LMFDB - Embedded newform 3328.2.a.bp.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,8,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.10323968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 25x^{2} - 8 $ x^6 - 10x^4 + 25x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.47817$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47817 q^{3} +4.14134 q^{5} +1.96435 q^{7} +3.14134 q^{9} -3.34225 q^{11} +1.00000 q^{13} -10.2629 q^{15} -4.86799 q^{17} +7.27095 q^{19} -4.86799 q^{21} -4.95634 q^{23} +12.1507 q^{25} -0.350255 q^{27} +2.00000 q^{29} -4.44252 q^{31} +8.28267 q^{33} +8.13503 q^{35} +2.58532 q^{37} -2.47817 q^{39} +11.7360 q^{41} -0.350255 q^{43} +13.0093 q^{45} +4.79278 q^{47} -3.14134 q^{49} +12.0637 q^{51} +13.7360 q^{53} -13.8414 q^{55} -18.0187 q^{57} +2.64174 q^{59} -1.45331 q^{61} +6.17068 q^{63} +4.14134 q^{65} -5.54279 q^{67} +12.2827 q^{69} -0.936701 q^{71} +0.829359 q^{73} -30.1114 q^{75} -6.56534 q^{77} -11.6408 q^{79} -8.55602 q^{81} +8.99911 q^{83} -20.1600 q^{85} -4.95634 q^{87} -6.28267 q^{89} +1.96435 q^{91} +11.0093 q^{93} +30.1114 q^{95} +14.5653 q^{97} -10.4991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{5} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 4 q^{21} + 14 q^{25} + 12 q^{29} + 16 q^{33} + 24 q^{37} + 20 q^{41} + 36 q^{45} - 2 q^{49} + 32 q^{53} - 24 q^{57} + 8 q^{61} + 8 q^{65} + 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) $ 6 * q + 8 * q^5 + 2 * q^9 + 6 * q^13 - 4 * q^17 - 4 * q^21 + 14 * q^25 + 12 * q^29 + 16 * q^33 + 24 * q^37 + 20 * q^41 + 36 * q^45 - 2 * q^49 + 32 * q^53 - 24 * q^57 + 8 * q^61 + 8 * q^65 + 40 * q^69 - 12 * q^73 + 28 * q^77 - 26 * q^81 - 20 * q^85 - 4 * q^89 + 24 * q^93 + 20 * 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 $

$2$	0	0
$2$	0	0
$3$	−2.47817	−1.43077	−0.715387	−	0.698729i	$-0.753749\pi$
$3$	−2.47817	−1.43077	−0.715387	+	0.698729i	$0.753749\pi$
$4$	0	0
$5$	4.14134	1.85206	0.926031	−	0.377448i	$-0.123198\pi$
$5$	4.14134	1.85206	0.926031	+	0.377448i	$0.123198\pi$
$6$	0	0
$7$	1.96435	0.742454	0.371227	−	0.928542i	$-0.378937\pi$
$7$	1.96435	0.742454	0.371227	+	0.928542i	$0.378937\pi$
$8$	0	0
$9$	3.14134	1.04711
$10$	0	0
$11$	−3.34225	−1.00773	−0.503863	−	0.863783i	$-0.668088\pi$
$11$	−3.34225	−1.00773	−0.503863	+	0.863783i	$0.668088\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−10.2629	−2.64988
$16$	0	0
$17$	−4.86799	−1.18066	−0.590331	−	0.807161i	$-0.701003\pi$
$17$	−4.86799	−1.18066	−0.590331	+	0.807161i	$0.701003\pi$
$18$	0	0
$19$	7.27095	1.66807	0.834035	−	0.551712i	$-0.186025\pi$
$19$	7.27095	1.66807	0.834035	+	0.551712i	$0.186025\pi$
$20$	0	0
$21$	−4.86799	−1.06228
$22$	0	0
$23$	−4.95634	−1.03347	−0.516735	−	0.856146i	$-0.672853\pi$
$23$	−4.95634	−1.03347	−0.516735	+	0.856146i	$0.672853\pi$
$24$	0	0
$25$	12.1507	2.43013
$26$	0	0
$27$	−0.350255	−0.0674066
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.44252	−0.797900	−0.398950	−	0.916973i	$-0.630625\pi$
$31$	−4.44252	−0.797900	−0.398950	+	0.916973i	$0.630625\pi$
$32$	0	0
$33$	8.28267	1.44183
$34$	0	0
$35$	8.13503	1.37507
$36$	0	0
$37$	2.58532	0.425024	0.212512	−	0.977158i	$-0.431836\pi$
$37$	2.58532	0.425024	0.212512	+	0.977158i	$0.431836\pi$
$38$	0	0
$39$	−2.47817	−0.396825
$40$	0	0
$41$	11.7360	1.83285	0.916426	−	0.400203i	$-0.131060\pi$
$41$	11.7360	1.83285	0.916426	+	0.400203i	$0.131060\pi$
$42$	0	0
$43$	−0.350255	−0.0534134	−0.0267067	−	0.999643i	$-0.508502\pi$
$43$	−0.350255	−0.0534134	−0.0267067	+	0.999643i	$0.508502\pi$
$44$	0	0
$45$	13.0093	1.93932
$46$	0	0
$47$	4.79278	0.699098	0.349549	−	0.936918i	$-0.386335\pi$
$47$	4.79278	0.699098	0.349549	+	0.936918i	$0.386335\pi$
$48$	0	0
$49$	−3.14134	−0.448762
$50$	0	0
$51$	12.0637	1.68926
$52$	0	0
$53$	13.7360	1.88678	0.943391	−	0.331682i	$-0.107616\pi$
$53$	13.7360	1.88678	0.943391	+	0.331682i	$0.107616\pi$
$54$	0	0
$55$	−13.8414	−1.86637
$56$	0	0
$57$	−18.0187	−2.38663
$58$	0	0
$59$	2.64174	0.343925	0.171963	−	0.985103i	$-0.444989\pi$
$59$	2.64174	0.343925	0.171963	+	0.985103i	$0.444989\pi$
$60$	0	0
$61$	−1.45331	−0.186078	−0.0930388	−	0.995662i	$-0.529658\pi$
$61$	−1.45331	−0.186078	−0.0930388	+	0.995662i	$0.529658\pi$
$62$	0	0
$63$	6.17068	0.777432
$64$	0	0
$65$	4.14134	0.513670
$66$	0	0
$67$	−5.54279	−0.677160	−0.338580	−	0.940938i	$-0.609946\pi$
$67$	−5.54279	−0.677160	−0.338580	+	0.940938i	$0.609946\pi$
$68$	0	0
$69$	12.2827	1.47866
$70$	0	0
$71$	−0.936701	−0.111166	−0.0555830	−	0.998454i	$-0.517702\pi$
$71$	−0.936701	−0.111166	−0.0555830	+	0.998454i	$0.517702\pi$
$72$	0	0
$73$	0.829359	0.0970692	0.0485346	−	0.998822i	$-0.484545\pi$
