LMFDB - Embedded newform 1008.3.cd.a.415.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(415,1008)] 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("1008.415"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 2])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cd (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			415.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1008.415
Dual form			1008.3.cd.a.991.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.50000 - 2.59808i) q^{7} -23.0000 q^{13} +(31.5000 + 18.1865i) q^{19} +(12.5000 + 21.6506i) q^{25} +(52.5000 - 30.3109i) q^{31} +(36.5000 - 63.2199i) q^{37} +60.6218i q^{43} +(35.5000 + 33.7750i) q^{49} +(-37.0000 + 64.0859i) q^{61} +(115.500 - 66.6840i) q^{67} +(-48.5000 - 84.0045i) q^{73} +(136.500 + 78.8083i) q^{79} +(149.500 + 59.7558i) q^{91} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 13 q^{7} - 46 q^{13} + 63 q^{19} + 25 q^{25} + 105 q^{31} + 73 q^{37} + 71 q^{49} - 74 q^{61} + 231 q^{67} - 97 q^{73} + 273 q^{79} + 299 q^{91} - 4 q^{97}+O(q^{100}) $ 2 * q - 13 * q^7 - 46 * q^13 + 63 * q^19 + 25 * q^25 + 105 * q^31 + 73 * q^37 + 71 * q^49 - 74 * q^61 + 231 * q^67 - 97 * q^73 + 273 * q^79 + 299 * q^91 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$7$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−23.0000			−1.76923			−0.884615	−	0.466321i	$-0.845579\pi$
$13$	−23.0000			−1.76923			−0.884615	+	0.466321i	$0.845579\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	31.5000	+	18.1865i	1.65789	+	0.957186i	0.973684	+	0.227901i	$0.0731864\pi$
$19$	31.5000	+	18.1865i	1.65789	+	0.957186i	0.684211	+	0.729285i	$0.260147\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	12.5000	+	21.6506i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	52.5000	−	30.3109i	1.69355	−	0.977771i	0.741935	−	0.670471i	$-0.233908\pi$
$31$	52.5000	−	30.3109i	1.69355	−	0.977771i	0.951613	−	0.307299i	$-0.0994253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	36.5000	−	63.2199i	0.986486	−	1.70864i	0.351351	−	0.936244i	$-0.385722\pi$
$37$	36.5000	−	63.2199i	0.986486	−	1.70864i	0.635135	−	0.772401i	$-0.280944\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$			60.6218i			1.40981i	0.709302	+	0.704904i	$0.249010\pi$
$43$			60.6218i			1.40981i	−0.709302	+	0.704904i	$0.750990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	35.5000	+	33.7750i	0.724490	+	0.689286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−37.0000	+	64.0859i	−0.606557	+	1.05059i	0.385246	+	0.922814i	$0.374117\pi$
$61$	−37.0000	+	64.0859i	−0.606557	+	1.05059i	−0.991803	+	0.127774i	$0.959217\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	115.500	−	66.6840i	1.72388	−	0.995283i	0.813433	−	0.581659i	$-0.197596\pi$
$67$	115.500	−	66.6840i	1.72388	−	0.995283i	0.910448	−	0.413624i	$-0.135737\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−48.5000	−	84.0045i	−0.664384	−	1.15075i	−0.979452	−	0.201677i	$-0.935361\pi$
$73$	−48.5000	−	84.0045i	−0.664384	−	1.15075i	0.315068	−	0.949069i	$-0.397973\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	136.500	+	78.8083i	1.72785	+	0.997574i	0.898734	+	0.438494i	$0.144488\pi$
$79$	136.500	+	78.8083i	1.72785	+	0.997574i	0.829114	+	0.559080i	$0.188845\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	149.500	+	59.7558i	1.64286	+	0.656657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.00000			−0.0206186			−0.0103093	−	0.999947i	$-0.503282\pi$
$97$	−2.00000			−0.0206186			−0.0103093	+	0.999947i	$0.503282\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	115.500	+	66.6840i	1.12136	+	0.647417i	0.941748	−	0.336321i	$-0.109183\pi$
