LMFDB - Embedded newform 6039.2.a.n.1.8

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:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-1.65004 q^{2} +0.722641 q^{4} +2.19434 q^{5} -0.319690 q^{7} +2.10770 q^{8} +O(q^{10})$	\(q-1.65004 q^{2} +0.722641 q^{4} +2.19434 q^{5} -0.319690 q^{7} +2.10770 q^{8} -3.62075 q^{10} -1.00000 q^{11} -1.63109 q^{13} +0.527501 q^{14} -4.92307 q^{16} +3.25344 q^{17} -7.30110 q^{19} +1.58572 q^{20} +1.65004 q^{22} +0.797447 q^{23} -0.184889 q^{25} +2.69137 q^{26} -0.231021 q^{28} -7.77212 q^{29} +8.65076 q^{31} +3.90788 q^{32} -5.36832 q^{34} -0.701506 q^{35} +8.96557 q^{37} +12.0471 q^{38} +4.62500 q^{40} +12.3092 q^{41} +12.1259 q^{43} -0.722641 q^{44} -1.31582 q^{46} -12.9527 q^{47} -6.89780 q^{49} +0.305075 q^{50} -1.17869 q^{52} -5.25605 q^{53} -2.19434 q^{55} -0.673809 q^{56} +12.8243 q^{58} -7.12892 q^{59} -1.00000 q^{61} -14.2741 q^{62} +3.39797 q^{64} -3.57916 q^{65} -12.2491 q^{67} +2.35107 q^{68} +1.15752 q^{70} +4.00849 q^{71} -6.19448 q^{73} -14.7936 q^{74} -5.27607 q^{76} +0.319690 q^{77} +7.48729 q^{79} -10.8029 q^{80} -20.3108 q^{82} -12.3101 q^{83} +7.13915 q^{85} -20.0082 q^{86} -2.10770 q^{88} -9.74291 q^{89} +0.521442 q^{91} +0.576268 q^{92} +21.3726 q^{94} -16.0211 q^{95} -14.3699 q^{97} +11.3817 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * 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 $

$2$	−1.65004	−1.16676	−0.583378	−	0.812201i	$-0.698269\pi$
$2$	−1.65004	−1.16676	−0.583378	+	0.812201i	$0.698269\pi$
$3$	0	0
$3$	0	0
$4$	0.722641	0.361320
$5$	2.19434	0.981337	0.490668	−	0.871346i	$-0.336753\pi$
$5$	2.19434	0.981337	0.490668	+	0.871346i	$0.336753\pi$
$6$	0	0
$7$	−0.319690	−0.120831	−0.0604156	−	0.998173i	$-0.519243\pi$
$7$	−0.319690	−0.120831	−0.0604156	+	0.998173i	$0.519243\pi$
$8$	2.10770	0.745183
$9$	0	0
$10$	−3.62075	−1.14498
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.63109	−0.452383	−0.226191	−	0.974083i	$-0.572628\pi$
$13$	−1.63109	−0.452383	−0.226191	+	0.974083i	$0.572628\pi$
$14$	0.527501	0.140981
$15$	0	0
$16$	−4.92307	−1.23077
$17$	3.25344	0.789076	0.394538	−	0.918880i	$-0.370905\pi$
$17$	3.25344	0.789076	0.394538	+	0.918880i	$0.370905\pi$
$18$	0	0
$19$	−7.30110	−1.67499	−0.837493	−	0.546448i	$-0.815980\pi$
$19$	−7.30110	−1.67499	−0.837493	+	0.546448i	$0.815980\pi$
$20$	1.58572	0.354577
$21$	0	0
$22$	1.65004	0.351790
$23$	0.797447	0.166279	0.0831396	−	0.996538i	$-0.473505\pi$
$23$	0.797447	0.166279	0.0831396	+	0.996538i	$0.473505\pi$
$24$	0	0
$25$	−0.184889	−0.0369779
$26$	2.69137	0.527821
$27$	0	0
$28$	−0.231021	−0.0436588
$29$	−7.77212	−1.44325	−0.721623	−	0.692286i	$-0.756604\pi$
$29$	−7.77212	−1.44325	−0.721623	+	0.692286i	$0.756604\pi$
$30$	0	0
$31$	8.65076	1.55372	0.776861	−	0.629672i	$-0.216811\pi$
$31$	8.65076	1.55372	0.776861	+	0.629672i	$0.216811\pi$
$32$	3.90788	0.690823
$33$	0	0
$34$	−5.36832	−0.920659
$35$	−0.701506	−0.118576
$36$	0	0
$37$	8.96557	1.47393	0.736966	−	0.675930i	$-0.236258\pi$
$37$	8.96557	1.47393	0.736966	+	0.675930i	$0.236258\pi$
$38$	12.0471	1.95430
$39$	0	0
$40$	4.62500	0.731276
$41$	12.3092	1.92238	0.961191	−	0.275885i	$-0.0889708\pi$
$41$	12.3092	1.92238	0.961191	+	0.275885i	$0.0889708\pi$
$42$	0	0
$43$	12.1259	1.84918	0.924590	−	0.380962i	$-0.124407\pi$
$43$	12.1259	1.84918	0.924590	+	0.380962i	$0.124407\pi$
$44$	−0.722641	−0.108942
$45$	0	0
$46$	−1.31582	−0.194007
$47$	−12.9527	−1.88935	−0.944675	−	0.328007i	$-0.893623\pi$
$47$	−12.9527	−1.88935	−0.944675	+	0.328007i	$0.893623\pi$
$48$	0	0
$49$	−6.89780	−0.985400
$50$	0.305075	0.0431442
$51$	0	0
$52$	−1.17869	−0.163455
$53$	−5.25605	−0.721975	−0.360987	−	0.932571i	$-0.617560\pi$
$53$	−5.25605	−0.721975	−0.360987	+	0.932571i	$0.617560\pi$
$54$	0	0
$55$	−2.19434	−0.295884
$56$	−0.673809	−0.0900415
$57$	0	0
$58$	12.8243	1.68392
$59$	−7.12892	−0.928106	−0.464053	−	0.885807i	$-0.653605\pi$
$59$	−7.12892	−0.928106	−0.464053	+	0.885807i	$0.653605\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−14.2741	−1.81282
$63$	0	0
$64$	3.39797	0.424746
$65$	−3.57916	−0.443940
$66$	0	0
$67$	−12.2491	−1.49647	−0.748233	−	0.663436i	$-0.769097\pi$
$67$	−12.2491	−1.49647	−0.748233	+	0.663436i	$0.769097\pi$
$68$	2.35107	0.285109
$69$	0	0
$70$	1.15752	0.138350
$71$	4.00849	0.475720	0.237860	−	0.971299i	$-0.423554\pi$
$71$	4.00849	0.475720	0.237860	+	0.971299i	$0.423554\pi$
$72$	0	0
$73$	−6.19448	−0.725009	−0.362504	−	0.931982i	$-0.618078\pi$
$73$	−6.19448	−0.725009	−0.362504	+	0.931982i	$0.618078\pi$
$74$	−14.7936	−1.71972
$75$	0	0
$76$	−5.27607	−0.605207
$77$	0.319690	0.0364320
$78$	0	0
$79$	7.48729	0.842386	0.421193	−	0.906971i	$-0.361611\pi$
$79$	7.48729	0.842386	0.421193	+	0.906971i	$0.361611\pi$
$80$	−10.8029	−1.20780
$81$	0	0
$82$	−20.3108	−2.24295
$83$	−12.3101	−1.35121	−0.675603	−	0.737265i	$-0.736117\pi$
$83$	−12.3101	−1.35121	−0.675603	+	0.737265i	$0.736117\pi$
$84$	0	0
$85$	7.13915	0.774349
$86$	−20.0082	−2.15754
$87$	0	0
$88$	−2.10770	−0.224681
