LMFDB - Embedded newform 6040.2.a.n.1.10

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 $ x^13 - 4x^12 - 14x^11 + 70x^10 + 41x^9 - 403x^8 + 109x^7 + 870x^6 - 444x^5 - 708x^4 + 373x^3 + 108x^2 - 82x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.14309$ 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)

$f(q)$	$=$	$q+1.14309 q^{3} -1.00000 q^{5} +3.09474 q^{7} -1.69334 q^{9} +O(q^{10})$	$q+1.14309 q^{3} -1.00000 q^{5} +3.09474 q^{7} -1.69334 q^{9} -1.73003 q^{11} -2.25696 q^{13} -1.14309 q^{15} -2.56723 q^{17} +2.85712 q^{19} +3.53757 q^{21} -2.45188 q^{23} +1.00000 q^{25} -5.36492 q^{27} +2.22403 q^{29} +6.27500 q^{31} -1.97758 q^{33} -3.09474 q^{35} -3.11378 q^{37} -2.57991 q^{39} +8.33290 q^{41} -6.35719 q^{43} +1.69334 q^{45} -13.5744 q^{47} +2.57741 q^{49} -2.93458 q^{51} +0.404612 q^{53} +1.73003 q^{55} +3.26595 q^{57} +5.72300 q^{59} +3.49523 q^{61} -5.24045 q^{63} +2.25696 q^{65} -9.88565 q^{67} -2.80272 q^{69} -0.871559 q^{71} -8.06656 q^{73} +1.14309 q^{75} -5.35399 q^{77} -14.8827 q^{79} -1.05257 q^{81} +11.5688 q^{83} +2.56723 q^{85} +2.54228 q^{87} -11.1370 q^{89} -6.98470 q^{91} +7.17290 q^{93} -2.85712 q^{95} +10.1341 q^{97} +2.92953 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * 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
$3$	1.14309	0.659964	0.329982	−	0.943987i	$-0.392957\pi$
$3$	1.14309	0.659964	0.329982	+	0.943987i	$0.392957\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.09474	1.16970	0.584851	−	0.811141i	$-0.301153\pi$
$7$	3.09474	1.16970	0.584851	+	0.811141i	$0.301153\pi$
$8$	0	0
$9$	−1.69334	−0.564447
$10$	0	0
$11$	−1.73003	−0.521624	−0.260812	−	0.965390i	$-0.583990\pi$
$11$	−1.73003	−0.521624	−0.260812	+	0.965390i	$0.583990\pi$
$12$	0	0
$13$	−2.25696	−0.625968	−0.312984	−	0.949758i	$-0.601329\pi$
$13$	−2.25696	−0.625968	−0.312984	+	0.949758i	$0.601329\pi$
$14$	0	0
$15$	−1.14309	−0.295145
$16$	0	0
$17$	−2.56723	−0.622646	−0.311323	−	0.950304i	$-0.600772\pi$
$17$	−2.56723	−0.622646	−0.311323	+	0.950304i	$0.600772\pi$
$18$	0	0
$19$	2.85712	0.655468	0.327734	−	0.944770i	$-0.393715\pi$
$19$	2.85712	0.655468	0.327734	+	0.944770i	$0.393715\pi$
$20$	0	0
$21$	3.53757	0.771961
$22$	0	0
$23$	−2.45188	−0.511252	−0.255626	−	0.966776i	$-0.582282\pi$
$23$	−2.45188	−0.511252	−0.255626	+	0.966776i	$0.582282\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.36492	−1.03248
$28$	0	0
$29$	2.22403	0.412993	0.206496	−	0.978447i	$-0.433794\pi$
$29$	2.22403	0.412993	0.206496	+	0.978447i	$0.433794\pi$
$30$	0	0
$31$	6.27500	1.12702	0.563512	−	0.826108i	$-0.309450\pi$
$31$	6.27500	1.12702	0.563512	+	0.826108i	$0.309450\pi$
$32$	0	0
$33$	−1.97758	−0.344253
$34$	0	0
$35$	−3.09474	−0.523106
$36$	0	0
$37$	−3.11378	−0.511902	−0.255951	−	0.966690i	$-0.582389\pi$
$37$	−3.11378	−0.511902	−0.255951	+	0.966690i	$0.582389\pi$
$38$	0	0
$39$	−2.57991	−0.413116
$40$	0	0
$41$	8.33290	1.30138	0.650690	−	0.759343i	$-0.274480\pi$
$41$	8.33290	1.30138	0.650690	+	0.759343i	$0.274480\pi$
$42$	0	0
$43$	−6.35719	−0.969462	−0.484731	−	0.874663i	$-0.661082\pi$
$43$	−6.35719	−0.969462	−0.484731	+	0.874663i	$0.661082\pi$
$44$	0	0
$45$	1.69334	0.252428
$46$	0	0
$47$	−13.5744	−1.98002	−0.990012	−	0.140981i	$-0.954974\pi$
$47$	−13.5744	−1.98002	−0.990012	+	0.140981i	$0.954974\pi$
$48$	0	0
$49$	2.57741	0.368201
$50$	0	0
$51$	−2.93458	−0.410924
$52$	0	0
$53$	0.404612	0.0555777	0.0277888	−	0.999614i	$-0.491153\pi$
$53$	0.404612	0.0555777	0.0277888	+	0.999614i	$0.491153\pi$
$54$	0	0
$55$	1.73003	0.233277
$56$	0	0
$57$	3.26595	0.432586
$58$	0	0
$59$	5.72300	0.745071	0.372535	−	0.928018i	$-0.378489\pi$
$59$	5.72300	0.745071	0.372535	+	0.928018i	$0.378489\pi$
$60$	0	0
$61$	3.49523	0.447518	0.223759	−	0.974644i	$-0.428167\pi$
$61$	3.49523	0.447518	0.223759	+	0.974644i	$0.428167\pi$
$62$	0	0
$63$	−5.24045	−0.660235
$64$	0	0
$65$	2.25696	0.279941
$66$	0	0
$67$	−9.88565	−1.20772	−0.603862	−	0.797089i	$-0.706372\pi$
$67$	−9.88565	−1.20772	−0.603862	+	0.797089i	$0.706372\pi$
$68$	0	0
$69$	−2.80272	−0.337408
$70$	0	0
$71$	−0.871559	−0.103435	−0.0517175	−	0.998662i	$-0.516470\pi$
$71$	−0.871559	−0.103435	−0.0517175	+	0.998662i	$0.516470\pi$
$72$	0	0
$73$	−8.06656	−0.944120	−0.472060	−	0.881567i	$-0.656489\pi$
$73$	−8.06656	−0.944120	−0.472060	+	0.881567i	$0.656489\pi$
$74$	0	0
$75$	1.14309	0.131993
$76$	0	0
$77$	−5.35399	−0.610144
$78$	0	0
$79$	−14.8827	−1.67444	−0.837219	−	0.546868i	$-0.815820\pi$
