LMFDB - Embedded newform 6040.2.a.s.1.20

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:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+2.57003 q^{3} +1.00000 q^{5} -3.15345 q^{7} +3.60506 q^{9} +O(q^{10})$	$q+2.57003 q^{3} +1.00000 q^{5} -3.15345 q^{7} +3.60506 q^{9} +2.12212 q^{11} +0.673513 q^{13} +2.57003 q^{15} +4.82334 q^{17} -1.64276 q^{19} -8.10446 q^{21} -1.16665 q^{23} +1.00000 q^{25} +1.55502 q^{27} +3.52118 q^{29} -9.60169 q^{31} +5.45392 q^{33} -3.15345 q^{35} +7.72286 q^{37} +1.73095 q^{39} -2.83247 q^{41} +6.32436 q^{43} +3.60506 q^{45} +11.5531 q^{47} +2.94423 q^{49} +12.3961 q^{51} +6.46363 q^{53} +2.12212 q^{55} -4.22195 q^{57} -6.30979 q^{59} +11.9769 q^{61} -11.3684 q^{63} +0.673513 q^{65} +14.4771 q^{67} -2.99832 q^{69} -2.78213 q^{71} +13.7688 q^{73} +2.57003 q^{75} -6.69199 q^{77} +4.22229 q^{79} -6.81872 q^{81} -1.91251 q^{83} +4.82334 q^{85} +9.04953 q^{87} +14.4986 q^{89} -2.12389 q^{91} -24.6767 q^{93} -1.64276 q^{95} +10.0326 q^{97} +7.65037 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	$ 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) $ 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * 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
$3$	2.57003	1.48381	0.741904	−	0.670506i	$-0.233923\pi$
$3$	2.57003	1.48381	0.741904	+	0.670506i	$0.233923\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.15345	−1.19189	−0.595945	−	0.803025i	$-0.703223\pi$
$7$	−3.15345	−1.19189	−0.595945	+	0.803025i	$0.703223\pi$
$8$	0	0
$9$	3.60506	1.20169
$10$	0	0
$11$	2.12212	0.639843	0.319922	−	0.947444i	$-0.396343\pi$
$11$	2.12212	0.639843	0.319922	+	0.947444i	$0.396343\pi$
$12$	0	0
$13$	0.673513	0.186799	0.0933994	−	0.995629i	$-0.470227\pi$
$13$	0.673513	0.186799	0.0933994	+	0.995629i	$0.470227\pi$
$14$	0	0
$15$	2.57003	0.663579
$16$	0	0
$17$	4.82334	1.16983	0.584915	−	0.811094i	$-0.301128\pi$
$17$	4.82334	1.16983	0.584915	+	0.811094i	$0.301128\pi$
$18$	0	0
$19$	−1.64276	−0.376876	−0.188438	−	0.982085i	$-0.560342\pi$
$19$	−1.64276	−0.376876	−0.188438	+	0.982085i	$0.560342\pi$
$20$	0	0
$21$	−8.10446	−1.76854
$22$	0	0
$23$	−1.16665	−0.243263	−0.121631	−	0.992575i	$-0.538813\pi$
$23$	−1.16665	−0.243263	−0.121631	+	0.992575i	$0.538813\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.55502	0.299265
$28$	0	0
$29$	3.52118	0.653866	0.326933	−	0.945048i	$-0.393985\pi$
$29$	3.52118	0.653866	0.326933	+	0.945048i	$0.393985\pi$
$30$	0	0
$31$	−9.60169	−1.72452	−0.862258	−	0.506470i	$-0.830950\pi$
$31$	−9.60169	−1.72452	−0.862258	+	0.506470i	$0.830950\pi$
$32$	0	0
$33$	5.45392	0.949405
$34$	0	0
$35$	−3.15345	−0.533030
$36$	0	0
$37$	7.72286	1.26963	0.634815	−	0.772664i	$-0.281076\pi$
$37$	7.72286	1.26963	0.634815	+	0.772664i	$0.281076\pi$
$38$	0	0
$39$	1.73095	0.277174
$40$	0	0
$41$	−2.83247	−0.442358	−0.221179	−	0.975233i	$-0.570990\pi$
$41$	−2.83247	−0.442358	−0.221179	+	0.975233i	$0.570990\pi$
$42$	0	0
$43$	6.32436	0.964455	0.482228	−	0.876046i	$-0.339828\pi$
$43$	6.32436	0.964455	0.482228	+	0.876046i	$0.339828\pi$
$44$	0	0
$45$	3.60506	0.537411
$46$	0	0
$47$	11.5531	1.68519	0.842596	−	0.538546i	$-0.181026\pi$
$47$	11.5531	1.68519	0.842596	+	0.538546i	$0.181026\pi$
$48$	0	0
$49$	2.94423	0.420604
$50$	0	0
$51$	12.3961	1.73580
$52$	0	0
$53$	6.46363	0.887848	0.443924	−	0.896064i	$-0.353586\pi$
$53$	6.46363	0.887848	0.443924	+	0.896064i	$0.353586\pi$
$54$	0	0
$55$	2.12212	0.286147
$56$	0	0
$57$	−4.22195	−0.559211
$58$	0	0
$59$	−6.30979	−0.821465	−0.410732	−	0.911756i	$-0.634727\pi$
$59$	−6.30979	−0.821465	−0.410732	+	0.911756i	$0.634727\pi$
$60$	0	0
$61$	11.9769	1.53349	0.766744	−	0.641953i	$-0.221876\pi$
$61$	11.9769	1.53349	0.766744	+	0.641953i	$0.221876\pi$
$62$	0	0
$63$	−11.3684	−1.43228
$64$	0	0
$65$	0.673513	0.0835390
$66$	0	0
$67$	14.4771	1.76866	0.884331	−	0.466860i	$-0.154615\pi$
$67$	14.4771	1.76866	0.884331	+	0.466860i	$0.154615\pi$
$68$	0	0
$69$	−2.99832	−0.360955
$70$	0	0
$71$	−2.78213	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$71$	−2.78213	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$72$	0	0
$73$	13.7688	1.61151	0.805757	−	0.592246i	$-0.201759\pi$
$73$	13.7688	1.61151	0.805757	+	0.592246i	$0.201759\pi$
$74$	0	0
$75$	2.57003	0.296762
$76$	0	0
$77$	−6.69199	−0.762623
$78$	0	0
$79$	4.22229	0.475045	0.237522	−	0.971382i	$-0.423665\pi$
$79$	4.22229	0.475045	0.237522	+	0.971382i	$0.423665\pi$
$80$	0	0
$81$	−6.81872	−0.757636
$82$	0	0
$83$	−1.91251	−0.209925	−0.104962	−	0.994476i	$-0.533472\pi$
