LMFDB - Embedded newform 6069.2.a.m.1.3

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)

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")

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);

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 = [3,-2,3,2,-3,-2,3] 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:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ 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)

$f(q)$	$=$	\(q+1.17009 q^{2} +1.00000 q^{3} -0.630898 q^{4} -0.460811 q^{5} +1.17009 q^{6} +1.00000 q^{7} -3.07838 q^{8} +1.00000 q^{9} -0.539189 q^{10} -0.709275 q^{11} -0.630898 q^{12} +3.87936 q^{13} +1.17009 q^{14} -0.460811 q^{15} -2.34017 q^{16} +1.17009 q^{18} -4.17009 q^{19} +0.290725 q^{20} +1.00000 q^{21} -0.829914 q^{22} -7.34017 q^{23} -3.07838 q^{24} -4.78765 q^{25} +4.53919 q^{26} +1.00000 q^{27} -0.630898 q^{28} +1.70928 q^{29} -0.539189 q^{30} +4.87936 q^{31} +3.41855 q^{32} -0.709275 q^{33} -0.460811 q^{35} -0.630898 q^{36} -1.90829 q^{37} -4.87936 q^{38} +3.87936 q^{39} +1.41855 q^{40} -4.17009 q^{41} +1.17009 q^{42} -8.75872 q^{43} +0.447480 q^{44} -0.460811 q^{45} -8.58864 q^{46} +6.72261 q^{47} -2.34017 q^{48} +1.00000 q^{49} -5.60197 q^{50} -2.44748 q^{52} -4.58864 q^{53} +1.17009 q^{54} +0.326842 q^{55} -3.07838 q^{56} -4.17009 q^{57} +2.00000 q^{58} +3.21953 q^{59} +0.290725 q^{60} +8.48133 q^{61} +5.70928 q^{62} +1.00000 q^{63} +8.68035 q^{64} -1.78765 q^{65} -0.829914 q^{66} -6.81432 q^{67} -7.34017 q^{69} -0.539189 q^{70} -10.7298 q^{71} -3.07838 q^{72} -5.23513 q^{73} -2.23287 q^{74} -4.78765 q^{75} +2.63090 q^{76} -0.709275 q^{77} +4.53919 q^{78} -14.1639 q^{79} +1.07838 q^{80} +1.00000 q^{81} -4.87936 q^{82} +0.496928 q^{83} -0.630898 q^{84} -10.2485 q^{86} +1.70928 q^{87} +2.18342 q^{88} +3.60197 q^{89} -0.539189 q^{90} +3.87936 q^{91} +4.63090 q^{92} +4.87936 q^{93} +7.86603 q^{94} +1.92162 q^{95} +3.41855 q^{96} -11.0784 q^{97} +1.17009 q^{98} -0.709275 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 5 q^{11} + 2 q^{12} - q^{13} - 2 q^{14} - 3 q^{15} + 4 q^{16} - 2 q^{18} - 7 q^{19} + 8 q^{20} + 3 q^{21} - 8 q^{22}+ \cdots + 5 q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 + 5 * q^11 + 2 * q^12 - q^13 - 2 * q^14 - 3 * q^15 + 4 * q^16 - 2 * q^18 - 7 * q^19 + 8 * q^20 + 3 * q^21 - 8 * q^22 - 11 * q^23 - 6 * q^24 - 4 * q^25 + 12 * q^26 + 3 * q^27 + 2 * q^28 - 2 * q^29 + 2 * q^31 - 4 * q^32 + 5 * q^33 - 3 * q^35 + 2 * q^36 - 8 * q^37 - 2 * q^38 - q^39 - 10 * q^40 - 7 * q^41 - 2 * q^42 - q^43 + 2 * q^44 - 3 * q^45 - 6 * q^46 + 14 * q^47 + 4 * q^48 + 3 * q^49 + 2 * q^50 - 8 * q^52 + 6 * q^53 - 2 * q^54 - 11 * q^55 - 6 * q^56 - 7 * q^57 + 6 * q^58 - 14 * q^59 + 8 * q^60 - 6 * q^61 + 10 * q^62 + 3 * q^63 + 4 * q^64 + 5 * q^65 - 8 * q^66 - 12 * q^67 - 11 * q^69 + 8 * q^71 - 6 * q^72 - 6 * q^73 + 16 * q^74 - 4 * q^75 + 4 * q^76 + 5 * q^77 + 12 * q^78 - 2 * q^79 + 3 * q^81 - 2 * q^82 - 16 * q^83 + 2 * q^84 - 22 * q^86 - 2 * q^87 + 2 * q^88 - 8 * q^89 - q^91 + 10 * q^92 + 2 * q^93 + 10 * q^94 + 9 * q^95 - 4 * q^96 - 30 * q^97 - 2 * q^98 + 5 * 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 $

$2$	1.17009	0.827376	0.413688	−	0.910419i	$-0.364240\pi$
$2$	1.17009	0.827376	0.413688	+	0.910419i	$0.364240\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.630898	−0.315449
$5$	−0.460811	−0.206081	−0.103041	−	0.994677i	$-0.532857\pi$
$5$	−0.460811	−0.206081	−0.103041	+	0.994677i	$0.532857\pi$
$6$	1.17009	0.477686
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.07838	−1.08837
$9$	1.00000	0.333333
$10$	−0.539189	−0.170506
$11$	−0.709275	−0.213855	−0.106927	−	0.994267i	$-0.534101\pi$
$11$	−0.709275	−0.213855	−0.106927	+	0.994267i	$0.534101\pi$
$12$	−0.630898	−0.182124
$13$	3.87936	1.07594	0.537971	−	0.842964i	$-0.319191\pi$
$13$	3.87936	1.07594	0.537971	+	0.842964i	$0.319191\pi$
$14$	1.17009	0.312719
$15$	−0.460811	−0.118981
$16$	−2.34017	−0.585043
$17$	0	0
$17$	0	0
$18$	1.17009	0.275792
$19$	−4.17009	−0.956683	−0.478342	−	0.878174i	$-0.658762\pi$
$19$	−4.17009	−0.956683	−0.478342	+	0.878174i	$0.658762\pi$
$20$	0.290725	0.0650080
$21$	1.00000	0.218218
$22$	−0.829914	−0.176938
$23$	−7.34017	−1.53053	−0.765266	−	0.643714i	$-0.777393\pi$
$23$	−7.34017	−1.53053	−0.765266	+	0.643714i	$0.777393\pi$
$24$	−3.07838	−0.628371
$25$	−4.78765	−0.957531
$26$	4.53919	0.890208
$27$	1.00000	0.192450
$28$	−0.630898	−0.119228
$29$	1.70928	0.317404	0.158702	−	0.987326i	$-0.449269\pi$
$29$	1.70928	0.317404	0.158702	+	0.987326i	$0.449269\pi$
$30$	−0.539189	−0.0984420
$31$	4.87936	0.876359	0.438180	−	0.898887i	$-0.355623\pi$
$31$	4.87936	0.876359	0.438180	+	0.898887i	$0.355623\pi$
$32$	3.41855	0.604320
$33$	−0.709275	−0.123469
$34$	0	0
$35$	−0.460811	−0.0778913
$36$	−0.630898	−0.105150
$37$	−1.90829	−0.313721	−0.156861	−	0.987621i	$-0.550137\pi$
$37$	−1.90829	−0.313721	−0.156861	+	0.987621i	$0.550137\pi$
$38$	−4.87936	−0.791537
$39$	3.87936	0.621195
$40$	1.41855	0.224293
$41$	−4.17009	−0.651258	−0.325629	−	0.945498i	$-0.605576\pi$
$41$	−4.17009	−0.651258	−0.325629	+	0.945498i	$0.605576\pi$
$42$	1.17009	0.180548
$43$	−8.75872	−1.33569	−0.667846	−	0.744299i	$-0.732784\pi$
$43$	−8.75872	−1.33569	−0.667846	+	0.744299i	$0.732784\pi$
$44$	0.447480	0.0674602
$45$	−0.460811	−0.0686937
$46$	−8.58864	−1.26633
$47$	6.72261	0.980593	0.490296	−	0.871556i	$-0.336889\pi$
$47$	6.72261	0.980593	0.490296	+	0.871556i	$0.336889\pi$
$48$	−2.34017	−0.337775
$49$	1.00000	0.142857
$50$	−5.60197	−0.792238
$51$	0	0
$52$	−2.44748	−0.339404
$53$	−4.58864	−0.630298	−0.315149	−	0.949042i	$-0.602054\pi$
$53$	−4.58864	−0.630298	−0.315149	+	0.949042i	$0.602054\pi$
$54$	1.17009	0.159229
$55$	0.326842	0.0440714
$56$	−3.07838	−0.411366
$57$	−4.17009	−0.552341
