LMFDB - Embedded newform 6039.2.a.p.1.15

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			1.15
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)

$f(q)$	$=$	$q+0.800654 q^{2} -1.35895 q^{4} +0.757549 q^{5} +0.484724 q^{7} -2.68936 q^{8} +O(q^{10})$	\(q+0.800654 q^{2} -1.35895 q^{4} +0.757549 q^{5} +0.484724 q^{7} -2.68936 q^{8} +0.606535 q^{10} -1.00000 q^{11} -5.01069 q^{13} +0.388096 q^{14} +0.564659 q^{16} -0.818362 q^{17} +1.42494 q^{19} -1.02947 q^{20} -0.800654 q^{22} +2.31191 q^{23} -4.42612 q^{25} -4.01183 q^{26} -0.658717 q^{28} -4.07528 q^{29} +8.39203 q^{31} +5.83082 q^{32} -0.655225 q^{34} +0.367202 q^{35} -5.04139 q^{37} +1.14089 q^{38} -2.03732 q^{40} -4.54679 q^{41} +10.5901 q^{43} +1.35895 q^{44} +1.85104 q^{46} +0.677261 q^{47} -6.76504 q^{49} -3.54379 q^{50} +6.80930 q^{52} +3.48120 q^{53} -0.757549 q^{55} -1.30360 q^{56} -3.26289 q^{58} +13.2967 q^{59} +1.00000 q^{61} +6.71911 q^{62} +3.53915 q^{64} -3.79585 q^{65} -1.60430 q^{67} +1.11212 q^{68} +0.294002 q^{70} -3.46463 q^{71} +9.56962 q^{73} -4.03641 q^{74} -1.93643 q^{76} -0.484724 q^{77} +2.71696 q^{79} +0.427757 q^{80} -3.64041 q^{82} +1.98699 q^{83} -0.619950 q^{85} +8.47901 q^{86} +2.68936 q^{88} -0.245057 q^{89} -2.42880 q^{91} -3.14178 q^{92} +0.542252 q^{94} +1.07946 q^{95} +10.4352 q^{97} -5.41646 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.800654	0.566148	0.283074	−	0.959098i	$-0.408646\pi$
$2$	0.800654	0.566148	0.283074	+	0.959098i	$0.408646\pi$
$3$	0	0
$3$	0	0
$4$	−1.35895	−0.679476
$5$	0.757549	0.338786	0.169393	−	0.985549i	$-0.445819\pi$
$5$	0.757549	0.338786	0.169393	+	0.985549i	$0.445819\pi$
$6$	0	0
$7$	0.484724	0.183208	0.0916042	−	0.995795i	$-0.470801\pi$
$7$	0.484724	0.183208	0.0916042	+	0.995795i	$0.470801\pi$
$8$	−2.68936	−0.950832
$9$	0	0
$10$	0.606535	0.191803
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.01069	−1.38972	−0.694858	−	0.719147i	$-0.744533\pi$
$13$	−5.01069	−1.38972	−0.694858	+	0.719147i	$0.744533\pi$
$14$	0.388096	0.103723
$15$	0	0
$16$	0.564659	0.141165
$17$	−0.818362	−0.198482	−0.0992410	−	0.995063i	$-0.531641\pi$
$17$	−0.818362	−0.198482	−0.0992410	+	0.995063i	$0.531641\pi$
$18$	0	0
$19$	1.42494	0.326904	0.163452	−	0.986551i	$-0.447737\pi$
$19$	1.42494	0.326904	0.163452	+	0.986551i	$0.447737\pi$
$20$	−1.02947	−0.230197
$21$	0	0
$22$	−0.800654	−0.170700
$23$	2.31191	0.482068	0.241034	−	0.970517i	$-0.422514\pi$
$23$	2.31191	0.482068	0.241034	+	0.970517i	$0.422514\pi$
$24$	0	0
$25$	−4.42612	−0.885224
$26$	−4.01183	−0.786785
$27$	0	0
$28$	−0.658717	−0.124486
$29$	−4.07528	−0.756760	−0.378380	−	0.925650i	$-0.623519\pi$
$29$	−4.07528	−0.756760	−0.378380	+	0.925650i	$0.623519\pi$
$30$	0	0
$31$	8.39203	1.50725	0.753626	−	0.657303i	$-0.228303\pi$
$31$	8.39203	1.50725	0.753626	+	0.657303i	$0.228303\pi$
$32$	5.83082	1.03075
$33$	0	0
$34$	−0.655225	−0.112370
$35$	0.367202	0.0620685
$36$	0	0
$37$	−5.04139	−0.828799	−0.414399	−	0.910095i	$-0.636008\pi$
$37$	−5.04139	−0.828799	−0.414399	+	0.910095i	$0.636008\pi$
$38$	1.14089	0.185076
$39$	0	0
$40$	−2.03732	−0.322129
$41$	−4.54679	−0.710090	−0.355045	−	0.934849i	$-0.615534\pi$
$41$	−4.54679	−0.710090	−0.355045	+	0.934849i	$0.615534\pi$
$42$	0	0
$43$	10.5901	1.61498	0.807488	−	0.589885i	$-0.200827\pi$
$43$	10.5901	1.61498	0.807488	+	0.589885i	$0.200827\pi$
$44$	1.35895	0.204870
$45$	0	0
$46$	1.85104	0.272922
$47$	0.677261	0.0987886	0.0493943	−	0.998779i	$-0.484271\pi$
$47$	0.677261	0.0987886	0.0493943	+	0.998779i	$0.484271\pi$
$48$	0	0
$49$	−6.76504	−0.966435
$50$	−3.54379	−0.501168
$51$	0	0
$52$	6.80930	0.944279
$53$	3.48120	0.478180	0.239090	−	0.970997i	$-0.423151\pi$
$53$	3.48120	0.478180	0.239090	+	0.970997i	$0.423151\pi$
$54$	0	0
$55$	−0.757549	−0.102148
$56$	−1.30360	−0.174200
$57$	0	0
$58$	−3.26289	−0.428438
$59$	13.2967	1.73108	0.865540	−	0.500840i	$-0.166976\pi$
$59$	13.2967	1.73108	0.865540	+	0.500840i	$0.166976\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	6.71911	0.853328
$63$	0	0
$64$	3.53915	0.442394
$65$	−3.79585	−0.470817
$66$	0	0
$67$	−1.60430	−0.195997	−0.0979984	−	0.995187i	$-0.531244\pi$
$67$	−1.60430	−0.195997	−0.0979984	+	0.995187i	$0.531244\pi$
$68$	1.11212	0.134864
$69$	0	0
$70$	0.294002	0.0351400
$71$	−3.46463	−0.411176	−0.205588	−	0.978639i	$-0.565911\pi$
$71$	−3.46463	−0.411176	−0.205588	+	0.978639i	$0.565911\pi$
$72$	0	0
$73$	9.56962	1.12004	0.560019	−	0.828479i	$-0.310794\pi$
$73$	9.56962	1.12004	0.560019	+	0.828479i	$0.310794\pi$
$74$	−4.03641	−0.469223
$75$	0	0
$76$	−1.93643	−0.222124
$77$	−0.484724	−0.0552394
$78$	0	0
$79$	2.71696	0.305682	0.152841	−	0.988251i	$-0.451158\pi$
$79$	2.71696	0.305682	0.152841	+	0.988251i	$0.451158\pi$
$80$	0.427757	0.0478247
$81$	0	0
$82$	−3.64041	−0.402016
$83$	1.98699	0.218100	0.109050	−	0.994036i	$-0.465219\pi$
