LMFDB - Embedded newform 1008.3.cg.e.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(145,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cg (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			577.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1008.577
Dual form			1008.3.cg.e.145.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.50000 - 4.33013i) q^{7} +O(q^{10})$	$q+(5.50000 - 4.33013i) q^{7} +25.9808i q^{13} +(7.50000 - 4.33013i) q^{19} +(-12.5000 + 21.6506i) q^{25} +(-16.5000 - 9.52628i) q^{31} +(23.5000 + 40.7032i) q^{37} +83.0000 q^{43} +(11.5000 - 47.6314i) q^{49} +(84.0000 - 48.4974i) q^{61} +(-54.5000 + 94.3968i) q^{67} +(25.5000 + 14.7224i) q^{73} +(65.5000 + 113.449i) q^{79} +(112.500 + 142.894i) q^{91} +193.990i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 11 q^{7}+O(q^{10}) $ 2 * q + 11 * q^7	$ 2 q + 11 q^{7} + 15 q^{19} - 25 q^{25} - 33 q^{31} + 47 q^{37} + 166 q^{43} + 23 q^{49} + 168 q^{61} - 109 q^{67} + 51 q^{73} + 131 q^{79} + 225 q^{91}+O(q^{100}) $ 2 * q + 11 * q^7 + 15 * q^19 - 25 * q^25 - 33 * q^31 + 47 * q^37 + 166 * q^43 + 23 * q^49 + 168 * q^61 - 109 * q^67 + 51 * q^73 + 131 * q^79 + 225 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	5.50000	−	4.33013i	0.785714	−	0.618590i
$7$	5.50000	−	4.33013i	0.785714	−	0.618590i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$			25.9808i			1.99852i	0.0384615	+	0.999260i	$0.487754\pi$
$13$			25.9808i			1.99852i	−0.0384615	+	0.999260i	$0.512246\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	7.50000	−	4.33013i	0.394737	−	0.227901i	−0.289474	−	0.957186i	$-0.593480\pi$
$19$	7.50000	−	4.33013i	0.394737	−	0.227901i	0.684211	+	0.729285i	$0.260147\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		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
$31$	−16.5000	−	9.52628i	−0.532258	−	0.307299i	0.209677	−	0.977771i	$-0.432759\pi$
$31$	−16.5000	−	9.52628i	−0.532258	−	0.307299i	−0.741935	+	0.670471i	$0.766092\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	23.5000	+	40.7032i	0.635135	+	1.10009i	0.986486	+	0.163843i	$0.0523889\pi$
$37$	23.5000	+	40.7032i	0.635135	+	1.10009i	−0.351351	+	0.936244i	$0.614278\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	83.0000			1.93023			0.965116	−	0.261822i	$-0.0843232\pi$
$43$	83.0000			1.93023			0.965116	+	0.261822i	$0.0843232\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	11.5000	−	47.6314i	0.234694	−	0.972069i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	84.0000	−	48.4974i	1.37705	−	0.795040i	0.385246	−	0.922814i	$-0.374117\pi$
$61$	84.0000	−	48.4974i	1.37705	−	0.795040i	0.991803	+	0.127774i	$0.0407833\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−54.5000	+	94.3968i	−0.813433	+	1.40891i	0.0970149	+	0.995283i	$0.469071\pi$
$67$	−54.5000	+	94.3968i	−0.813433	+	1.40891i	−0.910448	+	0.413624i	$0.864263\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	25.5000	+	14.7224i	0.349315	+	0.201677i	0.664384	−	0.747392i	$-0.268694\pi$
$73$	25.5000	+	14.7224i	0.349315	+	0.201677i	−0.315068	+	0.949069i	$0.602027\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	65.5000	+	113.449i	0.829114	+	1.43607i	0.898734	+	0.438494i	$0.144488\pi$
$79$	65.5000	+	113.449i	0.829114	+	1.43607i	−0.0696203	+	0.997574i	$0.522179\pi$
$80$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$	112.500	+	142.894i	1.23626	+	1.57027i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			193.990i			1.99989i	0.0103093	+	0.999947i	$0.496718\pi$
$97$			193.990i			1.99989i	−0.0103093	+	0.999947i	$0.503282\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−175.500	+	101.325i	−1.70388	+	0.983738i	−0.762136	+	0.647417i	$0.775849\pi$
$103$	−175.500	+	101.325i	−1.70388	+	0.983738i	−0.941748	+	0.336321i	$0.890817\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	71.5000	−	123.842i	0.655963	−	1.13616i	−0.325688	−	0.945477i	$-0.605596\pi$
