LMFDB - Embedded newform 2112.2.f.g.1057.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,2,Mod(1057,2112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2112.1057"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,-24,0,0,0, 0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)

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

Self dual:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1057.5
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2112.1057
Dual form			2112.2.f.g.1057.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.82843i q^{5} -1.00000 q^{9} -1.00000i q^{11} -0.635674i q^{13} +2.82843 q^{15} +2.89898 q^{17} +4.89898i q^{19} +0.635674 q^{23} -3.00000 q^{25} -1.00000i q^{27} -6.29253i q^{29} +2.19275 q^{31} +1.00000 q^{33} -6.92820i q^{37} +0.635674 q^{39} -1.10102 q^{41} +0.898979i q^{43} +2.82843i q^{45} +6.29253 q^{47} -7.00000 q^{49} +2.89898i q^{51} -4.09978i q^{53} -2.82843 q^{55} -4.89898 q^{57} -13.7980i q^{59} -11.9494i q^{61} -1.79796 q^{65} -5.79796i q^{67} +0.635674i q^{69} -5.02118 q^{71} -11.7980 q^{73} -3.00000i q^{75} +6.92820 q^{79} +1.00000 q^{81} -9.79796i q^{83} -8.19955i q^{85} +6.29253 q^{87} +3.79796 q^{89} +2.19275i q^{93} +13.8564 q^{95} -6.00000 q^{97} +1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 16 q^{17} - 24 q^{25} + 8 q^{33} - 48 q^{41} - 56 q^{49} + 64 q^{65} - 16 q^{73} + 8 q^{81} - 48 q^{89} - 48 q^{97}+O(q^{100}) $ 8 * q - 8 * q^9 - 16 * q^17 - 24 * q^25 + 8 * q^33 - 48 * q^41 - 56 * q^49 + 64 * q^65 - 16 * q^73 + 8 * q^81 - 48 * q^89 - 48 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	− 0.635674i	− 0.176304i	−0.996107	−	0.0881522i	$-0.971904\pi$
$13$	− 0.635674i	− 0.176304i	0.996107	−	0.0881522i	$-0.0280962\pi$
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	2.89898	0.703106	0.351553	−	0.936168i	$-0.385654\pi$
$17$	2.89898	0.703106	0.351553	+	0.936168i	$0.385654\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.635674	0.132547	0.0662736	−	0.997801i	$-0.478889\pi$
$23$	0.635674	0.132547	0.0662736	+	0.997801i	$0.478889\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 6.29253i	− 1.16849i	−0.811576	−	0.584247i	$-0.801390\pi$
$29$	− 6.29253i	− 1.16849i	0.811576	−	0.584247i	$-0.198610\pi$
$30$	0	0
$31$	2.19275	0.393830	0.196915	−	0.980421i	$-0.436908\pi$
$31$	2.19275	0.393830	0.196915	+	0.980421i	$0.436908\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.92820i	− 1.13899i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	− 6.92820i	− 1.13899i	0.821995	−	0.569495i	$-0.192861\pi$
$38$	0	0
$39$	0.635674	0.101789
$40$	0	0
$41$	−1.10102	−0.171951	−0.0859753	−	0.996297i	$-0.527401\pi$
$41$	−1.10102	−0.171951	−0.0859753	+	0.996297i	$0.527401\pi$
$42$	0	0
$43$	0.898979i	0.137093i	0.997648	+	0.0685465i	$0.0218362\pi$
$43$	0.898979i	0.137093i	−0.997648	+	0.0685465i	$0.978164\pi$
$44$	0	0
$45$	2.82843i	0.421637i
$46$	0	0
$47$	6.29253	0.917860	0.458930	−	0.888473i	$-0.348233\pi$
$47$	6.29253	0.917860	0.458930	+	0.888473i	$0.348233\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	2.89898i	0.405938i
$52$	0	0
$53$	− 4.09978i	− 0.563148i	−0.959540	−	0.281574i	$-0.909144\pi$
$53$	− 4.09978i	− 0.563148i	0.959540	−	0.281574i	$-0.0908564\pi$
$54$	0	0
$55$	−2.82843	−0.381385
$56$	0	0
$57$	−4.89898	−0.648886
$58$	0	0
$59$	− 13.7980i	− 1.79634i	−0.439647	−	0.898171i	$-0.644896\pi$
$59$	− 13.7980i	− 1.79634i	0.439647	−	0.898171i	$-0.355104\pi$
$60$	0	0
$61$	− 11.9494i	− 1.52996i	−0.644053	−	0.764981i	$-0.722748\pi$
$61$	− 11.9494i	− 1.52996i	0.644053	−	0.764981i	$-0.277252\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.79796	−0.223009
$66$	0	0
$67$	− 5.79796i	− 0.708333i	−0.935182	−	0.354167i	$-0.884765\pi$
$67$	− 5.79796i	− 0.708333i	0.935182	−	0.354167i	$-0.115235\pi$
$68$	0	0
$69$	0.635674i	0.0765262i
$70$	0	0
$71$	−5.02118	−0.595904	−0.297952	−	0.954581i	$-0.596304\pi$
$71$	−5.02118	−0.595904	−0.297952	+	0.954581i	$0.596304\pi$
$72$	0	0
$73$	−11.7980	−1.38085	−0.690423	−	0.723406i	$-0.742576\pi$
$73$	−11.7980	−1.38085	−0.690423	+	0.723406i	$0.742576\pi$
$74$	0	0
$75$	− 3.00000i	− 0.346410i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.92820	0.779484	0.389742	−	0.920924i	$-0.372564\pi$
$79$	6.92820	0.779484	0.389742	+	0.920924i	$0.372564\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 9.79796i	− 1.07547i	−0.843115	−	0.537733i	$-0.819281\pi$
