LMFDB - Embedded newform 6039.2.a.p.1.2

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

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.42221 q^{2} +3.86709 q^{4} +1.62300 q^{5} +2.13922 q^{7} -4.52249 q^{8} +O(q^{10})$	\(q-2.42221 q^{2} +3.86709 q^{4} +1.62300 q^{5} +2.13922 q^{7} -4.52249 q^{8} -3.93125 q^{10} -1.00000 q^{11} -5.27740 q^{13} -5.18165 q^{14} +3.22023 q^{16} +5.69421 q^{17} -3.03765 q^{19} +6.27630 q^{20} +2.42221 q^{22} +4.60146 q^{23} -2.36586 q^{25} +12.7830 q^{26} +8.27258 q^{28} +8.32621 q^{29} +2.27040 q^{31} +1.24492 q^{32} -13.7926 q^{34} +3.47197 q^{35} +0.699534 q^{37} +7.35781 q^{38} -7.34002 q^{40} -9.03931 q^{41} -5.76169 q^{43} -3.86709 q^{44} -11.1457 q^{46} +2.26259 q^{47} -2.42372 q^{49} +5.73061 q^{50} -20.4082 q^{52} -4.65998 q^{53} -1.62300 q^{55} -9.67462 q^{56} -20.1678 q^{58} -4.54214 q^{59} +1.00000 q^{61} -5.49938 q^{62} -9.45591 q^{64} -8.56523 q^{65} +4.47292 q^{67} +22.0200 q^{68} -8.40983 q^{70} +12.9682 q^{71} +7.59203 q^{73} -1.69442 q^{74} -11.7469 q^{76} -2.13922 q^{77} +8.14700 q^{79} +5.22644 q^{80} +21.8951 q^{82} -10.6217 q^{83} +9.24172 q^{85} +13.9560 q^{86} +4.52249 q^{88} +17.1770 q^{89} -11.2895 q^{91} +17.7943 q^{92} -5.48046 q^{94} -4.93011 q^{95} +11.2641 q^{97} +5.87076 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	$ 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) $ 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 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 $

$2$	−2.42221	−1.71276	−0.856380	−	0.516346i	$-0.827292\pi$
$2$	−2.42221	−1.71276	−0.856380	+	0.516346i	$0.827292\pi$
$3$	0	0
$3$	0	0
$4$	3.86709	1.93355
$5$	1.62300	0.725829	0.362914	−	0.931822i	$-0.381782\pi$
$5$	1.62300	0.725829	0.362914	+	0.931822i	$0.381782\pi$
$6$	0	0
$7$	2.13922	0.808551	0.404275	−	0.914637i	$-0.367524\pi$
$7$	2.13922	0.808551	0.404275	+	0.914637i	$0.367524\pi$
$8$	−4.52249	−1.59894
$9$	0	0
$10$	−3.93125	−1.24317
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.27740	−1.46369	−0.731843	−	0.681473i	$-0.761340\pi$
$13$	−5.27740	−1.46369	−0.731843	+	0.681473i	$0.761340\pi$
$14$	−5.18165	−1.38485
$15$	0	0
$16$	3.22023	0.805057
$17$	5.69421	1.38105	0.690525	−	0.723309i	$-0.257380\pi$
$17$	5.69421	1.38105	0.690525	+	0.723309i	$0.257380\pi$
$18$	0	0
$19$	−3.03765	−0.696884	−0.348442	−	0.937330i	$-0.613289\pi$
$19$	−3.03765	−0.696884	−0.348442	+	0.937330i	$0.613289\pi$
$20$	6.27630	1.40342
$21$	0	0
$22$	2.42221	0.516417
$23$	4.60146	0.959472	0.479736	−	0.877413i	$-0.340732\pi$
$23$	4.60146	0.959472	0.479736	+	0.877413i	$0.340732\pi$
$24$	0	0
$25$	−2.36586	−0.473172
$26$	12.7830	2.50694
$27$	0	0
$28$	8.27258	1.56337
$29$	8.32621	1.54614	0.773069	−	0.634322i	$-0.218721\pi$
$29$	8.32621	1.54614	0.773069	+	0.634322i	$0.218721\pi$
$30$	0	0
$31$	2.27040	0.407775	0.203888	−	0.978994i	$-0.434642\pi$
$31$	2.27040	0.407775	0.203888	+	0.978994i	$0.434642\pi$
$32$	1.24492	0.220073
$33$	0	0
$34$	−13.7926	−2.36541
$35$	3.47197	0.586869
$36$	0	0
$37$	0.699534	0.115003	0.0575013	−	0.998345i	$-0.481687\pi$
$37$	0.699534	0.115003	0.0575013	+	0.998345i	$0.481687\pi$
$38$	7.35781	1.19359
$39$	0	0
$40$	−7.34002	−1.16056
$41$	−9.03931	−1.41170	−0.705851	−	0.708360i	$-0.749435\pi$
$41$	−9.03931	−1.41170	−0.705851	+	0.708360i	$0.749435\pi$
$42$	0	0
$43$	−5.76169	−0.878650	−0.439325	−	0.898328i	$-0.644782\pi$
$43$	−5.76169	−0.878650	−0.439325	+	0.898328i	$0.644782\pi$
$44$	−3.86709	−0.582986
$45$	0	0
$46$	−11.1457	−1.64334
$47$	2.26259	0.330032	0.165016	−	0.986291i	$-0.447232\pi$
$47$	2.26259	0.330032	0.165016	+	0.986291i	$0.447232\pi$
$48$	0	0
$49$	−2.42372	−0.346246
$50$	5.73061	0.810431
$51$	0	0
$52$	−20.4082	−2.83011
$53$	−4.65998	−0.640097	−0.320049	−	0.947401i	$-0.603699\pi$
$53$	−4.65998	−0.640097	−0.320049	+	0.947401i	$0.603699\pi$
$54$	0	0
$55$	−1.62300	−0.218846
$56$	−9.67462	−1.29283
$57$	0	0
$58$	−20.1678	−2.64816
$59$	−4.54214	−0.591336	−0.295668	−	0.955291i	$-0.595542\pi$
$59$	−4.54214	−0.591336	−0.295668	+	0.955291i	$0.595542\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−5.49938	−0.698421
$63$	0	0
$64$	−9.45591	−1.18199
$65$	−8.56523	−1.06239
$66$	0	0
$67$	4.47292	0.546455	0.273227	−	0.961950i	$-0.411909\pi$
$67$	4.47292	0.546455	0.273227	+	0.961950i	$0.411909\pi$
$68$	22.0200	2.67032
$69$	0	0
$70$	−8.40983	−1.00517
$71$	12.9682	1.53904	0.769521	−	0.638621i	$-0.220495\pi$
$71$	12.9682	1.53904	0.769521	+	0.638621i	$0.220495\pi$
$72$	0	0
$73$	7.59203	0.888580	0.444290	−	0.895883i	$-0.353456\pi$
$73$	7.59203	0.888580	0.444290	+	0.895883i	$0.353456\pi$
$74$	−1.69442	−0.196972
$75$	0	0
$76$	−11.7469	−1.34746
$77$	−2.13922	−0.243787
$78$	0	0
$79$	8.14700	0.916609	0.458305	−	0.888795i	$-0.348457\pi$
$79$	8.14700	0.916609	0.458305	+	0.888795i	$0.348457\pi$
$80$	5.22644	0.584334
$81$	0	0
$82$	21.8951	2.41791
$83$	−10.6217	−1.16589	−0.582943	−	0.812513i	$-0.698099\pi$
$83$	−10.6217	−1.16589	−0.582943	+	0.812513i	$0.698099\pi$
$84$	0	0
