LMFDB - Embedded newform 6024.2.a.r.1.16

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			1.16
Root			$2.61062$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.61062 q^{5} +4.62670 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.61062 q^{5} +4.62670 q^{7} +1.00000 q^{9} -1.55991 q^{11} +3.43285 q^{13} +2.61062 q^{15} -6.11372 q^{17} +4.80516 q^{19} +4.62670 q^{21} +2.72306 q^{23} +1.81533 q^{25} +1.00000 q^{27} -5.64593 q^{29} +3.67521 q^{31} -1.55991 q^{33} +12.0785 q^{35} -6.00642 q^{37} +3.43285 q^{39} -7.07786 q^{41} +1.50848 q^{43} +2.61062 q^{45} +5.90026 q^{47} +14.4063 q^{49} -6.11372 q^{51} +8.58389 q^{53} -4.07232 q^{55} +4.80516 q^{57} +0.323464 q^{59} +14.0170 q^{61} +4.62670 q^{63} +8.96188 q^{65} +5.97595 q^{67} +2.72306 q^{69} -15.2027 q^{71} +5.82723 q^{73} +1.81533 q^{75} -7.21721 q^{77} -4.32167 q^{79} +1.00000 q^{81} -13.5942 q^{83} -15.9606 q^{85} -5.64593 q^{87} +11.0858 q^{89} +15.8828 q^{91} +3.67521 q^{93} +12.5444 q^{95} +17.2758 q^{97} -1.55991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.61062	1.16750	0.583752	−	0.811932i	$-0.301584\pi$
$5$	2.61062	1.16750	0.583752	+	0.811932i	$0.301584\pi$
$6$	0	0
$7$	4.62670	1.74873	0.874364	−	0.485271i	$-0.161279\pi$
$7$	4.62670	1.74873	0.874364	+	0.485271i	$0.161279\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.55991	−0.470329	−0.235165	−	0.971956i	$-0.575563\pi$
$11$	−1.55991	−0.470329	−0.235165	+	0.971956i	$0.575563\pi$
$12$	0	0
$13$	3.43285	0.952103	0.476051	−	0.879418i	$-0.342068\pi$
$13$	3.43285	0.952103	0.476051	+	0.879418i	$0.342068\pi$
$14$	0	0
$15$	2.61062	0.674059
$16$	0	0
$17$	−6.11372	−1.48279	−0.741397	−	0.671066i	$-0.765837\pi$
$17$	−6.11372	−1.48279	−0.741397	+	0.671066i	$0.765837\pi$
$18$	0	0
$19$	4.80516	1.10238	0.551189	−	0.834380i	$-0.314174\pi$
$19$	4.80516	1.10238	0.551189	+	0.834380i	$0.314174\pi$
$20$	0	0
$21$	4.62670	1.00963
$22$	0	0
$23$	2.72306	0.567797	0.283899	−	0.958854i	$-0.408372\pi$
$23$	2.72306	0.567797	0.283899	+	0.958854i	$0.408372\pi$
$24$	0	0
$25$	1.81533	0.363067
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.64593	−1.04842	−0.524212	−	0.851588i	$-0.675640\pi$
$29$	−5.64593	−1.04842	−0.524212	+	0.851588i	$0.675640\pi$
$30$	0	0
$31$	3.67521	0.660086	0.330043	−	0.943966i	$-0.392937\pi$
$31$	3.67521	0.660086	0.330043	+	0.943966i	$0.392937\pi$
$32$	0	0
$33$	−1.55991	−0.271545
$34$	0	0
$35$	12.0785	2.04165
$36$	0	0
$37$	−6.00642	−0.987450	−0.493725	−	0.869618i	$-0.664365\pi$
$37$	−6.00642	−0.987450	−0.493725	+	0.869618i	$0.664365\pi$
$38$	0	0
$39$	3.43285	0.549697
$40$	0	0
$41$	−7.07786	−1.10538	−0.552688	−	0.833388i	$-0.686398\pi$
$41$	−7.07786	−1.10538	−0.552688	+	0.833388i	$0.686398\pi$
$42$	0	0
$43$	1.50848	0.230041	0.115021	−	0.993363i	$-0.463307\pi$
$43$	1.50848	0.230041	0.115021	+	0.993363i	$0.463307\pi$
$44$	0	0
$45$	2.61062	0.389168
$46$	0	0
$47$	5.90026	0.860641	0.430321	−	0.902676i	$-0.358400\pi$
$47$	5.90026	0.860641	0.430321	+	0.902676i	$0.358400\pi$
$48$	0	0
$49$	14.4063	2.05805
$50$	0	0
$51$	−6.11372	−0.856092
$52$	0	0
$53$	8.58389	1.17909	0.589544	−	0.807736i	$-0.299308\pi$
$53$	8.58389	1.17909	0.589544	+	0.807736i	$0.299308\pi$
$54$	0	0
$55$	−4.07232	−0.549111
$56$	0	0
$57$	4.80516	0.636458
$58$	0	0
$59$	0.323464	0.0421114	0.0210557	−	0.999778i	$-0.493297\pi$
$59$	0.323464	0.0421114	0.0210557	+	0.999778i	$0.493297\pi$
$60$	0	0
$61$	14.0170	1.79469	0.897343	−	0.441333i	$-0.145494\pi$
$61$	14.0170	1.79469	0.897343	+	0.441333i	$0.145494\pi$
$62$	0	0
$63$	4.62670	0.582909
$64$	0	0
$65$	8.96188	1.11158
$66$	0	0
$67$	5.97595	0.730078	0.365039	−	0.930992i	$-0.381056\pi$
$67$	5.97595	0.730078	0.365039	+	0.930992i	$0.381056\pi$
$68$	0	0
$69$	2.72306	0.327818
$70$	0	0
$71$	−15.2027	−1.80423	−0.902117	−	0.431491i	$-0.857988\pi$
$71$	−15.2027	−1.80423	−0.902117	+	0.431491i	$0.857988\pi$
$72$	0	0
$73$	5.82723	0.682026	0.341013	−	0.940059i	$-0.389230\pi$
$73$	5.82723	0.682026	0.341013	+	0.940059i	$0.389230\pi$
$74$	0	0
$75$	1.81533	0.209617
$76$	0	0
$77$	−7.21721	−0.822477
$78$	0	0
$79$	−4.32167	−0.486226	−0.243113	−	0.969998i	$-0.578169\pi$
$79$	−4.32167	−0.486226	−0.243113	+	0.969998i	$0.578169\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.5942	−1.49215	−0.746077	−	0.665860i	$-0.768065\pi$
$83$	−13.5942	−1.49215	−0.746077	+	0.665860i	$0.768065\pi$
$84$	0	0
$85$	−15.9606	−1.73117
$86$	0	0
$87$	−5.64593	−0.605308
