LMFDB - Embedded newform 8020.2.a.f.1.20

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:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+0.353843 q^{3} +1.00000 q^{5} +1.60118 q^{7} -2.87480 q^{9} +O(q^{10})$	$q+0.353843 q^{3} +1.00000 q^{5} +1.60118 q^{7} -2.87480 q^{9} -5.86469 q^{11} -0.686513 q^{13} +0.353843 q^{15} -0.0112906 q^{17} +5.24856 q^{19} +0.566567 q^{21} +3.26657 q^{23} +1.00000 q^{25} -2.07876 q^{27} +1.70582 q^{29} +3.06549 q^{31} -2.07518 q^{33} +1.60118 q^{35} +10.1824 q^{37} -0.242918 q^{39} -0.405074 q^{41} -4.86212 q^{43} -2.87480 q^{45} +7.29020 q^{47} -4.43622 q^{49} -0.00399511 q^{51} -8.31594 q^{53} -5.86469 q^{55} +1.85717 q^{57} +3.40437 q^{59} -5.96211 q^{61} -4.60307 q^{63} -0.686513 q^{65} -6.10250 q^{67} +1.15585 q^{69} -5.70056 q^{71} -11.6323 q^{73} +0.353843 q^{75} -9.39043 q^{77} +3.61775 q^{79} +7.88883 q^{81} +9.53837 q^{83} -0.0112906 q^{85} +0.603591 q^{87} +13.3299 q^{89} -1.09923 q^{91} +1.08470 q^{93} +5.24856 q^{95} +6.80254 q^{97} +16.8598 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * 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
$3$	0.353843	0.204291	0.102146	−	0.994769i	$-0.467429\pi$
$3$	0.353843	0.204291	0.102146	+	0.994769i	$0.467429\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.60118	0.605190	0.302595	−	0.953119i	$-0.402147\pi$
$7$	1.60118	0.605190	0.302595	+	0.953119i	$0.402147\pi$
$8$	0	0
$9$	−2.87480	−0.958265
$10$	0	0
$11$	−5.86469	−1.76827	−0.884135	−	0.467231i	$-0.845252\pi$
$11$	−5.86469	−1.76827	−0.884135	+	0.467231i	$0.845252\pi$
$12$	0	0
$13$	−0.686513	−0.190404	−0.0952022	−	0.995458i	$-0.530350\pi$
$13$	−0.686513	−0.190404	−0.0952022	+	0.995458i	$0.530350\pi$
$14$	0	0
$15$	0.353843	0.0913619
$16$	0	0
$17$	−0.0112906	−0.00273838	−0.00136919	−	0.999999i	$-0.500436\pi$
$17$	−0.0112906	−0.00273838	−0.00136919	+	0.999999i	$0.500436\pi$
$18$	0	0
$19$	5.24856	1.20410	0.602051	−	0.798457i	$-0.294350\pi$
$19$	5.24856	1.20410	0.602051	+	0.798457i	$0.294350\pi$
$20$	0	0
$21$	0.566567	0.123635
$22$	0	0
$23$	3.26657	0.681127	0.340563	−	0.940222i	$-0.389382\pi$
$23$	3.26657	0.681127	0.340563	+	0.940222i	$0.389382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.07876	−0.400057
$28$	0	0
$29$	1.70582	0.316762	0.158381	−	0.987378i	$-0.449373\pi$
$29$	1.70582	0.316762	0.158381	+	0.987378i	$0.449373\pi$
$30$	0	0
$31$	3.06549	0.550578	0.275289	−	0.961362i	$-0.411226\pi$
$31$	3.06549	0.550578	0.275289	+	0.961362i	$0.411226\pi$
$32$	0	0
$33$	−2.07518	−0.361242
$34$	0	0
$35$	1.60118	0.270649
$36$	0	0
$37$	10.1824	1.67397	0.836984	−	0.547227i	$-0.184317\pi$
$37$	10.1824	1.67397	0.836984	+	0.547227i	$0.184317\pi$
$38$	0	0
$39$	−0.242918	−0.0388980
$40$	0	0
$41$	−0.405074	−0.0632620	−0.0316310	−	0.999500i	$-0.510070\pi$
$41$	−0.405074	−0.0632620	−0.0316310	+	0.999500i	$0.510070\pi$
$42$	0	0
$43$	−4.86212	−0.741467	−0.370733	−	0.928739i	$-0.620894\pi$
$43$	−4.86212	−0.741467	−0.370733	+	0.928739i	$0.620894\pi$
$44$	0	0
$45$	−2.87480	−0.428549
$46$	0	0
$47$	7.29020	1.06338	0.531692	−	0.846938i	$-0.321556\pi$
$47$	7.29020	1.06338	0.531692	+	0.846938i	$0.321556\pi$
$48$	0	0
$49$	−4.43622	−0.633746
$50$	0	0
$51$	−0.00399511	−0.000559427 0
$52$	0	0
$53$	−8.31594	−1.14228	−0.571141	−	0.820852i	$-0.693499\pi$
$53$	−8.31594	−1.14228	−0.571141	+	0.820852i	$0.693499\pi$
$54$	0	0
$55$	−5.86469	−0.790795
$56$	0	0
$57$	1.85717	0.245988
$58$	0	0
$59$	3.40437	0.443212	0.221606	−	0.975136i	$-0.428870\pi$
$59$	3.40437	0.443212	0.221606	+	0.975136i	$0.428870\pi$
$60$	0	0
$61$	−5.96211	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$61$	−5.96211	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$62$	0	0
$63$	−4.60307	−0.579932
$64$	0	0
$65$	−0.686513	−0.0851514
$66$	0	0
$67$	−6.10250	−0.745540	−0.372770	−	0.927924i	$-0.621592\pi$
$67$	−6.10250	−0.745540	−0.372770	+	0.927924i	$0.621592\pi$
$68$	0	0
$69$	1.15585	0.139148
$70$	0	0
$71$	−5.70056	−0.676532	−0.338266	−	0.941051i	$-0.609840\pi$
$71$	−5.70056	−0.676532	−0.338266	+	0.941051i	$0.609840\pi$
$72$	0	0
$73$	−11.6323	−1.36146	−0.680728	−	0.732536i	$-0.738336\pi$
$73$	−11.6323	−1.36146	−0.680728	+	0.732536i	$0.738336\pi$
$74$	0	0
$75$	0.353843	0.0408583
$76$	0	0
$77$	−9.39043	−1.07014
$78$	0	0
$79$	3.61775	0.407029	0.203514	−	0.979072i	$-0.434764\pi$
$79$	3.61775	0.407029	0.203514	+	0.979072i	$0.434764\pi$
$80$	0	0
$81$	7.88883	0.876537
$82$	0	0
$83$	9.53837	1.04697	0.523486	−	0.852034i	$-0.324631\pi$
