LMFDB - Embedded newform 8020.2.a.f.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:	$37$
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)

$f(q)$	$=$	$q-0.161183 q^{3} +1.00000 q^{5} -0.359157 q^{7} -2.97402 q^{9} +O(q^{10})$	$q-0.161183 q^{3} +1.00000 q^{5} -0.359157 q^{7} -2.97402 q^{9} +4.96439 q^{11} +5.13268 q^{13} -0.161183 q^{15} +4.76444 q^{17} +4.84823 q^{19} +0.0578901 q^{21} +7.93639 q^{23} +1.00000 q^{25} +0.962910 q^{27} -3.70872 q^{29} -0.925922 q^{31} -0.800176 q^{33} -0.359157 q^{35} -4.26654 q^{37} -0.827300 q^{39} +5.39484 q^{41} +11.4098 q^{43} -2.97402 q^{45} -9.45198 q^{47} -6.87101 q^{49} -0.767947 q^{51} +1.52213 q^{53} +4.96439 q^{55} -0.781452 q^{57} +13.9873 q^{59} +1.66475 q^{61} +1.06814 q^{63} +5.13268 q^{65} -13.0294 q^{67} -1.27921 q^{69} -0.552675 q^{71} -5.16060 q^{73} -0.161183 q^{75} -1.78300 q^{77} +8.46947 q^{79} +8.76686 q^{81} -9.93429 q^{83} +4.76444 q^{85} +0.597783 q^{87} -14.1499 q^{89} -1.84344 q^{91} +0.149243 q^{93} +4.84823 q^{95} +0.0798836 q^{97} -14.7642 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.161183	−0.0930590	−0.0465295	−	0.998917i	$-0.514816\pi$
$3$	−0.161183	−0.0930590	−0.0465295	+	0.998917i	$0.514816\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.359157	−0.135749	−0.0678744	−	0.997694i	$-0.521622\pi$
$7$	−0.359157	−0.135749	−0.0678744	+	0.997694i	$0.521622\pi$
$8$	0	0
$9$	−2.97402	−0.991340
$10$	0	0
$11$	4.96439	1.49682	0.748410	−	0.663236i	$-0.230817\pi$
$11$	4.96439	1.49682	0.748410	+	0.663236i	$0.230817\pi$
$12$	0	0
$13$	5.13268	1.42355	0.711774	−	0.702408i	$-0.247892\pi$
$13$	5.13268	1.42355	0.711774	+	0.702408i	$0.247892\pi$
$14$	0	0
$15$	−0.161183	−0.0416173
$16$	0	0
$17$	4.76444	1.15555	0.577774	−	0.816197i	$-0.303922\pi$
$17$	4.76444	1.15555	0.577774	+	0.816197i	$0.303922\pi$
$18$	0	0
$19$	4.84823	1.11226	0.556130	−	0.831095i	$-0.312286\pi$
$19$	4.84823	1.11226	0.556130	+	0.831095i	$0.312286\pi$
$20$	0	0
$21$	0.0578901	0.0126326
$22$	0	0
$23$	7.93639	1.65485	0.827426	−	0.561574i	$-0.189804\pi$
$23$	7.93639	1.65485	0.827426	+	0.561574i	$0.189804\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.962910	0.185312
$28$	0	0
$29$	−3.70872	−0.688692	−0.344346	−	0.938843i	$-0.611899\pi$
$29$	−3.70872	−0.688692	−0.344346	+	0.938843i	$0.611899\pi$
$30$	0	0
$31$	−0.925922	−0.166301	−0.0831503	−	0.996537i	$-0.526498\pi$
$31$	−0.925922	−0.166301	−0.0831503	+	0.996537i	$0.526498\pi$
$32$	0	0
$33$	−0.800176	−0.139293
$34$	0	0
$35$	−0.359157	−0.0607087
$36$	0	0
$37$	−4.26654	−0.701414	−0.350707	−	0.936485i	$-0.614059\pi$
$37$	−4.26654	−0.701414	−0.350707	+	0.936485i	$0.614059\pi$
$38$	0	0
$39$	−0.827300	−0.132474
$40$	0	0
$41$	5.39484	0.842532	0.421266	−	0.906937i	$-0.361586\pi$
$41$	5.39484	0.842532	0.421266	+	0.906937i	$0.361586\pi$
$42$	0	0
$43$	11.4098	1.73997	0.869986	−	0.493077i	$-0.164128\pi$
$43$	11.4098	1.73997	0.869986	+	0.493077i	$0.164128\pi$
$44$	0	0
$45$	−2.97402	−0.443341
$46$	0	0
$47$	−9.45198	−1.37871	−0.689356	−	0.724423i	$-0.742107\pi$
$47$	−9.45198	−1.37871	−0.689356	+	0.724423i	$0.742107\pi$
$48$	0	0
$49$	−6.87101	−0.981572
$50$	0	0
$51$	−0.767947	−0.107534
$52$	0	0
$53$	1.52213	0.209081	0.104541	−	0.994521i	$-0.466663\pi$
$53$	1.52213	0.209081	0.104541	+	0.994521i	$0.466663\pi$
$54$	0	0
$55$	4.96439	0.669399
$56$	0	0
$57$	−0.781452	−0.103506
$58$	0	0
$59$	13.9873	1.82099	0.910493	−	0.413523i	$-0.135702\pi$
$59$	13.9873	1.82099	0.910493	+	0.413523i	$0.135702\pi$
$60$	0	0
$61$	1.66475	0.213149	0.106575	−	0.994305i	$-0.466012\pi$
$61$	1.66475	0.213149	0.106575	+	0.994305i	$0.466012\pi$
$62$	0	0
$63$	1.06814	0.134573
$64$	0	0
$65$	5.13268	0.636630
$66$	0	0
$67$	−13.0294	−1.59179	−0.795897	−	0.605432i	$-0.793000\pi$
$67$	−13.0294	−1.59179	−0.795897	+	0.605432i	$0.793000\pi$
$68$	0	0
$69$	−1.27921	−0.153999
$70$	0	0
$71$	−0.552675	−0.0655904	−0.0327952	−	0.999462i	$-0.510441\pi$
$71$	−0.552675	−0.0655904	−0.0327952	+	0.999462i	$0.510441\pi$
$72$	0	0
$73$	−5.16060	−0.604002	−0.302001	−	0.953308i	$-0.597655\pi$
$73$	−5.16060	−0.604002	−0.302001	+	0.953308i	$0.597655\pi$
$74$	0	0
$75$	−0.161183	−0.0186118
$76$	0	0
$77$	−1.78300	−0.203192
$78$	0	0
$79$	8.46947	0.952890	0.476445	−	0.879204i	$-0.341925\pi$
$79$	8.46947	0.952890	0.476445	+	0.879204i	$0.341925\pi$
$80$	0	0
$81$	8.76686	0.974095
$82$	0	0
$83$	−9.93429	−1.09043	−0.545215	−	0.838296i	$-0.683552\pi$
$83$	−9.93429	−1.09043	−0.545215	+	0.838296i	$0.683552\pi$
