LMFDB - Embedded newform 2016.3.e.a.1007.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,3,Mod(1007,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.1007"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-184] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$54.9320212950$
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^{14} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1007.1
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2016.1007
Dual form			2016.3.e.a.1007.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.92820i q^{5} +(-4.89898 - 5.00000i) q^{7} +9.89949i q^{11} +19.5959 q^{13} +6.92820 q^{17} +9.79796i q^{19} +41.0122 q^{23} -23.0000 q^{25} -18.3848 q^{29} +39.1918 q^{31} +(-34.6410 + 33.9411i) q^{35} +26.0000i q^{37} +62.3538 q^{41} -48.0000 q^{43} +27.7128i q^{47} +(-1.00000 + 48.9898i) q^{49} +52.3259 q^{53} +68.5857 q^{55} +69.2820 q^{59} -29.3939 q^{61} -135.765i q^{65} +14.0000 q^{67} -41.0122 q^{71} -29.3939i q^{73} +(49.4975 - 48.4974i) q^{77} -48.0000i q^{79} -110.851 q^{83} -48.0000i q^{85} +145.492 q^{89} +(-96.0000 - 97.9796i) q^{91} +67.8823 q^{95} -166.565i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 184 q^{25} - 384 q^{43} - 8 q^{49} + 112 q^{67} - 768 q^{91}+O(q^{100}) $ 8 * q - 184 * q^25 - 384 * q^43 - 8 * q^49 + 112 * q^67 - 768 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	6.92820i		−	1.38564i	−0.721110	−	0.692820i	$-0.756368\pi$
$5$		−	6.92820i		−	1.38564i	0.721110	−	0.692820i	$-0.243632\pi$
$6$		0			0
$7$	−4.89898	−	5.00000i	−0.699854	−	0.714286i
$7$	−4.89898	−	5.00000i	−0.699854	−	0.714286i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			9.89949i			0.899954i	0.893040	+	0.449977i	$0.148568\pi$
$11$			9.89949i			0.899954i	−0.893040	+	0.449977i	$0.851432\pi$
$12$		0			0
$13$	19.5959			1.50738			0.753689	−	0.657231i	$-0.228272\pi$
$13$	19.5959			1.50738			0.753689	+	0.657231i	$0.228272\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6.92820			0.407541			0.203771	−	0.979019i	$-0.434680\pi$
$17$	6.92820			0.407541			0.203771	+	0.979019i	$0.434680\pi$
$18$		0			0
$19$			9.79796i			0.515682i	0.966187	+	0.257841i	$0.0830111\pi$
$19$			9.79796i			0.515682i	−0.966187	+	0.257841i	$0.916989\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	41.0122			1.78314			0.891569	−	0.452884i	$-0.149605\pi$
$23$	41.0122			1.78314			0.891569	+	0.452884i	$0.149605\pi$
$24$		0			0
$25$	−23.0000			−0.920000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−18.3848			−0.633958			−0.316979	−	0.948433i	$-0.602668\pi$
$29$	−18.3848			−0.633958			−0.316979	+	0.948433i	$0.602668\pi$
$30$		0			0
$31$	39.1918			1.26425			0.632126	−	0.774865i	$-0.282182\pi$
$31$	39.1918			1.26425			0.632126	+	0.774865i	$0.282182\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−34.6410	+	33.9411i	−0.989743	+	0.969746i
$36$		0			0
$37$			26.0000i			0.702703i	0.936244	+	0.351351i	$0.114278\pi$
$37$			26.0000i			0.702703i	−0.936244	+	0.351351i	$0.885722\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	62.3538			1.52083			0.760413	−	0.649440i	$-0.224997\pi$
$41$	62.3538			1.52083			0.760413	+	0.649440i	$0.224997\pi$
$42$		0			0
$43$	−48.0000			−1.11628			−0.558140	−	0.829747i	$-0.688485\pi$
$43$	−48.0000			−1.11628			−0.558140	+	0.829747i	$0.688485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			27.7128i			0.589634i	0.955554	+	0.294817i	$0.0952587\pi$
$47$			27.7128i			0.589634i	−0.955554	+	0.294817i	$0.904741\pi$
$48$		0			0
$49$	−1.00000	+	48.9898i	−0.0204082	+	0.999792i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	52.3259			0.987281			0.493641	−	0.869666i	$-0.335666\pi$
$53$	52.3259			0.987281			0.493641	+	0.869666i	$0.335666\pi$
$54$		0			0
$55$	68.5857			1.24701
$56$		0			0
$57$		0			0
$58$		0			0
$59$	69.2820			1.17427			0.587136	−	0.809488i	$-0.300255\pi$
$59$	69.2820			1.17427			0.587136	+	0.809488i	$0.300255\pi$
$60$		0			0
$61$	−29.3939			−0.481867			−0.240933	−	0.970542i	$-0.577454\pi$
$61$	−29.3939			−0.481867			−0.240933	+	0.970542i	$0.577454\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	135.765i		−	2.08868i
$66$		0			0
$67$	14.0000			0.208955			0.104478	−	0.994527i	$-0.466683\pi$
$67$	14.0000			0.208955			0.104478	+	0.994527i	$0.466683\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−41.0122			−0.577637			−0.288818	−	0.957384i	$-0.593262\pi$
$71$	−41.0122			−0.577637			−0.288818	+	0.957384i	$0.593262\pi$
$72$		0			0
$73$		−	29.3939i		−	0.402656i	−0.979524	−	0.201328i	$-0.935474\pi$
$73$		−	29.3939i		−	0.402656i	0.979524	−	0.201328i	$-0.0645257\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	49.4975	−	48.4974i	0.642824	−	0.629837i
$78$		0			0
$79$		−	48.0000i		−	0.607595i	−0.952737	−	0.303797i	$-0.901745\pi$
$79$		−	48.0000i		−	0.607595i	0.952737	−	0.303797i	$-0.0982546\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−110.851			−1.33556			−0.667779	−	0.744360i	$-0.732755\pi$
$83$	−110.851			−1.33556			−0.667779	+	0.744360i	$0.732755\pi$
$84$		0			0