$85$	9.24172	1.00241
$86$	13.9560	1.50492
$87$	0	0
$88$	4.52249	0.482099
$89$	17.1770	1.82076	0.910379	−	0.413775i	$-0.135790\pi$
$89$	17.1770	1.82076	0.910379	+	0.413775i	$0.135790\pi$
$90$	0	0
$91$	−11.2895	−1.18346
$92$	17.7943	1.85518
$93$	0	0
$94$	−5.48046	−0.565266
$95$	−4.93011	−0.505818
$96$	0	0
$97$	11.2641	1.14370	0.571850	−	0.820358i	$-0.306226\pi$
$97$	11.2641	1.14370	0.571850	+	0.820358i	$0.306226\pi$
$98$	5.87076	0.593036
$99$	0	0
$100$	−9.14901	−0.914901
$101$	2.35275	0.234107	0.117054	−	0.993126i	$-0.462655\pi$
$101$	2.35275	0.234107	0.117054	+	0.993126i	$0.462655\pi$
$102$	0	0
$103$	14.8087	1.45915	0.729574	−	0.683902i	$-0.239719\pi$
$103$	14.8087	1.45915	0.729574	+	0.683902i	$0.239719\pi$
$104$	23.8670	2.34035
$105$	0	0
$106$	11.2874	1.09633
$107$	−3.60228	−0.348245	−0.174123	−	0.984724i	$-0.555709\pi$
$107$	−3.60228	−0.348245	−0.174123	+	0.984724i	$0.555709\pi$
$108$	0	0
$109$	−8.82470	−0.845253	−0.422626	−	0.906304i	$-0.638892\pi$
$109$	−8.82470	−0.845253	−0.422626	+	0.906304i	$0.638892\pi$
$110$	3.93125	0.374830
$111$	0	0
$112$	6.88879	0.650929
$113$	−0.365897	−0.0344207	−0.0172103	−	0.999852i	$-0.505478\pi$
$113$	−0.365897	−0.0344207	−0.0172103	+	0.999852i	$0.505478\pi$
$114$	0	0
$115$	7.46819	0.696412
$116$	32.1982	2.98953
$117$	0	0
$118$	11.0020	1.01282
$119$	12.1812	1.11665
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.42221	−0.219296
$123$	0	0
$124$	8.77984	0.788453
$125$	−11.9548	−1.06927
$126$	0	0
$127$	0.278127	0.0246798	0.0123399	−	0.999924i	$-0.496072\pi$
$127$	0.278127	0.0246798	0.0123399	+	0.999924i	$0.496072\pi$
$128$	20.4143	1.80439
$129$	0	0
$130$	20.7468	1.81961
$131$	6.61226	0.577716	0.288858	−	0.957372i	$-0.406725\pi$
$131$	6.61226	0.577716	0.288858	+	0.957372i	$0.406725\pi$
$132$	0	0
$133$	−6.49821	−0.563466
$134$	−10.8344	−0.935946
$135$	0	0
$136$	−25.7520	−2.20822
$137$	21.1012	1.80279	0.901397	−	0.432993i	$-0.142542\pi$
$137$	21.1012	1.80279	0.901397	+	0.432993i	$0.142542\pi$
$138$	0	0
$139$	9.39646	0.796997	0.398499	−	0.917169i	$-0.369531\pi$
$139$	9.39646	0.796997	0.398499	+	0.917169i	$0.369531\pi$
$140$	13.4264	1.13474
$141$	0	0
$142$	−31.4117	−2.63601
$143$	5.27740	0.441318
$144$	0	0
$145$	13.5135	1.12223
$146$	−18.3895	−1.52192
$147$	0	0
$148$	2.70516	0.222363
$149$	−19.8700	−1.62782	−0.813908	−	0.580994i	$-0.802664\pi$
$149$	−19.8700	−1.62782	−0.813908	+	0.580994i	$0.802664\pi$
$150$	0	0
$151$	6.68370	0.543912	0.271956	−	0.962310i	$-0.412329\pi$
$151$	6.68370	0.543912	0.271956	+	0.962310i	$0.412329\pi$
$152$	13.7377	1.11428
$153$	0	0
$154$	5.18165	0.417549
$155$	3.68486	0.295975
$156$	0	0
$157$	13.5288	1.07972	0.539859	−	0.841756i	$-0.318478\pi$
$157$	13.5288	1.07972	0.539859	+	0.841756i	$0.318478\pi$
$158$	−19.7337	−1.56993
$159$	0	0
$160$	2.02051	0.159735
$161$	9.84356	0.775781
$162$	0	0
$163$	−20.5759	−1.61163	−0.805814	−	0.592168i	$-0.798272\pi$
$163$	−20.5759	−1.61163	−0.805814	+	0.592168i	$0.798272\pi$
$164$	−34.9559	−2.72959
$165$	0	0
$166$	25.7281	1.99688
$167$	9.81420	0.759446	0.379723	−	0.925100i	$-0.376019\pi$
$167$	9.81420	0.759446	0.379723	+	0.925100i	$0.376019\pi$
$168$	0	0
$169$	14.8509	1.14238
$170$	−22.3854	−1.71688
$171$	0	0
$172$	−22.2810	−1.69891
$173$	11.2041	0.851833	0.425917	−	0.904762i	$-0.359952\pi$
$173$	11.2041	0.851833	0.425917	+	0.904762i	$0.359952\pi$
$174$	0	0
$175$	−5.06111	−0.382584
$176$	−3.22023	−0.242734
$177$	0	0
$178$	−41.6063	−3.11852
$179$	−21.0468	−1.57311	−0.786555	−	0.617521i	$-0.788137\pi$
$179$	−21.0468	−1.57311	−0.786555	+	0.617521i	$0.788137\pi$
$180$	0	0
$181$	9.75194	0.724856	0.362428	−	0.932012i	$-0.381948\pi$
$181$	9.75194	0.724856	0.362428	+	0.932012i	$0.381948\pi$
$182$	27.3456	2.02699
$183$	0	0
$184$	−20.8101	−1.53414
$185$	1.13535	0.0834723
$186$	0	0
$187$	−5.69421	−0.416402
$188$	8.74964	0.638133
$189$	0	0
$190$	11.9418	0.866346
$191$	16.6429	1.20424	0.602118	−	0.798407i	$-0.294324\pi$
$191$	16.6429	1.20424	0.602118	+	0.798407i	$0.294324\pi$
$192$	0	0
$193$	−22.7709	−1.63908	−0.819542	−	0.573019i	$-0.805772\pi$
$193$	−22.7709	−1.63908	−0.819542	+	0.573019i	$0.805772\pi$
$194$	−27.2841	−1.95888
$195$	0	0
$196$	−9.37276	−0.669483
$197$	−0.822129	−0.0585743	−0.0292871	−	0.999571i	$-0.509324\pi$
$197$	−0.822129	−0.0585743	−0.0292871	+	0.999571i	$0.509324\pi$
$198$	0	0
$199$	2.47861	0.175704	0.0878519	−	0.996134i	$-0.472000\pi$
$199$	2.47861	0.175704	0.0878519	+	0.996134i	$0.472000\pi$
$200$	10.6996	0.756575
$201$	0	0
$202$	−5.69885	−0.400969
$203$	17.8116	1.25013
$204$	0	0
$205$	−14.6708	−1.02465
$206$	−35.8698	−2.49917
$207$	0	0
$208$	−16.9944	−1.17835
$209$	3.03765	0.210118
$210$	0	0
$211$	−7.84899	−0.540347	−0.270174	−	0.962812i	$-0.587081\pi$
$211$	−7.84899	−0.540347	−0.270174	+	0.962812i	$0.587081\pi$
$212$	−18.0206	−1.23766
$213$	0	0
$214$	8.72546	0.596460
$215$	−9.35125	−0.637750
$216$	0	0
$217$	4.85689	0.329707
$218$	21.3753	1.44772
$219$	0	0
$220$	−6.27630	−0.423148
$221$	−30.0506	−2.02142