$83$	− 9.79796i	− 1.07547i	0.843115	−	0.537733i	$-0.180719\pi$
$84$	0	0
$85$	− 8.19955i	− 0.889366i
$86$	0	0
$87$	6.29253	0.674630
$88$	0	0
$89$	3.79796	0.402583	0.201291	−	0.979531i	$-0.435486\pi$
$89$	3.79796	0.402583	0.201291	+	0.979531i	$0.435486\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.19275i	0.227378i
$94$	0	0
$95$	13.8564	1.42164
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	1.00000i	0.100504i
$100$	0	0
$101$	11.9494i	1.18901i	0.804093	+	0.594504i	$0.202652\pi$
$101$	11.9494i	1.18901i	−0.804093	+	0.594504i	$0.797348\pi$
$102$	0	0
$103$	−2.19275	−0.216058	−0.108029	−	0.994148i	$-0.534454\pi$
$103$	−2.19275	−0.216058	−0.108029	+	0.994148i	$0.534454\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	0	0
$109$	5.02118i	0.480942i	0.970656	+	0.240471i	$0.0773019\pi$
$109$	5.02118i	0.480942i	−0.970656	+	0.240471i	$0.922698\pi$
$110$	0	0
$111$	6.92820	0.657596
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	− 1.79796i	− 0.167661i
$116$	0	0
$117$	0.635674i	0.0587681i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 1.10102i	− 0.0992757i
$124$	0	0
$125$	− 5.65685i	− 0.505964i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−0.898979	−0.0791507
$130$	0	0
$131$	5.79796i	0.506570i	0.967392	+	0.253285i	$0.0815110\pi$
$131$	5.79796i	0.506570i	−0.967392	+	0.253285i	$0.918489\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.82843	−0.243432
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	− 7.10102i	− 0.602301i	−0.953577	−	0.301150i	$-0.902629\pi$
$139$	− 7.10102i	− 0.602301i	0.953577	−	0.301150i	$-0.0973706\pi$
$140$	0	0
$141$	6.29253i	0.529927i
$142$	0	0
$143$	−0.635674	−0.0531578
$144$	0	0
$145$	−17.7980	−1.47804
$146$	0	0
$147$	− 7.00000i	− 0.577350i
$148$	0	0
$149$	6.29253i	0.515504i	0.966211	+	0.257752i	$0.0829818\pi$
$149$	6.29253i	0.515504i	−0.966211	+	0.257752i	$0.917018\pi$
$150$	0	0
$151$	−6.92820	−0.563809	−0.281905	−	0.959442i	$-0.590966\pi$
$151$	−6.92820	−0.563809	−0.281905	+	0.959442i	$0.590966\pi$
$152$	0	0
$153$	−2.89898	−0.234369
$154$	0	0
$155$	− 6.20204i	− 0.498160i
$156$	0	0
$157$	− 5.65685i	− 0.451466i	−0.974189	−	0.225733i	$-0.927522\pi$
$157$	− 5.65685i	− 0.451466i	0.974189	−	0.225733i	$-0.0724777\pi$
$158$	0	0
$159$	4.09978	0.325133
$160$	0	0
$161$	0	0
$162$	0	0
$163$	21.7980i	1.70735i	0.520808	+	0.853674i	$0.325631\pi$
$163$	21.7980i	1.70735i	−0.520808	+	0.853674i	$0.674369\pi$
$164$	0	0
$165$	− 2.82843i	− 0.220193i
$166$	0	0
$167$	23.8988	1.84934	0.924671	−	0.380767i	$-0.124340\pi$
$167$	23.8988	1.84934	0.924671	+	0.380767i	$0.124340\pi$
$168$	0	0
$169$	12.5959	0.968917
$170$	0	0
$171$	− 4.89898i	− 0.374634i
$172$	0	0
$173$	− 18.8776i	− 1.43524i	−0.696437	−	0.717618i	$-0.745232\pi$
$173$	− 18.8776i	− 1.43524i	0.696437	−	0.717618i	$-0.254768\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.7980	1.03712
$178$	0	0
$179$	4.00000i	0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$	4.00000i	0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	− 23.8988i	− 1.77638i	−0.459475	−	0.888191i	$-0.651962\pi$
$181$	− 23.8988i	− 1.77638i	0.459475	−	0.888191i	$-0.348038\pi$
$182$	0	0
$183$	11.9494	0.883324
$184$	0	0
$185$	−19.5959	−1.44072
$186$	0	0
$187$	− 2.89898i	− 0.211994i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.90702	0.137987	0.0689937	−	0.997617i	$-0.478021\pi$
$191$	1.90702	0.137987	0.0689937	+	0.997617i	$0.478021\pi$
$192$	0	0
$193$	17.5959	1.26658	0.633291	−	0.773914i	$-0.281704\pi$
$193$	17.5959	1.26658	0.633291	+	0.773914i	$0.281704\pi$
$194$	0	0
$195$	− 1.79796i	− 0.128755i
$196$	0	0
$197$	14.4921i	1.03252i	0.856433	+	0.516259i	$0.172676\pi$
$197$	14.4921i	1.03252i	−0.856433	+	0.516259i	$0.827324\pi$
$198$	0	0
$199$	3.46410	0.245564	0.122782	−	0.992434i	$-0.460818\pi$
$199$	3.46410	0.245564	0.122782	+	0.992434i	$0.460818\pi$
$200$	0	0
$201$	5.79796	0.408956
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.11416i	0.217502i
$206$	0	0
$207$	−0.635674	−0.0441824
$208$	0	0
$209$	4.89898	0.338869
$210$	0	0
$211$	14.6969i	1.01178i	0.862598	+	0.505889i	$0.168836\pi$
$211$	14.6969i	1.01178i	−0.862598	+	0.505889i	$0.831164\pi$
$212$	0	0
$213$	− 5.02118i	− 0.344046i
$214$	0	0
$215$	2.54270	0.173411
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 11.7980i	− 0.797232i
$220$	0	0
$221$	− 1.84281i	− 0.123961i
$222$	0	0
