LMFDB - Embedded newform 6040.2.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 $ x^19 - 5x^18 - 29x^17 + 165x^16 + 325x^15 - 2208x^14 - 1891x^13 + 15895x^12 + 6652x^11 - 67665x^10 - 17345x^9 + 174105x^8 + 41499x^7 - 262172x^6 - 80919x^5 + 206783x^4 + 91643x^3 - 59750x^2 - 40224x - 5628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.90812$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90812 q^{3} +1.00000 q^{5} +1.15158 q^{7} +0.640908 q^{9} +O(q^{10})$	$q-1.90812 q^{3} +1.00000 q^{5} +1.15158 q^{7} +0.640908 q^{9} +4.77254 q^{11} +3.49113 q^{13} -1.90812 q^{15} -7.78011 q^{17} +1.91284 q^{19} -2.19736 q^{21} -6.30459 q^{23} +1.00000 q^{25} +4.50142 q^{27} -0.185654 q^{29} -9.77853 q^{31} -9.10657 q^{33} +1.15158 q^{35} +9.42468 q^{37} -6.66148 q^{39} -3.34035 q^{41} -1.84412 q^{43} +0.640908 q^{45} -1.21211 q^{47} -5.67385 q^{49} +14.8454 q^{51} -0.503670 q^{53} +4.77254 q^{55} -3.64993 q^{57} -14.1409 q^{59} -7.19463 q^{61} +0.738059 q^{63} +3.49113 q^{65} +1.44963 q^{67} +12.0299 q^{69} -12.3875 q^{71} +5.53104 q^{73} -1.90812 q^{75} +5.49599 q^{77} -8.71646 q^{79} -10.5120 q^{81} -5.43959 q^{83} -7.78011 q^{85} +0.354249 q^{87} +1.98863 q^{89} +4.02033 q^{91} +18.6586 q^{93} +1.91284 q^{95} +16.0631 q^{97} +3.05876 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.90812	−1.10165	−0.550826	−	0.834620i	$-0.685687\pi$
$3$	−1.90812	−1.10165	−0.550826	+	0.834620i	$0.685687\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.15158	0.435258	0.217629	−	0.976032i	$-0.430168\pi$
$7$	1.15158	0.435258	0.217629	+	0.976032i	$0.430168\pi$
$8$	0	0
$9$	0.640908	0.213636
$10$	0	0
$11$	4.77254	1.43898	0.719488	−	0.694505i	$-0.244377\pi$
$11$	4.77254	1.43898	0.719488	+	0.694505i	$0.244377\pi$
$12$	0	0
$13$	3.49113	0.968265	0.484132	−	0.874995i	$-0.339135\pi$
$13$	3.49113	0.968265	0.484132	+	0.874995i	$0.339135\pi$
$14$	0	0
$15$	−1.90812	−0.492674
$16$	0	0
$17$	−7.78011	−1.88695	−0.943477	−	0.331439i	$-0.892466\pi$
$17$	−7.78011	−1.88695	−0.943477	+	0.331439i	$0.892466\pi$
$18$	0	0
$19$	1.91284	0.438836	0.219418	−	0.975631i	$-0.429584\pi$
$19$	1.91284	0.438836	0.219418	+	0.975631i	$0.429584\pi$
$20$	0	0
$21$	−2.19736	−0.479503
$22$	0	0
$23$	−6.30459	−1.31460	−0.657299	−	0.753630i	$-0.728301\pi$
$23$	−6.30459	−1.31460	−0.657299	+	0.753630i	$0.728301\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.50142	0.866299
$28$	0	0
$29$	−0.185654	−0.0344750	−0.0172375	−	0.999851i	$-0.505487\pi$
$29$	−0.185654	−0.0344750	−0.0172375	+	0.999851i	$0.505487\pi$
$30$	0	0
$31$	−9.77853	−1.75628	−0.878138	−	0.478408i	$-0.841214\pi$
$31$	−9.77853	−1.75628	−0.878138	+	0.478408i	$0.841214\pi$
$32$	0	0
$33$	−9.10657	−1.58525
$34$	0	0
$35$	1.15158	0.194653
$36$	0	0
$37$	9.42468	1.54941	0.774704	−	0.632324i	$-0.217899\pi$
$37$	9.42468	1.54941	0.774704	+	0.632324i	$0.217899\pi$
$38$	0	0
$39$	−6.66148	−1.06669
$40$	0	0
$41$	−3.34035	−0.521675	−0.260838	−	0.965383i	$-0.583999\pi$
$41$	−3.34035	−0.521675	−0.260838	+	0.965383i	$0.583999\pi$
$42$	0	0
$43$	−1.84412	−0.281226	−0.140613	−	0.990065i	$-0.544907\pi$
$43$	−1.84412	−0.281226	−0.140613	+	0.990065i	$0.544907\pi$
$44$	0	0
$45$	0.640908	0.0955409
$46$	0	0
$47$	−1.21211	−0.176805	−0.0884024	−	0.996085i	$-0.528176\pi$
$47$	−1.21211	−0.176805	−0.0884024	+	0.996085i	$0.528176\pi$
$48$	0	0
$49$	−5.67385	−0.810550
$50$	0	0
$51$	14.8454	2.07877
$52$	0	0
$53$	−0.503670	−0.0691844	−0.0345922	−	0.999402i	$-0.511013\pi$
$53$	−0.503670	−0.0691844	−0.0345922	+	0.999402i	$0.511013\pi$
$54$	0	0
$55$	4.77254	0.643530
$56$	0	0
$57$	−3.64993	−0.483445
$58$	0	0
$59$	−14.1409	−1.84099	−0.920495	−	0.390754i	$-0.872214\pi$
$59$	−14.1409	−1.84099	−0.920495	+	0.390754i	$0.872214\pi$
$60$	0	0
$61$	−7.19463	−0.921178	−0.460589	−	0.887614i	$-0.652362\pi$
$61$	−7.19463	−0.921178	−0.460589	+	0.887614i	$0.652362\pi$
$62$	0	0
$63$	0.738059	0.0929867
$64$	0	0
$65$	3.49113	0.433021
$66$	0	0
$67$	1.44963	0.177101	0.0885505	−	0.996072i	$-0.471777\pi$
$67$	1.44963	0.177101	0.0885505	+	0.996072i	$0.471777\pi$
$68$	0	0
$69$	12.0299	1.44823
$70$	0	0
$71$	−12.3875	−1.47013	−0.735064	−	0.677998i	$-0.762848\pi$
$71$	−12.3875	−1.47013	−0.735064	+	0.677998i	$0.762848\pi$
$72$	0	0
$73$	5.53104	0.647359	0.323680	−	0.946167i	$-0.395080\pi$
$73$	5.53104	0.647359	0.323680	+	0.946167i	$0.395080\pi$
$74$	0	0
$75$	−1.90812	−0.220330
$76$	0	0
$77$	5.49599	0.626326
$78$	0	0
$79$	−8.71646	−0.980679	−0.490339	−	0.871532i	$-0.663127\pi$