$79$	−14.8827	−1.67444	−0.837219	+	0.546868i	$0.815820\pi$
$80$	0	0
$81$	−1.05257	−0.116952
$82$	0	0
$83$	11.5688	1.26984	0.634919	−	0.772578i	$-0.281033\pi$
$83$	11.5688	1.26984	0.634919	+	0.772578i	$0.281033\pi$
$84$	0	0
$85$	2.56723	0.278456
$86$	0	0
$87$	2.54228	0.272561
$88$	0	0
$89$	−11.1370	−1.18052	−0.590259	−	0.807214i	$-0.700974\pi$
$89$	−11.1370	−1.18052	−0.590259	+	0.807214i	$0.700974\pi$
$90$	0	0
$91$	−6.98470	−0.732195
$92$	0	0
$93$	7.17290	0.743795
$94$	0	0
$95$	−2.85712	−0.293134
$96$	0	0
$97$	10.1341	1.02896	0.514480	−	0.857502i	$-0.327985\pi$
$97$	10.1341	1.02896	0.514480	+	0.857502i	$0.327985\pi$
$98$	0	0
$99$	2.92953	0.294429
$100$	0	0
$101$	−12.4134	−1.23518	−0.617588	−	0.786502i	$-0.711890\pi$
$101$	−12.4134	−1.23518	−0.617588	+	0.786502i	$0.711890\pi$
$102$	0	0
$103$	−15.3075	−1.50830	−0.754148	−	0.656704i	$-0.771950\pi$
$103$	−15.3075	−1.50830	−0.754148	+	0.656704i	$0.771950\pi$
$104$	0	0
$105$	−3.53757	−0.345231
$106$	0	0
$107$	−6.18262	−0.597697	−0.298848	−	0.954301i	$-0.596603\pi$
$107$	−6.18262	−0.597697	−0.298848	+	0.954301i	$0.596603\pi$
$108$	0	0
$109$	7.59400	0.727374	0.363687	−	0.931521i	$-0.381518\pi$
$109$	7.59400	0.727374	0.363687	+	0.931521i	$0.381518\pi$
$110$	0	0
$111$	−3.55934	−0.337837
$112$	0	0
$113$	−0.985069	−0.0926675	−0.0463337	−	0.998926i	$-0.514754\pi$
$113$	−0.985069	−0.0926675	−0.0463337	+	0.998926i	$0.514754\pi$
$114$	0	0
$115$	2.45188	0.228639
$116$	0	0
$117$	3.82180	0.353326
$118$	0	0
$119$	−7.94492	−0.728310
$120$	0	0
$121$	−8.00699	−0.727908
$122$	0	0
$123$	9.52527	0.858864
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.53102	0.490799	0.245400	−	0.969422i	$-0.421081\pi$
$127$	5.53102	0.490799	0.245400	+	0.969422i	$0.421081\pi$
$128$	0	0
$129$	−7.26684	−0.639810
$130$	0	0
$131$	−18.2481	−1.59435	−0.797173	−	0.603751i	$-0.793672\pi$
$131$	−18.2481	−1.59435	−0.797173	+	0.603751i	$0.793672\pi$
$132$	0	0
$133$	8.84204	0.766702
$134$	0	0
$135$	5.36492	0.461739
$136$	0	0
$137$	−15.2304	−1.30122	−0.650611	−	0.759411i	$-0.725487\pi$
$137$	−15.2304	−1.30122	−0.650611	+	0.759411i	$0.725487\pi$
$138$	0	0
$139$	5.77978	0.490235	0.245117	−	0.969493i	$-0.421174\pi$
$139$	5.77978	0.490235	0.245117	+	0.969493i	$0.421174\pi$
$140$	0	0
$141$	−15.5167	−1.30675
$142$	0	0
$143$	3.90461	0.326520
$144$	0	0
$145$	−2.22403	−0.184696
$146$	0	0
$147$	2.94621	0.242999
$148$	0	0
$149$	−14.5360	−1.19084	−0.595419	−	0.803415i	$-0.703014\pi$
$149$	−14.5360	−1.19084	−0.595419	+	0.803415i	$0.703014\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	4.34721	0.351451
$154$	0	0
$155$	−6.27500	−0.504020
$156$	0	0
$157$	12.7092	1.01430	0.507151	−	0.861857i	$-0.330699\pi$
$157$	12.7092	1.01430	0.507151	+	0.861857i	$0.330699\pi$
$158$	0	0
$159$	0.462508	0.0366793
$160$	0	0
$161$	−7.58792	−0.598012
$162$	0	0
$163$	6.37443	0.499284	0.249642	−	0.968338i	$-0.419687\pi$
$163$	6.37443	0.499284	0.249642	+	0.968338i	$0.419687\pi$
$164$	0	0
$165$	1.97758	0.153955
$166$	0	0
$167$	−17.5902	−1.36117	−0.680583	−	0.732671i	$-0.738274\pi$
$167$	−17.5902	−1.36117	−0.680583	+	0.732671i	$0.738274\pi$
$168$	0	0
$169$	−7.90614	−0.608164
$170$	0	0
$171$	−4.83808	−0.369977
$172$	0	0
$173$	−17.6723	−1.34360	−0.671800	−	0.740733i	$-0.734479\pi$
$173$	−17.6723	−1.34360	−0.671800	+	0.740733i	$0.734479\pi$
$174$	0	0
$175$	3.09474	0.233940
$176$	0	0
$177$	6.54191	0.491720
$178$	0	0
$179$	7.42408	0.554901	0.277451	−	0.960740i	$-0.410510\pi$
$179$	7.42408	0.554901	0.277451	+	0.960740i	$0.410510\pi$
$180$	0	0
$181$	−3.55115	−0.263955	−0.131978	−	0.991253i	$-0.542133\pi$
$181$	−3.55115	−0.263955	−0.131978	+	0.991253i	$0.542133\pi$
$182$	0	0
$183$	3.99537	0.295346
$184$	0	0
$185$	3.11378	0.228930
$186$	0	0
$187$	4.44140	0.324787
$188$	0	0
$189$	−16.6030	−1.20769
$190$	0	0
$191$	9.39670	0.679922	0.339961	−	0.940440i	$-0.389586\pi$
$191$	9.39670	0.679922	0.339961	+	0.940440i	$0.389586\pi$
$192$	0	0
$193$	26.4123	1.90120	0.950600	−	0.310419i	$-0.100470\pi$
$193$	26.4123	1.90120	0.950600	+	0.310419i	$0.100470\pi$
$194$	0	0
$195$	2.57991	0.184751
$196$	0	0
$197$	6.97666	0.497067	0.248533	−	0.968623i	$-0.420051\pi$
$197$	6.97666	0.497067	0.248533	+	0.968623i	$0.420051\pi$
$198$	0	0
$199$	−16.0146	−1.13525	−0.567623	−	0.823288i	$-0.692137\pi$
$199$	−16.0146	−1.13525	−0.567623	+	0.823288i	$0.692137\pi$
$200$	0	0
$201$	−11.3002	−0.797055
$202$	0	0
$203$	6.88281	0.483078
$204$	0	0
$205$	−8.33290	−0.581995
$206$	0	0
$207$	4.15186	0.288575
