LMFDB - Embedded newform 8020.2.a.e.1.18

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			1.18
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)

$f(q)$	$=$	$q-0.167962 q^{3} -1.00000 q^{5} +2.31084 q^{7} -2.97179 q^{9} +O(q^{10})$	$q-0.167962 q^{3} -1.00000 q^{5} +2.31084 q^{7} -2.97179 q^{9} +2.88795 q^{11} +4.00711 q^{13} +0.167962 q^{15} +4.29329 q^{17} -0.543420 q^{19} -0.388133 q^{21} -1.32347 q^{23} +1.00000 q^{25} +1.00303 q^{27} -6.26176 q^{29} +0.454333 q^{31} -0.485066 q^{33} -2.31084 q^{35} -1.85666 q^{37} -0.673041 q^{39} +8.59884 q^{41} -7.75654 q^{43} +2.97179 q^{45} +12.0779 q^{47} -1.66001 q^{49} -0.721110 q^{51} +9.04827 q^{53} -2.88795 q^{55} +0.0912739 q^{57} +3.28580 q^{59} -2.38822 q^{61} -6.86733 q^{63} -4.00711 q^{65} +12.2530 q^{67} +0.222292 q^{69} +2.38396 q^{71} +12.9385 q^{73} -0.167962 q^{75} +6.67360 q^{77} +1.17724 q^{79} +8.74690 q^{81} -3.48118 q^{83} -4.29329 q^{85} +1.05174 q^{87} -16.1077 q^{89} +9.25978 q^{91} -0.0763106 q^{93} +0.543420 q^{95} +3.13575 q^{97} -8.58239 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.167962	−0.0969728	−0.0484864	−	0.998824i	$-0.515440\pi$
$3$	−0.167962	−0.0969728	−0.0484864	+	0.998824i	$0.515440\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.31084	0.873416	0.436708	−	0.899603i	$-0.356144\pi$
$7$	2.31084	0.873416	0.436708	+	0.899603i	$0.356144\pi$
$8$	0	0
$9$	−2.97179	−0.990596
$10$	0	0
$11$	2.88795	0.870751	0.435376	−	0.900249i	$-0.356616\pi$
$11$	2.88795	0.870751	0.435376	+	0.900249i	$0.356616\pi$
$12$	0	0
$13$	4.00711	1.11137	0.555686	−	0.831392i	$-0.312456\pi$
$13$	4.00711	1.11137	0.555686	+	0.831392i	$0.312456\pi$
$14$	0	0
$15$	0.167962	0.0433676
$16$	0	0
$17$	4.29329	1.04128	0.520638	−	0.853777i	$-0.325694\pi$
$17$	4.29329	1.04128	0.520638	+	0.853777i	$0.325694\pi$
$18$	0	0
$19$	−0.543420	−0.124669	−0.0623346	−	0.998055i	$-0.519855\pi$
$19$	−0.543420	−0.124669	−0.0623346	+	0.998055i	$0.519855\pi$
$20$	0	0
$21$	−0.388133	−0.0846976
$22$	0	0
$23$	−1.32347	−0.275962	−0.137981	−	0.990435i	$-0.544061\pi$
$23$	−1.32347	−0.275962	−0.137981	+	0.990435i	$0.544061\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00303	0.193034
$28$	0	0
$29$	−6.26176	−1.16278	−0.581390	−	0.813625i	$-0.697491\pi$
$29$	−6.26176	−1.16278	−0.581390	+	0.813625i	$0.697491\pi$
$30$	0	0
$31$	0.454333	0.0816006	0.0408003	−	0.999167i	$-0.487009\pi$
$31$	0.454333	0.0816006	0.0408003	+	0.999167i	$0.487009\pi$
$32$	0	0
$33$	−0.485066	−0.0844392
$34$	0	0
$35$	−2.31084	−0.390603
$36$	0	0
$37$	−1.85666	−0.305234	−0.152617	−	0.988285i	$-0.548770\pi$
$37$	−1.85666	−0.305234	−0.152617	+	0.988285i	$0.548770\pi$
$38$	0	0
$39$	−0.673041	−0.107773
$40$	0	0
$41$	8.59884	1.34291	0.671457	−	0.741044i	$-0.265669\pi$
$41$	8.59884	1.34291	0.671457	+	0.741044i	$0.265669\pi$
$42$	0	0
$43$	−7.75654	−1.18286	−0.591431	−	0.806356i	$-0.701437\pi$
$43$	−7.75654	−1.18286	−0.591431	+	0.806356i	$0.701437\pi$
$44$	0	0
$45$	2.97179	0.443008
$46$	0	0
$47$	12.0779	1.76175	0.880874	−	0.473352i	$-0.156956\pi$
$47$	12.0779	1.76175	0.880874	+	0.473352i	$0.156956\pi$
$48$	0	0
$49$	−1.66001	−0.237145
$50$	0	0
$51$	−0.721110	−0.100976
$52$	0	0
$53$	9.04827	1.24288	0.621438	−	0.783464i	$-0.286549\pi$
$53$	9.04827	1.24288	0.621438	+	0.783464i	$0.286549\pi$
$54$	0	0
$55$	−2.88795	−0.389412
$56$	0	0
$57$	0.0912739	0.0120895
$58$	0	0
$59$	3.28580	0.427775	0.213888	−	0.976858i	$-0.431387\pi$
$59$	3.28580	0.427775	0.213888	+	0.976858i	$0.431387\pi$
$60$	0	0
$61$	−2.38822	−0.305780	−0.152890	−	0.988243i	$-0.548858\pi$
$61$	−2.38822	−0.305780	−0.152890	+	0.988243i	$0.548858\pi$
$62$	0	0
$63$	−6.86733	−0.865202
$64$	0	0
$65$	−4.00711	−0.497020
$66$	0	0
$67$	12.2530	1.49694	0.748469	−	0.663169i	$-0.230789\pi$
$67$	12.2530	1.49694	0.748469	+	0.663169i	$0.230789\pi$
$68$	0	0
$69$	0.222292	0.0267608
$70$	0	0
$71$	2.38396	0.282924	0.141462	−	0.989944i	$-0.454820\pi$
$71$	2.38396	0.282924	0.141462	+	0.989944i	$0.454820\pi$
$72$	0	0
$73$	12.9385	1.51434	0.757169	−	0.653219i	$-0.226582\pi$
$73$	12.9385	1.51434	0.757169	+	0.653219i	$0.226582\pi$
$74$	0	0
$75$	−0.167962	−0.0193946
$76$	0	0
$77$	6.67360	0.760528
$78$	0	0
$79$	1.17724	0.132449	0.0662247	−	0.997805i	$-0.478905\pi$
$79$	1.17724	0.132449	0.0662247	+	0.997805i	$0.478905\pi$
$80$	0	0
$81$	8.74690	0.971877
$82$	0	0
$83$	−3.48118	−0.382109	−0.191055	−	0.981579i	$-0.561191\pi$
$83$	−3.48118	−0.382109	−0.191055	+	0.981579i	$0.561191\pi$