$79$	−8.71646	−0.980679	−0.490339	+	0.871532i	$0.663127\pi$
$80$	0	0
$81$	−10.5120	−1.16800
$82$	0	0
$83$	−5.43959	−0.597072	−0.298536	−	0.954398i	$-0.596498\pi$
$83$	−5.43959	−0.597072	−0.298536	+	0.954398i	$0.596498\pi$
$84$	0	0
$85$	−7.78011	−0.843871
$86$	0	0
$87$	0.354249	0.0379795
$88$	0	0
$89$	1.98863	0.210794	0.105397	−	0.994430i	$-0.466389\pi$
$89$	1.98863	0.210794	0.105397	+	0.994430i	$0.466389\pi$
$90$	0	0
$91$	4.02033	0.421445
$92$	0	0
$93$	18.6586	1.93480
$94$	0	0
$95$	1.91284	0.196254
$96$	0	0
$97$	16.0631	1.63096	0.815481	−	0.578783i	$-0.196472\pi$
$97$	16.0631	1.63096	0.815481	+	0.578783i	$0.196472\pi$
$98$	0	0
$99$	3.05876	0.307417
$100$	0	0
$101$	−5.80780	−0.577898	−0.288949	−	0.957345i	$-0.593306\pi$
$101$	−5.80780	−0.577898	−0.288949	+	0.957345i	$0.593306\pi$
$102$	0	0
$103$	16.2606	1.60221	0.801103	−	0.598526i	$-0.204247\pi$
$103$	16.2606	1.60221	0.801103	+	0.598526i	$0.204247\pi$
$104$	0	0
$105$	−2.19736	−0.214440
$106$	0	0
$107$	10.4039	1.00578	0.502889	−	0.864351i	$-0.332270\pi$
$107$	10.4039	1.00578	0.502889	+	0.864351i	$0.332270\pi$
$108$	0	0
$109$	8.68259	0.831641	0.415821	−	0.909447i	$-0.363494\pi$
$109$	8.68259	0.831641	0.415821	+	0.909447i	$0.363494\pi$
$110$	0	0
$111$	−17.9834	−1.70691
$112$	0	0
$113$	−7.72969	−0.727148	−0.363574	−	0.931565i	$-0.618444\pi$
$113$	−7.72969	−0.727148	−0.363574	+	0.931565i	$0.618444\pi$
$114$	0	0
$115$	−6.30459	−0.587906
$116$	0	0
$117$	2.23749	0.206856
$118$	0	0
$119$	−8.95945	−0.821312
$120$	0	0
$121$	11.7772	1.07065
$122$	0	0
$123$	6.37378	0.574704
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.9015	−1.32229	−0.661145	−	0.750258i	$-0.729929\pi$
$127$	−14.9015	−1.32229	−0.661145	+	0.750258i	$0.729929\pi$
$128$	0	0
$129$	3.51880	0.309813
$130$	0	0
$131$	15.9332	1.39209	0.696047	−	0.717996i	$-0.254940\pi$
$131$	15.9332	1.39209	0.696047	+	0.717996i	$0.254940\pi$
$132$	0	0
$133$	2.20280	0.191007
$134$	0	0
$135$	4.50142	0.387421
$136$	0	0
$137$	0.648293	0.0553874	0.0276937	−	0.999616i	$-0.491184\pi$
$137$	0.648293	0.0553874	0.0276937	+	0.999616i	$0.491184\pi$
$138$	0	0
$139$	2.77102	0.235035	0.117518	−	0.993071i	$-0.462506\pi$
$139$	2.77102	0.235035	0.117518	+	0.993071i	$0.462506\pi$
$140$	0	0
$141$	2.31285	0.194777
$142$	0	0
$143$	16.6616	1.39331
$144$	0	0
$145$	−0.185654	−0.0154177
$146$	0	0
$147$	10.8264	0.892944
$148$	0	0
$149$	7.92025	0.648852	0.324426	−	0.945911i	$-0.394829\pi$
$149$	7.92025	0.648852	0.324426	+	0.945911i	$0.394829\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−4.98633	−0.403121
$154$	0	0
$155$	−9.77853	−0.785430
$156$	0	0
$157$	6.40941	0.511527	0.255763	−	0.966739i	$-0.417673\pi$
$157$	6.40941	0.511527	0.255763	+	0.966739i	$0.417673\pi$
$158$	0	0
$159$	0.961061	0.0762171
$160$	0	0
$161$	−7.26026	−0.572189
$162$	0	0
$163$	−18.7773	−1.47075	−0.735374	−	0.677662i	$-0.762993\pi$
$163$	−18.7773	−1.47075	−0.735374	+	0.677662i	$0.762993\pi$
$164$	0	0
$165$	−9.10657	−0.708945
$166$	0	0
$167$	−8.35575	−0.646587	−0.323294	−	0.946299i	$-0.604790\pi$
$167$	−8.35575	−0.646587	−0.323294	+	0.946299i	$0.604790\pi$
$168$	0	0
$169$	−0.812022	−0.0624632
$170$	0	0
$171$	1.22596	0.0937512
$172$	0	0
$173$	11.9231	0.906498	0.453249	−	0.891384i	$-0.350265\pi$
$173$	11.9231	0.906498	0.453249	+	0.891384i	$0.350265\pi$
$174$	0	0
$175$	1.15158	0.0870516
$176$	0	0
$177$	26.9825	2.02813
$178$	0	0
$179$	−18.9922	−1.41954	−0.709771	−	0.704433i	$-0.751201\pi$
$179$	−18.9922	−1.41954	−0.709771	+	0.704433i	$0.751201\pi$
$180$	0	0
$181$	6.88300	0.511609	0.255805	−	0.966728i	$-0.417660\pi$
$181$	6.88300	0.511609	0.255805	+	0.966728i	$0.417660\pi$
$182$	0	0
$183$	13.7282	1.01482
$184$	0	0
$185$	9.42468	0.692916
$186$	0	0
$187$	−37.1309	−2.71528
$188$	0	0
$189$	5.18377	0.377064
$190$	0	0
$191$	−19.3701	−1.40157	−0.700787	−	0.713370i	$-0.747168\pi$
$191$	−19.3701	−1.40157	−0.700787	+	0.713370i	$0.747168\pi$
$192$	0	0
$193$	−12.2516	−0.881890	−0.440945	−	0.897534i	$-0.645357\pi$
$193$	−12.2516	−0.881890	−0.440945	+	0.897534i	$0.645357\pi$
$194$	0	0
$195$	−6.66148	−0.477038
$196$	0	0
$197$	−6.80002	−0.484482	−0.242241	−	0.970216i	$-0.577882\pi$
$197$	−6.80002	−0.484482	−0.242241	+	0.970216i	$0.577882\pi$
$198$	0	0
$199$	4.40229	0.312070	0.156035	−	0.987752i	$-0.450129\pi$
$199$	4.40229	0.312070	0.156035	+	0.987752i	$0.450129\pi$
$200$	0	0
$201$	−2.76607	−0.195104
$202$	0	0
$203$	−0.213796	−0.0150055
$204$	0	0
$205$	−3.34035	−0.233300
$206$	0	0
$207$	−4.04066	−0.280845
$208$	0	0
