LMFDB - Embedded newform 8020.2.a.e.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:	$0$
Dimension:	$35$
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)

$f(q)$	$=$	$q-1.51742 q^{3} -1.00000 q^{5} -1.44284 q^{7} -0.697439 q^{9} +O(q^{10})$	$q-1.51742 q^{3} -1.00000 q^{5} -1.44284 q^{7} -0.697439 q^{9} +6.35633 q^{11} -3.95743 q^{13} +1.51742 q^{15} -4.99012 q^{17} -3.38278 q^{19} +2.18939 q^{21} +2.61820 q^{23} +1.00000 q^{25} +5.61056 q^{27} -6.38382 q^{29} +7.93658 q^{31} -9.64522 q^{33} +1.44284 q^{35} -5.92546 q^{37} +6.00508 q^{39} +5.63069 q^{41} -0.271651 q^{43} +0.697439 q^{45} +3.69094 q^{47} -4.91822 q^{49} +7.57211 q^{51} -5.32053 q^{53} -6.35633 q^{55} +5.13310 q^{57} -10.6684 q^{59} -13.4265 q^{61} +1.00629 q^{63} +3.95743 q^{65} +5.91647 q^{67} -3.97291 q^{69} -12.1925 q^{71} -13.7425 q^{73} -1.51742 q^{75} -9.17115 q^{77} +15.3336 q^{79} -6.42126 q^{81} +0.961591 q^{83} +4.99012 q^{85} +9.68693 q^{87} +0.887393 q^{89} +5.70993 q^{91} -12.0431 q^{93} +3.38278 q^{95} +12.5839 q^{97} -4.43316 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.51742	−0.876082	−0.438041	−	0.898955i	$-0.644328\pi$
$3$	−1.51742	−0.876082	−0.438041	+	0.898955i	$0.644328\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.44284	−0.545341	−0.272671	−	0.962107i	$-0.587907\pi$
$7$	−1.44284	−0.545341	−0.272671	+	0.962107i	$0.587907\pi$
$8$	0	0
$9$	−0.697439	−0.232480
$10$	0	0
$11$	6.35633	1.91651	0.958253	−	0.285921i	$-0.0922996\pi$
$11$	6.35633	1.91651	0.958253	+	0.285921i	$0.0922996\pi$
$12$	0	0
$13$	−3.95743	−1.09759	−0.548797	−	0.835956i	$-0.684914\pi$
$13$	−3.95743	−1.09759	−0.548797	+	0.835956i	$0.684914\pi$
$14$	0	0
$15$	1.51742	0.391796
$16$	0	0
$17$	−4.99012	−1.21028	−0.605141	−	0.796118i	$-0.706883\pi$
$17$	−4.99012	−1.21028	−0.605141	+	0.796118i	$0.706883\pi$
$18$	0	0
$19$	−3.38278	−0.776063	−0.388032	−	0.921646i	$-0.626845\pi$
$19$	−3.38278	−0.776063	−0.388032	+	0.921646i	$0.626845\pi$
$20$	0	0
$21$	2.18939	0.477764
$22$	0	0
$23$	2.61820	0.545932	0.272966	−	0.962024i	$-0.411995\pi$
$23$	2.61820	0.545932	0.272966	+	0.962024i	$0.411995\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.61056	1.07975
$28$	0	0
$29$	−6.38382	−1.18545	−0.592723	−	0.805406i	$-0.701947\pi$
$29$	−6.38382	−1.18545	−0.592723	+	0.805406i	$0.701947\pi$
$30$	0	0
$31$	7.93658	1.42545	0.712726	−	0.701442i	$-0.247460\pi$
$31$	7.93658	1.42545	0.712726	+	0.701442i	$0.247460\pi$
$32$	0	0
$33$	−9.64522	−1.67902
$34$	0	0
$35$	1.44284	0.243884
$36$	0	0
$37$	−5.92546	−0.974140	−0.487070	−	0.873363i	$-0.661934\pi$
$37$	−5.92546	−0.974140	−0.487070	+	0.873363i	$0.661934\pi$
$38$	0	0
$39$	6.00508	0.961582
$40$	0	0
$41$	5.63069	0.879366	0.439683	−	0.898153i	$-0.355091\pi$
$41$	5.63069	0.879366	0.439683	+	0.898153i	$0.355091\pi$
$42$	0	0
$43$	−0.271651	−0.0414264	−0.0207132	−	0.999785i	$-0.506594\pi$
$43$	−0.271651	−0.0414264	−0.0207132	+	0.999785i	$0.506594\pi$
$44$	0	0
$45$	0.697439	0.103968
$46$	0	0
$47$	3.69094	0.538379	0.269189	−	0.963087i	$-0.413244\pi$
$47$	3.69094	0.538379	0.269189	+	0.963087i	$0.413244\pi$
$48$	0	0
$49$	−4.91822	−0.702603
$50$	0	0
$51$	7.57211	1.06031
$52$	0	0
$53$	−5.32053	−0.730831	−0.365415	−	0.930845i	$-0.619073\pi$
$53$	−5.32053	−0.730831	−0.365415	+	0.930845i	$0.619073\pi$
$54$	0	0
$55$	−6.35633	−0.857088
$56$	0	0
$57$	5.13310	0.679895
$58$	0	0
$59$	−10.6684	−1.38891	−0.694454	−	0.719537i	$-0.744354\pi$
$59$	−10.6684	−1.38891	−0.694454	+	0.719537i	$0.744354\pi$
$60$	0	0
$61$	−13.4265	−1.71909	−0.859545	−	0.511060i	$-0.829253\pi$
$61$	−13.4265	−1.71909	−0.859545	+	0.511060i	$0.829253\pi$
$62$	0	0
$63$	1.00629	0.126781
$64$	0	0
$65$	3.95743	0.490859
$66$	0	0
$67$	5.91647	0.722812	0.361406	−	0.932409i	$-0.382297\pi$
$67$	5.91647	0.722812	0.361406	+	0.932409i	$0.382297\pi$
$68$	0	0
$69$	−3.97291	−0.478282
$70$	0	0
$71$	−12.1925	−1.44699	−0.723493	−	0.690332i	$-0.757465\pi$
$71$	−12.1925	−1.44699	−0.723493	+	0.690332i	$0.757465\pi$
$72$	0	0
$73$	−13.7425	−1.60843	−0.804217	−	0.594335i	$-0.797415\pi$
$73$	−13.7425	−1.60843	−0.804217	+	0.594335i	$0.797415\pi$
$74$	0	0
$75$	−1.51742	−0.175216
$76$	0	0
$77$	−9.17115	−1.04515
$78$	0	0
$79$	15.3336	1.72516	0.862582	−	0.505917i	$-0.168846\pi$
$79$	15.3336	1.72516	0.862582	+	0.505917i	$0.168846\pi$
$80$	0	0
$81$	−6.42126	−0.713473
$82$	0	0
$83$	0.961591	0.105548	0.0527742	−	0.998606i	$-0.483194\pi$
$83$	0.961591	0.105548	0.0527742	+	0.998606i	$0.483194\pi$
