LMFDB - Embedded newform 4012.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75827 q^{3} +0.760103 q^{5} +0.427165 q^{7} +4.60804 q^{9} +O(q^{10})$	$q-2.75827 q^{3} +0.760103 q^{5} +0.427165 q^{7} +4.60804 q^{9} -3.25351 q^{11} -6.57262 q^{13} -2.09657 q^{15} +1.00000 q^{17} -0.578623 q^{19} -1.17823 q^{21} -2.85099 q^{23} -4.42224 q^{25} -4.43540 q^{27} -6.79984 q^{29} +9.48688 q^{31} +8.97406 q^{33} +0.324689 q^{35} +6.42717 q^{37} +18.1290 q^{39} -12.1378 q^{41} +0.645866 q^{43} +3.50259 q^{45} +12.2631 q^{47} -6.81753 q^{49} -2.75827 q^{51} +6.10341 q^{53} -2.47301 q^{55} +1.59600 q^{57} +1.00000 q^{59} -4.26500 q^{61} +1.96839 q^{63} -4.99587 q^{65} -10.8674 q^{67} +7.86378 q^{69} +16.0101 q^{71} -6.67123 q^{73} +12.1977 q^{75} -1.38979 q^{77} +0.0868161 q^{79} -1.59009 q^{81} -10.2997 q^{83} +0.760103 q^{85} +18.7558 q^{87} -4.56317 q^{89} -2.80759 q^{91} -26.1674 q^{93} -0.439814 q^{95} -10.9753 q^{97} -14.9923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	−2.75827	−1.59249	−0.796243	−	0.604977i	$-0.793182\pi$
$3$	−2.75827	−1.59249	−0.796243	+	0.604977i	$0.793182\pi$
$4$	0	0
$5$	0.760103	0.339929	0.169964	−	0.985450i	$-0.445635\pi$
$5$	0.760103	0.339929	0.169964	+	0.985450i	$0.445635\pi$
$6$	0	0
$7$	0.427165	0.161453	0.0807266	−	0.996736i	$-0.474276\pi$
$7$	0.427165	0.161453	0.0807266	+	0.996736i	$0.474276\pi$
$8$	0	0
$9$	4.60804	1.53601
$10$	0	0
$11$	−3.25351	−0.980971	−0.490485	−	0.871449i	$-0.663181\pi$
$11$	−3.25351	−0.980971	−0.490485	+	0.871449i	$0.663181\pi$
$12$	0	0
$13$	−6.57262	−1.82292	−0.911459	−	0.411392i	$-0.865043\pi$
$13$	−6.57262	−1.82292	−0.911459	+	0.411392i	$0.865043\pi$
$14$	0	0
$15$	−2.09657	−0.541332
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.578623	−0.132745	−0.0663727	−	0.997795i	$-0.521143\pi$
$19$	−0.578623	−0.132745	−0.0663727	+	0.997795i	$0.521143\pi$
$20$	0	0
$21$	−1.17823	−0.257112
$22$	0	0
$23$	−2.85099	−0.594472	−0.297236	−	0.954804i	$-0.596065\pi$
$23$	−2.85099	−0.594472	−0.297236	+	0.954804i	$0.596065\pi$
$24$	0	0
$25$	−4.42224	−0.884449
$26$	0	0
$27$	−4.43540	−0.853593
$28$	0	0
$29$	−6.79984	−1.26270	−0.631349	−	0.775499i	$-0.717498\pi$
$29$	−6.79984	−1.26270	−0.631349	+	0.775499i	$0.717498\pi$
$30$	0	0
$31$	9.48688	1.70389	0.851947	−	0.523628i	$-0.175422\pi$
$31$	9.48688	1.70389	0.851947	+	0.523628i	$0.175422\pi$
$32$	0	0
$33$	8.97406	1.56218
$34$	0	0
$35$	0.324689	0.0548825
$36$	0	0
$37$	6.42717	1.05662	0.528310	−	0.849051i	$-0.322826\pi$
$37$	6.42717	1.05662	0.528310	+	0.849051i	$0.322826\pi$
$38$	0	0
$39$	18.1290	2.90297
$40$	0	0
$41$	−12.1378	−1.89561	−0.947805	−	0.318849i	$-0.896704\pi$
$41$	−12.1378	−1.89561	−0.947805	+	0.318849i	$0.896704\pi$
$42$	0	0
$43$	0.645866	0.0984937	0.0492468	−	0.998787i	$-0.484318\pi$
$43$	0.645866	0.0984937	0.0492468	+	0.998787i	$0.484318\pi$
$44$	0	0
$45$	3.50259	0.522135
$46$	0	0
$47$	12.2631	1.78876	0.894378	−	0.447311i	$-0.147618\pi$
$47$	12.2631	1.78876	0.894378	+	0.447311i	$0.147618\pi$
$48$	0	0
$49$	−6.81753	−0.973933
$50$	0	0
$51$	−2.75827	−0.386235
$52$	0	0
$53$	6.10341	0.838367	0.419184	−	0.907901i	$-0.362316\pi$
$53$	6.10341	0.838367	0.419184	+	0.907901i	$0.362316\pi$
$54$	0	0
$55$	−2.47301	−0.333460
$56$	0	0
$57$	1.59600	0.211395
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−4.26500	−0.546077	−0.273039	−	0.962003i	$-0.588029\pi$
$61$	−4.26500	−0.546077	−0.273039	+	0.962003i	$0.588029\pi$
$62$	0	0
$63$	1.96839	0.247994
$64$	0	0
$65$	−4.99587	−0.619662
$66$	0	0
$67$	−10.8674	−1.32766	−0.663832	−	0.747882i	$-0.731071\pi$
$67$	−10.8674	−1.32766	−0.663832	+	0.747882i	$0.731071\pi$
$68$	0	0
$69$	7.86378	0.946688
$70$	0	0
$71$	16.0101	1.90005	0.950024	−	0.312176i	$-0.101058\pi$
$71$	16.0101	1.90005	0.950024	+	0.312176i	$0.101058\pi$
$72$	0	0
$73$	−6.67123	−0.780809	−0.390404	−	0.920644i	$-0.627665\pi$
$73$	−6.67123	−0.780809	−0.390404	+	0.920644i	$0.627665\pi$
$74$	0	0
$75$	12.1977	1.40847
$76$	0	0
$77$	−1.38979	−0.158381
$78$	0	0
$79$	0.0868161	0.00976758	0.00488379	−	0.999988i	$-0.498445\pi$
$79$	0.0868161	0.00976758	0.00488379	+	0.999988i	$0.498445\pi$
$80$	0	0
$81$	−1.59009	−0.176677
$82$	0	0
$83$	−10.2997	−1.13054	−0.565272	−	0.824905i	$-0.691229\pi$
$83$	−10.2997	−1.13054	−0.565272	+	0.824905i	$0.691229\pi$
$84$	0	0
$85$	0.760103	0.0824448
$86$	0	0
$87$	18.7558	2.01083
$88$	0	0
$89$	−4.56317	−0.483695	−0.241848	−	0.970314i	$-0.577753\pi$
