LMFDB - Embedded newform 1458.2.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1458,2,Mod(1,1458)] 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("1458.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1458, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1458.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,6,0,6,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.6421886147$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6357609.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 $ x^6 - 9x^4 - x^3 + 18x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.842316$ of defining polynomial
Character	$\chi$	$=$	1458.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.08802 q^{5} -4.01220 q^{7} +1.00000 q^{8} +2.08802 q^{10} +0.212447 q^{11} +5.19903 q^{13} -4.01220 q^{14} +1.00000 q^{16} +3.78552 q^{17} -1.27281 q^{19} +2.08802 q^{20} +0.212447 q^{22} +0.829442 q^{23} -0.640186 q^{25} +5.19903 q^{26} -4.01220 q^{28} +9.21123 q^{29} +3.53590 q^{31} +1.00000 q^{32} +3.78552 q^{34} -8.37755 q^{35} +3.55460 q^{37} -1.27281 q^{38} +2.08802 q^{40} +2.73940 q^{41} +6.89653 q^{43} +0.212447 q^{44} +0.829442 q^{46} -1.27727 q^{47} +9.09779 q^{49} -0.640186 q^{50} +5.19903 q^{52} -11.2992 q^{53} +0.443592 q^{55} -4.01220 q^{56} +9.21123 q^{58} -8.55985 q^{59} +2.83368 q^{61} +3.53590 q^{62} +1.00000 q^{64} +10.8557 q^{65} +9.76056 q^{67} +3.78552 q^{68} -8.37755 q^{70} -11.7241 q^{71} +0.750325 q^{73} +3.55460 q^{74} -1.27281 q^{76} -0.852380 q^{77} -2.36778 q^{79} +2.08802 q^{80} +2.73940 q^{82} -11.4005 q^{83} +7.90424 q^{85} +6.89653 q^{86} +0.212447 q^{88} +5.39625 q^{89} -20.8596 q^{91} +0.829442 q^{92} -1.27727 q^{94} -2.65765 q^{95} -12.5632 q^{97} +9.09779 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8} + 3 q^{10} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{23} + 15 q^{25} + 9 q^{26} + 6 q^{28} + 3 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8 + 3 * q^10 + 3 * q^11 + 9 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 9 * q^19 + 3 * q^20 + 3 * q^22 - 3 * q^23 + 15 * q^25 + 9 * q^26 + 6 * q^28 + 3 * q^29 + 12 * q^31 + 6 * q^32 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 9 * q^38 + 3 * q^40 + 3 * q^41 + 12 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 12 * q^49 + 15 * q^50 + 9 * q^52 - 6 * q^53 + 9 * q^55 + 6 * q^56 + 3 * q^58 - 3 * q^59 + 12 * q^61 + 12 * q^62 + 6 * q^64 - 3 * q^65 + 21 * q^67 + 6 * q^68 - 3 * q^70 - 12 * q^71 + 21 * q^73 + 15 * q^74 + 9 * q^76 - 3 * q^77 + 3 * q^80 + 3 * q^82 - 27 * q^83 + 12 * q^86 + 3 * q^88 - 12 * q^89 + 6 * q^91 - 3 * q^92 - 9 * q^94 - 30 * q^95 + 18 * q^97 + 12 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.08802	0.933790	0.466895	−	0.884313i	$-0.345373\pi$
$5$	2.08802	0.933790	0.466895	+	0.884313i	$0.345373\pi$
$6$	0	0
$7$	−4.01220	−1.51647	−0.758235	−	0.651981i	$-0.773938\pi$
$7$	−4.01220	−1.51647	−0.758235	+	0.651981i	$0.773938\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.08802	0.660289
$11$	0.212447	0.0640551	0.0320276	−	0.999487i	$-0.489804\pi$
$11$	0.212447	0.0640551	0.0320276	+	0.999487i	$0.489804\pi$
$12$	0	0
$13$	5.19903	1.44195	0.720975	−	0.692961i	$-0.243694\pi$
$13$	5.19903	1.44195	0.720975	+	0.692961i	$0.243694\pi$
$14$	−4.01220	−1.07231
$15$	0	0
$16$	1.00000	0.250000
$17$	3.78552	0.918124	0.459062	−	0.888404i	$-0.348186\pi$
$17$	3.78552	0.918124	0.459062	+	0.888404i	$0.348186\pi$
$18$	0	0
$19$	−1.27281	−0.292002	−0.146001	−	0.989284i	$-0.546640\pi$
$19$	−1.27281	−0.292002	−0.146001	+	0.989284i	$0.546640\pi$
$20$	2.08802	0.466895
$21$	0	0
$22$	0.212447	0.0452938
$23$	0.829442	0.172951	0.0864753	−	0.996254i	$-0.472440\pi$
$23$	0.829442	0.172951	0.0864753	+	0.996254i	$0.472440\pi$
$24$	0	0
$25$	−0.640186	−0.128037
$26$	5.19903	1.01961
$27$	0	0
$28$	−4.01220	−0.758235
$29$	9.21123	1.71048	0.855241	−	0.518230i	$-0.173409\pi$
$29$	9.21123	1.71048	0.855241	+	0.518230i	$0.173409\pi$
$30$	0	0
$31$	3.53590	0.635067	0.317534	−	0.948247i	$-0.397145\pi$
$31$	3.53590	0.635067	0.317534	+	0.948247i	$0.397145\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.78552	0.649212
$35$	−8.37755	−1.41606
$36$	0	0
$37$	3.55460	0.584373	0.292187	−	0.956361i	$-0.405617\pi$
$37$	3.55460	0.584373	0.292187	+	0.956361i	$0.405617\pi$
$38$	−1.27281	−0.206477
$39$	0	0
$40$	2.08802	0.330144
$41$	2.73940	0.427822	0.213911	−	0.976853i	$-0.431380\pi$
$41$	2.73940	0.427822	0.213911	+	0.976853i	$0.431380\pi$
$42$	0	0
$43$	6.89653	1.05171	0.525856	−	0.850574i	$-0.323745\pi$
$43$	6.89653	1.05171	0.525856	+	0.850574i	$0.323745\pi$
$44$	0.212447	0.0320276
$45$	0	0
$46$	0.829442	0.122295
$47$	−1.27727	−0.186309	−0.0931547	−	0.995652i	$-0.529695\pi$
$47$	−1.27727	−0.186309	−0.0931547	+	0.995652i	$0.529695\pi$
$48$	0	0
$49$	9.09779	1.29968
$50$	−0.640186	−0.0905359
$51$	0	0
$52$	5.19903	0.720975
$53$	−11.2992	−1.55207	−0.776036	−	0.630689i	$-0.782772\pi$
$53$	−11.2992	−1.55207	−0.776036	+	0.630689i	$0.782772\pi$
$54$	0	0
$55$	0.443592	0.0598140
$56$	−4.01220	−0.536153
$57$	0	0
$58$	9.21123	1.20949
$59$	−8.55985	−1.11440	−0.557199	−	0.830379i	$-0.688124\pi$
$59$	−8.55985	−1.11440	−0.557199	+	0.830379i	$0.688124\pi$
$60$	0	0
$61$	2.83368	0.362816	0.181408	−	0.983408i	$-0.441935\pi$
