LMFDB - Embedded newform 324.6.e.c.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,6,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, 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:	$51.9643576194$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.6.e.c.217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(12.5000 + 21.6506i) q^{7} +O(q^{10})$	$q+(12.5000 + 21.6506i) q^{7} +(213.500 - 369.793i) q^{13} -1711.00 q^{19} +(1562.50 + 2706.33i) q^{25} +(5162.00 - 8940.85i) q^{31} -6661.00 q^{37} +(1676.00 + 2902.92i) q^{43} +(8091.00 - 14014.0i) q^{49} +(-28463.5 - 49300.2i) q^{61} +(18969.5 - 32856.1i) q^{67} +79577.0 q^{73} +(-45428.5 - 78684.5i) q^{79} +10675.0 q^{91} +(-88862.5 - 153914. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 25 q^{7}+O(q^{10}) $ 2 * q + 25 * q^7	$ 2 q + 25 q^{7} + 427 q^{13} - 3422 q^{19} + 3125 q^{25} + 10324 q^{31} - 13322 q^{37} + 3352 q^{43} + 16182 q^{49} - 56927 q^{61} + 37939 q^{67} + 159154 q^{73} - 90857 q^{79} + 21350 q^{91} - 177725 q^{97}+O(q^{100}) $ 2 * q + 25 * q^7 + 427 * q^13 - 3422 * q^19 + 3125 * q^25 + 10324 * q^31 - 13322 * q^37 + 3352 * q^43 + 16182 * q^49 - 56927 * q^61 + 37939 * q^67 + 159154 * q^73 - 90857 * q^79 + 21350 * q^91 - 177725 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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
$7$	12.5000	+	21.6506i	0.0964195	+	0.167003i	0.910200	−	0.414169i	$-0.135928\pi$
$7$	12.5000	+	21.6506i	0.0964195	+	0.167003i	−0.813781	+	0.581172i	$0.802594\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	213.500	−	369.793i	0.350380	−	0.606876i	−0.635936	−	0.771742i	$-0.719386\pi$
$13$	213.500	−	369.793i	0.350380	−	0.606876i	0.986316	+	0.164866i	$0.0527191\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−1711.00			−1.08734			−0.543671	−	0.839299i	$-0.682966\pi$
$19$	−1711.00			−1.08734			−0.543671	+	0.839299i	$0.682966\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	1562.50	+	2706.33i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	5162.00	−	8940.85i	0.964748	−	1.67099i	0.254456	−	0.967084i	$-0.418103\pi$
$31$	5162.00	−	8940.85i	0.964748	−	1.67099i	0.710291	−	0.703908i	$-0.248563\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6661.00			−0.799899			−0.399949	−	0.916537i	$-0.630972\pi$
$37$	−6661.00			−0.799899			−0.399949	+	0.916537i	$0.630972\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	1676.00	+	2902.92i	0.138230	+	0.239422i	0.926827	−	0.375489i	$-0.122525\pi$
$43$	1676.00	+	2902.92i	0.138230	+	0.239422i	−0.788597	+	0.614911i	$0.789192\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	8091.00	−	14014.0i	0.481407	−	0.833821i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−28463.5	−	49300.2i	−0.979408	−	1.69638i	−0.664546	−	0.747247i	$-0.731375\pi$
$61$	−28463.5	−	49300.2i	−0.979408	−	1.69638i	−0.314862	−	0.949137i	$-0.601958\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	18969.5	−	32856.1i	0.516260	−	0.894189i	−0.483561	−	0.875310i	$-0.660657\pi$
$67$	18969.5	−	32856.1i	0.516260	−	0.894189i	0.999822	−	0.0188789i	$-0.00600969\pi$
$68$		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
$73$	79577.0			1.74775			0.873877	−	0.486147i	$-0.161598\pi$
$73$	79577.0			1.74775			0.873877	+	0.486147i	$0.161598\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−45428.5	−	78684.5i	−0.818956	−	1.41847i	−0.906452	−	0.422309i	$-0.861220\pi$
$79$	−45428.5	−	78684.5i	−0.818956	−	1.41847i	0.0874958	−	0.996165i	$-0.472114\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	10675.0			0.135134
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−88862.5	−	153914.i	−0.958935	−	1.66092i	−0.725095	−	0.688649i	$-0.758204\pi$
$97$	−88862.5	−	153914.i	−0.958935	−	1.66092i	−0.233840	−	0.972275i	$-0.575129\pi$
$98$		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
$103$	105738.	−	183144.i	0.982065	−	1.70099i	0.327748	−	0.944765i	$-0.393710\pi$
$103$	105738.	−	183144.i	0.982065	−	1.70099i	0.654317	−	0.756221i	$-0.272956\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	114482.			0.922935			0.461467	−	0.887157i	$-0.347323\pi$
