LMFDB - Embedded newform 3344.2.a.t.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3344,2,Mod(1,3344)] 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("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-1,0,-5,0,-6] 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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.15351$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26452 q^{3} +0.637602 q^{5} -2.66942 q^{7} +2.12805 q^{9} -1.00000 q^{11} -1.20725 q^{13} -1.44386 q^{15} +0.222022 q^{17} +1.00000 q^{19} +6.04495 q^{21} +9.48717 q^{23} -4.59346 q^{25} +1.97454 q^{27} +4.94518 q^{29} +3.05563 q^{31} +2.26452 q^{33} -1.70203 q^{35} +7.18015 q^{37} +2.73384 q^{39} -6.92650 q^{41} -1.53538 q^{43} +1.35685 q^{45} -5.94581 q^{47} +0.125785 q^{49} -0.502774 q^{51} -9.63879 q^{53} -0.637602 q^{55} -2.26452 q^{57} -2.65817 q^{59} -0.809792 q^{61} -5.68066 q^{63} -0.769744 q^{65} +2.22447 q^{67} -21.4839 q^{69} -2.58912 q^{71} +16.6108 q^{73} +10.4020 q^{75} +2.66942 q^{77} -5.69298 q^{79} -10.8556 q^{81} +2.93784 q^{83} +0.141562 q^{85} -11.1985 q^{87} -5.54136 q^{89} +3.22265 q^{91} -6.91954 q^{93} +0.637602 q^{95} +13.7728 q^{97} -2.12805 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.26452	−1.30742	−0.653711	−	0.756745i	$-0.726789\pi$
$3$	−2.26452	−1.30742	−0.653711	+	0.756745i	$0.726789\pi$
$4$	0	0
$5$	0.637602	0.285144	0.142572	−	0.989784i	$-0.454463\pi$
$5$	0.637602	0.285144	0.142572	+	0.989784i	$0.454463\pi$
$6$	0	0
$7$	−2.66942	−1.00894	−0.504472	−	0.863428i	$-0.668313\pi$
$7$	−2.66942	−1.00894	−0.504472	+	0.863428i	$0.668313\pi$
$8$	0	0
$9$	2.12805	0.709351
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.20725	−0.334831	−0.167415	−	0.985886i	$-0.553542\pi$
$13$	−1.20725	−0.334831	−0.167415	+	0.985886i	$0.553542\pi$
$14$	0	0
$15$	−1.44386	−0.372804
$16$	0	0
$17$	0.222022	0.0538483	0.0269242	−	0.999637i	$-0.491429\pi$
$17$	0.222022	0.0538483	0.0269242	+	0.999637i	$0.491429\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	6.04495	1.31912
$22$	0	0
$23$	9.48717	1.97821	0.989106	−	0.147206i	$-0.0470279\pi$
$23$	9.48717	1.97821	0.989106	+	0.147206i	$0.0470279\pi$
$24$	0	0
$25$	−4.59346	−0.918693
$26$	0	0
$27$	1.97454	0.380001
$28$	0	0
$29$	4.94518	0.918297	0.459148	−	0.888360i	$-0.348155\pi$
$29$	4.94518	0.918297	0.459148	+	0.888360i	$0.348155\pi$
$30$	0	0
$31$	3.05563	0.548808	0.274404	−	0.961615i	$-0.411520\pi$
$31$	3.05563	0.548808	0.274404	+	0.961615i	$0.411520\pi$
$32$	0	0
$33$	2.26452	0.394202
$34$	0	0
$35$	−1.70203	−0.287695
$36$	0	0
$37$	7.18015	1.18041	0.590205	−	0.807254i	$-0.299047\pi$
$37$	7.18015	1.18041	0.590205	+	0.807254i	$0.299047\pi$
$38$	0	0
$39$	2.73384	0.437765
$40$	0	0
$41$	−6.92650	−1.08174	−0.540869	−	0.841107i	$-0.681904\pi$
$41$	−6.92650	−1.08174	−0.540869	+	0.841107i	$0.681904\pi$
$42$	0	0
$43$	−1.53538	−0.234144	−0.117072	−	0.993123i	$-0.537351\pi$
$43$	−1.53538	−0.234144	−0.117072	+	0.993123i	$0.537351\pi$
$44$	0	0
$45$	1.35685	0.202267
$46$	0	0
$47$	−5.94581	−0.867285	−0.433642	−	0.901085i	$-0.642772\pi$
$47$	−5.94581	−0.867285	−0.433642	+	0.901085i	$0.642772\pi$
$48$	0	0
$49$	0.125785	0.0179692
$50$	0	0
$51$	−0.502774	−0.0704024
$52$	0	0
$53$	−9.63879	−1.32399	−0.661995	−	0.749509i	$-0.730290\pi$
$53$	−9.63879	−1.32399	−0.661995	+	0.749509i	$0.730290\pi$
$54$	0	0
$55$	−0.637602	−0.0859742
$56$	0	0
$57$	−2.26452	−0.299943
$58$	0	0
$59$	−2.65817	−0.346065	−0.173032	−	0.984916i	$-0.555357\pi$
$59$	−2.65817	−0.346065	−0.173032	+	0.984916i	$0.555357\pi$
$60$	0	0
$61$	−0.809792	−0.103683	−0.0518416	−	0.998655i	$-0.516509\pi$
$61$	−0.809792	−0.103683	−0.0518416	+	0.998655i	$0.516509\pi$
$62$	0	0
$63$	−5.68066	−0.715696
$64$	0	0
$65$	−0.769744	−0.0954750
$66$	0	0
$67$	2.22447	0.271763	0.135881	−	0.990725i	$-0.456613\pi$
$67$	2.22447	0.271763	0.135881	+	0.990725i	$0.456613\pi$
$68$	0	0
$69$	−21.4839	−2.58636
$70$	0	0
$71$	−2.58912	−0.307272	−0.153636	−	0.988127i	$-0.549098\pi$
$71$	−2.58912	−0.307272	−0.153636	+	0.988127i	$0.549098\pi$
$72$	0	0
$73$	16.6108	1.94414	0.972071	−	0.234687i	$-0.0754065\pi$
$73$	16.6108	1.94414	0.972071	+	0.234687i	$0.0754065\pi$
$74$	0	0
$75$	10.4020	1.20112
$76$	0	0
$77$	2.66942	0.304208
$78$	0	0
$79$	−5.69298	−0.640510	−0.320255	−	0.947331i	$-0.603769\pi$
$79$	−5.69298	−0.640510	−0.320255	+	0.947331i	$0.603769\pi$
$80$	0	0
$81$	−10.8556	−1.20617
$82$	0	0
$83$	2.93784	0.322470	0.161235	−	0.986916i	$-0.448452\pi$
$83$	2.93784	0.322470	0.161235	+	0.986916i	$0.448452\pi$
