LMFDB - Embedded newform 1664.2.b.k.833.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,-4] 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:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.426337261060096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 $ x^12 - 4x^9 - 3x^8 + 4x^7 + 8x^6 + 8x^5 - 12x^4 - 32*x^3 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.12
Root			$-1.35818 + 0.394157i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.k.833.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.47817i q^{3} +4.14134i q^{5} -1.96435 q^{7} -3.14134 q^{9} -3.34225i q^{11} -1.00000i q^{13} -10.2629 q^{15} -4.86799 q^{17} -7.27095i q^{19} -4.86799i q^{21} +4.95634 q^{23} -12.1507 q^{25} -0.350255i q^{27} -2.00000i q^{29} -4.44252 q^{31} +8.28267 q^{33} -8.13503i q^{35} +2.58532i q^{37} +2.47817 q^{39} -11.7360 q^{41} -0.350255i q^{43} -13.0093i q^{45} +4.79278 q^{47} -3.14134 q^{49} -12.0637i q^{51} +13.7360i q^{53} +13.8414 q^{55} +18.0187 q^{57} +2.64174i q^{59} +1.45331i q^{61} +6.17068 q^{63} +4.14134 q^{65} +5.54279i q^{67} +12.2827i q^{69} +0.936701 q^{71} -0.829359 q^{73} -30.1114i q^{75} +6.56534i q^{77} -11.6408 q^{79} -8.55602 q^{81} -8.99911i q^{83} -20.1600i q^{85} +4.95634 q^{87} +6.28267 q^{89} +1.96435i q^{91} -11.0093i q^{93} +30.1114 q^{95} +14.5653 q^{97} +10.4991i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) $ 12 * q - 4 * q^9 - 8 * q^17 - 28 * q^25 + 32 * q^33 - 40 * q^41 - 4 * q^49 + 48 * q^57 + 16 * q^65 + 24 * q^73 - 52 * q^81 + 8 * q^89 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.47817i	1.43077i	0.698729	+	0.715387i	$0.253749\pi$
$3$	2.47817i	1.43077i	−0.698729	+	0.715387i	$0.746251\pi$
$4$	0	0
$5$	4.14134i	1.85206i	0.377448	+	0.926031i	$0.376802\pi$
$5$	4.14134i	1.85206i	−0.377448	+	0.926031i	$0.623198\pi$
$6$	0	0
$7$	−1.96435	−0.742454	−0.371227	−	0.928542i	$-0.621063\pi$
$7$	−1.96435	−0.742454	−0.371227	+	0.928542i	$0.621063\pi$
$8$	0	0
$9$	−3.14134	−1.04711
$10$	0	0
$11$	− 3.34225i	− 1.00773i	−0.863783	−	0.503863i	$-0.831912\pi$
$11$	− 3.34225i	− 1.00773i	0.863783	−	0.503863i	$-0.168088\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	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
$19$	− 7.27095i	− 1.66807i	−0.551712	−	0.834035i	$-0.686025\pi$
$19$	− 7.27095i	− 1.66807i	0.551712	−	0.834035i	$-0.313975\pi$
$20$	0	0
$21$	− 4.86799i	− 1.06228i
$22$	0	0
$23$	4.95634	1.03347	0.516735	−	0.856146i	$-0.327147\pi$
$23$	4.95634	1.03347	0.516735	+	0.856146i	$0.327147\pi$
$24$	0	0
$25$	−12.1507	−2.43013
$26$	0	0
$27$	− 0.350255i	− 0.0674066i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	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
$35$	− 8.13503i	− 1.37507i
$36$	0	0
$37$	2.58532i	0.425024i	0.977158	+	0.212512i	$0.0681645\pi$
$37$	2.58532i	0.425024i	−0.977158	+	0.212512i	$0.931836\pi$
$38$	0	0
$39$	2.47817	0.396825
$40$	0	0
$41$	−11.7360	−1.83285	−0.916426	−	0.400203i	$-0.868940\pi$
$41$	−11.7360	−1.83285	−0.916426	+	0.400203i	$0.868940\pi$
$42$	0	0
$43$	− 0.350255i	− 0.0534134i	−0.999643	−	0.0267067i	$-0.991498\pi$
$43$	− 0.350255i	− 0.0534134i	0.999643	−	0.0267067i	$-0.00850202\pi$
$44$	0	0
$45$	− 13.0093i	− 1.93932i
$46$	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
$51$	− 12.0637i	− 1.68926i
$52$	0	0
$53$	13.7360i	1.88678i	0.331682	+	0.943391i	$0.392384\pi$
$53$	13.7360i	1.88678i	−0.331682	+	0.943391i	$0.607616\pi$
$54$	0	0
$55$	13.8414	1.86637
$56$	0	0
$57$	18.0187	2.38663
$58$	0	0
$59$	2.64174i	0.343925i	0.985103	+	0.171963i	$0.0550108\pi$
$59$	2.64174i	0.343925i	−0.985103	+	0.171963i	$0.944989\pi$
$60$	0	0
$61$	1.45331i	0.186078i	0.995662	+	0.0930388i	$0.0296581\pi$
$61$	1.45331i	0.186078i	−0.995662	+	0.0930388i	$0.970342\pi$
$62$	0	0
$63$	6.17068	0.777432
$64$	0	0
$65$	4.14134	0.513670
$66$	0	0
$67$	5.54279i	0.677160i	0.940938	+	0.338580i	$0.109946\pi$
$67$	5.54279i	0.677160i	−0.940938	+	0.338580i	$0.890054\pi$
$68$	0	0
$69$	12.2827i	1.47866i
$70$	0	0
$71$	0.936701	0.111166	0.0555830	−	0.998454i	$-0.482298\pi$
$71$	0.936701	0.111166	0.0555830	+	0.998454i	$0.482298\pi$