$84$	0	0
$85$	4.76444	0.516777
$86$	0	0
$87$	0.597783	0.0640890
$88$	0	0
$89$	−14.1499	−1.49989	−0.749943	−	0.661503i	$-0.769919\pi$
$89$	−14.1499	−1.49989	−0.749943	+	0.661503i	$0.769919\pi$
$90$	0	0
$91$	−1.84344	−0.193245
$92$	0	0
$93$	0.149243	0.0154758
$94$	0	0
$95$	4.84823	0.497418
$96$	0	0
$97$	0.0798836	0.00811095	0.00405547	−	0.999992i	$-0.498709\pi$
$97$	0.0798836	0.00811095	0.00405547	+	0.999992i	$0.498709\pi$
$98$	0	0
$99$	−14.7642	−1.48386
$100$	0	0
$101$	3.38761	0.337079	0.168540	−	0.985695i	$-0.446095\pi$
$101$	3.38761	0.337079	0.168540	+	0.985695i	$0.446095\pi$
$102$	0	0
$103$	−7.00036	−0.689766	−0.344883	−	0.938646i	$-0.612081\pi$
$103$	−7.00036	−0.689766	−0.344883	+	0.938646i	$0.612081\pi$
$104$	0	0
$105$	0.0578901	0.00564949
$106$	0	0
$107$	−13.9889	−1.35236	−0.676180	−	0.736737i	$-0.736366\pi$
$107$	−13.9889	−1.35236	−0.676180	+	0.736737i	$0.736366\pi$
$108$	0	0
$109$	7.02896	0.673252	0.336626	−	0.941638i	$-0.390714\pi$
$109$	7.02896	0.673252	0.336626	+	0.941638i	$0.390714\pi$
$110$	0	0
$111$	0.687693	0.0652729
$112$	0	0
$113$	−17.8824	−1.68224	−0.841118	−	0.540851i	$-0.818102\pi$
$113$	−17.8824	−1.68224	−0.841118	+	0.540851i	$0.818102\pi$
$114$	0	0
$115$	7.93639	0.740073
$116$	0	0
$117$	−15.2647	−1.41122
$118$	0	0
$119$	−1.71119	−0.156864
$120$	0	0
$121$	13.6452	1.24047
$122$	0	0
$123$	−0.869556	−0.0784052
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.25783	−0.111615	−0.0558073	−	0.998442i	$-0.517773\pi$
$127$	−1.25783	−0.111615	−0.0558073	+	0.998442i	$0.517773\pi$
$128$	0	0
$129$	−1.83906	−0.161920
$130$	0	0
$131$	−17.4033	−1.52053	−0.760265	−	0.649613i	$-0.774931\pi$
$131$	−17.4033	−1.52053	−0.760265	+	0.649613i	$0.774931\pi$
$132$	0	0
$133$	−1.74128	−0.150988
$134$	0	0
$135$	0.962910	0.0828741
$136$	0	0
$137$	3.36857	0.287797	0.143898	−	0.989592i	$-0.454036\pi$
$137$	3.36857	0.287797	0.143898	+	0.989592i	$0.454036\pi$
$138$	0	0
$139$	21.9614	1.86274	0.931369	−	0.364076i	$-0.118615\pi$
$139$	21.9614	1.86274	0.931369	+	0.364076i	$0.118615\pi$
$140$	0	0
$141$	1.52350	0.128302
$142$	0	0
$143$	25.4806	2.13080
$144$	0	0
$145$	−3.70872	−0.307993
$146$	0	0
$147$	1.10749	0.0913442
$148$	0	0
$149$	8.30385	0.680278	0.340139	−	0.940375i	$-0.389526\pi$
$149$	8.30385	0.680278	0.340139	+	0.940375i	$0.389526\pi$
$150$	0	0
$151$	−16.0822	−1.30875	−0.654375	−	0.756170i	$-0.727068\pi$
$151$	−16.0822	−1.30875	−0.654375	+	0.756170i	$0.727068\pi$
$152$	0	0
$153$	−14.1696	−1.14554
$154$	0	0
$155$	−0.925922	−0.0743719
$156$	0	0
$157$	−7.12329	−0.568501	−0.284250	−	0.958750i	$-0.591745\pi$
$157$	−7.12329	−0.568501	−0.284250	+	0.958750i	$0.591745\pi$
$158$	0	0
$159$	−0.245342	−0.0194569
$160$	0	0
$161$	−2.85041	−0.224644
$162$	0	0
$163$	21.3068	1.66888	0.834439	−	0.551101i	$-0.185792\pi$
$163$	21.3068	1.66888	0.834439	+	0.551101i	$0.185792\pi$
$164$	0	0
$165$	−0.800176	−0.0622936
$166$	0	0
$167$	4.21134	0.325884	0.162942	−	0.986636i	$-0.447902\pi$
$167$	4.21134	0.325884	0.162942	+	0.986636i	$0.447902\pi$
$168$	0	0
$169$	13.3444	1.02649
$170$	0	0
$171$	−14.4187	−1.10263
$172$	0	0
$173$	8.92819	0.678798	0.339399	−	0.940643i	$-0.389776\pi$
$173$	8.92819	0.678798	0.339399	+	0.940643i	$0.389776\pi$
$174$	0	0
$175$	−0.359157	−0.0271498
$176$	0	0
$177$	−2.25451	−0.169459
$178$	0	0
$179$	−10.3506	−0.773637	−0.386818	−	0.922156i	$-0.626426\pi$
$179$	−10.3506	−0.773637	−0.386818	+	0.922156i	$0.626426\pi$
$180$	0	0
$181$	21.6769	1.61123	0.805614	−	0.592440i	$-0.201835\pi$
$181$	21.6769	1.61123	0.805614	+	0.592440i	$0.201835\pi$
$182$	0	0
$183$	−0.268329	−0.0198355
$184$	0	0
$185$	−4.26654	−0.313682
$186$	0	0
$187$	23.6526	1.72965
$188$	0	0
$189$	−0.345836	−0.0251559
$190$	0	0
$191$	3.92452	0.283969	0.141984	−	0.989869i	$-0.454652\pi$
$191$	3.92452	0.283969	0.141984	+	0.989869i	$0.454652\pi$
$192$	0	0
$193$	21.2775	1.53159	0.765794	−	0.643086i	$-0.222346\pi$
$193$	21.2775	1.53159	0.765794	+	0.643086i	$0.222346\pi$
$194$	0	0
$195$	−0.827300	−0.0592442
$196$	0	0
$197$	20.5809	1.46633	0.733165	−	0.680051i	$-0.238042\pi$
$197$	20.5809	1.46633	0.733165	+	0.680051i	$0.238042\pi$
$198$	0	0
$199$	6.23071	0.441683	0.220842	−	0.975310i	$-0.429120\pi$
$199$	6.23071	0.441683	0.220842	+	0.975310i	$0.429120\pi$
$200$	0	0
$201$	2.10012	0.148131
$202$	0	0
$203$	1.33202	0.0934891
$204$	0	0
$205$	5.39484	0.376792
$206$	0	0
$207$	−23.6030	−1.64052
$208$	0	0
$209$	24.0685	1.66485
$210$	0	0
$211$	−25.5935	−1.76193	−0.880966	−	0.473179i	$-0.843106\pi$
