LMFDB - Embedded newform 6039.2.a.o.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:	$0$
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)

$f(q)$	$=$	$q-1.06662 q^{2} -0.862332 q^{4} +3.84706 q^{5} -1.30714 q^{7} +3.05301 q^{8} +O(q^{10})$	\(q-1.06662 q^{2} -0.862332 q^{4} +3.84706 q^{5} -1.30714 q^{7} +3.05301 q^{8} -4.10333 q^{10} +1.00000 q^{11} +1.88758 q^{13} +1.39422 q^{14} -1.53172 q^{16} -6.08208 q^{17} +5.65651 q^{19} -3.31744 q^{20} -1.06662 q^{22} +3.88616 q^{23} +9.79987 q^{25} -2.01332 q^{26} +1.12719 q^{28} +5.21411 q^{29} +4.43056 q^{31} -4.47226 q^{32} +6.48724 q^{34} -5.02865 q^{35} +0.333734 q^{37} -6.03332 q^{38} +11.7451 q^{40} +1.38003 q^{41} +2.89547 q^{43} -0.862332 q^{44} -4.14504 q^{46} -8.79970 q^{47} -5.29138 q^{49} -10.4527 q^{50} -1.62772 q^{52} -8.65075 q^{53} +3.84706 q^{55} -3.99071 q^{56} -5.56145 q^{58} +8.72137 q^{59} -1.00000 q^{61} -4.72570 q^{62} +7.83362 q^{64} +7.26162 q^{65} -13.6466 q^{67} +5.24478 q^{68} +5.36363 q^{70} +8.66429 q^{71} -3.51520 q^{73} -0.355965 q^{74} -4.87780 q^{76} -1.30714 q^{77} +12.5324 q^{79} -5.89261 q^{80} -1.47196 q^{82} +17.3477 q^{83} -23.3981 q^{85} -3.08836 q^{86} +3.05301 q^{88} +16.4189 q^{89} -2.46733 q^{91} -3.35116 q^{92} +9.38589 q^{94} +21.7609 q^{95} +0.868462 q^{97} +5.64387 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.06662	−0.754211	−0.377105	−	0.926170i	$-0.623081\pi$
$2$	−1.06662	−0.754211	−0.377105	+	0.926170i	$0.623081\pi$
$3$	0	0
$3$	0	0
$4$	−0.862332	−0.431166
$5$	3.84706	1.72046	0.860229	−	0.509908i	$-0.170321\pi$
$5$	3.84706	1.72046	0.860229	+	0.509908i	$0.170321\pi$
$6$	0	0
$7$	−1.30714	−0.494053	−0.247026	−	0.969009i	$-0.579453\pi$
$7$	−1.30714	−0.494053	−0.247026	+	0.969009i	$0.579453\pi$
$8$	3.05301	1.07940
$9$	0	0
$10$	−4.10333	−1.29759
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.88758	0.523519	0.261760	−	0.965133i	$-0.415697\pi$
$13$	1.88758	0.523519	0.261760	+	0.965133i	$0.415697\pi$
$14$	1.39422	0.372620
$15$	0	0
$16$	−1.53172	−0.382929
$17$	−6.08208	−1.47512	−0.737561	−	0.675281i	$-0.764023\pi$
$17$	−6.08208	−1.47512	−0.737561	+	0.675281i	$0.764023\pi$
$18$	0	0
$19$	5.65651	1.29769	0.648847	−	0.760919i	$-0.275252\pi$
$19$	5.65651	1.29769	0.648847	+	0.760919i	$0.275252\pi$
$20$	−3.31744	−0.741803
$21$	0	0
$22$	−1.06662	−0.227403
$23$	3.88616	0.810320	0.405160	−	0.914246i	$-0.367216\pi$
$23$	3.88616	0.810320	0.405160	+	0.914246i	$0.367216\pi$
$24$	0	0
$25$	9.79987	1.95997
$26$	−2.01332	−0.394844
$27$	0	0
$28$	1.12719	0.213019
$29$	5.21411	0.968236	0.484118	−	0.875003i	$-0.339140\pi$
$29$	5.21411	0.968236	0.484118	+	0.875003i	$0.339140\pi$
$30$	0	0
$31$	4.43056	0.795752	0.397876	−	0.917439i	$-0.369747\pi$
$31$	4.43056	0.795752	0.397876	+	0.917439i	$0.369747\pi$
$32$	−4.47226	−0.790591
$33$	0	0
$34$	6.48724	1.11255
$35$	−5.02865	−0.849997
$36$	0	0
$37$	0.333734	0.0548655	0.0274327	−	0.999624i	$-0.491267\pi$
$37$	0.333734	0.0548655	0.0274327	+	0.999624i	$0.491267\pi$
$38$	−6.03332	−0.978734
$39$	0	0
$40$	11.7451	1.85706
$41$	1.38003	0.215525	0.107762	−	0.994177i	$-0.465631\pi$
$41$	1.38003	0.215525	0.107762	+	0.994177i	$0.465631\pi$
$42$	0	0
$43$	2.89547	0.441556	0.220778	−	0.975324i	$-0.429140\pi$
$43$	2.89547	0.441556	0.220778	+	0.975324i	$0.429140\pi$
$44$	−0.862332	−0.130002
$45$	0	0
$46$	−4.14504	−0.611152
$47$	−8.79970	−1.28357	−0.641784	−	0.766886i	$-0.721805\pi$
$47$	−8.79970	−1.28357	−0.641784	+	0.766886i	$0.721805\pi$
$48$	0	0
$49$	−5.29138	−0.755912
$50$	−10.4527	−1.47823
$51$	0	0
$52$	−1.62772	−0.225724
$53$	−8.65075	−1.18827	−0.594136	−	0.804365i	$-0.702506\pi$
$53$	−8.65075	−1.18827	−0.594136	+	0.804365i	$0.702506\pi$
$54$	0	0
$55$	3.84706	0.518737
$56$	−3.99071	−0.533281
$57$	0	0
$58$	−5.56145	−0.730254
$59$	8.72137	1.13543	0.567713	−	0.823226i	$-0.307828\pi$
$59$	8.72137	1.13543	0.567713	+	0.823226i	$0.307828\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−4.72570	−0.600165
$63$	0	0
$64$	7.83362	0.979202
$65$	7.26162	0.900693
$66$	0	0
$67$	−13.6466	−1.66720	−0.833600	−	0.552369i	$-0.813724\pi$
$67$	−13.6466	−1.66720	−0.833600	+	0.552369i	$0.813724\pi$
$68$	5.24478	0.636023
$69$	0	0
$70$	5.36363	0.641077
$71$	8.66429	1.02826	0.514131	−	0.857712i	$-0.328115\pi$
$71$	8.66429	1.02826	0.514131	+	0.857712i	$0.328115\pi$
$72$	0	0
$73$	−3.51520	−0.411423	−0.205712	−	0.978613i	$-0.565951\pi$
$73$	−3.51520	−0.411423	−0.205712	+	0.978613i	$0.565951\pi$
$74$	−0.355965	−0.0413801
$75$	0	0
$76$	−4.87780	−0.559522
$77$	−1.30714	−0.148963
$78$	0	0
$79$	12.5324	1.41000	0.705000	−	0.709207i	$-0.250947\pi$
$79$	12.5324	1.41000	0.705000	+	0.709207i	$0.250947\pi$
$80$	−5.89261	−0.658814
$81$	0	0
$82$	−1.47196	−0.162551
$83$	17.3477	1.90415	0.952076	−	0.305861i	$-0.0989444\pi$
$83$	17.3477	1.90415	0.952076	+	0.305861i	$0.0989444\pi$