$83$	9.53837	1.04697	0.523486	+	0.852034i	$0.324631\pi$
$84$	0	0
$85$	−0.0112906	−0.00122464
$86$	0	0
$87$	0.603591	0.0647117
$88$	0	0
$89$	13.3299	1.41297	0.706485	−	0.707728i	$-0.250280\pi$
$89$	13.3299	1.41297	0.706485	+	0.707728i	$0.250280\pi$
$90$	0	0
$91$	−1.09923	−0.115231
$92$	0	0
$93$	1.08470	0.112478
$94$	0	0
$95$	5.24856	0.538491
$96$	0	0
$97$	6.80254	0.690694	0.345347	−	0.938475i	$-0.387761\pi$
$97$	6.80254	0.690694	0.345347	+	0.938475i	$0.387761\pi$
$98$	0	0
$99$	16.8598	1.69447
$100$	0	0
$101$	11.1053	1.10502	0.552510	−	0.833506i	$-0.313670\pi$
$101$	11.1053	1.10502	0.552510	+	0.833506i	$0.313670\pi$
$102$	0	0
$103$	−0.414220	−0.0408143	−0.0204072	−	0.999792i	$-0.506496\pi$
$103$	−0.414220	−0.0408143	−0.0204072	+	0.999792i	$0.506496\pi$
$104$	0	0
$105$	0.566567	0.0552913
$106$	0	0
$107$	−18.8681	−1.82405	−0.912024	−	0.410138i	$-0.865481\pi$
$107$	−18.8681	−1.82405	−0.912024	+	0.410138i	$0.865481\pi$
$108$	0	0
$109$	−7.84824	−0.751725	−0.375862	−	0.926675i	$-0.622654\pi$
$109$	−7.84824	−0.751725	−0.375862	+	0.926675i	$0.622654\pi$
$110$	0	0
$111$	3.60295	0.341977
$112$	0	0
$113$	7.53211	0.708561	0.354281	−	0.935139i	$-0.384726\pi$
$113$	7.53211	0.708561	0.354281	+	0.935139i	$0.384726\pi$
$114$	0	0
$115$	3.26657	0.304609
$116$	0	0
$117$	1.97358	0.182458
$118$	0	0
$119$	−0.0180783	−0.00165724
$120$	0	0
$121$	23.3946	2.12678
$122$	0	0
$123$	−0.143333	−0.0129239
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	7.97213	0.707412	0.353706	−	0.935357i	$-0.384921\pi$
$127$	7.97213	0.707412	0.353706	+	0.935357i	$0.384921\pi$
$128$	0	0
$129$	−1.72043	−0.151475
$130$	0	0
$131$	13.3810	1.16911	0.584553	−	0.811355i	$-0.301270\pi$
$131$	13.3810	1.16911	0.584553	+	0.811355i	$0.301270\pi$
$132$	0	0
$133$	8.40390	0.728711
$134$	0	0
$135$	−2.07876	−0.178911
$136$	0	0
$137$	21.1669	1.80841	0.904205	−	0.427098i	$-0.140464\pi$
$137$	21.1669	1.80841	0.904205	+	0.427098i	$0.140464\pi$
$138$	0	0
$139$	5.31747	0.451022	0.225511	−	0.974241i	$-0.427595\pi$
$139$	5.31747	0.451022	0.225511	+	0.974241i	$0.427595\pi$
$140$	0	0
$141$	2.57959	0.217240
$142$	0	0
$143$	4.02618	0.336686
$144$	0	0
$145$	1.70582	0.141660
$146$	0	0
$147$	−1.56973	−0.129469
$148$	0	0
$149$	16.4071	1.34412	0.672059	−	0.740497i	$-0.265410\pi$
$149$	16.4071	1.34412	0.672059	+	0.740497i	$0.265410\pi$
$150$	0	0
$151$	−4.18705	−0.340737	−0.170369	−	0.985380i	$-0.554496\pi$
$151$	−4.18705	−0.340737	−0.170369	+	0.985380i	$0.554496\pi$
$152$	0	0
$153$	0.0324582	0.00262409
$154$	0	0
$155$	3.06549	0.246226
$156$	0	0
$157$	−19.4003	−1.54831	−0.774156	−	0.632995i	$-0.781825\pi$
$157$	−19.4003	−1.54831	−0.774156	+	0.632995i	$0.781825\pi$
$158$	0	0
$159$	−2.94254	−0.233358
$160$	0	0
$161$	5.23037	0.412211
$162$	0	0
$163$	6.64019	0.520100	0.260050	−	0.965595i	$-0.416261\pi$
$163$	6.64019	0.520100	0.260050	+	0.965595i	$0.416261\pi$
$164$	0	0
$165$	−2.07518	−0.161553
$166$	0	0
$167$	13.2582	1.02595	0.512977	−	0.858402i	$-0.328543\pi$
$167$	13.2582	1.02595	0.512977	+	0.858402i	$0.328543\pi$
$168$	0	0
$169$	−12.5287	−0.963746
$170$	0	0
$171$	−15.0885	−1.15385
$172$	0	0
$173$	0.786765	0.0598166	0.0299083	−	0.999553i	$-0.490478\pi$
$173$	0.786765	0.0598166	0.0299083	+	0.999553i	$0.490478\pi$
$174$	0	0
$175$	1.60118	0.121038
$176$	0	0
$177$	1.20461	0.0905444
$178$	0	0
$179$	22.6173	1.69049	0.845247	−	0.534376i	$-0.179453\pi$
$179$	22.6173	1.69049	0.845247	+	0.534376i	$0.179453\pi$
$180$	0	0
$181$	11.7080	0.870246	0.435123	−	0.900371i	$-0.356705\pi$
$181$	11.7080	0.870246	0.435123	+	0.900371i	$0.356705\pi$
$182$	0	0
$183$	−2.10965	−0.155950
$184$	0	0
$185$	10.1824	0.748621
$186$	0	0
$187$	0.0662160	0.00484219
$188$	0	0
$189$	−3.32846	−0.242110
$190$	0	0
$191$	10.2247	0.739833	0.369917	−	0.929065i	$-0.379386\pi$
$191$	10.2247	0.739833	0.369917	+	0.929065i	$0.379386\pi$
$192$	0	0
$193$	19.1548	1.37880	0.689398	−	0.724383i	$-0.257875\pi$
$193$	19.1548	1.37880	0.689398	+	0.724383i	$0.257875\pi$
$194$	0	0
$195$	−0.242918	−0.0173957
$196$	0	0
$197$	5.43519	0.387241	0.193621	−	0.981077i	$-0.437977\pi$
$197$	5.43519	0.387241	0.193621	+	0.981077i	$0.437977\pi$
$198$	0	0
$199$	−18.8429	−1.33574	−0.667869	−	0.744279i	$-0.732793\pi$
$199$	−18.8429	−1.33574	−0.667869	+	0.744279i	$0.732793\pi$
$200$	0	0
$201$	−2.15933	−0.152307
$202$	0	0
$203$	2.73132	0.191701
$204$	0	0
$205$	−0.405074	−0.0282916
$206$	0	0
$207$	−9.39071	−0.652700
$208$	0	0
