LMFDB - Embedded newform 3969.2.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3969,2,Mod(1,3969)] 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("3969.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3969, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,-3,0,0,6,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53209 q^{2} +4.41147 q^{4} +0.879385 q^{5} +6.10607 q^{8} +2.22668 q^{10} +3.87939 q^{11} +5.45336 q^{13} +6.63816 q^{16} -1.65270 q^{17} -2.41147 q^{19} +3.87939 q^{20} +9.82295 q^{22} +3.16250 q^{23} -4.22668 q^{25} +13.8084 q^{26} -6.04963 q^{29} +4.55438 q^{31} +4.59627 q^{32} -4.18479 q^{34} -4.55438 q^{37} -6.10607 q^{38} +5.36959 q^{40} +1.18479 q^{41} +0.184793 q^{43} +17.1138 q^{44} +8.00774 q^{46} +1.02229 q^{47} -10.7023 q^{50} +24.0574 q^{52} +7.29086 q^{53} +3.41147 q^{55} -15.3182 q^{58} -6.66044 q^{59} +2.59627 q^{61} +11.5321 q^{62} -1.63816 q^{64} +4.79561 q^{65} -2.95811 q^{67} -7.29086 q^{68} -3.68004 q^{71} +12.7811 q^{73} -11.5321 q^{74} -10.6382 q^{76} -5.95811 q^{79} +5.83750 q^{80} +3.00000 q^{82} +0.218941 q^{83} -1.45336 q^{85} +0.467911 q^{86} +23.6878 q^{88} -11.0273 q^{89} +13.9513 q^{92} +2.58853 q^{94} -2.12061 q^{95} -12.5030 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{8} + 6 q^{11} + 3 q^{13} + 3 q^{16} - 6 q^{17} + 3 q^{19} + 6 q^{20} + 9 q^{22} + 12 q^{23} - 6 q^{25} + 3 q^{26} + 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} - 6 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^8 + 6 * q^11 + 3 * q^13 + 3 * q^16 - 6 * q^17 + 3 * q^19 + 6 * q^20 + 9 * q^22 + 12 * q^23 - 6 * q^25 + 3 * q^26 + 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 - 6 * q^38 + 9 * q^40 - 3 * q^43 + 15 * q^44 - 3 * q^47 - 6 * q^50 + 21 * q^52 + 6 * q^53 - 9 * q^58 + 3 * q^59 - 6 * q^61 + 30 * q^62 + 12 * q^64 + 15 * q^65 - 12 * q^67 - 6 * q^68 + 9 * q^71 + 21 * q^73 - 30 * q^74 - 15 * q^76 - 21 * q^79 + 15 * q^80 + 9 * q^82 + 18 * q^83 + 9 * q^85 + 6 * q^86 + 27 * q^88 - 12 * q^89 + 3 * q^92 + 18 * q^94 - 12 * q^95 + 3 * q^97

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 $

$2$	2.53209	1.79046	0.895229	−	0.445607i	$-0.147012\pi$
$2$	2.53209	1.79046	0.895229	+	0.445607i	$0.147012\pi$
$3$	0	0
$3$	0	0
$4$	4.41147	2.20574
$5$	0.879385	0.393273	0.196637	−	0.980476i	$-0.436998\pi$
$5$	0.879385	0.393273	0.196637	+	0.980476i	$0.436998\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	6.10607	2.15882
$9$	0	0
$10$	2.22668	0.704139
$11$	3.87939	1.16968	0.584839	−	0.811149i	$-0.301158\pi$
$11$	3.87939	1.16968	0.584839	+	0.811149i	$0.301158\pi$
$12$	0	0
$13$	5.45336	1.51249	0.756245	−	0.654288i	$-0.227032\pi$
$13$	5.45336	1.51249	0.756245	+	0.654288i	$0.227032\pi$
$14$	0	0
$15$	0	0
$16$	6.63816	1.65954
$17$	−1.65270	−0.400840	−0.200420	−	0.979710i	$-0.564231\pi$
$17$	−1.65270	−0.400840	−0.200420	+	0.979710i	$0.564231\pi$
$18$	0	0
$19$	−2.41147	−0.553230	−0.276615	−	0.960981i	$-0.589213\pi$
$19$	−2.41147	−0.553230	−0.276615	+	0.960981i	$0.589213\pi$
$20$	3.87939	0.867457
$21$	0	0
$22$	9.82295	2.09426
$23$	3.16250	0.659428	0.329714	−	0.944081i	$-0.393048\pi$
$23$	3.16250	0.659428	0.329714	+	0.944081i	$0.393048\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	13.8084	2.70805
$27$	0	0
$28$	0	0
$29$	−6.04963	−1.12339	−0.561694	−	0.827345i	$-0.689850\pi$
$29$	−6.04963	−1.12339	−0.561694	+	0.827345i	$0.689850\pi$
$30$	0	0
$31$	4.55438	0.817990	0.408995	−	0.912537i	$-0.365879\pi$
$31$	4.55438	0.817990	0.408995	+	0.912537i	$0.365879\pi$
$32$	4.59627	0.812513
$33$	0	0
$34$	−4.18479	−0.717686
$35$	0	0
$36$	0	0
$37$	−4.55438	−0.748735	−0.374368	−	0.927280i	$-0.622140\pi$
$37$	−4.55438	−0.748735	−0.374368	+	0.927280i	$0.622140\pi$
$38$	−6.10607	−0.990535
$39$	0	0
$40$	5.36959	0.849006
$41$	1.18479	0.185034	0.0925168	−	0.995711i	$-0.470509\pi$
$41$	1.18479	0.185034	0.0925168	+	0.995711i	$0.470509\pi$
$42$	0	0
$43$	0.184793	0.0281806	0.0140903	−	0.999901i	$-0.495515\pi$
$43$	0.184793	0.0281806	0.0140903	+	0.999901i	$0.495515\pi$
$44$	17.1138	2.58000
$45$	0	0
$46$	8.00774	1.18068
$47$	1.02229	0.149116	0.0745581	−	0.997217i	$-0.476245\pi$
$47$	1.02229	0.149116	0.0745581	+	0.997217i	$0.476245\pi$
$48$	0	0
$49$	0	0
$50$	−10.7023	−1.51354
$51$	0	0
$52$	24.0574	3.33616
$53$	7.29086	1.00148	0.500738	−	0.865599i	$-0.333062\pi$
$53$	7.29086	1.00148	0.500738	+	0.865599i	$0.333062\pi$
$54$	0	0
$55$	3.41147	0.460003
$56$	0	0
$57$	0	0
$58$	−15.3182	−2.01138
$59$	−6.66044	−0.867116	−0.433558	−	0.901126i	$-0.642742\pi$
$59$	−6.66044	−0.867116	−0.433558	+	0.901126i	$0.642742\pi$
$60$	0	0
$61$	2.59627	0.332418	0.166209	−	0.986091i	$-0.446847\pi$
$61$	2.59627	0.332418	0.166209	+	0.986091i	$0.446847\pi$
$62$	11.5321	1.46458
$63$	0	0
$64$	−1.63816	−0.204769
$65$	4.79561	0.594822
$66$	0	0
$67$	−2.95811	−0.361391	−0.180695	−	0.983539i	$-0.557835\pi$
$67$	−2.95811	−0.361391	−0.180695	+	0.983539i	$0.557835\pi$
$68$	−7.29086	−0.884147
$69$	0	0
$70$	0	0
$71$	−3.68004	−0.436741	−0.218370	−	0.975866i	$-0.570074\pi$
$71$	−3.68004	−0.436741	−0.218370	+	0.975866i	$0.570074\pi$