$89$	−9.74291	−1.03275	−0.516373	−	0.856364i	$-0.672718\pi$
$89$	−9.74291	−1.03275	−0.516373	+	0.856364i	$0.672718\pi$
$90$	0	0
$91$	0.521442	0.0546620
$92$	0.576268	0.0600801
$93$	0	0
$94$	21.3726	2.20441
$95$	−16.0211	−1.64373
$96$	0	0
$97$	−14.3699	−1.45904	−0.729521	−	0.683958i	$-0.760257\pi$
$97$	−14.3699	−1.45904	−0.729521	+	0.683958i	$0.760257\pi$
$98$	11.3817	1.14972
$99$	0	0
$100$	−0.133609	−0.0133609
$101$	0.386986	0.0385065	0.0192533	−	0.999815i	$-0.493871\pi$
$101$	0.386986	0.0385065	0.0192533	+	0.999815i	$0.493871\pi$
$102$	0	0
$103$	8.33215	0.820991	0.410495	−	0.911863i	$-0.365356\pi$
$103$	8.33215	0.820991	0.410495	+	0.911863i	$0.365356\pi$
$104$	−3.43784	−0.337108
$105$	0	0
$106$	8.67271	0.842368
$107$	6.16556	0.596047	0.298024	−	0.954558i	$-0.403673\pi$
$107$	6.16556	0.596047	0.298024	+	0.954558i	$0.403673\pi$
$108$	0	0
$109$	0.711842	0.0681821	0.0340911	−	0.999419i	$-0.489146\pi$
$109$	0.711842	0.0681821	0.0340911	+	0.999419i	$0.489146\pi$
$110$	3.62075	0.345225
$111$	0	0
$112$	1.57385	0.148715
$113$	6.32742	0.595233	0.297617	−	0.954685i	$-0.403808\pi$
$113$	6.32742	0.595233	0.297617	+	0.954685i	$0.403808\pi$
$114$	0	0
$115$	1.74987	0.163176
$116$	−5.61645	−0.521474
$117$	0	0
$118$	11.7630	1.08287
$119$	−1.04009	−0.0953450
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.65004	0.149388
$123$	0	0
$124$	6.25139	0.561392
$125$	−11.3774	−1.01762
$126$	0	0
$127$	4.79293	0.425304	0.212652	−	0.977128i	$-0.431790\pi$
$127$	4.79293	0.425304	0.212652	+	0.977128i	$0.431790\pi$
$128$	−13.4226	−1.18640
$129$	0	0
$130$	5.90577	0.517970
$131$	15.5675	1.36014	0.680068	−	0.733149i	$-0.261950\pi$
$131$	15.5675	1.36014	0.680068	+	0.733149i	$0.261950\pi$
$132$	0	0
$133$	2.33408	0.202391
$134$	20.2115	1.74601
$135$	0	0
$136$	6.85727	0.588006
$137$	−16.0484	−1.37111	−0.685555	−	0.728021i	$-0.740441\pi$
$137$	−16.0484	−1.37111	−0.685555	+	0.728021i	$0.740441\pi$
$138$	0	0
$139$	16.7035	1.41677	0.708386	−	0.705826i	$-0.249424\pi$
$139$	16.7035	1.41677	0.708386	+	0.705826i	$0.249424\pi$
$140$	−0.506937	−0.0428440
$141$	0	0
$142$	−6.61417	−0.555049
$143$	1.63109	0.136399
$144$	0	0
$145$	−17.0546	−1.41631
$146$	10.2212	0.845909
$147$	0	0
$148$	6.47889	0.532561
$149$	−9.45799	−0.774828	−0.387414	−	0.921906i	$-0.626632\pi$
$149$	−9.45799	−0.774828	−0.387414	+	0.921906i	$0.626632\pi$
$150$	0	0
$151$	5.58193	0.454251	0.227126	−	0.973865i	$-0.427067\pi$
$151$	5.58193	0.454251	0.227126	+	0.973865i	$0.427067\pi$
$152$	−15.3885	−1.24817
$153$	0	0
$154$	−0.527501	−0.0425073
$155$	18.9827	1.52472
$156$	0	0
$157$	−18.5888	−1.48354	−0.741772	−	0.670652i	$-0.766014\pi$
$157$	−18.5888	−1.48354	−0.741772	+	0.670652i	$0.766014\pi$
$158$	−12.3544	−0.982859
$159$	0	0
$160$	8.57521	0.677930
$161$	−0.254936	−0.0200917
$162$	0	0
$163$	3.69915	0.289739	0.144870	−	0.989451i	$-0.453724\pi$
$163$	3.69915	0.289739	0.144870	+	0.989451i	$0.453724\pi$
$164$	8.89516	0.694596
$165$	0	0
$166$	20.3121	1.57653
$167$	−0.0234996	−0.00181845	−0.000909226	−	1.00000i	$-0.500289\pi$
$167$	−0.0234996	−0.00181845	−0.000909226		1.00000i	$0.500289\pi$
$168$	0	0
$169$	−10.3395	−0.795350
$170$	−11.7799	−0.903477
$171$	0	0
$172$	8.76266	0.668147
$173$	−14.2664	−1.08466	−0.542328	−	0.840167i	$-0.682457\pi$
$173$	−14.2664	−1.08466	−0.542328	+	0.840167i	$0.682457\pi$
$174$	0	0
$175$	0.0591072	0.00446809
$176$	4.92307	0.371091
$177$	0	0
$178$	16.0762	1.20496
$179$	6.91982	0.517211	0.258606	−	0.965983i	$-0.416737\pi$
$179$	6.91982	0.517211	0.258606	+	0.965983i	$0.416737\pi$
$180$	0	0
$181$	−0.548195	−0.0407470	−0.0203735	−	0.999792i	$-0.506486\pi$
$181$	−0.548195	−0.0407470	−0.0203735	+	0.999792i	$0.506486\pi$
$182$	−0.860402	−0.0637772
$183$	0	0
$184$	1.68078	0.123909
$185$	19.6735	1.44642
$186$	0	0
$187$	−3.25344	−0.237915
$188$	−9.36017	−0.682661
$189$	0	0
$190$	26.4354	1.91783
$191$	−1.75781	−0.127191	−0.0635954	−	0.997976i	$-0.520257\pi$
$191$	−1.75781	−0.127191	−0.0635954	+	0.997976i	$0.520257\pi$
$192$	0	0
$193$	−21.9780	−1.58201	−0.791005	−	0.611810i	$-0.790442\pi$
$193$	−21.9780	−1.58201	−0.791005	+	0.611810i	$0.790442\pi$
$194$	23.7109	1.70235
$195$	0	0
$196$	−4.98463	−0.356045
$197$	−24.7936	−1.76647	−0.883235	−	0.468930i	$-0.844640\pi$
$197$	−24.7936	−1.76647	−0.883235	+	0.468930i	$0.844640\pi$
$198$	0	0
$199$	−12.0395	−0.853458	−0.426729	−	0.904380i	$-0.640334\pi$
$199$	−12.0395	−0.853458	−0.426729	+	0.904380i	$0.640334\pi$
$200$	−0.389691	−0.0275553
$201$	0	0
$202$	−0.638543	−0.0449277
$203$	2.48467	0.174389
$204$	0	0
$205$	27.0106	1.88650
$206$	−13.7484	−0.957896
$207$	0	0
$208$	8.02997	0.556778
$209$	7.30110	0.505027
$210$	0	0
$211$	12.6937	0.873873	0.436937	−	0.899492i	$-0.356063\pi$
$211$	12.6937	0.873873	0.436937	+	0.899492i	$0.356063\pi$
$212$	−3.79824	−0.260864
$213$	0	0
$214$	−10.1734	−0.695442
$215$	26.6083	1.81467
$216$	0	0
$217$	−2.76556	−0.187738
$218$	−1.17457	−0.0795519
$219$	0	0
$220$	−1.58572	−0.106909
$221$	−5.30666	−0.356964
$222$	0	0
$223$	14.6755	0.982747	0.491373	−	0.870949i	$-0.336495\pi$
