LMFDB - Embedded newform 6027.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{4} +3.46410 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{4} +3.46410 q^{5} +1.00000 q^{9} +5.73205 q^{11} -2.00000 q^{12} -5.46410 q^{13} +3.46410 q^{15} +4.00000 q^{16} -5.00000 q^{17} -0.535898 q^{19} -6.92820 q^{20} +5.46410 q^{23} +7.00000 q^{25} +1.00000 q^{27} -5.19615 q^{29} -5.73205 q^{31} +5.73205 q^{33} -2.00000 q^{36} -1.00000 q^{37} -5.46410 q^{39} +1.00000 q^{41} +9.92820 q^{43} -11.4641 q^{44} +3.46410 q^{45} +11.9282 q^{47} +4.00000 q^{48} -5.00000 q^{51} +10.9282 q^{52} +10.9282 q^{53} +19.8564 q^{55} -0.535898 q^{57} +6.92820 q^{59} -6.92820 q^{60} +1.73205 q^{61} -8.00000 q^{64} -18.9282 q^{65} +7.46410 q^{67} +10.0000 q^{68} +5.46410 q^{69} +13.1962 q^{71} +9.19615 q^{73} +7.00000 q^{75} +1.07180 q^{76} +0.928203 q^{79} +13.8564 q^{80} +1.00000 q^{81} -10.0000 q^{83} -17.3205 q^{85} -5.19615 q^{87} -8.92820 q^{89} -10.9282 q^{92} -5.73205 q^{93} -1.85641 q^{95} -2.39230 q^{97} +5.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9} + 8 q^{11} - 4 q^{12} - 4 q^{13} + 8 q^{16} - 10 q^{17} - 8 q^{19} + 4 q^{23} + 14 q^{25} + 2 q^{27} - 8 q^{31} + 8 q^{33} - 4 q^{36} - 2 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} - 16 q^{44} + 10 q^{47} + 8 q^{48} - 10 q^{51} + 8 q^{52} + 8 q^{53} + 12 q^{55} - 8 q^{57} - 16 q^{64} - 24 q^{65} + 8 q^{67} + 20 q^{68} + 4 q^{69} + 16 q^{71} + 8 q^{73} + 14 q^{75} + 16 q^{76} - 12 q^{79} + 2 q^{81} - 20 q^{83} - 4 q^{89} - 8 q^{92} - 8 q^{93} + 24 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9 + 8 * q^11 - 4 * q^12 - 4 * q^13 + 8 * q^16 - 10 * q^17 - 8 * q^19 + 4 * q^23 + 14 * q^25 + 2 * q^27 - 8 * q^31 + 8 * q^33 - 4 * q^36 - 2 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 - 16 * q^44 + 10 * q^47 + 8 * q^48 - 10 * q^51 + 8 * q^52 + 8 * q^53 + 12 * q^55 - 8 * q^57 - 16 * q^64 - 24 * q^65 + 8 * q^67 + 20 * q^68 + 4 * q^69 + 16 * q^71 + 8 * q^73 + 14 * q^75 + 16 * q^76 - 12 * q^79 + 2 * q^81 - 20 * q^83 - 4 * q^89 - 8 * q^92 - 8 * q^93 + 24 * q^95 + 16 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−2.00000	−1.00000
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.73205	1.72828	0.864139	−	0.503253i	$-0.167864\pi$
$11$	5.73205	1.72828	0.864139	+	0.503253i	$0.167864\pi$
$12$	−2.00000	−0.577350
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	4.00000	1.00000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−0.535898	−0.122944	−0.0614718	−	0.998109i	$-0.519579\pi$
$19$	−0.535898	−0.122944	−0.0614718	+	0.998109i	$0.519579\pi$
$20$	−6.92820	−1.54919
$21$	0	0
$22$	0	0
$23$	5.46410	1.13934	0.569672	−	0.821872i	$-0.307070\pi$
$23$	5.46410	1.13934	0.569672	+	0.821872i	$0.307070\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.19615	−0.964901	−0.482451	−	0.875923i	$-0.660253\pi$
$29$	−5.19615	−0.964901	−0.482451	+	0.875923i	$0.660253\pi$
$30$	0	0
$31$	−5.73205	−1.02951	−0.514753	−	0.857338i	$-0.672117\pi$
$31$	−5.73205	−1.02951	−0.514753	+	0.857338i	$0.672117\pi$
$32$	0	0
$33$	5.73205	0.997822
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	−5.46410	−0.874957
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	9.92820	1.51404	0.757018	−	0.653394i	$-0.226655\pi$
$43$	9.92820	1.51404	0.757018	+	0.653394i	$0.226655\pi$
$44$	−11.4641	−1.72828
$45$	3.46410	0.516398
$46$	0	0
$47$	11.9282	1.73991	0.869954	−	0.493134i	$-0.164149\pi$
$47$	11.9282	1.73991	0.869954	+	0.493134i	$0.164149\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	0	0
$51$	−5.00000	−0.700140
$52$	10.9282	1.51547
$53$	10.9282	1.50110	0.750552	−	0.660811i	$-0.229788\pi$
$53$	10.9282	1.50110	0.750552	+	0.660811i	$0.229788\pi$
$54$	0	0
$55$	19.8564	2.67744
$56$	0	0
$57$	−0.535898	−0.0709815
$58$	0	0
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	−6.92820	−0.894427
$61$	1.73205	0.221766	0.110883	−	0.993833i	$-0.464632\pi$
$61$	1.73205	0.221766	0.110883	+	0.993833i	$0.464632\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−18.9282	−2.34775
$66$	0	0
$67$	7.46410	0.911885	0.455943	−	0.890009i	$-0.349302\pi$
$67$	7.46410	0.911885	0.455943	+	0.890009i	$0.349302\pi$
$68$	10.0000	1.21268
$69$	5.46410	0.657801
$70$	0	0
$71$	13.1962	1.56610	0.783048	−	0.621962i	$-0.213664\pi$
$71$	13.1962	1.56610	0.783048	+	0.621962i	$0.213664\pi$
$72$	0	0
$73$	9.19615	1.07633	0.538164	−	0.842840i	$-0.319118\pi$
$73$	9.19615	1.07633	0.538164	+	0.842840i	$0.319118\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	1.07180	0.122944
$77$	0	0
$78$	0	0
$79$	0.928203	0.104431	0.0522155	−	0.998636i	$-0.483372\pi$
$79$	0.928203	0.104431	0.0522155	+	0.998636i	$0.483372\pi$
