LMFDB - Embedded newform 6039.2.a.j.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
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} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} + \cdots - 1 $ x^14 - x^13 - 21x^12 + 20x^11 + 167x^10 - 148x^9 - 627x^8 + 497x^7 + 1123x^6 - 745x^5 - 802x^4 + 386x^3 + 74x^2 - 15x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.67203$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67203 q^{2} +0.795679 q^{4} +1.45953 q^{5} -0.113904 q^{7} -2.01366 q^{8} +O(q^{10})$	\(q+1.67203 q^{2} +0.795679 q^{4} +1.45953 q^{5} -0.113904 q^{7} -2.01366 q^{8} +2.44037 q^{10} +1.00000 q^{11} -2.41931 q^{13} -0.190451 q^{14} -4.95825 q^{16} -4.33956 q^{17} +5.52946 q^{19} +1.16132 q^{20} +1.67203 q^{22} +5.61639 q^{23} -2.86978 q^{25} -4.04515 q^{26} -0.0906313 q^{28} +2.97696 q^{29} +3.23872 q^{31} -4.26302 q^{32} -7.25587 q^{34} -0.166247 q^{35} +5.47586 q^{37} +9.24541 q^{38} -2.93899 q^{40} +3.54936 q^{41} +3.64080 q^{43} +0.795679 q^{44} +9.39076 q^{46} +6.81812 q^{47} -6.98703 q^{49} -4.79835 q^{50} -1.92499 q^{52} +0.123290 q^{53} +1.45953 q^{55} +0.229364 q^{56} +4.97756 q^{58} +14.8361 q^{59} +1.00000 q^{61} +5.41523 q^{62} +2.78861 q^{64} -3.53104 q^{65} +9.16048 q^{67} -3.45290 q^{68} -0.277969 q^{70} +9.69135 q^{71} -1.05277 q^{73} +9.15579 q^{74} +4.39968 q^{76} -0.113904 q^{77} +10.0943 q^{79} -7.23671 q^{80} +5.93464 q^{82} +2.01215 q^{83} -6.33371 q^{85} +6.08752 q^{86} -2.01366 q^{88} -16.8141 q^{89} +0.275569 q^{91} +4.46885 q^{92} +11.4001 q^{94} +8.07040 q^{95} -4.43143 q^{97} -11.6825 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	$ 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) $ 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

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 $

$2$	1.67203	1.18230	0.591151	−	0.806561i	$-0.298674\pi$
$2$	1.67203	1.18230	0.591151	+	0.806561i	$0.298674\pi$
$3$	0	0
$3$	0	0
$4$	0.795679	0.397840
$5$	1.45953	0.652721	0.326360	−	0.945245i	$-0.394178\pi$
$5$	1.45953	0.652721	0.326360	+	0.945245i	$0.394178\pi$
$6$	0	0
$7$	−0.113904	−0.0430518	−0.0215259	−	0.999768i	$-0.506852\pi$
$7$	−0.113904	−0.0430518	−0.0215259	+	0.999768i	$0.506852\pi$
$8$	−2.01366	−0.711936
$9$	0	0
$10$	2.44037	0.771713
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.41931	−0.670995	−0.335497	−	0.942041i	$-0.608904\pi$
$13$	−2.41931	−0.670995	−0.335497	+	0.942041i	$0.608904\pi$
$14$	−0.190451	−0.0509002
$15$	0	0
$16$	−4.95825	−1.23956
$17$	−4.33956	−1.05250	−0.526249	−	0.850330i	$-0.676402\pi$
$17$	−4.33956	−1.05250	−0.526249	+	0.850330i	$0.676402\pi$
$18$	0	0
$19$	5.52946	1.26855	0.634273	−	0.773110i	$-0.281300\pi$
$19$	5.52946	1.26855	0.634273	+	0.773110i	$0.281300\pi$
$20$	1.16132	0.259678
$21$	0	0
$22$	1.67203	0.356478
$23$	5.61639	1.17110	0.585549	−	0.810637i	$-0.300879\pi$
$23$	5.61639	1.17110	0.585549	+	0.810637i	$0.300879\pi$
$24$	0	0
$25$	−2.86978	−0.573956
$26$	−4.04515	−0.793319
$27$	0	0
$28$	−0.0906313	−0.0171277
$29$	2.97696	0.552807	0.276404	−	0.961042i	$-0.410857\pi$
$29$	2.97696	0.552807	0.276404	+	0.961042i	$0.410857\pi$
$30$	0	0
$31$	3.23872	0.581691	0.290846	−	0.956770i	$-0.406063\pi$
$31$	3.23872	0.581691	0.290846	+	0.956770i	$0.406063\pi$
$32$	−4.26302	−0.753603
$33$	0	0
$34$	−7.25587	−1.24437
$35$	−0.166247	−0.0281008
$36$	0	0
$37$	5.47586	0.900225	0.450113	−	0.892972i	$-0.351384\pi$
$37$	5.47586	0.900225	0.450113	+	0.892972i	$0.351384\pi$
$38$	9.24541	1.49980
$39$	0	0
$40$	−2.93899	−0.464695
$41$	3.54936	0.554317	0.277159	−	0.960824i	$-0.410607\pi$
$41$	3.54936	0.554317	0.277159	+	0.960824i	$0.410607\pi$
$42$	0	0
$43$	3.64080	0.555217	0.277609	−	0.960694i	$-0.410458\pi$
$43$	3.64080	0.555217	0.277609	+	0.960694i	$0.410458\pi$
$44$	0.795679	0.119953
$45$	0	0
$46$	9.39076	1.38459
$47$	6.81812	0.994525	0.497263	−	0.867600i	$-0.334339\pi$
$47$	6.81812	0.994525	0.497263	+	0.867600i	$0.334339\pi$
$48$	0	0
$49$	−6.98703	−0.998147
$50$	−4.79835	−0.678589
$51$	0	0
$52$	−1.92499	−0.266948
$53$	0.123290	0.0169352	0.00846758	−	0.999964i	$-0.497305\pi$
$53$	0.123290	0.0169352	0.00846758	+	0.999964i	$0.497305\pi$
$54$	0	0
$55$	1.45953	0.196803
$56$	0.229364	0.0306501
$57$	0	0
$58$	4.97756	0.653585
$59$	14.8361	1.93149	0.965746	−	0.259488i	$-0.0835538\pi$
$59$	14.8361	1.93149	0.965746	+	0.259488i	$0.0835538\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	5.41523	0.687735
$63$	0	0
$64$	2.78861	0.348576
$65$	−3.53104	−0.437972
$66$	0	0
$67$	9.16048	1.11913	0.559566	−	0.828786i	$-0.310968\pi$
$67$	9.16048	1.11913	0.559566	+	0.828786i	$0.310968\pi$
$68$	−3.45290	−0.418726
$69$	0	0
$70$	−0.277969	−0.0332236
$71$	9.69135	1.15015	0.575076	−	0.818100i	$-0.304973\pi$
$71$	9.69135	1.15015	0.575076	+	0.818100i	$0.304973\pi$
$72$	0	0
$73$	−1.05277	−0.123217	−0.0616087	−	0.998100i	$-0.519623\pi$
$73$	−1.05277	−0.123217	−0.0616087	+	0.998100i	$0.519623\pi$
$74$	9.15579	1.06434
$75$	0	0
$76$	4.39968	0.504678
$77$	−0.113904	−0.0129806
$78$	0	0
$79$	10.0943	1.13570	0.567850	−	0.823132i	$-0.307775\pi$