$223$	−24.2487	−1.62381	−0.811907	−	0.583787i	$-0.801570\pi$
$223$	−24.2487	−1.62381	−0.811907	+	0.583787i	$0.801570\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	13.7980i	0.915803i	0.889003	+	0.457901i	$0.151399\pi$
$227$	13.7980i	0.915803i	−0.889003	+	0.457901i	$0.848601\pi$
$228$	0	0
$229$	− 15.1278i	− 0.999670i	−0.866121	−	0.499835i	$-0.833394\pi$
$229$	− 15.1278i	− 0.999670i	0.866121	−	0.499835i	$-0.166606\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.10102	0.0721303	0.0360651	−	0.999349i	$-0.488518\pi$
$233$	1.10102	0.0721303	0.0360651	+	0.999349i	$0.488518\pi$
$234$	0	0
$235$	− 17.7980i	− 1.16101i
$236$	0	0
$237$	6.92820i	0.450035i
$238$	0	0
$239$	12.5851	0.814060	0.407030	−	0.913415i	$-0.366564\pi$
$239$	12.5851	0.814060	0.407030	+	0.913415i	$0.366564\pi$
$240$	0	0
$241$	7.79796	0.502311	0.251155	−	0.967947i	$-0.419189\pi$
$241$	7.79796	0.502311	0.251155	+	0.967947i	$0.419189\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	19.7990i	1.26491i
$246$	0	0
$247$	3.11416	0.198149
$248$	0	0
$249$	9.79796	0.620920
$250$	0	0
$251$	21.7980i	1.37587i	0.725770	+	0.687937i	$0.241484\pi$
$251$	21.7980i	1.37587i	−0.725770	+	0.687937i	$0.758516\pi$
$252$	0	0
$253$	− 0.635674i	− 0.0399645i
$254$	0	0
$255$	8.19955	0.513476
$256$	0	0
$257$	23.7980	1.48448	0.742238	−	0.670136i	$-0.233764\pi$
$257$	23.7980	1.48448	0.742238	+	0.670136i	$0.233764\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.29253i	0.389498i
$262$	0	0
$263$	11.3137	0.697633	0.348817	−	0.937191i	$-0.386584\pi$
$263$	11.3137	0.697633	0.348817	+	0.937191i	$0.386584\pi$
$264$	0	0
$265$	−11.5959	−0.712332
$266$	0	0
$267$	3.79796i	0.232431i
$268$	0	0
$269$	12.8708i	0.784746i	0.919806	+	0.392373i	$0.128346\pi$
$269$	12.8708i	0.784746i	−0.919806	+	0.392373i	$0.871654\pi$
$270$	0	0
$271$	−6.92820	−0.420858	−0.210429	−	0.977609i	$-0.567486\pi$
$271$	−6.92820	−0.420858	−0.210429	+	0.977609i	$0.567486\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000i	0.180907i
$276$	0	0
$277$	− 17.6062i	− 1.05786i	−0.848667	−	0.528928i	$-0.822594\pi$
$277$	− 17.6062i	− 1.05786i	0.848667	−	0.528928i	$-0.177406\pi$
$278$	0	0
$279$	−2.19275	−0.131277
$280$	0	0
$281$	1.10102	0.0656814	0.0328407	−	0.999461i	$-0.489545\pi$
$281$	1.10102	0.0656814	0.0328407	+	0.999461i	$0.489545\pi$
$282$	0	0
$283$	11.1010i	0.659887i	0.944001	+	0.329944i	$0.107030\pi$
$283$	11.1010i	0.659887i	−0.944001	+	0.329944i	$0.892970\pi$
$284$	0	0
$285$	13.8564i	0.820783i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.59592	−0.505642
$290$	0	0
$291$	− 6.00000i	− 0.351726i
$292$	0	0
$293$	10.6780i	0.623817i	0.950112	+	0.311909i	$0.100968\pi$
$293$	10.6780i	0.623817i	−0.950112	+	0.311909i	$0.899032\pi$
$294$	0	0
$295$	−39.0265	−2.27221
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	− 0.404082i	− 0.0233687i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.9494	−0.686474
$304$	0	0
$305$	−33.7980	−1.93527
$306$	0	0
$307$	26.6969i	1.52367i	0.647768	+	0.761837i	$0.275702\pi$
$307$	26.6969i	1.52367i	−0.647768	+	0.761837i	$0.724298\pi$
$308$	0	0
$309$	− 2.19275i	− 0.124741i
$310$	0	0
$311$	−6.29253	−0.356817	−0.178408	−	0.983957i	$-0.557095\pi$
$311$	−6.29253	−0.356817	−0.178408	+	0.983957i	$0.557095\pi$
$312$	0	0
$313$	17.5959	0.994580	0.497290	−	0.867584i	$-0.334328\pi$
$313$	17.5959	0.994580	0.497290	+	0.867584i	$0.334328\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.82843i	− 0.158860i	−0.996840	−	0.0794301i	$-0.974690\pi$
$317$	− 2.82843i	− 0.158860i	0.996840	−	0.0794301i	$-0.0253101\pi$
$318$	0	0
$319$	−6.29253	−0.352314
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	14.2020i	0.790223i
$324$	0	0
$325$	1.90702i	0.105783i
$326$	0	0
$327$	−5.02118	−0.277672
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 33.3939i	− 1.83549i	−0.397166	−	0.917747i	$-0.630006\pi$
$331$	− 33.3939i	− 1.83549i	0.397166	−	0.917747i	$-0.369994\pi$
$332$	0	0
$333$	6.92820i	0.379663i
$334$	0	0
$335$	−16.3991	−0.895979
$336$	0	0
$337$	−31.7980	−1.73215	−0.866073	−	0.499918i	$-0.833363\pi$
$337$	−31.7980	−1.73215	−0.866073	+	0.499918i	$0.833363\pi$
$338$	0	0
$339$	6.00000i	0.325875i
$340$	0	0
$341$	− 2.19275i	− 0.118744i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.79796	0.0967989
$346$	0	0
$347$	− 25.7980i	− 1.38491i	−0.721463	−	0.692453i	$-0.756530\pi$
$347$	− 25.7980i	− 1.38491i	0.721463	−	0.692453i	$-0.243470\pi$