$211$	−25.5935	−1.76193	−0.880966	+	0.473179i	$0.843106\pi$
$212$	0	0
$213$	0.0890818	0.00610378
$214$	0	0
$215$	11.4098	0.778139
$216$	0	0
$217$	0.332552	0.0225751
$218$	0	0
$219$	0.831800	0.0562078
$220$	0	0
$221$	24.4544	1.64498
$222$	0	0
$223$	−10.6318	−0.711959	−0.355979	−	0.934494i	$-0.615853\pi$
$223$	−10.6318	−0.711959	−0.355979	+	0.934494i	$0.615853\pi$
$224$	0	0
$225$	−2.97402	−0.198268
$226$	0	0
$227$	18.1787	1.20656	0.603282	−	0.797528i	$-0.293859\pi$
$227$	18.1787	1.20656	0.603282	+	0.797528i	$0.293859\pi$
$228$	0	0
$229$	14.5153	0.959195	0.479597	−	0.877489i	$-0.340783\pi$
$229$	14.5153	0.959195	0.479597	+	0.877489i	$0.340783\pi$
$230$	0	0
$231$	0.287389	0.0189088
$232$	0	0
$233$	12.9826	0.850518	0.425259	−	0.905072i	$-0.360183\pi$
$233$	12.9826	0.850518	0.425259	+	0.905072i	$0.360183\pi$
$234$	0	0
$235$	−9.45198	−0.616579
$236$	0	0
$237$	−1.36513	−0.0886750
$238$	0	0
$239$	−24.4302	−1.58026	−0.790128	−	0.612942i	$-0.789986\pi$
$239$	−24.4302	−1.58026	−0.790128	+	0.612942i	$0.789986\pi$
$240$	0	0
$241$	−21.3287	−1.37390	−0.686950	−	0.726705i	$-0.741051\pi$
$241$	−21.3287	−1.37390	−0.686950	+	0.726705i	$0.741051\pi$
$242$	0	0
$243$	−4.30180	−0.275960
$244$	0	0
$245$	−6.87101	−0.438972
$246$	0	0
$247$	24.8844	1.58336
$248$	0	0
$249$	1.60124	0.101474
$250$	0	0
$251$	3.10482	0.195975	0.0979873	−	0.995188i	$-0.468760\pi$
$251$	3.10482	0.195975	0.0979873	+	0.995188i	$0.468760\pi$
$252$	0	0
$253$	39.3994	2.47702
$254$	0	0
$255$	−0.767947	−0.0480907
$256$	0	0
$257$	−13.8919	−0.866552	−0.433276	−	0.901261i	$-0.642642\pi$
$257$	−13.8919	−0.866552	−0.433276	+	0.901261i	$0.642642\pi$
$258$	0	0
$259$	1.53236	0.0952161
$260$	0	0
$261$	11.0298	0.682728
$262$	0	0
$263$	27.2730	1.68172	0.840861	−	0.541251i	$-0.182049\pi$
$263$	27.2730	1.68172	0.840861	+	0.541251i	$0.182049\pi$
$264$	0	0
$265$	1.52213	0.0935039
$266$	0	0
$267$	2.28072	0.139578
$268$	0	0
$269$	15.9425	0.972028	0.486014	−	0.873951i	$-0.338450\pi$
$269$	15.9425	0.972028	0.486014	+	0.873951i	$0.338450\pi$
$270$	0	0
$271$	−19.2210	−1.16760	−0.583798	−	0.811899i	$-0.698434\pi$
$271$	−19.2210	−1.16760	−0.583798	+	0.811899i	$0.698434\pi$
$272$	0	0
$273$	0.297131	0.0179832
$274$	0	0
$275$	4.96439	0.299364
$276$	0	0
$277$	−7.14563	−0.429339	−0.214670	−	0.976687i	$-0.568867\pi$
$277$	−7.14563	−0.429339	−0.214670	+	0.976687i	$0.568867\pi$
$278$	0	0
$279$	2.75371	0.164860
$280$	0	0
$281$	−16.6580	−0.993735	−0.496867	−	0.867827i	$-0.665516\pi$
$281$	−16.6580	−0.993735	−0.496867	+	0.867827i	$0.665516\pi$
$282$	0	0
$283$	14.1591	0.841671	0.420836	−	0.907137i	$-0.361737\pi$
$283$	14.1591	0.841671	0.420836	+	0.907137i	$0.361737\pi$
$284$	0	0
$285$	−0.781452	−0.0462892
$286$	0	0
$287$	−1.93760	−0.114373
$288$	0	0
$289$	5.69993	0.335290
$290$	0	0
$291$	−0.0128759	−0.000754797 0
$292$	0	0
$293$	−18.2531	−1.06636	−0.533179	−	0.846002i	$-0.679003\pi$
$293$	−18.2531	−1.06636	−0.533179	+	0.846002i	$0.679003\pi$
$294$	0	0
$295$	13.9873	0.814370
$296$	0	0
$297$	4.78026	0.277379
$298$	0	0
$299$	40.7349	2.35576
$300$	0	0
$301$	−4.09790	−0.236199
$302$	0	0
$303$	−0.546024	−0.0313683
$304$	0	0
$305$	1.66475	0.0953233
$306$	0	0
$307$	−18.9665	−1.08247	−0.541237	−	0.840870i	$-0.682044\pi$
$307$	−18.9665	−1.08247	−0.541237	+	0.840870i	$0.682044\pi$
$308$	0	0
$309$	1.12834	0.0641889
$310$	0	0
$311$	25.8966	1.46846	0.734232	−	0.678899i	$-0.237542\pi$
$311$	25.8966	1.46846	0.734232	+	0.678899i	$0.237542\pi$
$312$	0	0
$313$	1.21408	0.0686241	0.0343120	−	0.999411i	$-0.489076\pi$
$313$	1.21408	0.0686241	0.0343120	+	0.999411i	$0.489076\pi$
$314$	0	0
$315$	1.06814	0.0601830
$316$	0	0
$317$	5.81337	0.326512	0.163256	−	0.986584i	$-0.447800\pi$
$317$	5.81337	0.326512	0.163256	+	0.986584i	$0.447800\pi$
$318$	0	0
$319$	−18.4116	−1.03085
$320$	0	0
$321$	2.25477	0.125849
$322$	0	0
$323$	23.0991	1.28527
$324$	0	0
$325$	5.13268	0.284710
$326$	0	0
$327$	−1.13295	−0.0626522
$328$	0	0
$329$	3.39475	0.187158
$330$	0	0
$331$	−29.9851	−1.64813	−0.824066	−	0.566494i	$-0.808300\pi$
$331$	−29.9851	−1.64813	−0.824066	+	0.566494i	$0.808300\pi$
$332$	0	0
$333$	12.6888	0.695340
$334$	0	0
$335$	−13.0294	−0.711872
$336$	0	0
$337$	4.43430	0.241552	0.120776	−	0.992680i	$-0.461462\pi$
$337$	4.43430	0.241552	0.120776	+	0.992680i	$0.461462\pi$
$338$	0	0
$339$	2.88234	0.156547
$340$	0	0
$341$	−4.59664	−0.248922
$342$	0	0
$343$	4.98187	0.268996
$344$	0	0
$345$	−1.27921	−0.0688704
$346$	0	0
