LMFDB - Embedded newform 6024.2.a.q.1.17

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 $ x^18 - x^17 - 57x^16 + 51x^15 + 1328x^14 - 1116x^13 - 16275x^12 + 13699x^11 + 112394x^10 - 101250x^9 - 432956x^8 + 439806x^7 + 844117x^6 - 1010785x^5 - 593552x^4 + 965980x^3 - 88040x^2 - 184992x + 44032
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$3.61923$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.61923 q^{5} +0.0319200 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.61923 q^{5} +0.0319200 q^{7} +1.00000 q^{9} -0.312672 q^{11} -2.87518 q^{13} +3.61923 q^{15} +2.30414 q^{17} -5.22777 q^{19} +0.0319200 q^{21} +3.65809 q^{23} +8.09881 q^{25} +1.00000 q^{27} -1.69461 q^{29} +6.24383 q^{31} -0.312672 q^{33} +0.115526 q^{35} -8.58373 q^{37} -2.87518 q^{39} +5.46981 q^{41} +5.43178 q^{43} +3.61923 q^{45} +10.8461 q^{47} -6.99898 q^{49} +2.30414 q^{51} -1.61988 q^{53} -1.13163 q^{55} -5.22777 q^{57} +14.6009 q^{59} +9.00904 q^{61} +0.0319200 q^{63} -10.4059 q^{65} +8.45909 q^{67} +3.65809 q^{69} +3.00139 q^{71} -12.2053 q^{73} +8.09881 q^{75} -0.00998047 q^{77} +2.62332 q^{79} +1.00000 q^{81} +7.13603 q^{83} +8.33920 q^{85} -1.69461 q^{87} +5.64037 q^{89} -0.0917756 q^{91} +6.24383 q^{93} -18.9205 q^{95} +4.88534 q^{97} -0.312672 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * 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
$5$	3.61923	1.61857	0.809284	−	0.587417i	$-0.199855\pi$
$5$	3.61923	1.61857	0.809284	+	0.587417i	$0.199855\pi$
$6$	0	0
$7$	0.0319200	0.0120646	0.00603231	−	0.999982i	$-0.498080\pi$
$7$	0.0319200	0.0120646	0.00603231	+	0.999982i	$0.498080\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.312672	−0.0942741	−0.0471370	−	0.998888i	$-0.515010\pi$
$11$	−0.312672	−0.0942741	−0.0471370	+	0.998888i	$0.515010\pi$
$12$	0	0
$13$	−2.87518	−0.797431	−0.398715	−	0.917075i	$-0.630544\pi$
$13$	−2.87518	−0.797431	−0.398715	+	0.917075i	$0.630544\pi$
$14$	0	0
$15$	3.61923	0.934481
$16$	0	0
$17$	2.30414	0.558835	0.279418	−	0.960170i	$-0.409859\pi$
$17$	2.30414	0.558835	0.279418	+	0.960170i	$0.409859\pi$
$18$	0	0
$19$	−5.22777	−1.19933	−0.599667	−	0.800250i	$-0.704700\pi$
$19$	−5.22777	−1.19933	−0.599667	+	0.800250i	$0.704700\pi$
$20$	0	0
$21$	0.0319200	0.00696551
$22$	0	0
$23$	3.65809	0.762765	0.381383	−	0.924417i	$-0.375448\pi$
$23$	3.65809	0.762765	0.381383	+	0.924417i	$0.375448\pi$
$24$	0	0
$25$	8.09881	1.61976
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.69461	−0.314681	−0.157340	−	0.987544i	$-0.550292\pi$
$29$	−1.69461	−0.314681	−0.157340	+	0.987544i	$0.550292\pi$
$30$	0	0
$31$	6.24383	1.12142	0.560712	−	0.828011i	$-0.310527\pi$
$31$	6.24383	1.12142	0.560712	+	0.828011i	$0.310527\pi$
$32$	0	0
$33$	−0.312672	−0.0544292
$34$	0	0
$35$	0.115526	0.0195274
$36$	0	0
$37$	−8.58373	−1.41116	−0.705578	−	0.708632i	$-0.749313\pi$
$37$	−8.58373	−1.41116	−0.705578	+	0.708632i	$0.749313\pi$
$38$	0	0
$39$	−2.87518	−0.460397
$40$	0	0
$41$	5.46981	0.854241	0.427120	−	0.904195i	$-0.359528\pi$
$41$	5.46981	0.854241	0.427120	+	0.904195i	$0.359528\pi$
$42$	0	0
$43$	5.43178	0.828339	0.414169	−	0.910200i	$-0.364072\pi$
$43$	5.43178	0.828339	0.414169	+	0.910200i	$0.364072\pi$
$44$	0	0
$45$	3.61923	0.539523
$46$	0	0
$47$	10.8461	1.58207	0.791035	−	0.611770i	$-0.209542\pi$
$47$	10.8461	1.58207	0.791035	+	0.611770i	$0.209542\pi$
$48$	0	0
$49$	−6.99898	−0.999854
$50$	0	0
$51$	2.30414	0.322644
$52$	0	0
$53$	−1.61988	−0.222508	−0.111254	−	0.993792i	$-0.535487\pi$
$53$	−1.61988	−0.222508	−0.111254	+	0.993792i	$0.535487\pi$
$54$	0	0
$55$	−1.13163	−0.152589
$56$	0	0
$57$	−5.22777	−0.692436
$58$	0	0
$59$	14.6009	1.90087	0.950437	−	0.310918i	$-0.100636\pi$
$59$	14.6009	1.90087	0.950437	+	0.310918i	$0.100636\pi$
$60$	0	0
$61$	9.00904	1.15349	0.576745	−	0.816924i	$-0.304323\pi$
$61$	9.00904	1.15349	0.576745	+	0.816924i	$0.304323\pi$
$62$	0	0
$63$	0.0319200	0.00402154
$64$	0	0
$65$	−10.4059	−1.29070
$66$	0	0
$67$	8.45909	1.03344	0.516721	−	0.856154i	$-0.327152\pi$
$67$	8.45909	1.03344	0.516721	+	0.856154i	$0.327152\pi$
$68$	0	0
$69$	3.65809	0.440383
$70$	0	0
$71$	3.00139	0.356200	0.178100	−	0.984012i	$-0.443005\pi$
$71$	3.00139	0.356200	0.178100	+	0.984012i	$0.443005\pi$
$72$	0	0
$73$	−12.2053	−1.42852	−0.714261	−	0.699880i	$-0.753237\pi$
$73$	−12.2053	−1.42852	−0.714261	+	0.699880i	$0.753237\pi$
$74$	0	0
$75$	8.09881	0.935171
$76$	0	0
$77$	−0.00998047	−0.00113738
$78$	0	0
$79$	2.62332	0.295146	0.147573	−	0.989051i	$-0.452854\pi$