$89$	−4.56317	−0.483695	−0.241848	+	0.970314i	$0.577753\pi$
$90$	0	0
$91$	−2.80759	−0.294316
$92$	0	0
$93$	−26.1674	−2.71343
$94$	0	0
$95$	−0.439814	−0.0451239
$96$	0	0
$97$	−10.9753	−1.11437	−0.557185	−	0.830388i	$-0.688119\pi$
$97$	−10.9753	−1.11437	−0.557185	+	0.830388i	$0.688119\pi$
$98$	0	0
$99$	−14.9923	−1.50678
$100$	0	0
$101$	6.98621	0.695154	0.347577	−	0.937652i	$-0.387005\pi$
$101$	6.98621	0.695154	0.347577	+	0.937652i	$0.387005\pi$
$102$	0	0
$103$	12.0883	1.19110	0.595550	−	0.803318i	$-0.296934\pi$
$103$	12.0883	1.19110	0.595550	+	0.803318i	$0.296934\pi$
$104$	0	0
$105$	−0.895580	−0.0873997
$106$	0	0
$107$	−13.0235	−1.25903	−0.629514	−	0.776989i	$-0.716746\pi$
$107$	−13.0235	−1.25903	−0.629514	+	0.776989i	$0.716746\pi$
$108$	0	0
$109$	−1.44352	−0.138264	−0.0691322	−	0.997608i	$-0.522023\pi$
$109$	−1.44352	−0.138264	−0.0691322	+	0.997608i	$0.522023\pi$
$110$	0	0
$111$	−17.7279	−1.68265
$112$	0	0
$113$	18.4167	1.73250	0.866249	−	0.499613i	$-0.166524\pi$
$113$	18.4167	1.73250	0.866249	+	0.499613i	$0.166524\pi$
$114$	0	0
$115$	−2.16704	−0.202078
$116$	0	0
$117$	−30.2869	−2.80002
$118$	0	0
$119$	0.427165	0.0391581
$120$	0	0
$121$	−0.414659	−0.0376963
$122$	0	0
$123$	33.4794	3.01873
$124$	0	0
$125$	−7.16188	−0.640578
$126$	0	0
$127$	4.02392	0.357065	0.178533	−	0.983934i	$-0.442865\pi$
$127$	4.02392	0.357065	0.178533	+	0.983934i	$0.442865\pi$
$128$	0	0
$129$	−1.78147	−0.156850
$130$	0	0
$131$	4.88068	0.426427	0.213214	−	0.977006i	$-0.431607\pi$
$131$	4.88068	0.426427	0.213214	+	0.977006i	$0.431607\pi$
$132$	0	0
$133$	−0.247168	−0.0214321
$134$	0	0
$135$	−3.37136	−0.290161
$136$	0	0
$137$	−3.06002	−0.261435	−0.130718	−	0.991420i	$-0.541728\pi$
$137$	−3.06002	−0.261435	−0.130718	+	0.991420i	$0.541728\pi$
$138$	0	0
$139$	5.42667	0.460284	0.230142	−	0.973157i	$-0.426081\pi$
$139$	5.42667	0.460284	0.230142	+	0.973157i	$0.426081\pi$
$140$	0	0
$141$	−33.8249	−2.84857
$142$	0	0
$143$	21.3841	1.78823
$144$	0	0
$145$	−5.16858	−0.429227
$146$	0	0
$147$	18.8046	1.55097
$148$	0	0
$149$	16.8366	1.37931	0.689655	−	0.724139i	$-0.257762\pi$
$149$	16.8366	1.37931	0.689655	+	0.724139i	$0.257762\pi$
$150$	0	0
$151$	14.1625	1.15252	0.576262	−	0.817265i	$-0.304511\pi$
$151$	14.1625	1.15252	0.576262	+	0.817265i	$0.304511\pi$
$152$	0	0
$153$	4.60804	0.372538
$154$	0	0
$155$	7.21101	0.579202
$156$	0	0
$157$	3.05372	0.243714	0.121857	−	0.992548i	$-0.461115\pi$
$157$	3.05372	0.243714	0.121857	+	0.992548i	$0.461115\pi$
$158$	0	0
$159$	−16.8348	−1.33509
$160$	0	0
$161$	−1.21784	−0.0959793
$162$	0	0
$163$	9.30695	0.728977	0.364488	−	0.931208i	$-0.381244\pi$
$163$	9.30695	0.728977	0.364488	+	0.931208i	$0.381244\pi$
$164$	0	0
$165$	6.82121	0.531030
$166$	0	0
$167$	−0.909313	−0.0703647	−0.0351824	−	0.999381i	$-0.511201\pi$
$167$	−0.909313	−0.0703647	−0.0351824	+	0.999381i	$0.511201\pi$
$168$	0	0
$169$	30.1994	2.32303
$170$	0	0
$171$	−2.66632	−0.203899
$172$	0	0
$173$	22.2195	1.68932	0.844658	−	0.535307i	$-0.179804\pi$
$173$	22.2195	1.68932	0.844658	+	0.535307i	$0.179804\pi$
$174$	0	0
$175$	−1.88903	−0.142797
$176$	0	0
$177$	−2.75827	−0.207324
$178$	0	0
$179$	−16.3819	−1.22444	−0.612221	−	0.790687i	$-0.709724\pi$
$179$	−16.3819	−1.22444	−0.612221	+	0.790687i	$0.709724\pi$
$180$	0	0
$181$	−6.53079	−0.485430	−0.242715	−	0.970098i	$-0.578038\pi$
$181$	−6.53079	−0.485430	−0.242715	+	0.970098i	$0.578038\pi$
$182$	0	0
$183$	11.7640	0.869621
$184$	0	0
$185$	4.88532	0.359176
$186$	0	0
$187$	−3.25351	−0.237920
$188$	0	0
$189$	−1.89465	−0.137815
$190$	0	0
$191$	−25.6612	−1.85678	−0.928390	−	0.371607i	$-0.878807\pi$
$191$	−25.6612	−1.85678	−0.928390	+	0.371607i	$0.878807\pi$
$192$	0	0
$193$	15.6747	1.12829	0.564145	−	0.825676i	$-0.309206\pi$
$193$	15.6747	1.12829	0.564145	+	0.825676i	$0.309206\pi$
$194$	0	0
$195$	13.7799	0.986803
$196$	0	0
$197$	−3.20888	−0.228623	−0.114312	−	0.993445i	$-0.536466\pi$
$197$	−3.20888	−0.228623	−0.114312	+	0.993445i	$0.536466\pi$
$198$	0	0
$199$	11.8693	0.841394	0.420697	−	0.907201i	$-0.361785\pi$
$199$	11.8693	0.841394	0.420697	+	0.907201i	$0.361785\pi$
$200$	0	0
$201$	29.9752	2.11429
$202$	0	0
$203$	−2.90465	−0.203867
$204$	0	0
$205$	−9.22601	−0.644372
$206$	0	0
$207$	−13.1375	−0.913116
$208$	0	0
$209$	1.88256	0.130219
$210$	0	0
$211$	−11.9186	−0.820508	−0.410254	−	0.911971i	$-0.634560\pi$
$211$	−11.9186	−0.820508	−0.410254	+	0.911971i	$0.634560\pi$
$212$	0	0
$213$	−44.1601	−3.02580