$109$	71.5000	−	123.842i	0.655963	−	1.13616i	0.981651	−	0.190684i	$-0.0610707\pi$
$110$		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
$127$	253.000			1.99213			0.996063	−	0.0886483i	$-0.0282547\pi$
$127$	253.000			1.99213			0.996063	+	0.0886483i	$0.0282547\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	22.5000	−	56.2917i	0.169173	−	0.423245i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$		−	119.512i		−	0.859795i	−0.902878	−	0.429898i	$-0.858550\pi$
$139$		−	119.512i		−	0.859795i	0.902878	−	0.429898i	$-0.141450\pi$
$140$		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	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	143.000	−	247.683i	0.947020	−	1.64029i	0.195364	−	0.980731i	$-0.437411\pi$
$151$	143.000	−	247.683i	0.947020	−	1.64029i	0.751656	−	0.659556i	$-0.229256\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	252.000	+	145.492i	1.60510	+	0.926702i	0.990446	+	0.137902i	$0.0440359\pi$
$157$	252.000	+	145.492i	1.60510	+	0.926702i	0.614650	+	0.788800i	$0.289297\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−131.000	−	226.899i	−0.803681	−	1.39202i	−0.917178	−	0.398478i	$-0.869539\pi$
$163$	−131.000	−	226.899i	−0.803681	−	1.39202i	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
$169$	−506.000			−2.99408
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	25.0000	+	173.205i	0.142857	+	0.989743i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$			361.999i			1.99999i	0.00276243	+	0.999996i	$0.499121\pi$
$181$			361.999i			1.99999i	−0.00276243	+	0.999996i	$0.500879\pi$
$182$		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		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	71.5000	−	123.842i	0.370466	−	0.641666i	−0.619171	−	0.785256i	$-0.712531\pi$
$193$	71.5000	−	123.842i	0.370466	−	0.641666i	0.989637	+	0.143590i	$0.0458646\pi$
$194$		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	$-0.411696\pi$
$199$	−84.0000	−	48.4974i	−0.422111	−	0.243706i	−0.695980	+	0.718061i	$0.745030\pi$
$200$		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
$211$	−166.000			−0.786730			−0.393365	−	0.919382i	$-0.628689\pi$
$211$	−166.000			−0.786730			−0.393365	+	0.919382i	$0.628689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−132.000	+	19.0526i	−0.608295	+	0.0877998i
$218$		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	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	217.500	−	125.574i	0.949782	−	0.548357i	0.0567686	−	0.998387i	$-0.481920\pi$
$229$	217.500	−	125.574i	0.949782	−	0.548357i	0.893013	+	0.450031i	$0.148587\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−336.000	−	193.990i	−1.39419	−	0.804936i	−0.400415	−	0.916334i	$-0.631134\pi$
$241$	−336.000	−	193.990i	−1.39419	−	0.804936i	−0.993776	+	0.111397i	$0.964467\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	112.500	+	194.856i	0.455466	+	0.788890i
$248$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	305.500	+	122.110i	1.17954	+	0.471466i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−420.000	+	242.487i	−1.54982	+	0.894786i	−0.551661	+	0.834069i	$0.686006\pi$
$271$	−420.000	+	242.487i	−1.54982	+	0.894786i	−0.998155	+	0.0607176i	$0.980661\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	264.500	−	458.127i	0.954874	−	1.65389i	0.220217	−	0.975451i	$-0.429324\pi$
$277$	264.500	−	458.127i	0.954874	−	1.65389i	0.734657	−	0.678439i	$-0.237343\pi$
$278$		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
$283$	−487.500	−	281.458i	−1.72261	−	0.994552i	−0.913428	−	0.407001i	$-0.866574\pi$
$283$	−487.500	−	281.458i	−1.72261	−	0.994552i	−0.809187	−	0.587551i	$-0.800092\pi$
