LMFDB - Embedded newform 6069.2.a.z.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6069,2,Mod(1,6069)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6069.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6069, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6069 = 3 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6069.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,-9,6,3,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.4612089867$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 12x^{7} - 3x^{6} + 45x^{5} + 21x^{4} - 53x^{3} - 39x^{2} + 3 $ x^9 - 12x^7 - 3x^6 + 45x^5 + 21x^4 - 53x^3 - 39x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.61669$ of defining polynomial
Character	$\chi$	$=$	6069.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.61669 q^{2} -1.00000 q^{3} +4.84705 q^{4} +2.76381 q^{5} -2.61669 q^{6} -1.00000 q^{7} +7.44985 q^{8} +1.00000 q^{9} +7.23202 q^{10} +5.58538 q^{11} -4.84705 q^{12} -5.07348 q^{13} -2.61669 q^{14} -2.76381 q^{15} +9.79983 q^{16} +2.61669 q^{18} -5.85784 q^{19} +13.3963 q^{20} +1.00000 q^{21} +14.6152 q^{22} +7.19337 q^{23} -7.44985 q^{24} +2.63864 q^{25} -13.2757 q^{26} -1.00000 q^{27} -4.84705 q^{28} -0.275564 q^{29} -7.23202 q^{30} +0.624797 q^{31} +10.7434 q^{32} -5.58538 q^{33} -2.76381 q^{35} +4.84705 q^{36} +6.05974 q^{37} -15.3281 q^{38} +5.07348 q^{39} +20.5900 q^{40} +2.96242 q^{41} +2.61669 q^{42} +7.28189 q^{43} +27.0727 q^{44} +2.76381 q^{45} +18.8228 q^{46} +7.91595 q^{47} -9.79983 q^{48} +1.00000 q^{49} +6.90449 q^{50} -24.5914 q^{52} +13.2264 q^{53} -2.61669 q^{54} +15.4369 q^{55} -7.44985 q^{56} +5.85784 q^{57} -0.721064 q^{58} -9.62507 q^{59} -13.3963 q^{60} +2.87024 q^{61} +1.63490 q^{62} -1.00000 q^{63} +8.51243 q^{64} -14.0221 q^{65} -14.6152 q^{66} -6.08070 q^{67} -7.19337 q^{69} -7.23202 q^{70} -13.0963 q^{71} +7.44985 q^{72} +5.83555 q^{73} +15.8564 q^{74} -2.63864 q^{75} -28.3933 q^{76} -5.58538 q^{77} +13.2757 q^{78} +0.284512 q^{79} +27.0849 q^{80} +1.00000 q^{81} +7.75174 q^{82} +5.78939 q^{83} +4.84705 q^{84} +19.0544 q^{86} +0.275564 q^{87} +41.6103 q^{88} +12.5408 q^{89} +7.23202 q^{90} +5.07348 q^{91} +34.8667 q^{92} -0.624797 q^{93} +20.7136 q^{94} -16.1899 q^{95} -10.7434 q^{96} -0.129518 q^{97} +2.61669 q^{98} +5.58538 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} + 6 q^{4} + 3 q^{5} - 9 q^{7} + 9 q^{8} + 9 q^{9} + 12 q^{10} + 18 q^{11} - 6 q^{12} - 21 q^{13} - 3 q^{15} - 9 q^{19} + 15 q^{20} + 9 q^{21} + 6 q^{22} - 9 q^{24} + 3 q^{26} - 9 q^{27} - 6 q^{28}+ \cdots + 18 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 + 6 * q^4 + 3 * q^5 - 9 * q^7 + 9 * q^8 + 9 * q^9 + 12 * q^10 + 18 * q^11 - 6 * q^12 - 21 * q^13 - 3 * q^15 - 9 * q^19 + 15 * q^20 + 9 * q^21 + 6 * q^22 - 9 * q^24 + 3 * q^26 - 9 * q^27 - 6 * q^28 + 6 * q^29 - 12 * q^30 + 30 * q^31 + 3 * q^32 - 18 * q^33 - 3 * q^35 + 6 * q^36 + 12 * q^37 - 36 * q^38 + 21 * q^39 + 30 * q^40 + 9 * q^41 + 30 * q^44 + 3 * q^45 + 33 * q^46 - 9 * q^47 + 9 * q^49 - 12 * q^50 - 12 * q^52 + 30 * q^55 - 9 * q^56 + 9 * q^57 + 9 * q^58 - 3 * q^59 - 15 * q^60 + 33 * q^61 - 12 * q^62 - 9 * q^63 - 15 * q^64 - 15 * q^65 - 6 * q^66 - 15 * q^67 - 12 * q^70 + 9 * q^72 + 21 * q^73 + 6 * q^74 + 3 * q^76 - 18 * q^77 - 3 * q^78 + 30 * q^79 + 9 * q^81 + 9 * q^82 + 6 * q^84 + 72 * q^86 - 6 * q^87 + 48 * q^88 - 12 * q^89 + 12 * q^90 + 21 * q^91 + 48 * q^92 - 30 * q^93 + 48 * q^94 - 18 * q^95 - 3 * q^96 + 18 * q^99

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 $

$2$	2.61669	1.85028	0.925139	−	0.379629i	$-0.123948\pi$
$2$	2.61669	1.85028	0.925139	+	0.379629i	$0.123948\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.84705	2.42353
$5$	2.76381	1.23601	0.618006	−	0.786173i	$-0.287941\pi$
$5$	2.76381	1.23601	0.618006	+	0.786173i	$0.287941\pi$
$6$	−2.61669	−1.06826
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	7.44985	2.63392
$9$	1.00000	0.333333
$10$	7.23202	2.28697
$11$	5.58538	1.68406	0.842028	−	0.539434i	$-0.181362\pi$
$11$	5.58538	1.68406	0.842028	+	0.539434i	$0.181362\pi$
$12$	−4.84705	−1.39922
$13$	−5.07348	−1.40713	−0.703565	−	0.710631i	$-0.748410\pi$
$13$	−5.07348	−1.40713	−0.703565	+	0.710631i	$0.748410\pi$
$14$	−2.61669	−0.699339
$15$	−2.76381	−0.713612
$16$	9.79983	2.44996
$17$	0	0
$17$	0	0
$18$	2.61669	0.616759
$19$	−5.85784	−1.34388	−0.671940	−	0.740606i	$-0.734539\pi$
$19$	−5.85784	−1.34388	−0.671940	+	0.740606i	$0.734539\pi$
$20$	13.3963	2.99551
$21$	1.00000	0.218218
$22$	14.6152	3.11597
$23$	7.19337	1.49992	0.749961	−	0.661482i	$-0.230072\pi$
$23$	7.19337	1.49992	0.749961	+	0.661482i	$0.230072\pi$
$24$	−7.44985	−1.52069
$25$	2.63864	0.527727
$26$	−13.2757	−2.60358
$27$	−1.00000	−0.192450
$28$	−4.84705	−0.916007
$29$	−0.275564	−0.0511709	−0.0255855	−	0.999673i	$-0.508145\pi$
$29$	−0.275564	−0.0511709	−0.0255855	+	0.999673i	$0.508145\pi$
$30$	−7.23202	−1.32038
$31$	0.624797	0.112217	0.0561085	−	0.998425i	$-0.482131\pi$
$31$	0.624797	0.112217	0.0561085	+	0.998425i	$0.482131\pi$
$32$	10.7434	1.89918
$33$	−5.58538	−0.972290
$34$	0	0
$35$	−2.76381	−0.467169
$36$	4.84705	0.807842
$37$	6.05974	0.996215	0.498107	−	0.867115i	$-0.334029\pi$
$37$	6.05974	0.996215	0.498107	+	0.867115i	$0.334029\pi$
$38$	−15.3281	−2.48655
$39$	5.07348	0.812407
$40$	20.5900	3.25556
$41$	2.96242	0.462653	0.231326	−	0.972876i	$-0.425693\pi$
$41$	2.96242	0.462653	0.231326	+	0.972876i	$0.425693\pi$
$42$	2.61669	0.403764
$43$	7.28189	1.11048	0.555239	−	0.831691i	$-0.312627\pi$
$43$	7.28189	1.11048	0.555239	+	0.831691i	$0.312627\pi$
$44$	27.0727	4.08136
$45$	2.76381	0.412004
$46$	18.8228	2.77527
$47$	7.91595	1.15466	0.577330	−	0.816511i	$-0.304095\pi$
$47$	7.91595	1.15466	0.577330	+	0.816511i	$0.304095\pi$