$88$	0	0
$89$	11.0858	1.17510	0.587548	−	0.809190i	$-0.300093\pi$
$89$	11.0858	1.17510	0.587548	+	0.809190i	$0.300093\pi$
$90$	0	0
$91$	15.8828	1.66497
$92$	0	0
$93$	3.67521	0.381101
$94$	0	0
$95$	12.5444	1.28703
$96$	0	0
$97$	17.2758	1.75410	0.877048	−	0.480403i	$-0.159510\pi$
$97$	17.2758	1.75410	0.877048	+	0.480403i	$0.159510\pi$
$98$	0	0
$99$	−1.55991	−0.156776
$100$	0	0
$101$	−14.6460	−1.45733	−0.728665	−	0.684871i	$-0.759859\pi$
$101$	−14.6460	−1.45733	−0.728665	+	0.684871i	$0.759859\pi$
$102$	0	0
$103$	−1.20775	−0.119003	−0.0595016	−	0.998228i	$-0.518951\pi$
$103$	−1.20775	−0.119003	−0.0595016	+	0.998228i	$0.518951\pi$
$104$	0	0
$105$	12.0785	1.17875
$106$	0	0
$107$	−4.04768	−0.391304	−0.195652	−	0.980673i	$-0.562682\pi$
$107$	−4.04768	−0.391304	−0.195652	+	0.980673i	$0.562682\pi$
$108$	0	0
$109$	6.37285	0.610408	0.305204	−	0.952287i	$-0.401275\pi$
$109$	6.37285	0.610408	0.305204	+	0.952287i	$0.401275\pi$
$110$	0	0
$111$	−6.00642	−0.570104
$112$	0	0
$113$	5.11735	0.481399	0.240700	−	0.970600i	$-0.422623\pi$
$113$	5.11735	0.481399	0.240700	+	0.970600i	$0.422623\pi$
$114$	0	0
$115$	7.10887	0.662906
$116$	0	0
$117$	3.43285	0.317368
$118$	0	0
$119$	−28.2863	−2.59300
$120$	0	0
$121$	−8.56670	−0.778791
$122$	0	0
$123$	−7.07786	−0.638189
$124$	0	0
$125$	−8.31395	−0.743623
$126$	0	0
$127$	−12.4906	−1.10836	−0.554180	−	0.832397i	$-0.686968\pi$
$127$	−12.4906	−1.10836	−0.554180	+	0.832397i	$0.686968\pi$
$128$	0	0
$129$	1.50848	0.132815
$130$	0	0
$131$	13.0174	1.13734	0.568669	−	0.822566i	$-0.307459\pi$
$131$	13.0174	1.13734	0.568669	+	0.822566i	$0.307459\pi$
$132$	0	0
$133$	22.2320	1.92776
$134$	0	0
$135$	2.61062	0.224686
$136$	0	0
$137$	4.69994	0.401543	0.200772	−	0.979638i	$-0.435655\pi$
$137$	4.69994	0.401543	0.200772	+	0.979638i	$0.435655\pi$
$138$	0	0
$139$	3.75415	0.318423	0.159212	−	0.987244i	$-0.449105\pi$
$139$	3.75415	0.318423	0.159212	+	0.987244i	$0.449105\pi$
$140$	0	0
$141$	5.90026	0.496892
$142$	0	0
$143$	−5.35493	−0.447802
$144$	0	0
$145$	−14.7394	−1.22404
$146$	0	0
$147$	14.4063	1.18821
$148$	0	0
$149$	7.84696	0.642848	0.321424	−	0.946935i	$-0.395839\pi$
$149$	7.84696	0.642848	0.321424	+	0.946935i	$0.395839\pi$
$150$	0	0
$151$	−8.86870	−0.721725	−0.360862	−	0.932619i	$-0.617518\pi$
$151$	−8.86870	−0.721725	−0.360862	+	0.932619i	$0.617518\pi$
$152$	0	0
$153$	−6.11372	−0.494265
$154$	0	0
$155$	9.59456	0.770654
$156$	0	0
$157$	−14.0013	−1.11742	−0.558711	−	0.829363i	$-0.688704\pi$
$157$	−14.0013	−1.11742	−0.558711	+	0.829363i	$0.688704\pi$
$158$	0	0
$159$	8.58389	0.680747
$160$	0	0
$161$	12.5988	0.992923
$162$	0	0
$163$	9.70571	0.760210	0.380105	−	0.924943i	$-0.375888\pi$
$163$	9.70571	0.760210	0.380105	+	0.924943i	$0.375888\pi$
$164$	0	0
$165$	−4.07232	−0.317030
$166$	0	0
$167$	−12.4728	−0.965173	−0.482586	−	0.875848i	$-0.660303\pi$
$167$	−12.4728	−0.965173	−0.482586	+	0.875848i	$0.660303\pi$
$168$	0	0
$169$	−1.21551	−0.0935008
$170$	0	0
$171$	4.80516	0.367459
$172$	0	0
$173$	16.8096	1.27801	0.639005	−	0.769203i	$-0.279346\pi$
$173$	16.8096	1.27801	0.639005	+	0.769203i	$0.279346\pi$
$174$	0	0
$175$	8.39900	0.634905
$176$	0	0
$177$	0.323464	0.0243130
$178$	0	0
$179$	18.6607	1.39476	0.697382	−	0.716700i	$-0.254348\pi$
$179$	18.6607	1.39476	0.697382	+	0.716700i	$0.254348\pi$
$180$	0	0
$181$	−19.7249	−1.46614	−0.733070	−	0.680153i	$-0.761913\pi$
$181$	−19.7249	−1.46614	−0.733070	+	0.680153i	$0.761913\pi$
$182$	0	0
$183$	14.0170	1.03616
$184$	0	0
$185$	−15.6805	−1.15285
$186$	0	0
$187$	9.53682	0.697402
$188$	0	0
$189$	4.62670	0.336543
$190$	0	0
$191$	−0.253839	−0.0183671	−0.00918357	−	0.999958i	$-0.502923\pi$
$191$	−0.253839	−0.0183671	−0.00918357	+	0.999958i	$0.502923\pi$
$192$	0	0
$193$	−12.9346	−0.931053	−0.465527	−	0.885034i	$-0.654135\pi$
$193$	−12.9346	−0.931053	−0.465527	+	0.885034i	$0.654135\pi$
$194$	0	0
$195$	8.96188	0.641773
$196$	0	0
$197$	23.8451	1.69889	0.849447	−	0.527674i	$-0.176936\pi$
$197$	23.8451	1.69889	0.849447	+	0.527674i	$0.176936\pi$
$198$	0	0
$199$	3.86103	0.273701	0.136850	−	0.990592i	$-0.456302\pi$
$199$	3.86103	0.273701	0.136850	+	0.990592i	$0.456302\pi$
$200$	0	0
$201$	5.97595	0.421511
$202$	0	0
$203$	−26.1220	−1.83341
$204$	0	0
$205$	−18.4776	−1.29053
$206$	0	0
$207$	2.72306	0.189266
$208$	0	0
$209$	−7.49559	−0.518481
$210$	0	0
$211$	−23.8243	−1.64013	−0.820067	−	0.572268i	$-0.806064\pi$
$211$	−23.8243	−1.64013	−0.820067	+	0.572268i	$0.806064\pi$
