LMFDB - Embedded newform 6027.2.a.n.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{4} -3.46410 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{4} -3.46410 q^{5} +1.00000 q^{9} +2.26795 q^{11} -2.00000 q^{12} +1.46410 q^{13} -3.46410 q^{15} +4.00000 q^{16} -5.00000 q^{17} -7.46410 q^{19} +6.92820 q^{20} -1.46410 q^{23} +7.00000 q^{25} +1.00000 q^{27} +5.19615 q^{29} -2.26795 q^{31} +2.26795 q^{33} -2.00000 q^{36} -1.00000 q^{37} +1.46410 q^{39} +1.00000 q^{41} -3.92820 q^{43} -4.53590 q^{44} -3.46410 q^{45} -1.92820 q^{47} +4.00000 q^{48} -5.00000 q^{51} -2.92820 q^{52} -2.92820 q^{53} -7.85641 q^{55} -7.46410 q^{57} -6.92820 q^{59} +6.92820 q^{60} -1.73205 q^{61} -8.00000 q^{64} -5.07180 q^{65} +0.535898 q^{67} +10.0000 q^{68} -1.46410 q^{69} +2.80385 q^{71} -1.19615 q^{73} +7.00000 q^{75} +14.9282 q^{76} -12.9282 q^{79} -13.8564 q^{80} +1.00000 q^{81} -10.0000 q^{83} +17.3205 q^{85} +5.19615 q^{87} +4.92820 q^{89} +2.92820 q^{92} -2.26795 q^{93} +25.8564 q^{95} +18.3923 q^{97} +2.26795 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.26795	0.683812	0.341906	−	0.939734i	$-0.388927\pi$
$11$	2.26795	0.683812	0.341906	+	0.939734i	$0.388927\pi$
$12$	−2.00000	−0.577350
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	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
$19$	−7.46410	−1.71238	−0.856191	−	0.516659i	$-0.827175\pi$
$19$	−7.46410	−1.71238	−0.856191	+	0.516659i	$0.827175\pi$
$20$	6.92820	1.54919
$21$	0	0
$22$	0	0
$23$	−1.46410	−0.305286	−0.152643	−	0.988281i	$-0.548779\pi$
$23$	−1.46410	−0.305286	−0.152643	+	0.988281i	$0.548779\pi$
$24$	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	$-0.339747\pi$
$29$	5.19615	0.964901	0.482451	+	0.875923i	$0.339747\pi$
$30$	0	0
$31$	−2.26795	−0.407336	−0.203668	−	0.979040i	$-0.565286\pi$
$31$	−2.26795	−0.407336	−0.203668	+	0.979040i	$0.565286\pi$
$32$	0	0
$33$	2.26795	0.394799
$34$	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
$39$	1.46410	0.234444
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−3.92820	−0.599045	−0.299523	−	0.954089i	$-0.596827\pi$
$43$	−3.92820	−0.599045	−0.299523	+	0.954089i	$0.596827\pi$
$44$	−4.53590	−0.683812
$45$	−3.46410	−0.516398
$46$	0	0
$47$	−1.92820	−0.281257	−0.140629	−	0.990062i	$-0.544912\pi$
$47$	−1.92820	−0.281257	−0.140629	+	0.990062i	$0.544912\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	0	0
$51$	−5.00000	−0.700140
$52$	−2.92820	−0.406069
$53$	−2.92820	−0.402220	−0.201110	−	0.979569i	$-0.564455\pi$
$53$	−2.92820	−0.402220	−0.201110	+	0.979569i	$0.564455\pi$
$54$	0	0
$55$	−7.85641	−1.05936
$56$	0	0
$57$	−7.46410	−0.988644
$58$	0	0
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	6.92820	0.894427
$61$	−1.73205	−0.221766	−0.110883	−	0.993833i	$-0.535368\pi$
$61$	−1.73205	−0.221766	−0.110883	+	0.993833i	$0.535368\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−5.07180	−0.629079
$66$	0	0
$67$	0.535898	0.0654704	0.0327352	−	0.999464i	$-0.489578\pi$
$67$	0.535898	0.0654704	0.0327352	+	0.999464i	$0.489578\pi$
$68$	10.0000	1.21268
$69$	−1.46410	−0.176257
$70$	0	0
$71$	2.80385	0.332755	0.166378	−	0.986062i	$-0.446793\pi$
$71$	2.80385	0.332755	0.166378	+	0.986062i	$0.446793\pi$
$72$	0	0
$73$	−1.19615	−0.139999	−0.0699995	−	0.997547i	$-0.522300\pi$
$73$	−1.19615	−0.139999	−0.0699995	+	0.997547i	$0.522300\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	14.9282	1.71238
$77$	0	0
$78$	0	0
$79$	−12.9282	−1.45454	−0.727268	−	0.686353i	$-0.759210\pi$
$79$	−12.9282	−1.45454	−0.727268	+	0.686353i	$0.759210\pi$
$80$	−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
$89$	4.92820	0.522388	0.261194	−	0.965286i	$-0.415884\pi$
$89$	4.92820	0.522388	0.261194	+	0.965286i	$0.415884\pi$
$90$	0	0
$91$	0	0
$92$	2.92820	0.305286
$93$	−2.26795	−0.235175
$94$	0	0
$95$	25.8564	2.65281
$96$	0	0
$97$	18.3923	1.86746	0.933728	−	0.357984i	$-0.116536\pi$
$97$	18.3923	1.86746	0.933728	+	0.357984i	$0.116536\pi$
$98$	0	0
$99$	2.26795	0.227937
$100$	−14.0000	−1.40000
$101$	8.85641	0.881245	0.440623	−	0.897692i	$-0.354758\pi$
$101$	8.85641	0.881245	0.440623	+	0.897692i	$0.354758\pi$
$102$	0	0
$103$	−9.19615	−0.906124	−0.453062	−	0.891479i	$-0.649668\pi$
$103$	−9.19615	−0.906124	−0.453062	+	0.891479i	$0.649668\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	−2.00000	−0.192450