$84$	0	0
$85$	4.99012	0.541255
$86$	0	0
$87$	9.68693	1.03855
$88$	0	0
$89$	0.887393	0.0940635	0.0470317	−	0.998893i	$-0.485024\pi$
$89$	0.887393	0.0940635	0.0470317	+	0.998893i	$0.485024\pi$
$90$	0	0
$91$	5.70993	0.598563
$92$	0	0
$93$	−12.0431	−1.24881
$94$	0	0
$95$	3.38278	0.347066
$96$	0	0
$97$	12.5839	1.27771	0.638853	−	0.769329i	$-0.279409\pi$
$97$	12.5839	1.27771	0.638853	+	0.769329i	$0.279409\pi$
$98$	0	0
$99$	−4.43316	−0.445549
$100$	0	0
$101$	15.9011	1.58222	0.791111	−	0.611673i	$-0.209503\pi$
$101$	15.9011	1.58222	0.791111	+	0.611673i	$0.209503\pi$
$102$	0	0
$103$	−13.0266	−1.28355	−0.641775	−	0.766893i	$-0.721802\pi$
$103$	−13.0266	−1.28355	−0.641775	+	0.766893i	$0.721802\pi$
$104$	0	0
$105$	−2.18939	−0.213662
$106$	0	0
$107$	0.118069	0.0114142	0.00570709	−	0.999984i	$-0.498183\pi$
$107$	0.118069	0.0114142	0.00570709	+	0.999984i	$0.498183\pi$
$108$	0	0
$109$	20.6381	1.97678	0.988388	−	0.151954i	$-0.0485565\pi$
$109$	20.6381	1.97678	0.988388	+	0.151954i	$0.0485565\pi$
$110$	0	0
$111$	8.99141	0.853427
$112$	0	0
$113$	−6.51975	−0.613326	−0.306663	−	0.951818i	$-0.599212\pi$
$113$	−6.51975	−0.613326	−0.306663	+	0.951818i	$0.599212\pi$
$114$	0	0
$115$	−2.61820	−0.244148
$116$	0	0
$117$	2.76007	0.255168
$118$	0	0
$119$	7.19993	0.660017
$120$	0	0
$121$	29.4030	2.67300
$122$	0	0
$123$	−8.54412	−0.770397
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.85503	0.342079	0.171039	−	0.985264i	$-0.445287\pi$
$127$	3.85503	0.342079	0.171039	+	0.985264i	$0.445287\pi$
$128$	0	0
$129$	0.412209	0.0362930
$130$	0	0
$131$	−6.47666	−0.565869	−0.282934	−	0.959139i	$-0.591308\pi$
$131$	−6.47666	−0.565869	−0.282934	+	0.959139i	$0.591308\pi$
$132$	0	0
$133$	4.88080	0.423219
$134$	0	0
$135$	−5.61056	−0.482881
$136$	0	0
$137$	−9.59428	−0.819695	−0.409847	−	0.912154i	$-0.634418\pi$
$137$	−9.59428	−0.819695	−0.409847	+	0.912154i	$0.634418\pi$
$138$	0	0
$139$	−12.6740	−1.07499	−0.537497	−	0.843266i	$-0.680630\pi$
$139$	−12.6740	−1.07499	−0.537497	+	0.843266i	$0.680630\pi$
$140$	0	0
$141$	−5.60070	−0.471664
$142$	0	0
$143$	−25.1547	−2.10354
$144$	0	0
$145$	6.38382	0.530148
$146$	0	0
$147$	7.46300	0.615538
$148$	0	0
$149$	7.15976	0.586551	0.293275	−	0.956028i	$-0.405255\pi$
$149$	7.15976	0.586551	0.293275	+	0.956028i	$0.405255\pi$
$150$	0	0
$151$	13.8525	1.12730	0.563650	−	0.826014i	$-0.309397\pi$
$151$	13.8525	1.12730	0.563650	+	0.826014i	$0.309397\pi$
$152$	0	0
$153$	3.48031	0.281366
$154$	0	0
$155$	−7.93658	−0.637482
$156$	0	0
$157$	−7.41536	−0.591810	−0.295905	−	0.955217i	$-0.595621\pi$
$157$	−7.41536	−0.591810	−0.295905	+	0.955217i	$0.595621\pi$
$158$	0	0
$159$	8.07347	0.640268
$160$	0	0
$161$	−3.77764	−0.297719
$162$	0	0
$163$	3.49483	0.273736	0.136868	−	0.990589i	$-0.456296\pi$
$163$	3.49483	0.273736	0.136868	+	0.990589i	$0.456296\pi$
$164$	0	0
$165$	9.64522	0.750879
$166$	0	0
$167$	−10.6000	−0.820253	−0.410126	−	0.912029i	$-0.634515\pi$
$167$	−10.6000	−0.820253	−0.410126	+	0.912029i	$0.634515\pi$
$168$	0	0
$169$	2.66124	0.204711
$170$	0	0
$171$	2.35928	0.180419
$172$	0	0
$173$	−23.4026	−1.77926	−0.889632	−	0.456678i	$-0.849039\pi$
$173$	−23.4026	−1.77926	−0.889632	+	0.456678i	$0.849039\pi$
$174$	0	0
$175$	−1.44284	−0.109068
$176$	0	0
$177$	16.1884	1.21680
$178$	0	0
$179$	−11.5282	−0.861661	−0.430831	−	0.902433i	$-0.641779\pi$
$179$	−11.5282	−0.861661	−0.430831	+	0.902433i	$0.641779\pi$
$180$	0	0
$181$	14.6351	1.08782	0.543908	−	0.839145i	$-0.316944\pi$
$181$	14.6351	1.08782	0.543908	+	0.839145i	$0.316944\pi$
$182$	0	0
$183$	20.3737	1.50606
$184$	0	0
$185$	5.92546	0.435649
$186$	0	0
$187$	−31.7189	−2.31951
$188$	0	0
$189$	−8.09513	−0.588834
$190$	0	0
$191$	−5.11843	−0.370356	−0.185178	−	0.982705i	$-0.559286\pi$
$191$	−5.11843	−0.370356	−0.185178	+	0.982705i	$0.559286\pi$
$192$	0	0
$193$	−26.2991	−1.89305	−0.946524	−	0.322634i	$-0.895432\pi$
$193$	−26.2991	−1.89305	−0.946524	+	0.322634i	$0.895432\pi$
$194$	0	0
$195$	−6.00508	−0.430033
$196$	0	0
$197$	17.1392	1.22112	0.610558	−	0.791972i	$-0.290945\pi$
$197$	17.1392	1.22112	0.610558	+	0.791972i	$0.290945\pi$
$198$	0	0
$199$	1.03795	0.0735784	0.0367892	−	0.999323i	$-0.488287\pi$
$199$	1.03795	0.0735784	0.0367892	+	0.999323i	$0.488287\pi$
$200$	0	0
$201$	−8.97776	−0.633242
$202$	0	0
$203$	9.21082	0.646473
$204$	0	0
$205$	−5.63069	−0.393265
$206$	0	0
$207$	−1.82604	−0.126918
$208$	0	0
$209$	−21.5021	−1.48733
$210$	0	0