$48$	−9.79983	−1.41448
$49$	1.00000	0.142857
$50$	6.90449	0.976442
$51$	0	0
$52$	−24.5914	−3.41022
$53$	13.2264	1.81678	0.908391	−	0.418121i	$-0.137311\pi$
$53$	13.2264	1.81678	0.908391	+	0.418121i	$0.137311\pi$
$54$	−2.61669	−0.356086
$55$	15.4369	2.08151
$56$	−7.44985	−0.995529
$57$	5.85784	0.775889
$58$	−0.721064	−0.0946804
$59$	−9.62507	−1.25308	−0.626538	−	0.779391i	$-0.715529\pi$
$59$	−9.62507	−1.25308	−0.626538	+	0.779391i	$0.715529\pi$
$60$	−13.3963	−1.72946
$61$	2.87024	0.367496	0.183748	−	0.982973i	$-0.441177\pi$
$61$	2.87024	0.367496	0.183748	+	0.982973i	$0.441177\pi$
$62$	1.63490	0.207632
$63$	−1.00000	−0.125988
$64$	8.51243	1.06405
$65$	−14.0221	−1.73923
$66$	−14.6152	−1.79901
$67$	−6.08070	−0.742876	−0.371438	−	0.928458i	$-0.621135\pi$
$67$	−6.08070	−0.742876	−0.371438	+	0.928458i	$0.621135\pi$
$68$	0	0
$69$	−7.19337	−0.865981
$70$	−7.23202	−0.864392
$71$	−13.0963	−1.55424	−0.777122	−	0.629350i	$-0.783321\pi$
$71$	−13.0963	−1.55424	−0.777122	+	0.629350i	$0.783321\pi$
$72$	7.44985	0.877974
$73$	5.83555	0.682999	0.341499	−	0.939882i	$-0.389065\pi$
$73$	5.83555	0.682999	0.341499	+	0.939882i	$0.389065\pi$
$74$	15.8564	1.84327
$75$	−2.63864	−0.304683
$76$	−28.3933	−3.25693
$77$	−5.58538	−0.636513
$78$	13.2757	1.50318
$79$	0.284512	0.0320101	0.0160051	−	0.999872i	$-0.494905\pi$
$79$	0.284512	0.0320101	0.0160051	+	0.999872i	$0.494905\pi$
$80$	27.0849	3.02818
$81$	1.00000	0.111111
$82$	7.75174	0.856036
$83$	5.78939	0.635468	0.317734	−	0.948180i	$-0.397078\pi$
$83$	5.78939	0.635468	0.317734	+	0.948180i	$0.397078\pi$
$84$	4.84705	0.528857
$85$	0	0
$86$	19.0544	2.05469
$87$	0.275564	0.0295435
$88$	41.6103	4.43567
$89$	12.5408	1.32933	0.664663	−	0.747143i	$-0.268575\pi$
$89$	12.5408	1.32933	0.664663	+	0.747143i	$0.268575\pi$
$90$	7.23202	0.762322
$91$	5.07348	0.531845
$92$	34.8667	3.63510
$93$	−0.624797	−0.0647885
$94$	20.7136	2.13644
$95$	−16.1899	−1.66105
$96$	−10.7434	−1.09649
$97$	−0.129518	−0.0131506	−0.00657529	−	0.999978i	$-0.502093\pi$
$97$	−0.129518	−0.0131506	−0.00657529	+	0.999978i	$0.502093\pi$
$98$	2.61669	0.264325
$99$	5.58538	0.561352
$100$	12.7896	1.27896
$101$	−19.7540	−1.96560	−0.982798	−	0.184685i	$-0.940874\pi$
$101$	−19.7540	−1.96560	−0.982798	+	0.184685i	$0.940874\pi$
$102$	0	0
$103$	2.08651	0.205590	0.102795	−	0.994703i	$-0.467221\pi$
$103$	2.08651	0.205590	0.102795	+	0.994703i	$0.467221\pi$
$104$	−37.7967	−3.70627
$105$	2.76381	0.269720
$106$	34.6093	3.36155
$107$	−10.5516	−1.02006	−0.510029	−	0.860157i	$-0.670365\pi$
$107$	−10.5516	−1.02006	−0.510029	+	0.860157i	$0.670365\pi$
$108$	−4.84705	−0.466408
$109$	−13.1386	−1.25845	−0.629225	−	0.777223i	$-0.716628\pi$
$109$	−13.1386	−1.25845	−0.629225	+	0.777223i	$0.716628\pi$
$110$	40.3936	3.85138
$111$	−6.05974	−0.575165
$112$	−9.79983	−0.925997
$113$	−14.7507	−1.38763	−0.693815	−	0.720153i	$-0.744071\pi$
$113$	−14.7507	−1.38763	−0.693815	+	0.720153i	$0.744071\pi$
$114$	15.3281	1.43561
$115$	19.8811	1.85392
$116$	−1.33567	−0.124014
$117$	−5.07348	−0.469043
$118$	−25.1858	−2.31854
$119$	0	0
$120$	−20.5900	−1.87960
$121$	20.1965	1.83604
$122$	7.51051	0.679970
$123$	−2.96242	−0.267113
$124$	3.02843	0.271961
$125$	−6.52636	−0.583735
$126$	−2.61669	−0.233113
$127$	−11.4133	−1.01277	−0.506384	−	0.862308i	$-0.669018\pi$
$127$	−11.4133	−1.01277	−0.506384	+	0.862308i	$0.669018\pi$
$128$	0.787603	0.0696149
$129$	−7.28189	−0.641135
$130$	−36.6915	−3.21806
$131$	5.89815	0.515324	0.257662	−	0.966235i	$-0.417048\pi$
$131$	5.89815	0.515324	0.257662	+	0.966235i	$0.417048\pi$
$132$	−27.0727	−2.35637
$133$	5.85784	0.507939
$134$	−15.9113	−1.37453
$135$	−2.76381	−0.237871
$136$	0	0
$137$	−6.03719	−0.515792	−0.257896	−	0.966173i	$-0.583029\pi$
$137$	−6.03719	−0.515792	−0.257896	+	0.966173i	$0.583029\pi$
$138$	−18.8228	−1.60230
$139$	13.1383	1.11437	0.557186	−	0.830388i	$-0.311881\pi$
$139$	13.1383	1.11437	0.557186	+	0.830388i	$0.311881\pi$
$140$	−13.3963	−1.13220
$141$	−7.91595	−0.666643
$142$	−34.2689	−2.87578
$143$	−28.3373	−2.36969
$144$	9.79983	0.816653
$145$	−0.761605	−0.0632479
$146$	15.2698	1.26374
$147$	−1.00000	−0.0824786
$148$	29.3719	2.41435
$149$	−9.76050	−0.799611	−0.399805	−	0.916600i	$-0.630922\pi$
$149$	−9.76050	−0.799611	−0.399805	+	0.916600i	$0.630922\pi$
$150$	−6.90449	−0.563749
$151$	−6.38384	−0.519510	−0.259755	−	0.965675i	$-0.583642\pi$
$151$	−6.38384	−0.519510	−0.259755	+	0.965675i	$0.583642\pi$
$152$	−43.6400	−3.53967
$153$	0	0
$154$	−14.6152	−1.17773
$155$	1.72682	0.138702
$156$	24.5914	1.96889
$157$	−7.05575	−0.563110	−0.281555	−	0.959545i	$-0.590850\pi$
$157$	−7.05575	−0.563110	−0.281555	+	0.959545i	$0.590850\pi$
$158$	0.744480	0.0592276
$159$	−13.2264	−1.04892
$160$	29.6927	2.34741
$161$	−7.19337	−0.566917
$162$	2.61669	0.205586
$163$	6.11704	0.479124	0.239562	−	0.970881i	$-0.422996\pi$
$163$	6.11704	0.479124	0.239562	+	0.970881i	$0.422996\pi$
$164$	14.3590	1.12125
$165$	−15.4369	−1.20176
$166$	15.1490	1.17579
$167$	−11.6264	−0.899678	−0.449839	−	0.893110i	$-0.648519\pi$
$167$	−11.6264	−0.899678	−0.449839	+	0.893110i	$0.648519\pi$
$168$	7.44985	0.574769
$169$	12.7402	0.980016
$170$	0	0
$171$	−5.85784	−0.447960
$172$	35.2957	2.69127
$173$	17.4498	1.32668	0.663341	−	0.748317i	$-0.269138\pi$
$173$	17.4498	1.32668	0.663341	+	0.748317i	$0.269138\pi$
$174$	0.721064	0.0546637
$175$	−2.63864	−0.199462
$176$	54.7358	4.12587
$177$	9.62507	0.723464
$178$	32.8155	2.45962
$179$	4.87845	0.364632	0.182316	−	0.983240i	$-0.441641\pi$
$179$	4.87845	0.364632	0.182316	+	0.983240i	$0.441641\pi$
$180$	13.3963	0.998503
$181$	5.44597	0.404796	0.202398	−	0.979303i	$-0.435127\pi$
$181$	5.44597	0.404796	0.202398	+	0.979303i	$0.435127\pi$
$182$	13.2757	0.984061
