LMFDB - Embedded newform 6024.2.a.p.1.6

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.379598$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.379598 q^{5} -4.36268 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.379598 q^{5} -4.36268 q^{7} +1.00000 q^{9} +1.18831 q^{11} +3.35744 q^{13} +0.379598 q^{15} -1.21331 q^{17} +0.595416 q^{19} +4.36268 q^{21} -9.02398 q^{23} -4.85591 q^{25} -1.00000 q^{27} +7.97562 q^{29} +0.130338 q^{31} -1.18831 q^{33} +1.65606 q^{35} -7.46173 q^{37} -3.35744 q^{39} -2.90235 q^{41} -0.939239 q^{43} -0.379598 q^{45} -1.28457 q^{47} +12.0330 q^{49} +1.21331 q^{51} +3.77634 q^{53} -0.451079 q^{55} -0.595416 q^{57} +1.68235 q^{59} +11.4379 q^{61} -4.36268 q^{63} -1.27448 q^{65} +4.67971 q^{67} +9.02398 q^{69} -2.52321 q^{71} -7.93161 q^{73} +4.85591 q^{75} -5.18420 q^{77} -0.107292 q^{79} +1.00000 q^{81} -13.7284 q^{83} +0.460571 q^{85} -7.97562 q^{87} -4.75857 q^{89} -14.6474 q^{91} -0.130338 q^{93} -0.226019 q^{95} -5.58069 q^{97} +1.18831 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * 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
$5$	−0.379598	−0.169761	−0.0848807	−	0.996391i	$-0.527051\pi$
$5$	−0.379598	−0.169761	−0.0848807	+	0.996391i	$0.527051\pi$
$6$	0	0
$7$	−4.36268	−1.64894	−0.824469	−	0.565908i	$-0.808526\pi$
$7$	−4.36268	−1.64894	−0.824469	+	0.565908i	$0.808526\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.18831	0.358288	0.179144	−	0.983823i	$-0.442667\pi$
$11$	1.18831	0.358288	0.179144	+	0.983823i	$0.442667\pi$
$12$	0	0
$13$	3.35744	0.931187	0.465593	−	0.884999i	$-0.345841\pi$
$13$	3.35744	0.931187	0.465593	+	0.884999i	$0.345841\pi$
$14$	0	0
$15$	0.379598	0.0980117
$16$	0	0
$17$	−1.21331	−0.294271	−0.147136	−	0.989116i	$-0.547005\pi$
$17$	−1.21331	−0.294271	−0.147136	+	0.989116i	$0.547005\pi$
$18$	0	0
$19$	0.595416	0.136598	0.0682989	−	0.997665i	$-0.478243\pi$
$19$	0.595416	0.136598	0.0682989	+	0.997665i	$0.478243\pi$
$20$	0	0
$21$	4.36268	0.952014
$22$	0	0
$23$	−9.02398	−1.88163	−0.940815	−	0.338921i	$-0.889938\pi$
$23$	−9.02398	−1.88163	−0.940815	+	0.338921i	$0.889938\pi$
$24$	0	0
$25$	−4.85591	−0.971181
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.97562	1.48104	0.740518	−	0.672036i	$-0.234580\pi$
$29$	7.97562	1.48104	0.740518	+	0.672036i	$0.234580\pi$
$30$	0	0
$31$	0.130338	0.0234095	0.0117047	−	0.999931i	$-0.496274\pi$
$31$	0.130338	0.0234095	0.0117047	+	0.999931i	$0.496274\pi$
$32$	0	0
$33$	−1.18831	−0.206858
$34$	0	0
$35$	1.65606	0.279926
$36$	0	0
$37$	−7.46173	−1.22670	−0.613350	−	0.789811i	$-0.710179\pi$
$37$	−7.46173	−1.22670	−0.613350	+	0.789811i	$0.710179\pi$
$38$	0	0
$39$	−3.35744	−0.537621
$40$	0	0
$41$	−2.90235	−0.453272	−0.226636	−	0.973980i	$-0.572773\pi$
$41$	−2.90235	−0.453272	−0.226636	+	0.973980i	$0.572773\pi$
$42$	0	0
$43$	−0.939239	−0.143233	−0.0716163	−	0.997432i	$-0.522816\pi$
$43$	−0.939239	−0.143233	−0.0716163	+	0.997432i	$0.522816\pi$
$44$	0	0
$45$	−0.379598	−0.0565871
$46$	0	0
$47$	−1.28457	−0.187374	−0.0936868	−	0.995602i	$-0.529865\pi$
$47$	−1.28457	−0.187374	−0.0936868	+	0.995602i	$0.529865\pi$
$48$	0	0
$49$	12.0330	1.71899
$50$	0	0
$51$	1.21331	0.169898
$52$	0	0
$53$	3.77634	0.518720	0.259360	−	0.965781i	$-0.416488\pi$
$53$	3.77634	0.518720	0.259360	+	0.965781i	$0.416488\pi$
$54$	0	0
$55$	−0.451079	−0.0608235
$56$	0	0
$57$	−0.595416	−0.0788648
$58$	0	0
$59$	1.68235	0.219024	0.109512	−	0.993985i	$-0.465071\pi$
$59$	1.68235	0.219024	0.109512	+	0.993985i	$0.465071\pi$
$60$	0	0
$61$	11.4379	1.46448	0.732238	−	0.681049i	$-0.238476\pi$
$61$	11.4379	1.46448	0.732238	+	0.681049i	$0.238476\pi$
$62$	0	0
$63$	−4.36268	−0.549646
$64$	0	0
$65$	−1.27448	−0.158080
$66$	0	0
$67$	4.67971	0.571717	0.285859	−	0.958272i	$-0.407721\pi$
$67$	4.67971	0.571717	0.285859	+	0.958272i	$0.407721\pi$
$68$	0	0
$69$	9.02398	1.08636
$70$	0	0
$71$	−2.52321	−0.299450	−0.149725	−	0.988728i	$-0.547839\pi$
$71$	−2.52321	−0.299450	−0.149725	+	0.988728i	$0.547839\pi$
$72$	0	0
$73$	−7.93161	−0.928325	−0.464162	−	0.885750i	$-0.653645\pi$
$73$	−7.93161	−0.928325	−0.464162	+	0.885750i	$0.653645\pi$
$74$	0	0
$75$	4.85591	0.560712
$76$	0	0
$77$	−5.18420	−0.590795
$78$	0	0
$79$	−0.107292	−0.0120713	−0.00603565	−	0.999982i	$-0.501921\pi$
$79$	−0.107292	−0.0120713	−0.00603565	+	0.999982i	$0.501921\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.7284	−1.50688	−0.753442	−	0.657514i	$-0.771608\pi$