$72$	0	0
$73$	−0.829359	−0.0970692	−0.0485346	−	0.998822i	$-0.515455\pi$
$73$	−0.829359	−0.0970692	−0.0485346	+	0.998822i	$0.515455\pi$
$74$	0	0
$75$	− 30.1114i	− 3.47697i
$76$	0	0
$77$	6.56534i	0.748190i
$78$	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
$83$	− 8.99911i	− 0.987780i	−0.869524	−	0.493890i	$-0.835575\pi$
$83$	− 8.99911i	− 0.987780i	0.869524	−	0.493890i	$-0.164425\pi$
$84$	0	0
$85$	− 20.1600i	− 2.18666i
$86$	0	0
$87$	4.95634	0.531376
$88$	0	0
$89$	6.28267	0.665962	0.332981	−	0.942934i	$-0.391946\pi$
$89$	6.28267	0.665962	0.332981	+	0.942934i	$0.391946\pi$
$90$	0	0
$91$	1.96435i	0.205920i
$92$	0	0
$93$	− 11.0093i	− 1.14161i
$94$	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
$99$	10.4991i	1.05520i
$100$	0	0
$101$	− 6.01866i	− 0.598879i	−0.954115	−	0.299439i	$-0.903200\pi$
$101$	− 6.01866i	− 0.598879i	0.954115	−	0.299439i	$-0.0967996\pi$
$102$	0	0
$103$	−12.8137	−1.26257	−0.631287	−	0.775549i	$-0.717473\pi$
$103$	−12.8137	−1.26257	−0.631287	+	0.775549i	$0.717473\pi$
$104$	0	0
$105$	20.1600	1.96741
$106$	0	0
$107$	14.9417i	1.44446i	0.691651	+	0.722232i	$0.256884\pi$
$107$	14.9417i	1.44446i	−0.691651	+	0.722232i	$0.743116\pi$
$108$	0	0
$109$	11.4333i	1.09512i	0.836768	+	0.547558i	$0.184442\pi$
$109$	11.4333i	1.09512i	−0.836768	+	0.547558i	$0.815558\pi$
$110$	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
$115$	20.5259i	1.91405i
$116$	0	0
$117$	3.14134i	0.290417i
$118$	0	0
$119$	9.56243	0.876587
$120$	0	0
$121$	−0.170641	−0.0155128
$122$	0	0
$123$	− 29.0838i	− 2.62240i
$124$	0	0
$125$	− 29.6133i	− 2.64869i
$126$	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
$131$	− 14.1916i	− 1.23993i	−0.784630	−	0.619965i	$-0.787147\pi$
$131$	− 14.1916i	− 1.23993i	0.784630	−	0.619965i	$-0.212853\pi$
$132$	0	0
$133$	14.2827i	1.23846i
$134$	0	0
$135$	1.45052	0.124841
$136$	0	0
$137$	10.5653	0.902658	0.451329	−	0.892358i	$-0.350950\pi$
$137$	10.5653	0.902658	0.451329	+	0.892358i	$0.350950\pi$
$138$	0	0
$139$	0.277633i	0.0235486i	0.999931	+	0.0117743i	$0.00374796\pi$
$139$	0.277633i	0.0235486i	−0.999931	+	0.0117743i	$0.996252\pi$
$140$	0	0
$141$	11.8773i	1.00025i
$142$	0	0
$143$	−3.34225	−0.279493
$144$	0	0
$145$	8.28267	0.687838
$146$	0	0
$147$	− 7.78477i	− 0.642077i
$148$	0	0
$149$	18.5653i	1.52093i	0.649378	+	0.760466i	$0.275029\pi$
$149$	18.5653i	1.52093i	−0.649378	+	0.760466i	$0.724971\pi$
$150$	0	0
$151$	−12.9773	−1.05608	−0.528039	−	0.849220i	$-0.677073\pi$
$151$	−12.9773	−1.05608	−0.528039	+	0.849220i	$0.677073\pi$
$152$	0	0
$153$	15.2920	1.23629
$154$	0	0
$155$	− 18.3980i	− 1.47776i
$156$	0	0
$157$	7.73599i	0.617399i	0.951160	+	0.308699i	$0.0998937\pi$
$157$	7.73599i	0.617399i	−0.951160	+	0.308699i	$0.900106\pi$
$158$	0	0
$159$	−34.0401	−2.69956
$160$	0	0
$161$	−9.73599	−0.767303
$162$	0	0
$163$	− 14.2826i	− 1.11870i	−0.828932	−	0.559349i	$-0.811051\pi$
$163$	− 14.2826i	− 1.11870i	0.828932	−	0.559349i	$-0.188949\pi$
$164$	0	0
$165$	34.3013i	2.67035i
$166$	0	0
$167$	−18.6110	−1.44017	−0.720083	−	0.693888i	$-0.755896\pi$
$167$	−18.6110	−1.44017	−0.720083	+	0.693888i	$0.755896\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	22.8405i	1.74666i
$172$	0	0
$173$	− 14.8294i	− 1.12746i	−0.825961	−	0.563728i	$-0.809367\pi$
$173$	− 14.8294i	− 1.12746i	0.825961	−	0.563728i	$-0.190633\pi$
$174$	0	0
$175$	23.8681	1.80426
$176$	0	0
$177$	−6.54669	−0.492079
$178$	0	0
$179$	0.604770i	0.0452027i	0.999745	+	0.0226013i	$0.00719484\pi$
$179$	0.604770i	0.0452027i	−0.999745	+	0.0226013i	$0.992805\pi$
$180$	0	0
$181$	− 4.54669i	− 0.337953i	−0.985620	−	0.168976i	$-0.945954\pi$
$181$	− 4.54669i	− 0.337953i	0.985620	−	0.168976i	$-0.0540461\pi$
$182$	0	0
$183$	−3.60156	−0.266235
$184$	0	0
$185$	−10.7067	−0.787171
$186$	0	0
$187$	16.2701i	1.18978i
$188$	0	0
$189$	0.688023i	0.0500463i
$190$	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
$195$	10.2629i	0.734945i
$196$	0	0
$197$	0.849335i	0.0605126i	0.999542	+	0.0302563i	$0.00963235\pi$
$197$	0.849335i	0.0605126i	−0.999542	+	0.0302563i	$0.990368\pi$
$198$	0	0