$222$	0	0
$223$	12.4273	0.832192	0.416096	−	0.909321i	$-0.363398\pi$
$223$	12.4273	0.832192	0.416096	+	0.909321i	$0.363398\pi$
$224$	2.66316	0.177940
$225$	0	0
$226$	0.886278	0.0589543
$227$	17.6730	1.17300	0.586500	−	0.809949i	$-0.300505\pi$
$227$	17.6730	1.17300	0.586500	+	0.809949i	$0.300505\pi$
$228$	0	0
$229$	8.02626	0.530390	0.265195	−	0.964195i	$-0.414564\pi$
$229$	8.02626	0.530390	0.265195	+	0.964195i	$0.414564\pi$
$230$	−18.0895	−1.19279
$231$	0	0
$232$	−37.6552	−2.47218
$233$	−14.3495	−0.940065	−0.470032	−	0.882649i	$-0.655758\pi$
$233$	−14.3495	−0.940065	−0.470032	+	0.882649i	$0.655758\pi$
$234$	0	0
$235$	3.67219	0.239547
$236$	−17.5649	−1.14338
$237$	0	0
$238$	−29.5054	−1.91255
$239$	27.6846	1.79077	0.895385	−	0.445294i	$-0.146901\pi$
$239$	27.6846	1.79077	0.895385	+	0.445294i	$0.146901\pi$
$240$	0	0
$241$	25.1831	1.62219	0.811094	−	0.584916i	$-0.198872\pi$
$241$	25.1831	1.62219	0.811094	+	0.584916i	$0.198872\pi$
$242$	−2.42221	−0.155705
$243$	0	0
$244$	3.86709	0.247565
$245$	−3.93371	−0.251315
$246$	0	0
$247$	16.0309	1.02002
$248$	−10.2679	−0.652009
$249$	0	0
$250$	28.9571	1.83140
$251$	4.68631	0.295797	0.147899	−	0.989003i	$-0.452749\pi$
$251$	4.68631	0.295797	0.147899	+	0.989003i	$0.452749\pi$
$252$	0	0
$253$	−4.60146	−0.289292
$254$	−0.673681	−0.0422705
$255$	0	0
$256$	−30.5360	−1.90850
$257$	12.5006	0.779768	0.389884	−	0.920864i	$-0.372515\pi$
$257$	12.5006	0.779768	0.389884	+	0.920864i	$0.372515\pi$
$258$	0	0
$259$	1.49646	0.0929855
$260$	−33.1225	−2.05417
$261$	0	0
$262$	−16.0163	−0.989488
$263$	−6.65962	−0.410650	−0.205325	−	0.978694i	$-0.565825\pi$
$263$	−6.65962	−0.410650	−0.205325	+	0.978694i	$0.565825\pi$
$264$	0	0
$265$	−7.56316	−0.464601
$266$	15.7400	0.965082
$267$	0	0
$268$	17.2972	1.05660
$269$	28.2558	1.72279	0.861394	−	0.507938i	$-0.169592\pi$
$269$	28.2558	1.72279	0.861394	+	0.507938i	$0.169592\pi$
$270$	0	0
$271$	−4.38875	−0.266598	−0.133299	−	0.991076i	$-0.542557\pi$
$271$	−4.38875	−0.266598	−0.133299	+	0.991076i	$0.542557\pi$
$272$	18.3367	1.11182
$273$	0	0
$274$	−51.1114	−3.08775
$275$	2.36586	0.142667
$276$	0	0
$277$	−18.2457	−1.09628	−0.548139	−	0.836388i	$-0.684663\pi$
$277$	−18.2457	−1.09628	−0.548139	+	0.836388i	$0.684663\pi$
$278$	−22.7602	−1.36507
$279$	0	0
$280$	−15.7019	−0.938370
$281$	1.52769	0.0911341	0.0455670	−	0.998961i	$-0.485491\pi$
$281$	1.52769	0.0911341	0.0455670	+	0.998961i	$0.485491\pi$
$282$	0	0
$283$	−32.1884	−1.91340	−0.956701	−	0.291072i	$-0.905988\pi$
$283$	−32.1884	−1.91340	−0.956701	+	0.291072i	$0.905988\pi$
$284$	50.1493	2.97581
$285$	0	0
$286$	−12.7830	−0.755872
$287$	−19.3371	−1.14143
$288$	0	0
$289$	15.4240	0.907296
$290$	−32.7324	−1.92211
$291$	0	0
$292$	29.3591	1.71811
$293$	−15.1528	−0.885235	−0.442617	−	0.896711i	$-0.645950\pi$
$293$	−15.1528	−0.885235	−0.442617	+	0.896711i	$0.645950\pi$
$294$	0	0
$295$	−7.37190	−0.429209
$296$	−3.16364	−0.183883
$297$	0	0
$298$	48.1294	2.78806
$299$	−24.2837	−1.40437
$300$	0	0
$301$	−12.3256	−0.710433
$302$	−16.1893	−0.931591
$303$	0	0
$304$	−9.78191	−0.561031
$305$	1.62300	0.0929329
$306$	0	0
$307$	−5.45357	−0.311252	−0.155626	−	0.987816i	$-0.549739\pi$
$307$	−5.45357	−0.311252	−0.155626	+	0.987816i	$0.549739\pi$
$308$	−8.27258	−0.471374
$309$	0	0
$310$	−8.92550	−0.506935
$311$	4.96546	0.281565	0.140783	−	0.990041i	$-0.455038\pi$
$311$	4.96546	0.281565	0.140783	+	0.990041i	$0.455038\pi$
$312$	0	0
$313$	−7.33634	−0.414674	−0.207337	−	0.978270i	$-0.566480\pi$
$313$	−7.33634	−0.414674	−0.207337	+	0.978270i	$0.566480\pi$
$314$	−32.7696	−1.84930
$315$	0	0
$316$	31.5052	1.77231
$317$	20.3699	1.14409	0.572044	−	0.820223i	$-0.306150\pi$
$317$	20.3699	1.14409	0.572044	+	0.820223i	$0.306150\pi$
$318$	0	0
$319$	−8.32621	−0.466178
$320$	−15.3470	−0.857921
$321$	0	0
$322$	−23.8432	−1.32873
$323$	−17.2970	−0.962431
$324$	0	0
$325$	12.4856	0.692576
$326$	49.8391	2.76033
$327$	0	0
$328$	40.8802	2.25723
$329$	4.84018	0.266848
$330$	0	0
$331$	0.240068	0.0131953	0.00659767	−	0.999978i	$-0.497900\pi$
$331$	0.240068	0.0131953	0.00659767	+	0.999978i	$0.497900\pi$
$332$	−41.0752	−2.25430
$333$	0	0
$334$	−23.7720	−1.30075
$335$	7.25957	0.396633
$336$	0	0
$337$	−15.6600	−0.853054	−0.426527	−	0.904475i	$-0.640263\pi$
$337$	−15.6600	−0.853054	−0.426527	+	0.904475i	$0.640263\pi$
$338$	−35.9720	−1.95662
$339$	0	0
$340$	35.7386	1.93820
$341$	−2.27040	−0.122949
$342$	0	0
$343$	−20.1594	−1.08851
$344$	26.0572	1.40491
$345$	0	0
$346$	−27.1387	−1.45899
$347$	−1.44399	−0.0775175	−0.0387588	−	0.999249i	$-0.512340\pi$
$347$	−1.44399	−0.0775175	−0.0387588	+	0.999249i	$0.512340\pi$
$348$	0	0
$349$	17.5351	0.938633	0.469317	−	0.883030i	$-0.344500\pi$
$349$	17.5351	0.938633	0.469317	+	0.883030i	$0.344500\pi$
$350$	12.2591	0.655274
$351$	0	0
$352$	−1.24492	−0.0663544
$353$	−17.6287	−0.938280	−0.469140	−	0.883124i	$-0.655436\pi$
$353$	−17.6287	−0.938280	−0.469140	+	0.883124i	$0.655436\pi$
$354$	0	0
$355$	21.0474	1.11708
$356$	66.4251	3.52052
