LMFDB - Embedded newform 6024.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 9 x^{19} - 31 x^{18} + 471 x^{17} - 82 x^{16} - 9476 x^{15} + 12881 x^{14} + 94079 x^{13} + \cdots + 24832 $ x^20 - 9x^19 - 31x^18 + 471x^17 - 82x^16 - 9476x^15 + 12881x^14 + 94079x^13 - 181272x^12 - 504914x^11 + 1108012x^10 + 1591114x^9 - 3399127x^8 - 3224089x^7 + 5089722x^6 + 4287384x^5 - 2840216x^4 - 2820864x^3 - 208576x^2 + 204992*x + 24832
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.69479$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.69479 q^{5} -1.23254 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.69479 q^{5} -1.23254 q^{7} +1.00000 q^{9} +0.313298 q^{11} +6.23784 q^{13} -3.69479 q^{15} -6.66173 q^{17} +0.170818 q^{19} -1.23254 q^{21} -1.48449 q^{23} +8.65149 q^{25} +1.00000 q^{27} +1.35837 q^{29} +1.26428 q^{31} +0.313298 q^{33} +4.55398 q^{35} -1.64126 q^{37} +6.23784 q^{39} +0.180309 q^{41} -0.765853 q^{43} -3.69479 q^{45} -13.4564 q^{47} -5.48084 q^{49} -6.66173 q^{51} +2.25751 q^{53} -1.15757 q^{55} +0.170818 q^{57} +5.64566 q^{59} +6.33817 q^{61} -1.23254 q^{63} -23.0475 q^{65} -9.49476 q^{67} -1.48449 q^{69} -2.86093 q^{71} -2.30965 q^{73} +8.65149 q^{75} -0.386153 q^{77} +13.6931 q^{79} +1.00000 q^{81} +5.61239 q^{83} +24.6137 q^{85} +1.35837 q^{87} +9.48087 q^{89} -7.68839 q^{91} +1.26428 q^{93} -0.631137 q^{95} +17.1972 q^{97} +0.313298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	$ 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) $ 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.69479	−1.65236	−0.826181	−	0.563405i	$-0.809491\pi$
$5$	−3.69479	−1.65236	−0.826181	+	0.563405i	$0.809491\pi$
$6$	0	0
$7$	−1.23254	−0.465856	−0.232928	−	0.972494i	$-0.574831\pi$
$7$	−1.23254	−0.465856	−0.232928	+	0.972494i	$0.574831\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.313298	0.0944629	0.0472315	−	0.998884i	$-0.484960\pi$
$11$	0.313298	0.0944629	0.0472315	+	0.998884i	$0.484960\pi$
$12$	0	0
$13$	6.23784	1.73007	0.865033	−	0.501716i	$-0.167298\pi$
$13$	6.23784	1.73007	0.865033	+	0.501716i	$0.167298\pi$
$14$	0	0
$15$	−3.69479	−0.953991
$16$	0	0
$17$	−6.66173	−1.61571	−0.807853	−	0.589384i	$-0.799371\pi$
$17$	−6.66173	−1.61571	−0.807853	+	0.589384i	$0.799371\pi$
$18$	0	0
$19$	0.170818	0.0391884	0.0195942	−	0.999808i	$-0.493763\pi$
$19$	0.170818	0.0391884	0.0195942	+	0.999808i	$0.493763\pi$
$20$	0	0
$21$	−1.23254	−0.268962
$22$	0	0
$23$	−1.48449	−0.309538	−0.154769	−	0.987951i	$-0.549463\pi$
$23$	−1.48449	−0.309538	−0.154769	+	0.987951i	$0.549463\pi$
$24$	0	0
$25$	8.65149	1.73030
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.35837	0.252243	0.126122	−	0.992015i	$-0.459747\pi$
$29$	1.35837	0.252243	0.126122	+	0.992015i	$0.459747\pi$
$30$	0	0
$31$	1.26428	0.227072	0.113536	−	0.993534i	$-0.463782\pi$
$31$	1.26428	0.227072	0.113536	+	0.993534i	$0.463782\pi$
$32$	0	0
$33$	0.313298	0.0545382
$34$	0	0
$35$	4.55398	0.769763
$36$	0	0
$37$	−1.64126	−0.269821	−0.134911	−	0.990858i	$-0.543075\pi$
$37$	−1.64126	−0.269821	−0.134911	+	0.990858i	$0.543075\pi$
$38$	0	0
$39$	6.23784	0.998854
$40$	0	0
$41$	0.180309	0.0281595	0.0140798	−	0.999901i	$-0.495518\pi$
$41$	0.180309	0.0281595	0.0140798	+	0.999901i	$0.495518\pi$
$42$	0	0
$43$	−0.765853	−0.116792	−0.0583958	−	0.998294i	$-0.518599\pi$
$43$	−0.765853	−0.116792	−0.0583958	+	0.998294i	$0.518599\pi$
$44$	0	0
$45$	−3.69479	−0.550787
$46$	0	0
$47$	−13.4564	−1.96281	−0.981405	−	0.191947i	$-0.938520\pi$
$47$	−13.4564	−1.96281	−0.981405	+	0.191947i	$0.938520\pi$
$48$	0	0
$49$	−5.48084	−0.782978
$50$	0	0
$51$	−6.66173	−0.932829
$52$	0	0
$53$	2.25751	0.310093	0.155047	−	0.987907i	$-0.450447\pi$
$53$	2.25751	0.310093	0.155047	+	0.987907i	$0.450447\pi$
$54$	0	0
$55$	−1.15757	−0.156087
$56$	0	0
$57$	0.170818	0.0226254
$58$	0	0
$59$	5.64566	0.735002	0.367501	−	0.930023i	$-0.380213\pi$
$59$	5.64566	0.735002	0.367501	+	0.930023i	$0.380213\pi$
$60$	0	0
$61$	6.33817	0.811519	0.405760	−	0.913980i	$-0.367007\pi$
$61$	6.33817	0.811519	0.405760	+	0.913980i	$0.367007\pi$
$62$	0	0
$63$	−1.23254	−0.155285
$64$	0	0
$65$	−23.0475	−2.85869
$66$	0	0
$67$	−9.49476	−1.15997	−0.579985	−	0.814627i	$-0.696941\pi$
$67$	−9.49476	−1.15997	−0.579985	+	0.814627i	$0.696941\pi$
$68$	0	0
$69$	−1.48449	−0.178712
$70$	0	0
$71$	−2.86093	−0.339529	−0.169765	−	0.985485i	$-0.554301\pi$
$71$	−2.86093	−0.339529	−0.169765	+	0.985485i	$0.554301\pi$
$72$	0	0
$73$	−2.30965	−0.270324	−0.135162	−	0.990823i	$-0.543156\pi$
$73$	−2.30965	−0.270324	−0.135162	+	0.990823i	$0.543156\pi$
$74$	0	0
$75$	8.65149	0.998988
$76$	0	0
$77$	−0.386153	−0.0440062
$78$	0	0