$109$	9.46410	0.906497	0.453248	−	0.891384i	$-0.350265\pi$
$109$	9.46410	0.906497	0.453248	+	0.891384i	$0.350265\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	0	0
$113$	17.3205	1.62938	0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205	1.62938	0.814688	+	0.579899i	$0.196908\pi$
$114$	0	0
$115$	5.07180	0.472947
$116$	−10.3923	−0.964901
$117$	1.46410	0.135356
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.85641	−0.532401
$122$	0	0
$123$	1.00000	0.0901670
$124$	4.53590	0.407336
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−9.85641	−0.874615	−0.437307	−	0.899312i	$-0.644068\pi$
$127$	−9.85641	−0.874615	−0.437307	+	0.899312i	$0.644068\pi$
$128$	0	0
$129$	−3.92820	−0.345859
$130$	0	0
$131$	17.4641	1.52585	0.762923	−	0.646490i	$-0.223764\pi$
$131$	17.4641	1.52585	0.762923	+	0.646490i	$0.223764\pi$
$132$	−4.53590	−0.394799
$133$	0	0
$134$	0	0
$135$	−3.46410	−0.298142
$136$	0	0
$137$	13.1962	1.12742	0.563712	−	0.825972i	$-0.309373\pi$
$137$	13.1962	1.12742	0.563712	+	0.825972i	$0.309373\pi$
$138$	0	0
$139$	4.53590	0.384730	0.192365	−	0.981323i	$-0.438384\pi$
$139$	4.53590	0.384730	0.192365	+	0.981323i	$0.438384\pi$
$140$	0	0
$141$	−1.92820	−0.162384
$142$	0	0
$143$	3.32051	0.277675
$144$	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	$-0.307880\pi$
$149$	13.8564	1.13516	0.567581	+	0.823318i	$0.307880\pi$
$150$	0	0
$151$	−0.392305	−0.0319253	−0.0159627	−	0.999873i	$-0.505081\pi$
$151$	−0.392305	−0.0319253	−0.0159627	+	0.999873i	$0.505081\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	7.85641	0.631042
$156$	−2.92820	−0.234444
$157$	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
$159$	−2.92820	−0.232222
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−4.07180	−0.318928	−0.159464	−	0.987204i	$-0.550977\pi$
$163$	−4.07180	−0.318928	−0.159464	+	0.987204i	$0.550977\pi$
$164$	−2.00000	−0.156174
$165$	−7.85641	−0.611620
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−7.46410	−0.570794
$172$	7.85641	0.599045
$173$	−20.9282	−1.59114	−0.795571	−	0.605860i	$-0.792829\pi$
$173$	−20.9282	−1.59114	−0.795571	+	0.605860i	$0.792829\pi$
$174$	0	0
$175$	0	0
$176$	9.07180	0.683812
$177$	−6.92820	−0.520756
$178$	0	0
$179$	−15.5885	−1.16514	−0.582568	−	0.812782i	$-0.697952\pi$
$179$	−15.5885	−1.16514	−0.582568	+	0.812782i	$0.697952\pi$
$180$	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
$187$	−11.3397	−0.829244
$188$	3.85641	0.281257
$189$	0	0
$190$	0	0
$191$	10.3923	0.751961	0.375980	−	0.926628i	$-0.377306\pi$
$191$	10.3923	0.751961	0.375980	+	0.926628i	$0.377306\pi$
$192$	−8.00000	−0.577350
$193$	27.3205	1.96657	0.983287	−	0.182064i	$-0.0582779\pi$
$193$	27.3205	1.96657	0.983287	+	0.182064i	$0.0582779\pi$
$194$	0	0
$195$	−5.07180	−0.363199
$196$	0	0
$197$	7.46410	0.531795	0.265898	−	0.964001i	$-0.414332\pi$
$197$	7.46410	0.531795	0.265898	+	0.964001i	$0.414332\pi$
$198$	0	0
$199$	9.32051	0.660713	0.330357	−	0.943856i	$-0.392831\pi$
$199$	9.32051	0.660713	0.330357	+	0.943856i	$0.392831\pi$
$200$	0	0
$201$	0.535898	0.0377994
$202$	0	0
$203$	0	0
$204$	10.0000	0.700140
$205$	−3.46410	−0.241943
$206$	0	0
$207$	−1.46410	−0.101762
$208$	5.85641	0.406069
$209$	−16.9282	−1.17095
$210$	0	0
$211$	1.07180	0.0737855	0.0368928	−	0.999319i	$-0.488254\pi$
$211$	1.07180	0.0737855	0.0368928	+	0.999319i	$0.488254\pi$
$212$	5.85641	0.402220
$213$	2.80385	0.192116
$214$	0	0
$215$	13.6077	0.928037
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.19615	−0.0808285
$220$	15.7128	1.05936
$221$	−7.32051	−0.492431
$222$	0	0
$223$	−15.4641	−1.03555	−0.517776	−	0.855516i	$-0.673240\pi$
$223$	−15.4641	−1.03555	−0.517776	+	0.855516i	$0.673240\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	13.9282	0.924447	0.462224	−	0.886763i	$-0.347052\pi$
$227$	13.9282	0.924447	0.462224	+	0.886763i	$0.347052\pi$
$228$	14.9282	0.988644
$229$	4.92820	0.325665	0.162832	−	0.986654i	$-0.447937\pi$
$229$	4.92820	0.325665	0.162832	+	0.986654i	$0.447937\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.8564	0.907763	0.453882	−	0.891062i	$-0.350039\pi$
$233$	13.8564	0.907763	0.453882	+	0.891062i	$0.350039\pi$
$234$	0	0
$235$	6.67949	0.435722
$236$	13.8564	0.901975
$237$	−12.9282	−0.839777
$238$	0	0