$58$	2.00000	0.262613
$59$	3.21953	0.419148	0.209574	−	0.977793i	$-0.432792\pi$
$59$	3.21953	0.419148	0.209574	+	0.977793i	$0.432792\pi$
$60$	0.290725	0.0375324
$61$	8.48133	1.08592	0.542962	−	0.839758i	$-0.317303\pi$
$61$	8.48133	1.08592	0.542962	+	0.839758i	$0.317303\pi$
$62$	5.70928	0.725079
$63$	1.00000	0.125988
$64$	8.68035	1.08504
$65$	−1.78765	−0.221731
$66$	−0.829914	−0.102155
$67$	−6.81432	−0.832501	−0.416251	−	0.909250i	$-0.636656\pi$
$67$	−6.81432	−0.832501	−0.416251	+	0.909250i	$0.636656\pi$
$68$	0	0
$69$	−7.34017	−0.883653
$70$	−0.539189	−0.0644454
$71$	−10.7298	−1.27339	−0.636696	−	0.771115i	$-0.719699\pi$
$71$	−10.7298	−1.27339	−0.636696	+	0.771115i	$0.719699\pi$
$72$	−3.07838	−0.362790
$73$	−5.23513	−0.612726	−0.306363	−	0.951915i	$-0.599112\pi$
$73$	−5.23513	−0.612726	−0.306363	+	0.951915i	$0.599112\pi$
$74$	−2.23287	−0.259565
$75$	−4.78765	−0.552831
$76$	2.63090	0.301785
$77$	−0.709275	−0.0808294
$78$	4.53919	0.513962
$79$	−14.1639	−1.59357	−0.796784	−	0.604264i	$-0.793467\pi$
$79$	−14.1639	−1.59357	−0.796784	+	0.604264i	$0.793467\pi$
$80$	1.07838	0.120566
$81$	1.00000	0.111111
$82$	−4.87936	−0.538835
$83$	0.496928	0.0545450	0.0272725	−	0.999628i	$-0.491318\pi$
$83$	0.496928	0.0545450	0.0272725	+	0.999628i	$0.491318\pi$
$84$	−0.630898	−0.0688366
$85$	0	0
$86$	−10.2485	−1.10512
$87$	1.70928	0.183254
$88$	2.18342	0.232753
$89$	3.60197	0.381808	0.190904	−	0.981609i	$-0.438858\pi$
$89$	3.60197	0.381808	0.190904	+	0.981609i	$0.438858\pi$
$90$	−0.539189	−0.0568355
$91$	3.87936	0.406668
$92$	4.63090	0.482804
$93$	4.87936	0.505966
$94$	7.86603	0.811319
$95$	1.92162	0.197154
$96$	3.41855	0.348904
$97$	−11.0784	−1.12484	−0.562419	−	0.826852i	$-0.690129\pi$
$97$	−11.0784	−1.12484	−0.562419	+	0.826852i	$0.690129\pi$
$98$	1.17009	0.118197
$99$	−0.709275	−0.0712849
$100$	3.02052	0.302052
$101$	−14.0566	−1.39869	−0.699344	−	0.714785i	$-0.746524\pi$
$101$	−14.0566	−1.39869	−0.699344	+	0.714785i	$0.746524\pi$
$102$	0	0
$103$	−17.0566	−1.68064	−0.840320	−	0.542090i	$-0.817633\pi$
$103$	−17.0566	−1.68064	−0.840320	+	0.542090i	$0.817633\pi$
$104$	−11.9421	−1.17102
$105$	−0.460811	−0.0449706
$106$	−5.36910	−0.521493
$107$	0.973338	0.0940961	0.0470481	−	0.998893i	$-0.485019\pi$
$107$	0.973338	0.0940961	0.0470481	+	0.998893i	$0.485019\pi$
$108$	−0.630898	−0.0607082
$109$	2.40522	0.230378	0.115189	−	0.993344i	$-0.463253\pi$
$109$	2.40522	0.230378	0.115189	+	0.993344i	$0.463253\pi$
$110$	0.382433	0.0364636
$111$	−1.90829	−0.181127
$112$	−2.34017	−0.221126
$113$	16.9916	1.59843	0.799217	−	0.601042i	$-0.205248\pi$
$113$	16.9916	1.59843	0.799217	+	0.601042i	$0.205248\pi$
$114$	−4.87936	−0.456994
$115$	3.38243	0.315414
$116$	−1.07838	−0.100125
$117$	3.87936	0.358647
$118$	3.76713	0.346793
$119$	0	0
$120$	1.41855	0.129495
$121$	−10.4969	−0.954266
$122$	9.92389	0.898467
$123$	−4.17009	−0.376004
$124$	−3.07838	−0.276446
$125$	4.51026	0.403410
$126$	1.17009	0.104240
$127$	−4.21235	−0.373785	−0.186893	−	0.982380i	$-0.559842\pi$
$127$	−4.21235	−0.373785	−0.186893	+	0.982380i	$0.559842\pi$
$128$	3.31965	0.293419
$129$	−8.75872	−0.771163
$130$	−2.09171	−0.183455
$131$	16.0361	1.40108	0.700541	−	0.713612i	$-0.252942\pi$
$131$	16.0361	1.40108	0.700541	+	0.713612i	$0.252942\pi$
$132$	0.447480	0.0389481
$133$	−4.17009	−0.361592
$134$	−7.97334	−0.688791
$135$	−0.460811	−0.0396603
$136$	0	0
$137$	−6.43188	−0.549513	−0.274756	−	0.961514i	$-0.588597\pi$
$137$	−6.43188	−0.549513	−0.274756	+	0.961514i	$0.588597\pi$
$138$	−8.58864	−0.731113
$139$	12.4391	1.05507	0.527534	−	0.849534i	$-0.323117\pi$
$139$	12.4391	1.05507	0.527534	+	0.849534i	$0.323117\pi$
$140$	0.290725	0.0245707
$141$	6.72261	0.566146
$142$	−12.5548	−1.05357
$143$	−2.75154	−0.230095
$144$	−2.34017	−0.195014
$145$	−0.787653	−0.0654110
$146$	−6.12556	−0.506955
$147$	1.00000	0.0824786
$148$	1.20394	0.0989630
$149$	14.1256	1.15721	0.578605	−	0.815608i	$-0.303597\pi$
$149$	14.1256	1.15721	0.578605	+	0.815608i	$0.303597\pi$
$150$	−5.60197	−0.457399
$151$	−16.2907	−1.32572	−0.662860	−	0.748743i	$-0.730658\pi$
$151$	−16.2907	−1.32572	−0.662860	+	0.748743i	$0.730658\pi$
$152$	12.8371	1.04123
$153$	0	0
$154$	−0.829914	−0.0668763
$155$	−2.24846	−0.180601
$156$	−2.44748	−0.195955
$157$	−19.7721	−1.57798	−0.788991	−	0.614405i	$-0.789396\pi$
$157$	−19.7721	−1.57798	−0.788991	+	0.614405i	$0.789396\pi$
$158$	−16.5730	−1.31848
$159$	−4.58864	−0.363903
$160$	−1.57531	−0.124539
$161$	−7.34017	−0.578487
$162$	1.17009	0.0919307
$163$	5.10504	0.399858	0.199929	−	0.979810i	$-0.435929\pi$
$163$	5.10504	0.399858	0.199929	+	0.979810i	$0.435929\pi$
$164$	2.63090	0.205439
$165$	0.326842	0.0254446
$166$	0.581449	0.0451292
$167$	−1.64650	−0.127410	−0.0637048	−	0.997969i	$-0.520292\pi$
$167$	−1.64650	−0.127410	−0.0637048	+	0.997969i	$0.520292\pi$
$168$	−3.07838	−0.237502
$169$	2.04945	0.157650
$170$	0	0
$171$	−4.17009	−0.318894
$172$	5.52586	0.421343
$173$	16.7165	1.27093	0.635465	−	0.772130i	$-0.280809\pi$
$173$	16.7165	1.27093	0.635465	+	0.772130i	$0.280809\pi$
$174$	2.00000	0.151620
$175$	−4.78765	−0.361913
$176$	1.65983	0.125114
$177$	3.21953	0.241995
$178$	4.21461	0.315899
$179$	−24.0144	−1.79492	−0.897459	−	0.441097i	$-0.854589\pi$
$179$	−24.0144	−1.79492	−0.897459	+	0.441097i	$0.854589\pi$
$180$	0.290725	0.0216693
$181$	−19.5174	−1.45072	−0.725360	−	0.688369i	$-0.758327\pi$
$181$	−19.5174	−1.45072	−0.725360	+	0.688369i	$0.758327\pi$
$182$	4.53919	0.336467
$183$	8.48133	0.626958
$184$	22.5958	1.66579
$185$	0.879362	0.0646520
$186$	5.70928	0.418624
$187$	0	0
$188$	−4.24128	−0.309327
$189$	1.00000	0.0727393
$190$	2.24846	0.163121
$191$	−10.8371	−0.784145	−0.392073	−	0.919934i	$-0.628242\pi$
$191$	−10.8371	−0.784145	−0.392073	+	0.919934i	$0.628242\pi$