$208$	0	0
$209$	−4.94291	−0.341908
$210$	0	0
$211$	9.30596	0.640649	0.320324	−	0.947308i	$-0.396208\pi$
$211$	9.30596	0.640649	0.320324	+	0.947308i	$0.396208\pi$
$212$	0	0
$213$	−0.996272	−0.0682634
$214$	0	0
$215$	6.35719	0.433556
$216$	0	0
$217$	19.4195	1.31828
$218$	0	0
$219$	−9.22082	−0.623085
$220$	0	0
$221$	5.79414	0.389756
$222$	0	0
$223$	−3.80699	−0.254935	−0.127468	−	0.991843i	$-0.540685\pi$
$223$	−3.80699	−0.254935	−0.127468	+	0.991843i	$0.540685\pi$
$224$	0	0
$225$	−1.69334	−0.112889
$226$	0	0
$227$	−16.5848	−1.10077	−0.550386	−	0.834910i	$-0.685519\pi$
$227$	−16.5848	−1.10077	−0.550386	+	0.834910i	$0.685519\pi$
$228$	0	0
$229$	12.8939	0.852050	0.426025	−	0.904711i	$-0.359914\pi$
$229$	12.8939	0.852050	0.426025	+	0.904711i	$0.359914\pi$
$230$	0	0
$231$	−6.12010	−0.402673
$232$	0	0
$233$	−16.7221	−1.09550	−0.547751	−	0.836641i	$-0.684516\pi$
$233$	−16.7221	−1.09550	−0.547751	+	0.836641i	$0.684516\pi$
$234$	0	0
$235$	13.5744	0.885494
$236$	0	0
$237$	−17.0123	−1.10507
$238$	0	0
$239$	5.04047	0.326041	0.163021	−	0.986623i	$-0.447876\pi$
$239$	5.04047	0.326041	0.163021	+	0.986623i	$0.447876\pi$
$240$	0	0
$241$	24.5682	1.58258	0.791289	−	0.611443i	$-0.209411\pi$
$241$	24.5682	1.58258	0.791289	+	0.611443i	$0.209411\pi$
$242$	0	0
$243$	14.8916	0.955295
$244$	0	0
$245$	−2.57741	−0.164664
$246$	0	0
$247$	−6.44840	−0.410302
$248$	0	0
$249$	13.2242	0.838048
$250$	0	0
$251$	−12.7968	−0.807727	−0.403864	−	0.914819i	$-0.632333\pi$
$251$	−12.7968	−0.807727	−0.403864	+	0.914819i	$0.632333\pi$
$252$	0	0
$253$	4.24182	0.266681
$254$	0	0
$255$	2.93458	0.183771
$256$	0	0
$257$	−6.10848	−0.381036	−0.190518	−	0.981684i	$-0.561017\pi$
$257$	−6.10848	−0.381036	−0.190518	+	0.981684i	$0.561017\pi$
$258$	0	0
$259$	−9.63633	−0.598773
$260$	0	0
$261$	−3.76605	−0.233113
$262$	0	0
$263$	24.3889	1.50388	0.751941	−	0.659230i	$-0.229118\pi$
$263$	24.3889	1.50388	0.751941	+	0.659230i	$0.229118\pi$
$264$	0	0
$265$	−0.404612	−0.0248551
$266$	0	0
$267$	−12.7306	−0.779100
$268$	0	0
$269$	2.04476	0.124671	0.0623357	−	0.998055i	$-0.480145\pi$
$269$	2.04476	0.124671	0.0623357	+	0.998055i	$0.480145\pi$
$270$	0	0
$271$	14.0267	0.852061	0.426031	−	0.904709i	$-0.359912\pi$
$271$	14.0267	0.852061	0.426031	+	0.904709i	$0.359912\pi$
$272$	0	0
$273$	−7.98415	−0.483223
$274$	0	0
$275$	−1.73003	−0.104325
$276$	0	0
$277$	10.7187	0.644026	0.322013	−	0.946735i	$-0.395641\pi$
$277$	10.7187	0.644026	0.322013	+	0.946735i	$0.395641\pi$
$278$	0	0
$279$	−10.6257	−0.636145
$280$	0	0
$281$	−18.4936	−1.10324	−0.551618	−	0.834097i	$-0.685989\pi$
$281$	−18.4936	−1.10324	−0.551618	+	0.834097i	$0.685989\pi$
$282$	0	0
$283$	−5.25851	−0.312586	−0.156293	−	0.987711i	$-0.549954\pi$
$283$	−5.25851	−0.312586	−0.156293	+	0.987711i	$0.549954\pi$
$284$	0	0
$285$	−3.26595	−0.193458
$286$	0	0
$287$	25.7881	1.52223
$288$	0	0
$289$	−10.4093	−0.612312
$290$	0	0
$291$	11.5842	0.679077
$292$	0	0
$293$	10.6347	0.621286	0.310643	−	0.950527i	$-0.399456\pi$
$293$	10.6347	0.621286	0.310643	+	0.950527i	$0.399456\pi$
$294$	0	0
$295$	−5.72300	−0.333206
$296$	0	0
$297$	9.28148	0.538566
$298$	0	0
$299$	5.53378	0.320027
$300$	0	0
$301$	−19.6738	−1.13398
$302$	0	0
$303$	−14.1896	−0.815172
$304$	0	0
$305$	−3.49523	−0.200136
$306$	0	0
$307$	−10.9517	−0.625047	−0.312524	−	0.949910i	$-0.601174\pi$
$307$	−10.9517	−0.625047	−0.312524	+	0.949910i	$0.601174\pi$
$308$	0	0
$309$	−17.4979	−0.995422
$310$	0	0
$311$	−6.38807	−0.362234	−0.181117	−	0.983462i	$-0.557971\pi$
$311$	−6.38807	−0.362234	−0.181117	+	0.983462i	$0.557971\pi$
$312$	0	0
$313$	−9.56125	−0.540434	−0.270217	−	0.962799i	$-0.587095\pi$
$313$	−9.56125	−0.540434	−0.270217	+	0.962799i	$0.587095\pi$
$314$	0	0
$315$	5.24045	0.295266
$316$	0	0
$317$	8.16224	0.458437	0.229218	−	0.973375i	$-0.426383\pi$
$317$	8.16224	0.458437	0.229218	+	0.973375i	$0.426383\pi$
$318$	0	0
$319$	−3.84765	−0.215427
$320$	0	0
$321$	−7.06731	−0.394459
$322$	0	0
$323$	−7.33490	−0.408125
$324$	0	0
$325$	−2.25696	−0.125194
$326$	0	0
$327$	8.68064	0.480041
$328$	0	0
$329$	−42.0091	−2.31604
$330$	0	0
$331$	33.0925	1.81893	0.909464	−	0.415782i	$-0.136492\pi$
$331$	33.0925	1.81893	0.909464	+	0.415782i	$0.136492\pi$
$332$	0	0
$333$	5.27269	0.288942
$334$	0	0
$335$	9.88565	0.540111
$336$	0	0
$337$	−9.43652	−0.514040	−0.257020	−	0.966406i	$-0.582741\pi$
$337$	−9.43652	−0.514040	−0.257020	+	0.966406i	$0.582741\pi$
$338$	0	0
$339$	−1.12602	−0.0611572
$340$	0	0
$341$	−10.8559	−0.587883