$347$	22.1442	1.18876	0.594381	−	0.804184i	$-0.297397\pi$
$347$	22.1442	1.18876	0.594381	+	0.804184i	$0.297397\pi$
$348$	0	0
$349$	−16.9969	−0.909825	−0.454913	−	0.890536i	$-0.650330\pi$
$349$	−16.9969	−0.909825	−0.454913	+	0.890536i	$0.650330\pi$
$350$	0	0
$351$	4.94231	0.263801
$352$	0	0
$353$	−19.3789	−1.03143	−0.515716	−	0.856759i	$-0.672474\pi$
$353$	−19.3789	−1.03143	−0.515716	+	0.856759i	$0.672474\pi$
$354$	0	0
$355$	−0.552675	−0.0293329
$356$	0	0
$357$	0.275814	0.0145976
$358$	0	0
$359$	−23.3377	−1.23171	−0.615857	−	0.787858i	$-0.711190\pi$
$359$	−23.3377	−1.23171	−0.615857	+	0.787858i	$0.711190\pi$
$360$	0	0
$361$	4.50533	0.237123
$362$	0	0
$363$	−2.19937	−0.115437
$364$	0	0
$365$	−5.16060	−0.270118
$366$	0	0
$367$	30.8182	1.60870	0.804349	−	0.594158i	$-0.202515\pi$
$367$	30.8182	1.60870	0.804349	+	0.594158i	$0.202515\pi$
$368$	0	0
$369$	−16.0444	−0.835236
$370$	0	0
$371$	−0.546685	−0.0283825
$372$	0	0
$373$	−0.904749	−0.0468461	−0.0234231	−	0.999726i	$-0.507456\pi$
$373$	−0.904749	−0.0468461	−0.0234231	+	0.999726i	$0.507456\pi$
$374$	0	0
$375$	−0.161183	−0.00832345
$376$	0	0
$377$	−19.0357	−0.980387
$378$	0	0
$379$	−17.4273	−0.895179	−0.447590	−	0.894239i	$-0.647717\pi$
$379$	−17.4273	−0.895179	−0.447590	+	0.894239i	$0.647717\pi$
$380$	0	0
$381$	0.202741	0.0103867
$382$	0	0
$383$	−15.2270	−0.778061	−0.389030	−	0.921225i	$-0.627190\pi$
$383$	−15.2270	−0.778061	−0.389030	+	0.921225i	$0.627190\pi$
$384$	0	0
$385$	−1.78300	−0.0908700
$386$	0	0
$387$	−33.9328	−1.72490
$388$	0	0
$389$	20.5113	1.03997	0.519983	−	0.854177i	$-0.325938\pi$
$389$	20.5113	1.03997	0.519983	+	0.854177i	$0.325938\pi$
$390$	0	0
$391$	37.8125	1.91226
$392$	0	0
$393$	2.80511	0.141499
$394$	0	0
$395$	8.46947	0.426145
$396$	0	0
$397$	31.6098	1.58645	0.793224	−	0.608930i	$-0.208401\pi$
$397$	31.6098	1.58645	0.793224	+	0.608930i	$0.208401\pi$
$398$	0	0
$399$	0.280664	0.0140508
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−4.75246	−0.236737
$404$	0	0
$405$	8.76686	0.435629
$406$	0	0
$407$	−21.1808	−1.04989
$408$	0	0
$409$	−17.4760	−0.864130	−0.432065	−	0.901842i	$-0.642215\pi$
$409$	−17.4760	−0.864130	−0.432065	+	0.901842i	$0.642215\pi$
$410$	0	0
$411$	−0.542957	−0.0267821
$412$	0	0
$413$	−5.02363	−0.247197
$414$	0	0
$415$	−9.93429	−0.487655
$416$	0	0
$417$	−3.53980	−0.173345
$418$	0	0
$419$	6.30109	0.307828	0.153914	−	0.988084i	$-0.450812\pi$
$419$	6.30109	0.307828	0.153914	+	0.988084i	$0.450812\pi$
$420$	0	0
$421$	−5.33392	−0.259960	−0.129980	−	0.991517i	$-0.541491\pi$
$421$	−5.33392	−0.259960	−0.129980	+	0.991517i	$0.541491\pi$
$422$	0	0
$423$	28.1104	1.36677
$424$	0	0
$425$	4.76444	0.231109
$426$	0	0
$427$	−0.597907	−0.0289348
$428$	0	0
$429$	−4.10704	−0.198290
$430$	0	0
$431$	−4.45570	−0.214623	−0.107312	−	0.994225i	$-0.534224\pi$
$431$	−4.45570	−0.214623	−0.107312	+	0.994225i	$0.534224\pi$
$432$	0	0
$433$	31.9643	1.53611	0.768053	−	0.640387i	$-0.221226\pi$
$433$	31.9643	1.53611	0.768053	+	0.640387i	$0.221226\pi$
$434$	0	0
$435$	0.597783	0.0286615
$436$	0	0
$437$	38.4775	1.84063
$438$	0	0
$439$	−25.1169	−1.19876	−0.599382	−	0.800463i	$-0.704587\pi$
$439$	−25.1169	−1.19876	−0.599382	+	0.800463i	$0.704587\pi$
$440$	0	0
$441$	20.4345	0.973072
$442$	0	0
$443$	−7.39080	−0.351148	−0.175574	−	0.984466i	$-0.556178\pi$
$443$	−7.39080	−0.351148	−0.175574	+	0.984466i	$0.556178\pi$
$444$	0	0
$445$	−14.1499	−0.670769
$446$	0	0
$447$	−1.33844	−0.0633060
$448$	0	0
$449$	−16.3972	−0.773831	−0.386915	−	0.922115i	$-0.626459\pi$
$449$	−16.3972	−0.773831	−0.386915	+	0.922115i	$0.626459\pi$
$450$	0	0
$451$	26.7821	1.26112
$452$	0	0
$453$	2.59217	0.121791
$454$	0	0
$455$	−1.84344	−0.0864218
$456$	0	0
$457$	4.58996	0.214710	0.107355	−	0.994221i	$-0.465762\pi$
$457$	4.58996	0.214710	0.107355	+	0.994221i	$0.465762\pi$
$458$	0	0
$459$	4.58773	0.214137
$460$	0	0
$461$	−3.54715	−0.165207	−0.0826036	−	0.996582i	$-0.526324\pi$
$461$	−3.54715	−0.165207	−0.0826036	+	0.996582i	$0.526324\pi$
$462$	0	0
$463$	−14.1236	−0.656380	−0.328190	−	0.944612i	$-0.606439\pi$
$463$	−14.1236	−0.656380	−0.328190	+	0.944612i	$0.606439\pi$
$464$	0	0
$465$	0.149243	0.00692097
$466$	0	0
$467$	−5.21786	−0.241454	−0.120727	−	0.992686i	$-0.538523\pi$
$467$	−5.21786	−0.241454	−0.120727	+	0.992686i	$0.538523\pi$
$468$	0	0
$469$	4.67960	0.216084
$470$	0	0
$471$	1.14815	0.0529041
$472$	0	0
$473$	56.6425	2.60443
$474$	0	0
$475$	4.84823	0.222452
$476$	0	0
$477$	−4.52685	−0.207270
$478$	0	0