$79$	13.6931	1.54059	0.770295	−	0.637688i	$-0.220109\pi$
$79$	13.6931	1.54059	0.770295	+	0.637688i	$0.220109\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.61239	0.616040	0.308020	−	0.951380i	$-0.400334\pi$
$83$	5.61239	0.616040	0.308020	+	0.951380i	$0.400334\pi$
$84$	0	0
$85$	24.6137	2.66973
$86$	0	0
$87$	1.35837	0.145633
$88$	0	0
$89$	9.48087	1.00497	0.502485	−	0.864586i	$-0.332419\pi$
$89$	9.48087	1.00497	0.502485	+	0.864586i	$0.332419\pi$
$90$	0	0
$91$	−7.68839	−0.805962
$92$	0	0
$93$	1.26428	0.131100
$94$	0	0
$95$	−0.631137	−0.0647533
$96$	0	0
$97$	17.1972	1.74611	0.873056	−	0.487621i	$-0.162135\pi$
$97$	17.1972	1.74611	0.873056	+	0.487621i	$0.162135\pi$
$98$	0	0
$99$	0.313298	0.0314876
$100$	0	0
$101$	−0.183502	−0.0182592	−0.00912959	−	0.999958i	$-0.502906\pi$
$101$	−0.183502	−0.0182592	−0.00912959	+	0.999958i	$0.502906\pi$
$102$	0	0
$103$	13.6406	1.34405	0.672026	−	0.740528i	$-0.265424\pi$
$103$	13.6406	1.34405	0.672026	+	0.740528i	$0.265424\pi$
$104$	0	0
$105$	4.55398	0.444423
$106$	0	0
$107$	4.24174	0.410064	0.205032	−	0.978755i	$-0.434270\pi$
$107$	4.24174	0.410064	0.205032	+	0.978755i	$0.434270\pi$
$108$	0	0
$109$	12.4679	1.19421	0.597104	−	0.802164i	$-0.296318\pi$
$109$	12.4679	1.19421	0.597104	+	0.802164i	$0.296318\pi$
$110$	0	0
$111$	−1.64126	−0.155781
$112$	0	0
$113$	−16.6914	−1.57020	−0.785098	−	0.619371i	$-0.787388\pi$
$113$	−16.6914	−1.57020	−0.785098	+	0.619371i	$0.787388\pi$
$114$	0	0
$115$	5.48489	0.511469
$116$	0	0
$117$	6.23784	0.576688
$118$	0	0
$119$	8.21085	0.752687
$120$	0	0
$121$	−10.9018	−0.991077
$122$	0	0
$123$	0.180309	0.0162579
$124$	0	0
$125$	−13.4915	−1.20672
$126$	0	0
$127$	18.5639	1.64728	0.823639	−	0.567114i	$-0.191940\pi$
$127$	18.5639	1.64728	0.823639	+	0.567114i	$0.191940\pi$
$128$	0	0
$129$	−0.765853	−0.0674296
$130$	0	0
$131$	11.8256	1.03321	0.516603	−	0.856225i	$-0.327196\pi$
$131$	11.8256	1.03321	0.516603	+	0.856225i	$0.327196\pi$
$132$	0	0
$133$	−0.210540	−0.0182562
$134$	0	0
$135$	−3.69479	−0.317997
$136$	0	0
$137$	−12.3583	−1.05584	−0.527918	−	0.849295i	$-0.677027\pi$
$137$	−12.3583	−1.05584	−0.527918	+	0.849295i	$0.677027\pi$
$138$	0	0
$139$	−9.12703	−0.774144	−0.387072	−	0.922049i	$-0.626514\pi$
$139$	−9.12703	−0.774144	−0.387072	+	0.922049i	$0.626514\pi$
$140$	0	0
$141$	−13.4564	−1.13323
$142$	0	0
$143$	1.95430	0.163427
$144$	0	0
$145$	−5.01890	−0.416797
$146$	0	0
$147$	−5.48084	−0.452052
$148$	0	0
$149$	11.4050	0.934336	0.467168	−	0.884169i	$-0.345274\pi$
$149$	11.4050	0.934336	0.467168	+	0.884169i	$0.345274\pi$
$150$	0	0
$151$	6.32251	0.514519	0.257259	−	0.966342i	$-0.417181\pi$
$151$	6.32251	0.514519	0.257259	+	0.966342i	$0.417181\pi$
$152$	0	0
$153$	−6.66173	−0.538569
$154$	0	0
$155$	−4.67127	−0.375205
$156$	0	0
$157$	−23.1612	−1.84846	−0.924231	−	0.381834i	$-0.875293\pi$
$157$	−23.1612	−1.84846	−0.924231	+	0.381834i	$0.875293\pi$
$158$	0	0
$159$	2.25751	0.179032
$160$	0	0
$161$	1.82970	0.144200
$162$	0	0
$163$	−20.6081	−1.61415	−0.807076	−	0.590447i	$-0.798951\pi$
$163$	−20.6081	−1.61415	−0.807076	+	0.590447i	$0.798951\pi$
$164$	0	0
$165$	−1.15757	−0.0901168
$166$	0	0
$167$	24.5066	1.89638	0.948190	−	0.317703i	$-0.102911\pi$
$167$	24.5066	1.89638	0.948190	+	0.317703i	$0.102911\pi$
$168$	0	0
$169$	25.9106	1.99312
$170$	0	0
$171$	0.170818	0.0130628
$172$	0	0
$173$	−1.50511	−0.114432	−0.0572158	−	0.998362i	$-0.518222\pi$
$173$	−1.50511	−0.114432	−0.0572158	+	0.998362i	$0.518222\pi$
$174$	0	0
$175$	−10.6633	−0.806070
$176$	0	0
$177$	5.64566	0.424354
$178$	0	0
$179$	22.0126	1.64530	0.822649	−	0.568550i	$-0.192495\pi$
$179$	22.0126	1.64530	0.822649	+	0.568550i	$0.192495\pi$
$180$	0	0
$181$	22.0702	1.64046	0.820231	−	0.572032i	$-0.193845\pi$
$181$	22.0702	1.64046	0.820231	+	0.572032i	$0.193845\pi$
$182$	0	0
$183$	6.33817	0.468531
$184$	0	0
$185$	6.06411	0.445842
$186$	0	0
$187$	−2.08711	−0.152624
$188$	0	0
$189$	−1.23254	−0.0896541
$190$	0	0
$191$	−0.633039	−0.0458051	−0.0229025	−	0.999738i	$-0.507291\pi$
$191$	−0.633039	−0.0458051	−0.0229025	+	0.999738i	$0.507291\pi$
$192$	0	0
$193$	8.73825	0.628993	0.314496	−	0.949259i	$-0.398164\pi$
$193$	8.73825	0.628993	0.314496	+	0.949259i	$0.398164\pi$
$194$	0	0
$195$	−23.0475	−1.65047
$196$	0	0
$197$	23.0630	1.64317	0.821584	−	0.570088i	$-0.193091\pi$
$197$	23.0630	1.64317	0.821584	+	0.570088i	$0.193091\pi$
$198$	0	0
$199$	−7.34756	−0.520855	−0.260427	−	0.965493i	$-0.583863\pi$
$199$	−7.34756	−0.520855	−0.260427	+	0.965493i	$0.583863\pi$
$200$	0	0
$201$	−9.49476	−0.669709
$202$	0	0
$203$	−1.67425	−0.117509
$204$	0	0
$205$	−0.666204	−0.0465297
$206$	0	0
$207$	−1.48449	−0.103179
$208$	0	0