$84$	0	0
$85$	−4.29329	−0.465673
$86$	0	0
$87$	1.05174	0.112758
$88$	0	0
$89$	−16.1077	−1.70741	−0.853706	−	0.520755i	$-0.825651\pi$
$89$	−16.1077	−1.70741	−0.853706	+	0.520755i	$0.825651\pi$
$90$	0	0
$91$	9.25978	0.970689
$92$	0	0
$93$	−0.0763106	−0.00791305
$94$	0	0
$95$	0.543420	0.0557537
$96$	0	0
$97$	3.13575	0.318387	0.159194	−	0.987247i	$-0.449111\pi$
$97$	3.13575	0.318387	0.159194	+	0.987247i	$0.449111\pi$
$98$	0	0
$99$	−8.58239	−0.862563
$100$	0	0
$101$	−9.44651	−0.939962	−0.469981	−	0.882676i	$-0.655739\pi$
$101$	−9.44651	−0.939962	−0.469981	+	0.882676i	$0.655739\pi$
$102$	0	0
$103$	−3.45500	−0.340431	−0.170216	−	0.985407i	$-0.554446\pi$
$103$	−3.45500	−0.340431	−0.170216	+	0.985407i	$0.554446\pi$
$104$	0	0
$105$	0.388133	0.0378779
$106$	0	0
$107$	−14.7515	−1.42609	−0.713043	−	0.701120i	$-0.752684\pi$
$107$	−14.7515	−1.42609	−0.713043	+	0.701120i	$0.752684\pi$
$108$	0	0
$109$	−10.4354	−0.999534	−0.499767	−	0.866160i	$-0.666581\pi$
$109$	−10.4354	−0.999534	−0.499767	+	0.866160i	$0.666581\pi$
$110$	0	0
$111$	0.311849	0.0295994
$112$	0	0
$113$	−1.06724	−0.100397	−0.0501986	−	0.998739i	$-0.515985\pi$
$113$	−1.06724	−0.100397	−0.0501986	+	0.998739i	$0.515985\pi$
$114$	0	0
$115$	1.32347	0.123414
$116$	0	0
$117$	−11.9083	−1.10092
$118$	0	0
$119$	9.92112	0.909467
$120$	0	0
$121$	−2.65972	−0.241792
$122$	0	0
$123$	−1.44428	−0.130226
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.84622	0.430032	0.215016	−	0.976610i	$-0.431020\pi$
$127$	4.84622	0.430032	0.215016	+	0.976610i	$0.431020\pi$
$128$	0	0
$129$	1.30280	0.114705
$130$	0	0
$131$	−4.20761	−0.367621	−0.183810	−	0.982962i	$-0.558843\pi$
$131$	−4.20761	−0.367621	−0.183810	+	0.982962i	$0.558843\pi$
$132$	0	0
$133$	−1.25576	−0.108888
$134$	0	0
$135$	−1.00303	−0.0863273
$136$	0	0
$137$	−10.4180	−0.890069	−0.445034	−	0.895514i	$-0.646809\pi$
$137$	−10.4180	−0.890069	−0.445034	+	0.895514i	$0.646809\pi$
$138$	0	0
$139$	18.9487	1.60721	0.803603	−	0.595165i	$-0.202913\pi$
$139$	18.9487	1.60721	0.803603	+	0.595165i	$0.202913\pi$
$140$	0	0
$141$	−2.02863	−0.170842
$142$	0	0
$143$	11.5723	0.967728
$144$	0	0
$145$	6.26176	0.520011
$146$	0	0
$147$	0.278819	0.0229966
$148$	0	0
$149$	−15.9429	−1.30609	−0.653046	−	0.757319i	$-0.726509\pi$
$149$	−15.9429	−1.30609	−0.653046	+	0.757319i	$0.726509\pi$
$150$	0	0
$151$	2.61030	0.212423	0.106212	−	0.994344i	$-0.466128\pi$
$151$	2.61030	0.212423	0.106212	+	0.994344i	$0.466128\pi$
$152$	0	0
$153$	−12.7588	−1.03148
$154$	0	0
$155$	−0.454333	−0.0364929
$156$	0	0
$157$	16.6401	1.32802	0.664012	−	0.747722i	$-0.268853\pi$
$157$	16.6401	1.32802	0.664012	+	0.747722i	$0.268853\pi$
$158$	0	0
$159$	−1.51976	−0.120525
$160$	0	0
$161$	−3.05832	−0.241029
$162$	0	0
$163$	−19.6658	−1.54035	−0.770174	−	0.637834i	$-0.779831\pi$
$163$	−19.6658	−1.54035	−0.770174	+	0.637834i	$0.779831\pi$
$164$	0	0
$165$	0.485066	0.0377624
$166$	0	0
$167$	9.15274	0.708260	0.354130	−	0.935196i	$-0.384777\pi$
$167$	9.15274	0.708260	0.354130	+	0.935196i	$0.384777\pi$
$168$	0	0
$169$	3.05690	0.235146
$170$	0	0
$171$	1.61493	0.123497
$172$	0	0
$173$	23.5910	1.79359	0.896796	−	0.442445i	$-0.145889\pi$
$173$	23.5910	1.79359	0.896796	+	0.442445i	$0.145889\pi$
$174$	0	0
$175$	2.31084	0.174683
$176$	0	0
$177$	−0.551890	−0.0414826
$178$	0	0
$179$	5.99881	0.448372	0.224186	−	0.974546i	$-0.428028\pi$
$179$	5.99881	0.448372	0.224186	+	0.974546i	$0.428028\pi$
$180$	0	0
$181$	−1.27628	−0.0948648	−0.0474324	−	0.998874i	$-0.515104\pi$
$181$	−1.27628	−0.0948648	−0.0474324	+	0.998874i	$0.515104\pi$
$182$	0	0
$183$	0.401130	0.0296524
$184$	0	0
$185$	1.85666	0.136505
$186$	0	0
$187$	12.3988	0.906693
$188$	0	0
$189$	2.31785	0.168599
$190$	0	0
$191$	−21.5387	−1.55848	−0.779242	−	0.626723i	$-0.784396\pi$
$191$	−21.5387	−1.55848	−0.779242	+	0.626723i	$0.784396\pi$
$192$	0	0
$193$	−10.6259	−0.764871	−0.382436	−	0.923982i	$-0.624915\pi$
$193$	−10.6259	−0.764871	−0.382436	+	0.923982i	$0.624915\pi$
$194$	0	0
$195$	0.673041	0.0481975
$196$	0	0
$197$	20.5168	1.46176	0.730880	−	0.682506i	$-0.239110\pi$
$197$	20.5168	1.46176	0.730880	+	0.682506i	$0.239110\pi$
$198$	0	0
$199$	15.3366	1.08718	0.543591	−	0.839350i	$-0.317064\pi$
$199$	15.3366	1.08718	0.543591	+	0.839350i	$0.317064\pi$
$200$	0	0
$201$	−2.05803	−0.145162
$202$	0	0
$203$	−14.4699	−1.01559
$204$	0	0
$205$	−8.59884	−0.600569
$206$	0	0
$207$	3.93306	0.273366
$208$	0	0
$209$	−1.56937	−0.108556
