LMFDB - Embedded newform 117.6.g.a.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,6,Mod(55,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, 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("117.55");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	117.g (of order $3$, degree $2$, 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:	$18.7649069181$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	117.55
Dual form			117.6.g.a.100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(16.0000 - 27.7128i) q^{4} +(105.500 - 182.731i) q^{7} +O(q^{10})$	$q+(16.0000 - 27.7128i) q^{4} +(105.500 - 182.731i) q^{7} +(-387.500 + 470.252i) q^{13} +(-512.000 - 886.810i) q^{16} +(716.000 - 1240.15i) q^{19} -3125.00 q^{25} +(-3376.00 - 5847.40i) q^{28} +2723.00 q^{31} +(-8275.00 - 14332.7i) q^{37} +(9561.50 - 16561.0i) q^{43} +(-13857.0 - 24001.0i) q^{49} +(6832.00 + 18262.7i) q^{52} +(-28463.5 + 49300.2i) q^{61} -32768.0 q^{64} +(18969.5 + 32856.1i) q^{67} +79577.0 q^{73} +(-22912.0 - 39684.7i) q^{76} +90857.0 q^{79} +(45048.5 + 120420. i) q^{91} +(-88862.5 + 153914. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{4} + 211 q^{7}+O(q^{10}) $ 2 * q + 32 * q^4 + 211 * q^7	$ 2 q + 32 q^{4} + 211 q^{7} - 775 q^{13} - 1024 q^{16} + 1432 q^{19} - 6250 q^{25} - 6752 q^{28} + 5446 q^{31} - 16550 q^{37} + 19123 q^{43} - 27714 q^{49} + 13664 q^{52} - 56927 q^{61} - 65536 q^{64} + 37939 q^{67} + 159154 q^{73} - 45824 q^{76} + 181714 q^{79} + 90097 q^{91} - 177725 q^{97}+O(q^{100}) $ 2 * q + 32 * q^4 + 211 * q^7 - 775 * q^13 - 1024 * q^16 + 1432 * q^19 - 6250 * q^25 - 6752 * q^28 + 5446 * q^31 - 16550 * q^37 + 19123 * q^43 - 27714 * q^49 + 13664 * q^52 - 56927 * q^61 - 65536 * q^64 + 37939 * q^67 + 159154 * q^73 - 45824 * q^76 + 181714 * q^79 + 90097 * q^91 - 177725 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		0			0
$3$		0			0
$4$	16.0000	−	27.7128i	0.500000	−	0.866025i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	105.500	−	182.731i	0.813781	−	1.40951i	−0.0964195	−	0.995341i	$-0.530739\pi$
$7$	105.500	−	182.731i	0.813781	−	1.40951i	0.910200	−	0.414169i	$-0.135928\pi$
$8$		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
$13$	−387.500	+	470.252i	−0.635936	+	0.771742i
$13$	−387.500	+	470.252i	−0.635936	+	0.771742i
$14$		0			0
$15$		0			0
$16$	−512.000	−	886.810i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	716.000	−	1240.15i	0.455018	−	0.788115i	−0.543671	−	0.839299i	$-0.682966\pi$
$19$	716.000	−	1240.15i	0.455018	−	0.788115i	0.998689	+	0.0511835i	$0.0162993\pi$
$20$		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
$25$	−3125.00			−1.00000
$26$		0			0
$27$		0			0
$28$	−3376.00	−	5847.40i	−0.813781	−	1.40951i
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	2723.00			0.508913			0.254456	−	0.967084i	$-0.418103\pi$
$31$	2723.00			0.508913			0.254456	+	0.967084i	$0.418103\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−8275.00	−	14332.7i	−0.993719	−	1.72117i	−0.593770	−	0.804635i	$-0.702361\pi$
$37$	−8275.00	−	14332.7i	−0.993719	−	1.72117i	−0.399949	−	0.916537i	$-0.630972\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	9561.50	−	16561.0i	0.788597	−	1.36589i	−0.138230	−	0.990400i	$-0.544141\pi$
$43$	9561.50	−	16561.0i	0.788597	−	1.36589i	0.926827	−	0.375489i	$-0.122525\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−13857.0	−	24001.0i	−0.824478	−	1.42804i
$50$		0			0
$51$		0			0
$52$	6832.00	+	18262.7i	0.350380	+	0.936608i
$53$		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	−0.314862	+	0.949137i	$0.601958\pi$
$61$	−28463.5	+	49300.2i	−0.979408	+	1.69638i	−0.664546	+	0.747247i	$0.731375\pi$
$62$		0			0
$63$		0			0
$64$	−32768.0			−1.00000