$183$	−2.87024	−0.212174
$184$	53.5896	3.95068
$185$	16.7480	1.23133
$186$	−1.63490	−0.119877
$187$	0	0
$188$	38.3691	2.79835
$189$	1.00000	0.0727393
$190$	−42.3640	−3.07341
$191$	8.69999	0.629509	0.314755	−	0.949173i	$-0.398078\pi$
$191$	8.69999	0.629509	0.314755	+	0.949173i	$0.398078\pi$
$192$	−8.51243	−0.614332
$193$	−5.38757	−0.387806	−0.193903	−	0.981021i	$-0.562115\pi$
$193$	−5.38757	−0.387806	−0.193903	+	0.981021i	$0.562115\pi$
$194$	−0.338909	−0.0243322
$195$	14.0221	1.00415
$196$	4.84705	0.346218
$197$	−19.1210	−1.36232	−0.681158	−	0.732137i	$-0.738523\pi$
$197$	−19.1210	−1.36232	−0.681158	+	0.732137i	$0.738523\pi$
$198$	14.6152	1.03866
$199$	15.6272	1.10778	0.553892	−	0.832588i	$-0.313142\pi$
$199$	15.6272	1.10778	0.553892	+	0.832588i	$0.313142\pi$
$200$	19.6575	1.38999
$201$	6.08070	0.428900
$202$	−51.6900	−3.63690
$203$	0.275564	0.0193408
$204$	0	0
$205$	8.18757	0.571845
$206$	5.45975	0.380399
$207$	7.19337	0.499974
$208$	−49.7192	−3.44741
$209$	−32.7183	−2.26317
$210$	7.23202	0.499057
$211$	0.403460	0.0277754	0.0138877	−	0.999904i	$-0.495579\pi$
$211$	0.403460	0.0277754	0.0138877	+	0.999904i	$0.495579\pi$
$212$	64.1090	4.40302
$213$	13.0963	0.897343
$214$	−27.6101	−1.88739
$215$	20.1258	1.37256
$216$	−7.44985	−0.506898
$217$	−0.624797	−0.0424140
$218$	−34.3796	−2.32848
$219$	−5.83555	−0.394330
$220$	74.8236	5.04461
$221$	0	0
$222$	−15.8564	−1.06421
$223$	−14.9272	−0.999600	−0.499800	−	0.866141i	$-0.666593\pi$
$223$	−14.9272	−0.999600	−0.499800	+	0.866141i	$0.666593\pi$
$224$	−10.7434	−0.717823
$225$	2.63864	0.175909
$226$	−38.5980	−2.56750
$227$	21.6525	1.43713	0.718564	−	0.695461i	$-0.244800\pi$
$227$	21.6525	1.43713	0.718564	+	0.695461i	$0.244800\pi$
$228$	28.3933	1.88039
$229$	−7.26368	−0.479997	−0.239999	−	0.970773i	$-0.577147\pi$
$229$	−7.26368	−0.479997	−0.239999	+	0.970773i	$0.577147\pi$
$230$	52.0227	3.43027
$231$	5.58538	0.367491
$232$	−2.05291	−0.134780
$233$	12.4353	0.814663	0.407331	−	0.913280i	$-0.366459\pi$
$233$	12.4353	0.814663	0.407331	+	0.913280i	$0.366459\pi$
$234$	−13.2757	−0.867861
$235$	21.8782	1.42717
$236$	−46.6532	−3.03687
$237$	−0.284512	−0.0184811
$238$	0	0
$239$	−15.8342	−1.02423	−0.512114	−	0.858918i	$-0.671137\pi$
$239$	−15.8342	−1.02423	−0.512114	+	0.858918i	$0.671137\pi$
$240$	−27.0849	−1.74832
$241$	−15.5853	−1.00394	−0.501968	−	0.864886i	$-0.667391\pi$
$241$	−15.5853	−1.00394	−0.501968	+	0.864886i	$0.667391\pi$
$242$	52.8479	3.39719
$243$	−1.00000	−0.0641500
$244$	13.9122	0.890637
$245$	2.76381	0.176573
$246$	−7.75174	−0.494233
$247$	29.7196	1.89101
$248$	4.65465	0.295570
$249$	−5.78939	−0.366888
$250$	−17.0774	−1.08007
$251$	−2.55446	−0.161236	−0.0806181	−	0.996745i	$-0.525689\pi$
$251$	−2.55446	−0.161236	−0.0806181	+	0.996745i	$0.525689\pi$
$252$	−4.84705	−0.305336
$253$	40.1777	2.52595
$254$	−29.8651	−1.87390
$255$	0	0
$256$	−14.9640	−0.935247
$257$	−10.3379	−0.644859	−0.322430	−	0.946593i	$-0.604500\pi$
$257$	−10.3379	−0.644859	−0.322430	+	0.946593i	$0.604500\pi$
$258$	−19.0544	−1.18628
$259$	−6.05974	−0.376534
$260$	−67.9660	−4.21507
$261$	−0.275564	−0.0170570
$262$	15.4336	0.953492
$263$	−19.0391	−1.17400	−0.587000	−	0.809587i	$-0.699691\pi$
$263$	−19.0391	−1.17400	−0.587000	+	0.809587i	$0.699691\pi$
$264$	−41.6103	−2.56094
$265$	36.5552	2.24557
$266$	15.3281	0.939828
$267$	−12.5408	−0.767487
$268$	−29.4735	−1.80038
$269$	−12.4382	−0.758368	−0.379184	−	0.925321i	$-0.623795\pi$
$269$	−12.4382	−0.758368	−0.379184	+	0.925321i	$0.623795\pi$
$270$	−7.23202	−0.440127
$271$	19.0256	1.15572	0.577862	−	0.816135i	$-0.303887\pi$
$271$	19.0256	1.15572	0.577862	+	0.816135i	$0.303887\pi$
$272$	0	0
$273$	−5.07348	−0.307061
$274$	−15.7974	−0.954359
$275$	14.7378	0.888722
$276$	−34.8667	−2.09873
$277$	21.1859	1.27294	0.636469	−	0.771302i	$-0.280394\pi$
$277$	21.1859	1.27294	0.636469	+	0.771302i	$0.280394\pi$
$278$	34.3787	2.06190
$279$	0.624797	0.0374056
$280$	−20.5900	−1.23049
$281$	15.0367	0.897013	0.448506	−	0.893780i	$-0.351956\pi$
$281$	15.0367	0.897013	0.448506	+	0.893780i	$0.351956\pi$
$282$	−20.7136	−1.23348
$283$	22.9827	1.36618	0.683089	−	0.730335i	$-0.260636\pi$
$283$	22.9827	1.36618	0.683089	+	0.730335i	$0.260636\pi$
$284$	−63.4784	−3.76675
$285$	16.1899	0.959009
$286$	−74.1499	−4.38458
$287$	−2.96242	−0.174866
$288$	10.7434	0.633060
$289$	0	0
$290$	−1.99288	−0.117026
$291$	0.129518	0.00759249
$292$	28.2852	1.65527
$293$	−21.8382	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$293$	−21.8382	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$294$	−2.61669	−0.152608
$295$	−26.6018	−1.54882
$296$	45.1441	2.62395
$297$	−5.58538	−0.324097
$298$	−25.5402	−1.47950
$299$	−36.4954	−2.11059
$300$	−12.7896	−0.738409
$301$	−7.28189	−0.419721
$302$	−16.7045	−0.961237
$303$	19.7540	1.13484
$304$	−57.4058	−3.29245
$305$	7.93278	0.454230
$306$	0	0
$307$	−26.8841	−1.53436	−0.767180	−	0.641432i	$-0.778341\pi$
$307$	−26.8841	−1.53436	−0.767180	+	0.641432i	$0.778341\pi$
$308$	−27.0727	−1.54261
$309$	−2.08651	−0.118697
$310$	4.51855	0.256636
$311$	7.26449	0.411931	0.205966	−	0.978559i	$-0.433966\pi$
$311$	7.26449	0.411931	0.205966	+	0.978559i	$0.433966\pi$
$312$	37.7967	2.13982
$313$	−8.34124	−0.471475	−0.235737	−	0.971817i	$-0.575751\pi$
$313$	−8.34124	−0.471475	−0.235737	+	0.971817i	$0.575751\pi$
$314$	−18.4627	−1.04191
$315$	−2.76381	−0.155723
$316$	1.37905	0.0775774
$317$	8.02321	0.450628	0.225314	−	0.974286i	$-0.427659\pi$
$317$	8.02321	0.450628	0.225314	+	0.974286i	$0.427659\pi$
$318$	−34.6093	−1.94079
$319$	−1.53913	−0.0861747
$320$	23.5267	1.31518
$321$	10.5516	0.588930
$322$	−18.8228	−1.04895
$323$	0	0
$324$	4.84705	0.269281
$325$	−13.3871	−0.742581
$326$	16.0064	0.886512
$327$	13.1386	0.726566