$212$	0	0
$213$	−15.2027	−1.04168
$214$	0	0
$215$	3.93807	0.268574
$216$	0	0
$217$	17.0041	1.15431
$218$	0	0
$219$	5.82723	0.393768
$220$	0	0
$221$	−20.9875	−1.41177
$222$	0	0
$223$	−9.29951	−0.622741	−0.311371	−	0.950289i	$-0.600788\pi$
$223$	−9.29951	−0.622741	−0.311371	+	0.950289i	$0.600788\pi$
$224$	0	0
$225$	1.81533	0.121022
$226$	0	0
$227$	−23.5977	−1.56623	−0.783117	−	0.621875i	$-0.786371\pi$
$227$	−23.5977	−1.56623	−0.783117	+	0.621875i	$0.786371\pi$
$228$	0	0
$229$	−22.8025	−1.50683	−0.753416	−	0.657545i	$-0.771595\pi$
$229$	−22.8025	−1.50683	−0.753416	+	0.657545i	$0.771595\pi$
$230$	0	0
$231$	−7.21721	−0.474858
$232$	0	0
$233$	−21.2562	−1.39254	−0.696268	−	0.717782i	$-0.745158\pi$
$233$	−21.2562	−1.39254	−0.696268	+	0.717782i	$0.745158\pi$
$234$	0	0
$235$	15.4033	1.00480
$236$	0	0
$237$	−4.32167	−0.280723
$238$	0	0
$239$	21.0491	1.36156	0.680778	−	0.732490i	$-0.261642\pi$
$239$	21.0491	1.36156	0.680778	+	0.732490i	$0.261642\pi$
$240$	0	0
$241$	−9.14492	−0.589076	−0.294538	−	0.955640i	$-0.595166\pi$
$241$	−9.14492	−0.589076	−0.294538	+	0.955640i	$0.595166\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	37.6094	2.40278
$246$	0	0
$247$	16.4954	1.04958
$248$	0	0
$249$	−13.5942	−0.861495
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−4.24772	−0.267052
$254$	0	0
$255$	−15.9606	−0.999491
$256$	0	0
$257$	−6.62745	−0.413409	−0.206704	−	0.978403i	$-0.566274\pi$
$257$	−6.62745	−0.413409	−0.206704	+	0.978403i	$0.566274\pi$
$258$	0	0
$259$	−27.7899	−1.72678
$260$	0	0
$261$	−5.64593	−0.349475
$262$	0	0
$263$	−17.8474	−1.10052	−0.550258	−	0.834995i	$-0.685471\pi$
$263$	−17.8474	−1.10052	−0.550258	+	0.834995i	$0.685471\pi$
$264$	0	0
$265$	22.4093	1.37659
$266$	0	0
$267$	11.0858	0.678442
$268$	0	0
$269$	7.87669	0.480250	0.240125	−	0.970742i	$-0.422811\pi$
$269$	7.87669	0.480250	0.240125	+	0.970742i	$0.422811\pi$
$270$	0	0
$271$	8.95950	0.544251	0.272125	−	0.962262i	$-0.412273\pi$
$271$	8.95950	0.544251	0.272125	+	0.962262i	$0.412273\pi$
$272$	0	0
$273$	15.8828	0.961270
$274$	0	0
$275$	−2.83175	−0.170761
$276$	0	0
$277$	−24.1725	−1.45238	−0.726191	−	0.687492i	$-0.758711\pi$
$277$	−24.1725	−1.45238	−0.726191	+	0.687492i	$0.758711\pi$
$278$	0	0
$279$	3.67521	0.220029
$280$	0	0
$281$	13.6044	0.811573	0.405786	−	0.913968i	$-0.366998\pi$
$281$	13.6044	0.811573	0.405786	+	0.913968i	$0.366998\pi$
$282$	0	0
$283$	−26.0058	−1.54589	−0.772943	−	0.634476i	$-0.781216\pi$
$283$	−26.0058	−1.54589	−0.772943	+	0.634476i	$0.781216\pi$
$284$	0	0
$285$	12.5444	0.743068
$286$	0	0
$287$	−32.7471	−1.93300
$288$	0	0
$289$	20.3776	1.19868
$290$	0	0
$291$	17.2758	1.01273
$292$	0	0
$293$	24.1079	1.40840	0.704200	−	0.710002i	$-0.251306\pi$
$293$	24.1079	1.40840	0.704200	+	0.710002i	$0.251306\pi$
$294$	0	0
$295$	0.844440	0.0491652
$296$	0	0
$297$	−1.55991	−0.0905149
$298$	0	0
$299$	9.34787	0.540601
$300$	0	0
$301$	6.97929	0.402280
$302$	0	0
$303$	−14.6460	−0.841389
$304$	0	0
$305$	36.5929	2.09530
$306$	0	0
$307$	−31.2632	−1.78429	−0.892143	−	0.451752i	$-0.850799\pi$
$307$	−31.2632	−1.78429	−0.892143	+	0.451752i	$0.850799\pi$
$308$	0	0
$309$	−1.20775	−0.0687065
$310$	0	0
$311$	−7.04296	−0.399370	−0.199685	−	0.979860i	$-0.563992\pi$
$311$	−7.04296	−0.399370	−0.199685	+	0.979860i	$0.563992\pi$
$312$	0	0
$313$	24.6961	1.39591	0.697954	−	0.716143i	$-0.254094\pi$
$313$	24.6961	1.39591	0.697954	+	0.716143i	$0.254094\pi$
$314$	0	0
$315$	12.0785	0.680549
$316$	0	0
$317$	21.9471	1.23267	0.616337	−	0.787482i	$-0.288616\pi$
$317$	21.9471	1.23267	0.616337	+	0.787482i	$0.288616\pi$
$318$	0	0
$319$	8.80712	0.493104
$320$	0	0
$321$	−4.04768	−0.225920
$322$	0	0
$323$	−29.3774	−1.63460
$324$	0	0
$325$	6.23177	0.345677
$326$	0	0
$327$	6.37285	0.352419
$328$	0	0
$329$	27.2987	1.50503
$330$	0	0
$331$	21.8240	1.19956	0.599779	−	0.800166i	$-0.295255\pi$
$331$	21.8240	1.19956	0.599779	+	0.800166i	$0.295255\pi$
$332$	0	0
$333$	−6.00642	−0.329150
$334$	0	0
$335$	15.6009	0.852370
$336$	0	0
$337$	−6.30567	−0.343492	−0.171746	−	0.985141i	$-0.554941\pi$
$337$	−6.30567	−0.343492	−0.171746	+	0.985141i	$0.554941\pi$
$338$	0	0
$339$	5.11735	0.277936
$340$	0	0
$341$	−5.73297	−0.310458
$342$	0	0
$343$	34.2668	1.85024
$344$	0	0
$345$	7.10887	0.382729
$346$	0	0
$347$	−7.11536	−0.381973	−0.190986	−	0.981593i	$-0.561169\pi$
$347$	−7.11536	−0.381973	−0.190986	+	0.981593i	$0.561169\pi$
$348$	0	0