$61$	2.83368	0.362816	0.181408	+	0.983408i	$0.441935\pi$
$62$	3.53590	0.449060
$63$	0	0
$64$	1.00000	0.125000
$65$	10.8557	1.34648
$66$	0	0
$67$	9.76056	1.19244	0.596221	−	0.802820i	$-0.296668\pi$
$67$	9.76056	1.19244	0.596221	+	0.802820i	$0.296668\pi$
$68$	3.78552	0.459062
$69$	0	0
$70$	−8.37755	−1.00131
$71$	−11.7241	−1.39140	−0.695700	−	0.718333i	$-0.744906\pi$
$71$	−11.7241	−1.39140	−0.695700	+	0.718333i	$0.744906\pi$
$72$	0	0
$73$	0.750325	0.0878189	0.0439094	−	0.999036i	$-0.486019\pi$
$73$	0.750325	0.0878189	0.0439094	+	0.999036i	$0.486019\pi$
$74$	3.55460	0.413214
$75$	0	0
$76$	−1.27281	−0.146001
$77$	−0.852380	−0.0971377
$78$	0	0
$79$	−2.36778	−0.266396	−0.133198	−	0.991089i	$-0.542525\pi$
$79$	−2.36778	−0.266396	−0.133198	+	0.991089i	$0.542525\pi$
$80$	2.08802	0.233447
$81$	0	0
$82$	2.73940	0.302516
$83$	−11.4005	−1.25137	−0.625683	−	0.780077i	$-0.715180\pi$
$83$	−11.4005	−1.25137	−0.625683	+	0.780077i	$0.715180\pi$
$84$	0	0
$85$	7.90424	0.857335
$86$	6.89653	0.743672
$87$	0	0
$88$	0.212447	0.0226469
$89$	5.39625	0.572001	0.286001	−	0.958229i	$-0.407674\pi$
$89$	5.39625	0.572001	0.286001	+	0.958229i	$0.407674\pi$
$90$	0	0
$91$	−20.8596	−2.18668
$92$	0.829442	0.0864753
$93$	0	0
$94$	−1.27727	−0.131741
$95$	−2.65765	−0.272669
$96$	0	0
$97$	−12.5632	−1.27560	−0.637798	−	0.770204i	$-0.720154\pi$
$97$	−12.5632	−1.27560	−0.637798	+	0.770204i	$0.720154\pi$
$98$	9.09779	0.919015
$99$	0	0
$100$	−0.640186	−0.0640186
$101$	−2.89452	−0.288016	−0.144008	−	0.989577i	$-0.545999\pi$
$101$	−2.89452	−0.288016	−0.144008	+	0.989577i	$0.545999\pi$
$102$	0	0
$103$	16.6900	1.64452	0.822258	−	0.569114i	$-0.192714\pi$
$103$	16.6900	1.64452	0.822258	+	0.569114i	$0.192714\pi$
$104$	5.19903	0.509807
$105$	0	0
$106$	−11.2992	−1.09748
$107$	0.321371	0.0310681	0.0155341	−	0.999879i	$-0.495055\pi$
$107$	0.321371	0.0310681	0.0155341	+	0.999879i	$0.495055\pi$
$108$	0	0
$109$	0.568378	0.0544407	0.0272204	−	0.999629i	$-0.491334\pi$
$109$	0.568378	0.0544407	0.0272204	+	0.999629i	$0.491334\pi$
$110$	0.443592	0.0422949
$111$	0	0
$112$	−4.01220	−0.379118
$113$	−1.35981	−0.127921	−0.0639603	−	0.997952i	$-0.520373\pi$
$113$	−1.35981	−0.127921	−0.0639603	+	0.997952i	$0.520373\pi$
$114$	0	0
$115$	1.73189	0.161500
$116$	9.21123	0.855241
$117$	0	0
$118$	−8.55985	−0.787998
$119$	−15.1883	−1.39231
$120$	0	0
$121$	−10.9549	−0.995897
$122$	2.83368	0.256549
$123$	0	0
$124$	3.53590	0.317534
$125$	−11.7768	−1.05335
$126$	0	0
$127$	1.80401	0.160080	0.0800402	−	0.996792i	$-0.474495\pi$
$127$	1.80401	0.160080	0.0800402	+	0.996792i	$0.474495\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	10.8557	0.952104
$131$	−9.71443	−0.848754	−0.424377	−	0.905486i	$-0.639507\pi$
$131$	−9.71443	−0.848754	−0.424377	+	0.905486i	$0.639507\pi$
$132$	0	0
$133$	5.10677	0.442813
$134$	9.76056	0.843184
$135$	0	0
$136$	3.78552	0.324606
$137$	−9.06792	−0.774725	−0.387363	−	0.921927i	$-0.626614\pi$
$137$	−9.06792	−0.774725	−0.387363	+	0.921927i	$0.626614\pi$
$138$	0	0
$139$	−10.0409	−0.851656	−0.425828	−	0.904804i	$-0.640017\pi$
$139$	−10.0409	−0.851656	−0.425828	+	0.904804i	$0.640017\pi$
$140$	−8.37755	−0.708032
$141$	0	0
$142$	−11.7241	−0.983868
$143$	1.10452	0.0923643
$144$	0	0
$145$	19.2332	1.59723
$146$	0.750325	0.0620973
$147$	0	0
$148$	3.55460	0.292187
$149$	15.8424	1.29786	0.648931	−	0.760847i	$-0.275216\pi$
$149$	15.8424	1.29786	0.648931	+	0.760847i	$0.275216\pi$
$150$	0	0
$151$	5.30755	0.431922	0.215961	−	0.976402i	$-0.430712\pi$
$151$	5.30755	0.431922	0.215961	+	0.976402i	$0.430712\pi$
$152$	−1.27281	−0.103238
$153$	0	0
$154$	−0.852380	−0.0686867
$155$	7.38303	0.593019
$156$	0	0
$157$	−11.5170	−0.919159	−0.459580	−	0.888137i	$-0.652000\pi$
$157$	−11.5170	−0.919159	−0.459580	+	0.888137i	$0.652000\pi$
$158$	−2.36778	−0.188370
$159$	0	0
$160$	2.08802	0.165072
$161$	−3.32789	−0.262275
$162$	0	0
$163$	7.66336	0.600241	0.300120	−	0.953901i	$-0.402973\pi$
$163$	7.66336	0.600241	0.300120	+	0.953901i	$0.402973\pi$
$164$	2.73940	0.213911
$165$	0	0
$166$	−11.4005	−0.884850
$167$	6.10999	0.472805	0.236403	−	0.971655i	$-0.424032\pi$
$167$	6.10999	0.472805	0.236403	+	0.971655i	$0.424032\pi$
$168$	0	0
$169$	14.0299	1.07922
$170$	7.90424	0.606227
$171$	0	0
$172$	6.89653	0.525856
$173$	−11.3035	−0.859388	−0.429694	−	0.902975i	$-0.641379\pi$
$173$	−11.3035	−0.859388	−0.429694	+	0.902975i	$0.641379\pi$
$174$	0	0
$175$	2.56856	0.194165
$176$	0.212447	0.0160138
$177$	0	0
$178$	5.39625	0.404466
$179$	6.92990	0.517965	0.258982	−	0.965882i	$-0.416613\pi$
$179$	6.92990	0.517965	0.258982	+	0.965882i	$0.416613\pi$
$180$	0	0
$181$	3.03763	0.225785	0.112893	−	0.993607i	$-0.463988\pi$
$181$	3.03763	0.225785	0.112893	+	0.993607i	$0.463988\pi$
$182$	−20.8596	−1.54621
$183$	0	0
$184$	0.829442	0.0611473
$185$	7.42207	0.545681
$186$	0	0
$187$	0.804222	0.0588105
$188$	−1.27727	−0.0931547
$189$	0	0
$190$	−2.65765	−0.192806
$191$	−5.09718	−0.368819	−0.184409	−	0.982850i	$-0.559037\pi$
$191$	−5.09718	−0.368819	−0.184409	+	0.982850i	$0.559037\pi$
$192$	0	0
$193$	18.8683	1.35817	0.679084	−	0.734061i	$-0.262377\pi$
$193$	18.8683	1.35817	0.679084	+	0.734061i	$0.262377\pi$
$194$	−12.5632	−0.901982
$195$	0	0
$196$	9.09779	0.649842