$357$	0	0
$358$	50.9797	2.69436
$359$	16.9449	0.894316	0.447158	−	0.894455i	$-0.352436\pi$
$359$	16.9449	0.894316	0.447158	+	0.894455i	$0.352436\pi$
$360$	0	0
$361$	−9.77270	−0.514353
$362$	−23.6212	−1.24150
$363$	0	0
$364$	−43.6577	−2.28828
$365$	12.3219	0.644957
$366$	0	0
$367$	−6.61498	−0.345299	−0.172650	−	0.984983i	$-0.555233\pi$
$367$	−6.61498	−0.345299	−0.172650	+	0.984983i	$0.555233\pi$
$368$	14.8178	0.772429
$369$	0	0
$370$	−2.75004	−0.142968
$371$	−9.96873	−0.517551
$372$	0	0
$373$	20.4449	1.05859	0.529297	−	0.848436i	$-0.322456\pi$
$373$	20.4449	1.05859	0.529297	+	0.848436i	$0.322456\pi$
$374$	13.7926	0.713197
$375$	0	0
$376$	−10.2325	−0.527703
$377$	−43.9407	−2.26306
$378$	0	0
$379$	1.24256	0.0638258	0.0319129	−	0.999491i	$-0.489840\pi$
$379$	1.24256	0.0638258	0.0319129	+	0.999491i	$0.489840\pi$
$380$	−19.0652	−0.978024
$381$	0	0
$382$	−40.3125	−2.06257
$383$	12.1855	0.622651	0.311326	−	0.950303i	$-0.399227\pi$
$383$	12.1855	0.622651	0.311326	+	0.950303i	$0.399227\pi$
$384$	0	0
$385$	−3.47197	−0.176948
$386$	55.1558	2.80736
$387$	0	0
$388$	43.5595	2.21140
$389$	−9.87548	−0.500707	−0.250353	−	0.968155i	$-0.580547\pi$
$389$	−9.87548	−0.500707	−0.250353	+	0.968155i	$0.580547\pi$
$390$	0	0
$391$	26.2017	1.32508
$392$	10.9613	0.553627
$393$	0	0
$394$	1.99137	0.100324
$395$	13.2226	0.665302
$396$	0	0
$397$	39.0428	1.95950	0.979750	−	0.200223i	$-0.0641667\pi$
$397$	39.0428	1.95950	0.979750	+	0.200223i	$0.0641667\pi$
$398$	−6.00370	−0.300938
$399$	0	0
$400$	−7.61861	−0.380931
$401$	8.35542	0.417250	0.208625	−	0.977996i	$-0.433101\pi$
$401$	8.35542	0.417250	0.208625	+	0.977996i	$0.433101\pi$
$402$	0	0
$403$	−11.9818	−0.596855
$404$	9.09830	0.452657
$405$	0	0
$406$	−43.1434	−2.14117
$407$	−0.699534	−0.0346746
$408$	0	0
$409$	−13.9383	−0.689203	−0.344601	−	0.938749i	$-0.611986\pi$
$409$	−13.9383	−0.689203	−0.344601	+	0.938749i	$0.611986\pi$
$410$	35.5358	1.75499
$411$	0	0
$412$	57.2667	2.82133
$413$	−9.71665	−0.478125
$414$	0	0
$415$	−17.2391	−0.846234
$416$	−6.56993	−0.322117
$417$	0	0
$418$	−7.35781	−0.359882
$419$	−2.95724	−0.144471	−0.0722353	−	0.997388i	$-0.523013\pi$
$419$	−2.95724	−0.144471	−0.0722353	+	0.997388i	$0.523013\pi$
$420$	0	0
$421$	16.0226	0.780894	0.390447	−	0.920625i	$-0.372321\pi$
$421$	16.0226	0.780894	0.390447	+	0.920625i	$0.372321\pi$
$422$	19.0119	0.925485
$423$	0	0
$424$	21.0747	1.02348
$425$	−13.4717	−0.653474
$426$	0	0
$427$	2.13922	0.103524
$428$	−13.9303	−0.673348
$429$	0	0
$430$	22.6507	1.09231
$431$	13.7952	0.664491	0.332245	−	0.943193i	$-0.392194\pi$
$431$	13.7952	0.664491	0.332245	+	0.943193i	$0.392194\pi$
$432$	0	0
$433$	17.1111	0.822305	0.411153	−	0.911567i	$-0.365126\pi$
$433$	17.1111	0.822305	0.411153	+	0.911567i	$0.365126\pi$
$434$	−11.7644	−0.564709
$435$	0	0
$436$	−34.1259	−1.63434
$437$	−13.9776	−0.668640
$438$	0	0
$439$	−29.4753	−1.40678	−0.703391	−	0.710803i	$-0.748332\pi$
$439$	−29.4753	−1.40678	−0.703391	+	0.710803i	$0.748332\pi$
$440$	7.34002	0.349922
$441$	0	0
$442$	72.7888	3.46221
$443$	29.1223	1.38364	0.691820	−	0.722070i	$-0.256809\pi$
$443$	29.1223	1.38364	0.691820	+	0.722070i	$0.256809\pi$
$444$	0	0
$445$	27.8783	1.32156
$446$	−30.1014	−1.42534
$447$	0	0
$448$	−20.2283	−0.955697
$449$	−27.5692	−1.30107	−0.650536	−	0.759475i	$-0.725456\pi$
$449$	−27.5692	−1.30107	−0.650536	+	0.759475i	$0.725456\pi$
$450$	0	0
$451$	9.03931	0.425644
$452$	−1.41496	−0.0665539
$453$	0	0
$454$	−42.8077	−2.00907
$455$	−18.3229	−0.858993
$456$	0	0
$457$	7.02925	0.328814	0.164407	−	0.986393i	$-0.447429\pi$
$457$	7.02925	0.328814	0.164407	+	0.986393i	$0.447429\pi$
$458$	−19.4413	−0.908431
$459$	0	0
$460$	28.8802	1.34655
$461$	29.4761	1.37284	0.686419	−	0.727206i	$-0.259181\pi$
$461$	29.4761	1.37284	0.686419	+	0.727206i	$0.259181\pi$
$462$	0	0
$463$	−32.9605	−1.53180	−0.765902	−	0.642957i	$-0.777707\pi$
$463$	−32.9605	−1.53180	−0.765902	+	0.642957i	$0.777707\pi$
$464$	26.8123	1.24473
$465$	0	0
$466$	34.7574	1.61011
$467$	−18.7654	−0.868360	−0.434180	−	0.900826i	$-0.642962\pi$
$467$	−18.7654	−0.868360	−0.434180	+	0.900826i	$0.642962\pi$
$468$	0	0
$469$	9.56858	0.441836
$470$	−8.89481	−0.410287
$471$	0	0
$472$	20.5418	0.945512
$473$	5.76169	0.264923
$474$	0	0
$475$	7.18665	0.329746
$476$	47.1058	2.15909
$477$	0	0
$478$	−67.0579	−3.06716
$479$	12.6736	0.579071	0.289535	−	0.957167i	$-0.406499\pi$
$479$	12.6736	0.579071	0.289535	+	0.957167i	$0.406499\pi$
$480$	0	0
$481$	−3.69172	−0.168328
$482$	−60.9988	−2.77842
$483$	0	0
$484$	3.86709	0.175777
$485$	18.2817	0.830130
$486$	0	0
$487$	43.1499	1.95531	0.977654	−	0.210222i	$-0.0674186\pi$
$487$	43.1499	1.95531	0.977654	+	0.210222i	$0.0674186\pi$
$488$	−4.52249	−0.204724
$489$	0	0
$490$	9.52826	0.430443
$491$	14.3238	0.646423	0.323211	−	0.946327i	$-0.395238\pi$
$491$	14.3238	0.646423	0.323211	+	0.946327i	$0.395238\pi$
$492$	0	0
$493$	47.4112	2.13529
$494$	−38.8301	−1.74705
$495$	0	0
$496$	7.31120	0.328282
$497$	27.7419	1.24439
$498$	0	0