$79$	10.0943	1.13570	0.567850	+	0.823132i	$0.307775\pi$
$80$	−7.23671	−0.809089
$81$	0	0
$82$	5.93464	0.655371
$83$	2.01215	0.220862	0.110431	−	0.993884i	$-0.464777\pi$
$83$	2.01215	0.220862	0.110431	+	0.993884i	$0.464777\pi$
$84$	0	0
$85$	−6.33371	−0.686988
$86$	6.08752	0.656435
$87$	0	0
$88$	−2.01366	−0.214657
$89$	−16.8141	−1.78229	−0.891145	−	0.453719i	$-0.850097\pi$
$89$	−16.8141	−1.78229	−0.891145	+	0.453719i	$0.850097\pi$
$90$	0	0
$91$	0.275569	0.0288875
$92$	4.46885	0.465909
$93$	0	0
$94$	11.4001	1.17583
$95$	8.07040	0.828006
$96$	0	0
$97$	−4.43143	−0.449944	−0.224972	−	0.974365i	$-0.572229\pi$
$97$	−4.43143	−0.449944	−0.224972	+	0.974365i	$0.572229\pi$
$98$	−11.6825	−1.18011
$99$	0	0
$100$	−2.28342	−0.228342
$101$	9.12365	0.907837	0.453918	−	0.891043i	$-0.350026\pi$
$101$	9.12365	0.907837	0.453918	+	0.891043i	$0.350026\pi$
$102$	0	0
$103$	−4.46502	−0.439951	−0.219976	−	0.975505i	$-0.570598\pi$
$103$	−4.46502	−0.439951	−0.219976	+	0.975505i	$0.570598\pi$
$104$	4.87165	0.477705
$105$	0	0
$106$	0.206144	0.0200225
$107$	−3.58525	−0.346599	−0.173300	−	0.984869i	$-0.555443\pi$
$107$	−3.58525	−0.346599	−0.173300	+	0.984869i	$0.555443\pi$
$108$	0	0
$109$	3.16796	0.303435	0.151718	−	0.988424i	$-0.451520\pi$
$109$	3.16796	0.303435	0.151718	+	0.988424i	$0.451520\pi$
$110$	2.44037	0.232680
$111$	0	0
$112$	0.564766	0.0533654
$113$	−9.84126	−0.925788	−0.462894	−	0.886414i	$-0.653189\pi$
$113$	−9.84126	−0.925788	−0.462894	+	0.886414i	$0.653189\pi$
$114$	0	0
$115$	8.19728	0.764400
$116$	2.36870	0.219929
$117$	0	0
$118$	24.8063	2.28361
$119$	0.494295	0.0453119
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.67203	0.151378
$123$	0	0
$124$	2.57698	0.231420
$125$	−11.4862	−1.02735
$126$	0	0
$127$	7.25474	0.643754	0.321877	−	0.946782i	$-0.395686\pi$
$127$	7.25474	0.643754	0.321877	+	0.946782i	$0.395686\pi$
$128$	13.1887	1.16573
$129$	0	0
$130$	−5.90401	−0.517816
$131$	−5.18052	−0.452624	−0.226312	−	0.974055i	$-0.572667\pi$
$131$	−5.18052	−0.452624	−0.226312	+	0.974055i	$0.572667\pi$
$132$	0	0
$133$	−0.629829	−0.0546131
$134$	15.3166	1.32315
$135$	0	0
$136$	8.73840	0.749311
$137$	−15.9087	−1.35918	−0.679588	−	0.733594i	$-0.737841\pi$
$137$	−15.9087	−1.35918	−0.679588	+	0.733594i	$0.737841\pi$
$138$	0	0
$139$	17.4182	1.47740	0.738698	−	0.674037i	$-0.235441\pi$
$139$	17.4182	1.47740	0.738698	+	0.674037i	$0.235441\pi$
$140$	−0.132279	−0.0111796
$141$	0	0
$142$	16.2042	1.35983
$143$	−2.41931	−0.202312
$144$	0	0
$145$	4.34495	0.360829
$146$	−1.76026	−0.145680
$147$	0	0
$148$	4.35703	0.358145
$149$	20.0971	1.64642	0.823208	−	0.567740i	$-0.192182\pi$
$149$	20.0971	1.64642	0.823208	+	0.567740i	$0.192182\pi$
$150$	0	0
$151$	14.1784	1.15382	0.576910	−	0.816808i	$-0.304258\pi$
$151$	14.1784	1.15382	0.576910	+	0.816808i	$0.304258\pi$
$152$	−11.1344	−0.903123
$153$	0	0
$154$	−0.190451	−0.0153470
$155$	4.72700	0.379682
$156$	0	0
$157$	4.99242	0.398439	0.199219	−	0.979955i	$-0.436159\pi$
$157$	4.99242	0.398439	0.199219	+	0.979955i	$0.436159\pi$
$158$	16.8780	1.34274
$159$	0	0
$160$	−6.22200	−0.491892
$161$	−0.639731	−0.0504179
$162$	0	0
$163$	−5.98250	−0.468586	−0.234293	−	0.972166i	$-0.575277\pi$
$163$	−5.98250	−0.468586	−0.234293	+	0.972166i	$0.575277\pi$
$164$	2.82416	0.220529
$165$	0	0
$166$	3.36438	0.261126
$167$	−24.2227	−1.87441	−0.937203	−	0.348784i	$-0.886595\pi$
$167$	−24.2227	−1.87441	−0.937203	+	0.348784i	$0.886595\pi$
$168$	0	0
$169$	−7.14696	−0.549766
$170$	−10.5901	−0.812227
$171$	0	0
$172$	2.89691	0.220887
$173$	−11.4724	−0.872227	−0.436114	−	0.899892i	$-0.643645\pi$
$173$	−11.4724	−0.872227	−0.436114	+	0.899892i	$0.643645\pi$
$174$	0	0
$175$	0.326880	0.0247098
$176$	−4.95825	−0.373742
$177$	0	0
$178$	−28.1136	−2.10721
$179$	−16.7617	−1.25283	−0.626413	−	0.779492i	$-0.715477\pi$
$179$	−16.7617	−1.25283	−0.626413	+	0.779492i	$0.715477\pi$
$180$	0	0
$181$	23.4517	1.74315	0.871574	−	0.490264i	$-0.163100\pi$
$181$	23.4517	1.74315	0.871574	+	0.490264i	$0.163100\pi$
$182$	0.460760	0.0341538
$183$	0	0
$184$	−11.3095	−0.833747
$185$	7.99217	0.587596
$186$	0	0
$187$	−4.33956	−0.317340
$188$	5.42504	0.395662
$189$	0	0
$190$	13.4939	0.978953
$191$	8.77663	0.635054	0.317527	−	0.948249i	$-0.397148\pi$
$191$	8.77663	0.635054	0.317527	+	0.948249i	$0.397148\pi$
$192$	0	0
$193$	11.8741	0.854716	0.427358	−	0.904083i	$-0.359444\pi$
$193$	11.8741	0.854716	0.427358	+	0.904083i	$0.359444\pi$
$194$	−7.40948	−0.531970
$195$	0	0
$196$	−5.55943	−0.397102
$197$	6.68035	0.475956	0.237978	−	0.971271i	$-0.423515\pi$
$197$	6.68035	0.475956	0.237978	+	0.971271i	$0.423515\pi$
$198$	0	0
$199$	−12.2108	−0.865602	−0.432801	−	0.901490i	$-0.642475\pi$
$199$	−12.2108	−0.865602	−0.432801	+	0.901490i	$0.642475\pi$
$200$	5.77875	0.408620
$201$	0	0
$202$	15.2550	1.07334
$203$	−0.339088	−0.0237993
$204$	0	0
$205$	5.18039	0.361814
$206$	−7.46564	−0.520156
$207$	0	0
$208$	11.9955	0.831740
$209$	5.52946	0.382481
$210$	0	0
$211$	2.24877	0.154812	0.0774058	−	0.997000i	$-0.475336\pi$
$211$	2.24877	0.154812	0.0774058	+	0.997000i	$0.475336\pi$
$212$	0.0980991	0.00673748