$284$		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		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	456.500	−	359.401i	1.51661	−	1.19402i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			559.452i			1.82232i	0.412052	+	0.911160i	$0.364812\pi$
$307$			559.452i			1.82232i	−0.412052	+	0.911160i	$0.635188\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−370.500	+	213.908i	−1.18371	+	0.683413i	−0.956869	−	0.290520i	$-0.906172\pi$
$313$	−370.500	+	213.908i	−1.18371	+	0.683413i	−0.226837	+	0.973933i	$0.572838\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−562.500	−	324.760i	−1.73077	−	0.999260i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−149.500	−	258.942i	−0.451662	−	0.782301i	0.546828	−	0.837245i	$-0.315835\pi$
$331$	−149.500	−	258.942i	−0.451662	−	0.782301i	−0.998489	+	0.0549442i	$0.982502\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−167.000			−0.495549			−0.247774	−	0.968818i	$-0.579699\pi$
$337$	−167.000			−0.495549			−0.247774	+	0.968818i	$0.579699\pi$
$338$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$		−	484.974i		−	1.38961i	−0.719198	−	0.694805i	$-0.755490\pi$
$349$		−	484.974i		−	1.38961i	0.719198	−	0.694805i	$-0.244510\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−143.000	+	247.683i	−0.396122	+	0.686103i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−604.500	−	349.008i	−1.64714	−	0.950976i	−0.978202	−	0.207657i	$-0.933416\pi$
$367$	−604.500	−	349.008i	−1.64714	−	0.950976i	−0.668937	−	0.743319i	$-0.733251\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	60.5000	+	104.789i	0.162198	+	0.280936i	0.935657	−	0.352911i	$-0.114808\pi$
$373$	60.5000	+	104.789i	0.162198	+	0.280936i	−0.773458	+	0.633847i	$0.781475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	83.0000			0.218997			0.109499	−	0.993987i	$-0.465075\pi$
$379$	83.0000			0.218997			0.109499	+	0.993987i	$0.465075\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	34.5000	−	19.9186i	0.0869018	−	0.0501728i	−0.455919	−	0.890021i	$-0.650689\pi$
$397$	34.5000	−	19.9186i	0.0869018	−	0.0501728i	0.542821	+	0.839848i	$0.317356\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	247.500	−	428.683i	0.614144	−	1.06373i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−697.500	−	402.702i	−1.70538	−	0.984601i	−0.940098	−	0.340905i	$-0.889267\pi$
$409$	−697.500	−	402.702i	−1.70538	−	0.984601i	−0.765281	−	0.643696i	$-0.777400\pi$
$410$		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
$421$	839.000			1.99287			0.996437	−	0.0843398i	$-0.0268781\pi$
$421$	839.000			1.99287			0.996437	+	0.0843398i	$0.0268781\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	252.000	−	630.466i	0.590164	−	1.47650i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$			704.945i			1.62805i	0.580831	+	0.814024i	$0.302728\pi$
$433$			704.945i			1.62805i	−0.580831	+	0.814024i	$0.697272\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	756.000	−	436.477i	1.72210	−	0.994252i	0.807517	−	0.589844i	$-0.200811\pi$
$439$	756.000	−	436.477i	1.72210	−	0.994252i	0.914579	−	0.404408i	$-0.132522\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		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
$457$	23.5000	+	40.7032i	0.0514223	+	0.0890661i	0.890591	−	0.454805i	$-0.150291\pi$
$457$	23.5000	+	40.7032i	0.0514223	+	0.0890661i	−0.839168	+	0.543872i	$0.816958\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−923.000			−1.99352			−0.996760	−	0.0804300i	$-0.974371\pi$
$463$	−923.000			−1.99352			−0.996760	+	0.0804300i	$0.974371\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$	109.000	+	755.174i	0.232409	+	1.61018i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$			216.506i			0.455803i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	−1057.50	+	610.548i	−2.19854	+	1.26933i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	306.500	−	530.874i	0.629363	−	1.09009i	−0.358316	−	0.933600i	$-0.616649\pi$