$209$	9.12913	0.631475
$210$	0	0
$211$	0.143515	0.00987997	0.00493999	−	0.999988i	$-0.498428\pi$
$211$	0.143515	0.00987997	0.00493999	+	0.999988i	$0.498428\pi$
$212$	0	0
$213$	23.6368	1.61957
$214$	0	0
$215$	−1.84412	−0.125768
$216$	0	0
$217$	−11.2608	−0.764433
$218$	0	0
$219$	−10.5539	−0.713164
$220$	0	0
$221$	−27.1614	−1.82707
$222$	0	0
$223$	−19.0197	−1.27365	−0.636827	−	0.771007i	$-0.719753\pi$
$223$	−19.0197	−1.27365	−0.636827	+	0.771007i	$0.719753\pi$
$224$	0	0
$225$	0.640908	0.0427272
$226$	0	0
$227$	11.7645	0.780837	0.390419	−	0.920637i	$-0.372330\pi$
$227$	11.7645	0.780837	0.390419	+	0.920637i	$0.372330\pi$
$228$	0	0
$229$	21.6927	1.43349	0.716746	−	0.697334i	$-0.245630\pi$
$229$	21.6927	1.43349	0.716746	+	0.697334i	$0.245630\pi$
$230$	0	0
$231$	−10.4870	−0.689993
$232$	0	0
$233$	8.17820	0.535772	0.267886	−	0.963451i	$-0.413675\pi$
$233$	8.17820	0.535772	0.267886	+	0.963451i	$0.413675\pi$
$234$	0	0
$235$	−1.21211	−0.0790695
$236$	0	0
$237$	16.6320	1.08037
$238$	0	0
$239$	−20.1698	−1.30467	−0.652337	−	0.757929i	$-0.726211\pi$
$239$	−20.1698	−1.30467	−0.652337	+	0.757929i	$0.726211\pi$
$240$	0	0
$241$	17.9815	1.15829	0.579145	−	0.815224i	$-0.303386\pi$
$241$	17.9815	1.15829	0.579145	+	0.815224i	$0.303386\pi$
$242$	0	0
$243$	6.55378	0.420425
$244$	0	0
$245$	−5.67385	−0.362489
$246$	0	0
$247$	6.67798	0.424910
$248$	0	0
$249$	10.3794	0.657766
$250$	0	0
$251$	−4.51640	−0.285072	−0.142536	−	0.989790i	$-0.545526\pi$
$251$	−4.51640	−0.285072	−0.142536	+	0.989790i	$0.545526\pi$
$252$	0	0
$253$	−30.0889	−1.89167
$254$	0	0
$255$	14.8454	0.929652
$256$	0	0
$257$	−3.17303	−0.197928	−0.0989639	−	0.995091i	$-0.531553\pi$
$257$	−3.17303	−0.197928	−0.0989639	+	0.995091i	$0.531553\pi$
$258$	0	0
$259$	10.8533	0.674392
$260$	0	0
$261$	−0.118987	−0.00736510
$262$	0	0
$263$	−4.14288	−0.255461	−0.127730	−	0.991809i	$-0.540769\pi$
$263$	−4.14288	−0.255461	−0.127730	+	0.991809i	$0.540769\pi$
$264$	0	0
$265$	−0.503670	−0.0309402
$266$	0	0
$267$	−3.79454	−0.232222
$268$	0	0
$269$	−23.4001	−1.42673	−0.713363	−	0.700794i	$-0.752829\pi$
$269$	−23.4001	−1.42673	−0.713363	+	0.700794i	$0.752829\pi$
$270$	0	0
$271$	−23.3117	−1.41609	−0.708043	−	0.706170i	$-0.750422\pi$
$271$	−23.3117	−1.41609	−0.708043	+	0.706170i	$0.750422\pi$
$272$	0	0
$273$	−7.67126	−0.464285
$274$	0	0
$275$	4.77254	0.287795
$276$	0	0
$277$	5.61214	0.337201	0.168600	−	0.985684i	$-0.446075\pi$
$277$	5.61214	0.337201	0.168600	+	0.985684i	$0.446075\pi$
$278$	0	0
$279$	−6.26713	−0.375204
$280$	0	0
$281$	−18.9155	−1.12840	−0.564201	−	0.825637i	$-0.690816\pi$
$281$	−18.9155	−1.12840	−0.564201	+	0.825637i	$0.690816\pi$
$282$	0	0
$283$	−17.8090	−1.05863	−0.529317	−	0.848424i	$-0.677552\pi$
$283$	−17.8090	−1.05863	−0.529317	+	0.848424i	$0.677552\pi$
$284$	0	0
$285$	−3.64993	−0.216203
$286$	0	0
$287$	−3.84670	−0.227063
$288$	0	0
$289$	43.5301	2.56059
$290$	0	0
$291$	−30.6503	−1.79675
$292$	0	0
$293$	−19.9853	−1.16756	−0.583778	−	0.811913i	$-0.698426\pi$
$293$	−19.9853	−1.16756	−0.583778	+	0.811913i	$0.698426\pi$
$294$	0	0
$295$	−14.1409	−0.823316
$296$	0	0
$297$	21.4832	1.24658
$298$	0	0
$299$	−22.0101	−1.27288
$300$	0	0
$301$	−2.12366	−0.122406
$302$	0	0
$303$	11.0820	0.636642
$304$	0	0
$305$	−7.19463	−0.411963
$306$	0	0
$307$	4.11533	0.234874	0.117437	−	0.993080i	$-0.462532\pi$
$307$	4.11533	0.234874	0.117437	+	0.993080i	$0.462532\pi$
$308$	0	0
$309$	−31.0272	−1.76507
$310$	0	0
$311$	18.5633	1.05263	0.526315	−	0.850289i	$-0.323573\pi$
$311$	18.5633	1.05263	0.526315	+	0.850289i	$0.323573\pi$
$312$	0	0
$313$	−19.2697	−1.08919	−0.544595	−	0.838699i	$-0.683317\pi$
$313$	−19.2697	−1.08919	−0.544595	+	0.838699i	$0.683317\pi$
$314$	0	0
$315$	0.738059	0.0415849
$316$	0	0
$317$	14.6002	0.820029	0.410014	−	0.912079i	$-0.365524\pi$
$317$	14.6002	0.820029	0.410014	+	0.912079i	$0.365524\pi$
$318$	0	0
$319$	−0.886040	−0.0496087
$320$	0	0
$321$	−19.8518	−1.10802
$322$	0	0
$323$	−14.8821	−0.828064
$324$	0	0
$325$	3.49113	0.193653
$326$	0	0
$327$	−16.5674	−0.916179
$328$	0	0
$329$	−1.39585	−0.0769557
$330$	0	0
$331$	−23.4749	−1.29029	−0.645147	−	0.764058i	$-0.723204\pi$
$331$	−23.4749	−1.29029	−0.645147	+	0.764058i	$0.723204\pi$
$332$	0	0
$333$	6.04035	0.331009
$334$	0	0
$335$	1.44963	0.0792020
$336$	0	0
$337$	−6.72634	−0.366407	−0.183203	−	0.983075i	$-0.558647\pi$
$337$	−6.72634	−0.366407	−0.183203	+	0.983075i	$0.558647\pi$
$338$	0	0
$339$	14.7491	0.801064
$340$	0	0
$341$	−46.6684	−2.52724
$342$	0	0
$343$	−14.5950	−0.788057