$73$	0.829359	0.0970692	0.0485346	+	0.998822i	$0.484545\pi$
$74$	0	0
$75$	−30.1114	−3.47697
$76$	0	0
$77$	−6.56534	−0.748190
$78$	0	0
$79$	−11.6408	−1.30970	−0.654849	−	0.755760i	$-0.727268\pi$
$79$	−11.6408	−1.30970	−0.654849	+	0.755760i	$0.727268\pi$
$80$	0	0
$81$	−8.55602	−0.950668
$82$	0	0
$83$	8.99911	0.987780	0.493890	−	0.869524i	$-0.335575\pi$
$83$	8.99911	0.987780	0.493890	+	0.869524i	$0.335575\pi$
$84$	0	0
$85$	−20.1600	−2.18666
$86$	0	0
$87$	−4.95634	−0.531376
$88$	0	0
$89$	−6.28267	−0.665962	−0.332981	−	0.942934i	$-0.608054\pi$
$89$	−6.28267	−0.665962	−0.332981	+	0.942934i	$0.608054\pi$
$90$	0	0
$91$	1.96435	0.205920
$92$	0	0
$93$	11.0093	1.14161
$94$	0	0
$95$	30.1114	3.08937
$96$	0	0
$97$	14.5653	1.47889	0.739443	−	0.673219i	$-0.235089\pi$
$97$	14.5653	1.47889	0.739443	+	0.673219i	$0.235089\pi$
$98$	0	0
$99$	−10.4991	−1.05520
$100$	0	0
$101$	−6.01866	−0.598879	−0.299439	−	0.954115i	$-0.596800\pi$
$101$	−6.01866	−0.598879	−0.299439	+	0.954115i	$0.596800\pi$
$102$	0	0
$103$	12.8137	1.26257	0.631287	−	0.775549i	$-0.282527\pi$
$103$	12.8137	1.26257	0.631287	+	0.775549i	$0.282527\pi$
$104$	0	0
$105$	−20.1600	−1.96741
$106$	0	0
$107$	14.9417	1.44446	0.722232	−	0.691651i	$-0.243116\pi$
$107$	14.9417	1.44446	0.722232	+	0.691651i	$0.243116\pi$
$108$	0	0
$109$	−11.4333	−1.09512	−0.547558	−	0.836768i	$-0.684442\pi$
$109$	−11.4333	−1.09512	−0.547558	+	0.836768i	$0.684442\pi$
$110$	0	0
$111$	−6.40687	−0.608113
$112$	0	0
$113$	−6.82936	−0.642452	−0.321226	−	0.947003i	$-0.604095\pi$
$113$	−6.82936	−0.642452	−0.321226	+	0.947003i	$0.604095\pi$
$114$	0	0
$115$	−20.5259	−1.91405
$116$	0	0
$117$	3.14134	0.290417
$118$	0	0
$119$	−9.56243	−0.876587
$120$	0	0
$121$	0.170641	0.0155128
$122$	0	0
$123$	−29.0838	−2.62240
$124$	0	0
$125$	29.6133	2.64869
$126$	0	0
$127$	−16.2701	−1.44373	−0.721867	−	0.692032i	$-0.756716\pi$
$127$	−16.2701	−1.44373	−0.721867	+	0.692032i	$0.756716\pi$
$128$	0	0
$129$	0.867993	0.0764225
$130$	0	0
$131$	14.1916	1.23993	0.619965	−	0.784630i	$-0.287147\pi$
$131$	14.1916	1.23993	0.619965	+	0.784630i	$0.287147\pi$
$132$	0	0
$133$	14.2827	1.23846
$134$	0	0
$135$	−1.45052	−0.124841
$136$	0	0
$137$	−10.5653	−0.902658	−0.451329	−	0.892358i	$-0.649050\pi$
$137$	−10.5653	−0.902658	−0.451329	+	0.892358i	$0.649050\pi$
$138$	0	0
$139$	0.277633	0.0235486	0.0117743	−	0.999931i	$-0.496252\pi$
$139$	0.277633	0.0235486	0.0117743	+	0.999931i	$0.496252\pi$
$140$	0	0
$141$	−11.8773	−1.00025
$142$	0	0
$143$	−3.34225	−0.279493
$144$	0	0
$145$	8.28267	0.687838
$146$	0	0
$147$	7.78477	0.642077
$148$	0	0
$149$	18.5653	1.52093	0.760466	−	0.649378i	$-0.224971\pi$
$149$	18.5653	1.52093	0.760466	+	0.649378i	$0.224971\pi$
$150$	0	0
$151$	12.9773	1.05608	0.528039	−	0.849220i	$-0.322927\pi$
$151$	12.9773	1.05608	0.528039	+	0.849220i	$0.322927\pi$
$152$	0	0
$153$	−15.2920	−1.23629
$154$	0	0
$155$	−18.3980	−1.47776
$156$	0	0
$157$	−7.73599	−0.617399	−0.308699	−	0.951160i	$-0.599894\pi$
$157$	−7.73599	−0.617399	−0.308699	+	0.951160i	$0.599894\pi$
$158$	0	0
$159$	−34.0401	−2.69956
$160$	0	0
$161$	−9.73599	−0.767303
$162$	0	0
$163$	14.2826	1.11870	0.559349	−	0.828932i	$-0.311051\pi$
$163$	14.2826	1.11870	0.559349	+	0.828932i	$0.311051\pi$
$164$	0	0
$165$	34.3013	2.67035
$166$	0	0
$167$	18.6110	1.44017	0.720083	−	0.693888i	$-0.244104\pi$
$167$	18.6110	1.44017	0.720083	+	0.693888i	$0.244104\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	22.8405	1.74666
$172$	0	0
$173$	14.8294	1.12746	0.563728	−	0.825961i	$-0.309367\pi$
$173$	14.8294	1.12746	0.563728	+	0.825961i	$0.309367\pi$
$174$	0	0
$175$	23.8681	1.80426
$176$	0	0
$177$	−6.54669	−0.492079
$178$	0	0
$179$	−0.604770	−0.0452027	−0.0226013	−	0.999745i	$-0.507195\pi$
$179$	−0.604770	−0.0452027	−0.0226013	+	0.999745i	$0.507195\pi$
$180$	0	0
$181$	−4.54669	−0.337953	−0.168976	−	0.985620i	$-0.554046\pi$
$181$	−4.54669	−0.337953	−0.168976	+	0.985620i	$0.554046\pi$
$182$	0	0
$183$	3.60156	0.266235
$184$	0	0
$185$	10.7067	0.787171
$186$	0	0
$187$	16.2701	1.18978
$188$	0	0
$189$	−0.688023	−0.0500463
$190$	0	0
$191$	13.8414	1.00153	0.500764	−	0.865584i	$-0.333053\pi$
$191$	13.8414	1.00153	0.500764	+	0.865584i	$0.333053\pi$
$192$	0	0
$193$	−10.5653	−0.760510	−0.380255	−	0.924882i	$-0.624164\pi$
$193$	−10.5653	−0.760510	−0.380255	+	0.924882i	$0.624164\pi$
$194$	0	0
$195$	−10.2629	−0.734945
$196$	0	0
$197$	0.849335	0.0605126	0.0302563	−	0.999542i	$-0.490368\pi$
$197$	0.849335	0.0605126	0.0302563	+	0.999542i	$0.490368\pi$
$198$	0	0
$199$	3.55532	0.252030	0.126015	−	0.992028i	$-0.459781\pi$
$199$	3.55532	0.252030	0.126015	+	0.992028i	$0.459781\pi$