$214$	0	0
$215$	0.490925	0.0334808
$216$	0	0
$217$	4.05246	0.275099
$218$	0	0
$219$	18.4010	1.24343
$220$	0	0
$221$	−6.57262	−0.442122
$222$	0	0
$223$	9.42763	0.631321	0.315661	−	0.948872i	$-0.397774\pi$
$223$	9.42763	0.631321	0.315661	+	0.948872i	$0.397774\pi$
$224$	0	0
$225$	−20.3779	−1.35852
$226$	0	0
$227$	26.0310	1.72774	0.863870	−	0.503716i	$-0.168034\pi$
$227$	26.0310	1.72774	0.863870	+	0.503716i	$0.168034\pi$
$228$	0	0
$229$	15.1755	1.00282	0.501411	−	0.865209i	$-0.332814\pi$
$229$	15.1755	1.00282	0.501411	+	0.865209i	$0.332814\pi$
$230$	0	0
$231$	3.83340	0.252219
$232$	0	0
$233$	−10.4010	−0.681392	−0.340696	−	0.940173i	$-0.610663\pi$
$233$	−10.4010	−0.681392	−0.340696	+	0.940173i	$0.610663\pi$
$234$	0	0
$235$	9.32122	0.608049
$236$	0	0
$237$	−0.239462	−0.0155547
$238$	0	0
$239$	15.4247	0.997744	0.498872	−	0.866676i	$-0.333748\pi$
$239$	15.4247	0.997744	0.498872	+	0.866676i	$0.333748\pi$
$240$	0	0
$241$	9.03815	0.582198	0.291099	−	0.956693i	$-0.405979\pi$
$241$	9.03815	0.582198	0.291099	+	0.956693i	$0.405979\pi$
$242$	0	0
$243$	17.6921	1.13495
$244$	0	0
$245$	−5.18203	−0.331068
$246$	0	0
$247$	3.80307	0.241984
$248$	0	0
$249$	28.4094	1.80038
$250$	0	0
$251$	−23.2081	−1.46488	−0.732442	−	0.680830i	$-0.761619\pi$
$251$	−23.2081	−1.46488	−0.732442	+	0.680830i	$0.761619\pi$
$252$	0	0
$253$	9.27572	0.583159
$254$	0	0
$255$	−2.09657	−0.131292
$256$	0	0
$257$	21.0761	1.31469	0.657344	−	0.753590i	$-0.271680\pi$
$257$	21.0761	1.31469	0.657344	+	0.753590i	$0.271680\pi$
$258$	0	0
$259$	2.74546	0.170595
$260$	0	0
$261$	−31.3339	−1.93952
$262$	0	0
$263$	11.7525	0.724690	0.362345	−	0.932044i	$-0.381976\pi$
$263$	11.7525	0.724690	0.362345	+	0.932044i	$0.381976\pi$
$264$	0	0
$265$	4.63922	0.284985
$266$	0	0
$267$	12.5865	0.770278
$268$	0	0
$269$	−6.77343	−0.412984	−0.206492	−	0.978448i	$-0.566205\pi$
$269$	−6.77343	−0.412984	−0.206492	+	0.978448i	$0.566205\pi$
$270$	0	0
$271$	−19.5858	−1.18975	−0.594877	−	0.803817i	$-0.702799\pi$
$271$	−19.5858	−1.18975	−0.594877	+	0.803817i	$0.702799\pi$
$272$	0	0
$273$	7.74409	0.468694
$274$	0	0
$275$	14.3878	0.867618
$276$	0	0
$277$	20.9573	1.25920	0.629601	−	0.776919i	$-0.283218\pi$
$277$	20.9573	1.25920	0.629601	+	0.776919i	$0.283218\pi$
$278$	0	0
$279$	43.7159	2.61720
$280$	0	0
$281$	1.14610	0.0683708	0.0341854	−	0.999416i	$-0.489116\pi$
$281$	1.14610	0.0683708	0.0341854	+	0.999416i	$0.489116\pi$
$282$	0	0
$283$	10.0380	0.596695	0.298348	−	0.954457i	$-0.403565\pi$
$283$	10.0380	0.596695	0.298348	+	0.954457i	$0.403565\pi$
$284$	0	0
$285$	1.21312	0.0718592
$286$	0	0
$287$	−5.18485	−0.306052
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	30.2727	1.77462
$292$	0	0
$293$	−6.06981	−0.354602	−0.177301	−	0.984157i	$-0.556737\pi$
$293$	−6.06981	−0.354602	−0.177301	+	0.984157i	$0.556737\pi$
$294$	0	0
$295$	0.760103	0.0442549
$296$	0	0
$297$	14.4306	0.837350
$298$	0	0
$299$	18.7385	1.08367
$300$	0	0
$301$	0.275891	0.0159021
$302$	0	0
$303$	−19.2698	−1.10702
$304$	0	0
$305$	−3.24184	−0.185627
$306$	0	0
$307$	−21.7560	−1.24168	−0.620839	−	0.783938i	$-0.713208\pi$
$307$	−21.7560	−1.24168	−0.620839	+	0.783938i	$0.713208\pi$
$308$	0	0
$309$	−33.3429	−1.89681
$310$	0	0
$311$	29.4554	1.67026	0.835131	−	0.550051i	$-0.185392\pi$
$311$	29.4554	1.67026	0.835131	+	0.550051i	$0.185392\pi$
$312$	0	0
$313$	4.20696	0.237792	0.118896	−	0.992907i	$-0.462065\pi$
$313$	4.20696	0.237792	0.118896	+	0.992907i	$0.462065\pi$
$314$	0	0
$315$	1.49618	0.0843003
$316$	0	0
$317$	1.11028	0.0623594	0.0311797	−	0.999514i	$-0.490074\pi$
$317$	1.11028	0.0623594	0.0311797	+	0.999514i	$0.490074\pi$
$318$	0	0
$319$	22.1234	1.23867
$320$	0	0
$321$	35.9222	2.00498
$322$	0	0
$323$	−0.578623	−0.0321955
$324$	0	0
$325$	29.0657	1.61228
$326$	0	0
$327$	3.98162	0.220184
$328$	0	0
$329$	5.23836	0.288800
$330$	0	0
$331$	13.8251	0.759895	0.379947	−	0.925008i	$-0.375942\pi$
$331$	13.8251	0.759895	0.379947	+	0.925008i	$0.375942\pi$
$332$	0	0
$333$	29.6167	1.62298
$334$	0	0
$335$	−8.26034	−0.451311
$336$	0	0
$337$	21.3940	1.16540	0.582702	−	0.812686i	$-0.301996\pi$
$337$	21.3940	1.16540	0.582702	+	0.812686i	$0.301996\pi$
$338$	0	0
$339$	−50.7982	−2.75898
$340$	0	0
$341$	−30.8657	−1.67147
$342$	0	0
$343$	−5.90236	−0.318698
$344$	0	0
$345$	5.97729	0.321806
$346$	0	0
$347$	2.21422	0.118866	0.0594328	−	0.998232i	$-0.481071\pi$
$347$	2.21422	0.118866	0.0594328	+	0.998232i	$0.481071\pi$