$348$	0	0
$349$	− 30.1913i	− 1.61610i	−0.589112	−	0.808051i	$-0.700522\pi$
$349$	− 30.1913i	− 1.61610i	0.589112	−	0.808051i	$-0.299478\pi$
$350$	0	0
$351$	−0.635674	−0.0339298
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	14.2020i	0.753766i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.3137	0.597115	0.298557	−	0.954392i	$-0.403495\pi$
$359$	11.3137	0.597115	0.298557	+	0.954392i	$0.403495\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	33.3697i	1.74665i
$366$	0	0
$367$	−26.0915	−1.36197	−0.680983	−	0.732299i	$-0.738447\pi$
$367$	−26.0915	−1.36197	−0.680983	+	0.732299i	$0.738447\pi$
$368$	0	0
$369$	1.10102	0.0573168
$370$	0	0
$371$	0	0
$372$	0	0
$373$	21.4203i	1.10910i	0.832150	+	0.554550i	$0.187110\pi$
$373$	21.4203i	1.10910i	−0.832150	+	0.554550i	$0.812890\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	25.3939i	1.30440i	0.758049	+	0.652198i	$0.226153\pi$
$379$	25.3939i	1.30440i	−0.758049	+	0.652198i	$0.773847\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.7340	−1.67263	−0.836314	−	0.548250i	$-0.815294\pi$
$383$	−32.7340	−1.67263	−0.836314	+	0.548250i	$0.815294\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 0.898979i	− 0.0456977i
$388$	0	0
$389$	− 2.82843i	− 0.143407i	−0.997426	−	0.0717035i	$-0.977156\pi$
$389$	− 2.82843i	− 0.143407i	0.997426	−	0.0717035i	$-0.0228435\pi$
$390$	0	0
$391$	1.84281	0.0931948
$392$	0	0
$393$	−5.79796	−0.292468
$394$	0	0
$395$	− 19.5959i	− 0.985978i
$396$	0	0
$397$	16.9706i	0.851728i	0.904787	+	0.425864i	$0.140030\pi$
$397$	16.9706i	0.851728i	−0.904787	+	0.425864i	$0.859970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.20204	0.209840	0.104920	−	0.994481i	$-0.466541\pi$
$401$	4.20204	0.209840	0.104920	+	0.994481i	$0.466541\pi$
$402$	0	0
$403$	− 1.39388i	− 0.0694340i
$404$	0	0
$405$	− 2.82843i	− 0.140546i
$406$	0	0
$407$	−6.92820	−0.343418
$408$	0	0
$409$	8.20204	0.405565	0.202782	−	0.979224i	$-0.435002\pi$
$409$	8.20204	0.405565	0.202782	+	0.979224i	$0.435002\pi$
$410$	0	0
$411$	− 2.00000i	− 0.0986527i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−27.7128	−1.36037
$416$	0	0
$417$	7.10102	0.347738
$418$	0	0
$419$	21.7980i	1.06490i	0.846461	+	0.532450i	$0.178729\pi$
$419$	21.7980i	1.06490i	−0.846461	+	0.532450i	$0.821271\pi$
$420$	0	0
$421$	25.1701i	1.22672i	0.789805	+	0.613358i	$0.210182\pi$
$421$	25.1701i	1.22672i	−0.789805	+	0.613358i	$0.789818\pi$
$422$	0	0
$423$	−6.29253	−0.305953
$424$	0	0
$425$	−8.69694	−0.421863
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 0.635674i	− 0.0306907i
$430$	0	0
$431$	−12.5851	−0.606201	−0.303100	−	0.952959i	$-0.598022\pi$
$431$	−12.5851	−0.606201	−0.303100	+	0.952959i	$0.598022\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	− 17.7980i	− 0.853347i
$436$	0	0
$437$	3.11416i	0.148970i
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	− 10.7423i	− 0.509231i
$446$	0	0
$447$	−6.29253	−0.297626
$448$	0	0
$449$	11.7980	0.556780	0.278390	−	0.960468i	$-0.410199\pi$
$449$	11.7980	0.556780	0.278390	+	0.960468i	$0.410199\pi$
$450$	0	0
$451$	1.10102i	0.0518450i
$452$	0	0
$453$	− 6.92820i	− 0.325515i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.7980	1.30033	0.650167	−	0.759791i	$-0.274699\pi$
$457$	27.7980	1.30033	0.650167	+	0.759791i	$0.274699\pi$
$458$	0	0
$459$	− 2.89898i	− 0.135313i
$460$	0	0
$461$	40.9335i	1.90647i	0.302238	+	0.953233i	$0.402266\pi$
$461$	40.9335i	1.90647i	−0.302238	+	0.953233i	$0.597734\pi$
$462$	0	0
$463$	−31.7484	−1.47547	−0.737736	−	0.675089i	$-0.764105\pi$
$463$	−31.7484	−1.47547	−0.737736	+	0.675089i	$0.764105\pi$
$464$	0	0
$465$	6.20204	0.287613
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.65685	0.260654
$472$	0	0
$473$	0.898979	0.0413351
$474$	0	0
$475$	− 14.6969i	− 0.674342i
$476$	0	0
$477$	4.09978i	0.187716i
$478$	0	0
$479$	15.1278	0.691205	0.345602	−	0.938381i	$-0.387675\pi$
$479$	15.1278	0.691205	0.345602	+	0.938381i	$0.387675\pi$
$480$	0	0
$481$	−4.40408	−0.200809
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.9706i	0.770594i
$486$	0	0
$487$	−17.3205	−0.784867	−0.392434	−	0.919780i	$-0.628367\pi$
$487$	−17.3205	−0.784867	−0.392434	+	0.919780i	$0.628367\pi$
$488$	0	0
$489$	−21.7980	−0.985738
$490$	0	0
$491$	− 24.0000i	− 1.08310i	−0.840667	−	0.541552i	$-0.817837\pi$
$491$	− 24.0000i	− 1.08310i	0.840667	−	0.541552i	$-0.182163\pi$