$72$	0	0
$73$	12.7811	1.49591	0.747955	−	0.663750i	$-0.231036\pi$
$73$	12.7811	1.49591	0.747955	+	0.663750i	$0.231036\pi$
$74$	−11.5321	−1.34058
$75$	0	0
$76$	−10.6382	−1.22028
$77$	0	0
$78$	0	0
$79$	−5.95811	−0.670340	−0.335170	−	0.942158i	$-0.608794\pi$
$79$	−5.95811	−0.670340	−0.335170	+	0.942158i	$0.608794\pi$
$80$	5.83750	0.652652
$81$	0	0
$82$	3.00000	0.331295
$83$	0.218941	0.0240319	0.0120159	−	0.999928i	$-0.496175\pi$
$83$	0.218941	0.0240319	0.0120159	+	0.999928i	$0.496175\pi$
$84$	0	0
$85$	−1.45336	−0.157639
$86$	0.467911	0.0504562
$87$	0	0
$88$	23.6878	2.52513
$89$	−11.0273	−1.16890	−0.584448	−	0.811431i	$-0.698689\pi$
$89$	−11.0273	−1.16890	−0.584448	+	0.811431i	$0.698689\pi$
$90$	0	0
$91$	0	0
$92$	13.9513	1.45452
$93$	0	0
$94$	2.58853	0.266986
$95$	−2.12061	−0.217570
$96$	0	0
$97$	−12.5030	−1.26949	−0.634743	−	0.772723i	$-0.718894\pi$
$97$	−12.5030	−1.26949	−0.634743	+	0.772723i	$0.718894\pi$
$98$	0	0
$99$	0	0
$100$	−18.6459	−1.86459
$101$	9.71688	0.966866	0.483433	−	0.875381i	$-0.339390\pi$
$101$	9.71688	0.966866	0.483433	+	0.875381i	$0.339390\pi$
$102$	0	0
$103$	−6.59627	−0.649949	−0.324975	−	0.945723i	$-0.605356\pi$
$103$	−6.59627	−0.649949	−0.324975	+	0.945723i	$0.605356\pi$
$104$	33.2986	3.26520
$105$	0	0
$106$	18.4611	1.79310
$107$	2.38919	0.230971	0.115486	−	0.993309i	$-0.463158\pi$
$107$	2.38919	0.230971	0.115486	+	0.993309i	$0.463158\pi$
$108$	0	0
$109$	3.95811	0.379118	0.189559	−	0.981869i	$-0.439294\pi$
$109$	3.95811	0.379118	0.189559	+	0.981869i	$0.439294\pi$
$110$	8.63816	0.823616
$111$	0	0
$112$	0	0
$113$	16.4534	1.54780	0.773901	−	0.633306i	$-0.218303\pi$
$113$	16.4534	1.54780	0.773901	+	0.633306i	$0.218303\pi$
$114$	0	0
$115$	2.78106	0.259335
$116$	−26.6878	−2.47790
$117$	0	0
$118$	−16.8648	−1.55253
$119$	0	0
$120$	0	0
$121$	4.04963	0.368148
$122$	6.57398	0.595180
$123$	0	0
$124$	20.0915	1.80427
$125$	−8.11381	−0.725721
$126$	0	0
$127$	17.6536	1.56651	0.783253	−	0.621702i	$-0.213559\pi$
$127$	17.6536	1.56651	0.783253	+	0.621702i	$0.213559\pi$
$128$	−13.3405	−1.17914
$129$	0	0
$130$	12.1429	1.06500
$131$	19.1976	1.67730	0.838650	−	0.544670i	$-0.183345\pi$
$131$	19.1976	1.67730	0.838650	+	0.544670i	$0.183345\pi$
$132$	0	0
$133$	0	0
$134$	−7.49020	−0.647055
$135$	0	0
$136$	−10.0915	−0.865341
$137$	18.1557	1.55115	0.775573	−	0.631258i	$-0.217461\pi$
$137$	18.1557	1.55115	0.775573	+	0.631258i	$0.217461\pi$
$138$	0	0
$139$	−22.0574	−1.87088	−0.935441	−	0.353483i	$-0.884997\pi$
$139$	−22.0574	−1.87088	−0.935441	+	0.353483i	$0.884997\pi$
$140$	0	0
$141$	0	0
$142$	−9.31820	−0.781966
$143$	21.1557	1.76913
$144$	0	0
$145$	−5.31996	−0.441798
$146$	32.3628	2.67836
$147$	0	0
$148$	−20.0915	−1.65151
$149$	−15.1557	−1.24160	−0.620802	−	0.783968i	$-0.713193\pi$
$149$	−15.1557	−1.24160	−0.620802	+	0.783968i	$0.713193\pi$
$150$	0	0
$151$	−18.9564	−1.54265	−0.771323	−	0.636444i	$-0.780405\pi$
$151$	−18.9564	−1.54265	−0.771323	+	0.636444i	$0.780405\pi$
$152$	−14.7246	−1.19432
$153$	0	0
$154$	0	0
$155$	4.00505	0.321694
$156$	0	0
$157$	18.0574	1.44114	0.720568	−	0.693385i	$-0.243881\pi$
$157$	18.0574	1.44114	0.720568	+	0.693385i	$0.243881\pi$
$158$	−15.0865	−1.20021
$159$	0	0
$160$	4.04189	0.319539
$161$	0	0
$162$	0	0
$163$	0.958111	0.0750450	0.0375225	−	0.999296i	$-0.488053\pi$
$163$	0.958111	0.0750450	0.0375225	+	0.999296i	$0.488053\pi$
$164$	5.22668	0.408135
$165$	0	0
$166$	0.554378	0.0430280
$167$	−19.8384	−1.53514	−0.767572	−	0.640963i	$-0.778535\pi$
$167$	−19.8384	−1.53514	−0.767572	+	0.640963i	$0.778535\pi$
$168$	0	0
$169$	16.7392	1.28763
$170$	−3.68004	−0.282247
$171$	0	0
$172$	0.815207	0.0621590
$173$	−22.6827	−1.72454	−0.862268	−	0.506452i	$-0.830957\pi$
$173$	−22.6827	−1.72454	−0.862268	+	0.506452i	$0.830957\pi$
$174$	0	0
$175$	0	0
$176$	25.7520	1.94113
$177$	0	0
$178$	−27.9222	−2.09286
$179$	−7.34730	−0.549163	−0.274581	−	0.961564i	$-0.588539\pi$
$179$	−7.34730	−0.549163	−0.274581	+	0.961564i	$0.588539\pi$
$180$	0	0
$181$	3.44562	0.256111	0.128056	−	0.991767i	$-0.459126\pi$
$181$	3.44562	0.256111	0.128056	+	0.991767i	$0.459126\pi$
$182$	0	0
$183$	0	0
$184$	19.3105	1.42359
$185$	−4.00505	−0.294457
$186$	0	0
$187$	−6.41147	−0.468853
$188$	4.50980	0.328911
$189$	0	0
$190$	−5.36959	−0.389551
$191$	5.65776	0.409381	0.204690	−	0.978827i	$-0.434381\pi$
$191$	5.65776	0.409381	0.204690	+	0.978827i	$0.434381\pi$
$192$	0	0
$193$	9.59627	0.690754	0.345377	−	0.938464i	$-0.387751\pi$
$193$	9.59627	0.690754	0.345377	+	0.938464i	$0.387751\pi$
$194$	−31.6587	−2.27296
$195$	0	0
$196$	0	0
$197$	8.31996	0.592772	0.296386	−	0.955068i	$-0.404218\pi$
$197$	8.31996	0.592772	0.296386	+	0.955068i	$0.404218\pi$
$198$	0	0
$199$	−6.59627	−0.467597	−0.233798	−	0.972285i	$-0.575116\pi$
$199$	−6.59627	−0.467597	−0.233798	+	0.972285i	$0.575116\pi$
$200$	−25.8084	−1.82493
$201$	0	0
$202$	24.6040	1.73113
$203$	0	0
$204$	0	0
$205$	1.04189	0.0727687
$206$	−16.7023	−1.16371
$207$	0	0
$208$	36.2003	2.51004
$209$	−9.35504	−0.647101
$210$	0	0
$211$	−3.36959	−0.231972	−0.115986	−	0.993251i	$-0.537003\pi$