$84$	0	0
$85$	−23.3981	−2.53788
$86$	−3.08836	−0.333026
$87$	0	0
$88$	3.05301	0.325452
$89$	16.4189	1.74040	0.870200	−	0.492699i	$-0.163990\pi$
$89$	16.4189	1.74040	0.870200	+	0.492699i	$0.163990\pi$
$90$	0	0
$91$	−2.46733	−0.258646
$92$	−3.35116	−0.349383
$93$	0	0
$94$	9.38589	0.968081
$95$	21.7609	2.23263
$96$	0	0
$97$	0.868462	0.0881790	0.0440895	−	0.999028i	$-0.485961\pi$
$97$	0.868462	0.0881790	0.0440895	+	0.999028i	$0.485961\pi$
$98$	5.64387	0.570117
$99$	0	0
$100$	−8.45074	−0.845074
$101$	11.8709	1.18120	0.590600	−	0.806965i	$-0.298891\pi$
$101$	11.8709	1.18120	0.590600	+	0.806965i	$0.298891\pi$
$102$	0	0
$103$	−3.07827	−0.303311	−0.151655	−	0.988433i	$-0.548460\pi$
$103$	−3.07827	−0.303311	−0.151655	+	0.988433i	$0.548460\pi$
$104$	5.76278	0.565087
$105$	0	0
$106$	9.22702	0.896208
$107$	−14.9590	−1.44614	−0.723071	−	0.690773i	$-0.757270\pi$
$107$	−14.9590	−1.44614	−0.723071	+	0.690773i	$0.757270\pi$
$108$	0	0
$109$	18.4831	1.77036	0.885178	−	0.465253i	$-0.154037\pi$
$109$	18.4831	1.77036	0.885178	+	0.465253i	$0.154037\pi$
$110$	−4.10333	−0.391237
$111$	0	0
$112$	2.00217	0.189187
$113$	−14.8399	−1.39602	−0.698010	−	0.716088i	$-0.745931\pi$
$113$	−14.8399	−1.39602	−0.698010	+	0.716088i	$0.745931\pi$
$114$	0	0
$115$	14.9503	1.39412
$116$	−4.49630	−0.417471
$117$	0	0
$118$	−9.30235	−0.856351
$119$	7.95014	0.728788
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.06662	0.0965668
$123$	0	0
$124$	−3.82062	−0.343101
$125$	18.4654	1.65159
$126$	0	0
$127$	−0.322298	−0.0285993	−0.0142997	−	0.999898i	$-0.504552\pi$
$127$	−0.322298	−0.0285993	−0.0142997	+	0.999898i	$0.504552\pi$
$128$	0.589068	0.0520668
$129$	0	0
$130$	−7.74535	−0.679312
$131$	3.25289	0.284207	0.142103	−	0.989852i	$-0.454613\pi$
$131$	3.25289	0.284207	0.142103	+	0.989852i	$0.454613\pi$
$132$	0	0
$133$	−7.39386	−0.641129
$134$	14.5557	1.25742
$135$	0	0
$136$	−18.5686	−1.59225
$137$	5.25663	0.449104	0.224552	−	0.974462i	$-0.427908\pi$
$137$	5.25663	0.449104	0.224552	+	0.974462i	$0.427908\pi$
$138$	0	0
$139$	4.25725	0.361095	0.180548	−	0.983566i	$-0.442213\pi$
$139$	4.25725	0.361095	0.180548	+	0.983566i	$0.442213\pi$
$140$	4.33637	0.366490
$141$	0	0
$142$	−9.24146	−0.775526
$143$	1.88758	0.157847
$144$	0	0
$145$	20.0590	1.66581
$146$	3.74937	0.310300
$147$	0	0
$148$	−0.287789	−0.0236561
$149$	−9.95436	−0.815493	−0.407747	−	0.913095i	$-0.633685\pi$
$149$	−9.95436	−0.815493	−0.407747	+	0.913095i	$0.633685\pi$
$150$	0	0
$151$	−19.7417	−1.60656	−0.803278	−	0.595605i	$-0.796912\pi$
$151$	−19.7417	−1.60656	−0.803278	+	0.595605i	$0.796912\pi$
$152$	17.2694	1.40073
$153$	0	0
$154$	1.39422	0.112349
$155$	17.0446	1.36906
$156$	0	0
$157$	0.583696	0.0465840	0.0232920	−	0.999729i	$-0.492585\pi$
$157$	0.583696	0.0465840	0.0232920	+	0.999729i	$0.492585\pi$
$158$	−13.3672	−1.06344
$159$	0	0
$160$	−17.2051	−1.36018
$161$	−5.07976	−0.400341
$162$	0	0
$163$	−12.1555	−0.952092	−0.476046	−	0.879420i	$-0.657930\pi$
$163$	−12.1555	−0.952092	−0.476046	+	0.879420i	$0.657930\pi$
$164$	−1.19005	−0.0929270
$165$	0	0
$166$	−18.5033	−1.43613
$167$	−2.52429	−0.195335	−0.0976676	−	0.995219i	$-0.531138\pi$
$167$	−2.52429	−0.195335	−0.0976676	+	0.995219i	$0.531138\pi$
$168$	0	0
$169$	−9.43706	−0.725927
$170$	24.9568	1.91410
$171$	0	0
$172$	−2.49686	−0.190384
$173$	5.90731	0.449125	0.224562	−	0.974460i	$-0.427905\pi$
$173$	5.90731	0.449125	0.224562	+	0.974460i	$0.427905\pi$
$174$	0	0
$175$	−12.8098	−0.968330
$176$	−1.53172	−0.115458
$177$	0	0
$178$	−17.5126	−1.31263
$179$	−16.7922	−1.25511	−0.627554	−	0.778573i	$-0.715944\pi$
$179$	−16.7922	−1.25511	−0.627554	+	0.778573i	$0.715944\pi$
$180$	0	0
$181$	11.5242	0.856586	0.428293	−	0.903640i	$-0.359115\pi$
$181$	11.5242	0.856586	0.428293	+	0.903640i	$0.359115\pi$
$182$	2.63169	0.195074
$183$	0	0
$184$	11.8645	0.874660
$185$	1.28389	0.0943937
$186$	0	0
$187$	−6.08208	−0.444766
$188$	7.58826	0.553431
$189$	0	0
$190$	−23.2106	−1.68387
$191$	−25.9211	−1.87558	−0.937792	−	0.347197i	$-0.887133\pi$
$191$	−25.9211	−1.87558	−0.937792	+	0.347197i	$0.887133\pi$
$192$	0	0
$193$	−22.7774	−1.63955	−0.819777	−	0.572683i	$-0.805903\pi$
$193$	−22.7774	−1.63955	−0.819777	+	0.572683i	$0.805903\pi$
$194$	−0.926315	−0.0665055
$195$	0	0
$196$	4.56293	0.325924
$197$	3.30956	0.235796	0.117898	−	0.993026i	$-0.462384\pi$
$197$	3.30956	0.235796	0.117898	+	0.993026i	$0.462384\pi$
$198$	0	0
$199$	7.19908	0.510329	0.255165	−	0.966898i	$-0.417870\pi$
$199$	7.19908	0.510329	0.255165	+	0.966898i	$0.417870\pi$
$200$	29.9191	2.11560
$201$	0	0
$202$	−12.6617	−0.890873
$203$	−6.81558	−0.478360
$204$	0	0
$205$	5.30906	0.370801
$206$	3.28333	0.228760
$207$	0	0
$208$	−2.89123	−0.200471
$209$	5.65651	0.391269
$210$	0	0
$211$	0.394449	0.0271550	0.0135775	−	0.999908i	$-0.495678\pi$