$192$	8.68035	0.626450
$193$	7.72487	0.556049	0.278024	−	0.960574i	$-0.410320\pi$
$193$	7.72487	0.556049	0.278024	+	0.960574i	$0.410320\pi$
$194$	−12.9627	−0.930665
$195$	−1.78765	−0.128016
$196$	−0.630898	−0.0450641
$197$	13.2062	0.940903	0.470452	−	0.882426i	$-0.344091\pi$
$197$	13.2062	0.940903	0.470452	+	0.882426i	$0.344091\pi$
$198$	−0.829914	−0.0589794
$199$	17.5597	1.24477	0.622387	−	0.782709i	$-0.286163\pi$
$199$	17.5597	1.24477	0.622387	+	0.782709i	$0.286163\pi$
$200$	14.7382	1.04215
$201$	−6.81432	−0.480645
$202$	−16.4475	−1.15724
$203$	1.70928	0.119968
$204$	0	0
$205$	1.92162	0.134212
$206$	−19.9577	−1.39052
$207$	−7.34017	−0.510177
$208$	−9.07838	−0.629472
$209$	2.95774	0.204591
$210$	−0.539189	−0.0372076
$211$	−4.14957	−0.285668	−0.142834	−	0.989747i	$-0.545622\pi$
$211$	−4.14957	−0.285668	−0.142834	+	0.989747i	$0.545622\pi$
$212$	2.89496	0.198827
$213$	−10.7298	−0.735193
$214$	1.13889	0.0778529
$215$	4.03612	0.275261
$216$	−3.07838	−0.209457
$217$	4.87936	0.331233
$218$	2.81432	0.190609
$219$	−5.23513	−0.353758
$220$	−0.206204	−0.0139023
$221$	0	0
$222$	−2.23287	−0.149860
$223$	3.69594	0.247499	0.123749	−	0.992314i	$-0.460508\pi$
$223$	3.69594	0.247499	0.123749	+	0.992314i	$0.460508\pi$
$224$	3.41855	0.228412
$225$	−4.78765	−0.319177
$226$	19.8816	1.32251
$227$	11.2751	0.748356	0.374178	−	0.927357i	$-0.377925\pi$
$227$	11.2751	0.748356	0.374178	+	0.927357i	$0.377925\pi$
$228$	2.63090	0.174235
$229$	9.96493	0.658501	0.329250	−	0.944243i	$-0.393204\pi$
$229$	9.96493	0.658501	0.329250	+	0.944243i	$0.393204\pi$
$230$	3.95774	0.260966
$231$	−0.709275	−0.0466669
$232$	−5.26180	−0.345454
$233$	−25.9093	−1.69738	−0.848689	−	0.528893i	$-0.822607\pi$
$233$	−25.9093	−1.69738	−0.848689	+	0.528893i	$0.822607\pi$
$234$	4.53919	0.296736
$235$	−3.09785	−0.202082
$236$	−2.03120	−0.132220
$237$	−14.1639	−0.920047
$238$	0	0
$239$	−19.4524	−1.25827	−0.629136	−	0.777296i	$-0.716591\pi$
$239$	−19.4524	−1.25827	−0.629136	+	0.777296i	$0.716591\pi$
$240$	1.07838	0.0696090
$241$	−21.9421	−1.41342	−0.706709	−	0.707505i	$-0.749821\pi$
$241$	−21.9421	−1.41342	−0.706709	+	0.707505i	$0.749821\pi$
$242$	−12.2823	−0.789537
$243$	1.00000	0.0641500
$244$	−5.35085	−0.342553
$245$	−0.460811	−0.0294401
$246$	−4.87936	−0.311097
$247$	−16.1773	−1.02934
$248$	−15.0205	−0.953804
$249$	0.496928	0.0314916
$250$	5.27739	0.333772
$251$	7.00946	0.442433	0.221216	−	0.975225i	$-0.428997\pi$
$251$	7.00946	0.442433	0.221216	+	0.975225i	$0.428997\pi$
$252$	−0.630898	−0.0397428
$253$	5.20620	0.327311
$254$	−4.92881	−0.309261
$255$	0	0
$256$	−13.4764	−0.842276
$257$	9.71646	0.606096	0.303048	−	0.952975i	$-0.401996\pi$
$257$	9.71646	0.606096	0.303048	+	0.952975i	$0.401996\pi$
$258$	−10.2485	−0.638042
$259$	−1.90829	−0.118575
$260$	1.12783	0.0699448
$261$	1.70928	0.105801
$262$	18.7636	1.15922
$263$	−5.17727	−0.319244	−0.159622	−	0.987178i	$-0.551028\pi$
$263$	−5.17727	−0.319244	−0.159622	+	0.987178i	$0.551028\pi$
$264$	2.18342	0.134380
$265$	2.11450	0.129892
$266$	−4.87936	−0.299173
$267$	3.60197	0.220437
$268$	4.29914	0.262611
$269$	1.58864	0.0968609	0.0484305	−	0.998827i	$-0.484578\pi$
$269$	1.58864	0.0968609	0.0484305	+	0.998827i	$0.484578\pi$
$270$	−0.539189	−0.0328140
$271$	−19.9867	−1.21410	−0.607052	−	0.794662i	$-0.707648\pi$
$271$	−19.9867	−1.21410	−0.607052	+	0.794662i	$0.707648\pi$
$272$	0	0
$273$	3.87936	0.234790
$274$	−7.52586	−0.454654
$275$	3.39576	0.204772
$276$	4.63090	0.278747
$277$	−18.6947	−1.12326	−0.561628	−	0.827390i	$-0.689825\pi$
$277$	−18.6947	−1.12326	−0.561628	+	0.827390i	$0.689825\pi$
$278$	14.5548	0.872938
$279$	4.87936	0.292120
$280$	1.41855	0.0847746
$281$	22.4969	1.34205	0.671027	−	0.741433i	$-0.265853\pi$
$281$	22.4969	1.34205	0.671027	+	0.741433i	$0.265853\pi$
$282$	7.86603	0.468415
$283$	−2.06892	−0.122985	−0.0614923	−	0.998108i	$-0.519586\pi$
$283$	−2.06892	−0.122985	−0.0614923	+	0.998108i	$0.519586\pi$
$284$	6.76940	0.401690
$285$	1.92162	0.113827
$286$	−3.21953	−0.190375
$287$	−4.17009	−0.246152
$288$	3.41855	0.201440
$289$	0	0
$290$	−0.921622	−0.0541195
$291$	−11.0784	−0.649426
$292$	3.30283	0.193284
$293$	−29.6020	−1.72937	−0.864683	−	0.502318i	$-0.832481\pi$
$293$	−29.6020	−1.72937	−0.864683	+	0.502318i	$0.832481\pi$
$294$	1.17009	0.0682408
$295$	−1.48360	−0.0863784
$296$	5.87444	0.341445
$297$	−0.709275	−0.0411563
$298$	16.5281	0.957449
$299$	−28.4752	−1.64676
$300$	3.02052	0.174390
$301$	−8.75872	−0.504844
$302$	−19.0616	−1.09687
$303$	−14.0566	−0.807533
$304$	9.75872	0.559701
$305$	−3.90829	−0.223788
$306$	0	0
$307$	−3.28846	−0.187682	−0.0938411	−	0.995587i	$-0.529915\pi$
$307$	−3.28846	−0.187682	−0.0938411	+	0.995587i	$0.529915\pi$
$308$	0.447480	0.0254975
$309$	−17.0566	−0.970318
$310$	−2.63090	−0.149425
$311$	−19.2351	−1.09072	−0.545362	−	0.838201i	$-0.683608\pi$
$311$	−19.2351	−1.09072	−0.545362	+	0.838201i	$0.683608\pi$
$312$	−11.9421	−0.676091
$313$	−8.74927	−0.494538	−0.247269	−	0.968947i	$-0.579533\pi$
$313$	−8.74927	−0.494538	−0.247269	+	0.968947i	$0.579533\pi$
$314$	−23.1350	−1.30558
$315$	−0.460811	−0.0259638
$316$	8.93600	0.502689
$317$	18.2823	1.02684	0.513419	−	0.858138i	$-0.328379\pi$
$317$	18.2823	1.02684	0.513419	+	0.858138i	$0.328379\pi$
$318$	−5.36910	−0.301084
$319$	−1.21235	−0.0678784
$320$	−4.00000	−0.223607
$321$	0.973338	0.0543264
$322$	−8.58864	−0.478626
$323$	0	0
$324$	−0.630898	−0.0350499
$325$	−18.5730	−1.03025
$326$	5.97334	0.330833
$327$	2.40522	0.133009
$328$	12.8371	0.708810
$329$	6.72261	0.370629
$330$	0.382433	0.0210523
$331$	−27.7298	−1.52417	−0.762084	−	0.647479i	$-0.775824\pi$
$331$	−27.7298	−1.52417	−0.762084	+	0.647479i	$0.775824\pi$
$332$	−0.313511	−0.0172062
$333$	−1.90829	−0.104574
$334$	−1.92654	−0.105416
$335$	3.14011	0.171563