$83$	−13.7284	−1.50688	−0.753442	+	0.657514i	$0.771608\pi$
$84$	0	0
$85$	0.460571	0.0499559
$86$	0	0
$87$	−7.97562	−0.855077
$88$	0	0
$89$	−4.75857	−0.504408	−0.252204	−	0.967674i	$-0.581155\pi$
$89$	−4.75857	−0.504408	−0.252204	+	0.967674i	$0.581155\pi$
$90$	0	0
$91$	−14.6474	−1.53547
$92$	0	0
$93$	−0.130338	−0.0135155
$94$	0	0
$95$	−0.226019	−0.0231890
$96$	0	0
$97$	−5.58069	−0.566633	−0.283316	−	0.959027i	$-0.591435\pi$
$97$	−5.58069	−0.566633	−0.283316	+	0.959027i	$0.591435\pi$
$98$	0	0
$99$	1.18831	0.119429
$100$	0	0
$101$	0.762433	0.0758649	0.0379325	−	0.999280i	$-0.487923\pi$
$101$	0.762433	0.0758649	0.0379325	+	0.999280i	$0.487923\pi$
$102$	0	0
$103$	−9.23323	−0.909777	−0.454888	−	0.890548i	$-0.650321\pi$
$103$	−9.23323	−0.909777	−0.454888	+	0.890548i	$0.650321\pi$
$104$	0	0
$105$	−1.65606	−0.161615
$106$	0	0
$107$	−3.20746	−0.310076	−0.155038	−	0.987908i	$-0.549550\pi$
$107$	−3.20746	−0.310076	−0.155038	+	0.987908i	$0.549550\pi$
$108$	0	0
$109$	11.6658	1.11738	0.558691	−	0.829376i	$-0.311304\pi$
$109$	11.6658	1.11738	0.558691	+	0.829376i	$0.311304\pi$
$110$	0	0
$111$	7.46173	0.708236
$112$	0	0
$113$	10.2478	0.964032	0.482016	−	0.876162i	$-0.339905\pi$
$113$	10.2478	0.964032	0.482016	+	0.876162i	$0.339905\pi$
$114$	0	0
$115$	3.42548	0.319428
$116$	0	0
$117$	3.35744	0.310396
$118$	0	0
$119$	5.29329	0.485235
$120$	0	0
$121$	−9.58792	−0.871630
$122$	0	0
$123$	2.90235	0.261697
$124$	0	0
$125$	3.74128	0.334630
$126$	0	0
$127$	−9.77969	−0.867807	−0.433904	−	0.900959i	$-0.642864\pi$
$127$	−9.77969	−0.867807	−0.433904	+	0.900959i	$0.642864\pi$
$128$	0	0
$129$	0.939239	0.0826954
$130$	0	0
$131$	6.53248	0.570745	0.285373	−	0.958417i	$-0.407883\pi$
$131$	6.53248	0.570745	0.285373	+	0.958417i	$0.407883\pi$
$132$	0	0
$133$	−2.59761	−0.225241
$134$	0	0
$135$	0.379598	0.0326706
$136$	0	0
$137$	−9.15339	−0.782026	−0.391013	−	0.920385i	$-0.627875\pi$
$137$	−9.15339	−0.782026	−0.391013	+	0.920385i	$0.627875\pi$
$138$	0	0
$139$	7.65949	0.649669	0.324835	−	0.945771i	$-0.394691\pi$
$139$	7.65949	0.649669	0.324835	+	0.945771i	$0.394691\pi$
$140$	0	0
$141$	1.28457	0.108180
$142$	0	0
$143$	3.98967	0.333633
$144$	0	0
$145$	−3.02753	−0.251423
$146$	0	0
$147$	−12.0330	−0.992462
$148$	0	0
$149$	6.71326	0.549971	0.274986	−	0.961448i	$-0.411327\pi$
$149$	6.71326	0.549971	0.274986	+	0.961448i	$0.411327\pi$
$150$	0	0
$151$	18.2679	1.48662	0.743310	−	0.668947i	$-0.233255\pi$
$151$	18.2679	1.48662	0.743310	+	0.668947i	$0.233255\pi$
$152$	0	0
$153$	−1.21331	−0.0980905
$154$	0	0
$155$	−0.0494762	−0.00397402
$156$	0	0
$157$	2.57131	0.205213	0.102606	−	0.994722i	$-0.467282\pi$
$157$	2.57131	0.205213	0.102606	+	0.994722i	$0.467282\pi$
$158$	0	0
$159$	−3.77634	−0.299483
$160$	0	0
$161$	39.3687	3.10269
$162$	0	0
$163$	14.8920	1.16643	0.583215	−	0.812318i	$-0.301795\pi$
$163$	14.8920	1.16643	0.583215	+	0.812318i	$0.301795\pi$
$164$	0	0
$165$	0.451079	0.0351165
$166$	0	0
$167$	9.95093	0.770026	0.385013	−	0.922911i	$-0.374197\pi$
$167$	9.95093	0.770026	0.385013	+	0.922911i	$0.374197\pi$
$168$	0	0
$169$	−1.72758	−0.132891
$170$	0	0
$171$	0.595416	0.0455326
$172$	0	0
$173$	13.0662	0.993405	0.496702	−	0.867921i	$-0.334544\pi$
$173$	13.0662	0.993405	0.496702	+	0.867921i	$0.334544\pi$
$174$	0	0
$175$	21.1848	1.60142
$176$	0	0
$177$	−1.68235	−0.126454
$178$	0	0
$179$	12.1286	0.906532	0.453266	−	0.891375i	$-0.350259\pi$
$179$	12.1286	0.906532	0.453266	+	0.891375i	$0.350259\pi$
$180$	0	0
$181$	2.96918	0.220698	0.110349	−	0.993893i	$-0.464803\pi$
$181$	2.96918	0.220698	0.110349	+	0.993893i	$0.464803\pi$
$182$	0	0
$183$	−11.4379	−0.845516
$184$	0	0
$185$	2.83246	0.208246
$186$	0	0
$187$	−1.44179	−0.105434
$188$	0	0
$189$	4.36268	0.317338
$190$	0	0
$191$	−18.7841	−1.35917	−0.679585	−	0.733596i	$-0.737840\pi$
$191$	−18.7841	−1.35917	−0.679585	+	0.733596i	$0.737840\pi$
$192$	0	0
$193$	12.8445	0.924564	0.462282	−	0.886733i	$-0.347031\pi$
$193$	12.8445	0.924564	0.462282	+	0.886733i	$0.347031\pi$
$194$	0	0
$195$	1.27448	0.0912673
$196$	0	0
$197$	7.79838	0.555611	0.277806	−	0.960637i	$-0.410393\pi$
$197$	7.79838	0.555611	0.277806	+	0.960637i	$0.410393\pi$
$198$	0	0
$199$	−7.79124	−0.552306	−0.276153	−	0.961114i	$-0.589060\pi$
$199$	−7.79124	−0.552306	−0.276153	+	0.961114i	$0.589060\pi$
$200$	0	0
$201$	−4.67971	−0.330081
$202$	0	0
$203$	−34.7951	−2.44214
$204$	0	0
$205$	1.10173	0.0769480
$206$	0	0
$207$	−9.02398	−0.627210
$208$	0	0
$209$	0.707538	0.0489414