$479$	−41.2666	−1.88552	−0.942759	−	0.333475i	$-0.891779\pi$
$479$	−41.2666	−1.88552	−0.942759	+	0.333475i	$0.891779\pi$
$480$	0	0
$481$	−21.8988	−0.998497
$482$	0	0
$483$	0.459438	0.0209052
$484$	0	0
$485$	0.0798836	0.00362733
$486$	0	0
$487$	−14.1008	−0.638970	−0.319485	−	0.947591i	$-0.603510\pi$
$487$	−14.1008	−0.638970	−0.319485	+	0.947591i	$0.603510\pi$
$488$	0	0
$489$	−3.43429	−0.155304
$490$	0	0
$491$	−20.9403	−0.945023	−0.472512	−	0.881325i	$-0.656652\pi$
$491$	−20.9403	−0.945023	−0.472512	+	0.881325i	$0.656652\pi$
$492$	0	0
$493$	−17.6700	−0.795817
$494$	0	0
$495$	−14.7642	−0.663602
$496$	0	0
$497$	0.198497	0.00890382
$498$	0	0
$499$	37.1107	1.66130	0.830651	−	0.556794i	$-0.187969\pi$
$499$	37.1107	1.66130	0.830651	+	0.556794i	$0.187969\pi$
$500$	0	0
$501$	−0.678797	−0.0303264
$502$	0	0
$503$	−6.23320	−0.277924	−0.138962	−	0.990298i	$-0.544377\pi$
$503$	−6.23320	−0.277924	−0.138962	+	0.990298i	$0.544377\pi$
$504$	0	0
$505$	3.38761	0.150746
$506$	0	0
$507$	−2.15088	−0.0955241
$508$	0	0
$509$	38.7170	1.71610	0.858051	−	0.513564i	$-0.171675\pi$
$509$	38.7170	1.71610	0.858051	+	0.513564i	$0.171675\pi$
$510$	0	0
$511$	1.85347	0.0819925
$512$	0	0
$513$	4.66841	0.206115
$514$	0	0
$515$	−7.00036	−0.308473
$516$	0	0
$517$	−46.9233	−2.06369
$518$	0	0
$519$	−1.43907	−0.0631683
$520$	0	0
$521$	−13.2148	−0.578949	−0.289475	−	0.957186i	$-0.593481\pi$
$521$	−13.2148	−0.578949	−0.289475	+	0.957186i	$0.593481\pi$
$522$	0	0
$523$	−6.89588	−0.301536	−0.150768	−	0.988569i	$-0.548175\pi$
$523$	−6.89588	−0.301536	−0.150768	+	0.988569i	$0.548175\pi$
$524$	0	0
$525$	0.0578901	0.00252653
$526$	0	0
$527$	−4.41151	−0.192168
$528$	0	0
$529$	39.9863	1.73854
$530$	0	0
$531$	−41.5984	−1.80522
$532$	0	0
$533$	27.6900	1.19939
$534$	0	0
$535$	−13.9889	−0.604793
$536$	0	0
$537$	1.66833	0.0719939
$538$	0	0
$539$	−34.1104	−1.46924
$540$	0	0
$541$	15.0154	0.645562	0.322781	−	0.946474i	$-0.395382\pi$
$541$	15.0154	0.645562	0.322781	+	0.946474i	$0.395382\pi$
$542$	0	0
$543$	−3.49394	−0.149939
$544$	0	0
$545$	7.02896	0.301087
$546$	0	0
$547$	20.8343	0.890812	0.445406	−	0.895329i	$-0.353059\pi$
$547$	20.8343	0.890812	0.445406	+	0.895329i	$0.353059\pi$
$548$	0	0
$549$	−4.95100	−0.211303
$550$	0	0
$551$	−17.9807	−0.766005
$552$	0	0
$553$	−3.04187	−0.129354
$554$	0	0
$555$	0.687693	0.0291909
$556$	0	0
$557$	−7.09387	−0.300577	−0.150288	−	0.988642i	$-0.548020\pi$
$557$	−7.09387	−0.300577	−0.150288	+	0.988642i	$0.548020\pi$
$558$	0	0
$559$	58.5626	2.47693
$560$	0	0
$561$	−3.81239	−0.160959
$562$	0	0
$563$	43.9447	1.85205	0.926023	−	0.377467i	$-0.123205\pi$
$563$	43.9447	1.85205	0.926023	+	0.377467i	$0.123205\pi$
$564$	0	0
$565$	−17.8824	−0.752319
$566$	0	0
$567$	−3.14868	−0.132232
$568$	0	0
$569$	−26.1157	−1.09483	−0.547413	−	0.836862i	$-0.684387\pi$
$569$	−26.1157	−1.09483	−0.547413	+	0.836862i	$0.684387\pi$
$570$	0	0
$571$	19.2296	0.804732	0.402366	−	0.915479i	$-0.368188\pi$
$571$	19.2296	0.804732	0.402366	+	0.915479i	$0.368188\pi$
$572$	0	0
$573$	−0.632566	−0.0264258
$574$	0	0
$575$	7.93639	0.330970
$576$	0	0
$577$	−9.84706	−0.409939	−0.204969	−	0.978768i	$-0.565709\pi$
$577$	−9.84706	−0.409939	−0.204969	+	0.978768i	$0.565709\pi$
$578$	0	0
$579$	−3.42957	−0.142528
$580$	0	0
$581$	3.56797	0.148024
$582$	0	0
$583$	7.55647	0.312957
$584$	0	0
$585$	−15.2647	−0.631117
$586$	0	0
$587$	16.6132	0.685698	0.342849	−	0.939390i	$-0.388608\pi$
$587$	16.6132	0.685698	0.342849	+	0.939390i	$0.388608\pi$
$588$	0	0
$589$	−4.48908	−0.184970
$590$	0	0
$591$	−3.31729	−0.136455
$592$	0	0
$593$	13.0665	0.536578	0.268289	−	0.963338i	$-0.413542\pi$
$593$	13.0665	0.536578	0.268289	+	0.963338i	$0.413542\pi$
$594$	0	0
$595$	−1.71119	−0.0701518
$596$	0	0
$597$	−1.00428	−0.0411026
$598$	0	0
$599$	−12.5848	−0.514201	−0.257101	−	0.966385i	$-0.582767\pi$
$599$	−12.5848	−0.514201	−0.257101	+	0.966385i	$0.582767\pi$
$600$	0	0
$601$	−12.3353	−0.503166	−0.251583	−	0.967836i	$-0.580951\pi$
$601$	−12.3353	−0.503166	−0.251583	+	0.967836i	$0.580951\pi$
$602$	0	0
$603$	38.7497	1.57801
$604$	0	0
$605$	13.6452	0.554756
$606$	0	0
$607$	38.0106	1.54280	0.771402	−	0.636348i	$-0.219556\pi$
$607$	38.0106	1.54280	0.771402	+	0.636348i	$0.219556\pi$
$608$	0	0
$609$	−0.214698	−0.00870001
$610$	0	0
$611$	−48.5139	−1.96266
$612$	0	0
$613$	−33.5778	−1.35619	−0.678097	−	0.734972i	$-0.737195\pi$
$613$	−33.5778	−1.35619	−0.678097	+	0.734972i	$0.737195\pi$
$614$	0	0
$615$	−0.869556	−0.0350639
$616$	0	0