$200$	0	0
$201$	13.7360	0.968862
$202$	0	0
$203$	3.92870	0.275740
$204$	0	0
$205$	48.6027	3.39456
$206$	0	0
$207$	−15.5695	−1.08216
$208$	0	0
$209$	−24.3013	−1.68096
$210$	0	0
$211$	−10.6363	−0.732234	−0.366117	−	0.930569i	$-0.619313\pi$
$211$	−10.6363	−0.732234	−0.366117	+	0.930569i	$0.619313\pi$
$212$	0	0
$213$	2.32131	0.159053
$214$	0	0
$215$	−1.45052	−0.0989249
$216$	0	0
$217$	−8.72666	−0.592404
$218$	0	0
$219$	−2.05529	−0.138884
$220$	0	0
$221$	−4.86799	−0.327457
$222$	0	0
$223$	−7.22144	−0.483584	−0.241792	−	0.970328i	$-0.577735\pi$
$223$	−7.22144	−0.483584	−0.241792	+	0.970328i	$0.577735\pi$
$224$	0	0
$225$	38.1693	2.54462
$226$	0	0
$227$	25.0410	1.66203	0.831016	−	0.556249i	$-0.187760\pi$
$227$	25.0410	1.66203	0.831016	+	0.556249i	$0.187760\pi$
$228$	0	0
$229$	−6.15999	−0.407064	−0.203532	−	0.979068i	$-0.565242\pi$
$229$	−6.15999	−0.407064	−0.203532	+	0.979068i	$0.565242\pi$
$230$	0	0
$231$	16.2701	1.07049
$232$	0	0
$233$	−13.5946	−0.890615	−0.445308	−	0.895378i	$-0.646906\pi$
$233$	−13.5946	−0.890615	−0.445308	+	0.895378i	$0.646906\pi$
$234$	0	0
$235$	19.8485	1.29477
$236$	0	0
$237$	28.8480	1.87388
$238$	0	0
$239$	10.7041	0.692394	0.346197	−	0.938162i	$-0.387473\pi$
$239$	10.7041	0.692394	0.346197	+	0.938162i	$0.387473\pi$
$240$	0	0
$241$	−5.18930	−0.334272	−0.167136	−	0.985934i	$-0.553452\pi$
$241$	−5.18930	−0.334272	−0.167136	+	0.985934i	$0.553452\pi$
$242$	0	0
$243$	22.2540	1.42760
$244$	0	0
$245$	−13.0093	−0.831136
$246$	0	0
$247$	7.27095	0.462639
$248$	0	0
$249$	−22.3013	−1.41329
$250$	0	0
$251$	17.3703	1.09640	0.548202	−	0.836346i	$-0.315312\pi$
$251$	17.3703	1.09640	0.548202	+	0.836346i	$0.315312\pi$
$252$	0	0
$253$	16.5653	1.04145
$254$	0	0
$255$	49.9599	3.12861
$256$	0	0
$257$	−15.1507	−0.945073	−0.472536	−	0.881311i	$-0.656661\pi$
$257$	−15.1507	−0.945073	−0.472536	+	0.881311i	$0.656661\pi$
$258$	0	0
$259$	5.07847	0.315561
$260$	0	0
$261$	6.28267	0.388888
$262$	0	0
$263$	14.3967	0.887736	0.443868	−	0.896092i	$-0.353606\pi$
$263$	14.3967	0.887736	0.443868	+	0.896092i	$0.353606\pi$
$264$	0	0
$265$	56.8853	3.49444
$266$	0	0
$267$	15.5695	0.952840
$268$	0	0
$269$	−6.84802	−0.417531	−0.208765	−	0.977966i	$-0.566945\pi$
$269$	−6.84802	−0.417531	−0.208765	+	0.977966i	$0.566945\pi$
$270$	0	0
$271$	23.8186	1.44688	0.723439	−	0.690388i	$-0.242560\pi$
$271$	23.8186	1.44688	0.723439	+	0.690388i	$0.242560\pi$
$272$	0	0
$273$	−4.86799	−0.294624
$274$	0	0
$275$	−40.6106	−2.44891
$276$	0	0
$277$	8.56534	0.514642	0.257321	−	0.966326i	$-0.417160\pi$
$277$	8.56534	0.514642	0.257321	+	0.966326i	$0.417160\pi$
$278$	0	0
$279$	−13.9554	−0.835491
$280$	0	0
$281$	12.3013	0.733836	0.366918	−	0.930253i	$-0.380413\pi$
$281$	12.3013	0.733836	0.366918	+	0.930253i	$0.380413\pi$
$282$	0	0
$283$	−10.3124	−0.613011	−0.306506	−	0.951869i	$-0.599160\pi$
$283$	−10.3124	−0.613011	−0.306506	+	0.951869i	$0.599160\pi$
$284$	0	0
$285$	−74.6213	−4.42019
$286$	0	0
$287$	23.0536	1.36081
$288$	0	0
$289$	6.69735	0.393962
$290$	0	0
$291$	−36.0954	−2.11595
$292$	0	0
$293$	−15.9987	−0.934653	−0.467326	−	0.884085i	$-0.654783\pi$
$293$	−15.9987	−0.934653	−0.467326	+	0.884085i	$0.654783\pi$
$294$	0	0
$295$	10.9403	0.636971
$296$	0	0
$297$	1.17064	0.0679275
$298$	0	0
$299$	−4.95634	−0.286633
$300$	0	0
$301$	−0.688023	−0.0396570
$302$	0	0
$303$	14.9153	0.856860
$304$	0	0
$305$	−6.01866	−0.344627
$306$	0	0
$307$	−30.9260	−1.76504	−0.882520	−	0.470274i	$-0.844155\pi$
$307$	−30.9260	−1.76504	−0.882520	+	0.470274i	$0.844155\pi$
$308$	0	0
$309$	−31.7546	−1.80646
$310$	0	0
$311$	−2.20054	−0.124781	−0.0623905	−	0.998052i	$-0.519872\pi$
$311$	−2.20054	−0.124781	−0.0623905	+	0.998052i	$0.519872\pi$
$312$	0	0
$313$	21.3306	1.20568	0.602839	−	0.797863i	$-0.294036\pi$
$313$	21.3306	1.20568	0.602839	+	0.797863i	$0.294036\pi$
$314$	0	0
$315$	25.5549	1.43985
$316$	0	0
$317$	−17.4720	−0.981324	−0.490662	−	0.871350i	$-0.663245\pi$
$317$	−17.4720	−0.981324	−0.490662	+	0.871350i	$0.663245\pi$
$318$	0	0
$319$	−6.68450	−0.374260
$320$	0	0
$321$	−37.0280	−2.06670
$322$	0	0
$323$	−35.3949	−1.96943
$324$	0	0
$325$	12.1507	0.673998
$326$	0	0
$327$	28.3338	1.56686
$328$	0	0
$329$	9.41468	0.519048
$330$	0	0
$331$	4.74327	0.260714	0.130357	−	0.991467i	$-0.458388\pi$
$331$	4.74327	0.260714	0.130357	+	0.991467i	$0.458388\pi$
$332$	0	0
$333$	8.12136	0.445048
$334$	0	0
$335$	−22.9546	−1.25414
$336$	0	0
$337$	28.4427	1.54937	0.774685	−	0.632347i	$-0.217908\pi$
$337$	28.4427	1.54937	0.774685	+	0.632347i	$0.217908\pi$
$338$	0	0
$339$	16.9243	0.919203
$340$	0	0
$341$	14.8480	0.804065
$342$	0	0
$343$	−19.9211	−1.07564