$197$	−20.0440	−1.42807	−0.714036	−	0.700109i	$-0.753135\pi$
$197$	−20.0440	−1.42807	−0.714036	+	0.700109i	$0.753135\pi$
$198$	0	0
$199$	5.50212	0.390035	0.195018	−	0.980800i	$-0.437524\pi$
$199$	5.50212	0.390035	0.195018	+	0.980800i	$0.437524\pi$
$200$	−0.640186	−0.0452680
$201$	0	0
$202$	−2.89452	−0.203658
$203$	−36.9574	−2.59390
$204$	0	0
$205$	5.71990	0.399495
$206$	16.6900	1.16285
$207$	0	0
$208$	5.19903	0.360488
$209$	−0.270404	−0.0187043
$210$	0	0
$211$	−25.1451	−1.73106	−0.865531	−	0.500855i	$-0.833019\pi$
$211$	−25.1451	−1.73106	−0.865531	+	0.500855i	$0.833019\pi$
$212$	−11.2992	−0.776036
$213$	0	0
$214$	0.321371	0.0219685
$215$	14.4001	0.982077
$216$	0	0
$217$	−14.1868	−0.963061
$218$	0.568378	0.0384954
$219$	0	0
$220$	0.443592	0.0299070
$221$	19.6810	1.32389
$222$	0	0
$223$	−3.75239	−0.251279	−0.125639	−	0.992076i	$-0.540098\pi$
$223$	−3.75239	−0.251279	−0.125639	+	0.992076i	$0.540098\pi$
$224$	−4.01220	−0.268077
$225$	0	0
$226$	−1.35981	−0.0904535
$227$	18.7277	1.24300	0.621500	−	0.783414i	$-0.286524\pi$
$227$	18.7277	1.24300	0.621500	+	0.783414i	$0.286524\pi$
$228$	0	0
$229$	−4.14333	−0.273799	−0.136899	−	0.990585i	$-0.543714\pi$
$229$	−4.14333	−0.273799	−0.136899	+	0.990585i	$0.543714\pi$
$230$	1.73189	0.114197
$231$	0	0
$232$	9.21123	0.604747
$233$	−1.91711	−0.125594	−0.0627971	−	0.998026i	$-0.520002\pi$
$233$	−1.91711	−0.125594	−0.0627971	+	0.998026i	$0.520002\pi$
$234$	0	0
$235$	−2.66697	−0.173974
$236$	−8.55985	−0.557199
$237$	0	0
$238$	−15.1883	−0.984511
$239$	24.1993	1.56532	0.782660	−	0.622449i	$-0.213862\pi$
$239$	24.1993	1.56532	0.782660	+	0.622449i	$0.213862\pi$
$240$	0	0
$241$	13.2049	0.850605	0.425302	−	0.905051i	$-0.360168\pi$
$241$	13.2049	0.850605	0.425302	+	0.905051i	$0.360168\pi$
$242$	−10.9549	−0.704205
$243$	0	0
$244$	2.83368	0.181408
$245$	18.9963	1.21363
$246$	0	0
$247$	−6.61737	−0.421053
$248$	3.53590	0.224530
$249$	0	0
$250$	−11.7768	−0.744830
$251$	−8.65988	−0.546607	−0.273303	−	0.961928i	$-0.588116\pi$
$251$	−8.65988	−0.546607	−0.273303	+	0.961928i	$0.588116\pi$
$252$	0	0
$253$	0.176212	0.0110784
$254$	1.80401	0.113194
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.8308	1.36177	0.680883	−	0.732392i	$-0.261596\pi$
$257$	21.8308	1.36177	0.680883	+	0.732392i	$0.261596\pi$
$258$	0	0
$259$	−14.2618	−0.886185
$260$	10.8557	0.673239
$261$	0	0
$262$	−9.71443	−0.600159
$263$	−5.68753	−0.350708	−0.175354	−	0.984505i	$-0.556107\pi$
$263$	−5.68753	−0.350708	−0.175354	+	0.984505i	$0.556107\pi$
$264$	0	0
$265$	−23.5930	−1.44931
$266$	5.10677	0.313116
$267$	0	0
$268$	9.76056	0.596221
$269$	−22.7662	−1.38808	−0.694041	−	0.719936i	$-0.744171\pi$
$269$	−22.7662	−1.38808	−0.694041	+	0.719936i	$0.744171\pi$
$270$	0	0
$271$	−19.8340	−1.20483	−0.602414	−	0.798184i	$-0.705794\pi$
$271$	−19.8340	−1.20483	−0.602414	+	0.798184i	$0.705794\pi$
$272$	3.78552	0.229531
$273$	0	0
$274$	−9.06792	−0.547813
$275$	−0.136005	−0.00820143
$276$	0	0
$277$	−24.1368	−1.45024	−0.725121	−	0.688622i	$-0.758216\pi$
$277$	−24.1368	−1.45024	−0.725121	+	0.688622i	$0.758216\pi$
$278$	−10.0409	−0.602211
$279$	0	0
$280$	−8.37755	−0.500654
$281$	−23.6553	−1.41116	−0.705578	−	0.708632i	$-0.749312\pi$
$281$	−23.6553	−1.41116	−0.705578	+	0.708632i	$0.749312\pi$
$282$	0	0
$283$	−11.6242	−0.690984	−0.345492	−	0.938422i	$-0.612288\pi$
$283$	−11.6242	−0.690984	−0.345492	+	0.938422i	$0.612288\pi$
$284$	−11.7241	−0.695700
$285$	0	0
$286$	1.10452	0.0653114
$287$	−10.9910	−0.648779
$288$	0	0
$289$	−2.66981	−0.157048
$290$	19.2332	1.12941
$291$	0	0
$292$	0.750325	0.0439094
$293$	−10.7109	−0.625739	−0.312869	−	0.949796i	$-0.601290\pi$
$293$	−10.7109	−0.625739	−0.312869	+	0.949796i	$0.601290\pi$
$294$	0	0
$295$	−17.8731	−1.04061
$296$	3.55460	0.206607
$297$	0	0
$298$	15.8424	0.917728
$299$	4.31229	0.249386
$300$	0	0
$301$	−27.6703	−1.59489
$302$	5.30755	0.305415
$303$	0	0
$304$	−1.27281	−0.0730006
$305$	5.91677	0.338793
$306$	0	0
$307$	24.7876	1.41470	0.707351	−	0.706862i	$-0.249890\pi$
$307$	24.7876	1.41470	0.707351	+	0.706862i	$0.249890\pi$
$308$	−0.852380	−0.0485689
$309$	0	0
$310$	7.38303	0.419328
$311$	−21.7835	−1.23523	−0.617615	−	0.786481i	$-0.711901\pi$
$311$	−21.7835	−1.23523	−0.617615	+	0.786481i	$0.711901\pi$
$312$	0	0
$313$	−9.40246	−0.531459	−0.265729	−	0.964048i	$-0.585613\pi$
$313$	−9.40246	−0.531459	−0.265729	+	0.964048i	$0.585613\pi$
$314$	−11.5170	−0.649944
$315$	0	0
$316$	−2.36778	−0.133198
$317$	−14.5059	−0.814731	−0.407365	−	0.913265i	$-0.633552\pi$
$317$	−14.5059	−0.814731	−0.407365	+	0.913265i	$0.633552\pi$
$318$	0	0
$319$	1.95690	0.109565
$320$	2.08802	0.116724
$321$	0	0
$322$	−3.32789	−0.185456
$323$	−4.81825	−0.268095
$324$	0	0
$325$	−3.32834	−0.184623
$326$	7.66336	0.424434
$327$	0	0
$328$	2.73940	0.151258
$329$	5.12468	0.282533
$330$	0	0
$331$	0.806740	0.0443424	0.0221712	−	0.999754i	$-0.492942\pi$
$331$	0.806740	0.0443424	0.0221712	+	0.999754i	$0.492942\pi$
$332$	−11.4005	−0.625683
$333$	0	0
$334$	6.10999	0.334324
$335$	20.3802	1.11349
$336$	0	0
$337$	−26.6658	−1.45258	−0.726289	−	0.687389i	$-0.758757\pi$
$337$	−26.6658	−1.45258	−0.726289	+	0.687389i	$0.758757\pi$
$338$	14.0299	0.763125
$339$	0	0
$340$	7.90424	0.428667
$341$	0.751191	0.0406793
$342$	0	0
$343$	−8.41675	−0.454462