$211$	13.7374	0.945724	0.472862	−	0.881137i	$-0.343221\pi$
$211$	13.7374	0.945724	0.472862	+	0.881137i	$0.343221\pi$
$212$	0	0
$213$	18.5012	1.26768
$214$	0	0
$215$	0.271651	0.0185265
$216$	0	0
$217$	−11.4512	−0.777358
$218$	0	0
$219$	20.8531	1.40912
$220$	0	0
$221$	19.7481	1.32840
$222$	0	0
$223$	−12.8979	−0.863709	−0.431855	−	0.901943i	$-0.642141\pi$
$223$	−12.8979	−0.863709	−0.431855	+	0.901943i	$0.642141\pi$
$224$	0	0
$225$	−0.697439	−0.0464960
$226$	0	0
$227$	−0.642177	−0.0426228	−0.0213114	−	0.999773i	$-0.506784\pi$
$227$	−0.642177	−0.0426228	−0.0213114	+	0.999773i	$0.506784\pi$
$228$	0	0
$229$	17.2283	1.13848	0.569240	−	0.822172i	$-0.307238\pi$
$229$	17.2283	1.13848	0.569240	+	0.822172i	$0.307238\pi$
$230$	0	0
$231$	13.9165	0.915637
$232$	0	0
$233$	15.1462	0.992262	0.496131	−	0.868248i	$-0.334754\pi$
$233$	15.1462	0.992262	0.496131	+	0.868248i	$0.334754\pi$
$234$	0	0
$235$	−3.69094	−0.240770
$236$	0	0
$237$	−23.2675	−1.51139
$238$	0	0
$239$	22.1033	1.42975	0.714873	−	0.699254i	$-0.246484\pi$
$239$	22.1033	1.42975	0.714873	+	0.699254i	$0.246484\pi$
$240$	0	0
$241$	8.77406	0.565187	0.282594	−	0.959240i	$-0.408805\pi$
$241$	8.77406	0.565187	0.282594	+	0.959240i	$0.408805\pi$
$242$	0	0
$243$	−7.08795	−0.454692
$244$	0	0
$245$	4.91822	0.314214
$246$	0	0
$247$	13.3871	0.851802
$248$	0	0
$249$	−1.45914	−0.0924691
$250$	0	0
$251$	−6.66110	−0.420445	−0.210223	−	0.977654i	$-0.567419\pi$
$251$	−6.66110	−0.420445	−0.210223	+	0.977654i	$0.567419\pi$
$252$	0	0
$253$	16.6421	1.04628
$254$	0	0
$255$	−7.57211	−0.474184
$256$	0	0
$257$	12.3902	0.772882	0.386441	−	0.922314i	$-0.373704\pi$
$257$	12.3902	0.772882	0.386441	+	0.922314i	$0.373704\pi$
$258$	0	0
$259$	8.54948	0.531239
$260$	0	0
$261$	4.45233	0.275592
$262$	0	0
$263$	20.5449	1.26685	0.633426	−	0.773803i	$-0.281648\pi$
$263$	20.5449	1.26685	0.633426	+	0.773803i	$0.281648\pi$
$264$	0	0
$265$	5.32053	0.326837
$266$	0	0
$267$	−1.34655	−0.0824073
$268$	0	0
$269$	5.87510	0.358211	0.179105	−	0.983830i	$-0.442680\pi$
$269$	5.87510	0.358211	0.179105	+	0.983830i	$0.442680\pi$
$270$	0	0
$271$	−31.1429	−1.89180	−0.945899	−	0.324461i	$-0.894817\pi$
$271$	−31.1429	−1.89180	−0.945899	+	0.324461i	$0.894817\pi$
$272$	0	0
$273$	−8.66435	−0.524390
$274$	0	0
$275$	6.35633	0.383301
$276$	0	0
$277$	−4.27274	−0.256724	−0.128362	−	0.991727i	$-0.540972\pi$
$277$	−4.27274	−0.256724	−0.128362	+	0.991727i	$0.540972\pi$
$278$	0	0
$279$	−5.53528	−0.331389
$280$	0	0
$281$	−9.18669	−0.548032	−0.274016	−	0.961725i	$-0.588352\pi$
$281$	−9.18669	−0.548032	−0.274016	+	0.961725i	$0.588352\pi$
$282$	0	0
$283$	19.2505	1.14432	0.572162	−	0.820141i	$-0.306105\pi$
$283$	19.2505	1.14432	0.572162	+	0.820141i	$0.306105\pi$
$284$	0	0
$285$	−5.13310	−0.304058
$286$	0	0
$287$	−8.12417	−0.479555
$288$	0	0
$289$	7.90132	0.464784
$290$	0	0
$291$	−19.0951	−1.11938
$292$	0	0
$293$	8.72244	0.509570	0.254785	−	0.966998i	$-0.417995\pi$
$293$	8.72244	0.509570	0.254785	+	0.966998i	$0.417995\pi$
$294$	0	0
$295$	10.6684	0.621139
$296$	0	0
$297$	35.6626	2.06935
$298$	0	0
$299$	−10.3613	−0.599212
$300$	0	0
$301$	0.391949	0.0225915
$302$	0	0
$303$	−24.1287	−1.38616
$304$	0	0
$305$	13.4265	0.768801
$306$	0	0
$307$	11.9202	0.680325	0.340162	−	0.940367i	$-0.389518\pi$
$307$	11.9202	0.680325	0.340162	+	0.940367i	$0.389518\pi$
$308$	0	0
$309$	19.7668	1.12450
$310$	0	0
$311$	23.4994	1.33253	0.666265	−	0.745715i	$-0.267892\pi$
$311$	23.4994	1.33253	0.666265	+	0.745715i	$0.267892\pi$
$312$	0	0
$313$	−4.55693	−0.257573	−0.128786	−	0.991672i	$-0.541108\pi$
$313$	−4.55693	−0.257573	−0.128786	+	0.991672i	$0.541108\pi$
$314$	0	0
$315$	−1.00629	−0.0566981
$316$	0	0
$317$	−31.1453	−1.74929	−0.874647	−	0.484760i	$-0.838907\pi$
$317$	−31.1453	−1.74929	−0.874647	+	0.484760i	$0.838907\pi$
$318$	0	0
$319$	−40.5777	−2.27191
$320$	0	0
$321$	−0.179161	−0.00999977
$322$	0	0
$323$	16.8805	0.939256
$324$	0	0
$325$	−3.95743	−0.219519
$326$	0	0
$327$	−31.3167	−1.73182
$328$	0	0
$329$	−5.32542	−0.293600
$330$	0	0
$331$	16.4864	0.906173	0.453087	−	0.891466i	$-0.350323\pi$
$331$	16.4864	0.906173	0.453087	+	0.891466i	$0.350323\pi$
$332$	0	0
$333$	4.13265	0.226468
$334$	0	0
$335$	−5.91647	−0.323251
$336$	0	0
$337$	14.1599	0.771340	0.385670	−	0.922637i	$-0.373970\pi$
$337$	14.1599	0.771340	0.385670	+	0.922637i	$0.373970\pi$
$338$	0	0
$339$	9.89319	0.537324
$340$	0	0
$341$	50.4475	2.73189
$342$	0	0
$343$	17.1961	0.928500
$344$	0	0
$345$	3.97291	0.213894
$346$	0	0