$348$	0	0
$349$	0.699955	0.0374677	0.0187339	−	0.999825i	$-0.494036\pi$
$349$	0.699955	0.0374677	0.0187339	+	0.999825i	$0.494036\pi$
$350$	0	0
$351$	29.1522	1.55603
$352$	0	0
$353$	33.7170	1.79457	0.897286	−	0.441449i	$-0.145536\pi$
$353$	33.7170	1.79457	0.897286	+	0.441449i	$0.145536\pi$
$354$	0	0
$355$	12.1693	0.645881
$356$	0	0
$357$	−1.17823	−0.0623588
$358$	0	0
$359$	−6.45125	−0.340484	−0.170242	−	0.985402i	$-0.554455\pi$
$359$	−6.45125	−0.340484	−0.170242	+	0.985402i	$0.554455\pi$
$360$	0	0
$361$	−18.6652	−0.982379
$362$	0	0
$363$	1.14374	0.0600308
$364$	0	0
$365$	−5.07083	−0.265419
$366$	0	0
$367$	35.2564	1.84037	0.920184	−	0.391486i	$-0.128039\pi$
$367$	35.2564	1.84037	0.920184	+	0.391486i	$0.128039\pi$
$368$	0	0
$369$	−55.9316	−2.91168
$370$	0	0
$371$	2.60716	0.135357
$372$	0	0
$373$	33.5843	1.73893	0.869466	−	0.493993i	$-0.164463\pi$
$373$	33.5843	1.73893	0.869466	+	0.493993i	$0.164463\pi$
$374$	0	0
$375$	19.7544	1.02011
$376$	0	0
$377$	44.6928	2.30179
$378$	0	0
$379$	23.1217	1.18768	0.593841	−	0.804583i	$-0.297611\pi$
$379$	23.1217	1.18768	0.593841	+	0.804583i	$0.297611\pi$
$380$	0	0
$381$	−11.0991	−0.568622
$382$	0	0
$383$	−21.6744	−1.10751	−0.553755	−	0.832679i	$-0.686806\pi$
$383$	−21.6744	−1.10751	−0.553755	+	0.832679i	$0.686806\pi$
$384$	0	0
$385$	−1.05638	−0.0538382
$386$	0	0
$387$	2.97618	0.151288
$388$	0	0
$389$	14.0266	0.711177	0.355589	−	0.934643i	$-0.384280\pi$
$389$	14.0266	0.711177	0.355589	+	0.934643i	$0.384280\pi$
$390$	0	0
$391$	−2.85099	−0.144181
$392$	0	0
$393$	−13.4622	−0.679080
$394$	0	0
$395$	0.0659892	0.00332028
$396$	0	0
$397$	−37.0963	−1.86181	−0.930905	−	0.365260i	$-0.880980\pi$
$397$	−37.0963	−1.86181	−0.930905	+	0.365260i	$0.880980\pi$
$398$	0	0
$399$	0.681754	0.0341304
$400$	0	0
$401$	20.8503	1.04122	0.520608	−	0.853796i	$-0.325705\pi$
$401$	20.8503	1.04122	0.520608	+	0.853796i	$0.325705\pi$
$402$	0	0
$403$	−62.3537	−3.10606
$404$	0	0
$405$	−1.20864	−0.0600576
$406$	0	0
$407$	−20.9109	−1.03651
$408$	0	0
$409$	−12.2050	−0.603500	−0.301750	−	0.953387i	$-0.597571\pi$
$409$	−12.2050	−0.603500	−0.301750	+	0.953387i	$0.597571\pi$
$410$	0	0
$411$	8.44035	0.416332
$412$	0	0
$413$	0.427165	0.0210194
$414$	0	0
$415$	−7.82887	−0.384304
$416$	0	0
$417$	−14.9682	−0.732996
$418$	0	0
$419$	32.5656	1.59093	0.795467	−	0.605997i	$-0.207226\pi$
$419$	32.5656	1.59093	0.795467	+	0.605997i	$0.207226\pi$
$420$	0	0
$421$	−30.9372	−1.50779	−0.753894	−	0.656997i	$-0.771827\pi$
$421$	−30.9372	−1.50779	−0.753894	+	0.656997i	$0.771827\pi$
$422$	0	0
$423$	56.5088	2.74755
$424$	0	0
$425$	−4.42224	−0.214510
$426$	0	0
$427$	−1.82186	−0.0881659
$428$	0	0
$429$	−58.9831	−2.84773
$430$	0	0
$431$	−21.7133	−1.04589	−0.522946	−	0.852366i	$-0.675167\pi$
$431$	−21.7133	−1.04589	−0.522946	+	0.852366i	$0.675167\pi$
$432$	0	0
$433$	−23.6501	−1.13655	−0.568276	−	0.822838i	$-0.692389\pi$
$433$	−23.6501	−1.13655	−0.568276	+	0.822838i	$0.692389\pi$
$434$	0	0
$435$	14.2563	0.683538
$436$	0	0
$437$	1.64965	0.0789133
$438$	0	0
$439$	7.30540	0.348668	0.174334	−	0.984687i	$-0.444223\pi$
$439$	7.30540	0.348668	0.174334	+	0.984687i	$0.444223\pi$
$440$	0	0
$441$	−31.4154	−1.49597
$442$	0	0
$443$	−17.9412	−0.852414	−0.426207	−	0.904626i	$-0.640150\pi$
$443$	−17.9412	−0.852414	−0.426207	+	0.904626i	$0.640150\pi$
$444$	0	0
$445$	−3.46848	−0.164422
$446$	0	0
$447$	−46.4399	−2.19653
$448$	0	0
$449$	7.16896	0.338324	0.169162	−	0.985588i	$-0.445894\pi$
$449$	7.16896	0.338324	0.169162	+	0.985588i	$0.445894\pi$
$450$	0	0
$451$	39.4906	1.85954
$452$	0	0
$453$	−39.0639	−1.83538
$454$	0	0
$455$	−2.13406	−0.100046
$456$	0	0
$457$	−28.1037	−1.31464	−0.657319	−	0.753613i	$-0.728309\pi$
$457$	−28.1037	−1.31464	−0.657319	+	0.753613i	$0.728309\pi$
$458$	0	0
$459$	−4.43540	−0.207027
$460$	0	0
$461$	0.848995	0.0395416	0.0197708	−	0.999805i	$-0.493706\pi$
$461$	0.848995	0.0395416	0.0197708	+	0.999805i	$0.493706\pi$
$462$	0	0
$463$	36.8692	1.71345	0.856727	−	0.515769i	$-0.172494\pi$
$463$	36.8692	1.71345	0.856727	+	0.515769i	$0.172494\pi$
$464$	0	0
$465$	−19.8899	−0.922372
$466$	0	0
$467$	−17.9134	−0.828935	−0.414467	−	0.910064i	$-0.636032\pi$
$467$	−17.9134	−0.828935	−0.414467	+	0.910064i	$0.636032\pi$
$468$	0	0
$469$	−4.64217	−0.214355
$470$	0	0
$471$	−8.42298	−0.388111
$472$	0	0
$473$	−2.10133	−0.0966194
$474$	0	0
$475$	2.55881	0.117406
$476$	0	0
$477$	28.1247	1.28774
$478$	0	0