$344$	0	0
$345$	12.0299	0.647667
$346$	0	0
$347$	−1.14897	−0.0616800	−0.0308400	−	0.999524i	$-0.509818\pi$
$347$	−1.14897	−0.0616800	−0.0308400	+	0.999524i	$0.509818\pi$
$348$	0	0
$349$	−23.7212	−1.26977	−0.634885	−	0.772607i	$-0.718952\pi$
$349$	−23.7212	−1.26977	−0.634885	+	0.772607i	$0.718952\pi$
$350$	0	0
$351$	15.7150	0.838807
$352$	0	0
$353$	−21.8193	−1.16133	−0.580663	−	0.814144i	$-0.697207\pi$
$353$	−21.8193	−1.16133	−0.580663	+	0.814144i	$0.697207\pi$
$354$	0	0
$355$	−12.3875	−0.657461
$356$	0	0
$357$	17.0957	0.904799
$358$	0	0
$359$	−7.32958	−0.386841	−0.193420	−	0.981116i	$-0.561958\pi$
$359$	−7.32958	−0.386841	−0.193420	+	0.981116i	$0.561958\pi$
$360$	0	0
$361$	−15.3410	−0.807423
$362$	0	0
$363$	−22.4722	−1.17948
$364$	0	0
$365$	5.53104	0.289508
$366$	0	0
$367$	35.8361	1.87063	0.935315	−	0.353817i	$-0.115117\pi$
$367$	35.8361	1.87063	0.935315	+	0.353817i	$0.115117\pi$
$368$	0	0
$369$	−2.14086	−0.111449
$370$	0	0
$371$	−0.580019	−0.0301131
$372$	0	0
$373$	−1.19833	−0.0620473	−0.0310237	−	0.999519i	$-0.509877\pi$
$373$	−1.19833	−0.0620473	−0.0310237	+	0.999519i	$0.509877\pi$
$374$	0	0
$375$	−1.90812	−0.0985347
$376$	0	0
$377$	−0.648141	−0.0333810
$378$	0	0
$379$	−21.6068	−1.10986	−0.554932	−	0.831895i	$-0.687256\pi$
$379$	−21.6068	−1.10986	−0.554932	+	0.831895i	$0.687256\pi$
$380$	0	0
$381$	28.4337	1.45670
$382$	0	0
$383$	29.7618	1.52076	0.760378	−	0.649481i	$-0.225014\pi$
$383$	29.7618	1.52076	0.760378	+	0.649481i	$0.225014\pi$
$384$	0	0
$385$	5.49599	0.280101
$386$	0	0
$387$	−1.18191	−0.0600800
$388$	0	0
$389$	28.8316	1.46182	0.730909	−	0.682475i	$-0.239096\pi$
$389$	28.8316	1.46182	0.730909	+	0.682475i	$0.239096\pi$
$390$	0	0
$391$	49.0504	2.48058
$392$	0	0
$393$	−30.4025	−1.53360
$394$	0	0
$395$	−8.71646	−0.438573
$396$	0	0
$397$	−0.653539	−0.0328002	−0.0164001	−	0.999866i	$-0.505221\pi$
$397$	−0.653539	−0.0328002	−0.0164001	+	0.999866i	$0.505221\pi$
$398$	0	0
$399$	−4.20320	−0.210423
$400$	0	0
$401$	−15.7068	−0.784359	−0.392179	−	0.919889i	$-0.628279\pi$
$401$	−15.7068	−0.784359	−0.392179	+	0.919889i	$0.628279\pi$
$402$	0	0
$403$	−34.1381	−1.70054
$404$	0	0
$405$	−10.5120	−0.522344
$406$	0	0
$407$	44.9797	2.22956
$408$	0	0
$409$	31.8387	1.57432	0.787160	−	0.616748i	$-0.211550\pi$
$409$	31.8387	1.57432	0.787160	+	0.616748i	$0.211550\pi$
$410$	0	0
$411$	−1.23702	−0.0610176
$412$	0	0
$413$	−16.2845	−0.801306
$414$	0	0
$415$	−5.43959	−0.267019
$416$	0	0
$417$	−5.28743	−0.258927
$418$	0	0
$419$	−29.6514	−1.44857	−0.724284	−	0.689502i	$-0.757829\pi$
$419$	−29.6514	−1.44857	−0.724284	+	0.689502i	$0.757829\pi$
$420$	0	0
$421$	−28.2695	−1.37777	−0.688886	−	0.724870i	$-0.741900\pi$
$421$	−28.2695	−1.37777	−0.688886	+	0.724870i	$0.741900\pi$
$422$	0	0
$423$	−0.776852	−0.0377718
$424$	0	0
$425$	−7.78011	−0.377391
$426$	0	0
$427$	−8.28522	−0.400950
$428$	0	0
$429$	−31.7922	−1.53494
$430$	0	0
$431$	18.1568	0.874580	0.437290	−	0.899321i	$-0.355938\pi$
$431$	18.1568	0.874580	0.437290	+	0.899321i	$0.355938\pi$
$432$	0	0
$433$	14.0694	0.676134	0.338067	−	0.941122i	$-0.390227\pi$
$433$	14.0694	0.676134	0.338067	+	0.941122i	$0.390227\pi$
$434$	0	0
$435$	0.354249	0.0169849
$436$	0	0
$437$	−12.0597	−0.576893
$438$	0	0
$439$	5.94867	0.283914	0.141957	−	0.989873i	$-0.454660\pi$
$439$	5.94867	0.283914	0.141957	+	0.989873i	$0.454660\pi$
$440$	0	0
$441$	−3.63642	−0.173163
$442$	0	0
$443$	−20.8504	−0.990631	−0.495315	−	0.868713i	$-0.664947\pi$
$443$	−20.8504	−0.990631	−0.495315	+	0.868713i	$0.664947\pi$
$444$	0	0
$445$	1.98863	0.0942701
$446$	0	0
$447$	−15.1128	−0.714809
$448$	0	0
$449$	−0.972773	−0.0459080	−0.0229540	−	0.999737i	$-0.507307\pi$
$449$	−0.972773	−0.0459080	−0.0229540	+	0.999737i	$0.507307\pi$
$450$	0	0
$451$	−15.9420	−0.750678
$452$	0	0
$453$	−1.90812	−0.0896511
$454$	0	0
$455$	4.02033	0.188476
$456$	0	0
$457$	−0.139601	−0.00653026	−0.00326513	−	0.999995i	$-0.501039\pi$
$457$	−0.139601	−0.00653026	−0.00326513	+	0.999995i	$0.501039\pi$
$458$	0	0
$459$	−35.0216	−1.63467
$460$	0	0
$461$	39.7094	1.84945	0.924725	−	0.380636i	$-0.124295\pi$
$461$	39.7094	1.84945	0.924725	+	0.380636i	$0.124295\pi$
$462$	0	0
$463$	−39.2427	−1.82376	−0.911880	−	0.410456i	$-0.865370\pi$
$463$	−39.2427	−1.82376	−0.911880	+	0.410456i	$0.865370\pi$
$464$	0	0
$465$	18.6586	0.865270
$466$	0	0
$467$	13.0241	0.602682	0.301341	−	0.953516i	$-0.402566\pi$
$467$	13.0241	0.602682	0.301341	+	0.953516i	$0.402566\pi$
$468$	0	0
$469$	1.66938	0.0770847
$470$	0	0