$349$	−20.6428	−1.10498	−0.552492	−	0.833518i	$-0.686323\pi$
$349$	−20.6428	−1.10498	−0.552492	+	0.833518i	$0.686323\pi$
$350$	0	0
$351$	3.43285	0.183232
$352$	0	0
$353$	−3.15601	−0.167977	−0.0839887	−	0.996467i	$-0.526766\pi$
$353$	−3.15601	−0.167977	−0.0839887	+	0.996467i	$0.526766\pi$
$354$	0	0
$355$	−39.6886	−2.10645
$356$	0	0
$357$	−28.2863	−1.49707
$358$	0	0
$359$	−8.59760	−0.453764	−0.226882	−	0.973922i	$-0.572853\pi$
$359$	−8.59760	−0.453764	−0.226882	+	0.973922i	$0.572853\pi$
$360$	0	0
$361$	4.08952	0.215238
$362$	0	0
$363$	−8.56670	−0.449635
$364$	0	0
$365$	15.2127	0.796268
$366$	0	0
$367$	7.60507	0.396981	0.198491	−	0.980103i	$-0.436396\pi$
$367$	7.60507	0.396981	0.198491	+	0.980103i	$0.436396\pi$
$368$	0	0
$369$	−7.07786	−0.368459
$370$	0	0
$371$	39.7151	2.06190
$372$	0	0
$373$	25.4825	1.31943	0.659717	−	0.751514i	$-0.270676\pi$
$373$	25.4825	1.31943	0.659717	+	0.751514i	$0.270676\pi$
$374$	0	0
$375$	−8.31395	−0.429331
$376$	0	0
$377$	−19.3817	−0.998207
$378$	0	0
$379$	−14.1034	−0.724442	−0.362221	−	0.932092i	$-0.617981\pi$
$379$	−14.1034	−0.724442	−0.362221	+	0.932092i	$0.617981\pi$
$380$	0	0
$381$	−12.4906	−0.639912
$382$	0	0
$383$	−26.0803	−1.33264	−0.666321	−	0.745665i	$-0.732132\pi$
$383$	−26.0803	−1.33264	−0.666321	+	0.745665i	$0.732132\pi$
$384$	0	0
$385$	−18.8414	−0.960246
$386$	0	0
$387$	1.50848	0.0766805
$388$	0	0
$389$	−28.9148	−1.46604	−0.733018	−	0.680209i	$-0.761889\pi$
$389$	−28.9148	−1.46604	−0.733018	+	0.680209i	$0.761889\pi$
$390$	0	0
$391$	−16.6480	−0.841927
$392$	0	0
$393$	13.0174	0.656642
$394$	0	0
$395$	−11.2822	−0.567671
$396$	0	0
$397$	5.99671	0.300966	0.150483	−	0.988613i	$-0.451917\pi$
$397$	5.99671	0.300966	0.150483	+	0.988613i	$0.451917\pi$
$398$	0	0
$399$	22.2320	1.11299
$400$	0	0
$401$	−3.99574	−0.199538	−0.0997689	−	0.995011i	$-0.531810\pi$
$401$	−3.99574	−0.199538	−0.0997689	+	0.995011i	$0.531810\pi$
$402$	0	0
$403$	12.6164	0.628470
$404$	0	0
$405$	2.61062	0.129723
$406$	0	0
$407$	9.36945	0.464426
$408$	0	0
$409$	31.7596	1.57041	0.785205	−	0.619236i	$-0.212558\pi$
$409$	31.7596	1.57041	0.785205	+	0.619236i	$0.212558\pi$
$410$	0	0
$411$	4.69994	0.231831
$412$	0	0
$413$	1.49657	0.0736413
$414$	0	0
$415$	−35.4892	−1.74210
$416$	0	0
$417$	3.75415	0.183842
$418$	0	0
$419$	−15.5246	−0.758427	−0.379213	−	0.925309i	$-0.623805\pi$
$419$	−15.5246	−0.758427	−0.379213	+	0.925309i	$0.623805\pi$
$420$	0	0
$421$	−5.23045	−0.254917	−0.127458	−	0.991844i	$-0.540682\pi$
$421$	−5.23045	−0.254917	−0.127458	+	0.991844i	$0.540682\pi$
$422$	0	0
$423$	5.90026	0.286880
$424$	0	0
$425$	−11.0984	−0.538353
$426$	0	0
$427$	64.8522	3.13842
$428$	0	0
$429$	−5.35493	−0.258538
$430$	0	0
$431$	15.8842	0.765114	0.382557	−	0.923932i	$-0.375043\pi$
$431$	15.8842	0.765114	0.382557	+	0.923932i	$0.375043\pi$
$432$	0	0
$433$	37.4098	1.79780	0.898901	−	0.438151i	$-0.144367\pi$
$433$	37.4098	1.79780	0.898901	+	0.438151i	$0.144367\pi$
$434$	0	0
$435$	−14.7394	−0.706699
$436$	0	0
$437$	13.0847	0.625927
$438$	0	0
$439$	3.58905	0.171296	0.0856481	−	0.996325i	$-0.472704\pi$
$439$	3.58905	0.171296	0.0856481	+	0.996325i	$0.472704\pi$
$440$	0	0
$441$	14.4063	0.686016
$442$	0	0
$443$	4.37216	0.207728	0.103864	−	0.994592i	$-0.466879\pi$
$443$	4.37216	0.207728	0.103864	+	0.994592i	$0.466879\pi$
$444$	0	0
$445$	28.9409	1.37193
$446$	0	0
$447$	7.84696	0.371149
$448$	0	0
$449$	−7.70111	−0.363438	−0.181719	−	0.983350i	$-0.558166\pi$
$449$	−7.70111	−0.363438	−0.181719	+	0.983350i	$0.558166\pi$
$450$	0	0
$451$	11.0408	0.519891
$452$	0	0
$453$	−8.86870	−0.416688
$454$	0	0
$455$	41.4639	1.94386
$456$	0	0
$457$	27.1527	1.27015	0.635076	−	0.772450i	$-0.280969\pi$
$457$	27.1527	1.27015	0.635076	+	0.772450i	$0.280969\pi$
$458$	0	0
$459$	−6.11372	−0.285364
$460$	0	0
$461$	−32.8823	−1.53148	−0.765740	−	0.643150i	$-0.777627\pi$
$461$	−32.8823	−1.53148	−0.765740	+	0.643150i	$0.777627\pi$
$462$	0	0
$463$	3.87463	0.180070	0.0900348	−	0.995939i	$-0.471302\pi$
$463$	3.87463	0.180070	0.0900348	+	0.995939i	$0.471302\pi$
$464$	0	0
$465$	9.59456	0.444937
$466$	0	0
$467$	32.3220	1.49568	0.747842	−	0.663877i	$-0.231090\pi$
$467$	32.3220	1.49568	0.747842	+	0.663877i	$0.231090\pi$
$468$	0	0
$469$	27.6489	1.27671
$470$	0	0
$471$	−14.0013	−0.645144
$472$	0	0
$473$	−2.35309	−0.108195
$474$	0	0
$475$	8.72296	0.400237
$476$	0	0
$477$	8.58389	0.393029
$478$	0	0
$479$	−16.8865	−0.771565	−0.385783	−	0.922590i	$-0.626069\pi$