$109$	114482.			0.922935			0.461467	+	0.887157i	$0.347323\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$113$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	80525.5	−	139474.i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−267100.			−1.46948			−0.734742	−	0.678347i	$-0.762697\pi$
$127$	−267100.			−1.46948			−0.734742	+	0.678347i	$0.762697\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−21387.5	−	37044.2i	−0.104841	−	0.181590i
$134$		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
$139$	227328.	−	393745.i	0.997969	−	1.72853i	0.443812	−	0.896120i	$-0.353626\pi$
$139$	227328.	−	393745.i	0.997969	−	1.72853i	0.554157	−	0.832412i	$-0.313041\pi$
$140$		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
$151$	63813.5	+	110528.i	0.227756	+	0.394485i	0.957143	−	0.289616i	$-0.0935277\pi$
$151$	63813.5	+	110528.i	0.227756	+	0.394485i	−0.729387	+	0.684102i	$0.760194\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−54607.0	+	94582.1i	−0.176807	+	0.306239i	−0.940785	−	0.339004i	$-0.889910\pi$
$157$	−54607.0	+	94582.1i	−0.176807	+	0.306239i	0.763978	+	0.645242i	$0.223243\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−325873.			−0.960681			−0.480341	−	0.877082i	$-0.659487\pi$
$163$	−325873.			−0.960681			−0.480341	+	0.877082i	$0.659487\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	94482.0	+	163648.i	0.254467	+	0.440751i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−39062.5	+	67658.2i	−0.0964195	+	0.167003i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	619001.			1.40441			0.702207	−	0.711973i	$-0.252198\pi$
$181$	619001.			1.40441			0.702207	+	0.711973i	$0.252198\pi$
$182$		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
$193$	−328188.	+	568437.i	−0.634204	+	1.09847i	0.352480	+	0.935820i	$0.385339\pi$
$193$	−328188.	+	568437.i	−0.634204	+	1.09847i	−0.986683	+	0.162653i	$0.947995\pi$
$194$		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
$199$	−102493.			−0.183469			−0.0917343	−	0.995784i	$-0.529241\pi$
$199$	−102493.			−0.183469			−0.0917343	+	0.995784i	$0.529241\pi$
$200$		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
$211$	−618150.	+	1.07067e6i	−0.955845	+	1.65557i	−0.223422	+	0.974722i	$0.571723\pi$
$211$	−618150.	+	1.07067e6i	−0.955845	+	1.65557i	−0.732423	+	0.680850i	$0.761611\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	258100.			0.372082
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−152026.	−	263317.i	−0.204718	−	0.354582i	0.745325	−	0.666701i	$-0.232294\pi$
$223$	−152026.	−	263317.i	−0.204718	−	0.354582i	−0.950043	+	0.312120i	$0.898961\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−634909.	+	1.09969e6i	−0.800060	+	1.38575i	0.119515	+	0.992832i	$0.461866\pi$
$229$	−634909.	+	1.09969e6i	−0.800060	+	1.38575i	−0.919576	+	0.392913i	$0.871467\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	866776.	+	1.50130e6i	0.961311	+	1.66504i	0.719215	+	0.694788i	$0.244502\pi$
$241$	866776.	+	1.50130e6i	0.961311	+	1.66504i	0.242096	+	0.970252i	$0.422165\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−365298.	+	632716.i	−0.380983	+	0.659882i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		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
$259$	−83262.5	−	144215.i	−0.0771259	−	0.133586i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−365851.			−0.302608			−0.151304	−	0.988487i	$-0.548347\pi$
$271$	−365851.			−0.302608			−0.151304	+	0.988487i	$0.548347\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.24331e6	+	2.15347e6i	0.973596	+	1.68632i	0.684491	+	0.729021i	$0.260024\pi$
$277$	1.24331e6	+	2.15347e6i	0.973596	+	1.68632i	0.289105	+	0.957297i	$0.406642\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$	−976.000	+	1690.48i	−0.000724409	+	0.00125471i	−0.866387	−	0.499373i	$-0.833564\pi$
$283$	−976.000	+	1690.48i	−0.000724409	+	0.00125471i	0.865663	+	0.500627i	$0.166897\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.41986e6			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−41900.0	+	72572.9i	−0.0266562	+	0.0461698i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.30114e6			−1.39347			−0.696733	−	0.717331i	$-0.745364\pi$