$65$		0			0
$66$		0			0
$67$	18969.5	+	32856.1i	0.516260	+	0.894189i	0.999822	+	0.0188789i	$0.00600969\pi$
$67$	18969.5	+	32856.1i	0.516260	+	0.894189i	−0.483561	+	0.875310i	$0.660657\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		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
$76$	−22912.0	−	39684.7i	−0.455018	−	0.788115i
$77$		0			0
$78$		0			0
$79$	90857.0			1.63791			0.818956	−	0.573856i	$-0.194553\pi$
$79$	90857.0			1.63791			0.818956	+	0.573856i	$0.194553\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	45048.5	+	120420.i	0.570265	+	1.52439i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−88862.5	+	153914.i	−0.958935	+	1.66092i	−0.233840	+	0.972275i	$0.575129\pi$
$97$	−88862.5	+	153914.i	−0.958935	+	1.66092i	−0.725095	+	0.688649i	$0.758204\pi$
$98$		0			0
$99$		0			0
$100$	−50000.0	+	86602.5i	−0.500000	+	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	70577.0			0.655496			0.327748	−	0.944765i	$-0.393710\pi$
$103$	70577.0			0.655496			0.327748	+	0.944765i	$0.393710\pi$
$104$		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
$109$	133361.			1.07513			0.537567	−	0.843221i	$-0.319344\pi$
$109$	133361.			1.07513			0.537567	+	0.843221i	$0.319344\pi$
$110$		0			0
$111$		0			0
$112$	−216064.			−1.62756
$113$		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
$124$	43568.0	−	75462.0i	0.254456	−	0.440731i
$125$		0			0
$126$		0			0
$127$	40005.5	+	69291.6i	0.220095	+	0.381216i	0.954837	−	0.297131i	$-0.0960299\pi$
$127$	40005.5	+	69291.6i	0.220095	+	0.381216i	−0.734742	+	0.678347i	$0.762697\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−151076.	−	261671.i	−0.740570	−	1.28271i
$134$		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
$139$	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
$148$	−529600.			−1.98744
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$	−408724.			−1.45877			−0.729387	−	0.684102i	$-0.760194\pi$
$151$	−408724.			−1.45877			−0.729387	+	0.684102i	$0.760194\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	471911.			1.52796			0.763978	−	0.645242i	$-0.223243\pi$
$157$	471911.			1.52796			0.763978	+	0.645242i	$0.223243\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	176188.	−	305166.i	0.519405	−	0.899636i	−0.480341	−	0.877082i	$-0.659487\pi$
$163$	176188.	−	305166.i	0.519405	−	0.899636i	0.999746	−	0.0225538i	$-0.00717969\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−70980.5	−	364445.i	−0.191171	−	0.981557i
$170$		0			0
$171$		0			0
$172$	−305968.	−	529952.i	−0.788597	−	1.36589i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−329688.	+	571036.i	−0.813781	+	1.40951i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	234026.			0.530967			0.265484	−	0.964115i	$-0.414468\pi$
$181$	234026.			0.530967			0.265484	+	0.964115i	$0.414468\pi$
$182$		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	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	510588.	+	884365.i	0.986683	+	1.70899i	0.634204	+	0.773166i	$0.281328\pi$
$193$	510588.	+	884365.i	0.986683	+	1.70899i	0.352480	+	0.935820i	$0.385339\pi$
$194$		0			0
$195$		0			0
$196$	−886848.			−1.64896
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$	51246.5	−	88761.5i	0.0917343	−	0.158888i	−0.816507	−	0.577336i	$-0.804092\pi$
$199$	51246.5	−	88761.5i	0.0917343	−	0.158888i	0.908241	+	0.418448i	$0.137426\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	615424.	+	102870.i	0.986316	+	0.164866i
$209$		0			0
$210$		0			0
$211$	473662.	+	820406.i	0.732423	+	1.26859i	0.955845	+	0.293872i	$0.0949439\pi$
$211$	473662.	+	820406.i	0.732423	+	1.26859i	−0.223422	+	0.974722i	$0.571723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	287276.	−	497577.i	0.414143	−	0.717317i