$213$	0	0
$214$	−5.99464	−0.409785
$215$	5.31385	0.362402
$216$	0	0
$217$	−0.368904	−0.0250428
$218$	5.29692	0.358753
$219$	0	0
$220$	1.16132	0.0782959
$221$	10.4987	0.706221
$222$	0	0
$223$	25.2425	1.69036	0.845181	−	0.534480i	$-0.179493\pi$
$223$	25.2425	1.69036	0.845181	+	0.534480i	$0.179493\pi$
$224$	0.485577	0.0324440
$225$	0	0
$226$	−16.4549	−1.09456
$227$	15.9227	1.05682	0.528412	−	0.848988i	$-0.322788\pi$
$227$	15.9227	1.05682	0.528412	+	0.848988i	$0.322788\pi$
$228$	0	0
$229$	4.84415	0.320111	0.160055	−	0.987108i	$-0.448833\pi$
$229$	4.84415	0.320111	0.160055	+	0.987108i	$0.448833\pi$
$230$	13.7061	0.903752
$231$	0	0
$232$	−5.99457	−0.393563
$233$	6.87950	0.450691	0.225346	−	0.974279i	$-0.427649\pi$
$233$	6.87950	0.450691	0.225346	+	0.974279i	$0.427649\pi$
$234$	0	0
$235$	9.95124	0.649147
$236$	11.8048	0.768425
$237$	0	0
$238$	0.826475	0.0535724
$239$	−26.2207	−1.69607	−0.848037	−	0.529938i	$-0.822215\pi$
$239$	−26.2207	−1.69607	−0.848037	+	0.529938i	$0.822215\pi$
$240$	0	0
$241$	−21.9334	−1.41286	−0.706428	−	0.707785i	$-0.749694\pi$
$241$	−21.9334	−1.41286	−0.706428	+	0.707785i	$0.749694\pi$
$242$	1.67203	0.107482
$243$	0	0
$244$	0.795679	0.0509382
$245$	−10.1978	−0.651511
$246$	0	0
$247$	−13.3775	−0.851187
$248$	−6.52167	−0.414127
$249$	0	0
$250$	−19.2052	−1.21464
$251$	−24.9091	−1.57225	−0.786125	−	0.618068i	$-0.787916\pi$
$251$	−24.9091	−1.57225	−0.786125	+	0.618068i	$0.787916\pi$
$252$	0	0
$253$	5.61639	0.353099
$254$	12.1301	0.761112
$255$	0	0
$256$	16.4746	1.02966
$257$	−0.438875	−0.0273763	−0.0136881	−	0.999906i	$-0.504357\pi$
$257$	−0.438875	−0.0273763	−0.0136881	+	0.999906i	$0.504357\pi$
$258$	0	0
$259$	−0.623724	−0.0387563
$260$	−2.80958	−0.174243
$261$	0	0
$262$	−8.66198	−0.535139
$263$	−21.5826	−1.33084	−0.665421	−	0.746469i	$-0.731748\pi$
$263$	−21.5826	−1.33084	−0.665421	+	0.746469i	$0.731748\pi$
$264$	0	0
$265$	0.179945	0.0110539
$266$	−1.05309	−0.0645692
$267$	0	0
$268$	7.28881	0.445235
$269$	24.6911	1.50544	0.752720	−	0.658341i	$-0.228741\pi$
$269$	24.6911	1.50544	0.752720	+	0.658341i	$0.228741\pi$
$270$	0	0
$271$	−18.7879	−1.14128	−0.570641	−	0.821200i	$-0.693305\pi$
$271$	−18.7879	−1.14128	−0.570641	+	0.821200i	$0.693305\pi$
$272$	21.5167	1.30464
$273$	0	0
$274$	−26.5999	−1.60696
$275$	−2.86978	−0.173054
$276$	0	0
$277$	−1.86162	−0.111854	−0.0559270	−	0.998435i	$-0.517811\pi$
$277$	−1.86162	−0.111854	−0.0559270	+	0.998435i	$0.517811\pi$
$278$	29.1238	1.74673
$279$	0	0
$280$	0.334764	0.0200060
$281$	−12.0550	−0.719140	−0.359570	−	0.933118i	$-0.617077\pi$
$281$	−12.0550	−0.719140	−0.359570	+	0.933118i	$0.617077\pi$
$282$	0	0
$283$	17.4984	1.04017	0.520084	−	0.854115i	$-0.325900\pi$
$283$	17.4984	1.04017	0.520084	+	0.854115i	$0.325900\pi$
$284$	7.71121	0.457576
$285$	0	0
$286$	−4.04515	−0.239195
$287$	−0.404288	−0.0238644
$288$	0	0
$289$	1.83181	0.107754
$290$	7.26488	0.426609
$291$	0	0
$292$	−0.837667	−0.0490208
$293$	4.01414	0.234509	0.117254	−	0.993102i	$-0.462591\pi$
$293$	4.01414	0.234509	0.117254	+	0.993102i	$0.462591\pi$
$294$	0	0
$295$	21.6537	1.26073
$296$	−11.0265	−0.640903
$297$	0	0
$298$	33.6029	1.94656
$299$	−13.5878	−0.785801
$300$	0	0
$301$	−0.414703	−0.0239031
$302$	23.7066	1.36416
$303$	0	0
$304$	−27.4165	−1.57244
$305$	1.45953	0.0835723
$306$	0	0
$307$	15.3443	0.875746	0.437873	−	0.899037i	$-0.355732\pi$
$307$	15.3443	0.875746	0.437873	+	0.899037i	$0.355732\pi$
$308$	−0.0906313	−0.00516420
$309$	0	0
$310$	7.90368	0.448899
$311$	−28.3138	−1.60553	−0.802764	−	0.596297i	$-0.796638\pi$
$311$	−28.3138	−1.60553	−0.802764	+	0.596297i	$0.796638\pi$
$312$	0	0
$313$	26.2747	1.48513	0.742567	−	0.669771i	$-0.233608\pi$
$313$	26.2747	1.48513	0.742567	+	0.669771i	$0.233608\pi$
$314$	8.34748	0.471075
$315$	0	0
$316$	8.03185	0.451827
$317$	−31.0474	−1.74379	−0.871897	−	0.489689i	$-0.837110\pi$
$317$	−31.0474	−1.74379	−0.871897	+	0.489689i	$0.837110\pi$
$318$	0	0
$319$	2.97696	0.166678
$320$	4.07005	0.227523
$321$	0	0
$322$	−1.06965	−0.0596092
$323$	−23.9954	−1.33514
$324$	0	0
$325$	6.94287	0.385121
$326$	−10.0029	−0.554010
$327$	0	0
$328$	−7.14720	−0.394638
$329$	−0.776614	−0.0428161
$330$	0	0
$331$	−20.5558	−1.12985	−0.564924	−	0.825143i	$-0.691094\pi$
$331$	−20.5558	−1.12985	−0.564924	+	0.825143i	$0.691094\pi$
$332$	1.60103	0.0878679
$333$	0	0
$334$	−40.5010	−2.21612
$335$	13.3700	0.730480
$336$	0	0
$337$	−0.489461	−0.0266626	−0.0133313	−	0.999911i	$-0.504244\pi$
$337$	−0.489461	−0.0266626	−0.0133313	+	0.999911i	$0.504244\pi$
$338$	−11.9499	−0.649990
$339$	0	0
$340$	−5.03961	−0.273311
$341$	3.23872	0.175386
$342$	0	0
$343$	1.59318	0.0860238
$344$	−7.33133	−0.395279
$345$	0	0
$346$	−19.1821	−1.03124
$347$	8.54263	0.458593	0.229296	−	0.973357i	$-0.426358\pi$
$347$	8.54263	0.458593	0.229296	+	0.973357i	$0.426358\pi$
$348$	0	0
$349$	28.5046	1.52582	0.762909	−	0.646506i	$-0.223770\pi$
$349$	28.5046	1.52582	0.762909	+	0.646506i	$0.223770\pi$
$350$	0.546553	0.0292145
$351$	0	0
$352$	−4.26302	−0.227220
$353$	−15.4235	−0.820912	−0.410456	−	0.911880i	$-0.634630\pi$