$344$	0	0
$345$	50.8667	2.73857
$346$	0	0
$347$	8.13503	0.436711	0.218356	−	0.975869i	$-0.429931\pi$
$347$	8.13503	0.436711	0.218356	+	0.975869i	$0.429931\pi$
$348$	0	0
$349$	−26.0827	−1.39618	−0.698088	−	0.716012i	$-0.745966\pi$
$349$	−26.0827	−1.39618	−0.698088	+	0.716012i	$0.745966\pi$
$350$	0	0
$351$	−0.350255	−0.0186952
$352$	0	0
$353$	32.0187	1.70418	0.852091	−	0.523394i	$-0.175335\pi$
$353$	32.0187	1.70418	0.852091	+	0.523394i	$0.175335\pi$
$354$	0	0
$355$	−3.87919	−0.205886
$356$	0	0
$357$	23.6974	1.25420
$358$	0	0
$359$	−29.1252	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$359$	−29.1252	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$360$	0	0
$361$	33.8667	1.78246
$362$	0	0
$363$	−0.422877	−0.0221953
$364$	0	0
$365$	3.43466	0.179778
$366$	0	0
$367$	35.7221	1.86468	0.932338	−	0.361588i	$-0.117765\pi$
$367$	35.7221	1.86468	0.932338	+	0.361588i	$0.117765\pi$
$368$	0	0
$369$	36.8667	1.91920
$370$	0	0
$371$	26.9823	1.40085
$372$	0	0
$373$	8.30133	0.429827	0.214913	−	0.976633i	$-0.431053\pi$
$373$	8.30133	0.429827	0.214913	+	0.976633i	$0.431053\pi$
$374$	0	0
$375$	−73.3869	−3.78968
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−15.0821	−0.774716	−0.387358	−	0.921929i	$-0.626612\pi$
$379$	−15.0821	−0.774716	−0.387358	+	0.921929i	$0.626612\pi$
$380$	0	0
$381$	40.3200	2.06566
$382$	0	0
$383$	−35.7037	−1.82438	−0.912188	−	0.409773i	$-0.865608\pi$
$383$	−35.7037	−1.82438	−0.912188	+	0.409773i	$0.865608\pi$
$384$	0	0
$385$	−27.1893	−1.38569
$386$	0	0
$387$	−1.10027	−0.0559298
$388$	0	0
$389$	4.62395	0.234444	0.117222	−	0.993106i	$-0.462601\pi$
$389$	4.62395	0.234444	0.117222	+	0.993106i	$0.462601\pi$
$390$	0	0
$391$	24.1274	1.22018
$392$	0	0
$393$	−35.1693	−1.77406
$394$	0	0
$395$	−48.2087	−2.42564
$396$	0	0
$397$	−3.09337	−0.155252	−0.0776260	−	0.996983i	$-0.524734\pi$
$397$	−3.09337	−0.155252	−0.0776260	+	0.996983i	$0.524734\pi$
$398$	0	0
$399$	−35.3949	−1.77196
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	−4.44252	−0.221298
$404$	0	0
$405$	−35.4333	−1.76070
$406$	0	0
$407$	−8.64079	−0.428308
$408$	0	0
$409$	3.09337	0.152958	0.0764788	−	0.997071i	$-0.475632\pi$
$409$	3.09337	0.152958	0.0764788	+	0.997071i	$0.475632\pi$
$410$	0	0
$411$	26.1827	1.29150
$412$	0	0
$413$	5.18930	0.255349
$414$	0	0
$415$	37.2683	1.82943
$416$	0	0
$417$	−0.688023	−0.0336926
$418$	0	0
$419$	11.6903	0.571111	0.285555	−	0.958362i	$-0.407822\pi$
$419$	11.6903	0.571111	0.285555	+	0.958362i	$0.407822\pi$
$420$	0	0
$421$	27.6133	1.34579	0.672895	−	0.739738i	$-0.265051\pi$
$421$	27.6133	1.34579	0.672895	+	0.739738i	$0.265051\pi$
$422$	0	0
$423$	15.0557	0.732034
$424$	0	0
$425$	−59.1493	−2.86916
$426$	0	0
$427$	−2.85481	−0.138154
$428$	0	0
$429$	8.28267	0.399891
$430$	0	0
$431$	−12.9509	−0.623824	−0.311912	−	0.950111i	$-0.600969\pi$
$431$	−12.9509	−0.623824	−0.311912	+	0.950111i	$0.600969\pi$
$432$	0	0
$433$	−1.87732	−0.0902183	−0.0451092	−	0.998982i	$-0.514364\pi$
$433$	−1.87732	−0.0902183	−0.0451092	+	0.998982i	$0.514364\pi$
$434$	0	0
$435$	−20.5259	−0.984141
$436$	0	0
$437$	−36.0373	−1.72390
$438$	0	0
$439$	18.3716	0.876828	0.438414	−	0.898773i	$-0.355540\pi$
$439$	18.3716	0.876828	0.438414	+	0.898773i	$0.355540\pi$
$440$	0	0
$441$	−9.86799	−0.469904
$442$	0	0
$443$	2.03218	0.0965516	0.0482758	−	0.998834i	$-0.484627\pi$
$443$	2.03218	0.0965516	0.0482758	+	0.998834i	$0.484627\pi$
$444$	0	0
$445$	−26.0187	−1.23340
$446$	0	0
$447$	−46.0081	−2.17611
$448$	0	0
$449$	18.3599	0.866459	0.433230	−	0.901284i	$-0.357374\pi$
$449$	18.3599	0.866459	0.433230	+	0.901284i	$0.357374\pi$
$450$	0	0
$451$	−39.2246	−1.84701
$452$	0	0
$453$	−32.1600	−1.51101
$454$	0	0
$455$	8.13503	0.381376
$456$	0	0
$457$	−1.18930	−0.0556330	−0.0278165	−	0.999613i	$-0.508855\pi$
$457$	−1.18930	−0.0556330	−0.0278165	+	0.999613i	$0.508855\pi$
$458$	0	0
$459$	1.70504	0.0795844
$460$	0	0
$461$	−1.20927	−0.0563215	−0.0281608	−	0.999603i	$-0.508965\pi$
$461$	−1.20927	−0.0563215	−0.0281608	+	0.999603i	$0.508965\pi$
$462$	0	0
$463$	−35.2082	−1.63627	−0.818133	−	0.575030i	$-0.804991\pi$
$463$	−35.2082	−1.63627	−0.818133	+	0.575030i	$0.804991\pi$
$464$	0	0
$465$	45.5933	2.11434
$466$	0	0
$467$	−21.7714	−1.00746	−0.503730	−	0.863861i	$-0.668039\pi$
$467$	−21.7714	−1.00746	−0.503730	+	0.863861i	$0.668039\pi$
$468$	0	0
$469$	−10.8880	−0.502760
$470$	0	0
$471$	19.1711	0.883358
$472$	0	0
$473$	1.17064	0.0538261
$474$	0	0
$475$	88.3468	4.05363
$476$	0	0
$477$	43.1493	1.97567
$478$	0	0
$479$	5.16615	0.236047	0.118024	−	0.993011i	$-0.462344\pi$
$479$	5.16615	0.236047	0.118024	+	0.993011i	$0.462344\pi$
$480$	0	0