$80$	13.8564	1.54919
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	−17.3205	−1.87867
$86$	0	0
$87$	−5.19615	−0.557086
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	0	0
$92$	−10.9282	−1.13934
$93$	−5.73205	−0.594386
$94$	0	0
$95$	−1.85641	−0.190463
$96$	0	0
$97$	−2.39230	−0.242902	−0.121451	−	0.992597i	$-0.538755\pi$
$97$	−2.39230	−0.242902	−0.121451	+	0.992597i	$0.538755\pi$
$98$	0	0
$99$	5.73205	0.576093
$100$	−14.0000	−1.40000
$101$	−18.8564	−1.87628	−0.938141	−	0.346253i	$-0.887454\pi$
$101$	−18.8564	−1.87628	−0.938141	+	0.346253i	$0.887454\pi$
$102$	0	0
$103$	1.19615	0.117860	0.0589302	−	0.998262i	$-0.481231\pi$
$103$	1.19615	0.117860	0.0589302	+	0.998262i	$0.481231\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.46410	−0.334887	−0.167444	−	0.985882i	$-0.553551\pi$
$107$	−3.46410	−0.334887	−0.167444	+	0.985882i	$0.553551\pi$
$108$	−2.00000	−0.192450
$109$	2.53590	0.242895	0.121448	−	0.992598i	$-0.461246\pi$
$109$	2.53590	0.242895	0.121448	+	0.992598i	$0.461246\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	0	0
$115$	18.9282	1.76506
$116$	10.3923	0.964901
$117$	−5.46410	−0.505156
$118$	0	0
$119$	0	0
$120$	0	0
$121$	21.8564	1.98695
$122$	0	0
$123$	1.00000	0.0901670
$124$	11.4641	1.02951
$125$	6.92820	0.619677
$126$	0	0
$127$	17.8564	1.58450	0.792250	−	0.610197i	$-0.208910\pi$
$127$	17.8564	1.58450	0.792250	+	0.610197i	$0.208910\pi$
$128$	0	0
$129$	9.92820	0.874130
$130$	0	0
$131$	10.5359	0.920526	0.460263	−	0.887783i	$-0.347755\pi$
$131$	10.5359	0.920526	0.460263	+	0.887783i	$0.347755\pi$
$132$	−11.4641	−0.997822
$133$	0	0
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	2.80385	0.239549	0.119774	−	0.992801i	$-0.461783\pi$
$137$	2.80385	0.239549	0.119774	+	0.992801i	$0.461783\pi$
$138$	0	0
$139$	11.4641	0.972372	0.486186	−	0.873855i	$-0.338388\pi$
$139$	11.4641	0.972372	0.486186	+	0.873855i	$0.338388\pi$
$140$	0	0
$141$	11.9282	1.00454
$142$	0	0
$143$	−31.3205	−2.61915
$144$	4.00000	0.333333
$145$	−18.0000	−1.49482
$146$	0	0
$147$	0	0
$148$	2.00000	0.164399
$149$	−13.8564	−1.13516	−0.567581	−	0.823318i	$-0.692120\pi$
$149$	−13.8564	−1.13516	−0.567581	+	0.823318i	$0.692120\pi$
$150$	0	0
$151$	20.3923	1.65950	0.829751	−	0.558134i	$-0.188482\pi$
$151$	20.3923	1.65950	0.829751	+	0.558134i	$0.188482\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	−19.8564	−1.59490
$156$	10.9282	0.874957
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	10.9282	0.866663
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−17.9282	−1.40425	−0.702123	−	0.712056i	$-0.747764\pi$
$163$	−17.9282	−1.40425	−0.702123	+	0.712056i	$0.747764\pi$
$164$	−2.00000	−0.156174
$165$	19.8564	1.54582
$166$	0	0
$167$	18.9282	1.46471	0.732354	−	0.680924i	$-0.238422\pi$
$167$	18.9282	1.46471	0.732354	+	0.680924i	$0.238422\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	−0.535898	−0.0409812
$172$	−19.8564	−1.51404
$173$	−7.07180	−0.537659	−0.268829	−	0.963188i	$-0.586637\pi$
$173$	−7.07180	−0.537659	−0.268829	+	0.963188i	$0.586637\pi$
$174$	0	0
$175$	0	0
$176$	22.9282	1.72828
$177$	6.92820	0.520756
$178$	0	0
$179$	15.5885	1.16514	0.582568	−	0.812782i	$-0.302048\pi$
$179$	15.5885	1.16514	0.582568	+	0.812782i	$0.302048\pi$
$180$	−6.92820	−0.516398
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	1.73205	0.128037
$184$	0	0
$185$	−3.46410	−0.254686
$186$	0	0
$187$	−28.6603	−2.09585
$188$	−23.8564	−1.73991
$189$	0	0
$190$	0	0
$191$	−10.3923	−0.751961	−0.375980	−	0.926628i	$-0.622694\pi$
$191$	−10.3923	−0.751961	−0.375980	+	0.926628i	$0.622694\pi$
$192$	−8.00000	−0.577350
$193$	−7.32051	−0.526942	−0.263471	−	0.964667i	$-0.584867\pi$
$193$	−7.32051	−0.526942	−0.263471	+	0.964667i	$0.584867\pi$
$194$	0	0
$195$	−18.9282	−1.35548
$196$	0	0
$197$	0.535898	0.0381812	0.0190906	−	0.999818i	$-0.493923\pi$
$197$	0.535898	0.0381812	0.0190906	+	0.999818i	$0.493923\pi$
$198$	0	0
$199$	−25.3205	−1.79492	−0.897462	−	0.441093i	$-0.854591\pi$
$199$	−25.3205	−1.79492	−0.897462	+	0.441093i	$0.854591\pi$
$200$	0	0
$201$	7.46410	0.526477
$202$	0	0
$203$	0	0
$204$	10.0000	0.700140
$205$	3.46410	0.241943
$206$	0	0
$207$	5.46410	0.379781
$208$	−21.8564	−1.51547
$209$	−3.07180	−0.212481
$210$	0	0
$211$	14.9282	1.02770	0.513850	−	0.857880i	$-0.328219\pi$
$211$	14.9282	1.02770	0.513850	+	0.857880i	$0.328219\pi$
$212$	−21.8564	−1.50110
$213$	13.1962	0.904185
$214$	0	0
$215$	34.3923	2.34554
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.19615	0.621418
$220$	−39.7128	−2.67744
$221$	27.3205	1.83778
$222$	0	0
$223$	−8.53590	−0.571606	−0.285803	−	0.958288i	$-0.592260\pi$