$328$	22.0696	1.21859
$329$	−7.91595	−0.436421
$330$	−40.3936	−2.22359
$331$	−17.7841	−0.977504	−0.488752	−	0.872423i	$-0.662548\pi$
$331$	−17.7841	−0.977504	−0.488752	+	0.872423i	$0.662548\pi$
$332$	28.0615	1.54008
$333$	6.05974	0.332072
$334$	−30.4227	−1.66465
$335$	−16.8059	−0.918204
$336$	9.79983	0.534625
$337$	−19.3293	−1.05293	−0.526467	−	0.850195i	$-0.676484\pi$
$337$	−19.3293	−1.05293	−0.526467	+	0.850195i	$0.676484\pi$
$338$	33.3371	1.81330
$339$	14.7507	0.801149
$340$	0	0
$341$	3.48973	0.188980
$342$	−15.3281	−0.828850
$343$	−1.00000	−0.0539949
$344$	54.2490	2.92491
$345$	−19.8811	−1.07036
$346$	45.6607	2.45473
$347$	−10.7771	−0.578543	−0.289271	−	0.957247i	$-0.593413\pi$
$347$	−10.7771	−0.578543	−0.289271	+	0.957247i	$0.593413\pi$
$348$	1.33567	0.0715996
$349$	17.4483	0.933984	0.466992	−	0.884262i	$-0.345338\pi$
$349$	17.4483	0.933984	0.466992	+	0.884262i	$0.345338\pi$
$350$	−6.90449	−0.369060
$351$	5.07348	0.270802
$352$	60.0059	3.19833
$353$	−17.6615	−0.940028	−0.470014	−	0.882659i	$-0.655751\pi$
$353$	−17.6615	−0.940028	−0.470014	+	0.882659i	$0.655751\pi$
$354$	25.1858	1.33861
$355$	−36.1956	−1.92106
$356$	60.7862	3.22166
$357$	0	0
$358$	12.7654	0.674671
$359$	29.1158	1.53667	0.768337	−	0.640045i	$-0.221084\pi$
$359$	29.1158	1.53667	0.768337	+	0.640045i	$0.221084\pi$
$360$	20.5900	1.08519
$361$	15.3142	0.806013
$362$	14.2504	0.748985
$363$	−20.1965	−1.06004
$364$	24.5914	1.28894
$365$	16.1283	0.844195
$366$	−7.51051	−0.392581
$367$	−2.95843	−0.154429	−0.0772143	−	0.997015i	$-0.524603\pi$
$367$	−2.95843	−0.154429	−0.0772143	+	0.997015i	$0.524603\pi$
$368$	70.4939	3.67475
$369$	2.96242	0.154218
$370$	43.8242	2.27831
$371$	−13.2264	−0.686679
$372$	−3.02843	−0.157017
$373$	−21.2072	−1.09807	−0.549034	−	0.835800i	$-0.685004\pi$
$373$	−21.2072	−1.09807	−0.549034	+	0.835800i	$0.685004\pi$
$374$	0	0
$375$	6.52636	0.337020
$376$	58.9727	3.04128
$377$	1.39807	0.0720041
$378$	2.61669	0.134588
$379$	36.9808	1.89958	0.949789	−	0.312891i	$-0.101297\pi$
$379$	36.9808	1.89958	0.949789	+	0.312891i	$0.101297\pi$
$380$	−78.4735	−4.02561
$381$	11.4133	0.584721
$382$	22.7652	1.16477
$383$	16.5879	0.847600	0.423800	−	0.905756i	$-0.360696\pi$
$383$	16.5879	0.847600	0.423800	+	0.905756i	$0.360696\pi$
$384$	−0.787603	−0.0401922
$385$	−15.4369	−0.786738
$386$	−14.0976	−0.717549
$387$	7.28189	0.370159
$388$	−0.627782	−0.0318708
$389$	−2.29142	−0.116179	−0.0580897	−	0.998311i	$-0.518501\pi$
$389$	−2.29142	−0.116179	−0.0580897	+	0.998311i	$0.518501\pi$
$390$	36.6915	1.85795
$391$	0	0
$392$	7.44985	0.376274
$393$	−5.89815	−0.297522
$394$	−50.0337	−2.52066
$395$	0.786337	0.0395649
$396$	27.0727	1.36045
$397$	−13.8984	−0.697540	−0.348770	−	0.937208i	$-0.613401\pi$
$397$	−13.8984	−0.697540	−0.348770	+	0.937208i	$0.613401\pi$
$398$	40.8916	2.04971
$399$	−5.85784	−0.293259
$400$	25.8582	1.29291
$401$	7.95363	0.397185	0.198593	−	0.980082i	$-0.436363\pi$
$401$	7.95363	0.397185	0.198593	+	0.980082i	$0.436363\pi$
$402$	15.9113	0.793583
$403$	−3.16990	−0.157904
$404$	−95.7487	−4.76367
$405$	2.76381	0.137335
$406$	0.721064	0.0357858
$407$	33.8459	1.67768
$408$	0	0
$409$	−28.1423	−1.39155	−0.695773	−	0.718262i	$-0.744938\pi$
$409$	−28.1423	−1.39155	−0.695773	+	0.718262i	$0.744938\pi$
$410$	21.4243	1.05807
$411$	6.03719	0.297793
$412$	10.1134	0.498253
$413$	9.62507	0.473619
$414$	18.8228	0.925091
$415$	16.0008	0.785447
$416$	−54.5064	−2.67240
$417$	−13.1383	−0.643383
$418$	−85.6135	−4.18749
$419$	−9.30811	−0.454731	−0.227365	−	0.973809i	$-0.573011\pi$
$419$	−9.30811	−0.454731	−0.227365	+	0.973809i	$0.573011\pi$
$420$	13.3963	0.653674
$421$	−23.1810	−1.12977	−0.564887	−	0.825168i	$-0.691080\pi$
$421$	−23.1810	−1.12977	−0.564887	+	0.825168i	$0.691080\pi$
$422$	1.05573	0.0513921
$423$	7.91595	0.384887
$424$	98.5346	4.78526
$425$	0	0
$426$	34.2689	1.66033
$427$	−2.87024	−0.138901
$428$	−51.1440	−2.47214
$429$	28.3373	1.36814
$430$	52.6628	2.53963
$431$	−33.2822	−1.60315	−0.801574	−	0.597896i	$-0.796004\pi$
$431$	−33.2822	−1.60315	−0.801574	+	0.597896i	$0.796004\pi$
$432$	−9.79983	−0.471495
$433$	−1.07693	−0.0517542	−0.0258771	−	0.999665i	$-0.508238\pi$
$433$	−1.07693	−0.0517542	−0.0258771	+	0.999665i	$0.508238\pi$
$434$	−1.63490	−0.0784777
$435$	0.761605	0.0365162
$436$	−63.6835	−3.04989
$437$	−42.1376	−2.01572
$438$	−15.2698	−0.729619
$439$	−26.5568	−1.26749	−0.633743	−	0.773544i	$-0.718482\pi$
$439$	−26.5568	−1.26749	−0.633743	+	0.773544i	$0.718482\pi$
$440$	115.003	5.48254
$441$	1.00000	0.0476190
$442$	0	0
$443$	−11.6004	−0.551151	−0.275575	−	0.961279i	$-0.588868\pi$
$443$	−11.6004	−0.551151	−0.275575	+	0.961279i	$0.588868\pi$
$444$	−29.3719	−1.39393
$445$	34.6605	1.64306
$446$	−39.0599	−1.84954
$447$	9.76050	0.461656
$448$	−8.51243	−0.402175
$449$	−12.0212	−0.567314	−0.283657	−	0.958926i	$-0.591548\pi$
$449$	−12.0212	−0.567314	−0.283657	+	0.958926i	$0.591548\pi$
$450$	6.90449	0.325481
$451$	16.5463	0.779133
$452$	−71.4975	−3.36296
$453$	6.38384	0.299939
$454$	56.6579	2.65909
$455$	14.0221	0.657368
$456$	43.6400	2.04363
$457$	−26.7807	−1.25275	−0.626374	−	0.779523i	$-0.715462\pi$
$457$	−26.7807	−1.25275	−0.626374	+	0.779523i	$0.715462\pi$
$458$	−19.0068	−0.888128
$459$	0	0
$460$	96.3648	4.49303
$461$	10.3367	0.481430	0.240715	−	0.970596i	$-0.422618\pi$
$461$	10.3367	0.481430	0.240715	+	0.970596i	$0.422618\pi$
$462$	14.6152	0.679961
$463$	−29.3959	−1.36614	−0.683072	−	0.730351i	$-0.739357\pi$
$463$	−29.3959	−1.36614	−0.683072	+	0.730351i	$0.739357\pi$
$464$	−2.70048	−0.125367
$465$	−1.72682	−0.0800794
$466$	32.5393	1.50735
$467$	9.33117	0.431795	0.215898	−	0.976416i	$-0.430732\pi$
$467$	9.33117	0.431795	0.215898	+	0.976416i	$0.430732\pi$
$468$	−24.5914	−1.13674