$307$	−2.30114e6			−1.39347			−0.696733	+	0.717331i	$0.745364\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	1.65049e6	+	2.85873e6i	0.952250	+	1.64935i	0.740539	+	0.672014i	$0.234570\pi$
$313$	1.65049e6	+	2.85873e6i	0.952250	+	1.64935i	0.211712	+	0.977332i	$0.432096\pi$
$314$		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
$325$	1.33438e6			0.700760
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.84650e6	−	3.19823e6i	−0.926359	−	1.60450i	−0.789361	−	0.613929i	$-0.789588\pi$
$331$	−1.84650e6	−	3.19823e6i	−0.926359	−	1.60450i	−0.136998	−	0.990571i	$-0.543745\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2.05841e6	−	3.56526e6i	0.987316	−	1.71008i	0.356161	−	0.934424i	$-0.384085\pi$
$337$	2.05841e6	−	3.56526e6i	0.987316	−	1.71008i	0.631155	−	0.775657i	$-0.282581\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	824725.			0.378507
$344$		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
$349$	393978.	+	682391.i	0.173145	+	0.299895i	0.939518	−	0.342501i	$-0.111274\pi$
$349$	393978.	+	682391.i	0.173145	+	0.299895i	−0.766373	+	0.642396i	$0.777941\pi$
$350$		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			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	451422.			0.182312
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−2.58027e6	−	4.46916e6i	−1.00000	−	1.73205i	−0.500563	−	0.865700i	$-0.666874\pi$
$367$	−2.58027e6	−	4.46916e6i	−1.00000	−	1.73205i	−0.499437	−	0.866350i	$-0.666460\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−991938.	+	1.71809e6i	−0.369158	+	0.639400i	−0.989434	−	0.144983i	$-0.953687\pi$
$373$	−991938.	+	1.71809e6i	−0.369158	+	0.639400i	0.620276	+	0.784384i	$0.287021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	69893.0			0.0249940			0.0124970	−	0.999922i	$-0.496022\pi$
$379$	69893.0			0.0249940			0.0124970	+	0.999922i	$0.496022\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		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
$397$	5.78949e6			1.84359			0.921794	−	0.387681i	$-0.126724\pi$
$397$	5.78949e6			1.84359			0.921794	+	0.387681i	$0.126724\pi$
$398$		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
$403$	−2.20417e6	−	3.81774e6i	−0.676057	−	1.17096i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.58510e6	−	4.47753e6i	0.764134	−	1.32352i	−0.176569	−	0.984288i	$-0.556500\pi$
$409$	2.58510e6	−	4.47753e6i	0.764134	−	1.32352i	0.940703	−	0.339231i	$-0.110167\pi$
$410$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	−3.55595e6	−	6.15909e6i	−0.977801	−	1.69360i	−0.670364	−	0.742033i	$-0.733862\pi$
$421$	−3.55595e6	−	6.15909e6i	−0.977801	−	1.69360i	−0.307437	−	0.951568i	$-0.599471\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	711588.	−	1.23251e6i	0.188868	−	0.327129i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	1.85760e6			0.476138			0.238069	−	0.971248i	$-0.423486\pi$
$433$	1.85760e6			0.476138			0.238069	+	0.971248i	$0.423486\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.99667e6	−	3.45833e6i	−0.494475	−	0.856455i	0.505505	−	0.862824i	$-0.331306\pi$
$439$	−1.99667e6	−	3.45833e6i	−0.494475	−	0.856455i	−0.999980	+	0.00636830i	$0.997973\pi$
$440$		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
$457$	4.12872e6	+	7.15116e6i	0.924752	+	1.60172i	0.791959	+	0.610574i	$0.209061\pi$
$457$	4.12872e6	+	7.15116e6i	0.924752	+	1.60172i	0.132793	+	0.991144i	$0.457605\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	4.51954e6	−	7.82807e6i	0.979809	−	1.69708i	0.316755	−	0.948507i	$-0.397407\pi$
$463$	4.51954e6	−	7.82807e6i	0.979809	−	1.69708i	0.663054	−	0.748572i	$-0.269260\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	948475.			0.199110
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.67344e6	−	4.63053e6i	−0.543671	−	0.941665i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−1.42212e6	+	2.46319e6i	−0.280269	+	0.485440i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.30549e6			−1.58688			−0.793438	−	0.608652i	$-0.791711\pi$