$239$	−8.53590	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$239$	−8.53590	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$240$	−13.8564	−0.894427
$241$	−5.19615	−0.334714	−0.167357	−	0.985896i	$-0.553523\pi$
$241$	−5.19615	−0.334714	−0.167357	+	0.985896i	$0.553523\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	3.46410	0.221766
$245$	0	0
$246$	0	0
$247$	−10.9282	−0.695345
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	0.535898	0.0338256	0.0169128	−	0.999857i	$-0.494616\pi$
$251$	0.535898	0.0338256	0.0169128	+	0.999857i	$0.494616\pi$
$252$	0	0
$253$	−3.32051	−0.208759
$254$	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
$260$	10.1436	0.629079
$261$	5.19615	0.321634
$262$	0	0
$263$	4.12436	0.254319	0.127159	−	0.991882i	$-0.459414\pi$
$263$	4.12436	0.254319	0.127159	+	0.991882i	$0.459414\pi$
$264$	0	0
$265$	10.1436	0.623116
$266$	0	0
$267$	4.92820	0.301601
$268$	−1.07180	−0.0654704
$269$	10.5359	0.642385	0.321193	−	0.947014i	$-0.395916\pi$
$269$	10.5359	0.642385	0.321193	+	0.947014i	$0.395916\pi$
$270$	0	0
$271$	19.5885	1.18991	0.594957	−	0.803758i	$-0.297169\pi$
$271$	19.5885	1.18991	0.594957	+	0.803758i	$0.297169\pi$
$272$	−20.0000	−1.21268
$273$	0	0
$274$	0	0
$275$	15.8756	0.957337
$276$	2.92820	0.176257
$277$	−8.07180	−0.484987	−0.242494	−	0.970153i	$-0.577965\pi$
$277$	−8.07180	−0.484987	−0.242494	+	0.970153i	$0.577965\pi$
$278$	0	0
$279$	−2.26795	−0.135779
$280$	0	0
$281$	24.1244	1.43914	0.719569	−	0.694421i	$-0.244339\pi$
$281$	24.1244	1.43914	0.719569	+	0.694421i	$0.244339\pi$
$282$	0	0
$283$	−12.6603	−0.752574	−0.376287	−	0.926503i	$-0.622799\pi$
$283$	−12.6603	−0.752574	−0.376287	+	0.926503i	$0.622799\pi$
$284$	−5.60770	−0.332755
$285$	25.8564	1.53160
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	18.3923	1.07818
$292$	2.39230	0.139999
$293$	−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
$297$	2.26795	0.131600
$298$	0	0
$299$	−2.14359	−0.123967
$300$	−14.0000	−0.808290
$301$	0	0
$302$	0	0
$303$	8.85641	0.508787
$304$	−29.8564	−1.71238
$305$	6.00000	0.343559
$306$	0	0
$307$	17.7321	1.01202	0.506011	−	0.862527i	$-0.331120\pi$
$307$	17.7321	1.01202	0.506011	+	0.862527i	$0.331120\pi$
$308$	0	0
$309$	−9.19615	−0.523151
$310$	0	0
$311$	−1.85641	−0.105267	−0.0526336	−	0.998614i	$-0.516762\pi$
$311$	−1.85641	−0.105267	−0.0526336	+	0.998614i	$0.516762\pi$
$312$	0	0
$313$	−10.9282	−0.617699	−0.308849	−	0.951111i	$-0.599944\pi$
$313$	−10.9282	−0.617699	−0.308849	+	0.951111i	$0.599944\pi$
$314$	0	0
$315$	0	0
$316$	25.8564	1.45454
$317$	33.1962	1.86448	0.932241	−	0.361838i	$-0.117851\pi$
$317$	33.1962	1.86448	0.932241	+	0.361838i	$0.117851\pi$
$318$	0	0
$319$	11.7846	0.659811
$320$	27.7128	1.54919
$321$	3.46410	0.193347
$322$	0	0
$323$	37.3205	2.07657
$324$	−2.00000	−0.111111
$325$	10.2487	0.568496
$326$	0	0
$327$	9.46410	0.523366
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−32.2487	−1.77255	−0.886275	−	0.463160i	$-0.846716\pi$
$331$	−32.2487	−1.77255	−0.886275	+	0.463160i	$0.846716\pi$
$332$	20.0000	1.09764
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−1.85641	−0.101426
$336$	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
$341$	−5.14359	−0.278541
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.07180	0.273056
$346$	0	0
$347$	12.6603	0.679638	0.339819	−	0.940491i	$-0.389634\pi$
$347$	12.6603	0.679638	0.339819	+	0.940491i	$0.389634\pi$
$348$	−10.3923	−0.557086
$349$	−27.5885	−1.47678	−0.738388	−	0.674376i	$-0.764413\pi$
$349$	−27.5885	−1.47678	−0.738388	+	0.674376i	$0.764413\pi$
$350$	0	0
$351$	1.46410	0.0781480
$352$	0	0
$353$	−21.3205	−1.13478	−0.567388	−	0.823451i	$-0.692046\pi$
$353$	−21.3205	−1.13478	−0.567388	+	0.823451i	$0.692046\pi$
$354$	0	0
$355$	−9.71281	−0.515503
$356$	−9.85641	−0.522388
$357$	0	0
$358$	0	0
$359$	−23.7128	−1.25151	−0.625757	−	0.780018i	$-0.715210\pi$
$359$	−23.7128	−1.25151	−0.625757	+	0.780018i	$0.715210\pi$
$360$	0	0
$361$	36.7128	1.93225
$362$	0	0
$363$	−5.85641	−0.307382
$364$	0	0
$365$	4.14359	0.216886
$366$	0	0
$367$	23.0526	1.20333	0.601667	−	0.798747i	$-0.294503\pi$
$367$	23.0526	1.20333	0.601667	+	0.798747i	$0.294503\pi$
$368$	−5.85641	−0.305286
$369$	1.00000	0.0520579
$370$	0	0
$371$	0	0
$372$	4.53590	0.235175
$373$	34.8564	1.80480	0.902398	−	0.430903i	$-0.141805\pi$
$373$	34.8564	1.80480	0.902398	+	0.430903i	$0.141805\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	7.60770	0.391816