$84$	0	0
$85$	0.141562	0.0153545
$86$	0	0
$87$	−11.1985	−1.20060
$88$	0	0
$89$	−5.54136	−0.587383	−0.293692	−	0.955900i	$-0.594884\pi$
$89$	−5.54136	−0.587383	−0.293692	+	0.955900i	$0.594884\pi$
$90$	0	0
$91$	3.22265	0.337826
$92$	0	0
$93$	−6.91954	−0.717523
$94$	0	0
$95$	0.637602	0.0654166
$96$	0	0
$97$	13.7728	1.39842	0.699209	−	0.714917i	$-0.253536\pi$
$97$	13.7728	1.39842	0.699209	+	0.714917i	$0.253536\pi$
$98$	0	0
$99$	−2.12805	−0.213877
$100$	0	0
$101$	12.0308	1.19711	0.598555	−	0.801082i	$-0.295742\pi$
$101$	12.0308	1.19711	0.598555	+	0.801082i	$0.295742\pi$
$102$	0	0
$103$	−8.53068	−0.840553	−0.420276	−	0.907396i	$-0.638067\pi$
$103$	−8.53068	−0.840553	−0.420276	+	0.907396i	$0.638067\pi$
$104$	0	0
$105$	3.85427	0.376138
$106$	0	0
$107$	−1.64585	−0.159110	−0.0795552	−	0.996830i	$-0.525350\pi$
$107$	−1.64585	−0.159110	−0.0795552	+	0.996830i	$0.525350\pi$
$108$	0	0
$109$	16.8065	1.60977	0.804886	−	0.593430i	$-0.202226\pi$
$109$	16.8065	1.60977	0.804886	+	0.593430i	$0.202226\pi$
$110$	0	0
$111$	−16.2596	−1.54329
$112$	0	0
$113$	−11.2947	−1.06252	−0.531258	−	0.847210i	$-0.678280\pi$
$113$	−11.2947	−1.06252	−0.531258	+	0.847210i	$0.678280\pi$
$114$	0	0
$115$	6.04904	0.564076
$116$	0	0
$117$	−2.56909	−0.237512
$118$	0	0
$119$	−0.592670	−0.0543300
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.6852	1.41429
$124$	0	0
$125$	−6.11681	−0.547104
$126$	0	0
$127$	−18.5894	−1.64954	−0.824771	−	0.565467i	$-0.808696\pi$
$127$	−18.5894	−1.64954	−0.824771	+	0.565467i	$0.808696\pi$
$128$	0	0
$129$	3.47690	0.306124
$130$	0	0
$131$	3.09098	0.270060	0.135030	−	0.990842i	$-0.456887\pi$
$131$	3.09098	0.270060	0.135030	+	0.990842i	$0.456887\pi$
$132$	0	0
$133$	−2.66942	−0.231468
$134$	0	0
$135$	1.25897	0.108355
$136$	0	0
$137$	−8.67619	−0.741257	−0.370629	−	0.928781i	$-0.620858\pi$
$137$	−8.67619	−0.741257	−0.370629	+	0.928781i	$0.620858\pi$
$138$	0	0
$139$	−14.5426	−1.23348	−0.616742	−	0.787166i	$-0.711548\pi$
$139$	−14.5426	−1.23348	−0.616742	+	0.787166i	$0.711548\pi$
$140$	0	0
$141$	13.4644	1.13391
$142$	0	0
$143$	1.20725	0.100955
$144$	0	0
$145$	3.15306	0.261847
$146$	0	0
$147$	−0.284842	−0.0234933
$148$	0	0
$149$	−18.4716	−1.51325	−0.756625	−	0.653849i	$-0.773153\pi$
$149$	−18.4716	−1.51325	−0.756625	+	0.653849i	$0.773153\pi$
$150$	0	0
$151$	0.913992	0.0743796	0.0371898	−	0.999308i	$-0.488159\pi$
$151$	0.913992	0.0743796	0.0371898	+	0.999308i	$0.488159\pi$
$152$	0	0
$153$	0.472475	0.0381973
$154$	0	0
$155$	1.94828	0.156489
$156$	0	0
$157$	−2.39421	−0.191079	−0.0955395	−	0.995426i	$-0.530458\pi$
$157$	−2.39421	−0.191079	−0.0955395	+	0.995426i	$0.530458\pi$
$158$	0	0
$159$	21.8272	1.73101
$160$	0	0
$161$	−25.3252	−1.99591
$162$	0	0
$163$	−14.1986	−1.11212	−0.556061	−	0.831141i	$-0.687688\pi$
$163$	−14.1986	−1.11212	−0.556061	+	0.831141i	$0.687688\pi$
$164$	0	0
$165$	1.44386	0.112405
$166$	0	0
$167$	−9.25056	−0.715830	−0.357915	−	0.933754i	$-0.616512\pi$
$167$	−9.25056	−0.715830	−0.357915	+	0.933754i	$0.616512\pi$
$168$	0	0
$169$	−11.5426	−0.887888
$170$	0	0
$171$	2.12805	0.162736
$172$	0	0
$173$	5.39376	0.410080	0.205040	−	0.978754i	$-0.434268\pi$
$173$	5.39376	0.410080	0.205040	+	0.978754i	$0.434268\pi$
$174$	0	0
$175$	12.2619	0.926910
$176$	0	0
$177$	6.01949	0.452453
$178$	0	0
$179$	−1.30720	−0.0977048	−0.0488524	−	0.998806i	$-0.515556\pi$
$179$	−1.30720	−0.0977048	−0.0488524	+	0.998806i	$0.515556\pi$
$180$	0	0
$181$	−10.7624	−0.799961	−0.399980	−	0.916524i	$-0.630983\pi$
$181$	−10.7624	−0.799961	−0.399980	+	0.916524i	$0.630983\pi$
$182$	0	0
$183$	1.83379	0.135558
$184$	0	0
$185$	4.57808	0.336587
$186$	0	0
$187$	−0.222022	−0.0162359
$188$	0	0
$189$	−5.27088	−0.383400
$190$	0	0
$191$	−15.3041	−1.10737	−0.553684	−	0.832727i	$-0.686778\pi$
$191$	−15.3041	−1.10737	−0.553684	+	0.832727i	$0.686778\pi$
$192$	0	0
$193$	−18.3032	−1.31749	−0.658747	−	0.752364i	$-0.728913\pi$
$193$	−18.3032	−1.31749	−0.658747	+	0.752364i	$0.728913\pi$
$194$	0	0
$195$	1.74310	0.124826
$196$	0	0
$197$	−0.576171	−0.0410505	−0.0205252	−	0.999789i	$-0.506534\pi$
$197$	−0.576171	−0.0410505	−0.0205252	+	0.999789i	$0.506534\pi$
$198$	0	0
$199$	10.1678	0.720778	0.360389	−	0.932802i	$-0.382644\pi$
$199$	10.1678	0.720778	0.360389	+	0.932802i	$0.382644\pi$
$200$	0	0
$201$	−5.03736	−0.355308
$202$	0	0
$203$	−13.2007	−0.926510
$204$	0	0
$205$	−4.41635	−0.308451
$206$	0	0
$207$	20.1892	1.40325
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	7.25837	0.499687	0.249843	−	0.968286i	$-0.419621\pi$