$469$	6.08070	0.280781
$470$	57.2484	2.64067
$471$	7.05575	0.325112
$472$	−71.7053	−3.30051
$473$	40.6721	1.87011
$474$	−0.744480	−0.0341951
$475$	−15.4567	−0.709202
$476$	0	0
$477$	13.2264	0.605594
$478$	−41.4331	−1.89511
$479$	24.3969	1.11472	0.557360	−	0.830271i	$-0.311814\pi$
$479$	24.3969	1.11472	0.557360	+	0.830271i	$0.311814\pi$
$480$	−29.6927	−1.35528
$481$	−30.7440	−1.40180
$482$	−40.7818	−1.85756
$483$	7.19337	0.327310
$484$	97.8935	4.44970
$485$	−0.357963	−0.0162543
$486$	−2.61669	−0.118695
$487$	30.1340	1.36550	0.682750	−	0.730652i	$-0.260784\pi$
$487$	30.1340	1.36550	0.682750	+	0.730652i	$0.260784\pi$
$488$	21.3828	0.967956
$489$	−6.11704	−0.276622
$490$	7.23202	0.326710
$491$	−4.36531	−0.197004	−0.0985018	−	0.995137i	$-0.531405\pi$
$491$	−4.36531	−0.197004	−0.0985018	+	0.995137i	$0.531405\pi$
$492$	−14.3590	−0.647355
$493$	0	0
$494$	77.7670	3.49890
$495$	15.4369	0.693838
$496$	6.12291	0.274927
$497$	13.0963	0.587449
$498$	−15.1490	−0.678844
$499$	37.4564	1.67678	0.838390	−	0.545071i	$-0.183497\pi$
$499$	37.4564	1.67678	0.838390	+	0.545071i	$0.183497\pi$
$500$	−31.6336	−1.41470
$501$	11.6264	0.519429
$502$	−6.68423	−0.298332
$503$	−11.8848	−0.529918	−0.264959	−	0.964260i	$-0.585358\pi$
$503$	−11.8848	−0.529918	−0.264959	+	0.964260i	$0.585358\pi$
$504$	−7.44985	−0.331843
$505$	−54.5962	−2.42950
$506$	105.133	4.67371
$507$	−12.7402	−0.565812
$508$	−55.3209	−2.45447
$509$	−4.59528	−0.203682	−0.101841	−	0.994801i	$-0.532473\pi$
$509$	−4.59528	−0.203682	−0.101841	+	0.994801i	$0.532473\pi$
$510$	0	0
$511$	−5.83555	−0.258149
$512$	−40.7312	−1.80008
$513$	5.85784	0.258630
$514$	−27.0510	−1.19317
$515$	5.76672	0.254112
$516$	−35.2957	−1.55381
$517$	44.2136	1.94451
$518$	−15.8564	−0.696692
$519$	−17.4498	−0.765961
$520$	−104.463	−4.58100
$521$	15.8135	0.692804	0.346402	−	0.938086i	$-0.387403\pi$
$521$	15.8135	0.692804	0.346402	+	0.938086i	$0.387403\pi$
$522$	−0.721064	−0.0315601
$523$	36.0341	1.57566	0.787830	−	0.615893i	$-0.211204\pi$
$523$	36.0341	1.57566	0.787830	+	0.615893i	$0.211204\pi$
$524$	28.5887	1.24890
$525$	2.63864	0.115160
$526$	−49.8193	−2.17222
$527$	0	0
$528$	−54.7358	−2.38207
$529$	28.7446	1.24977
$530$	95.6535	4.15492
$531$	−9.62507	−0.417692
$532$	28.3933	1.23100
$533$	−15.0298	−0.651013
$534$	−32.8155	−1.42006
$535$	−29.1625	−1.26080
$536$	−45.3003	−1.95668
$537$	−4.87845	−0.210521
$538$	−32.5468	−1.40319
$539$	5.58538	0.240579
$540$	−13.3963	−0.576486
$541$	2.43128	0.104529	0.0522645	−	0.998633i	$-0.483356\pi$
$541$	2.43128	0.104529	0.0522645	+	0.998633i	$0.483356\pi$
$542$	49.7841	2.13841
$543$	−5.44597	−0.233709
$544$	0	0
$545$	−36.3126	−1.55546
$546$	−13.2757	−0.568148
$547$	−17.4796	−0.747375	−0.373687	−	0.927555i	$-0.621907\pi$
$547$	−17.4796	−0.747375	−0.373687	+	0.927555i	$0.621907\pi$
$548$	−29.2626	−1.25004
$549$	2.87024	0.122499
$550$	38.5642	1.64438
$551$	1.61421	0.0687675
$552$	−53.5896	−2.28092
$553$	−0.284512	−0.0120987
$554$	55.4370	2.35529
$555$	−16.7480	−0.710911
$556$	63.6819	2.70071
$557$	7.58526	0.321398	0.160699	−	0.987003i	$-0.448625\pi$
$557$	7.58526	0.321398	0.160699	+	0.987003i	$0.448625\pi$
$558$	1.63490	0.0692108
$559$	−36.9445	−1.56259
$560$	−27.0849	−1.14454
$561$	0	0
$562$	39.3463	1.65972
$563$	13.8322	0.582957	0.291479	−	0.956577i	$-0.405853\pi$
$563$	13.8322	0.582957	0.291479	+	0.956577i	$0.405853\pi$
$564$	−38.3691	−1.61563
$565$	−40.7681	−1.71513
$566$	60.1385	2.52781
$567$	−1.00000	−0.0419961
$568$	−97.5654	−4.09375
$569$	34.9751	1.46623	0.733116	−	0.680103i	$-0.238065\pi$
$569$	34.9751	1.46623	0.733116	+	0.680103i	$0.238065\pi$
$570$	42.3640	1.77443
$571$	14.5370	0.608356	0.304178	−	0.952615i	$-0.401618\pi$
$571$	14.5370	0.608356	0.304178	+	0.952615i	$0.401618\pi$
$572$	−137.353	−5.74300
$573$	−8.69999	−0.363447
$574$	−7.75174	−0.323551
$575$	18.9807	0.791550
$576$	8.51243	0.354685
$577$	−30.2940	−1.26115	−0.630577	−	0.776127i	$-0.717182\pi$
$577$	−30.2940	−1.26115	−0.630577	+	0.776127i	$0.717182\pi$
$578$	0	0
$579$	5.38757	0.223900
$580$	−3.69154	−0.153283
$581$	−5.78939	−0.240184
$582$	0.338909	0.0140482
$583$	73.8744	3.05956
$584$	43.4740	1.79897
$585$	−14.0221	−0.579744
$586$	−57.1438	−2.36059
$587$	31.9801	1.31996	0.659980	−	0.751283i	$-0.270565\pi$
$587$	31.9801	1.31996	0.659980	+	0.751283i	$0.270565\pi$
$588$	−4.84705	−0.199889
$589$	−3.65996	−0.150806
$590$	−69.6087	−2.86574
$591$	19.1210	0.786533
$592$	59.3844	2.44068
$593$	−5.32747	−0.218773	−0.109387	−	0.993999i	$-0.534889\pi$
$593$	−5.32747	−0.218773	−0.109387	+	0.993999i	$0.534889\pi$
$594$	−14.6152	−0.599669
$595$	0	0
$596$	−47.3097	−1.93788
$597$	−15.6272	−0.639580
$598$	−95.4972	−3.90517
$599$	−36.7506	−1.50159	−0.750794	−	0.660537i	$-0.770329\pi$
$599$	−36.7506	−1.50159	−0.750794	+	0.660537i	$0.770329\pi$
$600$	−19.6575	−0.802512
$601$	−16.4824	−0.672331	−0.336166	−	0.941803i	$-0.609130\pi$
$601$	−16.4824	−0.672331	−0.336166	+	0.941803i	$0.609130\pi$
$602$	−19.0544	−0.776601
$603$	−6.08070	−0.247625
$604$	−30.9428	−1.25905
$605$	55.8192	2.26937
$606$	51.6900	2.09976
$607$	−7.55618	−0.306696	−0.153348	−	0.988172i	$-0.549005\pi$
$607$	−7.55618	−0.306696	−0.153348	+	0.988172i	$0.549005\pi$
$608$	−62.9330	−2.55227
$609$	−0.275564	−0.0111664
$610$	20.7576	0.840451
$611$	−40.1614	−1.62476
$612$	0	0
$613$	0.616253	0.0248902	0.0124451	−	0.999923i	$-0.496038\pi$
$613$	0.616253	0.0248902	0.0124451	+	0.999923i	$0.496038\pi$
$614$	−70.3474	−2.83899
$615$	−8.18757	−0.330155
$616$	−41.6103	−1.67653
$617$	7.42430	0.298891	0.149446	−	0.988770i	$-0.452251\pi$
$617$	7.42430	0.298891	0.149446	+	0.988770i	$0.452251\pi$
$618$	−5.45975	−0.219623
$619$	4.87422	0.195912	0.0979558	−	0.995191i	$-0.468770\pi$