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	−51.7128	−2.65281
$381$	−9.85641	−0.504959
$382$	0	0
$383$	22.0718	1.12782	0.563908	−	0.825838i	$-0.309297\pi$
$383$	22.0718	1.12782	0.563908	+	0.825838i	$0.309297\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.92820	−0.199682
$388$	−36.7846	−1.86746
$389$	34.7846	1.76365	0.881825	−	0.471577i	$-0.156315\pi$
$389$	34.7846	1.76365	0.881825	+	0.471577i	$0.156315\pi$
$390$	0	0
$391$	7.32051	0.370214
$392$	0	0
$393$	17.4641	0.880947
$394$	0	0
$395$	44.7846	2.25336
$396$	−4.53590	−0.227937
$397$	−10.5359	−0.528782	−0.264391	−	0.964416i	$-0.585171\pi$
$397$	−10.5359	−0.528782	−0.264391	+	0.964416i	$0.585171\pi$
$398$	0	0
$399$	0	0
$400$	28.0000	1.40000
$401$	−30.9282	−1.54448	−0.772240	−	0.635330i	$-0.780864\pi$
$401$	−30.9282	−1.54448	−0.772240	+	0.635330i	$0.780864\pi$
$402$	0	0
$403$	−3.32051	−0.165406
$404$	−17.7128	−0.881245
$405$	−3.46410	−0.172133
$406$	0	0
$407$	−2.26795	−0.112418
$408$	0	0
$409$	−2.26795	−0.112143	−0.0560714	−	0.998427i	$-0.517857\pi$
$409$	−2.26795	−0.112143	−0.0560714	+	0.998427i	$0.517857\pi$
$410$	0	0
$411$	13.1962	0.650918
$412$	18.3923	0.906124
$413$	0	0
$414$	0	0
$415$	34.6410	1.70046
$416$	0	0
$417$	4.53590	0.222124
$418$	0	0
$419$	23.7128	1.15845	0.579223	−	0.815169i	$-0.303356\pi$
$419$	23.7128	1.15845	0.579223	+	0.815169i	$0.303356\pi$
$420$	0	0
$421$	18.5359	0.903384	0.451692	−	0.892174i	$-0.350821\pi$
$421$	18.5359	0.903384	0.451692	+	0.892174i	$0.350821\pi$
$422$	0	0
$423$	−1.92820	−0.0937524
$424$	0	0
$425$	−35.0000	−1.69775
$426$	0	0
$427$	0	0
$428$	−6.92820	−0.334887
$429$	3.32051	0.160316
$430$	0	0
$431$	9.46410	0.455870	0.227935	−	0.973676i	$-0.426803\pi$
$431$	9.46410	0.455870	0.227935	+	0.973676i	$0.426803\pi$
$432$	4.00000	0.192450
$433$	−0.411543	−0.0197775	−0.00988874	−	0.999951i	$-0.503148\pi$
$433$	−0.411543	−0.0197775	−0.00988874	+	0.999951i	$0.503148\pi$
$434$	0	0
$435$	−18.0000	−0.863034
$436$	−18.9282	−0.906497
$437$	10.9282	0.522767
$438$	0	0
$439$	−35.4641	−1.69261	−0.846305	−	0.532699i	$-0.821178\pi$
$439$	−35.4641	−1.69261	−0.846305	+	0.532699i	$0.821178\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.3923	1.15891	0.579457	−	0.815003i	$-0.303265\pi$
$443$	24.3923	1.15891	0.579457	+	0.815003i	$0.303265\pi$
$444$	2.00000	0.0949158
$445$	−17.0718	−0.809281
$446$	0	0
$447$	13.8564	0.655386
$448$	0	0
$449$	11.3205	0.534248	0.267124	−	0.963662i	$-0.413927\pi$
$449$	11.3205	0.534248	0.267124	+	0.963662i	$0.413927\pi$
$450$	0	0
$451$	2.26795	0.106794
$452$	−34.6410	−1.62938
$453$	−0.392305	−0.0184321
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.6410	1.43333	0.716663	−	0.697419i	$-0.245669\pi$
$457$	30.6410	1.43333	0.716663	+	0.697419i	$0.245669\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	−10.1436	−0.472947
$461$	24.2487	1.12938	0.564688	−	0.825305i	$-0.308997\pi$
$461$	24.2487	1.12938	0.564688	+	0.825305i	$0.308997\pi$
$462$	0	0
$463$	−12.3923	−0.575919	−0.287960	−	0.957643i	$-0.592977\pi$
$463$	−12.3923	−0.575919	−0.287960	+	0.957643i	$0.592977\pi$
$464$	20.7846	0.964901
$465$	7.85641	0.364332
$466$	0	0
$467$	−7.46410	−0.345397	−0.172699	−	0.984975i	$-0.555249\pi$
$467$	−7.46410	−0.345397	−0.172699	+	0.984975i	$0.555249\pi$
$468$	−2.92820	−0.135356
$469$	0	0
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	−8.90897	−0.409635
$474$	0	0
$475$	−52.2487	−2.39734
$476$	0	0
$477$	−2.92820	−0.134073
$478$	0	0
$479$	−14.7128	−0.672246	−0.336123	−	0.941818i	$-0.609116\pi$
$479$	−14.7128	−0.672246	−0.336123	+	0.941818i	$0.609116\pi$
$480$	0	0
$481$	−1.46410	−0.0667573
$482$	0	0
$483$	0	0
$484$	11.7128	0.532401
$485$	−63.7128	−2.89305
$486$	0	0
$487$	15.9282	0.721776	0.360888	−	0.932609i	$-0.382474\pi$
$487$	15.9282	0.721776	0.360888	+	0.932609i	$0.382474\pi$
$488$	0	0
$489$	−4.07180	−0.184133
$490$	0	0
$491$	−23.8564	−1.07662	−0.538312	−	0.842745i	$-0.680938\pi$
$491$	−23.8564	−1.07662	−0.538312	+	0.842745i	$0.680938\pi$
$492$	−2.00000	−0.0901670
$493$	−25.9808	−1.17011
$494$	0	0
$495$	−7.85641	−0.353119
$496$	−9.07180	−0.407336
$497$	0	0
$498$	0	0
$499$	5.46410	0.244607	0.122303	−	0.992493i	$-0.460972\pi$
$499$	5.46410	0.244607	0.122303	+	0.992493i	$0.460972\pi$
$500$	13.8564	0.619677
$501$	5.07180	0.226591