$209$	−30.7812	−2.12918
$210$	0	0
$211$	13.2296	0.910761	0.455381	−	0.890297i	$-0.349503\pi$
$211$	13.2296	0.910761	0.455381	+	0.890297i	$0.349503\pi$
$212$	0	0
$213$	−2.01710	−0.138210
$214$	0	0
$215$	−4.86212	−0.331594
$216$	0	0
$217$	4.90840	0.333204
$218$	0	0
$219$	−4.11600	−0.278134
$220$	0	0
$221$	0.00775116	0.000521399 0
$222$	0	0
$223$	−6.31789	−0.423077	−0.211539	−	0.977370i	$-0.567847\pi$
$223$	−6.31789	−0.423077	−0.211539	+	0.977370i	$0.567847\pi$
$224$	0	0
$225$	−2.87480	−0.191653
$226$	0	0
$227$	7.48526	0.496814	0.248407	−	0.968656i	$-0.420093\pi$
$227$	7.48526	0.496814	0.248407	+	0.968656i	$0.420093\pi$
$228$	0	0
$229$	14.4933	0.957747	0.478874	−	0.877884i	$-0.341045\pi$
$229$	14.4933	0.957747	0.478874	+	0.877884i	$0.341045\pi$
$230$	0	0
$231$	−3.32274	−0.218620
$232$	0	0
$233$	8.70438	0.570243	0.285121	−	0.958491i	$-0.407966\pi$
$233$	8.70438	0.570243	0.285121	+	0.958491i	$0.407966\pi$
$234$	0	0
$235$	7.29020	0.475560
$236$	0	0
$237$	1.28012	0.0831525
$238$	0	0
$239$	8.08004	0.522654	0.261327	−	0.965250i	$-0.415840\pi$
$239$	8.08004	0.522654	0.261327	+	0.965250i	$0.415840\pi$
$240$	0	0
$241$	11.3944	0.733979	0.366990	−	0.930225i	$-0.380388\pi$
$241$	11.3944	0.733979	0.366990	+	0.930225i	$0.380388\pi$
$242$	0	0
$243$	9.02768	0.579126
$244$	0	0
$245$	−4.43622	−0.283420
$246$	0	0
$247$	−3.60321	−0.229267
$248$	0	0
$249$	3.37509	0.213887
$250$	0	0
$251$	8.69288	0.548690	0.274345	−	0.961631i	$-0.411539\pi$
$251$	8.69288	0.548690	0.274345	+	0.961631i	$0.411539\pi$
$252$	0	0
$253$	−19.1574	−1.20442
$254$	0	0
$255$	−0.00399511	−0.000250183 0
$256$	0	0
$257$	8.15012	0.508390	0.254195	−	0.967153i	$-0.418189\pi$
$257$	8.15012	0.508390	0.254195	+	0.967153i	$0.418189\pi$
$258$	0	0
$259$	16.3038	1.01307
$260$	0	0
$261$	−4.90387	−0.303542
$262$	0	0
$263$	2.51567	0.155123	0.0775613	−	0.996988i	$-0.475287\pi$
$263$	2.51567	0.155123	0.0775613	+	0.996988i	$0.475287\pi$
$264$	0	0
$265$	−8.31594	−0.510844
$266$	0	0
$267$	4.71671	0.288658
$268$	0	0
$269$	20.9862	1.27955	0.639774	−	0.768563i	$-0.279028\pi$
$269$	20.9862	1.27955	0.639774	+	0.768563i	$0.279028\pi$
$270$	0	0
$271$	−1.95684	−0.118870	−0.0594348	−	0.998232i	$-0.518930\pi$
$271$	−1.95684	−0.118870	−0.0594348	+	0.998232i	$0.518930\pi$
$272$	0	0
$273$	−0.388956	−0.0235407
$274$	0	0
$275$	−5.86469	−0.353654
$276$	0	0
$277$	−26.6120	−1.59896	−0.799479	−	0.600695i	$-0.794891\pi$
$277$	−26.6120	−1.59896	−0.799479	+	0.600695i	$0.794891\pi$
$278$	0	0
$279$	−8.81265	−0.527599
$280$	0	0
$281$	12.7023	0.757755	0.378878	−	0.925447i	$-0.376310\pi$
$281$	12.7023	0.757755	0.378878	+	0.925447i	$0.376310\pi$
$282$	0	0
$283$	−2.21152	−0.131461	−0.0657307	−	0.997837i	$-0.520938\pi$
$283$	−2.21152	−0.131461	−0.0657307	+	0.997837i	$0.520938\pi$
$284$	0	0
$285$	1.85717	0.110009
$286$	0	0
$287$	−0.648597	−0.0382855
$288$	0	0
$289$	−16.9999	−0.999993
$290$	0	0
$291$	2.40703	0.141103
$292$	0	0
$293$	21.8083	1.27406	0.637028	−	0.770840i	$-0.280163\pi$
$293$	21.8083	1.27406	0.637028	+	0.770840i	$0.280163\pi$
$294$	0	0
$295$	3.40437	0.198210
$296$	0	0
$297$	12.1913	0.707408
$298$	0	0
$299$	−2.24254	−0.129690
$300$	0	0
$301$	−7.78514	−0.448728
$302$	0	0
$303$	3.92954	0.225746
$304$	0	0
$305$	−5.96211	−0.341389
$306$	0	0
$307$	12.8298	0.732236	0.366118	−	0.930568i	$-0.380687\pi$
$307$	12.8298	0.732236	0.366118	+	0.930568i	$0.380687\pi$
$308$	0	0
$309$	−0.146569	−0.00833802
$310$	0	0
$311$	0.435686	0.0247055	0.0123527	−	0.999924i	$-0.496068\pi$
$311$	0.435686	0.0247055	0.0123527	+	0.999924i	$0.496068\pi$
$312$	0	0
$313$	23.6483	1.33668	0.668341	−	0.743855i	$-0.267005\pi$
$313$	23.6483	1.33668	0.668341	+	0.743855i	$0.267005\pi$
$314$	0	0
$315$	−4.60307	−0.259353
$316$	0	0
$317$	7.08040	0.397675	0.198837	−	0.980032i	$-0.436283\pi$
$317$	7.08040	0.397675	0.198837	+	0.980032i	$0.436283\pi$
$318$	0	0
$319$	−10.0041	−0.560121
$320$	0	0
$321$	−6.67634	−0.372637
$322$	0	0
$323$	−0.0592596	−0.00329729
$324$	0	0
$325$	−0.686513	−0.0380809
$326$	0	0
$327$	−2.77705	−0.153571
$328$	0	0
$329$	11.6729	0.643549
$330$	0	0
$331$	−18.9483	−1.04149	−0.520746	−	0.853712i	$-0.674346\pi$
$331$	−18.9483	−1.04149	−0.520746	+	0.853712i	$0.674346\pi$
$332$	0	0
$333$	−29.2722	−1.60411
$334$	0	0
$335$	−6.10250	−0.333415
$336$	0	0
$337$	3.54553	0.193138	0.0965688	−	0.995326i	$-0.469213\pi$
$337$	3.54553	0.193138	0.0965688	+	0.995326i	$0.469213\pi$
$338$	0	0
$339$	2.66518	0.144753
$340$	0	0
$341$	−17.9781	−0.973570