$223$	−8.53590	−0.571606	−0.285803	+	0.958288i	$0.592260\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	0.0717968	0.00476532	0.00238266	−	0.999997i	$-0.499242\pi$
$227$	0.0717968	0.00476532	0.00238266	+	0.999997i	$0.499242\pi$
$228$	1.07180	0.0709815
$229$	−8.92820	−0.589992	−0.294996	−	0.955498i	$-0.595318\pi$
$229$	−8.92820	−0.589992	−0.294996	+	0.955498i	$0.595318\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.8564	−0.907763	−0.453882	−	0.891062i	$-0.649961\pi$
$233$	−13.8564	−0.907763	−0.453882	+	0.891062i	$0.649961\pi$
$234$	0	0
$235$	41.3205	2.69545
$236$	−13.8564	−0.901975
$237$	0.928203	0.0602933
$238$	0	0
$239$	−15.4641	−1.00029	−0.500145	−	0.865942i	$-0.666720\pi$
$239$	−15.4641	−1.00029	−0.500145	+	0.865942i	$0.666720\pi$
$240$	13.8564	0.894427
$241$	5.19615	0.334714	0.167357	−	0.985896i	$-0.446477\pi$
$241$	5.19615	0.334714	0.167357	+	0.985896i	$0.446477\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−3.46410	−0.221766
$245$	0	0
$246$	0	0
$247$	2.92820	0.186317
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	7.46410	0.471130	0.235565	−	0.971859i	$-0.424306\pi$
$251$	7.46410	0.471130	0.235565	+	0.971859i	$0.424306\pi$
$252$	0	0
$253$	31.3205	1.96910
$254$	0	0
$255$	−17.3205	−1.08465
$256$	16.0000	1.00000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	0	0
$260$	37.8564	2.34775
$261$	−5.19615	−0.321634
$262$	0	0
$263$	−20.1244	−1.24092	−0.620460	−	0.784238i	$-0.713054\pi$
$263$	−20.1244	−1.24092	−0.620460	+	0.784238i	$0.713054\pi$
$264$	0	0
$265$	37.8564	2.32550
$266$	0	0
$267$	−8.92820	−0.546397
$268$	−14.9282	−0.911885
$269$	17.4641	1.06481	0.532403	−	0.846491i	$-0.321289\pi$
$269$	17.4641	1.06481	0.532403	+	0.846491i	$0.321289\pi$
$270$	0	0
$271$	−11.5885	−0.703949	−0.351974	−	0.936010i	$-0.614490\pi$
$271$	−11.5885	−0.703949	−0.351974	+	0.936010i	$0.614490\pi$
$272$	−20.0000	−1.21268
$273$	0	0
$274$	0	0
$275$	40.1244	2.41959
$276$	−10.9282	−0.657801
$277$	−21.9282	−1.31754	−0.658769	−	0.752345i	$-0.728923\pi$
$277$	−21.9282	−1.31754	−0.658769	+	0.752345i	$0.728923\pi$
$278$	0	0
$279$	−5.73205	−0.343169
$280$	0	0
$281$	−0.124356	−0.00741844	−0.00370922	−	0.999993i	$-0.501181\pi$
$281$	−0.124356	−0.00741844	−0.00370922	+	0.999993i	$0.501181\pi$
$282$	0	0
$283$	4.66025	0.277023	0.138512	−	0.990361i	$-0.455768\pi$
$283$	4.66025	0.277023	0.138512	+	0.990361i	$0.455768\pi$
$284$	−26.3923	−1.56610
$285$	−1.85641	−0.109964
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.39230	−0.140239
$292$	−18.3923	−1.07633
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	0	0
$297$	5.73205	0.332607
$298$	0	0
$299$	−29.8564	−1.72664
$300$	−14.0000	−0.808290
$301$	0	0
$302$	0	0
$303$	−18.8564	−1.08327
$304$	−2.14359	−0.122944
$305$	6.00000	0.343559
$306$	0	0
$307$	14.2679	0.814315	0.407157	−	0.913358i	$-0.366520\pi$
$307$	14.2679	0.814315	0.407157	+	0.913358i	$0.366520\pi$
$308$	0	0
$309$	1.19615	0.0680467
$310$	0	0
$311$	25.8564	1.46618	0.733091	−	0.680130i	$-0.238077\pi$
$311$	25.8564	1.46618	0.733091	+	0.680130i	$0.238077\pi$
$312$	0	0
$313$	2.92820	0.165512	0.0827559	−	0.996570i	$-0.473628\pi$
$313$	2.92820	0.165512	0.0827559	+	0.996570i	$0.473628\pi$
$314$	0	0
$315$	0	0
$316$	−1.85641	−0.104431
$317$	22.8038	1.28079	0.640396	−	0.768045i	$-0.278770\pi$
$317$	22.8038	1.28079	0.640396	+	0.768045i	$0.278770\pi$
$318$	0	0
$319$	−29.7846	−1.66762
$320$	−27.7128	−1.54919
$321$	−3.46410	−0.193347
$322$	0	0
$323$	2.67949	0.149091
$324$	−2.00000	−0.111111
$325$	−38.2487	−2.12166
$326$	0	0
$327$	2.53590	0.140236
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.2487	0.893110	0.446555	−	0.894756i	$-0.352651\pi$
$331$	16.2487	0.893110	0.446555	+	0.894756i	$0.352651\pi$
$332$	20.0000	1.09764
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	25.8564	1.41269
$336$	0	0
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	0	0
$339$	−17.3205	−0.940721
$340$	34.6410	1.87867
$341$	−32.8564	−1.77927
$342$	0	0
$343$	0	0
$344$	0	0
$345$	18.9282	1.01906
$346$	0	0
$347$	−4.66025	−0.250176	−0.125088	−	0.992146i	$-0.539921\pi$
$347$	−4.66025	−0.250176	−0.125088	+	0.992146i	$0.539921\pi$
$348$	10.3923	0.557086
$349$	3.58846	0.192086	0.0960429	−	0.995377i	$-0.469381\pi$
$349$	3.58846	0.192086	0.0960429	+	0.995377i	$0.469381\pi$
$350$	0	0
$351$	−5.46410	−0.291652
$352$	0	0
$353$	13.3205	0.708979	0.354490	−	0.935060i	$-0.384655\pi$
$353$	13.3205	0.708979	0.354490	+	0.935060i	$0.384655\pi$
$354$	0	0
$355$	45.7128	2.42618
$356$	17.8564	0.946388
$357$	0	0
$358$	0	0
$359$	31.7128	1.67374	0.836869	−	0.547403i	$-0.184384\pi$
$359$	31.7128	1.67374	0.836869	+	0.547403i	$0.184384\pi$
$360$	0	0
$361$	−18.7128	−0.984885