$223$	14.6755	0.982747	0.491373	+	0.870949i	$0.336495\pi$
$224$	−1.24931	−0.0834730
$225$	0	0
$226$	−10.4405	−0.694492
$227$	5.08012	0.337180	0.168590	−	0.985686i	$-0.446079\pi$
$227$	5.08012	0.337180	0.168590	+	0.985686i	$0.446079\pi$
$228$	0	0
$229$	−9.37426	−0.619469	−0.309734	−	0.950823i	$-0.600240\pi$
$229$	−9.37426	−0.619469	−0.309734	+	0.950823i	$0.600240\pi$
$230$	−2.88736	−0.190387
$231$	0	0
$232$	−16.3813	−1.07548
$233$	2.93011	0.191958	0.0959791	−	0.995383i	$-0.469402\pi$
$233$	2.93011	0.191958	0.0959791	+	0.995383i	$0.469402\pi$
$234$	0	0
$235$	−28.4227	−1.85409
$236$	−5.15165	−0.335344
$237$	0	0
$238$	1.71620	0.111244
$239$	−19.1779	−1.24052	−0.620259	−	0.784397i	$-0.712972\pi$
$239$	−19.1779	−1.24052	−0.620259	+	0.784397i	$0.712972\pi$
$240$	0	0
$241$	26.3245	1.69571	0.847854	−	0.530231i	$-0.177895\pi$
$241$	26.3245	1.69571	0.847854	+	0.530231i	$0.177895\pi$
$242$	−1.65004	−0.106069
$243$	0	0
$244$	−0.722641	−0.0462623
$245$	−15.1361	−0.967009
$246$	0	0
$247$	11.9087	0.757735
$248$	18.2332	1.15781
$249$	0	0
$250$	18.7732	1.18732
$251$	−11.2266	−0.708616	−0.354308	−	0.935129i	$-0.615284\pi$
$251$	−11.2266	−0.708616	−0.354308	+	0.935129i	$0.615284\pi$
$252$	0	0
$253$	−0.797447	−0.0501351
$254$	−7.90854	−0.496226
$255$	0	0
$256$	15.3519	0.959491
$257$	8.75622	0.546198	0.273099	−	0.961986i	$-0.411951\pi$
$257$	8.75622	0.546198	0.273099	+	0.961986i	$0.411951\pi$
$258$	0	0
$259$	−2.86620	−0.178097
$260$	−2.58645	−0.160405
$261$	0	0
$262$	−25.6870	−1.58695
$263$	−16.2105	−0.999580	−0.499790	−	0.866147i	$-0.666589\pi$
$263$	−16.2105	−0.999580	−0.499790	+	0.866147i	$0.666589\pi$
$264$	0	0
$265$	−11.5335	−0.708500
$266$	−3.85134	−0.236141
$267$	0	0
$268$	−8.85170	−0.540704
$269$	27.7539	1.69219	0.846093	−	0.533036i	$-0.178949\pi$
$269$	27.7539	1.69219	0.846093	+	0.533036i	$0.178949\pi$
$270$	0	0
$271$	−16.6905	−1.01388	−0.506938	−	0.861982i	$-0.669223\pi$
$271$	−16.6905	−1.01388	−0.506938	+	0.861982i	$0.669223\pi$
$272$	−16.0169	−0.971169
$273$	0	0
$274$	26.4806	1.59975
$275$	0.184889	0.0111493
$276$	0	0
$277$	29.3377	1.76273	0.881365	−	0.472435i	$-0.156625\pi$
$277$	29.3377	1.76273	0.881365	+	0.472435i	$0.156625\pi$
$278$	−27.5615	−1.65303
$279$	0	0
$280$	−1.47856	−0.0883610
$281$	−14.6268	−0.872562	−0.436281	−	0.899811i	$-0.643705\pi$
$281$	−14.6268	−0.872562	−0.436281	+	0.899811i	$0.643705\pi$
$282$	0	0
$283$	−22.8643	−1.35914	−0.679570	−	0.733611i	$-0.737834\pi$
$283$	−22.8643	−1.35914	−0.679570	+	0.733611i	$0.737834\pi$
$284$	2.89670	0.171887
$285$	0	0
$286$	−2.69137	−0.159144
$287$	−3.93514	−0.232284
$288$	0	0
$289$	−6.41511	−0.377360
$290$	28.1409	1.65249
$291$	0	0
$292$	−4.47638	−0.261961
$293$	−7.55441	−0.441333	−0.220667	−	0.975349i	$-0.570823\pi$
$293$	−7.55441	−0.441333	−0.220667	+	0.975349i	$0.570823\pi$
$294$	0	0
$295$	−15.6432	−0.910785
$296$	18.8967	1.09835
$297$	0	0
$298$	15.6061	0.904036
$299$	−1.30071	−0.0752219
$300$	0	0
$301$	−3.87652	−0.223439
$302$	−9.21043	−0.530000
$303$	0	0
$304$	35.9438	2.06152
$305$	−2.19434	−0.125647
$306$	0	0
$307$	26.3232	1.50234	0.751171	−	0.660107i	$-0.229489\pi$
$307$	26.3232	1.50234	0.751171	+	0.660107i	$0.229489\pi$
$308$	0.231021	0.0131636
$309$	0	0
$310$	−31.3222	−1.77898
$311$	−16.8755	−0.956921	−0.478460	−	0.878109i	$-0.658805\pi$
$311$	−16.8755	−0.956921	−0.478460	+	0.878109i	$0.658805\pi$
$312$	0	0
$313$	29.0770	1.64353	0.821763	−	0.569829i	$-0.192991\pi$
$313$	29.0770	1.64353	0.821763	+	0.569829i	$0.192991\pi$
$314$	30.6722	1.73093
$315$	0	0
$316$	5.41062	0.304371
$317$	−17.9910	−1.01048	−0.505238	−	0.862980i	$-0.668595\pi$
$317$	−17.9910	−1.01048	−0.505238	+	0.862980i	$0.668595\pi$
$318$	0	0
$319$	7.77212	0.435155
$320$	7.45628	0.416819
$321$	0	0
$322$	0.420655	0.0234422
$323$	−23.7537	−1.32169
$324$	0	0
$325$	0.301571	0.0167282
$326$	−6.10375	−0.338055
$327$	0	0
$328$	25.9442	1.43253
$329$	4.14085	0.228293
$330$	0	0
$331$	1.80851	0.0994046	0.0497023	−	0.998764i	$-0.484173\pi$
$331$	1.80851	0.0994046	0.0497023	+	0.998764i	$0.484173\pi$
$332$	−8.89576	−0.488218
$333$	0	0
$334$	0.0387753	0.00212169
$335$	−26.8786	−1.46854
$336$	0	0
$337$	18.3441	0.999266	0.499633	−	0.866237i	$-0.333468\pi$
$337$	18.3441	0.999266	0.499633	+	0.866237i	$0.333468\pi$
$338$	17.0607	0.927979
$339$	0	0
$340$	5.15904	0.279788
$341$	−8.65076	−0.468465
$342$	0	0
$343$	4.44298	0.239898
$344$	25.5577	1.37798
$345$	0	0
$346$	23.5402	1.26553
$347$	−17.2356	−0.925257	−0.462629	−	0.886552i	$-0.653094\pi$
$347$	−17.2356	−0.925257	−0.462629	+	0.886552i	$0.653094\pi$
$348$	0	0
$349$	−10.0338	−0.537095	−0.268548	−	0.963266i	$-0.586544\pi$
$349$	−10.0338	−0.537095	−0.268548	+	0.963266i	$0.586544\pi$
$350$	−0.0975294	−0.00521317
$351$	0	0
$352$	−3.90788	−0.208291
$353$	−1.58685	−0.0844595	−0.0422297	−	0.999108i	$-0.513446\pi$
$353$	−1.58685	−0.0844595	−0.0422297	+	0.999108i	$0.513446\pi$
$354$	0	0
$355$	8.79597	0.466841
$356$	−7.04062	−0.373152
$357$	0	0
$358$	−11.4180	−0.603460