$344$	6.89653	0.371836
$345$	0	0
$346$	−11.3035	−0.607679
$347$	−29.1043	−1.56240	−0.781200	−	0.624281i	$-0.785392\pi$
$347$	−29.1043	−1.56240	−0.781200	+	0.624281i	$0.785392\pi$
$348$	0	0
$349$	−29.5485	−1.58170	−0.790848	−	0.612013i	$-0.790360\pi$
$349$	−29.5485	−1.58170	−0.790848	+	0.612013i	$0.790360\pi$
$350$	2.56856	0.137295
$351$	0	0
$352$	0.212447	0.0113235
$353$	6.25256	0.332790	0.166395	−	0.986059i	$-0.446787\pi$
$353$	6.25256	0.332790	0.166395	+	0.986059i	$0.446787\pi$
$354$	0	0
$355$	−24.4802	−1.29927
$356$	5.39625	0.286001
$357$	0	0
$358$	6.92990	0.366257
$359$	−2.58629	−0.136499	−0.0682495	−	0.997668i	$-0.521741\pi$
$359$	−2.58629	−0.136499	−0.0682495	+	0.997668i	$0.521741\pi$
$360$	0	0
$361$	−17.3800	−0.914735
$362$	3.03763	0.159654
$363$	0	0
$364$	−20.8596	−1.09334
$365$	1.56669	0.0820043
$366$	0	0
$367$	18.6812	0.975151	0.487575	−	0.873081i	$-0.337881\pi$
$367$	18.6812	0.975151	0.487575	+	0.873081i	$0.337881\pi$
$368$	0.829442	0.0432377
$369$	0	0
$370$	7.42207	0.385855
$371$	45.3349	2.35367
$372$	0	0
$373$	13.5261	0.700357	0.350178	−	0.936683i	$-0.386121\pi$
$373$	13.5261	0.700357	0.350178	+	0.936683i	$0.386121\pi$
$374$	0.804222	0.0415853
$375$	0	0
$376$	−1.27727	−0.0658703
$377$	47.8894	2.46643
$378$	0	0
$379$	30.2178	1.55218	0.776092	−	0.630620i	$-0.217199\pi$
$379$	30.2178	1.55218	0.776092	+	0.630620i	$0.217199\pi$
$380$	−2.65765	−0.136334
$381$	0	0
$382$	−5.09718	−0.260794
$383$	−38.5891	−1.97181	−0.985904	−	0.167310i	$-0.946492\pi$
$383$	−38.5891	−1.97181	−0.985904	+	0.167310i	$0.946492\pi$
$384$	0	0
$385$	−1.77978	−0.0907062
$386$	18.8683	0.960369
$387$	0	0
$388$	−12.5632	−0.637798
$389$	−5.62618	−0.285259	−0.142629	−	0.989776i	$-0.545556\pi$
$389$	−5.62618	−0.285259	−0.142629	+	0.989776i	$0.545556\pi$
$390$	0	0
$391$	3.13987	0.158790
$392$	9.09779	0.459508
$393$	0	0
$394$	−20.0440	−1.00980
$395$	−4.94397	−0.248758
$396$	0	0
$397$	7.71618	0.387264	0.193632	−	0.981074i	$-0.437973\pi$
$397$	7.71618	0.387264	0.193632	+	0.981074i	$0.437973\pi$
$398$	5.50212	0.275796
$399$	0	0
$400$	−0.640186	−0.0320093
$401$	36.3793	1.81670	0.908349	−	0.418213i	$-0.137343\pi$
$401$	36.3793	1.81670	0.908349	+	0.418213i	$0.137343\pi$
$402$	0	0
$403$	18.3833	0.915736
$404$	−2.89452	−0.144008
$405$	0	0
$406$	−36.9574	−1.83416
$407$	0.755164	0.0374321
$408$	0	0
$409$	5.86768	0.290138	0.145069	−	0.989422i	$-0.453660\pi$
$409$	5.86768	0.290138	0.145069	+	0.989422i	$0.453660\pi$
$410$	5.71990	0.282486
$411$	0	0
$412$	16.6900	0.822258
$413$	34.3439	1.68995
$414$	0	0
$415$	−23.8044	−1.16851
$416$	5.19903	0.254903
$417$	0	0
$418$	−0.270404	−0.0132259
$419$	21.1356	1.03254	0.516270	−	0.856426i	$-0.327320\pi$
$419$	21.1356	1.03254	0.516270	+	0.856426i	$0.327320\pi$
$420$	0	0
$421$	7.44291	0.362745	0.181373	−	0.983414i	$-0.441946\pi$
$421$	7.44291	0.362745	0.181373	+	0.983414i	$0.441946\pi$
$422$	−25.1451	−1.22405
$423$	0	0
$424$	−11.2992	−0.548740
$425$	−2.42344	−0.117554
$426$	0	0
$427$	−11.3693	−0.550199
$428$	0.321371	0.0155341
$429$	0	0
$430$	14.4001	0.694433
$431$	−13.0502	−0.628607	−0.314303	−	0.949323i	$-0.601771\pi$
$431$	−13.0502	−0.628607	−0.314303	+	0.949323i	$0.601771\pi$
$432$	0	0
$433$	0.143533	0.00689775	0.00344887	−	0.999994i	$-0.498902\pi$
$433$	0.143533	0.00689775	0.00344887	+	0.999994i	$0.498902\pi$
$434$	−14.1868	−0.680987
$435$	0	0
$436$	0.568378	0.0272204
$437$	−1.05572	−0.0505020
$438$	0	0
$439$	13.5565	0.647017	0.323509	−	0.946225i	$-0.395138\pi$
$439$	13.5565	0.647017	0.323509	+	0.946225i	$0.395138\pi$
$440$	0.443592	0.0211474
$441$	0	0
$442$	19.6810	0.936132
$443$	12.2813	0.583501	0.291750	−	0.956494i	$-0.405762\pi$
$443$	12.2813	0.583501	0.291750	+	0.956494i	$0.405762\pi$
$444$	0	0
$445$	11.2675	0.534129
$446$	−3.75239	−0.177681
$447$	0	0
$448$	−4.01220	−0.189559
$449$	40.6646	1.91908	0.959540	−	0.281573i	$-0.0908563\pi$
$449$	40.6646	1.91908	0.959540	+	0.281573i	$0.0908563\pi$
$450$	0	0
$451$	0.581976	0.0274042
$452$	−1.35981	−0.0639603
$453$	0	0
$454$	18.7277	0.878934
$455$	−43.5551	−2.04190
$456$	0	0
$457$	32.5910	1.52454	0.762271	−	0.647258i	$-0.224084\pi$
$457$	32.5910	1.52454	0.762271	+	0.647258i	$0.224084\pi$
$458$	−4.14333	−0.193605
$459$	0	0
$460$	1.73189	0.0807498
$461$	−27.3204	−1.27244	−0.636220	−	0.771508i	$-0.719503\pi$
$461$	−27.3204	−1.27244	−0.636220	+	0.771508i	$0.719503\pi$
$462$	0	0
$463$	−3.39919	−0.157974	−0.0789868	−	0.996876i	$-0.525168\pi$
$463$	−3.39919	−0.157974	−0.0789868	+	0.996876i	$0.525168\pi$
$464$	9.21123	0.427621
$465$	0	0
$466$	−1.91711	−0.0888085
$467$	32.9976	1.52695	0.763473	−	0.645840i	$-0.223493\pi$
$467$	32.9976	1.52695	0.763473	+	0.645840i	$0.223493\pi$
$468$	0	0
$469$	−39.1613	−1.80830
$470$	−2.66697	−0.123018
$471$	0	0
$472$	−8.55985	−0.393999
$473$	1.46515	0.0673675
$474$	0	0
$475$	0.814834	0.0373872
$476$	−15.1883	−0.696154
$477$	0	0
$478$	24.1993	1.10685
$479$	−23.0433	−1.05287	−0.526437	−	0.850214i	$-0.676472\pi$
$479$	−23.0433	−1.05287	−0.526437	+	0.850214i	$0.676472\pi$
$480$	0	0
$481$	18.4805	0.842637
$482$	13.2049	0.601468
$483$	0	0
$484$	−10.9549	−0.497948
$485$	−26.2321	−1.19114
$486$	0	0
$487$	26.9315	1.22038	0.610192	−	0.792254i	$-0.291092\pi$
$487$	26.9315	1.22038	0.610192	+	0.792254i	$0.291092\pi$