$211$	−3.36959	−0.231972	−0.115986	+	0.993251i	$0.537003\pi$
$212$	32.1634	2.20899
$213$	0	0
$214$	6.04963	0.413544
$215$	0.162504	0.0110827
$216$	0	0
$217$	0	0
$218$	10.0223	0.678795
$219$	0	0
$220$	15.0496	1.01465
$221$	−9.01279	−0.606266
$222$	0	0
$223$	6.27631	0.420293	0.210146	−	0.977670i	$-0.432606\pi$
$223$	6.27631	0.420293	0.210146	+	0.977670i	$0.432606\pi$
$224$	0	0
$225$	0	0
$226$	41.6614	2.77127
$227$	6.16250	0.409020	0.204510	−	0.978865i	$-0.434440\pi$
$227$	6.16250	0.409020	0.204510	+	0.978865i	$0.434440\pi$
$228$	0	0
$229$	−23.3851	−1.54533	−0.772664	−	0.634815i	$-0.781076\pi$
$229$	−23.3851	−1.54533	−0.772664	+	0.634815i	$0.781076\pi$
$230$	7.04189	0.464328
$231$	0	0
$232$	−36.9394	−2.42519
$233$	−8.52528	−0.558510	−0.279255	−	0.960217i	$-0.590087\pi$
$233$	−8.52528	−0.558510	−0.279255	+	0.960217i	$0.590087\pi$
$234$	0	0
$235$	0.898986	0.0586434
$236$	−29.3824	−1.91263
$237$	0	0
$238$	0	0
$239$	14.5621	0.941945	0.470973	−	0.882148i	$-0.343903\pi$
$239$	14.5621	0.941945	0.470973	+	0.882148i	$0.343903\pi$
$240$	0	0
$241$	5.40373	0.348085	0.174043	−	0.984738i	$-0.444317\pi$
$241$	5.40373	0.348085	0.174043	+	0.984738i	$0.444317\pi$
$242$	10.2540	0.659154
$243$	0	0
$244$	11.4534	0.733226
$245$	0	0
$246$	0	0
$247$	−13.1506	−0.836755
$248$	27.8093	1.76589
$249$	0	0
$250$	−20.5449	−1.29937
$251$	12.0669	0.761654	0.380827	−	0.924646i	$-0.375639\pi$
$251$	12.0669	0.761654	0.380827	+	0.924646i	$0.375639\pi$
$252$	0	0
$253$	12.2686	0.771318
$254$	44.7006	2.80476
$255$	0	0
$256$	−30.5030	−1.90644
$257$	−10.5662	−0.659104	−0.329552	−	0.944137i	$-0.606898\pi$
$257$	−10.5662	−0.659104	−0.329552	+	0.944137i	$0.606898\pi$
$258$	0	0
$259$	0	0
$260$	21.1557	1.31202
$261$	0	0
$262$	48.6100	3.00314
$263$	−28.3533	−1.74834	−0.874169	−	0.485622i	$-0.838593\pi$
$263$	−28.3533	−1.74834	−0.874169	+	0.485622i	$0.838593\pi$
$264$	0	0
$265$	6.41147	0.393854
$266$	0	0
$267$	0	0
$268$	−13.0496	−0.797133
$269$	−7.48339	−0.456271	−0.228135	−	0.973629i	$-0.573263\pi$
$269$	−7.48339	−0.456271	−0.228135	+	0.973629i	$0.573263\pi$
$270$	0	0
$271$	−13.6382	−0.828459	−0.414229	−	0.910172i	$-0.635949\pi$
$271$	−13.6382	−0.828459	−0.414229	+	0.910172i	$0.635949\pi$
$272$	−10.9709	−0.665209
$273$	0	0
$274$	45.9718	2.77726
$275$	−16.3969	−0.988772
$276$	0	0
$277$	−6.15064	−0.369556	−0.184778	−	0.982780i	$-0.559157\pi$
$277$	−6.15064	−0.369556	−0.184778	+	0.982780i	$0.559157\pi$
$278$	−55.8512	−3.34973
$279$	0	0
$280$	0	0
$281$	3.31221	0.197590	0.0987951	−	0.995108i	$-0.468501\pi$
$281$	3.31221	0.197590	0.0987951	+	0.995108i	$0.468501\pi$
$282$	0	0
$283$	−29.0232	−1.72525	−0.862626	−	0.505843i	$-0.831182\pi$
$283$	−29.0232	−1.72525	−0.862626	+	0.505843i	$0.831182\pi$
$284$	−16.2344	−0.963336
$285$	0	0
$286$	53.5681	3.16755
$287$	0	0
$288$	0	0
$289$	−14.2686	−0.839328
$290$	−13.4706	−0.791021
$291$	0	0
$292$	56.3833	3.29958
$293$	8.41921	0.491856	0.245928	−	0.969288i	$-0.420907\pi$
$293$	8.41921	0.491856	0.245928	+	0.969288i	$0.420907\pi$
$294$	0	0
$295$	−5.85710	−0.341013
$296$	−27.8093	−1.61638
$297$	0	0
$298$	−38.3756	−2.22304
$299$	17.2463	0.997378
$300$	0	0
$301$	0	0
$302$	−47.9992	−2.76204
$303$	0	0
$304$	−16.0077	−0.918107
$305$	2.28312	0.130731
$306$	0	0
$307$	12.6878	0.724130	0.362065	−	0.932153i	$-0.382072\pi$
$307$	12.6878	0.724130	0.362065	+	0.932153i	$0.382072\pi$
$308$	0	0
$309$	0	0
$310$	10.1411	0.575979
$311$	16.4902	0.935073	0.467537	−	0.883974i	$-0.345142\pi$
$311$	16.4902	0.935073	0.467537	+	0.883974i	$0.345142\pi$
$312$	0	0
$313$	−28.5185	−1.61196	−0.805980	−	0.591943i	$-0.798361\pi$
$313$	−28.5185	−1.61196	−0.805980	+	0.591943i	$0.798361\pi$
$314$	45.7229	2.58029
$315$	0	0
$316$	−26.2841	−1.47859
$317$	−25.8949	−1.45440	−0.727200	−	0.686425i	$-0.759179\pi$
$317$	−25.8949	−1.45440	−0.727200	+	0.686425i	$0.759179\pi$
$318$	0	0
$319$	−23.4688	−1.31400
$320$	−1.44057	−0.0805303
$321$	0	0
$322$	0	0
$323$	3.98545	0.221756
$324$	0	0
$325$	−23.0496	−1.27856
$326$	2.42602	0.134365
$327$	0	0
$328$	7.23442	0.399454
$329$	0	0
$330$	0	0
$331$	8.21894	0.451754	0.225877	−	0.974156i	$-0.427475\pi$
$331$	8.21894	0.451754	0.225877	+	0.974156i	$0.427475\pi$
$332$	0.965852	0.0530080
$333$	0	0
$334$	−50.2327	−2.74861
$335$	−2.60132	−0.142125
$336$	0	0
$337$	4.57129	0.249014	0.124507	−	0.992219i	$-0.460265\pi$
$337$	4.57129	0.249014	0.124507	+	0.992219i	$0.460265\pi$
$338$	42.3851	2.30544
$339$	0	0
$340$	−6.41147	−0.347711
$341$	17.6682	0.956786
$342$	0	0
$343$	0	0
$344$	1.12836	0.0608369
$345$	0	0
$346$	−57.4347	−3.08771
$347$	22.4662	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$347$	22.4662	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$348$	0	0
$349$	−26.0993	−1.39706	−0.698531	−	0.715580i	$-0.746162\pi$
$349$	−26.0993	−1.39706	−0.698531	+	0.715580i	$0.746162\pi$
$350$	0	0
$351$	0	0
$352$	17.8307	0.950379
$353$	0.355037	0.0188967	0.00944836	−	0.999955i	$-0.496992\pi$
$353$	0.355037	0.0188967	0.00944836	+	0.999955i	$0.496992\pi$
$354$	0	0
$355$	−3.23618	−0.171758