$619$	4.87422	0.195912	0.0979558	+	0.995191i	$0.468770\pi$
$620$	8.36999	0.336147
$621$	−7.19337	−0.288660
$622$	19.0089	0.762187
$623$	−12.5408	−0.502438
$624$	49.7192	1.99036
$625$	−31.2308	−1.24923
$626$	−21.8264	−0.872360
$627$	32.7183	1.30664
$628$	−34.1996	−1.36471
$629$	0	0
$630$	−7.23202	−0.288131
$631$	22.5285	0.896844	0.448422	−	0.893822i	$-0.351986\pi$
$631$	22.5285	0.896844	0.448422	+	0.893822i	$0.351986\pi$
$632$	2.11957	0.0843121
$633$	−0.403460	−0.0160361
$634$	20.9942	0.833787
$635$	−31.5442	−1.25179
$636$	−64.1090	−2.54209
$637$	−5.07348	−0.201019
$638$	−4.02742	−0.159447
$639$	−13.0963	−0.518081
$640$	2.17678	0.0860450
$641$	−21.6098	−0.853536	−0.426768	−	0.904361i	$-0.640348\pi$
$641$	−21.6098	−0.853536	−0.426768	+	0.904361i	$0.640348\pi$
$642$	27.6101	1.08968
$643$	−4.51189	−0.177931	−0.0889657	−	0.996035i	$-0.528356\pi$
$643$	−4.51189	−0.177931	−0.0889657	+	0.996035i	$0.528356\pi$
$644$	−34.8667	−1.37394
$645$	−20.1258	−0.792451
$646$	0	0
$647$	−28.9261	−1.13720	−0.568601	−	0.822613i	$-0.692515\pi$
$647$	−28.9261	−1.13720	−0.568601	+	0.822613i	$0.692515\pi$
$648$	7.44985	0.292658
$649$	−53.7597	−2.11025
$650$	−35.0298	−1.37398
$651$	0.624797	0.0244877
$652$	29.6497	1.16117
$653$	27.4859	1.07561	0.537803	−	0.843071i	$-0.319255\pi$
$653$	27.4859	1.07561	0.537803	+	0.843071i	$0.319255\pi$
$654$	34.3796	1.34435
$655$	16.3014	0.636947
$656$	29.0312	1.13348
$657$	5.83555	0.227666
$658$	−20.7136	−0.807499
$659$	−37.9793	−1.47946	−0.739731	−	0.672903i	$-0.765047\pi$
$659$	−37.9793	−1.47946	−0.739731	+	0.672903i	$0.765047\pi$
$660$	−74.8236	−2.91251
$661$	28.3154	1.10134	0.550672	−	0.834722i	$-0.314372\pi$
$661$	28.3154	1.10134	0.550672	+	0.834722i	$0.314372\pi$
$662$	−46.5355	−1.80865
$663$	0	0
$664$	43.1301	1.67377
$665$	16.1899	0.627819
$666$	15.8564	0.614425
$667$	−1.98223	−0.0767524
$668$	−56.3538	−2.18039
$669$	14.9272	0.577120
$670$	−43.9758	−1.69893
$671$	16.0314	0.618884
$672$	10.7434	0.414435
$673$	−26.5155	−1.02210	−0.511049	−	0.859552i	$-0.670743\pi$
$673$	−26.5155	−1.02210	−0.511049	+	0.859552i	$0.670743\pi$
$674$	−50.5788	−1.94822
$675$	−2.63864	−0.101561
$676$	61.7525	2.37510
$677$	−19.8325	−0.762227	−0.381113	−	0.924528i	$-0.624459\pi$
$677$	−19.8325	−0.762227	−0.381113	+	0.924528i	$0.624459\pi$
$678$	38.5980	1.48235
$679$	0.129518	0.00497045
$680$	0	0
$681$	−21.6525	−0.829726
$682$	9.13154	0.349665
$683$	−15.3010	−0.585476	−0.292738	−	0.956193i	$-0.594566\pi$
$683$	−15.3010	−0.585476	−0.292738	+	0.956193i	$0.594566\pi$
$684$	−28.3933	−1.08564
$685$	−16.6856	−0.637526
$686$	−2.61669	−0.0999056
$687$	7.26368	0.277127
$688$	71.3613	2.72062
$689$	−67.1038	−2.55645
$690$	−52.0227	−1.98047
$691$	−41.8654	−1.59263	−0.796317	−	0.604880i	$-0.793221\pi$
$691$	−41.8654	−1.59263	−0.796317	+	0.604880i	$0.793221\pi$
$692$	84.5801	3.21525
$693$	−5.58538	−0.212171
$694$	−28.2002	−1.07046
$695$	36.3116	1.37738
$696$	2.05291	0.0778153
$697$	0	0
$698$	45.6566	1.72813
$699$	−12.4353	−0.470346
$700$	−12.7896	−0.483402
$701$	28.4222	1.07349	0.536745	−	0.843744i	$-0.319654\pi$
$701$	28.4222	1.07349	0.536745	+	0.843744i	$0.319654\pi$
$702$	13.2757	0.501060
$703$	−35.4969	−1.33879
$704$	47.5452	1.79193
$705$	−21.8782	−0.823980
$706$	−46.2147	−1.73931
$707$	19.7540	0.742925
$708$	46.6532	1.75334
$709$	23.2643	0.873708	0.436854	−	0.899532i	$-0.356093\pi$
$709$	23.2643	0.873708	0.436854	+	0.899532i	$0.356093\pi$
$710$	−94.7127	−3.55450
$711$	0.284512	0.0106700
$712$	93.4274	3.50134
$713$	4.49440	0.168317
$714$	0	0
$715$	−78.3189	−2.92896
$716$	23.6461	0.883697
$717$	15.8342	0.591338
$718$	76.1870	2.84328
$719$	−14.5943	−0.544275	−0.272137	−	0.962258i	$-0.587731\pi$
$719$	−14.5943	−0.544275	−0.272137	+	0.962258i	$0.587731\pi$
$720$	27.0849	1.00939
$721$	−2.08651	−0.0777058
$722$	40.0726	1.49135
$723$	15.5853	0.579623
$724$	26.3969	0.981034
$725$	−0.727112	−0.0270043
$726$	−52.8479	−1.96137
$727$	0.762956	0.0282965	0.0141482	−	0.999900i	$-0.495496\pi$
$727$	0.762956	0.0282965	0.0141482	+	0.999900i	$0.495496\pi$
$728$	37.7967	1.40084
$729$	1.00000	0.0370370
$730$	42.2028	1.56200
$731$	0	0
$732$	−13.9122	−0.514210
$733$	17.8009	0.657492	0.328746	−	0.944418i	$-0.393374\pi$
$733$	17.8009	0.657492	0.328746	+	0.944418i	$0.393374\pi$
$734$	−7.74128	−0.285736
$735$	−2.76381	−0.101945
$736$	77.2812	2.84862
$737$	−33.9630	−1.25104
$738$	7.75174	0.285345
$739$	19.8524	0.730284	0.365142	−	0.930952i	$-0.381020\pi$
$739$	19.8524	0.730284	0.365142	+	0.930952i	$0.381020\pi$
$740$	81.1782	2.98417
$741$	−29.7196	−1.09178
$742$	−34.6093	−1.27055
$743$	−10.3252	−0.378794	−0.189397	−	0.981901i	$-0.560653\pi$
$743$	−10.3252	−0.378794	−0.189397	+	0.981901i	$0.560653\pi$
$744$	−4.65465	−0.170648
$745$	−26.9761	−0.988329
$746$	−55.4926	−2.03173
$747$	5.78939	0.211823
$748$	0	0
$749$	10.5516	0.385545
$750$	17.0774	0.623580
$751$	16.5138	0.602599	0.301299	−	0.953530i	$-0.402580\pi$
$751$	16.5138	0.602599	0.301299	+	0.953530i	$0.402580\pi$
$752$	77.5750	2.82887
$753$	2.55446	0.0930898
$754$	3.65831	0.133228
$755$	−17.6437	−0.642120
$756$	4.84705	0.176286
$757$	43.1759	1.56925	0.784627	−	0.619969i	$-0.212855\pi$
$757$	43.1759	1.56925	0.784627	+	0.619969i	$0.212855\pi$
$758$	96.7673	3.51475
$759$	−40.1777	−1.45836
$760$	−120.613	−4.37508
$761$	−2.70210	−0.0979510	−0.0489755	−	0.998800i	$-0.515596\pi$
$761$	−2.70210	−0.0979510	−0.0489755	+	0.998800i	$0.515596\pi$
$762$	29.8651	1.08190
$763$	13.1386	0.475649
$764$	42.1693	1.52563
$765$	0	0
$766$	43.4053	1.56830
$767$	48.8326	1.76324
$768$	14.9640	0.539965
$769$	−13.6795	−0.493295	−0.246647	−	0.969105i	$-0.579329\pi$
$769$	−13.6795	−0.493295	−0.246647	+	0.969105i	$0.579329\pi$
$770$	−40.3936	−1.45568
$771$	10.3379	0.372310