$211$	0.394449	0.0271550	0.0135775	+	0.999908i	$0.495678\pi$
$212$	7.45982	0.512343
$213$	0	0
$214$	15.9555	1.09070
$215$	11.1391	0.759678
$216$	0	0
$217$	−5.79137	−0.393143
$218$	−19.7143	−1.33522
$219$	0	0
$220$	−3.31744	−0.223662
$221$	−11.4804	−0.772255
$222$	0	0
$223$	−24.5186	−1.64189	−0.820943	−	0.571010i	$-0.806552\pi$
$223$	−24.5186	−1.64189	−0.820943	+	0.571010i	$0.806552\pi$
$224$	5.84587	0.390594
$225$	0	0
$226$	15.8284	1.05289
$227$	23.4419	1.55590	0.777948	−	0.628328i	$-0.216260\pi$
$227$	23.4419	1.55590	0.777948	+	0.628328i	$0.216260\pi$
$228$	0	0
$229$	10.0978	0.667285	0.333642	−	0.942700i	$-0.391722\pi$
$229$	10.0978	0.667285	0.333642	+	0.942700i	$0.391722\pi$
$230$	−15.9462	−1.05146
$231$	0	0
$232$	15.9187	1.04512
$233$	24.1101	1.57950	0.789751	−	0.613427i	$-0.210210\pi$
$233$	24.1101	1.57950	0.789751	+	0.613427i	$0.210210\pi$
$234$	0	0
$235$	−33.8530	−2.20832
$236$	−7.52072	−0.489557
$237$	0	0
$238$	−8.47974	−0.549660
$239$	10.7849	0.697615	0.348807	−	0.937194i	$-0.386587\pi$
$239$	10.7849	0.697615	0.348807	+	0.937194i	$0.386587\pi$
$240$	0	0
$241$	2.90158	0.186907	0.0934535	−	0.995624i	$-0.470209\pi$
$241$	2.90158	0.186907	0.0934535	+	0.995624i	$0.470209\pi$
$242$	−1.06662	−0.0685646
$243$	0	0
$244$	0.862332	0.0552052
$245$	−20.3563	−1.30051
$246$	0	0
$247$	10.6771	0.679368
$248$	13.5265	0.858935
$249$	0	0
$250$	−19.6954	−1.24565
$251$	−27.6225	−1.74352	−0.871759	−	0.489934i	$-0.837021\pi$
$251$	−27.6225	−1.74352	−0.871759	+	0.489934i	$0.837021\pi$
$252$	0	0
$253$	3.88616	0.244321
$254$	0.343768	0.0215699
$255$	0	0
$256$	−16.2955	−1.01847
$257$	29.5438	1.84289	0.921446	−	0.388506i	$-0.127009\pi$
$257$	29.5438	1.84289	0.921446	+	0.388506i	$0.127009\pi$
$258$	0	0
$259$	−0.436237	−0.0271064
$260$	−6.26193	−0.388348
$261$	0	0
$262$	−3.46959	−0.214352
$263$	2.19996	0.135655	0.0678276	−	0.997697i	$-0.478393\pi$
$263$	2.19996	0.135655	0.0678276	+	0.997697i	$0.478393\pi$
$264$	0	0
$265$	−33.2800	−2.04437
$266$	7.88640	0.483546
$267$	0	0
$268$	11.7679	0.718840
$269$	22.2392	1.35595	0.677975	−	0.735085i	$-0.262858\pi$
$269$	22.2392	1.35595	0.677975	+	0.735085i	$0.262858\pi$
$270$	0	0
$271$	12.9322	0.785576	0.392788	−	0.919629i	$-0.371511\pi$
$271$	12.9322	0.785576	0.392788	+	0.919629i	$0.371511\pi$
$272$	9.31604	0.564868
$273$	0	0
$274$	−5.60680	−0.338719
$275$	9.79987	0.590954
$276$	0	0
$277$	7.90696	0.475083	0.237542	−	0.971377i	$-0.423658\pi$
$277$	7.90696	0.475083	0.237542	+	0.971377i	$0.423658\pi$
$278$	−4.54085	−0.272342
$279$	0	0
$280$	−15.3525	−0.917487
$281$	30.8547	1.84064	0.920319	−	0.391169i	$-0.127929\pi$
$281$	30.8547	1.84064	0.920319	+	0.391169i	$0.127929\pi$
$282$	0	0
$283$	22.5319	1.33938	0.669691	−	0.742640i	$-0.266426\pi$
$283$	22.5319	1.33938	0.669691	+	0.742640i	$0.266426\pi$
$284$	−7.47150	−0.443352
$285$	0	0
$286$	−2.01332	−0.119050
$287$	−1.80390	−0.106481
$288$	0	0
$289$	19.9917	1.17598
$290$	−21.3952	−1.25637
$291$	0	0
$292$	3.03127	0.177392
$293$	−8.00446	−0.467626	−0.233813	−	0.972282i	$-0.575120\pi$
$293$	−8.00446	−0.467626	−0.233813	+	0.972282i	$0.575120\pi$
$294$	0	0
$295$	33.5516	1.95345
$296$	1.01889	0.0592218
$297$	0	0
$298$	10.6175	0.615054
$299$	7.33542	0.424218
$300$	0	0
$301$	−3.78479	−0.218152
$302$	21.0568	1.21168
$303$	0	0
$304$	−8.66418	−0.496925
$305$	−3.84706	−0.220282
$306$	0	0
$307$	6.75287	0.385407	0.192703	−	0.981257i	$-0.438274\pi$
$307$	6.75287	0.385407	0.192703	+	0.981257i	$0.438274\pi$
$308$	1.12719	0.0642276
$309$	0	0
$310$	−18.1801	−1.03256
$311$	0.555110	0.0314774	0.0157387	−	0.999876i	$-0.494990\pi$
$311$	0.555110	0.0314774	0.0157387	+	0.999876i	$0.494990\pi$
$312$	0	0
$313$	21.7300	1.22825	0.614126	−	0.789208i	$-0.289509\pi$
$313$	21.7300	1.22825	0.614126	+	0.789208i	$0.289509\pi$
$314$	−0.622579	−0.0351342
$315$	0	0
$316$	−10.8071	−0.607945
$317$	−8.37014	−0.470114	−0.235057	−	0.971982i	$-0.575528\pi$
$317$	−8.37014	−0.470114	−0.235057	+	0.971982i	$0.575528\pi$
$318$	0	0
$319$	5.21411	0.291934
$320$	30.1364	1.68468
$321$	0	0
$322$	5.41815	0.301941
$323$	−34.4034	−1.91426
$324$	0	0
$325$	18.4980	1.02608
$326$	12.9652	0.718078
$327$	0	0
$328$	4.21324	0.232638
$329$	11.5024	0.634150
$330$	0	0
$331$	−19.6955	−1.08256	−0.541281	−	0.840842i	$-0.682061\pi$
$331$	−19.6955	−1.08256	−0.541281	+	0.840842i	$0.682061\pi$
$332$	−14.9594	−0.821006
$333$	0	0
$334$	2.69244	0.147324
$335$	−52.4994	−2.86835
$336$	0	0
$337$	−26.8012	−1.45996	−0.729978	−	0.683471i	$-0.760470\pi$
$337$	−26.8012	−1.45996	−0.729978	+	0.683471i	$0.760470\pi$
$338$	10.0657	0.547502
$339$	0	0
$340$	20.1770	1.09425
$341$	4.43056	0.239928
$342$	0	0
$343$	16.0666	0.867513
$344$	8.83990	0.476616
$345$	0	0
$346$	−6.30083	−0.338735
$347$	−34.6310	−1.85909	−0.929545	−	0.368708i	$-0.879800\pi$
$347$	−34.6310	−1.85909	−0.929545	+	0.368708i	$0.879800\pi$