$487$	−8.30549e6			−1.58688			−0.793438	+	0.608652i	$0.791711\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4.18078e6	−	7.24133e6i	0.751634	−	1.30187i	−0.195397	−	0.980724i	$-0.562599\pi$
$499$	4.18078e6	−	7.24133e6i	0.751634	−	1.30187i	0.947030	−	0.321144i	$-0.104067\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		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
$511$	994712.	+	1.72289e6i	0.168518	+	0.291881i
$512$		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			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−1.17287e7			−1.87497			−0.937486	−	0.348023i	$-0.886853\pi$
$523$	−1.17287e7			−1.87497			−0.937486	+	0.348023i	$0.886853\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.21817e6	+	5.57404e6i	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
$541$	1.30081e7			1.91082			0.955410	−	0.295281i	$-0.0954134\pi$
$541$	1.30081e7			1.91082			0.955410	+	0.295281i	$0.0954134\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	5.65244e6	+	9.79032e6i	0.807733	+	1.39904i	0.914430	+	0.404744i	$0.132639\pi$
$547$	5.65244e6	+	9.79032e6i	0.807733	+	1.39904i	−0.106697	+	0.994292i	$0.534027\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.13571e6	−	1.96711e6i	0.157927	−	0.273537i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	1.43130e6			0.193732
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		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
$571$	−4.77499e6	+	8.27052e6i	−0.612889	+	1.06156i	0.377862	+	0.925862i	$0.376660\pi$
$571$	−4.77499e6	+	8.27052e6i	−0.612889	+	1.06156i	−0.990751	+	0.135693i	$0.956674\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.29099e7			−1.61430			−0.807150	−	0.590347i	$-0.798991\pi$
$577$	−1.29099e7			−1.61430			−0.807150	+	0.590347i	$0.798991\pi$
$578$		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		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$	−8.83218e6	+	1.52978e7i	−1.04901	+	1.81694i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		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
$601$	−3.01081e6	−	5.21488e6i	−0.340015	−	0.588923i	0.644420	−	0.764671i	$-0.277099\pi$
$601$	−3.01081e6	−	5.21488e6i	−0.340015	−	0.588923i	−0.984435	+	0.175749i	$0.943765\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	42180.5	−	73058.8i	0.00464665	−	0.00804823i	−0.863693	−	0.504019i	$-0.831854\pi$
$607$	42180.5	−	73058.8i	0.00464665	−	0.00804823i	0.868339	+	0.495970i	$0.165188\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.63902e7			1.76171			0.880853	−	0.473389i	$-0.156969\pi$
$613$	1.63902e7			1.76171			0.880853	+	0.473389i	$0.156969\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$	6.11302e6	+	1.05881e7i	0.641253	+	1.11068i	0.985153	+	0.171676i	$0.0549183\pi$
$619$	6.11302e6	+	1.05881e7i	0.641253	+	1.11068i	−0.343901	+	0.939006i	$0.611748\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4.88281e6	+	8.45728e6i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.98765e6			−0.198732			−0.0993658	−	0.995051i	$-0.531681\pi$
$631$	−1.98765e6			−0.198732			−0.0993658	+	0.995051i	$0.531681\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.45486e6	−	5.98399e6i	−0.337351	−	0.584308i
$638$		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
$643$	−1.03351e7	+	1.79009e7i	−0.985796	+	1.70745i	−0.347454	+	0.937697i	$0.612954\pi$
$643$	−1.03351e7	+	1.79009e7i	−0.985796	+	1.70745i	−0.638342	+	0.769753i	$0.720380\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		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		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−9.89638e6	+	1.71410e7i	−0.880993	+	1.52592i	−0.0307548	+	0.999527i	$0.509791\pi$
$661$	−9.89638e6	+	1.71410e7i	−0.880993	+	1.52592i	−0.850238	+	0.526398i	$0.823542\pi$
$662$		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
$673$	1.12094e7	+	1.94152e7i	0.953991	+	1.65236i	0.736661	+	0.676262i	$0.236401\pi$