$83$	−1.91251	−0.209925	−0.104962	+	0.994476i	$0.533472\pi$
$84$	0	0
$85$	4.82334	0.523164
$86$	0	0
$87$	9.04953	0.970212
$88$	0	0
$89$	14.4986	1.53685	0.768424	−	0.639942i	$-0.221041\pi$
$89$	14.4986	1.53685	0.768424	+	0.639942i	$0.221041\pi$
$90$	0	0
$91$	−2.12389	−0.222644
$92$	0	0
$93$	−24.6767	−2.55885
$94$	0	0
$95$	−1.64276	−0.168544
$96$	0	0
$97$	10.0326	1.01866	0.509329	−	0.860572i	$-0.329894\pi$
$97$	10.0326	1.01866	0.509329	+	0.860572i	$0.329894\pi$
$98$	0	0
$99$	7.65037	0.768891
$100$	0	0
$101$	1.46804	0.146075	0.0730376	−	0.997329i	$-0.476731\pi$
$101$	1.46804	0.146075	0.0730376	+	0.997329i	$0.476731\pi$
$102$	0	0
$103$	−15.7932	−1.55616	−0.778078	−	0.628168i	$-0.783805\pi$
$103$	−15.7932	−1.55616	−0.778078	+	0.628168i	$0.783805\pi$
$104$	0	0
$105$	−8.10446	−0.790914
$106$	0	0
$107$	14.8007	1.43084	0.715419	−	0.698696i	$-0.246236\pi$
$107$	14.8007	1.43084	0.715419	+	0.698696i	$0.246236\pi$
$108$	0	0
$109$	−8.99224	−0.861300	−0.430650	−	0.902519i	$-0.641716\pi$
$109$	−8.99224	−0.861300	−0.430650	+	0.902519i	$0.641716\pi$
$110$	0	0
$111$	19.8480	1.88389
$112$	0	0
$113$	−3.44800	−0.324360	−0.162180	−	0.986761i	$-0.551853\pi$
$113$	−3.44800	−0.324360	−0.162180	+	0.986761i	$0.551853\pi$
$114$	0	0
$115$	−1.16665	−0.108790
$116$	0	0
$117$	2.42805	0.224474
$118$	0	0
$119$	−15.2101	−1.39431
$120$	0	0
$121$	−6.49661	−0.590601
$122$	0	0
$123$	−7.27954	−0.656374
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.7396	−1.84034	−0.920171	−	0.391517i	$-0.871950\pi$
$127$	−20.7396	−1.84034	−0.920171	+	0.391517i	$0.871950\pi$
$128$	0	0
$129$	16.2538	1.43107
$130$	0	0
$131$	−19.1724	−1.67510	−0.837552	−	0.546357i	$-0.816014\pi$
$131$	−19.1724	−1.67510	−0.837552	+	0.546357i	$0.816014\pi$
$132$	0	0
$133$	5.18036	0.449195
$134$	0	0
$135$	1.55502	0.133835
$136$	0	0
$137$	−0.177174	−0.0151370	−0.00756849	−	0.999971i	$-0.502409\pi$
$137$	−0.177174	−0.0151370	−0.00756849	+	0.999971i	$0.502409\pi$
$138$	0	0
$139$	15.1528	1.28524	0.642622	−	0.766183i	$-0.277846\pi$
$139$	15.1528	1.28524	0.642622	+	0.766183i	$0.277846\pi$
$140$	0	0
$141$	29.6918	2.50050
$142$	0	0
$143$	1.42928	0.119522
$144$	0	0
$145$	3.52118	0.292418
$146$	0	0
$147$	7.56675	0.624095
$148$	0	0
$149$	−13.6810	−1.12079	−0.560395	−	0.828226i	$-0.689350\pi$
$149$	−13.6810	−1.12079	−0.560395	+	0.828226i	$0.689350\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	17.3884	1.40577
$154$	0	0
$155$	−9.60169	−0.771227
$156$	0	0
$157$	−9.74330	−0.777600	−0.388800	−	0.921322i	$-0.627110\pi$
$157$	−9.74330	−0.777600	−0.388800	+	0.921322i	$0.627110\pi$
$158$	0	0
$159$	16.6117	1.31740
$160$	0	0
$161$	3.67896	0.289943
$162$	0	0
$163$	−5.15277	−0.403596	−0.201798	−	0.979427i	$-0.564679\pi$
$163$	−5.15277	−0.403596	−0.201798	+	0.979427i	$0.564679\pi$
$164$	0	0
$165$	5.45392	0.424587
$166$	0	0
$167$	−2.43702	−0.188583	−0.0942913	−	0.995545i	$-0.530058\pi$
$167$	−2.43702	−0.188583	−0.0942913	+	0.995545i	$0.530058\pi$
$168$	0	0
$169$	−12.5464	−0.965106
$170$	0	0
$171$	−5.92226	−0.452886
$172$	0	0
$173$	−16.8647	−1.28220	−0.641101	−	0.767456i	$-0.721522\pi$
$173$	−16.8647	−1.28220	−0.641101	+	0.767456i	$0.721522\pi$
$174$	0	0
$175$	−3.15345	−0.238378
$176$	0	0
$177$	−16.2164	−1.21890
$178$	0	0
$179$	4.50950	0.337056	0.168528	−	0.985697i	$-0.446099\pi$
$179$	4.50950	0.337056	0.168528	+	0.985697i	$0.446099\pi$
$180$	0	0
$181$	16.6801	1.23982	0.619910	−	0.784673i	$-0.287169\pi$
$181$	16.6801	1.23982	0.619910	+	0.784673i	$0.287169\pi$
$182$	0	0
$183$	30.7811	2.27540
$184$	0	0
$185$	7.72286	0.567796
$186$	0	0
$187$	10.2357	0.748508
$188$	0	0
$189$	−4.90369	−0.356691
$190$	0	0
$191$	9.49673	0.687159	0.343580	−	0.939124i	$-0.388360\pi$
$191$	9.49673	0.687159	0.343580	+	0.939124i	$0.388360\pi$
$192$	0	0
$193$	21.3293	1.53532	0.767658	−	0.640860i	$-0.221422\pi$
$193$	21.3293	1.53532	0.767658	+	0.640860i	$0.221422\pi$
$194$	0	0
$195$	1.73095	0.123956
$196$	0	0
$197$	−16.0968	−1.14685	−0.573426	−	0.819257i	$-0.694386\pi$
$197$	−16.0968	−1.14685	−0.573426	+	0.819257i	$0.694386\pi$
$198$	0	0
$199$	18.1104	1.28381	0.641907	−	0.766782i	$-0.278143\pi$
$199$	18.1104	1.28381	0.641907	+	0.766782i	$0.278143\pi$
$200$	0	0
$201$	37.2067	2.62436
$202$	0	0
$203$	−11.1038	−0.779337
$204$	0	0
$205$	−2.83247	−0.197828
$206$	0	0
$207$	−4.20583	−0.292326
$208$	0	0
$209$	−3.48614	−0.241141
$210$	0	0
$211$	17.7004	1.21855	0.609273	−	0.792960i	$-0.291461\pi$
$211$	17.7004	1.21855	0.609273	+	0.792960i	$0.291461\pi$
$212$	0	0
$213$	−7.15015	−0.489921
$214$	0	0
$215$	6.32436	0.431318