$479$	−16.8865	−0.771565	−0.385783	+	0.922590i	$0.626069\pi$
$480$	0	0
$481$	−20.6192	−0.940153
$482$	0	0
$483$	12.5988	0.573264
$484$	0	0
$485$	45.1006	2.04791
$486$	0	0
$487$	−10.3466	−0.468850	−0.234425	−	0.972134i	$-0.575321\pi$
$487$	−10.3466	−0.468850	−0.234425	+	0.972134i	$0.575321\pi$
$488$	0	0
$489$	9.70571	0.438907
$490$	0	0
$491$	12.5473	0.566249	0.283125	−	0.959083i	$-0.408629\pi$
$491$	12.5473	0.566249	0.283125	+	0.959083i	$0.408629\pi$
$492$	0	0
$493$	34.5177	1.55460
$494$	0	0
$495$	−4.07232	−0.183037
$496$	0	0
$497$	−70.3385	−3.15511
$498$	0	0
$499$	−23.7685	−1.06403	−0.532013	−	0.846736i	$-0.678564\pi$
$499$	−23.7685	−1.06403	−0.532013	+	0.846736i	$0.678564\pi$
$500$	0	0
$501$	−12.4728	−0.557243
$502$	0	0
$503$	3.39319	0.151295	0.0756475	−	0.997135i	$-0.475898\pi$
$503$	3.39319	0.151295	0.0756475	+	0.997135i	$0.475898\pi$
$504$	0	0
$505$	−38.2351	−1.70144
$506$	0	0
$507$	−1.21551	−0.0539827
$508$	0	0
$509$	15.4323	0.684025	0.342012	−	0.939695i	$-0.388892\pi$
$509$	15.4323	0.684025	0.342012	+	0.939695i	$0.388892\pi$
$510$	0	0
$511$	26.9608	1.19268
$512$	0	0
$513$	4.80516	0.212153
$514$	0	0
$515$	−3.15298	−0.138937
$516$	0	0
$517$	−9.20385	−0.404785
$518$	0	0
$519$	16.8096	0.737859
$520$	0	0
$521$	−2.91857	−0.127865	−0.0639324	−	0.997954i	$-0.520364\pi$
$521$	−2.91857	−0.127865	−0.0639324	+	0.997954i	$0.520364\pi$
$522$	0	0
$523$	39.2883	1.71796	0.858979	−	0.512011i	$-0.171099\pi$
$523$	39.2883	1.71796	0.858979	+	0.512011i	$0.171099\pi$
$524$	0	0
$525$	8.39900	0.366562
$526$	0	0
$527$	−22.4692	−0.978773
$528$	0	0
$529$	−15.5849	−0.677606
$530$	0	0
$531$	0.323464	0.0140371
$532$	0	0
$533$	−24.2973	−1.05243
$534$	0	0
$535$	−10.5670	−0.456849
$536$	0	0
$537$	18.6607	0.805267
$538$	0	0
$539$	−22.4725	−0.967960
$540$	0	0
$541$	−41.7330	−1.79424	−0.897120	−	0.441786i	$-0.854345\pi$
$541$	−41.7330	−1.79424	−0.897120	+	0.441786i	$0.854345\pi$
$542$	0	0
$543$	−19.7249	−0.846477
$544$	0	0
$545$	16.6371	0.712654
$546$	0	0
$547$	−6.83861	−0.292398	−0.146199	−	0.989255i	$-0.546704\pi$
$547$	−6.83861	−0.292398	−0.146199	+	0.989255i	$0.546704\pi$
$548$	0	0
$549$	14.0170	0.598229
$550$	0	0
$551$	−27.1296	−1.15576
$552$	0	0
$553$	−19.9951	−0.850276
$554$	0	0
$555$	−15.6805	−0.665599
$556$	0	0
$557$	41.0402	1.73893	0.869464	−	0.493996i	$-0.164464\pi$
$557$	41.0402	1.73893	0.869464	+	0.493996i	$0.164464\pi$
$558$	0	0
$559$	5.17840	0.219023
$560$	0	0
$561$	9.53682	0.402645
$562$	0	0
$563$	−39.6739	−1.67205	−0.836027	−	0.548689i	$-0.815127\pi$
$563$	−39.6739	−1.67205	−0.836027	+	0.548689i	$0.815127\pi$
$564$	0	0
$565$	13.3594	0.562036
$566$	0	0
$567$	4.62670	0.194303
$568$	0	0
$569$	35.3873	1.48351	0.741757	−	0.670669i	$-0.233993\pi$
$569$	35.3873	1.48351	0.741757	+	0.670669i	$0.233993\pi$
$570$	0	0
$571$	20.1962	0.845185	0.422593	−	0.906320i	$-0.361120\pi$
$571$	20.1962	0.845185	0.422593	+	0.906320i	$0.361120\pi$
$572$	0	0
$573$	−0.253839	−0.0106043
$574$	0	0
$575$	4.94326	0.206148
$576$	0	0
$577$	41.4669	1.72629	0.863145	−	0.504956i	$-0.168491\pi$
$577$	41.4669	1.72629	0.863145	+	0.504956i	$0.168491\pi$
$578$	0	0
$579$	−12.9346	−0.537544
$580$	0	0
$581$	−62.8961	−2.60937
$582$	0	0
$583$	−13.3901	−0.554559
$584$	0	0
$585$	8.96188	0.370528
$586$	0	0
$587$	33.7689	1.39379	0.696897	−	0.717171i	$-0.254564\pi$
$587$	33.7689	1.39379	0.696897	+	0.717171i	$0.254564\pi$
$588$	0	0
$589$	17.6599	0.727665
$590$	0	0
$591$	23.8451	0.980857
$592$	0	0
$593$	−22.0275	−0.904560	−0.452280	−	0.891876i	$-0.649389\pi$
$593$	−22.0275	−0.904560	−0.452280	+	0.891876i	$0.649389\pi$
$594$	0	0
$595$	−73.8449	−3.02734
$596$	0	0
$597$	3.86103	0.158021
$598$	0	0
$599$	−29.9836	−1.22510	−0.612548	−	0.790434i	$-0.709855\pi$
$599$	−29.9836	−1.22510	−0.612548	+	0.790434i	$0.709855\pi$
$600$	0	0
$601$	15.3160	0.624753	0.312377	−	0.949958i	$-0.398875\pi$
$601$	15.3160	0.624753	0.312377	+	0.949958i	$0.398875\pi$
$602$	0	0
$603$	5.97595	0.243359
$604$	0	0
$605$	−22.3644	−0.909241
$606$	0	0
$607$	31.3005	1.27045	0.635224	−	0.772328i	$-0.280908\pi$
$607$	31.3005	1.27045	0.635224	+	0.772328i	$0.280908\pi$
$608$	0	0
$609$	−26.1220	−1.05852
$610$	0	0
$611$	20.2547	0.819419
$612$	0	0
$613$	30.6947	1.23975	0.619874	−	0.784701i	$-0.287184\pi$
$613$	30.6947	1.23975	0.619874	+	0.784701i	$0.287184\pi$
$614$	0	0
$615$	−18.4776	−0.745089
$616$	0	0
$617$	24.3758	0.981333	0.490667	−	0.871347i	$-0.336753\pi$