$502$	0	0
$503$	29.7846	1.32803	0.664015	−	0.747719i	$-0.268851\pi$
$503$	29.7846	1.32803	0.664015	+	0.747719i	$0.268851\pi$
$504$	0	0
$505$	−30.6795	−1.36522
$506$	0	0
$507$	−10.8564	−0.482150
$508$	19.7128	0.874615
$509$	21.9282	0.971951	0.485975	−	0.873973i	$-0.338465\pi$
$509$	21.9282	0.971951	0.485975	+	0.873973i	$0.338465\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.46410	−0.329548
$514$	0	0
$515$	31.8564	1.40376
$516$	7.85641	0.345859
$517$	−4.37307	−0.192327
$518$	0	0
$519$	−20.9282	−0.918646
$520$	0	0
$521$	−20.8564	−0.913736	−0.456868	−	0.889535i	$-0.651029\pi$
$521$	−20.8564	−0.913736	−0.456868	+	0.889535i	$0.651029\pi$
$522$	0	0
$523$	−31.1769	−1.36327	−0.681636	−	0.731692i	$-0.738731\pi$
$523$	−31.1769	−1.36327	−0.681636	+	0.731692i	$0.738731\pi$
$524$	−34.9282	−1.52585
$525$	0	0
$526$	0	0
$527$	11.3397	0.493967
$528$	9.07180	0.394799
$529$	−20.8564	−0.906800
$530$	0	0
$531$	−6.92820	−0.300658
$532$	0	0
$533$	1.46410	0.0634173
$534$	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
$541$	16.9282	0.727800	0.363900	−	0.931438i	$-0.381445\pi$
$541$	16.9282	0.727800	0.363900	+	0.931438i	$0.381445\pi$
$542$	0	0
$543$	20.0000	0.858282
$544$	0	0
$545$	−32.7846	−1.40434
$546$	0	0
$547$	−11.6077	−0.496309	−0.248155	−	0.968720i	$-0.579824\pi$
$547$	−11.6077	−0.496309	−0.248155	+	0.968720i	$0.579824\pi$
$548$	−26.3923	−1.12742
$549$	−1.73205	−0.0739221
$550$	0	0
$551$	−38.7846	−1.65228
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.46410	0.147043
$556$	−9.07180	−0.384730
$557$	10.5167	0.445605	0.222803	−	0.974864i	$-0.428479\pi$
$557$	10.5167	0.445605	0.222803	+	0.974864i	$0.428479\pi$
$558$	0	0
$559$	−5.75129	−0.243254
$560$	0	0
$561$	−11.3397	−0.478764
$562$	0	0
$563$	−5.92820	−0.249844	−0.124922	−	0.992167i	$-0.539868\pi$
$563$	−5.92820	−0.249844	−0.124922	+	0.992167i	$0.539868\pi$
$564$	3.85641	0.162384
$565$	−60.0000	−2.52422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.2487	−1.10040	−0.550202	−	0.835032i	$-0.685449\pi$
$569$	−26.2487	−1.10040	−0.550202	+	0.835032i	$0.685449\pi$
$570$	0	0
$571$	34.9282	1.46170	0.730850	−	0.682538i	$-0.239124\pi$
$571$	34.9282	1.46170	0.730850	+	0.682538i	$0.239124\pi$
$572$	−6.64102	−0.277675
$573$	10.3923	0.434145
$574$	0	0
$575$	−10.2487	−0.427401
$576$	−8.00000	−0.333333
$577$	−35.3205	−1.47041	−0.735206	−	0.677844i	$-0.762915\pi$
$577$	−35.3205	−1.47041	−0.735206	+	0.677844i	$0.762915\pi$
$578$	0	0
$579$	27.3205	1.13540
$580$	36.0000	1.49482
$581$	0	0
$582$	0	0
$583$	−6.64102	−0.275043
$584$	0	0
$585$	−5.07180	−0.209693
$586$	0	0
$587$	−41.9282	−1.73056	−0.865281	−	0.501287i	$-0.832860\pi$
$587$	−41.9282	−1.73056	−0.865281	+	0.501287i	$0.832860\pi$
$588$	0	0
$589$	16.9282	0.697514
$590$	0	0
$591$	7.46410	0.307032
$592$	−4.00000	−0.164399
$593$	42.7128	1.75400	0.877002	−	0.480486i	$-0.159540\pi$
$593$	42.7128	1.75400	0.877002	+	0.480486i	$0.159540\pi$
$594$	0	0
$595$	0	0
$596$	−27.7128	−1.13516
$597$	9.32051	0.381463
$598$	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
$601$	40.2487	1.64178	0.820890	−	0.571087i	$-0.193478\pi$
$601$	40.2487	1.64178	0.820890	+	0.571087i	$0.193478\pi$
$602$	0	0
$603$	0.535898	0.0218235
$604$	0.784610	0.0319253
$605$	20.2872	0.824791
$606$	0	0
$607$	−44.2487	−1.79600	−0.898000	−	0.439996i	$-0.854980\pi$
$607$	−44.2487	−1.79600	−0.898000	+	0.439996i	$0.854980\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.82309	−0.114210
$612$	10.0000	0.404226
$613$	−32.6410	−1.31836	−0.659179	−	0.751986i	$-0.729096\pi$
$613$	−32.6410	−1.31836	−0.659179	+	0.751986i	$0.729096\pi$
$614$	0	0
$615$	−3.46410	−0.139686
$616$	0	0
$617$	−19.8564	−0.799389	−0.399694	−	0.916648i	$-0.630884\pi$
$617$	−19.8564	−0.799389	−0.399694	+	0.916648i	$0.630884\pi$
$618$	0	0
$619$	37.1962	1.49504	0.747520	−	0.664240i	$-0.231245\pi$
$619$	37.1962	1.49504	0.747520	+	0.664240i	$0.231245\pi$
$620$	−15.7128	−0.631042
$621$	−1.46410	−0.0587524
$622$	0	0
$623$	0	0
$624$	5.85641	0.234444
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−16.9282	−0.676047
$628$	−20.0000	−0.798087
$629$	5.00000	0.199363
$630$	0	0
$631$	0.0717968	0.00285818	0.00142909	−	0.999999i	$-0.499545\pi$
$631$	0.0717968	0.00285818	0.00142909	+	0.999999i	$0.499545\pi$
$632$	0	0
$633$	1.07180	0.0426001
$634$	0	0
$635$	34.1436	1.35495