$479$	3.52295	0.160968	0.0804838	−	0.996756i	$-0.474353\pi$
$479$	3.52295	0.160968	0.0804838	+	0.996756i	$0.474353\pi$
$480$	0	0
$481$	−42.2434	−1.92613
$482$	0	0
$483$	3.35913	0.152846
$484$	0	0
$485$	−8.34234	−0.378806
$486$	0	0
$487$	30.6312	1.38803	0.694015	−	0.719960i	$-0.255840\pi$
$487$	30.6312	1.38803	0.694015	+	0.719960i	$0.255840\pi$
$488$	0	0
$489$	−25.6711	−1.16089
$490$	0	0
$491$	−29.7410	−1.34219	−0.671096	−	0.741371i	$-0.734176\pi$
$491$	−29.7410	−1.34219	−0.671096	+	0.741371i	$0.734176\pi$
$492$	0	0
$493$	−6.79984	−0.306249
$494$	0	0
$495$	−11.3957	−0.512199
$496$	0	0
$497$	6.83895	0.306769
$498$	0	0
$499$	−18.2622	−0.817528	−0.408764	−	0.912640i	$-0.634040\pi$
$499$	−18.2622	−0.817528	−0.408764	+	0.912640i	$0.634040\pi$
$500$	0	0
$501$	2.50813	0.112055
$502$	0	0
$503$	9.01103	0.401782	0.200891	−	0.979614i	$-0.435616\pi$
$503$	9.01103	0.401782	0.200891	+	0.979614i	$0.435616\pi$
$504$	0	0
$505$	5.31024	0.236303
$506$	0	0
$507$	−83.2979	−3.69939
$508$	0	0
$509$	21.2907	0.943693	0.471847	−	0.881681i	$-0.343588\pi$
$509$	21.2907	0.943693	0.471847	+	0.881681i	$0.343588\pi$
$510$	0	0
$511$	−2.84972	−0.126064
$512$	0	0
$513$	2.56643	0.113311
$514$	0	0
$515$	9.18839	0.404889
$516$	0	0
$517$	−39.8981	−1.75472
$518$	0	0
$519$	−61.2873	−2.69021
$520$	0	0
$521$	0.872248	0.0382139	0.0191069	−	0.999817i	$-0.493918\pi$
$521$	0.872248	0.0382139	0.0191069	+	0.999817i	$0.493918\pi$
$522$	0	0
$523$	−40.6937	−1.77941	−0.889705	−	0.456537i	$-0.849090\pi$
$523$	−40.6937	−1.77941	−0.889705	+	0.456537i	$0.849090\pi$
$524$	0	0
$525$	5.21044	0.227402
$526$	0	0
$527$	9.48688	0.413255
$528$	0	0
$529$	−14.8719	−0.646603
$530$	0	0
$531$	4.60804	0.199972
$532$	0	0
$533$	79.7774	3.45554
$534$	0	0
$535$	−9.89919	−0.427979
$536$	0	0
$537$	45.1857	1.94991
$538$	0	0
$539$	22.1809	0.955400
$540$	0	0
$541$	−22.9242	−0.985590	−0.492795	−	0.870145i	$-0.664025\pi$
$541$	−22.9242	−0.985590	−0.492795	+	0.870145i	$0.664025\pi$
$542$	0	0
$543$	18.0137	0.773040
$544$	0	0
$545$	−1.09723	−0.0470000
$546$	0	0
$547$	2.23912	0.0957379	0.0478689	−	0.998854i	$-0.484757\pi$
$547$	2.23912	0.0957379	0.0478689	+	0.998854i	$0.484757\pi$
$548$	0	0
$549$	−19.6533	−0.838782
$550$	0	0
$551$	3.93455	0.167617
$552$	0	0
$553$	0.0370848	0.00157701
$554$	0	0
$555$	−13.4750	−0.571982
$556$	0	0
$557$	45.0544	1.90902	0.954508	−	0.298185i	$-0.0963811\pi$
$557$	45.0544	1.90902	0.954508	+	0.298185i	$0.0963811\pi$
$558$	0	0
$559$	−4.24503	−0.179546
$560$	0	0
$561$	8.97406	0.378885
$562$	0	0
$563$	39.1546	1.65017	0.825084	−	0.565010i	$-0.191128\pi$
$563$	39.1546	1.65017	0.825084	+	0.565010i	$0.191128\pi$
$564$	0	0
$565$	13.9986	0.588925
$566$	0	0
$567$	−0.679233	−0.0285251
$568$	0	0
$569$	4.32609	0.181359	0.0906795	−	0.995880i	$-0.471096\pi$
$569$	4.32609	0.181359	0.0906795	+	0.995880i	$0.471096\pi$
$570$	0	0
$571$	−14.6179	−0.611738	−0.305869	−	0.952074i	$-0.598947\pi$
$571$	−14.6179	−0.611738	−0.305869	+	0.952074i	$0.598947\pi$
$572$	0	0
$573$	70.7805	2.95690
$574$	0	0
$575$	12.6078	0.525780
$576$	0	0
$577$	−6.73424	−0.280350	−0.140175	−	0.990127i	$-0.544767\pi$
$577$	−6.73424	−0.280350	−0.140175	+	0.990127i	$0.544767\pi$
$578$	0	0
$579$	−43.2350	−1.79679
$580$	0	0
$581$	−4.39969	−0.182530
$582$	0	0
$583$	−19.8575	−0.822414
$584$	0	0
$585$	−23.0212	−0.951808
$586$	0	0
$587$	37.2513	1.53752	0.768762	−	0.639535i	$-0.220873\pi$
$587$	37.2513	1.53752	0.768762	+	0.639535i	$0.220873\pi$
$588$	0	0
$589$	−5.48933	−0.226184
$590$	0	0
$591$	8.85094	0.364079
$592$	0	0
$593$	40.8684	1.67826	0.839131	−	0.543929i	$-0.183064\pi$
$593$	40.8684	1.67826	0.839131	+	0.543929i	$0.183064\pi$
$594$	0	0
$595$	0.324689	0.0133110
$596$	0	0
$597$	−32.7388	−1.33991
$598$	0	0
$599$	6.98169	0.285264	0.142632	−	0.989776i	$-0.454443\pi$
$599$	6.98169	0.285264	0.142632	+	0.989776i	$0.454443\pi$
$600$	0	0
$601$	−30.5956	−1.24802	−0.624011	−	0.781416i	$-0.714498\pi$
$601$	−30.5956	−1.24802	−0.624011	+	0.781416i	$0.714498\pi$
$602$	0	0
$603$	−50.0774	−2.03931
$604$	0	0
$605$	−0.315184	−0.0128140
$606$	0	0
$607$	−26.5685	−1.07838	−0.539191	−	0.842183i	$-0.681270\pi$
$607$	−26.5685	−1.07838	−0.539191	+	0.842183i	$0.681270\pi$
$608$	0	0
$609$	8.01181	0.324655
$610$	0	0
$611$	−80.6007	−3.26076
$612$	0	0
$613$	−27.5286	−1.11187	−0.555935	−	0.831226i	$-0.687639\pi$
$613$	−27.5286	−1.11187	−0.555935	+	0.831226i	$0.687639\pi$
$614$	0	0
$615$	25.4478	1.02615
$616$	0	0