$342$	0	0
$343$	−13.6868	−0.739016
$344$	0	0
$345$	2.80272	0.150893
$346$	0	0
$347$	−17.1789	−0.922213	−0.461106	−	0.887345i	$-0.652547\pi$
$347$	−17.1789	−0.922213	−0.461106	+	0.887345i	$0.652547\pi$
$348$	0	0
$349$	10.7442	0.575123	0.287561	−	0.957762i	$-0.407155\pi$
$349$	10.7442	0.575123	0.287561	+	0.957762i	$0.407155\pi$
$350$	0	0
$351$	12.1084	0.646299
$352$	0	0
$353$	−0.533413	−0.0283907	−0.0141953	−	0.999899i	$-0.504519\pi$
$353$	−0.533413	−0.0283907	−0.0141953	+	0.999899i	$0.504519\pi$
$354$	0	0
$355$	0.871559	0.0462576
$356$	0	0
$357$	−9.08177	−0.480658
$358$	0	0
$359$	36.1329	1.90702	0.953512	−	0.301356i	$-0.0974391\pi$
$359$	36.1329	1.90702	0.953512	+	0.301356i	$0.0974391\pi$
$360$	0	0
$361$	−10.8369	−0.570361
$362$	0	0
$363$	−9.15273	−0.480394
$364$	0	0
$365$	8.06656	0.422223
$366$	0	0
$367$	−20.8438	−1.08804	−0.544019	−	0.839073i	$-0.683098\pi$
$367$	−20.8438	−1.08804	−0.544019	+	0.839073i	$0.683098\pi$
$368$	0	0
$369$	−14.1104	−0.734560
$370$	0	0
$371$	1.25217	0.0650093
$372$	0	0
$373$	−23.0722	−1.19464	−0.597318	−	0.802005i	$-0.703767\pi$
$373$	−23.0722	−1.19464	−0.597318	+	0.802005i	$0.703767\pi$
$374$	0	0
$375$	−1.14309	−0.0590290
$376$	0	0
$377$	−5.01955	−0.258520
$378$	0	0
$379$	33.7566	1.73396	0.866979	−	0.498344i	$-0.166058\pi$
$379$	33.7566	1.73396	0.866979	+	0.498344i	$0.166058\pi$
$380$	0	0
$381$	6.32247	0.323910
$382$	0	0
$383$	4.88836	0.249784	0.124892	−	0.992170i	$-0.460142\pi$
$383$	4.88836	0.249784	0.124892	+	0.992170i	$0.460142\pi$
$384$	0	0
$385$	5.35399	0.272865
$386$	0	0
$387$	10.7649	0.547210
$388$	0	0
$389$	−6.21688	−0.315208	−0.157604	−	0.987502i	$-0.550377\pi$
$389$	−6.21688	−0.315208	−0.157604	+	0.987502i	$0.550377\pi$
$390$	0	0
$391$	6.29454	0.318329
$392$	0	0
$393$	−20.8593	−1.05221
$394$	0	0
$395$	14.8827	0.748831
$396$	0	0
$397$	3.93736	0.197610	0.0988051	−	0.995107i	$-0.468498\pi$
$397$	3.93736	0.197610	0.0988051	+	0.995107i	$0.468498\pi$
$398$	0	0
$399$	10.1073	0.505996
$400$	0	0
$401$	−2.11919	−0.105827	−0.0529135	−	0.998599i	$-0.516851\pi$
$401$	−2.11919	−0.105827	−0.0529135	+	0.998599i	$0.516851\pi$
$402$	0	0
$403$	−14.1624	−0.705481
$404$	0	0
$405$	1.05257	0.0523026
$406$	0	0
$407$	5.38694	0.267020
$408$	0	0
$409$	−15.8850	−0.785465	−0.392733	−	0.919653i	$-0.628470\pi$
$409$	−15.8850	−0.785465	−0.392733	+	0.919653i	$0.628470\pi$
$410$	0	0
$411$	−17.4097	−0.858760
$412$	0	0
$413$	17.7112	0.871510
$414$	0	0
$415$	−11.5688	−0.567889
$416$	0	0
$417$	6.60682	0.323537
$418$	0	0
$419$	−29.2644	−1.42966	−0.714829	−	0.699299i	$-0.753496\pi$
$419$	−29.2644	−1.42966	−0.714829	+	0.699299i	$0.753496\pi$
$420$	0	0
$421$	13.2827	0.647360	0.323680	−	0.946167i	$-0.395080\pi$
$421$	13.2827	0.647360	0.323680	+	0.946167i	$0.395080\pi$
$422$	0	0
$423$	22.9860	1.11762
$424$	0	0
$425$	−2.56723	−0.124529
$426$	0	0
$427$	10.8168	0.523462
$428$	0	0
$429$	4.46332	0.215491
$430$	0	0
$431$	13.6029	0.655226	0.327613	−	0.944812i	$-0.393756\pi$
$431$	13.6029	0.655226	0.327613	+	0.944812i	$0.393756\pi$
$432$	0	0
$433$	0.589866	0.0283472	0.0141736	−	0.999900i	$-0.495488\pi$
$433$	0.589866	0.0283472	0.0141736	+	0.999900i	$0.495488\pi$
$434$	0	0
$435$	−2.54228	−0.121893
$436$	0	0
$437$	−7.00531	−0.335109
$438$	0	0
$439$	−3.48615	−0.166385	−0.0831925	−	0.996533i	$-0.526512\pi$
$439$	−3.48615	−0.166385	−0.0831925	+	0.996533i	$0.526512\pi$
$440$	0	0
$441$	−4.36443	−0.207830
$442$	0	0
$443$	−11.1435	−0.529446	−0.264723	−	0.964325i	$-0.585280\pi$
$443$	−11.1435	−0.529446	−0.264723	+	0.964325i	$0.585280\pi$
$444$	0	0
$445$	11.1370	0.527944
$446$	0	0
$447$	−16.6160	−0.785911
$448$	0	0
$449$	−22.3743	−1.05591	−0.527955	−	0.849272i	$-0.677041\pi$
$449$	−22.3743	−1.05591	−0.527955	+	0.849272i	$0.677041\pi$
$450$	0	0
$451$	−14.4162	−0.678831
$452$	0	0
$453$	1.14309	0.0537071
$454$	0	0
$455$	6.98470	0.327448
$456$	0	0
$457$	−21.5785	−1.00940	−0.504699	−	0.863295i	$-0.668397\pi$
$457$	−21.5785	−1.00940	−0.504699	+	0.863295i	$0.668397\pi$
$458$	0	0
$459$	13.7730	0.642869
$460$	0	0
$461$	32.4730	1.51242	0.756208	−	0.654331i	$-0.227050\pi$
$461$	32.4730	1.51242	0.756208	+	0.654331i	$0.227050\pi$
$462$	0	0
$463$	13.8215	0.642338	0.321169	−	0.947022i	$-0.395924\pi$
$463$	13.8215	0.642338	0.321169	+	0.947022i	$0.395924\pi$
$464$	0	0
$465$	−7.17290	−0.332635
$466$	0	0
$467$	−19.5714	−0.905657	−0.452829	−	0.891598i	$-0.649585\pi$
$467$	−19.5714	−0.905657	−0.452829	+	0.891598i	$0.649585\pi$
$468$	0	0
$469$	−30.5935	−1.41268