$353$	−15.4235	−0.820912	−0.410456	+	0.911880i	$0.634630\pi$
$354$	0	0
$355$	14.1448	0.750728
$356$	−13.3786	−0.709066
$357$	0	0
$358$	−28.0260	−1.48122
$359$	30.6063	1.61534	0.807669	−	0.589637i	$-0.200729\pi$
$359$	30.6063	1.61534	0.807669	+	0.589637i	$0.200729\pi$
$360$	0	0
$361$	11.5749	0.609206
$362$	39.2118	2.06093
$363$	0	0
$364$	0.219265	0.0114926
$365$	−1.53655	−0.0804265
$366$	0	0
$367$	2.65622	0.138654	0.0693268	−	0.997594i	$-0.477915\pi$
$367$	2.65622	0.138654	0.0693268	+	0.997594i	$0.477915\pi$
$368$	−27.8475	−1.45165
$369$	0	0
$370$	13.3631	0.694716
$371$	−0.0140432	−0.000729089 0
$372$	0	0
$373$	−0.426449	−0.0220807	−0.0110404	−	0.999939i	$-0.503514\pi$
$373$	−0.426449	−0.0220807	−0.0110404	+	0.999939i	$0.503514\pi$
$374$	−7.25587	−0.375192
$375$	0	0
$376$	−13.7294	−0.708038
$377$	−7.20217	−0.370931
$378$	0	0
$379$	28.8875	1.48385	0.741925	−	0.670483i	$-0.233913\pi$
$379$	28.8875	1.48385	0.741925	+	0.670483i	$0.233913\pi$
$380$	6.42145	0.329414
$381$	0	0
$382$	14.6748	0.750827
$383$	29.4474	1.50469	0.752346	−	0.658768i	$-0.228922\pi$
$383$	29.4474	1.50469	0.752346	+	0.658768i	$0.228922\pi$
$384$	0	0
$385$	−0.166247	−0.00847271
$386$	19.8538	1.01053
$387$	0	0
$388$	−3.52600	−0.179006
$389$	11.6234	0.589329	0.294665	−	0.955601i	$-0.404792\pi$
$389$	11.6234	0.589329	0.294665	+	0.955601i	$0.404792\pi$
$390$	0	0
$391$	−24.3727	−1.23258
$392$	14.0695	0.710616
$393$	0	0
$394$	11.1697	0.562724
$395$	14.7330	0.741295
$396$	0	0
$397$	−25.8143	−1.29558	−0.647791	−	0.761818i	$-0.724307\pi$
$397$	−25.8143	−1.29558	−0.647791	+	0.761818i	$0.724307\pi$
$398$	−20.4168	−1.02340
$399$	0	0
$400$	14.2291	0.711454
$401$	6.33074	0.316142	0.158071	−	0.987428i	$-0.449472\pi$
$401$	6.33074	0.316142	0.158071	+	0.987428i	$0.449472\pi$
$402$	0	0
$403$	−7.83545	−0.390312
$404$	7.25950	0.361174
$405$	0	0
$406$	−0.566965	−0.0281380
$407$	5.47586	0.271428
$408$	0	0
$409$	−17.4823	−0.864446	−0.432223	−	0.901767i	$-0.642271\pi$
$409$	−17.4823	−0.864446	−0.432223	+	0.901767i	$0.642271\pi$
$410$	8.66177	0.427774
$411$	0	0
$412$	−3.55272	−0.175030
$413$	−1.68989	−0.0831542
$414$	0	0
$415$	2.93679	0.144161
$416$	10.3136	0.505664
$417$	0	0
$418$	9.24541	0.452208
$419$	−16.0311	−0.783171	−0.391585	−	0.920142i	$-0.628073\pi$
$419$	−16.0311	−0.783171	−0.391585	+	0.920142i	$0.628073\pi$
$420$	0	0
$421$	2.65713	0.129500	0.0647502	−	0.997902i	$-0.479375\pi$
$421$	2.65713	0.129500	0.0647502	+	0.997902i	$0.479375\pi$
$422$	3.76001	0.183034
$423$	0	0
$424$	−0.248263	−0.0120567
$425$	12.4536	0.604088
$426$	0	0
$427$	−0.113904	−0.00551222
$428$	−2.85271	−0.137891
$429$	0	0
$430$	8.88491	0.428468
$431$	16.3460	0.787358	0.393679	−	0.919248i	$-0.371202\pi$
$431$	16.3460	0.787358	0.393679	+	0.919248i	$0.371202\pi$
$432$	0	0
$433$	−19.2280	−0.924038	−0.462019	−	0.886870i	$-0.652875\pi$
$433$	−19.2280	−0.924038	−0.462019	+	0.886870i	$0.652875\pi$
$434$	−0.616818	−0.0296082
$435$	0	0
$436$	2.52068	0.120719
$437$	31.0556	1.48559
$438$	0	0
$439$	−1.73219	−0.0826728	−0.0413364	−	0.999145i	$-0.513162\pi$
$439$	−1.73219	−0.0826728	−0.0413364	+	0.999145i	$0.513162\pi$
$440$	−2.93899	−0.140111
$441$	0	0
$442$	17.5542	0.834967
$443$	−12.1970	−0.579497	−0.289749	−	0.957103i	$-0.593572\pi$
$443$	−12.1970	−0.579497	−0.289749	+	0.957103i	$0.593572\pi$
$444$	0	0
$445$	−24.5406	−1.16334
$446$	42.2062	1.99852
$447$	0	0
$448$	−0.317635	−0.0150068
$449$	17.4644	0.824195	0.412098	−	0.911140i	$-0.364796\pi$
$449$	17.4644	0.824195	0.412098	+	0.911140i	$0.364796\pi$
$450$	0	0
$451$	3.54936	0.167133
$452$	−7.83049	−0.368315
$453$	0	0
$454$	26.6232	1.24949
$455$	0.402201	0.0188555
$456$	0	0
$457$	15.5179	0.725897	0.362949	−	0.931809i	$-0.381770\pi$
$457$	15.5179	0.725897	0.362949	+	0.931809i	$0.381770\pi$
$458$	8.09956	0.378468
$459$	0	0
$460$	6.52241	0.304109
$461$	12.5703	0.585458	0.292729	−	0.956195i	$-0.405437\pi$
$461$	12.5703	0.585458	0.292729	+	0.956195i	$0.405437\pi$
$462$	0	0
$463$	12.0019	0.557778	0.278889	−	0.960323i	$-0.410034\pi$
$463$	12.0019	0.557778	0.278889	+	0.960323i	$0.410034\pi$
$464$	−14.7605	−0.685239
$465$	0	0
$466$	11.5027	0.532853
$467$	−37.4089	−1.73108	−0.865539	−	0.500842i	$-0.833024\pi$
$467$	−37.4089	−1.73108	−0.865539	+	0.500842i	$0.833024\pi$
$468$	0	0
$469$	−1.04342	−0.0481806
$470$	16.6388	0.767489
$471$	0	0
$472$	−29.8748	−1.37510
$473$	3.64080	0.167404
$474$	0	0
$475$	−15.8683	−0.728089
$476$	0.393300	0.0180269
$477$	0	0
$478$	−43.8417	−2.00527
$479$	17.7608	0.811514	0.405757	−	0.913981i	$-0.367008\pi$
$479$	17.7608	0.811514	0.405757	+	0.913981i	$0.367008\pi$
$480$	0	0
$481$	−13.2478	−0.604046
$482$	−36.6733	−1.67042
$483$	0	0
$484$	0.795679	0.0361672
$485$	−6.46780	−0.293688
$486$	0	0
$487$	−13.1113	−0.594129	−0.297064	−	0.954857i	$-0.596008\pi$
$487$	−13.1113	−0.594129	−0.297064	+	0.954857i	$0.596008\pi$
$488$	−2.01366	−0.0911540
$489$	0	0
$490$	−17.0509	−0.770283
$491$	22.4438	1.01287	0.506437	−	0.862277i	$-0.330962\pi$
$491$	22.4438	1.01287	0.506437	+	0.862277i	$0.330962\pi$
$492$	0	0
$493$	−12.9187	−0.581829
$494$	−22.3675	−1.00636
$495$	0	0
