.417579	+	0.908641i	$0.362879\pi$
$354$	0	0
$355$	−16.2918	−0.864677
$356$	0	0
$357$	−34.8010	−1.84186
$358$	0	0
$359$	−18.7331	−0.988695	−0.494347	−	0.869265i	$-0.664593\pi$
$359$	−18.7331	−0.988695	−0.494347	+	0.869265i	$0.664593\pi$
$360$	0	0
$361$	0.420464	0.0221297
$362$	0	0
$363$	6.61239	0.347061
$364$	0	0
$365$	−4.97205	−0.260249
$366$	0	0
$367$	17.1258	0.893960	0.446980	−	0.894544i	$-0.352499\pi$
$367$	17.1258	0.893960	0.446980	+	0.894544i	$0.352499\pi$
$368$	0	0
$369$	−1.62177	−0.0844258
$370$	0	0
$371$	−21.2701	−1.10429
$372$	0	0
$373$	−4.06658	−0.210559	−0.105280	−	0.994443i	$-0.533574\pi$
$373$	−4.06658	−0.210559	−0.105280	+	0.994443i	$0.533574\pi$
$374$	0	0
$375$	1.42561	0.0736181
$376$	0	0
$377$	20.9769	1.08037
$378$	0	0
$379$	−17.7387	−0.911177	−0.455588	−	0.890191i	$-0.650571\pi$
$379$	−17.7387	−0.911177	−0.455588	+	0.890191i	$0.650571\pi$
$380$	0	0
$381$	10.2979	0.527578
$382$	0	0
$383$	−28.7273	−1.46790	−0.733949	−	0.679205i	$-0.762325\pi$
$383$	−28.7273	−1.46790	−0.733949	+	0.679205i	$0.762325\pi$
$384$	0	0
$385$	9.65120	0.491871
$386$	0	0
$387$	−9.18636	−0.466969
$388$	0	0
$389$	−32.9678	−1.67153	−0.835766	−	0.549086i	$-0.814976\pi$
$389$	−32.9678	−1.67153	−0.835766	+	0.549086i	$0.814976\pi$
$390$	0	0
$391$	−6.43905	−0.325637
$392$	0	0
$393$	−19.6474	−0.991080
$394$	0	0
$395$	5.11619	0.257424
$396$	0	0
$397$	−0.434718	−0.0218179	−0.0109089	−	0.999940i	$-0.503472\pi$
$397$	−0.434718	−0.0218179	−0.0109089	+	0.999940i	$0.503472\pi$
$398$	0	0
$399$	−24.0395	−1.20348
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−29.7276	−1.48084
$404$	0	0
$405$	5.16076	0.256440
$406$	0	0
$407$	−29.6150	−1.46796
$408$	0	0
$409$	−23.1892	−1.14663	−0.573316	−	0.819334i	$-0.694343\pi$
$409$	−23.1892	−1.14663	−0.573316	+	0.819334i	$0.694343\pi$
$410$	0	0
$411$	2.16811	0.106945
$412$	0	0
$413$	−19.2722	−0.948325
$414$	0	0
$415$	1.92072	0.0942845
$416$	0	0
$417$	−3.95504	−0.193679
$418$	0	0
$419$	8.75794	0.427853	0.213927	−	0.976850i	$-0.431375\pi$
$419$	8.75794	0.427853	0.213927	+	0.976850i	$0.431375\pi$
$420$	0	0
$421$	30.2112	1.47240	0.736201	−	0.676763i	$-0.236618\pi$
$421$	30.2112	1.47240	0.736201	+	0.676763i	$0.236618\pi$
$422$	0	0
$423$	−9.19283	−0.446971
$424$	0	0
$425$	−6.37964	−0.309458
$426$	0	0
$427$	24.8766	1.20386
$428$	0	0
$429$	−25.6722	−1.23947
$430$	0	0
$431$	−15.0159	−0.723292	−0.361646	−	0.932316i	$-0.617785\pi$
$431$	−15.0159	−0.723292	−0.361646	+	0.932316i	$0.617785\pi$
$432$	0	0
$433$	12.3453	0.593277	0.296639	−	0.954990i	$-0.404134\pi$
$433$	12.3453	0.593277	0.296639	+	0.954990i	$0.404134\pi$
$434$	0	0
$435$	4.18857	0.200827
$436$	0	0
$437$	−4.44790	−0.212772
$438$	0	0
$439$	4.98867	0.238096	0.119048	−	0.992888i	$-0.462016\pi$