$218$		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	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	1.26982e6			1.60012			0.800060	−	0.599919i	$-0.204801\pi$
$229$	1.26982e6			1.60012			0.800060	+	0.599919i	$0.204801\pi$
$230$		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			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−648487.	+	1.12321e6i	−0.719215	+	1.24572i	0.242096	+	0.970252i	$0.422165\pi$
$241$	−648487.	+	1.12321e6i	−0.719215	+	1.24572i	−0.961311	+	0.275465i	$0.911168\pi$
$242$		0			0
$243$		0			0
$244$	910832.	+	1.57761e6i	0.979408	+	1.69638i
$245$		0			0
$246$		0			0
$247$	305732.	+	817258.i	0.318859	+	0.852347i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−524288.	+	908093.i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−3.49205e6			−3.23468
$260$		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
$268$	1.21405e6			1.03252
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	943500.	+	1.63419e6i	0.780403	+	1.35170i	0.931707	+	0.363210i	$0.118319\pi$
$271$	943500.	+	1.63419e6i	0.780403	+	1.35170i	−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	0.289105	−	0.957297i	$-0.406642\pi$
$277$	1.24331e6	−	2.15347e6i	0.973596	−	1.68632i	0.684491	−	0.729021i	$-0.260024\pi$
$278$		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
$283$	−1.16631e6	−	2.02011e6i	−0.865663	−	1.49937i	−0.866387	−	0.499373i	$-0.833564\pi$
$283$	−1.16631e6	−	2.02011e6i	−0.865663	−	1.49937i	0.000724409	−	1.00000i	$-0.499769\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	709928.	+	1.22963e6i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$	1.27323e6	−	2.20530e6i	0.873877	−	1.51360i
$293$		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
$301$	−2.01748e6	−	3.49437e6i	−1.28349	−	2.22307i
$302$		0			0
$303$		0			0
$304$	−1.46637e6			−0.910037
$305$		0			0
$306$		0			0
$307$	3.20232e6			1.93919			0.969593	−	0.244723i	$-0.0786971\pi$
$307$	3.20232e6			1.93919			0.969593	+	0.244723i	$0.0786971\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−3.30097e6			−1.90450			−0.952250	−	0.305318i	$-0.901237\pi$
$313$	−3.30097e6			−1.90450			−0.952250	+	0.305318i	$0.901237\pi$
$314$		0			0
$315$		0			0
$316$	1.45371e6	−	2.51790e6i	0.818956	−	1.41847i
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.21094e6	−	1.46954e6i	0.635936	−	0.771742i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.84650e6	+	3.19823e6i	−0.926359	+	1.60450i	−0.136998	+	0.990571i	$0.543745\pi$
$331$	−1.84650e6	+	3.19823e6i	−0.926359	+	1.60450i	−0.789361	+	0.613929i	$0.789588\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.11681e6			−1.97463			−0.987316	−	0.158767i	$-0.949248\pi$
$337$	−4.11681e6			−1.97463			−0.987316	+	0.158767i	$0.949248\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.30138e6			−1.05622
$344$		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
$361$	212738.	+	368472.i	0.0859164	+	0.148812i
$362$		0			0
$363$		0			0
$364$	4.05795e6	+	678299.i	1.60529	+	0.268329i
$365$		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
$373$	2.65864e6	−	4.60489e6i	0.989434	−	1.71375i	0.369158	−	0.929367i	$-0.379646\pi$
$373$	2.65864e6	−	4.60489e6i	0.989434	−	1.71375i	0.620276	−	0.784384i	$-0.287021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−34946.5	−	60529.1i	−0.0124970	−	0.0216454i	0.859709	−	0.510784i	$-0.170645\pi$
$379$	−34946.5	−	60529.1i	−0.0124970	−	0.0216454i	−0.872206	+	0.489138i	$0.837311\pi$
$380$		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
$388$	2.84360e6	+	4.92526e6i	0.958935	+	1.66092i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	393030.	−	680749.i	0.125156	−	0.216776i	−0.796638	−	0.604456i	$-0.793390\pi$