$496$	−16.0584	−0.721043
$497$	−1.10389	−0.0495161
$498$	0	0
$499$	17.9269	0.802517	0.401259	−	0.915965i	$-0.368573\pi$
$499$	17.9269	0.802517	0.401259	+	0.915965i	$0.368573\pi$
$500$	−9.13930	−0.408722
$501$	0	0
$502$	−41.6487	−1.85887
$503$	43.4638	1.93796	0.968978	−	0.247148i	$-0.0794932\pi$
$503$	43.4638	1.93796	0.968978	+	0.247148i	$0.0794932\pi$
$504$	0	0
$505$	13.3162	0.592564
$506$	9.39076	0.417470
$507$	0	0
$508$	5.77245	0.256111
$509$	−3.04838	−0.135117	−0.0675585	−	0.997715i	$-0.521521\pi$
$509$	−3.04838	−0.135117	−0.0675585	+	0.997715i	$0.521521\pi$
$510$	0	0
$511$	0.119915	0.00530473
$512$	1.16870	0.0516496
$513$	0	0
$514$	−0.733812	−0.0323670
$515$	−6.51682	−0.287165
$516$	0	0
$517$	6.81812	0.299861
$518$	−1.04288	−0.0458217
$519$	0	0
$520$	7.11032	0.311808
$521$	4.92667	0.215841	0.107921	−	0.994160i	$-0.465581\pi$
$521$	4.92667	0.215841	0.107921	+	0.994160i	$0.465581\pi$
$522$	0	0
$523$	1.37122	0.0599595	0.0299797	−	0.999551i	$-0.490456\pi$
$523$	1.37122	0.0599595	0.0299797	+	0.999551i	$0.490456\pi$
$524$	−4.12203	−0.180072
$525$	0	0
$526$	−36.0867	−1.57346
$527$	−14.0546	−0.612229
$528$	0	0
$529$	8.54384	0.371471
$530$	0.300873	0.0130691
$531$	0	0
$532$	−0.501142	−0.0217273
$533$	−8.58699	−0.371944
$534$	0	0
$535$	−5.23277	−0.226232
$536$	−18.4461	−0.796750
$537$	0	0
$538$	41.2842	1.77989
$539$	−6.98703	−0.300953
$540$	0	0
$541$	−37.0801	−1.59420	−0.797099	−	0.603848i	$-0.793633\pi$
$541$	−37.0801	−1.59420	−0.797099	+	0.603848i	$0.793633\pi$
$542$	−31.4139	−1.34934
$543$	0	0
$544$	18.4997	0.793167
$545$	4.62372	0.198059
$546$	0	0
$547$	11.1746	0.477790	0.238895	−	0.971045i	$-0.423215\pi$
$547$	11.1746	0.477790	0.238895	+	0.971045i	$0.423215\pi$
$548$	−12.6583	−0.540734
$549$	0	0
$550$	−4.79835	−0.204602
$551$	16.4610	0.701261
$552$	0	0
$553$	−1.14979	−0.0488939
$554$	−3.11268	−0.132245
$555$	0	0
$556$	13.8593	0.587766
$557$	15.3427	0.650092	0.325046	−	0.945698i	$-0.394620\pi$
$557$	15.3427	0.650092	0.325046	+	0.945698i	$0.394620\pi$
$558$	0	0
$559$	−8.80821	−0.372548
$560$	0.824292	0.0348327
$561$	0	0
$562$	−20.1563	−0.850241
$563$	−43.3954	−1.82890	−0.914449	−	0.404701i	$-0.867376\pi$
$563$	−43.3954	−1.82890	−0.914449	+	0.404701i	$0.867376\pi$
$564$	0	0
$565$	−14.3636	−0.604281
$566$	29.2577	1.22979
$567$	0	0
$568$	−19.5151	−0.818834
$569$	−1.47086	−0.0616615	−0.0308308	−	0.999525i	$-0.509815\pi$
$569$	−1.47086	−0.0616615	−0.0308308	+	0.999525i	$0.509815\pi$
$570$	0	0
$571$	−7.09968	−0.297112	−0.148556	−	0.988904i	$-0.547463\pi$
$571$	−7.09968	−0.297112	−0.148556	+	0.988904i	$0.547463\pi$
$572$	−1.92499	−0.0804879
$573$	0	0
$574$	−0.675981	−0.0282149
$575$	−16.1178	−0.672159
$576$	0	0
$577$	−20.6477	−0.859575	−0.429788	−	0.902930i	$-0.641412\pi$
$577$	−20.6477	−0.859575	−0.429788	+	0.902930i	$0.641412\pi$
$578$	3.06284	0.127397
$579$	0	0
$580$	3.45719	0.143552
$581$	−0.229193	−0.00950852
$582$	0	0
$583$	0.123290	0.00510614
$584$	2.11992	0.0877229
$585$	0	0
$586$	6.71176	0.277260
$587$	3.56601	0.147185	0.0735925	−	0.997288i	$-0.476554\pi$
$587$	3.56601	0.147185	0.0735925	+	0.997288i	$0.476554\pi$
$588$	0	0
$589$	17.9084	0.737901
$590$	36.2056	1.49056
$591$	0	0
$592$	−27.1507	−1.11589
$593$	−23.6716	−0.972077	−0.486038	−	0.873937i	$-0.661558\pi$
$593$	−23.6716	−0.972077	−0.486038	+	0.873937i	$0.661558\pi$
$594$	0	0
$595$	0.721437	0.0295760
$596$	15.9908	0.655010
$597$	0	0
$598$	−22.7191	−0.929054
$599$	−34.3309	−1.40272	−0.701360	−	0.712807i	$-0.747424\pi$
$599$	−34.3309	−1.40272	−0.701360	+	0.712807i	$0.747424\pi$
$600$	0	0
$601$	2.29548	0.0936346	0.0468173	−	0.998903i	$-0.485092\pi$
$601$	2.29548	0.0936346	0.0468173	+	0.998903i	$0.485092\pi$
$602$	−0.693395	−0.0282607
$603$	0	0
$604$	11.2814	0.459035
$605$	1.45953	0.0593382
$606$	0	0
$607$	−47.1071	−1.91202	−0.956010	−	0.293335i	$-0.905235\pi$
$607$	−47.1071	−1.91202	−0.956010	+	0.293335i	$0.905235\pi$
$608$	−23.5722	−0.955980
$609$	0	0
$610$	2.44037	0.0988078
$611$	−16.4951	−0.667321
$612$	0	0
$613$	19.9949	0.807586	0.403793	−	0.914850i	$-0.367692\pi$
$613$	19.9949	0.807586	0.403793	+	0.914850i	$0.367692\pi$
$614$	25.6561	1.03540
$615$	0	0
$616$	0.229364	0.00924135
$617$	−28.3395	−1.14090	−0.570452	−	0.821331i	$-0.693232\pi$
$617$	−28.3395	−1.14090	−0.570452	+	0.821331i	$0.693232\pi$
$618$	0	0
$619$	−18.4108	−0.739994	−0.369997	−	0.929033i	$-0.620641\pi$
$619$	−18.4108	−0.739994	−0.369997	+	0.929033i	$0.620641\pi$
$620$	3.76118	0.151053
$621$	0	0
$622$	−47.3415	−1.89822
$623$	1.91520	0.0767308
$624$	0	0
$625$	−2.41548	−0.0966192
$626$	43.9321	1.75588
$627$	0	0
$628$	3.97237	0.158515
$629$	−23.7628	−0.947486
$630$	0	0
$631$	7.42123	0.295435	0.147717	−	0.989030i	$-0.452807\pi$
$631$	7.42123	0.295435	0.147717	+	0.989030i	$0.452807\pi$
$632$	−20.3265	−0.808546
$633$	0	0
$634$	−51.9121	−2.06169
$635$	10.5885	0.420191
$636$	0	0
$637$	16.9038	0.669751
$638$	4.97756	0.197063
$639$	0	0
$640$	19.2492	0.760893
$641$	−6.86699	−0.271230	−0.135615	−	0.990762i	$-0.543301\pi$
$641$	−6.86699	−0.271230	−0.135615	+	0.990762i	$0.543301\pi$
$642$	0	0