$439$	4.98867	0.238096	0.119048	+	0.992888i	$0.462016\pi$
$440$	0	0
$441$	−7.39434	−0.352111
$442$	0	0
$443$	−14.6749	−0.697223	−0.348612	−	0.937267i	$-0.613347\pi$
$443$	−14.6749	−0.697223	−0.348612	+	0.937267i	$0.613347\pi$
$444$	0	0
$445$	16.4866	0.781539
$446$	0	0
$447$	27.4492	1.29830
$448$	0	0
$449$	25.1113	1.18508	0.592538	−	0.805543i	$-0.298126\pi$
$449$	25.1113	1.18508	0.592538	+	0.805543i	$0.298126\pi$
$450$	0	0
$451$	4.22729	0.199055
$452$	0	0
$453$	−19.8516	−0.932709
$454$	0	0
$455$	27.3194	1.28075
$456$	0	0
$457$	14.4500	0.675944	0.337972	−	0.941156i	$-0.390259\pi$
$457$	14.4500	0.675944	0.337972	+	0.941156i	$0.390259\pi$
$458$	0	0
$459$	−36.0852	−1.68431
$460$	0	0
$461$	−3.75446	−0.174863	−0.0874313	−	0.996171i	$-0.527866\pi$
$461$	−3.75446	−0.174863	−0.0874313	+	0.996171i	$0.527866\pi$
$462$	0	0
$463$	−0.405427	−0.0188418	−0.00942089	−	0.999956i	$-0.502999\pi$
$463$	−0.405427	−0.0188418	−0.00942089	+	0.999956i	$0.502999\pi$
$464$	0	0
$465$	−5.93586	−0.275269
$466$	0	0
$467$	2.43678	0.112761	0.0563804	−	0.998409i	$-0.482044\pi$
$467$	2.43678	0.112761	0.0563804	+	0.998409i	$0.482044\pi$
$468$	0	0
$469$	40.9573	1.89123
$470$	0	0
$471$	−11.9018	−0.548405
$472$	0	0
$473$	23.9451	1.10100
$474$	0	0
$475$	−4.40687	−0.202201
$476$	0	0
$477$	−5.37884	−0.246280
$478$	0	0
$479$	18.0431	0.824408	0.412204	−	0.911091i	$-0.364759\pi$
$479$	18.0431	0.824408	0.412204	+	0.911091i	$0.364759\pi$
$480$	0	0
$481$	−83.8301	−3.82233
$482$	0	0
$483$	5.50580	0.250523
$484$	0	0
$485$	−13.7260	−0.623266
$486$	0	0
$487$	27.7589	1.25788	0.628938	−	0.777455i	$-0.283490\pi$
$487$	27.7589	1.25788	0.628938	+	0.777455i	$0.283490\pi$
$488$	0	0
$489$	−12.7394	−0.576093
$490$	0	0
$491$	−34.5678	−1.56002	−0.780011	−	0.625765i	$-0.784787\pi$
$491$	−34.5678	−1.56002	−0.780011	+	0.625765i	$0.784787\pi$
$492$	0	0
$493$	−18.7440	−0.844186
$494$	0	0
$495$	2.44062	0.109698
$496$	0	0
$497$	−62.3394	−2.79630
$498$	0	0
$499$	−17.4337	−0.780440	−0.390220	−	0.920722i	$-0.627601\pi$
$499$	−17.4337	−0.780440	−0.390220	+	0.920722i	$0.627601\pi$
$500$	0	0
$501$	−15.6011	−0.697006
$502$	0	0
$503$	0.906638	0.0404250	0.0202125	−	0.999796i	$-0.493566\pi$
$503$	0.906638	0.0404250	0.0202125	+	0.999796i	$0.493566\pi$
$504$	0	0
$505$	3.05617	0.135998
$506$	0	0
$507$	−54.1366	−2.40429
$508$	0	0
$509$	10.0622	0.445998	0.222999	−	0.974819i	$-0.428415\pi$
$509$	10.0622	0.445998	0.222999	+	0.974819i	$0.428415\pi$
$510$	0	0
$511$	−19.0252	−0.841626
$512$	0	0
$513$	−24.9266	−1.10053
$514$	0	0
$515$	−9.88073	−0.435397
$516$	0	0
$517$	23.9620	1.05385
$518$	0	0
$519$	7.37613	0.323776
$520$	0	0
$521$	−36.0100	−1.57763	−0.788814	−	0.614632i	$-0.789305\pi$