$348$	0	0
$349$	14.2538	0.762989	0.381494	−	0.924371i	$-0.375410\pi$
$349$	14.2538	0.762989	0.381494	+	0.924371i	$0.375410\pi$
$350$	13.6631	0.730325
$351$	0	0
$352$	−4.47226	−0.238372
$353$	14.6186	0.778068	0.389034	−	0.921223i	$-0.372809\pi$
$353$	14.6186	0.778068	0.389034	+	0.921223i	$0.372809\pi$
$354$	0	0
$355$	33.3320	1.76908
$356$	−14.1585	−0.750401
$357$	0	0
$358$	17.9108	0.946616
$359$	−13.2948	−0.701672	−0.350836	−	0.936437i	$-0.614103\pi$
$359$	−13.2948	−0.701672	−0.350836	+	0.936437i	$0.614103\pi$
$360$	0	0
$361$	12.9962	0.684008
$362$	−12.2919	−0.646046
$363$	0	0
$364$	2.12766	0.111520
$365$	−13.5232	−0.707836
$366$	0	0
$367$	−28.3662	−1.48070	−0.740352	−	0.672219i	$-0.765341\pi$
$367$	−28.3662	−1.48070	−0.740352	+	0.672219i	$0.765341\pi$
$368$	−5.95250	−0.310296
$369$	0	0
$370$	−1.36942	−0.0711927
$371$	11.3078	0.587069
$372$	0	0
$373$	−11.4176	−0.591183	−0.295592	−	0.955314i	$-0.595517\pi$
$373$	−11.4176	−0.591183	−0.295592	+	0.955314i	$0.595517\pi$
$374$	6.48724	0.335447
$375$	0	0
$376$	−26.8655	−1.38548
$377$	9.84203	0.506891
$378$	0	0
$379$	−27.7493	−1.42539	−0.712693	−	0.701476i	$-0.752525\pi$
$379$	−27.7493	−1.42539	−0.712693	+	0.701476i	$0.752525\pi$
$380$	−18.7652	−0.962633
$381$	0	0
$382$	27.6478	1.41459
$383$	31.1416	1.59126	0.795630	−	0.605783i	$-0.207140\pi$
$383$	31.1416	1.59126	0.795630	+	0.605783i	$0.207140\pi$
$384$	0	0
$385$	−5.02865	−0.256284
$386$	24.2947	1.23657
$387$	0	0
$388$	−0.748903	−0.0380198
$389$	22.8498	1.15853	0.579265	−	0.815139i	$-0.303340\pi$
$389$	22.8498	1.15853	0.579265	+	0.815139i	$0.303340\pi$
$390$	0	0
$391$	−23.6359	−1.19532
$392$	−16.1546	−0.815932
$393$	0	0
$394$	−3.53003	−0.177840
$395$	48.2127	2.42585
$396$	0	0
$397$	9.46546	0.475058	0.237529	−	0.971380i	$-0.423663\pi$
$397$	9.46546	0.475058	0.237529	+	0.971380i	$0.423663\pi$
$398$	−7.67865	−0.384896
$399$	0	0
$400$	−15.0106	−0.750532
$401$	26.3660	1.31666	0.658328	−	0.752731i	$-0.271264\pi$
$401$	26.3660	1.31666	0.658328	+	0.752731i	$0.271264\pi$
$402$	0	0
$403$	8.36302	0.416592
$404$	−10.2367	−0.509293
$405$	0	0
$406$	7.26960	0.360784
$407$	0.333734	0.0165426
$408$	0	0
$409$	−28.3260	−1.40063	−0.700314	−	0.713835i	$-0.746957\pi$
$409$	−28.3260	−1.40063	−0.700314	+	0.713835i	$0.746957\pi$
$410$	−5.66273	−0.279662
$411$	0	0
$412$	2.65449	0.130777
$413$	−11.4001	−0.560960
$414$	0	0
$415$	66.7375	3.27601
$416$	−8.44173	−0.413890
$417$	0	0
$418$	−6.03332	−0.295100
$419$	2.35720	0.115157	0.0575783	−	0.998341i	$-0.481662\pi$
$419$	2.35720	0.115157	0.0575783	+	0.998341i	$0.481662\pi$
$420$	0	0
$421$	29.9303	1.45871	0.729357	−	0.684134i	$-0.239820\pi$
$421$	29.9303	1.45871	0.729357	+	0.684134i	$0.239820\pi$
$422$	−0.420726	−0.0204806
$423$	0	0
$424$	−26.4108	−1.28262
$425$	−59.6036	−2.89120
$426$	0	0
$427$	1.30714	0.0632570
$428$	12.8996	0.623528
$429$	0	0
$430$	−11.8811	−0.572957
$431$	21.4507	1.03325	0.516623	−	0.856213i	$-0.327189\pi$
$431$	21.4507	1.03325	0.516623	+	0.856213i	$0.327189\pi$
$432$	0	0
$433$	37.0433	1.78019	0.890094	−	0.455777i	$-0.150639\pi$
$433$	37.0433	1.78019	0.890094	+	0.455777i	$0.150639\pi$
$434$	6.17716	0.296513
$435$	0	0
$436$	−15.9385	−0.763317
$437$	21.9821	1.05155
$438$	0	0
$439$	−9.66718	−0.461389	−0.230695	−	0.973026i	$-0.574100\pi$
$439$	−9.66718	−0.461389	−0.230695	+	0.973026i	$0.574100\pi$
$440$	11.7451	0.559926
$441$	0	0
$442$	12.2452	0.582443
$443$	−14.7946	−0.702911	−0.351455	−	0.936205i	$-0.614313\pi$
$443$	−14.7946	−0.702911	−0.351455	+	0.936205i	$0.614313\pi$
$444$	0	0
$445$	63.1645	2.99428
$446$	26.1519	1.23833
$447$	0	0
$448$	−10.2396	−0.483777
$449$	−2.74268	−0.129435	−0.0647176	−	0.997904i	$-0.520615\pi$
$449$	−2.74268	−0.129435	−0.0647176	+	0.997904i	$0.520615\pi$
$450$	0	0
$451$	1.38003	0.0649831
$452$	12.7969	0.601916
$453$	0	0
$454$	−25.0035	−1.17347
$455$	−9.49196	−0.444990
$456$	0	0
$457$	−4.97569	−0.232753	−0.116376	−	0.993205i	$-0.537128\pi$
$457$	−4.97569	−0.232753	−0.116376	+	0.993205i	$0.537128\pi$
$458$	−10.7705	−0.503273
$459$	0	0
$460$	−12.8921	−0.601098
$461$	14.8152	0.690012	0.345006	−	0.938600i	$-0.387877\pi$
$461$	14.8152	0.690012	0.345006	+	0.938600i	$0.387877\pi$
$462$	0	0
$463$	−12.2719	−0.570325	−0.285163	−	0.958479i	$-0.592048\pi$
$463$	−12.2719	−0.570325	−0.285163	+	0.958479i	$0.592048\pi$
$464$	−7.98655	−0.370766
$465$	0	0
$466$	−25.7161	−1.19128
$467$	37.5019	1.73538	0.867691	−	0.497103i	$-0.165603\pi$
$467$	37.5019	1.73538	0.867691	+	0.497103i	$0.165603\pi$
$468$	0	0
$469$	17.8381	0.823685
$470$	36.1081	1.66554
$471$	0	0
$472$	26.6264	1.22558
$473$	2.89547	0.133134
$474$	0	0
$475$	55.4331	2.54344
$476$	−6.85566	−0.314229
$477$	0	0
$478$	−11.5033	−0.526148
$479$	−18.6945	−0.854173	−0.427087	−	0.904211i	$-0.640460\pi$
$479$	−18.6945	−0.854173	−0.427087	+	0.904211i	$0.640460\pi$