$342$	0	0
$343$	−18.3115	−0.988726
$344$	0	0
$345$	1.15585	0.0622290
$346$	0	0
$347$	−14.5406	−0.780578	−0.390289	−	0.920692i	$-0.627625\pi$
$347$	−14.5406	−0.780578	−0.390289	+	0.920692i	$0.627625\pi$
$348$	0	0
$349$	28.4097	1.52074	0.760368	−	0.649493i	$-0.225019\pi$
$349$	28.4097	1.52074	0.760368	+	0.649493i	$0.225019\pi$
$350$	0	0
$351$	1.42709	0.0761726
$352$	0	0
$353$	−30.4085	−1.61848	−0.809240	−	0.587478i	$-0.800121\pi$
$353$	−30.4085	−1.61848	−0.809240	+	0.587478i	$0.800121\pi$
$354$	0	0
$355$	−5.70056	−0.302554
$356$	0	0
$357$	−0.00639689	−0.000338560 0
$358$	0	0
$359$	29.2533	1.54393	0.771964	−	0.635666i	$-0.219275\pi$
$359$	29.2533	1.54393	0.771964	+	0.635666i	$0.219275\pi$
$360$	0	0
$361$	8.54741	0.449864
$362$	0	0
$363$	8.27801	0.434483
$364$	0	0
$365$	−11.6323	−0.608862
$366$	0	0
$367$	−28.2563	−1.47496	−0.737482	−	0.675366i	$-0.763986\pi$
$367$	−28.2563	−1.47496	−0.737482	+	0.675366i	$0.763986\pi$
$368$	0	0
$369$	1.16451	0.0606217
$370$	0	0
$371$	−13.3153	−0.691297
$372$	0	0
$373$	−21.8828	−1.13305	−0.566524	−	0.824045i	$-0.691712\pi$
$373$	−21.8828	−1.13305	−0.566524	+	0.824045i	$0.691712\pi$
$374$	0	0
$375$	0.353843	0.0182724
$376$	0	0
$377$	−1.17106	−0.0603129
$378$	0	0
$379$	−37.0103	−1.90109	−0.950545	−	0.310586i	$-0.899475\pi$
$379$	−37.0103	−1.90109	−0.950545	+	0.310586i	$0.899475\pi$
$380$	0	0
$381$	2.82088	0.144518
$382$	0	0
$383$	−0.767051	−0.0391945	−0.0195972	−	0.999808i	$-0.506238\pi$
$383$	−0.767051	−0.0391945	−0.0195972	+	0.999808i	$0.506238\pi$
$384$	0	0
$385$	−9.39043	−0.478581
$386$	0	0
$387$	13.9776	0.710522
$388$	0	0
$389$	9.05114	0.458911	0.229455	−	0.973319i	$-0.426305\pi$
$389$	9.05114	0.458911	0.229455	+	0.973319i	$0.426305\pi$
$390$	0	0
$391$	−0.0368816	−0.00186518
$392$	0	0
$393$	4.73479	0.238838
$394$	0	0
$395$	3.61775	0.182029
$396$	0	0
$397$	−3.32412	−0.166833	−0.0834163	−	0.996515i	$-0.526583\pi$
$397$	−3.32412	−0.166833	−0.0834163	+	0.996515i	$0.526583\pi$
$398$	0	0
$399$	2.97366	0.148869
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−2.10450	−0.104832
$404$	0	0
$405$	7.88883	0.391999
$406$	0	0
$407$	−59.7163	−2.96003
$408$	0	0
$409$	−8.43917	−0.417290	−0.208645	−	0.977991i	$-0.566905\pi$
$409$	−8.43917	−0.417290	−0.208645	+	0.977991i	$0.566905\pi$
$410$	0	0
$411$	7.48976	0.369443
$412$	0	0
$413$	5.45102	0.268227
$414$	0	0
$415$	9.53837	0.468220
$416$	0	0
$417$	1.88155	0.0921400
$418$	0	0
$419$	31.3373	1.53093	0.765463	−	0.643480i	$-0.222510\pi$
$419$	31.3373	1.53093	0.765463	+	0.643480i	$0.222510\pi$
$420$	0	0
$421$	−17.7578	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$421$	−17.7578	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$422$	0	0
$423$	−20.9578	−1.01900
$424$	0	0
$425$	−0.0112906	−0.000547676 0
$426$	0	0
$427$	−9.54642	−0.461984
$428$	0	0
$429$	1.42464	0.0687822
$430$	0	0
$431$	−24.6099	−1.18542	−0.592708	−	0.805417i	$-0.701941\pi$
$431$	−24.6099	−1.18542	−0.592708	+	0.805417i	$0.701941\pi$
$432$	0	0
$433$	4.85588	0.233359	0.116679	−	0.993170i	$-0.462775\pi$
$433$	4.85588	0.233359	0.116679	+	0.993170i	$0.462775\pi$
$434$	0	0
$435$	0.603591	0.0289400
$436$	0	0
$437$	17.1448	0.820147
$438$	0	0
$439$	33.7054	1.60867	0.804336	−	0.594175i	$-0.202521\pi$
$439$	33.7054	1.60867	0.804336	+	0.594175i	$0.202521\pi$
$440$	0	0
$441$	12.7532	0.607296
$442$	0	0
$443$	21.3243	1.01315	0.506574	−	0.862197i	$-0.330912\pi$
$443$	21.3243	1.01315	0.506574	+	0.862197i	$0.330912\pi$
$444$	0	0
$445$	13.3299	0.631899
$446$	0	0
$447$	5.80552	0.274592
$448$	0	0
$449$	39.3904	1.85895	0.929474	−	0.368887i	$-0.120261\pi$
$449$	39.3904	1.85895	0.929474	+	0.368887i	$0.120261\pi$
$450$	0	0
$451$	2.37563	0.111864
$452$	0	0
$453$	−1.48156	−0.0696097
$454$	0	0
$455$	−1.09923	−0.0515328
$456$	0	0
$457$	18.5377	0.867159	0.433579	−	0.901115i	$-0.357250\pi$
$457$	18.5377	0.867159	0.433579	+	0.901115i	$0.357250\pi$
$458$	0	0
$459$	0.0234705	0.00109551
$460$	0	0
$461$	−18.2037	−0.847830	−0.423915	−	0.905702i	$-0.639345\pi$
$461$	−18.2037	−0.847830	−0.423915	+	0.905702i	$0.639345\pi$
$462$	0	0
$463$	14.9394	0.694294	0.347147	−	0.937811i	$-0.387150\pi$
$463$	14.9394	0.694294	0.347147	+	0.937811i	$0.387150\pi$
$464$	0	0
$465$	1.08470	0.0503018
$466$	0	0
$467$	−11.3212	−0.523880	−0.261940	−	0.965084i	$-0.584362\pi$
$467$	−11.3212	−0.523880	−0.261940	+	0.965084i	$0.584362\pi$
$468$	0	0
$469$	−9.77121	−0.451193
$470$	0	0
$471$	−6.86466	−0.316307
$472$	0	0
$473$	28.5148	1.31111