$470$	0	0
$471$	14.5277	0.669403
$472$	0	0
$473$	10.9981	0.505694
$474$	0	0
$475$	2.85712	0.131094
$476$	0	0
$477$	−0.685146	−0.0313707
$478$	0	0
$479$	−31.0959	−1.42081	−0.710405	−	0.703793i	$-0.751488\pi$
$479$	−31.0959	−1.42081	−0.710405	+	0.703793i	$0.751488\pi$
$480$	0	0
$481$	7.02767	0.320434
$482$	0	0
$483$	−8.67368	−0.394666
$484$	0	0
$485$	−10.1341	−0.460165
$486$	0	0
$487$	−25.7819	−1.16829	−0.584144	−	0.811650i	$-0.698570\pi$
$487$	−25.7819	−1.16829	−0.584144	+	0.811650i	$0.698570\pi$
$488$	0	0
$489$	7.28656	0.329510
$490$	0	0
$491$	35.8094	1.61606	0.808028	−	0.589144i	$-0.200535\pi$
$491$	35.8094	1.61606	0.808028	+	0.589144i	$0.200535\pi$
$492$	0	0
$493$	−5.70962	−0.257148
$494$	0	0
$495$	−2.92953	−0.131673
$496$	0	0
$497$	−2.69725	−0.120988
$498$	0	0
$499$	−10.7746	−0.482338	−0.241169	−	0.970483i	$-0.577531\pi$
$499$	−10.7746	−0.482338	−0.241169	+	0.970483i	$0.577531\pi$
$500$	0	0
$501$	−20.1072	−0.898321
$502$	0	0
$503$	2.77797	0.123864	0.0619318	−	0.998080i	$-0.480274\pi$
$503$	2.77797	0.123864	0.0619318	+	0.998080i	$0.480274\pi$
$504$	0	0
$505$	12.4134	0.552387
$506$	0	0
$507$	−9.03744	−0.401367
$508$	0	0
$509$	−31.8113	−1.41001	−0.705004	−	0.709203i	$-0.749055\pi$
$509$	−31.8113	−1.41001	−0.705004	+	0.709203i	$0.749055\pi$
$510$	0	0
$511$	−24.9639	−1.10434
$512$	0	0
$513$	−15.3282	−0.676757
$514$	0	0
$515$	15.3075	0.674531
$516$	0	0
$517$	23.4841	1.03283
$518$	0	0
$519$	−20.2011	−0.886728
$520$	0	0
$521$	−20.0965	−0.880445	−0.440222	−	0.897889i	$-0.645100\pi$
$521$	−20.0965	−0.880445	−0.440222	+	0.897889i	$0.645100\pi$
$522$	0	0
$523$	38.1696	1.66904	0.834520	−	0.550977i	$-0.185745\pi$
$523$	38.1696	1.66904	0.834520	+	0.550977i	$0.185745\pi$
$524$	0	0
$525$	3.53757	0.154392
$526$	0	0
$527$	−16.1094	−0.701737
$528$	0	0
$529$	−16.9883	−0.738622
$530$	0	0
$531$	−9.69099	−0.420553
$532$	0	0
$533$	−18.8070	−0.814622
$534$	0	0
$535$	6.18262	0.267298
$536$	0	0
$537$	8.48640	0.366215
$538$	0	0
$539$	−4.45899	−0.192062
$540$	0	0
$541$	24.7563	1.06435	0.532177	−	0.846633i	$-0.321374\pi$
$541$	24.7563	1.06435	0.532177	+	0.846633i	$0.321374\pi$
$542$	0	0
$543$	−4.05929	−0.174201
$544$	0	0
$545$	−7.59400	−0.325291
$546$	0	0
$547$	38.7435	1.65655	0.828276	−	0.560321i	$-0.189322\pi$
$547$	38.7435	1.65655	0.828276	+	0.560321i	$0.189322\pi$
$548$	0	0
$549$	−5.91862	−0.252600
$550$	0	0
$551$	6.35434	0.270704
$552$	0	0
$553$	−46.0581	−1.95859
$554$	0	0
$555$	3.55934	0.151085
$556$	0	0
$557$	−8.43242	−0.357293	−0.178647	−	0.983913i	$-0.557172\pi$
$557$	−8.43242	−0.357293	−0.178647	+	0.983913i	$0.557172\pi$
$558$	0	0
$559$	14.3479	0.606852
$560$	0	0
$561$	5.07692	0.214348
$562$	0	0
$563$	−23.4123	−0.986711	−0.493356	−	0.869828i	$-0.664230\pi$
$563$	−23.4123	−0.986711	−0.493356	+	0.869828i	$0.664230\pi$
$564$	0	0
$565$	0.985069	0.0414422
$566$	0	0
$567$	−3.25743	−0.136799
$568$	0	0
$569$	3.01671	0.126467	0.0632335	−	0.997999i	$-0.479859\pi$
$569$	3.01671	0.126467	0.0632335	+	0.997999i	$0.479859\pi$
$570$	0	0
$571$	−16.0534	−0.671816	−0.335908	−	0.941895i	$-0.609043\pi$
$571$	−16.0534	−0.671816	−0.335908	+	0.941895i	$0.609043\pi$
$572$	0	0
$573$	10.7413	0.448724
$574$	0	0
$575$	−2.45188	−0.102250
$576$	0	0
$577$	−24.2259	−1.00854	−0.504269	−	0.863547i	$-0.668238\pi$
$577$	−24.2259	−1.00854	−0.504269	+	0.863547i	$0.668238\pi$
$578$	0	0
$579$	30.1917	1.25472
$580$	0	0
$581$	35.8024	1.48533
$582$	0	0
$583$	−0.699990	−0.0289906
$584$	0	0
$585$	−3.82180	−0.158012
$586$	0	0
$587$	−10.8024	−0.445862	−0.222931	−	0.974834i	$-0.571562\pi$
$587$	−10.8024	−0.445862	−0.222931	+	0.974834i	$0.571562\pi$
$588$	0	0
$589$	17.9284	0.738728
$590$	0	0
$591$	7.97497	0.328046
$592$	0	0
$593$	10.9062	0.447865	0.223932	−	0.974605i	$-0.428110\pi$
$593$	10.9062	0.447865	0.223932	+	0.974605i	$0.428110\pi$
$594$	0	0
$595$	7.94492	0.325710
$596$	0	0
$597$	−18.3062	−0.749222
$598$	0	0
$599$	−17.9711	−0.734279	−0.367140	−	0.930166i	$-0.619663\pi$
$599$	−17.9711	−0.734279	−0.367140	+	0.930166i	$0.619663\pi$
$600$	0	0
$601$	10.3666	0.422861	0.211430	−	0.977393i	$-0.432188\pi$
$601$	10.3666	0.422861	0.211430	+	0.977393i	$0.432188\pi$
$602$	0	0
$603$	16.7398	0.681697
$604$	0	0
$605$	8.00699	0.325531
$606$	0	0
$607$	48.0906	1.95194	0.975968	−	0.217913i	$-0.0699250\pi$
$607$	48.0906	1.95194	0.975968	+	0.217913i	$0.0699250\pi$
$608$	0	0
$609$	7.86768	0.318814
$610$	0	0
$611$	30.6368	1.23943
$612$	0	0
$613$	−4.60126	−0.185843	−0.0929215	−	0.995673i	$-0.529621\pi$