$79$	2.62332	0.295146	0.147573	+	0.989051i	$0.452854\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.13603	0.783281	0.391640	−	0.920118i	$-0.371908\pi$
$83$	7.13603	0.783281	0.391640	+	0.920118i	$0.371908\pi$
$84$	0	0
$85$	8.33920	0.904513
$86$	0	0
$87$	−1.69461	−0.181681
$88$	0	0
$89$	5.64037	0.597878	0.298939	−	0.954272i	$-0.403367\pi$
$89$	5.64037	0.597878	0.298939	+	0.954272i	$0.403367\pi$
$90$	0	0
$91$	−0.0917756	−0.00962069
$92$	0	0
$93$	6.24383	0.647455
$94$	0	0
$95$	−18.9205	−1.94120
$96$	0	0
$97$	4.88534	0.496031	0.248016	−	0.968756i	$-0.420222\pi$
$97$	4.88534	0.496031	0.248016	+	0.968756i	$0.420222\pi$
$98$	0	0
$99$	−0.312672	−0.0314247
$100$	0	0
$101$	15.7175	1.56395	0.781973	−	0.623313i	$-0.214214\pi$
$101$	15.7175	1.56395	0.781973	+	0.623313i	$0.214214\pi$
$102$	0	0
$103$	−13.4250	−1.32280	−0.661401	−	0.750032i	$-0.730038\pi$
$103$	−13.4250	−1.32280	−0.661401	+	0.750032i	$0.730038\pi$
$104$	0	0
$105$	0.115526	0.0112741
$106$	0	0
$107$	11.5777	1.11926	0.559632	−	0.828742i	$-0.310943\pi$
$107$	11.5777	1.11926	0.559632	+	0.828742i	$0.310943\pi$
$108$	0	0
$109$	19.1570	1.83491	0.917453	−	0.397845i	$-0.130242\pi$
$109$	19.1570	1.83491	0.917453	+	0.397845i	$0.130242\pi$
$110$	0	0
$111$	−8.58373	−0.814731
$112$	0	0
$113$	−3.06009	−0.287869	−0.143935	−	0.989587i	$-0.545976\pi$
$113$	−3.06009	−0.287869	−0.143935	+	0.989587i	$0.545976\pi$
$114$	0	0
$115$	13.2395	1.23459
$116$	0	0
$117$	−2.87518	−0.265810
$118$	0	0
$119$	0.0735480	0.00674213
$120$	0	0
$121$	−10.9022	−0.991112
$122$	0	0
$123$	5.46981	0.493196
$124$	0	0
$125$	11.2153	1.00313
$126$	0	0
$127$	3.33917	0.296303	0.148152	−	0.988965i	$-0.452668\pi$
$127$	3.33917	0.296303	0.148152	+	0.988965i	$0.452668\pi$
$128$	0	0
$129$	5.43178	0.478242
$130$	0	0
$131$	−5.96293	−0.520984	−0.260492	−	0.965476i	$-0.583885\pi$
$131$	−5.96293	−0.520984	−0.260492	+	0.965476i	$0.583885\pi$
$132$	0	0
$133$	−0.166870	−0.0144695
$134$	0	0
$135$	3.61923	0.311494
$136$	0	0
$137$	−4.35326	−0.371924	−0.185962	−	0.982557i	$-0.559540\pi$
$137$	−4.35326	−0.371924	−0.185962	+	0.982557i	$0.559540\pi$
$138$	0	0
$139$	3.71240	0.314882	0.157441	−	0.987528i	$-0.449676\pi$
$139$	3.71240	0.314882	0.157441	+	0.987528i	$0.449676\pi$
$140$	0	0
$141$	10.8461	0.913409
$142$	0	0
$143$	0.898987	0.0751770
$144$	0	0
$145$	−6.13317	−0.509332
$146$	0	0
$147$	−6.99898	−0.577266
$148$	0	0
$149$	−2.20913	−0.180979	−0.0904894	−	0.995897i	$-0.528843\pi$
$149$	−2.20913	−0.180979	−0.0904894	+	0.995897i	$0.528843\pi$
$150$	0	0
$151$	−1.07471	−0.0874590	−0.0437295	−	0.999043i	$-0.513924\pi$
$151$	−1.07471	−0.0874590	−0.0437295	+	0.999043i	$0.513924\pi$
$152$	0	0
$153$	2.30414	0.186278
$154$	0	0
$155$	22.5978	1.81510
$156$	0	0
$157$	−13.8800	−1.10774	−0.553872	−	0.832602i	$-0.686850\pi$
$157$	−13.8800	−1.10774	−0.553872	+	0.832602i	$0.686850\pi$
$158$	0	0
$159$	−1.61988	−0.128465
$160$	0	0
$161$	0.116766	0.00920247
$162$	0	0
$163$	11.3494	0.888956	0.444478	−	0.895790i	$-0.353389\pi$
$163$	11.3494	0.888956	0.444478	+	0.895790i	$0.353389\pi$
$164$	0	0
$165$	−1.13163	−0.0880973
$166$	0	0
$167$	21.4501	1.65986	0.829929	−	0.557869i	$-0.188381\pi$
$167$	21.4501	1.65986	0.829929	+	0.557869i	$0.188381\pi$
$168$	0	0
$169$	−4.73336	−0.364104
$170$	0	0
$171$	−5.22777	−0.399778
$172$	0	0
$173$	−19.7615	−1.50244	−0.751218	−	0.660054i	$-0.770533\pi$
$173$	−19.7615	−1.50244	−0.751218	+	0.660054i	$0.770533\pi$
$174$	0	0
$175$	0.258514	0.0195418
$176$	0	0
$177$	14.6009	1.09747
$178$	0	0
$179$	7.61796	0.569393	0.284696	−	0.958618i	$-0.408107\pi$
$179$	7.61796	0.569393	0.284696	+	0.958618i	$0.408107\pi$
$180$	0	0
$181$	5.37630	0.399617	0.199809	−	0.979835i	$-0.435968\pi$
$181$	5.37630	0.399617	0.199809	+	0.979835i	$0.435968\pi$
$182$	0	0
$183$	9.00904	0.665968
$184$	0	0
$185$	−31.0665	−2.28405
$186$	0	0
$187$	−0.720439	−0.0526837
$188$	0	0
$189$	0.0319200	0.00232184
$190$	0	0
$191$	21.2209	1.53549	0.767744	−	0.640757i	$-0.221379\pi$
$191$	21.2209	1.53549	0.767744	+	0.640757i	$0.221379\pi$
$192$	0	0
$193$	−16.7930	−1.20879	−0.604393	−	0.796686i	$-0.706584\pi$
$193$	−16.7930	−1.20879	−0.604393	+	0.796686i	$0.706584\pi$
$194$	0	0
$195$	−10.4059	−0.745184
$196$	0	0
$197$	−6.76278	−0.481828	−0.240914	−	0.970546i	$-0.577447\pi$
$197$	−6.76278	−0.481828	−0.240914	+	0.970546i	$0.577447\pi$
$198$	0	0
$199$	−19.4789	−1.38082	−0.690411	−	0.723417i	$-0.742570\pi$
$199$	−19.4789	−1.38082	−0.690411	+	0.723417i	$0.742570\pi$
$200$	0	0