$103$	115.500	+	66.6840i	1.12136	+	0.647417i	0.179612	+	0.983738i	$0.442516\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	35.5000	+	61.4878i	0.325688	+	0.564108i	0.981651	−	0.190684i	$-0.0610707\pi$
$109$	35.5000	+	61.4878i	0.325688	+	0.564108i	−0.655963	+	0.754793i	$0.727737\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	+	104.789i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	230.363i		−	1.81388i	−0.421260	−	0.906940i	$-0.638412\pi$
$127$		−	230.363i		−	1.81388i	0.421260	−	0.906940i	$-0.361588\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	−157.500	−	200.052i	−1.18421	−	1.50415i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$			157.617i			1.13393i	0.823741	+	0.566966i	$0.191883\pi$
$139$			157.617i			1.13393i	−0.823741	+	0.566966i	$0.808117\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	−84.0000	+	48.4974i	−0.556291	+	0.321175i	−0.751656	−	0.659556i	$-0.770744\pi$
$151$	−84.0000	+	48.4974i	−0.556291	+	0.321175i	0.195364	+	0.980731i	$0.437411\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.0000	+	102.191i	0.375796	+	0.650898i	0.990446	−	0.137902i	$-0.0440359\pi$
$157$	59.0000	+	102.191i	0.375796	+	0.650898i	−0.614650	+	0.788800i	$0.710703\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	168.000	+	96.9948i	1.03067	+	0.595060i	0.917178	−	0.398478i	$-0.130461\pi$
$163$	168.000	+	96.9948i	1.03067	+	0.595060i	0.113497	+	0.993538i	$0.463795\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	360.000			2.13018
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−25.0000	−	173.205i	−0.142857	−	0.989743i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	313.000			1.72928			0.864641	−	0.502390i	$-0.167546\pi$
$181$	313.000			1.72928			0.864641	+	0.502390i	$0.167546\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−119.500	−	206.980i	−0.619171	−	1.07244i	−0.989637	−	0.143590i	$-0.954135\pi$
$193$	−119.500	−	206.980i	−0.619171	−	1.07244i	0.370466	−	0.928846i	$-0.379198\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	84.0000	−	48.4974i	0.422111	−	0.243706i	−0.273869	−	0.961767i	$-0.588304\pi$
$199$	84.0000	−	48.4974i	0.422111	−	0.243706i	0.695980	+	0.718061i	$0.254970\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			387.979i			1.83876i	0.393365	+	0.919382i	$0.371311\pi$
$211$			387.979i			1.83876i	−0.393365	+	0.919382i	$0.628689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−420.000	+	60.6218i	−1.93548	+	0.279363i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	290.985i		−	1.30486i	−0.757848	−	0.652432i	$-0.773749\pi$
$223$		−	290.985i		−	1.30486i	0.757848	−	0.652432i	$-0.226251\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	−204.500	+	354.204i	−0.893013	+	1.54674i	−0.0567686	+	0.998387i	$0.518080\pi$
$229$	−204.500	+	354.204i	−0.893013	+	1.54674i	−0.836245	+	0.548357i	$0.815254\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−143.000	−	247.683i	−0.593361	−	1.02773i	−0.993776	−	0.111397i	$-0.964467\pi$
$241$	−143.000	−	247.683i	−0.593361	−	1.02773i	0.400415	−	0.916334i	$-0.368866\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−724.500	−	418.290i	−2.93320	−	1.69348i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−401.500	+	316.099i	−1.55019	+	1.22046i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−420.000	−	242.487i	−1.54982	−	0.894786i	−0.998155	−	0.0607176i	$-0.980661\pi$
$271$	−420.000	−	242.487i	−1.54982	−	0.894786i	−0.551661	−	0.834069i	$-0.686006\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	203.500	+	352.472i	0.734657	+	1.27246i	0.954874	+	0.297012i	$0.0959902\pi$
$277$	203.500	+	352.472i	0.734657	+	1.27246i	−0.220217	+	0.975451i	$0.570676\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	−199.500	+	115.181i	−0.704947	+	0.407001i	−0.809187	−	0.587551i	$-0.800092\pi$