$613$	−4.60126	−0.185843	−0.0929215	+	0.995673i	$0.529621\pi$
$614$	0	0
$615$	−9.52527	−0.384096
$616$	0	0
$617$	27.4517	1.10516	0.552582	−	0.833459i	$-0.313643\pi$
$617$	27.4517	1.10516	0.552582	+	0.833459i	$0.313643\pi$
$618$	0	0
$619$	20.0617	0.806349	0.403175	−	0.915123i	$-0.367907\pi$
$619$	20.0617	0.806349	0.403175	+	0.915123i	$0.367907\pi$
$620$	0	0
$621$	13.1541	0.527857
$622$	0	0
$623$	−34.4661	−1.38085
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.65019	−0.225647
$628$	0	0
$629$	7.99380	0.318734
$630$	0	0
$631$	40.2819	1.60360	0.801799	−	0.597594i	$-0.203876\pi$
$631$	40.2819	1.60360	0.801799	+	0.597594i	$0.203876\pi$
$632$	0	0
$633$	10.6376	0.422805
$634$	0	0
$635$	−5.53102	−0.219492
$636$	0	0
$637$	−5.81710	−0.230482
$638$	0	0
$639$	1.47585	0.0583836
$640$	0	0
$641$	19.3744	0.765242	0.382621	−	0.923905i	$-0.375022\pi$
$641$	19.3744	0.765242	0.382621	+	0.923905i	$0.375022\pi$
$642$	0	0
$643$	0.789978	0.0311537	0.0155768	−	0.999879i	$-0.495042\pi$
$643$	0.789978	0.0311537	0.0155768	+	0.999879i	$0.495042\pi$
$644$	0	0
$645$	7.26684	0.286132
$646$	0	0
$647$	−1.72893	−0.0679713	−0.0339857	−	0.999422i	$-0.510820\pi$
$647$	−1.72893	−0.0679713	−0.0339857	+	0.999422i	$0.510820\pi$
$648$	0	0
$649$	−9.90096	−0.388647
$650$	0	0
$651$	22.1983	0.870018
$652$	0	0
$653$	−19.7441	−0.772646	−0.386323	−	0.922364i	$-0.626255\pi$
$653$	−19.7441	−0.772646	−0.386323	+	0.922364i	$0.626255\pi$
$654$	0	0
$655$	18.2481	0.713013
$656$	0	0
$657$	13.6594	0.532906
$658$	0	0
$659$	11.5920	0.451560	0.225780	−	0.974178i	$-0.427507\pi$
$659$	11.5920	0.451560	0.225780	+	0.974178i	$0.427507\pi$
$660$	0	0
$661$	23.2154	0.902976	0.451488	−	0.892277i	$-0.350893\pi$
$661$	23.2154	0.902976	0.451488	+	0.892277i	$0.350893\pi$
$662$	0	0
$663$	6.62324	0.257225
$664$	0	0
$665$	−8.84204	−0.342880
$666$	0	0
$667$	−5.45306	−0.211143
$668$	0	0
$669$	−4.35174	−0.168248
$670$	0	0
$671$	−6.04685	−0.233436
$672$	0	0
$673$	−18.9013	−0.728590	−0.364295	−	0.931284i	$-0.618690\pi$
$673$	−18.9013	−0.728590	−0.364295	+	0.931284i	$0.618690\pi$
$674$	0	0
$675$	−5.36492	−0.206496
$676$	0	0
$677$	21.9983	0.845463	0.422732	−	0.906255i	$-0.361071\pi$
$677$	21.9983	0.845463	0.422732	+	0.906255i	$0.361071\pi$
$678$	0	0
$679$	31.3623	1.20358
$680$	0	0
$681$	−18.9580	−0.726470
$682$	0	0
$683$	−49.2371	−1.88401	−0.942003	−	0.335606i	$-0.891059\pi$
$683$	−49.2371	−1.88401	−0.942003	+	0.335606i	$0.891059\pi$
$684$	0	0
$685$	15.2304	0.581924
$686$	0	0
$687$	14.7389	0.562322
$688$	0	0
$689$	−0.913191	−0.0347898
$690$	0	0
$691$	20.5088	0.780192	0.390096	−	0.920774i	$-0.372442\pi$
$691$	20.5088	0.780192	0.390096	+	0.920774i	$0.372442\pi$
$692$	0	0
$693$	9.06614	0.344394
$694$	0	0
$695$	−5.77978	−0.219240
$696$	0	0
$697$	−21.3925	−0.810299
$698$	0	0
$699$	−19.1149	−0.722992
$700$	0	0
$701$	13.8236	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$701$	13.8236	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$702$	0	0
$703$	−8.89644	−0.335536
$704$	0	0
$705$	15.5167	0.584394
$706$	0	0
$707$	−38.4161	−1.44479
$708$	0	0
$709$	−42.7892	−1.60698	−0.803491	−	0.595317i	$-0.797026\pi$
$709$	−42.7892	−1.60698	−0.803491	+	0.595317i	$0.797026\pi$
$710$	0	0
$711$	25.2015	0.945131
$712$	0	0
$713$	−15.3855	−0.576193
$714$	0	0
$715$	−3.90461	−0.146024
$716$	0	0
$717$	5.76172	0.215175
$718$	0	0
$719$	10.0575	0.375082	0.187541	−	0.982257i	$-0.439948\pi$
$719$	10.0575	0.375082	0.187541	+	0.982257i	$0.439948\pi$
$720$	0	0
$721$	−47.3728	−1.76426
$722$	0	0
$723$	28.0837	1.04444
$724$	0	0
$725$	2.22403	0.0825986
$726$	0	0
$727$	−2.92139	−0.108348	−0.0541742	−	0.998531i	$-0.517253\pi$
$727$	−2.92139	−0.108348	−0.0541742	+	0.998531i	$0.517253\pi$
$728$	0	0
$729$	20.1801	0.747413
$730$	0	0
$731$	16.3204	0.603631
$732$	0	0
$733$	35.4987	1.31117	0.655587	−	0.755120i	$-0.272421\pi$
$733$	35.4987	1.31117	0.655587	+	0.755120i	$0.272421\pi$
$734$	0	0
$735$	−2.94621	−0.108673
$736$	0	0
$737$	17.1025	0.629978
$738$	0	0
$739$	8.47755	0.311852	0.155926	−	0.987769i	$-0.450164\pi$
$739$	8.47755	0.311852	0.155926	+	0.987769i	$0.450164\pi$
$740$	0	0
$741$	−7.37111	−0.270785
$742$	0	0
$743$	20.0783	0.736602	0.368301	−	0.929707i	$-0.379940\pi$
$743$	20.0783	0.736602	0.368301	+	0.929707i	$0.379940\pi$
$744$	0	0
$745$	14.5360	0.532559
$746$	0	0
$747$	−19.5899	−0.716757
$748$	0	0
$749$	−19.1336	−0.699127
$750$	0	0
$751$	31.8311	1.16153	0.580767	−	0.814070i	$-0.302753\pi$