$210$	0	0
$211$	9.25392	0.637066	0.318533	−	0.947912i	$-0.396810\pi$
$211$	9.25392	0.637066	0.318533	+	0.947912i	$0.396810\pi$
$212$	0	0
$213$	−0.400414	−0.0274359
$214$	0	0
$215$	7.75654	0.528992
$216$	0	0
$217$	1.04989	0.0712713
$218$	0	0
$219$	−2.17318	−0.146850
$220$	0	0
$221$	17.2037	1.15724
$222$	0	0
$223$	1.26478	0.0846957	0.0423478	−	0.999103i	$-0.486516\pi$
$223$	1.26478	0.0846957	0.0423478	+	0.999103i	$0.486516\pi$
$224$	0	0
$225$	−2.97179	−0.198119
$226$	0	0
$227$	18.0327	1.19687	0.598436	−	0.801170i	$-0.295789\pi$
$227$	18.0327	1.19687	0.598436	+	0.801170i	$0.295789\pi$
$228$	0	0
$229$	−19.9680	−1.31952	−0.659760	−	0.751476i	$-0.729342\pi$
$229$	−19.9680	−1.31952	−0.659760	+	0.751476i	$0.729342\pi$
$230$	0	0
$231$	−1.12091	−0.0737506
$232$	0	0
$233$	2.17750	0.142653	0.0713264	−	0.997453i	$-0.477277\pi$
$233$	2.17750	0.142653	0.0713264	+	0.997453i	$0.477277\pi$
$234$	0	0
$235$	−12.0779	−0.787877
$236$	0	0
$237$	−0.197731	−0.0128440
$238$	0	0
$239$	5.44743	0.352365	0.176183	−	0.984358i	$-0.443625\pi$
$239$	5.44743	0.352365	0.176183	+	0.984358i	$0.443625\pi$
$240$	0	0
$241$	17.2721	1.11259	0.556297	−	0.830984i	$-0.312222\pi$
$241$	17.2721	1.11259	0.556297	+	0.830984i	$0.312222\pi$
$242$	0	0
$243$	−4.47824	−0.287279
$244$	0	0
$245$	1.66001	0.106054
$246$	0	0
$247$	−2.17754	−0.138554
$248$	0	0
$249$	0.584706	0.0370542
$250$	0	0
$251$	2.24376	0.141625	0.0708124	−	0.997490i	$-0.477441\pi$
$251$	2.24376	0.141625	0.0708124	+	0.997490i	$0.477441\pi$
$252$	0	0
$253$	−3.82211	−0.240294
$254$	0	0
$255$	0.721110	0.0451576
$256$	0	0
$257$	8.13076	0.507183	0.253591	−	0.967311i	$-0.418388\pi$
$257$	8.13076	0.507183	0.253591	+	0.967311i	$0.418388\pi$
$258$	0	0
$259$	−4.29045	−0.266596
$260$	0	0
$261$	18.6086	1.15185
$262$	0	0
$263$	24.6796	1.52181	0.760904	−	0.648864i	$-0.224756\pi$
$263$	24.6796	1.52181	0.760904	+	0.648864i	$0.224756\pi$
$264$	0	0
$265$	−9.04827	−0.555831
$266$	0	0
$267$	2.70548	0.165573
$268$	0	0
$269$	28.3502	1.72854	0.864271	−	0.503027i	$-0.167780\pi$
$269$	28.3502	1.72854	0.864271	+	0.503027i	$0.167780\pi$
$270$	0	0
$271$	−7.86882	−0.477997	−0.238998	−	0.971020i	$-0.576819\pi$
$271$	−7.86882	−0.477997	−0.238998	+	0.971020i	$0.576819\pi$
$272$	0	0
$273$	−1.55529	−0.0941305
$274$	0	0
$275$	2.88795	0.174150
$276$	0	0
$277$	−12.8532	−0.772271	−0.386136	−	0.922442i	$-0.626190\pi$
$277$	−12.8532	−0.772271	−0.386136	+	0.922442i	$0.626190\pi$
$278$	0	0
$279$	−1.35018	−0.0808333
$280$	0	0
$281$	25.0601	1.49496	0.747479	−	0.664286i	$-0.231264\pi$
$281$	25.0601	1.49496	0.747479	+	0.664286i	$0.231264\pi$
$282$	0	0
$283$	29.3935	1.74726	0.873631	−	0.486589i	$-0.161759\pi$
$283$	29.3935	1.74726	0.873631	+	0.486589i	$0.161759\pi$
$284$	0	0
$285$	−0.0912739	−0.00540660
$286$	0	0
$287$	19.8706	1.17292
$288$	0	0
$289$	1.43237	0.0842569
$290$	0	0
$291$	−0.526687	−0.0308749
$292$	0	0
$293$	−5.38807	−0.314774	−0.157387	−	0.987537i	$-0.550307\pi$
$293$	−5.38807	−0.314774	−0.157387	+	0.987537i	$0.550307\pi$
$294$	0	0
$295$	−3.28580	−0.191307
$296$	0	0
$297$	2.89671	0.168084
$298$	0	0
$299$	−5.30327	−0.306696
$300$	0	0
$301$	−17.9241	−1.03313
$302$	0	0
$303$	1.58665	0.0911508
$304$	0	0
$305$	2.38822	0.136749
$306$	0	0
$307$	2.34557	0.133869	0.0669345	−	0.997757i	$-0.478678\pi$
$307$	2.34557	0.133869	0.0669345	+	0.997757i	$0.478678\pi$
$308$	0	0
$309$	0.580308	0.0330126
$310$	0	0
$311$	20.8479	1.18218	0.591089	−	0.806607i	$-0.298698\pi$
$311$	20.8479	1.18218	0.591089	+	0.806607i	$0.298698\pi$
$312$	0	0
$313$	21.0363	1.18904	0.594522	−	0.804079i	$-0.297341\pi$
$313$	21.0363	1.18904	0.594522	+	0.804079i	$0.297341\pi$
$314$	0	0
$315$	6.86733	0.386930
$316$	0	0
$317$	19.6536	1.10386	0.551928	−	0.833891i	$-0.313892\pi$
$317$	19.6536	1.10386	0.551928	+	0.833891i	$0.313892\pi$
$318$	0	0
$319$	−18.0837	−1.01249
$320$	0	0
$321$	2.47770	0.138292
$322$	0	0
$323$	−2.33306	−0.129815
$324$	0	0
$325$	4.00711	0.222274
$326$	0	0
$327$	1.75276	0.0969277
$328$	0	0
$329$	27.9102	1.53874
$330$	0	0
$331$	20.5821	1.13130	0.565649	−	0.824646i	$-0.308626\pi$
$331$	20.5821	1.13130	0.565649	+	0.824646i	$0.308626\pi$
$332$	0	0
$333$	5.51761	0.302363
$334$	0	0
$335$	−12.2530	−0.669451
$336$	0	0
$337$	18.7490	1.02132	0.510662	−	0.859781i	$-0.329400\pi$
$337$	18.7490	1.02132	0.510662	+	0.859781i	$0.329400\pi$
$338$	0	0
$339$	0.179255	0.00973579
$340$	0	0
$341$	1.31209	0.0710538
$342$	0	0
$343$	−20.0119	−1.08054