$617$	0.223894	0.00901364	0.00450682	−	0.999990i	$-0.498565\pi$
$617$	0.223894	0.00901364	0.00450682	+	0.999990i	$0.498565\pi$
$618$	0	0
$619$	33.9431	1.36429	0.682144	−	0.731218i	$-0.261048\pi$
$619$	33.9431	1.36429	0.682144	+	0.731218i	$0.261048\pi$
$620$	0	0
$621$	12.6453	0.507437
$622$	0	0
$623$	−1.94923	−0.0780941
$624$	0	0
$625$	16.6674	0.666698
$626$	0	0
$627$	−5.19260	−0.207372
$628$	0	0
$629$	6.42717	0.256268
$630$	0	0
$631$	−45.4662	−1.80998	−0.904990	−	0.425433i	$-0.860122\pi$
$631$	−45.4662	−1.80998	−0.904990	+	0.425433i	$0.860122\pi$
$632$	0	0
$633$	32.8746	1.30665
$634$	0	0
$635$	3.05860	0.121377
$636$	0	0
$637$	44.8090	1.77540
$638$	0	0
$639$	73.7751	2.91850
$640$	0	0
$641$	−20.0099	−0.790342	−0.395171	−	0.918608i	$-0.629315\pi$
$641$	−20.0099	−0.790342	−0.395171	+	0.918608i	$0.629315\pi$
$642$	0	0
$643$	30.9478	1.22046	0.610231	−	0.792224i	$-0.291077\pi$
$643$	30.9478	1.22046	0.610231	+	0.792224i	$0.291077\pi$
$644$	0	0
$645$	−1.35410	−0.0533177
$646$	0	0
$647$	−28.7133	−1.12884	−0.564418	−	0.825489i	$-0.690900\pi$
$647$	−28.7133	−1.12884	−0.564418	+	0.825489i	$0.690900\pi$
$648$	0	0
$649$	−3.25351	−0.127712
$650$	0	0
$651$	−11.1778	−0.438092
$652$	0	0
$653$	16.7582	0.655800	0.327900	−	0.944712i	$-0.393659\pi$
$653$	16.7582	0.655800	0.327900	+	0.944712i	$0.393659\pi$
$654$	0	0
$655$	3.70982	0.144955
$656$	0	0
$657$	−30.7413	−1.19933
$658$	0	0
$659$	−29.7995	−1.16083	−0.580413	−	0.814322i	$-0.697109\pi$
$659$	−29.7995	−1.16083	−0.580413	+	0.814322i	$0.697109\pi$
$660$	0	0
$661$	−21.9366	−0.853234	−0.426617	−	0.904432i	$-0.640295\pi$
$661$	−21.9366	−0.853234	−0.426617	+	0.904432i	$0.640295\pi$
$662$	0	0
$663$	18.1290	0.704074
$664$	0	0
$665$	−0.187873	−0.00728540
$666$	0	0
$667$	19.3862	0.750638
$668$	0	0
$669$	−26.0039	−1.00537
$670$	0	0
$671$	13.8762	0.535686
$672$	0	0
$673$	−7.91962	−0.305279	−0.152640	−	0.988282i	$-0.548777\pi$
$673$	−7.91962	−0.305279	−0.152640	+	0.988282i	$0.548777\pi$
$674$	0	0
$675$	19.6144	0.754959
$676$	0	0
$677$	13.4944	0.518632	0.259316	−	0.965793i	$-0.416503\pi$
$677$	13.4944	0.518632	0.259316	+	0.965793i	$0.416503\pi$
$678$	0	0
$679$	−4.68825	−0.179919
$680$	0	0
$681$	−71.8005	−2.75140
$682$	0	0
$683$	29.3040	1.12129	0.560643	−	0.828058i	$-0.310554\pi$
$683$	29.3040	1.12129	0.560643	+	0.828058i	$0.310554\pi$
$684$	0	0
$685$	−2.32593	−0.0888693
$686$	0	0
$687$	−41.8580	−1.59698
$688$	0	0
$689$	−40.1154	−1.52827
$690$	0	0
$691$	22.2135	0.845042	0.422521	−	0.906353i	$-0.361145\pi$
$691$	22.2135	0.845042	0.422521	+	0.906353i	$0.361145\pi$
$692$	0	0
$693$	−6.40419	−0.243275
$694$	0	0
$695$	4.12483	0.156464
$696$	0	0
$697$	−12.1378	−0.459753
$698$	0	0
$699$	28.6887	1.08511
$700$	0	0
$701$	−5.33906	−0.201654	−0.100827	−	0.994904i	$-0.532149\pi$
$701$	−5.33906	−0.201654	−0.100827	+	0.994904i	$0.532149\pi$
$702$	0	0
$703$	−3.71891	−0.140261
$704$	0	0
$705$	−25.7104	−0.968310
$706$	0	0
$707$	2.98426	0.112235
$708$	0	0
$709$	−28.5047	−1.07052	−0.535258	−	0.844689i	$-0.679785\pi$
$709$	−28.5047	−1.07052	−0.535258	+	0.844689i	$0.679785\pi$
$710$	0	0
$711$	0.400052	0.0150031
$712$	0	0
$713$	−27.0470	−1.01292
$714$	0	0
$715$	16.2541	0.607870
$716$	0	0
$717$	−42.5456	−1.58889
$718$	0	0
$719$	38.5941	1.43932	0.719658	−	0.694329i	$-0.244299\pi$
$719$	38.5941	1.43932	0.719658	+	0.694329i	$0.244299\pi$
$720$	0	0
$721$	5.16371	0.192307
$722$	0	0
$723$	−24.9296	−0.927143
$724$	0	0
$725$	30.0705	1.11679
$726$	0	0
$727$	22.5227	0.835320	0.417660	−	0.908603i	$-0.362850\pi$
$727$	22.5227	0.835320	0.417660	+	0.908603i	$0.362850\pi$
$728$	0	0
$729$	−44.0293	−1.63071
$730$	0	0
$731$	0.645866	0.0238882
$732$	0	0
$733$	35.4235	1.30840	0.654199	−	0.756322i	$-0.273006\pi$
$733$	35.4235	1.30840	0.654199	+	0.756322i	$0.273006\pi$
$734$	0	0
$735$	14.2934	0.527221
$736$	0	0
$737$	35.3572	1.30240
$738$	0	0
$739$	−5.60333	−0.206122	−0.103061	−	0.994675i	$-0.532864\pi$
$739$	−5.60333	−0.206122	−0.103061	+	0.994675i	$0.532864\pi$
$740$	0	0
$741$	−10.4899	−0.385356
$742$	0	0
$743$	−13.9189	−0.510634	−0.255317	−	0.966857i	$-0.582180\pi$
$743$	−13.9189	−0.510634	−0.255317	+	0.966857i	$0.582180\pi$
$744$	0	0
$745$	12.7976	0.468867
$746$	0	0
$747$	−47.4616	−1.73653
$748$	0	0
$749$	−5.56317	−0.203274
$750$	0	0
$751$	−22.6867	−0.827849	−0.413924	−	0.910311i	$-0.635842\pi$
$751$	−22.6867	−0.827849	−0.413924	+	0.910311i	$0.635842\pi$
$752$	0	0
$753$	64.0142	2.33281
$754$	0	0