$283$	−199.500	+	115.181i	−0.704947	+	0.407001i	0.104240	+	0.994552i	$0.466759\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	144.500	−	250.281i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	157.500	−	394.042i	0.523256	−	1.30911i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	60.6218i		−	0.197465i	−0.995114	−	0.0987325i	$-0.968521\pi$
$307$		−	60.6218i		−	0.197465i	0.995114	−	0.0987325i	$-0.0314788\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−299.500	+	518.749i	−0.956869	+	1.65735i	−0.226837	+	0.973933i	$0.572838\pi$
$313$	−299.500	+	518.749i	−0.956869	+	1.65735i	−0.730032	+	0.683413i	$0.760495\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−287.500	−	497.965i	−0.884615	−	1.53220i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−31.5000	−	18.1865i	−0.0951662	−	0.0549442i	0.451662	−	0.892189i	$-0.350831\pi$
$331$	−31.5000	−	18.1865i	−0.0951662	−	0.0549442i	−0.546828	+	0.837245i	$0.684165\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	649.000			1.92582			0.962908	−	0.269830i	$-0.0869675\pi$
$337$	649.000			1.92582			0.962908	+	0.269830i	$0.0869675\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−143.000	−	311.769i	−0.416910	−	0.908948i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−502.000			−1.43840			−0.719198	−	0.694805i	$-0.755490\pi$
$349$	−502.000			−1.43840			−0.719198	+	0.694805i	$0.755490\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	481.000	+	833.116i	1.33241	+	2.30780i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−472.500	+	272.798i	−1.28747	+	0.743319i	−0.978202	−	0.207657i	$-0.933416\pi$
$367$	−472.500	+	272.798i	−1.28747	+	0.743319i	−0.309264	+	0.950976i	$0.600083\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	288.500	−	499.697i	0.773458	−	1.33967i	−0.162198	−	0.986758i	$-0.551858\pi$
$373$	288.500	−	499.697i	0.773458	−	1.33967i	0.935657	−	0.352911i	$-0.114808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	448.601i		−	1.18364i	−0.806069	−	0.591822i	$-0.798409\pi$
$379$		−	448.601i		−	1.18364i	0.806069	−	0.591822i	$-0.201591\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−215.500	+	373.257i	−0.542821	+	0.940194i	0.455919	+	0.890021i	$0.349311\pi$
$397$	−215.500	+	373.257i	−0.542821	+	0.940194i	−0.998741	+	0.0501728i	$0.984023\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	−1207.50	+	697.150i	−2.99628	+	1.72990i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−384.500	−	665.974i	−0.940098	−	1.62830i	−0.765281	−	0.643696i	$-0.777400\pi$
$409$	−384.500	−	665.974i	−0.940098	−	1.62830i	−0.174817	−	0.984601i	$-0.555933\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	481.000			1.14252			0.571259	−	0.820770i	$-0.306455\pi$
$421$	481.000			1.14252			0.571259	+	0.820770i	$0.306455\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	407.000	−	320.429i	0.953162	−	0.750420i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	359.000			0.829099			0.414550	−	0.910027i	$-0.363939\pi$
$433$	359.000			0.829099			0.414550	+	0.910027i	$0.363939\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−756.000	−	436.477i	−1.72210	−	0.994252i	−0.914579	−	0.404408i	$-0.867478\pi$
$439$	−756.000	−	436.477i	−1.72210	−	0.994252i	−0.807517	−	0.589844i	$-0.799189\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	383.500	−	664.241i	0.839168	−	1.45348i	−0.0514223	−	0.998677i	$-0.516375\pi$
$457$	383.500	−	664.241i	0.839168	−	1.45348i	0.890591	−	0.454805i	$-0.150291\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	836.581i		−	1.80687i	−0.428726	−	0.903435i	$-0.641037\pi$