$211$	7.25837	0.499687	0.249843	+	0.968286i	$0.419621\pi$
$212$	0	0
$213$	5.86312	0.401734
$214$	0	0
$215$	−0.978963	−0.0667647
$216$	0	0
$217$	−8.15675	−0.553716
$218$	0	0
$219$	−37.6154	−2.54181
$220$	0	0
$221$	−0.268036	−0.0180301
$222$	0	0
$223$	−16.2772	−1.09000	−0.545000	−	0.838436i	$-0.683471\pi$
$223$	−16.2772	−1.09000	−0.545000	+	0.838436i	$0.683471\pi$
$224$	0	0
$225$	−9.77513	−0.651675
$226$	0	0
$227$	20.9260	1.38890	0.694452	−	0.719539i	$-0.255647\pi$
$227$	20.9260	1.38890	0.694452	+	0.719539i	$0.255647\pi$
$228$	0	0
$229$	−10.8083	−0.714234	−0.357117	−	0.934060i	$-0.616240\pi$
$229$	−10.8083	−0.714234	−0.357117	+	0.934060i	$0.616240\pi$
$230$	0	0
$231$	−6.04495	−0.397728
$232$	0	0
$233$	−25.4173	−1.66514	−0.832570	−	0.553919i	$-0.813132\pi$
$233$	−25.4173	−1.66514	−0.832570	+	0.553919i	$0.813132\pi$
$234$	0	0
$235$	−3.79106	−0.247301
$236$	0	0
$237$	12.8919	0.837417
$238$	0	0
$239$	−0.352238	−0.0227844	−0.0113922	−	0.999935i	$-0.503626\pi$
$239$	−0.352238	−0.0227844	−0.0113922	+	0.999935i	$0.503626\pi$
$240$	0	0
$241$	−15.6832	−1.01024	−0.505122	−	0.863048i	$-0.668552\pi$
$241$	−15.6832	−1.01024	−0.505122	+	0.863048i	$0.668552\pi$
$242$	0	0
$243$	18.6590	1.19697
$244$	0	0
$245$	0.0802004	0.00512382
$246$	0	0
$247$	−1.20725	−0.0768154
$248$	0	0
$249$	−6.65281	−0.421605
$250$	0	0
$251$	−1.16332	−0.0734282	−0.0367141	−	0.999326i	$-0.511689\pi$
$251$	−1.16332	−0.0734282	−0.0367141	+	0.999326i	$0.511689\pi$
$252$	0	0
$253$	−9.48717	−0.596453
$254$	0	0
$255$	−0.320570	−0.0200748
$256$	0	0
$257$	−15.0510	−0.938857	−0.469428	−	0.882971i	$-0.655540\pi$
$257$	−15.0510	−0.938857	−0.469428	+	0.882971i	$0.655540\pi$
$258$	0	0
$259$	−19.1668	−1.19097
$260$	0	0
$261$	10.5236	0.651394
$262$	0	0
$263$	20.6609	1.27400	0.637002	−	0.770862i	$-0.280174\pi$
$263$	20.6609	1.27400	0.637002	+	0.770862i	$0.280174\pi$
$264$	0	0
$265$	−6.14571	−0.377528
$266$	0	0
$267$	12.5485	0.767958
$268$	0	0
$269$	−2.82662	−0.172342	−0.0861711	−	0.996280i	$-0.527463\pi$
$269$	−2.82662	−0.172342	−0.0861711	+	0.996280i	$0.527463\pi$
$270$	0	0
$271$	−23.5970	−1.43341	−0.716707	−	0.697375i	$-0.754351\pi$
$271$	−23.5970	−1.43341	−0.716707	+	0.697375i	$0.754351\pi$
$272$	0	0
$273$	−7.29776	−0.441680
$274$	0	0
$275$	4.59346	0.276996
$276$	0	0
$277$	−18.7773	−1.12822	−0.564110	−	0.825700i	$-0.690781\pi$
$277$	−18.7773	−1.12822	−0.564110	+	0.825700i	$0.690781\pi$
$278$	0	0
$279$	6.50254	0.389297
$280$	0	0
$281$	27.2632	1.62639	0.813194	−	0.581993i	$-0.197727\pi$
$281$	27.2632	1.62639	0.813194	+	0.581993i	$0.197727\pi$
$282$	0	0
$283$	27.2601	1.62044	0.810221	−	0.586124i	$-0.199347\pi$
$283$	27.2601	1.62044	0.810221	+	0.586124i	$0.199347\pi$
$284$	0	0
$285$	−1.44386	−0.0855270
$286$	0	0
$287$	18.4897	1.09141
$288$	0	0
$289$	−16.9507	−0.997100
$290$	0	0
$291$	−31.1888	−1.82832
$292$	0	0
$293$	30.9996	1.81101	0.905507	−	0.424332i	$-0.139491\pi$
$293$	30.9996	1.81101	0.905507	+	0.424332i	$0.139491\pi$
$294$	0	0
$295$	−1.69486	−0.0986784
$296$	0	0
$297$	−1.97454	−0.114575
$298$	0	0
$299$	−11.4534	−0.662366
$300$	0	0
$301$	4.09858	0.236238
$302$	0	0
$303$	−27.2440	−1.56513
$304$	0	0
$305$	−0.516325	−0.0295647
$306$	0	0
$307$	18.2495	1.04156	0.520778	−	0.853692i	$-0.325642\pi$
$307$	18.2495	1.04156	0.520778	+	0.853692i	$0.325642\pi$
$308$	0	0
$309$	19.3179	1.09896
$310$	0	0
$311$	−30.8931	−1.75179	−0.875894	−	0.482503i	$-0.839728\pi$
$311$	−30.8931	−1.75179	−0.875894	+	0.482503i	$0.839728\pi$
$312$	0	0
$313$	28.7973	1.62772	0.813859	−	0.581062i	$-0.197363\pi$
$313$	28.7973	1.62772	0.813859	+	0.581062i	$0.197363\pi$
$314$	0	0
$315$	−3.62200	−0.204076
$316$	0	0
$317$	5.96040	0.334769	0.167385	−	0.985892i	$-0.446468\pi$
$317$	5.96040	0.334769	0.167385	+	0.985892i	$0.446468\pi$
$318$	0	0
$319$	−4.94518	−0.276877
$320$	0	0
$321$	3.72706	0.208024
$322$	0	0
$323$	0.222022	0.0123536
$324$	0	0
$325$	5.54545	0.307606
$326$	0	0
$327$	−38.0587	−2.10465
$328$	0	0
$329$	15.8718	0.875043
$330$	0	0
$331$	−33.4010	−1.83589	−0.917944	−	0.396711i	$-0.870151\pi$
$331$	−33.4010	−1.83589	−0.917944	+	0.396711i	$0.870151\pi$
$332$	0	0
$333$	15.2797	0.837325
$334$	0	0
$335$	1.41833	0.0774915
$336$	0	0
$337$	24.9988	1.36177	0.680885	−	0.732390i	$-0.261595\pi$
$337$	24.9988	1.36177	0.680885	+	0.732390i	$0.261595\pi$
$338$	0	0
$339$	25.5771	1.38916
$340$	0	0
$341$	−3.05563	−0.165472
$342$	0	0
$343$	18.3501	0.990815
$344$	0	0
$345$	−13.6982	−0.737485
$346$	0	0