$83$	1.98699	0.218100	0.109050	+	0.994036i	$0.465219\pi$
$84$	0	0
$85$	−0.619950	−0.0672430
$86$	8.47901	0.914315
$87$	0	0
$88$	2.68936	0.286687
$89$	−0.245057	−0.0259760	−0.0129880	−	0.999916i	$-0.504134\pi$
$89$	−0.245057	−0.0259760	−0.0129880	+	0.999916i	$0.504134\pi$
$90$	0	0
$91$	−2.42880	−0.254608
$92$	−3.14178	−0.327554
$93$	0	0
$94$	0.542252	0.0559290
$95$	1.07946	0.110751
$96$	0	0
$97$	10.4352	1.05953	0.529766	−	0.848144i	$-0.322280\pi$
$97$	10.4352	1.05953	0.529766	+	0.848144i	$0.322280\pi$
$98$	−5.41646	−0.547145
$99$	0	0
$100$	6.01489	0.601489
$101$	11.6900	1.16319	0.581597	−	0.813477i	$-0.302428\pi$
$101$	11.6900	1.16319	0.581597	+	0.813477i	$0.302428\pi$
$102$	0	0
$103$	−4.20936	−0.414761	−0.207381	−	0.978260i	$-0.566494\pi$
$103$	−4.20936	−0.414761	−0.207381	+	0.978260i	$0.566494\pi$
$104$	13.4756	1.32139
$105$	0	0
$106$	2.78724	0.270721
$107$	12.1254	1.17221	0.586104	−	0.810236i	$-0.300661\pi$
$107$	12.1254	1.17221	0.586104	+	0.810236i	$0.300661\pi$
$108$	0	0
$109$	2.63197	0.252097	0.126049	−	0.992024i	$-0.459770\pi$
$109$	2.63197	0.252097	0.126049	+	0.992024i	$0.459770\pi$
$110$	−0.606535	−0.0578309
$111$	0	0
$112$	0.273703	0.0258625
$113$	−10.0558	−0.945968	−0.472984	−	0.881071i	$-0.656823\pi$
$113$	−10.0558	−0.945968	−0.472984	+	0.881071i	$0.656823\pi$
$114$	0	0
$115$	1.75139	0.163318
$116$	5.53811	0.514200
$117$	0	0
$118$	10.6460	0.980047
$119$	−0.396680	−0.0363636
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.800654	0.0724878
$123$	0	0
$124$	−11.4044	−1.02414
$125$	−7.14075	−0.638688
$126$	0	0
$127$	−17.5055	−1.55336	−0.776680	−	0.629895i	$-0.783098\pi$
$127$	−17.5055	−1.55336	−0.776680	+	0.629895i	$0.783098\pi$
$128$	−8.82800	−0.780292
$129$	0	0
$130$	−3.03916	−0.266552
$131$	14.2701	1.24679	0.623393	−	0.781908i	$-0.285754\pi$
$131$	14.2701	1.24679	0.623393	+	0.781908i	$0.285754\pi$
$132$	0	0
$133$	0.690703	0.0598916
$134$	−1.28449	−0.110963
$135$	0	0
$136$	2.20087	0.188723
$137$	2.79459	0.238758	0.119379	−	0.992849i	$-0.461910\pi$
$137$	2.79459	0.238758	0.119379	+	0.992849i	$0.461910\pi$
$138$	0	0
$139$	14.9466	1.26776	0.633879	−	0.773433i	$-0.281462\pi$
$139$	14.9466	1.26776	0.633879	+	0.773433i	$0.281462\pi$
$140$	−0.499010	−0.0421741
$141$	0	0
$142$	−2.77397	−0.232786
$143$	5.01069	0.419015
$144$	0	0
$145$	−3.08722	−0.256380
$146$	7.66195	0.634108
$147$	0	0
$148$	6.85101	0.563149
$149$	−24.1051	−1.97477	−0.987383	−	0.158351i	$-0.949382\pi$
$149$	−24.1051	−1.97477	−0.987383	+	0.158351i	$0.949382\pi$
$150$	0	0
$151$	4.02632	0.327657	0.163828	−	0.986489i	$-0.447616\pi$
$151$	4.02632	0.327657	0.163828	+	0.986489i	$0.447616\pi$
$152$	−3.83218	−0.310831
$153$	0	0
$154$	−0.388096	−0.0312737
$155$	6.35737	0.510637
$156$	0	0
$157$	17.0904	1.36396	0.681979	−	0.731371i	$-0.261119\pi$
$157$	17.0904	1.36396	0.681979	+	0.731371i	$0.261119\pi$
$158$	2.17535	0.173061
$159$	0	0
$160$	4.41713	0.349205
$161$	1.12064	0.0883188
$162$	0	0
$163$	19.5568	1.53181	0.765903	−	0.642956i	$-0.222292\pi$
$163$	19.5568	1.53181	0.765903	+	0.642956i	$0.222292\pi$
$164$	6.17888	0.482489
$165$	0	0
$166$	1.59089	0.123477
$167$	15.3827	1.19035	0.595176	−	0.803596i	$-0.297082\pi$
$167$	15.3827	1.19035	0.595176	+	0.803596i	$0.297082\pi$
$168$	0	0
$169$	12.1070	0.931311
$170$	−0.496365	−0.0380695
$171$	0	0
$172$	−14.3914	−1.09734
$173$	−19.4959	−1.48225	−0.741125	−	0.671368i	$-0.765707\pi$
$173$	−19.4959	−1.48225	−0.741125	+	0.671368i	$0.765707\pi$
$174$	0	0
$175$	−2.14545	−0.162180
$176$	−0.564659	−0.0425627
$177$	0	0
$178$	−0.196206	−0.0147062
$179$	24.4374	1.82653	0.913267	−	0.407361i	$-0.133551\pi$
$179$	24.4374	1.82653	0.913267	+	0.407361i	$0.133551\pi$
$180$	0	0
$181$	−1.29165	−0.0960074	−0.0480037	−	0.998847i	$-0.515286\pi$
$181$	−1.29165	−0.0960074	−0.0480037	+	0.998847i	$0.515286\pi$
$182$	−1.94463	−0.144146
$183$	0	0
$184$	−6.21757	−0.458365
$185$	−3.81910	−0.280786
$186$	0	0
$187$	0.818362	0.0598446
$188$	−0.920366	−0.0671246
$189$	0	0
$190$	0.864278	0.0627013
$191$	16.9401	1.22575	0.612873	−	0.790181i	$-0.290014\pi$
$191$	16.9401	1.22575	0.612873	+	0.790181i	$0.290014\pi$
$192$	0	0
$193$	3.77042	0.271401	0.135701	−	0.990750i	$-0.456672\pi$
$193$	3.77042	0.271401	0.135701	+	0.990750i	$0.456672\pi$
$194$	8.35497	0.599852
$195$	0	0
$196$	9.19337	0.656670
$197$	0.0770406	0.00548891	0.00274446	−	0.999996i	$-0.499126\pi$
$197$	0.0770406	0.00548891	0.00274446	+	0.999996i	$0.499126\pi$
$198$	0	0
$199$	−23.7662	−1.68474	−0.842371	−	0.538898i	$-0.818841\pi$
$199$	−23.7662	−1.68474	−0.842371	+	0.538898i	$0.818841\pi$
$200$	11.9034	0.841699
$201$	0	0
$202$	9.35962	0.658540
$203$	−1.97538	−0.138645
$204$	0	0
$205$	−3.44442	−0.240569
$206$	−3.37025	−0.234816
$207$	0	0
$208$	−2.82933	−0.196179
$209$	−1.42494	−0.0985653
$210$	0	0
$211$	−22.3516	−1.53875	−0.769374	−	0.638798i	$-0.779432\pi$