$463$		−	836.581i		−	1.80687i	0.428726	−	0.903435i	$-0.358963\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	−924.000	+	133.368i	−1.97015	+	0.284367i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$			909.327i			1.91437i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	−839.500	+	1454.06i	−1.74532	+	3.02299i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	787.500	−	454.663i	1.61704	−	0.933600i	0.629363	−	0.777111i	$-0.283316\pi$
$487$	787.500	−	454.663i	1.61704	−	0.933600i	0.987680	−	0.156489i	$-0.0500176\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	451.500	+	260.674i	0.904810	+	0.522392i	0.878758	−	0.477269i	$-0.158373\pi$
$499$	451.500	+	260.674i	0.904810	+	0.522392i	0.0260521	+	0.999661i	$0.491706\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	97.0000	+	672.036i	0.189824	+	1.31514i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	892.500	+	515.285i	1.70650	+	0.985249i	0.938815	+	0.344423i	$0.111925\pi$
$523$	892.500	+	515.285i	1.70650	+	0.985249i	0.767686	+	0.640826i	$0.221408\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	−	458.127i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−120.500	+	208.712i	−0.222736	+	0.385790i	−0.955638	−	0.294545i	$-0.904832\pi$
$541$	−120.500	+	208.712i	−0.222736	+	0.385790i	0.732902	+	0.680334i	$0.238165\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			969.948i			1.77321i	0.462523	+	0.886607i	$0.346944\pi$
$547$			969.948i			1.77321i	−0.462523	+	0.886607i	$0.653056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−682.500	−	866.891i	−1.23418	−	1.56762i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		−	1394.30i		−	2.49428i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	976.500	−	563.783i	1.71016	−	0.987360i	0.775832	−	0.630940i	$-0.217330\pi$
$571$	976.500	−	563.783i	1.71016	−	0.987360i	0.934326	−	0.356420i	$-0.116003\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−35.5000	−	61.4878i	−0.0615251	−	0.106565i	0.833622	−	0.552335i	$-0.186263\pi$
$577$	−35.5000	−	61.4878i	−0.0615251	−	0.106565i	−0.895147	+	0.445770i	$0.852930\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	2205.00			3.74363
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	1199.00			1.99501			0.997504	−	0.0706077i	$-0.0224939\pi$
$601$	1199.00			1.99501			0.997504	+	0.0706077i	$0.0224939\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	220.500	+	127.306i	0.363262	+	0.209729i	0.670511	−	0.741900i	$-0.266075\pi$
$607$	220.500	+	127.306i	0.363262	+	0.209729i	−0.307249	+	0.951629i	$0.599408\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	563.000	+	975.145i	0.918434	+	1.59077i	0.801794	+	0.597600i	$0.203879\pi$
$613$	563.000	+	975.145i	0.918434	+	1.59077i	0.116639	+	0.993174i	$0.462788\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	367.500	−	212.176i	0.593700	−	0.342773i	−0.172859	−	0.984947i	$-0.555301\pi$
$619$	367.500	−	212.176i	0.593700	−	0.342773i	0.766559	+	0.642174i	$0.221967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	+	541.266i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			1066.94i			1.69088i	0.534073	+	0.845438i	$0.320661\pi$
$631$			1066.94i			1.69088i	−0.534073	+	0.845438i	$0.679339\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−816.500	−	776.825i	−1.28179	−	1.21951i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$			351.606i			0.546822i	0.961897	+	0.273411i	$0.0881518\pi$
$643$			351.606i			0.546822i	−0.961897	+	0.273411i	$0.911848\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−539.500	−	934.441i	−0.816188	−	1.41368i	−0.908472	−	0.417946i	$-0.862750\pi$
$661$	−539.500	−	934.441i	−0.816188	−	1.41368i	0.0922844	−	0.995733i	$-0.470583\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	23.0000			0.0341753			0.0170877	−	0.999854i	$-0.494561\pi$