$521$	−36.0100	−1.57763	−0.788814	+	0.614632i	$0.789305\pi$
$522$	0	0
$523$	28.4576	1.24436	0.622181	−	0.782873i	$-0.286247\pi$
$523$	28.4576	1.24436	0.622181	+	0.782873i	$0.286247\pi$
$524$	0	0
$525$	5.45500	0.238076
$526$	0	0
$527$	26.5632	1.15711
$528$	0	0
$529$	−21.9813	−0.955708
$530$	0	0
$531$	−4.87361	−0.211497
$532$	0	0
$533$	11.9661	0.518307
$534$	0	0
$535$	−8.34543	−0.360804
$536$	0	0
$537$	35.2881	1.52280
$538$	0	0
$539$	19.2740	0.830191
$540$	0	0
$541$	−30.8044	−1.32438	−0.662192	−	0.749334i	$-0.730374\pi$
$541$	−30.8044	−1.32438	−0.662192	+	0.749334i	$0.730374\pi$
$542$	0	0
$543$	5.47431	0.234925
$544$	0	0
$545$	−8.17421	−0.350145
$546$	0	0
$547$	−17.3332	−0.741113	−0.370556	−	0.928810i	$-0.620833\pi$
$547$	−17.3332	−0.741113	−0.370556	+	0.928810i	$0.620833\pi$
$548$	0	0
$549$	6.29086	0.268488
$550$	0	0
$551$	−12.9478	−0.551594
$552$	0	0
$553$	19.5768	0.832490
$554$	0	0
$555$	−16.7388	−0.710523
$556$	0	0
$557$	−11.8944	−0.503984	−0.251992	−	0.967729i	$-0.581086\pi$
$557$	−11.8944	−0.503984	−0.251992	+	0.967729i	$0.581086\pi$
$558$	0	0
$559$	67.7807	2.86682
$560$	0	0
$561$	22.9395	0.968505
$562$	0	0
$563$	−22.8529	−0.963136	−0.481568	−	0.876409i	$-0.659933\pi$
$563$	−22.8529	−0.963136	−0.481568	+	0.876409i	$0.659933\pi$
$564$	0	0
$565$	−10.7614	−0.452736
$566$	0	0
$567$	19.7473	0.829309
$568$	0	0
$569$	−35.2331	−1.47705	−0.738524	−	0.674227i	$-0.764477\pi$
$569$	−35.2331	−1.47705	−0.738524	+	0.674227i	$0.764477\pi$
$570$	0	0
$571$	27.3222	1.14340	0.571699	−	0.820463i	$-0.306284\pi$
$571$	27.3222	1.14340	0.571699	+	0.820463i	$0.306284\pi$
$572$	0	0
$573$	−11.8604	−0.495474
$574$	0	0
$575$	1.00931	0.0420912
$576$	0	0
$577$	24.5717	1.02293	0.511466	−	0.859304i	$-0.329103\pi$
$577$	24.5717	1.02293	0.511466	+	0.859304i	$0.329103\pi$
$578$	0	0
$579$	29.9774	1.24582
$580$	0	0
$581$	7.34952	0.304909
$582$	0	0
$583$	14.0204	0.580667
$584$	0	0
$585$	6.90859	0.285635
$586$	0	0
$587$	2.68394	0.110778	0.0553890	−	0.998465i	$-0.482360\pi$
$587$	2.68394	0.110778	0.0553890	+	0.998465i	$0.482360\pi$
$588$	0	0
$589$	18.3490	0.756059
$590$	0	0
$591$	37.3179	1.53505
$592$	0	0
$593$	−22.3433	−0.917529	−0.458764	−	0.888558i	$-0.651708\pi$
$593$	−22.3433	−0.917529	−0.458764	+	0.888558i	$0.651708\pi$
$594$	0	0
$595$	−24.4113	−1.00077
$596$	0	0
$597$	19.4898	0.797665
$598$	0	0
$599$	−14.9922	−0.612565	−0.306282	−	0.951941i	$-0.599085\pi$
$599$	−14.9922	−0.612565	−0.306282	+	0.951941i	$0.599085\pi$
$600$	0	0
$601$	−15.4886	−0.631793	−0.315896	−	0.948794i	$-0.602305\pi$
$601$	−15.4886	−0.631793	−0.315896	+	0.948794i	$0.602305\pi$
$602$	0	0
$603$	10.3574	0.421785
$604$	0	0
$605$	4.63829	0.188573
$606$	0	0
$607$	32.6772	1.32633	0.663164	−	0.748474i	$-0.269213\pi$