$617$	38.4508	1.54797	0.773986	−	0.633202i	$-0.218260\pi$
$617$	38.4508	1.54797	0.773986	+	0.633202i	$0.218260\pi$
$618$	0	0
$619$	23.4690	0.943298	0.471649	−	0.881786i	$-0.343659\pi$
$619$	23.4690	0.943298	0.471649	+	0.881786i	$0.343659\pi$
$620$	0	0
$621$	7.64203	0.306664
$622$	0	0
$623$	5.08204	0.203608
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.87943	−0.154930
$628$	0	0
$629$	−20.3277	−0.810517
$630$	0	0
$631$	17.9191	0.713347	0.356673	−	0.934229i	$-0.383911\pi$
$631$	17.9191	0.713347	0.356673	+	0.934229i	$0.383911\pi$
$632$	0	0
$633$	4.12524	0.163964
$634$	0	0
$635$	−1.25783	−0.0499156
$636$	0	0
$637$	−35.2667	−1.39732
$638$	0	0
$639$	1.64367	0.0650224
$640$	0	0
$641$	10.6180	0.419385	0.209692	−	0.977767i	$-0.432754\pi$
$641$	10.6180	0.419385	0.209692	+	0.977767i	$0.432754\pi$
$642$	0	0
$643$	−8.58734	−0.338651	−0.169326	−	0.985560i	$-0.554159\pi$
$643$	−8.58734	−0.338651	−0.169326	+	0.985560i	$0.554159\pi$
$644$	0	0
$645$	−1.83906	−0.0724129
$646$	0	0
$647$	−39.5403	−1.55449	−0.777245	−	0.629199i	$-0.783383\pi$
$647$	−39.5403	−1.55449	−0.777245	+	0.629199i	$0.783383\pi$
$648$	0	0
$649$	69.4383	2.72569
$650$	0	0
$651$	−0.0536017	−0.00210082
$652$	0	0
$653$	4.86092	0.190223	0.0951113	−	0.995467i	$-0.469679\pi$
$653$	4.86092	0.190223	0.0951113	+	0.995467i	$0.469679\pi$
$654$	0	0
$655$	−17.4033	−0.680002
$656$	0	0
$657$	15.3477	0.598771
$658$	0	0
$659$	8.48949	0.330703	0.165352	−	0.986235i	$-0.447124\pi$
$659$	8.48949	0.330703	0.165352	+	0.986235i	$0.447124\pi$
$660$	0	0
$661$	−4.45956	−0.173457	−0.0867284	−	0.996232i	$-0.527641\pi$
$661$	−4.45956	−0.173457	−0.0867284	+	0.996232i	$0.527641\pi$
$662$	0	0
$663$	−3.94162	−0.153080
$664$	0	0
$665$	−1.74128	−0.0675239
$666$	0	0
$667$	−29.4339	−1.13968
$668$	0	0
$669$	1.71367	0.0662542
$670$	0	0
$671$	8.26447	0.319046
$672$	0	0
$673$	−0.913315	−0.0352057	−0.0176029	−	0.999845i	$-0.505603\pi$
$673$	−0.913315	−0.0352057	−0.0176029	+	0.999845i	$0.505603\pi$
$674$	0	0
$675$	0.962910	0.0370624
$676$	0	0
$677$	22.8986	0.880066	0.440033	−	0.897982i	$-0.354967\pi$
$677$	22.8986	0.880066	0.440033	+	0.897982i	$0.354967\pi$
$678$	0	0
$679$	−0.0286908	−0.00110105
$680$	0	0
$681$	−2.93010	−0.112282
$682$	0	0
$683$	7.18500	0.274926	0.137463	−	0.990507i	$-0.456105\pi$
$683$	7.18500	0.274926	0.137463	+	0.990507i	$0.456105\pi$
$684$	0	0
$685$	3.36857	0.128707
$686$	0	0
$687$	−2.33961	−0.0892617
$688$	0	0
$689$	7.81262	0.297637
$690$	0	0
$691$	−24.2231	−0.921490	−0.460745	−	0.887532i	$-0.652418\pi$
$691$	−24.2231	−0.921490	−0.460745	+	0.887532i	$0.652418\pi$
$692$	0	0
$693$	5.30267	0.201432
$694$	0	0
$695$	21.9614	0.833042
$696$	0	0
$697$	25.7034	0.973586
$698$	0	0
$699$	−2.09257	−0.0791484
$700$	0	0
$701$	6.93735	0.262020	0.131010	−	0.991381i	$-0.458178\pi$
$701$	6.93735	0.262020	0.131010	+	0.991381i	$0.458178\pi$
$702$	0	0
$703$	−20.6851	−0.780155
$704$	0	0
$705$	1.52350	0.0573782
$706$	0	0
$707$	−1.21668	−0.0457581
$708$	0	0
$709$	12.0648	0.453102	0.226551	−	0.973999i	$-0.427255\pi$
$709$	12.0648	0.453102	0.226551	+	0.973999i	$0.427255\pi$
$710$	0	0
$711$	−25.1884	−0.944638
$712$	0	0
$713$	−7.34849	−0.275203
$714$	0	0
$715$	25.4806	0.952921
$716$	0	0
$717$	3.93772	0.147057
$718$	0	0
$719$	−26.1799	−0.976347	−0.488173	−	0.872747i	$-0.662337\pi$
$719$	−26.1799	−0.976347	−0.488173	+	0.872747i	$0.662337\pi$
$720$	0	0
$721$	2.51423	0.0936349
$722$	0	0
$723$	3.43782	0.127854
$724$	0	0
$725$	−3.70872	−0.137738
$726$	0	0
$727$	−9.30183	−0.344986	−0.172493	−	0.985011i	$-0.555182\pi$
$727$	−9.30183	−0.344986	−0.172493	+	0.985011i	$0.555182\pi$
$728$	0	0
$729$	−25.6072	−0.948414
$730$	0	0
$731$	54.3612	2.01062
$732$	0	0
$733$	34.5825	1.27733	0.638666	−	0.769484i	$-0.279486\pi$
$733$	34.5825	1.27733	0.638666	+	0.769484i	$0.279486\pi$
$734$	0	0
$735$	1.10749	0.0408503
$736$	0	0
$737$	−64.6830	−2.38263
$738$	0	0
$739$	−0.798408	−0.0293699	−0.0146850	−	0.999892i	$-0.504675\pi$
$739$	−0.798408	−0.0293699	−0.0146850	+	0.999892i	$0.504675\pi$
$740$	0	0
$741$	−4.01094	−0.147346
$742$	0	0
$743$	−35.2914	−1.29471	−0.647357	−	0.762187i	$-0.724126\pi$
$743$	−35.2914	−1.29471	−0.647357	+	0.762187i	$0.724126\pi$
$744$	0	0
$745$	8.30385	0.304229
$746$	0	0
$747$	29.5448	1.08099
$748$	0	0
$749$	5.02422	0.183581
$750$	0	0
$751$	−28.5746	−1.04270	−0.521351	−	0.853342i	$-0.674572\pi$
$751$	−28.5746	−1.04270	−0.521351	+	0.853342i	$0.674572\pi$
$752$	0	0
$753$	−0.500444	−0.0182372
$754$	0	0
$755$	−16.0822	−0.585291