$359$	2.09928	0.110796	0.0553979	−	0.998464i	$-0.482357\pi$
$359$	2.09928	0.110796	0.0553979	+	0.998464i	$0.482357\pi$
$360$	0	0
$361$	34.3060	1.80558
$362$	0.904546	0.0475419
$363$	0	0
$364$	0.376815	0.0197505
$365$	−13.5928	−0.711478
$366$	0	0
$367$	−7.76644	−0.405405	−0.202703	−	0.979240i	$-0.564972\pi$
$367$	−7.76644	−0.405405	−0.202703	+	0.979240i	$0.564972\pi$
$368$	−3.92589	−0.204651
$369$	0	0
$370$	−32.4621	−1.68762
$371$	1.68031	0.0872371
$372$	0	0
$373$	4.37473	0.226515	0.113257	−	0.993566i	$-0.463872\pi$
$373$	4.37473	0.226515	0.113257	+	0.993566i	$0.463872\pi$
$374$	5.36832	0.277589
$375$	0	0
$376$	−27.3004	−1.40791
$377$	12.6770	0.652900
$378$	0	0
$379$	−9.19599	−0.472366	−0.236183	−	0.971709i	$-0.575896\pi$
$379$	−9.19599	−0.472366	−0.236183	+	0.971709i	$0.575896\pi$
$380$	−11.5775	−0.593912
$381$	0	0
$382$	2.90046	0.148401
$383$	23.8951	1.22098	0.610490	−	0.792024i	$-0.290972\pi$
$383$	23.8951	1.22098	0.610490	+	0.792024i	$0.290972\pi$
$384$	0	0
$385$	0.701506	0.0357521
$386$	36.2646	1.84582
$387$	0	0
$388$	−10.3843	−0.527182
$389$	7.85375	0.398201	0.199100	−	0.979979i	$-0.436198\pi$
$389$	7.85375	0.398201	0.199100	+	0.979979i	$0.436198\pi$
$390$	0	0
$391$	2.59445	0.131207
$392$	−14.5385	−0.734304
$393$	0	0
$394$	40.9105	2.06104
$395$	16.4296	0.826665
$396$	0	0
$397$	11.8336	0.593909	0.296955	−	0.954892i	$-0.404029\pi$
$397$	11.8336	0.593909	0.296955	+	0.954892i	$0.404029\pi$
$398$	19.8657	0.995777
$399$	0	0
$400$	0.910224	0.0455112
$401$	−24.3913	−1.21804	−0.609021	−	0.793154i	$-0.708438\pi$
$401$	−24.3913	−1.21804	−0.609021	+	0.793154i	$0.708438\pi$
$402$	0	0
$403$	−14.1102	−0.702877
$404$	0.279652	0.0139132
$405$	0	0
$406$	−4.09980	−0.203470
$407$	−8.96557	−0.444407
$408$	0	0
$409$	−24.7258	−1.22261	−0.611306	−	0.791394i	$-0.709356\pi$
$409$	−24.7258	−1.22261	−0.611306	+	0.791394i	$0.709356\pi$
$410$	−44.5687	−2.20109
$411$	0	0
$412$	6.02115	0.296641
$413$	2.27904	0.112144
$414$	0	0
$415$	−27.0124	−1.32599
$416$	−6.37411	−0.312516
$417$	0	0
$418$	−12.0471	−0.589244
$419$	26.4067	1.29005	0.645026	−	0.764161i	$-0.276847\pi$
$419$	26.4067	1.29005	0.645026	+	0.764161i	$0.276847\pi$
$420$	0	0
$421$	−38.5056	−1.87665	−0.938323	−	0.345759i	$-0.887622\pi$
$421$	−38.5056	−1.87665	−0.938323	+	0.345759i	$0.887622\pi$
$422$	−20.9452	−1.01960
$423$	0	0
$424$	−11.0782	−0.538003
$425$	−0.601527	−0.0291784
$426$	0	0
$427$	0.319690	0.0154709
$428$	4.45549	0.215364
$429$	0	0
$430$	−43.9048	−2.11728
$431$	−8.57622	−0.413102	−0.206551	−	0.978436i	$-0.566224\pi$
$431$	−8.57622	−0.413102	−0.206551	+	0.978436i	$0.566224\pi$
$432$	0	0
$433$	−38.7809	−1.86369	−0.931846	−	0.362854i	$-0.881802\pi$
$433$	−38.7809	−1.86369	−0.931846	+	0.362854i	$0.881802\pi$
$434$	4.56329	0.219045
$435$	0	0
$436$	0.514406	0.0246356
$437$	−5.82224	−0.278515
$438$	0	0
$439$	−17.4458	−0.832643	−0.416322	−	0.909217i	$-0.636681\pi$
$439$	−17.4458	−0.832643	−0.416322	+	0.909217i	$0.636681\pi$
$440$	−4.62500	−0.220488
$441$	0	0
$442$	8.75621	0.416490
$443$	12.0252	0.571334	0.285667	−	0.958329i	$-0.407785\pi$
$443$	12.0252	0.571334	0.285667	+	0.958329i	$0.407785\pi$
$444$	0	0
$445$	−21.3792	−1.01347
$446$	−24.2153	−1.14663
$447$	0	0
$448$	−1.08629	−0.0513226
$449$	−12.9798	−0.612554	−0.306277	−	0.951942i	$-0.599083\pi$
$449$	−12.9798	−0.612554	−0.306277	+	0.951942i	$0.599083\pi$
$450$	0	0
$451$	−12.3092	−0.579620
$452$	4.57245	0.215070
$453$	0	0
$454$	−8.38242	−0.393406
$455$	1.14422	0.0536418
$456$	0	0
$457$	−28.3022	−1.32392	−0.661961	−	0.749538i	$-0.730276\pi$
$457$	−28.3022	−1.32392	−0.661961	+	0.749538i	$0.730276\pi$
$458$	15.4679	0.722769
$459$	0	0
$460$	1.26453	0.0589588
$461$	3.12025	0.145325	0.0726624	−	0.997357i	$-0.476850\pi$
$461$	3.12025	0.145325	0.0726624	+	0.997357i	$0.476850\pi$
$462$	0	0
$463$	−0.952248	−0.0442547	−0.0221274	−	0.999755i	$-0.507044\pi$
$463$	−0.952248	−0.0442547	−0.0221274	+	0.999755i	$0.507044\pi$
$464$	38.2627	1.77630
$465$	0	0
$466$	−4.83481	−0.223968
$467$	−10.9341	−0.505971	−0.252986	−	0.967470i	$-0.581413\pi$
$467$	−10.9341	−0.505971	−0.252986	+	0.967470i	$0.581413\pi$
$468$	0	0
$469$	3.91591	0.180820
$470$	46.8986	2.16327
$471$	0	0
$472$	−15.0256	−0.691609
$473$	−12.1259	−0.557549
$474$	0	0
$475$	1.34990	0.0619375
$476$	−0.751613	−0.0344501
$477$	0	0
$478$	31.6444	1.44738
$479$	−18.1504	−0.829313	−0.414657	−	0.909978i	$-0.636098\pi$
$479$	−18.1504	−0.829313	−0.414657	+	0.909978i	$0.636098\pi$
$480$	0	0
$481$	−14.6237	−0.666781
$482$	−43.4365	−1.97848
$483$	0	0
$484$	0.722641	0.0328473
$485$	−31.5324	−1.43181
$486$	0	0
$487$	−14.9241	−0.676276	−0.338138	−	0.941097i	$-0.609797\pi$
$487$	−14.9241	−0.676276	−0.338138	+	0.941097i	$0.609797\pi$
$488$	−2.10770	−0.0954110
$489$	0	0
$490$	24.9752	1.12826
$491$	−22.9234	−1.03452	−0.517259	−	0.855829i	$-0.673048\pi$
$491$	−22.9234	−1.03452	−0.517259	+	0.855829i	$0.673048\pi$
$492$	0	0
$493$	−25.2861	−1.13883
$494$	−19.6499	−0.884092
$495$	0	0
$496$	−42.5883	−1.91227
$497$	−1.28147	−0.0574818
$498$	0	0