$755$	10.7649	0.391776
$756$	0	0
$757$	−6.75507	−0.245517	−0.122759	−	0.992437i	$-0.539174\pi$
$757$	−6.75507	−0.245517	−0.122759	+	0.992437i	$0.539174\pi$
$758$	0	0
$759$	−25.5849	−0.928673
$760$	0	0
$761$	−4.77970	−0.173264	−0.0866321	−	0.996240i	$-0.527610\pi$
$761$	−4.77970	−0.173264	−0.0866321	+	0.996240i	$0.527610\pi$
$762$	0	0
$763$	−0.616622	−0.0223232
$764$	0	0
$765$	3.50259	0.126636
$766$	0	0
$767$	−6.57262	−0.237324
$768$	0	0
$769$	10.3944	0.374830	0.187415	−	0.982281i	$-0.439989\pi$
$769$	10.3944	0.374830	0.187415	+	0.982281i	$0.439989\pi$
$770$	0	0
$771$	−58.1334	−2.09362
$772$	0	0
$773$	−33.6724	−1.21111	−0.605555	−	0.795803i	$-0.707049\pi$
$773$	−33.6724	−1.21111	−0.605555	+	0.795803i	$0.707049\pi$
$774$	0	0
$775$	−41.9533	−1.50701
$776$	0	0
$777$	−7.57272	−0.271670
$778$	0	0
$779$	7.02323	0.251633
$780$	0	0
$781$	−52.0890	−1.86389
$782$	0	0
$783$	30.1600	1.07783
$784$	0	0
$785$	2.32114	0.0828452
$786$	0	0
$787$	−45.9696	−1.63864	−0.819319	−	0.573338i	$-0.805648\pi$
$787$	−45.9696	−1.63864	−0.819319	+	0.573338i	$0.805648\pi$
$788$	0	0
$789$	−32.4165	−1.15406
$790$	0	0
$791$	7.86697	0.279717
$792$	0	0
$793$	28.0322	0.995454
$794$	0	0
$795$	−12.7962	−0.453835
$796$	0	0
$797$	17.0426	0.603680	0.301840	−	0.953359i	$-0.402399\pi$
$797$	17.0426	0.603680	0.301840	+	0.953359i	$0.402399\pi$
$798$	0	0
$799$	12.2631	0.433837
$800$	0	0
$801$	−21.0273	−0.742962
$802$	0	0
$803$	21.7049	0.765950
$804$	0	0
$805$	−0.925685	−0.0326261
$806$	0	0
$807$	18.6829	0.657671
$808$	0	0
$809$	−3.49020	−0.122709	−0.0613544	−	0.998116i	$-0.519542\pi$
$809$	−3.49020	−0.122709	−0.0613544	+	0.998116i	$0.519542\pi$
$810$	0	0
$811$	34.3636	1.20667	0.603335	−	0.797488i	$-0.293838\pi$
$811$	34.3636	1.20667	0.603335	+	0.797488i	$0.293838\pi$
$812$	0	0
$813$	54.0229	1.89467
$814$	0	0
$815$	7.07425	0.247800
$816$	0	0
$817$	−0.373713	−0.0130746
$818$	0	0
$819$	−12.9375	−0.452073
$820$	0	0
$821$	−44.6635	−1.55877	−0.779383	−	0.626547i	$-0.784468\pi$
$821$	−44.6635	−1.55877	−0.779383	+	0.626547i	$0.784468\pi$
$822$	0	0
$823$	−5.50075	−0.191744	−0.0958721	−	0.995394i	$-0.530564\pi$
$823$	−5.50075	−0.191744	−0.0958721	+	0.995394i	$0.530564\pi$
$824$	0	0
$825$	−39.6855	−1.38167
$826$	0	0
$827$	0.742708	0.0258265	0.0129132	−	0.999917i	$-0.495889\pi$
$827$	0.742708	0.0258265	0.0129132	+	0.999917i	$0.495889\pi$
$828$	0	0
$829$	−50.6814	−1.76024	−0.880119	−	0.474753i	$-0.842537\pi$
$829$	−50.6814	−1.76024	−0.880119	+	0.474753i	$0.842537\pi$
$830$	0	0
$831$	−57.8058	−2.00526
$832$	0	0
$833$	−6.81753	−0.236213
$834$	0	0
$835$	−0.691172	−0.0239190
$836$	0	0
$837$	−42.0781	−1.45443
$838$	0	0
$839$	16.4361	0.567437	0.283718	−	0.958908i	$-0.408432\pi$
$839$	16.4361	0.567437	0.283718	+	0.958908i	$0.408432\pi$
$840$	0	0
$841$	17.2378	0.594407
$842$	0	0
$843$	−3.16126	−0.108880
$844$	0	0
$845$	22.9546	0.789663
$846$	0	0
$847$	−0.177128	−0.00608618
$848$	0	0
$849$	−27.6874	−0.950229
$850$	0	0
$851$	−18.3238	−0.628131
$852$	0	0
$853$	43.8399	1.50105	0.750524	−	0.660843i	$-0.229801\pi$
$853$	43.8399	1.50105	0.750524	+	0.660843i	$0.229801\pi$
$854$	0	0
$855$	−2.02668	−0.0693109
$856$	0	0
$857$	−11.6995	−0.399649	−0.199824	−	0.979832i	$-0.564037\pi$
$857$	−11.6995	−0.399649	−0.199824	+	0.979832i	$0.564037\pi$
$858$	0	0
$859$	29.6420	1.01137	0.505686	−	0.862717i	$-0.331239\pi$
$859$	29.6420	1.01137	0.505686	+	0.862717i	$0.331239\pi$
$860$	0	0
$861$	14.3012	0.487384
$862$	0	0
$863$	−12.6501	−0.430613	−0.215307	−	0.976547i	$-0.569075\pi$
$863$	−12.6501	−0.430613	−0.215307	+	0.976547i	$0.569075\pi$
$864$	0	0
$865$	16.8891	0.574247
$866$	0	0
$867$	−2.75827	−0.0936757
$868$	0	0
$869$	−0.282457	−0.00958171
$870$	0	0
$871$	71.4273	2.42022
$872$	0	0
$873$	−50.5745	−1.71169
$874$	0	0
$875$	−3.05930	−0.103423
$876$	0	0
$877$	−33.0665	−1.11658	−0.558289	−	0.829647i	$-0.688542\pi$
$877$	−33.0665	−1.11658	−0.558289	+	0.829647i	$0.688542\pi$
$878$	0	0
$879$	16.7422	0.564699
$880$	0	0
$881$	8.23457	0.277430	0.138715	−	0.990332i	$-0.455703\pi$
$881$	8.23457	0.277430	0.138715	+	0.990332i	$0.455703\pi$
$882$	0	0
$883$	−9.95095	−0.334876	−0.167438	−	0.985883i	$-0.553549\pi$
$883$	−9.95095	−0.334876	−0.167438	+	0.985883i	$0.553549\pi$
$884$	0	0
$885$	−2.09657	−0.0704754
$886$	0	0
$887$	37.2087	1.24935	0.624673	−	0.780886i	$-0.285232\pi$
$887$	37.2087	1.24935	0.624673	+	0.780886i	$0.285232\pi$
$888$	0	0
$889$	1.71888	0.0576493