$751$	31.8311	1.16153	0.580767	+	0.814070i	$0.302753\pi$
$752$	0	0
$753$	−14.6279	−0.533071
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	25.5507	0.928656	0.464328	−	0.885663i	$-0.346296\pi$
$757$	25.5507	0.928656	0.464328	+	0.885663i	$0.346296\pi$
$758$	0	0
$759$	4.84879	0.176000
$760$	0	0
$761$	−48.8844	−1.77206	−0.886029	−	0.463630i	$-0.846547\pi$
$761$	−48.8844	−1.77206	−0.886029	+	0.463630i	$0.846547\pi$
$762$	0	0
$763$	23.5015	0.850810
$764$	0	0
$765$	−4.34721	−0.157174
$766$	0	0
$767$	−12.9166	−0.466390
$768$	0	0
$769$	−13.7587	−0.496150	−0.248075	−	0.968741i	$-0.579798\pi$
$769$	−13.7587	−0.496150	−0.248075	+	0.968741i	$0.579798\pi$
$770$	0	0
$771$	−6.98255	−0.251470
$772$	0	0
$773$	−19.2813	−0.693500	−0.346750	−	0.937958i	$-0.612715\pi$
$773$	−19.2813	−0.693500	−0.346750	+	0.937958i	$0.612715\pi$
$774$	0	0
$775$	6.27500	0.225405
$776$	0	0
$777$	−11.0152	−0.395169
$778$	0	0
$779$	23.8081	0.853013
$780$	0	0
$781$	1.50782	0.0539542
$782$	0	0
$783$	−11.9318	−0.426407
$784$	0	0
$785$	−12.7092	−0.453610
$786$	0	0
$787$	44.9373	1.60184	0.800920	−	0.598771i	$-0.204344\pi$
$787$	44.9373	1.60184	0.800920	+	0.598771i	$0.204344\pi$
$788$	0	0
$789$	27.8787	0.992508
$790$	0	0
$791$	−3.04853	−0.108393
$792$	0	0
$793$	−7.88859	−0.280132
$794$	0	0
$795$	−0.462508	−0.0164035
$796$	0	0
$797$	−25.0401	−0.886967	−0.443483	−	0.896283i	$-0.646257\pi$
$797$	−25.0401	−0.886967	−0.443483	+	0.896283i	$0.646257\pi$
$798$	0	0
$799$	34.8486	1.23285
$800$	0	0
$801$	18.8587	0.666340
$802$	0	0
$803$	13.9554	0.492475
$804$	0	0
$805$	7.58792	0.267439
$806$	0	0
$807$	2.33735	0.0822786
$808$	0	0
$809$	35.5442	1.24967	0.624834	−	0.780757i	$-0.285166\pi$
$809$	35.5442	1.24967	0.624834	+	0.780757i	$0.285166\pi$
$810$	0	0
$811$	−17.1426	−0.601957	−0.300979	−	0.953631i	$-0.597313\pi$
$811$	−17.1426	−0.601957	−0.300979	+	0.953631i	$0.597313\pi$
$812$	0	0
$813$	16.0338	0.562330
$814$	0	0
$815$	−6.37443	−0.223287
$816$	0	0
$817$	−18.1632	−0.635451
$818$	0	0
$819$	11.8275	0.413285
$820$	0	0
$821$	17.3732	0.606328	0.303164	−	0.952938i	$-0.401957\pi$
$821$	17.3732	0.606328	0.303164	+	0.952938i	$0.401957\pi$
$822$	0	0
$823$	−19.4482	−0.677921	−0.338960	−	0.940801i	$-0.610075\pi$
$823$	−19.4482	−0.677921	−0.338960	+	0.940801i	$0.610075\pi$
$824$	0	0
$825$	−1.97758	−0.0688506
$826$	0	0
$827$	26.6625	0.927148	0.463574	−	0.886058i	$-0.346567\pi$
$827$	26.6625	0.927148	0.463574	+	0.886058i	$0.346567\pi$
$828$	0	0
$829$	−30.8948	−1.07302	−0.536510	−	0.843894i	$-0.680258\pi$
$829$	−30.8948	−1.07302	−0.536510	+	0.843894i	$0.680258\pi$
$830$	0	0
$831$	12.2525	0.425034
$832$	0	0
$833$	−6.61681	−0.229259
$834$	0	0
$835$	17.5902	0.608732
$836$	0	0
$837$	−33.6649	−1.16363
$838$	0	0
$839$	44.0266	1.51997	0.759984	−	0.649941i	$-0.225207\pi$
$839$	44.0266	1.51997	0.759984	+	0.649941i	$0.225207\pi$
$840$	0	0
$841$	−24.0537	−0.829437
$842$	0	0
$843$	−21.1399	−0.728097
$844$	0	0
$845$	7.90614	0.271979
$846$	0	0
$847$	−24.7795	−0.851435
$848$	0	0
$849$	−6.01096	−0.206295
$850$	0	0
$851$	7.63460	0.261711
$852$	0	0
$853$	−0.678353	−0.0232264	−0.0116132	−	0.999933i	$-0.503697\pi$
$853$	−0.678353	−0.0232264	−0.0116132	+	0.999933i	$0.503697\pi$
$854$	0	0
$855$	4.83808	0.165459
$856$	0	0
$857$	4.75770	0.162520	0.0812600	−	0.996693i	$-0.474106\pi$
$857$	4.75770	0.162520	0.0812600	+	0.996693i	$0.474106\pi$
$858$	0	0
$859$	−29.1672	−0.995170	−0.497585	−	0.867415i	$-0.665780\pi$
$859$	−29.1672	−0.995170	−0.497585	+	0.867415i	$0.665780\pi$
$860$	0	0
$861$	29.4782	1.00461
$862$	0	0
$863$	−21.2068	−0.721890	−0.360945	−	0.932587i	$-0.617546\pi$
$863$	−21.2068	−0.721890	−0.360945	+	0.932587i	$0.617546\pi$
$864$	0	0
$865$	17.6723	0.600876
$866$	0	0
$867$	−11.8988	−0.404104
$868$	0	0
$869$	25.7476	0.873427
$870$	0	0
$871$	22.3115	0.755996
$872$	0	0
$873$	−17.1605	−0.580793
$874$	0	0
$875$	−3.09474	−0.104621
$876$	0	0
$877$	−0.672441	−0.0227067	−0.0113534	−	0.999936i	$-0.503614\pi$
$877$	−0.672441	−0.0227067	−0.0113534	+	0.999936i	$0.503614\pi$
$878$	0	0
$879$	12.1564	0.410027
$880$	0	0
$881$	47.7395	1.60838	0.804192	−	0.594370i	$-0.202599\pi$
$881$	47.7395	1.60838	0.804192	+	0.594370i	$0.202599\pi$
$882$	0	0
$883$	−21.4329	−0.721274	−0.360637	−	0.932706i	$-0.617441\pi$
$883$	−21.4329	−0.721274	−0.360637	+	0.932706i	$0.617441\pi$
$884$	0	0
$885$	−6.54191	−0.219904
$886$	0	0
$887$	16.5339	0.555156	0.277578	−	0.960703i	$-0.410468\pi$
$887$	16.5339	0.555156	0.277578	+	0.960703i	$0.410468\pi$