$481$	2.58532	0.117880
$482$	0	0
$483$	24.1274	1.09784
$484$	0	0
$485$	60.3200	2.73899
$486$	0	0
$487$	−24.6413	−1.11660	−0.558301	−	0.829638i	$-0.688547\pi$
$487$	−24.6413	−1.11660	−0.558301	+	0.829638i	$0.688547\pi$
$488$	0	0
$489$	−35.3947	−1.60060
$490$	0	0
$491$	−16.5741	−0.747977	−0.373989	−	0.927433i	$-0.622010\pi$
$491$	−16.5741	−0.747977	−0.373989	+	0.927433i	$0.622010\pi$
$492$	0	0
$493$	−9.73599	−0.438487
$494$	0	0
$495$	−43.4804	−1.95430
$496$	0	0
$497$	−1.84001	−0.0825356
$498$	0	0
$499$	−14.1007	−0.631234	−0.315617	−	0.948887i	$-0.602211\pi$
$499$	−14.1007	−0.631234	−0.315617	+	0.948887i	$0.602211\pi$
$500$	0	0
$501$	−46.1214	−2.06055
$502$	0	0
$503$	20.3440	0.907094	0.453547	−	0.891232i	$-0.350158\pi$
$503$	20.3440	0.907094	0.453547	+	0.891232i	$0.350158\pi$
$504$	0	0
$505$	−24.9253	−1.10916
$506$	0	0
$507$	−2.47817	−0.110059
$508$	0	0
$509$	−0.341281	−0.0151270	−0.00756352	−	0.999971i	$-0.502408\pi$
$509$	−0.341281	−0.0151270	−0.00756352	+	0.999971i	$0.502408\pi$
$510$	0	0
$511$	1.62915	0.0720694
$512$	0	0
$513$	−2.54669	−0.112439
$514$	0	0
$515$	53.0660	2.33837
$516$	0	0
$517$	−16.0187	−0.704500
$518$	0	0
$519$	−36.7497	−1.61313
$520$	0	0
$521$	24.4427	1.07085	0.535426	−	0.844582i	$-0.320151\pi$
$521$	24.4427	1.07085	0.535426	+	0.844582i	$0.320151\pi$
$522$	0	0
$523$	32.4212	1.41768	0.708841	−	0.705368i	$-0.249218\pi$
$523$	32.4212	1.41768	0.708841	+	0.705368i	$0.249218\pi$
$524$	0	0
$525$	−59.1493	−2.58149
$526$	0	0
$527$	21.6262	0.942050
$528$	0	0
$529$	1.56534	0.0680585
$530$	0	0
$531$	8.29859	0.360128
$532$	0	0
$533$	11.7360	0.508342
$534$	0	0
$535$	61.8784	2.67524
$536$	0	0
$537$	1.49873	0.0646748
$538$	0	0
$539$	10.4991	0.452230
$540$	0	0
$541$	−26.5853	−1.14299	−0.571496	−	0.820605i	$-0.693637\pi$
$541$	−26.5853	−1.14299	−0.571496	+	0.820605i	$0.693637\pi$
$542$	0	0
$543$	11.2675	0.483534
$544$	0	0
$545$	−47.3493	−2.02822
$546$	0	0
$547$	−32.8177	−1.40319	−0.701593	−	0.712578i	$-0.747527\pi$
$547$	−32.8177	−1.40319	−0.701593	+	0.712578i	$0.747527\pi$
$548$	0	0
$549$	−4.56534	−0.194844
$550$	0	0
$551$	14.5419	0.619506
$552$	0	0
$553$	−22.8667	−0.972390
$554$	0	0
$555$	−26.5330	−1.12626
$556$	0	0
$557$	35.4333	1.50136	0.750679	−	0.660667i	$-0.229726\pi$
$557$	35.4333	1.50136	0.750679	+	0.660667i	$0.229726\pi$
$558$	0	0
$559$	−0.350255	−0.0148142
$560$	0	0
$561$	−40.3200	−1.70231
$562$	0	0
$563$	−0.750014	−0.0316093	−0.0158047	−	0.999875i	$-0.505031\pi$
$563$	−0.750014	−0.0316093	−0.0158047	+	0.999875i	$0.505031\pi$
$564$	0	0
$565$	−28.2827	−1.18986
$566$	0	0
$567$	−16.8070	−0.705827
$568$	0	0
$569$	−22.5454	−0.945151	−0.472576	−	0.881290i	$-0.656676\pi$
$569$	−22.5454	−0.945151	−0.472576	+	0.881290i	$0.656676\pi$
$570$	0	0
$571$	−7.76165	−0.324815	−0.162408	−	0.986724i	$-0.551926\pi$
$571$	−7.76165	−0.324815	−0.162408	+	0.986724i	$0.551926\pi$
$572$	0	0
$573$	−34.3013	−1.43296
$574$	0	0
$575$	−60.2229	−2.51147
$576$	0	0
$577$	2.07727	0.0864777	0.0432389	−	0.999065i	$-0.486232\pi$
$577$	2.07727	0.0864777	0.0432389	+	0.999065i	$0.486232\pi$
$578$	0	0
$579$	26.1827	1.08812
$580$	0	0
$581$	17.6774	0.733381
$582$	0	0
$583$	−45.9091	−1.90136
$584$	0	0
$585$	13.0093	0.537870
$586$	0	0
$587$	7.74333	0.319601	0.159801	−	0.987149i	$-0.448915\pi$
$587$	7.74333	0.319601	0.159801	+	0.987149i	$0.448915\pi$
$588$	0	0
$589$	−32.3013	−1.33095
$590$	0	0
$591$	−2.10480	−0.0865798
$592$	0	0
$593$	21.1493	0.868500	0.434250	−	0.900793i	$-0.357014\pi$
$593$	21.1493	0.868500	0.434250	+	0.900793i	$0.357014\pi$
$594$	0	0
$595$	−39.6012	−1.62349
$596$	0	0
$597$	−8.81070	−0.360598
$598$	0	0
$599$	41.2799	1.68665	0.843326	−	0.537403i	$-0.180595\pi$
$599$	41.2799	1.68665	0.843326	+	0.537403i	$0.180595\pi$
$600$	0	0
$601$	−25.0734	−1.02277	−0.511383	−	0.859353i	$-0.670866\pi$
$601$	−25.0734	−1.02277	−0.511383	+	0.859353i	$0.670866\pi$
$602$	0	0
$603$	−17.4118	−0.709062
$604$	0	0
$605$	0.706681	0.0287307
$606$	0	0
$607$	−1.17289	−0.0476062	−0.0238031	−	0.999717i	$-0.507577\pi$
$607$	−1.17289	−0.0476062	−0.0238031	+	0.999717i	$0.507577\pi$
$608$	0	0
$609$	−9.73599	−0.394522
$610$	0	0
$611$	4.79278	0.193895
$612$	0	0
$613$	11.6587	0.470891	0.235446	−	0.971888i	$-0.424345\pi$
$613$	11.6587	0.470891	0.235446	+	0.971888i	$0.424345\pi$
$614$	0	0
$615$	−120.446	−4.85684
$616$	0	0
$617$	−32.8667	−1.32316	−0.661581	−	0.749873i	$-0.730114\pi$
$617$	−32.8667	−1.32316	−0.661581	+	0.749873i	$0.730114\pi$
$618$	0	0
$619$	−20.2666	−0.814583	−0.407291	−	0.913298i	$-0.633527\pi$
$619$	−20.2666	−0.814583	−0.407291	+	0.913298i	$0.633527\pi$
$620$	0	0
$621$	1.73599	0.0696627
$622$	0	0