$480$	0	0
$481$	0.629948	0.0287231
$482$	−3.09487	−0.140967
$483$	0	0
$484$	−0.862332	−0.0391969
$485$	3.34103	0.151708
$486$	0	0
$487$	12.9351	0.586146	0.293073	−	0.956090i	$-0.405322\pi$
$487$	12.9351	0.586146	0.293073	+	0.956090i	$0.405322\pi$
$488$	−3.05301	−0.138203
$489$	0	0
$490$	21.7123	0.980862
$491$	−11.5844	−0.522796	−0.261398	−	0.965231i	$-0.584183\pi$
$491$	−11.5844	−0.522796	−0.261398	+	0.965231i	$0.584183\pi$
$492$	0	0
$493$	−31.7127	−1.42827
$494$	−11.3884	−0.512386
$495$	0	0
$496$	−6.78637	−0.304717
$497$	−11.3254	−0.508016
$498$	0	0
$499$	39.5938	1.77246	0.886231	−	0.463244i	$-0.153315\pi$
$499$	39.5938	1.77246	0.886231	+	0.463244i	$0.153315\pi$
$500$	−15.9233	−0.712111
$501$	0	0
$502$	29.4626	1.31498
$503$	8.64929	0.385653	0.192826	−	0.981233i	$-0.438235\pi$
$503$	8.64929	0.385653	0.192826	+	0.981233i	$0.438235\pi$
$504$	0	0
$505$	45.6681	2.03220
$506$	−4.14504	−0.184269
$507$	0	0
$508$	0.277928	0.0123311
$509$	−33.3433	−1.47792	−0.738959	−	0.673751i	$-0.764682\pi$
$509$	−33.3433	−1.47792	−0.738959	+	0.673751i	$0.764682\pi$
$510$	0	0
$511$	4.59486	0.203265
$512$	16.2029	0.716075
$513$	0	0
$514$	−31.5119	−1.38993
$515$	−11.8423	−0.521833
$516$	0	0
$517$	−8.79970	−0.387010
$518$	0.465297	0.0204440
$519$	0	0
$520$	22.1698	0.972209
$521$	9.91783	0.434508	0.217254	−	0.976115i	$-0.430290\pi$
$521$	9.91783	0.434508	0.217254	+	0.976115i	$0.430290\pi$
$522$	0	0
$523$	−29.4020	−1.28566	−0.642830	−	0.766009i	$-0.722240\pi$
$523$	−29.4020	−1.28566	−0.642830	+	0.766009i	$0.722240\pi$
$524$	−2.80508	−0.122540
$525$	0	0
$526$	−2.34651	−0.102313
$527$	−26.9470	−1.17383
$528$	0	0
$529$	−7.89776	−0.343381
$530$	35.4969	1.54189
$531$	0	0
$532$	6.37597	0.276433
$533$	2.60491	0.112831
$534$	0	0
$535$	−57.5482	−2.48803
$536$	−41.6632	−1.79958
$537$	0	0
$538$	−23.7207	−1.02267
$539$	−5.29138	−0.227916
$540$	0	0
$541$	−0.843378	−0.0362597	−0.0181298	−	0.999836i	$-0.505771\pi$
$541$	−0.843378	−0.0362597	−0.0181298	+	0.999836i	$0.505771\pi$
$542$	−13.7937	−0.592490
$543$	0	0
$544$	27.2007	1.16622
$545$	71.1054	3.04582
$546$	0	0
$547$	10.9337	0.467491	0.233745	−	0.972298i	$-0.424902\pi$
$547$	10.9337	0.467491	0.233745	+	0.972298i	$0.424902\pi$
$548$	−4.53296	−0.193639
$549$	0	0
$550$	−10.4527	−0.445704
$551$	29.4937	1.25647
$552$	0	0
$553$	−16.3816	−0.696615
$554$	−8.43368	−0.358313
$555$	0	0
$556$	−3.67116	−0.155692
$557$	10.2205	0.433058	0.216529	−	0.976276i	$-0.430526\pi$
$557$	10.2205	0.433058	0.216529	+	0.976276i	$0.430526\pi$
$558$	0	0
$559$	5.46543	0.231163
$560$	7.70247	0.325489
$561$	0	0
$562$	−32.9101	−1.38823
$563$	44.3040	1.86719	0.933595	−	0.358331i	$-0.116654\pi$
$563$	44.3040	1.86719	0.933595	+	0.358331i	$0.116654\pi$
$564$	0	0
$565$	−57.0899	−2.40179
$566$	−24.0329	−1.01018
$567$	0	0
$568$	26.4521	1.10991
$569$	17.1931	0.720773	0.360386	−	0.932803i	$-0.382645\pi$
$569$	17.1931	0.720773	0.360386	+	0.932803i	$0.382645\pi$
$570$	0	0
$571$	−33.5623	−1.40454	−0.702269	−	0.711912i	$-0.747829\pi$
$571$	−33.5623	−1.40454	−0.702269	+	0.711912i	$0.747829\pi$
$572$	−1.62772	−0.0680583
$573$	0	0
$574$	1.92406	0.0803088
$575$	38.0838	1.58821
$576$	0	0
$577$	−18.0110	−0.749808	−0.374904	−	0.927064i	$-0.622324\pi$
$577$	−18.0110	−0.749808	−0.374904	+	0.927064i	$0.622324\pi$
$578$	−21.3235	−0.886940
$579$	0	0
$580$	−17.2975	−0.718241
$581$	−22.6758	−0.940752
$582$	0	0
$583$	−8.65075	−0.358278
$584$	−10.7319	−0.444091
$585$	0	0
$586$	8.53768	0.352688
$587$	−2.32228	−0.0958506	−0.0479253	−	0.998851i	$-0.515261\pi$
$587$	−2.32228	−0.0958506	−0.0479253	+	0.998851i	$0.515261\pi$
$588$	0	0
$589$	25.0615	1.03264
$590$	−35.7867	−1.47331
$591$	0	0
$592$	−0.511186	−0.0210096
$593$	31.6520	1.29979	0.649896	−	0.760023i	$-0.274812\pi$
$593$	31.6520	1.29979	0.649896	+	0.760023i	$0.274812\pi$
$594$	0	0
$595$	30.5847	1.25385
$596$	8.58397	0.351613
$597$	0	0
$598$	−7.82407	−0.319950
$599$	30.3578	1.24038	0.620192	−	0.784450i	$-0.287055\pi$
$599$	30.3578	1.24038	0.620192	+	0.784450i	$0.287055\pi$
$600$	0	0
$601$	30.9164	1.26111	0.630553	−	0.776146i	$-0.282828\pi$
$601$	30.9164	1.26111	0.630553	+	0.776146i	$0.282828\pi$
$602$	4.03692	0.164532
$603$	0	0
$604$	17.0239	0.692692
$605$	3.84706	0.156405
$606$	0	0
$607$	9.78343	0.397097	0.198549	−	0.980091i	$-0.436377\pi$
$607$	9.78343	0.397097	0.198549	+	0.980091i	$0.436377\pi$
$608$	−25.2974	−1.02595
$609$	0	0
$610$	4.10333	0.166139
$611$	−16.6101	−0.671973
$612$	0	0
$613$	27.8982	1.12680	0.563399	−	0.826185i	$-0.309493\pi$
$613$	27.8982	1.12680	0.563399	+	0.826185i	$0.309493\pi$
$614$	−7.20272	−0.290678
$615$	0	0
$616$	−3.99071	−0.160790
$617$	−13.7708	−0.554392	−0.277196	−	0.960813i	$-0.589405\pi$
$617$	−13.7708	−0.554392	−0.277196	+	0.960813i	$0.589405\pi$
$618$	0	0
$619$	6.12746	0.246283	0.123142	−	0.992389i	$-0.460703\pi$