$499$	−28.1154	−1.25862	−0.629308	−	0.777156i	$-0.716662\pi$
$499$	−28.1154	−1.25862	−0.629308	+	0.777156i	$0.716662\pi$
$500$	−46.2304	−2.06749
$501$	0	0
$502$	−11.3512	−0.506630
$503$	4.93449	0.220018	0.110009	−	0.993931i	$-0.464912\pi$
$503$	4.93449	0.220018	0.110009	+	0.993931i	$0.464912\pi$
$504$	0	0
$505$	3.81852	0.169922
$506$	11.1457	0.495487
$507$	0	0
$508$	1.07554	0.0477195
$509$	−30.5093	−1.35230	−0.676150	−	0.736764i	$-0.736353\pi$
$509$	−30.5093	−1.35230	−0.676150	+	0.736764i	$0.736353\pi$
$510$	0	0
$511$	16.2411	0.718462
$512$	33.1358	1.46441
$513$	0	0
$514$	−30.2791	−1.33556
$515$	24.0346	1.05909
$516$	0	0
$517$	−2.26259	−0.0995085
$518$	−3.62474	−0.159262
$519$	0	0
$520$	38.7362	1.69869
$521$	30.6188	1.34143	0.670717	−	0.741713i	$-0.265986\pi$
$521$	30.6188	1.34143	0.670717	+	0.741713i	$0.265986\pi$
$522$	0	0
$523$	−27.4211	−1.19904	−0.599521	−	0.800359i	$-0.704642\pi$
$523$	−27.4211	−1.19904	−0.599521	+	0.800359i	$0.704642\pi$
$524$	25.5702	1.11704
$525$	0	0
$526$	16.1310	0.703344
$527$	12.9281	0.563158
$528$	0	0
$529$	−1.82652	−0.0794140
$530$	18.3195	0.795750
$531$	0	0
$532$	−25.1292	−1.08949
$533$	47.7040	2.06629
$534$	0	0
$535$	−5.84650	−0.252766
$536$	−20.2288	−0.873749
$537$	0	0
$538$	−68.4415	−2.95072
$539$	2.42372	0.104397
$540$	0	0
$541$	23.8988	1.02749	0.513744	−	0.857944i	$-0.328258\pi$
$541$	23.8988	1.02749	0.513744	+	0.857944i	$0.328258\pi$
$542$	10.6305	0.456618
$543$	0	0
$544$	7.08883	0.303931
$545$	−14.3225	−0.613509
$546$	0	0
$547$	26.8323	1.14727	0.573634	−	0.819112i	$-0.305533\pi$
$547$	26.8323	1.14727	0.573634	+	0.819112i	$0.305533\pi$
$548$	81.6002	3.48579
$549$	0	0
$550$	−5.73061	−0.244354
$551$	−25.2921	−1.07748
$552$	0	0
$553$	17.4283	0.741125
$554$	44.1949	1.87766
$555$	0	0
$556$	36.3370	1.54103
$557$	−40.7329	−1.72591	−0.862954	−	0.505282i	$-0.831389\pi$
$557$	−40.7329	−1.72591	−0.862954	+	0.505282i	$0.831389\pi$
$558$	0	0
$559$	30.4067	1.28607
$560$	11.1805	0.472463
$561$	0	0
$562$	−3.70037	−0.156091
$563$	−28.7604	−1.21211	−0.606053	−	0.795424i	$-0.707248\pi$
$563$	−28.7604	−1.21211	−0.606053	+	0.795424i	$0.707248\pi$
$564$	0	0
$565$	−0.593851	−0.0249835
$566$	77.9670	3.27720
$567$	0	0
$568$	−58.6486	−2.46084
$569$	−1.10759	−0.0464327	−0.0232163	−	0.999730i	$-0.507391\pi$
$569$	−1.10759	−0.0464327	−0.0232163	+	0.999730i	$0.507391\pi$
$570$	0	0
$571$	−30.4761	−1.27539	−0.637693	−	0.770291i	$-0.720111\pi$
$571$	−30.4761	−1.27539	−0.637693	+	0.770291i	$0.720111\pi$
$572$	20.4082	0.853309
$573$	0	0
$574$	46.8385	1.95500
$575$	−10.8864	−0.453995
$576$	0	0
$577$	21.1970	0.882444	0.441222	−	0.897398i	$-0.354545\pi$
$577$	21.1970	0.882444	0.441222	+	0.897398i	$0.354545\pi$
$578$	−37.3602	−1.55398
$579$	0	0
$580$	52.2578	2.16989
$581$	−22.7223	−0.942679
$582$	0	0
$583$	4.65998	0.192997
$584$	−34.3349	−1.42079
$585$	0	0
$586$	36.7032	1.51619
$587$	11.2509	0.464373	0.232186	−	0.972671i	$-0.425412\pi$
$587$	11.2509	0.464373	0.232186	+	0.972671i	$0.425412\pi$
$588$	0	0
$589$	−6.89667	−0.284172
$590$	17.8563	0.735132
$591$	0	0
$592$	2.25266	0.0925837
$593$	−7.84599	−0.322196	−0.161098	−	0.986938i	$-0.551504\pi$
$593$	−7.84599	−0.322196	−0.161098	+	0.986938i	$0.551504\pi$
$594$	0	0
$595$	19.7701	0.810495
$596$	−76.8393	−3.14746
$597$	0	0
$598$	58.8203	2.40534
$599$	6.40154	0.261560	0.130780	−	0.991411i	$-0.458252\pi$
$599$	6.40154	0.261560	0.130780	+	0.991411i	$0.458252\pi$
$600$	0	0
$601$	−14.2014	−0.579287	−0.289643	−	0.957135i	$-0.593537\pi$
$601$	−14.2014	−0.579287	−0.289643	+	0.957135i	$0.593537\pi$
$602$	29.8551	1.21680
$603$	0	0
$604$	25.8465	1.05168
$605$	1.62300	0.0659845
$606$	0	0
$607$	3.39879	0.137953	0.0689764	−	0.997618i	$-0.478027\pi$
$607$	3.39879	0.137953	0.0689764	+	0.997618i	$0.478027\pi$
$608$	−3.78162	−0.153365
$609$	0	0
$610$	−3.93125	−0.159172
$611$	−11.9406	−0.483064
$612$	0	0
$613$	17.2883	0.698266	0.349133	−	0.937073i	$-0.386476\pi$
$613$	17.2883	0.698266	0.349133	+	0.937073i	$0.386476\pi$
$614$	13.2097	0.533099
$615$	0	0
$616$	9.67462	0.389802
$617$	−2.93400	−0.118118	−0.0590591	−	0.998254i	$-0.518810\pi$
$617$	−2.93400	−0.118118	−0.0590591	+	0.998254i	$0.518810\pi$
$618$	0	0
$619$	−26.0844	−1.04842	−0.524210	−	0.851589i	$-0.675639\pi$
$619$	−26.0844	−1.04842	−0.524210	+	0.851589i	$0.675639\pi$
$620$	14.2497	0.572282
$621$	0	0
$622$	−12.0274	−0.482254
$623$	36.7455	1.47218
$624$	0	0
$625$	−7.57339	−0.302936
$626$	17.7701	0.710238
$627$	0	0
$628$	52.3172	2.08768
$629$	3.98329	0.158824
$630$	0	0
$631$	−1.44915	−0.0576898	−0.0288449	−	0.999584i	$-0.509183\pi$
$631$	−1.44915	−0.0576898	−0.0288449	+	0.999584i	$0.509183\pi$
$632$	−36.8447	−1.46561
$633$	0	0
$634$	−49.3402	−1.95955
$635$	0.451401	0.0179133
$636$	0	0
$637$	12.7909	0.506795
$638$	20.1678	0.798451
$639$	0	0
$640$	33.1325	1.30968
$641$	11.2625	0.444843	0.222421	−	0.974951i	$-0.428604\pi$
$641$	11.2625	0.444843	0.222421	+	0.974951i	$0.428604\pi$
$642$	0	0
$643$	−14.1625	−0.558513	−0.279257	−	0.960216i	$-0.590088\pi$