$607$	32.6772	1.32633	0.663164	+	0.748474i	$0.269213\pi$
$608$	0	0
$609$	16.0273	0.649459
$610$	0	0
$611$	67.8284	2.74404
$612$	0	0
$613$	−41.9453	−1.69415	−0.847077	−	0.531471i	$-0.821640\pi$
$613$	−41.9453	−1.69415	−0.847077	+	0.531471i	$0.821640\pi$
$614$	0	0
$615$	2.38933	0.0963469
$616$	0	0
$617$	19.5582	0.787383	0.393692	−	0.919243i	$-0.371198\pi$
$617$	19.5582	0.787383	0.393692	+	0.919243i	$0.371198\pi$
$618$	0	0
$619$	27.6626	1.11185	0.555926	−	0.831232i	$-0.312364\pi$
$619$	27.6626	1.11185	0.555926	+	0.831232i	$0.312364\pi$
$620$	0	0
$621$	5.70898	0.229093
$622$	0	0
$623$	63.0849	2.52744
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.8459	0.632824
$628$	0	0
$629$	74.9067	2.98672
$630$	0	0
$631$	18.7228	0.745342	0.372671	−	0.927964i	$-0.378442\pi$
$631$	18.7228	0.745342	0.372671	+	0.927964i	$0.378442\pi$
$632$	0	0
$633$	−14.4155	−0.572965
$634$	0	0
$635$	7.22352	0.286657
$636$	0	0
$637$	54.5584	2.16168
$638$	0	0
$639$	−15.7645	−0.623636
$640$	0	0
$641$	46.0681	1.81958	0.909789	−	0.415070i	$-0.136243\pi$
$641$	46.0681	1.81958	0.909789	+	0.415070i	$0.136243\pi$
$642$	0	0
$643$	2.44079	0.0962552	0.0481276	−	0.998841i	$-0.484675\pi$
$643$	2.44079	0.0962552	0.0481276	+	0.998841i	$0.484675\pi$
$644$	0	0
$645$	13.5341	0.532906
$646$	0	0
$647$	−38.1979	−1.50171	−0.750857	−	0.660465i	$-0.770359\pi$
$647$	−38.1979	−1.50171	−0.750857	+	0.660465i	$0.770359\pi$
$648$	0	0
$649$	12.7035	0.498657
$650$	0	0
$651$	−22.7132	−0.890201
$652$	0	0
$653$	−14.0090	−0.548216	−0.274108	−	0.961699i	$-0.588383\pi$
$653$	−14.0090	−0.548216	−0.274108	+	0.961699i	$0.588383\pi$
$654$	0	0
$655$	−13.7818	−0.538498
$656$	0	0
$657$	−4.81115	−0.187701
$658$	0	0
$659$	−4.04132	−0.157428	−0.0787138	−	0.996897i	$-0.525081\pi$
$659$	−4.04132	−0.157428	−0.0787138	+	0.996897i	$0.525081\pi$
$660$	0	0
$661$	−27.8755	−1.08423	−0.542115	−	0.840304i	$-0.682376\pi$
$661$	−27.8755	−1.08423	−0.542115	+	0.840304i	$0.682376\pi$
$662$	0	0
$663$	64.9340	2.52183
$664$	0	0
$665$	−16.8626	−0.653903
$666$	0	0
$667$	2.96545	0.114823
$668$	0	0
$669$	3.06574	0.118528
$670$	0	0
$671$	−16.3977	−0.633027
$672$	0	0
$673$	−19.3480	−0.745809	−0.372905	−	0.927870i	$-0.621638\pi$
$673$	−19.3480	−0.745809	−0.372905	+	0.927870i	$0.621638\pi$
$674$	0	0
$675$	5.65630	0.217711
$676$	0	0
$677$	30.6933	1.17964	0.589820	−	0.807535i	$-0.299199\pi$
$677$	30.6933	1.17964	0.589820	+	0.807535i	$0.299199\pi$
$678$	0	0
$679$	−52.5217	−2.01560
$680$	0	0
$681$	−4.53552	−0.173801
$682$	0	0
$683$	−6.40544	−0.245097	−0.122549	−	0.992463i	$-0.539107\pi$
$683$	−6.40544	−0.245097	−0.122549	+	0.992463i	$0.539107\pi$
$684$	0	0
$685$	1.52083	0.0581080
$686$	0	0
$687$	28.9199	1.10336
$688$	0	0