$619$	6.12746	0.246283	0.123142	+	0.992389i	$0.460703\pi$
$620$	−14.6981	−0.590291
$621$	0	0
$622$	−0.592089	−0.0237406
$623$	−21.4618	−0.859849
$624$	0	0
$625$	22.0381	0.881523
$626$	−23.1775	−0.926361
$627$	0	0
$628$	−0.503340	−0.0200855
$629$	−2.02980	−0.0809333
$630$	0	0
$631$	24.4768	0.974404	0.487202	−	0.873289i	$-0.338017\pi$
$631$	24.4768	0.974404	0.487202	+	0.873289i	$0.338017\pi$
$632$	38.2614	1.52196
$633$	0	0
$634$	8.92772	0.354565
$635$	−1.23990	−0.0492039
$636$	0	0
$637$	−9.98789	−0.395735
$638$	−5.56145	−0.220180
$639$	0	0
$640$	2.26618	0.0895787
$641$	−42.8010	−1.69054	−0.845269	−	0.534342i	$-0.820560\pi$
$641$	−42.8010	−1.69054	−0.845269	+	0.534342i	$0.820560\pi$
$642$	0	0
$643$	−5.18582	−0.204509	−0.102254	−	0.994758i	$-0.532606\pi$
$643$	−5.18582	−0.204509	−0.102254	+	0.994758i	$0.532606\pi$
$644$	4.38044	0.172613
$645$	0	0
$646$	36.6952	1.44375
$647$	3.40738	0.133958	0.0669790	−	0.997754i	$-0.478664\pi$
$647$	3.40738	0.133958	0.0669790	+	0.997754i	$0.478664\pi$
$648$	0	0
$649$	8.72137	0.342344
$650$	−19.7302	−0.773884
$651$	0	0
$652$	10.4821	0.410510
$653$	17.2888	0.676564	0.338282	−	0.941045i	$-0.390154\pi$
$653$	17.2888	0.676564	0.338282	+	0.941045i	$0.390154\pi$
$654$	0	0
$655$	12.5141	0.488965
$656$	−2.11382	−0.0825307
$657$	0	0
$658$	−12.2687	−0.478283
$659$	24.2273	0.943762	0.471881	−	0.881662i	$-0.343575\pi$
$659$	24.2273	0.943762	0.471881	+	0.881662i	$0.343575\pi$
$660$	0	0
$661$	−6.67103	−0.259473	−0.129737	−	0.991548i	$-0.541413\pi$
$661$	−6.67103	−0.259473	−0.129737	+	0.991548i	$0.541413\pi$
$662$	21.0075	0.816480
$663$	0	0
$664$	52.9625	2.05534
$665$	−28.4446	−1.10304
$666$	0	0
$667$	20.2629	0.784582
$668$	2.17677	0.0842219
$669$	0	0
$670$	55.9966	2.16334
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−21.3202	−0.821833	−0.410917	−	0.911673i	$-0.634791\pi$
$673$	−21.3202	−0.821833	−0.410917	+	0.911673i	$0.634791\pi$
$674$	28.5866	1.10111
$675$	0	0
$676$	8.13788	0.312995
$677$	32.3929	1.24496	0.622480	−	0.782636i	$-0.286125\pi$
$677$	32.3929	1.24496	0.622480	+	0.782636i	$0.286125\pi$
$678$	0	0
$679$	−1.13520	−0.0435651
$680$	−71.4347	−2.73939
$681$	0	0
$682$	−4.72570	−0.180956
$683$	11.0222	0.421753	0.210876	−	0.977513i	$-0.432368\pi$
$683$	11.0222	0.421753	0.210876	+	0.977513i	$0.432368\pi$
$684$	0	0
$685$	20.2226	0.772664
$686$	−17.1368	−0.654288
$687$	0	0
$688$	−4.43505	−0.169085
$689$	−16.3290	−0.622084
$690$	0	0
$691$	7.00689	0.266555	0.133277	−	0.991079i	$-0.457450\pi$
$691$	7.00689	0.266555	0.133277	+	0.991079i	$0.457450\pi$
$692$	−5.09407	−0.193647
$693$	0	0
$694$	36.9380	1.40215
$695$	16.3779	0.621249
$696$	0	0
$697$	−8.39347	−0.317925
$698$	−15.2033	−0.575454
$699$	0	0
$700$	11.0463	0.417511
$701$	24.9199	0.941213	0.470607	−	0.882343i	$-0.344035\pi$
$701$	24.9199	0.941213	0.470607	+	0.882343i	$0.344035\pi$
$702$	0	0
$703$	1.88777	0.0711986
$704$	7.83362	0.295240
$705$	0	0
$706$	−15.5924	−0.586827
$707$	−15.5169	−0.583575
$708$	0	0
$709$	12.5387	0.470900	0.235450	−	0.971887i	$-0.424344\pi$
$709$	12.5387	0.470900	0.235450	+	0.971887i	$0.424344\pi$
$710$	−35.5524	−1.33426
$711$	0	0
$712$	50.1270	1.87859
$713$	17.2179	0.644814
$714$	0	0
$715$	7.26162	0.271569
$716$	14.4805	0.541160
$717$	0	0
$718$	14.1804	0.529209
$719$	22.4892	0.838704	0.419352	−	0.907824i	$-0.362257\pi$
$719$	22.4892	0.838704	0.419352	+	0.907824i	$0.362257\pi$
$720$	0	0
$721$	4.02373	0.149852
$722$	−13.8619	−0.515886
$723$	0	0
$724$	−9.93768	−0.369331
$725$	51.0976	1.89772
$726$	0	0
$727$	−4.64876	−0.172413	−0.0862064	−	0.996277i	$-0.527474\pi$
$727$	−4.64876	−0.172413	−0.0862064	+	0.996277i	$0.527474\pi$
$728$	−7.53277	−0.279183
$729$	0	0
$730$	14.4240	0.533857
$731$	−17.6105	−0.651349
$732$	0	0
$733$	23.6684	0.874211	0.437106	−	0.899410i	$-0.356004\pi$
$733$	23.6684	0.874211	0.437106	+	0.899410i	$0.356004\pi$
$734$	30.2558	1.11676
$735$	0	0
$736$	−17.3799	−0.640632
$737$	−13.6466	−0.502680
$738$	0	0
$739$	−3.97656	−0.146280	−0.0731400	−	0.997322i	$-0.523302\pi$
$739$	−3.97656	−0.146280	−0.0731400	+	0.997322i	$0.523302\pi$
$740$	−1.10714	−0.0406994
$741$	0	0
$742$	−12.0610	−0.442774
$743$	−15.6103	−0.572686	−0.286343	−	0.958127i	$-0.592440\pi$
$743$	−15.6103	−0.572686	−0.286343	+	0.958127i	$0.592440\pi$
$744$	0	0
$745$	−38.2950	−1.40302
$746$	12.1782	0.445877
$747$	0	0
$748$	5.24478	0.191768
$749$	19.5535	0.714471
$750$	0	0
$751$	−26.6330	−0.971853	−0.485927	−	0.874000i	$-0.661518\pi$
$751$	−26.6330	−0.971853	−0.485927	+	0.874000i	$0.661518\pi$
$752$	13.4787	0.491516
$753$	0	0
$754$	−10.4977	−0.382302
$755$	−75.9474	−2.76401
$756$	0	0
$757$	−11.8154	−0.429437	−0.214718	−	0.976676i	$-0.568883\pi$
$757$	−11.8154	−0.429437	−0.214718	+	0.976676i	$0.568883\pi$
$758$	29.5978	1.07504
$759$	0	0
$760$	66.4363	2.40990