$216$	0	0
$217$	30.2784	2.05543
$218$	0	0
$219$	35.3862	2.39118
$220$	0	0
$221$	3.24858	0.218523
$222$	0	0
$223$	−13.4260	−0.899073	−0.449536	−	0.893262i	$-0.648411\pi$
$223$	−13.4260	−0.899073	−0.449536	+	0.893262i	$0.648411\pi$
$224$	0	0
$225$	3.60506	0.240337
$226$	0	0
$227$	6.90561	0.458342	0.229171	−	0.973386i	$-0.426399\pi$
$227$	6.90561	0.458342	0.229171	+	0.973386i	$0.426399\pi$
$228$	0	0
$229$	8.34735	0.551609	0.275804	−	0.961214i	$-0.411056\pi$
$229$	8.34735	0.551609	0.275804	+	0.961214i	$0.411056\pi$
$230$	0	0
$231$	−17.1986	−1.13159
$232$	0	0
$233$	−10.1267	−0.663425	−0.331713	−	0.943380i	$-0.607626\pi$
$233$	−10.1267	−0.663425	−0.331713	+	0.943380i	$0.607626\pi$
$234$	0	0
$235$	11.5531	0.753641
$236$	0	0
$237$	10.8514	0.704875
$238$	0	0
$239$	−24.7742	−1.60251	−0.801255	−	0.598323i	$-0.795834\pi$
$239$	−24.7742	−1.60251	−0.801255	+	0.598323i	$0.795834\pi$
$240$	0	0
$241$	17.3393	1.11692	0.558462	−	0.829530i	$-0.311392\pi$
$241$	17.3393	1.11692	0.558462	+	0.829530i	$0.311392\pi$
$242$	0	0
$243$	−22.1894	−1.42345
$244$	0	0
$245$	2.94423	0.188100
$246$	0	0
$247$	−1.10642	−0.0703999
$248$	0	0
$249$	−4.91520	−0.311488
$250$	0	0
$251$	15.7180	0.992113	0.496057	−	0.868290i	$-0.334781\pi$
$251$	15.7180	0.992113	0.496057	+	0.868290i	$0.334781\pi$
$252$	0	0
$253$	−2.47576	−0.155650
$254$	0	0
$255$	12.3961	0.776275
$256$	0	0
$257$	−23.4325	−1.46168	−0.730839	−	0.682550i	$-0.760871\pi$
$257$	−23.4325	−1.46168	−0.730839	+	0.682550i	$0.760871\pi$
$258$	0	0
$259$	−24.3536	−1.51326
$260$	0	0
$261$	12.6941	0.785742
$262$	0	0
$263$	−29.3523	−1.80994	−0.904972	−	0.425472i	$-0.860108\pi$
$263$	−29.3523	−1.80994	−0.904972	+	0.425472i	$0.860108\pi$
$264$	0	0
$265$	6.46363	0.397058
$266$	0	0
$267$	37.2618	2.28039
$268$	0	0
$269$	−15.4199	−0.940165	−0.470083	−	0.882622i	$-0.655776\pi$
$269$	−15.4199	−0.940165	−0.470083	+	0.882622i	$0.655776\pi$
$270$	0	0
$271$	−9.87590	−0.599918	−0.299959	−	0.953952i	$-0.596973\pi$
$271$	−9.87590	−0.599918	−0.299959	+	0.953952i	$0.596973\pi$
$272$	0	0
$273$	−5.45846	−0.330361
$274$	0	0
$275$	2.12212	0.127969
$276$	0	0
$277$	7.24182	0.435119	0.217559	−	0.976047i	$-0.430190\pi$
$277$	7.24182	0.435119	0.217559	+	0.976047i	$0.430190\pi$
$278$	0	0
$279$	−34.6147	−2.07233
$280$	0	0
$281$	27.3015	1.62867	0.814337	−	0.580393i	$-0.197101\pi$
$281$	27.3015	1.62867	0.814337	+	0.580393i	$0.197101\pi$
$282$	0	0
$283$	−0.859481	−0.0510909	−0.0255454	−	0.999674i	$-0.508132\pi$
$283$	−0.859481	−0.0510909	−0.0255454	+	0.999674i	$0.508132\pi$
$284$	0	0
$285$	−4.22195	−0.250087
$286$	0	0
$287$	8.93205	0.527242
$288$	0	0
$289$	6.26456	0.368504
$290$	0	0
$291$	25.7841	1.51149
$292$	0	0
$293$	5.53182	0.323172	0.161586	−	0.986859i	$-0.448339\pi$
$293$	5.53182	0.323172	0.161586	+	0.986859i	$0.448339\pi$
$294$	0	0
$295$	−6.30979	−0.367370
$296$	0	0
$297$	3.29995	0.191482
$298$	0	0
$299$	−0.785752	−0.0454412
$300$	0	0
$301$	−19.9435	−1.14953
$302$	0	0
$303$	3.77290	0.216748
$304$	0	0
$305$	11.9769	0.685796
$306$	0	0
$307$	24.2862	1.38609	0.693043	−	0.720896i	$-0.256269\pi$
$307$	24.2862	1.38609	0.693043	+	0.720896i	$0.256269\pi$
$308$	0	0
$309$	−40.5891	−2.30904
$310$	0	0
$311$	30.0049	1.70142	0.850710	−	0.525635i	$-0.176172\pi$
$311$	30.0049	1.70142	0.850710	+	0.525635i	$0.176172\pi$
$312$	0	0
$313$	−16.7606	−0.947364	−0.473682	−	0.880696i	$-0.657075\pi$
$313$	−16.7606	−0.947364	−0.473682	+	0.880696i	$0.657075\pi$
$314$	0	0
$315$	−11.3684	−0.640535
$316$	0	0
$317$	−4.92136	−0.276411	−0.138206	−	0.990404i	$-0.544133\pi$
$317$	−4.92136	−0.276411	−0.138206	+	0.990404i	$0.544133\pi$
$318$	0	0
$319$	7.47236	0.418372
$320$	0	0
$321$	38.0383	2.12309
$322$	0	0
$323$	−7.92359	−0.440881
$324$	0	0
$325$	0.673513	0.0373598
$326$	0	0
$327$	−23.1103	−1.27800
$328$	0	0
$329$	−36.4321	−2.00857
$330$	0	0
$331$	24.4220	1.34235	0.671177	−	0.741297i	$-0.265789\pi$
$331$	24.4220	1.34235	0.671177	+	0.741297i	$0.265789\pi$
$332$	0	0
$333$	27.8414	1.52570
$334$	0	0
$335$	14.4771	0.790970
$336$	0	0
$337$	−16.4813	−0.897796	−0.448898	−	0.893583i	$-0.648183\pi$
$337$	−16.4813	−0.897796	−0.448898	+	0.893583i	$0.648183\pi$
$338$	0	0
$339$	−8.86146	−0.481288
$340$	0	0
$341$	−20.3759	−1.10342
$342$	0	0
$343$	12.7897	0.690577
$344$	0	0
$345$	−2.99832	−0.161424
$346$	0	0
$347$	25.2903	1.35765	0.678827	−	0.734298i	$-0.262488\pi$
$347$	25.2903	1.35765	0.678827	+	0.734298i	$0.262488\pi$
$348$	0	0
$349$	−13.3083	−0.712378	−0.356189	−	0.934414i	$-0.615924\pi$
$349$	−13.3083	−0.712378	−0.356189	+	0.934414i	$0.615924\pi$