$344$	0	0
$345$	−0.222292	−0.0119678
$346$	0	0
$347$	−5.47946	−0.294153	−0.147076	−	0.989125i	$-0.546986\pi$
$347$	−5.47946	−0.294153	−0.147076	+	0.989125i	$0.546986\pi$
$348$	0	0
$349$	−12.5046	−0.669355	−0.334678	−	0.942333i	$-0.608627\pi$
$349$	−12.5046	−0.669355	−0.334678	+	0.942333i	$0.608627\pi$
$350$	0	0
$351$	4.01926	0.214532
$352$	0	0
$353$	11.8973	0.633228	0.316614	−	0.948555i	$-0.397454\pi$
$353$	11.8973	0.633228	0.316614	+	0.948555i	$0.397454\pi$
$354$	0	0
$355$	−2.38396	−0.126527
$356$	0	0
$357$	−1.66637	−0.0881936
$358$	0	0
$359$	−11.0091	−0.581038	−0.290519	−	0.956869i	$-0.593828\pi$
$359$	−11.0091	−0.581038	−0.290519	+	0.956869i	$0.593828\pi$
$360$	0	0
$361$	−18.7047	−0.984458
$362$	0	0
$363$	0.446731	0.0234473
$364$	0	0
$365$	−12.9385	−0.677233
$366$	0	0
$367$	−5.88379	−0.307131	−0.153566	−	0.988138i	$-0.549076\pi$
$367$	−5.88379	−0.307131	−0.153566	+	0.988138i	$0.549076\pi$
$368$	0	0
$369$	−25.5539	−1.33029
$370$	0	0
$371$	20.9091	1.08555
$372$	0	0
$373$	6.01102	0.311239	0.155619	−	0.987817i	$-0.450263\pi$
$373$	6.01102	0.311239	0.155619	+	0.987817i	$0.450263\pi$
$374$	0	0
$375$	0.167962	0.00867352
$376$	0	0
$377$	−25.0916	−1.29228
$378$	0	0
$379$	−0.784250	−0.0402842	−0.0201421	−	0.999797i	$-0.506412\pi$
$379$	−0.784250	−0.0402842	−0.0201421	+	0.999797i	$0.506412\pi$
$380$	0	0
$381$	−0.813980	−0.0417015
$382$	0	0
$383$	32.5353	1.66248	0.831239	−	0.555915i	$-0.187632\pi$
$383$	32.5353	1.66248	0.831239	+	0.555915i	$0.187632\pi$
$384$	0	0
$385$	−6.67360	−0.340118
$386$	0	0
$387$	23.0508	1.17174
$388$	0	0
$389$	−9.01777	−0.457219	−0.228610	−	0.973518i	$-0.573418\pi$
$389$	−9.01777	−0.457219	−0.228610	+	0.973518i	$0.573418\pi$
$390$	0	0
$391$	−5.68202	−0.287352
$392$	0	0
$393$	0.706719	0.0356492
$394$	0	0
$395$	−1.17724	−0.0592332
$396$	0	0
$397$	−38.1716	−1.91578	−0.957889	−	0.287138i	$-0.907296\pi$
$397$	−38.1716	−1.91578	−0.957889	+	0.287138i	$0.907296\pi$
$398$	0	0
$399$	0.210919	0.0105592
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	1.82056	0.0906886
$404$	0	0
$405$	−8.74690	−0.434637
$406$	0	0
$407$	−5.36196	−0.265783
$408$	0	0
$409$	1.60227	0.0792270	0.0396135	−	0.999215i	$-0.487387\pi$
$409$	1.60227	0.0792270	0.0396135	+	0.999215i	$0.487387\pi$
$410$	0	0
$411$	1.74982	0.0863125
$412$	0	0
$413$	7.59297	0.373626
$414$	0	0
$415$	3.48118	0.170884
$416$	0	0
$417$	−3.18266	−0.155855
$418$	0	0
$419$	33.5190	1.63751	0.818755	−	0.574143i	$-0.194664\pi$
$419$	33.5190	1.63751	0.818755	+	0.574143i	$0.194664\pi$
$420$	0	0
$421$	11.5038	0.560661	0.280330	−	0.959904i	$-0.409556\pi$
$421$	11.5038	0.560661	0.280330	+	0.959904i	$0.409556\pi$
$422$	0	0
$423$	−35.8931	−1.74518
$424$	0	0
$425$	4.29329	0.208255
$426$	0	0
$427$	−5.51880	−0.267073
$428$	0	0
$429$	−1.94371	−0.0938433
$430$	0	0
$431$	−24.5930	−1.18460	−0.592302	−	0.805716i	$-0.701781\pi$
$431$	−24.5930	−1.18460	−0.592302	+	0.805716i	$0.701781\pi$
$432$	0	0
$433$	−2.28160	−0.109647	−0.0548233	−	0.998496i	$-0.517460\pi$
$433$	−2.28160	−0.109647	−0.0548233	+	0.998496i	$0.517460\pi$
$434$	0	0
$435$	−1.05174	−0.0504270
$436$	0	0
$437$	0.719198	0.0344039
$438$	0	0
$439$	6.09489	0.290893	0.145447	−	0.989366i	$-0.453538\pi$
$439$	6.09489	0.290893	0.145447	+	0.989366i	$0.453538\pi$
$440$	0	0
$441$	4.93321	0.234915
$442$	0	0
$443$	−10.7843	−0.512376	−0.256188	−	0.966627i	$-0.582467\pi$
$443$	−10.7843	−0.512376	−0.256188	+	0.966627i	$0.582467\pi$
$444$	0	0
$445$	16.1077	0.763578
$446$	0	0
$447$	2.67780	0.126655
$448$	0	0
$449$	36.5705	1.72587	0.862933	−	0.505318i	$-0.168625\pi$
$449$	36.5705	1.72587	0.862933	+	0.505318i	$0.168625\pi$
$450$	0	0
$451$	24.8331	1.16934
$452$	0	0
$453$	−0.438431	−0.0205993
$454$	0	0
$455$	−9.25978	−0.434105
$456$	0	0
$457$	28.7005	1.34255	0.671276	−	0.741208i	$-0.265747\pi$
$457$	28.7005	1.34255	0.671276	+	0.741208i	$0.265747\pi$
$458$	0	0
$459$	4.30631	0.201002
$460$	0	0
$461$	20.9481	0.975651	0.487826	−	0.872941i	$-0.337790\pi$
$461$	20.9481	0.975651	0.487826	+	0.872941i	$0.337790\pi$
$462$	0	0
$463$	−13.4692	−0.625966	−0.312983	−	0.949759i	$-0.601328\pi$
$463$	−13.4692	−0.625966	−0.312983	+	0.949759i	$0.601328\pi$
$464$	0	0
$465$	0.0763106	0.00353882
$466$	0	0
$467$	−34.8944	−1.61472	−0.807359	−	0.590060i	$-0.799104\pi$
$467$	−34.8944	−1.61472	−0.807359	+	0.590060i	$0.799104\pi$
$468$	0	0
$469$	28.3147	1.30745
$470$	0	0
$471$	−2.79490	−0.128782
$472$	0	0
$473$	−22.4005	−1.02998