$85$		−	48.0000i		−	0.564706i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	145.492			1.63474			0.817372	−	0.576110i	$-0.195430\pi$
$89$	145.492			1.63474			0.817372	+	0.576110i	$0.195430\pi$
$90$		0			0
$91$	−96.0000	−	97.9796i	−1.05495	−	1.07670i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	67.8823			0.714550
$96$		0			0
$97$		−	166.565i		−	1.71717i	−0.512673	−	0.858584i	$-0.671345\pi$
$97$		−	166.565i		−	1.71717i	0.512673	−	0.858584i	$-0.328655\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			159.349i			1.57771i	0.614580	+	0.788855i	$0.289326\pi$
$101$			159.349i			1.57771i	−0.614580	+	0.788855i	$0.710674\pi$
$102$		0			0
$103$	−97.9796			−0.951258			−0.475629	−	0.879646i	$-0.657780\pi$
$103$	−97.9796			−0.951258			−0.475629	+	0.879646i	$0.657780\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			57.9828i			0.541895i	0.962594	+	0.270947i	$0.0873370\pi$
$107$			57.9828i			0.541895i	−0.962594	+	0.270947i	$0.912663\pi$
$108$		0			0
$109$		−	122.000i		−	1.11927i	−0.828741	−	0.559633i	$-0.810942\pi$
$109$		−	122.000i		−	1.11927i	0.828741	−	0.559633i	$-0.189058\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	66.4680i		−	0.588213i	−0.955773	−	0.294106i	$-0.904978\pi$
$113$		−	66.4680i		−	0.588213i	0.955773	−	0.294106i	$-0.0950220\pi$
$114$		0			0
$115$		−	284.141i		−	2.47079i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−33.9411	−	34.6410i	−0.285220	−	0.291101i
$120$		0			0
$121$	23.0000			0.190083
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	13.8564i		−	0.110851i
$126$		0			0
$127$		−	240.000i		−	1.88976i	−0.327411	−	0.944882i	$-0.606176\pi$
$127$		−	240.000i		−	1.88976i	0.327411	−	0.944882i	$-0.393824\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	27.7128			0.211548			0.105774	−	0.994390i	$-0.466268\pi$
$131$	27.7128			0.211548			0.105774	+	0.994390i	$0.466268\pi$
$132$		0			0
$133$	48.9898	−	48.0000i	0.368344	−	0.360902i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			35.3553i			0.258068i	0.991640	+	0.129034i	$0.0411877\pi$
$137$			35.3553i			0.258068i	−0.991640	+	0.129034i	$0.958812\pi$
$138$		0			0
$139$		−	19.5959i		−	0.140978i	−0.997513	−	0.0704889i	$-0.977544\pi$
$139$		−	19.5959i		−	0.140978i	0.997513	−	0.0704889i	$-0.0224559\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			193.990i			1.35657i
$144$		0			0
$145$			127.373i			0.878438i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−151.321			−1.01558			−0.507788	−	0.861482i	$-0.669537\pi$
$149$	−151.321			−1.01558			−0.507788	+	0.861482i	$0.669537\pi$
$150$		0			0
$151$			10.0000i			0.0662252i	0.999452	+	0.0331126i	$0.0105420\pi$
$151$			10.0000i			0.0662252i	−0.999452	+	0.0331126i	$0.989458\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	271.529i		−	1.75180i
$156$		0			0
$157$	205.757			1.31056			0.655278	−	0.755388i	$-0.272552\pi$
$157$	205.757			1.31056			0.655278	+	0.755388i	$0.272552\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−200.918	−	205.061i	−1.24794	−	1.27367i
$162$		0			0
$163$	−302.000			−1.85276			−0.926380	−	0.376589i	$-0.877097\pi$
$163$	−302.000			−1.85276			−0.926380	+	0.376589i	$0.877097\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	69.2820i		−	0.414862i	−0.978250	−	0.207431i	$-0.933490\pi$
$167$		−	69.2820i		−	0.414862i	0.978250	−	0.207431i	$-0.0665103\pi$
$168$		0			0
$169$	215.000			1.27219
$170$		0			0
$171$		0			0
$172$		0			0
$173$			297.913i			1.72204i	0.508572	+	0.861019i	$0.330174\pi$
$173$			297.913i			1.72204i	−0.508572	+	0.861019i	$0.669826\pi$
$174$		0			0
$175$	112.677	+	115.000i	0.643866	+	0.657143i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	111.723i		−	0.624150i	−0.950057	−	0.312075i	$-0.898976\pi$
$179$		−	111.723i		−	0.624150i	0.950057	−	0.312075i	$-0.101024\pi$
$180$		0			0
$181$	235.151			1.29918			0.649588	−	0.760286i	$-0.274941\pi$
$181$	235.151			1.29918			0.649588	+	0.760286i	$0.274941\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	180.133			0.973693
$186$		0			0
$187$			68.5857i			0.366769i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	74.9533			0.392426			0.196213	−	0.980561i	$-0.437136\pi$
$191$	74.9533			0.392426			0.196213	+	0.980561i	$0.437136\pi$
$192$		0			0
$193$	−336.000			−1.74093			−0.870466	−	0.492228i	$-0.836183\pi$
$193$	−336.000			−1.74093			−0.870466	+	0.492228i	$0.836183\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	188.090			0.954774			0.477387	−	0.878693i	$-0.341584\pi$
$197$	188.090			0.954774			0.477387	+	0.878693i	$0.341584\pi$
$198$		0			0
$199$	88.1816			0.443124			0.221562	−	0.975146i	$-0.428885\pi$
$199$	88.1816			0.443124			0.221562	+	0.975146i	$0.428885\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	90.0666	+	91.9239i	0.443678	+	0.452827i
$204$		0			0
$205$		−	432.000i		−	2.10732i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−96.9948			−0.464090
$210$		0			0
$211$	−144.000			−0.682464			−0.341232	−	0.939979i	$-0.610844\pi$
$211$	−144.000			−0.682464			−0.341232	+	0.939979i	$0.610844\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			332.554i			1.54676i