$210$	0	0
$211$	14.4230	0.992920	0.496460	−	0.868060i	$-0.334633\pi$
$211$	14.4230	0.992920	0.496460	+	0.868060i	$0.334633\pi$
$212$	0	0
$213$	2.52321	0.172887
$214$	0	0
$215$	0.356533	0.0243154
$216$	0	0
$217$	−0.568624	−0.0386007
$218$	0	0
$219$	7.93161	0.535968
$220$	0	0
$221$	−4.07363	−0.274022
$222$	0	0
$223$	10.8481	0.726441	0.363221	−	0.931703i	$-0.381677\pi$
$223$	10.8481	0.726441	0.363221	+	0.931703i	$0.381677\pi$
$224$	0	0
$225$	−4.85591	−0.323727
$226$	0	0
$227$	11.1128	0.737580	0.368790	−	0.929513i	$-0.379772\pi$
$227$	11.1128	0.737580	0.368790	+	0.929513i	$0.379772\pi$
$228$	0	0
$229$	3.74591	0.247537	0.123768	−	0.992311i	$-0.460502\pi$
$229$	3.74591	0.247537	0.123768	+	0.992311i	$0.460502\pi$
$230$	0	0
$231$	5.18420	0.341096
$232$	0	0
$233$	9.95027	0.651864	0.325932	−	0.945393i	$-0.394322\pi$
$233$	9.95027	0.651864	0.325932	+	0.945393i	$0.394322\pi$
$234$	0	0
$235$	0.487620	0.0318088
$236$	0	0
$237$	0.107292	0.00696937
$238$	0	0
$239$	10.2508	0.663072	0.331536	−	0.943443i	$-0.392433\pi$
$239$	10.2508	0.663072	0.331536	+	0.943443i	$0.392433\pi$
$240$	0	0
$241$	−7.19108	−0.463218	−0.231609	−	0.972809i	$-0.574399\pi$
$241$	−7.19108	−0.463218	−0.231609	+	0.972809i	$0.574399\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.56769	−0.291819
$246$	0	0
$247$	1.99908	0.127198
$248$	0	0
$249$	13.7284	0.870000
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−10.7233	−0.674166
$254$	0	0
$255$	−0.460571	−0.0288421
$256$	0	0
$257$	8.16309	0.509200	0.254600	−	0.967047i	$-0.418056\pi$
$257$	8.16309	0.509200	0.254600	+	0.967047i	$0.418056\pi$
$258$	0	0
$259$	32.5531	2.02275
$260$	0	0
$261$	7.97562	0.493679
$262$	0	0
$263$	−3.63039	−0.223859	−0.111930	−	0.993716i	$-0.535703\pi$
$263$	−3.63039	−0.223859	−0.111930	+	0.993716i	$0.535703\pi$
$264$	0	0
$265$	−1.43349	−0.0880586
$266$	0	0
$267$	4.75857	0.291220
$268$	0	0
$269$	11.1430	0.679401	0.339701	−	0.940534i	$-0.389674\pi$
$269$	11.1430	0.679401	0.339701	+	0.940534i	$0.389674\pi$
$270$	0	0
$271$	28.2618	1.71678	0.858390	−	0.512998i	$-0.171465\pi$
$271$	28.2618	1.71678	0.858390	+	0.512998i	$0.171465\pi$
$272$	0	0
$273$	14.6474	0.886503
$274$	0	0
$275$	−5.77031	−0.347963
$276$	0	0
$277$	26.9136	1.61708	0.808541	−	0.588440i	$-0.200258\pi$
$277$	26.9136	1.61708	0.808541	+	0.588440i	$0.200258\pi$
$278$	0	0
$279$	0.130338	0.00780316
$280$	0	0
$281$	27.1977	1.62248	0.811238	−	0.584716i	$-0.198794\pi$
$281$	27.1977	1.62248	0.811238	+	0.584716i	$0.198794\pi$
$282$	0	0
$283$	−3.63960	−0.216352	−0.108176	−	0.994132i	$-0.534501\pi$
$283$	−3.63960	−0.216352	−0.108176	+	0.994132i	$0.534501\pi$
$284$	0	0
$285$	0.226019	0.0133882
$286$	0	0
$287$	12.6620	0.747417
$288$	0	0
$289$	−15.5279	−0.913404
$290$	0	0
$291$	5.58069	0.327146
$292$	0	0
$293$	28.1973	1.64730	0.823651	−	0.567096i	$-0.191933\pi$
$293$	28.1973	1.64730	0.823651	+	0.567096i	$0.191933\pi$
$294$	0	0
$295$	−0.638618	−0.0371818
$296$	0	0
$297$	−1.18831	−0.0689526
$298$	0	0
$299$	−30.2975	−1.75215
$300$	0	0
$301$	4.09760	0.236182
$302$	0	0
$303$	−0.762433	−0.0438006
$304$	0	0
$305$	−4.34181	−0.248611
$306$	0	0
$307$	−5.51754	−0.314903	−0.157451	−	0.987527i	$-0.550328\pi$
$307$	−5.51754	−0.314903	−0.157451	+	0.987527i	$0.550328\pi$
$308$	0	0
$309$	9.23323	0.525260
$310$	0	0
$311$	−17.2248	−0.976728	−0.488364	−	0.872640i	$-0.662406\pi$
$311$	−17.2248	−0.976728	−0.488364	+	0.872640i	$0.662406\pi$
$312$	0	0
$313$	14.6048	0.825511	0.412755	−	0.910842i	$-0.364566\pi$
$313$	14.6048	0.825511	0.412755	+	0.910842i	$0.364566\pi$
$314$	0	0
$315$	1.65606	0.0933086
$316$	0	0
$317$	−3.71983	−0.208926	−0.104463	−	0.994529i	$-0.533312\pi$
$317$	−3.71983	−0.208926	−0.104463	+	0.994529i	$0.533312\pi$
$318$	0	0
$319$	9.47749	0.530638
$320$	0	0
$321$	3.20746	0.179023
$322$	0	0
$323$	−0.722426	−0.0401968
$324$	0	0
$325$	−16.3034	−0.904351
$326$	0	0
$327$	−11.6658	−0.645121
$328$	0	0
$329$	5.60416	0.308967
$330$	0	0
$331$	−17.1677	−0.943623	−0.471811	−	0.881700i	$-0.656400\pi$
$331$	−17.1677	−0.943623	−0.471811	+	0.881700i	$0.656400\pi$
$332$	0	0
$333$	−7.46173	−0.408900
$334$	0	0
$335$	−1.77641	−0.0970554
$336$	0	0
$337$	6.76245	0.368374	0.184187	−	0.982891i	$-0.441035\pi$
$337$	6.76245	0.368374	0.184187	+	0.982891i	$0.441035\pi$
$338$	0	0
$339$	−10.2478	−0.556584
$340$	0	0
$341$	0.154882	0.00838734
$342$	0	0
$343$	−21.9572	−1.18558
$344$	0	0
$345$	−3.42548	−0.184422
$346$	0	0
$347$	13.3348	0.715849	0.357925	−	0.933750i	$-0.383484\pi$