$397$	393030.	−	680749.i	0.125156	−	0.216776i	0.921794	+	0.387681i	$0.126724\pi$
$398$		0			0
$399$		0			0
$400$	1.60000e6	+	2.77128e6i	0.500000	+	0.866025i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−1.05516e6	+	1.28050e6i	−0.323636	+	0.392749i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−3.18244e6	+	5.51216e6i	−0.940703	+	1.62935i	−0.176569	+	0.984288i	$0.556500\pi$
$409$	−3.18244e6	+	5.51216e6i	−0.940703	+	1.62935i	−0.764134	+	0.645057i	$0.776834\pi$
$410$		0			0
$411$		0			0
$412$	1.12923e6	−	1.95589e6i	0.327748	−	0.567676i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−4.87580e6			−1.34073			−0.670364	−	0.742033i	$-0.733862\pi$
$421$	−4.87580e6			−1.34073			−0.670364	+	0.742033i	$0.733862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.00580e6	+	1.04023e7i	1.59405	+	2.76097i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	3.74596e6	−	6.48820e6i	0.960160	−	1.66305i	0.238069	−	0.971248i	$-0.423486\pi$
$433$	3.74596e6	−	6.48820e6i	0.960160	−	1.66305i	0.722091	−	0.691798i	$-0.243181\pi$
$434$		0			0
$435$		0			0
$436$	2.13378e6	−	3.69581e6i	0.537567	−	0.931093i
$437$		0			0
$438$		0			0
$439$	4.03787e6	+	6.99380e6i	0.999980	+	1.73202i	0.505505	+	0.862824i	$0.331306\pi$
$439$	4.03787e6	+	6.99380e6i	0.999980	+	1.73202i	0.494475	+	0.869192i	$0.335360\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−3.45702e6	+	5.98774e6i	−0.813781	+	1.40951i
$449$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$449$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−592880.	−	1.02690e6i	−0.132793	−	0.230005i	0.791959	−	0.610574i	$-0.209061\pi$
$457$	−592880.	−	1.02690e6i	−0.132793	−	0.230005i	−0.924752	+	0.380569i	$0.875728\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	−9.03907e6			−1.95962			−0.979809	−	0.199936i	$-0.935927\pi$
$463$	−9.03907e6			−1.95962			−0.979809	+	0.199936i	$0.935927\pi$
$464$		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
$469$	8.00513e6			1.68049
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.23750e6	+	3.87546e6i	−0.455018	+	0.788115i
$476$		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
$481$	9.94655e6	+	1.66260e6i	1.96024	+	0.327660i
$482$		0			0
$483$		0			0
$484$	−2.57682e6	−	4.46318e6i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−4.83518e6	+	8.37478e6i	−0.923827	+	1.60011i	−0.130389	+	0.991463i	$0.541623\pi$
$487$	−4.83518e6	+	8.37478e6i	−0.923827	+	1.60011i	−0.793438	+	0.608652i	$0.791711\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.39418e6	−	2.41478e6i	−0.254456	−	0.440731i
$497$		0			0
$498$		0			0
$499$	−8.36157e6			−1.50327			−0.751634	−	0.659581i	$-0.770734\pi$
$499$	−8.36157e6			−1.50327			−0.751634	+	0.659581i	$0.770734\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	2.56035e6			0.440190
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	8.39537e6	−	1.45412e7i	1.42229	−	2.46348i
$512$		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
$523$	−4.81752e6	−	8.34420e6i	−0.770140	−	1.33392i	−0.937486	−	0.348023i	$-0.886853\pi$
$523$	−4.81752e6	−	8.34420e6i	−0.770140	−	1.33392i	0.167346	−	0.985898i	$-0.446480\pi$
$524$		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
$532$	−9.66886e6			−1.48114
$533$		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
$547$	−1.13049e7			−1.61547			−0.807733	−	0.589548i	$-0.799306\pi$
$547$	−1.13049e7			−1.61547			−0.807733	+	0.589548i	$0.799306\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	9.58541e6	−	1.66024e7i	1.33290	−	2.30865i
$554$		0			0
$555$		0			0
$556$	−7.27451e6	−	1.25998e7i	−0.997969	−	1.72853i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$	4.08276e6	+	1.09137e7i	0.552617	+	1.47721i
$560$		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