$617$	24.3758	0.981333	0.490667	+	0.871347i	$0.336753\pi$
$618$	0	0
$619$	13.2595	0.532943	0.266472	−	0.963843i	$-0.414142\pi$
$619$	13.2595	0.532943	0.266472	+	0.963843i	$0.414142\pi$
$620$	0	0
$621$	2.72306	0.109273
$622$	0	0
$623$	51.2908	2.05492
$624$	0	0
$625$	−30.7812	−1.23125
$626$	0	0
$627$	−7.49559	−0.299345
$628$	0	0
$629$	36.7216	1.46419
$630$	0	0
$631$	−1.85870	−0.0739935	−0.0369968	−	0.999315i	$-0.511779\pi$
$631$	−1.85870	−0.0739935	−0.0369968	+	0.999315i	$0.511779\pi$
$632$	0	0
$633$	−23.8243	−0.946931
$634$	0	0
$635$	−32.6082	−1.29402
$636$	0	0
$637$	49.4548	1.95947
$638$	0	0
$639$	−15.2027	−0.601411
$640$	0	0
$641$	41.3889	1.63476	0.817382	−	0.576096i	$-0.195424\pi$
$641$	41.3889	1.63476	0.817382	+	0.576096i	$0.195424\pi$
$642$	0	0
$643$	−41.5777	−1.63967	−0.819833	−	0.572603i	$-0.805934\pi$
$643$	−41.5777	−1.63967	−0.819833	+	0.572603i	$0.805934\pi$
$644$	0	0
$645$	3.93807	0.155062
$646$	0	0
$647$	−3.58149	−0.140803	−0.0704015	−	0.997519i	$-0.522428\pi$
$647$	−3.58149	−0.140803	−0.0704015	+	0.997519i	$0.522428\pi$
$648$	0	0
$649$	−0.504573	−0.0198062
$650$	0	0
$651$	17.0041	0.666442
$652$	0	0
$653$	−42.7648	−1.67351	−0.836757	−	0.547574i	$-0.815551\pi$
$653$	−42.7648	−1.67351	−0.836757	+	0.547574i	$0.815551\pi$
$654$	0	0
$655$	33.9835	1.32785
$656$	0	0
$657$	5.82723	0.227342
$658$	0	0
$659$	−10.4723	−0.407942	−0.203971	−	0.978977i	$-0.565385\pi$
$659$	−10.4723	−0.407942	−0.203971	+	0.978977i	$0.565385\pi$
$660$	0	0
$661$	16.1404	0.627788	0.313894	−	0.949458i	$-0.398366\pi$
$661$	16.1404	0.627788	0.313894	+	0.949458i	$0.398366\pi$
$662$	0	0
$663$	−20.9875	−0.815087
$664$	0	0
$665$	58.0393	2.25067
$666$	0	0
$667$	−15.3742	−0.595292
$668$	0	0
$669$	−9.29951	−0.359540
$670$	0	0
$671$	−21.8651	−0.844093
$672$	0	0
$673$	−44.0010	−1.69611	−0.848056	−	0.529907i	$-0.822227\pi$
$673$	−44.0010	−1.69611	−0.848056	+	0.529907i	$0.822227\pi$
$674$	0	0
$675$	1.81533	0.0698722
$676$	0	0
$677$	−35.1448	−1.35072	−0.675362	−	0.737486i	$-0.736013\pi$
$677$	−35.1448	−1.35072	−0.675362	+	0.737486i	$0.736013\pi$
$678$	0	0
$679$	79.9301	3.06744
$680$	0	0
$681$	−23.5977	−0.904265
$682$	0	0
$683$	17.8879	0.684462	0.342231	−	0.939616i	$-0.388818\pi$
$683$	17.8879	0.684462	0.342231	+	0.939616i	$0.388818\pi$
$684$	0	0
$685$	12.2698	0.468803
$686$	0	0
$687$	−22.8025	−0.869970
$688$	0	0
$689$	29.4672	1.12261
$690$	0	0
$691$	8.05833	0.306553	0.153277	−	0.988183i	$-0.451017\pi$
$691$	8.05833	0.306553	0.153277	+	0.988183i	$0.451017\pi$
$692$	0	0
$693$	−7.21721	−0.274159
$694$	0	0
$695$	9.80066	0.371760
$696$	0	0
$697$	43.2721	1.63905
$698$	0	0
$699$	−21.2562	−0.803981
$700$	0	0
$701$	9.80756	0.370426	0.185213	−	0.982698i	$-0.440702\pi$
$701$	9.80756	0.370426	0.185213	+	0.982698i	$0.440702\pi$
$702$	0	0
$703$	−28.8618	−1.08854
$704$	0	0
$705$	15.4033	0.580123
$706$	0	0
$707$	−67.7625	−2.54847
$708$	0	0
$709$	3.74493	0.140644	0.0703219	−	0.997524i	$-0.477597\pi$
$709$	3.74493	0.140644	0.0703219	+	0.997524i	$0.477597\pi$
$710$	0	0
$711$	−4.32167	−0.162075
$712$	0	0
$713$	10.0078	0.374795
$714$	0	0
$715$	−13.9797	−0.522810
$716$	0	0
$717$	21.0491	0.786094
$718$	0	0
$719$	−4.77803	−0.178191	−0.0890953	−	0.996023i	$-0.528398\pi$
$719$	−4.77803	−0.178191	−0.0890953	+	0.996023i	$0.528398\pi$
$720$	0	0
$721$	−5.58790	−0.208104
$722$	0	0
$723$	−9.14492	−0.340103
$724$	0	0
$725$	−10.2493	−0.380648
$726$	0	0
$727$	−15.8061	−0.586217	−0.293109	−	0.956079i	$-0.594690\pi$
$727$	−15.8061	−0.586217	−0.293109	+	0.956079i	$0.594690\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.22244	−0.341104
$732$	0	0
$733$	−13.6377	−0.503721	−0.251861	−	0.967764i	$-0.581042\pi$
$733$	−13.6377	−0.503721	−0.251861	+	0.967764i	$0.581042\pi$
$734$	0	0
$735$	37.6094	1.38725
$736$	0	0
$737$	−9.32191	−0.343377
$738$	0	0
$739$	−38.3068	−1.40914	−0.704568	−	0.709636i	$-0.748859\pi$
$739$	−38.3068	−1.40914	−0.704568	+	0.709636i	$0.748859\pi$
$740$	0	0
$741$	16.4954	0.605974
$742$	0	0
$743$	−16.4693	−0.604199	−0.302100	−	0.953276i	$-0.597687\pi$
$743$	−16.4693	−0.604199	−0.302100	+	0.953276i	$0.597687\pi$
$744$	0	0
$745$	20.4854	0.750528
$746$	0	0
$747$	−13.5942	−0.497385
$748$	0	0
$749$	−18.7274	−0.684284
$750$	0	0
$751$	29.1397	1.06332	0.531660	−	0.846958i	$-0.321568\pi$
$751$	29.1397	1.06332	0.531660	+	0.846958i	$0.321568\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−23.1528	−0.842617
$756$	0	0