$689$	39.6872	1.51196
$690$	0	0
$691$	−43.2323	−1.64463	−0.822317	−	0.569030i	$-0.807319\pi$
$691$	−43.2323	−1.64463	−0.822317	+	0.569030i	$0.807319\pi$
$692$	0	0
$693$	9.33888	0.354755
$694$	0	0
$695$	−2.77428	−0.105234
$696$	0	0
$697$	−10.6923	−0.405000
$698$	0	0
$699$	30.4652	1.15230
$700$	0	0
$701$	−0.538908	−0.0203543	−0.0101771	−	0.999948i	$-0.503240\pi$
$701$	−0.538908	−0.0203543	−0.0101771	+	0.999948i	$0.503240\pi$
$702$	0	0
$703$	51.7433	1.95154
$704$	0	0
$705$	13.5437	0.510084
$706$	0	0
$707$	11.6942	0.439807
$708$	0	0
$709$	33.0003	1.23935	0.619676	−	0.784858i	$-0.287264\pi$
$709$	33.0003	1.23935	0.619676	+	0.784858i	$0.287264\pi$
$710$	0	0
$711$	4.95063	0.185663
$712$	0	0
$713$	−4.20251	−0.157385
$714$	0	0
$715$	−18.0079	−0.673457
$716$	0	0
$717$	43.4430	1.62241
$718$	0	0
$719$	19.4337	0.724754	0.362377	−	0.932032i	$-0.381965\pi$
$719$	19.4337	0.724754	0.362377	+	0.932032i	$0.381965\pi$
$720$	0	0
$721$	−37.8080	−1.40804
$722$	0	0
$723$	21.7661	0.809489
$724$	0	0
$725$	2.93809	0.109118
$726$	0	0
$727$	−2.10647	−0.0781246	−0.0390623	−	0.999237i	$-0.512437\pi$
$727$	−2.10647	−0.0781246	−0.0390623	+	0.999237i	$0.512437\pi$
$728$	0	0
$729$	29.1848	1.08092
$730$	0	0
$731$	−60.5656	−2.24010
$732$	0	0
$733$	−4.03179	−0.148918	−0.0744588	−	0.997224i	$-0.523723\pi$
$733$	−4.03179	−0.148918	−0.0744588	+	0.997224i	$0.523723\pi$
$734$	0	0
$735$	10.8940	0.401830
$736$	0	0
$737$	−26.9975	−0.994465
$738$	0	0
$739$	31.2105	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$739$	31.2105	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$740$	0	0
$741$	44.8545	1.64777
$742$	0	0
$743$	−23.8418	−0.874670	−0.437335	−	0.899299i	$-0.644078\pi$
$743$	−23.8418	−0.874670	−0.437335	+	0.899299i	$0.644078\pi$
$744$	0	0
$745$	19.2544	0.705425
$746$	0	0
$747$	1.85856	0.0680013
$748$	0	0
$749$	−31.9333	−1.16682
$750$	0	0
$751$	−34.3344	−1.25288	−0.626441	−	0.779469i	$-0.715489\pi$
$751$	−34.3344	−1.25288	−0.626441	+	0.779469i	$0.715489\pi$
$752$	0	0
$753$	24.0607	0.876819
$754$	0	0
$755$	−13.9250	−0.506782
$756$	0	0
$757$	−14.7732	−0.536942	−0.268471	−	0.963288i	$-0.586518\pi$
$757$	−14.7732	−0.536942	−0.268471	+	0.963288i	$0.586518\pi$
$758$	0	0
$759$	−3.62922	−0.131732
$760$	0	0
$761$	36.1973	1.31215	0.656075	−	0.754695i	$-0.272215\pi$
$761$	36.1973	1.31215	0.656075	+	0.754695i	$0.272215\pi$
$762$	0	0
$763$	−31.2781	−1.13234
$764$	0	0
$765$	−6.17319	−0.223192
$766$	0	0
$767$	35.9595	1.29842
$768$	0	0
$769$	−36.2936	−1.30878	−0.654390	−	0.756157i	$-0.727074\pi$
$769$	−36.2936	−1.30878	−0.654390	+	0.756157i	$0.727074\pi$
$770$	0	0
$771$	−7.61086	−0.274099
$772$	0	0
$773$	25.4127	0.914030	0.457015	−	0.889459i	$-0.348919\pi$