$362$	0	0
$363$	21.8564	1.14716
$364$	0	0
$365$	31.8564	1.66744
$366$	0	0
$367$	−15.0526	−0.785737	−0.392869	−	0.919595i	$-0.628517\pi$
$367$	−15.0526	−0.785737	−0.392869	+	0.919595i	$0.628517\pi$
$368$	21.8564	1.13934
$369$	1.00000	0.0520579
$370$	0	0
$371$	0	0
$372$	11.4641	0.594386
$373$	7.14359	0.369881	0.184941	−	0.982750i	$-0.440791\pi$
$373$	7.14359	0.369881	0.184941	+	0.982750i	$0.440791\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	28.3923	1.46228
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	3.71281	0.190463
$381$	17.8564	0.914811
$382$	0	0
$383$	35.9282	1.83585	0.917923	−	0.396759i	$-0.129865\pi$
$383$	35.9282	1.83585	0.917923	+	0.396759i	$0.129865\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.92820	0.504679
$388$	4.78461	0.242902
$389$	−6.78461	−0.343993	−0.171997	−	0.985098i	$-0.555022\pi$
$389$	−6.78461	−0.343993	−0.171997	+	0.985098i	$0.555022\pi$
$390$	0	0
$391$	−27.3205	−1.38166
$392$	0	0
$393$	10.5359	0.531466
$394$	0	0
$395$	3.21539	0.161784
$396$	−11.4641	−0.576093
$397$	−17.4641	−0.876498	−0.438249	−	0.898854i	$-0.644401\pi$
$397$	−17.4641	−0.876498	−0.438249	+	0.898854i	$0.644401\pi$
$398$	0	0
$399$	0	0
$400$	28.0000	1.40000
$401$	−17.0718	−0.852525	−0.426262	−	0.904600i	$-0.640170\pi$
$401$	−17.0718	−0.852525	−0.426262	+	0.904600i	$0.640170\pi$
$402$	0	0
$403$	31.3205	1.56019
$404$	37.7128	1.87628
$405$	3.46410	0.172133
$406$	0	0
$407$	−5.73205	−0.284127
$408$	0	0
$409$	−5.73205	−0.283432	−0.141716	−	0.989907i	$-0.545262\pi$
$409$	−5.73205	−0.283432	−0.141716	+	0.989907i	$0.545262\pi$
$410$	0	0
$411$	2.80385	0.138304
$412$	−2.39230	−0.117860
$413$	0	0
$414$	0	0
$415$	−34.6410	−1.70046
$416$	0	0
$417$	11.4641	0.561399
$418$	0	0
$419$	−31.7128	−1.54927	−0.774636	−	0.632407i	$-0.782067\pi$
$419$	−31.7128	−1.54927	−0.774636	+	0.632407i	$0.782067\pi$
$420$	0	0
$421$	25.4641	1.24104	0.620522	−	0.784189i	$-0.286921\pi$
$421$	25.4641	1.24104	0.620522	+	0.784189i	$0.286921\pi$
$422$	0	0
$423$	11.9282	0.579969
$424$	0	0
$425$	−35.0000	−1.69775
$426$	0	0
$427$	0	0
$428$	6.92820	0.334887
$429$	−31.3205	−1.51217
$430$	0	0
$431$	2.53590	0.122150	0.0610750	−	0.998133i	$-0.480547\pi$
$431$	2.53590	0.122150	0.0610750	+	0.998133i	$0.480547\pi$
$432$	4.00000	0.192450
$433$	−31.5885	−1.51804	−0.759022	−	0.651065i	$-0.774323\pi$
$433$	−31.5885	−1.51804	−0.759022	+	0.651065i	$0.774323\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	−5.07180	−0.242895
$437$	−2.92820	−0.140075
$438$	0	0
$439$	−28.5359	−1.36194	−0.680972	−	0.732309i	$-0.738443\pi$
$439$	−28.5359	−1.36194	−0.680972	+	0.732309i	$0.738443\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.60770	0.171407	0.0857034	−	0.996321i	$-0.472686\pi$
$443$	3.60770	0.171407	0.0857034	+	0.996321i	$0.472686\pi$
$444$	2.00000	0.0949158
$445$	−30.9282	−1.46614
$446$	0	0
$447$	−13.8564	−0.655386
$448$	0	0
$449$	−23.3205	−1.10056	−0.550281	−	0.834979i	$-0.685480\pi$
$449$	−23.3205	−1.10056	−0.550281	+	0.834979i	$0.685480\pi$
$450$	0	0
$451$	5.73205	0.269912
$452$	34.6410	1.62938
$453$	20.3923	0.958114
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.6410	−1.80755	−0.903775	−	0.428007i	$-0.859216\pi$
$457$	−38.6410	−1.80755	−0.903775	+	0.428007i	$0.859216\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	−37.8564	−1.76506
$461$	−24.2487	−1.12938	−0.564688	−	0.825305i	$-0.691003\pi$
$461$	−24.2487	−1.12938	−0.564688	+	0.825305i	$0.691003\pi$
$462$	0	0
$463$	8.39230	0.390023	0.195012	−	0.980801i	$-0.437526\pi$
$463$	8.39230	0.390023	0.195012	+	0.980801i	$0.437526\pi$
$464$	−20.7846	−0.964901
$465$	−19.8564	−0.920819
$466$	0	0
$467$	−0.535898	−0.0247984	−0.0123992	−	0.999923i	$-0.503947\pi$
$467$	−0.535898	−0.0247984	−0.0123992	+	0.999923i	$0.503947\pi$
$468$	10.9282	0.505156
$469$	0	0
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	56.9090	2.61668
$474$	0	0
$475$	−3.75129	−0.172121
$476$	0	0
$477$	10.9282	0.500368
$478$	0	0
$479$	40.7128	1.86022	0.930108	−	0.367286i	$-0.119713\pi$
$479$	40.7128	1.86022	0.930108	+	0.367286i	$0.119713\pi$
$480$	0	0
$481$	5.46410	0.249142
$482$	0	0
$483$	0	0
$484$	−43.7128	−1.98695
$485$	−8.28719	−0.376302
$486$	0	0
$487$	2.07180	0.0938821	0.0469410	−	0.998898i	$-0.485053\pi$
$487$	2.07180	0.0938821	0.0469410	+	0.998898i	$0.485053\pi$
$488$	0	0
$489$	−17.9282	−0.810741
$490$	0	0
$491$	3.85641	0.174037	0.0870186	−	0.996207i	$-0.472266\pi$
$491$	3.85641	0.174037	0.0870186	+	0.996207i	$0.472266\pi$
$492$	−2.00000	−0.0901670
$493$	25.9808	1.17011
$494$	0	0
$495$	19.8564	0.892479
$496$	−22.9282	−1.02951
$497$	0	0
$498$	0	0
$499$	−1.46410	−0.0655422	−0.0327711	−	0.999463i	$-0.510433\pi$