$492$	0	0
$493$	− 18.2419i	− 0.821574i
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	23.8988i	1.06772i
$502$	0	0
$503$	−27.7128	−1.23565	−0.617827	−	0.786314i	$-0.711987\pi$
$503$	−27.7128	−1.23565	−0.617827	+	0.786314i	$0.711987\pi$
$504$	0	0
$505$	33.7980	1.50399
$506$	0	0
$507$	12.5959i	0.559404i
$508$	0	0
$509$	− 18.5276i	− 0.821223i	−0.911810	−	0.410612i	$-0.865315\pi$
$509$	− 18.5276i	− 0.821223i	0.911810	−	0.410612i	$-0.134685\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.89898	0.216295
$514$	0	0
$515$	6.20204i	0.273295i
$516$	0	0
$517$	− 6.29253i	− 0.276745i
$518$	0	0
$519$	18.8776	0.828634
$520$	0	0
$521$	0.202041	0.00885158	0.00442579	−	0.999990i	$-0.498591\pi$
$521$	0.202041	0.00885158	0.00442579	+	0.999990i	$0.498591\pi$
$522$	0	0
$523$	28.4949i	1.24599i	0.782224	+	0.622997i	$0.214085\pi$
$523$	28.4949i	1.24599i	−0.782224	+	0.622997i	$0.785915\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.35674	0.276904
$528$	0	0
$529$	−22.5959	−0.982431
$530$	0	0
$531$	13.7980i	0.598780i
$532$	0	0
$533$	0.699891i	0.0303156i
$534$	0	0
$535$	−22.6274	−0.978269
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	7.00000i	0.301511i
$540$	0	0
$541$	41.5050i	1.78444i	0.451602	+	0.892220i	$0.350853\pi$
$541$	41.5050i	1.78444i	−0.451602	+	0.892220i	$0.649147\pi$
$542$	0	0
$543$	23.8988	1.02559
$544$	0	0
$545$	14.2020	0.608349
$546$	0	0
$547$	30.6969i	1.31251i	0.754541	+	0.656253i	$0.227860\pi$
$547$	30.6969i	1.31251i	−0.754541	+	0.656253i	$0.772140\pi$
$548$	0	0
$549$	11.9494i	0.509987i
$550$	0	0
$551$	30.8270	1.31327
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 19.5959i	− 0.831800i
$556$	0	0
$557$	− 37.1195i	− 1.57280i	−0.617715	−	0.786402i	$-0.711942\pi$
$557$	− 37.1195i	− 1.57280i	0.617715	−	0.786402i	$-0.288058\pi$
$558$	0	0
$559$	0.571458	0.0241701
$560$	0	0
$561$	2.89898	0.122395
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	− 16.9706i	− 0.713957i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.69694	0.364595	0.182297	−	0.983243i	$-0.441647\pi$
$569$	8.69694	0.364595	0.182297	+	0.983243i	$0.441647\pi$
$570$	0	0
$571$	10.6969i	0.447653i	0.974629	+	0.223826i	$0.0718549\pi$
$571$	10.6969i	0.447653i	−0.974629	+	0.223826i	$0.928145\pi$
$572$	0	0
$573$	1.90702i	0.0796670i
$574$	0	0
$575$	−1.90702	−0.0795284
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	17.5959i	0.731261i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.09978	−0.169795
$584$	0	0
$585$	1.79796	0.0743365
$586$	0	0
$587$	− 2.20204i	− 0.0908880i	−0.998967	−	0.0454440i	$-0.985530\pi$
$587$	− 2.20204i	− 0.0908880i	0.998967	−	0.0454440i	$-0.0144703\pi$
$588$	0	0
$589$	10.7423i	0.442627i
$590$	0	0
$591$	−14.4921	−0.596125
$592$	0	0
$593$	22.8990	0.940348	0.470174	−	0.882574i	$-0.344191\pi$
$593$	22.8990	0.940348	0.470174	+	0.882574i	$0.344191\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.46410i	0.141776i
$598$	0	0
$599$	−24.5344	−1.00245	−0.501225	−	0.865317i	$-0.667117\pi$
$599$	−24.5344	−1.00245	−0.501225	+	0.865317i	$0.667117\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	5.79796i	0.236111i
$604$	0	0
$605$	2.82843i	0.114992i
$606$	0	0
$607$	36.4838	1.48083	0.740416	−	0.672149i	$-0.234628\pi$
$607$	36.4838	1.48083	0.740416	+	0.672149i	$0.234628\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.00000i	− 0.161823i
$612$	0	0
$613$	9.40669i	0.379932i	0.981791	+	0.189966i	$0.0608378\pi$
$613$	9.40669i	0.379932i	−0.981791	+	0.189966i	$0.939162\pi$
$614$	0	0
$615$	−3.11416	−0.125575
$616$	0	0
$617$	41.1918	1.65832	0.829160	−	0.559011i	$-0.188819\pi$
$617$	41.1918	1.65832	0.829160	+	0.559011i	$0.188819\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	− 0.635674i	− 0.0255087i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	4.89898i	0.195646i
$628$	0	0
$629$	− 20.0847i	− 0.800830i
$630$	0	0
$631$	43.0621	1.71427	0.857137	−	0.515088i	$-0.172241\pi$
$631$	43.0621	1.71427	0.857137	+	0.515088i	$0.172241\pi$
$632$	0	0
$633$	−14.6969	−0.584151
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.44972i	0.176304i
$638$	0	0
$639$	5.02118	0.198635
$640$	0	0
$641$	33.5959	1.32696	0.663479	−	0.748194i	$-0.269079\pi$
$641$	33.5959	1.32696	0.663479	+	0.748194i	$0.269079\pi$
$642$	0	0
$643$	15.5959i	0.615043i	0.951541	+	0.307521i	$0.0994996\pi$
$643$	15.5959i	0.615043i	−0.951541	+	0.307521i	$0.900500\pi$