$643$	45.7495	1.80418	0.902092	−	0.431544i	$-0.142031\pi$
$643$	45.7495	1.80418	0.902092	+	0.431544i	$0.142031\pi$
$644$	−0.509021	−0.0200582
$645$	0	0
$646$	−40.1211	−1.57854
$647$	17.7282	0.696968	0.348484	−	0.937315i	$-0.386697\pi$
$647$	17.7282	0.696968	0.348484	+	0.937315i	$0.386697\pi$
$648$	0	0
$649$	14.8361	0.582367
$650$	11.6087	0.455330
$651$	0	0
$652$	−4.76015	−0.186422
$653$	−25.1767	−0.985239	−0.492619	−	0.870245i	$-0.663960\pi$
$653$	−25.1767	−0.985239	−0.492619	+	0.870245i	$0.663960\pi$
$654$	0	0
$655$	−7.56112	−0.295437
$656$	−17.5986	−0.687111
$657$	0	0
$658$	−1.29852	−0.0506216
$659$	−47.2222	−1.83951	−0.919757	−	0.392487i	$-0.871615\pi$
$659$	−47.2222	−1.83951	−0.919757	+	0.392487i	$0.871615\pi$
$660$	0	0
$661$	−10.8885	−0.423512	−0.211756	−	0.977323i	$-0.567918\pi$
$661$	−10.8885	−0.423512	−0.211756	+	0.977323i	$0.567918\pi$
$662$	−34.3698	−1.33582
$663$	0	0
$664$	−4.05179	−0.157240
$665$	−0.919253	−0.0356471
$666$	0	0
$667$	16.7198	0.647391
$668$	−19.2735	−0.745713
$669$	0	0
$670$	22.3550	0.863649
$671$	1.00000	0.0386046
$672$	0	0
$673$	−3.19833	−0.123287	−0.0616433	−	0.998098i	$-0.519634\pi$
$673$	−3.19833	−0.123287	−0.0616433	+	0.998098i	$0.519634\pi$
$674$	−0.818392	−0.0315233
$675$	0	0
$676$	−5.68669	−0.218719
$677$	20.9309	0.804439	0.402219	−	0.915543i	$-0.368239\pi$
$677$	20.9309	0.804439	0.402219	+	0.915543i	$0.368239\pi$
$678$	0	0
$679$	0.504759	0.0193709
$680$	12.7539	0.489091
$681$	0	0
$682$	5.41523	0.207360
$683$	−9.41323	−0.360187	−0.180094	−	0.983649i	$-0.557640\pi$
$683$	−9.41323	−0.360187	−0.180094	+	0.983649i	$0.557640\pi$
$684$	0	0
$685$	−23.2193	−0.887162
$686$	2.66385	0.101706
$687$	0	0
$688$	−18.0520	−0.688227
$689$	−0.298276	−0.0113634
$690$	0	0
$691$	34.9093	1.32801	0.664006	−	0.747727i	$-0.268855\pi$
$691$	34.9093	1.32801	0.664006	+	0.747727i	$0.268855\pi$
$692$	−9.12832	−0.347007
$693$	0	0
$694$	14.2835	0.542195
$695$	25.4224	0.964326
$696$	0	0
$697$	−15.4027	−0.583418
$698$	47.6606	1.80398
$699$	0	0
$700$	0.260092	0.00983055
$701$	21.4585	0.810478	0.405239	−	0.914211i	$-0.367188\pi$
$701$	21.4585	0.810478	0.405239	+	0.914211i	$0.367188\pi$
$702$	0	0
$703$	30.2785	1.14198
$704$	2.78861	0.105100
$705$	0	0
$706$	−25.7886	−0.970566
$707$	−1.03922	−0.0390840
$708$	0	0
$709$	14.7920	0.555527	0.277764	−	0.960649i	$-0.410407\pi$
$709$	14.7920	0.555527	0.277764	+	0.960649i	$0.410407\pi$
$710$	23.6505	0.887588
$711$	0	0
$712$	33.8578	1.26888
$713$	18.1899	0.681217
$714$	0	0
$715$	−3.53104	−0.132054
$716$	−13.3369	−0.498424
$717$	0	0
$718$	51.1746	1.90982
$719$	−3.91313	−0.145935	−0.0729676	−	0.997334i	$-0.523247\pi$
$719$	−3.91313	−0.145935	−0.0729676	+	0.997334i	$0.523247\pi$
$720$	0	0
$721$	0.508585	0.0189407
$722$	19.3536	0.720266
$723$	0	0
$724$	18.6600	0.693494
$725$	−8.54321	−0.317287
$726$	0	0
$727$	−22.4942	−0.834263	−0.417131	−	0.908846i	$-0.636964\pi$
$727$	−22.4942	−0.834263	−0.417131	+	0.908846i	$0.636964\pi$
$728$	−0.554903	−0.0205661
$729$	0	0
$730$	−2.56915	−0.0950885
$731$	−15.7995	−0.584365
$732$	0	0
$733$	19.0644	0.704161	0.352080	−	0.935970i	$-0.385474\pi$
$733$	19.0644	0.704161	0.352080	+	0.935970i	$0.385474\pi$
$734$	4.44128	0.163931
$735$	0	0
$736$	−23.9428	−0.882544
$737$	9.16048	0.337431
$738$	0	0
$739$	28.1238	1.03455	0.517275	−	0.855819i	$-0.326946\pi$
$739$	28.1238	1.03455	0.517275	+	0.855819i	$0.326946\pi$
$740$	6.35920	0.233769
$741$	0	0
$742$	−0.0234807	−0.000862003 0
$743$	−26.7820	−0.982535	−0.491268	−	0.871009i	$-0.663466\pi$
$743$	−26.7820	−0.982535	−0.491268	+	0.871009i	$0.663466\pi$
$744$	0	0
$745$	29.3322	1.07465
$746$	−0.713036	−0.0261061
$747$	0	0
$748$	−3.45290	−0.126251
$749$	0.408375	0.0149217
$750$	0	0
$751$	−48.7584	−1.77922	−0.889609	−	0.456722i	$-0.849023\pi$
$751$	−48.7584	−1.77922	−0.889609	+	0.456722i	$0.849023\pi$
$752$	−33.8060	−1.23278
$753$	0	0
$754$	−12.0422	−0.438552
$755$	20.6937	0.753122
$756$	0	0
$757$	19.1838	0.697246	0.348623	−	0.937263i	$-0.386649\pi$
$757$	19.1838	0.697246	0.348623	+	0.937263i	$0.386649\pi$
$758$	48.3007	1.75436
$759$	0	0
$760$	−16.2510	−0.589487
$761$	26.2237	0.950608	0.475304	−	0.879822i	$-0.342338\pi$
$761$	26.2237	0.950608	0.475304	+	0.879822i	$0.342338\pi$
$762$	0	0
$763$	−0.360844	−0.0130634
$764$	6.98338	0.252650
$765$	0	0
$766$	49.2369	1.77900
$767$	−35.8930	−1.29602
$768$	0	0
$769$	−34.4961	−1.24396	−0.621981	−	0.783032i	$-0.713672\pi$
$769$	−34.4961	−1.24396	−0.621981	+	0.783032i	$0.713672\pi$
$770$	−0.277969	−0.0100173
$771$	0	0
$772$	9.44797	0.340040
$773$	−36.7533	−1.32192	−0.660961	−	0.750420i	$-0.729851\pi$
$773$	−36.7533	−1.32192	−0.660961	+	0.750420i	$0.729851\pi$
$774$	0	0
$775$	−9.29441	−0.333865
$776$	8.92339	0.320331
$777$	0	0
$778$	19.4346	0.696766
$779$	19.6261	0.703177
$780$	0	0
$781$	9.69135	0.346784
$782$	−40.7518	−1.45728
$783$	0	0
$784$	34.6434	1.23727
$785$	7.28658	0.260069
$786$	0	0
$787$	−5.85918	−0.208857	−0.104429	−	0.994532i	$-0.533301\pi$
$787$	−5.85918	−0.208857	−0.104429	+	0.994532i	$0.533301\pi$
$788$	5.31542	0.189354
$789$	0	0
$790$	24.6339	0.876435