$350$	0	0
$351$	1.04733	0.0559023
$352$	0	0
$353$	7.68173	0.408857	0.204428	−	0.978882i	$-0.434466\pi$
$353$	7.68173	0.408857	0.204428	+	0.978882i	$0.434466\pi$
$354$	0	0
$355$	−2.78213	−0.147660
$356$	0	0
$357$	−39.0905	−2.06889
$358$	0	0
$359$	27.1124	1.43094	0.715470	−	0.698643i	$-0.246213\pi$
$359$	27.1124	1.43094	0.715470	+	0.698643i	$0.246213\pi$
$360$	0	0
$361$	−16.3013	−0.857965
$362$	0	0
$363$	−16.6965	−0.876338
$364$	0	0
$365$	13.7688	0.720691
$366$	0	0
$367$	−27.2416	−1.42200	−0.711001	−	0.703191i	$-0.751758\pi$
$367$	−27.2416	−1.42200	−0.711001	+	0.703191i	$0.751758\pi$
$368$	0	0
$369$	−10.2112	−0.531576
$370$	0	0
$371$	−20.3827	−1.05822
$372$	0	0
$373$	10.0894	0.522412	0.261206	−	0.965283i	$-0.415880\pi$
$373$	10.0894	0.522412	0.261206	+	0.965283i	$0.415880\pi$
$374$	0	0
$375$	2.57003	0.132716
$376$	0	0
$377$	2.37156	0.122141
$378$	0	0
$379$	31.3712	1.61143	0.805716	−	0.592303i	$-0.201781\pi$
$379$	31.3712	1.61143	0.805716	+	0.592303i	$0.201781\pi$
$380$	0	0
$381$	−53.3014	−2.73071
$382$	0	0
$383$	−20.2452	−1.03448	−0.517240	−	0.855840i	$-0.673041\pi$
$383$	−20.2452	−1.03448	−0.517240	+	0.855840i	$0.673041\pi$
$384$	0	0
$385$	−6.69199	−0.341056
$386$	0	0
$387$	22.7997	1.15897
$388$	0	0
$389$	24.8953	1.26224	0.631120	−	0.775685i	$-0.282595\pi$
$389$	24.8953	1.26224	0.631120	+	0.775685i	$0.282595\pi$
$390$	0	0
$391$	−5.62713	−0.284576
$392$	0	0
$393$	−49.2738	−2.48553
$394$	0	0
$395$	4.22229	0.212446
$396$	0	0
$397$	15.0767	0.756676	0.378338	−	0.925668i	$-0.376496\pi$
$397$	15.0767	0.756676	0.378338	+	0.925668i	$0.376496\pi$
$398$	0	0
$399$	13.3137	0.666519
$400$	0	0
$401$	−5.26584	−0.262963	−0.131482	−	0.991319i	$-0.541973\pi$
$401$	−5.26584	−0.262963	−0.131482	+	0.991319i	$0.541973\pi$
$402$	0	0
$403$	−6.46686	−0.322137
$404$	0	0
$405$	−6.81872	−0.338825
$406$	0	0
$407$	16.3888	0.812365
$408$	0	0
$409$	1.56608	0.0774379	0.0387190	−	0.999250i	$-0.487672\pi$
$409$	1.56608	0.0774379	0.0387190	+	0.999250i	$0.487672\pi$
$410$	0	0
$411$	−0.455342	−0.0224604
$412$	0	0
$413$	19.8976	0.979096
$414$	0	0
$415$	−1.91251	−0.0938812
$416$	0	0
$417$	38.9432	1.90706
$418$	0	0
$419$	−21.9043	−1.07010	−0.535048	−	0.844822i	$-0.679707\pi$
$419$	−21.9043	−1.07010	−0.535048	+	0.844822i	$0.679707\pi$
$420$	0	0
$421$	17.1295	0.834839	0.417420	−	0.908714i	$-0.362935\pi$
$421$	17.1295	0.834839	0.417420	+	0.908714i	$0.362935\pi$
$422$	0	0
$423$	41.6496	2.02507
$424$	0	0
$425$	4.82334	0.233966
$426$	0	0
$427$	−37.7686	−1.82775
$428$	0	0
$429$	3.67328	0.177348
$430$	0	0
$431$	−2.43878	−0.117472	−0.0587360	−	0.998274i	$-0.518707\pi$
$431$	−2.43878	−0.117472	−0.0587360	+	0.998274i	$0.518707\pi$
$432$	0	0
$433$	−34.4973	−1.65783	−0.828917	−	0.559371i	$-0.811043\pi$
$433$	−34.4973	−1.65783	−0.828917	+	0.559371i	$0.811043\pi$
$434$	0	0
$435$	9.04953	0.433892
$436$	0	0
$437$	1.91652	0.0916798
$438$	0	0
$439$	−31.8740	−1.52126	−0.760632	−	0.649183i	$-0.775111\pi$
$439$	−31.8740	−1.52126	−0.760632	+	0.649183i	$0.775111\pi$
$440$	0	0
$441$	10.6141	0.505434
$442$	0	0
$443$	−24.7438	−1.17562	−0.587808	−	0.809001i	$-0.700009\pi$
$443$	−24.7438	−1.17562	−0.587808	+	0.809001i	$0.700009\pi$
$444$	0	0
$445$	14.4986	0.687299
$446$	0	0
$447$	−35.1605	−1.66304
$448$	0	0
$449$	0.210009	0.00991093	0.00495546	−	0.999988i	$-0.498423\pi$
$449$	0.210009	0.00991093	0.00495546	+	0.999988i	$0.498423\pi$
$450$	0	0
$451$	−6.01085	−0.283040
$452$	0	0
$453$	2.57003	0.120751
$454$	0	0
$455$	−2.12389	−0.0995694
$456$	0	0
$457$	−25.7362	−1.20389	−0.601943	−	0.798539i	$-0.705607\pi$
$457$	−25.7362	−1.20389	−0.601943	+	0.798539i	$0.705607\pi$
$458$	0	0
$459$	7.50041	0.350089
$460$	0	0
$461$	−1.13244	−0.0527430	−0.0263715	−	0.999652i	$-0.508395\pi$
$461$	−1.13244	−0.0527430	−0.0263715	+	0.999652i	$0.508395\pi$
$462$	0	0
$463$	−22.3380	−1.03813	−0.519066	−	0.854734i	$-0.673720\pi$
$463$	−22.3380	−1.03813	−0.519066	+	0.854734i	$0.673720\pi$
$464$	0	0
$465$	−24.6767	−1.14435
$466$	0	0
$467$	27.2543	1.26118	0.630588	−	0.776117i	$-0.282814\pi$
$467$	27.2543	1.26118	0.630588	+	0.776117i	$0.282814\pi$
$468$	0	0
$469$	−45.6529	−2.10805
$470$	0	0
$471$	−25.0406	−1.15381
$472$	0	0
$473$	13.4210	0.617100
$474$	0	0
$475$	−1.64276	−0.0753751
$476$	0	0
$477$	23.3018	1.06692
$478$	0	0
$479$	18.2032	0.831726	0.415863	−	0.909427i	$-0.363480\pi$
$479$	18.2032	0.831726	0.415863	+	0.909427i	$0.363480\pi$
$480$	0	0
$481$	5.20145	0.237166
$482$	0	0
$483$	9.45504	0.430219
$484$	0	0
$485$	10.0326	0.455557
$486$	0	0