$673$	23.0000			0.0341753			0.0170877	+	0.999854i	$0.494561\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	13.0000	+	5.19615i	0.0191458	+	0.00765265i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	808.500	+	466.788i	1.17004	+	0.675525i	0.953690	−	0.300790i	$-0.0972504\pi$
$691$	808.500	+	466.788i	1.17004	+	0.675525i	0.216353	+	0.976315i	$0.430584\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	2299.50	−	1327.62i	3.27098	−	1.88850i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−467.000	+	808.868i	−0.658674	+	1.14086i	0.322285	+	0.946643i	$0.395549\pi$
$709$	−467.000	+	808.868i	−0.658674	+	1.14086i	−0.980959	+	0.194214i	$0.937784\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	−577.500	−	733.524i	−0.800971	−	1.01737i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			1103.32i			1.51763i	0.651307	+	0.758815i	$0.274221\pi$
$727$			1103.32i			1.51763i	−0.651307	+	0.758815i	$0.725779\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−708.500	+	1227.16i	−0.966576	+	1.67416i	−0.261255	+	0.965270i	$0.584136\pi$
$733$	−708.500	+	1227.16i	−0.966576	+	1.67416i	−0.705321	+	0.708888i	$0.749197\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	556.500	−	321.295i	0.753045	−	0.434771i	−0.0737483	−	0.997277i	$-0.523496\pi$
$739$	556.500	−	321.295i	0.753045	−	0.434771i	0.826793	+	0.562506i	$0.190163\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1291.50	+	745.648i	1.71971	+	0.992873i	0.919441	+	0.393229i	$0.128642\pi$
$751$	1291.50	+	745.648i	1.71971	+	0.992873i	0.800266	+	0.599645i	$0.204691\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	838.000			1.10700			0.553501	−	0.832849i	$-0.313292\pi$
$757$	838.000			1.10700			0.553501	+	0.832849i	$0.313292\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−71.0000	−	491.902i	−0.0930537	−	0.644695i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	863.000			1.12224			0.561118	−	0.827736i	$-0.310371\pi$
$769$	863.000			1.12224			0.561118	+	0.827736i	$0.310371\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	1312.50	+	757.772i	1.69355	+	0.977771i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	168.000	−	96.9948i	0.213469	−	0.123246i	−0.389454	−	0.921046i	$-0.627336\pi$
$787$	168.000	−	96.9948i	0.213469	−	0.123246i	0.602922	+	0.797800i	$0.294003\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	851.000	−	1473.98i	1.07314	−	1.85873i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$		−	581.969i		−	0.717594i	−0.933416	−	0.358797i	$-0.883187\pi$
$811$		−	581.969i		−	0.717594i	0.933416	−	0.358797i	$-0.116813\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1102.50	+	1909.59i	−1.34945	+	2.33731i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	−1092.00	+	630.466i	−1.32685	+	0.766059i	−0.984812	−	0.173626i	$-0.944452\pi$
$823$	−1092.00	+	630.466i	−1.32685	+	0.766059i	−0.342041	+	0.939685i	$0.611118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−804.500	−	1393.43i	−0.970446	−	1.68086i	−0.694210	−	0.719773i	$-0.744246\pi$
$829$	−804.500	−	1393.43i	−0.970446	−	1.68086i	−0.276236	−	0.961090i	$-0.589087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−841.000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	665.500	−	523.945i	0.785714	−	0.618590i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−481.000			−0.563892			−0.281946	−	0.959430i	$-0.590980\pi$
$853$	−481.000			−0.563892			−0.281946	+	0.959430i	$0.590980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−840.000	−	484.974i	−0.977881	−	0.564580i	−0.0762515	−	0.997089i	$-0.524295\pi$
$859$	−840.000	−	484.974i	−0.977881	−	0.564580i	−0.901630	+	0.432509i	$0.857629\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−2656.50	+	1533.73i	−3.04994	+	1.76089i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	299.000	−	517.883i	0.340935	−	0.590517i	−0.643672	−	0.765302i	$-0.722590\pi$