$644$	0	0
$645$	2.54270i	0.100119i
$646$	0	0
$647$	−29.6198	−1.16448	−0.582238	−	0.813018i	$-0.697823\pi$
$647$	−29.6198	−1.16448	−0.582238	+	0.813018i	$0.697823\pi$
$648$	0	0
$649$	−13.7980	−0.541617
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.9267i	1.36679i	0.730049	+	0.683395i	$0.239497\pi$
$653$	34.9267i	1.36679i	−0.730049	+	0.683395i	$0.760503\pi$
$654$	0	0
$655$	16.3991	0.640766
$656$	0	0
$657$	11.7980	0.460282
$658$	0	0
$659$	− 37.3939i	− 1.45666i	−0.685227	−	0.728329i	$-0.740297\pi$
$659$	− 37.3939i	− 1.45666i	0.685227	−	0.728329i	$-0.259703\pi$
$660$	0	0
$661$	44.6834i	1.73798i	0.494828	+	0.868991i	$0.335231\pi$
$661$	44.6834i	1.73798i	−0.494828	+	0.868991i	$0.664769\pi$
$662$	0	0
$663$	1.84281	0.0715687
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.00000i	− 0.154881i
$668$	0	0
$669$	− 24.2487i	− 0.937509i
$670$	0	0
$671$	−11.9494	−0.461301
$672$	0	0
$673$	9.59592	0.369895	0.184948	−	0.982748i	$-0.440788\pi$
$673$	9.59592	0.369895	0.184948	+	0.982748i	$0.440788\pi$
$674$	0	0
$675$	3.00000i	0.115470i
$676$	0	0
$677$	24.5344i	0.942935i	0.881883	+	0.471468i	$0.156276\pi$
$677$	24.5344i	0.942935i	−0.881883	+	0.471468i	$0.843724\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−13.7980	−0.528739
$682$	0	0
$683$	− 18.2020i	− 0.696482i	−0.937405	−	0.348241i	$-0.886779\pi$
$683$	− 18.2020i	− 0.696482i	0.937405	−	0.348241i	$-0.113221\pi$
$684$	0	0
$685$	5.65685i	0.216137i
$686$	0	0
$687$	15.1278	0.577160
$688$	0	0
$689$	−2.60612	−0.0992854
$690$	0	0
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.0847	−0.761857
$696$	0	0
$697$	−3.19184	−0.120899
$698$	0	0
$699$	1.10102i	0.0416444i
$700$	0	0
$701$	− 40.9335i	− 1.54604i	−0.634382	−	0.773019i	$-0.718746\pi$
$701$	− 40.9335i	− 1.54604i	0.634382	−	0.773019i	$-0.281254\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	0	0
$705$	17.7980	0.670310
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 37.7552i	− 1.41793i	−0.705246	−	0.708963i	$-0.749163\pi$
$709$	− 37.7552i	− 1.41793i	0.705246	−	0.708963i	$-0.250837\pi$
$710$	0	0
$711$	−6.92820	−0.259828
$712$	0	0
$713$	1.39388	0.0522011
$714$	0	0
$715$	1.79796i	0.0672399i
$716$	0	0
$717$	12.5851i	0.469998i
$718$	0	0
$719$	16.3349	0.609189	0.304594	−	0.952482i	$-0.401479\pi$
$719$	16.3349	0.609189	0.304594	+	0.952482i	$0.401479\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.79796i	0.290009i
$724$	0	0
$725$	18.8776i	0.701096i
$726$	0	0
$727$	−14.7778	−0.548079	−0.274039	−	0.961718i	$-0.588360\pi$
$727$	−14.7778	−0.548079	−0.274039	+	0.961718i	$0.588360\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	2.60612i	0.0963909i
$732$	0	0
$733$	− 15.0635i	− 0.556385i	−0.960525	−	0.278192i	$-0.910265\pi$
$733$	− 15.0635i	− 0.556385i	0.960525	−	0.278192i	$-0.0897352\pi$
$734$	0	0
$735$	−19.7990	−0.730297
$736$	0	0
$737$	−5.79796	−0.213571
$738$	0	0
$739$	− 46.6969i	− 1.71777i	−0.512165	−	0.858887i	$-0.671156\pi$
$739$	− 46.6969i	− 1.71777i	0.512165	−	0.858887i	$-0.328844\pi$
$740$	0	0
$741$	3.11416i	0.114401i
$742$	0	0
$743$	26.4415	0.970043	0.485022	−	0.874502i	$-0.338812\pi$
$743$	26.4415	0.970043	0.485022	+	0.874502i	$0.338812\pi$
$744$	0	0
$745$	17.7980	0.652067
$746$	0	0
$747$	9.79796i	0.358489i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−6.57826	−0.240044	−0.120022	−	0.992771i	$-0.538297\pi$
$751$	−6.57826	−0.240044	−0.120022	+	0.992771i	$0.538297\pi$
$752$	0	0
$753$	−21.7980	−0.794362
$754$	0	0
$755$	19.5959i	0.713168i
$756$	0	0
$757$	− 16.3991i	− 0.596036i	−0.954560	−	0.298018i	$-0.903675\pi$
$757$	− 16.3991i	− 0.596036i	0.954560	−	0.298018i	$-0.0963255\pi$
$758$	0	0
$759$	0.635674	0.0230735
$760$	0	0
$761$	48.6969	1.76526	0.882631	−	0.470066i	$-0.155770\pi$
$761$	48.6969	1.76526	0.882631	+	0.470066i	$0.155770\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.19955i	0.296455i
$766$	0	0
$767$	−8.77101	−0.316703
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	23.7980i	0.857063i
$772$	0	0
$773$	− 51.8973i	− 1.86662i	−0.359076	−	0.933308i	$-0.616908\pi$
$773$	− 51.8973i	− 1.86662i	0.359076	−	0.933308i	$-0.383092\pi$
$774$	0	0
$775$	−6.57826	−0.236298
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.39388i	− 0.193256i
$780$	0	0
$781$	5.02118i	0.179672i
$782$	0	0
$783$	−6.29253	−0.224877
$784$	0	0
$785$	−16.0000	−0.571064
$786$	0	0
$787$	22.2929i	0.794655i	0.917677	+	0.397327i	$0.130062\pi$