$209$	0.0535170	0.00370185
$210$	0	0
$211$	3.31331	0.228098	0.114049	−	0.993475i	$-0.463618\pi$
$211$	3.31331	0.228098	0.114049	+	0.993475i	$0.463618\pi$
$212$	0	0
$213$	−2.86093	−0.196027
$214$	0	0
$215$	2.82967	0.192982
$216$	0	0
$217$	−1.55828	−0.105783
$218$	0	0
$219$	−2.30965	−0.156072
$220$	0	0
$221$	−41.5548	−2.79528
$222$	0	0
$223$	−12.3079	−0.824199	−0.412100	−	0.911139i	$-0.635204\pi$
$223$	−12.3079	−0.824199	−0.412100	+	0.911139i	$0.635204\pi$
$224$	0	0
$225$	8.65149	0.576766
$226$	0	0
$227$	13.7350	0.911623	0.455811	−	0.890076i	$-0.349349\pi$
$227$	13.7350	0.911623	0.455811	+	0.890076i	$0.349349\pi$
$228$	0	0
$229$	26.1180	1.72592	0.862962	−	0.505269i	$-0.168607\pi$
$229$	26.1180	1.72592	0.862962	+	0.505269i	$0.168607\pi$
$230$	0	0
$231$	−0.386153	−0.0254070
$232$	0	0
$233$	−6.61492	−0.433358	−0.216679	−	0.976243i	$-0.569522\pi$
$233$	−6.61492	−0.433358	−0.216679	+	0.976243i	$0.569522\pi$
$234$	0	0
$235$	49.7184	3.24327
$236$	0	0
$237$	13.6931	0.889460
$238$	0	0
$239$	11.6101	0.750996	0.375498	−	0.926823i	$-0.377472\pi$
$239$	11.6101	0.750996	0.375498	+	0.926823i	$0.377472\pi$
$240$	0	0
$241$	6.47238	0.416923	0.208461	−	0.978031i	$-0.433154\pi$
$241$	6.47238	0.416923	0.208461	+	0.978031i	$0.433154\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	20.2506	1.29376
$246$	0	0
$247$	1.06554	0.0677984
$248$	0	0
$249$	5.61239	0.355671
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.465089	−0.0292399
$254$	0	0
$255$	24.6137	1.54137
$256$	0	0
$257$	4.66307	0.290875	0.145437	−	0.989367i	$-0.453541\pi$
$257$	4.66307	0.290875	0.145437	+	0.989367i	$0.453541\pi$
$258$	0	0
$259$	2.02292	0.125698
$260$	0	0
$261$	1.35837	0.0840811
$262$	0	0
$263$	28.2913	1.74452	0.872258	−	0.489046i	$-0.162655\pi$
$263$	28.2913	1.74452	0.872258	+	0.489046i	$0.162655\pi$
$264$	0	0
$265$	−8.34104	−0.512386
$266$	0	0
$267$	9.48087	0.580220
$268$	0	0
$269$	−12.0500	−0.734704	−0.367352	−	0.930082i	$-0.619736\pi$
$269$	−12.0500	−0.734704	−0.367352	+	0.930082i	$0.619736\pi$
$270$	0	0
$271$	2.31775	0.140794	0.0703968	−	0.997519i	$-0.477573\pi$
$271$	2.31775	0.140794	0.0703968	+	0.997519i	$0.477573\pi$
$272$	0	0
$273$	−7.68839	−0.465322
$274$	0	0
$275$	2.71050	0.163449
$276$	0	0
$277$	−2.53710	−0.152440	−0.0762199	−	0.997091i	$-0.524285\pi$
$277$	−2.53710	−0.152440	−0.0762199	+	0.997091i	$0.524285\pi$
$278$	0	0
$279$	1.26428	0.0756907
$280$	0	0
$281$	5.48547	0.327236	0.163618	−	0.986524i	$-0.447684\pi$
$281$	5.48547	0.327236	0.163618	+	0.986524i	$0.447684\pi$
$282$	0	0
$283$	−1.31767	−0.0783272	−0.0391636	−	0.999233i	$-0.512469\pi$
$283$	−1.31767	−0.0783272	−0.0391636	+	0.999233i	$0.512469\pi$
$284$	0	0
$285$	−0.631137	−0.0373854
$286$	0	0
$287$	−0.222238	−0.0131183
$288$	0	0
$289$	27.3786	1.61051
$290$	0	0
$291$	17.1972	1.00812
$292$	0	0
$293$	10.9497	0.639691	0.319845	−	0.947470i	$-0.396369\pi$
$293$	10.9497	0.639691	0.319845	+	0.947470i	$0.396369\pi$
$294$	0	0
$295$	−20.8595	−1.21449
$296$	0	0
$297$	0.313298	0.0181794
$298$	0	0
$299$	−9.26003	−0.535521
$300$	0	0
$301$	0.943945	0.0544081
$302$	0	0
$303$	−0.183502	−0.0105419
$304$	0	0
$305$	−23.4182	−1.34092
$306$	0	0
$307$	−14.0251	−0.800456	−0.400228	−	0.916416i	$-0.631069\pi$
$307$	−14.0251	−0.800456	−0.400228	+	0.916416i	$0.631069\pi$
$308$	0	0
$309$	13.6406	0.775989
$310$	0	0
$311$	−13.8046	−0.782784	−0.391392	−	0.920224i	$-0.628006\pi$
$311$	−13.8046	−0.782784	−0.391392	+	0.920224i	$0.628006\pi$
$312$	0	0
$313$	−13.4945	−0.762756	−0.381378	−	0.924419i	$-0.624550\pi$
$313$	−13.4945	−0.762756	−0.381378	+	0.924419i	$0.624550\pi$
$314$	0	0
$315$	4.55398	0.256588
$316$	0	0
$317$	28.4314	1.59686	0.798432	−	0.602085i	$-0.205663\pi$
$317$	28.4314	1.59686	0.798432	+	0.602085i	$0.205663\pi$
$318$	0	0
$319$	0.425576	0.0238277
$320$	0	0
$321$	4.24174	0.236751
$322$	0	0
$323$	−1.13794	−0.0633169
$324$	0	0
$325$	53.9666	2.99353
$326$	0	0
$327$	12.4679	0.689476
$328$	0	0
$329$	16.5855	0.914388
$330$	0	0
$331$	7.43854	0.408859	0.204429	−	0.978881i	$-0.434466\pi$
$331$	7.43854	0.408859	0.204429	+	0.978881i	$0.434466\pi$
$332$	0	0
$333$	−1.64126	−0.0899404
$334$	0	0
$335$	35.0812	1.91669
$336$	0	0
$337$	27.9276	1.52131	0.760655	−	0.649156i	$-0.224878\pi$
$337$	27.9276	1.52131	0.760655	+	0.649156i	$0.224878\pi$
$338$	0	0
$339$	−16.6914	−0.906554
$340$	0	0
$341$	0.396098	0.0214499
$342$	0	0
$343$	15.3831	0.830612
$344$	0	0
$345$	5.48489	0.295297
$346$	0	0
$347$	3.22493	0.173123	0.0865617	−	0.996246i	$-0.472412\pi$
$347$	3.22493	0.173123	0.0865617	+	0.996246i	$0.472412\pi$
$348$	0	0
$349$	−2.22899	−0.119315	−0.0596574	−	0.998219i	$-0.519001\pi$