$216$		0			0
$217$	−192.000	−	195.959i	−0.884793	−	0.903038i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	135.765			0.614319
$222$		0			0
$223$	107.778			0.483307			0.241654	−	0.970363i	$-0.422310\pi$
$223$	107.778			0.483307			0.241654	+	0.970363i	$0.422310\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	166.277			0.732497			0.366249	−	0.930517i	$-0.380642\pi$
$227$	166.277			0.732497			0.366249	+	0.930517i	$0.380642\pi$
$228$		0			0
$229$	−78.3837			−0.342287			−0.171143	−	0.985246i	$-0.554746\pi$
$229$	−78.3837			−0.342287			−0.171143	+	0.985246i	$0.554746\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			100.409i			0.430941i	0.976510	+	0.215470i	$0.0691284\pi$
$233$			100.409i			0.430941i	−0.976510	+	0.215470i	$0.930872\pi$
$234$		0			0
$235$	192.000			0.817021
$236$		0			0
$237$		0			0
$238$		0			0
$239$	264.458			1.10652			0.553259	−	0.833009i	$-0.313384\pi$
$239$	264.458			1.10652			0.553259	+	0.833009i	$0.313384\pi$
$240$		0			0
$241$			225.353i			0.935075i	0.883973	+	0.467537i	$0.154859\pi$
$241$			225.353i			0.935075i	−0.883973	+	0.467537i	$0.845141\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	339.411	+	6.92820i	1.38535	+	0.0282784i
$246$		0			0
$247$			192.000i			0.777328i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.8564			−0.0552048			−0.0276024	−	0.999619i	$-0.508787\pi$
$251$	−13.8564			−0.0552048			−0.0276024	+	0.999619i	$0.508787\pi$
$252$		0			0
$253$			406.000i			1.60474i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	242.487			0.943530			0.471765	−	0.881724i	$-0.343617\pi$
$257$	242.487			0.943530			0.471765	+	0.881724i	$0.343617\pi$
$258$		0			0
$259$	130.000	−	127.373i	0.501931	−	0.491789i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	94.7523			0.360275			0.180137	−	0.983641i	$-0.442346\pi$
$263$	94.7523			0.360275			0.180137	+	0.983641i	$0.442346\pi$
$264$		0			0
$265$		−	362.524i		−	1.36802i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	297.913i		−	1.10748i	−0.832689	−	0.553741i	$-0.813200\pi$
$269$		−	297.913i		−	1.10748i	0.832689	−	0.553741i	$-0.186800\pi$
$270$		0			0
$271$	254.747			0.940026			0.470013	−	0.882660i	$-0.344249\pi$
$271$	254.747			0.940026			0.470013	+	0.882660i	$0.344249\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	227.688i		−	0.827958i
$276$		0			0
$277$		−	336.000i		−	1.21300i	−0.795085	−	0.606498i	$-0.792574\pi$
$277$		−	336.000i		−	1.21300i	0.795085	−	0.606498i	$-0.207426\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			371.938i			1.32362i	0.749670	+	0.661812i	$0.230212\pi$
$281$			371.938i			1.32362i	−0.749670	+	0.661812i	$0.769788\pi$
$282$		0			0
$283$			460.504i			1.62722i	0.581409	+	0.813611i	$0.302502\pi$
$283$			460.504i			1.62722i	−0.581409	+	0.813611i	$0.697498\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−305.470	−	311.769i	−1.06436	−	1.08630i
$288$		0			0
$289$	−241.000			−0.833910
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	242.487i		−	0.827601i	−0.910368	−	0.413801i	$-0.864201\pi$
$293$		−	242.487i		−	0.827601i	0.910368	−	0.413801i	$-0.135799\pi$
$294$		0			0
$295$		−	480.000i		−	1.62712i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	803.672			2.68786
$300$		0			0
$301$	235.151	+	240.000i	0.781233	+	0.797342i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			203.647i			0.667694i
$306$		0			0
$307$		−	303.737i		−	0.989370i	−0.869072	−	0.494685i	$-0.835283\pi$
$307$		−	303.737i		−	0.989370i	0.869072	−	0.494685i	$-0.164717\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	235.559i		−	0.757424i	−0.925515	−	0.378712i	$-0.876367\pi$
$311$		−	235.559i		−	0.757424i	0.925515	−	0.378712i	$-0.123633\pi$
$312$		0			0
$313$		−	333.131i		−	1.06432i	−0.846645	−	0.532158i	$-0.821381\pi$
$313$		−	333.131i		−	1.06432i	0.846645	−	0.532158i	$-0.178619\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−222.032			−0.700415			−0.350207	−	0.936672i	$-0.613889\pi$
$317$	−222.032			−0.700415			−0.350207	+	0.936672i	$0.613889\pi$
$318$		0			0
$319$		−	182.000i		−	0.570533i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			67.8823i			0.210162i
$324$		0			0
$325$	−450.706			−1.38679
$326$		0			0
$327$		0			0
$328$		0			0
$329$	138.564	−	135.765i	0.421167	−	0.412658i
$330$		0			0
$331$	−144.000			−0.435045			−0.217523	−	0.976055i	$-0.569798\pi$
$331$	−144.000			−0.435045			−0.217523	+	0.976055i	$0.569798\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	96.9948i		−	0.289537i
$336$		0			0
$337$	−480.000			−1.42433			−0.712166	−	0.702011i	$-0.752286\pi$
$337$	−480.000			−1.42433			−0.712166	+	0.702011i	$0.752286\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			387.979i			1.13777i
$342$		0			0
$343$	249.848	−	235.000i	0.728420	−	0.685131i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			213.546i			0.615407i	0.951482	+	0.307704i	$0.0995605\pi$
$347$			213.546i			0.615407i	−0.951482	+	0.307704i	$0.900440\pi$
$348$		0			0