$347$	13.3348	0.715849	0.357925	+	0.933750i	$0.383484\pi$
$348$	0	0
$349$	6.37697	0.341351	0.170676	−	0.985327i	$-0.445405\pi$
$349$	6.37697	0.341351	0.170676	+	0.985327i	$0.445405\pi$
$350$	0	0
$351$	−3.35744	−0.179207
$352$	0	0
$353$	−3.05329	−0.162510	−0.0812552	−	0.996693i	$-0.525893\pi$
$353$	−3.05329	−0.162510	−0.0812552	+	0.996693i	$0.525893\pi$
$354$	0	0
$355$	0.957804	0.0508350
$356$	0	0
$357$	−5.29329	−0.280151
$358$	0	0
$359$	35.7773	1.88825	0.944127	−	0.329583i	$-0.106908\pi$
$359$	35.7773	1.88825	0.944127	+	0.329583i	$0.106908\pi$
$360$	0	0
$361$	−18.6455	−0.981341
$362$	0	0
$363$	9.58792	0.503236
$364$	0	0
$365$	3.01082	0.157594
$366$	0	0
$367$	24.3785	1.27255	0.636274	−	0.771463i	$-0.280475\pi$
$367$	24.3785	1.27255	0.636274	+	0.771463i	$0.280475\pi$
$368$	0	0
$369$	−2.90235	−0.151091
$370$	0	0
$371$	−16.4750	−0.855337
$372$	0	0
$373$	3.27960	0.169811	0.0849057	−	0.996389i	$-0.472941\pi$
$373$	3.27960	0.169811	0.0849057	+	0.996389i	$0.472941\pi$
$374$	0	0
$375$	−3.74128	−0.193199
$376$	0	0
$377$	26.7777	1.37912
$378$	0	0
$379$	−1.62759	−0.0836039	−0.0418019	−	0.999126i	$-0.513310\pi$
$379$	−1.62759	−0.0836039	−0.0418019	+	0.999126i	$0.513310\pi$
$380$	0	0
$381$	9.77969	0.501029
$382$	0	0
$383$	−4.81942	−0.246261	−0.123130	−	0.992391i	$-0.539293\pi$
$383$	−4.81942	−0.246261	−0.123130	+	0.992391i	$0.539293\pi$
$384$	0	0
$385$	1.96791	0.100294
$386$	0	0
$387$	−0.939239	−0.0477442
$388$	0	0
$389$	24.3797	1.23610	0.618049	−	0.786139i	$-0.287923\pi$
$389$	24.3797	1.23610	0.618049	+	0.786139i	$0.287923\pi$
$390$	0	0
$391$	10.9489	0.553710
$392$	0	0
$393$	−6.53248	−0.329520
$394$	0	0
$395$	0.0407278	0.00204924
$396$	0	0
$397$	−27.1121	−1.36072	−0.680358	−	0.732880i	$-0.738176\pi$
$397$	−27.1121	−1.36072	−0.680358	+	0.732880i	$0.738176\pi$
$398$	0	0
$399$	2.59761	0.130043
$400$	0	0
$401$	1.28179	0.0640095	0.0320047	−	0.999488i	$-0.489811\pi$
$401$	1.28179	0.0640095	0.0320047	+	0.999488i	$0.489811\pi$
$402$	0	0
$403$	0.437604	0.0217986
$404$	0	0
$405$	−0.379598	−0.0188624
$406$	0	0
$407$	−8.86683	−0.439512
$408$	0	0
$409$	−20.0177	−0.989813	−0.494907	−	0.868946i	$-0.664798\pi$
$409$	−20.0177	−0.989813	−0.494907	+	0.868946i	$0.664798\pi$
$410$	0	0
$411$	9.15339	0.451503
$412$	0	0
$413$	−7.33957	−0.361157
$414$	0	0
$415$	5.21126	0.255811
$416$	0	0
$417$	−7.65949	−0.375087
$418$	0	0
$419$	−25.4257	−1.24213	−0.621064	−	0.783760i	$-0.713299\pi$
$419$	−25.4257	−1.24213	−0.621064	+	0.783760i	$0.713299\pi$
$420$	0	0
$421$	−5.37016	−0.261726	−0.130863	−	0.991400i	$-0.541775\pi$
$421$	−5.37016	−0.261726	−0.130863	+	0.991400i	$0.541775\pi$
$422$	0	0
$423$	−1.28457	−0.0624579
$424$	0	0
$425$	5.89173	0.285791
$426$	0	0
$427$	−49.9000	−2.41483
$428$	0	0
$429$	−3.98967	−0.192623
$430$	0	0
$431$	26.6731	1.28480	0.642399	−	0.766370i	$-0.277939\pi$
$431$	26.6731	1.28480	0.642399	+	0.766370i	$0.277939\pi$
$432$	0	0
$433$	−10.4359	−0.501519	−0.250760	−	0.968049i	$-0.580680\pi$
$433$	−10.4359	−0.501519	−0.250760	+	0.968049i	$0.580680\pi$
$434$	0	0
$435$	3.02753	0.145159
$436$	0	0
$437$	−5.37302	−0.257026
$438$	0	0
$439$	−4.54219	−0.216787	−0.108393	−	0.994108i	$-0.534571\pi$
$439$	−4.54219	−0.216787	−0.108393	+	0.994108i	$0.534571\pi$
$440$	0	0
$441$	12.0330	0.572998
$442$	0	0
$443$	−10.6293	−0.505012	−0.252506	−	0.967595i	$-0.581255\pi$
$443$	−10.6293	−0.505012	−0.252506	+	0.967595i	$0.581255\pi$
$444$	0	0
$445$	1.80634	0.0856289
$446$	0	0
$447$	−6.71326	−0.317526
$448$	0	0
$449$	0.0378994	0.00178858	0.000894291	−	1.00000i	$-0.499715\pi$
$449$	0.0378994	0.00178858	0.000894291		1.00000i	$0.499715\pi$
$450$	0	0
$451$	−3.44889	−0.162402
$452$	0	0
$453$	−18.2679	−0.858300
$454$	0	0
$455$	5.56014	0.260663
$456$	0	0
$457$	−33.3898	−1.56191	−0.780954	−	0.624588i	$-0.785267\pi$
$457$	−33.3898	−1.56191	−0.780954	+	0.624588i	$0.785267\pi$
$458$	0	0
$459$	1.21331	0.0566326
$460$	0	0
$461$	12.8683	0.599338	0.299669	−	0.954043i	$-0.403124\pi$
$461$	12.8683	0.599338	0.299669	+	0.954043i	$0.403124\pi$
$462$	0	0
$463$	−26.4629	−1.22984	−0.614918	−	0.788591i	$-0.710811\pi$
$463$	−26.4629	−1.22984	−0.614918	+	0.788591i	$0.710811\pi$
$464$	0	0
$465$	0.0494762	0.00229440
$466$	0	0
$467$	37.6901	1.74409	0.872044	−	0.489428i	$-0.162794\pi$
$467$	37.6901	1.74409	0.872044	+	0.489428i	$0.162794\pi$
$468$	0	0
$469$	−20.4161	−0.942726
$470$	0	0
$471$	−2.57131	−0.118480
$472$	0	0
$473$	−1.11610	−0.0513186