$201$	8.45909	0.596658
$202$	0	0
$203$	−0.0540918	−0.00379650
$204$	0	0
$205$	19.7965	1.38265
$206$	0	0
$207$	3.65809	0.254255
$208$	0	0
$209$	1.63458	0.113066
$210$	0	0
$211$	−13.7976	−0.949864	−0.474932	−	0.880022i	$-0.657527\pi$
$211$	−13.7976	−0.949864	−0.474932	+	0.880022i	$0.657527\pi$
$212$	0	0
$213$	3.00139	0.205652
$214$	0	0
$215$	19.6589	1.34072
$216$	0	0
$217$	0.199303	0.0135296
$218$	0	0
$219$	−12.2053	−0.824757
$220$	0	0
$221$	−6.62480	−0.445633
$222$	0	0
$223$	−5.45235	−0.365117	−0.182558	−	0.983195i	$-0.558438\pi$
$223$	−5.45235	−0.365117	−0.182558	+	0.983195i	$0.558438\pi$
$224$	0	0
$225$	8.09881	0.539921
$226$	0	0
$227$	−24.7784	−1.64460	−0.822299	−	0.569055i	$-0.807309\pi$
$227$	−24.7784	−1.64460	−0.822299	+	0.569055i	$0.807309\pi$
$228$	0	0
$229$	−27.0729	−1.78903	−0.894514	−	0.447041i	$-0.852478\pi$
$229$	−27.0729	−1.78903	−0.894514	+	0.447041i	$0.852478\pi$
$230$	0	0
$231$	−0.00998047	−0.000656667 0
$232$	0	0
$233$	−5.78959	−0.379289	−0.189644	−	0.981853i	$-0.560734\pi$
$233$	−5.78959	−0.379289	−0.189644	+	0.981853i	$0.560734\pi$
$234$	0	0
$235$	39.2546	2.56069
$236$	0	0
$237$	2.62332	0.170403
$238$	0	0
$239$	−29.3455	−1.89820	−0.949100	−	0.314974i	$-0.898004\pi$
$239$	−29.3455	−1.89820	−0.949100	+	0.314974i	$0.898004\pi$
$240$	0	0
$241$	5.49426	0.353917	0.176958	−	0.984218i	$-0.443374\pi$
$241$	5.49426	0.353917	0.176958	+	0.984218i	$0.443374\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−25.3309	−1.61833
$246$	0	0
$247$	15.0308	0.956385
$248$	0	0
$249$	7.13603	0.452227
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−1.14378	−0.0719090
$254$	0	0
$255$	8.33920	0.522221
$256$	0	0
$257$	22.6505	1.41290	0.706451	−	0.707762i	$-0.250295\pi$
$257$	22.6505	1.41290	0.706451	+	0.707762i	$0.250295\pi$
$258$	0	0
$259$	−0.273992	−0.0170251
$260$	0	0
$261$	−1.69461	−0.104894
$262$	0	0
$263$	−19.7642	−1.21871	−0.609356	−	0.792897i	$-0.708572\pi$
$263$	−19.7642	−1.21871	−0.609356	+	0.792897i	$0.708572\pi$
$264$	0	0
$265$	−5.86272	−0.360144
$266$	0	0
$267$	5.64037	0.345185
$268$	0	0
$269$	3.79302	0.231265	0.115632	−	0.993292i	$-0.463111\pi$
$269$	3.79302	0.231265	0.115632	+	0.993292i	$0.463111\pi$
$270$	0	0
$271$	−22.6436	−1.37550	−0.687751	−	0.725947i	$-0.741402\pi$
$271$	−22.6436	−1.37550	−0.687751	+	0.725947i	$0.741402\pi$
$272$	0	0
$273$	−0.0917756	−0.00555451
$274$	0	0
$275$	−2.53227	−0.152702
$276$	0	0
$277$	11.6019	0.697089	0.348544	−	0.937292i	$-0.386676\pi$
$277$	11.6019	0.697089	0.348544	+	0.937292i	$0.386676\pi$
$278$	0	0
$279$	6.24383	0.373808
$280$	0	0
$281$	−4.79292	−0.285922	−0.142961	−	0.989728i	$-0.545662\pi$
$281$	−4.79292	−0.285922	−0.142961	+	0.989728i	$0.545662\pi$
$282$	0	0
$283$	−21.7215	−1.29121	−0.645605	−	0.763672i	$-0.723395\pi$
$283$	−21.7215	−1.29121	−0.645605	+	0.763672i	$0.723395\pi$
$284$	0	0
$285$	−18.9205	−1.12075
$286$	0	0
$287$	0.174596	0.0103061
$288$	0	0
$289$	−11.6910	−0.687703
$290$	0	0
$291$	4.88534	0.286384
$292$	0	0
$293$	6.54077	0.382116	0.191058	−	0.981579i	$-0.438808\pi$
$293$	6.54077	0.382116	0.191058	+	0.981579i	$0.438808\pi$
$294$	0	0
$295$	52.8439	3.07669
$296$	0	0
$297$	−0.312672	−0.0181431
$298$	0	0
$299$	−10.5177	−0.608252
$300$	0	0
$301$	0.173382	0.00999359
$302$	0	0
$303$	15.7175	0.902944
$304$	0	0
$305$	32.6058	1.86700
$306$	0	0
$307$	−17.3508	−0.990265	−0.495133	−	0.868817i	$-0.664881\pi$
$307$	−17.3508	−0.990265	−0.495133	+	0.868817i	$0.664881\pi$
$308$	0	0
$309$	−13.4250	−0.763721
$310$	0	0
$311$	−19.1714	−1.08711	−0.543555	−	0.839373i	$-0.682922\pi$
$311$	−19.1714	−1.08711	−0.543555	+	0.839373i	$0.682922\pi$
$312$	0	0
$313$	2.54517	0.143862	0.0719309	−	0.997410i	$-0.477084\pi$
$313$	2.54517	0.143862	0.0719309	+	0.997410i	$0.477084\pi$
$314$	0	0
$315$	0.115526	0.00650913
$316$	0	0
$317$	−21.1456	−1.18765	−0.593827	−	0.804592i	$-0.702384\pi$
$317$	−21.1456	−1.18765	−0.593827	+	0.804592i	$0.702384\pi$
$318$	0	0
$319$	0.529856	0.0296662
$320$	0	0
$321$	11.5777	0.646207
$322$	0	0
$323$	−12.0455	−0.670230
$324$	0	0
$325$	−23.2855	−1.29165
$326$	0	0
$327$	19.1570	1.05938
$328$	0	0
$329$	0.346208	0.0190871
$330$	0	0
$331$	−2.39249	−0.131503	−0.0657517	−	0.997836i	$-0.520945\pi$
$331$	−2.39249	−0.131503	−0.0657517	+	0.997836i	$0.520945\pi$
$332$	0	0
$333$	−8.58373	−0.470385
$334$	0	0
$335$	30.6154	1.67270
$336$	0	0
$337$	26.1592	1.42498	0.712490	−	0.701682i	$-0.247567\pi$
$337$	26.1592	1.42498	0.712490	+	0.701682i	$0.247567\pi$
$338$	0	0
$339$	−3.06009	−0.166201
$340$	0	0