$211$	−22.3516	−1.53875	−0.769374	+	0.638798i	$0.779432\pi$
$212$	−4.73079	−0.324912
$213$	0	0
$214$	9.70827	0.663644
$215$	8.02252	0.547132
$216$	0	0
$217$	4.06781	0.276141
$218$	2.10730	0.142724
$219$	0	0
$220$	1.02947	0.0694071
$221$	4.10056	0.275834
$222$	0	0
$223$	25.9726	1.73926	0.869628	−	0.493708i	$-0.164359\pi$
$223$	25.9726	1.73926	0.869628	+	0.493708i	$0.164359\pi$
$224$	2.82633	0.188842
$225$	0	0
$226$	−8.05120	−0.535558
$227$	−13.5983	−0.902550	−0.451275	−	0.892385i	$-0.649031\pi$
$227$	−13.5983	−0.902550	−0.451275	+	0.892385i	$0.649031\pi$
$228$	0	0
$229$	−3.21717	−0.212596	−0.106298	−	0.994334i	$-0.533900\pi$
$229$	−3.21717	−0.212596	−0.106298	+	0.994334i	$0.533900\pi$
$230$	1.40226	0.0924621
$231$	0	0
$232$	10.9599	0.719552
$233$	11.7553	0.770113	0.385056	−	0.922893i	$-0.374182\pi$
$233$	11.7553	0.770113	0.385056	+	0.922893i	$0.374182\pi$
$234$	0	0
$235$	0.513059	0.0334682
$236$	−18.0696	−1.17623
$237$	0	0
$238$	−0.317603	−0.0205872
$239$	2.97160	0.192217	0.0961083	−	0.995371i	$-0.469360\pi$
$239$	2.97160	0.192217	0.0961083	+	0.995371i	$0.469360\pi$
$240$	0	0
$241$	−24.0598	−1.54983	−0.774913	−	0.632067i	$-0.782207\pi$
$241$	−24.0598	−1.54983	−0.774913	+	0.632067i	$0.782207\pi$
$242$	0.800654	0.0514680
$243$	0	0
$244$	−1.35895	−0.0869980
$245$	−5.12485	−0.327415
$246$	0	0
$247$	−7.13995	−0.454304
$248$	−22.5692	−1.43314
$249$	0	0
$250$	−5.71727	−0.361592
$251$	21.8684	1.38032	0.690160	−	0.723657i	$-0.257540\pi$
$251$	21.8684	1.38032	0.690160	+	0.723657i	$0.257540\pi$
$252$	0	0
$253$	−2.31191	−0.145349
$254$	−14.0158	−0.879432
$255$	0	0
$256$	−14.1465	−0.884155
$257$	30.3572	1.89363	0.946815	−	0.321778i	$-0.104281\pi$
$257$	30.3572	1.89363	0.946815	+	0.321778i	$0.104281\pi$
$258$	0	0
$259$	−2.44368	−0.151843
$260$	5.15838	0.319909
$261$	0	0
$262$	11.4254	0.705866
$263$	9.83934	0.606720	0.303360	−	0.952876i	$-0.401892\pi$
$263$	9.83934	0.606720	0.303360	+	0.952876i	$0.401892\pi$
$264$	0	0
$265$	2.63718	0.162001
$266$	0.553015	0.0339075
$267$	0	0
$268$	2.18017	0.133175
$269$	−3.90386	−0.238023	−0.119011	−	0.992893i	$-0.537973\pi$
$269$	−3.90386	−0.238023	−0.119011	+	0.992893i	$0.537973\pi$
$270$	0	0
$271$	1.21908	0.0740541	0.0370270	−	0.999314i	$-0.488211\pi$
$271$	1.21908	0.0740541	0.0370270	+	0.999314i	$0.488211\pi$
$272$	−0.462095	−0.0280186
$273$	0	0
$274$	2.23750	0.135172
$275$	4.42612	0.266905
$276$	0	0
$277$	18.4956	1.11129	0.555645	−	0.831420i	$-0.312471\pi$
$277$	18.4956	1.11129	0.555645	+	0.831420i	$0.312471\pi$
$278$	11.9671	0.717738
$279$	0	0
$280$	−0.987539	−0.0590167
$281$	18.1883	1.08502	0.542510	−	0.840049i	$-0.317474\pi$
$281$	18.1883	1.08502	0.542510	+	0.840049i	$0.317474\pi$
$282$	0	0
$283$	22.3267	1.32718	0.663592	−	0.748095i	$-0.269031\pi$
$283$	22.3267	1.32718	0.663592	+	0.748095i	$0.269031\pi$
$284$	4.70827	0.279384
$285$	0	0
$286$	4.01183	0.237225
$287$	−2.20394	−0.130094
$288$	0	0
$289$	−16.3303	−0.960605
$290$	−2.47180	−0.145149
$291$	0	0
$292$	−13.0047	−0.761040
$293$	27.2758	1.59347	0.796736	−	0.604328i	$-0.206558\pi$
$293$	27.2758	1.59347	0.796736	+	0.604328i	$0.206558\pi$
$294$	0	0
$295$	10.0729	0.586466
$296$	13.5581	0.788048
$297$	0	0
$298$	−19.2998	−1.11801
$299$	−11.5843	−0.669937
$300$	0	0
$301$	5.13327	0.295877
$302$	3.22369	0.185502
$303$	0	0
$304$	0.804606	0.0461473
$305$	0.757549	0.0433772
$306$	0	0
$307$	12.8441	0.733052	0.366526	−	0.930408i	$-0.380547\pi$
$307$	12.8441	0.733052	0.366526	+	0.930408i	$0.380547\pi$
$308$	0.658717	0.0375339
$309$	0	0
$310$	5.09006	0.289096
$311$	−2.01297	−0.114145	−0.0570726	−	0.998370i	$-0.518177\pi$
$311$	−2.01297	−0.114145	−0.0570726	+	0.998370i	$0.518177\pi$
$312$	0	0
$313$	−18.3982	−1.03993	−0.519964	−	0.854188i	$-0.674055\pi$
$313$	−18.3982	−1.03993	−0.519964	+	0.854188i	$0.674055\pi$
$314$	13.6835	0.772203
$315$	0	0
$316$	−3.69223	−0.207704
$317$	−24.6038	−1.38189	−0.690943	−	0.722909i	$-0.742805\pi$
$317$	−24.6038	−1.38189	−0.690943	+	0.722909i	$0.742805\pi$
$318$	0	0
$319$	4.07528	0.228172
$320$	2.68108	0.149877
$321$	0	0
$322$	0.897245	0.0500015
$323$	−1.16612	−0.0648846
$324$	0	0
$325$	22.1779	1.23021
$326$	15.6582	0.867229
$327$	0	0
$328$	12.2280	0.675176
$329$	0.328284	0.0180989
$330$	0	0
$331$	−12.5325	−0.688850	−0.344425	−	0.938814i	$-0.611926\pi$
$331$	−12.5325	−0.688850	−0.344425	+	0.938814i	$0.611926\pi$
$332$	−2.70022	−0.148194
$333$	0	0
$334$	12.3162	0.673915
$335$	−1.21534	−0.0664010
$336$	0	0
$337$	−30.9041	−1.68345	−0.841727	−	0.539904i	$-0.818461\pi$
$337$	−30.9041	−1.68345	−0.841727	+	0.539904i	$0.818461\pi$
$338$	9.69356	0.527260
$339$	0	0
$340$	0.842483	0.0456900
$341$	−8.39203	−0.454454
$342$	0	0
$343$	−6.67224	−0.360267
$344$	−28.4806	−1.53557
$345$	0	0
$346$	−15.6095	−0.839172
$347$	22.1988	1.19169	0.595847	−	0.803098i	$-0.296816\pi$
$347$	22.1988	1.19169	0.595847	+	0.803098i	$0.296816\pi$
$348$	0	0