$636$	5.85641	0.232222
$637$	0	0
$638$	0	0
$639$	2.80385	0.110918
$640$	0	0
$641$	2.51666	0.0994021	0.0497011	−	0.998764i	$-0.484173\pi$
$641$	2.51666	0.0994021	0.0497011	+	0.998764i	$0.484173\pi$
$642$	0	0
$643$	10.2487	0.404170	0.202085	−	0.979368i	$-0.435228\pi$
$643$	10.2487	0.404170	0.202085	+	0.979368i	$0.435228\pi$
$644$	0	0
$645$	13.6077	0.535802
$646$	0	0
$647$	−10.9282	−0.429632	−0.214816	−	0.976655i	$-0.568915\pi$
$647$	−10.9282	−0.429632	−0.214816	+	0.976655i	$0.568915\pi$
$648$	0	0
$649$	−15.7128	−0.616782
$650$	0	0
$651$	0	0
$652$	8.14359	0.318928
$653$	4.66025	0.182370	0.0911849	−	0.995834i	$-0.470935\pi$
$653$	4.66025	0.182370	0.0911849	+	0.995834i	$0.470935\pi$
$654$	0	0
$655$	−60.4974	−2.36383
$656$	4.00000	0.156174
$657$	−1.19615	−0.0466664
$658$	0	0
$659$	−19.4641	−0.758214	−0.379107	−	0.925353i	$-0.623769\pi$
$659$	−19.4641	−0.758214	−0.379107	+	0.925353i	$0.623769\pi$
$660$	15.7128	0.611620
$661$	−38.6410	−1.50296	−0.751481	−	0.659755i	$-0.770660\pi$
$661$	−38.6410	−1.50296	−0.751481	+	0.659755i	$0.770660\pi$
$662$	0	0
$663$	−7.32051	−0.284305
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.60770	−0.294571
$668$	−10.1436	−0.392467
$669$	−15.4641	−0.597877
$670$	0	0
$671$	−3.92820	−0.151647
$672$	0	0
$673$	8.53590	0.329035	0.164517	−	0.986374i	$-0.447393\pi$
$673$	8.53590	0.329035	0.164517	+	0.986374i	$0.447393\pi$
$674$	0	0
$675$	7.00000	0.269430
$676$	21.7128	0.835108
$677$	23.8564	0.916876	0.458438	−	0.888726i	$-0.348409\pi$
$677$	23.8564	0.916876	0.458438	+	0.888726i	$0.348409\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	13.9282	0.533730
$682$	0	0
$683$	36.2487	1.38702	0.693509	−	0.720448i	$-0.256064\pi$
$683$	36.2487	1.38702	0.693509	+	0.720448i	$0.256064\pi$
$684$	14.9282	0.570794
$685$	−45.7128	−1.74660
$686$	0	0
$687$	4.92820	0.188023
$688$	−15.7128	−0.599045
$689$	−4.28719	−0.163329
$690$	0	0
$691$	21.1769	0.805608	0.402804	−	0.915286i	$-0.368036\pi$
$691$	21.1769	0.805608	0.402804	+	0.915286i	$0.368036\pi$
$692$	41.8564	1.59114
$693$	0	0
$694$	0	0
$695$	−15.7128	−0.596021
$696$	0	0
$697$	−5.00000	−0.189389
$698$	0	0
$699$	13.8564	0.524097
$700$	0	0
$701$	−8.53590	−0.322396	−0.161198	−	0.986922i	$-0.551536\pi$
$701$	−8.53590	−0.322396	−0.161198	+	0.986922i	$0.551536\pi$
$702$	0	0
$703$	7.46410	0.281514
$704$	−18.1436	−0.683812
$705$	6.67949	0.251564
$706$	0	0
$707$	0	0
$708$	13.8564	0.520756
$709$	−1.32051	−0.0495927	−0.0247964	−	0.999693i	$-0.507894\pi$
$709$	−1.32051	−0.0495927	−0.0247964	+	0.999693i	$0.507894\pi$
$710$	0	0
$711$	−12.9282	−0.484846
$712$	0	0
$713$	3.32051	0.124354
$714$	0	0
$715$	−11.5026	−0.430172
$716$	31.1769	1.16514
$717$	−8.53590	−0.318779
$718$	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
$727$	40.6410	1.50729	0.753646	−	0.657281i	$-0.228293\pi$
$727$	40.6410	1.50729	0.753646	+	0.657281i	$0.228293\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	19.6410	0.726449
$732$	3.46410	0.128037
$733$	21.9808	0.811878	0.405939	−	0.913900i	$-0.366945\pi$
$733$	21.9808	0.811878	0.405939	+	0.913900i	$0.366945\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.21539	0.0447695
$738$	0	0
$739$	−33.7846	−1.24279	−0.621393	−	0.783499i	$-0.713433\pi$
$739$	−33.7846	−1.24279	−0.621393	+	0.783499i	$0.713433\pi$
$740$	−6.92820	−0.254686
$741$	−10.9282	−0.401458
$742$	0	0
$743$	20.5359	0.753389	0.376695	−	0.926338i	$-0.377061\pi$
$743$	20.5359	0.753389	0.376695	+	0.926338i	$0.377061\pi$
$744$	0	0
$745$	−48.0000	−1.75858
$746$	0	0
$747$	−10.0000	−0.365881
$748$	22.6795	0.829244
$749$	0	0
$750$	0	0
$751$	24.2487	0.884848	0.442424	−	0.896806i	$-0.354119\pi$
$751$	24.2487	0.884848	0.442424	+	0.896806i	$0.354119\pi$
$752$	−7.71281	−0.281257
$753$	0.535898	0.0195292
$754$	0	0
$755$	1.35898	0.0494585
$756$	0	0
$757$	−11.8564	−0.430928	−0.215464	−	0.976512i	$-0.569126\pi$
$757$	−11.8564	−0.430928	−0.215464	+	0.976512i	$0.569126\pi$
$758$	0	0
$759$	−3.32051	−0.120527
$760$	0	0
$761$	−38.2487	−1.38651	−0.693257	−	0.720690i	$-0.743825\pi$
$761$	−38.2487	−1.38651	−0.693257	+	0.720690i	$0.743825\pi$
$762$	0	0
$763$	0	0
$764$	−20.7846	−0.751961
$765$	17.3205	0.626224
$766$	0	0
$767$	−10.1436	−0.366264
$768$	16.0000	0.577350
$769$	−13.1962	−0.475865	−0.237933	−	0.971282i	$-0.576470\pi$
$769$	−13.1962	−0.475865	−0.237933	+	0.971282i	$0.576470\pi$