$336$	−2.34017	−0.127667
$337$	22.6297	1.23272	0.616358	−	0.787466i	$-0.288607\pi$
$337$	22.6297	1.23272	0.616358	+	0.787466i	$0.288607\pi$
$338$	2.39803	0.130436
$339$	16.9916	0.922856
$340$	0	0
$341$	−3.46081	−0.187413
$342$	−4.87936	−0.263846
$343$	1.00000	0.0539949
$344$	26.9627	1.45373
$345$	3.38243	0.182104
$346$	19.5597	1.05154
$347$	1.00227	0.0538045	0.0269023	−	0.999638i	$-0.491436\pi$
$347$	1.00227	0.0538045	0.0269023	+	0.999638i	$0.491436\pi$
$348$	−1.07838	−0.0578071
$349$	10.4569	0.559747	0.279873	−	0.960037i	$-0.409707\pi$
$349$	10.4569	0.559747	0.279873	+	0.960037i	$0.409707\pi$
$350$	−5.60197	−0.299438
$351$	3.87936	0.207065
$352$	−2.42469	−0.129237
$353$	19.1617	1.01987	0.509937	−	0.860212i	$-0.329669\pi$
$353$	19.1617	1.01987	0.509937	+	0.860212i	$0.329669\pi$
$354$	3.76713	0.200221
$355$	4.94441	0.262422
$356$	−2.27247	−0.120441
$357$	0	0
$358$	−28.0989	−1.48507
$359$	−4.67316	−0.246640	−0.123320	−	0.992367i	$-0.539354\pi$
$359$	−4.67316	−0.246640	−0.123320	+	0.992367i	$0.539354\pi$
$360$	1.41855	0.0747642
$361$	−1.61038	−0.0847568
$362$	−22.8371	−1.20029
$363$	−10.4969	−0.550946
$364$	−2.44748	−0.128283
$365$	2.41241	0.126271
$366$	9.92389	0.518730
$367$	33.1917	1.73259	0.866295	−	0.499533i	$-0.166495\pi$
$367$	33.1917	1.73259	0.866295	+	0.499533i	$0.166495\pi$
$368$	17.1773	0.895427
$369$	−4.17009	−0.217086
$370$	1.02893	0.0534915
$371$	−4.58864	−0.238230
$372$	−3.07838	−0.159606
$373$	−5.02893	−0.260388	−0.130194	−	0.991489i	$-0.541560\pi$
$373$	−5.02893	−0.260388	−0.130194	+	0.991489i	$0.541560\pi$
$374$	0	0
$375$	4.51026	0.232909
$376$	−20.6947	−1.06725
$377$	6.63090	0.341509
$378$	1.17009	0.0601828
$379$	31.6742	1.62699	0.813497	−	0.581569i	$-0.197561\pi$
$379$	31.6742	1.62699	0.813497	+	0.581569i	$0.197561\pi$
$380$	−1.21235	−0.0621921
$381$	−4.21235	−0.215805
$382$	−12.6803	−0.648783
$383$	7.93108	0.405259	0.202630	−	0.979255i	$-0.435051\pi$
$383$	7.93108	0.405259	0.202630	+	0.979255i	$0.435051\pi$
$384$	3.31965	0.169405
$385$	0.326842	0.0166574
$386$	9.03877	0.460061
$387$	−8.75872	−0.445231
$388$	6.98932	0.354829
$389$	−16.5503	−0.839131	−0.419566	−	0.907725i	$-0.637818\pi$
$389$	−16.5503	−0.839131	−0.419566	+	0.907725i	$0.637818\pi$
$390$	−2.09171	−0.105918
$391$	0	0
$392$	−3.07838	−0.155482
$393$	16.0361	0.808915
$394$	15.4524	0.778481
$395$	6.52690	0.328404
$396$	0.447480	0.0224867
$397$	26.3701	1.32348	0.661740	−	0.749734i	$-0.269818\pi$
$397$	26.3701	1.32348	0.661740	+	0.749734i	$0.269818\pi$
$398$	20.5464	1.02990
$399$	−4.17009	−0.208765
$400$	11.2039	0.560197
$401$	−26.2534	−1.31103	−0.655516	−	0.755182i	$-0.727549\pi$
$401$	−26.2534	−1.31103	−0.655516	+	0.755182i	$0.727549\pi$
$402$	−7.97334	−0.397674
$403$	18.9288	0.942911
$404$	8.86830	0.441214
$405$	−0.460811	−0.0228979
$406$	2.00000	0.0992583
$407$	1.35350	0.0670907
$408$	0	0
$409$	15.2713	0.755115	0.377557	−	0.925986i	$-0.376764\pi$
$409$	15.2713	0.755115	0.377557	+	0.925986i	$0.376764\pi$
$410$	2.24846	0.111044
$411$	−6.43188	−0.317261
$412$	10.7610	0.530156
$413$	3.21953	0.158423
$414$	−8.58864	−0.422108
$415$	−0.228990	−0.0112407
$416$	13.2618	0.650213
$417$	12.4391	0.609144
$418$	3.46081	0.169274
$419$	9.57918	0.467974	0.233987	−	0.972240i	$-0.424823\pi$
$419$	9.57918	0.467974	0.233987	+	0.972240i	$0.424823\pi$
$420$	0.290725	0.0141859
$421$	33.6453	1.63977	0.819885	−	0.572528i	$-0.194037\pi$
$421$	33.6453	1.63977	0.819885	+	0.572528i	$0.194037\pi$
$422$	−4.85535	−0.236355
$423$	6.72261	0.326864
$424$	14.1256	0.685998
$425$	0	0
$426$	−12.5548	−0.608281
$427$	8.48133	0.410440
$428$	−0.614077	−0.0296825
$429$	−2.75154	−0.132845
$430$	4.72261	0.227744
$431$	15.6332	0.753023	0.376512	−	0.926412i	$-0.377124\pi$
$431$	15.6332	0.753023	0.376512	+	0.926412i	$0.377124\pi$
$432$	−2.34017	−0.112592
$433$	10.4524	0.502310	0.251155	−	0.967947i	$-0.419190\pi$
$433$	10.4524	0.502310	0.251155	+	0.967947i	$0.419190\pi$
$434$	5.70928	0.274054
$435$	−0.787653	−0.0377651
$436$	−1.51745	−0.0726725
$437$	30.6092	1.46423
$438$	−6.12556	−0.292691
$439$	−13.6586	−0.651890	−0.325945	−	0.945389i	$-0.605682\pi$
$439$	−13.6586	−0.651890	−0.325945	+	0.945389i	$0.605682\pi$
$440$	−1.00614	−0.0479660
$441$	1.00000	0.0476190
$442$	0	0
$443$	15.4836	0.735648	0.367824	−	0.929895i	$-0.380103\pi$
$443$	15.4836	0.735648	0.367824	+	0.929895i	$0.380103\pi$
$444$	1.20394	0.0571363
$445$	−1.65983	−0.0786833
$446$	4.32457	0.204775
$447$	14.1256	0.668116
$448$	8.68035	0.410108
$449$	16.9132	0.798184	0.399092	−	0.916911i	$-0.369325\pi$
$449$	16.9132	0.798184	0.399092	+	0.916911i	$0.369325\pi$
$450$	−5.60197	−0.264079
$451$	2.95774	0.139275
$452$	−10.7200	−0.504224
$453$	−16.2907	−0.765405
$454$	13.1929	0.619172
$455$	−1.78765	−0.0838065
$456$	12.8371	0.601152
$457$	−7.98545	−0.373543	−0.186772	−	0.982403i	$-0.559802\pi$
$457$	−7.98545	−0.373543	−0.186772	+	0.982403i	$0.559802\pi$
$458$	11.6598	0.544828
$459$	0	0
$460$	−2.13397	−0.0994968
$461$	28.4235	1.32381	0.661907	−	0.749586i	$-0.269748\pi$
$461$	28.4235	1.32381	0.661907	+	0.749586i	$0.269748\pi$
$462$	−0.829914	−0.0386111
$463$	2.89884	0.134720	0.0673602	−	0.997729i	$-0.478542\pi$
$463$	2.89884	0.134720	0.0673602	+	0.997729i	$0.478542\pi$
$464$	−4.00000	−0.185695
$465$	−2.24846	−0.104270
$466$	−30.3162	−1.40437
$467$	0.993857	0.0459902	0.0229951	−	0.999736i	$-0.492680\pi$
$467$	0.993857	0.0459902	0.0229951	+	0.999736i	$0.492680\pi$
$468$	−2.44748	−0.113135
$469$	−6.81432	−0.314656
$470$	−3.62475	−0.167197
$471$	−19.7721	−0.911048
$472$	−9.91094	−0.456188
$473$	6.21235	0.285644
$474$	−16.5730	−0.761225
$475$	19.9649	0.916054
$476$	0	0
$477$	−4.58864	−0.210099
$478$	−22.7610	−1.04106
$479$	−17.9288	−0.819188	−0.409594	−	0.912268i	$-0.634330\pi$