$791$	1.12096	0.0398568
$792$	0	0
$793$	−2.41931	−0.0859121
$794$	−43.1622	−1.53177
$795$	0	0
$796$	−9.71589	−0.344371
$797$	−19.2286	−0.681112	−0.340556	−	0.940224i	$-0.610615\pi$
$797$	−19.2286	−0.681112	−0.340556	+	0.940224i	$0.610615\pi$
$798$	0	0
$799$	−29.5877	−1.04674
$800$	12.2339	0.432535
$801$	0	0
$802$	10.5852	0.373776
$803$	−1.05277	−0.0371514
$804$	0	0
$805$	−0.933705	−0.0329088
$806$	−13.1011	−0.461466
$807$	0	0
$808$	−18.3719	−0.646321
$809$	−8.57652	−0.301535	−0.150767	−	0.988569i	$-0.548174\pi$
$809$	−8.57652	−0.301535	−0.150767	+	0.988569i	$0.548174\pi$
$810$	0	0
$811$	51.6913	1.81513	0.907564	−	0.419914i	$-0.137940\pi$
$811$	51.6913	1.81513	0.907564	+	0.419914i	$0.137940\pi$
$812$	−0.269806	−0.00946832
$813$	0	0
$814$	9.15579	0.320910
$815$	−8.73163	−0.305856
$816$	0	0
$817$	20.1317	0.704318
$818$	−29.2310	−1.02204
$819$	0	0
$820$	4.12193	0.143944
$821$	37.9548	1.32463	0.662316	−	0.749224i	$-0.269573\pi$
$821$	37.9548	1.32463	0.662316	+	0.749224i	$0.269573\pi$
$822$	0	0
$823$	23.9123	0.833531	0.416766	−	0.909014i	$-0.363164\pi$
$823$	23.9123	0.833531	0.416766	+	0.909014i	$0.363164\pi$
$824$	8.99102	0.313217
$825$	0	0
$826$	−2.82555	−0.0983134
$827$	−20.6669	−0.718658	−0.359329	−	0.933211i	$-0.616994\pi$
$827$	−20.6669	−0.718658	−0.359329	+	0.933211i	$0.616994\pi$
$828$	0	0
$829$	9.30470	0.323166	0.161583	−	0.986859i	$-0.448340\pi$
$829$	9.30470	0.323166	0.161583	+	0.986859i	$0.448340\pi$
$830$	4.91040	0.170443
$831$	0	0
$832$	−6.74650	−0.233893
$833$	30.3206	1.05055
$834$	0	0
$835$	−35.3537	−1.22346
$836$	4.39968	0.152166
$837$	0	0
$838$	−26.8045	−0.925945
$839$	−28.4780	−0.983170	−0.491585	−	0.870830i	$-0.663582\pi$
$839$	−28.4780	−0.983170	−0.491585	+	0.870830i	$0.663582\pi$
$840$	0	0
$841$	−20.1377	−0.694404
$842$	4.44279	0.153109
$843$	0	0
$844$	1.78930	0.0615902
$845$	−10.4312	−0.358844
$846$	0	0
$847$	−0.113904	−0.00391380
$848$	−0.611302	−0.0209922
$849$	0	0
$850$	20.8228	0.714214
$851$	30.7545	1.05425
$852$	0	0
$853$	−16.0864	−0.550789	−0.275395	−	0.961331i	$-0.588808\pi$
$853$	−16.0864	−0.550789	−0.275395	+	0.961331i	$0.588808\pi$
$854$	−0.190451	−0.00651711
$855$	0	0
$856$	7.21947	0.246756
$857$	−32.5019	−1.11024	−0.555122	−	0.831769i	$-0.687328\pi$
$857$	−32.5019	−1.11024	−0.555122	+	0.831769i	$0.687328\pi$
$858$	0	0
$859$	53.0060	1.80854	0.904270	−	0.426961i	$-0.140416\pi$
$859$	53.0060	1.80854	0.904270	+	0.426961i	$0.140416\pi$
$860$	4.22812	0.144178
$861$	0	0
$862$	27.3309	0.930895
$863$	−18.4211	−0.627061	−0.313530	−	0.949578i	$-0.601512\pi$
$863$	−18.4211	−0.627061	−0.313530	+	0.949578i	$0.601512\pi$
$864$	0	0
$865$	−16.7442	−0.569321
$866$	−32.1497	−1.09249
$867$	0	0
$868$	−0.293529	−0.00996304
$869$	10.0943	0.342427
$870$	0	0
$871$	−22.1620	−0.750931
$872$	−6.37919	−0.216027
$873$	0	0
$874$	51.9259	1.75642
$875$	1.30832	0.0442294
$876$	0	0
$877$	14.6497	0.494684	0.247342	−	0.968928i	$-0.420443\pi$
$877$	14.6497	0.494684	0.247342	+	0.968928i	$0.420443\pi$
$878$	−2.89627	−0.0977443
$879$	0	0
$880$	−7.23671	−0.243949
$881$	47.0593	1.58547	0.792733	−	0.609569i	$-0.208657\pi$
$881$	47.0593	1.58547	0.792733	+	0.609569i	$0.208657\pi$
$882$	0	0
$883$	16.8945	0.568544	0.284272	−	0.958744i	$-0.408248\pi$
$883$	16.8945	0.568544	0.284272	+	0.958744i	$0.408248\pi$
$884$	8.35362	0.280963
$885$	0	0
$886$	−20.3938	−0.685141
$887$	−38.0179	−1.27651	−0.638257	−	0.769823i	$-0.720344\pi$
$887$	−38.0179	−1.27651	−0.638257	+	0.769823i	$0.720344\pi$
$888$	0	0
$889$	−0.826346	−0.0277147
$890$	−41.0326	−1.37542
$891$	0	0
$892$	20.0849	0.672493
$893$	37.7005	1.26160
$894$	0	0
$895$	−24.4641	−0.817745
$896$	−1.50225	−0.0501866
$897$	0	0
$898$	29.2010	0.974448
$899$	9.64153	0.321563
$900$	0	0
$901$	−0.535024	−0.0178242
$902$	5.93464	0.197602
$903$	0	0
$904$	19.8169	0.659101
$905$	34.2283	1.13779
$906$	0	0
$907$	−3.40956	−0.113213	−0.0566063	−	0.998397i	$-0.518028\pi$
$907$	−3.40956	−0.113213	−0.0566063	+	0.998397i	$0.518028\pi$
$908$	12.6693	0.420447
$909$	0	0
$910$	0.672492	0.0222929
$911$	35.6182	1.18009	0.590043	−	0.807372i	$-0.299111\pi$
$911$	35.6182	1.18009	0.590043	+	0.807372i	$0.299111\pi$
$912$	0	0
$913$	2.01215	0.0665925
$914$	25.9464	0.858230
$915$	0	0
$916$	3.85439	0.127353
$917$	0.590084	0.0194863
$918$	0	0
$919$	−23.5214	−0.775899	−0.387950	−	0.921681i	$-0.626817\pi$
$919$	−23.5214	−0.775899	−0.387950	+	0.921681i	$0.626817\pi$
$920$	−16.5065	−0.544204
$921$	0	0
$922$	21.0179	0.692188
$923$	−23.4463	−0.771746
$924$	0	0
$925$	−15.7145	−0.516689
$926$	20.0676	0.659462
$927$	0	0
$928$	−12.6908	−0.416597
$929$	−17.9960	−0.590430	−0.295215	−	0.955431i	$-0.595391\pi$
$929$	−17.9960	−0.590430	−0.295215	+	0.955431i	$0.595391\pi$
$930$	0	0
$931$	−38.6345	−1.26619
$932$	5.47388	0.179303
$933$	0	0
$934$	−62.5487	−2.04666
$935$	−6.33371	−0.207135
$936$	0	0
$937$	−6.46056	−0.211057	−0.105529	−	0.994416i	$-0.533653\pi$
$937$	−6.46056	−0.211057	−0.105529	+	0.994416i	$0.533653\pi$
$938$	−1.74463	−0.0569640
$939$	0	0
$940$	7.91800	0.258257
$941$	−19.8660	−0.647614	−0.323807	−	0.946123i	$-0.604963\pi$