$341$	−1.95227	−0.105721
$342$	0	0
$343$	−0.446847	−0.0241275
$344$	0	0
$345$	13.2395	0.712789
$346$	0	0
$347$	−6.28915	−0.337619	−0.168810	−	0.985649i	$-0.553992\pi$
$347$	−6.28915	−0.337619	−0.168810	+	0.985649i	$0.553992\pi$
$348$	0	0
$349$	23.2444	1.24425	0.622123	−	0.782919i	$-0.286270\pi$
$349$	23.2444	1.24425	0.622123	+	0.782919i	$0.286270\pi$
$350$	0	0
$351$	−2.87518	−0.153466
$352$	0	0
$353$	−25.8126	−1.37387	−0.686933	−	0.726720i	$-0.741044\pi$
$353$	−25.8126	−1.37387	−0.686933	+	0.726720i	$0.741044\pi$
$354$	0	0
$355$	10.8627	0.576533
$356$	0	0
$357$	0.0735480	0.00389257
$358$	0	0
$359$	8.38391	0.442486	0.221243	−	0.975219i	$-0.428989\pi$
$359$	8.38391	0.442486	0.221243	+	0.975219i	$0.428989\pi$
$360$	0	0
$361$	8.32962	0.438401
$362$	0	0
$363$	−10.9022	−0.572219
$364$	0	0
$365$	−44.1737	−2.31216
$366$	0	0
$367$	29.7159	1.55116	0.775579	−	0.631251i	$-0.217458\pi$
$367$	29.7159	1.55116	0.775579	+	0.631251i	$0.217458\pi$
$368$	0	0
$369$	5.46981	0.284747
$370$	0	0
$371$	−0.0517066	−0.00268447
$372$	0	0
$373$	31.4949	1.63075	0.815373	−	0.578937i	$-0.196532\pi$
$373$	31.4949	1.63075	0.815373	+	0.578937i	$0.196532\pi$
$374$	0	0
$375$	11.2153	0.579157
$376$	0	0
$377$	4.87230	0.250936
$378$	0	0
$379$	−7.01234	−0.360200	−0.180100	−	0.983648i	$-0.557642\pi$
$379$	−7.01234	−0.360200	−0.180100	+	0.983648i	$0.557642\pi$
$380$	0	0
$381$	3.33917	0.171071
$382$	0	0
$383$	−7.19461	−0.367627	−0.183814	−	0.982961i	$-0.558844\pi$
$383$	−7.19461	−0.367627	−0.183814	+	0.982961i	$0.558844\pi$
$384$	0	0
$385$	−0.0361216	−0.00184093
$386$	0	0
$387$	5.43178	0.276113
$388$	0	0
$389$	3.40093	0.172434	0.0862170	−	0.996276i	$-0.472522\pi$
$389$	3.40093	0.172434	0.0862170	+	0.996276i	$0.472522\pi$
$390$	0	0
$391$	8.42875	0.426260
$392$	0	0
$393$	−5.96293	−0.300790
$394$	0	0
$395$	9.49438	0.477714
$396$	0	0
$397$	32.2333	1.61774	0.808872	−	0.587985i	$-0.200078\pi$
$397$	32.2333	1.61774	0.808872	+	0.587985i	$0.200078\pi$
$398$	0	0
$399$	−0.166870	−0.00835397
$400$	0	0
$401$	−19.9370	−0.995608	−0.497804	−	0.867290i	$-0.665860\pi$
$401$	−19.9370	−0.995608	−0.497804	+	0.867290i	$0.665860\pi$
$402$	0	0
$403$	−17.9521	−0.894258
$404$	0	0
$405$	3.61923	0.179841
$406$	0	0
$407$	2.68389	0.133035
$408$	0	0
$409$	−29.7786	−1.47246	−0.736229	−	0.676733i	$-0.763395\pi$
$409$	−29.7786	−1.47246	−0.736229	+	0.676733i	$0.763395\pi$
$410$	0	0
$411$	−4.35326	−0.214730
$412$	0	0
$413$	0.466060	0.0229333
$414$	0	0
$415$	25.8269	1.26779
$416$	0	0
$417$	3.71240	0.181797
$418$	0	0
$419$	6.00865	0.293542	0.146771	−	0.989171i	$-0.453112\pi$
$419$	6.00865	0.293542	0.146771	+	0.989171i	$0.453112\pi$
$420$	0	0
$421$	8.56632	0.417497	0.208748	−	0.977969i	$-0.433061\pi$
$421$	8.56632	0.417497	0.208748	+	0.977969i	$0.433061\pi$
$422$	0	0
$423$	10.8461	0.527357
$424$	0	0
$425$	18.6608	0.905181
$426$	0	0
$427$	0.287568	0.0139164
$428$	0	0
$429$	0.898987	0.0434035
$430$	0	0
$431$	−2.67518	−0.128859	−0.0644294	−	0.997922i	$-0.520523\pi$
$431$	−2.67518	−0.128859	−0.0644294	+	0.997922i	$0.520523\pi$
$432$	0	0
$433$	−14.0916	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$433$	−14.0916	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$434$	0	0
$435$	−6.13317	−0.294063
$436$	0	0
$437$	−19.1237	−0.914810
$438$	0	0
$439$	−36.1060	−1.72325	−0.861623	−	0.507550i	$-0.830551\pi$
$439$	−36.1060	−1.72325	−0.861623	+	0.507550i	$0.830551\pi$
$440$	0	0
$441$	−6.99898	−0.333285
$442$	0	0
$443$	17.9050	0.850692	0.425346	−	0.905031i	$-0.360152\pi$
$443$	17.9050	0.850692	0.425346	+	0.905031i	$0.360152\pi$
$444$	0	0
$445$	20.4138	0.967707
$446$	0	0
$447$	−2.20913	−0.104488
$448$	0	0
$449$	5.87971	0.277481	0.138740	−	0.990329i	$-0.455695\pi$
$449$	5.87971	0.277481	0.138740	+	0.990329i	$0.455695\pi$
$450$	0	0
$451$	−1.71025	−0.0805327
$452$	0	0
$453$	−1.07471	−0.0504945
$454$	0	0
$455$	−0.332157	−0.0155717
$456$	0	0
$457$	−11.4603	−0.536089	−0.268044	−	0.963407i	$-0.586377\pi$
$457$	−11.4603	−0.536089	−0.268044	+	0.963407i	$0.586377\pi$
$458$	0	0
$459$	2.30414	0.107548
$460$	0	0
$461$	−34.9422	−1.62742	−0.813711	−	0.581270i	$-0.802556\pi$
$461$	−34.9422	−1.62742	−0.813711	+	0.581270i	$0.802556\pi$
$462$	0	0
$463$	10.2941	0.478407	0.239203	−	0.970969i	$-0.423114\pi$
$463$	10.2941	0.478407	0.239203	+	0.970969i	$0.423114\pi$
$464$	0	0
$465$	22.5978	1.04795
$466$	0	0
$467$	14.9569	0.692124	0.346062	−	0.938212i	$-0.387519\pi$
$467$	14.9569	0.692124	0.346062	+	0.938212i	$0.387519\pi$
$468$	0	0