$499$	−8.90416	−0.398605	−0.199303	−	0.979938i	$-0.563868\pi$
$499$	−8.90416	−0.398605	−0.199303	+	0.979938i	$0.563868\pi$
$500$	−8.22177	−0.367689
$501$	0	0
$502$	18.5244	0.826783
$503$	−21.3591	−0.952357	−0.476179	−	0.879349i	$-0.657978\pi$
$503$	−21.3591	−0.952357	−0.476179	+	0.879349i	$0.657978\pi$
$504$	0	0
$505$	0.849176	0.0377878
$506$	1.31582	0.0584954
$507$	0	0
$508$	3.46357	0.153671
$509$	25.1922	1.11662	0.558311	−	0.829631i	$-0.311449\pi$
$509$	25.1922	1.11662	0.558311	+	0.829631i	$0.311449\pi$
$510$	0	0
$511$	1.98031	0.0876038
$512$	1.51389	0.0669053
$513$	0	0
$514$	−14.4481	−0.637280
$515$	18.2835	0.805669
$516$	0	0
$517$	12.9527	0.569661
$518$	4.72935	0.207796
$519$	0	0
$520$	−7.54378	−0.330817
$521$	−29.7190	−1.30201	−0.651006	−	0.759073i	$-0.725653\pi$
$521$	−29.7190	−1.30201	−0.651006	+	0.759073i	$0.725653\pi$
$522$	0	0
$523$	19.7284	0.862661	0.431330	−	0.902194i	$-0.358044\pi$
$523$	19.7284	0.862661	0.431330	+	0.902194i	$0.358044\pi$
$524$	11.2497	0.491445
$525$	0	0
$526$	26.7479	1.16627
$527$	28.1447	1.22600
$528$	0	0
$529$	−22.3641	−0.972351
$530$	19.0308	0.826647
$531$	0	0
$532$	1.68670	0.0731279
$533$	−20.0775	−0.869653
$534$	0	0
$535$	13.5293	0.584923
$536$	−25.8174	−1.11514
$537$	0	0
$538$	−45.7951	−1.97437
$539$	6.89780	0.297109
$540$	0	0
$541$	9.06280	0.389640	0.194820	−	0.980839i	$-0.437588\pi$
$541$	9.06280	0.389640	0.194820	+	0.980839i	$0.437588\pi$
$542$	27.5401	1.18295
$543$	0	0
$544$	12.7141	0.545112
$545$	1.56202	0.0669096
$546$	0	0
$547$	−6.14196	−0.262611	−0.131306	−	0.991342i	$-0.541917\pi$
$547$	−6.14196	−0.262611	−0.131306	+	0.991342i	$0.541917\pi$
$548$	−11.5973	−0.495410
$549$	0	0
$550$	−0.305075	−0.0130085
$551$	56.7450	2.41742
$552$	0	0
$553$	−2.39361	−0.101787
$554$	−48.4084	−2.05668
$555$	0	0
$556$	12.0706	0.511908
$557$	35.5642	1.50690	0.753452	−	0.657503i	$-0.228387\pi$
$557$	35.5642	1.50690	0.753452	+	0.657503i	$0.228387\pi$
$558$	0	0
$559$	−19.7784	−0.836538
$560$	3.45357	0.145940
$561$	0	0
$562$	24.1348	1.01807
$563$	−8.99620	−0.379145	−0.189572	−	0.981867i	$-0.560710\pi$
$563$	−8.99620	−0.379145	−0.189572	+	0.981867i	$0.560710\pi$
$564$	0	0
$565$	13.8845	0.584124
$566$	37.7270	1.58578
$567$	0	0
$568$	8.44868	0.354499
$569$	−33.8544	−1.41925	−0.709626	−	0.704578i	$-0.751136\pi$
$569$	−33.8544	−1.41925	−0.709626	+	0.704578i	$0.751136\pi$
$570$	0	0
$571$	45.3074	1.89605	0.948027	−	0.318189i	$-0.103075\pi$
$571$	45.3074	1.89605	0.948027	+	0.318189i	$0.103075\pi$
$572$	1.17869	0.0492836
$573$	0	0
$574$	6.49314	0.271019
$575$	−0.147440	−0.00614866
$576$	0	0
$577$	−31.4777	−1.31044	−0.655218	−	0.755440i	$-0.727423\pi$
$577$	−31.4777	−1.31044	−0.655218	+	0.755440i	$0.727423\pi$
$578$	10.5852	0.440287
$579$	0	0
$580$	−12.3244	−0.511742
$581$	3.93540	0.163268
$582$	0	0
$583$	5.25605	0.217684
$584$	−13.0561	−0.540265
$585$	0	0
$586$	12.4651	0.514929
$587$	35.3166	1.45767	0.728837	−	0.684687i	$-0.240061\pi$
$587$	35.3166	1.45767	0.728837	+	0.684687i	$0.240061\pi$
$588$	0	0
$589$	−63.1600	−2.60246
$590$	25.8120	1.06266
$591$	0	0
$592$	−44.1382	−1.81407
$593$	2.61073	0.107210	0.0536049	−	0.998562i	$-0.482929\pi$
$593$	2.61073	0.107210	0.0536049	+	0.998562i	$0.482929\pi$
$594$	0	0
$595$	−2.28231	−0.0935656
$596$	−6.83473	−0.279961
$597$	0	0
$598$	2.14622	0.0877656
$599$	−1.25833	−0.0514140	−0.0257070	−	0.999670i	$-0.508184\pi$
$599$	−1.25833	−0.0514140	−0.0257070	+	0.999670i	$0.508184\pi$
$600$	0	0
$601$	−29.6010	−1.20745	−0.603724	−	0.797193i	$-0.706317\pi$
$601$	−29.6010	−1.20745	−0.603724	+	0.797193i	$0.706317\pi$
$602$	6.39642	0.260699
$603$	0	0
$604$	4.03373	0.164130
$605$	2.19434	0.0892124
$606$	0	0
$607$	17.3055	0.702407	0.351203	−	0.936299i	$-0.385773\pi$
$607$	17.3055	0.702407	0.351203	+	0.936299i	$0.385773\pi$
$608$	−28.5318	−1.15712
$609$	0	0
$610$	3.62075	0.146600
$611$	21.1271	0.854710
$612$	0	0
$613$	15.0353	0.607269	0.303635	−	0.952789i	$-0.401800\pi$
$613$	15.0353	0.607269	0.303635	+	0.952789i	$0.401800\pi$
$614$	−43.4344	−1.75287
$615$	0	0
$616$	0.673809	0.0271485
$617$	38.6536	1.55614	0.778069	−	0.628179i	$-0.216200\pi$
$617$	38.6536	1.55614	0.778069	+	0.628179i	$0.216200\pi$
$618$	0	0
$619$	−30.7996	−1.23794	−0.618970	−	0.785415i	$-0.712450\pi$
$619$	−30.7996	−1.23794	−0.618970	+	0.785415i	$0.712450\pi$
$620$	13.7177	0.550914
$621$	0	0
$622$	27.8453	1.11649
$623$	3.11471	0.124788
$624$	0	0
$625$	−24.0414	−0.961655
$626$	−47.9782	−1.91760
$627$	0	0
$628$	−13.4330	−0.536035
$629$	29.1690	1.16304
$630$	0	0
$631$	−42.0830	−1.67530	−0.837649	−	0.546209i	$-0.816070\pi$
$631$	−42.0830	−1.67530	−0.837649	+	0.546209i	$0.816070\pi$
$632$	15.7809	0.627732
$633$	0	0
$634$	29.6860	1.17898
$635$	10.5173	0.417366
$636$	0	0
$637$	11.2509	0.445778
$638$	−12.8243	−0.507720
$639$	0	0
$640$	−29.4536	−1.16426
$641$	−30.0875	−1.18838	−0.594192	−	0.804323i	$-0.702528\pi$
$641$	−30.0875	−1.18838	−0.594192	+	0.804323i	$0.702528\pi$
$642$	0	0
$643$	6.57759	0.259395	0.129698	−	0.991554i	$-0.458599\pi$
$643$	6.57759	0.259395	0.129698	+	0.991554i	$0.458599\pi$