$890$	0	0
$891$	5.17339	0.173315
$892$	0	0
$893$	−7.09572	−0.237449
$894$	0	0
$895$	−12.4519	−0.416223
$896$	0	0
$897$	−51.6857	−1.72573
$898$	0	0
$899$	−64.5093	−2.15150
$900$	0	0
$901$	6.10341	0.203334
$902$	0	0
$903$	−0.760982	−0.0253239
$904$	0	0
$905$	−4.96407	−0.165011
$906$	0	0
$907$	−37.2590	−1.23717	−0.618583	−	0.785720i	$-0.712293\pi$
$907$	−37.2590	−1.23717	−0.618583	+	0.785720i	$0.712293\pi$
$908$	0	0
$909$	32.1927	1.06776
$910$	0	0
$911$	−33.4535	−1.10836	−0.554182	−	0.832396i	$-0.686969\pi$
$911$	−33.4535	−1.10836	−0.554182	+	0.832396i	$0.686969\pi$
$912$	0	0
$913$	33.5103	1.10903
$914$	0	0
$915$	8.94187	0.295609
$916$	0	0
$917$	2.08486	0.0688480
$918$	0	0
$919$	37.3165	1.23096	0.615478	−	0.788154i	$-0.288963\pi$
$919$	37.3165	1.23096	0.615478	+	0.788154i	$0.288963\pi$
$920$	0	0
$921$	60.0088	1.97736
$922$	0	0
$923$	−105.228	−3.46363
$924$	0	0
$925$	−28.4225	−0.934527
$926$	0	0
$927$	55.7035	1.82954
$928$	0	0
$929$	11.0174	0.361470	0.180735	−	0.983532i	$-0.442152\pi$
$929$	11.0174	0.361470	0.180735	+	0.983532i	$0.442152\pi$
$930$	0	0
$931$	3.94478	0.129285
$932$	0	0
$933$	−81.2458	−2.65987
$934$	0	0
$935$	−2.47301	−0.0808759
$936$	0	0
$937$	1.48755	0.0485960	0.0242980	−	0.999705i	$-0.492265\pi$
$937$	1.48755	0.0485960	0.0242980	+	0.999705i	$0.492265\pi$
$938$	0	0
$939$	−11.6039	−0.378680
$940$	0	0
$941$	−2.36189	−0.0769954	−0.0384977	−	0.999259i	$-0.512257\pi$
$941$	−2.36189	−0.0769954	−0.0384977	+	0.999259i	$0.512257\pi$
$942$	0	0
$943$	34.6048	1.12689
$944$	0	0
$945$	−1.44013	−0.0468474
$946$	0	0
$947$	28.9688	0.941361	0.470680	−	0.882304i	$-0.344009\pi$
$947$	28.9688	0.941361	0.470680	+	0.882304i	$0.344009\pi$
$948$	0	0
$949$	43.8475	1.42335
$950$	0	0
$951$	−3.06244	−0.0993065
$952$	0	0
$953$	−48.4499	−1.56945	−0.784723	−	0.619847i	$-0.787195\pi$
$953$	−48.4499	−1.56945	−0.784723	+	0.619847i	$0.787195\pi$
$954$	0	0
$955$	−19.5052	−0.631173
$956$	0	0
$957$	−61.0221	−1.97257
$958$	0	0
$959$	−1.30713	−0.0422095
$960$	0	0
$961$	59.0009	1.90326
$962$	0	0
$963$	−60.0127	−1.93388
$964$	0	0
$965$	11.9144	0.383538
$966$	0	0
$967$	−45.6956	−1.46947	−0.734736	−	0.678353i	$-0.762694\pi$
$967$	−45.6956	−1.46947	−0.734736	+	0.678353i	$0.762694\pi$
$968$	0	0
$969$	1.59600	0.0512708
$970$	0	0
$971$	12.6315	0.405364	0.202682	−	0.979245i	$-0.435034\pi$
$971$	12.6315	0.405364	0.202682	+	0.979245i	$0.435034\pi$
$972$	0	0
$973$	2.31808	0.0743143
$974$	0	0
$975$	−80.1711	−2.56753
$976$	0	0
$977$	−53.3017	−1.70527	−0.852636	−	0.522506i	$-0.824997\pi$
$977$	−53.3017	−1.70527	−0.852636	+	0.522506i	$0.824997\pi$
$978$	0	0
$979$	14.8463	0.474491
$980$	0	0
$981$	−6.65181	−0.212376
$982$	0	0
$983$	5.08685	0.162245	0.0811227	−	0.996704i	$-0.474149\pi$
$983$	5.08685	0.162245	0.0811227	+	0.996704i	$0.474149\pi$
$984$	0	0
$985$	−2.43908	−0.0777155
$986$	0	0
$987$	−14.4488	−0.459911
$988$	0	0
$989$	−1.84136	−0.0585517
$990$	0	0
$991$	3.34580	0.106283	0.0531414	−	0.998587i	$-0.483077\pi$
$991$	3.34580	0.106283	0.0531414	+	0.998587i	$0.483077\pi$
$992$	0	0
$993$	−38.1332	−1.21012
$994$	0	0
$995$	9.02192	0.286014
$996$	0	0
$997$	14.0436	0.444766	0.222383	−	0.974959i	$-0.428616\pi$
$997$	14.0436	0.444766	0.222383	+	0.974959i	$0.428616\pi$
$998$	0	0
$999$	−28.5071	−0.901924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.1	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.1	✓	21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75827 q^{3} +0.760103 q^{5} +0.427165 q^{7} +4.60804 q^{9} +O(q^{10})\)	\(q-2.75827 q^{3} +0.760103 q^{5} +0.427165 q^{7} +4.60804 q^{9} -3.25351 q^{11} -6.57262 q^{13} -2.09657 q^{15} +1.00000 q^{17} -0.578623 q^{19} -1.17823 q^{21} -2.85099 q^{23} -4.42224 q^{25} -4.43540 q^{27} -6.79984 q^{29} +9.48688 q^{31} +8.97406 q^{33} +0.324689 q^{35} +6.42717 q^{37} +18.1290 q^{39} -12.1378 q^{41} +0.645866 q^{43} +3.50259 q^{45} +12.2631 q^{47} -6.81753 q^{49} -2.75827 q^{51} +6.10341 q^{53} -2.47301 q^{55} +1.59600 q^{57} +1.00000 q^{59} -4.26500 q^{61} +1.96839 q^{63} -4.99587 q^{65} -10.8674 q^{67} +7.86378 q^{69} +16.0101 q^{71} -6.67123 q^{73} +12.1977 q^{75} -1.38979 q^{77} +0.0868161 q^{79} -1.59009 q^{81} -10.2997 q^{83} +0.760103 q^{85} +18.7558 q^{87} -4.56317 q^{89} -2.80759 q^{91} -26.1674 q^{93} -0.439814 q^{95} -10.9753 q^{97} -14.9923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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