$349$	−2.22899	−0.119315	−0.0596574	+	0.998219i	$0.519001\pi$
$350$	0	0
$351$	6.23784	0.332951
$352$	0	0
$353$	18.6138	0.990711	0.495356	−	0.868690i	$-0.335038\pi$
$353$	18.6138	0.990711	0.495356	+	0.868690i	$0.335038\pi$
$354$	0	0
$355$	10.5705	0.561025
$356$	0	0
$357$	8.21085	0.434564
$358$	0	0
$359$	−20.0189	−1.05656	−0.528279	−	0.849071i	$-0.677163\pi$
$359$	−20.0189	−1.05656	−0.528279	+	0.849071i	$0.677163\pi$
$360$	0	0
$361$	−18.9708	−0.998464
$362$	0	0
$363$	−10.9018	−0.572198
$364$	0	0
$365$	8.53368	0.446673
$366$	0	0
$367$	−24.0931	−1.25765	−0.628825	−	0.777547i	$-0.716464\pi$
$367$	−24.0931	−1.25765	−0.628825	+	0.777547i	$0.716464\pi$
$368$	0	0
$369$	0.180309	0.00938651
$370$	0	0
$371$	−2.78248	−0.144459
$372$	0	0
$373$	−11.0579	−0.572558	−0.286279	−	0.958146i	$-0.592418\pi$
$373$	−11.0579	−0.572558	−0.286279	+	0.958146i	$0.592418\pi$
$374$	0	0
$375$	−13.4915	−0.696697
$376$	0	0
$377$	8.47331	0.436397
$378$	0	0
$379$	2.68376	0.137856	0.0689279	−	0.997622i	$-0.478042\pi$
$379$	2.68376	0.137856	0.0689279	+	0.997622i	$0.478042\pi$
$380$	0	0
$381$	18.5639	0.951057
$382$	0	0
$383$	−18.8210	−0.961706	−0.480853	−	0.876801i	$-0.659673\pi$
$383$	−18.8210	−0.961706	−0.480853	+	0.876801i	$0.659673\pi$
$384$	0	0
$385$	1.42675	0.0727141
$386$	0	0
$387$	−0.765853	−0.0389305
$388$	0	0
$389$	−2.17566	−0.110310	−0.0551552	−	0.998478i	$-0.517565\pi$
$389$	−2.17566	−0.110310	−0.0551552	+	0.998478i	$0.517565\pi$
$390$	0	0
$391$	9.88929	0.500123
$392$	0	0
$393$	11.8256	0.596522
$394$	0	0
$395$	−50.5930	−2.54561
$396$	0	0
$397$	−10.6367	−0.533839	−0.266920	−	0.963719i	$-0.586006\pi$
$397$	−10.6367	−0.533839	−0.266920	+	0.963719i	$0.586006\pi$
$398$	0	0
$399$	−0.210540	−0.0105402
$400$	0	0
$401$	−6.38430	−0.318817	−0.159408	−	0.987213i	$-0.550959\pi$
$401$	−6.38430	−0.318817	−0.159408	+	0.987213i	$0.550959\pi$
$402$	0	0
$403$	7.88640	0.392850
$404$	0	0
$405$	−3.69479	−0.183596
$406$	0	0
$407$	−0.514203	−0.0254881
$408$	0	0
$409$	39.6769	1.96190	0.980949	−	0.194265i	$-0.0622320\pi$
$409$	39.6769	1.96190	0.980949	+	0.194265i	$0.0622320\pi$
$410$	0	0
$411$	−12.3583	−0.609588
$412$	0	0
$413$	−6.95850	−0.342405
$414$	0	0
$415$	−20.7366	−1.01792
$416$	0	0
$417$	−9.12703	−0.446952
$418$	0	0
$419$	−17.9507	−0.876949	−0.438475	−	0.898744i	$-0.644481\pi$
$419$	−17.9507	−0.876949	−0.438475	+	0.898744i	$0.644481\pi$
$420$	0	0
$421$	8.26079	0.402606	0.201303	−	0.979529i	$-0.435482\pi$
$421$	8.26079	0.402606	0.201303	+	0.979529i	$0.435482\pi$
$422$	0	0
$423$	−13.4564	−0.654270
$424$	0	0
$425$	−57.6339	−2.79565
$426$	0	0
$427$	−7.81205	−0.378052
$428$	0	0
$429$	1.95430	0.0943546
$430$	0	0
$431$	−30.0177	−1.44590	−0.722952	−	0.690898i	$-0.757215\pi$
$431$	−30.0177	−1.44590	−0.722952	+	0.690898i	$0.757215\pi$
$432$	0	0
$433$	−6.37088	−0.306165	−0.153082	−	0.988213i	$-0.548920\pi$
$433$	−6.37088	−0.306165	−0.153082	+	0.988213i	$0.548920\pi$
$434$	0	0
$435$	−5.01890	−0.240638
$436$	0	0
$437$	−0.253578	−0.0121303
$438$	0	0
$439$	28.1263	1.34240	0.671198	−	0.741278i	$-0.265780\pi$
$439$	28.1263	1.34240	0.671198	+	0.741278i	$0.265780\pi$
$440$	0	0
$441$	−5.48084	−0.260993
$442$	0	0
$443$	−30.1373	−1.43187	−0.715934	−	0.698168i	$-0.753999\pi$
$443$	−30.1373	−1.43187	−0.715934	+	0.698168i	$0.753999\pi$
$444$	0	0
$445$	−35.0298	−1.66057
$446$	0	0
$447$	11.4050	0.539439
$448$	0	0
$449$	32.0669	1.51333	0.756664	−	0.653804i	$-0.226828\pi$
$449$	32.0669	1.51333	0.756664	+	0.653804i	$0.226828\pi$
$450$	0	0
$451$	0.0564905	0.00266003
$452$	0	0
$453$	6.32251	0.297058
$454$	0	0
$455$	28.4070	1.33174
$456$	0	0
$457$	−1.75809	−0.0822400	−0.0411200	−	0.999154i	$-0.513093\pi$
$457$	−1.75809	−0.0822400	−0.0411200	+	0.999154i	$0.513093\pi$
$458$	0	0
$459$	−6.66173	−0.310943
$460$	0	0
$461$	8.53675	0.397596	0.198798	−	0.980040i	$-0.436296\pi$
$461$	8.53675	0.397596	0.198798	+	0.980040i	$0.436296\pi$
$462$	0	0
$463$	21.1385	0.982388	0.491194	−	0.871050i	$-0.336561\pi$
$463$	21.1385	0.982388	0.491194	+	0.871050i	$0.336561\pi$
$464$	0	0
$465$	−4.67127	−0.216625
$466$	0	0
$467$	−11.2275	−0.519549	−0.259774	−	0.965669i	$-0.583648\pi$
$467$	−11.2275	−0.519549	−0.259774	+	0.965669i	$0.583648\pi$
$468$	0	0
$469$	11.7027	0.540379
$470$	0	0
$471$	−23.1612	−1.06721
$472$	0	0
$473$	−0.239940	−0.0110325
$474$	0	0
$475$	1.47783	0.0678075
$476$	0	0
$477$	2.25751	0.103364
$478$	0	0
$479$	−14.5326	−0.664013	−0.332007	−	0.943277i	$-0.607726\pi$
$479$	−14.5326	−0.664013	−0.332007	+	0.943277i	$0.607726\pi$
$480$	0	0
$481$	−10.2379	−0.466808
$482$	0	0
$483$	1.82970	0.0832541
$484$	0	0
$485$	−63.5401	−2.88521
$486$	0	0
$487$	−3.73215	−0.169120	−0.0845599	−	0.996418i	$-0.526948\pi$