$347$	17.2505	0.926054	0.463027	−	0.886344i	$-0.346763\pi$
$347$	17.2505	0.926054	0.463027	+	0.886344i	$0.346763\pi$
$348$	0	0
$349$	9.13929	0.489215	0.244607	−	0.969622i	$-0.421341\pi$
$349$	9.13929	0.489215	0.244607	+	0.969622i	$0.421341\pi$
$350$	0	0
$351$	−22.2034	−1.18513
$352$	0	0
$353$	13.8868	0.739119	0.369559	−	0.929207i	$-0.379509\pi$
$353$	13.8868	0.739119	0.369559	+	0.929207i	$0.379509\pi$
$354$	0	0
$355$	12.1925	0.647112
$356$	0	0
$357$	−10.9253	−0.578229
$358$	0	0
$359$	−0.398521	−0.0210331	−0.0105166	−	0.999945i	$-0.503348\pi$
$359$	−0.398521	−0.0210331	−0.0105166	+	0.999945i	$0.503348\pi$
$360$	0	0
$361$	−7.55679	−0.397726
$362$	0	0
$363$	−44.6166	−2.34176
$364$	0	0
$365$	13.7425	0.719314
$366$	0	0
$367$	34.2981	1.79035	0.895173	−	0.445720i	$-0.147052\pi$
$367$	34.2981	1.79035	0.895173	+	0.445720i	$0.147052\pi$
$368$	0	0
$369$	−3.92707	−0.204435
$370$	0	0
$371$	7.67666	0.398552
$372$	0	0
$373$	14.9034	0.771670	0.385835	−	0.922568i	$-0.373913\pi$
$373$	14.9034	0.771670	0.385835	+	0.922568i	$0.373913\pi$
$374$	0	0
$375$	1.51742	0.0783592
$376$	0	0
$377$	25.2635	1.30114
$378$	0	0
$379$	5.56665	0.285939	0.142970	−	0.989727i	$-0.454335\pi$
$379$	5.56665	0.285939	0.142970	+	0.989727i	$0.454335\pi$
$380$	0	0
$381$	−5.84970	−0.299689
$382$	0	0
$383$	19.8814	1.01589	0.507946	−	0.861389i	$-0.330405\pi$
$383$	19.8814	1.01589	0.507946	+	0.861389i	$0.330405\pi$
$384$	0	0
$385$	9.17115	0.467405
$386$	0	0
$387$	0.189460	0.00963080
$388$	0	0
$389$	−20.1393	−1.02110	−0.510552	−	0.859847i	$-0.670559\pi$
$389$	−20.1393	−1.02110	−0.510552	+	0.859847i	$0.670559\pi$
$390$	0	0
$391$	−13.0651	−0.660732
$392$	0	0
$393$	9.82781	0.495748
$394$	0	0
$395$	−15.3336	−0.771517
$396$	0	0
$397$	9.33085	0.468302	0.234151	−	0.972200i	$-0.424769\pi$
$397$	9.33085	0.468302	0.234151	+	0.972200i	$0.424769\pi$
$398$	0	0
$399$	−7.40622	−0.370775
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	−31.4085	−1.56457
$404$	0	0
$405$	6.42126	0.319075
$406$	0	0
$407$	−37.6642	−1.86695
$408$	0	0
$409$	19.9758	0.987740	0.493870	−	0.869536i	$-0.335582\pi$
$409$	19.9758	0.987740	0.493870	+	0.869536i	$0.335582\pi$
$410$	0	0
$411$	14.5585	0.718120
$412$	0	0
$413$	15.3928	0.757429
$414$	0	0
$415$	−0.961591	−0.0472027
$416$	0	0
$417$	19.2318	0.941783
$418$	0	0
$419$	9.58067	0.468046	0.234023	−	0.972231i	$-0.424811\pi$
$419$	9.58067	0.468046	0.234023	+	0.972231i	$0.424811\pi$
$420$	0	0
$421$	−17.6549	−0.860447	−0.430224	−	0.902722i	$-0.641565\pi$
$421$	−17.6549	−0.860447	−0.430224	+	0.902722i	$0.641565\pi$
$422$	0	0
$423$	−2.57420	−0.125162
$424$	0	0
$425$	−4.99012	−0.242056
$426$	0	0
$427$	19.3723	0.937491
$428$	0	0
$429$	38.1703	1.84288
$430$	0	0
$431$	28.9132	1.39270	0.696351	−	0.717701i	$-0.254806\pi$
$431$	28.9132	1.39270	0.696351	+	0.717701i	$0.254806\pi$
$432$	0	0
$433$	18.9011	0.908330	0.454165	−	0.890918i	$-0.349938\pi$
$433$	18.9011	0.908330	0.454165	+	0.890918i	$0.349938\pi$
$434$	0	0
$435$	−9.68693	−0.464453
$436$	0	0
$437$	−8.85680	−0.423678
$438$	0	0
$439$	−1.07746	−0.0514244	−0.0257122	−	0.999669i	$-0.508185\pi$
$439$	−1.07746	−0.0514244	−0.0257122	+	0.999669i	$0.508185\pi$
$440$	0	0
$441$	3.43016	0.163341
$442$	0	0
$443$	−4.99573	−0.237354	−0.118677	−	0.992933i	$-0.537865\pi$
$443$	−4.99573	−0.237354	−0.118677	+	0.992933i	$0.537865\pi$
$444$	0	0
$445$	−0.887393	−0.0420665
$446$	0	0
$447$	−10.8644	−0.513867
$448$	0	0
$449$	34.1675	1.61246	0.806232	−	0.591600i	$-0.201503\pi$
$449$	34.1675	1.61246	0.806232	+	0.591600i	$0.201503\pi$
$450$	0	0
$451$	35.7905	1.68531
$452$	0	0
$453$	−21.0201	−0.987608
$454$	0	0
$455$	−5.70993	−0.267685
$456$	0	0
$457$	−8.05397	−0.376749	−0.188374	−	0.982097i	$-0.560322\pi$
$457$	−8.05397	−0.376749	−0.188374	+	0.982097i	$0.560322\pi$
$458$	0	0
$459$	−27.9974	−1.30681
$460$	0	0
$461$	−13.7947	−0.642485	−0.321242	−	0.946997i	$-0.604100\pi$
$461$	−13.7947	−0.642485	−0.321242	+	0.946997i	$0.604100\pi$
$462$	0	0
$463$	24.9367	1.15891	0.579454	−	0.815005i	$-0.303266\pi$
$463$	24.9367	1.15891	0.579454	+	0.815005i	$0.303266\pi$
$464$	0	0
$465$	12.0431	0.558486
$466$	0	0
$467$	−21.4891	−0.994396	−0.497198	−	0.867637i	$-0.665638\pi$
$467$	−21.4891	−0.994396	−0.497198	+	0.867637i	$0.665638\pi$
$468$	0	0
$469$	−8.53650	−0.394179
$470$	0	0
$471$	11.2522	0.518475
$472$	0	0
$473$	−1.72671	−0.0793940
$474$	0	0
$475$	−3.38278	−0.155213
$476$	0	0
$477$	3.71075	0.169903
$478$	0	0