$772$	−26.1139	−0.939859
$773$	1.25509	0.0451425	0.0225712	−	0.999745i	$-0.492815\pi$
$773$	1.25509	0.0451425	0.0225712	+	0.999745i	$0.492815\pi$
$774$	19.0544	0.684898
$775$	1.64861	0.0592199
$776$	−0.964892	−0.0346376
$777$	6.05974	0.217392
$778$	−5.99592	−0.214964
$779$	−17.3534	−0.621750
$780$	67.9660	2.43357
$781$	−73.1478	−2.61743
$782$	0	0
$783$	0.275564	0.00984785
$784$	9.79983	0.349994
$785$	−19.5007	−0.696011
$786$	−15.4336	−0.550499
$787$	−49.0395	−1.74807	−0.874034	−	0.485865i	$-0.838505\pi$
$787$	−49.0395	−1.74807	−0.874034	+	0.485865i	$0.838505\pi$
$788$	−92.6806	−3.30161
$789$	19.0391	0.677809
$790$	2.05760	0.0732061
$791$	14.7507	0.524475
$792$	41.6103	1.47856
$793$	−14.5621	−0.517115
$794$	−36.3677	−1.29064
$795$	−36.5552	−1.29648
$796$	75.7460	2.68475
$797$	−6.18255	−0.218997	−0.109499	−	0.993987i	$-0.534925\pi$
$797$	−6.18255	−0.218997	−0.109499	+	0.993987i	$0.534925\pi$
$798$	−15.3281	−0.542610
$799$	0	0
$800$	28.3479	1.00225
$801$	12.5408	0.443109
$802$	20.8122	0.734903
$803$	32.5937	1.15021
$804$	29.4735	1.03945
$805$	−19.8811	−0.700717
$806$	−8.29463	−0.292166
$807$	12.4382	0.437844
$808$	−147.164	−5.17722
$809$	−28.5917	−1.00523	−0.502615	−	0.864510i	$-0.667629\pi$
$809$	−28.5917	−1.00523	−0.502615	+	0.864510i	$0.667629\pi$
$810$	7.23202	0.254107
$811$	9.03830	0.317378	0.158689	−	0.987329i	$-0.449273\pi$
$811$	9.03830	0.317378	0.158689	+	0.987329i	$0.449273\pi$
$812$	1.33567	0.0468729
$813$	−19.0256	−0.667257
$814$	88.5643	3.10418
$815$	16.9063	0.592203
$816$	0	0
$817$	−42.6561	−1.49235
$818$	−73.6395	−2.57475
$819$	5.07348	0.177282
$820$	39.6856	1.38588
$821$	7.67418	0.267831	0.133915	−	0.990993i	$-0.457245\pi$
$821$	7.67418	0.267831	0.133915	+	0.990993i	$0.457245\pi$
$822$	15.7974	0.550999
$823$	−5.95277	−0.207500	−0.103750	−	0.994603i	$-0.533084\pi$
$823$	−5.95277	−0.207500	−0.103750	+	0.994603i	$0.533084\pi$
$824$	15.5442	0.541508
$825$	−14.7378	−0.513104
$826$	25.1858	0.876326
$827$	−41.3595	−1.43821	−0.719106	−	0.694901i	$-0.755448\pi$
$827$	−41.3595	−1.43821	−0.719106	+	0.694901i	$0.755448\pi$
$828$	34.8667	1.21170
$829$	15.5716	0.540825	0.270412	−	0.962745i	$-0.412840\pi$
$829$	15.5716	0.540825	0.270412	+	0.962745i	$0.412840\pi$
$830$	41.8690	1.45329
$831$	−21.1859	−0.734932
$832$	−43.1877	−1.49726
$833$	0	0
$834$	−34.3787	−1.19044
$835$	−32.1331	−1.11201
$836$	−158.587	−5.48485
$837$	−0.624797	−0.0215962
$838$	−24.3564	−0.841379
$839$	56.3582	1.94570	0.972851	−	0.231432i	$-0.0743412\pi$
$839$	56.3582	1.94570	0.972851	+	0.231432i	$0.0743412\pi$
$840$	20.5900	0.710421
$841$	−28.9241	−0.997382
$842$	−60.6575	−2.09040
$843$	−15.0367	−0.517890
$844$	1.95559	0.0673143
$845$	35.2115	1.21131
$846$	20.7136	0.712147
$847$	−20.1965	−0.693960
$848$	129.616	4.45104
$849$	−22.9827	−0.788763
$850$	0	0
$851$	43.5900	1.49424
$852$	63.4784	2.17473
$853$	24.7043	0.845859	0.422930	−	0.906163i	$-0.361002\pi$
$853$	24.7043	0.845859	0.422930	+	0.906163i	$0.361002\pi$
$854$	−7.51051	−0.257005
$855$	−16.1899	−0.553684
$856$	−78.6075	−2.68675
$857$	2.85535	0.0975371	0.0487685	−	0.998810i	$-0.484470\pi$
$857$	2.85535	0.0975371	0.0487685	+	0.998810i	$0.484470\pi$
$858$	74.1499	2.53144
$859$	18.0803	0.616893	0.308447	−	0.951242i	$-0.400191\pi$
$859$	18.0803	0.616893	0.308447	+	0.951242i	$0.400191\pi$
$860$	97.5506	3.32645
$861$	2.96242	0.100959
$862$	−87.0892	−2.96627
$863$	28.6098	0.973889	0.486945	−	0.873433i	$-0.338111\pi$
$863$	28.6098	0.973889	0.486945	+	0.873433i	$0.338111\pi$
$864$	−10.7434	−0.365498
$865$	48.2279	1.63980
$866$	−2.81800	−0.0957596
$867$	0	0
$868$	−3.02843	−0.102792
$869$	1.58911	0.0539068
$870$	1.99288	0.0675651
$871$	30.8503	1.04532
$872$	−97.8806	−3.31466
$873$	−0.129518	−0.00438353
$874$	−110.261	−3.72963
$875$	6.52636	0.220631
$876$	−28.2852	−0.955669
$877$	−46.7191	−1.57759	−0.788795	−	0.614656i	$-0.789295\pi$
$877$	−46.7191	−1.57759	−0.788795	+	0.614656i	$0.789295\pi$
$878$	−69.4908	−2.34520
$879$	21.8382	0.736585
$880$	151.279	5.09962
$881$	29.7130	1.00106	0.500528	−	0.865721i	$-0.333139\pi$
$881$	29.7130	1.00106	0.500528	+	0.865721i	$0.333139\pi$
$882$	2.61669	0.0881085
$883$	21.3317	0.717869	0.358935	−	0.933363i	$-0.383140\pi$
$883$	21.3317	0.717869	0.358935	+	0.933363i	$0.383140\pi$
$884$	0	0
$885$	26.6018	0.894211
$886$	−30.3546	−1.01978
$887$	30.5336	1.02522	0.512609	−	0.858622i	$-0.328679\pi$
$887$	30.5336	1.02522	0.512609	+	0.858622i	$0.328679\pi$
$888$	−45.1441	−1.51494
$889$	11.4133	0.382790
$890$	90.6957	3.04013
$891$	5.58538	0.187117
$892$	−72.3530	−2.42256
$893$	−46.3704	−1.55172
$894$	25.5402	0.854191
$895$	13.4831	0.450690
$896$	−0.787603	−0.0263120
$897$	36.4954	1.21855
$898$	−31.4557	−1.04969
$899$	−0.172171	−0.00574224
$900$	12.7896	0.426320
$901$	0	0
$902$	43.2964	1.44161
$903$	7.28189	0.242326
$904$	−109.891	−3.65491
$905$	15.0516	0.500333
$906$	16.7045	0.554970
$907$	16.9926	0.564230	0.282115	−	0.959381i	$-0.408964\pi$
$907$	16.9926	0.564230	0.282115	+	0.959381i	$0.408964\pi$
$908$	104.951	3.48292
$909$	−19.7540	−0.655199
$910$	36.6915	1.21631
$911$	−11.8657	−0.393127	−0.196564	−	0.980491i	$-0.562978\pi$
$911$	−11.8657	−0.393127	−0.196564	+	0.980491i	$0.562978\pi$
$912$	57.4058	1.90090
$913$	32.3360	1.07016
$914$	−70.0767	−2.31793
$915$	−7.93278	−0.262250
$916$	−35.2074	−1.16329
$917$	−5.89815	−0.194774
$918$	0	0
$919$	48.0482	1.58496	0.792482	−	0.609895i	$-0.208788\pi$
$919$	48.0482	1.58496	0.792482	+	0.609895i	$0.208788\pi$
$920$	148.111	4.88309
$921$	26.8841	0.885863
$922$	27.0480	0.890779
$923$	66.4438	2.18702
$924$	27.0727	0.890625
$925$	15.9894	0.525730
$926$	−76.9200	−2.52775
$927$	2.08651	0.0685300
$928$	−2.96049	−0.0971828
$929$	43.7755	1.43623	0.718115	−	0.695925i	$-0.245005\pi$