$487$	−6.76062	−0.306353	−0.153176	−	0.988199i	$-0.548950\pi$
$487$	−6.76062	−0.306353	−0.153176	+	0.988199i	$0.548950\pi$
$488$	0	0
$489$	−13.2428	−0.598860
$490$	0	0
$491$	20.5566	0.927707	0.463853	−	0.885912i	$-0.346466\pi$
$491$	20.5566	0.927707	0.463853	+	0.885912i	$0.346466\pi$
$492$	0	0
$493$	16.9838	0.764912
$494$	0	0
$495$	7.65037	0.343859
$496$	0	0
$497$	8.77329	0.393536
$498$	0	0
$499$	−29.0197	−1.29910	−0.649549	−	0.760320i	$-0.725042\pi$
$499$	−29.0197	−1.29910	−0.649549	+	0.760320i	$0.725042\pi$
$500$	0	0
$501$	−6.26323	−0.279820
$502$	0	0
$503$	−37.6058	−1.67676	−0.838380	−	0.545086i	$-0.816497\pi$
$503$	−37.6058	−1.67676	−0.838380	+	0.545086i	$0.816497\pi$
$504$	0	0
$505$	1.46804	0.0653268
$506$	0	0
$507$	−32.2446	−1.43203
$508$	0	0
$509$	6.59421	0.292283	0.146142	−	0.989264i	$-0.453314\pi$
$509$	6.59421	0.292283	0.146142	+	0.989264i	$0.453314\pi$
$510$	0	0
$511$	−43.4191	−1.92075
$512$	0	0
$513$	−2.55454	−0.112786
$514$	0	0
$515$	−15.7932	−0.695934
$516$	0	0
$517$	24.5171	1.07826
$518$	0	0
$519$	−43.3429	−1.90254
$520$	0	0
$521$	−33.5717	−1.47080	−0.735401	−	0.677632i	$-0.763006\pi$
$521$	−33.5717	−1.47080	−0.735401	+	0.677632i	$0.763006\pi$
$522$	0	0
$523$	−23.0858	−1.00947	−0.504735	−	0.863274i	$-0.668410\pi$
$523$	−23.0858	−1.00947	−0.504735	+	0.863274i	$0.668410\pi$
$524$	0	0
$525$	−8.10446	−0.353707
$526$	0	0
$527$	−46.3122	−2.01739
$528$	0	0
$529$	−21.6389	−0.940823
$530$	0	0
$531$	−22.7472	−0.987144
$532$	0	0
$533$	−1.90771	−0.0826319
$534$	0	0
$535$	14.8007	0.639890
$536$	0	0
$537$	11.5896	0.500126
$538$	0	0
$539$	6.24800	0.269121
$540$	0	0
$541$	1.40031	0.0602038	0.0301019	−	0.999547i	$-0.490417\pi$
$541$	1.40031	0.0602038	0.0301019	+	0.999547i	$0.490417\pi$
$542$	0	0
$543$	42.8683	1.83965
$544$	0	0
$545$	−8.99224	−0.385185
$546$	0	0
$547$	43.8802	1.87618	0.938091	−	0.346388i	$-0.112592\pi$
$547$	43.8802	1.87618	0.938091	+	0.346388i	$0.112592\pi$
$548$	0	0
$549$	43.1775	1.84277
$550$	0	0
$551$	−5.78446	−0.246426
$552$	0	0
$553$	−13.3148	−0.566202
$554$	0	0
$555$	19.8480	0.842501
$556$	0	0
$557$	33.7775	1.43120	0.715599	−	0.698511i	$-0.246154\pi$
$557$	33.7775	1.43120	0.715599	+	0.698511i	$0.246154\pi$
$558$	0	0
$559$	4.25954	0.180159
$560$	0	0
$561$	26.3061	1.11064
$562$	0	0
$563$	−5.32686	−0.224500	−0.112250	−	0.993680i	$-0.535806\pi$
$563$	−5.32686	−0.224500	−0.112250	+	0.993680i	$0.535806\pi$
$564$	0	0
$565$	−3.44800	−0.145058
$566$	0	0
$567$	21.5025	0.903019
$568$	0	0
$569$	−0.397426	−0.0166610	−0.00833048	−	0.999965i	$-0.502652\pi$
$569$	−0.397426	−0.0166610	−0.00833048	+	0.999965i	$0.502652\pi$
$570$	0	0
$571$	0.905592	0.0378978	0.0189489	−	0.999820i	$-0.493968\pi$
$571$	0.905592	0.0378978	0.0189489	+	0.999820i	$0.493968\pi$
$572$	0	0
$573$	24.4069	1.01961
$574$	0	0
$575$	−1.16665	−0.0486525
$576$	0	0
$577$	25.6857	1.06931	0.534655	−	0.845070i	$-0.320442\pi$
$577$	25.6857	1.06931	0.534655	+	0.845070i	$0.320442\pi$
$578$	0	0
$579$	54.8169	2.27811
$580$	0	0
$581$	6.03098	0.250207
$582$	0	0
$583$	13.7166	0.568084
$584$	0	0
$585$	2.42805	0.100388
$586$	0	0
$587$	−30.0125	−1.23875	−0.619374	−	0.785096i	$-0.712614\pi$
$587$	−30.0125	−1.23875	−0.619374	+	0.785096i	$0.712614\pi$
$588$	0	0
$589$	15.7733	0.649928
$590$	0	0
$591$	−41.3694	−1.70171
$592$	0	0
$593$	−38.6588	−1.58753	−0.793763	−	0.608227i	$-0.791881\pi$
$593$	−38.6588	−1.58753	−0.793763	+	0.608227i	$0.791881\pi$
$594$	0	0
$595$	−15.2101	−0.623555
$596$	0	0
$597$	46.5444	1.90493
$598$	0	0
$599$	−16.4952	−0.673977	−0.336989	−	0.941509i	$-0.609408\pi$
$599$	−16.4952	−0.673977	−0.336989	+	0.941509i	$0.609408\pi$
$600$	0	0
$601$	8.25124	0.336575	0.168288	−	0.985738i	$-0.446176\pi$
$601$	8.25124	0.336575	0.168288	+	0.985738i	$0.446176\pi$
$602$	0	0
$603$	52.1909	2.12538
$604$	0	0
$605$	−6.49661	−0.264125
$606$	0	0
$607$	−42.8956	−1.74108	−0.870539	−	0.492099i	$-0.836230\pi$
$607$	−42.8956	−1.74108	−0.870539	+	0.492099i	$0.836230\pi$
$608$	0	0
$609$	−28.5372	−1.15639
$610$	0	0
$611$	7.78116	0.314792
$612$	0	0
$613$	29.0439	1.17307	0.586537	−	0.809922i	$-0.300491\pi$
$613$	29.0439	1.17307	0.586537	+	0.809922i	$0.300491\pi$
$614$	0	0
$615$	−7.27954	−0.293539
$616$	0	0
$617$	39.6243	1.59522	0.797608	−	0.603177i	$-0.206099\pi$
$617$	39.6243	1.59522	0.797608	+	0.603177i	$0.206099\pi$
$618$	0	0
$619$	17.3939	0.699122	0.349561	−	0.936914i	$-0.386331\pi$
$619$	17.3939	0.699122	0.349561	+	0.936914i	$0.386331\pi$
$620$	0	0
$621$	−1.81416	−0.0727999
$622$	0	0
$623$	−45.7205	−1.83175