$643$	−14.1625	−0.558513	−0.279257	+	0.960216i	$0.590088\pi$
$644$	38.0660	1.50001
$645$	0	0
$646$	41.8969	1.64841
$647$	−4.33020	−0.170238	−0.0851188	−	0.996371i	$-0.527127\pi$
$647$	−4.33020	−0.170238	−0.0851188	+	0.996371i	$0.527127\pi$
$648$	0	0
$649$	4.54214	0.178295
$650$	−30.2427	−1.18622
$651$	0	0
$652$	−79.5689	−3.11616
$653$	−13.3283	−0.521575	−0.260788	−	0.965396i	$-0.583982\pi$
$653$	−13.3283	−0.521575	−0.260788	+	0.965396i	$0.583982\pi$
$654$	0	0
$655$	10.7317	0.419323
$656$	−29.1086	−1.13650
$657$	0	0
$658$	−11.7239	−0.457047
$659$	33.7120	1.31323	0.656616	−	0.754225i	$-0.271987\pi$
$659$	33.7120	1.31323	0.656616	+	0.754225i	$0.271987\pi$
$660$	0	0
$661$	39.3781	1.53163	0.765815	−	0.643061i	$-0.222336\pi$
$661$	39.3781	1.53163	0.765815	+	0.643061i	$0.222336\pi$
$662$	−0.581495	−0.0226004
$663$	0	0
$664$	48.0367	1.86419
$665$	−10.5466	−0.408980
$666$	0	0
$667$	38.3127	1.48348
$668$	37.9524	1.46842
$669$	0	0
$670$	−17.5842	−0.679336
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−14.0460	−0.541434	−0.270717	−	0.962659i	$-0.587261\pi$
$673$	−14.0460	−0.541434	−0.270717	+	0.962659i	$0.587261\pi$
$674$	37.9318	1.46108
$675$	0	0
$676$	57.4298	2.20884
$677$	17.7142	0.680812	0.340406	−	0.940279i	$-0.389436\pi$
$677$	17.7142	0.680812	0.340406	+	0.940279i	$0.389436\pi$
$678$	0	0
$679$	24.0965	0.924739
$680$	−41.7956	−1.60279
$681$	0	0
$682$	5.49938	0.210582
$683$	6.38935	0.244482	0.122241	−	0.992500i	$-0.460992\pi$
$683$	6.38935	0.244482	0.122241	+	0.992500i	$0.460992\pi$
$684$	0	0
$685$	34.2473	1.30852
$686$	48.8304	1.86435
$687$	0	0
$688$	−18.5540	−0.707363
$689$	24.5925	0.936901
$690$	0	0
$691$	27.4165	1.04297	0.521487	−	0.853259i	$-0.325377\pi$
$691$	27.4165	1.04297	0.521487	+	0.853259i	$0.325377\pi$
$692$	43.3274	1.64706
$693$	0	0
$694$	3.49765	0.132769
$695$	15.2505	0.578484
$696$	0	0
$697$	−51.4717	−1.94963
$698$	−42.4737	−1.60765
$699$	0	0
$700$	−19.5718	−0.739744
$701$	38.6143	1.45844	0.729222	−	0.684278i	$-0.239882\pi$
$701$	38.6143	1.45844	0.729222	+	0.684278i	$0.239882\pi$
$702$	0	0
$703$	−2.12494	−0.0801435
$704$	9.45591	0.356383
$705$	0	0
$706$	42.7004	1.60705
$707$	5.03305	0.189288
$708$	0	0
$709$	−0.467284	−0.0175492	−0.00877461	−	0.999962i	$-0.502793\pi$
$709$	−0.467284	−0.0175492	−0.00877461	+	0.999962i	$0.502793\pi$
$710$	−50.9813	−1.91329
$711$	0	0
$712$	−77.6828	−2.91129
$713$	10.4472	0.391249
$714$	0	0
$715$	8.56523	0.320321
$716$	−81.3898	−3.04168
$717$	0	0
$718$	−41.0440	−1.53175
$719$	−44.1210	−1.64544	−0.822718	−	0.568450i	$-0.807543\pi$
$719$	−44.1210	−1.64544	−0.822718	+	0.568450i	$0.807543\pi$
$720$	0	0
$721$	31.6792	1.17979
$722$	23.6715	0.880963
$723$	0	0
$724$	37.7117	1.40154
$725$	−19.6986	−0.731589
$726$	0	0
$727$	23.2335	0.861682	0.430841	−	0.902428i	$-0.358217\pi$
$727$	23.2335	0.861682	0.430841	+	0.902428i	$0.358217\pi$
$728$	51.0568	1.89229
$729$	0	0
$730$	−29.8462	−1.10466
$731$	−32.8083	−1.21346
$732$	0	0
$733$	0.978912	0.0361569	0.0180785	−	0.999837i	$-0.494245\pi$
$733$	0.978912	0.0361569	0.0180785	+	0.999837i	$0.494245\pi$
$734$	16.0229	0.591415
$735$	0	0
$736$	5.72845	0.211153
$737$	−4.47292	−0.164762
$738$	0	0
$739$	47.8798	1.76129	0.880643	−	0.473780i	$-0.157111\pi$
$739$	47.8798	1.76129	0.880643	+	0.473780i	$0.157111\pi$
$740$	4.39049	0.161398
$741$	0	0
$742$	24.1464	0.886441
$743$	−31.7292	−1.16403	−0.582016	−	0.813177i	$-0.697736\pi$
$743$	−31.7292	−1.16403	−0.582016	+	0.813177i	$0.697736\pi$
$744$	0	0
$745$	−32.2491	−1.18152
$746$	−49.5217	−1.81312
$747$	0	0
$748$	−22.0200	−0.805133
$749$	−7.70608	−0.281574
$750$	0	0
$751$	44.0450	1.60722	0.803612	−	0.595154i	$-0.202909\pi$
$751$	44.0450	1.60722	0.803612	+	0.595154i	$0.202909\pi$
$752$	7.28605	0.265695
$753$	0	0
$754$	106.433	3.87608
$755$	10.8477	0.394787
$756$	0	0
$757$	−36.4118	−1.32341	−0.661705	−	0.749764i	$-0.730167\pi$
$757$	−36.4118	−1.32341	−0.661705	+	0.749764i	$0.730167\pi$
$758$	−3.00973	−0.109318
$759$	0	0
$760$	22.2964	0.808774
$761$	25.9396	0.940311	0.470155	−	0.882584i	$-0.344198\pi$
$761$	25.9396	0.940311	0.470155	+	0.882584i	$0.344198\pi$
$762$	0	0
$763$	−18.8780	−0.683430
$764$	64.3595	2.32845
$765$	0	0
$766$	−29.5159	−1.06645
$767$	23.9707	0.865530
$768$	0	0
$769$	−21.3909	−0.771377	−0.385689	−	0.922629i	$-0.626036\pi$
$769$	−21.3909	−0.771377	−0.385689	+	0.922629i	$0.626036\pi$
$770$	8.40983	0.303069
$771$	0	0
$772$	−88.0571	−3.16925
$773$	2.21281	0.0795893	0.0397947	−	0.999208i	$-0.487330\pi$
$773$	2.21281	0.0795893	0.0397947	+	0.999208i	$0.487330\pi$
$774$	0	0
$775$	−5.37145	−0.192948
$776$	−50.9419	−1.82871
$777$	0	0
$778$	23.9205	0.857590
$779$	27.4582	0.983793
$780$	0	0
$781$	−12.9682	−0.464039
$782$	−63.4660	−2.26954
$783$	0	0
$784$	−7.80494	−0.278748
$785$	21.9573	0.783690
$786$	0	0
$787$	2.32024	0.0827077	0.0413539	−	0.999145i	$-0.486833\pi$
$787$	2.32024	0.0827077	0.0413539	+	0.999145i	$0.486833\pi$
$788$	−3.17925	−0.113256
$789$	0	0
$790$	−32.0279	−1.13950