$479$	2.01963	0.0922791	0.0461396	−	0.998935i	$-0.485308\pi$
$479$	2.01963	0.0922791	0.0461396	+	0.998935i	$0.485308\pi$
$480$	0	0
$481$	23.4496	1.06921
$482$	0	0
$483$	5.73226	0.260827
$484$	0	0
$485$	−12.5839	−0.571407
$486$	0	0
$487$	37.8448	1.71491	0.857456	−	0.514558i	$-0.172044\pi$
$487$	37.8448	1.71491	0.857456	+	0.514558i	$0.172044\pi$
$488$	0	0
$489$	−5.30312	−0.239815
$490$	0	0
$491$	21.9723	0.991595	0.495797	−	0.868438i	$-0.334876\pi$
$491$	21.9723	0.991595	0.495797	+	0.868438i	$0.334876\pi$
$492$	0	0
$493$	31.8560	1.43472
$494$	0	0
$495$	4.43316	0.199256
$496$	0	0
$497$	17.5918	0.789101
$498$	0	0
$499$	41.6287	1.86356	0.931779	−	0.363026i	$-0.118256\pi$
$499$	41.6287	1.86356	0.931779	+	0.363026i	$0.118256\pi$
$500$	0	0
$501$	16.0846	0.718609
$502$	0	0
$503$	43.2590	1.92882	0.964411	−	0.264409i	$-0.0851768\pi$
$503$	43.2590	1.92882	0.964411	+	0.264409i	$0.0851768\pi$
$504$	0	0
$505$	−15.9011	−0.707591
$506$	0	0
$507$	−4.03822	−0.179343
$508$	0	0
$509$	20.6270	0.914275	0.457137	−	0.889396i	$-0.348875\pi$
$509$	20.6270	0.914275	0.457137	+	0.889396i	$0.348875\pi$
$510$	0	0
$511$	19.8281	0.877146
$512$	0	0
$513$	−18.9793	−0.837957
$514$	0	0
$515$	13.0266	0.574021
$516$	0	0
$517$	23.4608	1.03181
$518$	0	0
$519$	35.5115	1.55878
$520$	0	0
$521$	24.3799	1.06810	0.534051	−	0.845452i	$-0.320669\pi$
$521$	24.3799	1.06810	0.534051	+	0.845452i	$0.320669\pi$
$522$	0	0
$523$	−32.4346	−1.41827	−0.709133	−	0.705074i	$-0.750914\pi$
$523$	−32.4346	−1.41827	−0.709133	+	0.705074i	$0.750914\pi$
$524$	0	0
$525$	2.18939	0.0955528
$526$	0	0
$527$	−39.6045	−1.72520
$528$	0	0
$529$	−16.1450	−0.701958
$530$	0	0
$531$	7.44057	0.322893
$532$	0	0
$533$	−22.2831	−0.965186
$534$	0	0
$535$	−0.118069	−0.00510458
$536$	0	0
$537$	17.4932	0.754886
$538$	0	0
$539$	−31.2618	−1.34654
$540$	0	0
$541$	16.7852	0.721651	0.360826	−	0.932633i	$-0.382495\pi$
$541$	16.7852	0.721651	0.360826	+	0.932633i	$0.382495\pi$
$542$	0	0
$543$	−22.2076	−0.953017
$544$	0	0
$545$	−20.6381	−0.884041
$546$	0	0
$547$	−35.0696	−1.49947	−0.749734	−	0.661740i	$-0.769818\pi$
$547$	−35.0696	−1.49947	−0.749734	+	0.661740i	$0.769818\pi$
$548$	0	0
$549$	9.36419	0.399654
$550$	0	0
$551$	21.5951	0.919981
$552$	0	0
$553$	−22.1239	−0.940803
$554$	0	0
$555$	−8.99141	−0.381664
$556$	0	0
$557$	41.9919	1.77925	0.889627	−	0.456689i	$-0.150965\pi$
$557$	41.9919	1.77925	0.889627	+	0.456689i	$0.150965\pi$
$558$	0	0
$559$	1.07504	0.0454694
$560$	0	0
$561$	48.1308	2.03209
$562$	0	0
$563$	4.84688	0.204272	0.102136	−	0.994770i	$-0.467432\pi$
$563$	4.84688	0.204272	0.102136	+	0.994770i	$0.467432\pi$
$564$	0	0
$565$	6.51975	0.274288
$566$	0	0
$567$	9.26483	0.389086
$568$	0	0
$569$	−40.6601	−1.70456	−0.852279	−	0.523087i	$-0.824780\pi$
$569$	−40.6601	−1.70456	−0.852279	+	0.523087i	$0.824780\pi$
$570$	0	0
$571$	−11.5809	−0.484644	−0.242322	−	0.970196i	$-0.577909\pi$
$571$	−11.5809	−0.484644	−0.242322	+	0.970196i	$0.577909\pi$
$572$	0	0
$573$	7.76680	0.324463
$574$	0	0
$575$	2.61820	0.109186
$576$	0	0
$577$	−21.6376	−0.900786	−0.450393	−	0.892831i	$-0.648716\pi$
$577$	−21.6376	−0.900786	−0.450393	+	0.892831i	$0.648716\pi$
$578$	0	0
$579$	39.9067	1.65847
$580$	0	0
$581$	−1.38742	−0.0575599
$582$	0	0
$583$	−33.8190	−1.40064
$584$	0	0
$585$	−2.76007	−0.114115
$586$	0	0
$587$	−12.1846	−0.502914	−0.251457	−	0.967868i	$-0.580910\pi$
$587$	−12.1846	−0.502914	−0.251457	+	0.967868i	$0.580910\pi$
$588$	0	0
$589$	−26.8477	−1.10624
$590$	0	0
$591$	−26.0073	−1.06980
$592$	0	0
$593$	−28.4088	−1.16661	−0.583304	−	0.812254i	$-0.698241\pi$
$593$	−28.4088	−1.16661	−0.583304	+	0.812254i	$0.698241\pi$
$594$	0	0
$595$	−7.19993	−0.295169
$596$	0	0
$597$	−1.57501	−0.0644607
$598$	0	0
$599$	7.21345	0.294734	0.147367	−	0.989082i	$-0.452920\pi$
$599$	7.21345	0.294734	0.147367	+	0.989082i	$0.452920\pi$
$600$	0	0
$601$	27.9309	1.13933	0.569663	−	0.821878i	$-0.307074\pi$
$601$	27.9309	1.13933	0.569663	+	0.821878i	$0.307074\pi$
$602$	0	0
$603$	−4.12638	−0.168039
$604$	0	0
$605$	−29.4030	−1.19540
$606$	0	0
$607$	13.9695	0.567004	0.283502	−	0.958972i	$-0.408504\pi$
$607$	13.9695	0.567004	0.283502	+	0.958972i	$0.408504\pi$
$608$	0	0
$609$	−13.9767	−0.566363
$610$	0	0
$611$	−14.6066	−0.590921
$612$	0	0
$613$	12.3879	0.500341	0.250170	−	0.968202i	$-0.419513\pi$
$613$	12.3879	0.500341	0.250170	+	0.968202i	$0.419513\pi$
$614$	0	0
$615$	8.54412	0.344532
$616$	0	0
$617$	−27.9018	−1.12328	−0.561642	−	0.827380i	$-0.689830\pi$