$349$	−68.5857			−0.196521			−0.0982603	−	0.995161i	$-0.531328\pi$
$349$	−68.5857			−0.196521			−0.0982603	+	0.995161i	$0.531328\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−256.344			−0.726186			−0.363093	−	0.931753i	$-0.618279\pi$
$353$	−256.344			−0.726186			−0.363093	+	0.931753i	$0.618279\pi$
$354$		0			0
$355$			284.141i			0.800397i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	637.810			1.77663			0.888315	−	0.459234i	$-0.151876\pi$
$359$	637.810			1.77663			0.888315	+	0.459234i	$0.151876\pi$
$360$		0			0
$361$	265.000			0.734072
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−203.647			−0.557936
$366$		0			0
$367$	−499.696			−1.36157			−0.680785	−	0.732484i	$-0.738361\pi$
$367$	−499.696			−1.36157			−0.680785	+	0.732484i	$0.738361\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−256.344	−	261.630i	−0.690953	−	0.705201i
$372$		0			0
$373$			576.000i			1.54424i	0.635479	+	0.772118i	$0.280803\pi$
$373$			576.000i			1.54424i	−0.635479	+	0.772118i	$0.719197\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−360.267			−0.955614
$378$		0			0
$379$	−48.0000			−0.126649			−0.0633245	−	0.997993i	$-0.520170\pi$
$379$	−48.0000			−0.126649			−0.0633245	+	0.997993i	$0.520170\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			221.703i			0.578858i	0.957200	+	0.289429i	$0.0934654\pi$
$383$			221.703i			0.578858i	−0.957200	+	0.289429i	$0.906535\pi$
$384$		0			0
$385$	−336.000	−	342.929i	−0.872727	−	0.890724i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	188.090			0.483523			0.241761	−	0.970336i	$-0.422275\pi$
$389$	188.090			0.483523			0.241761	+	0.970336i	$0.422275\pi$
$390$		0			0
$391$	284.141			0.726703
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−332.554			−0.841908
$396$		0			0
$397$	127.373			0.320840			0.160420	−	0.987049i	$-0.448715\pi$
$397$	127.373			0.320840			0.160420	+	0.987049i	$0.448715\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	137.179i		−	0.342092i	−0.985263	−	0.171046i	$-0.945285\pi$
$401$		−	137.179i		−	0.342092i	0.985263	−	0.171046i	$-0.0547146\pi$
$402$		0			0
$403$	768.000			1.90571
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−257.387			−0.632400
$408$		0			0
$409$		−	519.292i		−	1.26966i	−0.772651	−	0.634831i	$-0.781070\pi$
$409$		−	519.292i		−	1.26966i	0.772651	−	0.634831i	$-0.218930\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−339.411	−	346.410i	−0.821819	−	0.838766i
$414$		0			0
$415$			768.000i			1.85060i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.8564			0.0330702			0.0165351	−	0.999863i	$-0.494736\pi$
$419$	13.8564			0.0330702			0.0165351	+	0.999863i	$0.494736\pi$
$420$		0			0
$421$			432.000i			1.02613i	0.858350	+	0.513064i	$0.171490\pi$
$421$			432.000i			1.02613i	−0.858350	+	0.513064i	$0.828510\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−159.349			−0.374938
$426$		0			0
$427$	144.000	+	146.969i	0.337237	+	0.344191i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.07107			−0.0164062			−0.00820309	−	0.999966i	$-0.502611\pi$
$431$	−7.07107			−0.0164062			−0.00820309	+	0.999966i	$0.502611\pi$
$432$		0			0
$433$			764.241i			1.76499i	0.470321	+	0.882495i	$0.344138\pi$
$433$			764.241i			1.76499i	−0.470321	+	0.882495i	$0.655862\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			401.836i			0.919533i
$438$		0			0
$439$	−656.463			−1.49536			−0.747680	−	0.664059i	$-0.768832\pi$
$439$	−656.463			−1.49536			−0.747680	+	0.664059i	$0.768832\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	688.722i		−	1.55468i	−0.629082	−	0.777339i	$-0.716569\pi$
$443$		−	688.722i		−	1.55468i	0.629082	−	0.777339i	$-0.283431\pi$
$444$		0			0
$445$		−	1008.00i		−	2.26517i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			270.115i			0.601592i	0.953689	+	0.300796i	$0.0972523\pi$
$449$			270.115i			0.601592i	−0.953689	+	0.300796i	$0.902748\pi$
$450$		0			0
$451$			617.271i			1.36867i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−678.823	+	665.108i	−1.49192	+	1.46177i
$456$		0			0
$457$	96.0000			0.210066			0.105033	−	0.994469i	$-0.466505\pi$
$457$	96.0000			0.210066			0.105033	+	0.994469i	$0.466505\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			20.7846i			0.0450859i	0.999746	+	0.0225430i	$0.00717626\pi$
$461$			20.7846i			0.0450859i	−0.999746	+	0.0225430i	$0.992824\pi$
$462$		0			0
$463$		−	240.000i		−	0.518359i	−0.965829	−	0.259179i	$-0.916548\pi$
$463$		−	240.000i		−	0.518359i	0.965829	−	0.259179i	$-0.0834520\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−152.420			−0.326382			−0.163191	−	0.986594i	$-0.552179\pi$
$467$	−152.420			−0.326382			−0.163191	+	0.986594i	$0.552179\pi$
$468$		0			0
$469$	−68.5857	−	70.0000i	−0.146238	−	0.149254i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	475.176i		−	1.00460i
$474$		0			0
$475$		−	225.353i		−	0.474427i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			263.272i			0.549628i	0.961497	+	0.274814i	$0.0886162\pi$
$479$			263.272i			0.549628i	−0.961497	+	0.274814i	$0.911384\pi$
$480$		0			0