$471$	−12.2299	−0.563524
$472$	0	0
$473$	−8.80115	−0.404677
$474$	0	0
$475$	1.91284	0.0877673
$476$	0	0
$477$	−0.322806	−0.0147803
$478$	0	0
$479$	−24.7172	−1.12936	−0.564680	−	0.825310i	$-0.691000\pi$
$479$	−24.7172	−1.12936	−0.564680	+	0.825310i	$0.691000\pi$
$480$	0	0
$481$	32.9028	1.50024
$482$	0	0
$483$	13.8534	0.630353
$484$	0	0
$485$	16.0631	0.729389
$486$	0	0
$487$	−39.0981	−1.77170	−0.885852	−	0.463968i	$-0.846425\pi$
$487$	−39.0981	−1.77170	−0.885852	+	0.463968i	$0.846425\pi$
$488$	0	0
$489$	35.8292	1.62025
$490$	0	0
$491$	23.0824	1.04169	0.520847	−	0.853650i	$-0.325616\pi$
$491$	23.0824	1.04169	0.520847	+	0.853650i	$0.325616\pi$
$492$	0	0
$493$	1.44441	0.0650528
$494$	0	0
$495$	3.05876	0.137481
$496$	0	0
$497$	−14.2653	−0.639885
$498$	0	0
$499$	8.73690	0.391117	0.195559	−	0.980692i	$-0.437348\pi$
$499$	8.73690	0.391117	0.195559	+	0.980692i	$0.437348\pi$
$500$	0	0
$501$	15.9437	0.712314
$502$	0	0
$503$	16.8938	0.753255	0.376628	−	0.926365i	$-0.377084\pi$
$503$	16.8938	0.753255	0.376628	+	0.926365i	$0.377084\pi$
$504$	0	0
$505$	−5.80780	−0.258444
$506$	0	0
$507$	1.54943	0.0688127
$508$	0	0
$509$	−9.18885	−0.407289	−0.203644	−	0.979045i	$-0.565279\pi$
$509$	−9.18885	−0.407289	−0.203644	+	0.979045i	$0.565279\pi$
$510$	0	0
$511$	6.36946	0.281768
$512$	0	0
$513$	8.61052	0.380164
$514$	0	0
$515$	16.2606	0.716529
$516$	0	0
$517$	−5.78486	−0.254418
$518$	0	0
$519$	−22.7507	−0.998644
$520$	0	0
$521$	27.4061	1.20068	0.600342	−	0.799743i	$-0.295031\pi$
$521$	27.4061	1.20068	0.600342	+	0.799743i	$0.295031\pi$
$522$	0	0
$523$	3.95352	0.172875	0.0864377	−	0.996257i	$-0.472452\pi$
$523$	3.95352	0.172875	0.0864377	+	0.996257i	$0.472452\pi$
$524$	0	0
$525$	−2.19736	−0.0959005
$526$	0	0
$527$	76.0780	3.31401
$528$	0	0
$529$	16.7478	0.728166
$530$	0	0
$531$	−9.06302	−0.393302
$532$	0	0
$533$	−11.6616	−0.505120
$534$	0	0
$535$	10.4039	0.449798
$536$	0	0
$537$	36.2393	1.56384
$538$	0	0
$539$	−27.0787	−1.16636
$540$	0	0
$541$	−0.912930	−0.0392499	−0.0196250	−	0.999807i	$-0.506247\pi$
$541$	−0.912930	−0.0392499	−0.0196250	+	0.999807i	$0.506247\pi$
$542$	0	0
$543$	−13.1336	−0.563615
$544$	0	0
$545$	8.68259	0.371921
$546$	0	0
$547$	14.9716	0.640141	0.320070	−	0.947394i	$-0.396293\pi$
$547$	14.9716	0.640141	0.320070	+	0.947394i	$0.396293\pi$
$548$	0	0
$549$	−4.61109	−0.196797
$550$	0	0
$551$	−0.355126	−0.0151289
$552$	0	0
$553$	−10.0377	−0.426848
$554$	0	0
$555$	−17.9834	−0.763352
$556$	0	0
$557$	43.1682	1.82909	0.914547	−	0.404479i	$-0.132547\pi$
$557$	43.1682	1.82909	0.914547	+	0.404479i	$0.132547\pi$
$558$	0	0
$559$	−6.43807	−0.272301
$560$	0	0
$561$	70.8501	2.99129
$562$	0	0
$563$	−45.3966	−1.91324	−0.956618	−	0.291344i	$-0.905898\pi$
$563$	−45.3966	−1.91324	−0.956618	+	0.291344i	$0.905898\pi$
$564$	0	0
$565$	−7.72969	−0.325190
$566$	0	0
$567$	−12.1054	−0.508379
$568$	0	0
$569$	−33.3868	−1.39965	−0.699824	−	0.714315i	$-0.746738\pi$
$569$	−33.3868	−1.39965	−0.699824	+	0.714315i	$0.746738\pi$
$570$	0	0
$571$	8.12756	0.340128	0.170064	−	0.985433i	$-0.445603\pi$
$571$	8.12756	0.340128	0.170064	+	0.985433i	$0.445603\pi$
$572$	0	0
$573$	36.9605	1.54405
$574$	0	0
$575$	−6.30459	−0.262919
$576$	0	0
$577$	−0.873789	−0.0363763	−0.0181882	−	0.999835i	$-0.505790\pi$
$577$	−0.873789	−0.0363763	−0.0181882	+	0.999835i	$0.505790\pi$
$578$	0	0
$579$	23.3775	0.971535
$580$	0	0
$581$	−6.26415	−0.259881
$582$	0	0
$583$	−2.40379	−0.0995547
$584$	0	0
$585$	2.23749	0.0925089
$586$	0	0
$587$	36.0685	1.48871	0.744353	−	0.667787i	$-0.232758\pi$
$587$	36.0685	1.48871	0.744353	+	0.667787i	$0.232758\pi$
$588$	0	0
$589$	−18.7048	−0.770717
$590$	0	0
$591$	12.9752	0.533730
$592$	0	0
$593$	−44.1750	−1.81405	−0.907025	−	0.421076i	$-0.861653\pi$
$593$	−44.1750	−1.81405	−0.907025	+	0.421076i	$0.861653\pi$
$594$	0	0
$595$	−8.95945	−0.367302
$596$	0	0
$597$	−8.40008	−0.343792
$598$	0	0
$599$	−7.51214	−0.306938	−0.153469	−	0.988153i	$-0.549045\pi$
$599$	−7.51214	−0.306938	−0.153469	+	0.988153i	$0.549045\pi$
$600$	0	0
$601$	−35.3318	−1.44122	−0.720608	−	0.693343i	$-0.756137\pi$
$601$	−35.3318	−1.44122	−0.720608	+	0.693343i	$0.756137\pi$
$602$	0	0
$603$	0.929082	0.0378352
$604$	0	0
$605$	11.7772	0.478810
$606$	0	0
$607$	1.71787	0.0697261	0.0348631	−	0.999392i	$-0.488900\pi$
$607$	1.71787	0.0697261	0.0348631	+	0.999392i	$0.488900\pi$
$608$	0	0
$609$	0.407947	0.0165309
$610$	0	0
$611$	−4.23164	−0.171194
$612$	0	0
$613$	−2.78026	−0.112294	−0.0561469	−	0.998423i	$-0.517882\pi$