$941$	−19.8660	−0.647614	−0.323807	+	0.946123i	$0.604963\pi$
$942$	0	0
$943$	19.9346	0.649160
$944$	−73.5610	−2.39421
$945$	0	0
$946$	6.08752	0.197922
$947$	−53.8245	−1.74906	−0.874531	−	0.484970i	$-0.838831\pi$
$947$	−53.8245	−1.74906	−0.874531	+	0.484970i	$0.838831\pi$
$948$	0	0
$949$	2.54697	0.0826782
$950$	−26.5323	−0.860821
$951$	0	0
$952$	−0.995341	−0.0322592
$953$	−37.5503	−1.21637	−0.608186	−	0.793794i	$-0.708103\pi$
$953$	−37.5503	−1.21637	−0.608186	+	0.793794i	$0.708103\pi$
$954$	0	0
$955$	12.8097	0.414513
$956$	−20.8632	−0.674765
$957$	0	0
$958$	29.6966	0.959455
$959$	1.81207	0.0585149
$960$	0	0
$961$	−20.5107	−0.661636
$962$	−22.1507	−0.714166
$963$	0	0
$964$	−17.4520	−0.562090
$965$	17.3306	0.557891
$966$	0	0
$967$	29.7039	0.955211	0.477606	−	0.878574i	$-0.341505\pi$
$967$	29.7039	0.955211	0.477606	+	0.878574i	$0.341505\pi$
$968$	−2.01366	−0.0647214
$969$	0	0
$970$	−10.8143	−0.347228
$971$	−20.9546	−0.672464	−0.336232	−	0.941779i	$-0.609153\pi$
$971$	−20.9546	−0.672464	−0.336232	+	0.941779i	$0.609153\pi$
$972$	0	0
$973$	−1.98401	−0.0636045
$974$	−21.9224	−0.702440
$975$	0	0
$976$	−4.95825	−0.158710
$977$	29.9552	0.958353	0.479177	−	0.877718i	$-0.340935\pi$
$977$	29.9552	0.958353	0.479177	+	0.877718i	$0.340935\pi$
$978$	0	0
$979$	−16.8141	−0.537381
$980$	−8.11415	−0.259197
$981$	0	0
$982$	37.5266	1.19752
$983$	−52.2964	−1.66800	−0.833998	−	0.551767i	$-0.813954\pi$
$983$	−52.2964	−1.66800	−0.833998	+	0.551767i	$0.813954\pi$
$984$	0	0
$985$	9.75016	0.310666
$986$	−21.6004	−0.687898
$987$	0	0
$988$	−10.6442	−0.338636
$989$	20.4482	0.650214
$990$	0	0
$991$	23.5537	0.748209	0.374105	−	0.927387i	$-0.377950\pi$
$991$	23.5537	0.748209	0.374105	+	0.927387i	$0.377950\pi$
$992$	−13.8067	−0.438364
$993$	0	0
$994$	−1.84573	−0.0585430
$995$	−17.8220	−0.564996
$996$	0	0
$997$	38.4376	1.21733	0.608665	−	0.793427i	$-0.291705\pi$
$997$	38.4376	1.21733	0.608665	+	0.793427i	$0.291705\pi$
$998$	29.9743	0.948818
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.j.1.11		14
3.2	odd	2		2013.2.a.h.1.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.h.1.4	✓	14	3.2	odd	2
6039.2.a.j.1.11		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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67203 q^{2} +0.795679 q^{4} +1.45953 q^{5} -0.113904 q^{7} -2.01366 q^{8} +O(q^{10})\)	\(q+1.67203 q^{2} +0.795679 q^{4} +1.45953 q^{5} -0.113904 q^{7} -2.01366 q^{8} +2.44037 q^{10} +1.00000 q^{11} -2.41931 q^{13} -0.190451 q^{14} -4.95825 q^{16} -4.33956 q^{17} +5.52946 q^{19} +1.16132 q^{20} +1.67203 q^{22} +5.61639 q^{23} -2.86978 q^{25} -4.04515 q^{26} -0.0906313 q^{28} +2.97696 q^{29} +3.23872 q^{31} -4.26302 q^{32} -7.25587 q^{34} -0.166247 q^{35} +5.47586 q^{37} +9.24541 q^{38} -2.93899 q^{40} +3.54936 q^{41} +3.64080 q^{43} +0.795679 q^{44} +9.39076 q^{46} +6.81812 q^{47} -6.98703 q^{49} -4.79835 q^{50} -1.92499 q^{52} +0.123290 q^{53} +1.45953 q^{55} +0.229364 q^{56} +4.97756 q^{58} +14.8361 q^{59} +1.00000 q^{61} +5.41523 q^{62} +2.78861 q^{64} -3.53104 q^{65} +9.16048 q^{67} -3.45290 q^{68} -0.277969 q^{70} +9.69135 q^{71} -1.05277 q^{73} +9.15579 q^{74} +4.39968 q^{76} -0.113904 q^{77} +10.0943 q^{79} -7.23671 q^{80} +5.93464 q^{82} +2.01215 q^{83} -6.33371 q^{85} +6.08752 q^{86} -2.01366 q^{88} -16.8141 q^{89} +0.275569 q^{91} +4.46885 q^{92} +11.4001 q^{94} +8.07040 q^{95} -4.43143 q^{97} -11.6825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7}+O(q^{10}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7	\( 14 q + q^{2} + 15 q^{4} - q^{5} + 9 q^{7} + 6 q^{10} + 14 q^{11} + q^{13} + 7 q^{14} + 17 q^{16} + 9 q^{17} + 22 q^{19} - 23 q^{20} + q^{22} - q^{23} + 25 q^{25} - 4 q^{26} + 37 q^{28} + 6 q^{29} + 9 q^{31} - 4 q^{32} + 8 q^{34} - 18 q^{35} + 18 q^{37} - 8 q^{38} + 16 q^{40} + 25 q^{41} + 25 q^{43} + 15 q^{44} + 20 q^{46} - 36 q^{47} + 25 q^{49} - 2 q^{50} - 13 q^{52} - q^{55} + 40 q^{56} + 33 q^{58} - 17 q^{59} + 14 q^{61} + 13 q^{62} - 6 q^{64} + 61 q^{65} + 22 q^{67} - 66 q^{68} + 44 q^{70} + 13 q^{71} + 20 q^{73} + 12 q^{74} + 49 q^{76} + 9 q^{77} + 31 q^{79} - 88 q^{80} + 2 q^{82} - 32 q^{83} + 2 q^{85} + 14 q^{86} + 21 q^{89} + 45 q^{91} + 14 q^{92} - 31 q^{94} - 23 q^{95} + 37 q^{97} + 38 q^{98}+O(q^{100}) \) 14 * q + q^2 + 15 * q^4 - q^5 + 9 * q^7 + 6 * q^10 + 14 * q^11 + q^13 + 7 * q^14 + 17 * q^16 + 9 * q^17 + 22 * q^19 - 23 * q^20 + q^22 - q^23 + 25 * q^25 - 4 * q^26 + 37 * q^28 + 6 * q^29 + 9 * q^31 - 4 * q^32 + 8 * q^34 - 18 * q^35 + 18 * q^37 - 8 * q^38 + 16 * q^40 + 25 * q^41 + 25 * q^43 + 15 * q^44 + 20 * q^46 - 36 * q^47 + 25 * q^49 - 2 * q^50 - 13 * q^52 - q^55 + 40 * q^56 + 33 * q^58 - 17 * q^59 + 14 * q^61 + 13 * q^62 - 6 * q^64 + 61 * q^65 + 22 * q^67 - 66 * q^68 + 44 * q^70 + 13 * q^71 + 20 * q^73 + 12 * q^74 + 49 * q^76 + 9 * q^77 + 31 * q^79 - 88 * q^80 + 2 * q^82 - 32 * q^83 + 2 * q^85 + 14 * q^86 + 21 * q^89 + 45 * q^91 + 14 * q^92 - 31 * q^94 - 23 * q^95 + 37 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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