$487$	306.500	−	530.874i	0.629363	−	1.09009i	0.987680	−	0.156489i	$-0.0500176\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	438.500	+	759.504i	0.878758	+	1.52205i	0.852705	+	0.522392i	$0.174960\pi$
$499$	438.500	+	759.504i	0.878758	+	1.52205i	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		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	204.000	−	29.4449i	0.399217	−	0.0576220i
$512$		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		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−580.500	+	335.152i	−1.10994	+	0.640826i	−0.938815	−	0.344423i	$-0.888075\pi$
$523$	−580.500	+	335.152i	−1.10994	+	0.640826i	−0.171128	+	0.985249i	$0.554741\pi$
$524$		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
$541$	396.500	+	686.758i	0.732902	+	1.26942i	0.955638	+	0.294545i	$0.0951680\pi$
$541$	396.500	+	686.758i	0.732902	+	1.26942i	−0.222736	+	0.974879i	$0.571499\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−506.000			−0.925046			−0.462523	−	0.886607i	$-0.653056\pi$
$547$	−506.000			−0.925046			−0.462523	+	0.886607i	$0.653056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	851.500	+	340.348i	1.53978	+	0.615457i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$			2156.40i			3.85761i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	533.500	−	924.049i	0.934326	−	1.61830i	0.158494	−	0.987360i	$-0.449336\pi$
$571$	533.500	−	924.049i	0.934326	−	1.61830i	0.775832	−	0.630940i	$-0.217330\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−445.500	−	257.210i	−0.772097	−	0.445770i	0.0615251	−	0.998106i	$-0.480404\pi$
$577$	−445.500	−	257.210i	−0.772097	−	0.445770i	−0.833622	+	0.552335i	$0.813737\pi$
$578$		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
$589$	−165.000			−0.280136
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$		−	995.929i		−	1.65712i	−0.559900	−	0.828560i	$-0.689161\pi$
$601$		−	995.929i		−	1.65712i	0.559900	−	0.828560i	$-0.310839\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1000.50	−	577.639i	1.64827	−	0.951629i	0.670511	−	0.741900i	$-0.266075\pi$
$607$	1000.50	−	577.639i	1.64827	−	0.951629i	0.977759	−	0.209729i	$-0.0672583\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	563.000	−	975.145i	0.918434	−	1.59077i	0.116639	−	0.993174i	$-0.462788\pi$
$613$	563.000	−	975.145i	0.918434	−	1.59077i	0.801794	−	0.597600i	$-0.203879\pi$
$614$		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
$619$	688.500	+	397.506i	1.11228	+	0.642174i	0.939418	−	0.342773i	$-0.111366\pi$
$619$	688.500	+	397.506i	1.11228	+	0.642174i	0.172859	+	0.984947i	$0.444699\pi$
$620$		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
$631$	−674.000			−1.06815			−0.534073	−	0.845438i	$-0.679339\pi$
$631$	−674.000			−1.06815			−0.534073	+	0.845438i	$0.679339\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1237.50	+	298.779i	1.94270	+	0.469040i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$			895.470i			1.39264i	0.717729	+	0.696322i	$0.245181\pi$
$643$			895.470i			1.39264i	−0.717729	+	0.696322i	$0.754819\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	478.500	+	276.262i	0.723903	+	0.417946i	0.816188	−	0.577787i	$-0.196084\pi$
$661$	478.500	+	276.262i	0.723903	+	0.417946i	−0.0922844	+	0.995733i	$0.529417\pi$
$662$		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
$673$	1177.00			1.74889			0.874443	−	0.485129i	$-0.161227\pi$
$673$	1177.00			1.74889			0.874443	+	0.485129i	$0.161227\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	840.000	+	1066.94i	1.23711	+	1.57135i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1168.50	+	674.634i	−1.69103	+	0.976315i	−0.737337	+	0.675525i	$0.763917\pi$
$691$	−1168.50	+	674.634i	−1.69103	+	0.976315i	−0.953690	+	0.300790i	$0.902750\pi$