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.95949	−0.357807
$628$	0	0
$629$	37.2500	1.48525
$630$	0	0
$631$	−29.6058	−1.17859	−0.589295	−	0.807918i	$-0.700594\pi$
$631$	−29.6058	−1.17859	−0.589295	+	0.807918i	$0.700594\pi$
$632$	0	0
$633$	45.4906	1.80809
$634$	0	0
$635$	−20.7396	−0.823026
$636$	0	0
$637$	1.98297	0.0785683
$638$	0	0
$639$	−10.0297	−0.396770
$640$	0	0
$641$	−46.3159	−1.82937	−0.914684	−	0.404170i	$-0.867560\pi$
$641$	−46.3159	−1.82937	−0.914684	+	0.404170i	$0.867560\pi$
$642$	0	0
$643$	−44.1037	−1.73928	−0.869641	−	0.493685i	$-0.835650\pi$
$643$	−44.1037	−1.73928	−0.869641	+	0.493685i	$0.835650\pi$
$644$	0	0
$645$	16.2538	0.639993
$646$	0	0
$647$	26.9141	1.05810	0.529051	−	0.848590i	$-0.322548\pi$
$647$	26.9141	1.05810	0.529051	+	0.848590i	$0.322548\pi$
$648$	0	0
$649$	−13.3901	−0.525609
$650$	0	0
$651$	77.8165	3.04987
$652$	0	0
$653$	21.5398	0.842917	0.421459	−	0.906848i	$-0.361518\pi$
$653$	21.5398	0.842917	0.421459	+	0.906848i	$0.361518\pi$
$654$	0	0
$655$	−19.1724	−0.749129
$656$	0	0
$657$	49.6373	1.93654
$658$	0	0
$659$	−18.7396	−0.729990	−0.364995	−	0.931009i	$-0.618929\pi$
$659$	−18.7396	−0.729990	−0.364995	+	0.931009i	$0.618929\pi$
$660$	0	0
$661$	12.6837	0.493337	0.246669	−	0.969100i	$-0.420664\pi$
$661$	12.6837	0.493337	0.246669	+	0.969100i	$0.420664\pi$
$662$	0	0
$663$	8.34895	0.324246
$664$	0	0
$665$	5.18036	0.200886
$666$	0	0
$667$	−4.10797	−0.159061
$668$	0	0
$669$	−34.5053	−1.33405
$670$	0	0
$671$	25.4165	0.981192
$672$	0	0
$673$	4.58669	0.176804	0.0884020	−	0.996085i	$-0.471824\pi$
$673$	4.58669	0.176804	0.0884020	+	0.996085i	$0.471824\pi$
$674$	0	0
$675$	1.55502	0.0598529
$676$	0	0
$677$	−36.7919	−1.41403	−0.707014	−	0.707200i	$-0.749958\pi$
$677$	−36.7919	−1.41403	−0.707014	+	0.707200i	$0.749958\pi$
$678$	0	0
$679$	−31.6373	−1.21413
$680$	0	0
$681$	17.7476	0.680091
$682$	0	0
$683$	−41.0844	−1.57205	−0.786026	−	0.618193i	$-0.787865\pi$
$683$	−41.0844	−1.57205	−0.786026	+	0.618193i	$0.787865\pi$
$684$	0	0
$685$	−0.177174	−0.00676946
$686$	0	0
$687$	21.4530	0.818481
$688$	0	0
$689$	4.35334	0.165849
$690$	0	0
$691$	−27.5111	−1.04657	−0.523285	−	0.852158i	$-0.675294\pi$
$691$	−27.5111	−1.04657	−0.523285	+	0.852158i	$0.675294\pi$
$692$	0	0
$693$	−24.1250	−0.916435
$694$	0	0
$695$	15.1528	0.574779
$696$	0	0
$697$	−13.6620	−0.517484
$698$	0	0
$699$	−26.0261	−0.984396
$700$	0	0
$701$	19.1867	0.724673	0.362336	−	0.932047i	$-0.381979\pi$
$701$	19.1867	0.724673	0.362336	+	0.932047i	$0.381979\pi$
$702$	0	0
$703$	−12.6868	−0.478493
$704$	0	0
$705$	29.6918	1.11826
$706$	0	0
$707$	−4.62938	−0.174106
$708$	0	0
$709$	−19.2761	−0.723927	−0.361964	−	0.932192i	$-0.617894\pi$
$709$	−19.2761	−0.723927	−0.361964	+	0.932192i	$0.617894\pi$
$710$	0	0
$711$	15.2216	0.570855
$712$	0	0
$713$	11.2018	0.419510
$714$	0	0
$715$	1.42928	0.0534519
$716$	0	0
$717$	−63.6705	−2.37782
$718$	0	0
$719$	35.8025	1.33521	0.667604	−	0.744516i	$-0.267320\pi$
$719$	35.8025	1.33521	0.667604	+	0.744516i	$0.267320\pi$
$720$	0	0
$721$	49.8032	1.85477
$722$	0	0
$723$	44.5626	1.65730
$724$	0	0
$725$	3.52118	0.130773
$726$	0	0
$727$	4.82414	0.178917	0.0894587	−	0.995991i	$-0.471486\pi$
$727$	4.82414	0.178917	0.0894587	+	0.995991i	$0.471486\pi$
$728$	0	0
$729$	−36.5713	−1.35449
$730$	0	0
$731$	30.5045	1.12825
$732$	0	0
$733$	32.7102	1.20818	0.604089	−	0.796917i	$-0.293537\pi$
$733$	32.7102	1.20818	0.604089	+	0.796917i	$0.293537\pi$
$734$	0	0
$735$	7.56675	0.279104
$736$	0	0
$737$	30.7222	1.13167
$738$	0	0
$739$	5.89445	0.216831	0.108415	−	0.994106i	$-0.465422\pi$
$739$	5.89445	0.216831	0.108415	+	0.994106i	$0.465422\pi$
$740$	0	0
$741$	−2.84354	−0.104460
$742$	0	0
$743$	−30.1300	−1.10536	−0.552681	−	0.833393i	$-0.686395\pi$
$743$	−30.1300	−1.10536	−0.552681	+	0.833393i	$0.686395\pi$
$744$	0	0
$745$	−13.6810	−0.501232
$746$	0	0
$747$	−6.89470	−0.252264
$748$	0	0
$749$	−46.6732	−1.70540
$750$	0	0
$751$	−17.4824	−0.637942	−0.318971	−	0.947765i	$-0.603337\pi$
$751$	−17.4824	−0.637942	−0.318971	+	0.947765i	$0.603337\pi$
$752$	0	0
$753$	40.3958	1.47211
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	17.6604	0.641878	0.320939	−	0.947100i	$-0.396002\pi$
$757$	17.6604	0.641878	0.320939	+	0.947100i	$0.396002\pi$
$758$	0	0
$759$	−6.36279	−0.230955
$760$	0	0
$761$	27.4991	0.996842	0.498421	−	0.866935i	$-0.333913\pi$
$761$	27.4991	0.996842	0.498421	+	0.866935i	$0.333913\pi$
$762$	0	0
$763$	28.3565	1.02658
$764$	0	0
$765$	17.3884	0.628679
$766$	0	0
$767$	−4.24973	−0.153449
$768$	0	0