$770$	0	0
$771$	3.00000	0.108042
$772$	−54.6410	−1.96657
$773$	−30.8564	−1.10983	−0.554914	−	0.831908i	$-0.687249\pi$
$773$	−30.8564	−1.10983	−0.554914	+	0.831908i	$0.687249\pi$
$774$	0	0
$775$	−15.8756	−0.570270
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.46410	−0.267429
$780$	10.1436	0.363199
$781$	6.35898	0.227542
$782$	0	0
$783$	5.19615	0.185695
$784$	0	0
$785$	−34.6410	−1.23639
$786$	0	0
$787$	18.5167	0.660048	0.330024	−	0.943973i	$-0.392943\pi$
$787$	18.5167	0.660048	0.330024	+	0.943973i	$0.392943\pi$
$788$	−14.9282	−0.531795
$789$	4.12436	0.146831
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.53590	−0.0900524
$794$	0	0
$795$	10.1436	0.359756
$796$	−18.6410	−0.660713
$797$	7.85641	0.278288	0.139144	−	0.990272i	$-0.455565\pi$
$797$	7.85641	0.278288	0.139144	+	0.990272i	$0.455565\pi$
$798$	0	0
$799$	9.64102	0.341075
$800$	0	0
$801$	4.92820	0.174129
$802$	0	0
$803$	−2.71281	−0.0957331
$804$	−1.07180	−0.0377994
$805$	0	0
$806$	0	0
$807$	10.5359	0.370881
$808$	0	0
$809$	−54.9282	−1.93117	−0.965586	−	0.260083i	$-0.916250\pi$
$809$	−54.9282	−1.93117	−0.965586	+	0.260083i	$0.916250\pi$
$810$	0	0
$811$	1.73205	0.0608205	0.0304103	−	0.999538i	$-0.490319\pi$
$811$	1.73205	0.0608205	0.0304103	+	0.999538i	$0.490319\pi$
$812$	0	0
$813$	19.5885	0.686997
$814$	0	0
$815$	14.1051	0.494081
$816$	−20.0000	−0.700140
$817$	29.3205	1.02579
$818$	0	0
$819$	0	0
$820$	6.92820	0.241943
$821$	−43.3205	−1.51190	−0.755948	−	0.654632i	$-0.772824\pi$
$821$	−43.3205	−1.51190	−0.755948	+	0.654632i	$0.772824\pi$
$822$	0	0
$823$	−43.5692	−1.51873	−0.759364	−	0.650666i	$-0.774490\pi$
$823$	−43.5692	−1.51873	−0.759364	+	0.650666i	$0.774490\pi$
$824$	0	0
$825$	15.8756	0.552719
$826$	0	0
$827$	−34.8038	−1.21025	−0.605124	−	0.796131i	$-0.706877\pi$
$827$	−34.8038	−1.21025	−0.605124	+	0.796131i	$0.706877\pi$
$828$	2.92820	0.101762
$829$	−23.0526	−0.800648	−0.400324	−	0.916374i	$-0.631102\pi$
$829$	−23.0526	−0.800648	−0.400324	+	0.916374i	$0.631102\pi$
$830$	0	0
$831$	−8.07180	−0.280008
$832$	−11.7128	−0.406069
$833$	0	0
$834$	0	0
$835$	−17.5692	−0.608008
$836$	33.8564	1.17095
$837$	−2.26795	−0.0783918
$838$	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
$843$	24.1244	0.830887
$844$	−2.14359	−0.0737855
$845$	37.6077	1.29374
$846$	0	0
$847$	0	0
$848$	−11.7128	−0.402220
$849$	−12.6603	−0.434499
$850$	0	0
$851$	1.46410	0.0501888
$852$	−5.60770	−0.192116
$853$	3.71281	0.127124	0.0635621	−	0.997978i	$-0.479754\pi$
$853$	3.71281	0.127124	0.0635621	+	0.997978i	$0.479754\pi$
$854$	0	0
$855$	25.8564	0.884270
$856$	0	0
$857$	−14.2487	−0.486727	−0.243363	−	0.969935i	$-0.578251\pi$
$857$	−14.2487	−0.486727	−0.243363	+	0.969935i	$0.578251\pi$
$858$	0	0
$859$	13.1962	0.450247	0.225123	−	0.974330i	$-0.427721\pi$
$859$	13.1962	0.450247	0.225123	+	0.974330i	$0.427721\pi$
$860$	−27.2154	−0.928037
$861$	0	0
$862$	0	0
$863$	49.3205	1.67889	0.839445	−	0.543445i	$-0.182880\pi$
$863$	49.3205	1.67889	0.839445	+	0.543445i	$0.182880\pi$
$864$	0	0
$865$	72.4974	2.46499
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−29.3205	−0.994630
$870$	0	0
$871$	0.784610	0.0265855
$872$	0	0
$873$	18.3923	0.622485
$874$	0	0
$875$	0	0
$876$	2.39230	0.0808285
$877$	41.9282	1.41581	0.707907	−	0.706305i	$-0.249639\pi$
$877$	41.9282	1.41581	0.707907	+	0.706305i	$0.249639\pi$
$878$	0	0
$879$	−9.00000	−0.303562
$880$	−31.4256	−1.05936
$881$	33.4641	1.12743	0.563717	−	0.825968i	$-0.309371\pi$
$881$	33.4641	1.12743	0.563717	+	0.825968i	$0.309371\pi$
$882$	0	0
$883$	−43.5692	−1.46622	−0.733110	−	0.680110i	$-0.761932\pi$
$883$	−43.5692	−1.46622	−0.733110	+	0.680110i	$0.761932\pi$
$884$	14.6410	0.492431
$885$	24.0000	0.806751
$886$	0	0
$887$	6.85641	0.230216	0.115108	−	0.993353i	$-0.463279\pi$
$887$	6.85641	0.230216	0.115108	+	0.993353i	$0.463279\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.26795	0.0759792
$892$	30.9282	1.03555
$893$	14.3923	0.481620
$894$	0	0
$895$	54.0000	1.80502
$896$	0	0
$897$	−2.14359	−0.0715725
$898$	0	0
$899$	−11.7846	−0.393039
$900$	−14.0000	−0.466667
$901$	14.6410	0.487763
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−69.2820	−2.30301
$906$	0	0
$907$	−21.8564	−0.725730	−0.362865	−	0.931842i	$-0.618201\pi$
$907$	−21.8564	−0.725730	−0.362865	+	0.931842i	$0.618201\pi$