$487$	−3.73215	−0.169120	−0.0845599	+	0.996418i	$0.526948\pi$
$488$	0	0
$489$	−20.6081	−0.931931
$490$	0	0
$491$	9.14389	0.412658	0.206329	−	0.978483i	$-0.433848\pi$
$491$	9.14389	0.412658	0.206329	+	0.978483i	$0.433848\pi$
$492$	0	0
$493$	−9.04911	−0.407551
$494$	0	0
$495$	−1.15757	−0.0520290
$496$	0	0
$497$	3.52621	0.158172
$498$	0	0
$499$	9.88385	0.442462	0.221231	−	0.975221i	$-0.428993\pi$
$499$	9.88385	0.442462	0.221231	+	0.975221i	$0.428993\pi$
$500$	0	0
$501$	24.5066	1.09488
$502$	0	0
$503$	14.0939	0.628417	0.314208	−	0.949354i	$-0.398261\pi$
$503$	14.0939	0.628417	0.314208	+	0.949354i	$0.398261\pi$
$504$	0	0
$505$	0.678003	0.0301707
$506$	0	0
$507$	25.9106	1.15073
$508$	0	0
$509$	11.9811	0.531054	0.265527	−	0.964103i	$-0.414454\pi$
$509$	11.9811	0.531054	0.265527	+	0.964103i	$0.414454\pi$
$510$	0	0
$511$	2.84674	0.125932
$512$	0	0
$513$	0.170818	0.00754180
$514$	0	0
$515$	−50.3993	−2.22086
$516$	0	0
$517$	−4.21585	−0.185413
$518$	0	0
$519$	−1.50511	−0.0660671
$520$	0	0
$521$	43.3669	1.89994	0.949969	−	0.312344i	$-0.101114\pi$
$521$	43.3669	1.89994	0.949969	+	0.312344i	$0.101114\pi$
$522$	0	0
$523$	−23.5307	−1.02893	−0.514464	−	0.857512i	$-0.672009\pi$
$523$	−23.5307	−1.02893	−0.514464	+	0.857512i	$0.672009\pi$
$524$	0	0
$525$	−10.6633	−0.465385
$526$	0	0
$527$	−8.42232	−0.366882
$528$	0	0
$529$	−20.7963	−0.904186
$530$	0	0
$531$	5.64566	0.245001
$532$	0	0
$533$	1.12474	0.0487178
$534$	0	0
$535$	−15.6723	−0.677575
$536$	0	0
$537$	22.0126	0.949913
$538$	0	0
$539$	−1.71714	−0.0739624
$540$	0	0
$541$	11.4713	0.493188	0.246594	−	0.969119i	$-0.420689\pi$
$541$	11.4713	0.493188	0.246594	+	0.969119i	$0.420689\pi$
$542$	0	0
$543$	22.0702	0.947122
$544$	0	0
$545$	−46.0662	−1.97326
$546$	0	0
$547$	−41.1695	−1.76028	−0.880141	−	0.474713i	$-0.842552\pi$
$547$	−41.1695	−1.76028	−0.880141	+	0.474713i	$0.842552\pi$
$548$	0	0
$549$	6.33817	0.270506
$550$	0	0
$551$	0.232035	0.00988501
$552$	0	0
$553$	−16.8772	−0.717693
$554$	0	0
$555$	6.06411	0.257407
$556$	0	0
$557$	−35.8849	−1.52049	−0.760245	−	0.649636i	$-0.774921\pi$
$557$	−35.8849	−1.52049	−0.760245	+	0.649636i	$0.774921\pi$
$558$	0	0
$559$	−4.77727	−0.202057
$560$	0	0
$561$	−2.08711	−0.0881177
$562$	0	0
$563$	−31.0135	−1.30706	−0.653532	−	0.756899i	$-0.726714\pi$
$563$	−31.0135	−1.30706	−0.653532	+	0.756899i	$0.726714\pi$
$564$	0	0
$565$	61.6713	2.59453
$566$	0	0
$567$	−1.23254	−0.0517618
$568$	0	0
$569$	−1.85351	−0.0777031	−0.0388515	−	0.999245i	$-0.512370\pi$
$569$	−1.85351	−0.0777031	−0.0388515	+	0.999245i	$0.512370\pi$
$570$	0	0
$571$	−30.5843	−1.27991	−0.639957	−	0.768411i	$-0.721048\pi$
$571$	−30.5843	−1.27991	−0.639957	+	0.768411i	$0.721048\pi$
$572$	0	0
$573$	−0.633039	−0.0264456
$574$	0	0
$575$	−12.8431	−0.535593
$576$	0	0
$577$	35.2752	1.46853	0.734263	−	0.678865i	$-0.237528\pi$
$577$	35.2752	1.46853	0.734263	+	0.678865i	$0.237528\pi$
$578$	0	0
$579$	8.73825	0.363149
$580$	0	0
$581$	−6.91750	−0.286986
$582$	0	0
$583$	0.707275	0.0292923
$584$	0	0
$585$	−23.0475	−0.952897
$586$	0	0
$587$	37.5078	1.54811	0.774056	−	0.633117i	$-0.218225\pi$
$587$	37.5078	1.54811	0.774056	+	0.633117i	$0.218225\pi$
$588$	0	0
$589$	0.215963	0.00889859
$590$	0	0
$591$	23.0630	0.948683
$592$	0	0
$593$	40.2741	1.65386	0.826929	−	0.562306i	$-0.190086\pi$
$593$	40.2741	1.65386	0.826929	+	0.562306i	$0.190086\pi$
$594$	0	0
$595$	−30.3374	−1.24371
$596$	0	0
$597$	−7.34756	−0.300716
$598$	0	0
$599$	−4.86736	−0.198875	−0.0994374	−	0.995044i	$-0.531704\pi$
$599$	−4.86736	−0.198875	−0.0994374	+	0.995044i	$0.531704\pi$
$600$	0	0
$601$	19.0396	0.776640	0.388320	−	0.921525i	$-0.373056\pi$
$601$	19.0396	0.776640	0.388320	+	0.921525i	$0.373056\pi$
$602$	0	0
$603$	−9.49476	−0.386656
$604$	0	0
$605$	40.2800	1.63762
$606$	0	0
$607$	26.9814	1.09514	0.547570	−	0.836760i	$-0.315553\pi$
$607$	26.9814	1.09514	0.547570	+	0.836760i	$0.315553\pi$
$608$	0	0
$609$	−1.67425	−0.0678440
$610$	0	0
$611$	−83.9385	−3.39579
$612$	0	0
$613$	7.47153	0.301772	0.150886	−	0.988551i	$-0.451787\pi$
$613$	7.47153	0.301772	0.150886	+	0.988551i	$0.451787\pi$
$614$	0	0
$615$	−0.666204	−0.0268639
$616$	0	0
$617$	−36.7456	−1.47932	−0.739661	−	0.672980i	$-0.765014\pi$
$617$	−36.7456	−1.47932	−0.739661	+	0.672980i	$0.765014\pi$
$618$	0	0
$619$	−28.6055	−1.14975	−0.574877	−	0.818240i	$-0.694950\pi$
$619$	−28.6055	−1.14975	−0.574877	+	0.818240i	$0.694950\pi$
$620$	0	0
$621$	−1.48449	−0.0595707
$622$	0	0
$623$	−11.6855	−0.468172
$624$	0	0
$625$	6.59079	0.263632
$626$	0	0
$627$	0.0535170	0.00213726
$628$	0	0
$629$	10.9336	0.435952
$630$	0	0
$631$	−4.92365	−0.196008	−0.0980038	−	0.995186i	$-0.531246\pi$