$474$	0	0
$475$	−0.543420	−0.0249338
$476$	0	0
$477$	−26.8895	−1.23119
$478$	0	0
$479$	37.2537	1.70217	0.851083	−	0.525031i	$-0.175946\pi$
$479$	37.2537	1.70217	0.851083	+	0.525031i	$0.175946\pi$
$480$	0	0
$481$	−7.43985	−0.339228
$482$	0	0
$483$	0.513681	0.0233733
$484$	0	0
$485$	−3.13575	−0.142387
$486$	0	0
$487$	10.0541	0.455593	0.227796	−	0.973709i	$-0.426848\pi$
$487$	10.0541	0.455593	0.227796	+	0.973709i	$0.426848\pi$
$488$	0	0
$489$	3.30311	0.149372
$490$	0	0
$491$	14.0750	0.635197	0.317599	−	0.948225i	$-0.397124\pi$
$491$	14.0750	0.635197	0.317599	+	0.948225i	$0.397124\pi$
$492$	0	0
$493$	−26.8836	−1.21078
$494$	0	0
$495$	8.58239	0.385750
$496$	0	0
$497$	5.50894	0.247110
$498$	0	0
$499$	−32.8499	−1.47056	−0.735281	−	0.677762i	$-0.762950\pi$
$499$	−32.8499	−1.47056	−0.735281	+	0.677762i	$0.762950\pi$
$500$	0	0
$501$	−1.53731	−0.0686820
$502$	0	0
$503$	12.8816	0.574363	0.287181	−	0.957876i	$-0.407282\pi$
$503$	12.8816	0.574363	0.287181	+	0.957876i	$0.407282\pi$
$504$	0	0
$505$	9.44651	0.420364
$506$	0	0
$507$	−0.513443	−0.0228028
$508$	0	0
$509$	22.0378	0.976808	0.488404	−	0.872618i	$-0.337579\pi$
$509$	22.0378	0.976808	0.488404	+	0.872618i	$0.337579\pi$
$510$	0	0
$511$	29.8988	1.32265
$512$	0	0
$513$	−0.545068	−0.0240654
$514$	0	0
$515$	3.45500	0.152245
$516$	0	0
$517$	34.8805	1.53404
$518$	0	0
$519$	−3.96239	−0.173930
$520$	0	0
$521$	14.8989	0.652732	0.326366	−	0.945243i	$-0.394176\pi$
$521$	14.8989	0.652732	0.326366	+	0.945243i	$0.394176\pi$
$522$	0	0
$523$	−29.6264	−1.29547	−0.647736	−	0.761865i	$-0.724284\pi$
$523$	−29.6264	−1.29547	−0.647736	+	0.761865i	$0.724284\pi$
$524$	0	0
$525$	−0.388133	−0.0169395
$526$	0	0
$527$	1.95059	0.0849688
$528$	0	0
$529$	−21.2484	−0.923845
$530$	0	0
$531$	−9.76471	−0.423752
$532$	0	0
$533$	34.4565	1.49248
$534$	0	0
$535$	14.7515	0.637765
$536$	0	0
$537$	−1.00757	−0.0434799
$538$	0	0
$539$	−4.79405	−0.206494
$540$	0	0
$541$	−0.145378	−0.00625027	−0.00312514	−	0.999995i	$-0.500995\pi$
$541$	−0.145378	−0.00625027	−0.00312514	+	0.999995i	$0.500995\pi$
$542$	0	0
$543$	0.214366	0.00919931
$544$	0	0
$545$	10.4354	0.447005
$546$	0	0
$547$	−13.4593	−0.575477	−0.287739	−	0.957709i	$-0.592903\pi$
$547$	−13.4593	−0.575477	−0.287739	+	0.957709i	$0.592903\pi$
$548$	0	0
$549$	7.09729	0.302905
$550$	0	0
$551$	3.40277	0.144963
$552$	0	0
$553$	2.72040	0.115683
$554$	0	0
$555$	−0.311849	−0.0132372
$556$	0	0
$557$	21.0901	0.893616	0.446808	−	0.894630i	$-0.352561\pi$
$557$	21.0901	0.893616	0.446808	+	0.894630i	$0.352561\pi$
$558$	0	0
$559$	−31.0813	−1.31460
$560$	0	0
$561$	−2.08253	−0.0879246
$562$	0	0
$563$	−39.0266	−1.64477	−0.822387	−	0.568929i	$-0.807358\pi$
$563$	−39.0266	−1.64477	−0.822387	+	0.568929i	$0.807358\pi$
$564$	0	0
$565$	1.06724	0.0448990
$566$	0	0
$567$	20.2127	0.848853
$568$	0	0
$569$	−24.0462	−1.00807	−0.504034	−	0.863684i	$-0.668151\pi$
$569$	−24.0462	−1.00807	−0.504034	+	0.863684i	$0.668151\pi$
$570$	0	0
$571$	−18.1578	−0.759881	−0.379941	−	0.925011i	$-0.624056\pi$
$571$	−18.1578	−0.759881	−0.379941	+	0.925011i	$0.624056\pi$
$572$	0	0
$573$	3.61768	0.151131
$574$	0	0
$575$	−1.32347	−0.0551923
$576$	0	0
$577$	−6.66340	−0.277401	−0.138701	−	0.990334i	$-0.544293\pi$
$577$	−6.66340	−0.277401	−0.138701	+	0.990334i	$0.544293\pi$
$578$	0	0
$579$	1.78475	0.0741717
$580$	0	0
$581$	−8.04446	−0.333740
$582$	0	0
$583$	26.1310	1.08224
$584$	0	0
$585$	11.9083	0.492346
$586$	0	0
$587$	27.1626	1.12112	0.560561	−	0.828113i	$-0.310586\pi$
$587$	27.1626	1.12112	0.560561	+	0.828113i	$0.310586\pi$
$588$	0	0
$589$	−0.246894	−0.0101731
$590$	0	0
$591$	−3.44604	−0.141751
$592$	0	0
$593$	−14.0839	−0.578356	−0.289178	−	0.957275i	$-0.593382\pi$
$593$	−14.0839	−0.578356	−0.289178	+	0.957275i	$0.593382\pi$
$594$	0	0
$595$	−9.92112	−0.406726
$596$	0	0
$597$	−2.57596	−0.105427
$598$	0	0
$599$	27.7143	1.13237	0.566187	−	0.824277i	$-0.308418\pi$
$599$	27.7143	1.13237	0.566187	+	0.824277i	$0.308418\pi$
$600$	0	0
$601$	15.2083	0.620358	0.310179	−	0.950678i	$-0.399611\pi$
$601$	15.2083	0.620358	0.310179	+	0.950678i	$0.399611\pi$
$602$	0	0
$603$	−36.4132	−1.48286
$604$	0	0
$605$	2.65972	0.108133
$606$	0	0
$607$	−41.5541	−1.68663	−0.843315	−	0.537419i	$-0.819399\pi$
$607$	−41.5541	−1.68663	−0.843315	+	0.537419i	$0.819399\pi$
$608$	0	0
$609$	2.43040	0.0984847
$610$	0	0
$611$	48.3975	1.95796
$612$	0	0
$613$	−4.64646	−0.187669	−0.0938343	−	0.995588i	$-0.529912\pi$