$474$	0	0
$475$	5.24856	0.240821
$476$	0	0
$477$	23.9066	1.09461
$478$	0	0
$479$	−14.9717	−0.684075	−0.342038	−	0.939686i	$-0.611117\pi$
$479$	−14.9717	−0.684075	−0.342038	+	0.939686i	$0.611117\pi$
$480$	0	0
$481$	−6.99031	−0.318731
$482$	0	0
$483$	1.85073	0.0842111
$484$	0	0
$485$	6.80254	0.308888
$486$	0	0
$487$	−17.0402	−0.772165	−0.386083	−	0.922464i	$-0.626172\pi$
$487$	−17.0402	−0.772165	−0.386083	+	0.922464i	$0.626172\pi$
$488$	0	0
$489$	2.34959	0.106252
$490$	0	0
$491$	−24.9642	−1.12662	−0.563309	−	0.826246i	$-0.690472\pi$
$491$	−24.9642	−1.12662	−0.563309	+	0.826246i	$0.690472\pi$
$492$	0	0
$493$	−0.0192597	−0.000867414 0
$494$	0	0
$495$	16.8598	0.757791
$496$	0	0
$497$	−9.12763	−0.409430
$498$	0	0
$499$	−25.0826	−1.12285	−0.561425	−	0.827527i	$-0.689747\pi$
$499$	−25.0826	−1.12285	−0.561425	+	0.827527i	$0.689747\pi$
$500$	0	0
$501$	4.69134	0.209594
$502$	0	0
$503$	23.9538	1.06805	0.534024	−	0.845469i	$-0.320679\pi$
$503$	23.9538	1.06805	0.534024	+	0.845469i	$0.320679\pi$
$504$	0	0
$505$	11.1053	0.494180
$506$	0	0
$507$	−4.43319	−0.196885
$508$	0	0
$509$	6.07143	0.269111	0.134556	−	0.990906i	$-0.457039\pi$
$509$	6.07143	0.269111	0.134556	+	0.990906i	$0.457039\pi$
$510$	0	0
$511$	−18.6254	−0.823939
$512$	0	0
$513$	−10.9105	−0.481709
$514$	0	0
$515$	−0.414220	−0.0182527
$516$	0	0
$517$	−42.7547	−1.88035
$518$	0	0
$519$	0.278391	0.0122200
$520$	0	0
$521$	−40.8285	−1.78873	−0.894365	−	0.447339i	$-0.852372\pi$
$521$	−40.8285	−1.78873	−0.894365	+	0.447339i	$0.852372\pi$
$522$	0	0
$523$	17.2010	0.752146	0.376073	−	0.926590i	$-0.377274\pi$
$523$	17.2010	0.752146	0.376073	+	0.926590i	$0.377274\pi$
$524$	0	0
$525$	0.566567	0.0247270
$526$	0	0
$527$	−0.0346113	−0.00150769
$528$	0	0
$529$	−12.3295	−0.536067
$530$	0	0
$531$	−9.78688	−0.424714
$532$	0	0
$533$	0.278089	0.0120454
$534$	0	0
$535$	−18.8681	−0.815739
$536$	0	0
$537$	8.00296	0.345353
$538$	0	0
$539$	26.0170	1.12063
$540$	0	0
$541$	2.54630	0.109474	0.0547370	−	0.998501i	$-0.482568\pi$
$541$	2.54630	0.109474	0.0547370	+	0.998501i	$0.482568\pi$
$542$	0	0
$543$	4.14278	0.177784
$544$	0	0
$545$	−7.84824	−0.336182
$546$	0	0
$547$	−30.9803	−1.32462	−0.662311	−	0.749229i	$-0.730424\pi$
$547$	−30.9803	−1.32462	−0.662311	+	0.749229i	$0.730424\pi$
$548$	0	0
$549$	17.1398	0.731511
$550$	0	0
$551$	8.95308	0.381414
$552$	0	0
$553$	5.79267	0.246330
$554$	0	0
$555$	3.60295	0.152937
$556$	0	0
$557$	−31.4239	−1.33147	−0.665736	−	0.746188i	$-0.731882\pi$
$557$	−31.4239	−1.33147	−0.665736	+	0.746188i	$0.731882\pi$
$558$	0	0
$559$	3.33791	0.141179
$560$	0	0
$561$	0.0234301	0.000989219 0
$562$	0	0
$563$	−20.6659	−0.870963	−0.435481	−	0.900198i	$-0.643422\pi$
$563$	−20.6659	−0.870963	−0.435481	+	0.900198i	$0.643422\pi$
$564$	0	0
$565$	7.53211	0.316878
$566$	0	0
$567$	12.6314	0.530471
$568$	0	0
$569$	−16.5430	−0.693520	−0.346760	−	0.937954i	$-0.612718\pi$
$569$	−16.5430	−0.693520	−0.346760	+	0.937954i	$0.612718\pi$
$570$	0	0
$571$	−26.0683	−1.09092	−0.545462	−	0.838136i	$-0.683646\pi$
$571$	−26.0683	−1.09092	−0.545462	+	0.838136i	$0.683646\pi$
$572$	0	0
$573$	3.61794	0.151142
$574$	0	0
$575$	3.26657	0.136225
$576$	0	0
$577$	46.0706	1.91794	0.958971	−	0.283503i	$-0.0914965\pi$
$577$	46.0706	1.91794	0.958971	+	0.283503i	$0.0914965\pi$
$578$	0	0
$579$	6.77781	0.281676
$580$	0	0
$581$	15.2727	0.633617
$582$	0	0
$583$	48.7704	2.01986
$584$	0	0
$585$	1.97358	0.0815976
$586$	0	0
$587$	12.4610	0.514322	0.257161	−	0.966369i	$-0.417213\pi$
$587$	12.4610	0.514322	0.257161	+	0.966369i	$0.417213\pi$
$588$	0	0
$589$	16.0894	0.662952
$590$	0	0
$591$	1.92320	0.0791100
$592$	0	0
$593$	21.5727	0.885885	0.442943	−	0.896550i	$-0.353935\pi$
$593$	21.5727	0.885885	0.442943	+	0.896550i	$0.353935\pi$
$594$	0	0
$595$	−0.0180783	−0.000741140 0
$596$	0	0
$597$	−6.66743	−0.272880
$598$	0	0
$599$	27.1933	1.11109	0.555544	−	0.831487i	$-0.312510\pi$
$599$	27.1933	1.11109	0.555544	+	0.831487i	$0.312510\pi$
$600$	0	0
$601$	42.1972	1.72126	0.860630	−	0.509230i	$-0.170070\pi$
$601$	42.1972	1.72126	0.860630	+	0.509230i	$0.170070\pi$
$602$	0	0
$603$	17.5434	0.714424
$604$	0	0
$605$	23.3946	0.951125
$606$	0	0
$607$	1.61421	0.0655186	0.0327593	−	0.999463i	$-0.489571\pi$
$607$	1.61421	0.0655186	0.0327593	+	0.999463i	$0.489571\pi$
$608$	0	0
$609$	0.966458	0.0391629
$610$	0	0
$611$	−5.00481	−0.202473
$612$	0	0
$613$	32.9984	1.33279	0.666397	−	0.745597i	$-0.267836\pi$
$613$	32.9984	1.33279	0.666397	+	0.745597i	$0.267836\pi$