$791$	−0.782735	−0.0278308
$792$	0	0
$793$	−5.27740	−0.187406
$794$	−94.5697	−3.35615
$795$	0	0
$796$	9.58501	0.339732
$797$	−30.7167	−1.08804	−0.544021	−	0.839071i	$-0.683099\pi$
$797$	−30.7167	−1.08804	−0.544021	+	0.839071i	$0.683099\pi$
$798$	0	0
$799$	12.8837	0.455791
$800$	−2.94530	−0.104132
$801$	0	0
$802$	−20.2386	−0.714649
$803$	−7.59203	−0.267917
$804$	0	0
$805$	15.9761	0.563085
$806$	29.0224	1.02227
$807$	0	0
$808$	−10.6403	−0.374324
$809$	22.1560	0.778965	0.389483	−	0.921034i	$-0.372654\pi$
$809$	22.1560	0.778965	0.389483	+	0.921034i	$0.372654\pi$
$810$	0	0
$811$	1.43453	0.0503733	0.0251866	−	0.999683i	$-0.491982\pi$
$811$	1.43453	0.0503733	0.0251866	+	0.999683i	$0.491982\pi$
$812$	68.8792	2.41719
$813$	0	0
$814$	1.69442	0.0593893
$815$	−33.3947	−1.16977
$816$	0	0
$817$	17.5020	0.612317
$818$	33.7614	1.18044
$819$	0	0
$820$	−56.7334	−1.98122
$821$	5.11300	0.178445	0.0892225	−	0.996012i	$-0.471562\pi$
$821$	5.11300	0.178445	0.0892225	+	0.996012i	$0.471562\pi$
$822$	0	0
$823$	−53.1611	−1.85308	−0.926539	−	0.376199i	$-0.877231\pi$
$823$	−53.1611	−1.85308	−0.926539	+	0.376199i	$0.877231\pi$
$824$	−66.9723	−2.33309
$825$	0	0
$826$	23.5358	0.818914
$827$	−52.4477	−1.82378	−0.911892	−	0.410430i	$-0.865379\pi$
$827$	−52.4477	−1.82378	−0.911892	+	0.410430i	$0.865379\pi$
$828$	0	0
$829$	−9.24344	−0.321038	−0.160519	−	0.987033i	$-0.551317\pi$
$829$	−9.24344	−0.321038	−0.160519	+	0.987033i	$0.551317\pi$
$830$	41.7567	1.44940
$831$	0	0
$832$	49.9026	1.73006
$833$	−13.8012	−0.478183
$834$	0	0
$835$	15.9285	0.551228
$836$	11.7469	0.406274
$837$	0	0
$838$	7.16305	0.247443
$839$	−15.7224	−0.542796	−0.271398	−	0.962467i	$-0.587486\pi$
$839$	−15.7224	−0.542796	−0.271398	+	0.962467i	$0.587486\pi$
$840$	0	0
$841$	40.3257	1.39054
$842$	−38.8101	−1.33748
$843$	0	0
$844$	−30.3528	−1.04479
$845$	24.1031	0.829170
$846$	0	0
$847$	2.13922	0.0735046
$848$	−15.0062	−0.515315
$849$	0	0
$850$	32.6313	1.11924
$851$	3.21888	0.110342
$852$	0	0
$853$	37.2881	1.27672	0.638360	−	0.769738i	$-0.279613\pi$
$853$	37.2881	1.27672	0.638360	+	0.769738i	$0.279613\pi$
$854$	−5.18165	−0.177312
$855$	0	0
$856$	16.2913	0.556824
$857$	−40.5577	−1.38542	−0.692712	−	0.721214i	$-0.743584\pi$
$857$	−40.5577	−1.38542	−0.692712	+	0.721214i	$0.743584\pi$
$858$	0	0
$859$	21.5469	0.735169	0.367585	−	0.929990i	$-0.380185\pi$
$859$	21.5469	0.735169	0.367585	+	0.929990i	$0.380185\pi$
$860$	−36.1621	−1.23312
$861$	0	0
$862$	−33.4148	−1.13811
$863$	49.4450	1.68313	0.841564	−	0.540157i	$-0.181635\pi$
$863$	49.4450	1.68313	0.841564	+	0.540157i	$0.181635\pi$
$864$	0	0
$865$	18.1843	0.618285
$866$	−41.4466	−1.40841
$867$	0	0
$868$	18.7820	0.637504
$869$	−8.14700	−0.276368
$870$	0	0
$871$	−23.6054	−0.799838
$872$	39.9096	1.35151
$873$	0	0
$874$	33.8567	1.14522
$875$	−25.5740	−0.864560
$876$	0	0
$877$	46.5333	1.57132	0.785659	−	0.618660i	$-0.212324\pi$
$877$	46.5333	1.57132	0.785659	+	0.618660i	$0.212324\pi$
$878$	71.3954	2.40948
$879$	0	0
$880$	−5.22644	−0.176183
$881$	−19.6429	−0.661785	−0.330893	−	0.943668i	$-0.607350\pi$
$881$	−19.6429	−0.661785	−0.330893	+	0.943668i	$0.607350\pi$
$882$	0	0
$883$	16.8620	0.567452	0.283726	−	0.958905i	$-0.408429\pi$
$883$	16.8620	0.567452	0.283726	+	0.958905i	$0.408429\pi$
$884$	−116.209	−3.90852
$885$	0	0
$886$	−70.5402	−2.36984
$887$	−12.4306	−0.417380	−0.208690	−	0.977982i	$-0.566920\pi$
$887$	−12.4306	−0.417380	−0.208690	+	0.977982i	$0.566920\pi$
$888$	0	0
$889$	0.594976	0.0199548
$890$	−67.5271	−2.26351
$891$	0	0
$892$	48.0574	1.60908
$893$	−6.87294	−0.229994
$894$	0	0
$895$	−34.1590	−1.14181
$896$	43.6708	1.45894
$897$	0	0
$898$	66.7784	2.22843
$899$	18.9038	0.630477
$900$	0	0
$901$	−26.5349	−0.884006
$902$	−21.8951	−0.729027
$903$	0	0
$904$	1.65476	0.0550366
$905$	15.8274	0.526122
$906$	0	0
$907$	51.3925	1.70646	0.853230	−	0.521534i	$-0.174640\pi$
$907$	51.3925	1.70646	0.853230	+	0.521534i	$0.174640\pi$
$908$	68.3432	2.26805
$909$	0	0
$910$	44.3820	1.47125
$911$	21.3780	0.708286	0.354143	−	0.935191i	$-0.384773\pi$
$911$	21.3780	0.708286	0.354143	+	0.935191i	$0.384773\pi$
$912$	0	0
$913$	10.6217	0.351528
$914$	−17.0263	−0.563180
$915$	0	0
$916$	31.0383	1.02553
$917$	14.1451	0.467112
$918$	0	0
$919$	−3.46810	−0.114402	−0.0572010	−	0.998363i	$-0.518218\pi$
$919$	−3.46810	−0.114402	−0.0572010	+	0.998363i	$0.518218\pi$
$920$	−33.7748	−1.11352
$921$	0	0
$922$	−71.3973	−2.35134
$923$	−68.4383	−2.25268
$924$	0	0
$925$	−1.65500	−0.0544161
$926$	79.8372	2.62361
$927$	0	0
$928$	10.3654	0.340262
$929$	55.7219	1.82818	0.914088	−	0.405516i	$-0.132908\pi$
$929$	55.7219	1.82818	0.914088	+	0.405516i	$0.132908\pi$
$930$	0	0
$931$	7.36241	0.241293
$932$	−55.4907	−1.81766
$933$	0	0
$934$	45.4538	1.48729
$935$	−9.24172	−0.302237
$936$	0	0
$937$	26.0266	0.850253	0.425126	−	0.905134i	$-0.360230\pi$
$937$	26.0266	0.850253	0.425126	+	0.905134i	$0.360230\pi$
$938$	−23.1771	−0.756759
$939$	0	0
$940$	14.2007	0.463176