$613$	−4.64646	−0.187669	−0.0938343	+	0.995588i	$0.529912\pi$
$614$	0	0
$615$	1.44428	0.0582389
$616$	0	0
$617$	6.72209	0.270621	0.135311	−	0.990803i	$-0.456797\pi$
$617$	6.72209	0.270621	0.135311	+	0.990803i	$0.456797\pi$
$618$	0	0
$619$	−5.23891	−0.210570	−0.105285	−	0.994442i	$-0.533575\pi$
$619$	−5.23891	−0.210570	−0.105285	+	0.994442i	$0.533575\pi$
$620$	0	0
$621$	−1.32748	−0.0532699
$622$	0	0
$623$	−37.2223	−1.49128
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.263595	0.0105270
$628$	0	0
$629$	−7.97120	−0.317833
$630$	0	0
$631$	47.1669	1.87768	0.938842	−	0.344347i	$-0.111900\pi$
$631$	47.1669	1.87768	0.938842	+	0.344347i	$0.111900\pi$
$632$	0	0
$633$	−1.55431	−0.0617781
$634$	0	0
$635$	−4.84622	−0.192316
$636$	0	0
$637$	−6.65185	−0.263556
$638$	0	0
$639$	−7.08462	−0.280263
$640$	0	0
$641$	−29.4110	−1.16166	−0.580832	−	0.814024i	$-0.697273\pi$
$641$	−29.4110	−1.16166	−0.580832	+	0.814024i	$0.697273\pi$
$642$	0	0
$643$	30.4265	1.19991	0.599953	−	0.800035i	$-0.295186\pi$
$643$	30.4265	1.19991	0.599953	+	0.800035i	$0.295186\pi$
$644$	0	0
$645$	−1.30280	−0.0512978
$646$	0	0
$647$	−40.3503	−1.58633	−0.793167	−	0.609005i	$-0.791569\pi$
$647$	−40.3503	−1.58633	−0.793167	+	0.609005i	$0.791569\pi$
$648$	0	0
$649$	9.48925	0.372486
$650$	0	0
$651$	−0.176342	−0.00691138
$652$	0	0
$653$	−7.13103	−0.279059	−0.139529	−	0.990218i	$-0.544559\pi$
$653$	−7.13103	−0.279059	−0.139529	+	0.990218i	$0.544559\pi$
$654$	0	0
$655$	4.20761	0.164405
$656$	0	0
$657$	−38.4505	−1.50010
$658$	0	0
$659$	−1.28210	−0.0499436	−0.0249718	−	0.999688i	$-0.507950\pi$
$659$	−1.28210	−0.0499436	−0.0249718	+	0.999688i	$0.507950\pi$
$660$	0	0
$661$	26.1114	1.01562	0.507809	−	0.861470i	$-0.330456\pi$
$661$	26.1114	1.01562	0.507809	+	0.861470i	$0.330456\pi$
$662$	0	0
$663$	−2.88956	−0.112221
$664$	0	0
$665$	1.25576	0.0486962
$666$	0	0
$667$	8.28723	0.320883
$668$	0	0
$669$	−0.212434	−0.00821318
$670$	0	0
$671$	−6.89707	−0.266258
$672$	0	0
$673$	10.2642	0.395657	0.197828	−	0.980237i	$-0.436611\pi$
$673$	10.2642	0.395657	0.197828	+	0.980237i	$0.436611\pi$
$674$	0	0
$675$	1.00303	0.0386068
$676$	0	0
$677$	22.8310	0.877467	0.438734	−	0.898617i	$-0.355427\pi$
$677$	22.8310	0.877467	0.438734	+	0.898617i	$0.355427\pi$
$678$	0	0
$679$	7.24622	0.278084
$680$	0	0
$681$	−3.02881	−0.116064
$682$	0	0
$683$	2.18251	0.0835116	0.0417558	−	0.999128i	$-0.486705\pi$
$683$	2.18251	0.0835116	0.0417558	+	0.999128i	$0.486705\pi$
$684$	0	0
$685$	10.4180	0.398051
$686$	0	0
$687$	3.35386	0.127958
$688$	0	0
$689$	36.2574	1.38130
$690$	0	0
$691$	38.6866	1.47171	0.735854	−	0.677140i	$-0.236781\pi$
$691$	38.6866	1.47171	0.735854	+	0.677140i	$0.236781\pi$
$692$	0	0
$693$	−19.8325	−0.753376
$694$	0	0
$695$	−18.9487	−0.718765
$696$	0	0
$697$	36.9173	1.39834
$698$	0	0
$699$	−0.365737	−0.0138334
$700$	0	0
$701$	−11.0834	−0.418613	−0.209307	−	0.977850i	$-0.567121\pi$
$701$	−11.0834	−0.418613	−0.209307	+	0.977850i	$0.567121\pi$
$702$	0	0
$703$	1.00895	0.0380532
$704$	0	0
$705$	2.02863	0.0764027
$706$	0	0
$707$	−21.8294	−0.820978
$708$	0	0
$709$	−32.0549	−1.20385	−0.601924	−	0.798554i	$-0.705599\pi$
$709$	−32.0549	−1.20385	−0.601924	+	0.798554i	$0.705599\pi$
$710$	0	0
$711$	−3.49850	−0.131204
$712$	0	0
$713$	−0.601294	−0.0225186
$714$	0	0
$715$	−11.5723	−0.432781
$716$	0	0
$717$	−0.914961	−0.0341699
$718$	0	0
$719$	5.42238	0.202221	0.101110	−	0.994875i	$-0.467760\pi$
$719$	5.42238	0.202221	0.101110	+	0.994875i	$0.467760\pi$
$720$	0	0
$721$	−7.98395	−0.297338
$722$	0	0
$723$	−2.90105	−0.107891
$724$	0	0
$725$	−6.26176	−0.232556
$726$	0	0
$727$	−33.6332	−1.24739	−0.623694	−	0.781669i	$-0.714369\pi$
$727$	−33.6332	−1.24739	−0.623694	+	0.781669i	$0.714369\pi$
$728$	0	0
$729$	−25.4885	−0.944019
$730$	0	0
$731$	−33.3011	−1.23169
$732$	0	0
$733$	−3.78868	−0.139938	−0.0699690	−	0.997549i	$-0.522290\pi$
$733$	−3.78868	−0.139938	−0.0699690	+	0.997549i	$0.522290\pi$
$734$	0	0
$735$	−0.278819	−0.0102844
$736$	0	0
$737$	35.3860	1.30346
$738$	0	0
$739$	−16.1364	−0.593588	−0.296794	−	0.954941i	$-0.595918\pi$
$739$	−16.1364	−0.593588	−0.296794	+	0.954941i	$0.595918\pi$
$740$	0	0
$741$	0.365744	0.0134359
$742$	0	0
$743$	12.7626	0.468213	0.234107	−	0.972211i	$-0.424783\pi$
$743$	12.7626	0.468213	0.234107	+	0.972211i	$0.424783\pi$
$744$	0	0
$745$	15.9429	0.584102
$746$	0	0
$747$	10.3453	0.378516
$748$	0	0
$749$	−34.0885	−1.24557
$750$	0	0
$751$	−33.1559	−1.20988	−0.604938	−	0.796273i	$-0.706802\pi$