$199$	−3.55532	−0.252030	−0.126015	−	0.992028i	$-0.540219\pi$
$199$	−3.55532	−0.252030	−0.126015	+	0.992028i	$0.540219\pi$
$200$	0	0
$201$	−13.7360	−0.968862
$202$	0	0
$203$	3.92870i	0.275740i
$204$	0	0
$205$	− 48.6027i	− 3.39456i
$206$	0	0
$207$	−15.5695	−1.08216
$208$	0	0
$209$	−24.3013	−1.68096
$210$	0	0
$211$	10.6363i	0.732234i	0.930569	+	0.366117i	$0.119313\pi$
$211$	10.6363i	0.732234i	−0.930569	+	0.366117i	$0.880687\pi$
$212$	0	0
$213$	2.32131i	0.159053i
$214$	0	0
$215$	1.45052	0.0989249
$216$	0	0
$217$	8.72666	0.592404
$218$	0	0
$219$	− 2.05529i	− 0.138884i
$220$	0	0
$221$	4.86799i	0.327457i
$222$	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
$227$	− 25.0410i	− 1.66203i	−0.556249	−	0.831016i	$-0.687760\pi$
$227$	− 25.0410i	− 1.66203i	0.556249	−	0.831016i	$-0.312240\pi$
$228$	0	0
$229$	− 6.15999i	− 0.407064i	−0.979068	−	0.203532i	$-0.934758\pi$
$229$	− 6.15999i	− 0.407064i	0.979068	−	0.203532i	$-0.0652421\pi$
$230$	0	0
$231$	−16.2701	−1.07049
$232$	0	0
$233$	13.5946	0.890615	0.445308	−	0.895378i	$-0.353094\pi$
$233$	13.5946	0.890615	0.445308	+	0.895378i	$0.353094\pi$
$234$	0	0
$235$	19.8485i	1.29477i
$236$	0	0
$237$	− 28.8480i	− 1.87388i
$238$	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
$243$	− 22.2540i	− 1.42760i
$244$	0	0
$245$	− 13.0093i	− 0.831136i
$246$	0	0
$247$	−7.27095	−0.462639
$248$	0	0
$249$	22.3013	1.41329
$250$	0	0
$251$	17.3703i	1.09640i	0.836346	+	0.548202i	$0.184688\pi$
$251$	17.3703i	1.09640i	−0.836346	+	0.548202i	$0.815312\pi$
$252$	0	0
$253$	− 16.5653i	− 1.04145i
$254$	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
$259$	− 5.07847i	− 0.315561i
$260$	0	0
$261$	6.28267i	0.388888i
$262$	0	0
$263$	−14.3967	−0.887736	−0.443868	−	0.896092i	$-0.646394\pi$
$263$	−14.3967	−0.887736	−0.443868	+	0.896092i	$0.646394\pi$
$264$	0	0
$265$	−56.8853	−3.49444
$266$	0	0
$267$	15.5695i	0.952840i
$268$	0	0
$269$	6.84802i	0.417531i	0.977966	+	0.208765i	$0.0669445\pi$
$269$	6.84802i	0.417531i	−0.977966	+	0.208765i	$0.933055\pi$
$270$	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
$275$	40.6106i	2.44891i
$276$	0	0
$277$	8.56534i	0.514642i	0.966326	+	0.257321i	$0.0828397\pi$
$277$	8.56534i	0.514642i	−0.966326	+	0.257321i	$0.917160\pi$
$278$	0	0
$279$	13.9554	0.835491
$280$	0	0
$281$	−12.3013	−0.733836	−0.366918	−	0.930253i	$-0.619587\pi$
$281$	−12.3013	−0.733836	−0.366918	+	0.930253i	$0.619587\pi$
$282$	0	0
$283$	− 10.3124i	− 0.613011i	−0.951869	−	0.306506i	$-0.900840\pi$
$283$	− 10.3124i	− 0.613011i	0.951869	−	0.306506i	$-0.0991598\pi$
$284$	0	0
$285$	74.6213i	4.42019i
$286$	0	0
$287$	23.0536	1.36081
$288$	0	0
$289$	6.69735	0.393962
$290$	0	0
$291$	36.0954i	2.11595i
$292$	0	0
$293$	− 15.9987i	− 0.934653i	−0.884085	−	0.467326i	$-0.845217\pi$
$293$	− 15.9987i	− 0.934653i	0.884085	−	0.467326i	$-0.154783\pi$
$294$	0	0
$295$	−10.9403	−0.636971
$296$	0	0
$297$	−1.17064	−0.0679275
$298$	0	0
$299$	− 4.95634i	− 0.286633i
$300$	0	0
$301$	0.688023i	0.0396570i
$302$	0	0
$303$	14.9153	0.856860
$304$	0	0
$305$	−6.01866	−0.344627
$306$	0	0
$307$	30.9260i	1.76504i	0.470274	+	0.882520i	$0.344155\pi$
$307$	30.9260i	1.76504i	−0.470274	+	0.882520i	$0.655845\pi$
$308$	0	0
$309$	− 31.7546i	− 1.80646i
$310$	0	0
$311$	2.20054	0.124781	0.0623905	−	0.998052i	$-0.480128\pi$
$311$	2.20054	0.124781	0.0623905	+	0.998052i	$0.480128\pi$
$312$	0	0
$313$	−21.3306	−1.20568	−0.602839	−	0.797863i	$-0.705964\pi$
$313$	−21.3306	−1.20568	−0.602839	+	0.797863i	$0.705964\pi$
$314$	0	0
$315$	25.5549i	1.43985i
$316$	0	0
$317$	17.4720i	0.981324i	0.871350	+	0.490662i	$0.163245\pi$
$317$	17.4720i	0.981324i	−0.871350	+	0.490662i	$0.836755\pi$
$318$	0	0
$319$	−6.68450	−0.374260
$320$	0	0
$321$	−37.0280	−2.06670
$322$	0	0
$323$	35.3949i	1.96943i
$324$	0	0
$325$	12.1507i	0.673998i
$326$	0	0
$327$	−28.3338	−1.56686
$328$	0	0
$329$	−9.41468	−0.519048
$330$	0	0
$331$	4.74327i	0.260714i	0.991467	+	0.130357i	$0.0416123\pi$
$331$	4.74327i	0.260714i	−0.991467	+	0.130357i	$0.958388\pi$
$332$	0	0
$333$	− 8.12136i	− 0.445048i
$334$	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