$349$	−19.6704	−1.05293	−0.526467	−	0.850195i	$-0.676484\pi$
$349$	−19.6704	−1.05293	−0.526467	+	0.850195i	$0.676484\pi$
$350$	−1.71776	−0.0918181
$351$	0	0
$352$	−5.83082	−0.310784
$353$	−4.88476	−0.259989	−0.129995	−	0.991515i	$-0.541496\pi$
$353$	−4.88476	−0.259989	−0.129995	+	0.991515i	$0.541496\pi$
$354$	0	0
$355$	−2.62463	−0.139301
$356$	0.333021	0.0176501
$357$	0	0
$358$	19.5659	1.03409
$359$	−22.1631	−1.16972	−0.584861	−	0.811133i	$-0.698851\pi$
$359$	−22.1631	−1.16972	−0.584861	+	0.811133i	$0.698851\pi$
$360$	0	0
$361$	−16.9695	−0.893134
$362$	−1.03416	−0.0543544
$363$	0	0
$364$	3.30063	0.173000
$365$	7.24946	0.379454
$366$	0	0
$367$	−13.5132	−0.705386	−0.352693	−	0.935739i	$-0.614734\pi$
$367$	−13.5132	−0.705386	−0.352693	+	0.935739i	$0.614734\pi$
$368$	1.30544	0.0680509
$369$	0	0
$370$	−3.05778	−0.158966
$371$	1.68742	0.0876065
$372$	0	0
$373$	−5.15344	−0.266835	−0.133418	−	0.991060i	$-0.542595\pi$
$373$	−5.15344	−0.266835	−0.133418	+	0.991060i	$0.542595\pi$
$374$	0.655225	0.0338809
$375$	0	0
$376$	−1.82140	−0.0939314
$377$	20.4200	1.05168
$378$	0	0
$379$	−1.89547	−0.0973638	−0.0486819	−	0.998814i	$-0.515502\pi$
$379$	−1.89547	−0.0973638	−0.0486819	+	0.998814i	$0.515502\pi$
$380$	−1.46694	−0.0752525
$381$	0	0
$382$	13.5632	0.693954
$383$	38.0682	1.94519	0.972597	−	0.232499i	$-0.0746904\pi$
$383$	38.0682	1.94519	0.972597	+	0.232499i	$0.0746904\pi$
$384$	0	0
$385$	−0.367202	−0.0187144
$386$	3.01881	0.153653
$387$	0	0
$388$	−14.1809	−0.719927
$389$	35.2526	1.78738	0.893688	−	0.448688i	$-0.148109\pi$
$389$	35.2526	1.78738	0.893688	+	0.448688i	$0.148109\pi$
$390$	0	0
$391$	−1.89198	−0.0956817
$392$	18.1936	0.918917
$393$	0	0
$394$	0.0616829	0.00310754
$395$	2.05823	0.103561
$396$	0	0
$397$	−26.7733	−1.34371	−0.671857	−	0.740681i	$-0.734503\pi$
$397$	−26.7733	−1.34371	−0.671857	+	0.740681i	$0.734503\pi$
$398$	−19.0285	−0.953814
$399$	0	0
$400$	−2.49925	−0.124962
$401$	38.7029	1.93273	0.966366	−	0.257169i	$-0.0827899\pi$
$401$	38.7029	1.93273	0.966366	+	0.257169i	$0.0827899\pi$
$402$	0	0
$403$	−42.0499	−2.09465
$404$	−15.8861	−0.790363
$405$	0	0
$406$	−1.58160	−0.0784934
$407$	5.04139	0.249892
$408$	0	0
$409$	−10.0285	−0.495877	−0.247938	−	0.968776i	$-0.579753\pi$
$409$	−10.0285	−0.495877	−0.247938	+	0.968776i	$0.579753\pi$
$410$	−2.75779	−0.136198
$411$	0	0
$412$	5.72033	0.281820
$413$	6.44521	0.317148
$414$	0	0
$415$	1.50524	0.0738894
$416$	−29.2164	−1.43245
$417$	0	0
$418$	−1.14089	−0.0558026
$419$	35.7891	1.74841	0.874205	−	0.485557i	$-0.161383\pi$
$419$	35.7891	1.74841	0.874205	+	0.485557i	$0.161383\pi$
$420$	0	0
$421$	40.4213	1.97001	0.985007	−	0.172512i	$-0.0551885\pi$
$421$	40.4213	1.97001	0.985007	+	0.172512i	$0.0551885\pi$
$422$	−17.8959	−0.871160
$423$	0	0
$424$	−9.36220	−0.454669
$425$	3.62217	0.175701
$426$	0	0
$427$	0.484724	0.0234574
$428$	−16.4779	−0.796488
$429$	0	0
$430$	6.42327	0.309757
$431$	24.7614	1.19271	0.596357	−	0.802719i	$-0.296614\pi$
$431$	24.7614	1.19271	0.596357	+	0.802719i	$0.296614\pi$
$432$	0	0
$433$	24.9542	1.19922	0.599611	−	0.800292i	$-0.295322\pi$
$433$	24.9542	1.19922	0.599611	+	0.800292i	$0.295322\pi$
$434$	3.25691	0.156337
$435$	0	0
$436$	−3.57673	−0.171294
$437$	3.29435	0.157590
$438$	0	0
$439$	−10.7811	−0.514553	−0.257276	−	0.966338i	$-0.582825\pi$
$439$	−10.7811	−0.514553	−0.257276	+	0.966338i	$0.582825\pi$
$440$	2.03732	0.0971256
$441$	0	0
$442$	3.28313	0.156163
$443$	−31.2140	−1.48302	−0.741512	−	0.670939i	$-0.765891\pi$
$443$	−31.2140	−1.48302	−0.741512	+	0.670939i	$0.765891\pi$
$444$	0	0
$445$	−0.185643	−0.00880030
$446$	20.7951	0.984676
$447$	0	0
$448$	1.71551	0.0810502
$449$	−3.22305	−0.152105	−0.0760525	−	0.997104i	$-0.524232\pi$
$449$	−3.22305	−0.152105	−0.0760525	+	0.997104i	$0.524232\pi$
$450$	0	0
$451$	4.54679	0.214100
$452$	13.6653	0.642763
$453$	0	0
$454$	−10.8875	−0.510977
$455$	−1.83994	−0.0862576
$456$	0	0
$457$	−2.23263	−0.104438	−0.0522191	−	0.998636i	$-0.516629\pi$
$457$	−2.23263	−0.104438	−0.0522191	+	0.998636i	$0.516629\pi$
$458$	−2.57584	−0.120361
$459$	0	0
$460$	−2.38006	−0.110971
$461$	21.8889	1.01947	0.509733	−	0.860332i	$-0.329744\pi$
$461$	21.8889	1.01947	0.509733	+	0.860332i	$0.329744\pi$
$462$	0	0
$463$	6.69036	0.310927	0.155464	−	0.987842i	$-0.450313\pi$
$463$	6.69036	0.310927	0.155464	+	0.987842i	$0.450313\pi$
$464$	−2.30114	−0.106828
$465$	0	0
$466$	9.41190	0.435998
$467$	23.5598	1.09022	0.545108	−	0.838366i	$-0.316488\pi$
$467$	23.5598	1.09022	0.545108	+	0.838366i	$0.316488\pi$
$468$	0	0
$469$	−0.777644	−0.0359082
$470$	0.410782	0.0189480
$471$	0	0
$472$	−35.7595	−1.64597
$473$	−10.5901	−0.486933
$474$	0	0
$475$	−6.30696	−0.289383
$476$	0.539069	0.0247082
$477$	0	0
$478$	2.37922	0.108823
$479$	24.2892	1.10980	0.554900	−	0.831917i	$-0.312756\pi$
$479$	24.2892	1.10980	0.554900	+	0.831917i	$0.312756\pi$