$769$	36.2988	1.30897	0.654485	−	0.756075i	$-0.272886\pi$
$769$	36.2988	1.30897	0.654485	+	0.756075i	$0.272886\pi$
$770$	0	0
$771$	−60.2222	−2.16885
$772$	0	0
$773$	12.3583	0.444498	0.222249	−	0.974990i	$-0.428660\pi$
$773$	12.3583	0.444498	0.222249	+	0.974990i	$0.428660\pi$
$774$	0	0
$775$	−9.60169	−0.344903
$776$	0	0
$777$	−62.5896	−2.24539
$778$	0	0
$779$	4.65308	0.166714
$780$	0	0
$781$	−5.90401	−0.211262
$782$	0	0
$783$	5.47552	0.195679
$784$	0	0
$785$	−9.74330	−0.347753
$786$	0	0
$787$	−32.5120	−1.15893	−0.579463	−	0.814999i	$-0.696738\pi$
$787$	−32.5120	−1.15893	−0.579463	+	0.814999i	$0.696738\pi$
$788$	0	0
$789$	−75.4364	−2.68561
$790$	0	0
$791$	10.8731	0.386602
$792$	0	0
$793$	8.06661	0.286454
$794$	0	0
$795$	16.6117	0.589158
$796$	0	0
$797$	10.3303	0.365917	0.182958	−	0.983121i	$-0.441433\pi$
$797$	10.3303	0.365917	0.182958	+	0.983121i	$0.441433\pi$
$798$	0	0
$799$	55.7245	1.97139
$800$	0	0
$801$	52.2683	1.84681
$802$	0	0
$803$	29.2190	1.03112
$804$	0	0
$805$	3.67896	0.129666
$806$	0	0
$807$	−39.6295	−1.39502
$808$	0	0
$809$	−54.8007	−1.92669	−0.963344	−	0.268268i	$-0.913549\pi$
$809$	−54.8007	−1.92669	−0.963344	+	0.268268i	$0.913549\pi$
$810$	0	0
$811$	4.98429	0.175022	0.0875110	−	0.996164i	$-0.472109\pi$
$811$	4.98429	0.175022	0.0875110	+	0.996164i	$0.472109\pi$
$812$	0	0
$813$	−25.3814	−0.890164
$814$	0	0
$815$	−5.15277	−0.180494
$816$	0	0
$817$	−10.3894	−0.363480
$818$	0	0
$819$	−7.65674	−0.267548
$820$	0	0
$821$	−3.62129	−0.126384	−0.0631920	−	0.998001i	$-0.520128\pi$
$821$	−3.62129	−0.126384	−0.0631920	+	0.998001i	$0.520128\pi$
$822$	0	0
$823$	−14.6852	−0.511894	−0.255947	−	0.966691i	$-0.582387\pi$
$823$	−14.6852	−0.511894	−0.255947	+	0.966691i	$0.582387\pi$
$824$	0	0
$825$	5.45392	0.189881
$826$	0	0
$827$	−26.1470	−0.909222	−0.454611	−	0.890690i	$-0.650222\pi$
$827$	−26.1470	−0.909222	−0.454611	+	0.890690i	$0.650222\pi$
$828$	0	0
$829$	−25.3070	−0.878948	−0.439474	−	0.898255i	$-0.644835\pi$
$829$	−25.3070	−0.878948	−0.439474	+	0.898255i	$0.644835\pi$
$830$	0	0
$831$	18.6117	0.645632
$832$	0	0
$833$	14.2010	0.492035
$834$	0	0
$835$	−2.43702	−0.0843367
$836$	0	0
$837$	−14.9309	−0.516086
$838$	0	0
$839$	−7.77412	−0.268392	−0.134196	−	0.990955i	$-0.542845\pi$
$839$	−7.77412	−0.268392	−0.134196	+	0.990955i	$0.542845\pi$
$840$	0	0
$841$	−16.6013	−0.572459
$842$	0	0
$843$	70.1658	2.41664
$844$	0	0
$845$	−12.5464	−0.431609
$846$	0	0
$847$	20.4867	0.703931
$848$	0	0
$849$	−2.20889	−0.0758090
$850$	0	0
$851$	−9.00985	−0.308854
$852$	0	0
$853$	12.1755	0.416881	0.208440	−	0.978035i	$-0.433161\pi$
$853$	12.1755	0.416881	0.208440	+	0.978035i	$0.433161\pi$
$854$	0	0
$855$	−5.92226	−0.202537
$856$	0	0
$857$	6.01578	0.205495	0.102748	−	0.994707i	$-0.467237\pi$
$857$	6.01578	0.205495	0.102748	+	0.994707i	$0.467237\pi$
$858$	0	0
$859$	−36.7920	−1.25533	−0.627663	−	0.778485i	$-0.715988\pi$
$859$	−36.7920	−1.25533	−0.627663	+	0.778485i	$0.715988\pi$
$860$	0	0
$861$	22.9556	0.782326
$862$	0	0
$863$	14.4129	0.490620	0.245310	−	0.969445i	$-0.421110\pi$
$863$	14.4129	0.490620	0.245310	+	0.969445i	$0.421110\pi$
$864$	0	0
$865$	−16.8647	−0.573418
$866$	0	0
$867$	16.1001	0.546789
$868$	0	0
$869$	8.96021	0.303954
$870$	0	0
$871$	9.75053	0.330384
$872$	0	0
$873$	36.1682	1.22411
$874$	0	0
$875$	−3.15345	−0.106606
$876$	0	0
$877$	−10.8654	−0.366898	−0.183449	−	0.983029i	$-0.558726\pi$
$877$	−10.8654	−0.366898	−0.183449	+	0.983029i	$0.558726\pi$
$878$	0	0
$879$	14.2169	0.479526
$880$	0	0
$881$	−28.1746	−0.949228	−0.474614	−	0.880194i	$-0.657412\pi$
$881$	−28.1746	−0.949228	−0.474614	+	0.880194i	$0.657412\pi$
$882$	0	0
$883$	−5.94601	−0.200099	−0.100050	−	0.994982i	$-0.531900\pi$
$883$	−5.94601	−0.200099	−0.100050	+	0.994982i	$0.531900\pi$
$884$	0	0
$885$	−16.2164	−0.545107
$886$	0	0
$887$	−24.6887	−0.828967	−0.414483	−	0.910057i	$-0.636038\pi$
$887$	−24.6887	−0.828967	−0.414483	+	0.910057i	$0.636038\pi$
$888$	0	0
$889$	65.4012	2.19349
$890$	0	0
$891$	−14.4701	−0.484768
$892$	0	0
$893$	−18.9790	−0.635108
$894$	0	0
$895$	4.50950	0.150736
$896$	0	0
$897$	−2.01941	−0.0674260
$898$	0	0
$899$	−33.8092	−1.12760
$900$	0	0
$901$	31.1763	1.03863
$902$	0	0
$903$	−51.2555	−1.70568
$904$	0	0
$905$	16.6801	0.554464
$906$	0	0
$907$	−2.88031	−0.0956390	−0.0478195	−	0.998856i	$-0.515227\pi$
$907$	−2.88031	−0.0956390	−0.0478195	+	0.998856i	$0.515227\pi$
$908$	0	0
$909$	5.29236	0.175537
$910$	0	0
$911$	−10.9399	−0.362453	−0.181227	−	0.983441i	$-0.558007\pi$
$911$	−10.9399	−0.362453	−0.181227	+	0.983441i	$0.558007\pi$