$479$	−17.9288	−0.819188	−0.409594	+	0.912268i	$0.634330\pi$
$480$	−1.57531	−0.0719026
$481$	−7.40295	−0.337546
$482$	−25.6742	−1.16943
$483$	−7.34017	−0.333989
$484$	6.62249	0.301022
$485$	5.10504	0.231808
$486$	1.17009	0.0530762
$487$	−12.0410	−0.545632	−0.272816	−	0.962066i	$-0.587955\pi$
$487$	−12.0410	−0.545632	−0.272816	+	0.962066i	$0.587955\pi$
$488$	−26.1087	−1.18189
$489$	5.10504	0.230858
$490$	−0.539189	−0.0243581
$491$	−19.5753	−0.883421	−0.441711	−	0.897158i	$-0.645628\pi$
$491$	−19.5753	−0.883421	−0.441711	+	0.897158i	$0.645628\pi$
$492$	2.63090	0.118610
$493$	0	0
$494$	−18.9288	−0.851647
$495$	0.326842	0.0146905
$496$	−11.4186	−0.512708
$497$	−10.7298	−0.481297
$498$	0.581449	0.0260554
$499$	−29.9493	−1.34072	−0.670358	−	0.742038i	$-0.733859\pi$
$499$	−29.9493	−1.34072	−0.670358	+	0.742038i	$0.733859\pi$
$500$	−2.84551	−0.127255
$501$	−1.64650	−0.0735600
$502$	8.20167	0.366058
$503$	−13.9204	−0.620680	−0.310340	−	0.950626i	$-0.600443\pi$
$503$	−13.9204	−0.620680	−0.310340	+	0.950626i	$0.600443\pi$
$504$	−3.07838	−0.137122
$505$	6.47745	0.288243
$506$	6.09171	0.270809
$507$	2.04945	0.0910192
$508$	2.65756	0.117910
$509$	24.5380	1.08763	0.543813	−	0.839206i	$-0.316980\pi$
$509$	24.5380	1.08763	0.543813	+	0.839206i	$0.316980\pi$
$510$	0	0
$511$	−5.23513	−0.231589
$512$	−22.4079	−0.990297
$513$	−4.17009	−0.184114
$514$	11.3691	0.501470
$515$	7.85989	0.346348
$516$	5.52586	0.243262
$517$	−4.76818	−0.209704
$518$	−2.23287	−0.0981065
$519$	16.7165	0.733771
$520$	5.50307	0.241326
$521$	14.9350	0.654312	0.327156	−	0.944970i	$-0.393910\pi$
$521$	14.9350	0.654312	0.327156	+	0.944970i	$0.393910\pi$
$522$	2.00000	0.0875376
$523$	2.21008	0.0966400	0.0483200	−	0.998832i	$-0.484613\pi$
$523$	2.21008	0.0966400	0.0483200	+	0.998832i	$0.484613\pi$
$524$	−10.1171	−0.441970
$525$	−4.78765	−0.208950
$526$	−6.05786	−0.264135
$527$	0	0
$528$	1.65983	0.0722347
$529$	30.8781	1.34253
$530$	2.47414	0.107470
$531$	3.21953	0.139716
$532$	2.63090	0.114064
$533$	−16.1773	−0.700716
$534$	4.21461	0.182384
$535$	−0.448525	−0.0193914
$536$	20.9770	0.906070
$537$	−24.0144	−1.03630
$538$	1.85884	0.0801404
$539$	−0.709275	−0.0305507
$540$	0.290725	0.0125108
$541$	−9.75872	−0.419560	−0.209780	−	0.977749i	$-0.567275\pi$
$541$	−9.75872	−0.419560	−0.209780	+	0.977749i	$0.567275\pi$
$542$	−23.3861	−1.00452
$543$	−19.5174	−0.837574
$544$	0	0
$545$	−1.10835	−0.0474766
$546$	4.53919	0.194259
$547$	−4.61530	−0.197336	−0.0986680	−	0.995120i	$-0.531458\pi$
$547$	−4.61530	−0.197336	−0.0986680	+	0.995120i	$0.531458\pi$
$548$	4.05786	0.173343
$549$	8.48133	0.361974
$550$	3.97334	0.169424
$551$	−7.12783	−0.303656
$552$	22.5958	0.961742
$553$	−14.1639	−0.602312
$554$	−21.8744	−0.929356
$555$	0.879362	0.0373268
$556$	−7.84778	−0.332820
$557$	−13.4836	−0.571318	−0.285659	−	0.958331i	$-0.592213\pi$
$557$	−13.4836	−0.571318	−0.285659	+	0.958331i	$0.592213\pi$
$558$	5.70928	0.241693
$559$	−33.9783	−1.43713
$560$	1.07838	0.0455698
$561$	0	0
$562$	26.3234	1.11038
$563$	0.779243	0.0328412	0.0164206	−	0.999865i	$-0.494773\pi$
$563$	0.779243	0.0328412	0.0164206	+	0.999865i	$0.494773\pi$
$564$	−4.24128	−0.178590
$565$	−7.82991	−0.329407
$566$	−2.42082	−0.101755
$567$	1.00000	0.0419961
$568$	33.0304	1.38592
$569$	18.8638	0.790810	0.395405	−	0.918507i	$-0.370604\pi$
$569$	18.8638	0.790810	0.395405	+	0.918507i	$0.370604\pi$
$570$	2.24846	0.0941778
$571$	39.6697	1.66012	0.830062	−	0.557671i	$-0.188305\pi$
$571$	39.6697	1.66012	0.830062	+	0.557671i	$0.188305\pi$
$572$	1.73594	0.0725832
$573$	−10.8371	−0.452726
$574$	−4.87936	−0.203661
$575$	35.1422	1.46553
$576$	8.68035	0.361681
$577$	34.2072	1.42407	0.712033	−	0.702146i	$-0.247775\pi$
$577$	34.2072	1.42407	0.712033	+	0.702146i	$0.247775\pi$
$578$	0	0
$579$	7.72487	0.321035
$580$	0.496928	0.0206338
$581$	0.496928	0.0206161
$582$	−12.9627	−0.537320
$583$	3.25461	0.134792
$584$	16.1157	0.666873
$585$	−1.78765	−0.0739104
$586$	−34.6369	−1.43084
$587$	−26.1990	−1.08135	−0.540675	−	0.841232i	$-0.681831\pi$
$587$	−26.1990	−1.08135	−0.540675	+	0.841232i	$0.681831\pi$
$588$	−0.630898	−0.0260178
$589$	−20.3474	−0.838398
$590$	−1.73594	−0.0714674
$591$	13.2062	0.543231
$592$	4.46573	0.183540
$593$	0.749268	0.0307688	0.0153844	−	0.999882i	$-0.495103\pi$
$593$	0.749268	0.0307688	0.0153844	+	0.999882i	$0.495103\pi$
$594$	−0.829914	−0.0340518
$595$	0	0
$596$	−8.91178	−0.365041
$597$	17.5597	0.718671
$598$	−33.3184	−1.36249
$599$	5.67420	0.231842	0.115921	−	0.993258i	$-0.463018\pi$
$599$	5.67420	0.231842	0.115921	+	0.993258i	$0.463018\pi$
$600$	14.7382	0.601685
$601$	2.97826	0.121486	0.0607428	−	0.998153i	$-0.480653\pi$
$601$	2.97826	0.121486	0.0607428	+	0.998153i	$0.480653\pi$
$602$	−10.2485	−0.417696
$603$	−6.81432	−0.277500
$604$	10.2778	0.418197
$605$	4.83710	0.196656
$606$	−16.4475	−0.668133
$607$	−33.3328	−1.35294	−0.676468	−	0.736472i	$-0.736490\pi$
$607$	−33.3328	−1.35294	−0.676468	+	0.736472i	$0.736490\pi$
$608$	−14.2557	−0.578143
$609$	1.70928	0.0692633
$610$	−4.57304	−0.185157
$611$	26.0794	1.05506
$612$	0	0
$613$	9.65368	0.389909	0.194954	−	0.980812i	$-0.437544\pi$
$613$	9.65368	0.389909	0.194954	+	0.980812i	$0.437544\pi$
$614$	−3.84778	−0.155284
$615$	1.92162	0.0774873
$616$	2.18342	0.0879724
$617$	−15.7815	−0.635340	−0.317670	−	0.948201i	$-0.602900\pi$
$617$	−15.7815	−0.635340	−0.317670	+	0.948201i	$0.602900\pi$
$618$	−19.9577	−0.802818
$619$	−22.0566	−0.886531	−0.443266	−	0.896390i	$-0.646180\pi$
$619$	−22.0566	−0.886531	−0.443266	+	0.896390i	$0.646180\pi$
$620$	1.41855	0.0569704
$621$	−7.34017	−0.294551
$622$	−22.5068	−0.902439
$623$	3.60197	0.144310
$624$	−9.07838	−0.363426
$625$	21.8599	0.874396
$626$	−10.2374	−0.409169
$627$	2.95774	0.118121
$628$	12.4741	0.497772
$629$	0	0
$630$	−0.539189	−0.0214818