$571$	−1.54378e7			−1.98150			−0.990751	−	0.135693i	$-0.956674\pi$
$571$	−1.54378e7			−1.98150			−0.990751	+	0.135693i	$0.956674\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.72229e6			−0.215360			−0.107680	−	0.994186i	$-0.534342\pi$
$577$	−1.72229e6			−0.215360			−0.107680	+	0.994186i	$0.534342\pi$
$578$		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
$589$	1.94967e6	−	3.37692e6i	0.231565	−	0.401082i
$590$		0			0
$591$		0			0
$592$	−8.47360e6	+	1.46767e7i	−0.993719	+	1.72117i
$593$		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			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		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
$604$	−6.53958e6	+	1.13269e7i	−0.729387	+	1.26333i
$605$		0			0
$606$		0			0
$607$	−7.88245e6	+	1.36528e7i	−0.868339	+	1.50401i	−0.00464665	+	0.999989i	$0.501479\pi$
$607$	−7.88245e6	+	1.36528e7i	−0.868339	+	1.50401i	−0.863693	+	0.504019i	$0.831854\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−8.19511e6	−	1.41943e7i	−0.880853	−	1.52568i	−0.850394	−	0.526147i	$-0.823636\pi$
$613$	−8.19511e6	−	1.41943e7i	−0.880853	−	1.52568i	−0.0304599	−	0.999536i	$-0.509697\pi$
$614$		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
$619$	1.87828e7			1.97031			0.985153	−	0.171676i	$-0.0549183\pi$
$619$	1.87828e7			1.97031			0.985153	+	0.171676i	$0.0549183\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.76562e6			1.00000
$626$		0			0
$627$		0			0
$628$	7.55058e6	−	1.30780e7i	0.763978	−	1.32325i
$629$		0			0
$630$		0			0
$631$	993825.	−	1.72136e6i	0.0993658	−	0.172107i	−0.812057	−	0.583579i	$-0.801652\pi$
$631$	993825.	−	1.72136e6i	0.0993658	−	0.172107i	0.911422	+	0.411472i	$0.134985\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.66561e7	+	2.78412e6i	1.62639	+	0.271856i
$638$		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
$643$	3.64271e6	−	6.30936e6i	0.347454	−	0.601808i	−0.638342	−	0.769753i	$-0.720380\pi$
$643$	3.64271e6	−	6.30936e6i	0.347454	−	0.601808i	0.985796	+	0.167944i	$0.0537129\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−5.63800e6	−	9.76530e6i	−0.519405	−	0.899636i
$653$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	345474.	+	598379.i	0.0307548	+	0.0532688i	0.880993	−	0.473129i	$-0.156876\pi$
$661$	345474.	+	598379.i	0.0307548	+	0.0532688i	−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
$676$	−1.12355e7	−	3.86405e6i	−0.945639	−	0.325219i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	1.87500e7	+	3.24759e7i	1.56073	+	2.70326i
$680$		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
$688$	−1.95820e7			−1.57719
$689$		0			0
$690$		0			0
$691$	1.21911e7	+	2.11156e7i	0.971289	+	1.68232i	0.691673	+	0.722211i	$0.256874\pi$
$691$	1.21911e7	+	2.11156e7i	0.971289	+	1.68232i	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
$700$	1.05500e7	+	1.82731e7i	0.813781	+	1.40951i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−2.36996e7			−1.80864
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.18185e7	+	2.04702e7i	−0.882971	+	1.52935i	−0.0349502	+	0.999389i	$0.511127\pi$
$709$	−1.18185e7	+	2.04702e7i	−0.882971	+	1.52935i	−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		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	7.44587e6	−	1.28966e7i	0.533430	−	0.923928i
$722$		0			0
$723$		0			0
$724$	3.74442e6	−	6.48552e6i	0.265484	−	0.459831i
$725$		0			0
$726$		0			0
$727$	1.56893e7			1.10095			0.550474	−	0.834853i	$-0.314447\pi$
$727$	1.56893e7			1.10095			0.550474	+	0.834853i	$0.314447\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.75648e7			−1.20749			−0.603746	−	0.797177i	$-0.706326\pi$