$469$	0.270014	0.0124681
$470$	0	0
$471$	−13.8800	−0.639557
$472$	0	0
$473$	−1.69836	−0.0780909
$474$	0	0
$475$	−42.3388	−1.94264
$476$	0	0
$477$	−1.61988	−0.0741693
$478$	0	0
$479$	−4.32633	−0.197675	−0.0988376	−	0.995104i	$-0.531512\pi$
$479$	−4.32633	−0.197675	−0.0988376	+	0.995104i	$0.531512\pi$
$480$	0	0
$481$	24.6797	1.12530
$482$	0	0
$483$	0.116766	0.00531305
$484$	0	0
$485$	17.6812	0.802861
$486$	0	0
$487$	−36.4186	−1.65028	−0.825142	−	0.564926i	$-0.808905\pi$
$487$	−36.4186	−1.65028	−0.825142	+	0.564926i	$0.808905\pi$
$488$	0	0
$489$	11.3494	0.513239
$490$	0	0
$491$	−5.00968	−0.226084	−0.113042	−	0.993590i	$-0.536059\pi$
$491$	−5.00968	−0.226084	−0.113042	+	0.993590i	$0.536059\pi$
$492$	0	0
$493$	−3.90461	−0.175855
$494$	0	0
$495$	−1.13163	−0.0508630
$496$	0	0
$497$	0.0958043	0.00429741
$498$	0	0
$499$	10.2599	0.459298	0.229649	−	0.973273i	$-0.426242\pi$
$499$	10.2599	0.459298	0.229649	+	0.973273i	$0.426242\pi$
$500$	0	0
$501$	21.4501	0.958320
$502$	0	0
$503$	−22.3454	−0.996330	−0.498165	−	0.867082i	$-0.665993\pi$
$503$	−22.3454	−0.996330	−0.498165	+	0.867082i	$0.665993\pi$
$504$	0	0
$505$	56.8851	2.53135
$506$	0	0
$507$	−4.73336	−0.210216
$508$	0	0
$509$	7.60135	0.336924	0.168462	−	0.985708i	$-0.446120\pi$
$509$	7.60135	0.336924	0.168462	+	0.985708i	$0.446120\pi$
$510$	0	0
$511$	−0.389593	−0.0172346
$512$	0	0
$513$	−5.22777	−0.230812
$514$	0	0
$515$	−48.5881	−2.14105
$516$	0	0
$517$	−3.39128	−0.149148
$518$	0	0
$519$	−19.7615	−0.867432
$520$	0	0
$521$	−17.8583	−0.782386	−0.391193	−	0.920309i	$-0.627937\pi$
$521$	−17.8583	−0.782386	−0.391193	+	0.920309i	$0.627937\pi$
$522$	0	0
$523$	24.1303	1.05515	0.527573	−	0.849510i	$-0.323102\pi$
$523$	24.1303	1.05515	0.527573	+	0.849510i	$0.323102\pi$
$524$	0	0
$525$	0.258514	0.0112825
$526$	0	0
$527$	14.3866	0.626692
$528$	0	0
$529$	−9.61835	−0.418189
$530$	0	0
$531$	14.6009	0.633625
$532$	0	0
$533$	−15.7267	−0.681198
$534$	0	0
$535$	41.9025	1.81160
$536$	0	0
$537$	7.61796	0.328739
$538$	0	0
$539$	2.18838	0.0942603
$540$	0	0
$541$	−37.5545	−1.61460	−0.807298	−	0.590144i	$-0.799071\pi$
$541$	−37.5545	−1.61460	−0.807298	+	0.590144i	$0.799071\pi$
$542$	0	0
$543$	5.37630	0.230719
$544$	0	0
$545$	69.3335	2.96992
$546$	0	0
$547$	−7.22318	−0.308841	−0.154420	−	0.988005i	$-0.549351\pi$
$547$	−7.22318	−0.308841	−0.154420	+	0.988005i	$0.549351\pi$
$548$	0	0
$549$	9.00904	0.384497
$550$	0	0
$551$	8.85903	0.377407
$552$	0	0
$553$	0.0837361	0.00356082
$554$	0	0
$555$	−31.0665	−1.31870
$556$	0	0
$557$	−14.1204	−0.598302	−0.299151	−	0.954206i	$-0.596703\pi$
$557$	−14.1204	−0.598302	−0.299151	+	0.954206i	$0.596703\pi$
$558$	0	0
$559$	−15.6173	−0.660543
$560$	0	0
$561$	−0.720439	−0.0304169
$562$	0	0
$563$	−33.3897	−1.40721	−0.703603	−	0.710593i	$-0.748427\pi$
$563$	−33.3897	−1.40721	−0.703603	+	0.710593i	$0.748427\pi$
$564$	0	0
$565$	−11.0752	−0.465936
$566$	0	0
$567$	0.0319200	0.00134051
$568$	0	0
$569$	41.8562	1.75470	0.877351	−	0.479848i	$-0.159308\pi$
$569$	41.8562	1.75470	0.877351	+	0.479848i	$0.159308\pi$
$570$	0	0
$571$	27.5034	1.15098	0.575490	−	0.817809i	$-0.304811\pi$
$571$	27.5034	1.15098	0.575490	+	0.817809i	$0.304811\pi$
$572$	0	0
$573$	21.2209	0.886515
$574$	0	0
$575$	29.6262	1.23550
$576$	0	0
$577$	−19.8613	−0.826836	−0.413418	−	0.910541i	$-0.635665\pi$
$577$	−19.8613	−0.826836	−0.413418	+	0.910541i	$0.635665\pi$
$578$	0	0
$579$	−16.7930	−0.697893
$580$	0	0
$581$	0.227782	0.00944998
$582$	0	0
$583$	0.506491	0.0209767
$584$	0	0
$585$	−10.4059	−0.430232
$586$	0	0
$587$	29.2283	1.20638	0.603191	−	0.797597i	$-0.293896\pi$
$587$	29.2283	1.20638	0.603191	+	0.797597i	$0.293896\pi$
$588$	0	0
$589$	−32.6413	−1.34496
$590$	0	0
$591$	−6.76278	−0.278184
$592$	0	0
$593$	17.1789	0.705455	0.352727	−	0.935726i	$-0.385254\pi$
$593$	17.1789	0.705455	0.352727	+	0.935726i	$0.385254\pi$
$594$	0	0
$595$	0.266187	0.0109126
$596$	0	0
$597$	−19.4789	−0.797218
$598$	0	0
$599$	31.4703	1.28584	0.642921	−	0.765933i	$-0.277722\pi$
$599$	31.4703	1.28584	0.642921	+	0.765933i	$0.277722\pi$
$600$	0	0
$601$	−19.1378	−0.780648	−0.390324	−	0.920678i	$-0.627637\pi$
$601$	−19.1378	−0.780648	−0.390324	+	0.920678i	$0.627637\pi$
$602$	0	0
$603$	8.45909	0.344481
$604$	0	0
$605$	−39.4577	−1.60418
$606$	0	0
$607$	−4.65500	−0.188941	−0.0944704	−	0.995528i	$-0.530116\pi$
$607$	−4.65500	−0.188941	−0.0944704	+	0.995528i	$0.530116\pi$
$608$	0	0
$609$	−0.0540918	−0.00219191
$610$	0	0
$611$	−31.1846	−1.26159
$612$	0	0