$474$	0	0
$475$	−2.89128	−0.132661
$476$	0	0
$477$	3.77634	0.172907
$478$	0	0
$479$	42.6196	1.94734	0.973671	−	0.227960i	$-0.0732055\pi$
$479$	42.6196	1.94734	0.973671	+	0.227960i	$0.0732055\pi$
$480$	0	0
$481$	−25.0523	−1.14229
$482$	0	0
$483$	−39.3687	−1.79134
$484$	0	0
$485$	2.11842	0.0961923
$486$	0	0
$487$	34.6549	1.57036	0.785182	−	0.619265i	$-0.212569\pi$
$487$	34.6549	1.57036	0.785182	+	0.619265i	$0.212569\pi$
$488$	0	0
$489$	−14.8920	−0.673438
$490$	0	0
$491$	−15.7995	−0.713021	−0.356510	−	0.934291i	$-0.616034\pi$
$491$	−15.7995	−0.713021	−0.356510	+	0.934291i	$0.616034\pi$
$492$	0	0
$493$	−9.67692	−0.435827
$494$	0	0
$495$	−0.451079	−0.0202745
$496$	0	0
$497$	11.0079	0.493774
$498$	0	0
$499$	−0.288228	−0.0129029	−0.00645143	−	0.999979i	$-0.502054\pi$
$499$	−0.288228	−0.0129029	−0.00645143	+	0.999979i	$0.502054\pi$
$500$	0	0
$501$	−9.95093	−0.444575
$502$	0	0
$503$	−1.13702	−0.0506974	−0.0253487	−	0.999679i	$-0.508070\pi$
$503$	−1.13702	−0.0506974	−0.0253487	+	0.999679i	$0.508070\pi$
$504$	0	0
$505$	−0.289418	−0.0128789
$506$	0	0
$507$	1.72758	0.0767246
$508$	0	0
$509$	19.8077	0.877961	0.438981	−	0.898496i	$-0.355340\pi$
$509$	19.8077	0.877961	0.438981	+	0.898496i	$0.355340\pi$
$510$	0	0
$511$	34.6031	1.53075
$512$	0	0
$513$	−0.595416	−0.0262883
$514$	0	0
$515$	3.50491	0.154445
$516$	0	0
$517$	−1.52646	−0.0671338
$518$	0	0
$519$	−13.0662	−0.573543
$520$	0	0
$521$	23.2631	1.01917	0.509587	−	0.860419i	$-0.329798\pi$
$521$	23.2631	1.01917	0.509587	+	0.860419i	$0.329798\pi$
$522$	0	0
$523$	−23.9970	−1.04932	−0.524658	−	0.851313i	$-0.675807\pi$
$523$	−23.9970	−1.04932	−0.524658	+	0.851313i	$0.675807\pi$
$524$	0	0
$525$	−21.1848	−0.924578
$526$	0	0
$527$	−0.158141	−0.00688874
$528$	0	0
$529$	58.4322	2.54053
$530$	0	0
$531$	1.68235	0.0730080
$532$	0	0
$533$	−9.74449	−0.422081
$534$	0	0
$535$	1.21754	0.0526390
$536$	0	0
$537$	−12.1286	−0.523386
$538$	0	0
$539$	14.2989	0.615896
$540$	0	0
$541$	−40.3238	−1.73366	−0.866828	−	0.498607i	$-0.833845\pi$
$541$	−40.3238	−1.73366	−0.866828	+	0.498607i	$0.833845\pi$
$542$	0	0
$543$	−2.96918	−0.127420
$544$	0	0
$545$	−4.42832	−0.189688
$546$	0	0
$547$	26.4240	1.12981	0.564904	−	0.825157i	$-0.308913\pi$
$547$	26.4240	1.12981	0.564904	+	0.825157i	$0.308913\pi$
$548$	0	0
$549$	11.4379	0.488159
$550$	0	0
$551$	4.74881	0.202306
$552$	0	0
$553$	0.468081	0.0199048
$554$	0	0
$555$	−2.83246	−0.120231
$556$	0	0
$557$	−1.08977	−0.0461749	−0.0230874	−	0.999733i	$-0.507350\pi$
$557$	−1.08977	−0.0461749	−0.0230874	+	0.999733i	$0.507350\pi$
$558$	0	0
$559$	−3.15344	−0.133376
$560$	0	0
$561$	1.44179	0.0608723
$562$	0	0
$563$	−7.23204	−0.304794	−0.152397	−	0.988319i	$-0.548699\pi$
$563$	−7.23204	−0.304794	−0.152397	+	0.988319i	$0.548699\pi$
$564$	0	0
$565$	−3.89004	−0.163655
$566$	0	0
$567$	−4.36268	−0.183215
$568$	0	0
$569$	−11.8905	−0.498476	−0.249238	−	0.968442i	$-0.580180\pi$
$569$	−11.8905	−0.498476	−0.249238	+	0.968442i	$0.580180\pi$
$570$	0	0
$571$	−44.7484	−1.87266	−0.936330	−	0.351120i	$-0.885801\pi$
$571$	−44.7484	−1.87266	−0.936330	+	0.351120i	$0.885801\pi$
$572$	0	0
$573$	18.7841	0.784718
$574$	0	0
$575$	43.8196	1.82740
$576$	0	0
$577$	16.0727	0.669114	0.334557	−	0.942376i	$-0.391413\pi$
$577$	16.0727	0.669114	0.334557	+	0.942376i	$0.391413\pi$
$578$	0	0
$579$	−12.8445	−0.533797
$580$	0	0
$581$	59.8925	2.48476
$582$	0	0
$583$	4.48745	0.185851
$584$	0	0
$585$	−1.27448	−0.0526932
$586$	0	0
$587$	36.6318	1.51195	0.755977	−	0.654598i	$-0.227162\pi$
$587$	36.6318	1.51195	0.755977	+	0.654598i	$0.227162\pi$
$588$	0	0
$589$	0.0776056	0.00319768
$590$	0	0
$591$	−7.79838	−0.320782
$592$	0	0
$593$	−21.6185	−0.887765	−0.443883	−	0.896085i	$-0.646399\pi$
$593$	−21.6185	−0.887765	−0.443883	+	0.896085i	$0.646399\pi$
$594$	0	0
$595$	−2.00932	−0.0823742
$596$	0	0
$597$	7.79124	0.318874
$598$	0	0
$599$	6.28319	0.256724	0.128362	−	0.991727i	$-0.459028\pi$
$599$	6.28319	0.256724	0.128362	+	0.991727i	$0.459028\pi$
$600$	0	0
$601$	10.0721	0.410849	0.205424	−	0.978673i	$-0.434143\pi$
$601$	10.0721	0.410849	0.205424	+	0.978673i	$0.434143\pi$
$602$	0	0
$603$	4.67971	0.190572
$604$	0	0
$605$	3.63956	0.147969
$606$	0	0
$607$	−14.7383	−0.598210	−0.299105	−	0.954220i	$-0.596688\pi$
$607$	−14.7383	−0.598210	−0.299105	+	0.954220i	$0.596688\pi$
$608$	0	0
$609$	34.7951	1.40997
$610$	0	0
$611$	−4.31287	−0.174480
$612$	0	0
$613$	6.46510	0.261123	0.130562	−	0.991440i	$-0.458322\pi$