$761$	49.9914	1.81219	0.906094	−	0.423077i	$-0.139050\pi$
$761$	49.9914	1.81219	0.906094	+	0.423077i	$0.139050\pi$
$762$	0	0
$763$	−24.1600	−0.874649
$764$	22.3526	0.808689
$765$	0	0
$766$	−33.2161	−1.20015
$767$	16.4623	0.594418
$768$	0	0
$769$	−4.88686	−0.176225	−0.0881124	−	0.996111i	$-0.528083\pi$
$769$	−4.88686	−0.176225	−0.0881124	+	0.996111i	$0.528083\pi$
$770$	5.36363	0.193292
$771$	0	0
$772$	19.6417	0.706920
$773$	−38.7693	−1.39444	−0.697218	−	0.716859i	$-0.745579\pi$
$773$	−38.7693	−1.39444	−0.697218	+	0.716859i	$0.745579\pi$
$774$	0	0
$775$	43.4189	1.55965
$776$	2.65142	0.0951805
$777$	0	0
$778$	−24.3719	−0.873775
$779$	7.80617	0.279685
$780$	0	0
$781$	8.66429	0.310033
$782$	25.2105	0.901524
$783$	0	0
$784$	8.10491	0.289461
$785$	2.24551	0.0801458
$786$	0	0
$787$	10.9007	0.388568	0.194284	−	0.980945i	$-0.437762\pi$
$787$	10.9007	0.388568	0.194284	+	0.980945i	$0.437762\pi$
$788$	−2.85394	−0.101667
$789$	0	0
$790$	−51.4244	−1.82960
$791$	19.3978	0.689707
$792$	0	0
$793$	−1.88758	−0.0670298
$794$	−10.0960	−0.358294
$795$	0	0
$796$	−6.20800	−0.220037
$797$	−31.0261	−1.09900	−0.549501	−	0.835493i	$-0.685182\pi$
$797$	−31.0261	−1.09900	−0.549501	+	0.835493i	$0.685182\pi$
$798$	0	0
$799$	53.5205	1.89342
$800$	−43.8276	−1.54954
$801$	0	0
$802$	−28.1224	−0.993036
$803$	−3.51520	−0.124049
$804$	0	0
$805$	−19.5421	−0.688770
$806$	−8.92012	−0.314198
$807$	0	0
$808$	36.2419	1.27499
$809$	−21.9510	−0.771755	−0.385878	−	0.922550i	$-0.626101\pi$
$809$	−21.9510	−0.771755	−0.385878	+	0.922550i	$0.626101\pi$
$810$	0	0
$811$	−47.8404	−1.67990	−0.839952	−	0.542661i	$-0.817417\pi$
$811$	−47.8404	−1.67990	−0.839952	+	0.542661i	$0.817417\pi$
$812$	5.87730	0.206253
$813$	0	0
$814$	−0.355965	−0.0124766
$815$	−46.7629	−1.63803
$816$	0	0
$817$	16.3783	0.573004
$818$	30.2129	1.05637
$819$	0	0
$820$	−4.57818	−0.159877
$821$	44.3951	1.54940	0.774700	−	0.632330i	$-0.217901\pi$
$821$	44.3951	1.54940	0.774700	+	0.632330i	$0.217901\pi$
$822$	0	0
$823$	−33.4474	−1.16590	−0.582951	−	0.812507i	$-0.698102\pi$
$823$	−33.4474	−1.16590	−0.582951	+	0.812507i	$0.698102\pi$
$824$	−9.39798	−0.327394
$825$	0	0
$826$	12.1595	0.423082
$827$	7.12993	0.247932	0.123966	−	0.992286i	$-0.460439\pi$
$827$	7.12993	0.247932	0.123966	+	0.992286i	$0.460439\pi$
$828$	0	0
$829$	19.3702	0.672755	0.336378	−	0.941727i	$-0.390798\pi$
$829$	19.3702	0.672755	0.336378	+	0.941727i	$0.390798\pi$
$830$	−71.1832	−2.47080
$831$	0	0
$832$	14.7865	0.512631
$833$	32.1826	1.11506
$834$	0	0
$835$	−9.71108	−0.336066
$836$	−4.87780	−0.168702
$837$	0	0
$838$	−2.51422	−0.0868523
$839$	−9.21465	−0.318125	−0.159063	−	0.987269i	$-0.550847\pi$
$839$	−9.21465	−0.318125	−0.159063	+	0.987269i	$0.550847\pi$
$840$	0	0
$841$	−1.81303	−0.0625182
$842$	−31.9241	−1.10018
$843$	0	0
$844$	−0.340146	−0.0117083
$845$	−36.3049	−1.24893
$846$	0	0
$847$	−1.30714	−0.0449139
$848$	13.2505	0.455024
$849$	0	0
$850$	63.5741	2.18057
$851$	1.29694	0.0444586
$852$	0	0
$853$	20.4368	0.699741	0.349871	−	0.936798i	$-0.386226\pi$
$853$	20.4368	0.699741	0.349871	+	0.936798i	$0.386226\pi$
$854$	−1.39422	−0.0477091
$855$	0	0
$856$	−45.6700	−1.56097
$857$	1.77536	0.0606453	0.0303226	−	0.999540i	$-0.490347\pi$
$857$	1.77536	0.0606453	0.0303226	+	0.999540i	$0.490347\pi$
$858$	0	0
$859$	−18.9126	−0.645290	−0.322645	−	0.946520i	$-0.604572\pi$
$859$	−18.9126	−0.645290	−0.322645	+	0.946520i	$0.604572\pi$
$860$	−9.60558	−0.327547
$861$	0	0
$862$	−22.8797	−0.779284
$863$	−13.1643	−0.448120	−0.224060	−	0.974575i	$-0.571931\pi$
$863$	−13.1643	−0.448120	−0.224060	+	0.974575i	$0.571931\pi$
$864$	0	0
$865$	22.7258	0.772700
$866$	−39.5109	−1.34264
$867$	0	0
$868$	4.99408	0.169510
$869$	12.5324	0.425131
$870$	0	0
$871$	−25.7590	−0.872811
$872$	56.4289	1.91092
$873$	0	0
$874$	−23.4465	−0.793088
$875$	−24.1368	−0.815974
$876$	0	0
$877$	−14.3372	−0.484132	−0.242066	−	0.970260i	$-0.577825\pi$
$877$	−14.3372	−0.484132	−0.242066	+	0.970260i	$0.577825\pi$
$878$	10.3112	0.347985
$879$	0	0
$880$	−5.89261	−0.198640
$881$	−46.8047	−1.57689	−0.788446	−	0.615104i	$-0.789114\pi$
$881$	−46.8047	−1.57689	−0.788446	+	0.615104i	$0.789114\pi$
$882$	0	0
$883$	−9.75219	−0.328187	−0.164094	−	0.986445i	$-0.552470\pi$
$883$	−9.75219	−0.328187	−0.164094	+	0.986445i	$0.552470\pi$
$884$	9.89992	0.332970
$885$	0	0
$886$	15.7801	0.530143
$887$	4.10461	0.137819	0.0689096	−	0.997623i	$-0.478048\pi$
$887$	4.10461	0.137819	0.0689096	+	0.997623i	$0.478048\pi$
$888$	0	0
$889$	0.421289	0.0141296
$890$	−67.3722	−2.25832
$891$	0	0
$892$	21.1432	0.707926
$893$	−49.7756	−1.66568
$894$	0	0
$895$	−64.6006	−2.15936
$896$	−0.769995	−0.0257237
$897$	0	0
$898$	2.92539	0.0976215
$899$	23.1014	0.770476
$900$	0	0
$901$	52.6146	1.75285
$902$	−1.47196	−0.0490110
$903$	0	0
$904$	−45.3063	−1.50686
$905$	44.3342	1.47372
$906$	0	0