$356$	−48.6468	−2.57828
$357$	0	0
$358$	−18.6040	−0.983252
$359$	5.45605	0.287959	0.143980	−	0.989581i	$-0.454010\pi$
$359$	5.45605	0.287959	0.143980	+	0.989581i	$0.454010\pi$
$360$	0	0
$361$	−13.1848	−0.693936
$362$	8.72462	0.458556
$363$	0	0
$364$	0	0
$365$	11.2395	0.588301
$366$	0	0
$367$	−10.9240	−0.570226	−0.285113	−	0.958494i	$-0.592031\pi$
$367$	−10.9240	−0.570226	−0.285113	+	0.958494i	$0.592031\pi$
$368$	20.9932	1.09435
$369$	0	0
$370$	−10.1411	−0.527213
$371$	0	0
$372$	0	0
$373$	1.73143	0.0896500	0.0448250	−	0.998995i	$-0.485727\pi$
$373$	1.73143	0.0896500	0.0448250	+	0.998995i	$0.485727\pi$
$374$	−16.2344	−0.839462
$375$	0	0
$376$	6.24216	0.321915
$377$	−32.9908	−1.69911
$378$	0	0
$379$	−12.1334	−0.623251	−0.311626	−	0.950205i	$-0.600873\pi$
$379$	−12.1334	−0.623251	−0.311626	+	0.950205i	$0.600873\pi$
$380$	−9.35504	−0.479903
$381$	0	0
$382$	14.3259	0.732979
$383$	8.71183	0.445154	0.222577	−	0.974915i	$-0.428553\pi$
$383$	8.71183	0.445154	0.222577	+	0.974915i	$0.428553\pi$
$384$	0	0
$385$	0	0
$386$	24.2986	1.23677
$387$	0	0
$388$	−55.1566	−2.80015
$389$	3.64321	0.184718	0.0923590	−	0.995726i	$-0.470559\pi$
$389$	3.64321	0.184718	0.0923590	+	0.995726i	$0.470559\pi$
$390$	0	0
$391$	−5.22668	−0.264325
$392$	0	0
$393$	0	0
$394$	21.0669	1.06133
$395$	−5.23947	−0.263627
$396$	0	0
$397$	15.4456	0.775194	0.387597	−	0.921829i	$-0.373305\pi$
$397$	15.4456	0.775194	0.387597	+	0.921829i	$0.373305\pi$
$398$	−16.7023	−0.837212
$399$	0	0
$400$	−28.0574	−1.40287
$401$	18.4219	0.919946	0.459973	−	0.887933i	$-0.347859\pi$
$401$	18.4219	0.919946	0.459973	+	0.887933i	$0.347859\pi$
$402$	0	0
$403$	24.8367	1.23720
$404$	42.8658	2.13265
$405$	0	0
$406$	0	0
$407$	−17.6682	−0.875779
$408$	0	0
$409$	28.6364	1.41598	0.707989	−	0.706223i	$-0.249602\pi$
$409$	28.6364	1.41598	0.707989	+	0.706223i	$0.249602\pi$
$410$	2.63816	0.130289
$411$	0	0
$412$	−29.0993	−1.43362
$413$	0	0
$414$	0	0
$415$	0.192533	0.00945109
$416$	25.0651	1.22892
$417$	0	0
$418$	−23.6878	−1.15861
$419$	34.6955	1.69499	0.847494	−	0.530805i	$-0.178111\pi$
$419$	34.6955	1.69499	0.847494	+	0.530805i	$0.178111\pi$
$420$	0	0
$421$	−27.4020	−1.33549	−0.667745	−	0.744390i	$-0.732740\pi$
$421$	−27.4020	−1.33549	−0.667745	+	0.744390i	$0.732740\pi$
$422$	−8.53209	−0.415336
$423$	0	0
$424$	44.5185	2.16201
$425$	6.98545	0.338844
$426$	0	0
$427$	0	0
$428$	10.5398	0.509462
$429$	0	0
$430$	0.411474	0.0198430
$431$	26.5921	1.28090	0.640449	−	0.768000i	$-0.278748\pi$
$431$	26.5921	1.28090	0.640449	+	0.768000i	$0.278748\pi$
$432$	0	0
$433$	−37.1830	−1.78690	−0.893451	−	0.449160i	$-0.851723\pi$
$433$	−37.1830	−1.78690	−0.893451	+	0.449160i	$0.851723\pi$
$434$	0	0
$435$	0	0
$436$	17.4611	0.836235
$437$	−7.62630	−0.364815
$438$	0	0
$439$	−25.0746	−1.19675	−0.598373	−	0.801218i	$-0.704186\pi$
$439$	−25.0746	−1.19675	−0.598373	+	0.801218i	$0.704186\pi$
$440$	20.8307	0.993064
$441$	0	0
$442$	−22.8212	−1.08549
$443$	2.04458	0.0971408	0.0485704	−	0.998820i	$-0.484533\pi$
$443$	2.04458	0.0971408	0.0485704	+	0.998820i	$0.484533\pi$
$444$	0	0
$445$	−9.69728	−0.459695
$446$	15.8922	0.752516
$447$	0	0
$448$	0	0
$449$	−10.2344	−0.482992	−0.241496	−	0.970402i	$-0.577638\pi$
$449$	−10.2344	−0.482992	−0.241496	+	0.970402i	$0.577638\pi$
$450$	0	0
$451$	4.59627	0.216430
$452$	72.5836	3.41404
$453$	0	0
$454$	15.6040	0.732332
$455$	0	0
$456$	0	0
$457$	−42.5945	−1.99249	−0.996244	−	0.0865948i	$-0.972401\pi$
$457$	−42.5945	−1.99249	−0.996244	+	0.0865948i	$0.972401\pi$
$458$	−59.2131	−2.76684
$459$	0	0
$460$	12.2686	0.572025
$461$	−0.504748	−0.0235084	−0.0117542	−	0.999931i	$-0.503742\pi$
$461$	−0.504748	−0.0235084	−0.0117542	+	0.999931i	$0.503742\pi$
$462$	0	0
$463$	2.68004	0.124552	0.0622761	−	0.998059i	$-0.480164\pi$
$463$	2.68004	0.124552	0.0622761	+	0.998059i	$0.480164\pi$
$464$	−40.1584	−1.86431
$465$	0	0
$466$	−21.5868	−0.999988
$467$	31.4165	1.45378	0.726892	−	0.686752i	$-0.240964\pi$
$467$	31.4165	1.45378	0.726892	+	0.686752i	$0.240964\pi$
$468$	0	0
$469$	0	0
$470$	2.27631	0.104998
$471$	0	0
$472$	−40.6691	−1.87195
$473$	0.716881	0.0329622
$474$	0	0
$475$	10.1925	0.467666
$476$	0	0
$477$	0	0
$478$	36.8726	1.68651
$479$	16.4406	0.751189	0.375594	−	0.926784i	$-0.377439\pi$
$479$	16.4406	0.751189	0.375594	+	0.926784i	$0.377439\pi$
$480$	0	0
$481$	−24.8367	−1.13245
$482$	13.6827	0.623231
$483$	0	0
$484$	17.8648	0.812038
$485$	−10.9949	−0.499255
$486$	0	0
$487$	−2.97535	−0.134826	−0.0674129	−	0.997725i	$-0.521474\pi$
$487$	−2.97535	−0.134826	−0.0674129	+	0.997725i	$0.521474\pi$
$488$	15.8530	0.717631
$489$	0	0
$490$	0	0
$491$	−26.4861	−1.19530	−0.597650	−	0.801757i	$-0.703899\pi$
$491$	−26.4861	−1.19530	−0.597650	+	0.801757i	$0.703899\pi$
$492$	0	0
$493$	9.99825	0.450298
$494$	−33.2986	−1.49817
$495$	0	0
$496$	30.2327	1.35749
$497$	0	0
$498$	0	0
$499$	−13.4439	−0.601830	−0.300915	−	0.953651i	$-0.597292\pi$
$499$	−13.4439	−0.601830	−0.300915	+	0.953651i	$0.597292\pi$
$500$	−35.7939	−1.60075
$501$	0	0
$502$	30.5544	1.36371
$503$	22.6631	1.01050	0.505250	−	0.862973i	$-0.331400\pi$