$613$	27.8725	1.12576	0.562881	−	0.826538i	$-0.309693\pi$
$613$	27.8725	1.12576	0.562881	+	0.826538i	$0.309693\pi$
$614$	0	0
$615$	19.7965	0.798271
$616$	0	0
$617$	−18.5186	−0.745530	−0.372765	−	0.927926i	$-0.621590\pi$
$617$	−18.5186	−0.745530	−0.372765	+	0.927926i	$0.621590\pi$
$618$	0	0
$619$	38.7627	1.55801	0.779003	−	0.627021i	$-0.215726\pi$
$619$	38.7627	1.55801	0.779003	+	0.627021i	$0.215726\pi$
$620$	0	0
$621$	3.65809	0.146794
$622$	0	0
$623$	0.180040	0.00721317
$624$	0	0
$625$	0.0967290	0.00386916
$626$	0	0
$627$	1.63458	0.0652787
$628$	0	0
$629$	−19.7781	−0.788604
$630$	0	0
$631$	15.3782	0.612198	0.306099	−	0.952000i	$-0.400976\pi$
$631$	15.3782	0.612198	0.306099	+	0.952000i	$0.400976\pi$
$632$	0	0
$633$	−13.7976	−0.548404
$634$	0	0
$635$	12.0852	0.479587
$636$	0	0
$637$	20.1233	0.797315
$638$	0	0
$639$	3.00139	0.118733
$640$	0	0
$641$	23.2717	0.919177	0.459588	−	0.888132i	$-0.347997\pi$
$641$	23.2717	0.919177	0.459588	+	0.888132i	$0.347997\pi$
$642$	0	0
$643$	33.8846	1.33628	0.668139	−	0.744036i	$-0.267091\pi$
$643$	33.8846	1.33628	0.668139	+	0.744036i	$0.267091\pi$
$644$	0	0
$645$	19.6589	0.774067
$646$	0	0
$647$	26.0798	1.02530	0.512651	−	0.858597i	$-0.328664\pi$
$647$	26.0798	1.02530	0.512651	+	0.858597i	$0.328664\pi$
$648$	0	0
$649$	−4.56528	−0.179203
$650$	0	0
$651$	0.199303	0.00781129
$652$	0	0
$653$	29.3986	1.15046	0.575229	−	0.817993i	$-0.304913\pi$
$653$	29.3986	1.15046	0.575229	+	0.817993i	$0.304913\pi$
$654$	0	0
$655$	−21.5812	−0.843248
$656$	0	0
$657$	−12.2053	−0.476174
$658$	0	0
$659$	48.4354	1.88677	0.943387	−	0.331694i	$-0.107620\pi$
$659$	48.4354	1.88677	0.943387	+	0.331694i	$0.107620\pi$
$660$	0	0
$661$	15.5149	0.603459	0.301730	−	0.953394i	$-0.402436\pi$
$661$	15.5149	0.603459	0.301730	+	0.953394i	$0.402436\pi$
$662$	0	0
$663$	−6.62480	−0.257286
$664$	0	0
$665$	−0.603942	−0.0234199
$666$	0	0
$667$	−6.19903	−0.240028
$668$	0	0
$669$	−5.45235	−0.210800
$670$	0	0
$671$	−2.81687	−0.108744
$672$	0	0
$673$	1.87515	0.0722819	0.0361410	−	0.999347i	$-0.488493\pi$
$673$	1.87515	0.0722819	0.0361410	+	0.999347i	$0.488493\pi$
$674$	0	0
$675$	8.09881	0.311724
$676$	0	0
$677$	4.39080	0.168752	0.0843762	−	0.996434i	$-0.473110\pi$
$677$	4.39080	0.168752	0.0843762	+	0.996434i	$0.473110\pi$
$678$	0	0
$679$	0.155940	0.00598443
$680$	0	0
$681$	−24.7784	−0.949509
$682$	0	0
$683$	−13.1717	−0.504001	−0.252001	−	0.967727i	$-0.581088\pi$
$683$	−13.1717	−0.504001	−0.252001	+	0.967727i	$0.581088\pi$
$684$	0	0
$685$	−15.7554	−0.601984
$686$	0	0
$687$	−27.0729	−1.03290
$688$	0	0
$689$	4.65745	0.177435
$690$	0	0
$691$	26.1849	0.996120	0.498060	−	0.867143i	$-0.334046\pi$
$691$	26.1849	0.996120	0.498060	+	0.867143i	$0.334046\pi$
$692$	0	0
$693$	−0.00998047	−0.000379127 0
$694$	0	0
$695$	13.4360	0.509657
$696$	0	0
$697$	12.6032	0.477380
$698$	0	0
$699$	−5.78959	−0.218982
$700$	0	0
$701$	−20.5455	−0.775993	−0.387996	−	0.921661i	$-0.626833\pi$
$701$	−20.5455	−0.775993	−0.387996	+	0.921661i	$0.626833\pi$
$702$	0	0
$703$	44.8738	1.69245
$704$	0	0
$705$	39.2546	1.47841
$706$	0	0
$707$	0.501701	0.0188684
$708$	0	0
$709$	20.2601	0.760885	0.380443	−	0.924805i	$-0.375772\pi$
$709$	20.2601	0.760885	0.380443	+	0.924805i	$0.375772\pi$
$710$	0	0
$711$	2.62332	0.0983820
$712$	0	0
$713$	22.8405	0.855384
$714$	0	0
$715$	3.25364	0.121679
$716$	0	0
$717$	−29.3455	−1.09593
$718$	0	0
$719$	−46.1268	−1.72024	−0.860119	−	0.510093i	$-0.829611\pi$
$719$	−46.1268	−1.72024	−0.860119	+	0.510093i	$0.829611\pi$
$720$	0	0
$721$	−0.428525	−0.0159591
$722$	0	0
$723$	5.49426	0.204334
$724$	0	0
$725$	−13.7243	−0.509708
$726$	0	0
$727$	−15.7097	−0.582641	−0.291320	−	0.956626i	$-0.594095\pi$
$727$	−15.7097	−0.582641	−0.291320	+	0.956626i	$0.594095\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.5156	0.462905
$732$	0	0
$733$	1.61491	0.0596480	0.0298240	−	0.999555i	$-0.490505\pi$
$733$	1.61491	0.0596480	0.0298240	+	0.999555i	$0.490505\pi$
$734$	0	0
$735$	−25.3309	−0.934345
$736$	0	0
$737$	−2.64492	−0.0974268
$738$	0	0
$739$	10.8363	0.398619	0.199310	−	0.979937i	$-0.436130\pi$
$739$	10.8363	0.398619	0.199310	+	0.979937i	$0.436130\pi$
$740$	0	0
$741$	15.0308	0.552169
$742$	0	0
$743$	−16.4611	−0.603898	−0.301949	−	0.953324i	$-0.597637\pi$
$743$	−16.4611	−0.603898	−0.301949	+	0.953324i	$0.597637\pi$
$744$	0	0
$745$	−7.99534	−0.292927
$746$	0	0
$747$	7.13603	0.261094
$748$	0	0
$749$	0.369561	0.0135035
$750$	0	0
$751$	−0.666445	−0.0243189	−0.0121595	−	0.999926i	$-0.503871\pi$