$631$	39.4801	1.57168	0.785839	−	0.618431i	$-0.212231\pi$
$631$	39.4801	1.57168	0.785839	+	0.618431i	$0.212231\pi$
$632$	43.6020	1.73439
$633$	−4.14957	−0.164931
$634$	21.3919	0.849580
$635$	1.94110	0.0770301
$636$	2.89496	0.114793
$637$	3.87936	0.153706
$638$	−1.41855	−0.0561610
$639$	−10.7298	−0.424464
$640$	−1.52973	−0.0604680
$641$	45.2122	1.78577	0.892887	−	0.450281i	$-0.148676\pi$
$641$	45.2122	1.78577	0.892887	+	0.450281i	$0.148676\pi$
$642$	1.13889	0.0449484
$643$	30.3090	1.19527	0.597635	−	0.801769i	$-0.296107\pi$
$643$	30.3090	1.19527	0.597635	+	0.801769i	$0.296107\pi$
$644$	4.63090	0.182483
$645$	4.03612	0.158922
$646$	0	0
$647$	35.8576	1.40971	0.704854	−	0.709352i	$-0.251012\pi$
$647$	35.8576	1.40971	0.704854	+	0.709352i	$0.251012\pi$
$648$	−3.07838	−0.120930
$649$	−2.28354	−0.0896367
$650$	−21.7321	−0.852402
$651$	4.87936	0.191237
$652$	−3.22076	−0.126135
$653$	38.4452	1.50448	0.752239	−	0.658891i	$-0.228974\pi$
$653$	38.4452	1.50448	0.752239	+	0.658891i	$0.228974\pi$
$654$	2.81432	0.110048
$655$	−7.38962	−0.288736
$656$	9.75872	0.381014
$657$	−5.23513	−0.204242
$658$	7.86603	0.306650
$659$	−25.7392	−1.00266	−0.501329	−	0.865257i	$-0.667156\pi$
$659$	−25.7392	−1.00266	−0.501329	+	0.865257i	$0.667156\pi$
$660$	−0.206204	−0.00802647
$661$	−16.0589	−0.624619	−0.312309	−	0.949980i	$-0.601103\pi$
$661$	−16.0589	−0.624619	−0.312309	+	0.949980i	$0.601103\pi$
$662$	−32.4463	−1.26106
$663$	0	0
$664$	−1.52973	−0.0593652
$665$	1.92162	0.0745173
$666$	−2.23287	−0.0865218
$667$	−12.5464	−0.485798
$668$	1.03877	0.0401912
$669$	3.69594	0.142893
$670$	3.67420	0.141947
$671$	−6.01560	−0.232230
$672$	3.41855	0.131873
$673$	32.2388	1.24272	0.621358	−	0.783527i	$-0.286581\pi$
$673$	32.2388	1.24272	0.621358	+	0.783527i	$0.286581\pi$
$674$	26.4787	1.01992
$675$	−4.78765	−0.184277
$676$	−1.29299	−0.0497305
$677$	−35.5718	−1.36714	−0.683568	−	0.729887i	$-0.739573\pi$
$677$	−35.5718	−1.36714	−0.683568	+	0.729887i	$0.739573\pi$
$678$	19.8816	0.763549
$679$	−11.0784	−0.425149
$680$	0	0
$681$	11.2751	0.432064
$682$	−4.04945	−0.155061
$683$	−9.67647	−0.370260	−0.185130	−	0.982714i	$-0.559271\pi$
$683$	−9.67647	−0.370260	−0.185130	+	0.982714i	$0.559271\pi$
$684$	2.63090	0.100595
$685$	2.96388	0.113244
$686$	1.17009	0.0446741
$687$	9.96493	0.380186
$688$	20.4969	0.781438
$689$	−17.8010	−0.678163
$690$	3.95774	0.150669
$691$	−29.7887	−1.13322	−0.566608	−	0.823988i	$-0.691745\pi$
$691$	−29.7887	−1.13322	−0.566608	+	0.823988i	$0.691745\pi$
$692$	−10.5464	−0.400913
$693$	−0.709275	−0.0269431
$694$	1.17274	0.0445166
$695$	−5.73206	−0.217429
$696$	−5.26180	−0.199448
$697$	0	0
$698$	12.2355	0.463121
$699$	−25.9093	−0.979981
$700$	3.02052	0.114165
$701$	−4.39803	−0.166111	−0.0830557	−	0.996545i	$-0.526468\pi$
$701$	−4.39803	−0.166111	−0.0830557	+	0.996545i	$0.526468\pi$
$702$	4.53919	0.171321
$703$	7.95774	0.300132
$704$	−6.15676	−0.232041
$705$	−3.09785	−0.116672
$706$	22.4208	0.843819
$707$	−14.0566	−0.528654
$708$	−2.03120	−0.0763371
$709$	−17.3607	−0.651994	−0.325997	−	0.945371i	$-0.605700\pi$
$709$	−17.3607	−0.651994	−0.325997	+	0.945371i	$0.605700\pi$
$710$	5.78539	0.217122
$711$	−14.1639	−0.531189
$712$	−11.0882	−0.415549
$713$	−35.8154	−1.34130
$714$	0	0
$715$	1.26794	0.0474182
$716$	15.1506	0.566205
$717$	−19.4524	−0.726463
$718$	−5.46800	−0.204064
$719$	−24.9265	−0.929603	−0.464802	−	0.885415i	$-0.653874\pi$
$719$	−24.9265	−0.929603	−0.464802	+	0.885415i	$0.653874\pi$
$720$	1.07838	0.0401888
$721$	−17.0566	−0.635222
$722$	−1.88428	−0.0701257
$723$	−21.9421	−0.816037
$724$	12.3135	0.457628
$725$	−8.18342	−0.303924
$726$	−12.2823	−0.455839
$727$	−20.7115	−0.768149	−0.384074	−	0.923302i	$-0.625479\pi$
$727$	−20.7115	−0.768149	−0.384074	+	0.923302i	$0.625479\pi$
$728$	−11.9421	−0.442605
$729$	1.00000	0.0370370
$730$	2.82273	0.104474
$731$	0	0
$732$	−5.35085	−0.197773
$733$	2.81044	0.103806	0.0519030	−	0.998652i	$-0.483471\pi$
$733$	2.81044	0.103806	0.0519030	+	0.998652i	$0.483471\pi$
$734$	38.8371	1.43350
$735$	−0.460811	−0.0169973
$736$	−25.0928	−0.924931
$737$	4.83323	0.178034
$738$	−4.87936	−0.179612
$739$	41.2388	1.51699	0.758497	−	0.651676i	$-0.225934\pi$
$739$	41.2388	1.51699	0.758497	+	0.651676i	$0.225934\pi$
$740$	−0.554787	−0.0203944
$741$	−16.1773	−0.594287
$742$	−5.36910	−0.197106
$743$	52.1750	1.91412	0.957058	−	0.289898i	$-0.0936212\pi$
$743$	52.1750	1.91412	0.957058	+	0.289898i	$0.0936212\pi$
$744$	−15.0205	−0.550679
$745$	−6.50921	−0.238479
$746$	−5.88428	−0.215439
$747$	0.496928	0.0181817
$748$	0	0
$749$	0.973338	0.0355650
$750$	5.27739	0.192703
$751$	53.0544	1.93598	0.967991	−	0.250986i	$-0.0807548\pi$
$751$	53.0544	1.93598	0.967991	+	0.250986i	$0.0807548\pi$
$752$	−15.7321	−0.573689
$753$	7.00946	0.255439
$754$	7.75872	0.282556
$755$	7.50695	0.273206
$756$	−0.630898	−0.0229455
$757$	−34.7548	−1.26319	−0.631593	−	0.775300i	$-0.717599\pi$
$757$	−34.7548	−1.26319	−0.631593	+	0.775300i	$0.717599\pi$
$758$	37.0616	1.34614
$759$	5.20620	0.188973
$760$	−5.91548	−0.214577
$761$	1.60650	0.0582357	0.0291178	−	0.999576i	$-0.490730\pi$
$761$	1.60650	0.0582357	0.0291178	+	0.999576i	$0.490730\pi$
$762$	−4.92881	−0.178552
$763$	2.40522	0.0870748
$764$	6.83710	0.247358
$765$	0	0
$766$	9.28005	0.335302
$767$	12.4897	0.450978
$768$	−13.4764	−0.486288
$769$	40.9676	1.47733	0.738664	−	0.674073i	$-0.235457\pi$
$769$	40.9676	1.47733	0.738664	+	0.674073i	$0.235457\pi$
$770$	0.382433	0.0137819
$771$	9.71646	0.349930
$772$	−4.87360	−0.175405
$773$	6.66475	0.239714	0.119857	−	0.992791i	$-0.461756\pi$
$773$	6.66475	0.239714	0.119857	+	0.992791i	$0.461756\pi$
$774$	−10.2485	−0.368373
$775$	−23.3607	−0.839141
$776$	34.1034	1.22424
$777$	−1.90829	−0.0684596
$778$	−19.3652	−0.694277
$779$	17.3896	0.623048