$733$	−1.75648e7			−1.20749			−0.603746	+	0.797177i	$0.706326\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.48018e7	+	2.56374e7i	0.997017	+	1.72688i	0.565352	+	0.824850i	$0.308740\pi$
$739$	1.48018e7	+	2.56374e7i	0.997017	+	1.72688i	0.431665	+	0.902034i	$0.357926\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−569374.	−	986185.i	−0.0368381	−	0.0638055i	0.847019	−	0.531563i	$-0.178395\pi$
$751$	−569374.	−	986185.i	−0.0368381	−	0.0638055i	−0.883857	+	0.467758i	$0.845062\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−9.92629e6	−	1.71928e7i	−0.629575	−	1.09046i	−0.987637	−	0.156758i	$-0.949896\pi$
$757$	−9.92629e6	−	1.71928e7i	−0.629575	−	1.09046i	0.358062	−	0.933698i	$-0.383438\pi$
$758$		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
$763$	1.40696e7	−	2.43692e7i	0.874923	−	1.51541i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−2.94144e6	−	5.09473e6i	−0.179368	−	0.310674i	0.762296	−	0.647228i	$-0.224072\pi$
$769$	−2.94144e6	−	5.09473e6i	−0.179368	−	0.310674i	−0.941664	+	0.336554i	$0.890739\pi$
$770$		0			0
$771$		0			0
$772$	3.26777e7			1.97337
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−8.50938e6			−0.508913
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.41896e7	+	2.45771e7i	−0.824478	+	1.42804i
$785$		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
$793$	−1.21539e7	−	3.24889e7i	−0.686330	−	1.83464i
$794$		0			0
$795$		0			0
$796$	−1.63989e6	−	2.84037e6i	−0.0917343	−	0.158888i
$797$		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		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$		0			0
$811$	−3.71463e7			−1.98319			−0.991594	−	0.129386i	$-0.958699\pi$
$811$	−3.71463e7			−1.98319			−0.991594	+	0.129386i	$0.958699\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.36921e7	−	2.37154e7i	−0.717652	−	1.24301i
$818$		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
$823$	−1.11462e7	+	1.93057e7i	−0.573623	+	0.993543i	0.422567	+	0.906332i	$0.361129\pi$
$823$	−1.11462e7	+	1.93057e7i	−0.573623	+	0.993543i	−0.996190	+	0.0872118i	$0.972204\pi$
$824$		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
$829$	8.39040e6	+	1.45326e7i	0.424030	+	0.734441i	0.996329	−	0.0856034i	$-0.0272818\pi$
$829$	8.39040e6	+	1.45326e7i	0.424030	+	0.734441i	−0.572299	+	0.820045i	$0.693948\pi$
$830$		0			0
$831$		0			0
$832$	1.26976e7	−	1.54092e7i	0.635936	−	0.771742i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	1.02556e7	−	1.77632e7i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	3.03143e7			1.46485
$845$		0			0
$846$		0			0
$847$	−1.69909e7	−	2.94291e7i	−0.813781	−	1.40951i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−3.00842e6			−0.141568			−0.0707842	−	0.997492i	$-0.522550\pi$
$853$	−3.00842e6			−0.141568			−0.0707842	+	0.997492i	$0.522550\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−3.88974e7			−1.79861			−0.899307	−	0.437317i	$-0.855929\pi$
$859$	−3.88974e7			−1.79861			−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
$868$	−9.19285e6	−	1.59225e7i	−0.414143	−	0.717317i
$869$		0			0
$870$		0			0
$871$	−2.28013e7	−	3.81131e6i	−1.01839	−	0.170227i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.91966e6	−	3.32494e6i	0.0842800	−	0.145977i	−0.820804	−	0.571210i	$-0.806474\pi$
$877$	1.91966e6	−	3.32494e6i	0.0842800	−	0.145977i	0.905084	+	0.425232i	$0.139808\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$		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	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	1.68823e7			0.716437
$890$		0			0
$891$		0			0
$892$	−9.72966e6			−0.409436
$893$		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
$907$	−1.16551e7	−	2.01872e7i	−0.470433	−	0.814814i	0.528995	−	0.848625i	$-0.322569\pi$
$907$	−1.16551e7	−	2.01872e7i	−0.470433	−	0.814814i	−0.999428	+	0.0338109i	$0.989236\pi$