$613$	−2.78026	−0.112294	−0.0561469	+	0.998423i	$0.517882\pi$
$614$	0	0
$615$	6.37378	0.257016
$616$	0	0
$617$	−33.1898	−1.33617	−0.668086	−	0.744084i	$-0.732886\pi$
$617$	−33.1898	−1.33617	−0.668086	+	0.744084i	$0.732886\pi$
$618$	0	0
$619$	12.9037	0.518642	0.259321	−	0.965791i	$-0.416501\pi$
$619$	12.9037	0.518642	0.259321	+	0.965791i	$0.416501\pi$
$620$	0	0
$621$	−28.3796	−1.13883
$622$	0	0
$623$	2.29007	0.0917499
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.4194	−0.695665
$628$	0	0
$629$	−73.3250	−2.92366
$630$	0	0
$631$	−23.9375	−0.952936	−0.476468	−	0.879192i	$-0.658083\pi$
$631$	−23.9375	−0.952936	−0.476468	+	0.879192i	$0.658083\pi$
$632$	0	0
$633$	−0.273843	−0.0108843
$634$	0	0
$635$	−14.9015	−0.591346
$636$	0	0
$637$	−19.8082	−0.784828
$638$	0	0
$639$	−7.93925	−0.314072
$640$	0	0
$641$	3.44753	0.136169	0.0680845	−	0.997680i	$-0.478311\pi$
$641$	3.44753	0.136169	0.0680845	+	0.997680i	$0.478311\pi$
$642$	0	0
$643$	8.44353	0.332980	0.166490	−	0.986043i	$-0.446757\pi$
$643$	8.44353	0.332980	0.166490	+	0.986043i	$0.446757\pi$
$644$	0	0
$645$	3.51880	0.138553
$646$	0	0
$647$	0.116748	0.00458986	0.00229493	−	0.999997i	$-0.499270\pi$
$647$	0.116748	0.00458986	0.00229493	+	0.999997i	$0.499270\pi$
$648$	0	0
$649$	−67.4881	−2.64914
$650$	0	0
$651$	21.4869	0.842139
$652$	0	0
$653$	14.8110	0.579600	0.289800	−	0.957087i	$-0.406411\pi$
$653$	14.8110	0.579600	0.289800	+	0.957087i	$0.406411\pi$
$654$	0	0
$655$	15.9332	0.622563
$656$	0	0
$657$	3.54489	0.138299
$658$	0	0
$659$	39.6137	1.54313	0.771564	−	0.636151i	$-0.219475\pi$
$659$	39.6137	1.54313	0.771564	+	0.636151i	$0.219475\pi$
$660$	0	0
$661$	0.763064	0.0296797	0.0148399	−	0.999890i	$-0.495276\pi$
$661$	0.763064	0.0296797	0.0148399	+	0.999890i	$0.495276\pi$
$662$	0	0
$663$	51.8270	2.01280
$664$	0	0
$665$	2.20280	0.0854209
$666$	0	0
$667$	1.17047	0.0453208
$668$	0	0
$669$	36.2918	1.40312
$670$	0	0
$671$	−34.3367	−1.32555
$672$	0	0
$673$	−12.7674	−0.492146	−0.246073	−	0.969251i	$-0.579140\pi$
$673$	−12.7674	−0.492146	−0.246073	+	0.969251i	$0.579140\pi$
$674$	0	0
$675$	4.50142	0.173260
$676$	0	0
$677$	−16.6653	−0.640499	−0.320250	−	0.947333i	$-0.603767\pi$
$677$	−16.6653	−0.640499	−0.320250	+	0.947333i	$0.603767\pi$
$678$	0	0
$679$	18.4980	0.709889
$680$	0	0
$681$	−22.4480	−0.860210
$682$	0	0
$683$	4.96758	0.190079	0.0950395	−	0.995474i	$-0.469702\pi$
$683$	4.96758	0.190079	0.0950395	+	0.995474i	$0.469702\pi$
$684$	0	0
$685$	0.648293	0.0247700
$686$	0	0
$687$	−41.3922	−1.57921
$688$	0	0
$689$	−1.75838	−0.0669888
$690$	0	0
$691$	−45.1073	−1.71596	−0.857982	−	0.513680i	$-0.828282\pi$
$691$	−45.1073	−1.71596	−0.857982	+	0.513680i	$0.828282\pi$
$692$	0	0
$693$	3.52242	0.133806
$694$	0	0
$695$	2.77102	0.105111
$696$	0	0
$697$	25.9883	0.984377
$698$	0	0
$699$	−15.6050	−0.590234
$700$	0	0
$701$	−20.1108	−0.759573	−0.379786	−	0.925074i	$-0.624003\pi$
$701$	−20.1108	−0.759573	−0.379786	+	0.925074i	$0.624003\pi$
$702$	0	0
$703$	18.0279	0.679937
$704$	0	0
$705$	2.31285	0.0871070
$706$	0	0
$707$	−6.68817	−0.251535
$708$	0	0
$709$	−31.3240	−1.17640	−0.588199	−	0.808716i	$-0.700163\pi$
$709$	−31.3240	−1.17640	−0.588199	+	0.808716i	$0.700163\pi$
$710$	0	0
$711$	−5.58645	−0.209508
$712$	0	0
$713$	61.6496	2.30879
$714$	0	0
$715$	16.6616	0.623107
$716$	0	0
$717$	38.4862	1.43730
$718$	0	0
$719$	−30.0942	−1.12233	−0.561163	−	0.827706i	$-0.689646\pi$
$719$	−30.0942	−1.12233	−0.561163	+	0.827706i	$0.689646\pi$
$720$	0	0
$721$	18.7255	0.697373
$722$	0	0
$723$	−34.3108	−1.27603
$724$	0	0
$725$	−0.185654	−0.00689500
$726$	0	0
$727$	31.0306	1.15086	0.575430	−	0.817851i	$-0.304835\pi$
$727$	31.0306	1.15086	0.575430	+	0.817851i	$0.304835\pi$
$728$	0	0
$729$	19.0305	0.704834
$730$	0	0
$731$	14.3475	0.530660
$732$	0	0
$733$	17.0809	0.630897	0.315448	−	0.948943i	$-0.397845\pi$
$733$	17.0809	0.630897	0.315448	+	0.948943i	$0.397845\pi$
$734$	0	0
$735$	10.8264	0.399337
$736$	0	0
$737$	6.91844	0.254844
$738$	0	0
$739$	−5.00537	−0.184125	−0.0920627	−	0.995753i	$-0.529346\pi$
$739$	−5.00537	−0.184125	−0.0920627	+	0.995753i	$0.529346\pi$
$740$	0	0
$741$	−12.7424	−0.468103
$742$	0	0
$743$	−0.484143	−0.0177615	−0.00888074	−	0.999961i	$-0.502827\pi$
$743$	−0.484143	−0.0177615	−0.00888074	+	0.999961i	$0.502827\pi$
$744$	0	0
$745$	7.92025	0.290175
$746$	0	0
$747$	−3.48627	−0.127556
$748$	0	0
$749$	11.9809	0.437773
$750$	0	0
$751$	−7.69016	−0.280618	−0.140309	−	0.990108i	$-0.544810\pi$
$751$	−7.69016	−0.280618	−0.140309	+	0.990108i	$0.544810\pi$