$613$	6.46510	0.261123	0.130562	+	0.991440i	$0.458322\pi$
$614$	0	0
$615$	−1.10173	−0.0444259
$616$	0	0
$617$	24.8464	1.00028	0.500140	−	0.865945i	$-0.333282\pi$
$617$	24.8464	1.00028	0.500140	+	0.865945i	$0.333282\pi$
$618$	0	0
$619$	−6.50936	−0.261633	−0.130817	−	0.991407i	$-0.541760\pi$
$619$	−6.50936	−0.261633	−0.130817	+	0.991407i	$0.541760\pi$
$620$	0	0
$621$	9.02398	0.362120
$622$	0	0
$623$	20.7601	0.831737
$624$	0	0
$625$	22.8593	0.914374
$626$	0	0
$627$	−0.707538	−0.0282563
$628$	0	0
$629$	9.05341	0.360983
$630$	0	0
$631$	−29.8212	−1.18716	−0.593581	−	0.804774i	$-0.702286\pi$
$631$	−29.8212	−1.18716	−0.593581	+	0.804774i	$0.702286\pi$
$632$	0	0
$633$	−14.4230	−0.573263
$634$	0	0
$635$	3.71235	0.147320
$636$	0	0
$637$	40.4000	1.60071
$638$	0	0
$639$	−2.52321	−0.0998165
$640$	0	0
$641$	−5.15805	−0.203731	−0.101865	−	0.994798i	$-0.532481\pi$
$641$	−5.15805	−0.203731	−0.101865	+	0.994798i	$0.532481\pi$
$642$	0	0
$643$	−42.7171	−1.68460	−0.842298	−	0.539012i	$-0.818798\pi$
$643$	−42.7171	−1.68460	−0.842298	+	0.539012i	$0.818798\pi$
$644$	0	0
$645$	−0.356533	−0.0140385
$646$	0	0
$647$	8.12564	0.319452	0.159726	−	0.987161i	$-0.448939\pi$
$647$	8.12564	0.319452	0.159726	+	0.987161i	$0.448939\pi$
$648$	0	0
$649$	1.99915	0.0784737
$650$	0	0
$651$	0.568624	0.0222862
$652$	0	0
$653$	5.95951	0.233213	0.116607	−	0.993178i	$-0.462798\pi$
$653$	5.95951	0.233213	0.116607	+	0.993178i	$0.462798\pi$
$654$	0	0
$655$	−2.47971	−0.0968904
$656$	0	0
$657$	−7.93161	−0.309442
$658$	0	0
$659$	−18.6077	−0.724854	−0.362427	−	0.932012i	$-0.618052\pi$
$659$	−18.6077	−0.724854	−0.362427	+	0.932012i	$0.618052\pi$
$660$	0	0
$661$	−24.2797	−0.944373	−0.472186	−	0.881499i	$-0.656535\pi$
$661$	−24.2797	−0.944373	−0.472186	+	0.881499i	$0.656535\pi$
$662$	0	0
$663$	4.07363	0.158206
$664$	0	0
$665$	0.986047	0.0382373
$666$	0	0
$667$	−71.9718	−2.78676
$668$	0	0
$669$	−10.8481	−0.419411
$670$	0	0
$671$	13.5918	0.524705
$672$	0	0
$673$	11.9228	0.459592	0.229796	−	0.973239i	$-0.426194\pi$
$673$	11.9228	0.459592	0.229796	+	0.973239i	$0.426194\pi$
$674$	0	0
$675$	4.85591	0.186904
$676$	0	0
$677$	9.93980	0.382017	0.191009	−	0.981588i	$-0.438824\pi$
$677$	9.93980	0.382017	0.191009	+	0.981588i	$0.438824\pi$
$678$	0	0
$679$	24.3467	0.934342
$680$	0	0
$681$	−11.1128	−0.425842
$682$	0	0
$683$	31.8083	1.21711	0.608556	−	0.793511i	$-0.291749\pi$
$683$	31.8083	1.21711	0.608556	+	0.793511i	$0.291749\pi$
$684$	0	0
$685$	3.47461	0.132758
$686$	0	0
$687$	−3.74591	−0.142915
$688$	0	0
$689$	12.6788	0.483025
$690$	0	0
$691$	−31.4714	−1.19723	−0.598614	−	0.801037i	$-0.704282\pi$
$691$	−31.4714	−1.19723	−0.598614	+	0.801037i	$0.704282\pi$
$692$	0	0
$693$	−5.18420	−0.196932
$694$	0	0
$695$	−2.90753	−0.110289
$696$	0	0
$697$	3.52146	0.133385
$698$	0	0
$699$	−9.95027	−0.376354
$700$	0	0
$701$	−27.2879	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$701$	−27.2879	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$702$	0	0
$703$	−4.44283	−0.167565
$704$	0	0
$705$	−0.487620	−0.0183648
$706$	0	0
$707$	−3.32625	−0.125097
$708$	0	0
$709$	30.2672	1.13671	0.568355	−	0.822784i	$-0.307580\pi$
$709$	30.2672	1.13671	0.568355	+	0.822784i	$0.307580\pi$
$710$	0	0
$711$	−0.107292	−0.00402377
$712$	0	0
$713$	−1.17617	−0.0440479
$714$	0	0
$715$	−1.51447	−0.0566380
$716$	0	0
$717$	−10.2508	−0.382825
$718$	0	0
$719$	−42.1098	−1.57043	−0.785214	−	0.619224i	$-0.787447\pi$
$719$	−42.1098	−1.57043	−0.785214	+	0.619224i	$0.787447\pi$
$720$	0	0
$721$	40.2816	1.50017
$722$	0	0
$723$	7.19108	0.267439
$724$	0	0
$725$	−38.7289	−1.43835
$726$	0	0
$727$	4.56908	0.169458	0.0847288	−	0.996404i	$-0.472998\pi$
$727$	4.56908	0.169458	0.0847288	+	0.996404i	$0.472998\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.13959	0.0421493
$732$	0	0
$733$	11.2050	0.413867	0.206934	−	0.978355i	$-0.433652\pi$
$733$	11.2050	0.413867	0.206934	+	0.978355i	$0.433652\pi$
$734$	0	0
$735$	4.56769	0.168482
$736$	0	0
$737$	5.56093	0.204839
$738$	0	0
$739$	24.6663	0.907365	0.453683	−	0.891163i	$-0.350110\pi$
$739$	24.6663	0.907365	0.453683	+	0.891163i	$0.350110\pi$
$740$	0	0
$741$	−1.99908	−0.0734379
$742$	0	0
$743$	26.0756	0.956622	0.478311	−	0.878190i	$-0.341249\pi$
$743$	26.0756	0.956622	0.478311	+	0.878190i	$0.341249\pi$
$744$	0	0
$745$	−2.54834	−0.0933639
$746$	0	0
$747$	−13.7284	−0.502295
$748$	0	0
$749$	13.9931	0.511297
$750$	0	0
$751$	−32.3959	−1.18214	−0.591072	−	0.806618i	$-0.701295\pi$