$888$	0	0
$889$	17.1171	0.574088
$890$	0	0
$891$	1.82098	0.0610050
$892$	0	0
$893$	−38.7836	−1.29784
$894$	0	0
$895$	−7.42408	−0.248159
$896$	0	0
$897$	6.32562	0.211206
$898$	0	0
$899$	13.9558	0.465453
$900$	0	0
$901$	−1.03873	−0.0346052
$902$	0	0
$903$	−22.4890	−0.748387
$904$	0	0
$905$	3.55115	0.118044
$906$	0	0
$907$	46.0331	1.52850	0.764252	−	0.644918i	$-0.223108\pi$
$907$	46.0331	1.52850	0.764252	+	0.644918i	$0.223108\pi$
$908$	0	0
$909$	21.0201	0.697192
$910$	0	0
$911$	7.61588	0.252325	0.126163	−	0.992010i	$-0.459734\pi$
$911$	7.61588	0.252325	0.126163	+	0.992010i	$0.459734\pi$
$912$	0	0
$913$	−20.0144	−0.662378
$914$	0	0
$915$	−3.99537	−0.132083
$916$	0	0
$917$	−56.4732	−1.86491
$918$	0	0
$919$	−25.7233	−0.848534	−0.424267	−	0.905537i	$-0.639468\pi$
$919$	−25.7233	−0.848534	−0.424267	+	0.905537i	$0.639468\pi$
$920$	0	0
$921$	−12.5188	−0.412509
$922$	0	0
$923$	1.96707	0.0647470
$924$	0	0
$925$	−3.11378	−0.102380
$926$	0	0
$927$	25.9209	0.851354
$928$	0	0
$929$	16.2944	0.534604	0.267302	−	0.963613i	$-0.413868\pi$
$929$	16.2944	0.534604	0.267302	+	0.963613i	$0.413868\pi$
$930$	0	0
$931$	7.36396	0.241344
$932$	0	0
$933$	−7.30214	−0.239061
$934$	0	0
$935$	−4.44140	−0.145249
$936$	0	0
$937$	−5.64382	−0.184376	−0.0921878	−	0.995742i	$-0.529386\pi$
$937$	−5.64382	−0.184376	−0.0921878	+	0.995742i	$0.529386\pi$
$938$	0	0
$939$	−10.9294	−0.356667
$940$	0	0
$941$	40.8169	1.33059	0.665296	−	0.746579i	$-0.268305\pi$
$941$	40.8169	1.33059	0.665296	+	0.746579i	$0.268305\pi$
$942$	0	0
$943$	−20.4312	−0.665333
$944$	0	0
$945$	16.6030	0.540096
$946$	0	0
$947$	3.49611	0.113608	0.0568041	−	0.998385i	$-0.481909\pi$
$947$	3.49611	0.113608	0.0568041	+	0.998385i	$0.481909\pi$
$948$	0	0
$949$	18.2059	0.590988
$950$	0	0
$951$	9.33018	0.302552
$952$	0	0
$953$	−13.6014	−0.440593	−0.220297	−	0.975433i	$-0.570703\pi$
$953$	−13.6014	−0.440593	−0.220297	+	0.975433i	$0.570703\pi$
$954$	0	0
$955$	−9.39670	−0.304070
$956$	0	0
$957$	−4.39822	−0.142174
$958$	0	0
$959$	−47.1341	−1.52204
$960$	0	0
$961$	8.37566	0.270183
$962$	0	0
$963$	10.4693	0.337368
$964$	0	0
$965$	−26.4123	−0.850242
$966$	0	0
$967$	−34.3187	−1.10362	−0.551808	−	0.833971i	$-0.686062\pi$
$967$	−34.3187	−1.10362	−0.551808	+	0.833971i	$0.686062\pi$
$968$	0	0
$969$	−8.38446	−0.269348
$970$	0	0
$971$	−21.8011	−0.699632	−0.349816	−	0.936818i	$-0.613756\pi$
$971$	−21.8011	−0.699632	−0.349816	+	0.936818i	$0.613756\pi$
$972$	0	0
$973$	17.8869	0.573428
$974$	0	0
$975$	−2.57991	−0.0826233
$976$	0	0
$977$	4.93007	0.157727	0.0788635	−	0.996885i	$-0.474871\pi$
$977$	4.93007	0.157727	0.0788635	+	0.996885i	$0.474871\pi$
$978$	0	0
$979$	19.2673	0.615787
$980$	0	0
$981$	−12.8592	−0.410564
$982$	0	0
$983$	54.4487	1.73664	0.868322	−	0.496000i	$-0.165199\pi$
$983$	54.4487	1.73664	0.868322	+	0.496000i	$0.165199\pi$
$984$	0	0
$985$	−6.97666	−0.222295
$986$	0	0
$987$	−48.0203	−1.52850
$988$	0	0
$989$	15.5870	0.495639
$990$	0	0
$991$	35.3444	1.12275	0.561376	−	0.827561i	$-0.310272\pi$
$991$	35.3444	1.12275	0.561376	+	0.827561i	$0.310272\pi$
$992$	0	0
$993$	37.8278	1.20043
$994$	0	0
$995$	16.0146	0.507698
$996$	0	0
$997$	19.5927	0.620506	0.310253	−	0.950654i	$-0.399586\pi$
$997$	19.5927	0.620506	0.310253	+	0.950654i	$0.399586\pi$
$998$	0	0
$999$	16.7052	0.528528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.n.1.10	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.n.1.10	✓	13	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)

\(f(q)\)	\(=\)	\(q+1.14309 q^{3} -1.00000 q^{5} +3.09474 q^{7} -1.69334 q^{9} +O(q^{10})\)	\(q+1.14309 q^{3} -1.00000 q^{5} +3.09474 q^{7} -1.69334 q^{9} -1.73003 q^{11} -2.25696 q^{13} -1.14309 q^{15} -2.56723 q^{17} +2.85712 q^{19} +3.53757 q^{21} -2.45188 q^{23} +1.00000 q^{25} -5.36492 q^{27} +2.22403 q^{29} +6.27500 q^{31} -1.97758 q^{33} -3.09474 q^{35} -3.11378 q^{37} -2.57991 q^{39} +8.33290 q^{41} -6.35719 q^{43} +1.69334 q^{45} -13.5744 q^{47} +2.57741 q^{49} -2.93458 q^{51} +0.404612 q^{53} +1.73003 q^{55} +3.26595 q^{57} +5.72300 q^{59} +3.49523 q^{61} -5.24045 q^{63} +2.25696 q^{65} -9.88565 q^{67} -2.80272 q^{69} -0.871559 q^{71} -8.06656 q^{73} +1.14309 q^{75} -5.35399 q^{77} -14.8827 q^{79} -1.05257 q^{81} +11.5688 q^{83} +2.56723 q^{85} +2.54228 q^{87} -11.1370 q^{89} -6.98470 q^{91} +7.17290 q^{93} -2.85712 q^{95} +10.1341 q^{97} +2.92953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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