$751$	−0.666445	−0.0243189	−0.0121595	+	0.999926i	$0.503871\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	−3.88964	−0.141558
$756$	0	0
$757$	−5.18791	−0.188558	−0.0942790	−	0.995546i	$-0.530055\pi$
$757$	−5.18791	−0.188558	−0.0942790	+	0.995546i	$0.530055\pi$
$758$	0	0
$759$	−1.14378	−0.0415167
$760$	0	0
$761$	−10.1843	−0.369181	−0.184591	−	0.982815i	$-0.559096\pi$
$761$	−10.1843	−0.369181	−0.184591	+	0.982815i	$0.559096\pi$
$762$	0	0
$763$	0.611490	0.0221374
$764$	0	0
$765$	8.33920	0.301504
$766$	0	0
$767$	−41.9801	−1.51582
$768$	0	0
$769$	54.1609	1.95309	0.976545	−	0.215312i	$-0.0690768\pi$
$769$	54.1609	1.95309	0.976545	+	0.215312i	$0.0690768\pi$
$770$	0	0
$771$	22.6505	0.815739
$772$	0	0
$773$	41.3461	1.48711	0.743557	−	0.668672i	$-0.233137\pi$
$773$	41.3461	1.48711	0.743557	+	0.668672i	$0.233137\pi$
$774$	0	0
$775$	50.5676	1.81644
$776$	0	0
$777$	−0.273992	−0.00982942
$778$	0	0
$779$	−28.5949	−1.02452
$780$	0	0
$781$	−0.938450	−0.0335804
$782$	0	0
$783$	−1.69461	−0.0605603
$784$	0	0
$785$	−50.2349	−1.79296
$786$	0	0
$787$	31.6112	1.12682	0.563409	−	0.826178i	$-0.309490\pi$
$787$	31.6112	1.12682	0.563409	+	0.826178i	$0.309490\pi$
$788$	0	0
$789$	−19.7642	−0.703623
$790$	0	0
$791$	−0.0976781	−0.00347303
$792$	0	0
$793$	−25.9026	−0.919828
$794$	0	0
$795$	−5.86272	−0.207929
$796$	0	0
$797$	−9.07039	−0.321290	−0.160645	−	0.987012i	$-0.551357\pi$
$797$	−9.07039	−0.321290	−0.160645	+	0.987012i	$0.551357\pi$
$798$	0	0
$799$	24.9910	0.884117
$800$	0	0
$801$	5.64037	0.199293
$802$	0	0
$803$	3.81625	0.134673
$804$	0	0
$805$	0.422604	0.0148948
$806$	0	0
$807$	3.79302	0.133521
$808$	0	0
$809$	5.69535	0.200238	0.100119	−	0.994975i	$-0.468078\pi$
$809$	5.69535	0.200238	0.100119	+	0.994975i	$0.468078\pi$
$810$	0	0
$811$	5.83379	0.204852	0.102426	−	0.994741i	$-0.467340\pi$
$811$	5.83379	0.204852	0.102426	+	0.994741i	$0.467340\pi$
$812$	0	0
$813$	−22.6436	−0.794146
$814$	0	0
$815$	41.0762	1.43884
$816$	0	0
$817$	−28.3961	−0.993454
$818$	0	0
$819$	−0.0917756	−0.00320690
$820$	0	0
$821$	−50.9005	−1.77644	−0.888219	−	0.459420i	$-0.848057\pi$
$821$	−50.9005	−1.77644	−0.888219	+	0.459420i	$0.848057\pi$
$822$	0	0
$823$	11.7239	0.408668	0.204334	−	0.978901i	$-0.434497\pi$
$823$	11.7239	0.408668	0.204334	+	0.978901i	$0.434497\pi$
$824$	0	0
$825$	−2.53227	−0.0881623
$826$	0	0
$827$	−46.9637	−1.63309	−0.816544	−	0.577283i	$-0.804113\pi$
$827$	−46.9637	−1.63309	−0.816544	+	0.577283i	$0.804113\pi$
$828$	0	0
$829$	−29.0704	−1.00966	−0.504829	−	0.863219i	$-0.668444\pi$
$829$	−29.0704	−1.00966	−0.504829	+	0.863219i	$0.668444\pi$
$830$	0	0
$831$	11.6019	0.402464
$832$	0	0
$833$	−16.1266	−0.558754
$834$	0	0
$835$	77.6328	2.68659
$836$	0	0
$837$	6.24383	0.215818
$838$	0	0
$839$	−51.8653	−1.79059	−0.895294	−	0.445475i	$-0.853035\pi$
$839$	−51.8653	−1.79059	−0.895294	+	0.445475i	$0.853035\pi$
$840$	0	0
$841$	−26.1283	−0.900976
$842$	0	0
$843$	−4.79292	−0.165077
$844$	0	0
$845$	−17.1311	−0.589328
$846$	0	0
$847$	−0.347999	−0.0119574
$848$	0	0
$849$	−21.7215	−0.745480
$850$	0	0
$851$	−31.4001	−1.07638
$852$	0	0
$853$	7.55455	0.258663	0.129331	−	0.991601i	$-0.458717\pi$
$853$	7.55455	0.258663	0.129331	+	0.991601i	$0.458717\pi$
$854$	0	0
$855$	−18.9205	−0.647068
$856$	0	0
$857$	−55.0331	−1.87990	−0.939948	−	0.341319i	$-0.889126\pi$
$857$	−55.0331	−1.87990	−0.939948	+	0.341319i	$0.889126\pi$
$858$	0	0
$859$	−23.7928	−0.811800	−0.405900	−	0.913917i	$-0.633042\pi$
$859$	−23.7928	−0.811800	−0.405900	+	0.913917i	$0.633042\pi$
$860$	0	0
$861$	0.174596	0.00595022
$862$	0	0
$863$	−28.1670	−0.958816	−0.479408	−	0.877592i	$-0.659149\pi$
$863$	−28.1670	−0.958816	−0.479408	+	0.877592i	$0.659149\pi$
$864$	0	0
$865$	−71.5213	−2.43180
$866$	0	0
$867$	−11.6910	−0.397046
$868$	0	0
$869$	−0.820236	−0.0278246
$870$	0	0
$871$	−24.3214	−0.824099
$872$	0	0
$873$	4.88534	0.165344
$874$	0	0
$875$	0.357993	0.0121024
$876$	0	0
$877$	−45.9355	−1.55113	−0.775565	−	0.631267i	$-0.782535\pi$
$877$	−45.9355	−1.55113	−0.775565	+	0.631267i	$0.782535\pi$
$878$	0	0
$879$	6.54077	0.220615
$880$	0	0
$881$	40.5900	1.36751	0.683755	−	0.729712i	$-0.260346\pi$
$881$	40.5900	1.36751	0.683755	+	0.729712i	$0.260346\pi$
$882$	0	0
$883$	6.61802	0.222714	0.111357	−	0.993780i	$-0.464480\pi$
$883$	6.61802	0.222714	0.111357	+	0.993780i	$0.464480\pi$
$884$	0	0
$885$	52.8439	1.77633
$886$	0	0
$887$	−12.2848	−0.412483	−0.206241	−	0.978501i	$-0.566123\pi$
$887$	−12.2848	−0.412483	−0.206241	+	0.978501i	$0.566123\pi$
$888$	0	0