$481$			509.494i			1.05924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1154.00			−2.37938
$486$		0			0
$487$			278.000i			0.570842i	0.958402	+	0.285421i	$0.0921334\pi$
$487$			278.000i			0.570842i	−0.958402	+	0.285421i	$0.907867\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			77.7817i			0.158415i	0.996858	+	0.0792075i	$0.0252390\pi$
$491$			77.7817i			0.158415i	−0.996858	+	0.0792075i	$0.974761\pi$
$492$		0			0
$493$	−127.373			−0.258364
$494$		0			0
$495$		0			0
$496$		0			0
$497$	200.918	+	205.061i	0.404261	+	0.412598i
$498$		0			0
$499$	528.000			1.05812			0.529058	−	0.848586i	$-0.322545\pi$
$499$	528.000			1.05812			0.529058	+	0.848586i	$0.322545\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	166.277i		−	0.330570i	−0.986246	−	0.165285i	$-0.947146\pi$
$503$		−	166.277i		−	0.330570i	0.986246	−	0.165285i	$-0.0528544\pi$
$504$		0			0
$505$	1104.00			2.18614
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	644.323i		−	1.26586i	−0.774209	−	0.632930i	$-0.781852\pi$
$509$		−	644.323i		−	1.26586i	0.774209	−	0.632930i	$-0.218148\pi$
$510$		0			0
$511$	−146.969	+	144.000i	−0.287611	+	0.281800i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			678.823i			1.31810i
$516$		0			0
$517$	−274.343			−0.530644
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−187.061			−0.359043			−0.179522	−	0.983754i	$-0.557455\pi$
$521$	−187.061			−0.359043			−0.179522	+	0.983754i	$0.557455\pi$
$522$		0			0
$523$		−	548.686i		−	1.04911i	−0.851376	−	0.524556i	$-0.824231\pi$
$523$		−	548.686i		−	1.04911i	0.851376	−	0.524556i	$-0.175769\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	271.529			0.515235
$528$		0			0
$529$	1153.00			2.17958
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1221.88			2.29246
$534$		0			0
$535$	401.716			0.750872
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−484.974	−	9.89949i	−0.899767	−	0.0183664i
$540$		0			0
$541$		−	432.000i		−	0.798521i	−0.916837	−	0.399261i	$-0.869267\pi$
$541$		−	432.000i		−	0.798521i	0.916837	−	0.399261i	$-0.130733\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−845.241			−1.55090
$546$		0			0
$547$	−658.000			−1.20293			−0.601463	−	0.798901i	$-0.705415\pi$
$547$	−658.000			−1.20293			−0.601463	+	0.798901i	$0.705415\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	180.133i		−	0.326921i
$552$		0			0
$553$	−240.000	+	235.151i	−0.433996	+	0.425228i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−289.914			−0.520492			−0.260246	−	0.965542i	$-0.583804\pi$
$557$	−289.914			−0.520492			−0.260246	+	0.965542i	$0.583804\pi$
$558$		0			0
$559$	−940.604			−1.68265
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$	−460.504			−0.815051
$566$		0			0
$567$		0			0
$568$		0			0
$569$			239.002i			0.420039i	0.977697	+	0.210019i	$0.0673527\pi$
$569$			239.002i			0.420039i	−0.977697	+	0.210019i	$0.932647\pi$
$570$		0			0
$571$	130.000			0.227671			0.113835	−	0.993500i	$-0.463686\pi$
$571$	130.000			0.227671			0.113835	+	0.993500i	$0.463686\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−943.280			−1.64049
$576$		0			0
$577$			78.3837i			0.135847i	0.997691	+	0.0679235i	$0.0216374\pi$
$577$			78.3837i			0.135847i	−0.997691	+	0.0679235i	$0.978363\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	543.058	+	554.256i	0.934695	+	0.953969i
$582$		0			0
$583$			518.000i			0.888508i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	290.985			0.495715			0.247857	−	0.968797i	$-0.420274\pi$
$587$	290.985			0.495715			0.247857	+	0.968797i	$0.420274\pi$
$588$		0			0
$589$			384.000i			0.651952i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	76.2102			0.128516			0.0642582	−	0.997933i	$-0.479532\pi$
$593$	76.2102			0.128516			0.0642582	+	0.997933i	$0.479532\pi$
$594$		0			0
$595$	−240.000	+	235.151i	−0.403361	+	0.395212i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	244.659			0.408446			0.204223	−	0.978924i	$-0.434533\pi$
$599$	244.659			0.408446			0.204223	+	0.978924i	$0.434533\pi$
$600$		0			0
$601$			842.624i			1.40204i	0.713143	+	0.701019i	$0.247271\pi$
$601$			842.624i			1.40204i	−0.713143	+	0.701019i	$0.752729\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	159.349i		−	0.263386i
$606$		0			0
$607$	382.120			0.629523			0.314761	−	0.949171i	$-0.398075\pi$
$607$	382.120			0.629523			0.314761	+	0.949171i	$0.398075\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			543.058i			0.888802i
$612$		0			0
$613$			480.000i			0.783034i	0.920171	+	0.391517i	$0.128050\pi$
$613$			480.000i			0.783034i	−0.920171	+	0.391517i	$0.871950\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			442.649i			0.717421i	0.933449	+	0.358711i	$0.116783\pi$
$617$			442.649i			0.717421i	−0.933449	+	0.358711i	$0.883217\pi$
$618$		0			0
$619$		−	529.090i		−	0.854749i	−0.904075	−	0.427375i	$-0.859439\pi$
$619$		−	529.090i		−	0.854749i	0.904075	−	0.427375i	$-0.140561\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−712.764	−	727.461i	−1.14408	−	1.16767i
$624$		0			0
$625$	−671.000			−1.07360
$626$		0			0
$627$		0			0