$757$	−14.5967	−0.530527	−0.265264	−	0.964176i	$-0.585459\pi$
$757$	−14.5967	−0.530527	−0.265264	+	0.964176i	$0.585459\pi$
$758$	0	0
$759$	−4.24772	−0.154182
$760$	0	0
$761$	−27.8819	−1.01072	−0.505360	−	0.862909i	$-0.668640\pi$
$761$	−27.8819	−1.01072	−0.505360	+	0.862909i	$0.668640\pi$
$762$	0	0
$763$	29.4852	1.06744
$764$	0	0
$765$	−15.9606	−0.577057
$766$	0	0
$767$	1.11040	0.0400943
$768$	0	0
$769$	4.98954	0.179927	0.0899637	−	0.995945i	$-0.471325\pi$
$769$	4.98954	0.179927	0.0899637	+	0.995945i	$0.471325\pi$
$770$	0	0
$771$	−6.62745	−0.238682
$772$	0	0
$773$	13.6709	0.491710	0.245855	−	0.969307i	$-0.420931\pi$
$773$	13.6709	0.491710	0.245855	+	0.969307i	$0.420931\pi$
$774$	0	0
$775$	6.67172	0.239655
$776$	0	0
$777$	−27.7899	−0.996957
$778$	0	0
$779$	−34.0102	−1.21854
$780$	0	0
$781$	23.7148	0.848584
$782$	0	0
$783$	−5.64593	−0.201769
$784$	0	0
$785$	−36.5519	−1.30459
$786$	0	0
$787$	−32.5649	−1.16081	−0.580406	−	0.814327i	$-0.697106\pi$
$787$	−32.5649	−1.16081	−0.580406	+	0.814327i	$0.697106\pi$
$788$	0	0
$789$	−17.8474	−0.635383
$790$	0	0
$791$	23.6764	0.841836
$792$	0	0
$793$	48.1182	1.70873
$794$	0	0
$795$	22.4093	0.794775
$796$	0	0
$797$	16.6317	0.589127	0.294563	−	0.955632i	$-0.404826\pi$
$797$	16.6317	0.589127	0.294563	+	0.955632i	$0.404826\pi$
$798$	0	0
$799$	−36.0725	−1.27615
$800$	0	0
$801$	11.0858	0.391698
$802$	0	0
$803$	−9.08993	−0.320777
$804$	0	0
$805$	32.8906	1.15924
$806$	0	0
$807$	7.87669	0.277273
$808$	0	0
$809$	−4.95842	−0.174329	−0.0871644	−	0.996194i	$-0.527781\pi$
$809$	−4.95842	−0.174329	−0.0871644	+	0.996194i	$0.527781\pi$
$810$	0	0
$811$	−31.6217	−1.11039	−0.555194	−	0.831721i	$-0.687356\pi$
$811$	−31.6217	−1.11039	−0.555194	+	0.831721i	$0.687356\pi$
$812$	0	0
$813$	8.95950	0.314223
$814$	0	0
$815$	25.3379	0.887548
$816$	0	0
$817$	7.24849	0.253593
$818$	0	0
$819$	15.8828	0.554989
$820$	0	0
$821$	30.7246	1.07230	0.536149	−	0.844124i	$-0.319879\pi$
$821$	30.7246	1.07230	0.536149	+	0.844124i	$0.319879\pi$
$822$	0	0
$823$	39.2244	1.36728	0.683639	−	0.729820i	$-0.260396\pi$
$823$	39.2244	1.36728	0.683639	+	0.729820i	$0.260396\pi$
$824$	0	0
$825$	−2.83175	−0.0985888
$826$	0	0
$827$	21.1524	0.735539	0.367770	−	0.929917i	$-0.380121\pi$
$827$	21.1524	0.735539	0.367770	+	0.929917i	$0.380121\pi$
$828$	0	0
$829$	7.67091	0.266422	0.133211	−	0.991088i	$-0.457471\pi$
$829$	7.67091	0.266422	0.133211	+	0.991088i	$0.457471\pi$
$830$	0	0
$831$	−24.1725	−0.838534
$832$	0	0
$833$	−88.0763	−3.05166
$834$	0	0
$835$	−32.5617	−1.12684
$836$	0	0
$837$	3.67521	0.127034
$838$	0	0
$839$	18.0318	0.622526	0.311263	−	0.950324i	$-0.399248\pi$
$839$	18.0318	0.622526	0.311263	+	0.950324i	$0.399248\pi$
$840$	0	0
$841$	2.87658	0.0991924
$842$	0	0
$843$	13.6044	0.468562
$844$	0	0
$845$	−3.17323	−0.109163
$846$	0	0
$847$	−39.6355	−1.36189
$848$	0	0
$849$	−26.0058	−0.892518
$850$	0	0
$851$	−16.3559	−0.560671
$852$	0	0
$853$	40.1765	1.37562	0.687808	−	0.725892i	$-0.258573\pi$
$853$	40.1765	1.37562	0.687808	+	0.725892i	$0.258573\pi$
$854$	0	0
$855$	12.5444	0.429010
$856$	0	0
$857$	−10.3810	−0.354607	−0.177304	−	0.984156i	$-0.556737\pi$
$857$	−10.3810	−0.354607	−0.177304	+	0.984156i	$0.556737\pi$
$858$	0	0
$859$	−28.1734	−0.961262	−0.480631	−	0.876923i	$-0.659592\pi$
$859$	−28.1734	−0.961262	−0.480631	+	0.876923i	$0.659592\pi$
$860$	0	0
$861$	−32.7471	−1.11602
$862$	0	0
$863$	31.0727	1.05773	0.528864	−	0.848707i	$-0.322618\pi$
$863$	31.0727	1.05773	0.528864	+	0.848707i	$0.322618\pi$
$864$	0	0
$865$	43.8834	1.49208
$866$	0	0
$867$	20.3776	0.692059
$868$	0	0
$869$	6.74139	0.228686
$870$	0	0
$871$	20.5146	0.695109
$872$	0	0
$873$	17.2758	0.584699
$874$	0	0
$875$	−38.4661	−1.30039
$876$	0	0
$877$	−32.0878	−1.08353	−0.541764	−	0.840531i	$-0.682243\pi$
$877$	−32.0878	−1.08353	−0.541764	+	0.840531i	$0.682243\pi$
$878$	0	0
$879$	24.1079	0.813140
$880$	0	0
$881$	−36.6709	−1.23547	−0.617737	−	0.786385i	$-0.711950\pi$
$881$	−36.6709	−1.23547	−0.617737	+	0.786385i	$0.711950\pi$
$882$	0	0
$883$	−31.9194	−1.07417	−0.537086	−	0.843527i	$-0.680475\pi$
$883$	−31.9194	−1.07417	−0.537086	+	0.843527i	$0.680475\pi$
$884$	0	0
$885$	0.844440	0.0283856
$886$	0	0
$887$	12.9925	0.436245	0.218123	−	0.975921i	$-0.430007\pi$
$887$	12.9925	0.436245	0.218123	+	0.975921i	$0.430007\pi$
$888$	0	0
$889$	−57.7902	−1.93822
$890$	0	0
$891$	−1.55991	−0.0522588
$892$	0	0
$893$	28.3517	0.948752
$894$	0	0