$751$	−32.3959	−1.18214	−0.591072	+	0.806618i	$0.701295\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−6.93445	−0.252371
$756$	0	0
$757$	−9.70963	−0.352902	−0.176451	−	0.984309i	$-0.556462\pi$
$757$	−9.70963	−0.352902	−0.176451	+	0.984309i	$0.556462\pi$
$758$	0	0
$759$	10.7233	0.389230
$760$	0	0
$761$	14.5706	0.528183	0.264092	−	0.964498i	$-0.414928\pi$
$761$	14.5706	0.528183	0.264092	+	0.964498i	$0.414928\pi$
$762$	0	0
$763$	−50.8942	−1.84249
$764$	0	0
$765$	0.460571	0.0166520
$766$	0	0
$767$	5.64841	0.203952
$768$	0	0
$769$	−3.87482	−0.139729	−0.0698647	−	0.997556i	$-0.522257\pi$
$769$	−3.87482	−0.139729	−0.0698647	+	0.997556i	$0.522257\pi$
$770$	0	0
$771$	−8.16309	−0.293986
$772$	0	0
$773$	−17.3261	−0.623177	−0.311588	−	0.950217i	$-0.600861\pi$
$773$	−17.3261	−0.623177	−0.311588	+	0.950217i	$0.600861\pi$
$774$	0	0
$775$	−0.632911	−0.0227348
$776$	0	0
$777$	−32.5531	−1.16784
$778$	0	0
$779$	−1.72811	−0.0619159
$780$	0	0
$781$	−2.99835	−0.107289
$782$	0	0
$783$	−7.97562	−0.285026
$784$	0	0
$785$	−0.976062	−0.0348372
$786$	0	0
$787$	16.1312	0.575015	0.287507	−	0.957778i	$-0.407173\pi$
$787$	16.1312	0.575015	0.287507	+	0.957778i	$0.407173\pi$
$788$	0	0
$789$	3.63039	0.129245
$790$	0	0
$791$	−44.7079	−1.58963
$792$	0	0
$793$	38.4022	1.36370
$794$	0	0
$795$	1.43349	0.0508407
$796$	0	0
$797$	28.0334	0.992994	0.496497	−	0.868038i	$-0.334619\pi$
$797$	28.0334	0.992994	0.496497	+	0.868038i	$0.334619\pi$
$798$	0	0
$799$	1.55858	0.0551387
$800$	0	0
$801$	−4.75857	−0.168136
$802$	0	0
$803$	−9.42519	−0.332608
$804$	0	0
$805$	−14.9443	−0.526717
$806$	0	0
$807$	−11.1430	−0.392253
$808$	0	0
$809$	−7.70645	−0.270945	−0.135472	−	0.990781i	$-0.543255\pi$
$809$	−7.70645	−0.270945	−0.135472	+	0.990781i	$0.543255\pi$
$810$	0	0
$811$	20.3118	0.713245	0.356622	−	0.934249i	$-0.383928\pi$
$811$	20.3118	0.713245	0.356622	+	0.934249i	$0.383928\pi$
$812$	0	0
$813$	−28.2618	−0.991183
$814$	0	0
$815$	−5.65296	−0.198015
$816$	0	0
$817$	−0.559238	−0.0195653
$818$	0	0
$819$	−14.6474	−0.511823
$820$	0	0
$821$	37.6022	1.31233	0.656163	−	0.754619i	$-0.272178\pi$
$821$	37.6022	1.31233	0.656163	+	0.754619i	$0.272178\pi$
$822$	0	0
$823$	26.0144	0.906804	0.453402	−	0.891306i	$-0.350210\pi$
$823$	26.0144	0.906804	0.453402	+	0.891306i	$0.350210\pi$
$824$	0	0
$825$	5.77031	0.200896
$826$	0	0
$827$	−17.3507	−0.603343	−0.301672	−	0.953412i	$-0.597545\pi$
$827$	−17.3507	−0.603343	−0.301672	+	0.953412i	$0.597545\pi$
$828$	0	0
$829$	−26.1634	−0.908693	−0.454346	−	0.890825i	$-0.650127\pi$
$829$	−26.1634	−0.908693	−0.454346	+	0.890825i	$0.650127\pi$
$830$	0	0
$831$	−26.9136	−0.933623
$832$	0	0
$833$	−14.5997	−0.505851
$834$	0	0
$835$	−3.77735	−0.130721
$836$	0	0
$837$	−0.130338	−0.00450515
$838$	0	0
$839$	−25.9982	−0.897558	−0.448779	−	0.893643i	$-0.648141\pi$
$839$	−25.9982	−0.897558	−0.448779	+	0.893643i	$0.648141\pi$
$840$	0	0
$841$	34.6106	1.19347
$842$	0	0
$843$	−27.1977	−0.936737
$844$	0	0
$845$	0.655786	0.0225597
$846$	0	0
$847$	41.8290	1.43726
$848$	0	0
$849$	3.63960	0.124911
$850$	0	0
$851$	67.3345	2.30820
$852$	0	0
$853$	21.0137	0.719494	0.359747	−	0.933050i	$-0.382863\pi$
$853$	21.0137	0.719494	0.359747	+	0.933050i	$0.382863\pi$
$854$	0	0
$855$	−0.226019	−0.00772968
$856$	0	0
$857$	2.96330	0.101224	0.0506122	−	0.998718i	$-0.483883\pi$
$857$	2.96330	0.101224	0.0506122	+	0.998718i	$0.483883\pi$
$858$	0	0
$859$	−17.8080	−0.607601	−0.303800	−	0.952736i	$-0.598256\pi$
$859$	−17.8080	−0.607601	−0.303800	+	0.952736i	$0.598256\pi$
$860$	0	0
$861$	−12.6620	−0.431521
$862$	0	0
$863$	−9.63306	−0.327913	−0.163957	−	0.986468i	$-0.552426\pi$
$863$	−9.63306	−0.327913	−0.163957	+	0.986468i	$0.552426\pi$
$864$	0	0
$865$	−4.95990	−0.168642
$866$	0	0
$867$	15.5279	0.527354
$868$	0	0
$869$	−0.127496	−0.00432500
$870$	0	0
$871$	15.7118	0.532375
$872$	0	0
$873$	−5.58069	−0.188878
$874$	0	0
$875$	−16.3220	−0.551784
$876$	0	0
$877$	−32.4765	−1.09665	−0.548326	−	0.836264i	$-0.684735\pi$
$877$	−32.4765	−1.09665	−0.548326	+	0.836264i	$0.684735\pi$
$878$	0	0
$879$	−28.1973	−0.951071
$880$	0	0
$881$	25.9585	0.874565	0.437282	−	0.899324i	$-0.355941\pi$
$881$	25.9585	0.874565	0.437282	+	0.899324i	$0.355941\pi$
$882$	0	0
$883$	37.1262	1.24940	0.624698	−	0.780866i	$-0.285222\pi$
$883$	37.1262	1.24940	0.624698	+	0.780866i	$0.285222\pi$
$884$	0	0
$885$	0.638618	0.0214669
$886$	0	0
$887$	20.8797	0.701070	0.350535	−	0.936550i	$-0.386000\pi$