$503$	22.6631	1.01050	0.505250	+	0.862973i	$0.331400\pi$
$504$	0	0
$505$	8.54488	0.380242
$506$	31.0651	1.38101
$507$	0	0
$508$	77.8786	3.45530
$509$	−9.54757	−0.423189	−0.211594	−	0.977358i	$-0.567866\pi$
$509$	−9.54757	−0.423189	−0.211594	+	0.977358i	$0.567866\pi$
$510$	0	0
$511$	0	0
$512$	−50.5553	−2.23425
$513$	0	0
$514$	−26.7547	−1.18010
$515$	−5.80066	−0.255608
$516$	0	0
$517$	3.96585	0.174418
$518$	0	0
$519$	0	0
$520$	29.2823	1.28411
$521$	3.11287	0.136377	0.0681887	−	0.997672i	$-0.478278\pi$
$521$	3.11287	0.136377	0.0681887	+	0.997672i	$0.478278\pi$
$522$	0	0
$523$	16.1489	0.706142	0.353071	−	0.935597i	$-0.385137\pi$
$523$	16.1489	0.706142	0.353071	+	0.935597i	$0.385137\pi$
$524$	84.6897	3.69968
$525$	0	0
$526$	−71.7930	−3.13032
$527$	−7.52704	−0.327883
$528$	0	0
$529$	−12.9986	−0.565155
$530$	16.2344	0.705178
$531$	0	0
$532$	0	0
$533$	6.46110	0.279861
$534$	0	0
$535$	2.10101	0.0908348
$536$	−18.0624	−0.780178
$537$	0	0
$538$	−18.9486	−0.816933
$539$	0	0
$540$	0	0
$541$	−5.01548	−0.215632	−0.107816	−	0.994171i	$-0.534386\pi$
$541$	−5.01548	−0.215632	−0.107816	+	0.994171i	$0.534386\pi$
$542$	−34.5330	−1.48332
$543$	0	0
$544$	−7.59627	−0.325687
$545$	3.48070	0.149097
$546$	0	0
$547$	16.4780	0.704549	0.352275	−	0.935897i	$-0.385408\pi$
$547$	16.4780	0.704549	0.352275	+	0.935897i	$0.385408\pi$
$548$	80.0934	3.42142
$549$	0	0
$550$	−41.5185	−1.77035
$551$	14.5885	0.621492
$552$	0	0
$553$	0	0
$554$	−15.5740	−0.661675
$555$	0	0
$556$	−97.3055	−4.12667
$557$	−34.5631	−1.46448	−0.732242	−	0.681045i	$-0.761526\pi$
$557$	−34.5631	−1.46448	−0.732242	+	0.681045i	$0.761526\pi$
$558$	0	0
$559$	1.00774	0.0426229
$560$	0	0
$561$	0	0
$562$	8.38682	0.353777
$563$	37.2104	1.56823	0.784115	−	0.620615i	$-0.213117\pi$
$563$	37.2104	1.56823	0.784115	+	0.620615i	$0.213117\pi$
$564$	0	0
$565$	14.4688	0.608709
$566$	−73.4894	−3.08899
$567$	0	0
$568$	−22.4706	−0.942845
$569$	0.404667	0.0169645	0.00848226	−	0.999964i	$-0.497300\pi$
$569$	0.404667	0.0169645	0.00848226	+	0.999964i	$0.497300\pi$
$570$	0	0
$571$	−37.7793	−1.58101	−0.790507	−	0.612453i	$-0.790183\pi$
$571$	−37.7793	−1.58101	−0.790507	+	0.612453i	$0.790183\pi$
$572$	93.3278	3.90223
$573$	0	0
$574$	0	0
$575$	−13.3669	−0.557438
$576$	0	0
$577$	2.21120	0.0920535	0.0460267	−	0.998940i	$-0.485344\pi$
$577$	2.21120	0.0920535	0.0460267	+	0.998940i	$0.485344\pi$
$578$	−36.1293	−1.50278
$579$	0	0
$580$	−23.4688	−0.974491
$581$	0	0
$582$	0	0
$583$	28.2841	1.17141
$584$	78.0420	3.22940
$585$	0	0
$586$	21.3182	0.880647
$587$	−24.2098	−0.999244	−0.499622	−	0.866243i	$-0.666528\pi$
$587$	−24.2098	−0.999244	−0.499622	+	0.866243i	$0.666528\pi$
$588$	0	0
$589$	−10.9828	−0.452537
$590$	−14.8307	−0.610570
$591$	0	0
$592$	−30.2327	−1.24255
$593$	−12.2385	−0.502577	−0.251288	−	0.967912i	$-0.580854\pi$
$593$	−12.2385	−0.502577	−0.251288	+	0.967912i	$0.580854\pi$
$594$	0	0
$595$	0	0
$596$	−66.8590	−2.73865
$597$	0	0
$598$	43.6691	1.78576
$599$	39.6168	1.61870	0.809349	−	0.587328i	$-0.199820\pi$
$599$	39.6168	1.61870	0.809349	+	0.587328i	$0.199820\pi$
$600$	0	0
$601$	30.0077	1.22404	0.612021	−	0.790842i	$-0.290357\pi$
$601$	30.0077	1.22404	0.612021	+	0.790842i	$0.290357\pi$
$602$	0	0
$603$	0	0
$604$	−83.6255	−3.40267
$605$	3.56118	0.144783
$606$	0	0
$607$	19.4843	0.790844	0.395422	−	0.918499i	$-0.370598\pi$
$607$	19.4843	0.790844	0.395422	+	0.918499i	$0.370598\pi$
$608$	−11.0838	−0.449507
$609$	0	0
$610$	5.78106	0.234068
$611$	5.57491	0.225537
$612$	0	0
$613$	−18.5276	−0.748325	−0.374162	−	0.927363i	$-0.622070\pi$
$613$	−18.5276	−0.748325	−0.374162	+	0.927363i	$0.622070\pi$
$614$	32.1266	1.29652
$615$	0	0
$616$	0	0
$617$	27.8402	1.12080	0.560402	−	0.828221i	$-0.310647\pi$
$617$	27.8402	1.12080	0.560402	+	0.828221i	$0.310647\pi$
$618$	0	0
$619$	44.9813	1.80795	0.903976	−	0.427583i	$-0.140635\pi$
$619$	44.9813	1.80795	0.903976	+	0.427583i	$0.140635\pi$
$620$	17.6682	0.709571
$621$	0	0
$622$	41.7547	1.67421
$623$	0	0
$624$	0	0
$625$	13.9982	0.559930
$626$	−72.2113	−2.88614
$627$	0	0
$628$	79.6596	3.17877
$629$	7.52704	0.300123
$630$	0	0
$631$	9.43613	0.375646	0.187823	−	0.982203i	$-0.439857\pi$
$631$	9.43613	0.375646	0.187823	+	0.982203i	$0.439857\pi$
$632$	−36.3806	−1.44714
$633$	0	0
$634$	−65.5681	−2.60404
$635$	15.5243	0.616065
$636$	0	0
$637$	0	0
$638$	−59.4252	−2.35267
$639$	0	0
$640$	−11.7314	−0.463725
$641$	37.3901	1.47682	0.738410	−	0.674352i	$-0.235577\pi$
$641$	37.3901	1.47682	0.738410	+	0.674352i	$0.235577\pi$
$642$	0	0
$643$	−1.61175	−0.0635611	−0.0317806	−	0.999495i	$-0.510118\pi$
$643$	−1.61175	−0.0635611	−0.0317806	+	0.999495i	$0.510118\pi$
$644$	0	0
$645$	0	0
$646$	10.0915	0.397046
$647$	41.1762	1.61880	0.809402	−	0.587255i	$-0.199791\pi$
$647$	41.1762	1.61880	0.809402	+	0.587255i	$0.199791\pi$
$648$	0	0
$649$	−25.8384	−1.01425
$650$	−58.3637	−2.28921
$651$	0	0
$652$	4.22668	0.165530
$653$	3.05199	0.119434	0.0597169	−	0.998215i	$-0.480980\pi$
$653$	3.05199	0.119434	0.0597169	+	0.998215i	$0.480980\pi$
$654$	0	0
$655$	16.8821	0.659637
$656$	7.86484	0.307070