$756$	0	0
$757$	−6.50649	−0.236483	−0.118241	−	0.992985i	$-0.537726\pi$
$757$	−6.50649	−0.236483	−0.118241	+	0.992985i	$0.537726\pi$
$758$	0	0
$759$	−6.35051	−0.230509
$760$	0	0
$761$	−4.00041	−0.145015	−0.0725074	−	0.997368i	$-0.523100\pi$
$761$	−4.00041	−0.145015	−0.0725074	+	0.997368i	$0.523100\pi$
$762$	0	0
$763$	−2.52450	−0.0913931
$764$	0	0
$765$	−14.1696	−0.512301
$766$	0	0
$767$	71.7921	2.59226
$768$	0	0
$769$	22.8880	0.825362	0.412681	−	0.910876i	$-0.364592\pi$
$769$	22.8880	0.825362	0.412681	+	0.910876i	$0.364592\pi$
$770$	0	0
$771$	2.23914	0.0806405
$772$	0	0
$773$	−43.9050	−1.57915	−0.789576	−	0.613653i	$-0.789699\pi$
$773$	−43.9050	−1.57915	−0.789576	+	0.613653i	$0.789699\pi$
$774$	0	0
$775$	−0.925922	−0.0332601
$776$	0	0
$777$	−0.246990	−0.00886072
$778$	0	0
$779$	26.1554	0.937115
$780$	0	0
$781$	−2.74370	−0.0981771
$782$	0	0
$783$	−3.57117	−0.127623
$784$	0	0
$785$	−7.12329	−0.254241
$786$	0	0
$787$	−40.4216	−1.44088	−0.720438	−	0.693519i	$-0.756059\pi$
$787$	−40.4216	−1.44088	−0.720438	+	0.693519i	$0.756059\pi$
$788$	0	0
$789$	−4.39594	−0.156499
$790$	0	0
$791$	6.42261	0.228362
$792$	0	0
$793$	8.54462	0.303428
$794$	0	0
$795$	−0.245342	−0.00870138
$796$	0	0
$797$	20.6591	0.731783	0.365892	−	0.930657i	$-0.380764\pi$
$797$	20.6591	0.731783	0.365892	+	0.930657i	$0.380764\pi$
$798$	0	0
$799$	−45.0334	−1.59317
$800$	0	0
$801$	42.0821	1.48690
$802$	0	0
$803$	−25.6192	−0.904083
$804$	0	0
$805$	−2.85041	−0.100464
$806$	0	0
$807$	−2.56965	−0.0904560
$808$	0	0
$809$	−38.0421	−1.33749	−0.668745	−	0.743492i	$-0.733168\pi$
$809$	−38.0421	−1.33749	−0.668745	+	0.743492i	$0.733168\pi$
$810$	0	0
$811$	−6.80731	−0.239037	−0.119518	−	0.992832i	$-0.538135\pi$
$811$	−6.80731	−0.239037	−0.119518	+	0.992832i	$0.538135\pi$
$812$	0	0
$813$	3.09810	0.108655
$814$	0	0
$815$	21.3068	0.746345
$816$	0	0
$817$	55.3171	1.93530
$818$	0	0
$819$	5.48242	0.191571
$820$	0	0
$821$	1.22642	0.0428025	0.0214012	−	0.999771i	$-0.493187\pi$
$821$	1.22642	0.0428025	0.0214012	+	0.999771i	$0.493187\pi$
$822$	0	0
$823$	−18.5353	−0.646101	−0.323051	−	0.946382i	$-0.604708\pi$
$823$	−18.5353	−0.646101	−0.323051	+	0.946382i	$0.604708\pi$
$824$	0	0
$825$	−0.800176	−0.0278585
$826$	0	0
$827$	11.7504	0.408603	0.204301	−	0.978908i	$-0.434508\pi$
$827$	11.7504	0.408603	0.204301	+	0.978908i	$0.434508\pi$
$828$	0	0
$829$	1.94117	0.0674195	0.0337097	−	0.999432i	$-0.489268\pi$
$829$	1.94117	0.0674195	0.0337097	+	0.999432i	$0.489268\pi$
$830$	0	0
$831$	1.15175	0.0399539
$832$	0	0
$833$	−32.7365	−1.13425
$834$	0	0
$835$	4.21134	0.145740
$836$	0	0
$837$	−0.891580	−0.0308175
$838$	0	0
$839$	−29.3754	−1.01415	−0.507076	−	0.861901i	$-0.669274\pi$
$839$	−29.3754	−1.01415	−0.507076	+	0.861901i	$0.669274\pi$
$840$	0	0
$841$	−15.2454	−0.525703
$842$	0	0
$843$	2.68499	0.0924760
$844$	0	0
$845$	13.3444	0.459060
$846$	0	0
$847$	−4.90078	−0.168393
$848$	0	0
$849$	−2.28221	−0.0783251
$850$	0	0
$851$	−33.8609	−1.16074
$852$	0	0
$853$	44.9017	1.53741	0.768703	−	0.639606i	$-0.220902\pi$
$853$	44.9017	1.53741	0.768703	+	0.639606i	$0.220902\pi$
$854$	0	0
$855$	−14.4187	−0.493110
$856$	0	0
$857$	40.4810	1.38281	0.691403	−	0.722469i	$-0.256993\pi$
$857$	40.4810	1.38281	0.691403	+	0.722469i	$0.256993\pi$
$858$	0	0
$859$	1.32591	0.0452394	0.0226197	−	0.999744i	$-0.492799\pi$
$859$	1.32591	0.0452394	0.0226197	+	0.999744i	$0.492799\pi$
$860$	0	0
$861$	0.312308	0.0106434
$862$	0	0
$863$	−38.0907	−1.29662	−0.648310	−	0.761376i	$-0.724524\pi$
$863$	−38.0907	−1.29662	−0.648310	+	0.761376i	$0.724524\pi$
$864$	0	0
$865$	8.92819	0.303568
$866$	0	0
$867$	−0.918731	−0.0312017
$868$	0	0
$869$	42.0458	1.42631
$870$	0	0
$871$	−66.8757	−2.26600
$872$	0	0
$873$	−0.237575	−0.00804071
$874$	0	0
$875$	−0.359157	−0.0121417
$876$	0	0
$877$	−11.9447	−0.403342	−0.201671	−	0.979453i	$-0.564637\pi$
$877$	−11.9447	−0.403342	−0.201671	+	0.979453i	$0.564637\pi$
$878$	0	0
$879$	2.94209	0.0992342
$880$	0	0
$881$	−7.79684	−0.262682	−0.131341	−	0.991337i	$-0.541928\pi$
$881$	−7.79684	−0.262682	−0.131341	+	0.991337i	$0.541928\pi$
$882$	0	0
$883$	11.9488	0.402111	0.201055	−	0.979580i	$-0.435563\pi$
$883$	11.9488	0.402111	0.201055	+	0.979580i	$0.435563\pi$
$884$	0	0
$885$	−2.25451	−0.0757845
$886$	0	0
$887$	22.8733	0.768012	0.384006	−	0.923331i	$-0.374544\pi$
$887$	22.8733	0.768012	0.384006	+	0.923331i	$0.374544\pi$
$888$	0	0
$889$	0.451760	0.0151515
$890$	0	0
$891$	43.5221	1.45805
$892$	0	0
$893$	−45.8254	−1.53349