$631$	−4.92365	−0.196008	−0.0980038	+	0.995186i	$0.531246\pi$
$632$	0	0
$633$	3.31331	0.131692
$634$	0	0
$635$	−68.5897	−2.72190
$636$	0	0
$637$	−34.1886	−1.35460
$638$	0	0
$639$	−2.86093	−0.113176
$640$	0	0
$641$	−17.4185	−0.687988	−0.343994	−	0.938972i	$-0.611780\pi$
$641$	−17.4185	−0.687988	−0.343994	+	0.938972i	$0.611780\pi$
$642$	0	0
$643$	−34.3904	−1.35623	−0.678113	−	0.734957i	$-0.737202\pi$
$643$	−34.3904	−1.35623	−0.678113	+	0.734957i	$0.737202\pi$
$644$	0	0
$645$	2.82967	0.111418
$646$	0	0
$647$	2.09858	0.0825037	0.0412519	−	0.999149i	$-0.486865\pi$
$647$	2.09858	0.0825037	0.0412519	+	0.999149i	$0.486865\pi$
$648$	0	0
$649$	1.76877	0.0694305
$650$	0	0
$651$	−1.55828	−0.0610739
$652$	0	0
$653$	14.8464	0.580986	0.290493	−	0.956877i	$-0.406181\pi$
$653$	14.8464	0.580986	0.290493	+	0.956877i	$0.406181\pi$
$654$	0	0
$655$	−43.6931	−1.70723
$656$	0	0
$657$	−2.30965	−0.0901081
$658$	0	0
$659$	−1.48161	−0.0577154	−0.0288577	−	0.999584i	$-0.509187\pi$
$659$	−1.48161	−0.0577154	−0.0288577	+	0.999584i	$0.509187\pi$
$660$	0	0
$661$	−9.78293	−0.380512	−0.190256	−	0.981735i	$-0.560932\pi$
$661$	−9.78293	−0.380512	−0.190256	+	0.981735i	$0.560932\pi$
$662$	0	0
$663$	−41.5548	−1.61385
$664$	0	0
$665$	0.777902	0.0301658
$666$	0	0
$667$	−2.01649	−0.0780790
$668$	0	0
$669$	−12.3079	−0.475852
$670$	0	0
$671$	1.98574	0.0766585
$672$	0	0
$673$	37.2178	1.43464	0.717320	−	0.696744i	$-0.245369\pi$
$673$	37.2178	1.43464	0.717320	+	0.696744i	$0.245369\pi$
$674$	0	0
$675$	8.65149	0.332996
$676$	0	0
$677$	36.9487	1.42005	0.710027	−	0.704175i	$-0.248683\pi$
$677$	36.9487	1.42005	0.710027	+	0.704175i	$0.248683\pi$
$678$	0	0
$679$	−21.1962	−0.813437
$680$	0	0
$681$	13.7350	0.526326
$682$	0	0
$683$	3.07554	0.117682	0.0588412	−	0.998267i	$-0.481259\pi$
$683$	3.07554	0.117682	0.0588412	+	0.998267i	$0.481259\pi$
$684$	0	0
$685$	45.6612	1.74462
$686$	0	0
$687$	26.1180	0.996462
$688$	0	0
$689$	14.0820	0.536482
$690$	0	0
$691$	−27.5570	−1.04832	−0.524159	−	0.851620i	$-0.675620\pi$
$691$	−27.5570	−1.04832	−0.524159	+	0.851620i	$0.675620\pi$
$692$	0	0
$693$	−0.386153	−0.0146687
$694$	0	0
$695$	33.7225	1.27917
$696$	0	0
$697$	−1.20117	−0.0454975
$698$	0	0
$699$	−6.61492	−0.250199
$700$	0	0
$701$	−21.7338	−0.820873	−0.410436	−	0.911889i	$-0.634624\pi$
$701$	−21.7338	−0.820873	−0.410436	+	0.911889i	$0.634624\pi$
$702$	0	0
$703$	−0.280357	−0.0105738
$704$	0	0
$705$	49.7184	1.87250
$706$	0	0
$707$	0.226174	0.00850615
$708$	0	0
$709$	21.8037	0.818857	0.409428	−	0.912342i	$-0.365728\pi$
$709$	21.8037	0.818857	0.409428	+	0.912342i	$0.365728\pi$
$710$	0	0
$711$	13.6931	0.513530
$712$	0	0
$713$	−1.87682	−0.0702875
$714$	0	0
$715$	−7.22074	−0.270041
$716$	0	0
$717$	11.6101	0.433588
$718$	0	0
$719$	−21.4457	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$719$	−21.4457	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$720$	0	0
$721$	−16.8126	−0.626135
$722$	0	0
$723$	6.47238	0.240711
$724$	0	0
$725$	11.7519	0.436456
$726$	0	0
$727$	0.0965648	0.00358139	0.00179070	−	0.999998i	$-0.499430\pi$
$727$	0.0965648	0.00358139	0.00179070	+	0.999998i	$0.499430\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.10191	0.188701
$732$	0	0
$733$	31.1343	1.14997	0.574986	−	0.818163i	$-0.305007\pi$
$733$	31.1343	1.14997	0.574986	+	0.818163i	$0.305007\pi$
$734$	0	0
$735$	20.2506	0.746954
$736$	0	0
$737$	−2.97469	−0.109574
$738$	0	0
$739$	−25.8313	−0.950221	−0.475110	−	0.879926i	$-0.657592\pi$
$739$	−25.8313	−0.950221	−0.475110	+	0.879926i	$0.657592\pi$
$740$	0	0
$741$	1.06554	0.0391434
$742$	0	0
$743$	18.3156	0.671934	0.335967	−	0.941874i	$-0.390937\pi$
$743$	18.3156	0.671934	0.335967	+	0.941874i	$0.390937\pi$
$744$	0	0
$745$	−42.1392	−1.54386
$746$	0	0
$747$	5.61239	0.205347
$748$	0	0
$749$	−5.22812	−0.191031
$750$	0	0
$751$	42.9921	1.56881	0.784403	−	0.620252i	$-0.212970\pi$
$751$	42.9921	1.56881	0.784403	+	0.620252i	$0.212970\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−23.3604	−0.850171
$756$	0	0
$757$	32.6232	1.18571	0.592855	−	0.805309i	$-0.298001\pi$
$757$	32.6232	1.18571	0.592855	+	0.805309i	$0.298001\pi$
$758$	0	0
$759$	−0.465089	−0.0168817
$760$	0	0
$761$	3.81487	0.138289	0.0691444	−	0.997607i	$-0.477973\pi$
$761$	3.81487	0.138289	0.0691444	+	0.997607i	$0.477973\pi$
$762$	0	0
$763$	−15.3672	−0.556329
$764$	0	0
$765$	24.6137	0.889910
$766$	0	0
$767$	35.2167	1.27160
$768$	0	0
$769$	−18.7778	−0.677146	−0.338573	−	0.940940i	$-0.609944\pi$
$769$	−18.7778	−0.677146	−0.338573	+	0.940940i	$0.609944\pi$
$770$	0	0
$771$	4.66307	0.167936
$772$	0	0
$773$	−38.5414	−1.38624	−0.693118	−	0.720824i	$-0.743764\pi$