$787$	22.2929i	0.794655i	−0.917677	+	0.397327i	$0.869938\pi$
$788$	0	0
$789$	11.3137i	0.402779i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−7.59592	−0.269739
$794$	0	0
$795$	− 11.5959i	− 0.411265i
$796$	0	0
$797$	50.6260i	1.79326i	0.442777	+	0.896632i	$0.353993\pi$
$797$	50.6260i	1.79326i	−0.442777	+	0.896632i	$0.646007\pi$
$798$	0	0
$799$	18.2419	0.645352
$800$	0	0
$801$	−3.79796	−0.134194
$802$	0	0
$803$	11.7980i	0.416341i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.8708	−0.453073
$808$	0	0
$809$	−54.4949	−1.91594	−0.957969	−	0.286871i	$-0.907385\pi$
$809$	−54.4949	−1.91594	−0.957969	+	0.286871i	$0.907385\pi$
$810$	0	0
$811$	26.6969i	0.937456i	0.883342	+	0.468728i	$0.155288\pi$
$811$	26.6969i	0.937456i	−0.883342	+	0.468728i	$0.844712\pi$
$812$	0	0
$813$	− 6.92820i	− 0.242983i
$814$	0	0
$815$	61.6539	2.15964
$816$	0	0
$817$	−4.40408	−0.154079
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24.5344i	− 0.856258i	−0.903718	−	0.428129i	$-0.859173\pi$
$821$	− 24.5344i	− 0.856258i	0.903718	−	0.428129i	$-0.140827\pi$
$822$	0	0
$823$	−21.7060	−0.756624	−0.378312	−	0.925678i	$-0.623495\pi$
$823$	−21.7060	−0.756624	−0.378312	+	0.925678i	$0.623495\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	− 27.1918i	− 0.945553i	−0.881182	−	0.472776i	$-0.843252\pi$
$827$	− 27.1918i	− 0.945553i	0.881182	−	0.472776i	$-0.156748\pi$
$828$	0	0
$829$	20.0847i	0.697571i	0.937203	+	0.348786i	$0.113406\pi$
$829$	20.0847i	0.697571i	−0.937203	+	0.348786i	$0.886594\pi$
$830$	0	0
$831$	17.6062	0.610754
$832$	0	0
$833$	−20.2929	−0.703106
$834$	0	0
$835$	− 67.5959i	− 2.33925i
$836$	0	0
$837$	− 2.19275i	− 0.0757926i
$838$	0	0
$839$	4.44972	0.153621	0.0768107	−	0.997046i	$-0.475526\pi$
$839$	4.44972	0.153621	0.0768107	+	0.997046i	$0.475526\pi$
$840$	0	0
$841$	−10.5959	−0.365376
$842$	0	0
$843$	1.10102i	0.0379212i
$844$	0	0
$845$	− 35.6266i	− 1.22559i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−11.1010	−0.380986
$850$	0	0
$851$	− 4.40408i	− 0.150970i
$852$	0	0
$853$	28.9199i	0.990200i	0.868836	+	0.495100i	$0.164869\pi$
$853$	28.9199i	0.990200i	−0.868836	+	0.495100i	$0.835131\pi$
$854$	0	0
$855$	−13.8564	−0.473879
$856$	0	0
$857$	−2.89898	−0.0990273	−0.0495136	−	0.998773i	$-0.515767\pi$
$857$	−2.89898	−0.0990273	−0.0495136	+	0.998773i	$0.515767\pi$
$858$	0	0
$859$	− 21.7980i	− 0.743737i	−0.928285	−	0.371868i	$-0.878717\pi$
$859$	− 21.7980i	− 0.743737i	0.928285	−	0.371868i	$-0.121283\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.8617	−1.62923	−0.814616	−	0.580000i	$-0.803053\pi$
$863$	−47.8617	−1.62923	−0.814616	+	0.580000i	$0.803053\pi$
$864$	0	0
$865$	−53.3939	−1.81545
$866$	0	0
$867$	− 8.59592i	− 0.291933i
$868$	0	0
$869$	− 6.92820i	− 0.235023i
$870$	0	0
$871$	−3.68561	−0.124882
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 33.3055i	− 1.12465i	−0.826918	−	0.562323i	$-0.809908\pi$
$877$	− 33.3055i	− 1.12465i	0.826918	−	0.562323i	$-0.190092\pi$
$878$	0	0
$879$	−10.6780	−0.360161
$880$	0	0
$881$	−23.7980	−0.801774	−0.400887	−	0.916128i	$-0.631298\pi$
$881$	−23.7980	−0.801774	−0.400887	+	0.916128i	$0.631298\pi$
$882$	0	0
$883$	10.2020i	0.343326i	0.985156	+	0.171663i	$0.0549140\pi$
$883$	10.2020i	0.343326i	−0.985156	+	0.171663i	$0.945086\pi$
$884$	0	0
$885$	− 39.0265i	− 1.31186i
$886$	0	0
$887$	28.9842	0.973193	0.486596	−	0.873627i	$-0.338238\pi$
$887$	28.9842	0.973193	0.486596	+	0.873627i	$0.338238\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 1.00000i	− 0.0335013i
$892$	0	0
$893$	30.8270i	1.03159i
$894$	0	0
$895$	11.3137	0.378176
$896$	0	0
$897$	0.404082	0.0134919
$898$	0	0
$899$	− 13.7980i	− 0.460188i
$900$	0	0
$901$	− 11.8852i	− 0.395952i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−67.5959	−2.24696
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	0	0
$909$	− 11.9494i	− 0.396336i
$910$	0	0
$911$	−4.44972	−0.147426	−0.0737129	−	0.997280i	$-0.523485\pi$
$911$	−4.44972	−0.147426	−0.0737129	+	0.997280i	$0.523485\pi$
$912$	0	0
$913$	−9.79796	−0.324265
$914$	0	0
$915$	− 33.7980i	− 1.11733i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	52.8829	1.74445	0.872223	−	0.489108i	$-0.162677\pi$
$919$	52.8829	1.74445	0.872223	+	0.489108i	$0.162677\pi$
$920$	0	0
$921$	−26.6969	−0.879694
$922$	0	0
$923$	3.19184i	0.105061i
$924$	0	0
$925$	20.7846i	0.683394i
$926$	0	0
$927$	2.19275	0.0720194