$488$	2.83368	0.128275
$489$	0	0
$490$	18.9963	0.858167
$491$	−0.247993	−0.0111918	−0.00559589	−	0.999984i	$-0.501781\pi$
$491$	−0.247993	−0.0111918	−0.00559589	+	0.999984i	$0.501781\pi$
$492$	0	0
$493$	34.8693	1.57044
$494$	−6.61737	−0.297730
$495$	0	0
$496$	3.53590	0.158767
$497$	47.0397	2.11002
$498$	0	0
$499$	−26.7183	−1.19607	−0.598037	−	0.801468i	$-0.704053\pi$
$499$	−26.7183	−1.19607	−0.598037	+	0.801468i	$0.704053\pi$
$500$	−11.7768	−0.526675
$501$	0	0
$502$	−8.65988	−0.386509
$503$	26.1743	1.16705	0.583527	−	0.812094i	$-0.301672\pi$
$503$	26.1743	1.16705	0.583527	+	0.812094i	$0.301672\pi$
$504$	0	0
$505$	−6.04381	−0.268946
$506$	0.176212	0.00783359
$507$	0	0
$508$	1.80401	0.0800402
$509$	5.45781	0.241913	0.120957	−	0.992658i	$-0.461404\pi$
$509$	5.45781	0.241913	0.120957	+	0.992658i	$0.461404\pi$
$510$	0	0
$511$	−3.01046	−0.133175
$512$	1.00000	0.0441942
$513$	0	0
$514$	21.8308	0.962914
$515$	34.8490	1.53563
$516$	0	0
$517$	−0.271353	−0.0119341
$518$	−14.2618	−0.626627
$519$	0	0
$520$	10.8557	0.476052
$521$	−26.1192	−1.14430	−0.572151	−	0.820148i	$-0.693891\pi$
$521$	−26.1192	−1.14430	−0.572151	+	0.820148i	$0.693891\pi$
$522$	0	0
$523$	7.40765	0.323914	0.161957	−	0.986798i	$-0.448219\pi$
$523$	7.40765	0.323914	0.161957	+	0.986798i	$0.448219\pi$
$524$	−9.71443	−0.424377
$525$	0	0
$526$	−5.68753	−0.247988
$527$	13.3852	0.583071
$528$	0	0
$529$	−22.3120	−0.970088
$530$	−23.5930	−1.02482
$531$	0	0
$532$	5.10677	0.221407
$533$	14.2422	0.616898
$534$	0	0
$535$	0.671028	0.0290111
$536$	9.76056	0.421592
$537$	0	0
$538$	−22.7662	−0.981522
$539$	1.93280	0.0832514
$540$	0	0
$541$	11.3209	0.486725	0.243362	−	0.969935i	$-0.421750\pi$
$541$	11.3209	0.486725	0.243362	+	0.969935i	$0.421750\pi$
$542$	−19.8340	−0.851941
$543$	0	0
$544$	3.78552	0.162303
$545$	1.18678	0.0508362
$546$	0	0
$547$	−10.6377	−0.454834	−0.227417	−	0.973797i	$-0.573028\pi$
$547$	−10.6377	−0.454834	−0.227417	+	0.973797i	$0.573028\pi$
$548$	−9.06792	−0.387363
$549$	0	0
$550$	−0.136005	−0.00579929
$551$	−11.7241	−0.499465
$552$	0	0
$553$	9.50002	0.403982
$554$	−24.1368	−1.02548
$555$	0	0
$556$	−10.0409	−0.425828
$557$	−11.8477	−0.502003	−0.251001	−	0.967987i	$-0.580760\pi$
$557$	−11.8477	−0.502003	−0.251001	+	0.967987i	$0.580760\pi$
$558$	0	0
$559$	35.8553	1.51652
$560$	−8.37755	−0.354016
$561$	0	0
$562$	−23.6553	−0.997838
$563$	6.00795	0.253205	0.126602	−	0.991954i	$-0.459593\pi$
$563$	6.00795	0.253205	0.126602	+	0.991954i	$0.459593\pi$
$564$	0	0
$565$	−2.83932	−0.119451
$566$	−11.6242	−0.488600
$567$	0	0
$568$	−11.7241	−0.491934
$569$	−29.1884	−1.22364	−0.611820	−	0.790997i	$-0.709562\pi$
$569$	−29.1884	−1.22364	−0.611820	+	0.790997i	$0.709562\pi$
$570$	0	0
$571$	21.8141	0.912890	0.456445	−	0.889752i	$-0.349123\pi$
$571$	21.8141	0.912890	0.456445	+	0.889752i	$0.349123\pi$
$572$	1.10452	0.0461822
$573$	0	0
$574$	−10.9910	−0.458756
$575$	−0.530997	−0.0221441
$576$	0	0
$577$	40.0799	1.66855	0.834273	−	0.551351i	$-0.185888\pi$
$577$	40.0799	1.66855	0.834273	+	0.551351i	$0.185888\pi$
$578$	−2.66981	−0.111050
$579$	0	0
$580$	19.2332	0.798615
$581$	45.7411	1.89766
$582$	0	0
$583$	−2.40049	−0.0994181
$584$	0.750325	0.0310487
$585$	0	0
$586$	−10.7109	−0.442464
$587$	−26.8657	−1.10887	−0.554433	−	0.832228i	$-0.687065\pi$
$587$	−26.8657	−1.10887	−0.554433	+	0.832228i	$0.687065\pi$
$588$	0	0
$589$	−4.50053	−0.185441
$590$	−17.8731	−0.735825
$591$	0	0
$592$	3.55460	0.146093
$593$	30.4609	1.25088	0.625440	−	0.780272i	$-0.284919\pi$
$593$	30.4609	1.25088	0.625440	+	0.780272i	$0.284919\pi$
$594$	0	0
$595$	−31.7134	−1.30012
$596$	15.8424	0.648931
$597$	0	0
$598$	4.31229	0.176343
$599$	−29.1888	−1.19262	−0.596312	−	0.802753i	$-0.703368\pi$
$599$	−29.1888	−1.19262	−0.596312	+	0.802753i	$0.703368\pi$
$600$	0	0
$601$	−23.2218	−0.947238	−0.473619	−	0.880730i	$-0.657053\pi$
$601$	−23.2218	−0.947238	−0.473619	+	0.880730i	$0.657053\pi$
$602$	−27.6703	−1.12776
$603$	0	0
$604$	5.30755	0.215961
$605$	−22.8739	−0.929958
$606$	0	0
$607$	42.6713	1.73198	0.865988	−	0.500065i	$-0.166691\pi$
$607$	42.6713	1.73198	0.865988	+	0.500065i	$0.166691\pi$
$608$	−1.27281	−0.0516192
$609$	0	0
$610$	5.91677	0.239563
$611$	−6.64058	−0.268649
$612$	0	0
$613$	−13.6082	−0.549631	−0.274815	−	0.961497i	$-0.588617\pi$
$613$	−13.6082	−0.549631	−0.274815	+	0.961497i	$0.588617\pi$
$614$	24.7876	1.00035
$615$	0	0
$616$	−0.852380	−0.0343434
$617$	28.1285	1.13241	0.566206	−	0.824264i	$-0.308411\pi$
$617$	28.1285	1.13241	0.566206	+	0.824264i	$0.308411\pi$
$618$	0	0
$619$	−19.4969	−0.783645	−0.391822	−	0.920041i	$-0.628155\pi$
$619$	−19.4969	−0.783645	−0.391822	+	0.920041i	$0.628155\pi$
$620$	7.38303	0.296510
$621$	0	0
$622$	−21.7835	−0.873439
$623$	−21.6509	−0.867423
$624$	0	0
$625$	−21.3892	−0.855569
$626$	−9.40246	−0.375798
$627$	0	0
$628$	−11.5170	−0.459580
$629$	13.4560	0.536527
$630$	0	0
$631$	12.5494	0.499584	0.249792	−	0.968299i	$-0.419638\pi$
$631$	12.5494	0.499584	0.249792	+	0.968299i	$0.419638\pi$
$632$	−2.36778	−0.0941852
$633$	0	0
$634$	−14.5059	−0.576102
$635$	3.76681	0.149481
$636$	0	0
$637$	47.2996	1.87408
$638$	1.95690	0.0774743
$639$	0	0
$640$	2.08802	0.0825361
$641$	−13.9528	−0.551103	−0.275552	−	0.961286i	$-0.588861\pi$