$623$	−12.3414	−0.494446
$624$	0	0
$625$	61.8853	2.47541
$626$	0	0
$627$	60.2229	2.40507
$628$	0	0
$629$	−12.5853	−0.501810
$630$	0	0
$631$	−44.5888	−1.77505	−0.887525	−	0.460759i	$-0.847577\pi$
$631$	−44.5888	−1.77505	−0.887525	+	0.460759i	$0.847577\pi$
$632$	0	0
$633$	26.3586	1.04766
$634$	0	0
$635$	−67.3798	−2.67388
$636$	0	0
$637$	−3.14134	−0.124464
$638$	0	0
$639$	−2.94249	−0.116403
$640$	0	0
$641$	−39.4320	−1.55747	−0.778736	−	0.627352i	$-0.784139\pi$
$641$	−39.4320	−1.55747	−0.778736	+	0.627352i	$0.784139\pi$
$642$	0	0
$643$	−12.3725	−0.487925	−0.243963	−	0.969785i	$-0.578447\pi$
$643$	−12.3725	−0.487925	−0.243963	+	0.969785i	$0.578447\pi$
$644$	0	0
$645$	3.59465	0.141539
$646$	0	0
$647$	32.5401	1.27928	0.639642	−	0.768673i	$-0.279083\pi$
$647$	32.5401	1.27928	0.639642	+	0.768673i	$0.279083\pi$
$648$	0	0
$649$	−8.82936	−0.346583
$650$	0	0
$651$	21.6262	0.847596
$652$	0	0
$653$	50.3200	1.96917	0.984587	−	0.174898i	$-0.0559594\pi$
$653$	50.3200	1.96917	0.984587	+	0.174898i	$0.0559594\pi$
$654$	0	0
$655$	58.7723	2.29643
$656$	0	0
$657$	2.60530	0.101642
$658$	0	0
$659$	−16.9969	−0.662107	−0.331054	−	0.943612i	$-0.607404\pi$
$659$	−16.9969	−0.662107	−0.331054	+	0.943612i	$0.607404\pi$
$660$	0	0
$661$	−34.5653	−1.34444	−0.672218	−	0.740353i	$-0.734658\pi$
$661$	−34.5653	−1.34444	−0.672218	+	0.740353i	$0.734658\pi$
$662$	0	0
$663$	12.0637	0.468516
$664$	0	0
$665$	59.1493	2.29371
$666$	0	0
$667$	−9.91269	−0.383821
$668$	0	0
$669$	17.8960	0.691899
$670$	0	0
$671$	4.85734	0.187515
$672$	0	0
$673$	−31.9800	−1.23274	−0.616370	−	0.787457i	$-0.711397\pi$
$673$	−31.9800	−1.23274	−0.616370	+	0.787457i	$0.711397\pi$
$674$	0	0
$675$	−4.25583	−0.163807
$676$	0	0
$677$	7.71733	0.296601	0.148301	−	0.988942i	$-0.452620\pi$
$677$	7.71733	0.296601	0.148301	+	0.988942i	$0.452620\pi$
$678$	0	0
$679$	28.6114	1.09801
$680$	0	0
$681$	−62.0560	−2.37799
$682$	0	0
$683$	−50.7976	−1.94372	−0.971859	−	0.235565i	$-0.924306\pi$
$683$	−50.7976	−1.94372	−0.971859	+	0.235565i	$0.924306\pi$
$684$	0	0
$685$	−43.7546	−1.67178
$686$	0	0
$687$	15.2655	0.582416
$688$	0	0
$689$	13.7360	0.523299
$690$	0	0
$691$	−0.295958	−0.0112588	−0.00562939	−	0.999984i	$-0.501792\pi$
$691$	−0.295958	−0.0112588	−0.00562939	+	0.999984i	$0.501792\pi$
$692$	0	0
$693$	−20.6240	−0.783439
$694$	0	0
$695$	1.14977	0.0436134
$696$	0	0
$697$	−57.1307	−2.16398
$698$	0	0
$699$	33.6899	1.27427
$700$	0	0
$701$	−10.2054	−0.385453	−0.192726	−	0.981253i	$-0.561733\pi$
$701$	−10.2054	−0.385453	−0.192726	+	0.981253i	$0.561733\pi$
$702$	0	0
$703$	18.7977	0.708970
$704$	0	0
$705$	−49.1880	−1.85253
$706$	0	0
$707$	−11.8227	−0.444640
$708$	0	0
$709$	−33.4720	−1.25707	−0.628533	−	0.777783i	$-0.716344\pi$
$709$	−33.4720	−1.25707	−0.628533	+	0.777783i	$0.716344\pi$
$710$	0	0
$711$	−36.5678	−1.37140
$712$	0	0
$713$	22.0187	0.824605
$714$	0	0
$715$	−13.8414	−0.517638
$716$	0	0
$717$	−26.5267	−0.990658
$718$	0	0
$719$	−27.6365	−1.03067	−0.515334	−	0.856989i	$-0.672332\pi$
$719$	−27.6365	−1.03067	−0.515334	+	0.856989i	$0.672332\pi$
$720$	0	0
$721$	25.1706	0.937404
$722$	0	0
$723$	12.8600	0.478268
$724$	0	0
$725$	24.3013	0.902529
$726$	0	0
$727$	34.5492	1.28136	0.640679	−	0.767809i	$-0.278653\pi$
$727$	34.5492	1.28136	0.640679	+	0.767809i	$0.278653\pi$
$728$	0	0
$729$	−29.4813	−1.09190
$730$	0	0
$731$	1.70504	0.0630632
$732$	0	0
$733$	14.4427	0.533452	0.266726	−	0.963772i	$-0.414058\pi$
$733$	14.4427	0.533452	0.266726	+	0.963772i	$0.414058\pi$
$734$	0	0
$735$	32.2394	1.18917
$736$	0	0
$737$	18.5254	0.682392
$738$	0	0
$739$	−0.213072	−0.00783799	−0.00391899	−	0.999992i	$-0.501247\pi$
$739$	−0.213072	−0.00783799	−0.00391899	+	0.999992i	$0.501247\pi$
$740$	0	0
$741$	−18.0187	−0.661932
$742$	0	0
$743$	0.765071	0.0280678	0.0140339	−	0.999902i	$-0.495533\pi$
$743$	0.765071	0.0280678	0.0140339	+	0.999902i	$0.495533\pi$
$744$	0	0
$745$	76.8853	2.81686
$746$	0	0
$747$	28.2692	1.03432
$748$	0	0
$749$	29.3506	1.07245
$750$	0	0
$751$	−14.1158	−0.515091	−0.257546	−	0.966266i	$-0.582914\pi$
$751$	−14.1158	−0.515091	−0.257546	+	0.966266i	$0.582914\pi$
$752$	0	0
$753$	−43.0466	−1.56871
$754$	0	0
$755$	53.7434	1.95592
$756$	0	0
$757$	28.3200	1.02931	0.514654	−	0.857398i	$-0.327920\pi$
$757$	28.3200	1.02931	0.514654	+	0.857398i	$0.327920\pi$
$758$	0	0
$759$	−41.0518	−1.49008
$760$	0	0
$761$	23.9414	0.867875	0.433937	−	0.900943i	$-0.357124\pi$
$761$	23.9414	0.867875	0.433937	+	0.900943i	$0.357124\pi$
$762$	0	0
$763$	−22.4591	−0.813072
$764$	0	0
$765$	−63.3293	−2.28968
$766$	0	0
$767$	2.64174	0.0953877
$768$	0	0
$769$	−33.4320	−1.20559	−0.602795	−	0.797896i	$-0.705946\pi$