$692$		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
$703$	352.500	+	203.516i	0.501422	+	0.289496i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−467.000	−	808.868i	−0.658674	−	1.14086i	−0.980959	−	0.194214i	$-0.937784\pi$
$709$	−467.000	−	808.868i	−0.658674	−	1.14086i	0.322285	−	0.946643i	$-0.395549\pi$
$710$		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	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−526.500	+	1317.22i	−0.730236	+	1.82694i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	268.468i		−	0.369282i	−0.982806	−	0.184641i	$-0.940888\pi$
$727$		−	268.468i		−	0.369282i	0.982806	−	0.184641i	$-0.0591122\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1225.50	+	707.543i	−1.67190	+	0.965270i	−0.705321	+	0.708888i	$0.749197\pi$
$733$	−1225.50	+	707.543i	−1.67190	+	0.965270i	−0.966576	+	0.256381i	$0.917470\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−54.5000	+	94.3968i	−0.0737483	+	0.127736i	−0.900541	−	0.434771i	$-0.856829\pi$
$739$	−54.5000	+	94.3968i	−0.0737483	+	0.127736i	0.826793	+	0.562506i	$0.190163\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	690.500	+	1195.98i	0.919441	+	1.59252i	0.800266	+	0.599645i	$0.204691\pi$
$751$	690.500	+	1195.98i	0.919441	+	1.59252i	0.119174	+	0.992873i	$0.461975\pi$
$752$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−143.000	−	990.733i	−0.187418	−	1.29847i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$			1383.91i			1.79962i	0.436281	+	0.899811i	$0.356295\pi$
$769$			1383.91i			1.79962i	−0.436281	+	0.899811i	$0.643705\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	412.500	−	238.157i	0.532258	−	0.307299i
$776$		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	0.602922	−	0.797800i	$-0.294003\pi$
$787$	168.000	+	96.9948i	0.213469	+	0.123246i	−0.389454	+	0.921046i	$0.627336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1260.00	+	2182.38i	1.58890	+	2.75206i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		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	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$			581.969i			0.717594i	0.933416	+	0.358797i	$0.116813\pi$
$811$			581.969i			0.717594i	−0.933416	+	0.358797i	$0.883187\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	622.500	−	359.401i	0.761934	−	0.439903i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$	529.000	−	916.255i	0.642770	−	1.11331i	−0.342041	−	0.939685i	$-0.611118\pi$
$823$	529.000	−	916.255i	0.642770	−	1.11331i	0.984812	−	0.173626i	$-0.0555484\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−1033.50	−	596.692i	−1.24668	−	0.719773i	−0.276236	−	0.961090i	$-0.589087\pi$
$829$	−1033.50	−	596.692i	−1.24668	−	0.719773i	−0.970446	+	0.241317i	$0.922421\pi$
$830$		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
$847$	786.500	+	314.367i	0.928571	+	0.371154i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1234.95i			1.44778i	0.689918	+	0.723888i	$0.257647\pi$
$853$			1234.95i			1.44778i	−0.689918	+	0.723888i	$0.742353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	840.000	−	484.974i	0.977881	−	0.564580i	0.0762515	−	0.997089i	$-0.475705\pi$
$859$	840.000	−	484.974i	0.977881	−	0.564580i	0.901630	+	0.432509i	$0.142371\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−2452.50	−	1415.95i	−2.81573	−	1.62566i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	299.000	+	517.883i	0.340935	+	0.590517i	0.984607	−	0.174785i	$-0.0559231\pi$
$877$	299.000	+	517.883i	0.340935	+	0.590517i	−0.643672	+	0.765302i	$0.722590\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1259.00			1.42582			0.712911	−	0.701255i	$-0.247377\pi$
$883$	1259.00			1.42582			0.712911	+	0.701255i	$0.247377\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	1391.50	−	1095.52i	1.56524	−	1.23231i