$673$	1.12094e7	+	1.94152e7i	0.953991	+	1.65236i	0.217330	+	0.976098i	$0.430265\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	2.22156e6	−	3.84786e6i	0.184920	−	0.320291i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−8.68152e6	−	1.50368e7i	−0.691673	−	1.19801i	−0.971289	−	0.237901i	$-0.923541\pi$
$691$	−8.68152e6	−	1.50368e7i	−0.691673	−	1.19801i	0.279616	−	0.960112i	$-0.409793\pi$
$692$		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
$703$	1.13970e7			0.869763
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	467806.	+	810263.i	0.0349502	+	0.0605355i	0.882971	−	0.469427i	$-0.155539\pi$
$709$	467806.	+	810263.i	0.0349502	+	0.0605355i	−0.848021	+	0.529962i	$0.822206\pi$
$710$		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			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	5.28692e6			0.378761
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.42256e7	+	2.46395e7i	0.998240	+	1.72900i	0.550474	+	0.834853i	$0.314447\pi$
$727$	1.42256e7	+	2.46395e7i	0.998240	+	1.72900i	0.447767	+	0.894150i	$0.352219\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	5.65138e6	−	9.78847e6i	0.388503	−	0.672907i	−0.603746	−	0.797177i	$-0.706326\pi$
$733$	5.65138e6	−	9.78847e6i	0.388503	−	0.672907i	0.992248	+	0.124270i	$0.0396590\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−2.96035e7			−1.99403			−0.997017	−	0.0771842i	$-0.975407\pi$
$739$	−2.96035e7			−1.99403			−0.997017	+	0.0771842i	$0.975407\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.30916e7	+	2.26753e7i	−0.847019	+	1.46708i	0.0368381	+	0.999321i	$0.488271\pi$
$751$	−1.30916e7	+	2.26753e7i	−0.847019	+	1.46708i	−0.883857	+	0.467758i	$0.845062\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−3.11435e7			−1.97527			−0.987637	−	0.156758i	$-0.949896\pi$
$757$	−3.11435e7			−1.97527			−0.987637	+	0.156758i	$0.949896\pi$
$758$		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
$763$	1.43102e6	+	2.47861e6i	0.0889889	+	0.154133i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.54423e7	−	2.67468e7i	0.941664	−	1.63101i	0.179368	−	0.983782i	$-0.442595\pi$
$769$	1.54423e7	−	2.67468e7i	0.941664	−	1.63101i	0.762296	−	0.647228i	$-0.224072\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	3.22625e7			1.92950
$776$		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
$787$	1.28512e7	−	2.22589e7i	0.739616	−	1.28105i	−0.213052	−	0.977041i	$-0.568340\pi$
$787$	1.28512e7	−	2.22589e7i	0.739616	−	1.28105i	0.952668	−	0.304012i	$-0.0983263\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.43078e7			−1.37266
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$797$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$798$		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			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	2.27708e7			1.21570			0.607849	−	0.794053i	$-0.292033\pi$
$811$	2.27708e7			1.21570			0.607849	+	0.794053i	$0.292033\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.86764e6	−	4.96689e6i	−0.150303	−	0.260333i
$818$		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
$823$	1.93572e7	−	3.35276e7i	0.996190	−	1.72545i	0.422567	−	0.906332i	$-0.361129\pi$
$823$	1.93572e7	−	3.35276e7i	0.996190	−	1.72545i	0.573623	−	0.819120i	$-0.305538\pi$
$824$		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
$829$	−1.67808e7			−0.848060			−0.424030	−	0.905648i	$-0.639385\pi$
$829$	−1.67808e7			−0.848060			−0.424030	+	0.905648i	$0.639385\pi$
$830$		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		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	1.02556e7	−	1.77632e7i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4.02628e6			0.192839
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.91096e7	−	3.30987e7i	−0.899245	−	1.55754i	−0.828461	−	0.560047i	$-0.810783\pi$
$853$	−1.91096e7	−	3.30987e7i	−0.899245	−	1.55754i	−0.0707842	−	0.997492i	$-0.522550\pi$
$854$		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
$859$	−1.79149e7	+	3.10294e7i	−0.828381	+	1.43480i	0.0709259	+	0.997482i	$0.477405\pi$