$912$	0	0
$913$	−4.05857	−0.134319
$914$	0	0
$915$	30.7811	1.01759
$916$	0	0
$917$	60.4593	1.99654
$918$	0	0
$919$	31.9874	1.05517	0.527584	−	0.849503i	$-0.323098\pi$
$919$	31.9874	1.05517	0.527584	+	0.849503i	$0.323098\pi$
$920$	0	0
$921$	62.4163	2.05669
$922$	0	0
$923$	−1.87380	−0.0616768
$924$	0	0
$925$	7.72286	0.253926
$926$	0	0
$927$	−56.9356	−1.87001
$928$	0	0
$929$	−11.0462	−0.362414	−0.181207	−	0.983445i	$-0.558000\pi$
$929$	−11.0462	−0.362414	−0.181207	+	0.983445i	$0.558000\pi$
$930$	0	0
$931$	−4.83667	−0.158515
$932$	0	0
$933$	77.1135	2.52458
$934$	0	0
$935$	10.2357	0.334743
$936$	0	0
$937$	−30.1363	−0.984511	−0.492255	−	0.870451i	$-0.663827\pi$
$937$	−30.1363	−0.984511	−0.492255	+	0.870451i	$0.663827\pi$
$938$	0	0
$939$	−43.0752	−1.40571
$940$	0	0
$941$	23.3144	0.760028	0.380014	−	0.924981i	$-0.375919\pi$
$941$	23.3144	0.760028	0.380014	+	0.924981i	$0.375919\pi$
$942$	0	0
$943$	3.30449	0.107609
$944$	0	0
$945$	−4.90369	−0.159517
$946$	0	0
$947$	−14.0406	−0.456259	−0.228129	−	0.973631i	$-0.573261\pi$
$947$	−14.0406	−0.456259	−0.228129	+	0.973631i	$0.573261\pi$
$948$	0	0
$949$	9.27345	0.301029
$950$	0	0
$951$	−12.6480	−0.410141
$952$	0	0
$953$	46.0882	1.49294	0.746472	−	0.665417i	$-0.231746\pi$
$953$	46.0882	1.49294	0.746472	+	0.665417i	$0.231746\pi$
$954$	0	0
$955$	9.49673	0.307307
$956$	0	0
$957$	19.2042	0.620783
$958$	0	0
$959$	0.558708	0.0180416
$960$	0	0
$961$	61.1925	1.97395
$962$	0	0
$963$	53.3574	1.71942
$964$	0	0
$965$	21.3293	0.686614
$966$	0	0
$967$	−18.3476	−0.590019	−0.295009	−	0.955494i	$-0.595323\pi$
$967$	−18.3476	−0.590019	−0.295009	+	0.955494i	$0.595323\pi$
$968$	0	0
$969$	−20.3639	−0.654182
$970$	0	0
$971$	4.50747	0.144652	0.0723258	−	0.997381i	$-0.476958\pi$
$971$	4.50747	0.144652	0.0723258	+	0.997381i	$0.476958\pi$
$972$	0	0
$973$	−47.7836	−1.53187
$974$	0	0
$975$	1.73095	0.0554347
$976$	0	0
$977$	25.2629	0.808233	0.404117	−	0.914708i	$-0.367579\pi$
$977$	25.2629	0.808233	0.404117	+	0.914708i	$0.367579\pi$
$978$	0	0
$979$	30.7677	0.983341
$980$	0	0
$981$	−32.4176	−1.03501
$982$	0	0
$983$	−13.1415	−0.419150	−0.209575	−	0.977793i	$-0.567208\pi$
$983$	−13.1415	−0.419150	−0.209575	+	0.977793i	$0.567208\pi$
$984$	0	0
$985$	−16.0968	−0.512888
$986$	0	0
$987$	−93.6316	−2.98033
$988$	0	0
$989$	−7.37829	−0.234616
$990$	0	0
$991$	−29.2238	−0.928323	−0.464162	−	0.885751i	$-0.653644\pi$
$991$	−29.2238	−0.928323	−0.464162	+	0.885751i	$0.653644\pi$
$992$	0	0
$993$	62.7653	1.99180
$994$	0	0
$995$	18.1104	0.574139
$996$	0	0
$997$	−22.3106	−0.706584	−0.353292	−	0.935513i	$-0.614938\pi$
$997$	−22.3106	−0.706584	−0.353292	+	0.935513i	$0.614938\pi$
$998$	0	0
$999$	12.0092	0.379956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.s.1.20	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.s.1.20	✓	24	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)

\(f(q)\)	\(=\)	\(q+2.57003 q^{3} +1.00000 q^{5} -3.15345 q^{7} +3.60506 q^{9} +O(q^{10})\)	\(q+2.57003 q^{3} +1.00000 q^{5} -3.15345 q^{7} +3.60506 q^{9} +2.12212 q^{11} +0.673513 q^{13} +2.57003 q^{15} +4.82334 q^{17} -1.64276 q^{19} -8.10446 q^{21} -1.16665 q^{23} +1.00000 q^{25} +1.55502 q^{27} +3.52118 q^{29} -9.60169 q^{31} +5.45392 q^{33} -3.15345 q^{35} +7.72286 q^{37} +1.73095 q^{39} -2.83247 q^{41} +6.32436 q^{43} +3.60506 q^{45} +11.5531 q^{47} +2.94423 q^{49} +12.3961 q^{51} +6.46363 q^{53} +2.12212 q^{55} -4.22195 q^{57} -6.30979 q^{59} +11.9769 q^{61} -11.3684 q^{63} +0.673513 q^{65} +14.4771 q^{67} -2.99832 q^{69} -2.78213 q^{71} +13.7688 q^{73} +2.57003 q^{75} -6.69199 q^{77} +4.22229 q^{79} -6.81872 q^{81} -1.91251 q^{83} +4.82334 q^{85} +9.04953 q^{87} +14.4986 q^{89} -2.12389 q^{91} -24.6767 q^{93} -1.64276 q^{95} +10.0326 q^{97} +7.65037 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9	\( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 24 * q + 2 * q^3 + 24 * q^5 + 3 * q^7 + 40 * q^9 + 17 * q^11 + 16 * q^13 + 2 * q^15 + 22 * q^17 + 16 * q^19 - q^21 + 7 * q^23 + 24 * q^25 - 4 * q^27 + 25 * q^29 + 28 * q^31 + 11 * q^33 + 3 * q^35 + 26 * q^37 + 13 * q^39 + 38 * q^41 - 13 * q^43 + 40 * q^45 + 12 * q^47 + 61 * q^49 + 53 * q^53 + 17 * q^55 + 30 * q^57 + 35 * q^59 + 44 * q^61 - 9 * q^63 + 16 * q^65 - 15 * q^67 + 9 * q^69 + 22 * q^71 + 31 * q^73 + 2 * q^75 + 26 * q^77 + 20 * q^79 + 88 * q^81 - 14 * q^83 + 22 * q^85 - 18 * q^87 + 37 * q^89 - 26 * q^91 + 13 * q^93 + 16 * q^95 + 21 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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