$628$		0			0
$629$			180.133i			0.286380i
$630$		0			0
$631$		−	48.0000i		−	0.0760697i	−0.999276	−	0.0380349i	$-0.987890\pi$
$631$		−	48.0000i		−	0.0760697i	0.999276	−	0.0380349i	$-0.0121098\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1662.77			−2.61853
$636$		0			0
$637$	−19.5959	+	960.000i	−0.0307628	+	1.50706i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1.41421i			0.00220626i	0.999999	+	0.00110313i	$0.000351137\pi$
$641$			1.41421i			0.00220626i	−0.999999	+	0.00110313i	$0.999649\pi$
$642$		0			0
$643$			793.635i			1.23427i	0.786858	+	0.617134i	$0.211706\pi$
$643$			793.635i			1.23427i	−0.786858	+	0.617134i	$0.788294\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1274.79i		−	1.97031i	−0.171672	−	0.985154i	$-0.554917\pi$
$647$		−	1274.79i		−	1.97031i	0.171672	−	0.985154i	$-0.445083\pi$
$648$		0			0
$649$			685.857i			1.05679i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1172.38			−1.79538			−0.897690	−	0.440628i	$-0.854756\pi$
$653$	−1172.38			−1.79538			−0.897690	+	0.440628i	$0.854756\pi$
$654$		0			0
$655$		−	192.000i		−	0.293130i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			634.982i			0.963554i	0.876294	+	0.481777i	$0.160008\pi$
$659$			634.982i			0.963554i	−0.876294	+	0.481777i	$0.839992\pi$
$660$		0			0
$661$	342.929			0.518803			0.259401	−	0.965770i	$-0.416475\pi$
$661$	342.929			0.518803			0.259401	+	0.965770i	$0.416475\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−332.554	−	339.411i	−0.500081	−	0.510393i
$666$		0			0
$667$	−754.000			−1.13043
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	290.985i		−	0.433658i
$672$		0			0
$673$	912.000			1.35513			0.677563	−	0.735465i	$-0.263036\pi$
$673$	912.000			1.35513			0.677563	+	0.735465i	$0.263036\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	256.344i		−	0.378646i	−0.981915	−	0.189323i	$-0.939371\pi$
$677$		−	256.344i		−	0.378646i	0.981915	−	0.189323i	$-0.0606294\pi$
$678$		0			0
$679$	−832.827	+	816.000i	−1.22655	+	1.20177i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1299.66i			1.90287i	0.307845	+	0.951437i	$0.400392\pi$
$683$			1299.66i			1.90287i	−0.307845	+	0.951437i	$0.599608\pi$
$684$		0			0
$685$	244.949			0.357590
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1025.37			1.48821
$690$		0			0
$691$		−	117.576i		−	0.170153i	−0.996374	−	0.0850763i	$-0.972887\pi$
$691$		−	117.576i		−	0.170153i	0.996374	−	0.0850763i	$-0.0271134\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−135.765			−0.195345
$696$		0			0
$697$	432.000			0.619799
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−49.4975			−0.0706098			−0.0353049	−	0.999377i	$-0.511240\pi$
$701$	−49.4975			−0.0706098			−0.0353049	+	0.999377i	$0.511240\pi$
$702$		0			0
$703$	−254.747			−0.362371
$704$		0			0
$705$		0			0
$706$		0			0
$707$	796.743	−	780.646i	1.12694	−	1.10417i
$708$		0			0
$709$			502.000i			0.708039i	0.935238	+	0.354020i	$0.115185\pi$
$709$			502.000i			0.708039i	−0.935238	+	0.354020i	$0.884815\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1607.34			2.25434
$714$		0			0
$715$	1344.00			1.87972
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1344.07i		−	1.86936i	−0.355487	−	0.934681i	$-0.615685\pi$
$719$		−	1344.07i		−	1.86936i	0.355487	−	0.934681i	$-0.384315\pi$
$720$		0			0
$721$	480.000	+	489.898i	0.665742	+	0.679470i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	422.850			0.583241
$726$		0			0
$727$	470.302			0.646908			0.323454	−	0.946244i	$-0.395156\pi$
$727$	470.302			0.646908			0.323454	+	0.946244i	$0.395156\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−332.554			−0.454930
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			138.593i			0.188050i
$738$		0			0
$739$	−274.000			−0.370771			−0.185386	−	0.982666i	$-0.559353\pi$
$739$	−274.000			−0.370771			−0.185386	+	0.982666i	$0.559353\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	298.399			0.401614			0.200807	−	0.979631i	$-0.435644\pi$
$743$	298.399			0.401614			0.200807	+	0.979631i	$0.435644\pi$
$744$		0			0
$745$			1048.38i			1.40722i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	289.914	−	284.056i	0.387068	−	0.379247i
$750$		0			0
$751$			1094.00i			1.45672i	0.685192	+	0.728362i	$0.259718\pi$
$751$			1094.00i			1.45672i	−0.685192	+	0.728362i	$0.740282\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	69.2820			0.0917643
$756$		0			0
$757$		−	118.000i		−	0.155878i	−0.996958	−	0.0779392i	$-0.975166\pi$
$757$		−	118.000i		−	0.155878i	0.996958	−	0.0779392i	$-0.0248340\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−547.328			−0.719222			−0.359611	−	0.933102i	$-0.617091\pi$
$761$	−547.328			−0.719222			−0.359611	+	0.933102i	$0.617091\pi$
$762$		0			0
$763$	−610.000	+	597.675i	−0.799476	+	0.783323i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1357.65			1.77007
$768$		0			0
$769$		−	803.433i		−	1.04478i	−0.852708	−	0.522388i	$-0.825041\pi$
$769$		−	803.433i		−	1.04478i	0.852708	−	0.522388i	$-0.174959\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	561.184i		−	0.725982i	−0.931793	−	0.362991i	$-0.881755\pi$