$773$	25.4127	0.914030	0.457015	+	0.889459i	$0.348919\pi$
$774$	0	0
$775$	−4.16374	−0.149566
$776$	0	0
$777$	−64.0500	−2.29778
$778$	0	0
$779$	−7.38593	−0.264628
$780$	0	0
$781$	41.0918	1.47038
$782$	0	0
$783$	16.6187	0.593905
$784$	0	0
$785$	−8.34856	−0.297973
$786$	0	0
$787$	6.08274	0.216826	0.108413	−	0.994106i	$-0.465423\pi$
$787$	6.08274	0.216826	0.108413	+	0.994106i	$0.465423\pi$
$788$	0	0
$789$	−23.2564	−0.827948
$790$	0	0
$791$	−41.1779	−1.46412
$792$	0	0
$793$	−46.4165	−1.64830
$794$	0	0
$795$	7.92456	0.281055
$796$	0	0
$797$	−22.1851	−0.785837	−0.392919	−	0.919573i	$-0.628535\pi$
$797$	−22.1851	−0.785837	−0.392919	+	0.919573i	$0.628535\pi$
$798$	0	0
$799$	−60.6083	−2.14417
$800$	0	0
$801$	15.9531	0.563674
$802$	0	0
$803$	12.5407	0.442552
$804$	0	0
$805$	3.86207	0.136120
$806$	0	0
$807$	8.94164	0.314761
$808$	0	0
$809$	1.60973	0.0565951	0.0282975	−	0.999600i	$-0.490991\pi$
$809$	1.60973	0.0565951	0.0282975	+	0.999600i	$0.490991\pi$
$810$	0	0
$811$	−20.4204	−0.717057	−0.358528	−	0.933519i	$-0.616721\pi$
$811$	−20.4204	−0.717057	−0.358528	+	0.933519i	$0.616721\pi$
$812$	0	0
$813$	−34.8926	−1.22374
$814$	0	0
$815$	−8.93608	−0.313017
$816$	0	0
$817$	−41.8369	−1.46369
$818$	0	0
$819$	26.4353	0.923724
$820$	0	0
$821$	−28.6127	−0.998590	−0.499295	−	0.866432i	$-0.666408\pi$
$821$	−28.6127	−0.998590	−0.499295	+	0.866432i	$0.666408\pi$
$822$	0	0
$823$	−38.9332	−1.35713	−0.678563	−	0.734542i	$-0.737397\pi$
$823$	−38.9332	−1.35713	−0.678563	+	0.734542i	$0.737397\pi$
$824$	0	0
$825$	−3.59573	−0.125187
$826$	0	0
$827$	21.6386	0.752448	0.376224	−	0.926529i	$-0.377222\pi$
$827$	21.6386	0.752448	0.376224	+	0.926529i	$0.377222\pi$
$828$	0	0
$829$	−8.99005	−0.312237	−0.156119	−	0.987738i	$-0.549898\pi$
$829$	−8.99005	−0.312237	−0.156119	+	0.987738i	$0.549898\pi$
$830$	0	0
$831$	8.72493	0.302664
$832$	0	0
$833$	−48.7508	−1.68912
$834$	0	0
$835$	−10.9435	−0.378715
$836$	0	0
$837$	−23.5514	−0.814054
$838$	0	0
$839$	2.55082	0.0880640	0.0440320	−	0.999030i	$-0.485980\pi$
$839$	2.55082	0.0880640	0.0440320	+	0.999030i	$0.485980\pi$
$840$	0	0
$841$	−20.3676	−0.702331
$842$	0	0
$843$	15.2874	0.526526
$844$	0	0
$845$	−37.9743	−1.30636
$846$	0	0
$847$	17.7481	0.609833
$848$	0	0
$849$	25.5399	0.876526
$850$	0	0
$851$	−11.8509	−0.406242
$852$	0	0
$853$	−5.78746	−0.198159	−0.0990795	−	0.995080i	$-0.531590\pi$
$853$	−5.78746	−0.198159	−0.0990795	+	0.995080i	$0.531590\pi$
$854$	0	0
$855$	−4.26426	−0.145834
$856$	0	0
$857$	0.883123	0.0301669	0.0150834	−	0.999886i	$-0.495199\pi$
$857$	0.883123	0.0301669	0.0150834	+	0.999886i	$0.495199\pi$
$858$	0	0
$859$	−22.8753	−0.780494	−0.390247	−	0.920710i	$-0.627610\pi$
$859$	−22.8753	−0.780494	−0.390247	+	0.920710i	$0.627610\pi$
$860$	0	0