$617$	−27.9018	−1.12328	−0.561642	+	0.827380i	$0.689830\pi$
$618$	0	0
$619$	5.54966	0.223060	0.111530	−	0.993761i	$-0.464425\pi$
$619$	5.54966	0.223060	0.111530	+	0.993761i	$0.464425\pi$
$620$	0	0
$621$	14.6896	0.589473
$622$	0	0
$623$	−1.28036	−0.0512967
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	32.6277	1.30302
$628$	0	0
$629$	29.5688	1.17898
$630$	0	0
$631$	0.375760	0.0149588	0.00747938	−	0.999972i	$-0.497619\pi$
$631$	0.375760	0.0149588	0.00747938	+	0.999972i	$0.497619\pi$
$632$	0	0
$633$	−20.8454	−0.828532
$634$	0	0
$635$	−3.85503	−0.152982
$636$	0	0
$637$	19.4635	0.771172
$638$	0	0
$639$	8.50354	0.336395
$640$	0	0
$641$	18.3407	0.724416	0.362208	−	0.932097i	$-0.382023\pi$
$641$	18.3407	0.724416	0.362208	+	0.932097i	$0.382023\pi$
$642$	0	0
$643$	10.9983	0.433730	0.216865	−	0.976202i	$-0.430417\pi$
$643$	10.9983	0.433730	0.216865	+	0.976202i	$0.430417\pi$
$644$	0	0
$645$	−0.412209	−0.0162307
$646$	0	0
$647$	−31.4013	−1.23451	−0.617256	−	0.786762i	$-0.711756\pi$
$647$	−31.4013	−1.23451	−0.617256	+	0.786762i	$0.711756\pi$
$648$	0	0
$649$	−67.8119	−2.66185
$650$	0	0
$651$	17.3763	0.681029
$652$	0	0
$653$	13.2094	0.516925	0.258462	−	0.966021i	$-0.416784\pi$
$653$	13.2094	0.516925	0.258462	+	0.966021i	$0.416784\pi$
$654$	0	0
$655$	6.47666	0.253064
$656$	0	0
$657$	9.58454	0.373928
$658$	0	0
$659$	−10.0308	−0.390746	−0.195373	−	0.980729i	$-0.562592\pi$
$659$	−10.0308	−0.390746	−0.195373	+	0.980729i	$0.562592\pi$
$660$	0	0
$661$	4.48815	0.174569	0.0872845	−	0.996183i	$-0.472181\pi$
$661$	4.48815	0.174569	0.0872845	+	0.996183i	$0.472181\pi$
$662$	0	0
$663$	−29.9661	−1.16379
$664$	0	0
$665$	−4.88080	−0.189269
$666$	0	0
$667$	−16.7141	−0.647173
$668$	0	0
$669$	19.5716	0.756680
$670$	0	0
$671$	−85.3435	−3.29465
$672$	0	0
$673$	−29.5866	−1.14048	−0.570239	−	0.821479i	$-0.693150\pi$
$673$	−29.5866	−1.14048	−0.570239	+	0.821479i	$0.693150\pi$
$674$	0	0
$675$	5.61056	0.215951
$676$	0	0
$677$	−10.5115	−0.403989	−0.201995	−	0.979387i	$-0.564742\pi$
$677$	−10.5115	−0.403989	−0.201995	+	0.979387i	$0.564742\pi$
$678$	0	0
$679$	−18.1566	−0.696785
$680$	0	0
$681$	0.974451	0.0373411
$682$	0	0
$683$	50.7139	1.94051	0.970256	−	0.242081i	$-0.0778301\pi$
$683$	50.7139	1.94051	0.970256	+	0.242081i	$0.0778301\pi$
$684$	0	0
$685$	9.59428	0.366579
$686$	0	0
$687$	−26.1426	−0.997402
$688$	0	0
$689$	21.0556	0.802155
$690$	0	0
$691$	7.38050	0.280767	0.140384	−	0.990097i	$-0.455166\pi$
$691$	7.38050	0.280767	0.140384	+	0.990097i	$0.455166\pi$
$692$	0	0
$693$	6.39632	0.242976
$694$	0	0
$695$	12.6740	0.480752
$696$	0	0
$697$	−28.0978	−1.06428
$698$	0	0
$699$	−22.9832	−0.869303
$700$	0	0
$701$	−2.03868	−0.0769998	−0.0384999	−	0.999259i	$-0.512258\pi$
$701$	−2.03868	−0.0769998	−0.0384999	+	0.999259i	$0.512258\pi$
$702$	0	0
$703$	20.0445	0.755994
$704$	0	0
$705$	5.60070	0.210935
$706$	0	0
$707$	−22.9427	−0.862850
$708$	0	0
$709$	7.21856	0.271099	0.135549	−	0.990771i	$-0.456720\pi$
$709$	7.21856	0.271099	0.135549	+	0.990771i	$0.456720\pi$
$710$	0	0
$711$	−10.6943	−0.401066
$712$	0	0
$713$	20.7796	0.778201
$714$	0	0
$715$	25.1547	0.940734
$716$	0	0
$717$	−33.5400	−1.25258
$718$	0	0
$719$	−39.9193	−1.48874	−0.744370	−	0.667768i	$-0.767250\pi$
$719$	−39.9193	−1.48874	−0.744370	+	0.667768i	$0.767250\pi$
$720$	0	0
$721$	18.7953	0.699973
$722$	0	0
$723$	−13.3139	−0.495150
$724$	0	0
$725$	−6.38382	−0.237089
$726$	0	0
$727$	33.6782	1.24906	0.624528	−	0.781003i	$-0.285291\pi$
$727$	33.6782	1.24906	0.624528	+	0.781003i	$0.285291\pi$
$728$	0	0
$729$	30.0192	1.11182
$730$	0	0
$731$	1.35557	0.0501377
$732$	0	0
$733$	−1.97003	−0.0727645	−0.0363823	−	0.999338i	$-0.511583\pi$
$733$	−1.97003	−0.0727645	−0.0363823	+	0.999338i	$0.511583\pi$
$734$	0	0
$735$	−7.46300	−0.275277
$736$	0	0
$737$	37.6070	1.38527
$738$	0	0
$739$	−0.586737	−0.0215835	−0.0107917	−	0.999942i	$-0.503435\pi$
$739$	−0.586737	−0.0215835	−0.0107917	+	0.999942i	$0.503435\pi$
$740$	0	0
$741$	−20.3139	−0.746248
$742$	0	0
$743$	42.3282	1.55287	0.776435	−	0.630197i	$-0.217026\pi$
$743$	42.3282	1.55287	0.776435	+	0.630197i	$0.217026\pi$
$744$	0	0
$745$	−7.15976	−0.262313
$746$	0	0
$747$	−0.670652	−0.0245379
$748$	0	0
$749$	−0.170355	−0.00622463
$750$	0	0
$751$	−23.0620	−0.841544	−0.420772	−	0.907166i	$-0.638241\pi$
$751$	−23.0620	−0.841544	−0.420772	+	0.907166i	$0.638241\pi$
$752$	0	0
$753$	10.1077	0.368344
$754$	0	0
$755$	−13.8525	−0.504144
$756$	0	0