$908$		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
$916$	2.03171e7	−	3.51902e7i	0.800060	−	1.38575i
$917$		0			0
$918$		0			0
$919$	−2.31519e7	+	4.01003e7i	−0.904271	+	1.56624i	−0.0823777	+	0.996601i	$0.526251\pi$
$919$	−2.31519e7	+	4.01003e7i	−0.904271	+	1.56624i	−0.821893	+	0.569642i	$0.807082\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	2.58594e7	+	4.47898e7i	0.993719	+	1.72117i
$926$		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
$931$	−3.96864e7			−1.50061
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.36738e7			1.62507			0.812535	−	0.582913i	$-0.198087\pi$
$937$	4.36738e7			1.62507			0.812535	+	0.582913i	$0.198087\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		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
$949$	−3.08361e7	+	3.74212e7i	−1.11146	+	1.34882i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−2.12144e7			−0.741008
$962$		0			0
$963$		0			0
$964$	2.07516e7	+	3.59428e7i	0.719215	+	1.24572i
$965$		0			0
$966$		0			0
$967$	5.80289e7			1.99562			0.997811	−	0.0661347i	$-0.0210667\pi$
$967$	5.80289e7			1.99562			0.997811	+	0.0661347i	$0.0210667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$	−4.79663e7	−	8.30801e7i	−1.62426	−	2.81329i
$974$		0			0
$975$		0			0
$976$	5.82932e7			1.95882
$977$		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			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.75402e7	+	4.60343e6i	0.897584	+	0.150034i
$989$		0			0
$990$		0			0
$991$	2.05563e7	+	3.56046e7i	0.664908	+	1.15165i	0.979310	+	0.202365i	$0.0648626\pi$
$991$	2.05563e7	+	3.56046e7i	0.664908	+	1.15165i	−0.314402	+	0.949290i	$0.601804\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	7.82868e6	−	1.35597e7i	0.249431	−	0.432027i	−0.713937	−	0.700210i	$-0.753090\pi$
$997$	7.82868e6	−	1.35597e7i	0.249431	−	0.432027i	0.963368	+	0.268183i	$0.0864230\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.6.g.a.55.1	✓	2
3.2	odd	2	CM	117.6.g.a.55.1	✓	2
13.9	even	3	inner	117.6.g.a.100.1	yes	2
39.35	odd	6	inner	117.6.g.a.100.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.6.g.a.55.1	✓	2	1.1	even	1	trivial
117.6.g.a.55.1	✓	2	3.2	odd	2	CM
117.6.g.a.100.1	yes	2	13.9	even	3	inner
117.6.g.a.100.1	yes	2	39.35	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(16.0000 - 27.7128i) q^{4} +(105.500 - 182.731i) q^{7} +O(q^{10})\)	\(q+(16.0000 - 27.7128i) q^{4} +(105.500 - 182.731i) q^{7} +(-387.500 + 470.252i) q^{13} +(-512.000 - 886.810i) q^{16} +(716.000 - 1240.15i) q^{19} -3125.00 q^{25} +(-3376.00 - 5847.40i) q^{28} +2723.00 q^{31} +(-8275.00 - 14332.7i) q^{37} +(9561.50 - 16561.0i) q^{43} +(-13857.0 - 24001.0i) q^{49} +(6832.00 + 18262.7i) q^{52} +(-28463.5 + 49300.2i) q^{61} -32768.0 q^{64} +(18969.5 + 32856.1i) q^{67} +79577.0 q^{73} +(-22912.0 - 39684.7i) q^{76} +90857.0 q^{79} +(45048.5 + 120420. i) q^{91} +(-88862.5 + 153914. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{4} + 211 q^{7}+O(q^{10}) \) 2 * q + 32 * q^4 + 211 * q^7	\( 2 q + 32 q^{4} + 211 q^{7} - 775 q^{13} - 1024 q^{16} + 1432 q^{19} - 6250 q^{25} - 6752 q^{28} + 5446 q^{31} - 16550 q^{37} + 19123 q^{43} - 27714 q^{49} + 13664 q^{52} - 56927 q^{61} - 65536 q^{64} + 37939 q^{67} + 159154 q^{73} - 45824 q^{76} + 181714 q^{79} + 90097 q^{91} - 177725 q^{97}+O(q^{100}) \) 2 * q + 32 * q^4 + 211 * q^7 - 775 * q^13 - 1024 * q^16 + 1432 * q^19 - 6250 * q^25 - 6752 * q^28 + 5446 * q^31 - 16550 * q^37 + 19123 * q^43 - 27714 * q^49 + 13664 * q^52 - 56927 * q^61 - 65536 * q^64 + 37939 * q^67 + 159154 * q^73 - 45824 * q^76 + 181714 * q^79 + 90097 * q^91 - 177725 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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