$773$		−	561.184i		−	0.725982i	0.931793	−	0.362991i	$-0.118245\pi$
$774$		0			0
$775$	−901.412			−1.16311
$776$		0			0
$777$		0			0
$778$		0			0
$779$			610.940i			0.784262i
$780$		0			0
$781$		−	406.000i		−	0.519846i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	1425.53i		−	1.81596i
$786$		0			0
$787$		−	823.029i		−	1.04578i	−0.852400	−	0.522890i	$-0.824854\pi$
$787$		−	823.029i		−	1.04578i	0.852400	−	0.522890i	$-0.175146\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−332.340	+	325.626i	−0.420152	+	0.411663i
$792$		0			0
$793$	−576.000			−0.726356
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	630.466i		−	0.791050i	−0.918455	−	0.395525i	$-0.870563\pi$
$797$		−	630.466i		−	0.791050i	0.918455	−	0.395525i	$-0.129437\pi$
$798$		0			0
$799$			192.000i			0.240300i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	290.985			0.362372
$804$		0			0
$805$	−1420.70	+	1392.00i	−1.76485	+	1.72919i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1458.05i			1.80229i	0.433516	+	0.901146i	$0.357273\pi$
$809$			1458.05i			1.80229i	−0.433516	+	0.901146i	$0.642727\pi$
$810$		0			0
$811$		−	293.939i		−	0.362440i	−0.983443	−	0.181220i	$-0.941995\pi$
$811$		−	293.939i		−	0.362440i	0.983443	−	0.181220i	$-0.0580046\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			2092.32i			2.56726i
$816$		0			0
$817$		−	470.302i		−	0.575645i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−595.384			−0.725194			−0.362597	−	0.931946i	$-0.618110\pi$
$821$	−595.384			−0.725194			−0.362597	+	0.931946i	$0.618110\pi$
$822$		0			0
$823$		−	1296.00i		−	1.57473i	−0.616489	−	0.787363i	$-0.711446\pi$
$823$		−	1296.00i		−	1.57473i	0.616489	−	0.787363i	$-0.288554\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			485.075i			0.586548i	0.956028	+	0.293274i	$0.0947448\pi$
$827$			485.075i			0.586548i	−0.956028	+	0.293274i	$0.905255\pi$
$828$		0			0
$829$	293.939			0.354570			0.177285	−	0.984160i	$-0.443269\pi$
$829$	293.939			0.354570			0.177285	+	0.984160i	$0.443269\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.92820	+	339.411i	−0.00831717	+	0.407456i
$834$		0			0
$835$	−480.000			−0.574850
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1371.78i			1.63502i	0.575912	+	0.817511i	$0.304647\pi$
$839$			1371.78i			1.63502i	−0.575912	+	0.817511i	$0.695353\pi$
$840$		0			0
$841$	−503.000			−0.598098
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	1489.56i		−	1.76280i
$846$		0			0
$847$	−112.677	−	115.000i	−0.133030	−	0.135773i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1066.32i			1.25302i
$852$		0			0
$853$	−1185.55			−1.38986			−0.694931	−	0.719076i	$-0.744565\pi$
$853$	−1185.55			−1.38986			−0.694931	+	0.719076i	$0.744565\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−519.615			−0.606319			−0.303159	−	0.952940i	$-0.598041\pi$
$857$	−519.615			−0.606319			−0.303159	+	0.952940i	$0.598041\pi$
$858$		0			0
$859$			1459.90i			1.69953i	0.527162	+	0.849765i	$0.323256\pi$
$859$			1459.90i			1.69953i	−0.527162	+	0.849765i	$0.676744\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	60.8112			0.0704649			0.0352324	−	0.999379i	$-0.488783\pi$
$863$	60.8112			0.0704649			0.0352324	+	0.999379i	$0.488783\pi$
$864$		0			0
$865$	2064.00			2.38613
$866$		0			0
$867$		0			0
$868$		0			0
$869$	475.176			0.546808
$870$		0			0
$871$	274.343			0.314975
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−69.2820	+	67.8823i	−0.0791795	+	0.0775797i
$876$		0			0
$877$		−	672.000i		−	0.766249i	−0.923697	−	0.383124i	$-0.874848\pi$
$877$		−	672.000i		−	0.766249i	0.923697	−	0.383124i	$-0.125152\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1364.86			1.54921			0.774606	−	0.632444i	$-0.217948\pi$
$881$	1364.86			1.54921			0.774606	+	0.632444i	$0.217948\pi$
$882$		0			0
$883$	1392.00			1.57644			0.788222	−	0.615391i	$-0.211002\pi$
$883$	1392.00			1.57644			0.788222	+	0.615391i	$0.211002\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	956.092i		−	1.07789i	−0.842340	−	0.538947i	$-0.818822\pi$
$887$		−	956.092i		−	1.07789i	0.842340	−	0.538947i	$-0.181178\pi$
$888$		0			0
$889$	−1200.00	+	1175.76i	−1.34983	+	1.32256i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−271.529			−0.304064
$894$		0			0
$895$	−774.039			−0.864848
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−720.533			−0.801483
$900$		0			0
$901$	362.524			0.402358
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1629.17i		−	1.80019i
$906$		0			0
$907$	1392.00			1.53473			0.767365	−	0.641211i	$-0.221568\pi$
$907$	1392.00			1.53473			0.767365	+	0.641211i	$0.221568\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−875.398			−0.960920			−0.480460	−	0.877017i	$-0.659530\pi$
$911$	−875.398			−0.960920			−0.480460	+	0.877017i	$0.659530\pi$
$912$		0			0
$913$		−	1097.37i		−	1.20194i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−135.765	−	138.564i	−0.148053	−	0.151106i
$918$		0			0