$657$	0	0
$658$	0	0
$659$	41.6350	1.62187	0.810934	−	0.585138i	$-0.198959\pi$
$659$	41.6350	1.62187	0.810934	+	0.585138i	$0.198959\pi$
$660$	0	0
$661$	−20.3010	−0.789616	−0.394808	−	0.918764i	$-0.629189\pi$
$661$	−20.3010	−0.789616	−0.394808	+	0.918764i	$0.629189\pi$
$662$	20.8111	0.808846
$663$	0	0
$664$	1.33687	0.0518805
$665$	0	0
$666$	0	0
$667$	−19.1320	−0.740793
$668$	−87.5167	−3.38612
$669$	0	0
$670$	−6.58677	−0.254469
$671$	10.0719	0.388822
$672$	0	0
$673$	−0.830689	−0.0320207	−0.0160104	−	0.999872i	$-0.505096\pi$
$673$	−0.830689	−0.0320207	−0.0160104	+	0.999872i	$0.505096\pi$
$674$	11.5749	0.445849
$675$	0	0
$676$	73.8444	2.84017
$677$	−10.8672	−0.417660	−0.208830	−	0.977952i	$-0.566966\pi$
$677$	−10.8672	−0.417660	−0.208830	+	0.977952i	$0.566966\pi$
$678$	0	0
$679$	0	0
$680$	−8.87433	−0.340315
$681$	0	0
$682$	44.7374	1.71308
$683$	32.6946	1.25102	0.625512	−	0.780215i	$-0.284890\pi$
$683$	32.6946	1.25102	0.625512	+	0.780215i	$0.284890\pi$
$684$	0	0
$685$	15.9659	0.610024
$686$	0	0
$687$	0	0
$688$	1.22668	0.0467668
$689$	39.7597	1.51472
$690$	0	0
$691$	−14.9982	−0.570560	−0.285280	−	0.958444i	$-0.592087\pi$
$691$	−14.9982	−0.570560	−0.285280	+	0.958444i	$0.592087\pi$
$692$	−100.064	−3.80387
$693$	0	0
$694$	56.8863	2.15937
$695$	−19.3969	−0.735767
$696$	0	0
$697$	−1.95811	−0.0741687
$698$	−66.0856	−2.50138
$699$	0	0
$700$	0	0
$701$	26.4688	0.999714	0.499857	−	0.866108i	$-0.333386\pi$
$701$	26.4688	0.999714	0.499857	+	0.866108i	$0.333386\pi$
$702$	0	0
$703$	10.9828	0.414223
$704$	−6.35504	−0.239514
$705$	0	0
$706$	0.898986	0.0338338
$707$	0	0
$708$	0	0
$709$	15.3601	0.576860	0.288430	−	0.957501i	$-0.406867\pi$
$709$	15.3601	0.576860	0.288430	+	0.957501i	$0.406867\pi$
$710$	−8.19429	−0.307526
$711$	0	0
$712$	−67.3337	−2.52344
$713$	14.4032	0.539405
$714$	0	0
$715$	18.6040	0.695750
$716$	−32.4124	−1.21131
$717$	0	0
$718$	13.8152	0.515579
$719$	−26.7306	−0.996883	−0.498442	−	0.866923i	$-0.666094\pi$
$719$	−26.7306	−0.996883	−0.498442	+	0.866923i	$0.666094\pi$
$720$	0	0
$721$	0	0
$722$	−33.3851	−1.24246
$723$	0	0
$724$	15.2003	0.564914
$725$	25.5699	0.949641
$726$	0	0
$727$	45.6441	1.69285	0.846424	−	0.532510i	$-0.178751\pi$
$727$	45.6441	1.69285	0.846424	+	0.532510i	$0.178751\pi$
$728$	0	0
$729$	0	0
$730$	28.4593	1.05333
$731$	−0.305407	−0.0112959
$732$	0	0
$733$	−5.97502	−0.220693	−0.110346	−	0.993893i	$-0.535196\pi$
$733$	−5.97502	−0.220693	−0.110346	+	0.993893i	$0.535196\pi$
$734$	−27.6604	−1.02097
$735$	0	0
$736$	14.5357	0.535793
$737$	−11.4757	−0.422711
$738$	0	0
$739$	−35.5963	−1.30943	−0.654715	−	0.755876i	$-0.727211\pi$
$739$	−35.5963	−1.30943	−0.654715	+	0.755876i	$0.727211\pi$
$740$	−17.6682	−0.649495
$741$	0	0
$742$	0	0
$743$	−29.3087	−1.07523	−0.537616	−	0.843190i	$-0.680675\pi$
$743$	−29.3087	−1.07523	−0.537616	+	0.843190i	$0.680675\pi$
$744$	0	0
$745$	−13.3277	−0.488289
$746$	4.38413	0.160515
$747$	0	0
$748$	−28.2841	−1.03417
$749$	0	0
$750$	0	0
$751$	−17.3337	−0.632515	−0.316258	−	0.948673i	$-0.602426\pi$
$751$	−17.3337	−0.632515	−0.316258	+	0.948673i	$0.602426\pi$
$752$	6.78611	0.247464
$753$	0	0
$754$	−83.5357	−3.04219
$755$	−16.6699	−0.606681
$756$	0	0
$757$	−2.77156	−0.100734	−0.0503671	−	0.998731i	$-0.516039\pi$
$757$	−2.77156	−0.100734	−0.0503671	+	0.998731i	$0.516039\pi$
$758$	−30.7229	−1.11590
$759$	0	0
$760$	−12.9486	−0.469696
$761$	−7.50744	−0.272144	−0.136072	−	0.990699i	$-0.543448\pi$
$761$	−7.50744	−0.272144	−0.136072	+	0.990699i	$0.543448\pi$
$762$	0	0
$763$	0	0
$764$	24.9590	0.902987
$765$	0	0
$766$	22.0591	0.797029
$767$	−36.3218	−1.31150
$768$	0	0
$769$	−2.04364	−0.0736957	−0.0368478	−	0.999321i	$-0.511732\pi$
$769$	−2.04364	−0.0736957	−0.0368478	+	0.999321i	$0.511732\pi$
$770$	0	0
$771$	0	0
$772$	42.3337	1.52362
$773$	24.9418	0.897094	0.448547	−	0.893759i	$-0.351942\pi$
$773$	24.9418	0.897094	0.448547	+	0.893759i	$0.351942\pi$
$774$	0	0
$775$	−19.2499	−0.691477
$776$	−76.3441	−2.74059
$777$	0	0
$778$	9.22493	0.330730
$779$	−2.85710	−0.102366
$780$	0	0
$781$	−14.2763	−0.510847
$782$	−13.2344	−0.473262
$783$	0	0
$784$	0	0
$785$	15.8794	0.566760
$786$	0	0
$787$	−7.10700	−0.253337	−0.126669	−	0.991945i	$-0.540428\pi$
$787$	−7.10700	−0.253337	−0.126669	+	0.991945i	$0.540428\pi$
$788$	36.7033	1.30750
$789$	0	0
$790$	−13.2668	−0.472012
$791$	0	0
$792$	0	0
$793$	14.1584	0.502779
$794$	39.1097	1.38795
$795$	0	0
$796$	−29.0993	−1.03140
$797$	−33.6628	−1.19240	−0.596199	−	0.802837i	$-0.703323\pi$
$797$	−33.6628	−1.19240	−0.596199	+	0.802837i	$0.703323\pi$
$798$	0	0
$799$	−1.68954	−0.0597716
$800$	−19.4270	−0.686847
$801$	0	0
$802$	46.6459	1.64712
$803$	49.5827	1.74973
$804$	0	0
$805$	0	0
$806$	62.8887	2.21516
$807$	0	0
$808$	59.3319	2.08729
$809$	−12.8161	−0.450592	−0.225296	−	0.974290i	$-0.572335\pi$
$809$	−12.8161	−0.450592	−0.225296	+	0.974290i	$0.572335\pi$
$810$	0	0
$811$	26.1239	0.917335	0.458667	−	0.888608i	$-0.348327\pi$
$811$	26.1239	0.917335	0.458667	+	0.888608i	$0.348327\pi$
$812$	0	0
$813$	0	0
$814$	−44.7374	−1.56805
$815$	0.842549	0.0295132