$769$	−33.4320	−1.20559	−0.602795	+	0.797896i	$0.705946\pi$
$770$	0	0
$771$	37.5460	1.35218
$772$	0	0
$773$	26.1600	0.940910	0.470455	−	0.882424i	$-0.344090\pi$
$773$	26.1600	0.940910	0.470455	+	0.882424i	$0.344090\pi$
$774$	0	0
$775$	−53.9796	−1.93900
$776$	0	0
$777$	−12.5853	−0.451496
$778$	0	0
$779$	85.3317	3.05733
$780$	0	0
$781$	3.13069	0.112025
$782$	0	0
$783$	−0.700510	−0.0250342
$784$	0	0
$785$	−32.0373	−1.14346
$786$	0	0
$787$	27.8958	0.994379	0.497190	−	0.867642i	$-0.334365\pi$
$787$	27.8958	0.994379	0.497190	+	0.867642i	$0.334365\pi$
$788$	0	0
$789$	−35.6774	−1.27015
$790$	0	0
$791$	−13.4152	−0.476991
$792$	0	0
$793$	−1.45331	−0.0516087
$794$	0	0
$795$	−140.972	−4.99975
$796$	0	0
$797$	−21.1893	−0.750563	−0.375282	−	0.926911i	$-0.622454\pi$
$797$	−21.1893	−0.750563	−0.375282	+	0.926911i	$0.622454\pi$
$798$	0	0
$799$	−23.3312	−0.825398
$800$	0	0
$801$	−19.7360	−0.697337
$802$	0	0
$803$	−2.77193	−0.0978192
$804$	0	0
$805$	−40.3200	−1.42109
$806$	0	0
$807$	16.9706	0.597392
$808$	0	0
$809$	8.44267	0.296828	0.148414	−	0.988925i	$-0.452583\pi$
$809$	8.44267	0.296828	0.148414	+	0.988925i	$0.452583\pi$
$810$	0	0
$811$	4.41613	0.155071	0.0775357	−	0.996990i	$-0.475295\pi$
$811$	4.41613	0.155071	0.0775357	+	0.996990i	$0.475295\pi$
$812$	0	0
$813$	−59.0267	−2.07016
$814$	0	0
$815$	59.1490	2.07190
$816$	0	0
$817$	−2.54669	−0.0890973
$818$	0	0
$819$	6.17068	0.215621
$820$	0	0
$821$	−31.5360	−1.10062	−0.550308	−	0.834962i	$-0.685490\pi$
$821$	−31.5360	−1.10062	−0.550308	+	0.834962i	$0.685490\pi$
$822$	0	0
$823$	−16.1710	−0.563687	−0.281844	−	0.959460i	$-0.590946\pi$
$823$	−16.1710	−0.563687	−0.281844	+	0.959460i	$0.590946\pi$
$824$	0	0
$825$	100.640	3.50383
$826$	0	0
$827$	−36.5000	−1.26923	−0.634614	−	0.772829i	$-0.718841\pi$
$827$	−36.5000	−1.26923	−0.634614	+	0.772829i	$0.718841\pi$
$828$	0	0
$829$	−23.1893	−0.805398	−0.402699	−	0.915333i	$-0.631928\pi$
$829$	−23.1893	−0.805398	−0.402699	+	0.915333i	$0.631928\pi$
$830$	0	0
$831$	−21.2264	−0.736336
$832$	0	0
$833$	15.2920	0.529836
$834$	0	0
$835$	77.0746	2.66728
$836$	0	0
$837$	1.55602	0.0537838
$838$	0	0
$839$	−4.15203	−0.143344	−0.0716720	−	0.997428i	$-0.522833\pi$
$839$	−4.15203	−0.143344	−0.0716720	+	0.997428i	$0.522833\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−30.4848	−1.04995
$844$	0	0
$845$	4.14134	0.142466
$846$	0	0
$847$	0.335198	0.0115175
$848$	0	0
$849$	25.5560	0.877080
$850$	0	0
$851$	−12.8137	−0.439249
$852$	0	0
$853$	7.89598	0.270353	0.135177	−	0.990822i	$-0.456840\pi$
$853$	7.89598	0.270353	0.135177	+	0.990822i	$0.456840\pi$
$854$	0	0
$855$	94.5901	3.23491
$856$	0	0
$857$	−14.5653	−0.497543	−0.248771	−	0.968562i	$-0.580027\pi$
$857$	−14.5653	−0.497543	−0.248771	+	0.968562i	$0.580027\pi$
$858$	0	0
$859$	21.3980	0.730091	0.365046	−	0.930990i	$-0.381053\pi$
$859$	21.3980	0.730091	0.365046	+	0.930990i	$0.381053\pi$
$860$	0	0
$861$	−57.1307	−1.94701
$862$	0	0
$863$	14.0776	0.479206	0.239603	−	0.970871i	$-0.422983\pi$
$863$	14.0776	0.479206	0.239603	+	0.970871i	$0.422983\pi$
$864$	0	0
$865$	61.4134	2.08812
$866$	0	0
$867$	−16.5972	−0.563670
$868$	0	0
$869$	38.9066	1.31982
$870$	0	0
$871$	−5.54279	−0.187810
$872$	0	0
$873$	45.7546	1.54856
$874$	0	0
$875$	58.1708	1.96653
$876$	0	0
$877$	0.746632	0.0252120	0.0126060	−	0.999921i	$-0.495987\pi$
$877$	0.746632	0.0252120	0.0126060	+	0.999921i	$0.495987\pi$
$878$	0	0
$879$	39.6475	1.33728
$880$	0	0
$881$	−20.0759	−0.676376	−0.338188	−	0.941079i	$-0.609814\pi$
$881$	−20.0759	−0.676376	−0.338188	+	0.941079i	$0.609814\pi$
$882$	0	0
$883$	29.2888	0.985647	0.492823	−	0.870129i	$-0.335965\pi$
$883$	29.2888	0.985647	0.492823	+	0.870129i	$0.335965\pi$
$884$	0	0
$885$	−27.1120	−0.911361
$886$	0	0
$887$	33.7130	1.13197	0.565986	−	0.824415i	$-0.308496\pi$
$887$	33.7130	1.13197	0.565986	+	0.824415i	$0.308496\pi$
$888$	0	0
$889$	−31.9600	−1.07191
$890$	0	0
$891$	28.5964	0.958014
$892$	0	0
$893$	34.8480	1.16614
$894$	0	0
$895$	−2.50456	−0.0837181
$896$	0	0
$897$	12.2827	0.410106
$898$	0	0
$899$	−8.88504	−0.296333
$900$	0	0
$901$	−66.8667	−2.22765
$902$	0	0
$903$	1.70504	0.0567402
$904$	0	0
$905$	−18.8294	−0.625909
$906$	0	0
$907$	18.8208	0.624936	0.312468	−	0.949928i	$-0.398844\pi$
$907$	18.8208	0.624936	0.312468	+	0.949928i	$0.398844\pi$
$908$	0	0
$909$	−18.9066	−0.627093
$910$	0	0
$911$	−42.2709	−1.40050	−0.700249	−	0.713899i	$-0.746928\pi$
$911$	−42.2709	−1.40050	−0.700249	+	0.713899i	$0.746928\pi$
$912$	0	0
$913$	−30.0773	−0.995412
$914$	0	0
$915$	14.9153	0.493084
$916$	0	0
$917$	27.8773	0.920590
$918$	0	0
$919$	−8.23077	−0.271508	−0.135754	−	0.990743i	$-0.543346\pi$