$894$	0	0
$895$	−10.3506	−0.345981
$896$	0	0
$897$	−6.56578	−0.219225
$898$	0	0
$899$	3.43399	0.114530
$900$	0	0
$901$	7.25212	0.241603
$902$	0	0
$903$	0.660511	0.0219804
$904$	0	0
$905$	21.6769	0.720563
$906$	0	0
$907$	−0.348060	−0.0115571	−0.00577857	−	0.999983i	$-0.501839\pi$
$907$	−0.348060	−0.0115571	−0.00577857	+	0.999983i	$0.501839\pi$
$908$	0	0
$909$	−10.0748	−0.334160
$910$	0	0
$911$	38.6460	1.28040	0.640199	−	0.768209i	$-0.278852\pi$
$911$	38.6460	1.28040	0.640199	+	0.768209i	$0.278852\pi$
$912$	0	0
$913$	−49.3177	−1.63218
$914$	0	0
$915$	−0.268329	−0.00887069
$916$	0	0
$917$	6.25051	0.206410
$918$	0	0
$919$	−30.4024	−1.00288	−0.501442	−	0.865191i	$-0.667197\pi$
$919$	−30.4024	−1.00288	−0.501442	+	0.865191i	$0.667197\pi$
$920$	0	0
$921$	3.05707	0.100734
$922$	0	0
$923$	−2.83670	−0.0933712
$924$	0	0
$925$	−4.26654	−0.140283
$926$	0	0
$927$	20.8192	0.683792
$928$	0	0
$929$	45.4005	1.48954	0.744772	−	0.667319i	$-0.232558\pi$
$929$	45.4005	1.48954	0.744772	+	0.667319i	$0.232558\pi$
$930$	0	0
$931$	−33.3122	−1.09176
$932$	0	0
$933$	−4.17410	−0.136654
$934$	0	0
$935$	23.6526	0.773522
$936$	0	0
$937$	−3.84170	−0.125503	−0.0627515	−	0.998029i	$-0.519988\pi$
$937$	−3.84170	−0.125503	−0.0627515	+	0.998029i	$0.519988\pi$
$938$	0	0
$939$	−0.195690	−0.00638609
$940$	0	0
$941$	3.12903	0.102004	0.0510018	−	0.998699i	$-0.483759\pi$
$941$	3.12903	0.102004	0.0510018	+	0.998699i	$0.483759\pi$
$942$	0	0
$943$	42.8156	1.39427
$944$	0	0
$945$	−0.345836	−0.0112501
$946$	0	0
$947$	30.8473	1.00240	0.501201	−	0.865331i	$-0.332892\pi$
$947$	30.8473	1.00240	0.501201	+	0.865331i	$0.332892\pi$
$948$	0	0
$949$	−26.4877	−0.859826
$950$	0	0
$951$	−0.937017	−0.0303849
$952$	0	0
$953$	44.6924	1.44773	0.723864	−	0.689943i	$-0.242364\pi$
$953$	44.6924	1.44773	0.723864	+	0.689943i	$0.242364\pi$
$954$	0	0
$955$	3.92452	0.126995
$956$	0	0
$957$	2.96763	0.0959298
$958$	0	0
$959$	−1.20985	−0.0390680
$960$	0	0
$961$	−30.1427	−0.972344
$962$	0	0
$963$	41.6033	1.34065
$964$	0	0
$965$	21.2775	0.684947
$966$	0	0
$967$	0.380334	0.0122307	0.00611536	−	0.999981i	$-0.498053\pi$
$967$	0.380334	0.0122307	0.00611536	+	0.999981i	$0.498053\pi$
$968$	0	0
$969$	−3.72318	−0.119606
$970$	0	0
$971$	54.5227	1.74972	0.874858	−	0.484380i	$-0.160955\pi$
$971$	54.5227	1.74972	0.874858	+	0.484380i	$0.160955\pi$
$972$	0	0
$973$	−7.88759	−0.252864
$974$	0	0
$975$	−0.827300	−0.0264948
$976$	0	0
$977$	23.8047	0.761579	0.380789	−	0.924662i	$-0.375652\pi$
$977$	23.8047	0.761579	0.380789	+	0.924662i	$0.375652\pi$
$978$	0	0
$979$	−70.2456	−2.24506
$980$	0	0
$981$	−20.9043	−0.667422
$982$	0	0
$983$	10.0880	0.321758	0.160879	−	0.986974i	$-0.448567\pi$
$983$	10.0880	0.321758	0.160879	+	0.986974i	$0.448567\pi$
$984$	0	0
$985$	20.5809	0.655762
$986$	0	0
$987$	−0.547175	−0.0174168
$988$	0	0
$989$	90.5523	2.87940
$990$	0	0
$991$	22.4718	0.713840	0.356920	−	0.934135i	$-0.383827\pi$
$991$	22.4718	0.713840	0.356920	+	0.934135i	$0.383827\pi$
$992$	0	0
$993$	4.83309	0.153374
$994$	0	0
$995$	6.23071	0.197527
$996$	0	0
$997$	26.9528	0.853602	0.426801	−	0.904346i	$-0.359640\pi$
$997$	26.9528	0.853602	0.426801	+	0.904346i	$0.359640\pi$
$998$	0	0
$999$	−4.10829	−0.129981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.18	✓	37

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

\(f(q)\)	\(=\)	\(q-0.161183 q^{3} +1.00000 q^{5} -0.359157 q^{7} -2.97402 q^{9} +O(q^{10})\)	\(q-0.161183 q^{3} +1.00000 q^{5} -0.359157 q^{7} -2.97402 q^{9} +4.96439 q^{11} +5.13268 q^{13} -0.161183 q^{15} +4.76444 q^{17} +4.84823 q^{19} +0.0578901 q^{21} +7.93639 q^{23} +1.00000 q^{25} +0.962910 q^{27} -3.70872 q^{29} -0.925922 q^{31} -0.800176 q^{33} -0.359157 q^{35} -4.26654 q^{37} -0.827300 q^{39} +5.39484 q^{41} +11.4098 q^{43} -2.97402 q^{45} -9.45198 q^{47} -6.87101 q^{49} -0.767947 q^{51} +1.52213 q^{53} +4.96439 q^{55} -0.781452 q^{57} +13.9873 q^{59} +1.66475 q^{61} +1.06814 q^{63} +5.13268 q^{65} -13.0294 q^{67} -1.27921 q^{69} -0.552675 q^{71} -5.16060 q^{73} -0.161183 q^{75} -1.78300 q^{77} +8.46947 q^{79} +8.76686 q^{81} -9.93429 q^{83} +4.76444 q^{85} +0.597783 q^{87} -14.1499 q^{89} -1.84344 q^{91} +0.149243 q^{93} +4.84823 q^{95} +0.0798836 q^{97} -14.7642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 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

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Groups

Database

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