$752$	0	0
$753$	8.61781	0.314050
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	20.2578	0.736283	0.368141	−	0.929770i	$-0.379994\pi$
$757$	20.2578	0.736283	0.368141	+	0.929770i	$0.379994\pi$
$758$	0	0
$759$	57.4131	2.08397
$760$	0	0
$761$	5.25858	0.190623	0.0953117	−	0.995447i	$-0.469615\pi$
$761$	5.25858	0.190623	0.0953117	+	0.995447i	$0.469615\pi$
$762$	0	0
$763$	9.99874	0.361979
$764$	0	0
$765$	−4.98633	−0.180281
$766$	0	0
$767$	−49.3677	−1.78257
$768$	0	0
$769$	−27.3986	−0.988018	−0.494009	−	0.869457i	$-0.664469\pi$
$769$	−27.3986	−0.988018	−0.494009	+	0.869457i	$0.664469\pi$
$770$	0	0
$771$	6.05450	0.218048
$772$	0	0
$773$	−24.1021	−0.866892	−0.433446	−	0.901180i	$-0.642702\pi$
$773$	−24.1021	−0.866892	−0.433446	+	0.901180i	$0.642702\pi$
$774$	0	0
$775$	−9.77853	−0.351255
$776$	0	0
$777$	−20.7094	−0.742945
$778$	0	0
$779$	−6.38957	−0.228930
$780$	0	0
$781$	−59.1199	−2.11548
$782$	0	0
$783$	−0.835706	−0.0298657
$784$	0	0
$785$	6.40941	0.228762
$786$	0	0
$787$	22.8459	0.814367	0.407183	−	0.913346i	$-0.366511\pi$
$787$	22.8459	0.814367	0.407183	+	0.913346i	$0.366511\pi$
$788$	0	0
$789$	7.90509	0.281429
$790$	0	0
$791$	−8.90139	−0.316497
$792$	0	0
$793$	−25.1174	−0.891944
$794$	0	0
$795$	0.961061	0.0340853
$796$	0	0
$797$	14.3708	0.509039	0.254519	−	0.967068i	$-0.418083\pi$
$797$	14.3708	0.509039	0.254519	+	0.967068i	$0.418083\pi$
$798$	0	0
$799$	9.43036	0.333622
$800$	0	0
$801$	1.27453	0.0450332
$802$	0	0
$803$	26.3971	0.931534
$804$	0	0
$805$	−7.26026	−0.255891
$806$	0	0
$807$	44.6500	1.57176
$808$	0	0
$809$	−3.50159	−0.123109	−0.0615547	−	0.998104i	$-0.519606\pi$
$809$	−3.50159	−0.123109	−0.0615547	+	0.998104i	$0.519606\pi$
$810$	0	0
$811$	−23.0733	−0.810214	−0.405107	−	0.914269i	$-0.632766\pi$
$811$	−23.0733	−0.810214	−0.405107	+	0.914269i	$0.632766\pi$
$812$	0	0
$813$	44.4815	1.56003
$814$	0	0
$815$	−18.7773	−0.657738
$816$	0	0
$817$	−3.52752	−0.123412
$818$	0	0
$819$	2.57666	0.0900358
$820$	0	0
$821$	−4.92514	−0.171889	−0.0859443	−	0.996300i	$-0.527391\pi$
$821$	−4.92514	−0.171889	−0.0859443	+	0.996300i	$0.527391\pi$
$822$	0	0
$823$	20.5252	0.715465	0.357733	−	0.933824i	$-0.383550\pi$
$823$	20.5252	0.715465	0.357733	+	0.933824i	$0.383550\pi$
$824$	0	0
$825$	−9.10657	−0.317050
$826$	0	0
$827$	−23.6420	−0.822112	−0.411056	−	0.911610i	$-0.634840\pi$
$827$	−23.6420	−0.822112	−0.411056	+	0.911610i	$0.634840\pi$
$828$	0	0
$829$	5.90235	0.204997	0.102499	−	0.994733i	$-0.467316\pi$
$829$	5.90235	0.204997	0.102499	+	0.994733i	$0.467316\pi$
$830$	0	0
$831$	−10.7086	−0.371478
$832$	0	0
$833$	44.1432	1.52947
$834$	0	0
$835$	−8.35575	−0.289163
$836$	0	0
$837$	−44.0173	−1.52146
$838$	0	0
$839$	26.7355	0.923011	0.461505	−	0.887137i	$-0.347309\pi$
$839$	26.7355	0.923011	0.461505	+	0.887137i	$0.347309\pi$
$840$	0	0
$841$	−28.9655	−0.998811
$842$	0	0
$843$	36.0929	1.24311
$844$	0	0
$845$	−0.812022	−0.0279344
$846$	0	0
$847$	13.5624	0.466009
$848$	0	0
$849$	33.9816	1.16625
$850$	0	0
$851$	−59.4187	−2.03685
$852$	0	0
$853$	−41.1209	−1.40795	−0.703976	−	0.710224i	$-0.748594\pi$
$853$	−41.1209	−1.40795	−0.703976	+	0.710224i	$0.748594\pi$
$854$	0	0
$855$	1.22596	0.0419268
$856$	0	0
$857$	32.4116	1.10716	0.553580	−	0.832796i	$-0.313261\pi$
$857$	32.4116	1.10716	0.553580	+	0.832796i	$0.313261\pi$
$858$	0	0
$859$	5.88089	0.200653	0.100327	−	0.994955i	$-0.468011\pi$
$859$	5.88089	0.200653	0.100327	+	0.994955i	$0.468011\pi$
$860$	0	0
$861$	7.33994	0.250145
$862$	0	0
$863$	8.70645	0.296371	0.148185	−	0.988960i	$-0.452657\pi$
$863$	8.70645	0.296371	0.148185	+	0.988960i	$0.452657\pi$
$864$	0	0
$865$	11.9231	0.405398
$866$	0	0
$867$	−83.0605	−2.82088
$868$	0	0
$869$	−41.5997	−1.41117
$870$	0	0
$871$	5.06086	0.171481
$872$	0	0
$873$	10.2950	0.348432
$874$	0	0
$875$	1.15158	0.0389307
$876$	0	0
$877$	35.3638	1.19415	0.597075	−	0.802186i	$-0.296330\pi$
$877$	35.3638	1.19415	0.597075	+	0.802186i	$0.296330\pi$
$878$	0	0
$879$	38.1343	1.28624
$880$	0	0
$881$	−10.1564	−0.342178	−0.171089	−	0.985256i	$-0.554729\pi$
$881$	−10.1564	−0.342178	−0.171089	+	0.985256i	$0.554729\pi$
$882$	0	0
$883$	20.1784	0.679059	0.339529	−	0.940595i	$-0.389732\pi$
$883$	20.1784	0.679059	0.339529	+	0.940595i	$0.389732\pi$
$884$	0	0
$885$	26.9825	0.907007
$886$	0	0
$887$	31.1471	1.04582	0.522909	−	0.852389i	$-0.324847\pi$
$887$	31.1471	1.04582	0.522909	+	0.852389i	$0.324847\pi$
$888$	0	0
$889$	−17.1603	−0.575538
$890$	0	0
$891$	−50.1688	−1.68072
$892$	0	0