$890$		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
$907$	726.500	−	1258.33i	0.800992	−	1.38736i	−0.117971	−	0.993017i	$-0.537639\pi$
$907$	726.500	−	1258.33i	0.800992	−	1.38736i	0.918964	−	0.394342i	$-0.129028\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−485.500	−	840.911i	−0.528292	−	0.915028i	−0.999456	−	0.0329825i	$-0.989499\pi$
$919$	−485.500	−	840.911i	−0.528292	−	0.915028i	0.471164	−	0.882045i	$-0.343834\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1175.00			−1.27027
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−120.000	−	407.032i	−0.128894	−	0.437199i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	316.965i		−	0.338277i	−0.985592	−	0.169138i	$-0.945902\pi$
$937$		−	316.965i		−	0.338277i	0.985592	−	0.169138i	$-0.0540985\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	−382.500	+	662.509i	−0.403056	+	0.698113i
$950$		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
$961$	−299.000	−	517.883i	−0.311134	−	0.538900i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	253.000			0.261634			0.130817	−	0.991407i	$-0.458240\pi$
$967$	253.000			0.261634			0.130817	+	0.991407i	$0.458240\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$	−517.500	−	657.313i	−0.531860	−	0.675553i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−869.500	+	1506.02i	−0.877397	+	1.51970i	−0.0232089	+	0.999731i	$0.507388\pi$
$991$	−869.500	+	1506.02i	−0.877397	+	1.51970i	−0.854188	+	0.519965i	$0.825945\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1150.50	−	664.241i	−1.15396	−	0.666240i	−0.204112	−	0.978947i	$-0.565431\pi$
$997$	−1150.50	−	664.241i	−1.15396	−	0.666240i	−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.cg.e.577.1		2
3.2	odd	2	CM	1008.3.cg.e.577.1		2
4.3	odd	2		252.3.z.c.73.1	✓	2
7.5	odd	6	inner	1008.3.cg.e.145.1		2
12.11	even	2		252.3.z.c.73.1	✓	2
21.5	even	6	inner	1008.3.cg.e.145.1		2
28.3	even	6		1764.3.d.b.685.1		2
28.11	odd	6		1764.3.d.b.685.2		2
28.19	even	6		252.3.z.c.145.1	yes	2
28.23	odd	6		1764.3.z.c.901.1		2
28.27	even	2		1764.3.z.c.325.1		2
84.11	even	6		1764.3.d.b.685.2		2
84.23	even	6		1764.3.z.c.901.1		2
84.47	odd	6		252.3.z.c.145.1	yes	2
84.59	odd	6		1764.3.d.b.685.1		2
84.83	odd	2		1764.3.z.c.325.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.z.c.73.1	✓	2	4.3	odd	2
252.3.z.c.73.1	✓	2	12.11	even	2
252.3.z.c.145.1	yes	2	28.19	even	6
252.3.z.c.145.1	yes	2	84.47	odd	6
1008.3.cg.e.145.1		2	7.5	odd	6	inner
1008.3.cg.e.145.1		2	21.5	even	6	inner
1008.3.cg.e.577.1		2	1.1	even	1	trivial
1008.3.cg.e.577.1		2	3.2	odd	2	CM
1764.3.d.b.685.1		2	28.3	even	6
1764.3.d.b.685.1		2	84.59	odd	6
1764.3.d.b.685.2		2	28.11	odd	6
1764.3.d.b.685.2		2	84.11	even	6
1764.3.z.c.325.1		2	28.27	even	2
1764.3.z.c.325.1		2	84.83	odd	2
1764.3.z.c.901.1		2	28.23	odd	6
1764.3.z.c.901.1		2	84.23	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.50000 - 4.33013i) q^{7} +O(q^{10})\)	\(q+(5.50000 - 4.33013i) q^{7} +25.9808i q^{13} +(7.50000 - 4.33013i) q^{19} +(-12.5000 + 21.6506i) q^{25} +(-16.5000 - 9.52628i) q^{31} +(23.5000 + 40.7032i) q^{37} +83.0000 q^{43} +(11.5000 - 47.6314i) q^{49} +(84.0000 - 48.4974i) q^{61} +(-54.5000 + 94.3968i) q^{67} +(25.5000 + 14.7224i) q^{73} +(65.5000 + 113.449i) q^{79} +(112.500 + 142.894i) q^{91} +193.990i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 11 q^{7}+O(q^{10}) \) 2 * q + 11 * q^7	\( 2 q + 11 q^{7} + 15 q^{19} - 25 q^{25} - 33 q^{31} + 47 q^{37} + 166 q^{43} + 23 q^{49} + 168 q^{61} - 109 q^{67} + 51 q^{73} + 131 q^{79} + 225 q^{91}+O(q^{100}) \) 2 * q + 11 * q^7 + 15 * q^19 - 25 * q^25 - 33 * q^31 + 47 * q^37 + 166 * q^43 + 23 * q^49 + 168 * q^61 - 109 * q^67 + 51 * q^73 + 131 * q^79 + 225 * q^91

Introduction

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