$895$	48.7159	1.62839
$896$	0	0
$897$	9.34787	0.312116
$898$	0	0
$899$	−20.7500	−0.692050
$900$	0	0
$901$	−52.4795	−1.74835
$902$	0	0
$903$	6.97929	0.232256
$904$	0	0
$905$	−51.4942	−1.71173
$906$	0	0
$907$	38.8714	1.29070	0.645352	−	0.763886i	$-0.276711\pi$
$907$	38.8714	1.29070	0.645352	+	0.763886i	$0.276711\pi$
$908$	0	0
$909$	−14.6460	−0.485776
$910$	0	0
$911$	6.87682	0.227839	0.113920	−	0.993490i	$-0.463659\pi$
$911$	6.87682	0.227839	0.113920	+	0.993490i	$0.463659\pi$
$912$	0	0
$913$	21.2056	0.701803
$914$	0	0
$915$	36.5929	1.20972
$916$	0	0
$917$	60.2277	1.98889
$918$	0	0
$919$	−21.1537	−0.697795	−0.348897	−	0.937161i	$-0.613444\pi$
$919$	−21.1537	−0.697795	−0.348897	+	0.937161i	$0.613444\pi$
$920$	0	0
$921$	−31.2632	−1.03016
$922$	0	0
$923$	−52.1888	−1.71782
$924$	0	0
$925$	−10.9037	−0.358510
$926$	0	0
$927$	−1.20775	−0.0396677
$928$	0	0
$929$	−47.6468	−1.56324	−0.781620	−	0.623755i	$-0.785606\pi$
$929$	−47.6468	−1.56324	−0.781620	+	0.623755i	$0.785606\pi$
$930$	0	0
$931$	69.2247	2.26875
$932$	0	0
$933$	−7.04296	−0.230576
$934$	0	0
$935$	24.8970	0.814220
$936$	0	0
$937$	−40.5730	−1.32546	−0.662732	−	0.748857i	$-0.730603\pi$
$937$	−40.5730	−1.32546	−0.662732	+	0.748857i	$0.730603\pi$
$938$	0	0
$939$	24.6961	0.805928
$940$	0	0
$941$	0.778301	0.0253719	0.0126859	−	0.999920i	$-0.495962\pi$
$941$	0.778301	0.0253719	0.0126859	+	0.999920i	$0.495962\pi$
$942$	0	0
$943$	−19.2734	−0.627630
$944$	0	0
$945$	12.0785	0.392915
$946$	0	0
$947$	34.4735	1.12024	0.560119	−	0.828412i	$-0.310755\pi$
$947$	34.4735	1.12024	0.560119	+	0.828412i	$0.310755\pi$
$948$	0	0
$949$	20.0040	0.649358
$950$	0	0
$951$	21.9471	0.711685
$952$	0	0
$953$	6.89302	0.223287	0.111643	−	0.993748i	$-0.464389\pi$
$953$	6.89302	0.223287	0.111643	+	0.993748i	$0.464389\pi$
$954$	0	0
$955$	−0.662676	−0.0214437
$956$	0	0
$957$	8.80712	0.284694
$958$	0	0
$959$	21.7452	0.702189
$960$	0	0
$961$	−17.4929	−0.564286
$962$	0	0
$963$	−4.04768	−0.130435
$964$	0	0
$965$	−33.7673	−1.08701
$966$	0	0
$967$	−39.9113	−1.28346	−0.641730	−	0.766931i	$-0.721783\pi$
$967$	−39.9113	−1.28346	−0.641730	+	0.766931i	$0.721783\pi$
$968$	0	0
$969$	−29.3774	−0.943737
$970$	0	0
$971$	7.04334	0.226032	0.113016	−	0.993593i	$-0.463949\pi$
$971$	7.04334	0.226032	0.113016	+	0.993593i	$0.463949\pi$
$972$	0	0
$973$	17.3693	0.556835
$974$	0	0
$975$	6.23177	0.199577
$976$	0	0
$977$	−39.3996	−1.26051	−0.630253	−	0.776390i	$-0.717049\pi$
$977$	−39.3996	−1.26051	−0.630253	+	0.776390i	$0.717049\pi$
$978$	0	0
$979$	−17.2928	−0.552682
$980$	0	0
$981$	6.37285	0.203469
$982$	0	0
$983$	−2.08600	−0.0665329	−0.0332665	−	0.999447i	$-0.510591\pi$
$983$	−2.08600	−0.0665329	−0.0332665	+	0.999447i	$0.510591\pi$
$984$	0	0
$985$	62.2505	1.98347
$986$	0	0
$987$	27.2987	0.868928
$988$	0	0
$989$	4.10769	0.130617
$990$	0	0
$991$	−42.9348	−1.36387	−0.681935	−	0.731413i	$-0.738861\pi$
$991$	−42.9348	−1.36387	−0.681935	+	0.731413i	$0.738861\pi$
$992$	0	0
$993$	21.8240	0.692565
$994$	0	0
$995$	10.0797	0.319547
$996$	0	0
$997$	1.47643	0.0467591	0.0233796	−	0.999727i	$-0.492557\pi$
$997$	1.47643	0.0467591	0.0233796	+	0.999727i	$0.492557\pi$
$998$	0	0
$999$	−6.00642	−0.190035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.r.1.16	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.r.1.16	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.61062 q^{5} +4.62670 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.61062 q^{5} +4.62670 q^{7} +1.00000 q^{9} -1.55991 q^{11} +3.43285 q^{13} +2.61062 q^{15} -6.11372 q^{17} +4.80516 q^{19} +4.62670 q^{21} +2.72306 q^{23} +1.81533 q^{25} +1.00000 q^{27} -5.64593 q^{29} +3.67521 q^{31} -1.55991 q^{33} +12.0785 q^{35} -6.00642 q^{37} +3.43285 q^{39} -7.07786 q^{41} +1.50848 q^{43} +2.61062 q^{45} +5.90026 q^{47} +14.4063 q^{49} -6.11372 q^{51} +8.58389 q^{53} -4.07232 q^{55} +4.80516 q^{57} +0.323464 q^{59} +14.0170 q^{61} +4.62670 q^{63} +8.96188 q^{65} +5.97595 q^{67} +2.72306 q^{69} -15.2027 q^{71} +5.82723 q^{73} +1.81533 q^{75} -7.21721 q^{77} -4.32167 q^{79} +1.00000 q^{81} -13.5942 q^{83} -15.9606 q^{85} -5.64593 q^{87} +11.0858 q^{89} +15.8828 q^{91} +3.67521 q^{93} +12.5444 q^{95} +17.2758 q^{97} -1.55991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 9 q^{5} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 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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