$908$	−27.8564	−0.924447
$909$	8.85641	0.293748
$910$	0	0
$911$	24.7846	0.821151	0.410575	−	0.911827i	$-0.365328\pi$
$911$	24.7846	0.821151	0.410575	+	0.911827i	$0.365328\pi$
$912$	−29.8564	−0.988644
$913$	−22.6795	−0.750582
$914$	0	0
$915$	6.00000	0.198354
$916$	−9.85641	−0.325665
$917$	0	0
$918$	0	0
$919$	−18.5359	−0.611443	−0.305721	−	0.952121i	$-0.598898\pi$
$919$	−18.5359	−0.611443	−0.305721	+	0.952121i	$0.598898\pi$
$920$	0	0
$921$	17.7321	0.584291
$922$	0	0
$923$	4.10512	0.135122
$924$	0	0
$925$	−7.00000	−0.230159
$926$	0	0
$927$	−9.19615	−0.302041
$928$	0	0
$929$	−2.21539	−0.0726846	−0.0363423	−	0.999339i	$-0.511571\pi$
$929$	−2.21539	−0.0726846	−0.0363423	+	0.999339i	$0.511571\pi$
$930$	0	0
$931$	0	0
$932$	−27.7128	−0.907763
$933$	−1.85641	−0.0607760
$934$	0	0
$935$	39.2820	1.28466
$936$	0	0
$937$	18.9282	0.618357	0.309179	−	0.951004i	$-0.399946\pi$
$937$	18.9282	0.618357	0.309179	+	0.951004i	$0.399946\pi$
$938$	0	0
$939$	−10.9282	−0.356628
$940$	−13.3590	−0.435722
$941$	−1.32051	−0.0430473	−0.0215237	−	0.999768i	$-0.506852\pi$
$941$	−1.32051	−0.0430473	−0.0215237	+	0.999768i	$0.506852\pi$
$942$	0	0
$943$	−1.46410	−0.0476777
$944$	−27.7128	−0.901975
$945$	0	0
$946$	0	0
$947$	36.9282	1.20001	0.600003	−	0.799998i	$-0.295166\pi$
$947$	36.9282	1.20001	0.600003	+	0.799998i	$0.295166\pi$
$948$	25.8564	0.839777
$949$	−1.75129	−0.0568492
$950$	0	0
$951$	33.1962	1.07646
$952$	0	0
$953$	−55.3205	−1.79201	−0.896004	−	0.444047i	$-0.853542\pi$
$953$	−55.3205	−1.79201	−0.896004	+	0.444047i	$0.853542\pi$
$954$	0	0
$955$	−36.0000	−1.16493
$956$	17.0718	0.552141
$957$	11.7846	0.380942
$958$	0	0
$959$	0	0
$960$	27.7128	0.894427
$961$	−25.8564	−0.834078
$962$	0	0
$963$	3.46410	0.111629
$964$	10.3923	0.334714
$965$	−94.6410	−3.04660
$966$	0	0
$967$	12.2487	0.393892	0.196946	−	0.980414i	$-0.436898\pi$
$967$	12.2487	0.393892	0.196946	+	0.980414i	$0.436898\pi$
$968$	0	0
$969$	37.3205	1.19891
$970$	0	0
$971$	27.9282	0.896259	0.448129	−	0.893969i	$-0.352090\pi$
$971$	27.9282	0.896259	0.448129	+	0.893969i	$0.352090\pi$
$972$	−2.00000	−0.0641500
$973$	0	0
$974$	0	0
$975$	10.2487	0.328221
$976$	−6.92820	−0.221766
$977$	−37.9808	−1.21511	−0.607556	−	0.794277i	$-0.707850\pi$
$977$	−37.9808	−1.21511	−0.607556	+	0.794277i	$0.707850\pi$
$978$	0	0
$979$	11.1769	0.357216
$980$	0	0
$981$	9.46410	0.302166
$982$	0	0
$983$	26.7846	0.854296	0.427148	−	0.904182i	$-0.359518\pi$
$983$	26.7846	0.854296	0.427148	+	0.904182i	$0.359518\pi$
$984$	0	0
$985$	−25.8564	−0.823854
$986$	0	0
$987$	0	0
$988$	21.8564	0.695345
$989$	5.75129	0.182880
$990$	0	0
$991$	52.1051	1.65517	0.827587	−	0.561338i	$-0.189713\pi$
$991$	52.1051	1.65517	0.827587	+	0.561338i	$0.189713\pi$
$992$	0	0
$993$	−32.2487	−1.02338
$994$	0	0
$995$	−32.2872	−1.02357
$996$	20.0000	0.633724
$997$	−41.1769	−1.30409	−0.652043	−	0.758182i	$-0.726088\pi$
$997$	−41.1769	−1.30409	−0.652043	+	0.758182i	$0.726088\pi$
$998$	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.1.1	yes	2
7.6	odd	2		6027.2.a.l.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.l.1.2	✓	2	7.6	odd	2
6027.2.a.n.1.1	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{4} -3.46410 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{4} -3.46410 q^{5} +1.00000 q^{9} +2.26795 q^{11} -2.00000 q^{12} +1.46410 q^{13} -3.46410 q^{15} +4.00000 q^{16} -5.00000 q^{17} -7.46410 q^{19} +6.92820 q^{20} -1.46410 q^{23} +7.00000 q^{25} +1.00000 q^{27} +5.19615 q^{29} -2.26795 q^{31} +2.26795 q^{33} -2.00000 q^{36} -1.00000 q^{37} +1.46410 q^{39} +1.00000 q^{41} -3.92820 q^{43} -4.53590 q^{44} -3.46410 q^{45} -1.92820 q^{47} +4.00000 q^{48} -5.00000 q^{51} -2.92820 q^{52} -2.92820 q^{53} -7.85641 q^{55} -7.46410 q^{57} -6.92820 q^{59} +6.92820 q^{60} -1.73205 q^{61} -8.00000 q^{64} -5.07180 q^{65} +0.535898 q^{67} +10.0000 q^{68} -1.46410 q^{69} +2.80385 q^{71} -1.19615 q^{73} +7.00000 q^{75} +14.9282 q^{76} -12.9282 q^{79} -13.8564 q^{80} +1.00000 q^{81} -10.0000 q^{83} +17.3205 q^{85} +5.19615 q^{87} +4.92820 q^{89} +2.92820 q^{92} -2.26795 q^{93} +25.8564 q^{95} +18.3923 q^{97} +2.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) \) 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

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Varieties

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