$614$	0	0
$615$	−0.143333	−0.00577973
$616$	0	0
$617$	−1.61529	−0.0650290	−0.0325145	−	0.999471i	$-0.510352\pi$
$617$	−1.61529	−0.0650290	−0.0325145	+	0.999471i	$0.510352\pi$
$618$	0	0
$619$	14.8596	0.597259	0.298629	−	0.954369i	$-0.403471\pi$
$619$	14.8596	0.597259	0.298629	+	0.954369i	$0.403471\pi$
$620$	0	0
$621$	−6.79040	−0.272489
$622$	0	0
$623$	21.3436	0.855115
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−10.8917	−0.434973
$628$	0	0
$629$	−0.114965	−0.00458396
$630$	0	0
$631$	24.4984	0.975266	0.487633	−	0.873049i	$-0.337860\pi$
$631$	24.4984	0.975266	0.487633	+	0.873049i	$0.337860\pi$
$632$	0	0
$633$	4.68119	0.186061
$634$	0	0
$635$	7.97213	0.316364
$636$	0	0
$637$	3.04552	0.120668
$638$	0	0
$639$	16.3879	0.648297
$640$	0	0
$641$	38.6903	1.52817	0.764087	−	0.645113i	$-0.223190\pi$
$641$	38.6903	1.52817	0.764087	+	0.645113i	$0.223190\pi$
$642$	0	0
$643$	26.8608	1.05928	0.529642	−	0.848221i	$-0.322326\pi$
$643$	26.8608	1.05928	0.529642	+	0.848221i	$0.322326\pi$
$644$	0	0
$645$	−1.72043	−0.0677418
$646$	0	0
$647$	−33.9771	−1.33578	−0.667889	−	0.744261i	$-0.732802\pi$
$647$	−33.9771	−1.33578	−0.667889	+	0.744261i	$0.732802\pi$
$648$	0	0
$649$	−19.9656	−0.783718
$650$	0	0
$651$	1.73680	0.0680707
$652$	0	0
$653$	−47.6991	−1.86661	−0.933306	−	0.359082i	$-0.883090\pi$
$653$	−47.6991	−1.86661	−0.933306	+	0.359082i	$0.883090\pi$
$654$	0	0
$655$	13.3810	0.522840
$656$	0	0
$657$	33.4404	1.30464
$658$	0	0
$659$	−25.0260	−0.974875	−0.487438	−	0.873158i	$-0.662068\pi$
$659$	−25.0260	−0.974875	−0.487438	+	0.873158i	$0.662068\pi$
$660$	0	0
$661$	−45.5923	−1.77334	−0.886668	−	0.462406i	$-0.846986\pi$
$661$	−45.5923	−1.77334	−0.886668	+	0.462406i	$0.846986\pi$
$662$	0	0
$663$	0.00274269	0.000106517 0
$664$	0	0
$665$	8.40390	0.325889
$666$	0	0
$667$	5.57216	0.215755
$668$	0	0
$669$	−2.23554	−0.0864311
$670$	0	0
$671$	34.9659	1.34984
$672$	0	0
$673$	−9.22366	−0.355546	−0.177773	−	0.984072i	$-0.556889\pi$
$673$	−9.22366	−0.355546	−0.177773	+	0.984072i	$0.556889\pi$
$674$	0	0
$675$	−2.07876	−0.0800114
$676$	0	0
$677$	0.501049	0.0192569	0.00962844	−	0.999954i	$-0.496935\pi$
$677$	0.501049	0.0192569	0.00962844	+	0.999954i	$0.496935\pi$
$678$	0	0
$679$	10.8921	0.418001
$680$	0	0
$681$	2.64861	0.101495
$682$	0	0
$683$	10.7083	0.409742	0.204871	−	0.978789i	$-0.434322\pi$
$683$	10.7083	0.409742	0.204871	+	0.978789i	$0.434322\pi$
$684$	0	0
$685$	21.1669	0.808746
$686$	0	0
$687$	5.12837	0.195660
$688$	0	0
$689$	5.70900	0.217496
$690$	0	0
$691$	18.3475	0.697972	0.348986	−	0.937128i	$-0.386526\pi$
$691$	18.3475	0.697972	0.348986	+	0.937128i	$0.386526\pi$
$692$	0	0
$693$	26.9956	1.02548
$694$	0	0
$695$	5.31747	0.201703
$696$	0	0
$697$	0.00457354	0.000173235 0
$698$	0	0
$699$	3.07998	0.116496
$700$	0	0
$701$	18.2295	0.688519	0.344260	−	0.938874i	$-0.388130\pi$
$701$	18.2295	0.688519	0.344260	+	0.938874i	$0.388130\pi$
$702$	0	0
$703$	53.4427	2.01563
$704$	0	0
$705$	2.57959	0.0971528
$706$	0	0
$707$	17.7816	0.668747
$708$	0	0
$709$	−45.3815	−1.70434	−0.852169	−	0.523267i	$-0.824713\pi$
$709$	−45.3815	−1.70434	−0.852169	+	0.523267i	$0.824713\pi$
$710$	0	0
$711$	−10.4003	−0.390041
$712$	0	0
$713$	10.0136	0.375013
$714$	0	0
$715$	4.02618	0.150571
$716$	0	0
$717$	2.85907	0.106774
$718$	0	0
$719$	−30.8552	−1.15071	−0.575353	−	0.817905i	$-0.695135\pi$
$719$	−30.8552	−1.15071	−0.575353	+	0.817905i	$0.695135\pi$
$720$	0	0
$721$	−0.663242	−0.0247004
$722$	0	0
$723$	4.03184	0.149946
$724$	0	0
$725$	1.70582	0.0633524
$726$	0	0
$727$	0.823409	0.0305385	0.0152693	−	0.999883i	$-0.495139\pi$
$727$	0.823409	0.0305385	0.0152693	+	0.999883i	$0.495139\pi$
$728$	0	0
$729$	−20.4721	−0.758226
$730$	0	0
$731$	0.0548964	0.00203042
$732$	0	0
$733$	45.4361	1.67822	0.839111	−	0.543960i	$-0.183076\pi$
$733$	45.4361	1.67822	0.839111	+	0.543960i	$0.183076\pi$
$734$	0	0
$735$	−1.56973	−0.0579002
$736$	0	0
$737$	35.7893	1.31832
$738$	0	0
$739$	−36.7015	−1.35009	−0.675043	−	0.737779i	$-0.735875\pi$
$739$	−36.7015	−1.35009	−0.675043	+	0.737779i	$0.735875\pi$
$740$	0	0
$741$	−1.27497	−0.0468372
$742$	0	0
$743$	−52.5149	−1.92658	−0.963292	−	0.268455i	$-0.913487\pi$
$743$	−52.5149	−1.92658	−0.963292	+	0.268455i	$0.913487\pi$
$744$	0	0
$745$	16.4071	0.601108
$746$	0	0
$747$	−27.4209	−1.00328
$748$	0	0
$749$	−30.2112	−1.10389
$750$	0	0
$751$	−39.7125	−1.44913	−0.724565	−	0.689206i	$-0.757959\pi$
$751$	−39.7125	−1.44913	−0.724565	+	0.689206i	$0.757959\pi$