$751$	−33.1559	−1.20988	−0.604938	+	0.796273i	$0.706802\pi$
$752$	0	0
$753$	−0.376866	−0.0137338
$754$	0	0
$755$	−2.61030	−0.0949986
$756$	0	0
$757$	15.8089	0.574584	0.287292	−	0.957843i	$-0.407245\pi$
$757$	15.8089	0.574584	0.287292	+	0.957843i	$0.407245\pi$
$758$	0	0
$759$	0.641968	0.0233020
$760$	0	0
$761$	14.7026	0.532971	0.266485	−	0.963839i	$-0.414138\pi$
$761$	14.7026	0.532971	0.266485	+	0.963839i	$0.414138\pi$
$762$	0	0
$763$	−24.1147	−0.873009
$764$	0	0
$765$	12.7588	0.461294
$766$	0	0
$767$	13.1666	0.475417
$768$	0	0
$769$	47.4361	1.71059	0.855295	−	0.518141i	$-0.173376\pi$
$769$	47.4361	1.71059	0.855295	+	0.518141i	$0.173376\pi$
$770$	0	0
$771$	−1.36566	−0.0491829
$772$	0	0
$773$	3.72818	0.134093	0.0670466	−	0.997750i	$-0.478642\pi$
$773$	3.72818	0.134093	0.0670466	+	0.997750i	$0.478642\pi$
$774$	0	0
$775$	0.454333	0.0163201
$776$	0	0
$777$	0.720633	0.0258526
$778$	0	0
$779$	−4.67278	−0.167420
$780$	0	0
$781$	6.88476	0.246356
$782$	0	0
$783$	−6.28076	−0.224456
$784$	0	0
$785$	−16.6401	−0.593910
$786$	0	0
$787$	11.0447	0.393703	0.196851	−	0.980433i	$-0.436928\pi$
$787$	11.0447	0.393703	0.196851	+	0.980433i	$0.436928\pi$
$788$	0	0
$789$	−4.14523	−0.147574
$790$	0	0
$791$	−2.46621	−0.0876884
$792$	0	0
$793$	−9.56985	−0.339835
$794$	0	0
$795$	1.51976	0.0539005
$796$	0	0
$797$	43.9951	1.55839	0.779193	−	0.626784i	$-0.215629\pi$
$797$	43.9951	1.55839	0.779193	+	0.626784i	$0.215629\pi$
$798$	0	0
$799$	51.8541	1.83447
$800$	0	0
$801$	47.8687	1.69136
$802$	0	0
$803$	37.3658	1.31861
$804$	0	0
$805$	3.05832	0.107792
$806$	0	0
$807$	−4.76175	−0.167622
$808$	0	0
$809$	43.2065	1.51906	0.759529	−	0.650473i	$-0.225429\pi$
$809$	43.2065	1.51906	0.759529	+	0.650473i	$0.225429\pi$
$810$	0	0
$811$	7.34992	0.258091	0.129045	−	0.991639i	$-0.458809\pi$
$811$	7.34992	0.258091	0.129045	+	0.991639i	$0.458809\pi$
$812$	0	0
$813$	1.32166	0.0463527
$814$	0	0
$815$	19.6658	0.688864
$816$	0	0
$817$	4.21506	0.147466
$818$	0	0
$819$	−27.5181	−0.961561
$820$	0	0
$821$	10.5644	0.368701	0.184350	−	0.982861i	$-0.440982\pi$
$821$	10.5644	0.368701	0.184350	+	0.982861i	$0.440982\pi$
$822$	0	0
$823$	−11.8580	−0.413342	−0.206671	−	0.978410i	$-0.566263\pi$
$823$	−11.8580	−0.413342	−0.206671	+	0.978410i	$0.566263\pi$
$824$	0	0
$825$	−0.485066	−0.0168878
$826$	0	0
$827$	−5.74273	−0.199694	−0.0998471	−	0.995003i	$-0.531835\pi$
$827$	−5.74273	−0.199694	−0.0998471	+	0.995003i	$0.531835\pi$
$828$	0	0
$829$	−1.25331	−0.0435291	−0.0217646	−	0.999763i	$-0.506928\pi$
$829$	−1.25331	−0.0435291	−0.0217646	+	0.999763i	$0.506928\pi$
$830$	0	0
$831$	2.15884	0.0748894
$832$	0	0
$833$	−7.12693	−0.246933
$834$	0	0
$835$	−9.15274	−0.316744
$836$	0	0
$837$	0.455711	0.0157517
$838$	0	0
$839$	36.3941	1.25646	0.628231	−	0.778027i	$-0.283779\pi$
$839$	36.3941	1.25646	0.628231	+	0.778027i	$0.283779\pi$
$840$	0	0
$841$	10.2097	0.352059
$842$	0	0
$843$	−4.20913	−0.144970
$844$	0	0
$845$	−3.05690	−0.105161
$846$	0	0
$847$	−6.14618	−0.211185
$848$	0	0
$849$	−4.93699	−0.169437
$850$	0	0
$851$	2.45723	0.0842328
$852$	0	0
$853$	27.1617	0.930000	0.465000	−	0.885311i	$-0.346054\pi$
$853$	27.1617	0.930000	0.465000	+	0.885311i	$0.346054\pi$
$854$	0	0
$855$	−1.61493	−0.0552294
$856$	0	0
$857$	−25.2828	−0.863645	−0.431822	−	0.901959i	$-0.642129\pi$
$857$	−25.2828	−0.863645	−0.431822	+	0.901959i	$0.642129\pi$
$858$	0	0
$859$	−36.0130	−1.22875	−0.614373	−	0.789016i	$-0.710591\pi$
$859$	−36.0130	−1.22875	−0.614373	+	0.789016i	$0.710591\pi$
$860$	0	0
$861$	−3.33750	−0.113742
$862$	0	0
$863$	−23.7979	−0.810090	−0.405045	−	0.914297i	$-0.632744\pi$
$863$	−23.7979	−0.810090	−0.405045	+	0.914297i	$0.632744\pi$
$864$	0	0
$865$	−23.5910	−0.802118
$866$	0	0
$867$	−0.240583	−0.00817063
$868$	0	0
$869$	3.39980	0.115330
$870$	0	0
$871$	49.0990	1.66365
$872$	0	0
$873$	−9.31879	−0.315393
$874$	0	0
$875$	−2.31084	−0.0781207
$876$	0	0
$877$	10.1441	0.342542	0.171271	−	0.985224i	$-0.445213\pi$
$877$	10.1441	0.342542	0.171271	+	0.985224i	$0.445213\pi$
$878$	0	0
$879$	0.904990	0.0305245
$880$	0	0
$881$	−38.4315	−1.29479	−0.647395	−	0.762154i	$-0.724142\pi$
$881$	−38.4315	−1.29479	−0.647395	+	0.762154i	$0.724142\pi$
$882$	0	0
$883$	−0.307426	−0.0103457	−0.00517285	−	0.999987i	$-0.501647\pi$
$883$	−0.307426	−0.0103457	−0.00517285	+	0.999987i	$0.501647\pi$
$884$	0	0
$885$	0.551890	0.0185516
$886$	0	0
$887$	−52.7316	−1.77055	−0.885277	−	0.465064i	$-0.846031\pi$