$347$	8.35577	0.448561	0.224281	−	0.974525i	$-0.427997\pi$
$347$	8.35577	0.448561	0.224281	+	0.974525i	$0.427997\pi$
$348$	0	0
$349$	−28.3911	−1.51974	−0.759871	−	0.650074i	$-0.774738\pi$
$349$	−28.3911	−1.51974	−0.759871	+	0.650074i	$0.774738\pi$
$350$	0	0
$351$	−2.38377	−0.127236
$352$	0	0
$353$	−12.4082	−0.660423	−0.330212	−	0.943907i	$-0.607120\pi$
$353$	−12.4082	−0.660423	−0.330212	+	0.943907i	$0.607120\pi$
$354$	0	0
$355$	−1.65083	−0.0876169
$356$	0	0
$357$	1.34211	0.0710322
$358$	0	0
$359$	−12.1134	−0.639319	−0.319659	−	0.947533i	$-0.603568\pi$
$359$	−12.1134	−0.639319	−0.319659	+	0.947533i	$0.603568\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.26452	−0.118856
$364$	0	0
$365$	10.5911	0.554361
$366$	0	0
$367$	4.06064	0.211964	0.105982	−	0.994368i	$-0.466201\pi$
$367$	4.06064	0.211964	0.105982	+	0.994368i	$0.466201\pi$
$368$	0	0
$369$	−14.7399	−0.767331
$370$	0	0
$371$	25.7299	1.33583
$372$	0	0
$373$	35.4898	1.83759	0.918796	−	0.394732i	$-0.129163\pi$
$373$	35.4898	1.83759	0.918796	+	0.394732i	$0.129163\pi$
$374$	0	0
$375$	13.8516	0.715296
$376$	0	0
$377$	−5.97006	−0.307474
$378$	0	0
$379$	−13.5584	−0.696449	−0.348224	−	0.937411i	$-0.613215\pi$
$379$	−13.5584	−0.696449	−0.348224	+	0.937411i	$0.613215\pi$
$380$	0	0
$381$	42.0961	2.15665
$382$	0	0
$383$	23.8326	1.21779	0.608894	−	0.793251i	$-0.291613\pi$
$383$	23.8326	1.21779	0.608894	+	0.793251i	$0.291613\pi$
$384$	0	0
$385$	1.70203	0.0867432
$386$	0	0
$387$	−3.26737	−0.166090
$388$	0	0
$389$	−12.7567	−0.646789	−0.323395	−	0.946264i	$-0.604824\pi$
$389$	−12.7567	−0.646789	−0.323395	+	0.946264i	$0.604824\pi$
$390$	0	0
$391$	2.10636	0.106523
$392$	0	0
$393$	−6.99958	−0.353082
$394$	0	0
$395$	−3.62986	−0.182638
$396$	0	0
$397$	−30.3169	−1.52156	−0.760780	−	0.649009i	$-0.775184\pi$
$397$	−30.3169	−1.52156	−0.760780	+	0.649009i	$0.775184\pi$
$398$	0	0
$399$	6.04495	0.302626
$400$	0	0
$401$	32.6370	1.62982	0.814908	−	0.579591i	$-0.196788\pi$
$401$	32.6370	1.62982	0.814908	+	0.579591i	$0.196788\pi$
$402$	0	0
$403$	−3.68891	−0.183758
$404$	0	0
$405$	−6.92152	−0.343933
$406$	0	0
$407$	−7.18015	−0.355907
$408$	0	0
$409$	−23.1587	−1.14512	−0.572562	−	0.819862i	$-0.694050\pi$
$409$	−23.1587	−1.14512	−0.572562	+	0.819862i	$0.694050\pi$
$410$	0	0
$411$	19.6474	0.969135
$412$	0	0
$413$	7.09578	0.349160
$414$	0	0
$415$	1.87318	0.0919506
$416$	0	0
$417$	32.9319	1.61268
$418$	0	0
$419$	14.2213	0.694755	0.347377	−	0.937725i	$-0.387072\pi$
$419$	14.2213	0.694755	0.347377	+	0.937725i	$0.387072\pi$
$420$	0	0
$421$	15.9605	0.777866	0.388933	−	0.921266i	$-0.372844\pi$
$421$	15.9605	0.777866	0.388933	+	0.921266i	$0.372844\pi$
$422$	0	0
$423$	−12.6530	−0.615209
$424$	0	0
$425$	−1.01985	−0.0494701
$426$	0	0
$427$	2.16167	0.104611
$428$	0	0
$429$	−2.73384	−0.131991
$430$	0	0
$431$	−24.5100	−1.18061	−0.590304	−	0.807181i	$-0.700992\pi$
$431$	−24.5100	−1.18061	−0.590304	+	0.807181i	$0.700992\pi$
$432$	0	0
$433$	−19.1839	−0.921921	−0.460961	−	0.887421i	$-0.652495\pi$
$433$	−19.1839	−0.921921	−0.460961	+	0.887421i	$0.652495\pi$
$434$	0	0
$435$	−7.14016	−0.342344
$436$	0	0
$437$	9.48717	0.453833
$438$	0	0
$439$	−13.7961	−0.658450	−0.329225	−	0.944251i	$-0.606788\pi$
$439$	−13.7961	−0.658450	−0.329225	+	0.944251i	$0.606788\pi$
$440$	0	0
$441$	0.267676	0.0127465
$442$	0	0
$443$	−14.1259	−0.671140	−0.335570	−	0.942015i	$-0.608929\pi$
$443$	−14.1259	−0.671140	−0.335570	+	0.942015i	$0.608929\pi$
$444$	0	0
$445$	−3.53318	−0.167489
$446$	0	0
$447$	41.8292	1.97846
$448$	0	0
$449$	−21.0016	−0.991129	−0.495565	−	0.868571i	$-0.665039\pi$
$449$	−21.0016	−0.991129	−0.495565	+	0.868571i	$0.665039\pi$
$450$	0	0
$451$	6.92650	0.326156
$452$	0	0
$453$	−2.06975	−0.0972455
$454$	0	0
$455$	2.05477	0.0963290
$456$	0	0
$457$	−22.4621	−1.05073	−0.525367	−	0.850876i	$-0.676072\pi$
$457$	−22.4621	−1.05073	−0.525367	+	0.850876i	$0.676072\pi$
$458$	0	0
$459$	0.438393	0.0204624
$460$	0	0
$461$	−31.1213	−1.44946	−0.724732	−	0.689031i	$-0.758036\pi$
$461$	−31.1213	−1.44946	−0.724732	+	0.689031i	$0.758036\pi$
$462$	0	0
$463$	−31.6932	−1.47291	−0.736455	−	0.676487i	$-0.763502\pi$
$463$	−31.6932	−1.47291	−0.736455	+	0.676487i	$0.763502\pi$
$464$	0	0
$465$	−4.41191	−0.204597
$466$	0	0
$467$	1.75474	0.0811995	0.0405997	−	0.999175i	$-0.487073\pi$
$467$	1.75474	0.0811995	0.0405997	+	0.999175i	$0.487073\pi$
$468$	0	0
$469$	−5.93804	−0.274193
$470$	0	0
$471$	5.42174	0.249821
$472$	0	0
$473$	1.53538	0.0705970
$474$	0	0
$475$	−4.59346	−0.210763