$752$	0	0
$753$	3.07592	0.112093
$754$	0	0
$755$	−4.18705	−0.152382
$756$	0	0
$757$	53.0909	1.92962	0.964810	−	0.262947i	$-0.0846944\pi$
$757$	53.0909	1.92962	0.964810	+	0.262947i	$0.0846944\pi$
$758$	0	0
$759$	−6.77872	−0.246052
$760$	0	0
$761$	−5.65054	−0.204832	−0.102416	−	0.994742i	$-0.532657\pi$
$761$	−5.65054	−0.204832	−0.102416	+	0.994742i	$0.532657\pi$
$762$	0	0
$763$	−12.5665	−0.454936
$764$	0	0
$765$	0.0324582	0.00117353
$766$	0	0
$767$	−2.33715	−0.0843895
$768$	0	0
$769$	41.7902	1.50699	0.753497	−	0.657451i	$-0.228365\pi$
$769$	41.7902	1.50699	0.753497	+	0.657451i	$0.228365\pi$
$770$	0	0
$771$	2.88386	0.103860
$772$	0	0
$773$	17.4201	0.626557	0.313279	−	0.949661i	$-0.398573\pi$
$773$	17.4201	0.626557	0.313279	+	0.949661i	$0.398573\pi$
$774$	0	0
$775$	3.06549	0.110116
$776$	0	0
$777$	5.76898	0.206961
$778$	0	0
$779$	−2.12606	−0.0761739
$780$	0	0
$781$	33.4320	1.19629
$782$	0	0
$783$	−3.54597	−0.126723
$784$	0	0
$785$	−19.4003	−0.692426
$786$	0	0
$787$	12.8014	0.456321	0.228160	−	0.973624i	$-0.426729\pi$
$787$	12.8014	0.456321	0.228160	+	0.973624i	$0.426729\pi$
$788$	0	0
$789$	0.890151	0.0316902
$790$	0	0
$791$	12.0603	0.428814
$792$	0	0
$793$	4.09307	0.145349
$794$	0	0
$795$	−2.94254	−0.104361
$796$	0	0
$797$	26.8551	0.951256	0.475628	−	0.879647i	$-0.342221\pi$
$797$	26.8551	0.951256	0.475628	+	0.879647i	$0.342221\pi$
$798$	0	0
$799$	−0.0823109	−0.00291195
$800$	0	0
$801$	−38.3208	−1.35400
$802$	0	0
$803$	68.2197	2.40742
$804$	0	0
$805$	5.23037	0.184346
$806$	0	0
$807$	7.42581	0.261401
$808$	0	0
$809$	−47.7601	−1.67916	−0.839578	−	0.543239i	$-0.817198\pi$
$809$	−47.7601	−1.67916	−0.839578	+	0.543239i	$0.817198\pi$
$810$	0	0
$811$	−16.1185	−0.565998	−0.282999	−	0.959120i	$-0.591329\pi$
$811$	−16.1185	−0.565998	−0.282999	+	0.959120i	$0.591329\pi$
$812$	0	0
$813$	−0.692414	−0.0242840
$814$	0	0
$815$	6.64019	0.232596
$816$	0	0
$817$	−25.5192	−0.892802
$818$	0	0
$819$	3.16007	0.110422
$820$	0	0
$821$	−13.1180	−0.457822	−0.228911	−	0.973447i	$-0.573517\pi$
$821$	−13.1180	−0.457822	−0.228911	+	0.973447i	$0.573517\pi$
$822$	0	0
$823$	−35.6908	−1.24410	−0.622052	−	0.782976i	$-0.713701\pi$
$823$	−35.6908	−1.24410	−0.622052	+	0.782976i	$0.713701\pi$
$824$	0	0
$825$	−2.07518	−0.0722485
$826$	0	0
$827$	−50.3152	−1.74963	−0.874816	−	0.484455i	$-0.839018\pi$
$827$	−50.3152	−1.74963	−0.874816	+	0.484455i	$0.839018\pi$
$828$	0	0
$829$	−55.6374	−1.93237	−0.966183	−	0.257857i	$-0.916984\pi$
$829$	−55.6374	−1.93237	−0.966183	+	0.257857i	$0.916984\pi$
$830$	0	0
$831$	−9.41646	−0.326653
$832$	0	0
$833$	0.0500877	0.00173544
$834$	0	0
$835$	13.2582	0.458820
$836$	0	0
$837$	−6.37240	−0.220262
$838$	0	0
$839$	7.85103	0.271048	0.135524	−	0.990774i	$-0.456728\pi$
$839$	7.85103	0.271048	0.135524	+	0.990774i	$0.456728\pi$
$840$	0	0
$841$	−26.0902	−0.899662
$842$	0	0
$843$	4.49462	0.154803
$844$	0	0
$845$	−12.5287	−0.431000
$846$	0	0
$847$	37.4590	1.28711
$848$	0	0
$849$	−0.782532	−0.0268564
$850$	0	0
$851$	33.2613	1.14018
$852$	0	0
$853$	19.7190	0.675167	0.337583	−	0.941296i	$-0.390391\pi$
$853$	19.7190	0.675167	0.337583	+	0.941296i	$0.390391\pi$
$854$	0	0
$855$	−15.0885	−0.516017
$856$	0	0
$857$	−9.97110	−0.340606	−0.170303	−	0.985392i	$-0.554475\pi$
$857$	−9.97110	−0.340606	−0.170303	+	0.985392i	$0.554475\pi$
$858$	0	0
$859$	−17.4835	−0.596530	−0.298265	−	0.954483i	$-0.596408\pi$
$859$	−17.4835	−0.596530	−0.298265	+	0.954483i	$0.596408\pi$
$860$	0	0
$861$	−0.229502	−0.00782140
$862$	0	0
$863$	24.2756	0.826351	0.413176	−	0.910651i	$-0.364420\pi$
$863$	24.2756	0.826351	0.413176	+	0.910651i	$0.364420\pi$
$864$	0	0
$865$	0.786765	0.0267508
$866$	0	0
$867$	−6.01529	−0.204290
$868$	0	0
$869$	−21.2170	−0.719737
$870$	0	0
$871$	4.18945	0.141954
$872$	0	0
$873$	−19.5559	−0.661868
$874$	0	0
$875$	1.60118	0.0541298
$876$	0	0
$877$	50.3841	1.70135	0.850674	−	0.525693i	$-0.176194\pi$
$877$	50.3841	1.70135	0.850674	+	0.525693i	$0.176194\pi$
$878$	0	0
$879$	7.71673	0.260279
$880$	0	0
$881$	−13.1267	−0.442250	−0.221125	−	0.975246i	$-0.570973\pi$
$881$	−13.1267	−0.442250	−0.221125	+	0.975246i	$0.570973\pi$
$882$	0	0
$883$	0.609862	0.0205235	0.0102617	−	0.999947i	$-0.496734\pi$
$883$	0.609862	0.0205235	0.0102617	+	0.999947i	$0.496734\pi$
$884$	0	0
$885$	1.20461	0.0404927
$886$	0	0
$887$	−6.34353	−0.212995	−0.106497	−	0.994313i	$-0.533964\pi$
$887$	−6.34353	−0.212995	−0.106497	+	0.994313i	$0.533964\pi$
$888$	0	0