$641$	−13.9528	−0.551103	−0.275552	+	0.961286i	$0.588861\pi$
$642$	0	0
$643$	41.0511	1.61890	0.809449	−	0.587191i	$-0.199766\pi$
$643$	41.0511	1.61890	0.809449	+	0.587191i	$0.199766\pi$
$644$	−3.32789	−0.131137
$645$	0	0
$646$	−4.81825	−0.189571
$647$	0.303995	0.0119513	0.00597563	−	0.999982i	$-0.498098\pi$
$647$	0.303995	0.0119513	0.00597563	+	0.999982i	$0.498098\pi$
$648$	0	0
$649$	−1.81851	−0.0713829
$650$	−3.32834	−0.130548
$651$	0	0
$652$	7.66336	0.300120
$653$	47.1170	1.84383	0.921915	−	0.387393i	$-0.126624\pi$
$653$	47.1170	1.84383	0.921915	+	0.387393i	$0.126624\pi$
$654$	0	0
$655$	−20.2839	−0.792557
$656$	2.73940	0.106955
$657$	0	0
$658$	5.12468	0.199781
$659$	−42.9658	−1.67371	−0.836855	−	0.547425i	$-0.815608\pi$
$659$	−42.9658	−1.67371	−0.836855	+	0.547425i	$0.815608\pi$
$660$	0	0
$661$	27.5262	1.07064	0.535322	−	0.844648i	$-0.320190\pi$
$661$	27.5262	1.07064	0.535322	+	0.844648i	$0.320190\pi$
$662$	0.806740	0.0313548
$663$	0	0
$664$	−11.4005	−0.442425
$665$	10.6630	0.413494
$666$	0	0
$667$	7.64019	0.295829
$668$	6.10999	0.236403
$669$	0	0
$670$	20.3802	0.787356
$671$	0.602006	0.0232402
$672$	0	0
$673$	−33.6428	−1.29683	−0.648417	−	0.761285i	$-0.724569\pi$
$673$	−33.6428	−1.29683	−0.648417	+	0.761285i	$0.724569\pi$
$674$	−26.6658	−1.02713
$675$	0	0
$676$	14.0299	0.539611
$677$	34.0891	1.31015	0.655075	−	0.755564i	$-0.272637\pi$
$677$	34.0891	1.31015	0.655075	+	0.755564i	$0.272637\pi$
$678$	0	0
$679$	50.4060	1.93440
$680$	7.90424	0.303114
$681$	0	0
$682$	0.751191	0.0287646
$683$	−16.4530	−0.629557	−0.314778	−	0.949165i	$-0.601930\pi$
$683$	−16.4530	−0.629557	−0.314778	+	0.949165i	$0.601930\pi$
$684$	0	0
$685$	−18.9340	−0.723430
$686$	−8.41675	−0.321353
$687$	0	0
$688$	6.89653	0.262928
$689$	−58.7451	−2.23801
$690$	0	0
$691$	−14.2691	−0.542823	−0.271412	−	0.962463i	$-0.587490\pi$
$691$	−14.2691	−0.542823	−0.271412	+	0.962463i	$0.587490\pi$
$692$	−11.3035	−0.429694
$693$	0	0
$694$	−29.1043	−1.10478
$695$	−20.9655	−0.795267
$696$	0	0
$697$	10.3700	0.392793
$698$	−29.5485	−1.11843
$699$	0	0
$700$	2.56856	0.0970823
$701$	15.8891	0.600123	0.300062	−	0.953920i	$-0.402993\pi$
$701$	15.8891	0.600123	0.300062	+	0.953920i	$0.402993\pi$
$702$	0	0
$703$	−4.52433	−0.170638
$704$	0.212447	0.00800689
$705$	0	0
$706$	6.25256	0.235318
$707$	11.6134	0.436767
$708$	0	0
$709$	−8.24180	−0.309527	−0.154764	−	0.987952i	$-0.549462\pi$
$709$	−8.24180	−0.309527	−0.154764	+	0.987952i	$0.549462\pi$
$710$	−24.4802	−0.918726
$711$	0	0
$712$	5.39625	0.202233
$713$	2.93283	0.109835
$714$	0	0
$715$	2.30625	0.0862488
$716$	6.92990	0.258982
$717$	0	0
$718$	−2.58629	−0.0965194
$719$	21.8697	0.815601	0.407801	−	0.913071i	$-0.366296\pi$
$719$	21.8697	0.815601	0.407801	+	0.913071i	$0.366296\pi$
$720$	0	0
$721$	−66.9638	−2.49386
$722$	−17.3800	−0.646815
$723$	0	0
$724$	3.03763	0.112893
$725$	−5.89690	−0.219005
$726$	0	0
$727$	48.6581	1.80463	0.902314	−	0.431080i	$-0.141867\pi$
$727$	48.6581	1.80463	0.902314	+	0.431080i	$0.141867\pi$
$728$	−20.8596	−0.773107
$729$	0	0
$730$	1.56669	0.0579858
$731$	26.1070	0.965602
$732$	0	0
$733$	−26.1782	−0.966915	−0.483458	−	0.875368i	$-0.660619\pi$
$733$	−26.1782	−0.966915	−0.483458	+	0.875368i	$0.660619\pi$
$734$	18.6812	0.689536
$735$	0	0
$736$	0.829442	0.0305736
$737$	2.07360	0.0763820
$738$	0	0
$739$	2.57311	0.0946534	0.0473267	−	0.998879i	$-0.484930\pi$
$739$	2.57311	0.0946534	0.0473267	+	0.998879i	$0.484930\pi$
$740$	7.42207	0.272841
$741$	0	0
$742$	45.3349	1.66430
$743$	−48.9970	−1.79753	−0.898763	−	0.438434i	$-0.855533\pi$
$743$	−48.9970	−1.79753	−0.898763	+	0.438434i	$0.855533\pi$
$744$	0	0
$745$	33.0793	1.21193
$746$	13.5261	0.495227
$747$	0	0
$748$	0.804222	0.0294053
$749$	−1.28941	−0.0471139
$750$	0	0
$751$	8.51087	0.310566	0.155283	−	0.987870i	$-0.450371\pi$
$751$	8.51087	0.310566	0.155283	+	0.987870i	$0.450371\pi$
$752$	−1.27727	−0.0465774
$753$	0	0
$754$	47.8894	1.74403
$755$	11.0822	0.403324
$756$	0	0
$757$	17.3242	0.629658	0.314829	−	0.949148i	$-0.398053\pi$
$757$	17.3242	0.629658	0.314829	+	0.949148i	$0.398053\pi$
$758$	30.2178	1.09756
$759$	0	0
$760$	−2.65765	−0.0964030
$761$	10.4468	0.378694	0.189347	−	0.981910i	$-0.439363\pi$
$761$	10.4468	0.378694	0.189347	+	0.981910i	$0.439363\pi$
$762$	0	0
$763$	−2.28045	−0.0825577
$764$	−5.09718	−0.184409
$765$	0	0
$766$	−38.5891	−1.39428
$767$	−44.5029	−1.60691
$768$	0	0
$769$	39.4496	1.42259	0.711295	−	0.702894i	$-0.248109\pi$
$769$	39.4496	1.42259	0.711295	+	0.702894i	$0.248109\pi$
$770$	−1.77978	−0.0641389
$771$	0	0
$772$	18.8683	0.679084
$773$	−23.3425	−0.839572	−0.419786	−	0.907623i	$-0.637895\pi$
$773$	−23.3425	−0.839572	−0.419786	+	0.907623i	$0.637895\pi$
$774$	0	0
$775$	−2.26364	−0.0813122
$776$	−12.5632	−0.450991
$777$	0	0
$778$	−5.62618	−0.201708
$779$	−3.48673	−0.124925
$780$	0	0
$781$	−2.49076	−0.0891263
$782$	3.13987	0.112282
$783$	0	0
$784$	9.09779	0.324921
$785$	−24.0478	−0.858301
$786$	0	0
$787$	37.8895	1.35062	0.675308	−	0.737536i	$-0.264011\pi$
$787$	37.8895	1.35062	0.675308	+	0.737536i	$0.264011\pi$
$788$	−20.0440	−0.714036
$789$	0	0
$790$	−4.94397	−0.175898
$791$	5.45585	0.193988
$792$	0	0
$793$	14.7324	0.523162
$794$	7.71618	0.273837
$795$	0	0
$796$	5.50212	0.195018
$797$	−2.70448	−0.0957977	−0.0478989	−	0.998852i	$-0.515253\pi$