$476$	0	0
$477$	−20.5118	−0.939173
$478$	0	0
$479$	11.1294	0.508515	0.254257	−	0.967137i	$-0.418169\pi$
$479$	11.1294	0.508515	0.254257	+	0.967137i	$0.418169\pi$
$480$	0	0
$481$	−8.66823	−0.395237
$482$	0	0
$483$	57.3495	2.60949
$484$	0	0
$485$	8.78158	0.398751
$486$	0	0
$487$	−3.27007	−0.148181	−0.0740905	−	0.997252i	$-0.523605\pi$
$487$	−3.27007	−0.148181	−0.0740905	+	0.997252i	$0.523605\pi$
$488$	0	0
$489$	32.1531	1.45401
$490$	0	0
$491$	−29.2686	−1.32087	−0.660437	−	0.750882i	$-0.729629\pi$
$491$	−29.2686	−1.32087	−0.660437	+	0.750882i	$0.729629\pi$
$492$	0	0
$493$	1.09794	0.0494487
$494$	0	0
$495$	−1.35685	−0.0609859
$496$	0	0
$497$	6.91145	0.310021
$498$	0	0
$499$	1.04828	0.0469275	0.0234638	−	0.999725i	$-0.492531\pi$
$499$	1.04828	0.0469275	0.0234638	+	0.999725i	$0.492531\pi$
$500$	0	0
$501$	20.9481	0.935891
$502$	0	0
$503$	−28.4724	−1.26952	−0.634761	−	0.772709i	$-0.718901\pi$
$503$	−28.4724	−1.26952	−0.634761	+	0.772709i	$0.718901\pi$
$504$	0	0
$505$	7.67086	0.341349
$506$	0	0
$507$	26.1383	1.16084
$508$	0	0
$509$	39.6801	1.75879	0.879396	−	0.476091i	$-0.157947\pi$
$509$	39.6801	1.75879	0.879396	+	0.476091i	$0.157947\pi$
$510$	0	0
$511$	−44.3410	−1.96153
$512$	0	0
$513$	1.97454	0.0871782
$514$	0	0
$515$	−5.43918	−0.239679
$516$	0	0
$517$	5.94581	0.261496
$518$	0	0
$519$	−12.2143	−0.536147
$520$	0	0
$521$	−15.3168	−0.671043	−0.335522	−	0.942033i	$-0.608913\pi$
$521$	−15.3168	−0.671043	−0.335522	+	0.942033i	$0.608913\pi$
$522$	0	0
$523$	4.17850	0.182713	0.0913566	−	0.995818i	$-0.470880\pi$
$523$	4.17850	0.182713	0.0913566	+	0.995818i	$0.470880\pi$
$524$	0	0
$525$	−27.7672	−1.21186
$526$	0	0
$527$	0.678418	0.0295524
$528$	0	0
$529$	67.0064	2.91332
$530$	0	0
$531$	−5.65674	−0.245481
$532$	0	0
$533$	8.36201	0.362199
$534$	0	0
$535$	−1.04940	−0.0453694
$536$	0	0
$537$	2.96018	0.127741
$538$	0	0
$539$	−0.125785	−0.00541792
$540$	0	0
$541$	−14.6201	−0.628565	−0.314283	−	0.949329i	$-0.601764\pi$
$541$	−14.6201	−0.628565	−0.314283	+	0.949329i	$0.601764\pi$
$542$	0	0
$543$	24.3716	1.04589
$544$	0	0
$545$	10.7159	0.459017
$546$	0	0
$547$	34.7220	1.48461	0.742303	−	0.670064i	$-0.233733\pi$
$547$	34.7220	1.48461	0.742303	+	0.670064i	$0.233733\pi$
$548$	0	0
$549$	−1.72328	−0.0735478
$550$	0	0
$551$	4.94518	0.210672
$552$	0	0
$553$	15.1969	0.646240
$554$	0	0
$555$	−10.3672	−0.440061
$556$	0	0
$557$	−7.07603	−0.299821	−0.149910	−	0.988700i	$-0.547899\pi$
$557$	−7.07603	−0.299821	−0.149910	+	0.988700i	$0.547899\pi$
$558$	0	0
$559$	1.85359	0.0783985
$560$	0	0
$561$	0.502774	0.0212271
$562$	0	0
$563$	32.5712	1.37271	0.686357	−	0.727265i	$-0.259209\pi$
$563$	32.5712	1.37271	0.686357	+	0.727265i	$0.259209\pi$
$564$	0	0
$565$	−7.20152	−0.302970
$566$	0	0
$567$	28.9780	1.21696
$568$	0	0
$569$	32.6457	1.36858	0.684290	−	0.729210i	$-0.260112\pi$
$569$	32.6457	1.36858	0.684290	+	0.729210i	$0.260112\pi$
$570$	0	0
$571$	35.8059	1.49843	0.749216	−	0.662326i	$-0.230431\pi$
$571$	35.8059	1.49843	0.749216	+	0.662326i	$0.230431\pi$
$572$	0	0
$573$	34.6565	1.44780
$574$	0	0
$575$	−43.5790	−1.81737
$576$	0	0
$577$	15.7218	0.654509	0.327254	−	0.944936i	$-0.393877\pi$
$577$	15.7218	0.654509	0.327254	+	0.944936i	$0.393877\pi$
$578$	0	0
$579$	41.4480	1.72252
$580$	0	0
$581$	−7.84233	−0.325355
$582$	0	0
$583$	9.63879	0.399198
$584$	0	0
$585$	−1.63806	−0.0677253
$586$	0	0
$587$	−24.4810	−1.01044	−0.505220	−	0.862991i	$-0.668588\pi$
$587$	−24.4810	−1.01044	−0.505220	+	0.862991i	$0.668588\pi$
$588$	0	0
$589$	3.05563	0.125905
$590$	0	0
$591$	1.30475	0.0536703
$592$	0	0
$593$	−21.2995	−0.874666	−0.437333	−	0.899300i	$-0.644077\pi$
$593$	−21.2995	−0.874666	−0.437333	+	0.899300i	$0.644077\pi$
$594$	0	0
$595$	−0.377887	−0.0154919
$596$	0	0
$597$	−23.0253	−0.942361
$598$	0	0
$599$	8.90839	0.363987	0.181994	−	0.983300i	$-0.441745\pi$
$599$	8.90839	0.363987	0.181994	+	0.983300i	$0.441745\pi$
$600$	0	0
$601$	−8.36794	−0.341335	−0.170668	−	0.985329i	$-0.554592\pi$
$601$	−8.36794	−0.341335	−0.170668	+	0.985329i	$0.554592\pi$
$602$	0	0
$603$	4.73379	0.192775
$604$	0	0
$605$	0.637602	0.0259222
$606$	0	0
$607$	−43.2321	−1.75474	−0.877368	−	0.479819i	$-0.840702\pi$
$607$	−43.2321	−1.75474	−0.877368	+	0.479819i	$0.840702\pi$
$608$	0	0
$609$	29.8933	1.21134
$610$	0	0
$611$	7.17807	0.290394
$612$	0	0
$613$	−12.4893	−0.504438	−0.252219	−	0.967670i	$-0.581160\pi$
$613$	−12.4893	−0.504438	−0.252219	+	0.967670i	$0.581160\pi$
$614$	0	0
$615$	10.0009	0.403276
$616$	0	0