$644$	−0.184227	−0.00725955
$645$	0	0
$646$	39.1946	1.54209
$647$	−26.2306	−1.03123	−0.515616	−	0.856820i	$-0.672437\pi$
$647$	−26.2306	−1.03123	−0.515616	+	0.856820i	$0.672437\pi$
$648$	0	0
$649$	7.12892	0.279834
$650$	−0.497605	−0.0195177
$651$	0	0
$652$	2.67315	0.104689
$653$	1.76707	0.0691506	0.0345753	−	0.999402i	$-0.488992\pi$
$653$	1.76707	0.0691506	0.0345753	+	0.999402i	$0.488992\pi$
$654$	0	0
$655$	34.1603	1.33475
$656$	−60.5993	−2.36601
$657$	0	0
$658$	−6.83258	−0.266362
$659$	33.7679	1.31541	0.657705	−	0.753276i	$-0.271527\pi$
$659$	33.7679	1.31541	0.657705	+	0.753276i	$0.271527\pi$
$660$	0	0
$661$	5.25627	0.204445	0.102223	−	0.994762i	$-0.467405\pi$
$661$	5.25627	0.204445	0.102223	+	0.994762i	$0.467405\pi$
$662$	−2.98412	−0.115981
$663$	0	0
$664$	−25.9459	−1.00690
$665$	5.12176	0.198614
$666$	0	0
$667$	−6.19786	−0.239982
$668$	−0.0169817	−0.000657043 0
$669$	0	0
$670$	44.3509	1.71343
$671$	1.00000	0.0386046
$672$	0	0
$673$	−8.74520	−0.337103	−0.168551	−	0.985693i	$-0.553909\pi$
$673$	−8.74520	−0.337103	−0.168551	+	0.985693i	$0.553909\pi$
$674$	−30.2685	−1.16590
$675$	0	0
$676$	−7.47178	−0.287376
$677$	−17.8404	−0.685660	−0.342830	−	0.939397i	$-0.611386\pi$
$677$	−17.8404	−0.685660	−0.342830	+	0.939397i	$0.611386\pi$
$678$	0	0
$679$	4.59391	0.176298
$680$	15.0472	0.577032
$681$	0	0
$682$	14.2741	0.546584
$683$	−10.6453	−0.407332	−0.203666	−	0.979040i	$-0.565286\pi$
$683$	−10.6453	−0.407332	−0.203666	+	0.979040i	$0.565286\pi$
$684$	0	0
$685$	−35.2157	−1.34552
$686$	−7.33111	−0.279903
$687$	0	0
$688$	−59.6966	−2.27591
$689$	8.57310	0.326609
$690$	0	0
$691$	−9.44300	−0.359229	−0.179614	−	0.983737i	$-0.557485\pi$
$691$	−9.44300	−0.359229	−0.179614	+	0.983737i	$0.557485\pi$
$692$	−10.3095	−0.391909
$693$	0	0
$694$	28.4395	1.07955
$695$	36.6531	1.39033
$696$	0	0
$697$	40.0474	1.51690
$698$	16.5561	0.626659
$699$	0	0
$700$	0.0427133	0.00161441
$701$	−14.3538	−0.542136	−0.271068	−	0.962560i	$-0.587377\pi$
$701$	−14.3538	−0.542136	−0.271068	+	0.962560i	$0.587377\pi$
$702$	0	0
$703$	−65.4585	−2.46881
$704$	−3.39797	−0.128066
$705$	0	0
$706$	2.61837	0.0985436
$707$	−0.123715	−0.00465279
$708$	0	0
$709$	30.3842	1.14110	0.570550	−	0.821263i	$-0.306730\pi$
$709$	30.3842	1.14110	0.570550	+	0.821263i	$0.306730\pi$
$710$	−14.5137	−0.544690
$711$	0	0
$712$	−20.5351	−0.769585
$713$	6.89852	0.258352
$714$	0	0
$715$	3.57916	0.133853
$716$	5.00054	0.186879
$717$	0	0
$718$	−3.46390	−0.129272
$719$	−46.7518	−1.74355	−0.871774	−	0.489908i	$-0.837030\pi$
$719$	−46.7518	−1.74355	−0.871774	+	0.489908i	$0.837030\pi$
$720$	0	0
$721$	−2.66370	−0.0992014
$722$	−56.6064	−2.10667
$723$	0	0
$724$	−0.396148	−0.0147227
$725$	1.43698	0.0533682
$726$	0	0
$727$	42.1698	1.56399	0.781996	−	0.623284i	$-0.214202\pi$
$727$	42.1698	1.56399	0.781996	+	0.623284i	$0.214202\pi$
$728$	1.09904	0.0407332
$729$	0	0
$730$	22.4286	0.830122
$731$	39.4509	1.45914
$732$	0	0
$733$	21.2976	0.786645	0.393322	−	0.919401i	$-0.371326\pi$
$733$	21.2976	0.786645	0.393322	+	0.919401i	$0.371326\pi$
$734$	12.8150	0.473009
$735$	0	0
$736$	3.11633	0.114870
$737$	12.2491	0.451201
$738$	0	0
$739$	23.1327	0.850951	0.425475	−	0.904970i	$-0.360107\pi$
$739$	23.1327	0.850951	0.425475	+	0.904970i	$0.360107\pi$
$740$	14.2169	0.522622
$741$	0	0
$742$	−2.77258	−0.101784
$743$	−11.6309	−0.426697	−0.213348	−	0.976976i	$-0.568437\pi$
$743$	−11.6309	−0.426697	−0.213348	+	0.976976i	$0.568437\pi$
$744$	0	0
$745$	−20.7540	−0.760368
$746$	−7.21849	−0.264288
$747$	0	0
$748$	−2.35107	−0.0859636
$749$	−1.97107	−0.0720212
$750$	0	0
$751$	−0.112508	−0.00410547	−0.00205273	−	0.999998i	$-0.500653\pi$
$751$	−0.112508	−0.00410547	−0.00205273	+	0.999998i	$0.500653\pi$
$752$	63.7672	2.32535
$753$	0	0
$754$	−20.9176	−0.761775
$755$	12.2486	0.445773
$756$	0	0
$757$	−53.2014	−1.93364	−0.966820	−	0.255460i	$-0.917773\pi$
$757$	−53.2014	−1.93364	−0.966820	+	0.255460i	$0.917773\pi$
$758$	15.1738	0.551136
$759$	0	0
$760$	−33.7675	−1.22488
$761$	−26.8565	−0.973546	−0.486773	−	0.873528i	$-0.661826\pi$
$761$	−26.8565	−0.973546	−0.486773	+	0.873528i	$0.661826\pi$
$762$	0	0
$763$	−0.227569	−0.00823854
$764$	−1.27027	−0.0459566
$765$	0	0
$766$	−39.4279	−1.42459
$767$	11.6279	0.419859
$768$	0	0
$769$	−8.77563	−0.316457	−0.158229	−	0.987403i	$-0.550578\pi$
$769$	−8.77563	−0.316457	−0.158229	+	0.987403i	$0.550578\pi$
$770$	−1.15752	−0.0417140
$771$	0	0
$772$	−15.8822	−0.571612
$773$	−41.5686	−1.49512	−0.747559	−	0.664195i	$-0.768774\pi$
$773$	−41.5686	−1.49512	−0.747559	+	0.664195i	$0.768774\pi$
$774$	0	0
$775$	−1.59943	−0.0574534
$776$	−30.2874	−1.08725
$777$	0	0
$778$	−12.9590	−0.464604
$779$	−89.8710	−3.21996
$780$	0	0
$781$	−4.00849	−0.143435
$782$	−4.28095	−0.153087
$783$	0	0
$784$	33.9584	1.21280
$785$	−40.7900	−1.45586
$786$	0	0
$787$	23.5750	0.840359	0.420180	−	0.907441i	$-0.361967\pi$
$787$	23.5750	0.840359	0.420180	+	0.907441i	$0.361967\pi$
$788$	−17.9169	−0.638262
$789$	0	0
$790$	−27.1096	−0.964516