$889$	12.7648	0.428119
$890$	0	0
$891$	−46.2655	−1.54995
$892$	0	0
$893$	38.2631	1.28042
$894$	0	0
$895$	22.6173	0.756012
$896$	0	0
$897$	−0.793508	−0.0264945
$898$	0	0
$899$	5.22915	0.174402
$900$	0	0
$901$	0.0938922	0.00312800
$902$	0	0
$903$	−2.75472	−0.0916713
$904$	0	0
$905$	11.7080	0.389186
$906$	0	0
$907$	−45.6371	−1.51536	−0.757678	−	0.652629i	$-0.773666\pi$
$907$	−45.6371	−1.51536	−0.757678	+	0.652629i	$0.773666\pi$
$908$	0	0
$909$	−31.9255	−1.05890
$910$	0	0
$911$	4.85264	0.160775	0.0803875	−	0.996764i	$-0.474384\pi$
$911$	4.85264	0.160775	0.0803875	+	0.996764i	$0.474384\pi$
$912$	0	0
$913$	−55.9396	−1.85133
$914$	0	0
$915$	−2.10965	−0.0697429
$916$	0	0
$917$	21.4255	0.707531
$918$	0	0
$919$	−7.72269	−0.254748	−0.127374	−	0.991855i	$-0.540655\pi$
$919$	−7.72269	−0.254748	−0.127374	+	0.991855i	$0.540655\pi$
$920$	0	0
$921$	4.53974	0.149590
$922$	0	0
$923$	3.91351	0.128815
$924$	0	0
$925$	10.1824	0.334794
$926$	0	0
$927$	1.19080	0.0391110
$928$	0	0
$929$	19.8202	0.650281	0.325140	−	0.945666i	$-0.394588\pi$
$929$	19.8202	0.650281	0.325140	+	0.945666i	$0.394588\pi$
$930$	0	0
$931$	−23.2838	−0.763095
$932$	0	0
$933$	0.154164	0.00504712
$934$	0	0
$935$	0.0662160	0.00216549
$936$	0	0
$937$	−38.2577	−1.24982	−0.624912	−	0.780695i	$-0.714865\pi$
$937$	−38.2577	−1.24982	−0.624912	+	0.780695i	$0.714865\pi$
$938$	0	0
$939$	8.36780	0.273073
$940$	0	0
$941$	−0.293262	−0.00956007	−0.00478003	−	0.999989i	$-0.501522\pi$
$941$	−0.293262	−0.00956007	−0.00478003	+	0.999989i	$0.501522\pi$
$942$	0	0
$943$	−1.32320	−0.0430894
$944$	0	0
$945$	−3.32846	−0.108275
$946$	0	0
$947$	−30.0452	−0.976338	−0.488169	−	0.872749i	$-0.662335\pi$
$947$	−30.0452	−0.976338	−0.488169	+	0.872749i	$0.662335\pi$
$948$	0	0
$949$	7.98571	0.259227
$950$	0	0
$951$	2.50535	0.0812416
$952$	0	0
$953$	24.5452	0.795096	0.397548	−	0.917581i	$-0.369861\pi$
$953$	24.5452	0.795096	0.397548	+	0.917581i	$0.369861\pi$
$954$	0	0
$955$	10.2247	0.330863
$956$	0	0
$957$	−3.53987	−0.114428
$958$	0	0
$959$	33.8920	1.09443
$960$	0	0
$961$	−21.6028	−0.696864
$962$	0	0
$963$	54.2419	1.74792
$964$	0	0
$965$	19.1548	0.616616
$966$	0	0
$967$	−20.2859	−0.652349	−0.326175	−	0.945309i	$-0.605760\pi$
$967$	−20.2859	−0.652349	−0.326175	+	0.945309i	$0.605760\pi$
$968$	0	0
$969$	−0.0209686	−0.000673608 0
$970$	0	0
$971$	42.4843	1.36339	0.681693	−	0.731638i	$-0.261244\pi$
$971$	42.4843	1.36339	0.681693	+	0.731638i	$0.261244\pi$
$972$	0	0
$973$	8.51424	0.272954
$974$	0	0
$975$	−0.242918	−0.00777960
$976$	0	0
$977$	36.5447	1.16917	0.584584	−	0.811333i	$-0.301258\pi$
$977$	36.5447	1.16917	0.584584	+	0.811333i	$0.301258\pi$
$978$	0	0
$979$	−78.1759	−2.49851
$980$	0	0
$981$	22.5621	0.720352
$982$	0	0
$983$	10.4847	0.334409	0.167205	−	0.985922i	$-0.446526\pi$
$983$	10.4847	0.334409	0.167205	+	0.985922i	$0.446526\pi$
$984$	0	0
$985$	5.43519	0.173179
$986$	0	0
$987$	4.13038	0.131472
$988$	0	0
$989$	−15.8825	−0.505033
$990$	0	0
$991$	−27.4270	−0.871249	−0.435624	−	0.900129i	$-0.643472\pi$
$991$	−27.4270	−0.871249	−0.435624	+	0.900129i	$0.643472\pi$
$992$	0	0
$993$	−6.70472	−0.212768
$994$	0	0
$995$	−18.8429	−0.597360
$996$	0	0
$997$	−41.3901	−1.31084	−0.655418	−	0.755266i	$-0.727508\pi$
$997$	−41.3901	−1.31084	−0.655418	+	0.755266i	$0.727508\pi$
$998$	0	0
$999$	−21.1666	−0.669682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.20	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.20	✓	37	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)

\(f(q)\)	\(=\)	\(q+0.353843 q^{3} +1.00000 q^{5} +1.60118 q^{7} -2.87480 q^{9} +O(q^{10})\)	\(q+0.353843 q^{3} +1.00000 q^{5} +1.60118 q^{7} -2.87480 q^{9} -5.86469 q^{11} -0.686513 q^{13} +0.353843 q^{15} -0.0112906 q^{17} +5.24856 q^{19} +0.566567 q^{21} +3.26657 q^{23} +1.00000 q^{25} -2.07876 q^{27} +1.70582 q^{29} +3.06549 q^{31} -2.07518 q^{33} +1.60118 q^{35} +10.1824 q^{37} -0.242918 q^{39} -0.405074 q^{41} -4.86212 q^{43} -2.87480 q^{45} +7.29020 q^{47} -4.43622 q^{49} -0.00399511 q^{51} -8.31594 q^{53} -5.86469 q^{55} +1.85717 q^{57} +3.40437 q^{59} -5.96211 q^{61} -4.60307 q^{63} -0.686513 q^{65} -6.10250 q^{67} +1.15585 q^{69} -5.70056 q^{71} -11.6323 q^{73} +0.353843 q^{75} -9.39043 q^{77} +3.61775 q^{79} +7.88883 q^{81} +9.53837 q^{83} -0.0112906 q^{85} +0.603591 q^{87} +13.3299 q^{89} -1.09923 q^{91} +1.08470 q^{93} +5.24856 q^{95} +6.80254 q^{97} +16.8598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more