$617$	−18.4079	−0.741073	−0.370536	−	0.928818i	$-0.620826\pi$
$617$	−18.4079	−0.741073	−0.370536	+	0.928818i	$0.620826\pi$
$618$	0	0
$619$	7.30900	0.293773	0.146887	−	0.989153i	$-0.453075\pi$
$619$	7.30900	0.293773	0.146887	+	0.989153i	$0.453075\pi$
$620$	0	0
$621$	18.7328	0.751722
$622$	0	0
$623$	14.7922	0.592637
$624$	0	0
$625$	19.0672	0.762689
$626$	0	0
$627$	2.26452	0.0904362
$628$	0	0
$629$	1.59415	0.0635631
$630$	0	0
$631$	11.8397	0.471329	0.235665	−	0.971834i	$-0.424273\pi$
$631$	11.8397	0.471329	0.235665	+	0.971834i	$0.424273\pi$
$632$	0	0
$633$	−16.4367	−0.653301
$634$	0	0
$635$	−11.8526	−0.470357
$636$	0	0
$637$	−0.151853	−0.00601664
$638$	0	0
$639$	−5.50979	−0.217964
$640$	0	0
$641$	−36.5974	−1.44551	−0.722756	−	0.691104i	$-0.757125\pi$
$641$	−36.5974	−1.44551	−0.722756	+	0.691104i	$0.757125\pi$
$642$	0	0
$643$	14.0452	0.553888	0.276944	−	0.960886i	$-0.410678\pi$
$643$	14.0452	0.553888	0.276944	+	0.960886i	$0.410678\pi$
$644$	0	0
$645$	2.21688	0.0872896
$646$	0	0
$647$	43.5266	1.71121	0.855604	−	0.517631i	$-0.173186\pi$
$647$	43.5266	1.71121	0.855604	+	0.517631i	$0.173186\pi$
$648$	0	0
$649$	2.65817	0.104342
$650$	0	0
$651$	18.4711	0.723941
$652$	0	0
$653$	44.0328	1.72314	0.861568	−	0.507642i	$-0.169483\pi$
$653$	44.0328	1.72314	0.861568	+	0.507642i	$0.169483\pi$
$654$	0	0
$655$	1.97081	0.0770060
$656$	0	0
$657$	35.3486	1.37908
$658$	0	0
$659$	3.86784	0.150670	0.0753349	−	0.997158i	$-0.475997\pi$
$659$	3.86784	0.150670	0.0753349	+	0.997158i	$0.475997\pi$
$660$	0	0
$661$	7.96999	0.309997	0.154998	−	0.987915i	$-0.450463\pi$
$661$	7.96999	0.309997	0.154998	+	0.987915i	$0.450463\pi$
$662$	0	0
$663$	0.606973	0.0235729
$664$	0	0
$665$	−1.70203	−0.0660017
$666$	0	0
$667$	46.9158	1.81659
$668$	0	0
$669$	36.8600	1.42509
$670$	0	0
$671$	0.809792	0.0312617
$672$	0	0
$673$	21.7872	0.839835	0.419917	−	0.907562i	$-0.362059\pi$
$673$	21.7872	0.839835	0.419917	+	0.907562i	$0.362059\pi$
$674$	0	0
$675$	−9.06999	−0.349104
$676$	0	0
$677$	33.2004	1.27599	0.637997	−	0.770039i	$-0.279763\pi$
$677$	33.2004	1.27599	0.637997	+	0.770039i	$0.279763\pi$
$678$	0	0
$679$	−36.7654	−1.41093
$680$	0	0
$681$	−47.3873	−1.81588
$682$	0	0
$683$	34.9595	1.33769	0.668843	−	0.743403i	$-0.266790\pi$
$683$	34.9595	1.33769	0.668843	+	0.743403i	$0.266790\pi$
$684$	0	0
$685$	−5.53196	−0.211365
$686$	0	0
$687$	24.4757	0.933805
$688$	0	0
$689$	11.6364	0.443312
$690$	0	0
$691$	30.4626	1.15885	0.579426	−	0.815025i	$-0.303277\pi$
$691$	30.4626	1.15885	0.579426	+	0.815025i	$0.303277\pi$
$692$	0	0
$693$	5.68066	0.215790
$694$	0	0
$695$	−9.27236	−0.351721
$696$	0	0
$697$	−1.53784	−0.0582497
$698$	0	0
$699$	57.5579	2.17704
$700$	0	0
$701$	10.3329	0.390269	0.195135	−	0.980776i	$-0.437486\pi$
$701$	10.3329	0.390269	0.195135	+	0.980776i	$0.437486\pi$
$702$	0	0
$703$	7.18015	0.270805
$704$	0	0
$705$	8.58493	0.323327
$706$	0	0
$707$	−32.1152	−1.20782
$708$	0	0
$709$	−45.4499	−1.70691	−0.853454	−	0.521169i	$-0.825496\pi$
$709$	−45.4499	−1.70691	−0.853454	+	0.521169i	$0.825496\pi$
$710$	0	0
$711$	−12.1150	−0.454347
$712$	0	0
$713$	28.9893	1.08566
$714$	0	0
$715$	0.769744	0.0287868
$716$	0	0
$717$	0.797651	0.0297888
$718$	0	0
$719$	7.51239	0.280165	0.140082	−	0.990140i	$-0.455263\pi$
$719$	7.51239	0.280165	0.140082	+	0.990140i	$0.455263\pi$
$720$	0	0
$721$	22.7719	0.848071
$722$	0	0
$723$	35.5149	1.32081
$724$	0	0
$725$	−22.7155	−0.843632
$726$	0	0
$727$	−4.79898	−0.177984	−0.0889922	−	0.996032i	$-0.528365\pi$
$727$	−4.79898	−0.177984	−0.0889922	+	0.996032i	$0.528365\pi$
$728$	0	0
$729$	−9.68700	−0.358778
$730$	0	0
$731$	−0.340889	−0.0126082
$732$	0	0
$733$	−8.55077	−0.315830	−0.157915	−	0.987453i	$-0.550477\pi$
$733$	−8.55077	−0.315830	−0.157915	+	0.987453i	$0.550477\pi$
$734$	0	0
$735$	−0.181616	−0.00669899
$736$	0	0
$737$	−2.22447	−0.0819395
$738$	0	0
$739$	−38.8820	−1.43030	−0.715149	−	0.698972i	$-0.753641\pi$
$739$	−38.8820	−1.43030	−0.715149	+	0.698972i	$0.753641\pi$
$740$	0	0
$741$	2.73384	0.100430
$742$	0	0
$743$	−28.4350	−1.04318	−0.521589	−	0.853197i	$-0.674660\pi$
$743$	−28.4350	−1.04318	−0.521589	+	0.853197i	$0.674660\pi$
$744$	0	0
$745$	−11.7775	−0.431494
$746$	0	0
$747$	6.25189	0.228745
$748$	0	0
$749$	4.39346	0.160534
$750$	0	0
$751$	−18.2732	−0.666799	−0.333400	−	0.942786i	$-0.608196\pi$
$751$	−18.2732	−0.666799	−0.333400	+	0.942786i	$0.608196\pi$
$752$	0	0
$753$	2.63436	0.0960015
$754$	0	0
$755$	0.582763	0.0212089
$756$	0	0
$757$	1.10637	0.0402117	0.0201058	−	0.999798i	$-0.493600\pi$