$889$	0.106586	0.00357478
$890$	0	0
$891$	−0.312672	−0.0104749
$892$	0	0
$893$	−56.7011	−1.89743
$894$	0	0
$895$	27.5711	0.921601
$896$	0	0
$897$	−10.5177	−0.351175
$898$	0	0
$899$	−10.5808	−0.352891
$900$	0	0
$901$	−3.73243	−0.124345
$902$	0	0
$903$	0.173382	0.00576980
$904$	0	0
$905$	19.4581	0.646808
$906$	0	0
$907$	16.6766	0.553739	0.276869	−	0.960908i	$-0.410703\pi$
$907$	16.6766	0.553739	0.276869	+	0.960908i	$0.410703\pi$
$908$	0	0
$909$	15.7175	0.521315
$910$	0	0
$911$	−9.36655	−0.310328	−0.155164	−	0.987889i	$-0.549591\pi$
$911$	−9.36655	−0.310328	−0.155164	+	0.987889i	$0.549591\pi$
$912$	0	0
$913$	−2.23123	−0.0738431
$914$	0	0
$915$	32.6058	1.07791
$916$	0	0
$917$	−0.190337	−0.00628547
$918$	0	0
$919$	−25.4442	−0.839325	−0.419663	−	0.907680i	$-0.637852\pi$
$919$	−25.4442	−0.839325	−0.419663	+	0.907680i	$0.637852\pi$
$920$	0	0
$921$	−17.3508	−0.571730
$922$	0	0
$923$	−8.62953	−0.284045
$924$	0	0
$925$	−69.5180	−2.28574
$926$	0	0
$927$	−13.4250	−0.440934
$928$	0	0
$929$	16.9992	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$929$	16.9992	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$930$	0	0
$931$	36.5891	1.19916
$932$	0	0
$933$	−19.1714	−0.627644
$934$	0	0
$935$	−2.60743	−0.0852721
$936$	0	0
$937$	14.9314	0.487787	0.243893	−	0.969802i	$-0.421575\pi$
$937$	14.9314	0.487787	0.243893	+	0.969802i	$0.421575\pi$
$938$	0	0
$939$	2.54517	0.0830586
$940$	0	0
$941$	−28.3535	−0.924298	−0.462149	−	0.886802i	$-0.652921\pi$
$941$	−28.3535	−0.924298	−0.462149	+	0.886802i	$0.652921\pi$
$942$	0	0
$943$	20.0091	0.651585
$944$	0	0
$945$	0.115526	0.00375805
$946$	0	0
$947$	50.8110	1.65114	0.825568	−	0.564302i	$-0.190855\pi$
$947$	50.8110	1.65114	0.825568	+	0.564302i	$0.190855\pi$
$948$	0	0
$949$	35.0924	1.13915
$950$	0	0
$951$	−21.1456	−0.685693
$952$	0	0
$953$	0.534180	0.0173038	0.00865189	−	0.999963i	$-0.497246\pi$
$953$	0.534180	0.0173038	0.00865189	+	0.999963i	$0.497246\pi$
$954$	0	0
$955$	76.8032	2.48529
$956$	0	0
$957$	0.529856	0.0171278
$958$	0	0
$959$	−0.138956	−0.00448712
$960$	0	0
$961$	7.98538	0.257593
$962$	0	0
$963$	11.5777	0.373088
$964$	0	0
$965$	−60.7777	−1.95650
$966$	0	0
$967$	39.4291	1.26796	0.633978	−	0.773351i	$-0.281421\pi$
$967$	39.4291	1.26796	0.633978	+	0.773351i	$0.281421\pi$
$968$	0	0
$969$	−12.0455	−0.386957
$970$	0	0
$971$	−7.25015	−0.232668	−0.116334	−	0.993210i	$-0.537114\pi$
$971$	−7.25015	−0.232668	−0.116334	+	0.993210i	$0.537114\pi$
$972$	0	0
$973$	0.118500	0.00379893
$974$	0	0
$975$	−23.2855	−0.745734
$976$	0	0
$977$	−28.1656	−0.901099	−0.450549	−	0.892751i	$-0.648772\pi$
$977$	−28.1656	−0.901099	−0.450549	+	0.892751i	$0.648772\pi$
$978$	0	0
$979$	−1.76358	−0.0563644
$980$	0	0
$981$	19.1570	0.611635
$982$	0	0
$983$	−14.9195	−0.475858	−0.237929	−	0.971283i	$-0.576469\pi$
$983$	−14.9195	−0.475858	−0.237929	+	0.971283i	$0.576469\pi$
$984$	0	0
$985$	−24.4761	−0.779872
$986$	0	0
$987$	0.346208	0.0110199
$988$	0	0
$989$	19.8700	0.631828
$990$	0	0
$991$	52.2387	1.65942	0.829708	−	0.558198i	$-0.188507\pi$
$991$	52.2387	1.65942	0.829708	+	0.558198i	$0.188507\pi$
$992$	0	0
$993$	−2.39249	−0.0759235
$994$	0	0
$995$	−70.4986	−2.23496
$996$	0	0
$997$	5.30130	0.167894	0.0839469	−	0.996470i	$-0.473247\pi$
$997$	5.30130	0.167894	0.0839469	+	0.996470i	$0.473247\pi$
$998$	0	0
$999$	−8.58373	−0.271577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.q.1.17	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.q.1.17	✓	18	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.61923 q^{5} +0.0319200 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.61923 q^{5} +0.0319200 q^{7} +1.00000 q^{9} -0.312672 q^{11} -2.87518 q^{13} +3.61923 q^{15} +2.30414 q^{17} -5.22777 q^{19} +0.0319200 q^{21} +3.65809 q^{23} +8.09881 q^{25} +1.00000 q^{27} -1.69461 q^{29} +6.24383 q^{31} -0.312672 q^{33} +0.115526 q^{35} -8.58373 q^{37} -2.87518 q^{39} +5.46981 q^{41} +5.43178 q^{43} +3.61923 q^{45} +10.8461 q^{47} -6.99898 q^{49} +2.30414 q^{51} -1.61988 q^{53} -1.13163 q^{55} -5.22777 q^{57} +14.6009 q^{59} +9.00904 q^{61} +0.0319200 q^{63} -10.4059 q^{65} +8.45909 q^{67} +3.65809 q^{69} +3.00139 q^{71} -12.2053 q^{73} +8.09881 q^{75} -0.00998047 q^{77} +2.62332 q^{79} +1.00000 q^{81} +7.13603 q^{83} +8.33920 q^{85} -1.69461 q^{87} +5.64037 q^{89} -0.0917756 q^{91} +6.24383 q^{93} -18.9205 q^{95} +4.88534 q^{97} -0.312672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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