$893$	−2.31858	−0.0775883
$894$	0	0
$895$	−18.9922	−0.634838
$896$	0	0
$897$	41.9979	1.40227
$898$	0	0
$899$	1.81542	0.0605476
$900$	0	0
$901$	3.91861	0.130548
$902$	0	0
$903$	4.05219	0.134849
$904$	0	0
$905$	6.88300	0.228799
$906$	0	0
$907$	50.1859	1.66640	0.833198	−	0.552975i	$-0.186508\pi$
$907$	50.1859	1.66640	0.833198	+	0.552975i	$0.186508\pi$
$908$	0	0
$909$	−3.72226	−0.123460
$910$	0	0
$911$	−15.9822	−0.529513	−0.264757	−	0.964315i	$-0.585292\pi$
$911$	−15.9822	−0.529513	−0.264757	+	0.964315i	$0.585292\pi$
$912$	0	0
$913$	−25.9607	−0.859173
$914$	0	0
$915$	13.7282	0.453840
$916$	0	0
$917$	18.3485	0.605920
$918$	0	0
$919$	28.9576	0.955224	0.477612	−	0.878571i	$-0.341502\pi$
$919$	28.9576	0.955224	0.477612	+	0.878571i	$0.341502\pi$
$920$	0	0
$921$	−7.85252	−0.258749
$922$	0	0
$923$	−43.2464	−1.42347
$924$	0	0
$925$	9.42468	0.309882
$926$	0	0
$927$	10.4216	0.342289
$928$	0	0
$929$	13.9893	0.458973	0.229486	−	0.973312i	$-0.426295\pi$
$929$	13.9893	0.458973	0.229486	+	0.973312i	$0.426295\pi$
$930$	0	0
$931$	−10.8532	−0.355699
$932$	0	0
$933$	−35.4210	−1.15963
$934$	0	0
$935$	−37.1309	−1.21431
$936$	0	0
$937$	−40.0484	−1.30832	−0.654162	−	0.756354i	$-0.726979\pi$
$937$	−40.0484	−1.30832	−0.654162	+	0.756354i	$0.726979\pi$
$938$	0	0
$939$	36.7689	1.19991
$940$	0	0
$941$	−0.759012	−0.0247431	−0.0123715	−	0.999923i	$-0.503938\pi$
$941$	−0.759012	−0.0247431	−0.0123715	+	0.999923i	$0.503938\pi$
$942$	0	0
$943$	21.0595	0.685793
$944$	0	0
$945$	5.18377	0.168628
$946$	0	0
$947$	−10.8394	−0.352234	−0.176117	−	0.984369i	$-0.556354\pi$
$947$	−10.8394	−0.352234	−0.176117	+	0.984369i	$0.556354\pi$
$948$	0	0
$949$	19.3096	0.626815
$950$	0	0
$951$	−27.8589	−0.903386
$952$	0	0
$953$	24.6328	0.797936	0.398968	−	0.916965i	$-0.369368\pi$
$953$	24.6328	0.797936	0.398968	+	0.916965i	$0.369368\pi$
$954$	0	0
$955$	−19.3701	−0.626803
$956$	0	0
$957$	1.69067	0.0546515
$958$	0	0
$959$	0.746564	0.0241078
$960$	0	0
$961$	64.6196	2.08450
$962$	0	0
$963$	6.66791	0.214870
$964$	0	0
$965$	−12.2516	−0.394393
$966$	0	0
$967$	−5.46203	−0.175647	−0.0878236	−	0.996136i	$-0.527991\pi$
$967$	−5.46203	−0.175647	−0.0878236	+	0.996136i	$0.527991\pi$
$968$	0	0
$969$	28.3968	0.912238
$970$	0	0
$971$	34.7957	1.11665	0.558324	−	0.829623i	$-0.311444\pi$
$971$	34.7957	1.11665	0.558324	+	0.829623i	$0.311444\pi$
$972$	0	0
$973$	3.19107	0.102301
$974$	0	0
$975$	−6.66148	−0.213338
$976$	0	0
$977$	56.3630	1.80321	0.901606	−	0.432558i	$-0.142389\pi$
$977$	56.3630	1.80321	0.901606	+	0.432558i	$0.142389\pi$
$978$	0	0
$979$	9.49082	0.303328
$980$	0	0
$981$	5.56474	0.177669
$982$	0	0
$983$	3.05597	0.0974703	0.0487351	−	0.998812i	$-0.484481\pi$
$983$	3.05597	0.0974703	0.0487351	+	0.998812i	$0.484481\pi$
$984$	0	0
$985$	−6.80002	−0.216667
$986$	0	0
$987$	2.66344	0.0847783
$988$	0	0
$989$	11.6264	0.369699
$990$	0	0
$991$	−0.706158	−0.0224318	−0.0112159	−	0.999937i	$-0.503570\pi$
$991$	−0.706158	−0.0224318	−0.0112159	+	0.999937i	$0.503570\pi$
$992$	0	0
$993$	44.7927	1.42145
$994$	0	0
$995$	4.40229	0.139562
$996$	0	0
$997$	16.6064	0.525931	0.262966	−	0.964805i	$-0.415299\pi$
$997$	16.6064	0.525931	0.262966	+	0.964805i	$0.415299\pi$
$998$	0	0
$999$	42.4245	1.34225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.6	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.6	✓	19	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.90812 q^{3} +1.00000 q^{5} +1.15158 q^{7} +0.640908 q^{9} +O(q^{10})\)	\(q-1.90812 q^{3} +1.00000 q^{5} +1.15158 q^{7} +0.640908 q^{9} +4.77254 q^{11} +3.49113 q^{13} -1.90812 q^{15} -7.78011 q^{17} +1.91284 q^{19} -2.19736 q^{21} -6.30459 q^{23} +1.00000 q^{25} +4.50142 q^{27} -0.185654 q^{29} -9.77853 q^{31} -9.10657 q^{33} +1.15158 q^{35} +9.42468 q^{37} -6.66148 q^{39} -3.34035 q^{41} -1.84412 q^{43} +0.640908 q^{45} -1.21211 q^{47} -5.67385 q^{49} +14.8454 q^{51} -0.503670 q^{53} +4.77254 q^{55} -3.64993 q^{57} -14.1409 q^{59} -7.19463 q^{61} +0.738059 q^{63} +3.49113 q^{65} +1.44963 q^{67} +12.0299 q^{69} -12.3875 q^{71} +5.53104 q^{73} -1.90812 q^{75} +5.49599 q^{77} -8.71646 q^{79} -10.5120 q^{81} -5.43959 q^{83} -7.78011 q^{85} +0.354249 q^{87} +1.98863 q^{89} +4.02033 q^{91} +18.6586 q^{93} +1.91284 q^{95} +16.0631 q^{97} +3.05876 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

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Varieties

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Database

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