$757$	1.10637	0.0402117	0.0201058	+	0.999798i	$0.493600\pi$
$758$	0	0
$759$	21.4839	0.779816
$760$	0	0
$761$	1.10110	0.0399147	0.0199573	−	0.999801i	$-0.493647\pi$
$761$	1.10110	0.0399147	0.0199573	+	0.999801i	$0.493647\pi$
$762$	0	0
$763$	−44.8636	−1.62417
$764$	0	0
$765$	0.301251	0.0108918
$766$	0	0
$767$	3.20908	0.115873
$768$	0	0
$769$	30.2914	1.09234	0.546169	−	0.837675i	$-0.316086\pi$
$769$	30.2914	1.09234	0.546169	+	0.837675i	$0.316086\pi$
$770$	0	0
$771$	34.0833	1.22748
$772$	0	0
$773$	−43.9677	−1.58141	−0.790704	−	0.612199i	$-0.790285\pi$
$773$	−43.9677	−1.58141	−0.790704	+	0.612199i	$0.790285\pi$
$774$	0	0
$775$	−14.0359	−0.504186
$776$	0	0
$777$	43.4036	1.55710
$778$	0	0
$779$	−6.92650	−0.248168
$780$	0	0
$781$	2.58912	0.0926461
$782$	0	0
$783$	9.76447	0.348954
$784$	0	0
$785$	−1.52655	−0.0544851
$786$	0	0
$787$	37.8350	1.34867	0.674337	−	0.738424i	$-0.264430\pi$
$787$	37.8350	1.34867	0.674337	+	0.738424i	$0.264430\pi$
$788$	0	0
$789$	−46.7870	−1.66566
$790$	0	0
$791$	30.1502	1.07202
$792$	0	0
$793$	0.977621	0.0347163
$794$	0	0
$795$	13.9171	0.493588
$796$	0	0
$797$	11.2902	0.399918	0.199959	−	0.979804i	$-0.435919\pi$
$797$	11.2902	0.399918	0.199959	+	0.979804i	$0.435919\pi$
$798$	0	0
$799$	−1.32010	−0.0467018
$800$	0	0
$801$	−11.7923	−0.416661
$802$	0	0
$803$	−16.6108	−0.586181
$804$	0	0
$805$	−16.1474	−0.569121
$806$	0	0
$807$	6.40095	0.225324
$808$	0	0
$809$	17.8104	0.626180	0.313090	−	0.949723i	$-0.398636\pi$
$809$	17.8104	0.626180	0.313090	+	0.949723i	$0.398636\pi$
$810$	0	0
$811$	−28.1638	−0.988965	−0.494482	−	0.869188i	$-0.664642\pi$
$811$	−28.1638	−0.988965	−0.494482	+	0.869188i	$0.664642\pi$
$812$	0	0
$813$	53.4358	1.87408
$814$	0	0
$815$	−9.05307	−0.317115
$816$	0	0
$817$	−1.53538	−0.0537162
$818$	0	0
$819$	6.85797	0.239637
$820$	0	0
$821$	−44.2644	−1.54484	−0.772418	−	0.635114i	$-0.780953\pi$
$821$	−44.2644	−1.54484	−0.772418	+	0.635114i	$0.780953\pi$
$822$	0	0
$823$	22.4258	0.781716	0.390858	−	0.920451i	$-0.372178\pi$
$823$	22.4258	0.781716	0.390858	+	0.920451i	$0.372178\pi$
$824$	0	0
$825$	−10.4020	−0.362151
$826$	0	0
$827$	−19.7303	−0.686090	−0.343045	−	0.939319i	$-0.611458\pi$
$827$	−19.7303	−0.686090	−0.343045	+	0.939319i	$0.611458\pi$
$828$	0	0
$829$	5.70836	0.198259	0.0991297	−	0.995075i	$-0.468394\pi$
$829$	5.70836	0.198259	0.0991297	+	0.995075i	$0.468394\pi$
$830$	0	0
$831$	42.5216	1.47506
$832$	0	0
$833$	0.0279270	0.000967612 0
$834$	0	0
$835$	−5.89817	−0.204115
$836$	0	0
$837$	6.03348	0.208547
$838$	0	0
$839$	0.0666011	0.00229932	0.00114966	−	0.999999i	$-0.499634\pi$
$839$	0.0666011	0.00229932	0.00114966	+	0.999999i	$0.499634\pi$
$840$	0	0
$841$	−4.54521	−0.156731
$842$	0	0
$843$	−61.7381	−2.12637
$844$	0	0
$845$	−7.35955	−0.253176
$846$	0	0
$847$	−2.66942	−0.0917222
$848$	0	0
$849$	−61.7310	−2.11860
$850$	0	0
$851$	68.1193	2.33510
$852$	0	0
$853$	−14.8615	−0.508848	−0.254424	−	0.967093i	$-0.581886\pi$
$853$	−14.8615	−0.508848	−0.254424	+	0.967093i	$0.581886\pi$
$854$	0	0
$855$	1.35685	0.0464033
$856$	0	0
$857$	12.5169	0.427569	0.213785	−	0.976881i	$-0.431421\pi$
$857$	12.5169	0.427569	0.213785	+	0.976881i	$0.431421\pi$
$858$	0	0
$859$	21.2738	0.725854	0.362927	−	0.931818i	$-0.381777\pi$
$859$	21.2738	0.725854	0.362927	+	0.931818i	$0.381777\pi$
$860$	0	0
$861$	−41.8703	−1.42694
$862$	0	0
$863$	34.4402	1.17236	0.586179	−	0.810181i	$-0.300631\pi$
$863$	34.4402	1.17236	0.586179	+	0.810181i	$0.300631\pi$
$864$	0	0
$865$	3.43907	0.116932
$866$	0	0
$867$	38.3852	1.30363
$868$	0	0
$869$	5.69298	0.193121
$870$	0	0
$871$	−2.68549	−0.0909944
$872$	0	0
$873$	29.3093	0.991969
$874$	0	0
$875$	16.3283	0.551998
$876$	0	0
$877$	−0.910260	−0.0307373	−0.0153687	−	0.999882i	$-0.504892\pi$
$877$	−0.910260	−0.0307373	−0.0153687	+	0.999882i	$0.504892\pi$
$878$	0	0
$879$	−70.1991	−2.36776
$880$	0	0
$881$	33.7502	1.13707	0.568537	−	0.822658i	$-0.307510\pi$
$881$	33.7502	1.13707	0.568537	+	0.822658i	$0.307510\pi$
$882$	0	0
$883$	−39.8800	−1.34207	−0.671035	−	0.741426i	$-0.734150\pi$
$883$	−39.8800	−1.34207	−0.671035	+	0.741426i	$0.734150\pi$
$884$	0	0
$885$	3.83804	0.129014
$886$	0	0
$887$	16.1868	0.543500	0.271750	−	0.962368i	$-0.412398\pi$
$887$	16.1868	0.543500	0.271750	+	0.962368i	$0.412398\pi$
$888$	0	0
$889$	49.6228	1.66430
$890$	0	0
$891$	10.8556	0.363675
$892$	0	0
$893$	−5.94581	−0.198969
$894$	0	0
$895$	−0.833474	−0.0278599
$896$	0	0
$897$	25.9364	0.865991
$898$	0	0
$899$	15.1106	0.503968
$900$	0	0
$901$	−2.14003	−0.0712946