$859$	−1.79149e7	+	3.10294e7i	−0.828381	+	1.43480i	−0.899307	+	0.437317i	$0.855929\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−8.09998e6	−	1.40296e7i	−0.361775	−	0.626612i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.86956e7	−	3.23817e7i	0.820804	−	1.42167i	−0.0842800	−	0.996442i	$-0.526859\pi$
$877$	1.86956e7	−	3.23817e7i	0.820804	−	1.42167i	0.905084	−	0.425232i	$-0.139808\pi$
$878$		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
$883$	1.68468e7			0.727135			0.363567	−	0.931568i	$-0.381559\pi$
$883$	1.68468e7			0.727135			0.363567	+	0.931568i	$0.381559\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−3.33875e6	−	5.78288e6i	−0.141687	−	0.245409i
$890$		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
$907$	−1.31060e7	−	2.27002e7i	−0.528995	−	0.916247i	−0.999428	−	0.0338109i	$-0.989236\pi$
$907$	−1.31060e7	−	2.27002e7i	−0.528995	−	0.916247i	0.470433	−	0.882436i	$-0.344098\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	4.63039e7			1.80854			0.904271	−	0.426959i	$-0.140415\pi$
$919$	4.63039e7			1.80854			0.904271	+	0.426959i	$0.140415\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.04078e7	−	1.80269e7i	−0.399949	−	0.692733i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−1.38437e7	+	2.39780e7i	−0.523453	+	0.906648i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	5.29708e6			0.197100			0.0985501	−	0.995132i	$-0.468580\pi$
$937$	5.29708e6			0.197100			0.0985501	+	0.995132i	$0.468580\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		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
$949$	1.69897e7	−	2.94270e7i	0.612378	−	1.06067i
$950$		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
$961$	−3.89779e7	−	6.75117e7i	−1.36148	−	2.35815i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.61727e7	−	2.80119e7i	0.556180	−	0.963332i	−0.441631	−	0.897197i	$-0.645600\pi$
$967$	1.61727e7	−	2.80119e7i	0.556180	−	0.963332i	0.997811	−	0.0661347i	$-0.0210667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	1.13664e7			0.384895
$974$		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		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	6.05528e7			1.95862			0.979310	−	0.202365i	$-0.0648626\pi$
$991$	6.05528e7			1.95862			0.979310	+	0.202365i	$0.0648626\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	2.24077e7	+	3.88113e7i	0.713937	+	1.23658i	0.963368	+	0.268183i	$0.0864230\pi$
$997$	2.24077e7	+	3.88113e7i	0.713937	+	1.23658i	−0.249431	+	0.968393i	$0.580244\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.6.e.c.109.1		2
3.2	odd	2	CM	324.6.e.c.109.1		2
9.2	odd	6	inner	324.6.e.c.217.1		2
9.4	even	3		108.6.a.a.1.1	✓	1
9.5	odd	6		108.6.a.a.1.1	✓	1
9.7	even	3	inner	324.6.e.c.217.1		2
36.23	even	6		432.6.a.e.1.1		1
36.31	odd	6		432.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.a.a.1.1	✓	1	9.4	even	3
108.6.a.a.1.1	✓	1	9.5	odd	6
324.6.e.c.109.1		2	1.1	even	1	trivial
324.6.e.c.109.1		2	3.2	odd	2	CM
324.6.e.c.217.1		2	9.2	odd	6	inner
324.6.e.c.217.1		2	9.7	even	3	inner
432.6.a.e.1.1		1	36.23	even	6
432.6.a.e.1.1		1	36.31	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(12.5000 + 21.6506i) q^{7} +O(q^{10})\)	\(q+(12.5000 + 21.6506i) q^{7} +(213.500 - 369.793i) q^{13} -1711.00 q^{19} +(1562.50 + 2706.33i) q^{25} +(5162.00 - 8940.85i) q^{31} -6661.00 q^{37} +(1676.00 + 2902.92i) q^{43} +(8091.00 - 14014.0i) q^{49} +(-28463.5 - 49300.2i) q^{61} +(18969.5 - 32856.1i) q^{67} +79577.0 q^{73} +(-45428.5 - 78684.5i) q^{79} +10675.0 q^{91} +(-88862.5 - 153914. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 25 q^{7}+O(q^{10}) \) 2 * q + 25 * q^7	\( 2 q + 25 q^{7} + 427 q^{13} - 3422 q^{19} + 3125 q^{25} + 10324 q^{31} - 13322 q^{37} + 3352 q^{43} + 16182 q^{49} - 56927 q^{61} + 37939 q^{67} + 159154 q^{73} - 90857 q^{79} + 21350 q^{91} - 177725 q^{97}+O(q^{100}) \) 2 * q + 25 * q^7 + 427 * q^13 - 3422 * q^19 + 3125 * q^25 + 10324 * q^31 - 13322 * q^37 + 3352 * q^43 + 16182 * q^49 - 56927 * q^61 + 37939 * q^67 + 159154 * q^73 - 90857 * q^79 + 21350 * q^91 - 177725 * q^97

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