$499$	−1.46410	−0.0655422	−0.0327711	+	0.999463i	$0.510433\pi$
$500$	−13.8564	−0.619677
$501$	18.9282	0.845650
$502$	0	0
$503$	−11.7846	−0.525450	−0.262725	−	0.964871i	$-0.584621\pi$
$503$	−11.7846	−0.525450	−0.262725	+	0.964871i	$0.584621\pi$
$504$	0	0
$505$	−65.3205	−2.90672
$506$	0	0
$507$	16.8564	0.748619
$508$	−35.7128	−1.58450
$509$	8.07180	0.357776	0.178888	−	0.983869i	$-0.442750\pi$
$509$	8.07180	0.357776	0.178888	+	0.983869i	$0.442750\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−0.535898	−0.0236605
$514$	0	0
$515$	4.14359	0.182589
$516$	−19.8564	−0.874130
$517$	68.3731	3.00704
$518$	0	0
$519$	−7.07180	−0.310417
$520$	0	0
$521$	6.85641	0.300385	0.150192	−	0.988657i	$-0.452011\pi$
$521$	6.85641	0.300385	0.150192	+	0.988657i	$0.452011\pi$
$522$	0	0
$523$	31.1769	1.36327	0.681636	−	0.731692i	$-0.261269\pi$
$523$	31.1769	1.36327	0.681636	+	0.731692i	$0.261269\pi$
$524$	−21.0718	−0.920526
$525$	0	0
$526$	0	0
$527$	28.6603	1.24846
$528$	22.9282	0.997822
$529$	6.85641	0.298105
$530$	0	0
$531$	6.92820	0.300658
$532$	0	0
$533$	−5.46410	−0.236677
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	15.5885	0.672692
$538$	0	0
$539$	0	0
$540$	−6.92820	−0.298142
$541$	3.07180	0.132067	0.0660334	−	0.997817i	$-0.478966\pi$
$541$	3.07180	0.132067	0.0660334	+	0.997817i	$0.478966\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	8.78461	0.376291
$546$	0	0
$547$	−32.3923	−1.38499	−0.692497	−	0.721420i	$-0.743490\pi$
$547$	−32.3923	−1.38499	−0.692497	+	0.721420i	$0.743490\pi$
$548$	−5.60770	−0.239549
$549$	1.73205	0.0739221
$550$	0	0
$551$	2.78461	0.118628
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.46410	−0.147043
$556$	−22.9282	−0.972372
$557$	−34.5167	−1.46252	−0.731259	−	0.682100i	$-0.761067\pi$
$557$	−34.5167	−1.46252	−0.731259	+	0.682100i	$0.761067\pi$
$558$	0	0
$559$	−54.2487	−2.29448
$560$	0	0
$561$	−28.6603	−1.21004
$562$	0	0
$563$	7.92820	0.334134	0.167067	−	0.985946i	$-0.446570\pi$
$563$	7.92820	0.334134	0.167067	+	0.985946i	$0.446570\pi$
$564$	−23.8564	−1.00454
$565$	−60.0000	−2.52422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.2487	0.932714	0.466357	−	0.884596i	$-0.345566\pi$
$569$	22.2487	0.932714	0.466357	+	0.884596i	$0.345566\pi$
$570$	0	0
$571$	21.0718	0.881827	0.440914	−	0.897550i	$-0.354655\pi$
$571$	21.0718	0.881827	0.440914	+	0.897550i	$0.354655\pi$
$572$	62.6410	2.61915
$573$	−10.3923	−0.434145
$574$	0	0
$575$	38.2487	1.59508
$576$	−8.00000	−0.333333
$577$	−0.679492	−0.0282876	−0.0141438	−	0.999900i	$-0.504502\pi$
$577$	−0.679492	−0.0282876	−0.0141438	+	0.999900i	$0.504502\pi$
$578$	0	0
$579$	−7.32051	−0.304230
$580$	36.0000	1.49482
$581$	0	0
$582$	0	0
$583$	62.6410	2.59433
$584$	0	0
$585$	−18.9282	−0.782585
$586$	0	0
$587$	−28.0718	−1.15865	−0.579324	−	0.815098i	$-0.696683\pi$
$587$	−28.0718	−1.15865	−0.579324	+	0.815098i	$0.696683\pi$
$588$	0	0
$589$	3.07180	0.126571
$590$	0	0
$591$	0.535898	0.0220439
$592$	−4.00000	−0.164399
$593$	−12.7128	−0.522053	−0.261026	−	0.965332i	$-0.584061\pi$
$593$	−12.7128	−0.522053	−0.261026	+	0.965332i	$0.584061\pi$
$594$	0	0
$595$	0	0
$596$	27.7128	1.13516
$597$	−25.3205	−1.03630
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−8.24871	−0.336472	−0.168236	−	0.985747i	$-0.553807\pi$
$601$	−8.24871	−0.336472	−0.168236	+	0.985747i	$0.553807\pi$
$602$	0	0
$603$	7.46410	0.303962
$604$	−40.7846	−1.65950
$605$	75.7128	3.07816
$606$	0	0
$607$	4.24871	0.172450	0.0862249	−	0.996276i	$-0.472520\pi$
$607$	4.24871	0.172450	0.0862249	+	0.996276i	$0.472520\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−65.1769	−2.63678
$612$	10.0000	0.404226
$613$	36.6410	1.47992	0.739958	−	0.672653i	$-0.234845\pi$
$613$	36.6410	1.47992	0.739958	+	0.672653i	$0.234845\pi$
$614$	0	0
$615$	3.46410	0.139686
$616$	0	0
$617$	7.85641	0.316287	0.158144	−	0.987416i	$-0.449449\pi$
$617$	7.85641	0.316287	0.158144	+	0.987416i	$0.449449\pi$
$618$	0	0
$619$	26.8038	1.07734	0.538669	−	0.842518i	$-0.318927\pi$
$619$	26.8038	1.07734	0.538669	+	0.842518i	$0.318927\pi$
$620$	39.7128	1.59490
$621$	5.46410	0.219267
$622$	0	0
$623$	0	0
$624$	−21.8564	−0.874957
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−3.07180	−0.122676
$628$	−20.0000	−0.798087
$629$	5.00000	0.199363
$630$	0	0
$631$	13.9282	0.554473	0.277237	−	0.960802i	$-0.410581\pi$
$631$	13.9282	0.554473	0.277237	+	0.960802i	$0.410581\pi$
$632$	0	0
$633$	14.9282	0.593343
$634$	0	0
$635$	61.8564	2.45470
$636$	−21.8564	−0.866663
$637$	0	0
$638$	0	0
$639$	13.1962	0.522032
$640$	0	0
$641$	−42.5167	−1.67931	−0.839654	−	0.543122i	$-0.817242\pi$