$902$	0	0
$903$	−9.28131	−0.308863
$904$	0	0
$905$	−6.86211	−0.228104
$906$	0	0
$907$	14.3005	0.474839	0.237419	−	0.971407i	$-0.423698\pi$
$907$	14.3005	0.474839	0.237419	+	0.971407i	$0.423698\pi$
$908$	0	0
$909$	25.6022	0.849171
$910$	0	0
$911$	8.50108	0.281653	0.140827	−	0.990034i	$-0.455024\pi$
$911$	8.50108	0.281653	0.140827	+	0.990034i	$0.455024\pi$
$912$	0	0
$913$	−2.93784	−0.0972285
$914$	0	0
$915$	1.16923	0.0386535
$916$	0	0
$917$	−8.25111	−0.272476
$918$	0	0
$919$	26.0156	0.858177	0.429088	−	0.903263i	$-0.358835\pi$
$919$	26.0156	0.858177	0.429088	+	0.903263i	$0.358835\pi$
$920$	0	0
$921$	−41.3265	−1.36175
$922$	0	0
$923$	3.12571	0.102884
$924$	0	0
$925$	−32.9818	−1.08443
$926$	0	0
$927$	−18.1537	−0.596247
$928$	0	0
$929$	5.99773	0.196779	0.0983896	−	0.995148i	$-0.468631\pi$
$929$	5.99773	0.196779	0.0983896	+	0.995148i	$0.468631\pi$
$930$	0	0
$931$	0.125785	0.00412242
$932$	0	0
$933$	69.9581	2.29033
$934$	0	0
$935$	−0.141562	−0.00462957
$936$	0	0
$937$	−50.5847	−1.65253	−0.826265	−	0.563282i	$-0.809539\pi$
$937$	−50.5847	−1.65253	−0.826265	+	0.563282i	$0.809539\pi$
$938$	0	0
$939$	−65.2120	−2.12811
$940$	0	0
$941$	−21.8640	−0.712747	−0.356373	−	0.934344i	$-0.615987\pi$
$941$	−21.8640	−0.712747	−0.356373	+	0.934344i	$0.615987\pi$
$942$	0	0
$943$	−65.7129	−2.13991
$944$	0	0
$945$	−3.36072	−0.109324
$946$	0	0
$947$	33.1722	1.07795	0.538976	−	0.842321i	$-0.318812\pi$
$947$	33.1722	1.07795	0.538976	+	0.842321i	$0.318812\pi$
$948$	0	0
$949$	−20.0533	−0.650958
$950$	0	0
$951$	−13.4974	−0.437685
$952$	0	0
$953$	22.4494	0.727209	0.363604	−	0.931553i	$-0.381546\pi$
$953$	22.4494	0.727209	0.363604	+	0.931553i	$0.381546\pi$
$954$	0	0
$955$	−9.75794	−0.315760
$956$	0	0
$957$	11.1985	0.361995
$958$	0	0
$959$	23.1604	0.747887
$960$	0	0
$961$	−21.6631	−0.698810
$962$	0	0
$963$	−3.50246	−0.112865
$964$	0	0
$965$	−11.6702	−0.375676
$966$	0	0
$967$	0.790013	0.0254051	0.0127026	−	0.999919i	$-0.495957\pi$
$967$	0.790013	0.0254051	0.0127026	+	0.999919i	$0.495957\pi$
$968$	0	0
$969$	−0.502774	−0.0161514
$970$	0	0
$971$	−51.6970	−1.65904	−0.829518	−	0.558479i	$-0.811385\pi$
$971$	−51.6970	−1.65904	−0.829518	+	0.558479i	$0.811385\pi$
$972$	0	0
$973$	38.8201	1.24452
$974$	0	0
$975$	−12.5578	−0.402171
$976$	0	0
$977$	21.4536	0.686360	0.343180	−	0.939270i	$-0.388496\pi$
$977$	21.4536	0.686360	0.343180	+	0.939270i	$0.388496\pi$
$978$	0	0
$979$	5.54136	0.177103
$980$	0	0
$981$	35.7651	1.14189
$982$	0	0
$983$	42.9916	1.37122	0.685610	−	0.727969i	$-0.259536\pi$
$983$	42.9916	1.37122	0.685610	+	0.727969i	$0.259536\pi$
$984$	0	0
$985$	−0.367368	−0.0117053
$986$	0	0
$987$	−35.9421	−1.14405
$988$	0	0
$989$	−14.5664	−0.463186
$990$	0	0
$991$	−57.7319	−1.83392	−0.916958	−	0.398985i	$-0.869363\pi$
$991$	−57.7319	−1.83392	−0.916958	+	0.398985i	$0.869363\pi$
$992$	0	0
$993$	75.6373	2.40028
$994$	0	0
$995$	6.48303	0.205526
$996$	0	0
$997$	−30.3487	−0.961152	−0.480576	−	0.876953i	$-0.659572\pi$
$997$	−30.3487	−0.961152	−0.480576	+	0.876953i	$0.659572\pi$
$998$	0	0
$999$	14.1775	0.448557

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.t.1.1		5
4.3	odd	2		209.2.a.c.1.3	✓	5
12.11	even	2		1881.2.a.k.1.3		5
20.19	odd	2		5225.2.a.h.1.3		5
44.43	even	2		2299.2.a.n.1.3		5
76.75	even	2		3971.2.a.h.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.3	✓	5	4.3	odd	2
1881.2.a.k.1.3		5	12.11	even	2
2299.2.a.n.1.3		5	44.43	even	2
3344.2.a.t.1.1		5	1.1	even	1	trivial
3971.2.a.h.1.3		5	76.75	even	2
5225.2.a.h.1.3		5	20.19	odd	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26452 q^{3} +0.637602 q^{5} -2.66942 q^{7} +2.12805 q^{9} -1.00000 q^{11} -1.20725 q^{13} -1.44386 q^{15} +0.222022 q^{17} +1.00000 q^{19} +6.04495 q^{21} +9.48717 q^{23} -4.59346 q^{25} +1.97454 q^{27} +4.94518 q^{29} +3.05563 q^{31} +2.26452 q^{33} -1.70203 q^{35} +7.18015 q^{37} +2.73384 q^{39} -6.92650 q^{41} -1.53538 q^{43} +1.35685 q^{45} -5.94581 q^{47} +0.125785 q^{49} -0.502774 q^{51} -9.63879 q^{53} -0.637602 q^{55} -2.26452 q^{57} -2.65817 q^{59} -0.809792 q^{61} -5.68066 q^{63} -0.769744 q^{65} +2.22447 q^{67} -21.4839 q^{69} -2.58912 q^{71} +16.6108 q^{73} +10.4020 q^{75} +2.66942 q^{77} -5.69298 q^{79} -10.8556 q^{81} +2.93784 q^{83} +0.141562 q^{85} -11.1985 q^{87} -5.54136 q^{89} +3.22265 q^{91} -6.91954 q^{93} +0.637602 q^{95} +13.7728 q^{97} -2.12805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

Introduction

L-functions

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