$780$	1.12783	0.0403827
$781$	7.61038	0.272321
$782$	0	0
$783$	1.70928	0.0610845
$784$	−2.34017	−0.0835776
$785$	9.11118	0.325192
$786$	18.7636	0.669277
$787$	−12.5080	−0.445862	−0.222931	−	0.974834i	$-0.571562\pi$
$787$	−12.5080	−0.445862	−0.222931	+	0.974834i	$0.571562\pi$
$788$	−8.33176	−0.296807
$789$	−5.17727	−0.184316
$790$	7.63704	0.271714
$791$	16.9916	0.604151
$792$	2.18342	0.0775844
$793$	32.9021	1.16839
$794$	30.8554	1.09502
$795$	2.11450	0.0749934
$796$	−11.0784	−0.392663
$797$	−0.707008	−0.0250435	−0.0125218	−	0.999922i	$-0.503986\pi$
$797$	−0.707008	−0.0250435	−0.0125218	+	0.999922i	$0.503986\pi$
$798$	−4.87936	−0.172728
$799$	0	0
$800$	−16.3668	−0.578655
$801$	3.60197	0.127269
$802$	−30.7187	−1.08472
$803$	3.71315	0.131034
$804$	4.29914	0.151619
$805$	3.38243	0.119215
$806$	22.1483	0.780142
$807$	1.58864	0.0559227
$808$	43.2716	1.52229
$809$	8.20006	0.288299	0.144149	−	0.989556i	$-0.453955\pi$
$809$	8.20006	0.288299	0.144149	+	0.989556i	$0.453955\pi$
$810$	−0.539189	−0.0189452
$811$	−49.6697	−1.74414	−0.872069	−	0.489383i	$-0.837222\pi$
$811$	−49.6697	−1.74414	−0.872069	+	0.489383i	$0.837222\pi$
$812$	−1.07838	−0.0378436
$813$	−19.9867	−0.700963
$814$	1.58372	0.0555092
$815$	−2.35246	−0.0824030
$816$	0	0
$817$	36.5246	1.27784
$818$	17.8687	0.624764
$819$	3.87936	0.135556
$820$	−1.21235	−0.0423370
$821$	−3.29914	−0.115141	−0.0575703	−	0.998341i	$-0.518335\pi$
$821$	−3.29914	−0.115141	−0.0575703	+	0.998341i	$0.518335\pi$
$822$	−7.52586	−0.262494
$823$	−6.60092	−0.230094	−0.115047	−	0.993360i	$-0.536702\pi$
$823$	−6.60092	−0.230094	−0.115047	+	0.993360i	$0.536702\pi$
$824$	52.5068	1.82916
$825$	3.39576	0.118225
$826$	3.76713	0.131075
$827$	36.9383	1.28447	0.642235	−	0.766508i	$-0.278007\pi$
$827$	36.9383	1.28447	0.642235	+	0.766508i	$0.278007\pi$
$828$	4.63090	0.160935
$829$	−38.0288	−1.32079	−0.660397	−	0.750917i	$-0.729612\pi$
$829$	−38.0288	−1.32079	−0.660397	+	0.750917i	$0.729612\pi$
$830$	−0.267938	−0.00930027
$831$	−18.6947	−0.648513
$832$	33.6742	1.16744
$833$	0	0
$834$	14.5548	0.503991
$835$	0.758724	0.0262567
$836$	−1.86603	−0.0645380
$837$	4.87936	0.168655
$838$	11.2085	0.387190
$839$	51.0893	1.76380	0.881899	−	0.471439i	$-0.156265\pi$
$839$	51.0893	1.76380	0.881899	+	0.471439i	$0.156265\pi$
$840$	1.41855	0.0489446
$841$	−26.0784	−0.899254
$842$	39.3679	1.35671
$843$	22.4969	0.774835
$844$	2.61795	0.0901136
$845$	−0.944409	−0.0324886
$846$	7.86603	0.270440
$847$	−10.4969	−0.360679
$848$	10.7382	0.368751
$849$	−2.06892	−0.0710052
$850$	0	0
$851$	14.0072	0.480160
$852$	6.76940	0.231916
$853$	56.9081	1.94850	0.974248	−	0.225478i	$-0.0723943\pi$
$853$	56.9081	1.94850	0.974248	+	0.225478i	$0.0723943\pi$
$854$	9.92389	0.339589
$855$	1.92162	0.0657181
$856$	−2.99630	−0.102411
$857$	11.7731	0.402161	0.201081	−	0.979575i	$-0.435555\pi$
$857$	11.7731	0.402161	0.201081	+	0.979575i	$0.435555\pi$
$858$	−3.21953	−0.109913
$859$	19.0349	0.649462	0.324731	−	0.945806i	$-0.394726\pi$
$859$	19.0349	0.649462	0.324731	+	0.945806i	$0.394726\pi$
$860$	−2.54638	−0.0868307
$861$	−4.17009	−0.142116
$862$	18.2922	0.623033
$863$	−11.2858	−0.384173	−0.192087	−	0.981378i	$-0.561525\pi$
$863$	−11.2858	−0.384173	−0.192087	+	0.981378i	$0.561525\pi$
$864$	3.41855	0.116301
$865$	−7.70313	−0.261914
$866$	12.2302	0.415600
$867$	0	0
$868$	−3.07838	−0.104487
$869$	10.0461	0.340792
$870$	−0.921622	−0.0312459
$871$	−26.4352	−0.895722
$872$	−7.40417	−0.250737
$873$	−11.0784	−0.374946
$874$	35.8154	1.21147
$875$	4.51026	0.152475
$876$	3.30283	0.111592
$877$	−15.4257	−0.520890	−0.260445	−	0.965489i	$-0.583869\pi$
$877$	−15.4257	−0.520890	−0.260445	+	0.965489i	$0.583869\pi$
$878$	−15.9817	−0.539358
$879$	−29.6020	−0.998450
$880$	−0.764867	−0.0257837
$881$	30.1301	1.01511	0.507554	−	0.861620i	$-0.330550\pi$
$881$	30.1301	1.01511	0.507554	+	0.861620i	$0.330550\pi$
$882$	1.17009	0.0393989
$883$	−37.1217	−1.24924	−0.624622	−	0.780927i	$-0.714747\pi$
$883$	−37.1217	−1.24924	−0.624622	+	0.780927i	$0.714747\pi$
$884$	0	0
$885$	−1.48360	−0.0498706
$886$	18.1171	0.608657
$887$	−17.8033	−0.597775	−0.298887	−	0.954288i	$-0.596615\pi$
$887$	−17.8033	−0.597775	−0.298887	+	0.954288i	$0.596615\pi$
$888$	5.87444	0.197133
$889$	−4.21235	−0.141278
$890$	−1.94214	−0.0651007
$891$	−0.709275	−0.0237616
$892$	−2.33176	−0.0780732
$893$	−28.0338	−0.938117
$894$	16.5281	0.552783
$895$	11.0661	0.369899
$896$	3.31965	0.110902
$897$	−28.4752	−0.950759
$898$	19.7899	0.660398
$899$	8.34017	0.278160
$900$	3.02052	0.100684
$901$	0	0
$902$	3.46081	0.115232
$903$	−8.75872	−0.291472
$904$	−52.3065	−1.73969
$905$	8.99386	0.298966
$906$	−19.0616	−0.633278
$907$	−19.8576	−0.659361	−0.329681	−	0.944092i	$-0.606941\pi$
$907$	−19.8576	−0.659361	−0.329681	+	0.944092i	$0.606941\pi$
$908$	−7.11345	−0.236068
$909$	−14.0566	−0.466229
$910$	−2.09171	−0.0693395
$911$	37.1483	1.23078	0.615390	−	0.788223i	$-0.288999\pi$
$911$	37.1483	1.23078	0.615390	+	0.788223i	$0.288999\pi$
$912$	9.75872	0.323144
$913$	−0.352459	−0.0116647
$914$	−9.34366	−0.309061
$915$	−3.90829	−0.129204
$916$	−6.28685	−0.207723
$917$	16.0361	0.529559
$918$	0	0
$919$	−29.4268	−0.970700	−0.485350	−	0.874320i	$-0.661308\pi$
$919$	−29.4268	−0.970700	−0.485350	+	0.874320i	$0.661308\pi$
$920$	−10.4124	−0.343287
$921$	−3.28846	−0.108358
$922$	33.2579	1.09529
$923$	−41.6248	−1.37010
$924$	0.447480	0.0147210
$925$	9.13624	0.300398
$926$	3.39189	0.111464
$927$	−17.0566	−0.560213
$928$	5.84324	0.191814
$929$	−37.7864	−1.23973	−0.619866	−	0.784707i	$-0.712813\pi$
$929$	−37.7864	−1.23973	−0.619866	+	0.784707i	$0.712813\pi$
$930$	−2.63090	−0.0862705
$931$	−4.17009	−0.136669
$932$	16.3461	0.535436
$933$	−19.2351	−0.629730
$934$	1.16290	0.0380512
$935$	0	0
$936$	−11.9421	−0.390341