$816$	0	0
$817$	−0.445622	−0.0155904
$818$	72.5099	2.53525
$819$	0	0
$820$	4.59627	0.160509
$821$	27.6641	0.965483	0.482741	−	0.875763i	$-0.339641\pi$
$821$	27.6641	0.965483	0.482741	+	0.875763i	$0.339641\pi$
$822$	0	0
$823$	27.8324	0.970178	0.485089	−	0.874465i	$-0.338787\pi$
$823$	27.8324	0.970178	0.485089	+	0.874465i	$0.338787\pi$
$824$	−40.2772	−1.40312
$825$	0	0
$826$	0	0
$827$	4.65507	0.161873	0.0809363	−	0.996719i	$-0.474209\pi$
$827$	4.65507	0.161873	0.0809363	+	0.996719i	$0.474209\pi$
$828$	0	0
$829$	9.97359	0.346397	0.173199	−	0.984887i	$-0.444590\pi$
$829$	9.97359	0.346397	0.173199	+	0.984887i	$0.444590\pi$
$830$	0.487511	0.0169218
$831$	0	0
$832$	−8.93346	−0.309712
$833$	0	0
$834$	0	0
$835$	−17.4456	−0.603731
$836$	−41.2695	−1.42734
$837$	0	0
$838$	87.8522	3.03480
$839$	6.72967	0.232334	0.116167	−	0.993230i	$-0.462939\pi$
$839$	6.72967	0.232334	0.116167	+	0.993230i	$0.462939\pi$
$840$	0	0
$841$	7.59802	0.262001
$842$	−69.3842	−2.39114
$843$	0	0
$844$	−14.8648	−0.511669
$845$	14.7202	0.506390
$846$	0	0
$847$	0	0
$848$	48.3979	1.66199
$849$	0	0
$850$	17.6878	0.606686
$851$	−14.4032	−0.493737
$852$	0	0
$853$	−5.79055	−0.198265	−0.0991324	−	0.995074i	$-0.531607\pi$
$853$	−5.79055	−0.198265	−0.0991324	+	0.995074i	$0.531607\pi$
$854$	0	0
$855$	0	0
$856$	14.5885	0.498626
$857$	34.9077	1.19242	0.596211	−	0.802827i	$-0.296672\pi$
$857$	34.9077	1.19242	0.596211	+	0.802827i	$0.296672\pi$
$858$	0	0
$859$	12.6149	0.430416	0.215208	−	0.976568i	$-0.430957\pi$
$859$	12.6149	0.430416	0.215208	+	0.976568i	$0.430957\pi$
$860$	0.716881	0.0244455
$861$	0	0
$862$	67.3337	2.29339
$863$	−24.2053	−0.823959	−0.411979	−	0.911193i	$-0.635162\pi$
$863$	−24.2053	−0.823959	−0.411979	+	0.911193i	$0.635162\pi$
$864$	0	0
$865$	−19.9469	−0.678214
$866$	−94.1508	−3.19937
$867$	0	0
$868$	0	0
$869$	−23.1138	−0.784082
$870$	0	0
$871$	−16.1317	−0.546600
$872$	24.1685	0.818448
$873$	0	0
$874$	−19.3105	−0.653186
$875$	0	0
$876$	0	0
$877$	−1.12567	−0.0380111	−0.0190055	−	0.999819i	$-0.506050\pi$
$877$	−1.12567	−0.0380111	−0.0190055	+	0.999819i	$0.506050\pi$
$878$	−63.4911	−2.14272
$879$	0	0
$880$	22.6459	0.763393
$881$	4.38331	0.147678	0.0738388	−	0.997270i	$-0.476475\pi$
$881$	4.38331	0.147678	0.0738388	+	0.997270i	$0.476475\pi$
$882$	0	0
$883$	−6.88949	−0.231850	−0.115925	−	0.993258i	$-0.536983\pi$
$883$	−6.88949	−0.231850	−0.115925	+	0.993258i	$0.536983\pi$
$884$	−39.7597	−1.33726
$885$	0	0
$886$	5.17705	0.173926
$887$	−39.0752	−1.31202	−0.656009	−	0.754753i	$-0.727757\pi$
$887$	−39.0752	−1.31202	−0.656009	+	0.754753i	$0.727757\pi$
$888$	0	0
$889$	0	0
$890$	−24.5544	−0.823065
$891$	0	0
$892$	27.6878	0.927056
$893$	−2.46522	−0.0824955
$894$	0	0
$895$	−6.46110	−0.215971
$896$	0	0
$897$	0	0
$898$	−25.9145	−0.864777
$899$	−27.5523	−0.918921
$900$	0	0
$901$	−12.0496	−0.401431
$902$	11.6382	0.387508
$903$	0	0
$904$	100.465	3.34143
$905$	3.03003	0.100722
$906$	0	0
$907$	42.4938	1.41098	0.705492	−	0.708718i	$-0.250726\pi$
$907$	42.4938	1.41098	0.705492	+	0.708718i	$0.250726\pi$
$908$	27.1857	0.902190
$909$	0	0
$910$	0	0
$911$	−15.4935	−0.513322	−0.256661	−	0.966501i	$-0.582622\pi$
$911$	−15.4935	−0.513322	−0.256661	+	0.966501i	$0.582622\pi$
$912$	0	0
$913$	0.849356	0.0281096
$914$	−107.853	−3.56746
$915$	0	0
$916$	−103.163	−3.40859
$917$	0	0
$918$	0	0
$919$	6.52940	0.215385	0.107693	−	0.994184i	$-0.465654\pi$
$919$	6.52940	0.215385	0.107693	+	0.994184i	$0.465654\pi$
$920$	16.9813	0.559858
$921$	0	0
$922$	−1.27807	−0.0420909
$923$	−20.0686	−0.660567
$924$	0	0
$925$	19.2499	0.632933
$926$	6.78611	0.223005
$927$	0	0
$928$	−27.8057	−0.912767
$929$	−58.2772	−1.91201	−0.956007	−	0.293343i	$-0.905232\pi$
$929$	−58.2772	−1.91201	−0.956007	+	0.293343i	$0.905232\pi$
$930$	0	0
$931$	0	0
$932$	−37.6091	−1.23193
$933$	0	0
$934$	79.5494	2.60294
$935$	−5.63816	−0.184387
$936$	0	0
$937$	−32.4175	−1.05903	−0.529516	−	0.848300i	$-0.677626\pi$
$937$	−32.4175	−1.05903	−0.529516	+	0.848300i	$0.677626\pi$
$938$	0	0
$939$	0	0
$940$	3.96585	0.129352
$941$	27.3226	0.890693	0.445346	−	0.895358i	$-0.353081\pi$
$941$	27.3226	0.890693	0.445346	+	0.895358i	$0.353081\pi$
$942$	0	0
$943$	3.74691	0.122016
$944$	−44.2131	−1.43901
$945$	0	0
$946$	1.81521	0.0590175
$947$	−38.2131	−1.24176	−0.620879	−	0.783906i	$-0.713224\pi$
$947$	−38.2131	−1.24176	−0.620879	+	0.783906i	$0.713224\pi$
$948$	0	0
$949$	69.6998	2.26255
$950$	25.8084	0.837335
$951$	0	0
$952$	0	0
$953$	58.9377	1.90918	0.954590	−	0.297924i	$-0.0962943\pi$
$953$	58.9377	1.90918	0.954590	+	0.297924i	$0.0962943\pi$
$954$	0	0
$955$	4.97535	0.160998
$956$	64.2404	2.07768
$957$	0	0
$958$	41.6290	1.34497
$959$	0	0
$960$	0	0
$961$	−10.2576	−0.330892
$962$	−62.8887	−2.02761
$963$	0	0
$964$	23.8384	0.767784
$965$	8.43882	0.271655
$966$	0	0
$967$	24.7187	0.794901	0.397451	−	0.917624i	$-0.369895\pi$
$967$	24.7187	0.794901	0.397451	+	0.917624i	$0.369895\pi$
$968$	24.7273	0.794766
$969$	0	0
$970$	−27.8402	−0.893894
$971$	−8.17623	−0.262388	−0.131194	−	0.991357i	$-0.541881\pi$
$971$	−8.17623	−0.262388	−0.131194	+	0.991357i	$0.541881\pi$