$877$	299.000	−	517.883i	0.340935	−	0.590517i	0.984607	+	0.174785i	$0.0559231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		−	1709.53i		−	1.93605i	−0.250849	−	0.968026i	$-0.580710\pi$
$883$		−	1709.53i		−	1.93605i	0.250849	−	0.968026i	$-0.419290\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−598.500	+	1497.36i	−0.673228	+	1.68432i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	619.500	−	357.668i	0.683021	−	0.394342i	−0.117971	−	0.993017i	$-0.537639\pi$
$907$	619.500	−	357.668i	0.683021	−	0.394342i	0.800992	+	0.598675i	$0.204306\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	52.5000	+	30.3109i	0.0571273	+	0.0329825i	0.528292	−	0.849063i	$-0.322833\pi$
$919$	52.5000	+	30.3109i	0.0571273	+	0.0329825i	−0.471164	+	0.882045i	$0.656166\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1825.00			1.97297
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	504.000	+	1709.53i	0.541353	+	1.83623i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−649.000			−0.692636			−0.346318	−	0.938117i	$-0.612568\pi$
$937$	−649.000			−0.692636			−0.346318	+	0.938117i	$0.612568\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	1115.50	+	1932.10i	1.17545	+	2.03594i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1357.00	−	2350.39i	1.41207	−	2.44578i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	739.586i		−	0.764825i	−0.923992	−	0.382412i	$-0.875093\pi$
$967$		−	739.586i		−	0.764825i	0.923992	−	0.382412i	$-0.124907\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$	409.500	−	1024.51i	0.420863	−	1.05294i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	892.500	−	515.285i	0.900605	−	0.519965i	0.0232089	−	0.999731i	$-0.492612\pi$
$991$	892.500	−	515.285i	0.900605	−	0.519965i	0.877397	+	0.479766i	$0.159278\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−203.500	−	352.472i	−0.204112	−	0.353533i	0.745737	−	0.666240i	$-0.232097\pi$
$997$	−203.500	−	352.472i	−0.204112	−	0.353533i	−0.949850	+	0.312707i	$0.898764\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.cd.a.415.1	✓	2
3.2	odd	2	CM	1008.3.cd.a.415.1	✓	2
4.3	odd	2		1008.3.cd.d.415.1	yes	2
7.4	even	3		1008.3.cd.d.991.1	yes	2
12.11	even	2		1008.3.cd.d.415.1	yes	2
21.11	odd	6		1008.3.cd.d.991.1	yes	2
28.11	odd	6	inner	1008.3.cd.a.991.1	yes	2
84.11	even	6	inner	1008.3.cd.a.991.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1008.3.cd.a.415.1	✓	2	1.1	even	1	trivial
1008.3.cd.a.415.1	✓	2	3.2	odd	2	CM
1008.3.cd.a.991.1	yes	2	28.11	odd	6	inner
1008.3.cd.a.991.1	yes	2	84.11	even	6	inner
1008.3.cd.d.415.1	yes	2	4.3	odd	2
1008.3.cd.d.415.1	yes	2	12.11	even	2
1008.3.cd.d.991.1	yes	2	7.4	even	3
1008.3.cd.d.991.1	yes	2	21.11	odd	6

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cd (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-6.50000 - 2.59808i) q^{7} -23.0000 q^{13} +(31.5000 + 18.1865i) q^{19} +(12.5000 + 21.6506i) q^{25} +(52.5000 - 30.3109i) q^{31} +(36.5000 - 63.2199i) q^{37} +60.6218i q^{43} +(35.5000 + 33.7750i) q^{49} +(-37.0000 + 64.0859i) q^{61} +(115.500 - 66.6840i) q^{67} +(-48.5000 - 84.0045i) q^{73} +(136.500 + 78.8083i) q^{79} +(149.500 + 59.7558i) q^{91} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 13 q^{7} - 46 q^{13} + 63 q^{19} + 25 q^{25} + 105 q^{31} + 73 q^{37} + 71 q^{49} - 74 q^{61} + 231 q^{67} - 97 q^{73} + 273 q^{79} + 299 q^{91} - 4 q^{97}+O(q^{100}) \) 2 * q - 13 * q^7 - 46 * q^13 + 63 * q^19 + 25 * q^25 + 105 * q^31 + 73 * q^37 + 71 * q^49 - 74 * q^61 + 231 * q^67 - 97 * q^73 + 273 * q^79 + 299 * q^91 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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