$641$	−42.5167	−1.67931	−0.839654	+	0.543122i	$0.817242\pi$
$642$	0	0
$643$	−38.2487	−1.50838	−0.754191	−	0.656655i	$-0.771971\pi$
$643$	−38.2487	−1.50838	−0.754191	+	0.656655i	$0.771971\pi$
$644$	0	0
$645$	34.3923	1.35420
$646$	0	0
$647$	2.92820	0.115120	0.0575598	−	0.998342i	$-0.481668\pi$
$647$	2.92820	0.115120	0.0575598	+	0.998342i	$0.481668\pi$
$648$	0	0
$649$	39.7128	1.55886
$650$	0	0
$651$	0	0
$652$	35.8564	1.40425
$653$	−12.6603	−0.495434	−0.247717	−	0.968832i	$-0.579680\pi$
$653$	−12.6603	−0.495434	−0.247717	+	0.968832i	$0.579680\pi$
$654$	0	0
$655$	36.4974	1.42607
$656$	4.00000	0.156174
$657$	9.19615	0.358776
$658$	0	0
$659$	−12.5359	−0.488329	−0.244165	−	0.969734i	$-0.578514\pi$
$659$	−12.5359	−0.488329	−0.244165	+	0.969734i	$0.578514\pi$
$660$	−39.7128	−1.54582
$661$	30.6410	1.19180	0.595899	−	0.803060i	$-0.296796\pi$
$661$	30.6410	1.19180	0.595899	+	0.803060i	$0.296796\pi$
$662$	0	0
$663$	27.3205	1.06104
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−28.3923	−1.09935
$668$	−37.8564	−1.46471
$669$	−8.53590	−0.330017
$670$	0	0
$671$	9.92820	0.383274
$672$	0	0
$673$	15.4641	0.596097	0.298049	−	0.954551i	$-0.403664\pi$
$673$	15.4641	0.596097	0.298049	+	0.954551i	$0.403664\pi$
$674$	0	0
$675$	7.00000	0.269430
$676$	−33.7128	−1.29665
$677$	−3.85641	−0.148214	−0.0741069	−	0.997250i	$-0.523611\pi$
$677$	−3.85641	−0.148214	−0.0741069	+	0.997250i	$0.523611\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0.0717968	0.00275126
$682$	0	0
$683$	−12.2487	−0.468684	−0.234342	−	0.972154i	$-0.575294\pi$
$683$	−12.2487	−0.468684	−0.234342	+	0.972154i	$0.575294\pi$
$684$	1.07180	0.0409812
$685$	9.71281	0.371108
$686$	0	0
$687$	−8.92820	−0.340632
$688$	39.7128	1.51404
$689$	−59.7128	−2.27488
$690$	0	0
$691$	−41.1769	−1.56644	−0.783222	−	0.621742i	$-0.786425\pi$
$691$	−41.1769	−1.56644	−0.783222	+	0.621742i	$0.786425\pi$
$692$	14.1436	0.537659
$693$	0	0
$694$	0	0
$695$	39.7128	1.50639
$696$	0	0
$697$	−5.00000	−0.189389
$698$	0	0
$699$	−13.8564	−0.524097
$700$	0	0
$701$	−15.4641	−0.584071	−0.292036	−	0.956407i	$-0.594333\pi$
$701$	−15.4641	−0.584071	−0.292036	+	0.956407i	$0.594333\pi$
$702$	0	0
$703$	0.535898	0.0202118
$704$	−45.8564	−1.72828
$705$	41.3205	1.55622
$706$	0	0
$707$	0	0
$708$	−13.8564	−0.520756
$709$	33.3205	1.25138	0.625689	−	0.780073i	$-0.284818\pi$
$709$	33.3205	1.25138	0.625689	+	0.780073i	$0.284818\pi$
$710$	0	0
$711$	0.928203	0.0348103
$712$	0	0
$713$	−31.3205	−1.17296
$714$	0	0
$715$	−108.497	−4.05757
$716$	−31.1769	−1.16514
$717$	−15.4641	−0.577517
$718$	0	0
$719$	−17.0000	−0.633993	−0.316997	−	0.948427i	$-0.602674\pi$
$719$	−17.0000	−0.633993	−0.316997	+	0.948427i	$0.602674\pi$
$720$	13.8564	0.516398
$721$	0	0
$722$	0	0
$723$	5.19615	0.193247
$724$	−40.0000	−1.48659
$725$	−36.3731	−1.35086
$726$	0	0
$727$	−28.6410	−1.06224	−0.531118	−	0.847298i	$-0.678228\pi$
$727$	−28.6410	−1.06224	−0.531118	+	0.847298i	$0.678228\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−49.6410	−1.83604
$732$	−3.46410	−0.128037
$733$	−29.9808	−1.10736	−0.553682	−	0.832728i	$-0.686778\pi$
$733$	−29.9808	−1.10736	−0.553682	+	0.832728i	$0.686778\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42.7846	1.57599
$738$	0	0
$739$	7.78461	0.286361	0.143181	−	0.989697i	$-0.454267\pi$
$739$	7.78461	0.286361	0.143181	+	0.989697i	$0.454267\pi$
$740$	6.92820	0.254686
$741$	2.92820	0.107570
$742$	0	0
$743$	27.4641	1.00756	0.503780	−	0.863832i	$-0.331942\pi$
$743$	27.4641	1.00756	0.503780	+	0.863832i	$0.331942\pi$
$744$	0	0
$745$	−48.0000	−1.75858
$746$	0	0
$747$	−10.0000	−0.365881
$748$	57.3205	2.09585
$749$	0	0
$750$	0	0
$751$	−24.2487	−0.884848	−0.442424	−	0.896806i	$-0.645881\pi$
$751$	−24.2487	−0.884848	−0.442424	+	0.896806i	$0.645881\pi$
$752$	47.7128	1.73991
$753$	7.46410	0.272007
$754$	0	0
$755$	70.6410	2.57089
$756$	0	0
$757$	15.8564	0.576311	0.288155	−	0.957584i	$-0.406958\pi$
$757$	15.8564	0.576311	0.288155	+	0.957584i	$0.406958\pi$
$758$	0	0
$759$	31.3205	1.13686
$760$	0	0
$761$	10.2487	0.371515	0.185758	−	0.982596i	$-0.440526\pi$
$761$	10.2487	0.371515	0.185758	+	0.982596i	$0.440526\pi$
$762$	0	0
$763$	0	0
$764$	20.7846	0.751961
$765$	−17.3205	−0.626224
$766$	0	0
$767$	−37.8564	−1.36692
$768$	16.0000	0.577350
$769$	−2.80385	−0.101109	−0.0505547	−	0.998721i	$-0.516099\pi$
$769$	−2.80385	−0.101109	−0.0505547	+	0.998721i	$0.516099\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	14.6410	0.526942
$773$	−3.14359	−0.113067	−0.0565336	−	0.998401i	$-0.518005\pi$
$773$	−3.14359	−0.113067	−0.0565336	+	0.998401i	$0.518005\pi$
$774$	0	0
$775$	−40.1244	−1.44131
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.535898	−0.0192006