$773$	−38.5414	−1.38624	−0.693118	+	0.720824i	$0.743764\pi$
$774$	0	0
$775$	10.9379	0.392903
$776$	0	0
$777$	2.02292	0.0725717
$778$	0	0
$779$	0.0308000	0.00110353
$780$	0	0
$781$	−0.896323	−0.0320730
$782$	0	0
$783$	1.35837	0.0485443
$784$	0	0
$785$	85.5757	3.05433
$786$	0	0
$787$	43.1050	1.53653	0.768264	−	0.640133i	$-0.221121\pi$
$787$	43.1050	1.53653	0.768264	+	0.640133i	$0.221121\pi$
$788$	0	0
$789$	28.2913	1.00720
$790$	0	0
$791$	20.5729	0.731486
$792$	0	0
$793$	39.5365	1.40398
$794$	0	0
$795$	−8.34104	−0.295826
$796$	0	0
$797$	−30.4182	−1.07747	−0.538734	−	0.842476i	$-0.681097\pi$
$797$	−30.4182	−1.07747	−0.538734	+	0.842476i	$0.681097\pi$
$798$	0	0
$799$	89.6426	3.17133
$800$	0	0
$801$	9.48087	0.334990
$802$	0	0
$803$	−0.723610	−0.0255356
$804$	0	0
$805$	−6.76035	−0.238271
$806$	0	0
$807$	−12.0500	−0.424182
$808$	0	0
$809$	11.7646	0.413622	0.206811	−	0.978381i	$-0.433691\pi$
$809$	11.7646	0.413622	0.206811	+	0.978381i	$0.433691\pi$
$810$	0	0
$811$	33.0109	1.15917	0.579584	−	0.814912i	$-0.303215\pi$
$811$	33.0109	1.15917	0.579584	+	0.814912i	$0.303215\pi$
$812$	0	0
$813$	2.31775	0.0812872
$814$	0	0
$815$	76.1427	2.66716
$816$	0	0
$817$	−0.130822	−0.00457687
$818$	0	0
$819$	−7.68839	−0.268654
$820$	0	0
$821$	−8.29198	−0.289392	−0.144696	−	0.989476i	$-0.546220\pi$
$821$	−8.29198	−0.289392	−0.144696	+	0.989476i	$0.546220\pi$
$822$	0	0
$823$	24.9283	0.868946	0.434473	−	0.900685i	$-0.356935\pi$
$823$	24.9283	0.868946	0.434473	+	0.900685i	$0.356935\pi$
$824$	0	0
$825$	2.71050	0.0943673
$826$	0	0
$827$	−39.1755	−1.36227	−0.681133	−	0.732160i	$-0.738512\pi$
$827$	−39.1755	−1.36227	−0.681133	+	0.732160i	$0.738512\pi$
$828$	0	0
$829$	0.919374	0.0319312	0.0159656	−	0.999873i	$-0.494918\pi$
$829$	0.919374	0.0319312	0.0159656	+	0.999873i	$0.494918\pi$
$830$	0	0
$831$	−2.53710	−0.0880111
$832$	0	0
$833$	36.5119	1.26506
$834$	0	0
$835$	−90.5469	−3.13351
$836$	0	0
$837$	1.26428	0.0437001
$838$	0	0
$839$	−24.8070	−0.856434	−0.428217	−	0.903676i	$-0.640858\pi$
$839$	−24.8070	−0.856434	−0.428217	+	0.903676i	$0.640858\pi$
$840$	0	0
$841$	−27.1548	−0.936373
$842$	0	0
$843$	5.48547	0.188930
$844$	0	0
$845$	−95.7344	−3.29336
$846$	0	0
$847$	13.4370	0.461700
$848$	0	0
$849$	−1.31767	−0.0452222
$850$	0	0
$851$	2.43644	0.0835199
$852$	0	0
$853$	22.6437	0.775305	0.387652	−	0.921806i	$-0.373286\pi$
$853$	22.6437	0.775305	0.387652	+	0.921806i	$0.373286\pi$
$854$	0	0
$855$	−0.631137	−0.0215844
$856$	0	0
$857$	13.5042	0.461295	0.230647	−	0.973037i	$-0.425916\pi$
$857$	13.5042	0.461295	0.230647	+	0.973037i	$0.425916\pi$
$858$	0	0
$859$	14.4540	0.493166	0.246583	−	0.969122i	$-0.420692\pi$
$859$	14.4540	0.493166	0.246583	+	0.969122i	$0.420692\pi$
$860$	0	0
$861$	−0.222238	−0.00757385
$862$	0	0
$863$	−20.2900	−0.690680	−0.345340	−	0.938478i	$-0.612236\pi$
$863$	−20.2900	−0.690680	−0.345340	+	0.938478i	$0.612236\pi$
$864$	0	0
$865$	5.56108	0.189082
$866$	0	0
$867$	27.3786	0.929827
$868$	0	0
$869$	4.29001	0.145529
$870$	0	0
$871$	−59.2268	−2.00682
$872$	0	0
$873$	17.1972	0.582037
$874$	0	0
$875$	16.6288	0.562156
$876$	0	0
$877$	4.13850	0.139747	0.0698736	−	0.997556i	$-0.477740\pi$
$877$	4.13850	0.139747	0.0698736	+	0.997556i	$0.477740\pi$
$878$	0	0
$879$	10.9497	0.369326
$880$	0	0
$881$	−6.29330	−0.212027	−0.106013	−	0.994365i	$-0.533809\pi$
$881$	−6.29330	−0.212027	−0.106013	+	0.994365i	$0.533809\pi$
$882$	0	0
$883$	−33.6906	−1.13378	−0.566890	−	0.823793i	$-0.691854\pi$
$883$	−33.6906	−1.13378	−0.566890	+	0.823793i	$0.691854\pi$
$884$	0	0
$885$	−20.8595	−0.701185
$886$	0	0
$887$	25.2227	0.846895	0.423447	−	0.905921i	$-0.360820\pi$
$887$	25.2227	0.846895	0.423447	+	0.905921i	$0.360820\pi$
$888$	0	0
$889$	−22.8807	−0.767395
$890$	0	0
$891$	0.313298	0.0104959
$892$	0	0
$893$	−2.29859	−0.0769193
$894$	0	0
$895$	−81.3319	−2.71863
$896$	0	0
$897$	−9.26003	−0.309183
$898$	0	0
$899$	1.71737	0.0572775
$900$	0	0
$901$	−15.0389	−0.501020
$902$	0	0
$903$	0.943945	0.0314125
$904$	0	0
$905$	−81.5447	−2.71064
$906$	0	0
$907$	−40.4511	−1.34316	−0.671578	−	0.740934i	$-0.734383\pi$
$907$	−40.4511	−1.34316	−0.671578	+	0.740934i	$0.734383\pi$
$908$	0	0
$909$	−0.183502	−0.00608639
$910$	0	0
$911$	49.4775	1.63926	0.819632	−	0.572891i	$-0.194178\pi$
$911$	49.4775	1.63926	0.819632	+	0.572891i	$0.194178\pi$
$912$	0	0
$913$	1.75835	0.0581929
$914$	0	0
$915$	−23.4182	−0.774182
$916$	0	0
$917$	−14.5755	−0.481326
$918$	0	0
$919$	−11.4930	−0.379120	−0.189560	−	0.981869i	$-0.560706\pi$
$919$	−11.4930	−0.379120	−0.189560	+	0.981869i	$0.560706\pi$
$920$	0	0
$921$	−14.0251	−0.462143
$922$	0	0
$923$	−17.8460	−0.587408
$924$	0	0