$907$	13.5896	0.451234	0.225617	−	0.974216i	$-0.427560\pi$
$907$	13.5896	0.451234	0.225617	+	0.974216i	$0.427560\pi$
$908$	−20.2148	−0.670850
$909$	0	0
$910$	10.1243	0.335616
$911$	−9.74485	−0.322861	−0.161431	−	0.986884i	$-0.551611\pi$
$911$	−9.74485	−0.322861	−0.161431	+	0.986884i	$0.551611\pi$
$912$	0	0
$913$	17.3477	0.574123
$914$	5.30714	0.175545
$915$	0	0
$916$	−8.70770	−0.287711
$917$	−4.25199	−0.140413
$918$	0	0
$919$	10.3819	0.342467	0.171233	−	0.985230i	$-0.445225\pi$
$919$	10.3819	0.342467	0.171233	+	0.985230i	$0.445225\pi$
$920$	45.6433	1.50482
$921$	0	0
$922$	−15.8021	−0.520415
$923$	16.3545	0.538315
$924$	0	0
$925$	3.27054	0.107535
$926$	13.0894	0.430146
$927$	0	0
$928$	−23.3189	−0.765479
$929$	−35.2885	−1.15778	−0.578889	−	0.815406i	$-0.696514\pi$
$929$	−35.2885	−1.15778	−0.578889	+	0.815406i	$0.696514\pi$
$930$	0	0
$931$	−29.9308	−0.980942
$932$	−20.7909	−0.681028
$933$	0	0
$934$	−40.0001	−1.30884
$935$	−23.3981	−0.765201
$936$	0	0
$937$	−20.7523	−0.677948	−0.338974	−	0.940796i	$-0.610080\pi$
$937$	−20.7523	−0.677948	−0.338974	+	0.940796i	$0.610080\pi$
$938$	−19.0263	−0.621232
$939$	0	0
$940$	29.1925	0.952154
$941$	58.1983	1.89721	0.948605	−	0.316462i	$-0.102495\pi$
$941$	58.1983	1.89721	0.948605	+	0.316462i	$0.102495\pi$
$942$	0	0
$943$	5.36302	0.174644
$944$	−13.3587	−0.434788
$945$	0	0
$946$	−3.08836	−0.100411
$947$	23.9486	0.778226	0.389113	−	0.921190i	$-0.372781\pi$
$947$	23.9486	0.778226	0.389113	+	0.921190i	$0.372781\pi$
$948$	0	0
$949$	−6.63521	−0.215388
$950$	−59.1258	−1.91829
$951$	0	0
$952$	24.2718	0.786655
$953$	11.7283	0.379918	0.189959	−	0.981792i	$-0.439165\pi$
$953$	11.7283	0.379918	0.189959	+	0.981792i	$0.439165\pi$
$954$	0	0
$955$	−99.7200	−3.22686
$956$	−9.30014	−0.300788
$957$	0	0
$958$	19.9398	0.644227
$959$	−6.87115	−0.221881
$960$	0	0
$961$	−11.3701	−0.366779
$962$	−0.671912	−0.0216633
$963$	0	0
$964$	−2.50213	−0.0805880
$965$	−87.6261	−2.82078
$966$	0	0
$967$	−18.6819	−0.600768	−0.300384	−	0.953818i	$-0.597115\pi$
$967$	−18.6819	−0.600768	−0.300384	+	0.953818i	$0.597115\pi$
$968$	3.05301	0.0981274
$969$	0	0
$970$	−3.56359	−0.114420
$971$	42.8616	1.37549	0.687747	−	0.725951i	$-0.258600\pi$
$971$	42.8616	1.37549	0.687747	+	0.725951i	$0.258600\pi$
$972$	0	0
$973$	−5.56482	−0.178400
$974$	−13.7968	−0.442078
$975$	0	0
$976$	1.53172	0.0490291
$977$	−19.9817	−0.639271	−0.319636	−	0.947541i	$-0.603560\pi$
$977$	−19.9817	−0.639271	−0.319636	+	0.947541i	$0.603560\pi$
$978$	0	0
$979$	16.4189	0.524750
$980$	17.5539	0.560738
$981$	0	0
$982$	12.3561	0.394298
$983$	−52.3548	−1.66986	−0.834929	−	0.550358i	$-0.814491\pi$
$983$	−52.3548	−1.66986	−0.834929	+	0.550358i	$0.814491\pi$
$984$	0	0
$985$	12.7321	0.405678
$986$	33.8252	1.07721
$987$	0	0
$988$	−9.20721	−0.292920
$989$	11.2523	0.357802
$990$	0	0
$991$	−12.9868	−0.412539	−0.206269	−	0.978495i	$-0.566132\pi$
$991$	−12.9868	−0.412539	−0.206269	+	0.978495i	$0.566132\pi$
$992$	−19.8146	−0.629115
$993$	0	0
$994$	12.0799	0.383151
$995$	27.6953	0.878000
$996$	0	0
$997$	40.0519	1.26846	0.634228	−	0.773146i	$-0.281318\pi$
$997$	40.0519	1.26846	0.634228	+	0.773146i	$0.281318\pi$
$998$	−42.2313	−1.33681
$999$	0	0

Twists

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

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

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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)

\(f(q)\)	\(=\)	\(q-1.06662 q^{2} -0.862332 q^{4} +3.84706 q^{5} -1.30714 q^{7} +3.05301 q^{8} +O(q^{10})\)	\(q-1.06662 q^{2} -0.862332 q^{4} +3.84706 q^{5} -1.30714 q^{7} +3.05301 q^{8} -4.10333 q^{10} +1.00000 q^{11} +1.88758 q^{13} +1.39422 q^{14} -1.53172 q^{16} -6.08208 q^{17} +5.65651 q^{19} -3.31744 q^{20} -1.06662 q^{22} +3.88616 q^{23} +9.79987 q^{25} -2.01332 q^{26} +1.12719 q^{28} +5.21411 q^{29} +4.43056 q^{31} -4.47226 q^{32} +6.48724 q^{34} -5.02865 q^{35} +0.333734 q^{37} -6.03332 q^{38} +11.7451 q^{40} +1.38003 q^{41} +2.89547 q^{43} -0.862332 q^{44} -4.14504 q^{46} -8.79970 q^{47} -5.29138 q^{49} -10.4527 q^{50} -1.62772 q^{52} -8.65075 q^{53} +3.84706 q^{55} -3.99071 q^{56} -5.56145 q^{58} +8.72137 q^{59} -1.00000 q^{61} -4.72570 q^{62} +7.83362 q^{64} +7.26162 q^{65} -13.6466 q^{67} +5.24478 q^{68} +5.36363 q^{70} +8.66429 q^{71} -3.51520 q^{73} -0.355965 q^{74} -4.87780 q^{76} -1.30714 q^{77} +12.5324 q^{79} -5.89261 q^{80} -1.47196 q^{82} +17.3477 q^{83} -23.3981 q^{85} -3.08836 q^{86} +3.05301 q^{88} +16.4189 q^{89} -2.46733 q^{91} -3.35116 q^{92} +9.38589 q^{94} +21.7609 q^{95} +0.868462 q^{97} +5.64387 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 4 q^{5} + 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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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