$780$	37.8564	1.35548
$781$	75.6410	2.70665
$782$	0	0
$783$	−5.19615	−0.185695
$784$	0	0
$785$	34.6410	1.23639
$786$	0	0
$787$	−26.5167	−0.945217	−0.472608	−	0.881273i	$-0.656687\pi$
$787$	−26.5167	−0.945217	−0.472608	+	0.881273i	$0.656687\pi$
$788$	−1.07180	−0.0381812
$789$	−20.1244	−0.716446
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.46410	−0.336080
$794$	0	0
$795$	37.8564	1.34263
$796$	50.6410	1.79492
$797$	−19.8564	−0.703350	−0.351675	−	0.936122i	$-0.614388\pi$
$797$	−19.8564	−0.703350	−0.351675	+	0.936122i	$0.614388\pi$
$798$	0	0
$799$	−59.6410	−2.10995
$800$	0	0
$801$	−8.92820	−0.315463
$802$	0	0
$803$	52.7128	1.86019
$804$	−14.9282	−0.526477
$805$	0	0
$806$	0	0
$807$	17.4641	0.614765
$808$	0	0
$809$	−41.0718	−1.44401	−0.722004	−	0.691889i	$-0.756779\pi$
$809$	−41.0718	−1.44401	−0.722004	+	0.691889i	$0.756779\pi$
$810$	0	0
$811$	−1.73205	−0.0608205	−0.0304103	−	0.999538i	$-0.509681\pi$
$811$	−1.73205	−0.0608205	−0.0304103	+	0.999538i	$0.509681\pi$
$812$	0	0
$813$	−11.5885	−0.406425
$814$	0	0
$815$	−62.1051	−2.17545
$816$	−20.0000	−0.700140
$817$	−5.32051	−0.186141
$818$	0	0
$819$	0	0
$820$	−6.92820	−0.241943
$821$	−8.67949	−0.302916	−0.151458	−	0.988464i	$-0.548397\pi$
$821$	−8.67949	−0.302916	−0.151458	+	0.988464i	$0.548397\pi$
$822$	0	0
$823$	39.5692	1.37930	0.689648	−	0.724145i	$-0.257765\pi$
$823$	39.5692	1.37930	0.689648	+	0.724145i	$0.257765\pi$
$824$	0	0
$825$	40.1244	1.39695
$826$	0	0
$827$	−45.1962	−1.57162	−0.785812	−	0.618465i	$-0.787755\pi$
$827$	−45.1962	−1.57162	−0.785812	+	0.618465i	$0.787755\pi$
$828$	−10.9282	−0.379781
$829$	15.0526	0.522797	0.261398	−	0.965231i	$-0.415816\pi$
$829$	15.0526	0.522797	0.261398	+	0.965231i	$0.415816\pi$
$830$	0	0
$831$	−21.9282	−0.760681
$832$	43.7128	1.51547
$833$	0	0
$834$	0	0
$835$	65.5692	2.26912
$836$	6.14359	0.212481
$837$	−5.73205	−0.198129
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	−2.00000	−0.0689655
$842$	0	0
$843$	−0.124356	−0.00428304
$844$	−29.8564	−1.02770
$845$	58.3923	2.00876
$846$	0	0
$847$	0	0
$848$	43.7128	1.50110
$849$	4.66025	0.159940
$850$	0	0
$851$	−5.46410	−0.187307
$852$	−26.3923	−0.904185
$853$	−51.7128	−1.77061	−0.885306	−	0.465008i	$-0.846051\pi$
$853$	−51.7128	−1.77061	−0.885306	+	0.465008i	$0.846051\pi$
$854$	0	0
$855$	−1.85641	−0.0634878
$856$	0	0
$857$	34.2487	1.16991	0.584957	−	0.811064i	$-0.301111\pi$
$857$	34.2487	1.16991	0.584957	+	0.811064i	$0.301111\pi$
$858$	0	0
$859$	2.80385	0.0956660	0.0478330	−	0.998855i	$-0.484768\pi$
$859$	2.80385	0.0956660	0.0478330	+	0.998855i	$0.484768\pi$
$860$	−68.7846	−2.34554
$861$	0	0
$862$	0	0
$863$	14.6795	0.499696	0.249848	−	0.968285i	$-0.419619\pi$
$863$	14.6795	0.499696	0.249848	+	0.968285i	$0.419619\pi$
$864$	0	0
$865$	−24.4974	−0.832937
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	5.32051	0.180486
$870$	0	0
$871$	−40.7846	−1.38193
$872$	0	0
$873$	−2.39230	−0.0809673
$874$	0	0
$875$	0	0
$876$	−18.3923	−0.621418
$877$	28.0718	0.947917	0.473959	−	0.880547i	$-0.342825\pi$
$877$	28.0718	0.947917	0.473959	+	0.880547i	$0.342825\pi$
$878$	0	0
$879$	−9.00000	−0.303562
$880$	79.4256	2.67744
$881$	26.5359	0.894017	0.447009	−	0.894530i	$-0.352489\pi$
$881$	26.5359	0.894017	0.447009	+	0.894530i	$0.352489\pi$
$882$	0	0
$883$	39.5692	1.33161	0.665805	−	0.746126i	$-0.268088\pi$
$883$	39.5692	1.33161	0.665805	+	0.746126i	$0.268088\pi$
$884$	−54.6410	−1.83778
$885$	24.0000	0.806751
$886$	0	0
$887$	−20.8564	−0.700290	−0.350145	−	0.936696i	$-0.613868\pi$
$887$	−20.8564	−0.700290	−0.350145	+	0.936696i	$0.613868\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.73205	0.192031
$892$	17.0718	0.571606
$893$	−6.39230	−0.213910
$894$	0	0
$895$	54.0000	1.80502
$896$	0	0
$897$	−29.8564	−0.996876
$898$	0	0
$899$	29.7846	0.993372
$900$	−14.0000	−0.466667
$901$	−54.6410	−1.82036
$902$	0	0
$903$	0	0
$904$	0	0
$905$	69.2820	2.30301
$906$	0	0
$907$	5.85641	0.194459	0.0972294	−	0.995262i	$-0.469002\pi$
$907$	5.85641	0.194459	0.0972294	+	0.995262i	$0.469002\pi$
$908$	−0.143594	−0.00476532
$909$	−18.8564	−0.625428
$910$	0	0
$911$	−16.7846	−0.556099	−0.278049	−	0.960567i	$-0.589688\pi$
$911$	−16.7846	−0.556099	−0.278049	+	0.960567i	$0.589688\pi$
$912$	−2.14359	−0.0709815
$913$	−57.3205	−1.89703
$914$	0	0
$915$	6.00000	0.198354
$916$	17.8564	0.589992
$917$	0	0
$918$	0	0
$919$	−25.4641	−0.839983	−0.419992	−	0.907528i	$-0.637967\pi$
$919$	−25.4641	−0.839983	−0.419992	+	0.907528i	$0.637967\pi$
$920$	0	0
$921$	14.2679	0.470145
$922$	0	0
$923$	−72.1051	−2.37337
$924$	0	0
$925$	−7.00000	−0.230159
$926$	0	0
$927$	1.19615	0.0392868
$928$	0	0