$925$	−14.1993	−0.466871
$926$	0	0
$927$	13.6406	0.448017
$928$	0	0
$929$	20.5556	0.674407	0.337203	−	0.941432i	$-0.390519\pi$
$929$	20.5556	0.674407	0.337203	+	0.941432i	$0.390519\pi$
$930$	0	0
$931$	−0.936227	−0.0306836
$932$	0	0
$933$	−13.8046	−0.451941
$934$	0	0
$935$	7.71143	0.252191
$936$	0	0
$937$	9.53660	0.311547	0.155774	−	0.987793i	$-0.450213\pi$
$937$	9.53660	0.311547	0.155774	+	0.987793i	$0.450213\pi$
$938$	0	0
$939$	−13.4945	−0.440377
$940$	0	0
$941$	−30.2008	−0.984516	−0.492258	−	0.870449i	$-0.663828\pi$
$941$	−30.2008	−0.984516	−0.492258	+	0.870449i	$0.663828\pi$
$942$	0	0
$943$	−0.267667	−0.00871645
$944$	0	0
$945$	4.55398	0.148141
$946$	0	0
$947$	−7.16738	−0.232908	−0.116454	−	0.993196i	$-0.537153\pi$
$947$	−7.16738	−0.232908	−0.116454	+	0.993196i	$0.537153\pi$
$948$	0	0
$949$	−14.4072	−0.467679
$950$	0	0
$951$	28.4314	0.921950
$952$	0	0
$953$	−3.74465	−0.121301	−0.0606505	−	0.998159i	$-0.519318\pi$
$953$	−3.74465	−0.121301	−0.0606505	+	0.998159i	$0.519318\pi$
$954$	0	0
$955$	2.33895	0.0756866
$956$	0	0
$957$	0.425576	0.0137569
$958$	0	0
$959$	15.2320	0.491868
$960$	0	0
$961$	−29.4016	−0.948438
$962$	0	0
$963$	4.24174	0.136688
$964$	0	0
$965$	−32.2860	−1.03932
$966$	0	0
$967$	26.3715	0.848050	0.424025	−	0.905651i	$-0.360617\pi$
$967$	26.3715	0.848050	0.424025	+	0.905651i	$0.360617\pi$
$968$	0	0
$969$	−1.13794	−0.0365560
$970$	0	0
$971$	−34.2929	−1.10051	−0.550256	−	0.834996i	$-0.685470\pi$
$971$	−34.2929	−1.10051	−0.550256	+	0.834996i	$0.685470\pi$
$972$	0	0
$973$	11.2494	0.360640
$974$	0	0
$975$	53.9666	1.72831
$976$	0	0
$977$	−22.9452	−0.734082	−0.367041	−	0.930205i	$-0.619629\pi$
$977$	−22.9452	−0.734082	−0.367041	+	0.930205i	$0.619629\pi$
$978$	0	0
$979$	2.97034	0.0949324
$980$	0	0
$981$	12.4679	0.398069
$982$	0	0
$983$	−31.4734	−1.00385	−0.501923	−	0.864912i	$-0.667374\pi$
$983$	−31.4734	−1.00385	−0.501923	+	0.864912i	$0.667374\pi$
$984$	0	0
$985$	−85.2128	−2.71511
$986$	0	0
$987$	16.5855	0.527922
$988$	0	0
$989$	1.13690	0.0361514
$990$	0	0
$991$	−10.0655	−0.319741	−0.159870	−	0.987138i	$-0.551108\pi$
$991$	−10.0655	−0.319741	−0.159870	+	0.987138i	$0.551108\pi$
$992$	0	0
$993$	7.43854	0.236055
$994$	0	0
$995$	27.1477	0.860640
$996$	0	0
$997$	−19.4249	−0.615192	−0.307596	−	0.951517i	$-0.599525\pi$
$997$	−19.4249	−0.615192	−0.307596	+	0.951517i	$0.599525\pi$
$998$	0	0
$999$	−1.64126	−0.0519271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.2	✓	20	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.69479 q^{5} -1.23254 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.69479 q^{5} -1.23254 q^{7} +1.00000 q^{9} +0.313298 q^{11} +6.23784 q^{13} -3.69479 q^{15} -6.66173 q^{17} +0.170818 q^{19} -1.23254 q^{21} -1.48449 q^{23} +8.65149 q^{25} +1.00000 q^{27} +1.35837 q^{29} +1.26428 q^{31} +0.313298 q^{33} +4.55398 q^{35} -1.64126 q^{37} +6.23784 q^{39} +0.180309 q^{41} -0.765853 q^{43} -3.69479 q^{45} -13.4564 q^{47} -5.48084 q^{49} -6.66173 q^{51} +2.25751 q^{53} -1.15757 q^{55} +0.170818 q^{57} +5.64566 q^{59} +6.33817 q^{61} -1.23254 q^{63} -23.0475 q^{65} -9.49476 q^{67} -1.48449 q^{69} -2.86093 q^{71} -2.30965 q^{73} +8.65149 q^{75} -0.386153 q^{77} +13.6931 q^{79} +1.00000 q^{81} +5.61239 q^{83} +24.6137 q^{85} +1.35837 q^{87} +9.48087 q^{89} -7.68839 q^{91} +1.26428 q^{93} -0.631137 q^{95} +17.1972 q^{97} +0.313298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9} + 4 q^{11} + 21 q^{13} + 9 q^{15} + 10 q^{17} + 8 q^{19} + 9 q^{21} + 9 q^{23} + 43 q^{25} + 20 q^{27} + 18 q^{29} + 27 q^{31} + 4 q^{33} - 7 q^{35} + 33 q^{37} + 21 q^{39} + 14 q^{41} - 6 q^{43} + 9 q^{45} + 21 q^{47} + 47 q^{49} + 10 q^{51} + 23 q^{53} + 24 q^{55} + 8 q^{57} + 10 q^{59} + 28 q^{61} + 9 q^{63} + 2 q^{65} + 15 q^{67} + 9 q^{69} + 33 q^{71} + 50 q^{73} + 43 q^{75} + 20 q^{77} + 17 q^{79} + 20 q^{81} - 19 q^{83} + 41 q^{85} + 18 q^{87} + 21 q^{89} + 30 q^{91} + 27 q^{93} + 27 q^{95} + 47 q^{97} + 4 q^{99}+O(q^{100}) \) 20 * q + 20 * q^3 + 9 * q^5 + 9 * q^7 + 20 * q^9 + 4 * q^11 + 21 * q^13 + 9 * q^15 + 10 * q^17 + 8 * q^19 + 9 * q^21 + 9 * q^23 + 43 * q^25 + 20 * q^27 + 18 * q^29 + 27 * q^31 + 4 * q^33 - 7 * q^35 + 33 * q^37 + 21 * q^39 + 14 * q^41 - 6 * q^43 + 9 * q^45 + 21 * q^47 + 47 * q^49 + 10 * q^51 + 23 * q^53 + 24 * q^55 + 8 * q^57 + 10 * q^59 + 28 * q^61 + 9 * q^63 + 2 * q^65 + 15 * q^67 + 9 * q^69 + 33 * q^71 + 50 * q^73 + 43 * q^75 + 20 * q^77 + 17 * q^79 + 20 * q^81 - 19 * q^83 + 41 * q^85 + 18 * q^87 + 21 * q^89 + 30 * q^91 + 27 * q^93 + 27 * q^95 + 47 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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