$791$	−2.02281	−0.0719228
$792$	0	0
$793$	1.63109	0.0579217
$794$	−19.5259	−0.692947
$795$	0	0
$796$	−8.70023	−0.308372
$797$	21.7928	0.771940	0.385970	−	0.922511i	$-0.373867\pi$
$797$	21.7928	0.771940	0.385970	+	0.922511i	$0.373867\pi$
$798$	0	0
$799$	−42.1410	−1.49084
$800$	−0.722527	−0.0255452
$801$	0	0
$802$	40.2467	1.42116
$803$	6.19448	0.218598
$804$	0	0
$805$	−0.559414	−0.0197168
$806$	23.2824	0.820087
$807$	0	0
$808$	0.815648	0.0286944
$809$	−26.6020	−0.935277	−0.467639	−	0.883920i	$-0.654895\pi$
$809$	−26.6020	−0.935277	−0.467639	+	0.883920i	$0.654895\pi$
$810$	0	0
$811$	7.77300	0.272947	0.136473	−	0.990644i	$-0.456423\pi$
$811$	7.77300	0.272947	0.136473	+	0.990644i	$0.456423\pi$
$812$	1.79552	0.0630104
$813$	0	0
$814$	14.7936	0.518515
$815$	8.11717	0.284332
$816$	0	0
$817$	−88.5323	−3.09735
$818$	40.7986	1.42649
$819$	0	0
$820$	19.5190	0.681632
$821$	7.27693	0.253967	0.126983	−	0.991905i	$-0.459471\pi$
$821$	7.27693	0.253967	0.126983	+	0.991905i	$0.459471\pi$
$822$	0	0
$823$	−9.38569	−0.327164	−0.163582	−	0.986530i	$-0.552305\pi$
$823$	−9.38569	−0.327164	−0.163582	+	0.986530i	$0.552305\pi$
$824$	17.5616	0.611789
$825$	0	0
$826$	−3.76051	−0.130845
$827$	38.4908	1.33846	0.669228	−	0.743057i	$-0.266625\pi$
$827$	38.4908	1.33846	0.669228	+	0.743057i	$0.266625\pi$
$828$	0	0
$829$	36.9355	1.28282	0.641411	−	0.767197i	$-0.278349\pi$
$829$	36.9355	1.28282	0.641411	+	0.767197i	$0.278349\pi$
$830$	44.5717	1.54711
$831$	0	0
$832$	−5.54239	−0.192148
$833$	−22.4416	−0.777555
$834$	0	0
$835$	−0.0515659	−0.00178451
$836$	5.27607	0.182477
$837$	0	0
$838$	−43.5722	−1.50518
$839$	−17.3297	−0.598289	−0.299145	−	0.954208i	$-0.596701\pi$
$839$	−17.3297	−0.598289	−0.299145	+	0.954208i	$0.596701\pi$
$840$	0	0
$841$	31.4059	1.08296
$842$	63.5358	2.18959
$843$	0	0
$844$	9.17302	0.315748
$845$	−22.6884	−0.780506
$846$	0	0
$847$	−0.319690	−0.0109847
$848$	25.8759	0.888583
$849$	0	0
$850$	0.992545	0.0340440
$851$	7.14957	0.245084
$852$	0	0
$853$	13.5521	0.464016	0.232008	−	0.972714i	$-0.425470\pi$
$853$	13.5521	0.464016	0.232008	+	0.972714i	$0.425470\pi$
$854$	−0.527501	−0.0180507
$855$	0	0
$856$	12.9951	0.444165
$857$	33.1995	1.13407	0.567037	−	0.823693i	$-0.308090\pi$
$857$	33.1995	1.13407	0.567037	+	0.823693i	$0.308090\pi$
$858$	0	0
$859$	−24.2560	−0.827604	−0.413802	−	0.910367i	$-0.635799\pi$
$859$	−24.2560	−0.827604	−0.413802	+	0.910367i	$0.635799\pi$
$860$	19.2282	0.655677
$861$	0	0
$862$	14.1511	0.481990
$863$	−42.9445	−1.46185	−0.730923	−	0.682460i	$-0.760910\pi$
$863$	−42.9445	−1.46185	−0.730923	+	0.682460i	$0.760910\pi$
$864$	0	0
$865$	−31.3053	−1.06441
$866$	63.9902	2.17447
$867$	0	0
$868$	−1.99850	−0.0678337
$869$	−7.48729	−0.253989
$870$	0	0
$871$	19.9794	0.676976
$872$	1.50035	0.0508082
$873$	0	0
$874$	9.60694	0.324960
$875$	3.63723	0.122961
$876$	0	0
$877$	5.18051	0.174933	0.0874666	−	0.996167i	$-0.472123\pi$
$877$	5.18051	0.174933	0.0874666	+	0.996167i	$0.472123\pi$
$878$	28.7863	0.971492
$879$	0	0
$880$	10.8029	0.364165
$881$	12.8929	0.434375	0.217187	−	0.976130i	$-0.430312\pi$
$881$	12.8929	0.434375	0.217187	+	0.976130i	$0.430312\pi$
$882$	0	0
$883$	−13.4046	−0.451102	−0.225551	−	0.974231i	$-0.572418\pi$
$883$	−13.4046	−0.451102	−0.225551	+	0.974231i	$0.572418\pi$
$884$	−3.83481	−0.128979
$885$	0	0
$886$	−19.8421	−0.666608
$887$	−32.3517	−1.08626	−0.543132	−	0.839647i	$-0.682762\pi$
$887$	−32.3517	−1.08626	−0.543132	+	0.839647i	$0.682762\pi$
$888$	0	0
$889$	−1.53225	−0.0513900
$890$	35.2766	1.18247
$891$	0	0
$892$	10.6051	0.355087
$893$	94.5692	3.16464
$894$	0	0
$895$	15.1844	0.507559
$896$	4.29105	0.143354
$897$	0	0
$898$	21.4172	0.714702
$899$	−67.2347	−2.24240
$900$	0	0
$901$	−17.1003	−0.569693
$902$	20.3108	0.676275
$903$	0	0
$904$	13.3363	0.443558
$905$	−1.20292	−0.0399866
$906$	0	0
$907$	−48.8264	−1.62126	−0.810628	−	0.585562i	$-0.800874\pi$
$907$	−48.8264	−1.62126	−0.810628	+	0.585562i	$0.800874\pi$
$908$	3.67110	0.121830
$909$	0	0
$910$	−1.88801	−0.0625870
$911$	−26.6830	−0.884047	−0.442023	−	0.897003i	$-0.645739\pi$
$911$	−26.6830	−0.884047	−0.442023	+	0.897003i	$0.645739\pi$
$912$	0	0
$913$	12.3101	0.407404
$914$	46.6999	1.54469
$915$	0	0
$916$	−6.77423	−0.223827
$917$	−4.97676	−0.164347
$918$	0	0
$919$	25.4702	0.840183	0.420091	−	0.907482i	$-0.361998\pi$
$919$	25.4702	0.840183	0.420091	+	0.907482i	$0.361998\pi$
$920$	3.68819	0.121596
$921$	0	0
$922$	−5.14855	−0.169559
$923$	−6.53820	−0.215208
$924$	0	0
$925$	−1.65764	−0.0545029
$926$	1.57125	0.0516345
$927$	0	0
$928$	−30.3726	−0.997028
$929$	−28.6815	−0.941010	−0.470505	−	0.882397i	$-0.655928\pi$
$929$	−28.6815	−0.941010	−0.470505	+	0.882397i	$0.655928\pi$
$930$	0	0
$931$	50.3615	1.65053
$932$	2.11742	0.0693584
$933$	0	0
$934$	18.0418	0.590345
$935$	−7.13915	−0.233475
$936$	0	0
$937$	−19.4745	−0.636206	−0.318103	−	0.948056i	$-0.603046\pi$
$937$	−19.4745	−0.636206	−0.318103	+	0.948056i	$0.603046\pi$
$938$	−6.46142	−0.210973
$939$	0	0
$940$	−20.5394	−0.669920