$861$	9.14261	0.311579
$862$	0	0
$863$	32.2492	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$863$	32.2492	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$864$	0	0
$865$	5.17402	0.175922
$866$	0	0
$867$	−33.7866	−1.14745
$868$	0	0
$869$	−12.9043	−0.437748
$870$	0	0
$871$	−76.4209	−2.58942
$872$	0	0
$873$	−13.2818	−0.449521
$874$	0	0
$875$	3.82644	0.129357
$876$	0	0
$877$	22.5615	0.761849	0.380925	−	0.924606i	$-0.375606\pi$
$877$	22.5615	0.761849	0.380925	+	0.924606i	$0.375606\pi$
$878$	0	0
$879$	−34.0012	−1.14683
$880$	0	0
$881$	−21.0899	−0.710535	−0.355268	−	0.934765i	$-0.615610\pi$
$881$	−21.0899	−0.710535	−0.355268	+	0.934765i	$0.615610\pi$
$882$	0	0
$883$	33.1474	1.11550	0.557749	−	0.830009i	$-0.311665\pi$
$883$	33.1474	1.11550	0.557749	+	0.830009i	$0.311665\pi$
$884$	0	0
$885$	7.18022	0.241361
$886$	0	0
$887$	−29.2731	−0.982895	−0.491448	−	0.870907i	$-0.663532\pi$
$887$	−29.2731	−0.982895	−0.491448	+	0.870907i	$0.663532\pi$
$888$	0	0
$889$	27.6404	0.927028
$890$	0	0
$891$	−13.0167	−0.436075
$892$	0	0
$893$	−41.8664	−1.40101
$894$	0	0
$895$	24.7530	0.827402
$896$	0	0
$897$	−10.2731	−0.343009
$898$	0	0
$899$	−12.2335	−0.408009
$900$	0	0
$901$	−35.4626	−1.18143
$902$	0	0
$903$	51.7875	1.72338
$904$	0	0
$905$	3.83998	0.127645
$906$	0	0
$907$	44.7814	1.48694	0.743471	−	0.668768i	$-0.233178\pi$
$907$	44.7814	1.48694	0.743471	+	0.668768i	$0.233178\pi$
$908$	0	0
$909$	2.95727	0.0980863
$910$	0	0
$911$	−14.6694	−0.486020	−0.243010	−	0.970024i	$-0.578135\pi$
$911$	−14.6694	−0.486020	−0.243010	+	0.970024i	$0.578135\pi$
$912$	0	0
$913$	−4.84452	−0.160330
$914$	0	0
$915$	−9.26824	−0.306398
$916$	0	0
$917$	−52.7350	−1.74146
$918$	0	0
$919$	14.7674	0.487130	0.243565	−	0.969885i	$-0.421683\pi$
$919$	14.7674	0.487130	0.243565	+	0.969885i	$0.421683\pi$
$920$	0	0
$921$	40.0772	1.32059
$922$	0	0
$923$	116.317	3.82863
$924$	0	0
$925$	−11.7415	−0.386059
$926$	0	0
$927$	−9.56098	−0.314024
$928$	0	0
$929$	−13.1126	−0.430210	−0.215105	−	0.976591i	$-0.569009\pi$
$929$	−13.1126	−0.430210	−0.215105	+	0.976591i	$0.569009\pi$
$930$	0	0
$931$	−33.6756	−1.10367
$932$	0	0
$933$	26.5392	0.868853
$934$	0	0
$935$	16.0910	0.526232
$936$	0	0
$937$	15.9260	0.520279	0.260140	−	0.965571i	$-0.416231\pi$
$937$	15.9260	0.520279	0.260140	+	0.965571i	$0.416231\pi$
$938$	0	0
$939$	11.8818	0.387748
$940$	0	0
$941$	−22.8474	−0.744803	−0.372401	−	0.928072i	$-0.621465\pi$
$941$	−22.8474	−0.744803	−0.372401	+	0.928072i	$0.621465\pi$
$942$	0	0
$943$	1.69161	0.0550865
$944$	0	0
$945$	21.6435	0.704063
$946$	0	0
$947$	43.7130	1.42048	0.710241	−	0.703959i	$-0.248586\pi$
$947$	43.7130	1.42048	0.710241	+	0.703959i	$0.248586\pi$
$948$	0	0
$949$	35.4986	1.15233