$929$	43.7755	1.43623	0.718115	+	0.695925i	$0.245005\pi$
$930$	−4.51855	−0.148169
$931$	−5.85784	−0.191983
$932$	60.2745	1.97436
$933$	−7.26449	−0.237829
$934$	24.4168	0.798941
$935$	0	0
$936$	−37.7967	−1.23542
$937$	−37.1767	−1.21451	−0.607255	−	0.794507i	$-0.707729\pi$
$937$	−37.1767	−1.21451	−0.607255	+	0.794507i	$0.707729\pi$
$938$	15.9113	0.519522
$939$	8.34124	0.272206
$940$	106.045	3.45880
$941$	−38.5404	−1.25638	−0.628191	−	0.778059i	$-0.716204\pi$
$941$	−38.5404	−1.25638	−0.628191	+	0.778059i	$0.716204\pi$
$942$	18.4627	0.601547
$943$	21.3098	0.693943
$944$	−94.3240	−3.06998
$945$	2.76381	0.0899067
$946$	106.426	3.46022
$947$	−20.4311	−0.663921	−0.331961	−	0.943293i	$-0.607710\pi$
$947$	−20.4311	−0.663921	−0.331961	+	0.943293i	$0.607710\pi$
$948$	−1.37905	−0.0447893
$949$	−29.6065	−0.961068
$950$	−40.4454	−1.31222
$951$	−8.02321	−0.260170
$952$	0	0
$953$	−5.28068	−0.171058	−0.0855290	−	0.996336i	$-0.527258\pi$
$953$	−5.28068	−0.171058	−0.0855290	+	0.996336i	$0.527258\pi$
$954$	34.6093	1.12052
$955$	24.0451	0.778081
$956$	−76.7491	−2.48224
$957$	1.53913	0.0497530
$958$	63.8389	2.06254
$959$	6.03719	0.194951
$960$	−23.5267	−0.759322
$961$	−30.6096	−0.987407
$962$	−80.4473	−2.59373
$963$	−10.5516	−0.340019
$964$	−75.5428	−2.43307
$965$	−14.8902	−0.479333
$966$	18.8228	0.605614
$967$	21.0273	0.676191	0.338096	−	0.941112i	$-0.390217\pi$
$967$	21.0273	0.676191	0.338096	+	0.941112i	$0.390217\pi$
$968$	150.461	4.83600
$969$	0	0
$970$	−0.936679	−0.0300749
$971$	−6.39655	−0.205275	−0.102637	−	0.994719i	$-0.532728\pi$
$971$	−6.39655	−0.205275	−0.102637	+	0.994719i	$0.532728\pi$
$972$	−4.84705	−0.155469
$973$	−13.1383	−0.421193
$974$	78.8512	2.52655
$975$	13.3871	0.428729
$976$	28.1278	0.900350
$977$	0.806875	0.0258142	0.0129071	−	0.999917i	$-0.495891\pi$
$977$	0.806875	0.0258142	0.0129071	+	0.999917i	$0.495891\pi$
$978$	−16.0064	−0.511828
$979$	70.0454	2.23866
$980$	13.3963	0.427930
$981$	−13.1386	−0.419483
$982$	−11.4226	−0.364511
$983$	47.2420	1.50679	0.753394	−	0.657570i	$-0.228415\pi$
$983$	47.2420	1.50679	0.753394	+	0.657570i	$0.228415\pi$
$984$	−22.0696	−0.703554
$985$	−52.8468	−1.68384
$986$	0	0
$987$	7.91595	0.251968
$988$	144.053	4.58292
$989$	52.3814	1.66563
$990$	40.3936	1.28379
$991$	−25.4030	−0.806953	−0.403477	−	0.914990i	$-0.632198\pi$
$991$	−25.4030	−0.806953	−0.403477	+	0.914990i	$0.632198\pi$
$992$	6.71244	0.213120
$993$	17.7841	0.564362
$994$	34.2689	1.08694
$995$	43.1907	1.36924
$996$	−28.0615	−0.889163
$997$	−36.8561	−1.16725	−0.583623	−	0.812025i	$-0.698365\pi$
$997$	−36.8561	−1.16725	−0.583623	+	0.812025i	$0.698365\pi$
$998$	98.0118	3.10251
$999$	−6.05974	−0.191722

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.z.1.9	✓	9
17.16	even	2		6069.2.a.bc.1.9	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6069.2.a.z.1.9	✓	9	1.1	even	1	trivial
6069.2.a.bc.1.9	yes	9	17.16	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6069.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61669 q^{2} -1.00000 q^{3} +4.84705 q^{4} +2.76381 q^{5} -2.61669 q^{6} -1.00000 q^{7} +7.44985 q^{8} +1.00000 q^{9} +7.23202 q^{10} +5.58538 q^{11} -4.84705 q^{12} -5.07348 q^{13} -2.61669 q^{14} -2.76381 q^{15} +9.79983 q^{16} +2.61669 q^{18} -5.85784 q^{19} +13.3963 q^{20} +1.00000 q^{21} +14.6152 q^{22} +7.19337 q^{23} -7.44985 q^{24} +2.63864 q^{25} -13.2757 q^{26} -1.00000 q^{27} -4.84705 q^{28} -0.275564 q^{29} -7.23202 q^{30} +0.624797 q^{31} +10.7434 q^{32} -5.58538 q^{33} -2.76381 q^{35} +4.84705 q^{36} +6.05974 q^{37} -15.3281 q^{38} +5.07348 q^{39} +20.5900 q^{40} +2.96242 q^{41} +2.61669 q^{42} +7.28189 q^{43} +27.0727 q^{44} +2.76381 q^{45} +18.8228 q^{46} +7.91595 q^{47} -9.79983 q^{48} +1.00000 q^{49} +6.90449 q^{50} -24.5914 q^{52} +13.2264 q^{53} -2.61669 q^{54} +15.4369 q^{55} -7.44985 q^{56} +5.85784 q^{57} -0.721064 q^{58} -9.62507 q^{59} -13.3963 q^{60} +2.87024 q^{61} +1.63490 q^{62} -1.00000 q^{63} +8.51243 q^{64} -14.0221 q^{65} -14.6152 q^{66} -6.08070 q^{67} -7.19337 q^{69} -7.23202 q^{70} -13.0963 q^{71} +7.44985 q^{72} +5.83555 q^{73} +15.8564 q^{74} -2.63864 q^{75} -28.3933 q^{76} -5.58538 q^{77} +13.2757 q^{78} +0.284512 q^{79} +27.0849 q^{80} +1.00000 q^{81} +7.75174 q^{82} +5.78939 q^{83} +4.84705 q^{84} +19.0544 q^{86} +0.275564 q^{87} +41.6103 q^{88} +12.5408 q^{89} +7.23202 q^{90} +5.07348 q^{91} +34.8667 q^{92} -0.624797 q^{93} +20.7136 q^{94} -16.1899 q^{95} -10.7434 q^{96} -0.129518 q^{97} +2.61669 q^{98} +5.58538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} + 6 q^{4} + 3 q^{5} - 9 q^{7} + 9 q^{8} + 9 q^{9} + 12 q^{10} + 18 q^{11} - 6 q^{12} - 21 q^{13} - 3 q^{15} - 9 q^{19} + 15 q^{20} + 9 q^{21} + 6 q^{22} - 9 q^{24} + 3 q^{26} - 9 q^{27} - 6 q^{28}+ \cdots + 18 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 + 6 * q^4 + 3 * q^5 - 9 * q^7 + 9 * q^8 + 9 * q^9 + 12 * q^10 + 18 * q^11 - 6 * q^12 - 21 * q^13 - 3 * q^15 - 9 * q^19 + 15 * q^20 + 9 * q^21 + 6 * q^22 - 9 * q^24 + 3 * q^26 - 9 * q^27 - 6 * q^28 + 6 * q^29 - 12 * q^30 + 30 * q^31 + 3 * q^32 - 18 * q^33 - 3 * q^35 + 6 * q^36 + 12 * q^37 - 36 * q^38 + 21 * q^39 + 30 * q^40 + 9 * q^41 + 30 * q^44 + 3 * q^45 + 33 * q^46 - 9 * q^47 + 9 * q^49 - 12 * q^50 - 12 * q^52 + 30 * q^55 - 9 * q^56 + 9 * q^57 + 9 * q^58 - 3 * q^59 - 15 * q^60 + 33 * q^61 - 12 * q^62 - 9 * q^63 - 15 * q^64 - 15 * q^65 - 6 * q^66 - 15 * q^67 - 12 * q^70 + 9 * q^72 + 21 * q^73 + 6 * q^74 + 3 * q^76 - 18 * q^77 - 3 * q^78 + 30 * q^79 + 9 * q^81 + 9 * q^82 + 6 * q^84 + 72 * q^86 - 6 * q^87 + 48 * q^88 - 12 * q^89 + 12 * q^90 + 21 * q^91 + 48 * q^92 - 30 * q^93 + 48 * q^94 - 18 * q^95 - 3 * q^96 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more