$797$	−2.70448	−0.0957977	−0.0478989	+	0.998852i	$0.515253\pi$
$798$	0	0
$799$	−4.83515	−0.171055
$800$	−0.640186	−0.0226340
$801$	0	0
$802$	36.3793	1.28460
$803$	0.159404	0.00562525
$804$	0	0
$805$	−6.94870	−0.244909
$806$	18.3833	0.647523
$807$	0	0
$808$	−2.89452	−0.101829
$809$	−22.3977	−0.787460	−0.393730	−	0.919226i	$-0.628815\pi$
$809$	−22.3977	−0.787460	−0.393730	+	0.919226i	$0.628815\pi$
$810$	0	0
$811$	−35.0916	−1.23223	−0.616117	−	0.787655i	$-0.711295\pi$
$811$	−35.0916	−1.23223	−0.616117	+	0.787655i	$0.711295\pi$
$812$	−36.9574	−1.29695
$813$	0	0
$814$	0.755164	0.0264685
$815$	16.0012	0.560498
$816$	0	0
$817$	−8.77797	−0.307102
$818$	5.86768	0.205158
$819$	0	0
$820$	5.71990	0.199748
$821$	14.1679	0.494464	0.247232	−	0.968956i	$-0.420479\pi$
$821$	14.1679	0.494464	0.247232	+	0.968956i	$0.420479\pi$
$822$	0	0
$823$	−34.5999	−1.20608	−0.603038	−	0.797713i	$-0.706043\pi$
$823$	−34.5999	−1.20608	−0.603038	+	0.797713i	$0.706043\pi$
$824$	16.6900	0.581425
$825$	0	0
$826$	34.3439	1.19498
$827$	50.7694	1.76543	0.882713	−	0.469912i	$-0.155714\pi$
$827$	50.7694	1.76543	0.882713	+	0.469912i	$0.155714\pi$
$828$	0	0
$829$	54.5822	1.89572	0.947859	−	0.318691i	$-0.103243\pi$
$829$	54.5822	1.89572	0.947859	+	0.318691i	$0.103243\pi$
$830$	−23.8044	−0.826263
$831$	0	0
$832$	5.19903	0.180244
$833$	34.4399	1.19327
$834$	0	0
$835$	12.7578	0.441501
$836$	−0.270404	−0.00935213
$837$	0	0
$838$	21.1356	0.730117
$839$	−11.0785	−0.382472	−0.191236	−	0.981544i	$-0.561250\pi$
$839$	−11.0785	−0.382472	−0.191236	+	0.981544i	$0.561250\pi$
$840$	0	0
$841$	55.8468	1.92575
$842$	7.44291	0.256500
$843$	0	0
$844$	−25.1451	−0.865531
$845$	29.2946	1.00777
$846$	0	0
$847$	43.9532	1.51025
$848$	−11.2992	−0.388018
$849$	0	0
$850$	−2.42344	−0.0831232
$851$	2.94834	0.101068
$852$	0	0
$853$	−25.4380	−0.870981	−0.435490	−	0.900193i	$-0.643425\pi$
$853$	−25.4380	−0.870981	−0.435490	+	0.900193i	$0.643425\pi$
$854$	−11.3693	−0.389050
$855$	0	0
$856$	0.321371	0.0109842
$857$	47.1979	1.61225	0.806125	−	0.591745i	$-0.201561\pi$
$857$	47.1979	1.61225	0.806125	+	0.591745i	$0.201561\pi$
$858$	0	0
$859$	−24.5369	−0.837188	−0.418594	−	0.908173i	$-0.637477\pi$
$859$	−24.5369	−0.837188	−0.418594	+	0.908173i	$0.637477\pi$
$860$	14.4001	0.491039
$861$	0	0
$862$	−13.0502	−0.444492
$863$	28.4215	0.967479	0.483740	−	0.875212i	$-0.339278\pi$
$863$	28.4215	0.967479	0.483740	+	0.875212i	$0.339278\pi$
$864$	0	0
$865$	−23.6019	−0.802488
$866$	0.143533	0.00487744
$867$	0	0
$868$	−14.1868	−0.481530
$869$	−0.503027	−0.0170640
$870$	0	0
$871$	50.7454	1.71944
$872$	0.568378	0.0192477
$873$	0	0
$874$	−1.05572	−0.0357103
$875$	47.2509	1.59737
$876$	0	0
$877$	43.2512	1.46049	0.730244	−	0.683186i	$-0.239406\pi$
$877$	43.2512	1.46049	0.730244	+	0.683186i	$0.239406\pi$
$878$	13.5565	0.457510
$879$	0	0
$880$	0.443592	0.0149535
$881$	−20.4436	−0.688762	−0.344381	−	0.938830i	$-0.611911\pi$
$881$	−20.4436	−0.688762	−0.344381	+	0.938830i	$0.611911\pi$
$882$	0	0
$883$	−28.2937	−0.952160	−0.476080	−	0.879402i	$-0.657943\pi$
$883$	−28.2937	−0.952160	−0.476080	+	0.879402i	$0.657943\pi$
$884$	19.6810	0.661945
$885$	0	0
$886$	12.2813	0.412597
$887$	−57.4412	−1.92869	−0.964344	−	0.264652i	$-0.914743\pi$
$887$	−57.4412	−1.92869	−0.964344	+	0.264652i	$0.914743\pi$
$888$	0	0
$889$	−7.23808	−0.242757
$890$	11.2675	0.377686
$891$	0	0
$892$	−3.75239	−0.125639
$893$	1.62573	0.0544028
$894$	0	0
$895$	14.4697	0.483670
$896$	−4.01220	−0.134038
$897$	0	0
$898$	40.6646	1.35699
$899$	32.5700	1.08627
$900$	0	0
$901$	−42.7736	−1.42499
$902$	0.581976	0.0193777
$903$	0	0
$904$	−1.35981	−0.0452267
$905$	6.34263	0.210836
$906$	0	0
$907$	−28.4888	−0.945956	−0.472978	−	0.881074i	$-0.656821\pi$
$907$	−28.4888	−0.945956	−0.472978	+	0.881074i	$0.656821\pi$
$908$	18.7277	0.621500
$909$	0	0
$910$	−43.5551	−1.44384
$911$	−18.9786	−0.628789	−0.314394	−	0.949292i	$-0.601801\pi$
$911$	−18.9786	−0.628789	−0.314394	+	0.949292i	$0.601801\pi$
$912$	0	0
$913$	−2.42200	−0.0801564
$914$	32.5910	1.07801
$915$	0	0
$916$	−4.14333	−0.136899
$917$	38.9763	1.28711
$918$	0	0
$919$	−13.1481	−0.433715	−0.216857	−	0.976203i	$-0.569581\pi$
$919$	−13.1481	−0.433715	−0.216857	+	0.976203i	$0.569581\pi$
$920$	1.73189	0.0570987
$921$	0	0
$922$	−27.3204	−0.899751
$923$	−60.9541	−2.00633
$924$	0	0
$925$	−2.27561	−0.0748215
$926$	−3.39919	−0.111704
$927$	0	0
$928$	9.21123	0.302374
$929$	27.3820	0.898373	0.449187	−	0.893438i	$-0.351714\pi$
$929$	27.3820	0.898373	0.449187	+	0.893438i	$0.351714\pi$
$930$	0	0
$931$	−11.5797	−0.379511
$932$	−1.91711	−0.0627971
$933$	0	0
$934$	32.9976	1.07971
$935$	1.67923	0.0549167
$936$	0	0
$937$	40.0933	1.30979	0.654895	−	0.755720i	$-0.272713\pi$
$937$	40.0933	1.30979	0.654895	+	0.755720i	$0.272713\pi$
$938$	−39.1613	−1.27866
$939$	0	0
$940$	−2.66697	−0.0869869
$941$	−3.40906	−0.111132	−0.0555661	−	0.998455i	$-0.517696\pi$
$941$	−3.40906	−0.111132	−0.0555661	+	0.998455i	$0.517696\pi$
$942$	0	0
$943$	2.27217	0.0739921
$944$	−8.55985	−0.278600
$945$	0	0
$946$	1.46515	0.0476360
$947$	3.83506	0.124623	0.0623114	−	0.998057i	$-0.480153\pi$
$947$	3.83506	0.124623	0.0623114	+	0.998057i	$0.480153\pi$
$948$	0	0
$949$	3.90096	0.126630
$950$	0.814834	0.0264367