$339$	− 16.9243i	− 0.919203i
$340$	0	0
$341$	14.8480i	0.804065i
$342$	0	0
$343$	19.9211	1.07564
$344$	0	0
$345$	−50.8667	−2.73857
$346$	0	0
$347$	8.13503i	0.436711i	0.975869	+	0.218356i	$0.0700692\pi$
$347$	8.13503i	0.436711i	−0.975869	+	0.218356i	$0.929931\pi$
$348$	0	0
$349$	26.0827i	1.39618i	0.716012	+	0.698088i	$0.245966\pi$
$349$	26.0827i	1.39618i	−0.716012	+	0.698088i	$0.754034\pi$
$350$	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
$355$	3.87919i	0.205886i
$356$	0	0
$357$	23.6974i	1.25420i
$358$	0	0
$359$	29.1252	1.53717	0.768585	−	0.639747i	$-0.220961\pi$
$359$	29.1252	1.53717	0.768585	+	0.639747i	$0.220961\pi$
$360$	0	0
$361$	−33.8667	−1.78246
$362$	0	0
$363$	− 0.422877i	− 0.0221953i
$364$	0	0
$365$	− 3.43466i	− 0.179778i
$366$	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
$371$	− 26.9823i	− 1.40085i
$372$	0	0
$373$	8.30133i	0.429827i	0.976633	+	0.214913i	$0.0689469\pi$
$373$	8.30133i	0.429827i	−0.976633	+	0.214913i	$0.931053\pi$
$374$	0	0
$375$	73.3869	3.78968
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	− 15.0821i	− 0.774716i	−0.921929	−	0.387358i	$-0.873388\pi$
$379$	− 15.0821i	− 0.774716i	0.921929	−	0.387358i	$-0.126612\pi$
$380$	0	0
$381$	− 40.3200i	− 2.06566i
$382$	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
$387$	1.10027i	0.0559298i
$388$	0	0
$389$	4.62395i	0.234444i	0.993106	+	0.117222i	$0.0373989\pi$
$389$	4.62395i	0.234444i	−0.993106	+	0.117222i	$0.962601\pi$
$390$	0	0
$391$	−24.1274	−1.22018
$392$	0	0
$393$	35.1693	1.77406
$394$	0	0
$395$	− 48.2087i	− 2.42564i
$396$	0	0
$397$	3.09337i	0.155252i	0.996983	+	0.0776260i	$0.0247340\pi$
$397$	3.09337i	0.155252i	−0.996983	+	0.0776260i	$0.975266\pi$
$398$	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
$403$	4.44252i	0.221298i
$404$	0	0
$405$	− 35.4333i	− 1.76070i
$406$	0	0
$407$	8.64079	0.428308
$408$	0	0
$409$	−3.09337	−0.152958	−0.0764788	−	0.997071i	$-0.524368\pi$
$409$	−3.09337	−0.152958	−0.0764788	+	0.997071i	$0.524368\pi$
$410$	0	0
$411$	26.1827i	1.29150i
$412$	0	0
$413$	− 5.18930i	− 0.255349i
$414$	0	0
$415$	37.2683	1.82943
$416$	0	0
$417$	−0.688023	−0.0336926
$418$	0	0
$419$	− 11.6903i	− 0.571111i	−0.958362	−	0.285555i	$-0.907822\pi$
$419$	− 11.6903i	− 0.571111i	0.958362	−	0.285555i	$-0.0921781\pi$
$420$	0	0
$421$	27.6133i	1.34579i	0.739738	+	0.672895i	$0.234949\pi$
$421$	27.6133i	1.34579i	−0.739738	+	0.672895i	$0.765051\pi$
$422$	0	0
$423$	−15.0557	−0.732034
$424$	0	0
$425$	59.1493	2.86916
$426$	0	0
$427$	− 2.85481i	− 0.138154i
$428$	0	0
$429$	− 8.28267i	− 0.399891i
$430$	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
$435$	20.5259i	0.984141i
$436$	0	0
$437$	− 36.0373i	− 1.72390i
$438$	0	0
$439$	−18.3716	−0.876828	−0.438414	−	0.898773i	$-0.644460\pi$
$439$	−18.3716	−0.876828	−0.438414	+	0.898773i	$0.644460\pi$
$440$	0	0
$441$	9.86799	0.469904
$442$	0	0
$443$	2.03218i	0.0965516i	0.998834	+	0.0482758i	$0.0153726\pi$
$443$	2.03218i	0.0965516i	−0.998834	+	0.0482758i	$0.984627\pi$
$444$	0	0
$445$	26.0187i	1.23340i
$446$	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
$451$	39.2246i	1.84701i
$452$	0	0
$453$	− 32.1600i	− 1.51101i
$454$	0	0
$455$	−8.13503	−0.381376
$456$	0	0
$457$	1.18930	0.0556330	0.0278165	−	0.999613i	$-0.491145\pi$
$457$	1.18930	0.0556330	0.0278165	+	0.999613i	$0.491145\pi$
$458$	0	0
$459$	1.70504i	0.0795844i
$460$	0	0
$461$	1.20927i	0.0563215i	0.999603	+	0.0281608i	$0.00896503\pi$
$461$	1.20927i	0.0563215i	−0.999603	+	0.0281608i	$0.991035\pi$
$462$	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
$467$	21.7714i	1.00746i	0.863861	+	0.503730i	$0.168039\pi$
$467$	21.7714i	1.00746i	−0.863861	+	0.503730i	$0.831961\pi$
$468$	0	0
$469$	− 10.8880i	− 0.502760i
$470$	0	0
$471$	−19.1711	−0.883358
$472$	0	0
$473$	−1.17064	−0.0538261
$474$	0	0
$475$	88.3468i	4.05363i
$476$	0	0
$477$	− 43.1493i	− 1.97567i
$478$	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
$483$	− 24.1274i	− 1.09784i
$484$	0	0
$485$	60.3200i	2.73899i
$486$	0	0
$487$	24.6413	1.11660	0.558301	−	0.829638i	$-0.311453\pi$