$480$	0	0
$481$	25.2608	1.15180
$482$	−19.2636	−0.877431
$483$	0	0
$484$	−1.35895	−0.0617706
$485$	7.90516	0.358955
$486$	0	0
$487$	−26.9295	−1.22029	−0.610145	−	0.792289i	$-0.708889\pi$
$487$	−26.9295	−1.22029	−0.610145	+	0.792289i	$0.708889\pi$
$488$	−2.68936	−0.121742
$489$	0	0
$490$	−4.10324	−0.185365
$491$	−36.7607	−1.65899	−0.829493	−	0.558517i	$-0.811371\pi$
$491$	−36.7607	−1.65899	−0.829493	+	0.558517i	$0.811371\pi$
$492$	0	0
$493$	3.33505	0.150203
$494$	−5.71663	−0.257203
$495$	0	0
$496$	4.73863	0.212771
$497$	−1.67939	−0.0753308
$498$	0	0
$499$	29.7248	1.33067	0.665333	−	0.746547i	$-0.268290\pi$
$499$	29.7248	1.33067	0.665333	+	0.746547i	$0.268290\pi$
$500$	9.70394	0.433974
$501$	0	0
$502$	17.5090	0.781466
$503$	−26.0996	−1.16373	−0.581863	−	0.813287i	$-0.697676\pi$
$503$	−26.0996	−1.16373	−0.581863	+	0.813287i	$0.697676\pi$
$504$	0	0
$505$	8.85572	0.394074
$506$	−1.85104	−0.0822890
$507$	0	0
$508$	23.7891	1.05547
$509$	25.1275	1.11376	0.556878	−	0.830594i	$-0.311999\pi$
$509$	25.1275	1.11376	0.556878	+	0.830594i	$0.311999\pi$
$510$	0	0
$511$	4.63862	0.205201
$512$	6.32956	0.279730
$513$	0	0
$514$	24.3056	1.07207
$515$	−3.18880	−0.140515
$516$	0	0
$517$	−0.677261	−0.0297859
$518$	−1.95654	−0.0859655
$519$	0	0
$520$	10.2084	0.447668
$521$	27.1273	1.18847	0.594235	−	0.804291i	$-0.297455\pi$
$521$	27.1273	1.18847	0.594235	+	0.804291i	$0.297455\pi$
$522$	0	0
$523$	−37.4894	−1.63930	−0.819648	−	0.572867i	$-0.805831\pi$
$523$	−37.4894	−1.63930	−0.819648	+	0.572867i	$0.805831\pi$
$524$	−19.3924	−0.847162
$525$	0	0
$526$	7.87791	0.343493
$527$	−6.86772	−0.299162
$528$	0	0
$529$	−17.6551	−0.767611
$530$	2.11147	0.0917164
$531$	0	0
$532$	−0.938633	−0.0406949
$533$	22.7826	0.986823
$534$	0	0
$535$	9.18560	0.397128
$536$	4.31455	0.186360
$537$	0	0
$538$	−3.12565	−0.134756
$539$	6.76504	0.291391
$540$	0	0
$541$	24.8466	1.06824	0.534118	−	0.845410i	$-0.320644\pi$
$541$	24.8466	1.06824	0.534118	+	0.845410i	$0.320644\pi$
$542$	0.976065	0.0419256
$543$	0	0
$544$	−4.77172	−0.204586
$545$	1.99385	0.0854071
$546$	0	0
$547$	−23.7348	−1.01483	−0.507414	−	0.861702i	$-0.669399\pi$
$547$	−23.7348	−1.01483	−0.507414	+	0.861702i	$0.669399\pi$
$548$	−3.79772	−0.162230
$549$	0	0
$550$	3.54379	0.151108
$551$	−5.80703	−0.247388
$552$	0	0
$553$	1.31698	0.0560036
$554$	14.8085	0.629155
$555$	0	0
$556$	−20.3118	−0.861411
$557$	−15.4654	−0.655290	−0.327645	−	0.944801i	$-0.606255\pi$
$557$	−15.4654	−0.655290	−0.327645	+	0.944801i	$0.606255\pi$
$558$	0	0
$559$	−53.0637	−2.24436
$560$	0.207344	0.00876188
$561$	0	0
$562$	14.5625	0.614282
$563$	−13.7963	−0.581444	−0.290722	−	0.956808i	$-0.593896\pi$
$563$	−13.7963	−0.581444	−0.290722	+	0.956808i	$0.593896\pi$
$564$	0	0
$565$	−7.61775	−0.320481
$566$	17.8760	0.751382
$567$	0	0
$568$	9.31763	0.390959
$569$	−17.4535	−0.731688	−0.365844	−	0.930676i	$-0.619220\pi$
$569$	−17.4535	−0.731688	−0.365844	+	0.930676i	$0.619220\pi$
$570$	0	0
$571$	20.8933	0.874357	0.437178	−	0.899375i	$-0.355978\pi$
$571$	20.8933	0.874357	0.437178	+	0.899375i	$0.355978\pi$
$572$	−6.80930	−0.284711
$573$	0	0
$574$	−1.76459	−0.0736527
$575$	−10.2328	−0.426738
$576$	0	0
$577$	19.6912	0.819754	0.409877	−	0.912141i	$-0.365572\pi$
$577$	19.6912	0.819754	0.409877	+	0.912141i	$0.365572\pi$
$578$	−13.0749	−0.543845
$579$	0	0
$580$	4.19539	0.174204
$581$	0.963140	0.0399578
$582$	0	0
$583$	−3.48120	−0.144177
$584$	−25.7361	−1.06497
$585$	0	0
$586$	21.8385	0.902141
$587$	−29.6848	−1.22522	−0.612612	−	0.790384i	$-0.709881\pi$
$587$	−29.6848	−1.22522	−0.612612	+	0.790384i	$0.709881\pi$
$588$	0	0
$589$	11.9582	0.492727
$590$	8.06490	0.332027
$591$	0	0
$592$	−2.84666	−0.116997
$593$	21.3828	0.878085	0.439043	−	0.898466i	$-0.355318\pi$
$593$	21.3828	0.878085	0.439043	+	0.898466i	$0.355318\pi$
$594$	0	0
$595$	−0.300504	−0.0123195
$596$	32.7577	1.34181
$597$	0	0
$598$	−9.27502	−0.379284
$599$	45.7650	1.86991	0.934953	−	0.354772i	$-0.115441\pi$
$599$	45.7650	1.86991	0.934953	+	0.354772i	$0.115441\pi$
$600$	0	0
$601$	−9.41678	−0.384119	−0.192059	−	0.981383i	$-0.561517\pi$
$601$	−9.41678	−0.384119	−0.192059	+	0.981383i	$0.561517\pi$
$602$	4.10998	0.167510
$603$	0	0
$604$	−5.47157	−0.222635
$605$	0.757549	0.0307988
$606$	0	0
$607$	21.7086	0.881126	0.440563	−	0.897722i	$-0.354779\pi$
$607$	21.7086	0.881126	0.440563	+	0.897722i	$0.354779\pi$
$608$	8.30858	0.336957
$609$	0	0
$610$	0.606535	0.0245579
$611$	−3.39355	−0.137288
$612$	0	0
$613$	9.19714	0.371469	0.185735	−	0.982600i	$-0.440534\pi$
$613$	9.19714	0.371469	0.185735	+	0.982600i	$0.440534\pi$
$614$	10.2837	0.415016
$615$	0	0
$616$	1.30360	0.0525234
$617$	4.44983	0.179143	0.0895717	−	0.995980i	$-0.471450\pi$
$617$	4.44983	0.179143	0.0895717	+	0.995980i	$0.471450\pi$
$618$	0	0
$619$	−8.20985	−0.329982	−0.164991	−	0.986295i	$-0.552759\pi$