$887$	−52.7316	−1.77055	−0.885277	+	0.465064i	$0.846031\pi$
$888$	0	0
$889$	11.1988	0.375597
$890$	0	0
$891$	25.2606	0.846263
$892$	0	0
$893$	−6.56339	−0.219635
$894$	0	0
$895$	−5.99881	−0.200518
$896$	0	0
$897$	0.890746	0.0297412
$898$	0	0
$899$	−2.84493	−0.0948836
$900$	0	0
$901$	38.8469	1.29418
$902$	0	0
$903$	3.01057	0.100186
$904$	0	0
$905$	1.27628	0.0424248
$906$	0	0
$907$	3.47258	0.115305	0.0576526	−	0.998337i	$-0.481638\pi$
$907$	3.47258	0.115305	0.0576526	+	0.998337i	$0.481638\pi$
$908$	0	0
$909$	28.0730	0.931123
$910$	0	0
$911$	−50.7404	−1.68110	−0.840552	−	0.541731i	$-0.817769\pi$
$911$	−50.7404	−1.68110	−0.840552	+	0.541731i	$0.817769\pi$
$912$	0	0
$913$	−10.0535	−0.332722
$914$	0	0
$915$	−0.401130	−0.0132609
$916$	0	0
$917$	−9.72312	−0.321086
$918$	0	0
$919$	−24.8436	−0.819513	−0.409757	−	0.912195i	$-0.634386\pi$
$919$	−24.8436	−0.819513	−0.409757	+	0.912195i	$0.634386\pi$
$920$	0	0
$921$	−0.393967	−0.0129817
$922$	0	0
$923$	9.55277	0.314433
$924$	0	0
$925$	−1.85666	−0.0610467
$926$	0	0
$927$	10.2675	0.337230
$928$	0	0
$929$	39.5970	1.29913	0.649567	−	0.760304i	$-0.274950\pi$
$929$	39.5970	1.29913	0.649567	+	0.760304i	$0.274950\pi$
$930$	0	0
$931$	0.902085	0.0295646
$932$	0	0
$933$	−3.50166	−0.114639
$934$	0	0
$935$	−12.3988	−0.405485
$936$	0	0
$937$	−31.5826	−1.03176	−0.515879	−	0.856662i	$-0.672535\pi$
$937$	−31.5826	−1.03176	−0.515879	+	0.856662i	$0.672535\pi$
$938$	0	0
$939$	−3.53330	−0.115305
$940$	0	0
$941$	17.2419	0.562071	0.281036	−	0.959697i	$-0.409322\pi$
$941$	17.2419	0.562071	0.281036	+	0.959697i	$0.409322\pi$
$942$	0	0
$943$	−11.3803	−0.370592
$944$	0	0
$945$	−2.31785	−0.0753997
$946$	0	0
$947$	−47.1092	−1.53084	−0.765422	−	0.643528i	$-0.777470\pi$
$947$	−47.1092	−1.53084	−0.765422	+	0.643528i	$0.777470\pi$
$948$	0	0
$949$	51.8460	1.68299
$950$	0	0
$951$	−3.30106	−0.107044
$952$	0	0
$953$	19.4376	0.629646	0.314823	−	0.949150i	$-0.398055\pi$
$953$	19.4376	0.629646	0.314823	+	0.949150i	$0.398055\pi$
$954$	0	0
$955$	21.5387	0.696975
$956$	0	0
$957$	3.03737	0.0981843
$958$	0	0
$959$	−24.0743	−0.777400
$960$	0	0
$961$	−30.7936	−0.993341
$962$	0	0
$963$	43.8385	1.41268
$964$	0	0
$965$	10.6259	0.342061
$966$	0	0
$967$	−52.8772	−1.70042	−0.850208	−	0.526448i	$-0.823524\pi$
$967$	−52.8772	−1.70042	−0.850208	+	0.526448i	$0.823524\pi$
$968$	0	0
$969$	0.391865	0.0125885
$970$	0	0
$971$	−25.5589	−0.820225	−0.410113	−	0.912035i	$-0.634511\pi$
$971$	−25.5589	−0.820225	−0.410113	+	0.912035i	$0.634511\pi$
$972$	0	0
$973$	43.7874	1.40376
$974$	0	0
$975$	−0.673041	−0.0215546
$976$	0	0
$977$	11.0595	0.353824	0.176912	−	0.984227i	$-0.443389\pi$
$977$	11.0595	0.353824	0.176912	+	0.984227i	$0.443389\pi$
$978$	0	0
$979$	−46.5183	−1.48673
$980$	0	0
$981$	31.0119	0.990135
$982$	0	0
$983$	−44.5849	−1.42204	−0.711018	−	0.703173i	$-0.751766\pi$
$983$	−44.5849	−1.42204	−0.711018	+	0.703173i	$0.751766\pi$
$984$	0	0
$985$	−20.5168	−0.653719
$986$	0	0
$987$	−4.68785	−0.149216
$988$	0	0
$989$	10.2655	0.326424
$990$	0	0
$991$	−52.8690	−1.67944	−0.839719	−	0.543021i	$-0.817280\pi$
$991$	−52.8690	−1.67944	−0.839719	+	0.543021i	$0.817280\pi$
$992$	0	0
$993$	−3.45702	−0.109705
$994$	0	0
$995$	−15.3366	−0.486203
$996$	0	0
$997$	−32.6296	−1.03339	−0.516694	−	0.856170i	$-0.672838\pi$
$997$	−32.6296	−1.03339	−0.516694	+	0.856170i	$0.672838\pi$
$998$	0	0
$999$	−1.86230	−0.0589204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.18	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-0.167962 q^{3} -1.00000 q^{5} +2.31084 q^{7} -2.97179 q^{9} +O(q^{10})\)	\(q-0.167962 q^{3} -1.00000 q^{5} +2.31084 q^{7} -2.97179 q^{9} +2.88795 q^{11} +4.00711 q^{13} +0.167962 q^{15} +4.29329 q^{17} -0.543420 q^{19} -0.388133 q^{21} -1.32347 q^{23} +1.00000 q^{25} +1.00303 q^{27} -6.26176 q^{29} +0.454333 q^{31} -0.485066 q^{33} -2.31084 q^{35} -1.85666 q^{37} -0.673041 q^{39} +8.59884 q^{41} -7.75654 q^{43} +2.97179 q^{45} +12.0779 q^{47} -1.66001 q^{49} -0.721110 q^{51} +9.04827 q^{53} -2.88795 q^{55} +0.0912739 q^{57} +3.28580 q^{59} -2.38822 q^{61} -6.86733 q^{63} -4.00711 q^{65} +12.2530 q^{67} +0.222292 q^{69} +2.38396 q^{71} +12.9385 q^{73} -0.167962 q^{75} +6.67360 q^{77} +1.17724 q^{79} +8.74690 q^{81} -3.48118 q^{83} -4.29329 q^{85} +1.05174 q^{87} -16.1077 q^{89} +9.25978 q^{91} -0.0763106 q^{93} +0.543420 q^{95} +3.13575 q^{97} -8.58239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more