$919$	−8.23077	−0.271508	−0.135754	+	0.990743i	$0.543346\pi$
$920$	0	0
$921$	76.6400	2.52537
$922$	0	0
$923$	−0.936701	−0.0308319
$924$	0	0
$925$	31.4134	1.03286
$926$	0	0
$927$	40.2523	1.32206
$928$	0	0
$929$	0.623954	0.0204713	0.0102356	−	0.999948i	$-0.496742\pi$
$929$	0.623954	0.0204713	0.0102356	+	0.999948i	$0.496742\pi$
$930$	0	0
$931$	−22.8405	−0.748567
$932$	0	0
$933$	5.45331	0.178533
$934$	0	0
$935$	67.3798	2.20355
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	−52.8610	−1.72505
$940$	0	0
$941$	21.8773	0.713180	0.356590	−	0.934261i	$-0.383939\pi$
$941$	21.8773	0.713180	0.356590	+	0.934261i	$0.383939\pi$
$942$	0	0
$943$	−58.1676	−1.89420
$944$	0	0
$945$	−2.84934	−0.0926889
$946$	0	0
$947$	−9.42525	−0.306279	−0.153140	−	0.988205i	$-0.548938\pi$
$947$	−9.42525	−0.306279	−0.153140	+	0.988205i	$0.548938\pi$
$948$	0	0
$949$	0.829359	0.0269221
$950$	0	0
$951$	43.2985	1.40405
$952$	0	0
$953$	−7.99868	−0.259103	−0.129551	−	0.991573i	$-0.541354\pi$
$953$	−7.99868	−0.259103	−0.129551	+	0.991573i	$0.541354\pi$
$954$	0	0
$955$	57.3218	1.85489
$956$	0	0
$957$	16.5653	0.535482
$958$	0	0
$959$	−20.7540	−0.670182
$960$	0	0
$961$	−11.2640	−0.363355
$962$	0	0
$963$	46.9368	1.51252
$964$	0	0
$965$	−43.7546	−1.40851
$966$	0	0
$967$	21.2345	0.682854	0.341427	−	0.939908i	$-0.389090\pi$
$967$	21.2345	0.682854	0.341427	+	0.939908i	$0.389090\pi$
$968$	0	0
$969$	87.7147	2.81780
$970$	0	0
$971$	21.3485	0.685107	0.342553	−	0.939498i	$-0.388708\pi$
$971$	21.3485	0.685107	0.342553	+	0.939498i	$0.388708\pi$
$972$	0	0
$973$	0.545369	0.0174837
$974$	0	0
$975$	−30.1114	−0.964338
$976$	0	0
$977$	31.1307	0.995959	0.497979	−	0.867189i	$-0.334075\pi$
$977$	31.1307	0.995959	0.497979	+	0.867189i	$0.334075\pi$
$978$	0	0
$979$	20.9983	0.671108
$980$	0	0
$981$	−35.9160	−1.14671
$982$	0	0
$983$	−33.1497	−1.05731	−0.528655	−	0.848837i	$-0.677304\pi$
$983$	−33.1497	−1.05731	−0.528655	+	0.848837i	$0.677304\pi$
$984$	0	0
$985$	3.51738	0.112073
$986$	0	0
$987$	−23.3312	−0.742640
$988$	0	0
$989$	1.73599	0.0552011
$990$	0	0
$991$	−2.28342	−0.0725354	−0.0362677	−	0.999342i	$-0.511547\pi$
$991$	−2.28342	−0.0725354	−0.0362677	+	0.999342i	$0.511547\pi$
$992$	0	0
$993$	−11.7546	−0.373022
$994$	0	0
$995$	14.7238	0.466775
$996$	0	0
$997$	7.15198	0.226506	0.113253	−	0.993566i	$-0.463873\pi$
$997$	7.15198	0.226506	0.113253	+	0.993566i	$0.463873\pi$
$998$	0	0
$999$	−0.905522	−0.0286494

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bp.1.1		6
4.3	odd	2	inner	3328.2.a.bp.1.6		6
8.3	odd	2		3328.2.a.bo.1.1		6
8.5	even	2		3328.2.a.bo.1.6		6
16.3	odd	4		1664.2.b.k.833.11	yes	12
16.5	even	4		1664.2.b.k.833.12	yes	12
16.11	odd	4		1664.2.b.k.833.2	yes	12
16.13	even	4		1664.2.b.k.833.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.k.833.1	✓	12	16.13	even	4
1664.2.b.k.833.2	yes	12	16.11	odd	4
1664.2.b.k.833.11	yes	12	16.3	odd	4
1664.2.b.k.833.12	yes	12	16.5	even	4
3328.2.a.bo.1.1		6	8.3	odd	2
3328.2.a.bo.1.6		6	8.5	even	2
3328.2.a.bp.1.1		6	1.1	even	1	trivial
3328.2.a.bp.1.6		6	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47817 q^{3} +4.14134 q^{5} +1.96435 q^{7} +3.14134 q^{9} -3.34225 q^{11} +1.00000 q^{13} -10.2629 q^{15} -4.86799 q^{17} +7.27095 q^{19} -4.86799 q^{21} -4.95634 q^{23} +12.1507 q^{25} -0.350255 q^{27} +2.00000 q^{29} -4.44252 q^{31} +8.28267 q^{33} +8.13503 q^{35} +2.58532 q^{37} -2.47817 q^{39} +11.7360 q^{41} -0.350255 q^{43} +13.0093 q^{45} +4.79278 q^{47} -3.14134 q^{49} +12.0637 q^{51} +13.7360 q^{53} -13.8414 q^{55} -18.0187 q^{57} +2.64174 q^{59} -1.45331 q^{61} +6.17068 q^{63} +4.14134 q^{65} -5.54279 q^{67} +12.2827 q^{69} -0.936701 q^{71} +0.829359 q^{73} -30.1114 q^{75} -6.56534 q^{77} -11.6408 q^{79} -8.55602 q^{81} +8.99911 q^{83} -20.1600 q^{85} -4.95634 q^{87} -6.28267 q^{89} +1.96435 q^{91} +11.0093 q^{93} +30.1114 q^{95} +14.5653 q^{97} -10.4991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{5} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 4 q^{21} + 14 q^{25} + 12 q^{29} + 16 q^{33} + 24 q^{37} + 20 q^{41} + 36 q^{45} - 2 q^{49} + 32 q^{53} - 24 q^{57} + 8 q^{61} + 8 q^{65} + 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) 6 * q + 8 * q^5 + 2 * q^9 + 6 * q^13 - 4 * q^17 - 4 * q^21 + 14 * q^25 + 12 * q^29 + 16 * q^33 + 24 * q^37 + 20 * q^41 + 36 * q^45 - 2 * q^49 + 32 * q^53 - 24 * q^57 + 8 * q^61 + 8 * q^65 + 40 * q^69 - 12 * q^73 + 28 * q^77 - 26 * q^81 - 20 * q^85 - 4 * q^89 + 24 * q^93 + 20 * q^97

Introduction

L-functions

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Database

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