$619$	−8.20985	−0.329982	−0.164991	+	0.986295i	$0.552759\pi$
$620$	−8.63937	−0.346966
$621$	0	0
$622$	−1.61170	−0.0646231
$623$	−0.118785	−0.00475901
$624$	0	0
$625$	16.7211	0.668845
$626$	−14.7306	−0.588753
$627$	0	0
$628$	−23.2250	−0.926778
$629$	4.12568	0.164502
$630$	0	0
$631$	−19.9970	−0.796070	−0.398035	−	0.917370i	$-0.630308\pi$
$631$	−19.9970	−0.796070	−0.398035	+	0.917370i	$0.630308\pi$
$632$	−7.30689	−0.290653
$633$	0	0
$634$	−19.6991	−0.782352
$635$	−13.2613	−0.526257
$636$	0	0
$637$	33.8976	1.34307
$638$	3.26289	0.129179
$639$	0	0
$640$	−6.68764	−0.264352
$641$	26.1842	1.03422	0.517108	−	0.855920i	$-0.327009\pi$
$641$	26.1842	1.03422	0.517108	+	0.855920i	$0.327009\pi$
$642$	0	0
$643$	−44.3649	−1.74958	−0.874791	−	0.484501i	$-0.839001\pi$
$643$	−44.3649	−1.74958	−0.874791	+	0.484501i	$0.839001\pi$
$644$	−1.52290	−0.0600105
$645$	0	0
$646$	−0.933658	−0.0367343
$647$	−39.1228	−1.53808	−0.769039	−	0.639202i	$-0.779265\pi$
$647$	−39.1228	−1.53808	−0.769039	+	0.639202i	$0.779265\pi$
$648$	0	0
$649$	−13.2967	−0.521940
$650$	17.7568	0.696481
$651$	0	0
$652$	−26.5768	−1.04083
$653$	−21.9964	−0.860787	−0.430394	−	0.902641i	$-0.641625\pi$
$653$	−21.9964	−0.860787	−0.430394	+	0.902641i	$0.641625\pi$
$654$	0	0
$655$	10.8103	0.422394
$656$	−2.56739	−0.100240
$657$	0	0
$658$	0.262842	0.0102467
$659$	−44.8688	−1.74784	−0.873920	−	0.486070i	$-0.838430\pi$
$659$	−44.8688	−1.74784	−0.873920	+	0.486070i	$0.838430\pi$
$660$	0	0
$661$	9.24859	0.359728	0.179864	−	0.983691i	$-0.442434\pi$
$661$	9.24859	0.359728	0.179864	+	0.983691i	$0.442434\pi$
$662$	−10.0342	−0.389991
$663$	0	0
$664$	−5.34373	−0.207377
$665$	0.523242	0.0202905
$666$	0	0
$667$	−9.42169	−0.364809
$668$	−20.9044	−0.808816
$669$	0	0
$670$	−0.973066	−0.0375928
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	17.9441	0.691694	0.345847	−	0.938291i	$-0.387592\pi$
$673$	17.9441	0.691694	0.345847	+	0.938291i	$0.387592\pi$
$674$	−24.7435	−0.953084
$675$	0	0
$676$	−16.4529	−0.632804
$677$	−13.7921	−0.530072	−0.265036	−	0.964239i	$-0.585384\pi$
$677$	−13.7921	−0.530072	−0.265036	+	0.964239i	$0.585384\pi$
$678$	0	0
$679$	5.05818	0.194115
$680$	1.66727	0.0639368
$681$	0	0
$682$	−6.71911	−0.257288
$683$	−27.1498	−1.03886	−0.519429	−	0.854514i	$-0.673855\pi$
$683$	−27.1498	−1.03886	−0.519429	+	0.854514i	$0.673855\pi$
$684$	0	0
$685$	2.11704	0.0808879
$686$	−5.34216	−0.203965
$687$	0	0
$688$	5.97979	0.227977
$689$	−17.4432	−0.664534
$690$	0	0
$691$	16.9559	0.645035	0.322517	−	0.946564i	$-0.395471\pi$
$691$	16.9559	0.645035	0.322517	+	0.946564i	$0.395471\pi$
$692$	26.4941	1.00715
$693$	0	0
$694$	17.7736	0.674675
$695$	11.3228	0.429499
$696$	0	0
$697$	3.72092	0.140940
$698$	−15.7492	−0.596117
$699$	0	0
$700$	2.91556	0.110198
$701$	29.0054	1.09552	0.547760	−	0.836636i	$-0.315481\pi$
$701$	29.0054	1.09552	0.547760	+	0.836636i	$0.315481\pi$
$702$	0	0
$703$	−7.18368	−0.270938
$704$	−3.53915	−0.133387
$705$	0	0
$706$	−3.91100	−0.147192
$707$	5.66640	0.213107
$708$	0	0
$709$	−25.6560	−0.963533	−0.481766	−	0.876300i	$-0.660005\pi$
$709$	−25.6560	−0.963533	−0.481766	+	0.876300i	$0.660005\pi$
$710$	−2.10142	−0.0788649
$711$	0	0
$712$	0.659046	0.0246988
$713$	19.4016	0.726597
$714$	0	0
$715$	3.79585	0.141957
$716$	−33.2092	−1.24109
$717$	0	0
$718$	−17.7450	−0.662236
$719$	−27.3865	−1.02134	−0.510672	−	0.859775i	$-0.670603\pi$
$719$	−27.3865	−1.02134	−0.510672	+	0.859775i	$0.670603\pi$
$720$	0	0
$721$	−2.04038	−0.0759877
$722$	−13.5867	−0.505646
$723$	0	0
$724$	1.75529	0.0652348
$725$	18.0377	0.669902
$726$	0	0
$727$	29.0875	1.07879	0.539397	−	0.842051i	$-0.318652\pi$
$727$	29.0875	1.07879	0.539397	+	0.842051i	$0.318652\pi$
$728$	6.53192	0.242089
$729$	0	0
$730$	5.80431	0.214827
$731$	−8.66654	−0.320544
$732$	0	0
$733$	−46.1622	−1.70504	−0.852519	−	0.522696i	$-0.824926\pi$
$733$	−46.1622	−1.70504	−0.852519	+	0.522696i	$0.824926\pi$
$734$	−10.8194	−0.399353
$735$	0	0
$736$	13.4803	0.496892
$737$	1.60430	0.0590953
$738$	0	0
$739$	3.92378	0.144338	0.0721692	−	0.997392i	$-0.477008\pi$
$739$	3.92378	0.144338	0.0721692	+	0.997392i	$0.477008\pi$
$740$	5.18997	0.190787
$741$	0	0
$742$	1.35104	0.0495983
$743$	0.266844	0.00978956	0.00489478	−	0.999988i	$-0.498442\pi$
$743$	0.266844	0.00978956	0.00489478	+	0.999988i	$0.498442\pi$
$744$	0	0
$745$	−18.2608	−0.669024
$746$	−4.12613	−0.151068
$747$	0	0
$748$	−1.11212	−0.0406630
$749$	5.87748	0.214758
$750$	0	0
$751$	40.2514	1.46880	0.734398	−	0.678719i	$-0.237465\pi$
$751$	40.2514	1.46880	0.734398	+	0.678719i	$0.237465\pi$
$752$	0.382421	0.0139455
$753$	0	0
$754$	16.3493	0.595407
$755$	3.05013	0.111006
$756$	0	0
$757$	2.24790	0.0817012	0.0408506	−	0.999165i	$-0.486993\pi$
$757$	2.24790	0.0817012	0.0408506	+	0.999165i	$0.486993\pi$
$758$	−1.51762	−0.0551223
$759$	0	0
$760$	−2.90307	−0.105305
$761$	−24.1174	−0.874254	−0.437127	−	0.899400i	$-0.644004\pi$