$950$	0	0
$951$	−22.4167	−0.726911
$952$	0	0
$953$	−16.6640	−0.539799	−0.269900	−	0.962888i	$-0.586991\pi$
$953$	−16.6640	−0.539799	−0.269900	+	0.962888i	$0.586991\pi$
$954$	0	0
$955$	−8.31951	−0.269213
$956$	0	0
$957$	−10.5646	−0.341505
$958$	0	0
$959$	5.81937	0.187917
$960$	0	0
$961$	−13.6633	−0.440751
$962$	0	0
$963$	−8.07536	−0.260225
$964$	0	0
$965$	21.0278	0.676908
$966$	0	0
$967$	−10.9323	−0.351558	−0.175779	−	0.984430i	$-0.556244\pi$
$967$	−10.9323	−0.351558	−0.175779	+	0.984430i	$0.556244\pi$
$968$	0	0
$969$	−40.0799	−1.28755
$970$	0	0
$971$	−18.0693	−0.579871	−0.289935	−	0.957046i	$-0.593634\pi$
$971$	−18.0693	−0.579871	−0.289935	+	0.957046i	$0.593634\pi$
$972$	0	0
$973$	−10.6156	−0.340321
$974$	0	0
$975$	−10.1783	−0.325967
$976$	0	0
$977$	−17.9420	−0.574015	−0.287007	−	0.957928i	$-0.592660\pi$
$977$	−17.9420	−0.574015	−0.287007	+	0.957928i	$0.592660\pi$
$978$	0	0
$979$	−41.5832	−1.32900
$980$	0	0
$981$	−7.90969	−0.252537
$982$	0	0
$983$	−37.4902	−1.19575	−0.597876	−	0.801589i	$-0.703988\pi$
$983$	−37.4902	−1.19575	−0.597876	+	0.801589i	$0.703988\pi$
$984$	0	0
$985$	26.1768	0.834062
$986$	0	0
$987$	51.8240	1.64958
$988$	0	0
$989$	9.58199	0.304689
$990$	0	0
$991$	4.59427	0.145942	0.0729708	−	0.997334i	$-0.476752\pi$
$991$	4.59427	0.145942	0.0729708	+	0.997334i	$0.476752\pi$
$992$	0	0
$993$	25.4162	0.806560
$994$	0	0
$995$	13.6712	0.433407
$996$	0	0
$997$	45.5362	1.44215	0.721073	−	0.692859i	$-0.243649\pi$
$997$	45.5362	1.44215	0.721073	+	0.692859i	$0.243649\pi$
$998$	0	0
$999$	−66.4136	−2.10123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.10	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.10	✓	28	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)

\(f(q)\)	\(=\)	\(q-1.42561 q^{3} -1.00000 q^{5} -3.82644 q^{7} -0.967639 q^{9} +O(q^{10})\)	\(q-1.42561 q^{3} -1.00000 q^{5} -3.82644 q^{7} -0.967639 q^{9} +2.52224 q^{11} +7.13963 q^{13} +1.42561 q^{15} -6.37964 q^{17} -4.40687 q^{19} +5.45500 q^{21} +1.00931 q^{23} +1.00000 q^{25} +5.65630 q^{27} +2.93809 q^{29} -4.16374 q^{31} -3.59573 q^{33} +3.82644 q^{35} -11.7415 q^{37} -10.1783 q^{39} +1.67600 q^{41} +9.49358 q^{43} +0.967639 q^{45} +9.50027 q^{47} +7.64162 q^{49} +9.09487 q^{51} +5.55872 q^{53} -2.52224 q^{55} +6.28247 q^{57} +5.03660 q^{59} -6.50125 q^{61} +3.70261 q^{63} -7.13963 q^{65} -10.7038 q^{67} -1.43888 q^{69} +16.2918 q^{71} +4.97205 q^{73} -1.42561 q^{75} -9.65120 q^{77} -5.11619 q^{79} -5.16076 q^{81} -1.92072 q^{83} +6.37964 q^{85} -4.18857 q^{87} -16.4866 q^{89} -27.3194 q^{91} +5.93586 q^{93} +4.40687 q^{95} +13.7260 q^{97} -2.44062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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