$487$	24.6413	1.11660	0.558301	+	0.829638i	$0.311453\pi$
$488$	0	0
$489$	35.3947	1.60060
$490$	0	0
$491$	− 16.5741i	− 0.747977i	−0.927433	−	0.373989i	$-0.877990\pi$
$491$	− 16.5741i	− 0.747977i	0.927433	−	0.373989i	$-0.122010\pi$
$492$	0	0
$493$	9.73599i	0.438487i
$494$	0	0
$495$	−43.4804	−1.95430
$496$	0	0
$497$	−1.84001	−0.0825356
$498$	0	0
$499$	14.1007i	0.631234i	0.948887	+	0.315617i	$0.102211\pi$
$499$	14.1007i	0.631234i	−0.948887	+	0.315617i	$0.897789\pi$
$500$	0	0
$501$	− 46.1214i	− 2.06055i
$502$	0	0
$503$	−20.3440	−0.907094	−0.453547	−	0.891232i	$-0.649842\pi$
$503$	−20.3440	−0.907094	−0.453547	+	0.891232i	$0.649842\pi$
$504$	0	0
$505$	24.9253	1.10916
$506$	0	0
$507$	− 2.47817i	− 0.110059i
$508$	0	0
$509$	0.341281i	0.0151270i	0.999971	+	0.00756352i	$0.00240757\pi$
$509$	0.341281i	0.0151270i	−0.999971	+	0.00756352i	$0.997592\pi$
$510$	0	0
$511$	1.62915	0.0720694
$512$	0	0
$513$	−2.54669	−0.112439
$514$	0	0
$515$	− 53.0660i	− 2.33837i
$516$	0	0
$517$	− 16.0187i	− 0.704500i
$518$	0	0
$519$	36.7497	1.61313
$520$	0	0
$521$	−24.4427	−1.07085	−0.535426	−	0.844582i	$-0.679849\pi$
$521$	−24.4427	−1.07085	−0.535426	+	0.844582i	$0.679849\pi$
$522$	0	0
$523$	32.4212i	1.41768i	0.705368	+	0.708841i	$0.250782\pi$
$523$	32.4212i	1.41768i	−0.705368	+	0.708841i	$0.749218\pi$
$524$	0	0
$525$	59.1493i	2.58149i
$526$	0	0
$527$	21.6262	0.942050
$528$	0	0
$529$	1.56534	0.0680585
$530$	0	0
$531$	− 8.29859i	− 0.360128i
$532$	0	0
$533$	11.7360i	0.508342i
$534$	0	0
$535$	−61.8784	−2.67524
$536$	0	0
$537$	−1.49873	−0.0646748
$538$	0	0
$539$	10.4991i	0.452230i
$540$	0	0
$541$	26.5853i	1.14299i	0.820605	+	0.571496i	$0.193637\pi$
$541$	26.5853i	1.14299i	−0.820605	+	0.571496i	$0.806363\pi$
$542$	0	0
$543$	11.2675	0.483534
$544$	0	0
$545$	−47.3493	−2.02822
$546$	0	0
$547$	32.8177i	1.40319i	0.712578	+	0.701593i	$0.247527\pi$
$547$	32.8177i	1.40319i	−0.712578	+	0.701593i	$0.752473\pi$
$548$	0	0
$549$	− 4.56534i	− 0.194844i
$550$	0	0
$551$	−14.5419	−0.619506
$552$	0	0
$553$	22.8667	0.972390
$554$	0	0
$555$	− 26.5330i	− 1.12626i
$556$	0	0
$557$	− 35.4333i	− 1.50136i	−0.660667	−	0.750679i	$-0.729726\pi$
$557$	− 35.4333i	− 1.50136i	0.660667	−	0.750679i	$-0.270274\pi$
$558$	0	0
$559$	−0.350255	−0.0148142
$560$	0	0
$561$	−40.3200	−1.70231
$562$	0	0
$563$	0.750014i	0.0316093i	0.999875	+	0.0158047i	$0.00503099\pi$
$563$	0.750014i	0.0316093i	−0.999875	+	0.0158047i	$0.994969\pi$
$564$	0	0
$565$	− 28.2827i	− 1.18986i
$566$	0	0
$567$	16.8070	0.705827
$568$	0	0
$569$	22.5454	0.945151	0.472576	−	0.881290i	$-0.343324\pi$
$569$	22.5454	0.945151	0.472576	+	0.881290i	$0.343324\pi$
$570$	0	0
$571$	− 7.76165i	− 0.324815i	−0.986724	−	0.162408i	$-0.948074\pi$
$571$	− 7.76165i	− 0.324815i	0.986724	−	0.162408i	$-0.0519259\pi$
$572$	0	0
$573$	34.3013i	1.43296i
$574$	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
$579$	− 26.1827i	− 1.08812i
$580$	0	0
$581$	17.6774i	0.733381i
$582$	0	0
$583$	45.9091	1.90136
$584$	0	0
$585$	−13.0093	−0.537870
$586$	0	0
$587$	7.74333i	0.319601i	0.987149	+	0.159801i	$0.0510851\pi$
$587$	7.74333i	0.319601i	−0.987149	+	0.159801i	$0.948915\pi$
$588$	0	0
$589$	32.3013i	1.33095i
$590$	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
$595$	39.6012i	1.62349i
$596$	0	0
$597$	− 8.81070i	− 0.360598i
$598$	0	0
$599$	−41.2799	−1.68665	−0.843326	−	0.537403i	$-0.819405\pi$
$599$	−41.2799	−1.68665	−0.843326	+	0.537403i	$0.819405\pi$
$600$	0	0
$601$	25.0734	1.02277	0.511383	−	0.859353i	$-0.329134\pi$
$601$	25.0734	1.02277	0.511383	+	0.859353i	$0.329134\pi$
$602$	0	0
$603$	− 17.4118i	− 0.709062i
$604$	0	0
$605$	− 0.706681i	− 0.0287307i
$606$	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
$611$	− 4.79278i	− 0.193895i
$612$	0	0
$613$	11.6587i	0.470891i	0.971888	+	0.235446i	$0.0756550\pi$
$613$	11.6587i	0.470891i	−0.971888	+	0.235446i	$0.924345\pi$
$614$	0	0
$615$	120.446	4.85684
$616$	0	0
$617$	32.8667	1.32316	0.661581	−	0.749873i	$-0.269886\pi$
$617$	32.8667	1.32316	0.661581	+	0.749873i	$0.269886\pi$
$618$	0	0
$619$	− 20.2666i	− 0.814583i	−0.913298	−	0.407291i	$-0.866473\pi$