$761$	−24.1174	−0.874254	−0.437127	+	0.899400i	$0.644004\pi$
$762$	0	0
$763$	1.27578	0.0461863
$764$	−23.0209	−0.832866
$765$	0	0
$766$	30.4794	1.10127
$767$	−66.6256	−2.40571
$768$	0	0
$769$	31.4388	1.13371	0.566855	−	0.823817i	$-0.308160\pi$
$769$	31.4388	1.13371	0.566855	+	0.823817i	$0.308160\pi$
$770$	−0.294002	−0.0105951
$771$	0	0
$772$	−5.12383	−0.184411
$773$	−11.5078	−0.413906	−0.206953	−	0.978351i	$-0.566355\pi$
$773$	−11.5078	−0.413906	−0.206953	+	0.978351i	$0.566355\pi$
$774$	0	0
$775$	−37.1441	−1.33426
$776$	−28.0639	−1.00744
$777$	0	0
$778$	28.2251	1.01192
$779$	−6.47892	−0.232131
$780$	0	0
$781$	3.46463	0.123974
$782$	−1.51482	−0.0541700
$783$	0	0
$784$	−3.81994	−0.136426
$785$	12.9468	0.462091
$786$	0	0
$787$	−9.72240	−0.346566	−0.173283	−	0.984872i	$-0.555438\pi$
$787$	−9.72240	−0.346566	−0.173283	+	0.984872i	$0.555438\pi$
$788$	−0.104695	−0.00372959
$789$	0	0
$790$	1.64793	0.0586309
$791$	−4.87428	−0.173309
$792$	0	0
$793$	−5.01069	−0.177935
$794$	−21.4362	−0.760741
$795$	0	0
$796$	32.2972	1.14474
$797$	−44.9139	−1.59093	−0.795466	−	0.605999i	$-0.792774\pi$
$797$	−44.9139	−1.59093	−0.795466	+	0.605999i	$0.792774\pi$
$798$	0	0
$799$	−0.554245	−0.0196078
$800$	−25.8079	−0.912446
$801$	0	0
$802$	30.9877	1.09421
$803$	−9.56962	−0.337704
$804$	0	0
$805$	0.848940	0.0299212
$806$	−33.6674	−1.18588
$807$	0	0
$808$	−31.4385	−1.10600
$809$	−15.4892	−0.544570	−0.272285	−	0.962217i	$-0.587779\pi$
$809$	−15.4892	−0.544570	−0.272285	+	0.962217i	$0.587779\pi$
$810$	0	0
$811$	−19.5975	−0.688161	−0.344080	−	0.938940i	$-0.611809\pi$
$811$	−19.5975	−0.688161	−0.344080	+	0.938940i	$0.611809\pi$
$812$	2.68445	0.0942058
$813$	0	0
$814$	4.03641	0.141476
$815$	14.8152	0.518955
$816$	0	0
$817$	15.0903	0.527942
$818$	−8.02935	−0.280740
$819$	0	0
$820$	4.68080	0.163461
$821$	42.4380	1.48110	0.740549	−	0.672003i	$-0.234566\pi$
$821$	42.4380	1.48110	0.740549	+	0.672003i	$0.234566\pi$
$822$	0	0
$823$	43.0974	1.50228	0.751141	−	0.660142i	$-0.229504\pi$
$823$	43.0974	1.50228	0.751141	+	0.660142i	$0.229504\pi$
$824$	11.3205	0.394368
$825$	0	0
$826$	5.16039	0.179553
$827$	−9.38164	−0.326231	−0.163116	−	0.986607i	$-0.552154\pi$
$827$	−9.38164	−0.326231	−0.163116	+	0.986607i	$0.552154\pi$
$828$	0	0
$829$	−12.8922	−0.447765	−0.223883	−	0.974616i	$-0.571873\pi$
$829$	−12.8922	−0.447765	−0.223883	+	0.974616i	$0.571873\pi$
$830$	1.20518	0.0418323
$831$	0	0
$832$	−17.7336	−0.614802
$833$	5.53626	0.191820
$834$	0	0
$835$	11.6532	0.403275
$836$	1.93643	0.0669728
$837$	0	0
$838$	28.6547	0.989859
$839$	16.2624	0.561440	0.280720	−	0.959790i	$-0.409427\pi$
$839$	16.2624	0.561440	0.280720	+	0.959790i	$0.409427\pi$
$840$	0	0
$841$	−12.3921	−0.427314
$842$	32.3635	1.11532
$843$	0	0
$844$	30.3748	1.04554
$845$	9.17169	0.315516
$846$	0	0
$847$	0.484724	0.0166553
$848$	1.96569	0.0675021
$849$	0	0
$850$	2.90010	0.0994728
$851$	−11.6553	−0.399537
$852$	0	0
$853$	−44.2389	−1.51471	−0.757355	−	0.653003i	$-0.773509\pi$
$853$	−44.2389	−1.51471	−0.757355	+	0.653003i	$0.773509\pi$
$854$	0.388096	0.0132804
$855$	0	0
$856$	−32.6096	−1.11457
$857$	−16.1901	−0.553042	−0.276521	−	0.961008i	$-0.589181\pi$
$857$	−16.1901	−0.553042	−0.276521	+	0.961008i	$0.589181\pi$
$858$	0	0
$859$	−22.0520	−0.752405	−0.376203	−	0.926537i	$-0.622770\pi$
$859$	−22.0520	−0.752405	−0.376203	+	0.926537i	$0.622770\pi$
$860$	−10.9022	−0.371763
$861$	0	0
$862$	19.8253	0.675252
$863$	6.94346	0.236358	0.118179	−	0.992992i	$-0.462294\pi$
$863$	6.94346	0.236358	0.118179	+	0.992992i	$0.462294\pi$
$864$	0	0
$865$	−14.7691	−0.502166
$866$	19.9797	0.678937
$867$	0	0
$868$	−5.52797	−0.187631
$869$	−2.71696	−0.0921667
$870$	0	0
$871$	8.03867	0.272380
$872$	−7.07832	−0.239702
$873$	0	0
$874$	2.63763	0.0892192
$875$	−3.46129	−0.117013
$876$	0	0
$877$	−25.0036	−0.844312	−0.422156	−	0.906523i	$-0.638727\pi$
$877$	−25.0036	−0.844312	−0.422156	+	0.906523i	$0.638727\pi$
$878$	−8.63191	−0.291313
$879$	0	0
$880$	−0.427757	−0.0144197
$881$	11.3815	0.383452	0.191726	−	0.981448i	$-0.438591\pi$
$881$	11.3815	0.383452	0.191726	+	0.981448i	$0.438591\pi$
$882$	0	0
$883$	−4.34595	−0.146253	−0.0731264	−	0.997323i	$-0.523298\pi$
$883$	−4.34595	−0.146253	−0.0731264	+	0.997323i	$0.523298\pi$
$884$	−5.57247	−0.187422
$885$	0	0
$886$	−24.9917	−0.839611
$887$	26.1385	0.877644	0.438822	−	0.898574i	$-0.355396\pi$
$887$	26.1385	0.877644	0.438822	+	0.898574i	$0.355396\pi$
$888$	0	0
$889$	−8.48532	−0.284589
$890$	−0.148635	−0.00498227
$891$	0	0
$892$	−35.2956	−1.18178
$893$	0.965058	0.0322944
$894$	0	0
$895$	18.5125	0.618805
$896$	−4.27914	−0.142956
$897$	0	0
$898$	−2.58055	−0.0861139
$899$	−34.1998	−1.14063
$900$	0	0
$901$	−2.84888	−0.0949101
$902$	3.64041	0.121212
$903$	0	0
$904$	27.0436	0.899457
$905$	−0.978487	−0.0325260
$906$	0	0
$907$	−35.3582	−1.17405	−0.587024	−	0.809569i	$-0.699701\pi$