$941$	23.3524	0.761267	0.380634	−	0.924726i	$-0.375706\pi$
$941$	23.3524	0.761267	0.380634	+	0.924726i	$0.375706\pi$
$942$	0	0
$943$	9.81598	0.319652
$944$	35.0962	1.14228
$945$	0	0
$946$	20.0082	0.650524
$947$	55.2046	1.79391	0.896955	−	0.442123i	$-0.145774\pi$
$947$	55.2046	1.79391	0.896955	+	0.442123i	$0.145774\pi$
$948$	0	0
$949$	10.1038	0.327982
$950$	−2.22739	−0.0722659
$951$	0	0
$952$	−2.19220	−0.0710495
$953$	0.00964019	0.000312276 0	0.000156138	−	1.00000i	$-0.499950\pi$
$953$	0.00964019	0.000312276 0	0.000156138		1.00000i	$0.499950\pi$
$954$	0	0
$955$	−3.85723	−0.124817
$956$	−13.8588	−0.448224
$957$	0	0
$958$	29.9489	0.967606
$959$	5.13052	0.165673
$960$	0	0
$961$	43.8356	1.41405
$962$	24.1297	0.777971
$963$	0	0
$964$	19.0231	0.612694
$965$	−48.2271	−1.55248
$966$	0	0
$967$	22.6638	0.728818	0.364409	−	0.931239i	$-0.381271\pi$
$967$	22.6638	0.728818	0.364409	+	0.931239i	$0.381271\pi$
$968$	2.10770	0.0677440
$969$	0	0
$970$	52.0298	1.67058
$971$	−1.08081	−0.0346849	−0.0173425	−	0.999850i	$-0.505521\pi$
$971$	−1.08081	−0.0346849	−0.0173425	+	0.999850i	$0.505521\pi$
$972$	0	0
$973$	−5.33993	−0.171190
$974$	24.6254	0.789049
$975$	0	0
$976$	4.92307	0.157584
$977$	−23.4131	−0.749052	−0.374526	−	0.927216i	$-0.622195\pi$
$977$	−23.4131	−0.749052	−0.374526	+	0.927216i	$0.622195\pi$
$978$	0	0
$979$	9.74291	0.311385
$980$	−10.9380	−0.349400
$981$	0	0
$982$	37.8246	1.20703
$983$	23.3251	0.743956	0.371978	−	0.928242i	$-0.378680\pi$
$983$	23.3251	0.743956	0.371978	+	0.928242i	$0.378680\pi$
$984$	0	0
$985$	−54.4055	−1.73350
$986$	41.7232	1.32874
$987$	0	0
$988$	8.60574	0.273785
$989$	9.66976	0.307480
$990$	0	0
$991$	25.5519	0.811683	0.405841	−	0.913944i	$-0.366979\pi$
$991$	25.5519	0.811683	0.405841	+	0.913944i	$0.366979\pi$
$992$	33.8062	1.07335
$993$	0	0
$994$	2.11448	0.0670673
$995$	−26.4187	−0.837529
$996$	0	0
$997$	54.4150	1.72334	0.861670	−	0.507469i	$-0.169419\pi$
$997$	54.4150	1.72334	0.861670	+	0.507469i	$0.169419\pi$
$998$	14.6922	0.465075
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.n.1.8	✓	25
3.2	odd	2		6039.2.a.o.1.18	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.n.1.8	✓	25	1.1	even	1	trivial
6039.2.a.o.1.18	yes	25	3.2	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.65004 q^{2} +0.722641 q^{4} +2.19434 q^{5} -0.319690 q^{7} +2.10770 q^{8} +O(q^{10})\)	\(q-1.65004 q^{2} +0.722641 q^{4} +2.19434 q^{5} -0.319690 q^{7} +2.10770 q^{8} -3.62075 q^{10} -1.00000 q^{11} -1.63109 q^{13} +0.527501 q^{14} -4.92307 q^{16} +3.25344 q^{17} -7.30110 q^{19} +1.58572 q^{20} +1.65004 q^{22} +0.797447 q^{23} -0.184889 q^{25} +2.69137 q^{26} -0.231021 q^{28} -7.77212 q^{29} +8.65076 q^{31} +3.90788 q^{32} -5.36832 q^{34} -0.701506 q^{35} +8.96557 q^{37} +12.0471 q^{38} +4.62500 q^{40} +12.3092 q^{41} +12.1259 q^{43} -0.722641 q^{44} -1.31582 q^{46} -12.9527 q^{47} -6.89780 q^{49} +0.305075 q^{50} -1.17869 q^{52} -5.25605 q^{53} -2.19434 q^{55} -0.673809 q^{56} +12.8243 q^{58} -7.12892 q^{59} -1.00000 q^{61} -14.2741 q^{62} +3.39797 q^{64} -3.57916 q^{65} -12.2491 q^{67} +2.35107 q^{68} +1.15752 q^{70} +4.00849 q^{71} -6.19448 q^{73} -14.7936 q^{74} -5.27607 q^{76} +0.319690 q^{77} +7.48729 q^{79} -10.8029 q^{80} -20.3108 q^{82} -12.3101 q^{83} +7.13915 q^{85} -20.0082 q^{86} -2.10770 q^{88} -9.74291 q^{89} +0.521442 q^{91} +0.576268 q^{92} +21.3726 q^{94} -16.0211 q^{95} -14.3699 q^{97} +11.3817 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 4 q^{5} + 4 q^{7} - 15 q^{8} - 25 q^{11} + 4 q^{13} - 18 q^{14} + 21 q^{16} - 20 q^{17} + 14 q^{19} - 12 q^{20} + 5 q^{22} - 20 q^{23} + 13 q^{25} - 16 q^{26} - 14 q^{28} - 28 q^{29} - 12 q^{31} - 35 q^{32} + 6 q^{34} - 10 q^{35} - 8 q^{37} - 32 q^{38} + 24 q^{40} - 26 q^{41} + 18 q^{43} - 25 q^{44} + 4 q^{46} - 12 q^{47} + 23 q^{49} - 43 q^{50} + 22 q^{52} - 36 q^{53} + 4 q^{55} - 26 q^{56} - 20 q^{58} - 46 q^{59} - 25 q^{61} + 14 q^{62} - 13 q^{64} - 60 q^{65} - 20 q^{67} - 44 q^{68} - 20 q^{70} - 52 q^{71} + 6 q^{73} - 32 q^{74} - 4 q^{77} + 26 q^{79} - 52 q^{80} + 6 q^{82} - 38 q^{83} - 4 q^{85} - 34 q^{86} + 15 q^{88} - 82 q^{89} - 58 q^{91} - 36 q^{92} + 16 q^{94} - 30 q^{95} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 4 * q^5 + 4 * q^7 - 15 * q^8 - 25 * q^11 + 4 * q^13 - 18 * q^14 + 21 * q^16 - 20 * q^17 + 14 * q^19 - 12 * q^20 + 5 * q^22 - 20 * q^23 + 13 * q^25 - 16 * q^26 - 14 * q^28 - 28 * q^29 - 12 * q^31 - 35 * q^32 + 6 * q^34 - 10 * q^35 - 8 * q^37 - 32 * q^38 + 24 * q^40 - 26 * q^41 + 18 * q^43 - 25 * q^44 + 4 * q^46 - 12 * q^47 + 23 * q^49 - 43 * q^50 + 22 * q^52 - 36 * q^53 + 4 * q^55 - 26 * q^56 - 20 * q^58 - 46 * q^59 - 25 * q^61 + 14 * q^62 - 13 * q^64 - 60 * q^65 - 20 * q^67 - 44 * q^68 - 20 * q^70 - 52 * q^71 + 6 * q^73 - 32 * q^74 - 4 * q^77 + 26 * q^79 - 52 * q^80 + 6 * q^82 - 38 * q^83 - 4 * q^85 - 34 * q^86 + 15 * q^88 - 82 * q^89 - 58 * q^91 - 36 * q^92 + 16 * q^94 - 30 * q^95 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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