$619$	− 20.2666i	− 0.814583i	0.913298	−	0.407291i	$-0.133527\pi$
$620$	0	0
$621$	− 1.73599i	− 0.0696627i
$622$	0	0
$623$	−12.3414	−0.494446
$624$	0	0
$625$	61.8853	2.47541
$626$	0	0
$627$	− 60.2229i	− 2.40507i
$628$	0	0
$629$	− 12.5853i	− 0.501810i
$630$	0	0
$631$	44.5888	1.77505	0.887525	−	0.460759i	$-0.152423\pi$
$631$	44.5888	1.77505	0.887525	+	0.460759i	$0.152423\pi$
$632$	0	0
$633$	−26.3586	−1.04766
$634$	0	0
$635$	− 67.3798i	− 2.67388i
$636$	0	0
$637$	3.14134i	0.124464i
$638$	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
$643$	12.3725i	0.487925i	0.969785	+	0.243963i	$0.0784474\pi$
$643$	12.3725i	0.487925i	−0.969785	+	0.243963i	$0.921553\pi$
$644$	0	0
$645$	3.59465i	0.141539i
$646$	0	0
$647$	−32.5401	−1.27928	−0.639642	−	0.768673i	$-0.720917\pi$
$647$	−32.5401	−1.27928	−0.639642	+	0.768673i	$0.720917\pi$
$648$	0	0
$649$	8.82936	0.346583
$650$	0	0
$651$	21.6262i	0.847596i
$652$	0	0
$653$	− 50.3200i	− 1.96917i	−0.174898	−	0.984587i	$-0.555959\pi$
$653$	− 50.3200i	− 1.96917i	0.174898	−	0.984587i	$-0.444041\pi$
$654$	0	0
$655$	58.7723	2.29643
$656$	0	0
$657$	2.60530	0.101642
$658$	0	0
$659$	16.9969i	0.662107i	0.943612	+	0.331054i	$0.107404\pi$
$659$	16.9969i	0.662107i	−0.943612	+	0.331054i	$0.892596\pi$
$660$	0	0
$661$	− 34.5653i	− 1.34444i	−0.740353	−	0.672218i	$-0.765342\pi$
$661$	− 34.5653i	− 1.34444i	0.740353	−	0.672218i	$-0.234658\pi$
$662$	0	0
$663$	−12.0637	−0.468516
$664$	0	0
$665$	−59.1493	−2.29371
$666$	0	0
$667$	− 9.91269i	− 0.383821i
$668$	0	0
$669$	− 17.8960i	− 0.691899i
$670$	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
$675$	4.25583i	0.163807i
$676$	0	0
$677$	7.71733i	0.296601i	0.988942	+	0.148301i	$0.0473803\pi$
$677$	7.71733i	0.296601i	−0.988942	+	0.148301i	$0.952620\pi$
$678$	0	0
$679$	−28.6114	−1.09801
$680$	0	0
$681$	62.0560	2.37799
$682$	0	0
$683$	− 50.7976i	− 1.94372i	−0.235565	−	0.971859i	$-0.575694\pi$
$683$	− 50.7976i	− 1.94372i	0.235565	−	0.971859i	$-0.424306\pi$
$684$	0	0
$685$	43.7546i	1.67178i
$686$	0	0
$687$	15.2655	0.582416
$688$	0	0
$689$	13.7360	0.523299
$690$	0	0
$691$	0.295958i	0.0112588i	0.999984	+	0.00562939i	$0.00179190\pi$
$691$	0.295958i	0.0112588i	−0.999984	+	0.00562939i	$0.998208\pi$
$692$	0	0
$693$	− 20.6240i	− 0.783439i
$694$	0	0
$695$	−1.14977	−0.0436134
$696$	0	0
$697$	57.1307	2.16398
$698$	0	0
$699$	33.6899i	1.27427i
$700$	0	0
$701$	10.2054i	0.385453i	0.981253	+	0.192726i	$0.0617330\pi$
$701$	10.2054i	0.385453i	−0.981253	+	0.192726i	$0.938267\pi$
$702$	0	0
$703$	18.7977	0.708970
$704$	0	0
$705$	−49.1880	−1.85253
$706$	0	0
$707$	11.8227i	0.444640i
$708$	0	0
$709$	− 33.4720i	− 1.25707i	−0.777783	−	0.628533i	$-0.783656\pi$
$709$	− 33.4720i	− 1.25707i	0.777783	−	0.628533i	$-0.216344\pi$
$710$	0	0
$711$	36.5678	1.37140
$712$	0	0
$713$	−22.0187	−0.824605
$714$	0	0
$715$	− 13.8414i	− 0.517638i
$716$	0	0
$717$	26.5267i	0.990658i
$718$	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
$723$	− 12.8600i	− 0.478268i
$724$	0	0
$725$	24.3013i	0.902529i
$726$	0	0
$727$	−34.5492	−1.28136	−0.640679	−	0.767809i	$-0.721347\pi$
$727$	−34.5492	−1.28136	−0.640679	+	0.767809i	$0.721347\pi$
$728$	0	0
$729$	29.4813	1.09190
$730$	0	0
$731$	1.70504i	0.0630632i
$732$	0	0
$733$	− 14.4427i	− 0.533452i	−0.963772	−	0.266726i	$-0.914058\pi$
$733$	− 14.4427i	− 0.533452i	0.963772	−	0.266726i	$-0.0859419\pi$
$734$	0	0
$735$	32.2394	1.18917
$736$	0	0
$737$	18.5254	0.682392
$738$	0	0
$739$	0.213072i	0.00783799i	0.999992	+	0.00391899i	$0.00124746\pi$
$739$	0.213072i	0.00783799i	−0.999992	+	0.00391899i	$0.998753\pi$
$740$	0	0
$741$	− 18.0187i	− 0.661932i
$742$	0	0
$743$	−0.765071	−0.0280678	−0.0140339	−	0.999902i	$-0.504467\pi$
$743$	−0.765071	−0.0280678	−0.0140339	+	0.999902i	$0.504467\pi$
$744$	0	0
$745$	−76.8853	−2.81686
$746$	0	0
$747$	28.2692i	1.03432i
$748$	0	0
$749$	− 29.3506i	− 1.07245i
$750$	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
$755$	− 53.7434i	− 1.95592i
$756$	0	0
$757$	28.3200i	1.02931i	0.857398	+	0.514654i	$0.172080\pi$
$757$	28.3200i	1.02931i	−0.857398	+	0.514654i	$0.827920\pi$