$907$	−35.3582	−1.17405	−0.587024	+	0.809569i	$0.699701\pi$
$908$	18.4794	0.613262
$909$	0	0
$910$	−1.47315	−0.0488346
$911$	2.36119	0.0782297	0.0391149	−	0.999235i	$-0.487546\pi$
$911$	2.36119	0.0782297	0.0391149	+	0.999235i	$0.487546\pi$
$912$	0	0
$913$	−1.98699	−0.0657597
$914$	−1.78757	−0.0591275
$915$	0	0
$916$	4.37198	0.144454
$917$	6.91707	0.228422
$918$	0	0
$919$	−26.6464	−0.878984	−0.439492	−	0.898247i	$-0.644841\pi$
$919$	−26.6464	−0.878984	−0.439492	+	0.898247i	$0.644841\pi$
$920$	−4.71012	−0.155288
$921$	0	0
$922$	17.5254	0.577169
$923$	17.3602	0.571418
$924$	0	0
$925$	22.3138	0.733672
$926$	5.35666	0.176031
$927$	0	0
$928$	−23.7622	−0.780032
$929$	−44.5025	−1.46008	−0.730040	−	0.683405i	$-0.760498\pi$
$929$	−44.5025	−1.46008	−0.730040	+	0.683405i	$0.760498\pi$
$930$	0	0
$931$	−9.63980	−0.315932
$932$	−15.9748	−0.523273
$933$	0	0
$934$	18.8632	0.617224
$935$	0.619950	0.0202745
$936$	0	0
$937$	−11.1821	−0.365302	−0.182651	−	0.983178i	$-0.558468\pi$
$937$	−11.1821	−0.365302	−0.182651	+	0.983178i	$0.558468\pi$
$938$	−0.622624	−0.0203294
$939$	0	0
$940$	−0.697222	−0.0227409
$941$	−14.0890	−0.459287	−0.229644	−	0.973275i	$-0.573756\pi$
$941$	−14.0890	−0.459287	−0.229644	+	0.973275i	$0.573756\pi$
$942$	0	0
$943$	−10.5118	−0.342311
$944$	7.50808	0.244367
$945$	0	0
$946$	−8.47901	−0.275676
$947$	29.9028	0.971711	0.485856	−	0.874039i	$-0.338508\pi$
$947$	29.9028	0.971711	0.485856	+	0.874039i	$0.338508\pi$
$948$	0	0
$949$	−47.9504	−1.55654
$950$	−5.04970	−0.163834
$951$	0	0
$952$	1.06681	0.0345757
$953$	54.3151	1.75944	0.879719	−	0.475495i	$-0.157731\pi$
$953$	54.3151	1.75944	0.879719	+	0.475495i	$0.157731\pi$
$954$	0	0
$955$	12.8330	0.415266
$956$	−4.03826	−0.130607
$957$	0	0
$958$	19.4472	0.628311
$959$	1.35460	0.0437425
$960$	0	0
$961$	39.4261	1.27181
$962$	20.2252	0.652086
$963$	0	0
$964$	32.6961	1.05307
$965$	2.85628	0.0919470
$966$	0	0
$967$	−37.5687	−1.20813	−0.604064	−	0.796936i	$-0.706453\pi$
$967$	−37.5687	−1.20813	−0.604064	+	0.796936i	$0.706453\pi$
$968$	−2.68936	−0.0864393
$969$	0	0
$970$	6.32930	0.203222
$971$	3.59090	0.115237	0.0576187	−	0.998339i	$-0.481649\pi$
$971$	3.59090	0.115237	0.0576187	+	0.998339i	$0.481649\pi$
$972$	0	0
$973$	7.24499	0.232264
$974$	−21.5612	−0.690865
$975$	0	0
$976$	0.564659	0.0180743
$977$	−52.1987	−1.66998	−0.834992	−	0.550262i	$-0.814528\pi$
$977$	−52.1987	−1.66998	−0.834992	+	0.550262i	$0.814528\pi$
$978$	0	0
$979$	0.245057	0.00783205
$980$	6.96444	0.222471
$981$	0	0
$982$	−29.4326	−0.939232
$983$	37.3564	1.19149	0.595743	−	0.803175i	$-0.296858\pi$
$983$	37.3564	1.19149	0.595743	+	0.803175i	$0.296858\pi$
$984$	0	0
$985$	0.0583620	0.00185957
$986$	2.67022	0.0850373
$987$	0	0
$988$	9.70285	0.308689
$989$	24.4834	0.778527
$990$	0	0
$991$	46.5458	1.47858	0.739288	−	0.673389i	$-0.235162\pi$
$991$	46.5458	1.47858	0.739288	+	0.673389i	$0.235162\pi$
$992$	48.9324	1.55360
$993$	0	0
$994$	−1.34461	−0.0426484
$995$	−18.0041	−0.570768
$996$	0	0
$997$	−36.7392	−1.16354	−0.581772	−	0.813352i	$-0.697640\pi$
$997$	−36.7392	−1.16354	−0.581772	+	0.813352i	$0.697640\pi$
$998$	23.7993	0.753353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.p.1.15	yes	25
3.2	odd	2		6039.2.a.m.1.11	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.11	✓	25	3.2	odd	2
6039.2.a.p.1.15	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)

\(f(q)\)	\(=\)	\(q+0.800654 q^{2} -1.35895 q^{4} +0.757549 q^{5} +0.484724 q^{7} -2.68936 q^{8} +O(q^{10})\)	\(q+0.800654 q^{2} -1.35895 q^{4} +0.757549 q^{5} +0.484724 q^{7} -2.68936 q^{8} +0.606535 q^{10} -1.00000 q^{11} -5.01069 q^{13} +0.388096 q^{14} +0.564659 q^{16} -0.818362 q^{17} +1.42494 q^{19} -1.02947 q^{20} -0.800654 q^{22} +2.31191 q^{23} -4.42612 q^{25} -4.01183 q^{26} -0.658717 q^{28} -4.07528 q^{29} +8.39203 q^{31} +5.83082 q^{32} -0.655225 q^{34} +0.367202 q^{35} -5.04139 q^{37} +1.14089 q^{38} -2.03732 q^{40} -4.54679 q^{41} +10.5901 q^{43} +1.35895 q^{44} +1.85104 q^{46} +0.677261 q^{47} -6.76504 q^{49} -3.54379 q^{50} +6.80930 q^{52} +3.48120 q^{53} -0.757549 q^{55} -1.30360 q^{56} -3.26289 q^{58} +13.2967 q^{59} +1.00000 q^{61} +6.71911 q^{62} +3.53915 q^{64} -3.79585 q^{65} -1.60430 q^{67} +1.11212 q^{68} +0.294002 q^{70} -3.46463 q^{71} +9.56962 q^{73} -4.03641 q^{74} -1.93643 q^{76} -0.484724 q^{77} +2.71696 q^{79} +0.427757 q^{80} -3.64041 q^{82} +1.98699 q^{83} -0.619950 q^{85} +8.47901 q^{86} +2.68936 q^{88} -0.245057 q^{89} -2.42880 q^{91} -3.14178 q^{92} +0.542252 q^{94} +1.07946 q^{95} +10.4352 q^{97} -5.41646 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 5 q^{2} + 25 q^{4} + 12 q^{5} - 4 q^{7} + 15 q^{8}+O(q^{10}) \) 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

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Database

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