$757$	−36.7933	−1.33727	−0.668637	−	0.743589i	$-0.733122\pi$
$757$	−36.7933	−1.33727	−0.668637	+	0.743589i	$0.733122\pi$
$758$	0	0
$759$	−25.2531	−0.916630
$760$	0	0
$761$	14.6310	0.530375	0.265187	−	0.964197i	$-0.414566\pi$
$761$	14.6310	0.530375	0.265187	+	0.964197i	$0.414566\pi$
$762$	0	0
$763$	−29.7775	−1.07802
$764$	0	0
$765$	−3.48031	−0.125831
$766$	0	0
$767$	42.2195	1.52446
$768$	0	0
$769$	−12.6382	−0.455746	−0.227873	−	0.973691i	$-0.573177\pi$
$769$	−12.6382	−0.455746	−0.227873	+	0.973691i	$0.573177\pi$
$770$	0	0
$771$	−18.8012	−0.677108
$772$	0	0
$773$	37.5765	1.35153	0.675766	−	0.737116i	$-0.263813\pi$
$773$	37.5765	1.35153	0.675766	+	0.737116i	$0.263813\pi$
$774$	0	0
$775$	7.93658	0.285090
$776$	0	0
$777$	−12.9731	−0.465409
$778$	0	0
$779$	−19.0474	−0.682444
$780$	0	0
$781$	−77.4997	−2.77316
$782$	0	0
$783$	−35.8168	−1.27999
$784$	0	0
$785$	7.41536	0.264666
$786$	0	0
$787$	−41.6098	−1.48323	−0.741614	−	0.670827i	$-0.765939\pi$
$787$	−41.6098	−1.48323	−0.741614	+	0.670827i	$0.765939\pi$
$788$	0	0
$789$	−31.1752	−1.10987
$790$	0	0
$791$	9.40694	0.334472
$792$	0	0
$793$	53.1345	1.88686
$794$	0	0
$795$	−8.07347	−0.286337
$796$	0	0
$797$	−41.2597	−1.46149	−0.730747	−	0.682648i	$-0.760828\pi$
$797$	−41.2597	−1.46149	−0.730747	+	0.682648i	$0.760828\pi$
$798$	0	0
$799$	−18.4182	−0.651590
$800$	0	0
$801$	−0.618903	−0.0218679
$802$	0	0
$803$	−87.3517	−3.08257
$804$	0	0
$805$	3.77764	0.133144
$806$	0	0
$807$	−8.91498	−0.313822
$808$	0	0
$809$	32.6633	1.14838	0.574191	−	0.818721i	$-0.305317\pi$
$809$	32.6633	1.14838	0.574191	+	0.818721i	$0.305317\pi$
$810$	0	0
$811$	−41.6336	−1.46195	−0.730977	−	0.682402i	$-0.760935\pi$
$811$	−41.6336	−1.46195	−0.730977	+	0.682402i	$0.760935\pi$
$812$	0	0
$813$	47.2569	1.65737
$814$	0	0
$815$	−3.49483	−0.122419
$816$	0	0
$817$	0.918937	0.0321495
$818$	0	0
$819$	−3.98233	−0.139154
$820$	0	0
$821$	50.9446	1.77798	0.888990	−	0.457926i	$-0.151408\pi$
$821$	50.9446	1.77798	0.888990	+	0.457926i	$0.151408\pi$
$822$	0	0
$823$	27.6321	0.963195	0.481598	−	0.876392i	$-0.340057\pi$
$823$	27.6321	0.963195	0.481598	+	0.876392i	$0.340057\pi$
$824$	0	0
$825$	−9.64522	−0.335803
$826$	0	0
$827$	−16.9435	−0.589184	−0.294592	−	0.955623i	$-0.595184\pi$
$827$	−16.9435	−0.589184	−0.294592	+	0.955623i	$0.595184\pi$
$828$	0	0
$829$	3.55324	0.123409	0.0617046	−	0.998094i	$-0.480346\pi$
$829$	3.55324	0.123409	0.0617046	+	0.998094i	$0.480346\pi$
$830$	0	0
$831$	6.48354	0.224912
$832$	0	0
$833$	24.5425	0.850348
$834$	0	0
$835$	10.6000	0.366828
$836$	0	0
$837$	44.5287	1.53914
$838$	0	0
$839$	−17.6939	−0.610863	−0.305431	−	0.952214i	$-0.598801\pi$
$839$	−17.6939	−0.610863	−0.305431	+	0.952214i	$0.598801\pi$
$840$	0	0
$841$	11.7532	0.405282
$842$	0	0
$843$	13.9401	0.480121
$844$	0	0
$845$	−2.66124	−0.0915494
$846$	0	0
$847$	−42.4237	−1.45770
$848$	0	0
$849$	−29.2111	−1.00252
$850$	0	0
$851$	−15.5140	−0.531815
$852$	0	0
$853$	−11.2635	−0.385654	−0.192827	−	0.981233i	$-0.561766\pi$
$853$	−11.2635	−0.385654	−0.192827	+	0.981233i	$0.561766\pi$
$854$	0	0
$855$	−2.35928	−0.0806858
$856$	0	0
$857$	47.4127	1.61959	0.809794	−	0.586714i	$-0.199579\pi$
$857$	47.4127	1.61959	0.809794	+	0.586714i	$0.199579\pi$
$858$	0	0
$859$	48.2943	1.64778	0.823890	−	0.566750i	$-0.191800\pi$
$859$	48.2943	1.64778	0.823890	+	0.566750i	$0.191800\pi$
$860$	0	0
$861$	12.3278	0.420129
$862$	0	0
$863$	−22.0954	−0.752138	−0.376069	−	0.926592i	$-0.622724\pi$
$863$	−22.0954	−0.752138	−0.376069	+	0.926592i	$0.622724\pi$
$864$	0	0
$865$	23.4026	0.795711
$866$	0	0
$867$	−11.9896	−0.407189
$868$	0	0
$869$	97.4654	3.30629
$870$	0	0
$871$	−23.4140	−0.793353
$872$	0	0
$873$	−8.77653	−0.297041
$874$	0	0
$875$	1.44284	0.0487768
$876$	0	0
$877$	45.8336	1.54769	0.773845	−	0.633375i	$-0.218331\pi$
$877$	45.8336	1.54769	0.773845	+	0.633375i	$0.218331\pi$
$878$	0	0
$879$	−13.2356	−0.446425
$880$	0	0
$881$	−2.73920	−0.0922860	−0.0461430	−	0.998935i	$-0.514693\pi$
$881$	−2.73920	−0.0922860	−0.0461430	+	0.998935i	$0.514693\pi$
$882$	0	0
$883$	−23.1198	−0.778045	−0.389022	−	0.921228i	$-0.627187\pi$
$883$	−23.1198	−0.778045	−0.389022	+	0.921228i	$0.627187\pi$
$884$	0	0
$885$	−16.1884	−0.544169
$886$	0	0
$887$	44.8210	1.50494	0.752471	−	0.658625i	$-0.228862\pi$
$887$	44.8210	1.50494	0.752471	+	0.658625i	$0.228862\pi$
$888$	0	0
$889$	−5.56218	−0.186550
$890$	0	0
$891$	−40.8157	−1.36738
$892$	0	0
$893$	−12.4856	−0.417816
$894$	0	0
$895$	11.5282	0.385347
$896$	0	0