$972$	0	0
$973$	0	0
$974$	−7.53384	−0.241400
$975$	0	0
$976$	17.2344	0.551660
$977$	−15.8485	−0.507040	−0.253520	−	0.967330i	$-0.581588\pi$
$977$	−15.8485	−0.507040	−0.253520	+	0.967330i	$0.581588\pi$
$978$	0	0
$979$	−42.7793	−1.36723
$980$	0	0
$981$	0	0
$982$	−67.0651	−2.14013
$983$	−53.3063	−1.70021	−0.850104	−	0.526615i	$-0.823461\pi$
$983$	−53.3063	−1.70021	−0.850104	+	0.526615i	$0.823461\pi$
$984$	0	0
$985$	7.31645	0.233121
$986$	25.3164	0.806240
$987$	0	0
$988$	−58.0137	−1.84566
$989$	0.584407	0.0185831
$990$	0	0
$991$	40.2094	1.27730	0.638648	−	0.769499i	$-0.279494\pi$
$991$	40.2094	1.27730	0.638648	+	0.769499i	$0.279494\pi$
$992$	20.9331	0.664628
$993$	0	0
$994$	0	0
$995$	−5.80066	−0.183893
$996$	0	0
$997$	−28.7202	−0.909577	−0.454789	−	0.890599i	$-0.650285\pi$
$997$	−28.7202	−0.909577	−0.454789	+	0.890599i	$0.650285\pi$
$998$	−34.0411	−1.07755
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.q.1.3		3
3.2	odd	2		3969.2.a.l.1.1		3
7.6	odd	2		567.2.a.h.1.3		3
9.2	odd	6		1323.2.f.d.442.3		6
9.4	even	3		441.2.f.c.295.1		6
9.5	odd	6		1323.2.f.d.883.3		6
9.7	even	3		441.2.f.c.148.1		6
21.20	even	2		567.2.a.c.1.1		3
28.27	even	2		9072.2.a.ca.1.1		3
63.2	odd	6		1323.2.g.e.361.3		6
63.4	even	3		441.2.g.b.79.1		6
63.5	even	6		1323.2.h.c.802.1		6
63.11	odd	6		1323.2.h.b.226.1		6
63.13	odd	6		63.2.f.a.43.1	yes	6
63.16	even	3		441.2.g.b.67.1		6
63.20	even	6		189.2.f.b.64.3		6
63.23	odd	6		1323.2.h.b.802.1		6
63.25	even	3		441.2.h.e.373.3		6
63.31	odd	6		441.2.g.c.79.1		6
63.32	odd	6		1323.2.g.e.667.3		6
63.34	odd	6		63.2.f.a.22.1	✓	6
63.38	even	6		1323.2.h.c.226.1		6
63.40	odd	6		441.2.h.d.214.3		6
63.41	even	6		189.2.f.b.127.3		6
63.47	even	6		1323.2.g.d.361.3		6
63.52	odd	6		441.2.h.d.373.3		6
63.58	even	3		441.2.h.e.214.3		6
63.59	even	6		1323.2.g.d.667.3		6
63.61	odd	6		441.2.g.c.67.1		6
84.83	odd	2		9072.2.a.bs.1.3		3
252.83	odd	6		3024.2.r.k.1009.1		6
252.139	even	6		1008.2.r.h.673.3		6
252.167	odd	6		3024.2.r.k.2017.1		6
252.223	even	6		1008.2.r.h.337.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.a.22.1	✓	6	63.34	odd	6
63.2.f.a.43.1	yes	6	63.13	odd	6
189.2.f.b.64.3		6	63.20	even	6
189.2.f.b.127.3		6	63.41	even	6
441.2.f.c.148.1		6	9.7	even	3
441.2.f.c.295.1		6	9.4	even	3
441.2.g.b.67.1		6	63.16	even	3
441.2.g.b.79.1		6	63.4	even	3
441.2.g.c.67.1		6	63.61	odd	6
441.2.g.c.79.1		6	63.31	odd	6
441.2.h.d.214.3		6	63.40	odd	6
441.2.h.d.373.3		6	63.52	odd	6
441.2.h.e.214.3		6	63.58	even	3
441.2.h.e.373.3		6	63.25	even	3
567.2.a.c.1.1		3	21.20	even	2
567.2.a.h.1.3		3	7.6	odd	2
1008.2.r.h.337.3		6	252.223	even	6
1008.2.r.h.673.3		6	252.139	even	6
1323.2.f.d.442.3		6	9.2	odd	6
1323.2.f.d.883.3		6	9.5	odd	6
1323.2.g.d.361.3		6	63.47	even	6
1323.2.g.d.667.3		6	63.59	even	6
1323.2.g.e.361.3		6	63.2	odd	6
1323.2.g.e.667.3		6	63.32	odd	6
1323.2.h.b.226.1		6	63.11	odd	6
1323.2.h.b.802.1		6	63.23	odd	6
1323.2.h.c.226.1		6	63.38	even	6
1323.2.h.c.802.1		6	63.5	even	6
3024.2.r.k.1009.1		6	252.83	odd	6
3024.2.r.k.2017.1		6	252.167	odd	6
3969.2.a.l.1.1		3	3.2	odd	2
3969.2.a.q.1.3		3	1.1	even	1	trivial
9072.2.a.bs.1.3		3	84.83	odd	2
9072.2.a.ca.1.1		3	28.27	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53209 q^{2} +4.41147 q^{4} +0.879385 q^{5} +6.10607 q^{8} +2.22668 q^{10} +3.87939 q^{11} +5.45336 q^{13} +6.63816 q^{16} -1.65270 q^{17} -2.41147 q^{19} +3.87939 q^{20} +9.82295 q^{22} +3.16250 q^{23} -4.22668 q^{25} +13.8084 q^{26} -6.04963 q^{29} +4.55438 q^{31} +4.59627 q^{32} -4.18479 q^{34} -4.55438 q^{37} -6.10607 q^{38} +5.36959 q^{40} +1.18479 q^{41} +0.184793 q^{43} +17.1138 q^{44} +8.00774 q^{46} +1.02229 q^{47} -10.7023 q^{50} +24.0574 q^{52} +7.29086 q^{53} +3.41147 q^{55} -15.3182 q^{58} -6.66044 q^{59} +2.59627 q^{61} +11.5321 q^{62} -1.63816 q^{64} +4.79561 q^{65} -2.95811 q^{67} -7.29086 q^{68} -3.68004 q^{71} +12.7811 q^{73} -11.5321 q^{74} -10.6382 q^{76} -5.95811 q^{79} +5.83750 q^{80} +3.00000 q^{82} +0.218941 q^{83} -1.45336 q^{85} +0.467911 q^{86} +23.6878 q^{88} -11.0273 q^{89} +13.9513 q^{92} +2.58853 q^{94} -2.12061 q^{95} -12.5030 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 6 q^{8} + 6 q^{11} + 3 q^{13} + 3 q^{16} - 6 q^{17} + 3 q^{19} + 6 q^{20} + 9 q^{22} + 12 q^{23} - 6 q^{25} + 3 q^{26} + 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} - 6 q^{38}+ \cdots + 3 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 6 * q^8 + 6 * q^11 + 3 * q^13 + 3 * q^16 - 6 * q^17 + 3 * q^19 + 6 * q^20 + 9 * q^22 + 12 * q^23 - 6 * q^25 + 3 * q^26 + 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 - 6 * q^38 + 9 * q^40 - 3 * q^43 + 15 * q^44 - 3 * q^47 - 6 * q^50 + 21 * q^52 + 6 * q^53 - 9 * q^58 + 3 * q^59 - 6 * q^61 + 30 * q^62 + 12 * q^64 + 15 * q^65 - 12 * q^67 - 6 * q^68 + 9 * q^71 + 21 * q^73 - 30 * q^74 - 15 * q^76 - 21 * q^79 + 15 * q^80 + 9 * q^82 + 18 * q^83 + 9 * q^85 + 6 * q^86 + 27 * q^88 - 12 * q^89 + 3 * q^92 + 18 * q^94 - 12 * q^95 + 3 * q^97

Introduction

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