$758$	0	0
$759$	41.0518	1.49008
$760$	0	0
$761$	−23.9414	−0.867875	−0.433937	−	0.900943i	$-0.642876\pi$
$761$	−23.9414	−0.867875	−0.433937	+	0.900943i	$0.642876\pi$
$762$	0	0
$763$	− 22.4591i	− 0.813072i
$764$	0	0
$765$	63.3293i	2.28968i
$766$	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
$771$	− 37.5460i	− 1.35218i
$772$	0	0
$773$	26.1600i	0.940910i	0.882424	+	0.470455i	$0.155910\pi$
$773$	26.1600i	0.940910i	−0.882424	+	0.470455i	$0.844090\pi$
$774$	0	0
$775$	53.9796	1.93900
$776$	0	0
$777$	12.5853	0.451496
$778$	0	0
$779$	85.3317i	3.05733i
$780$	0	0
$781$	− 3.13069i	− 0.112025i
$782$	0	0
$783$	−0.700510	−0.0250342
$784$	0	0
$785$	−32.0373	−1.14346
$786$	0	0
$787$	− 27.8958i	− 0.994379i	−0.867642	−	0.497190i	$-0.834365\pi$
$787$	− 27.8958i	− 0.994379i	0.867642	−	0.497190i	$-0.165635\pi$
$788$	0	0
$789$	− 35.6774i	− 1.27015i
$790$	0	0
$791$	13.4152	0.476991
$792$	0	0
$793$	1.45331	0.0516087
$794$	0	0
$795$	− 140.972i	− 4.99975i
$796$	0	0
$797$	21.1893i	0.750563i	0.926911	+	0.375282i	$0.122454\pi$
$797$	21.1893i	0.750563i	−0.926911	+	0.375282i	$0.877546\pi$
$798$	0	0
$799$	−23.3312	−0.825398
$800$	0	0
$801$	−19.7360	−0.697337
$802$	0	0
$803$	2.77193i	0.0978192i
$804$	0	0
$805$	− 40.3200i	− 1.42109i
$806$	0	0
$807$	−16.9706	−0.597392
$808$	0	0
$809$	−8.44267	−0.296828	−0.148414	−	0.988925i	$-0.547417\pi$
$809$	−8.44267	−0.296828	−0.148414	+	0.988925i	$0.547417\pi$
$810$	0	0
$811$	4.41613i	0.155071i	0.996990	+	0.0775357i	$0.0247052\pi$
$811$	4.41613i	0.155071i	−0.996990	+	0.0775357i	$0.975295\pi$
$812$	0	0
$813$	59.0267i	2.07016i
$814$	0	0
$815$	59.1490	2.07190
$816$	0	0
$817$	−2.54669	−0.0890973
$818$	0	0
$819$	− 6.17068i	− 0.215621i
$820$	0	0
$821$	− 31.5360i	− 1.10062i	−0.834962	−	0.550308i	$-0.814510\pi$
$821$	− 31.5360i	− 1.10062i	0.834962	−	0.550308i	$-0.185490\pi$
$822$	0	0
$823$	16.1710	0.563687	0.281844	−	0.959460i	$-0.409054\pi$
$823$	16.1710	0.563687	0.281844	+	0.959460i	$0.409054\pi$
$824$	0	0
$825$	−100.640	−3.50383
$826$	0	0
$827$	− 36.5000i	− 1.26923i	−0.772829	−	0.634614i	$-0.781159\pi$
$827$	− 36.5000i	− 1.26923i	0.772829	−	0.634614i	$-0.218841\pi$
$828$	0	0
$829$	23.1893i	0.805398i	0.915333	+	0.402699i	$0.131928\pi$
$829$	23.1893i	0.805398i	−0.915333	+	0.402699i	$0.868072\pi$
$830$	0	0
$831$	−21.2264	−0.736336
$832$	0	0
$833$	15.2920	0.529836
$834$	0	0
$835$	− 77.0746i	− 2.66728i
$836$	0	0
$837$	1.55602i	0.0537838i
$838$	0	0
$839$	4.15203	0.143344	0.0716720	−	0.997428i	$-0.477167\pi$
$839$	4.15203	0.143344	0.0716720	+	0.997428i	$0.477167\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	− 30.4848i	− 1.04995i
$844$	0	0
$845$	− 4.14134i	− 0.142466i
$846$	0	0
$847$	0.335198	0.0115175
$848$	0	0
$849$	25.5560	0.877080
$850$	0	0
$851$	12.8137i	0.439249i
$852$	0	0
$853$	7.89598i	0.270353i	0.990822	+	0.135177i	$0.0431602\pi$
$853$	7.89598i	0.270353i	−0.990822	+	0.135177i	$0.956840\pi$
$854$	0	0
$855$	−94.5901	−3.23491
$856$	0	0
$857$	14.5653	0.497543	0.248771	−	0.968562i	$-0.419973\pi$
$857$	14.5653	0.497543	0.248771	+	0.968562i	$0.419973\pi$
$858$	0	0
$859$	21.3980i	0.730091i	0.930990	+	0.365046i	$0.118947\pi$
$859$	21.3980i	0.730091i	−0.930990	+	0.365046i	$0.881053\pi$
$860$	0	0
$861$	57.1307i	1.94701i
$862$	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
$867$	16.5972i	0.563670i
$868$	0	0
$869$	38.9066i	1.31982i
$870$	0	0
$871$	5.54279	0.187810
$872$	0	0
$873$	−45.7546	−1.54856
$874$	0	0
$875$	58.1708i	1.96653i
$876$	0	0
$877$	− 0.746632i	− 0.0252120i	−0.999921	−	0.0126060i	$-0.995987\pi$
$877$	− 0.746632i	− 0.0252120i	0.999921	−	0.0126060i	$-0.00401272\pi$
$878$	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
$883$	− 29.2888i	− 0.985647i	−0.870129	−	0.492823i	$-0.835965\pi$
$883$	− 29.2888i	− 0.985647i	0.870129	−	0.492823i	$-0.164035\pi$
$884$	0	0
$885$	− 27.1120i	− 0.911361i
$886$	0	0
$887$	−33.7130	−1.13197	−0.565986	−	0.824415i	$-0.691504\pi$
$887$	−33.7130	−1.13197	−0.565986	+	0.824415i	$0.691504\pi$
$888$	0	0
$889$	31.9600	1.07191
$890$	0	0
$891$	28.5964i	0.958014i
$892$	0	0
$893$	− 34.8480i	− 1.16614i
$894$	0	0