$929$	−43.7846	−1.43653	−0.718263	−	0.695771i	$-0.755063\pi$
$929$	−43.7846	−1.43653	−0.718263	+	0.695771i	$0.755063\pi$
$930$	0	0
$931$	0	0
$932$	27.7128	0.907763
$933$	25.8564	0.846501
$934$	0	0
$935$	−99.2820	−3.24687
$936$	0	0
$937$	5.07180	0.165688	0.0828442	−	0.996563i	$-0.473600\pi$
$937$	5.07180	0.165688	0.0828442	+	0.996563i	$0.473600\pi$
$938$	0	0
$939$	2.92820	0.0955583
$940$	−82.6410	−2.69545
$941$	33.3205	1.08622	0.543109	−	0.839662i	$-0.317247\pi$
$941$	33.3205	1.08622	0.543109	+	0.839662i	$0.317247\pi$
$942$	0	0
$943$	5.46410	0.177936
$944$	27.7128	0.901975
$945$	0	0
$946$	0	0
$947$	23.0718	0.749733	0.374866	−	0.927079i	$-0.377689\pi$
$947$	23.0718	0.749733	0.374866	+	0.927079i	$0.377689\pi$
$948$	−1.85641	−0.0602933
$949$	−50.2487	−1.63114
$950$	0	0
$951$	22.8038	0.739465
$952$	0	0
$953$	−20.6795	−0.669874	−0.334937	−	0.942240i	$-0.608715\pi$
$953$	−20.6795	−0.669874	−0.334937	+	0.942240i	$0.608715\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	30.9282	1.00029
$957$	−29.7846	−0.962800
$958$	0	0
$959$	0	0
$960$	−27.7128	−0.894427
$961$	1.85641	0.0598841
$962$	0	0
$963$	−3.46410	−0.111629
$964$	−10.3923	−0.334714
$965$	−25.3590	−0.816335
$966$	0	0
$967$	−36.2487	−1.16568	−0.582840	−	0.812587i	$-0.698059\pi$
$967$	−36.2487	−1.16568	−0.582840	+	0.812587i	$0.698059\pi$
$968$	0	0
$969$	2.67949	0.0860777
$970$	0	0
$971$	14.0718	0.451585	0.225793	−	0.974175i	$-0.427503\pi$
$971$	14.0718	0.451585	0.225793	+	0.974175i	$0.427503\pi$
$972$	−2.00000	−0.0641500
$973$	0	0
$974$	0	0
$975$	−38.2487	−1.22494
$976$	6.92820	0.221766
$977$	13.9808	0.447284	0.223642	−	0.974671i	$-0.428205\pi$
$977$	13.9808	0.447284	0.223642	+	0.974671i	$0.428205\pi$
$978$	0	0
$979$	−51.1769	−1.63562
$980$	0	0
$981$	2.53590	0.0809650
$982$	0	0
$983$	−14.7846	−0.471556	−0.235778	−	0.971807i	$-0.575764\pi$
$983$	−14.7846	−0.471556	−0.235778	+	0.971807i	$0.575764\pi$
$984$	0	0
$985$	1.85641	0.0591500
$986$	0	0
$987$	0	0
$988$	−5.85641	−0.186317
$989$	54.2487	1.72501
$990$	0	0
$991$	−24.1051	−0.765724	−0.382862	−	0.923805i	$-0.625062\pi$
$991$	−24.1051	−0.765724	−0.382862	+	0.923805i	$0.625062\pi$
$992$	0	0
$993$	16.2487	0.515637
$994$	0	0
$995$	−87.7128	−2.78068
$996$	20.0000	0.633724
$997$	21.1769	0.670680	0.335340	−	0.942097i	$-0.391149\pi$
$997$	21.1769	0.670680	0.335340	+	0.942097i	$0.391149\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.n.1.2	yes	2
7.6	odd	2		6027.2.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.l.1.1	✓	2	7.6	odd	2
6027.2.a.n.1.2	yes	2	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{4} +3.46410 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{4} +3.46410 q^{5} +1.00000 q^{9} +5.73205 q^{11} -2.00000 q^{12} -5.46410 q^{13} +3.46410 q^{15} +4.00000 q^{16} -5.00000 q^{17} -0.535898 q^{19} -6.92820 q^{20} +5.46410 q^{23} +7.00000 q^{25} +1.00000 q^{27} -5.19615 q^{29} -5.73205 q^{31} +5.73205 q^{33} -2.00000 q^{36} -1.00000 q^{37} -5.46410 q^{39} +1.00000 q^{41} +9.92820 q^{43} -11.4641 q^{44} +3.46410 q^{45} +11.9282 q^{47} +4.00000 q^{48} -5.00000 q^{51} +10.9282 q^{52} +10.9282 q^{53} +19.8564 q^{55} -0.535898 q^{57} +6.92820 q^{59} -6.92820 q^{60} +1.73205 q^{61} -8.00000 q^{64} -18.9282 q^{65} +7.46410 q^{67} +10.0000 q^{68} +5.46410 q^{69} +13.1962 q^{71} +9.19615 q^{73} +7.00000 q^{75} +1.07180 q^{76} +0.928203 q^{79} +13.8564 q^{80} +1.00000 q^{81} -10.0000 q^{83} -17.3205 q^{85} -5.19615 q^{87} -8.92820 q^{89} -10.9282 q^{92} -5.73205 q^{93} -1.85641 q^{95} -2.39230 q^{97} +5.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9} + 8 q^{11} - 4 q^{12} - 4 q^{13} + 8 q^{16} - 10 q^{17} - 8 q^{19} + 4 q^{23} + 14 q^{25} + 2 q^{27} - 8 q^{31} + 8 q^{33} - 4 q^{36} - 2 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} - 16 q^{44} + 10 q^{47} + 8 q^{48} - 10 q^{51} + 8 q^{52} + 8 q^{53} + 12 q^{55} - 8 q^{57} - 16 q^{64} - 24 q^{65} + 8 q^{67} + 20 q^{68} + 4 q^{69} + 16 q^{71} + 8 q^{73} + 14 q^{75} + 16 q^{76} - 12 q^{79} + 2 q^{81} - 20 q^{83} - 4 q^{89} - 8 q^{92} - 8 q^{93} + 24 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^4 + 2 * q^9 + 8 * q^11 - 4 * q^12 - 4 * q^13 + 8 * q^16 - 10 * q^17 - 8 * q^19 + 4 * q^23 + 14 * q^25 + 2 * q^27 - 8 * q^31 + 8 * q^33 - 4 * q^36 - 2 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 - 16 * q^44 + 10 * q^47 + 8 * q^48 - 10 * q^51 + 8 * q^52 + 8 * q^53 + 12 * q^55 - 8 * q^57 - 16 * q^64 - 24 * q^65 + 8 * q^67 + 20 * q^68 + 4 * q^69 + 16 * q^71 + 8 * q^73 + 14 * q^75 + 16 * q^76 - 12 * q^79 + 2 * q^81 - 20 * q^83 - 4 * q^89 - 8 * q^92 - 8 * q^93 + 24 * q^95 + 16 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more