$951$	0	0
$952$	−15.1883	−0.492255
$953$	−52.7929	−1.71013	−0.855065	−	0.518522i	$-0.826483\pi$
$953$	−52.7929	−1.71013	−0.855065	+	0.518522i	$0.826483\pi$
$954$	0	0
$955$	−10.6430	−0.344399
$956$	24.1993	0.782660
$957$	0	0
$958$	−23.0433	−0.744494
$959$	36.3824	1.17485
$960$	0	0
$961$	−18.4974	−0.596690
$962$	18.4805	0.595835
$963$	0	0
$964$	13.2049	0.425302
$965$	39.3973	1.26824
$966$	0	0
$967$	10.0437	0.322984	0.161492	−	0.986874i	$-0.448369\pi$
$967$	10.0437	0.322984	0.161492	+	0.986874i	$0.448369\pi$
$968$	−10.9549	−0.352103
$969$	0	0
$970$	−26.2321	−0.842262
$971$	4.06174	0.130348	0.0651738	−	0.997874i	$-0.479240\pi$
$971$	4.06174	0.130348	0.0651738	+	0.997874i	$0.479240\pi$
$972$	0	0
$973$	40.2860	1.29151
$974$	26.9315	0.862942
$975$	0	0
$976$	2.83368	0.0907039
$977$	18.2290	0.583198	0.291599	−	0.956541i	$-0.405813\pi$
$977$	18.2290	0.583198	0.291599	+	0.956541i	$0.405813\pi$
$978$	0	0
$979$	1.14642	0.0366396
$980$	18.9963	0.606816
$981$	0	0
$982$	−0.247993	−0.00791378
$983$	−42.0431	−1.34097	−0.670484	−	0.741924i	$-0.733913\pi$
$983$	−42.0431	−1.34097	−0.670484	+	0.741924i	$0.733913\pi$
$984$	0	0
$985$	−41.8521	−1.33352
$986$	34.8693	1.11047
$987$	0	0
$988$	−6.61737	−0.210527
$989$	5.72028	0.181894
$990$	0	0
$991$	−31.8731	−1.01248	−0.506241	−	0.862392i	$-0.668965\pi$
$991$	−31.8731	−1.01248	−0.506241	+	0.862392i	$0.668965\pi$
$992$	3.53590	0.112265
$993$	0	0
$994$	47.0397	1.49201
$995$	11.4885	0.364211
$996$	0	0
$997$	−31.9997	−1.01344	−0.506721	−	0.862110i	$-0.669143\pi$
$997$	−31.9997	−1.01344	−0.506721	+	0.862110i	$0.669143\pi$
$998$	−26.7183	−0.845753
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.2.a.g.1.4		6
3.2	odd	2		1458.2.a.f.1.3		6
9.2	odd	6		1458.2.c.g.973.4		12
9.4	even	3		1458.2.c.f.487.3		12
9.5	odd	6		1458.2.c.g.487.4		12
9.7	even	3		1458.2.c.f.973.3		12
27.2	odd	18		486.2.e.e.433.1		12
27.4	even	9		54.2.e.b.43.2	✓	12
27.5	odd	18		486.2.e.g.217.1		12
27.7	even	9		54.2.e.b.49.2	yes	12
27.11	odd	18		486.2.e.g.271.1		12
27.13	even	9		486.2.e.h.55.2		12
27.14	odd	18		486.2.e.e.55.1		12
27.16	even	9		486.2.e.f.271.2		12
27.20	odd	18		162.2.e.b.37.2		12
27.22	even	9		486.2.e.f.217.2		12
27.23	odd	18		162.2.e.b.127.2		12
27.25	even	9		486.2.e.h.433.2		12
108.7	odd	18		432.2.u.b.49.1		12
108.31	odd	18		432.2.u.b.97.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.e.b.43.2	✓	12	27.4	even	9
54.2.e.b.49.2	yes	12	27.7	even	9
162.2.e.b.37.2		12	27.20	odd	18
162.2.e.b.127.2		12	27.23	odd	18
432.2.u.b.49.1		12	108.7	odd	18
432.2.u.b.97.1		12	108.31	odd	18
486.2.e.e.55.1		12	27.14	odd	18
486.2.e.e.433.1		12	27.2	odd	18
486.2.e.f.217.2		12	27.22	even	9
486.2.e.f.271.2		12	27.16	even	9
486.2.e.g.217.1		12	27.5	odd	18
486.2.e.g.271.1		12	27.11	odd	18
486.2.e.h.55.2		12	27.13	even	9
486.2.e.h.433.2		12	27.25	even	9
1458.2.a.f.1.3		6	3.2	odd	2
1458.2.a.g.1.4		6	1.1	even	1	trivial
1458.2.c.f.487.3		12	9.4	even	3
1458.2.c.f.973.3		12	9.7	even	3
1458.2.c.g.487.4		12	9.5	odd	6
1458.2.c.g.973.4		12	9.2	odd	6

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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1458.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.08802 q^{5} -4.01220 q^{7} +1.00000 q^{8} +2.08802 q^{10} +0.212447 q^{11} +5.19903 q^{13} -4.01220 q^{14} +1.00000 q^{16} +3.78552 q^{17} -1.27281 q^{19} +2.08802 q^{20} +0.212447 q^{22} +0.829442 q^{23} -0.640186 q^{25} +5.19903 q^{26} -4.01220 q^{28} +9.21123 q^{29} +3.53590 q^{31} +1.00000 q^{32} +3.78552 q^{34} -8.37755 q^{35} +3.55460 q^{37} -1.27281 q^{38} +2.08802 q^{40} +2.73940 q^{41} +6.89653 q^{43} +0.212447 q^{44} +0.829442 q^{46} -1.27727 q^{47} +9.09779 q^{49} -0.640186 q^{50} +5.19903 q^{52} -11.2992 q^{53} +0.443592 q^{55} -4.01220 q^{56} +9.21123 q^{58} -8.55985 q^{59} +2.83368 q^{61} +3.53590 q^{62} +1.00000 q^{64} +10.8557 q^{65} +9.76056 q^{67} +3.78552 q^{68} -8.37755 q^{70} -11.7241 q^{71} +0.750325 q^{73} +3.55460 q^{74} -1.27281 q^{76} -0.852380 q^{77} -2.36778 q^{79} +2.08802 q^{80} +2.73940 q^{82} -11.4005 q^{83} +7.90424 q^{85} +6.89653 q^{86} +0.212447 q^{88} +5.39625 q^{89} -20.8596 q^{91} +0.829442 q^{92} -1.27727 q^{94} -2.65765 q^{95} -12.5632 q^{97} +9.09779 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 3 q^{5} + 6 q^{7} + 6 q^{8} + 3 q^{10} + 3 q^{11} + 9 q^{13} + 6 q^{14} + 6 q^{16} + 6 q^{17} + 9 q^{19} + 3 q^{20} + 3 q^{22} - 3 q^{23} + 15 q^{25} + 9 q^{26} + 6 q^{28} + 3 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 3 * q^5 + 6 * q^7 + 6 * q^8 + 3 * q^10 + 3 * q^11 + 9 * q^13 + 6 * q^14 + 6 * q^16 + 6 * q^17 + 9 * q^19 + 3 * q^20 + 3 * q^22 - 3 * q^23 + 15 * q^25 + 9 * q^26 + 6 * q^28 + 3 * q^29 + 12 * q^31 + 6 * q^32 + 6 * q^34 - 3 * q^35 + 15 * q^37 + 9 * q^38 + 3 * q^40 + 3 * q^41 + 12 * q^43 + 3 * q^44 - 3 * q^46 - 9 * q^47 + 12 * q^49 + 15 * q^50 + 9 * q^52 - 6 * q^53 + 9 * q^55 + 6 * q^56 + 3 * q^58 - 3 * q^59 + 12 * q^61 + 12 * q^62 + 6 * q^64 - 3 * q^65 + 21 * q^67 + 6 * q^68 - 3 * q^70 - 12 * q^71 + 21 * q^73 + 15 * q^74 + 9 * q^76 - 3 * q^77 + 3 * q^80 + 3 * q^82 - 27 * q^83 + 12 * q^86 + 3 * q^88 - 12 * q^89 + 6 * q^91 - 3 * q^92 - 9 * q^94 - 30 * q^95 + 18 * q^97 + 12 * q^98

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