$895$	−2.50456	−0.0837181
$896$	0	0
$897$	12.2827	0.410106
$898$	0	0
$899$	8.88504i	0.296333i
$900$	0	0
$901$	− 66.8667i	− 2.22765i
$902$	0	0
$903$	−1.70504	−0.0567402
$904$	0	0
$905$	18.8294	0.625909
$906$	0	0
$907$	18.8208i	0.624936i	0.949928	+	0.312468i	$0.101156\pi$
$907$	18.8208i	0.624936i	−0.949928	+	0.312468i	$0.898844\pi$
$908$	0	0
$909$	18.9066i	0.627093i
$910$	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
$915$	− 14.9153i	− 0.493084i
$916$	0	0
$917$	27.8773i	0.920590i
$918$	0	0
$919$	8.23077	0.271508	0.135754	−	0.990743i	$-0.456654\pi$
$919$	8.23077	0.271508	0.135754	+	0.990743i	$0.456654\pi$
$920$	0	0
$921$	−76.6400	−2.52537
$922$	0	0
$923$	− 0.936701i	− 0.0308319i
$924$	0	0
$925$	− 31.4134i	− 1.03286i
$926$	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
$931$	22.8405i	0.748567i
$932$	0	0
$933$	5.45331i	0.178533i
$934$	0	0
$935$	−67.3798	−2.20355
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	0	0
$939$	− 52.8610i	− 1.72505i
$940$	0	0
$941$	− 21.8773i	− 0.713180i	−0.934261	−	0.356590i	$-0.883939\pi$
$941$	− 21.8773i	− 0.713180i	0.934261	−	0.356590i	$-0.116061\pi$
$942$	0	0
$943$	−58.1676	−1.89420
$944$	0	0
$945$	−2.84934	−0.0926889
$946$	0	0
$947$	9.42525i	0.306279i	0.988205	+	0.153140i	$0.0489385\pi$
$947$	9.42525i	0.306279i	−0.988205	+	0.153140i	$0.951062\pi$
$948$	0	0
$949$	0.829359i	0.0269221i
$950$	0	0
$951$	−43.2985	−1.40405
$952$	0	0
$953$	7.99868	0.259103	0.129551	−	0.991573i	$-0.458646\pi$
$953$	7.99868	0.259103	0.129551	+	0.991573i	$0.458646\pi$
$954$	0	0
$955$	57.3218i	1.85489i
$956$	0	0
$957$	− 16.5653i	− 0.535482i
$958$	0	0
$959$	−20.7540	−0.670182
$960$	0	0
$961$	−11.2640	−0.363355
$962$	0	0
$963$	− 46.9368i	− 1.51252i
$964$	0	0
$965$	− 43.7546i	− 1.40851i
$966$	0	0
$967$	−21.2345	−0.682854	−0.341427	−	0.939908i	$-0.610910\pi$
$967$	−21.2345	−0.682854	−0.341427	+	0.939908i	$0.610910\pi$
$968$	0	0
$969$	−87.7147	−2.81780
$970$	0	0
$971$	21.3485i	0.685107i	0.939498	+	0.342553i	$0.111292\pi$
$971$	21.3485i	0.685107i	−0.939498	+	0.342553i	$0.888708\pi$
$972$	0	0
$973$	− 0.545369i	− 0.0174837i
$974$	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
$979$	− 20.9983i	− 0.671108i
$980$	0	0
$981$	− 35.9160i	− 1.14671i
$982$	0	0
$983$	33.1497	1.05731	0.528655	−	0.848837i	$-0.322696\pi$
$983$	33.1497	1.05731	0.528655	+	0.848837i	$0.322696\pi$
$984$	0	0
$985$	−3.51738	−0.112073
$986$	0	0
$987$	− 23.3312i	− 0.742640i
$988$	0	0
$989$	− 1.73599i	− 0.0552011i
$990$	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
$995$	− 14.7238i	− 0.466775i
$996$	0	0
$997$	7.15198i	0.226506i	0.993566	+	0.113253i	$0.0361270\pi$
$997$	7.15198i	0.226506i	−0.993566	+	0.113253i	$0.963873\pi$
$998$	0	0
$999$	0.905522	0.0286494

Twists

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

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

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:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.47817i q^{3} +4.14134i q^{5} -1.96435 q^{7} -3.14134 q^{9} -3.34225i q^{11} -1.00000i q^{13} -10.2629 q^{15} -4.86799 q^{17} -7.27095i q^{19} -4.86799i q^{21} +4.95634 q^{23} -12.1507 q^{25} -0.350255i q^{27} -2.00000i q^{29} -4.44252 q^{31} +8.28267 q^{33} -8.13503i q^{35} +2.58532i q^{37} +2.47817 q^{39} -11.7360 q^{41} -0.350255i q^{43} -13.0093i q^{45} +4.79278 q^{47} -3.14134 q^{49} -12.0637i q^{51} +13.7360i q^{53} +13.8414 q^{55} +18.0187 q^{57} +2.64174i q^{59} +1.45331i q^{61} +6.17068 q^{63} +4.14134 q^{65} +5.54279i q^{67} +12.2827i q^{69} +0.936701 q^{71} -0.829359 q^{73} -30.1114i q^{75} +6.56534i q^{77} -11.6408 q^{79} -8.55602 q^{81} -8.99911i q^{83} -20.1600i q^{85} +4.95634 q^{87} +6.28267 q^{89} +1.96435i q^{91} -11.0093i q^{93} +30.1114 q^{95} +14.5653 q^{97} +10.4991i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) \) 12 * q - 4 * q^9 - 8 * q^17 - 28 * q^25 + 32 * q^33 - 40 * q^41 - 4 * q^49 + 48 * q^57 + 16 * q^65 + 24 * q^73 - 52 * q^81 + 8 * q^89 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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