$887$	20.8797	0.701070	0.350535	+	0.936550i	$0.386000\pi$
$888$	0	0
$889$	42.6656	1.43096
$890$	0	0
$891$	1.18831	0.0398098
$892$	0	0
$893$	−0.764853	−0.0255948
$894$	0	0
$895$	−4.60398	−0.153894
$896$	0	0
$897$	30.2975	1.01160
$898$	0	0
$899$	1.03953	0.0346703
$900$	0	0
$901$	−4.58188	−0.152644
$902$	0	0
$903$	−4.09760	−0.136360
$904$	0	0
$905$	−1.12710	−0.0374659
$906$	0	0
$907$	−37.9522	−1.26018	−0.630091	−	0.776521i	$-0.716983\pi$
$907$	−37.9522	−1.26018	−0.630091	+	0.776521i	$0.716983\pi$
$908$	0	0
$909$	0.762433	0.0252883
$910$	0	0
$911$	3.07859	0.101998	0.0509991	−	0.998699i	$-0.483759\pi$
$911$	3.07859	0.101998	0.0509991	+	0.998699i	$0.483759\pi$
$912$	0	0
$913$	−16.3135	−0.539899
$914$	0	0
$915$	4.34181	0.143536
$916$	0	0
$917$	−28.4991	−0.941123
$918$	0	0
$919$	17.3790	0.573279	0.286640	−	0.958038i	$-0.407462\pi$
$919$	17.3790	0.573279	0.286640	+	0.958038i	$0.407462\pi$
$920$	0	0
$921$	5.51754	0.181809
$922$	0	0
$923$	−8.47152	−0.278844
$924$	0	0
$925$	36.2335	1.19135
$926$	0	0
$927$	−9.23323	−0.303259
$928$	0	0
$929$	−28.8779	−0.947452	−0.473726	−	0.880672i	$-0.657091\pi$
$929$	−28.8779	−0.947452	−0.473726	+	0.880672i	$0.657091\pi$
$930$	0	0
$931$	7.16462	0.234811
$932$	0	0
$933$	17.2248	0.563914
$934$	0	0
$935$	0.547300	0.0178986
$936$	0	0
$937$	−18.0332	−0.589118	−0.294559	−	0.955633i	$-0.595173\pi$
$937$	−18.0332	−0.589118	−0.294559	+	0.955633i	$0.595173\pi$
$938$	0	0
$939$	−14.6048	−0.476609
$940$	0	0
$941$	20.2316	0.659530	0.329765	−	0.944063i	$-0.393030\pi$
$941$	20.2316	0.659530	0.329765	+	0.944063i	$0.393030\pi$
$942$	0	0
$943$	26.1908	0.852889
$944$	0	0
$945$	−1.65606	−0.0538717
$946$	0	0
$947$	53.9195	1.75215	0.876074	−	0.482177i	$-0.160154\pi$
$947$	53.9195	1.75215	0.876074	+	0.482177i	$0.160154\pi$
$948$	0	0
$949$	−26.6299	−0.864444
$950$	0	0
$951$	3.71983	0.120624
$952$	0	0
$953$	−1.98241	−0.0642166	−0.0321083	−	0.999484i	$-0.510222\pi$
$953$	−1.98241	−0.0642166	−0.0321083	+	0.999484i	$0.510222\pi$
$954$	0	0
$955$	7.13041	0.230735
$956$	0	0
$957$	−9.47749	−0.306364
$958$	0	0
$959$	39.9333	1.28951
$960$	0	0
$961$	−30.9830	−0.999452
$962$	0	0
$963$	−3.20746	−0.103359
$964$	0	0
$965$	−4.87573	−0.156955
$966$	0	0
$967$	−9.29039	−0.298759	−0.149379	−	0.988780i	$-0.547728\pi$
$967$	−9.29039	−0.298759	−0.149379	+	0.988780i	$0.547728\pi$
$968$	0	0
$969$	0.722426	0.0232077
$970$	0	0
$971$	32.8179	1.05318	0.526589	−	0.850120i	$-0.323471\pi$
$971$	32.8179	1.05318	0.526589	+	0.850120i	$0.323471\pi$
$972$	0	0
$973$	−33.4159	−1.07126
$974$	0	0
$975$	16.3034	0.522127
$976$	0	0
$977$	2.64180	0.0845188	0.0422594	−	0.999107i	$-0.486544\pi$
$977$	2.64180	0.0845188	0.0422594	+	0.999107i	$0.486544\pi$
$978$	0	0
$979$	−5.65465	−0.180723
$980$	0	0
$981$	11.6658	0.372461
$982$	0	0
$983$	2.15010	0.0685775	0.0342887	−	0.999412i	$-0.489083\pi$
$983$	2.15010	0.0685775	0.0342887	+	0.999412i	$0.489083\pi$
$984$	0	0
$985$	−2.96025	−0.0943213
$986$	0	0
$987$	−5.60416	−0.178382
$988$	0	0
$989$	8.47567	0.269511
$990$	0	0
$991$	−31.2775	−0.993564	−0.496782	−	0.867875i	$-0.665485\pi$
$991$	−31.2775	−0.993564	−0.496782	+	0.867875i	$0.665485\pi$
$992$	0	0
$993$	17.1677	0.544801
$994$	0	0
$995$	2.95754	0.0937602
$996$	0	0
$997$	−2.78493	−0.0881996	−0.0440998	−	0.999027i	$-0.514042\pi$
$997$	−2.78493	−0.0881996	−0.0440998	+	0.999027i	$0.514042\pi$
$998$	0	0
$999$	7.46173	0.236079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.6	✓	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.379598 q^{5} -4.36268 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.379598 q^{5} -4.36268 q^{7} +1.00000 q^{9} +1.18831 q^{11} +3.35744 q^{13} +0.379598 q^{15} -1.21331 q^{17} +0.595416 q^{19} +4.36268 q^{21} -9.02398 q^{23} -4.85591 q^{25} -1.00000 q^{27} +7.97562 q^{29} +0.130338 q^{31} -1.18831 q^{33} +1.65606 q^{35} -7.46173 q^{37} -3.35744 q^{39} -2.90235 q^{41} -0.939239 q^{43} -0.379598 q^{45} -1.28457 q^{47} +12.0330 q^{49} +1.21331 q^{51} +3.77634 q^{53} -0.451079 q^{55} -0.595416 q^{57} +1.68235 q^{59} +11.4379 q^{61} -4.36268 q^{63} -1.27448 q^{65} +4.67971 q^{67} +9.02398 q^{69} -2.52321 q^{71} -7.93161 q^{73} +4.85591 q^{75} -5.18420 q^{77} -0.107292 q^{79} +1.00000 q^{81} -13.7284 q^{83} +0.460571 q^{85} -7.97562 q^{87} -4.75857 q^{89} -14.6474 q^{91} -0.130338 q^{93} -0.226019 q^{95} -5.58069 q^{97} +1.18831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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