$941$	22.0215	0.717879	0.358939	−	0.933361i	$-0.383138\pi$
$941$	22.0215	0.717879	0.358939	+	0.933361i	$0.383138\pi$
$942$	0	0
$943$	−41.5941	−1.35449
$944$	−14.6267	−0.476059
$945$	0	0
$946$	−13.9560	−0.453750
$947$	−57.9768	−1.88399	−0.941996	−	0.335625i	$-0.891053\pi$
$947$	−57.9768	−1.88399	−0.941996	+	0.335625i	$0.891053\pi$
$948$	0	0
$949$	−40.0662	−1.30060
$950$	−17.4076	−0.564776
$951$	0	0
$952$	−55.0893	−1.78546
$953$	39.9007	1.29251	0.646255	−	0.763121i	$-0.276334\pi$
$953$	39.9007	1.29251	0.646255	+	0.763121i	$0.276334\pi$
$954$	0	0
$955$	27.0114	0.874069
$956$	107.059	3.46254
$957$	0	0
$958$	−30.6981	−0.991809
$959$	45.1401	1.45765
$960$	0	0
$961$	−25.8453	−0.833719
$962$	8.94211	0.288305
$963$	0	0
$964$	97.3855	3.13658
$965$	−36.9572	−1.18969
$966$	0	0
$967$	9.61355	0.309151	0.154575	−	0.987981i	$-0.450599\pi$
$967$	9.61355	0.309151	0.154575	+	0.987981i	$0.450599\pi$
$968$	−4.52249	−0.145358
$969$	0	0
$970$	−44.2821	−1.42181
$971$	5.97114	0.191623	0.0958116	−	0.995399i	$-0.469455\pi$
$971$	5.97114	0.191623	0.0958116	+	0.995399i	$0.469455\pi$
$972$	0	0
$973$	20.1011	0.644413
$974$	−104.518	−3.34897
$975$	0	0
$976$	3.22023	0.103077
$977$	−20.1472	−0.644565	−0.322282	−	0.946644i	$-0.604450\pi$
$977$	−20.1472	−0.644565	−0.322282	+	0.946644i	$0.604450\pi$
$978$	0	0
$979$	−17.1770	−0.548979
$980$	−15.2120	−0.485930
$981$	0	0
$982$	−34.6952	−1.10717
$983$	27.6387	0.881537	0.440768	−	0.897621i	$-0.354706\pi$
$983$	27.6387	0.881537	0.440768	+	0.897621i	$0.354706\pi$
$984$	0	0
$985$	−1.33432	−0.0425149
$986$	−114.840	−3.65724
$987$	0	0
$988$	61.9928	1.97226
$989$	−26.5122	−0.843040
$990$	0	0
$991$	51.4791	1.63529	0.817644	−	0.575725i	$-0.195280\pi$
$991$	51.4791	1.63529	0.817644	+	0.575725i	$0.195280\pi$
$992$	2.82646	0.0897402
$993$	0	0
$994$	−67.1966	−2.13135
$995$	4.02279	0.127531
$996$	0	0
$997$	−59.6949	−1.89056	−0.945278	−	0.326265i	$-0.894210\pi$
$997$	−59.6949	−1.89056	−0.945278	+	0.326265i	$0.894210\pi$
$998$	68.1013	2.15571
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.2	yes	25
3.2	odd	2		6039.2.a.m.1.24	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.24	✓	25	3.2	odd	2
6039.2.a.p.1.2	yes	25	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)

\(f(q)\)	\(=\)	\(q-2.42221 q^{2} +3.86709 q^{4} +1.62300 q^{5} +2.13922 q^{7} -4.52249 q^{8} +O(q^{10})\)	\(q-2.42221 q^{2} +3.86709 q^{4} +1.62300 q^{5} +2.13922 q^{7} -4.52249 q^{8} -3.93125 q^{10} -1.00000 q^{11} -5.27740 q^{13} -5.18165 q^{14} +3.22023 q^{16} +5.69421 q^{17} -3.03765 q^{19} +6.27630 q^{20} +2.42221 q^{22} +4.60146 q^{23} -2.36586 q^{25} +12.7830 q^{26} +8.27258 q^{28} +8.32621 q^{29} +2.27040 q^{31} +1.24492 q^{32} -13.7926 q^{34} +3.47197 q^{35} +0.699534 q^{37} +7.35781 q^{38} -7.34002 q^{40} -9.03931 q^{41} -5.76169 q^{43} -3.86709 q^{44} -11.1457 q^{46} +2.26259 q^{47} -2.42372 q^{49} +5.73061 q^{50} -20.4082 q^{52} -4.65998 q^{53} -1.62300 q^{55} -9.67462 q^{56} -20.1678 q^{58} -4.54214 q^{59} +1.00000 q^{61} -5.49938 q^{62} -9.45591 q^{64} -8.56523 q^{65} +4.47292 q^{67} +22.0200 q^{68} -8.40983 q^{70} +12.9682 q^{71} +7.59203 q^{73} -1.69442 q^{74} -11.7469 q^{76} -2.13922 q^{77} +8.14700 q^{79} +5.22644 q^{80} +21.8951 q^{82} -10.6217 q^{83} +9.24172 q^{85} +13.9560 q^{86} +4.52249 q^{88} +17.1770 q^{89} -11.2895 q^{91} +17.7943 q^{92} -5.48046 q^{94} -4.93011 q^{95} +11.2641 q^{97} +5.87076 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8} - 12 q^{10} - 25 q^{11} - 4 q^{13} + 14 q^{14} + 21 q^{16} + 16 q^{17} - 18 q^{19} + 28 q^{20} - 5 q^{22} + 8 q^{23} + 29 q^{25} + 16 q^{26} + 18 q^{28} + 28 q^{29} - 8 q^{31} + 35 q^{32} + 6 q^{34} + 22 q^{35} + 4 q^{37} - 4 q^{38} - 12 q^{40} + 58 q^{41} - 26 q^{43} - 25 q^{44} + 8 q^{46} + 20 q^{47} + 23 q^{49} + 27 q^{50} - 2 q^{52} + 36 q^{53} - 12 q^{55} + 70 q^{56} + 12 q^{58} + 18 q^{59} + 25 q^{61} + 42 q^{62} + 35 q^{64} + 76 q^{65} - 8 q^{67} + 28 q^{68} + 76 q^{70} + 24 q^{71} + 2 q^{73} + 40 q^{74} - 64 q^{76} + 4 q^{77} - 22 q^{79} + 36 q^{80} + 30 q^{82} + 14 q^{83} + 70 q^{86} - 15 q^{88} + 82 q^{89} - 6 q^{91} + 48 q^{92} - 16 q^{94} + 34 q^{95} + 16 q^{97} + 35 q^{98}+O(q^{100}) \) 25 * q + 5 * q^2 + 25 * q^4 + 12 * q^5 - 4 * q^7 + 15 * q^8 - 12 * q^10 - 25 * q^11 - 4 * q^13 + 14 * q^14 + 21 * q^16 + 16 * q^17 - 18 * q^19 + 28 * q^20 - 5 * q^22 + 8 * q^23 + 29 * q^25 + 16 * q^26 + 18 * q^28 + 28 * q^29 - 8 * q^31 + 35 * q^32 + 6 * q^34 + 22 * q^35 + 4 * q^37 - 4 * q^38 - 12 * q^40 + 58 * q^41 - 26 * q^43 - 25 * q^44 + 8 * q^46 + 20 * q^47 + 23 * q^49 + 27 * q^50 - 2 * q^52 + 36 * q^53 - 12 * q^55 + 70 * q^56 + 12 * q^58 + 18 * q^59 + 25 * q^61 + 42 * q^62 + 35 * q^64 + 76 * q^65 - 8 * q^67 + 28 * q^68 + 76 * q^70 + 24 * q^71 + 2 * q^73 + 40 * q^74 - 64 * q^76 + 4 * q^77 - 22 * q^79 + 36 * q^80 + 30 * q^82 + 14 * q^83 + 70 * q^86 - 15 * q^88 + 82 * q^89 - 6 * q^91 + 48 * q^92 - 16 * q^94 + 34 * q^95 + 16 * q^97 + 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more