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	− 34.2929i	− 1.12390i
$932$	0	0
$933$	− 6.29253i	− 0.206008i
$934$	0	0
$935$	−8.19955	−0.268154
$936$	0	0
$937$	33.5959	1.09753	0.548765	−	0.835976i	$-0.315098\pi$
$937$	33.5959	1.09753	0.548765	+	0.835976i	$0.315098\pi$
$938$	0	0
$939$	17.5959i	0.574221i
$940$	0	0
$941$	− 10.6780i	− 0.348094i	−0.984737	−	0.174047i	$-0.944316\pi$
$941$	− 10.6780i	− 0.348094i	0.984737	−	0.174047i	$-0.0556845\pi$
$942$	0	0
$943$	−0.699891	−0.0227916
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.59592i	− 0.246834i	−0.992355	−	0.123417i	$-0.960615\pi$
$947$	− 7.59592i	− 0.246834i	0.992355	−	0.123417i	$-0.0393853\pi$
$948$	0	0
$949$	7.49966i	0.243449i
$950$	0	0
$951$	2.82843	0.0917180
$952$	0	0
$953$	36.2929	1.17564	0.587820	−	0.808991i	$-0.299986\pi$
$953$	36.2929	1.17564	0.587820	+	0.808991i	$0.299986\pi$
$954$	0	0
$955$	− 5.39388i	− 0.174542i
$956$	0	0
$957$	− 6.29253i	− 0.203409i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−26.1918	−0.844898
$962$	0	0
$963$	8.00000i	0.257796i
$964$	0	0
$965$	− 49.7688i	− 1.60211i
$966$	0	0
$967$	16.3991	0.527360	0.263680	−	0.964610i	$-0.415064\pi$
$967$	16.3991	0.527360	0.263680	+	0.964610i	$0.415064\pi$
$968$	0	0
$969$	−14.2020	−0.456235
$970$	0	0
$971$	7.59592i	0.243765i	0.992545	+	0.121882i	$0.0388930\pi$
$971$	7.59592i	0.243765i	−0.992545	+	0.121882i	$0.961107\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.90702	−0.0610736
$976$	0	0
$977$	−11.7980	−0.377450	−0.188725	−	0.982030i	$-0.560436\pi$
$977$	−11.7980	−0.377450	−0.188725	+	0.982030i	$0.560436\pi$
$978$	0	0
$979$	− 3.79796i	− 0.121383i
$980$	0	0
$981$	− 5.02118i	− 0.160314i
$982$	0	0
$983$	−34.0053	−1.08460	−0.542301	−	0.840184i	$-0.682447\pi$
$983$	−34.0053	−1.08460	−0.542301	+	0.840184i	$0.682447\pi$
$984$	0	0
$985$	40.9898	1.30604
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.571458i	0.0181713i
$990$	0	0
$991$	46.3047	1.47092	0.735458	−	0.677570i	$-0.236967\pi$
$991$	46.3047	1.47092	0.735458	+	0.677570i	$0.236967\pi$
$992$	0	0
$993$	33.3939	1.05972
$994$	0	0
$995$	− 9.79796i	− 0.310616i
$996$	0	0
$997$	52.8187i	1.67279i	0.548131	+	0.836393i	$0.315340\pi$
$997$	52.8187i	1.67279i	−0.548131	+	0.836393i	$0.684660\pi$
$998$	0	0
$999$	−6.92820	−0.219199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.f.g.1057.5	yes	8
3.2	odd	2		6336.2.f.l.3169.7		8
4.3	odd	2	inner	2112.2.f.g.1057.1	✓	8
8.3	odd	2	inner	2112.2.f.g.1057.8	yes	8
8.5	even	2	inner	2112.2.f.g.1057.4	yes	8
12.11	even	2		6336.2.f.l.3169.5		8
16.3	odd	4		8448.2.a.ct.1.2		4
16.5	even	4		8448.2.a.ct.1.3		4
16.11	odd	4		8448.2.a.cm.1.3		4
16.13	even	4		8448.2.a.cm.1.2		4
24.5	odd	2		6336.2.f.l.3169.2		8
24.11	even	2		6336.2.f.l.3169.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.f.g.1057.1	✓	8	4.3	odd	2	inner
2112.2.f.g.1057.4	yes	8	8.5	even	2	inner
2112.2.f.g.1057.5	yes	8	1.1	even	1	trivial
2112.2.f.g.1057.8	yes	8	8.3	odd	2	inner
6336.2.f.l.3169.2		8	24.5	odd	2
6336.2.f.l.3169.4		8	24.11	even	2
6336.2.f.l.3169.5		8	12.11	even	2
6336.2.f.l.3169.7		8	3.2	odd	2
8448.2.a.cm.1.2		4	16.13	even	4
8448.2.a.cm.1.3		4	16.11	odd	4
8448.2.a.ct.1.2		4	16.3	odd	4
8448.2.a.ct.1.3		4	16.5	even	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.82843i q^{5} -1.00000 q^{9} -1.00000i q^{11} -0.635674i q^{13} +2.82843 q^{15} +2.89898 q^{17} +4.89898i q^{19} +0.635674 q^{23} -3.00000 q^{25} -1.00000i q^{27} -6.29253i q^{29} +2.19275 q^{31} +1.00000 q^{33} -6.92820i q^{37} +0.635674 q^{39} -1.10102 q^{41} +0.898979i q^{43} +2.82843i q^{45} +6.29253 q^{47} -7.00000 q^{49} +2.89898i q^{51} -4.09978i q^{53} -2.82843 q^{55} -4.89898 q^{57} -13.7980i q^{59} -11.9494i q^{61} -1.79796 q^{65} -5.79796i q^{67} +0.635674i q^{69} -5.02118 q^{71} -11.7980 q^{73} -3.00000i q^{75} +6.92820 q^{79} +1.00000 q^{81} -9.79796i q^{83} -8.19955i q^{85} +6.29253 q^{87} +3.79796 q^{89} +2.19275i q^{93} +13.8564 q^{95} -6.00000 q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 16 q^{17} - 24 q^{25} + 8 q^{33} - 48 q^{41} - 56 q^{49} + 64 q^{65} - 16 q^{73} + 8 q^{81} - 48 q^{89} - 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^9 - 16 * q^17 - 24 * q^25 + 8 * q^33 - 48 * q^41 - 56 * q^49 + 64 * q^65 - 16 * q^73 + 8 * q^81 - 48 * q^89 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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