$897$	15.7225	0.524959
$898$	0	0
$899$	−50.6657	−1.68980
$900$	0	0
$901$	26.5501	0.884512
$902$	0	0
$903$	−0.594750	−0.0197920
$904$	0	0
$905$	−14.6351	−0.486487
$906$	0	0
$907$	39.1780	1.30089	0.650443	−	0.759555i	$-0.274583\pi$
$907$	39.1780	1.30089	0.650443	+	0.759555i	$0.274583\pi$
$908$	0	0
$909$	−11.0901	−0.367834
$910$	0	0
$911$	8.30976	0.275315	0.137657	−	0.990480i	$-0.456043\pi$
$911$	8.30976	0.275315	0.137657	+	0.990480i	$0.456043\pi$
$912$	0	0
$913$	6.11219	0.202284
$914$	0	0
$915$	−20.3737	−0.673533
$916$	0	0
$917$	9.34477	0.308592
$918$	0	0
$919$	18.2041	0.600498	0.300249	−	0.953861i	$-0.402930\pi$
$919$	18.2041	0.600498	0.300249	+	0.953861i	$0.402930\pi$
$920$	0	0
$921$	−18.0880	−0.596020
$922$	0	0
$923$	48.2510	1.58820
$924$	0	0
$925$	−5.92546	−0.194828
$926$	0	0
$927$	9.08527	0.298400
$928$	0	0
$929$	−24.9811	−0.819604	−0.409802	−	0.912174i	$-0.634402\pi$
$929$	−24.9811	−0.819604	−0.409802	+	0.912174i	$0.634402\pi$
$930$	0	0
$931$	16.6373	0.545264
$932$	0	0
$933$	−35.6585	−1.16741
$934$	0	0
$935$	31.7189	1.03732
$936$	0	0
$937$	−42.8060	−1.39841	−0.699206	−	0.714920i	$-0.746463\pi$
$937$	−42.8060	−1.39841	−0.699206	+	0.714920i	$0.746463\pi$
$938$	0	0
$939$	6.91477	0.225655
$940$	0	0
$941$	−10.0692	−0.328247	−0.164124	−	0.986440i	$-0.552480\pi$
$941$	−10.0692	−0.328247	−0.164124	+	0.986440i	$0.552480\pi$
$942$	0	0
$943$	14.7423	0.480075
$944$	0	0
$945$	8.09513	0.263335
$946$	0	0
$947$	−2.04096	−0.0663222	−0.0331611	−	0.999450i	$-0.510557\pi$
$947$	−2.04096	−0.0663222	−0.0331611	+	0.999450i	$0.510557\pi$
$948$	0	0
$949$	54.3848	1.76541
$950$	0	0
$951$	47.2605	1.53253
$952$	0	0
$953$	11.7916	0.381968	0.190984	−	0.981593i	$-0.438832\pi$
$953$	11.7916	0.381968	0.190984	+	0.981593i	$0.438832\pi$
$954$	0	0
$955$	5.11843	0.165628
$956$	0	0
$957$	61.5734	1.99038
$958$	0	0
$959$	13.8430	0.447013
$960$	0	0
$961$	31.9893	1.03191
$962$	0	0
$963$	−0.0823461	−0.00265357
$964$	0	0
$965$	26.2991	0.846596
$966$	0	0
$967$	−7.81675	−0.251370	−0.125685	−	0.992070i	$-0.540113\pi$
$967$	−7.81675	−0.251370	−0.125685	+	0.992070i	$0.540113\pi$
$968$	0	0
$969$	−25.6148	−0.822865
$970$	0	0
$971$	13.5603	0.435170	0.217585	−	0.976041i	$-0.430182\pi$
$971$	13.5603	0.435170	0.217585	+	0.976041i	$0.430182\pi$
$972$	0	0
$973$	18.2865	0.586239
$974$	0	0
$975$	6.00508	0.192316
$976$	0	0
$977$	10.1772	0.325598	0.162799	−	0.986659i	$-0.447948\pi$
$977$	10.1772	0.325598	0.162799	+	0.986659i	$0.447948\pi$
$978$	0	0
$979$	5.64056	0.180273
$980$	0	0
$981$	−14.3938	−0.459560
$982$	0	0
$983$	−37.0697	−1.18234	−0.591169	−	0.806548i	$-0.701333\pi$
$983$	−37.0697	−1.18234	−0.591169	+	0.806548i	$0.701333\pi$
$984$	0	0
$985$	−17.1392	−0.546099
$986$	0	0
$987$	8.08090	0.257218
$988$	0	0
$989$	−0.711237	−0.0226160
$990$	0	0
$991$	45.0057	1.42965	0.714827	−	0.699302i	$-0.246506\pi$
$991$	45.0057	1.42965	0.714827	+	0.699302i	$0.246506\pi$
$992$	0	0
$993$	−25.0168	−0.793883
$994$	0	0
$995$	−1.03795	−0.0329053
$996$	0	0
$997$	2.10086	0.0665350	0.0332675	−	0.999446i	$-0.489409\pi$
$997$	2.10086	0.0665350	0.0332675	+	0.999446i	$0.489409\pi$
$998$	0	0
$999$	−33.2452	−1.05183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.10	✓	35

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

\(f(q)\)	\(=\)	\(q-1.51742 q^{3} -1.00000 q^{5} -1.44284 q^{7} -0.697439 q^{9} +O(q^{10})\)	\(q-1.51742 q^{3} -1.00000 q^{5} -1.44284 q^{7} -0.697439 q^{9} +6.35633 q^{11} -3.95743 q^{13} +1.51742 q^{15} -4.99012 q^{17} -3.38278 q^{19} +2.18939 q^{21} +2.61820 q^{23} +1.00000 q^{25} +5.61056 q^{27} -6.38382 q^{29} +7.93658 q^{31} -9.64522 q^{33} +1.44284 q^{35} -5.92546 q^{37} +6.00508 q^{39} +5.63069 q^{41} -0.271651 q^{43} +0.697439 q^{45} +3.69094 q^{47} -4.91822 q^{49} +7.57211 q^{51} -5.32053 q^{53} -6.35633 q^{55} +5.13310 q^{57} -10.6684 q^{59} -13.4265 q^{61} +1.00629 q^{63} +3.95743 q^{65} +5.91647 q^{67} -3.97291 q^{69} -12.1925 q^{71} -13.7425 q^{73} -1.51742 q^{75} -9.17115 q^{77} +15.3336 q^{79} -6.42126 q^{81} +0.961591 q^{83} +4.99012 q^{85} +9.68693 q^{87} +0.887393 q^{89} +5.70993 q^{91} -12.0431 q^{93} +3.38278 q^{95} +12.5839 q^{97} -4.43316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 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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Representations

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Database

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