$919$			1738.00i			1.89119i	0.325352	+	0.945593i	$0.394517\pi$
$919$			1738.00i			1.89119i	−0.325352	+	0.945593i	$0.605483\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−803.672			−0.870717
$924$		0			0
$925$		−	598.000i		−	0.646486i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1600.41			−1.72273			−0.861364	−	0.507988i	$-0.830390\pi$
$929$	−1600.41			−1.72273			−0.861364	+	0.507988i	$0.830390\pi$
$930$		0			0
$931$	−480.000	−	9.79796i	−0.515575	−	0.0105241i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	475.176			0.508209
$936$		0			0
$937$		−	19.5959i		−	0.0209135i	−0.999945	−	0.0104567i	$-0.996671\pi$
$937$		−	19.5959i		−	0.0209135i	0.999945	−	0.0104567i	$-0.00332854\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	284.056i		−	0.301866i	−0.988544	−	0.150933i	$-0.951772\pi$
$941$		−	284.056i		−	0.301866i	0.988544	−	0.150933i	$-0.0482278\pi$
$942$		0			0
$943$	2557.27			2.71184
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1788.98i		−	1.88910i	−0.328364	−	0.944551i	$-0.606497\pi$
$947$		−	1788.98i		−	1.88910i	0.328364	−	0.944551i	$-0.393503\pi$
$948$		0			0
$949$		−	576.000i		−	0.606955i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	32.5269i		−	0.0341311i	−0.999854	−	0.0170655i	$-0.994568\pi$
$953$		−	32.5269i		−	0.0341311i	0.999854	−	0.0170655i	$-0.00543239\pi$
$954$		0			0
$955$		−	519.292i		−	0.543761i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	176.777	−	173.205i	0.184334	−	0.180610i
$960$		0			0
$961$	575.000			0.598335
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2327.88i			2.41231i
$966$		0			0
$967$		−	816.000i		−	0.843847i	−0.906631	−	0.421923i	$-0.861355\pi$
$967$		−	816.000i		−	0.843847i	0.906631	−	0.421923i	$-0.138645\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1163.94			−1.19870			−0.599350	−	0.800487i	$-0.704574\pi$
$971$	−1163.94			−1.19870			−0.599350	+	0.800487i	$0.704574\pi$
$972$		0			0
$973$	−97.9796	+	96.0000i	−0.100698	+	0.0986639i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1424.11i			1.45764i	0.684706	+	0.728819i	$0.259931\pi$
$977$			1424.11i			1.45764i	−0.684706	+	0.728819i	$0.740069\pi$
$978$		0			0
$979$			1440.30i			1.47120i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	900.666i		−	0.916243i	−0.888890	−	0.458121i	$-0.848523\pi$
$983$		−	900.666i		−	0.916243i	0.888890	−	0.458121i	$-0.151477\pi$
$984$		0			0
$985$		−	1303.13i		−	1.32297i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1968.59			−1.99048
$990$		0			0
$991$		−	1306.00i		−	1.31786i	−0.752204	−	0.658930i	$-0.771009\pi$
$991$		−	1306.00i		−	1.31786i	0.752204	−	0.658930i	$-0.228991\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	610.940i		−	0.614010i
$996$		0			0
$997$	−107.778			−0.108102			−0.0540509	−	0.998538i	$-0.517213\pi$
$997$	−107.778			−0.108102			−0.0540509	+	0.998538i	$0.517213\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.e.a.1007.1		8
3.2	odd	2	inner	2016.3.e.a.1007.5		8
4.3	odd	2		504.3.e.a.251.5	yes	8
7.6	odd	2	inner	2016.3.e.a.1007.7		8
8.3	odd	2	inner	2016.3.e.a.1007.8		8
8.5	even	2		504.3.e.a.251.2	yes	8
12.11	even	2		504.3.e.a.251.4	yes	8
21.20	even	2	inner	2016.3.e.a.1007.3		8
24.5	odd	2		504.3.e.a.251.7	yes	8
24.11	even	2	inner	2016.3.e.a.1007.4		8
28.27	even	2		504.3.e.a.251.6	yes	8
56.13	odd	2		504.3.e.a.251.1	✓	8
56.27	even	2	inner	2016.3.e.a.1007.2		8
84.83	odd	2		504.3.e.a.251.3	yes	8
168.83	odd	2	inner	2016.3.e.a.1007.6		8
168.125	even	2		504.3.e.a.251.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.e.a.251.1	✓	8	56.13	odd	2
504.3.e.a.251.2	yes	8	8.5	even	2
504.3.e.a.251.3	yes	8	84.83	odd	2
504.3.e.a.251.4	yes	8	12.11	even	2
504.3.e.a.251.5	yes	8	4.3	odd	2
504.3.e.a.251.6	yes	8	28.27	even	2
504.3.e.a.251.7	yes	8	24.5	odd	2
504.3.e.a.251.8	yes	8	168.125	even	2
2016.3.e.a.1007.1		8	1.1	even	1	trivial
2016.3.e.a.1007.2		8	56.27	even	2	inner
2016.3.e.a.1007.3		8	21.20	even	2	inner
2016.3.e.a.1007.4		8	24.11	even	2	inner
2016.3.e.a.1007.5		8	3.2	odd	2	inner
2016.3.e.a.1007.6		8	168.83	odd	2	inner
2016.3.e.a.1007.7		8	7.6	odd	2	inner
2016.3.e.a.1007.8		8	8.3	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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.92820i q^{5} +(-4.89898 - 5.00000i) q^{7} +9.89949i q^{11} +19.5959 q^{13} +6.92820 q^{17} +9.79796i q^{19} +41.0122 q^{23} -23.0000 q^{25} -18.3848 q^{29} +39.1918 q^{31} +(-34.6410 + 33.9411i) q^{35} +26.0000i q^{37} +62.3538 q^{41} -48.0000 q^{43} +27.7128i q^{47} +(-1.00000 + 48.9898i) q^{49} +52.3259 q^{53} +68.5857 q^{55} +69.2820 q^{59} -29.3939 q^{61} -135.765i q^{65} +14.0000 q^{67} -41.0122 q^{71} -29.3939i q^{73} +(49.4975 - 48.4974i) q^{77} -48.0000i q^{79} -110.851 q^{83} -48.0000i q^{85} +145.492 q^{89} +(-96.0000 - 97.9796i) q^{91} +67.8823 q^{95} -166.565i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 184 q^{25} - 384 q^{43} - 8 q^{49} + 112 q^{67} - 768 q^{91}+O(q^{100}) \) 8 * q - 184 * q^25 - 384 * q^43 - 8 * q^49 + 112 * q^67 - 768 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more