$937$	37.5981	1.22828	0.614138	−	0.789199i	$-0.289504\pi$
$937$	37.5981	1.22828	0.614138	+	0.789199i	$0.289504\pi$
$938$	−7.97334	−0.260339
$939$	−8.74927	−0.285522
$940$	1.95443	0.0637464
$941$	8.49239	0.276844	0.138422	−	0.990373i	$-0.455797\pi$
$941$	8.49239	0.276844	0.138422	+	0.990373i	$0.455797\pi$
$942$	−23.1350	−0.753779
$943$	30.6092	0.996771
$944$	−7.53427	−0.245220
$945$	−0.460811	−0.0149902
$946$	7.26898	0.236335
$947$	29.0166	0.942914	0.471457	−	0.881889i	$-0.343728\pi$
$947$	29.0166	0.942914	0.471457	+	0.881889i	$0.343728\pi$
$948$	8.93600	0.290228
$949$	−20.3090	−0.659257
$950$	23.3607	0.757921
$951$	18.2823	0.592845
$952$	0	0
$953$	52.5574	1.70250	0.851251	−	0.524758i	$-0.175844\pi$
$953$	52.5574	1.70250	0.851251	+	0.524758i	$0.175844\pi$
$954$	−5.36910	−0.173831
$955$	4.99386	0.161597
$956$	12.2725	0.396920
$957$	−1.21235	−0.0391896
$958$	−20.9783	−0.677777
$959$	−6.43188	−0.207696
$960$	−4.00000	−0.129099
$961$	−7.19183	−0.231994
$962$	−8.66209	−0.279277
$963$	0.973338	0.0313654
$964$	13.8432	0.445861
$965$	−3.55971	−0.114591
$966$	−8.58864	−0.276335
$967$	11.1795	0.359510	0.179755	−	0.983711i	$-0.442470\pi$
$967$	11.1795	0.359510	0.179755	+	0.983711i	$0.442470\pi$
$968$	32.3135	1.03860
$969$	0	0
$970$	5.97334	0.191792
$971$	15.7275	0.504720	0.252360	−	0.967633i	$-0.418793\pi$
$971$	15.7275	0.504720	0.252360	+	0.967633i	$0.418793\pi$
$972$	−0.630898	−0.0202361
$973$	12.4391	0.398778
$974$	−14.0891	−0.451442
$975$	−18.5730	−0.594813
$976$	−19.8478	−0.635312
$977$	−45.9688	−1.47067	−0.735336	−	0.677703i	$-0.762976\pi$
$977$	−45.9688	−1.47067	−0.735336	+	0.677703i	$0.762976\pi$
$978$	5.97334	0.191006
$979$	−2.55479	−0.0816514
$980$	0.290725	0.00928686
$981$	2.40522	0.0767928
$982$	−22.9048	−0.730922
$983$	−14.6670	−0.467805	−0.233903	−	0.972260i	$-0.575150\pi$
$983$	−14.6670	−0.467805	−0.233903	+	0.972260i	$0.575150\pi$
$984$	12.8371	0.409232
$985$	−6.08557	−0.193902
$986$	0	0
$987$	6.72261	0.213983
$988$	10.2062	0.324703
$989$	64.2905	2.04432
$990$	0.382433	0.0121545
$991$	17.5343	0.556994	0.278497	−	0.960437i	$-0.410164\pi$
$991$	17.5343	0.556994	0.278497	+	0.960437i	$0.410164\pi$
$992$	16.6803	0.529602
$993$	−27.7298	−0.879978
$994$	−12.5548	−0.398214
$995$	−8.09171	−0.256524
$996$	−0.313511	−0.00993398
$997$	−57.1506	−1.80998	−0.904989	−	0.425435i	$-0.860121\pi$
$997$	−57.1506	−1.80998	−0.904989	+	0.425435i	$0.860121\pi$
$998$	−35.0433	−1.10928
$999$	−1.90829	−0.0603757

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.m.1.3		3
17.4	even	4		357.2.f.a.169.1	✓	6
17.13	even	4		357.2.f.a.169.2	yes	6
17.16	even	2		6069.2.a.k.1.3		3
51.38	odd	4		1071.2.f.a.883.5		6
51.47	odd	4		1071.2.f.a.883.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.f.a.169.1	✓	6	17.4	even	4
357.2.f.a.169.2	yes	6	17.13	even	4
1071.2.f.a.883.5		6	51.38	odd	4
1071.2.f.a.883.6		6	51.47	odd	4
6069.2.a.k.1.3		3	17.16	even	2
6069.2.a.m.1.3		3	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.17009 q^{2} +1.00000 q^{3} -0.630898 q^{4} -0.460811 q^{5} +1.17009 q^{6} +1.00000 q^{7} -3.07838 q^{8} +1.00000 q^{9} -0.539189 q^{10} -0.709275 q^{11} -0.630898 q^{12} +3.87936 q^{13} +1.17009 q^{14} -0.460811 q^{15} -2.34017 q^{16} +1.17009 q^{18} -4.17009 q^{19} +0.290725 q^{20} +1.00000 q^{21} -0.829914 q^{22} -7.34017 q^{23} -3.07838 q^{24} -4.78765 q^{25} +4.53919 q^{26} +1.00000 q^{27} -0.630898 q^{28} +1.70928 q^{29} -0.539189 q^{30} +4.87936 q^{31} +3.41855 q^{32} -0.709275 q^{33} -0.460811 q^{35} -0.630898 q^{36} -1.90829 q^{37} -4.87936 q^{38} +3.87936 q^{39} +1.41855 q^{40} -4.17009 q^{41} +1.17009 q^{42} -8.75872 q^{43} +0.447480 q^{44} -0.460811 q^{45} -8.58864 q^{46} +6.72261 q^{47} -2.34017 q^{48} +1.00000 q^{49} -5.60197 q^{50} -2.44748 q^{52} -4.58864 q^{53} +1.17009 q^{54} +0.326842 q^{55} -3.07838 q^{56} -4.17009 q^{57} +2.00000 q^{58} +3.21953 q^{59} +0.290725 q^{60} +8.48133 q^{61} +5.70928 q^{62} +1.00000 q^{63} +8.68035 q^{64} -1.78765 q^{65} -0.829914 q^{66} -6.81432 q^{67} -7.34017 q^{69} -0.539189 q^{70} -10.7298 q^{71} -3.07838 q^{72} -5.23513 q^{73} -2.23287 q^{74} -4.78765 q^{75} +2.63090 q^{76} -0.709275 q^{77} +4.53919 q^{78} -14.1639 q^{79} +1.07838 q^{80} +1.00000 q^{81} -4.87936 q^{82} +0.496928 q^{83} -0.630898 q^{84} -10.2485 q^{86} +1.70928 q^{87} +2.18342 q^{88} +3.60197 q^{89} -0.539189 q^{90} +3.87936 q^{91} +4.63090 q^{92} +4.87936 q^{93} +7.86603 q^{94} +1.92162 q^{95} +3.41855 q^{96} -11.0784 q^{97} +1.17009 q^{98} -0.709275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 5 q^{11} + 2 q^{12} - q^{13} - 2 q^{14} - 3 q^{15} + 4 q^{16} - 2 q^{18} - 7 q^{19} + 8 q^{20} + 3 q^{21} - 8 q^{22}+ \cdots + 5 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^5 - 2 * q^6 + 3 * q^7 - 6 * q^8 + 3 * q^9 + 5 * q^11 + 2 * q^12 - q^13 - 2 * q^14 - 3 * q^15 + 4 * q^16 - 2 * q^18 - 7 * q^19 + 8 * q^20 + 3 * q^21 - 8 * q^22 - 11 * q^23 - 6 * q^24 - 4 * q^25 + 12 * q^26 + 3 * q^27 + 2 * q^28 - 2 * q^29 + 2 * q^31 - 4 * q^32 + 5 * q^33 - 3 * q^35 + 2 * q^36 - 8 * q^37 - 2 * q^38 - q^39 - 10 * q^40 - 7 * q^41 - 2 * q^42 - q^43 + 2 * q^44 - 3 * q^45 - 6 * q^46 + 14 * q^47 + 4 * q^48 + 3 * q^49 + 2 * q^50 - 8 * q^52 + 6 * q^53 - 2 * q^54 - 11 * q^55 - 6 * q^56 - 7 * q^57 + 6 * q^58 - 14 * q^59 + 8 * q^60 - 6 * q^61 + 10 * q^62 + 3 * q^63 + 4 * q^64 + 5 * q^65 - 8 * q^66 - 12 * q^67 - 11 * q^69 + 8 * q^71 - 6 * q^72 - 6 * q^73 + 16 * q^74 - 4 * q^75 + 4 * q^76 + 5 * q^77 + 12 * q^78 - 2 * q^79 + 3 * q^81 - 2 * q^82 - 16 * q^83 + 2 * q^84 - 22 * q^86 - 2 * q^87 + 2 * q^88 - 8 * q^89 - q^91 + 10 * q^92 + 2 * q^93 + 10 * q^94 + 9 * q^95 - 4 * q^96 - 30 * q^97 - 2 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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